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In many introductory courses to quantum mechanics, we see $\delta$-functions all over the place. For example when expressing an arbitrary wave function $\psi(x)$ in the basis of eigenfunctions of the position operator $\hat x$ as $$ \psi(x) = \int\mathrm d\xi\, \delta(x-\xi)\, \psi(\xi). $$ In bra-ket notation this corresponds to $$ \left|\psi\right\rangle = \int\mathrm d\xi\,\left|\,\xi\,\right\rangle\!\left\langle\,\xi\,\middle|\,\psi\,\right\rangle, $$ where $\left|\,\xi\,\right\rangle$ is the state corresponding to the wavefunction $x\mapsto\delta(x-\xi)$. Now the $\delta$-function is really not a function, but a distribution, that's defined by how it acts on test-functions, i.e. $\delta[\varphi] = \varphi(0)$.
Do you know of an introductory text on quantum mechanics that stresses this point and uses the language of distributions properly, avoding any functions with seamingly infinite peaks?
Perhaps you are starting by the wrong end. Your concern seems to be related in the first term with the totally misleading notation of integrals in quantum mechanics, and this is more related with the spectral theorem than with distributions itself. Distributions only appear in Quantum mechanics when certain operators has empty spectrum in the usual Hilber space. Then, you need to consider a bigger underlying space.
Once you have the mathematical background and feel totally comfortable with the integrals of quantum mechanics as (which is nothing more tan spectral thorem) you can move onto distributions in quantum mechanics, which are developed in the context of Gelfand triplets. An excellent reference is "The role of the rigged Hilbert space in Quantum Mechanics" from Rafael de la Madrid. It is freely available in the web.
Brian Hall "Quantum Theory for Mathematicians" is a recent nice book that presents the basics of QM with mathematical rigor, as suggested by the title. It covers a fair amount of topics, and seems suitable for an undergraduate level. The short book of Mackey "Mathematical foundations of Quantum Mechanics" is also a very nice book on the axiomatization of QM, but may be difficult for an undergraduate student.
Not the answer you're looking for? Browse other questions tagged quantum-mechanics mathematical-physics resource-recommendations dirac-delta-distributions or ask your own question.
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\begin{document}
\maketitle
\begin{abstract} We give a finite axiomatization for the variety generated by relational, integral ordered monoids. As a corollary we get a finite axiomatization for the language interpretation as well. \\ {\sc Keywords:} finite axiomatizability, algebras of relations, language algebras \end{abstract}
\section{Introduction}
We will focus on algebras of similarity type $\Lambda\subseteq (\join,\meet,\comp,0,\ide)$ where $\join,\meet,\comp$ are binary operations and $0,\ide$ are constants. We will consider two types of representations: as families of binary relations and as families of languages. Our main concern is whether the varieties generated by these interpretations are finitely axiomatizable.
\begin{definition}[Language algebras] Let $\mathfrak{A}=(A,\Lambda)$ be an algebra of similarity type $\Lambda\subseteq (\join,\meet,\comp,0,\ide)$. We say that $\mathfrak{A}$ is a \emph{language algebra} if the following holds. There is an alphabet $\Sigma$ such that $A$ is a family of languages over $\Sigma$, i.e., $A\subseteq\wp(\Sigma^*)$ where $\Sigma^*$ denotes the set of words (finite strings) over $\Sigma$, and the operations in $\Lambda$ are interpreted as follows: join $\join$ is union, meet $\meet$ is intersection, composition $\comp$ is concatenation \begin{equation*} a\comp b=\{ st: s\in a\mbox{ and } t\in b\} \end{equation*} $0$ is the empty language $\emptyset$ and $\ide$ is the singleton language
consisting of the empty word.
\end{definition}
We will denote the class of language algebras of similarity type $\Lambda$ by $\Lang(\Lambda)$.
\begin{definition}[Relation algebras] Let $\mathfrak{A}=(A,\Lambda)$ be an algebra of similarity type $\Lambda\subseteq (\join,\meet,\comp,0,\ide)$. We say that $\mathfrak{A}$ is a \emph{relation algebra} if the following holds. There is a set $U$, the \emph{base} of $\mathfrak{A}$, such that $A$ is a family of binary relations on $U$, i.e., $A\subseteq \wp(U\times U)$, and the operations in $\Lambda$ are interpreted as follows: join $\join$ is union, meet $\cdot$ is intersection, composition $\comp$ is relation composition \begin{align*} a\comp b&=\{(u,v)\in U\times U: (u,w)\in a \mbox{ and } (w,v)\in b \mbox{ for some } w\}\\ \intertext{$\ide$ is the identity relation on $U$} \ide&=\{(u,v)\in U\times U: u=v\} \end{align*} and $0$ is the empty relation $\emptyset$. \end{definition}
We will denote the class of relation algebras of similarity type $\Lambda$ by $\Rel(\Lambda)$.
In passing we note that the representation classes $\Lang(\Lambda)$ and $\Rel(\Lambda)$ are not finitely axiomatizable whenever $(\meet,\comp,\ide)\subseteq \Lambda$. Indeed, the class of language algebras is not closed under products (see below), while the quasivariety $\Rel(\meet,\comp,\ide)$ has no finite base \cite{HM-rep-07}. This is one of the main reasons why we concentrate on the generated varieties below.
Assume that $0,\ide\in \Lambda$ and let $\mathfrak{A}$ be a $\Lambda$-algebra. We say that $\mathfrak{A}$ is \emph{integral} if $\ide$ is an atom of $\mathfrak{A}$, i.e., $\ide$ is a minimal non-zero element. For a class $\K(\Lambda)$ of $\Lambda$-algebras, let $\K^i(\Lambda)$ denote the subclass of integral $\Lambda$-algebras. Observe that every language algebra is integral, $\Lang(\Lambda)=\Lang^i(\Lambda)$, while there are non-integral relation algebras, $\Rel(\Lambda)\supset\Rel^i(\Lambda)$.
For a class $\K(\Lambda)$ of $\Lambda$-algebras, let $\V(\K(\Lambda))$ denote the variety generated by $\K(\Lambda)$. Note that the variety $\V(\K^i(\Lambda))$ generated by integral algebras may contain non-integral algebras, since neither products nor homomorphisms preserve the property that $\ide$ is an atom.
\section{Main results}
First we look at $\Lambda=(\meet,\comp,0,\ide)$. As usual $x\le y$ is defined by $x\meet y=x$ and we assume that $\comp$ binds closer than $\meet$, e.g., we write $x\meet y\comp z$ for $x\meet(y\comp z)$.
We define $\mbox{\rm Ax}(\meet,\comp,0,\ide)$ as the collection of the following axioms. \begin{itemize} \item[] Semilattice axioms (for $\meet$) \item[] Monoid axioms (for $\comp$ and $\ide$) \item[] Monotonicity: \begin{equation}\label{eq:mon} (x\meet x')\comp (y\meet y')\le x\comp y \end{equation} \item[] Axioms for $0$: \begin{align} 0&=0\meet x\\ 0&=0\comp x= x\comp 0 \end{align} \item[]
Subidentity axioms: \begin{align} (\ide\meet x)\comp(\ide\meet y)&=\ide\meet x\meet y \label{eq:fun}\\ (\ide\meet x)\comp(y\meet z)&=(\ide\meet x)\comp y \meet z \label{eq:fun1l}\\ (x\meet y)\comp(\ide\meet z)&=x \meet y\comp(\ide\meet z) \label{eq:fun1r}
\end{align} \end{itemize} It is easily checked that all these axioms are valid in both relation and language algebras.
We will need additional axioms that are valid in language algebras and in integral relation algebras. We define $\mbox{\rm Ax}^i(\meet,\comp,0,\ide)$ as $\mbox{\rm Ax}(\meet,\comp,0,\ide)$ augmented with the integral axioms~\eqref{eq:int1} and~\eqref{eq:int2} below. \begin{itemize} \item[] ``Integrality'': \begin{align} \ide\meet x\comp y &= \ide\meet y\comp x \label{eq:int1} \\ (\ide\meet x)\comp y&= y\comp(\ide\meet x) \label{eq:int2}
\end{align} \end{itemize} Note that $\ide\meet x\comp y=0$ iff $\ide\meet y\comp x=0$ in a relation algebra $\mathfrak{A}$.
Hence, if $\mathfrak{A}$ is integral, then it follows that the two terms in~\eqref{eq:int1} are either $0$ or $\ide$ at the same time. Checking validity of the other integral axiom in integral algebras is similar.
The set $\mbox{\rm Ax}^i(\meet,\comp,0,\ide)$ is not independent, e.g., we can derive~\eqref{eq:fun1r} from~\eqref{eq:fun1l} with the use of~\eqref{eq:int2}.
\subsection{Axiomatizations for integral relation algebras}
\begin{theorem}\label{thm:main}
The variety $\V(\Rel^i(\meet,\comp,0,\ide))$ generated by $\Rel^i(\meet,\comp,0,\ide)$ is axiomatized by $\mbox{\rm Ax}^i(\meet,\comp,0,\ide)$. \end{theorem}
\begin{proof} By the validity of the axioms in integral relation algebras we get that equations derivable using equational logic from $\mbox{\rm Ax}^i(\meet,\comp,0,\ide)$ are valid in $\V(\Rel^i(\meet,\comp,0,\ide))$. We will prove the other direction by showing that, for every non-derivable equation, there is an algebra in~$\Rel^i(\meet,\comp,0,\ide)$
witnessing that the equation is not valid, see Lemma~\ref{lem:wit}. \end{proof}
Next we include union into the signature. Define $\mbox{\rm Ax}^i(\join,\meet,\comp,0,\ide)$ as $\mbox{\rm Ax}^i(\meet,\comp,0,\ide)$ augmented with the distributive lattice axioms (for $\meet$ and $\join$)
and that the operation $\comp$ is additive: \begin{align}
(x\join y)\comp z&= x\comp z \join y\comp z\\ x\comp (y\join z)&= x\comp y \join x\comp z
\end{align} for every $x,y,z$.
\begin{theorem}\label{thm:dl} The variety $\V(\Rel^i(\join,\meet,\comp,0,\ide))$ generated by $\Rel^i(\join,\meet,\comp,0,\ide)$ is axiomatized by $\mbox{\rm Ax}^i(\join,\meet,\comp,0,\ide)$. \end{theorem}
\begin{proof} This follows from \cite[Corollary~2]{bre:93} (see also \cite[Proposition~4.4]{AM-axi-11}), we just give a sketch here.
By the validity of the axioms we get that derivable equations are valid. For the other direction assume that $\V(\Rel^i(\join,\meet,\comp,0,\ide))\models a = b$. Thus we have both $\V(\Rel^i(\join,\meet,\comp,0,\ide))\models a\le b$ and $\V(\Rel^i(\join,\meet,\comp,0,\ide))\models b\le a$, where $x \le y$ is defined by $x\meet y=x$. Note that, using the additivity of the operations in relation algebras, we can rewrite every term as a join of join-free terms. Thus $a\le b$ can be equivalently rewritten as $a_1\join\dots\join a_n \le b_1\join\dots \join b_m$ where $a_i$ and $b_j$ do not contain $\join$. Furthermore, using the term graphs of \cite{AB-equ-95} one can show that $a_1\join\dots\join a_n\le b_1\join\dots \join b_m$ is valid in $\V(\Rel^i(\join,\meet,\comp,0,\ide))$ iff, for every $i$, there is $j$ such that $a_i\le b_j$ is valid in $\V(\Rel^i(\join,\meet,\comp,0,\ide))$. Thus $\V(\Rel^i(\meet,\comp,0,\ide))\models a_i\le b_j$ (since $a_i$ and $b_j$ do not contain $\join$). By Theorem~\ref{thm:main} we get $\mbox{\rm Ax}^i(\meet,\comp,0,\ide)\vdash a_i\le b_j$, whence $\mbox{\rm Ax}^i(\join,\meet,\comp,0,\ide) \vdash a_i\le b_j$. Since the distributive lattice axioms ensure that the ordering $x\le y$ can be equivalently defined by either $x\meet y=x$ or $x\join y=y$, we get $\mbox{\rm Ax}^i(\join,\meet,\comp,0,\ide) \vdash a_i\le b_1\join\dots \join b_m$. Since this holds for every $i$, using the additivity axioms we get $\mbox{\rm Ax}^i(\join,\meet,\comp,0,\ide) \vdash a\le b$. By an identical argument we get $\mbox{\rm Ax}^i(\join,\meet,\comp,0,\ide) \vdash b\le a$ as well, whence $\mbox{\rm Ax}^i(\join,\meet,\comp,0,\ide) \vdash a= b$, as desired. \end{proof}
\subsection{Axiomatization for language algebras}
\begin{theorem}\label{thm:lang} The variety $\V(\Lang(\join,\meet,\comp,0,\ide))$ generated by $\Lang(\join,\meet,\comp,0,\ide)$ is axiomatized by \begin{equation}\label{eq:empty} x\comp y\meet \ide = (x\meet\ide)\comp(y\meet\ide) \end{equation} together with $\mbox{\rm Ax}^i(\join,\meet,\comp,0,\ide)$. \end{theorem}
\begin{proof} In \cite{AMN-equ-11} it is shown that the equational theory of $\Lang(\join,\meet,\comp,0,\ide)$ is finitely axiomatizable over the equational theory of $\Rel(\join,\meet,\comp,0,\ide)$ by
the equations~\eqref{eq:empty} and~\eqref{eq:int2}.
By a minimal and straightforward modification of the proof of \cite[Corollary~3.7]{AMN-equ-11} we get that the equational theory of $\Lang(\join,\meet,\comp,0,\ide)$ is finitely axiomatizable by~\eqref{eq:empty} over the equational theory of $\Rel^i(\join,\meet,\comp,0,\ide)$, which is axiomatized by $\mbox{\rm Ax}^i(\join,\meet,\comp,0,\ide)$. \end{proof}
\section{Relational representation of the free algebra}
In this section we make the proof of Theorem~\ref{thm:main} complete. We will need the following easy consequences of $\mbox{\rm Ax}(\join,\meet,\comp,0,\ide)$: \begin{equation} (e_1\comp x\comp e_2)\meet(e_3\comp x \comp e_4)= (e_1\meet e_3)\comp x\comp (e_2\meet e_4) \label{eq:con1} \end{equation} \begin{equation} e\comp(x\meet y)=e\comp x\meet e\comp y= e\comp x\meet y= x\meet e\comp y \label{eq:con2} \end{equation} where $e,e_1,\ldots , e_4$ are subidentity terms (have the form $z\meet\ide$).
\subsection{Step-by-step construction}
Fix a countable set $X$ of variables. Let $T_X$ be the set of $(\meet,\comp,0,\ide)$-terms using the variables from $X$ and ${\mathfrak F}_X=(F_X,\meet,\comp,0,\ide)$ be the free algebra of the variety defined by $\mbox{\rm Ax}^i(\meet,\comp,0,\ide)$ which is freely generated by $X$. That is, $\mathfrak{F}_X$ is given by factoring the absolutely free algebra by the congruence of derivability from $\mbox{\rm Ax}^i(\meet,\comp,0,\ide)$ in equational logic.
Thus $\free_X\models\tau\le\sigma$ (under the natural valuation of evaluating every variable to itself) iff $\mbox{\rm Ax}^i(\meet,\comp,0,\ide)\vdash \tau\le\sigma$, where $\vdash$ denotes derivability in equational logic.
By a filter ${\mathcal F}$ of ${\mathfrak F}_X$ we mean a subset closed upward and under meet $\meet$, that is, $\tau,\sigma\in {\mathcal F}$ iff
$\tau\meet\sigma\in {\mathcal F}$.
For a $S\subseteq F_X$, let $\mathcal{F}(S)$ denote the filter generated by $S$. In particular, for a term $\tau$, $\mathcal{F}(\{\tau\})$ denotes the principal filter generated by $\{\tau\}$, i.e., the upward closure $\{\tau\}^\uparrow$ of the singleton set $\{\tau\}$. We extend the operation $\comp$ to subsets of elements as follows: \begin{align*}
X\comp Y &=\set{x\comp y:x\in X, y\in Y}
\end{align*} for subsets $X,Y$. When $X=\{ x\}$ is a singleton, we will also use the notation $x\comp Y=\{ x\comp y:y\in Y\}$, $\{ x\}^\uparrow= x^\uparrow$, $\mathcal{F}(\{\tau\})=\mathcal{F}(\tau)$, etc.
For the whole rest of the section we fix a term $\theta$ such that $\mbox{\rm Ax}^i(\meet,\comp,0,\ide)\not\vdash\theta=0$.
Next we define special filters that are generated by some set of subidentity elements.
We define \begin{align*} \mathcal{E}(\tau):&=\mathcal{F}(\{\epsilon\le\ide:\epsilon\comp\tau\comp \epsilon=\tau\})&&\\ &=\mathcal{F}(\{\epsilon\le\ide:\tau\comp \epsilon=\tau\})&&\mbox{by~\eqref{eq:int2},\eqref{eq:fun}}\\ &=\mathcal{F}(\{\epsilon\le\ide:\epsilon\comp\tau=\tau\})&&\mbox{by~\eqref{eq:int2},\eqref{eq:fun}} \end{align*} for each element $\tau$. It is worth noting that \begin{equation}\label{eq:emon} \tau\le\sigma\mbox{ implies }\mathcal{E}(\sigma)\subseteq\mathcal{E}(\tau) \end{equation} since $\epsilon\comp\tau=\epsilon\comp(\tau\meet\sigma)=\tau\meet\epsilon\comp\sigma\ge \tau\meet\sigma=\tau$ by \eqref{eq:con2} whenever $\epsilon\in\mathcal{E}(\sigma)$. We denote $\mathcal{E}:=\mathcal{E}(\theta)$ for our fixed term $\theta$. We say that the filter $\mathcal{F}$ is \emph{fundamental} if the following holds: there is an element $\tau$ such that
$\mathcal{F}=\mathcal{F}(\mathcal{E}\comp\tau\comp\mathcal{E})$. Observe that $\mathcal{E}=\mathcal{F}(\mathcal{E}\comp\ide\comp\mathcal{E})$ is fundamental, since for subidentity elements $\epsilon_1,\epsilon_2$, we have $\epsilon_1\comp\ide\comp \epsilon_2=\epsilon_1\comp \epsilon_2=\epsilon_1\meet \epsilon_2$ by~\eqref{eq:fun}.
Also note that $\mathcal{F}(\mathcal{E}\comp\tau\comp\mathcal{E})= (\mathcal{E}\comp\tau\comp\mathcal{E})^\uparrow$, since $(\epsilon_1\comp\tau\comp \epsilon_2)\meet(\epsilon_3\comp\tau\comp \epsilon_4)= (\epsilon_1\meet \epsilon_3)\comp\tau\comp(\epsilon_2\meet \epsilon_4)$ by~\eqref{eq:con1}.
We will define a chain of labelled, directed graphs $G_n= (U_n,\ell_n,E_n,W_n)$ for $n\in\omega$, where \begin{itemize} \item $U_n$ is the set of nodes, \item $\ell_n\colon U_n\times U_n\to\wp(F_X)$ is a labelling of edges, \item $E_n:=\{(u,v)\in U_n\times U_n: \ell_n(u,v)\ne\emptyset\}$ is the set of edges with non-empty labels, \item $W_n\subseteq E_n$ is a distinguished set of \emph{witness edges}. \end{itemize} We will make sure that the following inductive conditions are maintained during the construction: \begin{description} \item[RT] $E_n$ is reflexive and transitive. \item[Gen] $W_n$ generates $E_n$ by closing under transitivity. \item[Fun] for every $(u,v)\in E_n$, $\ell_n(u,v)$ is a proper, fundamental filter: there is $\tau$ such that $\ell_n(u,v)=\mathcal{F}(\mathcal{E}\comp\tau\comp\mathcal{E})$. \item[DR] for every $(u,v)\in E_n$,
$\mathcal{E}(\sigma)\subseteq\ell_n(u,u)=\ell_n(v,v)=\mathcal{E}$ for every $\sigma\in\ell_n(u,v)$. \item[Comp] for every $(u,v),(u,w),(w,v)\in E_n$, we have $\ell_n(u,w)\comp\ell_n(w,v)\subseteq\ell_n(u,v)$.
\item[Ide] for every $(u,v)\in E_n$, if $\ide\in\ell_n(u,v)$, then $u=v$. \end{description} Observe that DR implies $\ide\in\ell_n(u,u)=\ell_n(v,v)$ for every $u$ and $v$.
These conditions, with the exception of Ide, will be easily seen to be maintained during the construction. We will check that Ide holds for witness edges $(u,v)\in W_n$ as well, but we will establish the general case for Ide only at the end of the section.
The construction will terminate in $\omega$ steps, yielding $G_\omega=(U_\omega,\ell_\omega,E_\omega,W_\omega)$ where $U_\omega=\bigcup_n U_n$, $\ell_\omega=\bigcup_n \ell_n$, $E_\omega=\bigcup_n E_n$ and $W_\omega=\bigcup_n W_n$.
By the end of the construction we will achieve the following \emph{saturation} condition:
\begin{description}
\item[Sat] for every $(u,v)\in E_\omega$ and $\tau\comp\sigma\in\ell_\omega(u,v)$, we have $\tau\in\ell_\omega(u,w)$ and $\sigma\in\ell_\omega(w,v)$ for some $w\in U_\omega$.
\end{description}
In the 0th step of the step-by-step construction we define $G_0$ by creating a witness edge for our fixed term $\theta$. We choose $u_0,v_0\in\omega$ such that $u_0=v_0$ iff $\theta\le\ide$ is derivable.
We define \begin{align*} \ell_0(u_0,v_0)&=\mathcal{F}(\mathcal{E}\comp\theta\comp\mathcal{E})=\mathcal{F}(\theta)\\ \ell_0(u_0,u_0)&=\ell_0(v_0,v_0)=\mathcal{E}
\end{align*} and we label $(v_0,u_0)$ by $\emptyset$ in case $u_0\ne v_0$. Observe that $\ell_0$ is well defined. Indeed, in case $\theta\le\ide$, we have that $\mathcal{E}=\theta^\uparrow=\mathcal{F}(\theta)$.
All non-empty edges constructed so far will be witness edges: $W_0=E_0$.
It is easy to see that the inductive conditions are true. For Comp we note that, for every $\epsilon,\epsilon'\in\ell_0(u_0,u_0)=\ell_0(v_0,v_0)=\mathcal{E}$, we have $\epsilon\comp\epsilon'=\epsilon\meet\epsilon'\in\mathcal{E}$ by \eqref{eq:fun},
and $\epsilon\comp\theta\comp\epsilon'=\theta\in\ell_0(u_0,v_0)$ by
the definition of $\mathcal{E}$. DR easily follows from \eqref{eq:emon}.
Note that $\theta'\notin\ell_0(u_0,v_0)$ whenever $\mbox{\rm Ax}^i(\meet,\comp,0,\ide)\not\vdash\theta\le\theta'$, since $\theta'\notin\mathcal{F}(\theta)=\theta^\uparrow$.
In the $(n+1)$th step we assume inductively that $G_n=(U_n,\ell_n,E_n,W_n)$ with $U_n\subset \omega$ and $W_n\subseteq E_n\subseteq U_n\times U_n$ has been constructed and that $G_n$ satisfies the inductive conditions.
We also assume that there is a fair scheduling function $\Sigma\colon\omega\to\omega\times\omega\times F_X$, i.e., every element of $\omega\times\omega\times F_X$ appears infinitely often in the range of $\Sigma$.
Assume that $\Sigma(n+1)=(u,v,\tau\comp\sigma)$, $(u,v)\in E_n$ and $\tau\comp\sigma\in\ell_n(u,v)$, otherwise we define $G_{n+1}= G_n$.
Recall that $\ell_n(u,u)=\ell_n(v,v)=\mathcal{E}$ by DR. Let $\rho$ be such that $\ell_n(u,v)=\mathcal{F}(\mathcal{E}\comp\rho\comp\mathcal{E})$, and $\rho'$ be such that $\ell_n(v,u)=\mathcal{F}(\mathcal{E}\comp\rho'\comp\mathcal{E})$ in case $(v,u)\in E_n$ as well.
Thus $\epsilon_0\comp\rho\comp\epsilon_0\le\tau\comp\sigma$
for some subidentity $\epsilon_0\in\mathcal{E}$. Observe that, for any subidentity $\epsilon$, \begin{equation} (\epsilon_0\meet\epsilon)\comp\rho\comp(\epsilon_0\meet\epsilon)\meet\tau\comp\sigma= (\epsilon_0\meet\epsilon)\comp\rho\comp(\epsilon_0\meet\epsilon)\meet (\epsilon_0\meet\epsilon)\comp\tau\comp(\epsilon_0\meet\epsilon)\comp\sigma \label{eq:eps} \end{equation} by using $\epsilon_0\meet\epsilon=(\epsilon_0\meet\epsilon)\comp(\epsilon_0\meet\epsilon)$, \eqref{eq:int2} and~\eqref{eq:con2}.
We will assume that $\ide\notin\mathcal{F}(\mathcal{E}\comp\tau\comp\mathcal{E})$ and $\ide\notin\mathcal{F}(\mathcal{E}\comp\sigma\comp\mathcal{E})$, since we would have the required edges otherwise. Indeed, we have \begin{equation}\label{eq:nonide} \ide\in\mathcal{F}(\mathcal{E}\comp\tau\comp\mathcal{E})\mbox{ implies } \tau\in\ell_n(u,u)\mbox{ and }\sigma\in\ell_n(u,v) \end{equation} because of the following. Assume that $\epsilon'\comp\tau\comp\epsilon'\le\ide$ for some subidentity $\epsilon'\in\mathcal{E}$. Let $\epsilon=\epsilon_0\meet\epsilon'$. Then $\epsilon\comp\rho\comp\epsilon\in\ell_\omega(u,v)$ and
\begin{align*} \epsilon\comp\rho\comp\epsilon&=(\epsilon\comp\rho\comp\epsilon)\meet(\tau\comp\sigma)\\ &=(\epsilon\comp\rho\comp\epsilon)\meet (\epsilon\comp\tau\comp\epsilon\comp\sigma) &&\mbox{by }\eqref{eq:eps}\\ &\le\rho\meet\sigma &&\mbox{by }\epsilon\comp\tau\comp\epsilon\le\ide\\ &\le\sigma&& \end{align*} whence $\sigma\in\ell_n(u,v)$. Next we claim that $\epsilon\comp\tau\comp\epsilon\in\ell_n(u,u)$. Indeed, by using~\eqref{eq:eps},~\eqref{eq:con2}
we get \begin{align*} (\epsilon\comp\tau\comp\epsilon)\comp(\epsilon\comp\rho\comp\epsilon)&= (\epsilon\comp\tau\comp\epsilon)\comp(\tau\comp\sigma\meet \epsilon\comp\rho\comp\epsilon)\\ &=(\epsilon\comp\tau\comp\epsilon)\comp ((\epsilon\comp\tau\comp\epsilon\comp\sigma)\meet \epsilon\comp\rho\comp\epsilon)\\
&=(\epsilon\comp\tau\comp\epsilon\comp\sigma) \meet(\epsilon\comp\rho\comp\epsilon)\\ &=(\tau\comp\sigma) \meet(\epsilon\comp\rho\comp\epsilon)\\ &=\epsilon\comp\rho\comp\epsilon \end{align*} whence $\tau\ge\epsilon\comp\tau\comp\epsilon\in\ell_n(u,u)$ by DR. The case $\ide\in\mathcal{F}(\mathcal{E}\comp\sigma\comp\mathcal{E})$ is treated similarly.
Take a point $w\in\omega\smallsetminus U_n$ and define \begin{align*} \ell_{n+1}(w,w)&=\mathcal{E}\\ \ell_{n +1}(t,w)&=\mathcal{F}(\ell_n(t,u)\comp\tau\comp\mathcal{E})\\ \ell_{n +1}(w,s)&=\mathcal{F}(\mathcal{E}\comp\sigma\comp\ell_n(v,s)) \end{align*} whenever $(t,u),(v,s)\in E_n$. In particular, $$ \ell_{n +1}(u,w)= \mathcal{F}(\mathcal{E}\comp\tau\comp\mathcal{E}) \quad\mbox{and}\quad \ell_{n +1}(w,v)= \mathcal{F}(\mathcal{E}\comp\sigma\comp\mathcal{E}) $$ since $\ell_n(u,u)=\ell_n(v,v)=\mathcal{E}$. By Fun and DR there are $\xi$ and $\chi$ such that $\ell_n(t,u)=\mathcal{F}(\mathcal{E}\comp\xi\comp\mathcal{E})$ and $\ell_n(v,s)=\mathcal{F}(\mathcal{E}\comp\chi\comp\mathcal{E})$. Then using~\eqref{eq:int2} \begin{align*} \ell_{n +1}(t,w)&=\mathcal{F}(\mathcal{E}\comp\xi\comp\mathcal{E}\comp\tau\comp\mathcal{E}) =\mathcal{F}(\mathcal{E}\comp\xi\comp\tau\comp\mathcal{E})\\ \ell_{n +1}(w,s)&=\mathcal{F}(\mathcal{E}\comp\sigma\comp\mathcal{E}\comp\chi\comp\mathcal{E}) =\mathcal{F}(\mathcal{E}\comp\sigma\comp\chi\comp\mathcal{E}) \intertext{and similarly} \ell_{n +1}(w,u)&= \mathcal{F}(\mathcal{E}\comp\sigma\comp\rho'\comp\mathcal{E})\\ \ell_{n +1}(v,w)&= \mathcal{F}(\mathcal{E}\comp\rho'\comp\tau\comp\mathcal{E}) \end{align*} when $(v,u)\in E_n$. Thus the labels are fundamental filters.
For all other edges, we let $\ell_{n+1}(x,y)=\ell_{n}(x,y)$ if $(x,y)\in E_n$, and $\ell_{n+1}(x,y)=\emptyset$ otherwise. The witness edges are those of $G_n$ (i.e., $W_n$)
and $(u,w)$, $(w,v)$ and $(w,w)$. See Figure~\ref{fig:comp3}, where we show typical elements of the labels.
\begin{figure}
\caption{Successor step}
\label{fig:comp3}
\end{figure}
Observe that $E_{n+1}$ is reflexive, transitive and generated by $W_{n+1}$ (e.g., in case $(v,u)\in E_n$, the edge $(w,u)$ can be ``decomposed'' to $(w,v)\in W_{n+1}$ and $(v,u)$ which is generated by $W_n$ by the inductive condition). Ide holds for witness edges, since the new irreflexive witness edges $(u,w)$ and $(w,v)$ avoid $\ide$ by the assumption on $\tau$ and $\sigma$.
Next we check that Comp is maintained as well. Both $\ell_{n+1}(w,w)\comp\ell_{n+1}(w,s)\subseteq\ell_{n+1}(w,s)$ and $\ell_{n+1}(t,w)\comp\ell_{n+1}(w,w)\subseteq\ell_{n+1}(t,w)$ easily follow from the definition of the labels. Similarly we get $\ell_{n+1}(w,s)\comp\ell_{n+1}(s,s)\subseteq\ell_{n+1}(w,s)$ and $\ell_{n+1}(t,t)\comp\ell_{n+1}(t,w)\subseteq\ell_{n+1}(t,w)$ using that $G_n$ satisfies DR and Comp. If $(w,t),(t,w)\in E_{n+1}$, we need $\ell_{n+1}(w,t)\comp\ell_{n+1}(t,w)\subseteq\ell_{n+1}(w,w)$ as well. Let $\chi\in\ell_{n+1}(v,t)$ and $\xi\in\ell_{n+1}(t,u)$ so that $\epsilon\comp\sigma\comp\chi\comp\epsilon\in\ell_{n+1}(w,t)$ and $\epsilon\comp\xi\comp\tau\comp\epsilon\in\ell_{n+1}(t,w)$ for some subidentity $\epsilon\in\mathcal{E}$. Then \begin{align*} \epsilon\comp\sigma\comp\chi\comp\epsilon\comp\xi\comp\tau\comp\epsilon\meet\epsilon& \ge\epsilon\comp\sigma\comp\rho'\comp\tau\comp\epsilon\meet\epsilon\\ &=\epsilon\comp\sigma\comp\tau\comp\rho'\comp\epsilon\meet\epsilon\\ &\ge\epsilon\comp\rho\comp\rho'\comp\epsilon\meet\epsilon\\ &\in\ell_n(u,u)=\mathcal{E} \end{align*} by \eqref{eq:int1} and Comp for $G_n$. Thus $\epsilon\comp\sigma\comp\chi\comp\epsilon\comp\xi\comp\tau\comp\epsilon \in\mathcal{E}=\ell_{n+1}(w,w)$ as desired. The proof of $\ell_{n+1}(t,w)\comp\ell_{n+1}(w,t)\subseteq\ell_{n+1}(t,t)$ is similar. Next we show $\ell_{n+1}(t,w)\comp\ell_{n+1}(w,s)\subseteq\ell_{n+1}(t,s)$. Indeed, we have \begin{align*} \mathcal{F}(\ell_{n}(t,u)\comp\tau\comp\mathcal{E})\comp \mathcal{F}(\mathcal{E}\comp\sigma\comp\ell_{n}(v,s)) &\subseteq \mathcal{F}(\ell_{n}(t,u)\comp\tau\comp\sigma\comp\ell_{n}(v,s))\\ &\subseteq \mathcal{F}(\ell_{n}(t,u)\comp\rho\comp\ell_{n}(v,s))\\ &\subseteq \ell_{n}(t,s)=\ell_{n+1}(t,s) \end{align*} by Comp for $G_n$. For $\ell_{n+1}(s,t)\comp\ell_{n+1}(t,w)\subseteq\ell_{n+1}(s,w)$ we have \begin{align*} \ell_n(s,t)\comp\mathcal{F}(\ell_n(t,u)\comp\tau\comp\mathcal{E}) &\subseteq \mathcal{F}(\ell_{n}(s,t)\comp\ell_n(t,u)\comp\tau\comp\mathcal{E})\\ &\subseteq \mathcal{F}(\ell_{n}(s,u)\comp\tau\comp\mathcal{E})\\ &=\ell_{n+1}(s,w) \end{align*} by Comp for $G_n$, and similarly for $\ell_{n+1}(w,s)\comp\ell_{n+1}(s,t)\subseteq\ell_{n+1}(w,t)$.
Finally we check DR. First let $\xi\in\ell_{n+1}(t,u)$ and $\epsilon\in\mathcal{E}$ so that $\xi\comp\tau\comp\epsilon\in \mathcal{F}(\ell_{n+1}(t,u)\comp\tau\comp\mathcal{E})=\ell_{n+1}(t,w)$, and assume that $\epsilon'\le\ide$ such that $\xi\comp\tau\comp\epsilon\comp\epsilon'= \xi\comp\tau\comp\epsilon$. We show that $\epsilon'\in\mathcal{E}=\ell_{n+1}(w,w)$. We have $\xi\comp\tau\comp\sigma\comp\epsilon\comp\epsilon'= \xi\comp\tau\comp\epsilon\comp\epsilon'\comp\sigma= \xi\comp\tau\comp\epsilon\comp\sigma= \xi\comp\tau\comp\sigma\comp\epsilon\in\ell_{n}(t,v)$, whence $\epsilon'\in\ell_n(v,v)=\mathcal{E}=\ell_{n+1}(w,w)$ by DR for $G_n$. That is, $\mathcal{E}(\xi\comp\tau\comp\epsilon)\subseteq\ell_{n+1}(w,w)$. A similar proof shows DR for $\ell_{n+1}(w,s)$.
This finishes the successor step yielding $G_{n+1}=(U_{n+1},\ell_{n+1},E_{n+1},W_{n+1})$.
After the construction terminates we end up with a labelled structure $G_\omega=(U_\omega,\ell_\omega, E_\omega,W_\omega)$ such that $U_\omega=\bigcup_n U_n$, $\ell_\omega=\bigcup_n\ell_n$, $E_\omega=\bigcup_n E_n$ and $W_\omega=\bigcup_n W_n\subseteq E_\omega$. Observe that $G_{\omega}$ satisfies Sat, since the fair scheduling function $\Sigma$ ensures that every possible composition ``defect'' has been taken care of. But we have not showed yet that Ide is satisfied in general, we will do this shortly.
$G_\omega$ satisfies the other inductive conditions
(since so does every $G_n$).
\subsection{Graphs and algebras}
We define a valuation $\iota$ of variables on $U_\omega\times U_\omega$: \begin{equation}\label{eq:val} \iota(x)=\{(u,v)\in E_\omega: x\in\ell_\omega(u,v)\} \end{equation} for every variable $x\in X$. Let $\mathfrak{A}_\theta$ be the subalgebra of $(\wp(U_\omega\times U_\omega),\meet,\comp,0,\ide)$ generated by $\{\iota(x):x\in X\}$. Next we show that $\mathfrak{A}_\theta\in\R^i(\meet,\comp,0,\ide)$.
Let $\mathfrak{A}_\theta(u,v)$ denote the set of those elements (a filter) that hold at $(u,v)$.
\begin{lemma}\label{lem:gen} For every $u\in U_\omega$ and term $\tau$, \begin{equation}\label{eq:refl} \tau\in\ell_\omega(u,u)\text{ implies }\tau\in\mathfrak{A}_\theta(u,u). \end{equation} \end{lemma}
\begin{proof} We will prove the lemma together with \begin{equation}\label{eq:nrefl} \tau\not\le\ide\text{ and }\tau\in\ell_\omega(u,v)\text{ imply } \tau\in\mathfrak{A}_\theta(u,v) \end{equation} for every $(u,v)\in U_\omega\times U_\omega$. We will use simultaneous induction on the complexity of terms.
The base case, when $\tau$ is a variable, the identity constant $\ide$, or $0$ is straightforward by the definition of the valuation $\iota$ and the fact that the labels are proper filters.
Next assume that $\tau=\sigma\meet\rho$. Recall that $\ell_\omega(u,v)$ is a filter, whence $\tau\in\ell_\omega(u,v)$ implies $\sigma,\rho\in\ell_\omega(u,v)$. First consider the statement~\eqref{eq:nrefl}. If $\tau\not\le\ide$, then also $\sigma,\rho\not\le\ide$, whence we can apply the induction hypothesis (IH) for \eqref{eq:nrefl}, yielding $\sigma,\rho\in\mathfrak{A}_\theta(u,v)$ and $\sigma\meet\rho\in\mathfrak{A}_\theta(u,v)$, since $\mathfrak{A}_\theta(u,v)$ is a filter. The proof for~\eqref{eq:refl} is analogous.
Finally assume that $\tau=\sigma\comp\rho$. We start with showing~\eqref{eq:nrefl}. First assume that $\sigma\le\ide$, whence $\rho\not\le\ide$ (by $\tau\not\le\ide$). Then using~\eqref{eq:nonide} we get that $\sigma\in\ell_\omega(u,u)$ and $\rho\in\ell_\omega(u,v)$. By the IH for~\eqref{eq:refl} and~\eqref{eq:nrefl} we have $\sigma\in\mathfrak{A}_\theta(u,u)$ and $\rho\in\mathfrak{A}_\theta(u,v)$. Thus $\sigma\comp\rho\in\mathfrak{A}_\theta(u,v)$. The case $\rho\le\ide$ is treated similarly. The last case is when $\sigma,\rho\not\le\ide$. By Sat we have $w$ such that $\sigma\in\ell_\omega(u,w)$ and $\rho\in\ell_\omega(w,v)$. Using the IH for~\eqref{eq:nrefl} we get $\sigma\in\mathfrak{A}_\theta(u,w)$ and $\rho\in\mathfrak{A}_\theta(w,v)$, whence $\sigma\comp\rho\in\mathfrak{A}_\theta(u,v)$.
For~\eqref{eq:refl} we argue as follows. First assume that $\sigma\le\ide$. Using~\eqref{eq:nonide} we get that $\sigma,\rho\in\ell_\omega(u,u)$.
By the IH for~\eqref{eq:refl} we get $\sigma,\rho\in\mathfrak{A}_\theta(u,u)$, whence $\sigma\comp\rho\in\mathfrak{A}_\theta(u,u)$. The case for $\rho\le\ide$ is similar. Finally, the case $\sigma,\rho\not\le\ide$ is treated precisely like for~\eqref{eq:nrefl} at the end of the previous paragraph. \end{proof}
\begin{lemma}\label{lem:w1} For every term $\tau$ and for every edge $(u,v)\in U_\omega\times U_\omega$, \begin{equation*}
\tau\in\mathfrak{A}_\theta(u,v)\mbox{ implies }\tau\in\ell_\omega(u,v). \end{equation*} \end{lemma}
\begin{proof} This is an easy induction on the complexity of terms (using Comp for composition). \end{proof}
\begin{lemma}\label{lem:int}
$\mathfrak{A}_\theta\in \R^i(\meet,\comp,0,\ide)$. \end{lemma}
\begin{proof} Clearly, $\mathfrak{A}_\theta$ is an algebra of relations. Recall that we have $\ell_\omega(u,u)=\mathcal{E}(\theta)=\mathcal{E}$ for every $u\in U_\omega$. Then by Lemma~\ref{lem:gen} and Lemma~\ref{lem:w1} we get \begin{equation}\label{eq:uni} \mathfrak{A}_\theta(u,u)=\mathfrak{A}_\theta(v,v)\text{, for every }u,v\in U_\omega. \end{equation} Recall that $\mathfrak{A}_\theta$ is generated by $\{\iota(x): x\in X\}$, thus every element of $\mathfrak{A}_\theta$ is the interpretation of a term. Now assume indirectly that the interpretation of the identity constant $\ide$ in $\mathfrak{A}_\theta$ is not an atom, i.e., there is a term $\tau$ such that its interpretation is a proper, nonempty subset $T$ of $\{(u,u):u\in U_\omega\}$. Then $\tau\in\mathfrak{A}_\theta(u,u)$ iff $(u,u)\in T$, contradicting to \eqref{eq:uni}. Thus $\mathfrak{A}_\theta$ is an integral algebra. \end{proof}
Next we show that $\mathfrak{A}_\theta$ is a witness for the non-derivable equations of the form $\theta\le\theta'$ for our fixed $\theta$.
\begin{lemma}\label{lem:w} For every term $\tau$ and for every edge $(x,y)\in U_\omega\times U_\omega$, \begin{equation*}
\tau\in\ell_\omega(x,y)\mbox{ implies }\tau\in\mathfrak{A}_\theta(x,y). \end{equation*} \end{lemma}
\begin{proof} We already showed the lemma for the case $x=y$ and for the general case with the restriction that $\tau\not\le\ide$, see the proof of Lemma~\ref{lem:gen} above. Then it would suffice to show that $\ide$ cannot occur in the label of the edge $(x,y)$ whenever $x\ne y$. Indeed, our proof will show this (see~\eqref{eq:ind2} below), but we will go through all the cases for the sake of clarity.
We use induction over the ``distance'' between $x$ and $y$. To this end we define $\path(x,y)$ for $(x,y)\in E_\omega$ by recursion. For $(x,y)\in E_0$, we let $\path(x,y)=(x,y)$. Now assume that $(x,y)\in E_{n+1}\smallsetminus E_n$ and we already defined $\path$ for the elements of $E_n$. Assume that $U_{n+1}=U_n\cup \{w\}$ and that we expanded $U_n$ by $w$ because of some $\sigma\comp\rho\in\ell_n(u,v)$. Again we let $\path(x,y)=(x,y)$ for $x,y\in\{u,w,v\}$. Now assume that $t\ne u$, $(t,u)\in E_n$ and $\path(t,u)=(t,z_0,\dots,z_k,u)$. Then we let \begin{equation}\label{eq:path} \path(t,w)=(t,z_0,\dots,z_k,u,w) \end{equation} and define $\path(w,s)$ for $(v,s)\in E_n$ analogously. Finally, $d(x,y)$ is defined as the length of $\path(x,y)$.
The base case is when $(x,y)\in W_\omega$ is a witness edge: $d(x,y)=1$.
The base case will be established by induction on terms. The case of a variable is straightforward by the definition~\eqref{eq:val} of the valuation $\iota$. The case of $0$ follows from the fact that we used proper filters as labels. Next assume that $\tau$ is the constant $\ide$. Observe that $\ide\in\ell_\omega(x,y)$ implies $x=y$, since irreflexive witness edges avoid $\ide$. Then $\ide\in\mathfrak{A}_\theta(x,x)$, since $\mathfrak{A}_\theta$ is representable. The case for $\meet$ follows from the fact that we used filters as labels. For the case of $\comp$ assume that $\tau$ is $\sigma\comp\rho$ and that $\sigma\comp\rho\in\ell_\omega(x,y)$. Recall that we had either $\sigma\in\ell_\omega(x,x)$ and $\rho\in\ell_\omega(x,y)$, or $\sigma\in\ell_\omega(x,y)$ and $\rho\in\ell_\omega(y,y)$, or we constructed witness edges $(x,z)$ and $(z,y)$ such that $\sigma\in\ell_\omega(x,z)$ and $\rho\in\ell_\omega(z,y)$.
Thus $\sigma\in\mathfrak{A}_\theta(x,z)$ and $\rho\in\mathfrak{A}_\theta(z,y)$ by the IH, whence $\sigma\comp\rho\in\mathfrak{A}_\theta(x,y)$ as desired.
For the inductive case assume that $d(x,y)=k+1>1$. Again we use induction on terms. The cases for variables and $0$ are as in the base step $(x,y)\in W_\omega$. For the case of $\ide$ we show that $G_\omega$ in fact satisfies Ide:
\begin{equation}\label{eq:ind2} x\ne y\mbox{ implies }\ide\notin\ell_\omega(x,y). \end{equation}
Assume that $(x,y)$ was created in the $(n+1)$th step of the construction. Recall that during the construction we defined $\ell_{n+1}(x,y)$ as $\mathcal{F}(\ell_{n+1}(x,z)\comp\ell_{n+1}(z,y))$ for some $z$ such that either $(x,z)\in E_n$ and $(z,y)\in W_{n+1}$, or $(x,z)\in W_{n+1}$ and $(z,y)\in E_n$. Wlog assume the former.
Then, using the notation in Figure~\ref{fig:comp3}, we have $x=t$, $y=w$ and $z=u$. Recall that $\path(t,w)$ is defined by adding the step $(u,w)$ at the end of $\path(t,u)$, see~\eqref{eq:path}. Hence $d(x,z)=d(t,u) < d(t,w)=d(x,y)$.
Let $\ell_{n+1}(x,z)=\mathcal{F}(\mathcal{E}\comp\rho_1\comp\mathcal{E})$ and $\ell_{n+1}(z,y)=\mathcal{F}(\mathcal{E}\comp\rho_2\comp\mathcal{E})$, whence $\ell_{n+1}(x,y)=\mathcal{F}(\mathcal{E}\comp\rho_1\comp\rho_2\comp\mathcal{E})$ by the construction.
Since $d(x,z)<d(x,y)$ and $(z,y)\in W_{n+1}$, we can apply the induction hypothesis: $\epsilon\comp\rho_1\comp\epsilon\in\mathfrak{A}_\theta(x,z)$ and $\epsilon\comp\rho_2\comp\epsilon\in\mathfrak{A}_\theta(z,y)$ for every $\epsilon\in\mathcal{E}$. Thus we get $\epsilon\comp\rho_1\comp\rho_2\comp\epsilon\in\mathfrak{A}_\theta(x,y)\not\ni\ide$. Assume, for a contradiction, that $\ide\in\ell_{n+1}(x,y)$. Then $\epsilon\comp\rho_1\comp\rho_2\comp\epsilon\le\ide$ is derivable from $\mbox{\rm Ax}^i(\meet,\comp,0,\ide)$ for some $\epsilon\in\mathcal{E}$. But $\epsilon\comp\rho_1\comp\rho_2\comp\epsilon\le\ide$ is not valid in $\R^i(\meet,\comp,0,\ide)$ as witnessed by $\mathfrak{A}_\theta$, a contradiction. Hence~\eqref{eq:ind2} holds. Thus we have that $\ide\in\ell_\omega(x,y)$ only if $x=y$, and then $\ide\in\mathfrak{A}_\theta(x,y)$ as required.
The case of $\meet$ follows as above. Finally assume that $\tau=\sigma\comp\rho$ and $\sigma\comp\rho\in\ell_{n}(x,y)$. We have to consider three cases. The first is when $\sigma\in\ell_{n}(x,x)$ and $\rho\in\ell_{n}(x,y)$. Then $\sigma\in\mathfrak{A}_\theta(x,x)$ (since $(x,x)\in W_\omega$) and $\rho\in\mathfrak{A}_\theta(x,y)$ (since $\rho$ is a simpler term than $\sigma\comp\rho$). Thus $\sigma\comp\rho\in\mathfrak{A}_\theta(x,y)$. The case $\sigma\in\ell_{n}(x,y)$ and $\rho\in\ell_{n}(y,y)$ is similar. Finally, if neither of the above cases apply, then we created witness edges $(x,z)$ and $(z,y)$ such that $\sigma\in\ell_{\omega}(x,z)$ and $\rho\in\ell_{\omega}(z,y)$. By the base case we get $\sigma\in\mathfrak{A}_\theta(x,z)$ and $\rho\in\mathfrak{A}_\theta(z,y)$, whence $\sigma\comp\rho\in\mathfrak{A}_\theta(x,y)$ as desired. \end{proof}
\begin{lemma}\label{lem:wit} For every $\theta'$ such that $\mbox{\rm Ax}^i(\meet,\comp,0,\ide)\not\vdash \theta\le\theta'$, we have $\R^i(\meet,\comp,0,\ide)\not\models\theta\le\theta'$. \end{lemma}
\begin{proof} By Lemma~\ref{lem:int} we have $\mathfrak{A}_\theta\in \R^i(\meet,\comp,0,\ide)$. By the initial step of the step-by-step construction we have $\theta\in\ell_\omega(u_0,v_0)=\mathcal{F}(\theta)$ and $\theta'\notin\ell_\omega(u_0,v_0)$. So by Lemma~\ref{lem:w} we get $\theta\in\mathfrak{A}_\theta(u_0,v_0)$ and by Lemma~\ref{lem:w1} we have $\theta'\notin\mathfrak{A}_\theta(u_0,v_0)$. \end{proof}
\begin{remark}[Representing the free algebra] In the above construction we fixed a term $\theta$ and constructed an algebra $\mathfrak{A}_\theta\in\R^i(\meet,\comp,0,\ide)$. We can repeat the same construction for every non-zero element $\theta$ of the free algebra $\mathfrak{F}_X$, resulting in $\mathfrak{A}_\theta\in\R^i(\meet,\comp,0,\ide)$. It is not difficult to show that $\mathfrak{F}_X$ can be embedded into $\prod_{\theta\ne 0}\mathfrak{A}_\theta\in\V(\R^i(\meet,\comp,0,\ide))$. \end{remark}
\section{Conclusions}
The varieties $\V(\R(\Lambda))$ generated by algebras of binary relations of the similarity types $\Lambda=(\meet,\comp,\ide)$ and $\Lambda=(\join,\meet,\comp,\ide)$ were stated to be finitely axiomatizable in \cite{AM-axi-11} (Theorem~4.3 and Theorem~4.1(1), respectively), but their proofs relied on false lemmas. See \cite{AM-err-14}.
In more detail, the third case of Definition~4.6 is ambiguous, and Lemmas~4.7 and~4.8 are not true. These lemmas are used in the proof of Theorem~4.3. As a consequence, the proof of Theorem~4.3 breaks down for the equation $\ide\meet x\comp y\le x\comp(\ide\meet y\comp x)\comp y$. This equation is easily seen to be valid, but so far we did not manage to derive it from the axioms of Theorem~4.3. In fact, we conjecture that this equation does not follow from the axioms presented in \cite{AM-axi-11}. Theorem~4.3 is used in the proof of Theorem~4.1(1).
Since the main results of this paper, Theorem~\ref{thm:main} and~\ref{thm:dl}, give only a solution for the special case of integral algebras, we state the following as an open problem.
\begin{problem}\label{prob} Are the varieties generated by algebras of binary relations of the similarity types $(\meet,\comp,\ide)$ and $(\join,\meet,\comp,\ide)$ finitely axiomatizable? \end{problem}
In \cite{AMN-equ-11}, we stated that $\V(\Lang(\join,\meet,\comp 0, \ide))$ is finitely axiomatizable by a certain set of axioms, Corollary~3.7. The proof was based on the theorems of \cite{AM-axi-11} mentioned above, hence it is not correct. Luckily Theorem~\ref{thm:lang} above provides a satisfactory solution in this case.
Finally we mention a corollary related to commutative algebras. Let $\R^c(\Lambda)$ denote the class of relation algebras of signature $\Lambda$ that satisfy the commutativity axiom \begin{equation}\label{eq:comm} x\comp y = y\comp x \end{equation} for every $x$ and $y$. It is easy to see that $\V(\R^c(\meet,\comp, 0, \ide))$ satisfies $\mbox{\rm Ax}^i(\meet,\comp,0,\ide)$. Thus our construction can be applied in this case as well.
\begin{corollary} The varieties $\V(\R^c(\meet,\comp, 0, \ide))$, $\V(\R^c(\join,\meet,\comp, 0, \ide))$ are finitely axiomatized by commutativity~\eqref{eq:comm} and $\mbox{\rm Ax}(\meet,\comp,0,\ide)$, $\mbox{\rm Ax}(\join,\meet,\comp,0,\ide)$, respectively. \end{corollary}
\paragraph{Acknowledgements} The author is grateful to Hajnal Andr\'eka, Robin Hirsch and Istv\'an N\'emeti for helpful comments.
\end{document} | arXiv |
OSA Publishing > Optica > Volume 7 > Issue 12 > Page 1820
Prem Kumar, Editor-in-Chief
Free-electron shaping using quantum light
Valerio Di Giulio and F. Javier García de Abajo
Valerio Di Giulio1 and F. Javier García de Abajo1,2,*
1ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
2ICREA-Institució Catalana de Recerca i Estudis Avançats, Passeig Llus Companys 23, 08010 Barcelona, Spain
*Corresponding author: [email protected]
Valerio Di Giulio https://orcid.org/0000-0002-0948-4625
F. Javier García de Abajo https://orcid.org/0000-0002-4970-4565
V Di Giulio
F García de Abajo
•https://doi.org/10.1364/OPTICA.404598
Valerio Di Giulio and F. Javier García de Abajo, "Free-electron shaping using quantum light," Optica 7, 1820-1830 (2020)
Probing quantum optical excitations with fast electrons (OPTICA)
Quantum correlations in electron microscopy (OPTICA)
Temporal shaping and time-varying orbital angular momentum of displaced vortices (OPTICA)
In field scattering
Laser scattering
Talbot effect
Ultrashort pulses
Original Manuscript: August 4, 2020
Revised Manuscript: November 7, 2020
Manuscript Accepted: November 11, 2020
ELECTRON DENSITY MATRIX PRODUCED UPON PINEM INTERACTION
ELECTRON PULSE COMPRESSION WITH DIFFERENT OPTICAL MODE STATISTICS
EFFECT OF THE ELECTRON DENSITY MATRIX ON THE EXCITATION OF A SAMPLE
Controlling the wave function of free electrons is important to improve the spatial resolution of electron microscopes, the efficiency of electron interaction with sample modes of interest, and our ability to probe ultrafast materials dynamics at the nanoscale. In this context, attosecond electron compression has been recently demonstrated through interaction with the near fields created by scattering of ultrashort laser pulses at nanostructures followed by free-electron propagation. Here, we show that control over electron pulse shaping, compression, and statistics can be improved by replacing coherent laser excitation by interaction with quantum light. We find that compression is accelerated for fixed optical intensity by using phase-squeezed light, while amplitude squeezing produces ultrashort double-pulse profiles. The generated electron pulses exhibit periodic revivals in complete analogy to the optical Talbot effect. We further reveal that the coherences created in a sample by interaction with the modulated electron are strongly dependent on the statistics of the modulating light, while the diagonal part of the sample density matrix reduces to a Poissonian distribution regardless of the type of light used to shape the electron. The present study opens a new direction toward the generation of free-electron pulses with additional control over duration, shape, and statistics, which directly affect their interaction with a sample.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
The exploration of ultrafast phenomena generally relies on the use of short probe pulses, such as those provided by femtosecond visible-infrared lasers and attosecond x-ray sources [1 –3]. Electrons can potentially reach much shorter durations than light for typical beam energies in the ${10^2} - {10^5} \text{eV}$ range, as they are characterized by oscillation periods of 20–0.02 as. Electron pulse compression is also capital for free-electron lasers [4], relying on the $\propto {N^2}$ superradiance emission produced by $N$ electrons when acting as a single point charge. With applications such as imaging, spectroscopy, and light generation in view, strong interest has arisen in manipulating the free-electron density matrix using light.
Triggered by the advent of the so-called photon-induced near-field electron microscopy (PINEM) [5], a long series of experimental [5 –25] and theoretical [26 –34] studies has demonstrated that interaction with the optical near fields scattered from illuminated nanostructures provides an efficient way to manipulate the temporal and spatial distribution of free electrons. In PINEM, electron and light pulses are made to interact in the presence of a sample, giving rise to multiple photon exchanges between the optical field and the electron, and leading to comb-like energy spectra characterized by sidebands that are associated with different numbers of exchanged photons and separated from the incident electron energy by a multiple of the photon energy. Recent experiments have measured hundreds of such sidebands produced through suitable combinations of sample geometry and illumination conditions [23,24]. Additionally, electron pulse compression has been observed by free propagation of PINEM-modulated electrons over a sufficiently long distance [14,17,20,21]. The electron transforms into a series of pulses with duration down to the attosecond regime [14,17], which can be made even smaller by increasing the strength of the PINEM light [29].
While this type of electron–light interaction affects only the longitudinal part of the electron wave function, lateral control can be achieved either by the use of electron phase masks [35 –38] or through modulating the optical field with a transverse spatial resolution limited by the light wavelength and, more generally, by the polariton wavelength when relying on the excitation of optical modes in material surfaces. By analogy to elastic electron diffraction by light gratings in free space (the Kapitza–Dirac effect [39 –41]), which has been shown to also enable the formation of vortex beams [42], surface-plasmon standing waves can produce intense inelastic electron diffraction [30], as confirmed by the observation of discrete electron beam deflection upon absorption or emission of a given number of photons reflected from a thin metal plate [19]. Similarly, optical near fields can transfer orbital angular momentum [31], also demonstrated through the synthesis and observation of vortex electron beams produced by inelastic interaction with chiral near fields [22]. As a practical application of these phenomena, lateral phase imprinting on electron beams through optical fields has been recently proposed to provide a viable approach to aberration correction and lateral electron beam profiling [43].
By sweeping the photon energy of the light used for PINEM interaction, the near field experienced by the electrons undergoes amplitude modulations that map the optical response of the sample. This strategy has been proposed as a form of spectrally resolved microscopy that can combine the subnanometer spatial focusing of electron beams [44] with an excellent energy resolution limited by the spectral width of the light source [45,46]. A first demonstration of this possibility has enabled spatial mapping of plasmons in silver nanowires with ${\sim}20\; \text{meV}$ energy resolution without any need for electron monochromators [18], a result that is rivaling the energy resolution achieved through state-of-the-art electron energy-loss spectroscopy [47].
The above studies rely on coherent light, such as that generated by laser sources, while an extension to quantum optical fields has been recently predicted to introduce quantum effects in the electron spectra [48]. Quantum light thus presents an opportunity to further manipulate the electron wave function in applications such as pulse compression and modulation of the electron statistics.
Here, we show that a wide range of electron statistics can be reached through interaction of free electrons with quantum light. Besides changing the focusing properties of the optically modulated electrons, this interaction reveals a strong dependence of the electron density matrix on the statistics of the light field, which can be observed in a self-interference configuration setup. Specifically, we show that interaction with phase-squeezed and minimum-phase-uncertainty (MPU) light sources produces faster compression of the electron, while amplitude-squeezed light gives rise ultrashort double-pulse electron profiles. Additionally, we find that the interaction of the modulated electron with a target produces a Poissonian distribution of sample excitations with off-diagonal coherences that are strongly dependent on the statistics of the light used to modulate the electron. Besides the fundamental interest of this wealth of phenomena, we envision applications in the control of electron compression and in the generation of light with nontrivial statistics.
2. ELECTRON DENSITY MATRIX PRODUCED UPON PINEM INTERACTION
A. Quantum PINEM Interaction
Free-electron–light interaction has been extensively studied under the assumption of classical illumination [26,27]. An extension to describe the quantum evolution of the joint electron–light state has been recently presented [48], which we use here to investigate the modification produced in the electron density profile following propagation after PINEM interaction with nonclassical light. We first provide a succinct summary of this quantum formalism.
We consider the sample response to be dominated by a single bosonic optical mode oscillating at frequency ${\omega _0}$ and characterized by an electric-field distribution ${\vec {\cal E}_0}({\bf r})$ defined as either a normal [49] or a quasi-normal [50] bosonic mode. In addition, we assume that the electron always consists of a superposition of states with relativistic momentum and energy tightly focused around $\hbar {{\bf k}_0}$ and ${E_0}$ (i.e., having small uncertainties compared with $\hbar {\omega _0}/v$ and $\hbar {\omega _0}$, respectively, where $v$ is the electron velocity). Also, we ignore nonunitary elements in the dynamics by considering that the electron–light interaction happens on a fast time scale compared with the decay of the bosonic mode. These assumptions allow us to linearize the electron kinetic energy operator (nonrecoil approximation). Starting from the Dirac equation [51] and following an approach inspired by quantum optics methods [52] with an electromagnetic gauge in which the scalar potential is zero, the effective Hamiltonian of the system can be approximated by the noninteraction and interaction pieces [48],
(1a)$${\hat {\cal H}_0} = \hbar {\omega _0}{a^\dagger}a + {E_0} - \hbar {\bf v} \cdot (\text{i}\nabla + {{\bf k}_0}),$$
(1b)$${\hat {\cal H}_1} = - \text{i}(e{\bf v}/{\omega _0}) \cdot \left[{{{\vec {\cal E}}_0}({\bf r})a - \vec {\cal E}_0^*({\bf r}){a^\dagger}} \right],$$
respectively, where $a$ and ${a^\dagger}$ are annihilation and creation operators of the bosonic optical mode, and ${\bf v} = \hbar {{\bf k}_0}/{E_0} = v \hat{\bf z}$ is the electron velocity vector, taken to be along $\hat{\bf z}$. We remark that the aforementioned QED model accurately reproduces the electron-field dynamics when spin-flips, ponderomotive forces, and electron recoil can be safely disregarded. However, in situations departing from these conditions, the full minimal-coupling Hamiltonian has to be considered, and thus, numerical integration provides a more suitable method to explore the resulting physics [53 –55]. We can then write the solution for the electron-optical mode wave function as a sum of energy sidebands, each of them describing the amplitude associated with a net exchange of $\ell$ quanta with the optical mode ($\ell \gt 0$ for electron energy gain and $\ell \lt 0$ for loss). More precisely, we have (see Ref. [48] and Appendix A)
(2)$$|\psi ({\bf r},t)\rangle = {\psi _{\text{inc}}}({\bf r},t)\sum\limits_{\ell = - \infty}^\infty \sum\limits_{n = 0}^\infty {\text{e}^{\text{i}{\omega _0}[\ell (z/v - t) - nt]}}f_\ell ^n({\bf r})|n\rangle ,$$
where ${\bf r}$ denotes the electron coordinate, $|n\rangle$ runs over Fock states of the optical field, ${\psi _{\text{inc}}}({\bf r},t)$ is the incident electron wave function, and the amplitude coefficients admit the closed-form expression
(3)$$\begin{split}f_\ell ^n & = {\text{e}^{\text{i}(\chi + \ell \text{arg}\{- {\beta _0}\})}} {\alpha _{n + \ell}} F_\ell ^n, \\ F_\ell ^n & = |{\beta _0}{|^\ell}{\text{e}^{- |{\beta _0}{|^2}/2}}\sqrt {(n + \ell)!n!} \sum\limits_{n^\prime = \text{max}\{0, - \ell \}}^n \frac{{{{(- |{\beta _0}{|^2})}^{{n^\prime}}}}}{{n^\prime !(\ell + n^\prime)!(n - n^\prime)!}},\end{split}$$
$${\beta _0}({\bf R},z) = \frac{e}{{\hbar {\omega _0}}}\int_{- \infty}^z \text{d}z^\prime {{\cal E}_{0,z}}({\bf R},z^\prime){\text{e}^{- \text{i}{\omega _0}z^\prime /v}}$$
acting as a single-mode coupling coefficient and $\chi = (- e/\hbar {\omega _0})\int_{- \infty}^z \text{d}z^\prime \text{Im}\{\beta _0^*({\bf R},z^\prime){{\cal E}_{0,z}}({\bf R},z^\prime){\text{e}^{- \text{i}{\omega _0}z^\prime /v}}\}$ representing a global phase that is irrelevant in the present study. A dependence on lateral coordinates ${\bf R} = (x,y)$ is imprinted by the spatial distribution of the optical mode field. In the initial state (i.e., before quanta exchanges), only $\ell = 0$ terms are present, so we can write $f_\ell ^n(z \to - \infty) = {\delta _{\ell 0}}{\alpha _n}$, where the amplitudes ${\alpha _n}$ define the starting optical boson field, which must satisfy the normalization condition
(4)$$\sum\limits_n |{\alpha _n}{|^2} = 1.$$
Interestingly, the number of excitations $n^\prime = n + \ell$ is conserved along the temporal evolution of the system [48], thus allowing us to propagate each initial $n^\prime $ component separately and multiply it by the initial boson amplitude ${\alpha _{n + \ell}}$ when writing Eq. (3). Because the expansion coefficients defined in this equation are obtained from the evolution operator [48], they satisfy the normalization condition, $\sum\nolimits_{\ell n} |f_\ell ^n{|^2} = \sum\nolimits_{\ell n^\prime} |{\alpha _{{n^\prime}}}F_\ell ^{n^\prime - \ell}{|^2} = 1$, for any optical field, which leads to the following condition:
(5)$$\sum\limits_\ell {(F_\ell ^{n - \ell})^2} = 1,$$
satisfied for any $n$.
Electron propagation prior to interaction is described through the linearized Hamiltonian ${\hat {\cal H}_0}$, which essentially assumes that the electron beam is well collimated and energy dispersion is negligible in the PINEM interaction region, such that we can write
$${\psi _{\text{inc}}}({\bf r},t) = {\text{e}^{\text{i}{{\bf k}_0} \cdot {\bf r} - \text{i}{E_0}t/\hbar}}\phi ({\bf r} - {\bf v}t),$$
where $\phi$ is a slowly varying function of relative position ${\bf r} - {\bf v}t$. Importantly, Eq. (3) prescribes that the evolution of the electron-boson system is uniquely determined by the nondimensional coupling parameter ${\beta _0}$ in combination with the amplitudes ${\alpha _n}$ defining the initial optical wave function. In what follows, we assume no dependence on ${\bf R}$ (see below) and set ${\beta _0} \equiv {\beta _0}(z \to \infty)$ because we are interested in studying free-electron propagation after PINEM interaction has taken place, even though this dependence plays a fundamental role in the observed transfer of orbital angular momentum between photons and electrons [22], and in addition, it could be useful to correct electron beam aberrations [43]. Nevertheless, the coefficients of the quantum light state in Eq. (3) could provide an additional knob to further intertwine longitudinal and transverse electron degrees of freedom beyond what is possible using classical light. Additionally, they could affect the maximum achievable probability associated with specific PINEM sidebands, as well as the dependence on pulse duration, which also deserve further study.
B. Effect of Free Propagation
Our purpose is to investigate the electron characteristics after free propagation over a macroscopic distance of several mm from the PINEM interaction region [see Fig. 1(a)]. We identify in Eq. (2) a propagation phase ${\text{e}^{\text{i}{k_\ell}z}}$ associated with each $\ell$ sideband, in which the electron wave vector is replaced by its linearized nonrecoil version ${k_\ell} \approx {k_0} + \ell {\omega _0}/v$. While this approximation does accurately describe propagation over the relatively small extension of the PINEM interaction region, the exact expression
(6)$$\begin{split}{k_\ell} &= {\hbar ^{- 1}}\sqrt {E_\ell ^2/{c^2} - m_{\text{e}}^2{c^2}} \\ &\approx {k_0} + \ell {\omega _0}/v - 2\pi {\ell ^2}/{z_T} + \cdots \end{split}$$
needs to be used to deal with arbitrarily long propagation distances $z$, where the second-order correction, characterized by a distance
(7)$${z_T} = 4\pi {m_{\text{e}}}{v^3}{\gamma ^3}/\hbar \omega _0^2$$
(e.g., ${z_T} \approx 159\;\text{mm} $ for $\hbar {\omega _0} = 1.5\;\text{eV} $ and 100 keV electrons), is sufficiently accurate under the conditions here considered, giving rise to numerical results that are indistinguishable from the full expression in the examples shown below.
Fig. 1. Talbot effect and electron compression with classical light. (a) An electron Gaussian wave packet (green) is transformed through PINEM interaction followed by propagation along a distance $z$ into a substantially modified electron density profile in the propagation-distance-shifted time $\tau = t - t_p -z/v$ due to superposition of different energy components. (b) Electron density profile (vertical $\tau$ coordinate) as a function of propagation distance $z$ (horizontal axis) after PINEM interaction with coherent light. We consider $100\;\text{keV} $ electrons, a photon energy $\hbar {\omega _0} = 1.5\;\text{eV} $, and a coupling coefficient ${|\beta | = 5}$. Trains of compressed electron pulses are periodically observed at discrete multiple values of the Talbot propagation distance ${z_T}$. (c)–(e) Details of the $\tau - z$ map in (b) corresponding to the color-matched square regions of $z$ width $\Delta = 4\;\text{mm} $. (f) Same as (e), but for $z$ near $2{z_{{T}}}$.
Our purpose is to study electron propagating and dismiss any entanglement with the PINEM optical field. We thus consider the electron density matrix, obtained from the pure-joint-state density matrix $|\psi (z,t)\rangle \langle \psi (z^\prime ,t)|$ by tracing out the optical degrees of freedom,
(8)$$\rho (z,z^\prime ,t) = \sum\limits_{n = 0}^\infty {\psi _n}(z,t)\psi _n^*(z^\prime ,t),$$
$${\psi _n}(z,t) = \phi (z - vt)\sum\limits_{\ell = - \infty}^\infty {\alpha _{n + \ell}} F_\ell ^n{\text{e}^{\text{i}{k_\ell}z - \text{i}\ell {\omega _0}(t - {t_p})}},$$
where the phase of ${\beta _0}$ enters only through a time shift ${t_p} = \text{arg}\{- {\beta _0}\} /{\omega _0}$. We remark here that the mathematical operation of tracing out the degrees of freedom associated with the photonic mode to obtain a density matrix for the electron subsystem is physically justified by the fact that this operation ensures the correct measurement statistics if one only needs to measure electron properties (i.e., without performing any measurement on the rest of the system) [56].
We note that diffraction effects involving the transverse evolution of the wave function are disregarded. Under attainable experimental conditions, an initial 100 keV electron beam with $\varphi \sim 50\; \unicode{x00B5}\text{rad}$ divergence, focused to a $2/{k_0}\varphi \sim 25\;\text{nm} $ spot over the PINEM interaction region, becomes just a factor ${\sim}2$ wider after free propagation over a distance $z \sim 1\;\text{mm} $ due to diffraction. In addition, the results here presented are valid under the assumption that $\phi (z - vt)$ involves a sufficiently narrow wave vector decomposition to neglect corrections beyond the linear energy dependence of the wave vector during the propagation distances under consideration, so $\phi$ enters the electron density matrix just as a broad envelope factor. However, we note that these assumptions may break in scenarios involving slow electrons (${E_0} \mathbin{\lower.3ex\hbox{$\buildrel \lt \over{\smash{\scriptstyle\sim}\vphantom{_x}}$}} {10^2} \;\text{eV}$) or very strong electron-field coupling, in which the ponderomotive force can lead to a non-negligible beam spreading after interaction with the sample [55].
C. Talbot Effect and Periodicity of the Density Matrix
Retaining just up to ${\ell ^2}$ corrections in Eq. (6) for ${k_\ell}$ and considering relative positions $|z - z^\prime | \ll {z_T}$, we can recast the electron density matrix [Eq. (8)] as
$$\rho (z,z^\prime ,t) = {\text{e}^{\text{i}{k_0}(z - z^\prime)}}\phi (z - vt){\phi ^*}(z^\prime - vt)\tilde \rho (z,\tau ,\tau ^\prime),$$
(9)$$\begin{split}\tilde \rho (z,\tau ,\tau ^\prime)& = \sum\limits_{n\ell \ell ^\prime} {\alpha _{n + \ell}}\alpha _{n + \ell ^\prime}^*\;F_\ell ^nF_{\ell ^\prime}^n \\ &\quad \times {\text{e}^{2\pi \text{i}\left[{({{\ell ^\prime}^2} - {\ell ^2})z/{z_T} + (\ell ^\prime \tau ^\prime - \ell \tau)/{\tau _0}} \right]}},\end{split}$$
$\tau = t - {t_p} - z/v$, and $\tau ^\prime = t - {t_p} - z^\prime /v$. Disregarding the trivial phase propagation factor ${\text{e}^{\text{i}{k_0}(z - z^\prime)}}$ and the slowly varying envelope introduced by $\phi$, the density matrix is periodic in both of the time-shifted coordinates $\tau$ and $\tau ^\prime $ with the same period as the light optical cycle ${\tau _0} = 2\pi /{\omega _0}$. Additionally, we find that $\tilde \rho (z,\tau ,\tau ^\prime)$ portrays a periodic pattern as a function of propagation distance $z$ similar to the Talbot effect [57 –61], with a period given by ${z_T}$ [Eq. (7)].
To illustrate this effect, we plot in Fig. 1(b) the diagonal elements $\rho (z,z,t) = \sum\nolimits_{n = 0}^\infty {| {{\psi _n}(z,t)} |^2}$ normalized to the envelope density $|\phi (z - vt{)|^2}$ for coherent light illumination, which represent the scaled electron density profile as a function of time and propagation distance $z$ from the PINEM interaction region, calculated in the high-fluence classical limit (see below). Incidentally, off-diagonal elements are also considered and represented below in Fig. 4. The plot clearly reveals a train of temporally focused electron pulses at $z \sim 1.5\;\text{mm} $, followed by a series of focusing revivals at intervals of ${z_T} \approx 159\;\text{mm} $ and accompanied by temporally shifted revivals at fractional values of the Talbot distance ${z_T}$ [62] [see Fig. 1(c-f)].
3. ELECTRON PULSE COMPRESSION WITH DIFFERENT OPTICAL MODE STATISTICS
Before analyzing the effect of light statistics in the evolution of the electron after PINEM interaction, we remark that the previous formalism is only valid for pure initial optical states, whose density matrix is given by $\sum\nolimits_{{nn^\prime}} {\alpha _n}\alpha _{{n^\prime}}^*|n\rangle \langle n^\prime |$. In contrast, for a perfect mixture (i.e., an initial optical density matrix $\sum\nolimits_n |{\alpha _n}{|^2}|n\rangle \langle n|$ with no coherences), the outcome of interaction and propagation has to be separately calculated for each Fock state $|n\rangle$ and then averaged incoherently. Using the normalization conditions of Eqs. (4) and (5), we find an electron density matrix $\tilde \rho (z,\tau ,\tau) = 1$, which is not altered due to interference between different energy components after PINEM interaction. We note that a well-defined optical Fock state belongs to this category and thus does not produce changes in the electron density matrix either.
A. High-Fluence and Classical Limits
Electron coupling to a single optical mode is generally weak and therefore characterized by a small coupling coefficient $|{\beta _0}| \ll 1$ (e.g., we set $|{\beta _0}| = 0.2$ here, as a feasible value for coupling to Mie and plasmon modes in nanoparticles [48]). Still, a strong PINEM effect can be produced with a high average number of photons $\bar n = \sum\nolimits_n n|{\alpha _n}{|^2}$, while only sidebands $|\ell | \ll \bar n$ can then be efficiently populated. In this limit, using the Stirling formula to approximate the factorials containing $n$ in Eq. (3), we find (see Appendix B)
(10)$$F_\ell ^n \approx {J_\ell}(2\sqrt n |{\beta _0}|).$$
Additionally, if the optical mode is prepared in a coherent state (e.g., by exciting it with laser light), its population follows a Poissonian distribution $|{\alpha _n}{|^2} = {\text{e}^{- \bar n}} {\bar n^n}/n!$, which approaches a normal distribution [63] $|{\alpha _n}{|^2} \approx {\text{e}^{- {{(n - \bar n)}^2}/2\bar n}}/\sqrt {2\pi \bar n}$ for $\bar n \gg 1$. Introducing this expression in Eq. (9), approximating $n \approx \bar n$ in Eq. (10), neglecting $\ell $ and $\ell^\prime $ in front of $n$, and using the normalization condition $\sum\nolimits_n |{\alpha _n}{|^2} = 1$, we can write the density matrix in the high-fluence classical limit as
$$\tilde \rho (z,\tau ,\tau ^\prime) \approx {\psi _{\text{cl}}}(z,\tau)\psi _{\text{cl}}^*(z,\tau ^\prime),$$
$${\psi _{\text{cl}}}(z,\tau) = \sum\limits_\ell {J_\ell}(2|\beta |){\text{e}^{- 2\pi \text{i}({\ell ^2}z/{z_T} + \ell \tau /{\tau _0})}}$$
(11)$$\beta = \sqrt {\bar n} {\beta _0}$$
is the effective coupling coefficient, which is proportional to the light amplitude used to excite the optical mode. This result is consistent with previous theoretical [8,29] and experimental [17,21] studies of free propagation after high-fluence classical PINEM interaction. Electron compression and Talbot revivals in this limit are shown in Fig. 1(b) for coherent illumination with $|{\beta _0}| = 0.2$ and $|\beta| = 5$, while zooms of the focal regions are presented in Figs. 1(c)1(f) and 2(a).
Fig. 2. Electron compression using squeezed light. (a)–(d) Evolution of the electron density profile following PINEM interaction with (a) classical, (b) MPU, (c) phase-squeezed, and (d) amplitude-squeezed light using a single-mode coupling coefficient $|{\beta _0}| = 0.2$ and average population $\bar n = 625$ (i.e., $|\beta | = \sqrt {\bar n} |{\beta _0}| = 5$). (e) FWHM [see (a)] of the compressed electron density in (a)–(d) as a function of propagation distance $z$. (f) Minimum in the FWHM along the curves in (e) as a function of coupling coefficient $|\beta |$ (varying $|{\beta _0}|$ and keeping $\bar n = 625$). We consider $100\;\text{keV} $ electrons and a $1.5\;\text{eV} $ photon energy.
Interestingly, for any population of the optical mode that is smooth and strongly peaked around ${\bar n}\gg 1$, we can approximate $\alpha_{n + \ell} \approx \alpha_n $ for $|\ell| \ll n$, so the wave function completely separates into light and electron components in Eq. (2) for the PINEM interaction region, thus becoming $|\psi({\bf r}, t) \rangle \approx \{ \sum_{n-0}^{\infty} \alpha_n {\text{e}}^{-\text{i}n \omega_0 t}|n\rangle \} \times \{\psi_{\text{inc}}({\bf r}, t) \sum_{\ell= -\infty}^{\infty} {\text{e}}^{{\text{i}}(\chi + \ell_{\text{arg}}\{-\beta\})} J_\ell (2|\beta | ) {\text{e}}^{{\text{i}} \ell \omega_0 (z/v - t)}\}$, in agreement with a well-known expression for PINEM with classical light [19].
B. Coherent Squeezed Light
We now explore squeezed light as an experimentally feasible alternative to classical laser light to excite the PINEM optical mode. Single-mode coherent squeezed states $D(g)S(\zeta)|0\rangle$ are defined by applying the displacement and squeezing operators, $D(g) = \exp (g{a^\dagger} - {g^*}a)$ and $S(\zeta) = \exp [{({\zeta ^*}aa - \zeta {a^\dagger}{a^\dagger})/2}]$, to the optical vacuum [64]. Writing the squeezing parameter as $\zeta = {\text{e}^{\text{i}\theta}}s$, one can express the expansion coefficients of these states in the number basis representation as
$${\alpha _n} = \frac{{{{\left({\xi /2} \right)}^{n/2}}}}{{\sqrt {n!\cosh s}}}\;{\text{e}^{- (|g{|^2} + {g^{*2}}\xi)/2}}{H_n}\left[{\frac{{g + {g^*}\xi}}{{\sqrt {2\xi}}}} \right],$$
where $\xi = {\text{e}^{\text{i}\theta}}\tanh s$ and ${H_n}$ is the Hermite polynomial of order $n$. These coefficients reduce to those of a coherent state for $s = 0$. The average photon number is given by $\bar n = |g{|^2} + \mathop {\sinh}\nolimits^2 s$, while ${\alpha _n}$ depends on the phases of $g$ and $\zeta$ through the combination $\varphi = \theta /2 - \arg\{g\}$. In particular, the variance takes minimum and maximum values for $\varphi = 0$ and $\pi/2$, corresponding to amplitude- and phase-squeezed states, respectively [64].
We consider the two extreme possibilities of PINEM interaction with purely phase- and amplitude-squeezed light in Figs. 2(c) and 2(d), where we plot the density profile $ \tilde \rho (z,\tau ,\tau)$ as a function of propagation distance $z$ for fixed coupling strength [$|\beta | = 5$, obtained with $\bar n = 625$ and $|{\beta _0}| = 0.2$; see Eq. (11)]. Electron focusing takes place at a similar propagation distance $z \sim 2\;\text{mm} $ for all light statistics under consideration. When the illumination has classical [Fig. 2(a)] or amplitude-squeezed [Fig. 2(d)] statistics, the density shows oscillations as a function of distance $z$ after focusing. These oscillations disappear with phase-squeezed light [Fig. 2(c)]. Additionally, the latter produces a focal spot spanning a larger interval of propagation distances $z$ and emerging at a shorter value of $z$ in comparison with classical light [Fig. 2(e)]. The behavior with amplitude-squeezed light is the opposite, and in particular, the minimum full width at half-maximum (FWHM) of the focal spot is approximately twice larger than the result obtained with phase-squeezed or classical light. As already discussed for classical light [29], the degree of compression increases with increasing coupling $|\beta |$ [Fig. 2(f)].
Incidentally, upon visual inspection of the $z - \tau$ pattern for coherent-state illumination in Fig. 2(a), smoothing along $z$ would lead to vertical elongation of the density features, similar to those obtained using phase-squeezed light [Fig. 2(b)]; in contrast, smoothing along $\tau$ would produce a pattern more similar to that of amplitude-squeezed illumination [Fig. 2(d)]. This is consistent with the intuitive picture that phase-squeezing should generate sharper features in the wave function snapshots (i.e., narrower peaks as a function of $\tau$, accompanied by broadening along $z$ in order to preserve the total electron probability); conversely, amplitude-squeezed light should produce the opposite effect (broadening along $\tau$ and sharpening along $z$).
Fig. 3. Tailoring the electron wave packet with amplitude-squeezed light. (a)–(c) Electron density profile produced by PINEM interaction with classical (dashed curves) and amplitude-squeezed (solid curves) light after a propagation distance $z$ as indicated by labels. The electron–light coupling coefficient is assumed to be $|\beta | = 5$ with $|{\beta _0}| = 0.2$ and $\bar n = 625$. (d) Evolution of the density profile using amplitude-squeezed light for different coupling strengths $|\beta |$ obtained by varying $|{\beta _0}|$ with $\bar n = 625$. We consider $100\;\text{keV} $ electrons, a photon energy of $1.5\;\text{eV} $, and a single-mode coupling coefficient $|{\beta _0}| = 0.2$ in all cases.
1. Synthesis of Double-Peak Electron Pulses
Although PINEM interaction with amplitude-squeezed light renders comparatively poorer focusing, it shows an interesting double-peak pattern for $z$ after the focal spot. This effect, which is already observed in Fig. 2(d), is analyzed in more detail in Fig. 3 for different degrees of squeezing. We also show in the same figure the profiles obtained with classical light, revealing amplitude squeezing as a better strategy to produce such double-pulse pattern. We remark that the width and distance between the two pulses can be controlled by varying the coupling strength parameter $|\beta |$ [Fig. 3(d)]. Related to this, we note that a recent experiment [65] has shown that a single double-peak electron density profile can be achieved by exploiting classical midinfrared single-cycle laser pulses.
Fig. 4. Measuring the electron density matrix through self-interference. (a) Sketch of an experimental arrangement to explore electron auto-correlation by means of a beam splitter and different lengths ($z$ and $z^\prime $) along the two electron paths before recombination at the detection region. (b)–(i) Real (left panels) and imaginary (right panels) parts of the electron density matrix as a function of shifted times $\tau$ and $\tau ^\prime $ for $z = 1.6\;\text{mm} $ and different statistics of the PINEM light, as indicated by labels. We consider 100 keV electrons, 1.5 eV PINEM photons, a squeezing parameter $s = 2$, and coupling parameters $|{\beta _0}| = 0.2$ and $|\beta | = 5$.
C. Electron Compression with Minimum-Phase-Uncertainty Light
One expects that better focusing can be achieved by reducing phase uncertainty in the optical field. In the limit of large average photon number $\bar n \gg 1$, the state that produces a MPU has been shown to be given by [66]
$${\alpha _n} \approx \frac{C}{{\sqrt {\bar n}}}\text{Ai}\left[{{s_1}(1 - 2n/3\bar n)} \right],$$
where $\text{Ai}$ is the Airy function, ${s_1} \approx - 2.3381$ is its first zero, $C = \sqrt {2|{s_1}|/3} /\text{Ai}^\prime ({s_1}) \approx 1.7805$, and $\text{Ai}^\prime ({s_1})$ is the derivative of $\text{Ai}$. PINEM focusing with MPU light is illustrated in Fig. 2(b). In contrast to classical light, the Rabi-like oscillations along $z$ are now replaced by a well-defined short-period comb of electron density peaks. This is similar to what we obtain with phase-squeezed light [Fig. 2(c)], but the pattern with MPU light becomes more pronounced. Further deviations from coherent illumination are found in the speed at which compression is achieved: among the statistics under consideration, the shortest FWHM pulse with fixed light intensity and propagation distance is obtained when using MPU light [Fig. 2(e)]. Nevertheless, after a sufficiently large distance $z$, the FWHM reaches similar values with MPU, coherent, and phase-squeezed light, while amplitude-squeezed light systematically leads to lower compression, and this effect becomes more dramatic when increasing the coupling coefficient $|\beta |$ [Fig. 2(f)].
D. Electron Self-Interference
We can further modify the focal properties of the electron by mixing it with a delayed version of itself, using, for example, a beam splitter and different lengths $z$ and $z^\prime $ of the two electron paths converging at the observation region, as sketched in Fig. 4(a). We assume that $z - z^\prime $ is tuned to be a multiple of the electron wavelength, thus rendering $\rho \propto \tilde \rho$ [see Eq. (10)], considering for simplicity an incident electron plane wave [i.e., $\phi (z - vt) = 1/\sqrt L$, where $L$ is a quantization length]. Using the notation of Eq. (8), the electron density profile obtained in this way then results from the superposition $(L/2)\sum\nolimits_n |{\psi _n}(z,t) + {\text{e}^{\text{i}\varphi}}{\psi _n}(z^\prime ,t)|^2 \;\;=\;\; \tilde \rho (z,\tau ,\,\tau)/2 \;+\; \tilde \rho (z,\,\tau ^\prime ,\,\tau ^\prime)/2 \;+\; \text{Re}\{{\text{e}^{- \text{i}\varphi}}\tilde \rho $ $(z,\tau ,\tau ^\prime)\}$, where an overall phase $\varphi$ is introduced (e.g., by means of electrostatic elements along one of the electron arms [37]) to allow us to switch between the real and imaginary parts of $\tilde \rho (z,\tau ,\tau ^\prime)$. An example of how this quantity depends on PINEM-light statistics is shown in Figs. 4(b)–4(i), plotted over a discrete dense sampling of $\tau$ and $\tau ^\prime $ points satisfying the condition that $v(\tau - \tau ^\prime)$ represents multiples of the electron wavelength. Interestingly, we observe a rotation of the focal spot feature when going from classical to amplitude-squeezed light. This is consistent with the poorer focusing properties observed for the latter. Through the proposed electron self-interference, the focal spot profile can be modified to cover a wide variety of patterns observed for different light statistics. In particular, phase-squeezed and MPU light produces a radical departure in $\tilde \rho (z,\tau ,\tau ^\prime)$ relative to classical coherent light.
Fig. 5. Dependence of sample polarization on electron density matrix. (a) Sketch of an electron wave packet undergoing PINEM modulation, followed by propagation along a distance $d$, and interaction with a single-mode sample of frequency ${\omega _{0}^\prime} = m{\omega _0}$ that is a harmonic $m$ of the PINEM photon frequency. (b)–(e) Amplitude ${\Delta _m}$ of the oscillation at frequency ${\omega _{0}^\prime}$ displayed by the sample polarization after interaction with the electron. We plot $|{\Delta _m}|$ for a few values of $m$ as a function of PINEM-sample distance $d$ and different PINEM-light statistics. All parameters are the same as in Fig. 4.
4. EFFECT OF THE ELECTRON DENSITY MATRIX ON THE EXCITATION OF A SAMPLE
A commonly asked question relates to how the probability and distribution of excitations produced in a sample are affected by the profile of the beam in an electron microscope. The dependence on the transverse component of the electron wave function has been shown to reduce to a trivial average of the excitation produced by line-like beams over the lateral electron density profile [67,68]. In the present study, we concentrate instead on the longitudinal electron wave function (i.e., along the beam direction). Within first-order Born approximation, the excitation probability is known to be independent of the longitudinal electron wave function when the initial states of the sample and the electron are not phase-correlated [68,69], although a dependence has been shown to arise when the sample state is a coherent superposition of ground and excited states that is phase-locked with respect to the electron arrival time [69], and, for example, this effect is actually observed in double-PINEM experiments [17]. Here, we concentrate on the common scenario of a sample prepared in its ground state before interaction with the electron. Remarkably, even when considering higher-order interactions, the number of excitations created by the electron has been shown to still remain independent of the longitudinal wave function [70], which incidentally implies that the cathodoluminescence intensity is also independent. We generalize this result below by calculating the full density matrix of the bosonic mode, which turns out to have a Poissonian diagonal part equally independent of electron wave function, although the coherences exhibit a dependence on the quantum state of light used in the PINEM interaction to modulate the electron.
For simplicity, we consider a single sample bosonic mode of frequency ${\omega _{0}^\prime}$ interacting with an incident PINEM-modulated electron wave packet [Fig. 5(a)]. We can then treat the electron-sample interaction using the same formalism as in Section 2 by just iterating Eq. (2). We find the expression
(12)$$\begin{split}|\Psi (z,t)\rangle& = {\text{e}^{\text{i}{k_0}z - \text{i}{E_0}t/\hbar}}\phi (z - vt)\sum\limits_{\ell = - \infty}^\infty \sum\limits_{n = 0}^\infty \sum\limits_{n^\prime = 0}^\infty f_\ell ^n f_{- n^{\prime}}^{\prime n^\prime} \\&\quad \times {\text{e}^{\text{i}{\omega _0}[\ell (z/v - t) - nt] - 2\pi \text{i}{\ell ^2}d/{z_T} - \text{i}n^\prime {{\omega_{0}^\prime}}z/v}}|nn^\prime \rangle \end{split}$$
for the wave function of the entire system, comprising the electron, as well as the PINEM and sample bosonic modes, the Fock states of which are labeled by their respective occupation numbers $n$ and $n^\prime $. Primed quantities are reserved here for the sample [i.e., $f_\ell ^n$ refers to the first PINEM interaction, while $f_{\ell ^\prime}^{\prime n^{\prime}}$ describes the coupling to the sample in Eq. (12)], and in particular the condition $\ell ^\prime = - n^\prime $ (i.e., sample initially prepared in its ground state $|0\rangle$) is used to write the coefficients $f_{- n^\prime}^{\prime n^\prime}$. Additionally, we introduce a phase correction $\propto {\ell ^2}$ accounting for propagation over a macroscopic distance $d$ separating the PINEM and sample interaction regions, but we neglect this type of correction for relatively short propagation along the extension of the envelope function $\phi (z)$ and within the sample interaction region [see Fig. 5(a)]. The density matrix of the sample mode after interaction with the electron,
$${\rho ^{\text{sample}}} = \sum\limits_{{n^\prime_1}{n^\prime_2}} \rho _{{n^\prime_1}{n^\prime_2}}^{\text{sample}}{\text{e}^{- \text{i}({n^\prime_1} - {n^\prime_2}){{\omega_{0}^\prime}}t}}|{n^\prime _1}\rangle \langle {n^\prime _2}|,$$
is then obtained by tracing out electron (integral over $z$) and PINEM boson (sum over $n$) degrees of freedom. More precisely, we find the coefficients
(13)$$\begin{split}\rho _{{n^\prime_1}{n^\prime_2}}^{\text{sample}}& = {\text{e}^{\text{i}({n^\prime_1} - {n^\prime_2}){{\omega_{0}^\prime}}t}}\int \text{d}z\sum\limits_n \langle n{n^\prime _1}|\Psi (z,t)\rangle \langle \Psi (z,t)|n{n^\prime _2}\rangle \\ & = f_{- {n^\prime_1}}^{\prime {n^\prime_1}}f_{- {n^\prime_2}}^{\prime {n^\prime_2}*}\sum\limits_{{\ell _1} = - \infty}^\infty \sum\limits_{{\ell _2} = - \infty}^\infty {\phi _{{\ell _1}{\ell _2}{n^\prime_1}{n^\prime_2}}}\sum\limits_{n = 0}^\infty f_{{\ell _1}}^nf{_{{\ell _2}}^{{n}^*}},\end{split}$$
(14)$$\begin{split}{\phi _{{\ell _1}{\ell _2}{n^\prime_1}{n^\prime_2}}} &= {\text{e}^{2\pi \text{i}(\ell _2^2 - \ell _1^2)d/{z_T}}} \\&\quad \times \int \text{d}z |\phi (z{)|^2} {\text{e}^{\text{i}[({\ell _1} - {\ell _2}){\omega _0} - ({n^\prime_1} - {n^\prime_2}){{\omega_{0}^\prime}}]z/v}}.\end{split}$$
Incidentally, further electron propagation beyond the sample should also involve corrections to the linearized momentum $n^\prime {\omega_{0}^\prime}/v$, on which we are not interested here.
We note that the momentum decomposition of $\phi$ involves small wave vectors compared with $\omega /v$, so its role in the integral of Eq. (14) consists in introducing some broadening with respect to the perfect phase-matching condition
(15)$$({\ell _1} - {\ell _2}){\omega _0} = ({n^\prime _1} - {n^\prime _2}){\omega_{0}^\prime}.$$
Such broadening produces nonzero (but small) values of ${\phi _{{\ell _1}{\ell _2}{n^\prime_1}{n^\prime_2}}}$ even when ${\omega _0}/{\omega_{0}^\prime}$ is not a rational number. For simplicity, we consider ${\omega _0}/{\omega_{0}^\prime}$ to be a rational number and further assume the spectral width of the sample mode to also be small compared with ${\omega _0}$; the coefficients of Eq. (14) then reduce to
$${\phi _{{\ell _1}{\ell _2}{n^\prime_1}{n^\prime_2}}} = {\text{e}^{2\pi \text{i}(\ell _2^2 - \ell _1^2)d/{z_T}}},$$
subject to the condition given by Eq. (15).
We note that the diagonal elements $\rho _{n^\prime n^\prime}^{\text{sample}}$ involve just ${\ell _1} = {\ell _2}$ terms in virtue of Eq. (15), so the only nonzero coefficients in Eq. (13) for those elements are ${\phi _{\ell \ell n^\prime n^\prime}} = 1$, and, using the normalization condition $\sum\nolimits_{\ell n} |f_\ell ^n{|^2} = 1$, we find $\rho _{n^\prime n^\prime}^{\text{sample}} = |f_{- n^\prime}^{\prime n^\prime}{|^2}$, which does not depend on the PINEM coefficients $f_\ell ^n$: we corroborate that the number of excitations created in the sample is independent of how the incident PINEM electron is prepared [70]; additionally, the distribution of those excitations is also independent. More specifically, upon inspection of Eq. (3), we find $f_{- n^\prime}^{\prime n^\prime} = {\text{e}^{\text{i}\chi ^\prime}}{\text{e}^{- |{{\beta_{0} ^\prime}}{|^2}/2}}{\beta _{0}^\prime}^{*n^\prime}/\sqrt {n^\prime !}$; therefore,
$$\rho _{n^\prime n^\prime}^{\text{sample}} = {\left| {f_{- n^\prime}^{\prime n^\prime}} \right|^2} = {\text{e}^{- |{{\beta_{0} ^\prime}}{|^2}}}\frac{{|{{\beta_{0} ^\prime}}{|^{2n^\prime}}}}{{n^\prime !}}$$
reduces to a Poissonian distribution regardless of the quantum state of the incident electron, with average $|{\beta _{0}^\prime}{|^2}$ corresponding to the contribution of the mode under consideration to the electron energy-loss spectroscopy (EELS) probability. This result, which was found for excitation by an electron treated as a classical probe [71,72], is now generalized to a quantum treatment of the electron. We remark that this conclusion is in essence a result of the nonrecoil approximation.
Combining the above results, the elements of the sample density matrix can be written as
$$\begin{split}\rho _{{n^\prime_1}{n^\prime_2}}^{\text{sample}}& = {\text{e}^{- |{{\beta_{0} ^\prime}}{|^2}}}\frac{{{{( {{\beta_{0}^{\prime *}}})}^{{n^\prime_1}}}{{( {{\beta_{0} ^\prime}})}^{{n^\prime_2}}}}}{{\sqrt {{n^\prime_1}!{n^\prime_2}!}}} \\ &\quad\times {\sum_{\ell_1\ell_2}}^{\prime} {\text{e}^{2\pi \text{i}(\ell _2^2 - \ell _1^2)d/{z_T}}}\sum\limits_{n = 0}^\infty f_{{\ell _1}}^nf_{{\ell _2}}^{n*},\end{split}$$
where the sum is subject to the condition imposed by Eq. (15). The symmetry property $\rho _{{n^\prime_1}{n^\prime_2}}^{\text{sample}} = \rho _{{n^\prime_2}{n^\prime_1}}^{\text{sample}*}$ is easily verified from this expression. We can now calculate different observables involving the sample mode, as for example $({a^{\prime \dagger}} + a^\prime)$. The expectation value of this quantity, which vanishes unless the ratio of sample-to-PINEM mode frequencies ${\omega _{0}^\prime }/{\omega _0} = m$ is an integer, only involves terms in which ${n^\prime _1}$ and ${n^\prime _2}$ differ by 1. A straightforward calculation leads to the result
$$\langle ({a^{\prime \dagger}} + a^\prime)\rangle = 2\text{Re}\{ {\beta _{0}^\prime}{\Delta _m}{\text{e}^{\text{i}{{\omega _{0} ^\prime}}t}}\} ,$$
(16)$${\Delta _m} = {\text{e}^{2\pi \text{i}{m^2}d/{z_T}}}\sum\limits_{\ell = - \infty}^\infty {\text{e}^{4\pi \text{i}\ell md/{z_T}}}\sum\limits_{n = 0}^\infty f_\ell ^nf_{\ell + m}^{n*}.$$
This polarization matrix element has been recently shown to exhibit some degree of coherence with the light used to modulate the electron in the first PINEM interaction [70]. We show in Figs. 5(b)–5(e) the dependence of $|{\Delta _m}|$ on PINEM-sample separation $d$ for a few values of $m$ and different PINEM statistics. This quantity is periodic in $d$ with a period ${z_T}/2m$, as it is clear from the exponential inside the sum of Eq. (16). Dramatic differences are observed in $|{\Delta _m}|$ for different PINEM statistics; in particular, a clear trend is observed toward concentration of ${\Delta _m}$ at specific distances $d$ when the uncertainty in the light coherence is reduced (i.e., when moving from coherent or amplitude-squeezed light to phase-squeezed light, and eventually to MPU light).
Incidentally, a similar analysis for the ${N^{\text{th}}}$ moment $\propto {({a^{\prime \dagger}} + a^\prime)^N}$ leads to a contribution that includes a maximum oscillation frequency $N{\omega_{0}^\prime }$ with a coefficient $\beta_{0}^{\prime N} {\Delta _{\textit{mN}}}$. An effect at that order is produced if $mN$ is an integer, a condition that can be met for noninteger values of the sample-PINEM frequency ratio ${\omega_{0}^\prime }/{\omega _0} = m$; for example, an oscillation with frequency ${\omega _0}$ is induced in $\propto {({a^{\prime \dagger}} + a^\prime)^2}$ after electron-sample interaction if the sample mode frequency is half of the PINEM photon frequency.
The time dependence of the off-diagonal sample density matrix components under discussion could be measured through attosecond streaking [73,74], as a function of the delay between the times of arrival of the electron and an x-ray pulse, giving rise to oscillations in the energy of photoelectrons produced by the latter as a function of such delay. For low-frequency sample modes, a direct measurement could be based on time-resolved quantum tomography of the sample state; this strategy could benefit from low-frequency beatings resulting from the combination of multiple sample modes of similar frequency. More direct evidence should be provided by the nontrivial interference that has been shown to emerge when mixing the PINEM light with cathodoluminescence emission from the sample [70].
We have demonstrated that the interaction of free electrons with quantum light opens a new direction for modulating the longitudinal electron profile, the degree and duration of electron pulse compression, and the statistics associated with this compression. By squeezing the interacting light in phase, the formation of electron pulses is accelerated, and this effect is maximized when using optical fields with an Airy number distribution that minimizes phase uncertainty. Interestingly, amplitude-squeezed light leads to the emergence of double-pulse electron profiles, which could be useful to investigate dynamical processes in a sample. The influence of light statistics becomes more dramatic when examining the electron density matrix after interaction, a quantity that can be accessed through our proposed self-interference experiment. Additionally, we have shown that the excitation of a sample by the electron is affected by how the latter is modulated and, in particular, by the statistics of the modulating light. Indeed, although no dependence is predicted in the probability of exciting sample modes, the temporal evolution of the electron-induced off-diagonal sample density matrix elements shows a dramatic departure from the results observed with laser-modulated electrons when considering instead electrons that have interacted with quantum light. Besides their practical interest to shape and temporally compress free electrons, the results here presented reveal a wealth of fundamental phenomena emerging from the interaction with nonclassical light. We further anticipate potential application in the creation of light sources with nontrivial statistics through electron-induced optical emission using gratings and undulators.
APPENDIX A: DERIVATION OF EQ. (3)
We review a derivation of Eq. (3) presented elsewhere [48] to describe the evolution of an electron interacting with a dominant quantized electromagnetic mode in a quantum optics framework [52,75]. A generalization to multiple modes has also been reported [76]. Under the assumptions discussed in Section 2.A, inserting the ansatz solution for the wave function given in Eq. (2) into the Schrödinger equation $\text{i}\hbar {\partial _t}|\psi ({\bf r},t)\rangle = ({\hat {\cal H}_0} + {\hat {\cal H}_1})|\psi ({\bf r},t)\rangle$ for the Hamiltonian defined in Eq. (1), we find the differential equation
(A1)$${\partial _z}f_\ell ^n = \sqrt n {u^*} f_{\ell + 1}^{n - 1} - \sqrt {n + 1} u f_{\ell - 1}^{n + 1}$$
for the expansion coefficients, where ${u_j}(z) = (e/\hbar {\omega _0}){{\cal E}_{0,z}}(z){\text{e}^{- \text{i}{\omega _0}z/v}}$. We note that Eq. (A1) preserves the sum $n + \ell$, thus guaranteeing that the number of excitations in the electron-boson system is conserved along its evolution. This property ensures that the problem can be mapped onto a classically driven quantum harmonic oscillator (QHO), which admits an analytical solution [77]. The connection between the two systems is made clear by writing the Hamiltonian $\hat {\cal H} = \hbar \omega {a^\dagger}a + g(t)a + {g^*}(t){a^\dagger}$ for the QHO, along with its wave function $|\psi (t)\rangle = \sum\nolimits_n {c_n}(t){\text{e}^{- \text{i}n\omega t}}|n\rangle$, whose coefficients follow the equation of motion $\text{i}\hbar {\partial _t}{c_n} = [{\sqrt n {g^*} {c_{n - 1}}{\text{e}^{\text{i}\omega t}} + {\text{e}^{- \text{i}\omega t}}\sqrt {n + 1} g {c_{n + 1}}}]$. For an initial state written as $|{\psi ^{\text{I}}}({t_0})\rangle = \sum\nolimits_n {c_n}({t_0})|n\rangle$ in the interaction picture, the solution at later times can be expressed as $\langle n|{\psi ^{\text{I}}}(t)\rangle = \langle n|\hat {\cal S}(t,{t_0})|{\psi ^{\text{I}}}({t_0})\rangle$ in terms of the scattering operator [77],
(A2)$$\hat {\cal S}(t,{t_0}) = {\text{e}^{\text{i}\chi}}{\text{e}^{\beta _0^*{a^\dagger} - {\beta _0}a}},$$
where ${\beta _0}(t,{t_0}) = \frac{\text{i}}{\hbar}\int_{{t_0}}^t \text{d}t^\prime g(t^\prime){\text{e}^{- \text{i}\omega t^\prime}}$ is the coupling coefficient and $\chi = - \frac{1}{\hbar}\int_{{t_0}}^t \text{d}t^\prime \text{Re}\{{\beta _0}(t^\prime ,{t_0}){g^*}(t^\prime){\text{e}^{\text{i}\omega t^\prime}}\}$ is a global phase; we obtain ${c_n}(t) = \sum\nolimits_{m = 0}^\infty {c_m}({t_0})\langle n|\hat {\cal S}(t,{t_0})|m\rangle$. Then, an explicit expression for the matrix elements of the evolution operator [Eq. (A2)] can be obtained by applying the Baker–Campbell–Hausdorff formula,
$$\begin{split}\langle n|\hat {\cal S}(t,{t_0})|m\rangle & = {\text{e}^{{\text{i}} \chi}} \sqrt {m!n!}\,{\text{e}^{- |{\beta _0}{|^2}/2}}{(- {\beta _0})^{m - n}} \\ &\quad \times \sum\limits_{n^\prime = \max \{0,n - m\}}^n \frac{{{{(- |{\beta _0}{|^2})}^{{n^\prime}}}}}{{n^\prime !(m - n + n^\prime)!(n - n^\prime)!}}.\end{split}$$
We now connect these results with the solution of our electron-boson system by exploiting the mapping $f_{{n_0} - n}^n = {c_n}$ enabled by the conservation of ${n_0} = n + \ell$. Making the substitutions $\omega \to \omega_0 $, $g{\text{e}^{- \text{i}\omega t}} \to - \text{i}\hbar vu$, and $t \to z/v$, and imposing the condition $f_\ell ^n(z \to - \infty) = {\delta _{\ell ,0}}{\alpha _n}$ to Eq. (A1), where ${\alpha _n}$ represents the initial coefficients of the initial quantum light state $\sum\nolimits_n {\alpha _n}|n\rangle$, we readily find Eq. (3).
APPENDIX B: DERIVATION OF EQ. (10)
Starting from Eq. (3) and considering $n\gg1$ and $|\ell|\ll n$ for typical single-mode coupling conditions $|\beta_0|\ll 1$, the dominant contribution to the sum comes from $n^\prime \ll n$ terms, so we can approximate $F_\ell^n\approx\sum_{n^\prime =0}^\infty\frac{(-1)^{n^\prime}|\beta_0|^{2n^\prime +\ell}}{n^\prime!(\ell+n^\prime)!} \frac{\sqrt{(n+\ell)!n!}}{(n-n^\prime)!}$. We now apply the Stiling formula $n!\approx\sqrt{2\pi n}\,(n/e)^n$ to the factorials in the rightmost fraction and neglect $\ell$ and $n^\prime$ in front of $n$ in the factors that are not affected by an exponent $n$. This allows us to approximate $\sqrt{(n+\ell)!n!}/(n-n^\prime)!\approx (n/e)^{n^\prime+\ell/2}{\text{e}} ^M$, where $M=n\,\ln\left[\sqrt{1+\ell/n}/(1-n^\prime/n)\right]$. We then retain only terms up to first order in the Taylor expansion of the logarithm to find $M\approx n^\prime +\ell/2$. Upon insertion of this result into the above expression for $F_\ell^n$, we find $F_\ell^n\approx\sum_{n^\prime =0}^\infty(-1)^{n^\prime}|\sqrt{n}\beta_0|^{2n^\prime +\ell}/[n^\prime!(\ell+n^\prime)!]$, which directly yields Eq. (10) by identifying the sum as the Taylor expansion of the Bessel function $J_n$ with argument $2\sqrt{n}|\beta_0|$.
Ministerio de Economía y Competitividad (MAT2017-88492-R, SEV2015-0522); European Research Council (Advanced Grant 789104-eNANO); Centres de Recerca de Catalunya; Fundación Cellex; European Commission (Marie Sklodowska-Curie Grant 713729).
We thank Ofer Kfir and Claus Ropers for many helpful and enjoyable discussions. V. D. G. acknowledges financial support from the EU.
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Aflatooni, K.
Agostini, P.
Aoki, T.
Arie, A.
Asenjo Garcia, A.
Augé, F.
Bach, N.
Balcou, P.
Barwick, B.
Batelaan, H.
Batson, P. E.
Baum, P.
Be'er, O.
Béché, A.
Berruto, G.
Berry, M. V.
Biagioni, P.
Bormann, R.
Boxem, R. V.
Breger, P.
Busch, K.
Byer, R. L.
Cai, W.
Carbone, F.
Carpenter, R. W.
Chuang, I. L.
Clark, L.
Corkum, P. B.
Crozier, P. A.
Dahan, R.
Dellby, N.
Deng, H.
Dezfouli, M. K.
Di Giulio, V.
Dirac, P. A. M.
Domräse, T.
Dunin-Borkowski, R. E.
Echternkamp, K. E.
Eckstein, T.
Egerton, R. F.
Ehberger, D.
Feist, A.
Feller, W.
Fitzpatrick, A. W. P.
Flannigan, D. J.
Franke, S.
Freimund, D. L.
García de Abajo, F. J.
García de Abajo, V. D. F. J.
Gatzmann, J. G.
Glauber, R. J.
Gliserin, A.
Gover, A.
Grillo, V.
Guzzinati, G.
Handali, J.
Harris, J. S.
Hartl, I.
Harvey, T. R.
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Hertog, M. D.
Hohage, T.
Hommelhoff, P.
Howie, A.
Hughes, S.
Itatani, J.
Ivanov, M.
Ivanov, M. Y.
Juchtmans, R.
Kaminer, I.
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Kealhofer, C.
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Khalili, F. Y.
Kippenberg, T. J.
Kirchner, F. O.
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Fig. 1. Talbot effect and electron compression with classical light. (a) An electron Gaussian wave packet (green) is transformed through PINEM interaction followed by propagation along a distance $z$ into a substantially modified electron density profile in the propagation-distance-shifted time $\tau = t - t_p -z/v$ due to superposition of different energy components. (b) Electron density profile (vertical $\tau$ coordinate) as a function of propagation distance $z$ (horizontal axis) after PINEM interaction with coherent light. We consider $100\;\text{keV} $ electrons, a photon energy $\hbar {\omega _0} = 1.5\;\text{eV} $ , and a coupling coefficient ${|\beta | = 5}$ . Trains of compressed electron pulses are periodically observed at discrete multiple values of the Talbot propagation distance ${z_T}$ . (c)–(e) Details of the $\tau - z$ map in (b) corresponding to the color-matched square regions of $z$ width $\Delta = 4\;\text{mm} $ . (f) Same as (e), but for $z$ near $2{z_{{T}}}$ .
Fig. 2. Electron compression using squeezed light. (a)–(d) Evolution of the electron density profile following PINEM interaction with (a) classical, (b) MPU, (c) phase-squeezed, and (d) amplitude-squeezed light using a single-mode coupling coefficient $|{\beta _0}| = 0.2$ and average population $\bar n = 625$ (i.e., $|\beta | = \sqrt {\bar n} |{\beta _0}| = 5$ ). (e) FWHM [see (a)] of the compressed electron density in (a)–(d) as a function of propagation distance $z$ . (f) Minimum in the FWHM along the curves in (e) as a function of coupling coefficient $|\beta |$ (varying $|{\beta _0}|$ and keeping $\bar n = 625$ ). We consider $100\;\text{keV} $ electrons and a $1.5\;\text{eV} $ photon energy.
Fig. 3. Tailoring the electron wave packet with amplitude-squeezed light. (a)–(c) Electron density profile produced by PINEM interaction with classical (dashed curves) and amplitude-squeezed (solid curves) light after a propagation distance $z$ as indicated by labels. The electron–light coupling coefficient is assumed to be $|\beta | = 5$ with $|{\beta _0}| = 0.2$ and $\bar n = 625$ . (d) Evolution of the density profile using amplitude-squeezed light for different coupling strengths $|\beta |$ obtained by varying $|{\beta _0}|$ with $\bar n = 625$ . We consider $100\;\text{keV} $ electrons, a photon energy of $1.5\;\text{eV} $ , and a single-mode coupling coefficient $|{\beta _0}| = 0.2$ in all cases.
Fig. 4. Measuring the electron density matrix through self-interference. (a) Sketch of an experimental arrangement to explore electron auto-correlation by means of a beam splitter and different lengths ( $z$ and $z^\prime $ ) along the two electron paths before recombination at the detection region. (b)–(i) Real (left panels) and imaginary (right panels) parts of the electron density matrix as a function of shifted times $\tau$ and $\tau ^\prime $ for $z = 1.6\;\text{mm} $ and different statistics of the PINEM light, as indicated by labels. We consider 100 keV electrons, 1.5 eV PINEM photons, a squeezing parameter $s = 2$ , and coupling parameters $|{\beta _0}| = 0.2$ and $|\beta | = 5$ .
Fig. 5. Dependence of sample polarization on electron density matrix. (a) Sketch of an electron wave packet undergoing PINEM modulation, followed by propagation along a distance $d$ , and interaction with a single-mode sample of frequency ${\omega _{0}^\prime} = m{\omega _0}$ that is a harmonic $m$ of the PINEM photon frequency. (b)–(e) Amplitude ${\Delta _m}$ of the oscillation at frequency ${\omega _{0}^\prime}$ displayed by the sample polarization after interaction with the electron. We plot $|{\Delta _m}|$ for a few values of $m$ as a function of PINEM-sample distance $d$ and different PINEM-light statistics. All parameters are the same as in Fig. 4.
(1) $${\hat {\cal H}_0} = \hbar {\omega _0}{a^\dagger}a + {E_0} - \hbar {\bf v} \cdot (\text{i}\nabla + {{\bf k}_0}),$$
(2) $${\hat {\cal H}_1} = - \text{i}(e{\bf v}/{\omega _0}) \cdot \left[{{{\vec {\cal E}}_0}({\bf r})a - \vec {\cal E}_0^*({\bf r}){a^\dagger}} \right],$$
(3) $$|\psi ({\bf r},t)\rangle = {\psi _{\text{inc}}}({\bf r},t)\sum\limits_{\ell = - \infty}^\infty \sum\limits_{n = 0}^\infty {\text{e}^{\text{i}{\omega _0}[\ell (z/v - t) - nt]}}f_\ell ^n({\bf r})|n\rangle ,$$
(4) $$\begin{split}f_\ell ^n & = {\text{e}^{\text{i}(\chi + \ell \text{arg}\{- {\beta _0}\})}} {\alpha _{n + \ell}} F_\ell ^n, \\ F_\ell ^n & = |{\beta _0}{|^\ell}{\text{e}^{- |{\beta _0}{|^2}/2}}\sqrt {(n + \ell)!n!} \sum\limits_{n^\prime = \text{max}\{0, - \ell \}}^n \frac{{{{(- |{\beta _0}{|^2})}^{{n^\prime}}}}}{{n^\prime !(\ell + n^\prime)!(n - n^\prime)!}},\end{split}$$
(5) $${\beta _0}({\bf R},z) = \frac{e}{{\hbar {\omega _0}}}\int_{- \infty}^z \text{d}z^\prime {{\cal E}_{0,z}}({\bf R},z^\prime){\text{e}^{- \text{i}{\omega _0}z^\prime /v}}$$
(6) $$\sum\limits_n |{\alpha _n}{|^2} = 1.$$
(7) $$\sum\limits_\ell {(F_\ell ^{n - \ell})^2} = 1,$$
(8) $${\psi _{\text{inc}}}({\bf r},t) = {\text{e}^{\text{i}{{\bf k}_0} \cdot {\bf r} - \text{i}{E_0}t/\hbar}}\phi ({\bf r} - {\bf v}t),$$
(9) $$\begin{split}{k_\ell} &= {\hbar ^{- 1}}\sqrt {E_\ell ^2/{c^2} - m_{\text{e}}^2{c^2}} \\ &\approx {k_0} + \ell {\omega _0}/v - 2\pi {\ell ^2}/{z_T} + \cdots \end{split}$$
(10) $${z_T} = 4\pi {m_{\text{e}}}{v^3}{\gamma ^3}/\hbar \omega _0^2$$
(11) $$\rho (z,z^\prime ,t) = \sum\limits_{n = 0}^\infty {\psi _n}(z,t)\psi _n^*(z^\prime ,t),$$
(12) $${\psi _n}(z,t) = \phi (z - vt)\sum\limits_{\ell = - \infty}^\infty {\alpha _{n + \ell}} F_\ell ^n{\text{e}^{\text{i}{k_\ell}z - \text{i}\ell {\omega _0}(t - {t_p})}},$$
(13) $$\rho (z,z^\prime ,t) = {\text{e}^{\text{i}{k_0}(z - z^\prime)}}\phi (z - vt){\phi ^*}(z^\prime - vt)\tilde \rho (z,\tau ,\tau ^\prime),$$
(14) $$\begin{split}\tilde \rho (z,\tau ,\tau ^\prime)& = \sum\limits_{n\ell \ell ^\prime} {\alpha _{n + \ell}}\alpha _{n + \ell ^\prime}^*\;F_\ell ^nF_{\ell ^\prime}^n \\ &\quad \times {\text{e}^{2\pi \text{i}\left[{({{\ell ^\prime}^2} - {\ell ^2})z/{z_T} + (\ell ^\prime \tau ^\prime - \ell \tau)/{\tau _0}} \right]}},\end{split}$$
(15) $$F_\ell ^n \approx {J_\ell}(2\sqrt n |{\beta _0}|).$$
(16) $$\tilde \rho (z,\tau ,\tau ^\prime) \approx {\psi _{\text{cl}}}(z,\tau)\psi _{\text{cl}}^*(z,\tau ^\prime),$$
(17) $${\psi _{\text{cl}}}(z,\tau) = \sum\limits_\ell {J_\ell}(2|\beta |){\text{e}^{- 2\pi \text{i}({\ell ^2}z/{z_T} + \ell \tau /{\tau _0})}}$$
(18) $$\beta = \sqrt {\bar n} {\beta _0}$$
(19) $${\alpha _n} = \frac{{{{\left({\xi /2} \right)}^{n/2}}}}{{\sqrt {n!\cosh s}}}\;{\text{e}^{- (|g{|^2} + {g^{*2}}\xi)/2}}{H_n}\left[{\frac{{g + {g^*}\xi}}{{\sqrt {2\xi}}}} \right],$$
(20) $${\alpha _n} \approx \frac{C}{{\sqrt {\bar n}}}\text{Ai}\left[{{s_1}(1 - 2n/3\bar n)} \right],$$
(21) $$\begin{split}|\Psi (z,t)\rangle& = {\text{e}^{\text{i}{k_0}z - \text{i}{E_0}t/\hbar}}\phi (z - vt)\sum\limits_{\ell = - \infty}^\infty \sum\limits_{n = 0}^\infty \sum\limits_{n^\prime = 0}^\infty f_\ell ^n f_{- n^{\prime}}^{\prime n^\prime} \\&\quad \times {\text{e}^{\text{i}{\omega _0}[\ell (z/v - t) - nt] - 2\pi \text{i}{\ell ^2}d/{z_T} - \text{i}n^\prime {{\omega_{0}^\prime}}z/v}}|nn^\prime \rangle \end{split}$$
(22) $${\rho ^{\text{sample}}} = \sum\limits_{{n^\prime_1}{n^\prime_2}} \rho _{{n^\prime_1}{n^\prime_2}}^{\text{sample}}{\text{e}^{- \text{i}({n^\prime_1} - {n^\prime_2}){{\omega_{0}^\prime}}t}}|{n^\prime _1}\rangle \langle {n^\prime _2}|,$$
(23) $$\begin{split}\rho _{{n^\prime_1}{n^\prime_2}}^{\text{sample}}& = {\text{e}^{\text{i}({n^\prime_1} - {n^\prime_2}){{\omega_{0}^\prime}}t}}\int \text{d}z\sum\limits_n \langle n{n^\prime _1}|\Psi (z,t)\rangle \langle \Psi (z,t)|n{n^\prime _2}\rangle \\ & = f_{- {n^\prime_1}}^{\prime {n^\prime_1}}f_{- {n^\prime_2}}^{\prime {n^\prime_2}*}\sum\limits_{{\ell _1} = - \infty}^\infty \sum\limits_{{\ell _2} = - \infty}^\infty {\phi _{{\ell _1}{\ell _2}{n^\prime_1}{n^\prime_2}}}\sum\limits_{n = 0}^\infty f_{{\ell _1}}^nf{_{{\ell _2}}^{{n}^*}},\end{split}$$
(24) $$\begin{split}{\phi _{{\ell _1}{\ell _2}{n^\prime_1}{n^\prime_2}}} &= {\text{e}^{2\pi \text{i}(\ell _2^2 - \ell _1^2)d/{z_T}}} \\&\quad \times \int \text{d}z |\phi (z{)|^2} {\text{e}^{\text{i}[({\ell _1} - {\ell _2}){\omega _0} - ({n^\prime_1} - {n^\prime_2}){{\omega_{0}^\prime}}]z/v}}.\end{split}$$
(25) $$({\ell _1} - {\ell _2}){\omega _0} = ({n^\prime _1} - {n^\prime _2}){\omega_{0}^\prime}.$$
(26) $${\phi _{{\ell _1}{\ell _2}{n^\prime_1}{n^\prime_2}}} = {\text{e}^{2\pi \text{i}(\ell _2^2 - \ell _1^2)d/{z_T}}},$$
(27) $$\rho _{n^\prime n^\prime}^{\text{sample}} = {\left| {f_{- n^\prime}^{\prime n^\prime}} \right|^2} = {\text{e}^{- |{{\beta_{0} ^\prime}}{|^2}}}\frac{{|{{\beta_{0} ^\prime}}{|^{2n^\prime}}}}{{n^\prime !}}$$
(28) $$\begin{split}\rho _{{n^\prime_1}{n^\prime_2}}^{\text{sample}}& = {\text{e}^{- |{{\beta_{0} ^\prime}}{|^2}}}\frac{{{{( {{\beta_{0}^{\prime *}}})}^{{n^\prime_1}}}{{( {{\beta_{0} ^\prime}})}^{{n^\prime_2}}}}}{{\sqrt {{n^\prime_1}!{n^\prime_2}!}}} \\ &\quad\times {\sum_{\ell_1\ell_2}}^{\prime} {\text{e}^{2\pi \text{i}(\ell _2^2 - \ell _1^2)d/{z_T}}}\sum\limits_{n = 0}^\infty f_{{\ell _1}}^nf_{{\ell _2}}^{n*},\end{split}$$
(29) $$\langle ({a^{\prime \dagger}} + a^\prime)\rangle = 2\text{Re}\{ {\beta _{0}^\prime}{\Delta _m}{\text{e}^{\text{i}{{\omega _{0} ^\prime}}t}}\} ,$$
(30) $${\Delta _m} = {\text{e}^{2\pi \text{i}{m^2}d/{z_T}}}\sum\limits_{\ell = - \infty}^\infty {\text{e}^{4\pi \text{i}\ell md/{z_T}}}\sum\limits_{n = 0}^\infty f_\ell ^nf_{\ell + m}^{n*}.$$
(31) $${\partial _z}f_\ell ^n = \sqrt n {u^*} f_{\ell + 1}^{n - 1} - \sqrt {n + 1} u f_{\ell - 1}^{n + 1}$$
(32) $$\hat {\cal S}(t,{t_0}) = {\text{e}^{\text{i}\chi}}{\text{e}^{\beta _0^*{a^\dagger} - {\beta _0}a}},$$
(33) $$\begin{split}\langle n|\hat {\cal S}(t,{t_0})|m\rangle & = {\text{e}^{{\text{i}} \chi}} \sqrt {m!n!}\,{\text{e}^{- |{\beta _0}{|^2}/2}}{(- {\beta _0})^{m - n}} \\ &\quad \times \sum\limits_{n^\prime = \max \{0,n - m\}}^n \frac{{{{(- |{\beta _0}{|^2})}^{{n^\prime}}}}}{{n^\prime !(m - n + n^\prime)!(n - n^\prime)!}}.\end{split}$$ | CommonCrawl |
Tagged: vector
Are these vectors in the Nullspace of the Matrix?
Let $A=\begin{bmatrix}
1 & 0 & 3 & -2 \\
0 &3 & 1 & 1 \\
\end{bmatrix}$. For each of the following vectors, determine whether the vector is in the nullspace $\calN(A)$.
(a) $\begin{bmatrix}
-3 \\
\end{bmatrix}$
(b) $\begin{bmatrix}
(c) $\begin{bmatrix}
(d) $\begin{bmatrix}
Then, describe the nullspace $\calN(A)$ of the matrix $A$.
Spanning Sets for $\R^2$ or its Subspaces
In this problem, we use the following vectors in $\R^2$.
\[\mathbf{a}=\begin{bmatrix}
\end{bmatrix}, \mathbf{b}=\begin{bmatrix}
\end{bmatrix}, \mathbf{c}=\begin{bmatrix}
\end{bmatrix}, \mathbf{d}=\begin{bmatrix}
\end{bmatrix}, \mathbf{e}=\begin{bmatrix}
\end{bmatrix}, \mathbf{f}=\begin{bmatrix}
\end{bmatrix}.\] For each set $S$, determine whether $\Span(S)=\R^2$. If $\Span(S)\neq \R^2$, then give algebraic description for $\Span(S)$ and explain the geometric shape of $\Span(S)$.
(a) $S=\{\mathbf{a}, \mathbf{b}\}$
(b) $S=\{\mathbf{a}, \mathbf{c}\}$
(c) $S=\{\mathbf{c}, \mathbf{d}\}$
(d) $S=\{\mathbf{a}, \mathbf{f}\}$
(e) $S=\{\mathbf{e}, \mathbf{f}\}$
(f) $S=\{\mathbf{a}, \mathbf{b}, \mathbf{c}\}$
(g) $S=\{\mathbf{e}\}$
Find all Column Vector $\mathbf{w}$ such that $\mathbf{v}\mathbf{w}=0$ for a Fixed Vector $\mathbf{v}$
Let $\mathbf{v} = \begin{bmatrix} 2 & -5 & -1 \end{bmatrix}$.
Find all $3 \times 1$ column vectors $\mathbf{w}$ such that $\mathbf{v} \mathbf{w} = 0$.
Prove that the Dot Product is Commutative: $\mathbf{v}\cdot \mathbf{w}= \mathbf{w} \cdot \mathbf{v}$
Let $\mathbf{v}$ and $\mathbf{w}$ be two $n \times 1$ column vectors.
(a) Prove that $\mathbf{v}^\trans \mathbf{w} = \mathbf{w}^\trans \mathbf{v}$.
(b) Provide an example to show that $\mathbf{v} \mathbf{w}^\trans$ is not always equal to $\mathbf{w} \mathbf{v}^\trans$.
Calculate the following expressions, using the following matrices:
\[A = \begin{bmatrix} 2 & 3 \\ -5 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 0 & -1 \\ 1 & -1 \end{bmatrix}, \qquad \mathbf{v} = \begin{bmatrix} 2 \\ -4 \end{bmatrix}\]
(a) $A B^\trans + \mathbf{v} \mathbf{v}^\trans$.
(b) $A \mathbf{v} – 2 \mathbf{v}$.
(c) $\mathbf{v}^{\trans} B$.
(d) $\mathbf{v}^\trans \mathbf{v} + \mathbf{v}^\trans B A^\trans \mathbf{v}$.
Determine a Condition on $a, b$ so that Vectors are Linearly Dependent
\[\mathbf{v}_1=\begin{bmatrix}
a \\
\end{bmatrix}\] be vectors in $\R^3$.
Determine a condition on the scalars $a, b$ so that the set of vectors $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is linearly dependent.
The Matrix $[A_1, \dots, A_{n-1}, A\mathbf{b}]$ is Always Singular, Where $A=[A_1,\dots, A_{n-1}]$ and $\mathbf{b}\in \R^{n-1}$.
Let $A$ be an $n\times (n-1)$ matrix and let $\mathbf{b}$ be an $(n-1)$-dimensional vector.
Then the product $A\mathbf{b}$ is an $n$-dimensional vector.
Set the $n\times n$ matrix $B=[A_1, A_2, \dots, A_{n-1}, A\mathbf{b}]$, where $A_i$ is the $i$-th column vector of $A$.
Prove that $B$ is a singular matrix for any choice of $\mathbf{b}$.
Prove $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ and determine those $\mathbf{x}$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$
For each of the following matrix $A$, prove that $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ for all vectors $\mathbf{x}$ in $\R^2$. Also, determine those vectors $\mathbf{x}\in \R^2$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$.
4 & 2\\
2& 1
(b) $A=\begin{bmatrix}
Eigenvalues of Orthogonal Matrices Have Length 1. Every $3\times 3$ Orthogonal Matrix Has 1 as an Eigenvalue
(a) Let $A$ be a real orthogonal $n\times n$ matrix. Prove that the length (magnitude) of each eigenvalue of $A$ is $1$.
(b) Let $A$ be a real orthogonal $3\times 3$ matrix and suppose that the determinant of $A$ is $1$. Then prove that $A$ has $1$ as an eigenvalue.
If Every Vector is Eigenvector, then Matrix is a Multiple of Identity Matrix
Let $A$ be an $n\times n$ matrix. Assume that every vector $\mathbf{x}$ in $\R^n$ is an eigenvector for some eigenvalue of $A$.
Prove that there exists $\lambda\in \R$ such that $A=\lambda I$, where $I$ is the $n\times n$ identity matrix.
Quiz 5: Example and Non-Example of Subspaces in 3-Dimensional Space
Problem 1 Let $W$ be the subset of the $3$-dimensional vector space $\R^3$ defined by
\[W=\left\{ \mathbf{x}=\begin{bmatrix}
x_1 \\
x_3
\end{bmatrix}\in \R^3 \quad \middle| \quad 2x_1x_2=x_3 \right\}.\]
(a) Which of the following vectors are in the subset $W$? Choose all vectors that belong to $W$.
\[(1) \begin{bmatrix}
\end{bmatrix} \qquad(2) \begin{bmatrix}
\end{bmatrix} \qquad(3)\begin{bmatrix}
1 &2 &4
(b) Determine whether $W$ is a subspace of $\R^3$ or not.
Problem 2 Let $W$ be the subset of $\R^3$ defined by
\end{bmatrix} \in \R^3 \quad \middle| \quad x_1=3x_2 \text{ and } x_3=0 \right\}.\] Determine whether the subset $W$ is a subspace of $\R^3$ or not.
If a Matrix $A$ is Singular, then Exists Nonzero $B$ such that $AB$ is the Zero Matrix
Let $A$ be a $3\times 3$ singular matrix.
Then show that there exists a nonzero $3\times 3$ matrix $B$ such that
\[AB=O,\] where $O$ is the $3\times 3$ zero matrix.
Solve the System of Linear Equations and Give the Vector Form for the General Solution
Solve the following system of linear equations and give the vector form for the general solution.
\begin{align*}
x_1 -x_3 -2x_5&=1 \\
x_2+3x_3-x_5 &=2 \\
2x_1 -2x_3 +x_4 -3x_5 &= 0
\end{align*}
(The Ohio State University, linear algebra midterm exam problem)
Every Plane Through the Origin in the Three Dimensional Space is a Subspace
Prove that every plane in the $3$-dimensional space $\R^3$ that passes through the origin is a subspace of $\R^3$.
The Subset Consisting of the Zero Vector is a Subspace and its Dimension is Zero
Let $V$ be a subset of the vector space $\R^n$ consisting only of the zero vector of $\R^n$. Namely $V=\{\mathbf{0}\}$.
Then prove that $V$ is a subspace of $\R^n$.
Determine Whether a Set of Functions $f(x)$ such that $f(x)=f(1-x)$ is a Subspace
Let $V$ be the vector space over $\R$ of all real valued function on the interval $[0, 1]$ and let
\[W=\{ f(x)\in V \mid f(x)=f(1-x) \text{ for } x\in [0,1]\}\] be a subset of $V$. Determine whether the subset $W$ is a subspace of the vector space $V$.
Linearly Independent vectors $\mathbf{v}_1, \mathbf{v}_2$ and Linearly Independent Vectors $A\mathbf{v}_1, A\mathbf{v}_2$ for a Nonsingular Matrix
Let $\mathbf{v}_1$ and $\mathbf{v}_2$ be $2$-dimensional vectors and let $A$ be a $2\times 2$ matrix.
(a) Show that if $\mathbf{v}_1, \mathbf{v}_2$ are linearly dependent vectors, then the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly dependent.
(b) If $\mathbf{v}_1, \mathbf{v}_2$ are linearly independent vectors, can we conclude that the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly independent?
(c) If $\mathbf{v}_1, \mathbf{v}_2$ are linearly independent vectors and $A$ is nonsingular, then show that the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly independent.
Quiz 3. Condition that Vectors are Linearly Dependent/ Orthogonal Vectors are Linearly Independent
(a) For what value(s) of $a$ is the following set $S$ linearly dependent?
\[ S=\left \{\,\begin{bmatrix}
\end{bmatrix}, \begin{bmatrix}
a^2 \\
a^3
\end{bmatrix} \, \right\}.\]
(b) Let $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a set of nonzero vectors in $\R^m$ such that the dot product
\[\mathbf{v}_i\cdot \mathbf{v}_j=0\] when $i\neq j$.
Prove that the set is linearly independent.
Determine Linearly Independent or Linearly Dependent. Express as a Linear Combination
Determine whether the following set of vectors is linearly independent or linearly dependent. If the set is linearly dependent, express one vector in the set as a linear combination of the others.
\[\left\{\, \begin{bmatrix}
\end{bmatrix},
\end{bmatrix}\, \right\}.\]
The Union of Two Subspaces is Not a Subspace in a Vector Space
Let $U$ and $V$ be subspaces of the vector space $\R^n$.
If neither $U$ nor $V$ is a subset of the other, then prove that the union $U \cup V$ is not a subspace of $\R^n$.
A Group is Abelian if and only if Squaring is a Group Homomorphism
Three Equivalent Conditions for an Ideal is Prime in a PID
A ring is Local if and only if the set of Non-Units is an Ideal
Linear Algebra Midterm 1 at the Ohio State University (1/3)
Linear Properties of Matrix Multiplication and the Null Space of a Matrix
The Matrix for the Linear Transformation of the Reflection Across a Line in the Plane
Determine the Number of Elements of Order 3 in a Non-Cyclic Group of Order 57
Find an Orthonormal Basis of the Given Two Dimensional Vector Space | CommonCrawl |
\begin{document}
\title[Quasiconformal extension for harmonic mappings]{Quasiconformal extension for harmonic mappings on finitely connected domains}
\author[I. Efraimidis]{Iason Efraimidis} \address{Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, TX 79409, United States} \email{[email protected]}
\subjclass[2010]{30C55, 30C62, 31A05}
\keywords{Harmonic mapping, Schwarzian derivative, quasiconformal extension, quasiconformal decomposition}
\maketitle
\begin{abstract} We prove that a harmonic quasiconformal mapping defined on a finitely connected domain in the plane, all of whose boundary components are either points or quasicircles, admits a quasiconformal extension to the whole plane if its Schwarzian derivative is small. We also make the observation that a univalence criterion for harmonic mappings holds on uniform domains. \end{abstract}
\section{ Introduction}
Let $f$ be a harmonic mapping in a planar domain $D$ and let $\om=\overline{f_{\overline{z}}} / f_z$ be its dilatation. According to Lewy's theorem the mapping $f$ is locally univalent if and only if its Jacobian $J_f=|f_z|^2-|f_{\overline{z}}|^2$ does not vanish. Duren's book \cite{Du} contains valuable information about the theory of planar harmonic mappings.
The Schwarzian derivative of $f$ was defined by Hern\'andez and Mart\'in \cite{HM15} as \begin{equation} \label{Schw-def} S_f \, = \, \rho_{zz} - \tfrac{1}{2} (\rho_z)^2, \qquad \text{where} \qquad \rho =\log J_f. \end{equation} When $f$ is holomorphic this reduces to the classical Schwarzian derivative. Another definition, introduced by Chuaqui, Duren and Osgood \cite{CDO03}, applies to harmonic mappings which admit a lift to a minimal surface via the Weierstrass-Enneper formulas. However, focusing on the planar theory in this note we adopt the definition \eqref{Schw-def}.
We assume that $\overline{\SC}\backslash D$ contains at least three points, so that $D$ is equipped with the hyperbolic metric, defined by $$
\la_D \big(\pi(z)\big) |\pi'(z)| |dz| \, = \, \la_\SD(z) |dz|\, = \,\frac{|dz|}{1-|z|^2}, \qquad z\in \SD, $$ where $\SD$ is the unit disk and $\pi:\SD\to D$ is a universal covering map. The size of the Schwarzian derivative of a mapping $f$ in $D$ is measured by the norm $$
\|S_f\|_D \, = \, \sup_{z\in D} \, \la_D(z)^{-2} |S_f(z)|. $$
A domain $D$ in $\overline{\SC}$ is a $K$-quasidisk if it is the image of the unit disk under a $K$-quasiconformal self-map of $\overline{\SC}$, for some $K\geq 1$. The boundary of a quasidisk is called a quasicircle.
According to a theorem of Ahlfors~\cite{Ah}, if $D$ is a $K$-quasidisk then there exists a constant $c>0$, depending only on $K$, such that if $f$ is analytic in $D$ with $\|S_f\|_D \leq c$ then $f$ is univalent in $D$ and has a quasiconformal extension to $\overline{\SC}$. This has been generalized by Osgood~\cite{O80} to the case when $D$ is a finitely connected domain whose boundary components are either points or quasicircles. Further, the univalence criterion was generalized to uniform domains (see Section~\ref{sect-uniform} for a definition) by Gehring and Osgood~\cite{GO79} and, subsequently, the quasiconformal extension criterion was generalized to uniform domains by Astala and Heinonen~\cite{AH88}.
For harmonic mappings and the definition \eqref{Schw-def} of the Schwarzian derivative, a univalence and quasiconformal extension criterion in the unit disk $\SD$ was proved by Hern\'andez and Mart\'in~\cite{HM15-2}. This was recently generalized to quasidisks by the present author in \cite{Ef}. Moreover, in \cite{Ef} it was shown that if all boundary components of a finitely connected domain $D$ are either points or quasicircles then any harmonic mapping in $D$ with sufficiently small Schwarzian derivative is injective. The main purpose of this note is to prove the following theorem.
\begin{theorem}\label{main-thm}
Let $D$ be a finitely connected domain whose boundary components are either points or quasicircles and let also $d\in[0,1)$. Then there exists a constant $c>0$, depending only on the domain $D$ and the constant $d$, such that if $f$ is harmonic in $D$ with $\|S_f\|_D \leq c$ and with dilatation $\om$ satisfying $|\om(z)| \leq d$ for all $z\in D$ then $f$ admits a quasiconformal extension to $\overline{\SC}$. \end{theorem}
As mentioned above, for the case when $D$ is a (simply connected) quasidisk this was shown in \cite{Ef} while, on the other hand, for the case $d=0$ (when $f$ is analytic) this was proved by Osgood in \cite{O80}. Osgood's proof amounts to proving a univalence criterion in $f(D)$. Such an approach does not seem to work here since for a holomorphic $\phi$ on $f(D)$ the composition $\phi\circ f$ is not, in general, harmonic.
Since isolated boundary points are removable for quasiconformal mappings (see \cite[Ch.I, \S8.1]{LV}), we may assume for the proof of Theorem~\ref{main-thm} that $\partial D$ consists of $n$ non-degenerate quasicircles. Our proof will be based on the following theorem of Springer~\cite{Sp64} (see also \cite[Ch.II, \S8.3]{LV}).
\begin{theoremO}[\cite{Sp64}]\label{Springer-thm} Let $D$ and $D'$ be two $n$-tuply connected domains whose boundary curves are quasicircles. Then every quasiconformal mapping of $D$ onto $D'$ can be extended to a quasiconformal mapping of the whole plane. \end{theoremO}
Hence, to prove Theorem~\ref{main-thm} it suffices that we show that the boundary components of $f(D)$ are quasicircles. We prove this in Section~\ref{sect-proof}. It relies on Osgood's \cite{O80} quasiconformal decomposition, which we briefly present in Section~\ref{sect-QC-decomp}. In Section~\ref{sect-uniform} we give a univalence criterion on uniform domains.
\section{Quasiconformal Decomposition} \label{sect-QC-decomp}
Let $D$ be a domain in $\overline{\SC}$. A collection $\mathfrak{D}$ of domains $\De \subset D$ is called a $K$-quasiconformal decomposition of $D$ if each $\De$ is a $K$-quasidisk and any two points $z_1,z_2\in D$ lie in the closure of some $\De\in\mathfrak{D}$. This definition was introduced by Osgood in \cite{O80}, along with the following lemma.
\begin{lemmO}[\cite{O80}] \label{lem-qc-decomp} If $D$ is a finitely connected domain and each component of $\partial D$ is either a point or a quasicircle then $D$ is quasiconformally decomposable. \end{lemmO}
We now present, almost verbatim, the construction proving Lemma~\ref{lem-qc-decomp}. We focus on the parts of the construction we will be needing, maintaining the notation of \cite{O80} and skipping all the relevant proofs. The interested reader should consult \cite{O80} for further details.
As we mentioned earlier, we may assume that $\partial D$ consists of non-degenerate quasicircles $C_0, C_1, \ldots, C_{n-1}$, for $n\geq2$. Let $F$ be a conformal mapping of $D$ onto a circle domain $D'$. Then, with an application of Theorem~\ref{Springer-thm} to $F^{-1}$, it will be sufficient to find a quasiconformal decomposition of $D'$. Hence we may assume that $D$ itself is a circle domain with boundary circles $C_j, \, j=0,\ldots,n-1$.
If $n=2$ then we may assume that $D$ is the annulus $1<|z|<R$. Then the domains $$ \De_1=\{z\in D \,:\, 0<{\rm arg}(z)<\tfrac{4\pi}{3}\}, \qquad \De_2=e^{2\pi i /3} \De_1, \qquad \De_3=e^{4\pi i /3} \De_1 $$ make a quasiconformal decomposition of $D$.
Let $n\geq3$. Then there exists a conformal mapping $\Psi$ of the circle domain $D$ onto a domain $D'$ consisting of the entire plane minus $n$ finite rectilinear slits lying on rays emanating from the origin. The mapping can be chosen so that no two distinct slits lie on the same ray. The boundary behavior of $\Psi$ is the following: it can be analytically extended to $\overline{D}$, and the two endpoints of the slit $C_j'=\Psi(C_j)$ correspond to two points on the circle $C_j$ which partition $C_j$ into two arcs, each of which is mapped onto $C_j'$ in a one-to-one fashion.
Let $\xi_j$ be the endpoint of $C_j'$ furthest from the origin and let $Q_j'$ be the part of the ray that joins $\xi_j$ to infinity. Let also $S_j'$ be the sector between $C_j'$ and $C_{j+1}'$. Let $\om_j'$ be the midpoint of $C_j'$ and let $P_j'$ be a polygonal arc joining $\om_j'$ to $\om_{j+1}'$ that, except for its endpoints, lies completely in $S_j'$. Then $$ P'=\bigcup_{j=0}^{n-1} P_j' $$ is a closed polygon separating $0$ from $\infty$ that does not intersect any of the $Q_j'$. Let $G_0'$ and $G_1'$ be the components of $D'\backslash P'$ that contain $0$ and $\infty$, respectively. Now define $$ \Delta_{0j}' = G_0' \cup S_j', \qquad \Delta_j' = D' \backslash \Delta_{0j}' $$ and $$ \mathfrak{D}' = \{\Delta_{0j}' \, : \, j=0,1,\ldots,n-1 \} \cup \{\Delta_j' \, : \, j=0,1,\ldots,n-1 \}. $$ This collection has the covering property for $D'$.
We denote the various parts of $D$ corresponding under $\Psi^{-1}$ to those of $D'$ by the same symbol without the prime. Then $$ \mathfrak{D} = \{\Delta_{0j} \, : \, j=0,1,\ldots,n-1 \} \cup \{\Delta_j \, : \, j=0,1,\ldots,n-1 \} $$ is a quasiconformal decomposition of $D$.
\section{Proof of Theorem~\ref{main-thm}} \label{sect-proof}
Let $f$ be a mapping in $D$ as in Theorem~\ref{main-thm}. By Theorem~2 in \cite{Ef}, $f$ is injective if $c$ is sufficiently small. Also, $f$ extents continuously to $\partial D$ since every boundary point of $D$ belongs to $\partial \Delta$ for some $\Delta$ in the collection $\mathfrak{D}$ and, by Theorem~1 in \cite{Ef}, the restriction of $f$ on $\Delta$ admits a homeomorphic extension to $\overline{\SC}$.
Let $\Psi$ be a conformal mapping of $D$ onto the slit domain $D'$ of the previous section. Let $C_j$ be a boundary quasicircle of $D$. We first prove that $f(C_j)$ is a Jordan curve. The slit $C_j'$ is divided by its midpoint $\om_j'$ into two line segments, which we denote by $\Sigma_j'(m), m=1,2$, so that $$
\Sigma_j'(1)=\{z\in C_j' : |z|\leq|\om_j'|\} \qquad \text{and} \qquad \Sigma_j'(2)=\{z\in C_j' : |z|\geq|\om_j'|\}. $$ Let $\Sigma_j'(m)^{\pm}$ denote the two sides of $\Sigma_j'(m)$, so that a point $z_0$ on $\Sigma_j'(m)^-$ is reached only by points $z \in S_{j-1}'$, meaning that $\arg z \to (\arg z_0)^-$ when $z\to z_0$. Similarly, a point $z_0$ on $\Sigma_j'(m)^+$ is reached only by points $z \in S_j'$, so that $\arg z \to (\arg z_0)^+$ when $z\to z_0$. Corresponding under $\Psi^{-1}$ are four disjoint -except for their endpoints- arcs on the quasicircle $C_j$, denoted without the prime by $\Sigma_j(m)^{\pm}, m=1,2$. Now consider the domains $\Delta_{0,j-1}, \Delta_{0j}$ and $\Delta_k$ in the collection $\mathfrak{D}$, for some $k\neq j-1, j$; see Figure~\ref{fig} for their images under $\Psi$. By Theorem~1 in \cite{Ef} $f$ is injective up to the boundary of each $\Delta\in\mathfrak{D}$. Note that the arcs $\Sigma_j(1)^-, \Sigma_j(1)^+$ and $\Sigma_j(2)^-$ are subsets of $\partial\Delta_{0,j-1}$, so that their images under $f$, except for their endpoints, are disjoint. It remains to show that the images of these three arcs under $f$ are not intersected by the remaining image $f(\Sigma_j(2)^+)$. Note that the arcs $\Sigma_j(1)^-, \Sigma_j(1)^+$ and $\Sigma_j(2)^+$ are subsets of $\partial\Delta_{0j}$, so that $f(\Sigma_j(2)^+)$ does not intersect $f(\Sigma_j(1)^-)$ nor $f(\Sigma_j(1)^+)$. What remains to be seen is that $f(\Sigma_j(2)^-)$ and $f(\Sigma_j(2)^+)$ are disjoint and this follows from the fact that the arcs $\Sigma_j(2)^-$ and $\Sigma_j(2)^+$ are subsets of $\partial\Delta_k$.
\begin{figure}
\caption{The slit $C_j'$ and the three distinguished domains.}
\label{fig}
\end{figure}
To see that the Jordan curve $f(C_j)$ is actually a quasicircle note that each point of $f(C_j)$ belongs to some open subarc of $f(C_j)$ which is entirely included in the boundary of either $f(\Delta_{0,j-1}), f(\Delta_{0j})$ or $f(\Delta_k)$. These three domains are quasidisks by Theorem~3 in \cite{Ef}. Now the assertion that $f(C_j)$ is a quasicircle follows by an application of Theorem~8.7 in \cite[Ch.II, \S8.9]{LV}.
\section{Remarks on uniform domains} \label{sect-uniform}
A domain $D$ in $\SC$ is called uniform if there exist positive constants $a$ and $b$ such that each pair of points $z_1,z_2\in D$ can be joined by an arc $\ga \subset D$ so that for each $z\in \ga$ it holds $$
\ell(\ga) \leq a \, |z_1-z_2| $$ and $$ \min_{j=1,2}\ell(\ga_j) \leq b \, {\rm dist}(z,\partial D), $$ where $\ga_1, \ga_2$ are the components of $\ga\backslash\{z\}$, ${\rm dist}(z,\partial D)$ denotes the euclidean distance from $z$ to the boundary of $D$ and $\ell(\cdot)$ denotes euclidean length. Uniform domains were introduced by Martio and Sarvas \cite{MS79}; see also, \emph{e.g.}, \cite{GO79} for this equivalent definition. In \cite{MS79} it was shown that all boundary components of a uniform domain are either points or quasicircles. The converse of this is also true for finitely connected domains, but not, in general, for domains of infinite connectivity; see \cite[\S3.5]{GeHa}. The following univalence criterion was proved in \cite{MS79}.
\begin{theoremO}[\cite{MS79},\cite{GO79}] \label{Thm-uniform-1-1}
If $D$ is a uniform domain then there exists a constant $c>0$ such that every analytic function $f$ in $D$ with $\|S_f\|_D\leq c$ is injective. \end{theoremO}
Gehring and Osgood \cite{GO79} gave a different proof of Theorem~\ref{Thm-uniform-1-1} by providing a characterization of uniform domains. They showed that a domain $D$ is uniform if and only if it is quasiconformally decomposable in the following weaker (than the one we saw in Section~\ref{sect-QC-decomp}) sense: there exists a constant $K$ with the property that for each $z_1,z_2\in D$ there exists a $K$-quasidisk $\De\subset D$ for which $z_1,z_2\in\overline{\De}$. Note that, in contrast to Osgood's \cite{O80} decomposition, here $\De$ depends on the points $z_1,z_2$. However, this can readily be used to generalize the implication (i) $\Rightarrow$ (iii) of Theorem~2 in \cite{Ef}, according to which a univalence criterion for harmonic mappings holds on finitely connected uniform domains. The following theorem extends it to all uniform domains.
\begin{theorem}
Let $D$ be a uniform domain in $\SC$. Then there exists a constant $c>0$ such that if $f$ is harmonic in $D$ with $\|S_f\|_D \leq c$ then $f$ is injective. \end{theorem} \begin{proof} Assume that there exist distinct points $z_1,z_2\in D$ for which $f(z_1)=f(z_2)$. By \cite{GO79}, there exists a $K$-quasidisk $\De\subset D$ for which $z_1,z_2\in\overline{\De}$. The domain monotonicity for the hyperbolic metric shows that $$
\|S_f\|_\De \leq \|S_f\|_D \leq c. $$ But the homeomorphic extension of Theorem~1 in \cite{Ef} shows that if $c$ is sufficiently small then $f$ is injective up to the boundary of $\De$, a contradiction. \end{proof}
Regarding quasiconformal extension, Astala and Heinonen \cite{AH88} proved the following theorem.
\begin{theoremO}[\cite{AH88}] \label{Thm-uniform-qc-ext}
If $D$ is a uniform domain then there exists a constant $c>0$ such that every analytic function $f$ in $D$ with $\|S_f\|_D\leq c$ admits a quasiconformal extension to $\overline{\SC}$. \end{theoremO}
This evidently implies Theorem~\ref{Thm-uniform-1-1} and was also proved in substantially greater generality, but we omit it here. It is not clear how to generalize Theorem~\ref{Thm-uniform-qc-ext} to the setting of harmonic mappings. Therefore, we propose the following problem.
\begin{problem}
Let $D$ be a uniform domain. Does there exist a constant $c>0$ such that if $f$ is harmonic in $D$ with $\|S_f\|_D \leq c$ and with dilatation $\om$ satisfying $\sup_{z\in D}|\om(z)| <1$ then $f$ admits a quasiconformal extension to $\overline{\SC}$? \end{problem}
\end{document} | arXiv |
Suppose that $(u_n)$ is a sequence of real numbers satisfying
\[u_{n+2}=2u_{n+1}+u_n\]and that $u_3=9$ and $u_6=128$. What is $u_5$?
Let $u_4 = a.$ Then $u_5 = 2u_4 + u_3 = 2a + 9$ and $u_6 = 2u_5 + u_4 = 2(2a + 9) + a = 5a + 18 = 128.$ Solving for $a,$ we find $a = 22,$ so $u_5 = 2 \cdot 22 + 9 = \boxed{53}.$ | Math Dataset |
EURASIP Journal on Advances in Signal Processing
Consistent independent low-rank matrix analysis for determined blind source separation
Daichi Kitamura1 na1 &
Kohei Yatabe2 na1
EURASIP Journal on Advances in Signal Processing volume 2020, Article number: 46 (2020) Cite this article
Independent low-rank matrix analysis (ILRMA) is the state-of-the-art algorithm for blind source separation (BSS) in the determined situation (the number of microphones is greater than or equal to that of source signals). ILRMA achieves a great separation performance by modeling the power spectrograms of the source signals via the nonnegative matrix factorization (NMF). Such a highly developed source model can solve the permutation problem of the frequency-domain BSS to a large extent, which is the reason for the excellence of ILRMA. In this paper, we further improve the separation performance of ILRMA by additionally considering the general structure of spectrograms, which is called consistency, and hence, we call the proposed method Consistent ILRMA. Since a spectrogram is calculated by an overlapping window (and a window function induces spectral smearing called main- and side-lobes), the time-frequency bins depend on each other. In other words, the time-frequency components are related to each other via the uncertainty principle. Such co-occurrence among the spectral components can function as an assistant for solving the permutation problem, which has been demonstrated by a recent study. On the basis of these facts, we propose an algorithm for realizing Consistent ILRMA by slightly modifying the original algorithm. Its performance was extensively evaluated through experiments performed with various window lengths and shift lengths. The results indicated several tendencies of the original and proposed ILRMA that include some topics not fully discussed in the literature. For example, the proposed Consistent ILRMA tends to outperform the original ILRMA when the window length is sufficiently long compared to the reverberation time of the mixing system.
Blind source separation (BSS) is a technique for separating individual sources from an observed mixture without knowing how they were mixed. BSS for multichannel audio signals observed by multiple microphones has been particularly studied [1–13]. The BSS problem can be divided into two situations: underdetermined (the number of microphones is less than the number of sources) and (over-)determined (the number of microphones is greater than or equal to the number of sources) cases. This paper focuses on the determined BSS problem, as high-quality separation can be achieved compared with the underdetermined BSS methods.
Independent component analysis (ICA) is the most popular and successful algorithm for solving the determined BSS problem [1]. It estimates a demixing matrix (the inverse system of the mixing process) by assuming statistical independence between the sources. For a mixture of audio signals, ICA is usually applied in the time-frequency domain via the short-time Fourier transform (STFT) because the sources are mixed up by convolution. This strategy is called frequency-domain ICA (FDICA) [2] and independently applies ICA to the complex-valued signals in each frequency. Then, the estimated frequency-wise demixing matrices must be aligned over all frequencies so that the frequency components of the same source are grouped together. Such alignment of the frequency components is called a permutation problem [3–6], and a complete solution to it has not been established. Therefore, a great deal of research has tackled this problem.
To avoid the permutation misalignment as much as possible, various sophisticated source models have been proposed. Independent vector analysis (IVA) [7–10] is one of the most successful methods in the early stage of the development. It assumes higher-order dependencies (co-occurrence among the frequency components) of each source by utilizing a spherical generative model of the source frequency vector. This assumption enables IVA to simultaneously estimate the frequency-wise demixing matrices and solve the permutation problem to a large extent using only one objective function. It has been further developed by improving its source model. One natural and powerful extension of IVA is independent low-rank matrix analysis (ILRMA) [11, 12], which integrates the source model of nonnegative matrix factorization (NMF) [14, 15] based on the Itakura–Saito divergence (IS-NMF) [16] into IVA. This extension has greatly improved the performance of separation by taking the low-rank time-frequency structure (co-occurrence among the time-frequency bins) of the source signals into account. ILRMA has achieved the state-of-the-art performance and been further developed by several researchers [17–29]. In this respect, ILRMA can be considered the new standard of the determined BSS algorithms. However, the separation performance of IVA and ILRMA is still inferior compared to the ideal performance of ICA-based frequency-domain BSS. In [30], the performances of IVA and ILRMA were compared with that of FDICA with perfect permutation alignment using reference sources (ideal permutation solver), and it was confirmed that there is still a noticeable room for improvement of ILRMA-based BSS. In fact, IVA and ILRMA often encounter the block permutation problem, that is, group-wise permutation misalignment of components between sources [31].
The consistency of a spectrogram is another promising approach for solving the permutation problem. A recent study has shown that STFT can provide some effective information related to the co-occurrence among the time-frequency bins [32]. Since an overlapping window is utilized in STFT, the time-frequency bins are related to each other based on the overlapping segments. The frequency components within a segment are also related to each other because of the spectral smearing called main- and side-lobes of the window. In other words, the time-frequency components are not independent but related to each other via the uncertainty principle of time-frequency representation. Such relations have been well-studied in phase-aware signal processing [33–43] by the name of spectrogram consistency [44–47]. In the previous study [32], the spectrogram consistency was imposed on BSS to help the algorithm solve the permutation problem. This is an approach very different from the conventional studies of determined BSS because it utilizes the general property of STFT independent of the source model (in contrast to the abovementioned methods that focused on modeling of the source signals without considering the property of STFT). As the spectrogram consistency can be incorporated with any source model, its combination with the state-of-the-art algorithm should achieve a high separation performance.
However, the paper that proposed the combination of consistency and determined BSS [32] only showed the potential of consistency in an experiment using FDICA and IVA. The paper claimed that it was a first step of incorporating the spectrogram consistency with determined BSS, and no advanced method was tested. In particular, ILRMA was not considered because its algorithm is far more complicated than that derived in [32], and thus, it is not clear whether (and how much) the spectrogram consistency might improve the state-of-the-art BSS algorithm.
In this paper, we propose a new variant of ILRMA called Consistent ILRMA that considers the spectrogram consistency within the algorithm of ILRMA. The combination of IS-NMF and spectral smoothing of the inverse STFT (see Figs. 1 and 2 in Section 2.3) achieves the source modeling for a complex spectrogram. In particular, the spectral smearing in the frequency direction ties the adjacent frequency bins together, and this effect of spectrogram consistency helps ILRMA to solve the permutation problem. Since consistency is a concept depending on the parameters related to a window function, we extensively tested the separation performance of Consistent ILRMA through experiments with various window lengths and shift lengths. The results clarified several tendencies of the conventional and proposed methods, including that the proposed method outperforms the original ILRMA when the window length is sufficiently long compared to the reverberation time of the mixing system.
Inconsistent power spectrograms |Sart|2 (left column) and their consistent version (right column) obtained by applying inverse STFT and STFT. The top-left spectrogram is artificially produced with random phase. The middle-left and the bottom-left spectrograms are music and speech signals with random dropout. Enforcing spectrogram consistency can be viewed as a smoothing process of the inconsistent spectrogram along both time and frequency axes
Smoothing effect of spectrogram consistency applied to permutation misaligned signals: a music and b speech. The left column shows the original source signals |Sn|2, and the center column shows their randomly permuted versions, which simulates the permutation problem and is denoted as \(\boldsymbol {S}_{n}^{(\text {perm})}\). The right column shows the consistent versions of \(\boldsymbol {S}_{n}^{\mathrm {(perm)}}\). The smoothing effect mixes up the signals
Permutation problem of frequency-domain BSS and spectrogram consistency
Formulation of frequency-domain BSS
Let the lth sample of a time-domain signal be denoted as x[l], and N source signals be observed by M microphones. Then, the lth samples of the multichannel source, observed, and separated signals are respectively denoted as:
$$\begin{array}{*{20}l} {}\boldsymbol{s}[l] &= \left[\, s_{1}[l], s_{2}[l], \cdots, s_{n}[l], \cdots s_{N}[l] \,\right]^{\mathrm{T}} \in \mathbb{R}^{N}, \end{array} $$
$$\begin{array}{*{20}l} {}\boldsymbol{x}[l] &= \left[\, x_{1}[l], x_{2}[l], \cdots, x_{m}[l], \cdots x_{M}[l] \,\right]^{\mathrm{T}} \in \mathbb{R}^{M}, \end{array} $$
$$\begin{array}{*{20}l} {} \boldsymbol{y}[l] &= \left[\, y_{1}[l], y_{2}[l], \cdots, y_{n}[l], \cdots y_{N}[l] \,\right]^{\mathrm{T}} \in \mathbb{R}^{N}, \end{array} $$
where n=1,⋯,N,m=1,⋯,M, and l=1,⋯,L are the indexes of sources, microphones (channels), and discrete time, respectively, and ·T denotes the transpose. BSS aims at recovering the source signal s from the observed signal x, i.e., making y as close to s as possible.
In the frequency-domain BSS, those signals are handled in the time-frequency domain via STFT. Let the window length and shifting step of STFT be denoted as Q and τ, respectively. Then, the jth segment of a signal z[l] is defined as:
$$\begin{array}{*{20}l} {}\boldsymbol{z}^{[j]} &\,=\, \left[ z\left[(j\,-\,1)\tau \,+\,1\right]\!, z\left[(j\,-\,1)\tau \,+\,2\right], \cdots, z\!\left[\!(j\,-\,1)\tau \,+\,Q\right]\! \,\right]^{\mathrm{T}}\!, \\ &=\! \left[ z^{[j]}[1], z^{[j]}[2], \cdots, z^{[j]}[q], \cdots, z^{[j]}[Q] \,\right]^{\mathrm{T}} \in \mathbb{R}^{Q}, \end{array} $$
where j=1,⋯,J and q=1,⋯,Q are the indexes of the segments and in-segment samples, respectively, and the number of segments is given by J=L/τ with some zero-padding for adjusting the signal length L if necessary. STFT of a signal \(\boldsymbol {z} =\ [\,z[1], \,z[2],\cdots,z[L]\,]^{\mathrm {T}}\in \mathbb {R}^{L}\) is denoted by:
$$ \boldsymbol{Z} = \text{STFT}_{\boldsymbol{\omega}}(\boldsymbol{z}) \;\;\in\mathbb{C}^{I\times J}, $$
where the (i,j)th bin of the spectrogram Z is given as:
$$ z_{ij} = \sum\limits_{q=1}^{Q} \omega[q]\,z^{[j]}[q]\;\mathrm{e}^{-\imath2\pi(q-1)(i-1)/F}, $$
i=1,⋯,I is the index of frequency bins, F is an integer satisfying ⌊F/2⌋+1=I,⌊·⌋ is the floor function, ı denotes the imaginary unit, and ω is an analysis window. The inverse STFT with a synthesis window \(\widetilde {\boldsymbol {\omega }}\) is also defined in the usual way and denoted as \(\text {ISTFT}_{\widetilde {\boldsymbol {\omega }}}(\cdot)\). In this paper, we assume that the window pair satisfies the following perfect reconstruction condition:
$$ \boldsymbol{z} = \text{ISTFT}_{\widetilde{\boldsymbol{\omega}}}\left(\text{STFT}_{\boldsymbol{\omega}}(\boldsymbol{z})\right)\qquad\forall \boldsymbol{z}\in\mathbb{R}^{L}. $$
By applying STFT, the (i,j)th bin of the spectrograms of source, observed, and separated signals can be written as:
$$\begin{array}{*{20}l} \boldsymbol{s}_{ij} &= \left[\, s_{ij1}, s_{ij2}, \cdots, s_{ijn}, \cdots s_{ijN} \,\right]^{\mathrm{T}} \in \mathbb{C}^{N}, \end{array} $$
$$\begin{array}{*{20}l} \boldsymbol{x}_{ij} &= \left[\, x_{ij1}, x_{ij2}, \cdots, x_{ijm}, \cdots x_{ijM} \,\right]^{\mathrm{T}} \in \mathbb{C}^{M}, \end{array} $$
$$\begin{array}{*{20}l} \boldsymbol{y}_{ij} &= \left[\, y_{ij1}, y_{ij2}, \cdots, y_{ijn}, \cdots y_{ijN} \,\right]^{\mathrm{T}} \in \mathbb{C}^{N}. \end{array} $$
We also denote the spectrograms corresponding to the nth or mth signals in (8)–(10) as \(\boldsymbol {S}_{n}\in \mathbb {C}^{I\times J}, \boldsymbol {X}_{m}\in \mathbb {C}^{I\times J}\), and \(\boldsymbol {Y}_{n}\in \mathbb {C}^{I\times J}\), whose elements are sijn,xijm, and yijn, respectively. In the ordinary frequency-domain BSS, an instantaneous mixing process for each frequency bin is assumed:
$$\begin{array}{*{20}l} \boldsymbol{x}_{ij} = \boldsymbol{A}_{i}\boldsymbol{s}_{ij}, \end{array} $$
where \(\boldsymbol {A}_{i}\in \mathbb {C}^{M\times N}\) is a frequency-wise mixing matrix. The mixture model (11) is approximately valid when the reverberation time is sufficiently shorter than the length of the analysis window used in STFT [48].
Hereafter, we consider the determined case, i.e., M=N. In this case, BSS can be achieved by estimating the inverse of Ai for all frequency bins. By denoting an approximate inverse as \(\boldsymbol {W}_{i}\approx \boldsymbol {A}_{i}^{-1}\), the separation process can be written as:
$$\begin{array}{*{20}l} \boldsymbol{y}_{ij} = \boldsymbol{W}_{i}\boldsymbol{x}_{ij}, \end{array} $$
where \(\boldsymbol {W}_{i}=\left [\boldsymbol {w}_{i1},\boldsymbol {w}_{i2},\cdots,\boldsymbol {w}_{iN}\right ]^{\mathrm {H}}\in \mathbb {C}^{N\times M}\) is a frequency-wise demixing matrix and ·H denotes the Hermitian transpose. The aim of a determined BSS algorithm is to find the demixing matrices for all frequency bins so that the separated signals approximate the source signals.
Permutation problem in determined BSS
In practice, the scale and permutation of the separated signals are unknown because the information of the mixing process is missing. That is, when the separation is correctly performed by some demixing matrix Wi as in (12), the following signal is also a solution to the BSS problem:
$$ \hat{\boldsymbol{y}}_{ij} = \hat{\boldsymbol{W}}_{i}\boldsymbol{x}_{ij} \qquad \left(\hat{\boldsymbol{W}}_{i} = \boldsymbol{D}_{i}\boldsymbol{P}_{i}\boldsymbol{W}_{i}\right), $$
where \(\boldsymbol {D}_{i}\in \mathbb {C}^{N\times N}\) and Pi∈{0,1}N×N are arbitrary diagonal and permutation matrices, respectively. While the signal scale can easily be recovered by applying the back projection [49], the permutation of the estimated signals \(\hat {\boldsymbol {y}}_{ij}\) must be aligned for all frequency bins, i.e., Pi must be the same for all i. This alignment of the permutation of estimated signals is the permutation problem, which is the main obstacle of the frequency-domain determined BSS.
In FDICA, a permutation solver (realignment process of Pi) is utilized as a post-processing applied to the frequency-wise separated signals \(\hat {\boldsymbol {y}}_{ij}\) [4–6]. In recent frequency-domain BSS methods, an additional assumption on sources (or source model) is introduced to circumvent the permutation problem. For example, IVA assumes simultaneous co-occurrence of all frequency components in the same source, and ILRMA assumes a low-rank structure of the power spectrogram Yn. Other source models have also been proposed for improving the separation performance [50–52]. These source models can avoid the permutation problem to some extent during the estimation of \(\hat {\boldsymbol {W}}_{i}\). Recent developments of determined BSS have been achieved via the quest to find a better source model that represents the source signals more precisely.
Solving permutation problem by spectrogram consistency
A recent paper reported another approach for solving the permutation problem based on the general property of STFT called spectrogram consistency [32]. The consistency is a fundamental property of a spectrogram. Since any time-frequency representation has a theoretical limitation called the uncertainty principle, the time-frequency bins of a spectrogram are not independent but related to each other. The inverse STFT always modifies the spectrogram Zn that violates this kind of inter-time-frequency relation so that the relation is recovered. That is, a spectrogram Zn properly retains the inter-time-frequency relation if and only if
$$ \mathcal{E}(\boldsymbol{Z}_{n}) = \boldsymbol{Z}_{n} - \text{STFT}_{\boldsymbol{\omega}}\left(\text{ISTFT}_{\widetilde{\boldsymbol{\omega}}}(\boldsymbol{Z}_{n})\right) $$
is zero, i.e., \(\|\mathcal {E}(\boldsymbol {Z}_{n})\|=0\) for a norm ∥·∥. Such spectrogram Zn satisfying \(\|\mathcal {E}(\boldsymbol {Z}_{n})\|=0\) is said to be consistent.
Figure 1 demonstrates the effect of spectrogram consistency, where \(\boldsymbol {S}_{\text {art}}\in \mathbb {C}^{I\times J}\) is an artificially produced complex-valued spectrogram and |Sart|2 is its power spectrogram. The notation |·|2 for a matrix input represents the element-wise squared absolute value. By applying \(\text {STFT}_{\boldsymbol {\omega }}(\text {ISTFT}_{\widetilde {\boldsymbol {\omega }}}(\cdot))\), the inconsistent spectrogram Sart shown in the left column of Fig. 1 is converted into the corresponding consistent spectrogram, which is a smoothed version of Sart, as shown in the right column. This smoothing process occurs because the main- and side-lobes of the window function (and the overlap-add process) spread the energy of a time-frequency bin.
Since the inverse STFT is a process of recovering the consistency (the inter-time-frequency relation), it has the capability of aligning the frequency components. This is also demonstrated in Fig. 2. As a simulation of the permutation problem, the frequency bins in S1 and S2 were randomly shuffled to obtain the spectrogram with permutation misalignment, \(\boldsymbol {S}_{n}^{\mathrm {(perm)}}\) (the center column in the figure), which is a typical output signal of FDICA. Note that these misaligned spectrograms are perfectly separated for each frequency because each time-frequency bin contains only one of the two sources. By enforcing spectrogram consistency, the smoothing process spreads the time-frequency components as shown in the right column of Fig. 2. In other words, the inverse STFT mixes up the separated signals if the frequency-wise permutation is not aligned correctly. Therefore, enforcing consistency within a BSS algorithm by applying \(\text {STFT}_{\boldsymbol {\omega }}(\text {ISTFT}_{\widetilde {\boldsymbol {\omega }}}(\cdot))\) can improve the separation performance to some extent [32].
Proposed method
By incorporating spectrogram consistency into ILRMA, we propose a novel BSS method named Consistent ILRMA. In this section, after stating our motivation and contributions, we first review the standard ILRMA introduced in [11, 12] and then propose the consistent version of ILRMA with an algorithm that achieves Consistent ILRMA and is openly available on the web.
Motivations and contributions
The previous paper [32] only reported that the performances of traditional BSS algorithms, FDICA and IVA, were improved by enforcing consistency during the estimation of the demixing matrix Wi. In addition, no detailed experimental analysis related to STFT parameters was provided, even though the parameters of window functions in the STFT and inverse STFT directly affect the smoothing effect of spectrogram consistency.
The spectrogram consistency is a general property of STFT, and therefore, it can be combined with any source model for determined BSS. Its combination with state-of-the-art models, including ILRMA, is of great interest because the current mainstream algorithm for determined audio source separation is centered on ILRMA, which is based on an NMF-based richer time-frequency source model. Indeed, many recent papers are based on the framework of ILRMA [17–29]. Even though combining ILRMA with the spectrogram consistency should be able to exceed the limit of existing BSS algorithms, no such method has been investigated in the literature.
In this paper, we propose a new BSS algorithm that combines ILRMA and spectrogram consistency. Our first contribution is an algorithm that achieves Consistent ILRMA by inserting \(\text {STFT}_{\boldsymbol {\omega }}(\text {ISTFT}_{\widetilde {\boldsymbol {\omega }}}(\cdot))\) into the iterative optimization algorithm of ILRMA. The second contribution is to apply a scale-aligning process called iterative back projection within the iterative algorithm. This process enhances the separation performance when it is combined with spectrogram consistency. The third contribution is an experimental finding that spectrogram consistency can work properly with the iterative back projection. We found that both Consistent IVA and Consistent ILRMA require iterative back projection to achieve a good performance. Our fourth contribution is to provide the massive experimental results for several window functions, window lengths, shift lengths, reverberation times, and source types. We also provide discussions for clarifying the tendency of ILRMA with spectrogram consistency.
Standard ILRMA [12]
The original ILRMA [12] was derived from the following generative model of the spectrograms of the separated signals:
$$ {} \boldsymbol{Y}_{n} \sim p(\boldsymbol{Y}_{n}) = \prod\limits_{i,j} \mathcal{N}_{\mathrm{c}}\left(0,r_{ijn}\right) = \prod\limits_{i,j} \frac{ 1 }{ \pi r_{ijn}} \exp{\left(-\frac{ |y_{ijn}|^{2} }{ r_{ijn}} \right)}, $$
where \(\mathcal {N}_{\mathrm {c}}\left (\mu, r\right)\) is the circularly symmetric complex Gaussian distribution with mean μ and variance r. In this model, the source component yijn is assumed to obey a zero-mean and isotropic distribution, i.e., the phase of yijn is generated from the uniform distribution in the range [0,2π) and the real and imaginary parts of yijn are mutually independent. The validity of this assumption is shown in the Appendix. The variance rijn can be viewed as an expectation value of |yijn|2. This variance rijn as a two-dimensional array indexed by (i,j) is denoted as \(\boldsymbol {R}_{n}\in \mathbb {R}_{> 0}^{I\times J}\), which is called the variance spectrogram corresponding to the nth source. In ILRMA, the variance matrix Rn is modeled using the rank-K NMF, as:
$$\begin{array}{*{20}l} \boldsymbol{R}_{n} = \boldsymbol{T}_{n}\boldsymbol{V}_{n}, \end{array} $$
where \(\boldsymbol {T}_{n}\in \mathbb {R}_{> 0}^{I\times K}\) and \(\boldsymbol {V}_{n}\in \mathbb {R}_{> 0}^{K\times J}\) are the basis and activation matrices in NMF. The basis vectors in Tn, which represent spectral patterns of the nth source signal, are indexed by k=1,⋯,K. As in FDICA, statistical independence between the source signals is also assumed in ILRMA:
$$ p(\boldsymbol{Y}_{1}, \boldsymbol{Y}_{2}, \cdots, \boldsymbol{Y}_{N}) = \prod\limits_{n} p(\boldsymbol{Y}_{n}). $$
ILRMA estimates the demixing matrix Wi so that the power spectrograms of the separated signals |Yn|2 have a low-rank structure that can be well-approximated by TnVn with small K. This BSS principle of ILRMA is illustrated in Fig. 3. When the low-rank source model can appropriately fit to the power spectrograms of the original source signals |Sn|2, ILRMA provides an excellent separation performance without explicitly solving the permutation problem afterward.
BSS principle of standard ILRMA
The demixing matrix Wi and the nonnegative matrices Tn and Vn can be obtained through maximum likelihood estimation. The negative log-likelihood to be minimized, denoted by \(\mathcal {L}\), is given as [12]:
$$ {}\begin{aligned} \mathcal{L} &= - \log p(\boldsymbol{X}_{1}, \boldsymbol{X}_{2}, \cdots, \boldsymbol{X}_{M}), \\ &= -\sum\limits_{i,j} \log \left|\det \boldsymbol{W}_{i}\right|^{2} - \log p(\boldsymbol{Y}_{1}, \boldsymbol{Y}_{2}, \cdots, \boldsymbol{Y}_{N}), \\ &\stackrel{\mathrm{c}}{=} -2J\sum\limits_{i} |\det \boldsymbol{W}_{i}| \,+\, \sum\limits_{i,j,n} \!\left(\!\frac{ \left|\boldsymbol{w}_{in}^{\mathrm{H}}\boldsymbol{x}_{ij}\right|^{2} }{ {\sum\nolimits}_{k} t_{ikn}v_{kjn}} \!+ \!\log \sum\limits_{k} t_{ikn}v_{kjn} \!\right), \end{aligned} $$
where =c denotes equality up to constant factors, and tikn>0 and vkjn>0 are the elements of Tn and Vn, respectively. The minimization of (18) can be performed by iterating the following update rules for the spatial model parameters,
$$\begin{array}{*{20}l} \boldsymbol{U}_{in} &\leftarrow \frac{1}{J} \sum\limits_{j} \frac{1}{{\sum\nolimits}_{k} t_{ikn}v_{kjn}}\boldsymbol{x}_{ij}\boldsymbol{x}_{ij}^{\mathrm{H}}, \end{array} $$
$$\begin{array}{*{20}l} \boldsymbol{w}_{in} &\leftarrow \left(\boldsymbol{W}_{i}\boldsymbol{U}_{in} \right)^{-1}\boldsymbol{e}_{n}, \end{array} $$
$$\begin{array}{*{20}l} \boldsymbol{w}_{in} &\leftarrow \boldsymbol{w}_{in} \left(\boldsymbol{w}_{in}^{\mathrm{H}}\boldsymbol{U}_{in}\boldsymbol{w}_{in} \right)^{-\frac{1}{2}}, \end{array} $$
$$\begin{array}{*{20}l} y_{ijn} &\leftarrow \boldsymbol{w}_{in}^{\mathrm{H}}\boldsymbol{x}_{ij}, \end{array} $$
and for the source model parameters,
$$\begin{array}{*{20}l} t_{ikn} &\leftarrow t_{ikn} \sqrt{ \frac{ {\sum\nolimits}_{j} \left|y_{ijn}\right|^{2} \left(\sum_{k'} t_{ik'n}v_{k'jn} \right)^{-2} v_{kjn} }{ {\sum\nolimits}_{j} \left(\sum_{k'} t_{ik'n}v_{k'jn} \right)^{-1} v_{kjn}} }, \end{array} $$
$$\begin{array}{*{20}l} v_{kjn} &\leftarrow v_{kjn} \sqrt{ \frac{ {\sum\nolimits}_{i} \left|y_{ijn}\right|^{2} \left({\sum\nolimits}_{k'} t_{ik'n}v_{k'jn} \right)^{-2} t_{ikn} }{ {\sum\nolimits}_{i} \left({\sum\nolimits}_{k'} t_{ik'n}v_{k'jn} \right)^{-1} t_{ikn}} }, \end{array} $$
where en∈{0,1}N is the unit vector with the nth element equal to unity. Update rules (19)–(24) ensure the monotonic non-increase of the negative log-likelihood function \(\mathcal {L}\). After iterative calculations of updates (19)–(24), the separated signal can be obtained by (12).
Equation 22 is equivalent to beamforming [53] to xij with the beamformer coefficients win. Thus, FDICA, IVA, and ILRMA can be interpreted as an adaptive estimation process of beamforming coefficients without having to know the geometry of microphones and sources [54]. For this reason, the estimated signal Yn obtained by (22) is a complex-valued spectrogram, and we do not need to recover its phase components using, for example, Griffin–Lim algorithm-based techniques [37–40, 43, 55–59]. Both the amplitude and phase components of each source are recovered by the complex-valued linear separation filter win.
Proposed Consistent ILRMA
To further improve the separation performance of the standard ILRMA, we introduce the spectrogram consistency into the parameter update procedure. In the proposed Consistent ILRMA, the following combination of forward and inverse STFT is performed at the beginning of each iteration of parameter updates:
$$\begin{array}{*{20}l} \boldsymbol{Y}_{n} \leftarrow \text{STFT}_{\boldsymbol{\omega}}(\text{ISTFT}_{\widetilde{\boldsymbol{\omega}}}(\boldsymbol{Y}_{n})). \end{array} $$
This procedure is the projection of the spectrogram of a separated signal Yn onto the set of consistent spectrograms [32]. That is, \(\text {STFT}_{\boldsymbol {\omega }}(\text {ISTFT}_{\widetilde {\boldsymbol {\omega }}}(\boldsymbol {Y}_{n}))\) performs nothing if Yn is consistent, but otherwise, it smooths the complex spectrogram Yn, by going through the time domain, so that the uncertainty principle is satisfied.
In Consistent ILRMA, the calculation of (25) is performed in each iteration of parameter updates based on (19)–(24). Enforcing the spectrogram consistency for the temporary separated signal Yn in each iteration guides the parameters Wi,Tn, and Vn to better solutions, which results in higher separation performance compared to that of conventional ILRMA.
Note that this simple update (25) may increase the value of the negative log-likelihood function (18), and therefore, the monotonicity of the algorithm is no longer guaranteed. However, we will see later in the experiments that the value of the negative log-likelihood function stably decreases as in the standard ILRMA. The amount of the inconsistent component (14) also settles down to some specific value after several iterations.
Iterative back projection
Since frequency-domain BSS cannot determine the scales of estimated signals (represented by Di in (13)), the spectrogram of a separated signal Yn after an iteration is inconsistent due to the scale irregularity. To take full advantage of the projection enforcing spectrogram consistency in (25), we also propose applying the following back projection at the end of each iteration so that the frequency-wise scales are aligned.
In determined BSS, the back projection is a standard procedure for recovering the frequency-wise scales. It can be written as [49]:
$$ \tilde{\boldsymbol{y}}_{ijn} = \boldsymbol{W}_{i}^{-1} \left(\boldsymbol{e}_{n} \circ \boldsymbol{y}_{ij} \right) = y_{ijn}\boldsymbol{\lambda}_{in}, $$
where \(\tilde {\boldsymbol {y}}_{ijn} = \left [\, \tilde {y}_{ijn1}, \tilde {y}_{ijn2}, \cdots, \tilde {y}_{ijnM} \right ]^{\mathrm {T}}\in \mathbb {C}^{M}\) is the (i,j)th bin of the scale-fitted spectrogram of the nth separated signal, \(\boldsymbol {\lambda }_{in} = \left [\,\lambda _{in1}, \lambda _{in2}, \cdots, \lambda _{inM}\right ]^{\mathrm {T}}\in \mathbb {C}^{M}\) is a coefficient vector of back projection for the nth signal at the ith frequency, and ∘ denotes the element-wise multiplication. In the proposed method, this update (26) is performed at the end of each iteration so that the projection (25) at the beginning of the next iteration properly smooths the spectrograms without the effect of scale indeterminacy.
One side effect of this back projection is that the value of the negative log-likelihood function (18) is also changed due to the scale modification. In IVA, this problem cannot be avoided because the only parameter in IVA is the demixing matrix Wi. However, in ILRMA, since both the demixing matrix Wi and the source model parameter TnVn can determine the scale of estimated signal Yn, the likelihood variation can be avoided by appropriately adjusting win and Tn after the back projection. To prevent the likelihood variation, the following updates are required after performing (26):
$$\begin{array}{*{20}l} \boldsymbol{w}_{in} &\leftarrow \boldsymbol{w}_{in} \lambda_{inm_{\text{ref}}}, \end{array} $$
$$\begin{array}{*{20}l} t_{ikn} &\leftarrow t_{ikn} \left|\lambda_{inm_{\text{ref}}}\right|^{2}, \end{array} $$
where mref is the index of the reference channel utilized in the back projection.
The overall algorithm of the proposed Consistent ILRMA is summarized in Algorithm 1. The iterative loop for the parameter optimization appears in the second to eighth lines. The spectrogram consistency of the temporary separated signal Yn is ensured in the third line, and the iterative back projection is applied in the sixth and seventh lines. Note that an algorithm for the conventional ILRMA can be obtained by performing only the fourth and fifth lines (i.e., ignoring the third, sixth, and seventh lines). A Python code of the conventional ILRMA is openly available online (https://pyroomacoustics.readthedocs.io/en/pypi-release/pyroomacoustics.bss.ilrma.html), and therefore, the proposed Consistent ILRMA with Python can be easily implemented by slightly modifying the codes. A MATLAB code of Consistent ILRMA is also available online (https://github.com/d-kitamura/ILRMA/blob/master/consistentILRMA.m).
In this section, we conducted two experiments using synthesized and real-recorded mixtures. The synthesized mixtures were produced by convoluting the impulse responses to dry audio sources, while the real-recorded mixtures were actually recorded by using a microphone array in an ordinary room with ambient noise.
BSS of synthesized mixtures
We conducted determined BSS experiments using synthesized music and speech mixtures with two sources and two microphones (N = M = 2). The dry sources of music and speech signals, listed in Table 1, were respectively obtained from professionally produced music and underdetermined separation tasks provided as a part of SiSEC2011 [60]. They were convoluted with the impulse response E2A (T60=300 ms) or JR2 (T60=470 ms), obtained from the RWCP database [61], to simulate the multichannel observation signals. The recording conditions of these impulse responses are shown in Fig. 4.
Recording conditions of impulse responses: a E2A and b JR2
Table 1 Music and speech dry sources obtained from SiSEC2011
In this experiment, we compared the performance of six methods: three conventional and three proposed. The conventional methods were the standard IVA [10], Consistent IVA [32], and standard ILRMA [11]. The proposed methods were Consistent IVA with iterative back projection (Consistent IVA+BP), Consistent ILRMA, and Consistent ILRMA with iterative back projection (Consistent ILRMA+BP). For all methods, the initial demixing matrix was set to an identity matrix. For the ILRMA-based methods, the nonnegative matrices Tn and Vn were initialized using uniformly distributed random values in the range (0,1). Five trials were performed for each condition using different pseudorandom seeds. The number of bases for each source, K, was set to 10 for music mixtures and 2 for speech mixtures, where it was experimentally confirmed that these conditions provide the best performance for the conventional ILRMA [11]. To satisfy the perfect reconstruction condition (7), the inverse STFT was implemented by the canonical dual of the analysis window. For both Consistent IVA+BP and Consistent ILRMA+BP, the iterative back projection was applied, where the reference channel was set to mref = 1. Since the property of spectrogram consistency depends on the window length, shift length, and type of window function, various combinations of them were tested. The experimental conditions are summarized in Table 2.
Table 2 Experimental conditions
For quantitative evaluation of the separation performance, we measured the source-to-distortion ratio (SDR), source-to-interference ratio (SIR), and source-to-artifact ratio (SAR). In a noiseless situation, SDR, SIR, and SAR are defined as follows [62]:
$$\begin{array}{*{20}l} \text{SDR} &= 10\log_{10} \frac{ {\sum\nolimits}_{l} |s_{\mathrm{t}}[l]|^{2} }{ {\sum\nolimits}_{l} |e_{\mathrm{i}}[l] + e_{\mathrm{a}}[l] |^{2} }, \end{array} $$
$$\begin{array}{*{20}l} \text{SIR} &= 10\log_{10} \frac{ {\sum\nolimits}_{l} |s_{\mathrm{t}}[l]|^{2} }{ {\sum\nolimits}_{l} |e_{\mathrm{i}}[l]|^{2} }, \end{array} $$
$$\begin{array}{*{20}l} \text{SAR} &= 10\log_{10} \frac{ {\sum\nolimits}_{l} |s_{\mathrm{t}}[l]+e_{\mathrm{i}}[l]|^{2} }{ {\sum\nolimits}_{l} |e_{\mathrm{a}}[l]|^{2} }, \end{array} $$
where st[l],ei[l], and ea[l] are the lth sample of target signal, interference, and artificial components of the estimated signal, respectively, in the time domain. SIR and SAR are used to quantify the amount of interference rejection and the absence of artificial distortion of the estimated signal, respectively. SDR is used to quantify the overall separation performance, as SDR is in good agreement with both SIR and SAR for determined BSS.
In this experiment, the energy of sources was not adjusted, i.e., the energy ratio of sources (source-to-source ratio) was automatically determined by the initial volume of the dry sources and the level of the impulse responses. That is, the source-to-source ratio of each mixture signal is different from the others. To equally evaluate the performances of different mixtures, we calculated SDR improvement (ΔSDR) and SIR improvement (ΔSIR) defined as:
$$\begin{array}{*{20}l} \Delta\text{SDR} &= \text{SDR}_{\text{sep}} - \text{SDR}_{\text{input}}, \end{array} $$
$$\begin{array}{*{20}l} \Delta\text{SIR} &= \text{SIR}_{\text{sep}} - \text{SIR}_{\text{input}}, \end{array} $$
where SDRsep and SIRsep are the SDR and SIR of the separated signal, and SDRinput and SIRinput are the SDR and SIR of the initial mixture signal input to the BSS methods. Note that SAR improvement cannot be defined because its value of the signal without artificial processing cannot be defined (SARinput=∞).
Results and discussions
Figure 5 shows examples of the value of the negative log-likelihood function (18) of Consistent ILRMA+BP. Although the algorithmic convergence of the proposed method has not been theoretically justified because of the additional projection (25), we experimentally confirmed a smooth decrease of the cost function. We also confirmed that such behavior was common for the other experimental conditions and mixtures. This result indicates that the additional procedure in the proposed method does not have a harmful effect on the behavior of the overall algorithm.
Values of negative log-likelihood function (18) of Consistent ILRMA+BP (window length, 256 ms; shift length, 32 ms)
Figures 6 and 7 show examples of the energy of the inconsistent components (14) of standard ILRMA and Consistent ILRMA+BP. The energy was normalized by that of the initial spectrograms in order to align the vertical axis. Note that the energy of inconsistency components is not directly related to the degree of permutation misalignment or the separation performance. These figures are shown to confirm whether the proposed algorithm can properly reduce the degree of inconsistency. These values are completely zero when the separated spectrograms are consistent, and hence, those at the 0th iteration (the leftmost values) are zero because no processing is performed at that point. By iterating the algorithms, this energy rapidly increased because the demixing matrix for each frequency independently tried to process and separate the signals. However, the normalized energy tended toward some specific values after several iterations. We confirmed that the converged values of Consistent ILRMA+BP were always lower than those of standard ILRMA. This result indicates that Consistent ILRMA+BP reduces the amount of the inconsistent components and tries to make the separated spectrogram more consistent. In addition, similar to Fig. 5, the algorithmic stability of Consistent ILRMA+BP can be confirmed from Figs. 6 and 7.
Examples of normalized energy of inconsistent components \(\left (\|\mathcal {E}(\mathsf {Y})\|_{2}^{2} / \|\mathsf {X}\|_{2}^{2}\right)\) of ILRMA and Consistent ILRMA+BP for music 1: a 256-ms-long window and 32-ms shifting and b 1024-ms-long window and 512-ms shifting, where X=[X1,X2],Y= [Y1,Y2], and \(\mathcal {E}(\cdot)\) is in (14)
Examples of normalized energy of inconsistent components \((\|\mathcal {E}(\mathsf {Y})\|_{2}^{2} / \|\mathsf {X}\|_{2}^{2})\) of ILRMA and Consistent ILRMA+BP for speech 1: a 256-ms-long window and 32-ms shifting and b 1024-ms-long window and 512-ms shifting, where X= [X1,X2],Y= [Y1,Y2], and \(\mathcal {E}(\cdot)\) is in (14)
Figures 8 and 9 summarize the SDR improvements for the music mixtures and speech mixtures, respectively. The window function was the Hann window. Each box contains 50 results (i.e., 5 pseudorandom seeds × 10 mixtures in Table 1), where ΔSDRs of the two separated sources in each mixture were averaged. The central lines of the box plots indicate the median, and the bottom and top edges of the box indicate the 25th and 75th percentiles, respectively. Each row corresponds to the same window length, while each column corresponds to the same shift length. As we conducted the experiment for six window lengths, four shift lengths, and two impulse responses, each figure consists of 6×4×2 subfigures. In each subfigure, six boxes are shown to illustrate the results of (1) IVA, (2) Consistent IVA, (3) Consistent IVA+BP, (4) ILRMA, (5) Consistent ILRMA, and (6) Consistent ILRMA+BP. Since the tendency of the results was the same as Figs. 8 and 9, we provide the SDR improvements for the other windows (Hamming and Blackman) in the Appendix. The SIR improvement and SAR are also given in the Appendix.
Average SDR improvements for synthesized music mixtures (music 1–10) with aE2A and bJR2, where Hann window is used in STFT
Average SDR improvements for synthesized speech mixtures (speech 1–10) with aE2A and bJR2, where Hann window is used in STFT
Since IVA and ILRMA assume the instantaneous mixing model (11) for each frequency in the time-frequency domain, the window length should be long relative to the reverberation time to achieve accurate separation. At the same time, too long a window degrades the separation performance of IVA and ILRMA, as discussed in [30]. This is because capturing the source activity and spectral patterns becomes difficult for IVA and ILRMA as the time resolution of the spectrograms becomes low due to a long window. The robustness of IVA and ILRMA is also deteriorated by a long window because the effective number of time segments is decreased. This trade-off of the separation performance caused by window length in STFT can be easily confirmed from the results for both music (Fig. 8) and speech (Fig. 9) mixtures, which is consistent with the results in [30]. As shown in the figures, the performance was poor for the shorter windows (≤128 ms), and the performance for the longer windows (≥768 ms) was more varied than that of the shorter ones. The window length best suited for these conditions (combinations of source signals and impulse responses) seems to be around 256 ms or 512 ms. While the maximum achievable performance becomes higher as the window length becomes longer due to the mixing model (11), these results indicate that the source modeling becomes difficult for both IVA and ILRMA when the window length is too long. This trade-off should be important for discussing the results further.
By comparing the performances of the conventional (IVA, Consistent IVA, and ILRMA) and proposed (Consistent IVA+BP, Consistent ILRMA, and Consistent ILRMA+BP) methods, we can see that the proposed methods tend to outperform the conventional ones. Some comparisons are made as follows: ∙ Conventional and proposed IVAs. The proposed Consistent IVA+BP performed better than the conventional IVAs (IVA and Consistent IVA) in Figs. 8b and 9b when the window length was sufficiently long (≥256 ms). In those cases, the conventional Consistent IVA resulted in a worse performance than IVA, which indicates that just using spectrogram consistency cannot improve the performance of IVA. This demonstrates the importance of the iterative back projection when spectrogram consistency is considered within determined BSS. ∙ Conventional and proposed ILRMAs. The proposed Consistent ILRMA without BP performed comparably to the conventional ILRMA. In Figs. 8a and 9a, Consistent ILRMA performed better than ILRMA when the window length was long (≥768 ms). In contrast, in Figs. 8b and 9b, Consistent ILRMA performed worse than ILRMA. This is presumably because the scale ambiguity prevented the spectrogram consistency from working properly. By incorporating iterative back projection into Consistent ILRMA, the proposed Consistent ILRMA+BP performed better than the conventional ILRMA. In the best situation (the top-left subfigure of Fig. 8), Consistent ILRMA+BP performed 8 dB better than ILRMA by bringing out the potential of spectrogram consistency in determined BSS.
To further explain the experimental results, some notable tendencies are summarized as follows: ∙ Short window. When the window length was short (64 ms), all methods performed similarly in terms of ΔSDR. This is because the achievable performance was already limited by the window length that was shorter than the reverberation time. This result contradicted our expectation before performing the experiment. Since enforcing the consistency spreads the frequency components based on the main-lobe of the window function, we expected that the ability to solve the permutation problem would be higher when the window length was shorter because of the wider main-lobe. In reality, we found that the spectrogram consistency could assist IVA and ILRMA except for the cases where the window length was short (≤128 ms in this experiment) compared to the reverberation time. ∙ Large window shift. When the shift length was 1/2 of the window length, the performance of ILRMA significantly dropped compared to smaller shift lengths (1/4,1/8, and 1/16), especially when the window length was long (e.g., 1024 ms). This is presumably because the number of time segments was small, i.e., NMF in ILRMA failed to model the source signals from the given amount of data. In addition, for a larger window, distinguishing spectral patterns of the sources became difficult for ILRMA due to the time-directional blurring effect caused by a longer window. Such performance degradation was alleviated for Consistent ILRMA+BP. This might be because the smoothing process of the inverse STFT provides some additional information for the source modeling from the adjacent bins. ∙ Length of boxes. When the length of the box of ILRMA was long, as in Figs. 8a and 9a, Consistent ILRMA+BP was able to improve the performance. Conversely, when the length of the box of ILRMA was short, as in Figs. 8b and 9b, Consistent ILRMA+BP was only able to slightly improve the performance. Note that the vertical axes are different. This result indicates that the achievable performance decided by the mixing model (11) limits the improvement obtained by spectrogram consistency. Since consistency is the characteristic of a spectrogram, it cannot manage the mixing process. The demixing-filter update of ILRMA, which is the same for the conventional and proposed methods, manages the mixing process. Hence, when the mixing model has a mismatch with the observed condition, there is less room for spectrogram consistency to improve the performance. ∙ Improvement by consistency. The proposed method tended to achieve a good performance when the conventional ILRMA also worked well, e.g., Figs. 8a and 9a. This tendency indicates that the spectrogram consistency effectively promotes the separation when the estimated source Yn accurately approaches the original source Sn during the optimization, as Sn is naturally a consistent spectrogram. This is the reason we feel that the consistency can be an assistant of the frequency-domain BSS. An important aspect is that the source model (e.g., NMF in ILRMA) actually informs the separation cue, and the spectrogram consistency enhances the separation performance when the source modeling functions correctly.
BSS of real-recorded mixtures
Next, we evaluated the conventional and proposed methods using live-recorded music and speech mixtures obtained from underdetermined separation tasks in SiSEC2011 [60], where only two sources were mixed to make the BSS problem determined (M=N=2). The signals used in this experiment are listed in Table 3. The reverberation time of these signals was 250 ms, and the microphone spacing was 1 m (see [60]). Since these source signals were actually recorded using a microphone array in an ordinary room with ambient noise, the observed signals are more realistic compared to those in Section 4.1.
Table 3 Live-recorded music and speech signals obtained from SiSEC2011
For simplicity, in this experiment, we used STFT with a fixed condition, the 512-ms-long Hann window with 1/4 shifting. The experimental conditions other than the window were the same as those in Section 4.1.1.
Figure 10 shows the results of live-recorded music and speech mixtures. The absolute scores were lower than those for the synthesized mixtures discussed in Section 4.1.2 due to the existence of ambient noise. Still, we can confirm the improvements of the proposed Consistent IVA+BP and Consistent ILRMA+BP compared to the conventional IVA and ILRMA, respectively, for both the music (upper row) and speech (lower row) mixtures. In particular, Consistent IVA+BP improved more than 4 dB over IVA in terms of the median of the ΔSDR of speech mixtures. Consistent ILRMA+BP achieved the highest performance in terms of the median of the SDR improvement for both music and speech mixtures. These results confirm that the combination of spectrogram consistency and iterative back projection can assist the separation of determined BSS for a more realistic situation.
SDR improvements (left column), SIR improvements (center column), and SAR (right column) for live-recorded music and speech mixtures, where STFT is performed using the 512-ms-long Hann window with 1/4 shifting. Top row shows the performances for music mixtures, and bottom row shows the performances for speech mixtures
In this paper, we have proposed a new variant of the state-of-the-art determined BSS algorithm called Consistent ILRMA. It utilizes the smoothing effect of the inverse STFT in order to assist the separation and enhance the performance. Experimental results showed that the proposed method can improve the separation performance when the window length is sufficiently large (≥256 ms in the experimental condition of this paper). These results demonstrate the potential of considering spectrogram consistency within the state-of-the-art determined BSS algorithm. In addition, we experimentally confirmed the importance of iterative back projection for considering spectrogram consistency within determined BSS. It should be possible to construct a new source model in consideration of the spectrogram consistency, which can pave the way for the next direction of research on determined BSS.
Independence between real and imaginary parts of spectrogram
The source generative model (15) assumes that the real and imaginary parts of a source in the time-frequency domain are mutually independent because the generative model has a zero-mean and circularly symmetric shape in the complex plane. The independence between real and imaginary parts or amplitude and phase has been investigated, but its validity may depend on the parameters of STFT. Independence can be measured by a symmetric uncertainty coefficient [63–65]:
$$\begin{array}{*{20}l} C(q_{1}, q_{2}) = 2\frac{ H(q_{1}) + H(q_{2}) - H(q_{1}, q_{2}) }{ H(q_{1}) + H(q_{2}) }, \end{array} $$
where q1 and q2 are random variables, H(q1) and H(q2) are their entropy, and H(q1,q2) is the joint entropy of q1 and q2. Since the numerator of (35) corresponds to the mutual information of q1 and q2, the symmetric uncertainty coefficient can be interpreted as normalized mutual information. When q1 and q2 are mutually independent, (35) becomes zero. In contrast, when q1 and q2 are completely dependent, (35) becomes one.
Symmetric uncertainty coefficient between real and imaginary parts for music and speech sources, where a Hann, b Hamming, or c Blackman window is used in STFT. Left and right columns correspond to the music sources and speech sources, respectively
We calculated the symmetric uncertainty coefficient (35) between the real and imaginary parts of a time-frequency bin obtained by applying STFT to music or speech sources. Let s be a complex-valued time-frequency bin of a source (the indexes of frequency and time are omitted here). The independence between the real and imaginary parts can be measured by C(Re(s),Im(s)), where Re(·) and Im(·) return the real and imaginary parts of an input complex value, respectively. Here, H(Re(s)),H(Im(s)), and H(Re(s),Im(s)) were approximately obtained by calculating the histograms of Re(s) and Im(s). The number of bins in the histograms was set to 10,000. We used the dry sources listed in Table 1: 15 music (instrumental) and eight speech sources. The parameters of STFT were the same as those in Section 4.
Figure 11 shows the symmetric uncertainty coefficients averaged over all bins and sources. Their values C(Re(s),Im(s)) were almost zero for all STFT conditions and source types (music or speech), and thus, the assumption of independence between real and imaginary parts is valid for music and speech sources. This fact leads to the generative model assumed in ILRMA. Note that those symmetric uncertainty coefficients validated the independence of real and imaginary parts at each time-frequency bin. That is, the inter-bin relation is not considered here. The proposed method captures such inter-bin relations imposed by the spectrogram consistency, which is not apparent in these bin-wise assessments of independence.
Additional experimental results for synthesized mixtures
Figures 12–15, 16–21, and 22–27 show the SDR improvements, SIR improvements, and SAR, respectively, for synthesized music and speech mixtures. These figures correspond to the results and discussions in Section 4.1.2.
Average SDR improvements for synthesized music mixtures (music 1–10) with aE2A and bJR2, where Hamming window is used in STFT
Average SDR improvements for synthesized music mixtures (music 1–10) with aE2A and bJR2, where Blackman window is used in STFT
Average SDR improvements for synthesized speech mixtures (speech 1–10) with aE2A and bJR2, where Hamming window is used in STFT
Average SDR improvements for synthesized speech mixtures (speech 1–10) with aE2A and bJR2, where Blackman window is used in STFT
Average SIR improvements for synthesized music mixtures (music 1–10) with aE2A and bJR2, where Hann window is used in STFT
Average SIR improvements for synthesized music mixtures (music 1–10) with aE2A and bJR2, where Hamming window is used in STFT
Average SIR improvements for synthesized music mixtures (music 1–10) with aE2A and bJR2, where Blackman window is used in STFT
Average SIR improvements for synthesized speech mixtures (speech 1–10) with aE2A and bJR2, where Hann window is used in STFT
Average SIR improvements for synthesized speech mixtures (speech 1–10) with aE2A and bJR2, where Hamming window is used in STFT
Average SIR improvements for synthesized speech mixtures (speech 1–10) with aE2A and bJR2, where Blackman window is used in STFT
Average SAR for synthesized music mixtures (music 1–10) with aE2A and bJR2, where Hann window is used in STFT
Average SAR for synthesized music mixtures (music 1–10) with aE2A and bJR2, where Hamming window is used in STFT
Average SAR for synthesized music mixtures (music 1–10) with aE2A and bJR2, where Blackman window is used in STFT
Average SAR for synthesized speech mixtures (speech 1–10) with aE2A and bJR2, where Hann window is used in STFT
Average SAR for synthesized speech mixtures (speech 1–10) with aE2A and bJR2, where Hamming window is used in STFT
Average SAR for synthesized speech mixtures (speech 1–10) with aE2A and bJR2, where Blackman window is used in STFT
The datasets used for the experiments in this paper are openly available: SiSEC 2011 (http://sisec2011.wiki.irisa.fr/) and RWCP-SSD (http://research.nii.ac.jp/src/en/RWCP-SSD.html). Our MATLAB implementation of the proposed method is also openly available at the following site: https://github.com/d-kitamura/ILRMA/blob/master/ consistentILRMA.m
"In the original publication, the webpage includes Section "AAA" between "Abstract" and "Introduction (Sect.1). The article has been updated to delete this empty section."
BSS:
Blind source separation
ICA:
Independent component analysis
STFT:
Short-time Fourier transform
FDICA:
Frequency-domain independent component analysis
Independent vector analysis
ILRMA:
Independent low-rank matrix analysis
NMF:
Nonnegative matrix factorization
IS-NMF:
Nonnegative matrix factorization based on the Itakura–Saito divergence
SDR:
Source-to-distortion ratio
SIR:
Source-to-interference ratio
Source-to-artifact ratio
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The authors would like to thank Nao Toshima for his support on the experiment. Also, the authors would like to thank the anonymous reviewers for their valuable comments and suggestions that helped improve the quality of this manuscript.
This work was partially supported by JSPS Grants-in-Aid for Scientific Research 19K20306 and 19H01116.
Daichi Kitamura and Kohei Yatabe contributed equally to this work.
National Institute of Technology, Kagawa College, 355 Chokushi, Takamatsu, 761-8058, Kagawa, Japan
Daichi Kitamura
Waseda University, 3-4-1 Okubo, Shinjuku-ku, 169-8555, Tokyo, Japan
Kohei Yatabe
DK derived the algorithm, performed the experiment, drafted the manuscript for initial submission, and revised the manuscript. KY proposed the main idea, gave advice, mainly wrote the manuscript for initial submission, and corrected the draft of revised manuscript. The authors read and approved the final manuscript.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Kitamura, D., Yatabe, K. Consistent independent low-rank matrix analysis for determined blind source separation. EURASIP J. Adv. Signal Process. 2020, 46 (2020). https://doi.org/10.1186/s13634-020-00704-4
Audio source separation
Convolutive mixture
Demixing filter estimation
Phase-aware signal processing
Spectrogram consistency | CommonCrawl |
Higgs-Mechanism: Why are gauge boson masses not protected by gauge symmetry
In non-spontaneously broken QFT like QED the gauge bosons cannot have a mass due to gauge symmetry (follows from Ward identity). Also they have only 2 polarizations.
However in a spontaneously broken gauge theory the gauge boson becomes massive after the symmetry breaking. For example consider the standard U(1) example: $$L = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + D_\mu\phi^\dagger D^\mu\phi - V(\phi)$$ where $V(\phi)$ is a Mexican hat potential. After the symmetry breaking the photon is massive and has 3 polarizations.
What is the difference between both cases? Why is it relevant whether the theory is spontaneously broken or not (I guess it has something to do with the Ward identity)? And why are we so sure that this cannot happen in our 'normal' theories like QED (i.e. that in the end the photon becomes massive).
Edit: Maybe I should reformulate the question a bit. The definition of a spontaneous broken theory is that $Q|\Omega\rangle \neq 0$, where $Q$ is the charge operator as given by the symmetry and $|\Omega\rangle$ the groundstate. Why does the charge of the groundstate of my theory matter at all? I can formulate gauge invariance only as a Heisenberg operator symmetry: $\hat{A^\mu} \sim \hat{A^\mu} + \partial^\mu \hat{\alpha}$. Therefore the photon should still be massless. Obviously this cannot be true because we have plenty of examples where gauge bosons become massive (e.g. W boson or superconductors).
I guess that something goes wrong with the argument that the photon cannot be massive by gauge invariance. Since I don't know a nice proof of this (Only argumentations like this by resumming feynman diagrams) I guess something goes wrong there. The question is about what exactly goes wrong? And more important: Why is the mass of the photon protected in the normal case, but not in spontaneous symmetry breaking case. What's the crucial difference?
Edit 2: It took me a while to figure it out, but I think I found a suitable nonperturbative argumentation why the photon cannot become massive. Instead of unitarity gauge work in the usual Lorentz gauge (and Feynman gauge). Then the equal time commutation relations read $[A^\mu(x), A^\nu(0)] = 0$ and $[\dot{A}^\mu(x), A^\nu(0)] = i \eta^{\mu\nu}$. Also the equation of motion is given by $\partial_\alpha\partial^\alpha A^\mu = J^\mu$. Therefore apply the equation of motion on the propagator $\langle 0|TA^\mu(x)A^\nu(0)|0\rangle$:
\begin{align} \partial_\alpha \partial^\alpha \langle 0|TA^\mu(x)A^\nu(0)|0\rangle &= \partial_\alpha \langle 0|T(\partial^\alpha A^\mu(x))A^\nu(0)|0\rangle + \partial_\alpha \left(\delta(t) \delta_{0\alpha} \langle 0|[A^\mu(x),A^\nu(0)]|0\rangle\right)\\ &= \langle 0|T(\partial_\alpha\partial^\alpha A^\mu(x))A^\nu(0)|0\rangle + \delta(t) \langle 0|[\partial_0 A^\mu(x),A^\nu(0)]|0\rangle\\ &= \langle 0|TJ^\mu(x)A^\nu(0)|0\rangle + i\eta^{\mu\nu}\delta^4(x) \end{align}
The commutator come from time ordered product and can be replaced by there equal time value because of the $\delta(t)$ in front. After a Fourier transformation this reads (call the Fourier transformed propagator $D^{\mu\nu}(p)$):
$$-p^2 D^{\mu\nu}(p) = \int d^4x e^{-ipx} \langle 0|TJ^\mu(x)A^\nu(0)|0\rangle + i\eta^{\mu\nu}$$
For $J^\mu = 0$ (free case) this simply gives the free photon propagator $D^{\mu\nu}(p) = \frac{-i\eta^{\mu\nu}}{p^2}$. Now if $J^\mu \neq 0$ multiply both sides with $p_\mu$ (which translates to $\partial_\mu$ for $J^\mu$)
\begin{align}-p^2 p_\mu D^{\mu\nu}(p) &= -\int d^4x e^{-ipx} \langle 0|T(\partial_\mu J^\mu(x))A^\nu(0)|0\rangle - \int d^3x e^{-ipx} \langle 0|[J^0(x),A^\nu(0)]|0\rangle + ip^\nu\\ &= - \int d^3x e^{-ipx} \langle 0|[J^0(0,\vec{x}),A^\nu(0)]|0\rangle + ip^\nu \end{align}
here I used $\partial_\mu J^\mu = 0$. Now as long as $[J^0(0,\vec{x}),A^\nu(0)] = 0$ (this is for example the case in the theory above since $J^0$ does not contain any $\partial_0 A^0$) we have $$-p^2 p_\mu D^{\mu\nu}(p) = ip^\nu$$ or $$p_\mu D^{\mu\nu}(p) = -i\frac{p^\nu}{p^2}$$ which means that $D^{\mu\nu}(p)$ must have a pole at $p^2 = 0$ (i.e. the photon is massless). Clearly this it not satisfied by the propagator for a massive particle $\sim \frac{1}{p^2 - m^2}$
Note that this argumentation never used any property of the ground state of our theory. In the Higgs mechanism one expands the field around its vev and ignores higher orders. After that the photon is massive (in lowest order of perturbation theory) which is a clear contradiction to my derivation above. So my first guess would be that the mass of the photon will again cancel when taking into account all feynman diagrams.
quantum-field-theory gauge-theory higgs
toastertoaster
$\begingroup$ I very much does happen in QED, inside a superconductor where the U(1) group is SSBroken , and the photon becomes effectively massive through the Higgs mechanism: the resulting magnetic flux exclusion is called the Meissner effect. $\endgroup$ – Cosmas Zachos Feb 8 '19 at 23:55
$\begingroup$ Near duplicate. $\endgroup$ – Cosmas Zachos Feb 8 '19 at 23:57
$\begingroup$ Viola that's a quite nice example. So let's say I have a superconductor in the non superconducting phase. There I have a massless photon (right?). Then I cool it below the critical temperature. From the QFT principles the photon cannot obtain mass during the whole process (because it is protected by the ward identity). But if I just treat the final state with the spontaneous symmetry breaking formalism I find a massive photon. So in order for the symmetry breaking formalism to be correct there must be something that gives mass to the photon at some point during the process. $\endgroup$ – toaster Feb 9 '19 at 0:31
$\begingroup$ Whether or not a particle is massive or not cannot depend on the particular choice of gauge. During performing the spontaneous symmetry breaking formalism one expands the Higgs field around the VEV. What tells me that I am not doing something wrong here? In particular basic QFT principles tell me that the photon cannot obtain mass whatever interaction I introduce! On the other hand side I have the Higgs mechanism which is an approximation but gives me a massive photon. So either the first or the second statement must be wrong in general! I guess it's the first one but I would like to know why. $\endgroup$ – toaster Feb 9 '19 at 1:18
$\begingroup$ Related. $\endgroup$ – Cosmas Zachos Feb 9 '19 at 1:47
Why does the charge of the groundstate of my theory matter at all?
Because it dictates the coupling form of all states with vanishing v.e.v. to it, and hence the realization of the relevant symmetry, from which all else follows. It makes all the difference. Gauge invariance is always there.
You may be falling victim to chronic abuse of language plausibly invented for this very purpose. "Protection" has a technical connotation involving arbitrary radiative corrections, irrelevant in this context.
The assumption behind the Higgs mechanism, or the lack thereof, is that all quantum calculations have been carried out or guessed, to produce a low energy effective theory. This theory has or lacks SSB, the latter case being your "normal" theory.
Now, I could not, of course, improve on Kibble's near-perfect explanation of his work, beyond vulgarizing it with cartoons and annotations, possibly steering you to "follow the math".
If the theory has no SSB, as in conventional QED, the vacuum is symmetric, so U(1) rotations leave it alone, $e^{i\alpha Q} |0\rangle = |0\rangle$. Consequently $Q|0\rangle = 0$.
By contrast, inside a medium such as a superconductor's, $Q|0\rangle \neq 0$.
Kibble illustrate's Goldstone's celebrated (1961) sombrero potential, a gimmick allowing ready grasp of the group theory (but Landau & Ginzburg did that earlier, 1950, without covariance). All fields are defined with zero v.e.v. s as they should, at the true vacuum, at the minimum of the sombrero.
Specifically, it enforces that the bottom of the potential, the vacuum, has a massive "Higgs mode", $\varphi _1$; and a massless "Goldstone mode", $\varphi_2$.
The conserved EM current (13) is then $$ j_\mu= \frac{v}{2} \partial_\mu \varphi_2 + ..., $$ ignoring irrelevant multi-field terms. $v$ is a dimension-one parameter quantifying symmetry breaking in the sombrero, historically deducted from the higgs to ensure $\langle 0|\varphi_1|0\rangle=0$. The linear term, which makes all the difference, is the hallmark of the SSB, or "Nambu-Goldstone" realization of the symmetry.
Firstly, it ensures that, given (14), $$ \langle \varphi_2 | \int d^3x \frac{v}{2} \partial_0 \varphi_2(x) |0\rangle \neq 0, $$ that is Q cannot annihilate the vacuum, but, semiclassically, pumps zero energy goldstons in or out of it, to yield another vacuum degenerate with it.
By contrast, in the normal, "unbroken" phase, for $v=0$, there is no such chance, and the vacuum is annihilated.
The second, related, phenomenon, is that a crucial piece of the covariantly completed kinetic term for the goldston and higgs may be declared as a B field mass term, $$\tfrac{1}{2} \left ( \partial_\mu \varphi_2 + ev A_\mu \right )^2\equiv \frac{e^2 v^2 }{2} B_\mu^2 .$$
In the $\varphi_2=0$ gauge, where the goldston was all pumped into the "higgs" $\varphi_1$, we can afford to ignore it. (This is easier in polar field coordinates.)
You may then, if you are inclined, call the B a "massive photon", or whatever. In the above gauge, it identifies with the photon. The crucial physics, however, is that, at the level of this effective lagrangian, there are no massless modes, photon or Goldstone, but only massive modes (B), instead. You might go wild changing gauges and fixing them and identifying modes, but that's just language games. (The SM pros, of course, cannot live without them.)
In the "normal" case without SSB, v=0, no such luck and the photon is visibly massless, no matter what you do, while $\varphi_2$ is visibly massive and not a goldston. The current starts with the usual bilinear rotation term, dubbed "Wigner-Weyl" realization.
The theory is gauge invariant for vanishing or non-vanishing v, and you may well have escaped from the clutches of slippery "protection" talk. There has been lots of work anticipating whether a theory will, as a result of quantum interactions, end up as a zero or nonzero v effective lagrangian, but there are no foolproof guidelines.
Cosmas ZachosCosmas Zachos
$\begingroup$ Thanks for your answer. I have read your answer and Kibbles paper several times now and I am still not convinced. In the first orders of perturbation theory the photon looks massive. But what tells me that there is no cancellation of the photon mass at higher orders? I think I found a argumentation why the photon is massless in general (The only difference is a different gauge). I added it to my question, maybe you could take a look at it. $\endgroup$ – toaster Feb 12 '19 at 17:11
$\begingroup$ That was the basic starting assumption here: that, instead of connecting tree level with loops, one has computed all loops, and encoded them in an effective action with or without nontrivial vacuum. Promoting the argument to a Ward-identity or Slavnov-Taylor identity, better, is a messy story, and I thought, irrelevant here. I suspect you are heading headlong for Coleman-Weinberg technology, where radiative corrections do all the damage, but then you should express your question in its language, surely with an answer in its reviews. $\endgroup$ – Cosmas Zachos Feb 12 '19 at 18:37
$\begingroup$ Possibly related , if that's where you really wish to go. $\endgroup$ – Cosmas Zachos Feb 12 '19 at 19:45
I'll try to give my undergrad understanding of spontaneous symmetry breaking. Additions and corrections are welcome.
The term spontaneously broken is a bit misleading. One of my lecturers even calls it "the worst terminology in physics"... The whole theory is still symmetric even after spontaneous symmetry breaking. Only in the ground state the symmetry is let's call it "hidden" such that seemingly non gauge invariant terms like in your case a photon mass term can appear when we expand the scalar field $\phi$ around its minimum.
Regarding QED. We are 100% sure that the same thing could occur in some extension of QED. There exist several Toy models that lead to massive Photons hence "breaking" the U(1) gauge symmetry. However experiments tell us that if the photon has a mass it is very small...
KatermickieKatermickie
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\begin{document}
\title{Linear statistics of matrix ensembles in classical background } \author{Yang Chen and Chao Min\\ ([email protected], [email protected])\\ Department of Mathematics, University of Macau,\\ Avenida da Universidade, Taipa, Macau, China}
\date{\today}
\maketitle \begin{abstract} Given a joint probability density function of $N$ real random variables, $\{x_j\}_{j=1}^{N},$ obtained from the eigenvector-eigenvalue decomposition of
$N\times N$ random matrices, one constructs a random variable, the linear statistics, defined by the sum of smooth functions evaluated at the eigenvalues or singular values of the random matrix, namely, $\sum_{j=1}^{N}F(x_j).$
For the jpdfs obtained from the Gaussian and Laguerre ensembles, we compute, in this paper the moment generating function $\mathbb{E}_{\beta}({\rm exp}(-\lambda\sum_{j}F(x_j))),$ where $\mathbb{E}_{\beta}$ denotes expectation value over the Orthogonal ($\beta=1$) and Symplectic ($\beta=4)$ ensembles, in the form one plus a Schwartz function, none vanishing over $\mathbb{R}$ for the Gaussian ensembles and $\mathbb{R}^+$ for the Laguerre ensembles.
These are ultimately expressed in the form of the determinants of identity plus a scalar operator, from which we obtained the large $N$ asymptotic of the linear statistics from suitably scaled $F(\cdot).$ \end{abstract}
\section{Introduction} The well-known joint probability density function for the eigenvalues $\{x_j\}_{j=1}^{N}$ of $N\times N$ Hermitian matrices from an orthogonal ensemble ($\beta=1$), unitary ensemble ($\beta=2$) or symplectic ensemble ($\beta=4$) is given by \cite{Mehta} $$
P^{(\beta)}(x_{1},x_{2},\ldots,x_{N})\prod_{j=1}^{N}dx_j=C_{N}^{(\beta)}\prod_{1\leq j<k\leq N}\left|x_{j}-x_{k}\right|^{\beta} \prod_{j=1}^{N}w(x_{j})dx_j, $$ where $w(x)$ is a weight function or a probability density supported on $[a,b]$. In this paper, we require that the moments of $w$, namely, $$s_j:=\int_{a}^{b}x^j\;w(x)dx,$$ to exist for $j=0,1,2,\ldots.$ Here the normalization constant $C_{N}^{(\beta)}$ reads, $$
C_{N}^{(\beta)}=\frac{1}{\int_{[a,b]^{N}}\prod_{1\leq j<k\leq N}\left|x_{j}-x_{k}\right|^{\beta} \prod_{j=1}^{N}w(x_{j})dx_{j}}. $$ Many years ago, Selberg \cite{Selberg}, obtained closed form expression for $C_{N}^{(\beta)}$, where\\ $w(x)=x^{a}(1-x)^{b},\;a>-1,\;b>-1,\;x\in [0,1]$, the Jacobi weight. The constant $C_N^{(\beta)}$ with the Gaussian weight $w(x)={\rm e}^{-x^2}$,$\;x\in \mathbb{R}$ and Laguerre weight, $w(x)=x^{\alpha}\:{\rm e}^{-x},\;\alpha>-1,\;\;x\in \mathbb{R}^+$, can be found in \cite{Mehta}.
If we take $w(x)=\mathrm{e}^{-x^{2}},\; x\in \mathbb{R}$, these are known as the Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE) and Gaussian symplectic ensemble (GSE). If\\ $w(x)=x^{\alpha}\mathrm{e}^{-x},\;\;\alpha>-1,\;\;x\in \mathbb{R}^+,$ then we have analogously the LOE, LUE and LSE.
The moment generating function, $$ G_{N}^{(\beta)}(f):=\mathbb{E}_{\beta}\left({\rm e}^{-\lambda\:\sum_{j=1}^{N}F(x_j)}\right)
:=\frac{\int_{[a,b]^{N}}\prod_{1\leq j<k\leq N}\left|x_{j}-x_{k}\right|^{\beta}\prod_{j=1}^{N}w(x_{j})\:{\rm e}^{-\lambda\:F(x_j)}dx_{j}}{\int_{[a,b]^{N}}\prod_{1\leq j<k\leq N}\left|x_{j}-x_{k}\right|^{\beta}\prod_{j=1}^{N}w(x_{j})dx_{j}}, $$ is given by
\begin{equation} G_{N}^{(\beta)}(f)=C_{N}^{(\beta)}\int_{[a,b]^{N}}\prod_{1\leq j<k\leq N}\left|x_{j}-x_{k}\right|^{\beta} \prod_{j=1}^{N}w(x_{j})\left[1+f(x_{j})\right]dx_{j},\label{gnb} \end{equation} where $\beta=1, 2, 4$. Here ${\rm e}^{-\lambda\:F(x)}:=1+f(x)$, and we do not indicate that $f(\cdot)$ also depends on $\lambda.$ Throughout this paper, we assume $f(x)$ lies in the Schwartz space, and $1+f(x)\neq 0$ over $[a,b]$.
The simplest, very well-studied, unitary case corresponds to $\beta=2$, see \cite{Adler, Basor, Basor1993, Basor1997, Dieng, Tracy1998}, and the references therein.
We state here, for later development, facts on orthogonal polynomials. Let\\ $\varphi_{j}(x):=P_{j}(x)\sqrt{w(x)}$, $j=0,1,2,\ldots$, where $P_{j}(x)$ are the orthonormal polynomials of
degree $j$ with respect to the weight $w(x)$ supported on $[a,b],$ $$ \int_{a}^{b} P_j(x)P_k(x)w(x)dx=\delta_{j,k},\;\;j,k=0,1,2,\ldots. $$ It is a well-known fact that, $$ G_{N}^{(2)}(f)=\det\left(I+K_{N}^{(2)}f\right), $$ where $K_{N}^{(2)}f$ is an integral operator with kernel $K_{N}^{(2)}(x,y)f(y)$. Here $K_{N}^{(2)}(x,y):=\sum_{j=0}^{N-1}\varphi_{j}(x)\varphi_{j}(y)$, can be evaluated via the Christoffel-Darboux formula.
With Gaussian background, where $w(x)=\mathrm{e}^{-x^{2}},\; x\in \mathbb{R}$, we have $$P_{j}(x):=\frac{H_{j}(x)}{c_{j}},\;\;\;c_{j}=\pi^{\frac{1}{4}}2^{\frac{j}{2}}\sqrt{\Gamma(j+1)},\;\;j=0,1,2,...,$$ where $H_{j}(x)$ are the Hermite polynomials of degree $j$.
With Laguerre background, where $w(x)=x^{\alpha}\mathrm{e}^{-x},\;\alpha>-1,\;x\in \mathbb{R}^{+}$, we have $$P_{j}(x):=\frac{L_{j}^{(\alpha)}(x)}{c_{j}^{(\alpha)}},\;\;\;c_{j}^{(\alpha)}=\sqrt{\frac{\Gamma(j+\alpha+1)}{\Gamma(j+1)}},\;\;j=0,1,2,...,$$ where $L_{j}^{(\alpha)}(x)$ are the Laguerre polynomials of degree $j$. Properties of the Hermite and Laguerre polynomials can be found in \cite{Szego}.
We shall mainly deal with the $\beta=4$ and $\beta=1$ cases, which are more complicated than $\beta=2$ situation, especially so for $\beta=1$.
We always begin with the general case, and then apply the results obtained to the two classical ensembles, i.e., the Gaussian ensembles and Laguerre ensembles.
Furthermore, in $\beta=1$ situation, for convenience, $N$ is taken to be even, and it is expeditious to make use of the square root of the Gaussian weight, ${\rm e}^{-x^2/2},\;x\in \mathbb{R}$ and the square root of the Laguerre weight, $x^{\alpha/2}{\rm e}^{-x/2},\;\alpha>-2,\;x\in\mathbb{R}^+$ in later discussion.
In this paper, we shall be concerned with the large $N$ behavior of $G_{N}^{(\beta)}(f)$ for the Gaussian ensembles and Laguerre ensembles. For the Gaussian ensembles, we replace $f(x)$ by $f\left(\sqrt{2N}x\right),$ in the orthogonal case, and $f\left(\sqrt{4N}x\right)$, in the symplectic case. For the Laguerre ensembles, we replace $f(x)$ by $f\left(\sqrt{4Nx}\right)$, in the orthogonal case and $f\left(\sqrt{8Nx}\right)$, in the symplectic case.
We will ultimately give the mean and variance of the linear statistics, as $N\rightarrow\infty$, together with leading correction terms.
For comparison purposes, we write down results in the GUE, where, $w(x)=\mathrm{e}^{-x^{2}},\;x\in \mathbb{R}$. Denote by $\mu_{N}^{(GUE)}$ and $\mathcal{V}_{N}^{(GUE)}$ the mean and variance of the linear statistics
$\sum_{j=1}^{N}F\left(\sqrt{2N}x_{j}\right)$. It is shown in \cite{Basor1997}, that, \begin{equation} \mu_{N}^{(GUE)}\rightarrow\frac{1}{\pi}\int_{-\infty}^{\infty}F(x)dx,\;\;N\rightarrow\infty,\label{guem} \end{equation} \begin{equation} \mathcal{V}_{N}^{(GUE)}\rightarrow\frac{1}{\pi}\int_{-\infty}^{\infty}F^{2}(x)dx -\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\left[\frac{\sin(x-y)}{\pi(x-y)}\right]^{2}F(x)F(y)dx dy,\;\;N\rightarrow\infty.\label{guev} \end{equation} For the LUE, where $w(x)=x^{\alpha}\mathrm{e}^{-x},\;\alpha>-1,\;x\in \mathbb{R}^{+}$. Denote by $\mu_{N}^{(LUE,\:\alpha)}$ and $\mathcal{V}_{N}^{(LUE,\:\alpha)}$ the mean and variance of the linear statistics $\sum_{j=1}^{N}F\left(\sqrt{4N x_{j}}\right)$. It is shown in\cite{Basor1993}, that, \begin{equation} \mu_{N}^{(LUE,\:\alpha)}\rightarrow\int_{0}^{\infty}B^{(\alpha)}(x,x)F(x)dx,\;\;N\rightarrow\infty,\label{luem} \end{equation} \begin{equation} \mathcal{V}_{N}^{(LUE,\:\alpha)}\rightarrow\int_{0}^{\infty}B^{(\alpha)}(x,x)F^{2}(x)dx-\int_{0}^{\infty} \int_{0}^{\infty}B^{(\alpha)}(x,y)B^{(\alpha)}(y,x)F(x)F(y)dx dy,\;\;N\rightarrow\infty,\label{luev} \end{equation} where \begin{equation} B^{(\alpha)}(x,y):=\frac{J_{\alpha}(x)y J_{\alpha}'(y)-J_{\alpha}(y)x J_{\alpha}'(x)}{x^{2}-y^{2}}x,\label{bxy} \end{equation} \begin{equation} B^{(\alpha)}(x,x):=\frac{J_{\alpha}^{2}(x)-J_{\alpha-1}(x)J_{\alpha+1}(x)}{2}x.\label{bxx} \end{equation} Here $J_{\alpha}(\cdot)$ is the Bessel function of order $\alpha$.
We want to point out, the motivation of this paper comes from \cite{Dieng}, which provided results both for symplectic ensembles and orthogonal ensembles, and specialize to the Gaussian case, i.e., GSE and GOE. The \cite{Dieng} dealt with the situation where $f(.)$ is the characteristic function of an interval (or the union of disjoint intervals), and focus on the distribution of the $m$th largest eigenvalue in the GSE and GOE, while we are interested for "smooth" $f$, and we also consider the Laguerre case, i.e., LSE and LOE.
This paper is organized as follows. In Section 2, we recall a number of theorems, the operators $D$ and $\varepsilon,$ and end with two Lemmas relevant for later development. Section 3 begins with a general discussion of the Symplectic ensembles in a general setting, followed by detailed discussions on the GSE and LSE cases and ends with the computation of the mean and variance of linear statistics for large $N.$ Section 4 repeats the development in Section 3 but for the Orthogonal ensembles, which is harder. We conclude in Section 5.
\section{Preliminaries} For orientation purposes, we introduce here a number of results, which will be used throughout this paper.
\begin{theorem} The Stirling's formula \cite{Lebedev} \begin{equation} \Gamma(n)=\sqrt{2\pi}\:n^{n-\frac{1}{2}}\:\mathrm{e}^{-n}\left[1+O\left(n^{-1}\right)\right],\;\;n\rightarrow\infty. \label{stirling} \end{equation} \end{theorem}
\begin{lemma} $$ _{2}F_{1}\left(-N,1;\frac{3}{2};2\right)=(-1)^{N}\sqrt{\frac{\pi}{8N}}+O\left(N^{-1}\right),\;\;N\rightarrow\infty.\label{2f1} $$ \end{lemma}
\begin{proof} From the integral representation of the hypergeometric function, \begin{equation} _{2}F_{1}(\alpha,\beta;\gamma;z)=\frac{\Gamma(\gamma)}{\Gamma(\beta)\Gamma(\gamma-\beta)} \int_{0}^{1}t^{\beta-1}(1-t)^{\gamma-\beta-1}(1-tz)^{-\alpha}dt,\;\; \mathrm{Re}\gamma>\mathrm{Re}\beta>0, \label{ir} \end{equation} we obtain, $$ _{2}F_{1}\left(-N,1;\frac{3}{2};2\right)=\frac{1}{2}\int_{0}^{1}\frac{(1-2t)^{N}}{\sqrt{1-t}}dt. $$ Let $$ x=\sqrt{1-t}, $$ then \begin{eqnarray} _{2}F_{1}\left(-N,1;\frac{3}{2};2\right) &=&\int_{0}^{1}(2x^{2}-1)^{N}dx\nonumber\\ &=&\int_{0}^{\frac{\sqrt{2}}{2}}(2x^{2}-1)^{N}dx+\int_{\frac{\sqrt{2}}{2}}^{1}(2x^{2}-1)^{N}dx\nonumber\\ &=&(-1)^{N}\int_{0}^{\frac{\sqrt{2}}{2}}(1-2x^{2})^{N}dx+\int_{\frac{\sqrt{2}}{2}}^{1}(2x^{2}-1)^{N}dx.\label{two} \end{eqnarray} Consider the first integral in (\ref{two}). Let $$ x=\frac{\sqrt{2}}{2}\cos\theta, \;\;\theta\in\left[0,\frac{\pi}{2}\right], $$ then \begin{eqnarray} \int_{0}^{\frac{\sqrt{2}}{2}}(1-2x^{2})^{N}dx &=&\frac{\sqrt{2}}{2}\int_{0}^{\frac{\pi}{2}}(\sin\theta)^{2N+1}d\theta\nonumber\\ &=&\frac{\sqrt{2}}{2}\frac{\left[2^{N}\Gamma(N+1)\right]^{2}}{\Gamma(2N+2)}\nonumber\\ &=&\sqrt{\frac{\pi}{8N}}\left[1+O\left(N^{-1}\right)\right],\;\;N\rightarrow\infty,\nonumber \end{eqnarray} where use has been made of the Stirling's formula (\ref{stirling}), in the last equality.
Now consider the second integral in (\ref{two}), \begin{eqnarray} \int_{\frac{\sqrt{2}}{2}}^{1}(2x^{2}-1)^{N}dx &\leq&\int_{\frac{\sqrt{2}}{2}}^{1}(2x-1)^{N}dx\nonumber\\ &=&\frac{1-(\sqrt{2}-1)^{N+1}}{2(N+1)}.\nonumber \end{eqnarray} Hence $$ _{2}F_{1}\left(-N,1;\frac{3}{2};2\right)=(-1)^{N}\sqrt{\frac{\pi}{8N}}+O\left(N^{-1}\right),\;\;N\rightarrow\infty. $$ \end{proof}
\begin{lemma} If $\alpha>0$, then $$ \sum_{m=0}^{2n}\frac{(-1)^{m}}{m!\prod_{l=m}^{2n}(\alpha+2l)}{2n+\alpha \choose 2n-m+1}=\frac{1}{(2n+1)!}, \;\;n=0,1,2,\ldots,\label{bi} $$ where ${j \choose k}:=\frac{\Gamma(j+1)}{\Gamma(k+1)\Gamma(j-k+1)}$. \end{lemma}
\begin{proof} Firstly, \begin{eqnarray} &&\sum_{m=0}^{2n}\frac{(-1)^{m}}{m!\prod_{l=m}^{2n}(\alpha+2l)}{2n+\alpha \choose 2n-m+1}\nonumber\\ &=&\frac{1}{(2n+1)!}+\frac{\Gamma(2n+\alpha+1)\Gamma\big(\frac{\alpha}{2}\big)\ _{2}F_{1}\big(-2n-1,\frac{\alpha}{2};\alpha;2\big)}{2^{2n+1}\Gamma(2n+2) \Gamma\big(2n+\frac{\alpha}{2}+1\big)\Gamma(\alpha)}.\label{hy} \end{eqnarray} From (\ref{ir}), we see that, $$ _{2}F_{1}\Big(-2n-1,\frac{\alpha}{2};\alpha;2\Big) =\frac{\Gamma(\alpha)}{\Gamma^{2}\big(\frac{\alpha}{2}\big)}\int_{0}^{1}t^{\frac{\alpha}{2}-1}(1-t)^{\frac{\alpha}{2}-1}(1-2t)^{2n+1}dt. $$ Let $$ t=\cos^{2}\bigg(\frac{\theta}{2}\bigg),\;\; \theta\in[0,\pi], $$ then $$ \int_{0}^{1}t^{\frac{\alpha}{2}-1}(1-t)^{\frac{\alpha}{2}-1}(1-2t)^{2n+1}dt =-\left(\frac{1}{2}\right)^{\alpha-1}\int_{0}^{\pi}(\sin\theta)^{\alpha-1}(\cos\theta)^{2n+1} d\theta. $$ We carry out mathematical induction in $n$ to prove that for $\alpha>0,\;$ $\int_{0}^{\pi}(\sin\theta)^{\alpha-1}(\cos\theta)^{2n+1} d\theta=0.$ \\ Setting if $n=0$, in the integral above, we have, \begin{eqnarray} &&\int_{0}^{\pi}(\sin\theta)^{\alpha-1}\cos\theta d\theta\nonumber\\ &=&\int_{0}^{\pi}(\sin\theta)^{\alpha-1} d\sin\theta\nonumber\\
&=&\frac{(\sin\theta)^{\alpha}}{\alpha}\bigg|_{0}^{\pi}\nonumber\\ &=&0.\nonumber \end{eqnarray} Suppose $\int_{0}^{\pi}(\sin\theta)^{\alpha-1}(\cos\theta)^{2k+1} d\theta=0$, for $\alpha>0$, then \begin{eqnarray} &&\int_{0}^{\pi}(\sin\theta)^{\alpha-1}(\cos\theta)^{2k+3} d\theta\nonumber\\ &=&\int_{0}^{\pi}(\sin\theta)^{\alpha-1}(\cos\theta)^{2k+1}\cos^{2}\theta d\theta\nonumber\\ &=&\int_{0}^{\pi}(\sin\theta)^{\alpha-1}(\cos\theta)^{2k+1}\left(1-\sin^{2}\theta\right) d\theta\nonumber\\ &=&\int_{0}^{\pi}(\sin\theta)^{\alpha-1}(\cos\theta)^{2k+1}d\theta-\int_{0}^{\pi}(\sin\theta)^{\alpha+1}(\cos\theta)^{2k+1} d\theta\nonumber\\ &=&0.\nonumber \end{eqnarray} Hence, $$ \int_{0}^{\pi}(\sin\theta)^{\alpha-1}(\cos\theta)^{2n+1} d\theta=0,\;\;\alpha>0. $$ It follows that, for $\alpha>0,$ $$ _{2}F_{1}\Big(-2n-1,\frac{\alpha}{2};\alpha;2\Big)=0, $$ and from (\ref{hy}), the lemma follows. \end{proof}
\begin{theorem} $$ \prod_{1\leq j<k\leq N}\left(x_{j}-x_{k}\right)^{4}=\det\left(x_{k}^{j},jx_{k}^{j-1}\right)_{{j=0,\ldots,2N-1}\atop{k=1,\ldots,N}}, \label{det} $$ where the determinant on the right is a $2N\times2N$ determinant with alternating columns\\
$\{x_{1}^{j}\},\; \{jx_{1}^{j-1}\},\; \{x_{2}^{j}\},\; \{jx_{2}^{j-1}\}, \ldots$ \cite{Kuramoto}. \end{theorem} The next theorem, due to de Bruijn \cite{de Bruijn}, is instrumental for the finite $N$ computations in Sections 3 and 4.
\begin{theorem} For any integrable functions $p_{j}(x)$ and $q_{j}(x), j=1,2,\ldots$, we have $$ \left(\int_{[a,b]^{N}}\det\left(p_{j}(x_{k}),q_{j}(x_{k})\right)_{{j=1,\ldots,2N}\atop{k=1,\ldots,N}}dx_{1}\cdots dx_{N}\right)^{2} =(N!)^{2}\det\left(\int_{a}^{b}\left(p_{j}(x)q_{k}(x)-p_{k}(x)q_{j}(x)\right)dx\right)_{j,k=1}^{2N}, $$
$$ \left(\int_{a\leq x_{1}\leq\cdots\leq x_{N}\leq b}\det\left(p_{j}(x_{k})\right)_{j,k=1}^{N}dx_{1}\cdots dx_{N}\right)^{2} =\det\left(\int_{a}^{b}\int_{a}^{b}\mathrm{sgn}(y-x)p_{j}(x)p_{k}(y)dxdy\right)_{j,k=1}^{N},\label{pf} $$ where the determinant on the left of the first equality is a $2N\times2N$ determinant with alternating columns
$\{p_{j}(x_{1})\},\; \{q_{j}(x_{1})\},\; \{p_{j}(x_{2})\},\; \{q_{j}(x_{2})\}, \ldots$. In addition, $N$ is even in the second equality. \end{theorem}
\begin{theorem} If $A, B$ are Hilbert-Schmidt operators on a Hilbert space $\mathcal{H}$, then \cite{Gohberg} $$ \det(I+AB)=\det(I+BA).\label{hs} $$ \end{theorem}
Following \cite{Tracy1996, Tracy1998, Widom}, we introduce here operators $\varepsilon$ and $D$ which will be crucial for later development. Let $\varepsilon$ be the integral operator with kernel $$ \varepsilon(x,y):=\frac{1}{2}\mathrm{sgn}(x-y), $$ then for any integrable function $g$ defined on $[a,b],$ $$ \varepsilon g(x)=\int_{a}^{b}\varepsilon(x,t)g(t)dt=\frac{1}{2}\left(\int_{a}^{x}g(t)dt-\int_{x}^{b}g(t)dt\right),\;\;x\in[a,b]. $$ It is clear that $\varepsilon(y,x)=-\varepsilon(x,y)$, i.e., $\varepsilon^{t}=-\varepsilon$, where $^{t}$ denotes transpose.
Let $D$ be the operator that acts by differentiation, thus for any differentiable function $g$ defined on $[a,b],$ $$ Dg(x)=\frac{dg(x)}{dx}=g'(x). $$ With further conditions on $g(x)$, we prove an easy lemma on the commutator $[D,\varepsilon].$ \begin{lemma} For any function $g\in C^{1}[a,b]$ and $g(a)=g(b)=0$, $D\varepsilon g(x)=\varepsilon D g(x)=g(x)$, i.e., $D\varepsilon=\varepsilon D=I$.\label{de} \end{lemma}
\begin{proof} For any function $g\in C^{1}[a,b]$ and $g(a)=g(b)=0$, we have \begin{eqnarray} (D\varepsilon)g(x) &=&\frac{d}{dx}\int_{a}^{b}\varepsilon(x,t)g(t)dt\nonumber\\ &=&\frac{1}{2}\frac{d}{dx}\left[\int_{a}^{x}g(t)dt-\int_{x}^{b}g(t)dt\right]\nonumber\\ &=&\frac{1}{2}g(x)+\frac{1}{2}g(x)=g(x),\nonumber \end{eqnarray} and \begin{eqnarray} (\varepsilon D)g(x) &=&\int_{a}^{b}\varepsilon(x,t)g'(t)dt\nonumber\\ &=&\frac{1}{2}\int_{a}^{x}g'(t)dt-\frac{1}{2}\int_{x}^{b}g'(t)dt\nonumber\\ &=&\frac{1}{2}[g(x)-g(a)]-\frac{1}{2}[g(b)-g(x)]\nonumber\\ &=&g(x)-\frac{1}{2}[g(a)+g(b)]\nonumber\\ &=&g(x).\nonumber \end{eqnarray} The proof is complete. \end{proof}
Denote by $u\otimes v$ the integral operator with kernel $(u\otimes v)(x,y):=u(x)v(y)$. We have the following lemma. \begin{lemma} If $A$ is an integral operator with kernel $A(x,y)$, then $$ A(u\otimes v)=(Au)\otimes v,\;\; (u\otimes v)A=u\otimes(A^{t}v).\label{ab} $$ \end{lemma}
\section{The symplectic ensembles} \subsection{General case} Taking $\beta=4$ in (\ref{gnb}) gives the symplectic ensembles and generating function becomes, $$ G_{N}^{(4)}(f)=C_{N}^{(4)}\int_{[a,b]^{N}}\prod_{1\leq j<k\leq N}\left(x_{j}-x_{k}\right)^{4}\prod_{j=1}^{N} w(x_{j})\left[1+f(x_{j})\right]dx_{j}, $$ where $$ C_{N}^{(4)}=\frac{1}{\int_{[a,b]^N}\prod_{1\leq j<k\leq N}\left(x_{j}-x_{k}\right)^{4}\prod_{j=1}^{N}w(x_{j})dx_{j}} $$ is a constant depending on $N$.
We follow closely the computations of Dieng \cite{Dieng}, and Tracy and Widom \cite{Tracy1998}.
From Theorem \ref{det} and Theorem \ref{pf} and some linear algebra, we get $$ \left[G_{N}^{(4)}(f)\right]^{2}=\widehat{C_{N}^{(4)}}\det\left(\int_{a}^{b}\left[\pi_{j}(x)\pi_{k}'(x)-\pi_{j}'(x)\pi_{k}(x)\right] w(x)(1+f(x))dx\right)_{j,k=0}^{2N-1}, $$ where $\widehat{C_{N}^{(4)}}$ is a $N$ dependent constant, and $\pi_j(x)$ is any polynomial of degree $j$. Let \begin{equation} \psi_{j}(x)=\pi_{j}(x)\sqrt{w(x)},\label{psij4} \end{equation} and following \cite{Dieng, Tracy1998}, we see that, $$ \left[G_{N}^{(4)}(f)\right]^{2}=\det\left(I+\left(M^{(4)}\right)^{-1}L^{(4)}\right), $$ where $M^{(4)}, L^{(4)}$ are matrices given by $$ M^{(4)}=\left(\int_{a}^{b}\left(\psi_{j}(x)\psi_{k}'(x)-\psi_{j}'(x)\psi_{k}(x)\right)dx\right)_{j,k=0}^{2N-1}, $$ $$ L^{(4)}=\left(\int_{a}^{b}\left(\psi_{j}(x)\psi_{k}'(x)-\psi_{j}'(x)\psi_{k}(x)\right)f(x)dx\right)_{j,k=0}^{2N-1}. $$ With the notation, $$ \left(M^{(4)}\right)^{-1}=:(\mu_{jk})_{j,k=0}^{2N-1}, $$ we obtain, finally, \begin{equation} \left[G_{N}^{(4)}(f)\right]^{2}=\det\left(I+K_{N}^{(4)}f\right),\label{gn42} \end{equation} where $K_{N}^{(4)}$ is the integral operator $$ K_{N}^{(4)}=\begin{pmatrix} \sum_{j,k=0}^{2N-1}\mu_{jk}\psi_{j}'\otimes\psi_{k}&-\sum_{j,k=0}^{2N-1}\mu_{jk}\psi_{j}'\otimes\psi_{k}'\\ \sum_{j,k=0}^{2N-1}\mu_{jk}\psi_{j}\otimes\psi_{k}&-\sum_{j,k=0}^{2N-1}\mu_{jk}\psi_{j}\otimes\psi_{k}' \end{pmatrix} =:\begin{pmatrix} K_{N,4}^{(1,1)}&K_{N,4}^{(1,2)}\\ K_{N,4}^{(2,1)}&K_{N,4}^{(2,2)} \end{pmatrix}, $$ In the next theorem, we obtain further relations on $K_{N,4}^{(i,j)}, i,j=1,2$, which ultimately expresses $K_N^{(4)}$ in terms of $K_{N,4}^{(2,2)}.$
\begin{theorem} $$ K_{N,4}^{(2,1)}=K_{N,4}^{(2,2)}\varepsilon,\;\;K_{N,4}^{(1,1)}=DK_{N,4}^{(2,2)}\varepsilon,\;\;K_{N,4}^{(1,2)}=DK_{N,4}^{(2,2)}, \;\;K_{N,4}^{(2,2)}\varepsilon D=K_{N,4}^{(2,2)}. \label{re} $$ \end{theorem}
\begin{proof} First of all, \begin{eqnarray} K_{N,4}^{(2,2)}\varepsilon &=&-\sum_{j,k=0}^{2N-1}(\mu_{jk}\psi_{j}\otimes\psi_{k}')\varepsilon\nonumber\\ &=&\sum_{j,k=0}^{2N-1}\mu_{jk}\psi_{j}\otimes(\varepsilon\psi_{k}')\nonumber\\ &=&\sum_{j,k=0}^{2N-1}\mu_{jk}\psi_{j}\otimes\psi_{k}\nonumber\\ &=&K_{N,4}^{(2,1)}.\nonumber \end{eqnarray} Secondly, for any integrable function $g(x)$ defined on $[a,b]$, we have \begin{eqnarray} D K_{N,4}^{(2,1)}g(x) &=&\frac{d}{dx}\int_{a}^{b}K_{N,4}^{(2,1)}(x,y)g(y)dy\nonumber\\ &=&\int_{a}^{b}\frac{\partial}{\partial x}K_{N,4}^{(2,1)}(x,y)g(y)dy\nonumber\\ &=&\int_{a}^{b}K_{N,4}^{(1,1)}(x,y)g(y)dy\nonumber\\ &=&K_{N,4}^{(1,1)}g(x),\nonumber \end{eqnarray} i.e., $$ D K_{N,4}^{(2,1)}=K_{N,4}^{(1,1)}. $$ It follows that $$ K_{N,4}^{(1,1)}=DK_{N,4}^{(2,2)}\varepsilon. $$ Similarly, \begin{eqnarray} D K_{N,4}^{(2,2)}g(x) &=&\frac{d}{dx}\int_{a}^{b}K_{N,4}^{(2,2)}(x,y)g(y)dy=\int_{a}^{b}\frac{\partial}{\partial x}K_{N,4}^{(2,2)}(x,y)g(y)dy\nonumber\\ &=&\int_{a}^{b}K_{N,4}^{(1,2)}(x,y)g(y)dy\nonumber\\ &=&K_{N,4}^{(1,2)}g(x),\nonumber \end{eqnarray} i.e., $$ K_{N,4}^{(1,2)}=D K_{N,4}^{(2,2)}. $$ Finally, we find \begin{eqnarray} K_{N,4}^{(2,2)}\varepsilon D g(x) &=&K_{N,4}^{(2,1)} D g(x)=\int_{a}^{b}K_{N,4}^{(2,1)}(x,y)g'(y)dy\nonumber\\
&=&K_{N,4}^{(2,1)}(x,y)g(y)|_{a}^{b}-\int_{a}^{b}g(y)\frac{\partial}{\partial y}K_{N,4}^{(2,1)}(x,y)dy.\nonumber \end{eqnarray} Note that \begin{eqnarray} K_{N,4}^{(2,1)}(x,y): &=&\sum_{j,k=0}^{2N-1}\psi_{j}(x)\mu_{jk}\psi_{k}(y)\nonumber\\ &=&\sum_{j,k=0}^{2N-1}\psi_{j}(x)\mu_{jk}\pi_{k}(y)\sqrt{w(y)}\nonumber \end{eqnarray} and $$ w(a)=w(b)=0, $$ hence $$ K_{N,4}^{(2,1)}(x,a)=K_{N,4}^{(2,1)}(x,b)=0. $$ Continuing, \begin{eqnarray} K_{N,4}^{(2,2)}\varepsilon D g(x) &=&-\int_{a}^{b}g(y)\frac{\partial}{\partial y}K_{N,4}^{(2,1)}(x,y)dy=\int_{a}^{b}K_{N,4}^{(2,2)}(x,y)g(y)dy\nonumber\\ &=&K_{N,4}^{(2,2)}g(x),\nonumber \end{eqnarray} i.e., $$ K_{N,4}^{(2,2)}\varepsilon D=K_{N,4}^{(2,2)}. $$ The proof is complete. \end{proof} According to Theorem \ref{re}, $K_{N}^{(4)}$ can be written as \begin{eqnarray} K_{N}^{(4)} &=&\begin{pmatrix} D K_{N,4}^{(2,2)}\varepsilon&D K_{N,4}^{(2,2)}\\ K_{N,4}^{(2,2)}\varepsilon&K_{N,4}^{(2,2)} \end{pmatrix}= \begin{pmatrix} D&0\\ 0&I \end{pmatrix} \begin{pmatrix} K_{N,4}^{(2,2)}\varepsilon&K_{N,4}^{(2,2)}\\ K_{N,4}^{(2,2)}\varepsilon&K_{N,4}^{(2,2)} \end{pmatrix}\nonumber\\ &=&:\widetilde{A}\widetilde{B},\nonumber \end{eqnarray} where $$ \widetilde{A}=\begin{pmatrix} D&0\\ 0&I \end{pmatrix},\;\;\;\; \widetilde{B}=\begin{pmatrix} K_{N,4}^{(2,2)}\varepsilon&K_{N,4}^{(2,2)}\\ K_{N,4}^{(2,2)}\varepsilon&K_{N,4}^{(2,2)} \end{pmatrix}. $$ From (\ref{gn42}) and using Theorem \ref{hs}, we have $$ \left[G_{N}^{(4)}(f)\right]^{2}=\det\bigg(I+\Big(\widetilde{A}\widetilde{B}\Big)f\bigg) =\det\bigg(I+\widetilde{A}\Big(\widetilde{B}f\Big)\bigg)=\det\bigg(I+\Big(\widetilde{B}f\Big)\widetilde{A}\bigg)= \det\bigg(I+\widetilde{B}\Big(f\widetilde{A}\Big)\bigg). $$ Since \begin{eqnarray} \widetilde{B}\left(f\widetilde{A}\right) &=& \begin{pmatrix} K_{N,4}^{(2,1)}&K_{N,4}^{(2,2)}\\ K_{N,4}^{(2,1)}&K_{N,4}^{(2,2)} \end{pmatrix} \left(f \begin{pmatrix} D&0\\ 0&I \end{pmatrix}\right) \nonumber\\ &=& \begin{pmatrix} K_{N,4}^{(2,2)}\varepsilon&K_{N,4}^{(2,2)}\\ K_{N,4}^{(2,2)}\varepsilon&K_{N,4}^{(2,2)} \end{pmatrix} \begin{pmatrix} f D&0\\ 0&f \end{pmatrix} \nonumber\\ &=&\begin{pmatrix} K_{N,4}^{(2,2)}\varepsilon f D&K_{N,4}^{(2,2)}f\\ K_{N,4}^{(2,2)}\varepsilon f D&K_{N,4}^{(2,2)}f \end{pmatrix},\nonumber \end{eqnarray} then $$ \left[G_{N}^{(4)}(f)\right]^{2}=\det \begin{pmatrix} I+K_{N,4}^{(2,2)}\varepsilon f D&K_{N,4}^{(2,2)}f\\ K_{N,4}^{(2,2)}\varepsilon f D&I+K_{N,4}^{(2,2)}f \end{pmatrix}. $$
The computation below reduces the above into a determinant of scalar operators. We subtract row 1 from row 2, $$ \left[G_{N}^{(4)}(f)\right]^{2}=\det \begin{pmatrix} I+K_{N,4}^{(2,2)}\varepsilon f D&K_{N,4}^{(2,2)}f\\ -I&I \end{pmatrix}. $$ Next, add column 2 to column 1, \begin{eqnarray} \left[G_{N}^{(4)}(f)\right]^{2} &=&\det \begin{pmatrix} I+K_{N,4}^{(2,2)}\varepsilon f D+K_{N,4}^{(2,2)}f&K_{N,4}^{(2,2)}f\\ 0&I \end{pmatrix}\nonumber\\ &=&\det\left(I+K_{N,4}^{(2,2)}\varepsilon f D+K_{N,4}^{(2,2)}f\right).\nonumber \end{eqnarray} $\mathbf{Remark}.$ The above result agrees with \cite{Dieng} for the GUE case if we take $f=-\mu\:\chi_{J}$, where $\chi_{J}$ is the characteristic function of the interval $J$.
Now we use the commutator $[D,f]:=Df-fD$ to obtain a better suited result for our purpose. For a given function {\it smooth} $g(x)$, we have \begin{eqnarray} [D,f]g(x) &=&D f g(x)-f D g(x)\nonumber\\ &=&(f(x)g(x))'-f(x)g'(x)\nonumber\\ &=&f'(x)g(x),\nonumber \end{eqnarray} this is, $$ [D,f]=f'. $$ It follows that \begin{equation} fD=Df-[D,f]=Df-f'.\label{fd} \end{equation} Taking this into account, we have the following theorem. \begin{theorem} $$ \left[G_{N}^{(4)}(f)\right]^{2} =\det\left(I+2K_{N,4}^{(2,2)}f-K_{N,4}^{(2,2)}\varepsilon f'\right),\label{gn4s} $$ where the kernel of $K_{N,4}^{(2,2)}$ reads, $$ K_{N,4}^{(2,2)}(x,y)=-\sum_{j,k=0}^{2N-1}\psi_{j}(x)\mu_{jk}\psi_{k}'(y). $$ \end{theorem}
\subsection{GSE} In the case of the Gaussian weight $w(x)=\mathrm{e}^{-x^{2}},\;\;x\in \mathbb{R}$, we again follow the discussions \cite{Dieng, Tracy1998}, and choose a special $\psi_{j}$ to simplify $M^{(4)}$ as much as possible. To proceed, let
\begin{equation} \psi_{2j+1}(x):=\frac{1}{\sqrt{2}}\varphi_{2j+1}(x),\;\; \psi_{2j}(x):=-\frac{1}{\sqrt{2}}\varepsilon\varphi_{2j+1}(x),\;\;j=0,1,2,\ldots, \label{psi2} \end{equation} where $\varphi_{j}(x)$ is given by \begin{equation} \varphi_{j}(x)=\frac{H_{j}(x)}{c_{j}}\mathrm{e}^{-\frac{x^{2}}{2}},\;\;c_{j}=\pi^{\frac{1}{4}}2^{\frac{j}{2}}\sqrt{\Gamma(j+1)},\label{gue} \end{equation} and $H_{j}(x), j=0,1,\ldots$ are the usual Hermite polynomials with the orthogonality condition $$ \int_{-\infty}^{\infty}H_{m}(x)H_{n}(x)\mathrm{e}^{-x^{2}}dx=c_{n}^{2}\:\delta_{mn}. $$ We show in the next lemma that, this definition satisfies (\ref{psij4}), i.e., $\psi_{j}(x)=\pi_{j}(x)\mathrm{e}^{-\frac{x^{2}}{2}}, j=0,1,2,\ldots$, where $\pi_j(x)$ is a polynomial of degree $j.$ \begin{lemma} $\psi_{j}(x)\mathrm{e}^{\frac{x^{2}}{2}},\:j=0,1,2,\ldots$ is a polynomial of degree $j$. \end{lemma}
\begin{proof} If the index is odd, then it is clear that $\psi_{2j+1}(x)\mathrm{e}^{\frac{x^{2}}{2}}$ is a polynomial of degree $2j+1$. For even index, \begin{eqnarray} \psi_{2j}(x)\mathrm{e}^{\frac{x^{2}}{2}} &=&-\frac{1}{\sqrt{2}}\left(\varepsilon\varphi_{2j+1}(x)\right)\mathrm{e}^{\frac{x^{2}}{2}}\nonumber\\ &=&-\frac{1}{\sqrt{2}}\mathrm{e}^{\frac{x^{2}}{2}}\varepsilon\left(\frac{1}{c_{2j+1}}H_{2j+1}(x)\mathrm{e}^{-\frac{x^{2}}{2}}\right)\nonumber\\ &=&-\frac{1}{2\sqrt{2}c_{2j+1}}\mathrm{e}^{\frac{x^{2}}{2}}\left[\int_{-\infty}^{x}H_{2j+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy -\int_{x}^{\infty}H_{2j+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy\right],\;\;j=0,1,2,\cdots.\nonumber \end{eqnarray} Since $H_{2j+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}, j=0,1,\ldots$ is an odd function, we have $$ 0=\int_{-\infty}^{\infty}H_{2j+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy= \int_{-\infty}^{x}H_{2j+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy+\int_{x}^{\infty}H_{2j+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy. $$ Hence $$ \int_{x}^{\infty}H_{2j+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy=-\int_{-\infty}^{x}H_{2j+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy, $$ and we find, $$ \psi_{2j}(x)\mathrm{e}^{\frac{x^{2}}{2}}=-\frac{1}{\sqrt{2}c_{2j+1}}\mathrm{e}^{\frac{x^{2}}{2}}\int_{-\infty}^{x}H_{2j+1}(y) \mathrm{e}^{-\frac{y^{2}}{2}}dy. $$ From mathematical induction, it follows that \begin{eqnarray} \int_{-\infty}^{x}y^{k}\mathrm{e}^{-\frac{y^{2}}{2}}dy &=&-\mathrm{e}^{-\frac{x^{2}}{2}}\left[x^{k-1}+(k-1)x^{k-3}+(k-1)(k-3)x^{k-5}\right.\nonumber\\ &+&\left.\cdots+(k-1)(k-3)\cdots2\right],\;\;k=1,3,5,\cdots.\nonumber \end{eqnarray} Since $H_{2j+1}(y)$ is a linear combination of $y, y^{3},\ldots, y^{2j+1}$,
we see that $\int_{-\infty}^{x}H_{2j+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy$ is equal to $\mathrm{e}^{-\frac{x^{2}}{2}}$ multiplying
a polynomial of degree $2j$. It follows that $\psi_{2j}(x)\mathrm{e}^{\frac{x^{2}}{2}}$ is a polynomial of degree $2j$. The proof is complete. \end{proof}
Using (\ref{psi2}) to compute $$ M^{(4)} :=\left(\int_{-\infty}^{\infty}\left(\psi_{j}(x)\psi_{k}'(x)-\psi_{j}'(x)\psi_{k}(x)\right)dx\right)_{j,k=0}^{2N-1}, $$ we obtain the following lemma. \begin{lemma}$ \mathrm{\mathbf{(Dieng, Tracy-Widom)}}$ $$ M^{(4)}=\begin{pmatrix} 0&1&0&0&\cdots&0&0\\ -1&0&0&0&\cdots&0&0\\ 0&0&0&1&\cdots&0&0\\ 0&0&-1&0&\cdots&0&0\\ \vdots&\vdots&\vdots&\vdots&&\vdots&\vdots\\ 0&0&0&0&\cdots&0&1\\ 0&0&0&0&\cdots&-1&0 \end{pmatrix}_{2N\times 2N}. $$ \end{lemma}
It is clear that $\left(M^{(4)}\right)^{-1}=-M^{(4)}$, so $\mu_{2j,2j+1}=-1, \mu_{2j+1,2j}=1$, and $\mu_{jk}=0$ for other cases. Hence \begin{eqnarray} K_{N,4}^{(2,2)}(x,y) &=&-\sum_{j,k=0}^{2N-1}\psi_{j}(x)\mu_{jk}\psi_{k}'(y)\nonumber\\ &=&\sum_{j=0}^{N-1}\psi_{2j}(x)\psi_{2j+1}'(y)-\sum_{j=0}^{N-1}\psi_{2j+1}(x)\psi_{2j}'(y)\nonumber\\ &=&\frac{1}{2}\left[\sum_{j=0}^{N-1}\varphi_{2j+1}(x)\varphi_{2j+1}(y)-\sum_{j=0}^{N-1}\varepsilon\varphi_{2j+1}(x)\varphi_{2j+1}'(y)\right]. \label{sn} \end{eqnarray}
Recall that the Hermite polynomials $H_{j}$ satisfy the differentiation formulas \cite{Lebedev} \begin{equation} H_{j}'(x)=2xH_{j}(x)-H_{j+1}(x),\;\;j=0,1,2,\ldots, \label{rh} \end{equation} \begin{equation} H_{j}'(x)=2jH_{j-1}(x),\;\;j=0,1,2,\ldots. \label{lh} \end{equation} Using the fact that $H_{j}(x)=c_{j}\:\varphi_{j}(x)\:{\rm e}^{\frac{x^{2}}{2}}$, (\ref{rh}) becomes \begin{equation} \varphi_{j}'(x)=x\varphi_{j}(x)-\sqrt{2(j+1)}\varphi_{j+1}(x),\;\;j=0,1,2,\ldots, \label{rphi} \end{equation} and similarly, (\ref{lh}) becomes, \begin{equation} \varphi_{j}'(x)=-x\varphi_{j}(x)+\sqrt{2j}\varphi_{j-1}(x),\;\;j=0,1,2,\ldots. \label{lphi} \end{equation} Combining (\ref{rphi}) and (\ref{lphi}), to eliminate $\varphi_j(x),$ we obtain \begin{equation} \varphi_{j}'(x)=\sqrt{\frac{j}{2}}\varphi_{j-1}(x)-\sqrt{\frac{j}{2}+\frac{1}{2}}\varphi_{j+1}(x),\;\;j=0,1,2,\ldots. \label{phid} \end{equation} Using (\ref{phid}) to replace $\varphi_{2j+1}'(y)$, we find, \begin{eqnarray} &&\sum_{j=0}^{N-1}\varepsilon\varphi_{2j+1}(x)\varphi_{2j+1}'(y)\nonumber\\ &=&\sum_{j=0}^{N-1}\varepsilon\varphi_{2j+1}(x)\left[\sqrt{j+\frac{1}{2}}\varphi_{2j}(y)-\sqrt{j+1}\varphi_{2j+2}(y)\right]\nonumber\\ &=&\sum_{j=0}^{N-1}\sqrt{j+\frac{1}{2}}\varepsilon\varphi_{2j+1}(x)\varphi_{2j}(y) -\sum_{j=0}^{N-1}\sqrt{j+1}\varepsilon\varphi_{2j+1}(x)\varphi_{2j+2}(y)\nonumber\\ &=&\sum_{j=0}^{N}\sqrt{j+\frac{1}{2}}\varepsilon\varphi_{2j+1}(x)\varphi_{2j}(y) -\sum_{j=0}^{N}\sqrt{j}\varepsilon\varphi_{2j-1}(x)\varphi_{2j}(y)-\sqrt{N+\frac{1}{2}}\varepsilon\varphi_{2N+1}(x)\varphi_{2N}(y)\nonumber\\ &=&-\sum_{j=0}^{N}\left[\sqrt{j}\varepsilon\varphi_{2j-1}(x)-\sqrt{j+\frac{1}{2}}\varepsilon\varphi_{2j+1}(x)\right]\varphi_{2j}(y) -\sqrt{N+\frac{1}{2}}\varepsilon\varphi_{2N+1}(x)\varphi_{2N}(y).\label{phi2j} \end{eqnarray} To proceed further, using Lemma \ref{de},
together with (\ref{phid}) we find \begin{eqnarray} \varphi_{2j}(x) &=&\varepsilon D\varphi_{2j}(x)=\varepsilon\:\varphi_{2j}'(x)\nonumber\\
&=&\sqrt{j}\;\varepsilon\varphi_{2j-1}(x)-\sqrt{j+\frac{1}{2}}\;\varepsilon\varphi_{2j+1}(x).\label{phi2jy} \end{eqnarray} Substituting (\ref{phi2jy}) into (\ref{phi2j}), it follows that $$ \sum_{j=0}^{N-1}\varepsilon\varphi_{2j+1}(x)\varphi_{2j+1}'(y)=-\sum_{j=0}^{N}\varphi_{2j}(x)\varphi_{2j}(y) -\sqrt{N+\frac{1}{2}}\varepsilon\varphi_{2N+1}(x)\varphi_{2N}(y). $$ Hence, (\ref{sn}) becomes, \begin{eqnarray} K_{N,4}^{(2,2)}(x,y) &=&\frac{1}{2}\left[\sum_{j=0}^{N-1}\varphi_{2j+1}(x)\varphi_{2j+1}(y)+\sum_{j=0}^{N}\varphi_{2j}(x)\varphi_{2j}(y) +\sqrt{N+\frac{1}{2}}\varepsilon\varphi_{2N+1}(x)\varphi_{2N}(y)\right]\nonumber\\ &=&\frac{1}{2}\left[\sum_{j=0}^{2N}\varphi_{j}(x)\varphi_{j}(y)+\sqrt{N+\frac{1}{2}}\varepsilon\varphi_{2N+1}(x)\varphi_{2N}(y)\right]\nonumber\\ &=&\frac{1}{2}\left[S_N(x,y)+\sqrt{N+\frac{1}{2}}\varepsilon\varphi_{2N+1}(x)\varphi_{2N}(y)\right],\nonumber \end{eqnarray} where $$ S_{N}(x,y):=\sum_{j=0}^{2N}\varphi_{j}(x)\varphi_{j}(y)=\sqrt{N+\frac{1}{2}}\:\frac{\varphi_{2N+1}(x)\varphi_{2N}(y)-\varphi_{2N+1}(y)\varphi_{2N}(x)}{x-y}, $$ and here the last equality comes from the Christoffel-Darboux formula.
By Theorem \ref{gn4s}, we have the following theorem. \begin{theorem} $$ \left[G_{N}^{(4)}(f)\right]^{2} =\det\left(I+S_{N}f-\frac{1}{2}S_{N}\varepsilon f'+\sqrt{N+\frac{1}{2}}\left(\varepsilon\varphi_{2N+1}\otimes\varphi_{2N}f\right) +\frac{1}{2}\sqrt{N+\frac{1}{2}}\left(\varepsilon\varphi_{2N+1}\otimes\varepsilon\varphi_{2N}\right) f'\right). $$ \end{theorem}
\subsection{Large $N$ behavior of the GSE moment generating function} To proceed with the large $N$ investigation, write, $\left[G_{N}^{(4)}(f)\right]^{2}$, as $$ \left[G_{N}^{(4)}(f)\right]^{2}=:\det(I+T), $$ where \begin{equation} T:=S_{N}f-\frac{1}{2}S_{N}\varepsilon f'+\sqrt{N+\frac{1}{2}}\left(\varepsilon\varphi_{2N+1}\otimes\varphi_{2N}f\right) +\frac{1}{2}\sqrt{N+\frac{1}{2}}\left(\varepsilon\varphi_{2N+1}\otimes\varepsilon\varphi_{2N}\right) f'.\label{ct} \end{equation} We find, $$ \log\det(I+T)=\mathrm{Tr}\log(I+T)=\mathrm{Tr}\:T-\frac{1}{2}\mathrm{Tr}\:T^{2}+\frac{1}{3}\mathrm{Tr}\:T^{3}-\cdots. $$ This is obtained by the trace-log expansion of $\log\det(I+\nu\:T)$, where $0<\nu<1,$ follow by a continuation to $\nu=1$. We assume that similar continuation also holds in other cases.
\begin{theorem} $$ \lim_{N\rightarrow\infty}\frac{1}{\sqrt{4N}}S_{N}\left(\frac{x}{\sqrt{4N}},\frac{y}{\sqrt{4N}}\right)=\frac{\sin(x-y)}{\pi(x-y)}, $$
$$ \lim_{N\rightarrow\infty}\frac{1}{\sqrt{4N}}S_{N}\left(\frac{x}{\sqrt{4N}},\frac{x}{\sqrt{4N}}\right)=\frac{1}{\pi}.\label{sn4xx} $$ \end{theorem}
The next theorem characterizes the large $N$ asymptotic of various ``scaled" quantities. \begin{theorem} $$ \varphi_{2N}\left(\frac{x}{\sqrt{4N}}\right)=(-1)^{N}\pi^{-\frac{1}{2}}N^{-\frac{1}{4}}\cos x+O\left(N^{-\frac{3}{4}}\right),\;\;N\rightarrow\infty, $$
$$ \varepsilon\varphi_{2N}\left(\frac{x}{\sqrt{4N}}\right)=2^{-1}(-1)^{N}\pi^{-\frac{1}{2}}N^{-\frac{3}{4}}\sin x+O\left(N^{-\frac{5}{4}}\right),\;\;N\rightarrow\infty, $$
$$ \varepsilon\varphi_{2N+1}\left(\frac{x}{\sqrt{4N}}\right)=-2^{-\frac{1}{2}}N^{-\frac{1}{4}}+O\left(N^{-\frac{3}{4}}\right),\;\;N\rightarrow\infty.\label{gse} $$ \end{theorem}
\begin{proof} From the asymptotic expansion of Hermite polynomials \cite{Szego}, \begin{equation} H_{n}(x)\:\mathrm{e}^{-\frac{x^{2}}{2}}=\lambda_{n}\left[\cos\left(\sqrt{2n+1}x-\frac{n\pi}{2}\right)+O\left(n^{-\frac{1}{2}}\right)\right],\;\;n\rightarrow\infty, \label{asy} \end{equation} where $$ \lambda_{2n}=\frac{\Gamma(2n+1)}{\Gamma\left(n+1\right)}\qquad \mathrm{and} \qquad \lambda_{2n+1}=\frac{\Gamma(2n+3)}{\Gamma(n+2)} \left(4n+3\right)^{-\frac{1}{2}}. $$
We have \begin{eqnarray} \varphi_{2N}\left(\frac{x}{\sqrt{4N}}\right) &=&\frac{1}{\pi^{\frac{1}{4}}2^{N}\sqrt{\Gamma(2N+1)}}H_{2N}\left(\frac{x}{\sqrt{4N}}\right)\mathrm{e}^{-\frac{x^{2}}{8N}}\nonumber\\ &=&\frac{1}{\pi^{\frac{1}{4}}2^{N}\sqrt{\Gamma(2N+1)}}\cdot\frac{\Gamma(2N+1)}{\Gamma(N+1)}\left[\cos\left(\sqrt{\frac{4N+1}{4N}}x-N\pi\right)+O\left(N^{-\frac{1}{2}}\right)\right] \nonumber\\ &=&\frac{1}{\pi^{\frac{1}{4}}2^{N}}\cdot\frac{\sqrt{\Gamma(2N+1)}}{\Gamma(N+1)}\left[\cos(x-N\pi)+O\left(N^{-\frac{1}{2}}\right)\right] \nonumber\\ &=&(-1)^{N}\pi^{-\frac{1}{2}}N^{-\frac{1}{4}}\cos x+O\left(N^{-\frac{3}{4}}\right),\;\;N\rightarrow\infty,\nonumber \end{eqnarray} where we have used the Stirling's formula (\ref{stirling}).
A straightforward computation gives, \begin{eqnarray} \varepsilon\varphi_{2N}(x) &=&\frac{1}{2}\left(\int_{-\infty}^{x}\varphi_{2N}(y)dy-\int_{x}^{\infty}\varphi_{2N}(y)dy\right)\nonumber\\ &=&\frac{1}{2}\left(\int_{-\infty}^{0}\varphi_{2N}(y)dy+\int_{0}^{x}\varphi_{2N}(y)dy-\int_{x}^{0}\varphi_{2N}(y)dy-\int_{0}^{\infty}\varphi_{2N}(y)dy\right)\nonumber\\ &=&\frac{1}{2}\left(2\int_{0}^{x}\varphi_{2N}(y)dy+\int_{-\infty}^{0}\varphi_{2N}(y)dy-\int_{0}^{\infty}\varphi_{2N}(y)dy\right)\nonumber\\ &=&\frac{1}{\pi^{\frac{1}{4}}2^{N+1}\sqrt{\Gamma(2N+1)}}\left(2\int_{0}^{x}H_{2N}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy+\int_{-\infty}^{0}H_{2N}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy -\int_{0}^{\infty}H_{2N}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy\right)\nonumber\\ &=&\frac{1}{\pi^{\frac{1}{4}}2^{N}\sqrt{\Gamma(2N+1)}}\int_{0}^{x}H_{2N}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy.\nonumber \end{eqnarray} So we find, for large $N$, \begin{eqnarray} \varepsilon\varphi_{2N}\left(\frac{x}{\sqrt{4N}}\right) &=&\frac{1}{\pi^{\frac{1}{4}}2^{N}\sqrt{\Gamma(2N+1)}}\int_{0}^{\frac{x}{\sqrt{4N}}}H_{2N}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy\nonumber\\ &=&\frac{1}{\pi^{\frac{1}{4}}2^{N}\sqrt{\Gamma(2N+1)}}\cdot\frac{1}{\sqrt{4N}}\int_{0}^{x}H_{2N}\left(\frac{y}{\sqrt{4N}}\right)\mathrm{e}^{-\frac{y^{2}}{8N}}dy\nonumber\\ &=&\frac{1}{\pi^{\frac{1}{4}}2^{N}\sqrt{\Gamma(2N+1)}}\cdot\frac{1}{\sqrt{4N}}\cdot\frac{\Gamma(2N+1)}{\Gamma(N+1)} \int_{0}^{x}\left[(-1)^{N}\cos y+O\left(N^{-\frac{1}{2}}\right)\right]dy\nonumber\\ &=&\frac{\sqrt{\Gamma(2N+1)}}{\pi^{\frac{1}{4}}2^{N}\sqrt{4N}\:\Gamma(N+1)}\left[(-1)^{N}\sin x+O\left(N^{-\frac{1}{2}}\right)\right]\nonumber\\ &=&2^{-1}(-1)^{N}\pi^{-\frac{1}{2}}N^{-\frac{3}{4}}\sin x+O\left(N^{-\frac{5}{4}}\right),\;\;N\rightarrow\infty,\nonumber \end{eqnarray} where we have used the Stirling's formula (\ref{stirling}) in the last step.
Finally, a straightforward computation gives, \begin{eqnarray} \varepsilon\varphi_{2N+1}(x) &=&\frac{1}{2}\left[\int_{-\infty}^{x}\varphi_{2N+1}(y)dy-\int_{x}^{\infty}\varphi_{2N+1}(y)dy\right]\nonumber\\ &=&\frac{1}{\pi^{\frac{1}{4}}2^{N+\frac{3}{2}}\sqrt{\Gamma(2N+2)}}\left[\int_{-\infty}^{x}H_{2N+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy -\int_{x}^{\infty}H_{2N+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy\right]\nonumber\\ &=&\frac{1}{\pi^{\frac{1}{4}}2^{N+\frac{3}{2}}\sqrt{\Gamma(2N+2)}}\left[\int_{-\infty}^{0}H_{2N+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy +\int_{0}^{x}H_{2N+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy\right.\nonumber\\ &-&\left.\int_{x}^{0}H_{2N+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy-\int_{0}^{\infty}H_{2N+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy\right]\nonumber\\ &=&\frac{1}{\pi^{\frac{1}{4}}2^{N+\frac{3}{2}}\sqrt{\Gamma(2N+2)}}\left[2\int_{0}^{x}H_{2N+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy -2\int_{0}^{\infty}H_{2N+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy\right]\nonumber\\ &=&\frac{1}{\pi^{\frac{1}{4}}2^{N+\frac{1}{2}}\sqrt{\Gamma(2N+2)}}\left[\int_{0}^{x}H_{2N+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy -\int_{0}^{\infty}H_{2N+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy\right],\nonumber \end{eqnarray} continuing, we see that, \begin{eqnarray} \varepsilon\varphi_{2N+1}\left(\frac{x}{\sqrt{4N}}\right) &=&\frac{1}{\pi^{\frac{1}{4}}2^{N+\frac{1}{2}}\sqrt{\Gamma(2N+2)}}\int_{0}^{\frac{x}{\sqrt{4N}}}H_{2N+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy\nonumber\\ &-&\frac{1}{\pi^{\frac{1}{4}}2^{N+\frac{1}{2}}\sqrt{\Gamma(2N+2)}}\int_{0}^{\infty}H_{2N+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy\nonumber\\ &=&\frac{1}{\pi^{\frac{1}{4}}2^{N+\frac{1}{2}}\sqrt{\Gamma(2N+2)}}\cdot\frac{1}{\sqrt{4N}}\int_{0}^{x}H_{2N+1}\left(\frac{y}{\sqrt{4N}}\right)\mathrm{e}^{-\frac{y^{2}}{8N}}dy \nonumber\\ &-&\frac{1}{\pi^{\frac{1}{4}}2^{N+\frac{1}{2}}\sqrt{\Gamma(2N+2)}}\int_{0}^{\infty}H_{2N+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy.\nonumber \end{eqnarray} By (\ref{asy}), we have $$ H_{2N+1}\left(\frac{y}{\sqrt{4N}}\right)\mathrm{e}^{-\frac{y^{2}}{8N}}=\frac{\Gamma(2N+3)}{\Gamma(N+2)}(4N+3)^{-\frac{1}{2}} \left[(-1)^{N}\sin y+O\left(N^{-\frac{1}{2}}\right)\right],\;\;N\rightarrow\infty. $$ It follows that $$ \int_{0}^{x}H_{2N+1}\left(\frac{y}{\sqrt{4N}}\right)\mathrm{e}^{-\frac{y^{2}}{8N}}dy=\frac{\Gamma(2N+3)}{\Gamma(N+2)}(4N+3)^{-\frac{1}{2}} \left[(-1)^{N}(1-\cos x)+O\left(N^{-\frac{1}{2}}\right)\right],\;\;N\rightarrow\infty. $$ On the other hand, from \cite{Gradshteyn}, $$ \int_{0}^{\infty}H_{2N+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy=\frac{2\:(-1)^{N}\Gamma(2N+2)}{\Gamma(N+1)} \:_{2}F_{1}\left(-N,1;\frac{3}{2};2\right). $$ It follows from Lemma \ref{2f1}, that, $$ \int_{0}^{\infty}H_{2N+1}(y)\mathrm{e}^{-\frac{y^{2}}{2}}dy=\frac{2\:(-1)^{N}\Gamma(2N+2)}{\Gamma(N+1)}\left[(-1)^{N}\sqrt{\frac{\pi}{8N}} +O\left(N^{-1}\right)\right],\;\;N\rightarrow\infty. $$ By the Stirling's formula (\ref{stirling}), we obtain \begin{eqnarray} \varepsilon\varphi_{2N+1}\left(\frac{x}{\sqrt{4N}}\right) &=&\left[2^{-1}(-1)^{N}\pi^{-\frac{1}{2}}N^{-\frac{3}{4}}(1-\cos x)+O\left(N^{-\frac{5}{4}}\right)\right] -\left[2^{-\frac{1}{2}}N^{-\frac{1}{4}}+O\left(N^{-\frac{3}{4}}\right)\right]\nonumber\\ &=&-2^{-\frac{1}{2}}N^{-\frac{1}{4}}+O\left(N^{-\frac{3}{4}}\right),\;\;N\rightarrow\infty.\nonumber \end{eqnarray} \end{proof}
We are now in a position to compute $\mathrm{Tr}\:T$ and $\mathrm{Tr}\:T^{2}$ as $N\rightarrow\infty$, using Theorem \ref{sn4xx} and Theorem \ref{gse}. The estimates provided by Theorem \ref{gse} are instrumental in the large $N$ computations that follows.
In what follows, we replace $f(x)$ by $f\left(\sqrt{4N}x\right)$, and note that $$ f'(\sqrt{4N}x)=\frac{1}{\sqrt{4N}}\:\frac{d}{dx}f(\sqrt{4N}x). $$ The $f'$ that appears in the trace will be accordingly interpreted.
We first consider $\mathrm{Tr}\:T$, which reads, $$ \mathrm{Tr}\:T=\mathrm{Tr}\:S_{N}f-\mathrm{Tr}\: \frac{1}{2}S_{N}\varepsilon f'+\mathrm{Tr}\:\sqrt{N+\frac{1}{2}} \left(\varepsilon\varphi_{2N+1}\otimes\varphi_{2N}f\right) +\mathrm{Tr}\:\frac{1}{2}\sqrt{N+\frac{1}{2}}(\varepsilon\varphi_{2N+1}\otimes\varepsilon\varphi_{2N})f'. $$
$\mathrm{Tr}\:T$ has four parts.
\\ First of all, we find \begin{eqnarray} \mathrm{Tr}\:S_{N}f &=&\int_{-\infty}^{\infty}S_{N}(x,x)f\left(\sqrt{4N}x\right)dx\nonumber\\ &=&\int_{-\infty}^{\infty}\frac{1}{\sqrt{4N}}S_{N}\left(\frac{x}{\sqrt{4N}},\frac{x}{\sqrt{4N}}\right)f(x)dx\nonumber\\ &\rightarrow&\frac{1}{\pi}\int_{-\infty}^{\infty}f(x)dx,\;\;N\to\infty.\nonumber \end{eqnarray} The second term reads, \begin{eqnarray} \mathrm{Tr}\:\frac{1}{2}S_{N}\varepsilon f' &=&\frac{1}{2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}S_{N}(x,y)\varepsilon(y,x)f'\left(\sqrt{4N}x\right)dx dy\nonumber\\ &=&\frac{1}{4}\int_{-\infty}^{\infty}\left[\int_{x}^{\infty}S_{N}(x,y)dy-\int_{-\infty}^{x}S_{N}(x,y)dy\right]f'\left(\sqrt{4N}x\right)dx. \nonumber \end{eqnarray} To proceed further, let $$ u=\sqrt{4N}x,\;\;v=\sqrt{4N}y, $$ it follows that, \begin{eqnarray} \mathrm{Tr}\:\frac{1}{2}S_{N}\varepsilon f' &=&\frac{1}{4\sqrt{4N}}\int_{-\infty}^{\infty}\left[\int_{u}^{\infty}\frac{1}{\sqrt{4N}}S_{N}\left(\frac{u}{\sqrt{4N}},\frac{v}{\sqrt{4N}}\right)dv -\int_{-\infty}^{u}\frac{1}{\sqrt{4N}}S_{N}\left(\frac{u}{\sqrt{4N}},\frac{v}{\sqrt{4N}}\right)dv\right]f'(u)du\nonumber\\ &\rightarrow&\frac{1}{4\sqrt{4N}\pi}\int_{-\infty}^{\infty}\left[\int_{u}^{\infty}\frac{\sin(u-v)}{u-v}dv-\int_{-\infty}^{u}\frac{\sin(u-v)}{u-v}dv\right]f'(u)du, \;\;N\to\infty.\nonumber \end{eqnarray} Now let $t=u-v$, it follows that, \begin{eqnarray} \mathrm{Tr}\:\frac{1}{2}S_{N}\varepsilon f' &\rightarrow&\frac{1}{4\sqrt{4N}\pi}\int_{-\infty}^{\infty}\left[\int_{-\infty}^{0}\frac{\sin t}{t}dt -\int_{0}^{\infty}\frac{\sin t}{t}dt\right]f'(u)du\nonumber\\ &=&\frac{1}{4\sqrt{4N}\pi}\int_{-\infty}^{\infty}\left(\frac{\pi}{2}-\frac{\pi}{2}\right)f'(u)du\nonumber\\ &=&0,\;\;N\rightarrow\infty.\nonumber \end{eqnarray} For the third term, we have, \begin{eqnarray} \mathrm{Tr}\:\sqrt{N+\frac{1}{2}}\left(\varepsilon\varphi_{2N+1}\otimes\varphi_{2N}f\right) &=&\sqrt{N+\frac{1}{2}}\int_{-\infty}^{\infty}\varepsilon\varphi_{2N+1}(x)\varphi_{2N}(x)f\left(\sqrt{4N}x\right)dx\nonumber\\ &=&\sqrt{N+\frac{1}{2}}\cdot\frac{1}{\sqrt{4N}}\int_{-\infty}^{\infty}\varepsilon\varphi_{2N+1}\left(\frac{x}{\sqrt{4N}}\right) \varphi_{2N}\left(\frac{x}{\sqrt{4N}}\right)f(x)dx \nonumber\\ &=&-\frac{(-1)^{N}}{2\sqrt{2\pi N}}\int_{-\infty}^{\infty}\cos x\: f(x)dx+O\left(N^{-1}\right),\;\;N\to\infty,\nonumber \end{eqnarray} where use has been made of Theorem \ref{gse}.
Finally to the fourth term, and take note of Theorem \ref{gse}, \begin{eqnarray} \mathrm{Tr}\:\frac{1}{2}\sqrt{N+\frac{1}{2}}(\varepsilon\varphi_{2N+1}\otimes\varepsilon\varphi_{2N})f' &=&\frac{1}{2}\sqrt{N+\frac{1}{2}}\int_{-\infty}^{\infty}\varepsilon\varphi_{2N+1}(x)\varepsilon\varphi_{2N}(x) f'\left(\sqrt{4N}x\right)dx\nonumber\\ &=&\frac{1}{2}\sqrt{N+\frac{1}{2}}\cdot\frac{1}{\sqrt{4N}}\int_{-\infty}^{\infty}\varepsilon\varphi_{2N+1}\left(\frac{x}{\sqrt{4N}}\right) \varepsilon\varphi_{2N}\left(\frac{x}{\sqrt{4N}}\right)f'(x)dx\nonumber\\ &=&O\left(N^{-1}\right),\;\;N\to\infty.\nonumber \end{eqnarray}
Therefore, the large $N$ expansion of ${\mathrm{Tr}}T$, reads, $$ \mathrm{Tr}\:T=\frac{1}{\pi}\int_{-\infty}^{\infty}f(x)dx -\frac{(-1)^{N}}{2\sqrt{2\pi N}}\int_{-\infty}^{\infty}\cos x\: f(x)dx+O\left(N^{-1}\right),\;\;N\rightarrow\infty. $$
Working out $\mathrm{Tr}\:T^{2}$, with $T$ given by, (\ref{ct}), there are 10 traces: \begin{eqnarray} \mathrm{Tr}\:T^{2} &=&\mathrm{Tr}\:S_{N}f S_{N}f-\mathrm{Tr}\:S_{N}f S_{N}\varepsilon f'+\mathrm{Tr}\:\sqrt{4N+2}S_{N}f(\varepsilon\varphi_{2N+1} \otimes\varphi_{2N}f)\nonumber\\ &+&\mathrm{Tr}\:\sqrt{N+\frac{1}{2}}S_{N}f(\varepsilon\varphi_{2N+1}\otimes\varepsilon\varphi_{2N})f' +\mathrm{Tr}\:\frac{1}{4}S_{N}\varepsilon f'S_{N}\varepsilon f'\nonumber\\ &-&\mathrm{Tr}\:\sqrt{N+\frac{1}{2}}S_{N}\varepsilon f'(\varepsilon\varphi_{2N+1}\otimes\varphi_{2N}f) -\mathrm{Tr}\:\frac{1}{2}\sqrt{N+\frac{1}{2}}S_{N}\varepsilon f'(\varepsilon\varphi_{2N+1}\otimes\varepsilon\varphi_{2N})f'\nonumber\\ &+&\mathrm{Tr}\:\left(N+\frac{1}{2}\right)(\varepsilon\varphi_{2N+1}\otimes\varphi_{2N}f)(\varepsilon\varphi_{2N+1}\otimes\varphi_{2N}f) \nonumber\\ &+&\mathrm{Tr}\:\left(N+\frac{1}{2}\right)(\varepsilon\varphi_{2N+1}\otimes\varphi_{2N}f)(\varepsilon\varphi_{2N+1}\otimes\varepsilon \varphi_{2N})f'\nonumber\\ &+&\mathrm{Tr}\:\frac{1}{4}\left(N+\frac{1}{2}\right)(\varepsilon\varphi_{2N+1}\otimes\varepsilon\varphi_{2N})f'(\varepsilon\varphi_{2N+1} \otimes\varepsilon\varphi_{2N})f'.\label{t2} \end{eqnarray} In the following, we calculate the trace on the right side of (\ref{t2}), term by term. \\ The first term: \begin{eqnarray} \mathrm{Tr}\:S_{N}f S_{N}f &=&\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}S_{N}(x,y)f\left(\sqrt{4N}y\right)S_{N}(y,x)f\left(\sqrt{4N}x\right)dy dx\nonumber\\ &=&\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\left[\frac{1}{\sqrt{4N}}S_{N}\left(\frac{x}{\sqrt{4N}},\frac{y}{\sqrt{4N}}\right)\right]^{2} f(x)f(y)dx dy\nonumber\\ &\rightarrow&\frac{1}{\pi^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\left[\frac{\sin(x-y)}{x-y}\right]^{2}f(x)f(y)dx dy, \;\;N\to\infty.\nonumber \end{eqnarray} The second term reads, \begin{eqnarray} \mathrm{Tr}\:S_{N}f S_{N}\varepsilon f' &=&\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}S_{N}(x,y)f\left(\sqrt{4N}y\right)S_{N}(y,z)\varepsilon(z,x) f'\left(\sqrt{4N}x\right)dx dy dz\nonumber\\ &=&\frac{1}{2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}S_{N}(x,y)\left[\int_{x}^{\infty}S_{N}(y,z)dz-\int_{-\infty}^{x}S_{N}(y,z)dz\right] f'\left(\sqrt{4N}x\right)f\left(\sqrt{4N}y\right)dx dy.\nonumber \end{eqnarray} A change of variables, $$ u=\sqrt{4N}x,\;\;v=\sqrt{4N}y,\;\;w=\sqrt{4N}z, $$ give \begin{eqnarray} \mathrm{Tr}\:S_{N}f S_{N}\varepsilon f' &=&\frac{1}{2\sqrt{4N}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{1}{\sqrt{4N}}S_{N} \left(\frac{u}{\sqrt{4N}},\frac{v}{\sqrt{4N}} \right)\nonumber\\ &&\left[\int_{u}^{\infty}\frac{1}{\sqrt{4N}}S_{N}\left(\frac{v}{\sqrt{4N}},\frac{w}{\sqrt{4N}}\right)dw -\int_{-\infty}^{u}\frac{1}{\sqrt{4N}}S_{N}\left(\frac{v}{\sqrt{4N}},\frac{w}{\sqrt{4N}}\right)dw\right]f'(u)f(v)du dv\nonumber\\ &\rightarrow&\frac{1}{2\sqrt{4N}\pi^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\sin(u-v)}{u-v} \left[\int_{u}^{\infty}\frac{\sin(v-w)}{v-w}dw -\int_{-\infty}^{u}\frac{\sin(v-w)}{v-w}dw\right]\nonumber\\ &&f'(u)f(v)du dv,\;\;N\to\infty.\nonumber \end{eqnarray} To proceed further, let $t=v-w$, we see that, \begin{eqnarray} &&\int_{u}^{\infty}\frac{\sin(v-w)}{v-w}dw-\int_{-\infty}^{u}\frac{\sin(v-w)}{v-w}dw =\int_{-\infty}^{v-u}\frac{\sin t}{t}dt-\int_{v-u}^{\infty}\frac{\sin t}{t}dt =2\int_{0}^{v-u}\frac{\sin t}{t}dt\nonumber\\ &=&2\:\mathrm{Si}(v-u),\nonumber \end{eqnarray} where $\mathrm{Si}(x)$ is the sine integral $$ \mathrm{Si}(x):=\int_{0}^{x}\frac{\sin t}{t}dt. $$ Since $\mathrm{Si}(-x)=-\mathrm{Si}(x),$ it follows that \begin{eqnarray} \mathrm{Tr}\:S_{N}f S_{N}\varepsilon f' &\rightarrow&\frac{1}{\sqrt{4N}\pi^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\sin(u-v)}{u-v}\:\mathrm{Si}(v-u)f'(u)f(v)dudv\nonumber\\ &=&-\frac{1}{2\pi^{2}\sqrt{N}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\sin(u-v)}{u-v}\:\mathrm{Si}(u-v)f'(u)f(v)du dv,\;N\to\infty.\nonumber \end{eqnarray} The third term: \begin{eqnarray} &&\mathrm{Tr}\:\sqrt{4N+2}S_{N}f(\varepsilon\varphi_{2N+1}\otimes\varphi_{2N}f)\nonumber\\ &=&\sqrt{4N+2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}S_{N}(x,y)f\left(\sqrt{4N}y\right)\varepsilon\varphi_{2N+1}(y) \varphi_{2N}(x)f\left(\sqrt{4N}x\right)dxdy\nonumber\\ &=&\sqrt{4N+2}\cdot\frac{1}{\sqrt{4N}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{1}{\sqrt{4N}}S_{N} \left(\frac{x}{\sqrt{4N}},\frac{y}{\sqrt{4N}}\right) f(y)\varepsilon\varphi_{2N+1}\left(\frac{y}{\sqrt{4N}}\right)\varphi_{2N}\left(\frac{x}{\sqrt{4N}}\right)f(x)dxdy\nonumber\\ &=&-\frac{(-1)^{N}}{\pi\sqrt{2\pi N}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\sin(x-y)}{x-y}\cos x\:f(x)f(y)dx dy +O\left(N^{-1}\right),\;N\to\infty.\nonumber \end{eqnarray} The fourth term: \begin{eqnarray} &&\mathrm{Tr}\:\sqrt{N+\frac{1}{2}}S_{N}f(\varepsilon\varphi_{2N+1}\otimes\varepsilon\varphi_{2N})f'\nonumber\\ &=&\sqrt{N+\frac{1}{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}S_{N}(x,y)f\left(\sqrt{4N}y\right)\varepsilon\varphi_{2N+1}(y) \varepsilon\varphi_{2N}(x)f'\left(\sqrt{4N}x\right)dx dy\nonumber\\ &=&\sqrt{N+\frac{1}{2}}\cdot\frac{1}{\sqrt{4N}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{1}{\sqrt{4N}}S_{N} \left(\frac{x}{\sqrt{4N}},\frac{y}{\sqrt{4N}}\right)f(y) \varepsilon\varphi_{2N+1}\left(\frac{y}{\sqrt{4N}}\right)\varepsilon\varphi_{2N}\left(\frac{x}{\sqrt{4N}}\right)f'(x)dx dy\nonumber\\ &=&O\left(N^{-1}\right),\;N\to\infty.\nonumber \end{eqnarray} The fifth term, with the change of variables, $$ u=\sqrt{4N}x,\;\;v=\sqrt{4N}y,\;\;w=\sqrt{4N}z,\;\;\tau=\sqrt{4N}t, $$ we see that, \begin{eqnarray} &&\mathrm{Tr}\:\frac{1}{4}S_{N}\varepsilon f'S_{N}\varepsilon f'\nonumber\\ &=&\frac{1}{64N}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\left[\int_{w}^{\infty}\frac{1}{\sqrt{4N}}S_{N}\left(\frac{u}{\sqrt{4N}},\frac{v}{\sqrt{4N}}\right)dv -\int_{-\infty}^{w}\frac{1}{\sqrt{4N}}S_{N}\left(\frac{u}{\sqrt{4N}},\frac{v}{\sqrt{4N}}\right)dv\right]\nonumber\\ &&\left[\int_{u}^{\infty}\frac{1}{\sqrt{4N}}S_{N}\left(\frac{w}{\sqrt{4N}},\frac{\tau}{\sqrt{4N}}\right)d\tau -\int_{-\infty}^{u}\frac{1}{\sqrt{4N}}S_{N}\left(\frac{w}{\sqrt{4N}},\frac{\tau}{\sqrt{4N}}\right)d\tau\right]f'(u)f'(w)dudw\nonumber\\ &=&O\left(N^{-1}\right),\;N\to\infty.\nonumber \end{eqnarray} The sixth term, \begin{eqnarray} &&\mathrm{Tr}\:\sqrt{N+\frac{1}{2}}S_{N}\varepsilon f'(\varepsilon\varphi_{2N+1}\otimes\varphi_{2N}f)\nonumber\\ &=&\frac{1}{8N}\sqrt{N+\frac{1}{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\left[\int_{w}^{\infty}\frac{1}{\sqrt{4N}}S_{N}\left(\frac{u}{\sqrt{4N}},\frac{v}{\sqrt{4N}}\right)dv -\int_{-\infty}^{w}\frac{1}{\sqrt{4N}}S_{N}\left(\frac{u}{\sqrt{4N}},\frac{v}{\sqrt{4N}}\right)dv\right]\nonumber\\ &&\varphi_{2N}\left(\frac{u}{\sqrt{4N}}\right)\varepsilon\varphi_{2N+1}\left(\frac{w}{\sqrt{4N}}\right)f(u)f'(w)du dw\nonumber\\ &=&O\left(N^{-1}\right),\;N\to\infty.\nonumber \end{eqnarray} The seventh term, becomes, \begin{eqnarray} &&\mathrm{Tr}\:\frac{1}{2}\sqrt{N+\frac{1}{2}}S_{N}\varepsilon f'(\varepsilon\varphi_{2N+1}\otimes\varepsilon\varphi_{2N})f'\nonumber\\
&=&\frac{1}{16N}\sqrt{N+\frac{1}{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\left[\int_{w}^{\infty}\frac{1}{\sqrt{4N}}S_{N}\left(\frac{u}{\sqrt{4N}},\frac{v}{\sqrt{4N}}\right)dv -\int_{-\infty}^{w}\frac{1}{\sqrt{4N}}S_{N}\left(\frac{u}{\sqrt{4N}},\frac{v}{\sqrt{4N}}\right)dv\right]\nonumber\\ &&\varepsilon\varphi_{2N}\left(\frac{u}{\sqrt{4N}}\right)\varepsilon\varphi_{2N+1}\left(\frac{w}{\sqrt{4N}}\right)f'(u)f'(w)dudw\nonumber\\ &=&O\left(N^{-\frac{3}{2}}\right),\;N\to\infty.\nonumber \end{eqnarray} The eighth term can be computed in a similar manner, and we have,
\begin{eqnarray} &&\mathrm{Tr}\:\left(N+\frac{1}{2}\right)(\varepsilon\varphi_{2N+1}\otimes\varphi_{2N}f)(\varepsilon\varphi_{2N+1}\otimes\varphi_{2N}f)\nonumber\\ &=&\frac{N+\frac{1}{2}}{4N}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\varepsilon\varphi_{2N+1}\left(\frac{u}{\sqrt{4N}}\right) \varepsilon\varphi_{2N+1}\left(\frac{v}{\sqrt{4N}}\right)\varphi_{2N}\left(\frac{u}{\sqrt{4N}}\right)\varphi_{2N}\left(\frac{v}{\sqrt{4N}}\right) f(u)f(v)du dv\nonumber\\ &=&O\left(N^{-1}\right),\;N\to\infty.\nonumber \end{eqnarray} Proceeding in a similar manner with the ninth term, we have,
\begin{eqnarray} &&\mathrm{Tr}\:\left(N+\frac{1}{2}\right)(\varepsilon\varphi_{2N+1}\otimes\varphi_{2N}f)(\varepsilon\varphi_{2N+1}\otimes\varepsilon\varphi_{2N})f'\nonumber\\ &=&\frac{N+\frac{1}{2}}{4N}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\varepsilon\varphi_{2N+1}\left(\frac{u}{\sqrt{4N}}\right)\varepsilon\varphi_{2N+1} \left(\frac{v}{\sqrt{4N}}\right)\varepsilon\varphi_{2N}\left(\frac{u}{\sqrt{4N}}\right)\varphi_{2N}\left(\frac{v}{\sqrt{4N}}\right)f'(u)f(v)du dv\nonumber\\ &=&O\left(N^{-\frac{3}{2}}\right),\;N\to\infty.\nonumber \end{eqnarray} The tenth and last term in ${\rm Tr}T^2$, becomes
\begin{eqnarray} &&\mathrm{Tr}\:\frac{1}{4}\left(N+\frac{1}{2}\right)(\varepsilon\varphi_{2N+1}\otimes\varepsilon\varphi_{2N})f'(\varepsilon\varphi_{2N+1}\otimes\varepsilon\varphi_{2N})f'\nonumber\\ &=&\frac{N+\frac{1}{2}}{16N}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\varepsilon\varphi_{2N+1}\left(\frac{u}{\sqrt{4N}}\right) \varepsilon\varphi_{2N+1}\left(\frac{v}{\sqrt{4N}}\right)\varepsilon\varphi_{2N}\left(\frac{u}{\sqrt{4N}}\right) \varepsilon\varphi_{2N}\left(\frac{v}{\sqrt{4N}}\right)f'(u)f'(v)du dv\nonumber\\ &=&O\left(N^{-2}\right),\;N\to\infty.\nonumber \end{eqnarray} Hence, the large $N$ behavior of (\ref{t2}), reads, \begin{eqnarray} \mathrm{Tr}\:T^{2} &=&\frac{1}{\pi^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\left[\frac{\sin(x-y)}{x-y}\right]^{2}f(x)f(y)dx dy\nonumber\\ &+&\frac{1}{2\pi^{2}\sqrt{N}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\sin(x-y)}{x-y}\:\mathrm{Si}(x-y)f'(x)f(y)dx dy\nonumber\\ &-&\frac{(-1)^{N}}{\pi\sqrt{2\pi N}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\sin(x-y)}{x-y}\cos x\:f(x)f(y)dx dy +O\left(N^{-1}\right),\;N\to\infty.\nonumber \end{eqnarray} We are now in a position to compute the mean and variance of the (scaled) linear statistics $\sum_{j=1}^{N}F\left(\sqrt{4N}x_{j}\right)$, which are obtained as the coefficients of $\lambda$ and $\lambda^{2},$ of $\log\det(I+T).$
Since $$ f\left(\sqrt{4N}x\right)\approx-\lambda F\left(\sqrt{4N}x\right)+\frac{\lambda^{2}}{2}F^{2}\left(\sqrt{4N}x\right), $$ we replace $f$ with $-\lambda F+\frac{\lambda^{2}}{2}F^{2}$ in the expression of $\mathrm{Tr}\:T$ and $\mathrm{Tr}\:T^{2}$. A minor rearrangement gives, \begin{eqnarray} &&\log\det\left(I+T\right)\nonumber\\ &=&-\lambda\Bigg\{\frac{1}{\pi}\int_{-\infty}^{\infty}F(x)dx-\frac{(-1)^{N}}{2\sqrt{2\pi N}}\int_{-\infty}^{\infty}\cos x\: F(x)dx +O\left(N^{-1}\right)\Bigg\}\nonumber\\ &+&\frac{\lambda^{2}}{2}\Bigg\{\frac{1}{\pi}\int_{-\infty}^{\infty}F^{2}(x)dx -\frac{1}{\pi^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\left[\frac{\sin(x-y)}{x-y}\right]^{2}F(x)F(y)dx dy\nonumber\\ &-&\frac{1}{2\pi^{2}\sqrt{N}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\sin(x-y)}{x-y}\:\mathrm{Si}(x-y)F'(x)F(y)dx dy\nonumber\\ &-&\frac{(-1)^{N}}{2\sqrt{2\pi N}}\left[\int_{-\infty}^{\infty}\cos x\: F^{2}(x)dx- \frac{2}{\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\sin(x-y)}{x-y}\cos x\:F(x)F(y)dx dy\right]+O\left(N^{-1}\right)\Bigg\},\;\;N\rightarrow\infty.\nonumber \end{eqnarray} Denote by $\mu_{N}^{(GSE)}$ and $\mathcal{V}_{N}^{(GSE)}$ the mean and variance of the linear statistics $\sum_{j=1}^{N}F\left(\sqrt{4N}x_{j}\right)$, then we have obtained, the large $N$ corrections of these quantities. \begin{theorem} As $N\rightarrow\infty$, $$ \mu_{N}^{(GSE)}=\frac{1}{2}\:\mu_{N}^{(GUE)}-\frac{(-1)^{N}}{4\sqrt{2\pi N}}\int_{-\infty}^{\infty}\cos x\: F(x)dx+O\left(N^{-1}\right), $$ \begin{eqnarray} \mathcal{V}_{N}^{(GSE)} &=&\frac{1}{2}\:\mathcal{V}_{N}^{(GUE)}-\frac{1}{4\pi^{2}\sqrt{N}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\sin(x-y)}{x-y}\:\mathrm{Si}(x-y)F'(x)F(y)dx dy\nonumber\\ &-&\frac{(-1)^{N}}{4\sqrt{2\pi N}}\left[\int_{-\infty}^{\infty}\cos x\: F^{2}(x)dx- \frac{2}{\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\sin(x-y)}{x-y}\cos x\:F(x)F(y)dx dy\right]+O\left(N^{-1}\right),\nonumber \end{eqnarray} where $\mu_{N}^{(GUE)}$ and $\mathcal{V}_{N}^{(GUE)}$ for $N\rightarrow\infty$ are given in (\ref{guem}) and (\ref{guev}) respectively. \end{theorem}
\subsection{LSE} We study the case with the Laguerre background, namely, the weight,\\ $w(x)=x^{\alpha}\mathrm{e}^{-x},\;\;\alpha>0,\;\;x\in \mathbb{R}^+.$
The idea is to choose special $\psi_{j}$ so that $M^{(4)}$ takes on the simplest possible form. To this end, let \begin{equation} \psi_{2j+1}(x):=\frac{1}{\sqrt{2}}\varphi_{2j+1}^{(\alpha-1)}(x),\;\;\;j=0,1,2,\ldots,\label{psia1} \end{equation} \begin{equation} \psi_{2j}(x):=-\frac{1}{\sqrt{2}}\varepsilon\widetilde{\varphi}_{2j+1}^{(\alpha-1)}(x),\;\;\;j=0,1,2,\ldots,\label{psia2} \end{equation} where $\varphi_{j}^{(\alpha-1)}(x)$ and $\widetilde{\varphi}_{j}^{(\alpha-1)}(x)$ are given by \begin{equation} \varphi_{j}^{(\alpha-1)}(x)=\frac{L_{j}^{(\alpha-1)}(x)}{c_{j}^{(\alpha-1)}} x^{\frac{\alpha}{2}}\mathrm{e}^{-\frac{x}{2}},\;\;j=0,1,2,\ldots,\label{phij} \end{equation} $$ \widetilde{\varphi}_{j}^{(\alpha-1)}(x)= \frac{L_{j}^{(\alpha-1)}(x)}{c_{j}^{(\alpha-1)}}x^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{x}{2}}, \;\;j=0,1,2,\ldots. $$ Here $L_{j}^{(\alpha)}(x),\:j=0,1,2,\ldots, $ are the Laguerre polynomials, with the orthogonality condition, $$ \int_{0}^{\infty}L_{j}^{(\alpha)}(x)L_{k}^{(\alpha)}(x)x^{\alpha}\mathrm{e}^{-x}dx=\left(c_{j}^{(\alpha)}\right)^{2}\delta_{jk}, \;\;\;\;c_{j}^{(\alpha)}=\sqrt{\frac{\Gamma(j+\alpha+1)}{\Gamma(j+1)}}. $$ It is easy to see that $$ \int_{0}^{\infty}\varphi_{j}^{(\alpha-1)}(x)\widetilde{\varphi}_{k}^{(\alpha-1)}(x)dx=\delta_{jk},\;\;j,k=0,1,2,\ldots. $$ We now prove that (\ref{psia1}) and (\ref{psia2}) satisfy (\ref{psij4}), i.e., $\psi_{j}(x)=\pi_{j}(x)x^{\frac{\alpha}{2}}\mathrm{e}^{-\frac{x}{2}},\: j=0,1,2,\ldots$, where $\pi_{j}(x)$ is a polynomial of degree $j$.
\begin{theorem} $\psi_{j}(x)=\pi_{j}(x)x^{\frac{\alpha}{2}}\mathrm{e}^{-\frac{x}{2}},\: j=0,1,2,\ldots$, where $\pi_{j}(x)$ is a polynomial of degree $j$. \end{theorem}
\begin{proof} We prove this by considering two cases, $j$ odd, and $j$ even. If $j=2n+1$, then by (\ref{psia1}), $$ \pi_{2n+1}(x)=\frac{1}{\sqrt{2}c_{2n+1}^{(\alpha-1)}}L_{2n+1}^{(\alpha-1)}(x). $$
Let $j=2n$, then $$ \psi_{2n}(x)=-\frac{1}{\sqrt{2}}\varepsilon\widetilde{\varphi}_{2n+1}(x) =-\frac{1}{\sqrt{2}c_{2n+1}^{(\alpha-1)}}\int_{0}^{x}L_{2n+1}^{(\alpha-1)}(y)y^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{y}{2}}dy, $$ and we have used the fact $\int_{0}^{\infty}L_{2n+1}^{(\alpha-1)}(y)y^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{y}{2}}dy=0,\: n=0,1,2,\ldots.$
Let us rewrite the above as \begin{equation} \int_{0}^{x}L_{2n+1}^{(\alpha-1)}(y)y^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{y}{2}}dy =\widehat{\pi}_{2n}(x)x^{\frac{\alpha}{2}}\mathrm{e}^{-\frac{x}{2}},\;\;n=0,1,2,\ldots,\label{eq2} \end{equation} where $$ \widehat{\pi}_{2n}(x):=-\sqrt{2}c_{2n+1}^{(\alpha-1)}\pi_{2n}(x). $$ Take a derivative on both sides, $$ L_{2n+1}^{(\alpha-1)}(x)x^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{x}{2}}=\left(x\widehat{\pi}_{2n}'(x)+\frac{\alpha}{2}\widehat{\pi}_{2n}(x) -\frac{1}{2}x\widehat{\pi}_{2n}(x)\right)x^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{x}{2}}, $$
which becomes, \begin{equation} L_{2n+1}^{(\alpha-1)}(x)=x\widehat{\pi}_{2n}'(x)+\frac{\alpha}{2}\widehat{\pi}_{2n}(x)-\frac{1}{2}x\widehat{\pi}_{2n}(x).\label{pe} \end{equation} Note that equation (\ref{eq2}) is equivalent to (\ref{pe}). Now we seek to solve (\ref{pe}).
Suppose $$ \widehat{\pi}_{2n}(x):=a_{0}(n)+a_{1}(n)x+a_{2}(n)x^{2}+\cdots+a_{2n-1}(n)x^{2n-1}+a_{2n}(n)x^{2n}, $$ we see that the right side of (\ref{pe}) is equal to \begin{eqnarray} &&x\widehat{\pi}_{2n}'(x)+\frac{\alpha}{2}\widehat{\pi}_{2n}(x)-\frac{1}{2}x\widehat{\pi}_{2n}(x)\nonumber\\ &=&\frac{\alpha}{2}a_{0}(n)+\left[\left(1+\frac{\alpha}{2}\right)a_{1}(n)-\frac{1}{2}a_{0}(n)\right]x+\left[\left(2+\frac{\alpha}{2}\right)a_{2}(n) -\frac{1}{2}a_{1}(n)\right]x^{2}+\cdots\nonumber\\ &+&\left[\left(2n-1+\frac{\alpha}{2}\right)a_{2n-1}(n)-\frac{1}{2}a_{2n-2}(n)\right]x^{2n-1}+\left[\left(2n+\frac{\alpha}{2}\right)a_{2n}(n) -\frac{1}{2}a_{2n-1}(n)\right]x^{2n}\nonumber\\ &-&\frac{1}{2}a_{2n}(n)x^{2n+1}.\label{rhs} \end{eqnarray} On the other hand, the left side of (\ref{pe}) is equal to \begin{eqnarray} L_{2n+1}^{(\alpha-1)}(x) &=&{2n+\alpha \choose 2n+1}-{2n+\alpha \choose 2n}x+\frac{1}{2!}{2n+\alpha \choose 2n-1}x^{2}-\frac{1}{3!}{2n+\alpha \choose 2n-2}x^{3}\nonumber\\ &+&\cdots+\frac{1}{(2n)!}{2n+\alpha \choose 1}x^{2n}-\frac{1}{(2n+1)!}{2n+\alpha \choose 0}x^{2n+1}.\label{lhs} \end{eqnarray} Compare the coefficients of (\ref{rhs}) and (\ref{lhs}), we have the equations \begin{equation} \begin{cases} \frac{\alpha}{2}a_{0}(n)={2n+\alpha \choose 2n+1}\\ \left(1+\frac{\alpha}{2}\right)a_{1}(n)-\frac{1}{2}a_{0}(n)=-{2n+\alpha \choose 2n}\\ \left(2+\frac{\alpha}{2}\right)a_{2}(n)-\frac{1}{2}a_{1}(n)=\frac{1}{2!}{2n+\alpha \choose 2n-1}\\ \left(3+\frac{\alpha}{2}\right)a_{3}(n)-\frac{1}{2}a_{2}(n)=-\frac{1}{3!}{2n+\alpha \choose 2n-2}\\ \qquad\qquad\qquad\vdots\\ \left(2n-1+\frac{\alpha}{2}\right)a_{2n-1}(n)-\frac{1}{2}a_{2n-2}(n)=-\frac{1}{(2n-1)!}{2n+\alpha \choose 2}\\ \left(2n+\frac{\alpha}{2}\right)a_{2n}(n)-\frac{1}{2}a_{2n-1}(n)=-\frac{1}{(2n)!}{2n+\alpha \choose 1}\\ -\frac{1}{2}a_{2n}(n)=-\frac{1}{(2n+1)!}{2n+\alpha \choose 0}. \end{cases}\label{eq} \end{equation} By solving the first $2n+1$ equations in (\ref{eq}), we find, $$ \begin{cases} \frac{1}{2}a_{0}(n)=\frac{1}{\alpha}{2n+\alpha \choose 2n+1}\\ \frac{1}{2}a_{1}(n)=\frac{1}{\alpha(\alpha+2)}{2n+\alpha \choose 2n+1}-\frac{1}{\alpha+2}{2n+\alpha \choose 2n}\\ \frac{1}{2}a_{2}(n)=\frac{1}{\alpha(\alpha+2)(\alpha+4)}{2n+\alpha \choose 2n+1}-\frac{1}{(\alpha+2)(\alpha+4)}{2n+\alpha \choose 2n} +\frac{1}{2!}\frac{1}{\alpha+4}{2n+\alpha \choose 2n-1}\\ \frac{1}{2}a_{3}(n)=\frac{1}{\alpha(\alpha+2)(\alpha+4)(\alpha+6)}{2n+\alpha \choose 2n+1}-\frac{1}{(\alpha+2)(\alpha+4)(\alpha+6)}{2n+\alpha \choose 2n} +\frac{1}{2!}\frac{1}{(\alpha+4)(\alpha+6)}{2n+\alpha \choose 2n-1}-\frac{1}{3!}\frac{1}{\alpha+6}{2n+\alpha \choose 2n-2}\\ \qquad\qquad\qquad\vdots\\ \frac{1}{2}a_{2n}(n)=\frac{1}{\alpha(\alpha+2)(\alpha+4)\cdots(\alpha+4n)}{2n+\alpha \choose 2n+1}-\frac{1}{(\alpha+2)(\alpha+4)\cdots(\alpha+4n)}{2n+\alpha \choose 2n}+\frac{1}{2!}\frac{1}{(\alpha+4)(\alpha+6)\cdots(\alpha+4n)}{2n+\alpha \choose 2n-1}\\ \qquad\quad-\frac{1}{3!}\frac{1}{(\alpha+6)\cdots(\alpha+4n)}{2n+\alpha \choose 2n-2}+\cdots-\frac{1}{(2n-1)!}\frac{1}{(\alpha+4n-2)(\alpha+4n)}{2n+\alpha \choose 2}+\frac{1}{(2n)!}\frac{1}{\alpha+4n}{2n+\alpha \choose 1}. \end{cases} $$ The last equation of (\ref{eq}), simplifies to, $$ a_{2n}(n)=\frac{2}{(2n+1)!}. $$ By Lemma \ref{bi}, we see that linear system (\ref{eq}) is solvable. Hence $\widehat{\pi}_{2n}(x)$ is a polynomial of degree $2n.$
The proof is complete. \end{proof}
With (\ref{psia1}) and (\ref{psia2}), we compute $M^{(4)}:=\left(\int_{0}^{\infty}\left(\psi_{j}(x)\psi_{k}'(x)-\psi_{j}'(x)\psi_{k}(x)\right)dx\right)_{j,k=0}^{2N-1}$, resulting in the following theorem. \begin{theorem} $$ M^{(4)}=\begin{pmatrix} 0&1&0&0&\cdots&0&0\\ -1&0&0&0&\cdots&0&0\\ 0&0&0&1&\cdots&0&0\\ 0&0&-1&0&\cdots&0&0\\ \vdots&\vdots&\vdots&\vdots&&\vdots&\vdots\\ 0&0&0&0&\cdots&0&1\\ 0&0&0&0&\cdots&-1&0 \end{pmatrix}_{2N\times 2N}. $$ \end{theorem}
\begin{proof} Let $m_{jk}$ be the $(j,k)$-entry of $M^{(4)}$, i.e., $$ m_{jk}:=\int_{0}^{\infty}\left(\psi_{j}(x)\psi_{k}'(x)-\psi_{j}'(x)\psi_{k}(x)\right)dx,\;\; j,k=0,1,\ldots,2N-1. $$ We compute $m_{j,k}$ by considering four cases: $(j,k)=({\rm even},{\rm odd}),\;\;({\rm odd},{\rm even}), \;\;({\rm even},{\rm even}),\;\;({\rm odd},{\rm odd}).$
For the $({\rm even},{\rm odd})$ case, \begin{eqnarray} m_{2j, 2k+1} &=&\int_{0}^{\infty}\left(\psi_{2j}(x)\psi_{2k+1}'(x)-\psi_{2j}'(x)\psi_{2k+1}(x)\right)dx\nonumber\\ &=&\int_{0}^{\infty}\psi_{2j}(x)\psi_{2k+1}'(x)dx-\int_{0}^{\infty}\psi_{2j}'(x)\psi_{2k+1}(x)dx\nonumber\\
&=&\left(\psi_{2j}(x)\psi_{2k+1}(x)|_{0}^{\infty}-\int_{0}^{\infty}\psi_{2j}'(x)\psi_{2k+1}(x)dx\right) -\int_{0}^{\infty}\psi_{2j}'(x)\psi_{2k+1}(x)dx\nonumber\\ &=&-2\int_{0}^{\infty}\psi_{2j}'(x)\psi_{2k+1}(x)dx\nonumber\\ &=&-2\int_{0}^{\infty}\left(-\frac{1}{\sqrt{2}}\widetilde{\varphi}_{2j+1}^{(\alpha-1)}(x)\right) \left(\frac{1}{\sqrt{2}}\varphi_{2k+1}^{(\alpha-1)}(x)\right)dx\nonumber\\ &=&\int_{0}^{\infty}\widetilde{\varphi}_{2j+1}^{(\alpha-1)}(x)\varphi_{2k+1}^{(\alpha-1)}(x)dx\nonumber\\ &=&\delta_{jk}.\nonumber \end{eqnarray} For the $({\rm odd},{\rm even})$ case, since $M^{(4)}$ is antisymmetric, we have \begin{eqnarray} m_{2j+1,2k} &=&-m_{2k, 2j+1}\nonumber\\ &=&-\delta_{j,k}.\nonumber \end{eqnarray} For the $({\rm even},{\rm even})$ case and $j>k$, \begin{eqnarray} m_{2j,2k} &=&\int_{0}^{\infty}\left(\psi_{2j}(x)\psi_{2k}'(x)-\psi_{2j}'(x)\psi_{2k}(x)\right)dx\nonumber\\ &=&\int_{0}^{\infty}\psi_{2j}(x)\psi_{2k}'(x)dx-\int_{0}^{\infty}\psi_{2j}'(x)\psi_{2k}(x)dx\nonumber\\
&=&\psi_{2j}(x)\psi_{2k}(x)|_{0}^{\infty}-2\int_{0}^{\infty}\psi_{2j}'(x)\psi_{2k}(x)dx\nonumber\\ &=&-2\int_{0}^{\infty}\psi_{2j}'(x)\psi_{2k}(x)dx\nonumber\\ &=&-2\int_{0}^{\infty}\left(-\frac{1}{\sqrt{2}c_{2j+1}^{(\alpha-1)}}L_{2j+1}^{(\alpha-1)}(x)x^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{x}{2}}\right) \left(\pi_{2k}(x)x^{\frac{\alpha}{2}}\mathrm{e}^{-\frac{x}{2}}\right)dx\nonumber\\ &=&\frac{\sqrt{2}}{c_{2j+1}^{(\alpha-1)}}\int_{0}^{\infty}L_{2j+1}^{(\alpha-1)}(x)\pi_{2k}(x)x^{\alpha-1}\mathrm{e}^{-x}dx\nonumber\\ &=&0,\nonumber \end{eqnarray} since $\pi_{2k}(x)$ is a polynomial of degree $2k$ which is less than $2j+1$.
For the $({\rm odd},{\rm odd})$ case and $j>k$, \begin{eqnarray} m_{2j+1,2k+1} &=&\int_{0}^{\infty}\left(\psi_{2j+1}(x)\psi_{2k+1}'(x)-\psi_{2j+1}'(x)\psi_{2k+1}(x)\right)dx\nonumber\\ &=&\int_{0}^{\infty}\psi_{2j+1}(x)\psi_{2k+1}'(x)dx-\int_{0}^{\infty}\psi_{2j+1}'(x)\psi_{2k+1}(x)dx\nonumber\\
&=&\int_{0}^{\infty}\psi_{2j+1}(x)\psi_{2k+1}'(x)dx-\left(\psi_{2j+1}(x)\psi_{2k+1}(x)|_{0}^{\infty} -\int_{0}^{\infty}\psi_{2j+1}(x)\psi_{2k+1}'(x)dx\right)\nonumber\\ &=&2\int_{0}^{\infty}\psi_{2j+1}(x)\psi_{2k+1}'(x)dx\nonumber\\ &=&\frac{1}{c_{2j+1}^{(\alpha-1)}c_{2k+1}^{(\alpha-1)}}\int_{0}^{\infty}L_{2j+1}^{(\alpha-1)}(x)x^{\frac{\alpha}{2}}\mathrm{e}^{-\frac{x}{2}} \left(x\left(L_{2k+1}^{(\alpha-1)}(x)\right)'+\frac{\alpha}{2}L_{2k+1}^{(\alpha-1)}(x)-\frac{1}{2}x L_{2k+1}^{(\alpha-1)}(x)\right)x^{\frac{\alpha}{2}-1} \mathrm{e}^{-\frac{x}{2}}dx\nonumber\\ &=&\frac{1}{c_{2j+1}^{(\alpha-1)}c_{2k+1}^{(\alpha-1)}}\int_{0}^{\infty}L_{2j+1}^{(\alpha-1)}(x)\left(x\left(L_{2k+1}^{(\alpha-1)}(x)\right)' +\frac{\alpha}{2}L_{2k+1}^{(\alpha-1)}(x)-\frac{1}{2}x L_{2k+1}^{(\alpha-1)}(x)\right)x^{\alpha-1}\mathrm{e}^{-x}dx\nonumber\\ &=&0,\nonumber \end{eqnarray} since $x\left(L_{2k+1}^{(\alpha-1)}(x)\right)'+\frac{\alpha}{2}L_{2k+1}^{(\alpha-1)}(x)-\frac{1}{2}x L_{2k+1}^{(\alpha-1)}(x)$ is a polynomial of degree $2k+2$ which is less than $2j+1$.
If $j<k$, due to the fact that $M^{(4)}$ is antisymmetric, \begin{eqnarray} m_{2j,2k} &=&-m_{2k,2j}\nonumber\\ &=&0,\nonumber \end{eqnarray} \begin{eqnarray} m_{2j+1,2k+1} &=&-m_{2k+1,2j+1}\nonumber\\ &=&0.\nonumber \end{eqnarray} Thus $$ m_{2j,2k}=m_{2j+1,2k+1}=0,\;\;j,k=0,1,\ldots,N-1. $$ This is just the desired form of $M^{(4)}$. \end{proof}
It's clear that $\left(M^{(4)}\right)^{-1}=-M^{(4)}$, so $\mu_{2j,2j+1}=-1, \mu_{2j+1,2j}=1$, and $\mu_{jk}=0$ for other cases. The rest of this subsection is devoted to the determination of $K_{N,4}^{(2,2)}(x,y)$, \begin{eqnarray} K_{N,4}^{(2,2)}(x,y) &=&-\sum_{j,k=0}^{2N-1}\psi_{j}(x)\mu_{jk}\psi_{k}'(y)\nonumber\\ &=&\sum_{j=0}^{N-1}\psi_{2j}(x)\psi_{2j+1}'(y)-\sum_{j=0}^{N-1}\psi_{2j+1}(x)\psi_{2j}'(y)\nonumber\\ &=&\frac{1}{2}\sum_{j=0}^{N-1}\varphi_{2j+1}^{(\alpha-1)}(x)\widetilde{\varphi}_{2j+1}^{(\alpha-1)}(y) -\frac{1}{2}\sum_{j=0}^{N-1}\varepsilon\widetilde{\varphi}_{2j+1}^{(\alpha-1)}(x)\left[\varphi_{2j+1}^{(\alpha-1)}(y)\right]'.\nonumber \end{eqnarray} From (\ref{phij}), we see that, \begin{equation} \left[\varphi_{j}^{(\alpha-1)}(x)\right]'=\frac{1}{c_{j}^{(\alpha-1)}}x^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{x}{2}}\left[x\left(L_{j}^{(\alpha-1)}(x)\right)' +\frac{\alpha-x}{2} L_{j}^{(\alpha-1)}(x)\right].\label{sign} \end{equation} Recall that the Laguerre polynomials $L_{j}^{(\alpha-1)}(x),$ satisfy the differentiation formulas \cite{Gradshteyn}, \begin{equation} x\left(L_{j}^{(\alpha-1)}(x)\right)'=j L_{j}^{(\alpha-1)}(x)-(j+\alpha-1)L_{j-1}^{(\alpha-1)}(x),\;\;j=0,1,2,\ldots,\label{ldf1} \end{equation} \begin{equation} x\left(L_{j}^{(\alpha-1)}(x)\right)'=(j+1)L_{j+1}^{(\alpha-1)}(x)+(x-j-\alpha)L_{j}^{(\alpha-1)}(x),\;\;j=0,1,2,\ldots.\label{ldf2} \end{equation} Summing (\ref{ldf1}) and (\ref{ldf2}), and divide by 2, gives, $$ x\left(L_{j}^{(\alpha-1)}(x)\right)'=\frac{j+1}{2}L_{j+1}^{(\alpha-1)}(x)-\frac{j+\alpha-1}{2}L_{j-1}^{(\alpha-1)}(x) +\frac{x-\alpha}{2}L_{j}^{(\alpha-1)}(x),\;\;j=0,1,2,\ldots, $$ or $$ x\left(L_{j}^{(\alpha-1)}(x)\right)'+\frac{\alpha-x}{2}L_{j}^{(\alpha-1)}(x)=\frac{j+1}{2}L_{j+1}^{(\alpha-1)}(x)-\frac{j+\alpha-1}{2}L_{j-1}^{(\alpha-1)}(x) ,\;\;j=0,1,2,\ldots. $$ Hence (\ref{sign}) becomes, \begin{eqnarray} \left[\varphi_{j}^{(\alpha-1)}(x)\right]' &=&\frac{1}{c_{j}^{(\alpha-1)}}x^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{x}{2}}\left[\frac{j+1}{2}L_{j+1}^{(\alpha-1)}(x)-\frac{j+\alpha-1}{2} L_{j-1}^{(\alpha-1)}(x)\right]\nonumber\\ &=&\frac{1}{2}\sqrt{(j+1)(j+\alpha)}\:\widetilde{\varphi}_{j+1}^{(\alpha-1)}(x)-\frac{1}{2}\sqrt{j(j+\alpha-1)}\:\widetilde{\varphi}_{j-1}^{(\alpha-1)}(x).\label{phijpx} \end{eqnarray} We see that $\left[\varphi_{j}^{(\alpha-1)}\right]'$ is a linear combination of $\widetilde{\varphi}_{j+1}^{(\alpha-1)}$ and $\widetilde{\varphi}_{j-1}^{(\alpha-1)}$, just like the GSE case studied in the last section. Replacing $j$ by $2j+1$ in (\ref{phijpx}), to find, $$ \left[\varphi_{2j+1}^{(\alpha-1)}(y)\right]'=\sqrt{(j+1)\left(j+\frac{\alpha+1}{2}\right)}\widetilde{\varphi}_{2j+2}^{(\alpha-1)}(y)-\sqrt{\left(j+\frac{1}{2}\right) \left(j+\frac{\alpha}{2}\right)}\widetilde{\varphi}_{2j}^{(\alpha-1)}(y). $$ A straightforward computation shows that, \begin{eqnarray} &&\sum_{j=0}^{N-1}\varepsilon\widetilde{\varphi}_{2j+1}^{(\alpha-1)}(x)\left[\varphi_{2j+1}^{(\alpha-1)}(y)\right]'\nonumber\\ &=&\sum_{j=0}^{N-1}\varepsilon\widetilde{\varphi}_{2j+1}^{(\alpha-1)}(x)\sqrt{(j+1)\left(j+\frac{\alpha+1}{2}\right)}\widetilde{\varphi}_{2j+2}^{(\alpha-1)}(y) -\sum_{j=0}^{N-1}\varepsilon\widetilde{\varphi}_{2j+1}^{(\alpha-1)}(x)\sqrt{\left(j+\frac{1}{2}\right)\left(j+\frac{\alpha}{2}\right)} \widetilde{\varphi}_{2j}^{(\alpha-1)}(y)\nonumber\\ &=&\sum_{j=0}^{N}\sqrt{j\left(j+\frac{\alpha-1}{2}\right)}\varepsilon\widetilde{\varphi}_{2j-1}^{(\alpha-1)}(x)\widetilde{\varphi}_{2j}^{(\alpha-1)}(y) -\sum_{j=0}^{N}\sqrt{\left(j+\frac{1}{2}\right)\left(j+\frac{\alpha}{2}\right)}\varepsilon\widetilde{\varphi}_{2j+1}^{(\alpha-1)}(x)\widetilde{\varphi}_{2j}^{(\alpha-1)}(y) \nonumber\\ &+&\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)}\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}(x)\widetilde{\varphi}_{2N}^{(\alpha-1)}(y)\nonumber\\ &=&\sum_{j=0}^{N}\left[\sqrt{j\left(j+\frac{\alpha-1}{2}\right)}\varepsilon\widetilde{\varphi}_{2j-1}^{(\alpha-1)}(x) -\sqrt{\left(j+\frac{1}{2}\right)\left(j+\frac{\alpha}{2}\right)}\varepsilon\widetilde{\varphi}_{2j+1}^{(\alpha-1)}(x)\right] \widetilde{\varphi}_{2j}^{(\alpha-1)}(y)\nonumber\\ &+&\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)}\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}(x)\widetilde{\varphi}_{2N}^{(\alpha-1)}(y).\nonumber \end{eqnarray} From (\ref{phijpx}) and Theorem \ref{de}, \begin{eqnarray} \varphi_{2j}^{(\alpha-1)}(x) &=&\varepsilon\:D\:\varphi_{2j}^{(\alpha-1)}(x)\nonumber\\ &=&\varepsilon\left[\varphi_{2j}^{(\alpha-1)}(x)\right]'\nonumber\\ &=&\varepsilon\left[\frac{1}{2}\sqrt{(2j+1)(2j+\alpha)}\widetilde{\varphi}_{2j+1}^{(\alpha-1)}(x)-\frac{1}{2}\sqrt{2j(2j+\alpha-1)} \widetilde{\varphi}_{2j-1}^{(\alpha-1)}(x)\right]\nonumber\\ &=&\sqrt{\left(j+\frac{1}{2}\right)\left(j+\frac{\alpha}{2}\right)}\varepsilon\widetilde{\varphi}_{2j+1}^{(\alpha-1)}(x) -\sqrt{j\left(j+\frac{\alpha-1}{2}\right)}\varepsilon\widetilde{\varphi}_{2j-1}^{(\alpha-1)}(x).\nonumber \end{eqnarray} Hence, the sum, $\sum_{j=0}^{N-1}\varepsilon\widetilde{\varphi}_{2j+1}^{(\alpha-1)}(x)\left[\varphi_{2j+1}^{(\alpha-1)}(y)\right]'$ simplifies immediately, and leads to, $$ \sum_{j=0}^{N-1}\varepsilon\widetilde{\varphi}_{2j+1}^{(\alpha-1)}(x)\left[\varphi_{2j+1}^{(\alpha-1)}(y)\right]' =-\sum_{j=0}^{N}\varphi_{2j}^{(\alpha-1)}(x)\widetilde{\varphi}_{2j}^{(\alpha-1)}(y) +\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)}\:\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}(x)\widetilde{\varphi}_{2N}^{(\alpha-1)}(y). $$ It follows that, \begin{eqnarray} K_{N,4}^{(2,2)}(x,y) &=&\frac{1}{2}\sum_{j=0}^{N-1}\varphi_{2j+1}^{(\alpha-1)}(x)\widetilde{\varphi}_{2j+1}^{(\alpha-1)}(y)+\frac{1}{2}\sum_{j=0}^{N}\varphi_{2j}^{(\alpha-1)}(x) \widetilde{\varphi}_{2j}^{(\alpha-1)}(y)\nonumber\\ &-&\frac{1}{2}\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)}\:\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}(x)\widetilde{\varphi}_{2N}^{(\alpha-1)}(y)\nonumber\\ &=&\frac{1}{2}\sum_{j=0}^{2N}\varphi_{j}^{(\alpha-1)}(x)\widetilde{\varphi}_{j}^{(\alpha-1)}(y)-\frac{1}{2}\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)} \:\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}(x)\widetilde{\varphi}_{2N}^{(\alpha-1)}(y)\nonumber\\ &=&\frac{1}{2}S_{N}(x,y)-\frac{1}{2}\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)} \:\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}(x)\widetilde{\varphi}_{2N}^{(\alpha-1)}(y),\nonumber \end{eqnarray} where $$ S_{N}(x,y):=\sum_{j=0}^{2N}\varphi_{j}^{(\alpha-1)}(x)\widetilde{\varphi}_{j}^{(\alpha-1)}(y) =-\sqrt{(2N+1)(2N+\alpha)}\:\frac{\varphi_{2N+1}^{(\alpha-1)}(x)\widetilde{\varphi}_{2N}^{(\alpha-1)}(y) -\widetilde{\varphi}_{2N+1}^{(\alpha-1)}(y)\varphi_{2N}^{(\alpha-1)}(x)}{x-y}. $$ Here we used the Christoffel-Darboux formula in the last equality.
By Theorem \ref{gn4s}, we have the following theorem. \begin{theorem} \begin{eqnarray} \left[G_{N}^{(4)}(f)\right]^{2} &=&\det\Bigg(I+S_{N}f-\frac{1}{2}S_{N}\varepsilon f'-\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)} \left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\widetilde{\varphi}_{2N}^{(\alpha-1)}f\right)\nonumber\\ &-&\frac{1}{2}\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)} \left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\right)f'\Bigg).\nonumber \end{eqnarray} \end{theorem}
\subsection{Large $N$ behavior of the LSE moment generating function} Now consider the scaling limit of $\left[G_{N}^{(4)}(f)\right]^{2}$, write $$ \left[G_{N}^{(4)}(f)\right]^{2}=:\det(I+T), $$ where \begin{eqnarray} T:&=&S_{N}f-\frac{1}{2}S_{N}\varepsilon f'-\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)} \left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\widetilde{\varphi}_{2N}^{(\alpha-1)}f\right)\nonumber\\ &-&\frac{1}{2}\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)} \left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\right)f'.\nonumber \end{eqnarray}
\begin{theorem} $$ \lim_{N\rightarrow\infty}\frac{y}{4N}S_{N}\left(\frac{x^{2}}{8N},\frac{y^{2}}{8N}\right)=B^{(\alpha-1)}(x,y), $$
$$ \lim_{N\rightarrow\infty}\frac{x}{4N}S_{N}\left(\frac{x^{2}}{8N},\frac{x^{2}}{8N}\right)=B^{(\alpha-1)}(x,x),\label{ba4xx} $$ where $B^{(\alpha-1)}(x,y)$ and $B^{(\alpha-1)}(x,x)$ are given by (\ref{bxy}) and (\ref{bxx}) with $\alpha$ replaced by $\alpha-1$. \end{theorem}
\begin{theorem} $$ \widetilde{\varphi}_{2N}^{(\alpha-1)}\left(\frac{x^{2}}{8N}\right)= 2\:(2N)^{\frac{1}{2}}\:\frac{J_{\alpha-1}(x)}{x}+O\left(N^{-\frac{3}{2}}\right),\;\;N\rightarrow\infty, $$
$$ \varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\left(\frac{x^{2}}{8N}\right)= (2N)^{-\frac{1}{2}}\left[\int_{0}^{x}J_{\alpha-1}(y)dy-1\right]+O\left(N^{-\frac{5}{2}}\right), \;\;N\rightarrow\infty, $$
$$ \varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\left(\frac{x^{2}}{8N}\right)= (2N)^{-\frac{1}{2}}\int_{0}^{x}J_{\alpha-1}(y)dy+O\left(N^{-\frac{5}{2}}\right), \;\;N\rightarrow\infty.\label{lse} $$ \end{theorem}
\begin{proof} Recall the asymptotic formula of the Laguerre polynomials \cite{Szego}, $$ L_{N}^{(\alpha)}(x)x^{\frac{\alpha}{2}}\mathrm{e}^{-\frac{x}{2}}=\frac{\Gamma(N+\alpha+1)}{\Gamma(N+1)}\left(N+\frac{\alpha+1}{2}\right)^{-\frac{\alpha}{2}} J_{\alpha}\left(\sqrt{(4N+2\alpha+2)x}\right)+x^{\frac{5}{4}}O\left(N^{\frac{\alpha}{2}-\frac{3}{4}}\right),\;\;N\rightarrow\infty. $$ We find, \begin{eqnarray} \widetilde{\varphi}_{2N}^{(\alpha-1)}\left(\frac{x^{2}}{8N}\right) &=&\sqrt{\frac{\Gamma(2N+1)}{\Gamma(2N+\alpha)}}L_{2N}^{(\alpha-1)}\left(\frac{x^{2}}{8N}\right)\left(\frac{x^{2}}{8N}\right)^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{x^{2}}{16N}} \nonumber\\ &=&\sqrt{\frac{\Gamma(2N+1)}{\Gamma(2N+\alpha)}}\cdot\left(\frac{x^{2}}{8N}\right)^{-\frac{1}{2}}\left[\frac{\Gamma(2N+\alpha)}{\Gamma(2N+1)} \left(2N+\frac{\alpha}{2}\right)^{-\frac{\alpha-1}{2}}J_{\alpha-1}(x)+O\left(N^{\frac{\alpha}{2}-\frac{5}{2}}\right)\right]\nonumber\\ &=&2\:(2N)^{\frac{1}{2}}\:\frac{J_{\alpha-1}(x)}{x}+O\left(N^{-\frac{3}{2}}\right),\;\;N\rightarrow\infty,\nonumber \end{eqnarray} where we have used the formula, $$ \frac{\Gamma(n+a)}{\Gamma(n+b)}= n^{a-b}\left[1+O\left(n^{-1}\right)\right],\;\;n\rightarrow\infty. $$ Proceeding to $\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}(x)$, we have, \begin{eqnarray} &&\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}(x)\nonumber\\ &=&\frac{1}{2}\left[\int_{0}^{x}\widetilde{\varphi}_{2N}^{(\alpha-1)}(y)dy-\int_{x}^{\infty}\widetilde{\varphi}_{2N}^{(\alpha-1)}(y)dy\right]\nonumber\\ &=&\frac{1}{2}\sqrt{\frac{\Gamma(2N+1)}{\Gamma(2N+\alpha)}}\left[\int_{0}^{x}L_{2N}^{(\alpha-1)}(y)y^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{y}{2}}dy -\int_{x}^{\infty}L_{2N}^{(\alpha-1)}(y)y^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{y}{2}}dy\right]\nonumber\\ &=&\frac{1}{2}\sqrt{\frac{\Gamma(2N+1)}{\Gamma(2N+\alpha)}}\left[2\int_{0}^{x}L_{2N}^{(\alpha-1)}(y)y^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{y}{2}}dy -\int_{0}^{\infty}L_{2N}^{(\alpha-1)}(y)y^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{y}{2}}dy\right]\nonumber\\ &=&\frac{1}{2}\sqrt{\frac{\Gamma(2N+1)}{\Gamma(2N+\alpha)}}\left[2\int_{0}^{x}L_{2N}^{(\alpha-1)}(y)y^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{y}{2}}dy -\frac{2^{\frac{\alpha}{2}}\Gamma\left(N+\frac{\alpha}{2}\right)}{\Gamma(N+1)}\right],\nonumber \end{eqnarray} where we have used the fact that $$ \int_{0}^{\infty}L_{2N}^{(\alpha-1)}(y)y^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{y}{2}}dy=\frac{2^{\frac{\alpha}{2}}\Gamma\left(N+\frac{\alpha}{2}\right)}{\Gamma(N+1)}. $$ Continuing, \begin{eqnarray} &&\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\left(\frac{x^{2}}{8N}\right)\nonumber\\ &=&\frac{1}{2}\sqrt{\frac{\Gamma(2N+1)}{\Gamma(2N+\alpha)}}\left[\frac{1}{2N}\int_{0}^{x}L_{2N}^{(\alpha-1)}\left(\frac{y^{2}}{8N}\right) \left(\frac{y^{2}}{8N}\right)^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{y^{2}}{16N}}ydy -\frac{2^{\frac{\alpha}{2}}\Gamma\left(N+\frac{\alpha}{2}\right)}{\Gamma(N+1)}\right]\nonumber\\ &=&(2N)^{-\frac{1}{2}}\left(2N+\frac{\alpha}{2}\right)^{-\frac{\alpha-1}{2}}\sqrt{\frac{\Gamma(2N+\alpha)}{\Gamma(2N+1)}} \int_{0}^{x}J_{\alpha-1}(y)dy-2^{\frac{\alpha}{2}-1}\sqrt{\frac{\Gamma(2N+1)}{\Gamma(2N+\alpha)}}\cdot\frac{\Gamma\left(N+\frac{\alpha}{2}\right)}{\Gamma(N+1)} +O\left(N^{-\frac{5}{2}}\right)\nonumber\\ &=&(2N)^{-\frac{1}{2}}\left[\int_{0}^{x}J_{\alpha-1}(y)dy-1\right]+O\left(N^{-\frac{5}{2}}\right),\;\;N\rightarrow\infty.\nonumber \end{eqnarray} Similarly, \begin{eqnarray} &&\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}(x)\nonumber\\ &=&\frac{1}{2}\left[\int_{0}^{x}\widetilde{\varphi}_{2N+1}^{(\alpha-1)}(y)dy-\int_{x}^{\infty}\widetilde{\varphi}_{2N+1}^{(\alpha-1)}(y)dy\right]\nonumber\\ &=&\frac{1}{2}\sqrt{\frac{\Gamma(2N+2)}{\Gamma(2N+\alpha+1)}}\left[\int_{0}^{x}L_{2N+1}^{(\alpha-1)}(y)y^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{y}{2}}dy -\int_{x}^{\infty}L_{2N+1}^{(\alpha-1)}(y)y^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{y}{2}}dy\right]\nonumber\\ &=&\frac{1}{2}\sqrt{\frac{\Gamma(2N+2)}{\Gamma(2N+\alpha+1)}}\left[2\int_{0}^{x}L_{2N+1}^{(\alpha-1)}(y)y^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{y}{2}}dy -\int_{0}^{\infty}L_{2N+1}^{(\alpha-1)}(y)y^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{y}{2}}dy\right]\nonumber\\ &=&\sqrt{\frac{\Gamma(2N+2)}{\Gamma(2N+\alpha+1)}}\int_{0}^{x}L_{2N+1}^{(\alpha-1)}(y)y^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{y}{2}}dy,\nonumber \end{eqnarray} where we have used the fact that $$ \int_{0}^{\infty}L_{2N+1}^{(\alpha-1)}(y)y^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{y}{2}}dy=0. $$ It follows that \begin{eqnarray} &&\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\left(\frac{x^{2}}{8N}\right)\nonumber\\ &=&\sqrt{\frac{\Gamma(2N+2)}{\Gamma(2N+\alpha+1)}}\int_{0}^{\frac{x^{2}}{8N}}L_{2N+1}^{(\alpha-1)}(y)y^{\frac{\alpha}{2}-1}\mathrm{e}^{-\frac{y}{2}}dy\nonumber\\ &=&\frac{1}{4N}\sqrt{\frac{\Gamma(2N+2)}{\Gamma(2N+\alpha+1)}}\int_{0}^{x}L_{2N+1}^{(\alpha-1)}\left(\frac{y^{2}}{8N}\right)\left(\frac{y^{2}}{8N}\right)^{\frac{\alpha}{2}-1} \mathrm{e}^{-\frac{y^{2}}{16N}}ydy\nonumber\\ &=&(2N)^{-\frac{1}{2}}\left(2N+1+\frac{\alpha}{2}\right)^{-\frac{\alpha-1}{2}}\sqrt{\frac{\Gamma(2N+\alpha+1)}{\Gamma(2N+2)}}\int_{0}^{x}J_{\alpha-1}(y)dy +O\left(N^{-\frac{5}{2}}\right)\nonumber\\ &=&(2N)^{-\frac{1}{2}}\int_{0}^{x}J_{\alpha-1}(y)dy+O\left(N^{-\frac{5}{2}}\right),\;\;N\rightarrow\infty.\nonumber \end{eqnarray} \end{proof}
We now use Theorem \ref{ba4xx} and Theorem \ref{lse} to compute $\mathrm{Tr}\:T$ and $\mathrm{Tr}\:T^{2}$ as $N\rightarrow\infty$. In the computations below, we replace $f(x)$ by $f\left(\sqrt{8Nx}\right)$ and $f'(x)$ by $$ f'\left(\sqrt{8Nx}\right)=\frac{d}{\sqrt{8N}d\sqrt{x}}f\left(\sqrt{8Nx}\right). $$
Consider $\mathrm{Tr}\:T$, which reads, \begin{eqnarray} \mathrm{Tr}\:T&=&\mathrm{Tr}\:S_{N}f-\mathrm{Tr}\:\frac{1}{2}S_{N}\varepsilon f'-\mathrm{Tr}\:\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)} \left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\widetilde{\varphi}_{2N}^{(\alpha-1)}f\right)\nonumber\\ &-&\mathrm{Tr}\:\frac{1}{2}\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)} \left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\right)f'.\nonumber \end{eqnarray} So we compute $\mathrm{Tr}\:T$ by calculating the four terms in the right side. The first term, \begin{eqnarray} \mathrm{Tr}\:S_{N}f &=&\int_{0}^{\infty}S_{N}(x,x)f\left(\sqrt{8Nx}\right)dx\nonumber\\ &=&\int_{0}^{\infty}\frac{x}{4N}S_{N}\left(\frac{x^{2}}{8N},\frac{x^{2}}{8N}\right)f(x)dx\nonumber\\ &\rightarrow&\int_{0}^{\infty}B^{(\alpha-1)}(x,x)f(x)dx,\;\;N\rightarrow\infty.\nonumber \end{eqnarray}
The second term, \begin{eqnarray} \mathrm{Tr}\:\frac{1}{2}S_{N}\varepsilon f' &=&\frac{1}{2}\int_{0}^{\infty}\int_{0}^{\infty}S_{N}(x,y)\varepsilon(y,x)f'\left(\sqrt{8Nx}\right)dx dy\nonumber\\ &=&\frac{1}{4}\int_{0}^{\infty}\left[\int_{x}^{\infty}S_{N}(x,y)dy-\int_{0}^{x}S_{N}(x,y)dy\right]f'\left(\sqrt{8Nx}\right)dx.\nonumber \end{eqnarray} Let $$ u=\sqrt{8Nx},\;\;v=\sqrt{8Ny}, $$ then \begin{eqnarray} \mathrm{Tr}\:\frac{1}{2}S_{N}\varepsilon f' &=&\frac{1}{16N}\int_{0}^{\infty}\left[\int_{u}^{\infty}\frac{1}{4N}S_{N}\left(\frac{u^{2}}{8N},\frac{v^{2}}{8N}\right)vdv -\int_{0}^{u}\frac{1}{4N}S_{N}\left(\frac{u^{2}}{8N},\frac{v^{2}}{8N}\right)vdv\right]u\:f'(u)du\nonumber\\ &\rightarrow&\frac{1}{16N}\int_{0}^{\infty}\left[\int_{u}^{\infty}B^{(\alpha-1)}(u,v)dv -\int_{0}^{u}B^{(\alpha-1)}(u,v)dv\right]u\:f'(u)du,\;\;N\rightarrow\infty.\nonumber \end{eqnarray}
The third term, \begin{eqnarray} &&\mathrm{Tr}\:\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)} \left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\widetilde{\varphi}_{2N}^{(\alpha-1)}f\right)\nonumber\\ &=&\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)} \int_{0}^{\infty}\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}(x)\widetilde{\varphi}_{2N}^{(\alpha-1)}(x)f\left(\sqrt{8Nx}\right)dx\nonumber\\ &=&\frac{\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)}}{4N} \int_{0}^{\infty}\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\left(\frac{x^{2}}{8N}\right)\widetilde{\varphi}_{2N}^{(\alpha-1)}\left(\frac{x^{2}}{8N}\right)x f(x)dx\nonumber\\ &=&\frac{1}{2}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha-1}(y)dy\right]J_{\alpha-1}(x)f(x)dx+O\left(N^{-2}\right),\;\;N\rightarrow\infty.\nonumber \end{eqnarray}
The fourth term, \begin{eqnarray} &&\mathrm{Tr}\:\frac{1}{2}\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)} \left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\right)f'\nonumber\\ &=&\frac{1}{2}\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)} \int_{0}^{\infty}\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}(x)\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}(x)f'\left(\sqrt{8Nx}\right)dx\nonumber\\ &=&\frac{\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)}}{8N} \int_{0}^{\infty}\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\left(\frac{x^{2}}{8N}\right)\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\left(\frac{x^{2}}{8N}\right) x\:f'(x)dx\nonumber\\ &=&\frac{1}{16N}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha-1}(y)dy\right]\left[\int_{0}^{x}J_{\alpha-1}(y)dy-1\right]x\:f'(x)dx+O\left(N^{-3}\right),\;\;N\rightarrow\infty.\nonumber \end{eqnarray}
Therefore, \begin{eqnarray} \mathrm{Tr}\:T &=&\int_{0}^{\infty}B^{(\alpha-1)}(x,x)f(x)dx-\frac{1}{2}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha-1}(y)dy\right]J_{\alpha-1}(x)f(x)dx\nonumber\\ &-&\frac{1}{16N}\int_{0}^{\infty}\left[\int_{x}^{\infty}B^{(\alpha-1)}(x,y)dy-\int_{0}^{x}B^{(\alpha-1)}(x,y)dy\right]x\:f'(x)dx\nonumber\\ &-&\frac{1}{16N}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha-1}(y)dy\right]\left[\int_{0}^{x}J_{\alpha-1}(y)dy-1\right]x\:f'(x)dx+O\left(N^{-2}\right),\;\;N\rightarrow\infty.\nonumber \end{eqnarray}
Next, we compute $\mathrm{Tr}\:T^{2}$, where there are 10 traces, \begin{eqnarray} \mathrm{Tr}\:T^{2} &=&\mathrm{Tr}\:S_{N}f S_{N}f-\mathrm{Tr}\:S_{N}f S_{N}\varepsilon f'-\mathrm{Tr}\:2\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)} S_{N}f\left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\widetilde{\varphi}_{2N}^{(\alpha-1)}f\right)\nonumber\\ &-&\mathrm{Tr}\:\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)}S_{N}f\left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)} \otimes\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\right)f'+\mathrm{Tr}\:\frac{1}{4}S_{N}\varepsilon f'S_{N}\varepsilon f'\nonumber\\ &+&\mathrm{Tr}\:\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)} S_{N}\varepsilon f'\left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\widetilde{\varphi}_{2N}^{(\alpha-1)}f\right)\nonumber\\ &+&\mathrm{Tr}\:\frac{1}{2}\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)} S_{N}\varepsilon f'\left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\right)f'\nonumber\\ &+&\mathrm{Tr}\:\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right) \left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\widetilde{\varphi}_{2N}^{(\alpha-1)}f\right) \left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\widetilde{\varphi}_{2N}^{(\alpha-1)}f\right)\nonumber\\ &+&\mathrm{Tr}\:\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right) \left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\widetilde{\varphi}_{2N}^{(\alpha-1)}f\right) \left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\right)f'\nonumber\\ &+&\mathrm{Tr}\:\frac{1}{4}\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right) \left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\right)f' \left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\right)f'.\nonumber \end{eqnarray} In the following, we need to calculate the traces on the right side term by term. The first term, \begin{eqnarray} \mathrm{Tr}\:S_{N}f S_{N}f &=&\int_{0}^{\infty}\int_{0}^{\infty}S_{N}(x,y)f\left(\sqrt{8Ny}\right)S_{N}(y,x)f\left(\sqrt{8Nx}\right)dxdy\nonumber\\ &=&\int_{0}^{\infty}\int_{0}^{\infty}\frac{y}{4N}S_{N}\left(\frac{x^{2}}{8N},\frac{y^{2}}{8N}\right)\frac{x}{4N}S_{N}\left(\frac{y^{2}}{8N},\frac{x^{2}}{8N}\right) f(x)f(y)dxdy\nonumber\\ &\rightarrow&\int_{0}^{\infty}\int_{0}^{\infty}B^{(\alpha-1)}(x,y)B^{(\alpha-1)}(y,x)f(x)f(y)dxdy,\;\;N\rightarrow\infty.\nonumber \end{eqnarray}
The second term, \begin{eqnarray} &&\mathrm{Tr}\:S_{N}f S_{N}\varepsilon f'\nonumber\\ &=&\frac{1}{8N}\int_{0}^{\infty}\int_{0}^{\infty}\frac{v}{4N}S_{N}\left(\frac{u^{2}}{8N},\frac{v^{2}}{8N}\right) \left[\int_{u}^{\infty}\frac{w}{4N}S_{N}\left(\frac{v^{2}}{8N},\frac{w^{2}}{8N}\right)dw -\int_{0}^{u}\frac{w}{4N}S_{N}\left(\frac{v^{2}}{8N},\frac{w^{2}}{8N}\right)dw\right]\nonumber\\ &&u\:f'(u)f(v)du dv\nonumber\\ &\rightarrow&\frac{1}{8N}\int_{0}^{\infty}\int_{0}^{\infty}B^{(\alpha-1)}(u,v) \left[\int_{u}^{\infty}B^{(\alpha-1)}(v,w)dw-\int_{0}^{u}B^{(\alpha-1)}(v,w)dw\right]u\:f'(u)f(v)du dv,\;\;N\rightarrow\infty.\nonumber \end{eqnarray}
The third term, \begin{eqnarray} &&\mathrm{Tr}\:2\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)} S_{N}f\left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\widetilde{\varphi}_{2N}^{(\alpha-1)}f\right)\nonumber\\ &=&\frac{1}{2N}\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)}\int_{0}^{\infty}\int_{0}^{\infty} \frac{1}{4N}S_{N}\left(\frac{u^{2}}{8N},\frac{v^{2}}{8N}\right)v\:\widetilde{\varphi}_{2N}^{(\alpha-1)}\left(\frac{u^{2}}{8N}\right) \varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\left(\frac{v^{2}}{8N}\right)u f(u)f(v)dudv\nonumber\\ &=&\int_{0}^{\infty}\int_{0}^{\infty}B^{(\alpha-1)}(u,v)J_{\alpha-1}(u)\left[\int_{0}^{v}J_{\alpha-1}(t)dt\right]f(u)f(v)dudv+O\left(N^{-2}\right),\;\;N\rightarrow\infty.\nonumber \end{eqnarray}
The fourth term, \begin{eqnarray} &&\mathrm{Tr}\:\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)}S_{N}f\left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)} \otimes\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\right)f'\nonumber\\ &=&\frac{1}{4N}\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)}\int_{0}^{\infty}\int_{0}^{\infty}\frac{1}{4N}S_{N}\left(\frac{u^{2}}{8N},\frac{v^{2}}{8N}\right)v\:f(v) \varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\left(\frac{v^{2}}{8N}\right)\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\left(\frac{u^{2}}{8N}\right)u\:f'(u)dudv\nonumber\\ &=&\frac{1}{8N}\int_{0}^{\infty}\int_{0}^{\infty}B^{(\alpha-1)}(u,v)\left[\int_{0}^{u}J_{\alpha-1}(t)dt-1\right]\left[\int_{0}^{v}J_{\alpha-1}(t)dt\right]u\:f'(u)f(v)dudv +O\left(N^{-3}\right),\;\;N\rightarrow\infty.\nonumber \end{eqnarray}
The fifth term, \begin{eqnarray} &&\mathrm{Tr}\:\frac{1}{4}S_{N}\varepsilon f'S_{N}\varepsilon f'\nonumber\\ &=&\frac{1}{4}\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}S_{N}(x,y)\varepsilon(y,z)f'\left(\sqrt{8Nz}\right)S_{N}(z,t)\varepsilon(t,x) f'\left(\sqrt{8Nx}\right)dxdydzdt\nonumber\\ &=&\frac{1}{16}\int_{0}^{\infty}\int_{0}^{\infty}\left[\int_{z}^{\infty}S_{N}(x,y)dy-\int_{0}^{z}S_{N}(x,y)dy\right] \left[\int_{x}^{\infty}S_{N}(z,t)dt-\int_{0}^{x}S_{N}(z,t)dt\right]\nonumber\\ &&f'\left(\sqrt{8Nx}\right)f'\left(\sqrt{8Nz}\right)dxdz\nonumber\\ &=&\frac{1}{256N^{2}}\int_{0}^{\infty}\int_{0}^{\infty}\left[\int_{w}^{\infty}\frac{1}{4N}S_{N}\left(\frac{u^{2}}{8N},\frac{v^{2}}{8N}\right)vdv -\int_{0}^{w}\frac{1}{4N}S_{N}\left(\frac{u^{2}}{8N},\frac{v^{2}}{8N}\right)vdv\right]\nonumber\\ &&\left[\int_{u}^{\infty}\frac{1}{4N}S_{N}\left(\frac{w^{2}}{8N},\frac{\tau^{2}}{8N}\right)\tau d\tau -\int_{0}^{u}\frac{1}{4N}S_{N}\left(\frac{w^{2}}{8N},\frac{\tau^{2}}{8N}\right)\tau d\tau\right]u\:w\:f'(u)f'(w)dudw\nonumber\\ &=&O\left(\frac{1}{N^{2}}\right),\;\;N\rightarrow\infty.\nonumber \end{eqnarray}
The sixth term, \begin{eqnarray} &&\mathrm{Tr}\:\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)} S_{N}\varepsilon f'\left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\widetilde{\varphi}_{2N}^{(\alpha-1)}f\right)\nonumber\\ &=&\frac{1}{32N^{2}}\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)}\int_{0}^{\infty}\int_{0}^{\infty} \left[\int_{w}^{\infty}\frac{v}{4N}S_{N}\left(\frac{u^{2}}{8N},\frac{v^{2}}{8N}\right)dv-\int_{0}^{w}\frac{v}{4N}S_{N}\left(\frac{u^{2}}{8N},\frac{v^{2}}{8N}\right)dv\right] \nonumber\\ &&f'(w)\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\left(\frac{w^{2}}{8N}\right)\widetilde{\varphi}_{2N}^{(\alpha-1)}\left(\frac{u^{2}}{8N}\right)u\:w\:f(u)dudw\nonumber\\ &=&\frac{1}{16N}\int_{0}^{\infty}\int_{0}^{\infty} \left[\int_{w}^{\infty}B^{(\alpha-1)}(u,v)dv-\int_{0}^{w}B^{(\alpha-1)}(u,v)dv\right]\left[\int_{0}^{w}J_{\alpha-1}(t)dt\right]J_{\alpha-1}(u)w\:f'(w)f(u)dudw\nonumber\\ &+&O\left(N^{-3}\right),\;\;N\rightarrow\infty.\nonumber \end{eqnarray}
The seventh term, \begin{eqnarray} &&\mathrm{Tr}\:\frac{1}{2}\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)} S_{N}\varepsilon f'\left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\right)f'\nonumber\\ &=&\frac{1}{64N^{2}}\sqrt{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)}\int_{0}^{\infty}\int_{0}^{\infty}\left[\int_{w}^{\infty} \frac{v}{4N}S_{N}\left(\frac{u^{2}}{8N},\frac{v^{2}}{8N}\right)dv-\int_{0}^{w}\frac{v}{4N}S_{N}\left(\frac{u^{2}}{8N},\frac{v^{2}}{8N}\right)dv\right] \nonumber\\ &&f'(w)\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\left(\frac{w^{2}}{8N}\right)\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\left(\frac{u^{2}}{8N}\right) f'(u)uwdudw\nonumber\\ &=&O\left(\frac{1}{N^{2}}\right),\;\;N\rightarrow\infty.\nonumber \end{eqnarray}
The eighth term, \begin{eqnarray} &&\mathrm{Tr}\:\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right) \left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\widetilde{\varphi}_{2N}^{(\alpha-1)}f\right) \left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\widetilde{\varphi}_{2N}^{(\alpha-1)}f\right)\nonumber\\ &=&\frac{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)}{16N^{2}}\int_{0}^{\infty}\int_{0}^{\infty} \varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\left(\frac{u^{2}}{8N}\right)\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\left(\frac{v^{2}}{8N}\right) \widetilde{\varphi}_{2N}^{(\alpha-1)}\left(\frac{u^{2}}{8N}\right)\widetilde{\varphi}_{2N}^{(\alpha-1)}\left(\frac{v^{2}}{8N}\right) uvf(u)f(v)dudv\nonumber\\ &=&\frac{1}{4}\int_{0}^{\infty}\int_{0}^{\infty}\left[\int_{0}^{u}J_{\alpha-1}(t)dt\right]\left[\int_{0}^{v}J_{\alpha-1}(t)dt\right]J_{\alpha-1}(u)J_{\alpha-1}(v)f(u)f(v)dudv +O\left(N^{-2}\right),\;\;N\rightarrow\infty.\nonumber \end{eqnarray}
The ninth term, \begin{eqnarray} &&\mathrm{Tr}\:\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right) \left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\widetilde{\varphi}_{2N}^{(\alpha-1)}f\right) \left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\right)f'\nonumber\\ &=&\frac{\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)}{16N^{2}}\int_{0}^{\infty}\int_{0}^{\infty} \varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\left(\frac{u^{2}}{8N}\right)\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\left(\frac{v^{2}}{8N}\right) \varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\left(\frac{u^{2}}{8N}\right)\widetilde{\varphi}_{2N}^{(\alpha-1)}\left(\frac{v^{2}}{8N}\right)\nonumber\\ &&uv\: f'(u)f(v)dudv\nonumber\\ &=&\frac{1}{16N}\int_{0}^{\infty}\int_{0}^{\infty}\left[\int_{0}^{u}J_{\alpha-1}(t)dt\right]\left[\int_{0}^{v}J_{\alpha-1}(t)dt\right] \left[\int_{0}^{u}J_{\alpha-1}(t)dt-1\right]J_{\alpha-1}(v)u\:f'(u)f(v)dudv\nonumber\\ &+&O\left(N^{-3}\right),\;\;N\rightarrow\infty.\nonumber \end{eqnarray}
The tenth term, \begin{eqnarray} &&\mathrm{Tr}\:\frac{1}{4}\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right) \left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\right)f' \left(\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\otimes\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\right)f'\nonumber\\ &=&\frac{1}{64N^{2}}\left(N+\frac{1}{2}\right)\left(N+\frac{\alpha}{2}\right)\int_{0}^{\infty}\int_{0}^{\infty} \varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\left(\frac{u^{2}}{8N}\right)\varepsilon\widetilde{\varphi}_{2N+1}^{(\alpha-1)}\left(\frac{v^{2}}{8N}\right) \varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\left(\frac{u^{2}}{8N}\right)\varepsilon\widetilde{\varphi}_{2N}^{(\alpha-1)}\left(\frac{v^{2}}{8N}\right)\nonumber\\ &&uv\:f'(u)f'(v)dudv\nonumber\\ &=&O\left(\frac{1}{N^{2}}\right),\;\;N\rightarrow\infty.\nonumber \end{eqnarray}
Therefore, \begin{eqnarray} &&\mathrm{Tr}\:T^{2}\nonumber\\ &=&\int_{0}^{\infty}\int_{0}^{\infty}B^{(\alpha-1)}(x,y)B^{(\alpha-1)}(y,x)f(x)f(y)dxdy\nonumber\\ &-&\int_{0}^{\infty}\int_{0}^{\infty}B^{(\alpha-1)}(x,y)J_{\alpha-1}(x)\left[\int_{0}^{y}J_{\alpha-1}(z)dz\right]f(x)f(y)dxdy\nonumber\\ &+&\frac{1}{4}\int_{0}^{\infty}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha-1}(z)dz\right]\left[\int_{0}^{y}J_{\alpha-1}(z)dz\right]J_{\alpha-1}(x)J_{\alpha-1}(y)f(x)f(y)dxdy\nonumber\\ &-&\frac{1}{8N}\int_{0}^{\infty}\int_{0}^{\infty}B^{(\alpha-1)}(x,y)\left[\int_{x}^{\infty}B^{(\alpha-1)}(y,z)dz-\int_{0}^{x}B^{(\alpha-1)}(y,z)dz\right]x\:f'(x)f(y)dx dy\nonumber\\ &-&\frac{1}{8N}\int_{0}^{\infty}\int_{0}^{\infty}B^{(\alpha-1)}(x,y)\left[\int_{0}^{x}J_{\alpha-1}(z)dz-1\right]\left[\int_{0}^{y}J_{\alpha-1}(z)dz\right]x\:f'(x)f(y)dxdy\nonumber\\ &+&\frac{1}{16N}\int_{0}^{\infty}\int_{0}^{\infty} \left[\int_{x}^{\infty}B^{(\alpha-1)}(y,z)dz-\int_{0}^{x}B^{(\alpha-1)}(y,z)dz\right]\left[\int_{0}^{x}J_{\alpha-1}(z)dz\right]J_{\alpha-1}(y)x\:f'(x)f(y)dxdy\nonumber\\ &+&\frac{1}{16N}\int_{0}^{\infty}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha-1}(z)dz\right]\left[\int_{0}^{y}J_{\alpha-1}(z)dz\right] \left[\int_{0}^{x}J_{\alpha-1}(z)dz-1\right]J_{\alpha-1}(y)x\:f'(x)f(y)dxdy\nonumber\\ &+&O\left(N^{-2}\right),\;\;N\rightarrow\infty.\nonumber \end{eqnarray}
Now we want to see the mean and variance of the linear statistics $\sum_{j=1}^{N}F\left(\sqrt{8Nx_{j}}\right)$, so we need to obtain the coefficients of $\lambda$ and $\lambda^{2}$, firstly we know $$ f\left(\sqrt{8Nx}\right)\approx-\lambda F\left(\sqrt{8Nx}\right)+\frac{\lambda^{2}}{2}F^{2}\left(\sqrt{8Nx}\right), $$ then we replace $f$ with $-\lambda F+\frac{\lambda^{2}}{2}F^{2}$ in the expression of $\mathrm{Tr}\:T$ and $\mathrm{Tr}\:T^{2}$, similar to previous\\ discussions, denote by $\mu_{N}^{(LSE,\:\alpha)}$ and $\mathcal{V}_{N}^{(LSE,\:\alpha)}$ the mean and variance of the linear statistics $\sum_{j=1}^{N}F\left(\sqrt{8Nx_{j}}\right)$, we have the following theorem. \begin{theorem} As $N\rightarrow\infty$, \begin{eqnarray} \mu_{N}^{(LSE,\:\alpha)} &=&\frac{1}{2}\:\mu_{N}^{(LUE,\:\alpha-1)} -\frac{1}{4}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha-1}(y)dy\right]J_{\alpha-1}(x)F(x)dx\nonumber\\ &-&\frac{1}{32N}\int_{0}^{\infty}\left[\int_{x}^{\infty}B^{(\alpha-1)}(x,y)dy-\int_{0}^{x}B^{(\alpha-1)}(x,y)dy\right]x\:F'(x)dx\nonumber\\ &-&\frac{1}{32N}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha-1}(y)dy\right]\left[\int_{0}^{x}J_{\alpha-1}(y)dy-1\right]x\:F'(x)dx+O\left(N^{-2}\right),\nonumber \end{eqnarray} \begin{eqnarray} \mathcal{V}_{N}^{(LSE,\:\alpha)} &=&\frac{1}{2}\:\mathcal{V}_{N}^{(LUE,\:\alpha-1)}-\frac{1}{4}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha-1}(y)dy\right]J_{\alpha-1}(x)F^{2}(x)dx\nonumber\\ &+&\frac{1}{2}\int_{0}^{\infty}\int_{0}^{\infty}B^{(\alpha-1)}(x,y)J_{\alpha-1}(x)\left[\int_{0}^{y}J_{\alpha-1}(z)dz\right]F(x)F(y)dxdy\nonumber\\ &-&\frac{1}{8}\int_{0}^{\infty}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha-1}(z)dz\right]\left[\int_{0}^{y}J_{\alpha-1}(z)dz\right]J_{\alpha-1}(x)J_{\alpha-1}(y)F(x)F(y)dxdy\nonumber\\ &-&\frac{1}{16N}\int_{0}^{\infty}\left[\int_{x}^{\infty}B^{(\alpha-1)}(x,y)dy-\int_{0}^{x}B^{(\alpha-1)}(x,y)dy\right]x\:F(x)F'(x)dx\nonumber\\ &-&\frac{1}{16N}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha-1}(y)dy\right]\left[\int_{0}^{x}J_{\alpha-1}(y)dy-1\right]x\:F(x)F'(x)dx\nonumber\\ &+&\frac{1}{16N}\int_{0}^{\infty}\int_{0}^{\infty}B^{(\alpha-1)}(x,y)\left[\int_{x}^{\infty}B^{(\alpha-1)}(y,z)dz-\int_{0}^{x}B^{(\alpha-1)}(y,z)dz\right]x\:F'(x)F(y)dx dy\nonumber\\ &+&\frac{1}{16N}\int_{0}^{\infty}\int_{0}^{\infty}B^{(\alpha-1)}(x,y)\left[\int_{0}^{x}J_{\alpha-1}(z)dz-1\right]\left[\int_{0}^{y}J_{\alpha-1}(z)dz\right]x\:F'(x)F(y)dxdy\nonumber\\ &-&\frac{1}{32N}\int_{0}^{\infty}\int_{0}^{\infty} \left[\int_{x}^{\infty}B^{(\alpha-1)}(y,z)dz-\int_{0}^{x}B^{(\alpha-1)}(y,z)dz\right]\left[\int_{0}^{x}J_{\alpha-1}(z)dz\right]J_{\alpha-1}(y)\nonumber\\ &&x\:F'(x)F(y)dxdy\nonumber\\ &-&\frac{1}{32N}\int_{0}^{\infty}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha-1}(z)dz\right]\left[\int_{0}^{y}J_{\alpha-1}(z)dz\right] \left[\int_{0}^{x}J_{\alpha-1}(z)dz-1\right]J_{\alpha-1}(y)x\:F'(x)F(y)dxdy\nonumber\\ &+&O\left(N^{-2}\right),\nonumber \end{eqnarray} where $\mu_{N}^{(LUE,\:\alpha-1)}$ and $\mathcal{V}_{N}^{(LUE,\:\alpha-1)}$ for $N\rightarrow\infty$ are given in (\ref{luem}) and (\ref{luev}) respectively with $\alpha$ replaced by $\alpha-1$. \end{theorem}
\section{The orthogonal ensembles} \subsection{General case} For the orthogonal ensembles, $\beta=1$, the equation (\ref{gnb}), becomes, $$
G_{N}^{(1)}(f)=C_{N}^{(1)}\int_{[a,b]^n}\prod_{1\leq j<k\leq N}|x_{j}-x_{k}| \prod_{j=1}^{N}w(x_{j})\left[1+f(x_{j})\right]dx_{j}, $$ and we assume $N$ is even. Here, $$
C_{N}^{(1)}=\frac{1}{\int_{(a,b)^N}\prod_{1\leq j<k\leq N}\left|x_{j}-x_{k}\right|\prod_{j=1}^{N}w(x_{j})dx_{j}} $$ depends on $N$.
We also follow the treatment of \cite{Dieng, Tracy1998}, firstly using Theorem \ref{pf} and some computations, we find $$ \left[G_{N}^{(1)}(f)\right]^{2}=\widehat{C_{N}^{(1)}}\det\left(\int_{a}^{b}\int_{a}^{b}\varepsilon(x,y)\pi_{j}(x)\pi_{k}(y)w(x)w(y) (1+f(x))(1+f(y))dxdy\right)_{j,k=0}^{N-1}, $$ where $\widehat{C_{N}^{(1)}}$ is a constant depending on $N$ and $\pi_{j}(x)$ is an arbitrary polynomial of degree $j$. Let \begin{equation} \psi_{j}(x)=\pi_{j}(x)w(x),\label{psijx} \end{equation} it follows that $$ \left[G_{N}^{(1)}(f)\right]^{2}=\det\left(I+\left(M^{(1)}\right)^{-1}L^{(1)}\right), $$ where $$ M^{(1)}=\left(\int_{a}^{b}\psi_{j}(x)\varepsilon\psi_{k}(x)dx\right)_{j,k=0}^{N-1}, $$ $$ L^{(1)}=\left(\int_{a}^{b}\left(f(x)\psi_{j}(x)\varepsilon\psi_{k}(x)-f(x)\psi_{k}(x)\varepsilon\psi_{j}(x) -f(x)\psi_{k}(x)\varepsilon(f\psi_{j})(x)\right)dx\right)_{j,k=0}^{N-1}. $$ If $\left(M^{(1)}\right)^{-1}=(\mu_{jk})_{j,k=0}^{N-1}$, then $$ \left[G_{N}^{(1)}(f)\right]^{2}=\det(I+K_{N}^{(1)}f), $$ where $K_{N}^{(1)}$ is an integral operator $$ K_{N}^{(1)}= \begin{pmatrix} -\sum_{j,k=0}^{N-1}\mu_{jk}\psi_{j}\otimes\varepsilon\psi_{k}&\sum_{j,k=0}^{N-1}\mu_{jk}\psi_{j}\otimes\psi_{k}\\ -\sum_{j,k=0}^{N-1}\mu_{jk}\varepsilon\psi_{j}\otimes\varepsilon\psi_{k}-\varepsilon&\sum_{j,k=0}^{N-1}\mu_{jk}\varepsilon\psi_{j}\otimes\psi_{k}\end{pmatrix} =:\begin{pmatrix} K_{N,1}^{(1,1)}&K_{N,1}^{(1,2)}\\ K_{N,1}^{(2,1)}&K_{N,1}^{(2,2)} \end{pmatrix}. $$ In the next theorem, we obtain relations on $K_{N,1}^{(i,j)}, i,j=1,2$.
\begin{theorem} $$ K_{N,1}^{(1,2)}=DK_{N,1}^{(2,2)},\;\;K_{N,1}^{(2,1)}=K_{N,1}^{(2,2)}\varepsilon-\varepsilon,\;\;K_{N,1}^{(1,1)}=D K_{N,1}^{(2,2)}\varepsilon. \label{re1} $$ \end{theorem}
\begin{proof} For any integrable function $g(x)$ supported on $[a,b]$, we have \begin{eqnarray} D K_{N,1}^{(2,2)}g(x) &=&\frac{d}{dx}\int_{a}^{b}K_{N,1}^{(2,2)}(x,y)g(y)dy\nonumber\\ &=&\int_{a}^{b}\frac{\partial}{\partial x}K_{N,1}^{(2,2)}(x,y)g(y)dy\nonumber\\ &=&\int_{a}^{b}\frac{\partial}{\partial x}\sum_{j,k=0}^{N-1}\varepsilon\psi_{j}(x)\mu_{jk}\psi_{k}(y)g(y)dy\nonumber\\ &=&\int_{a}^{b}\sum_{j,k=0}^{N-1}(D\varepsilon\psi_{j}(x))\mu_{jk}\psi_{k}(y)g(y)dy\nonumber\\ &=&\int_{a}^{b}\sum_{j,k=0}^{N-1}\psi_{j}(x)\mu_{jk}\psi_{k}(y)g(y)dy\nonumber\\ &=&\int_{a}^{b}K_{N,1}^{(1,2)}(x,y)g(y)dy\nonumber\\ &=&K_{N,1}^{(1,2)}g(x),\nonumber \end{eqnarray} which implies, $$ K_{N,1}^{(1,2)}=D K_{N,1}^{(2,2)}. $$ Note that \begin{eqnarray} K_{N,1}^{(2,2)}\varepsilon &=&\sum_{j,k=0}^{N-1}\mu_{jk}\varepsilon\psi_{j}\otimes\psi_{k}\varepsilon\nonumber\\ &=&-\sum_{j,k=0}^{N-1}\mu_{jk}\varepsilon\psi_{j}\otimes\varepsilon\psi_{k} \label{ot}\\ &=&K_{N,1}^{(2,1)}+\varepsilon,\nonumber \end{eqnarray} that is, $$ K_{N,1}^{(2,1)}=K_{N,1}^{(2,2)}\varepsilon-\varepsilon. $$ Moreover, from (\ref{ot}), \begin{eqnarray} D K_{N,1}^{(2,2)}\varepsilon &=&-\sum_{j,k=0}^{N-1}\mu_{jk}(D\varepsilon\psi_{j})\otimes\varepsilon\psi_{k}\nonumber\\ &=&-\sum_{j,k=0}^{N-1}\mu_{jk}\psi_{j}\otimes\varepsilon\psi_{k}\nonumber\\ &=&K_{N,1}^{(1,1)}.\nonumber \end{eqnarray} The proof is complete. \end{proof}
The series of computations presented below takes the determinant into the form of identity plus scalar operators. From Theorem \ref{re1}, $K_{N}^{(1)}$ can be written as \begin{eqnarray} K_{N}^{(1)} &=&\begin{pmatrix} D K_{N,1}^{(2,2)}\varepsilon&D K_{N,1}^{(2,2)}\\ K_{N,1}^{(2,2)}\varepsilon-\varepsilon&K_{N,1}^{(2,2)} \end{pmatrix}\nonumber\\ &=& \begin{pmatrix} D&0\\ 0&I \end{pmatrix} \begin{pmatrix} K_{N,1}^{(2,2)}\varepsilon&K_{N,1}^{(2,2)}\\ K_{N,1}^{(2,2)}\varepsilon-\varepsilon&K_{N,1}^{(2,2)} \end{pmatrix}\nonumber\\ &=&:\widetilde{A}\widetilde{B},\nonumber \end{eqnarray} where $$ \widetilde{A}:=\begin{pmatrix} D&0\\ 0&I \end{pmatrix},\;\;\;\; \widetilde{B}:=\begin{pmatrix} K_{N,1}^{(2,2)}\varepsilon&K_{N,1}^{(2,2)}\\ K_{N,1}^{(2,2)}\varepsilon-\varepsilon&K_{N,1}^{(2,2)} \end{pmatrix}. $$ By Theorem \ref{hs}, we have $$ \left[G_{N}^{(1)}(f)\right]^{2}=\det\Big(I+\widetilde{A}\widetilde{B}f\Big)=\det\Big(I+\widetilde{B}f\widetilde{A}\Big). $$ Since \begin{eqnarray} \widetilde{B}f\widetilde{A} &=& \begin{pmatrix} K_{N,1}^{(2,2)}\varepsilon&K_{N,1}^{(2,2)}\\ K_{N,1}^{(2,2)}\varepsilon-\varepsilon&K_{N,1}^{(2,2)} \end{pmatrix} f \begin{pmatrix} D&0\\ 0&I \end{pmatrix} \nonumber\\ &=& \begin{pmatrix} K_{N,1}^{(2,2)}\varepsilon&K_{N,1}^{(2,2)}\\ K_{N,1}^{(2,2)}\varepsilon-\varepsilon&K_{N,1}^{(2,2)} \end{pmatrix} \begin{pmatrix} f D&0\\ 0&f \end{pmatrix} \nonumber\\ &=&\begin{pmatrix} K_{N,1}^{(2,2)}\varepsilon f D&K_{N,1}^{(2,2)}f\\ K_{N,1}^{(2,2)}\varepsilon f D-\varepsilon f D&K_{N,1}^{(2,2)}f \end{pmatrix},\nonumber \end{eqnarray} then $$ \left[G_{N}^{(1)}(f)\right]^{2}=\det\begin{pmatrix} I+K_{N,1}^{(2,2)}\varepsilon f D&K_{N,1}^{(2,2)}f\\ K_{N,1}^{(2,2)}\varepsilon f D-\varepsilon f D&I+K_{N,1}^{(2,2)}f \end{pmatrix}. $$
The computation below reduces the above into a determinant of scalar operators. We subtract row 1 from row 2, $$ \left[G_{N}^{(1)}(f)\right]^{2}=\det \begin{pmatrix} I+K_{N,1}^{(2,2)}\varepsilon f D&K_{N,1}^{(2,2)}f\\ -I-\varepsilon f D&I \end{pmatrix}. $$ Next, add column 2 times $I+\varepsilon f D$ to column 1, \begin{eqnarray} \left[G_{N}^{(1)}(f)\right]^{2} &=&\det \begin{pmatrix} I+K_{N,1}^{(2,2)}\varepsilon f D+K_{N,1}^{(2,2)}f(I+\varepsilon f D)&K_{N,1}^{(2,2)}f\\ 0&I \end{pmatrix}\nonumber\\ &=&\det\left(I+K_{N,1}^{(2,2)}(\varepsilon f D+f+f\varepsilon f D)\right).\nonumber \end{eqnarray} $\mathbf{Remark}.$ The above result agrees with \cite{Dieng} for the GOE case if we take $f=-\mu\:\chi_{J}$, where $\chi_{J}$ is the characteristic function of the interval $J$.
Using (\ref{fd}), Lemma \ref{de} and note that $f$ is a Schwartz function, then we have the following theorem. \begin{theorem} $$ \left[G_{N}^{(1)}(f)\right]^{2} =\det\left(I+K_{N,1}^{(2,2)}\left(f^{2}+2f\right)-K_{N,1}^{(2,2)}\varepsilon f'-K_{N,1}^{(2,2)}f\varepsilon f'\right), \label{sca1} $$ where the kernel of $K_{N,1}^{(2,2)}$ reads $$ K_{N,1}^{(2,2)}(x,y)=\sum_{j,k=0}^{N-1}\mu_{jk}\varepsilon\psi_{j}(x)\psi_{k}(y). $$ \end{theorem}
\subsection{GOE} It is convenient in this case to choose $$ w(x)=\mathrm{e}^{-\frac{x^{2}}{2}},\;\;x\in \mathbb{R}, $$ following Dieng and Tracy-Widom's discussion \cite{Dieng, Tracy1998}. We want to choose $\psi_{j}$ to make $M^{(1)}$ simplest possible. Define \begin{equation} \psi_{2n+1}(x):=\frac{d}{dx}\varphi_{2n}(x),\;\;\psi_{2n}(x):=\varphi_{2n}(x),\;\;n=0,1,2,\ldots,\label{psi3} \end{equation} where $\varphi_{j}(x)$ is given by (\ref{gue}), $$ \varphi_{j}(x)=\frac{H_{j}(x)}{c_{j}}\mathrm{e}^{-\frac{x^{2}}{2}},\;\;c_{j}=\pi^{\frac{1}{4}}2^{\frac{j}{2}}\sqrt{\Gamma(j+1)}. $$
We can check that this definition satisfies (\ref{psijx}), i.e., $\psi_{j}(x)=\pi_{j}(x)\mathrm{e}^{-\frac{x^{2}}{2}}, j=0,1,2,\ldots$, where $\pi_{j}(x)$ is a polynomial of degree $j$. The matrix $M^{(1)}:=\left(\int_{-\infty}^{\infty}\psi_{j}(x)\varepsilon\psi_{k}(x)dx\right)_{j,k=0}^{N-1}$ is computed below from (\ref{psi3}). \begin{lemma} $\mathrm{\mathbf{(Dieng, Tracy-Widom)}}$ $$ M^{(1)}=\begin{pmatrix} 0&1&0&0&\cdots&0&0\\ -1&0&0&0&\cdots&0&0\\ 0&0&0&1&\cdots&0&0\\ 0&0&-1&0&\cdots&0&0\\ \vdots&\vdots&\vdots&\vdots&&\vdots&\vdots\\ 0&0&0&0&\cdots&0&1\\ 0&0&0&0&\cdots&-1&0 \end{pmatrix}_{N\times N}. $$ \end{lemma}
It's obvious that $\left(M^{(1)}\right)^{-1}=-M^{(1)}$, so $\mu_{2j,2j+1}=-1, \mu_{2j+1,2j}=1$, and $\mu_{jk}=0$ for other cases, hence \begin{eqnarray} K_{N,1}^{(2,2)}(x,y) &=&\sum_{j,k=0}^{N-1}\mu_{jk}\varepsilon\psi_{j}(x)\psi_{k}(y)\nonumber\\ &=&-\sum_{j=0}^{\frac{N}{2}-1}\varepsilon\psi_{2j}(x)\psi_{2j+1}(y)+\sum_{j=0}^{\frac{N}{2}-1}\varepsilon\psi_{2j+1}(x)\psi_{2j}(y)\nonumber\\ &=&\sum_{j=0}^{\frac{N}{2}-1}\varphi_{2j}(x)\varphi_{2j}(y)-\sum_{j=0}^{\frac{N}{2}-1}\varepsilon\varphi_{2j}(x)\varphi_{2j}'(y).\nonumber \end{eqnarray} By (\ref{phid}), we have \begin{eqnarray} \sum_{j=0}^{\frac{N}{2}-1}\varepsilon\varphi_{2j}(x)\varphi_{2j}'(y) &=&\sum_{j=0}^{\frac{N}{2}-1}\varepsilon\varphi_{2j}(x)\left(\sqrt{j}\varphi_{2j-1}(y)-\sqrt{j+\frac{1}{2}}\varphi_{2j+1}(y)\right)\nonumber\\ &=&\sum_{j=0}^{\frac{N}{2}-1}\sqrt{j}\varepsilon\varphi_{2j}(x)\varphi_{2j-1}(y)-\sum_{j=0}^{\frac{N}{2}-1} \sqrt{j+\frac{1}{2}}\varepsilon\varphi_{2j}(x)\varphi_{2j+1}(y)\nonumber\\ &=&\sum_{j=1}^{\frac{N}{2}-1}\sqrt{j}\varepsilon\varphi_{2j}(x)\varphi_{2j-1}(y)-\sum_{j=1}^{\frac{N}{2}} \sqrt{j-\frac{1}{2}}\varepsilon\varphi_{2j-2}(x)\varphi_{2j-1}(y)\nonumber\\ &=&\sum_{j=1}^{\frac{N}{2}}\sqrt{j}\varepsilon\varphi_{2j}(x)\varphi_{2j-1}(y)-\sqrt{\frac{N}{2}}\varepsilon\varphi_{N}(x)\varphi_{N-1}(y) -\sum_{j=1}^{\frac{N}{2}}\sqrt{j-\frac{1}{2}}\varepsilon\varphi_{2j-2}(x)\varphi_{2j-1}(y)\nonumber\\ &=&-\sum_{j=1}^{\frac{N}{2}}\left(\sqrt{j-\frac{1}{2}}\varepsilon\varphi_{2j-2}(x)-\sqrt{j}\varepsilon\varphi_{2j}(x)\right)\varphi_{2j-1}(y) -\sqrt{\frac{N}{2}}\varepsilon\varphi_{N}(x)\varphi_{N-1}(y).\nonumber \end{eqnarray} Using (\ref{phid}) again, we find $$ \varphi_{2j-1}'(x)=\sqrt{j-\frac{1}{2}}\varphi_{2j-2}(x)-\sqrt{j}\varphi_{2j}(x), $$ then according to Lemma \ref{de}, $$ \varphi_{2j-1}(x)=\varepsilon\varphi_{2j-1}'(x)=\sqrt{j-\frac{1}{2}}\varepsilon\varphi_{2j-2}(x)-\sqrt{j}\varepsilon\varphi_{2j}(x). $$ Hence $$ \sum_{j=0}^{\frac{N}{2}-1}\varepsilon\varphi_{2j}(x)\varphi_{2j}'(y) =-\sum_{j=1}^{\frac{N}{2}}\varphi_{2j-1}(x)\varphi_{2j-1}(y)-\sqrt{\frac{N}{2}}\varepsilon\varphi_{N}(x)\varphi_{N-1}(y). $$ It follows that \begin{eqnarray} K_{N,1}^{(2,2)}(x,y) &=&\sum_{j=0}^{\frac{N}{2}-1}\varphi_{2j}(x)\varphi_{2j}(y)+\sum_{j=1}^{\frac{N}{2}}\varphi_{2j-1}(x)\varphi_{2j-1}(y) +\sqrt{\frac{N}{2}}\varepsilon\varphi_{N}(x)\varphi_{N-1}(y)\nonumber\\ &=&\sum_{j=0}^{N-1}\varphi_{j}(x)\varphi_{j}(y)+\sqrt{\frac{N}{2}}\varepsilon\varphi_{N}(x)\varphi_{N-1}(y)\nonumber\\ &=&S_{N}(x,y)+\sqrt{\frac{N}{2}}\varepsilon\varphi_{N}(x)\varphi_{N-1}(y),\nonumber \end{eqnarray} where $$ S_{N}(x,y):=\sum_{j=0}^{N-1}\varphi_{j}(x)\varphi_{j}(y)=\sqrt{\frac{N}{2}}\:\frac{\varphi_{N}(x)\varphi_{N-1}(y)-\varphi_{N}(y)\varphi_{N-1}(x)}{x-y}. $$ Here we have used the Christoffel-Darboux formula in the last equality. Note that\\ $S_{N}(x,y)=K_{N}^{(2)}(x,y)$.
By Theorem \ref{sca1}, we have the following theorem. \begin{theorem} \begin{eqnarray} \left[G_{N}^{(1)}(f)\right]^{2} &=&\det\bigg(I+S_{N}(f^{2}+2f)-S_{N}\varepsilon f'-S_{N}f\varepsilon f'+\sqrt{\frac{N}{2}}\varepsilon\varphi_{N}\otimes\varphi_{N-1}(f^{2}+2f)\nonumber\\ &+&\sqrt{\frac{N}{2}}(\varepsilon\varphi_{N}\otimes\varepsilon\varphi_{N-1})f'+\sqrt{\frac{N}{2}}\left(\varepsilon\varphi_{N}\otimes\varepsilon(\varphi_{N-1}f)\right) f'\bigg).\nonumber \end{eqnarray} \end{theorem}
\subsection{Large $N$ behavior of the GOE moment generating function} Now we consider the scaling limit of $\left[G_{N}^{(1)}(f)\right]^{2}$. Write $$ \left[G_{N}^{(1)}(f)\right]^{2}=:\det(I+T), $$ where \begin{eqnarray} T: &=&S_{N}(f^{2}+2f)-S_{N}\varepsilon f'-S_{N}f\varepsilon f'+\sqrt{\frac{N}{2}}\varepsilon\varphi_{N}\otimes\varphi_{N-1}(f^{2}+2f)\nonumber\\ &+&\sqrt{\frac{N}{2}}(\varepsilon\varphi_{N}\otimes\varepsilon\varphi_{N-1})f'+\sqrt{\frac{N}{2}}\left(\varepsilon\varphi_{N}\otimes\varepsilon(\varphi_{N-1}f)\right)f'.\nonumber \end{eqnarray} In the computations below, we replace $f(x)$ by $f\left(\sqrt{2N}x\right)$ and $f'(x)$ by $$ f'\left(\sqrt{2N}x\right)=\frac{d}{\sqrt{2N}dx}f\left(\sqrt{2N}x\right). $$ Similarly to the GSE case, we have the following theorems. \begin{theorem} $$ \lim_{N\rightarrow\infty}\frac{1}{\sqrt{2N}}S_{N}\left(\frac{x}{\sqrt{2N}},\frac{y}{\sqrt{2N}}\right)=\frac{\sin(x-y)}{\pi(x-y)}, $$
$$ \lim_{N\rightarrow\infty}\frac{1}{\sqrt{2N}}S_{N}\left(\frac{x}{\sqrt{2N}},\frac{x}{\sqrt{2N}}\right)=\frac{1}{\pi}.\label{sn1} $$ \end{theorem}
\begin{theorem} $$ \varphi_{N-1}\left(\frac{x}{\sqrt{2N}}\right)=-(-1)^{\frac{N}{2}}2^{\frac{1}{4}}\pi^{-\frac{1}{2}}N^{-\frac{1}{4}}\sin x+O\left(N^{-\frac{3}{4}}\right),\;\;N\rightarrow\infty, $$
$$ \varepsilon\varphi_{N}\left(\frac{x}{\sqrt{2N}}\right)=(-1)^{\frac{N}{2}}2^{-\frac{1}{4}}\pi^{-\frac{1}{2}}N^{-\frac{3}{4}}\sin x+O\left(N^{-\frac{5}{4}}\right), \;\;N\rightarrow\infty, $$
$$ \varepsilon\varphi_{N-1}\left(\frac{x}{\sqrt{2N}}\right)=-2^{-\frac{1}{4}}N^{-\frac{1}{4}}+O\left(N^{-\frac{3}{4}}\right),\;\;N\rightarrow\infty, $$
$$ \varepsilon\left(\varphi_{N-1}f\right)\left(\frac{x}{\sqrt{2N}}\right)=-(-1)^{\frac{N}{2}}2^{-\frac{5}{4}}\pi^{-\frac{1}{2}}N^{-\frac{3}{4}} \left[\int_{-\infty}^{x}\sin y\: f(y)dy-\int_{x}^{\infty}\sin y\: f(y)dy\right]+O\left(N^{-\frac{5}{4}}\right),\;\;N\rightarrow\infty.\label{goe} $$ \end{theorem}
Using Theorem \ref{sn1} and Theorem \ref{goe} to compute $\mathrm{Tr}\:T$ and $\mathrm{Tr}\:T^{2}$ as $N\rightarrow\infty$. we have \begin{eqnarray} \mathrm{Tr}\:T &=&\frac{1}{\pi}\int_{-\infty}^{\infty}\left[f^{2}(x)+2f(x)\right]dx\nonumber\\ &-&\frac{1}{2\pi\sqrt{2N}}\int_{-\infty}^{\infty}\left[\int_{x}^{\infty}\frac{\sin(x-y)}{x-y}f(y)dy-\int_{-\infty}^{x}\frac{\sin(x-y)}{x-y}f(y)dy\right]f'(x)dx\nonumber\\ &-&\frac{1}{2\pi N}\int_{-\infty}^{\infty}\sin^{2}x\left[f^{2}(x)+2f(x)\right]dx -\frac{(-1)^{\frac{N}{2}}}{2\sqrt{2\pi}N}\int_{-\infty}^{\infty}\sin x\:f'(x)dx+O\left(N^{-\frac{3}{2}}\right),\;\;N\rightarrow\infty,\nonumber \end{eqnarray} and \begin{eqnarray} \mathrm{Tr}\:T^{2} &=&\frac{1}{\pi^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\left[\frac{\sin(x-y)}{x-y}\right]^{2}\left[f^{2}(x)+2f(x)\right]\left[f^{2}(y)+2f(y)\right]dxdy\nonumber\\ &+&\frac{2}{\pi^{2}\sqrt{2N}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\sin(x-y)}{x-y}\:\mathrm{Si}(x-y)f'(x)\left[f^{2}(y)+2f(y)\right]dxdy\nonumber\\ &-&\frac{1}{\pi^{2}\sqrt{2N}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \left[\int_{x}^{\infty}\frac{\sin(y-z)}{y-z}f(z)dz-\int_{-\infty}^{x}\frac{\sin(y-z)}{y-z}f(z)dz\right]\frac{\sin(x-y)}{x-y}\nonumber\\ &&f'(x)\left[f^{2}(y)+2f(y)\right]dxdy+O\left(N^{-1}\right),\;\;N\rightarrow\infty.\nonumber \end{eqnarray}
Now we want to find the mean and variance of the linear statistics $\sum_{j=1}^{N}F\left(\sqrt{2N}x_{j}\right)$, since $$ f\left(\sqrt{2N}x\right)\approx-\lambda F\left(\sqrt{2N}x\right)+\frac{\lambda^{2}}{2}F^{2}\left(\sqrt{2N}x\right), $$ we replace $f$ with $-\lambda F+\frac{\lambda^{2}}{2}F^{2}$ in the expression of $\mathrm{Tr}\:T$ and $\mathrm{Tr}\:T^{2}$, we have \begin{eqnarray} &&\log\det\left(I+T\right)\nonumber\\ &=&-\lambda\Bigg\{\frac{2}{\pi}\int_{-\infty}^{\infty}F(x)dx -\frac{1}{\pi N}\int_{-\infty}^{\infty}\sin^{2}x\:F(x)dx -\frac{(-1)^{\frac{N}{2}}}{2\sqrt{2\pi}N}\int_{-\infty}^{\infty}\sin x\:F'(x)dx+O\left(N^{-\frac{3}{2}}\right)\Bigg\}\nonumber\\ &+&\frac{\lambda^{2}}{2}\Bigg\{\frac{4}{\pi}\int_{-\infty}^{\infty}F^{2}(x)dx -\frac{4}{\pi^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\left[\frac{\sin(x-y)}{x-y}\right]^{2}F(x)F(y)dxdy\nonumber\\ &-&\frac{1}{\pi\sqrt{2N}}\int_{-\infty}^{\infty}\left[\int_{x}^{\infty}\frac{\sin(x-y)}{x-y}F(y)dy-\int_{-\infty}^{x}\frac{\sin(x-y)}{x-y}F(y)dy\right]F'(x)dx\nonumber\\ &-&\frac{4}{\pi^{2}\sqrt{2N}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\sin(x-y)}{x-y}\:\mathrm{Si}(x-y)F'(x)F(y)dxdy+O\left(N^{-1}\right)\Bigg\}, \;\;N\rightarrow\infty.\nonumber \end{eqnarray} Denote by $\mu_{N}^{(GOE)}$ and $\mathcal{V}_{N}^{(GOE)}$ the mean and variance of the linear statistics $\sum_{j=1}^{N}F\left(\sqrt{2N}x_{j}\right)$, then we have the following theorem. \begin{theorem} As $N\rightarrow\infty$, $$ \mu_{N}^{(GOE)}=\mu_{N}^{(GUE)} -\frac{1}{2\pi N}\int_{-\infty}^{\infty}\sin^{2}x\:F(x)dx -\frac{(-1)^{\frac{N}{2}}}{4\sqrt{2\pi}N}\int_{-\infty}^{\infty}\sin x\:F'(x)dx+O\left(N^{-\frac{3}{2}}\right), $$ \begin{eqnarray} \mathcal{V}_{N}^{(GOE)} &=&2\mathcal{V}_{N}^{(GUE)} -\frac{1}{2\pi\sqrt{2N}}\int_{-\infty}^{\infty}\left[\int_{x}^{\infty}\frac{\sin(x-y)}{x-y}F(y)dy-\int_{-\infty}^{x}\frac{\sin(x-y)}{x-y}F(y)dy\right]F'(x)dx\nonumber\\ &-&\frac{2}{\pi^{2}\sqrt{2N}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\sin(x-y)}{x-y}\:\mathrm{Si}(x-y)F'(x)F(y)dxdy+O\left(N^{-1}\right),\nonumber \end{eqnarray} where $\mu_{N}^{(GUE)}$ and $\mathcal{V}_{N}^{(GUE)}$ for $N\rightarrow\infty$ are given in (\ref{guem}) and (\ref{guev}) respectively. \end{theorem}
\subsection{LOE} We now specialize results obtained in Section 4.1 to the situation where $w$ is taken to be the {\it square root} of the Laguerre weight, namely, $$ w(x)=x^{\frac{\alpha}{2}}\mathrm{e}^{-\frac{x}{2}},\;\;\alpha>-2,\;x\in \mathbb{R}^+. $$ Again we choose a special $\psi_{j}$ so that $M^{(1)}$ is as simple as possible. Let \begin{equation} \psi_{2n+1}(x):=\frac{d}{dx}\varphi_{2n}^{(\alpha+1)}(x),\;\;\psi_{2n}(x):=\widetilde{\varphi}_{2n}^{(\alpha+1)}(x),\;\;n=0,1,2,\ldots,\label{psila} \end{equation} where $\varphi_{j}^{(\alpha+1)}(x)$ and $\widetilde{\varphi}_{j}^{(\alpha+1)}(x)$ are given by \begin{equation} \varphi_{j}^{(\alpha+1)}(x):=\frac{L_{j}^{(\alpha+1)}(x)}{c_{j}^{(\alpha+1)}}x^{\frac{\alpha}{2}+1}\mathrm{e}^{-\frac{x}{2}},\;\;j=0,1,2,\ldots,\label{phijx} \end{equation} $$ \widetilde{\varphi}_{j}^{(\alpha+1)}(x):=\frac{L_{j}^{(\alpha+1)}(x)}{c_{j}^{(\alpha+1)}}x^{\frac{\alpha}{2}}\mathrm{e}^{-\frac{x}{2}},\;\;j=0,1,2,\ldots. $$ It's easy to see that $$ \int_{0}^{\infty}\varphi_{j}^{(\alpha+1)}(x)\widetilde{\varphi}_{k}^{(\alpha+1)}(x)dx=\delta_{jk},\;\;j,k=0,1,2,\ldots. $$ The next theorem shows that (\ref{psila}) satisfies (\ref{psijx}),
namely, $\psi_{j}(x)x^{-\frac{\alpha}{2}}\mathrm{e}^{\frac{x}{2}}$ is a polynomial of degree $j$. \begin{theorem} $\psi_{j}(x)x^{-\frac{\alpha}{2}}\mathrm{e}^{\frac{x}{2}},\: j=0,1,2,\ldots$ is a polynomial of degree $j$. \end{theorem}
\begin{proof} We prove this by considering two cases, $j$ even and $j$ odd. It is clear for even $j$, that $\psi_{2j}\:x^{-\alpha/2}\:{\rm e}^{x/2}$ is up to a constant multiple of $L_{2j}^{(\alpha+1)}(x).$
If $j$ is odd, then we find, \begin{eqnarray} \psi_{2n+1}(x)x^{-\frac{\alpha}{2}}\mathrm{e}^{\frac{x}{2}} &=&\left[\varphi_{2n}^{(\alpha+1)}(x)\right]'x^{-\frac{\alpha}{2}}\mathrm{e}^{\frac{x}{2}}\nonumber\\ &=&\left(\frac{1}{c_{2n}^{(\alpha+1)}}L_{2n}^{(\alpha+1)}(x)x^{\frac{\alpha}{2}+1}\mathrm{e}^{-\frac{x}{2}}\right)'x^{-\frac{\alpha}{2}}\mathrm{e}^{\frac{x}{2}} \nonumber\\ &=&\frac{1}{c_{2n}^{(\alpha+1)}}\left[x \left(L_{2n}^{(\alpha+1)}(x)\right)'+\left(\frac{\alpha}{2}+1\right)L_{2n}^{(\alpha+1)}(x)-\frac{1}{2}x L_{2n}^{(\alpha+1)}(x)\right],\;\;n=0,1,2,\ldots.\nonumber \end{eqnarray} Thus $\psi_{2n+1}(x)x^{-\alpha/2}\:\mathrm{e}^{\frac{x^{2}}{2}}$ is a polynomial of degree $2n+1$. The proof is complete. \end{proof} We use (\ref{psila}) to compute $M^{(1)}:=\left(\int_{0}^{\infty}\psi_{j}(x)\varepsilon\psi_{k}(x)dx\right)_{j,k=0}^{N-1}$, resulting in the following theorem. \begin{theorem} $$ M^{(1)}=\begin{pmatrix} 0&1&0&0&\cdots&0&0\\ -1&0&0&0&\cdots&0&0\\ 0&0&0&1&\cdots&0&0\\ 0&0&-1&0&\cdots&0&0\\ \vdots&\vdots&\vdots&\vdots&&\vdots&\vdots\\ 0&0&0&0&\cdots&0&1\\ 0&0&0&0&\cdots&-1&0 \end{pmatrix}_{N\times N}. $$ \end{theorem}
\begin{proof} Let $m_{jk}$ be the $(j,k)$-entry of $M^{(1)}$,
again separating into four cases: $(j,k)=({\rm even}, {\rm odd})$, $({\rm odd}, {\rm even})$,
$(\rm {even}, {\rm even})$ and $({\rm odd},{\rm odd})$. We find,
for the $({\rm even},{\rm odd})$ case, \begin{eqnarray} m_{2j,2k+1} &=&\int_{0}^{\infty}\psi_{2j}(x)\varepsilon\psi_{2k+1}(x)dx\nonumber\\ &=&\int_{0}^{\infty}\widetilde{\varphi}_{2j}^{(\alpha+1)}(x)\left(\varepsilon D\varphi_{2k}^{(\alpha+1)}(x)\right)dx\nonumber\\ &=&\int_{0}^{\infty}\widetilde{\varphi}_{2j}^{(\alpha+1)}(x)\varphi_{2k}^{(\alpha+1)}(x)dx\nonumber\\ &=&\delta_{jk}.\nonumber \end{eqnarray} For the $({\rm odd},{\rm even})$ case, note that $M^{(1)}$ is antisymmetric, then \begin{eqnarray} m_{2j+1,2k} &=&-m_{2k,2j+1}\nonumber\\ &=&-\delta_{jk}.\nonumber \end{eqnarray} The $({\rm even},{\rm even})$ case, \begin{eqnarray} m_{2j,2k} &=&\int_{0}^{\infty}\psi_{2j}(x)\varepsilon\psi_{2k}(x)dx\nonumber\\ &=&\int_{0}^{\infty}\widetilde{\varphi}_{2j}^{(\alpha+1)}(x)\varepsilon\widetilde{\varphi}_{2k}^{(\alpha+1)}(x)dx\nonumber\\ &=&\frac{1}{2c_{2j}^{(\alpha+1)}c_{2k}^{(\alpha+1)}}\int_{0}^{\infty}L_{2j}^{(\alpha+1)}(x)x^{\frac{\alpha}{2}}\mathrm{e}^{-\frac{x}{2}}\left(\int_{0}^{x} L_{2k}^{(\alpha+1)}(y)y^{\frac{\alpha}{2}}\mathrm{e}^{-\frac{y}{2}}dy-\int_{x}^{\infty}L_{2k}^{(\alpha+1)}(y)y^{\frac{\alpha}{2}}\mathrm{e}^{-\frac{y}{2}}dy\right)dx\nonumber\\ &=&\frac{1}{2c_{2j}^{(\alpha+1)}c_{2k}^{(\alpha+1)}}\int_{0}^{\infty}L_{2j}^{(\alpha+1)}(x)x^{\frac{\alpha}{2}}\mathrm{e}^{-\frac{x}{2}}\left(2\int_{0}^{x} L_{2k}^{(\alpha+1)}(y)y^{\frac{\alpha}{2}}\mathrm{e}^{-\frac{y}{2}}dy-\int_{0}^{\infty}L_{2k}^{(\alpha+1)}(y)y^{\frac{\alpha}{2}}\mathrm{e}^{-\frac{y}{2}}dy\right)dx\nonumber\\ &=&\frac{1}{2c_{2j}^{(\alpha+1)}c_{2k}^{(\alpha+1)}}\bigg(2\int_{0}^{\infty}L_{2j}^{(\alpha+1)}(x)x^{\frac{\alpha}{2}}\mathrm{e}^{-\frac{x}{2}}dx\int_{0}^{x} L_{2k}^{(\alpha+1)}(y)y^{\frac{\alpha}{2}}\mathrm{e}^{-\frac{y}{2}}dy\nonumber\\ &-&\int_{0}^{\infty}L_{2j}^{(\alpha+1)}(x)x^{\frac{\alpha}{2}}\mathrm{e}^{-\frac{x}{2}}dx\int_{0}^{\infty}L_{2k}^{(\alpha+1)}(y)y^{\frac{\alpha}{2}} \mathrm{e}^{-\frac{y}{2}}dy\bigg)\nonumber\\ &=&\frac{1}{2c_{2j}^{(\alpha+1)}c_{2k}^{(\alpha+1)}}\left[\frac{2^{\alpha+2}\Gamma\left(j+1+\frac{\alpha}{2}\right)\Gamma\left(k+1+\frac{\alpha}{2}\right)}{j!\:k!} -\frac{2^{\alpha+2}\Gamma\left(j+1+\frac{\alpha}{2}\right)\Gamma\left(k+1+\frac{\alpha}{2}\right)}{j!\:k!}\right]\nonumber\\ &=&0,\nonumber \end{eqnarray} where we have used the fact $$ \int_{0}^{\infty}L_{2j}^{(\alpha+1)}(x)x^{\frac{\alpha}{2}}\mathrm{e}^{-\frac{x}{2}}dx=\frac{2^{\frac{\alpha}{2}+1}\Gamma\left(j+1+\frac{\alpha}{2}\right)}{j!},\;\;j=0,1,2,\ldots. $$ Finally, the $({\rm odd},{\rm odd})$ case. If $j<k$, \begin{eqnarray} m_{2j+1,2k+1} &=&\int_{0}^{\infty}\psi_{2j+1}(x)\varepsilon\psi_{2k+1}(x)dx\nonumber\\ &=&\int_{0}^{\infty}\left[\varphi_{2j}^{(\alpha+1)}(x)\right]'\varepsilon D\varphi_{2k}^{(\alpha+1)}(x)dx\nonumber\\ &=&\int_{0}^{\infty}\left[\varphi_{2j}^{(\alpha+1)}(x)\right]'\varphi_{2k}^{(\alpha+1)}(x)dx\nonumber\\ &=&\int_{0}^{\infty}\frac{1}{c_{2j}^{(\alpha+1)}}\left[x \left(L_{2j}^{(\alpha+1)}(x)\right)'+\left(\frac{\alpha}{2}+1\right)L_{2j}^{(\alpha+1)}(x)-\frac{1}{2}x L_{2j}^{(\alpha+1)}(x)\right]x^{\frac{\alpha}{2}}\mathrm{e}^{-\frac{x}{2}}\nonumber\\ &&\frac{1}{c_{2k}^{(\alpha+1)}}L_{2k}^{(\alpha+1)}(x)x^{\frac{\alpha}{2}+1}\mathrm{e}^{-\frac{x}{2}}dx\nonumber\\ &=&\frac{1}{c_{2j}^{(\alpha+1)}c_{2k}^{(\alpha+1)}}\int_{0}^{\infty}L_{2k}^{(\alpha+1)}(x)\left[x \left(L_{2j}^{(\alpha+1)}(x)\right)'+\left(\frac{\alpha}{2} +1\right)L_{2j}^{(\alpha+1)}(x)-\frac{1}{2}x L_{2j}^{(\alpha+1)}(x)\right]x^{\alpha+1}\mathrm{e}^{-x}dx\nonumber\\ &=&0,\nonumber \end{eqnarray} since, $2j+1$, the degree of the polynomials $x \left(L_{2j}^{(\alpha+1)}(x)\right)'+\left(\frac{\alpha}{2}+1\right)L_{2j}^{(\alpha+1)}(x)-\frac{1}{2}x L_{2j}^{(\alpha+1)} (x)$, is less than $2k$.
If $j>k$, due to the fact that $M^{(1)}$ is antisymmetric, \begin{eqnarray} m_{2j+1,2k+1} &=&-m_{2k+1,2j+1}\nonumber\\ &=&0,\nonumber \end{eqnarray} hence $$ m_{2j+1,2k+1}=0,\;\;j,k=0,1,\ldots. $$ It is the desired result for $M^{(1)}$. The proof is complete. \end{proof}
We begin here a series of computations analogues to those in derivation of the LSE problem, ending up with an expression for $\left[G_N^{(1)}(f)\right]^2$ as a scalar Fredholm determinant.
It's obvious that $\left(M^{(1)}\right)^{-1}=-M^{(1)}$, so $\mu_{2j,2j+1}=-1, \mu_{2j+1,2j}=1$, and $\mu_{jk}=0$ for other cases, hence \begin{eqnarray} K_{N,1}^{(2,2)}(x,y) &=&\sum_{j,k=0}^{N-1}\mu_{jk}\varepsilon\psi_{j}(x)\psi_{k}(y)\nonumber\\ &=&-\sum_{j=0}^{\frac{N}{2}-1}\varepsilon\psi_{2j}(x)\psi_{2j+1}(y)+\sum_{j=0}^{\frac{N}{2}-1}\varepsilon\psi_{2j+1}(x)\psi_{2j}(y)\nonumber\\ &=&\sum_{j=0}^{\frac{N}{2}-1}\varphi_{2j}^{(\alpha+1)}(x)\widetilde{\varphi}_{2j}^{(\alpha+1)}(y)-\sum_{j=0}^{\frac{N}{2}-1}\varepsilon\widetilde{\varphi}_{2j}^{(\alpha+1)}(x) \left[\varphi_{2j}^{(\alpha+1)}(y)\right]'. \nonumber \end{eqnarray} Differentiating (\ref{phijx}) with respect to $x$, we find, $$ \left[\varphi_{j}^{(\alpha+1)}(x)\right]'=\frac{1}{c_{j}^{(\alpha+1)}} \left[x\left(L_{j}^{(\alpha+1)}(x)\right)'+\frac{\alpha+2-x}{2}L_{j}^{(\alpha+1)}(x)\right]x^{\frac{\alpha}{2}} \mathrm{e}^{-\frac{x}{2}}. $$ Recall that the Laguerre polynomials $L_{j}^{(\alpha+1)}$ satisfy the differentiation formulas \cite{Gradshteyn} \begin{equation} x\left(L_{j}^{(\alpha+1)}(x)\right)'=j L_{j}^{(\alpha+1)}(x)-(j+\alpha+1)L_{j-1}^{(\alpha+1)}(x),\;\;j=0,1,2,\ldots,\label{la1} \end{equation} \begin{equation} x\left(L_{j}^{(\alpha+1)}(x)\right)'=(j+1)L_{j+1}^{(\alpha+1)}(x)-(j+\alpha+2-x)L_{j}^{(\alpha+1)}(x),\;\;j=0,1,2,\ldots.\label{la2} \end{equation} The sum of (\ref{la1}) and (\ref{la2}) divided by 2, gives, $$ x\left(L_{j}^{(\alpha+1)}(x)\right)'=\frac{j+1}{2}L_{j+1}^{(\alpha+1)}(x)-\frac{j+\alpha+1}{2} L_{j-1}^{(\alpha+1)}(x)-\frac{\alpha+2-x}{2}L_{j}^{(\alpha+1)}(x), \;\;j=0,1,2,\ldots, $$ which is the same as, $$ x\left(L_j^{(\alpha+1)}(x)\right)'+\frac{\alpha+2-x}{2}L_j^{(\alpha+1)}(x)=\frac{j+1}{2} \:L_{j+1}^{(\alpha+1)}(x)-\frac{j+\alpha+1}{2}\:L_{j-1}^{(\alpha+1)}(x). $$ Hence the derivative of $\varphi_{j}^{(\alpha+1)}(x)$ becomes, \begin{eqnarray} \left[\varphi_{j}^{(\alpha+1)}(x)\right]' &=&\frac{1}{c_{j}^{(\alpha+1)}}\left[\frac{j+1}{2}L_{j+1}^{(\alpha+1)}(x)-\frac{j+\alpha+1}{2}L_{j-1}^{(\alpha+1)}(x) \right]x^{\frac{\alpha}{2}} \mathrm{e}^{-\frac{x}{2}}\nonumber\\ &=&\frac{1}{2}\sqrt{(j+1)(j+\alpha+2)}\:\widetilde{\varphi}_{j+1}^{(\alpha+1)}(x)-\frac{1}{2}\sqrt{j(j+\alpha+1)}\:\widetilde{\varphi}_{j-1}^{(\alpha+1)}(x). \label{phijp} \end{eqnarray} So replacing $j$ by $2j$, we see that, $$ \left[\varphi_{2j}^{(\alpha+1)}(y)\right]'=\sqrt{\left(j+\frac{1}{2}\right)\left(j+\frac{\alpha+2}{2}\right)}\:\widetilde{\varphi}_{2j+1}^{(\alpha+1)}(y) -\sqrt{j\left(j+\frac{\alpha+1}{2}\right)}\:\widetilde{\varphi}_{2j-1}^{(\alpha+1)}(y). $$ Continuing, we compute the summation to give, \begin{eqnarray} &&\sum_{j=0}^{\frac{N}{2}-1}\varepsilon\widetilde{\varphi}_{2j}^{(\alpha+1)}(x)\left[\varphi_{2j}^{(\alpha+1)}(y)\right]'\nonumber\\ &=&\sum_{j=0}^{\frac{N}{2}-1}\sqrt{\left(j+\frac{1}{2}\right)\left(j+\frac{\alpha+2}{2}\right)}\:\varepsilon\widetilde{\varphi}_{2j}^{(\alpha+1)}(x) \widetilde{\varphi}_{2j+1}^{(\alpha+1)}(y)\nonumber\\ &-&\sum_{j=1}^{\frac{N}{2}-1}\sqrt{j\left(j+\frac{\alpha+1}{2}\right)}\:\varepsilon\widetilde{\varphi}_{2j}^{(\alpha+1)}(x)\widetilde{\varphi}_{2j-1}^{(\alpha+1)}(y)\nonumber\\ &=&\sum_{j=1}^{\frac{N}{2}}\left[\sqrt{\left(j-\frac{1}{2}\right)\left(j+\frac{\alpha}{2}\right)}\:\varepsilon \widetilde{\varphi}_{2j-2}^{(\alpha+1)}(x)-\sqrt{j\left(j+\frac{\alpha+1}{2}\right)}\:\varepsilon\widetilde{\varphi}_{2j}^{(\alpha+1)}(x)\right] \widetilde{\varphi}_{2j-1}^{(\alpha+1)}(y)\nonumber\\ &+&\frac{1}{2}\sqrt{N(N+\alpha+1)}\:\varepsilon\widetilde{\varphi}_{N}^{(\alpha+1)}(x)\widetilde{\varphi}_{N-1}^{(\alpha+1)}(y).\nonumber \end{eqnarray} From (\ref{phijp}), and using Lemma \ref{de}, we have
\begin{eqnarray} \varphi_{2j-1}^{(\alpha+1)}(x) &=&\varepsilon\left[\varphi_{2j-1}^{(\alpha+1)}(x)\right]'\nonumber\\ &=&\sqrt{j\left(j+\frac{\alpha+1}{2}\right)}\:\varepsilon\widetilde{\varphi}_{2j}^{(\alpha+1)}(x)-\sqrt{\left(j-\frac{1}{2}\right) \left(j+\frac{\alpha}{2}\right)}\:\varepsilon\widetilde{\varphi}_{2j-2}^{(\alpha+1)}(x).\nonumber \end{eqnarray} The sum simplifies to $$ \sum_{j=0}^{\frac{N}{2}-1}\varepsilon\widetilde{\varphi}_{2j}^{(\alpha+1)}(x)\left[\varphi_{2j}^{(\alpha+1)}(y)\right]' =-\sum_{j=1}^{\frac{N}{2}}\varphi_{2j-1}^{(\alpha+1)}(x)\widetilde{\varphi}_{2j-1}^{(\alpha+1)}(y) +\frac{1}{2}\sqrt{N(N+\alpha+1)}\:\varepsilon\widetilde{\varphi}_{N}^{(\alpha+1)}(x)\widetilde{\varphi}_{N-1}^{(\alpha+1)}(y). $$ The computations above gives a compact form for $K_{N,1}^{(2,2)}(x,y):$ \begin{eqnarray} K_{N,1}^{(2,2)}(x,y) &=&\sum_{j=0}^{\frac{N}{2}-1}\varphi_{2j}^{(\alpha+1)}(x)\widetilde{\varphi}_{2j}^{(\alpha+1)}(y) +\sum_{j=1}^{\frac{N}{2}}\varphi_{2j-1}^{(\alpha+1)}(x)\widetilde{\varphi}_{2j-1}^{(\alpha+1)}(y) -\frac{1}{2}\sqrt{N(N+\alpha+1)}\:\varepsilon\widetilde{\varphi}_{N}^{(\alpha+1)}(x)\widetilde{\varphi}_{N-1}^{(\alpha+1)}(y)\nonumber\\ &=&\sum_{j=0}^{N-1}\varphi_{j}^{(\alpha+1)}(x)\widetilde{\varphi}_{j}^{(\alpha+1)}(y)-\frac{1}{2}\sqrt{N(N+\alpha+1)}\:\varepsilon \widetilde{\varphi}_{N}^{(\alpha+1)}(x)\widetilde{\varphi}_{N-1}^{(\alpha+1)}(y)\nonumber\\ &=&S_{N}(x,y)-\frac{1}{2}\sqrt{N(N+\alpha+1)}\:\varepsilon \widetilde{\varphi}_{N}^{(\alpha+1)}(x)\widetilde{\varphi}_{N-1}^{(\alpha+1)}(y),\nonumber \end{eqnarray}
where $$ S_{N}(x,y):=\sum_{j=0}^{N-1}\varphi_{j}^{(\alpha+1)}(x)\widetilde{\varphi}_{j}^{(\alpha+1)}(y) =-\sqrt{N(N+\alpha+1)}\frac{\varphi_{N}^{(\alpha+1)}(x)\widetilde{\varphi}_{N-1}^{(\alpha+1)}(y)-\widetilde{\varphi}_{N}^{(\alpha+1)}(y)\varphi_{N-1}^{(\alpha+1)}(x)}{x-y}. $$ Here we have used the Christoffel-Darboux formula in the last equality.
By Theorem \ref{sca1}, we have the following theorem. \begin{theorem} \begin{eqnarray} &&\left[G_{N}^{(1)}(f)\right]^{2}\nonumber\\ &=&\det\bigg(I+S_{N}(f^{2}+2f)-S_{N}\varepsilon f'-S_{N}f\varepsilon f' -\frac{1}{2}\sqrt{N(N+\alpha+1)}\Big[\varepsilon\widetilde{\varphi}_{N}^{(\alpha+1)}\otimes\widetilde{\varphi}_{N-1}^{(\alpha+1)} \left(f^{2}+2f\right)\Big]\nonumber\\ &-&\frac{1}{2}\sqrt{N(N+\alpha+1)}\left(\varepsilon\widetilde{\varphi}_{N}^{(\alpha+1)}\otimes\varepsilon\widetilde{\varphi}_{N-1}^{(\alpha+1)}\right)f' -\frac{1}{2}\sqrt{N(N+\alpha+1)}\left[\varepsilon\widetilde{\varphi}_{N}^{(\alpha+1)}\otimes\varepsilon\left(\widetilde{\varphi}_{N-1}^{(\alpha+1)}f\right)\right]f'\bigg).\nonumber \end{eqnarray} \end{theorem}
\subsection{Large $N$ behavior of the LOE moment generating function} Now we consider the scaling limit of $\left[G_{N}^{(1)}(f)\right]^{2}$. We write $$ \left[G_{N}^{(1)}(f)\right]^{2}=:\det(I+T), $$ where \begin{eqnarray} T: &=&S_{N}(f^{2}+2f)-S_{N}\varepsilon f'-S_{N}f\varepsilon f' -\frac{1}{2}\sqrt{N(N+\alpha+1)}\Big[\varepsilon\widetilde{\varphi}_{N}^{(\alpha+1)}\otimes\widetilde{\varphi}_{N-1}^{(\alpha+1)} \left(f^{2}+2f\right)\Big]\nonumber\\ &-&\frac{1}{2}\sqrt{N(N+\alpha+1)}\left(\varepsilon\widetilde{\varphi}_{N}^{(\alpha+1)}\otimes\varepsilon\widetilde{\varphi}_{N-1}^{(\alpha+1)}\right)f'\nonumber\\ &-&\frac{1}{2}\sqrt{N(N+\alpha+1)}\left[\varepsilon\widetilde{\varphi}_{N}^{(\alpha+1)}\otimes\varepsilon\left(\widetilde{\varphi}_{N-1}^{(\alpha+1)}f\right)\right]f'.\nonumber \end{eqnarray} In the computations below, we replace $f(x)$ by $f\left(\sqrt{4Nx}\right)$ and $f'(x)$ by $$f'\left(\sqrt{4Nx}\right)=\frac{d}{\sqrt{4N}d\sqrt{x}}f\left(\sqrt{4Nx}\right).$$ Similarly to the LSE case, we have the following two theorems. \begin{theorem} $$ \lim_{N\rightarrow\infty}\frac{y}{2N}S_{N}\left(\frac{x^{2}}{4N},\frac{y^{2}}{4N}\right)=B^{(\alpha+1)}(x,y), $$
$$ \lim_{N\rightarrow\infty}\frac{x}{2N}S_{N}\left(\frac{x^{2}}{4N},\frac{x^{2}}{4N}\right)=B^{(\alpha+1)}(x,x),\label{loe1} $$ where $B^{(\alpha+1)}(x,y)$ and $B^{(\alpha+1)}(x,x)$ are given by (\ref{bxy}) and (\ref{bxx}) with $\alpha$ replaced by $\alpha+1$. \end{theorem}
\begin{theorem} $$ \widetilde{\varphi}_{N-1}^{(\alpha+1)}\left(\frac{x^{2}}{4N}\right)= 2\: N^{\frac{1}{2}}\:\frac{J_{\alpha+1}(x)}{x}+O\left(N^{-\frac{3}{2}}\right),\;\;N\rightarrow\infty, $$
$$ \varepsilon\widetilde{\varphi}_{N}^{(\alpha+1)}\left(\frac{x^{2}}{4N}\right)= N^{-\frac{1}{2}}\left[\int_{0}^{x}J_{\alpha+1}(y)dy-1\right]+O\left(N^{-\frac{5}{2}}\right), \;\;N\rightarrow\infty, $$
$$ \varepsilon\widetilde{\varphi}_{N-1}^{(\alpha+1)}\left(\frac{x^{2}}{4N}\right)= N^{-\frac{1}{2}}\int_{0}^{x}J_{\alpha+1}(y)dy+O\left(N^{-\frac{5}{2}}\right),\;\;N\rightarrow\infty, $$
$$ \varepsilon\left(\widetilde{\varphi}_{N-1}^{(\alpha+1)}f\right)\left(\frac{x^{2}}{4N}\right)= \frac{1}{2}N^{-\frac{1}{2}} \left[\int_{0}^{x}J_{\alpha+1}(y)f(y)dy-\int_{x}^{\infty}J_{\alpha+1}(y)f(y)dy\right]+O\left(N^{-\frac{5}{2}}\right),\;\;N\rightarrow\infty.\label{loe2} $$ \end{theorem}
Using Theorem \ref{loe1} and Theorem \ref{loe2} to compute $\mathrm{Tr}\:T$ and $\mathrm{Tr}\:T^{2}$ as $N\rightarrow\infty$. We obtain \begin{eqnarray} \mathrm{Tr}\:T &=&\int_{0}^{\infty}B^{(\alpha+1)}(x,x)\left[f^{2}(x)+2f(x)\right]dx\nonumber\\ &-&\frac{1}{2}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha+1}(y)dy-1\right]J_{\alpha+1}(x)\left[f^{2}(x)+2f(x)\right]dx\nonumber\\ &-&\frac{1}{4N}\int_{0}^{\infty}\left[\int_{x}^{\infty}B^{(\alpha+1)}(x,y)dy-\int_{0}^{x}B^{(\alpha+1)}(x,y)dy\right]x\:f'(x)dx\nonumber\\ &-&\frac{1}{4N}\int_{0}^{\infty}\left[\int_{x}^{\infty}B^{(\alpha+1)}(x,y)f(y)dy-\int_{0}^{x}B^{(\alpha+1)}(x,y)f(y)dy\right]x\:f'(x)dx\nonumber\\ &-&\frac{1}{4N}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha+1}(y)dy-1\right]\left[\int_{0}^{x}J_{\alpha+1}(y)dy\right]x\:f'(x)dx\nonumber\\ &-&\frac{1}{8N}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha+1}(y)dy-1\right]\left[\int_{0}^{x}J_{\alpha+1}(y)f(y)dy-\int_{x}^{\infty}J_{\alpha+1}(y)f(y)dy\right]x\:f'(x)dx\nonumber\\ &+&O\left(N^{-2}\right),\;\;N\rightarrow\infty,\nonumber \end{eqnarray} and \begin{eqnarray} &&\mathrm{Tr}\:T^{2}\nonumber\\ &=&\int_{0}^{\infty}\int_{0}^{\infty}B^{(\alpha+1)}(x,y)B^{(\alpha+1)}(y,x)\left[f^{2}(x)+2f(x)\right]\left[f^{2}(y)+2f(y)\right]dxdy\nonumber\\ &-&\int_{0}^{\infty}\int_{0}^{\infty}B^{(\alpha+1)}(x,y)J_{\alpha+1}(x)\left[\int_{0}^{y}J_{\alpha+1}(z)dz-1\right]\left[f^{2}(x)+2f(x)\right] \left[f^{2}(y)+2f(y)\right]dxdy\nonumber\\ &+&\frac{1}{4}\int_{0}^{\infty}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha+1}(z)dz-1\right]\left[\int_{0}^{y}J_{\alpha+1}(z)dz-1\right] J_{\alpha+1}(x)J_{\alpha+1}(y)\nonumber\\ &&\left[f^{2}(x)+2f(x)\right]\left[f^{2}(y)+2f(y)\right]dxdy\nonumber\\ &-&\frac{1}{2N}\int_{0}^{\infty}\int_{0}^{\infty}B^{(\alpha+1)}(x,y)\left[\int_{x}^{\infty} B^{(\alpha+1)}(y,z)dz-\int_{0}^{x}B^{(\alpha+1)}(y,z)dz\right]x\:f'(x)\left[f^{2}(y)+2f(y)\right]dxdy\nonumber\\ &-&\frac{1}{2N}\int_{0}^{\infty}\int_{0}^{\infty}B^{(\alpha+1)}(x,y)\left[\int_{x}^{\infty} B^{(\alpha+1)}(y,z)f(z)dz-\int_{0}^{x}B^{(\alpha+1)}(y,z)f(z)dz\right]\nonumber\\ &&x\:f'(x)\left[f^{2}(y)+2f(y)\right]dxdy\nonumber\\ &-&\frac{1}{2N}\int_{0}^{\infty}\int_{0}^{\infty}B^{(\alpha+1)}(x,y)\left[\int_{0}^{x}J_{\alpha+1}(z)dz\right]\left[\int_{0}^{y}J_{\alpha+1}(z)dz-1\right] x\:f'(x)\left[f^{2}(y)+2f(y)\right]dxdy\nonumber\\ &-&\frac{1}{4N}\int_{0}^{\infty}\int_{0}^{\infty}B^{(\alpha+1)}(x,y)\left[\int_{0}^{x}J_{\alpha+1}(z)f(z)dz-\int_{x}^{\infty}J_{\alpha+1}(z)f(z)dz\right] \left[\int_{0}^{y}J_{\alpha+1}(z)dz-1\right]\nonumber\\ &&x\:f'(x)\left[f^{2}(y)+2f(y)\right]dxdy\nonumber\\ &+&\frac{1}{4N}\int_{0}^{\infty}\int_{0}^{\infty}\left[\int_{x}^{\infty}B^{(\alpha+1)}(y,z)dz -\int_{0}^{x}B^{(\alpha+1)}(y,z)dz\right]\left[\int_{0}^{x}J_{\alpha+1}(z)dz-1\right]J_{\alpha+1}(y)\nonumber\\ &&x\:f'(x)\left[f^{2}(y)+2f(y)\right]dxdy\nonumber\\ &+&\frac{1}{4N}\int_{0}^{\infty}\int_{0}^{\infty}\left[\int_{x}^{\infty}B^{(\alpha+1)}(y,z)f(z)dz -\int_{0}^{x}B^{(\alpha+1)}(y,z)f(z)dz\right]\left[\int_{0}^{x}J_{\alpha+1}(z)dz-1\right]J_{\alpha+1}(y)\nonumber\\ &&x\:f'(x)\left[f^{2}(y)+2f(y)\right]dxdy\nonumber\\ &+&\frac{1}{4N}\int_{0}^{\infty}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha+1}(z)dz-1\right]\left[\int_{0}^{y}J_{\alpha+1}(z)dz-1\right]\left[\int_{0}^{x}J_{\alpha+1}(z)dz\right] J_{\alpha+1}(y)\nonumber\\ &&x\:f'(x)\left[f^{2}(y)+2f(y)\right]dxdy\nonumber\\ &+&\frac{1}{8N}\int_{0}^{\infty}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha+1}(z)dz-1\right]\left[\int_{0}^{y}J_{\alpha+1}(z)dz-1\right]\nonumber\\ &&\left[\int_{0}^{x}J_{\alpha+1}(z)f(z)dz-\int_{x}^{\infty}J_{\alpha+1}(z)f(z)dz\right]J_{\alpha+1}(y)x\:f'(x)\left[f^{2}(y)+2f(y)\right]dxdy\nonumber\\ &+&O\left(N^{-2}\right),\;\;N\rightarrow\infty.\nonumber \end{eqnarray}
Now we want to see the mean and variance of the linear statistics $\sum_{j=1}^{N}F\left(\sqrt{4Nx_{j}}\right)$, so we need to obtain the coefficients of $\lambda$ and $\lambda^{2}$, firstly we know $$ f\left(\sqrt{4Nx}\right)\approx-\lambda F\left(\sqrt{4Nx}\right)+\frac{\lambda^{2}}{2}F^{2}\left(\sqrt{4Nx}\right), $$ then we replace $f$ with $-\lambda F+\frac{\lambda^{2}}{2}F^{2}$ in the expression of $\mathrm{Tr}\:T$ and $\mathrm{Tr}\:T^{2}$, similar to previous\\ discussions, denote by $\mu_{N}^{(LOE,\:\alpha)}$ and $\mathcal{V}_{N}^{(LOE,\:\alpha)}$ the mean and variance of the linear statistics $\sum_{j=1}^{N}F\left(\sqrt{4Nx_{j}}\right)$, then we have the following theorem. \begin{theorem} As $N\rightarrow\infty$, \begin{eqnarray} \mu_{N}^{(LOE,\:\alpha)} &=&\mu_{N}^{(LUE,\:\alpha+1)}-\frac{1}{2}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha+1}(y)dy-1\right]J_{\alpha+1}(x)F(x)dx\nonumber\\ &-&\frac{1}{8N}\int_{0}^{\infty}\left[\int_{x}^{\infty}B^{(\alpha+1)}(x,y)dy-\int_{0}^{x}B^{(\alpha+1)}(x,y)dy\right]x\:F'(x)dx\nonumber\\ &-&\frac{1}{8N}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha+1}(y)dy-1\right]\left[\int_{0}^{x}J_{\alpha+1}(y)dy\right]x\:F'(x)dx +O\left(N^{-2}\right),\nonumber \end{eqnarray} \begin{eqnarray} \mathcal{V}_{N}^{(LOE,\:\alpha)} &=&2\:\mathcal{V}_{N}^{(LUE,\:\alpha+1)}-\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha+1}(y)dy-1\right]J_{\alpha+1}(x)F^{2}(x)dx\nonumber\\ &+&2\int_{0}^{\infty}\int_{0}^{\infty}B^{(\alpha+1)}(x,y)J_{\alpha+1}(x)\left[\int_{0}^{y}J_{\alpha+1}(z)dz-1\right]F(x)F(y)dxdy\nonumber\\ &-&\frac{1}{2}\int_{0}^{\infty}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha+1}(z)dz-1\right]\left[\int_{0}^{y}J_{\alpha+1}(z)dz-1\right] J_{\alpha+1}(x)J_{\alpha+1}(y)F(x)F(y)dxdy\nonumber\\ &-&\frac{1}{4N}\int_{0}^{\infty}\left[\int_{x}^{\infty}B^{(\alpha+1)}(x,y)dy-\int_{0}^{x}B^{(\alpha+1)}(x,y)dy\right]x\:F(x)F'(x)dx\nonumber\\ &-&\frac{1}{4N}\int_{0}^{\infty}\left[\int_{x}^{\infty}B^{(\alpha+1)}(x,y)F(y)dy-\int_{0}^{x}B^{(\alpha+1)}(x,y)F(y)dy\right]x\:F'(x)dx\nonumber\\ &-&\frac{1}{4N}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha+1}(y)dy-1\right]\left[\int_{0}^{x}J_{\alpha+1}(y)dy\right]x\:F(x)F'(x)dx\nonumber\\ &-&\frac{1}{8N}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha+1}(y)dy-1\right]\left[\int_{0}^{x}J_{\alpha+1}(y)F(y)dy-\int_{x}^{\infty}J_{\alpha+1}(y)F(y)dy\right] x\:F'(x)dx\nonumber\\ &+&\frac{1}{2N}\int_{0}^{\infty}\int_{0}^{\infty}B^{(\alpha+1)}(x,y)\left[\int_{x}^{\infty} B^{(\alpha+1)}(y,z)dz-\int_{0}^{x}B^{(\alpha+1)}(y,z)dz\right]x\:F'(x)F(y)dxdy\nonumber\\ &+&\frac{1}{2N}\int_{0}^{\infty}\int_{0}^{\infty}B^{(\alpha+1)}(x,y)\left[\int_{0}^{x}J_{\alpha+1}(z)dz\right]\left[\int_{0}^{y}J_{\alpha+1}(z)dz-1\right] x\:F'(x)F(y)dxdy\nonumber\\ &-&\frac{1}{4N}\int_{0}^{\infty}\int_{0}^{\infty}\left[\int_{x}^{\infty}B^{(\alpha+1)}(y,z)dz -\int_{0}^{x}B^{(\alpha+1)}(y,z)dz\right]\left[\int_{0}^{x}J_{\alpha+1}(z)dz-1\right]J_{\alpha+1}(y)\nonumber\\ &&x\:F'(x)F(y)dxdy\nonumber\\ &-&\frac{1}{4N}\int_{0}^{\infty}\int_{0}^{\infty}\left[\int_{0}^{x}J_{\alpha+1}(z)dz-1\right]\left[\int_{0}^{y}J_{\alpha+1}(z)dz-1\right]\left[\int_{0}^{x}J_{\alpha+1}(z)dz\right] J_{\alpha+1}(y)\nonumber\\ &&x\:F'(x)F(y)dxdy+O\left(N^{-2}\right),\nonumber \end{eqnarray} where $\mu_{N}^{(LUE,\:\alpha+1)}$ and $\mathcal{V}_{N}^{(LUE,\:\alpha+1)}$ for $N\rightarrow\infty$ are given in (\ref{luem}) and (\ref{luev}) respectively
with $\alpha$ replaced by $\alpha+1$. \end{theorem}
\section{Conclusion} In this paper, we study the moment generating function of linear statistics of the form $\sum_{j=1}F(x_j),$ namely, the expectation value or average of $${\rm exp}\left[-\lambda\:\sum_{j=1}^{N}F(x_j)\right]$$ computed in the ``background" of the symplectic and orthogonal ensembles. We then specialize to the Gaussian case, where the background weight is the normal distribution and the Laguerre case, where the background weight is the gamma distribution.
Finally, we compute the large $N$ behavior of the moment generating, where $F(.)$ is suitably scaled and obtained the
mean and variance of the corresponding linear statistics in the four cases. The more complex situation with the Jacobi background will be studied
in the future, in which case the polynomials depend on two parameters $a$ and $b$.
\section{Acknowledgment}
The Authors would like to thank the Macau Science Foundation for generous support, awarding the grant FDCT/2012/A3.
\end{document} | arXiv |
\begin{document}
\baselineskip 13pt \pagestyle{myheadings} \thispagestyle{empty} \setcounter{page}{1}
\title[]{A note on vanishing of equivariant differentiable cohomology of proper actions and application to CR-automorphism and conformal groups} \author[]{Oliver Baues} \address{Department of Mathematics\\ University of Fribourg\\ Chemin du Mus\' ee 23\\ CH-1700 Fribourg, Switzerland} \email{[email protected]}
\author[]{Yoshinobu Kamishima} \address{Department of Mathematics, Josai University\\ Keyaki-dai 1-1, Sakado, Saitama 350-0295, Japan} \email{[email protected]} \keywords{$CR$-structure, Pseudo-Hermitian structure, Conformal structure, Lie groups, Differentiable cohomology, Equivariant cohomology} \subjclass[2010]{22E41, 53C10, 57S20, 53C55}
\begin{abstract} We establish that for any proper action of a Lie group on a manifold the associated equivariant differentiable cohomology groups with coefficients in modules of $\mathcal{C}^{\infty}$-functions vanish in all degrees except than zero. Furthermore let $G$ be a Lie group of $CR$-automorphisms of a strictly pseudo-convex $CR$-manifold $M$. We associate to $G$ a canonical class in the first differential cohomology of $G$ with coefficients in the $\mathcal{C}^{\infty}$-functions on $M$. This class is non-zero if and only if $G$ is essential in the sense that there does not exist a $CR$-compatible strictly pseudo-convex pseudo-Hermitian structure on $M$ which is preserved by $G$. We prove that a closed Lie subgroup $G$ of $CR$-automorphisms acts properly on $M$ if and only if its canonical class vanishes.
As a consequence of Schoen's theorem, it follows that for any strictly pseudo-convex $CR$-manifold $M$, there exists a compatible strictly pseudo-convex pseudo-Hermitian structure such that the CR-automorphism group for $M$ and the group of pseudo-Hermitian transformations coincide, except for two kinds of spherical $CR$-manifolds. Similar results hold for conformal Riemannian and K\"ahler manifolds. \end{abstract}
\date{January 27, 2021} \thanks{This work was partially supported by JSPS grant No 18K03284}
\maketitle
\thispagestyle{empty}
\section{Introduction}
The original motivation for this article arises from the observation that the dynamics of certain groups of automorphisms of geometric structures, for example in the cases of $CR$- and conformal manifolds, are to a large extent controlled by a one-cocycle for the \emph{differentiable cohomology} of those groups with coefficients in the $\mathcal{C}^{\infty}$-functions. In fact, as we will show, automorphism groups of such geometric structures act properly if and only if their \emph{canonical cohomology class}, which is defined by this one-cocycle, vanishes.
Behind the vanishing of the canonical class for proper actions in the above special cases there is a general vanishing principle for the equivariant cohomology groups of function spaces related to proper actions of Lie groups. This vanishing principle not only holds in degree one, but for arbitrary degree different from zero. The considerably more simple special case concerning only the first differentiable cohomology suffices for the original applications for $CR$ and conformal manifolds though.
In this article, we will work out the details of the theory of \emph{equivariant} continuous and differentiable cohomology groups associated to actions of Lie groups on manifolds and we will prove the vanishing theorem for proper actions. This material is harking back and based on original work of Van Est, Mostow, Mostow-Hochschild \cite{VanEst, Mostow, HM}.
Our exposition, as presented in Sections \ref{sec:cont_cohomology} and \ref{sec:smooth_coefficients}, will employ the language and techniques of sheaf cohomology to develop the necessary steps of the proofs in a structured and concise way.
This introduction is organized as follows. Before diving into the presentation of the theory of differentiable cohomology, we start by discussing the aforementioned applications in the context of automorphism groups of \emph{$CR$- and conformal manifolds} in Section \ref{sec1.1}. The structure of proofs and the relation with differentiable cohomology will be discussed in Section \ref{sec1.2}. Finally Section \ref{sec1.3} gives a general account on equivariant differentiable cohomology groups associated to proper actions.
\subsection{Automorphisms of $CR$ and conformal manifolds} \label{sec1.1} Let $\omega$ be a contact form on a connected smooth manifold $M$. Assume further that there exists a complex structure $J$ on the contact bundle $\ker \omega$ which is compatible with $\omega$ in the sense that the Levi form $d\omega \circ J$ is a positive definite Hermitian form. Then $$(M,\{\omega,J\})$$ is called a \emph{pseudo-Hermitian manifold}. Putting ${\mathsf{D}} = \ker\, \omega$, the pseudo-Hermitian structure $\{\omega,J\}$ induces a \emph{$CR$-structure} $\{{\mathsf{D}},J\}$ on $M$, which is then a \emph{strictly pseudo-convex} $CR$-structure.
These structures naturally associate to $M$ two important groups. Namely the group of pseudo-Hermitian transformations $$ \operatorname*{Psh}\,\left(M,\{\omega,J\}\right)
\; , $$
and the group of $CR$-automorphisms $$ \mathop{\rm Aut}\nolimits_{CR}\left(M, \{{\mathsf{D}},J\} \right) \, .
$$
The isometry group of a Riemannian manifold $M$ and also any closed subgroup of the isometry group itself are Lie groups, by the theorem of Myers and Steenrod \cite{MS}. Moreover, the isometry group acts properly on $M$. The pseudo-Hermitian group $\operatorname*{Psh}\,\left(M,\{\omega,J\}\right)$ is a Lie group, since it preserves the associated contact Riemannian metric $$ g = \omega \cdot \omega + d\omega \circ J \, . $$
The automorphism group of a strictly pseudo-convex $CR$-manifold is also a Lie group which follows by \cite{Tanaka, WE}.
\paragraph{\em Strictly pseudo-convex $CR$-manifolds}
For any strictly pseudo-convex $CR$-manifold
$(M, \{{\mathsf{D}},J\})$, there is, in general, no canonical choice in the conformal class of one-forms $$[\omega] = \, \{ \, f \cdot \omega \mid f \in \mathcal{C}^\infty(M, {\mathbb R}^{>0})\, \} $$ which are representing ${\mathsf{D}}$. Remark that the pseudo-Hermitian group $$\operatorname*{Psh}\,\left(M, \{ f \cdot \omega, J \}\right) $$
is always contained in the group $\mathop{\rm Aut}\nolimits_{CR}(M, \{{\mathsf{D}},J\})$, but it may vary considerably with the choice of $f$. Moreover, the Lie groups $\operatorname*{Psh}\,(M,\{f \cdot \omega,J\})$ act properly on $M$, whereas in special cases of certain spherical $CR$-manifolds $$\mathop{\rm Aut}\nolimits_{CR}(M, \{{\mathsf{D}},J\})$$ is too large and cannot act properly on $M$ (compare Theorem \ref{thm:schoen} below).
Now let $M$ be a strictly pseudo-convex $CR$-manifold. Assuming that $\mathop{\rm Aut}\nolimits_{CR}(M)$ is acting properly on $M$, we shall prove that there exists a pseudo-Hermitian structure on $M$ compatible with its $CR$-structure
such that
$ \operatorname*{Psh}\,(M)$ and $\mathop{\rm Aut}\nolimits_{CR}(M)$ coincide. This fact is already mentioned in the literature (see for example the remark following Conjecture 1.4 in \cite{JL}) but it is hard to locate a concise proof.
In the light of the celebrated result of R.\ Schoen \cite{SC} on the properness of $CR$-automorphism groups the following thus holds:
\begin{thm:intro}\label{thm:main_CR} Let $(M,\{{\mathsf{D}},J\})$ be a strictly pseudo-convex $CR$-manifold. Then either one of the following holds: \begin{enumerate} \item[(i)] There exists a pseudo-Hermitian structure $\{\eta,J\}$, with $D= \ker \eta $, such that \[ \mathop{\rm Aut}\nolimits_{CR}(M,\{{\mathsf{D}},J\})=\operatorname*{Psh}\,(M,\{\eta,J\}) \; .\]
\item[(ii)]
$M$ has a spherical $CR$-structure isomorphic to either the standard sphere $S^{2n+1}$ or the Heisenberg Lie group ${\mathcal{N}} = {\mathcal{N}}_{2n+1}$. \end{enumerate} In case (ii) the following holds:
\[\left(\operatorname*{Psh}\,(M),\mathop{\rm Aut}\nolimits_{CR}(M)\right)=\begin{cases} \left({\rm U}(n+1), {\rm PU}(n+1,1)\right) & (M=S^{2n+1}) \, ,\\ \left( {\mathcal{N}} \rtimes {\rm U}(n), {\mathcal{N}}\rtimes ({\rm U}(n)\times {\mathbb R}^{>0}) \right) & (M={\mathcal{N}}_{}) \, . \end{cases}\] \end{thm:intro}
For a \emph{compact} strictly pseudo-convex $CR$-manifold $M$, this result is originally due to Webster \cite{WE}, see also \cite{KA1}, and \cite[Proposition 4.4]{BGS} for a related result in the context of compact Sasaki manifolds, compare also \cite{OK}. For further background on spherical $CR$-manifolds see \cite{BS,KA3}.
\paragraph{\em Conformal Riemannian manifolds} When we replace $\omega$ by a Riemannian metric $g$ on $M$, there is a conformal analogue of Theorem \ref{thm:main_CR}. For this recall that a diffeomorphism $\alpha: M \to M$, which satisfies $$ \alpha^*g \, = \, f \cdot g \; , $$ for some positive function $f \in \mathcal{C}^{\infty}(M,{\mathbb R}^{>0})$, is called a conformal automorphism. Let $ \mathop{\rm Iso}\nolimits(M,g)$ denote the group of isometries and ${\rm Conf}(M,g)$ the group of conformal automorphisms for the metric $g$. The conformal automorphism group is a Lie group, since the $G$-structure underlying conformal geometry has its second prolongation vanishing \cite{Kob}, but in general it doesn't act properly on $M$.
Similarly as in the $CR$-case, we will prove that if ${\rm Conf}(M,g)$ acts properly on $M$ then there exists a Riemannian metric $h$ conformal to $g$ such that ${\rm Conf}(M,g)=\mathop{\rm Iso}\nolimits(M,h)$. In fact, this was first observed long time ago by D.V.\ Alekseevsky, see \cite[proof of Theorem 1]{Al}.
Together with the classification of conformal Riemannian manifolds with non-proper automorphisms groups completed by the proofs of J.\ Ferrand \cite{Ferrand} and R.\ Schoen \cite{SC}, this gives rise to the following (see also \cite{Al}):
\begin{thm:intro}\label{thm:main_conformal} Let $(M,g)$ be a Riemannian manifold. Then either one of the following holds: \begin{enumerate} \item[(i)] There exists a Riemannian metric $h$ conformal to $g$ such that \[ {\rm Conf}(M,g)=\mathop{\rm Iso}\nolimits(M,h).\] \item[(ii)]
$M$ is conformal to either the standard sphere $S^{n}$ or the Euclidean flat space ${\mathbb R}^n$. \end{enumerate} For {\rm (ii)}, it occurs $$ \left( \mathop{\rm Iso}\nolimits(M),{\rm Conf}(M) \right)= \begin{cases} \left( {\rm O}(n+1), {\rm PO}(n+1,1) \right) & (M=S^{n}) \, ,\\ \left({\mathbb R}^n\rtimes {\rm O}(n), {\mathbb R}^n\rtimes ({\rm O}(n)\times {\mathbb R}^{>0}) \right) & (M={\mathbb R}^n) \, . \end{cases} $$ \end{thm:intro}
\paragraph{\em Conformal K\"ahler manifolds} We shall also be concerned with the
holomorphic conformal deformation of \emph{K\"ahler manifolds}. Let $ (X,\{g,J\})$ be a K\"ahler manifold, where $g$ is a K\"ahler metric and $J$ denotes the complex structure on $X$. We let $\mathop{\rm Iso}\nolimits\left(X,\{g,J \}\right)$ denote the associated group of \emph{holomorphic} isometries and
$\mathrm{Conf}(X,\{g,J\})$ the holomorphic conformal group. In this context, we obtain:
\begin{thm:intro} \label{thm:main_lcK} Let $(X,\{g,J\})$ be a K\"ahler manifold, $\dim_{{\mathbb R}} X = 2n \geq 4$. Then either one of the following holds: \begin{enumerate} \item[(i)] There exists a Hermitian manifold $(X, \{h, J\})$ with the Hermitian metric $h$ conformal to the K\"ahler metric $g$ and
$$ \mathrm{Conf}(X,\{g,J\}) \, = \, \mathop{\rm Iso}\nolimits\left(X,\{h,J \}\right) \, . $$
\item[(ii)] $X$ is holomorphically isometric to ${\mathbb C}^n$. In this case, $$ \left( \mathop{\rm Iso}\nolimits(X,\{g,J\}),{\rm Conf}(X,\{g,J\}) \right) = \left({\mathbb C}^n\rtimes {\rm U}(n), {\mathbb C}^n\rtimes ({\rm U}(n)\times {\mathbb R}^{>0})\right). $$
\end{enumerate}
\end{thm:intro}
Note that Theorem \ref{thm:main_lcK} gives a strong \emph{metric} rigidity property while in Theorem \ref{thm:main_CR} and Theorem \ref{thm:main_conformal} we have rigidity only up to a conformal map.
The Hermitian metric $h$ in Theorem \ref{thm:main_lcK} is globally conformal to a K\"ahler metric. There is an important class of non-K\"ahler complex manifolds which carry Hermitian metrics \emph{locally} conformal to K\"ahler metrics, see \cite{DO,OV,Va}, for example Hopf manifolds $S^{3} \times S^{1}$ fall into this class. We apply Theorem \ref{thm:main_lcK} to prove the existence of {locally conformal K\"ahler metrics whose holomorphic isometry group is maximal among all Hermitian metrics in a given conformal class, see the discussion in Section \ref{sec:lcK} for further details.
\subsection {Structure of proofs} \label{sec1.2}
Theorem \ref{thm:main_CR}, respectively Theorem \ref{thm:main_conformal}, are based mainly on two ingredients. One corner stone are the following celebrated rigidity results which were obtained in their final form by R. Schoen \cite{SC} (see also J. Lee \cite{JL}) and J.\ Ferrand \cite{Ferrand}:
\subsubsection{Schoen's rigidity theorem and related results}
\begin{thm:intro}\label{thm:schoen} Let $(M,\{{\mathsf{D}},J\})$ be a strictly pseudo-convex $CR$-manifold of dimension $2n+1$. Then $\mathop{\rm Aut}\nolimits_{CR}(M,\{{\mathsf{D}},J\})$ acts properly on $M$ unless
$M$ has a spherical $CR$-structure isomorphic to either the standard sphere $S^{2n+1}$ or the Heisenberg Lie group ${\mathcal{N}}_{2n+1}$. \end{thm:intro}
A result analogous to Theorem \ref{thm:schoen} holds for conformal Riemannian manifolds, see \cite[Theorems 3.3, 3.4]{SC} and \cite{Ferrand}:
\begin{thm:intro}\label{thm:schoen2} Let $(M,g)$ be a Riemannian manifold. Then $\mathop{\rm Conf}\nolimits(M,g)$ acts properly on $M$, unless $M$ is conformally diffeomorphic to either the standard round sphere $S^{n}$ or Euclidean flat space ${\mathbb R}^{n}$. \end{thm:intro}
In the case of K\"ahler manifolds of complex dimension at least two, we strengthen
the above conformal rigidity to the following holomorphic metric rigidity:
\begin{thm:intro} \label{thm:schoenhol} Let $(X,\{g,J\})$ be a K\"ahler manifold, with $\dim X=2n\geq 4$. Then the holomorphic conformal group ${\mathop{\rm Conf}\nolimits}(X,\{g,J\})$ acts properly on $X$, unless $X$ is holomorphically isometric to the complex space ${\mathbb C}^n$. \end{thm:intro}
The proof of Theorem \ref{thm:schoenhol} can be found in Appendix \ref{proofA2}. It combines Schoen's conformal rigidity with classical rigidity results on conformally flat K\"ahler manifolds (see Appendix \ref{A1}).
\subsubsection{Differentiable cohomology of Lie groups associated to $CR$ and conformal actions} \label{subsec:cohomology_crc} Schoen's theorems setting the stage, the main additional ingredient used in the proofs of Theorem \ref{thm:main_CR} and Theorem \ref{thm:main_conformal} is the following vanishing result for the \emph{first differentiable cohomology group} of Lie groups:
\begin{thm:intro} \label{thm:main_h1} Let $G$ be a Lie group that acts smoothly and properly on a manifold $M$. Then $$ H^1_{d}\left(G, \mathcal{C}^{\infty}(M, {\mathbb R})\right) = \, \{0\} \, . $$ \end{thm:intro}
In Theorem \ref{thm:main_h1} the coefficients for cohomology are taken in the \emph{differentiable $G$-module} of smooth functions $\mathcal{C}^{\infty}(M, {\mathbb R})$, which has the $\mathcal{C}^{\infty}$-topology of maps and with the natural $G$-action on functions. The equivariant differentiable cohomology group
$H^1_{d}\left(G, \mathcal{C}^{\infty}(M, {\mathbb R})\right)$ is then a topological vector space, whose elements may be represented by certain smooth maps from $G$ to the locally convex vector space $\mathcal{C}^{\infty}(M, {\mathbb R})$.
\paragraph{\em Canonical cohomology class associated to $CR$-actions} To each $CR$-action of a Lie group $G$
there exists in the differentiable cohomology of $G$
a natural associated class $$ \mu_{\mathrm{CR}} = [\lambda_{\mathrm{CR}}] \; \in \, H^1_d\left(G, \mathcal{C}^\infty(M,{\mathbb R}^{>0})\right)$$ which is induced by the $CR$-structure on $M$. The class $\mu_{\mathrm{CR}}$ vanishes if and only if there exists a contact form $\eta$ compatible with the $CR$-structure, such that $G$ is contained in the group of pseudo-Hermitian transformations $\operatorname*{Psh}\,(M,\{\eta,J\})$. See Proposition \ref{pro:can_class_CR} in Section \ref{sec:cr} for further details.
Together with Theorem \ref{thm:main_h1} this reasoning reveals the following strong relationship between the above type of equivariant cohomology group and the properness of actions on $CR$- and conformal manifolds:
\begin{thm:intro} \label{thm:main_cr_and_h1} Let $G$ be a closed Lie subgroup of diffeomorphisms that preserves either a strictly pseudo-convex $CR$-structure or a conformal Riemannian structure on $M$. Then $G$ acts properly on $M$ if and only if the first differentiable cohomology group $H^1_{d}\left(G, \mathcal{C}^{\infty}(M, {\mathbb R})\right)$ vanishes. \end{thm:intro}
Conversely, taking into account that $\mathop{\rm SU}\nolimits(n+1,1) = \mathop{\rm Aut}\nolimits_{CR}(S^{2n+1})$ does not act properly on $S^{2n+1}$,
and similarly for the other geometric group actions appearing in (ii) of Theorem \ref{thm:main_CR} and Theorem \ref{thm:main_conformal}, it follows that these give examples of actions of semisimple Lie groups whose associated first equivariant differentiable cohomology groups are non-vanishing:
\begin{cor:intro} \label{cor:main_cr_and_h1} The following hold: \begin{eqnarray*} & H^1_{d}\left(\mathop{\rm SU}\nolimits(n+1,1), \mathcal{C}^{\infty}(S^{2n+1}, {\mathbb R})\right)\, \neq \; \{0\} \, , \\ & H^1_{d}\left(\mathop{\rm SO}\nolimits(n+1,1), \mathcal{C}^{\infty}(S^{n}, {\mathbb R})\right)\, \neq \; \{0\} \, . \end{eqnarray*} \end{cor:intro}
The proof of Theorem \ref{thm:main_cr_and_h1} is given in Section \ref{sec:proper_and_coho}. In fact, Theorem \ref{thm:proper_and_coho}} which can be found there gives a much stronger characterization of proper actions which is also involving vanishing of higher cohomology.
\subsection{Actions on manifolds and differentiable cohomology of Lie groups}
The methods which apply to prove Theorem \ref{thm:main_h1}, are based on foundational ideas on continuous and differentiable cohomology originally developed in the works of Mostow and Mostow-Hochschild \cite{Mostow,HM}, and also on somewhat later expositions \cite{Blanc, CW}. In particular, we use the slice theory of proper actions, integration over compact groups, and Shapiro Lemma type results for equivariant differentiable cohomology of groups with coefficients in function spaces. Finally, the language of sheaf cohomology is applied to pursue the necessary patching of local data to a global vanishing theorem, see Section \ref{sec:cont_cohomology} and Section \ref{sec:smooth_coefficients} for details.
\subsubsection{Vanishing of differentiable cohomology associated to proper actions} \label{sec1.3}
In the most general form presented in Section \ref{sec:smooth_coefficients}, the vanishing theorem
asserts, that, for any smooth and proper action of $G$ on $X$ and any differentiable $G$-module $V$, the differentiable cohomology module $$H^{*}_{d}\left(G, \mathcal{C}^{\infty}(X, V)\right)$$
is acyclic, that is, $H^{r}_{d}\left(G, \mathcal{C}^{\infty}(X, V)\right) = \{0\}$, $r \geq 1$.
With respect to the trivial $G$-module $V= {\mathbb R}$, this amounts to:
\begin{thm:intro} \label{thm:main_cohomology} Let $G$ be a Lie group which acts smoothly and properly on a manifold $X$. Then the differentiable cohomology groups of $G$ satisfy $$ H^{r}_{d} \left( G, \mathcal{C}^{\infty}(X, {\mathbb R}) \right) \, = \, \{0\}\, \text{ , for all $r \geq 1$.} $$ \end{thm:intro} The proof of Theorem \ref{thm:main_cohomology} and its further generalizations will be given in Section \ref{sec:smooth_coefficients}.
\paragraph{\em Further applications} We would like to point out that vanishing results of the above type are interesting in their own right and potentially bear many applications to the study of proper actions of Lie groups on geometric manifolds. For example, note that we do not assume that $G$ is connected, so that the theorems also apply to properly discontinuous actions and the covering theory of manifolds. If $G$ is discrete, the differentiable cohomology groups of $G$ with coefficients in a differentiable module $V$ reduce to the ordinary discrete cohomology groups of $G$ with coefficients $V$. Note further that, in case $G$ is acting properly discontinuously on $X$, Theorem \ref{thm:main_cohomology} is known due to seminal work of Conner-Raymond \cite{CR}, see also \cite[Chapter 7.5]{LR}. The equivariant cohomology theory associated with properly discontinuous actions of groups and applications of the corresponding vanishing results to the topology of manifolds are discussed and surveyed in the book \cite{LR}.
\section{Equivariant continuous cohomology} \label{sec:cont_cohomology} In this section
we start by reviewing several facts about continuous cohomology of locally compact groups (compare \cite{CW}, and also \cite[Chapter IX]{BW}).
Based on this we establish vanishing results for certain equivariant continuous cohomology groups related to proper actions on locally compact spaces. In the proof of the vanishing result we take a sheaf theoretic approach to equivariant cohomology.
\paragraph{\em Conventions} All spaces $X$ and $Y$ are assumed to be locally compact (and Hausdorff). We let $\mathcal{C}(X,Y)$ denote the space of continuous maps with the compact open topology.
\subsection{Continuous cohomology of locally compact groups} Let $G$ be a locally compact topological group and $V$ a continuous $G$-module. By definition, $V$ is a topological abelian group with a continuous action of $G$ by automorphisms. In addition we shall always assume that \emph{$V$ is a Hausdorff locally convex topological real vector space.} An isomorphism of continuous
$G$-modules is a $G$-equivariant isomorphism (linear homeomorphism) of topological vector spaces.
\subsubsection{Definition of continuous cohomology groups} \label{sect:cc_groups} Denote by $$ C^r(G;V) := \; \mathcal{C}(G^r,V)$$
the $G$-module of continuous (inhomogeneous) $r$-cochains of $G$ into $V$, which consists of continuous maps of the $r$-fold product $G^r=G\times\cdots\times G$ into $V$.
The inhomogeneous coboundary operator \begin{equation*} \partial^r: C^r(G;V) \; {\longrightarrow} \; C^{r+1}(G;V) \, , \end{equation*}satisfying $\partial^{r+1}\circ \partial^r=0$, is defined via: \begin{equation}\label{eq:partial} \begin{split} \partial^0 (v)(\alpha)&=\alpha\cdot v-v\ \, (\alpha\in G, v\in V) \, , \\
\partial^r \lambda(\alpha_1,\dots,\alpha_{r+1})&= \alpha_1 \cdot \lambda (\alpha_2,\dots,\alpha_{r+1})\\ &\ \ \ \ +\sum_{i=1}^r(-1)^i \lambda(\alpha_1,\dots, \alpha_i\alpha_{i+1},\dots,\alpha_{r+1})\\ & \ \ \ \ +(-1)^{r+1} \lambda(\alpha_1,\dots,\alpha_{r}). \end{split}\end{equation} Define the \emph{continuous cohomology groups} $$ \displaystyle H^r(G,V) := \, {\rm ker}(\partial^r)\big/\, {\rm im}(\partial^{r-1}) \, , \,r \geq 1$$ and put $$ H^0(G,V) := \, {\rm ker}(\partial^0) = V^{G} \; . $$
\subsubsection{Cohomology of compact groups and integration}
Recall that the locally convex vector space $V$ is called \emph{quasi-complete} if all closed bounded subsets of $V$ are complete. For quasi-complete $V$ integration of compactly supported continuous functions in $\mathcal{C}(G,V)$ is defined with respect to a left-invariant measure on $G$. That is, $V$ is a $G$-integrable module in the sense of \cite[\S 3]{HM} (see also \cite[2.13]{Mostow}).
\paragraph{\em Remark} For a survey concerning the existence of vector valued integrals refer to
\cite{Cass}.
\begin{pro}[\mbox{ \cite[Lemma 7]{CW}}] \label{compsta} Suppose that $G$ is compact and\/ let $V$ be a quasi-complete $G$-module. Then $$ \displaystyle H^r \! \left(G,V \right) \,= \, \{0\} \, , \; r\geq 1\; .$$
\end{pro} \begin{proof} Let $\lambda \in C^{r}\left(G; V \right)$ satisfy $\partial^{r} \lambda = 0$. With respect to a normalized finite Haar measure $d\alpha$ on $G$ define $$ \tau (\alpha_1,\dots,\alpha_{r-1}) = (-1)^{r} \int_{G} \lambda(\alpha_1,\dots,\alpha_{r-1}, \alpha) \, d \alpha \; , $$ to obtain $\tau \in C^{r-1}(G; V)$, which satisfies \[ \partial^{r-1} \tau = \lambda \; . \qedhere \]
\end{proof}
\subsubsection{Shapiro's lemma} Let $H$ be a closed subgroup of $G$ and let $W$ be an $H$-module. Put $$ {\rm Ind}_H^G \, W = \; \{f\in \mathcal{C}(G,W)\mid f(g h^{-1})=h \cdot f(g), \ g\in G, h\in H\}.$$ Then $ {\rm Ind}_H^G \, W$ turns into a continuous $G$-module, by the action $$ (\alpha \cdot f) \left(g\right) = f \left(\alpha^{-1} g \right) \; . $$
\noindent Remark that $G/H$ is paracompact \cite[Ch III \S6, Proposition 13]{bourbaki}. Suppose that the quotient map $$ G \, \to \, G \big/ H $$ admits continuous local cross sections. Then we have:
\begin{pro}[Shapiro Lemma, {cf$.$\,} \mbox{ \cite[Propositions 3,4]{CW}}] \label{Shapi}
$$ H^r \! \left(H,W\right) \; \cong \; H^r \! \left(G, \, {\rm Ind \,}_H^G \, W \right) \, , \, r\geq 0 . $$
\end{pro}
\subsection{Equivariant cohomology groups}
Let $X$ be a $G$-space and $V$ a continuous $G$-module. We consider the space of maps $$\mathcal{C}(X,V)$$ as a continuous $G$-module, where, for all $f \in \mathcal{C}(X,V)$, $\alpha \in G$, we declare the action of $G$ as \begin{equation} \label{eq:induced_action}
(\alpha \cdot f) \left(x\right) = \alpha \cdot \left(f \left(\alpha^{-1} \cdot x \right) \right) \; . \end{equation} We are interested in the properties of the continuous cohomology of $G$ with coefficients in the $G$-module $\mathcal{C}(X,V)$.
\subsubsection{Associated bundle over $\, G/H$} Let $H$ be a closed subgroup of $G$. For any $H$-space $Y$ the diagonal action of $H$ on $G \times Y$ is \begin{equation} \label{eq:diagonal}
h \cdot (g, y) = (g \, h^{-1}, \, h \cdot y) \, , \; \text{ for } h \in H. \end{equation} We then have the associated bundle with fiber $Y$ \begin{equation} \label{eq:abundle}
G \times_{H} Y \to \, G/H \, . \end{equation} The space $G \times_{H} Y$ is defined by taking the quotient of $G \times Y$ by the action \eqref{eq:diagonal}.
Observe that $G$ acts on $G \times_{H} Y$ by left-multiplication on the first factor, which turns $G \times_{H} Y$ into a $G$-space and \eqref{eq:abundle} into a $G$-map.
\paragraph{\em Local cross sections} In order that \eqref{eq:abundle} is a fiber bundle we need to require that $G \to G/H$ is a fiber bundle: \begin{lemma} \label{lem:cfiberbundles} Suppose that $G/H$ admits local cross sections. Then: \begin{enumerate} \item The map $G \to G/H$ is a fiber bundle. \item The map $G \times_{H} Y \to \, G/H$ is a fiber bundle. \item The quotient map $G \times Y \to G \times_{H} Y$ admits local cross sections. \end{enumerate} \end{lemma} \begin{proof} Let $s:U \to G$ be a local section for $\pi: G \to G/H$. Then the map $g \mapsto (g H), s(gH )^{-1} g)$, $\pi^{-1}(U) \to U \times H$ is an $H$-equivariant homeomorphism. Hence, $\pi$ is an $H$-principal bundle over $G/H$.
In the view of (1), (2) is implied by \cite[II Theorem 2.4]{Bredon_compact_trans}. In fact, for any section $s$ of $\pi$, the map $U \times Y \to G \times_{H} Y$, $(gH, y) \mapsto [s(gH), y]$ defines a local bundle chart for \eqref{eq:abundle}.
Furthermore, in such bundle chart $(gH, y) \mapsto (s(gH), y)$, $U \times Y \to G \times Y$ defines a local cross section for $G \times Y \to G \times_{H} Y $, showing (3). \end{proof}
Put $ \mathcal{C} \left(G \times Y,V\right)^{H} = \{ f: G \times Y \to V \mid f(g h^{-1}, h \cdot y) = f(g,y) \}$ for the subspace of $H$-invariant functions in $ \mathcal{C} \left(G \times Y, V\right)$.
\begin{lemma} \label{lem:cpushfw} Suppose that $G/H$ admits local cross sections. Then the natural map $$ \iota: \: \mathcal{C} \left(G \times Y,V\right)^{H} \to \mathcal{C} \left(G \times_{H} Y,V\right) $$ is a homeomorphism. \end{lemma} \begin{proof} Given $ \tilde f \in \mathcal{C} \left(G \times Y,V\right)^{H} \!$, $\iota(\tilde f) = f$
is the unique map satisfying the relation $\tilde f = f \circ \mathsf{p}$, where $\mathsf{p}: G \times Y \to G \times_{H} Y $ is the quotient map. Now, for any $C \subseteq G \times_{H} Y$ compact and $U \subseteq V$ open with $f(C) \subseteq U$, consider $${\mathcal{N}}(C,U) = \{ \ell : G \times_{H} Y \to V \mid \ell(C) \subseteq U \}$$ which is an open neighborhood of $f$ in $\mathcal{C} \left(G \times_{H} Y,V\right)$. By (3) of Lemma \ref{lem:cfiberbundles}, $\mathsf{p}: G \times Y \to G \times_{H} Y$ admits local cross sections. Since $G \times_{H} Y$ is locally compact, we may cover $C$ with finitely many compact neighborhoods $C_{i}$ such that $\mathsf{p}$ admits a section $s_{i}$ over $C_{i}$. Then put $\tilde C = \bigcup_{i} s_{i}(C_i \cap C)$. Hence, $\tilde f \in {\mathcal{N}}(\tilde C, U)$ and, for any $\tilde \ell \in {\mathcal{N}}(\tilde C, U) \cap \mathcal{C} \left(G \times Y,V\right)^{H}$, we have that $\iota(\tilde \ell) \in {\mathcal{N}}(C,U)$. Hence, $\iota$ is continuous.
Since $G \times_{H} Y$ is locally compact Hausdorff, the inverse map $$ \iota^{-1} \! : \; \mathcal{C} \left(G \times_{H} Y,V\right) \to \mathcal{C} \left(G \times Y,V\right)^{H}, \; f \mapsto \tilde f = f \circ \pi$$ is clearly continuous (with respect to the compact open topology). \end{proof}
\subsubsection{Adjointness properties of $\mathrm{Ind}_{H}^{G}$} Let $V$ be a continuous $G$-module. Since $H$ is a subgroup of $G$, the $G$-module $V$ is an $H$-module by restriction. The latter will be denoted by $\mathrm{res}_H^G \, V$. Similarly, if $X$ is a $G$-space, we let $\mathrm{res}_{H}^{G} X$ denote the restricted $H$-space.
\begin{lemma} \label{lem:ind} Let $V$ be a continuous $G$-module. There is a natural isomorphism of continuous $G$-modules \begin{equation} \label{eq:ind} \mu: \, {\rm Ind}_{H}^{G} \, \mathcal{C} \left(Y,\mathrm{res}_H^G \, V\right) \, \to \; \mathcal{C} \left(G \times_{H} Y,V\right) \; . \end{equation} Similarly, if $X$ is a $G$-space and $W$ an $H$-module, then there is a natural isomorphism of continuous $G$-modules \begin{equation} \label{eq:ind2} \nu: \mathcal{C}\left( X, \mathrm{Ind}^{G}_{H} \, W \right) \, \to \, \mathrm{Ind}^{G}_{H} \; \mathcal{C} \left( \mathrm{res}_H^G \, X, W \right) \; . \end{equation} \end{lemma}
\begin{proof}
Given $f \in {\rm Ind}_H^G\, \mathcal{C}(Y, \mathrm{res}_{H}^{G} V) \subseteq \mathcal{C}(G, C(Y,V))$, we define \begin{equation} \label{eq:corresp_maps} \bar \mu(f) \in \mathcal{C}(G \times Y, V) \; \text{, via } \bar \mu(f) (g,y) = g \cdot \left( \, f(g) \left(y\right) \, \right) \; . \end{equation} Since $f$ is $H$-equivariant (by definition of ${\rm Ind}_{H}^{G}$), it follows $$ \bar \mu(f) \left( h \cdot (g, y) \right) = \; \bar \mu(f) \left( g, y \right) \; . $$ That is, $\bar \mu(f)$ is $H$-invariant for the diagonal action \eqref{eq:diagonal} and descends to $$\mu(f)\in \mathcal{C} \left( G \times_{H} Y, V \right). $$ Conversely, every element of $\mathcal{C} \left( G \times_{H} Y, V \right)$ arises in this way. One can verify easily that $\mu$ is $G$-equivariant, and also that it is a linear map. It remains to show that $\mu$ is a homeomorphism.
Note first, since $G$ is Hausdorff and $Y$ is locally compact, the natural map \begin{equation} \label{eq:adjointc} \mathcal{C}(G, \mathcal{C}(Y,V)) \to \mathcal{C}(G \times Y, V) \end{equation}
is a homeomorphism of function spaces with respect to the compact open topologies \cite[Ch X \S 3, Corollaire 2]{bourbaki}. It follows that $$ \bar \mu: {\rm Ind}_H^G\, \mathcal{C}(Y,V) \to \mathcal{C} \left(G \times Y,V\right)^{H}$$ gives a homeomorphism. Finally, the push forward map $ \mathcal{C} \left(G \times Y,V\right)^{H} \to \mathcal{C} \left( G \times_{H} Y, V \right)$ is a homeomorphism, by Lemma \ref{lem:cpushfw}. Therefore, $\mu$ is a homeomorphism.
This proves \eqref{eq:ind}.
The proof of \eqref{eq:ind2} follows similarly, using the adjointness \eqref{eq:adjointc}. \end{proof}
\subsubsection{Slices, tubes and induced representations} Let $X$ be a $G$-space and $p \in X$. Put $G_p =\{ g \in G \mid g \cdot p = p \}$. Let $S_{p}$ be a $G_{p}$-invariant locally closed subset of $X$ with $p \in S_{p}$. Then $S_{p}$ is called a \emph{slice} for the $G$-action on $X$ if $$ U_{p} = G \cdot S_{p}$$ is an open subset of $X$ and the map \begin{equation} \label{eq:tube}
G \times_{G_{p}} S_{p} \, \to \, U_{p} \; , \; (g, s) \mapsto g \cdot s \end{equation} is a homeomorphism. If a slice exists then $U_{p}$ is called a \emph{tube} around the orbit $G \cdot p$.
For any \emph{$G$-invariant open subset} $ U \subseteq U_{p}$, $p \in U$, we define $$ S_{U} = U \cap S_{p} \, . $$ Then $S_U$ is a $G_p$-space. For any $G$-module $V$, since $U$ is a $G$-space, $\mathcal{C}(U,V)$ is a $G$-module.
\begin{lemma}[$G$-tubes] \label{lem:tubesc} There is a natural isomorphism of continuous $G$-modules \begin{equation} \label{eq:ind_tube} {\rm Ind}_{G_{p}}^G \mathcal{C}(S_{U}, \, \mathrm{res}_{G_p}^G \! V) \, {\longrightarrow} \, \; \mathcal{C}(U,V) \, . \end{equation} \end{lemma}
\begin{proof} Since $S_{p}$ is a slice at $p$, the natural map \eqref{eq:tube} is a $G$-equivariant homeomorphism,
and so are the restricted maps \begin{equation} \label{eq:slice_homeo} G \times_{G_{p}} \, S_{U} \to U \; . \end{equation} In particular, $U$ is a $G$-tube around $G \cdot p$. In view of the homeomorphism \eqref{eq:slice_homeo}, the claim follows from Lemma \ref{lem:ind}, by taking $H = G_p$.
\end{proof}
We conclude:
\begin{lemma}[Local vanishing] \label{lem:local_vanish}
Suppose there exists a slice $S_{p}$ for $p \in X$, such that $G_{p}$ is compact and $G \to G/G_{p}$ has local cross sections. Let $V$ be a quasi-complete continuous $G$-module. Then, for all sufficiently small $G$-invariant neighbourhoods $U$ of $p$, \begin{equation}\label{eq:smallvanish} H^r \! \left(G,\mathcal{C}(U,V)\right)= \{0\} \, , \; \,r\geq 1 \, . \end{equation} \end{lemma}
\begin{proof} By Lemma \ref{lem:tubesc}, $\mathcal{C}(U,V) = {\rm Ind}_{G_{p}}^G \mathcal{C}(S_{U} , \mathrm{res}_{G_p}^G V)$.
Shapiro's lemma (Proposition \ref{Shapi}) states that \begin{equation*}
H^r \! \left( G, \, {\rm Ind}_{G_{p}}^G \mathcal{C}(S_{U},\mathrm{res}_{G_p}^G V) \right) = \, H^r \! \left (G_p, \, \mathcal{C}(S_{U},\mathrm{res}_{G_p}^G V) \right) \; . \end{equation*}
As $V$ is quasi-complete also $\mathcal{C}(S_{U},V)$ is quasi-complete.
The module $\mathcal{C}(S_{U},V)$ is therefore $G_{p}$-integrable
(see in particular \cite[Proposition 3.1]{HM}). As $G_p$ is compact it follows from Proposition \ref{compsta} that
\[ \displaystyle H^r\! \left(G_p, \, \mathcal{C}(S_{U},\mathrm{res}_{G_p}^G V) \right)= \{0\} \, , \; r\geq 1. \qedhere \] \end{proof}
\subsection{Sheaf theoretic interpretation of equivariant cohomology} \label{sect:sheaves_c} Let $X$ be a $G$-space and let $V$ be a continuous $G$-module. We let
$$\pi: X \to X/G$$ denote the quotient map for the $G$-action.
Furthermore, let
$$ \mathcal{C}_{X,V} = \mathcal{C}( \cdot, V)$$ denote the sheaf of continuous functions on $X$ with values in $V$, as well as, $$ \mathcal{C}_{X} \text{ and } \mathcal{C}_{X/G}$$ the structure sheaves of continuous real valued functions on $X$ and $X/G$, respectively. Since $G$ acts on $X$ and $V$, $\mathcal{C}_{X,V}$ is a $G$-sheaf. That is, the sheaf $\mathcal{C}_{X,V}$ has an action of $G$ by co-morphisms which is defined by \eqref{eq:induced_action}. Remark further that $ \mathcal{C}_{X,V}$ is also a sheaf of $\mathcal{C}_{X}$-modules. (For general background on sheaf theory, see eg.\ \cite[Chapter II]{Wells} or \cite{Bredon_sheaves}. For the notion of $G$-sheaves, see \cite[Chapitre V]{tohoku2}.)
\paragraph{\em Direct image sheaves} Let ${\mathcal{A}}$ denote any sheaf on $X$. The direct image $\pi_{*} {\mathcal{A}}$ of ${\mathcal{A}}$ is the sheaf on $X/G$, where, for any open subset $U$ of $X/G$,
$$\pi_{*} {\mathcal{A}} \left(U\right ) \, = \, {\mathcal{A}} \left(\pi^{-1}(U)\right) \, . $$ We may view $\pi_{*}$ as a left-exact functor taking $G$-sheaves on $X$ to $G$-sheaves on $X/G$ (where $X/G$ has the trivial $G$-action). Of particular interest is the direct image sheaf of rings $\pi_{*} \, \mathcal{C}_{X}$. Its subsheaf $$\pi_{*}^{G} \, \mathcal{C}_{X}$$ of
$G$-invariant functions is called the \emph{equivariant direct image} of $\mathcal{C}_{X}$. Note that
the sheaf $\pi_{*}^{G} \, \mathcal{C}_{X}$ is canonically isomorphic to the sheaf $\mathcal{C}_{X/G}$
of continuous functions on $X/G$.
\paragraph{\em Resolution of structure sheaves on $X/G$}
We declare a differential sheaf \begin{equation} \label{eq:barresolution_c}
C^{0}_{X,V} \stackrel{\partial^{0}}{{\longrightarrow}} \, C^{1}_{X,V} \stackrel{\partial^{1}}{{\longrightarrow}}\, C^{2}_{X,V} \stackrel{\partial^{2}}{{\longrightarrow}} \ldots \; , \end{equation}
of $\pi_{*} \, \mathcal{C}_{X}$-modules on $X/G$. The module of sections of $C^{r}_{X,V}$ over an open subset $U \subseteq X/G$ is defined as $$ C^{r}_{X,V}\left( U \right) = \, C^{r} \! \left(G; \, \mathcal{C}(\pi^{-1}(U), V) \right) \, . $$ Here $\mathcal{C}(\pi^{-1}(U),V)$ is a $G$-module and, by definition, $C^{r}_{X,V}\left( U \right)$ consists of the inhomogeneous $r$-cochains with coefficients $\mathcal{C}(\pi^{-1}(U),V)$. The differential $$ C^{r}_{X,V}\left( U \right) \, \stackrel{\partial^{r}}{{\longrightarrow}}\, C^{r+1}_{X,V}\left( U \right) $$ is obtained by the usual formula \eqref{eq:partial} for inhomogeneous cochains. It is trivial to verify that these local maps patch together to define a homomorphism of sheaves $$ \partial^{r}: C^{r}_{X,V} \, {{\longrightarrow}}\, C^{r+1}_{X,V} \, . $$
\paragraph{\em Equivariant direct image of\/ $\mathcal{C}_{X,V}$} We remark that $C^{0}_{X,V} = \pi_{*} \, \mathcal{C}_{X,V}$ is a sheaf of $G$-modules, since $ C^{0}_{X,V}(U) = \mathcal{C}_{X,V}(\pi^{-1}(U))$.
The subsheaf $$\pi_{*}^G \, \mathcal{C}_{X,V} $$
of \emph{$G$-invariant} functions is generated by the presheaf
$$ U \, \mapsto \; \mathcal{C}(\pi^{-1}(U),V)^{G} \; . $$ Thus, we note that $$ \pi_{*}^G \, \mathcal{C}_{X,V} = \ker \partial^{0} \; . $$
\paragraph{\em Remark} Observe that $\pi_{*}^G \, \mathcal{C}_{X,V}$ is not necessarily a sheaf of functions on $X/G$, unless the action of $G$ on $V$ is trivial or $G$ acts freely on $X$. (In case $V$ is a trivial $G$-module, we may identify $\pi_{*}^G \, \mathcal{C}_{X,V}$ with the structure sheaf $ \mathcal{C}_{X/G,V}$ of continuous functions on $X/G$ taking values in $V$.)
\paragraph{\em Local vanishing of continuous cohomology} Suppose that, for all sufficiently small open neighbourhoods $U$ on $X/G$, we have \begin{equation} \label{eq:local_v}
H^r \left(G,\, \mathcal{C}(\pi^{-1}(U),V)\right)= \{0\} \, , \, \text{$r\geq 1$} \, .
\end{equation} Then we shall say that the continuous cohomology groups of $G$ with coefficients $\pi_{*} \, \mathcal{C}_{X,V}$ \emph{vanish locally}. This is clearly equivalent to the condition that the sequence \eqref{eq:barresolution_c} is an exact sequence of sheaves:
\begin{lemma} \label{lem:exact_c} If the continuous cohomology groups of $G$ with coefficients $\pi_{*} \, \mathcal{C}_{X,V} $ vanish locally then \begin{equation} \label{eq:resolution} \{0\} \, {\longrightarrow} \, \pi_{*}^G \, \mathcal{C}_{X,V} {\longrightarrow} \, C^{0}_{X,V} \stackrel{\partial^{0}}{{\longrightarrow}} \, C^{1}_{X,V} \stackrel{\partial^{1}}{{\longrightarrow}}\, C^{2}_{X,V} \stackrel{\partial^{2}}{{\longrightarrow}} \ldots \end{equation} is an exact sequence of sheaves on $X/G$. \end{lemma}
\paragraph{\em Cohomology of the equivariant direct image sheaf $\pi_{*}^G \, \mathcal{C}_{X,V}$}
By a standard argument on double complexes (compare \cite[Th\'eor\`eme 2.4.1]{tohoku1}), we can compute the sheaf cohomology groups $H^*_{X/G}(\pi_{*}^G \, \mathcal{C}_{X,V})$ as follows:
\begin{pro} \label{pro:ssequence} Suppose that the continuous cohomology groups of $G$ with coefficients $\pi_{*} \, \mathcal{C}_{X,V} $ vanish locally. Then there exists a spectral sequence converging to the sheaf cohomology $H^*_{X/G}(\pi_{*}^G \, \mathcal{C}_{X,V})$ with $$E^{p,q}_2 = H^p \left(H^q_{X/G}(C^*_{X,V})\right) . $$ \end{pro}
Note that the induced complex of global sections for the resolution \eqref{eq:resolution} takes the form \begin{equation} \label{eq:sections_c}
0 {\longrightarrow} \, \mathcal{C}(X,V)^{G} \, {\longrightarrow} \, \mathcal{C}(X,V) \stackrel{\partial^{0}}{{\longrightarrow}}\, C^{1}(G; \mathcal{C}(X,V) ) \stackrel{\partial^{1}}{{\longrightarrow}} \ldots \; .
\end{equation} By construction this is the inhomogeneous bar complex for the continuous cohomology of $G$ with coefficients in the $G$-module $ \mathcal{C}(X,V) $. Therefore $$ E^{p,0}_2 = H^p \! \left(H^0_{X/G}(C^*_{X,V}) \right)= H^p \left(C^*(G;\mathcal{C}(X,V))\right) = H^p\left(G, \mathcal{C}(X ,V)\right). $$
\paragraph{\em Paracompact quotient space $X/G$} Suppose that $X/G$ is a paracompact Hausdorff space. Then \eqref{eq:resolution} gives a resolution of $\pi_{*}^G \, \mathcal{C}_{X,V}$ by \emph{fine} sheaves:
\begin{lemma} \label{lem:fine_c} If $X/G$ is a paracompact Hausdorff space, then the sheaves
\begin{enumerate} \item $C^{r}_{X,V}$, $r \geq 0$, and \item $\pi_{*}^G \, \mathcal{C}_{X,V}$ \end{enumerate} are fine sheaves. \end{lemma} \begin{proof} Since $X/G$ is a paracompact Hausdorff space, any locally finite covering
by open sets admits a subordinate partition of unity.
This implies that the structure sheaf $\mathcal{C}_{X/G}$ is a fine sheaf of rings. In particular, any sheaf of $\mathcal{C}_{X/G}$-modules is a fine sheaf (see \cite[Theorem 9.16]{Bredon_sheaves}). Now $C^{r}_{X,V}$ is a sheaf of $\mathcal{C}_{X/G}$-modules, where, given $\epsilon \in \mathcal{C}_{X/G}$, an open subset $U \subseteq X/G$ and $ c \, \in \, C^{r}_{X,V}\left( U \right) \, $, we declare $$ \epsilon \cdot c \; \left(g_{1}, \ldots, g_{r} \right) = (\epsilon \circ \pi) \cdot c \left( g_{1}, \ldots, g_{r} \right) \; . $$ It follows that $C^{r}_{X,V}$ is a fine sheaf, hence (1).
Since $\pi_{*}^G \, \mathcal{C}_{X,V}$ is a sheaf of $\pi_{*}^{G} \, \mathcal{C}_{X} = \mathcal{C}_{X/G}$ -modules on $X/G$, it is a fine sheaf as well. Thus (2) holds. \end{proof}
In this situation the equivariant continuous cohomology groups of $G$ may be expressed in terms of sheaf cohomology on $X/G$:
\begin{corollary} \label{cor:groupc=sheafc} Suppose that $X/G$ is a paracompact Hausdorff space and that \eqref{eq:resolution} is exact. Then, for all $r \geq 0$, there is a natural isomorphism $$ H^{r} \left(G, \mathcal{C}(X,V) \right) \, \cong \, H^{r} \left( \pi_{*}^G \, \mathcal{C}_{X,V} \right)
\; .$$ \end{corollary} \begin{proof} The homology of the complex of global sections for any resolution of $\pi_{*}^G \, \mathcal{C}_{X,V}$ by fine sheaves is isomorphic to the sheaf cohomology $H^{*} \left( \pi_{*}^G \, \mathcal{C}_{X,V} \right)$ (see, for example \cite[Section II.3]{Wells}). By Lemma \ref{lem:fine_c} part (1), the resolution \eqref{eq:resolution} of $\pi_{*}^G \, \mathcal{C}_{X,V}$ is fine. Thus the cohomology of the sheaf $\pi_{*}^G \, \mathcal{C}_{X,V}$ is isomorphic to the homology of the complex \eqref{eq:sections_c}. (In particular, the spectral sequence in Proposition \ref{pro:ssequence} collapses at $E_2$. That is $E_2^{p,q} = \{0\}$, $q>0$.) \end{proof}
\subsection{Vanishing of equivariant continuous cohomology} If $X/G$ is a paracompact Hausdorff space, then $\pi_{*}^G \, \mathcal{C}_{X,V}$ is a fine sheaf (by Lemma \ref{lem:fine_c} part (2)). Therefore its sheaf cohomology must be acyclic. In the view of Corollary \ref{cor:groupc=sheafc} this proves:
\begin{theorem}[Vanishing theorem, continuous case] \label{thm:cont_vanish} \
Suppose that the continuous cohomology groups of $G$ with coefficients $\pi_* \, \mathcal{C}_{X,V}$ vanish locally and that $X/G$ is a paracompact Hausdorff space. Then $$H^r \left(G , \mathcal{C}(X,V) \right)= \{0\}\, \text{, for all $r \geq 1$} . $$ \end{theorem}
\paragraph{\em Proper actions of Lie groups} A typical case for application arises in smooth actions on manifolds. Let\/ $G$ be a Lie group and let $V$ be a quasi-complete continuous $G$-module. Then we have: \begin{corollary}[Vanishing theorem, smooth manifolds] \label{cor:smooth_man} Let $G$ be a Lie group and $X$ a $G$-manifold on which $G$ acts smoothly and properly. Then $$ H^{r}\left(G, \mathcal{C}(X,V) \right) = 0 \; , \text{ for all } r \geq 1 . $$ \end{corollary} \begin{proof} Since $X$ is paracompact and $G$ acts properly, $X/G$ is a paracompact Hausdorff space. By the differentiable slice theorem (see \cite[\S 4, Lemma 4]{JK}, for example), every point $p \in X$ admits a $G$-tube $U_{p}$. Moreover, since $G$ is a Lie group, for any closed subgroup $H$ of $G$, $G/H$ is a manifold and admits local (smooth) sections. Since $V$ is assumed quasi-complete, it follows by Lemma \ref{lem:local_vanish} that the continuous cohomology of $G$ with coefficients $\mathcal{C}(X,V)$ vanishes locally. Therefore, Theorem \ref{thm:cont_vanish} applies. \end{proof}
\section{Smooth coefficients and vanishing of equivariant differentiable cohomology} \label{sec:smooth_coefficients} Let $G$ be a Lie group
and $X$ a smooth manifold on which $G$ acts smoothly. Such $X$ will be called a \emph{differentiable $G$-space}. For any continuous $G$-module $V$, where $V$ is a locally convex topological vector space (as in Section \ref{sec:cont_cohomology}), let $$\mathcal{C}^{\infty}(X,V)$$ denote the vector space of smooth functions on $X$ with values in $V$. Endowed with the $\mathcal{C}^{\infty}$-topology of maps, $\mathcal{C}^{\infty}(X,V)$ is a locally convex vector space and (quasi-) complete if $V$ is (quasi-) complete \cite[Chapter 1, \S 10]{Groth_tvs}, see also \cite[\S 3]{Cass} and \cite{Schwartz}.
Moreover, since $G$ acts smoothly on $X$, with respect to \eqref{eq:induced_action}, $\mathcal{C}^{\infty}(X,V) $ becomes a continuous $G$-module in an obvious way.
Van Est and Mostow-Hochschild \cite{HM} introduced the notion of \emph{differentiable $G$-modules} and differentiable cohomology groups $ H^{r}_{d} \! \left(G, \cdot\right)$ based on smooth co\-chains (see Section \ref{sec:diff_cohomology} below for definitions). In particular, if $X$ is a differentiable $G$-space and $V$ a differentiable $G$-module then $\mathcal{C}^{\infty}(X,V)$ is a differentiable $G$-module.
The main result for this section will be:
\begin{theorem}[Vanishing theorem, smooth case] \label{thm:smooth_vanish} Let $X$ be a differentiable $G$-space on which $G$ acts properly, and let\/ $V$ be a differentiable $G$-module. Then $$ H^{r}_{d} \left(G, \mathcal{C}^{\infty}(X,V) \right) \, = \, H^{r} \left(G, \mathcal{C}^{\infty}(X,V) \right) \, = \{0\} \, , \; r \geq 1. $$ \end{theorem}
This result implies Theorem \ref{thm:main_cohomology} in the introduction.
Note that Theorem \ref{thm:smooth_vanish} is a differentiable version of Corollary \ref{cor:smooth_man}.
Here we are dealing with smooth functions as coefficients instead of continuous functions. Likewise, the differentiable cohomology groups $ H^{*}_{d}\left(G, \cdot \right)$ are using smooth cochains in their definition, and it is required that the coefficient modules for the differentiable cohomology functor $H_d^*(G, \cdot)$ are differentiable $G$-modules. We shall explain these notions right away in the following Section \ref{sec:diff_cohomology}.
\subsection{Differentiable cohomology groups} \label{sec:diff_cohomology} Since $G$ is a Lie group we can introduce a smooth analogue of the continuous cohomology theory, which was first systematically studied by Van Est \cite{VanEst}. Its foundations were further developed by Mostow and Hochschild, see \cite{Mostow, HM}. Another good reference is \cite{Blanc}. The differentiable cohomology of $G$ is a functor defined on the category of differentiable $G$-modules.
\paragraph{\em Differentiable $G$-modules} Let $V$ be a differentiable $G$-module. By this we mean a continuous $G$-module $V$ (with all the assumptions of Section \ref{sec:cont_cohomology} in place, in particular $V$ is a quasi-complete Hausdorff locally convex topological real vector space) that satisfies (see \cite{Blanc,HM}) that, for all $v \in V$, \begin{enumerate} \item the orbit map $G \to V$, $o_{v} \! : g \mapsto g \cdot v$ is smooth. \item the map $V \to \mathcal{C}^{\infty}(G, V)$, $v \mapsto o_{v}$ is smooth. \end{enumerate} An isomorphism of differentiable $G$-modules is a $G$-equivariant isomorphism of topological vector spaces.
\paragraph{\em Smooth functions on $G$-spaces} A differentiable $G$-space is a smooth manifold $X$ on which $G$ acts smoothly. In this situation, $$ \mathcal{C}^{\infty}(X,V)$$ with the usual action (defined by \eqref{eq:induced_action}) is a differentiable $G$-module. (Compare \cite[\mbox{$8^{\circ})$} Proposition]{Blanc}.)
\paragraph{\em Smooth cochains} The differentiable cohomology groups of $G$ with coefficients in the differentiable module $V$
are defined by using differentiable cochains instead of continuous cochains.
For this, we consider in the complex of continuous inhomogeneous cochains $$\left(C^{*}(G;V), \partial \right)$$ (see Section \ref{sect:cc_groups})
the subcomplex of differentiable inhomogeneous cochains $$\left( \, C^{*}_{d}(G;V), \partial \, \right) \, , \; \text{ where } \, C^{r}_{d}(G;V) := \, \mathcal{C}^{\infty}(G^{r},V) \, .$$ We put $B^r_{d}(G;V) = \partial \left(C^{r-1}_{d}(G;V)\right)$ and $Z^*_d(G;V) = \ker \partial \cap C^*_d(G;V)$. This defines the differentiable cohomology groups $$ H^{r}_{d}(G,V) = \, Z^r_d(G;V) \big/ B^r_d(G,V) \, . $$
We mention the following comparison result with continuous cohomology.
\begin{theorem}[Hochschild, Mostow \mbox{\cite[Theorem 5.1]{HM}}] \label{thm:comparison}
Let $G$ be a real Lie group and $V$ a differentiable $G$-module. Then the natural
map $$ H^{*}_{d}(G,V) \, \to \, H^{*}(G,V)$$
is an isomorphism of (topological) vector spaces.
\end{theorem}
\subsubsection{Compact Lie groups} Let $V$ be a differentiable $G$-module. By Proposition \ref{compsta}, vanishing of continous cohomology with coefficients in $V$ follows for all compact Lie groups $G$.
In fact, the same proof (or application of Theorem \ref{thm:smooth_vanish}) and Theorem \ref{thm:comparison}) shows:
\begin{lemma} \label{lem:smooth_van_compact} Let $G$ be a compact Lie group and let $V$ be a differentiable $G$-module. Then $ H_{d}^{r} \left(G, V\right) = \{0\}$, for all $r \geq 1$. \end{lemma}
\subsubsection{Smooth Shapiro lemma} \label{sec:smoothShapi} Let $H$ be a closed subgroup of $G$ and $W$ a differentiable $H$-module. Put $$ \Ind^{\infty} \,_H^G \, W = \; \{f\in \mathcal{C}^{\infty}(G,W)\mid f(g h^{-1})=h \cdot f(g), \ g\in G, \, h\in H\}.$$ Then the space $\Ind^{\infty} \,_H^G \, W$ turns into a differentiable $G$-module, by declaring $$ (\alpha \cdot f) \left(g\right) = f \left(\alpha^{-1} g\right) \; . $$ As it turns out the usual proof of Shapiro's lemma (Proposition \ref{Shapi}) works for differentiable cochains with respect to the functor $ \Ind^{\infty} \,_H^G$, compare \cite{Blanc}. It relies on a differentiable equivariant version of Frobenius reciprocity related to Lemma \ref{lem:ind2_smooth} below.
Let $W$ be a differentiable $H$-module. \begin{pro} [Smooth Shapiro lemma, see \mbox{ \cite[Th\'eor\`eme 11]{Blanc}}] \label{Shapi_smooth} There is a natural isomorphism $$ H^r _{d} \left(G, \Ind^{\infty} \,_H^G \, W \right) \; \cong \; H^r_{d} \left(H, W\right) \, , \, r \geq 0 . $$ \end{pro}
\paragraph{\em Remark} Note that, in the view of Theorem \ref{thm:comparison}, Proposition \ref{Shapi_smooth} implies that the continuous cohomology groups $H^r \! \left(H,W \right)$ and $H^r \! \left(G, \Ind^{\infty} \,_H^G \, W \right)$ are isomorphic.
\paragraph{\em Associated bundles over $\, G/H$} Let $Y$ be a differentiable $H$-space. Then the associated bundle $$ G \times_{H} Y$$
is a smooth $G$-manifold, and a locally trivial smooth fiber bundle over $G/H$ with fiber $Y$. The following is the differentiable analogue of Lemma \ref{lem:ind}.
\begin{lemma} \label{lem:ind_smooth} Let $V$ be a differentiable $G$-module. There is a natural isomorphism of differentiable $G$-modules \begin{equation} \label{eq:inds} \mu: \, \Ind^{\infty} \,_{H}^{G} \; \mathcal{C}^{\infty}(Y,\mathrm{res}_H^G \, V) \, \to \, \mathcal{C}^{\infty}(G \times_{H} Y,V) \; . \end{equation} \end{lemma} \begin{proof} First note that for any two smooth manifolds $A, B$, we have ({cf$.$\,} \cite[Ch. 1, \S 10 and Ch. 3. \S 8]{Groth_tvs}) the adjoint formula \begin{equation} \label{eq:adjoints}
\mathcal{C}^{\infty}(A, \mathcal{C}^{\infty}(B,V)) \, = \, \mathcal{C}^{\infty}(A \times B,V ) \; . \end{equation}
Also, since $H$ acts properly and freely on $G \times Y$, the differentiable slice theorem (compare \eqref{eq:stube}), shows that the quotient map $$G \times Y \to G \times_{H} Y$$ is a smooth $H$-principal bundle map which is locally trivial.
Now, for any trivial $H$-principal bundle $A \times H$, $\mathcal{C}^{\infty}(A \times H,V)^{H} = \mathcal{C}^{\infty}(A,V)$. This allows to show the homeomorphism \begin{equation} \label{eq:spushfw} \mathcal{C}^{\infty}(G \times Y,V)^{H} \to \mathcal{C}^{\infty}(G \times_{H} Y,V) \end{equation}
(compare Lemma \ref{lem:cpushfw}). The proof of Lemma \ref{lem:ind_smooth} carries now through analogously to the one of Lemma \ref{lem:ind} (1), using \eqref{eq:adjoints} and \eqref{eq:spushfw}. \end{proof}
\begin{corollary} [Smooth Shapiro lemma for actions on functions] \label{cor:Shapi_smooth} There is a natural isomorphism $$ H^r_{d} \left(H, \mathcal{C}^{\infty}(Y,\mathrm{res}_H^G V) \right) \, \to \, H^{r}_{d} \left(G, \mathcal{C}^{\infty}(G \times_{H} Y,V) \right) \; . $$ \end{corollary}
\paragraph{\em Equivariant reciprocity lemma} Let $X$ be a differentiable $G$-space. Then $X$ is also a differentiable $H$-space by restriction. We denote this space by $\mathrm{res}_H^G \, X $.
We mention the analogue of Lemma \ref{lem:ind} (2):
\begin{lemma} \label{lem:ind2_smooth} There is a natural isomorphism of differentiable $G$-modules $$ \nu: \mathcal{C}^{\infty}\left( X, \Ind^{\infty} \,^{G}_{H} \, W \right) \, \to \, \Ind^{\infty} \,^{G}_{H} \; \, \mathcal{C}^{\infty} \left( \mathrm{res}_H^G \, X, W \right) \; . $$ \end{lemma}
\subsubsection{Local vanishing for smooth coefficients} \hspace{1cm}
\paragraph{\em Differentiable slices and smooth $G$-tubes} Let $S_{p}$ be a differentiable slice at $p \in X$, and $U_{p} = G \cdot S_{p}$ the corresponding smooth $G$-tube. By definition a differentiable slice $S_{p}$ is a submanifold such that the natural map \begin{equation} \label{eq:stube} G \times_{G_{p}} S_{p} \to U_{p} \end{equation} is a $G$-equivariant diffeomorphism (see \cite[\S2 Lemma 4]{JK}). As before, for any, $G$-invariant open subset $U \subseteq U_{p}$ of $X$, $p \in U$, define $ S_{U} = S_{p} \cap U$, which is an open submanifold of the manifold $ S_{p}$. Moreover, then $ S_{U}$ is a differentiable slice with tube $U$. In fact, since \eqref{eq:stube} is a $G$-equivariant diffeomorphism,
so are the restricted maps $G \times_{G_{p}} S_{U} \to U$. In particular, if a tube $U_{p}$, exists we may always find a differentiable slice $S_{U}$ near $p$ which has compact closure. That is, by taking $U = G \cdot S$, where $S$ is a $G_{p}$-invariant neighborhood of $p$ in $S_{p}$ with compact closure in $S_{p}$.
\begin{lemma}[Smooth $G$-tubes] \label{lem:tubesc_smooth} There is a natural isomorphism of differentiable $G$-modules \begin{equation} \label{eq:ind_tubes} {\mathrm{Ind}_{G_{p}}^G \, \mathcal{C}^\infty}(S_{U},\mathrm{res}^G_{G_p} V) \, \to \, \mathcal{C}^{\infty}(U,V) \, \text{ and } \end{equation}
\begin{equation}\label{eq:smallvanish2} H^r _{d}\left(G,\mathcal{C}^{\infty}(U, V)\right)= \{0\} , \; \text{ for all } r \geq 1. \end{equation} \end{lemma} \begin{proof}
By Lemma \ref{lem:ind_smooth}, $ \Ind^{\infty} \,_{G_{p}}^G \, \mathcal{C}^{\infty}(S_{U} \, , \mathrm{res}^G_{G_p} V) = \mathcal{C}^{\infty}\left (G \times_{G_{p}} S_{U} \,,\, V \right)$. Using the differentiable Shapiro's lemma (Proposition \ref{Shapi_smooth}) we conclude \begin{equation*}
H^r_{d} \left( G, {\Ind^{\infty} \,}_{G_{p}}^G \, \mathcal{C}^{\infty}(S_{U},V) \right) = H^r_{d}\left (G_p, \mathcal{C}^{\infty}(S_{U}, V) \right) \; . \end{equation*} Since $G_p$ is compact, we have $H^r \left(G_p,\mathcal{C}^{\infty}(S_{U}, V) \right) = \{0\}$, $r\geq 1$. \end{proof}
\subsection{Proof of Theorem \ref{thm:smooth_vanish}} \hspace{1ex}
\noindent Let $X$ be a differentiable $G$-space and $V$ a differentiable $G$-module. We are also assuming that $X$ is a proper $G$-space. As before we let
$$\pi: X \to X/G$$ denote the quotient map for the $G$-action. Then $X/G$ is a locally compact, paracompact Hausdorff space, since $G$ acts properly. Moreover, by the differentiable slice theorem \cite[\S2 Lemma 4]{JK} differentiable slices do exist for every $p \in X$. Therefore, the local vanishing property for the differentiable cohomology of $G$ with coefficients in the function space $ \mathcal{C}^{\infty}(X, V)$ is satisfied by application of Lemma \ref{lem:tubesc_smooth}.
\paragraph{\em Differentiable structure sheaves} Following Section \ref{sect:sheaves_c}, we are now considering the properties of various sheaves of functions on $X$ and $X/G$ which are associated to the action of $G$. First let
$$ \mathcal{C}^{\infty}_{X,V} = \mathcal{C}^{\infty}( \cdot, V)$$ denote the sheaf of smooth functions on $X$ with values in $V$. Since $G$ acts on $X$ and $V$, the sheaf $\mathcal{C}^{\infty}_{X,V}$ has an action of $G$ by co-morphisms which is defined by \eqref{eq:induced_action}. Furthermore, let $$ \mathcal{C}^{\infty}_{X} \text{ and } \mathcal{C}^{\infty}_{X/G}$$ denote the structure sheaves of smooth real valued functions on $X$ and $X/G$, respectively. Note that $X/G$ is in general not a smooth manifold, but we can define $$ \mathcal{C}^{\infty}_{X/G} = \pi_{*}^{G} \, \mathcal{C}^{\infty}_{X} \; . $$ Similarly, for any differentiable $G$-module $V$, we have the equivariant direct image sheaf $\pi_{*}^G \, \mathcal{C}^{\infty}_{X,V}$ (compare Section \ref{sect:sheaves_c}).
\begin{lemma} \label{lem:fine_s} The sheaves $\pi_{*}^G \, \mathcal{C}^{\infty}_{X,V}$ are fine sheaves. \end{lemma} \begin{proof} Since $\pi_{*}^G \, \mathcal{C}^{\infty}_{X,V}$ is a sheaf of $\mathcal{C}^{\infty}_{X/G} = \pi_{*}^{G} \, \mathcal{C}^{\infty}_{X} $-modules, it is sufficient to show that the latter is a fine sheaf of rings (see for example \cite[\S 9]{Bredon_sheaves}). This amounts to constructing, for any covering of the form $\{ \pi^{-1}(U_{j})\}$ of $X$, where $\{U_{j}\}$ is a locally finite covering of $X/G$, an associated subordinate partition of unity by $G$-invariant smooth functions $\{ \tilde \eta_{j} \in \mathcal{C}^{\infty}({X})^{G}\}$.
To this end, it is sufficient to construct a partition of unity subordinate to any locally finite covering of $X$ by $G$-invariant open subsets that refines the covering $\{ \pi^{-1}(U_{j})\}$. Therefore, since $X$ is covered by $G$-tubes, we may assume that $\pi^{-1}(U_{j}) = G \times_{G_{j}} S_{j}$ is a differentiable $G$-tube and that $S_{j}$ has compact closure. In particular, $U_{j}$ has compact closure in $X/G$.
Since $X/G$ is paracompact, the covering $\{U_{j}\}$ has a subordinate continuous partition of unity $\epsilon_{j}$. Therefore the functions $\epsilon_{j} \circ \pi \in \mathcal{C}(X)^{G}$ give a continuous $G$-invariant partition of unity subordinate to $\{\pi^{-1}(U_{j})\}$.
Moreover, we may arrange things that there exist open $G$-invariant subsets $W^{1} \subset W^{2} \subset S_{j}$, which in some coordinate system are diffeomorphic to Euclidean balls of radius $1$, respectively $2$, such that
$$K_j = \mathop{\mathrm{supp}}{\; {(\epsilon_j \circ \pi)}_{| S_j}} \subset W^{1} . $$
Using the same technique as employed in the proof of \cite[Ch. VI, \S4 4.2 Theorem]{Bredon_compact_trans} (for reference on approximation of continuous functions by smooth functions, see
\cite{MI} and \cite[Theorem 2.2]{Hirsch}), we may approximate $\epsilon'_j$ by a smooth function $$ \eta'_j:S_j\,{\rightarrow} \, {\mathbb R}$$ such that \begin{enumerate} \item[(1)] $\eta'_j$ is a smooth (positive) function on $K_{j}$,
\item[(2)] $\eta'_j=0$ on $S_j - \overline{W}^{1}$ (where $ \overline{W}^{1}$ denotes the closure of $W^{1}$).
\end{enumerate}
Next let $\eta''_j \in \mathcal{C}^{\infty}(S_{j}, {\mathbb R})$ be defined by \[ \eta''_j(x)=\int_{G_j}\eta'_j(gx) \, dg.\] Then $\eta''_j$ is a $G_j$-invariant nonnegative smooth function. Since $W^{1}$ is $G_j$-invariant, $\eta''_j$ also satisfies $(1)$, $(2)$. In particular, note that \[ \mathop{\mathrm{supp}} {\epsilon'_j} \, \subseteq \, \mathop{\mathrm{supp}} {\; \eta''_j} \; . \] As the restriction map $\displaystyle \mathcal{C}^{\infty}(\pi^{-1}(U_j),{\mathbb R})^G\, \to \, \mathcal{C}^{\infty}(S_j,{\mathbb R})^{G_j}$ is bijective (as follows from \eqref{eq:ind_tubes}, for example), we obtain a $G$-invariant smooth function $\eta_j$ on $\pi^{-1}(U_j)$, which restricts to $\eta''_j$ on $S_j$.
With the above provisions in place, and taking into account that $\pi: S_{j} \to U_{j}$ is a quotient map (see \cite[Ch. VI, Proposition 3.3]{Bredon_compact_trans}), it follows that $G \cdot (S_j - \overline{W}^{1})$ is an open subset of $\pi^{-1}(U_j)$ and a neighborhood of the boundary of the open subset $G \cdot W_{2}$. Therefore, extension by $0$ outside $\pi^{-1}(U_j)$ shows that $\eta_j$ arises as the restriction of a smooth $G$-invariant function $\eta_{j}$ defined on $X$ with support in $\pi^{-1}(U_j)$.
By construction, $ a = \sum_{j} \eta_j \in \mathcal{C}^{\infty}(X,{\mathbb R})$,
is everywhere positive on $X$ and $G$-invariant. Thus $$ \tilde \eta_j =\frac 1{a} \, \eta_j$$ defines a partition of unity by $G$-invariant smooth functions subordinate to the covering by $G$-neighborhoods $\{ \pi^{-1}(U_j) \}$. \end{proof}
We conclude the proof of Theorem \ref{thm:smooth_vanish} using the reasoning developed in Section \ref{sect:sheaves_c} as follows:
First of all, in the view of \eqref{eq:smallvanish2} we have a resolution of sheaves \begin{equation} \label{eq:resolution_s} \{0\} \, {\longrightarrow} \, \pi_{*}^G \, \mathcal{C}^{\infty}_{X,V} {\longrightarrow} \, (C^\infty_{X,V})^0 \stackrel{\partial^{0}}{{\longrightarrow}} \, (C^\infty_{X,V})^1 \stackrel{\partial^{1}}{{\longrightarrow}}\, (C^{\infty}_{X,V})^2 \stackrel{\partial^{2}}{{\longrightarrow}} \ldots \, \; \; , \end{equation} where $(C^\infty_{X,V})^r$ are sheaves on $X/G$ whose sections over $U$ are differentiable $r$-cochains of $G$ with coefficients in smooth functions $\mathcal{C}^{\infty}(\pi^{-1}(U),V)$. Since the sheaves $(C^\infty_{X,V})^r$ are sheaves of $\mathcal{C}^{\infty}_{X/G}$ modules (which is a fine sheaf of rings by Lemma \ref{lem:fine_s}) these are fine sheaves. It follows \begin{equation} \label{eq:coho_ident_s}
H^{r}_d \left(G, \mathcal{C}^{\infty}(X,V) \right) \, \cong \, H^{r} \left( \pi_{*}^G \, \mathcal{C}^{\infty}_{X,V} \right) \, . \end{equation} Finally, by Lemma \ref{lem:fine_s}, $\pi_{*}^G \, \mathcal{C}^{\infty}_{X,V}$ is a fine sheaf. It follows that the right hand cohomology in \eqref{eq:coho_ident_s} is acyclic. Therefore, $$ H^{r}_d \left(G, \mathcal{C}^{\infty}(X,V) \right) = \{0\}\, , \, r \geq 1 \, . $$ This finishes the proof of Theorem \ref{thm:smooth_vanish}.
\section{Proof of Theorems \ref{thm:main_CR} and Theorem \ref{thm:main_conformal}} \label{sec:proofs_main}
Given a pseudo-Hermitian structure $\{ \omega, J \}$ on $M$, the distribution $$ D = \ker \omega$$ defines a strictly pseudo-convex $CR$-structure $\{D, J \}$ on $M$. We have the following naturally associated transformation groups, namely the group $$ \operatorname*{Psh}\,(M,\{\omega,J\}) =\{\, \alpha\in\operatorname*{Diff}\,(M)\mid \alpha^{*}\omega=\omega, \; \alpha_*\circ J=J\circ \alpha_*|_{{\mathsf{D}}} \,\} $$ of pseudo-Hermitian transformations and the group of $CR$- automorphisms
$$ \mathop{\rm Aut}\nolimits_{CR}(M, \{{\mathsf{D}},J\}) =\{ \, \alpha\in\operatorname*{Diff}\,(M)\mid \alpha_*{\mathsf{D}}={\mathsf{D}}, \; \alpha_*\circ J=J\circ \alpha_*|_{{\mathsf{D}}} \, \} \; \; . $$ In general, the inclusion $$\operatorname*{Psh}\,(M,\{\omega,J\}) \, \leq \, \mathop{\rm Aut}\nolimits_{CR}(M, \{{\mathsf{D}},J\})$$ is strict, since the contact form $\omega$ is determined by $D$ only up to conformal equivalence. Moreover, the Lie group $\operatorname*{Psh}\,(M,\{\omega,J\})$ always acts properly on $M$, whereas in some cases (as detailed in Theorem \ref{thm:schoen}) the $CR$-automorphism group $\mathop{\rm Aut}\nolimits_{CR}(M, \{{\mathsf{D}},J\})$ is too large and doesn't act properly on $M$.
About the possible relations of the group $\operatorname*{Psh}\,(M,\{\omega,J\})$ and the $CR$-automorphism group $\mathop{\rm Aut}\nolimits_{CR}(M, \{{\mathsf{D}},J\})$, we shall prove:
\begin{theorem} \label{thm:proper_cr} Let $(M,\{{\mathsf{D}},J\})$ be a strictly pseudo-convex $CR$-manifold and let $ G \, \leq \, \mathop{\rm Aut}\nolimits_{CR}(M, \{D,J\})$ be a subgroup of $CR$-automorphisms that acts properly on $M$. Then there exists on $M$ a pseudo-Hermitian structure $\{ \eta, J\}$ compatible with $\{{\mathsf{D}},J\}$, such that $ G \, \leq \, \operatorname*{Psh}\,(M, \{ \eta, J\} ) \, . $ \end{theorem}
Together with Schoen's theorem (Theorem \ref{thm:schoen}), this result obviously implies Theorem \ref{thm:main_CR} in the introduction.
This section is organized as follows: In subsection \ref{sec:crossed_hom} we prepare the proof of Theorem \ref{thm:proper_cr} with a brief discussion of equivariant crossed homomorphisms which are associated to group actions on manifolds. Following that we construct the canonical cohomology class associated to the action of $CR$-automorphisms. The proof of Theorem \ref{thm:proper_cr} is based on the vanishing of this class, see section \ref{sec:cr}. Analogous results for the conformal case and also for locally conformal K\"ahler manifolds will be discussed in subsections \ref{sec:conf} and \ref{sec:lcK}.
\subsection{Equivariant crossed homomorphisms} \label{sec:crossed_hom} Let $G$ be a Lie group which acts smoothly on the manifold $X$. We let ${\mathbb R}^{>0}$ denote the multiplicative group of positive real numbers. Next consider $$\mathcal{C}^{\infty}(X,{\mathbb R}^{>0})$$ the group of all smooth maps from $X$ into ${\mathbb R}^{>0}$ endowed with its natural $G$-module structure, where, for $\alpha\in G$, $f\in \mathcal{C}^{\infty}(X,{\mathbb R}^{>0})$, \begin{equation}\label{eq:mapusua} (\alpha \cdot f)(x)= f(\alpha^{-1}x) \, . \end{equation} In fact, taking the $\mathcal{C}^{\infty}$-topology of maps,
$\mathcal{C}^{\infty}(X,{\mathbb R}^{>0})$ is a differentiable $G$-module in the sense of Section \ref{sec:diff_cohomology}, and it is isomorphic to the $G$-module $\mathcal{C}^\infty(X,{\mathbb R})$.
Concerning the associated \emph{differentiable cohomology} groups of $G$,
we note:
\begin{theorem} \label{thm:vanish} Suppose that $G$ acts properly on $X$. Then, for all $r \geq 1$, we have $ \displaystyle H^r_d \left(G , \mathcal{C}^{\infty}(X,{\mathbb R}^{>0})\right) = \{0\}$.
\end{theorem} \begin{proof} The map $\exp: \mathcal{C}^{\infty}(X,{\mathbb R}) \, {\rightarrow} \, \mathcal{C}^{\infty}(X,{\mathbb R}^{>0})$ defined by $\displaystyle u=\exp f$, that is, \[ u(x)=e^{f(x)}\, , \; x \in X \] is easily seen to be an isomorphism of differentiable $G$-modules. In fact, the correspondence is $G$-equivariant, since, for all $\alpha \in G$, \begin{equation*} \exp{(\alpha\cdot f)(x)} =e^{f(\alpha^{-1}x)} =u(\alpha^{-1}x)=(\alpha\cdot u)(x) \, . \end{equation*} Hence, for all $r$, there is an isomorphism of differentiable cohomology groups \begin{equation*} H^r_d\left(G, \mathcal{C}^{\infty}(X,{\mathbb R})\right)\cong H^r_d\left(G, \mathcal{C}^{\infty}(X,{\mathbb R}^{>0})\right) \, . \end{equation*} Since $G$ acts properly, Theorem \ref{thm:vanish} implies \begin{equation*} H^r_d\left(G, \mathcal{C}^{\infty}(X,{\mathbb R}^{>0})\right)= \{0\} \, , \ r\geq 1. \qedhere \end{equation*} \end{proof}
\subsubsection{Differentiable crossed homomorphisms} Smooth one-cocycles $$\lambda \in \mathcal{C}^{\infty}\left(G, \mathcal{C}^{\infty}(X,{\mathbb R}^{>0})\right) \, , \; \partial^1 \lambda = 0$$ are representing the elements of the first \emph{differentiable} cohomology group $$ H^1_{d}\left(G, \mathcal{C}^{\infty}(X,{\mathbb R}^{>0})\right) \, . $$ The condition $\partial^1 \lambda = 0$ amounts to the requirement that, for all $\alpha,\beta\in G$, \begin{equation} \label{eq:crossed_hom} \lambda(\alpha \beta) \, (x)= \lambda (\beta)\,(\alpha^{-1} x ) \cdot \lambda(\alpha) \, (x) \ \, . \end{equation} Denoting by $\alpha_*$ the covariant map on forms which is induced by $\alpha$, this relation can be written in the concise form \begin{equation} \label{eq:crossed_homc} \lambda(\alpha \beta) = \, \alpha_* \, \lambda(\beta)\,\cdot \lambda(\alpha) . \end{equation} Such $\lambda$ are called \emph{differentiable crossed homomorphisms}.
\paragraph{\em Exact crossed homomorphisms} For any crossed homomorphism $\lambda$, the cohomology class $$[\, \lambda \, ] \, \in \, H^1_{d}\left(G, \mathcal{C}^{\infty}(X,{\mathbb R}^{>0})\right)$$ vanishes if there exists $ v\in \mathcal{C}^{\infty}(X,{\mathbb R}^{>0})$, with $ \partial^0 v= \lambda$. That is, if \begin{equation} \label{eq:coboundary} (\alpha_{*} v) \cdot v^{-1} =\lambda(\alpha) \, , \text{ for all } \alpha \in G , \end{equation} or, equivalently, $\lambda(\alpha)(x) = v(\alpha^{-1} x) \cdot v^{-1}(x)$, for all $x \in X$. Moreover, two crossed homomorphisms $\lambda$ and $\lambda'$ represent the same class in the group $H^1_{d}\left(G, \mathcal{C}^{\infty}(X,{\mathbb R}^{>0})\right)$ if and only if $ \lambda' = \lambda \cdot \partial^0 v$, for some $v \in \mathcal{C}^{\infty}(X,{\mathbb R}^{>0})$.
\subsection{$CR$-automorphisms and canonical class} \label{sec:cr} By definition, for any pseudo-Hermitian manifold $(M,\{\omega,J\})$,
the Levi form of the underlying $CR$-structure $$ B= d\omega \left(J \cdot ,\cdot \right)$$ is \emph{positive definite} on $D = \ker \omega$. For any $CR$-automorphism $$ \alpha \in\mathop{\rm Aut}\nolimits_{CR}(M,\{{\mathsf{D}},J\}) \, , $$ the equation $\alpha_*({{\mathsf{D}}})={{\mathsf{D}}}$ implies that there exists $ u_\alpha \in \mathcal{C}^\infty(M, {\mathbb R}^{>0})$ with \begin{equation} \label{eq:posi} \alpha^*\omega= u_\alpha\cdot\omega \; . \end{equation} (In fact, to show $u_\alpha>0$, note that, for any $0 \neq v \in D= \ker \omega$, we have $$ B(\alpha_* v, \alpha_* v) = d\omega (J \alpha_* v, \alpha_* v) = d \, \alpha^* \! \omega \left(Jv, v \right) = u_\alpha B(v, v) \, > 0 .\; ) $$
\subsubsection{Associated crossed homomorphism and canonical cohomology class}
Let $$ G \, \leq \, \mathop{\rm Aut}\nolimits_{CR}(M,\{{\mathsf{D}},J\})$$ be a group of $CR$-automorphisms. Next choose a compatible pseudo-Hermitian structure $\{\omega,J\}$ for the $CR$-structure on $M$. Define a map $$ \lambda_{\mathrm{CR}}: G \, \to \, \mathcal{C}^{\infty}(M,{\mathbb R}^{>0})$$ by declaring \begin{equation}\label{crossedhomo_cr}
\lambda_{\mathrm{CR}}(\alpha)\, = \, \alpha_* u_{\alpha}\, , \; \, \alpha\in \mathop{\rm Aut}\nolimits_{CR}(M) \, , \end{equation} where $u_{\alpha}$ is as in \eqref{eq:posi} defined relative to $\omega$. (More explicitly, \eqref{crossedhomo_cr} amounts to
$ \lambda_{\mathrm{CR}}(\alpha)\, (x)= u_{\alpha}\, (\alpha^{-1}x) $, for all $x \in M$.)
We claim that $\displaystyle \lambda_{\mathrm{CR}}$ is a crossed homomorphism for $G$.
\begin{definition} The map $ \lambda_{\mathrm{CR}}$ is called {crossed homomorphism associated to $G$} and the pseudo-Hermitian structure $\{\omega,J\}$. Its cohomology class $\mu_{\mathrm{CR}}$ is defined by the underlying $CR$-structure only and it is called the $\emph{canonical class}$ associated to the $CR$-action of $G$. \end{definition}
The geometric meaning of the canonical cohomology class is given by:
\begin{pro}[Cohomology class of $CR$-transformation group] \label{pro:can_class_CR} Suppose that $M$ is a stric\-tly pseudo-convex $CR$-manifold and let $G$ be a Lie subgroup of the group of $CR$-automorphisms of $M$. Then: \begin{enumerate} \item There exists in the differentiable cohomology of $G$ a natural associated class $$ \mu_{\mathrm{CR}} = [\lambda_{\mathrm{CR}}] \; \in \, H^1_d\left(G, \mathcal{C}^\infty(M,{\mathbb R}^{>0})\right)$$ which is induced by the $CR$-structure on $M$. \item Moreover, the class $\mu_{\mathrm{CR}}$ vanishes if and only if there exists a contact form $\eta$ compatible with the $CR$-structure, such that $G$ is contained in the group of pseudo-Hermitian transformations $\operatorname*{Psh}\,(M,\{\eta,J\})$. \end{enumerate} \end{pro} \begin{proof} To show that $\lambda_{CR}$ is a crossed homomorphism, we calculate \begin{equation*}\begin{split} (\alpha \cdot \beta)^*\, \omega&= \beta^* (\alpha^*\omega)=\beta^*\, (u_\alpha\, \omega)= (\beta^*u_\alpha) \cdot u_\beta \; \omega. \end{split} \end{equation*}
As, according to \eqref{eq:posi}, $(\alpha \cdot \beta)^*\omega= u_{\alpha\beta}\,\omega$, for some $u_{\alpha\beta}\in \mathcal{C}(X,{\mathbb R}^{>0})$, we have $ u_{\alpha\beta}\, = \beta^* u_\alpha \cdot u_\beta$. In particular, \begin{equation*}\begin{split}
\lambda_{\mathrm{CR}}(\alpha\beta) &= (\alpha \cdot \beta)_* \, u_{\alpha\beta} = \alpha_* u_\alpha \cdot (\alpha \cdot \beta)_* \, u_\beta = \lambda_{\mathrm{CR}}(\alpha) \cdot \alpha_* \lambda_{\mathrm{CR}}(\beta) \, . \end{split} \end{equation*} So $\lambda_{\mathrm{CR}}$ is a crossed homomorphism as in \eqref{eq:crossed_homc}.
Suppose next that $\omega = v \, \omega'$, for some $v \in \mathcal{C}^\infty(M,{\mathbb R}^{>0})$. From \begin{equation*} u_{\alpha} \, \omega = \alpha^{*} \omega = \alpha^{*} (v \, \omega') = \alpha^{*} v \cdot u'_{\alpha} \, \omega' = \alpha^{*} v \cdot u'_{\alpha} \cdot v^{-1} \, \omega
\end{equation*} we deduce $u_{\alpha} = \alpha^{*} v \cdot u'_{\alpha} \cdot v^{-1}$. Thus $\lambda_{\mathrm{CR}}(\alpha) = \alpha_{*} u_{\alpha} = v \cdot \alpha_{*} v^{-1} \cdot \alpha_{*} u'_{\alpha} = (\partial^{0} v^{-1}) (\alpha) \cdot \lambda'_{\mathrm{CR}}(\alpha) $, showing that the associated crossed homomorphisms $\lambda_{\mathrm{CR}}$ and $\lambda'_{\mathrm{CR}}$ are representing the same cohomology class $\mu_{\mathrm{CR}}$.
Next let $\{\omega,J\}$ be a pseudo-Hermitian structure on $M$ compatible with the $CR$-structure and $\lambda_{\mathrm{CR}}$ be the associated crossed homomorphism defined by $\omega$. Suppose that $[ \lambda_{\mathrm{CR}} ] = 0$. Therefore, as in \eqref{eq:coboundary}, there exists $v\in \mathcal{C}^{\infty}(M,{\mathbb R}^{>0})$ such that $ \displaystyle \alpha_* v =\lambda(\alpha) \cdot v \, , \text{ for all } \alpha \in G$, which amounts to
\begin{equation}\label{eq:cobound_cr} \begin{split} v &= \alpha^* \, (\lambda(\alpha) \cdot v ) = u_\alpha \cdot \alpha^*v \, . \end{split} \end{equation} Put a $1$-form $
\eta=v\cdot \omega $.
Then $$ \alpha^*\eta=\alpha^*v\cdot \alpha^*\omega=\alpha^*v \cdot u_\alpha \, \omega=v\cdot \omega=\eta \, .$$ Hence, $ \alpha \, \in \, \operatorname*{Psh}\,(M,\{\eta,J\})$ is equivalent to \eqref{eq:cobound_cr}. This shows (2). \end{proof}
\subsubsection{Proof of Theorem \ref{thm:proper_cr}} Since $G$ acts properly, Theorem \ref{thm:vanish} shows that the canonical crossed homomorphism $\lambda_{\mathrm{CR}}$ for the group of $CR$-\-auto\-mor\-phisms $G$, as defined in \eqref{crossedhomo_cr}, is exact. In the view of (2) of Proposition \ref{pro:can_class_CR}, this proves Theorem \ref{thm:proper_cr}.
\qed
\subsection{Conformal case} \label{sec:conf} Replacing the role of $\omega$ in $CR$-geometry by a Riemannian metric $g$ on $M$, the conformal class of $g$ is said to establish a \emph{conformal structure} on $M$. Every diffeomorphism $\alpha : M{\rightarrow} M$ that satisfies $\alpha^*g= u_\alpha \cdot g$ for some positive smooth function $u_\alpha \in \mathcal{C}^\infty(M,{\mathbb R}^{>0})$ is correspondingly called a \emph{conformal automorphism} of $(M,g)$.
As in the $CR$ case, Proposition \ref{pro:can_class_CR}, to any Lie group $G$ of conformal automorphisms there is a natural associated cohomology class $ \mu_{\mathrm{Conf}}$, which is an obstruction for $G$ being a group of isometries:
\begin{pro}[Cohomology class of conformal transformation group] \label{pro:can_class_Conf} Let $(M,g)$ be a Riemannian manifold and let $G$ be a Lie subgroup of the group of conformal automorphisms of $M$. Then: \begin{enumerate} \item There exists in the differentiable cohomology of $G$ a natural associated class $$ \mu_{\mathrm{Conf}} = [\lambda_{\mathrm{Conf}}] \; \in \, H^1_d\left(G, \mathcal{C}^\infty(M,{\mathbb R}^{>0})\right)$$ which is induced by the conformal structure on $M$. \item Moreover, the class $\mu_{\mathrm{Conf}}$ vanishes if and only if there exists a Riemannian metric $h$ conformal to $g$, such that $ G$ is contained in the group of isometries $ \mathop{\rm Iso}\nolimits(M, h) $. \end{enumerate} \end{pro} \begin{proof} We define the cocycle $ \lambda_{\mathrm{conf}}: G \, \to \, \mathcal{C}^{\infty}(M,{\mathbb R}^{>0})$ by declaring \begin{equation}\label{crossedhomo_conf}
\lambda_{\mathrm{conf}}(\alpha)\, = \, \alpha_* u_{\alpha}\, , \; \, \alpha\in \mathop{\rm Aut}\nolimits_{CR}(M) \, , \end{equation} where $u_\alpha \in \mathcal{C}^\infty(M,{\mathbb R}^{>0})$ is defined as above by the relation $\alpha^*g= u_\alpha \cdot g$. As in the proof of Proposition \ref{pro:can_class_CR} it can be verified that $\displaystyle \lambda_{\mathrm{conf}}$ is a crossed homomorphism for $G$, and its class $ \mu_{\mathrm{Conf}}$ in $H^1_d\left(G, \mathcal{C}^\infty(M,{\mathbb R}^{>0})\right)$ depends only on the conformal class of $g$. This shows (1).
The condition that $[\lambda_{\mathrm{Conf}}] \in H^1_d \left(G , \mathcal{C}^{\infty}(M,{\mathbb R}^{>0})\right)$ vanishes means that there exists $v\in \mathcal{C}^{\infty}(M,{\mathbb R}^{>0})$ with $\partial^{0} v = \lambda_{\mathrm{Conf}}$, that is, by \eqref{eq:coboundary}, \begin{equation} \label{eq:cobound_conf} (\alpha_{*} v) \cdot v^{-1} =\lambda_{\mathrm{Conf}}(\alpha) = \alpha_{*} u_{\alpha} \; \text{, for all $\alpha \in G$} . \end{equation}
On the other hand, putting $h = v \cdot g$, we have $\alpha^{*} h = \alpha^{*} v \cdot \alpha^{*} g = \alpha^{*} v \cdot u_{\alpha} g$. Therefore $\alpha^{*} h = h$ is equivalent to \eqref{eq:cobound_conf},
which is equivalent $(\partial^{0} v) (\alpha) = \lambda_{\mathrm{Conf}} (\alpha)$. This implies (2). \end{proof}
We also obtain the following analogue of Theorem \ref{thm:proper_cr}:
\begin{theorem} \label{thm:proper_conformal} Let $(M,g)$ be a Riemannian manifold. Suppose that $ G \, \leq \, \mathop{\rm Conf}\nolimits(M,g)$ is a subgroup of conformal automorphisms that acts properly on $M$. Then there exists on $M$ a Riemannian metric $h$ conformal to $g$, such that $ G \, \leq \, \mathop{\rm Iso}\nolimits(M, h) \, . $ \end{theorem} \begin{proof} Since $G$ acts properly, Theorem \ref{thm:vanish} implies that the associated differentiable cohomology class $ \mu_{\mathrm{Conf}}$ vanishes. Hence, there exists on $M$ a Riemannian metric $h = v \cdot g $, such that $ G \, \leq \, \mathop{\rm Iso}\nolimits(M, h) \, . $ \end{proof}
\subsection{Cohomological characterization of proper actions} \label{sec:proper_and_coho}
The following theorem shows that the properness of $CR$- and conformal actions is basically a vanishing property for differentiable cohomology. This also implies Corollary \ref{cor:main_cr_and_h1} in the introduction:
\begin{theorem} \label{thm:proper_and_coho} Let $G$ be a Lie group of diffeomorphisms of the smooth manifold $M$ that preserves either a strictly pseudo-convex $CR$-structure or a conformal Riemannian structure on $M$. If $G$ is closed in the group of all diffeomorphisms of $M$, then the following are equivalent: \begin{enumerate} \item $G$ acts properly on $M$. \item $H^1_{d}\left(G, \mathcal{C}^{\infty}(M, {\mathbb R})\right) = \{0\}$. \item $H^{r}_{d}\left(G, \mathcal{C}^{\infty}(M, V)\right) = \{0\}$, for all $r>0$, and any differentiable $G$-module $V$. \end{enumerate}
\end{theorem}
\begin{proof} Suppose that $G$ acts properly, then by Theorem \ref{thm:smooth_vanish}, (3) is satisfied. Now (3) clearly implies (2). Finally, if (2) is satisfied, the canonical class $\mu \in H^{1}_{d}\left(G , \mathcal{C}^{\infty}(M,{\mathbb R}^{>0})\right)$ associated to either the $CR$- or conformal structure on $M$ vanishes. In the first case, as is implied by Proposition \ref{pro:can_class_CR}, $G$ preserves an associated contact Riemannian metric on $M$, respectively in the case of Proposition \ref{pro:can_class_Conf}, the group $G$ preserves a Riemannian metric in the given conformal class. Since the group of isometries of a Riemannian manifold acts properly by the theorem of Myers and Steenrod \cite{MS}, and $G$ is a closed group of isometries, $G$ acts properly on $M$. \end{proof}
\subsection{Locally conformal K\"ahler metrics} \label{sec:lcK}
In this subsection we let $(X,J)$ denote a connected complex manifold satisfying ${\rm dim}_{\mathbb R}\, X=2n\geq 4$. Then a Hermitian metric $h$ for $X$ is called a \emph{locally conformal K\"ahler} metric if it is \emph{locally} conformal to a K\"ahler metric. (This means that
there exists an open covering $\{ U_\ell \}$ of $X$ and functions $u_\ell: U_\ell {\rightarrow}{\mathbb R}^{>0}$ such that $h=u_\ell \cdot g_{\ell}$, where $g_{\ell}$ is a K\"ahler metric on $(U_\ell, J)$, compare \cite{Va}, \cite{DO}.)
\subsubsection{Conformal automorphisms of K\"ahler metrics} Our first result concerns $lcK$-metrics which admit a K\"ahler metric in their conformal class:
\begin{theorem}\label{thm:hollcK} Let $(X,\{J,g\})$ be a K\"ahler manifold which is not holomorphically isometric to ${\mathbb C}^n$. Then the following hold:
{\bf (1)}\, There exists an $lcK$ manifold $(X, \{h,J\})$ such that the $lcK$ metric $h$ is conformal to the K\"ahler metric $g$ and satisfying
$$ \mathop{\rm Iso}\nolimits\left(X,\{h,J \}\right) \, = \, \mathrm{Conf}(X,\{g,J\})
\, . $$
{\bf (2)}\, Furthermore, the holomorphic isometry group $\mathop{\rm Iso}\nolimits\left(X,\{h,J \}\right)$ is maximal among all isometry groups of Hermitian metrics conformal to $g$. \end{theorem} \begin{proof}
Since $X$ is not holomorphically isometric to ${\mathbb C}^{n}$, we infer from Theorem \ref{thm:schoenhol} that the Lie group $ \mathrm{Conf}(X,\{g,J\})$ is acting properly on $X$.
Therefore, according to Theorem \ref{thm:proper_conformal}, there exists a Riemannian metric $h$ conformal to $g$, such that $ \mathrm{Conf}(X,\{g,J\}) \leq \mathop{\rm Iso}\nolimits(X, h)$. Since $h$ is conformal to the K\"ahler metric $g$, $h$ is Hermitian for the complex structure $J$ and $ \mathrm{Conf}(X,\{g,J\}) \leq
\mathop{\rm Iso}\nolimits\left(X,\{h,J \}\right)$. From the fact that $\mathrm{Conf}(X,\{g,J\}) = \mathrm{Conf}(X,\{h,J\})$,
we deduce that in fact $\mathrm{Conf}(X,\{g,J\}) =
\mathop{\rm Iso}\nolimits\left(X,\{h,J \}\right)$. This proves (1).
For the proof of (2), note that
$\operatorname*{Iso} \left(X,\{ h',J \}\right) \leq \, \mathrm{Conf}(X,\{h',J\}) = \mathrm{Conf}(X,\{g,J\}) = \operatorname*{Iso}\left(X,\{ h,J \}\right)$, by (1).
This shows (2). \end{proof}
\paragraph{\em Remark} For a K\"ahler manifold $(X, \{g,J\})$ with ${\rm dim}_{\mathbb R}\, X=2n\geq 4$,
the group of holomorphic conformal diffeomorphisms $$ {\mathop{\rm Conf}\nolimits}(X,\{g,J\})$$ coincides with the group of holomorphic homothetic transformations \begin{equation*} \begin{aligned} {\rm Hoth} & (X,\{g,J\}) = \\ & \{ \alpha \in \operatorname*{Diff}\,(X)\mid \alpha^* g= c\cdot g,\, \alpha_*J=J \alpha_* \text{, for some } c \in {\mathbb R}^{>0} \} \; . \end{aligned} \end{equation*} In fact, any biholomorphic conformal map between K\"ahler manifolds is easily seen to be an isometry up to constant scaling of the metrics, compare \cite[Theorem 6.5, p. 66]{YANO}. Therefore, (1) of Theorem \ref{thm:hollcK} also asserts the equality \begin{equation} \tag{{\bf 1'}}
\mathop{\rm Iso}\nolimits\left(X,\{h,J \}\right) \, = \, \mathrm{Hoth} (X,\{h,J\})
\, = \, \mathop{\rm Hoth}\nolimits\left(X,\{g,J\}\right) \; .
\end{equation}
\subsubsection{Conformal automorphisms of $lcK$-metrics}
We are now looking at the conformal class of $lcK$-metrics and their conformal automorphisms in general. First we recall that $lcK$-manifolds admit K\"ahler coverings (compare \cite[p. 65]{Va}):
\begin{lemma}[K\"ahler covering] Let $X$ be a simply connected complex manifold and $h$ any $lcK$-metric on $X$. Then \begin{enumerate} \item The $lcK$-metric $h$ is conformal to a K\"ahler metric $g$ (which is unique up to a constant factor). \item The $lcK$-metric $h$ is locally holomorphically conformal to ${\mathbb C}^{n}$ if and only if $g$ is a flat K\"ahler metric (that is, $g$ is locally holomorphically isometric to ${\mathbb C}^{n}$). \item The $lcK$-manifold $(X,\{h,J\})$ is holomorphically conformal to ${\mathbb C}^{n}$ if and only if the K\"ahler metric $(X,\{g,J\})$ is holomorphically isometric to ${\mathbb C}^{n}$. \end{enumerate} \end{lemma} \begin{proof} Let $\Theta = h(J \cdot, \cdot)$ be the fundamental two-form of the $lcK$-metric $h$. There exists a closed (global) one-form $\theta$ on $X$, called Lee form, such that $$ \displaystyle d\Theta=\theta\wedge \Theta \, . $$ Indeed, by the definition of $lcK$ metric, $\theta$ is constructed as $\theta = d \log u_{\ell}$ on $U_{\ell}$, where $h=u_\ell \cdot g_{\ell}$ and $g_{\ell}$ is a K\"ahler metric on $U_\ell$, for some covering of $X$. Furthermore, assuming that $X$ is simply connected,
$\theta=df$ for some function $f$ on $X$. Then $ \Omega=e^{-f}\cdot \Theta$ is a K\"ahler form on $X$. This proves (1).
The $lcK$-metric $h$ being locally holomorphically conformal to ${\mathbb C}^{n}$ means that there exists locally a holomorphic conformal map to ${\mathbb C}^{n}$. Since the K\"ahler metric $g$ is conformal to $h$, this map is also locally conformal for $g$, so $g$ is locally holomorphically conformal to the standard flat K\"ahler space ${\mathbb C}^{n}$. By the above remark following Theorem \ref{thm:hollcK}, $g$ is actually locally homothetic to the standard complex space ${\mathbb C}^{n}$, which also implies that $g$ flat and locally holomorphically isometric to ${\mathbb C}^{n}$, proving (2). The remark also implies (3). \end{proof}
The following is now a consequence of Theorem \ref{thm:hollcK}:
\begin{corollary}\label{cor:hollcK} Let $(X, \{h,J \})$ be an $lcK$-manifold whose universal covering manifold is not holomorphically conformal to ${\mathbb C}^n$. Then there exists an $lcK$-metric $h'$ conformal to $h$ which is satisfying
$$ \mathop{\rm Iso}\nolimits\left(X,\{h',J \}\right) \, = \, \mathrm{Conf}(X,\{h,J\}) \; .
$$
\end{corollary} \begin{proof} Let $\mathsf{p}: (\tilde X, \tilde h) \to (X,h)$ be the universal covering $lcK$-manifold. Let $g$ be the K\"ahler metric on $\tilde X$ conformal to $\tilde h$. Then the group of decktransformations $\Gamma$ for the covering $\mathsf{p}$ is contained in $\mathrm{Hoth} (\tilde X,\{g,J\})$. Since $(X,g)$ is not holomorphically isometric to ${\mathbb C}^{n}$, Theorem \ref{thm:hollcK} implies that there exists a Hermitian metric $\tilde h'$ conformal to $g$ such that $\mathop{\rm Iso}\nolimits(\tilde X,\{ \tilde h',J \}) \, = \, \mathrm{Conf}(\tilde X,\{g,J\}) = \mathrm{Conf}(\tilde X,\{\tilde h,J\})$. Since $\Gamma \leq \mathop{\rm Iso}\nolimits(\tilde X,\{ \tilde h',J \})$, there exists a unique $lcK$-metric $h'$ on $X$, such that $\mathsf{p}: (\tilde X, \tilde h') \to (X,h')$ is a holomorphic Riemannian covering, and this metric is conformal to $h$. Moreover, $\mathop{\rm Iso}\nolimits(X,\{ h',J \}) = \mathrm{Conf}(X,\{h,J\})$. \end{proof}
\appendix
\section{Other related results}
\subsection{Conformally flat K\"ahler manifolds} \label{A1}
The following results were prov\-ed by K.\ Yano and I.\ Mogi \cite[Theorem 4.1]{MK} in the case $\dim_{{\mathbb R}} X \geq 6$, and by S. Tanno \cite{Tano} for $\dim_{{\mathbb R}} X =4$.
\begin{theorem}\label{YM} Any conformally flat K\"ahler manifold $X$ of dimension $n\geq 6$ is locally flat, that is, it is \emph{locally holomorphically isometric} to ${\mathbb C}^n$. \end{theorem} Here the complex space ${\mathbb C}^n$ carries the standard flat K\"ahler metric.
\begin{theorem}\label{Tanno} Any\/ $4$-dimensional conformally flat K\"ahler manifold $X$ is \emph{locally holomorphically isometric} to either ${\mathbb C}^2$ or the product of $2$-dimensional surfaces ${\mathbb H}^2_{\mathbb R}\times S^2$ with constant opposite sign. \end{theorem}
Recall that the universal covering of any conformally flat $m$-dimensional manifold admits a conformal development map into the sphere $S^{m}$ and this map is unique up to composition with an element of $\mathop{\rm Conf}\nolimits(S^{m}) =\mathop{\rm PO}\nolimits(m+1,1)$, see for example \cite[Theorem 4]{Kui}. In the case of the two theorems, the universal covering space $\tilde X$ is thus mapped to $S^4-\{\infty\}= {\mathbb C}^2$ or the sphere complement $S^4-S^1={\mathbb H}^2_{\mathbb R}\times S^2$ (compare \cite{KA2}) through this developing map.
\subsection{Proof of Theorem \ref{thm:schoenhol}} \label{proofA2}
We assume that $ \mathop{\rm Conf}\nolimits(X, \{g,J\}) \leq \mathop{\rm Conf}\nolimits(X,g)$ does not act properly on $X$, where $g$ is the K\"ahler metric.
Therefore, Schoen's theorem \cite{SC} implies that there exists a conformal diffeomorphism from $X$ to either the sphere $S^{2n}$ or ${\mathbb R}^{2n}$. Since $X$ is assumed to be K\"ahler, $H^2(X,{\mathbb R}) \neq \{0\}$, in case $X$ is compact. Therefore, $S^{2n}$, $n>1$, does not occur.
Thus, in the case $n>2$, there exists a conformal diffeomorphism $\psi: X \to {\mathbb R}^{2n}$. In particular, $(X,g)$ is conformally flat and $X$ is simply connected.
\begin{proof}[Proof of Theorem \ref{thm:schoenhol}] \hspace{1cm}
(i)\, When $\dim_{{\mathbb R}} X\geq 6$, by the result of Yano and Mogi Theorem \ref{YM}, the K\"ahler manifold $X$ has everywhere {holomorphic sectional curvature $0$}. In particular, it is locally holomorphically isometric to the standard flat complex space ${\mathbb C}^{n}$ (compare \cite[IX, Theorem 7.9]{KN2}). Since $X$ is simply connected, the usual monodromy argument (see \cite{Kul}) shows that there is a holomorphic map $\varphi: X\,{\rightarrow}\, {\mathbb C}^n$, that is also an isometric immersion.
Up to a conformal map we may identify both domains ${\mathbb R}^{2n}$ and ${\mathbb C}^n$ with an open subset $U = S^{2n} - \{ p \}$ contained in $S^{2n}$. Thus $\varphi, \psi$ correspond to maps $$\bar \varphi, \bar \psi: \; X \to U \subset S^{2n}$$ and both $\bar \varphi$, $\bar \psi$ are developing maps for the locally flat conformal structure associated with $(X,g)$.
The uniformization theorem for locally flat conformal structures \cite{Kui} implies that the two developing maps $\bar \varphi$, $\bar \psi$
are \emph{equivalent} by an element $\alpha$ of $\mathop{\rm Conf}\nolimits(S^{2n})=\mathop{\rm PO}\nolimits(2n+1,1)$, so that $\alpha \circ \bar \psi= \bar \varphi$ on $X$. Clearly, since $\bar \psi$ is a diffeomorphism, this shows that $\bar \varphi$ is an injective embedding of $X$ into $U$. Note that every conformal embedding of Euclidean space ${\mathbb R}^{2n}$ into $S^{2n}$ has as image $S^{2n}$ with a point removed. Since the image of $\alpha \circ \bar \psi$ is contained in $U$, we conclude that $\alpha \circ \bar \psi$ is, in fact, surjective onto $U$. Thus the holomorphic isometric immersion $\varphi: X{\rightarrow}\, {\mathbb C}^n$ turns out to be an isometry.
(ii)\, $\dim_{{\mathbb R}} X=4$. By Tanno's result, Theorem \ref{Tanno}, since $X$ is conformally flat K\"ahler,
as above, the simply connected K\"ahler manifold $X$ admits a holomorphic immersion $\varphi: X{\rightarrow} {\mathbb C}^2$ or $\varphi: X{\rightarrow} {\mathbb H}^2_{\mathbb R}\times S^2$, respectively. Recall that there is a developing diffeomorphism $\psi: X{\rightarrow} {\mathbb R}^{4}$. As in part (i), by the uniformization theorem for locally flat conformal structures, there exists $\alpha \in\mathop{\rm PO}\nolimits(5,1)$ with $\alpha \circ \bar \psi = \bar \varphi$. Since the image of $\alpha \circ \bar \psi$ is $S^{4} - \{ p\}$, it cannot occur that $\bar \varphi$ takes values in ${\mathbb H}^2_{\mathbb R}\times S^2 = S^{4} -S^{1}$. As in part (i), we conclude that $\varphi: X{\rightarrow} {\mathbb C}^2$ is a holomorphic isometry.
(iii) $\dim_{{\mathbb R}} X = 2$. By Schoen's theorem \cite{SC}, $X$ is simply connected and there exists a conformal diffeomorphism to either $S^{2}$ or ${\mathbb R}^{2}$. In terms of uniformization of Riemann surfaces we conclude that $X$ is biholomorphic to $S^{2}$ or ${\mathbb C}$. \end{proof}
\end{document} | arXiv |
Asymptotic behavior of solutions of a nonlinear degenerate chemotaxis model
DCDS-B Home
Analysis of nanofluid flow past a permeable stretching/shrinking sheet
November 2020, 25(11): 4127-4164. doi: 10.3934/dcdsb.2020091
Analyzing plasmid segregation: Existence and stability of the eigensolution in a non-compact case
Eva Stadler 1,2,, and Johannes Müller 1,3,
Department of Mathematics, Technical University of Munich, Boltzmannstr. 3, 85748 Garching, Germany
Present address: Infection Analytics Program, Kirby Institute, UNSW Sydney, Wallace Wurth Building, High St, Kensington NSW 2052, Australia
Institute of Computational Biology, HelmholtzZentrum München - German Research Center for, Environmental Health, Ingolstädter Landstr. 1, 85764 Neuherberg, Germany
* Corresponding author: Eva Stadler
Received May 2019 Revised November 2019 Published November 2020 Early access April 2020
Fund Project: This work is part of the published dissertation thesis "Transport equations and plasmid-induced cellular heterogeneity" by ES (https://mediatum.ub.tum.de/1469742?id=1469742).
This work was funded by the German Research Foundation (DFG) priority program SPP1617 "Phenotypic heterogeneity and sociobiology of bacterial populations" (DFG MU 2339/2-2).
We study the distribution of autonomously replicating genetic elements, so-called plasmids, in a bacterial population. When a bacterium divides, the plasmids are segregated between the two daughter cells. We analyze a model for a bacterial population structured by their plasmid content. The model contains reproduction of both plasmids and bacteria, death of bacteria, and the distribution of plasmids at cell division. The model equation is a growth-fragmentation-death equation with an integral term containing a singular kernel. As we are interested in the long-term distribution of the plasmids, we consider the associated eigenproblem. Due to the singularity of the integral kernel, we do not have compactness. Thus, standard approaches to show the existence of an eigensolution like the Theorem of Krein-Rutman cannot be applied. We show the existence of an eigensolution using a fixed point theorem and the Laplace transform. The long-term dynamics of the model is analyzed using the Generalized Relative Entropy method.
Keywords: Plasmid dynamics, Growth-fragmentation-death equation, Eigenproblem, Non-compactness, Generalized Relative Entropy.
Mathematics Subject Classification: 92D25, 35Q92, 35L02, 35B40, 44A10.
Citation: Eva Stadler, Johannes Müller. Analyzing plasmid segregation: Existence and stability of the eigensolution in a non-compact case. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4127-4164. doi: 10.3934/dcdsb.2020091
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34] as described in [36,Section 5]">Figure 1. Numerically constructed eigenfunctions for $ \Phi(\xi) = 6\,\xi\,(1-\xi) $, $ \mu = 0.1/h $, $ b(z) = z(1-z)/h $, and different $ \beta $, viz. $ \beta = 0.45/h $ (black), $ 0.5/h $ (dark gray), and $ 0.55/h $ (light gray). The different cell division rates lead to different behavior of the eigenfunction $ \mathcal{U}(z) $ at the maximal plasmid number $ z_0 = 1 $. The eigenfunction was numerically constructed using the software R [34] as described in [36,Section 5]
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Eva Stadler Johannes Müller | CommonCrawl |
Many years ago I finished my PhD, entitled "The Mathematicization of Nature" (1998, LSE), in which I discussed the applicability of mathematics, the Quine-Putnam indispensability argument and considered a number of nominalist responses to it, in the end rejecting them all. The monograph Burgess & Rosen 1997, A Subject with No Object, had appeared a year earlier. At the time, I'd considered the issue definitively settled. And so I decided not to bother publishing anything in the area, as it would be pointless. (I did publish Ch. 5, which was about truth theories and deflationism.)
Jeez was I wrong! In the last fourteen years, the debate about the indispensability argument has continued, taking off in many different directions. And I'm pretty baffled at the whole thing. Even the formulation of the Indispensability Argument often given is incorrect, as far as I can see. So, here is mine, and I think it is reasonably faithful to the intentions of both Quine and Putnam.
1. Nominalism
Nominalism (in mathematics) is the claim that there are no numbers, sets, functions, and so on. (In addition, nominalism normally implies also that there are no syntactical types: i.e., finite sequences of symbols. Consequently there is a problem for nominalism at the level of syntax, a problem discussed long ago by Quine & Goodman 1947, "Steps Toward a Constructive Nominalism".) In particular, there are no mixed sets and no mixed functions. A mixed set is a set of non-mathematical entities, and a mixed function is a function whose domain or range includes some non-mathematical entities.
However, modern science is up-to-its-neck in mixed sets and functions. All the various quantities invoked in science are mixed functions. Laws of nature express properties of such mixed functions, and express relations between them. A differential equation in physics usually expresses some property of some mixed function(s). For example, it might say that a function defined on time instants has a certain property.
2. The Quine-Putnam Indispensability Argument
Quine and Putnam both gave versions of an argument, which I formulate like this:
(1) Mathematicized theories are inconsistent with nominalism.
(2) Our best scientific theories are mathematicized.
(C) So, if one accepts our best scientific theories, one must reject nominalism.
(The name "Quine-Putnam Indispensability Argument" derives, I believe, from Hartry Field.)
The argument for the first premise (1) is based on the following kind of example. Maxwell's Laws include the mathematicized law:
At any spacetime point $p$, $(\underline{\nabla} \cdot \underline{B})(p) = 0$.
This is often abbreviated "$(\underline{\nabla} \cdot \underline{B}) = 0$", but it is clear that quantification over spacetime points is implicitly intended.
Since $\underline{B}$ is a vector field on spacetime, it is a mixed function, whose domain is spacetime, and whose range is some vector space (one that is isomorphic to $\mathbb{R}^3$). If nominalism is true, it follows that $\underline{B}$ does not exist, and therefore that Maxwell's Law, "$(\underline{\nabla} \cdot \underline{B}) = 0$", is false. (A slightly fancier version of this would refer instead to the electromagnetic field tensor $F_{ab}$, whose components unify the $\underline{B}$-field and the $\underline{E}$-field; but the considerations are more or less the same.) In general, if nominalism is true, then any such mathematicized theory is false. This establishes (1).
If this is right, then we have a major worry: this shows that a certain philosophical theory (nominalism) contradicts science. This is probably the central reason I am suspicious of nominalism.
The argument for the second premise (2) requires one to compare our working mathematicized theories (Maxwell's theory; Schroedinger equation; Einstein's field equations; Yang-Mills gauge theories, etc.) with proposed nominalistic replacements. Having done this, one then concludes that either there are insuperable technical obstacles to the nominalization of such theories; or, though there may be, for certain mathematicized theories, nominalized replacements, even so, the mathematicized original is always a scientifically better theory, by scientific standards. (This is the sort of point emphasized by John Burgess, who semi-hemi-demi-jokingly suggested that nominalists might submit articles with their replacement theories to The Physical Review.)
So, our best scientific theories are mathematicized and are inconsistent with nominalism. Hence, if one accepts such theories, one must reject nominalism. This conclusion is epistemic only in a conditional sense. It simply says that one cannot have one's cake and eat it. One cannot be a nominalist and a scientific realist.
3. Responses
3.1 Rejecting (1): The rough idea is that mathematicized theories are consistent with nominalism. So, such theories may be true even though there are no mathematical entities. So, the magnetic field $\underline{B}$ doesn't exist, but, even so, Maxwell's Laws are true. This kind of view is advocated by Jody Azzouni (2004, Deflating Existential Consequence: A Case for Nominalism), but I'm not sure I quite understand it.
3.2 Rejecting (2): Our working scientific theories can be nominalized, and such theories are epistemically better. The betterness consists in the advantage that issues from the elimination of mathematicalia. This is essentially Hartry Field's approach (Field 1980, Science Without Numbers).
3.3 Accepting, but living with, the conclusion: a nominalist might accept the Quine-Putnam argument, conceding the premises, but insist that one may "accept" mathematicized scientific theories in a weaker sense, which involves only accepting their nominalistic content. This is essentially Mary Leng's and Joseph Melia's approach (Leng 2010, Mathematics and Reality; and Melia 2000, "Weaseling Aaway the Indispensability Argument" (Mind)).
Published by Jeffrey Ketland at 4:46 am
joseph 20 January 2013 at 11:41
Thanks Jeff for your concise and clear summary. Could I ask, not having read Field, is the 'advantage that issues from the elimination of mathematicalia" basically that science then becomes nominalism friendly? and is the appeal of nominalism for the would be scientific realist partly explained by the worry that realism about abstracta and mathematicalia seem to make a thoroughgoing materialism impossible?
apologies for any naiveté, not a professional philosopher and all that.
Colin 20 January 2013 at 15:05
Ad 3.1, I think the basic idea in it's negative form is simple enough. It is just a rejection of the Quinean criterion of ontological commitment. One can quantify-over without being-committed-to. This naturally raises a lot of questions about what the criteria for commitment *are*. I think Azzouni's answer is unsatisfying in this respect, but one can imagine how this project might go, e.g. it might have something to do with believer/speaker intentions.
Jeffrey Ketland 20 January 2013 at 16:05
"is the 'advantage that issues from the elimination of mathematicalia" basically that science then becomes nominalism friendly? and is the appeal of nominalism for the would be scientific realist partly explained by the worry that realism about abstracta and mathematicalia seem to make a thoroughgoing materialism impossible?"
The advantage is ontological parsimony; so a nominalized scientific theory doesn't require the existence of abstract entities (like vector fields, or vectors). So one can defuse the epistemological problem of "access": how we "know" about abstract entities, given that they're non-causal.
Field also highlights another advantage, namely that a nominalized theory (of the kind he gives in his 1980 monograph) explains the conventional role played by the mathematical aspects of usual scientific theories. Roughly, the mathematics is only increasing the conceptual simplicity of theories (which is why it is useful), and not contributing to its genuine physical content.
Nathan Coppedge 26 January 2013 at 22:06
Here is a contention and agreement. One extension of Quine's argument is that all forms of abstracta reduce to 'applications' which only have validity through pragmatic reference. Then not only does math get caught up in a vast contingency of conflating validity with usefulness or vice versa (a kind of Jacob's Ladder problem), but there is an appealing argument for the universalism of applications that may open mathematics to what you call 'mixed functions'.
For example, if a set is not a universal set in terms of its contents, what is it saying about its usefulness? Although this may reduce adequately to a claim that a function is an application, it would do not not make assumptions about what this means---since 'application' suddenly may mean 'mathematics' to the mathemician---it does not mitigate arguments that 'other applications' could be equally useful. Perhaps this amounts to the claim that mathematicians are attracted to an 'illusory' usefulness much in the way that statisticians sometimes become poor economists.
I find it appealing that mathematics may be 'just one form' of usefulness, and I think there is no implicit problem in widening the field of potential quasi-mathematical applications. The question is really one of standardization, once it is accepted---I think it is obvious to accept---that mathematics is a form of nominalism. And amongst other questions is the question of whether math has been 'synched' to real cognitive processes, or instead merely taps into strengths and weaknesses, proving things that are already true about the mind, yet remain trivial.
Yes, in a nutshell, that sounds right.
But I think Azzouni formulates matters *epistemically* (in terms of beliefs, etc.), whereas Quine's analysis is semantic, and concerns the existential implications of sentences. That is, if a sentence (e.g., a natural language sentence) $S$ is regimented as $\exists x Fx$, then $S$ implies that there are $F$s.
There is wriggle room here at the "regimentation" stage; so Quine gives lots of examples of eliminating apparent existential implications in Word & Object.
But not so much wriggle room, I think, at the semantic implication stage, because it's hard to see what $\exists x Fx$ could mean except that there are $F$. For the semantic clause defining $\exists x Fx$ is: it's true iff there are $F$s. I suppose one could have say a free logic, or a number of different quantifiers in operation, with inner and outer domains and whatnot. This is a sort of Graham Priest direction. But I think this isn't what Azzouni has in mind, and it's connected to speaker intentions and beliefs, as you say.
So, as far as I understand it, Azzouni notion of "ontological commitment" is an epistemic notion, rather than Quine's semantic one.
I think Quine's argument is syntactic, while his style is semantic.
Nathan Coppedge 8 June 2014 at 00:14
To be clear, my view of nominalism DOES support the view that nominalism can make statements, or prove entities, albeit these are not entities in the sense of mathematics. For example, the categorical-deductive sets {A-B:C-D, A-D:C-B} can refer to any appropriate object in space, although with some level of metaphoricalization, or at least some statement about its intended functionality.
More about this in my book, The Dimensional Philosopher's Toolkit (2013, 2014), not to be confused with Baggini and Fosl's classic.
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According to you, if an thing referred to by a mathematical equation does not "exist", then the equation is false. This seems to me to be a self-serving claim. After all, if "2" does not exist, that would mean that "2 + 2 = 4" is necessarily false.
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Alternative splicing links histone modifications to stem cell fate decision
Yungang Xu ORCID: orcid.org/0000-0002-9834-30061,2 na1,
Weiling Zhao1,2 na1,
Scott D. Olson3,
Karthik S. Prabhakara3 &
Xiaobo Zhou1,2
Understanding the embryonic stem cell (ESC) fate decision between self-renewal and proper differentiation is important for developmental biology and regenerative medicine. Attention has focused on mechanisms involving histone modifications, alternative pre-messenger RNA splicing, and cell-cycle progression. However, their intricate interrelations and joint contributions to ESC fate decision remain unclear.
We analyze the transcriptomes and epigenomes of human ESC and five types of differentiated cells. We identify thousands of alternatively spliced exons and reveal their development and lineage-dependent characterizations. Several histone modifications show dynamic changes in alternatively spliced exons and three are strongly associated with 52.8% of alternative splicing events upon hESC differentiation. The histone modification-associated alternatively spliced genes predominantly function in G2/M phases and ATM/ATR-mediated DNA damage response pathway for cell differentiation, whereas other alternatively spliced genes are enriched in the G1 phase and pathways for self-renewal. These results imply a potential epigenetic mechanism by which some histone modifications contribute to ESC fate decision through the regulation of alternative splicing in specific pathways and cell-cycle genes. Supported by experimental validations and extended datasets from Roadmap/ENCODE projects, we exemplify this mechanism by a cell-cycle-related transcription factor, PBX1, which regulates the pluripotency regulatory network by binding to NANOG. We suggest that the isoform switch from PBX1a to PBX1b links H3K36me3 to hESC fate determination through the PSIP1/SRSF1 adaptor, which results in the exon skipping of PBX1.
We reveal the mechanism by which alternative splicing links histone modifications to stem cell fate decision.
Embryonic stem cells (ESCs), the pluripotent stem cells derived from the inner cell mass of a blastocyst, provide a vital tool for studying the regulation of early embryonic development and cell fate decision and hold the promise for regenerative medicine [1]. The past few years have witnessed remarkable progress in understanding the ESC fate decision, i.e. either pluripotency maintenance (self-renewal) or proper differentiation [2]. The underlying mechanisms have been largely expanded from the core pluripotent transcription factors (TFs) [3], signaling pathways [4,5,6,7,8,9], specific microRNAs [10, 11], and long non-coding RNAs [12] to alternative pre-messenger RNA (mRNA) splicing (AS) [13, 14], histone modifications (HMs) [15,16,17,18,19], and cell-cycle machinery [20]. These emerging mechanisms suggest their intricate interrelations and potential joint contributions to ESC pluripotency and differentiation, which, however, remain unknown.
Alternative splicing (AS) is one of the most important pre-mRNA processing to increase the diversity of transcriptome and proteome in tissue-dependent and development-dependent manners [21]. The estimates based on RNA-sequencing (RNA-seq) revealed that up to 94%, 60%, and 25% of genes in human, Drosophila melanogaster, and Caenorhabditis elegans, respectively, undergo AS [21,22,23,24,25]. AS also provides a powerful mechanism to control the developmental decision in ESCs [26,27,28]. Specific isoforms are necessary to maintain both the identity and activity of stem cells and switching to different isoforms ensures proper differentiation [29]. In particular, the AS of TFs plays major roles in ESC fate determination, such as FGF4 [30] and FOXP1 [13] for hESC, and Tcf3 [14] and Sall4 [31] for mouse ESCs (mESCs). Understanding the precise regulations on AS would contribute to the elucidation of ESC fate decision and has attracted extensive efforts [32]. For many years, studies aiming to shed light on this process focused on the RNA level, characterizing the manner by which splicing factors (SFs) and auxiliary proteins interact with splicing signals, thereby enabling, facilitating, and regulating RNA splicing. These cis-acting RNA elements and trans-acting SFs have been assembled into splicing code [33], revealing a number of AS regulators critical for ESC differentiation, such as MBNL [34] and SON [28]. However, these genetic controls are far from sufficient to explain the faithful regulation of AS [35], especially in some cases that tissue-specific AS patterns exist despite the identity in sequences and ubiquitous expression of involved SFs [36, 37], indicating additional regulatory layers leading to specific AS patterns. As expected, we are increasingly aware that splicing is not an isolated process; rather, it occurs co-transcriptionally and is presumably also regulated by transcription-related processes. Emerging provocative studies have unveiled that AS is subject to extensive controls not only from genetic but also epigenetic mechanisms due to its co-transcriptional occurrence [38]. The epigenetic mechanisms, such as HMs, benefit ESCs by providing an epigenetic memory for splicing decisions so that the splicing pattern could be passed on during self-renewal and be modified during differentiation without the requirement of establishing new AS rules [38].
HMs have long been thought to play crucial roles in ESC maintenance and differentiation by determining what parts of the genome are expressed. Specific genomic regulatory regions, such as enhancers and promoters, undergo dynamic changes in HMs during ESC differentiation to transcriptionally permit or repress the expression of genes required for cell fate decision [15]. For example, the co-occurrence of the active (H3K4me3) and repressive (H3K27me3) HMs at the promoters of developmentally regulated genes defines the bivalent domains, resulting in the poised states of these genes [39]. These poised states will be dissolved upon differentiation to allow these genes to be active or more stably repressed depending on the lineage being specified, which enables the ESCs to change their identities [40]. In addition to above roles in determining transcripts abundance, HMs are emerging as major regulators to define the transcripts structure by determining how the genome is spliced when being transcribed, adding another layer of regulatory complexity beyond the genetic splicing code [41]. A number of HMs, such as H3K4me3 [42], H3K9me3 [43], H3K36me3 [44, 45], and hyperacetylation of H3 and H4 [46,47,48,49,50], have been proven to regulate AS by either directly recruiting chromatin-associated factors and SFs or indirectly modulating transcriptional elongation rate [38]. Together, these studies reveal that HMs determine not only what parts of the genome are expressed, but also how they are spliced. However, few studies focused on the detailed mechanisms, i.e. epigenetic regulations on AS in the context of cell fate decision.
Additionally, cell-cycle machinery dominates the mechanisms underlying ESC pluripotency and differentiation [20, 51]. Changes of cell fates require going through the cell-cycle progression. Studies in mESCs [52] and hESCs [53, 54] found that the cell fate specification starts in the G1 phase when ESCs can sense differentiation signals. Cell fate commitment is only achieved in G2/M phases when pluripotency is dissolved through cell-cycle-dependent mechanisms. However, whether the HMs and AS and their interrelations are involved in these cell-cycle-dependent mechanisms remains unclear. Therefore, it is intuitive to expect that HMs could contribute to ESC pluripotency and differentiation by regulating the AS of genes required for specific processes, such cell-cycle progression. Nevertheless, we do not even have a comprehensive view of how HMs relate to AS outcome at a genome-wide level during ESC differentiation. Therefore, further studies are required to elucidate the extent to which the HMs are associated with specific splicing repertoire and their joint contributions to ESC fate decision between self-renewal and proper differentiation.
To address these gaps in current knowledge, we performed genome-wide association studies between transcriptome and epigenome of the differentiation from the hESCs (H1 cell line) to five differentiated cell types [15]. These cells cover three germ layers for embryogenesis, adult stem cells, and adult somatic cells, representing multiple lineages of different developmental levels (Additional file 1: Figure S1A). This carefully selected dataset enabled our understanding of AS epigenetic regulations in the context of cell fate decision. First, we identified several thousands of AS events that are differentially spliced between the hESCs and differentiated cells, including 3513 mutually exclusive exons (MXE) and 3678 skipped exons (SE) which were used for further analyses. These hESC differentiation-related AS events involve ~ 20% of expressed genes and characterize the multiple lineage differentiation. Second, we profiled 16 HMs with chromatin immunoprecipitation sequencing (ChIP-seq) data available for all six cell types, including nine types of acetylation and seven types of methylation. Following the observation that the dynamic changes of most HMs are enriched in AS exons and significantly different between inclusion-gain and inclusion-loss exons, we found that three of the 16 investigated HMs (H3K36me3, H3K27ac, and H4K8ac) are strongly associated with 52.8% of hESC differentiation-related AS exons. We then linked the association between HMs and AS to cell-cycle progression based on the additional discovery that the AS genes predominantly function in cell-cycle progression. More intriguingly, we found that HMs and AS are associated in G2/M phases and involved in ESC fate decision through promoting pluripotency state dissolution, repressing self-renewal, or both. In particular, with experimental valuations, we demonstrated an H3K36me3-regulated isoform switch from PBX1a to PBX1b, which is implicated in hESC differentiation by attenuating the activity of the pluripotency regulatory network. Collectively, we presented a mechanism conveying the HM information into cell fate decision through the regulation of AS, which will drive extensive studies on the involvements of HMs in cell fate decision via determining the transcript structure rather than only the transcript abundance.
AS characterizes hESC differentiation
The role of AS in the regulation of ES cell fates adds another notable regulatory layer to the known mechanisms that govern stemness and differentiation [55]. To screen the AS events associated with ES cell fate decision, we investigated a panel of RNA-seq data during hESC (H1 cell line) differentiation [15]. We considered four cell types directly differentiated from H1 cells, including trophoblast-like cells (TBL), mesendoderm (ME), neural progenitor cells (NPC), and mesenchymal stem cells (MSC). We also considered IMR90, a cell line for primary human fetal lung fibroblast, as an example of terminally differentiated cells. These cells represent five cell lineages of different developmental levels (Additional file 1: Figure S1A). We identified thousands of AS events of all types with their changes of "per spliced in" (ΔPSIs) are > 0.1 (inclusion-loss) or < − 0.1 (inclusion-gain), and with the false discovery rates (FDRs) are < 0.05 based on the measurement used by rMATS [56] (Additional file 1: Figure S1B and Table S1, see "Methods"). We implemented further analyses only on the most common AS events, including 3513 MXEs and 3678 SEs, which are referred to as hESC differentiation-associated AS exons (Additional file 1: Figure S1C and Additional file 2: Table S2).
These hESC differentiation-related AS exons possess typical properties, as previously described [57, 58], as follows: (1) most of their hosting genes are not differentially expressed between hESCs and differentiated cells (Additional file 1: Figure S1D); (2) they tend to be shorter with much longer flanking introns compared to the average length of all exons and introns (RefSeq annotation), respectively (Additional file 1: Figure S2A, B); (3) the arrangement of shorter AS exons surrounded by longer introns is consistent across cell lineages and AS types (Additional file 1: Figure S2C, D); and (4) the lengths of AS exons are more often divisible by three to preserve the reading frame (Additional file 1: Figure S2E).
During hESC differentiation, about 20% of expressed genes undergo AS (2257 genes for SE and 2489 genes for MXE), including previously known ESC-specific AS genes, such as the pluripotency factor FOXP1 [13] (Fig. 1a) and the Wnt/β-catenin signalling component CTNND1 [14] (Fig. 1b). These hESC differentiation-related AS genes include many TFs, transcriptional co-factors, chromatin remodelling factors, housekeeping genes, and bivalent domain genes implicated in ESC pluripotency and development [39] (Fig. 1c and Additional file 1: Figure S1C). Enrichment analysis based on a stemness gene set [59] also shows that hESC differentiation-related AS genes are enriched in the regulators or markers that are most significantly associated with stemness signatures of ESCs (Additional file 1: Figure S3A, see "Methods").
AS characterizes the hESC differentiation. a, b Sashimi plots show two AS events of previously known ESC-specific AS events, FOXP1 (a) and CTNND1 (b). Inset histograms show the PSIs (Ψ) of the AS exons in all cell types based on the MISO estimation. c The bar graph shows that the number of total AS events and lineage-specific AS events increase coordinately with the developmental levels. Higher developmental level induces more (lineage-specific) AS events. MXE.sp. and SE.sp. indicate the percentage of lineage-specific AS events. d Heat maps show the differential "percent splice in" (ΔPSIs) of SE (left) and MXE (right) AS events (rows) for each cell lineage (columns). For MXE event, the ΔPSIs are of the upstream exons. e, f The hosting genes of MXE (e) and SE (f) AS events characterize cell lineages. Black and white bars refer to the common AS genes shared by all cell lineages, while the colour bars indicate the lineage-specific AS genes. The length of the colour bars is proportional to the percentage of lineage-specific genes. Dark fills indicate the inclusion-gain events, while light fills indicate the inclusion-loss events. The numbers in the bars are the proportion of corresponding parts; the numbers in the parentheses are the numbers of common AS genes or lineage-specific AS genes of each lineage. Gain or loss for MXE events refers to the upstream exons. Also see Additional file 1: Figures S1–S3
Clustering on AS events across cell lineages show lineage-dependent splicing patterns (Fig. 1d). Upon hESC differentiation, the SE exons tend to lose their inclusion levels (inclusion-loss), while the upstream exons of MXE events are likely to gain their inclusion levels (inclusion-gain) (Fisher's exact test, p = 3.83E-107). The numbers of AS events increase accordingly with the developmental level following hESC differentiation (Fig. 1c). For example, the differentiation to ME involves the fewest AS events and ME presents the most stem-cell-like AS profiles, while the IMR90 has the most AS events and exhibits the most similar AS profiles to adult cells (Fig. 1c, d). Inter-lineage comparisons show, on average, that 42.0% of SE and 56.4% of MXE events (Fig. 1c, d and Additional file 1: Figure S3B, C), involved in 29.6% and 38.6% of AS hosting genes (Fig. 1e, f and Additional file 1: Figure S3D, E), are lineage-specific. In contrast, only 0.65% of SE and 0.14% of MEX events (Additional file 1: Figure S3B, C), involved in 0.49% and 1.52% of AS hosting genes, are shared by all lineages (Fig. 1e, f and Additional file 1: Figure S3D, E). Similar trends are observed from pairwise comparisons (Additional file 1: Figure S3F). Furthermore, one-third of AS genes (n = 881) have both MXE and SE events (Additional file 1: Figure S3G). Only four genes are common across all cell lineages and AS types, of which the AS events of Ctnnd1 and Mbd1 have been reported to regulate mESC differentiation [14]. Together, these results demonstrate that AS depicts lineage-dependent and developmental level-dependent characterizations of hESC differentiation.
Dynamic changes of HMs predominantly occur in AS exons
In ESCs, epigenetic mechanisms contribute mainly to maintaining the expression of pluripotency genes and the repression of lineage-specific genes in order to avoid exiting from stemness. Upon differentiation, epigenetic mechanisms orchestrate the expression of developmental programs spatiotemporally to ensure the heritability of existing or newly acquired phenotypic states. Though epigenetic signatures are mainly found to be enriched in promoters and enhancers, it has become increasingly clear that they are also present in gene bodies, especially in exon regions, implying a potential link of epigenetic regulation to pre-mRNA splicing [60, 61]. Consistent with previous reports [36, 37, 62], we also observed that few involved SFs are differentially expressed during H1 cells differentiation (Additional file 1: Figure S3H, see "Methods"), which confirms the existence of an additional layer of epigenetic regulations on AS. However, the extents to which the AS is epigenetically regulated and how these AS genes contribute to the cell fate decision are poorly understood. We focused on 16 HMs, including nine histone acetylation and seven histone methylation that have available data in all six cell types (see "Methods") and aimed to reveal their associations with AS genes during hESC differentiation.
To investigate whether the dynamic changes of these HMs upon cell differentiation prefer the AS exons consistently (Fig. 2a, b), we profiled the differential HM patterns of around the hESC differentiation-associated AS exons and the same number of randomly selected constitutive splicing (CS) exons of the same AS genes for each differentiation lineage. We compared the changes of ChIP-seq reads count (normalized Δ reads count, see "Methods") in ± 150-bp regions around the splice sites upon hESC differentiation (Fig. 2c and Additional file 1: Figure S4, see "Methods"). Except for a small part of cases (with black dots or boxes in Fig. 2d), most HMs changed more significantly around AS exons than around constitutive exons upon hESC differentiation (Mann–Whitney–Wilcoxon test, p ≤ 0.05, Fig. 2d and Additional file 1: Figure S4). Nevertheless, some HMs displayed strong links to AS, such as H3K79me1 and H3K36me3, while others only had weak link strengths, such as H3K27me3 and H3K9me3 (Fig. 2d). This result is consistent with the fact that the former are involved in active expression and AS regulation [38, 44, 63], while the latter are the epigenetic marks of repressed regions and heterochromatin [64]. The link strengths are presented as the -log10 p values to test whether the HM changes consistently prefer the AS exons across different cell lineages and AS types (Fig. 2d sidebar graph, see "Methods"). Taken together, these results, from a global view, revealed a potential regulatory link from HMs to RNA splicing, of which some are strong while the others are weak.
Dynamic changes of HMs predominantly occur in AS exons. a, b Genome browser views of representative H3K36me3 changes in MXE (exemplified as FGFR2) and SE (exemplified as CDC27) events, respectively, showing that the changes of H3K36me3 around the AS exons (blue shading) are more significant than around the flanking constitutive exons (gray shading) in 4 H1-derived cell types and IMR90. The tracks of H1 are duplicated as yellow shadings overlapping with other tracks of the derived cells (green) for a better comparison. c Representative profiles of HM changes (normalized Δ reads number) around the AS exons and randomly selected constitutive splicing (CS) exons upon hESC differentiation, shown as the average of all cell lineages pooled together. The ± 150-bp regions (exons and flanking introns) of the splice sites were considered and 15 bp-binned to produce the curves. It shows that the changes of HMs are more significant around AS exons than around constitutive exons, especially in exonic regions (gray shading). The p values, Mann–Whitney–Wilcoxon test. d The statistic significances for changes of all 16 HMs in all cell lineages and pooling them together (pooled), represented as the -log10 p values based on Mann–Whitney–Wilcoxon test. The detailed profiles are provided in Additional file 1: Figure S4. Black boxes indicate the cases that HMs around constitutive exons change more significantly than around AS exons, corresponding to the red-shaded panels in Additional file 1: Figure S4. Sidebars represent the significances whether the changes of HMs are consistently enriched in AS exons across cell lineages, showing the link strength between AS and HMs and represented as the -log10 p value based on Fisher's exact test. The yellow vertical line indicates the significance cutoff of 0.05. Also see Additional file 1: Figure S4
Three HMs are significantly associated with AS upon hESC differentiation
To quantitatively associate the HMs with AS, all ChIP-seq data were processed for narrow peak calling using MACS2 [65]. For each AS exon of each differentiation lineage, we then quantified the differential inclusion levels, i.e. the changes of "percent splice in" (ΔPSIs, Additional file 1: Figure S1B), and the differential HMs signals, i.e. the changes of normalized narrow peak height of ChIP-seq (ΔHMs, Additional file 1: Figure S5A, see "Methods") between H1 and differentiated cells. We observed significant differences in all HM profiles (except H3K27me3, Additional file 1: Figure S5B) between the inclusion-gain and inclusion-loss exons across cell lineages and AS types (Mann–Whitney–Wilcoxon test, p ≤ 0.05) (Fig. 3a and Additional file 1: Figure S5B). However, three independent correlation tests showed only weak global quantitative associations between the ΔPSIs and ΔHMs for some HMs (Fig. 3c and Additional file 1: Figure S5C), including eight HMs for MXE AS exons and eight HMs for SE AS exons. The weak associations may indicate that only subsets of AS exons are strongly associated with HMs and vice versa, which is consistent with a recent report [66].
A subset of HMs and AS are strongly associated upon hESC differentiation. a Representative profiles of HM (H3K36me3) changes (normalized Δ reads number) around the inclusion-gain (red lines) and inclusion-loss (blue lines) AS exons, as well as randomly selected constitutive splicing (CS) exons (black lines) for both MXE (left) and SE (right) AS events. It shows that HM changes are significantly different between inclusion-gain and inclusion-loss AS exons (p values, Mann–Whitney–Wilcoxon test). Additional file 1: Figure S5B provides the whole significances of all HMs across AS types and cell lineages. b Pearson correlation test between differential HM signals (ΔHMs) and differential inclusion levels (ΔPSIs), taking H3k36me3 as an example. Additional file 1: Figure S5C provides the correlation test results of other HMs based on two more tests. c A representative k-means cluster shows a subset of SE AS events having a negative correlation between the ΔPSIs and the ΔHMs of H3K36me3. Additional file 1: Figures S5D and S6 provide all the clustering results. d Scatter plot shows that HM-associated AS events display significant correlations between the ΔPSIs and the ΔHMs upon hESC differentiation, taking H3K27ac–associated (positively) MXE events as an example. Also see Additional file 1: Figures S5, S6
To explore the subsets of highly associated AS exons and corresponding HMs, we performed k-means clustering on the sets of inclusion-gain and inclusion-loss exons of SE and MXE events, separately, taking the ΔHMs of eight identified HMs as epigenetic features (Fig. 3c and Additional file 1: Figures S5D and S6, see "Methods"). We obtained three subsets of HM-associated SE exons and three subsets of HM-associated MXE exons (Additional file 3: Table S3). The three HM-associated SE subsets include 180, 664, and 1062 exons and are negatively associated with H4K8ac (Additional file 1: Figure S6), negatively associated with H3K36me3 (Fig. 3c), and positively associated with H3K36me3 (Additional file 1: Figure S6), respectively. The three HM-associated MXE subsets include 99, 821, and 971 exons and are positively associated with H3K27ac (Fig. 3d), negatively associated with H3K36me3 (Additional file 1: Figure S6), and positively associated with H3K36me3 (Additional file 1: Figure S6), respectively. The exons of each subset show significant correlations between their ΔPSIs and ΔHMs upon hESC differentiation (Fig. 3d). These HM-associated AS exons account for an average of 52.8% of hESC differentiation-related AS events, on average (Additional file 1: Figure S5E).
Of the three AS-associated HMs, H3K36me3 has both positive and negative correlations with AS exons. This is consistent with the fact that H3K36me3 has dual functions in regulating AS through two different chromatin-adapter systems, PSIP1/SRSF1 [45] and MRG15/PTBP1 [44]. The former increases the inclusion levels of targeting AS exons, whereas the latter decreases the inclusion levels [38]. As expected, 139 and 11 of our identified H3K36me3-associated AS genes have been reported to be regulated by SRSF1 [67, 68] (Additional file 1: Figure S5F) and PTBP1 [69] (Additional file 1: Figure S5G), respectively. Taken together, our analysis showed that more than half (52.8%) of hESC differentiation-associated AS events are significantly associated with three of 16 HMs during hESC differentiation, including H3K36me3, H3K27ac, and H4K8ac.
HM-associated AS genes predominantly function in G2/M phases to facilitate hESC differentiation
Epigenetic mechanisms have been proposed to be dynamic and play crucial roles in human ESC differentiation [15, 16]. Given the aforementioned associations between HMs and AS, and the well-established links between AS and hESC differentiation, we hypothesized that the three HMs (H3K36me3, H3K27ac, and H4K8ac) may contribute to stem cell differentiation through their associated AS events. To test our hypothesis and gain more insights into the differences between the HM-associated and HM-unassociated AS events, we performed comparative function analyses between their hosting genes, revealing that HMs are involved in alternatively splicing the core components of cell-cycle machinery and related pathways to regulate stem cell pluripotency and differentiation.
We found that HMs prefer to be associated with even shorter AS exons (Additional file 1: Figure S7A, p < 0.001, Student's t-test), though AS exons are shorter than the average length of all exons (Additional file 1: Figure S2A). HM-associated genes (n = 2125) show more lineage specificity, i.e. more genes (49.76% vs 29.6% of MXE or 38.6% of SE genes) are lineage-specific (Additional file 1: Figures S7B and S3D, E), regardless of whether IMR90 is included or not (Additional file 1: Figure S7C). Only a few HM-associated genes are shared by different cell lineages, even in pairwise comparisons (Additional file 1: Figure S7D); the most common shared genes are lineage-independent housekeeping genes (Additional file 1: Figure S7E). These suggest that HM-associated AS genes contribute more to lineage specificity. In addition, the HM-associated AS genes (966 of 2125) are more enriched in stemness signatures than unassociated AS genes (429 of 1057) (Fig. 4a). TF binding enrichment analysis shows that HM-associated AS genes are likely to be regulated by TFs involved in cell differentiation, whereas HM-unassociated AS genes are likely to be regulated by TFs involved in cell proliferation and growth (Fig. 4b). All these results suggest that HM-associated and HM-unassociated AS genes function differently during hESC differentiation.
HM-associated AS genes predominantly function in G2/M cell-cycle phases contributing to hESC differentiation. a HM-associated AS genes are enriched more significantly in stemness signatures than HM-unassociated AS genes. b TF binding enrichment shows that HM-associated AS genes prefer to be regulated by TFs involved in cell differentiation, while the HM-unassociated AS genes are prone to be regulated by TFs involved in cell proliferation and growth. c GO enrichment analysis shows that HM-associated AS genes are enriched more significantly in cell-cycle progression than HM-unassociated AS genes, shown as the -log10 p values after FDR (≤ 0.05) adjustment. d The significant enrichment of HM-associated AS genes in the cell cycle are consistent across cell lineages, with the MSC as an exception that no significant enrichment was observed. e The top 20 enriched functions show that HM-associated AS genes involved in cell-cycle progression prefer to function in G2/M phases and DNA damage response. f The canonical pathway enrichment shows that AMT/ATR-mediated DNA damage response is the top enriched pathway of HM-associated AS genes. The vertical lines (yellow) indicate the significance cutoff of 0.05. Also see Additional file 1: Figures S7, S8
Gene Ontology (GO) enrichment analysis shows that more than half of the HM-associated AS genes (1120 of 2125) function in cell-cycle progression and exhibit more significant enrichment than do HM-unassociated AS genes (376 of 1057, Fig. 4c, d and Additional file 1: Figure S8A). The significance of the top enriched GO term (GO:0007049, cell cycle) is consistent across cell lineages, although HM-associated AS genes exhibit more lineage specificity and few of them are shared among lineages (Additional file 1: Figures S7B–D and S8B). These results suggest the involvement of HMs in AS regulation of the cell-cycle machinery that has been reported to be exploited by stem cells to control their cell fate decision [20].
Further study of the top enriched cell-cycle AS genes (Fig. 4d and Additional file 1: Figure S8A) shows that HM-associated (n = 282) and HM-unassociated AS genes (n = 150) play roles in different cell-cycle phases and related pathways. The former is prone to function in G2/M phases and DNA damage response (Fig. 4e, f). This indicates that HMs contribute to cell differentiation, at least partially, via AS regulations in these phases, which is consistent with the fact that inheritance of HMs in daughter cells occurs during the G2 phases [20]. The latter play roles in G1 phase, cell-cycle arrest, and Wnt/β-catenin signalling (Additional file 1: Figure S8C, D). Since cell fate choices seem to occur or at least be initiated during G1/S transition [53], while cell fate commitment is achieved in G2/M [54], it could be rational for stem cells to change their identity during the G2 phase when HMs are reprogrammed [20].
Intriguingly, the top enriched pathway of HM-associated AS genes is "ATM/ATR-mediated DNA damage response," which is activated in S/G2 phases and has been recently reported as a gatekeeper of the pluripotency state dissolution (PSD) that participates in allowing hESC differentiation [54]. Together with our previous results [19], it suggests the presence of a combinational mechanism involving HMs and AS, wherein HMs facilitate the PSD and cell fate commitment by alternatively splicing the key components of the ATM/ATR pathway. Additionally, many cell-cycle TF genes are involved in the top enriched HM-associated AS gene set. The pre-B-cell leukaemia transcription factor 1 (PBX1) is one of these genes that contribute to cell-cycle progression and is discussed later in next section. Taken together, we suggest that three of 16 HMs function in positive or negative ways affect the AS of subsets of genes and further contribute to hESC differentiation in a cell-cycle phase-dependent manner. The results suggest a potential mechanistic model connecting the HMs, AS regulations, and cell-cycle progression with the cell fate decision.
Splicing of PBX1 links H3K36me3 to hESC fate decision
The past few years have identified key factors required for maintaining the pluripotent state of hESCs [70, 71], including NANOG, OCT4 (POU5F1), SOX2, KLF4, and c-MYC, the combination of which was called Yamanaka factors and sufficient to reprogram somatic cells into induced pluripotent stem cells (iPSCs) [72]. These factors appear to activate a transcriptional network that endows cells with pluripotency [73]. The above integrative analyses showed strong links between three HMs and RNA splicing, revealing a group of epigenetic regulated AS genes involved in cell-cycle machinery. PBX1 was one of the genes that their ASs are positively associated with H3K36me3 (Fig. 5a, b). Its protein is a member of the TALE (three-amino acid loop extension) family homeodomain transcription factors [74, 75] and well-known for its functions in lymphoblastic leukaemia [76,77,78,79] and several cancers [80,81,82,83,84,85,86,87,88,89]. PBX1 also plays roles in regulating developmental gene expression [90], maintaining stemness and self-renewal [80, 91, 92], and promoting the cell-cycle transition to the S phase [93]. Additionally, multiple lines of evidence obtained from in vivo and in vitro highlighted its functions as a pioneer factor [86, 94]. However, few studies have distinguished the distinct functions of its different isoforms.
Isoform switch from PBX1a and PBX1b during hESC differentiation. a Genome browser view shows the AS event and H3K36me3 signals of PBX1 upon hESC differentiation. The green horizontal bars below the ChIP-seq tracks indicate the narrow peaks called by MACS2. b The inclusion level for exon 7 of PBX1 is significantly correlated to the H3K36me3 signals over this exon across cell lineages. c The sequence difference of three protein isoforms of PBX1 and the main functional domains. d The relative expressions of PBX1a and PBX1b in 56 cells/tissues, representing the differential expressions of two isoforms in three groups based on their developmental states. e The expression levels of NANOG and OCT4 genes are negatively correlated with the expression of PBX1b. f The expression levels of PSIP1 and SRSF1 show significant positive correlations with the expression level of PBX1a. Also see Additional file 1: Figures S9, S10
PBX1 has three isoforms [95], including PBX1a, PBX1b, and PBX1c (Fig. 5c and Additional file 1: Figure S9A). PBX1a and PBX1b are produced by the AS of exon 7 (Fig. 5a) and attract most of the research attention of PBX1. PBX1b retains the same DNA-binding homeodomain as PBX1a, but changes 14 amino acids (from 334 to 347) and truncates the remaining 83 amino acids at the C-terminus of PBX1a (Fig. 5c and Additional file 1: Figure S9A). This C-terminal alteration of PBX1a has been reported to affect its cooperative interactions with HOX partners [96], which may impart different functions to these two isoforms. We here revealed its H3K36me3-regulated isoform switch between PBX1a and PBX1b, which functions at the upstream of pluripotency transcriptional network to link H3K36me3 with ESC fate decision.
We first observed differential transcript expressions of these two isoforms between the hESCs and differentiated cells, wherein PBX1a was predominantly transcribed in hESCs, while PBX1b was predominantly induced in differentiated cells (Fig. 5a and Additional file 1: Figure S9B). The same trend was also observed in an extended dataset of 56 human cell lines/tissues (Fig. 5d) from the Roadmap [97] and ENCODE [98] projects (Additional file 4: Table S4). Additionally, we did not observe significantly different expression of the total PBX1 and three other PBX family members across cell types (Additional file 1: Figure S9C, fold change < 2), indicating that the isoform switch of PBX1, rather than the differential expression of its family members, plays more important roles during hESC differentiation. To further test the possible mechanism by which PBX1b contributes to stem cell differentiation, we investigated the transcription levels of Yamanaka factors. Of these TFs, the NANOG is activated by direct promoter binding of PBX1 and KLF4, which is essential for stemness maintenance [91, 99]. Consistently, all these core factors are repressed in differentiated cells where PBX1b is highly expressed (Additional file 1: Figure S9D–G), even though the PBX1a is expressed. Based on the 56 human cell lines/tissues, we also observed significant negative correlations between expression of most important pluripotent factors (NANOG and OCT4) and PBX1b (Fig. 5e), as well as positive correlations between these two factors and PBX1a (or inclusion level of exon 7, Additional file 1: Figure S10A, B). Consistent with previous reports showing that the PBX1a and PBX1b differ in their ability to activate or repress the expression of reporter genes [100, 101], we hypothesize that PBX1a promotes the activity of the pluripotent regulatory network by promoting the expression of NANOG, whereas PBX1b may attenuate this activity by competitively binding and regulating the same target gene, since PBX1b retains the same DNA-binding domain as PBX1a. These observations are strongly suggestive that the switch from PBX1a to PBX1b is a mechanism by which PBX1 contributes to hESC differentiation via regulating the pluripotency regulatory network.
Exon 7 of PBX1 shows significantly positive correlations between its inclusion levels (PSIs) and the surrounding epigenetic signals of H3K36me3 in hESCs and differentiated cells (Fig. 5b). It suggests a potential role of H3K36me3 in regulating the isoform switch between PBX1a and PBX1b. To investigate the regulatory role of H3K36me3, we focused on two previously proved chromatin-adaptor complexes, MRG15/PTBP1 [44] and PSIP1/SRSF1 [45], which endow dual functions to H3K36me3 in AS regulation [38]. Based on the 56 cell lines/tissues from the Roadmap/ENCODE projects, we first found significant positive correlations between the expressions of PBX1a (or inclusion level of exon 7) and PSIP1/SRSF1 (Fig. 5f), but not with MRG15/PTBP1 (Additional file 1: Figure S10C, D). This result suggests that the AS of PBX1 is epigenetic regulated by H3K36me3 through the PSIP1/SRSF1 adaptor system, which was strongly supported by a recent report using the HeLa cell lines [67]. The overexpression of SRSF1 in Hela cells introduces a PSI increase of 0.18 for exon 7 of PBX1 (chr1: 164789308–164,789,421 based on NCBI37/hg19 genome assembly) based on the RNA-seq (Table S1 of [67]). Additionally, this exon was one of the 104 AS exons that were further validated using radioactive reverse transcription polymerase chain reaction (RT-PCR) (Table S2 of [67]). Their results showed that exon 7 of PBX1 is indeed a splicing target of SRSF1, supporting our conclusions.
We then validated the above hypotheses on MSCs and IM90 cells, since these two cells types show the most significant difference from H1 cells regarding our hypotheses (Fig. 5b). We cultured H1 cells, IMR90 cells, and induced H1 cells to differentiate into MSCs (H1-MSCs, see "Methods" for details). Additionally, we also included other two sources of MSCs, including one derived from human bone marrow (hBM-MSCs) and the other derived from adipose tissue (hAT-MSCs) (see "Methods" for details). Consistent with the results from RNA-seq, the same expression patterns of Yamanaka factors in H1, MSCs, and IMR90 cells were observed using quantitative RT-PCR (qRT-PCR) and western blot (Fig. 6a), which confirmed the pluripotent state of H1 cells and the differentiated states of other cell types. We then detected the isoform switch from PBX1a to PBX1b in our cultured cells, which are consistent both in mRNA and protein levels (Fig. 6b and Additional file 1: Figure S10E) and further confirmed by the western blot using PBX1b-specific antibody (anti-PBX1b) (Fig. 6b bottom and Additional file 1: Figure S10E iii). These results have verified that the PBX1b was significantly induced in differentiated cells, where the PBX1a was significantly reduced.
Isoform switch of PBX1 links H3K36me3 to hESC fate decision. a qRT-PCR and western blot show the expression levels of Yamanaka factors in H1, MSC, and IMR90 cells. Whiskers denote the standard deviations of three replicates. b RT-PCR and western blot show the isoform switches between PBX1a and PBX1b from H1 cells to differentiated cells. c i. ChIP-PCR shows the differential binding of PBX1b to NANOG promoter in H1 cells and differentiated cells; ii. ChIP-PCR shows the reduced H3K36me3 signal in differentiated cells; iii. ChIP-PCR shows the differential recruitment of PSIP1 to exon 7 of PBX1. d RIP-PCR show the differential recruitment of SRSF1 around exon 7 of PBX1. e Co-IP shows the overall physical interaction between PSIP1 and SRSF1 in all studied cell types. f The mechanism by which H3K36me3 is linked to cell fate decision by regulating the isoform switch of PBX1, which functions upstream of the pluripotency regulatory network. Also see Additional file 1: Figures S9, S10
We also validated the mechanism by which the splicing of PBX1 links H3K36me3 to stem cell fate decision. We first confirmed that PBX1b also binds to the promoter of NANOG at the same region where PBX1a binds to and the binding signals (ChIP-PCR) were high in the differentiated cells but very low in H1 stem cells (Fig. 6c i and Additional file 1: Figure S10F i). Consistent with the results from ChIP-seq, we also observed reduced H3K36 tri-methylation around exon 7 of PBX1 based on ChIP-PCR assay (Fig. 6c ii and Additional file 1: Figure S10F ii). Furthermore, the chromatin factor PSIP1 only showed high binding signal in H1 stem cells (Fig. 6c iii and Additional file 1: Figure S10F iii), which recruit the SF SRSF1 to the PBX1 exclusively in H1 stem cells (Fig. 6d and Additional file 1: Figure S10G) even though the physical binding between these two factors were universally detected in all cell types (Fig. 6e). All these experimental results suggested that, upon differentiation, stem cells reduced the H3K36 tri-methylation and may attenuate the recruitment of PSIP1/SRSF1 adaptor around exon 7 of PBX1, leading to the exclusion of exon 7 and highly expressed PBX1b in differentiated cells. High expression of PBX1b may attenuate the activity of PBX1a in promoting the pluripotency regulatory network.
Taken together, we suggested that H3K36me3 regulates the AS of PBX1 via the PSIP1/SRSF1 adaptor system, leading the isoform switch from PBX1a to PBX1b during hESC differentiation. Subsequently, PBX1b competitively binds to NANOG and abolishes the bindings of PBX1a. This competitive binding attenuates the pluripotency regulatory network to repress self-renewal and consequently facilitate differentiation (Fig. 6f). These findings revealed how the PBX1 contributes to cell fate decision and exemplify the mechanism by which AS links HMs to stem cell fate decision.
ESCs provide a vital tool for studying the regulation of early embryonic development and cell fate decision. [1]. In addition to the core pluripotency regulatory network, emerging evidence revealed other processes regulating ESC pluripotency and differentiation, including HMs, AS, cell-cycle machinery, and signalling pathways [54]. Here, we connected these previously separate avenues of investigations, beginning with the discovery that three of 16 HMs are significantly associated with more than half of AS events upon hESC differentiation. Further analyses implicated the association of HMs, AS regulation, and cell-cycle progression with hESC fate decision. Specifically, HMs orchestrate a subset of AS outcomes that play critical roles in cell-cycle progression via the related pathways, such as ATM/ATR-mediated DNA response [19], and TFs, such as PBX1 (Additional file 1: Figure S10H). In this way, HMs, AS regulation, and signalling pathways are converged into the cell-cycle machinery that has been claimed to rule the pluripotency [20].
Although epigenetic signatures, such as HMs, are mainly enriched in promoters and other intergenic regions, it has become increasingly clear that they are also present in the gene body, especially in exon regions. This indicates a potential link between epigenetic regulation and the predominantly co-transcriptional occurrence of AS. Thus far, H3K36me3 [44, 45], H3K4me3 [42], H3K9me3 [43], and the acetylation of H3 and H4 [46,47,48,49,50] have been revealed to regulate AS, either via the chromatin-adapter systems or by altering Pol II elongation rate. Here, we investigated the extent to which the HMs could associate with AS by integrative analyses on both transcriptome and epigenome data during hESC differentiation. We found that three HMs are significantly associated with about half of AS events. By contrast, a recent report showed that only about 4% of differentially regulated exons among five human cell lines are positively associated with three promoter-like epigenetic signatures, including H3K9ac, H3K27ac, and H3K4m3 [66]. Like that report, we also found a positive association of H3K27ac with a subset of AS events. However, our results differ regarding the other two HMs that we identified to be associated with AS.
In our study, H3K36me3 is associated with the most identified HM-associated AS events, either positively or negatively. It is reasonable since H3K36me3 is a mark for actively expressed genomes [63] and it has been reported to have dual roles in AS regulations through two different chromatin-adapter systems, PSIP1/SRSF1 [44] and MRG15/PTBP1 [45]. SRSF1 is a SF which will increase the inclusion of targeted AS exons and PTBP1 will decrease the inclusion levels of the regulated AS exons. Therefore, the exons that are regulated by the PSIP1/SRSF1 adapter system will show positive correlations with the H3K36me3, while the exons regulated by MRG15/PTBP1 will show negative correlations. Our results are consistent with this fact and show both direction correlations between different sets of AS events and H3K36me3. Many of these AS events in our study have been validated by other studies (Additional file 1: Figure S5F, G).
H4K8ac is associated with the fewest number of AS events in our results. Although rarely studied, H4K8ac is known to act as a transcriptional activator in both the promoters and transcribed regions [102]. Its negative association with AS is supported by the finding that it recruits proteins involved in increasing the Pol II elongation rate [103]. This suggests that H4K8ac may function in AS regulation by altering the Pol II elongation rate, rather than via the chromatin-adaptor systems. However, further studies are required. Collectively, possible reasons for the different results between this study and others [66] could be that: (1) we considered differentiation from hESCs to five different cell types, which covered more inter-lineage AS events than the previous report; or (2) different sets of epigenetic signatures were considered, which may lead to relatively biased results in both studies. Obviously, the inclusion of more cell lineages and epigenetic signatures may reduce this bias. Therefore, an extended dataset of 56 cell lines/tissues was included in our study and the observations support our results.
Our study also extended the understanding that HMs contribute to cell fate decision via determining not only what parts of the genome are expressed, but also how they are spliced [38]. We demonstrated that the HM-associated AS events have a significant impact on cell fate decision in a cell-cycle-dependent manner. The most intriguing discovery is that the HM-associated genes are enriched in G2/M phases and predominantly function in ATM/ATR-mediated DNA response. Evidentially, the ATM/ATR-mediated checkpoint has been recently revealed to attenuate pluripotency state dissolution and serves as a gatekeeper of the pluripotency state through the cell cycle [54]. The cell cycle has been considered the hub machinery for cell fate decision [20] since all commitments will go through the cell-cycle progression. Our study expanded such machinery by linking the HMs and AS regulation to cell-cycle pathways and TFs, which, together, contribute to cell fate decision (Additional file 1: Figure S10H).
We also exemplified our hypothesized mechanism by an H3K36me3-regulated isoforms switch of PBX1. In addition to its well-known functions in lymphoblastic leukaemia [76,77,78,79] and a number of cancers [80,81,82,83,84,85,86,87,88,89], PBX1 was also found to promote hESC self-renewal by corporately binding to the regulatory elements of NANOG with KLF4 [99]. We found that the transcriptions of two isoforms of PBX1, PBX1a and PBX1b, are regulated by H3K36me3 during hESC differentiation. Their protein isoforms competitively bind NANOG and the binding of PBX1b will abolish the binding of PBX1a, which further attenuates the activity of the core pluripotency regulatory network composed of Yamanaka factors. The switch from PBX1a to PBX1b is modulated by H3K36me3 via the PSIP1/SRSF1 adapter system [45]. Our results were also supported by an extended dataset of 56 cell lines/tissues from the Roadmap/ENCODE projects. Collectively, our findings expanded understanding of the core transcriptional network by adding a regulatory layer of HM-associated AS (Fig. 6f).
A very recent report showed that the switch in Pbx1 isoforms was regulated by Ptbp1 during neuronal differentiation in mice [104], indicating a contradiction that the AS of PBX1 should be negatively regulated by H3K36me3 via the MRG15/PTBP1 [44]. Our study also included the neuronal lineage and showed that differentiation to NPC is an exception, distinct from other lineages. If NPC is considered separately, the results are consistent with the recent report [104] showing that NPCs and mature neurons express increasing levels of PBX1a rather than PBX1b (Additional file 1: Figure S9B). Another recent report showed that PBX1 was a splicing target of SRSF1 in the HeLa cell line [67], which strongly supports our findings. Taken together, these evidence suggests that there are two parallel mechanisms regulating PBX1 isoforms in embryonic development, in which neuronal differentiation adopts a mechanism that is different from other lineages.
Finally, it is worth noting that both our work and other studies [66] reported that HMs cannot explain all AS events identified either during ESC differentiation or based on pairwise comparisons between cell types. Moreover, bidirectional communication between HMs and AS has been widely reported. For instance, the AS can enhance the recruitment of H3K36 methyltransferase HYPB/Set2, resulting in significant differences in H3K36me3 around the AS exons [66]. These findings increased the complexity of defining the cause and effect between HMs and AS. Nevertheless, our findings suggest that at least a subset of AS events are regulated epigenetically, similar to the way that epigenetic states around the transcription start sites define what parts of the genome are expressed. Additionally, as we described in our previous study, the AS outcomes may be estimated more precisely by combining splicing cis-elements and trans-factors (i.e. genetic splicing code) and HMs (i.e. epigenetic splicing code), as an "extended splicing code" [19]. Taken together, we presented a mechanism conveying the HM information into cell fate decision through the AS of cell-cycle factors or the core components of pathways that controlling cell-cycle progression (Additional file 1: Figure S10H).
We performed integrative analyses on transcriptome and epigenome data of the hESCs, H1 cell line, and five differentiated cell types, demonstrating that three of 16 HMs were strongly associated with half of AS events upon hESC differentiation. We proposed a potential epigenetic mechanism by which some HMs contribute to ESC fate decision through the AS regulation of specific pathways and cell-cycle genes. We experimentally validated one cell-cycle-related transcription factor, PBX1, which demonstrated that AS provides a mechanism conveying the HM information into the regulation of cell fate decisions (Fig. 6f). Our study will have a broad impact on the field of epigenetic reprogramming in mammalian development involving splicing regulations and cell-cycle machinery.
Identification of AS exons upon hESC differentiation
Aligned BAM files (hg18) for all six cell types (H1, ME, TBL, MSC, NPC, and IMR90) were downloaded from the source provided in reference. Two BAM files (replicates) of each cell type were analyzed using rMATS (version 3.0.9) [56] and MISO (version 0.4.6) [105]. The rMATS was used to identify AS exons based on the differential PSI (Ψ) values between each differentiated cell type and H1 cells (Additional file 1: Figure S1B). The splicing changes (ΔPSIs or ΔΨ) are used to identify the AS events between H1 and other cell types. A higher cutoff is always helpful in reducing the false positive while compromising the sensitivity. The cutoff, |ΔΨ| ≥ 0.1 or |ΔΨ| ≥ 10%, is widely accepted and used in AS identification [25, 106,107,108]. Many other studies even used 0.05 as the cutoff [109,110,111,112,113]. We did additional correlation analyses based on different ΔPSI cutoffs (0.1, 0.2, 0.3, 0.4, and 0.5). With the increase of the cutoffs, the number of AS events was significantly reduced (Additional file 1: Figure S11A); however, the correlations ware only slightly increased between AS and some HMs (Additional file 1: Figure S11B, C, upper panels), i.e. no consistent impacts of the cutoffs on the correlations were observed. Similarly, the correlation significances were also not consistently affected (Additional file 1: Figure S11B, C, lower panels). Therefore, in our study, only AS exons which hold the |ΔPSI| ≥ 0.1, p value ≤ 0.01, and FDR ≤ 5%, were considered as final hESC differentiation-related AS exons.
All identified AS event types are summarized in Additional file 1: Table S1. Finally, two types of AS exons, namely SE and MXE, which are most common and account for major AS events, were used for subsequent analysis (Additional file 2: Table S2). MISO was used to estimate the confidence interval of each PSI and generate Sashimi graphs [114] (see Figs. 1a, b and 5a). To match the ChIP-seq analysis, genomic coordinates of identified AS events were lifted over to hg19 using LiftOver tool downloaded from UCSC.
ChIP-seq data process and HM profiling
ChIP-seq data (aligned BAM files, hg19) were downloaded from Gene Expression Omnibus (GEO, accession ID: GSE16256). This dataset includes the ChIP-seq reads of up to 24 types of HMs for six cell types (H1, ME, TBL, MSC, NPC, and IMR90). Among these, nine histone acetylation modifications (H2AK5ac, H2BK120ac, H2BK5ac, H3K18ac, H3K23ac, H3K27ac, H3K4ac, H3K9ac, and H4K8ac) and seven histone methylation modifications (H3K27me3, H3K36me3, H3K4me1, H3K4me2, H3K4me3, H3K79me1, and H3K9me3) are available for all six cell types and therefore were used for our analyses.
To generate global differential profiles of HM changes between AS exons and constitutive exons upon hESC differentiation, for each MXE and SE AS events, we first randomly selected the constitutive splicing (CS) exons from the same genes, composing a set of CS exons. We then considered the HM changes in a ± 150-bp region flanking both splice sites of each AS and CS exon, i.e. a 300-bp exon-intron boundary region. Each region was 15 bp-binned. Alternatively, for a few cases where the exon or intron is < 150 bps, the entire exonic or intronic region was evenly divided into 10 bins. This scaling allows combining all regions of different lengths to generate uniform profiles of HM changes around the splice sites (see Fig. 2c and Additional file 1: Figure S4). To this end, we calculated the sequencing depth-normalized Δ reads number for each binned region between H1 cells and differentiated cells as follows:
$$ \varDelta\ reads\ number=\frac{\mathrm{reads}\ \mathrm{number}\ \mathrm{of}\ \mathrm{H}1\ \mathrm{cells}-\mathrm{reads}\ \mathrm{number}\ \mathrm{of}\ \mathrm{differentiated}\ \mathrm{cells}}{\mathrm{bin}\ \mathrm{size}\ \mathrm{in}\ \mathrm{bps}} $$
Each region is assigned a value representing the average Δ reads number between H1 cells and differentiated cells for each HM. We also compared HM profiles between the inclusion-gain and inclusion-loss exons (Fig. 3a and Additional file 1: Figure S5B) using the same strategy. The statistical results (Fig. 2c and Additional file 1: Figure S5B) are presented as the p values based on Mann–Whitney–Wilcoxon tests (using the R package).
To quantitatively link HMs to AS upon hESC differentiation, the ChIP-seq data were further processed by narrow peak calling. For each histone ChIP-seq dataset, the MACS v2.0.10 peak caller (https://github.com/taoliu/MACS/) [65, 115] was used to compare ChIP-seq signal to a corresponding whole cell extract (WCE) sequenced control to identify narrow regions of enrichment (narrow peak) that pass a Poisson p value threshold of 0.01. All other parameters (options) were set as default. We then compared the HM signals between H1 cells and differentiated cells. We defined the "differential HM signals (ΔHMs)" as the difference of the normalized peak signals (i.e. the heights of the narrow peaks) between H1 and the differentiated cells. Because the 3′ splice sites (3′ ends of the introns) determine the inclusion of the downstream AS exons [116] and the distances from the peaks to their target sites affect their regulatory effects [117], we normalize the peak signals against the distance (in kb) between the peak summits and 3′ splice sites (Additional file 1: Figure S5A). Since there is no evidence showing that distal HMs could regulate the AS, we only considered local peaks with at least 1 bp overlapping on either side of the AS exon. For exons without overlapping peaks, peak signals of these exons were set to zero. For the exons there are more than one overlapping peaks, the peak signals of these exons were set to the greater ones. For MXE events, only the upstream AS exons were considered due to their exclusiveness in inclusion level between these two exons, i.e. the sum of the PSIs for 2 exons of an MXE event is always 1.
Association studies and k-means clustering
To quantitatively estimate associations between HMs and AS, we first used three independent correlation tests, including Pearson correlation (PC), multiple linear regression (MLR), and logistic regression (LLR), to test global correlations between AS events and each of 16 HMs based on differential inclusions (ΔPSIs) and differential HM signals (ΔHMs). PC was performed using the R package (stats, cor.test(),method = 'pearson'). MLR and LLR were calculated using Weka 3 [118], wherein the ΔHMs are independent variables and ΔPSIs are response variables. The results show that only some HMs correlate with AS, and most correlations are weak (Additional file 1: Figure S5C). HMs that have significant correlations (p ≤ 0.05) with AS were used for further clustering analysis, through which we identified six subsets of HM-associated AS events (Additional file 3: Table S3).
K-means clustering was performed separately on inclusion-gain and inclusion-loss AS events of MXE and SE, based on the selected HM signatures (Additional file 1: Figure S5C, checked with a "√"). K was set to 6 for all clustering processes (Additional file 1: Figure S5D), which produced the minimal root mean square error (RMSE) for most cases based on a series of clustering with k in the range of 2–8 (data not shown). Then the two clusters that generate mirror patterns, of which one was from inclusion-gain events and one was from inclusion-loss events, were combined to be considered as a subset of HM-associated AS events (Additional file 1: Figure S6). Finally, we identified six subsets of HM-associated AS events displaying significantly positive or negative correlations with three HMs, respectively.
Gene expression quantification
For each cell type, two aligned BAM files (replicates) were used to estimate the expression level of genes using Cufflinks [119]. Default parameters (options) were used. The expression level of each gene was presented as FPKM for each cell type. Differentially expressed genes (DEGs) were defined as those genes whose fold changes between the differentiated cells and hESCs are > 2. Specifically for DEG analysis of SF genes, we collected a list of 47 ubiquitously expressed SFs with "RNA splicing" in their GO annotations from AmiGO 2 [120]. The enrichment significances in Additional file 1: Figures S1D and S3H are shown as the p values based on hypergeometric tests, using the DEGs of all expressed genes (with the FPKM ≥ 1 in at least hESCs or one of the differentiated cell types) as background. We found that the AS genes are generally not differentially expressed between hESCs and differentiated cells, indicating that they function in hESC differentiation via isoform switches rather than expression changes. Few SF genes show differential expression between hESCs and differentiated cells, indicating the existence of epigenetic control of AS, rather than the direct control on SFs expression.
Genome annotations
Since the RNA-seq reads (BAM) files and ChIP-seq read (BAM) files downloaded from the public sources were mapped to different human genome assemblies, NCBI36/hg18 (Mar. 2006) and GRCh37/hg19 (February 2009), respectively, we downloaded two version of gene annotations (in GTF formats) from the UCSC Table Browser [121]. The hg18 GTF file was used for rMATS and MISO to identify AS during the differentiation from H1 ESCs into five differentiated cells. The hg19 GTF file was used to define the genome coordinates of AS exons and further for ChIP-seq profile analysis (Figs. 2a–c, 3a, and Additional file 1: Figures S4 and S5A). We compared exonic and intronic lengths based on hg18 annotation (Additional file 1: Figure S2).
Gene Ontology enrichment analysis
The GO enrichment analysis was performed using ToppGene [122] by searching the HGNC Symbol database under default parameters (p value method: probability density function). Overrepresented GO terms for the GO domain "biological process" were used to generate data shown in Fig. 4c–e and Additional file 1: Figure S8A–C using either the FDR (0.05) adjusted p value or the enriched gene numbers (Additional file 1: Figure S8A).
Canonic pathway enrichment analysis
Both the HM-associated (n = 282) and HM-unassociated (150) AS genes from the top enriched GO term (GO:0007049) were used to perform canonic pathway enrichment (Fig. 4f and Additional file 1: Figure S8D) analysis through Ingenuity Pathway Analysis (IPA, http://www.ingenuity.com/products/ipa).
Stemness signature and TF binding enrichment analysis
StemChecker [59], a web-based tool to discover and explore stemness signatures in gene sets, was used to calculate the enrichment of AS genes in stemness signatures. Significances were tested (hypergeometric test) using the gene set from human (Homo sapiens) of this database as the background. For all AS genes (n = 3865), 2979 genes were found in this dataset. Of 2979 genes, 1395 were found in the stemness signature gene set, most of which (n = 813) are ESC signature genes. Additional file 1: Figure S3A shows the enrichment significance as the -log10 p values (Bonferroni adjusted). For HM-associated AS genes (n = 2125), 1992 genes were found in this dataset. Of 1992 genes, 966 were found in the stemness signature gene set, most of which (n = 562) are ESC signature genes. For HM-unassociated genes (n = 1057), 987 genes were found in this dataset. Of 987 genes, 429 were found in the stemness signature gene set, most of which (n = 251) are ESC signature genes. The significances are shown as -log10 p values (Bonferroni adjusted) in Fig. 4a.
FunRich (v2.1.2) [123, 124], a stand-alone functional enrichment analysis tool, was used for TF binding enrichment analysis to get the enriched TFs that may regulate the query genes. The top six enriched TFs of HM-associated and HM-unassociated AS genes are presented and shown as the proportion of enriched AS genes. It shows that HM-associated AS genes are more likely to be regulated by TFs involved in cell differentiation and development, while the HM-unassociated AS genes are more likely to be regulated by TFs involved in cell proliferation and renewal (Fig. 4b).
Roadmap and ENCODE data analysis
All raw data are available from the GEO accession IDs GSE18927 and GSE16256. The individual sources of RNA-seq data for 56 cell lines/tissues from Roadmap/ENCODE projects are listed in Additional file 4: Table S4. The RNA-seq data (BAM files) were used to calculate the PSI of exon 7 for PBX1 in each cell line/tissue and to estimate the expression levels of all gene (FPKM), based on aforementioned strategies. The relative expression levels of PBX1a and PBX1b shown in Fig. 5 and Additional file 1: Figure S10 were calculated as the individual PFKM value of each divided by their total FPKM values.
Statistical analyses and tests
Levels of significance were calculated with the Mann–Whitney–Wilcoxon test for Figs. 2c, d, 3a, and Additional file 1: Figure S4 and S5B, with Fisher's exact test for Figs. 1d, 2d, and Additional file 1: Figure S5B, with Student's t-test for Fig. 5d and Additional file 1: Figure S7A, and with a hypergeometric test for Additional file 1: Figures S1D and S3H. Levels of correlation significance were calculated with PC, MLR, and LLR for Fig. 3c, d, and Additional file 1: Figure S5C. MLR and LLR were performed using Weka 3 [118], whereas all other tests were performed using R packages. The p values for the enrichment analyses (Fig. 4, Additional file 1: Figures S3A and S8) were adjusted either by PDR or Bonferroni (refer to the corresponding method sections for details). The statistical analyses of the ChIP, RIP, and western blotting assays were shown in Additional file 1: Figure S10E–G. Three replicates were conducted for each assay and the quantitatively statistical analyses were performed on the relative band intensities normalized by control genes (ß-Actin) or input signals. ImageJ (https://imagej.nih.gov/ij/) was used to quantify the band intensities. ANOVA was used for statistical significance tests.
Cell culture and MSC induction
Human embryonic stem cell (H1cell line) line was purchased from the WiCell Research Institute (product ID: WA01). H1 cells were cultured on the matrigel-coated six-well culture plates (BD Bioscience) in the defined mTeSR1 culture medium (STEMCELL Technologies). The culture medium was refreshed daily. The H1-derived mesenchymal stem cells (H1-MSC) were differentiated from H1 cells as described previously [125]. Briefly, small H1 cell aggregates were cultured on a monolayer of mouse OP9 bone-marrow stromal cell line (ATCC) for nine days. After depleting the OP9 cell layer, the cells were then cultured in semi-solid colony-forming serum-free medium supplemented with fibroblast growth factor 2 and platelet-derived growth factor BB for two weeks. The mesodermal colonies were selected and cultured in mesenchymal serum-free expansion medium with FGF2 to generate and expand H1-MSCs. hAT-MSCs were derived from the subcutaneous fats provided by the National Disease Research Interchange (NDRI) using the protocol described previously [126]. Briefly, the adipose tissue was mechanically minced and digested with collagenase Type II (Worthington Bio, Lakewood, NJ, USA). The resulted single cell suspension was cultured in α-minimal essential medium with 5% human platelet lysate (Cook Regentec, Indianapolis, IN, USA), 10 μg/mL gentamicin (Life Technologies, CA, USA), and 1× Glutamax (Life Technologies). After reaching ~ 70% confluence, the adherent cells were harvested at the passage 1 (P1) hAT-MSC/AEY239. hBM-MSCs were isolated from commercially available fresh human bone marrow (hBM) aspirates (AllCells, Emeryville, CA, USA) and expanded following a standard protocol [127]. Briefly, hBM-MSC were cultured in α-minimal essential medium supplemented with 17% fetal bovine serum (FBS; lot-selected for rapid growth of MSC; Atlanta Biologicals, Norcross, GA, USA), 100 units/mL penicillin (Life Technologies, CA, USA), 100 μg/mL streptomycin (Life Technologies), and 2 mM L-glutamine (Life Technologies). After reaching ~ 70% confluence, the adherent cells were harvested at the passage 1 (P1) hBM-MSC-5204(G). The human IMR-90 cell line was purchased from the American Tissue Type Culture Collection (Manassas, VA, USA) and cultured in Eagle's Minimum Essential Medium (Thermo Scientific, Logan, UT, USA) supplemented 10% fetal calf serum, 2 mM L-glutamine, 100 U/mL penicillin, and 100 μg/mL streptomycin at 37 °C in humidified 5% CO2.
RNA isolation and RT-PCR
Total RNA was extracted from cells using the RNeasy Mini plus kit (Qiagen, Valencia, CA, USA) according to the manufacturer's instructions. RT reactions for first-strand complementary DNA (cDNA) synthesis were performed with 2 μg of total RNA in 25 uL reaction mixture containing 5 μL of 5× first-strand synthesis buffer, 0.5 mM dNTP, 0.5 μg oligo(dT)12-18mer (Invitrogen), 200 units of M-MLV reverse transcriptase (Promega, Madison, WI, USA), and 25 units of RNase inhibitor (Invitrogen). The mixture was then incubated at 42 °C for 50 min and 52 °C for 10 min. The reaction was stopped by heating at 70 °C for 15 min. The PCR amplifications were carried out in 50 μL reaction solution containing 1 μL of RT product, 5 μL of 10× PCR buffer, 0.15 mM MgCl2, 0.2 mM dNTP, 0.2 mM sense and antisense primers, and 2.5 U Taq polymerase (Bohringer, Mannheim, Germany). The sequences for the upstream and downstream primers of PBX1 and β-actin are listed in Additional file 1: Table S5. PCR reaction solution was denatured initially at 95 °C for 3 min, followed by 35 cycles at 95 °C for 1 min, 55 °C for 40 s, and 72 °C 40 s. The final extension step was at 72 °C for 10 min. The PCR products were resolved in a 2% ethidium bromide-containing agarose gel and visualized using ChemiDoc MP Imager (Bio-Rad).
Quantitative RT-PCR
The qPCR amplification was done in a 20-uL reaction mixture containing 100 ng of cDNA, 10 uL 2× All-in-One™ qPCR mix (GeneCopoeia, Rockville, MD, USA), 0.3 mM of upstream and downstream primers, and nuclear-free water. The PCR reaction was conducted with 1 cycle at 95 °C for 10 min, 40 cycles at 95 °C for 15 s, 40 °C for 30 s, and 60 °C for 1 min, followed by dissociation curve analysis distinguishing PCR products. The expression level of a gene was normalized with the endogenous control gene β-actin. The relative expressions of genes were calculated using the 2-ΔΔCT method, normalized by β-actin and presented as mean ± SD (n = 3) (Fig. 6a). The sequences of the paired sense and antisense primers for human SOX2, NANOG, OCT4, c-MYC, KLF4, and β-actin are listed in Additional file 1: Table S5.
Cells were lysed with 1× RIPA buffer supplemented with protease and phosphatase inhibitor cocktail (Roche Applied Science, Indianapolis, IN, USA) and stored in aliquots at − 20 °C until use. Twenty micrograms of cell lysates were mixed with an equal volume of Laemmli sample buffer, denatured by boiling, and separated by SDS-PAGE. The separated proteins were then transferred to PVDF membranes (Bio-Rad, Hercules, CA, USA). The membranes were blocked using 5% BSA for 1 h at room temperature and incubated with the first antibodies against 1% BSA overnight. SOX2, OCT4, NANOG, KLF4, c-MYC, and β-actin antibodies were from Cell Signalling Technology (Beverly, MA, USA). PBX1 antibody was from Abcam (Cambridge, MA, USA) and PBX1b antibody from Santa Cruz Technology (Dallas, TX, USA). After incubation with IgG horseradish peroxidase-conjugated secondary antibodies (Cell Signalling) for 2 h at room temperature, the immunoblots were developed using SuperSignal West Pico PLUS Chemiluminescent reagent (Thermo Fisher Scientific, Waltham, MA, USA) and imaged using ChemiDoc MP Imager (Bio-Rad).
Co-immunoprecipitation (Co-IP)
Co-IP was used to validate the PSIP1-SRSF1 interaction (Fig. 6e). Cells were lysed with ice-cold non-denaturing lysis buffer containing 20 mM Tris HCl (pH 8), 137 mM NaCl, 1% Nonidet P-40, 2 mM EDTA, and proteinase inhibitor cocktails. A total of 200 μg protein were pre-cleared with 30 uL protein G magnetic beads (Cell Signalling) for 30 min at 4 °C on a rotator to reduce non-specific protein binding. The pre-cleared lysate was immunoprecipitated with the 5 μL anti-SRSF1 antibody (Invitrogen) overnight and then incubated with protein G magnetic beads for 4 h at 4 °C. Beads without anti-SRSF1 antibody were used as an IP control. The protein G magnetic beads were washed five times with lysis buffer and the precipitated protein collected. The SRSF1-bound PSIP1 protein (also known as LEDGF) level was determined with the PSIP1 antibody (Human LEDGF Antibody, R&D Systems) using western blot as described above. Three specific bands (one for p75 and two for p52) were detected (Fig. 6e) as indicated on the R&D Systems website (https://www.rndsystems.com/products/human-ledgf-antibody-762815_mab3468).
ChIP assay was performed using a SimpleChIP Enzymatic Chromatin IP Kit (Cell Signaling Technology) according to the manufacturer's instructions. Briefly, 2 × 107 cells were cross-linked with 1% formaldehyde and lysed with lysis buffer. Chromatin was fragmented by partial digestion with Micrococcal Nuclease Chromatin. The protein–DNA complex was precipitated with ChIP-Grade Protein G Magnetic Beads (Cell Signalling) and ChIP-validated antibodies against H3K36me3 (Abcam), PSIP1 (Novus), and PBX1b (Santa Cruz). Normal mouse IgG and normal rabbit IgG were used as negative controls. After reversal of protein–DNA cross-links, the DNA was purified using DNA purification spin columns. The purified DNA fragments were then amplified with the appropriate primers on T100 thermal cycler (Bio-Rad). The primer pairs used for PCR are listed in Additional file 1: Table S5. The H3K36me3-immunoprecipitated and PSIP1-immunoprecipitated DNA fragments surrounding exon 7 of PBX1b and the promoter region of NANOG in the PBX1b-immunoprecipitated DNA fragments were PCR-amplified. The ChIP-PCR products were revealed by electrophoresis on a 2% agarose gel (Fig. 6c).
RIP assay was performed using Magna RIP™ RNA-Binding Protein Immunoprecipitation Kit (Millipore Sigma) following the manufacturer's instructions. Briefly, the cells were lysed with RIP lysis buffer with RNase and protease inhibitors. In total, 500 μg of total protein was precleared with protein G magnetic beads for 30 min. The protein G magnetic beads were preincubated with 5 μg of mouse monoclonal anti-SRSP1 antibody or normal mouse IgG for 2 h at 4 °C. The antibody-coated beads were then incubated with precleared cell lysates at 4 °C overnight with rotation. The RNA/protein/beads conjugates were washed five times with RIP was buffer and the RNA–protein complexes were eluted from the protein G magnetic beads on the magnetic separator. The SRSP1-bound RNA was extracted using acid phenol-chloroform and precipitated with ethanol. The RNA was then reverse-transcribed and the expression levels of PBX1a and PBX1b in the immunoprecipitated and non-immunoprecipitated (input) samples were analyzed using RT-PCR (Fig. 6d).
Alternative (pre-mRNA) splicing
CS:
Constitutive splicing
Embryonic stem cell
H2AK5ac:
H2A acetylated lysine 5
H2BK120ac:
H2B acetylated lysine120
H2BK5ac:
H2B acetylated lysine 5
H3K18ac:
H3 acetylated lysine 18
H3K27me3:
H3 tri-methylated lysine 27
H3K4ac:
H3 acetylated lysine 4
H3K4me1:
H3 mono-methylated lysine 4
H3 di-methylated lysine 4
H3 tri-methylated lysine 4
H3 mono-methylated lysine 79
HM:
MXE:
Mutually excluded exon
Percent splice in
Skipped exon
Splicing factor
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The authors thank Drs Guangxu Jin, Liang Liu, and Dongmin Guo from the Wake Forest School of Medicine, who gave a lot of comments and suggestions on the data analyses and interpretations. They also acknowledge the editorial assistance of Karen Klein, MA, from the Wake Forest Clinical and Translational Science Institute (UL1 TR001420; PI: McClain).
This work was funded by the National Institutes of Health (NIH) [1R01GM123037 and AR069395].
All RNA-seq and 16 HMs ChIP-seq data of H1 and five other differentiated cells are available in Gene Expression Omnibus (GEO) under accession number GSE16256 [128]. The BAM files of the RNA-seq data (two replicates for each, aligned to human genome hg18) are alternatively available at http://renlab.sdsc.edu/differentiation/download.html. Both RNA-seq and ChIP-seq data of 56 cell lines/tissues from the Roadmap/ENCODE projects [97, 98] are available on their official website (RoadMap: ftp://ftp.ncbi.nlm.nih.gov/pub/geo/DATA/roadmapepigenomics/by_sample/; ENCODE: ftp://hgdownload.cse.ucsc.edu/goldenPath/hg19/encodeDCC/) and all raw files are also available at GEO under the accession numbers GSE18927 [128] and GSE16256 [129]. Additional file 4: Table S4 provides the detailed information of these data.
Yungang Xu and Weiling Zhao contributed equally to this work.
Center for Computational Systems Medicine, School of Biomedical Informatics, The University of Texas Health Science Center at Houston, Houston, TX, 77030, USA
Yungang Xu
, Weiling Zhao
& Xiaobo Zhou
Center for Bioinformatics and Systems Biology, Wake Forest School of Medicine, Winston-Salem, NC, 27157, USA
Department of Pediatric Surgery, McGovern Medical School, The University of Texas Health Science Center at Houston, Houston, TX, 77030, USA
Scott D. Olson
& Karthik S. Prabhakara
Search for Yungang Xu in:
Search for Weiling Zhao in:
Search for Scott D. Olson in:
Search for Karthik S. Prabhakara in:
Search for Xiaobo Zhou in:
YX and XZ conceived the study. YX carried out the sequencing data analysis, interpreted the results, and proposed the mechanisms. WZ conducted the experimental validations. WZ, SO, and KP worked on growing and characterizing the MSCs. YX wrote the first draft of the manuscript and all authors revised it. All authors read and approved the final version of the manuscript.
Correspondence to Xiaobo Zhou.
Figure S1. Identifying hESC differentiation-related AS exons. Figure S2. The hESC differentiation-related AS exons possess the typical properties of AS exons. Figure S3. AS profiles upon hESC differentiation show lineage-specific splicing pattern. Figure S4. HMs change significantly around the alternatively spliced exons upon hESC differentiation. Figure S5. A subset of AS events are significantly associated with some HMs upon hESC differentiation. Figure S6. K-means clustering based on selected epigenetic features of eight HMs for MXE and SE AS exons. Figure S7. HM-associated AS genes are more lineage-specific. Figure S8. HM-unassociated AS genes are enriched in G1 cell-cycle phase and pathways for self-renewal. Figure S9. Isoform switch from PBX1a and PBX1b during hESC differentiation. Figure S10. Isoform switch of PBX1 links H3K36me3 to hESC fate decision. Figure S11. The effect of ΔPSI cutoffs for AS-HM correlations. Table S1. The number of all AS events identified during hESC differentiation. Table S5. The PCR primers used in this study. (PDF 1917 kb)
Table S2. AS events (AS exons) during the differentiation from H1 cells to differentiated cells. (XLSX 1852 kb)
Table S3. HM-associated AS exons based on k-means clustering. (XLSX 1088 kb)
Table S4. 56 cell lines/tissues and their corresponding RNA-seq data sources from ENCODE and Roadmap projects. (XLSX 14 kb)
Xu, Y., Zhao, W., Olson, S.D. et al. Alternative splicing links histone modifications to stem cell fate decision. Genome Biol 19, 133 (2018). https://doi.org/10.1186/s13059-018-1512-3
Alternative splicing
Cell fate decision
Cell cycle machinery | CommonCrawl |
\begin{document}
\title{Multistep Multiappliance Load Prediction}
\begin{abstract}
A well-performing prediction model is vital for a recommendation system suggesting actions for energy-efficient consumer behavior. However, reliable and accurate predictions depend on informative features and a suitable model design to perform well and robustly across different households and appliances. Moreover, customers' unjustifiably high expectations to accurate predictions may discourage them from using the system in the long-term. In this paper, we design a three-step forecasting framework to assess predictability, engineering features, and deep learning architectures to forecast 24 hourly load values. First, our predictability analysis provides a tool for expectation management to cushion customers' anticipations. Second, we design several new weather-, time- and appliance-related parameters for the modeling procedure and test their contribution to the model’s prediction performance. Third, we examine six deep learning techniques and compare them to tree- and support vector regression benchmarks. We develop a robust and accurate model for the appliance-level load prediction based on four datasets from four different regions (US, UK, Austria, and Canada) with an equal set of appliances. The empirical results show that cyclical encoding of time features and weather indicators alongside a long-short term memory (LSTM) model offer the optimal performance.
\end{abstract}
\keywords{Multivariate \and Multistep \and Time Series \and Prediction \and Appliance Level \and Electricity Load }
\section{Introduction}\label{introduction}
Soaring energy costs and awareness of personal responsibility to reduce carbon emissions, especially in Europe \citep{EUCommission_2021}, fuel demand for reliable and detailed energy profiling. Recommender systems that visualize upcoming costs and carbon emissions rely on energy consumption forecasts. Thus, reliable prediction of daily appliance energy profiles with hourly consumption values facilitates consumers' and technologies' endeavors towards a sustainable everyday life.
Appliance load modeling suffers from high volatility and uncertainty in underlying data. Reliable and accurate predictions depend on informative features and a suitable model design to perform well and robustly across different households and appliances. Further, efficient prediction parameters largely depend on the time horizon and forecast granularity of the prediction problem. Deep learning addresses these challenges by efficiently processing highly variable data sources and flexibly adapting to large feature dimensions.
Many smart home applications in the private sector struggle with interoperability, usability and satisfying consumer expectations \citep{iew_intelligent_energy}. Especially preset and high expectations to fast response time and accurate prediction outcomes demand for quick responses and transparent results of forecasters, as well as direct expectation management to cushion anticipations. Additionally, prediction frameworks should produce reliable results to support smart home- or recommendation applications. This study, therefore, designs a three-step forecasting framework assessing the predictability of underlying data, verifying potent feature groups and evaluating reliable modeling architectures for the day ahead device level load profiling in one-hour time intervals.
Until recently, existing approaches failed to simultaneously capitalize on additional information sources and efficient modeling structures. Comparative studies evaluate the predictability of appliance data leaving out an assessment of model performances on multistep forecasting tasks. Statistical approaches rely on separately modeling appliance on/off states and usage duration, while they fall short of reporting exact hourly usages. Most frameworks prove efficiency only in an aggregated setting for multiple appliances. Deep learning solutions concentrate on shorter prediction periods covering one hour ahead and higher data resolutions in minutes. Presented approaches fail to formulate longer (practical) forecasting horizons and largely ignore the potential lying in additional feature engineering.
This work addresses these shortcomings by setting up a suitable prediction problem of forecasting the next 24 hourly load values of typical appliances with different usage structures (i.e., fridge, washing machine, dryer, dishwasher, and television). The presented three-step framework evaluates data predictability, the impact of feature engineering and the performance of deep learning architectures across four different data sets from different geographical regions. Analyzed feature groups include additional information (environmental, date-time and appliance features) as well as engineered features using auto-correlation, statistical summary and phase space reconstruction techniques. Suitable model architectures includes new and existing approaches applied to related load prediction tasks. Inter alia the \gls{CNN-LSTM} and \gls{S2S reversed} architectures, as well as the \gls{MSVR} are applied to device level prediction for the first time. The code for the proposed prediction framework is available on GitHub.
The remainder of this paper is organized as follows: The subsequent subsection provides an overview of related literature. Section \ref{seq:methods} introduces the feature engineering and prediction methods as well as evaluation metrics. Section \ref{seq:exp_res} presents the datasets, preprocessing and experimental design. Section \ref{seq:results} shows the results of the different prediction methods and analyzes their implications, while their limitations and an outlook on future research follow in section \ref{seq:discussion}. Finally, section \ref{seq:conclusion} concludes the presented work.
\section{Related Literature} \label{seq:rel_literature}
Modeling aggregated residential energy consumption has been studied extensively with large success. However, few studies look at individual appliance load profiles. Despite similarities in types of prediction problems and applicable forecasting methods, individual device load forecasts are much more susceptible to uncertainty from human usage decisions. Without aggregation, as in household level (aggregated) load forecasting, random estimation errors do not cancel each other out. Additionally, individual appliances more selectively depend on influential factors such as environmental conditions or time factors. Table 2.1 , therefore, summarizes existing work on individual appliance level load predictions and applied features.
In detail, initial work modeling appliance electric profiles originates from bottom-up aggregated load profiling. \cite{capassoBottomupApproachResidential1994} analyze socioeconomic and demographic characteristics as well as average load profiles of household appliances from survey data and field measurements utilizing probability functions and a Monte Carlo extraction process. \cite{paateromodel2006} expanded this approach by formulating stochastic processes looking at collections of appliances from Finish households, separately simulating the use of each appliance.
With the advancement of data collection technologies, the most recent works build on more precise and larger datasets. Statistical approaches often indirectly estimate device-level electric consumption by modeling finite sets of operating states and the operating duration within each state to derive total load consumption profiles. Following this approach, \cite{Jin2020} utilize past on and off states for statistical analysis and probability simulations of daily television profiles. In \cite{Gao2018}'s work external environmental factors, time indicators and family internal characteristics identify similar past load patterns to resemble future patterns. A more complex approach in \cite{Yuting20} models operating states and duration times through K-means clustering combined with random sampling from state-specific probability distributions fed to a Conditional Hidden Semi-Markov model. The latter two studies predict aggregated values for groups of similar appliances.
In contrast, \cite{SinghAbdulsalam18} use a more data storage-intensive approach to profiling appliances. Based on frequent appliance usage patterns stored in an incrementally updated Database Management System their Bayesian Network predicts appliance activity. To derive total load consumption profiles for various time periods (hour, day, week, etc.), they combine the predicted activity state with the average appliance electric usage and operating duration.
These indirect and empirical methods highly depend on the frequency of observations, which increases the importance of frequently observed and former values over more recent load values less frequently observed. This leads to the lagged response problem, where sudden changes in appliance usage profiles influence predictions only after a certain lag of time. In the context of user-centered applications, long adaption periods of models and predictions discourage utilization and satisfaction.
To overcome this problem, direct approaches use time series forecasting methods to obtain the next load value from the most recent lagged time steps. In this field, research focuses on neural networks as a promising alternative capable of learning complex, nonlinear relationships between appliances and their environment. In single-step forecasting studied models include a nonlinear auto-regressive neural network predicting the next six-second time interval \citep{Laouali2022} and a linearly activated multilayer perceptron (linear regression) forecasting daily consumption \citep{Hossen2018} including weather features.
Reverting to machine learning solutions applied to appliance load forecasting \cite{lachutPredictabilityEnergyUse2014} focus on the predictability of appliance loads and compares prediction accuracies across models for single-step time horizons from one hour up to one week ahead. The authors include date-time features and achieve accuracies up to 89\% for hourly and 74\% for daily predictions. Results on predictability across different homes show large variations.
\cite{Ra20} extend the univariate single time step approach and formulate a multivariate prediction problem by estimating load values of multiple appliances simultaneously. They design a \gls{LSTM} to predict the load values of multiple appliances for one time step ahead, comparing performances against a \gls{FFNN} and a random forest model. They additionally introduce \textit{Last_seen_on} and \textit{Last_seen_off} states as generated input features to reduce the overall errors. This multivariate prediction approach successfully captures dependencies between multiple output values of different appliances. This is an advantage over fitting separate algorithms for each output step.
In contrast and the most similar approach to the one proposed in this work, the same authors propose a framework to predict multiple future consumption values of single appliances, conditioning the model to learn dependencies among multiple time steps. \cite{Razghandi2021_1} and \cite{Razghandi2021_2} utilize an encoder-decoder type network called sequence-to-sequence model. The first variation uses long-short term memory layers, while the latter relies on bidirectional long-short term memory layers to predict load one hour ahead in 10 minute time steps.
The overview presented and outlined in Table 2.1 concludes the following: Firstly, non-deep learning approaches heavily rely on including important features for predictions while deep learning methods so far fail to utilize this additional potential. Secondly, reported deep learning approaches are limited to data granularity either smaller than 15 minutes or as aggregated as daily load values, with max forecasting horizons of six time steps resembling one hour. Assuming that individuals plan their day at least the day before and only once, practical models need larger prediction horizons covered with a more informative granularity of load estimates. Finally and regarding real-world applications, no study considers more than one data source for an equal set of appliances.
\begin{figure}\label{tab:lit_review}
\end{figure}
\section{Methodology} \label{seq:methods} This section describes a three-step framework to assess predictability, the engineered features and the deep learning architectures to forecast 24 load values. The predictability analysis forms expectations of model performance and quantifies the performance of feature groups and forecasting algorithms. The feature engineering step generates and groups similar features to distinguish the contribution of the data sources to predictions. The term features refers to variables containing either supplementary information or a reformulation of existing data (engineered features). The impact of different types of features is assessed and compared across four different datasets. The main step compares performances among different deep learning architectures and with non-deep learning (benchmark) algorithms. The following subsections introduce the methods used in this study.
\subsection{Predictability} Predictability quantifies inherent information contained in a time series and assists in evaluating the predictive power of different forecasting methods. Model performance measures the probability of success yet it cannot provide an understanding for whether predictions improve. Intuitively, predictability estimates the highest level of performance possible for a time series and specifies whether the system is unpredictable or the model choice is poor.
A well-established measure of complexity in time series data is the \gls{PE} proposed by \cite{Bandt2002PermutationEA}. \gls{PE} captures the order relations between values and extracts a probability distribution of the ordinal patterns. \cite{AQUINO2017277} applies this \gls{PE} to electric appliance loads mapping load histograms onto a Causality Complexity-Entropy Plane to contextualize electric load behavior.\footnote{The work in \citep{AQUINO2017277} uses the REDD data.} This work relies on a modified version of the \gls{PE} the \gls{wPE} measure formulated by \cite{article13_fadlallah}. \gls{wPE} incorporates amplitude information to improve handling abrupt signal changes and more accurately assessing regular as well as noisy and (linearly) distorted data segments. The \gls{wPE} is applicable to regular, chaotic, noisy or real-world time series and fit the volatile appliance load data better than the regular \gls{PE}. \gls{wPE} uses weighted relative frequencies defined by equation \ref{eq:wpe_1} to incorporate the amplitude information into the Shannon entropy formula \ref{eq:wpe_2}.
\begin{equation}\label{eq:wpe_1}
p_w(\pi_i^{m, \tau}) = \frac{\sum_{j\leq N} 1_{u:\textrm{type}(u)=\pi_i}(X_{j}^{m, \tau}) * w_j} {\sum_{j\leq N} 1_{u:\textrm{type}(u) \in \Pi}(X_{j}^{m, \tau}) * w_j} \end{equation} \begin{equation}\label{eq:wpe_2}
H_w(m,\tau) = - \sum_{i:\pi_{i}^{m,\tau} \in \Pi}p_w(\pi_i^{m, \tau}) \ln p_w(\pi_i^{m, \tau}) \end{equation}
The choice of weight values $w_j$ reflects a specific feature or a combination of multiple features from each vector $X_{j}^{m, \tau}$. Following \cite{article13_fadlallah} the weights are computed by the variance of each neighbors vector of $X_{j}^{m, \tau}$ in equation \ref{eq:wpe_3} with $\bar{X}_{j}^{m, \tau}$ denoting its mean (equation \ref{eq:wpe_4}).
\begin{equation}\label{eq:wpe_3}
w_j = \frac{1}{m} \sum_{k=1}{m}(x_{j+(k-1)\tau} - \bar{X}_{j}^{m, \tau})^2 \end{equation}
with \begin{equation}\label{eq:wpe_4}
\bar{X}_{j}^{m, \tau} = \frac{1}{m} \sum_{k=1}{m}(x_{j+(k+1)\tau} \end{equation}
The work in \cite{Riedly_2013} gives a guideline on how to choose parameters optimally. The final parameter sets are reported in Appendix \ref{app:wpe}.
\subsection{Feature Engineering} The inclusion of features and their contribution to prediction accuracy within deep learning is hardly studied in the context of appliance level load modeling. Some approaches include numerical or one-hot encoded time features without reporting their influence on predictions. Research in other areas vastly confirms large benefits from including additional information, especially in deep learning. Presumably, additional features likewise improve forecasters in the appliance load domain. The following subsections describe included feature groups proven informative in the literature on aggregated household energy consumption. In total 10 feature groups are defined: date-time, weather, appliance, \gls{ls-on/ls-off}, auto-regressive, interaction, \gls{VEST} and phase space features as well as the aggregate feature group "\gls{w+dt}" and "all" (including all features).
\subsubsection{Date-Time Features} Opinions and practices on whether and how to include date-time features vary substantially. Common variations in load forecasting include one-hot encoded, numerical encoded and sine cosine transformed time features \citep{candanedo_data_2017}, \citep{khatoon_effects_2014}, \citep{en6031385}. Some, amongst others \cite{Razghandi2021_1}, argue against explicitly including date-time features in \gls{LSTM} modeling as the positioning of the values within the input sequence already carries the time point information. Consequently, they exclusively include a numerical representation of weekday features, but no feature with the hour and minute of the day. Though sequentially ordered input structures of \gls{LSTM} layers and numerically encoded time inputs fail to represent the cyclical nature of most date-time variables. A cyclical representation takes into account that the end and beginning of a sequence are numerically as close as time values in between, i.e. 23rd hour of the day is as close to hour zero as hour one to hour two \citep{1630255}.
In other domains of energy forecasting, such as forecasting load at electric charging sites \citep{Unterluggauer2021} or forecasting power grid states \citep{9207536}, date-time features transformed with the sine and cosine function successfully improve prediction outcomes. Following listed examples, the hour of the day, the day of the week, the week of the month and, whenever more than a year of data is available, the month will each be represented by sine and cosine transformed values calculated as in equation \ref{eq:sine_cosine}. Further, the date-time feature group includes binary workday, holiday and weekend indicators \citep{9006868}, \citep{en11071636}.
\begin{equation}\label{eq:sine_cosine}
\textrm{Sin}_{fe} = \sin(2\pi*n) \quad \textrm{and} \quad \textrm{Cos}_{fe} = \cos(2\pi*n) \end{equation} with, \begin{equation}
n =
\begin{cases}
\textrm{hour}/24 & fe=\textrm{hour} \\
\textrm{weekday}/7 & fe=\textrm{day of week} \\
\textrm{monthweek}/5 & fe=\textrm{week of month} \\
\textrm{month}/12 & fe=\textrm{month and traindata > 1 year} \\
\end{cases} \end{equation}
\subsubsection{Weather Features} Weather features have been frequently proven to improve (non-deep learning) load predictors for appliance and aggregated electricity consumption of households \citep{Sinimaa21}, \citep{Gao2018} and \citep{Hossen2018}. Weather is expected to influence appliance use as humans differ their behavior accordingly and some appliance electric consumption directly depend on environmental conditions such as temperature f.e. fridge or air conditioners. External factors influence the dependencies between load consumption and weather variables, i.e. thermostats. To assess the impact of weather features and their potential contribution to accurately predict energy usage this study includes the most commonly used weather features temperature, humidity and wind speed \citep{8905698}, \citep{MUGHEES2021114844}.
\subsubsection{Appliance Loads Features} Especially non-deep learning approaches concentrate on the dependencies between appliances and their potential to predict single appliance load profiles. \cite{SinghAbdulsalam18} find specific appliance combinations to occur often. Intuitively specific appliances tend to be used in combination such as washing machines and dryers. However, with longer prediction horizons load values of accompanying appliances precede with a time lag, i.e. information from the washing machine one hour ago is not available for the prediction of a dryers activity. Therefore, the predictive power of other appliance loads when modeling a specific load profile 24 hours ahead must be assessed carefully. This study integrates the load measures of selected appliances and defines their influence on a forecaster's performance.
\subsubsection{Engineered Features} Auto-regressive features \citep{LI2021116509} and statistical summary features such as mean and median \citep{smithMachineLearningFast2020} as well as average and standard deviation of load consumption over k-time steps \citep{9467267} positively impact aggregated load forecasting quality. Features especially summarizing a larger past time window might especially help predict appliances used less frequently than on a daily basis. Therefore the moving average and moving max value of the past 12, 24, 36 and 72 hours compose the auto-regressive feature group. Further, four simple interaction variables between each appliance pair (sum, product, mean and standard deviation) as well as the mean and standard deviation across all appliance loads form the interaction feature group to capture more complex dependencies among appliances.
\cite{cerqueira_vest_2020} develop an automatic feature engineering package called \gls{VEST} to build auto-regressive and summary features from time series data, which covers a more extensive set of engineered features. To profit from this research and available feature engineering packages, the study uses a feature set selected by the \gls{VEST} feature engineering package for time series analysis in the \gls{VEST} feature group. Equally, the study includes \gls{ls-on/ls-off} states for the target appliance as proposed in \citep{razghandi_residential_2020}. This \gls{ls-on/ls-off} feature group useful in a multivariate prediction setting might similarly contribute to performances in a multistep prediction setting.
\subsubsection{Phase Space Reconstruction Features} \cite{Drezga1998InputVS} initially used features containing reconstructed dynamics of a chaotic system as an embedding method for load forecasting. These features require phase space reconstruction\footnote{Phase space reconstruction is the foundation of nonlinear time series analysis describing the reconstruction of complete system dynamics using a single time series \citep{second_psr_def} from \citep{PSR_def_2015}.} techniques utilized for aggregated load predictions \citep{4746520}, \citep{FAN201813} and \citep{en12224349}. Several undefined exogenous factors influence the electric use of appliances directly such as human activities or environmental conditions and indirectly via factors steering human behavior. As a consequence appliance load data shows complex characteristics such as multidimensional nonlinearity and high grades of uncertainty typical for dynamic systems \citep{FAN201813}. Therefore, the motivation to transfer phase space reconstruction techniques to appliance load prediction assumes that the new phase space features reconstruct the more complex dynamics of a target appliance's usage and represent immeasurable influences on load usage profiles.
Phase space reconstruction maps the observed time series into an embedding space that preserves the structure (topology) of the underlying dynamical system. To construct phase space reconstruction features, the target values are embedded in the space of their temporal lags using the Taken embedding \citep{takens_taken_1981}. \footnote{Takens' theorem, also named the delay embedding theorem, shows that a time series of measurements of a single observable can be used to reconstruct qualitative features of the underlying phase space system \citep{Huke2006EmbeddingND}. A phase space formulates the space in which all possible states of a system are represented. In this application, a phase space representation of the target appliance loads describes all possible target values and their complex relation to each other.} Appendix \ref{app:wpe} specifies software and parameters for the construction of the phase space features.
\subsection{Prediction Algorithms} A vast selection of (hybrid) neural network architectures has proven to accurately predict multistep or multivariate sequential data. Subsequent subsections describe the deep learning architectures including \gls{LSTM}, \gls{BiLSTM}, encoder-decoder networks, \gls{FFNN} and \gls{CNN-LSTM}.
\subsubsection{Long-Short Term Memory Network} \label{LSTM} \gls{LSTM} networks, initially proposed by \cite{LSTM_basics}, are a type of recurrent neural network often used in time series prediction. They remember information from several past input steps while sequentially calculating outputs capturing the time dependencies of input variables. The architectural unique characteristics of \gls{LSTM} Networks are the \gls{LSTM} cell memory states that convey information across a chain of \gls{LSTM} cell states and update new information only if considered important. \gls{LSTM} cells process their own previous cell output $h_{t-1}$ together with new input values $x_t$ at time $t$ and update cell memory states $C_{t-1} \rightarrow C_t$ to calculate the new output value $h_t$. In this process previous cell outputs $h_{t-1}$ are filtered through input gates ${i_t}$, forget gates ${f_t}$, update gates $\widetilde{C}_t$ ($g_t$) and output gates ${o_t}$ to find the important information for updating cell memory state and calculating cell outputs. The stepwise calculations of \gls{LSTM} cell states are as follows:
\begin{equation}\label{eq:lstm1}
f_t = \sigma_{f_t} (W_f \cdot [h_{t-1}, x_t] + b_f). \end{equation} \begin{equation}\label{eq:lstm2}
i_t = \sigma_{i_t} (W_i \cdot [h_{t-1}, x_t] + b_i). \end{equation} \begin{equation}\label{eq:lstm3}
\widetilde{C}_t = tanh(W_C \cdot [h_{t-1}, x_t] + b_C). \end{equation} \begin{equation}\label{eq:lstm4}
C_t = f_t \cdot C_{t-1} + i_t \cdot \widetilde{C}_t. \end{equation} \begin{equation}\label{eq:lstm5}
o_t = \sigma_{o_t}(W_o \cdot [h_{t-1}, x_t] + b_o). \end{equation} \begin{equation}\label{eq:lstm6}
h_t = o_t*tanh(C_t). \end{equation}
As demonstrated in Figure \ref{fig:lstm_unit}, first the input and the previous cell output are pushed through the sigmoid forget layer \ref{eq:lstm1} which determines what information from the previous cell state to keep or not to keep. Secondly, the sigmoid input layer \ref{eq:lstm2} decides which values of the cell state to update and the tanh update layer \ref{eq:lstm3} calculates candidate values for updating cell states. Equation \ref{eq:lstm4} calculates the updated cell states. Another layer, the sigmoid output layer \ref{eq:lstm5}, decides which parts of the cell state to output. The updated cell states are then pushed through a $tanh$ layer and multiplied by the output filter in Equation \ref{eq:lstm6}. This cell output together with new input values initiates the next update process of the cell state and repeats itself iteratively. This structure remembers long-term dependencies and due to the multiple gate calculations form an additive structure of the gradient term. The gradient term updates layer weights by calculating derivatives in the backward propagation process. In this context, an additive structure of the gradient term reduces the likelihood of exploding and vanishing gradients, which leads to a more stable network training making \gls{LSTM} networks a reliable choice for time series predictions \citep{LSTM_basics}.
\subsubsection{Bidirectional Long-Short Term Memory Network} \gls{LSTM} layers forward pass input values sequentially. Consequently the first output of an \gls{LSTM} cell is based solely on the first input and fails to use all values within the input sequence. A \gls{BiLSTM}, proposed by \cite{bilstm_basic}, enables all \gls{LSTM} cell states to use information from the complete input sequence, passing it forward and backward through the network (Figure \ref{fig:bilstm_layer}). To add a backward pass the \gls{BiLSTM} adapts the standard \gls{LSTM} network by adding a separated hidden layer processing sequences in reverse. The forward layer processes the input sequence from beginning to end the backward hidden layer from end to beginning. Every \gls{BiLSTM} layer contains double the number of memory cells. The information of the forward and backward pass is stored in separate hidden states ($\overleftarrow{h_t}$ and $\overrightarrow{h_t}$) which are concatenated to produce the final hidden state $h_t$. At any time step t, the forward and backward layer outputs of a \gls{BiLSTM} cell are computed using the standard \gls{LSTM} unit’s operating equations \ref{eq:lstm1}–\ref{eq:lstm5}. Then the final hidden state vector is computed by combining the hidden state sequences in \ref{eq:bilstm1}:
\begin{equation}\label{eq:bilstm1}
y_t = \oplus(\overrightarrow{h_t}, \overleftarrow{h_t}). \end{equation}
where $\oplus$ is a concatenate function. It should be noted that other operations, such as summation, multiplication, or averages, can be used instead. \gls{BiLSTM} Networks initially improved long-range context processing in Natural Language Processing \citep{bilstm_basic}. Momentarily, they frequently deliver state of the art performance in (load) time series predictions \citep{mugheesDeepSequenceSequence2021} and outperform deep learning architectures such as the \gls{LSTM} and \gls{CNN} \citep{article13_fadlallah}.
\begin{figure}
\caption{\\ Functionality of the LSTM-Unit (Image Source: \cite{blog_lstm_fig})}
\label{fig:lstm_unit}
\caption{\\ Functionality of the BiLSTM-Layer (Image Source: \cite{blog_bilstm_fig})}
\label{fig:bilstm_layer}
\end{figure}
\subsubsection{Encoder-Decoder Networks} Encoder-decoder models, such as the sequence-to-sequence model, likewise originate from language translation \citep{NIPS2014_a14ac55a} and prove powerful in time series forecasting \citep{8701741}, \citep{sehovac_deep_2020} and \citep{article13_fadlallah}. An encoder-decoder architecture comprises two network parts, an encoder network and a decoder network. The encoder calculates a hidden representation of the inputs encoded in hidden states and passes them on to the decoder network for calculating predictions. The difference to standard neural network training is the transfer of states instead of layer outputs between the encoder and decoder and often the two parts are pre-trained separately before the final prediction stage. This study includes two variations of the sequence-to-sequence network. The sequence-to-sequence model used by \cite{Razghandi2021_2} further called \gls{S2S reversed} and a variation of the sequence-to-sequence model proposed in the winning solution to the web traffic time series forecasting challenge introduced by \cite{blog_s2s} called \gls{S2S context}.
The \gls{S2S reversed} uses two decoders shown in Figure \ref{fig:ed_ra}. In the first step the encoder trains together with a decoder that learns the reversed representation of the input sequence. This forces the encoder to include a backward representation of the input window in its hidden states. In a second training step, the weights of the pre-trained encoder initialize the training of the second decoder, which learns to predict the next output sequence. As shown in section \ref{seq:results} the training time is prolonged substantially by this double decoder network setup.
The second variation \gls{S2S context} uses no pre-training, instead, input information is separated into past observations of the target sequence and additional features. The encoder calculates a hidden state representation of the additional features as a sort of context. Transferring these states initializes the decoder network. The decoder uses the past target sequence as input together with the encoder weights to predict the next output sequence. This variation trains only one network and reduces computation times compared to \gls{S2S reversed}. Both networks use standard LSTM-layers within the encoder and decoder parts.
\begin{figure}
\caption{\\ S2S-Model with Reversed Sequence Decoder (Image Source: \cite{Razghandi2021_2})}
\label{fig:ed_ra}
\caption{\\ Functionality of 1D-CNN-Layer (Image Source: \cite{blog_lstm_fig})}
\label{fig:cnn_layer}
\end{figure}
\subsubsection{Feed Forward Neural Network} The simplest deep learning architecture is the feed forward neural network, where information flows only in one direction (forward). The network consists of multiple layers where each node is connected to all the following nodes in the next layer. The simple structure gives this type of network an advantage in computational speed. However, its simplicity potentially results in a less complex representation of dependencies among variables reducing performance. In detail a feed forward network defines a mapping of inputs $x$ on outputs $y$ in the form of $y = f(x; \theta)$. It learns the value of the parameters $\theta$ that result in the best function approximation $f$ of $y$. In \gls{FFNN} there are no feedback connections as in recurrent neural networks like \gls{LSTM}s \citep{Goodfellow_2016}.
\subsubsection{Convolutional Long-Short Term Memory Network} Another deep learning architecture proposed for time series prediction is the \gls{CNN}. Originally used for image processing by \cite{6795724} this architecture uses convolutional layers consisting of kernel matrices that convolve the time series information in each layer and thereby extract more complex features. This process has proven to extract important features, especially in multivariate prediction problems such as load forecasting \citep{amarasinghe_deep_2017}.
Figure \ref{fig:cnn_layer} depicts the architecture of the \gls{CNN}-layer. Typically, time series problems use $1D$ convolutions. They mimic moving average computations and extract features across time steps and time series (spacial hierarchies). When using a $1D$ convolutional layer, a one-dimensional kernel with size $k$ functions like a weight mask that multiplied with the input layer fold inputs together to form the layer output. Often the convolutional layer output is passed on through an activation function \gls{LeakyReLU} and through a pooling layer. The pooling layer (i.e. max pooling) summarizes the feature map into a lower-dimensional representation, also called smoothing, thereby reducing the influence of small data fluctuations. The final output is flattened to either produce predictions or to be passed on to a fully connected layer. As proven by \cite{yan_multi_step_2018} the predictions benefit from adding a final \gls{LSTM} layer to predict final sequential output windows forming a \gls{CNN-LSTM} model.
\subsection{Benchmark Models} Two prediction algorithms, one tree-based and one based on \gls{SVR} benchmark the deep learning performances. \gls{XGBoost} is a widely used and stable performing machine learning method while the \gls{MSVR} is a multioutput adaption of the \gls{SVR}. The single \gls{SVR} frequently serves as a well-performing benchmark in single output time series problems.
\subsubsection{XGBoost} \gls{XGBoost} first proposed by \cite{ChenG16} is a classic tree-based algorithm using a gradient boosting framework. The intuition behind boosting is adding new decision trees predicting the values better where the initial model failed to give good results. Adding a multiple of these boosting trees to an ensemble of trees is expected to improve predictions by a multiple. There is no ready implemented version for multistep regression problems using the \gls{XGBoost} algorithm. This implies that one model for each time step must be fitted.
\subsubsection{Multiple-Output Support Vector Regression} Even though benchmarking often uses \gls{SVR}, in multistep prediction problems the basic \gls{SVR} structure restricts the algorithm to singular output values. This requires either to use predicted values iteratively as input to get the next prediction step or to fit one \gls{SVR} for each output step. In their work \cite{bao_multi-step-ahead_2014} further develop the initial version of the \gls{MSVR} first proposed by \cite{perezcruz_msvr} and design a \gls{SVR} for multistep time series prediction problems. \cite{bao_multi-step-ahead_2014} prove the \gls{MSVR} to outperform the iterative and multi-model \gls{SVR} approach in multistep time series prediction while keeping computational costs low. The \gls{MSVR} preserves the stochastic dependencies within the time series data by estimating a multidimensional output. This facilitates mapping the underlying time series dynamics by estimating a multidimensional output \citep{perezcruz_msvr}. For the full derivation of the \gls{MSVR} solution see \cite{bao_multi-step-ahead_2014}. Pre-tests of all three prediction strategies on the \gls{REFIT} data in this study confirms the effectiveness and superior performance of the \gls{MSVR} over the direct and iterative standard \gls{SVR} replicating results in \cite{bao_multi-step-ahead_2014}.
\section{Data and Experimental Design} \label{seq:exp_res}
All models are fitted to data from four homes, taken from datasets covering different geographic locations in Europe and North America. Table \ref{tab:datasets} summarizes details on the datasets and indicates whether the homes use a thermostat. The \gls{AMPds2} \citep{makelectr2016} is the largest dataset covering two full years of data. The \gls{REFIT} \citep{Murray2017AnEL} spans a period of almost two years but contains a wider gap of six weeks of missing data. \gls{GREEND} \citep{7007698} and \gls{PecanSD} \cite{7418187} are smaller datasets with data from 10 months and six months respectively with the most recent data reported from 2019 in Pecan Street. If available, the study uses the weather features published alongside the datasets. Otherwise, the preprocessing step merged historic weather data from a local weather station \footnote{obtained from https://openweathermap.org/}. The distance to the nearest weather stations is maximum 10 kilometers and the main pool of appliances selected contains appliances responsible for a large part of total appliance electric usages such as fridge, washing machine, dryer, dishwasher and television.
{\scriptsize \begin{xltabular}{\linewidth}{@{}
>{\hsize=0.75\hsize}C
>{\hsize=1\hsize}C
>{\hsize=1\hsize}C
>{\hsize=0.75\hsize}C
>{\hsize=1.5\hsize}C
>{\hsize=0.75\hsize}C
>{\hsize=1.25\hsize}C
@{}}
\caption{Datasets}
\label{tab:datasets}\\
\toprule
\toprule Dataset & Country
& Years
& WS Dist.
& Appliances
& Thermostat
& Train/Validation/Test \\
\midrule \endfirsthead
\midrule[\heavyrulewidth] \multicolumn{7}{r}{\textit{Continue on the next page}} \endfoot \endlastfoot REFIT* & UK, Leicester & 2013-11-04/2015-05-09 & 3 km & fridge, dryer, washing machine, dishwasher, television & no & 7,540/1,884/2,807
\\ \addlinespace
AMPds2 & Canada, British Columbia & 2012-04-01/2014-03-31 & < 1 km & fridge, dryer, washing machine, dishwasher, television & yes & 11,710/2,927/2,883
\\ \addlinespace GREEND* & Austria, Kärnten & 2013-12-07/2014-10-13 & < 10 km** & fridge, washing machine, dishwasher, television & n.a. & 4,116/1,029/600
\\ \addlinespace PecanSD* & USA, New York & 2019-05-01/2019-10-31 & < 5 km** & fridge, dryer, washing machine, dishwasher & n.a. & 2,353/588/1,447
\\ \midrule[\heavyrulewidth]
\multicolumn{7}{l}{\scriptsize* Selected houses: GREEND - building 0; Pecan Street - house ID 3996; REFIT - house 1;} \\ \multicolumn{7}{l}{** historic weather data from a nearby weather station} \\ \normalsize \end{xltabular}}
\subsection{Preprocessing} The first step aggregates all original load measurements to hourly measurements and split into train, test and validation sets. Table \ref{tab:datasets} describes training, validation and test split sizes. Test data sizes vary due to testing periods starting at the beginning of a month but roughly lie between 20\% and 10\% of the full data set size. In all cases, training algorithms use 20\% of training data samples for validation.
The preprocessing pipeline imputes missing date-time values, not exceeding a gap larger than three days. The mean of the same hour on the same weekday at previous time steps imputes the missing load values to maintain presumably regular consumption patterns for weekdays. Weather variables use the mean of the corresponding week to impute values resembling values close to the same point in time. For longer time gaps this method preserves the data structure, as opposed to, i.e. forward filling introducing significantly different data patterns. The wide data gap in the \gls{REFIT} dataset was kept unimputed.
To detect outliers, the 90\% winsorization calculates the 5th and 95th percentiles of the data and deletes all values below and above the lower and upper bounds. The subsequent step imputes the deleted measures by forward filling the previous value. In this case, forward imputation has only minor effects on the data structure and is a highly efficient method. Only fridge usage profiles in \gls{REFIT} and \gls{PecanSD} justified using outlier replacements in seven and three cases respectively. Similarly, among weather variables only wind speed in \gls{REFIT} contained extreme values. A check on historic wind speed data for reported date and location additionally verified the identified values as outliers.
Further, the Standardscaler function from sklearn standardizes features by removing the mean and scaling to unit variance with the formula $z = (x - u) / s$. Standardization of data identifies the output range beforehand and reduces the impact of larger numbers to stabilize the training process. Note that to normalize training values only the first 80\% of observations within the time series data were used to fit the Standardscaler to avoid information overspill.
After computing the features on the imputed and normalized data, rolling windows select input and output windows with 24 values (hours) comprising the training dataset. In a rolling window approach, the next data sample shifts by one value. For testing data, the experiment uses a slicing windows approach, which copies the real-world scenario where the next new 24 observations are presented all at once, not each hour. Hence the next data sample is the former sample shifted by 24 time values. Validation sets are constructed from the training set after the full preprocessing procedure.
\subsection{Model Setup} Model input shapes after data preparation are ($N, ts, f$) with $N$ equal to the number of samples, $ts$ the number of time steps (24) and $f$ the number of features. \gls{XGBoost}, \gls{MSVR} and \gls{FFNN} require a flattened input shape, where the feature and time step dimension is flattened out to take the form ($N, ts\cdot f$). To fit input into the \gls{CNN-LSTM} model original inputs are extended to a fourth dimension to ($N, ts, f, 1$) to satisfy CNN input requirements. Most of the feature groups fit well into this shape. However, selected \gls{VEST} features and phase space transformations of the target variable change shapes to (70, 1) and (23, 2) respectively. As these transformations break up the sequential character of the input data the seq2seq architectures and the \gls{MSVR} are not considered when looking at the impact of \gls{VEST} and phase space features. The adaptations of the architecture needed to extract correct output from the adapted input dimensions make the models incomparable.
Model architectures use the TensorFlow framework with GPU support for tuning and training. For detailed parameter settings and tuning spaces see Table \ref{tab:model_params} in Appendix \ref{app:TunParams}. All models were trained using a Google Colab Pro account with priority access to high-memory virtual machines (32 GB RAM) and GPU (T4 or P100 GPU) support. Unfortunately, as Colab resources are distributed among users, training times might vary on equal training tasks depending on the computing capacity provided.
\subsection{Evaluation Metrics} Mainly the \gls{RMSE}, \gls{nRMSE} and a \gls{acc95} evaluate the presented results. The \gls{RMSE} measures the average magnitude of prediction errors calculated with a quadratic scoring rule that penalizes larger results more and is defined in \ref{eq:RMSE}. The \gls{nRMSE} is a scale-independent metric used by other comparable studies. It relates the \gls{RMSE} to the observed value range by the Formula in \ref{eq:nrmse}.
The accuracy at the 95\% level \gls{acc95} (see equation \ref{eq:acc95}) calculates the sum of all predictions deviating less than 5\% from the true value and reports the share of non-deviating values in percent off all values. \cite{lachutPredictabilityEnergyUse2014} also report the \gls{acc95} as an indicator for predictability. The \gls{acc95} serves as a proxy to report the number of correctly predicted values and sets the performance indicated by the \gls{RMSE} and the \gls{nRMSE} into relation. Additionally and to ensure comparability to related literature, the results on \gls{GREEND} report the \gls{MAE}. The \gls{MAE} is defined in \ref{eq:MAE} and measures the average magnitude of prediction errors.
\begin{equation}\label{eq:nrmse}
nRMSE = \frac{nRMSE}{max_{y_j}-min_{y_j}}. \end{equation} \begin{equation}\label{eq:RMSE}
RMSE = \sqrt{(\frac{1}{n}) \sum_{i=1}^{n} (y_j - \hat{y}_j)^2}. \end{equation} \begin{equation}\label{eq:acc95}
acc95 =
\begin{cases}
0 & \frac{|\hat{y}_j| - |y_j| * 100}{|y_j|} > 5 \\
1 & \frac{|\hat{y}_j| - |y_j| * 100}{|y_j|} \leq 5 \\
\end{cases} \end{equation} \begin{equation}\label{eq:MAE}
MAE = \frac{1}{n} \sum_{i=1}^{n} |y_j - \hat{y}_j|. \end{equation}
Additionally, the \gls{MASE} defined by equation \ref{eq:MASE} reports the effectiveness of the forecasting algorithm with respect to a seasonal naïve forecaster \citep{HYNDMAN2006679}. A \gls{MASE} lower than one indicates better performance, f.e. a \gls{MASE} of 0.5 implies that a model would have double the predictive power of a naïve forecaster. In the case of a large percentage of zero values in the data, the \gls{MASE} more reliably measures the predictive capacity of models than the combination of the \gls{nRMSE} and the \gls{acc95}.
\begin{equation} \label{eq:MASE}
MASE = \frac{MAE}{MAE_{naive}} =
\frac{MAE}{\frac{1}{n} \sum_{i=1}^{n} |y_j - x_j|} \end{equation}
\section{Results} \label{seq:results}
This chapter groups results in three parts. The first two subsections report on data predictability and important features. The third section elaborates on the highest performing models, prediction stability across datasets, reliable model-feature combinations and suitability of \gls{CNN-LSTM} and \gls{S2S context} models for the task at hand. All subsections will first focus on results for the fridge appliance across all data sources and second compare these results to performances on other appliance types. For appliance types except for fridge the presented results include only outcomes of \gls{REFIT}. Additionally, Table \ref{tab:fe_perform_datasets} and Figures \ref{fig:changes_model_nrmse} and \ref{fig:changes_model_acc} compare training duration across models to verify the applicability of presented approaches for customer-oriented solutions.
\subsection{Predictability}
The \gls{wPE} measures a considerable amount of noise in the reported appliance load data similar to results in the related literature \citep{AQUINO2017277}. Further, data predictabilities of fridge profiles show higher accuracies on \gls{REFIT} in comparison to \cite{lachutPredictabilityEnergyUse2014} who reports accuracies between 50\% and 74\% a day ahead. The combination of these findings validates the \gls{wPE} as an effective indicator for predictability and verifies the effectiveness of proposed models over simpler statistical approaches in \cite{lachutPredictabilityEnergyUse2014}. The \gls{wPE} approach is suitable to pre-assess potential of prediction modeling and the amount of predictable data.
Overall the \gls{wPE} reliable assesses, which appliances and datasets are easier to predict. Models reach higher accuracies on prediction tasks specified as easier to predict and lower otherwise. Entropy values above 0.5 indicate a significant amount of randomness in the data \citep{Bandt2002PermutationEA}. According to Figure \ref{fig:wpe_point} the dishwasher and the dryer as well as fridge profiles on \gls{REFIT} show lower complexity compared to the fridge and the television and fridge profiles on the other datasets. The \gls{wPE} analysis indicates larger divergence between the television and washing machine than measured accuracies confirm. This indicates that models better fit television profiles with a higher data complexity. Further in Table \ref{tab:mase_dataset} the predictive power of models measured in terms of the \gls{MASE} values is higher for \gls{GREEND} and \gls{AMPds2} as well as for fridge and washing machine. In combination, both measures confirm that deep learning approaches perform stronger whenever data complexity is higher.
\begin{figure}
\caption{\\ Weighted Permutation Entropy for All Appliances and Datasets}
\label{fig:wpe_point}
\end{figure}
\begin{table} \centering \caption{MASE Values Across Datasets} \label{tab:mase_dataset}\ \footnotesize\setlength{\tabcolsep}{1pt} \begin{tabularx}{\textwidth} { p{2 cm} p{1.5 cm} p{1.5 cm} p{2 cm} p{2 cm} } \\ \hline Data & Max & Min & Model Min & FE Min \\ \hline
GREEND & 0.9362 & \textbf{0.6792} & LSTM & date-time
\\ \addlinespace
PecanSD & 0.9656 & \textbf{0.7968} & S2S context & weather and date-time
\\ \addlinespace
REFIT & 0.8755 & \textbf{0.7927} & CNN-LSTM & phase space
\\ \addlinespace
AMPds2 & 0.9208 & \textbf{0.7057} & CNN-LSTM & date-time
\\
\end{tabularx} \end{table}
\begin{table} \centering \caption{MASE Values Across Appliances on REFIT} \label{tab:mase_appliances}\ \footnotesize\setlength{\tabcolsep}{1pt} \begin{tabularx}{\textwidth} { p{2 cm} p{1.5 cm} p{1.5 cm} p{2 cm} p{2 cm} } \\ \hline Appliance & Max & Min & Model Min & FE Min \\ \hline
fridge & 0.8755 & \textbf{0.7948} & CNN-LSTM & weather
\\ \addlinespace
washing machine & 1.5525 & \textbf{0.7595} & LSTM & w + dt
\\ \addlinespace
television & 1.3974 & \textbf{1.0758} & LSTM & w + dt
\\ \addlinespace
dishwasher & 2.0960 & \textbf{0.8204} & LSTM & appliances
\\ \addlinespace
dryer & 6.1995 & \textbf{1.2514} & LSTM & appliances
\\
\end{tabularx} \end{table}
\subsection{Important Feature Groups} Figures \ref{fig:changes_fe_nrmse} and \ref{fig:changes_fe_acc} visualize the impact of features on predictions. It becomes clear that sine cosine encoded date-time features and the holiday indicator are the most important features, followed by weather features and \gls{ls-on/ls-off} indicators. The impact of engineered features remains mixed across the datasets.
Date-time features positively impact prediction quality most frequently across all datasets and provide the largest improvements compared to other feature groups. Especially in combination with weather features, models highly profit from the cyclical information added to the sequential input improving prediction accuracy and error margins. This underlines the importance of including date-time features. Both numerical encoding of date-time features and one-hot encoded time features yielded worse results in comparison to predicting the target from its past values across all datasets.
\begin{figure}
\caption{\\ Changes in Mean nRMSE Scores per Feature Group}
\label{fig:changes_fe_nrmse}
\caption{\\ Changes in Mean Accuracy per Feature Group}
\label{fig:changes_fe_acc}
\end{figure}
The second most important features are weather indicators, despite their less consistent contribution across datasets. Weather features perform stronger in overall error on the \gls{nRMSE} across datasets, while gains in \gls{acc95} depend on the underlying data with the risk of worsening the performance. For example, on the \gls{AMPds2} dataset performance drops significantly when including weather features. The household in the \gls{AMPds2} uses thermostats in each room. This most likely corrupts the potential information contained in outdoor environmental variables, when no thermostat is in place. Although weather features seem to be an influential factor for load predictions, their utility for predictive performance depends on other external factors, possibly not always available to application providers.
Phase space features mostly influence the overall error margin, without a clear tendency across models and datasets. However, looking closer at the results for individual model-feature combinations, interestingly, phase space features combined with \gls{CNN-LSTM} model consistently show a small improvement of the \gls{nRMSE} mostly without decreasing accuracy. This holds true across datasets and appliance types. The phase space reconstruction breaks up the sequential structure of the inputs complicating the sequential processing of values. Conclusively, the convolutional processing provides a better fit to the new data structure than sequential layer designs.
All other feature groups show small irregular effects across datasets. Their explanatory value for the target appliance loads depend on individual household properties and are not universal explanatory. Similarly, increasing performance through engineering past, summarizing and auto-correlated features, depends on the underlying data and requires a more in-depth feature selection on individual case basis.
\subsection{Model Evaluation}\label{seq:model_results} According to Figure \ref{fig:changes_model_nrmse} and \ref{fig:changes_model_acc} nearly all deep learning models outperform the benchmarks in accuracy, except for \gls{REFIT} where the \gls{XGBoost} scores the highest. However, the high accuracy of the \gls{XGBoost} approach evidently requires significant concessions to higher error scores, while deep learning alternatives reach comparably high accuracy with lower error scores. This proves the consistency of deep learning performance across data sources and their good fit to complex appliance profiles.
Comparing the best performing model-feature combinations referenced in Table \ref{tab:fe_perform_datasets} the \gls{LSTM} delivers high and stable performances across all datasets when combined with varying feature combinations. The \gls{LSTM} always ranks among the highest predictive performances. Especially for data with lower predictability, the \gls{LSTM} heavily draws information from date-time features.
\begin{figure}
\caption{\\ Changes in Mean nRMSE Scores per Models}
\label{fig:changes_model_nrmse}
\caption{\\ Changes in Mean Accuracy per Models}
\label{fig:changes_model_acc}
\end{figure}
The \gls{S2S reversed} shows a solid performance across feature groups on average. However, the \gls{S2S reversed} is computationally expensive, resulting in longer training times This can be seen in Table \ref{tab:com_time}. Comparatively, the less complex \gls{S2S context} architecture outperforms the \gls{S2S reversed} on most of the predictions while maintaining a faster runtime (3 min 52 seconds on average). Further results indicate that the \gls{S2S context} seems to perform the best on television and \gls{AMPds2} hinting at a good fit on irregular data and appliance types. On the \gls{AMPds2}, marked as the least predictable dataset, the \gls{S2S reversed} reaches the highest performance in terms of accuracies (see Table \ref{tab:fe_perform_datasets}).
The \gls{MSVR} generally performs very strongly in terms of overall errors alongside \gls{LSTM} and \gls{S2S reversed} especially for appliances with many zero values. Less consistent results for the \gls{MSVR} on the \gls{acc95} score show performance beneath deep learning alternatives. Lastly, the \gls{MSVR} outperforms the \gls{XGBoost} benchmark as well as the random forest algorithm\footnote{Results obtained with a random forest algorithm are not reported as exponentially growing prediction times disqualify it as a candidate for user-centric applications.}.
\begin{figure}\label{tab:fe_perform_datasets}
\end{figure}
\subsubsection{Performance on Different Appliance Types}
Results for other appliance types reported in Tables \ref{tab:fe_perform_datasets} and \ref{tab:mase_appliances} confirm the significance of date-time and weather as well as appliance features. When evaluating performance on other appliances it is important to keep in mind the high percentage of zero values. These appliances stay turned off most of the time with zero value percentages in the training data above 63\% for television and above 80\% for the rest in comparison to the fridge with only 25\% of zero values. High accuracies on these appliances are equally attainable by a predictor issuing zero all the time. In this case, reverting to the more reliant \gls{MASE} metric in Table \ref{tab:mase_appliances} consistently reports the \gls{LSTM} to provide the highest predictive power among algorithms and across different appliance types. The results confirm the superiority of the \gls{LSTM} model alongside weather, date-time and appliance features.
Interestingly, \gls{MASE} values for television and dryer are higher than 1 indicating that models struggle to predict better than a naive forecaster. Especially for the dryer, this result contradicts the the high predictability measured in the weighted permutation entropy. Interpreting these results, concludes that even though predictability of data is high, the models struggle to learn a proper representation, when a large share of the data contains equal values. For models to learn a representation regardless, either requires a larger ground truth database for training or specialized forecasting algorithms detecting anomalies.
\subsubsection{Training Time} Deployment of user friendly applications requires fast and scalable algorithms. Training duration of an algorithm on differently sized feature sets approximates both performance indicators. Table \ref{tab:com_time} reports summarized training duration of presented model architectures across different sets of features. With the second shortest duration, the \gls{LSTM} confirms its fitness as a practical solution. In general training times with deep learning (except \gls{CNN-LSTM}) vary little among different feature groups confirming their scalability to higher feature dimensions. Using \gls{S2S reversed} for production, requires higher engineering efforts for efficient computation or more powerful computation resources, as computation times are by far the highest.
\begin{table}[!htbp] \caption{Training Time in Seconds} \label{tab:com_time}\ \begin{tabularx}{\linewidth}{@{}
>{\hsize=1.55\hsize}C
>{\hsize=1\hsize}C
>{\hsize=0.75\hsize}C
>{\hsize=0.75\hsize}C
>{\hsize=1\hsize}C
>{\hsize=1.1\hsize}C
>{\hsize=1.1\hsize}C
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@{}} \hline computation time & XGBoost & MSVR & LSTM & BiLSTM & S2S reversed & S2S context & FFNN & CNN-LSTM \\ \hline
mean & 15 & 63 & 24 & 22 & 266 & 37 & 6 & 24
\\
max & 66 & 73 & 26 & 24 & 298 & 40 & 8 & 55
\\
min & 8 & 45 & 22 & 20 & 243 & 34 & \textbf{5} & 20
\\ \hline
\end{tabularx} \end{table}
\section{Discussion} \label{seq:discussion}
This section embeds reported results into related work and discusses implications and importance of results for industry and the research community.
\cite{Ra20} demonstrate the case in which important features for multivariate appliance load prediction apply less strongly to multistep output. In their work, including \gls{ls-on/ls-off} features largely improves the reported \gls{nRMSE} of their \gls{LSTM} network (ca. +80\%). The results in this study show only a small positive impact across datasets. Importantly, the referenced authors use a different dataset\footnote{The authors use the DRED dataset.} and forecast one time step for multiple appliances. Naturally, the influence of the observed \gls{ls-on/ls-off} states 24 hours ago are not as informative as the observation of the last hour.
The \gls{LSTM} model proves itself as a strong alternative to the \gls{S2S reversed} proposed for shorter prediction tasks in Table \ref{tab:lit_review}. On the same data, the presented \gls{LSTM} model including weather features outperforms their model on all three metrics. Subsequently in both, performance and training duration, the simpler \gls{LSTM} network is preferable over the \gls{S2S reversed}. However, the comparison should be taken carefully, as models in \cite{Razghandi2021_2} predict a 10-min resolution as opposed to one hour in this application. A more coarse forecasting granularity might be easier to predict.
The presented \gls{LSTM} outperforms the Hidden Semi Markov Model of \cite{Yuting20} that predicts the next 60 time steps of one-minute data. The differences become more evident when considering the focus of \cite{Yuting20} on groups of similar appliances including up to 50 appliances in reported target values. In previous research, single appliance loads are associated with higher error margins. Conclusively, the presented models show competitive results to the statistical approach in \cite{Yuting20}.
\subsection{Contributions}
Deep learning approaches for appliance level load prediction significantly improve with environmental and cyclical time-related information. Especially, the impact of sine cosine encoded features contradicts research assuming the sequential ordering of input values to suffice in \gls{LSTM} feature modeling and adds to the debate on efficiently encoded time features. Less complex statistical summary features, like the auto-regressive and interaction features, show lower impact and confirm the high capacity of deep learning models to autonomously extract this information from feature sets. On the other hand, chosen architectures demonstrate difficulties extracting valuable information from highly complex features, such as phase space reconstruction variables. Most likely, breaking up the sequential character of inputs disrupts efficient information extraction by models that rely on this structure. Conclusively, sequential models are no good fit for phase space reconstruction features. Correlations presented in appliance features seem of reduced importance in a multistep single appliance prediction task.
Further, the results confirm that deep learning approach are superior to alternative time series prediction methods, particularly when dealing with irregular data structures. Simpler architectures such as the \gls{LSTM} and the \gls{S2S context} showed consistently higher performance over more complex design variations such as the \gls{BiLSTM} or \gls{S2S reversed}. In conclusion, higher architectural complexity based on similar single-layer designs have only limited potential to improve predictions. Nevertheless, the potential and flexibility of new deep learning techniques and their expected development support deep learning as a trustworthy approach for modeling single appliance load profiles.
\gls{MSVR} outperforms tree-based alternatives frequently applied in related literature. This application therefore serves as a benchmark for future research work. Benchmarking with the \gls{MSVR} challenges deep learning algorithms in terms of \gls{RMSE} and \gls{nRMSE}, especially on less variable data. With a consistently low \gls{nRMSE} this model might even be a justified choice, whenever applications strongly rely on a lower error margin over higher accuracies. An example might be when high prediction errors are associated with high costs. In this case, outperforming the \gls{MSVR} indicates a good model fit to the data.
Expectations of superior performance from the \gls{CNN-LSTM} were not confirmed. Nevertheless, the model shows the best abilities to process highly complex non-sequential features such as the phase space indicators. This promising solution from aggregated load profiling specifically extracts information from input features well. Hence, the moderate performance ranking questions the importance of an additional feature extraction layer for the task at hand.
The newly applied architecture, \gls{S2S context}, succeeds in outperforming its counterpart the \gls{S2S reversed}, with significantly less computation time. Accuracy rates fluctuate less across feature groups and indicate a lower dependence of the encoder-decoder structure on the selection of input features. Conclusively, the sequence-to-sequence structure might be preferred whenever less capacity for detailed feature engineering and selection is available.
Overall and in comparison to the existing literature on appliance profiling, transferring the feature engineering from non-deep learning approaches to the more efficient deep learning method largely improves performances. Reevaluating existing deep learning approaches for the more practical task of forecasting 24 hourly load values, contradict the proven superiority of sequence-to-sequence modeling over simpler \gls{LSTM} models in \cite{Razghandi2021_2}. Seemingly, the advantage of reverse sequence modeling diminishes with a more coarse forecasting granularity. Instead, additional information from date-time and weather features become important.
\subsection{Implications} The presented three step framework is directly deployable in industry applications. Incorporating the presented framework would enable \gls{HEMS} and recommendation applications to transparently provide consumers with insights into their consumption profiles and showcase potential energy-savings. The predictability analysis actively manages expectations by pre-reporting expected error margins and the complexity of appliances loads chosen for predictions. The forecasting step enhanced by cyclical features and weather indicators improve the quality of existing applications and thereby comply with consumers' expectations. Further, adequately addressing consumers expectations increases satisfaction and participation, the key indicator of success in any energy-saving program \citep{SUN2017383}.
Secondly, conducting a predictability analysis prior to deployment of a modeling framework effectively assesses data potential and pre-identifies correct model structures. This speeds up application engineering and saves costs for smart home application providers, ultimately reducing entry costs of new app providers to design solutions. Further, easily accessible and low-cost engineering further nurtures innovation and development for applications tailored to consumers' needs motivating new consumers to engage in energy-saving behaviors.
\subsection{Limitations} \label{seq:limitation} Selected deep learning architectures show restricted capacity to model all types of appliances usage profiles. Profiles of seldomly used appliances require modeling designs specialized in detecting the few positive states or larger data samples to correctly learn a profile's specificity. An interesting approach responding to a higher demand for information within these tasks extends the multistep approach by a multivariate dimension. Simultaneously modeling multiple appliance profiles in one model could ascertain whether dependencies among appliances and the prediction values of related devices provide the additional information needed to correctly predict seldom values.
The use of different geographical locations verifies the robustness of the presented results. However, the scope of this study is limited to the western hemisphere and leaves additional verification for other regions pending. Similarly, due to restrictions in available data highly promising influential features could not be tested. As an example, smart home applications might have local access to a family calendar through connected smartphone applications. The availability of this data limits a complete analysis of the predictive power of important features in appliance load modeling. Nevertheless, this work succeeds in guiding intuitions for the importance of not available features. Additionally, the predictability analysis provides fast testing for new data to form expectations for new appliance data sources.
\subsection{Recommendations}
In deployment, developers of smart home applications reliant on appliance load prediction would benefit from conducting a predictability analysis prior to implementation. This can guide suitable model choice and points to appliances requiring additional components capturing seldom behavior. A predictability step as conducted in the presented framework identifies data with a good fit to the model especially when the data structure is noisy and appliances are hardly used. In deployment this will save engineering costs and avoid promising model application performances that cannot be met.
secondly, implementation of smart meter systems and recommender applications should time label smart meter readings and connect the HEMS system to weather data either from the home itself or from a local weather station to enhance their system robustness. Accompanying options to specify household default characteristics, such as thermostat usage, delivers additional insights and prevents corrupted results from external influences. Generally advanced engineering and additional processing of input features is not needed. Rather efforts might go into more data collection of promising features not included in this study and described in the following section.
\subsection{Future Work} Future research on multistep, multi-appliance profiling can address some of the limitations stated in Section \ref{seq:limitation}. This task comprises a higher output complexity and requires high capacity deep learning architectures such as transformers. Interestingly, transformers process complete input sequences without relying on past hidden states and sequential processing. This could preserve the long-term dependencies within time-ordered sequences and capture the cyclical characteristics important within the presented topic. Equally, mechanisms such as multi-head attention and positional embedding designed to capture the multivariate dependencies might effectively predict different appliance usage profiles simultaneously and thereby improve the difficulty in seldom used appliance profiling.
Another interesting future path would collect additional data more directly representing human behavior. Humans are the most influential impact factor and hence occupancy, current well-being, family planners and attitudes towards smart home topics form highly interesting data sources that might further close the gap in accuracy between predictions and the ground truth. All research from how to collect this data up to its impact and predictability would significantly contribute to appliance level load applications.
\section{Conclusion}\label{seq:conclusion} This paper examines various deep learning architectures along with important feature groups for appliance level load prediction. It shows that deep learning consistently provides more accurate predictions than tree-based and multistep \gls{SVR} benchmarks. The contributions of the paper are twofold. First, the study demonstrates the robustness of the \gls{LSTM} network across different data sets and appliance types. Secondly, it identifies cyclical encoded time features as a highly important feature group alongside weather features to enhance prediction performance. The findings additionally contribute to the distinct usage of time features in sequential forecasters demonstrating the positive impact of cyclical encoding.
\appendix \appendixpage \renewcommand{\Alph{subsection}}{\Alph{subsection}} \renewcommand{\thesubsection.\arabic{equation}}{\Alph{subsection}.\arabic{equation}} \renewcommand{\thesubsection.\arabic{table}}{\Alph{table}} \renewcommand{\thesubsection.\arabic{table}}{\Alph{subsection}.\arabic{table}}
\subsection{Parameters}
\subsubsection{Predictability and Feature Parameters}\label{app:wpe}
\begin{table}[!htbp] \centering \caption{Predictability and Feature Parameters} \label{tab:wpe_fe_params}\ \footnotesize\setlength{\tabcolsep}{1pt} \begin{tabularx}{\linewidth}{@{}
>{\hsize=1.25\hsize}C
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@{}} \midrule parameters & setting & software (packages) \\ \midrule
order: 7 & weighted & pyentrp \\ delay: 1 & permutation &\\ normalize: False & entropy &\\ \midrule
time_delay: 1 & Taken & giotto-tda \\ dimension: 2 & embedding & \citep{tauzin2020giottotda} \\ flatten: True & &\\
\end{tabularx} \end{table}
\subsubsection{Model Parameters} \label{app:TunParams}
\begin{figure}\label{tab:model_params}
\end{figure}
\begin{figure}\label{tab:results_fridge}
\end{figure}
\begin{figure}\label{tab:results_appliances}
\end{figure}
\end{document} | arXiv |
\begin{document}
\title{Family Column Generation: A Principled Stabilized Column Generation Approach} \begin{abstract} We tackle the problem of accelerating column generation (CG) approaches to set cover formulations in operations research. At each iteration of CG we generate a dual solution that approximately solves the LP over all columns consisting of a subset of columns in the nascent set. We refer to this linear program (LP) as the Family Restricted Master Problem (FRMP), which provides a tighter bound on the master problem at each iteration of CG, while preserving efficient inference. For example, in the single source capacitated facility location problem (SSCFLP) the family of a column $l$ associated with facility $f$ and customer set $N_l$ contains the set of columns associated with $f$ and the customer set that lies in the power set of $N_l$. The solution to FRMP optimization is attacked with a coordinate ascent method in the dual. The generation of direction of travel corresponds to solving the restricted master problem over columns corresponding to the reduced lowest cost column in each family given specific dual variables based on the incumbent dual, and is easily generated without resolving complex pricing problems. We apply our algorithm to the SSCFLP and demonstrate improved performance over two relevant baselines. \end{abstract} Keywords: Column Generation, Stabilization, Mixed Integer Programming
\section{Introduction} \doublespacing Expanded linear programming (LP) relaxations provide much tighter bounds than compact relaxations in many problem domains in operations research \citep{lubbecke2005selected,desrosiers2005primer}, and more recently in computer vision \cite{yarkony2020data, FlexDOIArticle}. Column generation \citep{barnprice}(CG) is used to solve these expanded formulations. CG proceeds by relaxing the binary valued constraint in the expanded integer linear programming formulation then constructing a sufficient set of the columns to exactly solve the LP relaxation. CG achieves this by iterating between \textbf{(1)} solving the LP over a limited subset of the variables (called the nascent set) and \textbf{(2)} computing negative reduced cost columns. The LP is called the restricted master problem (RMP) while the mechanism to generate negative reduced cost columns is called pricing. The problem of generating these reduced cost columns is problem domain specific; but typically involves solving a small scale integer program, often via a dynamic program \citep{lubbecke2005selected}. CG terminates when no negative reduced cost columns exist, at which point the solution is provably optimal for the expanded LP relaxation.
CG can be accelerated by dual stabilization methods \citep{ben2006dual,haghani2020integer,du1999stabilized,FlexDOIArticle,Pessoa2018Automation}. Stabilization methods select the dual solution to do pricing on in such a manner as to decrease the number of iterations of CG needed to solve the problem.
In this paper we introduce a novel dual stabilization approach which we refer to as Family Column Generation (FCG). At each iteration of FCG we generate a dual solution, which solves the RMP over all columns consisting of a subset of columns in the nascent set. However this does not require additional expensive calls to pricing. A special LP solver is developed to solve this LP exploiting its structural properties. To use this LP solver we require an oracle that can generate the lowest reduced cost column over a subset of the elements in an existing column and obeying the structural properties or that existing column. Such optimization is far easier than standard pricing.
We organize this document as follows. In Section \ref{CG_review} we review the classic column generation formulation of expanded set cover problems. In Section \ref{litRev} we review the existing literature in dual stabilization. In section \ref{Implementation} we discuss implementation details that are necessary for our exposition of the Family CG approach in section \ref{sec_family_RMP}. In Section \ref{exper} we provide experimental validation of our approach on single source capacitated facility location (SSCFLP) \citep{diaz2002branch}. In Section \ref{FutureWork} we discuss an alternative construction of families of a column as well as our plans to apply this technique to other combinatorial optimization problems which are of significant current interest. Section \ref{conc} concludes this paper.
\section{Column Generation Review} \label{CG_review}
In this section we review the core concepts in Column Generation (CG) required to understand our Family Column Generation (FCG) algorithm.
\subsection{Basic Column Generation} \label{rev_CG} We now consider the basic CG algorithm in the context of the following broad class of set cover problems. We use $N$ to denote the set of items to be covered which we index by $u$. We use $F$ to denote the set of facilities which we index by $f$. We use $\Omega$ which we index by $l$ to denote the set of feasible pairs of subset of $N$, and a single member of $f$. We use $a_{ul}\in \{0,1\},a_{fl} \in \{0,1\}$ to describe $\Omega$. Here $a_{ul}=1$ IFF $l$ includes item $u$ and otherwise $a_{ul}=0$. Similarly $a_{fl}=1$ IFF $l$ includes uses facility $f$ and otherwise $a_{fl}=0$. We associate each $l\in \Omega$ with a cost $c_l$. We frame set cover as the following integer linear program which enforces that each facility is used no more than once. Here $\theta_l \in \{0,1\}$ is a binary variable used to describe the solution to set cover. Here $\theta_l=1$ IFF $l$ is included in the solution and otherwise is zero.
\begin{subequations} \label{basicILP} \begin{align} \label{eq_basic_obj}
\min_{\theta_l \in \{0,1\} }\sum_{l \in \Omega}c_l \theta_l\\
\label{eq_basic_cover}
\sum_{l \in \Omega}a_{ul}\theta_l \geq 1 \quad \forall u \in N\\
\label{eq_basic_pack}
\sum_{l \in \Omega}a_{fl}\theta_l \leq 1 \quad \forall f \in F \end{align} \end{subequations}
We now describe the components of \eqref{basicILP}. In \eqref{eq_basic_obj} we seek to minimize the total cost of elements in $\Omega$ selected. In \eqref{eq_basic_cover} we enforce that each item is included in at least one column. In most applications \eqref{eq_basic_cover} is tight for all $u \in N$ in any optimal solution \cite{barnprice}. In \eqref{eq_basic_pack} we enforce that each facility is used no more than once. Often we may enforce that the total number of columns selected does not exceed a particular value. So as to not over-complicate the write up we do not include this but not that $c_l$ can be offset by a constant for all $l$ to encourage/discourage the use of more columns.
Problems of the form \eqref{basicILP} are common in the operations research literature and include the single source capacitated facility location problem (SSCFLP) \citep{diaz2002branch} and the vehicle routing problem (VRP) \citep{costa2019,Desrochers1992}.
Solving \eqref{basicILP} is NP-hard however efficient exact or approximate algorithms can be built using the column generation (CG) framework \citep{barnprice}. CG solves the LP relaxation of \eqref{basicILP}, which in many applications well approximates \eqref{basicILP}.
CG operates as follows. CG generates a sufficient subset of $\Omega$ denoted $\Omega_R$ to provably solve optimization over all $\Omega$. This is done by iterativley solving the LP over $\Omega_R$ and adding columns with negative reduced cost to $\Omega_R$. CG terminates when no column in $\Omega_R$ has negative reduced cost. We write the primal and dual LP relaxations below over $\Omega_R$ with dual variables denoted in [\ref{basicLDual}] by their associated constraints.
\begin{subequations} \label{basicLPrimal} \begin{align}
\mbox{Primal Optimization: }\min_{\theta \geq 0}\sum_{l \in \Omega_R}c_l \theta_l\\
\sum_{l \in \Omega_R}a_{ul}\theta_l \geq 1 \quad \forall u \in N \quad [\pi_u]\\
\sum_{l \in \Omega_R}a_{fl}\theta_l \leq 1 \quad \forall f \in F \quad [\pi_f] \end{align} \end{subequations}
\begin{subequations} \label{basicLDual} \begin{align}
\mbox{Dual Optimization: }\max_{\pi \geq 0}\sum_{u \in N}\pi_u-\sum_{f \in F}\pi_f\\
c_l+\sum_{f \in F}a_{fl}\pi_f-\sum_{l \in \Omega}a_{ul}\pi_u \geq 0 \forall l \in \Omega_R, [\theta_l] \end{align} \end{subequations} CG optimization is initialized with columns $\Omega_R$ describing a feasible solution to \eqref{basicLPrimal}. Next CG iterates between the following two steps until no column has negative reduced cost. \textbf{(1)} Solve \eqref{basicLPrimal} providing a primal and a dual solution. \textbf{(2)} Find the lowest reduced cost column associated with each $f \in F$, and add the negative reduced cost columns identified to $\Omega_R$. This step is referred to as pricing. CG terminates when pricing fails to identify a negative reduced cost column. The corresponding primal solution $\theta$ is provably optimal for optimization over $\Omega$ at termination of CG.
Pricing is written below given fixed $f$ using $\Omega_f$ to denote the subset of columns associated with $f$. \begin{subequations} \label{pricer} \begin{align}
\min_{l \in \Omega_f}\bar{c}_l\\
\bar{c}_l=c_l+\pi_f-\sum_{u \in N} a_{ul}\pi_u \end{align} \end{subequations} Solving pricing is typically a combinatorial optimization problem that is problem domain specific. Most commonly it is solved as a dynamic program or resource constrained shortest path problem \citep{lubbecke2005selected} though can be a small scale ILP \citep{FlexDOIArticle,zhang2017efficient}.
The solution at termination of optimization may still be fractional. Multiple mechanisms can be used to tackle this issue while achieving exact optimization. The set cover LP relaxation can be further tightened in a cutting plane manner by valid inequalities such as subset-row inequalities \citep{jepsen2008subset,wang2017tracking}. The use of such valid inequalities permits optimization with CG. CG can be built into an exact branch-bound search procedure using branch-price \citep{barnprice}.
\section{Related Literature} \label{litRev}
The literature on column generation techniques is vast. Here we review the most relevant material, namely research on trust region based methods and dual optimal inequalities used to accelerate CG. CG suffers from slow convergence when the number of items in a column becomes large. This tends to produce intermediate dual solutions which are sparse and do not share properties with those of known dual optimal solutions. We now consider some methods designed to circumvent this difficulty.
\subsection{Trust Region Based Methods} Trust regions based methods discourage \citep{du1999stabilized} or prevent \citep{marsten1975boxstep} the next dual solution from leaving the area around the best dual solution generated thus far (in terms of Lagrangian relaxation). This is done since the current set of columns provides little information regarding the Lagrangian relaxation of dual solutions dis-similar to a previously generated dual solution.
Smoothing based approaches \citep{Pessoa2018Automation} are a simple class of trust region approaches that achieve excellent results in practice. Smoothing based approaches only differ from standard CG in the selection of the dual variables terms which pricing is done on. Specifically they use a convex combination of: \\ 1) The dual solution of the current restricted master problem and \\ 2) The dual solution generated thus far with greatest Lagrangian relaxation.
\subsection{Dual Optimal Inequalities} Dual Optimal Inequalities (DOI)\citep{ben2006dual} provide provable bounds on the optimal dual solution and ensure that the each step dual solution lies in the corresponding space. The primal DOI correspond to slack variables in the primal such as providing rewards for over-covering items or costs for swapping one item for another \citep{Gschwind2016Dual}. DOI are provably inactive at termination of CG but may or may not be active in intermediate steps.
\subsection{Flexible Dual Optimal Inequalities} We now consider the Flexible Dual Optimal Inequalities (F-DOI) \citep{FlexDOIArticle,haghani2020relaxed,haghani2020smooth}. F-DOI exploit the following observation: the change in the cost of a column induced by removing a small number of items is often small and can be easily bounded. Such bounds are column specific and exploit properties of the problem domain. In the primal form additional variables which provide reward for over-covering items are included. These rewards are set such that at optimality they are not used, but prior to termination of CG they have an important role. This can be understood as adding to the RMP all columns consisting of subsets of columns in the RMP. The costs of these columns provide upper bounds on the true cost of the column. F-DOI and their predecessors, varying and invariant DOI \citep{yarkony2020data}, provide considerable speed-ups and also make CG more robust to the specific selection of optimization parameters.
\section{Implementation Details} \label{Implementation} Here we review Lagrangian Relaxation and our test problem, the single source capacitated facility location problem. Further, we review the box-step method because we will need it in our FCG approach.
\subsection{Lagrangian Relaxation} \label{rev_Lag}
The Lagrangian relaxation \citep{desrosiers2005primer} is smooth, convex and has identical value to the MP at the optimizing $\pi$ for the MP. At any given point in CG optimization a lower bound on the optimal solution can be generated using the Lagrangian relaxation. The Lagrangian relaxation can be used to provide as stopping criteria for CG. Specifically when the difference between the objective of the RMP solution and the Lagrangian relaxation is sufficiently small (according to a user defined criteria) we terminate CG optimization. We write the Lagrangian relaxation at a given dual solution $\pi$ as $\ell_{\pi}$, which we define below.
\begin{subequations} \label{ellDef} \begin{align} \label{ellDefBasic}
\mbox{MP}\geq \ell_{\pi}=\sum_{u \in N} \pi_u -\sum_{f \in F}\pi_f+\sum_{f \in F} \min(0,\min_{l \in \Omega_{f}}\bar{c}_l)
\end{align} \end{subequations}
Observe that any non-negative solution $\pi$ can be projected to one that is a feasible solution to the dual master problem (\eqref{basicLDual} with $\Omega_R \leftarrow \Omega$) with objective identical to the Lagrangian relaxation in \eqref{ellDefBasic} by setting $\pi_f\leftarrow \pi_f+ \min(0,\min_{l \in \Omega_{f}}\bar{c}_l)$.
\subsection{Application: Single Source Capacitated Facility Location} \label{rev_sscflp} To provide a meaningful example we consider the classical single source capacitated facility location problem (SSCFLP) which is often used to explore column generation techniques \citep{diaz2002branch,haghani2020smooth}.
In the SSCFLP, we are given a set of customers $N$ and a set of potential facilities $F$, such that $F\cap N=\emptyset$. Each facility $f\in F$ is associated a fixed opening cost $c_f$ and a capacity $K_f$. Each customer $u\in N$ is associated a demand $d_u\geq 0$. Each pair $(f, u)\in F\times N$ is associated an assignment cost $c_{fu}\geq 0$. Without loss of generality, we may assume that all parameters are integer-valued. The SSCFLP consists in selecting a subset of facilities to open, and to assign every customer to exactly one open facility in such a way that the capacities of the selected facilities are respected, at minimum total cost (fixed costs + assignment costs).
The pricing subproblem is a 0-1 knapsack problem which is easy to solve in practice\citep{cuttingstock,gilmore1965multistage}. Specifically we condition on facility $f$ and solve the knapsack problem below. We use $x_{u} \in \{0,1\}$ to denote the decision variable for customer $u$; where $x_u=1$ IFF customer $u$ is included. \begin{subequations} \label{pricingKnap} \begin{align} \min_{x_u \in \{0,1\} \quad \forall u \in N}c_f+\pi_f+\sum_{u \in N}x_u(c_{fu}-\pi_u)\\ \sum_{u \in N}x_u d_u \leq K_f \end{align} \end{subequations}
\subsection{Box-Step Method for CG Optimization} \label{rev_box}
The box-step method is a classic mechanism to accelerate CG \citep{marsten1975boxstep} which lies in the family of trust region methods \citep{du1999stabilized}. Trust region methods broadly can be understood from the dual perspective as follows. The current set of columns $\Omega_R$ provide a good estimate of the Lagrangian relaxation for dual solutions $\pi$ that are very similar to previously generated dual solutions over the course of CG optimization. However $\Omega_R$ does not provide a good estimate of the Lagrangian relaxation for highly distinct points, thus grossly overestimating the dual objective for solutions which are dissimilar to previous solutions. Trust region based methods ensure or encourage the new solution to lie near the previously generated optimal dual solution. Trust region methods thus make CG approaches operate more like gradient ascent methods. By this we mean that that $\Omega_R$ and the dual solution change gradually based on each-other. We now describe a trivial variant of the box-step.
For efficiency of notation we use $\Psi_{\pi^+,\pi^-,\hat{\Omega}}$ to be the value of the LP over the box defined by $\pi^+$ and $\pi^-$ over a set $\hat{\Omega}\subseteq \Omega$. \begin{subequations} \label{boundRMP} \begin{align}
\Psi_{\pi^+,\pi^-,\hat{\Omega}}=\max_{\substack{\pi \geq 0 }} \sum_{u \in N} \pi_u-\sum_{f \in F}\pi_f
\\
c_l+\sum_{f \in F}a_{fl}\pi_f-\sum_{u \in N}a_{ul}\pi_u \geq 0 \ \quad \forall l \in \hat{\Omega}\\
\label{boundPI}
\pi^+_u\geq \pi_u \geq \pi^-_u \quad \forall u \in N \end{align} \end{subequations}
Note that \eqref{boundRMP} always has a dual feasible solution since $\pi_f$ is not restricted.
The primal form of optimization in \eqref{boundRMP} is written as follows using $\delta^+_u,\delta^-_u$ to denote the primal variables associated with \eqref{boundPI}. So as to avoid confusion with the dual problem we use $\Delta^+_u\leftarrow \pi^+_u$ and $\Delta^-_u\leftarrow \max(0,\pi^-_u)$.
\begin{subequations} \label{primal_box} \begin{align} \min_{\substack{\theta \geq 0\\ \delta \geq 0}}\sum_{l \in \hat{\Omega}} c_l \theta_l+\sum_{u \in N}\Delta^+_u\delta^+_u-\Delta^-_u\delta^-_u\\ \delta_u^+-\delta_{u}^- +\sum_{l \in \hat{\Omega}}a_{ul}\theta_l\geq 1 \quad \forall u \in N\\ \sum_{l \in \hat{\Omega}}a_{fl}\theta_l\leq 1 \quad \forall f \in F \end{align} \end{subequations} The trivial box-step method proceeds iterates between the following two steps. \textbf{(1)} We solve \eqref{boundRMP} around the solution $\pi^*$ which is the dual solution with highest Lagrangian relaxation generated thus far. The box around $\pi^*$ is parameterized by a fixed constant $\nu \in \mathbb{R}_{+}$; where $\pi^+_u\leftarrow \pi^*_u +\nu$ and $\pi^-_u\leftarrow \max(0,\pi^*_u -\nu)$ for all $u \in N$. (\textbf{2})is performed as in standard CG. We iterate between solving \eqref{primal_box}, and pricing until no column has negative reduced cost and the $\delta$ terms are inactive. This certifies that an optimal solution has been produced to the master problem (MP). The selection of $\nu$ can be understood in terms of the following trade off. Smaller values of $\nu$ means each iteration of box-step makes less progress towards solving the MP. However if $\nu$ is smaller then any given iteration is more inclined to improve the Lagrangian relaxation and hence update $\pi^*$. Sophisticated box-step methods can employ schedules and mechanisms for changing $\nu$ over the course of optimization. We write the box-step method formally in Alg \ref{basicBOx}, which we annotate below.
\begin{algorithm}[!b]
\caption{Box-Step Method} \begin{algorithmic}[1] \State $\Omega_R,\nu,\pi^* \leftarrow $ from user \label{Zline_rec_input_start}
\Repeat \label{Zline_outer_start}
\State $\Delta^+_u, \leftarrow \pi^*_u+\nu$ for all $u \in N$ \label{Zset_delta_1} \State $\Delta^-_u,\leftarrow \max(0,\pi^*_u -\nu)$ for all $u \in N$ \label{Zset_delta_2} \State Solve for $\theta,\bar{\pi},\delta$ using \eqref{primal_box} over $\Omega_R$ \label{Zline_solve_FRMP}
\For{$f \in F$}
\label{Zline_pricing_Start}
\State $l^*_f\leftarrow \mbox{arg}\min_{l \in \Omega_f} c_l-\bar{\pi}_f-\sum_{u \in N}\bar{\pi}_u a_{ul}$
\If {$0>c_{l^*_f}+\bar{\pi}_f-\sum_{u \in N}\bar{\pi}_u a_{ul^*_f}$}
\State $\Omega_R \leftarrow \Omega_R \cup l^*_f$
\EndIf
\EndFor
\label{Zline_pricing_End}
\If{$\ell_{\pi^*}\leq \ell_{\bar{\pi}}$}
\label{Zstore_best_start}
\State $\pi^*\leftarrow \bar{\pi}$
\EndIf
\label{Zstore_best_end}
\Until{$c_{l^*_f}+\bar{\pi}_f-\sum_{u \in N}\bar{\pi}_u a_{ul^*_f} \geq 0$ for all $f \in F$ and $\delta^+_u=\delta^-_u=0$ for all $u \in N$}
\label{Zline_outer_end}
\State Return last $\theta$ generated. \label{ZreturnSol} \end{algorithmic} \label{basicBOx} \end{algorithm}
\begin{itemize}
\item Line \ref{Zline_rec_input_start}: We receive the input $\Omega_R$ which provides for a feasible solution for the MP. We also receive step size $\nu$ and initial $\pi^*$ which can be set trivially to the RMP solution over $\Omega_R$ or any other mechanism such as the zero vector.
\item Lines \ref{Zline_outer_start}-\ref{Zline_outer_end}: Solve the MP over $\Omega$. We certify that we have solved optimization by terminating when no column has negative reduced cost and $\delta$ is zero valued. We should note that if $\Delta^-_u=0$ then we do not include the primal variable $\delta^-_u$ as the corresponding dual constraint is redundant. We thus set $\delta^-_u=0$ by force when $\Delta^-_u=0$.
\begin{enumerate}
\item Line \ref{Zline_solve_FRMP}: Receive solution to RMP over $\Omega_R$ in box around $\pi^*$.
\item Lines \ref{Zline_pricing_Start}-\ref{Zline_pricing_End}: Compute the lowest reduced cost column associated with each $f \in F$. Then add any negative reduced cost columns computed to $\Omega_R$.
\item Lines \ref{Zstore_best_start}-\ref{Zstore_best_end}: We store the best solution found thus far. Here best means the one which maximizes the Lagrangian relaxation.
\end{enumerate}
\item Line\ref{ZreturnSol}: Return the last solution generated $\theta$ which is provably optimal and feasible for the MP. \end{itemize}
\section{Family Column Generation} \label{sec_family_RMP} In this section we introduce our Family Column Generation (FCG) algorithm. FCG differs from standard CG only in the mechanism to generate dual solutions, upon which pricing is performed. FCG (approximately) solves the Family Restricted Master Problem (FRMP) at each at iteration of CG; where the FRMP better approximates the MP at each iteration of CG than the standard RMP, while preserving efficient inference. The FRMP describes the LP over the set of all columns that lie in the family of any member of $l \in \Omega_R$, which we denote $\Omega^+_R$. We define $\Omega_{\hat{l}}$ to be the family of columns corresponding to subsets of the items that compose $l$, denoted $N_{\hat{l}}$ ($N_{\hat{l}}=\{u \in N \mbox{, s.t. } a_{ul}=1\}$) and preserve structural properties of $l$ (such as $f_l$ and ordering of items if an ordering exists). Thus $\Omega_R^+=\cup_{\hat{l} \in \Omega_R}\cup_{l \in \Omega_{\hat{l}}}$.
We now describe $\Omega_{\hat{l}}$ for two applications. \\
Consider in SSCFLP a column $\hat{l}$ that is associated with facility $f$ and items $N_{\hat{l}}$. The set $\Omega_{\hat{l}}$ is the set of columns each associated with facility $f$ and a customer set that lies in the power set of $N_{\hat{l}}$. \\
In the case of capacitated vehicle routing problem (CVRP) $\Omega_{\hat{l}}$ contains the set of columns in which the order of customers visited is the same as $l$ but a subset (perhaps empty) of those customers are removed.
The solution to FRMP optimization is attacked with a coordinate ascent method in the dual. In this method we generate a direction of travel then travel in that direction the optimal amount. We repeat this until we no longer improve the FRMP objective. FCG terminates only when FRMP is solved exactly and no column has negative reduced cost certifying that the solution is optimal for the MP.
We organize this section as follows. In Section \ref{genDirec} we consider the generation of directions of travel. In Section \ref{optTrav} we consider the determination of the optimal travel distance. In Section \ref{sec_algForm} formalize FCG in algorithmic form and provide analysis. In Section \ref{convergAnal} we show that FCG optimally solves the MP. In Section \ref{speedInner} we accelerate FCG by permitting early termination of FRMP solution without compromising convergence of FCG to the optimal solution of the MP. In Section \ref{applicationSSCFLP} we discuss the use of FCG for SSCFLP.
\subsection{Generation of Direction of Travel} \label{genDirec} We now consider the generation of directions to improve an incumbent dual solution denoted $\pi^0$. This solution is based on creating an approximation to the dual FRMP around $\pi^0$ where $\nu$ describes the size of the window over which our approximation is constructed. The value $\nu$ trades off the size of the space the approximation is done over and the quality of the approximation. Thus $\nu+\pi^0_u=\pi_u^+\geq \pi_u \geq \pi^-_u=\max(0,\pi^0_u-\nu)$ for all $u \in N$ and $\pi_f$ unrestricted. We then optimize an upper bound on the MP over this local approximation providing a solution $\bar{\pi}$. Then we travel from $\pi^0$ in the direction of $\vec{\pi}$ where $\vec{\pi}\leftarrow \bar{\pi}-\pi^0$.
We seek to construct a set $\Omega_{R\pi^+}$ so as to provide a good approximation to $\Psi_{\pi^+,\pi^-,\Omega^+_R}$ that is no worse than that of $\Psi_{\pi^+,\pi^-,\Omega_R}$ at any point in the window of ($\pi^+,\pi^-$). Thus the following should be satisfied. $\Psi_{\pi^+,\pi^-,\Omega_R}\geq \Psi_{\pi^+,\pi^-,\Omega_{R\pi^+}}\approx \Psi_{\pi^+,\pi^-,\Omega^+_R} $.
Since we have to construct many approximations in our procedure to solve FRMP; the construction of $\Omega_{R\pi^+}$ must not use expensive operations such as calls to non-trivial pricing problems, or calls to an LP solver, and must not result in an explosion in the number of columns. Our construction is motivated by the following equivalent way of writing optimization over $\Omega^+_R$ over the window defined by $\pi^+,\pi^-$.
\begin{subequations} \label{dual_FMP2} \begin{align} \max_{\substack{\pi \geq 0 }} \sum_{u \in N} \pi_u-\sum_{f \in F}\pi_f
\\
\label{minTerm}
\min_{l \in \Omega_{\hat{l}}}c_l+\sum_{f \in F}a_{fl}\pi_f-\sum_{u \in N}a_{ul}\pi_u \geq 0 \quad \forall \hat{l} \in \Omega_R\\
\pi^+_u \geq \pi_u \geq \pi^-_u \quad \forall u \in N
\end{align} \end{subequations}
Optimization over \eqref{dual_FMP2} is intractable but can be efficiently approximated using the following upper bound. We replace the $\pi_u$ terms in \eqref{minTerm} with $\pi^+_u$. Thus there is only one constraint for each $l \in \Omega_R$. Furthermore in many applications solving the optimization in \eqref{minTerm} is as easy as shown in our application; No expensive calls to NP-hard pricing oracles are required. Using \eqref{minTerm} we define $\Omega_{R\pi^+}$ below.
\begin{subequations} \label{projSet} \begin{align}
\Omega_{R\pi^+}=\cup_{\hat{l} \in \Omega_R}\hat{l}_{\pi^+}\\
\hat{l}_{\pi^+}= \mbox{arg}\min_{l \in \Omega_{\hat{l}}}c_{l}+\sum_{f \in F}a_{fl}\pi_f-\sum_{u \in N_{\hat{l}}}a_{ul}\pi^+_u
\label{projection_eq} \end{align} \end{subequations}
We now show that $\Psi_{\pi^+,\pi^-,\Omega_R}\geq \Psi_{\pi^+,\pi^-,\Omega_{R\pi^+}}$
\begin{theorem}
Consider that the claim is false. Thus there must exist a solution $\pi$ which is feasible for the optimization over $\Psi_{\pi^+,\pi^-,\Omega_{R\pi^+}}$ but not for the optimization over $\Psi_{\pi^+,\pi^-,\Omega_R}$ since the objective function of the two functions is identical. There must exist a constraint $l\in \Omega_R$ for which the following holds.
\begin{align} \label{contradictEQ} -c_{\hat{l}}-\sum_{f \in F}a_{f\hat{l}}\pi_f+\sum_{u \in N} a_{u\hat{l}}\pi_u> -c_{\hat{l}_{\pi^+}}-\sum_{f \in F}a_{f\hat{l}_{\pi^+}}\pi_f+\sum_{u \in N} a_{u\hat{l}_{\pi^+}}\pi_u
\end{align}
By construction in \eqref{projection_eq} we know that for any $\hat{l} \in \Omega_R$ \begin{align} \label{boundRQ}
c_{\hat{l}}+\sum_{f \in F}a_{f\hat{l}}\pi^+_f-\sum_{u \in N_{\hat{l}}}a_{ul}\pi^+_u\geq c_{\hat{l}_{\pi^+}}+\sum_{f \in F}a_{f\hat{l}_{\pi^+}}\pi^+_f-\sum_{u \in N_{\hat{l}}}a_{u\hat{l}_{\pi^+}}\pi^+_u \end{align} Now we add \eqref{contradictEQ} and \eqref{boundRQ} creating the following inequality; which we then simplify on the succeeding line. \begin{subequations}
\begin{align}
\sum_{u \in N}a_{u\hat{l}}(\pi_u-\pi^+_u)> \sum_{u \in N}a_{u \hat{l}_{\pi^+}}(\pi_u-\pi^+_u)\\
\label{mySumEQ}
\sum_{u \in N}(a_{u\hat{l}}-a_{u \hat{l}_{\pi^+}})(\pi_u-\pi^+_u)> 0 \end{align} \end{subequations}
Observe that in \eqref{mySumEQ} that $(a_{u\hat{l}}-a_{u\hat{l}_{\pi^+}})$ is strictly non-negative since $ \hat{l}_{\pi^+}$ lies in $\Omega_{\hat{l}}$. Observe also that $(\pi_u-\pi^+_u)$ is strictly non-positive since $\pi^+_u\geq \pi_u$. Thus $(a_{u\hat{l}}-a_{u\hat{l}_{\pi^+}})(\pi_u-\pi^+_u)$ is non-positive for every $u \in N$. The sum of non-positive terms on the LHS of \eqref{mySumEQ} is thus non-positive. Thus we have created a contradiction and hence proved the claim. \textbf{QED}.
\end{theorem}
Thus in this section we have provided a mechanism to produce directions of travel that locally approximate the FRMP over the area around an incumbent solution $\pi^0$.
\subsection{Determining the Optimal Travel Distance} \label{optTrav}
In this section we determine the optimal distance to travel along the ray starting at $\pi^0$ and traveling in direction $\vec{\pi}$ (where the ray is denoted $(\pi^0,\vec{\pi})$) so as to approximately maximize the Lagrangian relaxation (as written in \eqref{ellDefBasic}).
Since evaluating the Lagrangian relaxation requires a call to pricing, which may be expensive, we use a convex approximation to \eqref{ellDefBasic} that like the Lagrangian relaxation is equal to the MP at an optimizing $\pi$ (for MP). This bound denoted $\ell^{\Omega^+_{R}}_{\pi}$ considers only columns in $\Omega^+_R$ and uses \eqref{projection_eq}, which is easy to compute. Below we define $\ell^{\Omega^+_R}_{\pi}$ as follows using helper term $\ell^{\hat{l}}_{\pi}$, which is the reduced cost of the lowest reduced cost column in the family of $\hat{l}$.
\begin{subequations} \label{ellDef2} \begin{align}
\label{ellDefADV}
\ell_{\pi} \leq \ell^{\Omega^+_R}_{\pi}=\sum_{u \in N} \pi_u-\sum_{f \in F}\pi_f+\sum_{f \in F}\min_{\hat{l} \in \Omega_R \cap \Omega_f} \min(0,\ell^{\hat{l}}_{\pi})\\
\ell^{\hat{l}}_{\pi}=\min_{l \in \Omega_{\hat{l}}}\bar{c}_l
\end{align} \end{subequations}
Since $\pi$ is non-negative the maximum travel distance $m$ satisfies the following inequality.
\begin{align} \label{maxMval}
m\leq \frac{-\vec{\pi}_z}{\pi^0_z} \quad \forall z \in N \cup F, \vec{\pi}_z <0 \end{align} We set $m$ to the max possible value as described by \eqref{maxMval} unless it is unrestricted by \eqref{maxMval} and hence we set it to a large positive number ($m \leftarrow 100$ in experiments).
For any $\eta \in [0,1]$ we use $\phi_{\eta}$ to denote $\ell^{\Omega^+_{R}}_{\pi+\eta m\vec{\pi}}$). Since $\ell^{\hat{\Omega}}_{\pi}$ is a concave function of $\pi$ for any $\hat{\Omega} \subseteq \Omega$ we use the following binary search style procedure to maximize $\phi_{\eta}$ with respect to $\eta$. At any given point in our procedure we preserve the following invariant $0\leq \eta^-\leq \eta^*\leq \eta^+\leq 1$ where $\eta^*=\mbox{arg}\max_{\eta \in [0,1]}\phi_{\eta}$.
We use $\zeta^-\leftarrow\frac{3\eta^-}{4}+\frac{\eta^+}{4},\zeta \leftarrow \frac{2\eta^+}{4}+\frac{2\eta^+}{4},\zeta^+\leftarrow\frac{1\eta^-}{4}+\frac{3\eta^+}{4}$ to be the one quarter,one half and three quarter way points between $\eta^-$ and $\eta^+$. At any given step of the iteration we evaluate $\phi_{\zeta^-},\phi_{\zeta},\phi_{\zeta^+}$. We then update $\nu^+$,$\nu^-$ so as to preserve the invariant $ \eta^-\leq \eta^*\leq \eta^+$ while decreasing $\eta^+-\eta^-$. At any given point in optimization there are three cases which describe the corresponding property of $\eta^*$.
\begin{enumerate} \item $\phi_{\zeta}=\max(\phi_{\zeta},\phi_{\zeta^+},\phi_{\zeta^-})$: Then $\eta^*$ lies in $[\zeta^-,\zeta^+]$
\item $\phi_{\zeta^-}=\max(\phi_{\zeta},\phi_{\zeta^+},\phi_{\zeta^-})$: Then $\eta^*$ lies in $[\eta^-,\zeta]$
\item $\phi_{\zeta^+}=\max(\phi_{\zeta},\phi_{\zeta^+},\phi_{\zeta^-})$: Then $\eta^*$ lies in $[\zeta,\eta^+]$ \end{enumerate} We formalize the search in Alg \ref{searchProc}. We terminate when we are within a user defined tolerance $\epsilon$ ($\epsilon=10^{-5}$ in our experiments) then return the best point found thus far.
\begin{algorithm}[!b]
\caption{Optimization to determine $\eta^*$} \begin{algorithmic}[1] \State $\eta^-\leftarrow 0$ \State $\eta^+ \leftarrow 1$ \While {$\eta^+-\eta^->\epsilon$} \State $\zeta \leftarrow \frac{2\eta^-}{4}+\frac{2\eta^+}{4}$ \State $\zeta^+\leftarrow \frac{1\eta^-}{4}+\frac{3\eta^+}{4}$ \State $\zeta^-\leftarrow \frac{3\eta^-}{4}+\frac{\eta^+}{4}$ \If{$\phi_{\zeta}=\max(\phi_{\zeta},\phi_{\zeta^+},\phi_{\zeta^-})$} \State $\eta^- \leftarrow \zeta^-$ \State $\eta^+ \leftarrow \zeta^+$ \State Continue \EndIf \If{$\phi_{\zeta^-}=\max(\phi_{\zeta},\phi_{\zeta^+},\phi_{\zeta^-})$} \State $\eta^+\leftarrow \zeta $ \State Continue \EndIf \If{$\phi_{\zeta^+}=\max(\phi_{\zeta},\phi_{\zeta^+},\phi_{\zeta^-})$} \State $\eta^-\leftarrow \zeta $ \State Continue \EndIf \EndWhile
\State Return $\mbox{arg}\max_{\eta \in \{ \eta^-,\eta^+\}}\phi_{\eta}$
\end{algorithmic} \label{searchProc} \end{algorithm}
\subsection{Algorithmic Formulation of Family-CG} \label{sec_algForm} We now consider an algorithmic formulation which we refer to as Family Column Generation (FCG). At each iteration of FCG we solve the Family RMP approximately then do pricing. The FCG is parameterized by one parameter only which is the step size $\nu$. The solution of the FRMP is found using the coordinate ascent approach of alternating between generating directions and going the optimal amount in that direction as measured by an approximation to the Lagrangian relaxation.
We terminate FRMP optimization when the current solution $\pi$ satisfies one of the following:
\\ \textbf{(1)} $\pi$ describes the optimal solution to the RMP over $\Omega^+_R$ or \\ \textbf{(2)} If the optimal distance to travel does not equal or exceed the minimum amount $\frac{1}{m}$. In Section \ref{convergAnal} we show that to ensure converge of FCG we must travel at least $\frac{1}{m}$ times the maximum possible distance.
\begin{itemize}
\item Line \ref{line_rec_input_start}: We receive the input $\Omega_R$ which provides for a feasible though not optimal solution for the MP. We also receive step size $\nu$ and initial $\pi^*$ which can be set trivially to the RMP solution over $\Omega_R$ or any other mechanism such as the zero vector.
\item Line \ref{line_outer_start}-\ref{line_outer_end}: Solve the MP over $\Omega$. We certify that we have solved optimization by terminating when no column has negative reduced cost. Termination can only happen in this algorithm if FRMP is solved optimally.
\begin{enumerate}
\item Line \ref{line_solve_FRMP}: Receive approximate solution to FRMP over $\Omega_R$.
\item Line \ref{line_pricing_Start}-\ref{line_pricing_End}: Compute the lowest reduced cost column associated with each $f \in F$. Then add any negative reduced cost columns computed to $\Omega_R$.
\item Line \ref{store_best_start}-\ref{store_best_end}: We store the best solution found thus far. Here best means the one which maximizes the Lagrangian relaxation.
\end{enumerate}
\item Line \ref{returnSol}:Return the last solution generated $\theta$ which is provably optimal and feasible for the MP. \end{itemize}
\begin{algorithm}[!b]
\caption{Family Column Generation} \begin{algorithmic}[1] \State $\Omega_R,\nu,\pi^* \leftarrow $ from user \label{line_rec_input_start}
\Repeat \label{line_outer_start}
\State $\bar{\pi},\theta \leftarrow $ Solve FRMP approximately using Alg \ref{ezVer_master} given $\Omega_R,\pi^*$ and $\nu$. \label{line_solve_FRMP}
\For{$f \in F$}
\label{line_pricing_Start}
\State $l^*_f\leftarrow \mbox{arg}\min_{l \in \Omega_f} c_l+\bar{\pi}_f-\sum_{u \in N}\bar{\pi}_u a_{ul}$
\If {$0>c_{l^*_f}+\bar{\pi}_f-\sum_{u \in N}\bar{\pi}_u a_{ul^*_f}$}
\State $\Omega_R \leftarrow \Omega_R \cup l^*_f$
\EndIf
\EndFor
\label{line_pricing_End}
\If{$\ell^{\Omega_R}_{\pi^*}\leq \ell^{\Omega_R}_{\bar{\pi}}$}
\label{store_best_start}
\State $\pi^*\leftarrow \bar{\pi}$
\EndIf
\label{store_best_end}
\Until{$c_{l^*_f}+\bar{\pi}_f-\sum_{u \in N}\bar{\pi}_u a_{ul^*_f} \geq 0$ for all $f \in F$}
\label{line_outer_end}
\State Return $\theta$ for last $\theta$ generated. \label{returnSol} \end{algorithmic} \label{ezVer} \end{algorithm}
We now consider the solution to the FRMP problem.
\begin{itemize}
\item Line \ref{line_rec_input_start2}: Receive input feasible solution, step size and initial solution $\pi^0$
\item Line \ref{step_inner_start}-\ref{step_inner_end}: Solve the FRMP approximately using coordinate ascent. Terminate when either FRMP is optimally solved or there exists an $l \in \Omega^+_R-\Omega_R$ with negative reduced cost meaning that the updated step did not improve the Lagrangian relaxation over $\Omega^+_R$.
\begin{algorithm}[!b]
\caption{Solve Family RMP} \begin{algorithmic}[1] \State $\Omega_R,\nu,\pi^0 \leftarrow $ from user \label{line_rec_input_start2}
\While{True} \label{step_inner_start}
\State $\Omega_{R\pi^+}\leftarrow \cup_{\hat{l} \in \Omega_R} \mbox{arg}\min_{l \in \Omega_{\hat{l}}}c_l+\pi^0_{f_l}-\sum_{u \in N}a_{ul}(\pi^0_u+\nu)$ \label{line_Gen_Q} \State $\Delta^+_u, \leftarrow \pi^0_u+\nu$ for all $u \in N$ \label{set_delta_1} \State $\Delta^-_u,\leftarrow \max(0,\pi^0_u -\nu)$ for all $u \in N$ \label{set_delta_2}
\State $\bar{\pi},\theta \leftarrow $ Solve \eqref{primal_box} in primal/dual form over $\Omega_{R\pi^+}$
\If {$\ell^{\Omega^+_R}_{\bar{\pi}}\leq \ell^{\Omega^+_R}_{\pi^{0}}$}
\label{checkCondStart}
\State Break
\label{BreakLine}
\EndIf
\label{checkCondEnd}
\label{line_gen_Dual}
\State $\vec{\pi} \leftarrow \bar{\pi}-\pi^{0}$ \State $m \leftarrow \min_{\substack{z \in N\cup F\\ \vec{\pi}_z<0}}\frac{-\vec{\pi}_z}{\pi^{0}_z}$ Compute max feasible step size. \label{compute_step_size}
\label{compute_Direction}
\State $\eta \leftarrow \max_{\eta \in [\frac{1}{m},1]}\phi_{\eta}$ via Alg \ref{searchProc} \label{compute_step_length}
\State $\pi^0 \leftarrow \pi^{0}+\eta m\vec{\pi}$ \label{update_pi_1}
\EndWhile
\label{step_inner_end}
\State Return $\bar{\pi},\theta$
\label{return_pi} \end{algorithmic} \label{ezVer_master} \end{algorithm}
\begin{enumerate}
\item Line \ref{line_Gen_Q}: Grab $\Omega_{R\pi^+}$. Here $f_l$ refers to the facility associated with column $l$ meaning $f_l \leftarrow \mbox{arg}\max_{f \in F}a_{fl}$
\item Line \ref{set_delta_1}-\ref{set_delta_2}: Determine $\Delta^+,\Delta^-$ terms
\item Line \ref{line_gen_Dual}: Solve \eqref{primal_box} providing optimal primal/dual solution pair over the box described by $\Delta$.
\item Line \ref{checkCondStart}-\ref{checkCondEnd}: If $\bar{\pi}$ is associated with inferior approximate Lagrangian relaxation relative to $\pi^0$ we break and return $\bar{\pi}$. This assures that a new column is generated very close to the current incumbent dual maximizing solution. Note that $\bar{\pi}$ corresponds to the minimum possible step-size.
\item Line \ref{compute_step_size}: Compute maximum feasible step size as described by \eqref{maxMval}. If no components appear under the min then this is unbounded and use a large positive number.
\item Line \ref{compute_Direction}: Compute the point $\hat{\pi}$ corresponding to traveling the maximum possible distance in the vector corresponding to starting at $\pi^0$ and traveling towards $\pi$.
\item Line \ref{compute_step_length}: Compute the optimal travel distance using Alg \ref{searchProc}. Note that we need only search range $\eta \in [\frac{1}{m},1]$ since the minimum step size is $\frac{1}{m}$.
\item Line \ref{update_pi_1}: Update $\pi^0$ to be optima
\end{enumerate}
\item Line \ref{return_pi}: Return approximate solution to FRMP. \end{itemize}
\subsection{Proof of Convergence} \label{convergAnal} We now show that at termination of Alg \ref{ezVer} that the solution $\theta$ is optimal for the dual master problem. We do this as follows.
Via \eqref{boundRQ} we know that $c_{l}+\pi_{f_l}+\sum_{u \in N}a_{ul}\pi_u\geq 0$ for each $l \in \Omega_R$. Thus pricing can never generate a column already in $\Omega_R$.
If Line \ref{checkCondStart} terminates with an inequality then there must be a column with negative reduced cost and hence Alg \ref{ezVer} does not terminate that iteration. If Line \ref{checkCondStart} terminates with an equality then the solution to the FRMP is solved exactly over $\Omega^+_R$ and thus the objective is less than than the master problem. Hence a column with negative reduced cost will be generated if the FRMP objective is greater than the master problem objective.
\subsection{Limiting the number of inner loop iterations}
\label{speedInner}
In this section we accelerate optimization by not solving the FRMP exactly in Alg \ref{ezVer_master} when to do so would require a large number of iterations. We only run the loop solving the FRMP in Alg \ref{ezVer_master} (Lines:\ref{step_inner_start}-\ref{step_inner_end}) up to a finite number of iterations; where the number of iterations is a fixed user defined parameter. If Lines: \ref{step_inner_start}-\ref{step_inner_end} do not terminate in that period via Line \ref{BreakLine} we do not terminate optimization Alg \ref{ezVer} if no negative reduced cost column is found in Line \ref{line_outer_end} but instead continue Alg \ref{ezVer}. The use of this does not interfere with the convergence guarantees in Section \ref{convergAnal}. In fact it merely consists of generating a column when we have only partially completed optimization over the FRMP and FRMP optimization continues unmodified if no column with negative reduced cost is identified.
\section{Experimental Analysis on the SSCFLP} \label{exper} \subsection{Column Projection} \label{applicationSSCFLP}
Family CG requires in Algorithm \ref{ezVer_master} that we can solve \eqref{projection_eq}. We now show that this is easy for our application of SSCFLP. We write \eqref{projection_eq} for SSCFLP below.
\begin{subequations} \label{pricingKnap2} \begin{align} \min_{x_u \in \{0,1\} \quad \forall u \in N}c_f+\pi_f+\sum_{u \in N}x_u(c_{fu}-\pi_u)\\ \sum_{u \in N}x_u d_u \leq K_f \label{enfkap2} \\ x_u=0 \quad \forall u \notin N_l \label{enfBinary} \end{align} \end{subequations}
Observe that any binary valued solution satisfying \eqref{enfBinary} satisfies \eqref{enfkap2} and thus \eqref{enfkap2} can be ignored. Thus an optimizer of \eqref{pricingKnap2} is written as follows $x_u\leftarrow [u \in N_l][c_{fu}<\pi^+_u]$ for all $u \in N$ where $[]$ is the binary indicator function.
\subsection{Numerical Experiments} \label{}
In this section we provide numerical experiments demonstrating the FRMP's effectiveness in accelerating CG convergence when compared to baseline methods. We apply FRMP to the SSCFLP and evaluate its convergence time and iterations required when solving the linear relaxation. We compare against unstabilized CG and against smoothing, which has shown to offer significant speedups on the SSCFLP. We test on 50 randomly generated instances. In each instance we have 50 facilities and 250 customers. Each facility has a capacity of 150 and a fixed opening cost of 5. Each customer has a demand which is randomly generated uniformly over the set $\{1,2,3,4,5\}$. To generate random customer service costs, each customer along with each facility is randomly given a position uniformly on the unit square. Service cost for each facility to each customer are set as the distance from that facility's position to that particular customer.
For smoothing we initialize our center dual solution $\pi^c$ to 0. We send to pricing a convex combination of our center dual solution and the current dual solution to the RMP $\pi_{pricing}=\lambda\pi^c+(1-\lambda)\pi$. We initially set $\lambda$ to 0 and reduce it by 0.1 every time there is a misprice. Following a misprice, when pricing finally produces a negative reduced cost column, we return $\lambda$ back to its initial value of 0.9 and proceed as before. Runtime and iteration count results for the FRMP and smoothing are shown in Table \ref{table:structured}. We show the number of iterations required and the total time required. As well we show the total time required for only solving the LP in the primal. This is presented since the FRMP requires some notable overhead in primal portion of the algorithm that lies outside solving the linear program. This overhead can be vary depending on problem characteristics or implementation so we provide some perspective on performance when its effects are excluded.
\begin{table}[!hbtp]
\centering
\scalebox{0.9}{
\begin{tabular}{|c|c|c|c|}
\hline
\multirow{1}{*}{ } & \multicolumn{1}{c|}{\bf Unstabilized} & \multicolumn{1}{c|}{\bf Smoothing} & \multicolumn{1}{c|}{\bf Family }\\
\hline
mean total iterations & 1736.2 & 465.3 & 175.3 \\
\hline
median total iterations & 1212.5 & 373.5 & 148.5 \\
\hline
mean total runtime & 2750.0 & 79.7 & 125.1 \\
\hline
median total runtime & 521.7 & 67.9 & 78.6 \\
\hline
mean total LP runtime & 2735.6 & 76.5 & 40.0 \\
\hline
median total LP runtime & 513.3 & 65.3 & 24.0\\
\hline
\end{tabular}}
\caption{SSCFLP results.}
\label{table:structured} \end{table}
The FRMP shows significant reductions in the average number of iterations required (and consequently the number of calls to pricing) when compared to smoothing. Relevant plots for iteration counts are shown in Figure \ref{fig:iter}. When considering total runtime, the FRMP falls short of outperforming smoothing on average. Plots for total runtime are shown in Figure \ref{fig:time}. Looking only at total LP solver time, the FRMP again offers vast improvements over smoothing. Relevant plots for LP runtime are shown in Figure \ref{fig:LPtime}.
\begin{figure}
\caption{Aggregate results as a function of iteration count.
\textbf{(Left):} Average relative gap for the lower bound as a function of iteration.
\textbf{(Right):} Iteration count comparison of Family RMP vs smoothing over 50 instances.}
\label{fig:iter}
\end{figure}
\begin{figure}
\caption{Aggregate results as a function of total runtime.
\textbf{(Left):} Average relative gap for the lower bound as a function of total runtime.
\textbf{(Right):} Total runtime comparison of Family RMP vs smoothing over 50 instances.}
\label{fig:time}
\end{figure}
\begin{figure}
\caption{Aggregate results as a function of total LP time.
\textbf{(Left):} Average relative gap for the lower bound as a function of total LP time.
\textbf{(Right):} Total LP time comparison of Family RMP vs smoothing over 50 instances.}
\label{fig:LPtime}
\end{figure}
\section{Future Research} \label{FutureWork}
First we describe an enhancement to our Family RMP method. Then we outline future analysis of these methods on other important combinatorial optimization problems.
\subsection{Alternative Families} In this case we will define the family $\Omega^+_R$ as the same begin $\cup_{l \in R}\Omega_{\hat{l}}$. However we will change the definition of $\Omega_{\hat{l}}$ to be a superset of what it was before. The only requirement is that we can solve for the lowest reduced cost member of $\Omega_{\hat{l}}$ easily.
For the case of SSCFLP we write this set as follows. Let $a_{dl}$ be the number of items in $l$ containing at least $d$ unit of demand meaning $a_{dl}=\sum_{u \in N;d\leq d_u}a_{ul}$. We define the $\Omega_{\hat{l}}$ to be the set of columns\\ $\hat{l} \in \Omega $ $a_{d\hat{l}}\leq a_{dl}$ for all $d \in [0,1,...,\max_{u \in N}d_u]$.
We write projection pricing below given column $\hat{l}$ associated with facility $f$
$\min_{l \in \Omega_{\hat{l}}}c_l+\pi_f-\sum_{u \in N}a_{ul}\pi_u$
We write this as an optimization problem as follows. \begin{subequations} \label{priceEZ} \begin{align}
\min_{x \in \{0,1\} }c_f+\pi_f+\sum_{u \in N}(c_{fu}-\pi_u)x_u
\sum_{u \in N}x_ud_u\leq K_f\\
\sum_{u \in N}x_u [d_u\geq d]\leq a_{dl} \end{align}
\end{subequations} Observe that we can solve \eqref{priceEZ} by first sorting $(c_{fu}-\pi_u)$ from order of smallest to largest and excluding the positive values. Let $u_i$ be the i'th item in the list . We can greedily and optimally construct the solution to \eqref{priceEZ} as follows. \begin{align}
x_{u_i} \leftarrow 1 \quad IFF \quad
\sum_{j<i}x_{u_j}[d_j\leq d]<a_{dl} \forall d\leq d_u \end{align}
We need only one additional tool to apply the technique of Family RMP. We need to produce $\Omega_{R\pi^+}$. We use upper bound terms for $u \in N_{\hat{l}}$ and lower bound terms for $u \notin N_{\hat{l}}$. The replacement equation for \eqref{projSet} is written below.
\begin{subequations} \label{projSet2} \begin{align}
\Omega_{R\pi^+}=\cup_{\hat{l} \in \Omega_R}\hat{l}_{\pi^+}\\
\hat{l}_{\pi^+}= \mbox{arg}\min_{l \in \Omega_{\hat{l}}}c_{l}+\sum_{f \in F}a_{fl}\pi_f-\sum_{u \in N_{\hat{l}}}a_{ul}\pi^+_u-\sum_{u \notin N_{\hat{l}}}a_{ul}\pi^-_u \end{align} \end{subequations} This satisfies the properties discussed in Section \ref{genDirec}.
\subsection{Additional Problems} In the future we plan to explore additional problems such as those discussed in \citep{Pessoa2018Automation}. That paper examines nine classical combinatorial optimization problems. Of the problems discussed in that paper, we are most concerned with capaciated vehicle routing and multi-activity shift scheduling. In addition, we plan to immediately turn our attention to applying these techniques to the various optimization problems arising in automated warehousing operations (see for example \citep{haghani2021multi}). Variations on multi-robot routing and product placement in shelf-to-picker and picker-to shelf problems in these operations are of keen current interest.
\section{Conclusion} \label{conc} We have introduced a new methodology to accelerate the convergence of column generation when applied to set-covering-based formulations by stabilizing dual optimization. Our methodology can be applied on any set cover-CG based formulation and we believe that it will accelerate the adoption of column generation methods for optimization problems that must be solved in real-time.
Our approach seeks to solve a restricted master problem over the set of all columns in the family of columns in the restricted master problem(RMP). Here the family of a column $l$is simply all columns with item set that is a subset of those in $l$ and for which any structural properties are preserved. In vehicle routing the structure describes the order of the items visited while in facility location it describes the facility associated with the column.
Our approach is identical to standard CG with regard to pricing though it differs with respect to the solution to the restricted master problem. Since optimization over the family can not be done explicitly due to combinatorial explosion, a coordinate ascent approach is employed. Directions are generated via solving over an upper bound over a finite box and the optimal amount of travel distance is determined by binary search. The optimization over the finite box uses columns that remove items for which the dual variable range does not justify the inclusion of.
FRMP CG circumvents the difficulties in the box-step method on which it is based. First it circumvents the need to choose a complex schedule for step size $\nu$ by following the direction produced so as to maximize an approximation to the Lagrangian relaxation. Thus a small fixed $\nu$ can be used and large moves in the dual space can still be achieved. The projection step at CG ensures that each column in the RMP is mapped to a column providing a constraint to the dual solution, ensuring that the intermediate dual solutions are not bound by only a small number of columns.
We demonstrate the efficacy of our approach on the SSCFLP. In future work we intend to explore vehicle routing problems and related multiple robot routing problems. In vehicle routing problems, pricing over the family of a given column computes the lowest reduced cost column containing a subset of the items in column $l$ and obeying the ordering of the items in $l$. Thus pricing involves solving a shortest path problem and not a resource constrained shortest path problem. That difference results in an immense improvement in efficiency. We also intend to explore adaptive step-size changing mechanisms to improve our current approach which uses a fixed step size throughout optimization.
\singlespacing
\appendix
\end{document} | arXiv |
Fundamental lemma of the calculus of variations
In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point. Accordingly, the necessary condition of extremum (functional derivative equal zero) appears in a weak formulation (variational form) integrated with an arbitrary function δf. The fundamental lemma of the calculus of variations is typically used to transform this weak formulation into the strong formulation (differential equation), free of the integration with arbitrary function. The proof usually exploits the possibility to choose δf concentrated on an interval on which f keeps sign (positive or negative). Several versions of the lemma are in use. Basic versions are easy to formulate and prove. More powerful versions are used when needed.
Basic version
If a continuous function $f$ on an open interval $(a,b)$ satisfies the equality
$\int _{a}^{b}f(x)h(x)\,\mathrm {d} x=0$
for all compactly supported smooth functions $h$ on $(a,b)$, then $f$ is identically zero.[1][2]
Here "smooth" may be interpreted as "infinitely differentiable",[1] but often is interpreted as "twice continuously differentiable" or "continuously differentiable" or even just "continuous",[2] since these weaker statements may be strong enough for a given task. "Compactly supported" means "vanishes outside $[c,d]$ for some $c$, $d$ such that $a<c<d<b$";[1] but often a weaker statement suffices, assuming only that $h$ (or $h$ and a number of its derivatives) vanishes at the endpoints $a$, $b$;[2] in this case the closed interval $[a,b]$ is used.
Version for two given functions
If a pair of continuous functions f, g on an interval (a,b) satisfies the equality
$\int _{a}^{b}(f(x)\,h(x)+g(x)\,h'(x))\,\mathrm {d} x=0$
for all compactly supported smooth functions h on (a,b), then g is differentiable, and g' = f everywhere.[3][4]
The special case for g = 0 is just the basic version.
Here is the special case for f = 0 (often sufficient).
If a continuous function g on an interval (a,b) satisfies the equality
$\int _{a}^{b}g(x)\,h'(x)\,\mathrm {d} x=0$
for all smooth functions h on (a,b) such that $h(a)=h(b)=0$, then g is constant.[5]
If, in addition, continuous differentiability of g is assumed, then integration by parts reduces both statements to the basic version; this case is attributed to Joseph-Louis Lagrange, while the proof of differentiability of g is due to Paul du Bois-Reymond.
Versions for discontinuous functions
The given functions (f, g) may be discontinuous, provided that they are locally integrable (on the given interval). In this case, Lebesgue integration is meant, the conclusions hold almost everywhere (thus, in all continuity points), and differentiability of g is interpreted as local absolute continuity (rather than continuous differentiability).[6][7] Sometimes the given functions are assumed to be piecewise continuous, in which case Riemann integration suffices, and the conclusions are stated everywhere except the finite set of discontinuity points.[4]
Higher derivatives
If a tuple of continuous functions $f_{0},f_{1},\dots ,f_{n}$ on an interval (a,b) satisfies the equality
$\int _{a}^{b}(f_{0}(x)\,h(x)+f_{1}(x)\,h'(x)+\dots +f_{n}(x)\,h^{(n)}(x))\,\mathrm {d} x=0$
for all compactly supported smooth functions h on (a,b), then there exist continuously differentiable functions $u_{0},u_{1},\dots ,u_{n-1}$ on (a,b) such that
${\begin{aligned}f_{0}&=u'_{0},\\f_{1}&=u_{0}+u'_{1},\\f_{2}&=u_{1}+u'_{2}\\\vdots \\f_{n-1}&=u_{n-2}+u'_{n-1},\\f_{n}&=u_{n-1}\end{aligned}}$
everywhere.[8]
This necessary condition is also sufficient, since the integrand becomes $(u_{0}h)'+(u_{1}h')'+\dots +(u_{n-1}h^{(n-1)})'.$
The case n = 1 is just the version for two given functions, since $f=f_{0}=u'_{0}$ and $f_{1}=u_{0},$ thus, $f_{0}-f'_{1}=0.$
In contrast, the case n=2 does not lead to the relation $f_{0}-f'_{1}+f''_{2}=0,$ since the function $f_{2}=u_{1}$ need not be differentiable twice. The sufficient condition $f_{0}-f'_{1}+f''_{2}=0$ is not necessary. Rather, the necessary and sufficient condition may be written as $f_{0}-(f_{1}-f'_{2})'=0$ for n=2, $f_{0}-(f_{1}-(f_{2}-f'_{3})')'=0$ for n=3, and so on; in general, the brackets cannot be opened because of non-differentiability.
Vector-valued functions
Generalization to vector-valued functions $(a,b)\to \mathbb {R} ^{d}$ is straightforward; one applies the results for scalar functions to each coordinate separately,[9] or treats the vector-valued case from the beginning.[10]
Multivariable functions
If a continuous multivariable function f on an open set $\Omega \subset \mathbb {R} ^{d}$ satisfies the equality
$\int _{\Omega }f(x)\,h(x)\,\mathrm {d} x=0$
for all compactly supported smooth functions h on Ω, then f is identically zero.
Similarly to the basic version, one may consider a continuous function f on the closure of Ω, assuming that h vanishes on the boundary of Ω (rather than compactly supported).[11]
Here is a version for discontinuous multivariable functions.
Let $\Omega \subset \mathbb {R} ^{d}$ be an open set, and $f\in L^{2}(\Omega )$ satisfy the equality
$\int _{\Omega }f(x)\,h(x)\,\mathrm {d} x=0$
for all compactly supported smooth functions h on Ω. Then f=0 (in L2, that is, almost everywhere).[12]
Applications
This lemma is used to prove that extrema of the functional
$J[y]=\int _{x_{0}}^{x_{1}}L(t,y(t),{\dot {y}}(t))\,\mathrm {d} t$
are weak solutions $y:[x_{0},x_{1}]\to V$ (for an appropriate vector space $V$) of the Euler–Lagrange equation
${\partial L(t,y(t),{\dot {y}}(t)) \over \partial y}={\mathrm {d} \over \mathrm {d} t}{\partial L(t,y(t),{\dot {y}}(t)) \over \partial {\dot {y}}}.$
The Euler–Lagrange equation plays a prominent role in classical mechanics and differential geometry.
Notes
1. Jost & Li-Jost 1998, Lemma 1.1.1 on p.6
2. Gelfand & Fomin 1963, Lemma 1 on p.9 (and Remark)
3. Gelfand & Fomin 1963, Lemma 4 on p.11
4. Hestenes 1966, Lemma 15.1 on p.50
5. Gelfand & Fomin 1963, Lemma 2 on p.10
6. Jost & Li-Jost 1998, Lemma 1.2.1 on p.13
7. Giaquinta & Hildebrandt 1996, section 2.3: Mollifiers
8. Hestenes 1966, Lemma 13.1 on p.105
9. Gelfand & Fomin 1963, p.35
10. Jost & Li-Jost 1998
11. Gelfand & Fomin 1963, Lemma on p.22; the proof applies in both situations.
12. Jost & Li-Jost 1998, Lemma 3.2.3 on p.170
References
• Jost, Jürgen; Li-Jost, Xianqing (1998), Calculus of variations, Cambridge University
• Gelfand, I.M.; Fomin, S.V. (1963), Calculus of variations, Prentice-Hall (transl. from Russian).
• Hestenes, Magnus R. (1966), Calculus of variations and optimal control theory, John Wiley
• Giaquinta, Mariano; Hildebrandt, Stefan (1996), Calculus of Variations I, Springer
| Wikipedia |
Pyramidal Cells are present in the cerebral cortex and the hippocampus. They exhibit non linear summation of inputs, with dendritic compartments acting as individual subunits capable of producing their own spikes. These dendrites then project to the cell body, where all dendritic signals are summed and integrated to determine the spiking behavior of the Pyramidal Cell body. Thus, Pyramidal Cells are multi-subunit structures, capable of computations which are far more complex than those of a linear point neuron. Properties of Pyramidal Cells useful for modeling are outlined below. Note that there are several different types of pyramidal cell depending on which region of the brain one is considering. Although they are expected to behave similarly, they do have some differences. It is for this reason that information will be given for individual pyramidal cell types.
All pyramidal cells contain the same basic structure. They are composed of a pyramidal shaped cell body, and a single, heavily branching axon projecting from the base.
The dendrites of a pyramidal neuron can be divided into two domains: basal and apical.
The basal dendritic tree is composed of 3-5 primary dendrites. Each of these divides to form branches of progressively thinning length.
The apical tree, which also splits several times, ends in a largely branched section known as the apical tuft.
The dendrites of pyramidal cells are covered with tiny branches known as dendritic spines. As one moves distal to the soma, the number of spines increases. These spines increase the surface area, and are believed to be where the majority of synapses arrive at the dendrites.
From a modeling perspective, both basal and apical branches contain proximal, medial, and distal compartments. These can be viewed as individual computational subunits. In addition, CA1 pyramidal neurons contain oblique medial and oblique distal branches, which arise from the medial apical and medial distal dendrites respectively.
Leads to an EPSP that is 1.28 $\pm$ 0.16 times as strong as the expected linear response.
Similar to the two layer neural network, this model treats pyramidal neurons as cells composed of two layers. The first layer is composed of many individual subunits (the dendrites), in which a set of terms is calculated based on the input vector. These terms are then summed up in the second layer, giving the cells overall sub threshold activity level.
In addition, an output nonlinearity $g$ may be applied to $a(x)$, giving $y = g(a)$. While this model is more realistic than a point neuron, it only characterizes subthreshold activity, and is therefore not useful when analyzing spiking behaviour.
This model is nearly identical to the two layer model, however now the Apical Tuft is being taken into consideration. The Apical Tuft serves as a third layer of computation which calculates a gain factor that is transmitted to the soma. This gain factor is simply a multiplier to the somatic output calculated in the Two Layer Model. At the Apical Tuft Itself, dendritic branches calculate a sigmoid function of their inputs, acting much like the typical dendritic subunits of the Two Layer Model. These responses are then summed at an integrating center, and converted into a gain factor, which is then transmitted to the soma.
This model came about from studies showing that the largest postsynaptic response in a pyramidal cell occured when activated synapses were all located within clusters of an intermediate size. In the model, these clusters are treated as neuron-like subunits called clusterons. Each synapse has a region of distance $D$ centered over it. If two synapses are activated, and the distance between them is less than $D/2$, then they are considered to be in the same subunit, and a multiplicative interaction occurs between the two of them. In particular, consider an input $x_j$ with a region of $D_j$ surrounding it.
where $w_j$ is the weight of the synapse.
A major drawback to this model is that it only considers excitatory inputs, and does not take into consideration spatial characteristics of the branching dendritic tree.
Coincidence detection is believed to be important in the induction of synaptic plasticity and long term potentiation (LTP). Following a somatic action potential, a Back Propagating Action Potential (BPAP) sends a wave of depolarization towards the distal dendrites of the cell. If the BPAP reaches the dendrite at the same time as an EPSP is induced (or slightly before), the depolarization caused by the BPAP and the EPSP becomes much greater than their expected linear sum. This depolarization is mediated by dendritic sodium channels which open when EPSP's bring the dendritic membrane potential into a range where a BPAP will cause a threshold to be crossed.This wave of depolarization can then travel back to the soma, and while it will not affect the initial action potential amplitude, it causes a large afterdepolarization which may induce the soma to fire another action potential (leading to burst firing). The time window between the arrival of a BPAP and the EPSP for coincidence detection to occur appears to be similar to the time window between pre and post synaptic neuron firing in the induction of LTP (the strengthening of a synapse when pre and postsynaptic neurons fire simultaneously). This implies that coincidence detection may be important in LTP, synaptic plasticity, and long term memory.
EPSP must occur simultaneously with, or less than 10 - 15 ms before somatic action potential for coincidence detection to occur.
Maximal amplification of BPAP occurs when EPSP occurs less than 3 ms before a somatic action potential.
BPAP amplification was only observed at dendritic distances greater than 450 μm.
Gasparini, S., Migliore, M., and Magee., J.C. (2004). On the Initiation and Propagation of Dendritic Spikes in CA1 Pyramidal Neurons. The Journal of Neuroscience, 24(49):11046-11056.
Megias, M., Emri, ZS., Freund, T.F., and Guly, A. i. (2001) Total number and distribution of inhibitory and excitatory synapses on hippocampal CA1 pyramidal cells. Neuroscience, 102, 3:527-540.
Poirazi, P., Brannon, T., and Mel, B.W. (2003). Pyramidal Neuron as Two-Layer Neural Network. Neuron, 37, 989-999.
Polsky, A., Mel, B.W., and Schiller, J. (2004). Computational subunits in thin dendrites of pyramidal cells. Nature neuroscience, 7, 6:621-627.
Spruston, N. (2008). Pyramidal neurons: dendritic structure and synaptic integration. Nature neuroscience reviews. 9:206-221. | CommonCrawl |
Contact, Hours, etc.
Trace: • filters-lab
Messy, Non-linear, Non-normal Model
System or Transition Model
Observation or Sensor Model
True values and input data
In this lab you will implement a simple particle filter and test your implementation. There will be no report, just print out a few runs. Please do not take the time to do anything fancy. Just hard code the model and run it a few times.
Please send me your code and the output from 4 runs.
This test data is based on the robot navigation systems we talked about in class. The robot's true position is given in x, y coordinates. The robot moves by specifying a direction (in radians from vertical) and a distance. The robot senses its position as the distance from 2 beacons ($d_{A}$ the distance from beacon A and $d_{B}$ the distance from beacon B)
$x_{0} \sim N(0,1) \!$
$y_{0} \sim N(0,1) \!$
The movement of the robot is random in $d$, the distance travelled, and $\theta$, the angle chosen. The intended distance (5) and direction $\left(\frac{\pi}{5}\right)$ are fixed.
$ d \sim N(5,1)\!$
<!– Warning this line is broken. The current version of the wiki will not parse it. I have commented it out. The line following it seems to work. $ \theta \sim \mathit{Uniform} ((\pi/5-\pi/36), (\pi/5+\pi/36))\!$ –> $ \theta \sim \mathit{Uniform} \left(\frac{\pi}{5}-\frac{\pi}{36},\frac{\pi}{5}+\frac{\pi}{36}\right)$
$x_{t+1}|x_{t} = x_{t} + d \cdot \cos \left(\theta\right)$
$y_{t+1}|y_{t} = y_{t} + d \cdot \sin \left(\theta\right)$
The robot senses its position as the distance from beacon A ($d_{A}$) and the distance from beacon B($d_{B}$). Both are imperfect measures as shown below:
$d_{A} = \sqrt{(-100-x)^{2}+(100-y)^{2}} $
$d_{B} = \sqrt{(150-x)^{2}+(90-y)^{2}} $
$r_{A} \sim N (d_{A}, 1)\!$
$r_{B} \sim N (d_{B}, 1)\!$
Your test data is given below. Columns 1 and 2 are the TRUE x and y (the output should be close to this). Columns 3 and 4 are the input data.
6.23506410875 & 3.47428776487 & 143.69345025 & 166.824055471 \\
10.1040856095 & 6.77685868518 & 145.664295064 & 164.501752829 \\
14.0195541954 & 10.0769108846 & 144.591594567 & 157.474359865 \\
18.0100054682 & 13.0579715054 & 146.50065381 & 152.469730508 \\
33.479183402 & 25.2895006721 & 152.367509975 & 133.667699492 \\
38.1615374 & 28.4013522307 & 155.708667162 & 126.696182098 \\
51.2820776483 & 38.372507025 & 164.570101706 & 111.140841502 \\
54.1752013771 & 40.6643076262 & 164.408405506 & 107.94706493 \\
62.951248513 & 46.4661999927 & 171.489734721 & 96.7475872389 \\
67.0126051217 & 49.4875173971 & 173.631500934 & 91.6295157657 \\
Use the noisy input values given above (NOT THE TRUE VALUES GIVEN ABOVE! They are included for reference only!) to infer the location of the robot. Report you findings in the form of a mean and a variance for x and a mean and a variance for y for each time step.
Given below is sample output from five separate runs of the particle filter. Note that columns 1 and 2 are the mean x and mean y of the particles. Columns 3 and 4 are the variances of the x and y values of the particles. These tests were run with 100 particles.
These answers are based on simulation. As with any simulation based approach, it is highly unlikely that you will get the same answer. It is also highly unlikely that you will get very different answers. We have listed results for 5 runs so that you can get a feel for how widely the results are varying.
Run #1
5.88606083114 & 3.82965157521 & 0.406162538 & 0.595028642435 \\
9.5895510708 & 5.96734479066 & 0.238958305025 & 0.511479417689 \\
13.9570561381 & 9.51319610885 & 0.389455944163 & 0.403564484278 \\
18.2636571862 & 12.7184190759 & 0.386009572459 & 0.377006008423 \\
23.4404892475 & 16.5467159461 & 0.24622051089 & 0.295367855259 \\
26.5832671078 & 19.14311067 & 0.396216711862 & 0.319908203942 \\
33.7744730723 & 24.8619576825 & 0.3590174446 & 0.472543788055 \\
46.663814675 & 34.3545973368 & 0.303326033837 & 0.481012206298 \\
54.4571680282 & 40.1109639611 & 0.503938147108 & 0.45904147466 \\
63.525034013 & 46.5594323783 & 0.361265534216 & 0.51375962392 \\
5.90120838885 & 3.48596079416 & 0.410220615472 & 0.640784190638 \\
13.9047771233 & 9.24301171167 & 0.46877884686 & 0.393973997427 \\
23.3499557467 & 16.5337033997 & 0.51289009183 & 0.55295880543 \\
42.4892746032 & 31.160354343 & 0.283057889157 & 0.549393655945 \\
5.88624337826 & 3.7439412516 & 0.310305563726 & 0.799427213409 \\
46.72466167 & 34.7980049482 & 0.290479450733 & 0.399775602304 \\
63.3400429555 & 46.8806733312 & 0.241533135065 & 0.422926971 \\
23.3517720275 & 16.701293073 & 0.39343636963 & 0.534702420785 \\
cs-677sp2010/filters-lab.txt · Last modified: 2014/12/12 20:37 by ryancha | CommonCrawl |
\begin{document}
\title{Geometric Spanning Cycles in Bichromatic Point Sets}
\begin{abstract} Given a set of points in the plane each colored either red or blue, we find non-self-intersecting geometric spanning cycles of the red points and of the blue points such that each edge of the red spanning cycle is crossed at most three times by the blue spanning cycle and vice-versa. \end{abstract}
\section{Introduction} A \emph{geometric graph} is a graph embedded in the plane with edges that are straight-line segments. A set of points is in general position if no three points of the set are collinear. In this paper, a bichromatic point set is a finite set of points $S$ in general position, partitioned into two disjoint color classes $S_R$ and $S_B$ (red and blue.)
Several problems have been studied that involve finding geometric graphs on sets of red and blue points. Alternating paths in bichromatic point sets in convex position were studied in \cite{akiyama1990simple}. Alternating paths in general position were studied in \cite{abellanas1999bipartite}. Alternating paths in points with more than two colors were studied in \cite{merino2006length}. \cite{tokunaga1996intersection} examined non-self-intersecting geometric spanning trees of the red points and the blue points and found a tight bound on the minimum number of intersection points between the red and blue spanning trees. \cite{Kano2005301} considered the case of more than two colors, and studied the
number of intersections for monochromatic spanning trees and for monochromatic spanning cycles. \cite{merino2005intersection} obtained a tight bound on the number of intersections in monochromatic perfect matchings. \cite{kano2013discrete} looked at points and lines in the plane lattice, and studied the number of crossings for alternating matchings and monochromatic spanning trees.
In \cite{tokunaga1996intersection}, Tokunaga also showed that there exist non-intersecting geometric spanning paths $P_R$ and $P_B$ of the red and blue points respectively, such that each edge of $P_R$ is crossed at most once by $P_B$, and vice-versa.
One may then wonder if a similar result is possible for cycles; i.e., is it possible to construct spanning cycles with ``few'' intersections on bichromatic point sets. In particular, we wonder for what values of $k$ does the following statement hold: there exist non-intersecting geometric spanning cycles $C_R$ and $C_B$ of the red and blue points respectively, such that each edge of $C_R$ is crossed at most $k$ times by $C_B$, and vice-versa. In \cite{tokunaga1996intersection}, Tokunaga conjectured that this statement is true when $k=2$. It is easy to see that the statement is false with $k=1$. We show here that the statement is true when $k=3$.
\begin{theorem} \label{thm:main} Given any bichromatic point set in general position, there exists a non-self-intersecting geometric spanning cycle of the red points and a non-self-intersecting geometric spanning cycle of the blue points such that each edge of the red spanning cycle is crossed by at most three edges of the blue spanning cycle, and vice-versa. \end{theorem}
\subsection{Definitions and Notation} If $X$ is a set of points and $y$ is a point outside the convex hull of $X$, we say that $y$ \emph{sees} a point $x\in X$ (with respect to $X$), if the line segment $(x,y)$ intersects the convex hull of $X$ only at $x$. In other words, if the convex hull of $X$ were opaque, then $y$ could see $x$. In particular, $x$ must be a vertex of the convex hull in order for $y$ to see it. If $X$ and $Y$ are two sets of points with disjoint convex hulls, and $x\in X$ and $y\in Y$, we say that $x$ and $y$ \emph{see each other} (with respect to $X$ and $Y$) if $y$ sees $x$ with respect to $X$ and $x$ sees $y$ with respect to $Y$.
Let $X$ be a set of points in the plane and $p\not\in X$. The \emph{radial order} of $X$ about $p$ is the cyclic list $(x_1, x_2, \dots, x_n)$ taking all the elements $x\in X$ ordered clockwise around $p$. The \emph{interval between $x_i$ and $x_j$ in the radial order} is the set consisting of $\{x_i,x_{i+1},\ldots,x_j\}$, if $i\leq j$, or $\{x_i,x_{i+1},\ldots,x_n,x_1,\ldots,x_j\}$ if $j<i$. If $y$ is in the interval between $x_i$ and $x_j$ in the radial order, then we say that $y$ lies \emph{between $x_i$ and $x_j$ in the radial order}.
If $R$ and $B$ are disjoint sets of red and blue points, respectively, then a \emph{blob} in the radial order of $R\cup B$ about $p$ is a maximal monochromatic set of consecutive elements in the radial order. Because the blobs are monochromatic, it is natural to speak of \emph{red blobs} and \emph{blue blobs}, containing red and blue points respectively.
Given a radial order $(x_1,\ldots,x_n)$ of $R\cup B$ with respect to a point $p\not\in R\cup B$, for any $i\in\{1,\ldots,n\}$, $x_{i-1}$ is called the point \emph{before} $x_i$ and $x_{i+1}$ is called the point \emph{after} $x_i$, where the indices are taken modulo $n$. For any blob $X$ in this radial ordering, the \emph{first point of $X$} is the point $x_i\in X$ such that $x_{i-1}\not\in X$. Similarly, the \emph{last point of $X$} is the point $x_i \in X$ such that $x_{i+1}\not\in X$. We say that $Y$ is the \emph{previous red (resp.\ blue) blob before} $X$ if $Y$ is a blue blob and there is no red (resp.\ blue) point between the last point of $Y$ and the first point of $X$ in the radial order. Similarly, we say that $Z$ is the \emph{next red (resp.\ blue) blob after} $X$ if there is no red (resp.\ blue) point between the the last point of $X$ and the first point of $Z$ in the radial order.
\subsection{Jump Configurations} If $X$ is a blob, and $Y$ is the next blob after $X$ of the same color as $X$, then a \emph{jump edge} from $X$ to $Y$ is a line segment $(x,y)$ where $x\in X$, $y\in Y$ such that $x$ and $y$ see each other with respect to $X$ and $Y$ and the angle between $x$ and $y$ with respect to the point $p$ is at most $\pi$. This last condition ensures that each point $z$ on the line segment $(x,y)$ would, if added to the radial order, lie between $x$ and $y$. See Figure \ref{fig:jump-edges}.
\begin{figure}
\caption{Jump edges from a blob $X$ to the next blob of the same color, $Y$. $x_1$ to $y_1$ is a jump edge. $x_2$ to $y_2$ is not a jump edge; it intersects the interior of the convex hulls of the blobs containing $x_2$ and $y_2$. $x_3$ to $y_3$ is not a jump edge because the angle from $x_3$ to $y_3$ is greater than $\pi$.}
\label{fig:jump-edges}
\end{figure}
A \emph{jump configuration} $J$ is a collection of jump edges, one blue edge from each blue blob to the next blue blob and one red edge from each red blob to the next red blob, such that the blue edges do not cross each other, the red edges do not cross each other, and, for each blob $X$, the two edges with endpoints in $X$ do not share a common endpoint unless $X$ contains only one point.
In this section we show that a jump configuration can be completed to a pair of non-intersecting spanning cycles of the two color classes by adding a spanning path within each blob, such that each edge of these cycles will be crossed at most three times, except where a specific structure, called a 4-forcing, appears. 4-forcings will be defined later in this section.
\begin{lemma}
\label{lem:jump-config-blob-order}
If there is a jump configuration $J$ in the radial order with respect to $p$, then for all blobs $X$, the angle from the first point in $X$ to the last point in $X$ is strictly less than $\pi$.
Hence, for any $z$ in the convex hull of $X$, if $z$ were added to the radial order, it would lie in the interval from the first point in $X$ to the last point in $X$, which implies that the convex hulls of the blobs are disjoint.
\begin{proof}
Let $(b,a)$ be the jump edge in $J$ between the blob before $X$ and the blob after $X$.
By the definition of a jump edge, the angle from $b$ to $a$ is strictly less than $\pi$.
Also, $X$ is contained in the interval between $b$ and $a$ in the radial order, so the angle from the first point in $X$ to the last point in $X$ is strictly less than the angle from $b$ to $a$, which is less than $\pi$.
\end{proof} \end{lemma}
The following two lemmas describe how jump edges in a jump configuration can intersect, and how jump edges can intersect the convex hulls of blobs.
\begin{lemma}
\label{lem:jump-config-crossing}
If $X_0$ is a blob, and, for $i\in\{1,2,3,4\}$, $X_i$ is the next blob after $X_{i-1}$, then for any jump configuration $J$, the only jump edges in $J$ that can cross the jump edge from $X_1$ to $X_3$ are the jump edges from $X_0$ to $X_2$ and from $X_2$ to $X_4$.
In particular, no jump edges of the same color cross, and each jump edge is crossed by at most two jump edges of the opposite color.
\begin{proof}
Let $(x,y)$ be the jump edge from $X_1$ to $X_3$ in the jump configuration $J$, and let $(x',y')$ be any jump edge in $J$ such that $(x,y)$ and $(x',y')$ meet at a point $z$.
By construction, if added to the radial order, $z$ would lie between $x$ and $y$, and between $x'$ and $y'$.
Therefore the interval $I$ between $x$ and $y$ and the interval $I'$ between $x'$ and $y'$ intersect.
If $I'\supseteq I$, then $(x',y')$ must be a jump edge from $X_1$ to $X_3$, so $(x',y')=(x,y)$.
Otherwise, either $x'\in I$ or $y'\in I$, so, without loss of generality, $x'$ lies between $x$ and $y$ in the radial order.
Therefore $x'\in X_1 \cup X_2 \cup X_3$.
If $x'\in X_1$ and $y'$ is in the previous blob of the same color, then $(x,y)$ and $(x',y')$ do not cross by construction, and similarly if $x'\in X_3$ and $y'$ is in the next blob of the same color.
If $x'\in X_2$, then $(x',y')$ is either the jump edge from $X_0$ to $X_2$ or the jump edge from $X_2$ to $X_4$.
\end{proof} \end{lemma}
\begin{lemma}
\label{lem:jump-config-cross-blob}
If $X_0$ is a blob, and, for $i\in\{1,2,3,4\}$, $X_i$ is the next blob after $X_{i-1}$, then for any jump configuration $J$, the only jump edges that can intersect the convex hull of $X_2$ are the jump edges from $X_0$ to $X_2$, from $X_1$ to $X_3$, and from $X_2$ to $X_4$.
\begin{proof}
Let $(y_1,y_2)$ be the jump edge between blobs $Y_1$ and $Y_2$ in the jump configuration $J$ such that $(y_1,y_2)$ intersects the convex hull of $X_1$ at a point $z$.
Then $z$, if added to the radial order, would lie between the first point of $X_1$ and the last point of $X_1$.
But $z$ lies between a point in $Y_1$ and a point in $Y_2$ in the radial order, so $X_1$ must be $Y_1$, $Y_2$, or the blob between $Y_1$ and $Y_2$ in the radial order.
\end{proof} \end{lemma}
When a jump edge passes through the convex hull of a blob $X$ (which can only happen when it is the jump edge from the previous blob before $X$ to the next blob after $X$), we want to find a spanning path within $X$ that crosses the jump edge as few times as possible. The following lemma gives a construction for such a path.
\begin{lemma}
\label{lem:spanning-path-crossing}
If $X$ is a finite set of points in general position, $x,y\in X$ are distinct points, and $\ell$ is a line, then there exists a non-self-intersecting spanning path $P$ of $X$ such that
\begin{enumerate}[(i)]
\item if $X$ lies entirely on one side of $\ell$, then $P$ does not cross $\ell$;
\item if $x$ and $y$ lie on opposite sides of $\ell$, then $P$ crosses $\ell$ exactly once;
\item if $x$ and $y$ lie on the same side of $\ell$, and $X$ contains points on both sides of $\ell$, then $P$ crosses $\ell$ exactly twice.
\end{enumerate}
\begin{proof}
By induction on $n=|X|$.
If $n=2$, then $X=\{x,y\}$, and so the one-edge path from $x$ to $y$ crosses $\ell$ zero times, if $x$ and $y$ are on the same side of $\ell$; or once if $x$ and $y$ are on opposite sides of $\ell$.
If $n>2$, then $X\setminus\{x\}$ contains at least two points, and so $x$ sees at least two points of $X\setminus\{x\}$.
One of these points, $x'$, is not $y$.
If there are multiple choices for $x'$, choose $x'$ to lie on the same side of $\ell$ as $x$.
By the induction hypothesis, there exists a spanning path $P'$ of $X\setminus\{v\}$ from $x'$ to $y$ satisfying (i) (ii) and (iii).
$P'$ lies entirely within the convex hull of $X\setminus \{x\}$, and the edge from $x$ to $x'$ intersects the convex hull only at $x'$, so $P=P'\cup(x,x')$ is a non-self-intersecting spanning path of $X$ from $x$ to $y$.
If $X$ lies entirely on one side of $\ell$, then $P$ lies entirely on one side of $\ell$ because it is contained in the convex hull of $X$, so (i) holds.
Now suppose $x$ and $y$ lie on opposite sides of $\ell$.
If $x'$ lies on the same side of $\ell$ as $x$ then $P$ crosses $\ell$ as many times as $P'$, which is once.
If $x'$ lies on the opposite side of $\ell$ from $x$, then $X\setminus\{x\}$ cannot contain any point on the same side of $\ell$ as $x$, or else $x$ would see some point $x''$ on the same side of $\ell$, and $x''\neq y$ because $y$ is on the opposite side of $\ell$, so $x''$ would have been chosen over $x'$.
Therefore when $x'$ lies on the opposite side of $\ell$ from $x$, $P'$ does not cross $\ell$, so $P$ crosses $\ell$ exactly once.
Finally, suppose $x$ and $y$ lie on the same side of $\ell$, and that $X$ contains some point $z$ on the opposite side of $\ell$ from $x$ and $y$.
If $x'$ lies on the same side of $\ell$ as $x$ and $y$, then $P$ crosses $\ell$ as many times as $P'$, which is twice.
If $x'$ lies on the opposite side of $\ell$ from $x$ and $y$, then $P'$ crosses $\ell$ once, so $P$ crosses $\ell$ twice.
\end{proof} \end{lemma}
We now show how to construct a pair of monochromatic spanning cycles from a jump configuration by adding spanning paths within each blob.
The next lemma shows that there is exactly one case in which adding spanning paths can force our monochromatic spanning cycles to have edges that are crossed 4 times. We call this case a \emph{4-forcing} and define it as follows: Suppose we are given a jump configuration $J$. Let $X_0,Y_1,X_1,Y_2,X_2$ be consecutive blobs such that the jump edge $(y_1,y_2)$ between $Y_1$ and $Y_2$ in $J$ intersects the convex hull of $X_1$ and crosses both the jump edge from $X_0$ to $X_1$ and the jump edge from $X_1$ to $X_2$, and such that the endpoints in $X_1$ of the jump edges from $X_0$ and to $X_2$ both lie on the same side of $(y_1,y_2)$. Then this is called a \emph{4-forcing} in $J$, and $(y_1,y_2)$ is called the \emph{center edge} of the 4-forcing. See Figure \ref{fig:4-forcing}.
\begin{figure}
\caption{4-forcing}
\label{fig:4-forcing}
\end{figure}
\begin{lemma}
\label{lem:complete-jump-config}
Given any jump configuration $J$,
spanning paths of each blob can be added to $J$ to construct a pair of non-self-intersecting spanning cycles of the red and blue points respectively such that, if an edge $e$ is crossed more than three times by the opposite color cycle, then $e$ is in $J$ (and not in one of the spanning paths), and $e$ is the center edge of a 4-forcing.
\begin{proof}
For each blob $X$, let $a_X\in X$ be the point incident with the jump edge to $X$ from the previous blob of the same color, and let $b_X\in X$ be the point incident with the jump edge from $X$ to the next blob of the same color.
If $W$ is the previous blob before $X$ and $Y$ is the next blob after $X$, let $\ell_X$ be the line through $b_W$ and $a_Y$, and note that $(b_W,a_Y)$ is the only jump edge that can cross through the convex hull of $X$.
Let $P_X$ be a spanning path of $X$ from $a_X$ to $b_X$ minimizing the number of crossings of the $\ell_X$.
By Lemma \ref{lem:jump-config-cross-blob}, no jump edge of the same color as $X$ can cross the convex hull of $X$, and hence no jump edge can cross $P_X$.
Also, for any blob $X'\neq X$, the convex hulls of $X'$ and $X$ are disjoint, so $P_X$ and $P_{X'}$ cannot cross.
By Lemma \ref{lem:jump-config-crossing}, two jump edges of the same color cannot cross, so the edges of $J\cup \bigcup\limits_{\emph{all blobs $X$}}P_X$ form a pair of non-self-intersecting spanning-cycles.
Let $e$ be an edge of one of these spanning cycles which is crossed at least 4 times by the other cycle.
By Lemma \ref{lem:jump-config-cross-blob}, any edge of $P_X$ can only be crossed by the jump edge $(b_W,a_Y)$, and hence cannot be crossed 4 times, so $e$ is a jump edge.
Without loss of generality, suppose $e$ is the jump edge between $W$ and $Y$.
By Lemma \ref{lem:jump-config-crossing}, $e$ is crossed by at most two other jump edges (the two jump edges to and from $X$).
The only blob for which $e$ could cross the convex hull is $X$, so the only blob whos spanning-path $e$ could cross is $P_X$.
By Lemma \ref{lem:spanning-path-crossing}, $P_X$ crosses $e$ at most two times, with equality only if $a_X$ and $b_X$ lie on the same side of $\ell_X$.
Therefore $e$ must be crossed by both of the jump edges incident with $X$, so $e$ is the center edge of a 4-forcing.
\end{proof} \end{lemma}
In the remainder of the paper, we focus on finding a good jump configuration such that when we add spanning cycles of each blob, as in the previous lemma, we have no 4-crossings \textemdash this will result in a pair of monochromatic spanning cycles in which each edge is crossed at most 3 times. In the following sections we show that it is possible to avoid 4-crossings by choosing $p$ carefully.
\section{Monster-Jumps}
If $B_1,R_1,B_2,R_2$ are consecutive blobs such that $B_1$ and $B_2$ are blue and $R_1$ and $R_2$ are red, then we say that $R_1$ to $R_2$ is a \emph{red monster-jump} if $|R_1|>1$ and the angle from the \emph{second} point in $R_1$ to the \emph{first} point in $R_2$ is at least $\pi$, and the line segment between the last point in $B_1$ and the first point in $B_2$ intersects the convex hull of $R_1$. See Figure \ref{fig:red-monster-jump}.
If $B_1,R_1,B_2,R_2$ are consecutive blobs such that $B_1$ and $B_2$ are blue and $R_1$ and $R_2$ are red, then we say that $B_1$ to $B_2$ is a \emph{blue monster-jump} if $|B_2|>1$ and the angle from the \emph{last} point in $B_1$ to the \emph{second to last} point in $B_2$ (i.e., the point before the last point in $B_2$) is at least $\pi$, and the line segment between the last point in $R_1$ and the first point in $R_2$ intersects the convex hull of $B_2$. See Figure \ref{fig:blue-monster-jump}.
Note the slight asymmetry between the definition of red and blue monster-jump. In particular, if the colors are reversed and the plane reflected, then red monster-jumps become blue monster-jumps, and vice-versa.
\begin{figure}
\caption{Red monster-jump between $R_1$ and $R_2$.}
\label{fig:red-monster-jump}
\caption{Blue monster-jump between $B_1$ and $B_2$.}
\label{fig:blue-monster-jump}
\caption{Red and blue monster-jumps}
\label{fig:monster-jumps}
\end{figure}
In this section we show that if $p$ is chosen such that the radial ordering about $p$ has no monster-jumps, then we can construct a jump configuration which can be completed to a pair of monochromatic spanning cycles with no 4-crossings.
\subsection{Monster-Jumps and 4-forcings} \begin{lemma}
\label{lem:jump-config-exists}
Suppose that, for each blob $X$, the angle from the last point in $X$ to the first point in the next blob of the same color is less than $\pi$. Then the collection of jump-edges consisting of, for each blob $X$, the edge from the last point of $X$ to the first point of the next blob of the same color, is a valid jump configuration.
\begin{proof}
For each blob $X$, let $a_X$ be the first point in $X$, and $b_X$ be the last point in $X$.
Then for each blob $X$, if $X'$ is the next blob of the same color, then the angle from $b_X$ to $a_{X'}$ is less than $\pi$.
Therefore every point on the line segment $(b_X,a_{X'})$ would, if added to the radial order, lie in the interval between $b_X$ and $a_{X'}$.
If $Y$ is the blob before $X$ and $Y'$ is the blob after $X$, then $X$ is contained in the interval between the last point of $Y$ and the first point of $Y'$, which has angle less than $\pi$, so, for any point $z$ in the convex hull of $X$, if $z$ was added to the radial order, $z$ would lie between the first point of $X$ and the last point of $X$.
Hence, $(b_X,a_{X'})$ can only intersect the convex hull of $X$ and $b_X$.
Similarly, $(b_X,a_{X'})$ can only intersect the convex hull of $X'$ at $a_{X'}$.
Therefore $(b_X,a_{X'})$ is a valid jump edge.
If $X''$ is the next blob of the same color after $X'$, then the interval between $b_X$ and $a_{X'}$ and the interval between $b_{X'}$ and $a_{X''}$ are either disjoint, if $|X'|>1$, or meet at the point $a_{X'}=b_{X'}$, if $|X'|=1$.
Therefore the line segments $(b_{X},a_{X'})$ and $(b_{X'},a_{X''})$ are either disjoint, if $|X'|>1$; or meet at $a_{X'}=b_{X'}$, if $|X'|=1$.
Thus, the collection $J$ of jump edges consisting of, for each blob $X$ with next blob of the same color $X'$, $(b_X,a_{X'})$, is a valid jump configuration.
\end{proof} \end{lemma}
\begin{figure}
\caption{In the proof of Lemma \ref{lem:no-4-forcing}, if there is a 4-forcing on the red jump edge between $R_1$ and $R_2$, then the blue-red crossing between the jump edge from $B_1$ to $B_2$ and the jump edge from $R_1$ to $R_2$ can be removed by replacing $b_{R_1}$ by $b'_{R_1}$ and replacing $a_{B_2}$ by $a'_{B_2}$.}
\label{fig:no-4-forcing-proof}
\end{figure}
\begin{lemma}
\label{lem:no-4-forcing}
If a radial order contains no red monster-jump and no blue monster-jump and, for each blob $X$, the angle from the last point in $X$ to the first point in the next blob of the same color is less than $\pi$, then there exists a jump configuration which contains no 4-forcing.
\begin{proof}
A \emph{blue-red crossing} in a jump configuration occurs when there are consecutive blobs $B_1,R_1,B_2,R_2$ such that $B_1$ and $B_2$ are blue and $R_1$ and $R_2$ are red, and the jump edge from $B_1$ to $B_2$ crosses the jump edge from $R_1$ to $R_2$.
For each red blob $R$, let $a_R$ be the first point in $R$.
For each blue blob $B$, let $b_B$ be the last point in $B$.
For each red blob $R$ choose $b_R\in R$, and for each blue blob $B$ choose $a_B\in B$ such that the collection $J$ of edges consisting of, for each blob $X$, $(b_X,a_{X'})$, where $X'$ is the next blob of the same color, is a valid jump configuration, and that the number of blue-red crossings is minimized, with ties broken by minimizing the number of 4-crossings.
Note that such a jump configuration exists by Lemma \ref{lem:jump-config-exists}.
Suppose that the jump configuration $J$ contains a 4-forcing.
By swapping the colors and reflecting the plane, if necessary (so that we preserve the fact that there are no red or blue monster-jumps), we may assume that there are consecutive blobs $B_1,R_1,B_2,R_2,B_3$ such that $B_1,B_2,B_3$ are blue and $R_1,R_2$ are red, and that the red jump edge $(b_{R_1},a_{R_2})$ from $R_1$ to $R_2$ is the center edge of a 4-forcing.
Then $(b_{B_1},a_{B_2})$ and $(b_{B_2},a_{B_3})$ both cross $(b_{R_1},a_{R_2})$, $a_{B_2}$ and $b_{B_2}$ are on the same side of the line $\ell$ through $b_{R_1}$ and $a_{R_2}$, and $B_2$ contains points on both sides of $\ell$.
If $b'_{R_1}$ is the last point in $R_1$, then replacing $(b_{R_1},a_{R_2})$ with $(b'_{R_1},a_{R_2})$ in $J$ will give another valid jump configuration $J'$, and cannot increase the number of blue-red crossings, because the only blue-red crossing that a jump edge from $R_1$ to $R_2$ could be involved in is with the jump edge from $B_1$ to $B_2$, which $(b_{R_1},a_{R_2})$ already crosses.
Therefore, by minimality of $J$, $(b'_{R_1},a_{R_2})$ must be the center edge of a 4-forcing in $J'$.
This means that $a_{B_2}$ and $b_{B_2}$ are both on the same side of the line $\ell'$ through $b'_{R_1}$ and $a_{R_2}$, and $B_2$ contains points on both sides of $\ell'$, and that $b_{B_1}$ and $a_{B_3}$ are on the opposite side of $\ell'$ from $a_{B_2}$ and $b_{B_2}$.
There is some point $a'_{B_2}\in B_2$ on the same side of $\ell'$ as $b_{B_1}$, which $b_{B_1}$ sees with respect to $B_2$.
The angle from $b_{B_1}$ to $a'_{B_2}$ is at most the angle from $b_{B_1}$ to the second to last point in $B_2$, which is less than $\pi$ because $B_1$ to $B_2$ is not a blue monster-jump.
Thus $a'_{B_2}$ sees $b_{B_1}$ with respect to $B_1$, so $(b_{B_1},a'_{B_2})$ is a valid jump edge, and replacing $(b_{B_1},a_{B_2})$ by $(b_{B_1},a'_{B_2})$ in $J'$ gives a valid jump configuration $J''$.
The only blue-red crossing which $(b_{B_1},a'_{B_2})$ can be involved in is with a jump edge from $R_1$ to $R_2$, and $(b_{B_1},a'_{B_2})$ does not cross $(b'_{R_1},a_{R_2})$, so $J''$ has fewer blue-red crossings than the chosen jump configuration $J$, a contradiction.
See Figure \ref{fig:no-4-forcing-proof}.
\end{proof} \end{lemma}
\subsection{Avoiding Monster-Jumps} We have shown that if we choose $p$ such that the radial order about $p$ gives us a jump configuration with no monster-jumps, then we can complete this jump configuration to a pair of monochromatic spanning cycles with no 4-crossings. In this section we show that it is possible to choose a point $p$ that avoids monster-jumps. This is broken down into two cases: when the blue and red convex hulls properly overlap (Lemma \ref{lem:no-monster-jump-overlap}) and when the red convex hull contains the blue convex hull (Lemma \ref{lem:no-monster-jump-contains}). The only other alternative is that the red and blue convex hulls are disjoint, in which case the desired spanning cycles exist trivially.
\begin{figure}
\caption{The choice of $p$ when the red and blue convex hulls properly overlap (Lemma \ref{lem:no-monster-jump-overlap}).}
\label{fig:overlap-case}
\end{figure}
\begin{lemma}
\label{lem:no-monster-jump-overlap}
Suppose the bichromatic point set $S$ contains at least three red points and at least three blue points and that the convex hulls of the red points and of the blue points properly overlap; i.e., the intersection of the convex hulls is non-empty, and the blue convex hull is not contained in the red convex hull, nor vice-versa.
Then after possibly swapping the color classes, there exists a point $p$ in the intersection of the interior of the convex hulls such that the radial order about $p$ contains neither a red nor a blue monster-jump.
\begin{proof}
Note that because the convex hulls properly overlap, the boundaries of the convex hulls must intersect at some point $q$.
Let $(r_1,r_2)$ be the red segment containing $q$, such that $r_2$ follows $r_1$ in the clockwise ordering of the vertices of the red convex hull.
Similarly, let $(b_1,b_2)$ be the blue segment containing $q$, such that $b_2$ follows $b_1$ in the clockwise ordering of the vertices of the blue convex hull.
By swapping colors if necessary, we may assume that $b_1,r_1,b_2,r_2$ appear in that order in the (clockwise) radial order about $q$.
Therefore $r_1$ is outside the blue convex hull and $b_2$ is outside the red convex hull.
For all $\epsilon>0$, there is a point $p$ in the intersection of the interiors of the red and blue convex hulls such that $|p-q|<\epsilon$.
We will show that for $\epsilon$ sufficiently small, there is no red or blue monster-jump in the radial order about $p$.
If $\epsilon$ is sufficiently small, then the radial orders of the bichromatic point set with respect to $p$ and with respect to $q$ coincide.
Any point between $b_1$ and $b_2$ in the radial order with respect to $p$ is outside the blue convex hull, and hence must be red.
Therefore there is exactly one red blob $R_1$ between $b_1$ and $b_2$, and $r_1\in R_1$.
Similarly there is exactly one blue blob, $B_2$, between $r_1$ and $r_2$, and $b_2\in B_2$.
Let $B_1$ be the blue blob containing $b_1$ and $R_2$ be the red blob containing $r_2$.
Note that $b_1$ is the last point in $B_1$ and $b_2$ is the first point in $B_2$, and the blob $R_1$ lies entirely on one side of the line through $b_1$ and $b_2$, so $R_1$ to $R_2$ is not a red monster-jump.
Similarly, $r_1$ is the last point in $R_1$, $r_2$ is the first point in $R_2$, and $B_2$ lies entirely on one side of the line between $r_1$ and $r_2$, so $B_1$ to $B_2$ is not a blue monster-jump.
Let $B$ be any blue blob which is not $B_1$, and let $B'$ be the next blue blob after $B$.
If $|B'|=1$, then $B$ to $B'$ is not a blue monster-jump, so suppose $|B'|>1$.
Then the last point, $b$, in $B$ and the second to last point, $b'$, in $B'$ lie in the interval between $b_2$ and $b_1$ in the radial order with respect to $p$.
Note that $b\neq b_1$ because $B\neq B_1$ and $b'\neq b_1$, because $b_1$ is the not the second to last point in $B_1$.
Therefore if $b'_1$ is the point after $b_1$ in the radial order with respect to $p$, then the angle from $b$ to $b'$ is at most the angle from $b'_1$ to $b_2$, which, for $\epsilon$ sufficiently small, is less than $\pi$.
So, $B$ to $B'$ is not a blue monster-jump.
By a symmetric argument, if $R$ is a red blob which is not $R_1$ and $R'$ is the next red blob after $R$ and $\epsilon$ is sufficiently small, then $R$ to $R'$ is not a red monster-jump.
Therefore, if $\epsilon$ is sufficiently small, then the radial order of the bichromatic point set with respect to $p$ contains no red or blue monster-jump.
\end{proof} \end{lemma}
\begin{lemma}
\label{lem:no-monster-jump-contains}
Suppose the bichromatic point set $S$ contains at least three red points and at least three blue points and that the red convex hull contains the blue convex hull.
Then there exists a point $p$ in the interior of the blue convex hull such that the radial order of the bichromatic point set with respect to $p$ contains no red or blue monster-jump.
\begin{proof}
Let $(b_1,\ldots,b_k)$ be the vertices of the blue convex hull in clockwise order.
For $i\in\{1,\ldots,k\}$, let $H_i$ be the open half-plane bounded by the line through $b_i$ and $b_{i+1\pmod{k}}$ which contains no blue points.
Note that $\bigcup_{i=1}^k H_i\setminus H_{i+1\pmod{k}}$ is the complement of the blue convex hull, and so, for some $i\in\{1,\ldots,k\}$, $H_i\setminus H_{i+1\pmod{k}}$ contains a red point.
Without loss of generality, there is some red point $r_1\in H_k \setminus H_1$.
Because $b_1$ is in the interior of the red convex hull, there is some red point in $H_1$; choose a red point $r_2$ in $H_1$ minimizing the angle from $r_2$ to $b_2$ with respect to $b_1$.
\begin{figure}
\caption{Case 1: The blue convex hull is contained in the red convex hull, and there are red points $r_1$ in $H_k\setminus H_1$ and $r_2$ in $H_1\setminus H_k$.}
\label{fig:containment-case-1}
\end{figure}
\textbf{Case 1:} $r_2\in H_1\setminus H_k$ (see Figure \ref{fig:containment-case-1}). For $\delta>0$, let $v_\delta$ be the vector $b_k-b_1$ rotated counter-clockwise by $\delta$, and, for $\epsilon>0$, let $p=b_1 + \epsilon v_\delta$, and consider the radial order of the bichromatic points with respect to $p$.
For $i\in\{1,2,k\}$, let $B_i$ be the blue blob containing $b_i$.
For $i\in\{1,2\}$, let $R_i$ be the red blob containing $r_i$.
If $\epsilon$ and $\delta$ are sufficiently small, $R_1$ lies entirely on one side of the line through $b_1$ and $b_k$, which are the first point of $B_1$ and last point of $B_k$ respectively, so $R_1$ to $R_2$ is not a red monster-jump.
Again, if $\epsilon$ and $\delta$ are sufficiently small, $R_2$ lies entirely on one side of the line through $b_1$ and $b_2$, so $R_2$ to the next red blob is not a red monster-jump.
If $R$ is a red blob that is not $R_1$ or $R_2$ and $R'$ is the next red blob and $\epsilon$ and $\delta$ are sufficiently small, then $R$ lies entirely on the same side of the line through $b_1$ and $b_2$ as $r_1$, so the angle between the second point of $R$ and the first point of $R'$ is at most the angle between the second point of $R$ and $r_1$, which is less than $\pi$.
Therefore the radial order with respect to $p$ contains no red monster-jump.
For $\epsilon$ and $\delta$ sufficiently small, the points before and after $b_1$ are both red, so $B_1=\{b_1\}$.
Therefore $B_k$ to $B_1$ is not a blue monster-jump.
Let $b_k'$ be the previous blue point before $b_k$ in the radial order.
For any blue blob $B$ that is not $B_k$, if $B'$ is the next blue blob, then the angle between the last point $B$ and the second last point of $B'$ is at most the angle between $b_1$ and $b'_k$, which is less than $\pi$ if $\epsilon$ and $\delta$ are sufficiently small.
Therefore the radial order with respect to $p$ contains no blue monster-jump.
\begin{figure}
\caption{Case 2: The blue convex hull is contained in the red convex hull and there are red points $r_1$ in $H_k\setminus H_1$ and $r_2$ in $H_1\cap H_k$.}
\label{fig:containment-case-2}
\end{figure}
\textbf{Case 2:} $r_2\in H_1\cap H_k$ (see Figure \ref{fig:containment-case-2}).
Let $\ell$ be the line through $r_2$ and $b_1$.
Because $b_1$ is in the interior of the red convex hull, there is a red point $r_3$ on the opposite side of $\ell$ from $r_1$.
However, $r_3$ cannot lie between $r_2$ and $b_2$ in the radial order with respect to $b_1$, so $r_3$ lies between $b_2$ and $b_k$ in the radial order with respect to $b_1$, and the angle from $r_2$ to $r_3$ with respect to $b_1$ is less than $\pi$.
For $\delta>0$, let $v_\delta$ be the vector from $r_1$ to $b_1$ rotated counter-clockwise by $\delta$.
As in case 1, for $\epsilon>0$, let $p=b_1+\epsilon v_\delta$, and consider the radial order of the bichromatic points with respect to $p$.
For $i\in\{1,2,k\}$, let $B_i$ be the blue blob containing $b_i$, and, for $i\in\{1,2\}$, let $R_i$ be the red blob containing $r_i$.
If $\epsilon$ and $\delta$ are sufficiently small, then $R_1$ lies entirely on one side of the line between $b_k$ and $b_1$, which are the last point of $B_k$ and first point of $B_1$ respectively, so $R_1$ to the next red blob is not a red monster-jump.
For $\epsilon$ and $\delta$ sufficiently small, $b_1$ is the point before $r_2$ and $b_2$ is the point after $r_2$ so $R_2=\{r_2\}$. Therefore, $R_2$ to the next red blob is not a red monster-jump.
If $R$ is a red blob that is neither $R_1$ nor $R_2$, and $R'$ is the next red blob, then $R$ lies entirely on the same side of the line through $b_1$ and $b_2$ as $r_1$, so the angle from the second point in $R$ to the first point in $R'$ is at most the angle from $b_2$ to $r_1$, which is less than $\pi$.
Therefore the radial order with respect to $p$ contains no blue monster-jump.
If $\epsilon$ and $\delta$ are sufficiently small, then the points before and after $b_1$ are both red, so $B_1=\{b_1\}$.
Therefore $B_k$ to $B_1$ is not a blue monster-jump.
If $\epsilon$ and $\delta$ are sufficiently small, then $r_3$ lies between $b_2$ and $b_k$, and the angle from $b_1$ to $r_3$ is less than $\pi$.
The blob $B_2$ ends before $r_3$, so the angle from $b_1$ to any point in $B_2$ is less than $\pi$, and $B_1$ to $B_2$ is not a blue monster-jump.
If $B$ is a blue blob that is not $B_k$ or $B_1$, and $B'$ is the next blue blob, then $B$ and $B'$ lie between $b_2$ and $b_k$ in the radial order, so the angle from any point in $B$ to any point in $B'$ is less than $\pi$, and hence $B$ to $B'$ is not a blue monster-jump.
Therefore the radial order with respect to $p$ contains no blue monster-jump.
In both cases, $p$ can be chosen such that the radial order with respect to $p$ contains no red monster-jump and no blue monster-jump.
\end{proof} \end{lemma}
\begin{proof}[Proof of Theorem \ref{thm:main}] If the red convex hull and the blue convex hull are disjoint, then any pair of red and blue spanning cycle will be disjoint, so assume the red and blue convex hulls intersect. By Lemma \ref{lem:complete-jump-config}, it suffices to show that for some point $p$, the radial order about $p$ contains a jump-configuration with no 4-forcing. By Lemma \ref{lem:no-4-forcing}, it suffices to show that there exists a point $p$ such that the radial order about $p$ contains no red or blue monster-jump.
Either the red and blue convex hulls properly overlap, or the blue convex hull is contained in the red convex hull, or the red convex hull is contained in the blue convex hull. If the red and blue convex hulls properly overlap, then by Lemma \ref{lem:no-monster-jump-overlap}, there is a point $p$ such that the radial order with respect to $p$ contains no red or blue monster-jump. If the red convex hull is contained in the blue convex hull, we may swap the colors, so that the blue convex hull is contained in the red convex hull, and in that case, by Lemma \ref{lem:no-monster-jump-contains}, there is a point $p$ such that the radial order with respect to $p$ contains no red or blue monster-jump. \end{proof}
{}
\end{document} | arXiv |
Zariski's main theorem
In algebraic geometry, Zariski's main theorem, proved by Oscar Zariski (1943), is a statement about the structure of birational morphisms stating roughly that there is only one branch at any normal point of a variety. It is the special case of Zariski's connectedness theorem when the two varieties are birational.
Zariski's main theorem can be stated in several ways which at first sight seem to be quite different, but are in fact deeply related. Some of the variations that have been called Zariski's main theorem are as follows:
• The total transform of a normal fundamental point of a birational map has positive dimension. This is essentially Zariski's original form of his main theorem.
• A birational morphism with finite fibers to a normal variety is an isomorphism to an open subset.
• The total transform of a normal point under a proper birational morphism is connected.
• A closely related theorem of Grothendieck describes the structure of quasi-finite morphisms of schemes, which implies Zariski's original main theorem.
• Several results in commutative algebra that imply the geometric form of Zariski's main theorem.
• A normal local ring is unibranch, which is a variation of the statement that the transform of a normal point is connected.
• The local ring of a normal point of a variety is analytically normal. This is a strong form of the statement that it is unibranch.
The name "Zariski's main theorem" comes from the fact that Zariski labelled it as the "MAIN THEOREM" in Zariski (1943).
Zariski's main theorem for birational morphisms
Let f be a birational mapping of algebraic varieties V and W. Recall that f is defined by a closed subvariety $\Gamma \subset V\times W$ (a "graph" of f) such that the projection on the first factor $p_{1}$ induces an isomorphism between an open $U\subset V$ and $p_{1}^{-1}(U)$, and such that $p_{2}\circ p_{1}^{-1}$ is an isomorphism on U too. The complement of U in V is called a fundamental variety or indeterminacy locus, and the image of a subset of V under $p_{2}\circ p_{1}^{-1}$ is called a total transform of it.
The original statement of the theorem in (Zariski 1943, p. 522) reads:
MAIN THEOREM: If W is an irreducible fundamental variety on V of a birational correspondence T between V and V′ and if T has no fundamental elements on V′ then — under the assumption that V is locally normal at W — each irreducible component of the transform T[W] is of higher dimension than W.
Here T is essentially a morphism from V′ to V that is birational, W is a subvariety of the set where the inverse of T is not defined whose local ring is normal, and the transform T[W] means the inverse image of W under the morphism from V′ to V.
Here are some variants of this theorem stated using more recent terminology. Hartshorne (1977, Corollary III.11.4) calls the following connectedness statement "Zariski's Main theorem":
If f:X→Y is a birational projective morphism between noetherian integral schemes, then the inverse image of every normal point of Y is connected.
The following consequence of it (Theorem V.5.2,loc.cit.) also goes under this name:
If f:X→Y is a birational transformation of projective varieties with Y normal, then the total transform of a fundamental point of f is connected and of dimension at least 1.
Examples
• Suppose that V is a smooth variety of dimension greater than 1 and V′ is given by blowing up a point W on V. Then V is normal at W, and the component of the transform of W is a projective space, which has dimension greater than W as predicted by Zariski's original form of his main theorem.
• In the previous example the transform of W was irreducible. It is easy to find examples where the total transform is reducible by blowing up other points on the transform. For example, if V′ is given by blowing up a point W on V and then blowing up another point on this transform, the total transform of W has 2 irreducible components meeting at a point. As predicted by Hartshorne's form of the main theorem, the total transform is connected and of dimension at least 1.
• For an example where W is not normal and the conclusion of the main theorem fails, take V′ to be a smooth variety, and take V to be given by identifying two distinct points on V′, and take W to be the image of these two points. Then W is not normal, and the transform of W consists of two points, which is not connected and does not have positive dimension.
Zariski's main theorem for quasifinite morphisms
In EGA III, Grothendieck calls the following statement which does not involve connectedness a "Main theorem" of Zariski Grothendieck (1961, Théorème 4.4.3):
If f:X→Y is a quasi-projective morphism of Noetherian schemes then the set of points that are isolated in their fiber is open in X. Moreover the induced scheme of this set is isomorphic to an open subset of a scheme that is finite over Y.
In EGA IV, Grothendieck observed that the last statement could be deduced from a more general theorem about the structure of quasi-finite morphisms, and the latter is often referred to as the "Zariski's main theorem in the form of Grothendieck". It is well known that open immersions and finite morphisms are quasi-finite. Grothendieck proved that under the hypothesis of separatedness all quasi-finite morphisms are compositions of such Grothendieck (1966, Théorème 8.12.6):
if Y is a quasi-compact separated scheme and $f:X\to Y$ is a separated, quasi-finite, finitely presented morphism then there is a factorization into $X\to Z\to Y$, where the first map is an open immersion and the second one is finite.
The relation between this theorem about quasi-finite morphisms and Théorème 4.4.3 of EGA III quoted above is that if f:X→Y is a projective morphism of varieties, then the set of points that are isolated in their fiber is quasifinite over Y. Then structure theorem for quasi-finite morphisms applies and yields the desired result.
Zariski's main theorem for commutative rings
Zariski (1949) reformulated his main theorem in terms of commutative algebra as a statement about local rings. Grothendieck (1961, Théorème 4.4.7) generalized Zariski's formulation as follows:
If B is an algebra of finite type over a local Noetherian ring A, and n is a maximal ideal of B which is minimal among ideals of B whose inverse image in A is the maximal ideal m of A, then there is a finite A-algebra A′ with a maximal ideal m′ (whose inverse image in A is m) such that the localization Bn is isomorphic to the A-algebra A′m′.
If in addition A and B are integral and have the same field of fractions, and A is integrally closed, then this theorem implies that A and B are equal. This is essentially Zariski's formulation of his main theorem in terms of commutative rings.
Zariski's main theorem: topological form
A topological version of Zariski's main theorem says that if x is a (closed) point of a normal complex variety it is unibranch; in other words there are arbitrarily small neighborhoods U of x such that the set of non-singular points of U is connected (Mumford 1999, III.9).
The property of being normal is stronger than the property of being unibranch: for example, a cusp of a plane curve is unibranch but not normal.
Zariski's main theorem: power series form
A formal power series version of Zariski's main theorem says that if x is a normal point of a variety then it is analytically normal; in other words the completion of the local ring at x is a normal integral domain (Mumford 1999, III.9).
See also
• Deligne's connectedness theorem
• Fulton–Hansen connectedness theorem
• Grothendieck's connectedness theorem
• Stein factorization
• Theorem on formal functions
References
• Danilov, V.I. (2001) [1994], "Zariski theorem", Encyclopedia of Mathematics, EMS Press
• Grothendieck, Alexandre (1961), Eléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : III. Étude cohomologique des faisceaux cohérents, Première partie, Publications Mathématiques de l'IHÉS, vol. 11, pp. 5–167
• Grothendieck, Alexandre (1966), Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Troisième partie, Publications Mathématiques de l'IHÉS, vol. 28, pp. 43–48
• Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
• Mumford, David (1999) [1988], The red book of varieties and schemes, Lecture Notes in Mathematics, vol. 1358 (expanded, Includes Michigan Lectures (1974) on Curves and their Jacobians ed.), Berlin, New York: Springer-Verlag, doi:10.1007/b62130, ISBN 978-3-540-63293-1, MR 1748380
• Peskine, Christian (1966), "Une généralisation du main theorem de Zariski", Bull. Sci. Math. (2), 90: 119–127
• Raynaud, Michel (1970), Anneaux locaux henséliens, Lecture Notes in Mathematics, vol. 169, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0069571, ISBN 978-3-540-05283-8, MR 0277519
• Zariski, Oscar (1943), "Foundations of a general theory of birational correspondences.", Trans. Amer. Math. Soc., 53 (3): 490–542, doi:10.2307/1990215, JSTOR 1990215, MR 0008468
• Zariski, Oscar (1949), "A simple analytical proof of a fundamental property of birational transformations.", Proc. Natl. Acad. Sci. U.S.A., 35 (1): 62–66, Bibcode:1949PNAS...35...62Z, doi:10.1073/pnas.35.1.62, JSTOR 88284, MR 0028056, PMC 1062959, PMID 16588856
External links
• Is there an intuitive reason for Zariski's main theorem?
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For spiked population model, we investigate the large dimension $N$ and large sample size $M$ asymptotic behavior of the Support Vector Machine (SVM) classification method in the limit of $N,M\rightarrow\infty$ at fixed $\alpha=M/N$. We focus on the generalization performance by analytically evaluating the angle between the normal direction vectors of SVM separating hyperplane and corresponding Bayes optimal separating hyperplane. This is an analogous result to the one shown in Paul (2007) and Nadler (2008) for the angle between the sample eigenvector and the population eigenvector in random matrix theorem. We provide not just bound, but sharp prediction of the asymptotic behavior of SVM that can be determined by a set of nonlinear equations. Based on the analytical results, we propose a new method of selecting tuning parameter which significantly reduces the computational cost. A surprising finding is that SVM achieves its best performance at small value of the tuning parameter under spiked population model. These results are confirmed to be correct by comparing with those of numerical simulations on finite-size systems. We also apply our formulas to an actual dataset of breast cancer and find agreement between analytical derivations and numerical computations based on cross validation. | CommonCrawl |
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Nonmonotone retrospective conic trust region method for unconstrained optimization
2013, 3(2): 327-345. doi: 10.3934/naco.2013.3.327
An adaptive wavelet method and its analysis for parabolic equations
Qiang Guo 1, and Dong Liang 1,
Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada
Received April 2012 Revised March 2013 Published April 2013
In this paper, we analyze an adaptive wavelet method with variable time step sizes and space refinement for parabolic equations. The advantages of multi-resolution wavelet processes combined with certain equivalences involving weighted sequence norms of wavelet coefficients allow us to set up an efficient adaptive algorithm producing locally refined spaces for each time step. Reliable and efficient a posteriori error estimate is derived, which assesses the discretization error with respect to a given quantity of physical interest. The influence of the time and space discretization errors is separated into different error indicators. We prove that the proposed adaptive wavelet algorithm terminates in a finite number of iterations for any given accuracy.
Keywords: a posteriori estimate, adaptive wavelet method, multi-resolution, Parabolic equations, convergence..
Mathematics Subject Classification: Primary: 65M60, 65N60; Secondary: 35K1.
Citation: Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327
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Qiang Guo Dong Liang | CommonCrawl |
\begin{document}
\setlength{\abovedisplayskip}{1.0ex} \setlength{\abovedisplayshortskip}{0.8ex}
\setlength{\belowdisplayskip}{1.0ex} \setlength{\belowdisplayshortskip}{0.8ex} \title{\large \textsc{Set-oriented numerical computation of rotation sets}
\begin{abstract}
\noindent
We establish a set-oriented algorithm for the numerical approximation of the
rotation set of homeomorphisms of the two-torus homotopic to the identity. A
theoretical background is given by the concept of $\ensuremath{\varepsilon}$-rotation sets. These
are obtained by replacing orbits with $\ensuremath{\varepsilon}$-pseudo-orbits in the definition of
the Misiurewicz-Ziemian rotation set and are shown to converge to the latter as
$\ensuremath{\varepsilon}$ decreases to zero. Based on this result, we prove the convergence of the
numerical approximations as precision and iteration time tend to
infinity. Further, we provide analytic error estimates for the algorithm under
an additional boundedness assumption, which is known to hold in many relevant
cases and in particular for non-empty interior rotation sets.
\noindent{\em 2010 MSC numbers. Primary: 65P99, Secondary: 37M25, 37E45 } \end{abstract}
\section{Introduction}
\noindent Rotation theory for orientation-preserving homeomorphisms on the circle was established by H. Poincar\'{e} in 1885 \cite{poincare:1885}, who showed that the long-term behaviour of such maps can be classified by the rotation number. This topological invariant provides a dichotomy for the dynamics depending on whether it is a rational or an irrational number, corresponding to periodic or quasiperiodic motion, respectively.
For homeomorphisms of higher dimensional tori, it is well-known that a unique rotation vector does not need to exist anymore. In \cite{MisiurewiczZiemian1989RotationSets}, Misiurewicz and Ziemian therefore introduce the \textit{rotation set} of a torus homeomorphism, which collects all possible asymptotic rotation vectors and carries strong information about the system's asymptotic behaviour. For homeomorphisms on the two-torus, this compact non-empty set is always convex \cite{MisiurewiczZiemian1989RotationSets}. If it has non-empty interior, then the system has positive topological entropy \cite{llibre/mackay:1991} and all rational points in the interior of the rotation set are realised by periodic orbits \cite{franks:1989}. Apart from these nowadays classical results, considerable progress has been made in the last years on the rotation theory of surface homeomorphisms (e.g.\ \cite{beguin/crovisier/leroux:2007}--\nocite{Jaeger2011EllipticStars,KoropeckiTal2012StrictlyToral,KoropeckiTal2012BoundedandUnbounded,Davalos2013SublinearDiffusion}\cite{LeCalvezZanata2015RationalModeLocking}), including partial results on the well-known Franks-Misiurewicz Conjecture \cite{franks/misiurewicz:1990}--\nocite{LeCalvezTal2015ForcingTheory,JaegerPasseggi2015SemiconjugateToIrrational,JaegerTal2016IrrationalRotationFactors,Kocsard2016MinimalTH,KoropeckiPasseggiSambarino2016FMC}\cite{AvilaLiuXu2017FMCcounterexamples} that excludes the existence of certain line segments as rotation sets.
Concerning the shape of the rotation set, it is further known that generically -- in the $\mathcal{C}^0$-topology -- it is a polygon with rational vertices \cite{Passeggi2013RationalRotationSets}. For generic area-preserving torus homeomorphisms, this polygon is non-degenerate (has non-empty interior) \cite{Guiheneuf2015RotationSetComputation}. Moreover, all rational polygons can be realised as the rotation set of a torus homeomorphism \cite{kwapisz:1992}. For a particular parameter family of such maps, derived, through some elaborate inverse limit construction, from symbolic systems related to beta expansions, bifurcations of the rotation set involving new types of convex sets have been described in \cite{BoylandDeCarvalhoHall2016NewRotationSets} (see also \cite{kwapisz:1995}). Apart from that, however, there is still little knowledge concerning the question which compact convex subsets of the plane may appear. Likewise, there is still not much insight into the behaviour of the rotation set in `natural' parameter families of torus diffeomorphisms, such as those studied by theoretical physicists in \cite{LeboeufKurchanFeingoldArovas1990PhaseSpaceLocalization}.
One serious obstruction for further progress in this direction is the fact that, except for some particular cases, it is usually not possible to analytically determine the rotation set. Moreover, due to the inherent nonlinearity of the problem, the numerical computation has proven to be difficult as well. In \cite{Guiheneuf2015RotationSetComputation}, pointwise approaches to the numerical approximation of the rotation set are discussed. Based on the detection of periodic orbits of the system on the one hand, and of a discretisation of the system by the projection to a finite lattice on the other hand, the complexity of this numerical task as well as the restrictions of the proposed approaches are discussed. It turns out that an accurate computation is already difficult in the case where the rotation set is still a rational polygon, but its vertices correspond to periodic orbits of larger periods. When the rotation set is not a rational polygon at all, it has to be expected that the situation is even worse.
Thus, our aim here is to establish a more reliable algorithm for the numerical computation of rotation sets based on set-oriented methods. On the theoretical level, this corresponds to considering $\varepsilon$-pseudo-orbits reflecting the inaccuracies of numerical calculations. This approach leads to the definition of an $\varepsilon$\textit{-rotation set} in Section~\ref{sec_epsrotset}. In the two-dimensional case, we prove that as the perturbation size $\varepsilon > 0$ decreases to zero the respective $\varepsilon$-rotation sets converge to the original rotation set (Theorem \ref{main}).
In Section~\ref{sec_algo}, we formulate our set-oriented algorithm for the numerical approximation of the rotation set of a homeomorphism $f: \tm{2} \longrightarrow \tm{2}$ homotopic to the identity, which is defined via a lift $F: \mathbb{R}^2 \longrightarrow \mathbb{R}^2$ of $f$. Based on the fact that the rotation set is approximated with arbitrary precision in the Hausdorff metric by the normalised iterates $\frac{1}{n}F^n([0,1]^2)$ of the unit square (Corollary~\ref{Col_Fn_convH}), our algorithm aims to provide a good visualisation of the latter sets for large $n \in \mathbb{N}$. This is backed up by rigorous error estimates and a result on the convergence of the approximations to the true rotation set as the precision and iteration time tend to infinity (Theorem~\ref{Th_box_err_asympt}). If the considered system has a shadowing property, this leads to a further considerable improvement of the results on the convergence rate (Theorem \ref{t.shadowing}). By a systematic overestimation of the system's orbits, we further ensure that the numerical algorithm always yields a super-set of the actual rotation set. This is particularly important since, as mentioned, previous approaches tend to `miss out' the vertices of polygonal rotation sets if these are realised by periodic orbits of large period \cite{Guiheneuf2015RotationSetComputation}.
Numerical results for some parameter families inspired by \cite{MisiurewiczZiemian1989RotationSets,
Guiheneuf2015RotationSetComputation,LeboeufKurchanFeingoldArovas1990PhaseSpaceLocalization} are then presented in Section \ref{sec_nums}. It turns out that in most cases the algorithm performs much better than predicted by the numerical error estimates. The reason for this is possibly the fact that the considered systems possess a shadowing property. As mentioned, this leads to a faster convergence of the approximations.
\noindent{\bf Acknowledgements.} TJ has been supported by a Heisenberg professorship of the German Research Council (grant OE 538/6-1). TJ and KPG acknowledge funding from EU Marie-Sk{\l}odowska-Curie Innovative Training Network {\em Critical Transitions in Complex Systems} (H2020-MSCA-2014-ITN 643073 CRITICS).
\section{Preliminaries} \label{sec_prelims}
\noindent By $\ensuremath{\mathrm{Conv}}(A)$ we denote the closed convex hull of a set $A\ensuremath{\subseteq}\ensuremath{\mathbb{R}}^m$. The open $\ensuremath{\varepsilon}$-neighbourhood of points or sets in $\ensuremath{\mathbb{R}}^m$ will be denoted by $B_\ensuremath{\varepsilon}(x)$, respectively $B_\ensuremath{\varepsilon}(A)$. The closure of a set $A$ is denoted by $\overline{A}$. By $\mathrm{d}_{\mathrm{H}}(A,B)$ we denote the Hausdorff distance between two subsets $A,B$ of a metric space and also write $A_n\rightarrow_{\mathrm{H}} A$ if a sequence $(A_n)_{n\in\mathbb{N}}$ converges to $A$ in the Hausdorff topology as $n\to\infty$.
For $m \in \mathbb{N}$, we let $\tm{m} =\mathbb{R}^m / \mathbb{Z}^m$ denote the $m$-dimensional torus. The set of all lifts $F:\ensuremath{\mathbb{R}}^m\to\ensuremath{\mathbb{R}}^m$ of continuous maps $f: \tm{m} \longrightarrow \tm{m}$ homotopic to the identity is denoted by $\cm{m}$. The subset of $\cm{m}$ consisting of lifts of torus homeomorphisms is denoted by~$\hm{m}$. Note that $\cm{m}$ consists of all continuous functions $F: \mathbb{R}^m \longrightarrow \mathbb{R}^m$ that satisfy \begin{align}
F(x+k) \ = \ F(x) + k
\label{Eaddintvect} \end{align} for all $x \in \mathbb{R}^m, k\in \mathbb{Z}^m$, and $\hm{m}$ is the set of all such $F$ which are in addition homeomorphisms of the plane. In their seminal paper \cite{MisiurewiczZiemian1989RotationSets} Misiurewicz and Ziemian introduced the rotation set as
\begin{align*}
\rs{F} \ = \ \left\{ v \in \mathbb{R}^m \; \Big| \; \exists n_i
\nearrow\infty, x_i \in\mathbb{R}^m :\ \ensuremath{\lim_{i\rightarrow\infty}} \frac{F^{n_i}(x_i)-x_i}{n_i}
\to v \right\}. \end{align*} Thus, $\rs{F}$ can be viewed as the collection of all possible rotation vectors of the system. By writing \begin{align}
K_k(F) \ = \
\left\{ \frac{F^k(x) - x}{k} \; \bigg| \; x \in [0,1]^m \right\}
\label{def_K_k} \end{align} for $k \in \mathbb{N}$, we alternatively obtain the rotation set as the upper Hausdorff limit of the sequence $\left( K_n(F) \right)_{n \in \mathbb{N}}$, that is, \begin{align}
\rs{F} \ = \ \bigcap\limits_{n \geq 1}
\overline{\bigcup\limits_{k \geq n} K_k(F)} \ .
\label{def_rot_set_sets} \end{align}
\noindent Moreover, in the two-dimensional case, the approximating sets $K_n(F)$ converge to the rotation set in the Hausdorff distance.
\begin{lemma}[\cite{MisiurewiczZiemian1989RotationSets}] \label{main_Kn_convH}
Let $F \in \hm{2}$. Then
$K_n(F) \rightarrow_{\mathrm{H}} \rs{F} \text{ as } n \to \infty.$
\end{lemma}
\noindent Furthermore, we have \begin{theorem}[\cite{MisiurewiczZiemian1989RotationSets}] \label{Thm_convexity} Let $F \in \hm{2}$. Then the
rotation set $\rs{F}$ is convex. \end{theorem} \begin{lemma}[\cite{MisiurewiczZiemian1989RotationSets}] \label{LProper3}
Let $F \in \cm{m}$. Then $\rs{F} \subseteq \ensuremath{\mathrm{Conv}}(K_n(F))$ for all $n\in\mathbb{N}$. \end{lemma}
\noindent A generalisation of Lemma \ref{LProper3} to $\varepsilon$-rotation sets is given by Lemma \ref{difficultest} below. An important technical estimate in \cite{MisiurewiczZiemian1989RotationSets} is the following. \begin{lemma}[\cite{MisiurewiczZiemian1989RotationSets}]\label{l.MZ-estimate}
Let $G\in\hm{2}$. Then $\ensuremath{\mathrm{Conv}} \left( G([0,1]^2) \right) \subseteq
\overline{B_{\sqrt{2}}(G([0,1]^2))}$. \end{lemma}
\noindent Applied to $G=F^n$ and taking into account that \[ \mathrm{d}_{\mathrm{H}} \left( K_n(F), \frac{F^n([0,1]^2)}{n} \right) \ <\ \frac{\sqrt{2}}{n} \ , \]
this immediately implies the following consequence. \begin{corollary}\label{c.MZ-estimate}
If $F\in\hm{2}$ and $n\in\ensuremath{\mathbb{N}}$, then \[
\ensuremath{\mathrm{Conv}}(K_n(F))\ \ensuremath{\subseteq} \ \overline{B_{\frac{3\sqrt{2}}{n}}(K_n(F))} \ . \] \end{corollary}
\section{The $\boldsymbol{\varepsilon}$-rotation set} \label{sec_epsrotset}
\noindent In order to model numerical approximations of rotation sets, we take into account the fact that inaccuracies are inherent to computer simulations. While the definition of the rotation set is based on proper orbits of the underlying torus map, we introduce an alternative by allowing perturbations of the system's orbits. This corresponds to the well-known concept of $\varepsilon$-pseudo-orbits and leads to the notion of $\ensuremath{\varepsilon}$-rotation sets. Since the start of this project, these have also been described independently by Guiheneuf and Koropecki in \cite{GuiheneufKoropecki2016RotationSetStability}.
Let $(X,\mathrm{d})$ be a metric space and $F: X \longrightarrow X$ a self-map. For $n \in \mathbb{N}$ and $\varepsilon \geq 0$, an $(n+1)$-tuple $(\xi_j)_{j=0}^{n}$ of points $\xi_j \in X$ is called an~$\varepsilon$\textit{-pseudo-orbit} of length $n\!+\!1$ if
\begin{align}
\mathrm{d}\left(F(\xi_j),\xi_{j+1}\right) \ \leq \ \varepsilon
\label{def_eps_orbit}
\end{align}
for all $j \in \{0,\ldots,n-1\}$. In the same way infinite sequences
$(\xi_j)_{j=0}^{\infty}$ or $(\xi_j)_{j=-\infty}^{\infty}$ satisfying
(\ref{def_eps_orbit}) for all $j \in \mathbb{N}_0$ or $j\in\mathbb{Z}$,
respectively, are called (infinite) $\varepsilon$-pseudo-orbits. Obviously, every
proper orbit is an $\ensuremath{\varepsilon}$-pseudo-orbit for all $\ensuremath{\varepsilon}>0$. Note that we slightly
deviate from the standard definition by not requiring strictness of the
inequality in (\ref{def_eps_orbit}). This will be very convenient later on for
technical reasons, but has no significance on a conceptual level.
For $F \in \cm{m}$ and $\varepsilon \geq 0$, we now define the $\varepsilon$\textit{-rotation set} to be the set of all accumulation points of sequences of the form \begin{align*}
\left(\frac{\xi_{n_i}^i - \xi_0^i}{n_i}\right)_{i \in \mathbb{N}} \end{align*} where $n_i \to \infty$ as $i \to \infty$ and $(\xi^i_j)_{j=0}^{n_i}$ is an $\varepsilon$-pseudo-orbit of $F$ of length $n_i+1$ for each $i \in \mathbb{N}$. Analogous to (\ref{def_rot_set_sets}), we let \begin{align*}
K_k^\varepsilon(F)
\ = \ \left\{ \frac{\xi_k-\xi_0}{k} \; \bigg| \;
\left( \xi_j \right)_{j=0}^{k} \text{ is an }
\varepsilon\text{-pseudo-orbit of } F
\text{ with } \xi_0 \in [0,1]^m
\right\}
\end{align*} for $k \in \mathbb{N}$ and can alternatively define the $\varepsilon$-rotation set by \begin{align}
\epsrs{F}{\varepsilon}
\ = \ \bigcap\limits_{n \geq 1}
\overline{\bigcup\limits_{k \geq n} K_k^\varepsilon(F)}\ .
\label{intersecteps} \end{align} Note that since $F$ commutes with integer translations, one could also replace $\xi_0 \in [0,1]^m$ by $\xi_0\in\ensuremath{\mathbb{R}}^m$ in the definition of $K_k^\ensuremath{\varepsilon}(F)$. For every $k \in \mathbb{N}$, the inclusion $K_k(F) \subseteq K_k^{\varepsilon}(F)$ is apparent, so that due to (\ref{def_rot_set_sets}) and (\ref{intersecteps}), we obtain \begin{align}
\rs{F} \ \subseteq \ \epsrs{F}{\varepsilon}
\label{Erset_subs_epsrs} \end{align}
for all $\varepsilon \geq 0$. We omit the elementary proof of the following lemma. \begin{lemma}\label{props}
Let $F \in \cm{m}, \varepsilon > 0$ and $M = \max\nolimits_{x\in [0,1]^m} \|F(x)-x\|$. Then
\begin{itemize}
\item[(i)] for each $k \in \mathbb{N}$ the set $K_k^{\varepsilon}(F)$ is non-empty
and compact, and thus the same holds for $\epsrs{F}{\varepsilon}$;
\item[(ii)] \label{props2}
$\rs{F} \subseteq \overline{B_M(0)}$
and $\epsrs{F}{\varepsilon} \subseteq \overline{B_{M+\varepsilon}(0)}$.
\end{itemize} \end{lemma}
\noindent We now aim to show convergence of the $\ensuremath{\varepsilon}$-rotation sets as $\ensuremath{\varepsilon}$ decreases to zero. This presents the main result of this section. An alternative proof can be found in \cite{GuiheneufKoropecki2016RotationSetStability}. However, we include the proof both for the convenience of the reader and because the employed arguments and estimates will become crucial again in Section~\ref{sec_algo}.
\begin{theorem} \label{main}
Let $F \in \hm{2}$. Then
$\epsrs{F}{\varepsilon} \rightarrow_{\mathrm{H}} \rs{F} \text{ as } \varepsilon \to 0.$
\end{theorem}
\noindent Since for $\rs{F} \ensuremath{\subseteq} \epsrs{F}{\varepsilon}$ for all $\ensuremath{\varepsilon}>0$ by (\ref{Erset_subs_epsrs}), it remains to show that for every $\delta > 0$ there is an $\varepsilon > 0$ such that \begin{align}
\epsrs{F}{\varepsilon} \ \subseteq \ B_\delta(\rs{F}) \ .
\label{main2} \end{align} We split the proof into Lemmas~\ref{difficultest} and \ref{Prop_knconv}, beginning with a formulation of Lemma~\ref{LProper3} in terms of $\varepsilon$-rotation sets.
\begin{lemma} \label{difficultest}
Let $F \in \cm{m}$ and $\varepsilon \geq 0$. Then
$\epsrs{F}{\varepsilon} \subseteq \ensuremath{\mathrm{Conv}} (K_n^{\varepsilon}(F))$
for all $n \in \mathbb{N}$. \end{lemma}
\begin{proof}
Let $k,n\in \mathbb{N}$. For an $\varepsilon$-pseudo-orbit $(\xi_j)_{j=0}^{kn}$
with $\frac{1}{kn} \left( \xi_{kn} -\xi_0 \right) \in K_{kn}^\varepsilon(F)$
we~have
\begin{align*}
\frac{1}{kn} \left( \xi_{kn} -\xi_0 \right)
\ = \ \frac{1}{k} \sum\limits_{i=0}^{k-1}
\frac{1}{n} \left( \xi_{(i+1)n} -\xi_{in} \right)
\ \in \ \ensuremath{\mathrm{Conv}} \left( K_n^\varepsilon(F) \right)\ ,
\end{align*}
since the tuples $( \xi_{in}, \xi_{in+1}, \ldots,\xi_{(i+1)n})$ are
$\varepsilon$-pseudo-orbits of $F$ of length $n+1$ and hence $\frac{1}{n}(
\xi_{(i+1)n} -\xi_{in} ) \in K_n^{\varepsilon}(F)$ for each $i \in
\{0,\ldots,k-1\}$. Consequently, for all $k,n \in \mathbb{N}$ we have
\begin{align}
\label{Econv_eps}
K_{kn}^{\varepsilon}(F) \ \subseteq \ \ensuremath{\mathrm{Conv}}(K_n^{\varepsilon}(F))\ .
\end{align}
Now let $k,n\in\mathbb{N}$ with $k\geq n$. Each number $k$ can uniquely be
split into $k = m_k n + r_k$, where $m_k \in \mathbb{N}$ is the integer part
of $\frac{k}{n}$ and $r_k \in \{0,\ldots,n-1\}$. If $(\xi_j)_{j=0}^ k$ is
an~$\varepsilon$-pseudo-orbit of length $k+1$, we find
\begin{align}
\label{rest}
K_k^\varepsilon(F)
\ \ni \ \frac{1}{k} \left( \xi_k - \xi_0 \right)
\ = \ \frac{1}{k} \left( \xi_{m_k n+r_k} - \xi_{m_k n} \right)
+\frac{1}{k} \left( \xi_{m_k n} - \xi_0 \right) \ .
\end{align}
Since the tuple $(\xi_{m_k n+j})_{j=0}^{r_k}$ is an $\varepsilon$-pseudo-orbit of $F$
of length $r_k+1$, we obtain~that
\begin{eqnarray*}
\lefteqn{\left\| \xi_{m_k n+r_k} - \xi_{m_k n} \right\|
\ = \ \left\| \sum\limits_{i=0}^{r_k-1}
\xi_{m_k n+i+1} - \xi_{m_k n +i} \right\| } \\
&\leq & \sum\limits_{i=0}^{r_k-1} \left( \left\|
F(\xi_{m_k n +i})-\xi_{m_k n +i} \right\| + \varepsilon \right)
\ \leq\ r_k (M + \varepsilon) \ .
\end{eqnarray*}
Thus, the norm of the first summand in (\ref{rest}) is bounded by
$\frac{r_k}{k}(M+\varepsilon)$. For the second summand, we have
\begin{align*}
\frac{1}{k}\left(\xi_{m_k}-\xi_0\right)
\ = \ \frac{m_k n}{k} \cdot \frac{1}{m_k n} \left( \xi_{m_k n} - \xi_0 \right)
\ \in \ \frac{m_k n}{k} \cdot K_{m_k n}^\varepsilon(F) \ .
\end{align*}
By (\ref{Econv_eps}), $K_{m_k n}^\varepsilon(F)$ is included in $\ensuremath{\mathrm{Conv}} \left( K_n^\varepsilon(F) \right)$.
Altogether, we obtain
\begin{align}
K_k^\varepsilon(F) &\ \subseteq \ \overline{B_{\frac{r_k}{k}(M +
\varepsilon)}\left(\frac{m_k n}{k} \cdot K_{m_k n}^\varepsilon(F)\right)}
\nonumber \\
&\ \subseteq \ \overline{B_{\frac{r_k}{k}(M + \varepsilon)}\left(\left( 1 -
\frac{r_k}{k} \right)\ensuremath{\mathrm{Conv}}\left( K_n^\varepsilon(F) \right)\right)} \ .
\label{diffproof1}
\end{align}
For a fixed value of $n$, the inclusion (\ref{diffproof1}) is valid for all $k
\geq n$. Therefore, for each $n \in \mathbb{N}$, the representation
(\ref{intersecteps}) of the $\varepsilon$-rotation set yields
\begin{align*}
\epsrs{F}{\varepsilon}
& \ = \ \bigcap\limits_{l \geq 1}
\overline{\bigcup\limits_{k \geq l} K_k^\varepsilon(F)}
\ \subseteq\
\bigcap\limits_{l \geq n} \overline{\bigcup\limits_{k \geq l}
K_k^\varepsilon(F)} \\
&\ \subseteq \ \bigcap\limits_{l \geq n}
\overline{\bigcup\limits_{k \geq l} B_{\frac{r_k}{k}(M +
\varepsilon)}\left(\left(1 - \frac{r_k}{k} \right) \ensuremath{\mathrm{Conv}} \left(
K_n^\varepsilon(F) \right)\right)}
\ = \ \ensuremath{\mathrm{Conv}} \left( K_n^\varepsilon(F) \right)\ ,
\end{align*}
due to the convergence $\frac{r_k}{k} \to 0$ as $k \to \infty$. \end{proof}
\begin{lemma} \label{Prop_knconv}
Let $F \in \cm{m}$ and $n \in \mathbb{N}$. Then
$K_n^\varepsilon(F) \rightarrow_{\mathrm{H}} K_n(F) \text{ as } \varepsilon \to 0.$
\end{lemma}
\begin{proof}
The inclusion $K_n(F) \subseteq K_n^{\varepsilon}(F)$ is evident for every
$\varepsilon > 0$. Let $\delta > 0$. Due to the continuity of $F$, there exists an
$\varepsilon_0 > 0$ such that $\|F^n(\xi_0)-\xi_n\| < n \delta$ for every
$\varepsilon$-pseudo-orbit $(\xi_j)_{j=0}^n$ of $F$ with $0<\varepsilon \leq
\varepsilon_0$. Therefore, we obtain
\begin{displaymath}
\frac{\xi_n - \xi_0}{n}
\ = \ \frac{F^n(\xi_0)-\xi_0}{n} + \frac{\xi_n-F^n(\xi_0)}{n}
\ \in\ B_\delta(K_n(F)) \ .
\end{displaymath}
As $\delta > 0$ was chosen arbitrarily and $n\in\ensuremath{\mathbb{N}}$ is fixed, this proves the
convergence $K_n^{\varepsilon}(F) \rightarrow_{\mathrm{H}} K_n(F)$ as $\varepsilon \to 0$. \end{proof}
\noindent {\em Proof of Theorem \ref{main}.}\quad Recall that in order to prove Theorem \ref{main}, we have to show the inclusion (\ref{main2}). Let $\delta > 0$. By Lemma \ref{main_Kn_convH}, the rotation set $\rs{F}$ can be approximated up to an arbitrary precision with respect to the Hausdorff distance by the sets $K_n(F)$. Therefore, we choose $n_0 \in \mathbb{N}$ sufficiently large, so that \begin{align}
\label{proof1}
K_{n_0}(F) \ \subseteq \ \overline{B_{\frac{\delta}{2}}(\rs{F})} \ . \end{align} Furthermore, by Lemma \ref{Prop_knconv} there exists an $\varepsilon_0 > 0$ sufficiently small, such that \begin{align}
\label{proof2}
K_{n_0}^{\varepsilon}(F) \ \subseteq \ \overline{B_{\frac{\delta}{2}}(K_{n_0}(F))} \end{align} for all $0 < \varepsilon \leq \varepsilon_0$. Applying Lemma \ref{difficultest}, together with (\ref{proof1}) and (\ref{proof2}) we obtain \begin{align*}
\epsrs{F}{\varepsilon} &
\ \subseteq \
\ensuremath{\mathrm{Conv}} \left( K_{n_0}^{\varepsilon}(F) \right)
\ \subseteq \
\ensuremath{\mathrm{Conv}} \left( B_{\frac{\delta}{2}}(K_{n_0}(F) \right) \\ & \
\subseteq \
\ensuremath{\mathrm{Conv}} \left(B_\delta(\rs{F}) \right)
\ = \ \overline{B_\delta(\rs{F})} \end{align*} for every $0 < \varepsilon \leq \varepsilon_0$. $\overline{B_\delta(\rs{F})}$ is convex itself, since the rotation set is convex by Theorem~\ref{Thm_convexity}. As $\delta > 0$ was arbitrary, this shows the asserted convergence $\epsrs{F}{\varepsilon} \rightarrow_{\mathrm{H}} \rs{F}$ as $\varepsilon \to 0$. \qed
\section{Set-oriented computation of rotation sets} \label{sec_algo}
\subsection{Why set-oriented numerics -- shortcomings of the direct approach.}
Before we turn to our algorithm for the set-oriented computation of rotation sets, we briefly want to comment on the problems that arise with a more conventional approach.
The easiest way to compute the rotation set of a torus homeomorphism numerically would be to fix a standard grid $\Gamma\ensuremath{\subseteq}[0,1]^2$ of $N\times N$ points, to compute the normalised displacement vectors $\frac{1}{n}(F^n(x)-x)$ for all $x\in\Gamma$ and some large $n\in\ensuremath{\mathbb{N}}$ and to plot the collection of these vectors in order to obtain an approximation of the rotation set. However, one may now consider the following situation, which actually turns out to be generic (with respect to $\mathcal{C}^0$-topology, see \cite{Guiheneuf2015RotationSetComputation}): Suppose that an area-preserving torus homeomorphism $f$ has a rotation set with non-empty interior, and that the Lebesgue measure $\ensuremath{\mathrm{Leb}}_{\ensuremath{\mathbb{T}}^2}$ on $\ensuremath{\mathbb{T}^2}$ is ergodic with respect to $f$. Then there exists a well-defined rotation vector \[ \varrho_f(\ensuremath{\mathrm{Leb}}_{\ensuremath{\mathbb{T}}^2}) \ = \ \int_{\ensuremath{\mathbb{T}}^2} \varphi(x) \ dx , \] where $\varphi(x) = F(x)-x$ is interpreted as a function on the torus, and we have \[ \ensuremath{\lim_{n\rightarrow\infty}} \frac{F^n(x)-x}{n} \ = \ \ntel \ensuremath{\sum_{i=0}^{n-1}} (\varphi\circ f^i)(x) \ = \ \varrho_f(\ensuremath{\mathrm{Leb}}_{\ensuremath{\mathbb{T}}^2}) \] for Lebesgue-a.e.\ $x\in\ensuremath{\mathbb{T}^2}$. Hence, if $n$ above is chosen too large, the rotation vectors coming from starting points in the grid $\Gamma$ will almost surely be arbitrarily close to $\varrho_f(\ensuremath{\mathrm{Leb}}_{\ensuremath{\mathbb{T}}^2})$. In this case, the numerical approximation of the rotation set will show the singleton $\varrho_f(\ensuremath{\mathrm{Leb}}_{\ensuremath{\mathbb{T}}^2})$, or something very close to it, whereas the true rotation set is much bigger due to its non-empty interior. Figure \ref{Fig_dir_mthd_a1b1} illustrates this numerical phenomenon for a standard example $f_{1,1}$ (see Section \ref{sec_nums}) of a torus diffeomorphism given by the lift \begin{equation*}
F_{1,1}:\mathbb{R}^2 \to\mathbb{R}^2 \ , \quad (x,y) \mapsto \big{(}x+\sin(2\pi(y+\sin(2\pi x))),\ y+\sin(2\pi x)\big{)}, \end{equation*} which exhibits $\varrho_{f_{1,1}}(\ensuremath{\mathrm{Leb}}_{\ensuremath{\mathbb{T}}^2}) = (0,0)$ and is known to have the rotation set $\rs{F_{1,1}} = [-1,1]^2$ (see Lemma \ref{l.rotset_F11}); as the number of iterations increases the majority of the approximate rotation vectors based on the iteration of grid points tend towards the origin and only the fixed points at the vertices are detected.
\begin{figure}\label{Fig_dir_mthd_a1b1}
\end{figure}
\noindent It is effects like this which prevent a direct approach from producing reasonable results \cite{Guiheneuf2015RotationSetComputation}, and frequently lead to an underestimation of the actual rotation set. In order to counter this, one would have to choose the grid constant $N$ exponentially large with respect to $n$ (of magnitude $N\sim L^n$, where $L$ is a Lipschitz constant of $f$), which is not feasible for efficient computations.
Moreover, the above discussion shows that in the direct approach there is a critical dependency between the parameters $N$ (grid size) and $n$ (number of iterates), which cannot be chosen independent of each other. In contrast to this, it is surely desirable to have an algorithm whose precision increases in both parameters independently, so that computation capacities can be pushed to their limit without having to worry about the precise relation of the parameters. This is exactly what the set-oriented method will allow us to do.
\subsection{The set-oriented approach -- description of the algorithm.}
\label{algorithm} We start with a basic observation. By Lemma~\ref{main_Kn_convH}, the sets $K_n(F)$ converge to the rotation set $\rs{F}$, and moreover we have the elementary estimate \begin{align}
\mathrm{d}_{\mathrm{H}}\left(\frac{1}{n}F^n([0,1]^2),K_n(F) \right)\ \leq \ \frac{\sqrt{2}}{n}
\label{E_Fn_convH2} \ . \end{align} Thus, we obtain \begin{corollary} \label{Col_Fn_convH}
Let $F \in \hm{2}$. Then
$\frac{1}{n}F^n([0,1]^2) \rightarrow_{\mathrm{H}} \rs{F}$ as $n \to \infty$. \end{corollary}
\noindent This allows to focus on the sets $\frac{1}{n}F^n([0,1]^2)$ in order to compute the rotation sets, which is quite convenient from a practical viewpoint. Hence, we aim for an accurate numerical approximation of the sets $\frac{1}{n}F^n([0,1]^2)$ for some large $n \in \mathbb{N}$. Inspired by the software package GAIO (Global Analysis of Invariant Objects), we apply the concept of \textit{box coverings} to formulate our algorithm. This library, developed by Dellnitz, Froyland and Junge \cite{DellnitzFroylandJunge2001GAIO}, provides numerical methods for the analysis of dynamical systems by set-oriented techniques. As mentioned above, on the theoretical level this corresponds to considering $\ensuremath{\varepsilon}$-pseudo-orbits.
\noindent For a given maximal diameter $\varepsilon > 0$ let $\mathcal{B}_0$ be a collection of compact sets $B \subset [0,1]^2$ with \begin{align}
\sup\limits_{B \in \mathcal{B}_0}
\ensuremath{\mathrm{diam}}(B)\ \leq \ \varepsilon
\quad \text{ and } \quad
\bigcup\limits _{B \in \mathcal{B}_0} B \ = \ [0,1]^2 \ .
\label{E_def_covI2} \end{align} Note that $\mathcal{B}_0$ can be considered as a covering of $\ensuremath{\mathbb{T}}^2$, whose lift to $\ensuremath{\mathbb{R}}^2$ is then given~by \begin{align*}
\mathcal{B} \ = \ \left\{ B + t \; \big| \; B \in \mathcal{B}_0,\ t \in
\mathbb{Z}^2 \right\} \ . \end{align*} The elements of the collections $\mathcal{B}_0$ and $\mathcal{B}$ will be referred to as \textit{boxes}. Further, given $\eta>0$, for each box $B\in\mathcal{B}$, let $\Gamma_B\ensuremath{\subseteq} B$ be a finite set of points which is $\eta$-dense in $B$ and chosen such that $\Gamma_{B+t}=\Gamma_B+t$ for every integer vector $t\in\ensuremath{\mathbb{Z}}^2$. Finally, fix $R>0$. Then, for $F \in \hm{2}$, we now generate a sequence of sets $\left( Q_n^\ast\right)_{n\in\mathbb{N}}$ (approximations of the rotation set) according to the following algorithm.
\begin{minipage}[t]{0.18\textwidth}
\raggedright {\em Initialisation:} \end{minipage} \begin{minipage}[t]{0.15\textwidth}
\raggedleft $Q_0$ \end{minipage} \begin{minipage}[t]{0.5\textwidth}
\raggedright
$\ =\ [0,1]^2 $ \end{minipage}
\begin{minipage}[t]{0.1\textwidth} \, \end{minipage}
\begin{minipage}{0.18\textwidth} \raggedright {\em Box images:} \end{minipage} \begin{minipage}{0.7\textwidth} \raggedright
Given $B\in\mathcal{B}$, let the {\em box image} of $B$ be defined as \end{minipage}
\begin{minipage}[t]{0.18\textwidth}
\quad \end{minipage} \begin{minipage}[t]{0.15\textwidth}
\raggedleft $\mathcal{I}(B)$ \end{minipage} \begin{minipage}[t]{0.5\textwidth}
\raggedright
$\ =\ \{B'\in\mathcal{B} \mid \exists x\in\Gamma_B: d(F(x),B')\leq R\} $ \end{minipage}
\begin{minipage}[t]{0.1\textwidth}
\raggedleft
\begin{align} \, \label{e.box_images} \end{align} \end{minipage}
\begin{minipage}[t]{0.18\textwidth}
\quad \end{minipage} \begin{minipage}{0.7\textwidth} \raggedright
(see Fig. \ref{f.box_covering}). \end{minipage}
\begin{minipage}{0.18\textwidth} \raggedright {\em Iteration:} \end{minipage} \begin{minipage}{0.7\textwidth} \raggedright
For $k = 0, \ldots, n-1$ generate the {\em box coverings} \end{minipage}
\begin{minipage}[t]{0.18\textwidth}
\quad \end{minipage} \begin{minipage}[t]{0.15\textwidth}
\raggedleft $\mathcal{B}_{k+1}$ \end{minipage} \begin{minipage}[t]{0.45\textwidth}
\raggedright
$\ = \ \left\{ B' \in \mathcal{B}\mid \exists B \in \mathcal{B}_k: B' \in \mathcal{I}(B) \right\}, $ \end{minipage}
\begin{minipage}[t]{0.1\textwidth}
\raggedleft
\begin{align} \, \label{E_box_coll_Bn} \end{align} \end{minipage}
\begin{minipage}[t]{0.18\textwidth}
\quad \end{minipage} \begin{minipage}[t]{0.15\textwidth}
\raggedleft $ Q_{k+1}$ \end{minipage} \begin{minipage}[t]{0.45\textwidth}
\raggedright
$\ = \ \bigcup\nolimits_{B \in \mathcal{B}_{k+1}} \!\!B. $ \end{minipage}
\begin{minipage}[t]{0.1\textwidth}
\raggedleft
\begin{align} \, \label{E_def_Qn} \end{align} \end{minipage}
\begin{minipage}{0.18\textwidth} \raggedright {\em Normalisation:} \end{minipage} \begin{minipage}{0.15\textwidth} \raggedleft $Q_n^\ast$ \end{minipage} \begin{minipage}{0.45\textwidth} \raggedright
$\ = \ \frac{1}{n}Q_{n} ,\ n \in \mathbb{N}$ \end{minipage}
\begin{minipage}{0.1\textwidth} \raggedleft
\begin{align} \, \label{e.Qnstar} \end{align} \end{minipage}
\begin{figure}
\caption{Box image of a box $B$ in the box covering $\mathcal{B}_k$, one test point, exemplarily.}
\label{f.box_covering}
\end{figure}
\noindent The starting point of our algorithm is the unit square $[0,1]^2$. Instead of iterating single test points, we consider collections of boxes $B \in \mathcal{B}$ and always collect those which are hit by an image of one of the previous boxes.
By choosing the parameter $R$ for the numerical calculation of the box images sufficiently large, we can ensure that no boxes are missed out and the images are always overestimated (see Lemma~\ref{l.box_overestimation}). This leads to a systematic overestimation of the sets $\frac{1}{n}F^n\left([0,1]^2\right)$, which will eventually lead to the fact that the algorithm always yields a superset of the actual rotation set (up to a small shift of order $\nicefrac{2\sqrt{2}}{n}$, see Lemma~\ref{L_dH_Kn_Kneps}). Hence, the effect of underestimation described for the direct approach in the previous subsection can be excluded (see Theorem~\ref{Th_box_err_asympt} below).
Note also that for the implementation of the algorithm, the box images only have to be computed once of each box $B\in\mathcal{B}_0$ at the beginning. This immediately provides the box images for the respective integer translates as well, and the information can then be used throughout the whole iterative procedure.
\subsection{Convergence of the approximations -- qualitative results and error
bounds.}
Relations between the normalised approximations $Q_n^\ast$ and the rotation set $\rs{F}$ are established by the following two results. Theorem~\ref{Th_box_err_asympt} is qualitative in nature and ensures convergence, whereas Theorem~\ref{Thm_main_alg_err} uses an additional boundedness assumption to provide error estimates.
\begin{theorem} \label{Th_box_err_asympt} Suppose that $F \in \hm{2}$ is
Lipschitz continuous with Lipschitz constant $L> 1$ and for each $\ensuremath{\varepsilon}>0$
the constants $\eta,R>0$ are chosen such that $L\eta\leq R\leq \ensuremath{\varepsilon}$. Then
\begin{equation}
\label{eq:Qn-convergence}
\lim_{\substack{n\to\infty\\\ensuremath{\varepsilon}\to 0}} \ \mathrm{d}_{\mathrm{H}}(Q_n^\ast,\rs{F}) \ = \ 0 \ ,
\end{equation}
in the sense that for all $\delta>0$ there exist $\ensuremath{\varepsilon}_0>0$ and $n_0\in\ensuremath{\mathbb{N}}$
such that if $\ensuremath{\varepsilon}$ in (\ref{E_def_covI2}) satisfies $\ensuremath{\varepsilon} \in(0,\ensuremath{\varepsilon}_0]$ and
$n\geq n_0$, then $\mathrm{d}_{\mathrm{H}}(Q_n^\ast,\rs{F}) < \delta$. \end{theorem}
\begin{rem}
For theoretical purposes, one may also want to ignore the fact that the images
of the boxes $B\in\mathcal{B}$ can only be approximated via test points, and define
alternative sequences $\big{(} \tilde Q_n \big{)}_{n \in \mathbb{N}},\big{(} \tilde Q_n^* \big{)}_{n \in \mathbb{N}}$ by using the precise images via
\begin{align*}
\tilde Q_0 &\ = \ Q_0 \ ,\\
\tilde Q_n &\ =\ \bigcup \left\{ B\in \mathcal{B} \mid B\cap F(\tilde Q_{n-1})\neq \emptyset \right\} \ ,\\
\tilde Q_n^* &\ =\ \frac{1}{n} \tilde Q_n \ .
\end{align*}
This corresponds to setting the parameters $\eta$ and $R$ to zero. Then the
above convergence result is still valid for the new sequence, that is,
$\lim_{\substack{n\to\infty\\\ensuremath{\varepsilon}\to 0}} \ \mathrm{d}_{\mathrm{H}}(\tilde
Q_n^\ast,\rs{F})=0$. Moreover, the assumption of Lipschitz continuity is not
needed in this case, but it would still be required in the respective analogue
of the error estimates given in Theorem~\ref{Thm_main_alg_err} below. \end{rem}
\noindent For the proof of Theorem~\ref{Th_box_err_asympt}, we will need the following lemma.
\begin{lemma}
\label{l.box_overestimation} Suppose that $F \in \hm{2}$ is
Lipschitz continuous with Lipschitz constant $L>1$ and for each $\ensuremath{\varepsilon}>0$ the
constants $\eta,R>0$ are chosen such that $L\eta\leq R\leq \ensuremath{\varepsilon}$. Then
\begin{align*}
\frac{F^n\left([0,1]^2\right)}{n} \ \ensuremath{\subseteq} \ Q^*_n
\ \ensuremath{\subseteq} \ \left\{\frac{\xi_n}{n}\; \bigg| \;
(\xi_j)_{j=0}^n \textrm{ is a } 2\ensuremath{\varepsilon}\textrm{-pseudo-orbit of } F \textrm{
with } \xi_0\in[0,1]^2\right\} \ .
\end{align*} \end{lemma} \begin{proof}
First, suppose that $v=\frac{1}{n} F^n(x_0)\in \frac{1}{n}F^n\left([0,1]^2\right)$ and let
$x_j=F^j(x_0)$. Choose boxes $B_j\in\mathcal{B}$ such that $x_j\in B_j$. We claim that
$B_j\ensuremath{\subseteq} Q_j$ for $j=0\ensuremath{,\ldots,} n$, so that finally $x_n=nv\in B_n\ensuremath{\subseteq} Q_n$ and
hence $v\in Q_n^*$.
For $j=0$ the claim is obvious. Therefore, suppose that $B_j\ensuremath{\subseteq} Q_j$. Then
there exists a test point $y\in\Gamma_{B_j}$ $\eta$-close to $x_j$, so that
$d(F(y),x_{j+1})\leq L\eta\leq R$. Since $x_{j+1}\in B_{j+1}$, this implies
$B_{j+1}\in \mathcal{I}(B)$ and thus $B_{j+1}\in Q_{j+1}$.
This shows the first inclusion $\frac{1}{n} F^n\left([0,1]^2\right)\ensuremath{\subseteq} Q^*_n$. In order
to show the second inclusion, suppose that $v\in Q^*_n$. Then, by definition
of $Q_n$, there exists a sequence of boxes $B_0\ensuremath{,\ldots,} B_{n}$ such that $nv\in B_n$
and $B_{j+1}\in\mathcal{I}(B_j)$ for all $j=0\ensuremath{,\ldots,} n-1$. By definition of the box
images, there exists a sequence of test points $\xi_j\in \Gamma_{B_j}$, $j=0\ensuremath{,\ldots,} n-1$,
such that $d(F(\xi_j),B_{j+1})\leq R$. Hence
\[
d(F(\xi_j),\xi_{j+1}) \ \leq \ R+\ensuremath{\varepsilon} \ \leq 2\ensuremath{\varepsilon} \ .
\]
for all $j=0\ensuremath{,\ldots,} n-2$, and this remains true for $j=n-1$ if we let $\xi_n=nv\in
B_n$. This means that $(\xi_j)_{j=0}^n$ is a $2\ensuremath{\varepsilon}$-pseudo-orbit and thus
yields the required second inclusion. \end{proof}
\begin{corollary} \label{c.box-overestimation}
In the situation of Lemma~\ref{l.box_overestimation}, we have
\[
K_n(F) \ \ensuremath{\subseteq} \
B_{\frac{\sqrt{2}}{n}}\left(Q_n^*\right) \ensuremath{\quad \textrm{and} \quad}
Q_n^* \ \ensuremath{\subseteq} \ B_{\frac{\sqrt{2}}{n}}\left(K_n^{2\ensuremath{\varepsilon}}(F)\right) \ .
\] \end{corollary}
\begin{proof}[\textbf{\textit{Proof of Theorem~\ref{Th_box_err_asympt}}}]
First, by Theorem~\ref{main} and the convexity of the rotation set, there
exists $\ensuremath{\varepsilon}_0>0$ such that \[ \mathrm{d}_{\mathrm{H}}\left(\ensuremath{\mathrm{Conv}}\left(\epsrs{F}{2\ensuremath{\varepsilon}_0}\right),\rs{F}\right) \ < \ \frac{\delta}{3} \ . \] Further, by the definition of $\epsrs{F}{2\ensuremath{\varepsilon}_0}$, there exists $n_0\in\ensuremath{\mathbb{N}}$ such that for all $n\geq n_0$ we have $K_n^{2\ensuremath{\varepsilon}_0}(F) \ensuremath{\subseteq} B_{\nicefrac{\delta}{3}}\left(\epsrs{F}{2\ensuremath{\varepsilon}_0}\right)$ and hence \[ \ensuremath{\mathrm{Conv}}\left(K_n^{2\ensuremath{\varepsilon}_0}(F)\right) \ \ensuremath{\subseteq} \ B_{\frac{\delta}{3}}\left(\ensuremath{\mathrm{Conv}}\left(\epsrs{F}{2\ensuremath{\varepsilon}_0}\right)\right) \ . \] As for all $n\in\ensuremath{\mathbb{N}}$ we have $\rs{F}\ensuremath{\subseteq} \ensuremath{\mathrm{Conv}}\left(K_n(F)\right)\ensuremath{\subseteq}\ensuremath{\mathrm{Conv}}\left(K_n^{2\ensuremath{\varepsilon}_0}(F)\right)$ (see Lemma~\ref{LProper3}), we obtain that \begin{equation}\label{e.qualitative_proof_1} \mathrm{d}_{\mathrm{H}}\left(\ensuremath{\mathrm{Conv}}\left(K^{2\ensuremath{\varepsilon}_0}_n(F)\right),\rs{F}\right) \ < \frac{2\delta}{3} \ . \end{equation} At the same time, Lemma~\ref{main_Kn_convH} implies that we can choose $n_0$ such that \begin{equation}\label{e.qualitative_proof_2}
\mathrm{d}_{\mathrm{H}}\left(K_n(F),\rs{F}\right) \ < \ \frac{2\delta}{3} \end{equation} for all $n\geq n_0$. If $n_0$ is chosen such that $\nicefrac{\sqrt{2}}{n_0} < \nicefrac{\delta}{3}$ and $n\geq n_0$, then using (\ref{e.qualitative_proof_2}) together with the first inclusion in Corollary~\ref{c.box-overestimation}, we obtain \[ \rs{F} \ \ensuremath{\subseteq} \ B_{\frac{2\delta}{3}}(K_n(F)) \ \ensuremath{\subseteq} \ B_\delta(Q^*_n) \ . \] Conversely, (\ref{e.qualitative_proof_1}) together with the second inclusion in Corollary~\ref{c.box-overestimation} yield \[ Q^*_n \ \ensuremath{\subseteq} \ B_{\frac{\delta}{3}}\! \left(K_n^{2\ensuremath{\varepsilon}_0}(F)\right) \ \ensuremath{\subseteq} \ B_\delta(\rs{F}) \ . \] Together with the fact that $K_n^{2\ensuremath{\varepsilon}}(F)\ensuremath{\subseteq} K_n^{2\ensuremath{\varepsilon}_0}(F)$ for all $\ensuremath{\varepsilon}<\ensuremath{\varepsilon}_0$, this means that \[ \mathrm{d}_{\mathrm{H}}(Q^\ast_n,\rs{F}) \ < \ \delta \] whenever $n\geq n_0$ and $\ensuremath{\varepsilon}\in(0,\ensuremath{\varepsilon}_0]$, as required. \end{proof}
\noindent In general, it is not possible to give quantitative error estimates for the numerical computation of rotation sets. The reason is that there are no general apriori bounds for the convergence of the sets $K_n(F)$ to $\rs{F}$. However, it turns out that in many situations, and in particular whenever the rotation set has non-empty interior, there exists a positive constant $c>0$ such that \begin{align}
\mathrm{d}_{\mathrm{H}} (K_n(F),\rs{F}) \leq \frac{c}{n} \quad \textrm{ for all } n\in\ensuremath{\mathbb{N}}\ .
\label{E_err_plausible}
\tag{BD} \end{align} This fact has been proven for diffeomorphisms in \cite{AddasZanata2015BoundedMeanMotionDiffeos} and the result was later generalised to homeomorphisms in \cite{LeCalvezTal2015ForcingTheory}. This is also referred to as {\em bounded deviation property}. In our context, (\ref{E_err_plausible}) together with the existence of a Lipschitz constant allows to provide the following quantitative estimates.
\begin{theorem} \label{Thm_main_alg_err} Suppose $F \in \hm{2}$ is Lipschitz continuous with Lipschitz constant $L> 1$, let $\varepsilon > 0$, $\eta L\leq R\leq \ensuremath{\varepsilon}$ and $n \in \mathbb{N}$. Further, assume that (\ref{E_err_plausible}) holds for
$c>0$. Then \begin{align}
\mathrm{d}_{\mathrm{H}} \left(Q_n^\ast,\rs{F}\right)\ \leq \
\max \left\{ \frac{2\sqrt{2}}{n},
\frac{\sqrt{2}}{n} + \gamma_{\varepsilon,n} \right\} \ ,
\label{E_main_err} \end{align} where \begin{align}
\gamma_{\varepsilon,n} \ = \
\frac{2r_n}{n}(M+\varepsilon) + \left( 1- \frac{r_n}{n} \right)
\min\limits_{1\leq k\leq n} \frac{1}{k} \left( c + 2\varepsilon \frac{L^k-1}{L-1}
\right) \ .
\label{E_min_gamma} \end{align}
Here $M =\max_{x\in [0,1]^2} \|F(x)-x\|$, $k_n$ is the number between $1$ and $n$ for which the minimum on the right is attained and $r_n = n \bmod k_n$. \end{theorem}
\noindent The proof is given in Section~\ref{QuantitativeEstimates} below.
\begin{rem} \label{Rem_err_nstar}
\begin{itemize}
\item[(a)] It should be noted that the above estimate is rather of theoretical
than practical interest. This is exemplified in part (b) of this remark
below. We include it nevertheless, since on the one hand it demonstrates
what is possible on the analytic side, and on the other hand the proof
reflects and demonstrates very well how and why the nonlinearity of the
dynamics complicates the efficient computation of rotation sets.
\item[(b)] In order to see why the above estimates are hardly relevant for the
numerical implementation, suppose that the constant $c$ in
(\ref{E_err_plausible}) is known (which is usually not the case) and
relatively small, say, equal to $1$. Even in this case, in order to achieve
an apriori error bound of order $10^{-2}$, the integer $k$ in the term
$\frac{1}{k} \big{(} c + 2\varepsilon \frac{L^k-1}{L-1} \big{)}$ in
(\ref{E_min_gamma}) would have to be at least $100$ -- otherwise $\nicefrac{c}{k}> 100$
-- but then at the same time $\ensuremath{\varepsilon}$ would need to be smaller than
$\big{(}\frac{2(L^k-1)}{L-1}\big{)}^{{ }_{-1}}$. Hence, even if $L$ is only $2$,
this would require to work with a box diameter of $\ensuremath{\varepsilon}\simeq 2^{-100}$,
which is hardly possible with contemporary computers.
\item[(c)] Fortunately, it turns out that in the actual implementation the
convergence is usually much faster than indicated by the above error
estimates. One possible reason is the fact that $f$ may possess a shadowing
property. This leads to improved error bounds, where essentially the
exponential term $\ensuremath{\varepsilon}\frac{L^k-1}{L-1}$ can be dropped altogether. We
discuss this in detail in the following subsection.
\end{itemize} \end{rem}
\subsection{Implications of shadowing.}\label{Shadowing}
Given a self map $g:X\to X$ of some metric space $X$, we say an orbit $\nfolge{x_n}$ of $g$ {\em $\delta$-shadows} a sequence $\nfolge{\xi_n}$ if $d(x_n,\xi_n)<\delta$ for all $n\in\ensuremath{\mathbb{N}}$. Given $\delta,\ensuremath{\varepsilon}>0$, we say $g$ has the {\em $\delta$-shadowing property with constant $\ensuremath{\varepsilon}$} if all $\ensuremath{\varepsilon}$-pseudo-orbits of $g$ are $\delta$-shadowed by some orbit of $g$. If such a constant $\ensuremath{\varepsilon}=\ensuremath{\varepsilon}(\delta)$ exists for all $\delta>0$, we simply say that $g$ has the {\em shadowing property}.
\begin{theorem} \label{t.shadowing} Suppose $f$ is a torus homeomorphism
homotopic to the identity with lift $F\in\hm{2}$ and $\delta,\gamma\in (0,\nicefrac{1}{2})$
are such that $d(x,y)<\delta$ implies $d(f(x),f(y))<\nicefrac{1}{2}-\gamma$ for all
$x,y\in\ensuremath{\mathbb{T}}^2$. Further, assume that $f$ has the $\delta$-shadowing property
with constant $\ensuremath{\varepsilon}\in(0,\gamma)$. Then
\[
\epsrs{F}{\ensuremath{\varepsilon}} \ = \ \rs{F} \ .
\] \end{theorem} \begin{corollary}
If a torus homeomorphism $f$ homotopic to the identity has the shadowing
property, then there exists $\ensuremath{\varepsilon}_0>0$ such that $\epsrs{F}{\ensuremath{\varepsilon}}=\rs{F}$ for
any $\ensuremath{\varepsilon}\in(0,\ensuremath{\varepsilon}_0]$ and any lift $F$ of $f$. \end{corollary}
\noindent For the proof of Theorem~\ref{t.shadowing}, we need the following elementary statement. \begin{lemma}
Suppose $f,F,\delta,\gamma,\ensuremath{\varepsilon}$ are chosen as in
Theorem~\ref{t.shadowing}. Then $F$ has the $\delta$-shadowing property with
constant $\ensuremath{\varepsilon}$. \end{lemma} \begin{proof}
Suppose $\nfolge{\hat\xi_n}$ is an $\ensuremath{\varepsilon}$-pseudo-orbit for $F$. Then
$\xi_n=\pi(\hat\xi_n)$ defines an $\ensuremath{\varepsilon}$-pseudo-orbit of $f$, and we can
therefore find some $x_0\in\ensuremath{\mathbb{T}}^2$ such that $d\left(f^n(x_0),\xi_n\right)<\delta$ for all
$n\geq 0$. Let $\hat x_0$ be the unique lift of $x_0$ in $B_\delta\big{(}\hat
\xi_0\big{)}$. We claim that $d(F^n(\hat x_0),\hat\xi_n)<\delta$ for all $n\geq 0$, so that the orbit of $\hat x_0$ is the required $\delta$-shadowing~orbit.
For $n=0$ there is nothing to prove. If $d(F^n(\hat x_0),\hat\xi_n)<\delta$
for some $n\geq 0$, then $F^{n+1}(\hat x_0)\in
B_{1/2-\gamma}\big{(}F(\hat\xi_n)\big{)}\ensuremath{\subseteq}
B_{1/2}\big{(}\hat\xi_{n+1}\big{)}$. However, as
$B_{1/2}\big{(}\hat\xi_{n+1}\big{)}$ projects injectively to $\ensuremath{\mathbb{T}}^2$ and
$d(\pi(F^{n+1}(\hat x_0),\pi(\hat\xi_{n+1})) =
d(f^{n+1}(x_0),\xi_{n+1})<\delta$, we also obtain
$d(F^{n+1}(\hat x_0),\hat\xi_{n+1})) <\delta$ as required. \end{proof} \begin{proof}[\textbf{\textit{Proof of Theorem~\ref{t.shadowing}}}]
Under the assumptions of the theorem, any finite $\ensuremath{\varepsilon}$-pseudo-orbit $(\hat
\xi_j)_{j=0}^n$ of $F$ is $\delta$-shadowed by some $F$-orbit $(x_j)_{j=0}^n$, so
that
\[
\left| \frac{\hat\xi_n-\hat\xi_0}{n} - \frac{F^n(x_0)-x_0}{n}\right| \ \leq \
\frac{2\delta}{n} \ .
\]
Therefore $\mathrm{d}_{\mathrm{H}}(K_n(F),K^\ensuremath{\varepsilon}_n(F)) \leq \nicefrac{2\delta}{n}$, and due to the definition
of the sets $K_n(F)$, $K^\ensuremath{\varepsilon}_n(F)$ and $\epsrs{F}{\ensuremath{\varepsilon}}$ and the convergence
$K_n(F) \rightarrow_{\mathrm{H}} \rs{F}$ as $n \to \infty$ by Lemma~\ref{main_Kn_convH}, this implies
$\epsrs{F}{\ensuremath{\varepsilon}}=\rs{F}$. \end{proof}
\noindent As mentioned before, the shadowing property leads to improved error estimates for the set-oriented computation of the rotation set, and in particular allows to eliminate the exponential term $\gamma_{\ensuremath{\varepsilon},n}$ in Theorem~\ref{Thm_main_alg_err}. \begin{theorem}\label{t.shadowing_error}
Suppose that $f$ satisfies the assumptions of
Theorem~\ref{Thm_main_alg_err} and the bounded deviations hypothesis
(\ref{E_err_plausible}) with constant $c>0$. Let $F\in\hm{2}$ be a lift of $f$
and $Q_n^\ast$ be defined by (\ref{E_def_Qn}). Then
\[
\mathrm{d}_{\mathrm{H}}(Q^\ast_n,\rs{F}) \ \leq \ \frac{\sqrt{2}+1+c}{n} \ .
\] \end{theorem} \noindent The proof is postponed until the end of the next subsection.
\subsection{Quantitative estimates -- proofs of Theorems~\ref{Thm_main_alg_err}
and \ref{t.shadowing_error}.} \label{QuantitativeEstimates}
Throughout this section, we assume that $F\in\hm{2}$ and $Q_n,\ Q_n^\ast$ are defined as in the preceding section. Let
\begin{align*}
P_n^\varepsilon(F) \ = \ \left\{ \xi_n \in \mathbb{R}^2 \mid (\xi_j)_{j=0}^n
\text{ is an } \varepsilon\text{-pseudo-orbit of } F \text{ with } \xi_0 \in
[0,1]^2 \right\} \end{align*} and note that $P^0_n(F)=F^n\left([0,1]^2\right)$. Then the statement of Lemma~\ref{l.box_overestimation} can be rewritten as \begin{align}
P^0_n(F) \ \ensuremath{\subseteq} \ Q_n \ \subseteq \ P_n^{2\varepsilon}(F) \ .
\label{E_def_Pneps} \end{align} This leads to the following initial estimate.
\begin{lemma} \label{Prop_mainalg_1} Suppose that $F \in \hm{2}$ is
Lipschitz continuous with Lipschitz constant $L>1$ and for each $\ensuremath{\varepsilon}>0$ the
constants $\eta,R>0$, are chosen such that $L\eta\leq R\leq \ensuremath{\varepsilon}$. Then, for
every $n\in\mathbb{N}$, we have
\begin{displaymath}
\rs{F} \ \subseteq \ \overline{B_{\frac{2\sqrt{2}}{n}}(Q_n^\ast)} \ .
\end{displaymath} \end{lemma}
\begin{proof}
We have
\begin{eqnarray*}
\rs{F} & \stackrel{\textrm{Lemma~\ref{LProper3}}}{\ensuremath{\subseteq}} & \ensuremath{\mathrm{Conv}}(K_n(F)) \ \ensuremath{\subseteq}
\ \overline{B_{\frac{\sqrt{2}}{n}}\left(\frac{1}{n}\ensuremath{\mathrm{Conv}}\big{(}F^n([0,1]^2\big{)}\right)} \\ & \stackrel{\textrm{Lemma~\ref{l.MZ-estimate}}}{\ensuremath{\subseteq}} &
\overline{B_{\frac{2\sqrt{2}}{n}}\left(\frac{1}{n}\big{(}F^n([0,1]^2\big{)}\right)} \ \stackrel{\textrm{Lemma}~\ref{l.box_overestimation}}{\ensuremath{\subseteq}} \
\overline{B_{\frac{2\sqrt{2}}{n}}(Q_n^\ast)} \ .
\end{eqnarray*} \end{proof}
\begin{lemma} \label{L_dH_Kn_Kneps} Suppose that $F \in \hm{2}$ is
Lipschitz continuous with Lipschitz constant $L>1$ and $\ensuremath{\varepsilon}>0$. Then, for
all $n\in\ensuremath{\mathbb{N}}$, we have
\begin{align*}
\mathrm{d}_{\mathrm{H}} \big{(}K_n^{2\varepsilon}(F), K_n(F)\big{)} \ \leq \ \frac{2\varepsilon(L^n-1)}{n(L-1)} \
\eqqcolon \kappa_{\varepsilon,n}\ .
\end{align*} \end{lemma}
\begin{proof}
Let $(\hat\xi_j)_{j=0}^ {n}$ be an $2\varepsilon$-pseudo-orbit of $F$ with
$\hat\xi_0 \in [0,1]^2$. Using the fact that $\|\hat\xi_1 - F(\hat\xi_0)\|
\leq 2\varepsilon$ and
\begin{align*}
\|\hat\xi_n-F^j(\hat\xi_0)\|
\ \leq \ \|F(\hat\xi_{j-1}) - F(F^{j-1}(\hat\xi_0))\| + 2\varepsilon
\ \leq \ L \|\hat\xi_{j-1}-F^{j-1}(\hat\xi_0)\| + 2\varepsilon,
\end{align*}
for all $j=1\ensuremath{,\ldots,} n-1$, we recursively obtain the estimate
\begin{align*}
\| \hat\xi_n - F^n(\hat\xi_0)\| \ \leq \ 2\varepsilon \sum\limits_{i=0}^ {n-1} L^i \ = \
2\varepsilon \frac{L^n -1}{L-1} \ .
\end{align*}
Thus, for any $v=\frac{1}{n}(\xi_n - \xi_0) \in
K_n^{2\varepsilon}(F)$
we have
\begin{align*}
\frac{\xi_n-\xi_0}{n} \ = \ \frac{\xi_n - F^n(\xi_0)}{n} +
\underbrace{\frac{F^n(\xi_0)-\xi_0}{n}}_{\in K_n(F)} \ ,
\end{align*}
so that
\[
K^{2\ensuremath{\varepsilon}}_n(F) \ \ensuremath{\subseteq} \ \overline{B_{\frac{2\varepsilon(L^n-1)}{n(L-1)}}(K_n(F))} \ .
\]
Since conversely we always have $K_n(F)\ensuremath{\subseteq} K^{2\ensuremath{\varepsilon}}_n(F)$, this proves the
statement. \end{proof}
\begin{lemma} \label{Prop_Kneps_incl_rs} Suppose that $F \in \hm{2}$ is
Lipschitz continuous with Lipschitz constant $L>1$ and $\ensuremath{\varepsilon}>0$. Further,
assume that $F$ additionally satisfies (\ref{E_err_plausible}) with
$c>0$. Then for all $n\geq 0$ we have
\begin{align*}
K_n^{2\varepsilon}(F) \ \subseteq \ \overline{B_{\gamma_{\ensuremath{\varepsilon},n}}(\rs{F})} \ ,
\end{align*}
where $\gamma_{\varepsilon,n}$ is defined as in (\ref{E_min_gamma}). \end{lemma}
\begin{proof}
Let $k\in\mathbb{N}$. Applying the estimate for the Hausdorff distance between
the sets~$K_k^{2\varepsilon}(F)$ and $K_k(F)$, denoted by $\kappa_{\varepsilon,k}$ in
Lemma \ref{L_dH_Kn_Kneps}, and the assumption (\ref{E_err_plausible}), we
obtain \begin{equation}\begin{split}
\lefteqn{ \ensuremath{\mathrm{Conv}}\left(K_k^{2\varepsilon}(F)\right)
\ \subseteq \ \ensuremath{\mathrm{Conv}} \left(B_{\kappa_{\varepsilon,k}}(K_k(F)) \right) }
\\
&\subseteq \ \ensuremath{\mathrm{Conv}} \left(B_{\frac{c}{k}+\kappa_{\varepsilon,k}}(\rs{F}) \right)
\label{E_Kneps_incl1} \
= \ \overline{B_{\frac{c}{k}+\kappa_{\varepsilon,k}}(\rs{F})}\ .
\end{split} \end{equation} For the last equality, note that $\rs{F}$ is convex. Let $k_n \in \{1,\ldots,n\}$ be the natural number for which the minimum in the definition of~$\gamma_{\varepsilon,n}$ in~(\ref{E_min_gamma}) is attained. Further, let $m_n\in\ensuremath{\mathbb{N}},\ r_n \in \{0,\ldots,k_n-1\}$ be such that $n = m_n k_n + r_n$. By the inclusion (\ref{diffproof1}) in the proof of Lemma~\ref{difficultest}, since $n \geq k_n$, we know that
\begin{align}
K_n^{2\ensuremath{\varepsilon}}(F) &\subseteq \ \overline{B_{\frac{r_n}{n}(M +
2\varepsilon)}\left(\left( 1 - \frac{r_n}{n} \right) \ensuremath{\mathrm{Conv}} \left(
K_{k_n}^{2\varepsilon}(F) \right)\right)} \ .
\label{E_Kneps_incl2}
\end{align}
Combining (\ref{E_Kneps_incl1}) and (\ref{E_Kneps_incl2}) and by the choice of $k_n$ and the definition (\ref{E_min_gamma})
of~$\gamma_{\varepsilon,n}$, we obtain
\begin{eqnarray*}
K_n^{2\ensuremath{\varepsilon}}(F) & \subseteq &
\overline{B_{\frac{r_n}{n}(M+2\varepsilon)}\left(\left(1-\frac{r_n}{n}\right)
B_{\frac{c}{k}+\kappa_{\ensuremath{\varepsilon},k_n}}(\rs{F})\right)} \\
& = & \overline{B_{\frac{r_n}{n}(M+2\varepsilon)+
\left(1-\frac{r_n}{n}\right)\left(\frac{c}{k_n}+\kappa_{\varepsilon,k_n} \right)}
\left(\left(1-\frac{r_n}{n}\right)\rs{F}\right)} \\
&\subseteq & \overline{B_{\frac{r_n}{n}(2M+2\varepsilon) + \left(1-\frac{r_n}{n}\right)
\left(\frac{c}{k_n}+\kappa_{\varepsilon,k_n}\right)}(\rs{F})} \\
& = &\overline{B_{\gamma_{\varepsilon,n}}(\rs{F})} \ .
\end{eqnarray*}
For the inclusion from the second to the third line,
note that $\rs{F}\ensuremath{\subseteq} \overline{B_M(0)}$, so that $\left(1-\frac{r_n}{n}\right)\rs{F}\ensuremath{\subseteq}
\overline{B_{\frac{r_n}{n}M}\left(\rs{F}\right)}$. \end{proof}
\begin{proof}[\textbf{\textit{Proof of Theorem \ref{Thm_main_alg_err}}}.]
In Lemma \ref{Prop_mainalg_1} we deduced that \begin{align*}
\rs{F} \subseteq \overline{B_{\frac{2\sqrt{2}}{n}}(Q_n^\ast)} \ . \end{align*} Conversely, from (\ref{E_def_Pneps}) and Lemma~\ref{Prop_Kneps_incl_rs} we obtain \begin{align*}
Q_n^\ast \ =\ \frac{1}{n} Q_n
\ \subseteq\ \frac{1}{n} P_n^{2\varepsilon}(F)
\ \subseteq\ \overline{B_{\frac{\sqrt{2}}{n}}\left(K_n^{2\varepsilon}(F)\right)}
\ \subseteq \ \overline{B_{\frac{\sqrt{2}}{n}+\gamma_{\varepsilon,n}}(\rs{F})} \ .
\end{align*} Altogether, we obtain the error estimate (\ref{E_main_err}). \ \end{proof}
\begin{proof}[\textbf{\textit{Proof of Theorem~\ref{t.shadowing_error}.}}]
On the one hand, we have
\begin{equation}
\rs{F} \ \ensuremath{\subseteq} \ B_{\frac{2\sqrt{2}}{n}}(Q^\ast_n) \
\end{equation}
by Lemma~\ref{Prop_mainalg_1}. On the other hand, we have shown in the proof of
Theorem~\ref{t.shadowing} that $K_n^{2\ensuremath{\varepsilon}}(F) \ensuremath{\subseteq} B_{\nicefrac{2\delta}{n}}(K_n(F))
\ensuremath{\subseteq} B_{\nicefrac{1}{n}}(K_n(F))$ (note that $\delta<\nicefrac{1}{2}$ by assumption) and thus~obtain
\begin{eqnarray*}
Q^*_n & \ensuremath{\subseteq} & \frac{1}{n} P^{2\ensuremath{\varepsilon}}_n(F) \ \ensuremath{\subseteq} \ B_{\frac{\sqrt{2}}{n}}(K^{2\ensuremath{\varepsilon}_n}(F)) \\ &
\ensuremath{\subseteq} & B_{\frac{\sqrt{2}+1}{n}}(K_n(F)) \ensuremath{\subseteq} B_{\frac{\sqrt{2}+1+c}{n}}(\rs{F}) \ .
\end{eqnarray*} This shows the required estimate. \end{proof}
\section{Numerical implementation and results} \label{sec_nums}
\noindent In order to implement the above algorithm and to apply it to some specific examples, we consider a standard family of (lifts of) torus diffeomorphisms given by \begin{equation} \label{e.standard_family}
F_{\alpha,\beta}:\mathbb{R}^2 \to\mathbb{R}^2 \ , \quad (x,y) \mapsto \big{(}x+\alpha\sin(2\pi(y+\beta\sin(2\pi x))),\ y+\beta\sin(2\pi x)\big{)}\ , \end{equation} where $\alpha,\beta\in\ensuremath{\mathbb{R}}$. Note that $F_{\alpha,\beta}$ is obtained as the composition of two skew shifts, $F_{\alpha,\beta}=F_\alpha\circ F_\beta$, where \begin{align*}
F_\alpha(x,y) = \big{(}x+\alpha\sin(2\pi y),\ y\big{)} \ensuremath{\quad \textrm{and} \quad}
F_\beta(x,y) = \big{(}x,\ y+\beta\sin(2\pi x)\big{)}\ . \end{align*} See \cite{LeboeufKurchanFeingoldArovas1990PhaseSpaceLocalization,Jaeger2011EllipticStars} for previous numerical studies and \cite{MisiurewiczZiemian1989RotationSets} for structurally similar examples. For specific parameter values, the rotation set of $F_{\alpha,\beta}$ can easily be determined analytically, which allows to test the numerical algorithm in a controlled setting.
\begin{lemma} \label{l.rotset_F11}
$\rs{F_{1,1}}=[-1,1]^2. $ \end{lemma} \begin{proof} This follows directly from two elementary observations. First, we have the general estimate $\rs{F_{\alpha,\beta}}\ensuremath{\subseteq} [-\alpha,\alpha]\times [-\beta,\beta]$, as $\alpha$
and $\beta$ are the maximal step sizes in the horizontal and vertical
direction. Hence, $\rs{F_{1,1}} \ensuremath{\subseteq} [-1,1]^2$.
Conversely, it is easily checked that the rotation vectors $(1,1),(-1,1),(1,-1)$ and $(-1,-1)$ are realised by the fixed points $(\nicefrac{1}{4},\nicefrac{1}{4}),(\nicefrac{3}{4},\nicefrac{1}{4}),(\nicefrac{1}{4},\nicefrac{3}{4})$ and $(\nicefrac{3}{4},\nicefrac{3}{4})$. By convexity, this means that $[-1,1]^2\ensuremath{\subseteq}\rs{F_{1,1}}$. \end{proof}
\noindent For the initialisation of the algorithm in Section~\ref{algorithm} , we choose $\mathcal{B}_0$ to be the standard covering of $[0,1]^2$ by $k^2$ squares of side length $\nicefrac{1}{k}$, $k \in \mathbb{N}$. Note that we can thus choose $\ensuremath{\varepsilon}=\nicefrac{\sqrt{2}}{k}$ in (\ref{E_def_covI2}). We fix a Lipschitz constant $L$ of $F_{\alpha,\beta}$ (for example, $L=1+4\pi^2$ works for all $(\alpha,\beta)\in[0,1]^2$) and set $R=\ensuremath{\varepsilon}$ in (\ref{e.box_images}). Moreover, for each $B\in\mathcal{B}_0$ we choose $\Gamma_B$ as a standard grid of $m^2$ points in $B$, so that $\Gamma_B$ is $\nicefrac{\sqrt{2}}{(k(m-1))}$-dense in $B$ (see Fig. \ref{f.box_covering}). Thereby, we choose $m=m(k)$ such that $\eta=\nicefrac{\sqrt{2}}{(k(m-1))}< \nicefrac{\ensuremath{\varepsilon}}{L}$.
\noindent In order to keep the dependence on $k$ explicit, we will from now on write $Q_{k,n}^*$, instead of $Q_n^*$, for the approximation defined in (\ref{e.Qnstar}). Then the assumptions of Theorem~\ref{Thm_main_alg_err} with $\ensuremath{\varepsilon}=\nicefrac{\sqrt{2}}{k}$ are satisfied and we obtain that $\lim_{k,n\to\infty} Q_{k,n}^*=\rs{F_{\alpha,\beta}}$, with error bound provided by (\ref{E_main_err}) (and by Theorem~\ref{t.shadowing_error} if $F_{\alpha,\beta}$ has the shadowing property). Figure~\ref{f.iterates_for_F11_1} shows $Q_{k,n}^*$ for $F_{1,1}$ for $k=8$ and different values of $n$.
\begin{figure}
\caption{Approximations $Q^*_{k,n}$ for the rotation set of the map $F_{1,1}$ with $k=8$ and $n = 1,2,5,10,25,50,100,200$ (from top left to bottom right).}
\label{f.iterates_for_F11_1}
\end{figure}
\noindent Zooming in (Fig. \ref{f.F11_zoom_ul}) on the boundary of $Q_{8,100}^*$ and $Q_{8,200}^*$ of Figure~\ref{f.iterates_for_F11_1} reveals the difference that exists between these approximations and $\rs{F_{1,1}}=[0,1]^2$, which is of a magnitude of $10^{-2}$ (and hence significantly smaller than the theoretical error bound in (\ref{E_main_err})). By Lemma \ref{Prop_mainalg_1}, the $\nicefrac{2\sqrt{2}}{n}$-neighbourhood of $Q_{k,n}^\ast$ covers the rotation~set.
\begin{figure}
\caption{Zoom on top left area of the approximations $Q_{8,100}^\ast$ (left) and
$Q_{8,200}^\ast$ (right) for the rotation set of the map $F_{1,1}$.}
\label{f.F11_zoom_ul}
\end{figure}
\noindent As further examples, we consider the maps $F_{\nicefrac{1}{2},\nicefrac{1}{2}},F_{\nicefrac{3}{5},\nicefrac{3}{5}}$ and $F_{\nicefrac{3}{4},1}$. In the first case, we still have an a priori lower bound for the rotation set: $\rs{F_{\nicefrac{1}{2},\nicefrac{1}{2}}}$ contains the square spanned by the points $(\pm \nicefrac{1}{2},0)$ and $(0,\pm \nicefrac{1}{2})$, since these rotation vectors are realised by the two-periodic points $(0,\nicefrac{1}{4}),(\nicefrac{1}{4},0)$,$(0,\nicefrac{3}{4})$ and $(\nicefrac{3}{4},0)$. The numerical approximation in Figure \ref{f.further_RS} indicates that this square indeed is the rotation set of $F_{\nicefrac{1}{2},\nicefrac{1}{2}}$ (recall here that our algorithm never underestimates).
\begin{figure}\label{f.further_RS}
\end{figure}
\noindent In the examples $F_{1,1}$ and $F_{\nicefrac{1}{2},\nicefrac{1}{2}}$ above, the vertices of the polygonal rotation set are realised by periodic orbits of very low period (1, respectively 2). As discussed in \cite{Guiheneuf2015RotationSetComputation}, such rotation sets can still be accurately predicted by conventional direct approaches, but these tend to fail if the vertices correspond to periodic points of higher periods. For this reason, we next consider the map $G = g_3 \circ g_2 \circ g_1: \mathbb{R}^2 \longrightarrow \mathbb{R}^2$ with \begin{align*}
g_1(x,y) &= \left( x,\ y + \textstyle{\frac{1}{8}} \sin (5 \cdot 2 \pi x) \right), \\
g_2(x,y) &=\left( x + \textstyle{\frac{2}{5}} \sin (8 \cdot 2 \pi y), \ y \right)
\text{ and } \\
g_3(x,y) &=\left( x - \textstyle{\frac{1}{5}},\ y + \textstyle{\frac{2}{8}} \right) \ . \end{align*}
The related rotation set can be determined analytically as the rectangle $\rs{G} = [-\nicefrac{3}{5},\nicefrac{1}{5}] \times [\nicefrac{1}{8},\nicefrac{3}{8}]$ since its vertices correspond to elliptic periodic orbits of period $40$ (those of the points $(\nicefrac{3}{20},\nicefrac{3}{32}),(\nicefrac{1}{20},\nicefrac{3}{32}), (\nicefrac{1}{20},\nicefrac{1}{32})$ and $(\nicefrac{3}{20},\nicefrac{1}{32})$). Figure \ref{f.high_period} shows the approximate rotation set $Q_{60,130}^*$ for $G$.
\begin{figure}
\caption{Approximation $Q_{60,130}^\ast$ for the rotation set of the map $G$.}
\label{f.high_period}
\end{figure}
\noindent Finally, we consider a slightly perturbed version $\bar{F} = R \circ F$ of the above examples by introducing a slight additional rotation \[ R:\mathbb{R}^2 \to \mathbb{R}^2, (x,y) \mapsto (x+r_1,y+r_2), r_1,r_2\in \mathbb{R} \ . \] In Table \ref{t.table} we collect the specific parameter values for both $k$ and $n$ and the perturbations $r_1$ and $r_2$, on which we base our approximations of the related rotation sets of the perturbed maps.
Although it is difficult to check rigorously, we expect that for small perturbations these modifications should not alter the rotation sets of the above examples due to the generic structural stability of the dynamics \cite{Passeggi2013RationalRotationSets}. Moreover, we expect that the vertices of the rotation sets are still realised by periodic orbits that lie close to the original ones. This fact could in principle be checked by a qualitative index argument. However, we refrain from going into detail and content ourselves with the numerical confirmation of the stability provided by Figure \ref{f.further_RS_pert}.
\begin{table}[!ht] \centering
\begin{tabular}{|c|c|c|c|c|c|c|} \hline
& $\bar{F}_{\frac{1}{2},\frac{1}{2}}$
& $\bar{G}$
& $\bar{F}_{1,\frac{1}{4}}$
& $\bar{F}_{\frac{3}{5},\frac{3}{5}}$
& $\bar{F}_{\frac{3}{4},1}$
& $\bar{F}_{1,1}$ \rule{0pt}{13pt} \\
\hline
$k$ &$50$ &$60$ &$16$ &$50$ &$45$ &$8$ \\
$n$ & $130$ &$130$ &$140$ &$100$ &$80$ &$100$ \\
$r_1$ & $0.012$&$0.008$& $0.012$&$0.01$ &$0.002$ &$0.022$ \\
$r_2$ & $0.014$&$0.001$&$0.002$ &$0.011$&$0.013$ &$0.015$ \\
\hline \end{tabular}
\rule{0pt}{7pt} \caption{Parameter values for the approximations shown in Figure \ref{f.further_RS_pert}.} \label{t.table} \end{table}
\begin{figure}\label{f.further_RS_pert}
\end{figure}
\section{Conclusion}
\noindent In conclusion, the set-oriented approach to the computation of rotation sets provides better and more stable results than conventional direct approaches. Moreover, it can at least partially be backed up by rigorous convergence results, even if the theoretical error estimates are not useful in practice. The much better performance of the algorithm for specific examples finds a possible explanation in the likely presence of a shadowing property, which can again be backed up by rigorous results.
What remains is to use this new numerical method in order to perform a systematic and detailed study of the behaviour and bifurcations of rotation sets in standard parameter families, as the one given by (\ref{e.standard_family}). Of course, it is highly likely that the performance of the algorithm becomes increasingly worse as bifurcation parameters are approached, at which the rotation set changes and structural stability and shadowing therefore have to break down. Therefore, it seems feasible to carry out such investigations in collaboration with experts on scientific computing and access to high-performance computing facilities, so that at least the limits of contemporary computing capacities can be exhausted to partially counter these effects. We leave this as a task for future research.
\begin{center}
{\bf Matlab codes will be made available on the authors' homepages.} \end{center}
\end{document} | arXiv |
\begin{document}
\title{On the Existential Fragments of Local First-Order Logics with Data}
\begin{abstract}We study first-order logic over unordered structures
whose elements carry a finite number of data values from an infinite domain which can be compared wrt. equality. As the satisfiability problem for this logic is undecidable in general, in a previous work, we have introduced a family of local fragments that restrict quantification to neighbourhoods of a given reference point. We provide here the precise complexity characterisation of the satisfiability problem for the existential fragments of this local logic depending on the number of data values carried by each element and the radius of the considered neighbourhoods. \end{abstract}
\section{Introduction}
First-order data logic has emerged to specify properties involving infinite data domains. Potential applications include XML reasoning and the specification of concurrent systems and distributed algorithms. The idea is to extend classic mathematical structures by a mapping that associates with every element of the universe a value from an infinite domain. When comparing data values only for equality, this view is equivalent to extending the underlying signature by a binary relation symbol whose interpretation is restricted to an equivalence relation.
Data logics over word and tree structures were studied in \cite{BojanczykMSS09,BojanczykDMSS11}. In particular, the authors showed that two-variable first-order logic on words has a decidable satisfiability problem. Other types of data logics allow \emph{two} data values to be associated with an element \cite{Kieronski05,KieronskiT09}, though they do not assume a linearly ordered or tree-like universe. Again, satisfiability turned out to be decidable for the two-variable fragment of first-order logic. Other notable extensions, either to multiple data values or to totally ordered data domains, include \cite{KaraSZ10,DeckerHLT14,ManuelZ13,Tan14}.
When considering an arbitrary number of first-order variables, which we do in this paper, the decidability frontier is quickly crossed without further constraints as soon as the number of allowed data in gretar then two \cite{Janiczak-Undecidability-fm53}. One of the restrictions we consider here is locality, an essential concept in first-order logic. It is well known that first-order logic is only able to express local properties: a first-order formula can always be written as a combination of properties of elements that have limited, i.e., bounded by a given radius, distance from some reference points \cite{Han65,Gai82}. In the presence of (several) data values, imposing a corresponding locality restriction on a logic can help ensuring decidability of its satisfiability problem.
In previous work, we considered a local fragment of first-order data logic over structures whose elements (i) are unordered (as opposed to, e.g., words or trees), and (ii) each carries two data values. We showed that the fragment has a decidable satisfiability problem when restricting local properties to radius~1, while it is undecidable for any radius greater than 1.
In the present paper, we study orthogonal local fragments where global quantification is restricted to being existential (while quantification inside a local property is still unrestricted). We obtain decidability for (i) radius 1 and an arbitrary number of data values, and for (ii) radius 2 and two data values. In all cases, we provide tight complexity upper and lower bounds. Moreover, these results mark the exact decidability frontier: satisfiability is undecidable as soon as we consider radius 3 in presence of two data values, or radius 2 together with three data values.
To give a possible application domain of our logic, consider distributed algorithms running on a cloud of processes. Those algorithms are usually designed to be correct independently of the number of processes executing them. Every process gets some inputs and produces some outputs, usually from an infinite domain. These may include process identifiers, nonces, etc. Inputs and outputs together determine the behavior of a distributed algorithm. A simple example is leader election, where every process gets a unique id, whereas the output should be the id of the elected leader and so be the same for all processes. To formalize correctness properties and to define the intended input-output relation, it is hence essential to have suitable data logics at hand.
\paragraph{Outline.} The paper is structured as follows. In Section~\ref{sec:preliminaries}, we recall important notions such as structures and first-order logic, and we introduce the local fragments considered in this paper. Section~\ref{sec:decidability} presents the decidable cases, whereas, in Section~\ref{sec:undecidability}, we show that all remaining cases lead to undecidability.
\noindent This work was partly supported by the project ANR FREDDA (ANR-17-CE40-0013).
\section{Structures and first-order logic} \label{sec:preliminaries}
\subsection{Data Structures}
We define here the class of models we are interested in. It consists of sets of nodes containing data values with the assumption that each node is labeled by a set of predicates and carries the same number of values. We consider hence $\Sigma$ a finite set of unary relation symbols (sometimes called unary predicates) and an integer $D \geq 0$. A \emph{$D$-data structure} over $\Sigma$ is a tuple $\AA=(A,(P_{\sigma})_{\sigma \in \Sigma},\f{1},\ldots,\f{D})$
(in the following, we simply write $(A,(P_{\sigma}),\f{1},\ldots,\f{D})$) where $A$ is a nonempty finite set, $P_\sigma \subseteq A$ for all $\sigma \in \Sigma$, and $\f{i}$s are mappings $A \to \N$. Intuitively $A$ represents the set of nodes and $f_i(a)$ is the $i$-th data value carried by $a$ for each node $a \in A$.
For $X\subseteq A$, we let $\Valuessub{\AA}{X} = \{\f{i}(a) \mid a \in X, i\in\{1,\ldots,D\}\}$.
The set of all $D$-data structures over $\Sigma$ is denoted by $\nData{D}{\Sigma}$.
While this representation is often very convenient to represent data values, a more standard way of representing mathematical structures is in terms of binary relations. For every $(i,j) \in \{1,\ldots,D\} \times \{1,\ldots,D\}$, the mappings $\f{1},\ldots,\f{D}$ determine a binary relation ${\relsaaord{i}{j}{\AA}} \subseteq A \times A$ as follows: $\relsaa{i}{j}{\AA}{a}{b}$ iff $\funct{i}(a) = \funct{j}(b)$. We may omit the superscript $\AA$ if it is clear from the context and if $D=1$, as there will be only one relation, we way may write $\sim$ for $\relsaord{1}{1}$.
\subsection{First-Order Logic}
Let $\mathcal{V} = \{x,y,\ldots\}$ be a countably infinite set of variables. The set $\ndFO{D}{\Sigma}$ of first-order formulas interpreted over $D$-data structures over $\Sigma$ is inductively given by the grammar $\varphi ::= \Pform{\sigma}{x} \mid \rels{i}{j}{x}{y} \mid x=y \mid \varphi \vee \varphi \mid \neg \varphi \mid \exists x.\varphi$, where $x$ and $y$ range over $\mathcal{V}$, $\sigma$ ranges over $\Sigma$, and $i,j \in \{1,\ldots,D\}$. We use standard abbreviations such as $\wedge$ for conjunction and $\Rightarrow$ for implication. We write $\varphi(x_1,\ldots,x_k)$ to indicate that the free variables of $\varphi$ are among $x_1,\ldots,x_k$. We call $\varphi$ a \emph{sentence} if it does not contain free variables.
For $\AA=(A,(P_{\sigma}),\f{1},\ldots,\f{D}) \in \nData{D}{\Sigma}$ and a formula $\varphi\in\ndFO{D}{\Sigma}$, the satisfaction relation $\AA \models_I \varphi$ is defined wrt.\ an interpretation function $I: \mathcal{V} \to A$. The purpose of $I$ is to assign an interpretation to every (free) variable of $\varphi$ so that $\varphi$ can be assigned a truth value. For $x \in \mathcal{V}$ and $a \in A$, the interpretation function $\Intrepl{x}{a}$ maps $x$ to $a$ and coincides with $I$ on all other variables. We then define: \begin{center} \begin{tabular}{ll} $\AA \models_I \Pform{\sigma}{x}$ if $I(x) \in P_{\sigma}$ & $\AA \models_I \varphi_1 \vee \varphi_2$ if $\AA \models_I \varphi_1$ or $\AA \models_I \varphi_2$\\
$\AA \models_I \rels{i}{j}{x}{y}$ if $\relsaa{i}{j}{\AA}{I(x)}{I(y)}$~~~ & $\AA \models_I \neg \varphi$ if $\AA \not\models_I \varphi$\\
$\AA \models_I x = y$ if $I(x) = I(y)$ & $\AA \models_I \exists x.\varphi$ if there is $a \in A$ s.t. $\AA \models_{\Intrepl{x}{a}} \varphi$ \end{tabular} \end{center}
Finally, for a data structure $\AA=(A,(P_{\sigma}),\f{1},\ldots,\f{D})$, a formula $\varphi(x_1,\ldots,x_k)$ and $a_1,\ldots,a_k\in A$, we write $\AA\models\varphi(a_1,\ldots a_k)$ if there exists an interpretation function $I$ such that $\AA\models_{I[x_1/a_1]\ldots[x_k/a_k]} \varphi$. In particular, for a sentence $\varphi$, we write $\AA\models\varphi$ if there exists an interpretation function $I$ such that $\AA \models_I \varphi$.
\begin{example}\label{ex:leader-election} Assume a unary predicate $\mathrm{leader} \in \Sigma$. The following formula from $\ndFO{2}{\Sigma}$ expresses correctness of a leader-election algorithm: (i)~there is a unique process that has been elected leader, and (ii)~all processes agree, in terms of their output values (their second data), on the identity (the first data) of the leader: $ \exists x. (\mathrm{leader}(x) \et \forall y. \big(\mathrm{leader}(y) \Rightarrow y=x)\big) \et \forall y. \exists x. (\mathrm{leader}(x) \et \rels{1}{2}{x}{y})$. \end{example}
We are interested here in the satisfiability problem for these logics. Let $\mathcal{F}$ denote a generic class of first-order formulas, parameterized by $\Sigma$ and $D$. In particular, for $\mathcal{F} = \textup{dFO}$, we have that $\mathcal{F}[\Sigma,D]$ is the class $\ndFO{D}{\Sigma}$. The satisfiability problem for $\mathcal{F}$ wrt.\ $D$-data structures is defined as follows:
\begin{center} \begin{decproblem}
\problemtitle{\nDataSat{\mathcal{F}}{D}}
\probleminput{A finite set $\Sigma$ and a sentence $\varphi \in \mathcal{F}[\Sigma,D]$.}
\problemquestion{Is there $\AA \in \nData{D}{\Sigma}$ such that $\AA \models \varphi$\,?} \end{decproblem} \end{center}
The following negative result (see \cite[Theorem~1]{Janiczak-Undecidability-fm53}) calls for restrictions of the general logic. \begin{theorem}\cite{Janiczak-Undecidability-fm53}\label{thm:undec-general}
The problem $\nDataSat{\textup{dFO}}{2}$ is undecidable, even when we require that $\Sigma = \emptyset$ and we do not use $\relsaord{1}{2}$ and $\relsaord{2}{1}$ in the considered formulas. \end{theorem}
\subsection{Local First-Order Logic and its existential fragment}
We are interested in logics combining the advantages of $\ndFO{D}{\Sigma}$,
while preserving decidability. With this in mind, we have introduced in \cite{bollig-local-fsttcs21}, for the case of two data values, a \emph{local} restriction, where the scope of quantification in the presence of free variables is restricted to the view of a given element. We present now the defintion of such restrictions adapted to the case of many data values.
First, the view of a node $a$ includes all elements whose distance to $a$ is bounded by a given radius. It is formalized using the notion of a Gaifman graph (for an introduction, see~\cite{Libkin04}). We use here a variant that is suitable for our setting and that we call \emph{data graph}. Given a data structure $\AA=(A,(P_{\sigma}),\f{1},\ldots,\f{D}) \in \nData{D}{\Sigma}$, we define its \emph{data graph} $\gaifmanish{\AA}=(\Vertex{\gaifmanish{\AA}},\Edge{\gaifmanish{\AA}})$ with set of vertices $\Vertex{\gaifmanish{\AA}} = A \times\{1,\ldots,D\}$ and set of edges $\Edge{\gaifmanish{\AA}} = \{ ((a,i),(b,j)) \in \Vertex{\gaifmanish{\AA}} \times \Vertex{\gaifmanish{\AA}} \mid a=b$ or $\rels{i}{j}{a}{b} \}$.
Figure \ref{fig:gaifman-a} provides an example of the graph $\gaifmanish{\AA}$ for a data structure with $2$ data values.
\newcommand{\selfconnectionright}[1]{\draw[<->, line width=0.7pt] (#1.one east) .. controls +(.4,0) and +(.4,0) .. (#1.two east);} \newcommand{\selfconnectionleft}[1]{\draw[<->, line width=0.7pt] (#1.one west) .. controls +(-.4,0) and +(-.4,0) .. (#1.two west);}
\begin{figure*}
\caption{A data structure $\AA$ and $\gaifmanish{\AA}$.}
\label{fig:gaifman-a}
\caption{$\vprojr{\AA}{a}{2}$: the $2$ view of $a$}
\label{fig:gaifman-b}
\caption{
}
\label{fig:gaifman}
\end{figure*}
We then define the distance $\distaa{(a,i)}{(b,j)}{\AA} \in \N \cup \{\infty\}$ between two elements $(a,i)$ and $(b,j)$ from $A \times\{1,\ldots,D\}$ as the length of the shortest path from $(a,i)$ to $(b,j)$ in $\gaifmanish{\AA}$.
For $a \in A$ and $r \in \N$, the \emph{radius-$r$-ball around} $a$ is the set $\Ball{r}{a}{\AA} = \{ (b,j)\in\Vertex{\gaifmanish{\AA}} \mid \distaa{(a,i)}{(b,j)}{\AA}\leq r $ for some $i \in \{1,\ldots,D\}\}$. This ball contains the elements of $\Vertex{\gaifmanish{\AA}}$ that can be reached from $(a,1),\ldots,(a,D)$ through a path of length at most $r$. On Figure~\ref{fig:gaifman-a} the blue nodes represent $\Ball{2}{a}{\AA}$.
We now define the $r$-view of an element $a$ in the $D$-data structure $\AA$. Intuitively it is a $D$-data structure with the same elements as $\AA$ but where the data values which are not in the radius-$r$-ball around $a$ are changed with new values all different one from each other. Let $f_{\textup{new}}: A \times \{1,\ldots,D\} \to \N \setminus \Values{\AA}$ be an injective mapping. The \emph{$r$-view of $a$ in $\AA$} is the structure $\vprojr{\AA}{a}{r} = (A,(P_{\sigma}),\f{1}',\ldots,\f{n}') \in \nData{D}{\Sigma}$ where its universe is the same as the one of $\AA$ and the unary predicates stay the same and $\funct{i}'(b)= \funct{i}(b)$ if $(b,i) \in\Ball{r}{a}{\AA}$, and $\funct{i}'(b)= f_{\textup{new}}((b,i))$ otherwise.
On Figure~\ref{fig:gaifman-b}, the structure $\vprojr{\AA}{a}{2}$ is depicted where the values of the red nodes, not belonging to $\Ball{2}{a}{\AA}$ have been replaced by fresh values not in $\{1,\ldots,7\}$.
We are now ready to present the logic $\rndFO{D}{\Sigma}{r}$, where $r \in \N$, interpreted over structures from $\nData{D}{\Sigma}$. It is given by the grammar \begin{align*}
\varphi ~&::=~ \locformr{\psi}{x}{r} \;\mid\; x=y \;\mid\; \exists x.\varphi \;\mid\; \varphi \vee \varphi \;\mid\; \neg \varphi \end{align*} where $\psi$ is a formula from $\ndFO{D}{\Sigma}$ with (at most) one free variable $x$. This logic uses the \emph{local modality} $\locformr{\psi}{x}{r}$ to specify that the formula $\psi$ should be interpreted over the $r$-view of the element associated to the variable $x$.
For $\AA \in \nData{D}{\Sigma}$ and an interpretation function $I$, we have indeed $\AA \models_I \locformr{\psi}{x}{r}$ iff $\vprojr{\AA}{I(x)}{r} \models_I \psi$.
\begin{example} We now illustrate what can be specified by formulas in the logic $\rndFO{2}{\Sigma}{1}$. We can rewrite the formula from Example~\ref{ex:leader-election} so that it falls into our fragment as follows: $\exists x. (\locformr{\mathrm{leader}(x)}{x}{1} \et \forall y. \linebreak[0](\locformr{\mathrm{leader}(y)}{y}{1} \Rightarrow x=y)) \et \forall y. \linebreak[0] \locformr{\exists x. \mathrm{leader}(x) \et \rels{2}{1}{y}{x} }{y}{1} $. The next formula specifies an algorithm in which all processes suggest a value and then choose a new value among those that have been suggested at least twice:
$\forall x.\locformr{\exists
y.\exists z. y \neq z \et \rels{2}{1}{x}{y} \et \rels{2}{1}{x}{z} }{x}{1} $. We can also specify partial renaming, i.e., two output values agree only if their input values are the same:
$\forall x.\locformr{\forall y.(\rels{2}{2}{x}{y}\donc\rels{1}{1}{x}{y}}{x}{1}$. Conversely, the formula $\forall x.\locformr{\forall y.(\rels{1}{1}{x}{y}\donc\rels{2}{2}{x}{y}}{x}{1}$ specifies partial fusion of equivalences classes.
\end{example}
In \cite{bollig-local-fsttcs21}, we have studied the decidability status of the satisfiability problem for $\rndFO{2}{\Sigma}{r}$ with $r \geq 1$ and we have shown that \nDataSat{$\rndFOr{2}$}{2} is undecidable and that \nDataSat{$\rndFOr{1}$}{2} is decidable when restricting the formulas (and the view of elements) to binary relations belonging to the set $\{\rels{1}{1}{}{},\rels{2}{2}{}{},\rels{1}{2}{}{}\}$. Whether \nDataSat{$\rndFOr{1}$}{2} in its full generality is decidable or not remains an open problem.
We wish to study here the existential fragment of $\rndFO{D}{\Sigma}{r}$ (with $r \geq 1$ and $D \geq 1$) and establish when its satisfiability problem is decidable. This fragment, denoted by $\eFO{D}{\Sigma}{r}$, is given by the grammar \[\varphi ~::=~ \locformr{\psi}{x}{r} \;\mid\; x=y \;\mid\; \neg(x=y) \;\mid\; \exists x.\varphi \;\mid\; \varphi\ou\varphi \;\mid\; \varphi\et\varphi \] where $\psi$ is a formula from $\ndFO{D}{\Sigma}$ with (at most) one free variable $x$. The quantifier free fragment $\qfFO{D}{\Sigma}{r}$ is defined by the grammar $\varphi ~::=~ \locformr{\psi}{x}{r} \;\mid\; x=y \;\mid\; \neg(x=y) \;\mid\; \varphi\ou\varphi \;\mid\; \varphi\et\varphi $.
\begin{remark}
Note that for both these fragments, we do not impose any restrictions on the use of quantifiers in the formula $\psi$ located under the local modality $\locformr{\psi}{x}{r}$. \end{remark}
\section{Decidability results} \label{sec:decidability} We show here decidability of $\nDataSat{\eFOr{2}}{2}$ and, for all $D \geq 0$, $\nDataSat{\eFOr{1}}{D}$.
\subsection{Preliminary results: 0 and 1 data values}
We introduce two preliminary results we shall use in this section to obtain new decidability results. First, note that formulas in $\ndFO{0}{\Sigma}$ (i.e. where no data is considered) correspond to first order logic formulas with a set of predicates and equality test as a unique relation. As mentioned in Chapter 6.2.1 of \cite{borger-classical-springer97}, these formulas belong to the \emph{L\"owenheim class with equality} also called as the relational monadic formulas, and their satisfiability problem is in \textsc{NEXP}. Furthermore, thanks to \cite{etessami-first-ic02} (Theorem 11), we know that this latter problem is \textsc{NEXP}-hard even if one considers formulas which use only two variables.
\begin{theorem}\label{thm:0fo}
$\nDataSat{\textup{dFO}}{0}$ is \textsc{NEXP}-complete. \end{theorem}
In \cite{Mundhenk09}, the authors study the satisfiability problem for Hybrid logic over Kripke structures where the transition relation is an equivalence relation, and they show that it is \textsc{N2EXP}-complete. Furthermore in \cite{Fitting12}, it is shown that Hybrid logic can be translated to first-order logic in polynomial time and this holds as well for the converse translation. Since $1$-data structures can be interpreted as Kripke structures with one equivalence relation, altogether this allows us to obtain the following preliminary result about the satisfiability problem of $\ndFO{1}{\Sigma}$.
\begin{theorem}\label{thm:1fo}
$\nDataSat{\textup{dFO}}{1}$ is \textsc{N2EXP}-complete. \end{theorem}
\subsection{Two data values and balls of radius 2}
In this section, we prove that the satisfiability problem for the existential fragment of local first-order logic with two data values and balls of radius two is decidable.
To obtain this result we provide a reduction to the satisfiability problem for first-order logic over $1$-data structures. Our reduction is based on the following intuition. Consider a $2$-data structure $\AA=(A,(P_{\sigma}),\f{1},\f{2}) \in \nData{2}{\Sigma}$ and an element $a \in A$. If we take an element $b$ in $\Ball{2}{a}{\AA}$, the radius-2-ball around $a$, we know that either $\f{1}(b)$ or $\f{2}(b)$ is a common value with $a$. In fact, if $b$ is at distance $1$ of $a$, this holds by definition and if $b$ is at distance $2$ then $b$ shares an element with $c$ at distance $1$ of $a$ and this element has to be shared with $a$ as well so $b$ ends to be at distance $1$ of $a$. The trick consists then in using extra-labels for elements sharing a value with $a$ that can be forgotten and to keep only the value of $b$ not present in $a$, this construction leading to a $1$-data structure. It remains to show that we can ensure that a $1$-data structure is the fruit of this construction in a formula of $\ndFO{1}{\Sigma'}$ (where $\Sigma'$ is obtained from $\Sigma$ by adding extra predicates).\\
The first step for our reduction consists in providing a characterisation for the elements located in the radius-1-ball and the radius-2-ball around another element.
\begin{lemma}\label{lem:shape-balls}
Let $\AA=(A,(P_{\sigma}),\f{1},\f{2}) \in
\nData{2}{\Sigma}$ and $a,b\in A$ and $j \in \{1,2\}$. We have: \begin{enumerate} \item $(b,j)\in \Ball{1}{a}{\AA}$ iff there is $i\in \{1,2\}$ such that $\relsaa{i}{j}{\AA}{a}{b}$. \item $(b,j)\in \Ball{2}{a}{\AA}$ iff there exists $i,k\in \{1,2\}$ such that $\relsaa{i}{k}{\AA}{a}{b}$. \end{enumerate} \end{lemma}
\begin{proof} We show both statements: \begin{enumerate} \item Since $(b,j)\in \Ball{1}{a}{\AA}$, by definition we have either $b=a$ and in that case $\relsaa{j}{j}{\AA}{a}{b}$ holds, or $b \neq a$ and necessarily there exists $i\in \{1,2\}$ such that $\relsaa{i}{j}{\AA}{a}{b}$.
\item First, if there exists $i,k\in \{1,2\}$ such that $\relsaa{i}{k}{\AA}{a}{b}$, then $(b,k)\in \Ball{1}{a}{\AA}$ and $(b,j)\in \Ball{2}{a}{\AA}$ by definition. Assume now that $(b,j)\in \Ball{2}{a}{\AA}$. Hence there exists $i\in \{1,2\}$ such that $\distaa{(a,i)}{(b,j)}{\AA}\leq 2$. We perform a case analysis on the value of $\distaa{(a,i)}{(b,j)}{\AA}$. \begin{itemize}
\item \textbf{Case $\distaa{(a,i)}{(b,j)}{\AA}=0$}. In that case
$a=b$ and $i=j$ and we have $\relsaa{i}{i}{\AA}{a}{b}$.
\item \textbf{Case $\distaa{(a,i)}{(b,j)}{\AA}=1$}. In that case,
$((a,i),(b,j))$ is an edge in the data graph $\gaifmanish{\AA}$
of $\AA$ which means that $\relsaa{i}{j}{\AA}{a}{b}$ holds.
\item \textbf{Case $\distaa{(a,i)}{(b,j)}{\AA}=2$}. Note that
we have by definition $a \neq b$. Furthermore, in that case, there is
$(c,k)\in A\times\{1,2\}$ such that $((a,i),(c,k))$ and
$((c,k),(b,j))$ are edges in $\gaifmanish{\AA}$. If $c\neq a$ and
$c\neq b$, this implies that $\relsaa{i}{k}{\AA}{a}{c}$ and
$\relsaa{k}{j}{\AA}{c}{b}$, so $\relsaa{i}{j}{\AA}{a}{b}$ and
$\distaa{(a,i)}{(b,j)}{\AA}=1$ which is a contradiction.
If $c=a$ and $c\neq b$, this implies that $\relsaa{k}{j}{\AA}{a}{b}$.
If $c\neq a$ and $c = b$, this implies that $\relsaa{i}{k}{\AA}{a}{b}$. \end{itemize} \end{enumerate} \end{proof}
We consider a formula $\varphi=\exists x_1\ldots\exists x_n.\varphi_{qf}(x_1,\ldots,x_n)$ of $\eFO{2}{\Sigma}{2}$ in prenex normal form, i.e., such that $\varphi_{qf}(x_1,\ldots,x_n)\in\qfFO{2}{\Sigma}{2}$. We know that there is a structure $\AA=(A,(P_{\sigma})_{\sigma \in \Sigma},\linebreak[0]\f{1},\f{2})$ in $\nData{2}{\Sigma}$ such that $\AA\models\varphi$ if and only if there are $a_1,\ldots,a_n \in A $ such that $\AA\models\varphi_{qf}(a_1,\ldots,a_n)$.
Let $\AA=(A,(P_{\sigma})_{\sigma \in \Sigma},\f{1},\f{2})$ be a structure in $\nData{2}{\Sigma}$ and a tuple $\tuple{a} = (a_1,\ldots,a_n)$ of elements in $A^n$. We shall present the construction of a $1$-data structure $\sem{\AA}_{\tuple{a}}$ in $\nData{1}{\Unary'}$ (with $\Sigma \subseteq \Unary'$) with the same set of nodes as $\AA$, but where each node carries a single data value. In order to retrieve the data relations that hold in $\AA$ while reasoning over $\sem{\AA}_{\tuple{a}}$, we introduce extra-predicates in $\Unary'$ to establish whether a node shares a common value with one of the nodes among $a_1,\ldots,a_n$ in $\AA$.
\begin{figure*}
\caption{A data structure $\AA$ and $\gaifmanish{\AA}$.}
\label{fig:abstract-a}
\caption{$\sem{\AA}_{(a)}$.}
\label{fig:abstract-b}
\caption{ }
\label{fig:abstract}
\end{figure*}
We now explain formally how we build $\sem{\AA}_{\tuple{a}}$. Let $\Udeci{n}=\{\udd{p}{i}{j}\mid p\in\{1,\ldots,n\}, i,j\in\{1,2\}\}$ be a set of new unary predicates and $\Unary' = \Sigma \cup \Udeci{n}$. For every element $b\in A$, the predicates in $\Udeci{n}$ are used to keep track of the relation between the data values of $b$ and the one of $a_1,\ldots,a_n$ in $\AA$. Formally, we define $\uP{\udd{p}{i}{j}}=\{b\in A\mid \AA\models \rels{i}{j}{a_p}{b}\}$. We now define a data function $f:A\to \N$. We recall for this matter that $\Valuessub{\AA}{\tuple{a}} = \{f_1(a_1),f_2(a_1),\ldots,f_1(a_n),f_2(a_n)\}$ and let $f_{\textup{new}}:A\to\N\setminus \Values{\AA}$ be an injection. For every $b \in A$, we set: \[
f(b) = \begin{cases}
f_2(b) \text{ if } f_1(b)\in \Valuessub{\AA}{\tuple{a}} \text{ and } f_2(b)\notin \Valuessub{\AA}{\tuple{a}}\\
f_1(b) \text{ if } f_1(b)\notin \Valuessub{\AA}{\tuple{a}} \text{ and } f_2(b)\in \Valuessub{\AA}{\tuple{a}}\\
f_{\textup{new}}(b) \text{ otherwise}
\end{cases} \] Hence depending if $f_1(b)$ or $f_2(b)$ is in $\Valuessub{\AA}{\tuple{a}}$, it splits the elements of $\AA$ in four categories. If $f_1(b)$ and $f_2(b)$ are in $\Valuessub{\AA}{\tuple{a}}$, the predicates in $\Udeci{n}$ allow us to retrieve all the data values of $b$. Given $j\in\{1,2\}$, if $f_j(b)$ is in $\Valuessub{\AA}{\tuple{a}}$ but $f_{3-j}(b)$ is not, the new predicates will give us the $j$-th data value of $b$ and we have to keep track of the $(3-j)$-th one, so we save it in $f(b)$. Lastly, if neither $f_1(b)$ nor $f_2(b)$ is in $\Valuessub{\AA}{\tuple{a}}$, we will never be able to see the data values of $b$ in $\varphi_{q_f}$ (thanks to Lemma \ref{lem:shape-balls}), so they do not matter to us. Finally, we have $\sem{\AA}_{\tuple{a}} = (A, (\uP{\sigma})_{\sigma\in\Unary'}, f) $. Figure \ref{fig:abstract-b} provides an example of $\Valuessub{\AA}{\tuple{a}}$ for the data structures depicted on Figure \ref{fig:abstract-a} and $\tuple{a}=(a)$.
The next lemma formalizes the connection existing between $\AA$ and $\sem{\AA}_{\tuple{a}}$ with $\tuple{a} = (a_1,\ldots,a_n)$.
\begin{lemma}\label{lem:r2dv2-semantique}
Let $b,c\in A$ and $j,k\in\{1,2\}$ and $p\in\{1,\ldots,n\}$. The following statements then hold.
\begin{enumerate}
\item If $(b,j)\in\Ball{1}{a_p}{\AA}$ and $(c,k)\in\Ball{1}{a_p}{\AA}$ then $\vprojr{\AA}{a_p}{2}\models\rels{j}{k}{b}{c}$ iff there is $i\in\{1,2\}$ s.t. $b \in \uP{\udd{p}{i}{j}}$ and $c \in \uP{\udd{p}{i}{k}}$.
\item If $(b,j)\in\Ball{2}{a_p}{\AA}\setminus\Ball{1}{a_p}{\AA}$ and $(c,k)\in\Ball{1}{a_p}{\AA}$ then $\vprojr{\AA}{a_p}{2}\nvDash\rels{j}{k}{b}{c}$
\item If $(b,j),(c,k) \in\Ball{2}{a_p}{\AA}\setminus\Ball{1}{a_p}{\AA}$ then $\vprojr{\AA}{a_p}{2}\models\rels{j}{k}{b}{c}$ iff either $\relsaa{1}{1}{\sem{\AA}_{\tuple{a}}}{b}{c}$ or there exists $p' \in \{1,\ldots,n\}$ and $\ell \in \{1,2\}$ such that $b \in \uP{\udd{p'}{\ell}{j}}$ and $c \in \uP{\udd{p'}{\ell}{k}}$ .
\item If $(b,j)\notin\Ball{2}{a_p}{\AA}$ and $(c,k)\in\Ball{2}{a_p}{\AA}$ then $\vprojr{\AA}{a_p}{2}\nvDash\rels{j}{k}{b}{c}$
\item If $(b,j)\notin\Ball{2}{a_p}{\AA}$ and $(c,k)\notin\Ball{2}{a_p}{\AA}$ then $\vprojr{\AA}{a_p}{2}\models\rels{j}{k}{b}{c}$ iff $b=c$ and $j=k$.
\end{enumerate} \end{lemma}
\begin{proof}
We suppose that $\vprojr{\AA}{a_p}{2} = (A,(\uP{\sigma})_\sigma,f^p_1,f^p_2)$.
\begin{enumerate}
\item Assume that $(b,j)\in\Ball{1}{a_p}{\AA}$ and $(c,k)\in\Ball{1}{a_p}{\AA}$.
It implies that $f^p_j(b)=f_j(b)$ and $f^p_k(c)=f_k(c)$.
Then assume that $\vprojr{\AA}{a_p}{2}\models\rels{j}{k}{b}{c}$.
As $(b,j)\in\Ball{1}{a_p}{\AA}$, thanks to Lemma \ref{lem:shape-balls}.1 it means that there is a $i\in\{1,2\}$ such that $\relsaa{i}{j}{\AA}{a_p}{b}$.
So we have $f_k(c)=f^p_k(c)=f^p_j(b)=f_j(b)=f_i(a_p)$, that is $\relsaa{i}{k}{\AA}{a_p}{c}$. Hence by definition, $b \in \uP{\udd{p}{i}{j}}$ and $c \in \uP{\udd{p}{i}{k}}$.
Conversely, let $i\in\{1,2\}$ such that $b \in \uP{\udd{p}{i}{j}}$ and $c \in \uP{\udd{p}{i}{k}}$. This means that $\relsaa{i}{j}{\AA}{a_p}{b}$ and $\relsaa{i}{k}{\AA}{a_p}{c}$.
So $f^p_j(b)=f_j(b)=f_i(a_p)=f_k(c)=f^p_k(c)$, that is $\vprojr{\AA}{a_p}{2}\models\rels{j}{k}{b}{c}$.
\item Assume that $(b,j)\in\Ball{2}{a_p}{\AA}\setminus\Ball{1}{a_p}{\AA}$ and $(c,k)\in\Ball{1}{a_p}{\AA}$.
It implies that $f^p_j(b)=f_j(b)$ and $f^p_k(c)=f_k(c)$.
Thanks to Lemma \ref{lem:shape-balls}.1, $(c,k)\in\Ball{1}{a_p}{\AA}$ implies that $f_k(c)\in\{f_1(a_p),f_2(a_p)\}$ and $(b,j)\notin\Ball{1}{a_p}{\AA}$ implies that $f_j(b)\notin\{f_1(a_p),f_2(a_p)\}$.
So $\vprojr{\AA}{a_p}{2}\not \models\rels{j}{k}{b}{c}$.
\item Assume that $(b,j), (c,k) \in\Ball{2}{a_p}{\AA}\setminus\Ball{1}{a_p}{\AA}$. As previously, we have that $f_j(b)\notin\{f_1(a_p),f_2(a_p)\}$ and $f_k(c)\notin\{f_1(a_p),f_2(a_p)\}$, and thanks to Lemma \ref{lem:shape-balls}.2, we have $f_{3-j}(b) \in \{f_1(a_p),f_2(a_p)\}$ and $f_{3-k}(b) \in \{f_1(a_p),f_2(a_p)\}$. There is then two cases:
\begin{itemize}
\item Suppose there does not exists $p' \in \{1,\ldots,n\}$ such that $f_{j}(b) \in \{f_1(a_{p'}),f_2(a_{p'})\}$ .This allows us to deduce that $f^p_j(b)=f_j(b)=f(b)$ and $f^p_k(c)=f_k(c)$. If $\vprojr{\AA}{a_p}{2}\models\rels{j}{k}{b}{c}$, then necessarily there does not exists $p' \in \{1,\ldots,n\}$ such that $f_{k}(c) \in \{f_1(a_{p'}),f_2(a_{p'})\}$ so we have $f^p_k(c)=f_k(c)=f(c)$ and $f(b)=f(c)$, consequently $\relsaa{1}{1}{\sem{\AA}_{\tuple{a}}}{b}{c}$. Similarly assume that $\relsaa{1}{1}{\sem{\AA}_{\tuple{a}}}{b}{c}$, this means that $f(b)=f(c)$ and either $b=c$ and $k=j$ or $b \neq c$ and by injectivity of $f$,we have $f_j(b)=f(b)=f_k(c)$. This allows us to deduce that $\vprojr{\AA}{a_p}{2}\models\rels{j}{k}{b}{c}$.
\item If there exists $p' \in \{1,\ldots,n\}$ such that $f_{j}(b) = f_\ell(a_{p'})$ for some $\ell \in \{1,2\}$. Then we have $b \in \uP{\udd{p'}{\ell}{j}}$. Consequently, we have $\vprojr{\AA}{a_p}{2}\models\rels{j}{k}{b}{c}$ iff $c \in \uP{\udd{p'}{\ell}{k}}$.
\end{itemize}
\item We prove the case 4 and 5 at the same time.
Assume that $(b,j)\notin\Ball{2}{a_p}{\AA}$.
It means that in order to have $f^p_j(b)=f^p_k(c)$, we must have $(b,j)=(c,k)$.
So if $(c,k)\in\Ball{2}{a_p}{\AA}$, we can not have $\vprojr{\AA}{a_p}{2}\models\rels{j}{k}{b}{c}$ which ends case 4.
And if $(c,k)\notin\Ball{2}{a_p}{\AA}$, we have that $\vprojr{\AA}{a_p}{2}\models\rels{j}{k}{b}{c}$ iff $b=c$ and $j=k$.
\end{enumerate} \end{proof}
We shall now see how we translate the formula $\varphi_{qf}(x_1,\ldots,x_n)$ into a formula $\phit{\varphi_{qf}}(x_1,\ldots,x_n)$ in $\ndFO{1}{\Unary'}$ such that $\AA$ satisfies $\varphi_{qf}(a_1,\ldots,a_n)$ if, and only if, $\sem{\AA}_{\tuple{a}}$ satisfies $\phit{\varphi_{qf}}(a_1,\ldots,a_n)$. Thanks to the previous lemma we know that if $\vprojr{\AA}{a_p}{2}\models\rels{j}{k}{b}{c}$ then $(b,j)$ and $(c,k)$ must belong to the same set among $\Ball{1}{a_p}{\AA}$, $\Ball{2}{a_p}{\AA}\setminus\Ball{1}{a_p}{\AA}$ and $\comp{\Ball{2}{a_p}{\AA}}$ and we can test in $\sem{\AA}_{\tuple{a}}$ whether $(b,j)$ is a member of $\Ball{1}{a_p}{\AA}$ or $\Ball{2}{a_p}{\AA}$. Indeed, thanks to Lemmas \ref{lem:shape-balls}.1 and \ref{lem:shape-balls}.2, we have $(b,j) \in \Ball{1}{a_p}{\AA}$ iff $b\in\bigcup_{i=1,2}\uP{\udd{p}{i}{j}}$ and $(b,j) \in \Ball{2}{a_p}{\AA}$ iff $b\in\bigcup_{i=1,2}^{j'=1,2} \uP{\udd{p}{i}{j'}}$. This reasoning leads to the following formulas in $\ndFO{1}{\Unary'}$ with $p \in \{1,\ldots,n\}$ and $j \in \{1,2\}$: \begin{itemize} \item $\phiBun{j}(y) := \udd{p}{1}{j}(y) \ou \udd{p}{2}{j}(y)$ to test if the $j$-th field of an element belongs to $\Ball{1}{a_p}{\AA}$ \item $\phiBdeux(y) := \phiBun{1}(y) \ou \phiBun{2}(y)$ to test if a field of an element belongs to $\Ball{2}{a_p}{\AA}$ \item $\phiBdsu{j}(y) := \phiBdeux(y) \et \neg\phiBun{j}(y)$ to test that the $j$-th field of an element belongs to $\Ball{2}{a_p}{\AA}\setminus\Ball{1}{a_p}{\AA}$ \end{itemize}
We shall now present how we use these formulas to translate atomic formulas of the form $\rels{j}{k}{y}{z}$ under some $\locformr{-}{x_p}{2}$. For this matter, we rely on the three following formulas of $\ndFO{1}{\Unary'}$: \begin{itemize} \item The first formula asks for $(y,j)$ and $(z,k)$ to be in $\Ball{1}{a_p}{1}$ (where here we abuse notations, using variables for the elements they represent) and for these two data values to coincide with one data value of $a_p$, it corresponds to Lemma \ref{lem:r2dv2-semantique}.1:
$$
\varphi_{j,k,a_p}^{r=1}(y,z) := \phiBun{j}(y) \et \phiBun{k}(z) \et \Ou_ {i=1,2}\udd{p}{i}{j}(y)\et\udd{p}{i}{k}(z)
$$ \item The second formula asks for $(y,j)$ and $(z,k)$ to be in $\Ball{2}{a_p}{\AA}\setminus\Ball{1}{a_p}{\AA}$ and checks either whether the data values of $y$ and $z$ in $\sem{\AA}_{\tuple{a}}$ are equal or whether there exist $p'$ and $\ell$ such that $y$ belongs to $\udd{p'}{\ell}{j}(y)$ and $z$ belongs to $\udd{p'}{\ell}{k}(z)$, it corresponds to Lemma \ref{lem:r2dv2-semantique}.3:
$$
\varphi_{j,k,a_p}^{r=2}(y,z) := \phiBdsu{j}(y) \et \phiBdsu{k}(z) \et \big (y\sim z \ou\big(\Ou^n_{p'=1}\Ou^2_ {\ell=1}\udd{p'}{\ell}{j}(y)\et\udd{p'}{\ell}{k}(z)\big)\big)
$$ \item The third formula asks for $(y,j)$ and $(z,k)$ to not belong to $\Ball{2}{a_p}{\AA}$ and for $y=z$, it corresponds to Lemma \ref{lem:r2dv2-semantique}.5:
$$
\varphi_{j,k,a_p}^{r>2}(y,z) := \begin{cases}
\neg \phiBdeux(y) \et \neg\phiBdeux(z) \et y=z &\text{ if } j=k \\
\bot &\text{ otherwise}
\end{cases}
$$ \end{itemize}
Finally, here is the inductive definition of the translation $\T{-}$ which uses sub transformations $\Tp{-}$ in order to remember the centre of the ball and leads to the construction of $\phit{\varphi_{qf}}(x_1,\ldots,x_n)$: \[ \begin{array}{rcl}
\T{\varphi\ou\varphi'} &=& \T{\varphi} \ou \T{\varphi'}\\
\T{x_p=x_p'} &=& x_p=x_p' \\
\T{\neg\varphi} &=& \neg\T{\varphi} \\
\T{\locformr{\psi}{x_p}{2}} &=& \Tp{\psi} \\
\Tp{\rels{j}{k}{y}{z}} &=&\varphi_{j,k,a_p}^{r=1}(y,z) \ou \varphi_{j,k,a_p}^{r=2}(y,z) \ou \varphi_{j,k,a_p}^{r>2}(y,z)\\
\Tp{\sigma(x)} &=& \sigma(x) \\
\Tp{x=y} &=& x=y \\
\Tp{\varphi\ou\varphi'}&=& \Tp{\varphi} \ou \Tp{\varphi'} \\
\Tp{\neg\varphi} &=& \neg\Tp{\varphi}\\
\Tp{\exists x. \varphi} &=& \exists x.\Tp{\varphi}\\
\end{array}\]
\begin{lemma} \label{lem:correct}
We have $\AA\models\varphi_{qf}(\tuple{a})$ iff $\sem{\AA}_{\tuple{a}}\models\phit{\varphi_{qf}}(\tuple{a})$. \end{lemma} \begin{proof}
Because of the inductive definition of $\T{\varphi}$ and that only the atomic formulas $\rels{j}{k}{y}{z}$ change, we only have to prove that given $b,c\in A$, we have $\vprojr{\AA}{a_p}{2}\models\rels{j}{k}{b}{c}$ iff $\sem{\AA}_{\tuple{a}}\models \Tp{\rels{j}{k}{y}{z}}(b,c)$.
We first suppose that $\vprojr{\AA}{a_p}{2}\models\rels{j}{k}{b}{c}$.
Using Lemma \ref{lem:r2dv2-semantique}, it implies that $(b,j)$ and $(c,k)$ belong to same set between $\Ball{1}{a_p}{\AA}$, $\Ball{2}{a_p}{\AA} \setminus \Ball{1}{a_p}{\AA}$ and $\comp{\Ball{2}{a_p}{\AA}}$. We proceed by a case analysis.
\begin{itemize}
\item If $(b,j),(c,k)\in\Ball{1}{a_p}{\AA}$ then by lemma \ref{lem:r2dv2-semantique}.1 we have that $\sem{\AA}_{\tuple{a}}\models\varphi_{j,k,a_p}^{r=1}(b,c)$ and thus $\sem{\AA}_{\tuple{a}}\models \Tp{\rels{j}{k}{y}{z}}(b,c)$.
\item If $(b,j),(c,k)\in\Ball{2}{a_p}{\AA} \setminus \Ball{1}{a_p}{\AA}$ then by lemma \ref{lem:r2dv2-semantique}.3 we have that $\sem{\AA}_{\tuple{a}}\models\varphi_{j,k,a_p}^{r=2}(b,c)$ and thus $\sem{\AA}_{\tuple{a}}\models \Tp{\rels{j}{k}{y}{z}}(b,c)$.
\item If $(b,j),(c,k)\in\comp{\Ball{2}{a_p}{\AA}}$ then by lemma \ref{lem:r2dv2-semantique}.5 we have that $\sem{\AA}_{\tuple{a}}\models\varphi_{j,k,a_p}^{r>2}(b,c)$ and thus $\sem{\AA}_{\tuple{a}}\models \Tp{\rels{j}{k}{y}{z}}(b,c)$.
\end{itemize}
We now suppose that $\sem{\AA}_{\tuple{a}}\models \Tp{\rels{j}{k}{y}{z}}(b,c)$.
It means that $\sem{\AA}_{\tuple{a}}$ satisfies at least $\varphi_{j,k,a_p}^{r=1}(b,c)$, $\varphi_{j,k,a_p}^{r=2}(b,c)$ or $\varphi_{j,k,a_p}^{r>2}(b,c)$.
If $\sem{\AA}_{\tuple{a}}\models\varphi_{j,k,a_p}^{r=1}(b,c)$, it implies that $(b,j)$ and $(c,k)$ are in $\Ball{1}{a_p}{\AA}$, and we can then apply lemma \ref{lem:r2dv2-semantique}.1 to deduce that $\vprojr{\AA}{a_p}{2}\models\rels{j}{k}{b}{c}$.
If $\sem{\AA}_{\tuple{a}}\models\varphi_{j,k,a_p}^{r=2}(b,c)$, it implies that $(b,j)$ and $(c,k)$ are in $\Ball{2}{a_p}{\AA} \setminus \Ball{1}{a_p}{\AA}$, and we can then apply lemma \ref{lem:r2dv2-semantique}.3 to deduce that $\vprojr{\AA}{a_p}{2}\models\rels{j}{k}{b}{c}$.
If $\sem{\AA}_{\tuple{a}}\models\varphi_{j,k,a_p}^{r>2}(b,c)$, it implies that $(b,j)$ and $(c,k)$ are in $\comp{\Ball{2}{a_p}{\AA}}$, and we can then apply lemma \ref{lem:r2dv2-semantique}.5 to deduce that $\vprojr{\AA}{a_p}{2}\models\rels{j}{k}{b}{c}$.
\end{proof}
To provide a reduction from $\nDataSat{\eFOr{2}}{2}$ to $\nDataSat{\textup{dFO}}{1}$, having the formula $\phit{\varphi_{qf}}(x_1,\ldots,x_n)$ is not enough because to use the result of the previous Lemma, we need to ensure that there exists a model $\BB$ and a tuple of elements $(a_1,\ldots,a_n)$ such that $\BB \models\ \phit{\varphi_{qf}}(a_1,\ldots,a_n)$ and as well that there exists $\AA\in \nData{2}{\Sigma}$ such that $ \BB = \sem{\AA}_{\tuple{a}}$. We explain now how we can ensure this last point.
Now, we want to characterize the structures of the form $\sem{\AA}_{\tuple{a}}$. Given $\BB = (A,(\uP{\sigma})_{\sigma\in\Unary'},f)\in\nData{1}{\Unary'}$ and $\tuple{a}\in A$, we say that $(\BB,\tuple{a})$ is \emph{well formed} iff there exists a structure $\AA\in \nData{2}{\Sigma}$ such that $ \BB = \sem{\AA}_{\tuple{a}}$. Hence $(\BB,\tuple{a})$ is \emph{well formed} iff there exist two functions $f_1,f_2:A\to\N$ such that $\sem{\AA}_{\tuple{a}}=\sem{(A,(\uP{\sigma})_{\sigma\in\Sigma}, f_1,f_2)}_{\tuple{a}}$. We state three properties on $(\BB,\tuple{a})$, and we will show that they characterize being well formed. \begin{enumerate}
\item (Transitivity) For all $b,c\in A$, $p,q \in\{1,\ldots,n\}$,
$i,j,k,\ell \in\{1,2\}$ if $b\in\uP{\udd{p}{i}{j}}$, $c\in\uP{\udd{p}{i}{\ell}}$ and $b\in\uP{\udd{q}{k}{j}}$ then $c\in\uP{\udd{q}{k}{\ell}}$.
\item (Reflexivity) For all $p$ and $i$, we have $a_p\in\uP{\udd{p}{i}{i}}$
\item (Uniqueness) For all $b\in A$, if $b\in\bigcap_{j=1,2}\bigcup_{p=1,\ldots,n}^{i=1,2} \uP{\udd{p}{i}{j}}$ or $b\notin\bigcup_{j=1,2}\bigcup_{p=1,\ldots,n}^{i=1,2} \uP{\udd{p}{i}{j}}$ then for any $c\in B$ such that $f(c)=f(b)$ we have $c=b$. \end{enumerate} Each property can be expressed by a first order logic formula, which we respectively name $\varphi_{\mathit{tran}}$, $\varphi_{\mathit{refl}}$ and $\varphi_{\mathit{uniq}}$ and we denote by $\varphi_{\mathit{wf}}$ their conjunction: $$
\begin{array}{ll} \varphi_{\mathit{tran}} &= \forall y \forall z.\Et_{p,q=1}^{n}\Et_{i,j,k,\ell=1}^2 \Big(\udd{p}{i}{j}(y) \et \udd{p}{i}{\ell}(z) \et \udd{q}{k}{j}(y) \donc \udd{q}{k}{\ell}(z)\Big) \\ \varphi_{\mathit{refl}}(x_1,\ldots,x_n) &=\Et_{p=1}^n\Et_{i=1}^2 \udd{p}{i}{i}(x_p) \\ \varphi_{\mathit{uniq}} &= \forall y. \Big(\Et_{j=1}^2 \Ou^n_{p=1} \Ou_{i=1}^2 \udd{p}{i}{j}(y) \ou \Et_{j=1}^2 \Et^n_{p=1}\Et^2_{i=1} \neg\udd{p}{i}{j}(y)\Big) \donc (\forall z. y\sim z \donc y=z)\\ \varphi_{\mathit{wf}}(x_1,\ldots,x_n) &=\varphi_{\mathit{tran}} \et \varphi_{\mathit{refl}}(x_1,\ldots,x_n) \et
\varphi_{\mathit{uniq}}
\end{array}
$$
The next lemma expresses that the formula $\varphi_{\mathit{wf}}$ allows to characterise precisely the $1$-data structures in $\nData{1}{\Unary'}$ which are well-formed.
\begin{lemma}\label{lem:well-formed}
Let $\BB\in\nData{1}{\Unary'}$ and $a_1,\ldots,a_n$ elements of
$\BB$, then $(\BB,\tuple{a})$ is well formed iff $\BB\models\varphi_{\mathit{wf}}(\tuple{a})$.
\end{lemma}
\begin{proof}
First, if $(\BB,\tuple{a})$ is well formed, then there there exists $\AA\in \nData{2}{\Sigma}$ such that $ \BB = \sem{\AA}_{\tuple{a}}$ and by construction we have $\sem{\AA}_{\tuple{a}} \models\varphi_{\mathit{wf}}(\tuple{a})$. We now suppose that $\BB=(A,(\uP{\sigma})_{\sigma\in\Unary'},f)$ and $\BB\models\varphi_{\mathit{wf}}(\tuple{a})$.
In order to define the functions $f_1,f_2:A\to\N$, we need
to introduce some objects.
We first define a function $g :
\{1,\ldots,n\} \times \{1,2\} \to \N\setminus \im{f}$ (where
$\im{f}$ is the image of $f$ in $\BB$) which
verifies the following properties:
\begin{itemize}
\item for all $p \in \{1,\ldots,n\}$ and $i \in \{1,2\}$, we
have
$a_p \in \uP{\udd{p}{i}{3-i}} $ iff $g(p,1)=g(p,2)$;
\item for all $p, q \in \{1,\ldots,n\}$ and $i,j \in \{1,2\}$,
we have $a_q \in \uP{\udd{p}{i}{j}} $ iff $g(p,i)=g(q,j)$.
\end{itemize} We use this function to fix the two data values carried by the elements in $\{a_1,\ldots,a_m\}$. We now explain why this function is well founded, it is due to the fact that $\BB\models\varphi_{\mathit{tran}} \et \varphi_{\mathit{refl}}(a_1,\ldots,a_n)$. In fact, since $\BB \models \varphi_{\mathit{refl}}(a_1,\ldots,a_n)$, we have for all $p \in \{1,\ldots,n\}$ and $i \in \{1,2\}$, $a_p \in \uP{\udd{p}{i}{i}} $. Furthermore if $a_p \in \uP{\udd{p}{i}{j}}$ then $a_p \in \uP{\udd{p}{j}{i}}$ thanks to the formula $\varphi_{\mathit{tran}}$; indeed since we have $a_p \in \uP{\udd{p}{i}{j}}$ and $a_p \in \uP{\udd{p}{i}{i}}$ and $a_p \in \uP{\udd{p}{j}{j}}$, we obtain $a_p \in \uP{\udd{p}{j}{i}}$. Next, we also have that if $a_q \in \uP{\udd{p}{i}{j}}$ then $a_p \in \uP{\udd{q}{j}{i}}$ again thanks to $\varphi_{\mathit{tran}}$; indeed since we have $a_q \in \uP{\udd{p}{i}{j}}$ and $a_p \in \uP{\udd{p}{i}{i}}$ and $a_q \in \uP{\udd{q}{j}{j}}$, we obtain $a_p \in \uP{\udd{q}{j}{i}}$.
We also need a natural $d_{\mathit{out}}$ belonging to $\N\setminus
(\im{g}\cup\im{f})$. For $j \in
\{1,2\}$, we define $f_j$ as follows for all $b \in A$:
\[f_j(b) = \left\{\begin{array}{ll}
g(p,i) & \text{if for some } p,i \text{ we have } b\in\uP{\udd{p}{i}{j}} \\
f(b) &\text{if for all $p,i$ we have $b\notin\uP{\udd{p}{i}{j}}$ and for some $p,i$ we have $b\in\uP{\udd{p}{i}{3-j}}$} \\
d_{\mathit{out}} &\text{if for all $p,i,j'$, we have } b\notin\uP{\udd{p}{i}{j'}}
\end{array}\right.
\]
Here again, we can show that since $\BB\models\varphi_{\mathit{tran}} \et \varphi_{\mathit{refl}}(a_1,\ldots,a_n)$, the functions $f_1$ and $f_2$ are well founded. Indeed, assume that $b\in\uP{\udd{p}{i}{j}} \cap
\uP{\udd{q}{k}{j}}$, then we have necessarily that
$g(p,i)=g(q,k)$. For this we need to show that $a_p \in
\udd{q}{k}{i}$ and we use again the formula $\varphi_{\mathit{tran}}$. This can be
obtained because we have $b\in\uP{\udd{p}{i}{j}}$ and
$a_p\in\uP{\udd{p}{i}{i}}$ and $b \in \uP{\udd{q}{k}{j}}$.
We then define $\AA$ as the $2$-data-structures $(A,(P_{\sigma})_{\sigma \in \Sigma},\f{1},\f{2})$. It remains to prove that $\BB = \sem{\AA}_{\tuple{a}}$.
First, note that for all $b\in A$, $p \in \{1,\ldots,n\}$ and $i,j\in\{1,2\}$, we have $b\in\uP{\udd{p}{i}{j}}$ iff $\relsaa{i}{j}{\AA}{a_p}{b}$. Indeed, we have $b\in\uP{\udd{p}{i}{j}}$, we have that $f_j(b)=g(p,i)$ and since $a_p \in \uP{\udd{p}{i}{j}}$ we have as well that $f_i(a_p)=g(p,i)$, as a consequence $\relsaa{i}{j}{\AA}{a_p}{b}$. In the other direction, if $\relsaa{i}{j}{\AA}{a_p}{b}$, it means that $f_j(b)=f_i(a_p)=g(p,i)$ and thus $b\in\uP{\udd{p}{i}{j}}$. Now to have $\BB = \sem{\AA}_{\tuple{a}}$, one has only to be careful in the choice of function $f_{\textup{new}}$ while building $\sem{\AA}_{\tuple{a}}$. We recall that this function is injective and is used to give a value to the elements $b \in A$ such that neither $f_1(b)\in \Valuessub{\AA}{\tuple{a}} \text{ and } f_2(b)\notin \Valuessub{\AA}{\tuple{a}}$ nor $ f_1(b)\notin \Valuessub{\AA}{\tuple{a}} \text{ and } f_2(b)\in \Valuessub{\AA}{\tuple{a}}$. For these elements, we make $f_{\textup{new}}$ matches with the function $f$ and the fact that we define an injection is guaranteed by the formula $\varphi_{\mathit{uniq}}$. \end{proof}
Using the results of Lemma \ref{lem:correct} and \ref{lem:well-formed}, we deduce that the formula $\varphi=\exists x_1\ldots\exists x_n.\varphi_{qf}(x_1,\ldots,x_n)$ of $\eFO{2}{\Sigma}{2}$ is satisfiable iff the formula $\psi=\exists x_1\ldots\exists x_n.\phit{\varphi_{qf}}(x_1,\ldots,x_n) \wedge \varphi_{\mathit{wf}}(x_1,\ldots,x_n) $ is satisfiable. Note that $\psi$ can be built in polynomial time from $\varphi$ and that it belongs to $\ndFO{1}{\Unary'}$. Hence, thanks to Theorem \ref{thm:1fo}, we obtain that $\nDataSat{\eFOr{2}}{2}$ is in \textsc{N2EXP}.
We can as well obtain a matching lower bound thanks to a reduction from $\nDataSat{\textup{dFO}}{1}$. For this matter we rely on two crucial points. First in the formulas of $\eFO{2}{\Sigma}{2}$, there is no restriction on the use of quantifiers for the formulas located under the scope of the $\locformr{\cdot}{x}{2}$ modality and consequently we can write inside this modality a formula of $\ndFO{1}{\Sigma}$ without any modification. Second we can extend a model $\ndFO{1}{\Sigma}$ into a $2$-data structure such that all elements and their values are located in the same radius-$2$-ball by adding everywhere a second data value equal to $0$. More formally, let $\varphi$ be a formula in $\ndFO{1}{\Sigma}$ and consider the formula $\exists x.\locformr{\varphi}{x}{2}$ where we interpret $\varphi$ over $2$-data structures (this formula simply never mentions the values located in the second fields). We have then the following lemma.
\begin{lemma} \label{lem:hardness-radius2-2}
There exists $\AA \in \nData{1}{\Sigma}$ such that $\AA \models \varphi$
if and only if there exists $\BB \in \nData{2}{\Sigma}$ such that
$\BB \models \exists x.\locformr{\varphi}{x}{2}$. \end{lemma}
\begin{proof}
Assume that there exists $\AA=(A,(P_{\sigma})_{\sigma \in
\Sigma},\f{1})$ in $\nData{1}{\Sigma}$ such that $\AA \models
\varphi$. Consider the $2$-data structure $\BB=(A,(P_{\sigma})_{\sigma \in
\Sigma},\f{1},\f{2})$ such that $\f{2}(a)=0$ for all $a\in
A$. Let $a \in A$. It is clear that we have $\vprojr{\BB}{a}{2}=\BB$
and that $\vprojr{\BB}{a}{2} \models \varphi$ (because $\AA \models
\varphi$ and $\varphi$ never mentions the second values of the elements
since it is a formula in $\ndFO{1}{\Sigma}$ ). Consequently $\BB \models \exists
x.\locformr{\varphi}{x}{2}$.
Assume now that there exists $\BB=(A,(P_{\sigma})_{\sigma \in
\Sigma},\f{1},\f{2})$ in $ \nData{2}{\Sigma}$ such that $\BB \models \exists x.\locformr{\varphi}{x}{2}$. Hence there exists $a \in A$ such that $\vprojr{\BB}{a}{2} \models \varphi$, but then by forgetting the second value in $\vprojr{\BB}{a}{2}$ we obtain a model in $\nData{1}{\Sigma}$ which satisfies $\varphi$. \end{proof}
Since $\nDataSat{\textup{dFO}}{1}$ is \textsc{N2EXP}-hard (see Theorem \ref{thm:1fo}), we obtain the desired lower bound.
\begin{theorem}\label{thm:radius2-2}
The problem $\nDataSat{\eFOr{2}}{2}$ is \textsc{N2EXP}-complete. \end{theorem}
\subsection{Balls of radius 1 and any number of data values }
Let $D \geq 1$. We first show that $\nDataSat{\eFOr{1}}{D}$ is in \textsc{NEXP} by providing a reduction towards $\nDataSat{\textup{dFO}}{0}$. This reduction uses the characterisation of the radius-1-ball provided by Lemma \ref{lem:shape-balls} and is very similar to the reduction provided in the previous section. In fact,
for an element $b$ located in the radius-1-ball of another element $a$, we use extra unary predicates to explicit which are the values of $b$ that are common with the values of $a$. We provide here the main step of this reduction whose proof follows the same line as the one of Theorem \ref{thm:radius2-2}.
We consider a formula $\varphi=\exists x_1\ldots\exists x_n.\varphi_{qf}(x_1,\ldots,x_n)$ of $\eFO{D}{\Sigma}{1}$ in prenex normal form, i.e., such that $\varphi_{qf}(x_1,\ldots,x_n)\in\qfFO{D}{\Sigma}{1}$. We know that there is a structure $\AA=(A,(P_{\sigma})_{\sigma \in
\Sigma},\linebreak[0]\f{1},\f{2},\ldots,\f{D})$ in $\nData{D}{\Sigma}$ such that $\AA\models\varphi$ if and only if there are $a_1,\ldots,a_n \in A $ such that $\AA\models\varphi_{qf}(a_1,\ldots,a_n)$. Let then $\AA=(A,(P_{\sigma})_{\sigma \in \Sigma},\f{1},\f{2},\ldots,\f{D})$ in $\nData{D}{\Sigma}$ and a tuple $\tuple{a} = (a_1,\ldots,a_n)$ of elements in $A^n$. Let $\Omega_n=\{\udd{p}{i}{j}\mid p\in\{1,\ldots,n\}, i,j\in\{1,\ldots,D\}\}$ be a set of new unary predicates and $\Unary' = \Sigma \cup \Omega_n$. For every element $b\in A$, the predicates in $\Omega_n$ are used to keep track of the relation between the data values of $b$ and the one of $a_1,\ldots,a_n$ in $\AA$. Formally, we have $\uP{\udd{p}{i}{j}}=\{b\in A\mid \AA\models \rels{i}{j}{a_p}{b}\}$. Finally, we build the $0$-data-structure $\sem{\AA}'_{\tuple{a}}= (A, (\uP{\sigma})_{\sigma\in\Unary'}) $. Similarly to Lemma \ref{lem:r2dv2-semantique}, we have the following connection between $\AA$ and $\sem{\AA}'_{\tuple{a}}$.
\begin{lemma}\label{lem:r1-semantique}
Let $b,c\in A$ and $j,k\in\{1,\ldots,D\}$ and $p\in\{1,\ldots,n\}$. The following statements hold:
\begin{enumerate}
\item If $(b,j)\in\Ball{1}{a_p}{\AA}$ and $(c,k)\in\Ball{1}{a_p}{\AA}$ then $\vprojr{\AA}{a_p}{1}\models\rels{j}{k}{b}{c}$ iff there is $i\in\{1,2\}$ s.t. $b \in \uP{\udd{p}{i}{j}}$ and $c \in \uP{\udd{p}{i}{k}}$.
\item If $(b,j)\notin\Ball{1}{a_p}{\AA}$ and $(c,k)\in\Ball{1}{a_p}{\AA}$ then $\vprojr{\AA}{a_p}{1}\nvDash\rels{j}{k}{b}{c}$
\item If $(b,j)\notin\Ball{1}{a_p}{\AA}$ and $(c,k)\notin\Ball{1}{a_p}{\AA}$ then $\vprojr{\AA}{a_p}{1}\models\rels{j}{k}{b}{c}$ iff $b=c$ and $j=k$.
\end{enumerate} \end{lemma}
We shall now see how we translate the formula $\varphi_{qf}(x_1,\ldots,x_n)$ into a formula $\phit{\varphi_{qf}}'(x_1,\ldots,x_n)$ in $\ndFO{0}{\Unary'}$ such that $\AA$ satisfies $\varphi_{qf}(a_1,\ldots,a_n)$ if, and only if, $\sem{\AA}'_{\tuple{a}}$ satisfies $\phit{\varphi_{qf}}(a_1,\ldots,a_n)$. As in the previous section, we introduce the following formula in $\ndFO{0}{\Unary'}$ with $p \in \{1,\ldots,n\}$ and $j \in \{1,\ldots,D\}$ to test if the $j$-th field of an element belongs to $\Ball{1}{a_p}{\AA}$: $$ \phiBun{j}(y) := \bigvee_{i \in \{1,\ldots,D\}}\udd{p}{i}{j}(y) $$
We now present how we translate atomic formulas of the form $\rels{j}{k}{y}{z}$ under some $\locformr{-}{x_p}{1}$. For this matter, we rely on two formulas of $\ndFO{0}{\Unary'}$ which can be described as follows: \begin{itemize} \item The first formula asks for $(y,j)$ and $(z,k)$ to be in $\Ball{1}{a_p}{1}$ (here we abuse notations, using variables for the elements they represent) and for these two data values to coincide with one data value of $a_p$, it corresponds to Lemma \ref{lem:r1-semantique}.1:
$$
\psi_{j,k,a_p}^{r=1}(y,z) := \phiBun{j}(y) \et \phiBun{k}(z) \et \Ou^D_ {i=1}\udd{p}{i}{j}(y)\et\udd{p}{i}{k}(z)
$$ \item The second formula asks for $(y,j)$ and $(z,k)$ to not belong to $\Ball{1}{a_p}{\AA}$ and for $y=z$, it corresponds to Lemma \ref{lem:r1-semantique}.3:
$$
\psi_{j,k,a_p}^{r>1}(y,z) := \begin{cases}
\bigwedge^D_{i=1} (\neg
\phiBun{i}(y) \wedge \neg \phiBun{i}(z)) \et y=z &\text{ if } j=k \\
\bot &\text{ otherwise}
\end{cases}
$$ \end{itemize}
Finally, as before we provide an inductive definition of the
translation $\Tbis{-}$ which uses subtransformations $\Tpbis{-}$ in
order to remember the centre of the ball and leads to the
construction of $\phit{\varphi_{qf}}'(x_1,\ldots,x_n)$. We only
detail the case
$$
\Tpbis{\rels{j}{k}{y}{z}} =\psi_{j,k,a_p}^{r=1}(y,z) \ou
\psi_{j,k,a_p}^{r>1}(y,z)
$$
as the other cases are identical as for the
translation $\T{-}$ shown in the previous section. This leads to
the following lemma (which is the pendant of Lemma
\ref{lem:correct}).
\begin{lemma} \label{lem:correct2}
We have $\AA\models\varphi_{qf}(\tuple{a})$ iff $\sem{\AA}'_{\tuple{a}}\models\phit{\varphi_{qf}}'(\tuple{a})$. \end{lemma}
As we had to characterise the well-formed $1$-data structure, a similar trick is necessary here. For this matter, we use the following formulas: $$
\begin{array}{ll} \psi_{\mathit{tran}} &= \forall y \forall z.\Et_{p,q=1}^{n}\Et_{i,j,k,\ell=1}^D \Big(\udd{p}{i}{j}(y) \et \udd{p}{i}{\ell}(z) \et \udd{q}{k}{j}(y) \donc \udd{q}{k}{\ell}(z)\Big) \\ \psi_{\mathit{refl}}(x_1,\ldots,x_n) &=\Et_{p=1}^n\Et_{i=1}^D \udd{p}{i}{i}(x_p) \\
\psi_{\mathit{wf}}(x_1,\ldots,x_n) &=\psi_{\mathit{tran}} \et \psi_{\mathit{refl}}(x_1,\ldots,x_n)
\end{array}
$$
Finally with the same reasoning as the one given in the previous section, we can show that the formula $\varphi=\exists x_1\ldots\exists x_n.\linebreak[0]\varphi_{qf}(x_1,\ldots,x_n)$ of $\eFO{D}{\Sigma}{1}$ is satisfiable iff the formula $\exists x_1\ldots\exists x_n.\linebreak[0]\phit{\varphi_{qf}}'(x_1,\ldots,x_n) \wedge \psi_{\mathit{wf}}(x_1,\ldots,x_n) $ is satisfiable. Note that this latter formula can be built in polynomial time from $\varphi$ and that it belongs to $\ndFO{0}{\Unary'}$. Hence, thanks to Theorem \ref{thm:0fo}, we obtain that $\nDataSat{\eFOr{1}}{D}$ is in \textsc{NEXP}. The matching lower bound is as well obtained the same way by reducing $\nDataSat{\textup{dFO}}{0}$ to $\nDataSat{\eFOr{1}}{D}$ showing that a formula $\varphi$ in $\ndFO{0}{\Sigma}$ is satisfiable iff the formula $\exists x.\locformr{\varphi}{x}{1}$ in $\eFO{1}{\Sigma}{D}$ is satisfiable.
\begin{theorem}
For all $D \geq 1$, the problem $\nDataSat{\eFOr{1}}{D}$ is \textsc{NEXP}-complete. \end{theorem}
\section{Undecidability results} \label{sec:undecidability} We show here $\nDataSat{\eFOr{3}}{2}$ and $\nDataSat{\eFOr{2}}{3}$ are undecidable. To obtain this we provide reductions from $\nDataSat{\textup{dFO}}{2}$ and we use the fact that any 2-data structure can be interpreted as a radius-3-ball of a 2-data structure or respectively as a radius-2-ball of a 3-data structure.
\subsection{Radius 3 and two data values}
In order to reduce $\nDataSat{\textup{dFO}}{2}$ to $\nDataSat{\eFOr{3}}{2}$, we show that we can transform slightly any $2$-data structure $\AA$ into an other 2-data structure $\addge{\AA}$ such that $\addge{\AA}$ corresponds to the radius-3-ball of any element of $\addge{\AA}$ and this transformation has some kind of inverse. Furthermore, given a formula $\varphi \in \ndFO{2}{\Sigma}$, we transform it into a formula $T(\varphi)$ in $\eFO{2}{\Sigma'}{3}$ such that $\AA$ satisfies $\varphi$ iff $\addge{\AA}$ satisfies $T(\varphi)$ . What follows is the formalisation of this reasoning.
Let $\AA=(A,(\uP{\sigma})_{\sigma},\ifunct,\ofunct)$ be a $2$-data structure in $\nData{2}{\Sigma}$ and $\mathsf{ge}$ be a fresh unary predicate not in $\Sigma$. From $\AA$ we build the following $2$-data structure $\addge{\AA}=(A',(\uP{\sigma}')_{\sigma},\ifunct',\ofunct')\in\nData{2}{\Sigma\cup\{\mathsf{ge}\}}$ such that: \begin{itemize}
\item $A' = A \uplus \Values{\AA}\times\Values\AA$,
\item for $i\in\{1,2\}$ and $a\in A$, $f_i'(a)=f_i(a)$ and for $(d_1,d_2)\in \Values{\AA}\times\Values\AA$, $f_i((d_1,d_2))=d_i$,
\item for $\sigma\in\Sigma$, $\uP{\sigma}'=\uP{\sigma}$,
\item $\uP{\mathsf{ge}}=\Values{\AA}\times\Values\AA$.
\end{itemize} Hence to build $\addge{\AA}$ from $\AA$ we have added to the elements of $\AA$ all pairs of data presented in $\AA$ and in order to recognise these new elements in the structure we use the new unary predicate $\mathsf{ge}$. We add these extra elements to ensure that all the elements of the structure are located in the radius-3-ball of any element of $\addge{\AA}$. We have then the following property.
\begin{lemma}\label{lem:ge-has-radius-3}
$\vprojr{\addge{\AA}}{a}{3}=\addge{\AA}$ for all $a \in A'$. \end{lemma} \begin{proof}
Let $b\in A'$ and $i,j \in \{1,2\}$. We show that
$\distaa{(a,i)}{(b,j)}{\addge{\AA}}\leq 3$. i.e. that there is a path of length at most 3 from $(a,i)$ to $(b,j)$ in the data graph $\gaifmanish{\addge{\AA}}$.
By construction of $\addge{\AA}$, there is an element $c\in A'$ such that $f_1(c)=f_i(a)$ and $f_2(c)=f_j(b)$.
So we have the path $(a,i),(c,1),(c,2),(b,j)$ of length at most 3 from $(a,i)$ to $(b,j)$ in $\gaifmanish{\addge{\AA}}$. \end{proof}
Conversely, to $\AA=(A,(\uP{\sigma})_{\sigma},\ifunct,\ofunct)\in\nData{2}{\Sigma\cup\{\mathsf{ge}\}}$, we associate $\minusge{\AA}=(A',(\uP{\sigma}')_{\sigma},\ifunct',\ofunct')\in\nData{2}{\Sigma}$ where: \begin{itemize}
\item $A' = A \setminus \uP{\mathsf{ge}}$,
\item for $i\in\{1,2\}$ and $a\in A'$, $f_i'(a)=f_i(a)$,
\item for $\sigma\in\Sigma$, $\uP{\sigma}'=\uP{\sigma}'\setminus \uP{\mathsf{ge}}$. \end{itemize}
Finally we inductively translate any formula $\varphi\in\ndFO{2}{\Sigma}$ into $T(\varphi)\in\ndFO{2}{\Sigma\cup\{\mathsf{ge}\}}$ by making it quantify over elements not labeled with $\mathsf{ge}$: $T(\sigma(x)) = \sigma(x)$, $T(\rels{i}{j}{x}{y})=\rels{i}{j}{x}{y}$, $T( x=y )= (x=y) $, $T(\exists x.\varphi)=\exists x. \neg \mathsf{ge}(x) \wedge T(\varphi)$, $T( \varphi \vee \varphi')=T(\varphi) \vee T(\varphi')$ and $T(\neg \varphi)=\neg T(\varphi)$.
\begin{lemma}\label{lem:ge-vs-without}
Let $\varphi$ be a sentence in $\ndFO{2}{\Sigma}$,
$\AA\in\nData{2}{\Sigma}$ and $\BB \in
\nData{2}{\Sigma\cup\{\mathsf{ge}\}}$. The two following properties hold:
\begin{itemize}
\item $\AA\models\varphi$ iff $\addge{\AA}\models T(\varphi)$
\item $\minusge{\BB} \models\varphi$ iff $\BB\models T(\varphi)$.
\end{itemize} \end{lemma}
\begin{proof}
As for any $\AA\in\nData{2}{\Sigma}$ we have $\minusge{(\addge{\AA})} = \AA$, it is sufficient to prove the second point.
We reason by induction on $\varphi$.
Let $\AA=(A,(\uP{\sigma})_{\sigma},\ifunct,\ofunct)\in\nData{2}{\Sigma\cup\{\mathsf{ge}\}}$ and let $\minusge{\AA}=(A',(\uP{\sigma}')_{\sigma},\ifunct',\ofunct')\in\nData{2}{\Sigma}$.
The inductive hypothesis is that for any formula $\varphi\in\ndFO{2}{\Sigma}$ (closed or not) and any context interpretation function $I: \mathcal{V} \to A'$ we have $\minusge{\AA} \models_I \varphi \text{ iff } \AA \models_I T(\varphi)$.
Note that the inductive hypothesis is well founded in the sense that the interpretation $I$ always maps variables to elements of the structures.
We prove two cases: when $\varphi$ is a unary predicate and when
$\varphi$ starts by an existential quantification, the other cases
being similar. First, assume that $\varphi = \sigma(x)$ where $\sigma\in\Sigma$.
$\minusge{\AA} \models_I \sigma(x)$ holds iff $I(x)\in\uP{\sigma}'$.
As $I(x)\in A\setminus \uP{\mathsf{ge}}$, we have $I(x)\in\uP{\sigma}'$ iff $I(x)\in\uP{\sigma}$, which is equivalent to $\AA \models_I T(\sigma(x))$ .
Second assume $\varphi = \exists x.\varphi'$.
Suppose that $\minusge{\AA} \models_I \exists x.\varphi'$.
Thus, there is a $a\in A'$ such that $\minusge{\AA} \models_\Intrepl{x}{a} \varphi'$.
By inductive hypothesis, we have $\AA\models_\Intrepl{x}{a} T(\varphi')$.
As $a\in A' = A\setminus \uP{\mathsf{ge}}$, we have $\AA\models_\Intrepl{x}{a} \neg\mathsf{ge}(x)$, so $\AA\models_I \exists x. \neg\mathsf{ge}(x)\et T(\varphi')$ as desired.
Conversely, suppose that $\AA \models_I T(\exists x.\varphi') $.
It means that there is a $a\in A$ such that $\AA \models_\Intrepl{x}{a}\neg\mathsf{ge}(x)\et T(\varphi')$.
So we have that $a\in A'=A\setminus \uP{\mathsf{ge}}$, which means that $\Intrepl{x}{a}$ takes values in $A$ and we can apply the inductive hypothesis to get that
$\minusge{\AA} \models_\Intrepl{x}{a} \varphi'$.
So we have $\minusge{\AA} \models_I \exists x.\varphi'$. \end{proof}
From Theorem \ref{thm:undec-general}, we know that $\nDataSat{\textup{dFO}}{2}$ is undecidable. From a closed formula $\varphi\in\ndFO{2}{\Sigma}$, we build the formula $\exists x.\locformr{T(\varphi)}{x}{3}\in\eFO{2}{\Sigma\cup\{\mathsf{ge}\}}{3}$. Now if $\varphi$ is satisfiable, it means that there exists $\AA\in \nData{2}{\Sigma}$ such that $\AA\models\varphi$. By Lemma \ref{lem:ge-vs-without}, $\addge{\AA}\models T(\varphi)$. Let $a$ be an element of $\AA$, then thanks to Lemma \ref{lem:ge-has-radius-3}, we have $\vprojr{\addge{\AA}}{a}{3}\models T(\varphi)$.
Finally by definition of our logic, $\addge{\AA}\models\exists x.\locformr{T(\varphi)}{x}{3}$.
So $\exists x.\locformr{T(\varphi}{x}{3}$ is satisfiable. Now assume
that $\exists x.\locformr{T(\varphi)}{x}{3}$ is satisfiable. So there
exist $\AA \in \nData{2}{\Sigma\cup\{\mathsf{ge}\}}$ and an element $a$ of
$\AA$ such that $\vprojr{\AA}{a}{3}\models T(\varphi)$.
Using Lemma \ref{lem:ge-vs-without}, we obtain
$(\vprojr{\AA}{a}{3})_{\setminus\mathsf{ge}}\models\varphi$. Hence $\varphi$ is
satisfiable. This shows that we can reduce $\nDataSat{\textup{dFO}}{2}$ to $\nDataSat{\eFOr{3}}{2}$ .
\begin{theorem}\label{thm:undec-existential-r3}
The problem $\nDataSat{\eFOr{3}}{2}$ is undecidable. \end{theorem}
\subsection{Radius 2 and three data values}
We provide here a reduction from $\nDataSat{\textup{dFO}}{2}$ to $\nDataSat{\eFOr{2}}{3}$. The idea is similar to the one used in the proof of Lemma \ref{lem:hardness-radius2-2} to show that $\nDataSat{\eFOr{2}}{2}$ is \textsc{N2EXP}-hard by reducing $\nDataSat{\textup{dFO}}{1}$. Indeed we have the following Lemma.
\begin{lemma}
Let $\varphi$ be a formula in $\ndFO{2}{\Sigma}$. There exists $\AA \in \nData{2}{\Sigma}$ such that $\AA \models \varphi$
if and only if there exists $\BB \in \nData{3}{\Sigma}$ such that
$\BB \models \exists x.\locformr{\varphi}{x}{2}$. \end{lemma}
\begin{proof}
Assume that there exists $\AA=(A,(P_{\sigma})_{\sigma \in
\Sigma},\f{1},\f{2})$ in $\nData{2}{\Sigma}$ such that $\AA \models
\varphi$.Consider the $3$-data structure $\BB=(A,(P_{\sigma})_{\sigma \in
\Sigma},\f{1},\f{2},\f{3})$ such that $\f{3}(a)=0$ for all $a\in
A$. Let $a \in A$. It is clear that we have $\vprojr{\BB}{a}{2}=\BB$
and that $\vprojr{\BB}{a}{2} \models \varphi$ (because $\AA \models
\varphi$ and $\varphi$ never mentions the third values of the elements
since it is a formula in $\ndFO{1}{\Sigma}$). Consequently $\BB
\models \exists
x.\locformr{\varphi}{x}{2}$.
Assume now that there exists $\BB=(A,(P_{\sigma})_{\sigma \in
\Sigma},\f{1},\f{2},\f{3})$ in $ \nData{3}{\Sigma}$ such that $\BB \models \exists x.\locformr{\varphi}{x}{2}$. Hence there exists $a \in A$ such that $\vprojr{\BB}{a}{2} \models \varphi$, but then by forgetting the third value in $\vprojr{\BB}{a}{2}$ we obtain a model in $\nData{3}{\Sigma}$ which satisfies $\varphi$. \end{proof}
Using Theorem \ref{thm:undec-general}, we obtain the following result. \begin{theorem}\label{thm:undec-existential-r2}
The problem $\nDataSat{\eFOr{2}}{3}$ is undecidable. \end{theorem}
\end{document} | arXiv |
Exponential factorial
The exponential factorial is a positive integer n raised to the power of n − 1, which in turn is raised to the power of n − 2, and so on in a right-grouping manner. That is,
$n^{(n-1)^{(n-2)\cdots }}$
The exponential factorial can also be defined with the recurrence relation
$a_{1}=1,\quad a_{n}=n^{a_{n-1}}$
The first few exponential factorials are 1, 2, 9, 262144, ... (OEIS: A049384 or OEIS: A132859). For example, 262144 is an exponential factorial since
$262144=4^{3^{2^{1}}}$
Using the recurrence relation, the first exponential factorials are:
1
21 = 2
32 = 9
49 = 262144
5262144 = 6206069878...8212890625 (183231 digits)
The exponential factorials grow much more quickly than regular factorials or even hyperfactorials. The number of digits in the exponential factorial of 6 is approximately 5 × 10183 230.
The sum of the reciprocals of the exponential factorials from 1 onwards is the following transcendental number:
${\frac {1}{1}}+{\frac {1}{2^{1}}}+{\frac {1}{3^{2^{1}}}}+{\frac {1}{4^{3^{2^{1}}}}}+{\frac {1}{5^{4^{3^{2^{1}}}}}}+{\frac {1}{6^{5^{4^{3^{2^{1}}}}}}}+\ldots =1.611114925808376736\underbrace {111111111111\ldots 111111111111} _{183212}272243682859\ldots $
This sum is transcendental because it is a Liouville number.
Like tetration, there is currently no accepted method of extension of the exponential factorial function to real and complex values of its argument, unlike the factorial function, for which such an extension is provided by the gamma function. But it is possible to expand it if it is defined in a strip width of 1.
Similarly, there is disagreement about the appropriate value at 0; any value would be consistent with the recursive definition. A smooth extension to the reals would satisfy $f(0)=f'(1)$, which suggests a value strictly between 0 and 1.
Related functions, notation and conventions
References
• Jonathan Sondow, "Exponential Factorial" From Mathworld, a Wolfram Web resource
| Wikipedia |
For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \ldots\}$. Find the number of positive integers less than or equal to $10,000$ which are simultaneously $7$-safe, $11$-safe, and $13$-safe.
We see that a number $n$ is $p$-safe if and only if the residue of $n \mod p$ is greater than $2$ and less than $p-2$; thus, there are $p-5$ residues $\mod p$ that a $p$-safe number can have. Therefore, a number $n$ satisfying the conditions of the problem can have $2$ different residues $\mod 7$, $6$ different residues $\mod 11$, and $8$ different residues $\mod 13$. The Chinese Remainder Theorem states that for a number $x$ that is $a$ (mod b) $c$ (mod d) $e$ (mod f) has one solution if $gcd(b,d,f)=1$. For example, in our case, the number $n$ can be: 3 (mod 7) 3 (mod 11) 7 (mod 13) so since $gcd(7,11,13)$=1, there is 1 solution for n for this case of residues of $n$.
This means that by the Chinese Remainder Theorem, $n$ can have $2\cdot 6 \cdot 8 = 96$ different residues mod $7 \cdot 11 \cdot 13 = 1001$. Thus, there are $960$ values of $n$ satisfying the conditions in the range $0 \le n < 10010$. However, we must now remove any values greater than $10000$ that satisfy the conditions. By checking residues, we easily see that the only such values are $10006$ and $10007$, so there remain $\boxed{958}$ values satisfying the conditions of the problem. | Math Dataset |
Genome-wide association and epidemiological analyses reveal common genetic origins between uterine leiomyomata and endometriosis
C. S. Gallagher1 na1,
N. Mäkinen2 na1,
H. R. Harris3 na1,
N. Rahmioglu4 na1,
O. Uimari ORCID: orcid.org/0000-0002-8954-29005,6,
J. P. Cook7,
N. Shigesi5,
T. Ferreira4,8,
D. R. Velez-Edwards9,
T. L. Edwards ORCID: orcid.org/0000-0003-4318-611910,
S. Mortlock11,
Z. Ruhioglu ORCID: orcid.org/0000-0002-0256-522X2,
F. Day ORCID: orcid.org/0000-0003-3789-765112,
C. M. Becker ORCID: orcid.org/0000-0002-9870-95815,
V. Karhunen13,14,15,
H. Martikainen6,
M.-R. Järvelin ORCID: orcid.org/0000-0002-2149-063013,14,15,16,17,
R. M. Cantor18,
P. M. Ridker19,
K. L. Terry20,21,
J. E. Buring19,
S. D. Gordon ORCID: orcid.org/0000-0001-7623-328X22,
S. E. Medland ORCID: orcid.org/0000-0003-1382-380X23,
G. W. Montgomery ORCID: orcid.org/0000-0002-4140-813911,22,
D. R. Nyholt ORCID: orcid.org/0000-0001-7159-304022,24,
D. A. Hinds ORCID: orcid.org/0000-0002-4911-803X25,
J. Y. Tung25,
the 23andMe Research Team,
J. R. B. Perry12,
P. A. Lind ORCID: orcid.org/0000-0002-3887-259823,
J. N. Painter23,
N. G. Martin ORCID: orcid.org/0000-0003-4069-802022,
A. P. Morris4,7,
D. I. Chasman19 na2,
S. A. Missmer21,26 na2,
K. T. Zondervan ORCID: orcid.org/0000-0002-0275-99054,5 na2 &
C. C. Morton ORCID: orcid.org/0000-0003-2198-67562,27,28,29 na2
Nature Communications volume 10, Article number: 4857 (2019) Cite this article
Genome-wide association studies
Uterine leiomyomata (UL) are the most common neoplasms of the female reproductive tract and primary cause for hysterectomy, leading to considerable morbidity and high economic burden. Here we conduct a GWAS meta-analysis in 35,474 cases and 267,505 female controls of European ancestry, identifying eight novel genome-wide significant (P < 5 × 10−8) loci, in addition to confirming 21 previously reported loci, including multiple independent signals at 10 loci. Phenotypic stratification of UL by heavy menstrual bleeding in 3409 cases and 199,171 female controls reveals genome-wide significant associations at three of the 29 UL loci: 5p15.33 (TERT), 5q35.2 (FGFR4) and 11q22.3 (ATM). Four loci identified in the meta-analysis are also associated with endometriosis risk; an epidemiological meta-analysis across 402,868 women suggests at least a doubling of risk for UL diagnosis among those with a history of endometriosis. These findings increase our understanding of genetic contribution and biology underlying UL development, and suggest overlapping genetic origins with endometriosis.
Uterine leiomyomata (UL), also known as uterine fibroids, are hormone-driven tumors with an estimated prevalence ranging from 20–77%1,2. Although the majority of UL are asymptomatic, about 25% of women with UL are symptomatic, and may experience heavy menstrual bleeding (HMB), abdominal pain, infertility, and increased risk of miscarriage3. Currently, the only essentially curative treatment is uterine extirpation via total hysterectomy. Known risk factors for UL include increasing age up to menopause, ethnicity (particularly African ancestry), family history of UL, nulliparity, and increased body mass index (BMI)4. Studies of familial aggregation and twins, as well as racial differences in prevalence and morbidity, suggest heritable factors influence risk for developing UL5,6,7,8,9,10. Recent GWAS have identified 26 loci significantly associated (P < 5 × 10−8) with UL: 10q24.33, 11p15.5, and 22q13.1 in Japanese women11, 25 loci in white women of European ancestry12,13,14,15, including the three previously identified loci in Japanese women, and a distinct region at 22q13.1 in African American women16.
To define further the genetic architecture of UL, we perform a discovery meta-analysis of GWAS on UL across a total of 35,474 cases and 267,505 female controls of white European ancestry, which more than doubles the case sample size of previously reported GWAS11,12,14,15,16. The meta-analysis identifies eight novel loci significantly associated with UL (P < 5 × 10−8) and confirms 21 previously reported European risk loci. Interestingly, HMB-limited UL GWAS reveals three of the 29 independent loci to be significantly associated with the co-occurrence of UL and HMB. Four loci identified in the meta-analysis are also reported to be associated with risk for endometriosis, which together with an epidemiological meta-analysis indicating an association between endometriosis and diagnosis of UL suggest overlapping genetic origins between the two highly common gynecologic diseases.
UL GWAS meta-analysis
Our discovery meta-analysis of GWAS on UL includes four population-based cohorts and one direct-to-consumer cohort of white European ancestry: Women's Genome Health Study (WGHS), Northern Finnish Birth Cohort (NFBC), QIMR Berghofer Medical Research Institute (QIMR), UK Biobank (UKBB), and 23andMe (Supplementary Methods, Supplementary Table 1). Imputation of genotypes was carried out using 1000 Genomes Project Phase 3 and Haplotype Reference Consortium (HRC) reference panels. UL phenotype in each cohort was analyzed in a logistic regression or linear mixed model assuming additive genetic effects with multivariate adjustment for age, BMI, and/or correction for population structure. After quality control metrics were applied, including exclusion of non-informative (MAF < 0.01) and poorly imputed (r2 < 0.4) SNPs, we performed a fixed-effects, inverse-variance-weighted (IVW) meta-analysis across 35,474 cases with a clinical or self-reported history of UL and 267,505 unaffected female controls. Altogether 8,662,096 biallelic SNPs were analyzed and adjustments for genomic inflation performed (Supplementary Fig. 1, Supplementary Table 2). Through linkage disequilibrium score (LDSC) regression analysis, an estimated 89.5% of the genomic inflation factor (λGC) of 1.12 was attributable to polygenic heritability (intercept = 1.02, s.e. = 0.0081). Overall, individual SNP-based heritability (h2) was estimated to be 0.0281 (s.e. = 0.0029) on the liability scale.
Risk loci associated with UL
We observe genome-wide significant associations (P < 5 × 10−8) at 2505 SNPs across 29 independent loci (Table 1, Supplementary Fig. 2, Supplementary Table 3). The Manhattan plot is shown in Fig. 1. We identify eight novel loci associated with UL (2p23.2, 4q22.3, 6p21.31, 7q31.2, 10p11.22, 11p14.1, 12q15, and 12q24.31), which include the following candidate genes of interest: HMGA1, BABAM2, and WNT2. HMGA1 is a member of the high mobility group proteins and is involved in regulation of gene transcription17. Somatic rearrangements of HMGA1 at 6p21 have been recurrently documented in UL, albeit at a much lower frequency than those of HMGA2—another member of the high mobility group protein family18,19,20. BABAM2 at 2p23.2 encodes a death receptor-associating intracellular protein that promotes tumor growth by suppressing apoptosis21. Associations at 7q31.2 containing WNT2, a member of the Wnt gene family, provide support for the previously suggested role of Wnt signaling in UL22,23.
Table 1 Overview of lead SNPs with significant associations at 29 independent loci in UL GWAS meta-analysis
Manhattan plot for UL GWAS meta-analysis across all cohorts. Meta-analysis of GWAS including 302,979 women of white European ancestry across all cohorts identified 29 independent loci associated with UL. Red and blue horizontal lines indicate genome-wide significant (P < 5 × 10−8) and suggestive (P < 1 × 10−5) thresholds, respectively
Among 29 independent loci are 21 loci previously reported to be significantly associated with UL11,12,13,14,15,16. A number of identified loci harbor genes previously implicated in cell growth and cancer risk in different tissue types, including cervical cancer24, epithelial ovarian cancer25,26, breast cancer27,28, glioma29,30, bladder cancer31, and pancreatic cancer32,33,34. Specifically, seven independent loci contain well-characterized oncogenes and tumor suppressor genes from the Cancer Gene Census list in COSMIC35: PDGFRA, TERT, ESR1, WT1, ATM, FOXO1, and TP53.
Using approximate conditional analysis, we identify multiple distinct association signals for UL at 10 loci (at locus-wide significance, P < 1 × 10−5, Bonferroni correction) (Supplementary Table 4). Fine-mapping was conducted on all 43 distinct association signals arising from the 29 detected UL loci, revealing three association signals with a single variant in the 99% credible set (Fig. 2, Supplementary Table 5). The missense variant at 20p12.3 (rs16991615; E341K) maps to MCM8, a gene that encodes a protein involved in DNA double-strand break repair36. MCM8 has also been implicated in length of reproductive lifespan, menopause, and premature ovarian failure37,38. Another variant (rs78378222) resides in the 3'UTR of TP53 at 17p13.1, and has been shown to disturb 3'-end processing of TP53 mRNA39. This variant has been associated with both malignant and benign tumor types39,40,41.
Fine-mapping reveals three association signals with a single driver in 99% credible set. Association with UL is expressed as −log10(P value) for the three signals on chromosomes: (a) 13q14.11, (b) 17p13.1, and (c) 20p12.3. The labeled SNP represents the most significant SNP for each locus. SNP association P-value is shown on the y axis, while SNP position (with gene annotation) appears on the x axis. Each SNP is colored according to the strength of LD with the lead SNP. Regional association plots were produced in LocusZoom
UL GWAS limited by HMB
HMB, one of the major symptoms of UL, is estimated to affect up to 30% of reproductive-aged women, having a considerable impact on a woman's quality of life. Thus, variants specifically associated with this symptom are of particular interest for drug target development. We performed a GWAS on UL limited by HMB using a linear mixed model across 3409 cases and 199,171 unaffected female controls from the UKBB (Supplementary Methods, Supplementary Fig. 3). We observe genome-wide significant associations (P < 5 × 10−8) at three of the 29 independent UL loci: 5p15.33 (rs72709458, OR [95% CI] = 0.86 [0.81–0.91], P = 3.50 × 10−8), 5q35.2 (rs2456181, OR [95% CI] = 0.87 [0.83–0.91], P = 4.20 × 10−10), and 11q22.3 (rs1800057, OR [95% CI] = 0.66 [0.58–0.76], P = 2.80 x 10−9) (Fig. 3, Supplementary Fig. 4, Supplementary Table 6). The lead SNP at 11q22.3, a missense variant in ATM, has been associated with increased risk of various cancers, such as breast cancer42,43, while the lead SNP at 5p15.33, an intronic TERT variant, has previously been implicated in gliomas44. The lead SNP rs2456181 at 5q35.2 resides near FGFR4, a gene encoding a cell-surface receptor for fibroblast growth factors involved in regulation of several pathways, including cell proliferation, differentiation, and migration.
Manhattan plot for GWAS on UL limited by heavy menstrual bleeding. GWAS across 202,580 women of white European ancestry identified three independent loci associated with UL limited by heavy menstrual bleeding. Red and blue horizontal lines indicate genome-wide significant (P < 5 × 10−8) and suggestive (P < 1 × 10−5) thresholds, respectively
HMB GWAS
A GWAS based solely on HMB across 9813 cases and 210,946 female controls reveals one genome-wide significant association at 11p14.1, one of the eight novel loci associated with UL (Supplementary Figs. 5 and 6). The lead SNPs for UL and HMB at 11p14.1 are in high LD, and the direction of the effect is the same (Supplementary Fig. 7). This locus has previously been associated with endometriosis, age at menarche, and follicle-stimulating/luteinizing hormone levels45,46,47. According to GTEx (v7), the lead SNP for HMB (rs11031005) is a potential expression quantitative trait locus (eQTL) for ARL14EP in several tissue types, such as testis and thyroid. Mendelian randomization (MR) was used to assess the causality of genetic association between UL (exposure) and HMB (outcome). Interestingly, MR reveals that genetic predisposition to UL is causally linked to an increased risk of HMB, with the β estimate of 0.26 being significant in the IVW model (P = 1.2 × 10−12) in the absence of heterogeneity (P = 0.13) (Supplementary Table 7). The MR Egger regression shows no significant directional pleiotropy (intercept = 0.01, P = 0.36) supporting a causal relationship.
Overlap of UL and endometriosis
Interestingly, significant association signals are observed at several loci previously associated with endometriosis: 1p36.12 (rs7412010, OR [95% CI] = 1.13 [1.11–1.16], P = 2.43 × 10−29), 2p25.1 (rs35417544, OR [95% CI] = 1.09 [1.07–1.10], P = 2.32 × 10−19), 6q25.2 (rs58415480, OR [95% CI] = 1.19 [1.17–1.22], P = 1.86 × 10−54), and 11p14.1 (rs11031006, OR [95% CI] = 1.10 [1.07–1.12], P = 5.65 × 10−15)45,48,49,50. LD is strong between UL and previously reported endometriosis lead SNPs45 at all except one locus, 2p25.1 (Supplementary Table 8). In addition, the direction of effect is the same between the lead SNPs at 1p36.12. Using LDSC regression, we observe a moderate genetic correlation between UL and endometriosis in women with European ancestry (rg = 0.39, s.e = 0.05, P = 9.77 × 10−13). Endometriosis has an earlier age-of-onset than UL, with a mean age of 25–29 years and 35 years, respectively. MR suggests that genetic predisposition to endometriosis (exposure) is causally linked to an increased risk of UL (outcome); the β of 0.36 is significant (P = 3.7 × 10−3) in the IVW model (heterogeneity P = 9.5 × 10−68) (Supplementary Table 7). Leave-one-out sensitivity analysis reveals that no single SNP alone drives the significant relationship between endometriosis and UL, but instead the relationship is accounted for by contributions from multiple variants across the genome (Supplementary Fig. 8). Given the high degree of heterogeneity, the effect sizes were estimated in a minimal set of SNPs that when used as a genetic instrument eliminate the heterogeneity (Supplementary Fig. 9). The effect size estimate (β = 0.12) from the minimal set of variants remains significant (P = 4.3 × 10−3) in the IVW model in the absence of heterogeneity (P = 0.23). We also applied the MR pleiotropy residual sum and outlier (MR-PRESSO) global and distortion tests to adjust for variants causing significant bias in the estimates through horizontal pleiotropy. Outlier-adjusted estimates still provide significant evidence for a causal estimate of endometriosis on UL (β = 0.29, P = 0.002) (Supplementary Table 7).
Endometriosis, defined by ectopic growth of endometrial-like tissue outside the uterus, is a common inflammatory hormone-dependent disease that affects reproductive-aged women51. Although functional studies of relevant tissue are needed to confirm consequences of the variants in regulation of gene expression, each of the four observed overlapping genomic loci contains a gene(s) known to be involved in progesterone or estrogen signaling. WNT4 at 1p36.12 encodes a secreted signaling factor that promotes female sex development, and regulates both postnatal uterine development and progesterone signaling during decidualization52,53. Recently, SNPs at 1p36.12 associated with a greater endometriosis risk have been suggested to act through CDC42, a gene that encodes a small GTPase of the Rho family54. GREB1 at 2p25.1 is an early response gene in the estrogen receptor (ER)-regulated pathway, and promotes growth of breast and pancreatic cancer cells55,56. ESR1 at 6q25.2 encodes the alpha subunit of the ligand-activated nuclear ER that regulates cell proliferation in the uterus57. FSHB at 11p14.1 encodes the biologically active subunit of follicle-stimulating hormone, which regulates maturation of ovarian follicles and release of ova during menstruation58,59.
Epidemiological meta-analysis
Given shared risk loci and genetic correlation of UL and endometriosis, we conducted an epidemiological meta-analysis including 402,868 women from three population-based cohorts: Nurses' Health Study II (NHSII), Women's Health Study (WHS), and UKBB (Supplementary Methods, Supplementary Table 9), to assess the likelihood of UL diagnosis among women who had or had not been diagnosed with endometriosis. Women with endometriosis had a significantly higher likelihood of UL diagnosis (multivariable-adjusted summary relative risk (RR) [95% CI] = 2.17 [1.48–3.19]) (Fig. 4). All cohort-specific analyses demonstrated at least a doubling of risk, suggesting a robust association (Table 2). However, biologically and statistically significant heterogeneity was observed in the pooling of effect size estimates in the meta-analysis (P < 1 × 10−4) (Fig. 4). Therefore, absolute effect estimates need to be interpreted in the context of source populations. Heterogeneity could reflect various different population sampling and data collection factors among the three cohorts. First, the validity of self-reported diagnosis of endometriosis and to a lesser extent UL are known to be <75% in general population cohorts, such as UKBB, compared to more highly validated self-assessment in the medical professional NHSII and WHS cohorts7. Second, endometriosis clinical definitions prior to the 1990s were more restrictive—often limited to the presence of endometrioma and/or "powderburn" superficial peritoneal lesions among adult women60. Subsequently, definitions have expanded to recognize a wide range of superficial peritoneal phenotypic presentations, as well as incidence among adolescents and young women61. It may be impactful therefore that the WHS participants were ≥ 45 years of age in 1992, while NHSII participants were ≥ 25 years of age in 1989, and UKBB participants were aged 40 to 69 in 2006. Thus, disease definitions varied during the peak calendar years of incidence among the cohorts, and in addition, while the NHSII were queried about endometriosis prospectively during their reproductive years, the WHS and UKBB cohorts were cross-sectionally asked to recall their gynecologic health experience decades earlier. It is also important to note that while WHS and NHSII participants were asked specifically about endometriosis diagnosis via questionnaire, the UKBB data collection included qualitative interviews during which endometriosis would be documented only when the participant herself raised it as a health issue. Those with mild symptoms or those past their reproductive years and thus past the moderate to severe life-impacting symptoms of the disease may have been less likely to offer endometriosis among the list of their health issues. This is supported by the low prevalence of endometriosis reported within the UKBB compared to WHS, NHSII, and other population-based estimates62. However, the UKBB participants (due to the qualitative interview structure and recall bias) and the WHS participants (due to recall bias) could have been more likely to choose to report endometriosis if they also suffered from UL together, resulting in diagnostic bias and consequently an inflation of effect estimates. Indeed, the population heterogeneity and differing potential for diagnostic bias by cohort fits with the observed differences among effect estimate magnitudes with the RRs and CI widths ordered from NHSII (RR = 1.56) to WHS (RR = 1.96) to UKBB (RR = 3.50) (Fig. 4).
Epidemiologic meta-analysis demonstrates endometriosis is associated with UL. Random-effects, inverse-variance-weighted meta-analysis was performed across the effect sizes and standard errors in 402,868 women from three cohorts (NHSII, WHS, and UKBB). Squares represent point estimates from individual studies, whiskers correspond to the 95% CIs, and the diamond represents results from the meta-analysis. There was evidence of significant heterogeneity based on Cochran's Q statistic (P < 1 × 10−4)
Table 2 Multivariable-adjusted effect estimates of the association between endometriosis and UL among women in NHSII, WHS, and UKBB cohorts
Bioinformatic analyses of UL risk SNPs and loci
To estimate the genetic correlation between UL and various reproductive traits, as well as cardiometabolic traits/diseases, we performed LD Hub analysis for a total of 21 traits/diseases (Supplementary Data 1). We observe significant correlations between increased risk of UL and earlier age of menarche (rg = −0.16, P = 3.7 × 10−6), earlier age of first birth (rg = −0.14, P = 1.0 × 10−3), increased levels of triglycerides (rg = 0.13, P = 1.9 × 10−3), and increased BMI (rg = 0.11, P = 2.0 × 10−3), as previously suggested by epidemiological studies63,64, illustrating that common genetic factors can predispose women to both risk factors related to, for example, adverse metabolic and cardiovascular disease risk and UL. Gene-set and tissue enrichment analyses across 8971 SNPs with suggestive (P < 1 × 10−5) or significant (P < 5 × 10−8) UL associations using DEPICT65 reveal enrichments (false discovery rate (FDR) < 0.05) in gene sets, such as steroid hormone receptor (GO:0035258; P = 1.03 × 10−5), hormone receptor binding (GO:0051427; P = 9.07 × 10−5), and nuclear hormone receptor binding (GO:0035257; P = 1.53 × 10−4) (Supplementary Data 2 and 3). The results are concordant with the hormone-driven nature of UL. We did not observe any cell/tissue types significantly enriched for the expression of the genes in the associated loci (Supplementary Fig. 10). To identify potential causal genes at UL risk loci, we used a summary-data based MR (SMR) method, including both eQTL and mQTL data from peripheral blood66,67. We identify 18 potential causal genes showing no significant heterogeneity in SMR (PHEIDI > 5 × 10−3), including WNT4 (rs55938609, PSMR = 6.92 × 10−15), GREB1 (rs35417544, PSMR = 3.93 × 10−19), WT1 (rs12280757, PSMR = 1.87 × 10−18), and FOXO1 (rs3924478, PSMR = 5.76 × 10−10) (Supplementary Data 4 and 5).
FOXO1 expression in UL
To explore potential functional significance, we examined expression of the FOXO1 protein, a transcription factor that plays an important role in cell proliferation, apoptosis, DNA repair, and stress response68. Interestingly, inactivation of FOXO1 promotes cell proliferation and tumorigenesis in several hormone-regulated malignancies, such as prostate, breast, cervical, and endometrial cancers69,70,71,72. Conversely, we observe a significant increase in nuclear FOXO1 protein expression in UL compared to myometrial samples using immunohistochemistry on tissue microarrays (Supplementary Fig. 11). Patient-matched tumor-normal pairs show 1.69-fold higher (P = 0.01; paired t-test) nuclear FOXO1 expression in UL, while the expression is as much as 2.32-fold greater (P = 1.52 × 10−9; Welch's t-test) when all 335 UL are considered (Supplementary Fig. 12). These results are consistent with a previous study73, which showed phosphorylated (p) FOXO1 (pSer256) to be predominantly present in the nucleus in UL, but sequestered in the cytoplasm of myometrium. The concomitant increase of p-FOXO1 and reduced expression of its interaction partner 14-3-3\({\mathrm{\gamma }}\) in UL has been suggested to lead to impaired nuclear/cytoplasmic shuttling of p-FOXO1, which promotes cell survival73,74,75. We performed stratification of samples by genotype, revealing a statistically significant increase in FOXO1 levels of UL harboring the risk allele for rs6563799 (allelic dosage, P = 0.047; homozygosity for risk allele, P = 0.035) (Supplementary Figs. 11 and 13). An increase in FOXO1 levels of UL with the rs7986407 risk allele is also observed; however, the change is not statistically significant (Supplementary Figs. 11 and 13).
In our meta-analysis of GWAS on UL, we identify 29 genomic loci to be significantly associated with UL in women of white European ancestry, including eight novel and 21 previously reported loci. Candidate genes in the identified loci implicate pathways of estrogen and progesterone signaling (ESR1, FSHB, GREB1, WNT2, and WNT4), as well as cell growth (FOXO1, PDGFRA, TERT, TERC, and TP53) in predisposing women to UL. We do not confirm five of 26 previously identified loci reported to be significantly associated with UL12,14,15,16. Two of these loci, 3p24.1 and 16q12.1, are nominally significant (P < 1 × 10−5) in our GWAS meta-analysis, but the remaining three loci (3q29, 17q25.3 and a distinct region at 22q13.1) do not reach nominal significance. Ancestral differences may explain the absence of the association originally identified in African American women in the genomic region at 22q13.1, while variation in phenotypic definitions12 may underlie the two other loci.
Discovery of eight novel loci significantly associated with UL reveals several candidate genes of particular interest: BABAM2, FSHB, HMGA1, and WNT2. Because UL are benign tumors that rarely, if ever, develop into malignancy, the association between UL and multiple loci harboring well-known oncogenes and tumor suppressor genes is also worthy of note. Fine-mapping of the TP53 locus identifies rs78378222 to be the most probable causal variant, which has been shown to disrupt the polyadenylation sequence in the 3'UTR of TP53 and result in reduced expression of mRNA39. We also observe nuclear FOXO1 levels to be significantly elevated in UL when compared to myometrium. FOXO1 is a downstream target of the Akt signaling pathway that responds to hormone signaling through the progesterone receptor in UL and activates proliferative responses76.
HMB is one of the major debilitating symptoms of UL and can have a substantial impact on a woman's quality of life. Here, we report GWAS on both UL limited by HMB and solely on HMB, revealing potential targets for pharmacologic intervention: ARL14EP, ATM, TERT, and FGFR4. In addition, MR analyses suggest that genetic predisposition to UL is causally linked to an increased risk of HMB. These results form a solid basis for further work to elucidate the mechanisms underlying UL-related HMB and towards tailored treatments of UL and HMB.
Biological overlap between UL and endometriosis, two highly common gynecologic diseases has long been suspected due to similarities in molecular mechanisms and progenitor cells. Our UL GWAS meta-analysis indicates that genes previously associated with endometriosis and involved in hormone-signaling pathways are also associated with UL (WNT4/CDC42, GREB1, ESR1, and FSHB). Overlap observed in the genetic etiology of endometriosis and UL led us to epidemiologically quantify the co-occurrence of these two diseases across three independent cohorts. The epidemiological meta-analysis indicates that women with a history of endometriosis are at elevated risk for reporting UL. Results from our MR analyses suggest that genetic predisposition to endometriosis is causally linked to increased risk of UL. Alternatively, given the discordance in the direction of allelic effects for the UL and endometriosis loci, our MR results may indicate a significant overlap in the underlying biology of the two diseases. Additional work is needed to better quantify the contribution of genetic effects to the directional relationship between endometriosis and UL. Results of which will enable us to quantify what portion of the MR results reflect the fundamental pathobiological overlap in these two diseases of the uterus. Further characterization of the mutual pathogenic mechanisms of UL and endometriosis has the capacity to discover not only a deeper understanding of the underlying biology, but also treatments for two diseases that cause significant morbidity in roughly one-third of the world's population.
For UL GWAS meta-analysis, four population-based cohorts (WGHS, NFBC, QIMR and UKBB) and one direct-to-consumer cohort (23andMe) from the FibroGENE consortium were included (Supplementary Table 1), resulting in 35,474 UL cases and 267,505 female controls of white European ancestry. Sample sizes were maximized using a basic, harmonizing phenotype definition to separate cases and controls solely based on either self-report or clinically documented UL history. Our large-scale epidemiologic analysis was comprised of three population-based cohorts (NHSII, WHS, and UKBB), totaling 402,869 women. HMB GWAS included the UKBB cohort, consisting of 220,759 women. Detailed descriptions of cohorts and sample selections are available in Supplementary Methods. All participants provided informed consent in accordance with the processes approved by the relevant jurisdiction for human subject research for each cohort: the Partners HealthCare System Human Research Committee (WHS/WGHS), the Ethical Committee of the Northern Ostrobothnia Hospital District (NFBC), the Human Research Ethics Committee at the QIMR Berghofer Medical Research Institute and the Australian Twin Registry (QIMR), the North West Multi-centre Research Ethics Committee (UKBB), Ethical and Independent Review Services (an external institutional review board; 23andMe), and the Institutional Review Boards at Harvard T.H. Chan School of Public Health and Brigham and Women's Hospital (Partners Human Research Committee) (NHSII).
Several different Illumina-based genotyping platforms (Illumina Inc., San Diego, CA, USA) were used: HumanHap300 Duo'+' chips or the combination of the Human-Hap300 Duo and iSelect chips (WGHS), Infinium 370cnvDuo array (NFBC), 317 K, 370 K, or 610 K SNP platforms (QIMR). Genotyping of participants in the UKBB was performed either on the Affymetrix UK BiLEVE or Affymetrix UK Biobank Axiom® array with over 95% similarity. Genotyping of participants in the 23andMe cohort was performed on various versions of Illumina-based BeadChips.
Quality control and imputation
Each cohort conducted quality control measures and imputation for their data. For WGHS, NFBC, QIMR, and 23andMe, all cases and controls with a genotyping call rate <0.98 were excluded from the study. Imputation was performed on both autosomal and sex chromosomes using the reference panel from the 1000 Genomes Project European dataset (1000 G EUR) Phase 3. Imputation was carried out using ShapeIt2 and IMPUTE2 softwares77,78. SNPs with call rates of <99% or with deviation from Hardy-Weinberg equilibrium (P ≤ 1 × 10−6) were excluded from further analyses. Population stratification for the data was examined with principal component analysis (PCA) using EIGENSTRAT79. The four HapMap populations were used as reference groups: Europeans (CEU), Africans (YRI), Japanese (JPT), and Chinese (CHB). All observed outliers were removed from the study. UKBB data QC and imputation were performed centrally, prior to public release of the data80. Genotype data used in the present analyses were imputed up to the Haplotype Reference Consortium (HRC) panel. We applied additional quality control filters to exclude poorly imputed SNPs (r2 < 0.4) and SNPs with a MAF of <1%.
Association analyses
Using additive encoding of genotypes and adjusting for age, BMI, and/or the first five PCs, logistic regression analysis was performed in WGHS, NFBC, QIMR, and 23andMe cohorts and summary statistics were provided, including beta coefficients, χ2 values, and standard errors, for genotyped and imputed SNPs. The UKBB association analyses were conducted using a linear mixed model (BOLT-LMM v.2.3.2)81 adjusting for the two array types used, age and BMI (fixed effects), and a random effect accounting for relatedness between women. Effect size estimates (β and SE) from the linear mixed-model were converted to log-odds scale prior to meta-analysis. A fixed-effects, inverse-variance-weighted (IVW) meta-analysis on summary statistics was conducted using METAL82 across all cohorts (Supplementary Data 6). A total of 8,662,096 SNPs were available from at least two of the five cohorts. A quantile-quantile plot of the results from meta-analysis across all GWAS cohorts is shown in Supplementary Fig. 1. Details on the overall genomic inflation factor and number of analyzed SNPs for each cohort are provided in Supplementary Table 2. For GWAS meta-analysis, independence of genetic association with UL was defined as SNPs in low LD (r2 < 0.1) with nearby (≤500 kb) significantly associated SNPs. Individual loci correspond to regions of the genome containing all SNPs in LD (r2 > 0.6) with index SNPs. Any adjacent regions within 250 kb of one another were combined and classified as a single locus of association. All associated genomic regions were confirmed to have lead SNPs that were either directly genotyped or that met a rigorously high quality imputation threshold (INFO > 0.9) in at least two cohorts.
Linkage disequilibrium score regression (LDSC)
Analysis of residual inflation in test statistics was conducted using univariate LDSC regression. Individual χ2 values for each SNP analyzed in the GWAS meta-analysis was regressed onto LD scores estimated from the 1000 G EUR panel. Heritability calculations can be derived from analyzing the slope and y-axis intercept of the slope of the regression line. Percent impact of confounders, such as population stratification, on test statistic inflation are quantified as the LDSC ratio [((intercept–1))/((mean χ2–1))] × 100%. Remaining effects [(1–LDSC ratio) × 100%] represent the percentage of inflation attributed to polygenic heritability. Univariate LDSC regression was conducted using the LDSC software (https://github.com/bulik/ldsc.git). Adjustment of heritability (h2) calculations to the liability scale were performed by accounting for the prevalence of UL in the sample (~0.132) compared to the general population (~0.300). LDSC software was also used to estimate the genetic correlation between UL and endometriosis (Endo) using endometriosis GWA meta-analysis summary data from Sapkota et al.45 consisting of only European cohorts. The heritability and LD score intercepts for both traits were computed, in this analysis with SNPs present in both datasets for LDSC regression again using LD scores from the 1000 G EUR panel. Genetic correlation between traits was estimated as the genetic covariance among SNPs / √ h2UL × h2Endo.
Approximate conditional analysis
Approximate conditional analysis, implemented in GCTA83, was conducted to dissect distinct signals of association at each locus. Of note, where lead SNPs at adjacent loci mapped within 1 Mb of each other, loci were combined as a single region for conditional analysis, to account for potential LD between SNPs in different loci. GCTA makes use of meta-analysis association summary statistics (log-OR and corresponding standard error) and a reference panel of individual-level genotype data to obtain LD between all pairs of SNPs at a locus (or region) that approximates the covariance in effect estimates in a joint model. For these analyses, we made use of 5000 randomly selected white British women (of European descent) as reference. We used the -cojo-slct option to select index variants for each distinct association signal, at a locus-wide significance threshold of P < 10−5, which is a conservative Bonferroni correction for the number of SNPs mapping to a locus. For loci with multiple distinct association signals, we obtained the conditional association summary statistics for each by conditioning on all other index SNPs at the locus (or region) using the -cojo-cond option.
Fine-mapping distinct association signals
For each distinct association signal, association summary statistics (log-OR and corresponding standard error) were extracted from the meta-analysis for all SNPs at the locus (or region). For loci with a single signal of association, we made use of association summary statistics from the unconditional meta-analysis. For loci with multiple signals of association, we made use of association summary statistics from the approximate conditional analysis. For each SNP j, we calculated an approximate Bayes' factor in favor of association84, given by
$$\Lambda _j = \sqrt {\frac{{V_j}}{{V_j + \omega }}} {\mathrm{exp}}\left[ {\frac{{\omega \beta _j^2}}{{2V_j\left( {V_j + \omega } \right)}}} \right],$$
where βj and Vj denote the estimated log-OR and corresponding variance from the meta-analysis. The parameter ω denotes the prior variance in allelic effects, taken here to be 0.04 for a disease outcome84. We then calculated the posterior probability, πj, that the jth SNP is causal for the association signal, given by
$$\pi _{Cj} = \frac{{{\it{\Lambda }}_j}}{{\mathop {\sum}\nolimits_k {{\it{\Lambda }}_k} }},$$
where the summation is over all retained variants in the locus (or region). The 99% credible set for each signal was then constructed by: (i) ranking all variants according to their Bayes' factor, Λj; and (ii) including ranked variants until their cumulative posterior probability of causality is at least 0.99.
Heavy menstrual bleeding (HMB) GWAS
The HMB GWAS was conducted using data from the UKBB cohort (Supplementary Methods). Both hospital-linked medical records and self-report were considered to identify women with a history of UL, while for HMB only hospital-linked medical records were taken into account. Controls had no previous history of either UL or HMB. Association analyses were performed using a linear mixed model (BOLT-LMM v.2.3.2)81. Effect size estimates (β and SE) from the linear mixed-model were converted to log-odds scale.
Mendelian randomization (MR)
MR analyses were performed using the Two Sample Mendelian Randomization R package. GWAS summary statistics on HMB from the UKBB cohort were used to create outcome data for MR between UL (exposure) and HMB (outcome). To avoid overlap between samples in the exposure and outcome cohorts, we performed UL GWAS excluding all the HMB cases85. LD pruning was performed to confirm no duplication of exposure haplotypes or SNPs. Subsequently, data were harmonized to ensure the same reference alleles were used in exposure and outcome GWAS and that the variants were present in both GWAS datasets. Thirteen independent SNPs associated with UL from our GWAS meta-analysis were available in the HMB GWAS summary data to test for a causal effect of UL on HMB. There were too few significant SNPs available for HMB to test for a causal effect of HMB on UL.
GWAS summary statistics on endometriosis (with laparoscopy, without laparoscopy, and all self-reported endometriosis cases) from the WHS cohort were used to create outcome data for MR between UL (exposure) and endometriosis (outcome). To avoid overlap between samples in the exposure and outcome cohorts, WGHS was excluded from the UL GWAS for MR analysis. Twenty-two independent SNPs associated with UL were available in the endometriosis GWAS summary data to test for a causal effect of UL on endometriosis. For reverse causation model, summary statistics from seven GWAS listing 'endometriosis' as the phenotype of interest were available from the EMBL-EBI NHGRI GWAS catalog (Study Accession: GCST000797, GCST001894, GCST001720, GCST005906, GCST000916, GCST004549, GCST004873). Due to a low number of cases/controls or insufficient number of SNPs after LD pruning and data harmonizing, only one of the studies (GCST004549) was included in the analysis. Sixteen independent SNPs associated with endometriosis were available in our UL GWAS summary data to test for a causal effect of endometriosis on UL. The IVW model was used to test causality between exposure and outcome. In addition, the IVW (Q) method was used to test for heterogeneity, leave-one-out sensitivity analysis to identify the effect of individual SNPs, and MR Egger for horizontal pleiotropy. Due to heterogeneity in our initial MR estimates, we have now leveraged a similar approach to the one published in Corbin et al., 2016, to identify the minimum set of variants that when used as a genetic instrument eliminate heterogeneity86. We also conducted the MR-PRESSO test to identify and adjust for variants causing significant bias through horizontal pleiotropy87. MR-PRESSO method (1) applies a global test to evaluate whether horizontal pleiotropy is present, (2) calculates the causal estimates incorporating correction for the detected horizontal pleiotropy, and (3) applies a distortion test to evaluate if the causal estimate is significantly different after adjustment for outliers. We have reported the initial estimates along with the outlier-adjusted estimates as both the global and distortion tests showed significant results.
Co-morbidity analyses
Each cohort was analyzed individually with study-specific models chosen and covariates coded as appropriate for each cohort's data structure (Supplementary Methods). The study-specific effect estimates were combined using meta-analysis to obtain a summary RR. Between study heterogeneity was assessed with Cochran Q statistic and the I2 statistic88. Because heterogeneity among the studies was identified, we reported a random-effects IVW effect estimate based on the DerSimonian and Laird method89.
LD Hub, gene-set, cell/tissue enrichment, and SMR analyses
LD Hub analysis90 was conducted using summary-level results data of UL GWAS meta-analysis to estimate the genetic correlation between UL and 21 different traits/diseases, including various reproductive traits and cardiometabolic traits/diseases that have publicly available GWAS results on the LD Hub repository. Multiple-testing correction was performed (0.05/21 = 2.4 × 10−3). For gene-set and cell/tissue enrichment, summary statistics from the set of 8971 SNPs with suggestive (P < 1 × 10−5) or significant associations (P < 5 × 10−8) were analyzed using the Data-driven Expression-Prioritized Integration for Complex Traits (DEPICT) software65. Using the 1000 G EUR panel as a reference for LD calculations and the 'clumping' algorithm in PLINK91, we identified 104 independent loci at the suggestive threshold for DEPICT analyses (Supplementary Data 2). FDR < 0.05 was considered statistically significant. For SMR analysis, SNPs present in at least two studies in the summary statistics were considered. The analysis was run using eQTL data from the CAGE blood dataset66 and mQTLs from the LBC_BSGS blood dataset67.
FOXO1 immunohistochemistry and genotyping
FOXO1 immunostaining was performed on two replicate tissue microarrays (TMAs) containing 335 UL and 36 myometrium tissue samples from 200 white women of European ancestry obtained from myomectomies and hysterectomies. Tissue cores on the replicate TMAs represent different regions of the same samples, which include corresponding tumor-normal tissue pairs from 35 women. Immunohistochemistry was carried out using the BOND staining system (Leica Biosystems, Buffalo Grove, IL) with a primary antibody dilution 1:100 (clone C29H4, Cell Signaling Technology, Danvers, MA) and hematoxylin as the counterstain. Immunostaining was analyzed using Aperio ImageScope software (Leica Biosystems). Each core was evaluated for the ratio of stain to counterstain taking into account variable cellularity between cores. Only nuclear labeling of the protein was evaluated. The average stain-to-counterstain ratio was compared between patient-matched UL and myometrium samples using a paired t-test (two-tailed), while an unpaired t-test (Welch's t-test, two-tailed) was applied to compare all UL and myometrium samples. Genomic DNA from 109 UL on the TMA was available for genotyping. These UL were genotyped for two SNPs with genome-wide significance at the 13q14.11 locus: rs6563799 and rs7986407. For each SNP, the average FOXO1 stain-to-counterstain ratio was compared across increasing dosage of the risk allele using a one-way analysis of variance test (two-tailed). We also performed an unpaired t-test to compare mean expression of UL homozygous for the risk variant against the other genotypes (Welch's t-test, two-tailed). P-values < 0.05 were considered statistically significant.
For WHS see http://whs.bwh.harvard.edu/; for NFBC see http://www.oulu.fi/nfbc/; for QIMR see http://www.qimrberghofer.edu.au/; for UK Biobank see http://www.ukbiobank.ac.uk/; for 23andMe see https://research.23andme.com/; for METAL see http://csg.sph.umich.edu/abecasis/metal/; for LDSC see https://github.com/bulik/ldsc.git; for DEPICT see https://data.broadinstitute.org/mpg/depict/; for SMR see http://cnsgenomics.com/software/smr/; and for PLINK see http://pngu.mgh.harvard.edu/purcell/plink/.
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
The authors declare that the data supporting the findings of this study are available within the article and its Supplementary Information files. Summary statistics for the top 10,000 UL GWAS meta-analysis variants are provided in Supplementary Data 6. UL GWAS meta-analysis summary statistics (without 23andMe), UL GWAS limited by HMB and HMB GWAS summary statistics will be made available through the NHGRI-EBI GWAS Catalog https://www.ebi.ac.uk/gwas/downloads/summary-statistics. To request access to 23andMe GWAS summary statistics, please visit https://research.23andme.com/dataset-access/.
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The authors thank all of the women and their families who participated in WGHS, NFBC, QIMR, UK Biobank, 23andMe, and NHSII, and acknowledge the Channing Division of Network Medicine, Department of Medicine, Brigham and Women's Hospital and Harvard Medical School. This study was supported by the U.S. National Institutes of Health (NIH)/Eunice Kennedy Shriver National Institute of Child Health and Human Development (NICHD) grant HD060530 to C.C.M. C.C.M. is also supported by the NIHR Manchester Biomedical Research Centre. N.M. acknowledges support from the Academy of Finland (295693) and Orion Research Foundation. H.R.H. is supported by NIH K22 CA193860. T.F. is supported by the NIHR Biomedical Research Centre, Oxford. S.E.M. is supported by the National Health and Medical Research Council (NHMRC) Fellowship Scheme (1103623). We thank the Specialized Histopathology Core of the Dana-Farber/Harvard Cancer Center for FOXO1 immunostaining. The Dana-Farber/Harvard Cancer Center is supported in part by an NCI Cancer Center Support Grant P30 CA06516. Further acknowledgements are provided in Supplementary Note 1.
These authors contributed equally: C.S. Gallagher, N. Mäkinen, H.R. Harris, N. Rahmioglu.
These authors jointly supervised this work: D.I. Chasman, S.A. Missmer, K.T. Zondervan, C.C. Morton.
Department of Genetics, Harvard Medical School, Boston, MA, 02115, USA
C. S. Gallagher
Department of Obstetrics and Gynecology, Brigham and Women's Hospital, Harvard Medical School, Boston, MA, 02115, USA
N. Mäkinen, Z. Ruhioglu & C. C. Morton
Program in Epidemiology, Division of Public Health Sciences, Fred Hutchinson Cancer Research Center, Seattle, WA, 98109, USA
H. R. Harris
Wellcome Centre for Human Genetics, University of Oxford, Oxford, OX3 7BN, UK
N. Rahmioglu, T. Ferreira, A. P. Morris & K. T. Zondervan
Endometriosis CaRe Centre, Nuffield Department of Women's and Reproductive Health, University of Oxford, John Radcliffe Hospital, Oxford, OX3 9DU, UK
O. Uimari, N. Shigesi, C. M. Becker & K. T. Zondervan
Department of Obstetrics and Gynecology, Oulu University Hospital and PEDEGO Research Unit & Medical Research Center Oulu, University of Oulu and Oulu University Hospital, 90220, Oulu, Finland
O. Uimari & H. Martikainen
Department of Biostatistics, University of Liverpool, Liverpool, L69 3GL, UK
J. P. Cook & A. P. Morris
Big Data Institute, Li Ka Shing Center for Health Information and Discovery, Oxford University, Oxford, OX3 7LF, UK
T. Ferreira
Vanderbilt Genetics Institute, Vanderbilt Epidemiology Center, Institute for Medicine and Public Health, Department of Obstetrics and Gynecology, Vanderbilt University Medical Center, Nashville, TN, 37203, USA
D. R. Velez-Edwards
Division of Epidemiology, Department of Medicine, Institute for Medicine and Public Health, Vanderbilt Genetics Institute, Vanderbilt University Medical Center, Nashville, TN, 37203, USA
T. L. Edwards
Institute for Molecular Bioscience, University of Queensland, Brisbane, QLD, 4072, Australia
S. Mortlock & G. W. Montgomery
MRC Epidemiology Unit, University of Cambridge School of Clinical Medicine, Institute of Metabolic Science, Cambridge Biomedical Campus, Cambridge, CB2 0QQ, UK
F. Day & J. R. B. Perry
Center for Life Course Health Research, Faculty of Medicine, University of Oulu, 90220, Oulu, Finland
V. Karhunen & M.-R. Järvelin
Unit of Primary Health Care, Oulu University Hospital, 90220, Oulu, Finland
Department of Epidemiology and Biostatistics, MRC-PHE Centre for Environment and Health, School of Public Health, Imperial College London, London, W2 1PG, UK
Biocenter Oulu, University of Oulu, 90220, Oulu, Finland
M.-R. Järvelin
Department of Life Sciences, College of Health and Life Sciences, Brunel University London, Uxbridge, Middlesex, UB8 3PH, UK
Department of Human Genetics, David Geffen School of Medicine, University of California at Los Angeles, Los Angeles, CA, 90095, USA
R. M. Cantor
Division of Preventative Medicine, Brigham and Women's Hospital, Harvard Medical School, Boston, MA, USA
P. M. Ridker, J. E. Buring & D. I. Chasman
Obstetrics and Gynecology Epidemiology Center, Brigham and Women's Hospital and Harvard Medical School, Boston, MA, 02115, USA
K. L. Terry
Department of Epidemiology, Harvard T.H. Chan School of Public Health, Boston, MA, 02115, USA
K. L. Terry & S. A. Missmer
Genetic Epidemiology, QIMR Berghofer Medical Research Institute, Brisbane, QLD, 4006, Australia
S. D. Gordon, G. W. Montgomery, D. R. Nyholt & N. G. Martin
Psychiatric Genetics, QIMR Berghofer Medical Research Institute, Brisbane, QLD, 4006, Australia
S. E. Medland, P. A. Lind & J. N. Painter
Institute of Health and Biomedical Innovation and School of Biomedical Science, Queensland University of Technology, Brisbane, QLD, 4059, Australia
D. R. Nyholt
23andMe, Mountain View, CA, 94041, USA
D. A. Hinds, J. Y. Tung, Michelle Agee, Babak Alipanahi, Adam Auton, Robert K. Bell, Katarzyna Bryc, Sarah L. Elson, Pierre Fontanillas, Nicholas A. Furlotte, Karen E. Huber, Aaron Kleinman, Nadia K. Litterman, Matthew H. McIntyre, Joanna L. Mountain, Elizabeth S. Noblin, Carrie A. M. Northover, Steven J. Pitts, J. Fah Sathirapongsasuti, Olga V. Sazonova, Janie F. Shelton, Suyash Shringarpure, Chao Tian, Vladimir Vacic & Catherine H. Wilson
Department of Obstetrics, Gynecology, and Reproductive Biology, College of Human Medicine, Michigan State University, Grand Rapids, MI, 49503, USA
S. A. Missmer
Department of Pathology, Brigham and Women's Hospital, Harvard Medical School, Boston, MA, 02115, USA
C. C. Morton
Broad Institute of MIT and Harvard, Cambridge, MA, 02142, USA
Manchester Centre for Audiology and Deafness, Manchester Academic Health Science Center, University of Manchester, Manchester, M13 9PL, UK
N. Mäkinen
N. Rahmioglu
O. Uimari
J. P. Cook
N. Shigesi
S. Mortlock
Z. Ruhioglu
F. Day
C. M. Becker
V. Karhunen
H. Martikainen
P. M. Ridker
J. E. Buring
S. D. Gordon
S. E. Medland
G. W. Montgomery
D. A. Hinds
J. Y. Tung
J. R. B. Perry
P. A. Lind
J. N. Painter
N. G. Martin
A. P. Morris
D. I. Chasman
K. T. Zondervan
the 23andMe Research Team
Michelle Agee
, Babak Alipanahi
, Adam Auton
, Robert K. Bell
, Katarzyna Bryc
, Sarah L. Elson
, Pierre Fontanillas
, Nicholas A. Furlotte
, Karen E. Huber
, Aaron Kleinman
, Nadia K. Litterman
, Matthew H. McIntyre
, Joanna L. Mountain
, Elizabeth S. Noblin
, Carrie A. M. Northover
, Steven J. Pitts
, J. Fah Sathirapongsasuti
, Olga V. Sazonova
, Janie F. Shelton
, Suyash Shringarpure
, Chao Tian
, Vladimir Vacic
& Catherine H. Wilson
C.S.G., S.A.M., K.T.Z. and C.C.M. designed the study. O.U., C.M.B., H.M., M.-R.J., J.E.B, S.E.M., D.R.N., P.A.L., J.N.P. and the 23andMe Research team contributed to phenotypic/clinical aspects of the cohorts. O.U., J.P.C., N.R., T.F., D.R.V.-E., T.L.E., F.D., V.K., P.M.R., S.D.G., S.E.M., G.W.M., D.R.N., D.A.H., J.Y.T., the 23andMe Research team, J.R.B.P., P.A.L., J.N.P., N.G.M., A.P.M., D.I.C. and K.T.Z. contributed to genotyping, quality control, imputation, and/or association analysis of the genotyping data. C.S.G. and N.R. performed the UL GWAS meta-analysis. N.R. conducted the HB GWAS. R.M.C., A.P.M. and D.I.C. provided statistical genetics advice. C.S.G., N.M., N.R., Z.R., S.M., G.W.M. and A.P.M. carried out or assisted with GWAS downstream analyses. C.S.G., H.R.H., O.U., N.S., N.R., K.L.T, J.E.B, S.A.M. and K.T.Z. contributed to large-scale epidemiologic analysis. N.M., C.S.G. and H.R.H. drafted the paper. G.W.M., N.G.M., A.P.M., D.I.C., S.A.M., K.T.Z and C.C.M provided critical comments on the paper, draft, and analysis. All authors read and approved the final paper.
Correspondence to N. Mäkinen or C. C. Morton.
K.T.Z and C.M.B through Oxford University have research collaborations in benign gynecology with Bayer AG, Roche Diagnostics, Volition UK, and M DNA Life Sciences. D.A.H., J.Y.T., and members of the 23andMe Research Team are employees of 23andMe, Inc., and hold stock or stock options in 23andMe. The remaining authors declare no competing interests.
Peer review information Nature Communications thanks Siddhartha Kar and Joellen Schildkraut for their contribution to the peer review of this work. Peer reviewer reports are available.
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Supplementary Data 1
Gallagher, C.S., Mäkinen, N., Harris, H.R. et al. Genome-wide association and epidemiological analyses reveal common genetic origins between uterine leiomyomata and endometriosis. Nat Commun 10, 4857 (2019). https://doi.org/10.1038/s41467-019-12536-4
Polymorphic loci of the <i>ESR1<i/> gene are associated with the risk of developing preeclampsia with fetal growth retardation
O. V. Golovchenko
, M. Yu. Abramova
, I. V. Ponomarenko
& M. I. Churnosov
Obstetrics, Gynecology and Reproduction (2021)
Integration of genome-wide association study and expression quantitative trait locus mapping for identification of endometriosis-associated genes
Ya-Ching Chou
, Ming-Jer Chen
, Pi-Hua Chen
, Ching-Wen Chang
, Mu-Hsien Yu
, Yi-Jen Chen
, Eing-Mei Tsai
, Shih-Feng Tsai
, Wun-Syuan Kuo
& Chii-Ruey Tzeng
Scientific Reports (2021)
Risk factors for non‐response and discontinuation of Dienogest in endometriosis patients: A cohort study
Konstantinos Nirgianakis
, Cloé Vaineau
, Lia Agliati
, Brett McKinnon
, Maria L. Gasparri
& Michael D. Mueller
Acta Obstetricia et Gynecologica Scandinavica (2021)
Epidemiology of Adenomyosis
Kristen Upson
& Stacey A. Missmer
Seminars in Reproductive Medicine (2020)
Hereditary risk factors for uterine leiomyoma: a search for marker SNPs
K.A. Svirepova
, M.V. Kuznetsova
, N.S. Sogoyan
, D.V. Zelensky
, E.A. Lolomadze
, G.V. Mikhailovskaya
, N.D. Mishina
, A.E. Donnikov
& D.Yu. Trofimov
Bulletin of Russian State Medical University (2020)
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\begin{document}
\title[]{An infinite-dimensional affine stochastic volatility model} \author{Sonja Cox, Sven Karbach, Asma Khedher}
\thanks{This research is partially funded by The Dutch Research Council
(NWO) (Grant No: C.2327.0099)}
\begin{abstract} We introduce a flexible and tractable infinite-dimensional stochastic volatility model. More specifically, we consider a Hilbert space valued Ornstein--Uhlenbeck-type process, whos instantaneous covariance is given by a pure-jump stochastic process taking values in the cone of positive self-adjoint Hilbert-Schmidt operators. The tractability of our model lies in the fact that the two processes involved are jointly \emph{affine}, i.e., we show that their characteristic function can be given explicitely in terms of the solutions to a set of generalised Riccati equations. The flexibility lies in the fact that we allow multiple modeling options for the instantaneous covariance process, including state-dependent jump intensity.\par
Infinite dimensional volatility models arise e.g.\ when considering the dynamics of forward
rate functions in the Heath-Jarrow-Morton-Musiela modeling framework using the Filipovi\'c space. In this setting we
discuss various examples: an infinite-dimensional version of the
Barndorff-Nielsen--Shephard stochastic volatility model, as well as a model involving self-exciting volatility.
\end{abstract}
\keywords{Stochastic volatility, infinite-dimensional affine
processes, Heath-Jarrow-Morton-Musiela framework, forward price dynamics, Riccati equations,
state-dependent jump intensity} \maketitle
\section{Introduction} In this paper we propose a new class of \emph{affine} stochastic volatility models $(Y_t,X_t)_{t\geq 0}$, where $(Y_t)_{t\geq 0}$ takes values in a real separable Hilbert space $(H, \langle \cdot, \cdot \rangle_H)$ and $(X_t)_{t\geq 0}$ is a time-homogeneous \emph{affine} Markov process taking values in $\mathcal{H}^+ = \mathcal{L}_2^+(H)$, the cone of positive self-adjoint Hilbert-Schmidt operators on $H$. The process $X$ is taken from a class of affine processes introduced in~\cite{CKK20}. The process $(Y_t)_{t\geq 0}$ is modeled by the following stochastic differential equation \begin{align}\label{eq:stochastic-vola-model}
\D Y_{t}=\mathcal{A} Y_{t}\,\D t+ X_t^{1/2} \, \D W^{Q}_{t}, \quad t\geq 0, \quad Y_0=y \in H\,, \end{align} where $\mathcal{A}\colon\dom(\mathcal{A})\subseteq H\to H$ is a possibly unbounded operator with dense domain $\dom(\mathcal{A})$ and $(W^{Q}_{t})_{t\geq 0}$ is a $Q$-Brownian motion independent of $X$, with $Q$ a positive self-adjoint trace-class operator on $H$. Assuming that $X$ is progressively measurable and using moment bounds on $X$ established in~\cite{CKK20}, the existence of a solution to~\eqref{eq:stochastic-vola-model} is straightforward (see Lemma~\ref{lem:integrand} below).\par
In Section~\ref{sec:affine-vari-proc} we show that under the assumption that the Markov process $(X_t)_{t\geq 0}$ has c\`adl\`ag paths, it is a square-integrable semimartingale. This follows from the formulation of an associated martingale problem in terms of what we call {\it a weak} generator (see Definition \ref{def:weak-generator}) of the Markov process $(X_t)_{t\geq 0}$ and yields the explicit representation of $(X_t)_{t\geq 0}$ as
\begin{align}\label{eq:canonical-rep-X-intro} X_{t}=x+\int_{0}^{t}\Big(b+B(X_{s})+\int_{\cH^{+}\cap \set{
\norm{\xi}> 1}}\xi\,M(X_{s},\D\xi)\Big) \D s
+ J_{t},\quad t\geq 0,
\end{align} where $x, b\in\cH^{+}$, $B \in \mathcal{L}(\mathcal{H})$ is a bounded linear operator, given $y\in \mathcal{H}^+$ the measure $M(y,\cdot)\colon \mathcal{B}(\cHplus\setminus \{0\}) \rightarrow \mathbb{R}$ is such that $\nu^{X}(\D t,\D\xi)=M(X_{t},\D\xi)\D t$ is the predictable compensator of the jump-measure of $(X_t)_{t\geq
0}$, and $(J_t)_{t\geq 0}$ is a purely discontinuous $\cH^{+}$-valued square integrable martingale. Moreover, by exploiting the results in \cite{CKK20} and \cite{Me82}, we adapt the proof of \cite[Theorem II.2.42]{JS03} to our infinite-dimensional setting to obtain the characteristic triplet (see Definition~\ref{def:semimartingale-characteristics}) of $(X_t)_{t\geq 0}$ explicitly and show its affine form (see Proposition~\ref{prop:affine-semimartingale}). The detailed parameter specifications are given in Assumption~\ref{def:admissibility} below.\par
Our main motivation for studying Hilbert space-valued stochastic volatility models
is the modeling of forward prices in commodity or fixed-income markets under the Heath-Jarrow-Morton-Musiela (HJMM) modeling paradigm (see for example, \cite{BK14, BK15, Fil01, CT06}). In finite dimensions, multivariate stochastic volatility models with state dependent volatility dynamics driven by Brownian noise and jumps are considered for example in \cite{gourieroux2010derivative, caversaccio2014pricing,
leippold2008asset}. The variance process $X$ that we consider generalises the L\'evy driven case considered in~\cite{BRS15} to a model allowing for state-dependent jump intensities, while maintaining the desired {\it affine} property which makes these models tractable. Stochastic volatilities with jumps describe the financial time series in energy and fixed-income markets well, as it is illustrated, e.g., in \cite{eydeland2002energy, benth2012modeling, cont2001empirical,
leippold2008asset}. We refer in particular to \cite{leippold2008asset} in which the authors discussed convincing empirical evidence for state dependent-jumps in the volatility.
Our {\it main contribution} lies in showing that our stochastic volatility model $(Y,X)$ has the affine property, that is, we prove for all $t\geq 0$ that the mixed Fourier-Laplace transform of $(Y_t, X_t)$ is exponentially affine in the initial value $(y,x) \in H\times \cH^{+}$ and has a quasi-explicit formula in terms of a solution to generalised Riccati equations that are written in terms of the parameters of the model, see Theorem \ref{thm:joint_process_affine} below. For more on affine processes in various \emph{finite dimensional} state spaces, see, e.g.,~\cite{cuchiero2011affine, DFS03, KST13, KM15,
spreij2011affine, kallsen2010exponentially, CFMT11}. In particular,~\cite{CFMT11} considers affine processes in the space of positive self-adjoint matrices, i.e., they consider the finite-dimensional analogue of our variance process $X$. Infinite-dimensional affine stochastic processes have been considered in e.g.~\cite{STY20, Gra16, CT20, BRS15, benth2018heston, benth2021barndorff}. In particular,~\cite{BRS15, benth2018heston, benth2021barndorff} consider infinite-dimensional affine volatility models, however, they do not include state-dependent jump intensities.
The proof Theorem~\ref{thm:joint_process_affine}, i.e., of the affine property of our stochastic volatility model $(Y_t,X_t)_{t\geq 0}$, is in Section~\ref{sec:affine-property}. It involves considering an approximation $(Y_t^{(n)}, X_t)_{t\geq 0}$ of $(Y_t, X_t)_{t\geq 0}$ obtained by replacing $\mathcal{A}$ in \eqref{eq:stochastic-vola-model} by its Yosida approximation. The use of the approximation allows us to exploit the semimartingale theory and standard techniques in order to show that the approximating process is affine. To show that the affine property holds for the limiting process $(Y_t, X_t)_{t\geq 0}$, we study the convergence of the generalised Riccati equations associated with $(Y_t^{(n)}, X_t)_{t\geq 0}$ to those associated with $(Y_t,X_t)_{t\geq
0}$. We prove the existence of a unique solution to these generalised Ricatti equations by exploiting infinite dimensional ODE results and using the quasi-monotonicity argument to show that the solution stays in the cone $\cH^{+}$, see \cite{Dei77} and \cite{Mar76}. In order for the approach described above to succeed, we impose a commutativity-type condition on the covariance operator of the $Q$-Wiener process $(W^{Q}_t)_{t\geq
0}$ and the stochastic volatility $(X_t^{1/2})_{t\geq 0}$ (see Assumption~\ref{def:joint-assumption} below). This condition is also imposed in \cite{BRS15} and is rather limiting. However, we show that it can be avoided by considering a slightly different stochastic volatility model, see Remark \ref{rem:joint_model_alt} and the example in Section~\ref{sec:state-depend-stoch-general}.
In Section~\ref{sec:examples} we consider a number of examples. For the process $Y$ we assume the setting proposed in \cite{Fil01, BK14}, which can be used to model arbitrage-free forward prices at time $t \geq 0$ of a contract delivering an asset (commodity) or a stock at time $t + x$. In this case the operator $\mathcal{A}$ in \eqref{eq:stochastic-vola-model} is given by $\mathcal{A}=\partial/\partial x$ and the space $H$ is given by a Filipovi\'c space. For the process $(X_t)_{t\geq 0}$, we construct several examples in which we specify the drift and the jump parameters. We first show that the infinite dimensional lift of the multivariate Barndorff-Nielsen--Shephard model introduced in \cite{BRS15} is a particular example of our model class. The stochastic variance process $(X_t)_{t\geq 0}$ in this example is a stochastic differential equation driven by a \emph{L\'evy subordinator} in the space of self-adjoint Hilbert-Schmidt operators, as we show in Section~\ref{sec:compare_BRS}. As mentioned above, this example does not involve state-dependent jump intensities. However, Sections~\ref{sec:state-depend-stoch-simple},~\ref{sec:state-depend-stoch-fixedONB}, and~\ref{sec:state-depend-stoch-general} provide explicit paramater choices that \emph{do} involve state-dependent jump intensities. In Section~\ref{sec:state-depend-stoch-simple} we construct a variance process which is essentially one-dimensional; evolving along a fixed vector $z \in \cH^{+}$. In Section~\ref{sec:state-depend-stoch-fixedONB}, we construct a truly infinite-dimensional variance process $X$. In this example both $X_t$, $t\geq 0$, and $Q$ share a fixed orthonormal basis of eigenvectors. This is imposed to ensure that the commutativity condition given by Assumption~\ref{def:joint-assumption} is satisfied. In Section~\ref{sec:state-depend-stoch-general}, we avoid this commutativity condition by considering an example involving the alternative model discussed in Remark \ref{rem:joint_model_alt}. In a subsequent article we plan to compute option prices on forwards in commodity markets based on the models introduced here. In practice, these computations require the study of finite dimensional approximations of the variance process and its associated Ricatti equations, which is being tackled in the working paper \cite{Kar21}.
\subsection{Layout of the article} In Section~\ref{sec:joint-volat-model} we give an in-depth analysis of our stochastic volatility model and introduce sufficient parameter assumptions that ensure the well-posedness of our proposed model. Subsequently, in Section~\ref{sec:affine-property} we prove the affine-property of our joint model $(Y_t,X_t)_{t\geq 0}$. We split the proof into two parts, first in Section~\ref{sec:assoc-gener-ricc} we show the existence and uniqueness of solutions to the associated generalised Riccati equations under admissible parameter assumptions, thereafter in Section~\ref{sec:stoch-volat-models} we prove the affine transform formula. In Section~\ref{sec:examples}, we give several examples of stochastic volatility models included in our model class by specifying various variance processes $(X_t)_{t\geq0}$.
\subsection{Notation} For $(X,\tau)$ a topological space and $S \subset X$ we let $\mathcal{B}(S)$ denote the Borel-$\sigma$-algebra generated by the relative topology on $S$. We denote by $C^k([0,T];S)$ the space of $S$-valued $k$-times continuously differentiable functions on $[0,T]$. \par Throughout this article we fix a separable, infinite-dimensional real Hilbert space $(H,\langle\cdot,\cdot \rangle_H)$. The space of bounded linear operators from $H$ to $H$ is denoted by $\mathcal{L}(H)$. The adjoint of an operator $A \in \mathcal{L}(H)$ is denoted by $A^*$. We let $\mathcal{L}_{1}(H)\subseteq \mathcal{L}(H)$ and $\mathcal{L}_{2}(H)\subseteq \mathcal{L}(H)$ denote respectively the space of \emph{trace class operators} and the space of \emph{Hilbert-Schmidt operators} on $H$. Recall that $\mathcal{L}_{1}(H)$ is a Banach space with the norm \begin{align*}
\|A\|_{\mathcal{L}_{1}(H)} = \sum_{n=1}^{\infty} \langle (A^*A)^{1/2} e_n, e_n \rangle_H, \end{align*}
where $(e_n)_{n\in \mathbb{N}}$ is an orthonormal basis for $H$. Moreover, $\mathcal{L}_{2}(H)$ is a Hilbert space when endowed with the inner product \begin{align*}
\langle A, B \rangle_{\mathcal{L}_{2}(H)} = \sum_{n=1}^{\infty} \langle A e_n, B e_n \rangle_H. \end{align*}
Recall that for $A \in \mathcal{L}(H)$ and $B \in \mathcal{L}_{2}(H)$ we have $AB\in \mathcal{L}_{2}(H)$ and \begin{equation}\label{eq:L2L}
\|AB\|_{\mathcal{L}_{2}(H)} \leq \|A\|_{\mathcal{L}(H)} \|B\|_{\mathcal{L}_{2}(H)}\,. \end{equation}\par We define $\mathcal{H}$ to be the space of all self-adjoint Hilbert-Schmidt operators on $H$ and $\mathcal{H}^+$ to be the cone of all positive operators in $\mathcal{H}$: \begin{equation*}
\mathcal{H} := \{ A \in \mathcal{L}_{2}(H) \colon A = A^* \}, \ \text{and}\
\mathcal{H}^{+} := \{ A \in \mathcal{H} \colon \langle Ah, h\rangle_H \geq 0 \text{ for all } h\in H \}. \end{equation*}
For notational brevity we reserve $\langle \cdot, \cdot \rangle$ to denote the inner product on $\mathcal{L}_{2}(H)$, and $\| \cdot \|$ for the norm induced by $\langle \cdot, \cdot \rangle$. Note that $\mathcal{H}$ is a closed subspace of $\mathcal{L}_{2}(H)$, and that $\mathcal{H}^{+}$ is a self-dual cone in $\mathcal{H}$. For $x,y\in \mathcal{H}$ we write $x \leq_{\mathcal{H}^+} y$ if $y-x\in \cH^{+}$ (and $x\geq_{\mathcal{H}^+} y$ if $x-y\in \cH^{+}$). For $a,b\in H$, we let $a\otimes b$ be the linear operator defined by $a\otimes b (h)=\langle a, h\rangle_{H} b$ for every $h\in H$. Note that $a\otimes a\in\cH^{+}$ for every $a\in H$. When space is scarce, we shall write $a^{\otimes 2}\coloneqq a\otimes a$ .\par Finally, throughout this article we let $\chi\colon \mathcal{H}\rightarrow \mathcal{H}$
denote the truncation function given by $\chi(x) = x1_{\{\| x \| \leq 1\}}$.
\subsubsection{Hilbert valued semimartingales} We let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\geq 0},\mathbb{P})$ be a filtered probability space and let $(\mathcal{H},\langle \cdot,\cdot\rangle)$ be a separable Hilbert space. Let $M=(M_t)_{t\geq 0}$ be an $\mathcal{H}$-valued locally square-integrable martingale. Then we know from \cite[Theorem 21.6 and Section 23.3]{Me82} that there exists a unique (up to a $\mathbb{P}$-null set) c\`adl\`ag predictable process $\langle\langle M\rangle\rangle$ of finite variation taking values in the set of positive self-adjoint elements of $\mathcal{L}_1(\mathcal{H})$ such that $\langle\langle M\rangle\rangle_0 =0$ and $M\otimes M- \langle\langle M\rangle\rangle$ is an $\mathcal{L}_{1}(\mathcal{H})$-valued local martingale.\par Following~\cite[Definition 23.7]{Me82}, an $\mathcal{H}$-valued process $X= (X_t)_{t\geq 0}$ is called {\it a semimartingale} if \begin{align}\label{eq:semimartingale} X_t = X_0 + M_t+A_t, \qquad t\geq 0, \end{align}
where $X_0$ is $\mathcal{H}$-valued and $\mathcal{F}_0$-measurable, $M$ is a $\mathcal{H}$-valued
locally square-integrable martingale with c\`adl\`ag paths such that $M_0 =0$
and $A$ is an adapted $\mathcal{H}$-valued c\`adl\`ag process of finite variation with $A_0=0$.
When the process $A$ in \eqref{eq:semimartingale} is predictable, then $X$ is said to be {\it a special semimartingale}. The decomposition \eqref{eq:semimartingale} in this case is unique (see \cite[Theorem 23.6]{Me82}) and is called {\it the canonical decomposition} of
$X$. For a semimartingale $X$, we write $\Delta X_{t}=X_{t}-X_{t-}$, where $X_{t-}=\lim_{s\to t-}X_{s}$. Notice that when $\|\Delta X \|$ is bounded, then $X$ is a special semimartingale (see \cite[Chapter 4, Exercise 11]{Me82}).
Two $\mathcal{H}$-valued locally square-integrable martingales $M$ and $N$ are called {\it orthogonal} if the real-valued process $(\langle M_{t},N_{t}\rangle)_{t\geq 0}$ is a local martingale. Further we call $M$ a {\it purely discontinuous} local martingale if it is orthogonal to all continuous local martingales. An $\mathcal{H}$-valued semimartingale can be written as (see \cite[Theorem 20.2]{Me82}) \begin{align}\label{eq:LM-decomposition} X_t=X_0 + X_t^c + M_t^d+ A_t, \quad t\geq 0, \end{align} where $X_0$ is $\mathcal{F}_0$-measurable, $X^c$ is a continuous local martingale with $X^c_0 =0$, $M^d$ is a locally square integrable martingale orthogonal to $X^c$ with $M^d_0=0$, and $A$ is a c\`adl\`ag process of finite variation with $A_0=0$. The process $X^c$ in~\eqref{eq:LM-decomposition} is unique (up to a $\mathbb{P}$ null set), see~\cite[Chapter 4, Exercise 13]{Me82}.
We associate with the $\mathcal{H}$-valued semimartingale $X$, the integer-valued random measure $\mu^X\colon \mathcal{B}([0,\infty) \times \mathcal{H}) \rightarrow \mathbb{N}$ given by \begin{equation}\label{eq:def_random_jump_measure} \mu^X(\D t, \D \xi) = \sum_{s\geq 0}\mathbf{1}_{\{\Delta X_s \neq 0\}} \delta_{(s, \Delta X_s)} (\D t, \D \xi), \end{equation} where $\delta_a$denotes the Dirac measure at point $a$. Recall from \cite[Theorem II.1.8]{JS03}, the existence and uniqueness (up to a $\mathbb{P}$-null set) of {\it the predictable compensator} $\nu^X$ of $\mu^X$.
Given a semimartingale $X$ we define the `large jumps' process $\check{X}$ by $$\check{X}\coloneqq\sum_{s\leq \cdot}\Delta X_{s}\mathbf{1}_{\{\| \Delta X_{s} \| > 1 \}},$$ and we define the `small jumps' process \begin{equation}\label{eq:decomposition-chi} \hat{X} = X - \check{X}. \end{equation}
Since $\| \Delta \hat{X} \|
\leq 1$, $\hat{X}$ is a special semimartingale and hence it admits the unique decomposition
\begin{equation}\label{eq:canonical-decomposition}
\hat{X}_{t}=X_0+M_{t}^{\hat{X}}+A_{t}^{\hat{X}},\quad t\geq 0,
\end{equation}
where $X_0$ is $\mathcal{F}_0$-measurable, $M^{\hat{X}}$ is a local martingale with $M^{\hat{X}}_0=0$, and $A^{\hat{X}}$ is a predictable process of finite variation with $A^{\hat{X}}_0=0$.
We are ready to introduce {\it the characteristic triplet} of an $\mathcal{H}$-valued semimartingale $X$: \begin{definition}\label{def:semimartingale-characteristics}
Let $X$ be an $\mathcal{H}$-valued semimartingale, let $A^{\hat{X}}$ be the predictable process of finite variation from decomposition~\eqref{eq:canonical-decomposition}, let $X^c$ be the continuous martingale part of $X$ as provided by~\eqref{eq:LM-decomposition}, and let $\nu^X$ be the predictable compensator of $\mu^{X}$, where $\mu^X$ is defined by~\eqref{eq:def_random_jump_measure}. Then we call the triplet $({A^{\hat{X}}}, \langle\langle X^c \rangle\rangle, \nu^X)$ the \emph{characteristic triplet} of $X$. Note that the characteristic triplet consists of a predictable c\`adl\`ag $\mathcal{H}$-valued process of finite variation, a predictable c\`adl\`ag $\mathcal{L}_1(\mathcal{H})$-valued process of finite variation, and a predictable random measure on $\mathcal{B}([0,\infty) \times \mathcal{H})$.
\end{definition}
\section{The stochastic volatility model}\label{sec:joint-volat-model} In this section we specify our stochastic volatility model. First, in Subsection~\ref{sec:affine-vari-proc}, we introduce the stochastic variance process $X$, which is an affine Markov process on the cone of positive self-adjoint Hilbert-Schmidt operators, the existence of which is established in \cite{CKK20}. We show that whenever the process $X$ admits for a version with c\`adl\`ag paths, this version is actually a Markov semimartingale with characteristic triplet of an affine form and the representation~\eqref{eq:canonical-rep-X-intro} holds true. Subsequently, in Subsection~\ref{sec:joint_volatility}, we show that given such a stochastic variance process $X$ there exists a mild solution $Y$ to equation~\eqref{eq:stochastic-vola-model} with initial value $y\in H$, which enables us to introduce our joint stochastic volatility model $Z=(Y,X)$ (see Definition~\ref{def:joint_model} below).
\subsection{The affine variance process}\label{sec:affine-vari-proc} We model the stochastic variance process $(X_{t})_{t\geq 0}$ as a time-homogeneous \emph{affine} Markov process on the state space $\cH^{+}$ in the sense of \cite{CKK20}.
Recall that $\chi\colon \mathcal{H} \rightarrow \mathcal{H}$, $\chi(x) = x1_{\{ \| x \|
\leq 1\}}$ is our truncation function. Assume $(b,B,m,\mu)$ to be an admissible parameter set in the following sense \begin{assumption}{}{A}\label{def:admissibility}
An \emph{admissible parameter set} consists of
\begin{enumerate}
\item \label{item:drift} $b \in \mathcal{H}^+$,
\item \label{item:m-2moment} a measure
$m\colon\mathcal{B}(\cHplus\setminus \{0\})\to [0,\infty]$ such that
$\int_{\cHplus\setminus \{0\}} \| \xi \|^2 \,m(\D\xi) < \infty$ and there exists an element
$I_{m}\in \mathcal{H}$ such that
\begin{align*}
\int_{\cHplus\setminus \{0\} }|\langle
\chi(\xi),h\rangle|\,m(\D\xi)<\infty,\quad\text{for all }h\in\mathcal{H},
\end{align*}
and $\langle I_{m},h\rangle=\int_{\cHplus\setminus \{0\} }\langle \chi(\xi),h\rangle\, m(\D\xi)$ for every $h\in\mathcal{H}$. Moreover, it holds that
\begin{align*}
\langle b, v\rangle - \int_{\cHplus\setminus \{0\}} \langle
\chi(\xi), v\rangle \,m(\D\xi) \geq 0\, \quad\text{for all}\;v\in\cH^{+}.
\end{align*}
\item \label{item:affine-kernel} a $\mathcal{H}^{+}$-valued measure
$\mu \colon \mathcal{B}(\cHplus\setminus \{0\}) \rightarrow \mathcal{H}^+$ such that
\begin{align*}
\int_{\cHplus\setminus \{0\}} \langle \chi(\xi), u\rangle \frac{\langle \mu(\D\xi), x \rangle}{\| \xi \|^2 }< \infty,
\end{align*}
for all $u,x\in \mathcal{H}^{+}$ satisfying $\langle u,x \rangle = 0$\,,
\item \label{item:linear-operator} an operator $B\in \mathcal{L}(\mathcal{H})$
with adjoint $B^{*}$ satisfying
\begin{align*}
\left\langle B^{*}(u) , x \right\rangle
-
\int_{\cHplus\setminus \{0\}}
\langle \chi(\xi),u\rangle
\frac{\langle \mu(\D\xi), x \rangle}{\| \xi\|^2 }
\geq 0,
\end{align*}
for all $x,u \in \cH^{+}$ satisfying $\langle u,x\rangle=0$. \end{enumerate} \end{assumption} Given an admissible parameter set, the main result in \cite[Theorem 2.8]{CKK20} ensures the existence of a square-integrable time-homogeneous $\cH^{+}$-valued affine Markov process $X$. More specifically,~\cite[Theorem 2.8 and Proposition 4.17]{CKK20} imply Theorem~\ref{thm:existence-affine-process} below, which we need in our derivations later. In order to state this result we introduce our concept of a \emph{weak generator}\footnote{Alternatively, we could work in the framework of generalised Feller semigroups and their generators, as we did in~\cite{CKK20}, but this would require us to introduce more concepts.}, which is a minor modification of~\cite[Definition 9.36]{PZ07}.
\begin{definition}[Weak generator]\label{def:weak-generator} Let $X$ be a square-integrable time-homogeneous $\mathcal{H}^+$-valued Markov process with transition semigroup $(P_t)_{t\geq 0}$ acting on the space $C_{\textnormal{w}}(\mathcal{H}^+,\mathbb{R}) := \{ f \in C(\mathcal{H}^+,\mathbb{R}) \colon \sup_{x\in \mathcal{H}^+}
\frac{f(x)}{\|x\|^2+1} <\infty\}$. Then the \emph{weak generator} $\mathcal{G} \colon \dom(\mathcal{G})\subseteq C_{\textnormal{w}}(\mathcal{H}^+;\mathbb{R})\rightarrow C_{\textnormal{w}}(\mathcal{H}^+;\mathbb{R})$ of $(P_{t})_{t\geq 0}$ is defined as follows: $f \in \dom(\mathcal{G})$ if and only if there exists a $g\in C_{\textnormal{w}}(\mathcal{H}^+,\mathbb{R})$ such that \begin{align*} g(x) = \lim_{t\downarrow 0}\tfrac{P_{t}f(x)-f(x)}{t} \end{align*} and \begin{align*}
P_{t}f(x)=f(x)+\int_{0}^{t}P_{s}g(x)\D s \end{align*} for all $x\in \mathcal{H}^+$, and in this case we define $\mathcal{G} f := g$. \end{definition}
\begin{theorem}\label{thm:existence-affine-process} Let $(b, B, m, \mu)$ be an admissible parameter set conform Assumption~\ref{def:admissibility}). Then there exist constants $M,\omega\in [1,\infty)$ and a square-integrable time-homogeneous $\mathcal{H}^+$-valued Markov process $X$ with transition semigroup $(P_{t})_{t\geq 0}$, acting on functions $f \in C_{\textnormal{w}}(\mathcal{H}^+,\mathbb{R})$, and weak generator $(\mathcal{G},\dom(\mathcal{G}))$ such that the following holds:
\begin{enumerate}
\item \label{it:exp_bounds} $\mathbb{E}[ \| X_t \|^2 | X_0 = x ] \leq M e^{\omega t} (\| x \|^2 +1)$ for all $t\geq 0$,
\item $\lin\set{\E^{-\langle \cdot,
u\rangle}:\,u\in\cH^{+}} \cup \{ \langle \cdot , u\rangle \colon u \in \cH^{+}\} \subseteq \dom(\mathcal{G})$, and
\item for every $f\in \lin\set{\E^{-\langle \cdot, u\rangle} \colon u\in\cH^{+}}\cup \{ \langle \cdot , u\rangle \colon u \in \cH^{+}\} $ we have: \begin{align}\label{eq:affine-generator-form} \mathcal{\mathcal{G}} f(x) &= \langle b +B(x) , f'(x) \rangle + \int_{\cHplus\setminus \{0\}}
\left(f(x+\xi) -f(x)-\langle \chi(\xi), f'(x)\rangle\right)\,M(x,\D\xi), \end{align} where $M(x,\D\xi)\coloneqqm(\D\xi) +\frac{\langle\mu(\D\xi),x\rangle}{\norm{\xi}^2}$. \end{enumerate}
\end{theorem}
An additional assumption we want to impose on the affine variance processes under consideration is the requirement, that $X$ must admit for a version with c\`adl\`ag paths. \begin{assumption}{}{B}\label{assumption:cadlag-paths} The time-homogeneous Markov process $X$ associated with the parameters $(b,B,m,\mu)$ of Assumption \ref{def:admissibility} has c\`adl\`ag paths. \end{assumption} Unfortunately, in the setting of generalized Feller semigroups (which we used to establish Theorem~\ref{thm:existence-affine-process}), it is not immediate that the Markov process that is constructed has c\`adl\`ag paths (but see \cite[Theorem 2.13]{CT20} for a positive result). Some (rather limiting) conditions that ensure that Assumption~\ref{assumption:cadlag-paths} is satisfied are provided in the lemma below. In ongoing work~\cite{Kar21}, we hope to establish that in fact, Assumption~\ref{assumption:cadlag-paths} is always satisfied. \begin{lemma}\label{prop:cadlag-version}
Assume that $(b,B,m,\mu)$ is an admissible parameter set that fulfill either one of the following two cases:
\begin{enumerate}
\item \label{item:cadlag-1} (the L\'evy-driven case) $\mu(\D\xi)=0$,
\item \label{item:cadlag-3} (finite activity jumps) $m(\cHplus\setminus \{0\})<\infty$ and
$\int_{\cHplus\setminus \{0\}}\langle x,\frac{\mu(\D\xi)}{\norm{\xi}^{2}}\rangle<\infty$
for all $x\in\cH^{+}$.
\end{enumerate}
Then the affine Markov process $(X_{t})_{t\geq 0}$ associated to
$(b,B,m,\mu)$ admits for a version with c\`adl\`ag paths. \end{lemma} \begin{proof} To prove \ref{item:cadlag-1}, observe that the weak generator \eqref{eq:affine-generator-form} associated to the admissible parameters $(b,B,m,0)$ is a weak generator of a L\'evy driven SDE as described for example in \cite[equation 9.37]{PZ07}) and hence the assertion follows from \cite[Theorem 4.3]{PZ07}. In case of~\ref{item:cadlag-3}, the assertion follows from~\cite[Proposition 4.19]{CKK20}.
\end{proof}
We show in the next proposition that the version of $X$ with c\`adl\`ag paths is in fact a Markovian semimartingale: \begin{proposition}\label{prop:affine-semimartingale}
Suppose that $(b,B,m,\mu)$ is an admissible parameter set conform Assumption~\ref{def:admissibility} and such that the associated affine Markov
process $X$ satisfies Assumption \ref{assumption:cadlag-paths}. Then there
exists a version of $(X_{t})_{t\geq 0}$ which is a
$\cH^{+}$-valued semimartingale with semimartingale characteristics
$(A,C,\nu^X)$
of the form:
\begin{align}
A_{t}&=\int_{0}^{t}b+B(X_{s})\D s\\
C_{t}&=0, \label{eq:continuous-char-X}\\ \nu^X(\D t,\D\xi)&=M(X_t,\D \xi) \D t = \Big(m(\D\xi)+\langle X_{t},\frac{\mu(\D\xi)}{\norm{\xi}^{2}}\rangle\Big)\D t.\label{eq:jump-char-X}
\end{align}
Moreover, the following representation holds
\begin{align}\label{eq:canonical-rep-X}
X_{t}=X_0+\int_{0}^{t}\Big(b+B(X_{s})+\int_{{\cH^{+}\cap \set{
\norm{\xi}> 1}}}\xi \,M(X_s,\D\xi)\Big)\D s+ J_{t},\quad t\geq 0,
\end{align}
where $J$ is a purely discontinuous square integrable martingale.
\end{proposition}
In order to prove Proposition~\ref{prop:affine-semimartingale}, we need the following result, which can be obtained by mimicking the proof of~\cite[Proposition 9.38]{PZ07}:
\begin{proposition}\label{prop:in_weak_gen_gives_martingale} Let $X$ be a square-integrable time-homogeneous c\`adl\`ag Markov process on $\cH^{+}$ with transition semigroup $(P_t)_{t\geq 0}$ acting on $C_{\textnormal{w}}(\mathcal{H}^+,\mathbb{R})$,
let $\mathcal{G}$ be its weak generator and let $f\in \dom(\mathcal{G})$. Define $M_t = f(X_t) - f(X_0) - \int_0^{t} (\mathcal{G} f) (X_s) \D s$. Then $(M_t)_{t\geq 0}$ is a real-valued martingale. \end{proposition}
\begin{proof}[Proof of Proposition~\ref{prop:affine-semimartingale}]
Let $(e_{n})_{n\in\mathbb{N}}$
be an orthonormal basis of $\mathcal{H}$, then for every $n\in\mathbb{N}$, we have $e_{n}=e_{n}^{+}-e_{n}^{-}$, for $e_{n}^{+},e_{n}^{-}\in \cH^{+}$. By Theorem~\ref{thm:existence-affine-process} and Proposition~\ref{prop:in_weak_gen_gives_martingale}
applied to $f = \langle \cdot, e_n \rangle$ there exists a square-integrable martingale $J^{(n)}$ such that
\begin{align*}
\langle X_t , e_n\rangle &= \langle X_0 , e_n\rangle + \int_0^t\Big(\langle b+B(X_s) , e_n\rangle + \int_{\cH^{+} \cap \{\|\xi\|>1\}}\langle \xi, e_n\rangle M(X_s, \D \xi) \Big)\D s\\
& \qquad + J_t^{(n)}, \qquad t\geq 0\,.
\end{align*}
Noting that $X=\sum_{n=1}^{\infty}\langle X,e_{n} \rangle e_{n}$, we infer that $X$ is an $\cH^{+}$-valued semimartinagle with the decomposition in \eqref{eq:canonical-rep-X}, where $J = \sum_{n=1}^{\infty} J^{(n)} e_n$ is a square integrable $\mathcal{H}$-valued martingale.
We are left to show that $J$ is purely discontinuous and to make the characteristic triplet of $X$ explicit. These are known results in the finite-dimensional setting (see for instance \cite[Theorem II.2.42]{JS03}). Below, we adapt the proof of~\cite[Theorem II.2.42]{JS03} to our setting.
For that we decompose $X =A^{\hat{X}} + N^{\hat{X}} + \check{X}$ as in
\eqref{eq:decomposition-chi} and \eqref{eq:canonical-decomposition}.
Denote by $(A^{\hat{X}},C,\nu^X)$ the characteristic triplet of the
semimartingale $X$. Let $u\in\cH^{+}$ be arbitrary and
consider the function $g_{u} = \E^{-\langle \cdot, u\rangle}$, $u\in \cH^{+}$.
On the one hand, applying the It\^o formula to $g_{u}(X)$ (see for instance,
\cite[Theorem 27.2]{Me82}), yields that $g_{u}(X)$ is a real-valued semimartingale and
\begin{align}\label{eq:Ito-semimartingale-char}
&\E^{-\langle X_{t},u \rangle}\nonumber\\
&\quad = \E^{-\langle X_0 ,u \rangle}
-\int_{0}^{t}\E^{-\langle X_{s-},u \rangle}
\langle u, \D A_{s}^{ \hat{X} }\rangle
-\int_{0}^{t}\E^{-\langle X_{s-},u \rangle}
\langle u, \D N_{s}^{\hat{X}}\rangle
\nonumber\\
&\qquad
+\tfrac{1}{2}\int_{0}^{t}
\E^{-\langle X_{s-},u \rangle}
\langle u\otimes u,\D C_{s}\rangle_{\mathcal{L}_{2}(\mathcal{H})}
+ \int_0^t\int_{\cHplus\setminus \{0\}}
\E^{-\langle X_{s-},u \rangle} K(\xi,u) \nu^X(\D s,\D\xi)
\nonumber\\
&\qquad
+\int_0^t\int_{\cHplus\setminus \{0\}}
\E^{-\langle X_{s-},u \rangle}K(\xi, u)
(\mu^{X}(\D s, \D\xi) - \nu^X(\D s,\D\xi))\,,
\end{align}
where $K(\xi, u) = \E^{-\langle
\xi,u \rangle}-1+\langle
\chi(\xi),u\rangle$.
On the other hand, by Proposition~\ref{prop:in_weak_gen_gives_martingale} there exists a real-valued martingale $I^u$ such that
\begin{align}\label{eq:ito-generator}
\E^{-\langle X_{t},u \rangle}&=\E^{-\langle X_0,u
\rangle}+I^u_{t}-\int_{0}^{t}\E^{-\langle
X_{s},u \rangle}\big(\langle
b+B(X_{s}),u\rangle\big)\D s \nonumber\\
&\quad+\int_{0}^{t}\int_{\cHplus\setminus \{0\}} \E^{-\langle
X_{s},u \rangle}K(\xi,u) M(X_{s},\D\xi)\D s\,, \quad t\geq 0\,.
\end{align}
Note that for every $t\geq 0$, the integrals with respect to $\D s$
on the right-hand side of \eqref{eq:ito-generator} remain unchanged if we
take the left-limits $X_{s-}$ instead of $X_{s}$, as the number of jumps on
$[0,t]$ is at most countable. Moreover, as $X$ takes values in $\cH^{+}$, we have that $g_{u}(X)$ is bounded and hence it is a special semimartingale and its canonical decomposition is unique. Therefore the finite
variation part in formulas~\eqref{eq:Ito-semimartingale-char}
and~\eqref{eq:ito-generator} must coincide, i.e.,
\begin{align}\label{eq:semimartingale-compare}
& -\int_{0}^{t}\E^{-\langle X_{s-},u \rangle}\big(\langle u, \D A_{s}^{\hat{X}}\rangle
+\tfrac{1}{2}\langle u\otimes u, \D
C_{s}\rangle_{\mathcal{L}^2(\mathcal{H})}
+\int_{\cHplus\setminus \{0\}}K(\xi,u)\nu^X(\D s,\D\xi)\big)\nonumber\\
&\quad =
-\int_{0}^{t}\E^{-\langle X_{s},u \rangle}\big(\langle
b+B(X_{s}),u\rangle+\int_{\cHplus\setminus \{0\}}K(\xi,u)M(X_{s},\D\xi)\big)\D s,
\end{align}
must hold for all $ t\geq 0$ almost surely. Now, by integrating $\E^{\langle X_{s-},u\rangle}$ with
respect to both sides of~\eqref{eq:semimartingale-compare} over
$[0,t]$, we obtain
\begin{align*}
& -\langle u, A_{t}^{\hat{X}}\rangle+\tfrac{1}{2}\langle u\otimes u,
C_{t}\rangle_{\mathcal{L}^2(\mathcal{H})}+\int_{\cHplus\setminus \{0\}}K(\xi,u)\nu^X(
[0,t],\D\xi)\\
&\qquad =-\langle
u,\int_{0}^{t}b+B(X_{s})\D
s\rangle+\int_{0}^{t}\int_{\cHplus\setminus \{0\}}K(\xi,u)M(X_{s},\D\xi)\D s, \quad
\forall t \geq 0 \text{ a.s.}
\end{align*}
Now, following similar steps as in the proof of \cite[Theorem II.2.42]{JS03}
we conclude that
$C_{t}=0$, $\nu^X([0,t],\D\xi)=\int_{0}^{t}M(X_{s},\D\xi)\D s$ and
$A^{\hat{X}}_{t}=\int_{0}^{t}b+B(X_{s})\, \D s$, $t\geq 0$, and the
statements of the proposition follow.
\end{proof}
\subsection{The joint stochastic volatility model}\label{sec:joint_volatility} In this section we present our joint model, see Definition~\ref{def:joint_model} below, which involves taking the square root $X^{1/2}$ of the process $X$ from Theorem~\ref{thm:existence-affine-process} as volatility for the $H$-valued process $Y$ given by equation~\eqref{eq:Y} below.\par
Throughout this section we consider the following setting: let $(b,B,m,\mu)$ be a parameter set satisfying Assumption~\ref{def:admissibility}, let $x\in \mathcal{H}^+$ and $y\in H$, and let $Q \in \mathcal{L}_1(H)$ be self-adjoint and positive. Next, let $X$ be the square-integrable time-homogeneous Markov process associated with the parameter set $(b,B,m,\mu)$ the existence of which is guaranteed by Theorem~\ref{thm:existence-affine-process}; we denote the filtered probability space on which $X$ is defined by $(\Omega^1,\mathcal{F}^1,(\mathcal{F}_t^1)_{t\ge 0}, \mathbb{P}^1)$ and assume $\mathbb{P}^1(X_0=x)=1$. In addition, we let $(\Omega^2,\mathcal{F}^2, (\mathcal{F}_t^2)_{t\ge 0}, \mathbb{P}^2)$ be another filtered probability space, which satisfies the usual conditions and allows for a $Q$-Wiener process $W^{Q}\colon [0,\infty)\times \Omega \rightarrow H$. Now set \begin{align*} (\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})\coloneqq(\Omega^{1}\times\Omega^{2}, (\mathcal{F}^{1}\otimes \mathcal{F}^{2}), (\mathcal{F}^{1}_{t}\otimes \mathcal{F}_{t}^{2})_{t\geq 0}, \mathbb{P}^{1}\otimes \mathbb{P}^{2})\,, \end{align*} and denote the expectation with respect to $\mathbb{P}$ by $\mathbb{E}$. With slight abuse of notation we consider $X$ and $W^{Q}$ to be processes on $(\Omega,\mathcal{F},\mathbb{F})$ (note that they are independent).\par In addition, we assume $(\mathcal{A}, \dom(\mathcal{A}))$ to be the generator of a strongly continuous semigroup $(S(t))_{t\geq 0}$ on $H$.
Now consider the following SDE, for which Lemma~\ref{lem:integrand} below establishes the existence of a mild solution: \begin{align}\label{eq:Y}
\begin{cases}
\D Y_{t}=\mathcal{A} Y_{t}\,\D t+X_{t}^{1/2}\,\D W^{Q}_{t}\,, \quad t\geq 0,\\
Y_0 =y.
\end{cases} \end{align}
\begin{lemma}\label{lem:integrand} Assume the setting described above, in particular, let $(b,B,m,\mu)$ satisfy Assumption~\ref{def:admissibility} and let $X$ be the associated affine process. Moreover, let Assumption~\ref{assumption:cadlag-paths} hold. Then $X$ is progressive,
\begin{align}\label{eq:int-condition-X}
\EX{\int_0^t\norm{X^{1/2}_{s}Q^{1/2}}^2 \D s}<\infty\,, \end{align} and moreover \begin{align}\label{eq:solution-Y}
Y_{t}=S(t)y+\int_{0}^{t}S(t-s)X_{s}^{1/2}\D W^{Q}_{s}\,, \quad t\geq 0\,, \end{align} is the unique mild solution to \eqref{eq:Y}. \end{lemma} \begin{proof} The fact that $X$ is progressive follows from the $\mathbb{F}$-adaptedness of $X$ and Assumption \ref{assumption:cadlag-paths}. Moreover, it follows from Theorem~\ref{thm:existence-affine-process} \ref{it:exp_bounds} and H\"older's inequality that \begin{align*}
\mathbb{E} \| X_t^{1/2} Q^{1/2} \|^2
&\leq \| Q \|_{\mathcal{L}_1(H)}
\mathbb{E} \| X_t^{1/2} \|_{\mathcal{L}(H)}^2
\leq \| Q \|_{\mathcal{L}_1(H)}
\mathbb{E} \| X_t \|
\\
& \leq \sqrt{M} \| Q \|_{\mathcal{L}_1(H)} \operatorname{e}^{\omega t /2}
\sqrt{ \mathbb{E}\| X_0\|^2 +1}. \end{align*} Standard theory on infinite dimensional SDEs (see for instance~\cite[Section 6.1]{DZ92}) now yields the existence of a unique mild solution to \eqref{eq:Y} given by \eqref{eq:solution-Y}. \end{proof}
\begin{definition}\label{def:joint_model} Assume the setting described above, in particular, let $(b,B,m,\mu)$ satisfy Assumption~\ref{def:admissibility} and let $X$ be the associated affine process. Moreover, let Assumption~\ref{assumption:cadlag-paths} hold and let $Y$ be given by~\eqref{eq:solution-Y}. Then we refer to the $H\times \mathcal{H}^+$-valued process $Z=(Y,X)$ as the \emph{joint stochastic volatility model with affine pure-jump variance} (and with parameters $(b,B,m,\mu,Q,\mathcal{A})$ and initial value $(x,y)$). Note that the process $(Z,(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P}))$ is a (stochastically) weak solution to the following SDE in $H\times \mathcal{H}$: \begin{align}\label{eq:Z}
\begin{cases}
\D Z_{t}&= (\mathbf{b}+\mathbf{A} Z_{t})\, \D t+\mathbf{\Sigma}(Z_{t})\D \mathbf{W}_{t}+\D \mathbf{J}_{t}\,, \quad t\geq 0\,,\\
Z_{0}&=(y,x)\in H\times \mathcal{H}^+\,,
\end{cases} \end{align} where $\mathbf{b}, \mathbf{A}, \mathbf{\Sigma}, \mathbf{B}$, and $\mathbf{J}$ are as follows \begin{align*} \mathbf{b}\coloneqq\begin{bmatrix}
0 \\
b + \int_{\cH^{+} \cap \{\|\xi\| >1\}} \xi \,m (\D \xi) \end{bmatrix}, \quad
\mathbf{A}
\begin{bmatrix}
z_1 \\
z_2
\end{bmatrix}
\coloneqq
\begin{bmatrix}
\mathcal{A} z_1 \\
B(z_2)+\int_{\cH^{+}\cap \set{\norm{\xi}> 1}}\xi\,\frac{\langle z_{2}, \mu(\D\xi)\rangle}{\norm{\xi}^{2}}
\end{bmatrix}, \end{align*} \begin{align*} \mathbf{\Sigma}(z)\coloneqq\begin{bmatrix}
(z_{2})^{1/2} & 0 \\
0 & 0
\end{bmatrix},\quad \D\mathbf{W}\coloneqq\begin{bmatrix}
\D W^{Q} \\
0
\end{bmatrix}\,, \quad \text{ and }\quad \D\mathbf{J}\coloneqq\begin{bmatrix}
0 \\
\D J
\end{bmatrix}, \end{align*} where $J$ is the purely discontinuous square-integrable martingale obtained from Proposition~\ref{prop:affine-semimartingale}. \end{definition}
\begin{remark}\label{rem:rougher_noise} The assumption that $W^{Q}$ is a $Q$-Wiener process can be weakend whilst maintaining all results presented in this article. Indeed, as $X$ itself is already $\mathcal{H}^+$ valued, it suffices to assume that $Q\in \mathcal{L}_2(H)$ (instead of $Q\in \mathcal{L}_1(H)$) (see also the proof of Lemma~\ref{lem:integrand}). \end{remark}
In order to show that our joint model is affine (see Theorem~\ref{thm:joint_process_affine} below), we need one further assumption. This assumption is also imposed in~\cite{BRS15}, see Proposition 3.2 of that article.
\begin{assumption}{}{C}\label{def:joint-assumption}
There exists a positive and self-adjoint operator $D\in\mathcal{L}(H)$ such that
\begin{align*}
X_{t}^{1/2}QX_{t}^{1/2}=D^{1/2}X_{t}D^{1/2}\,,\quad \text{for all}\; t\geq 0.
\end{align*}
\end{assumption}
To the best of our knowledge, all examples for which Assumption~\ref{def:joint-assumption} holds are such that $Q$ and $X_t$ commute for all $t\geq 0$. In fact, as commuting self-adjoint and compact operators are jointly diagonizable, this is difficult to ensure without assuming there exists a fixed orthonormal basis $(e_n)_{n\in \MN}$ of $H$ that forms the eigenvectors of $Q$ and of $X_t$, $t\geq 0$. Note that this essentially reduces the state space of $X$ to the cone of positive, square integrable sequences $\ell_{2}^{+}$, i.e., we only model the eigenvalues of $X$, as the eigenvectors are fixed, see also Section~\ref{sec:state-depend-stoch-fixedONB}. In conclusion, Assumption~\ref{def:joint-assumption} is rather limiting. However, it can be circumvented if one considers a slightly different model, see Remarks~\ref{rem:joint_model_alt} and~\ref{rem:joint_model_whitenoise} below.
\begin{remark}\label{rem:joint_model_alt} Assumption~\ref{def:joint-assumption} can be omitted if, instead of equation~\eqref{eq:Y}, one assumes that the process $Y$ in the joint model satisfies the following stochastic differential equation: \begin{align}\label{eq:Y_alt}
\begin{cases}
\D Y_{t}=\mathcal{A} Y_{t}\,\D t+D^{1/2} X_{t}^{1/2}\,\D W_{t}\,, \quad t\geq 0,\\
Y_0 =y,
\end{cases} \end{align} where $W$ is an $H$-cylindrical Brownian motion (i.e., $\D W_t$ is white noise) and $D\in \mathcal{L}_1(H)$ is positive and self-adjoint (in fact, $D \in \cH^{+}$ suffices, see Remark~\ref{rem:rougher_noise}). In this case, provided Assumptions~\ref{def:admissibility} and~\ref{assumption:cadlag-paths} hold, we have
\begin{align}\label{eq:int-condition-X_alt}
\EX{\int_0^t\norm{D^{1/2} X^{1/2}_{s} }^2\D s}<\infty\,, \end{align} and \begin{align}\label{eq:solution-Y_alt}
Y_{t}=S(t)y+\int_{0}^{t}S(t-s) D^{1/2} X_{s}^{1/2}\D W_{s}\,, \quad t\geq 0\,, \end{align} is the unique mild solution to \eqref{eq:Y_alt}, see also \cite[Chapter 4, Section 3]{DZ92}. Moreover, Theorem~\ref{thm:joint_process_affine} remains valid: if $Y$ is given by~\eqref{eq:Y_alt} and Assumptions~\ref{def:admissibility} and~\ref{assumption:cadlag-paths} hold, we obtain \emph{exactly} the same expression for $\EX{\E^{\langle Y_t, u_1 \rangle_{H} - \langle X_t, u_2
\rangle }}$. In particular the joint model involving~\eqref{eq:Y_alt} under Assumptions~\ref{def:admissibility} and~\ref{assumption:cadlag-paths} coincides with the joint model involving~\eqref{eq:Y} under Assumptions~\ref{def:admissibility},~\ref{assumption:cadlag-paths}, and~\ref{def:joint-assumption}, in the sense that for every fixed time $t\geq 0$ the distribution of $(Y_{t},X_{t})$ is the same. We refer to Subsection~\ref{sec:state-depend-stoch-general} for an example of a joint model involving~\eqref{eq:Y_alt}. \end{remark}
\begin{remark}\label{rem:joint_model_whitenoise} If $(\mathcal{A},\dom(\mathcal{A}))$ is the generator of an analytic semigroup and moreover $\mathcal{A}^{-\alpha}\in \mathcal{L}_4(H)$ (equivalently, $\mathcal{A}^{-2\alpha} \in \mathcal{H}$) for some $\alpha \in [0,\frac{1}{2})$, then a mild solution to~\eqref{eq:Y} exists even if $W^{Q}$ is an $H$-cylindrical Brownian motion. These conditions are satisfied e.g.\ when $\mathcal{A}$ is the Laplacian on $\mathbb{R}^d$ for $d\in \{1,2,3\}$. We refer to~\cite{DZ92} for details. \par Although this provides another way to circumvent Assumption~\ref{def:joint-assumption} (as $Q$ is the identity in this case), we will not investigate this setting any further: for the applications we have in mind $(\mathcal{A},\dom(\mathcal{A}))$ fails to be the generator of an analytic semigroup. Note that to obtain the assertions of Theorem~\ref{thm:joint_process_affine} in this setting, one would have to adapt its proof: one would not only have to approximate the operator $\mathcal{A}$ but also the noise. \end{remark}
\section{The joint stochastic volatility model is affine} \label{sec:affine-property}
In this section we present our main result, namely that the stochastic volatility model $Z=(Y,X)$ conform Definition~\ref{def:joint_model} has the \emph{affine property}, see Theorem~\ref{thm:joint_process_affine}. In particular, this means that we can express the mixed Fourier-Laplace transform $\mathbb{E}[ \operatorname{e}^{i\langle Y_t, u
\rangle_H - \langle X_t, v\rangle}]$ ($u\in H, v\in \mathcal{H}^+$) in terms of the solution to \textit{generalised Riccati equations} associated to the model parameters $(b,B,m,\mu)$, $\mathcal{A}$ and $Q$ (respectively $D$). In the upcoming subsection we discuss the well-posedness of these generalised Ricatti equations. Our main result, Theorem~\ref{thm:joint_process_affine}, is contained and proven in Subsection~\ref{sec:stoch-volat-models}.
\subsection{Analysis of the associated generalised Riccati equations}\label{sec:assoc-gener-ricc} Let us fix an admissible parameter set $(b,B,m,\mu)$ conform~Assumptions~\ref{def:admissibility} and a positive self-adjoint $D \in \mathcal{L}(H)$. Define $F\colon \cH^{+}\to \mathbb{R}$ and $R\colon \I H \times \cH^{+} \to \mathcal{H}$, respectively as \begin{align}
F(u)&=\langle b, u\rangle-\int_{\cHplus\setminus \{0\}}\big(\E^{-\langle \xi, u\rangle}-1 +\langle \chi(\xi), u\rangle\big)m(\D \xi), \label{eq:F}\\
R(h,u)&= B^{*}(u)-\tfrac{1}{2}D^{1/2}h\otimes
D^{1/2}h-\int_{\cHplus\setminus \{0\}}\big(\E^{-\langle
\xi,u\rangle}-1+\langle \chi(\xi), u\rangle\big)\frac{\mu(\D \xi)}{\norm{\xi}^{2}}. \label{eq:R-intro} \end{align}
Let $(\mathcal{A},\dom(\mathcal{A}))$ be the generator of a strongly continuous semigroup $(S(t))_{t\geq 0}$ and let $(\mathcal{A}^*, \dom(\mathcal{A}^*))$ be its adjoint. It is well known that $(\mathcal{A}^{*},\dom(\mathcal{A}^{*}))$
generates the strongly continuous semigroup $(S^{*}(t))_{t\geq 0}$ on $H$, see for instance \cite[Theorem 4.3]{Gol85}.
Let $T \in \mathbb{R}^{+}$, $u_{1}\in \I H$ and $u_{2}\in\cH^{+}$. We consider the following system of differential equations, known as \textit{generalised Riccati equations} \begin{subequations}
\begin{align}
\,\frac{\partial\Phi}{\partial t}(t,u)&=F(\psi_{2}(t,u)), &\, 0< t \leq T, &\quad\Phi(0,u)=0,\label{eq:Riccati-phi-psi-1-1}\\
\,\psi_{1}(t,u)&=u_{1}-\I \mathcal{A}^{*}\left(\I\int_{0}^{t}\psi_{1}(s,u)\D
s\right), &\, 0< t \leq T, &\quad\psi_{1}(0,u)=u_{1}, \label{eq:Riccati-phi-psi-1-2}\\
\,\frac{\partial \psi_{2}}{\partial t}(t,u)&=R(\psi_{1}(t,u),
\psi_{2}(t,u)), &\, 0< t \leq T, & \quad
\psi_{2}(0,u)=u_{2}.\label{eq:Riccati-phi-psi-1-3}
\end{align} \end{subequations}
\begin{definition} Let $u=(u_{1},u_{2})\in \I H\times \cH^{+}$. We say that $(\Phi(\cdot, u),\Psi(\cdot, u)) \coloneqq (\Phi(\cdot, u),(\psi_{1}(\cdot, u),\psi_{2}(\cdot, u)))\colon [0,T] \to \mathbb{R} \times \I H\times \mathcal{H}$ is a \emph{mild solution} to ~\eqref{eq:Riccati-phi-psi-1-1}-\eqref{eq:Riccati-phi-psi-1-3} if $\Phi(\cdot,u)\in C^{1}([0,T];\mathbb{R}^{+})$,
$\psi_{1}(\cdot,u)\in C([0,T];\I H)$, $\psi_{2}(\cdot,u)\in
C^{1}([0,T];\cH^{+})$ and the map $(\Phi(\cdot,u),\Psi(\cdot,u))$ satisfies \eqref{eq:Riccati-phi-psi-1-1}-\eqref{eq:Riccati-phi-psi-1-3}.
\end{definition}
In the following proposition we show for every $u=(u_{1},u_{2})\in \I H\times\cH^{+}$ the existence of a unique mild solution $(\Phi(\cdot,u),\Psi(\cdot,u))$ to \eqref{eq:Riccati-phi-psi-1-1}-\eqref{eq:Riccati-phi-psi-1-3}.
\begin{proposition}\label{prop:existence-mild-sol} Let $(b,B,m,\mu)$ be an admissible parameter set conform~Assumption~\ref{def:admissibility}, let $(\mathcal{A},\operatorname{dom}(\mathcal{A}))$ be the generator of a strongly continuous semigroup, and let $D\in\mathcal{L}(H)$ be positive and self-adjoint.
Then for every $u \in \I H \times \mathcal{H}^+$ and $T\geq 0$ there exists a unique mild solution $(\Phi(\cdot, u),\Psi(\cdot, u))$ to \eqref{eq:Riccati-phi-psi-1-1}-\eqref{eq:Riccati-phi-psi-1-3} on $[0,T]$. \end{proposition}
\begin{proof} We set for $k \in \mathbb{N}$,
$$m^{(k)}(\D \xi) = \mathbf{1}_{\{\|\xi\|> 1/k\}} m(\D \xi) \quad \mbox{and} \quad \mu^{(k)}(\D \xi) = \mathbf{1}_{\{\|\xi\|> 1/k\}} \mu(\D \xi)\,.$$ Then for each $k\in\mathbb{N}$ we introduce $F^{(k)}\colon \mathcal{H}^+ \to \mathbb{R}$ and $R^{(k)}\colon \I H \times \cH^{+} \to \mathcal{H}$ defined respectively as \begin{align}
F^{(k)}(u)&=\langle b, u\rangle-\int_{\cHplus\setminus \{0\}}\big(\E^{-\langle \xi, u\rangle}-1 +\langle \chi(\xi), u\rangle\big)m^{(k)}(\D \xi), \label{eq:Fk}\\
R^{(k)}(h,u)&=\tilde{R}^{(k)}(u) -\tfrac{1}{2}D^{1/2}h\otimes D^{1/2}h\,, \label{eq:Rk} \end{align} where $\tilde{R}^{(k)}(u) =B^{*}(u) -\int_{\cHplus\setminus \{0\}}\big(\E^{-\langle
\xi,u\rangle}-1+\langle \chi(\xi), u\rangle\big)\frac{\mu^{(k)}(\D \xi)}{\norm{\xi}^{2}}$ , $u\in \cH^{+}$. Consider for $t\geq 0$, \begin{subequations}\label{eq:Riccati-phi-psi-k}
\begin{align}
\frac{\partial\Phi^{(k)}}{\partial t}(t,u)&=F^{(k)}(\psi^{(k)}_{2}(t,u)), & 0< t \leq T,
&\quad \Phi^{(k)}(0,u)=0, \label{eq:Riccati-phi-k}\\
\psi_{1}(t,u)&=u_{1}-\I\mathcal{A}^{*}\left(\I\int_{0}^{t}\psi_{1}(s,u)\D s\right), & 0< t \leq T, &\quad \psi_{1}(0,u)=u_{1},\label{eq:Riccati-psi1-k}\\
\frac{\partial \psi^{(k)}_{2}}{\partial t}(t,u)&=R^{(k)}(\psi_{1}(t,u), \psi^{(k)}_{2}(t,u)), & 0< t \leq T, &\quad
\psi^{(k)}_{2}(0,u)=u_{2}\,. \label{eq:Riccati-psi2-k}
\end{align}
\end{subequations} Standard semigroup theory (see, e.g.,~\cite[Chapter II, Lemma 1.3]{EN00}) ensures that the unique mild solution to \eqref{eq:Riccati-psi1-k} is given by $$\psi_1(t,(u_1,u_2)) =- \I S^*(t) (\I u_1)\,, \qquad t \in [0,T]$$ and $\psi_1(\cdot,u) \in C([0,T];\I H)$. Plugging $\psi_1(t,u)$ into \eqref{eq:Riccati-psi2-k}, yields \begin{align*}
\frac{\partial \psi^{(k)}_{2}}{\partial t}(t,u) &= \tilde{R}^{(k)}(\psi_2^{(k)}(t,u))
+
\tfrac{1}{2}D^{1/2} S^*(t) (\I u_1)\otimes
D^{1/2} S^*(t) (\I u_1)\,. \end{align*} For $k\in \MN$, $u_1 \in \I H$, $t\in [0,T]$, define $\mathcal{R}_{u_1}^{(k)} (t, \cdot)\colon \mathcal{H}^+ \to \mathcal{H}$, by $$\mathcal{R}^{(k)}_{u_1}(t,h) = \tilde{R}^{(k)}(h) + \tfrac{1}{2}D^{1/2}S^*(t) (\I u_1)\otimes D^{1/2}S^*(t)(\I u_1).$$ By \cite[Lemma 3.3]{CKK20} the function $\tilde{R}^{(k)}$ is Lipschitz continuous on $\cH^{+}$ and since the term $\frac{1}{2}D^{1/2}S^*(t) (\I u_1)\otimes D^{1/2}S^*(t)(\I u_1)$ does not depend on $h$, we conclude that for every $t\in [0,T]$ and $u_{1}\in \I H$ the function $\mathcal{R}^{(k)}_{u_1}(t, \cdot)$ is Lipschitz continuous on $\cH^{+}$ as well, with the same Lipschitz constant as $\tilde{R}^{(k)}$. By \cite[Lemma 3.2]{CKK20}, for every $k\in\mathbb{N}$ the function $\tilde{R}^{(k)}$ is quasi-monotone with respect to $\cH^{+}$ (see also \cite[Definition 3.1]{CKK20} for the notion of quasi-monotonicity, and see~\cite[Lemma 4.1 and Example 4.1]{Dei77} for relevant equivalent definitions).
From this we conclude that $\mathcal{R}^{(k)}_{u_{1}}(t,\cdot)$ is also quasi-monotone for every $t\in[0,T]$ and $u_{1}\in \I H$. Moreover, the growth condition \begin{align*}
\norm{\mathcal{R}^{(k)}_{u_{1}}(t,u_{2})}\leq \left( \| B
\|_{\mathcal{L}(\mathcal{H})}+ 2k\| \mu(\mathcal{H}^+ \setminus \{0\})\|
\right)\norm{u_2}+\tfrac{1}{2} M^2\E^{2 w t}\norm{D^{1/2}}_{\mathcal{L}(H)}^2\norm{u_{1}}_{H}^{2}, \end{align*} for every $t\in [0,T]$,$u_{1}\in \I H$ holds, where the constants $M\geq 1$ and $w\in\mathbb{R}$ are such that $\norm{S^{*}(t)}_{\mathcal{L}(H)}\leq M\E^{w t}$, for all $t\geq 0$ which exist for every strongly continuous semigroup, see \cite[Chapter I, Proposition 5.5]{EN00}. Thus the conditions of \cite[Chapter 6, Theorem 3.1 and Proposition 3.2]{Mar76} are satisfied and we conclude from this the existence of a unique solution $\psi_{2}^{(k)}(\cdot,u)$ on $[0,T]$ to the equation \begin{align*} \frac{\partial \psi^{(k)}_{2}}{\partial t}(t,(u_{1},u_{2}))&=\mathcal{R}^{(k)}_{u_{1}}(t,\psi^{(k)}_{2}(t,u)), \end{align*} such that $\psi^{(k)}_{2}(0,(u_{1},u_{2}))=u_{2}$, hence $\psi^{(k)}_{2}(\cdot,u)$ is the unique solution to equation~\eqref{eq:Riccati-psi2-k}. By setting $\Phi^{(k)}(t,u)= \int_{0}^{t}F^{(k)}(\psi_{2}^{(k)}(s,u))\D s$ and the continuity of $F^{(k)}$ it follows that $(\Phi^{(k)}(\cdot,u),\psi_{1}(\cdot,u),\psi_{2}^{(k)}(\cdot,u))$ is the unique mild solution to equations~\eqref{eq:Riccati-phi-k}-\eqref{eq:Riccati-psi2-k} on $[0,T]$.\par{}
Now, let $\mathcal{R}_{u_{1}}\colon [0,T]\times \cH^{+}\to \mathcal{H}$ be defined as the $\mathcal{R}^{(k)}_{u_{1}}$ above, only with $\tilde{R}^{(k)}$ replaced by $\tilde{R}$. By a similar reasoning as above and by \cite[Lemma 3.2 and Remark 3.4]{CKK20}, we conclude that $\mathcal{R}_{u_{1}}(t,\cdot)$ is locally Lipschitz continuous on $\cH^{+} $ and quasi-monotone with respect to $\cH^{+}$ for every $t\in [0,T]$ and $u_{1}\in\I H$. Thus by \cite[Chapter 6, Theorem 3.1]{Mar76} for every $t_{0}\leq T$ and $u_{2}\in\cH^{+}$, there exists a $t_{0}<t_{\max}\leq T$ and a mapping $\psi_{2,t_{0}}(\cdot,u)\colon [t_{0},t_{\max})\to\cH^{+}$ such that \begin{align*}
\frac{\partial \psi_{2,t_{0}}}{\partial
t}(t,(u_{1},u_{2}))&=\mathcal{R}_{u_{1}}(t,\psi_{2}(t,(u_{1},u_{2}))),\quad\text{for } t\in [t_{0},t_{\max}), \end{align*} and $\psi_{2,t_{0}}(t_{0},(u_{1},u_{2}))=u_{2}$. The function $\mathcal{R}_{u_{1}}$ maps bounded sets of $[0,\infty)\times \cH^{+}$ into bounded sets of $\mathcal{H}$, thus by \cite[Chapter 6, Proposition 1.1]{Mar76} it suffices to show that $t\mapsto\psi_{2}(t,u)$ is bounded throughout its lifetime, to conclude that $t_{\max}=T$. By arguing as in the proof of \cite[Proposition 3.7]{CKK20} we conclude that for every $t\geq 0$ and $(u_{1},u_{2})\in\I H\times\cH^{+}$ the sequence $(\psi_{2}^{(k)}(t,u))_{k\in\mathbb{N}}$ is a non-increasing sequence in $\cH^{+}$ converging to $\psi_{2}(t,u)\geq 0$ for $t\in [0,t_{\max})$, hence $$\norm{\psi_{2}(t,u)}\leq\norm{\psi^{(k)}_{2}(t,u)}\leq\norm{\psi^{(1)}_{2}(t,u)},$$ where the right-hand side is bounded on the whole $[0,T]$. Thus we conclude that $t_{\max}=T$ and $\psi_{2}(\cdot,u)$ is the unique solution to~\eqref{eq:Riccati-phi-psi-1-3}. Then again by inserting $\psi_{2}(\cdot,u)$ into~\eqref{eq:Riccati-phi-psi-1-1} and the continuity of $F$, we conclude the existence of a unique solution $\Phi(\cdot,u)$ of~\eqref{eq:Riccati-phi-psi-1-1} on $[0,T]$, and thus also of $(\Phi(\cdot,u),\Psi(\cdot,u))$, the unique mild solution to \eqref{eq:Riccati-phi-psi-1-1}-\eqref{eq:Riccati-phi-psi-1-3} on $[0,T]$. \end{proof}
\subsection{The affine property of our joint stochastic volatility model}\label{sec:stoch-volat-models} Exploiting the existence of a solution to the generalised Riccati equations \eqref{eq:Riccati-phi-psi-1-1}-\eqref{eq:Riccati-phi-psi-1-3}, we show in the following theorem that our joint stochastic volatility model $Z=(X,Y)$ conform Definition \ref{def:joint_model} has indeed the affine property. \begin{theorem}\label{thm:joint_process_affine}
Let $Z=(Y,X)$ be the stochastic volatility model conform
Definition~\ref{def:joint_model} and let
Assumption~\ref{def:joint-assumption} hold. Moreover, let
$u=(u_{1},u_{2})\in \I H\times\mathcal{H}$ and $(\Phi(\cdot,u),(\psi_1(\cdot,u),
\psi_2(\cdot,u)))$ be the mild solution to the generalised Riccati equations~\eqref{eq:Riccati-phi-psi-1-1}-\eqref{eq:Riccati-phi-psi-1-3}, the existence of which is guaranteed by Proposition~\ref{prop:existence-mild-sol}. Then for all $t\in \mathbb{R}^+$, it holds that
\begin{align}
\label{eq:extended-affine-formula}
\mathbb{E} \left[{\E^{\langle Y_{t}, u_{1}\rangle_{H}-\langle X_{t},
u_{2}\rangle}}\right]=\E^{-\Phi(t,u)+\langle y, \psi_{1}(t,u)\rangle_{H}-\langle x, \psi_{2}(t,u)\rangle}.
\end{align} \end{theorem}
In applications, we are usually interested in distributional properties of the process $(Y_{t})_{t\geq 0}$. Setting $u_{2}=0$ in equation~\eqref{eq:extended-affine-formula} we obtain a quasi-explicit formula for the characteristic function of $Y_{t}$ for $t\geq 0$. Due to its importance we state it as a (trivial) corollary of Proposition~\ref{thm:joint_process_affine}: \begin{corollary}\label{cor:solution-weak-strong} Let the assumption of Theorem~\ref{thm:joint_process_affine} hold. Then the characteristic function of the process $Y$ is exponential-affine in its
initial value $y\in H$ and the initial value $x\in\cH^{+}$ of the variance
process $X$, more specifically, for all $t\geq 0$ and $u_{1}\in \I H$ we have:
\begin{align}\label{eq:affine-formula-Y}
\EX{\E^{\langle Y_{t},u_{1}\rangle_{H}}}=\E^{-\Phi(t,(u_{1},0))+\langle y,\psi_{1}(t,(u_{1},0))\rangle_{H}-\langle
x, \psi_{2}(t,(u_{1},0))\rangle}.
\end{align} \end{corollary}
In order to prove Theorem~\ref{thm:joint_process_affine}, we first consider the joint process $(Y^{(n)},X)$ obtained by replacing $\mathcal{A}$ in~\eqref{eq:Y} by its Yosida approximation $\mathcal{A}^{(n)}:=n\mathcal{A}(nI-\mathcal{A})^{-1}$. The use of the approximation will allow us to exploit the semimartingale theory and to apply the It\^o formula and standard techniques in order to show that the approximating process $(Y^{(n)}, X)$ is affine. Then we study the affine property for the limiting process (see \eqref{eq:Yosida_OU_converges} below), when $n$ goes to $\infty$.
Given the assumptions of Lemma \ref{lem:integrand}, we know that inequality \eqref{eq:int-condition-X} holds. Therefore from standard theory on infinite dimensional SDEs (\cite[Proposition 6.4]{DZ92}) we know there exists a continuous adapted process $Y^{(n)}\colon [0,\infty)\times \Omega \rightarrow H$ such that \begin{align}\label{eq:approximating-Y} Y^{(n)}_{t}=y+\int_{0}^{t} \mathcal{A}^{(n)} Y^{(n)}_{s}\,\D s+ \int_{0}^{t} X_{s}^{1/2}\,\D W^{Q}_{s}\,, \quad t\geq 0. \end{align} Moreover,~\cite[Proposition 7.5]{DZ92} ensures that \begin{equation}\label{eq:Yosida_OU_converges} \lim_{n \rightarrow \infty} \mathbb{E} \left[
\sup_{0 \leq t \leq T} \| Y^{(n)}_t -Y_t \|^2_{H} \right]= 0\,. \end{equation} See also \cite[Theorem 5.1, Definition 2.6]{CoxHausenblas} where convergence rates are obtained for Yosida approximations of SPDEs in the case the linear part of the drift is the generator of an analytic semigroup, e.g., a Laplacian.\par Regarding the corresponding Riccati equations, we have the following result: \begin{proposition}\label{prop:uniform_conv_Yosida_Riccati} Let $(b,B,m,\mu)$ satisfy Assumption~\ref{def:admissibility}, let $(\mathcal{A},\dom(\mathcal{A}))$ be the generator of a strongly continuous semigroup, let $D\in \mathcal{L}(H)$ be a positive self-adjoint operator, and let $u\in iH\times \mathcal{H}^+$. Moreover, let $(\Phi(\cdot,u),(\psi_1(\cdot,u), \psi_2(\cdot,u)))$ be the mild solution to the generalised Riccati equation~\eqref{eq:Riccati-phi-psi-1-1}-\eqref{eq:Riccati-phi-psi-1-3}, and for $n\in\mathbb{N}$, let $(\Phi^{(n)}(\cdot,u),(\psi_1^{(n)}(\cdot,u), \psi_2^{(n)}(\cdot,u)))$ be the solution to~\eqref{eq:Riccati-phi-psi-1-1}-\eqref{eq:Riccati-phi-psi-1-3} with $\mathcal{A}=\mathcal{A}^{(n)}$. Then
$$\lim_{n\rightarrow \infty} \sup_{t\in [0,T]} | \Phi^{(n)}(t,u) - \Phi(t,u) | = 0$$ and
$$\lim_{n\rightarrow \infty} \Big(\sup_{t\in [0,T]} \| \psi_1^{(n)}(t,u) - \psi_1(t,u) \|_{H } + \sup_{t\in [0,T]} \| \psi_2^{(n)}(t,u) - \psi_2(t,u) \|\Big)= 0\,,$$ \end{proposition}
\begin{proof}
The uniform convergence of $\psi_1^{(n)}(\cdot,u)$ to $\psi_1(\cdot,u)$ on $[0,T]$ is a well-known property of the Yosida approximation, see, e.g.~\cite[Proof of Theorem I.3.1]{Paz83}.
Once this is established, the uniform convergence of $\psi_2^{(n)}(\cdot,u)$ to $\psi_2(\cdot,u)$ follows from~\cite[Chapter 6, Theorem 3.4]{Mar76}.
The uniform convergence of $\Phi^{(n)}(\cdot,u)$ to $\Phi(\cdot,u)$ follows from the uniform convergence of $\psi_i^{(n)}(\cdot, u)$ to $\psi_i(\cdot, u)$, $i\in \{1,2\}$. Hence the statement of the proposition is proved. \end{proof} With Proposition~\ref{prop:uniform_conv_Yosida_Riccati} and classical stochastic calculus we can now prove Theorem \ref{thm:joint_process_affine}: \begin{proof}[Proof of Theorem~\ref{thm:joint_process_affine}] Let $T\geq 0$ and $u=(u_{1},u_{2})\in \I H\times\cH^{+}$ be arbitrary. Moreover, let $(\Phi^{(n)}(\cdot,u), \Psi^{(n)}(\cdot, u))$, $n\in\mathbb{N}$, be the solution to \eqref{eq:Riccati-phi-psi-1-1}-\eqref{eq:Riccati-phi-psi-1-3} with $\mathcal{A}=\mathcal{A}^{(n)}$ (the $n^{\text{th}}$ Yosida approximation). Note that as $\mathcal{A}^{(n)}$ is bounded, $\Psi^{(n)}(\cdot,u)=(\psi_{1}(\cdot,u),\psi_{2}(\cdot,u))$ is differentiable. Define the function $f_u^{(n)}(t,y,x)\colon [0,T] \times H \times \cH^{+} \to \mathbb{C}$ as follows \begin{align*} f_u^{(n)}(t,y,x) = \E^{-\Phi^{(n)}(T-t,u)+\langle
y,\psi_{1}^{(n)}(T-t, u)\rangle_{H}-\langle x, \psi_{2}^{(n)}(T-t,u)\rangle}\,. \end{align*} Observe that $f_u^{(n)} \in C_b^{1,2,1}([0,T] \times H \times \cH^{+})$ and it holds \begin{align}\label{eq:time-dependent-1}
&\frac{\partial}{\partial t}f^{(n)}_{u}(t,y,x)\nonumber\\
&\quad=\Big(\frac{\partial \Phi^{(n)}}{\partial
t}(T-t,u)-\langle y,\frac{\partial \psi^{(n)}_{1}}{\partial
t}(T-t,u)\rangle_{H}+\langle x,\frac{\partial \psi^{(n)}_{2}}{\partial t}(T-t,u)\rangle
\Big)f^{(n)}_{u}(t,y,x)\nonumber\\
&\quad = \left( F(\psi^{(n)}_{2}(T-t,u))-\langle
y,(\mathcal{A}^{(n)})^*\psi_{1}^{(n)}(T-t,u)\rangle_{H}\right. \nonumber\\
&\quad \qquad +\left.\langle x, R(\psi_{1}^{(n)}(T-t,u),\psi_{2}^{(n)}(T-t,u))\rangle\right) f^{(n)}_{u}(t,y,x)\,. \end{align}
As before we write $K\colon \mathcal{H}\times \mathcal{H}\rightarrow \mathbb{R}$ for the function
$K(u,v) = \E^{-\langle u,v\rangle}-1 + \langle \chi(u),v\rangle$ and also
$\tilde{K}\colon \mathcal{H}\times \mathcal{H}\rightarrow \mathbb{R}$ for $\tilde{K}(u,v) = \E^{-\langle u,v\rangle}-1 + \langle u,v\rangle$. Then applying the It\^o formula to $(f^{(n)}_{u}(t,Y^{(n)}_{t},X_{t}))_{0\leq t \leq T}$, yields \allowdisplaybreaks \begin{align} &f^{(n)}_{u}(t,Y_{t}^{(n)},
X_{t})\nonumber\\
&\quad =f^{(n)}_{u}(0,Y_0,X_0)+\int_{0}^{t}\frac{\partial}{\partial
t}f^{(n)}_{u}(s,Y_{s}^{(n)},X_{s-})\D s\nonumber\\
&\qquad-\int_{0}^{t}f^{(n)}_{u}(s,Y^{(n)}_{s},X_{s-})\langle b+B(X_{s-}), \psi_2^{(n)}(T-s,u)\rangle
\D s\nonumber\\
&\qquad +\int_{0}^{t}f^{(n)}_{u}(s,Y^{(n)}_{s},X_{s-})\langle \mathcal{A}^{(n)} Y^{(n)}_{s},\psi_{1}^{(n)}(T-s,u)\rangle_{H}\D s \nonumber\\
&\qquad + \tfrac{1}{2}\int_{0}^{t}f^{(n)}_{u}(s,
Y^{(n)}_{s},X_{s-})\langle X_{s-}^{1/2}QX_{s-}^{1/2},
\psi^{(n)}_{1}(T-s, u)\otimes \psi^{(n)}_{1}(T-s, u) \rangle \D s \nonumber\\
&\qquad
+\int_0^t\int_{\cHplus\setminus \{0\}}
f^{(n)}_{u}(s,Y_{s}^{(n)},X_{s-})
K(\xi,
\psi_2^{(n)}(T-s,u))
M(X_{s}, \D \xi)\D s\nonumber\\
&\qquad +\int_{0}^{t} f^{(n)}_{u}(s,Y^{(n)}_{s},X_{s-})\langle \psi_1^{(n)}(T-s, u),X_{s-}^{1/2} \D W^{Q}_{s}\rangle_{H}
\nonumber\\
&\qquad +\int_0^t\int\limits_{\cHplus\setminus \{0\}}
f^{(n)}_{u}(s,Y_{s}^{(n)},X_{s-})\tilde{K}(\xi,
\psi_2^{(n)}(T-s,u))(\mu^X(\D s, \D\xi) - M(X_{s}, \D
\xi)\D s)\nonumber\\
&\qquad - \int_0^t f^{(n)}_{u}(s,Y^{(n)}_{s},X_{s-})
\langle \psi_2^{(n)}(T-s,u), \D J_{s}\rangle\,. \end{align} From \eqref{eq:time-dependent-1}, we infer \begin{align} &f^{(n)}_{u}(t,Y_{t}^{(n)},
X_{t})\nonumber\\
&\quad =\int_{0}^{t}
f^{(n)}_{u}(s,Y^{(n)}_{s},X_{s-})\langle
\psi_1^{(n)}(T-s, u),X_{s-}^{1/2} \D W^{Q}_{s}\rangle_{H}
\nonumber\\
&\qquad +\int_0^t\int\limits_{\cHplus\setminus \{0\}}
f^{(n)}_{u}(s,Y_{s}^{(n)},X_{s-}) \tilde{K}(\xi,
\psi_2^{(n)}(T-s,u))(\mu^X(\D s, \D\xi) - M(X_s, \D
\xi)\D s)\,\nonumber\\
&\qquad - \int_0^t f^{(n)}_{u}(s,Y^{(n)}_{s},X_{s-})
\langle \psi_2^{(n)}(T-s,u), \D J_s\rangle\,. \end{align}
We hence conclude that the process
$f_{u}^{(n)}(t, Y_{t}^{(n)},X_{t})$, $t\in [0,T]$
is a local martingale. Furthermore, since it is bounded on $[0,T]$, it is a martingale and it holds \begin{align*}
\EX{\E^{\langle Y^{(n)}_{T}, u_{1}\rangle_{H}-\langle X_{T},u_{2}\rangle}}
&=\EX{\E^{-\Phi^{(n)}(T,u)+\langle Y_{0}^{(n)},\psi_{1}^{(n)}(T,u)\rangle_{H}-\langle X_{0},
\psi_{2}^{(n)}(T,u)\rangle}}\\
&=\E^{-\Phi^{(n)}(T,u)+\langle
y,\psi_{1}^{(n)}(T,u)\rangle_{H}-\langle x, \psi_{2}^{(n)}(T,u)\rangle}. \end{align*} Now taking limits for $n\rightarrow \infty$, envoking~\eqref{eq:Yosida_OU_converges} and Proposition~\ref{prop:uniform_conv_Yosida_Riccati} and since $T\geq 0$ was arbitrary, we conclude the proof. \end{proof}
\section{Examples}\label{sec:examples} In this section we discuss several examples that are included in our class of joint stochastic volatility models with affine pure-jump variance. In all the examples we assume that the first component $Y$ is modeled in the abstract setting of Definition~\ref{def:joint_model}, that means we do not specify $Q$ or $\mathcal{A}$ any further, however we stress here that the HJMM modeling framework as described in \cite{Fil01, BK14}, where $H$ is the Filipovi\'c space and $\mathcal{A}=\partial/\partial x$, serves as the main example. Thus our focus here is on correct specifications of the parameter set $(b,B,m,\mu)$ and the initial value $X_{0}=x\in\cH^{+}$ such that Assumption~\ref{def:admissibility} holds and the associated process $(X_{t})_{t\geq 0}$ satisfies Assumption~\ref{assumption:cadlag-paths} as well as the joint process $(Y,X)$ satisfies Assumption~\ref{def:joint-assumption}.\par{} In Section~\ref{sec:operator-valued-bns} we show that an Ornstein-Uhlenbeck process driven by a L\'evy subordinator in $\cH^{+}$ is included in our model class for the variance process $X$, which is implied by the parameter choice $\mu=0$. Consequently, in Section~\ref{sec:compare_BRS} we conclude that our class of stochastic volatility models extends the infinite-dimensional lift of the BNS stochastic volatility model introduced in \cite{BRS15}. In the subsequent examples we focus on variance processes admitting for state-dependent jump intensities. Indeed, in Section~\ref{sec:state-depend-stoch-simple} we present a variance process $X$ which is essentially one-dimensional as the process evolves along a fixed vector $z\in\cH^{+}$. In Section~\ref{sec:state-depend-stoch-fixedONB} we consider a truly infinite-dimensional variance process $X$. However, to ensure that Assumption~\ref{def:joint-assumption} is satisfied, we assume that both $Q$ and $X_t$, $t\geq 0$, are diagonizable with respect to the same fixed orthonormal basis. We close this section with Section~\ref{sec:state-depend-stoch-general} in which we show the benefits of the model discussed in Remark~\ref{rem:joint_model_alt}, which does not require Assumption~\ref{def:joint-assumption} and thus allows for a more general variance process. \subsection{The operator-valued BNS SV model}\label{sec:operator-valued-bns} In~\cite{BRS15} the authors introduced an operator-valued volatility model that is an extension of the finite-dimensional model introduced in~\cite{BNS06} (and thus they named it the operator-valued BNS SV model). In their model, it is assumed that the volatility process $X$ is driven by a L\'evy process $(L_t)_{t\geq 0}$. In order to ensure that $X$ is positive, they assume $t \mapsto L_t$ is almost surely increasing with respect to $\mathcal{H}^+$, i.e. that $L$ is an $\mathcal{H}^+$\emph{-subordinator}. This holds if and only if for any fixed $t\geq 0$ we have $\mathbb{P}(L_t\in \mathcal{H}^+)=1$, (see also~\cite[Proposition 9]{PR03}). Roughly speaking, the model considered in~\cite{BRS15} amounts to taking $\mu\equiv 0$ in our setting (i.e, to considering a stochastic volatility model $Z=(Y, X)$ conform Definition \ref{def:joint_model} with parameters $(b,B, m,0,Q,A)$). Indeed, in Subsection~\ref{sec:compare_BRS} below we demonstrate that the model introduced in~\cite{BRS15} is fully contained in our setting. \par First, however, we show for this stochastic volatility model that the characteristic function of $Y_t$, $t \in [0,T]$, can be made explicit up to the Laplace exponent of the driving L\'evy subordinator, see Proposition~\ref{prop:BNS-SV} below.
\begin{proposition}\label{prop:BNS-SV} Let $(b,B,m,0)$ satisfy Assumption~\ref{def:admissibility} and let $X$ be the associated affine process with $X_{0}=x\in\cH^{+}$. Moreover, let $Q\in\mathcal{L}_{1}(H)$ be positive and self-adjoint such that Assumption~\ref{def:joint-assumption} holds and $\mathcal{A}\colon \dom(\mathcal{A})\subseteq H\to H$ be the generator of the strongly continuous semigroup $(S(t))_{t\geq
0}$. Then for every $y\in H$, the mild solution $Y$ of \eqref{eq:Y} exists and for all $v_1\in H$ and $t\geq 0$ it holds that \begin{align}\label{eq:affine-transform-levy} \EX{\E^{\I \langle Y_{t},v_{1}\rangle_{H}}}&=\exp \left(\I\langle y, S^{*}(t)v_{1}\rangle_{H}\right)\nonumber\\
&\quad\times\exp
\left(-\int_{0}^{t}\varphi_{L}\left(\tfrac{1}{2}\int_{0}^{s}\operatorname{e}^{(s-\tau)B^*}
(D^{1/2}S^{*}(\tau) v_{1})^{\otimes 2}\D\tau\right)\D s\right)\nonumber\\
&\quad\times\exp\left(-\tfrac{1}{2}\langle
x,\int_{0}^{t}\operatorname{e}^{\tau B^*} (D^{1/2}S^{*}(t-\tau)v_1)^{\otimes 2 }\D \tau\rangle\right), \end{align} where $\varphi_{L}\colon \mathcal{H}\to \mathbb{C}$ denotes the Laplace exponent of the L\'evy process $L$ with characteristics $(b,0,m)$ and is given by \begin{align}\label{eq:Laplace-Levysubordinator}
\varphi_{L}(u)= \langle b, u\rangle-\int_{\cHplus\setminus \{0\}}\E^{-\langle \xi,
u\rangle}-1+\langle \chi(\xi), u\rangle\, m(\D \xi)\, ,\quad u\in \cH^{+}. \end{align} \end{proposition}
\begin{proof}
The admissible parameter set $(b,B,m,0)$ corresponds to the solution $X$ of
a linear stochastic differential equation driven by a L\'evy process
$(L_{t})_{t\geq 0}$ with characteristics $(b,0,m)$. It is easy to see that $X$ has c\`adl\`ag
paths and hence Assumption~\ref{assumption:cadlag-paths} is satisfied. Thus
we are in the situation of Corollary~\ref{cor:solution-weak-strong} and
conclude that the affine transform formula~\eqref{eq:affine-formula-Y} holds
with
$(\Phi(\cdot,v),(\psi_{1}(\cdot,v),\psi_{2}(\cdot,v)))$ being the mild
solution to the generalised Riccati equations associated with $(b,B,m,0)$
and initial value $v=(v_{1},0)$ for $v_{1}\in H$. Hence, it is left to show that
the solutions have the explicit form as indicated by
formula~\eqref{eq:affine-transform-levy}. Indeed, observe that the unique mild solution to
equation~\eqref{eq:Riccati-phi-psi-1-2} is given by $\psi_{1}(t,(v_1,0))=\I
S^{*}(t)v_{1}$. Then inserting $\psi_{1}(\cdot,(v_{1},0))$ into
\eqref{eq:Riccati-phi-psi-1-3} and recalling that $\mu=0$ yields \begin{align*}
\frac{\partial \psi_{2}}{\partial s}(s,(v_1,0))
&=B^{*}(\psi_{2}(s,(v_1,0)))
+\tfrac{1}{2}D^{1/2} S^{*}(t) v_{1}\otimes
D^{1/2} S^{*}(t) v_{1}\,.
\end{align*}
By the variation of constant formula and recalling that $\psi_{2}(0,(v_{1},0))=0$, we conclude that the unique solution $\psi_{2}(\cdot,(v_{1},0))$ is given by \begin{align*}
\psi_{2}(t,(v_1,0))&=\tfrac{1}{2}\int_{0}^{t}\operatorname{e}^{(t-s)B^*}\big(D^{1/2}
S^{*}(s)v_{1}\otimes D^{1/2}S^{*}(s)v_{1}\big)\D s\\
&=\tfrac{1}{2}\int_{0}^{t}\operatorname{e}^{\tau B^*}\big(D^{1/2}
S^{*}(t-\tau)v_{1}\otimes D^{1/2}S^{*}(t-\tau)v_{1}\big)\D \tau. \end{align*} Lastly, by inserting $\psi_{2}(\cdot,(v_{1},0))$ into \eqref{eq:Riccati-phi-psi-1-1} and since $F$ is a continuous function, integrating \eqref{eq:Riccati-phi-psi-1-1} with respect to $t$ gives \begin{align*}
\Phi(t,(v_1,0))&=\int_{0}^{t} \Big(\langle b,\psi_{2}(s,(v_1,0))\rangle\\ &\qquad -\int_{\cHplus\setminus \{0\}}\E^{-\langle
\xi,\psi_{2}(s,(v_1,0))\rangle}-1+\langle \chi(\xi),
\psi_{2}(s,(v_1,0))\rangle
m(\D
\xi)\Big)\D
s\\
&=\int_{0}^{t}\varphi_{L}(\psi_{2}(s,(v_1,0))\D s. \end{align*}
Now, by inserting those formulas of $\Phi(t,(v_{1},0))$, $\psi_{1}(t,(v_{1},0))$ and $\psi_{2}(t,(v_{1},0))$ into~\eqref{eq:affine-formula-Y} we obtain the desired formula.
\end{proof}
\subsubsection{Comparison with the model introduced
in~\texorpdfstring{\cite{BRS15}}{Benth-Rüdiger-Süß}}\label{sec:compare_BRS}
In~\cite{BRS15} the following infinite dimensional volatility model is considered for $t\geq 0$: \begin{equation}\label{eq:Levy_volatility}
\begin{cases}
\D Y_t &= \mathcal{A} Y_t \D t + \sqrt{X_t} \D W^{Q}_t,\\
\D X_t &= B (X_t) \D t + \D L_t, \end{cases} \end{equation} where $(L_t)_{t\geq 0}$ is an $\mathcal{L}_2(H)$-valued L\'evy process satisfying $\mathbb{P}(L_t\in \mathcal{H}^+)=1$ for every $t\geq 0$. Moreover, it is assumed that $B\colon \mathcal{L}_2(H)\rightarrow \mathcal{L}_2(H)$ is of the form $Bv= cvc^*$ or $Bv= cv+ vc^*$ for some $c\in \mathcal{L}(H)$. Finally, $\mathcal{A}\colon \operatorname{dom}(\mathcal{A})\subseteq H \rightarrow H$ is assumed to be an unbounded operator generating a strongly continuous semigroup and $(W_t)_{t\geq 0}$ is assumed to be a $H$-valued Brownian motion which (at least, in the part of~\cite{BRS15} involving the affine property of $(Y,X)$) is assumed to be independent of $(L_t)_{t \geq 0}$ and with a covariance operator $Q$ that satisfies Assumption~\ref{def:joint-assumption}.\par
In this section we show that the joint volatility model~\eqref{eq:Levy_volatility} is a special case of our model in the case that $\mu\equiv 0$, more specifically, that~\cite[Proposition 3.2]{BRS15} is a special case of Proposition~\ref{prop:BNS-SV} above. To this end, we first remark that if $\gamma \in \mathcal{L}_2(H)$, $C\in\mathcal{L}_1(\mathcal{L}_2(H))$, and $\eta \colon \mathcal{B}(\mathcal{L}_2(H)) \rightarrow [0,\infty]$ are the characteristics of $L$, then $C|_{\mathcal{H}}\equiv 0$ thanks to~\cite[Proposition 2.10]{BRS15}. Moreover, in view of Lemma~\ref{lem:Levy_self-adjoint_char}, we have that $\gamma \in \mathcal{H}$, $C=0$, and $\supp(\eta)\subseteq \mathcal{H}$ (this answers an open question in~\cite{BRS15}: see the discussion prior to Proposition 2.11 in that article). Finally, it is easily verified that $B(\mathcal{H})\subset \mathcal{H}$ in both cases described above, so although the `ambient' space for $X$ is $\mathcal{L}_2(H)$ in~\cite{BRS15}, one can, without loss of generality, take $\mathcal{H}$ as ambient space for $X$. \par Next, note that the process $X$ in~\eqref{eq:Levy_volatility} has c\`adl\`ag paths by construction (see also Lemma~\ref{prop:cadlag-version}), so Assumption~\ref{assumption:cadlag-paths} is satisfied. It remains to verify that Assumption~\ref{def:admissibility} is met. Note that Assumption~\ref{def:admissibility}~\ref{item:affine-kernel} is immediately satisfied as $\mu\equiv 0$. To verify that the two choices for $B$ described above satisfy Assumption~\ref{def:admissibility}~\ref{item:linear-operator}, we recall from~\cite[Lemma 2.2]{BRS15} that in these cases one has $\E^{tB}(\mathcal{H}^+)\subseteq \mathcal{H}^+$ for all $t\geq 0$, which, by~\cite[Theorem 1]{LemmertVolkmann:1998}, implies that $B$ is quasi-monotone. Finally, Assumptions~\ref{def:admissibility}~\ref{item:drift} and~\ref{item:m-2moment} hold due to the following result from~\cite{PR03}:\par
\begin{theorem} Let $(L_t)_{t\geq 0}$ be an $\mathcal{H}$-valued L\'evy proces with characteristic triplet $(\gamma,C,\eta)$. Then the following two statements are equivalent: \begin{enumerate}
\item\label{it:Levy_in_cone} for all $t\geq 0$ we have $\mathbb{P}(L_t \in \mathcal{H}^+)=1$;
\item\label{it:Levy_characteristics_on_cone} $C=0$, $\supp(\eta)\subseteq \mathcal{H}^+$ and there exists an $I_{\eta}\in \mathcal{H}$ such that $\xi \mapsto |\langle \chi(\xi),h\rangle|$ is $\eta$-integrable and $\int_{\cHplus\setminus \{0\}} \langle \chi(\xi),h\rangle \,\eta(\D \xi) = \langle I_{\eta}, h \rangle$ for all $h\in \mathcal{H}$, and such that $\gamma - I_{\eta} \in \mathcal{H}^+$.
\end{enumerate}
\end{theorem}
\begin{proof} First, note that $\mathcal{H}^+$ is \emph{regular} (see, e.g., \cite[Theorem 1]{Kar59}), i.e., any sequence $(A_n)_{n\in \MN}$ in $\mathcal{H}$ satisfying $A_1 \leq_{\mathcal{H}^+} A_2 \leq_{\mathcal{H}^+} \ldots \leq_{\mathcal{H}^+} A$ for some $A\in \mathcal{H}$ is convergent in $\mathcal{H}$. The cone is also \emph{normal}: its dual $\mathcal{H}^+$ is generating for $\mathcal{H}$. Thus $\mathcal{H}^+$ is a regular normal proper cone in the terminology of~\cite{PR03}. Now, note that the implication ``\ref{it:Levy_in_cone}$\Rightarrow$\ref{it:Levy_characteristics_on_cone}'' follows from~\cite[Theorem 18]{PR03}, and reverse implication follows from~\cite[Theorem 10]{PR03}. \end{proof}
\subsection{An essentially one-dimensional variance
process}\label{sec:state-depend-stoch-simple} We now present a simple example of a pure-jump affine process $(X_{t})_{t\geq 0}$ on $\cH^{+}$ with state-dependent jump intensity. Starting from its initial value $X_{0}=x\in\cH^{+}$ this process moves along a single vector $z\in\cHplus\setminus \{0\}$ and is thus essentially one-dimensional. For this case we specify an admissible parameter set $(b,B,m,\mu)$ such that the associated affine process $X$ has c\`adl\`ag paths and is driven by a pure-jump process $(J_{t})_{t\geq 0}$ with jumps of size $\xi\in (0,\infty)$ in the single direction $z\in\cH^{+}$ with $\norm{z}=1$ and such that the jump-intensity depends on the current state of the process $X$. For the sake of simplicity, we let the constant parameters $b$ and $m$ be zero. Moreover, we shall fix the dependency structure by means of a fixed vector $g\in\cHplus\setminus \{0\}$. We then take a measure $\eta\colon \mathcal{B}((0,\infty))\to [0,\infty)$ such that $\int_{0}^{\infty}\lambda^{-2}\eta(\D\lambda)<\infty$ and define the vector valued measure $\mu\colon\mathcal{B}(\cHplus\setminus \{0\})\to \cH^{+}$ by \begin{align*} \mu(A)\coloneqq g\eta(\set{\lambda\in\mathbb{R}^{+} \colon \lambda z\in A}). \end{align*} From the assumption that $\int_{0}^{\infty}\lambda^{-2}\eta(\D\lambda)<\infty$ it follows that for every $x\in\cH^{+}$ the measure $M(x,\D\xi)$ on $\mathcal{B}(\cHplus\setminus \{0\})$ defined by \begin{align*} M(x,\D\xi)\coloneqq\frac{\langle x,g \rangle}{\norm{\xi}^{2}}\mu(\D\xi) \end{align*} is finite and thus also \begin{align*} \int_{\cHplus\setminus \{0\}}\langle \chi(\xi),u\rangle\frac{\langle \mu(\D\xi),
x\rangle}{\norm{\xi}^{2}}=\int_{0}^{1}\lambda^{-1}\eta(\D\lambda)\langle
z,u\rangle\langle g, x\rangle<\infty,\quad \forall u,x\in\cH^{+}. \end{align*}
We now must find a linear operator $B\colon \mathcal{H}\to \mathcal{H}$ such that \begin{align}\label{eq:example-2-quasi-monotone}
\langle B^{*}(u),x\rangle-\int_{\cHplus\setminus \{0\}}\langle
\chi(\xi),u\rangle\frac{\langle \mu(\D\xi), x\rangle}{\norm{\xi}^{2}}\geq 0,
\end{align} whenever $\langle x, u\rangle=0$ for $x,u\in\cH^{+}$. The simplest example is obtained by taking \begin{align*} B(u)\coloneqq\int_{\cHplus\setminus \{0\}}\chi(\xi) \frac{\langle u,
\mu(\D\xi)\rangle}{\norm{\xi}^{2}},\quad u\in\mathcal{H}. \end{align*}
From this we see that $B$ and $\mu$ indeed satisfy condition~\eqref{eq:example-2-quasi-monotone} and conclude that the parameter set $(0,B,0,\mu)$ is an admissible parameter set conform Definition~\ref{def:admissibility}. Thus the existence of an associated affine process $X$ on $\cH^{+}$ is guaranteed by Theorem~\ref{thm:existence-affine-process}. Since $\int_{\cHplus\setminus \{0\}}\norm{\xi}^{-2}\langle x, \mu(\D\xi)\rangle<\infty$ for all $x\in\cH^{+}$, it follows from Proposition~\ref{prop:cadlag-version} that Assumption~\ref{assumption:cadlag-paths} is satisfied as well. It remains to ensure that Assumption~\ref{def:joint-assumption} is satisfied. For this purpose it suffices to assume that $x$ and $z$ commute with $Q$. Indeed, note that for $u\in\set{x+\lambda z\colon \lambda\in
[0,\infty)}$ we have $B(u)\in\set{\lambda z\colon \lambda\in[0,\infty)}$. Thus from the semimartingale representation~\eqref{eq:canonical-rep-X}, we see that $X_{t}\in \set{x+\lambda z\colon \lambda\in [0,\infty)}$ for all $t\geq 0$, that means $X_{t}$ commutes with $Q$ for all $t\geq 0$ and therefore Assumption~\ref{def:joint-assumption} is satisfied. \subsection{A state-dependent stochastic volatility model on a fixed
ONB}\label{sec:state-depend-stoch-fixedONB} In this example we specify an admissible parameter set $(b,B,m,\mu)$ giving more general affine dynamics of the associated variance process $X$ on $\cH^{+}$. In the previous Section~\ref{sec:state-depend-stoch-simple} we imposed additional commutativity assumptions on the initial value $X_{0}=x\in\cH^{+}$, the jump direction $z$ and the covariance operator $Q$. In this example we allow for a more general jump behavior, while maintaining Assumption~\ref{def:joint-assumption}. To do so, we pick up the discussion preceding Remark~\ref{rem:joint_model_alt} and note here that Assumption~\ref{def:joint-assumption} is satisfied, whenever $Q$ and $X_{t}$ commute for all $t\geq 0$. Recall that $Q$ and $(X_{t})_{t\geq 0}$ commute if and only if they are jointly diagonizable. This motivates the consideration of a variance process $X$ that is diagonizable with respect to a fixed ONB.\\ More concretely, let $(e_{n})_{n\in\mathbb{N}}$ be an ONB of eigenvectors of the operator $Q$. We model $X$ such that $X_{t}$ $(t\geq 0)$ is diagonizable with respect to the ONB $(e_{n})_{n\in\mathbb{N}}$, i.e. \begin{align*} X_{t}=\sum_{i\in\mathbb{N}}\lambda_{i}(t)e_{n}\otimes e_{n},\quad t\geq 0, \end{align*} for the sequence of eigenvalues $(\lambda_{i}(t))_{i\in\mathbb{N}}$ of $X_{t}$ in $\ell^{+}_{2}$. Concerning the modeling of the dynamics of $(X_{t})_{t\geq
0}$, this essentially means that we model the dynamics of the sequence of eigenvalues $(\lambda_{i}(t))_{i\in\mathbb{N}}$ in $\ell^{+}_{2}$ only.\\ We now come to a specification of the parameters $(b,B,m,\mu)$ such that Assumption~\ref{def:admissibility} is satisfied and moreover such that $X_{t}$ is indeed diagonizable with respect to $(e_{n})_{n\in\mathbb{N}}$ for all $t\geq 0$. Let the measure $m\colon \mathcal{B}(\cHplus\setminus \{0\})\to [0,\infty)$ be such that for $A\in \mathcal{B}(\cHplus\setminus \{0\})$ we have \begin{align}\label{eq:example-m} m(A)\coloneqq\sum_{n\in\mathbb{N}}m_{n}(\set{\lambda\in (0,\infty)\colon \lambda (e_{n}\otimes e_{n})\in A}), \end{align} for a sequence $(m_{n})_{n\in\mathbb{N}}$ of finite measures on $\mathcal{B}((0,\infty))$ such that \begin{align}\label{eq:example-jump-cond-m} \sum_{n\in\mathbb{N}}m_{n}((0,\infty))<\infty\quad\text{and}\quad\sum_{n\in\mathbb{N}}\int_{1}^{\infty}\lambda^{2}m_{n}(\D\lambda)<\infty. \end{align} Then let $\tilde{b}\in\cH^{+}$ be diagonizable with respect to $(e_{n})_{n\in\mathbb{N}}$ and set \begin{align*} b=\tilde{b}+\int_{\cHplus\setminus \{0\}}\chi(\xi)m(\D\xi)=\tilde{b}+\sum_{n\in\mathbb{N}}\int_{0}^{1}\lambda\,m_{n}(\D\lambda)e_{n}\otimes
e_{n}. \end{align*} We see that $b$ and $m$ satisfy their respective conditions in Assumption~\ref{def:admissibility}.
Now, let $(g_{n})_{n\in\mathbb{N}}\subseteq\cH^{+}$ and define
$\mu(\D\xi)\colon \mathcal{B}(\cHplus\setminus \{0\})\to \cH^{+}$ by \begin{align}\label{eq:example-mu} \mu(A)=\sum_{n\in\mathbb{N}}g_{n}\mu_{n}(\set{\lambda\in (0,\infty)\colon \lambda (e_{n}\otimes e_{n})\in A}), \end{align} for a sequence of finite measures $(\mu_{n})_{n\in\mathbb{N}}$ on $\mathcal{B}((0,\infty))$ such that \begin{align}\label{eq:example-jump-cond-mu} \sum_{n\in\mathbb{N}}g_{n}\mu_{n}((0,\infty))\in \cH^{+} \quad\text{and}\quad
\sum_{n\in\mathbb{N}}\int_{0}^{1}\lambda^{-2}\mu_{n}(\D\lambda)\langle
g_{n},x\rangle<\infty,\quad\forall x\in\cH^{+}. \end{align} Moreover, let $G\in\mathcal{H}$ be diagonizable with respect to $(e_{n})_{n\in\mathbb{N}}$, note that this implies that for any $x\in\cH^{+}$ that is diagonizable with respect to $(e_{n})_{n\in\mathbb{N}}$, we have that $Gx+xG^{*}$ is diagonizable with respect to $(e_{n})_{n\in\mathbb{N}}$ as well. We thus define the linear operator $B\colon \mathcal{H}\to\mathcal{H}$ by \begin{align*} B(u)=Gu+uG^{*}+\int_{\cHplus\setminus \{0\}}\chi(\xi) \frac{\langle
\mu(\D\xi),u\rangle}{\norm{\xi}^{2}},\quad u\in\mathcal{H}. \end{align*} Now, one can check that $B$ and $\mu$ indeed satisfy their respective conditions in Assumption~\ref{def:admissibility}. Due to the first condition on $m$ in~\eqref{eq:example-jump-cond-m} and the second on $\mu$ in~\eqref{eq:example-jump-cond-mu}, it follows from Proposition~\ref{prop:cadlag-version} that Assumption~\ref{assumption:cadlag-paths} is satisfied.\\ Again from the semimartingale representation~\eqref{eq:canonical-rep-X} we conclude that for all $t\geq 0$ the operator $X_{t}$ is diagonizable with respect to $(e_{n})_{n\in\mathbb{N}}$ and thus Assumption~\ref{def:joint-assumption} is satisfied as well. \subsection{A general
state-dependent stochastic
volatility
model}\label{sec:state-depend-stoch-general} In this example we show that modeling under the alternative formulation of the model $(Y,X)$ provided by Remark~\ref{rem:joint_model_alt} gives considerably more freedom in the model parameter specification. Indeed, for the stochastic volatility model $(Y,X)$ given by the SDE \begin{align*}
\D (Y_{t},X_{t})&= \begin{bmatrix}
0 \\
b \end{bmatrix}+\begin{bmatrix}
\mathcal{A} Y_{t} \\
B(X_{t})
\end{bmatrix}\, \D t+\begin{bmatrix}
D^{1/2}X_{t}^{1/2} & 0 \\
0 & 0
\end{bmatrix}\D \begin{bmatrix}
W_{t} \\
0
\end{bmatrix}+\D \begin{bmatrix}
0 \\
J_{t}
\end{bmatrix}\,, \quad t\geq 0\,, \end{align*} with $(Y_{0},X_{0})=(y,x)\in H\times \mathcal{H}^+\,,$ and $W=(W_{t})_{t\geq 0}$ a cylindrical Brownian motion, the Assumption~\ref{def:joint-assumption} can be dropped. Therefore, every admissible parameter set $(b,B,m,\mu)$, such that the associated affine process $X$ satisfies Assumption~\ref{assumption:cadlag-paths} is a valid parameter choice. To emphasize the gained flexibility, we compare it with the example in Section~\ref{sec:state-depend-stoch-fixedONB}. For simplicity, we let $(e_{n})_{n\in\mathbb{N}}$ be some ONB of $H$ and specify $m$ and $\mu$ as in \eqref{eq:example-m} and \eqref{eq:example-mu}, respectively, with respect to this ONB. This means that the noise in the variance process $X$ again occurs on the diagonal only. However, $Q$ need not be diagonizable with respect to $(e_{n})_{\in\mathbb{N}}$ and instead of taking $b$ to be diagonizable with respect to the ONB $(e_{n})_{\in\mathbb{N}}$ and $B$ of the particular form above, we allow for a general drift $b\in\mathcal{H}$ such that $b-\int_{\cHplus\setminus \{0\}}\chi(\xi)m(\D\xi)\geq 0$. Moreover, let $C$ be a bounded linear operator on $H$ and define $\tilde{B}\in\mathcal{L}(\mathcal{H})$ by \begin{align*} B(u)=Cu+uC^{*}+\Gamma(u), \end{align*} for some $\Gamma\in\mathcal{L}(\mathcal{H})$ with $\Gamma(\cH^{+})\subseteq \cH^{+}$ and such that \begin{align*} \langle \Gamma(x), u\rangle-\int_{\cHplus\setminus \{0\}}\langle \chi(\xi), u \rangle\frac{\langle x, \mu(\D\xi)\rangle}{\norm{\xi}^{2}}\geq 0. \end{align*} We can again check, that $(b,B,m,\mu)$ satisfies the Assumptions~\ref{def:admissibility} and the associated affine process $X$ Assumption~\ref{assumption:cadlag-paths}. Then according to \eqref{eq:canonical-rep-X} the variance process $X$ has the representation \begin{align*} X_{t}= b+CX_{t}+X_{t}C^{*}+\Gamma(X_{t})+ J_{t}, \end{align*} which resembles the pure-jump affine dynamics of covariance processes in finite dimensions as in \cite[equation 1.2]{CFMT11}. \section{Conclusion and Outlook}\label{sec:conclusion}
In Section~\ref{sec:joint-volat-model} we introduce an infinite dimensional stochastic volatility model. More specifically, we consider a process $Y$ that solves a linear SDE in a Hilbert space $H$ with additive noise, where the variance of the noise is dictated by a process $X$ taking values in the space of self-adjoint Hilbert-Schmidt operators on $H$. The process $X$ is assumed to be an affine pure-jump process that allows for state-dependent jump intensities; its existence has been established in the previous work \cite{CKK20} under certain admissibility conditions on the parameters involved (see Assumption~\ref{def:admissibility}). \par In the derivation of the affine transform formula, we make use of Hilbert valued semimartingale calculus, for this reason we must assume that $X$ has c\`adl\`ag paths (see Assumption~\ref{assumption:cadlag-paths}). Currently, we establish existence of c\`adl\`ag paths under limited conditions (see Proposition~\ref{prop:cadlag-version}). Relaxing these conditions is one of the aims of the working paper~\cite{Kar21} where the author considers finite-dimensional approximations (in particular, Galerkin approximations of the associated generalised Riccati equations are considered) and studies convergence of the variance process in the Skorohod topology. \par Having introduced the joint model, we prove that it is affine (see Theorem~\ref{thm:joint_process_affine}). To this end, we need an additional `commutativity'-type assumption, see Assumption~\ref{def:joint-assumption}. This assumption is avoided by considering a slightly different model, see Remark~\ref{rem:joint_model_alt} and Subsection~\ref{sec:state-depend-stoch-general}. \par Our model extends the model introduced in~\cite{BRS15}, where the authors assume that $X$ is driven by a suitably chosen L\'evy process (see Subsection~\ref{sec:compare_BRS}). In Section~\ref{sec:examples} we also discuss other concrete examples of our model.\par Another way to avoid Assumption~\ref{def:joint-assumption} would be to construct a variance process $X$ that takes values in the space of self-adjoint \emph{trace class} operators. Indeed, in this case we can assume that the noise $\D W^{Q}$ driving $Y$ is white (i.e., $Q$ is the identity). However, taking the trace class operators as a state space is not trivial as this is a non-reflexive Banach space. We aim to pursue this direction of research in a forthcoming work.\par Finally, in a subsequent work, we plan to consider the dynamics of forward rates in commodity markets modeled by our proposed stochastic volatility dynamics. Then study the problem of computing option prices on these forwards. In practice, these computations require finite-rank approximations of the associated generalised Riccati equations as being considered in \cite{Kar21}.
\appendix \section{Auxiliary results}\label{sec:auxiliary-results}
\begin{lemma}\label{lem:cone_characteristic_fnc}
Let $(\mathcal{H}, \left\| \cdot \right\|, \langle \cdot , \cdot \rangle)$ be a separable real Hilbert space, let $K\subseteq \mathcal{H}$ be a cone such that $\mathcal{H} = K \oplus -K$ and let $\mu_1,\mu_2\colon \mathcal{B}(\mathcal{H}) \rightarrow \mathbb{R}$ be measures such that $\supp(\mu_1),\supp(\mu_2)\subseteq K$ and $\int_{\mathcal{H}} \operatorname{e}^{-\langle
x,y \rangle} \,\mu_1(\D x) = \int_{\mathcal{H}} \operatorname{e}^{-\langle x,y \rangle} \,\mu_2(\D x)$ for all $y\in K$. Then $\mu_1 = \mu_2$. \end{lemma}
\begin{proof} Let $(e_n)_{n\in \MN}$ be an orthonormal basis for $\mathcal{H}$ and let $e_n^+, e_n^{-} \in K$ be such that $e_n = e_n^+ - e_n^{-}$, $n\in \MN$. By Dynkin's lemma it suffices to prove that $\mu_1$ and $\mu_2$ coincide on sets of the type $\cap_{i=1}^{n} \{ \langle \cdot, e_i^+ \rangle \in B_i^+, \langle \cdot, e_i^{-}, \rangle \in B_i^{-}\}$, $B_1^+,B_1^-,\ldots,B_n^+,B_n^- \in \mathcal{B}(\mathbb{R})$ and $n\in \MN$. This implies that it suffices to prove the lemma for the case $\mathcal{H}=\mathbb{R}^n$ and $K=[0,\infty)^{n}\subseteq \mathbb{R}^n$, $n \in \MN$. In this case, the result follows by a standard Stone-Weierstrass argument, see, e.g.~\cite[Theorem E.1.14]{HNVW:2017}.
\end{proof}
\begin{lemma}\label{lem:Levy_self-adjoint_char} Let $(H,\langle \cdot,\cdot \rangle)$ be a Hilbert space, let $U\subseteq H$ be a closed linear subspace and let $P_U\colon H\rightarrow U$ be the orthogonal projection of $H$ onto $U$.
Moreover, let $(L_t)_{t\geq 0}$ be a $H$-valued L\'evy process satisfying $\mathbb{P}(L_1 \in U)=1$ and let $\gamma \in H$, $C\in \mathcal{L}_1(H)$ and $\eta\colon \mathcal{B}(H\setminus\set{0})\rightarrow [0,\infty]$ be its characteristics. In addition, let $\gamma_s\in U$, $C_s\in \mathcal{L}_1(U)$ and $\eta_s\colon \mathcal{B}(U\setminus\set{0})\rightarrow [0,\infty]$ be the characteristics of $L$ when interpreted as a $(U,\langle \cdot, \cdot\rangle)$-valued process. Then $\gamma = \gamma_s$, $C = C_s P_U$, and $\eta(A) = \eta_s(A\cap U)$ for all $A \in \mathcal{B}(H\setminus\set{0})$. In particular, $C h = 0$ whenever $h\in U^{\perp}$ and $\supp(\eta)\subseteq U$. \end{lemma}
\begin{proof} Define $\tilde{\eta} \colon \mathcal{B}(H\setminus\set{0})\rightarrow [0,\infty]$ by $\tilde{\eta}(A) = \eta_s(A\cap U)$, $A\in \mathcal{B}(H\setminus\set{0})$. Then for all $h\in H$ and $t\geq 0$ we have, using that $L_t \in U$ a.s.: \begin{equation*}\begin{aligned} & \mathbb{E}( \operatorname{e}^{i\langle L_t, h \rangle })
=
\mathbb{E}( \operatorname{e}^{i\langle L_t, P_U h \rangle })
\\ &
= \exp\left(
t\left(
i\langle \gamma_s, P_U h \rangle_H
-
\langle C_s P_U h, P_U h \rangle
\right)\right)
\\ & \quad
\times
\exp\left( t
\int_{U\setminus \{0\}}
\E^{i\langle \xi,P_U h\rangle}
-1
+ i\langle \xi , P_U h \rangle 1_{\{\|\xi\|<1\}}
\,\eta_s(\D \xi)
\right) \\& = \exp\left(
t\left(
i\langle \gamma_s, h \rangle_H
-
\langle C_s P_U h, h \rangle
+
\int_{H\setminus \{0\}}
\E^{i\langle \xi,h\rangle}
-1
+ i\langle \xi , h \rangle 1_{\{\|\xi\|<1\}}
\,\tilde{\eta}(\D \xi)
\right)\right). \end{aligned} \end{equation*} The result now follows from the uniqueness of the characteristic triplet. \end{proof}
\section*{Data Availability Statement} Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
\end{document} | arXiv |
OSA Publishing > Optical Materials Express > Volume 10 > Issue 11 > Page 2834
Alexandra Boltasseva, Editor-in-Chief
Collective phenomena in Dy-doped silver halides in the near- and mid-IR
Andrey G. Okhrimchuk, Andrey D. Pryamikov, Kirill N. Boldyrev, Leonid N. Butvina, and Evgeni Sorokin
Andrey G. Okhrimchuk,1,* Andrey D. Pryamikov,1 Kirill N. Boldyrev,2 Leonid N. Butvina,1 and Evgeni Sorokin1,3
1Prokhorov General Physics Institute of Russian Academy of Sciences, Dianov Fiber Optics Research Center, 38 Vavilova Street, Moscow 119333, Russia
2Institute of Spectroscopy of Russian Academy of Sciences, Troitsk, Moscow, Russia
3Photonics Institute, TU Wien, Gusshausstr. 27/387, 1040 Vienna, Austria
*Corresponding author: [email protected]
Andrey G. Okhrimchuk https://orcid.org/0000-0001-7837-158X
Evgeni Sorokin https://orcid.org/0000-0002-4703-9653
A Okhrimchuk
A Pryamikov
K Boldyrev
L Butvina
E Sorokin
•https://doi.org/10.1364/OME.406187
Andrey G. Okhrimchuk, Andrey D. Pryamikov, Kirill N. Boldyrev, Leonid N. Butvina, and Evgeni Sorokin, "Collective phenomena in Dy-doped silver halides in the near- and mid-IR," Opt. Mater. Express 10, 2834-2848 (2020)
Mid-infrared luminescence properties of Dy-doped silver halide crystals (AO)
Dy3+:Lu2O3 as a novel crystalline oxide for mid-infrared laser applications (OME)
Mid-infrared emission in Dy:YAlO3 crystal (OME)
Rare-earth-doped Materials
Diode lasers
Diode pumped lasers
Fluoride fibers
Pulsed operation
Original Manuscript: August 24, 2020
Growth of silver halide single crystals AgClBr:Dy
Experimental technique
Judd-Ofelt analysis
Discussions and theory of relaxation kinetics
The kinetics of the electronic transitions within the f-shell of Dy3+ ions were studied with monitoring near- and mid-IR luminescence decay under pulsed laser excitation at 1.3 µm. The luminescence decay curves were found to be profoundly non-exponential in all bands in the range between 1.3-5.5 µm. Such behavior is attributed to cross-relaxation and up-conversion processes dominating in relaxation of Dy3+ ions from the laser-excited multiplet 6H9/2+6F11/2. We suggest that strong collective phenomena occurring under relatively low concentrations are due to anomalous clustering of Dy3+ ions. The cross-relaxation enables an efficient population of 6H13/2 and 6H11/2 multiplets, offering this material as an active medium for a 3-µm and 4.3-µm lasers.
Bright light sources in the mid-infrared spectral region 3–6 µm are demanded for medical diagnostics, gas sensing in the environmental protecting and chemical industry. The high selectivity and sensitivity of such sensing is provided by the unique vibrational absorption spectra of different molecules. There has been a significant progress in the recent years has been achieved with fiber lasers operating in the 2.7–4 µm wavelength range, based on fluoride glasses, doped with Er3+, Ho3+ and Dy3+ ions. These fiber lasers demonstrate high output power [1–3], broadband gain for mode-locking [4], and tuning capability [5–7]. The longest lasing wavelength of 3.92 µm has been demonstrated at the ${}_{}^5{I_5} \to {}_{}^5{I_6}\; $ transition of the Но3+ ion [8]. The ${}_{}^4{F_{9/2}} \to {}_{}^4{I_{9/2}}$ transition of Er3+ and the ${}_{}^6{H_{13/2}} \to {}_{}^6{H_{15/2}}$ transition of Dy3+ provide tuning ranges from 3.35 µm to 3.8 µm and from 2.8 µm to 3.4 µm respectively, covering a significant part of the absorption lines for gases of interest [7]. We should mention however, that Dy3+-doped fibers have an advantage of being pumped at one wavelength, while Er3+ ions require two-wavelength pumping to operate on the above transition. Being doped to the chloride crystal the Dy3+ ion can also be used for generation in the 5.5 µm and 4.3 µm regions using the ${}_{}^6{H_{9/2}},{}_{}^6{F_{11/2}} \to {}_{}^6{H_{11/2}}$ and ${}_{}^6{H_{11/2}} \to {}_{}^6{H_{13/2}}\; $transitions correspondingly [9,10]. Unfortunately, these radiative transitions are quenched by phonons in the fluoride host glasses at room temperature.
In this work, we study Dy3+-doped crystalline silver halides AgCl0.5Br0.5. This host possess extremely narrow phonon spectra not exceeding 160 cm-1 [11] and are therefore prospective for obtaining laser action on transitions up to 5.5 µm wavelength [12]. Moreover, this host is suitable for fiber technology by hot extrusion method, but opposite to fluoride fiber is moister resistant. In comparison with previous study of AgCl0.5Br0.5:Dy [12], we were able to find collective phenomena in relaxations processes of excited states of Dy3+ ions by detailed studying of luminescence kinetics and spectral shapes for three near- and mid-IR bands with high time resolution in crystals with different dopant concentrations. This finding allowed us to conclude that AgCl0.5Br0.5:Dy crystal is perspective for generation of oscillations in mid-IR, because an Dy3+ ion excited to at ${}_{}^6{H_{9/2}},{}_{}^6{F_{11/2}}$ multiplet under 1.3-µm pumping gives higher than one ion at the ${}_{}^6{H_{13/2}}$ multiplet enabling emission at 3 µm, and following up-conversion efficiently populates ${}_{}^6{H_{11/2}}$ multiplet enabling emission at 4.3 µm.
2. Growth of silver halide single crystals AgCl0.5Br0.5:Dy
The single crystals have been grown using the Bridgeman-Stockbarger technique in the sealed fused-silica ampoules. For the charge preparation, we have first chemically synthesized single halide AgCl and AgBr in the powder form. These powders have then been purified by zone recrystallization in the sealed fused-silica ampoules for 15-20 times at the scanning speed of 30 mm/hour, producing purified salts in the form of transparent cylinders. After addition of dry dysprosium halides DyCl3 by Aldrich to the mixture of AgCl and AgBr with molar ratio 1:1 the multicomponent powder was loaded into cylindrical fused-silica ampoule, which was then evacuated and sealed. The Dy concentration in the charge varied from 0.05 to 0.3 wt.%. The growth took place at 0.8–1 mm/hour speed with temperature gradient at crystallization front about 20 K/cm (the melting point is at 412 °C). The grown crystals were cooled for 5 days together with the oven. All operations with silver halides have been performed under red light, to avoid building of silver clusters.
The doped boules have the form of 30–50 mm long cylinders with 8–10 mm diameter and conical ends. They are transparent in the head and scatter light in the tail part. This is obviously due to the low solubility of dysprosium in silver halides, causing its accumulation in the melt during growth and multiphase crystallization in the tail part.
Dy concentration in the boules has been measured by inductively coupled plasma mass spectrometry (ICM-MS), giving concentration distribution along the boule and along the cross-section radius. The Dy concentration increases from the center to the surface of the boule. The highest gradient is observed near the side surface at about 1 mm depth. The concentration variation in the inner part of the boule did not exceed 30%. All spectroscopic measurements, described below, have been performed with the inner part of the boule, where Dy distribution is homogeneous and the concentration is well defined.
Figure 1 shows the dysprosium concentration dependence on DyCl3 content in the charge. The concentration NDy has been measured in the head or middle part of the boule, far from the crystal surface. Being measured in these boule parts the dysprosium concentration NDy correlates with the DyCl3 content in the charge, while in the boule tail the concentration significantly varied towards higher or lower levels. Table 1 summarizes the samples used for spectroscopic measurements. The samples were cut from a head or middle part of boules, and the concentration was determined by the ICP-MS method.
Fig. 1. Dysprosium concentration in the crystal as a function of the DyCl3 content in the charge.
Table 1. Dysprosium concentrations in the studied samples.
View Table | View all tables in this article
3. Experimental technique
Absorption and luminescence spectra were investigated with FTIR spectrometer Bruker IFS 125 HR with resolution of 2 cm-1 at the temperatures 5K–300 K. The near-IR absorption spectra have been also recorded using the double-channel Perkin-Elmer-λ9 spectrophotometer at T=300 K. To excite the luminescence spectra, we used either a laser diode operating at 1.312 µm with 4 nm linewidth, or tunable femtosecond optical parametric amplifier (Orpheus, Light Conversion) with 40 nm linewidth and 10 kHz rep rate, or a home-made pulsed Nd:YAG laser operating at 1.319 µm and 5 Hz rep rate. The latter had 50 µs pulse duration in the free-running mode, and 30 ns in the Q-switched mode. Wavelength selection between 1319 nm and 1338 nm of the Nd:YAG laser was done with an intracavity Lyot filter. In order to apply the lock-in amplifier for luminescence detection, the laser diode pump current was modulated at 33 Hz with 15 ms pulses, which is longer, than average IR luminescence lifetime of Dy3+ ions, so that the excitation regime for all emitting transition was equivalent to continuous-wave (CW). The luminescence spectra and kinetics have been recorded with the grating spectrometer ИКС-31 (LOMO) using InGaAsP and InAs (Hamamatsu P7163) photodiodes in the 1–3 µm region (rise time is 0.1 µs), and the photoconductive MCT detector (InfraRed FTIR-16-1.0) in the 3–6 µm region, both with liquid nitrogen cooling. A germanium plate with thickness of 4 mm was used to reject high order diffraction when luminescence in the 4–5 µm region was detected, and an interference filter with cut-off wavelength of 5 µm was used for detecting luminescence in the 5-6 µm region. No filters were used when detecting luminescence in the 1.2-3 µm region.
4.1. Absorption
Figure 2(a) shows an overall room temperature and helium absorption spectra of the sample #4, which has the highest Dy concentration. At T=300 K samples with lower Dy concentration have analogous spectral positions of bands, but with slightly different peak height ratios. Overall, the absorption bands in the near- and mid-IR can be unambiguously assigned to the $f - f$ transitions in Dy3+ [9,12,13] (Fig. 3). The absorption band corresponding to transition from ground state to 6F3/2 multiplet has two lines at 13219 cm-1 and at 13241 cm-1 at T=5 K [Fig. 2(b)]. Number of the observed lines coincides with highest possible number of transitions from the lowest Stark sub-level of the ground state. Thus, we conclude that there is predominantly one type of Dy3+ centers in the crystal with maybe moderate number of other centers, indicated by the small tail on the high energy side of the most intense line. Positions of all Starks sublevels of the dominating centers were derived from the low temperature absorption and luminescence spectra of the sample #4 and are shown in Table 2.
Fig. 2. (a) The overall absorption spectra of the sample #4 at T=300 K (red line) and T=5 K (blue line). Terminal multiplets for absorption transitions are shown in the graph. (b) Absorption band of the sample #4 for transition from the ground state to the 6F3/2 multiplet at T=5 K.
Fig. 3. Energy level scheme of Dy3+ ions in AgCl0.5Br0.5 crystal. Energies of the lowest Stark sublevels are shown under each multiplet. Black arrows denote registered luminescence transitions. Solid red and blue arrows denote cross-relaxations {1} and {2}, and the green arrows denote up-conversion {3}. The dashed black arrow denotes an un-registered transition in luminescence.
The 1.3 µm band dominates in the absorption spectra of all samples. The structure and form of this band depends on the dopant concentration (Fig. 4.). The spectra of cross-section have been calculated as $\sigma (\lambda )= \alpha (\lambda )/{N_{\textrm{Dy}}}$ where $\alpha (\lambda )$ is the absorption coefficient and the dysprosium concentration ${N_{\textrm{Dy}}}$ has been determined by ICP-MS for each sample separately. One can see reduction of the bandwidth with the concentration increase. The central peak, hardly visible at low concentration, dominates in stronger doped samples. While the absorption band is almost unstructured at the lowest concentration, one can observe appearance of narrow feature at 1324 nm and more structured short-wavelength side at increasing concentrations. The intensities of other absorption bands also depend on the doping level. Figure 5 summarizes the influence of Dy concentration on the integrated absorption cross-section of the bands (indexed by $k$) ${\mathrm{\Xi }_k}$, calculated from absorption spectra by the formula [13]:
(1)$${\mathrm{\Xi }_\textrm{k}} = \frac{1}{{{N_{Dy}}}}\int {\frac{{\alpha \textrm{(}\lambda \textrm{)}}}{\lambda }} d\lambda .$$
Fig. 4. (a) Absorption cross-section spectra for the 1.3 µm band. (b) The same normalized absorption spectra. Dysprosium concentrations ${N_{\textrm{Dy}}}$ in a crystal are coded in the legend and applied to both plots.
Fig. 5. (a) Integrated absorption cross-sections Ξk for bands centered at 908 nm (blue), 1113 nm (red), 1295 nm (black), 1668 nm (green) as function of the dysprosium concentration ${N_{\textrm{Dy}}}$ in a crystal. (b) Plot of the crystal field parameters calculated by Judd-Ofelt analysis against the dysprosium concentration in a crystal.
Table 2. Energies of Stark's sub-levels of the Dy3+ ion in the AgCl0.5Br0.5:Dy single crystals determined from the temperature dependences of the luminescence and absorption spectra.
4.2. Luminescence spectra
Under quasi-CW excitation with the 1.3 µm laser diode we observe three intense emission bands for all samples: around 1.8 µm, 2.9 µm and 4.4 µm (Fig. 6). These bands unambiguously match those observed in KPb2Cl5:Dy [13], RbPb2Cl5:Dy [9] and AgClxBr1-x:Dy [12], and are identified as electronic f – f transitions ${}_{}^6{H_{11/2}} \to {}_{}^6{H_{15/2}}$ ("2"$\; \to $"0"), ${}_{}^6{H_{13/2}} \to {}_{}^6{H_{15/2}}$ ("1"$\; \to $"0"), and ${}_{}^6{H_{11/2}} \to {}_{}^6{H_{13/2}}$ ("2"$\; \to $"1"), respectively. The gap near 4.3 µm in the middle of the 4.4 µm band is caused by the CO2 absorption in the atmosphere. The strong narrow 2.6 µm line is the laser diode line in the second order of the diffraction grating. The luminescence bands near 1.3 µm, 2.4 µm, and 5.5 µm, corresponding to the ${}_{}^6{F_{11/2}},{}^6{H_{9/2}} \to {}_{}^6{H_{15/2}}$ ("3"$\; \to $"0"), ${}_{}^6{F_{11/2}},{}^6{H_{9/2}} \to {}_{}^6{H_{13/2}}$ ("3"$\; \to $"1"), and ${}_{}^6{F_{11/2}},{}^6{H_{9/2}} \to {}_{}^6{H_{11/2}}$ ("3"$\; \to $"2") transitions, were too weak to be observed under quasi-CW excitation.
Fig. 6. Luminescence spectra of crystals with different dysprosium concentrations under quasi-CW excitation at 1312 nm recorded with InAs detector (a, c) and MCT detector (b). (c) Luminescence spectra of the crystal 4 under tunable excitation into the 1.3 µm band. The excitation wavelengths are shown in the plot.
Fig. 7. Luminescence spectrum of the sample #1 under pulsed excitation at 1.319 µm.
The spectral form of each luminescence band does not depend on dysprosium concentration, but their relative intensities do [Figs. 6(a), 6(b)]. The spectra in Fig. 6(a) are normalized to keep the maximum at 1.8 µm the same for all samples. One can clearly see the increase of the relative strength of the 2.9 µm band with growing concentration.
Figure 6(c) shows luminescence spectra for the 2.9 µm band of the heavily doped sample #4 when it is excited at different wavelengths across the 1.3 µm absorption band by the tunable OPA source. The spectra slightly differ in the region of the dip at 2.9 µm, which becomes less deep when excited in the center of the 1.3 µm absorption band, and this change is beyond the recording noise. This is a sign of a certain degree of inhomogeneous broadening of the absorption spectra.
Using the pulsed excitation by the Nd:YAG laser at 1.319 µm, we were able to observe five luminescence bands in all samples. Namely, three bands described above and two additional bands in the 1.3 and 5.5 µm regions. Intensities of the latter bands are much smaller in comparison to others for all samples, especially for #4, and it increases at smaller Dy concentrations. Spectrum of the 5.5 µm luminescence band is detected with good signal/noise ratio only for the sample #1 (Fig. 7). This band belongs to the "3"$\; \to $"2" transition [9,12,13].
Luminescence corresponding to "3"$\; \to $"1" transition, which according to the level separation scheme in Fig. 3 should be centered around 2.4 µm, could not be observed in our samples under any excitation to the 1.3 µm band.
4.3. Luminescence kinetics
The luminescence decays of 1.3 µm, 1.79 µm, 3 µm and 4.4 µm bands were selectively detected through a grating monochromator under excitation at 1.319 µm by Q-switch pulses. All decay curves are profoundly non-exponential (Fig. 8). The decay curves for samples with varying concentrations do not differ significantly. At 3 µm, the luminescence decays with a time constant of 0.42 ms at the initial stage, and 25 ms at the decay tail. At 1.79 µm and 4.4 µm, the luminescence first grows up to 2.3 ms, and then decays with the time constant of nearly 10 ms.
Fig. 8. Normalized luminescence kinetics for the sample #4 under excitation at 1.319 µm by the Q-switch laser pulse. The luminescence was recorded at 1.295 µm (violet), 3.00 µm (green), and 1.79 µm (blue). The red lines are theoretical fittings. In the inset the vertical solid line t = 0.0213 ms shows coincidence of peaks in the 3.00 µm and 1.79 µm kinetics.
Figure 9 shows the luminescence kinetics for crystals with different dysprosium concentrations. Luminescence decay for the 1.3 µm band was detected through monochromator at 1.295 µm. Luminescence decays for all crystals was found to be strongly non-exponential with slight difference in time constants at the initial stage. For example, at times shorter than 0.1 ms the decay time of the sample #1 is about 73 µs, and of the sample #2 it is about 60 µs. At the longer times, the time constant is about 1.3 ms for both crystals.
Fig. 9. Luminescence kinetics detected at 1.295 µm (a) and in overall 5.5 µm band (b) under 30 ns excitation at 1.319 µm. The green curves: sample #1, the blue curves: sample #2, the magenta curve: sample #4. The red dashed line shows the extrapolated exponential decay with 60 µs time constant in the initial kinetics stage for the sample #2. The detector time constant is ∼2 µs.
The signal of 5.5 µm luminescence was collected in the broad spectral range using a low-pass filter with a cut-off at 5 µm [Fig. 9(b)]. At the initial stage, the luminescence decays is about 60 µs for sample #1, and 50 µs for samples #2 and #3. At longer times, the decay time is about 1.3–2 ms.
5. Judd-Ofelt analysis
We have performed Judd-Ofelt analysis of the absorption spectra for three samples. The starting data were the integrated absorption cross-sections, calculated for the absorption bands at 0.9, 1.1, 1.3 and 1.7 µm using the formula (1) [Fig. 5(a)]. The 2.8 µm band has been excluded from consideration, as it may overlap with the absorption of the impurity OH group. According to the theory, the integral absorption cross-section Ξk is connected with the crystal field parameters ${\mathrm{\Omega }_2},{\; }{\mathrm{\Omega }_4},{\; }{\mathrm{\Omega }_6}$ and reduced-matrix elements of the $f - f$ electric dipole transitions ${U_2},{U_4},{U_6}$ by the formula [13]:
(2)$${\Xi _k} = \frac{{4{\pi ^2}}}{3}\frac{{{e^2}}}{{\hbar c}}\frac{1}{{2J + 1}}\left[ {\frac{{{{({{n^2} + 2} )}^2}}}{{9n}}\sum\limits_{t = 2,4,6} {{\Omega _t}{U_t} + n{S^{MD}}} } \right],$$
where n is the refractive index, J is total angular momentum of the initial state, and ${S^{MD}}$ is the line strength of the magnetic dipole transition. The latter vanishes according to the selection rules for transitions from the ground state ${H_{15/2}}$ to the excited states ${}^6{H_{9/2}},{}^6{F_{11/2}}$ and ${}^6{H_{11/2}}$, which correspond to bands that are included into analysis. The experimental integral cross-section data were fitted to formula (2) by varying the crystal field parameters ${\Omega _t}$. The matrix elements ${U_t}$ were taken from the Ref. [14], $n = 2.2$. The results for crystal field parameters are shown in Fig. 5(b). Further we have calculated the radiative lifetimes for three lower multiplets (Table 3). For thermalized ${}^6{H_{9/2}}$ and ${F_{11/2}}$ multiplets we accounted for the Boltzmann population distribution.
Table 3. Radiative lifetimes of the Dy3+ multiplets calculated by Judd-Ofelt analysis for different dysprosium concentrations in AgCl0.5Br0.5:Dy3+.
6. Discussions and theory of relaxation kinetics
The observed absorption and luminescence spectra fit well to the energy level scheme as shown in Fig. 3, where the horizontal solid lines correspond to the lowest Stark sublevels of a multiplet and dashed lines correspond to the highest Starks sublevels. Analysis of the helium absorption spectra, made in the Section 4.1, allowed us to conclude that there is predominantly one type of Dy3+ centers in the AgCl0.5Br0.5:Dy crystal.
Decrease of the 1.3 µm bandwidth and increase of the integrated absorption cross-section ${\mathrm{\Xi }_k}$ with dysprosium concentration indicate changes in the local crystal field that take place under increase of dysprosium doping (Figs. 4, 5). These changes also presume strong mutual interaction of Dy3+ ions that is obviously should affect relaxation of the excited states.
The electron paramagnetic resonance analysis of the AgCl:Dy crystal (${N_{Dy}} \sim {10^{18}}$ cm-3) suggests, that the charge compensation of the Dy3+ ion occurs locally by two Ag+ vacancies in the nearest coordination sphere, and that the angle between the directions from Dy3+ ion to the two vacancies is 90° [15]. Thus, Dy3+ ion forms an associate ${V^{\prime}_{Ag}} - \textrm{D}{\textrm{y}^{ {\bullet}{\bullet} }} - {V^{\prime}_{Ag}}$, that possess a nonzero dipole moment. Since the AgCl0.5Br0,5 single crystal has the same space group and very close lattice parameter to AgCl, we may assume, that the charge compensation follows the same pattern. At significantly higher dysprosium concentrations, as in our case, such associates may cluster due to dipole attraction that will change the local field symmetry. The non-monotonic change of the spectral form [Figs. 6(a),6(b)] and integral cross-section (Fig. 5) may reflect change of the preferred cluster size or structure. Small change of the 2.9 µm band shape [Fig. 6(c)] under excitation into the different parts of the absorption band at 1.3 µm may reflects variation of the crystal field around Dy3+ ions due to cluster size change (including un-clustered ions).
Since only one type of Dy3+ centers dominates in the crystal, the strongly non-exponential luminescence decay, which we observed for all samples and all bands, cannot be explained by a luminescence decay time difference between Dy3+ ions. Thus, we conclude that it is due to collective relaxation processes with strong dependence of relaxation probability on distance between the interacting centers. Increase of the relative luminescence intensity in the 2.9 µm band with increasing of dysprosium concentration ${N_{\textrm{Dy}}}$ [Fig. 6(a)] indicates a significant role of the cross-relaxational interaction of the excited (donor) and ground-state (acceptor) Dy3+ ions according to the scheme ${}_{}^6{H_{15/2}},\; {}_{}^6{F_{11/2}} + {}_{}^6{H_{9/2}} \to 2{}_{}^6{H_{13/2}}$ ({1} in Fig. 3), because probability of such interaction is proportional to the acceptor concentration, i.e. to ${N_{\textrm{Dy}}}$. Assumption of the cross-relaxation path dominance at initial stages of the level "3" decay is supported by rapid growth of strong luminescence signal at 3 µm with the simultaneous decay at 1.3 µm. In order to explain this strong cross-relaxation under relatively low concentration of Dy3+ ions, we should suggest that the ${V^{\prime}_{Ag}} - \textrm{D}{\textrm{y}^{ {\bullet}{\bullet} }} - {V^{\prime}_{Ag}}$ associates tend to assemble to clusters due to the dipole interaction that is mentioned above. Moreover, peaks at 0.0213 ms on the decay curves for 3 µm and 1.8 µm luminescence bands indicate contribution of the second cross-relaxation on the scheme ${}_{}^6{H_{9/2}}+{}_{}^6{F_{11/2}}, {}_{}^6{H_{13/2}} \to 2{}_{}^6{H_{11/2}}$ ({2} in Fig. 3). Further growth of the level "2" population up to the maximum at 2.3 ms would be due to the $2\;{}_{}^6{H_{13/2}} \to {}_{}^6{H_{11/2}}, {}_{}^6{H_{15/2}}$ up-conversion, analogously to the up-conversion population of the ${}_{}^3{H_5}$ level in Tm3+ ions, which have a similar energy level scheme [16].
Summarizing, we accept the following model for explanation of the observed luminescence decay. We suggest that Dy3+ ions enter AgCl0.5Br0.5 crystal lattice in two different ways. First, they form clustered centers, where two or more Dy3+ ions or associates are located in the nearest lattice positions. Their interaction causes fast cross-relaxation {1} (${}_{}^6{H_{15/2}},\; {}_{}^6{H_{9/2}} + {}_{}^6{F_{11/2}} \to 2{}_{}^6{H_{13/2}}\; $) leading to rapid depopulation of the level "3" and population of the level "1", and then to cross-relaxation {2} (${}_{}^6{H_{9/2}}+\; {}_{}^6{F_{11/2}}, {}_{}^6{H_{13/2}} \to 2{}_{}^6{H_{11/2}}$) partially populating level "2" (Fig. 3). Further population of level "2" goes through up-conversion {3}. Second, they stochastically enter the lattice far from each other. Interaction with them is weaker than for the clustered ions and occurs according to the Förster scheme [16]. For the sake of simplicity, we include only the clearly visible Förster-type cross-relaxation {1} into the model. The cross relaxation {2} and up-conversion {3} processes are neglected for the stand-alone ions, because their probabilities turn out to be much lower than that of the process {1} even for the clustered ions (Table 4). Thus, in our model the level "2" is populated only for the clustered ions. In the following, we will denote parameters of the clustered and stand-alone ions as prime and double-prime, respectively. For populations, ${n_i} = {n^{\prime}_i} + {n^{\prime\prime}_i}$, where ${n_i}$ is the total population of the level #i, as shown by red numbers in parentheses in Fig. 3.
Population decay of the level "3" for clustered ions obey the rate equation:
(3)$$\frac{{d{{n^{\prime}_3}}}}{{dt}} ={-} {k_{30}}{n^{\prime}_0}{n^{\prime}_3} - \frac{{{{n^{\prime}_3}}}}{{{{\tau ^{\prime}_3}}}} - {k_{31}}{n_1}^\prime {n_3}^\prime ,$$
where ${k_{30}}$ is the {1} cross-relaxation coefficient, ${\tau ^{\prime}_3}$ is the radiative lifetime of the clustered ions on level "3", and ${k_{31}}$ is the {2} cross-relaxation coefficient. We consider that cross-relaxation {2} plays a smaller role in depopulation of level "3" in comparison with cross-relaxation {1}, because it starts with delay, when level {1} is populated (Fig. 3), but level "3" has already been partially depopulated. Thus, we neglect the third term in the Eq. (3). We thus obtain a simple solution for decay of the level "3" of the associated ions under the assumption that the ground state population change during relaxation processes is negligible:
(4)$${n^{\prime}_3}(t )= \eta {n_{30}}\exp ( - t/{\tau _{30}}),$$
where $\eta \; $ is the fraction of the clustered ions within total concentration of the excited Dy3+ ions, ${\tau _{30}}$ accounts both for the non-radiative cross-relaxation and radiative decays: $\tau _{30}^{ - 1} = \tau _{3c}^{ - 1} + \tau ^{{\prime}{ - 1}}_3,\; \; \tau _{3c}^{ - 1} = \eta {n_0}{k_{30}}$, ${n_{30}}$ is the total initial population of the level "3" just after the end of pump pulse.
Next, we observe that cross-relaxation of stand-alone ions after the time $t \approx 0.05$ ms obeys the Förster decay [17]:
(5)$${n^{\prime\prime}_3}(t )= ({1 - \eta } ){n_{30}}\exp ( - \gamma \sqrt t - t/{\tau ^{\prime\prime}_3}),$$
where $\gamma $ is the Förster coefficient, ${\tau ^{\prime\prime}_3}$ is the radiative lifetime of the stand-alone ions on level "3". Thus, decay of the total population at the level "3" after 0.05 ms reads as:
(6)$${n_3}(t )= \eta {n_{30}}\exp ( - t/{\tau _{30}}) + \; ({1 - \eta } ){n_{30}}\exp ( - \gamma \sqrt t - t/{\tau ^{\prime\prime}_3})$$
Luminescence decay curve of the 1.3 μm band for $t > 0.05$ ms was fitted by Eq. (6) by varying parameters $\eta ,\; \gamma ,\; {\tau _{30}}$ and ${\tau ^{\prime\prime}_3}$ (Fig. 8, Table 4). The fit quality hardly depends on ${\tau ^{\prime\prime}_3}$ value, and we can only set the lower limit of ${\tau ^{\prime\prime}_3} > 3$ ms from our data. Note that we cannot isolate radiative lifetime for the clustered ions ${\tau ^{\prime}_3}$ from the combined constant ${\tau _{30}}$ accounting for both the cross-relaxation and the radiative transition. By the reasons explained at the end of this Section, ${\tau ^{\prime}_3}$ is estimated by Judd-Ofelt analysis, so ${\tau ^{\prime}_3} \simeq 0.8$ ms ${\gg} {\tau _{30}}$ (Tables 3, 4), and ${\tau _{3c}} \simeq {\tau _{30}}$.
Table 4. Parameters calculated while fitting the luminescence decay curves for sample #4 according to Eq. (4) and (13)–(15) and resulted from Judd-Ofelt analysis.
Populations at levels "2" and "1" obey the following rate equations:
(7)$$\frac{{d{{n^{\prime}_2}}}}{{dt}} = \frac{{{{n^{\prime}_1}}}}{{{{\tau ^{\prime}_{1u}}}}} - \frac{{{{n^{\prime}_2}}}}{{{\tau _2}}} + \frac{{{{n^{\prime}_3}}}}{{{{\tau ^{\prime}_{32}}}}} + 2{k_{31}}{n^{\prime}_1}{n^{\prime}_3}\; ,$$
(8)$$\frac{{dn^{\prime\prime}_2{}}}{{dt}} ={-} \frac{{n^{\prime\prime}_2{}}}{{{\tau _2}}} + \frac{{n^{\prime\prime}_3{}}}{{{{\tau ^{\prime\prime}_{32}}}}}\; ,$$
(9)$$\frac{{d{{n^{\prime}_1}}}}{{dt}} = 2\eta {k_{30}}{n_0}{n^{\prime}_3} - \frac{{2{{n^{\prime}_1}}}}{{{{\tau ^{\prime}_{1u}}}}} - \frac{{{{n^{\prime}_1}}}}{{{\tau _{1r}}}} + \frac{{{{n^{\prime}_2}}}}{{{\tau _{21}}}} - {k_{31}}{n^{\prime}_1}{n^{\prime}_3},$$
(10)$$\frac{{d{{n^{\prime\prime}_1}}}}{{dt}} ={-} 2\left( {\frac{{d{{n^{\prime\prime}_3}}}}{{dt}} + \; \frac{{{{n^{\prime\prime}_3}}}}{{{{\tau^{\prime\prime}_3}}}}} \right) - \frac{{{{n^{\prime\prime}_1}}}}{{{\tau _{1r}}}} + \frac{{{{n^{\prime\prime}_2}}}}{{{\tau _{21}}}}\; ,$$
The first term in Eq. (7) and the second term in Eq. (9) are responsible for the up-conversion path {3}. We see that the up-conversion prevails in relaxation of the level "1". There are two reasons: 1) 1.8 µm luminescence growth, controlled by the up-conversion, is much faster than 3 µm luminescence decrease at the tail, which is controlled by radiative decay of level "1" (third terms in Eqs. (9) and (10); 2) cross-relaxation {2} has small contribution, because the corresponding peak at 0.0213 ms in the 1.8 µm luminescence signal is very small in comparison with the main maximum at 2.3 ms, which is due to the up-conversion {3} (Fig. 8). Thus, the up-conversion is a main relaxation channel followed immediately after cross-relaxation {1}. Since the cross-relaxation {1} always gives two paired excited ions at level "2", the up-conversion term is linear with respect to population ${n^{\prime}_1}$. In the Eq. (10) two terms in brackets are responsible for population of the level "1" of the stand-alone ions through cross-relaxation according to an equation analogous Eq.(3). Substituting Eq. (4) into Eqs. (7) and (9), then Eq. (5) into Eqs. (8) and (10) we get rate equations:
(11)$$\frac{{d{{n^{\prime}_2}}}}{{dt}} = \frac{{{{n^{\prime}_1}}}}{{{{\tau ^{\prime}_{1u}}}}} - \frac{{{{n^{\prime}_2}}}}{{{\tau _2}}} + \eta \frac{{{n_{30}}}}{{{{\tau ^{\prime}_{32}}}}}\exp ( - t/{\tau _{30}}) + 2{k_{31}}{n^{\prime}_3}{n^{\prime}_1}\; ,$$
(12)$$\frac{{d{{n^{\prime\prime}_2}}}}{{dt}} ={-} \frac{{{{n^{\prime\prime}_2}}}}{{{\tau _2}}} + ({1 - \eta } )\frac{{{n_{30}}}}{{{{\tau ^{\prime\prime}_{32}}}}}\; \exp ( - \gamma \sqrt t - t/{\tau ^{\prime\prime}_3}),$$
(13)$$\frac{{d{{n^{\prime}_1}}}}{{dt}} = 2\eta \frac{{{n_{30}}}}{{{\tau _{3c}}}}\exp ( - t/{\tau _{30}}) - \frac{{2{{n^{\prime}_1}}}}{{{{\tau ^{\prime}_{1u}}}}} - \frac{{{{n^{\prime}_1}}}}{{{\tau _{1r}}}} + \frac{{{{n^{\prime}_2}}}}{{{\tau _{21}}}} - {k_{31}}{n^{\prime}_1}{n^{\prime}_3}\; ,$$
(14)$$\frac{{d{{n^{\prime\prime}_1}}}}{{dt}} = \; \; \gamma ({1 - \eta } )\frac{{{n_{30}}}}{{\sqrt t }}\exp ( - \gamma \sqrt t - t/{\tau ^{\prime\prime}_3}) - \frac{{{{n^{\prime\prime}_1}}}}{{{\tau _{1r}}}} + \frac{{{{n^{\prime\prime}_2}}}}{{{\tau _{21}}}}\; .$$
Large difference in time scales allows fitting the decay for levels "1" and "2" separately from level "3" and further simplifying the model. During the first 0.45 ms, the level "3" population decreases tenfold, so for $t > 0.45$ ms kinetics of the levels "1" and "2" can be considered independently from that of level "3", starting from initial excitation of levels "1" and "2" by cross-relaxations {1} and {2} respectively (Fig. 3). Now level "2" of the stand-alone ions is not populated, because we excluded from consideration the radiative relaxation from level "3"to level "2" due to weakness of the 5.5 μm luminescence, and it is assumed that the up-conversion does not work for the stand-alone ions. Thus Eq. (12) can be omitted, and Eq. (13), Eq. (14), and Eq. (11) are respectively transformed to the rate equations:
(15)$$\frac{{d{{n^{\prime}_1}}}}{{dt}} ={-} \frac{{2{{n^{\prime}_1}}}}{{{{\tau ^{\prime}_{1u}}}}} - \frac{{{{n^{\prime}_1}}}}{{{\tau _{1r}}}} + \frac{{{{n^{\prime}_2}}}}{{{\tau _{21}}}} + {n^{\prime}_{10}}\delta ({{t_0}} ),$$
(16)$$\frac{{d{{n^{\prime\prime}_1}}}}{{dt}} = \; - \frac{{{{n^{\prime\prime}_1}}}}{{{\tau _{1r}}}} + \frac{{{{n^{\prime\prime}_2}}}}{{{\tau _{21}}}} + {n^{\prime\prime}_{10}}\delta ({{t_0}} ),$$
(17)$$\frac{{d{{n^{\prime}_2}}}}{{dt}} = \frac{{{{n^{\prime}_1}}}}{{{{\tau ^{\prime}_{1u}}}}} - \frac{{{{n^{\prime}_2}}}}{{{\tau _2}}} + {n^{\prime}_{20}}\delta ({{t_0}} ),$$
where $\delta (t )$ is a delta-function, ${n^{\prime}_{10}},\; {n^{\prime\prime}_{10}},\; {n^{\prime}_{20}}$ are the populations at levels "1" and "2", which result from the cross-relaxations {1} and {2} at time t0 = 0.45 ms. Using the pulsed excitation into the 1.3 µm band, we solved rate Eqs. (15)–(17) numerically and fitted the calculated dependencies ${n_2}(t )= {n^{\prime}_2}(t )$ and ${n_1}(t )= {n^{\prime}_1}(t )+ {n^{\prime\prime}_1}(t )$ to the experimental kinetics of the 1.8 and 3 µm luminescence on the time interval 0.45–55 ms (Fig. 8) by varying the parameters ${\tau ^{\prime}_{1u}},\; {\tau _{1r}},\; {\tau _2},\; {\tau _{21}},\; {n^{\prime}_{10}},\; {n^{\prime\prime}_{10}},$ and ${n^{\prime}_{20}}$ with an additional initial condition for population in the form ${n^{\prime}_{10}}({{t_0}} )+ {n^{\prime\prime}_{10}}({{t_0}} )+ {n^{\prime\prime}_{20}}({{t_0}} )= 1$. The calculated parameters are summarized in Table 4. Note good coincidence of the obtained radiative constants ${\tau _{1r}},{\tau _2}$ with measured ones in [12].
Judd-Ofelt analysis gives from two to three times lower values for the radiative lifetimes (Table 3) than values obtained from kinetics analysis. We explain this difference by a difference of radiative probabilities for the clustered and stand-alone ions. The clustered ions have higher absorption cross-section, and they dominate in absorption spectra, so that Judd-Ofelt analysis provides radiative lifetimes that are closer to those of the clustered ions (Table 3). The stand-alone ions dominate in the kinetics curve tails, providing the values for radiative lifetimes. In the Table 4 both values are shown. In Ref. [12] the radiative lifetimes obtained by Judd-Ofelt analysis are several times higher than in our work, while dysprosium concentration was nearly the same. This discrepancy could be due to lower clustering in samples investigated in [12] due to difference in the crystal growth regime.
Fig. 10. Spectra of absorption (black) and emission (red) cross-sections, calculated by the reciprocity method for the sample #4.
Since ${\tau _{3c}} < < {\tau ^{\prime}_3},{\tau ^{\prime\prime}_3}$, almost all excited ions relax from the level "3" to levels "1" and "2" during the first 0.45 ms. The obtained relaxation parameters allow calculating kinetics of populations of the levels "1" and "2" relative to the initial population of the level "3". We found that the peak population at level "1" reaches 77% of the initial population at level "3" (at t=0.45 ms), and the peak population at level "2" reaches 31% (at t=2.3 ms). It is important to note, that under CW excitation the relative population of the level "2" should be even higher, especially for samples with lower Dy3+ concentration, because relative intensity of the 1.8 μm band is higher for samples with lower Dy3+ concentration (Fig. 6(a)). This can be explained by the relative increase of the level "3" population for stand-alone ions, due to their longer lifetime. As a result, additional radiative channel "3" $\to $ "2" for population of level "2" comes into play under CW excitation.
Emission cross-section for the 3 µm emission band for sample #4 was calculated from the absorption spectrum by reciprocity method in the high temperature limit (Fig. 10) [18]. By the reason stated above this emission cross section spectrum is mostly related to the clustered ions, and these ions will be predominantly excited by pumping to 1.3 μm absorption band. Maximum emission cross section for stand-alone ions is obtained with Füchtbauer–Ladenburg formula using ${\tau ^{\prime\prime}_{1r}}$ and luminescence spectra [Fig. 6(a)], it was found as much as $5 \cdot {10^{ - 21}}c{m^2}$ at 3.0 μm, which is expectedly three times lower than that for the clustered ions. However, this value is very similar to the value obtained in Ref. [12], again indicating that the results of Ref. [12] refer to stand-alone ions.
Summarizing, these results make AgCl0.5Br0.5:Dy crystal promising for laser applications at 3 μm and 4.3 μm bands under 1.3 µm pumping.
AgCl0.5Br0,5:Dy crystals with Dy concentrations in the $({2 - 8} )\; \cdot {10^{18}}$ cm-3 range demonstrate strong interionic interactions during the decay of excited multiplets with emission in the near- and mid-infrared spectral regions. The dominant relaxation process under excitation to the most strong 1.3 µm absorption band is the cross-relaxation according to the scheme ${}_{}^6{H_{15/2}},\; {}_{}^6{H_{9/2}} + {}_{}^6{F_{11/2}} \to 2{}_{}^6{H_{13/2}}$, resulting in efficient population of the ${}_{}^6{H_{13/2}}$ multiplet. The ${}_{}^6{H_{11/2}}\; $multiplet is efficiently populated through cross-relaxation ${}_{}^6{H_{9/2}} + {}_{}^6{F_{11/2}}, {}_{}^6{H_{13/2}},\; \to 2{}_{}^6{H_{11/2}}$ and up-conversion $2{}_{}^6{H_{13/2}} \to {}_{}^6{H_{11/2}} + {}_{}^6{H_{15/2}}$. These processes make the AgCl0.5Br0,5:Dy crystal a perspective active medium for oscillation in the regions of 3 µm and 4.3 µm under pumping at 1.3 µm. Further on, polycrystalline silver halides fibers are manufactured by the hot extrusion. Thus, a mid-IR fiber laser based on these water-stable crystals and pumped by commercially available 1.3 µm diode lasers is feasible.
Russian Science Foundation (grant #19-12-00134).
1. Y. O. Aydin, V. Fortin, R. Vallée, and M. Bernier, "Towards power scaling of 2.8 μm fiber lasers," Opt. Lett. 43(18), 4542–4545 (2018). [CrossRef]
2. F. Maes, V. Fortin, M. Bernier, and R. Vallée, "5.6 W monolithic fiber laser at 3.55 μm," Opt. Lett. 42(11), 2054–2057 (2017). [CrossRef]
3. V. Fortin, M. Bernier, S. T. Bah, and R. Vallée, "30 W fluoride glass all-fiber laser at 2.94 μm," Opt. Lett. 40(12), 2882–2885 (2015). [CrossRef]
4. Y. Wang, F. Jobin, S. Duval, V. Fortin, P. Laporta, M. Bernier, G. Galzerano, and R. Vallée, "Ultrafast Dy3+:-fluoride fiber laser beyond 3 μm," Opt. Lett. 44(2), 395–398 (2019). [CrossRef]
5. O. Henderson-Sapir, S. D. Jackson, and D. J. Ottaway, "Versatile and widely tunable mid-infrared erbium doped ZBLAN fiber laser," Opt. Lett. 41(7), 1676–1679 (2016). [CrossRef]
6. M. R. Majewski and S. D. Jackson, "Tunable dysprosium laser," Opt. Lett. 41(19), 4496–4498 (2016). [CrossRef]
7. M. R. Majewski, R. I. Woodward, and S. D. Jackson, "Dysprosium-doped ZBLAN fiber laser tunable from 2.8 um to 3.4 µm, pumped at 1.7 µm," Opt. Lett. 43(5), 971–974 (2018). [CrossRef]
8. V. Fortin, R. Vallée, M. Poulain, S. Poulain, F. Maes, M. Bernier, and J.-Y. Carrée, "Room-temperature fiber laser at 3.92 μm," Optica 5(7), 761–764 (2018). [CrossRef]
9. A. G. Okhrimchuk, L. N. Butvina, E. M. Dianov, I. A. Shestakova, N. V Lichkova, V. N. Zagorodnev, and A. V Shestakov, "Optical spectroscopy of the RbPb2Cl5: Dy3+ laser crystal and oscillation at 5.5 µm at room temperature," J. Opt. Soc. Am. 24(10), 2690–2695 (2007). [CrossRef]
10. H. Jelínková, M. E. Doroshenko, M. Jelínek, J. Šulc, V. V. Osiko, V. V. Badikov, and D. V. Badikov, "Dysprosium-Doped PbGa2S4 Laser Generating at 4.3 µm Directly Pumped by 1.7 µm Laser Diode," Opt. Lett. 38(16), 3040–3043 (2013). [CrossRef]
11. A. Fujii, H. Stolz, and W. Von Der Osten, "Excitons and phonons in mixed silver halides studied by resonant Raman scattering," J. Phys. C: Solid State Phys. 16(9), 1713–1728 (1983). [CrossRef]
12. I. Shafir, A. Nause, L. Nagli, M. Rosenbluh, and A. Katzir, "Mid-infrared luminescence properties of Dy-doped silver halide crystals," Appl. Opt. 50(11), 1625–1630 (2011). [CrossRef]
13. M. C. Nostrand, R. H. Page, and S. A. Payne, "Optical properties of Dy3+ and Nd3+ doped KPb2Cl5," J. Opt. Soc. Am. B 18(3), 264–276 (2001). [CrossRef]
14. A. A. Kaminskii, Crystalline Lasers: Physical Processes and Operating Schemes (CRC Press, 1996).
15. F. B. I. Cook and M. J. A. Smith, "Electron paramagnetic resonance of trivalent chromium in silver chloride and silver bromide," J. Phys. C: Solid State Phys. 7(13), 2353–2364 (1974). [CrossRef]
16. G. Rustad and K. Stenersen, "Modeling of laser-pumped Tm and Ho lasers accounting for upconversion and ground-state depletion," IEEE J. Quantum Electron. 32(9), 1645–1656 (1996). [CrossRef]
17. T. Förster, "Experimentelle und theoretische Untersuchung des zwischenmolekularen Übergangs von Elektronenanregungsenergie," Z. Naturforsch. 4(5), 321–327 (1949). [CrossRef]
18. S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Wyers, "Infrared Cross-Section Measurements for Crystals Doped with Er3+, Tm3+, and Ho3+," IEEE J. Quantum Electron. 28(11), 2619–2630 (1992). [CrossRef]
Y. O. Aydin, V. Fortin, R. Vallée, and M. Bernier, "Towards power scaling of 2.8 μm fiber lasers," Opt. Lett. 43(18), 4542–4545 (2018).
F. Maes, V. Fortin, M. Bernier, and R. Vallée, "5.6 W monolithic fiber laser at 3.55 μm," Opt. Lett. 42(11), 2054–2057 (2017).
V. Fortin, M. Bernier, S. T. Bah, and R. Vallée, "30 W fluoride glass all-fiber laser at 2.94 μm," Opt. Lett. 40(12), 2882–2885 (2015).
Y. Wang, F. Jobin, S. Duval, V. Fortin, P. Laporta, M. Bernier, G. Galzerano, and R. Vallée, "Ultrafast Dy3+:-fluoride fiber laser beyond 3 μm," Opt. Lett. 44(2), 395–398 (2019).
O. Henderson-Sapir, S. D. Jackson, and D. J. Ottaway, "Versatile and widely tunable mid-infrared erbium doped ZBLAN fiber laser," Opt. Lett. 41(7), 1676–1679 (2016).
M. R. Majewski and S. D. Jackson, "Tunable dysprosium laser," Opt. Lett. 41(19), 4496–4498 (2016).
M. R. Majewski, R. I. Woodward, and S. D. Jackson, "Dysprosium-doped ZBLAN fiber laser tunable from 2.8 um to 3.4 µm, pumped at 1.7 µm," Opt. Lett. 43(5), 971–974 (2018).
V. Fortin, R. Vallée, M. Poulain, S. Poulain, F. Maes, M. Bernier, and J.-Y. Carrée, "Room-temperature fiber laser at 3.92 μm," Optica 5(7), 761–764 (2018).
A. G. Okhrimchuk, L. N. Butvina, E. M. Dianov, I. A. Shestakova, N. V Lichkova, V. N. Zagorodnev, and A. V Shestakov, "Optical spectroscopy of the RbPb2Cl5: Dy3+ laser crystal and oscillation at 5.5 µm at room temperature," J. Opt. Soc. Am. 24(10), 2690–2695 (2007).
H. Jelínková, M. E. Doroshenko, M. Jelínek, J. Šulc, V. V. Osiko, V. V. Badikov, and D. V. Badikov, "Dysprosium-Doped PbGa2S4 Laser Generating at 4.3 µm Directly Pumped by 1.7 µm Laser Diode," Opt. Lett. 38(16), 3040–3043 (2013).
A. Fujii, H. Stolz, and W. Von Der Osten, "Excitons and phonons in mixed silver halides studied by resonant Raman scattering," J. Phys. C: Solid State Phys. 16(9), 1713–1728 (1983).
I. Shafir, A. Nause, L. Nagli, M. Rosenbluh, and A. Katzir, "Mid-infrared luminescence properties of Dy-doped silver halide crystals," Appl. Opt. 50(11), 1625–1630 (2011).
M. C. Nostrand, R. H. Page, and S. A. Payne, "Optical properties of Dy3+ and Nd3+ doped KPb2Cl5," J. Opt. Soc. Am. B 18(3), 264–276 (2001).
A. A. Kaminskii, Crystalline Lasers: Physical Processes and Operating Schemes (CRC Press, 1996).
F. B. I. Cook and M. J. A. Smith, "Electron paramagnetic resonance of trivalent chromium in silver chloride and silver bromide," J. Phys. C: Solid State Phys. 7(13), 2353–2364 (1974).
G. Rustad and K. Stenersen, "Modeling of laser-pumped Tm and Ho lasers accounting for upconversion and ground-state depletion," IEEE J. Quantum Electron. 32(9), 1645–1656 (1996).
T. Förster, "Experimentelle und theoretische Untersuchung des zwischenmolekularen Übergangs von Elektronenanregungsenergie," Z. Naturforsch. 4(5), 321–327 (1949).
S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. F. Wyers, "Infrared Cross-Section Measurements for Crystals Doped with Er3+, Tm3+, and Ho3+," IEEE J. Quantum Electron. 28(11), 2619–2630 (1992).
Aydin, Y. O.
Badikov, D. V.
Badikov, V. V.
Bah, S. T.
Bernier, M.
Butvina, L. N.
Carrée, J.-Y.
Chase, L. L.
Cook, F. B. I.
Dianov, E. M.
Doroshenko, M. E.
Duval, S.
Förster, T.
Fortin, V.
Fujii, A.
Galzerano, G.
Henderson-Sapir, O.
Jackson, S. D.
Jelínek, M.
Jelínková, H.
Jobin, F.
Kaminskii, A. A.
Katzir, A.
Kway, W. L.
Laporta, P.
Lichkova, N. V
Maes, F.
Majewski, M. R.
Nagli, L.
Nause, A.
Nostrand, M. C.
Okhrimchuk, A. G.
Osiko, V. V.
Ottaway, D. J.
Page, R. H.
Payne, S. A.
Poulain, M.
Poulain, S.
Rosenbluh, M.
Rustad, G.
Shafir, I.
Shestakov, A. V
Shestakova, I. A.
Smith, L. K.
Smith, M. J. A.
Stenersen, K.
Stolz, H.
Šulc, J.
Vallée, R.
Von Der Osten, W.
Woodward, R. I.
Wyers, W. F.
Zagorodnev, V. N.
Appl. Opt. (1)
IEEE J. Quantum Electron. (2)
J. Opt. Soc. Am. (1)
J. Opt. Soc. Am. B (1)
J. Phys. C: Solid State Phys. (2)
Opt. Lett. (8)
Z. Naturforsch. (1)
Equations on this page are rendered with MathJax. Learn more.
(1) Ξ k = 1 N D y ∫ α ( λ ) λ d λ .
(2) Ξ k = 4 π 2 3 e 2 ℏ c 1 2 J + 1 [ ( n 2 + 2 ) 2 9 n ∑ t = 2 , 4 , 6 Ω t U t + n S M D ] ,
(3) d n 3 ′ d t = − k 30 n 0 ′ n 3 ′ − n 3 ′ τ 3 ′ − k 31 n 1 ′ n 3 ′ ,
(4) n 3 ′ ( t ) = η n 30 exp ( − t / τ 30 ) ,
(5) n 3 ′ ′ ( t ) = ( 1 − η ) n 30 exp ( − γ t − t / τ 3 ′ ′ ) ,
(6) n 3 ( t ) = η n 30 exp ( − t / τ 30 ) + ( 1 − η ) n 30 exp ( − γ t − t / τ 3 ′ ′ )
(7) d n 2 ′ d t = n 1 ′ τ 1 u ′ − n 2 ′ τ 2 + n 3 ′ τ 32 ′ + 2 k 31 n 1 ′ n 3 ′ ,
(8) d n 2 ′ ′ d t = − n 2 ′ ′ τ 2 + n 3 ′ ′ τ 32 ′ ′ ,
(9) d n 1 ′ d t = 2 η k 30 n 0 n 3 ′ − 2 n 1 ′ τ 1 u ′ − n 1 ′ τ 1 r + n 2 ′ τ 21 − k 31 n 1 ′ n 3 ′ ,
(10) d n 1 ′ ′ d t = − 2 ( d n 3 ′ ′ d t + n 3 ′ ′ τ 3 ′ ′ ) − n 1 ′ ′ τ 1 r + n 2 ′ ′ τ 21 ,
(11) d n 2 ′ d t = n 1 ′ τ 1 u ′ − n 2 ′ τ 2 + η n 30 τ 32 ′ exp ( − t / τ 30 ) + 2 k 31 n 3 ′ n 1 ′ ,
(12) d n 2 ′ ′ d t = − n 2 ′ ′ τ 2 + ( 1 − η ) n 30 τ 32 ′ ′ exp ( − γ t − t / τ 3 ′ ′ ) ,
(13) d n 1 ′ d t = 2 η n 30 τ 3 c exp ( − t / τ 30 ) − 2 n 1 ′ τ 1 u ′ − n 1 ′ τ 1 r + n 2 ′ τ 21 − k 31 n 1 ′ n 3 ′ ,
(14) d n 1 ′ ′ d t = γ ( 1 − η ) n 30 t exp ( − γ t − t / τ 3 ′ ′ ) − n 1 ′ ′ τ 1 r + n 2 ′ ′ τ 21 .
(15) d n 1 ′ d t = − 2 n 1 ′ τ 1 u ′ − n 1 ′ τ 1 r + n 2 ′ τ 21 + n 10 ′ δ ( t 0 ) ,
(16) d n 1 ′ ′ d t = − n 1 ′ ′ τ 1 r + n 2 ′ ′ τ 21 + n 10 ′ ′ δ ( t 0 ) ,
(17) d n 2 ′ d t = n 1 ′ τ 1 u ′ − n 2 ′ τ 2 + n 20 ′ δ ( t 0 ) ,
Dysprosium concentrations in the studied samples.
Crystal #
N Dy ( × 1018 cm-3) 2.30 4.05 4.73 7.16
Energies of Stark's sub-levels of the Dy3+ ion in the AgCl0.5Br0.5:Dy single crystals determined from the temperature dependences of the luminescence and absorption spectra.
2S+1LJ
E (cm-1)
6H15/2 1 0 6F11/2 1 7826
2 20.5 2 7908
3 52 a 3 7928
4 159 4 7946
5 176 a 5 7998
6 280 6 8033 a
7 308 a 6H7/2 1 8996.3
8 359 a 2 9021.6
6H13/2 1 3568 a 3 9149
2 3587 4 9158
3 3593.5 6F9/2 1 9185
4 3600.6 2 9195
5 - 3 9209
6H11/2 1 5921 6H5/2 1 10242
2 5937 a 2 10362
3 6021 3 -
4 6075 a 6F7/2 1 11020
6 6120 3 11073.5
6H9/2 1 7698.5 4 11108
2 7704 6H5/2 1 12385
3 7722 2 12398 a
4 7737 3 12472
5 7754 6F3/2 1 13241.3
? 7772 2 13253.8 a
a Not clearly observed or weak shoulder
Radiative lifetimes of the Dy3+ multiplets calculated by Judd-Ofelt analysis for different dysprosium concentrations in AgCl0.5Br0.5:Dy3+.
Lifetime τ rad (ms)
Crystal # 1 2 4
NDy, 1018 cm-3 2.3 4.05 7.16
6H9/2+6F11/2 1.2 0.55 0.80
6H11/2 8.7 4.1 5.5
6H13/2 12.3 6.6 8.8
Parameters calculated while fitting the luminescence decay curves for sample #4 according to Eq. (4) and (13)–(15) and resulted from Judd-Ofelt analysis.
Part of the clustered ions, excited to level "3", η 73%
Förster decay parameter, γ 1.1 ms-0.5
Constant for cross-relaxation {1} for the clustered ions, τ 3 c 62 µs
Radiative relaxation time of level "3" of stand-alone ions, τ 3 ′ ′ >3 ms
Radiative relaxation time of level "3" of clustered ions, τ 3 ′ 0.80 ms
Radiative relaxation time of level "2"of stand-alone ions, 9.9 ms
Radiative relaxation time of level "2" of clustered ions, τ 2 ′ 5.5 ms
Inverse of transition probability "2" → "1" for clustered ions, τ 21 >200 ms
Inverse of the up-conversion {3} probability τ 1 u 2.2 ms
Radiative relaxation time of level "1"of stand-alone ions, τ 1 r ′ ′ 26 ms
Radiative relaxation time of level "1" of clustered ions, τ 1 r ′ 8.8 ms
Initial population of the clustered ions at level "1", n 10 ′ 53%
Initial population of the stand-alone ions at level "1" n 10 ′ ′ 24%
Initial population of the stand-alone ions at level "2", n 20 ′ ′ 0 | CommonCrawl |
# Understanding the Discrete Fourier Transform
The DFT is based on the Fourier series, which is a mathematical representation of a periodic function. The Fourier series decomposes a periodic function into its constituent frequencies. In the context of audio processing, the DFT is used to analyze the frequency content of a sound signal.
To calculate the DFT of a signal, we multiply the signal values with the complex exponential function, which is defined as:
$$e^{j\omega t}$$
where $j$ is the imaginary unit, $\omega$ is the angular frequency, and $t$ is the time variable. The DFT is then calculated by summing the products over all time samples.
In JavaScript, we can use the Fast Fourier Transform (FFT) algorithm to efficiently compute the DFT of a signal. The FFT algorithm is an efficient way to calculate the DFT, as it reduces the time complexity from $O(n^2)$ to $O(n\log n)$.
## Exercise
Calculate the DFT of the following signal:
$$x(t) = \begin{cases} 1, & 0 \leq t < \frac{1}{2} \\ -1, & \frac{1}{2} \leq t < 1 \end{cases}$$
# Setting up the JavaScript environment for audio processing
To get started with audio processing in JavaScript, we need to include the Web Audio API in our HTML file. The Web Audio API provides a powerful and flexible system for controlling audio playback and processing.
Once the Web Audio API is included, we can create an instance of the `AudioContext` class, which serves as the main interface for audio processing in JavaScript.
There are several libraries and tools available for audio processing in JavaScript, such as Tone.js, Howler.js, and Web Audio API Extensions. These libraries provide a set of functions and utilities that simplify the process of creating and manipulating audio in JavaScript.
## Exercise
Include the Web Audio API in your HTML file and create an instance of the `AudioContext` class.
# Using the Web Audio API for real-time effects
The Web Audio API provides a set of audio nodes that can be connected to form an audio graph. The audio graph is a representation of the signal flow between different audio nodes. In JavaScript, we can create audio nodes using the `AudioContext` class.
To create a real-time audio effect, we can connect an audio source (such as a microphone input or an audio file) to an audio destination (such as an audio output device). We can then process the audio signal in real-time by applying various audio effects to the audio nodes in the audio graph.
## Exercise
Create an audio graph using the Web Audio API that connects an audio source to an audio destination. Apply a basic audio effect, such as a gain change, to the audio graph.
# Applying the Discrete Fourier Transform to audio data
To calculate the DFT of an audio signal, we first need to obtain the audio data from the audio source. We can use the `ScriptProcessorNode` interface in the Web Audio API to process the audio data in real-time.
Once we have the audio data, we can calculate the DFT using a JavaScript library, such as the `dsp.js` library. The `dsp.js` library provides functions for audio processing, including the DFT calculation.
To visualize the frequency content of the audio signal, we can use a JavaScript library, such as the `pizzicato.js` library. The `pizzicato.js` library provides functions for creating and manipulating audio effects in JavaScript.
## Exercise
Calculate the DFT of an audio signal using the Web Audio API and the `dsp.js` library. Visualize the frequency content of the audio signal using the `pizzicato.js` library.
# Implementing basic audio effects using the Discrete Fourier Transform
To create a basic audio effect, we first need to calculate the DFT of the audio signal using the Web Audio API and the `dsp.js` library. We can then manipulate the frequency components of the audio signal to create the desired audio effect.
For example, we can create a low-pass filter by setting all frequency components above a certain threshold to zero. This will attenuate the high-frequency components of the audio signal, resulting in a lower-pitched output.
## Exercise
Implement a low-pass filter using the DFT and apply it to an audio signal.
# Creating custom audio effects with the Discrete Fourier Transform
To create a custom audio effect, we first need to calculate the DFT of the audio signal using the Web Audio API and the `dsp.js` library. We can then design the audio effect by manipulating the frequency components of the audio signal based on a set of rules or algorithms.
For example, we can create a frequency-domain compression effect by compressing the amplitude of the frequency components based on a compression ratio. This will result in a more evenly-distributed frequency content in the audio signal, with a reduced dynamic range.
## Exercise
Design a frequency-domain compression effect using the DFT and apply it to an audio signal.
# Optimizing real-time audio processing for web applications
To optimize real-time audio processing for web applications, we can use techniques such as buffering and parallel processing. We can also use JavaScript libraries, such as the `dsp.js` library, which provide optimized functions for audio processing.
To ensure smooth and responsive audio processing, we can use the `requestAnimationFrame` function in JavaScript to schedule the processing tasks at the optimal time. This will prevent the audio processing from interfering with other tasks, such as rendering the user interface.
## Exercise
Optimize the real-time audio processing for a web application using the techniques and best practices discussed in this section.
# Advanced topics in real-time audio processing with JavaScript
To implement more complex audio effects, we can use the Discrete Fourier Transform (DFT) and the Web Audio API to analyze and manipulate the frequency content of the audio signal. We can also use JavaScript libraries, such as the `dsp.js` library, which provide advanced functions for audio processing.
For example, we can create a pitch-shifting effect by manipulating the frequency components of the audio signal based on a pitch-shifting algorithm. This will result in a higher or lower-pitched output.
## Exercise
Implement a pitch-shifting effect using the DFT and apply it to an audio signal.
# Conclusion and future developments in real-time audio effects
The real-time audio processing with JavaScript has a bright future, with advancements in web technologies and the increasing demand for immersive and interactive experiences on the web. As the Web Audio API and JavaScript libraries continue to evolve, we can expect more powerful and flexible audio processing capabilities for web applications.
In conclusion, creating real-time audio effects with the Discrete Fourier Transform in JavaScript offers a powerful and flexible way to analyze and manipulate audio signals in web applications. With the advancements in web technologies, we can expect even more exciting developments in real-time audio processing with JavaScript. | Textbooks |
\begin{document}
\title{Trees of fusion systems} \thanks{Accepted for publication subject to revisions by Journal of Algebra, October 2013}
\author{Jason Semeraro} \address{Institute of Mathematics, University of Oxford, 24-29 St Giles, Oxford} \email{[email protected]} \maketitle
\begin{abstract} We define a `tree of fusion systems' and give a sufficient condition for its completion to be saturated. We apply this result to enlarge an arbitrary fusion system by extending the automorphism groups of certain of its subgroups. \end{abstract}
Saturated fusion systems have come into prominence during the course of the last two decades and may be viewed as a convenient language in which to study some types of algebraic objects at a particular prime. The original idea is due to Puig in \cite{Puig}, although our notation and terminology more closely follows that of Broto, Levi and Oliver in \cite{BLO1}. The area is now well established and has attracted wide interest from researchers in group theory, topology and representation theory. Formally, a fusion system on a finite $p$-group $S$ is a category $\mathcal{F}$ whose objects consist of all subgroups of $S$ and whose morphisms a certain group monomorphisms between objects. Saturation is an additional property, which is satisfied by many `naturally occurring' fusion systems, including those induced by finite groups.
Recall that a tree of groups consists of a tree $\mathcal{T}$ with an assignment of groups to vertices and edges, and monomorphisms from `edge groups' to incident `vertex groups'. The completion of a tree of groups is the free product of all vertex groups modulo relations determined by the monomorphisms. The theory of trees of groups is a special case of Bass--Serre theory and this paper is an attempt to extend this theory to fusion systems (over a fixed prime $p$) by attaching fusion systems to vertices and edges of some fixed tree, and injective morphisms from edge to vertex fusion systems. As in the case for groups, the completion of a `tree of fusion systems' is a colimit for the natural diagram, and we find a condition which renders this the fusion system generated by each of those attached to a vertex.
One naturally obtains a tree of fusion systems from a tree of finite groups by replacing each group in the tree by its fusion systems at the prime $p$. Two questions arise at this point: \begin{itemize} \item[(1)] Is the completion of this tree of fusion systems the fusion system of the completion of the underlying tree of groups? \item[(2)] Is it possible to prove that the completion is saturated `fusion theoretically,' i.e. without reference to the groups themselves? \end{itemize} We answer (1) in the affirmative essentially by applying a straightforward lemma of Robinson concerning conjugacy relations in completions of trees of groups:
\begin{thma} Let $(\mathcal{T},\mathcal{G})$ be a tree of finite groups and write $\mathcal{G}_\mathcal{T}$ for the completion of $(\mathcal{T},\mathcal{G})$. Let $(\mathcal{T},\mathcal{F},\mathcal{S})$ be a tree of fusion systems induced by $(\mathcal{T},\mathcal{G})$ which satisfies $(H)$ so that there exists a completion $\mathcal{F}_\mathcal{T}$ for $(\mathcal{T},\mathcal{F},\mathcal{S})$. The following hold: \begin{itemize} \item[(a)] $\mathcal{S}(v_*)$ is a Sylow $p$-subgroup of $\mathcal{G}_\mathcal{T}$. \item[(b)] $\mathcal{F}_{\mathcal{S}(v_*)}(\mathcal{G}_\mathcal{T})= \mathcal{F}_\mathcal{T}$. \end{itemize} In particular, $\mathcal{F}_\mathcal{T}$ is independent of the choice of tree of fusion systems $(\mathcal{T},\mathcal{F},\mathcal{S})$ induced by $(\mathcal{T},\mathcal{G})$. \end{thma} In this theorem, $(H)$ is a condition on $(\mathcal{T},\mathcal{F},\mathcal{S})$ which forces the existence of a completion $\mathcal{F}_\mathcal{T}$ on the $p$-group $\mathcal{S}(v_*)$ associated to a fixed vertex $v_*$ of the underlying tree $\mathcal{T}$. To answer question (2) above, we require the introduction of some more terminology. Alperin's Theorem asserts that conjugacy in a saturated fusion system $\mathcal{F}$ on $S$ is determined by the `$\mathcal{F}$-essential' subgroups and $S$. Conversely a deep result of Puig asserts that whenever conjugacy in $\mathcal{F}$ is determined by the `$\mathcal{F}$-centric' subgroups, $\mathcal{F}$ is saturated if the saturation axioms hold between such subgroups. We exploit both of these facts in the proof of the following theorem:
\begin{thmb} Let $(\mathcal{T},\mathcal{F},\mathcal{S})$ be a tree of fusion systems which satisfies $(H)$ and assume that $\mathcal{F}(v)$ is saturated for each vertex $v$ of $\mathcal{T}.$ Write $S:=\mathcal{S}(v_*)$ and $\mathcal{F}_\mathcal{T}$ for the completion of $(\mathcal{T},\mathcal{F},\mathcal{S})$. Assume that the following hold for each $P \leqslant S$: \begin{itemize} \item[(a)] If $P$ is $\mathcal{F}_\mathcal{T}$-conjugate to an $\mathcal{F}(v)$-essential subgroup or $P=\mathcal{S}(v)$ then $P$ is $\mathcal{F}_\mathcal{T}$-centric. \item[(b)] If $P$ is $\mathcal{F}_\mathcal{T}$-centric then $\operatorname{Rep}_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$ is a tree. \end{itemize} Then $\mathcal{F}_\mathcal{T}$ is a saturated fusion system on $S$. \end{thmb} Theorem B in the case where $(\mathcal{T},\mathcal{F},\mathcal{S})$ is induced by a tree of groups $(\mathcal{T},\mathcal{G})$ is \cite[Theorem 4.2]{BLO4}, and can be deduced from Theorem B by applying Theorem A. The novelty of our approach is the introduction of the graph $\operatorname{Rep}_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$ called the `$P$-orbit graph' (defined for each $P \leqslant \mathcal{S}(v_*)$) which gives detailed information about the way in which $P$ `acts' on $\mathcal{F}_\mathcal{T}$. Condition (a) ensures that conjugacy in $\mathcal{F}_\mathcal{T}$ is determined by the $\mathcal{F}_\mathcal{T}$-centric subgroups and (b) ensures that the saturation axioms hold between such subgroups.
Theorem B is useful in determining fusion systems over specific (families of) $p$-groups. For example, in \cite{O5}, Oliver applies \cite[Theorem 5.1]{BLO4} (which follows from \cite[Theorem 4.2]{BLO4}) to prove that saturated fusion systems over $p$-groups with abelian subgroup of index $p$ are uniquely determined by their essential subgroups. Our next result is a generalisation of \cite[Theorem 5.1]{BLO4} to arbitrary fusion systems.
\begin{thmc} Let $\mathcal{F}_0$ be a saturated fusion system on a finite $p$-group $S$. For $1 \leqslant i \leqslant m$, let $Q_i \leqslant S$ be fully $\mathcal{F}_0$-normalised subgroups with $Q_i\varphi \nleq Q_j$ for each $\varphi \in \operatorname{Hom}_{\mathcal{F}_0}(Q_i,S)$ and $i \neq j$. Set $K_i:=\operatorname{Out}_{\mathcal{F}_0}(Q_i)$ and choose $\Delta_i \leqslant \operatorname{Out}(Q_i)$ so that $K_i$ is a strongly $p$-embedded subgroup of $\Delta_i$. Write $$\mathcal{F}= \langle \{\operatorname{Hom}_{\mathcal{F}_0}(P,S) \mid P \leqslant S\} \cup \{\Delta_i \mid 1 \leqslant i \leqslant m \} \rangle_S.$$ Assume further that for each $1 \leqslant i \leqslant m$, \begin{itemize} \item[(a)] $Q_i$ is $\mathcal{F}_0$-centric (hence $\mathcal{F}$-centric) and minimal (under inclusion) amongst all $\mathcal{F}$-centric subgroups; and \item[(b)] no proper subgroup of $Q_i$ is $\mathcal{F}_0$-essential. \end{itemize} Then $\mathcal{F}$ is saturated. \end{thmc}
Recall that a subgroup $H$ of a finite group $G$ is \textit{strongly $p$-embedded} if $H < G$, $H$ contains a Sylow $p$-subgroup of $G$ and $H \cap H^g$ is a $p'$-group for each $g \in G \backslash H$. The proof Theorem C is similar in structure to that of \cite[Theorem 5.1]{BLO4}, but uses less group theory.
The paper is structured as follows. In Section \ref{section1}, we introduce fusion systems, saturation, Alperin's theorem and $\mathcal{F}$-normaliser subsystems. We then introduce the main objects of study - trees of fusion systems - in Section \ref{treesfus}, along with their completions and $P$-orbit graphs. In Section \ref{orbfus}, we describe the relationship between trees of groups and trees of fusion systems and prove Theorem A. Theorem B will be proved in Section \ref{compfus} before being applied in Section \ref{extconst} to prove Theorem C.
Finally, we make some important remarks regarding notation. Elements of a finite group $G$ typically act on the right. We reserve the superscript notation $x^g$ for image of the conjugation action $g^{-1}xg$ of $G$ on itself for each $x,g \in G$. Similarly if $H$ is a subgroup of $G$ we write $H^g$ for the group $\{x^g \mid x \in H\}$. We will also frequently write $c_g$ for the group homomorphism induced by conjugation by $g$, stating the domain where appropriate. Note that group homomorphisms will always act on the right and composition read from left to right.
\section{Background}\label{section1} \subsection{Saturated Fusion Systems} We begin with a precise definition of what is meant by a fusion system. For any group $S$ and $P,Q \leqslant S$ we write Hom$_S(P,Q)$ for the set of homomorphisms from $P$ to $Q$ induced by conjugation by elements of $S$ and Inj$(P,Q)$ for the set of monomorphisms from $P$ to $Q$.
\begin{Def}\label{fus} Let $S$ be a finite $p$-group. A \textit{fusion system on $S$} is a category $\mathcal{F}$ where Ob$(\mathcal{F}):=\{P \mid P \leqslant S \}$, and where for each $P,Q \in$ Ob$(\mathcal{F})$ \begin{itemize} \item[(a)] Hom$_S(P,Q) \subseteq$ Hom$_\mathcal{F}(P,Q) \subseteq$ Inj$(P,Q)$; and \item[(b)] each $\varphi \in$ Hom$_\mathcal{F}(P,Q)$ factorises as an $\mathcal{F}$-isomorphism $\begin{CD} P @>>> P\varphi \\ \end{CD}$ followed by an \textit{inclusion} $\begin{CD} \iota_{P\varphi}^Q:P\varphi @>>> Q. \\ \end{CD}$ \end{itemize} \end{Def}
Given an arbitrary collection $X_{P,Q} \subseteq$ Inj$(P,Q)$ of morphisms which contains Hom$_S(P,Q)$ for each $P,Q \leqslant S$, we can always construct a fusion system $\mathcal{F}$ on $S$ with $X_{P,Q} \subseteq$ Hom$_\mathcal{F}(P,Q)$ and where Hom$_\mathcal{F}(P,Q)$ $\backslash$ $X_{P,Q}$ only consists of $\mathcal{F}$-isomorphisms. In particular, $\mathcal{F}$ is minimal (with the respect to the number of morphisms) amongst all fusion systems $\mathcal{G}$ on $S$ with the property that $X_{P,Q} \subseteq$ Hom$_\mathcal{G}(P,Q)$ for each $P,Q \leqslant S$. Next we define morphisms between fusion systems.
\begin{Def}
Let $\mathcal{F}$ and $\mathcal{E}$ be fusion systems on finite $p$-groups $S$ and $T$ respectively. $\varphi \in \operatorname{Hom}(S,T)$ is a \textit{morphism} from $\mathcal{F}$ to $\mathcal{E}$ if for each $P,R \leqslant S$ and each $\alpha \in \operatorname{Hom}_\mathcal{F}(P,R)$, there exists $\beta \in \mbox{Hom}_\mathcal{E}(P\varphi,R\varphi)$ such that $$\alpha \circ \varphi|_R=\varphi|_P \circ \beta.$$ Hence $\varphi$ induces a functor from $\mathcal{F}$ to $\mathcal{E}$. \end{Def}
In particular, fusion systems form a category which we denote by $\mathfrak{Fus}$. The following definition collects some important notions to which we will constantly refer, all of which are needed to define saturation. The reader is referred to \cite[Section I.2]{AKO} and \cite[Section 1.5]{CR} for a more thorough introduction to these ideas.
\begin{Def} Let $\mathcal{F}$ be a fusion system on a finite $p$-group $S$ and let $P,Q \leqslant S$. \begin{itemize} \item[(a)] Iso$_\mathcal{F}(P,Q)$ denotes the set of $\mathcal{F}$-isomorphisms from $P$ to $Q$ and Aut$_\mathcal{F}(P) $:= Iso$_\mathcal{F}(P,P)$. \item[(b)] $P$ and $Q$ are \textit{$\mathcal{F}$-conjugate} whenever Iso$_\mathcal{F}(P,Q) \neq \emptyset$ and $P^\mathcal{F}$ denotes the set of all $\mathcal{F}$-conjugates of $P$.
\item[(c)] $P$ is \textit{fully $\mathcal{F}$-normalised} respectively \textit{fully $\mathcal{F}$-centralised} if for each $R \in P^\mathcal{F}$, the inequality $$|N_S(P)| \geqslant |N_S(R)| \mbox{ respectively } |C_S(P)| \geqslant |C_S(R)|$$ holds. \item[(d)] $P$ is \textit{fully $\mathcal{F}$-automised} if Aut$_S(P) \in$ Syl$_p($Aut$_\mathcal{F}(P)).$ \item[(e)] For each $\varphi \in$ Hom$_\mathcal{F}(P,S)$, $$N_\varphi=N_\varphi^\mathcal{F}:=\{g \in N_S(P) \mid \varphi^{-1} \circ c_g \circ \varphi \in \mbox{Aut}_S(P\varphi) \},$$ where $c_g \in$ Aut$_S(P)$ is the automorphism induced by conjugation by $g$. \end{itemize} \end{Def} There are a number of equivalent ways to define saturation and we refer the reader to \cite[Section I.9]{AKO} or \cite[Section 4.3]{CR} for a comparison of these. The definition we choose is listed among the definitions in these references and also appears in \cite[Definition 1.2]{BLO1}.
\begin{Def}\label{sat} Let $\mathcal{F}$ be a fusion system on a finite $p$-group $S$. We say that $\mathcal{F}$ is \textit{saturated} if the following hold: \begin{itemize} \item[(a)] Whenever $P \leqslant S$ is fully $\mathcal{F}$-normalised, it is fully $\mathcal{F}$-centralised and fully $\mathcal{F}$-automised.
\item[(b)] For all $P \leqslant S$ and $\varphi \in$ Hom$_\mathcal{F}(P,S)$ such that $P\varphi$ is fully $\mathcal{F}$-centralised, there is $\overline{\varphi} \in $ Hom$_\mathcal{F}(N_\varphi, S)$ such that $\overline{\varphi}|_P=\varphi$. \end{itemize} \end{Def}
We observe that a natural class of examples of saturated fusion systems is provided by groups. A finite $p$-subgroup $S$ of a group $G$ is a \textit{Sylow $p$-subgroup of $G$} if every finite $p$-subgroup of $G$ is $G$-conjugate to a subgroup of $S$. Let $\mathcal{F}_S(G)$ denote the fusion system on $S$ where for each $P,Q \leqslant S$, Hom$_{\mathcal{F}_S(G)}(P,Q):=$ Hom$_G(P,Q).$
\begin{Thm}\label{fsgsat} Let $G$ be a finite group and $S$ be a Sylow $p$-subgroup of $G$. The fusion system $\mathcal{F}_S(G)$ is saturated. \end{Thm}
\subsection{Alperin's Theorem} We now concern ourselves with `generation' of saturated fusion systems, starting with some more definitions: \begin{Def} Let $\mathcal{F}$ be a fusion system on a finite $p$-group $S$ and let $P \leqslant S.$ \begin{itemize} \item[(a)] $P$ is \textit{$S$-centric} if $C_S(P)=Z(P)$ and $P$ is \textit{$\mathcal{F}$-centric} if $Q$ is $S$-centric for each $Q \in P^\mathcal{F}.$ \item[(b)] Write $\operatorname{Out}_\mathcal{F}(P):=\operatorname{Aut}_\mathcal{F}(P)/\operatorname{Inn}(P)$. $P$ is \textit{$\mathcal{F}$-essential} if $P$ is $\mathcal{F}$-centric and $\operatorname{Out}_\mathcal{F}(P)$ contains a strongly $p$-embedded subgroup. \item[(c)] $P$ is \textit{$\mathcal{F}$-radical} if $O_p(\operatorname{Out}_\mathcal{F}(P))=1$. \end{itemize} \end{Def} The following lemma concerning $\mathcal{F}$-centric subgroups is extremely useful.
\begin{Lem}\label{centlem} Let $\mathcal{F}$ be a fusion system on a finite $p$-group $S$. The following hold for each $P,Q \leqslant S$: \begin{itemize} \item[(a)] If $P$ is fully $\mathcal{F}$-centralised then $PC_S(P)$ is $\mathcal{F}$-centric. \item[(b)] If $P \leqslant Q$ and $P$ is $\mathcal{F}$-centric, then $Q$ is $\mathcal{F}$-centric. \end{itemize} \end{Lem}
\begin{proof} Writing $R=PC_S(P)$, we see immediately that $C_S(R) \leqslant R$. If $\varphi \in$ Hom$_\mathcal{F}(R,S)$ then $R\varphi \leqslant P\varphi C_S(P\varphi)$ and since $P$ is fully $\mathcal{F}$-centralised, necessarily $R\varphi = P\varphi C_S(P\varphi).$ Now $C_S(R\varphi) \leqslant R\varphi$ which proves (a). Part (b) is trivial. \end{proof} We introduce some notation which makes precise the notion of a `generating' set in our context.
\begin{Def} Let $S$ be a finite $p$-group and let $\mathcal{C}$ be a collection of injective maps between subgroups of $S$. Denote by $\langle \mathcal{C} \rangle_S$ the smallest fusion system on $S$ containing all maps which lie in $\mathcal{C}$. If $\mathcal{F}$ is a fusion system on $S$ and $\mathcal{F}=\langle \mathcal{C} \rangle_S$ then the set $\mathcal{C}$ is said to \textit{generate} $\mathcal{F}$. \end{Def} Observe that whenever $\mathcal{C}$ generates $\mathcal{F}$, each morphism in $\mathcal{F}$ can be written as a composite of restrictions of elements of $\mathcal{C}$. \begin{Def} A set of subgroups $\mathcal{X}$ of a finite $p$-group $S$ to be a \textit{conjugation family} for a fusion system $\mathcal{F}$ on $S$ if $\langle\operatorname{Aut}_\mathcal{F}(P) \mid P \in \mathcal{X} \rangle_S=\mathcal{F}$. \end{Def} We may now state Alperin's theorem for saturated fusion systems which provides a very useful conjugation family for any saturated fusion system $\mathcal{F}$. \begin{Thm}\label{alpthm} Let $\mathcal{F}$ be a saturated fusion system on a finite $p$-group $S$. $$\{S\} \cup \{P \mid \mbox{$P$ is fully $\mathcal{F}$-normalised and $\mathcal{F}$-essential} \}$$ is a conjugation family for $\mathcal{F}$. \end{Thm}
\begin{proof} See \cite[Theorem I.3.5]{AKO}. \end{proof}
Proving that a fusion system $\mathcal{F}$ is saturated can often be a difficult task, since there may be many subgroups for which the saturation axioms must be checked. Fortunately, this job is occasionally made easier when an $\mathcal{F}$-conjugation family is known to exist.
\begin{Thm}\label{alpthmconv} Let $\mathcal{F}$ be a fusion system on a finite $p$-group $S$. If $\mathcal{F}$-centric subgroups form a conjugation family then $\mathcal{F}$ is saturated if (a) and (b) in Definition \ref{fus} hold for all such subgroups. \end{Thm}
\begin{proof} See \cite[Theorem 3.8]{Puig}. \end{proof} We think of Theorem \ref{alpthmconv} as a partial converse to Theorem \ref{alpthm} and it is a fundamental tool in our argument to prove Theorem B.
\subsection{$\mathcal{F}$-normalisers and Constrained Fusion Systems} This section introduces two concepts which will be used in the proof of Theorem C. We begin with the analogue for fusion systems of the ordinary normaliser of a subgroup of a finite group. \begin{Def}
Let $\mathcal{F}$ be a fusion system on a finite $p$-group $S$ and let $Q \leqslant S$. The \textit{$\mathcal{F}$-normaliser of $Q$}, $N_\mathcal{F}(Q)$ is the fusion system on $N_S(Q)$ where for each $P,R \leqslant N_S(Q)$, Hom$_{N_\mathcal{F}(Q)}(P,R)$ is the set $$\{\varphi \in \mbox{Hom}_\mathcal{F}(P,R) \mid \mbox{ $\exists$ } \overline{\varphi} \in \mbox{Hom}_\mathcal{F}(PQ,RQ) \mbox{ s.t. } \overline{\varphi}|_P=\varphi \mbox{ and } \overline{\varphi}|_Q \in \operatorname{Aut}(Q)\}.$$ \end{Def}
\begin{Thm}\label{nfkq} Let $\mathcal{F}$ be a saturated fusion system on a finite $p$-group $S$ and let $Q \leqslant S$. If $Q$ is fully $\mathcal{F}$-normalised then $N_\mathcal{F}(Q)$ is saturated. \end{Thm}
\begin{proof} See \cite[Theorem I.5.5]{AKO}. \end{proof} The notion of an $\mathcal{F}$-normaliser naturally gives rise to the notion of a normal subgroup of a fusion system as follows. \begin{Def}\label{normdef} Let $\mathcal{F}$ be a fusion system on a finite $p$-group $S$ and let $Q \leqslant S$. $Q$ is \textit{normal} in $\mathcal{F}$ if $N_\mathcal{F}(Q)=\mathcal{F}$. Write $O_p(\mathcal{F})$ for the maximal normal subgroup of $\mathcal{F}.$ \end{Def} The following lemma provides a characterisation of $O_p(\mathcal{F})$ for a saturated fusion system $\mathcal{F}$ in terms of its fully $\mathcal{F}$-normalised, $\mathcal{F}$-essential subgroups. \begin{Lem}\label{esscont} Let $\mathcal{F}$ be a saturated fusion system on a finite $p$-group $S$ and let $Q \leqslant S$. If $Q$ is normal in $\mathcal{F}$ then $Q \leqslant R$ for each $\mathcal{F}$-centric $\mathcal{F}$-radical subgroup $R$ of $S$. Conversely, if $Q \leqslant R$ for each fully $\mathcal{F}$-normalised, $\mathcal{F}$-essential subgroup $R$ of $S$, then $Q$ is normal in $\mathcal{F}$. \end{Lem}
\begin{proof} See \cite[Theorem 4.61]{CR}. \end{proof}
The notion of a normal subgroup can also be applied to define the analogue for fusion systems of a $p$-constrained finite group. Recall that a finite group $G$ with $O_{p'}(G)=1$ is \textit{$p$-constrained} if there exists a normal subgroup $R \unlhd G$ with $C_G(R) \leqslant R$. \begin{Def} A saturated fusion system $\mathcal{F}$ on a finite $p$-group $S$ is \textit{constrained} if there exists an $\mathcal{F}$-centric subgroup of $S$ which is normal in $\mathcal{F}$. \end{Def}
The final result of this section asserts that every constrained fusion system arises as the fusion system of a $p$-constrained group. \begin{Thm}\label{const} Let $\mathcal{F}$ be a fusion system on a finite $p$-group $S$ and suppose that there exists an $\mathcal{F}$-centric subgroup $R$ of $S$ which is normal in $\mathcal{F}$. There exists a unique finite group $G$ with $S \in \operatorname{Syl}_p(G)$ with the properties that $$\mathcal{F}=\mathcal{F}_S(G), \mbox{ } R \unlhd G, \mbox{ } \operatorname{Out}_G(R) \cong G/R, \mbox{ and } O_{p'}(G)=1.$$ \end{Thm}
\begin{proof} See, for example \cite[Theorem I.5.10]{AKO}. \end{proof}
\section{Trees of Fusion Systems}\label{treesfus} In this section, we will carefully define what we mean by a tree of fusion systems and the completion of such an object. We then find a natural condition which ensures that the completion of a tree of fusion systems exists. We warn the reader that from now on the symbol $`\mathcal{F}$' will frequently be used to denote a functor with values in the category of fusion systems, rather than just a single fusion system. \subsection{Trees of Groups} We begin by introducing some notation. If $\mathcal{T}$ is a simple, undirected graph, write $V(\mathcal{T})$ and $E(\mathcal{T})$ for the sets of vertices and edges of $\mathcal{T}$ respectively. Each edge $e \in E(\mathcal{T})$ is regarded as an unordered pair of vertices, $(v,w)$ say, and $v$ and $w$ are said to be \textit{incident} on $e$. $\mathcal{T}$ gives rise to a category (also called $\mathcal{T}$) where Ob($\mathcal{T}$) is the disjoint union of the sets $V(\mathcal{T})$ and $E(\mathcal{T})$ and where for each edge $(v,w) \in E(\mathcal{T})$ there exists a pair of morphisms $$ \begin{CD} e @>>> v\\ @VVV \\ w \end{CD}$$ in $\mathcal{T}$. We denote the unique morphism in Hom$_\mathcal{T}(e,v)$ by $f_{ev}$ and write $\mathfrak{Grp}$ for the category of groups and group homomorphisms. \begin{Def} A \textit{tree of groups} is a pair $(\mathcal{T},\mathcal{G})$ where $\mathcal{T}$ is a tree and $\mathcal{G}$ is a functor from $\mathcal{T}$ (regarded as a category) to $\mathfrak{Grp}$ which sends $f_{ev}$ to group monomorphism from $\mathcal{G}(e)$ to $\mathcal{G}(v)$. The \textit{completion} $\mathcal{G}_\mathcal{T}$ of $(\mathcal{T},\mathcal{G})$ is the group $ \underrightarrow{\mbox{colim}}_{
\substack{
\mathcal{T}}} \mathcal{G}.$ \end{Def}
\begin{Lem} Each tree of groups $(\mathcal{T},\mathcal{G})$ has a completion which is unique up to group isomorphism. \end{Lem}
Let $(\mathcal{T},\mathcal{G})$ be a tree of groups with completion $G:=\mathcal{G}_\mathcal{T}$. Define $G/\mathcal{G}(-)$ to be the functor from $\mathcal{T}$ to $\mathfrak{Set}$ which sends each $v$ and $e \in$ Ob$(\mathcal{T})$ respectively to the sets of left cosets $G/\mathcal{G}(v)$ and $G/\mathcal{G}(e)$ and which sends $f_{ev} \in$ Hom$_\mathcal{T}(e,v)$ to the map from $G/\mathcal{G}(e)$ to $G/\mathcal{G}(v)$ given by sending left cosets $g\mathcal{G}(e)$ to $g\mathcal{G}(v)$. Define the \textit{orbit graph} $\tilde{\mathcal{T}}$ to be the space $$ \underrightarrow{\mbox{hocolim}}_{
\substack{
\mathcal{T}}} G/\mathcal{G}(-).$$ Equivalently, we may think of $\tilde{\mathcal{T}}$ as being a graph whose vertices and edges are labelled by the sets $\{g\mathcal{G}(v) \mid g \in G, v \in V(\mathcal{T})\}$ and $\{i\mathcal{G}(e) \mid i \in G, e \in E(\mathcal{T})\}$ respectively and where two vertices $g\mathcal{G}(v)$ and $h\mathcal{G}(w)$ are connected via an edge $i\mathcal{G}(e)$ if and only if $i\mathcal{G}(v)=g\mathcal{G}(v)$ and $i\mathcal{G}(w)=h\mathcal{G}(w)$. This is obviously a graph on which $G$ acts by left multiplication. Denote by $\tilde{\mathcal{T}}/G$ the graph with vertex set given by the set of orbits of $V(\tilde{\mathcal{T}})$ under the action of $G$ on $V(\tilde{\mathcal{T}})$ and likewise for the edges. We have the following theorem:
\begin{Thm}\label{fundbass} Let $(\mathcal{T},\mathcal{G})$ be a tree of groups with completion $G:=\mathcal{G}_\mathcal{T}$. Then $\tilde{\mathcal{T}}$ is a tree and $\tilde{\mathcal{T}}/G \simeq \mathcal{T}.$ \end{Thm}
\begin{proof} See \cite[I.4.5]{Serre} \end{proof}
\subsection{Trees of Fusion Systems} Suppose that $\mathcal{F}$ and $\mathcal{E}$ are fusion systems on finite $p$-groups $S$ and $T$ respectively. A morphism $\alpha \in$ Hom$(S,T)$ from $\mathcal{F}$ to $\mathcal{E}$ is \textit{injective} if it induces an injective map $$\begin{CD} \mbox{Hom}_\mathcal{F}(P,S) @>>> \mbox{Hom}_\mathcal{E}(P\alpha,T)\\ \end{CD}$$ for each $P \leqslant S$.
\begin{Def} A \textit{tree of $p$-fusion systems} is a triple $(\mathcal{T},\mathcal{F},\mathcal{S})$ where $(\mathcal{T},\mathcal{S})$ is a tree of finite $p$-groups and $\mathcal{F}$ is a functor from $\mathcal{T}$ to $\mathfrak{Fus}$ such that the following hold: \begin{itemize} \item[(a)] $\mathcal{F}(v)$ is a fusion system on $\mathcal{S}(v)$ and $\mathcal{F}(e)$ is a fusion system on $\mathcal{S}(e)$, and \item[(b)] $\mathcal{F}$ sends $f_{ev} \in$ Hom$_\mathcal{T}(e,v)$ to an injective morphism from $\mathcal{F}(e)$ to $\mathcal{F}(v).$ \end{itemize} \end{Def}
\begin{Def} Let $(\mathcal{T},\mathcal{F},\mathcal{S})$ be a tree of $p$-fusion systems. The \textit{completion} of $(\mathcal{T},\mathcal{F},\mathcal{S})$ is a colimit for $\mathcal{F}$. \end{Def}
Of course, we need conditions on $(\mathcal{T},\mathcal{F},\mathcal{S})$ which imply that a colimit for $\mathcal{F}$ exists, since this is no longer guaranteed as it was in the category of groups. Indeed, any colimit for $\mathcal{F}$ must be a fusion system on the completion $S_\mathcal{T}$ of $(\mathcal{T},\mathcal{S})$ and this may not be a $p$-group\footnote{Consider, for example the amalgam $C_2 * C_2 \cong D_{\infty}$ when $p=2$.}. The following is a very simple (and natural) condition to impose: \newline \newline \textbf{Hypothesis} \textit{$(H)$: There exists a vertex $v_* \in V(\mathcal{T})$ with the property that whenever $v \in V(\mathcal{T})$, $\mathcal{S}(e) \cong \mathcal{S}(v)$ where $e$ is the edge incident to $v$ in the unique minimal path from $v$ to $v_*$.} \newline \newline We will say that a tree of $p$-fusion systems $(\mathcal{T},\mathcal{F},\mathcal{S})$ satisfies $(H)$ if $(\mathcal{T},\mathcal{S})$ satisfies Hypothesis $(H)$. Let $(\mathcal{T},\mathcal{F},\mathcal{S})$ be a tree of $p$-fusion systems which satisfies $(H)$ and write $S:=\mathcal{S}(v_*)$. It is clear that $\mathcal{S}_\mathcal{T}$ is a finite group isomorphic to $\mathcal{S}(v_*)$, so that we may view $\mathcal{S}(e)$ and $\mathcal{S}(v)$ as subgroups of $\mathcal{S}(v_*)$ by identifying them with their images in the completion. Also, $\mathcal{S}(v) \cap \mathcal{S}(w)=\mathcal{S}(e)$ in $\mathcal{S}(v_*)$ whenever $v$ and $w$ are vertices of $\mathcal{T}$ incident on $e$. Furthermore, when $v$ is a vertex incident to an edge $e$, this identification allows us to embed each fusion system $\mathcal{F}(e)$ as a subsystem of $\mathcal{F}(v)$ by identifying each morphism $\alpha \in$ Hom$_{\mathcal{F}(e)}(P,\mathcal{S}(e))$ with its image under the functor from $\mathcal{F}(e)$ to $\mathcal{F}(v)$ induced by $f_{ev} \in$ Hom$_\mathcal{T}(e,v).$
\begin{Lem}\label{compft} Let $(\mathcal{T},\mathcal{F},\mathcal{S})$ be a tree of $p$-fusion systems which satisfies $(H)$, set $v_0:=v_*$ and write $S:=\mathcal{S}(v_0)=\mathcal{S}_\mathcal{T}$. Define $$\mathcal{F}_\mathcal{T}:=\langle \operatorname{Hom}_{\mathcal{F}(v)}(P,\mathcal{S}(v)) \mid P \leqslant \mathcal{S}(v),v \in V(\mathcal{T}) \rangle_S,$$ the fusion system generated by the $\mathcal{F}(v)$ for each $v \in V(\mathcal{T})$. $\mathcal{F}_\mathcal{T}$ is a colimit for $\mathcal{F}$ and each $\alpha \in \operatorname{Hom}_{\mathcal{F}_\mathcal{T}}(P,S)$ may be written as a composite
$$\begin{CD} P=P_0 @>\alpha_0>> P_1 @>\alpha_1>> P_2 @>\alpha_2>> \cdots @>\alpha_{n-1}>> P_n=P\alpha\\ \end{CD}$$
where for $0 \leqslant i \leqslant n-1$, $P_i \leqslant \mathcal{S}(v_{i-1}) \cap \mathcal{S}(v_i)$, $\alpha_i \in \operatorname{Mor}(\mathcal{F}(v_i))$ for some $v_i \in V(\mathcal{T})$ and $(v_i,v_{i+1})$ is an edge in $\mathcal{T}$. \end{Lem}
\begin{proof} The fact that $\mathcal{F}_\mathcal{T}$ is a colimit for $\mathcal{F}$ follows immediately from the fact that it is unique (up to an isomorphism of categories) amongst all fusion systems on $S$ which contain $\mathcal{F}(v)$ for each $v \in V(\mathcal{T})$\footnote{More information concerning this characterisation of $\mathcal{F}_\mathcal{T}$ is provided by the remarks which follow Definition \ref{fus}}. To see the second statement, clearly any such composite of morphisms lies in $\mathcal{F}_\mathcal{T}$. Conversely let $$\begin{CD} P=P_0 @>\alpha_0>> P_1 @>\alpha_1>> P_2 @>\alpha_2>> \cdots @>\alpha_{n-1}>> P_n=P\alpha\\ \end{CD}$$ be a representation of $\alpha \in$ Hom$_{\mathcal{F}_\mathcal{T}}(P,S)$ where $\alpha_i \in$ Mor$(\mathcal{F}(v_i))$ for $0 \leqslant i \leqslant n-1$. If $(v_{i-1},v_i)$ is not an edge then let $\eta$ be a minimal path in $\mathcal{T}$ from $v_{i-1}$ to $v_i$. Since $(\mathcal{T},S)$ satisfies $(H)$, $P_i$ is contained in $\mathcal{S}(w)$ for each vertex $w \in \eta$ so that by inserting identity morphisms, the above sequence of morphisms can be refined so that $(v_{i-1},v_i)$ is an edge for each $i$. This completes the proof of the lemma. \end{proof} One observes that by Lemma \ref{compft}, the competion $\mathcal{F}_\mathcal{T}$ of a tree of $p$-fusion systems $(\mathcal{T},\mathcal{F},\mathcal{S})$ which satisfies $(H)$ is independent of where $\mathcal{F}$ sends the edges $e$ of $\mathcal{T}$. We will make heavy use of this fact later in the proof of Theorem B.
\subsection{The $P$-orbit Graph} We now turn to the definition of a certain graph constructed from a tree of $p$-fusion systems (now simply referred to as a tree of fusion systems) and an arbitrary finite $p$-group $P$. Our discussion culminates in the proof of an important result, Proposition \ref{phipft}, which will allow us to describe morphisms in the completion combinatorially.
We begin by introducing some notation. Let $\mathcal{K} $ be a fusion system on a finite $p$-group $S$ and let $P$ be any other finite $p$-group. We define an equivalence relation $\sim$ on the set of homomorphisms $\operatorname{Hom}(P,S)$ as follows. For $\alpha, \beta \in \operatorname{Hom}(P,S)$ define $\alpha \sim \beta$ if and only if there is some $\gamma \in \operatorname{Iso}_\mathcal{K} (P\alpha,P\beta)$ such that $\alpha \circ \gamma = \beta$. The fact that $\sim$ is an equivalence relation follows from the axioms for a fusion system. Write $$\mbox{Rep}(P,\mathcal{K} ):= \mbox{Hom}(P,S)/\sim$$ for the set of all equivalence classes, and for each $\alpha \in $ Hom$(P,S)$ let $[\alpha]_\mathcal{K} $ denote the class of $\alpha$ in Rep$(P,\mathcal{K} )$. If $\widehat{\mathcal{K} }$ is a fusion system containing $\mathcal{K} $, set Rep$_{\widehat{\mathcal{K} }}(P,\mathcal{K} ):=$ Hom$_{\widehat{\mathcal{K} }}(P,S)/\sim$.
Let $(\mathcal{T},\mathcal{F},\mathcal{S})$ be a tree of fusion systems, and $P$ be a finite $p$-group. Define Rep$(P,\mathcal{F}(-))$ to be the functor from $\mathcal{T}$ to $\mathfrak{Set}$ which sends vertices $v$ and edges $e$ of $\mathcal{T}$ respectively to Rep$(P,\mathcal{F}(v))$ and Rep$(P,\mathcal{F}(e))$ and which sends $f_{ev} \in$ Hom$_\mathcal{T}(e,v)$ to the map from Rep$(P,\mathcal{F}(e))$ to Rep$(P,\mathcal{F}(v))$ given by $$[\alpha]_{\mathcal{F}(e)} \longmapsto [\alpha \circ \iota_{\mathcal{S}(e)}^{\mathcal{S}(v)}]_{\mathcal{F}(v)}.$$ Observe that this mapping is independent of the choice of $\alpha$. Using this definition we introduce the following space. \begin{Def} Let $P$ be a finite $p$-group and $(\mathcal{T},\mathcal{F},\mathcal{S})$ be a tree of fusion systems. The \textit{$P$-orbit graph}, Rep$(P,\mathcal{F})$ is the homotopy colimit $$ \underrightarrow{\mbox{hocolim}}_{
\substack{
\mathcal{T}}} \operatorname{Rep}(P,\mathcal{F}(-)).$$ \end{Def} Since there are no $n$-simplices in $ \underrightarrow{\mbox{hocolim}}_{
\substack{
\mathcal{T}}} \operatorname{Rep}(P,\mathcal{F}(-))$ for $n \geqslant 2$, Rep$(P,\mathcal{F})$ can be described as the geometric realisation of a graph as follows.
\begin{Lem}\label{repfgraph} Let $P$ be an arbitrary $p$-group and $(\mathcal{T},\mathcal{F},\mathcal{S})$ be a tree of fusion systems. Then $R:=$ Rep$(P,\mathcal{F})$ may be regarded as a graph with $$V(R) = \bigcup_{v \in V(\mathcal{T})} \operatorname{Rep}(P,\mathcal{F}(v)) \mbox{ and } E(R) = \bigcup_{e \in E(\mathcal{T})} \operatorname{Rep}(P,\mathcal{F}(e)),$$ where $[\alpha]_{\mathcal{F}(v)}$ and $[\beta]_{\mathcal{F}(w)}$ are connected via an edge $[\gamma]_{\mathcal{F}(e)}$ if and only if $v$ and $w$ are both incident on $e$ in $\mathcal{T}$ and the identities $$[\gamma \circ \iota_{\mathcal{S}(e)}^{\mathcal{S}(v)}]_{\mathcal{F}(v)}=[\alpha]_{\mathcal{F}(v)} \mbox{ and } [\gamma \circ \iota_{\mathcal{S}(e)}^{\mathcal{S}(v)}]_{\mathcal{F}(w)}=[\beta]_{\mathcal{F}(w)}$$ hold. \end{Lem}
\begin{proof} This follows immediately from the definition of the homotopy colimit. \end{proof}
If $(\mathcal{T},\mathcal{F},\mathcal{S})$ is a tree of fusion systems which satisfies $(H)$, then by Lemma \ref{compft}, $(\mathcal{T},\mathcal{F},\mathcal{S})$ has a completion which we denote by $\mathcal{F}_\mathcal{T}.$ Let Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F}(-))$ be the subfunctor of Rep$(P,\mathcal{F}(-))$ which sends vertices $v$ and edges $e$ of $\mathcal{T}$ respectively to Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F}(v))$ and Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F}(e))$ and which sends $f_{ev}$ to the map from Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F}(e))$ to Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F}(v))$ given by $$[\alpha]_{\mathcal{F}(e)} \longmapsto [\alpha \circ \iota_{\mathcal{S}(e)}^{\mathcal{S}(v)}]_{\mathcal{F}(v)}.$$ (Note that this map is well-defined since $\mathcal{F}_\mathcal{T}$ is closed under composition with inclusion morphisms). Let $$\mbox{Rep}_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F}):= \underrightarrow{\mbox{hocolim}}_{
\substack{
\mathcal{T}}} \operatorname{Rep}_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F}(-)).$$ We observe that the obvious analogue of Lemma \ref{repfgraph} holds for Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$, and that (for this reason) Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$ may be embedded as a subgraph of Rep$(P,\mathcal{F})$ in the obvious way. We end this section with an important result which provides us with a precise description of this embedding.
\begin{Prop}\label{phipft} Let $(\mathcal{T},\mathcal{F},\mathcal{S})$ be a tree of fusion systems which satisfies $(H)$ and let $\mathcal{F}_\mathcal{T}$ be its completion. Write $S:=\mathcal{S}(v_*)$ and fix $P \leqslant S$. The following hold: \begin{itemize} \item[(a)] The connected component of the vertex $[\iota_P^S]_{\mathcal{F}(v_*)}$ in $\operatorname{Rep}(P,\mathcal{F})$ is isomorphic to Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$. \item[(b)] The natural map $$\begin{CD} \Phi_P: \pi_0(\operatorname{Rep}(P,\mathcal{F})) @>>> \operatorname{Rep}(P,\mathcal{F}_\mathcal{T})\\ \end{CD}$$ which sends the connected component of a vertex $[\alpha]_{\mathcal{F}(v)}$ in $\operatorname{Rep}(P,\mathcal{F})$ to $[\alpha \circ \iota_{\mathcal{S}(v)}^S]_{\mathcal{F}_\mathcal{T}}$ is a bijection. \end{itemize} \end{Prop} \begin{proof} Since $\mathcal{T}$ has finitely many vertices and $\mathcal{S}(v)$ is finite for each $v \in V(\mathcal{T})$, the graph Rep$(P,\mathcal{F})$ contains finitely many vertices and edges. We may identify $\mathcal{F}(v)$ and $\mathcal{F}(e)$ with their images in $\mathcal{F}_\mathcal{T}$ by Lemma \ref{compft}. Set $v_0:=v_*$ and $\alpha_0:= \iota_P^{\mathcal{S}(v_0)}$ and let $$[\alpha_0]_{\mathcal{F}(v_0)}, [\alpha_1]_{\mathcal{F}(v_1)}, \ldots, [\alpha_n]_{\mathcal{F}(v_n)}$$ be a path in Rep$(P,\mathcal{F})$ and assume that $[\beta_i]_{\mathcal{F}(e_i)}$ is an edge from $[\alpha_{i-1}]_{\mathcal{F}(v_0)}$ to $[\alpha_i]_{\mathcal{F}(v_1)}$ for each $1 \leqslant i \leqslant n.$
We need to show that each vertex $[\alpha_i]_{\mathcal{F}(v_i)}$ lies in Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$. Clearly $[\alpha_0]_{\mathcal{F}(v_0)} \in$ Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$. Assume that $n \geqslant 1$, and that $[\alpha_i]_{\mathcal{F}(v_i)} \in$ Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$ for some $i < n$. If $[\beta_{i+1}]_{\mathcal{F}(e_{i+1})}$ is an edge from
$[\alpha_i]_{\mathcal{F}(v_i)}$ to $[\alpha_{i+1}]_{\mathcal{F}(v_{i+1})}$ then there exist maps $$\gamma \in \mbox{Hom}_{\mathcal{F}(v_i)}(P\alpha_i,P\beta_{i+1}) \mbox{ and } \delta \in \mbox{Hom}_{\mathcal{F}(v_{i+1})}(P\beta_{i+1},P\alpha_{i+1})$$ such that $\alpha_i \circ \gamma = \beta_{i+1}$ and $\beta_{i+1} \circ \delta=\alpha_{i+1}$. Hence $\gamma \circ \delta \in$ Hom$_{\mathcal{F}_\mathcal{T}}(P\alpha_i,P\alpha_{i+1}) $ and $[\alpha_{i+1}]_{\mathcal{F}(v_{i+1})} \in$ Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$ and by induction, $[\alpha_i]_{\mathcal{F}(v_i)} \in$ Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$ for all $0 \leqslant i \leqslant n.$
Conversely suppose that $\alpha \in$ Hom$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{S}(v))$ and that $[\alpha]_{\mathcal{F}(v)}$ is a vertex in Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$. By the definition of $\mathcal{F}_\mathcal{T}$, there exists a path $v_0,v_1, \ldots, v_n=v$ in $\mathcal{T}$ and for $1 \leqslant i \leqslant n$, maps $\alpha_i \in$ Hom$_{\mathcal{F}(v_i)}(P\alpha_{i-1}, P\alpha_i)$ with $\alpha_0=\iota_P^S$ such that $\alpha=\alpha_0 \circ \cdots \circ \alpha_n.$ This implies that there exists a path in Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$, $$[\alpha_0]_{\mathcal{F}(v_0)}, [\alpha_0 \circ \alpha_1]_{\mathcal{F}(v_1)} \ldots, [\alpha_0 \circ \alpha_1 \circ \cdots \circ \alpha_{n-1}]_{\mathcal{F}(v_{n-1})},[\alpha]_{\mathcal{F}(v_n)},$$ and completes the proof of (a).
Next, we prove (b). It suffices to show that two vertices $[\alpha]_{\mathcal{F}(v)}$ and $[\beta]_{\mathcal{F}(w)}$ are connected in Rep$(P,\mathcal{F})$ if and only if $[\alpha \circ \iota_{\mathcal{S}(v)}^S]_{\mathcal{F}_\mathcal{T}}$=$[\beta \circ \iota_{\mathcal{S}(w)}^S]_{\mathcal{F}_\mathcal{T}}$ in Rep$(P,\mathcal{F}_\mathcal{T}).$ It is enough to prove this when $[\alpha]_{\mathcal{F}(v)}$ and $[\beta]_{\mathcal{F}(w)}$ are connected via a single edge $[\gamma]_{\mathcal{F}(e)}$.
We prove the `only if' direction first. Thus we suppose that $[\alpha]_{\mathcal{F}(v)}$ and $[\beta]_{\mathcal{F}(w)}$ are connected via an edge $[\gamma]_{\mathcal{F}(e)}$ in Rep$(P,\mathcal{F})$ so that there exist morphisms $\gamma_1 \in$ Hom$_{\mathcal{F}(v)}(P\alpha,P\gamma)$ and $\gamma_2 \in$ Hom$_{\mathcal{F}(w)}(P\beta,P\gamma)$ with $$P\alpha\gamma_1=P\gamma=P\beta\gamma_2.$$ Then $\gamma_1\gamma_2^{-1} \in$ Hom$_{\mathcal{F}_\mathcal{T}}(P\alpha,P\beta)$ and $[\beta \circ \iota_{\mathcal{S}(w)}^S]=[\alpha\gamma_1\gamma_2^{-1} \circ \iota_{\mathcal{S}(w)}^S]=[\alpha \circ \iota_{\mathcal{S}(v)}^S] \in$ Rep$(P,\mathcal{F}_\mathcal{T})$. Conversely, suppose that $[\alpha \circ \iota_{\mathcal{S}(v)}^S]_{\mathcal{F}_\mathcal{T}}=[\beta \circ \iota_{\mathcal{S}(w)}^S]_{\mathcal{F}_\mathcal{T}}=[\delta]_{\mathcal{F}_\mathcal{T}}$, for some $\delta \in$ Hom$_{\mathcal{F}_\mathcal{T}}(P,S)$. Then $[\delta^{-1}\alpha]_{\mathcal{F}(v)}$, $[\delta^{-1}\beta]_{\mathcal{F}(v)}$ are vertices in Rep$_{\mathcal{F}_\mathcal{T}}(P\delta,\mathcal{F})$ and by part (a), this graph is connected so that there must exist a path joining $[\delta^{-1}\alpha]_{\mathcal{F}(v)}$ to $[\delta^{-1}\beta]_{\mathcal{F}(v)}$ in Rep$_{\mathcal{F}_\mathcal{T}}(P\delta,\mathcal{F})$. This is easily seen to be equivalent to the existence of a path joining $[\alpha]_{\mathcal{F}(v)}$ to $[\beta]_{\mathcal{F}(w)}$ in Rep$(P,\mathcal{F})$, completing the proof of the lemma. \end{proof}
\section{Trees of Group Fusion Systems}\label{orbfus} In this section we will give a precise description of the relationship between trees of groups $(\mathcal{T},\mathcal{G})$ and trees of fusion systems $(\mathcal{T},\mathcal{F},\mathcal{S})$ by considering what happens when the latter is induced by the former. It will turn out that both the completion and $P$-orbit graph of $(\mathcal{T},\mathcal{F},\mathcal{S})$ have group-theoretic descriptions in this case, which will allow us to give an entirely group theoretic interpretation of Theorem B, in Section \ref{compfus}.
\subsection{The Completion} We start by giving a precise explanation of the word `induced' above. The following lemma is a trivial consequence of Sylow's Theorem.
\begin{Lem}\label{induce} Let $(\mathcal{T},\mathcal{G})$ be a tree of finite groups. For any choice of Sylow $p$-subgroups $\mathcal{S}(v) \in \operatorname{Syl}_p(\mathcal{G}(v))$ and $\mathcal{S}(e) \in \operatorname{Syl}_p(\mathcal{G}(e))$, there exists a tree of finite $p$-groups $(\mathcal{T},\mathcal{S})$ and a tree of fusion systems $(\mathcal{T},\mathcal{F}_{\mathcal{S}(-)}(\mathcal{G}(-)),\mathcal{S}(-))$ \mbox{induced by} $(\mathcal{T},\mathcal{G})$. \end{Lem}
\begin{proof}
Fix a choice of Sylow $p$-subgroups, $\mathcal{S}(v) \in$ Syl$_p(\mathcal{G}(v))$ and $\mathcal{S}(e) \in$ Syl$_p(\mathcal{G}(e))$ for each vertex $v$ and edge $e$ of $\mathcal{T}$. If $v$ is incident to $e$ then by Sylow's Theorem there exists an element $g_{ev} \in \mathcal{G}(v)$ such that $(\mathcal{S}(e)\mathcal{G}(f_{ev}))^{g_{ev}} \leqslant \mathcal{S}(v).$ Let $\mathcal{S}$ be the functor from $\mathcal{T}$ to $\mathfrak{Grp}$ which sends $e$ and $v$ respectively to $\mathcal{S}(e)$ and $\mathcal{S}(v)$ and $f_{ev} \in$ Hom$_\mathcal{T}(e,v)$ to $\mathcal{G}(f_{ev}) \circ c_{g_{ev}} \in$ Hom$(\mathcal{S}(e),\mathcal{S}(v))$. Then $f_{ev}$ determines a tree of finite $p$-groups $(\mathcal{T},\mathcal{S})$. Now let $\mathcal{F}_{\mathcal{S}(-)}(\mathcal{G}(-))$ be the functor from $\mathcal{T}$ to $\mathfrak{Fus}$ which sends $e$ and $v$ respectively to $\mathcal{F}_{\mathcal{S}(e)}(\mathcal{G}(e))$ and $\mathcal{F}_{\mathcal{S}(v)}(\mathcal{G}(v))$ and $f_{ev}$ to the homomorphism $\Phi_{ev}:=\mathcal{G}(f_{ev}) \circ c_{g_{ev}}|_{\mathcal{S}(e)} \in$ Hom$(\mathcal{S}(e),\mathcal{S}(v)).$ Clearly $\Phi_{ev}$ is a morphism of fusion systems and hence determines a tree of fusion systems $(\mathcal{T},\mathcal{F}_{\mathcal{S}(-)}(\mathcal{G}(-)),\mathcal{S}(-))$, as required. \end{proof} The proof of Lemma \ref{induce} shows that there may be many trees of fusion systems $(\mathcal{T},\mathcal{F}_{\mathcal{S}(-)}(\mathcal{G}(-)),\mathcal{S}(-))$ induced by a tree of groups $(\mathcal{T},\mathcal{G})$, since a choice for the functor $\mathcal{S}(-)$ is made when Sylow's Theorem is applied. We need to show that this choice does not interfere with the isomorphism type of the completion $\mathcal{F}_\mathcal{T}$. To do this, we first isolate precisely how conjugation takes place in $\mathcal{G}_\mathcal{T}$ by proving the following lemma of Robinson (\cite[Lemma 1]{Rob}).
\begin{Lem}\label{roblem} Let $\mathcal{T}$ be a tree consisting of two vertices $v$ and $w$ with a single edge $e$ connecting them. Let $(\mathcal{T},\mathcal{G})$ be a tree of groups with completion $\mathcal{G}_\mathcal{T}$ and write $A:=\mathcal{G}(v)$, $B:=\mathcal{G}(w)$ and $C:=\mathcal{G}(e).$ The following hold: \begin{itemize} \item[(a)] Any product of elements whose successive terms lie alternately in the sets $A \backslash C$ and $B \backslash C$, lies outside of $C$, and outside at least one of $A$ and $B$. \item[(b)] Each element $g \in G \backslash A$ may be written as a product $g=a_0\omega b_{\infty}$ with $a_0 \in A$ and $b_{\infty} \in B$ so that either $\omega=1$ or $$\omega=\prod_{i=1}^s b_ia_i,$$ where $a_i \in A \backslash C$ and $b_i \in B \backslash C$ for $1 \leqslant i \leqslant s$. \end{itemize} Consequently, if $X \leqslant A$, $g \in G \backslash A$ and $X^g \leqslant A$ or $X^g \leqslant B$, then writing $g$ as a product $g=a_0b_1a_1 \ldots b_sa_sb_{\infty}$ as in (b), we have $$\langle X_0,X_i,Y_i \mid 1 \leqslant i \leqslant s \rangle \leqslant C$$ where $X_0=X^{a_0},Y_1=X_0^{b_1},X_1=Y_1^{a_1},$ and so on. \end{Lem}
\begin{proof} To see (a), let $w=g_0\ldots g_n$ be a product of elements whose successive terms lie alternately in the sets $A \backslash C$ and $B \backslash C$. Then $w$ is a reduced word in $\mathcal{G}_\mathcal{T}$ so if $w$ lies in $C$, it is no longer reduced, (being representable by an element of $C$). Observe that $g_0$ and $g_n$ dictate where $w$ lies. To see this, note that if $g_0,g_n \in A$ then $w \notin B \backslash C$, since otherwise $w=b \in B \backslash C$ implies that $$1=wb^{-1}=g_0,\ldots g_nb^{-1} \notin C,$$ a contradiction. Similarly if $g_0,g_n \in B$ then $w \notin A \backslash C$ and in the remaining cases (where $g_0$ and $g_n$ lie in different sets), $w \notin (A \cup B) \backslash C$. This proves (a). To see (b), note that certainly any element $g \in G \backslash A$ may be written in the stated way, since (by (a)) such a representation allows for all possibilities for the set in which the element $g$ lies. Finally, we prove the last assertion of the lemma. If $b_\infty \in C$ and equals $c$ say, then $X^{gc^{-1}}=(X^g)^{c^{-1}} \in A \cup B$ if and only if $X^g \in A \cup B.$ This proves that we may assume without loss of generality that either $b_\infty = 1$ (if $X^g \leqslant B$) or $b_\infty \in B \backslash C$ (if $X^g \leqslant A$.) In either case, suppose that there exists $u \in X^{a_0} \backslash C$. Then $$b_\infty^{-1}a_s^{-1} \ldots, b_1^{-1}ub_1\ldots a_sb_{\infty}$$ lies in $A$ or $B$ by assumption which contradicts (a). We obtain a similar contradiction if $X_0^{b_1}$ is not contained in $C$. Inductively, we arrive at the stated result. \end{proof} We can now apply Lemma \ref{roblem} to prove Theorem A, relating the two ways in which one can construct a fusion system from a tree of groups.
\begin{Thm}\label{treesgroupthm} Let $(\mathcal{T},\mathcal{G})$ be a tree of finite groups and write $\mathcal{G}_\mathcal{T}$ for the completion of $(\mathcal{T},\mathcal{G})$. Let $(\mathcal{T},\mathcal{F},\mathcal{S})$ be a tree of fusion systems induced by $(\mathcal{T},\mathcal{G})$ which satisfies $(H)$ so that there exists a completion $\mathcal{F}_\mathcal{T}$ for $(\mathcal{T},\mathcal{F},\mathcal{S})$. The following hold: \begin{itemize} \item[(a)] $\mathcal{S}(v_*)$ is a Sylow $p$-subgroup of $\mathcal{G}_\mathcal{T}$. \item[(b)] $\mathcal{F}_{\mathcal{S}(v_*)}(\mathcal{G}_\mathcal{T})= \mathcal{F}_\mathcal{T}$. \end{itemize} In particular, $\mathcal{F}_\mathcal{T}$ is independent of the choice of tree of fusion systems $(\mathcal{T},\mathcal{F},\mathcal{S})$ induced by $(\mathcal{T},\mathcal{G})$. \end{Thm}
\begin{proof} We first prove that $S:=S(v_*)$ is a Sylow $p$-subgroup of $G_\mathcal{T}$. Let $P$ be a finite $p$-subgroup of $\mathcal{G}_\mathcal{T}$ and consider the image $\mathcal{I} $ of $\tilde{\mathcal{T}}^P$ in $\mathcal{T}$ under the composite $$\begin{CD} \tilde{\mathcal{T}} @>>> \tilde{\mathcal{T}}/{\mathcal{G}_\mathcal{T}} @>\simeq>> \mathcal{T}.\\ \end{CD}$$
Since $\tilde{\mathcal{T}}$ is a tree, $\tilde{\mathcal{T}}^P$ is also a tree, so that $\mathcal{I} $ must be connected. Let $v$ be a vertex in $\mathcal{I} $ which is of minimal distance from $v_*$ and assume that $v \neq v_*$. This implies that there is some $g \in \mathcal{G}_\mathcal{T}$ such that $g\mathcal{G}(v) \in (\mathcal{G}_\mathcal{T}/\mathcal{G}(v))^P$ or equivalently such that $g^{-1}Pg \leqslant \mathcal{G}(v)$. Let $e$ be the edge $(v,w)$ incident to $v$ in the unique minimal path from $v$ to $v_*$. Since $(\mathcal{T},\mathcal{F},\mathcal{S})$ satisfies $(H)$, $p \nmid |\mathcal{G}(v):\mathcal{G}(e)|$, so by Sylow's Theorem there is $g' \in \mathcal{G}_\mathcal{T}$ such that $g'^{-1}Pg' \leqslant \mathcal{G}(e)$. Since $\mathcal{G}(e) \leqslant \mathcal{G}(w)$, we also have $g'\mathcal{G}(w) \in (\mathcal{G}_\mathcal{T}/\mathcal{G}(w))^P$, so that $w$ is a vertex closer to $v_*$ than $v$ in $\mathcal{I} $, a contradiction. Hence $v=v_*$, $g^{-1}Pg \leqslant \mathcal{G}(v_*)$ and by Sylow's Theorem there is some $g'' \in \mathcal{G}_\mathcal{T}$ with $g''^{-1}Pg'' \leqslant S$, as needed.
Next we prove that (b) holds. Observe that $\mathcal{F}_\mathcal{T} \subseteq \mathcal{F}_S(\mathcal{G}_\mathcal{T})$ since (by definition) each morphism in $\mathcal{F}_\mathcal{T}$ is a composite of restrictions of morphisms in $\mathcal{F}(v)$, each of which clearly lies in $\mathcal{F}_S(\mathcal{G}_\mathcal{T})$. Hence it remains to prove that $\mathcal{F}_S(\mathcal{G}_\mathcal{T}) \subseteq \mathcal{F}_\mathcal{T}$. We proceed by induction on the number of vertices $n:=|V(\mathcal{T})|$ of $\mathcal{T}$, the result being clear in the case where $n=1$. Suppose that $n > 1$ and fix an extremal vertex, $v$ of $\mathcal{T}$ not equal to $v_*$. Let $\mathcal{T}'$ be the tree obtained from $\mathcal{T}$ by removing $v$ and the unique edge $e$ to which $v$ is incident. Then $(\mathcal{T}',\mathcal{G})$ is a graph of groups and $(\mathcal{T}',\mathcal{F},\mathcal{S})$ is a tree of fusion systems induced by $(\mathcal{T}',\mathcal{G})$ which satisfies $(H)$. Furthermore, $S \in$ Syl$_p(\mathcal{G}_{\mathcal{T}'})$ and by induction, the completion $\mathcal{F}_{\mathcal{T}'}$ of $(\mathcal{T}',\mathcal{F},\mathcal{S})$ is the fusion system $\mathcal{F}_S(\mathcal{G}_{\mathcal{T}'})$. Since $$\mathcal{G}_{\mathcal{T}}=\mathcal{G}_{\mathcal{T}'} *_{\mathcal{G}(e)} \mathcal{G}(v) \mbox{ and } \mathcal{F}_\mathcal{T}=\langle \mathcal{F}_{\mathcal{T}'}, \mathcal{F}_{\mathcal{S}(v)}(\mathcal{G}(v)) \rangle,$$ it will be enough to show that $$\mathcal{F}_S(\mathcal{G}_{\mathcal{T}'} *_{\mathcal{G}(e)} \mathcal{G}(v)) \subseteq \langle \mathcal{F}_{\mathcal{T}'}, \mathcal{F}_{\mathcal{S}(v)}(\mathcal{G}(v)) \rangle. $$
Let $X \leqslant S$ and suppose that $\langle X,X^g \rangle \leqslant S$ with $g \in \mathcal{G}_\mathcal{T}.$ If $g \in \mathcal{G}_{\mathcal{T}'}$ then we are done, so suppose that $g \notin \mathcal{G}_{\mathcal{T}'}$ and write $$g=g_{-\infty}h_1g_1 \ldots h_rg_rh_{r+1}g_{\infty}$$ where $g_{-\infty},g_{\infty} \in \mathcal{G}_{\mathcal{T}'}$, $g_i \in \mathcal{G}_{\mathcal{T}'} \backslash \mathcal{G}(e)$ and $h_i \in \mathcal{G}(v) \backslash \mathcal{G}(e).$ Set $X^*:=X^{g_{-\infty}}.$ Then $\langle X^*,(X^*)^{h_1g_1 \ldots h_rg_rh_{r+1}} \rangle \leqslant \mathcal{G}_{\mathcal{T}'}$ so that by Lemma $\ref{roblem}$, $$X^*, (X^*)^{h_1}, (X^*)^{h_1g_1},\ldots,(X^*)^{h_1g_1\ldots h_rg_rh_{r+1}} \leqslant \mathcal{G}(e).$$
By Sylow's Theorem there exists $k_0 \in \mathcal{G}(e)$ such that $(X^*)^{k_0} \leqslant \mathcal{S}(e)$, $l_1 \in \mathcal{G}(e)$ such that $(X^*)^{h_1l_1} \leqslant \mathcal{S}(e)$, $k_1 \in \mathcal{G}(e)$ such that $(X^*)^{h_1g_1k_1} \leqslant \mathcal{S}(e)$, and so on. Furthermore the sequence of elements $$k_0, k_0^{-1}h_1l_1, l_1^{-1}g_1k_1,k_1^{-1}h_2l_2, \ldots, k_r^{-1}h_{r+1}l_{r+1},l_{r+1}^{-1}g_\infty$$ lie alternately in the groups $\mathcal{G}_{\mathcal{T}'}$ and $\mathcal{G}(v)$ and multiply to give $h_1g_1\ldots h_rg_rh_{r+1}g_{\infty}$. Now, for $1 \leqslant i \leqslant r,$ conjugation from $(X^*)^{h_1...h_il_i}$ to $(X^*)^{h_1...h_ig_ik_i}$ is carried out in the fusion system of $\mathcal{G}_{\mathcal{T}'}$ and conjugation from $(X^*)^{h_1...h_ig_ik_i}$ to $(X^*)^{h_1...g_ih_{i+1}l_{i+1}}$ occurs in the fusion system of $\mathcal{G}(v).$ Lastly, conjugation from $(X^*)^{h_1...h_rg_rh_{r+1}l_{r+1}}$ to $(X^*)^{h_1...h_rg_rh_{r+1}g_{\infty}}$ is carried out in the fusion system of $\mathcal{G}_{\mathcal{T}'}$. Since $\mathcal{S}(e)=\mathcal{S}(v)$, the result follows. \end{proof}
We end this section with a short example which illustrates how Theorem \ref{treesgroupthm} can be applied to produce fusion systems from group amalgams.
\begin{Ex}\label{ftex} Let $H \cong$ Sym$(4)$, $S \cong D_8$ be a Sylow 2-subgroup of $H$ and let $V_1$ and $V_2$ be the two non-cyclic non-conjugate subgroups of order 4 in $S$. Also, let $\mathcal{T}$ be a tree consisting of two vertices $v_1$ and $v_2$ with a single edge $e$ connecting them. Form a tree of groups $(\mathcal{T},\mathcal{G})$ where $\mathcal{G}(v_1) =\mathcal{G}(v_2) = H$ and $\mathcal{G}(e)= S$ and where $V_i f_{ev}$ is normal in $\mathcal{G}(v_i)$ but not normal in $\mathcal{G}(v_{3-i})$ for $i=1,2.$ Let $(\mathcal{T},\mathcal{F},\mathcal{S})$ be a tree of fusion systems induced by $(\mathcal{T},\mathcal{G})$ and observe that $(\mathcal{T},\mathcal{F},\mathcal{S})$ satisfies $(H)$ so that it has a completion $\mathcal{F}_\mathcal{T}$ by Lemma \ref{compft}. It is well known that the group $PSL_3(2)$ is a faithful, finite completion of $(\mathcal{T},\mathcal{G})$ (see, for example, \cite[Theorem A]{Gol}) so that by Theorem \ref{treesgroupthm} we must have that $\mathcal{F}_\mathcal{T}$ is the fusion system of $PSL_3(2)$. \end{Ex}
\subsection{The $P$-orbit Graph} We next consider what can said about the $P$-orbit graph in the special case where a tree of fusion systems $(\mathcal{T},\mathcal{F},\mathcal{S})$ which satisfies $(H)$ is induced by a tree of groups $(\mathcal{T},\mathcal{G}).$ It is shown that like the completion $\mathcal{F}_\mathcal{T}$, the isomorphism type of Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$ is independent of the choice of $(\mathcal{T},\mathcal{F},\mathcal{S})$ since it may described in terms of the orbit graph $\tilde{\mathcal{T}}$ of $(\mathcal{T},\mathcal{G})$.
We start by introducing some more notation. If $A$ and $B$ are groups, let Rep$(A,B)$ denote the set of orbits of the action of Inn$(B)$ on Hom$(A,B)$ by right composition. For each $\alpha \in$ Hom$(A,B)$, let $[\alpha]_B$ denote its class modulo Inn$(B)$. If $A$ and $B$ are subgroups of some group $C$, write Rep$_C(A,B)$ for the set of orbits of the action of Inn$(B)$ on Hom$_C(A,B) \cong N_C(A,B)/C_C(A)$ (where $N_C(A,B)=\{g \in C \mid A^g \leqslant B\}$ and $C_C(A)$ acts on $N_C(A,B)$ via left multiplication).
Now let $(\mathcal{T},\mathcal{G})$ be a tree of groups and fix a vertex $v_*$ of $\mathcal{T}$. For each $P \leqslant \mathcal{G}(v_*)$ let Rep$(P,\mathcal{G}(-))$ be the functor from $\mathcal{T}$ to $\mathfrak{Set}$ which sends vertices $v$ and edges $e$ respectively to Rep$(P,\mathcal{G}(e))$ and Rep$(P,\mathcal{G}(e))$ and which sends $f_{ev}$ to the map from Rep$(P,\mathcal{G}(e))$ to Rep$(P,\mathcal{G}(v))$ given by $$[\alpha]_{\mathcal{G}(e)} \longmapsto [\alpha \circ \iota_{\mathcal{G}(e)}^{\mathcal{G}(v)}]_{\mathcal{G}(v)}.$$ Similarly if $\mathcal{G}_\mathcal{T}$ is the completion of $(\mathcal{T},\mathcal{G})$ let Rep$_{\mathcal{G}_\mathcal{T}}(P,\mathcal{G}(-))$ be the functor from $\mathcal{T}$ to $\mathfrak{Set}$ which sends vertices $v$ and edges $e$ respectively to Rep$_{\mathcal{G}_\mathcal{T}}(P,\mathcal{G}(v))$ and Rep$_{\mathcal{G}_\mathcal{T}}(P,\mathcal{G}(e))$ and which sends $f_{ev}$ to the map given by $$[\alpha]_{\mathcal{G}(e)} \longmapsto [\alpha \circ \iota_{\mathcal{G}(e)}^{\mathcal{G}(v)}]_{\mathcal{G}(v)}.$$ (Observe that this well-defined). Now define: $$\mbox{Rep}(P,\mathcal{G}):= \underrightarrow{\mbox{hocolim}}_{
\substack{
\mathcal{T}}} \operatorname{Rep}(P,\mathcal{G}(-)) \mbox{, } \mbox{Rep}_{\mathcal{G}_\mathcal{T}}(P,\mathcal{G}):= \underrightarrow{\mbox{hocolim}}_{
\substack{
\mathcal{T}}} \operatorname{Rep}_{\mathcal{G}_\mathcal{T}}(P,\mathcal{G}(-))$$ both regarded as graphs in the usual way. The next proposition allows us to compare Rep$_{\mathcal{G}_\mathcal{T}}(P,\mathcal{G})$ with $\tilde{\mathcal{T}}$. \begin{Prop}\label{phip} Let $(\mathcal{T},\mathcal{G})$ be a tree of finite groups. Fix a vertex $v_*$ of $\mathcal{T}$ and a subgroup $P \leqslant \mathcal{G}(v_*)$ and write $\mathcal{G}_\mathcal{T}$ for the completion of $(\mathcal{T},\mathcal{G})$. There exists a graph isomorphism $$\begin{CD} \tilde{\mathcal{T}}^P/C_{\mathcal{G}_\mathcal{T}}(P) @>\sim>> \operatorname{Rep}_{\mathcal{G}_\mathcal{T}}(P,\mathcal{G}),\\ \end{CD}$$ where $\tilde{\mathcal{T}}^P$ is the subgraph of $\tilde{\mathcal{T}}$ fixed under the action of $P$. \end{Prop}
\begin{proof} Define a map $$\begin{CD} \Phi_P:\tilde{\mathcal{T}}^P @>>> \mbox{Rep}_{\mathcal{G}_\mathcal{T}}(P,\mathcal{G})\\ \end{CD}$$ by sending a vertex $g\mathcal{G}(v)$ to $[c_g]_{\mathcal{G}(v)}$ where $c_g \in$ Hom$_{\mathcal{G}_\mathcal{T}}(P,\mathcal{G}(v))$ and similarly by sending an edge $g\mathcal{G}(e)$ to $[c_g]_{\mathcal{G}(e)}$ where $c_g \in$ Hom$_{\mathcal{G}_\mathcal{T}}(P,\mathcal{G}(e)).$ Note that this makes sense: if $g\mathcal{G}(v)$ is a vertex in $\tilde{\mathcal{T}}^P$ then for each $x \in P$, $xg\mathcal{G}(v)=g\mathcal{G}(v)$ which is equivalent to $P^g \leqslant \mathcal{G}(v)$. The same argument works for edges. To see that this map defines a homomorphism of graphs, note that if $g\mathcal{G}(v)$ and $h\mathcal{G}(w)$ are connected via an edge $i\mathcal{G}(e)$ in $\tilde{\mathcal{T}}^P$ then $g\mathcal{G}(v)=i\mathcal{G}(v)$ and $h\mathcal{G}(w) \in i\mathcal{G}(w)$ so that [$c_g]_{\mathcal{G}(v)}=[c_i \circ \iota_{\mathcal{G}(e)}^{\mathcal{G}(v)}]_{\mathcal{G}(v)}$ and $[c_h]_{\mathcal{G}(w)}=[c_i \circ \iota_{\mathcal{G}(e)}^{\mathcal{G}(v)}]_{\mathcal{G}(w)}.$ It is clear from the definition that $\Phi_P$ is surjective. If $[c_g]_{\mathcal{G}(v)}=[c_h]_{\mathcal{G}(w)}$ then $v=w$ and there is some $r \in \mathcal{G}(v)$ such that $c_g=c_h \circ c_r \in$ Hom$_{\mathcal{G}_\mathcal{T}}(P, \mathcal{G}(v))$. This is equivalent to the orbits of $g$ and $hr$ under the action of $C_{\mathcal{G}_\mathcal{T}}(P)$ on $N_{\mathcal{G}_\mathcal{T}}(P,\mathcal{G}(v))$ being equivalent so that there is some $x \in C_{\mathcal{G}_\mathcal{T}}(P)$ with $xg=hr$. Hence $h^{-1}xg \in \mathcal{G}(v)$, $h\mathcal{G}(v)=xg\mathcal{G}(v)$ and the vertices $h\mathcal{G}(v)$ and $g\mathcal{G}(v)$ lie in the same $C_{\mathcal{G}_\mathcal{T}}(P)$-orbit, as needed. \end{proof}
We can now compare the graphs Rep$(P,\mathcal{G})$ and Rep$(P,\mathcal{F})$ when $\mathcal{F}$ is a choice of functor $\mathcal{F}_{\mathcal{S}(-)}(\mathcal{G}(-))$ associated to a tree of finite groups $(\mathcal{T},\mathcal{G})$ (see Lemma \ref{induce}). We need the following simple consequence of Sylow's Theorem.
\begin{Lem}\label{ftgt} Let $G$ be a finite group and let $S$ be a Sylow $p$-subgroup of $G$. Then for each $p$-group $P$, there exists a bijection $$\begin{CD} \operatorname{Rep}(P,\mathcal{F}_S(G)) @>\sim>> \operatorname{Rep}(P,G)\\ \end{CD}$$ given by the map $\Psi$ which sends $[\alpha]_{\mathcal{F}_S(G)}$ to $[\alpha \circ \iota_S^G]_G$ for each $\alpha \in \operatorname{Hom}(P,S).$ \end{Lem}
\begin{proof}
Let $\alpha,\beta \in$ Hom$(P,S)$ and suppose that $[\alpha \circ \iota_S^G]_G=[\beta \circ \iota_S^G]_G$. Then there exists a map $c_g \in$ Inn$(G)$ such that $\alpha \circ \iota_S^G \circ c_g=\beta \circ \iota_S^G$. This implies that $c_g|_S \in$ Hom$_{\mathcal{F}_S(G)}(P\alpha,P\beta)$ and $\alpha \circ c_g|_S = \beta$ so that $[\alpha]_{\mathcal{F}_S(G)}=[\beta]_{\mathcal{F}_S(G)}$ and $\Psi$ is injective. To see that $\Psi$ is surjective, let $[\gamma]_G \in$ Rep$(P,G)$. By Sylow's Theorem there exists $g \in G$ such that $(P\gamma)^g \leqslant S$. Set $\varphi:=\gamma \circ c_g \in$ Hom$(P,S)$ and observe that $[\varphi \circ \iota_S^G]_G=[\gamma]_G,$ as required. \end{proof}
\begin{Cor}\label{corftgt} Let $(\mathcal{T},\mathcal{G})$ be a tree of finite groups with completion $\mathcal{G}_\mathcal{T}$ and let $(\mathcal{T},\mathcal{F},\mathcal{S})$ be a tree of fusion systems induced by $(\mathcal{T},\mathcal{G})$. Fix a vertex $v_*$ of $\mathcal{T}$ and a subgroup $P \leqslant \mathcal{S}(v_*)$. Then there exists a natural isomorphism of functors $$\begin{CD} \operatorname{Rep}(P,\mathcal{F}(-)) @>\sim>> \operatorname{Rep}(P,\mathcal{G}(-))\\ \end{CD},$$ which induces a homotopy equivalence $\begin{CD} \operatorname{Rep}(P,\mathcal{F}) @>\simeq>> \operatorname{Rep}(P,\mathcal{G})\\ \end{CD}.$ Furthermore, if $(\mathcal{T},\mathcal{F},\mathcal{S})$ satisfies $(H)$ and has completion $\mathcal{F}_\mathcal{T}$ then $$\begin{CD} \operatorname{Rep}_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F}) @>\simeq>> \operatorname{Rep}_{\mathcal{G}_\mathcal{T}}(P,\mathcal{G})\\ \end{CD},$$ so that $\operatorname{Rep}_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$ is independent of the choice of tree of fusion systems $(\mathcal{T},\mathcal{F},\mathcal{S})$. \end{Cor}
\begin{proof} The existence of a natural isomorphism of functors $$\begin{CD} \eta:\mbox{Rep}(P,\mathcal{F}(-)) @>\sim>> \mbox{Rep}(P,\mathcal{G}(-))\\ \end{CD}$$ is an immediate consequence of Lemma \ref{ftgt}. By \cite[Proposition IV.1.9]{GJ} this natural isomorphism induces a homotopy equivalence $$\begin{CD} \mbox{hocolim}(\eta): \mbox{Rep}(P,\mathcal{F}) @>\simeq>> \mbox{Rep}(P,\mathcal{G})\\ \end{CD}.$$ If $(\mathcal{T},\mathcal{F},\mathcal{S})$ satisfies $(H)$ then the above equivalence sends the vertex $[\iota_P^{\mathcal{S}(v_*)}]_{\mathcal{F}(v_*)}$ to $[\iota_P^{\mathcal{G}(v_*)}]_{\mathcal{G}(v_*)}.$ Clearly $[\iota_P^{\mathcal{G}(v_*)}]_{\mathcal{G}(v_*)}$ lies in Rep$_{\mathcal{G}_\mathcal{T}}(P,\mathcal{G})$ (it is the image of the coset $\mathcal{G}(v_*)$ under $\Phi_P$). Since Rep$_{\mathcal{G}_\mathcal{T}}(P,\mathcal{G})$ is connected (being the image under $\Phi_P$ of $\tilde{\mathcal{T}}^P$ which is connected), hocolim($\eta$) must restrict to a homotopy equivalence $$\begin{CD} \mbox{Rep}_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F}) @>\simeq>> \mbox{Rep}_{\mathcal{G}_\mathcal{T}}(P,\mathcal{G})\\ \end{CD}$$ by Proposition \ref{phipft}. This completes the proof. \end{proof}
\section{The Completion of a Tree of Fusion Systems}\label{compfus} In this section, we prove our second main result, Theorem B, which gives conditions for a tree of fusion systems $(\mathcal{T},\mathcal{F},\mathcal{S})$ to have a saturated completion $\mathcal{F}_\mathcal{T}$:
\begin{Thm}\label{completionsat} Let $(\mathcal{T},\mathcal{F},\mathcal{S})$ be a tree of fusion systems which satisfies $(H)$ and assume that $\mathcal{F}(v)$ is saturated for each vertex $v$ of $\mathcal{T}.$ Write $S:=\mathcal{S}(v_*)$ and $\mathcal{F}_\mathcal{T}$ for the completion of $(\mathcal{T},\mathcal{F},\mathcal{S})$. Assume that the following hold for each $P \leqslant S$. \begin{itemize} \item[(a)] If $P$ is $\mathcal{F}_\mathcal{T}$-conjugate to an $\mathcal{F}(v)$-essential subgroup or $P=\mathcal{S}(v)$ then $P$ is $\mathcal{F}_\mathcal{T}$-centric. \item[(b)] If $P$ is $\mathcal{F}_\mathcal{T}$-centric then $\operatorname{Rep}_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$ is a tree. \end{itemize} Then $\mathcal{F}_\mathcal{T}$ is a saturated fusion system on $S$. \end{Thm} Conditions (a) and (b) are respectively motivated by Theorems \ref{alpthm} and \ref{alpthmconv}. The former condition implies that $\mathcal{F}_\mathcal{T}$ is generated by its $\mathcal{F}_\mathcal{T}$-centric subgroups, while the latter ensures that the saturation axioms (Definition \ref{sat} (a) and (b)) hold for all such subgroups. Thus the saturation of $\mathcal{F}_\mathcal{T}$ will ultimately follow from Theorem \ref{alpthmconv}. Theorem \ref{completionsat} was inspired by a theorem of Broto, Levi and Oliver (\cite[Theorem 4.2]{BLO4}) and we will deduce their result from ours in Corollary \ref{completionsatgroups}.
\subsection{Finite Group Actions on Trees} Before embarking on the proof of Theorem \ref{completionsat}, it will be important to understand the way in which the automiser Aut$_{\mathcal{F}_\mathcal{T}}(P)$ can act on the $P$-orbit graph Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$. Note that we may draw an analogy here with the situation for trees of groups $(\mathcal{T},\mathcal{G})$ where there exists an action of the completion $\mathcal{G}_\mathcal{T}$ on the orbit tree $\tilde{\mathcal{T}}$. We begin with two important concepts which play a pivotal role in the proof of Theorem \ref{completionsat}.
\begin{Def} Let $(\mathcal{T},\mathcal{F},\mathcal{S})$ be a tree of fusion systems which satisfies $(H)$ and write $\mathcal{F}_\mathcal{T}$ for the completion of $(\mathcal{T},\mathcal{F},\mathcal{S})$.
\begin{itemize} \item[(a)] For each pair of $p$-groups $P \leqslant Q$ define the \textit{restriction map} from $Q$ to $P$: $$\begin{CD} \mbox{res}^Q_P:\mbox{Rep}_{\mathcal{F}_\mathcal{T}}(Q,\mathcal{F}) @>>> \mbox{Rep}_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})\\ \end{CD}$$
to be the map which sends a vertex $[\varphi]_{\mathcal{F}(v)}$ in Rep$_{\mathcal{F}_\mathcal{T}}(Q,\mathcal{F})$ to $[\varphi|_P]_{\mathcal{F}(v)}$ in Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$. \item[(b)] For each $P \leqslant \mathcal{S}(v_*)$ the \textit{action} of $\mathcal{F}_\mathcal{T}$ on Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$ is given by the group action of Aut$_{\mathcal{F}_\mathcal{T}}(P)$ on Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$ which sends a vertex $[\varphi]_{\mathcal{F}(v)}$ to $[\psi \circ \varphi]_{\mathcal{F}(v)}$ for each $\psi \in$ Aut$_{\mathcal{F}_\mathcal{T}}(P)$. \end{itemize} \end{Def}
We quickly check that this definition make sense. Firstly, one observes that res$^Q_P$ defines a homomorphism of graphs. To see this, note that if $[\alpha]_{\mathcal{F}(v)}$ is incident to some edge $[\beta]_{\mathcal{F}(e)}$ in Rep$_{\mathcal{F}_\mathcal{T}}(Q,\mathcal{F})$, then there exists $\psi \in$ Hom$_{\mathcal{F}(v)}(Q\alpha,\mathcal{S}(v))$ such that $\beta \circ \iota_{\mathcal{S}(e)}^{\mathcal{S}(v)}=\alpha \circ \psi$. Hence $\beta|_P \circ \iota_{\mathcal{S}(e)}^{\mathcal{S}(v)}=\alpha|_P \circ \psi|_{P\alpha}$ and $[\alpha|_P]_{\mathcal{F}(v)}$ is incident to $[\beta|_P]_{\mathcal{F}(e)}$ in Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$. Secondly, note the described action of Aut$_{\mathcal{F}_\mathcal{T}}(P)$ on Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$ makes sense: if $[\varphi]_{\mathcal{F}(v)}$ is a vertex of Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$ and $\psi \in$ Aut$_{\mathcal{F}_\mathcal{T}}(P)$, then $\psi \circ \varphi \in$ Hom$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{S}(v))$ so that $[\psi \circ \varphi]_{\mathcal{F}(v)}$ is also a vertex of Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$.
The following proposition gives conditions on $(\mathcal{T},\mathcal{F},\mathcal{S})$ which will allow us to calculate the image of res$^Q_P$ in terms of certain fixed points of the action of $\mathcal{F}_\mathcal{T}$ on Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$.
\begin{Prop}\label{resprop} Let $(\mathcal{T},\mathcal{F},\mathcal{S})$ be a tree of fusion systems which satisfies $(H)$ and assume that the following hold: \begin{itemize} \item[(a)] $\mathcal{F}(v)$ is saturated for each vertex $v$ of $\mathcal{T};$ and \item[(b)] $\mathcal{F}(e)$ is a trivial fusion system $\mathcal{F}_{\mathcal{S}(e)}(\mathcal{S}(e))$ for each edge $e$ of $\mathcal{T}$. \end{itemize} Let $P$ and $Q$ be $p$-groups with $P$ $\unlhd$ $Q \leqslant S(v_*)$ and assume that $P$ is $\mathcal{F}_\mathcal{T}$-centric, where $\mathcal{F}_\mathcal{T}$ is the completion of $(\mathcal{T},\mathcal{F},\mathcal{S})$. If $\operatorname{Rep}_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$ is a tree, then the image of the restriction homomorphism $$\begin{CD} \operatorname{res}^Q_P:\operatorname{Rep}_{\mathcal{F}_\mathcal{T}}(Q,\mathcal{F}) @>>> \operatorname{Rep}_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})\\ \end{CD}$$ is equal to $\operatorname{Rep}_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})^{\operatorname{Aut}_Q(P)}$. \end{Prop}
\begin{proof}
Set $K:=$ Aut$_Q(P)$. If $[\alpha]$ is a vertex or edge in Rep$_{\mathcal{F}_\mathcal{T}}(Q, \mathcal{F})$ then $[\alpha|_P] \in$ Rep$_{\mathcal{F}_\mathcal{T}}(P, \mathcal{F})^K$ since $$c_g \circ \alpha|_P = \alpha|_P \circ (\alpha^{-1}|_{P\alpha} \circ c_g \circ \alpha|_P)$$ for each $g \in Q$. It remains to prove that each vertex or edge of Rep$_{\mathcal{F}_\mathcal{T}}(P, \mathcal{F})^K$ is the image of \textit{some} vertex or edge of Rep$_{\mathcal{F}_\mathcal{T}}(Q, \mathcal{F})$. Since Rep$_{\mathcal{F}_\mathcal{T}}(P, \mathcal{F})$ is a tree and $K$ is finite, Rep$_{\mathcal{F}_\mathcal{T}}(P, \mathcal{F})^K$ is also a tree. In particular Rep$_{\mathcal{F}_\mathcal{T}}(P, \mathcal{F})^K$ is connected. Hence to complete the proof, it will be enough to show that for each edge $[\beta]_{\mathcal{F}(e)}$ in Rep$_{\mathcal{F}_\mathcal{T}}(P, \mathcal{F})^K$ which is connected to the image $[\alpha|_P]_{\mathcal{F}(v)}$ of some vertex $[\alpha]_{\mathcal{F}(v)}$ under res$_P^Q$, there exists an edge $[\overline{\beta}]_{\mathcal{F}(e)}$ in Rep$_{\mathcal{F}_\mathcal{T}}(Q,\mathcal{F})$ connected to $[\alpha]_{\mathcal{F}(v)}$ whose image under res$_P^Q$ is $[\beta]_{\mathcal{F}(e)}$. Since $[\iota_Q^S]_{\mathcal{F}(v_*)}$ gets sent to $[\iota_P^S]_{\mathcal{F}(v_*)}$ under res$_P^Q$, we may assume (inductively) that we have chosen $e$ to be the edge to which $v$ is incident in the unique minimal path from $v$ to $v_*$. In particular we may assume that $\mathcal{S}(e)=\mathcal{S}(v)$. By assumption, there exists $\psi \in$ Hom$_{\mathcal{F}(v)}(P\beta,P\alpha)$ such that $\beta \circ \psi=\alpha|_P$. Set $R:=N_{\mathcal{S}(e)}^{K^\beta}(P\beta)$. Since $[\beta]_{\mathcal{F}(e)}$ is fixed by $K$, $K^\beta \leqslant$ Aut$_{\mathcal{S}(e)}(P\beta)$ so Aut$_R(P\beta)=K^{\beta}$. Now, Aut$_R(P\beta)^\psi=K^{\beta\psi}=K^\alpha=$ Aut$_{Q\alpha}(P\alpha)$ and hence $$Q\alpha \leqslant N_{\psi^{-1}}=\{g \in N_{\mathcal{S}(v)}(P\alpha) \mid (c_g)^{\psi^{-1}} \in \mbox{ Aut}_S(P\beta)\}.$$ Since $\mathcal{F}(v)$ is saturated and $P\alpha$ is $\mathcal{F}(v)$-centric (and therefore fully $\mathcal{F}(v)$-centralised), $\psi^{-1}$ extends to a map $\rho \in $ Hom$_{\mathcal{F}(v)}(Q\alpha, \mathcal{S}(v))$. Set $\overline{\beta}:=\alpha \circ \rho \in$ Hom$_{\mathcal{F}(v)}(Q,\mathcal{S}(v))$ so that res$_P^Q([\overline{\beta}]_{\mathcal{F}(e)})= [\overline{\beta}|_{P}]_{\mathcal{F}(e)}=[\beta]_{\mathcal{F}(e)}$ and $[\alpha]_{\mathcal{F}(v)}$ is incident to $[\overline{\beta}]_{\mathcal{F}(e)}$ in Rep$_{\mathcal{F}_\mathcal{T}}(Q,\mathcal{F})$. This completes the proof. \end{proof}
\subsection{Proof of Theorem B} We now have all of the tools necessary to prove Theorem B. We remind the reader that this theorem was originally conjectured based on the corresponding statement for trees of group fusion systems, \cite[Theorem 4.2]{BLO4}, and the proof in that case may be seen to rely on some fairly deep homotopy theory. No deep results are required in our proof and for this reason we feel the statement is more naturally understood in the context of trees of fusion systems. Nevertheless, we will present \cite[Theorem 4.2]{BLO4} as Corollary \ref{completionsatgroups} below.
A key step in our proof is the observation that the completion $\mathcal{F}_\mathcal{T}$ of a tree of fusion systems $(\mathcal{T},\mathcal{F},\mathcal{S})$ is independent of where $\mathcal{F}$ sends edges $e$ of $\mathcal{T}$, provided $\mathcal{F}_{\mathcal{S}(e)}(\mathcal{S}(e)) \subseteq \mathcal{F}(e) \subseteq \mathcal{F}(v)$ (inside $\mathcal{F}_\mathcal{T}$). This means that we are able to assume that $\mathcal{F}(e)$ is the trivial fusion system $\mathcal{F}_{\mathcal{S}(e)}(\mathcal{S}(e))$.
\begin{proof}[Proof of Theorem B] Let $(\mathcal{T},\mathcal{F}',\mathcal{S})$ be the tree of fusion systems obtained from $(\mathcal{T},\mathcal{F},\mathcal{S})$ by setting $\mathcal{F}'(v):=\mathcal{F}(v)$ for each vertex $v$ of $\mathcal{T}$ and $\mathcal{F}'(e):=\mathcal{F}_{\mathcal{S}(e)}(\mathcal{S}(e))$ for each edge $e$ of $\mathcal{T}.$ Then $(\mathcal{T},\mathcal{F}',\mathcal{S})$ satisfies $(H)$ and has completion $\mathcal{F}_\mathcal{T}$. Also, since any cycle in Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F}')$ is a cycle in Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$, hypothesis (b) in the theorem remains true for $(\mathcal{T},\mathcal{F}',\mathcal{S}).$ Hence we may assume from now on that $(\mathcal{T},\mathcal{F},\mathcal{S})=(\mathcal{T},\mathcal{F}',\mathcal{S})$.
Since $\mathcal{F}(v)$ is a saturated fusion system on $\mathcal{S}(v)$ for each vertex $v$ in $\mathcal{T}$, by Theorem \ref{alpthm} $$\mathcal{F}(v)=\langle \{\mbox{Aut} _{\mathcal{F}(v)}(P) \mid P \mbox{ is $\mathcal{F}(v)$-essential }\}\cup \{\mbox{Aut}_{\mathcal{F}(v)}(\mathcal{S}(v)) \} \rangle_{\mathcal{S}(v)}.$$ By hypothesis, each $\mathcal{F}(v)$-essential subgroup and each $\mathcal{S}(v)$ is an $\mathcal{F}_\mathcal{T}$-centric subgroup so by the definition of $\mathcal{F}_\mathcal{T}$ we have $$\mathcal{F}_\mathcal{T}=\langle \mbox{Aut} _{\mathcal{F}(v)}(P) \mid v \in V(\mathcal{T}), P \mbox{ is $\mathcal{F}_\mathcal{T}$-centric} \rangle_S.$$ Hence by Theorem \ref{alpthmconv} it suffices to verify that axioms (a) and (b) in Definition \ref{fus} hold for $\mathcal{F}_\mathcal{T}$-centric subgroups.
First we prove Definition \ref{fus} (a) holds. Let $P$ be a fully $\mathcal{F}_\mathcal{T}$-normalised, $\mathcal{F}_\mathcal{T}$-centric subgroup of $S$. Then since $|C_S(P\varphi)|=|Z(P\varphi)|=|Z(P)|$ for each $\varphi \in$ Hom$_{\mathcal{F}_\mathcal{T}}(P,S)$, $P$ is certainly fully $\mathcal{F}_\mathcal{T}$-centralised. It remains to prove that $P$ is fully $\mathcal{F}_\mathcal{T}$-automised. Each $\varphi \in$ Aut$_{\mathcal{F}_\mathcal{T}}(P)$ acts on Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$ by sending a vertex $[\alpha]_{\mathcal{F}(v)}$ to $[\varphi \circ \alpha]_{\mathcal{F}(v)}$. Since Aut$_{\mathcal{F}_\mathcal{T}}(P)$ is finite and Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$ is a tree, there exists a vertex or edge $[\alpha]_{\mathcal{F}(r)}$ in Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$ which is fixed under the action of Aut$_{\mathcal{F}_\mathcal{T}}(P)$ ($r \in V(\mathcal{T}) \cup E(\mathcal{T})$). This means that Aut$_{\mathcal{F}(r)}(P\alpha)=$ Aut$_{\mathcal{F}_\mathcal{T}}(P)^\alpha$.
Choose $\beta \in$ Hom$_{\mathcal{F}(r)}(P\alpha, S(r))$ so that $P\alpha\beta$ is fully $\mathcal{F}(r)$-normalised. Since $\mathcal{F}(r)$ is saturated, Aut$_{S(r)}(P\alpha\beta) \in$ Syl$_p($Aut$_{\mathcal{F}(r)}(P\alpha\beta))$. Now $$|\mbox{Aut}_S(P)|=|N_S(P)|/|Z(P)| \geqslant |N_S(P\alpha\beta)|/|Z(P\alpha\beta)| \geqslant |\mbox{Aut}_{S(r)}(P\alpha\beta)|,$$ where the first inequality follows from the fact that $P$ is fully $\mathcal{F}_\mathcal{T}$-normalised. Since $$|\mbox{Aut}_{\mathcal{F}(r)}(P\alpha\beta)|=|\mbox{Aut}_{\mathcal{F}(r)}(P\alpha)|=|\mbox{Aut}_{\mathcal{F}_\mathcal{T}}(P)|,$$ we must have Aut$_S(P) \in$ Syl$_p($Aut$_{\mathcal{F}_\mathcal{T}}(P))$, so that $P$ is fully $\mathcal{F}_\mathcal{T}$-automised, as needed.
Next we prove that (b) holds in Definition \ref{fus}. Fix $\varphi \in$ Hom$_{\mathcal{F}_\mathcal{T}}(P,S)$ where $P$ is $\mathcal{F}_\mathcal{T}$-centric. We need to show that there is some $\overline{\varphi} \in$ Hom$_{\mathcal{F}_\mathcal{T}}(N_\varphi, S)$ which extends $\varphi$. Set $K:=$ Aut$_{N_\varphi}(P)=$ Aut$_S(P)$ $\cap$ Aut$_S(P\varphi)^{\varphi^{-1}}.$ If $c_g \in K$ then $c_g \circ \varphi=\varphi \circ c_h$ for some $h \in$ Aut$_S(P\varphi)$ and this show that the vertex $[\varphi]_{\mathcal{F}(v_*)}$ in Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F})$ is fixed by the action of $K$. By Proposition \ref{resprop} applied with $Q=N_\varphi$ there is some $[\psi]_{\mathcal{F}(v_*)} \in$ Rep$_{\mathcal{F}_\mathcal{T}}(N_\varphi, \mathcal{F})$ with $[\psi|_P]_{\mathcal{F}(v_*)}=[\varphi]_{\mathcal{F}(v_*)}$. This implies that there is $\rho \in$ Iso$_{\mathcal{F}(v_*)}(P\varphi,P\psi)$ such that $\psi|_P=\varphi \circ \rho$. Since $\mathcal{F}(v_*)$ is saturated, $\rho^{-1}|_{P\psi}= (\psi|_P)^{-1} \circ \varphi$ extends to a map $\chi \in $ Hom$_{\mathcal{F}(v_*)}((N_\varphi)\psi, S)$. To see this, observe that $(N_\varphi)\psi \leqslant N_{\rho^{-1}}$ since for $g \in N_\varphi$, $(c_{g\psi})^{\rho^{-1}}=(c_g)^\varphi \in$ Aut$_S(P\varphi)=$ Aut$_S(P\psi\rho^{-1}).$ Hence we set $\overline{\varphi}:= \psi \circ \chi \in$ Hom$_\mathcal{F}(N_\varphi,S)$ so that $\overline{\varphi}$ extends $\varphi$, as needed. This completes the proof of (b) in Definition \ref{fus} and hence the proof of the theorem. \end{proof}
\begin{Cor}\label{completionsatgroups}
Let $(\mathcal{T},\mathcal{G})$ be a tree of groups with completion $\mathcal{G}_\mathcal{T}$ and assume that there exists a vertex $v_*$ of $\mathcal{T}$ for which the following is true: for each $v \in V(\mathcal{T})$ not equal to $v_*$, $p \nmid |\mathcal{G}(v):\mathcal{G}(e)|$ where $e$ is the to which $v$ is incident in the unique minimal path from $v$ to $v_*$. Write $S:=\mathcal{S}(v_*)$ and assume that for each $P \leqslant S$, \begin{itemize} \item[(a)] if $P$ is $\mathcal{G}_\mathcal{T}$-conjugate to an $\mathcal{F}_{\mathcal{S}(v)}(\mathcal{G}(v))$-essential subgroup then $P$ is $\mathcal{F}_S(G_\mathcal{T})$-centric; and \item[(b)] if $P$ is $\mathcal{F}_S(G_\mathcal{T})$-centric then $\tilde{\mathcal{T}}^P/C_{\mathcal{G}_\mathcal{T}}(P)$ is a tree. \end{itemize} Then $\mathcal{F}_S(\mathcal{G}_\mathcal{T})$ is saturated. \end{Cor}
\begin{proof} Let $(\mathcal{T},\mathcal{F},\mathcal{S})$ be a tree of fusion systems induced by $(\mathcal{T},\mathcal{G})$. The hypothesis of the corollary implies that $(\mathcal{T},\mathcal{F},\mathcal{S})$ satisfies $(H)$, so there exists a completion $\mathcal{F}_\mathcal{T}$ of $(\mathcal{T},\mathcal{F},\mathcal{S})$ by Lemma \ref{compft}. By Theorem \ref{treesgroupthm}, $\mathcal{F}_\mathcal{T} \cong \mathcal{F}_S(\mathcal{G}_\mathcal{T})$, so it suffices to verify conditions (a) and (b) in Theorem \ref{completionsat} hold. Clearly (a) holds by assumption and (b) holds by Proposition \ref{phip} and Corollary \ref{corftgt}. \end{proof}
\begin{Ex}
Let $(\mathcal{T},\mathcal{F},\mathcal{S})$ be the tree of fusion systems constructed in Example \ref{ftex}. Clearly (a) holds in the statement of Theorem \ref{completionsat} since $S, V_1$ and $V_2$ are the only $\mathcal{F}_\mathcal{T}$-centric subgroups. It remains to check that (b) holds for these subgroups. Firstly, we show that $|$Rep$_{\mathcal{F}_\mathcal{T}}(V_i,\mathcal{F}(e))|=3$. To see this, for $i=1,2$ write $Z(S)=\langle z \rangle$, choose $x_i$ such that $V_i=\langle z, x_i \rangle$ and notice that $V_i\varphi=V_i$ for each $\varphi \in$ Hom$_{\mathcal{F}_\mathcal{T}}(V_i,S)$. Since Aut$_{\mathcal{F}(e)}(V_i)$ interchanges $x_i$ and $x_iz$, the class of $\varphi$ is determined by which of $x_i,z$ and $x_iz$ gets mapped to $z$ under $\varphi$ yielding exactly $3$ classes of embeddings. By similar arguments the facts that Aut$_{\mathcal{F}(v_i)}(V_i) \cong$ Sym$(3)$ and Aut$_{\mathcal{F}(v_{3-i})}(V_i) \cong C_2$ imply that $|$Rep$_{\mathcal{F}_\mathcal{T}}(V_i,\mathcal{F}(v_i))|=1$ and $|$Rep$_{\mathcal{F}_\mathcal{T}}(V_i,\mathcal{F}(v_{3-i}))|=3$ respectively for $i=1,2$. Since $4=3+1$, the Euler characteristic of Rep$_{\mathcal{F}_\mathcal{T}}(V_i,\mathcal{F})$ is 0 for $i=1,2$ so this graph is a tree. In fact, we can see directly that Rep$_{\mathcal{F}_\mathcal{T}}(V_i,\mathcal{F})$ is a 3 pointed star. Finally, the graph Rep$_{\mathcal{F}_\mathcal{T}}(S,\mathcal{F})$ obviously consists of a single vertex and so the saturation of $\mathcal{F}_\mathcal{T}$ follows from Theorem \ref{completionsat}. \end{Ex}
\section{Attaching $p'$-automorphisms to Fusion Systems}\label{extconst} In this section, we apply Theorem B to prove Theorem C which is a general procedure for extending a fusion system by $p'$-automorphisms of its centric subgroups. Theorem C is a generalisation of \cite[Proposition 5.1]{BLO4} to arbitrary saturated fusion systems.
\subsection{Extensions of $p$-constrained Groups} We appeal to some cohomological machinery which will allow us to prove a lemma which gives conditions for a $p$-constrained group to be extended by a $p'$-group of automorphisms of a normal centric subgroup. First recall the following characterisation of extensions of non-abelian groups.
\begin{Thm}\label{nonabext} Let $G$ and $N$ be finite groups. Each homomorphism $$\begin{CD} \psi_E: G @>>> \operatorname{Out}(N) \end{CD}$$ determines an obstruction in $H^3(G,Z(N))$ which vanishes if and only if there exists an extension $$\begin{CD} 1 @>>> N @>>> E @>>> G @>>> 1 \end{CD}$$ which gives rise to $\psi_E$ via the conjugation action of $E$ on $N$. Furthermore, the group $H^2(G, Z(N))$ acts freely and transitively on the set of (suitably defined) equivalence classes of such extensions. \end{Thm}
\begin{proof} See \cite[Theorems IV.6.6 and IV.6.7]{Br}. \end{proof} When $H \leqslant G$, there is a \textit{restriction map} $$\begin{CD} H^n(G,M) @>\mbox{res}_H^G>> H^n(H,M) \end{CD}$$ for each $G$-module $M$ and $n \geqslant 0$ (see \cite[Section III.9]{Br}). Each $g \in G$ induces a well-defined map $\begin{CD} H^n(H,M) @>g>> H^n(H^g,M) \end{CD}$ and we define $z \in H^*(H,M)$ to be \textit{$G$-invariant} if $$\mbox{res}_{H \cap H^g}^H z = \mbox{res}_{H \cap H^g}^{H^g} zg $$ for each $g \in G$. The following result characterises $G$-invariant elements when $M$ is an $\mathbb{F}_pG$-module for some prime $p$.
\begin{Thm}
Let $G$ be a finite group, $p$ be prime and $M$ be an $\mathbb{F}_pG$-module. For each $H \leqslant G$ with $p \nmid |G:H|$ and $n \geqslant 0$, res$_H^G$ maps $H^n(G,M)$ isomorphically onto the set of $G$-invariant elements of $H^n(H,M)$. \end{Thm}
\begin{proof} This follows from \cite[Theorem III.10.3]{Br}. \end{proof} As an immediate consequence we have:
\begin{Cor}\label{cohcor} Let $G$ be a finite group and $M$ be an $\mathbb{F}_pG$-module. If $H$ is a strongly $p$-embedded subgroup of $G$ then res$_H^G$ induces an isomorphism $$\begin{CD} H^*(G,M) @>\mbox{res}_H^G>> H^*(H,M). \end{CD}$$ \end{Cor}
Theorem \ref{nonabext} and Corollary \ref{cohcor} may be applied to prove the following result which is a key ingredient in the proof of Theorem C.
\begin{Lem}\label{constex} Let $H$ be a finite group with $O_{p'}(H)=1$ and let $Q$ be a normal $p$-subgroup of $H$ with $C_H(Q) \leqslant Q$. Assume $\Delta \leqslant \operatorname{Out}(Q)$ is chosen such that $\operatorname{Out}_H(Q)$ is a strongly $p$-embedded subgroup of $\Delta$. There exists a finite group $G$ containing $H$ as a $p'$-index subgroup with $Q \unlhd G$ and $\operatorname{Out}_G(Q) \cong \Delta$. \end{Lem}
\begin{proof} Write $K:=$ Out$_H(Q)$. Since $K$ is strongly $p$-embedded in $\Delta$ and $Z(Q)$ may be regarded as an $\mathbb{F}_p \Delta$-module, Corollary \ref{cohcor} implies that $H^j(\Delta; Z(Q)) \cong H^j(K;Z(Q))$ for each $j > 0$. Since this holds when $j=3$, there exists a finite group $G$ which fits into a diagram
$$\begin{CD} 1 @>>> Q @>>> G @>>> \Delta @>>> 1\\
@. @| @AAA @AAA \\ 1 @>>> Q @>>> H @>>> K @>>> 1, \end{CD}$$ by Theorem \ref{nonabext}. Since $H^2(\Delta; Z(Q)) \cong H^2(K;Z(Q))$ acts freely and transitively on the set of all such extensions of $Q$ by $\Delta$, we may choose $G$ such that $H \leqslant G$. In particular, $\Delta \cong G/Q=$ Out$_G(Q)$, as required. \end{proof}
\subsection{Proof of Theorem C}\label{proofsat} Let $\mathcal{F}_0$ be a saturated fusion system on a finite $p$-group $S$. Our goal is to apply Lemma \ref{constex} to find conditions for there to exist a saturated fusion system $\mathcal{F}$ containing $\mathcal{F}_0$ with the property that Aut$_{\mathcal{F}_0}(Q)$ is a strongly $p$-embedded subgroup of Aut$_\mathcal{F}(Q)$ whenever $Q \leqslant S$ is $\mathcal{F}$-essential. The technique we use will involve the construction of a star of fusion systems $(\mathcal{T},\mathcal{F}(-),\mathcal{S}(-))$ with $\mathcal{F}_0$ associated to the central vertex and the fusion systems of certain $p'$-extensions of constrained models of the $N_{\mathcal{F}_0}(Q)$ associated to the outer vertices. The saturation of the completion of $(\mathcal{T},\mathcal{F}(-),\mathcal{S}(-))$ will follow from Theorem \ref{completionsat}.
\begin{Thm}\label{proofsatthm} Let $\mathcal{F}_0$ be a saturated fusion system on a finite $p$-group $S$. For $1 \leqslant i \leqslant m$, let $Q_i \leqslant S$ be fully $\mathcal{F}_0$-normalised subgroups with $Q_i\varphi \nleq Q_j$ for each $\varphi \in \operatorname{Hom}_{\mathcal{F}_0}(Q_i,S)$ and $i \neq j$. Set $K_i:=\operatorname{Out}_{\mathcal{F}_0}(Q_i)$ and choose $\Delta_i \leqslant \operatorname{Out}(Q_i)$ so that $K_i$ is a strongly $p$-embedded subgroup of $\Delta_i$. Write $$\mathcal{F}= \langle \{\operatorname{Hom}_{\mathcal{F}_0}(P,S) \mid P \leqslant S\} \cup \{\Delta_i \mid 1 \leqslant i \leqslant m \} \rangle_S.$$ Assume further that for each $1 \leqslant i \leqslant m$, \begin{itemize} \item[(a)] $Q_i$ is $\mathcal{F}_0$-centric (hence $\mathcal{F}$-centric) and minimal (under inclusion) amongst all $\mathcal{F}$-centric subgroups, and \item[(b)] no proper subgroup of $Q_i$ is $\mathcal{F}_0$-essential. \end{itemize} Then $\mathcal{F}$ is saturated. \end{Thm}
\begin{proof}
Since $Q_i$ is fully $\mathcal{F}_0$-normalised and $\mathcal{F}_0$-centric, $N_{\mathcal{F}_0}(Q_i)$ is a constrained fusion system. Hence by Theorem \ref{const} there exists a unique finite group $H_i$ with $Q_i \unlhd H_i$, $C_{H_i}(Q_i) \leqslant Q_i$ and $N_{\mathcal{F}_0}(Q_i)=\mathcal{F}_{N_S(Q_i)}(H_i)$. Since Out$_{\mathcal{F}_0}(Q_i)=$ Out$_{H_i}(Q_i)$, Lemma \ref{constex} implies that there is some finite group $G_i$ containing $H_i$ with $p \nmid |G_i: H_i|$ and which satisfies Out$_{G_i}(Q_i)=\Delta_i$. Now let $\mathcal{T}$ be the star formed by a central vertex $v_0$ and $m$ vertices $v_i$ each incident to a unique edge $e_i=(v_0,v_i)$ for $1 \leqslant i \leqslant m$. Let $(\mathcal{T},\mathcal{F}(-),\mathcal{S}(-))$ be the tree of fusion systems formed by setting $\mathcal{F}(v_0):=\mathcal{F}_0$, $\mathcal{F}(v_i):=\mathcal{F}_{N_S(Q_i)}(G_i)$ and $\mathcal{F}(e_i):=\mathcal{F}_{N_S(Q_i)}(H_i)=N_{\mathcal{F}_0}(Q_i)$ for $1 \leqslant i \leqslant m$. Then $(\mathcal{T},\mathcal{F}(-),\mathcal{S}(-))$ satisfies $(H)$, $\mathcal{F}_\mathcal{T}=\mathcal{F}$ by Lemma \ref{compft} and it suffices to verify that conditions (a) and (b) of Theorem \ref{completionsat} hold.
Suppose that $P=\mathcal{S}(v_i)$ or $P$ is $\mathcal{F}$-conjugate to an $\mathcal{F}(v_i)$-essential subgroup. We need to show that $P$ is $\mathcal{F}$-centric. In the first case, since $Q_i$ is $\mathcal{F}$-centric (by assumption), $P=N_S(Q_i)$ is also $\mathcal{F}$-centric by Lemma \ref{centlem} (b). In the second case, there exists some $\varphi \in$ Hom$_\mathcal{F}(P,S)$ such that $P\varphi \leqslant \mathcal{S}(v_i)$ is $\mathcal{F}(v_i)$-essential. If $i \neq 0$ then since $Q_i \unlhd \mathcal{F}(v_i)$, Lemma \ref{esscont} implies that $Q_i \leqslant P\varphi$ so that since $Q_i$ is $\mathcal{F}$-centric also $P\varphi$ (and hence $P$) is $\mathcal{F}$-centric by Lemma \ref{centlem} (b) again. If $i = 0$ then since $P\varphi$ is not contained in $Q_i$ (by condition (b) in the theorem) for all $1 \leqslant i \leqslant m$ and $P\varphi$ is $\mathcal{F}_0$-centric, $P\varphi$ is also $\mathcal{F}$-centric, by the definition of $\mathcal{F}_\mathcal{T}$.
It remains to prove that (b) holds in Theorem \ref{completionsat}.
It will be enough to show directly that the connected component of the vertex $[\iota_P^S]_{\mathcal{F}(v_0)}$ in Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F}(-))$ is a tree for each $\mathcal{F}_\mathcal{T}$-centric subgroup $P \leqslant S$. Suppose $[\beta]_{\mathcal{F}(v_0)}$ is connected to $[\beta']_{\mathcal{F}(v_i)}$ via an edge $[\alpha]_{\mathcal{F}(e_i)}$ in Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F}(-))$. If $[\beta']_{\mathcal{F}(v_i)}$ is incident to another edge $[\alpha']_{\mathcal{F}(e_i)}$ then there exists $\rho \in$ Iso$_{\mathcal{F}(v_i)}(P\alpha,P\alpha')$ such that $\alpha'=\alpha \circ \rho$. Since $Q_i \unlhd \mathcal{F}(v_i)$, $\rho$ extends to a map $\overline{\rho} \in$ Hom$_{\mathcal{F}(v_i)}((P\alpha) Q_i,(P\alpha') Q_i)$ with $\overline{\rho}|_{Q_i} \in $ Aut$_{\mathcal{F}(v_i)}(Q_i)$. Set $\chi:=[\rho|_{Q_i}] \in$ Out$_{\mathcal{F}(v_i)}(Q_i)=\Delta_i$ for the class of $\rho|_{Q_i}$ in $\Delta_i$ and note that $\chi \notin K_i$, since otherwise $[\alpha]_{\mathcal{F}(e_i)}=[\alpha']_{\mathcal{F}(e_i)}$. Now $$\chi^{-1} \circ \mbox{Out}_{(P\alpha) Q_i}(Q_i) \circ \chi = \mbox{Out}_{(P\alpha') Q_i}(Q_i) \leqslant K_i$$ and $$\mbox{Out}_{(P\alpha) Q_i}(Q_i) \leqslant K_i \cap \chi \circ K_i \circ \chi^{-1},$$ which is a $p'$-group by hypothesis. Since $Q_i$ is $\mathcal{F}$-centric, we have Out$_{(P\alpha) Q_i}(Q_i) \cong (P\alpha) Q_i/Q_i$ so that $P\alpha \leqslant Q_i$. But $P$ is $\mathcal{F}$-centric and $Q_i$ is minimal amongst all $\mathcal{F}$-centric subgroups (by assumption) and so $P\alpha=Q_i$. Furthermore since $Q_i$ is not $\mathcal{F}_0$-conjugate to $Q_j$ for $i \neq j$, this must occur for some unique $i$.
Setting $\beta=\iota_P^S$ in the previous paragraph we see that either Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F}(-))$ is a tree or $P$ is $\mathcal{F}_0$-conjugate to a unique $Q_i$. Suppose we are in this latter situation and let $\varGamma_0 \subset$ Rep$_\mathcal{F}(P,\mathcal{T})$ be the connected subgraph consisting of all vertices of the form $[\beta]_{\mathcal{F}(v_0)}$ or $[\beta']_{\mathcal{F}(v_i)}$ and edges of the form $[\alpha]_{\mathcal{F}(e_i)}$. If $[\gamma]_{\mathcal{F}(v_j)}$ is a vertex not in $\varGamma_0$ then $j \neq i$ and a minimal path from $[\gamma]_{\mathcal{F}(v_j)}$ to $\varGamma_0$ must consist of a single edge $[\delta]_{\mathcal{F}(e_j)}$, otherwise by the argument in the previous paragraph, we would have that $P$ is $\mathcal{F}_0$-conjugate to $Q_j$, a contradiction (see Figure 4.1 below). Hence $\varGamma_0$ is a deformation retract of Rep$_{\mathcal{F}_\mathcal{T}}(P,\mathcal{F}(-))$ and it is enough to prove that $\varGamma_0$ is a tree. In fact we prove that $\varGamma_0$ is a star. Suppose that a vertex $[\beta]_{\mathcal{F}(v_0)} \in \varGamma_0$ is incident to two (different) edges $[\alpha]_{\mathcal{F}(e_i)}$ and $[\alpha']_{\mathcal{F}(e_i)}$. The argument in the previous paragraph shows that $\alpha, \alpha' \in$ Iso$_{\mathcal{F}(e_i)}(P,Q_i)$ and there exists $\rho \in$ Aut$_{\mathcal{F}(v_i)}(Q_i)=\Delta_i$ such that $\alpha \circ \rho = \alpha'$. Hence $[\alpha]_{\mathcal{F}(e_i)} = [\alpha']_{\mathcal{F}(e_i)}$ and each $[\alpha]_{\mathcal{F}(v_0)} \in \varGamma_0$ is incident to a unique edge. This establishes the claim, and finishes the proof of the theorem. \end{proof}
\begin{figure}\label{treefig}
\end{figure}
\end{document} | arXiv |
Reproducible quantification of cardiac sympathetic innervation using graphical modeling of carbon-11-meta-hydroxyephedrine kinetics with dynamic PET-CT imaging
Tong Wang†1, 2,
Kai Yi Wu†1,
Robert C. Miner1,
Jennifer M. Renaud1,
Rob S. B. Beanlands1 and
Robert A. deKemp1Email authorView ORCID ID profile
Received: 21 March 2018
Graphical methods of radiotracer kinetic modeling in PET are ideal for parametric imaging and data quality assurance but can suffer from noise bias. This study compared the Logan and Multilinear Analysis-1 (MA1) graphical models to the standard one-tissue-compartment (1TC) model, including correction for partial-volume effects, in dynamic PET-CT studies of myocardial sympathetic innervation in the left ventricle (LV) using [11C]HED.
Test and retest [11C]HED PET imaging (47 ± 22 days apart) was performed in 18 subjects with heart failure symptoms. Myocardial tissue volume of distribution (VT) was estimated using Logan and MA1 graphical methods and compared to the 1TC standard model values using intraclass correlation (ICC) and Bland-Altman analysis of the non-parametric reproducibility coefficient (NPC).
A modeling start-time of t* = 5 min gave the best fit for both Logan and MA1 (R2 = 0.95) methods. Logan slightly underestimated VT relative to 1TC (p = 0.002), whereas MA1 did not (p = 0.96). Both the MA1 and Logan models exhibited good-to-excellent agreement with the 1TC (MA1-1TC ICC = 0.96; Logan-1TC ICC = 0.93) with no significant differences in NPC between the two comparisons (p = 0.92). All methods exhibited good-to-excellent test-retest repeatability with no significant differences in NPC (p = 0.57).
Logan and MA1 models exhibited similar agreement and variability compared to the 1TC for modeling of [11C]HED kinetics. Using t* = 5 min and partial-volume correction produced accurate estimates of VT as an index of myocardial sympathetic innervation.
One tissue compartment
Sympathetic nervous system
Developed as a positron emission tomography (PET) imaging agent to target the cardiac sympathetic nervous system, carbon-11-labeled meta-hydroxyephedrine ([11C]HED) is a norepinephrine analog that is taken up by nerve terminal varicosities in the myocardium, and used to assess sympathetic nerve function [1]. Since its genesis, it has been the cornerstone PET tracer for cardiac sympathetic innervation, employed in determination of neuronal-based defects leading to improved diagnosis and prognosis for pathologies such as heart failure, arrhythmia, and cardiomyopathy, in which cardiac neuronal function is often compromised, leading to decreased catecholamine sensitivity and lowered beta adrenergic receptor density [1]. Using PET [11C]HED imaging of cardiac tissues, the volume of distribution (VT) of the injected radiotracer is an invaluable metric that quantifies the uptake and retention of tracer, providing an index of sympathetic nerve density and reuptake-1 transporter activity. For cardiac PET applications especially, VT and other kinetic modeling parameters measured in the myocardium may be used to aid in the diagnosis of various innervation and perfusion-based pathologies.
In PET imaging studies, VT is defined as the equilibrium ratio of tracer concentration in tissue to that of unmetabolized parent tracer in plasma, but this direct measurement is typically not feasible due to the long time needed to reach equilibrium. Alternatively, kinetic modeling is commonly used to determine VT from a significantly shorter temporal sample following tracer injection [2]. While the physiological kinetics of [11C]HED may be modeled using a two-tissue-compartment model, the one-tissue-compartment (1TC) model has been shown to provide a robust representation with optimal clinical reproducibility in myocardial uptake studies, without sacrificing the accuracy of VT quantification [3]. Two graphical methods reported in the literature for kinetic modeling of reversible-binding tracers are the Logan [4] and Multilinear Analysis-1 (MA1) models [5], which are both computationally simpler than non-graphical (compartmental) methods [6], while being able to provide visual representations of kinetic parameters. The Logan method has been established as the standard graphical model to estimate VT in a wide range of PET applications in the brain and heart, while MA1 was proposed as an alternative numerical formulation to estimate VT with lower noise bias compared to Logan estimates [5]. Although [11C]HED is a widely used tracer, a comprehensive evaluation of the performance of the graphical and non-graphical methods to quantify its kinetics has not been performed. Furthermore, the effects of partial-volume losses on quantification of VT have not been well defined in the context of graphical kinetic modeling in the heart, where the effects of blood-pool spillover and motion are more apparent compared to the brain. The goal of this study was to determine a method of partial-volume correction applicable to graphical kinetic modeling and to compare the Logan and MA1 models to the standard 1TC kinetic model for accurate quantification of myocardial sympathetic innervation using dynamic [11C]HED PET-CT studies.
Patient study design
Twenty-three heart failure patients were recruited as control subjects for a previous study (PET-OSA: NCT00756366) investigating the effects of continuous positive airway pressure (CPAP) on sympathetic nerve function and cardiac energetics in heart failure patients with obstructive sleep apnea (OSA) [7]. These control patients had the same inclusion and exclusion criteria as the PET-OSA study, except they did not have OSA. Patient demographics were collected at baseline and follow-up visits.
Three of the 23 patients were missing baseline or follow-up PET scans and were excluded. Two other patients were also excluded: one with atrial fibrillation at baseline that was treated before the follow-up scan and the other with uncorrectable severe motion artifact, leaving N = 18 subjects included in the final analysis. All patients provided written informed consent, according to the research protocol approved by the Human Research Ethics Board at the University of Ottawa Heart Institute.
[11C]HED PET imaging
[11C]HED was synthesized from [11C]methyl-iodide and metaraminol-free base, with the use of standard methods for high purity and specific activity [8]. Images were obtained at baseline and follow-up (47 ± 22 days apart) on the ECAT-ART PET (Siemens/CTI, Knoxville, TN) or Discovery RX PET-VCT (GE Healthcare, Waukesha, WI) scanner, with ECG, heart rate, and blood pressure monitored at regular intervals. A transmission scan for attenuation correction was performed using Cs-137 isotope or X-ray CT [9], immediately after which 10–15 mCi (370–550 MBq) of [11C]HED was injected over 30 s and a dynamic PET series was acquired over a 40-min period (10 × 10 s; 1 × 60 s; 5 × 100 s; 3 × 180 s; 4 × 300 s) [10]. Image reconstruction was performed using filtered-back-projection with a 12-mm Hann filter and all corrections enabled for quantification of radioactivity concentration [11].
Tracer kinetic modeling
Blood metabolites correction
Quantitative analysis of [11C]HED kinetics requires correction for radiolabeled metabolites that accumulate over time in the bloodstream, which are not present in the myocardium [12]. The arterial whole-blood tracer concentration CWB(t) is typically measured using an image-derived region of interest (ROI) placed in the LV cavity and must be differentiated from the unchanged parent tracer concentration in plasma Cp(t) as defined using the standardized nomenclature of Innis et al. [13]. The relation between Cp(t) and CWB(t) was characterized as a time-varying function of plasma-to-whole-blood and unchanged parent-to-metabolized radiotracer in the bloodstream and expressed as a combined parent fraction in plasma pfp(t) function (Additional file 1: Figure S1) derived from studies performed previously in humans [3]:
$$ {C}_p(t)={C}_{WB}(t)\times pfp(t) $$
Compartment modeling and partial volume correction
The tracer volume of distribution VT in the myocardium is defined as the ratio of the concentration of tracer in tissue divided by the concentration of tracer in arterial plasma, after the system has reached equilibrium (at t ≥ TE) [13].
$$ {V}_T=\frac{C_T\left({T}_E\right)}{C_p\left({T}_E\right)},\mathrm{when}\ \frac{d{C}_T}{dt}=0 $$
Where CT is the concentration of tracer in myocardial tissue and Cp is the concentration of tracer in plasma. For a reversible 1TC kinetic model, the rate-of-change of tracer concentration in myocardial tissue is defined using the rate of influx from arterial plasma-to-tissue (K1) and the rate of efflux from the tissue compartment (k2) according to Eq. 3.
$$ \frac{d{C}_T(t)}{dt}={K}_1{C}_p(t)-{k}_2{C}_T(t) $$
At equilibrium (t ≥ TE), the rate-of-change of tracer concentration in tissue is equal to zero (\( \frac{d{C}_T(t)}{dt}=0\Big) \) [14]. Combining (2) and (3), the volume of distribution may be expressed as:
$$ {V}_T=\frac{K_1}{k_2}=\frac{C_T\left({T}_E\right)}{C_{p\left({T}_E\right)}} $$
This simple derivation is applied widely in the analysis of neuro-PET imaging studies. However, in cardiac PET applications, additional image blurring due to cardiac contractile and respiratory motion makes it difficult to deduce the exact boundaries of myocardial tissue based on the measured ROI. There is also a 10–15% fraction of blood volume within normal myocardial tissue that must be considered. These effects may be lessened with modern PET-CT hybrid scanners with improved spatial and contrast resolution, but for cardiac imaging, these effects are still pronounced, necessitating implementation of partial-volume corrections [15, 16]. With partial-volume spillover considered, our model of the imaging process becomes:
$$ {C}_{ROI}(t)= RC\times {C}_T(t)+{F}_{WB}\times {C}_{WB}(t) $$
where CROI(t) is the measured tracer concentration in the PET myocardial image ROI. FWB is the fraction of whole-blood signal CWB(t) contained in the measured ROI curve due to imaging spillover effects and anatomical blood volume in the myocardial tissue [11]. CT(t) is the tracer concentration in the myocardial tissue (excluding blood), and RC is the partial-volume recovery coefficient describing the fractional underestimation of CT(t) due to limited spatial resolution and myocardial motion blurring. In this study, the value of RC was estimated regionally as 1 − FWB, according to the method of Hutchins et al. [15] used commonly in the compartmental analysis of cardiac PET dynamic imaging studies. Blood spillover from the right ventricle cavity to the interventricular septum was not modeled explicitly. Spillover from the myocardium to blood-pool was not corrected, which might affect CWB(t) at later time points. Isolating for CT(t) we have:
$$ {C}_T(t)=\frac{C_{ROI}(t)-{F}_{WB}\times {C}_{WB}(t)}{RC} $$
Substituting CT from (5) into the definition of \( {V}_T=\frac{C_T\left({T}_E\right)}{C_p\left({T}_E\right)} \) in (1) and assuming t ≥ TE, we obtain:
$$ \frac{\left[\frac{C_{ROI}\left({T}_E\right)-{F}_{WB}\times {C}_{WB}\left({T}_E\right)}{RC}\right]}{C_p\left({T}_E\right)}={V}_T $$
$$ \frac{\left[{C}_{ROI}\left({T}_E\right)-{F}_{WB}\times {C}_{WB}\left({T}_E\right)\right]}{C_p\left({T}_E\right)}={V}_T\times RC $$
CWB may be expressed in terms of Cp from (1), only considering t ≥ TE:
$$ \frac{C_{ROI}\left({T}_E\right)-{F}_{WB}\times \left(\frac{C_p\left({T}_E\right)}{pfp\left({T}_E\right)}\right)}{C_p\left({T}_E\right)}={V}_T\times RC $$
Maintaining the same logic as (2) and distinguishing between VT for the volume of distribution that corresponds to the true myocardial tissue compartment (CT) and VROI for the volume of distribution that corresponds to the measured region of interest (CROI), VROI may be expressed as the ratio: \( \frac{C_{ROI}\left({T}_E\right)}{C_p\left({T}_E\right)}={V}_{ROI} \). This can hence be substituted into (9); then, Cp(t) may also be canceled from the second term of (9), yielding:
$$ {V}_{ROI}-\frac{F_{WB}}{pfp\left({T}_E\right)}={V}_T\times RC $$
where pfp(TE) represents the equilibrium value of pfp(t). Then, (10) can be rearranged to isolate VT as:
$$ {V}_T=\frac{V_{ROI}-\left(\frac{F_{WB}}{pfp\left({T}_E\right)}\right)}{RC} $$
From (11), we propose that VT may be estimated from VROI, with plasma-to-whole-blood and metabolite corrections as well as partial-volume effects considered explicitly.
Graphical kinetic modeling
The Logan model (12) was derived from the first-order differential equations for general compartment models. Its purpose was to create a graphical method of quantifying VT, while making kinetic modeling more mathematically and computationally simple and robust. The base equation, adapted from the original formulation [5] to fit the reversible 1TC model in the context of cardiac PET, is:
$$ \frac{\int_0^T{C}_{ROI}(t) dt}{C_{ROI}(T)}=\left({V}_{ROI}\right)\frac{\int_0^T{C}_p(t) dt}{C_{ROI}(T)}+ Int $$
CROI(t) and Cp(t) time-activity curves are used as measured input data. At a certain time (t*), the intercept term (Int) will become a constant value [17], at which point the equation becomes a linear system where the slope represents the volume of distribution in the ROI. Since the measured tissue curve CROI(t) is subject to blood spillover and partial-volume losses, only VROI may be obtained from the graphical model directly. Previous applications of this model have estimated VROI without explicit correction factors for partial-volume effects, which is required for cardiac applications. In our proposed model, Eq. (11) may be used to determine VT from the slope determined by the Logan model.
The second graphical method investigated is the MA1 model, originally formulated as a more numerically stable alternative to the Logan model [6]:
$$ {C}_{ROI}(T)=\frac{1}{Int}{\int}_0^T{C}_{ROI}(t) dt-\frac{\left({V}_{ROI}\right)}{\mathrm{I} nt}{\int}_0^T{C}_p(t) dt $$
As with the Logan model, only T > t* are used for MA1 analysis. MA1 is a multilinear equation with two independent variables, and the corresponding Logan slope VROI is equal to the negative ratio of the two coefficients, such that:
$$ {V}_{ROI}=-\left(\frac{-\frac{\left({V}_{ROI}\right)}{Int}}{\frac{1}{Int}}\right) $$
VTmay again be determined from the VROI value estimated using MA1, according to the relation defined in (11).
Determination of t* for graphical models
The estimation start-time (t*) was varied systematically from 1.5 to 20 min for a subset of five [11C]HED studies to determine the optimal value to be used for the main analysis. Goodness-of-fit was evaluated on the Logan plot as the Pearson correlation (r2) of the points from t* to 40 min, indicating the subset of points best described by a line. Since the r2 is not effective to assess goodness-of-fit of the near-horizontal fitted plane on the MA1 plots, an alternative metric was computed using the relative standard error of the estimate (rSEE) as 1 − SEE/mean. The optimal t* was determined by comparing VT values from the graphical methods to the 1TC model standard. Then, all subsequent analysis was performed using the same start-time for both Logan and MA1 models.
PET image analysis
The compartmental and graphical analysis models were implemented in the FlowQuant® analysis software (University of Ottawa Heart Institute, ON). The operator reliability of this automated software has been reported previously [18]. Briefly, the left ventricle (LV) myocardium was segmented automatically and partitioned into voxels using a 2D polar-map representation, with each voxel representing a transmural sub-region of the LV myocardial tissue. The arterial whole-blood (WB) ROI was positioned automatically at the center of the left atrioventricular valve plane. Time-activity curves were generated based on measured tracer activity in the LV cavity CWB(t) and myocardial tissue CT(t) ROIs, as input to the tracer kinetic models.
In each polar-map voxel, the 1TC model rate constants K1 and k2, as well as VT and the blood spillover fraction FWB, were estimated using weighted least-squares regression, according to Eqs. (3), (4), and (5). The Logan and MA1 graphical models in Eqs. (12) and (13) were used to calculate LV polar-maps of VROI. Scan-specific spillover values were calculated as the polar-map median FWB and the corresponding partial-volume recovery coefficient RC (1 − FWB), which were then used to estimate VT from the graphical model estimates of VROI according to Eq. (11). Image and data analyses were performed using MATLAB 2013b (The Mathworks, Natick, MA).
LV median VT values obtained from the 1TC, Logan, and MA1 methods were tabulated. Inter-model and test-retest mean effects were evaluated using two-way repeated measures ANOVA. Bland-Altman analyses and Intra-class correlation (ICC) were employed to evaluate the inter-model (MA1 vs 1TC, and Logan vs 1TC) and test-retest (baseline vs follow-up) reliability [19, 20]. Absolute-agreement ICC with two-way mixed effects was used for the inter-model reproducibility and test-retest repeatability [21]. To correct for skew in the VT distributions, VT values were logarithmically transformed before the ANOVA and ICC analyses. ICC values were categorized as: ICC > 0.90 excellent, > 0.75 very good, > 0.40 good, and ≤ 0.40 poor [22]. The limits-of-agreement of repeated measures were estimated using the following: (i) median difference ± non-parametric repeatability coefficient (NPC = 1.45 × IQR) to account for the variable effect of outliers and (ii) mean difference ± coefficient-of-repeatability (CR = 1.96 × SD). √(3/N) × SDdifference(t95%, n − 1) was used to calculate the 95% confidence intervals on the limits-of-agreement, where N is the number of pairs being analyzed [20]. Differences in VT values were divided by the mean VT to account for the increased variability of differences associated with increased mean VT. The NPC was also reported as it is a more robust measure of repeatability [23]. Non-parametric Levene's test was used to assess the equality of variance between groups. Bias in the Bland-Altman plots was assessed using the one-sample Wilcoxon Signed Ranked test against zero. A 2-tailed p value < 0.05 was considered statistically significant for all tests. Statistical analyses were performed using Excel 2016 (Microsoft) and SPSS 20.0 (IBM).
Baseline patient demographics are listed in Table 1. Most patients in this study were male (66.5%) with mean age of 66.5 ± 9.3 years. The majority (83.3%) were classified as having NYHA Class II heart failure and were taking one or more cardiac medications. The demographics were stable at follow-up compared those reported at baseline.
Patient characteristics (N = 18)
66.5 ± 9.3
Left ventricular ejection fraction (%)
Dyslipidemia
Beta blocker
New York Heart Association (NYHA)
Ischemic cardiomyopathy
Previous PCI or CABG
Previous MI
Values are mean ± standard deviation or number (percent) of patients
Adjustment of start-time (t*) for graphical models
Table 2 shows the goodness-of-fit metrics for the five patients randomly sampled from the entire cohort used for this study. These ranged from 0.80 to 0.99 across all scans and t* values of 1.5–20 min, as summarized in Fig. 1. Corresponding VT values ranged from 9 to 21 mL/cm3 for Logan and 10–28 mL/cm3 for MA1. The Logan VT increased systematically up to approximately 5–15 min, interpreted as the start of the steady-state (linear) phase. There was no t* value with Logan VT estimates equal to the 1TC reference value (20 mL/cm3); therefore, t* = 5 min was selected as the optimal start-time based on agreement of the MA1 VT values with the reference 1TC model. This value of t* also demonstrated the highest Logan r2 value (0.96), suggesting the best fit of a line was obtained for the points starting at 5 min. MA1 plots exhibited a steady increase in goodness-of-fit up to 5 min (0.97) with relatively little improvement at later start times.
Effect of graphical modeling start-times (t*) on measured VT and goodness-of-fit values (N = 5)
t* (min)
Logan VT
Logan r2
MA1 VT
MA1 1 − rSEE
The start-time of 5 min (values shown in italics) was selected with the MA1 VT value closest to the 1TC model reference value of 20 mL/cm3 and the highest Logan r2
Adjustment of MA1 and Logan model start-time t*, showing the goodness-of-fit metrics for Logan (Pearson r2) and MA1 (1 − rSEE) as well as the trend of VT values (blue and green lines) for both models compared to the 1TC reference (red line). t* = 5 min produced the highest Logan r2 with corresponding VT = 18.2 mL/cm3, and MA1 VT = 19.8 mL/cm3 which was close to the reference 1TC value of 20 mL/cm3 (black diamond)
Figure 2 shows VT polar-maps from a single patient scan using all 3 models, including graphical representations of the Logan and MA1 plots as well as the 1TC modeling results in Fig. 3. VT polar-maps were found to show very similar spatial distributions for all three kinetic models, as expected.
Example polar-maps of VROI estimated from a single patient scan using the one-tissue-compartment (1TC), Logan, and MA1 models. The polar-maps demonstrate similar regional patterns and global median values
Model fitting results for the same patient scan shown in Fig. 2. a One-tissue-compartment (1TC) analysis of [11C]HED PET data showing time-activity curves in arterial whole-blood (red) and metabolite-corrected arterial plasma (dotted red), as well as LV myocardial ROI (dark blue) and myocardial tissue alone (cyan). Residuals (measured–modeled PET data) are shown in green. b Logan and c MA1 plots of LV myocardial uptake demonstrating that steady-state (linear response) is reached after approximately 5 min post-injection (frame #15)
Comparison of Logan and MA1 versus 1TC
At baseline, VT values were 20 ± 8 mL/cm3 for 1TC, 17 ± 8.0 mL/cm3 for Logan, and 20 ± 16 mL/cm3 for MA1, as shown in Table 3. At follow-up, VT values were 21 ± 11 mL/cm3 for 1TC, 19 ± 12 mL/cm3 for Logan, and 23 ± 16 mL/cm3 for MA1. Intra-model comparison of the VT values at baseline vs. follow-up revealed that there was no significant difference between baseline and follow-up VT values for any of the models (p = 0.379). However, on average, VT values generated by Logan were 15% lower than those generated by MA1 (p = 0.002) and 12% lower than the 1TC values (p = 0.002). VT values generated by MA1 were not significantly different from those of 1TC (p = 0.958).
[11C]HED PET VT Measurements (N = 18)
Logan*
19.7 ± 7.8
21.3 ± 11.5
*Significant differences between Logan vs 1TC (p = 0.002) and Logan vs MA1 (p = 0.002)
To evaluate reproducibility between the three models, the 1TC model was taken as the reference standard for comparison with the Logan and MA1 models. Of the 36 scans compared in the inter-model analysis (Table 4, Fig. 4), the Logan-vs-1TC and MA1-vs-1TC comparisons exhibited similar reproducibility with NPC ~ 26.5%. However, the VT values generated from the Logan model were systematically lower than those generated from the 1TC model (median bias = − 14.5% and mean bias = − 16.3%, p < 0.001), but there was no systematic difference in VT when comparing MA1-vs-1TC models, p = 0.2). There was excellent agreement between MA1-vs-1TC values (ICC = 0.955, 95% CI [0.915, 0.977]) and good-to-excellent agreement between Logan-vs-1TC (ICC = 0.928, 95% CI [0.432, 0.978]). There was no difference in reproducibility between the MA1-vs-1TC and Logan-vs-1TC NPC values (nonparametric Levene's test, p = 0.915).
Inter-model reproducibility of VT measurements (N = 36)
Models compared
ICC [95%CI]
Average Delta ± RPC
Median Delta ± NPC
Logan vs 1TC
0.928 [0.432, 0.978]
− 16.3 ± 27.8% †
− 14.5 ± 26.6%†
MA1 vs 1TC
− 2.3 ± 34.2%
†Significant bias vs zero (p < 0.001)
Inter-model reproducibility of [11C]HED PET measurements of VT using MA1 (a, b) and Logan (c, d) models. Scatter-plots (a, c) show excellent correlation of the graphical model values versus the 1TC standard with R2 > 0.95. Bland Altman plots (b, d) show the 95% limits-of-agreement (dotted lines) and confidence intervals (shaded areas) at Delta = mean ± 1.96 × SD. [Values] are Delta = median ± 1.45 × IQR, p value assessed by the Wilcoxon Signed Rank Test
Test-retest repeatability of kinetic models
All models demonstrated very good repeatability (Table 5) with consistent ICC values = 0.837–0.852. The mean test-retest differences were all < 2% without any systematic bias observed between baseline and follow-up (Fig. 5), but this could be the result of relatively small sample size with fewer points (N = 18) compared to the inter-method analysis (N = 36). There was no difference in the test-retest reproducibility (NPC) values between the three methods (non-parametric Levene's test, p = 0.57).
Test-retest repeatability of VT measurements (N = 18)
Average Delta ± RPC
Median Delta ± NPC
1.3 ± 56.2%
− 10.4 ± 68.1%
Test-retest repeatability of [11C]HED PET measurements (N = 18 scans) for 1TC (a), MA1 (b), and Logan (c) methods. 95% limits-of-agreement (solid lines) and their confidence intervals (dotted lines) are shown at Delta = mean ± 1.96 × SD. [Values] are Delta = median ± 1.45 × IQR, p value assessed by the Wilcoxon Signed Rank Test. Average RPC and NPC values in d are 64 and 48%. F-test showed no differences in the baseline-follow-up variability between the three models (p < 0.05)
In an effort to improve and expand the use of kinetic modeling in cardiac PET studies of sympathetic innervation, we sought to evaluate multiple kinetic models for the analysis of [11C]HED studies. This was achieved by comparing the inter-method differences in VT quantified by the Logan and MA1 graphical models compared to the reference 1TC model in a sample of heart failure patients and assessing the test-retest repeatability between baseline and follow-up scans. HED PET is often used to evaluate therapy or disease progression in heart failure patients; therefore, evaluation of the test-retest repeatability is most relevant in this same population, as opposed to healthy normal subjects who generally have lower sympathetic tone. The patients' heart failure symptoms and medications were stable over the test-retest interval; therefore, any impact on the repeatability data should be minimal.
The MA1 model exhibited excellent agreement with 1TC, the Logan model exhibited good-to-excellent agreement with 1TC, and all models had good-to-excellent test-retest repeatability. Logan VT values were significantly lower than MA1 and 1TC VT values, while MA1 VT values were not significantly different from those obtained using the 1TC model (Table 3). While 1TC is the reference standard kinetic model in this instance, graphical models such as the Logan and MA1 are computationally simpler alternatives that allow for linearized visualization and analysis of tracer kinetic data. Our findings support the reliable use of both graphical analysis methods in addition to the standard 1TC model for tracer kinetic analysis of VT. These findings agree with previous studies using other PET tracers that compared various graphical models, including the Logan method, finding the results to be in agreement with standard compartment models, but computationally simpler, and potentially more robust [24–27].
In the present cardiac PET study, partial volume and spillover corrections were critical to implement into the graphical modeling calculations to avoid misinterpretation. The commonly used Logan and MA1 methods (Eqs. 12 and 13) only estimate the volume of distribution in the PET image region (VROI) as opposed to the myocardial tissue of interest (VT). Compared to PET measurements in other organ systems such as the brain, in cardiac studies, the measured ROI region contains much more spillover of blood signal within and adjacent to the myocardial tissues. Our implementation of a partial-volume correction method based on estimated recovery coefficients and whole-blood spillover fractions allowed accurate measurement of myocardial VT values using Logan and MA1 graphical models on a scan-specific basis. In this validation study, FWB was estimated first using the 1TC with spillover model, and then used to calculate the corresponding RC values for consistent partial-volume and spillover correction of the graphical model VROI estimates. It is clear that independent estimates of RC and FWB are required to determine VT from VROI as shown in Eq. 11; therefore, any error in the estimation of these correction factors in practice will be propagated directly into the corresponding values of VT. In the present study, the average FWB value was 0.37 ± 0.07, which could be used to estimate RC and hence VT in similar patient population studies with minimal added variability.
We investigated the effect of varying t* on the graphical model results (Table 2), which quantified VT using the plotted values at t ≥ t*. It has been reported that t* may be deduced directly from kinetic modeling data for some tracers [5], but the method we presented used a simpler and systematic approach to determine the t* which produced the same VT values on average compared to the MA1 plots. This approach is beneficial for tracers for which it is more difficult to estimate t* directly from the study data, such as those with relatively slower kinetics [28]. It also removes the need to estimate t* for each individual scan, which may be subject to variable noise effects. We propose t* = 5 min as an effective start-time for cardiac studies employing [11C]HED as it also gave the highest quality of linear fit (r2 > 0.95) using the Logan model, in addition to MA1 estimates of VT that were equal to the 1TC reference value on average. This start time was shown with our comparison of the three models to be robust, producing results for VT with excellent goodness-of-fit to the graphical models and inter-method agreement. It is worth noting that a slightly later start time of 10–15 min may have provided Logan VT values that correspond better with 1TC and MA1 (Fig. 1), but at the cost of a lower quality fit of the linear model and wider variability due to fewer fitted points.
Interestingly, the VT values determined by Logan were significantly lower than those determined by both MA1 and 1TC, while VT values determined by MA1 did not show a significant difference to those obtained from 1TC. More precisely, Logan exhibited a greater negative bias where VT was underestimated relative to 1TC, whereas a bias was not present between MA1 and 1TC (Table 4). In a similar kinetic model comparison using [18F]FCWAY and [11C]MDL neurological tracers, Ichise et al. [6] demonstrated that the MA1 model generated higher VT estimates than Logan, and that MA1 exhibited less bias compared to Logan at multiple imaging noise levels. Our results are consistent with these findings, affirming the original report of MA1 as a method to reduce the magnitude of bias induced by noise when using the Logan model [6]. Although Logan seemed to underestimate VT in our study population, it should be realized that the median bias of − 14.5% relative to the 1TC gold standard did not greatly affect the inter-model reproducibility of the models, which exhibited good to excellent agreement despite the bias that was present.
The use of [11C]HED to examine sympathetic function in cardiac PET is becoming increasingly widespread. Recently, it has been shown to be a powerful diagnostic and prognostic tool for patients with heart failure, arrhythmias, flow-innervation mismatches, and microvascular dysfunction in both infarcted and non-infarcted tissues [1, 29–33]. This field continues to be improved and shows promise for a wider variety of applications [34]. As cardiac innervation tracers increase in prevalence, the optimization and validation of kinetic modeling techniques becomes more important; extensions of the current study may be anticipated, such as those investigating the use of a two-tissue-compartment model to quantify cardiac NET re-uptake function more specifically. Moreover, comparisons of multiple kinetic modeling options, in particular those of a graphical nature as presented here, are possible with other cardiac innervation-based tracers such as the [18F]-labeled sympathetic innervation tracers MFBG, MHPG, LMI1195, etc., for more detailed evaluation of their kinetics [35, 36].
A few limitations were present in this study. The current study is a retrospective, single-center study that examined stable heart failure patients only from the PET-OSA trial. The results may be limited by the relatively small sample size (N = 18). Larger prospective studies would be beneficial to further validate the performance of the kinetic models as proposed.
A start time of 5 min was found to provide the best fit for Logan and MA1 models. The MA1-1TC comparison demonstrated excellent agreement while Logan-1TC and test-retest comparisons demonstrated good-to-excellent agreement when quantifying VT with partial volume correction. Although Logan underestimated VT due to the recognized noise bias, Logan and MA1 both exhibited similar test-retest variability, suggesting that they may be used in addition to 1TC in the modeling of [11C]HED kinetics, with benefits of greater computational simplicity and the ability to mathematically visualize kinetic parameters for better quality assurance.
Tong Wang and Kai Yi Wu contributed equally to this work.
Networks of Centres of Excellence of Canada (NCE-15-P06-001), Ontario Research Foundation (ORF-RE07-021).
The data will not be shared because it will be used in other upcoming studies.
TW performed kinetic analysis, created figures, and wrote the manuscript. KYW performed statistical analysis, created figures and tables, and wrote the manuscript with TW. TW and RdK formulated partial volume correction. RCM processed clinical studies and assisted in creation of figures. JMR assisted in implementation of kinetic analysis tools. RSB and RdK supervised project development and analysis. All authors were involved in the editing process. All authors read and approved the final manuscript.
All research subjects provided written informed consent, as approved by the Human Research Ethics Board at the University of Ottawa Heart Institute.
Consent has been obtained from participants to publish this work.
RSB and RdK have received unrestricted university-industry grant funding from the Ontario Research Fund and Lantheus Medical Imaging.
Additional file 1: Figure S1. Unchanged parent fraction in plasma (cyan) is calculated as the product of the plasma-to-whole blood fraction (green) times the unchanged parent fraction (blue) curves, derived from the human data presented in Harms et al. [3]. (DOCX 61 kb)
National Cardiac PET Centre, University of Ottawa Heart Institute, 40 Ruskin St, Ottawa, ON, K1Y 4W7, Canada
Department of Physiology, University of Toronto, 1 Kings College Circle, Toronto, ON, M5S 1A8, Canada
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\begin{definition}[Definition:Ultrafilter on Set/Definition 4]
Let $S$ be a non-empty set.
Let $\FF$ be a non-empty set of subsets of $S$.
Then $\FF$ is an '''ultrafilter''' on $S$ {{iff}} both of the following hold:
:$\FF$ has the finite intersection property
:For all $U \subseteq S$, either $U \in \FF$ or $U^\complement \in \FF$
where $U^\complement$ is the complement of $U$ in $S$.
\end{definition} | ProofWiki |
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Transient reductions in milk fat synthesis and their association with the ruminal and metabolic profile in dairy cows fed high-starch, low-fat diets
E. C. Sandri, J. Lévesque, A. Marco, Y. Couture, R. Gervais, D. E. Rico
Journal: animal / Volume 14 / Issue 12 / December 2020
Published online by Cambridge University Press: 23 June 2020, pp. 2523-2534
Sub-acute ruminal acidosis (SARA) is sometimes observed along with reduced milk fat synthesis. Inconsistent responses may be explained by dietary fat levels. Twelve ruminally cannulated cows were used in a Latin square design investigating the timing of metabolic and milk fat changes during Induction and Recovery from SARA by altering starch levels in low-fat diets. Treatments were (1) SARA Induction, (2) Recovery and (3) Control. Sub-acute ruminal acidosis was induced by feeding a diet containing 29.4% starch, 24.0% NDF and 2.8% fatty acids (FAs), whereas the Recovery and Control diets contained 19.9% starch, 31.0% NDF and 2.6% FA. Relative to Control, DM intake (DMI) and milk yield were higher in SARA from days 14 to 21 and from days 10 to 21, respectively (P < 0.05). Milk fat content was reduced from days 3 to 14 in SARA (P < 0.05) compared with Control, while greater protein and lactose contents were observed from days 14 to 21 and 3 to 21, respectively (P < 0.05). Milk fat yield was reduced by SARA on day 3 (P < 0.05), whereas both protein and lactose yields were higher on days 14 and 21 (P < 0.05). The ruminal acetate-to-propionate ratio was lower, and the concentrations of propionate and lactate were higher in the SARA treatment compared with Control on day 21 (P < 0.05). Plasma insulin increased during SARA, whereas plasma non-esterified fatty acids and milk β-hydroxybutyrate decreased (P < 0.05). Similarly to fat yield, the yield of milk preformed FA (>16C) was lower on day 3 (P < 0.05) and tended to be lower on day 7 in SARA cows (P < 0.10), whereas yield of de novo FA (<16C) was higher on day 21 (P < 0.01) in the SARA group relative to Control. The t10- to t11-18:1 ratio increased during the SARA Induction period (P < 0.05), but the concentration of t10-18:1 remained below 0.5% of milk fat, and t10,c12 conjugated linoleic acid remained below detection levels. Odd-chain FA increased, whereas branched-chain FA was reduced during SARA Induction from days 3 to 21 (P < 0.05). Sub-acute ruminal acidosis reduced milk fat synthesis transiently. Such reduction was not associated with ruminal biohydrogenation intermediates but rather with a transient reduction in supply of preformed FA. Subsequent rescue of milk fat synthesis may be associated with higher availability of substrates due to increased DMI during SARA.
1849 – Young And Suicide Prevention Programs Through Internet And Media: Supreme
M. D'aulerio, V. Carli, M. Iosue, F. Basilico, A.M. De Marco, L. Recchia, J. Balazs, A. Germanavicius, R. Hamilton, C. Masip, N. Mschin, A. Varnik, C. Wasserman, C. Hoven, M. Sarchiapone, D. Wasserman
Journal: European Psychiatry / Volume 28 / Issue S1 / 2013
Published online by Cambridge University Press: 15 April 2020, p. 1
The researches show a rapid growth of mental disorders among adolescents and young adults that often cooccurs with risk behaviours, such as suicide, which is one of the leading cause of death among young ages 15-34. Therefore it's necessary to use some tools that can promote mental health getting to young lives such as Internet and media.
SUPREME (Suicide Prevention by Internet and Media Based Mental Health Promotion) is aimed to increasing the prevention of risk behaviours and mental health promotion through the use of mass media and Internet.
The main expected outcome is to improve mental health among European adolescents.
In each European countries a sample of 300 students (average age of 15 years) will be selected. The prevention program will be a highly interactive website that which will address topics such as raising awareness about mental health and suicide, combating stigma, and stimulate peer help. The program will use different means of referral to the intervention website: "Adolescent related" and "Professional related". A questionnaire will be administered to the pupils for require the data on lifestyles, values and attitudes, psychological well-being, familiar relationship and friendship.
Some web-sites, managed by mental health professionals, produced encouraging results about their use in prevention of risk behaviours and in increase well-being, especially in youth with low self-esteem and low life-satisfaction. With the implementation of the SUPREME project we will be able to identify best practices for promoting mental health through the Internet and the media.
EPA-0325 – Serum s100b Protein Levels in First-episode Psychosis
S. Yelmo-Cruz, A.L. Morera-Fumero, G. Díaz-Marrero, J. Suárez-Jesús, D. Paico-Rodríguez, M. Henry-Benítez, P. Abreu-González, R. Gracia-Marco
S100B is a calcium-binding protein produced by the astrocytes that has been used as a biomarker of brain inflammation. S100B has been involved in the schizophrenia pathophysiology, being considered a marker of state and prognosis.
Studying the relationship between serum S100B levels and psychopathology in first-episode psychosis (FEP).
At admission and discharge, serum S100B levels were measured in 20 never-medicated FEP in-patients and 20 healthy controls. Psychopathology was assessed with the PANSS (Positive and Negative Syndrome Scale). The total, positive, negative and general psychopathology scores were assessed. Results are presented as mean±sd. and S100B levels in pg./ml.
At admission, patients had significantly higher serum S100B concentrations than healthy subjects (39.2±6.4 vs. 33.3±0.98, p<0.02). S100B levels increased from admission to discharge (39.2±6.4 vs. 40.0±6.8, p=0.285) but they do not reach statistical significance. There were no correlations between PANSS (total, positive, negative and general) scores and S100B at admission and discharge. Individual item by item PANSS correlations with S100B elicited a positive correlation with P5 (grandiosity) (r=0.486, p=0.030) and G5 (mannerisms/posturing) (r=0.514; p=0.02) at discharge. There also was a positive trend with G7 (motor retardation) (r=0.409; p=0.073) at discharge.
FEP in-patients have significantly increased serum levels of S100B proteins, suggesting an activation of glial cells that may be associated with a neurodegenerative/inflammatory process. Apart from the study of total scale scores, the analysis of individual item is also recommended. The long-term treatment effect (one year or more) may be relevant to see their relationship to S100B levels.
EPA-0289 – Psychopathology Sex Differences in Asthmatics
M. Henry, A. Morera, A. Henry, E. Diaz-Mesa, S. Yelmo-Cruz, J. Suarez-Jesus, D. Paico-Rodriguez, G. Diaz-Marrero, R. Gracia-Marco, I. Gonzalez-Martin
Although asthma has been one of the most investigated topics in psychosomatics, studies and papers on psychopathology in asthma are fairly scarce and of diverse meaning. Furthermore, psychopathology acoording to sex in asthma is not a common research topic.
Aim This study aims at analyzing psychopathology sex differences in asthmatics.
The psychopathology profile in a sample of 84 adult asthmatics in a hospital outpatient facility, mean age 34.62 (s.d.12.78), 36 male / 48 female, is studied. The Symptom Checklist-90-R (SCL-90-R) Self-Report Questionnaire was administered.
The symptomatic profile is characterized by higher scores in women, with a main elevation in the dimensions of Somatization (1.92), Depression (1.66), Obsession-Compulsion (1.62) and Anxiety (1.44) whereas lower scores are recorded in men, with a profile dominated by Hostility (1.70), Anxiety (1.68), Interpersonal Sensitivity (1.58) and Depression (1.44). These scores mainly contribute to the psychopathology pattern according to sex.
The possible clinical implications of the observed psychopathology sex differences should be taken into account in the management of these patients.
EPA-0356 – Tuberous Esclerosis Complex and Psychiatric Comorbidity: Two Case Reports
J. Suarez-Jesus, S. Yelmo-Cruz, D. Paico-Rodriguez, N. Suarez-Benitez, G. Diaz-Marrero, M. Henry-Benitez, R. Gracia-Marco
Tuberous Sclerosis Complex (TSC) is a genetic inherited disease characterized by hamartomatous growths in several organs as brain, skin, kidneys, hearth and eyes. The estimated incidence is approximately 1:6000 live births. The diagnosis is made clinically. Seizures are present in 87% of patients. Psychiatric comorbidity has been reported.
We report the clinical course of two patients with previous diagnosis of TSC. Psychiatric symptoms start in the adulthood without seizures history and absence of Subependimal Giant Cells Tumor (SGCT). The evolution and clinical features are described.
Married 33-years-old woman with two children affected with TSC. She was diagnosed after headache presentation in 2011. Initial MRI showed periventricular glioneuronal hamartomas. In January 2013 start with self-injurious (swallowing of objects) and autistic behaviours as well as several hospital urgency room visits. In addition, the patient presented with dull mood, emotional indifference and intellectual impairment, with no response to medication.
Married 43-years-old woman with a daughter affected with TSC. Diagnosis was made in 1999 and psychotic symptoms (delusional beliefs and auditory hallucinations) started in 2011 without previous psychiatric history. The MRI in 2013 shown subependymal nodules. Treatment with risperidone was effective.
Psychiatric symptoms are very often associated to the physical findings on TSC, even in adulthood diagnoses.
Psychiatric comorbidities are well described in literature. about 10-20% adult patients with TSC present clinically significant behavioral problems as self-injuries, frequently associated with SGCT. The European Expert Panel recommended regular assessment of cognitive development and behaviour and symptomatic treatment.
12 - From Parts to a Whole? Exploring Changes in Funerary Practices at Çatalhöyük
from Part IV - Greater Awareness of an Integrated Personal Self
By Scott D. Haddow, Eline M. J. Schotsmans, Marco Milella, Marin A. Pilloud, Belinda Tibbetts, Christopher J. Knüsel
Edited by Ian Hodder, Stanford University, California
Book: Consciousness, Creativity, and Self at the Dawn of Settled Life
Published online: 22 February 2020
Print publication: 05 March 2020, pp 250-272
View extract
Death is a universal and profoundly emotive human experience with social and economic implications that extend to communities as a whole. As such, the act of disposing of the dead is typically laden with deep meaning and significance. Archaeological investigations of funerary practices are thus important sources of information on the social contexts and worldviews of ancient societies. Changes in funerary practices are often thought to reflect organisational or cosmological transformations within a society (Carr 1995; Robb 2013). The focus of this volume is the role of cognition and consciousness in the accelerated sociocultural developments of the Neolithic Period in the Near East. In the introduction to this volume, Hodder identifies three commonly cited cognitive changes that can be measured against various archaeological datasets from Çatalhöyük. The funerary remains at Çatalhöyük are an obvious source of data for validating Hodder's third measure of change: a shift from a fluid and fragmented conception of the body and of selfhood to a greater awareness of an integrated, bounded personal self.
Equivalency of the diagnostic accuracy of the PHQ-8 and PHQ-9: a systematic review and individual participant data meta-analysis – ERRATUM
Yin Wu, Brooke Levis, Kira E. Riehm, Nazanin Saadat, Alexander W. Levis, Marleine Azar, Danielle B. Rice, Jill Boruff, Pim Cuijpers, Simon Gilbody, John P.A. Ioannidis, Lorie A. Kloda, Dean McMillan, Scott B. Patten, Ian Shrier, Roy C. Ziegelstein, Dickens H. Akena, Bruce Arroll, Liat Ayalon, Hamid R. Baradaran, Murray Baron, Charles H. Bombardier, Peter Butterworth, Gregory Carter, Marcos H. Chagas, Juliana C. N. Chan, Rushina Cholera, Yeates Conwell, Janneke M. de Manvan Ginkel, Jesse R. Fann, Felix H. Fischer, Daniel Fung, Bizu Gelaye, Felicity Goodyear-Smith, Catherine G. Greeno, Brian J. Hall, Patricia A. Harrison, Martin Härter, Ulrich Hegerl, Leanne Hides, Stevan E. Hobfoll, Marie Hudson, Thomas Hyphantis, Masatoshi Inagaki, Nathalie Jetté, Mohammad E. Khamseh, Kim M. Kiely, Yunxin Kwan, Femke Lamers, Shen-Ing Liu, Manote Lotrakul, Sonia R. Loureiro, Bernd Löwe, Anthony McGuire, Sherina Mohd-Sidik, Tiago N. Munhoz, Kumiko Muramatsu, Flávia L. Osório, Vikram Patel, Brian W. Pence, Philippe Persoons, Angelo Picardi, Katrin Reuter, Alasdair G. Rooney, Iná S. Santos, Juwita Shaaban, Abbey Sidebottom, Adam Simning, Lesley Stafford, Sharon Sung, Pei Lin Lynnette Tan, Alyna Turner, Henk C. van Weert, Jennifer White, Mary A. Whooley, Kirsty Winkley, Mitsuhiko Yamada, Andrea Benedetti, Brett D. Thombs
Journal: Psychological Medicine / Volume 50 / Issue 16 / December 2020
Published online by Cambridge University Press: 19 August 2019, p. 2816
Association of a priori dietary patterns with depressive symptoms: a harmonised meta-analysis of observational studies
Mary Nicolaou, Marco Colpo, Esther Vermeulen, Liset E. M. Elstgeest, Mieke Cabout, Deborah Gibson-Smith, Anika Knuppel, Giovana Sini, Danielle A. J. M. Schoenaker, Gita D. Mishra, Anja Lok, Brenda W. J. H. Penninx, Stefania Bandinelli, Eric J. Brunner, Aiko H. Zwinderman, Ingeborg A. Brouwer, Marjolein Visser,
Journal: Psychological Medicine / Volume 50 / Issue 11 / August 2020
Published online by Cambridge University Press: 14 August 2019, pp. 1872-1883
Print publication: August 2020
Review findings on the role of dietary patterns in preventing depression are inconsistent, possibly due to variation in assessment of dietary exposure and depression. We studied the association between dietary patterns and depressive symptoms in six population-based cohorts and meta-analysed the findings using a standardised approach that defined dietary exposure, depression assessment and covariates.
Included were cross-sectional data from 23 026 participants in six cohorts: InCHIANTI (Italy), LASA, NESDA, HELIUS (the Netherlands), ALSWH (Australia) and Whitehall II (UK). Analysis of incidence was based on three cohorts with repeated measures of depressive symptoms at 5–6 years of follow-up in 10 721 participants: Whitehall II, InCHIANTI, ALSWH. Three a priori dietary patterns, Mediterranean diet score (MDS), Alternative Healthy Eating Index (AHEI-2010), and the Dietary Approaches to Stop Hypertension (DASH) diet were investigated in relation to depressive symptoms. Analyses at the cohort-level adjusted for a fixed set of confounders, meta-analysis used a random-effects model.
Cross-sectional and prospective analyses showed statistically significant inverse associations of the three dietary patterns with depressive symptoms (continuous and dichotomous). In cross-sectional analysis, the association of diet with depressive symptoms using a cut-off yielded an adjusted OR of 0.87 (95% confidence interval 0.84–0.91) for MDS, 0.93 (0.88–0.98) for AHEI-2010, and 0.94 (0.87–1.01) for DASH. Similar associations were observed prospectively: 0.88 (0.80–0.96) for MDS; 0.95 (0.84–1.06) for AHEI-2010; 0.90 (0.84–0.97) for DASH.
Population-scale observational evidence indicates that adults following a healthy dietary pattern have fewer depressive symptoms and lower risk of developing depressive symptoms.
POSITIVE GROUND STATES FOR A CLASS OF SUPERLINEAR $(p,q)$ -LAPLACIAN COUPLED SYSTEMS INVOLVING SCHRÖDINGER EQUATIONS
Elliptic equations and systems
J. C. DE ALBUQUERQUE, JOÃO MARCOS DO Ó, EDCARLOS D. SILVA
Journal: Journal of the Australian Mathematical Society / Volume 109 / Issue 2 / October 2020
Published online by Cambridge University Press: 29 July 2019, pp. 193-216
Print publication: October 2020
We study the existence of positive ground state solutions for the following class of $(p,q)$ -Laplacian coupled systems
$$\begin{eqnarray}\left\{\begin{array}{@{}lr@{}}-\unicode[STIX]{x1D6E5}_{p}u+a(x)|u|^{p-2}u=f(u)+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}(x)|u|^{\unicode[STIX]{x1D6FC}-2}u|v|^{\unicode[STIX]{x1D6FD}}, & x\in \mathbb{R}^{N},\\ -\unicode[STIX]{x1D6E5}_{q}v+b(x)|v|^{q-2}v=g(v)+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D706}(x)|v|^{\unicode[STIX]{x1D6FD}-2}v|u|^{\unicode[STIX]{x1D6FC}}, & x\in \mathbb{R}^{N},\end{array}\right.\end{eqnarray}$$
where $1<p\leq q<N$ . Here the coefficient $\unicode[STIX]{x1D706}(x)$ of the coupling term is related to the potentials by the condition $|\unicode[STIX]{x1D706}(x)|\leq \unicode[STIX]{x1D6FF}a(x)^{\unicode[STIX]{x1D6FC}/p}b(x)^{\unicode[STIX]{x1D6FD}/q}$ , where $\unicode[STIX]{x1D6FF}\in (0,1)$ and $\unicode[STIX]{x1D6FC}/p+\unicode[STIX]{x1D6FD}/q=1$ . Using a variational approach based on minimization over the Nehari manifold, we establish the existence of positive ground state solutions for a large class of nonlinear terms and potentials.
Equivalency of the diagnostic accuracy of the PHQ-8 and PHQ-9: a systematic review and individual participant data meta-analysis
Yin Wu, Brooke Levis, Kira E. Riehm, Nazanin Saadat, Alexander W. Levis, Marleine Azar, Danielle B. Rice, Jill Boruff, Pim Cuijpers, Simon Gilbody, John P.A. Ioannidis, Lorie A. Kloda, Dean McMillan, Scott B. Patten, Ian Shrier, Roy C. Ziegelstein, Dickens H. Akena, Bruce Arroll, Liat Ayalon, Hamid R. Baradaran, Murray Baron, Charles H. Bombardier, Peter Butterworth, Gregory Carter, Marcos H. Chagas, Juliana C. N. Chan, Rushina Cholera, Yeates Conwell, Janneke M. de Man-van Ginkel, Jesse R. Fann, Felix H. Fischer, Daniel Fung, Bizu Gelaye, Felicity Goodyear-Smith, Catherine G. Greeno, Brian J. Hall, Patricia A. Harrison, Martin Härter, Ulrich Hegerl, Leanne Hides, Stevan E. Hobfoll, Marie Hudson, Thomas Hyphantis, Masatoshi Inagaki, Nathalie Jetté, Mohammad E. Khamseh, Kim M. Kiely, Yunxin Kwan, Femke Lamers, Shen-Ing Liu, Manote Lotrakul, Sonia R. Loureiro, Bernd Löwe, Anthony McGuire, Sherina Mohd-Sidik, Tiago N. Munhoz, Kumiko Muramatsu, Flávia L. Osório, Vikram Patel, Brian W. Pence, Philippe Persoons, Angelo Picardi, Katrin Reuter, Alasdair G. Rooney, Iná S. Santos, Juwita Shaaban, Abbey Sidebottom, Adam Simning, Lesley Stafford, Sharon Sung, Pei Lin Lynnette Tan, Alyna Turner, Henk C. van Weert, Jennifer White, Mary A. Whooley, Kirsty Winkley, Mitsuhiko Yamada, Andrea Benedetti, Brett D. Thombs
Journal: Psychological Medicine / Volume 50 / Issue 8 / June 2020
Published online by Cambridge University Press: 12 July 2019, pp. 1368-1380
Print publication: June 2020
Item 9 of the Patient Health Questionnaire-9 (PHQ-9) queries about thoughts of death and self-harm, but not suicidality. Although it is sometimes used to assess suicide risk, most positive responses are not associated with suicidality. The PHQ-8, which omits Item 9, is thus increasingly used in research. We assessed equivalency of total score correlations and the diagnostic accuracy to detect major depression of the PHQ-8 and PHQ-9.
We conducted an individual patient data meta-analysis. We fit bivariate random-effects models to assess diagnostic accuracy.
16 742 participants (2097 major depression cases) from 54 studies were included. The correlation between PHQ-8 and PHQ-9 scores was 0.996 (95% confidence interval 0.996 to 0.996). The standard cutoff score of 10 for the PHQ-9 maximized sensitivity + specificity for the PHQ-8 among studies that used a semi-structured diagnostic interview reference standard (N = 27). At cutoff 10, the PHQ-8 was less sensitive by 0.02 (−0.06 to 0.00) and more specific by 0.01 (0.00 to 0.01) among those studies (N = 27), with similar results for studies that used other types of interviews (N = 27). For all 54 primary studies combined, across all cutoffs, the PHQ-8 was less sensitive than the PHQ-9 by 0.00 to 0.05 (0.03 at cutoff 10), and specificity was within 0.01 for all cutoffs (0.00 to 0.01).
PHQ-8 and PHQ-9 total scores were similar. Sensitivity may be minimally reduced with the PHQ-8, but specificity is similar.
A 2D scintillator-based proton detector for high repetition rate experiments
M. Huault, D. De Luis, J. I. Apiñaniz, M. De Marco, C. Salgado, N. Gordillo, C. Gutiérrez Neira, J. A. Pérez-Hernández, R. Fedosejevs, G. Gatti, L. Roso, L. Volpe
Journal: High Power Laser Science and Engineering / Volume 7 / 2019
Published online by Cambridge University Press: 02 December 2019, e60
Print publication: 2019
We present a scintillator-based detector able to measure the proton energy and the spatial distribution with a relatively simple design. It has been designed and built at the Spanish Center for Pulsed Lasers (CLPU) in Salamanca and tested in the proton accelerator at the Centro de Micro-Análisis de Materiales (CMAM) in Madrid. The detector is capable of being set in the high repetition rate (HRR) mode and reproduces the performance of the radiochromic film detector. It represents a new class of online detectors for laser–plasma physics experiments in the newly emerging high power laser laboratories working at HRR.
Scenarios of Land Use and Land Cover Change and Their Multiple Impacts on Natural Capital in Tanzania
Claudia Capitani, Arnout van Soesbergen, Kusaga Mukama, Isaac Malugu, Boniface Mbilinyi, Nurdin Chamuya, Bas Kempen, Rogers Malimbwi, Rebecca Mant, Panteleo Munishi, Marco Andrew Njana, Antonia Ortmann, Philip J. Platts, Lisen Runsten, Marieke Sassen, Philippina Sayo, Deo Shirima, Elikamu Zahabu, Neil D. Burgess, Rob Marchant
Journal: Environmental Conservation / Volume 46 / Issue 1 / March 2019
Published online by Cambridge University Press: 18 September 2018, pp. 17-24
Print publication: March 2019
Reducing emissions from deforestation and forest degradation plus the conservation of forest carbon stocks, sustainable management of forests and enhancement of forest carbon stocks in developing countries (REDD+) requires information on land-use and land-cover changes (LULCCs) and carbon emission trends from the past to the present and into the future. Here, we use the results of participatory scenario development in Tanzania to assess the potential interacting impacts on carbon stock, biodiversity and water yield of alternative scenarios where REDD+ is or is not effectively implemented by 2025, a green economy (GE) scenario and a business as usual (BAU) scenario, respectively. Under the BAU scenario, LULCCs will cause 296 million tonnes of carbon (MtC) national stock loss by 2025, reduce the extent of suitable habitats for endemic and rare species (mainly in encroached protected mountain forests) and change water yields. In the GE scenario, national stock loss decreases to 133 MtC. In this scenario, consistent LULCC impacts occur within small forest patches with high carbon density, water catchment capacity and biodiversity richness. Opportunities for maximizing carbon emission reductions nationally are largely related to sustainable woodland management, but also contain trade-offs with biodiversity conservation and changes in water availability.
Probability of major depression diagnostic classification using semi-structured versus fully structured diagnostic interviews
Brooke Levis, Andrea Benedetti, Kira E. Riehm, Nazanin Saadat, Alexander W. Levis, Marleine Azar, Danielle B. Rice, Matthew J. Chiovitti, Tatiana A. Sanchez, Pim Cuijpers, Simon Gilbody, John P. A. Ioannidis, Lorie A. Kloda, Dean McMillan, Scott B. Patten, Ian Shrier, Russell J. Steele, Roy C. Ziegelstein, Dickens H. Akena, Bruce Arroll, Liat Ayalon, Hamid R. Baradaran, Murray Baron, Anna Beraldi, Charles H. Bombardier, Peter Butterworth, Gregory Carter, Marcos H. Chagas, Juliana C. N. Chan, Rushina Cholera, Neerja Chowdhary, Kerrie Clover, Yeates Conwell, Janneke M. de Man-van Ginkel, Jaime Delgadillo, Jesse R. Fann, Felix H. Fischer, Benjamin Fischler, Daniel Fung, Bizu Gelaye, Felicity Goodyear-Smith, Catherine G. Greeno, Brian J. Hall, John Hambridge, Patricia A. Harrison, Ulrich Hegerl, Leanne Hides, Stevan E. Hobfoll, Marie Hudson, Thomas Hyphantis, Masatoshi Inagaki, Khalida Ismail, Nathalie Jetté, Mohammad E. Khamseh, Kim M. Kiely, Femke Lamers, Shen-Ing Liu, Manote Lotrakul, Sonia R. Loureiro, Bernd Löwe, Laura Marsh, Anthony McGuire, Sherina Mohd Sidik, Tiago N. Munhoz, Kumiko Muramatsu, Flávia L. Osório, Vikram Patel, Brian W. Pence, Philippe Persoons, Angelo Picardi, Alasdair G. Rooney, Iná S. Santos, Juwita Shaaban, Abbey Sidebottom, Adam Simning, Lesley Stafford, Sharon Sung, Pei Lin Lynnette Tan, Alyna Turner, Christina M. van der Feltz-Cornelis, Henk C. van Weert, Paul A. Vöhringer, Jennifer White, Mary A. Whooley, Kirsty Winkley, Mitsuhiko Yamada, Yuying Zhang, Brett D. Thombs
Journal: The British Journal of Psychiatry / Volume 212 / Issue 6 / June 2018
Published online by Cambridge University Press: 02 May 2018, pp. 377-385
Different diagnostic interviews are used as reference standards for major depression classification in research. Semi-structured interviews involve clinical judgement, whereas fully structured interviews are completely scripted. The Mini International Neuropsychiatric Interview (MINI), a brief fully structured interview, is also sometimes used. It is not known whether interview method is associated with probability of major depression classification.
To evaluate the association between interview method and odds of major depression classification, controlling for depressive symptom scores and participant characteristics.
Data collected for an individual participant data meta-analysis of Patient Health Questionnaire-9 (PHQ-9) diagnostic accuracy were analysed and binomial generalised linear mixed models were fit.
A total of 17 158 participants (2287 with major depression) from 57 primary studies were analysed. Among fully structured interviews, odds of major depression were higher for the MINI compared with the Composite International Diagnostic Interview (CIDI) (odds ratio (OR) = 2.10; 95% CI = 1.15–3.87). Compared with semi-structured interviews, fully structured interviews (MINI excluded) were non-significantly more likely to classify participants with low-level depressive symptoms (PHQ-9 scores ≤6) as having major depression (OR = 3.13; 95% CI = 0.98–10.00), similarly likely for moderate-level symptoms (PHQ-9 scores 7–15) (OR = 0.96; 95% CI = 0.56–1.66) and significantly less likely for high-level symptoms (PHQ-9 scores ≥16) (OR = 0.50; 95% CI = 0.26–0.97).
The MINI may identify more people as depressed than the CIDI, and semi-structured and fully structured interviews may not be interchangeable methods, but these results should be replicated.
Drs Jetté and Patten declare that they received a grant, outside the submitted work, from the Hotchkiss Brain Institute, which was jointly funded by the Institute and Pfizer. Pfizer was the original sponsor of the development of the PHQ-9, which is now in the public domain. Dr Chan is a steering committee member or consultant of Astra Zeneca, Bayer, Lilly, MSD and Pfizer. She has received sponsorships and honorarium for giving lectures and providing consultancy and her affiliated institution has received research grants from these companies. Dr Hegerl declares that within the past 3 years, he was an advisory board member for Lundbeck, Servier and Otsuka Pharma; a consultant for Bayer Pharma; and a speaker for Medice Arzneimittel, Novartis, and Roche Pharma, all outside the submitted work. Dr Inagaki declares that he has received grants from Novartis Pharma, lecture fees from Pfizer, Mochida, Shionogi, Sumitomo Dainippon Pharma, Daiichi-Sankyo, Meiji Seika and Takeda, and royalties from Nippon Hyoron Sha, Nanzando, Seiwa Shoten, Igaku-shoin and Technomics, all outside of the submitted work. Dr Yamada reports personal fees from Meiji Seika Pharma Co., Ltd., MSD K.K., Asahi Kasei Pharma Corporation, Seishin Shobo, Seiwa Shoten Co., Ltd., Igaku-shoin Ltd., Chugai Igakusha and Sentan Igakusha, all outside the submitted work. All other authors declare no competing interests. No funder had any role in the design and conduct of the study; collection, management, analysis and interpretation of the data; preparation, review or approval of the manuscript; and decision to submit the manuscript for publication.
Sunspot data collection of Specola Solare Ticinese in Locarno
Renzo Ramelli, Marco Cagnotti, Sergio Cortesi, Michele Bianda, Andrea Manna
Journal: Proceedings of the International Astronomical Union / Volume 13 / Issue S340 / February 2018
Published online by Cambridge University Press: 27 November 2018, pp. 129-132
Print publication: February 2018
Sunspot observations and counting are carried out at the Specola Solare Ticinese in Locarno since 1957 when it was built as an external observing station of the Zurich observatory. When in 1980 the data center responsibility was transferred from ETH Zurich to the Royal Observatory of Belgium in Brussels, the observations in Locarno continued and Specola Solare Ticinese got the role of pilot station. The data collected at Specola cover now the last 6 solar cycles.
The aim of this presentation is to discuss and give an overview about the Specola data collection, the applied counting method and the future archiving projects. The latter includes the publication of all data and drawings in digital form in collaboration with the ETH Zurich University Archives, where a parallel digitization project is ongoing for the document of the former Swiss Federal Observatory in Zurich collected since the time of Rudolph Wolf.
Folate and vitamin B12 concentrations are associated with plasma DHA and EPA fatty acids in European adolescents: the Healthy Lifestyle in Europe by Nutrition in Adolescence (HELENA) study
I. Iglesia, I. Huybrechts, M. González-Gross, T. Mouratidou, J. Santabárbara, V. Chajès, E. M. González-Gil, J. Y. Park, S. Bel-Serrat, M. Cuenca-García, M. Castillo, M. Kersting, K. Widhalm, S. De Henauw, M. Sjöström, F. Gottrand, D. Molnár, Y. Manios, A. Kafatos, M. Ferrari, P. Stehle, A. Marcos, F. J. Sánchez-Muniz, L. A. Moreno
Journal: British Journal of Nutrition / Volume 117 / Issue 1 / 14 January 2017
Print publication: 14 January 2017
This study aimed to examine the association between vitamin B6, folate and vitamin B12 biomarkers and plasma fatty acids in European adolescents. A subsample from the Healthy Lifestyle in Europe by Nutrition in Adolescence study with valid data on B-vitamins and fatty acid blood parameters, and all the other covariates used in the analyses such as BMI, Diet Quality Index, education of the mother and physical activity assessed by a questionnaire, was selected resulting in 674 cases (43 % males). B-vitamin biomarkers were measured by chromatography and immunoassay and fatty acids by enzymatic analyses. Linear mixed models elucidated the association between B-vitamins and fatty acid blood parameters (changes in fatty acid profiles according to change in 10 units of vitamin B biomarkers). DHA, EPA) and n-3 fatty acids showed positive associations with B-vitamin biomarkers, mainly with those corresponding to folate and vitamin B12. Contrarily, negative associations were found with n-6:n-3 ratio, trans-fatty acids and oleic:stearic ratio. With total homocysteine (tHcy), all the associations found with these parameters were opposite (for instance, an increase of 10 nmol/l in red blood cell folate or holotranscobalamin in females produces an increase of 15·85 µmol/l of EPA (P value <0·01), whereas an increase of 10 nmol/l of tHcy in males produces a decrease of 2·06 µmol/l of DHA (P value <0·05). Positive associations between B-vitamins and specific fatty acids might suggest underlying mechanisms between B-vitamins and CVD and it is worth the attention of public health policies.
Targets for high repetition rate laser facilities: needs, challenges and perspectives
On the Cover of HPL
Target Fabrication (2017)
I. Prencipe, J. Fuchs, S. Pascarelli, D. W. Schumacher, R. B. Stephens, N. B. Alexander, R. Briggs, M. Büscher, M. O. Cernaianu, A. Choukourov, M. De Marco, A. Erbe, J. Fassbender, G. Fiquet, P. Fitzsimmons, C. Gheorghiu, J. Hund, L. G. Huang, M. Harmand, N. J. Hartley, A. Irman, T. Kluge, Z. Konopkova, S. Kraft, D. Kraus, V. Leca, D. Margarone, J. Metzkes, K. Nagai, W. Nazarov, P. Lutoslawski, D. Papp, M. Passoni, A. Pelka, J. P. Perin, J. Schulz, M. Smid, C. Spindloe, S. Steinke, R. Torchio, C. Vass, T. Wiste, R. Zaffino, K. Zeil, T. Tschentscher, U. Schramm, T. E. Cowan
Published online by Cambridge University Press: 24 July 2017, e17
A number of laser facilities coming online all over the world promise the capability of high-power laser experiments with shot repetition rates between 1 and 10 Hz. Target availability and technical issues related to the interaction environment could become a bottleneck for the exploitation of such facilities. In this paper, we report on target needs for three different classes of experiments: dynamic compression physics, electron transport and isochoric heating, and laser-driven particle and radiation sources. We also review some of the most challenging issues in target fabrication and high repetition rate operation. Finally, we discuss current target supply strategies and future perspectives to establish a sustainable target provision infrastructure for advanced laser facilities.
Biogeophysical properties of an expansive Antarctic supraglacial stream
Michael D. SanClements, Heidi J. Smith, Christine M. Foreman, Marco Tedesco, Yu-Ping Chin, Christopher Jaros, Diane M. McKnight
Journal: Antarctic Science / Volume 29 / Issue 1 / February 2017
Published online by Cambridge University Press: 20 October 2016, pp. 33-44
Supraglacial streams are important hydrologic features in glaciated environments as they are conduits for the transport of aeolian debris, meltwater, solutes and microbial communities. We characterized the basic geomorphology, hydrology and biogeochemistry of the Cotton Glacier supraglacial stream located in the McMurdo Dry Valleys of Antarctica. The distinctive geomorphology of the stream is driven by accumulated aeolian sediment from the Transantarctic Mountains, while solar radiation and summer temperatures govern melt in the system. The hydrologic functioning of the Cotton Glacier stream is largely controlled by the formation of ice dams that lead to vastly different annual flow regimes and extreme flushing events. Stream water is chemically dilute and lacks a detectable humic signature. However, the fluorescent signature of dissolved organic matter (DOM) in the stream does demonstrate an extremely transitory red-shifted signal found only in near-stream sediment leachates and during the initial flushing of the system at the onset of flow. This suggests that episodic physical flushing drives pulses of DOM with variable quality in this stream. This is the first description of a large Antarctic supraglacial stream and our results provide evidence that the hydrology and geomorphology of supraglacial streams drive resident microbial community composition and biogeochemical cycling.
Post-common envelope PN, fundamental or irrelevant?
Orsola De Marco, T. Reichardt, R. Iaconi, T. Hillwig, G. H. Jacoby, D. Keller, R. G. Izzard, J. Nordhaus, E. G. Blackman
Journal: Proceedings of the International Astronomical Union / Volume 12 / Issue S323 / October 2016
Published online by Cambridge University Press: 08 August 2017, pp. 213-217
One in 5 PN are ejected from common envelope binary interactions but Kepler results are already showing this proportion to be larger. Their properties, such as abundances can be starkly different from those of the general population, so they should be considered separately when using PN as chemical or population probes. Unfortunately post-common envelope PN cannot be discerned using only their morphologies, but this will change once we couple our new common envelope simulations with PN formation models.
Detection of secondary eclipses of WASP-10b and Qatar-1b in the Ks band and the correlation between Ks-band temperature and stellar activity.
Patricia Cruz, David Barrado, Jorge Lillo-Box, Marcos Diaz, Mercedes López-Morales, Jayne Birkby, Jonathan J. Fortney, Simon Hodgkin
Published online by Cambridge University Press: 12 September 2017, pp. 363-370
The Calar Alto Secondary Eclipse study was a program dedicated to observe secondary eclipses in the near-IR of two known close-orbiting exoplanets around K-dwarfs: WASP-10b and Qatar-1b. Such observations reveal hints on the orbital configuration of the system and on the thermal emission of the exoplanet, which allows the study of the brightness temperature of its atmosphere. The observations were performed at the Calar Alto Observatory (Spain). We used the OMEGA2000 instrument (Ks band) at the 3.5m telescope. The data was acquired with the telescope strongly defocused. The differential light curve was corrected from systematic effects using the Principal Component Analysis (PCA) technique. The final light curve was fitted using an occultation model to find the eclipse depth and a possible phase shift by performing a MCMC analysis. The observations have revealed a secondary eclipse of WASP-10b with depth of 0.137%, and a depth of 0.196% for Qatar-1b. The observed phase offset from expected mid-eclipse was of −0.0028 for WASP-10b, and of −0.0079 for Qatar-1b. These measured offsets led to a value for |ecosω| of 0.0044 for the WASP-10b system, leading to a derived eccentricity which was too small to be of any significance. For Qatar-1b, we have derived a |ecosω| of 0.0123, however, this last result needs to be confirmed with more data. The estimated Ks-band brightness temperatures are of 1647 K and 1885 K for WASP-10b and Qatar-1b, respectively. We also found an empirical correlation between the (R′HK) activity index of planet hosts and the Ks-band brightness temperature of exoplanets, considering a small number of systems.
The TcTASV proteins are novel promising antigens to detect active Trypanosoma cruzi infection in dogs
N. FLORIDIA-YAPUR, M. MONJE RUMI, P. RAGONE, J. J. LAUTHIER, N. TOMASINI, A. ALBERTI D'AMATO, P. DIOSQUE, R. CIMINO, J. D. MARCO, P. BARROSO, D. O. SANCHEZ, J. R. NASSER, V. TEKIEL
Journal: Parasitology / Volume 143 / Issue 11 / September 2016
Published online by Cambridge University Press: 13 May 2016, pp. 1382-1389
Print publication: September 2016
In regions where Chagas disease is endemic, canine Trypanosoma cruzi infection is highly correlated with the risk of transmission of the parasite to humans. Herein we evaluated the novel TcTASV protein family (subfamilies A, B, C), differentially expressed in bloodstream trypomastigotes, for the detection of naturally infected dogs. A gene of each TcTASV subfamily was cloned and expressed. Indirect enzyme-linked immunosorbent assays (ELISA) were developed using recombinant antigens individually or mixed together. Our results showed that dogs with active T. cruzi infection differentially reacted against the TcTASV-C subfamily. The use of both TcTASV-C plus TcTASV-A proteins (Mix A+C-ELISA) enhanced the reactivity of sera from dogs with active infection, detecting 94% of the evaluated samples. These findings agree with our previous observations, where the infected animals exhibited a quick anti-TcTASV-C antibody response, coincident with the beginning of parasitaemia, in a murine model of the disease. Results obtained in the present work prove that the Mix A+C-ELISA is a specific, simple and cheap technique to be applied in endemic areas in screening studies. The Mix A+C-ELISA could help to differentially detect canine hosts with active infection and therefore with high impact in the risk of transmission to humans. | CommonCrawl |
\begin{definition}[Definition:Involution (Mapping)/Definition 2]
Let $A$ be a set.
Let $f: A \to A$ be a mapping on $A$.
$f$ is an '''involution''' {{iff}}:
:$\forall x, y \in A: \map f x = y \implies \map f y = x$
\end{definition} | ProofWiki |
\begin{document}
\title{Fault-tolerant quantum computing with color codes}
\author{Andrew J. \surname{Landahl}} \email[]{[email protected]} \affiliation{Advanced Device Technologies,
Sandia National Laboratories,
Albuquerque, NM, 87185, USA} \affiliation{Center for Quantum Information and Control,
University of New Mexico,
Albuquerque, NM, 87131, USA} \author{Jonas T. \surname{Anderson}} \email[]{[email protected]} \affiliation{Center for Quantum Information and Control,
University of New Mexico,
Albuquerque, NM, 87131, USA} \author{Patrick R. \surname{Rice}} \email[]{[email protected]} \affiliation{Center for Quantum Information and Control,
University of New Mexico,
Albuquerque, NM, 87131, USA} \affiliation{Quantum Institute,
Los Alamos National Laboratories,
Los Alamos, NM, 87545, USA}
\begin{abstract}
We present and analyze protocols for fault-tolerant quantum computing using color codes. To process these codes, no qubit movement is necessary; nearest-neighbor gates in two spatial dimensions suffices. Our focus is on the color codes defined by the 4.8.8 semiregular lattice, as they provide the best error protection per physical qubit among color codes. We present circuit-level schemes for extracting the error syndrome of these codes fault-tolerantly. We further present an integer-program-based decoding algorithm for identifying the most likely error given the (possibly faulty) syndrome. We simulated our syndrome extraction and decoding algorithms against three physically-motivated noise models using Monte Carlo methods, and used the simulations to estimate the corresponding accuracy thresholds for fault-tolerant quantum error correction. We also used a self-avoiding walk analysis to lower-bound the accuracy threshold for two of these noise models. We present two methods for fault-tolerantly computing with these codes. In the first, many of the operations are transversal and therefore spatially local if two-dimensional arrays of qubits are stacked atop each other. In the second, code deformation techniques are used so that all quantum processing is spatially local in just two dimensions. In both cases, the accuracy threshold for computation is comparable to that for error correction. Our analysis demonstrates that color codes perform slightly better than Kitaev's surface codes when circuit details are ignored. When these details are considered, we estimate that color codes achieve a threshold of 0.082(3)\%, which is higher than the threshold of $1.3 \times 10^{-5}$ achieved by concatenated coding schemes restricted to nearest-neighbor gates in two dimensions [Spedalieri and Roychowdhury, Quant.\ Inf.\ Comp.\ \textbf{9}, 666 (2009)] but lower than the threshold of $0.75\%$ to $1.1\%$ reported for the Kitaev codes subject to the same restrictions [Raussendorf and Harrington, Phys.\ Rev.\ Lett.\ \textbf{98}, 190504 (2007); Wang \textit{et~al.}, Phys. Rev. A \textbf{83}, 020302(R) (2011)]. Finally, because the behavior of our decoder's performance for two of the noise models we consider maps onto an order-disorder phase transition in the three-body random-bond Ising model in 2D and the corresponding random-plaquette gauge model in 3D, our results also answer the Nishimori conjecture for these models in the negative: the statistical-mechanical classical spin systems associated to the 4.8.8 color codes are counterintuitively more ordered at positive temperature than at zero temperature.
\end{abstract}
\maketitle
\section{Introduction}
The promise of fault-tolerant quantum computing is a crowning achievement of quantum information science \cite{Shor:1996a, Aharonov:1997a, Aharonov:1999a, Kitaev:1997b, Steane:1997a, Knill:1998a, Preskill:1998a, Preskill:1998c}. Under a specific set of noise and control assumptions, the promise is that any ideal quantum circuit of size $L$ can be simulated to any desired precision $\varepsilon$ by a faulty quantum circuit whose size is at most ${\cal O}(\varepsilon^{-1} L \log^a L)$ for some (small) constant $a$. Fault-tolerant quantum computing protocols are judged by the resources they employ in the course of a simulation. Examples of such resources include the constant $a$, the hidden constant in the big-${\cal O}$ notation, and the requirements imposed by the noise and control assumptions. Often protocols are compared by a requirement encapsulated in a single number, the \emph{accuracy threshold}, which is an upper bound on the error probability per elementary operation that a faulty circuit must satisfy for the protocol to work. A variety of fault-tolerant quantum computing protocols have been developed, with threshold estimates ranging from as low as $10^{-6}$ \cite{Stephens:2008a} to as high as $3\%$ \cite{Knill:2004a, Knill:2004b, Knill:2004c}, depending on the protocol and the noise and control assumptions.
An important control constraint relevant for several quantum computing technologies is that the only multi-qubit gates that are possible are those between nearest-neighbor qubits, where the qubits are laid out in some 2D geometry in which each qubit neighbors a constant number of other qubits. Fault-tolerant quantum computing protocols based on concatenated quantum error-correcting codes have a fractal structure that is not commensurate with such a geometry. Indeed, forcing such codes into a semiregular 2D geometry requires that one introduce a substantial number of additional qubit-movement operations that expose the protocol to more errors, thereby diminishing its accuracy threshold. The largest accuracy threshold of which we are aware for a concatenated-coding protocol in a semiregular 2D geometry is $1.3 \times 10^{-5}$ \cite{Spedalieri:2009a}; that protocol is based on the concatenated nine-qubit Bacon-Shor code \cite{Bacon:2006a} embedded in the 2D square lattice.
Cognizant of the constraints imposed by 2D geometry, Kitaev introduced a family of quantum error-correcting codes called \emph{surface codes} that require only local quantum processing, where locality is defined by a graph embedded in a surface \cite{Kitaev:1996a}. Several fault-tolerant quantum computing protocols have been developed around surface codes \cite{Dennis:2002a, Raussendorf:2007a, Fowler:2008a}, and these protocols have significantly higher accuracy thresholds than their concatenated-coding counterparts. Numerical threshold estimates for surface-code protocols range from 0.75\% to 1.1\% \cite{Raussendorf:2007a, Fowler:2008a, Wang:2011a}; an analytic proof in Ref.~\cite{Dennis:2002a} guarantees that it is no less than $1.7 \times 10^{-4}$.
Recently Bombin and Martin-Delgado proposed a new family of quantum error-correcting codes they call \emph{color codes} which are also defined to be local relative to a graph embedded in a surface \cite{Bombin:2006b}. Specifically, they are defined by face-three-colorable trivalent graphs in the following way: on each vertex of the graph lies a qubit, and for each face $f$ of the graph, one defines two ``stabilizer generators'' or ``checks,'' $X_f$ and $Z_f$. $X_f$ is the tensor product of Pauli $X$ operators on each qubit incident on face $f$, while $Z_f$ is the tensor product of Pauli $Z$ operators on each qubit incident on face $f$. The color code's codespace is defined as the simultaneous $+1$ eigenspace of each of the check operators.
A fault-tolerant quantum computing protocol based on color codes requires an infinite family of color codes of increasing size in order to be able to simulate arbitrarily large ideal quantum circuits to increasing precision. A natural source for an infinite color-code family is a uniform tiling of the plane by a trivalent face-three-colorable lattice. Such a lattice can be embedded in any orientable surface, although later we will restrict attention to embeddings in planar discs. These ``semiregular'' or ``Archimedean'' lattices are described in \emph{vertex notation} as $r.s.t$, where each vertex is locally surrounded by an $r$-gon, an $s$-gon, and a $t$-gon. The only possible trivalent face-three-colorable tilings of the plane are the $4.8.8$ lattice, the $6.6.6$ (hex) lattice, and the $4.6.12$ lattice, depicted in Fig.~\ref{fig:trivalent-convex-uniform-tilings} \cite{Wiki-Uniform-Tilings:2011a}.
\begin{figure}
\caption{The three possible face-three-colorable trivalent uniform tilings of the plane.}
\label{fig:trivalent-convex-uniform-tilings}
\end{figure}
Accuracy thresholds for fault-tolerant quantum computing have been estimated for color codes in several highly idealized noise models numerically. The values of these thresholds are summarized in Table \ref{tab:code-thresholds}, along with analogous estimates for a well-studied surface code and two recently-proposed topological subsystem codes. This table contains numerous gaps, some of which we fill in with the results of this Article---the entries containing our results are highlighted in bold. The most significant gap, which we fill, is an estimate of the accuracy threshold for noise that afflicts the individual quantum circuit elements used in a fault-tolerant color-code-based quantum computing protocol. The accuracy threshold for noise afflicting the circuit model is perhaps the most instructive of all table entries. This is because this threshold establishes the target error rate per elementary operation that a quantum technology must meet to admit fault-tolerant quantum computation using these codes. It also allows for a fair ``apples-to-apples'' comparison to the high thresholds estimated for Kitaev's surface codes in the circuit model.
\begin{table*}[ht]
\centering
\begin{tabular}{c|r@{.}l|r@{.}l|r@{.}l|r@{.}l|r@{.}l|c|l} \hline \hline
& \multicolumn{6}{c|}{Code Capacity} & \multicolumn{4}{c|}{Phenomenological} & \multicolumn{2}{c}{Circuit-based} \\ \hline
Code & \multicolumn{2}{c|}{Other} & \multicolumn{2}{c|}{MLE} &
\multicolumn{2}{c|}{Optimal} & \multicolumn{2}{c|}{MLE} &
\multicolumn{2}{c|}{Optimal} & Other & \multicolumn{1}{c}{MLE} \\ \hline
\multirow{2}{*}{4.8.8} & 8&87\,\%\footnote[1]{Reference computes threshold against DP channel, not BP channel. For non-circuit-based noise models, the decoder used does not account for correlations between bit flips and phase flips in DP channel. In these models, we reported the result for the equivalent effective BP channel of strength $\frac{2}{3}p$.}$^,$\footnote[2]{Decoder based on hypergraph matching heuristic.} \cite{Wang:2009b} & \textbf{10}&\textbf{56(1)}\,\% & 10&9(2)\,\% \cite{Katzgraber:2009a} & \textbf{3}&\textbf{05(4)\,\%} &
\multicolumn{2}{c|}{} & ``$\sim 0.1$\,\%''$^{a,b,}$\footnote[3]{{Limited numerics only weakly suggest this value.}} \cite{Wang:2009b} & \textbf{0.082(3)\,\%} \\
& 8&7\,\%\footnote[4]{Decoder based on mapping to two Kitaev codes.}
\cite{Duclos-Cianci:2011a} & \multicolumn{2}{c|}{(Our result)} &
10&925(5)\,\% \cite{Ohzeki:2009b} & \multicolumn{2}{c|}{(Our result)} &
\multicolumn{2}{c|}{} & & \multicolumn{1}{c}{(Our result)} \\ \hline
\multirow{2}{*}{6.6.6} & \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} &
10&9(2)\,\% \cite{Katzgraber:2009a} & \multicolumn{2}{c|}{} & 4&5(2)\,\% \cite{Andrist:2010a} & & \\
& \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & 10&97(1)\,\%
\cite{Ohzeki:2009b} & \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & & \\ \hline
4.6.12 & \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} &
\multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & & \\ \hline
\multirow{2}{*}{4.4.4.4 Kitaev} & \multicolumn{2}{c|}{} & 10&31(1)\,\% \cite{Wang:2003a} & 10&9187\,\% \cite{Ohzeki:2009a} & 2&93(2)\,\% \cite{Wang:2003a} & 3&3\,\% \cite{Ohno:2004a} & & 0.75\,\%$^{a}$ \cite{Raussendorf:2007a} \\
& \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & 10&939(6)\,\%
\cite{deQueiroz:2009a} & \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & & 1.1\,\%$^{a}$ \cite{Wang:2011a} \\ \hline
3.4.6.4 TSCC & 1&3\,\%$^{a,c}$ \cite{Duclos-Cianci:2011a} & \multicolumn{2}{c|}{} &
\multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & & \\ \hline
``SBT'' \cite{Suchara:2010a} & 1&3\,\%$^{a,c}$ \cite{Suchara:2010a} & \multicolumn{2}{c|}{} &
\multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & & \\ \hline \hline
\end{tabular}
\caption{Numerically-estimated accuracy thresholds for several topological quantum error-correcting codes, noise models, and decoding algorithms. The first three codes (4.8.8, 6.6.6, 4.6.12) are the color codes described in Fig.~\ref{fig:trivalent-convex-uniform-tilings} and its preceding text. The last three codes are the Kitaev surface code on the square lattice \cite{Kitaev:1996a}, a topological subsystem color code on the 3.4.6.4 lattice \cite{Bombin:2010a}, and a hypergraph-based topological subsystem code proposed by Suchara, Bravyi, and Terhal \cite{Suchara:2010a}. The details of the noise models (code capacity, phenomenological, and circuit-based) and decoders (MLE, optimal, and other) are discussed in the text; when possible, results from other references have been translated into one of these models. The notation ``$x.y_1\cdots y_k(z)\%$'' means $x.y_1\cdots y_k\% \pm (z \times 10^{-k})\%$. When such notation is not used, it means that the no error analysis was reported in the reference from which the value was drawn.}
\label{tab:code-thresholds}
\end{table*}
In this Article, we analyze the accuracy threshold of the 4.8.8 color codes for fault-tolerant quantum computation under several noise and control models. We have restricted our analysis to protocols which use the decoder that identifies the most likely error (MLE) given the error syndrome. We formulate the MLE decoder as an integer program (IP), which in general is NP-hard to solve \cite{Berlekamp:1978a}. Although the decoder is inefficient, it establishes a threshold that we expect is close the the maximum threshold possible for these codes, namely the one obtainable by an optimal decoder, which identifies the most likely logical operation given the error syndrome. For small codes, the MLE IP can be solved ``offline'' ahead of time to generate a lookup table that can be used during the course of a ``live'' fault-tolerant quantum computing protocol. Our results comprise both numerical estimates of the accuracy threshold achieved via Monte Carlo simulations and a rigorous lower bound on the accuracy threshold that we prove using combinatorial counting arguments.
The remainder of this Article is organized as follows. In Sec.~\ref{sec:noise-and-control-model}, we lay out the control model and the three noise models we consider. In Sec.~\ref{sec:fault-tolerant-error-correction}, we summarize the properties of the 4.8.8 triangular color codes we study, present two circuit schedules for extracting the error syndrome in these codes, and formulate MLE decoders for these codes as integer programs for each of the noise models that we consider. In Sec.~\ref{sec:numerical-estimate-of-the-accuracy-threshold}, we report our numerical estimates for the accuracy threshold for fault-tolerant quantum error correction of these codes for each of the noise models that we consider. In Sec.~\ref{sec:analytic-bound}, we use a self-avoiding-walk analysis to prove rigorous lower bounds for the accuracy thresholds of fault-tolerant quantum error correction of these codes against two of the noise models that we consider. In Sec.~\ref{sec:ftqc-with-color-codes}, we relate the quantum error correction accuracy threshold to the quantum computation accuracy threshold for two scenarios: one in which logical qubits are associated with 2D planes that are stacked atop one another like pancakes and the other in which logical qubits are associated with ``defects'' in a single 2D substrate. In Sec.~\ref{sec:conclusions} we conclude, summarizing and interpreting our results both in terms of the accuracy thresholds we report and in terms of their consequences for ``re-entrant behavior'' of an order-disorder phase transition in two associated classical statistical-mechanical models. We cap off our conclusions with some parting thoughts about future directions that we believe are worthy of study.
\section{Noise and control model}
\label{sec:noise-and-control-model}
The performance of a fault-tolerant quantum computing (FTQC) protocol is strongly influenced by underlying architectural assumptions, so it is important to clearly list what they are. Indeed, when those assumptions are not borne out in real quantum information technologies, an FTQC protocol may fail entirely~\cite{Levy:2009a, Levy:2011a}.
Every existing FTQC protocol makes the following architectural assumptions---assumptions which appear to be necessary:
\begin{enumerate}
\item \textbf{\textsl{Nonincreasing error rate}}. The asymptotic scaling of the error rate as a function of the circuit's size is nonincreasing. This allows the performance of fault-tolerant circuits to increase asymptotically.
\item \textbf{\textsl{Parallel operation}}. The asymptotic parallel-processing rate is larger than a constant times the asymptotic error rate. This allows error correction to keep ahead of the errors.
\item \textbf{\textsl{Reusable memory}}. The asymptotic rate at which one can erase or replace qubits is larger than a constant times the asymptotic error rate. This allows entropy to be flushed from the computer faster than it is generated by errors.
\end{enumerate}
Some FTQC protocols also make the following architectural assumptions, which generally lead to higher accuracy thresholds; we make these assumptions here:
\begin{enumerate}
\setcounter{enumi}{3}
\item \textbf{\textsl{Reliable classical computation}}. Classical computations always return the correct result.
\item \textbf{\textsl{Fast classical computation}}. Classical computations are instantaneous.
\item \textbf{\textsl{No qubit leakage}}. Qubits never ``leak'' out of the computational Hilbert space.
\item \textbf{\textsl{Uncorrelated noise}}. Each qubit and gate is afflicted by an independent noise source.
\end{enumerate}
Some additional architectural assumptions, which have a less clear impact on the accuracy threshold, are frequently made as well; we also make these assumptions:
\begin{enumerate}
\setcounter{enumi}{7}
\item \textbf{\textsl{Standard gate basis}}. The set of (faulty) quantum gates (including preparation and measurement) available consists of $|0\rangle$,
$|+\rangle$, $I$, $X$, $Z$, $T$, $S$, $\CNOT$, $M_Z$, and $M_X$. The definition of what these gates are can be found in standard textbooks, \textit{e.g.}, in Refs.~\cite{Nielsen:2000a, Preskill:1998b}.
\item \textbf{\textsl{Equal-time gates}}. Each gate, including preparations and measurements, takes the same amount of time to complete.
\item \textbf{\textsl{\label{eq:uniformly-faulty-gates}Uniformly faulty gates}}. Each $k$-qubit gate, including preparations and measurements, is as equally as faulty as every other $k$-qubit gate.
\end{enumerate}
Inspired by the limitations of 2D geometry for some quantum computing technologies, we also make the following assumptions:
\begin{enumerate}
\setcounter{enumi}{10}
\item \textbf{\textsl{2D layout}}. Qubits are laid out on a structure describable by a graph embedded in a two-dimensional surface.
\item \textbf{\textsl{Local quantum processing}}. Gates can only couple nearest-neighbor qubits in the graph describing their layout.
\end{enumerate}
Finally, we make the following three variants of a thirteenth assumption about the noise model afflicting each gate. Of all the assumptions we make, we have found that this one is most likely to vary in the literature. Commonly-studied alternatives for this assumption include stochastic adversarial noise \cite{Aliferis:2006a, Aliferis:2007a, Aliferis:2008a, Aliferis:2009a}, purely depolarizing noise \cite{Cross:2009a}, and noise that has a strong bias, such as having phase flips significantly more probable than bit flips \cite{Aliferis:2008b}.
\begin{enumerate}[{13(}$a$).]
\item \textbf{\textsl{Circuit-level noise}}.\label{item:circuit-level-noise} Each faulty single-qubit preparation and faulty coherent single-qubit gate
($|0\rangle$, $|+\rangle$, $I$, $X$, $Z$, $H$, $T$, $S$) acts ideally, followed by the bit-flip channel of strength $p$, which applies bit flips (Pauli $X$ operators) with probability $p$, followed by the phase-flip channel of strength $p$, which applies phase flips (Pauli $Z$ operators) with probability $p$. We call this channel the \emph{BP channel}. Each faulty single-qubit measurement ($M_X$, $M_Z$) acts as the BP channel of probability $p$ followed by a measurement that returns the incorrect result with probability $p$. Importantly, this noise model assumes that the state after a measurement is in an eigenstate of the observable measured, just perhaps not the eigenstate that the measurement indicates. Each $\CNOT$ gate acts ideally followed by a channel in which each of the 16 two-factor Pauli products ($II$, $IX$, $XI$, $XY$, \textit{etc.}) is applied with probability $p/16$. We call this channel the \emph{DP channel}. This model differs slightly from a frequently-studied variant in the literature in which each of the 15 nontrivial two-factor Pauli products is applied with probability $p/15$ and the identity is applied with probability $1-p$.
\item \textbf{\textsl{Phenomenological noise}}. This noise model is the same as the circuit-level noise model (13($\ref{item:circuit-level-noise}$)), except that the circuit for syndrome extraction (described later) is modeled ``phenomenologically,'' having a probability $p$ for returning the wrong syndrome bit value. In this model, the propagation of errors between data qubits and between data and ancilla qubits induced by the syndrome extraction circuit are ignored. Single-qubit and two-qubit gates on the data qubits in circuits other than those used for syndrome extraction (\textit{e.g.}, for encoded computation) are still subject to the BP and DP channels, respectively, as before.
\item \textbf{\textsl{Code capacity noise}}. This model is the same as the phenomenological noise model, except that the syndrome-bit error rate is assumed to be zero. Because there is no need to repeat syndrome measurements in this model, and because the accuracy threshold for ``defect-braided'' quantum computation is the same as that for quantum memory (as argued later), the accuracy threshold for this noise model is the same as what in quantum information theory is called the single-shot, single-letter quantum capacity for color codes subject to the BP channel.
\end{enumerate}
\section{Fault-tolerant error correction of color codes}
\label{sec:fault-tolerant-error-correction}
\subsection{Code family}
We confine our analysis of color codes to the 4.8.8 color codes; our choice is motivated by two factors. First, of the three color codes on semiregular 2D lattices, the 4.8.8 code uses the fewest qubits per code distance. Second, the 4.8.8 code is the only one of the three which can realize encoded versions of the entire ``Clifford group'' \cite{Nielsen:2000a} of quantum gates, namely the gates which conjugate Pauli operators to Pauli operators in the Heisenberg picture, in a transversal fashion \cite{Bombin:2006b}, \textit{i.e.}, by applying the same operation to every qubit in a code block or between corresponding qubits in two code blocks. In particular, the gates $X$, $Z$, $H$, $S$, and $\CNOT$ have transversal encoded implementations for these codes. When encoded gates are implemented transversally, fault-tolerant quantum computing protocols for simulating these gates are generally simpler, leading to more favorable accuracy thresholds. The Clifford group of gates is an important group of gates for stabilizer codes such as the color codes, since error correction can be carried out solely using those gates \cite{Gottesman:1999b}.
We further restrict our analysis to \emph{planar} color codes, namely those which are embedded in the disc (a sphere with one puncture). We do this because, for all quantum-computing technologies of which we are aware, arranging qubits on a flat disc is more plausible than arranging them on a more general surface like a torus. The graph constraints defining color codes require that planar color codes have a boundary shaped like a polygon having $3m$ sides for some positive integer $m$. A $3m$-sided planar color code encodes $m$ logical qubits; we restrict attention to the simplest case in which $m=1$. In other words, our focus on this paper is on \emph{triangular} color codes. Examples of three different triangular color codes are depicted in Fig.~\ref{fig:triangular-codes}.
The code distance of a triangular color code is equal to its side length, namely the number of qubits along a side of the defining triangle. To see this, notice that the logical $X$ and $Z$ operators for the logical qubit are transversal because they are encoded Clifford gates. Thus, when one multiplies a logical $X$ or $Z$ operator by all checks of the same Pauli type, except the checks incident on a specified side, one obtains an equivalent logical operator whose Pauli-weight is equal to the that side's length. The family of 4.8.8 triangular codes we study is generated according to the pattern depicted in Fig.~\ref{fig:4.8.8-triangular-codes}. Note that the smallest triangular code (for any of three triangular code families depicted in Fig.~\ref{fig:triangular-codes}) is equivalent to the well-known Steane $[\![7, 1, 3]\!]$ code \cite{Steane:1996a}; triangular codes offer a way to generate an infinite code family from the Steane code by a means other than concatenation \footnote{Similarly, the three-dimensional color codes \cite{Bombin:2007b} offer a way to generate an infinite code family from the fifteen-qubit Reed-Muller code by a means other than concatenation.}.
\begin{figure}\label{fig:triangular-codes}
\end{figure}
\begin{figure}\label{fig:4.8.8-triangular-codes}
\end{figure}
Although the colors of the faces in a color code have no intrinsic meaning for the algebraic structure of the code other than constraining the class of graphs on which color codes are defined, it is useful to use the colors as placeholders in discussions from time to time. To that end, we will refer to the colors of the faces as ``red,'' ``green,'' and ``blue.'' We will further assign a color to each edge so that an edge's color is complementary to the colors of the two faces upon which it is incident. We will call a set of vertices lying on a collection of edges of the same color connected by faces also having that color a ``colored chain;'' an example of a colored chain is depicted in Fig.~\ref{fig:colored-chain}. We will assign colors to each side of a triangular code so that the color of the side is complementary to the colors of the faces terminating on that side; for example, in Figs.~\ref{fig:triangular-codes} and \ref{fig:4.8.8-triangular-codes}, the left sides of the triangles are blue, the right sides are green, and the bottoms are red. These side colors are indicated explicitly in Fig.~\ref{fig:colored-chain}.
\begin{figure}\label{fig:colored-chain}
\end{figure}
\subsection{Syndrome extraction}
To record each error-syndrome bit, the relevant data qubits interact with one or more ancilla qubits and the ancilla qubits are then measured. Shor \cite{Shor:1996a}, Steane \cite{Steane:1998a}, and Knill \cite{Knill:2004a} have devised elaborate methods for extracting an error syndrome to minimize the impact of ancilla-qubit errors spreading to the data qubits. For topological codes, however, such elaborate schemes are not necessary; a single ancilla qubit per syndrome bit suffices. This is because, by choosing an appropriate order in which data qubits interact with the ancilla qubit, the locality properties of the code will limit propagation of errors to a constant-distance spread. Using more elaborate ancillas is possible, and in general there is a tradeoff in the resulting accuracy threshold one must examine between the reduction in error propagation complexity offered versus the additional verification procedures required. Here, we examine the simplest case, with one ancilla qubit per syndrome bit. By placing two syndrome qubits at the center of each face $f$ (one for the $X_f$ measurement and one for the $Z_f$ measurement), the syndrome extraction process can be made spatially local, in keeping with the spirit of the semiregular 2D geometry constraints we are imposing.
Because color codes are Calderbank-Shor-Steane (CSS) codes \cite{Calderbank:1996a,Steane:1996b}, syndrome bits can be separated into those which identify $Z$ errors (phase flips) and those which identify $X$ errors (bit flips). These correspond to the bits coming from measuring the $X_f$ and $Z_f$ operators respectively. The circuit for measuring an operator $X_f$ is identical to the one for measuring the operator $Z_f$, except with the basis conjugated by a Hadamard gate; examples of bit-flip and phase-flip extraction circuits for the square faces in the 4.8.8 color code are depicted in Fig.~\ref{fig:synd-extract-circuit}.
\begin{figure}
\caption{Six-step circuits for measuring $X^{\otimes 4}$ and $Z^{\otimes 4}$.}
\label{fig:synd-extract-circuit}
\end{figure}
In a full round of syndrome extraction, both $X_f$ and $Z_f$ must be measured for each face $f$. One way of scheduling this is to perform all $X_f$ measurements in parallel followed by all $Z_f$ measurements in parallel. The minimal number of steps (ignoring preparation and measurement) for parallel $X_f$ measurements is eight; an example of such a schedule is depicted in Fig.~\ref{fig:naive-schedule}. The $Z_f$ measurements can be carried out by the same schedule, but in the Hadamard-conjugated basis as depicted in Fig.~\ref{fig:synd-extract-circuit}. A complete syndrome extraction round using this schedule then takes 20 steps: 10 for the $X_f$ measurement and 10 for the $Z_f$ measurement. For this schedule, one only needs to have one, not two, syndrome qubits at the center of each face.
\begin{figure}
\caption{Simple syndrome extraction circuit schedule. A round of $X$ checks is followed by a round of $Z$ checks. The number at each vertex corresponds to the discrete time step in which the physical qubit at that vertex interacts with the syndrome qubit at the face's center via a $\CNOT$ gate. The same schedule is used for both $X$ and $Z$ checks, but with the direction of the $\CNOT$ gates reversed.}
\label{fig:naive-schedule}
\end{figure}
The circuit for a full syndrome extraction round can be optimized to use fewer time steps when both syndrome qubits in a face can be processed in parallel. An example of an ``interleaved'' schedule that uses ten steps is depicted in Fig.~\ref{fig:interleaved-schedule}.
\begin{figure}
\caption{Schedule with $X$ and $Z$ syndromes measured concurrently, in ``interleaved'' fashion. This schedule takes $8$ steps, plus an extra step for syndrome qubit preparations, plus an extra step for syndrome qubit measurements. The label $m,n$ at a vertex indicates that at time step $m$ the qubit at that vertex interacts with the $X$-syndrome qubit via a $\CNOT$ gate and at time step $n$ the qubit at that vertex interacts with the $Z$-syndrome qubit via a $\CNOT$ gate.}
\label{fig:interleaved-schedule}
\end{figure}
We calculate estimates for the accuracy threshold for both schedules, to assess the impact of compressing the schedule. Some authors who have reported improved thresholds for concatenated-coding schemes using Bacon-Shor codes attribute the improvement in large part to the simplicity of the fault-tolerant Bacon-Shor-code syndrome-extraction circuit \cite{Aliferis:2007a}. For color codes, \textit{a priori}, it is not clear that using a simpler syndrome-extraction circuit will yield an analogous improvement. This is because these circuits are not constructed using any fault-tolerant design principles---catastrophic error propagation is halted by the codes' structure, not by circuit-design principles. It may be the case, in fact, that a simpler circuit will allow errors to propagate to a larger set of qubits than a less simple one. The set of errors to which individual errors are propagated by a syndrome-extraction circuit are called ``hooks'' in Ref.~\cite{Dennis:2002a}. An example of how an error can propagate to a ``hook'' using the schedule of Fig.~\ref{fig:interleaved-schedule} is depicted in Fig.~\ref{fig:interleaved-schedule-error-propagation}.
Neither the 20-step nor the 10-step schedule is necessarily optimal in the sense of yielding the highest threshold for a fixed number of time steps; we leave that optimization to others. Indeed any schedule that satisfies two constraints is valid: (1) no qubit can be acted upon by two gates at the same time and (2) any stabilizer generator for an error-free input state (including ancilla syndrome qubits) must propagate to an element of the stabilizer group for an error-free output state. Satisfying this second criterion is not trivial; for example, an ``obvious'' schedule that acts on each face in a clockwise fashion in a manner obeying constraint (1) will not satisfy constraint (2).
\begin{figure}
\caption{A single $X$ error that occurs between time steps five and six on the syndrome qubit for measuring $X^{\otimes 8}$, indicated by the small red circle, will propagate to other $X$ errors according to the arrows. Note that an even number of $X$ flips is equivalent to no flip at all. Errors that propagate to other syndrome qubits will not propagate further because the syndrome qubits are refreshed before each syndrome extraction round. This particular error causes three data qubits to flip. These flips are correctly detected by the yellow-colored syndrome bits.}
\label{fig:interleaved-schedule-error-propagation}
\end{figure}
The number of steps in the syndrome extraction round can be reduced further to eight steps if we prepare the ancillas for the octagon measurements not in single-qubit states but in cat-states $(|0\rangle^{\otimes 8} + |1\rangle^{\otimes 8})/\sqrt{2}$ and use Shor's method of syndrome extraction \cite{Shor:1996a}. (One can also use four-qubit cat states and create an eight-step schedule, as demonstrated in Ref.~\cite{Fowler:2008c}.) Eight steps is the absolute minimum possible for syndrome extraction, since each qubit must be checked by six different syndrome bits, which must also be prepared and measured. While using cat states reduces the circuit depth, the cat states need to be verified. We opted not to study this schedule because the verification is stochastic, which would lead to a difficult synchronization problem for a large-sized code. That said, such a schedule has the potential to offer a larger accuracy threshold.
Because there is an inherent asymmetry in the order in which we choose to perform $X_f$ and $Z_f$ measurements, we will report two threshold results, one for the $X_f$ measurements and one for the $Z_f$ measurements. When we only report one value, we are reporting the lower of the two threshold values. For the phenomenological noise model, we choose to model the $X_f$ and $Z_f$ syndrome extraction processes as occurring synchronously rather than one followed by the other, since so many of the details of the circuit are washed away in the model anyway. This has the advantage of enabling the accuracy threshold in the phenomenological model to be identified with a phase transition in an associated random-bond Ising model, as described in Ref.~\cite{Katzgraber:2009a}. We will discuss this connection in more detail in Secs.~\ref{sec:code-capacity-noise-model-threshold} and \ref{sec:RBIM-conclusions}.
Finally, it is worth reminding that the entire syndrome extraction round is repeated a number of times equal to the distance of the code when measurements are allowed to be faulty, such as in the circuit-level and phenomenological noise models that we study. This ensures that errors in the syndrome bit values can be suppressed as well as errors in the data qubits can be suppressed.
\subsection{Decoding algorithm}
The process of \emph{decoding} refers to a classical algorithm for identifying a recovery operation given an error syndrome, regardless of whether the code from which the syndrome was derived is classical or quantum. Importantly, decoding does not refer to ``unencoding,'' or performing the inverse of encoding. For classical linear codes, the optimal decoding algorithm is the Most Likely Error (MLE) algorithm, which identifies the recovery operation to be the most likely pattern of bit-flip errors given the syndrome. In general, this algorithm is NP-hard \cite{Berlekamp:1978a}, but there are many families of codes for which the algorithm is known to be efficient.
For quantum stabilizer codes, MLE decoding identifies the recovery operation to be the most likely $n$-qubit Pauli-group error given the syndrome. (The process of extracting the syndrome forces every error to ``collapse'' onto a definite $n$-qubit Pauli-group operator, which is why it is sufficient to restrict to this family of operators.) MLE decoding is not necessarily optimal for quantum stabilizer codes. This is because quantum error-correcting codes can be \emph{degenerate}, meaning that two distinct correctable errors can map to the same error syndrome. Color codes are examples of highly degenerate codes. The optimal decoding algorithm for quantum stabilizer codes instead identifies the recovery operation to be one that causes the most likely logical operator to be applied after recovery. This is akin to a doctor prescribing medicine that is most likely to cure the ailment rather than prescribing medicine that cures the most likely ailment.
Once a decoding algorithm has identified a recovery operation, which is some $n$-qubit Pauli-group operator, it need not necessarily be applied. Because the process of applying the recovery operation is subject to faults, it is wiser to wait until the end of the computation and apply the net recovery operation rather than apply it after each decoding step. One can even propagate the correction past the final qubit measurements at the end of the quantum computation, where the recovery operation becomes completely classical and fault-free. The catch is that one must (classically) adaptively update one's ``Pauli frame'' after each decoding iteration by permuting the interpretation of the Pauli operators $X$, $Y$, and $Z$ on each qubit as suggested by the recovery operation. (The Pauli operators get conjugated by the Pauli error identified by the decoder.)
For fault-tolerant quantum error correction and a number of interesting encoded quantum circuits, only Clifford gates are required. Since Clifford gates propagate Pauli operators to Pauli operators in the Heisenberg picture, one can efficiently track the changing Pauli frame through these gates, as guaranteed by the Gottesman-Knill theorem \cite{Gottesman:1997a}. One can safely defer applying recovery operations until after final measurement in each of these circuits. However, for universal quantum computation, at least one non-Clifford gate is required. In our protocols, the only such gate we use is the classically-controlled $S^\dagger$ gate, depicted later in the circuit of Fig.~\ref{fig:T-circuit}. Because this gate propagates a Pauli error to a sum of Pauli errors, it is necessary to actually apply the recovery operation before all but a constant number of these gates in order to prevent the number of terms required to track one's ``Heisenberg frame'' from growing exponentially.
We develop MLE decoders for triangular 4.8.8 color codes for the three noise model settings we study: code capacity, phenomenological, and circuit-based. For the code-capacity and phenomenological settings, the only operations are single-qubit measurements and identity gates. This means that they involve no circuitry that could map $X$ errors to $Z$ errors or vice-versa. Because of this, and because our noise model is one in which single-qubit operations are subject to BP channel noise (which applies $X$ errors and $Z$ errors independently), decoding can factor into bit-flip decoding and phase-flip decoding separately. Because color codes are also ``strong'' CSS codes \cite{Preskill:1998b}, the MLE decoders for bit-flip and phase-flip errors are in fact identical; for concreteness, we formulate the decoder for $Z_f$ syndrome bits here.
\subsubsection{Code capacity MLE decoder}
In the code-capacity setting, we have a single error-free $m$-bit syndrome $\mathbf{s} = (s_1, \ldots, s_m)^T$ where $s_f = 0$ when $Z_f$ is measured to have eigenvalue $+1$ and $s_f = 1$ when $Z_f$ is measured to have eigenvalue $-1$. (The value of $m$ is a function of the code size; for the triangular $n$-qubit distance-$d$ 4.8.8 color code, $m = (d+1)^2/4-1$ and $n= (d+1)^2/2 - 1$.) We assign a binary variable $x_v$ to each vertex $v$ indicating whether or not the recovery operation calls for the qubit at vertex $v$ to be bit-flipped (have Pauli $X$ applied). The objective of MLE decoding is to minimize the number of $x_v$ variables that are assigned the value $1$ subject to the constraint that the parity of the $x_v$ variables on each face is consistent with the observed syndrome. This can be expressed as the following mathematical optimization problem:
\begin{align}
\label{eq:2D_optimization_problem}
\text{min}&\ \sum_v x_v \\
\text{sto}&\ \bigoplus_{v \in f} x_v = s_f \qquad \forall f \\
x_v &\in {\mathds{B}} := \{0, 1\}.
\end{align}
This optimization problem can be expressed as a linear binary integer program (IP) over the finite field $GF(2)$ as follows:
\begin{align}
\label{eq:2D_GF2_LP}
\text{min}&\ \mathbf{1}^T \mathbf{x} \\
\label{eq:2D_GF2_LP_constraint}
\text{sto}&\ H\mathbf{x} = \mathbf{s} \bmod 2 \\
\mathbf{x} &\in {\mathds{B}}^n,
\end{align}
where $\mathbf{1}$ denotes the all-ones vector and $H$ is the parity check matrix associated with the $Z_f$-checks. (For color codes, this is the face-vertex incidence matrix.)
To take advantage of well-developed numerical optimization software, it is helpful to replace the linear algebra over $GF(2)$ in this mathematical program with linear algebra over ${\mathds{R}}$. One way to do this is to introduce ``slack variables'' into the optimization problem. Because each check operator in the code has Pauli weight four or Pauli weight eight, each row of $H$ has Hamming weight four or Hamming weight eight. This means that the $f$th component of the vector on the left hand side of constraint (\ref{eq:2D_GF2_LP_constraint}) is a sum of four or eight binary $x_v$ variables that must equal $s_f$ modulo 2. The modulo 2 restriction can be dropped by replacing $\mathbf{s}$ by $\mathbf{s} + 2\mathbf{z}_1 + 4\mathbf{z}_2 + 8\mathbf{z}_3$ in the constraint, where the $\mathbf{z}_i$ are binary ``slack variable'' vectors that allow the LHS to sum to any integer from $0\ldots 15$. While there can be many degenerate solutions to this revised optimization problem having different $\mathbf{z}_i$ values, any solution generates the same optimal $\mathbf{x}$ as before. By combining the $\mathbf{z}_i$ variables and the $\mathbf{x}$ variables into a single vector $\mathbf{y} = (\mathbf{x}^T, \mathbf{z}_1^T, \mathbf{z}_2^T, \mathbf{z}_3^T)^T$, the slack-variable version of the program becomes the following linear binary integer program in which the variables are restricted to be binary but in which the linear algebra is over ${\mathds{R}}$:
\begin{align}
\text{min}&\ \mathbf{c}^T \mathbf{y} \\
\text{sto}&\ A\mathbf{y} = \mathbf{s} \\
\mathbf{y} &\in {\mathds{B}}^n, \end{align}
where $c$ is a vector containing $n$ ones followed by $3m$ zeros and $A$ is the matrix generated by adjoining matrices to $H$ as
\begin{align}
\label{eq:A-matrix1}
A := \begin{pmatrix}
H\, |\, {-}2I\, |\, {-}4I\, |\, {-}8I
\end{pmatrix},
\end{align}
in which each $I$ denotes the $m \times m$ identity matrix.
There are a number of symmetries that color codes possess which allow one to significantly reduce the complexity of this binary IP. For example, if $\mathbf{y}$ satisfies the constraints of the IP, then so does $\mathbf{y}$ with any number of faces complemented. Since complementing the face of any optimal solution will not reduce its weight, we know that each face's sum will never be more than half the weight of that face. This means that for any particular instance of the IP specified by the syndrome vector $\mathbf{s}$, the sums for the octagon and square faces can only take the syndrome-dependent values listed in Table~\ref{tab:2D-IP}, thereby reducing the number of slack variables required. We take advantage of these kind of symmetries in the software we developed code for estimating the code capacity of 4.8.8 triangular color codes. For example, we never need to use three slack variables and some times we need none at all.
\begin{table}[ht]
\centering
\begin{tabular}{c|c|c} \hline \hline
& Octagon & Square \\ [0.5ex] \hline
$s=0$ & 0, 2, 4 & 0, 2 \\
$s=1$ & 1, 3 & 1 \\ [1ex] \hline
\end{tabular}
\caption{Possible values octagonal and square face check sums can take for an optimal IP solution if the face check sum's parity $s$ is fixed.}
\label{tab:2D-IP}
\end{table}
Maximum likelihood decoding is generally an NP-hard problem, and the color codes do not appear to fall into an ``easy'' subset of instances. This is unfortunate because their close cousins, the surface codes, do have efficient MLE decoders that can be solved as a minimum-weight perfect matching problem \cite{Dennis:2002a}. Nevertheless, we can solve the associated IP for reasonably small instance sizes.
\subsubsection{Phenomenological noise MLE decoder}
In the phenomenological noise model, the syndrome values themselves can be faulty so we repeat the syndrome extraction process a number of times equal to the distance of the code. In this setting, it is the \emph{difference} in syndrome bit values from one time step to the next rather than the absolute values at particular times step that indicate data errors. This is because a single data error at one time step will lead to flipped syndrome bits for all future time steps (assuming that the syndrome extraction is not faulty), and such a syndrome-bit history should not imply that data errors occurred at each time step---it should imply that a data error occurred only at the time step when the syndrome bit first changed its value. The difference in persistence between data and syndrome errors is depicted in Fig.~\ref{fig:3D-errors}. The input to a MLE decoder is therefore the collection of syndrome \emph{difference} vectors for all time steps, namely
\begin{align}
\Delta \mathbf{s}_t = \mathbf{s}_t - \mathbf{s}_{t-1} = (\mathbf{s}_t + \mathbf{s}_{t-1}) \bmod 2 \qquad \forall t,
\end{align}
where $\mathbf{s}_0 := \mathbf{0}$.
\begin{figure}
\caption{If syndrome qubits are also allowed to be in error, we repeat syndrome measurements. Time advances from bottom to top. Yellow circles indicate syndrome bits with the value 1. Solid yellow circles indicate bit-flip errors.}
\label{fig:3D-errors}
\end{figure}
For a distance $d$ color code, the optimization problem to solve is again to minimize the number of errors given the observed syndrome, except we now have $d$ time steps' worth of data-error vectors, $\mathbf{x}_1, \ldots, \mathbf{x}_d$, and $d$ time steps' worth of syndrome-error vectors, $\mathbf{r}_1, \ldots, \mathbf{r}_d$, as variables in the optimization problem. Mathematically, we can write the optimization problem as
\begin{gather}
\text{min}\ \sum_t \mathbf{1}^T \mathbf{x}_t \\
\label{eq:3D-constraints}
\text{sto}\ (H\mathbf{x}_t + \mathbf{r}_t + \mathbf{r}_{t-1}) \bmod 2 = \Delta \mathbf{s}_t \bmod 2 \quad \forall t \\
\mathbf{x} \in {\mathds{B}}^n.
\end{gather}
As we did for the code-capacity scenario, we can collect these constraints into a single constraint and add slack variables to make the problem a linear binary IP over the reals. Because the left-hand side of the constraints in Eq.~(\ref{eq:3D-constraints}) can sum to up to ten for octagon constraints and up to six for square constraints, three slack variables again suffice, allowing us to formulate the optimization problem as
\begin{align}
\text{min}&\ \mathbf{c}^T \mathbf{y} \\
\text{sto}&\ A\mathbf{y} = \Delta \mathbf{s} \\
\mathbf{y} &\in {\mathds{B}}^n, \end{align}
where $\mathbf{c}$ is a vector containing $(n+m)d$ ones followed by $3md$ zeros, $\Delta \mathbf{s}$ is the vector $(\Delta \mathbf{s}_1^T, \ldots, \Delta \mathbf{s}_d^T)^T$, $\mathbf{y}$ is the vector $(\mathbf{x}_1^T, \ldots, \mathbf{x}_d^T, \mathbf{r}_1^T, \ldots, \mathbf{r}_d^T, \mathbf{z}_1^T, \mathbf{z}_2^T, \mathbf{z}_3^T)^T$ and $A$ is the matrix
\begin{align}
\label{eq:A-matrix2}
A &= \left(\!\!\!\begin{array}{c|c|c|c|c}
\begin{matrix}
H & & & \\
& H & & \\
& & \ddots & \\
& & & H
\end{matrix}
&
\begin{matrix}
I & & & & \\
I & I & & & \\
& & \ddots & & \\
& & & I & I
\end{matrix}
&
-2I & -4I & -8I
\end{array}\!\!\! \right).
\end{align}
Finally, as we did for the code capacity setting, we can use symmetries to reduce the complexity of solving this IP; Table~\ref{tab:3D-IP} summarizes what the possible values are for the square-faced and octagonal-faced constraints.
\begin{table}[ht]
\centering
\begin{tabular}{c|c|c} \hline \hline
& Octagon & Square \\ [0.5ex] \hline
$s=0$ & 0, 2, 4, 6 & 0, 2, 4 \\
$s=1$ & 1, 3, 5 & 1, 3 \\ [1ex] \hline
\end{tabular}
\caption{Possible values octagonal and square face check sums can take for an optimal IP solution if the face check sum's parity $s$ is fixed.}
\label{tab:3D-IP}
\end{table}
\subsubsection{Circuit-level decoder}
In the circuit-level noise model, each component of the syndrome extraction circuit can fail with a probability that is a function of a parameter $p$, so that the overall probability of a syndrome bit being in error, $p_s$, is a complicated function of $p$. Even more dauntingly, the circuits can induce correlated errors between syndrome bits and between syndrome bits and data qubits. The phenomenological-noise model does not capture these noise correlations.
We developed an MLE decoder for the circuit-level noise model that accounts for both these induced error correlations and the fact that in this noise model, single-qubit operations are subject to BP-channel noise while $\CNOT$ gates are subject to DP-channel noise. However, this decoder uses exponentially many more constraints than the phenomenological decoder as a function of code size. Because the IP decoder is already NP-hard, we opted not to study this truly MLE decoder but rather use the phenomenological-noise MLE decoder, which ignores these subtleties. Taking correlations into account will likely boost the accuracy threshold, but probably not by large factors \cite{Fowler:2011a}. By way of comparison, the threshold for the square-lattice surface code in the circuit-level noise model is $0.68\%$ when the phenomenological decoder is used \cite{Harrington:2008a} ($0.75\%$ \cite{Raussendorf:2007a} when using a non-MLE decoder that takes into account some entropic effects), a threshold value that has recently been boosted to $1.1\%$ \cite{Wang:2011a} by accounting for some of the correlations in the noise. We leave the refinement of true MLE decoding of this noise model to others.
\section{Numerical estimate of the accuracy threshold for fault-tolerant quantum error correction}
\label{sec:numerical-estimate-of-the-accuracy-threshold}
\subsection{Code capacity noise model\label{sec:code-capacity-noise-model-threshold}}
Because the $[\![n, 1, d]\!]$ triangular 4.8.8 color codes are CSS codes, when they are subject to BP-channel noise of strength $p$, their code capacity is the same as their bit-flip or phase-flip capacity; we focus on the bit-flip capacity here for definiteness. The number of distinct bit-flip syndromes is $2^{(n-1)/2}$ and the number of distinct bit-flip errors is $2^n$. For small $n$, one can pre-solve the MLE decoding IP for each of the $2^{(n-1)/2}$ distinct bit-flip syndromes. One can then iterate through each of the $2^n$ distinct error patterns, compute its syndrome, and determine whether the combination of the error pattern plus the inferred correction by the IP leads to a logical operator, indicating failure of the decoding algorithm. Since error-correction is assumed to be error-free in this noise model, the corrected state is guaranteed to be in the codespace. Because ($a$) the logical bit-flip operator is transversal, ($b$) all stabilizer group elements have even weight, and ($c$) there are an odd number of qubits in every triangular code, it follows that one can identify a decoding failure quickly by computing whether the parity of the error pattern equals the parity of its IP-inferred correction; this means that is suffices to just store the parity of the inferred correction for each pre-computed IP instance. The probability of failure, $p_{\text{fail}}$ is therefore
\begin{align}
p_{\text{fail}}
&= \sum_{\text{failing patterns $E$}} p^{|E|}(1-p)^{n - |E|},
\end{align}
where $|E|$ denotes the Hamming weight of the bit-flip error pattern $E$.
We carried out this tabulation for the smallest triangular 4.8.8 color codes of distances 1, 3, 5, and 7 (corresponding to 1, 7, 17, and 31 qubits respectively) and computed the corresponding exact polynomials. To speed up the computation, we used several symmetries. For example, it suffices to examine only half of the error patterns because if the decoding algorithm succeeds on an error pattern, it fails on its complement and vice versa. Also, up to overall complementation, every error pattern can be uniquely expressed as the modulo-2 sum of an IP-inferred minimal-weight error pattern and a pattern where a bit-flip stabilizer group element has support. Finally, the decoding algorithm is guaranteed to work on all errors whose weight is less than the code's distance, so those error patterns do not need to be examined.
The formulas we obtained for the smallest codes of distance $1$, $3$, and $5$ (code sizes 1, 7, and 17) are:
\begin{align}
p_{\text{fail}}^{(1)} &= p \\
p_{\text{fail}}^{(3)} &= p^7 + 7p^6(1-p) + 28p^4(1-p)^3 \\
&\phantom{= }+ 7p^3(1-p)^4 + 21p^2(1-p)^5 \\
p_{\text{fail}}^{(5)} &=
p^{17} +
17p^{16}(1-p) +
136p^{15}(1-p)^2 \\
&\phantom{= }+
348p^{14}(1-p)^3 +
725p^{13}(1-p)^4 \\
&\phantom{= }+
3861p^{12}(1-p)^5 +
4764p^{11}(1-p)^6 \\
&\phantom{= }+
12136p^{10}(1-p)^7 +
9747p^9(1-p)^8 \\
&\phantom{= }+
14563p^8(1-p)^9 +
7312p^7(1-p)^{10} \\
&\phantom{= }+
7612p^6(1-p)^{11} +
2327p^5(1-p)^{12} \\
&\phantom{= }+
1655p^4(1-p)^{13} +
332p^3(1-p)^{14}.
\end{align}
The formula we obtained for the distance-7 triangular 4.8.8 color code (31 qubits) is a bit more hefty:
\begin{widetext}
\begin{align}
p_{\text{fail}}^{(7)} &=
p^{31} +
31 p^{30} (1-p) +
465 p^{29} (1-p)^2 +
4495 p^{28} (1-p)^3 +
25658 p^{27} (1-p)^4 +
96790 p^{26} (1-p)^5 \\ \nonumber
&\phantom{= }+
344858 p^{25} (1-p)^6 +
1288630 p^{24} (1-p)^7 +
3742943 p^{23} (1-p)^8 +
10488241 p^{22} (1-p)^9 \\ \nonumber
&\phantom{= }+
21436239 p^{21} (1-p)^{10} +
44259329 p^{20} (1-p)^{11} +
67781868 p^{19} (1-p)^{12} +
106951476p^{18} (1-p)^{13} \\ \nonumber
&\phantom{= }+
127137964p^{17} (1-p)^{14} +
155845748p^{16} (1-p)^{15} +
144694447p^{15} (1-p)^{16} +
138044561p^{14} (1-p)^{17} \\ \nonumber
&\phantom{= }+
99301599 p^{13} (1-p)^{18} +
73338657 p^{12} (1-p)^{19} +
40412986 p^{11} (1-p)^{20} +
22915926 p^{10} (1-p)^{21} \\ \nonumber
&\phantom{= }+
9671834 p^9 (1-p)^{22} +
4145782 p^8 (1-p)^{23} +
1340945 p^7 (1-p)^{24} +
391423 p^6 (1-p)^{25} +
73121 p^5 (1-p)^{26} \\ \nonumber
&\phantom{= }+
5807 p^4 (1-p)^{27}.
\end{align}
\end{widetext}
Our computing resources did not allow us to compute the exact polynomial for the next-sized code (distance 9 code on 49 qubits), so we resorted to a Monte Carlo estimate for $p_{\text{fail}}(p)$. We did this by first selecting three values of $p$ near where we believed the threshold to be. For each $p$, we generated $N$ trial error patterns drawn from the Bernoulli distribution, namely in which we applied a bit-flip on each of the $n$ qubits with probability $p$. We then inferred the syndrome for each error pattern and checked whether or not it led to a decoding failure for the MLE decoder. The optimal unbiased estimator for $p_{\text{fail}}$ that we used is
\begin{align}
\label{eq:p-fail-estimator-mean}
p_{\text{fail}}^{(\text{est})} = \frac{N_{\text{fail}}}{N}
\end{align}
with a variance of
\begin{align}
\label{eq:p-fail-estimator-variance}
({\sigma^2_{\text{fail}}})^{(\text{est})} = \frac{p_{\text{fail}}^{(\text{est})}\left(1-p_{\text{fail}}^{(\text{est})}\right)}{N}.
\end{align}
To get reasonably small error bars in these estimates, given where we believed the threshold to be, we chose $N = 10^5$. The polynomials for $p_{\text{fail}}(p)$ are plotted in Fig.~\ref{fig:code-capacity}, including our three points of Monte Carlo data. From these plots, we estimate the accuracy threshold for this noise model to be $10.56(1)\%$. The error we report in this value comes from the error analysis method we describe in detail in the next section.
\begin{figure}
\caption{Code capacity for the 4.8.8 triangular color codes. $p_{th} = 10.56(1) \%$. Error bars on Monte Carlo data reflect $10^5$ instances studied at each of the three corresponding values of $p$. The inset figures are zoom-ins near the crossing point to show greater resolution there.}
\label{fig:code-capacity}
\end{figure}
To put our result in context, we reference Table~\ref{tab:code-thresholds}. The threshold value of 10.56(1)\% we find is is slightly higher than the corresponding MLE threshold for the code capacity 10.31(1)\% of 4.4.4.4 surface codes. Intuitively this makes sense, as the 4.8.8 color code has both weight-8 and weight-4 stabilizer generators, both of which are modeled as being measured instantaneously and ideally. Being able to measure high-weight generators quickly should improve the performance of a code, which is the effect we observe.
Our threshold is also less than the threshold value of 10.925(5)\% for optimal decoding, which is also not surprising. As with the 4.4.4.4 surface codes, the reduction in threshold is not very significant. For both the surface codes and the 4.8.8 color codes, the accuracy threshold in the code capacity noise model corresponds to a phase transition in a random-bond Ising model (RBIM) of classical spins \cite{Dennis:2002a, Katzgraber:2009a}. For the color codes, the Ising model features 3-body interactions, whereas for the surface codes, the Ising model features 2-body interactions. The MLE decoder in both settings corresponds to the order-disorder transition in the spin model at zero temperature, whereas the optimal decoder corresponds to the order-disorder transition at the temperature along the so-called ``Nishimori line,'' where the randomness in the bond couplings equals the randomness in the state arising from finite temperature fluctuations. In both the surface-code and color-code settings, the small decrease in accuracy threshold when going from optimal to MLE decoding reflects that the phase-boundary in these models is re-entrant, but only by a small amount. Our results therefore imply a violation of the so-called Nishimori conjecture \cite{Nishimori:1981a, Nishimori:1986a}, which conjectures that the spin model shouldn't become more ordered as the temperature increases. The violation that our results imply is depicted in cartoon fashion in Fig.~\ref{fig:Nishimori}. To our knowledge, the violation of the Nishimori conjecture for the 3-body RBIM is unknown before our work. We expand more on this connection in Sec.~\ref{sec:RBIM-conclusions}.
\begin{figure}
\caption{ Phase diagram for 3-body random-bond Ising model. The dark circle is called the Nishimori point. The dotted line is the expected phase boundary given by the Nishimori conjecture. Our value of code capacity ($10.56(1)\%$) establishes that the $T=0$ intercept is $P_{c,0}$, while results of Ohzeki \cite{Ohzeki:2009b} ($10.925(5)\%$) establish that the Nishimori point occurs at $P_{c}$. Because $P{c} \neq P_{c,0}$, the Nishimori conjecture for this model is false.}
\label{fig:Nishimori}
\end{figure}
\subsection{Phenomenological noise model\label{sec:phenomenological-noise-model-threshold}}
In the phenomenological noise model, our fault-tolerant quantum error correction protocol repeats syndrome extraction multiple times to increase the reliability of the syndrome bits. This causes the number of possible error patterns for a given code size to grow so rapidly that obtaining exact curves for $p_{\text{fail}}(p)$ even for small code sizes is intractable. We therefore resorted to Monte Carlo estimates for these curves for even the smallest code sizes. The specific Monte Carlo algorithm we used for computing $p_{\text{fail}}$ at a fixed value of $p$ is listed in Algorithm \ref{alg:MC-estimator}.
In words, Algorithm \ref{alg:MC-estimator} creates an estimator for $p_{\text{fail}}$ by assessing the performance of many simulated trials of faulty quantum error correction. In each trial, errors are laid down, giving rise to an observed syndrome history. From the syndrome history, a correction is inferred. The actual error history and the inferred error history are XORed onto a single effective time slice, but the state in this effective time slice is not necessarily in the codespace. To achieve this, a fictional ideal (error-free) round of error correction is simulated. If this succeeds (\textit{i.e.}, if it does not generate a logical bit-flip operation), then the trial is deemed a success; otherwise it is deemed a failure. By repeating many trials, one obtains an optimal unbiased estimator for the failure probability $p_{\text{fail}}$, with mean and variance given by Eqs.~(\ref{eq:p-fail-estimator-mean}--\ref{eq:p-fail-estimator-variance}), identical to the formulas relevant in the code capacity noise model setting.
\renewcommand{\algorithmiccomment}[1]{// \textsl{#1}}
\algsetup{indent=1em} \begin{algorithm}[h!]
\caption{: $p_{\text{fail}}(p)$ by Monte Carlo\label{alg:MC-estimator}}
\begin{algorithmic}[1]
\STATE $n_{\text{faces}} \leftarrow \tfrac{1}{4}(d+1)^2 - 1$.
\FOR{ $i = 1$ \TO $N$}
\STATE \COMMENT{Generate data and syndrome errors for $d$ time slices.}
\FOR{$t = 1$ \TO $d$}
\FOR{$j = 1$ \TO $n$}
\STATE $E[t,j] \leftarrow 1$ with probability $p$. \COMMENT{Data errors.}
\ENDFOR
\FOR{$j = n+1$ \TO $n + 1 + n_{\text{faces}}$}
\STATE $E[t,j] \leftarrow 1$ with probability $p$. \COMMENT{Synd. errors.}
\ENDFOR
\ENDFOR
\STATE $E_{\text{min}} \leftarrow \text{Decode}(\text{Syndrome}(E))$. \COMMENT{3D error volume.}
\STATE $E' \leftarrow \bigoplus_{t} E[t] \oplus E_{\text{min}}[t]$. \COMMENT{2D error plane.}
\STATE $E'_{\text{min}} \leftarrow \text{Decode}(\text{Syndrome}(E'))$. \COMMENT{Ideal decoding}.
\IF{ $(\bigoplus_i E'[i] \oplus E'_{\text{min}}[i] = 1)$ }
\STATE $N_{\text{fail}} \leftarrow N_{\text{fail}} + 1$.
\ENDIF
\ENDFOR
\RETURN $p_{\text{fail}}^{(\text{est})} = N_{\text{fail}}/N$.
\end{algorithmic}
\end{algorithm}
Our plots of $p_{\text{fail}}$ versus $p$ for small-distance color codes are depicted in Fig.~\ref{fig:phenom-plots}. Just as for surface codes, the phenomenological noise MLE decoder can be mapped to a random-plaquette gauge model (RPGM) on classical spins such that the zero-temperature order-disorder phase transition in the spin model corresponds to the accuracy threshold of the color codes. Because of this, as argued in Ref.~\cite{Wang:2003a}, the mutual intersection of the curves in Fig.~\ref{fig:phenom-plots} at the threshold $p_{c}$ corresponds to critical behavior in the spin model such that the spin correlation length $\xi$ scales as
\begin{align}
\xi \sim |p - p_{c}|^{-\nu_0},
\end{align}
where $\nu_0$ is a critical exponent set by the universality class of the spin model.
\begin{figure}
\caption{Monte Carlo data used to estimate the accuracy threshold in the phenomenological noise model.}
\label{fig:phenom-plots}
\end{figure}
For a sufficiently large code distance $d$, then, the failure probability should scale as
\begin{align}
p_{\text{fail}} = (p - p_{c})d^{1/\nu_0}.
\end{align}
We use our Monte Carlo data to fit to this form, but as in Ref.~\cite{Wang:2003a}, we allow for systematic corrections coming from finite-size effects that create a constant offset. Specifically, we use the method of differential corrections \cite{Pezzullo:2011a} to fit the curves to the form
\begin{align}
\label{eq:pf-pc-curve-fit}
p_{\text{fail}} = A + B(p - p_c)d^{1/\nu_0}.
\end{align}
The linear fits to our data are plotted in Fig.~\ref{fig:phenom-line-fits}. Using the software of Ref.~\cite{Pezzullo:2011a}, we found the following values for $p_c$ and $\nu_0$:
\begin{align}
p_c &= 0.030\,534 \pm 0.000\,385 \\
\nu_0 &= 1.486\,681 \pm 0.166\,837.
\end{align}
\begin{figure}\label{fig:phenom-line-fits}
\end{figure}
To put our results in context, as we did in the code capacity setting, we reference Table~\ref{tab:code-thresholds}. For the same reasons as in the code capacity noise model setting, the threshold we compute is larger than the MLE decoder's threshold for the 4.4.4.4 surface codes. We conjecture that is it also measurably less than the threshold for the optimal color-code decoder, as is the case for optimal vs. MLE decoding for surface codes. So far, the threshold for optimal decoding of 4.8.8 color codes has not been estimated, but the analysis for optimal decoding of 6.6.6 color codes suggests that the threshold will be near $4.5\%$. If true, our data would signal a violation of the Nishimori conjecture for the RPGM associated with the 4.8.8 color code, something we are not aware of being reported elsewhere.
Finally, we note that while the value of $\nu_0$ is consistent with value of $\nu_0 = 1.463(6)$ obtained for the 4.4.4.4 surface code \cite{Wang:2003a} and the 6.6.6 color code, the uncertainty in the value we obtained is too high to draw any meaningful conclusions.
\subsection{Circuit-level noise model}
As with the phenomenological noise model, computing $p_{\text{fail}}(p)$ exactly even for small code sizes is intractable, so we again appeal to Monte Carlo estimation. Our Monte Carlo simulation algorithm is similar to Algorithm \ref{alg:MC-estimator}, except the manner in which the error pattern $E$ is generated is different. To generate $E$, we simulate BP and DP channel noise as described by the noise model on the explicit circuit given for syndrome extraction. This results in a correlated error model for syndrome and data qubits. We then use the phenomenological noise MLE decoder and assess success or failure as we did for that noise model.
We estimated the $p_{\text{fail}}(p)$ curves for several small 4.8.8 triangular color codes for both the $X$-then-$Z$ schedule of Fig.~\ref{fig:naive-schedule} and the interleaved $X$-$Z$ schedule of Fig.~\ref{fig:interleaved-schedule}. Our results are plotted in Figs.~\ref{fig:XZ-sched-plot} and \ref{fig:XZ-interleaved-plot}.
\begin{figure}
\caption{Monte Carlo data used to estimate accuracy threshold in the circuit-based noise model in which the noninterleaved syndrome extraction circuit is used.}
\label{fig:XZ-sched-plot}
\end{figure}
\begin{figure}
\caption{Monte Carlo data used to estimate the accuracy threshold in the circuit-based noise model in which the interleaved syndrome extraction circuit is used.}
\label{fig:XZ-interleaved-plot}
\end{figure}
To compute the accuracy thresholds from our data, we again fit our data near the crossings to an equations whose form is similar to that of by Eq.~(\ref{eq:pf-pc-curve-fit}). However, the motivation for such a fit is a bit more tenuous in this case because while the MLE decoder we are using maps to a RPGM, the noise model which generates it is correlated. For this reason, as also found in Ref.~\cite{Wang:2003a}, we found it necessary to include a quadratic term, unlike the case for the pure phenomenological noise model. In other words, we fit our data to an equation of the form
\begin{align}
p_{\text{fail}} = A + B(p - p_c)d^{1/\nu_0} + C(p-p_c)^2 d^{2/\nu_0}.
\end{align}
The quadratic fits to our data for the $X$-then-$Z$ schedule are plotted in Fig.~\ref{fig:quadratic-fits}. Again using the software of Ref.~\cite{Pezzullo:2011a}, we found the following values for $p_c$ and $\nu_0$ for the $X$-then-$Z$ schedule:
\begin{align}
p_c &= 0.000\,820 \pm 0.000\,022 \\
\nu_0 &= 1.350\,954 \pm 0.079\,188.
\end{align}
To be clear, there is both a $Z$-error and an $X$-error accuracy threshold; we report the smaller of the two here.
\begin{figure}\label{fig:quadratic-fits}
\end{figure}
Similarly, for the $XZ$-interleaved schedule we found
\begin{align}
p_c &= 0.000\,800 \pm 0.000\,037 \\
\nu_0 &= 1.509\,871 \pm 0.151\,690.
\end{align}
To remind, our results are for the smaller of the $X$-error and $Z$-error thresholds.
Our results show that despite our efforts to shorten the schedule of the syndrome extraction circuit, the impact on the resulting accuracy threshold is essentially indistinguishable. The value of $0.082(3)\%$ for the accuracy threshold for MLE decoding of the 4.8.8 color codes in the circuit-level noise model is about a factor of ten less than the the corresponding $0.68\%$ accuracy threshold for MLE decoding of 4.4.4.4 surface codes in the circuit-level noise model. We believe that the difference comes from the fact that the 4.8.8 codes have some weight-8 stabilizer generators while the 4.4.4.4 codes only have weight-4 stabilizer generators. This causes the circuits for extracting the syndrome for the weight-8 generators in the 4.8.8 codes to be larger, inviting more avenues for failure. Indeed, we have investigated the finite-sized error-propagation patterns for the 4.8.8 codes such as the one depicted in Fig.~\ref{fig:interleaved-schedule-error-propagation}, and they are significantly larger and more complex than the corresponding patterns for the 4.4.4.4 surface codes. Expanding this line of reasoning, we predict that the 6.6.6 color codes will have an MLE-decoded accuracy threshold in the circuit-based noise model that is somewhere between the 4.8.8 and 4.4.4.4 accuracy thresholds in this noise model.
\section{Analytic bound on the accuracy threshold for fault-tolerant quantum error correction}
\label{sec:analytic-bound}
While numerical estimates of the accuracy threshold are valuable, equally valuable are analytic proofs that the accuracy threshold is no smaller than a given value. One method of obtaining such a lower bound is to use the self-avoiding walk (SAW) method, first proposed in Ref.~\cite{Dennis:2002a}. The idea behind this method begins with the observation that our goal is to lower-bound the failure probability of decoding, which is the probability that the actual errors plus the inferred correction (modulo 2) lead to an error chain that corresponds to a logical operator. For color codes, logical operators can be not only string-like but also string-net like, as described in the original paper on color codes \cite{Bombin:2006b}. They must also have a Pauli-weight at least as large as the distance of the code. The probability that a logical operator is present in the post-corrected state is therefore at least as large as the probability that an error-chain string of Pauli-weight equal to the code distance is present. Certainly this is a very pessimistic bound; there are many error chain strings and string-nets of this Pauli weight that do not result in failure!
The SAW lower-bound method can be applied relatively straightforwardly to the code-capacity and phenomenological noise models with MLE decoding. The method begins to break down when applied to the circuit-level noise model with phenomenological MLE decoding. One reason for this is that the circuit introduces correlated errors, called ``hooks'' in Ref.~\cite{Dennis:2002a}, which suggest that the SAW bounding the failure probability should be allowed to sometimes take more than one step in a single iteration. With some finesse, this can be accounted for and bounded as in Ref.~\cite{Dennis:2002a}. However, for the color codes, the steps need not be path-connected either. For example, the circuit may create three separated errors on a single octagon plaquette. Calling such a process a ``walk'' or attempting to bound the behavior of the process by a true SAW method is dubious at best. For this reason, we have chosen to omit bounding the accuracy threshold in the circuit-level noise model and instead have bounded the accuracy threshold only for the other two noise models, as described below.
\subsection{Code capacity noise model}
As argued by Dennis \textit{et~al.}\ in Ref.~\cite{Dennis:2002a}, the probability that an $[\![n, k, d]\!]$ topological code decoded by an error-free MLE decoder fails is upper-bounded by the probability that a self-avoiding walk creates a closed path (\textit{i.e.}, a self-avoiding polygon or SAP) of length $d$ or greater:
\begin{align}
\label{eq:SAP-sum}
p_{\text{fail}} &\leq \sum_{L \geq d} \text{Prob}_{\text{SAP}}(d) \\
&\leq n\sum_{L \geq d} n_{\text{SAP}}(L)\,(4p(1-p))^{L/2}.
\end{align}
Self-avoiding walks on the 4.8.8 lattice have been studied, and it is known that the number of self-avoiding polygons of length $L$ on the lattice scales asymptotically as \cite{Jensen:1998a}
\begin{align}
n_{\text{SAP}}(L) \leq P(L)\mu_{4.8.8}^L, \quad \mu_{4.8.8} \approx 1.808\,830\,01(6),
\end{align}
where $P$ is a polynomial and $\mu_{4.8.8}$ is the so-called \emph{connective constant} for the $4.8.8$ lattice. (The value $\mu_{4.8.8}$ has been rigorously bounded to be $1.804\,596 \leq \mu_{4.8.8} \leq 1.829\,254$ \cite{Jensen:2004a, Alm:2005a}.) For small $p$, each summand in Eq.~(\ref{eq:SAP-sum}) is upper-bounded by the term with $L = d$, and the number of summands is at most a polynomial in $d$, so that $p_{\text{fail}} \to 0$ as $d \to \infty$ as long as
\begin{align}
\label{eq:p-transcendental}
p(1-p) \leq \frac{1}{4\mu_{4.8.8}^2}.
\end{align}
Solving this equation for $p$, we find that the code capacity threshold is at least
\begin{align}
p_{c} \geq 8.335\,745(1)\%.
\end{align}
Despite the crudeness of the SAW bound, it comes surprisingly close to the numerical value of $10.56(1)$ that we estimate in Sec.~\ref{sec:code-capacity-noise-model-threshold}.
\subsection{Phenomenological noise model}
The SAW bound method is essentially the same as for the code capacity noise model, except now errors can happen on syndrome qubits as well as data qubits and the set of all relevant qubits forms a three-dimensional volume. The relevant SAW traverses a 3D lattice that connects syndrome qubits and data qubits both with themselves and each other as dictated by the color code; the corresponding nonregular prismatic lattice is depicted in Fig.~\ref{fig:4.8.8-pizza-prism}. To our knowledge, the connective constant for this lattice is not known, but it could be computed in principle using standard methods, \textit{e.g.}, those outlined in Refs.~\cite{Jensen:1998a, Jensen:2004a, Alm:2005a}. We opted to bypass this analysis and instead compute a coarser bound on the failure probability.
\begin{figure}\label{fig:4.8.8-pizza-prism}
\end{figure}
Because the lattice in Fig.~\ref{fig:4.8.8-pizza-prism} has vertices of degree $\Delta$ equal to $6$, $8$, and $10$, we can bound the number of SAPs of length $L$ by
\begin{align}
n_{\text{SAP}}(L) \leq 2\Delta_{\text{max}}(2\Delta_{\text{max}} - 1)^{L - 1}.
\end{align}
Using $\Delta_{\text{max}} = 10$, we obtain a formula similar to that of Eq.~(\ref{eq:p-transcendental}), namely
\begin{align}
p(1-p) \leq \frac{1}{4(9)^2} = \frac{1}{324}.
\end{align}
Solving this equation for $p$, we find that the phenomenological noise threshold is at least
\begin{align}
p_{c} \geq \frac{9 - 4\sqrt{5}}{18} \approx 0.3096\%.
\end{align}
This bound is nearly a factor of ten less than the value of $p_c = 3.05(4)\%$ that we estimate in Sec.~\ref{sec:phenomenological-noise-model-threshold}. With further computational effort in determining the connective constant of the governing lattice, we suspect that the SAW bound will still be below our numerical estimate, but significantly closer, in analogy with the relationship between our SAW bound for the code capacity and the value we estimate numerically. We leave this analysis to others wishing to tighten this bound.
\section{Fault-tolerant computation with color codes}
\label{sec:ftqc-with-color-codes}
To establish a threshold for fault-tolerant quantum computation, it is sufficient to establish three things: 1) a threshold for fault-tolerant quantum error correction, 2) a procedure for performing a universal set of gates in encoded form, and 3) that a failure in an encoded gate that occurs with probability $p$ leads to failures in each output codeword with probability at most $p$. These three ingredients establish that each gate in a quantum circuit can be simulated fault-tolerantly by performing it in encoded form followed by fault-tolerant quantum error correction. We previously established the first criterion in Sec.~\ref{sec:fault-tolerant-error-correction}. We establish the second two criteria here for two possible computer architectures.
In the first, which we call the ``pancake architecture,'' each logical qubit is stored in its own triangular 4.8.8 color code and the logical qubits are stacked atop one another. This architecture is essentially the same as the one proposed in Ref.~\cite{Dennis:2002a}. Almost all encoded operations are implemented transversally in this model, acting on single ``logical qubit pancakes'' or between two such ``pancakes.'' In the second, which we call the ``defect architecture,'' each logical qubit is stored as a connected collection of missing check operators, which we call a ``defect,'' in a single 2D 4.8.8 substrate. This architecture is essentially the same as the one proposed in Ref.~\cite{Raussendorf:2007a}. Almost all encoded operations are performed in one of two ways: encoded single-qubit gates are performed by disconnecting a region containing the defect, operating transversally on the region, and reconnecting the region, while the encoded CNOT gate is performed by a sequence of local measurements that cause one defect to circulate around another.
\subsection{Fault-tolerance by transversal gates}
\label{sec:fault-tolerance-by-transversal-gates}
In this section, we compute the threshold for fault-tolerant quantum computation with triangular 4.8.8 color codes when (almost) all encoded gates are implemented \emph{transversally}. To remind, by calling a gate ``transversal,'' we mean that it acts identically on all physical qubits in a code block. For example a two-qubit transversal gate between two triangular codes acts as the same two-qubit physical gate between corresponding physical qubits in each code block. Some authors refer to this notion of transversality as \emph{strong} transversality \cite{Eastin:2007a}.
\subsubsection{Identity gate}
The accuracy threshold for the identity gate is exactly the same as the accuracy threshold for fault-tolerant quantum error correction, by definition. Schematically, Fig.~\ref{fig:noisy-I} depicts the noisy identity gate circuit.
\begin{figure}\label{fig:noisy-I}
\end{figure}
Formally, we can express the equivalence between the accuracy threshold for the identity gate and the accuracy threshold for fault-tolerant quantum error correction as
\begin{align}
p_{th}^{(I)} = p_{th}^{(\text{QEC})}.
\end{align}
\subsubsection{$\CNOT$ gate}
The color codes are Calderbank-Shor-Steane (CSS) codes \cite{Calderbank:1996a, Steane:1996b}, and for all such codes, the encoded controlled-NOT ($\CNOT$) gate can be implemented \emph{transversally}, namely by applying $\CNOT$ gates between corresponding pairs of physical qubits in two color codes. (For color codes, fewer $\CNOT$ gates than a fully transversal set also suffice.) Schematically, Fig.~\ref{fig:noisy-CNOT} depicts a noisy $\CNOT$ gate. Each physical $\CNOT$ gate propagates the BP channel on its control to the BP channel on its target and vice versa, so that the effective noise model seen by the fault-tolerant quantum error correction procedure on each code block after the encoded $\CNOT$ gate is the BP channel followed by the projection of the two-qubit DP channel onto a single qubit. Although the DP channel can create correlated errors between output code blocks, it will never cause a correlated error within a code block. Since our decoder treats the noise model phenomenologically, it does not account for DP-channel features such as the fact that in the DP channel a $Y$ error is more probable than the combination of separate $X$ and $Z$ errors. For this reason, since half of the DP-channel errors act as a bit-flip on a given code block and half of them act as a phase-flip on a given code block, our decoder interprets the post-$\CNOT$ noise model as a BP channel with an effective error rate of $p + p/2$ for bit flips and $p + p/2$ for phase flips. This means that the accuracy threshold for the $\CNOT$ gate is actually $2/3$ of the value for the identity gate. The $\CNOT$ gates used in an encoded $\CNOT$ gate must therefore meet a more stringent requirement than the identity gate to be implemented transversally fault-tolerantly. (However, the $\CNOT$ gates used in fault-tolerant quantum error correction still only need to meet the threshold for the encoded identity gate.)
\begin{align}
p_{th}^{(\CNOT)} = \frac{2}{3}p_{th}^{(I)}.
\end{align}
\begin{figure}\label{fig:noisy-CNOT}
\end{figure}
\subsubsection{Hadamard gate}
The color codes are \emph{strong} CSS codes, meaning that the $X$-type and $Z$-type stabilizer generators have the same structure. As with all strong CSS codes, the encoded Hadamard gate ($H$) can be implemented transversally.
Like the $\CNOT$ gate, the Hadamard gate propagates the BP channel to the BP channel. However, since faults in the Hadamard gate are modeled as an ideal Hadamard gate followed by the BP channel, the effective noise model is not one but \emph{two} actions of the BP channel, as depicted in Fig.~\ref{fig:noisy-H}.
\begin{figure}\label{fig:noisy-H}
\end{figure}
It is straightforward to show that two successive applications of the BP channel with probability $p$ is equivalent to one application of BP channel with probability $2p(1-p)$. This is therefore the effective post-Hadamard noise channel, so that the threshold for the Hadamard gate is about half of that for fault-tolerant quantum error correction:
\begin{align}
p_{th}^{(H)} = \frac{1}{2} - \frac{1}{2}\sqrt{1 - 2 p_{th}^{(I)}}
&\approx \frac{1}{2}{p_{th}^{(I)}}.
\end{align}
\subsubsection{Phase gate}
The color codes have the feature that each stabilizer generator for the code has a Pauli weight equal to $0 \bmod 4$ and each pair of generators are incident on $0 \bmod 2$ qubits. One can show that because of this, the encoded phase gate ($S$) has a transversal implementation \cite{Knill:1996a, Bombin:2006b}. (Technically, it is the transversal $S^\dagger$ operation that acts as an encoded $S$.)
While a faulty phase gate acts as an ideal phase gate followed by a BP channel, the phase gate itself does not propagate the BP channel preceding it symmetrically for bit flips and phase flips. This follows from the conjugation actions
\begin{align}
SXS^{\dagger} &= Y = iXZ & SZS^{\dagger} &= Z.
\end{align}
The phase gate therefore propagates a phase flip to a phase flip and a bit flip to both a bit-flip and a phase flip, as depicted in Fig.~\ref{fig:noisy-S}.
\begin{figure}\label{fig:noisy-S}
\end{figure}
The phase gate is correspondingly more sensitive to phase-flip noise because the effective phase-flip strength is $p^3 + 3p(1-p)^2$. The phase gate thus has separate thresholds for bit-flip and phase-flip noise. For bit-flip noise, the threshold is
\begin{align}
p_{th}^{(S, \text{bit-flip})} &= \frac{1}{2} - \frac{1}{2}\sqrt{1 - 2 p_{th}^{(I)}}
\approx \frac{1}{2}{p_{th}^{(I)}}.
\end{align}
For phase-flip noise, one must solve a cubic equation to get a closed-form solution for the threshold as a function of the threshold for the identity gate. While this is possible in principle, to save space we simply state the cubic equation in the variable $x = p_{th}^{(S, \text{phase-flip})}$ that must be solved and its approximate solution, which we can estimate because we know that the accuracy threshold is very close to 0:
\begin{align}
x^3 + 3x(1-x)^2 &= {p_{th}^{(I)}}, \\
x \approx \frac{1}{3}{p_{th}^{(I)}}.
\end{align}
\subsubsection{Single-qubit measurements}
To \emph{destructively} apply the encoded single-qubit measurements $M_X$ and $M_Z$, we transversally measure $X$ or $Z$ on each of the qubits in the code block. We then perform classical error correction on the measurement outcomes (because they may be faulty) to infer the outcome of the encoded measurement, as depicted schematically in Fig.~\ref{fig:noisy-measurements}.
\begin{figure}\label{fig:noisy-measurements}
\end{figure}
The correctness of this procedure follows from the fact that $X$ and $Z$ operators can be expressed as $Z = S^2$ and $X = HZH$, and the encoded operations $H$ and $S$ have previously been demonstrated to have transversal encoded implementations. Bit or phase errors (as relevant) before a measurement then map to bit errors on the observed classical bit pattern.
The reason the measurement is destructive is that after the measurement, the qubits are no longer in the codespace of the color code; the post-measured state is not projected onto an $X$ or $Z$ eigenstate in the codespace. However, as pointed out by Steane \cite{Steane:1998a}, given the ability to prepare encoded $|+\rangle$ states, a circuit composed of transversal ${\CNOT}$ and transversal destructive ${M}_X$ measurements can implement
\emph{nondestructive} $M_X$ measurements transversally. A similar story holds for encoded $|0\rangle$ states and $M_Z$ measurements. The circuits for generating these nondestructive measurements transversally are depicted in Fig.~\ref{fig:MZ-MX-transversal}. Because the encoded $|0\rangle$ and $|+\rangle$ states are being used to enable gates, namely nondestructive encoded measurements, these states are called ``magic states'' for the gates \cite{Bravyi:2005a}. Ordinarily, quantum error correction would follow not just one, but both of the outputs of the encoded $\CNOT$ gate in these circuits, but because one of the encoded qubits is destructively measured immediately after the $\CNOT$ gate, that encoded qubit does not require quantum error correction; it will be effectively performed by the classical error correction process occurring after the destructive measurement.
\begin{figure}
\caption{Circuits for nondestructive encoded
${M}_X$ and ${M}_Z$, using the states $|0\rangle$ and $|+\rangle$ as ``magic states.''}
\label{fig:MZ-MX-transversal}
\end{figure}
The threshold for destructive $M_Z$ and $M_X$ measurements is the same as the code capacity threshold for the code, regardless of which noise model we are considering. This is because the physical measurements are made only once, as repetition cannot improve their effective error rate. The (flawless) classical error correction performed in post-processing has a threshold equal to the code capacity threshold. Hence, we have the result that
\begin{align}
p_{th}^{(M_X, \text{destructive})} &= p_{th}^{(I, \text{code capacity})}, \\
p_{th}^{(M_Z, \text{destructive})} &= p_{th}^{(I, \text{code capacity})}.
\end{align}
Although these measurements need only be smaller than the code capacity threshold to implement the encoded measurement, when these measurements are used in the fault-tolerant quantum error correction protocol, they must be smaller than the threshold set by the prevailing noise model---a threshold that may be significantly lower.
To compute the threshold for nondestructive $M_Z$ and $M_X$ measurements, we examine how errors propagate through the circuits in Fig.~\ref{fig:MZ-MX-transversal}. As with the analysis of Fig.~\ref{fig:noisy-CNOT}, the effective noise channel we need to consider after the $\CNOT$ gate is the BP channel followed by the DP channel on each output. One of these enters a destructive measurement, which, as we found in the analysis of Fig.~\ref{fig:noisy-measurements}, has a rather high threshold equal to the code capacity even in the circuit-level noise model. However, it is lowered slightly by the fact the effective error rate is $\frac{2}{3}p$, as discussed in the analysis of the encoded $\CNOT$ gate. The other output enters a standard quantum error correction circuit, also subject to noise of strength $\frac{2}{3}p$. Since the lowest threshold of these two thresholds is this one, the overall threshold for an encoded nondestructive measurement is the same as the threshold for the encoded $\CNOT$ gate. Namely, we have the result that
\begin{align}
p_{th}^{(M_X, \text{nondestructive})} &= p_{th}^{(\CNOT)} = \frac{2}{3}p_{th}^{(I)}, \\
p_{th}^{(M_Z, \text{nondestructive})} &= p_{th}^{(\CNOT)} = \frac{2}{3}p_{th}^{(I)}.
\end{align}
\subsubsection{$|0\rangle$ and $|+\rangle$ preparation}
It is tempting to assert that the way to fault-tolerantly prepare the encoded $|0\rangle$ state is to perform an encoded nondestructive $M_Z$ measurement. The flaw with this reasoning is that the nondestructive $M_Z$
measurement requires the encoded $|+\rangle$ state as a magic state, and the analogous way of preparing a $|+\rangle$ state requires a $|0\rangle$ state.
To get out of this chicken-and-egg cycle, one must use an independent process. We describe a two-step process that works for preparation of an encoded $|0\rangle$ state; the process for preparing an encoded $|+\rangle$ state is similar.
The first step is to prepare the product state $|0\rangle^{\otimes n}$ by transversally measuring $M_Z$ on each physical qubit. This state is a stabilizer state, having $n$ check operators, with check operator $i$ being
$Z$ on qubit $i$ for $i = 1, \ldots n$. The second step is to fault-tolerantly measure the $X$ checks for the color code. Because the only $Z$-type operators consistent with all the $X$ checks are the color codes' $Z$ checks for the color code and the logical $Z$ operator, these measurements will transform the state into the logical $|0\rangle$ state.
It turns out that it is not necessary to also fault-tolerantly measure the $Z$ checks for the color code. The state is already in an eigenstate of these operators at this point, so all the measurements can do is yield syndrome bits. Had one obtained these bits and processed them, the post-corrected state would still have been subject to $X$ errors drawn from the same distribution as the $X$ errors afflicting the initial
$|0\rangle^{\otimes n}$ preparation---fault-tolerant error correction doesn't suppress the final error rate to zero, it only keeps it at the same rate one started with.
The threshold for preparation of encoded $|0\rangle$ and $|+\rangle$ states is therefore the same as the threshold for fault-tolerant quantum error correction, namely,
\begin{align}
p_{th}^{(|0\rangle)} &= p_{th}^{(|+\rangle)} = p_{th}^{(I)}.
\end{align}
It is worth noting that while the process for fault-tolerantly preparing
$|0\rangle$ and $|+\rangle$ states is not strictly transversal, the only nontransversal operation is fault-tolerant quantum error correction, a process that is required in addition to transversal operations in any event in order to achieve fault-tolerant quantum computation.
\subsubsection{$T$ gate}
\label{sec:T-gate}
Another gate that admits a transversal implementation with a magic state is the $T$ gate, also called the $\pi/8$ gate, defined as
\begin{align}
T := \begin{bmatrix}1&0\\0&e^{-i\pi/4}\end{bmatrix}
= e^{-i\pi/8} \begin{bmatrix}e^{i\pi/8}&0\\0&e^{-i\pi/8}\end{bmatrix}.
\end{align}
If we we have an encoded version of the state
\begin{align}
|\pi/4\rangle &:= TH|0\rangle \\
&= \frac{1}{\sqrt{2}}\left(|0\rangle + e^{i\pi/4}|1\rangle\right),
\end{align}
also called $|A\rangle$ and $|A_{\pi/4}\rangle$ in the literature, we can implement the $T$ gate transversally using the circuit of Fig.~\ref{fig:T-circuit}. This circuit is not a Clifford circuit, because the classically-controlled $S$ gate is not a Clifford gate. Nevertheless, it only uses gates that we have previously shown how to implement in encoded form by purely transversal operations.
\begin{figure}\label{fig:T-circuit}
\end{figure}
To compute the $T$ gate threshold, we again study error propagation through its defining circuit, \textit{viz.} the circuit in Fig.~\ref{fig:T-circuit}. As shown previously, the $\CNOT$ gate creates an input to the first QEC cycle that has a threshold of $2/3$ of the standard QEC threshold. The $S$ gate creates an input to the second QEC cycle which splits the threshold into bit-flip and phase-flip thresholds approximately equal to $1/2$ and $1/3$ of the standard QEC threshold. The threshold for the $T$ gate is set by the smallest of these, namely the $S$ gate threshold, which is
\begin{align}
p_{th}^{(T, \text{bit-flip})} &= \frac{1}{2} - \frac{1}{2}\sqrt{1 - 2 p_{th}^{(I)}}
\approx \frac{1}{2}{p_{th}^{(I)}}, \\
p_{th}^{(T, \text{phase-flip})} &= x \approx \frac{1}{3}{p_{th}^{(I)}}.
\end{align}
\subsubsection{$|\pi/4\rangle$ preparation}
\label{sec:T-state}
There are two alternatives for preparing encoded $|\pi/4\rangle$ states fault-tolerantly described in the literature. In the first, low-fidelity
$|\pi/4\rangle$ states are ``injected'' into the code by teleportation, using the circuit in Fig.~\ref{fig:M-injection} \cite{Knill:2004a}, and then
``distilled'' using encoded gates until the resultant $|\pi/4\rangle$ states have an error below the accuracy threshold. In the second, high-quality
$|\pi/4\rangle$ states are first distilled and then injected into the code.
\begin{figure}\label{fig:M-injection}
\end{figure}
The circuit depicted in Fig.~\ref{fig:M-injection} is not fault-tolerant, but faults are already suppressed by the code on the encoded qubits; only operations from the latter-half of the decoding circuit onwards are unprotected.
Unlike all of the previous encoded gates, this method for implementing an encoded $|\pi/4\rangle$ preparation requires an operation which is neither transversal nor fault-tolerant quantum error correction. The ``unencoding'' portion of the circuit is the time-reversed coherent circuit for encoding a state in the color code, derivable via standard stabilizer codes as shown in Ref.~\cite{Gottesman:1997a}. This unencoding circuit does not appear to have a transversal implementation. While the Eastin-Knill theorem \cite{Eastin:2008a} asserts that at least one nontransversal operation is required to generate a universal set of encoded gates, it does not guarantee that no transversal implementation of this circuit exists. That is because the process of fault-tolerant quantum error correction used to prepare
$|0\rangle$ and $|+\rangle$ states is not transversal. For 3D color codes
\cite{Bombin:2007b}, in which $T$ is intrinsically transversal and in which encoded $|0\rangle$ and $|+\rangle$ states still require fault-tolerant quantum error correction for preparation, only transversal and FTQEC operations are needed, for example. It would be interesting to develop a variant of the circuit in Fig.~\ref{fig:M-injection} which only uses transversal operations and possibly fault-tolerant quantum error correction to inject a $|\pi/4\rangle$ state into 2D color codes. We leave that for others to explore.
While the portion of the circuit in Fig.~\ref{fig:M-injection} in which the physical $|M\rangle$ state interacts with the unencoded qubit via a $\CNOT$ appears to also not be transversal, it can be made so with slight modification. In principle, one could prepare $n$ states of the form
$|M\rangle$ and transversally apply the $\CNOT$ gate between these and the code block, but only the one qubit corresponding to the unencoded state will be used to classically control the $\overline{X}$ and $\overline{Z}$ gates that are used to inject the correct state. As usual, these corrections do not need to be actually implemented, only used to update the Pauli frame.
Both alternatives for preparing high-quality encoded $|\pi/4\rangle$ states require a procedure for magic-state distillation. One option is to use the encoding circuit for the 15-qubit Reed-Muller code \cite{Knill:1998a} (also the smallest 3D color code \cite{Bombin:2007b}) run in reverse, as depicted in Fig.~\ref{fig:state-distillation}. For it to work, the initial states must have an error less than the $|\pi/4\rangle$ distillation threshold. For the circuit depicted in Fig.~\ref{fig:state-distillation}, the distillation threshold for independent, identically distributed (iid) depolarizing noise is $(6-2\sqrt{2})/7 \approx 45.3\%$ \cite{Reichardt:2006a, Buhrman:2004a}, for dephasing iid noise is $(\sqrt{2}-1)/\sqrt{2} \approx 29.3\%$ \cite{Reichardt:2006a, Virmani:2004a}, and for worst-case iid noise is $(\sqrt{2}-1)/2\sqrt{2} \approx 14.6\%$ \cite{Reichardt:2006a, Virmani:2004a}. The entire circuit must be run ${\cal O}(\mathop{\mathrm{poly}}(\varepsilon^{-1}))$ times to achieve an output error less than $\varepsilon$; convergence should be quite rapid in practice given the actual polynomial \cite{Reichardt:2006a}. Various tricks can be used to boost the distillation threshold and reduce the resources required to achieve high-fidelity states; any of these can be readily adapted to this setting.
\begin{figure}\label{fig:state-distillation}
\end{figure}
\subsubsection{Synthesis}
It is well-known result the gate basis $\{H, S, \CNOT, M_X, M_Z, |0\rangle, |+\rangle,
|\pi/4\rangle\}$ is universal for quantum computation \cite{Nielsen:2000a} (in fact, it is even overcomplete). We have presented transversal methods for performing color-code encoded versions of each of these except for the state preparations. By the Eastin-Knill theorem \cite{Eastin:2008a}, it is impossible to generate a complete universal encoded gate basis in transversal form. However, color codes offer a particularly gentle way around this theorem. There are only two nontransversal operations used. The first is fault-tolerant quantum error correction, a process that is required in addition to encoded computations in any event for the entire protocol to be fault tolerant. The second is the time-reversed coherent encoding circuit for color codes. Such a circuit is useful for encoding unknown quantum states, but in an actual quantum computation, the input state is known so it is not needed for this purpose. Whether this ``unencoding circuit'' can be replaced with another operation which uses only transversal operations and fault-tolerant quantum error correction is an interesting open question. For 3D color codes, we know that the answer is ``yes.''
The ``pancake architecture,'' described in Ref.~\cite{Dennis:2002a} for the Kitaev surface-codes, realizes the encoded gate set we described using only gates between spatially neighboring qubits. One difference in our analysis from that performed in Ref.~\cite{Dennis:2002a} is that we have analyzed the accuracy threshold not only for fault-tolerant quantum memory but also for fault-tolerant quantum computation, a feat made tractable by the strong CSS nature of the color codes.
\subsection{Fault-tolerance by code deformation}
The method of fault-tolerance described in Sec.~\ref{sec:fault-tolerance-by-transversal-gates} requires a three-dimensional architecture to allow the transversal $\CNOT$ gates to remain spatially local. This violates the spirit of using two-dimensional codes in the first place. Fortunately, it is possible to use \emph{code deformation} to achieve fault-tolerance in a strictly two-dimensional architecture. Our construction here mirrors that of Raussendorf \textit{et~al.}'s construction for surface codes \cite{Raussendorf:2007a, Raussendorf:2007b}. Fowler has independently constructed a method for using code deformation in 4.8.8 color codes that is similar to ours \cite{Fowler:2008c}. Some salient differences between our method and Fowler's are that ($i$) Fowler's logical qubits are always encoded in a triple of defects whereas ours are encoded in single defects except during certain logical gates, and ($ii$) Fowler's scheme disallows different defect types from occupying the same plaquette location while ours does not. Each of these differences allows our scheme to encode a higher density of information. Specifically, our scheme allows a six-fold increase in logical qubit density over the Fowler scheme.
To begin, we generate a sufficiently large 4.8.8 triangular color code by performing fault-tolerant quantum error correction on a collection of qubits. We are not interested in what state the triangular code encodes---all we require is that the state is in the codespace with arbitrarily high fidelity. We consider any logical qubits associated with the entire surface to be ``gauge'' qubits in the language of subsystem stabilizer code theory \cite{Poulin:2005a}. We will use this state as a substrate for generating and manipulating encoded qubits.
Each element of the standard set of stabilizer generators for a color code can be labeled by a face of a definite color (red, green, or blue) and an operator of a definite Pauli type ($X$ or $Z$). Notationally, we will refer to a generator as a $(c, P)$ generator if it is of color $c$ and Pauli type $P$. To prepare an encoded qubit in our color code substrate, we remove a connected product of stabilizer generators of the same color and type. (Generally removal of any element of the stabilizer group will yield a logical qubit; we restrict attention to this class for simplicity.) We call this removed region a \emph{defect} in analogy with the language used by Raussendorf \textit{et~al.}\ in Ref.~\cite{Raussendorf:2007a}. This removal is entirely passive---we simply cease measuring this product of stabilizer generators in future quantum error correction rounds. For this reason, it is manifestly a fault-tolerant process.
In the following sections, we describe how to perform a universal repertoire of encoded logic gates on defect-based logical qubits, with arbitrarily high fidelity. This is therefore a prescription for fault-tolerant quantum computation using code deformation.
\subsubsection{Preparing a defect in $|0\rangle$ or $|+\rangle$}
In principle, the generator removed to form a defect qubit can be identified with any element of the encoded Pauli group for that encoded qubit. For concreteness, we make the choice of calling the removed generator a logical $Z$ when it is $Z$-type defect (also called a `primal' or `smooth' defect in the language of Ref.~\cite{Raussendorf:2007a}) and a logical $X$ when it is $X$-type defect (also called a `dual' or `rough' defect in the language of Ref.~\cite{Raussendorf:2007a}). Thus removing a $c$-colored $X$- or
$Z$-type generator corresponds to preparing a logical $|+\rangle_{(c,X)}$ or
$|0\rangle_{(c,Z)}$ state respectively.
The logical $Z$ operator for a $(c, X)$ defect acts as $Z$ on a $c$-colored chain of qubits connecting the defect to another $c$-colored boundary, which may itself be another defect. If no such other boundary exists, then the defect fails to encode a logical qubit. To avoid this complication, we have chosen our substrate to be a triangular code, having boundaries of each of the three colors. Similarly, the logical $X$ operator for a $(c, Z)$ defect acts as $X$ on a $c$-colored chain of qubits connecting the defect to a $c$-colored boundary.
Preparing a $|+\rangle_{(c, Z)}$ or $|0\rangle_{(c, X)}$ state requires more care. To do this, we measure $M_X$ or $M_Z$ respectively along a $c$-colored chain of qubits from the plaquette we wish to store the logical qubit in and the nearest $c$-colored boundary. This projects each qubit along the chain into either $|+\rangle$ or $|-\rangle$ (resp. $|0\rangle$ or $|1\rangle$), which we can interpret as
$|+\rangle$ (resp. $|0\rangle$) for each qubit by changing local Pauli bases. We then measure the $Z$-checks (resp. $X$-checks) incident on this chain except the one at the defect location and correct any errors, which places the defect back into the substrate in the desired state.
An arbitrarily large $c$-colored defect can be prepared in a single step by ceasing to measure a collection of $c$-connected defects by a similar process, enabling the preparation process to be made arbitrarily reliable. This introduces a number of ``gauge'' qubits in the interior of the defect that can be ignored; the details of this are described in the next section.
\subsubsection{Growing, shrinking, and moving defects}
We grow a $(c, P)$ defect qubit on region $q$ in the following way. Suppose we would like to extend the defect so that it includes an adjacent region $q'$ of the same color and type. (By adjacent, we mean that the regions can be connected by a single two-qubit $c$-colored link.) To do this, we first perform the following conditional operation. If $P = X$, then we measure $ZZ$ on a $c$-colored link connecting the regions, while if $P = Z$, then we measure $XX$ on a $c$-colored link connecting the regions. Examples of how this works for octagonal and square defects are depicted in Figs.~\ref{fig:defect-growth1} and \ref{fig:defect-growth2}; the circuit in Fig.~\ref{fig:growth-circuit} implements this transformation. A $YY$ operator can be used to grow a $X$ and $Z$-type defect at the same time.
\begin{figure}
\caption{Growth of an octagonal green $Z$ defect by one site.}
\label{fig:defect-growth1}
\end{figure}
\begin{figure}
\caption{Growth of a square red $Z$ defect by one site.}
\label{fig:defect-growth2}
\end{figure}
\begin{figure}
\caption{Measurement of $XX$ to grow a $Z$-type defect. The measurement can be performed with existing circuitry already in place for syndrome extraction. }
\label{fig:growth-circuit}
\end{figure}
After this measurement, the new collective defect operator is the product of the $q$ and $q'$ defect operators. The $\pm XX$ or $\pm ZZ$ operator has also been added to the list of stabilizer generators. As usual, we do not need to actually correct the result to a $+1$ outcome: it suffices to update the Pauli frames of the stabilizer generators incident on these two interior qubits.
Because we will no longer use the weight-two operator, we may consider it to also be a ``gauge'' operator in the language of subsystem stabilizer codes \cite{Poulin:2005a}. This also makes its anticommuting partner a gauge operator, which we may interpret to be either of the original defect operators (on $q$ or $q'$). By introducing these two new gauge operators, we may reinterpret the defect logical operator on the collective $q$ and $q'$ region as acting solely on its boundary. In particular, the interior of the collective $q$ and $q'$ region need never be involved in future syndrome extractions.
An important question is whether the defect growth process is fault-tolerant. The simplest circuit for measuring $XX$ or $ZZ$ would perform $\CNOT$ gates into or out of an ancilla qubit to each of the two relevant qubits, as depicted in Fig.~\ref{fig:growth-circuit}. Although a single error in this ancilla qubit could propagate to two errors on the two interior qubits, because we subsequently treat these qubits as encoding a gauge qubit, we do not worry about errors on these. It could still be the case that the value of the measurement obtained is incorrect, which impacts the update of the Pauli frame of the two adjacent stabilizer generators in a correlated way. Thus a single syndrome measurement error would propagate to two syndrome-bit errors. To prevent this happening to first order in the error probability, we repeat the $XX$ or $ZZ$ measurement twice and use the majority vote of the three outcomes to update the Pauli frame.
Compared to the process of defect growth, defect contraction is much simpler: to shrink a defect by a single-plaquette, one simply measures that plaquette operator in the next round of fault-tolerant quantum error correction.
By a combination of local growth and shrinking processes, one can deform the code with a $(c, P)$ defect at one plaquette to a code with a $(c, P)$ defect anywhere else. In other words, the \emph{move} operation for a defect can be decomposed into a sequence of more elementary \emph{grow} and \emph{shrink} operations.
\subsubsection{Measuring a defect}
To destructively measure the logical operator encircling a defect, one first shrinks the defect to size of a single plaquette. Then one measures the defect with the existing circuitry at that plaquette as though it were a local stabilizer generator. The shrunken defect will have a significantly lower tolerance to one type of Pauli error but that error type is in the basis being measured in and will not disturb the measurement outcome. To destructively measure the string-like logical operator connecting two defects, one brings the two operators as close together as possible. One then measures the weight-two operator connecting the defects using the circuitry used to grow a defect from one site to encompass the other. Again, the tolerance to errors of one Pauli type will be significantly lower, but this will not be of the type that disturbs the measurement.
To nondestructively measure a defect, one uses the circuit of Fig.~\ref{fig:MZ-MX-transversal}, which uses destructive measurement of
$M_Z$ or $M_X$, preparation of $|0\rangle$ or $|+\rangle$, and the $\CNOT$ gate described in the next section.
\subsubsection{$\CNOT$ gate between defects}
It is straightforward to show that moving a $(c, Z)$ defect qubit around a $(c', X)$ defect qubit (or vice-versa) generates an encoded $\CNOT$ gate controlled by the $(c, Z)$ defect when $c$ and $c'$ are different colors; the construction is essentially the same as that in Refs.~\cite{Raussendorf:2007a, Raussendorf:2007b, Fowler:2008c}. Since this process traces out a braid in spacetime, we call this process ``braiding defects.'' Also drawing upon Refs.~\cite{Raussendorf:2007a, Raussendorf:2007b}, one can generate a $\CNOT$ gate between two $Z$-type defects or two $X$-type defects, whether they are the same color or not. The circuit for doing this between two $Z$-type defects is depicted in Fig.~\ref{fig:CNOT-same-type}; the circuit for doing this between two $X$-type defects is similar.
\begin{figure}
\caption{Circuit for braiding a $\CNOT$ gate between $Z$-type defects. The colors $c$ and $c'$ may be the same or different, but the color $c''$ is a color different from these. The circuit for braiding a $\CNOT$ gate between $X$-type defects is similar: the $\CNOT$
gate directions are reversed, the types of the defects and the types of the measurements have their Pauli types swapped from $X$ to $Z$ and vice-versa, and the $|0\rangle$ state becomes a $|+\rangle$ state and vice-versa.}
\label{fig:CNOT-same-type}
\end{figure}
One can convert an $X$-type defect into a $Z$-type defect, or vice-versa, (changing its color as a side effect) using one of the circuits in Fig.~\ref{fig:X-to-Z}. In conjunction with the other type of $\CNOT$ gates mentioned, this allows $\CNOT$ gates between two defects regardless of the colors or Pauli types they have.
\begin{figure}
\caption{Circuits for converting a $Z$-type defect into an $X$-type defect and vice-versa.}
\label{fig:X-to-Z}
\end{figure}
\subsubsection{Phase gate on a defect}
To perform an $S$ (phase) gate on a $(c, P)$ defect, we prepare two more qubits of $(c', P)$ and $(c'', P)$ type, each in the state $|0\rangle$ and use $\CNOT$ gates to put the defect into a three-defect repetition code. This maps our single-defect logical qubits into the three-defect logical qubits Fowler uses in his construction \cite{Fowler:2008c}. We then grow the defects and connect them so that they separate an interior triangular region from an exterior region, just as described in Fowler's construction. If $P = Z$, then as Fowler noted, it suffices to apply $S$ transversally (actually, $S^\dagger$ must be applied transversally) to generate a logical $S$ on the triple-defect qubit. However, if $P = X$, then Fowler's construction fails, because the ``exterior trees'' in his language fail to undergo the action $SXS^\dagger = Y$. ``Pruning'' the exterior tree as Fowler suggests for his implementation of the Hadamard gate fails as well, because such an operation yields only the ``byproduct operator'' for logical $X$ or $Z$ on the triple-defect qubit, but not both. To perform the $S$ gate on $X$-type defects, we propose the following two-step procedure. First, we arrange the defects to separate a triangular interior from the exterior and apply $S^\dagger$ transversally on the interior. Second, we rearrange the defects so that part of what was the exterior becomes the new interior, and perform $S^\dagger$ transversally on this new triangular interior. In this way, both the interior and exterior trees experience the $S$ gate. Once the gate is complete, we run the three-defect encoding circuit in reverse and absorb the two ancilla qubit regions back into the substrate in subsequent quantum error correction rounds.
Of course, the $S$ gate can also be achieved via magic states of the form
$|\pi/2\rangle := \frac{1}{\sqrt{2}}\left(|0\rangle + e^{i\pi/2}|1\rangle\right)$ (also called $|Y\rangle$ or $|+i\rangle$ in the literature) in a manner similar to what is done for surface codes. But this is one of the great benefits of 4.8.8 color codes---no magic state distillation and usage is required to realize this gate in encoded form. It may well be worth the lower accuracy threshold of color codes relative to surface codes in order to achieve the resource reduction for performing encoded $S$ gates.
\subsubsection{Hadamard gate on a defect}
From one point of view, a logical Hadamard gate is unnecessary because it can be implemented using the gates we have previously described, for example by the circuit of Fig.~\ref{fig:H-circuit}.
\begin{figure}\label{fig:H-circuit}
\end{figure}
However, we have developed a more resource-efficient way to perform this gate that we describe here.
If we want to perform a Hadamard gate on a $(c, P)$ defect, we first prepare an ancilliary $(c,P)$ defect in the state $|0\rangle$ and perform a $\CNOT$ gate from the defect qubit to this defect ancilla using the circuit of Fig.~\ref{fig:CNOT-same-type}. This encodes the original defect qubit into the two-qubit bit-flip repetition code across the two defects. The $ZZ$ operator for the two defect qubits is in the stabilizer group of this repetition code, so we can measure $ZZ$ without disturbing the encoded qubit. The logical $Z$ operator is a $c'$-colored chain of $Z$ operators around either of the defects and the logical $X$ operator is a $c$-colored chain of $X$ operators connecting the defects, where $c' \neq c$. This encoding is the one used at all times in the Raussendorf \textit{et~al.}\ scheme \cite{Raussendorf:2007a, Raussendorf:2007b}, but here we only use it to perform the Hadamard gate and go back to our original single-defect encoding once the gate is completed.
After we've encoded the defect qubit into two, we then perform individual $M_Z$ measurements on a $c'$-colored chain of qubits surrounding both defects, where $c' \neq c$. This separates the region of the two qubits from the substrate, so we can then apply $H$ transversally on the cut-out region without influencing the external substrate. This operation applies a logical Hadamard gate to the two defect qubits in the interior, but also turns them into $P'$-type defect qubits in the process, where $P'$ is conjugate to $P$ (\textit{i.e.}, $P' = Z$ if $P = X$, and $P' = X$ if $P = Z$). We then stitch the cut out region back into the code by measuring the $P$-check operators incident on the cut. The encoding circuit for the repetition code is run in reverse, and the resulting defect can be converted back to its original type and color using circuits of the form depicted in Fig.~\ref{fig:X-to-Z}.
\subsubsection{Injecting $|\pi/4\rangle$ into a defect}
To perform universal encoded quantum computation with defects, our approach requires defects encoded into the state $|\pi/4\rangle$ with an error below its distillation threshold, as discussed in the previous ``pancake'' architecture. We therefore need a method for injecting magic states into defects such that the injection process introduces errors at a rate below the distillation threshold. The single-qubit preparation threshold for a magic state is therefore the difference between its distillation threshold and the error introduced by its injection process.
It is worth remarking that this kind of injection process is used in defect-based surface code schemes as well \cite{Raussendorf:2006a, Raussendorf:2007a, Fowler:2008c}. In these schemes, one must not only inject $|\pi/4\rangle$ states, but also inject $|\pi/2\rangle$ states as well. However, the impact of errors introduced by errant injection has not been studied to our knowledge. It is unclear whether considering it will significantly alter the high threshold values numerically estimated for surface codes---the difference between a $1\%$ accuracy threshold and a $14\%$ distillation threshold is not that great, so it is reasonable to expect that it may be quite important, especially because injection generates small-sized defects that are not arbitrarily well-protected from noise at first. We do not investigate the impact of the injection process on the threshold for color codes here either, but we expect that it will be less consequential because the value of the color-code accuracy threshold is much lower than that for the surface codes.
To inject into a $(c, Z)$ defect, we identify the corner of the triangular substrate containing the $c$-colored plaquette and measure $M_Z$ on the qubit in the corner, isolating it from the code. We then apply $TH$ to the corner qubit and then measure the weight-four $X$ check in the corner, bringing the corner qubit back into the code. We then cease measuring the $Z$ check in the standard way, creating a single-plaquette $Z$-type defect in the corner. This defect is not well protected from noise, so we move it from the corner and grow it as fast as we can, so that the ambient noise doesn't degrade the fidelity of the encoded state.
\subsubsection{$T$ gate on a defect}
Given $|\pi/4\rangle$ defect qubits, we can distill them and use them to perform the $T$ gate in the same way as described in Secs.~\ref{sec:T-gate} and \ref{sec:T-state} for the ``pancake'' architecture.
\section{Conclusions}
\label{sec:conclusions}
\subsection{Fault-tolerant quantum computation}
We studied fault-tolerant quantum computation using color codes, inspired by ($a$) the need to minimize qubit transport in real technologies having 2D layouts and ($b$) the high accuracy thresholds reported for similar topological codes. We framed our study with a well-defined quantum control model and three physically-motivated noise models of increasing realism which we call the code-capacity noise model, the phenomenological noise model, and the circuit-based noise model.
The strategy behind our study was to first understand how to fault-tolerantly simulate the identity gate via fault-tolerant quantum error correction and then extend this understanding to how to fault-tolerantly simulate a universal set of quantum gates capable of general-purpose quantum computation.
In the course of studying fault-tolerant quantum error correction, we formulated most-likely-error decoding for color codes as a mathematical optimization problem known as an integer program. We also developed feasible schedules for parallelized syndrome extraction for the most efficient family of color codes, the 4.8.8 color codes. To better understand the performance of our decoder, we elaborated a previously-established connection between the performance of our decoder and some statistical-mechanical classical spin models.
Our numerically-estimated value for most-likely-error fault-tolerant quantum error correction for 4.8.8 color codes in the code-capacity noise model is 10.56(1)\%. This is not significantly different from what had previously been estimated for optimal decoding of these and the 6.6.6 color codes, or most-likely-error or optimal decoding of Kitaev's 4.4.4.4 surface codes. Indeed, the upper bound for any CSS code is slightly more than 11\%, so all of these codes perform close to optimally in this noise model. To support our numerical estimate, we proved that the threshold is at least $8.335\,745\,(1)\%$ using a self-avoiding walk technique.
Our numerically-estimated value for the accuracy threshold of most-likely-error fault-tolerant quantum error correction for 4.8.8 color codes in the phenomenological noise model is 3.05(4)\%. Again, this is not significantly different from what had previously been estimated for optimal decoding of the 6.6.6 color codes, or most-likely-error or optimal decoding of Kitaev's 4.4.4.4 surface codes. We attribute the nominal improvement we find relative to Kitaev's surface codes for both this and the previous noise model to the fact that the color codes have higher-weight stabilizer generators, which should be modeled as more errant, but which aren't in these noise models. To support our numerical estimate, we proved that the threshold is at least $0.3096\%$ using a self-avoiding walk technique.
Our numerically-estimated value for most-likely-error fault-tolerant quantum error correction for 4.8.8 color codes in the circuit-based noise model is 0.082(3)\%. By attempting to optimize the syndrome extraction circuit by hand, we ended up surprisingly \emph{decreasing} our threshold estimate to 0.080(3)\%, suggesting that optimizing the syndrome extraction circuit to find the highest threshold is a nontrivial task. Unlike our findings for the previous two noise models, our accuracy-threshold estimate is in fact significantly different from what had previously been estimated for most-likely-error decoding of Kitaev's 4.4.4.4 surface codes---it is nearly a tenth the comparable value of $0.68\%$. That said, it is consistent with the value of ``about 0.1\%'' estimated using a different suboptimal decoder for these codes considered in Ref.~\cite{Wang:2009b}. However, the estimate in Ref.~\cite{Wang:2009b} lacked any error analysis, so it is hard to determine how consistent these results truly are. We believe that the reduction in threshold relative to the surface code threshold comes from the increased weight of the stabilizer generators for the 4.8.8 color code. Based on this, we predict that the 6.6.6 color codes will have a quantum error-correction accuracy threshold for this noise model somewhere between 0.082(3)\% and 0.68\% without any additional optimizations. We did not prove a lower bound on the threshold in this noise model, as the self-avoiding walk technique breaks down for this noise model.
To extend our results to general-purpose fault-tolerant quantum computing, we considered two different approaches. In the first, the architecture consisted of 2D surfaces stacked like pancakes in which each surface corresponded to a logical qubit and almost all operations were either global transversal operations or local syndrome extraction operations. In the second, the architecture consisted of an extended 2D surface in which logical qubits were associated with ``defects'' and almost all operations were either defect braiding by local measurements or local syndrome extraction operations.
In the ``pancake'' architecture, we showed that encoded universal quantum computation was possible using only local stabilizer measurements, global transversal operations, and the time-reversed coherent encoding circuit for the color code, which was used to inject magic states. Each gate in this architecture has its own accuracy threshold that is a significant fraction of the quantum error correction (memory) threshold.
In the ``defect'' architecture, we showed that encoded universal quantum computation was possible using only local stabilizer measurements, code deformation, and transversal operations on isolated regions. These deformations came in different forms, including growing small defects into large ones, braiding defects around each other for encoded $\CNOT$ gates, and isolating defects from the rest of the code. Each gate has the same accuracy threshold as the quantum error correction (memory) threshold, although errors afflicting injected magic state defects before they are grown to full size may dominate the threshold for the less realistic code-capacity and phenomenological noise models.
Because the defect architecture has a higher threshold and is more consistent with the original motivation for our study---namely that many technologies are restricted to a single 2D layout---we believe the defect-based approach to be the most practical. To that end, we extended some of the defect-based approach for color codes presented in Ref.~\cite{Fowler:2008c} so that a significantly higher density of defects can be stored and processed in the surface.
\subsection{Relation to statistical-mechanical phase transitions}
\label{sec:RBIM-conclusions}
It has been previously established that there is a mapping between quantum color codes and a classical statistical-mechanical model known as the three-body random-bond Ising model (3BRBIM). In this mapping, each check maps to a classical $\pm 1$ spin and each qubit maps to a three-body interaction, with the interaction being ferromagnetic if the qubit is not in error and antiferromagnetic if it is. Specifically, the Hamiltonian constructed by this mapping is
\begin{align}
H &= \sum_{\text{qubits } q} J_q \prod_{\text{checks }c \ni q} S_c,
\end{align}
where $J_q \in \pm 1$ indicates a flip on qubit $q$ and $S_c \in \pm 1$ indicates the eigenvalue of the check $c$.
A feature of the mapping is that the code capacity for any particular decoding algorithm represents a point on the boundary of the order-disorder transition of the associated 3BRBIM. Our integer-programming decoder is an ``energy-minimizing'' decoder in this paradigm, corresponding to the phase boundary at zero temperature. Because our code-capacity value of 10.56(1)\% is lower than the code capacity of 10.925(5)\% of a ``free-energy-minimizing'' decoder implicitly explored by Ohzeki \cite{Ohzeki:2009b}, this demonstrates that the phase boundary of the 3BRBIM is ``re-entrant'' as depicted in Fig.~\ref{fig:Nishimori}, violating the so-called Nishimori conjecture for this system. This result is counterintuitive because it states that the 3BRBIM can become \emph{more} ordered by increasing the temperature, depending on the system's quenched disorder parameter. It would be exciting to see experimental confirmation of this effect.
\subsection{Future directions}
While we have been able to answer many questions about fault-tolerant quantum computing using color codes, practicalities have necessarily limited the focus of our analysis, leaving other related questions open. Our results also also raise new questions that we believe are worthy of study.
One future direction we mentioned is optimizing the syndrome extraction circuit. One could also examine using more elaborate ancilla states in the circuit, such as those used in the schemes proposed by Shor \cite{Shor:1996a}, Steane \cite{Steane:1998a}, and Knill \cite{Knill:2004a}. In any scheme one chooses, further improvement may still be possible by transforming the circuit used in an implementation.
Another future direction we alluded to is optimizing the decoding algorithm. One could examine the performance of the truly optimal decoder for the circuit model which accounts for the correlations in the noise induced by the syndrome extraction circuit. This will yield an upper bound on the accuracy threshold for the noise model(s) studied. On the other end of the spectrum, it would be useful to explore the performance of faster decoders which don't yield as high a threshold as the MLE decoder but which may be more valuable in practice. The renormalization group decoder \cite{Duclos-Cianci:2009a} and minimum-weight perfect matching decoder \cite{Dennis:2002a} (using a mapping of one color code to two Kitaev surface codes \cite{Duclos-Cianci:2011a}) are examples of this. Another alternative is to generalize the results of Feldman \textit{et~al.}, who developed an efficient linear-program decoder for binary codes based on an integer-program-based decoder similar to the one we developed here \cite{Feldman:2005a}.
The lower bound technique of self-avoiding walks that we used is certainly not the tightest, and it may be of interest to establish tighter lower bounds. For tighter bounds, it may be possible to use different techniques. In the case of the circuit-based noise model, the self-avoiding walk bound technique breaks down dramatically, and it would be worth exploring other lower-bound techniques in this setting.
While we believe the noise and control model that we studied is reasonable, it is certainly not unique and can be improved upon with more experimental input. As shown by Levy \textit{et~al.}, \cite{Levy:2009a, Levy:2011a}, when more realistic models are included, conclusions regarding fault tolerance can change dramatically. Even at an abstract level, one could modify our depolarizing noise model for $\CNOT$ gates so that it acted ideally with probability $1-p$ and applied one of the fifteen nontrivial Pauli operators with probability $p/15$ each rather than acting ideally with probability $1-p$ and applying one of the sixteen Pauli operators with probability $p/16$.
While we gave a prescription for injecting magic states into the color code for both the pancake and 2D defect-based architectures, we did not carefully study the threshold of the circuits used for injection. To our knowledge, this type of study has not been performed for Kitaev's surface codes either. Such studies would be valuable, as it could be the case that the magic state preparation threshold is actually less than the accuracy threshold reported for all of the other gates, even though the distillation threshold for the magic states is higher than the accuracy threshold for the other gates.
Finally, the connection between color codes and the three-body random-bond Ising model allowed us to explore the structure of order-disorder transition in the latter model by studying the former. This is one of the rare examples where a purely quantum information theoretic result has led to greater understanding of a classical system. Kitaev's surface codes and the two-body random-bond Ising model have a similar connection and have admitted a similar study \cite{Dennis:2002a, Wang:2003a}. It is clear that it is the CSS structure of these codes that admits these studies; one could argue that every CSS code is a topological code for some topology, having an associated classical statistical-mechanical model for a given quantum noise model. It might be interesting to use the fault-tolerant decoding of CSS codes generally as a tool to explore related statistical-mechanical systems with quenched disorder.
\begin{acknowledgments}
We would like to thank the following individuals for helpful discussions: Hector Bombin, Bob Carr, Chris Cesare, Guillaume Duclos-Cianci, Bryan Eastin, Austin Fowler, Anand Ganti, Peter Groszkowski, Jim Harrington, Charles Hill, Lloyd Hollenberg, Uzoma Onunkwo, Cindy Phillips, David Poulin, Robert Raussendorf, and David Wang. We would also like to thank Dave Gay for use of AMPL mathematical programming language. PRR was supported by the Quantum Institute at Los Alamos National Laboratories. AJL and JTA were supported in part by the National Science Foundation through Grant 0829944. JTA and AJL were supported in part by the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.
\end{acknowledgments}
\end{document} | arXiv |
Gerrymandering and the whole community
The US Supreme Court's decision to hear the Gill v. Whitford gerrymandering case is raising hopes that the judicial system will reign in the worst abuses of partisan electoral districting. Academics are partly to thank for the success of the legal challenge so far: the initial District Court ruling striking down the Republican-drawn map relied on the concept of the electoral "efficiency gap" metric developed by the legal scholar Nicholas Stephanopoplous and the political scientist Eric McGhee. It's a clever and (hopefully) legally-defensible method for proving that gerrymandering schemes are not just unfair but also unconstitutional. But if the "efficiency gap" is useful in quantifying just how much of a partisan advantage is being squeezed out of tricky borderlines, it's less useful as a method for drawing good districts.
That's because in both the minds of ordinary voters as well as in the theoretical underpinning of the Constitution, what matters most about an electoral unit is that it should represent a group of voters who feel like they have something in common—that is, districts and geographic communities should match as best as possible. When we look at a gerrymandered map, what makes it seem so offensive is that its bizarrely-shaped geographies don't follow the lines of any recognizably coherent cities or regions.
How do you statistically measure whether a map "looks weird" or not, though? And what exactly constitutes a geographically bound-together community of people, anyway? That's a question for geographers, not for political scientists—and although it seems simple at first, it turns out to be surprisingly challenging.
One major wrinkle of the problem lies in the fact that every electoral district must have the same number of people. But states aren't neatly divided into organic, self-contained cells each with the exact population necessary to form an electoral unit. For instance, imagine a state with one big city that has a slightly larger population than the necessary size of an electoral district. Some of that city is going to have to get lopped off and included with one of the neighboring rural electoral districts. But which part should get cut—and how do we choose in a way that doesn't cross the line into devious political advantage-seeking?
Geographers have many methods for taking the complex weave of spatial interactions and empirically dividing them into segments—a process called regional delineation. You can look at commutes, banking and commodity flows, watersheds, or even sports franchise fandoms in order to divide territory up into areas that seem like they belong together. But because those areas of common interest may have considerably different populations, they need to be sliced up in a way that produces equally-sized districts. That's a challenge that no purely-statistical approach has yet been able to unproblematically solve.
Nevertheless, the criteria of geographic likeness—essentially, a kind of neighborliness extended over large areas—ought to be the fundamental principle for combining people together for the function of electing their representatives. It's a somewhat different criteria, however, than the one drawn from political science which emphasizes partisan symmetry as the highest principle. The goal of drawing districts to match real-life communities of interest doesn't necessarily ensure that the districts match the overall partisan composition of the state. In fact, the opposite can sometimes be the case. Consider, for example, this imaginary state with four counties and two electoral districts.
Right now, this state could be accused of a partisan gerrymander in favor of the Democrats. Although the state is close to even in its partisan makeup, with 53% Democrats and 47% Republicans, the Democrats win both districts. Here's how the efficiency-gap calculation would break down:
R Wasted
D wasted
District 1 550 700 550 75
$$G = \frac{(550+615)-(75+10)}{550+700+615+635} = 0.432$$
The Republicans here have "wasted" all of their votes—leading to a colossal efficiency gap metric of 43%.
It would seemingly be much fairer, then, to draw the districts this way, so that the Democrats win the first district, while the Republicans win the second.
In this case, the efficiency gap is much smaller—just under 7%.
District 1 505 745 505 120
District 2 660 590 35 590
But so far the only information we've been working with is the raw, abstract partisan breakdown of the four counties. What if we added in some qualitative geography, and I told you that the imaginary state looked like this?
Suddenly the original electoral map—the one that seemed unfair at first—seems to make a lot more sense. Counties A and B seem to have a lot in common with each other in terms of economic and environmental relationships, and so do Counties C and D. Joining County A and C together, and County B and D together, violates the principle that places which are related to one another should vote together.
If the problem looks difficult in this highly-stylized imaginary map, imagine how much harder it gets when dealing with a real state, full of complicated cities and towns of every size, and overlapping, blurry regions of common interest. There are no perfect breaks like the range of hills in our fictional example above. No matter where a line is drawn, it will be cutting across somebody's sense of their home region. The best geographers can do is to try to put the lines where they disrupt the fewest number of connections as possible.
One radical way of getting around this problem is to abolish the single-member district system altogether, and replace it with a system of proportional party-list voting like the kind found in many parliamentary systems. An even more speculatively experimental approach would be to organize "districts" on a basis totally separate from geography, like the Vocational Panels of the Irish Seanad. For instance, in an election for 100 seats, you could divide the electorate into income percentiles, so that the richest 1% formed a "district" with one representative, the poorest 1% formed another "district" with one representative, and so on for each 1% slice of the population. (This would have the extra positive effect of dramatically curtailing the top 1%'s disproportionate chokehold on Congress.)
But so long as we have voting aggregated into geographic containers—which will almost certainly continue to be the case in the US—we should make sure that those containers feel like they make geographic sense, regardless of their partisan composition. Empirical studies of spatial relationships can provide a first step for determining what places belong together. After that, one useful process would be to ask people to draw the outlines of their own communities on a map, and try as best as possible to make electoral borders follow the lines where people imagine a break in the human landscape.
Whether it's statistical data or participatory research that drives the delineation of sensible regions, though, one seemingly-trivial reform could help force districting commissions to justify the logic of their line drawing: giving districts regionally identifiable names. Instead of a colorless administrative term like MO-4, why not call it, say, the Osage Plain District? It would be almost impossible to think of a regional term to capture the regional identity of an extreme gerrymander like PA-7. That in itself might not be enough to stop craven politicians, but it would at least start getting citizens and voters to think about their electoral constituencies and their imagined regional communities in the same context.
For all its legal utility in possibly convincing the courts to strike down gerrymandering, the "efficiency gap" doesn't actually tell us much about whether an electoral district coheres together geographically or not. The cultural geographer Carl Sauer wrote a critique of gerrymandering in 1918 which focused on how crazy electoral lines broke apart communities and thus crippled their ability to act as political entities. "The interests of representative government demand that such a crystallized opinion be given a voice," Sauer insisted, and "that it be not concealed by the division of the natural unit."
Sauer was wrong to think that there are perfect "natural units" that correspond in size to electoral districts. But he was right to say that one crucial function of an electoral district in a democratic system is not merely its role in partisan politics, but in the way that it gives a local community representation in the larger political entity. To preserve that function, we need an understanding of geography that is made up of something more than just red and blue dots.
27 June 2017, Fairchild Hall | CommonCrawl |
\begin{document}
\title[Average number of zeros of characters]{Average number of zeros of characters of finite groups}
\author{Sesuai Yash Madanha} \address{Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, 0002, South Africa} \email{[email protected]}
\thanks{}
\subjclass[2010]{Primary 20C15, 20D10}
\date{\today}
\keywords{zeros of characters, solvable groups, supersolvable groups, nilpotent groups, abelian groups}
\begin{abstract} There has been some interest on how the average character degree affects the structure of a finite group. We define, and denote by $ \mathrm{anz}(G) $, the average number of zeros of characters of a finite group $ G $ as the number of zeros in the character table of $ G $ divided by the number of irreducible characters of $ G $. We show that if $ \mathrm{anz}(G) < 1 $, then the group $ G $ is solvable and also that if $ \mathrm{anz}(G) < \frac{1}{2} $, then $ G $ is supersolvable. We characterise abelian groups by showing that $ \mathrm{anz}(G) < \frac{1}{3} $ if and only if $ G $ is abelian. \end{abstract}
\maketitle \begin{center} \textit{Dedicated to the memory of Kay Magaard} \end{center}
\section{Introduction}\label{s:intro}
Let $ G $ be a finite group and $ {\mathrm {Irr}} (G) $ be the set of complex irreducible characters of $G$. Let $T(G)$ be the sum of degrees of complex irreducible characters of $G$, that is, $ T(G)=\sum _{\chi \in {\mathrm {Irr}} (G)}\chi (1) $. Denote by $k(G)$ the number of conjugacy classes of $G$. Then $ k(G)=|{\mathrm {Irr}} (G)| $. Define the average character degree of $ G $ by \begin{center}
$ \mathrm{acd}(G):=\dfrac{T(G)}{|{\mathrm {Irr}} (G)|} $. \end{center} Recently, a lot of authors have investigated the average character degree of finite groups and how it influences the structure of the groups (see \cite{MT-V11,MN14,MN15}). In fact, Magaard and Tong-Viet \cite{MT-V11} proved that $ G $ is solvable whenever $ \mathrm{acd}(G) < 2 $ and they conjectured that their result still holds if $ \mathrm{acd}(G) \leqslant 3 $. This conjecture was settled by Isaacs, Loukaki and Moret\'o \cite[Theorem A]{ILM13} and they obtained some sufficient conditions for a group to be supersolvable and also to be nilpotent. In the same article, the authors of \cite{ILM13} conjectured that the best possible bound is when $ \mathrm{acd}(G) < \frac{16}{5} $ for $ G $ to be solvable. Moret\'o and Nguyen showed in \cite[Theorem A]{MN14}, that indeed this was the best bound. We shall state the results on average character degrees with the best bounds below.
\begin{theorem} \cite[Theorem A]{MN14}\label{MN14TheoremA} Let $ G $ be a finite group. If $ \mathrm{acd}(G) < \frac{16}{5} $, then $ G $ is solvable. \end{theorem}
\begin{theorem} \cite[Theorem B]{ILM13}\label{ILM13TheoremB} Let $ G $ be a finite group. If $ \mathrm{acd}(G) < \frac{3}{2} $, then $ G $ is supersolvable. \end{theorem}
\begin{theorem} \cite[Theorem C]{ILM13}\label{ILM13TheoremC} Let $ G $ be a finite group. If $ \mathrm{acd}(G) < \frac{4}{3} $, then $ G $ is nilpotent. \end{theorem} More work on average character degrees of finite groups is found in \cite{HT17, Hun17, HT20}. Another invariant that has been studied is the so-called average class size. We shall refer the reader to \cite{GR06, ILM13, Qia15} for more bibliographic information.
In this article, we investigate the corresponding problem for zeros of characters of finite groups. We have to define a new invariant first. Recall that if $ \chi (g)=0 $ for some $ g\in G $ and $ \chi \in {\mathrm {Irr}} (G) $, we say $ \chi $ vanishes on $ g $. So $ \chi $ vanishes on conjugacy classes. In the same spirit, we define $ \mathrm{nz}(G) $ to be the number of zeros in the character table of $ G $ and the \textit{average number of zeros of characters} of $ G $ by: \begin{center}
$ \mathrm{anz}(G):= \dfrac{\mathrm{nz}(G)}{|{\mathrm {Irr}} (G)|} $. \end{center} Since linear characters do not vanish on any conjugacy class, $ \mathrm{anz}(G)=0 $ for an abelian group $ G $. A classical theorem of Burnside \cite[Theorem 3.15]{Isa06} shows that $ \chi (g)=0 $ for some $ g\in G $ and a non-linear irreducible character $ \chi $, that is, $ \chi $ vanishes on some conjugacy class. This means $ \mathrm{anz}(G) > 0 $ for non-abelian groups. We show that zeros of characters influence the structure of a finite group.
We begin by proving a result analogous to \cite[Theorem 1.1]{MT-V11}:
\begin{thmA} If $ N $ is a minimal non-abelian normal subgroup of $ G $, then there exists a non-linear character $ \chi \in {\mathrm {Irr}}(G) $ such that $ \chi _{N} $ is irreducible and $ \chi $ vanishes on at least two conjugacy classes of $ G $. \end{thmA}
We need Theorem A to prove our second result below which corresponds to Theorem \ref{MN14TheoremA}:
\begin{thmB} Let $ G $ be a finite group. If $ \mathrm{anz}(G) < 1 $, then $ G $ is solvable. \end{thmB} Note that since $ \mathrm{anz}(\mathrm{A}_{5})=1 $, this bound is sharp.
We show that the converse of Theorem B does not necessarily hold. Let $ Q $ be a Sylow $ 3 $-subgroup of $ S= {\mathrm {PSL}}_{3}(7) $ and suppose $ G=N_{S}(Q) $. We have that $ \mathrm{anz}(G)= 7/6>1 $.
In Theorems \ref{ILM13TheoremB} and \ref{ILM13TheoremC}, the bounds are optimal since $ \mathrm{acd}(\mathrm{A}_{4})=\frac{3}{2} $ and $ \mathrm{acd}(\mathrm{S}_{3})=\frac{4}{3} $. Since $ \mathrm{anz}(\mathrm{A}_{4})=\frac{1}{2} $ and $ \mathrm{anz}(\mathrm{S}_{3})=\frac{1}{3} $, the following corresponding results hold: \begin{thmC} Let $ G $ be a finite group. If $ \mathrm{anz}(G) < \frac{1}{2} $, then $ G $ is supersolvable. \end{thmC}
\begin{thmD} Let $ G $ be a finite group. If $ \mathrm{anz}(G) < \frac{1}{3} $, then $ G $ is nilpotent. \end{thmD}
Note that for an abelian group $ G $, $ \mathrm{acd}(G) = 1 $ and $ \mathrm{anz}(G)=0 $. We show in Remark \ref{Abeliangroups} that there is no number $ c > 1 $ such that $ \mathrm{acd}(G) < c $ implies that $ G $ is abelian. Contrary to $ \mathrm{acd}(G) $, we show that there exists a bound $ c > 0 $, such that $ \mathrm{anz}(G) < c $ implies that $ G $ is abelian. Hence we obtain a non-trivial characterisation of abelian groups in terms of zeros of characters:
\begin{thmE} Let $ G $ be a finite group. Then $ G $ is abelian if and only if $ \mathrm{anz}(G) < \frac{1}{3} $. \end{thmE} When $ G $ is of odd order we obtain a bound that is better than that in Theorem C. The following result is analogous to \cite[Theorem D(a)]{ILM13}.
\begin{thmF} Let $ G $ be a finite group of odd order. If $ \mathrm{anz}(G) < 1 $, then $ G $ is supersolvable. \end{thmF} However, the bound in Theorem F might not be optimal. Indeed, in \cite[Theorem D(a)]{ILM13}, the bound is best possible since there exists a group $ G $ of odd order which is not supersolvable such that $ \mathrm{acd}(G)=\frac{27}{11} $ and also $ \mathrm{anz}(G)=\frac{16}{11} $ ($ G $ is the unique non-abelian group of order $ 75 $). Hence we propose the following conjecture:
\begin{conj1} Let $ G $ be a finite group of odd order. If $ \mathrm{anz}(G) < \frac{16}{11} $, then $ G $ is supersolvable. \end{conj1}
The article is organized as follows. In Section \ref{pre}, we list some preliminary results. In Section \ref{solvableandnonsolvable}, we prove Theorems A and B and in Section \ref{supersolvableuptoabelian} we prove Theorems C to F. We also show that there exist an infinite family of non-abelian nilpotent groups $ G $ such that $ \mathrm{acd}(G) < \frac{4}{3} $ in this last section.
\section{Preliminaries}\label{pre}
\begin{lemma}\label{extendible} Suppose that $ N $ is a minimal normal non-abelian subgroup of a group $ G $. Then there exists an irreducible character $ \theta $ of $ N $ such that $ \theta $ is extendible to $ G $ with $ \theta (1)\geqslant 5 $. In particular, if $ N $ is simple such that \begin{itemize} \item[(a)] $ N $ is isomorphic to $ \mathrm{A}_{5} $, then $ \theta(1)=5 $ \item[(b)] $ N $ is isomorphic to $ \mathrm{A}_{n} $, $ n\geqslant 7 $, then $ \theta(1)=n-1 $ \item[(c)] $ N $ is isomorphic to a finite group of Lie type defined over a finite field distinct from the Tits group $ ^{2}\mathrm{F}_{4}(2)' $ and $ {\mathrm {PSL}}_{2}(5) $, then $ \theta $ is the Steinberg character. \end{itemize} \end{lemma} \begin{proof} The first assertion is the statement of \cite[Theorem 1.1]{MT-V11} and the second assertion follows from the proof of \cite[Theorem 1.1]{MT-V11}. \end{proof}
\begin{lemma}\cite[Corollary 6.17]{Isa06}\label{injectivemap} Let $ N $ be a normal subgroup of $ G $ and let $ \chi \in {\mathrm {Irr}}(G) $ be such that $ \chi _{N}=\theta \in {\mathrm {Irr}}(N) $. Then the characters $ \beta \chi $ for $ \beta \in {\mathrm {Irr}} (G/N) $ are irreducible, distinct for distinct $ \beta $ and are all of the irreducible constituents of $ \theta ^{G} $. \end{lemma} Recall that $ {\mathrm {Irr}}(G{\mid} K)= \{\chi \in {\mathrm {Irr}}(G): K\nsubseteq \ker \chi\} $. For $ \chi \in {\mathrm {Irr}} (G) $ we write $ n\upsilon(\chi )$ for the number of conjugacy classes of $G$ on which $\chi$ vanishes. Then $ \mathrm{nz}(G{\mid} K):=\sum _{\chi \in {\mathrm {Irr}}(G\mid K)}n\upsilon(\chi) $.
\begin{lemma}\label{noofzerosfactorgroup} Let $ K $ be a normal subgroup of $ G $ such that $ K\leqslant G' $. If $ \mathrm{anz}(G) < 1 $, then $ \mathrm{anz}(G/K)\leqslant \mathrm{anz}(G) $. \end{lemma} \begin{proof}
Let $ a=\mathrm{anz}(G/K) $. Note that $ k(G)=|{\mathrm {Irr}} (G)|=|{\mathrm {Irr}}(G/K)|+|{\mathrm {Irr}}(G{\mid} K)| $. If $ |{\mathrm {Irr}}(G/K)|=c $, $ |{\mathrm {Irr}}(G{\mid} K)|=d $, $ \mathrm{nz}(G/K)=m $ and $ \mathrm{nz}(G{\mid} K)=n $, then $ a=\frac{m}{c} $. Since $ K\leqslant G' $, $ {\mathrm {Irr}}(G{\mid} K) $ is a set of non-linear irreducible characters of $ G $ and also using Burnside's Theorem, $ |{\mathrm {Irr}}(G{\mid} K)|= d\leqslant n=\mathrm{nz}(G{\mid} K) $. Since the character table of $ G/K $ is a sub-table of the character of $ G $, we have that $ \mathrm{nz}(G/K)\leqslant \mathrm{nz}(G) $. Hence $ 1 > \mathrm{anz}(G)=\frac{m + n}{c + d}\geqslant \frac{m + d}{c + d}\geqslant \frac{m}{c}=a $ as required. \end{proof}
We show that Theorem B holds for perfect groups.
\begin{lemma}\label{G=G'holds} Let $ G $ be a finite group such that $ G=G' $. Then $ \mathrm{anz}(G)\geqslant 1 $. \end{lemma} \begin{proof} Since $ G $ has one linear character, it is sufficient to show that there exists a non-linear irreducible character of $ G $ that vanishes on at least two conjugacy classes. Suppose the contrary, that is, every non-linear irreducible character of $ G $ vanishes on exactly one conjugacy class. Then by \cite[Proposition 2.7]{Chi99}, $ G $ is a Frobenius group with a complement of order $2$ and an abelian odd-order kernel, that is, $ G $ is solvable, contradicting the hypothesis that $ G $ is perfect. Hence the result follows. \end{proof}
Let $ G $ be a finite group and $ \chi \in {\mathrm {Irr}}(G) $. Recall that $\upsilon(\chi ):= \{x\in G\mid \chi (x)=0\} $.
\begin{lemma}\label{ZSS10Lemma2.1} Let $ G $ be a non-abelian finite group and $ \chi \in {\mathrm {Irr}}(G) $ be non-linear. Suppose that $ N $ is a normal subgroup of $ G $ such that $ G'\leqslant N < G $. If $ \chi_{N} $ is not irreducible, then the following two statements hold: \begin{itemize} \item[(a)] There exists a normal subgroup $ K $ of $ G $ such that $ N\leqslant K < G $ and $ G{\setminus} K \subseteq \upsilon(\chi) $. \item[(b)] If $ (G{\setminus} G')\cap \upsilon(\chi) $ consists of $ n $ conjugacy classes of $ G $, then \begin{center}
$ |G{:}G'|-|K{:}G'|\leqslant n $. \end{center} \end{itemize} \end{lemma} \begin{proof} For (a), note that since $ G/N $ is abelian, it follows that $ \chi $ is a relative $ M $-character with respect to $ N $ by \cite[Theorem 6.22]{Isa06}. This means that there exists $ K $ with $ N\leqslant K \leqslant G $ and $ \psi \in {\mathrm {Irr}}(K) $ such that $ \chi =\psi ^{G} $ and $ \psi _{N}\in {\mathrm {Irr}}(N) $. Hence $ G{\setminus} K \subseteq \upsilon(\chi) $, Since $ \chi _{N} $ is not irreducible we have that $ K < G $ and the result follows.
For (b), let $ g_{1}, g_{2}, \dots, g_{m} $ be a complete set of representatives of the cosets of $ G' $ in $ G $, with $ g_{1}, g_{2}, \dots ,g_{k} $ a complete set of representatives of the cosets of $ G' $ in $ K $ so that $ m=|G:G'| $ and $ k=|K:G'| $. Since $ \chi $ vanishes on $ G{\setminus} K $, we have that $ \chi $ vanishes on $ g_{k+1}, g_{k+2}, \dots ,g_{m} $. But these elements are not conjugate and so $ m-k\leq n $ as required. \end{proof}
In the following results we will consider groups with an irreducible character that vanishes on one conjugacy class. We denote by $ k_{G}(N) $, the number of conjugacy classes of $ G $ in $ N $ where $ N $ is a subset of $ G $. We first prove an easy lemma:
\begin{lemma}\label{Lemma2(2)}
Let $ N $ be a normal subgroup of $ G $. If $ k_{G}(G{\setminus} N)=1 $, then $ |G{:}N|=2 $ and $ G $ is a Frobenius group with an abelian kernel $ N $ of odd order. \end{lemma} \begin{proof}
Since $ k_{G}(G{\setminus} N)=1 $, all the elements in $ G{\setminus} N $ are conjugate. It follows that if $ x\in G{\setminus} N $, then the size of the conjugacy class containing $ x $ , say $ c $, is $ |G|-|N| $. Since $ N $ is a proper subgroup of $ G $, $ c\geq |G|-|G|/2=|G|/2 $. But the size of any conjugacy class of a non-trivial group is a proper divisor of the group, so we deduce that $ |N|=|G|/2 $. Then $ C_{G}(x)=\langle x\rangle $ is of order $ 2 $, so $ G=\langle x \rangle N $ is a Frobenius group with a complement $ \langle x \rangle $ of order $ 2 $ and hence the Frobenius kernel $ N $ is abelian of odd order. \end{proof}
We also need this result:
\begin{proposition}\label{classificationoneclass} Let $ G $ be an almost simple group. If $ \chi \in {\mathrm {Irr}} (G) $ vanishes on exactly one conjugacy class, then one of the following holds: \begin{itemize} \item[(a)] $ G= {\mathrm {PSL}}_{2}(5) $, $ \chi (1)=3 $ or $ \chi (1)=4 $; \item[(b)] $ G\in \{\mathrm{A}_{6}{:}2_{2},~ \mathrm{A}_{6}{:}2_{3}\} $, $ \chi(1)=9 $ for all such $ \chi\in {\mathrm {Irr}} (G) $; \item[(c)] $ G={\mathrm {PSL}}_{2}(7) $, $ \chi (1)= 3 $; \item[(d)] $ G={\mathrm {PSL}}_{2}(8){:}3 $, $ \chi (1)=7 $; \item[(e)] $ G={\mathrm {PGL}}_{2}(q) $, $ \chi (1)=q $, where $ q\geq 5 $; \item[(f)] $ G=$ $^{2}\rm{B_{2}}(8){:}3 $, $ \chi(1)=14 $. \end{itemize} \end{proposition}
\begin{proof} It was shown in \cite[Theorem 5.2]{Mad19q} and the proof of \cite[Theorem 1.2]{Mad19zp} that all irreducible characters of $ G $ that vanishes on exactly one conjugacy class are primitive, the result follows. Hence the list in \cite[Theorem 1.2]{Mad19zp} is a complete one. The result then follows. \end{proof}
\begin{lemma}\label{Sei68} A finite group $ G $ has exactly one non-linear irreducible character if and only if $ G $ is isomorphic to one of the following: \begin{itemize} \item[(a)] $ G $ is an extra-special $2$-group. \item[(b)] $G$ is a Frobenius group with an elementary abelian kernel of order $ p^{n} $ and a cyclic complement of order $ p^{n}-1 $, where $ p $ is prime and $ n $ a positive integer. \end{itemize} Moreover, $ \mathrm{anz}(G)=\frac{m - 1}{ m + 1} $ for some integer $ m\geqslant 2 $. \end{lemma} \begin{proof}
The statements (a) and (b) follows from \cite[Theorem]{Sei68}. Let $ G $ be an extra-special $ 2 $-group of order $ 2^{2k+1} $ with $ |G/G'|=|G/Z(G)|=2^{2k} $. Then $ G $ has $ 2^{2k} $ linear characters and so $ 2^{2k} + 1 $ irreducible characters. Note that the non-linear irreducible character $ \chi $ of $ G $ is fully ramified with respect to $ Z(G) $. In particular, $ \chi $ vanishes on $ G{\setminus} Z(G) $. Hence $ \chi $ vanishes on at least $ 2^{2k} - 1 $ conjugacy classes. Since $ G $ has the identity and a non-trivial central element which are non-vanishing elements, $ \chi $ vanishes on exactly $ 2^{2k} - 1 $ conjugacy classes. Then $ \mathrm{anz}(G) = \frac{2^{2k} - 1}{2^{2k} + 1} $ and the result follows.
Suppose $ G $ is a Frobenius group with an elementary abelian kernel $ G' $ of order $ p^{n} $ and a cyclic complement of order $ p^{n} - 1 $. Note that $ |G/G'|=p^{n} - 1 $ and so $ G $ has $ p^{n} - 1 $ linear characters. Hence $ G $ has $ p^{n} $ irreducible characters. Since the non-linear irreducible character $ \chi $ of $ G $ is induced from irreducible character of $ G' $, $ \chi $ vanishes on $ G{\setminus}G' $. It follows that $ \chi $ vanishes on at least $ p^{n} - 2 $ conjugacy classes. Since $ G' $ has one non-trivial conjugacy class of $ G $, $ \chi $ does not vanish on $ G' $. Otherwise, $ \chi $ vanishes on $ G{\setminus} \{1\}$, a contradiction. Since $ G $ has $ p^{n} $ conjugacy classes, we have that $ \chi $ vanishes on exactly $ p^{n} - 2 $ conjugacy classes and so $ \mathrm{anz}(G)=\frac{p^{n} - 2}{p^{n}} $ as required. \end{proof}
We shall need the following two results to prove Theorem F.
\begin{lemma}\cite{Pal81}\label{Pal81} A finite group $ G $ has exactly two non-linear irreducible characters if and only if $ G $ is isomorphic to one of the following: \begin{itemize} \item[(a)] $ G $ is an extra-special $3$-group.
\item[(b)] $ G $ is a $ 2 $-group of order $2^{2k+2} $, $|G'| = 2 $, $ |Z(G)| = 4 $, and $ G $ has two non-linear irreducible characters with equal degree $ 2^{k} $. \item[(c)] $ G $ is a Frobenius group with an elementary abelian kernel of order $ 9 $ and Frobenius complement $ Q_{8} $. \item[(d)] $ G $ is a Frobenius group with an elementary abelian kernel of order $ p^{k} $ and a cyclic Frobenius complement of order $ (p^{k} - 1)/2 $, where $ p $ is odd prime.
\item[(e)] $ G/Z(G) $ is a Frobenius group with an elementary abelian kernel of order $ p^{k} $ and a cyclic Frobenius complement of order $ p^{k} - 1 $, where $ p $ is prime and $ |Z(G)|=2 $. \end{itemize} \end{lemma}
\begin{lemma}\cite[Theorem 2.6]{Qia07}\label{Qia07Theorem2.6} Let $ G $ be a finite group of odd order. Then $ G $ has an irreducible character that vanishes on exactly two conjugacy classes if and only if $ G $ is one of the following groups: \begin{itemize} \item[(a)] $ G $ is a Frobenius group with a complement of order $ 3 $. \item[(b)] There are normal subgroups $ M $ and $ N $ of $ G $ such that: $ M $ is a Frobenius group with the kernel $ N $; $ G/N $ is a Frobenius group of order $ p(p - 1)/2 $ with the kernel $ M/N $ and a cyclic complement of order $ (p - 1)/2 $ for some odd prime $ p $. In this case, $ \chi_{M} $ is irreducible.
\end{itemize}
\end{lemma}
\begin{lemma}\label{numbertheoryresult} Let $ \ell, m, n, b $ be positive integers with $ \ell,n \geqslant 2 $ and $ b \geqslant 2\ell + 1 $. Suppose that $ b= mn $. Then \begin{center} $ b - m \geqslant \ell + 1 $. \end{center} \end{lemma} \begin{proof} Suppose $ b $ is odd. If $ b $ is prime, then $ b - 1\geqslant 2\ell + 1 - 1 > \ell + 1 $. If $ b $ is not prime, then the greatest $ m $ is when $ n=3 $ and $ b - m \geqslant b - \frac{b}{3}\geqslant \frac{2}{3}b > \frac{b}{2}=\ell $. If $ b $ is even, then $ b \geqslant 2\ell + 2 $ and the greatest $ m $ is when $ n=2 $. Hence $ b - m\geqslant 2\ell + 2 -(\ell + 1)=\ell + 1 $ as required. \end{proof}
\section{Non-solvable and solvable groups}\label{solvableandnonsolvable}
\begin{proof}[{\textbf{Proof of Theorem A}}] We begin the proof by showing that we may assume that $ N $ is simple. For if $ N=T_{1}\times T_{2}\cdots \times T_{k} $, where $ T_{i}\cong T $, $ T $ is a non-abelian simple group, for $ i=1,2, \dots ,k $ and $ k\geqslant 2 $, then by Lemma \ref{extendible}, there exists $ \theta_{i} \in {\mathrm {Irr}}(T_{i}) $ such that $ \chi _{N}=\theta_{1}\times \theta _{2}\times \cdots \times \theta_{k} $, where $ \chi \in {\mathrm {Irr}}(G) $. Since $ \theta_{1}(1)> 1 $, there exists $ x\in T_{1} $ such that $ \theta_{1}(x)=0 $ by Burnside's Theorem. But $ \chi $ vanishes on $ (x,1,\dots , 1) $ and $ (x,x,1,\dots, 1) $. Since these two elements are not conjugate in $ G $, $ \chi $ vanishes on at least two conjugacy classes of $ G $.
We may assume that $ N $ is simple. Let $ C=C_{G}(N) $. We claim that $ C=1 $. Otherwise, $ N\times C $ is a normal subgroup of $ G $. There exists a non-linear $ \theta \in {\mathrm {Irr}}(N) $ such that $ \theta _{N}=\chi $ by Lemma \ref{extendible}. There exists $ x\in N $ such that $ \theta (x)=0 $. Then $ \chi(xc)=0 $ for any $ c\in C $. Let $ c\in C{\setminus}\{1\} $. Since $ x $ and $ xc $ are not conjugate in $ G $, $ \chi $ vanishes on at least two conjugacy classes. Hence $ C=1 $ and we have that $ G $ is almost simple.
By Lemma \ref{extendible}, there exists $ \chi $ such that $ \chi_{N} $ is irreducible and $ \chi(1)\geqslant 5 $. If $ \chi $ vanishes on two conjugacy classes, then the result follows. We may assume that $ \chi $ vanishes on one conjugacy class. By Proposition \ref{classificationoneclass} and considering the choice of $ \chi $ from Lemma \ref{extendible}, we are left with the following cases: $ G\in \{ {\mathrm {PGL}}_{2}(q), \mathrm{A}_{6}{:}2_{2}, \mathrm{A}_{6}{:}2_{3}\} $(note that $ {\mathrm {PSL}}_{2}(9)\cong \mathrm{A}_{6} $). Note that $ N={\mathrm {PSL}}_{2}(q) $. We will choose another irreducible character of $ N $ that extends to $ G $. Obviously that alternative character vanishes on at least two conjugacy classes. For $ G=\mathrm{S}_{5} $, let $ \theta\in {\mathrm {Irr}}(\mathrm{A}_{5}) $ be such that $ \theta(1)=4 $. Then $ \theta $ is extendible to $ {\mathrm {Aut}}(\mathrm{A}_{5})=\mathrm{S}_{5} $. Using the {\sf Atlas} {} \cite{CCNPW85}, we have that $ \chi_{G} $ vanishes on two elements of distinct orders. Hence the result follows.
For $ G\in \{ \mathrm{A}_{6}{:}2_{2}, \mathrm{A}_{6}{:}2_{3} \}$, using the {\sf Atlas}{} \cite{CCNPW85}, we can choose an irreducible character $ \theta $ of $ N $ of degree $ 10 $ that extends to $ {\mathrm {Aut}}(\mathrm{A}_{6}) $. Lastly, $ \chi $ vanishes on more than two conjugacy classes of $ G $. Hence the result follows.
Suppose $ G = {\mathrm {PGL}}_{2}(q) $, where $ q\geqslant 7 $. It is well known that $ {\mathrm {PSL}}_{2}(q) $ has irreducible characters of degree $ q-1 $ and $ q + 1 $. By \cite[Theorem A]{Whi13}, an irreducible character $ \theta $ of degree $ q + 1 $ is extendible to $ {\mathrm {Aut}}({\mathrm {PSL}}_{2}(q)) $ except when $ N={\mathrm {PSL}}_{2}(3^{f}) $ and $ G={\mathrm {PGL}}_{2}(3^{f}) $, with $ f $ an odd positive integer (case (iii) of \cite[Theorem A]{Whi13}). If $ N={\mathrm {PSL}}_{2}(3^{f}) $ and $ G={\mathrm {PGL}}_{2}(3^{f}) $ with $ f $ an odd positive integer, then we choose an irreducible character $ \theta $ of $ {\mathrm {PSL}}_{2}(3^{f}) $ of degree $ q - 1 $ which is extendible to $ {\mathrm {Aut}}({\mathrm {PSL}}_{2}(3^{f})) $. Thus the result follows. \end{proof}
\begin{proof}[{\textbf{Proof of Theorem B}}] We shall prove our result by induction on $ |G| $. We may assume that $ G $ is non-solvable. If $ G=G' $, then the result follows by Lemma \ref{G=G'holds}, so $ G\neq G' $.
Let $ G^{\infty} $ be the solvable residual of $ G $. Note that $ G^{\infty} $ is perfect. If $ N < G^{\infty} \leqslant G' $ is a minimal normal subgroup of $ G $, then $ G^{\infty}/N $ is perfect and so $ G/N $ is non-solvable. But $ \mathrm{anz}(G/N) < 1 $ by Lemma \ref{noofzerosfactorgroup} and hence $ G/N $ is solvable by induction, a contradiction.
We may assume that $ N=G^{\infty} $ is a non-abelian minimal normal subgroup of $ G $. By Theorem A, there exists $ \chi \in {\mathrm {Irr}}(G) $ such that $ \chi_{N} $ is irreducible and $ \chi $ vanishes on two conjugacy classes. Suppose the two conjugacy classes are represented by elements $ g_{1} $ and $ g_{2} $ of $ G $. Since $ \chi_{G'} $ is irreducible, we have that the character $ \beta\chi $ is irreducible for every linear $ \beta $ of $ G $ by Lemma \ref{injectivemap}. The $ \beta\chi $'s are distinct for distinct characters $ \beta $. We show that every character of the form $ \beta\chi $ also vanishes on $ g_{1} $ and $ g_{2} $. Then $ \beta\chi(g_{i})=\beta(g_{i})\chi({g_{i}})=\beta(g_{i})\cdot 0=0 $, where $ i\in \{ 1, 2\} $.
Hence for every linear character $ \beta $ of $ G $, there is a corresponding non-linear irreducible character of $ G $ of the form $ \beta\chi $ that vanishes on two conjugacy classes of $ G $. Let $ a $ be the number of linear characters of $ G $ and $ b $ be the number of non-linear irreducible characters not of the form $ \beta\chi $ (note that these irreducible characters may vanish on more than one conjugacy class). Then $ |{\mathrm {Irr}}(G)|=2a + b $ and $ \mathrm{nz}(G)=2a + b + c $, where $ c $ is a non-negative integer. Therefore $ \mathrm{anz}(G)=\frac{2a + b + c}{2a + b}\geqslant \frac{2a + b}{2a + b}\geqslant \frac{2a}{2a}= 1 $, concluding our argument. \end{proof}
\section{Supersolvable, nilpotent and abelian groups}\label{supersolvableuptoabelian}
\begin{proof}[{\textbf{Proof of Theorem C}}]
Suppose $ G $ is non-abelian, that is, $ G' > 1 $. We shall use induction on $ |G| $ to show that $ G $ is supersolvable. By Theorem B, $ G $ is solvable since $ \mathrm{anz}(G) < \frac{1}{2} < 1 $. Let $ N\leqslant G' $ be a minimal normal subgroup of $ G $. By Lemma \ref{noofzerosfactorgroup}, $ \mathrm{anz}(G/N)\leqslant \mathrm{anz}(G) < \frac{1}{2} $. Using the inductive hypothesis, $ G/N $ is supersolvable. If $ N $ is cyclic or $ N\leqslant \Phi(G) $, the Frattini subgroup, then $ G $ is supersolvable and the result follows.
Suppose that $ \chi \in {\mathrm {Irr}}(G) $ be non-linear such that $ \chi_{G'} $ is irreducible. By Lemma \ref{injectivemap}, for every linear character $ \beta_{i} \in {\mathrm {Irr}}(G/G') $, there exists a non-linear $ \beta_{i}\chi \in {\mathrm {Irr}}(G) $, that is, the number of linear characters of $ G $ is less than or equal to the number of non-linear irreducible characters of $ G $. Every non-linear irreducible character of $ G $ vanishes on at least one conjugacy class by \cite[Theorem 3.15]{Isa06}. Let $ |{\mathrm {Irr}}(G)|=a + b $, where $ a $ is the number of the non-linear characters and $ b $ is the number of linear characters and $ \mathrm{nz}(G)=a + c $, where $ c $ is non-negative integer. Then $ \mathrm{anz}(G)=\frac{a + c}{a + b}\geqslant \frac{a + c}{2a} \geqslant\frac{1}{2} $, since $ a\geqslant b $. This contradicts our hypothesis.
Hence every non-linear irreducible character $ \chi $ of $ G $ is such that $ \chi_{G'} $ is not irreducible. Note that $ G $ is an $ M $-group by \cite[Theorems 6.22 and 6.23 ]{Isa06} since $ G/N $ is supersolvable and $ N $ is abelian. By Lemma \ref{ZSS10Lemma2.1}, there exists a normal subgroup $ K $ of $ G $ such that $ G' \leqslant K < G $ and $ G{\setminus} K \subseteq \upsilon(\chi) $. If $ \upsilon(\chi) $ is a conjugacy class, then $ G $ is a Frobenius group with an abelian kernel and a complement of order two by Lemma \ref{Lemma2(2)}. Thus by \cite[Proposition 2.7]{Chi99}, every irreducible character of $ G $ vanishes on at most one conjugacy class. Note that $ G $ has two linear characters. Since $ \mathrm{anz}(G) < \frac{1}{2} $, $ G $ can only have one non-linear character. It follows that $ G $ has three conjugacy classes. This implies that $ G\cong \mathrm{S}_{3} $. Now $ \mathrm{S}_{3} $ is supersolvable group and we are done.
Therefore every non-linear irreducible character of $ G $ vanishes on at least $ \ell $ conjugacy classes, where $ \ell \geqslant 2 $. Suppose $ G $ has an irreducible character $ \chi $ that vanishes on exactly $ \ell $ conjugacy classes. Let $ b $ be the number of linear characters of $ G $ and $ m $ be the number of non-linear irreducible characters of $ G $. If $ b \leqslant 2\ell -1 $, then $ \mathrm{anz}(G)\geqslant \frac{m\ell}{b + m}\geqslant \frac{1}{2} $, contradicting our hypothesis. We may assume that $ b = 2\ell $. Then $ G $ has only one non-linear irreducible character $ \chi $. By Lemma \ref{Sei68}, $ G $ is a Frobenius group with an elementary abelian kernel of order $ p^{n} $ and cyclic complement of order $ p^{n}-1 $ for some prime $ p $. Also note that $ |{\mathrm {Irr}}(G)|=2\ell + 1 $. By Lemma \ref{Sei68}, $ \mathrm{anz}(G)=\frac{2\ell - 1}{2\ell + 1} \geqslant \frac{1}{2} $ since $ \ell \geqslant 2 $, a contradiction.
Suppose that $ b \geqslant 2\ell + 1 $. Then by Lemma \ref{ZSS10Lemma2.1}, there exists a normal subgroup $ K $ of $ G $ such that $ G' \leqslant K < G $ and $ G{\setminus} K \subseteq \upsilon(\varphi) $ for every non-linear irreducible character $ \varphi $ of $ G $. We also have that $ |G{:}G'| - |K{:}G'| = b - | K{:}G'| \geqslant \ell + 1 $ by Lemma \ref{numbertheoryresult}, again contradicting our hypothesis that $ G $ has an irreducible character that vanishes on exactly $ \ell $ conjugacy classes. This concludes our proof. \end{proof}
\begin{proof}[{\textbf{Proof of Theorem D}}] Suppose $ \mathrm{anz}(G)<\frac{1}{3} $. By Theorem C, $ G $ is supersolvable group. In particular, $ G $ is an $ M $-group. Suppose that $ \chi \in {\mathrm {Irr}}(G) $ be non-linear such that $ \chi_{G'} $ is irreducible. Then by the argument in the second paragraph of the proof of Theorem C, $ \mathrm{anz}(G)\geqslant\frac{1}{2} > \frac{1}{3} $, a contradiction. Hence every non-linear irreducible character $ \chi $ of $ G $ is such that $ \chi_{G'} $ is not irreducible. There exists a normal subgroup $ K $ of $ G $ such that $ G' \leqslant K < G $ and $ G{\setminus} K \subseteq \upsilon(\chi) $ by Lemma \ref{ZSS10Lemma2.1}. If $ \upsilon(\chi) $ is a conjugacy class, then $ G $ is a Frobenius group with an abelian kernel and a complement of order two using Lemma \ref{Lemma2(2)}. By \cite[Proposition 2.7]{Chi99}, we have that every irreducible character of $ G $ vanishes on at most one conjugacy class. Since $ G $ has two linear characters, $ \mathrm{anz}(G) \geqslant \frac{1}{3} $, contradicting our hypothesis.
Hence every non-linear irreducible character of $ G $ vanishes on at least $ \ell $ conjugacy classes, where $ \ell \geqslant 2 $. Suppose $ G $ has an irreducible character $ \chi $ that vanishes on exactly $ \ell $ conjugacy classes. Let $ b $ be the number of linear characters of $ G $. If $ b \leqslant 3\ell -1 $, then $ \mathrm{anz}(G)\geqslant \frac{1}{3} $, a contradiction. We may assume that $ b = 3\ell $. Then $ G $ has only one non-linear irreducible character $ \chi $. By Lemma \ref{Sei68}, $ G $ is a Frobenius group with an elementary abelian kernel of order $ p^{n} $ and a cyclic complement of order $ p^{n}-1 $ for some prime $ p $. By Lemma \ref{Sei68}, $ \mathrm{anz}(G)=\frac{2\ell - 1}{2\ell + 1} > \frac{1}{3} $ since $ \ell \geqslant 2 $, a contradiction.
Suppose that $ b \geqslant 3\ell + 1 $. Then by Lemma \ref{ZSS10Lemma2.1}, there exists a normal subgroup $ K $ of $ G $ such that $ G' \leqslant K < G $ and $ G{\setminus} K \subseteq \upsilon(\varphi) $ for every non-linear irreducible character $ \varphi $ of $ G $. We also have that $ |G{:}G'| - |K{:}G'| = b - | K{:}G'| \geqslant \ell + 1 $ by Lemma \ref{numbertheoryresult}, contradicting our hypothesis that $ G $ has an irreducible character that vanishes on exactly $ \ell $ conjugacy classes. This concludes our proof. \end{proof}
\begin{rem}\label{Abeliangroups}
Let $ \mathcal{L} =\{ G \mid G $ is an extra-special $ 2 $-group of order $ 2^{2k + 1} $ for some positive integer $ k \} $ and $ G\in \mathcal{L} $. Since $ |G/G'|=2^{2k} $ and $ G $ has only one non-linear irreducible character, we have $ \mathrm{acd}(G)=\frac{2^{2k} + 2^{k}}{2^{2k} + 1} $. Note that $ \mathcal{L} $ is an infinite family of nilpotent groups $ G $ such that $ \mathrm{acd}(G) < \frac{4}{3} $. Hence $ \mathrm{acd}(G) \rightarrow 1 $ and $ G\rightarrow \infty $. In other words there does not exist $ c > 1 $ such that $ \mathrm{acd}(G) < c $ implies that $ G $ is abelian. \end{rem}
Our last result shows that there does not exist a non-abelian nilpotent group $ G $ such that $ \mathrm{anz}(G) < \frac{1}{3} $. We shall restate Theorem E below.
\begin{theorem} Let $ G $ be a finite group. Then $ G $ is abelian if and only if $ \mathrm{anz}(G) < \frac{1}{3} $. \end{theorem}
\begin{proof} If $ G $ is abelian, then $ \mathrm{anz}(G)=0 < \frac{1}{3} $. Suppose that $ \mathrm{anz}(G) < \frac{1}{3} $. Then $ G $ is nilpotent group by Theorem D. Using induction on $ |G| $, we have that $ G/N $ is abelian for some minimal normal subgroup $ N $ of $ G $, that is, $ N=G' $. If $ \chi_{G'} $ is irreducible, then $ \chi $ is linear. Hence every non-linear irreducible character $ \chi $ is such that $ \chi _{G'} $ is not irreducible.
By Lemma \ref{ZSS10Lemma2.1}, there exists a normal subgroup $ K $ of $ G $ such that $ G' \leqslant K < G $ and $ G{\setminus} K \subseteq \upsilon(\chi) $. If $ \upsilon(\chi) $ is a conjugacy class, then $ G $ is a Frobenius group with an abelian kernel and a complement of order two by Lemma \ref{Lemma2(2)}, a contradiction since $ G $ is nilpotent.
Hence every non-linear irreducible character of $ G $ vanishes on at least $ \ell $ conjugacy classes, where $ \ell \geqslant 2 $. Suppose $ G $ has an irreducible character $ \chi $ that vanishes on $ \ell $ conjugacy classes. Let $ b $ be the number of linear characters of $ G $. If $ b \leqslant 3\ell -1 $, then $ \mathrm{anz}(G)\geqslant \frac{1}{3} $, a contradiction. We may assume that $ b = 3\ell $. Then $ G $ has only one non-linear irreducible character $ \chi $. By Lemma \ref{Sei68}, $ G $ is an extra-special group $ 2 $-group. Using the second part of Lemma \ref{Sei68}, we have that $ \mathrm{anz}(G)=\frac{2\ell - 1}{2\ell + 1} > \frac{1}{3} $ since $ \ell \geqslant 2 $, a contradiction.
Suppose that $ b \geqslant 3\ell + 1 $. Then by Lemma \ref{ZSS10Lemma2.1}, there exists a normal subgroup $ K $ of $ G $ such that $ G' \leqslant K < G $ and $ G\setminus K \subseteq \upsilon(\varphi) $ for every non-linear irreducible character $ \varphi $ of $ G $. We have that $ |G{:}G'| - |K{:}G'| = b - | K{:}G'| \geqslant \ell + 1 $ by Lemma \ref{numbertheoryresult}, contradicting our hypothesis that $ G $ has an irreducible character that vanishes on exactly $ \ell $ conjugacy classes. This concludes our proof. \end{proof}
\begin{proof}[\textbf{Proof of Theorem F}] Suppose that $ G $ is non-abelian. Note that $ G $ is solvable. Also note that every non-linear irreducible character of $ G $ vanishes on at least $ \ell $ conjugacy classes, $ \ell \geqslant 2 $. Suppose that $ G $ has an irreducible character $ \chi $ that vanishes on two conjugacy classes. By Lemma \ref{Qia07Theorem2.6}, either $ G $ is a Frobenius group with a complement of order $ 3 $ or there are normal subgroups $ M $ and $ N $ of $ G $ such that: $ M $ is a Frobenius group with the kernel $ N $; $ G/N $ is a Frobenius group of order $ p(p - 1)/2 $ with the kernel $ M/N $ and a cyclic complement of order $ (p - 1)/2 $ for some odd prime $ p $. If $ G $ is a Frobenius group with a complement of order $ 3 $, then $ G $ can only have at most two non-linear irreducible characters. Otherwise $ \mathrm{anz}(G)\geqslant \frac{8}{7} $, a contradiction (note that a group of odd order has an even number of non-linear irreducible characters). By Lemma \ref{Pal81}, $ G $ is the group in the case (d) of Lemma \ref{Pal81} with $ (p^{k} - 1)/2=3 $, that is, $ p^{k} = 7 $ and $ |G|=21 $. Hence $ G $ is supersolvable. We may assume that $ G $ is the group in the second case above. Then $ M=G' $, $ \chi_{M} $ is irreducible and by Lemma \ref{injectivemap}, for every linear character $ \alpha $ of $ G $, there exists a corresponding $ \alpha \chi \in {\mathrm {Irr}}(G) $ and hence $ \mathrm{anz}(G) \geqslant 1 $, contradicting our hypothesis.
Hence every non-linear irreducible character of $ G $ vanishes on at least $ \ell $ conjugacy classes, $ \ell \geqslant 3 $. Suppose that $ G $ has irreducible character vanishing on exactly $ \ell $ conjugacy classes. If there exists $ \chi $ such that $ \chi_{G'} $ is irreducible, then for every linear character $ \alpha $ of $ G $, there exists a corresponding $ \alpha \chi \in {\mathrm {Irr}}(G) $ and hence $ \mathrm{anz}(G) \geqslant 1 $, contradicting our hypothesis. We may assume that for every irreducible character $ \chi $ of $ G $, $ \chi_{G'} $ is reducible. By Lemma \ref{ZSS10Lemma2.1}, there exists a normal subgroup $ K $ of $ G $ such that $ G' \leqslant K < G $ and $ G{\setminus} K \subseteq \upsilon(\varphi) $ for every non-linear irreducible character $ \varphi $ of $ G $. If $ b \leqslant 2\ell - 1 $, then $ G $ has at most two non-linear irreducible characters. Otherwise, $ \mathrm{anz}(G)\geqslant \frac{3\ell}{2\ell + 2} > 1 $, a contradiction. Hence $ G $ has exactly two non-linear irreducible characters. By Lemma \ref{Pal81}, $ G $ is a Frobenius group with an elementary abelian kernel $ G' $ of order $ p^{k} $ and a cyclic Frobenius complement of order $ (p^{k} - 1)/2 $, where $ p $ is odd prime. Note that every non-linear irreducible character of $ G $ vanishes on $ G{\setminus} G' $. Thus $ b=(p^{k} - 1)/2 $ and since $ |G{:}G'|=b $, we have that $ b - 1 \leqslant \ell $ using Lemma \ref{ZSS10Lemma2.1}. Hence $ \mathrm{anz}(G)\geqslant \frac{2(b-1)}{b + 2} > 1 $, a contradiction to our hypothesis.
If $ b \geqslant 2\ell + 1 $, we have that $ |G{:}G'| - |K{:}G'| = b - | K{:}G'| \geqslant \ell + 1 $ by Lemma \ref{numbertheoryresult}, a contradiction to the hypothesis that $ G $ has an irreducible character that vanishes on exactly $ \ell $ conjugacy classes. \end{proof}
\end{document} | arXiv |
Estimating $\sum_{x_i < X} \prod_i \phi(x_i)/ \mathrm{lcm}(x_i)^a$
Asked 1 year, 6 months ago
I would like to estimate from above the following sum
$$ \sum_{1 \leq x_1 < X} .. . \sum_{1 \leq x_n < X} \frac{\prod_{1 \leq i \leq n } \phi(x_i)}{\mathrm{lcm}(x_1, .., x_n)^a}. $$ $\phi$ is the Euler totient function and $a$ is a positive integer less than $2n$. A trivial estimate would be $\ll X^{2n - a}$. Is there a way to get a better bound? Thank you!
nt.number-theory
analytic-number-theory
José Hdz. Stgo.
8,34244 gold badges6565 silver badges100100 bronze badges
Johnny T.Johnny T.
$\begingroup$ You may also want to look at mathoverflow.net/q/288837/160943 $\endgroup$
– Hhhhhhhhhhh
One can improve on $X^{2n-a}$ as long as $(a,n)\neq(1,1)$ (for $a=n=1$, the sum grows like $X/\zeta(2)$ so there's no room for improvement). Let us introduce $$f_n(m) := \# \{ (x_1,\ldots,x_n) : \mathrm{lcm}(x_1,\ldots,x_n) = m\},$$ which satisfies $f_n(m) \le \tau(m)^n \ll_{n,\varepsilon} m^{\varepsilon}$ where $\tau$ is the usual divisor function. First suppose that $a >n$. We have $$\frac{\prod_{i=1}^{n} \phi(x_i)}{\mathrm{lcm}(x_1,\ldots,x_n)^a} \le \frac{\prod_{i=1}^{n} \phi(x_i)}{\max_{1\le i \le n} x_i^n} \frac{1}{\mathrm{lcm}(x_1,\ldots,x_n)} \le \frac{1}{\mathrm{lcm}(x_1,\ldots,x_n)}$$ which gives the upper bound $$< \sum_{1 \le m < X^{na}} \frac{f_n(m)}{m} = X^{o(1)}.$$ This is optimal since we have the lower bound $\ge 1$. We may assume $n \ge a$ from now on. Your sum is $$< X^n \sum_{1 \le m < X^{na}} \frac{f_n(m)}{m^a} \ll_{a,n} X^{n+o(1)}.$$ If $n\neq a$ this already beats $X^{2n-a}$. We now sketch how one can do better than $X^{n+o(1)}$. In section 3 of R. R. Hall's ``The distribution of squarefree numbers'' (Reine Angew. Math. 394 (1989), 107–117), the author introduces `total decomposition sets', which help him study a sum related to yours with $a=2$ and $n \ge 2$ (see his Lemma 3). Modifying the proof of Lemma 3 slightly, we obtain the bound $$\ll_{a,n} \begin{cases} X^{n-\frac{n(a-1)}{n-1}+o(1)} & \text{if }n > a,\\ X^{1+o(1)} & \text{if }n=a,\end{cases}$$ which beats $2n-a$ as long as $(a,n) \neq (1,1)$. The dependence on $a,n$ can be made explicit but is quite horrible. To see that $n=a$ and $a=1$ are optimal consider the contribution of $x_1=x_2=\ldots=x_n$.
Ofir GorodetskyOfir Gorodetsky
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How to bound $\sum_{1 \leq x_1, ..., x_n \leq N} lcm(x_1, ..., x_n)^{- \delta}$?
Estimate sum with Euler function
A sum involving Euler totient function
Rate of decay of the tail of Dirichlet series for Euler's totient function
How to obtain an upper bound for $\sum_{x < X} \frac{\mu^2(x) \tau_k(x)}{\phi(x)}$?
Estimating the sum $\sum_{1\leq x,y\leq n} \frac{x}{ \mathrm{lcm}(x,y)}$
Non-negativity of an infinite absolutely convergent sum | CommonCrawl |
# Fundamentals of probability theory
Probability distributions are used to describe the likelihood of different outcomes in a random experiment. The most common probability distributions include the binomial distribution, Poisson distribution, and normal distribution.
Conditional probability is the probability of an event given that another event has occurred. It is calculated using the formula:
$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$
Bayes' theorem is a fundamental result in probability theory that allows us to update our beliefs in light of new evidence. It is stated as:
$$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$
Independence is a key concept in probability theory. Two events are independent if the occurrence of one event does not affect the probability of the other event. The independence of two events can be checked using the formula:
$$P(A \cap B) = P(A)P(B)$$
Mutual information is a measure of the dependence between two random variables. It quantifies the amount of information gained about one random variable through observing the other random variable. The mutual information between two events A and B is calculated as:
$$I(A; B) = \sum_{x \in X} \sum_{y \in Y} P(x, y) \log \frac{P(x, y)}{P(x)P(y)}$$
## Exercise
Calculate the conditional probability of event A given event B using the formula:
$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$
# Basics of Python programming
Variables in Python are used to store data. They are created when you assign a value to them. Python has several built-in data types, including integers, floats, strings, and booleans.
Control structures in Python allow you to control the flow of your program. The most common control structures are if-else statements and for loops.
Functions in Python are reusable pieces of code that perform a specific task. They can be defined using the `def` keyword, and can take input parameters and return values.
## Exercise
Write a Python function that calculates the factorial of a given number.
# Representation of Bayesian networks in Python
To represent Bayesian networks in Python, we can use the `pgmpy` library. This library provides a flexible and user-friendly interface for creating, manipulating, and analyzing Bayesian networks.
First, we need to install the `pgmpy` library using pip:
```
pip install pgmpy
```
Once the library is installed, we can create a Bayesian network using the following code:
```python
from pgmpy.models import BayesianModel
# Create an empty Bayesian network
model = BayesianModel()
# Add nodes to the network
model.add_nodes_from(['A', 'B', 'C'])
# Add edges to the network
model.add_edges_from([('A', 'B'), ('B', 'C')])
# Print the network
print(model)
```
## Exercise
Create a Bayesian network with three nodes: A, B, and C. Add edges between A and B, and B and C.
# Inference in Bayesian networks
Inference in Bayesian networks involves computing the probability of a specific event or the probability of a set of events. We can use the `pgmpy` library to perform inference in our Bayesian networks.
To compute the joint probability distribution of a set of events, we can use the `pgmpy.inference` module. For example, to compute the joint probability of events A and B, we can use the following code:
```python
from pgmpy.inference import Inference
# Compute the joint probability distribution
infer = Inference(model)
joint_prob = infer.joint_probability(['A', 'B'])
# Print the joint probability distribution
print(joint_prob)
```
## Exercise
Compute the joint probability distribution of events A and B in the Bayesian network created in the previous section.
# Bayesian reasoning and decision-making
Bayesian networks can be used to perform Bayesian reasoning and decision-making. This involves updating our beliefs based on new evidence and making decisions that maximize the expected utility.
To perform Bayesian reasoning, we can use the `pgmpy.inference` module. For example, to update our beliefs about node A given evidence that node B occurred, we can use the following code:
```python
from pgmpy.inference import Inference
# Update our beliefs about node A given evidence for node B
infer = Inference(model)
posterior_prob = infer.posterior_probability({'B': True}, {'A': True})
# Print the updated beliefs
print(posterior_prob)
```
## Exercise
Update your beliefs about node A given evidence that node B occurred in the Bayesian network created in the previous section.
# Applications of Bayesian networks
Bayesian networks have numerous applications in various fields, including artificial intelligence, machine learning, and natural language processing.
Some common applications of Bayesian networks include:
- Spam detection in email
- Medical diagnosis
- Speech recognition
- Recommender systems
- Anomaly detection
## Exercise
Research and describe a specific application of Bayesian networks in a field of your choice.
# Real-world examples of Bayesian networks
To better understand the practical applications of Bayesian networks, let's consider a real-world example. Suppose we want to predict the probability of a customer churning based on their demographic information and purchase history. We can represent this problem as a Bayesian network and use the `pgmpy` library to perform inference and make predictions.
First, we need to define the nodes and edges in our Bayesian network. For example, we can have nodes for customer demographic information (age, income, and education) and purchase history (number of purchases and average purchase amount). We can also have a node for customer churn (churn = True or False).
Next, we can use the `pgmpy.inference` module to compute the joint probability distribution of the churn node and the demographic and purchase history nodes. We can then use Bayes' theorem to compute the conditional probability of churn given the demographic and purchase history information.
## Exercise
Create a Bayesian network for the customer churn prediction problem described above. Use the `pgmpy` library to compute the joint probability distribution of the churn node and the demographic and purchase history nodes.
# Advanced topics in Bayesian networks and Python
- Parameter estimation and learning algorithms
- Bayesian network structure learning
- Causal inference in Bayesian networks
- Dynamic Bayesian networks
Parameter estimation involves estimating the parameters of a Bayesian network from data. This can be done using various algorithms, such as maximum likelihood estimation and Bayesian estimation.
Bayesian network structure learning involves learning both the parameters and structure of a Bayesian network from data. This can be done using algorithms such as score-based structure learning and feature-based structure learning.
Causal inference in Bayesian networks involves determining the causal relationships between variables in a network. This can be done using algorithms such as causal discovery and causal inference.
Dynamic Bayesian networks are used to model time-varying relationships between variables. This can be useful in applications such as time series prediction and sensor networks.
## Exercise
Research and describe one advanced topic in Bayesian networks and Python.
# Evaluation and comparison of Bayesian network algorithms
To evaluate the performance of Bayesian network algorithms, we can use various metrics, such as accuracy, precision, recall, and F1 score. These metrics are commonly used in machine learning and natural language processing tasks.
For example, we can use the accuracy metric to evaluate the performance of a spam detection system based on a Bayesian network. We can compute the accuracy by comparing the predicted churn probabilities with the actual churn outcomes in a dataset.
We can also use the precision and recall metrics to evaluate the performance of a recommendation system based on a Bayesian network. These metrics quantify the trade-off between the number of true positive predictions and the number of false positive predictions.
## Exercise
Evaluate the performance of a spam detection system based on a Bayesian network using the accuracy metric.
# Integrating Bayesian networks with other machine learning techniques
Bayesian networks can be integrated with other machine learning techniques to create powerful predictive models. Some common integration techniques include:
- Combining Bayesian networks with decision trees
- Combining Bayesian networks with support vector machines
- Combining Bayesian networks with neural networks
For example, we can combine a Bayesian network with a decision tree to create a hybrid model that can learn both the probabilistic dependencies between variables and the decision rules for making predictions.
## Exercise
Research and describe one integration technique for Bayesian networks and another machine learning technique.
# Future trends and developments in Bayesian networks
The field of Bayesian networks is constantly evolving, with new techniques and applications being developed all the time. Some future trends and developments in Bayesian networks include:
- Deep learning for Bayesian networks
- Bayesian networks for big data
- Bayesian networks for explainable artificial intelligence
For example, deep learning techniques can be used to learn complex probabilistic dependencies between variables in Bayesian networks. These techniques can capture non-linear relationships and hierarchical structures, making them more powerful than traditional Bayesian network learning algorithms.
## Exercise
Research and describe one future trend or development in Bayesian networks. | Textbooks |
Abstract: NI2.00002 : Improving Cryogenic-DT Implosion Performance on OMEGA
T.C. Sangster
(Laboratory for Laser Energetics, U. of Rochester)
Although cryogenic-DT implosion performance has improved both in absolute terms and relative to hydro simulations, a number of long-standing discrepancies remain unresolved. Absolute yield performance increased with higher-quality capsule and ice surfaces, routine delivery of low-adiabat ($\alpha$ $\sim $ 2) laser pulses at specification, and more-accurate target alignment with respect to the beam pointing (typically less than 10-$\mu$m rms for all 60 beams). Higher implosion velocities using thinner ice and constant mass ablators have resulted in additional increases in measured yields and ion temperatures. However, ion temperatures remain systematically below the hydro predictions suggesting higher-than-predicted imprint levels (note that $T_{i} \sim T_{e}$ for all implosions except for cases where fuel motion artificially enhances $T_{i})$. Imprint reduction is being addressed using dopants (small at.{\%} of silicon) in the outer part of the ablator. To preserve the ablator mass, doped shells are necessarily thinner than undoped shells and recent compression results show a clear inverse relation between the inferred areal density and the measured yields. This suggests more radiative preheat with the thinner ablators (the areal densities are about 70{\%} of predictions---below what is expected based on burn truncation). While improved nonlocal thermal transport and cross-beam energy transfer models resolved a persistent discrepancy between predicted and measured bang times, the measured burn width is longer than predicted. Furthermore, core x-ray emission below 2.5 keV is consistently higher than predictions. These discrepancies, combined with improved modeling, implicate shell stability and suggest that thicker ablators and thinner ice (to preserve the overall payload mass) may lead to improved ignition hydro equivalency. This talk will show the latest experimental results using thicker ablators and ablators doped with silicon, and compare these results with the latest hydro simulations.\\[4pt] This work was supported by the U.S. Department of Energy Office of Inertial Confinement Fusion under Cooperative Agreement No. DE-FC52-08NA28302. In collaboration with V. N. Goncharov, R. Betti, T. R. Boehly, R. Epstein, C. Forrest, V. Yu. Glebov, S. X. Hu, I. V. Igumenshchev, D. H. Froula, R. L. McCrory, D. D. Meyerhofer, P. B. Radha, W. Seka, W. T. Shmayda, S. Skupsky, C. Stoeckl (Laboratory for Laser Energetics, U. of Rochester), J. A. Frenje, D. T. Casey, and M. Gatu-Johnson (Plasma Science and Fusion Center, MIT). | CommonCrawl |
Experimental investigation on a novel approach for laser surface hardening modelling
L. Orazi1,2,
A. Rota3,4 &
B. Reggiani1,4
Laser surface hardening is rapidly growing in industrial applications due to its high flexibility, accuracy, cleanness and energy efficiency. However, the experimental process optimization can be a tricky task due to the number of involved parameters, thus suggesting for alternative approaches such as reliable numerical simulations. Conventional laser hardening models compute the achieved hardness on the basis of microstructure predictions due to carbon diffusion during the process heat thermal cycle. Nevertheless, this approach is very time consuming and not allows to simulate real complex products during laser treatments. To overcome this limitation, a novel simplified approach for laser surface hardening modelling is presented and discussed. The basic assumption consists in neglecting the austenite homogenization due to the short time and the insufficient carbon diffusion during the heating phase of the process. In the present work, this assumption is experimentally verified through nano-hardness measurements on C45 carbon steel samples both laser and oven treated by means of atomic force microscopy (AFM) technique.
Laser hardening is a surface process with some peculiar characteristics such as that it does not require a quenching medium and it can be more selective if compared to the classical treatments carried out in an oven or by induction hardening. Furthermore, concave features like gear teeth root and caves are more easily processed compared to induction technique, and different shapes do not require the manufacturing of new customized induction wires. With the use of high-efficiency diode laser (Ion 2002), also the overall efficiency of the process could be sensibly higher than induction hardening that, given the lower power density, normally requires applied power of many teens of kW (Mühl et al. 2020). A detailed and up-to-date review about the process can be found in (Babu and Marimuthu 2019).
The effective applicability of laser transformation hardening depends on two basic aspects that determine the most important challenges in the development of this manufacturing technology: an accurate determination and control of the thermal field occurring onto the workpiece and the optimization of the laser scanning strategy when the part surface is larger than the beam spot size and thus multiple tracks have to be carried out (Tani et al. 2008). Concerning the first aspect, due to the fast heat cycle occurring when the laser beam irradiates the target material, the hysteresis in the austenization of the steel is very high and the actual transformation occurs at temperatures very near to the solidus one. This consideration leads to conclude that an inaccurate control of the surface temperature can easily determine the partial melting of the target material and consequently the inevitable impairment of the workpiece quality. Concerning the second aspect, the matter is probably even more serious: if the surface to be hardened is large and a single track is not sufficient to cover it, several adjacent tracks have to be performed determining an inevitable tempering of the previously hardened material due to its interaction with the thermal field caused by the later tracks. On the basis of the abovementioned considerations, setting up an effective laser surface hardening treatment emerges to be a complex activity often involving a costly and time expensive trial-and-error approach.
A possible approach to estimate the outcome of the laser hardening process is through the use of design of experiment techniques that permit to generate models correlating operating parameters with hardened thickness and/or hardened width. Examples applied to low-carbon steels can be found in (Babu et al. 2013; Chen et al. 2020).
Another approach is based on the numerical simulation of the phenomena involved in the process in order to allow a deep understanding of the process mechanisms while reducing and minimizing the experimental activities.
Many researchers stressed on the importance of laser hardening process simulation and many different modelling approaches were proposed in the last couple of decades. Theoretically, the process simulation requires the prediction of the thermal field in the target part, of the austenite formation, carbon diffusion and austenite homogenization and, finally, the microstructure prediction during the cooling. The thermal field can be predicted by solving the Fourier equation in the target domain, while, more attention needs the austenite formation prediction. The canonical model for the austenization of hypo-eutectoid steels in quasi-static conditions considers two distinct processes: the perlitic colony and ferritic grains transformations, due to the change from BCC to FCC of the iron lattice, and the homogenization of the obtained austenitic grains. These two processes are both considered in determining the set up for the proper conduction of the heat treatments in an oven. One of the first simulation model for the laser hardening can be found in (Ashby and Easterling 1984; Li et al. 1986), and it is based on a microstructural approach according to the heat treatments in an oven. This approach was then applied by other authors as in (Ohmura and Inoue 1989) in which it is supposed that the pearlite to austenite transformation terminates at the AC3 temperature for a pure iron considering a fixed and very high heating rate.
The availability of new, simple, reliable and efficient diode laser sources is probably the key factor that pushed several authors to reinvestigate laser surface hardening in the last years: the works in (Skvarenina and Shin 2006; Patwa and Shin 2006) are all based on the same microstructural approach presented in (Ashby and Easterling 1984; Li et al. 1986) and on the hypotheses formulated in (Jacot and Rappaz 1997; Jacot and Rappaz 1998). Another model based on a commercial Finite Element Method software was presented in (Miokovic et al. 2006; Miokovic et al. 2007).
In all the cited works, a heat conduction model with non-constant parameters is coupled with a phase transformation model evaluated at the microscale. All those approaches predict both the resulting phases and the hardness at the cost of a very long calculation time: the time step that guarantees the stability of the numerical integration is inversely proportional to the absolute values of the diffusivity and to the grid dimension and both of them are very small when applied to the microstructure in solid phases as shown in Eq. (1) where ρ is the density in \( \frac{\mathbf{kg}}{{\mathbf{m}}^{\mathbf{3}}} \), cp is the specific heat in \( \frac{\mathbf{J}}{\mathbf{kg}\ \boldsymbol{K}} \), κ the material conductivity in \( \frac{\boldsymbol{W}}{\boldsymbol{m}\ \boldsymbol{K}} \) and Δx, Δy and Δz are the dimensions of the 3D cell calculation.
$$ \Delta t\le \frac{1}{2}\frac{\rho\ {c}_p}{k}\frac{1}{\left(\frac{1}{\Delta {x}^2}+\frac{1}{\Delta {y}^2}+\frac{1}{\Delta {z}^2}\right)} $$
Small Δt and consequently long calculation time makes these models useless to simulate real complex applications, such as multi passes processes on real 3D objects.
An attempt to overcome these limitations was previously presented by the authors by introducing the hypothesis that carbon diffusion can be neglected during the austenization in the modelling of laser surface hardening process of hypo-eutectoid steels due the short time and the insufficient carbon diffusion during the heating phase of the process (Orazi et al. 2010). In more detail, it was shown that the accuracy in the hardness prediction as achieved by the numerical model remains high but the calculation time can be sensibly reduced and a real process optimization can be carried out in industrial components (Tani et al. 2010). However, in the aforementioned contributions, the neglecting of the austenite homogenization (carbon diffusion) was assumed as starting hypothesis however without any specific experimental validation. In the present work, experiments are presented to prove that austenite homogenization in laser surface hardening does not take place and it can be therefore neglected in process modelling and optimization. To this aim, a novel indirect approach to evaluate carbon diffusion during laser hardening and conventional oven hardening is used and discussed based on hardness measurements obtained by means of atomic force microscopy (AFM) technique applied on specimens made of C45 EN 10277-2 steel whose chemical composition is reported in Table 1.
Table 1 Chemical composition of C45 steel
The novel approach
From a physical point of view, the laser hardening process differs from the oven process for some important aspects:
The quenching (cooling) is performed by conduction in the bulk volume.
The process is very fast: fraction of seconds compared to fraction of hours.
Heating and cooling rate are very high.
Solid phase transformations happen with significant over and under heating due to the process hysteresis.
According to all these reasons, during laser surface hardening of thick, bulk parts with high thermal inertial almost all the austenized material become martensite after cooling and attention can be focused in obtaining austenite during the heating stage.
In (Jacot and Rappaz 1997; Jacot and Rappaz 1998), it is clearly demonstrated that, in case of high overheating, as in the laser hardening process, the carbide diffusion in the austenite mainly happens in the lateral side of the carbide plates as in Fig. 1, leading to a change in the cross-section of the carbon flux from λL as (Ashby and Easterling 1984; Li et al. 1986) to L2, where L is the characteristic size of the perlite colony, having order of several μm, and λ is the thickness of the lamina that is normally a fraction of μm.
The carbon flux in the cementite lamellae during austenization. On the left, the model by (Ashby and Easterling 1984; Li et al. 1986). On the right, what proposed in (Orazi et al. 2010; Tani et al. 2010)
Following this track, authors proposed in (Orazi et al. 2010), successively extended in (Fortunato et al. 2013), a model to simulate the effect of laser surface hardening process in terms of achieved hardness and microstructure. An integral transformation parameter Ip → a has been introduced and expressed in Eq. (2):
$$ {I}_{p\to a}=\underset{t_{A_{c1}}}{\overset{t_{A_{r1}}}{\int }}\exp\;\left(-\frac{Q_{p\to a}}{RT(t)}\right)\; dt $$
in which t is the time in s, Q is the activation energy in J, R the gas constant in \( \frac{J}{K} \), T the temperature of the material in K, \( {t}_{A_{c1}} \) is the time at which the steel overcome temperature Ac1 and \( {t}_{A_{r1}} \) is the moment when it descends below Ar1. In the proposed model, it is supposed that austenization completes when the value of the parameter Ip → a overcomes a threshold, Ip → a, th, experimentally determined.
$$ {I}_{p\to a}\ge {I}_{p\to a, th} $$
The previous model is based on the hypothesis that, due to the short interaction time in laser hardening, carbon diffusion can be neglected and no homogenization occurs in the austenitic grains. According to this new process simulation frame, laser surface hardening can be modelled in the following 3 steps:
Thermal field evaluation into the working part according to the laser parameters and scanning strategies.
Evaluation of Eq. (2) into the working part.
Declare austenite and consequently martensite formation where Eq. (3) is verified.
The different approaches between the traditional process modelling, based on the microstructure prediction, and the proposed new one is presented in Fig. 2.
a The classic model presented by Ashby (Ashby and Easterling 1984; Li et al. 1986) compared to b, the evaluation scheme proposed by authors in (Orazi et al. 2010; Fortunato et al. 2013)
As previously stated, the present work was aimed at verifying the assumption of austenite homogenization according to the carbon diffusivity at the high temperature achieved during laser hardening process. However, if it is easy to experimentally observe or calculate that during laser hardening the temperature, in the proximity of the surface, can be very close to the melting temperature, unfortunately, there are not so many information about carbon diffusivity in these extreme conditions. A brief state of the art on this aspect is then presented in the next section with the aim to introduce the novel adopted experimental approach.
Evaluation of carbon diffusivity
First historical measurements on carbon diffusivity at high temperature were done in (Weels and Mehl 1940; Weels et al. 1950), in which several tests by coupling steel disks with different carbon concentration and annealing at temperature between 750 and 1305 °C were executed.
In (Tibbetts 1980), different tests of plasma carburizing were performed in order to measure carbon diffusivity at high temperature for ferrous alloys. The author used the steady-state method previously applied by (Smith 1953) at ambient temperature.
In (Bhadesia 1981), the author used data from the previous works in order to obtain a more accurate estimation of the diffusivity that is nevertheless only expressed in graphical form.
In (Karabelchtchikova and Sisson Jr 2006), an estimation of the carbon diffusivity in austenite during gas carburizing is obtained by fitting a numerical model of the carbon diffusivity with the surface carbon content at different process parameters. With this approach, the diffusivity appears to range between 1.68·10−7 cm2/s at 880 °C and 5.06·10−8 cm2/s at 980 °C.
In (Lee et al. 2011), a critical review of twelve models for the carbon diffusivity in austenite is performed by evaluating their fitting with different sets of experimental data collected from literature. As expected, the models that generate the best numerical-experimental agreement were those including the influence on the diffusivity of the carbon content, the temperature and the alloying elements both on the diffusivity coefficient D and on the activation energy Q. The best equation obtained from the whole dataset is reported in Eq. (2):
$$ D\;\left(T,C\right)=\left(0.146-0.036C\right)\cdot \exp\;\left(-\frac{144.3-15.0C+0.37{C}^2}{R}\right) $$
where D is the carbon diffusivity in cm2/s, C the carbon content in mass percentage, T the temperature in °K, and R is the gas constant. As an example, a plot of the model as applied to C45 with a carbon content of 0.45% is reported in Fig. 3.
Carbon diffusivity as a function of temperature for the C45 as predicted by the analytical model reported in Eq. (4)
The reported carbon diffusivity models proposed in (Karabelchtchikova and Sisson Jr 2006; Lee et al. 2011) are used as references to estimate the carbon diffusion length lc given by Eq. (5):
$$ {l}_c=\sqrt{D\ \tau } $$
where D is the diffusivity and τ the time spent at a given temperature. Table 2 shows the diffusion length lc(1s) calculated at 1 s of interaction.
Table 2 Diffusion length evaluated at different temperatures (interaction time 1 s)
As can be seen, some discrepancy between data from different sources are present but the diffusion length lc(1 s) appears quite small compared to the average size of perlite colonies. This substantially confirms the hypothesis underlying the model proposed in (Orazi et al. 2010; Fortunato et al. 2013), but it also suggests how difficult is to obtain reliable data about carbon diffusivity at high temperature,
For these reasons, in the present work, it is proposed to evaluate the homogeneity of the carbon into the structure by an indirect experimental method consisting in the post hardening microstructure analysis via nano indentations. As already reported in literature (Furuhara et al. 2003), the hardness increasing in metallurgic tempering processes are related to intra-granular carbon diffusion in the pearlitic structures and inter-granular carbon diffusion between pearlitic and ferritic grains. To identify these mechanisms and to relate them to carbon diffusion, a high spatially resolved indentation technique is necessary. Conventional indentation techniques are not able to satisfy this request, due to the large size of the indenter (compared to the grain structure of the sample). For this purpose, an alternative technique is represented by atomic force microscopy (AFM) equipped with a diamond tip as an indenter and imaging probe at the same time. With this technique, it is possible to select the desired area of the sample and to proceed with indentation directly with the AFM tip, taking advantage of the nanometric spatial resolution typical of this kind of apparatus. The indentation marks can be imaged immediately after using the same indentation tip (Bhushan and Koinkar 1994; Butt et al. 2005). In addition, the use of AFM as indenter allows the application of a smaller load with respect to conventional indention, down to pN values. Such a reduced load generates very small marks and, consequently, high-density indentation matrices, increasing the resolution of the final hardness map. This technique enables the determination of hardness by a direct method, namely calculating the ratio L/A, where L is the applied load and A is the projected area. The alternative and most used indirect method to determine hardness consists of analysing the approach-retract indentation curve (Oliver-Pharr method (Oliver and Pharr 1992)). As already reported in literature (Miyake et al. 2004; Liang and Yao 2007), the Oliver-Pharr indirect method overestimates the hardness, because the generated pile-up cannot be taken into account. Therefore, the use of AFM as hardness tester appears powerful and attractive, both for the spatial resolution and for its sensitivity to pile-up generation. On the other hand, AFM indentation suffers from non-central geometry of the indenter, which induces the in-plane sliding of the tip during indentation (Kempf et al. 1998; Calabri et al. 2007). This effect, together with creep and hysteresis of the piezo-actuators, could introduce a systematic error to the hardness absolute value.
In this study, AFM equipped with a diamond tip was used to generate a high-resolution hardness map of the steel surfaces. The abovementioned drawback related to AFM indentation is not critical for the purpose of the present study, where the relative spatial variation and distribution of hardness are the key quantities.
The experimental set-up
In the present work, the negligible austenite homogenization during laser hardening process has been experimentally verified through hardness measurements on C45 specimens laser and oven treated obtained by means of atomic force microscopy (AFM) technique.
Specimens for laser treatments were 155 × 80 × 15 mm in order to guarantee an adequate thermal inertia for the cooling while the samples oven treated were smaller, roughly 30 × 20 × 15, all the material was annealed in oven obtaining pure perlite/ferritic mixture. Oven treated sample was heated at 840 °C for with low and controlled temperature gradient in the oven (austenitization), maintained at this temperature for 3 h (carbon homogenization in austenite) and quenched in water. Laser hardened sample was treated with a EL.EN FAF 3 kW continuous wave CO2 laser focused at 6 mm, operating at a power between 1100 and 1360 W, and scanning speed between 0.6 and 0.3 m/min resulting in a range of influence between 2.9 and 5.7 kJ/cm2. For a better coupling between the 10.6 μm wavelength of the CO2 laser and the material surface, a thin coating of graphite was applied in spray form. During the experiments, the surface under treatment was protected supplied through a lateral nozzle in order to decrease the effects of oxidation.
Given similar overall results for the two treatments, the AFM measurements were conducted on both the oven treated and laser-treated samples (Fig. 4).
Specimens geometry and position of the AFM measured area: a and b laser treated sample; c oven treated one. The grey areas represent the martensitic zones
The hardness of the laser-exposed region was investigated in a perpendicular cross-section of the sample, at about 24.5 μm from the treated surface to avoid the effects due to surface oxidation (the laser effect is expected to extend at least 1 mm from the surface) (Fig. 4a). For comparison, the pristine hardness was measured in an untreated region of the same sample, at a distance of 10 mm from the exposed surface (Fig. 4b). The analysis of the indentation matrices enables the drawing of the corresponding hardness maps for the laser-treated and untreated regions. For comparison, the same procedure was repeated on a standard heat-tempered sample.
The hardness maps in different regions of the sample (Fig. 4) were obtained by making regular indentation matrices using a Veeco DI-EnviroScope AFM, working in nano-indentation mode. The indenter consisted in a diamond Berkovich tip glued on a sapphire cantilever.
The elastic constant of the cantilever, according to Company calibration, was 5085 N/m and the applied load during indentation was between 1.23 and 1.25 μN. The angle between the cantilever and the surface was 12°. The hardness in each position of the matrix was obtained using the direct method, namely dividing the applied load by the projected area of indentation, measured in tapping mode AFM using the same indentation tip.
The hardness absolute values have been reported, even if, as already discussed, the reliability of AFM indentation is still debated, related to the non-central geometry of the indenter and the hysteresis and creep of the piezo-actuator. Anyway, for the purpose of the present study, the reader must focus on the hardness gradient of the investigated surface and on its distribution rather than on the absolute value.
Full martensite structures were obtained on the oven treated and in the track of the laser patterned samples with an average HV,1000 microhardness of 670. A representative image of the section of the patterned track is shown in Fig. 5. No appreciable melted and re-casted phases were observed during the experimental campaign and for the tested conditions.
Section of a laser hardened track. Power = 1.36 kW, scanning speed = 0.6 m/min
In Fig. 6a, the AFM image of a typical indentation matrix is reported. As can be seen, the distance among the indentations in the x and y direction is about 2.6 μm. To measure the projected area of each mark, high-resolved AFM images were acquired, so to improve the spatial resolution. The axis length of the triangular projected area is between 300 and 600 nm, while the depth goes from 10 to 100 nm. As an example, in Fig. 6b, the AFM high-resolution magnification of two adjacent indentation marks is reported. The two marks are characterized by different projected area and depth, evident from the corresponding distance–height graph (Fig. 6c). The measurement of the projected area related to the applied load enables the drawing of the hardness map of the selected region.
a AFM image of an indentation matrix on the sample treated with laser hardening with power = 1.36 kW and scanning speed 0.6 m/min; b high-resolution AFM image of two neighbouring indentations, used for the estimation of the projected indentation area; and c height profile of the indentations reported in b
In Fig. 7, three different hardness maps are reported, obtained by means of AFM nano-indentations. Figure 7a, b represents the hardness maps in two different regions closed to the laser-treated surface (24.5 μm from the surface) while Fig. 7c is related to a portion of the sample that was not influenced by laser exposure.
AFM nano-indentation maps corresponding to a, b different regions treated with laser tempering and c region not interested by laser irradiation, in the bulk of the sample. Colour scale is the same for all the maps
The colour scale is the same in all the maps for a better comparison, while the lateral dimensions are different. In both maps, a and b colour spreading is large, corresponding to a hardness variation from 3 to 16 GPa. Domains of different hardness are recognizable in the maps, with a mean linear grain size between 7 and 9 μm, calculated by a standard self-correlation function. In the map c, which corresponds to the untreated bulk portion of the specimen, the hardness appears almost uniformly distributed between 3 and 7 GPa, and no domains are evident. The hardness domain size in a and b is similar to the mean size of the steel grains. The fact that after laser exposure high-hardness domains appeared with the same dimensions of steel grains indicates that C atoms did not completely diffuse in the material, but remained confined in perlitic grains.
To quantify the statistical distribution of hardness in the tested regions of the specimen, the corresponding histograms are reported in Fig. 8.
Hardness distribution obtained by AFM nano-indentation for a untreated portion of the sample (bulk); b, c different regions interested by laser hardening; and d sample treated with standard thermic hardening. The continuous lines in b and c are the Gaussian fits supposing a bimodal hardness distribution
In the histogram in Fig. 8a, which corresponds to the untreated region, the hardness distribution is single-peaked between 3.5 and 5.0 GPa, asymmetric toward larger value (lognormal distribution), confirming the absence of multiple hardness domains. The hardness distributions corresponding to the laser-treated region are significantly different from that of the untreated region (Fig. 8b, c). As already mentioned, the spread of values is very large without observing the presence of a single peak. We can speculate the occurrence of a bimodal distribution in both, with peaks at about 4.5 and 8.3 GPa for histogram a, and 4.7 and 8.4 GPa for histogram b. This thesis is supported by the agreement between the related Gaussian fits and the corresponding hardness distribution suggesting that part of the treated material undergoes the transformation to austenite, but a relevant portion has the same hardness of the untreated region. This is in good agreement with what proposed in (Orazi et al. 2010; Fortunato et al. 2013) where, due to the short interaction time of laser hardening, C intra-diffusion takes place in the perlite colony, but inter-diffusion and homogenisation of ferrite grains are negligible. Considering the domain size calculated from map a and b, it follows that the C mean diffusion length is not larger than 10 μm. This result is in agreement with previous findings on laser-treated steel analysed by classical indentation taking into account the large size of the indenter, which reported a progressive decrease of hardness with the distance from the exposed area (Orazi et al. 2010). In Fig. 8d, the hardness distribution of a sample treated with standard tempering is reported. As expected, in this case the distribution is single-peaked, confirming a homogenous diffusion of C in all the austenitic grains. The lower absolute value measured in this case with respect to laser-treated sample is related to the difficulty in maintaining exactly the same set-up parameters (laser alignment on the AFM cantilever ) and other systematic error intrinsic to AFM indentation (non-central geometry of the indenter, creep and hysteresis of the piezo-actuators), as briefly discussed in the previous section.
In the present work, a novel simplified approach for laser surface hardening modelling is presented and discussed based on the assumption of neglecting the austenite homogenization during the heating phase of the process. The aim of the work was the experimental validation of this postulation through hardness measurements on C45 specimens laser and oven treated obtained by means of atomic force microscopy (AFM) technique. The achieved uneven AFM nano-indentation maps of laser hardened portions of the specimens indicated that carbon atoms did not completely diffuse in the material but remained confined in the perlitic grains. A hardness spread of 13 GPa was acquire for the laser-treated regions, significantly greater than the 4 GPa of the untreated region. A statistical elaboration of the acquired data showed a bimodal hardness distribution of the laser hardened portions once more suggesting that only a part of the treated material transformed to austenite homogenized its carbon content; this could moreover explain the slightly higher hardness of the laser hardened material compared to heat-treated one.
The raw/processed data required to reproduce these findings cannot be shared at this time due to technical or time limitations.
AFM:
BCC:
Body-centred cubic
Face-centred cubic
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No founding have been received for this research.
DISMI - Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, Reggio Emilia, Italy
L. Orazi & B. Reggiani
EN&TEC - Centro Interdipartimentale per la Ricerca Industriale ed il Trasferimento Tecnologico nel Settore delle Tecnologie Integrate per la Ricerca Sostenibile, della Conversione Efficiente dell'Energia, l'Efficienza Energetica degli Edifici, l'Illuminazione e la Domotica, University of Modena and Reggio Emilia, Reggio Emilia, Italy
L. Orazi
FIM – Department of Physics, Informatics and Mathematics, University of Modena and Reggio Emilia, Modena, Italy
A. Rota
INTERMECH - Centro Interdipartimentale per la Ricerca Applicata e i Servizi nel settore della Meccanica Avanzata e della Motoristica, University of Modena and Reggio Emilia, Modena, Italy
A. Rota & B. Reggiani
B. Reggiani
Leonardo Orazi: L.O., Alberto Rota A.R., Barbara Reggiani B.R. L.O. and A.R. were involved in planning and supervised the work. L.O., A.R. and B.R. processed the experimental data, performed the analysis, drafted the manuscript and designed the figures. L.O., A.R. performed the hardness and AFM measurements. All authors aided in interpreting the results, worked on the manuscript, discussed the results and commented on the manuscript. The authors read and approved the final manuscript.
Correspondence to L. Orazi.
Orazi, L., Rota, A. & Reggiani, B. Experimental investigation on a novel approach for laser surface hardening modelling. Int J Mech Mater Eng 16, 2 (2021). https://doi.org/10.1186/s40712-020-00124-0
Carbon diffusion
Austenite homogenization
Numerical model
Carbon steels | CommonCrawl |
Performance Modeling and Analysis
Performance Modeling and Analysis - History
Performance modeling and analysis has been and continues to be of great practical and theoretical importance in research labs in the design development and optimization of computer and communication systems and applications. This includes a broad spectrum of research activities from the use of more empirical methods (ranging from experimental tweaking of simple existing models up to building and experimenting with prototype implementations) through the use of simulation to more sophisticated mathematical methods. IBM Research has a long and rich history in this area of research. A small subset of examples include: Time-Sharing Computer Model, Token Ring Local Area Networks, Product Form Networks and RESQ Package, Computer Scheduling, S/390 Sysplex, Traffic Management, Parallel Scheduling. For more information, refer to S.S. Lavenberg and M.S. Squillante, Performance Evaluation in Industry: A Personal Perspective, in "Performance Evaluation -- Origins and Directions", G. Haring, C. Lindemann, M. Reiser (eds.), Springer-Verlag, 1999, and the references cited therein.
A Simple Queueing Network Model of a Time Sharing System One of the earliest documented successful applications of analytical computer performance modeling in industry is due to Lassettre and Scherr (1972) at IBM in the late 1960s during the development of IBM's OS/360 Time Sharing Option (TSO). The machine repairman model, originally developed in the operations research literature to model a single repairman servicing machines that break down, was used to represent OS/360 TSO running with a single memory partition and supporting multiple terminal attached users via time sharing the execution of their programs. With a single memory partition, only one user's program could be resident in memory and the system was time shared by swapping programs into and out of memory. The machine repairman model has a very simple analytical solution which was used to estimate the number of users the system could support without exceeding a specified average response time. The model was used in conjunction with measurements. Average program execution time was estimated from measurements on a TSO system with a second computer system simulating the users by running benchmark scripts and generating exponentially distributed user think times. The measured average response time was compared with the mean response time computed using the model under the assumption of exponentially distributed execution times with mean equal to the measured average. It is interesting to note that a substantial difference between the measured and predicted response time was usually due to bugs in the operating system. Once the bugs were removed the measured and predicted results tracked closely. In approximately 75% of the test cases, the prediction error was less than 10% with a maximum error of 24%. This was surprising due to the simplistic model assumptions. In particular, program execution times were assumed to be exponentially distributed, although measurements showed they were not, and programs were assumed to be executed first come first served, rather than via time sharing. Unknown at the time, this queueing model is an example of a product form queueing network. Product form queueing network results of the 1970s showed invariance of performance measures including mean response time when a first-come-first-served queue with exponential service times is replaced by a processor sharing queue with general service times. A processor sharing queue with general service times would have been a more realistic model of the TSO system, but the model's predictions would not have been different, thus helping to explain the model's accuracy.
The Performance of Token Ring Local Area Networks One of the most influential analytical performance modeling studies in IBM was done by Bux (1981). This study compared the analytically derived delay-throughput characteristics of local area networks based on ring and bus topologies. Included were the token ring, a ring network with access controlled by a single circulating token that was being built at IBM's Research Lab in Zurich, and the CSMA-CD (carrier sensing, multiple access with collision detection) bus that was the basis of Ethernet, originally developed by Xerox in the 1970s. The study primarily used existing analytical queueing results, modifying them as required to capture the essential characteristics of the networks being modeled. For example, the key to analyzing token ring performance was recognizing that a token ring functioned like a single server that served multiple queues by round robin polling. An elegant discrete time queueing analysis of such a polling system had appeared in Konheim and Meister (1974). (The discrete time results were converted to continuous time by letting the discrete time interval approach zero.) The study showed that the delay-throughput characteristics of the token ring and CSMA-CD bus were comparable at low transmission speeds, e.g. 1 Mb/sec, but the token ring was superior at higher speeds, e.g. 10 Mb/sec. (The key parameter affecting the relative performance of the token ring and CSMA-CD bus is the ratio of propagation delay to packet transmission time, with higher ratios favoring the token ring.) While many factors influenced IBM's decision to develop a token ring local area network product, the performance of the token ring as demonstrated in this study was a key factor.
Product Form Networks and The Research Queueing (RESQ) Package The discovery of product form queueing networks and their properties and the development of efficient computational algorithms for product form networks was a breakthrough in analytical performance modeling. Within the computer science literature, the classic paper that first defined product form networks and gave their properties is Baskett, Chandy, Muntz and Palacios-Gomez (1975). Researchers in IBM developed the main computational algorithms for solving product form networks, first the convolution algorithm, Reiser and Kobayashi (1975), and later the Mean Value Analysis (MVA) algorithm, Reiser and Lavenberg (1980), and they incorporated these algorithms in performance modeling software packages to make them available to performance modeling practitioners. The first such package was QNET4, software for specifying and solving closed multichain product form queueing networks. QNET4 provided a textual interface for the user to specify the queues and routing chains and their associated parameter values. Performance measures were computed using the convolution algorithm. Shortly after QNET4 was developed, it was integrated into a more general performance modeling package, the Research Queueing (RESQ) package.
RESQ allowed a user to specify and solve product form networks (initially using the convolution algorithm; the MVA algorithm was added later), but it also allowed a user to specify more general ``extended queueing networks'' and use discrete event simulation to estimate performance measures. The then recently developed regenerative method for estimating confidence intervals and controlling simulation run length was incorporated in RESQ. One of the key extensions was the inclusion of passive queues, which provide a convenient way to model simultaneous resource possession. QNET4's textual user interface was extended in a natural way to allow specification of extended queueing networks. The modeling level of abstraction provided by extended queueing networks and the implementation in RESQ proved very useful. It allowed the rapid development of simulation models without the programming required with a simulation programming language. It helped guard against the pitfall of developing overly detailed simulation models by forcing a higher level of abstraction. It included modern statistical simulation techniques and made them easy to use. It also helped bridge the gap between analytical modeling and simulation modeling by incorporating both product form networks and extended queueing networks. RESQ was a major success in IBM. Developed by researchers, it began to be widely used in product development groups in IBM in the late 1970s to model computer systems and subsystems and local and wide area computer networks. It was enhanced over time with additional computational algorithms and statistical methods, a graphical user interface, simulation animation and other features, and its widespread use in IBM continued into the 1990s. It was also made available for use in research and teaching at universities.
Computer System Scheduling The performance modeling and related stochastic literature over the past five decades is rich with studies of scheduling optimization problems. This includes optimal scheduling results for minimizing a weighted sum of the per-class mean response times, as well as for achieving a given vector of per-class mean response times, in a single queue or a queueing network, with or without side constraints (i.e., a per-class performance constraint that must be satisfied in addition to the global objective function). These results have basically established that, in many cases, the space of achievable performance measures is a polymatroid, or extended polymatroid, whose vertices correspond to the performance of the system under all possible fixed priority rules. Furthermore, the optimal or desired performance vector is a vertex or an interior point of this performance polytope, and the scheduling strategies which satisfy these classes of objective functions are some form or mixture of fixed priority policies (dynamic priority policies are considered below).
As computer technology advanced and the complexity of computer systems and applications continued to grow, new customer and/or user requirements arose that were not fully addressed by previous classes of scheduling objective functions. For this reason, research studies at IBM investigated specific scheduling optimization problems to consider the needs of certain IBM computer platforms. Two such studies became the basis for the processor scheduling algorithms in the Application System/400 (AS/400) and System/390 (S/390) computer systems.
Basis for the AS/400 Processor Scheduler:
Much of the previous scheduling research considered scheduling strategies for minimizing a weighted sum of the per-class mean response times. An important additional objective is to maintain a low variance of response times for each class. Two related research studies at IBM that investigated this problem resulted in the concepts of Delay Cost Scheduling, due to Franaszek and Nelson (1995), and Time-Function Scheduling, due to Fong and Squillante (1995). These two studies considered different forms of objective functions that are based on minimizing a weighted sum of per-class second moment measures of response time, and they used different approaches to establish certain structural properties for the respective optimal solutions.
One scheduling strategy with the structural properties for a particular instance of the scheduling objectives considered by Fong and Squillante is based on the use of general time-based functions to obtain effective and flexible control over the allocation of resources. This scheduling strategy is in part a generalization of the linear time-dependent priority discipline due to Kleinrock (1964) in which the priority of each job increases (linearly) according to a per-class function of some measure of time and the job with the highest instantaneous priority value in the queue is selected for execution at each scheduling epoch. In its most general setting, the time parameter for each per-class function can include any measure of the time spent waiting for a resource or set of resources (response mode), any measure of the time spent using a resource or set of resources (usage mode), and any combination of such modes, as developed by Fong, Hough and Squillante (1997). An adaptive feedback mechanism is used together with mathematical control formulas to adjust these per-class time functions, as well as to migrate each job to a different class upon the occurrence of certain events or upon the job exceeding some criteria, in order to satisfy the scheduling objective function across all and/or varying workloads. These control formulas can also be used to obtain a scheduling strategy that realizes the optimal or desired performance vector which is a (interior) point in the performance space, while providing better per-class response time variance properties. A number of important scheduling issues can be easily accommodated in the per-class time functions and/or addressed by the use of these time functions to control resource scheduling decisions; examples include priority inversion and processor-cache affinities. The theoretical properties of time-function scheduling can be further exploited to obtain very efficient implementations.
The above scheduling strategy is a fundamental aspect of the dynamic priority scheduling algorithms employed in AS/400 systems. Based on the control formulas mentioned above and a general set of assumptions regarding workloads and performance objectives, this scheduling technology offering has proven to be a great success with AS/400 customers providing efficient and effective control over resource management to address various customer requirements across numerous diverse workloads without any perceived increase in complexity by the system administrator and users.
Basis for the S/390 Processor Scheduler:
Instead of minimizing a weighted sum of the per-class mean response times, it can be more natural to associate a mean response time goal with each class and to consider the performance of the class relative to its goal. The corresponding objective function then is to minimize a vector of the per-class ratios of the response time mean to the response time goal. This problem is studied within the context of a multi-class M/GI/1 queue, with and without feedback, by Bhattacharya et al. (1993,1995) where adaptive scheduling strategies are presented and shown to lexicographically minimize the vector of per-class performance ratios (exactly or approximately). The results also apply to other systems including certain multi-class Jackson networks and multi-class M/GI/c queues.
Consider a $K$-class system at a scheduling decision time epoch in which the mean response time for class $i$ realized over the previous scheduling time interval(s) is $x_i$ and the specified mean response time goal for this class is $g_i$. A fixed scheduling policy that gives priority to jobs of class $i$ over jobs of class $j$ if $x_i/g_i \geq x_j/g_j$ is then used for the next scheduling time interval. In other words, priority is given to the class that has received the worse performance, relative to its goal, over the previous time interval(s). The time intervals between scheduling decision epochs, in which priorities are updated, can be arbitrary provided that they are bounded above by a finite measure. This scheduling strategy is proven optimal in the sense that it converges asymptotically in the number of time intervals to the optimal solution which lexicographically minimizes the vector of ratios $x_i/g_i$ arranged in non-increasing order $x_1/g_1 \geq x_2/g_2 \geq \cdots \geq x_K/g_K$ (with probability 1).
The above scheduling strategy is an important aspect of the processor scheduling algorithms employed in S/390 systems, where $x_i$ and $g_i$ can be functions of performance metrics other than just mean response time -- in particular, response time percentiles and/or velocity goals can be used together with or instead of mean response time goals. This S/390 concept is commonly referred to as goal-oriented scheduling. It is a key component of the workload management system provided on S/390 computers, which has been a great success for IBM in the control and management of mainframes and mainframe clusters.
Cluster Architectures and S/390 Parallel Sysplex As commercial workloads expand and evolve, clusters of multiple computers are required to support high transaction processing rates and high availability in large-scale commercial computing environments, which include on-line transaction processing systems and parallel database systems. The principal cluster architectures proposed to support these scalable commercial application environments are the shared-nothing (or partitioned), the shared-disk, the virtual shared-disk, and the Parallel Sysplex} models. The shared-nothing architecture consists of partitioning the database and disks among the nodes, where either function-shipping (i.e., a remote function call to be executed by the remote node with the results, if any, returned to the local node) or I/O shipping (i.e., a remote I/O request to fetch the required data from the remote node) is used when a local transaction needs to access data located at a remote node in the cluster. Advantages of this architecture include a higher local buffer hit ratio and no need for global concurrency control, whereas the main disadvantages center around the various additional costs for remote requests to access non-local databases and load balancing and availability problems. The shared-disk architecture consists of essentially having all nodes directly access the disks on which shared data is located, where each node has a local database buffer cache and a global concurrency control protocol is used to maintain consistency among the local caches as well as the database. This architecture has advantages with respect to load balancing and availability, but it can suffer from additional overhead to acquire and release global locks, as well as large overhead, latency and increased I/O activity for hot shared data (so-called ping-ponging). The virtual shared-disk architecture is functionally identical to the shared-nothing architecture with I/O shipping, while providing the view of a shared-disk model to the database (i.e., the partitioned disks are transparent to the database). The Parallel Sysplex architecture consists of the shared-disk model together with a shared coupling facility that provides a shared database buffer and highly-efficient support for locking, cache coherency and general-purpose queueing.
Various aspects of each of these principal cluster architecture alternatives have been analyzed with performance models, many of which have been formulated by decomposing the original problem into more tractable parts. The solutions of these hierarchical models have often involved a combination of different methods including analytical, mathematical optimization and simulation. The results of this analysis of the principal cluster architectures by researchers at IBM, such as King et al. (1997), and elsewhere influenced the design of the IBM S/390 Parallel Sysplex architecture, and was used by IBM to demonstrate the key advantages of this design over alternative cluster architectures. In particular, the Parallel Sysplex architecture provides the benefits of shared-disk architectures and exploits the coupling facility services to obtain very efficient intertransaction concurrency control, buffer cache coherency control, shared buffer management, and shared job queues. This results in transaction rate scaling which is close to linear in the number of nodes, high shared buffer hit ratios which can reduce the I/O rate per node, and excellent dynamic load balancing even in systems with heterogeneous nodes.
The Parallel Sysplex technology provides the fundamental infrastructure for IBM's large-scale enterprise server environments, and the above performance modeling studies played a key role in its design.
Network Traffic Management The allocation and management of shared network resources among different classes of traffic streams with a wide range of performance requirements and traffic characteristics in high-speed packet-switched network architectures, such as ATM, is more complex than in traditional networks. An important aspect of this traffic management problem is to characterize the effective bandwidth requirement of both individual connections and the aggregate bandwidth usage of multiple connections statistically multiplexed on a given network link, as a function of their statistical properties and the desired level of service. These metrics can then be used for efficient bandwidth management and traffic control in order to achieve high utilization of network resources while maintaining the desired level of service for all connections. Guerin et al. (1991) proposed a methodology for the computation of the effective bandwidth requirement of individual and multiplexed connections based on a combination of two complementary approaches, namely a fluid model to estimate the bandwidth requirement when the impact of individual connection characteristics is critical, and the stationary bit rate distribution to estimate the bandwidth requirement when the effect of statistical multiplexing is significant. While the fluid model and its extension to capture the impact of multiplexing can be used to obtain an exact computation of the bandwidth requirement, the computational complexity involved is too high in general, and particularly for real-time network traffic management. Guerin et al. therefore used the proposed methodology to develop a computationally simple approximate expression for the effective bandwidth requirement of individual and multiplexed connections, which is shown to have sufficiently good accuracy across the range of possible connection characteristics in comparison with both exact computations and simulation results. This approximation has proven quite successful as one of the key mechanisms used for traffic management in IBM's Networking BroadBand Services (NBBS) architecture, as well as in similar offerings from other companies.
Parallel Scheduling A significant body of the performance modeling research literature has focused on various aspects of the parallel computer scheduling problem -- i.e., the allocation of computing resources among the parallel jobs submitted for execution. Several classes of scheduling strategies have been proposed for such computing environments, each differing in the way the parallel resources are shared among the jobs. This includes the class of space-sharing strategies that share the processors in space by partitioning them among different parallel jobs, the class of time-sharing strategies that share the processors by rotating them among a set of jobs in time, and the class of gang-scheduling strategies that combine both space-sharing and time-sharing. The numerous performance modeling and optimization studies in this area by researchers at IBM and elsewhere have played a fundamental and important role in the design, development and implementation of different forms of space-sharing and gang-scheduling strategies in commercial parallel supercomputing systems. In fact, the IBM SP family of parallel computers supports various forms of space-sharing and gang-scheduling; support for space-sharing and/or gang-scheduling is also included in many other commercial parallel supercomputers, such as the Cray T3E, the Intel Paragon, the Meiko CS-2 and the SGI Origin.
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Cauchy principal value
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. In this method, a singularity on an integral interval is avoided by limiting the integral interval to the singularity (so the singularity is not covered by the integral).
This article is about a method for assigning values to improper integrals. For the values of a complex function associated with a single branch, see Principal value. For the negative-power portion of a Laurent series, see Principal part.
Formulation
Depending on the type of singularity in the integrand f, the Cauchy principal value is defined according to the following rules:
For a singularity at a finite number b
$\lim _{\;\varepsilon \to 0^{+}\;}\,\left[\,\int _{a}^{b-\varepsilon }f(x)\,\mathrm {d} x~+~\int _{b+\varepsilon }^{c}f(x)\,\mathrm {d} x\,\right]$
with $a<b<c$ and where b is the difficult point, at which the behavior of the function f is such that
$\int _{a}^{b}f(x)\,\mathrm {d} x=\pm \infty \quad $
for any $a<b$ and
$\int _{b}^{c}f(x)\,\mathrm {d} x=\mp \infty \quad $
for any $b<c.$
(See plus or minus for the precise use of notations ± and ∓.)
For a singularity at infinity ($\infty $)
$\lim _{a\to \infty }\,\int _{-a}^{a}f(x)\,\mathrm {d} x$
where
$\int _{-\infty }^{0}f(x)\,\mathrm {d} x=\pm \infty $
and
$\int _{0}^{\infty }f(x)\,\mathrm {d} x=\mp \infty .$
In some cases it is necessary to deal simultaneously with singularities both at a finite number b and at infinity. This is usually done by a limit of the form
$\lim _{\;\eta \to 0^{+}}\,\lim _{\;\varepsilon \to 0^{+}}\,\left[\,\int _{b-{\frac {1}{\eta }}}^{b-\varepsilon }f(x)\,\mathrm {d} x\,~+~\int _{b+\varepsilon }^{b+{\frac {1}{\eta }}}f(x)\,\mathrm {d} x\,\right].$
In those cases where the integral may be split into two independent, finite limits,
$\lim _{\;\varepsilon \to 0^{+}\;}\,\left|\,\int _{a}^{b-\varepsilon }f(x)\,\mathrm {d} x\,\right|\;<\;\infty $
and
$\lim _{\;\eta \to 0^{+}}\;\left|\,\int _{b+\eta }^{c}f(x)\,\mathrm {d} x\,\right|\;<\;\infty ,$
then the function is integrable in the ordinary sense. The result of the procedure for principal value is the same as the ordinary integral; since it no longer matches the definition, it is technically not a "principal value".
The Cauchy principal value can also be defined in terms of contour integrals of a complex-valued function $f(z):z=x+i\,y\;,$ with $x,y\in \mathbb {R} \;,$ with a pole on a contour C. Define $C(\varepsilon )$ to be that same contour, where the portion inside the disk of radius ε around the pole has been removed. Provided the function $f(z)$ is integrable over $C(\varepsilon )$ no matter how small ε becomes, then the Cauchy principal value is the limit:[1]
$\operatorname {p.\!v.} \int _{C}f(z)\,\mathrm {d} z=\lim _{\varepsilon \to 0^{+}}\int _{C(\varepsilon )}f(z)\,\mathrm {d} z.$
In the case of Lebesgue-integrable functions, that is, functions which are integrable in absolute value, these definitions coincide with the standard definition of the integral.
If the function $f(z)$ is meromorphic, the Sokhotski–Plemelj theorem relates the principal value of the integral over C with the mean-value of the integrals with the contour displaced slightly above and below, so that the residue theorem can be applied to those integrals.
Principal value integrals play a central role in the discussion of Hilbert transforms.[2]
Distribution theory
Let ${C_{c}^{\infty }}(\mathbb {R} )$ be the set of bump functions, i.e., the space of smooth functions with compact support on the real line $\mathbb {R} $. Then the map
$\operatorname {p.\!v.} \left({\frac {1}{x}}\right)\,:\,{C_{c}^{\infty }}(\mathbb {R} )\to \mathbb {C} $
defined via the Cauchy principal value as
$\left[\operatorname {p.\!v.} \left({\frac {1}{x}}\right)\right](u)=\lim _{\varepsilon \to 0^{+}}\int _{\mathbb {R} \setminus [-\varepsilon ,\varepsilon ]}{\frac {u(x)}{x}}\,\mathrm {d} x=\int _{0}^{+\infty }{\frac {u(x)-u(-x)}{x}}\,\mathrm {d} x\quad {\text{for }}u\in {C_{c}^{\infty }}(\mathbb {R} )$
is a distribution. The map itself may sometimes be called the principal value (hence the notation p.v.). This distribution appears, for example, in the Fourier transform of the sign function and the Heaviside step function.
Well-definedness as a Distribution
To prove the existence of the limit
$\int _{0}^{+\infty }{\frac {u(x)-u(-x)}{x}}\,\mathrm {d} x$
for a Schwartz function $u(x)$, first observe that ${\frac {u(x)-u(-x)}{x}}$ is continuous on $[0,\infty ),$ as
$\lim _{\,x\searrow 0\,}\;{\Bigl [}u(x)-u(-x){\Bigr ]}~=~0~$
and hence
$\lim _{x\searrow 0}\,{\frac {u(x)-u(-x)}{x}}~=~\lim _{\,x\searrow 0\,}\,{\frac {u'(x)+u'(-x)}{1}}~=~2u'(0)~,$
since $u'(x)$ is continuous and L'Hopital's rule applies.
Therefore, $\int _{0}^{1}\,{\frac {u(x)-u(-x)}{x}}\,\mathrm {d} x$ exists and by applying the mean value theorem to $u(x)-u(-x),$ we get:
$\left|\,\int _{0}^{1}\,{\frac {u(x)-u(-x)}{x}}\,\mathrm {d} x\,\right|\;\leq \;\int _{0}^{1}{\frac {{\bigl |}u(x)-u(-x){\bigr |}}{x}}\,\mathrm {d} x\;\leq \;\int _{0}^{1}\,{\frac {\,2x\,}{x}}\,\sup _{x\in \mathbb {R} }\,{\Bigl |}u'(x){\Bigr |}\,\mathrm {d} x\;\leq \;2\,\sup _{x\in \mathbb {R} }\,{\Bigl |}u'(x){\Bigr |}~.$
And furthermore:
$\left|\,\int _{1}^{\infty }{\frac {\;u(x)-u(-x)\;}{x}}\,\mathrm {d} x\,\right|\;\leq \;2\,\sup _{x\in \mathbb {R} }\,{\Bigl |}x\cdot u(x){\Bigr |}~\cdot \;\int _{1}^{\infty }{\frac {\mathrm {d} x}{\,x^{2}\,}}\;=\;2\,\sup _{x\in \mathbb {R} }\,{\Bigl |}x\cdot u(x){\Bigr |}~,$
we note that the map
$\operatorname {p.v.} \;\left({\frac {1}{\,x\,}}\right)\,:\,{C_{c}^{\infty }}(\mathbb {R} )\to \mathbb {C} $
is bounded by the usual seminorms for Schwartz functions $u$. Therefore, this map defines, as it is obviously linear, a continuous functional on the Schwartz space and therefore a tempered distribution.
Note that the proof needs $u$ merely to be continuously differentiable in a neighbourhood of 0 and $x\,u$ to be bounded towards infinity. The principal value therefore is defined on even weaker assumptions such as $u$ integrable with compact support and differentiable at 0.
More general definitions
The principal value is the inverse distribution of the function $x$ and is almost the only distribution with this property:
$xf=1\quad \Leftrightarrow \quad \exists K:\;\;f=\operatorname {p.\!v.} \left({\frac {1}{x}}\right)+K\delta ,$
where $K$ is a constant and $\delta $ the Dirac distribution.
In a broader sense, the principal value can be defined for a wide class of singular integral kernels on the Euclidean space $\mathbb {R} ^{n}$. If $K$ has an isolated singularity at the origin, but is an otherwise "nice" function, then the principal-value distribution is defined on compactly supported smooth functions by
$[\operatorname {p.\!v.} (K)](f)=\lim _{\varepsilon \to 0}\int _{\mathbb {R} ^{n}\setminus B_{\varepsilon }(0)}f(x)K(x)\,\mathrm {d} x.$
Such a limit may not be well defined, or, being well-defined, it may not necessarily define a distribution. It is, however, well-defined if $K$ is a continuous homogeneous function of degree $-n$ whose integral over any sphere centered at the origin vanishes. This is the case, for instance, with the Riesz transforms.
Examples
Consider the values of two limits:
$\lim _{a\to 0+}\left(\int _{-1}^{-a}{\frac {\mathrm {d} x}{x}}+\int _{a}^{1}{\frac {\mathrm {d} x}{x}}\right)=0,$
This is the Cauchy principal value of the otherwise ill-defined expression
$\int _{-1}^{1}{\frac {\mathrm {d} x}{x}},{\text{ (which gives }}{-\infty }+\infty {\text{)}}.$
Also:
$\lim _{a\to 0+}\left(\int _{-1}^{-2a}{\frac {\mathrm {d} x}{x}}+\int _{a}^{1}{\frac {\mathrm {d} x}{x}}\right)=\ln 2.$
Similarly, we have
$\lim _{a\to \infty }\int _{-a}^{a}{\frac {2x\,\mathrm {d} x}{x^{2}+1}}=0,$
This is the principal value of the otherwise ill-defined expression
$\int _{-\infty }^{\infty }{\frac {2x\,\mathrm {d} x}{x^{2}+1}}{\text{ (which gives }}{-\infty }+\infty {\text{)}}.$
but
$\lim _{a\to \infty }\int _{-2a}^{a}{\frac {2x\,\mathrm {d} x}{x^{2}+1}}=-\ln 4.$
Notation
Different authors use different notations for the Cauchy principal value of a function $f$, among others:
$PV\int f(x)\,\mathrm {d} x,$
$\mathrm {p.v.} \int f(x)\,\mathrm {d} x,$
$\int _{L}^{*}f(z)\,\mathrm {d} z,$
$-\!\!\!\!\!\!\int f(x)\,\mathrm {d} x,$
as well as $P,$ P.V., ${\mathcal {P}},$ $P_{v},$ $(CPV),$ and V.P.
See also
• Hadamard finite part integral
• Hilbert transform
• Sokhotski–Plemelj theorem
References
1. Kanwal, Ram P. (1996). Linear Integral Equations: Theory and technique (2nd ed.). Boston, MA: Birkhäuser. p. 191. ISBN 0-8176-3940-3 – via Google Books.
2. King, Frederick W. (2009). Hilbert Transforms. Cambridge, UK: Cambridge University Press. ISBN 978-0-521-88762-5.
| Wikipedia |
\begin{document}
\begin{CJK*}{GBK}{song}
\title{Quantum discord and its asymptotic behaviors in a time-dependent $XY$ spin chain}
\author{Jian Zhang} \email{[email protected]} \affiliation{Key Laboratory of Cluster Science of Ministry of Education and School of Chemistry, Beijing Institute of Technology, Beijing 100081, People's Republic of China} \affiliation{School of Electronic and Information Engineering, Hefei Normal University, Hefei 230601, People's Republic of China} \author{Bin Shao} \email{[email protected]} \affiliation{Key Laboratory of Cluster Science of Ministry of Education and School of Chemistry, Beijing Institute of Technology, Beijing 100081, People's Republic of China}
\author{Lian-Ao Wu} \email{lianao\[email protected]} \affiliation{Department of Theoretical Physics and History of Science, The Basque Country University (EHU/UPV),Post Office Box 644, 48080 Bilbao, Spain} \affiliation{IKERBASQUE, Basque Foundation for Science, 48011, Bilbao, Spain}
\author{Jian Zou} \affiliation{Key Laboratory of Cluster Science of Ministry of Education and School of Chemistry, Beijing Institute of Technology, Beijing 100081, People's Republic of China}
\date{\today}
\begin{abstract} We study the dynamics and the asymptotic behaviors of quantum discord in a one-dimensional $XY$ model coupled through time-dependent nearest-neighbor interactions and in the presence of a time-dependent magnetic field. We find that the time evolution of the nearest-neighbor quantum discord in this system shows non-ergodic behaviors but is asymptotic to its steady value at the long-time limit. The zero-temperature asymptotic behaviors of quantum discord is only determined by the ratio between the coupling parameter and magnetic field, whereas the finite-temperature asymptotic behaviors determined by both of them. These asymptotic behaviors are sensitive not only to the initial values of the coupling parameter and magnetic field, but also to the final values. It is interesting to note that quantum discords are more robust than entanglement against the effect of temperature. We also find particular parameter regimes, where the nearest-neighbor quantum discord is enhanced significantly.
\end{abstract} \pacs{03.65.Ta,75.10.Pq} \maketitle
\section{Introduction} Entanglement \cite{RevModPhys.81.865} is a quantum correlation without classical counterpart and has been widely believed as the main reason for the computational advantage of quantum over classical algorithms. However, it has been found recently that there exist strong indications \cite{PhysRevLett.100.050502} that an more general quantum correlation, quantum discord (QD) \cite{PhysRevLett.88.017901,0305-4470-34-35-315}, is the resource responsible for the speed up in the deterministic quantum computation with one quantum bit (DQC1) \cite{PhysRevLett.81.5672}. Moreover, quantum discord is more robust than the entanglement against decoherence such that quantum algorithms based only on quantum discord might be more robust than those on entanglement. Quantum discord \cite{PhysRevLett.88.017901,0305-4470-34-35-315}, as a up-and-coming quantum correlation, often arises as a consequence of coherence between different partitions in a quantum system, being present even in separable states. This quantum correlation has received much attention in quantum computation \cite{PhysRevLett.100.050502, PhysRevLett.101.200501}, quantum communication \cite{PhysRevA.82.032340,PhysRevA.82.012338, PhysRevA.84.012327}, dynamics of quantum discord \cite{PhysRevA.80.044102,PhysRevA.81.052107, PhysRevLett.105.150501,PhysRevLett.104.200401,PhysRevLett.106.050403}, quantum phase transitions(QPTs) \cite{PhysRevLett.105.095702,PhysRevA.80.022108,PhysRevA.82.042316}, witnessing \cite{PhysRevA.81.062102} etc.
On the other hand, the spin chain model have been proposed as a very reliable model for the future quantum-computing technology in different solid-state systems and a rich model for studying the novel physics of localized spin systems. This spin chain can be experimentally realized as trapped ions \cite{PhysRevA.63.012306,nphys2252}, superconducting junctions, coupled quantum dots \cite{PhysRevA.57.120,PhysRevB.59.2070}, ultracold quantum gases \cite{nphys2259} et al. The experimental progresses have triggered intensive theoretical research on entanglement and quantum discord in the one-dimensional spin chains \cite{PhysRevLett.93.250404,PhysRevLett.98.230503,RevModPhys.80.517,PhysRevA.84.012319,PhysRevA.71.022315}. Specifically, entanglement and quantum discord in the time-dependent spin chains is investigated \cite{PhysRevLett.107.010403,1742-5468-2011-08-P08026,0295-5075-98-3-30013,HuangPhysRev,Driven_xy_model,Sadiek_2010}. The dynamics of entanglement in an $XY$ and Ising spin chains has been studied by considering a constant nearest-neighbor coupling and in presence of a time varying magnetic field \cite{HuangPhysRev}. The entanglement dynamics in a time-dependent anisotropic $XY$ model with a small number of spins is studied numerically at zero temperature \cite{Driven_xy_model}. The time-dependent spin-spin coupling was represented by a dc part and a sinusoidal ac part. Recently, Ref. \cite{Sadiek_2010} carefully analyzes the time evolution of entanglement in a one-dimensional spin chain in presence of a time dependent magnetic field $h(t)$ and by considering a time dependent coupling parameter $J(t)$. Both $h(t)$ and $J(t)$ are step functions of time \cite{Sadiek_2010}. The entanglement undergoes a nonergodic behavior. The zero-temperature asymptotic behaviors of entanglement depend only on the ratio between the coupling parameter and magnetic field, whereas the finite-temperature asymptotic behaviors depend on both the coupling parameter and magnetic field. The aim of this work is to discuss what the advantages of quantum discord is in the time-dependent spin chain, comparing with entanglement and to explore the relation between the asymptotic behaviors of quantum discord and the parameter setting.
This paper attempts to study the dynamics of quantum discord in the time-dependent $XY$ model, and to explore the interesting asymptotic behaviors of quantum discord. In Sec. II, we introduce the time-dependent $XY$ spin chain and describe the general solution for this model. Sec. III analyzes the discord dynamics. The asymptotic behaviors of quantum discord are analyzed carefully in Sec. III. We conclude our work in Sec. IV.
\section{QUANTUM DISCORD IN THE TIME DEPENDENT XY MODEL}
In this section, we briefly review the exact solution for the spin $XY$ model of a one-dimensional lattices with $N$ sites coupled through time-dependent couplings $J(t)$ and subject to an external time-dependent magnetic field $h(t)$, and introduce the geometric measure of quantum discord.
The Hamiltonian for such a system is given by \cite{Sadiek_2010} (setting $\hbar=1$) \begin{equation} H=-\frac{J(t)}{2} \sum_{i=1}^{N}( (1+\gamma)\sigma_{i}^{x} \sigma_{i+1}^{x}+(1-\gamma)\sigma_{i}^{y} \sigma_{i+1}^{y})- h(t)\sum_{i=1}^{N}\sigma_{i}^{z}, \label{eq:H} \end{equation} where $\sigma^{\alpha}_{i},\alpha=\{x,y,z\}$ are the Pauli matrices and $\gamma$ is the anisotropy parameter.
The coupling and magnetic field are represented respectively by \begin{eqnarray} \nonumber J(t)&=&J_0 + (J_1 - J_0) \theta(t) \\
h(t)&=&h_0 + (h_1 - h_0) \theta(t), \end{eqnarray} where $\theta(t)$ is the mathematical step function \begin{equation} \theta(t)=\left\{ \begin{array}{lr} 0 & \qquad t\leq 0 \\ 1 & \qquad t>0 \end{array}. \right. \end{equation} We assume that the system is initially in thermal equilibrium. The reduced two-spin density matrix $\rho^{ij}(t)$ for this system therefore is \begin{equation} \rho^{ij}(t)=\left(\begin{array}{cccc}
\rho_{11} &0 &0 & \rho_{14} \\
0 & \rho_{22} & \rho_{23} &0 \\
0 & \rho_{23}^{\ast} & \rho_{33} &0 \\
\rho_{14}^{\ast} &0 &0 &\rho_{44} \\ \end{array}\right)\, , \label{eq:rhot}\end{equation} where matrix elements can be written in terms of one- and two-point correlation functions \cite{Sadiek_2010} \begin{eqnarray} \nonumber \rho_{11}&=&\left\langle M^z_{l}\right\rangle+\left\langle S^{z}_{l} S^{z}_{m} \right\rangle+\frac{1}{4} ,\\ \nonumber \rho_{22}&=&\rho_{33}=-\left\langle S^{z}_{l} S^{z}_{m} \right\rangle+\frac{1}{4} ,\\ \nonumber \rho_{44}&=&- \left\langle M^z_{l}\right\rangle+\left\langle S^{z}_{l} S^{z}_{m} \right\rangle+\frac{1}{4} ,\\ \nonumber \rho_{23}&=&\left\langle S^{x}_{l} S^{x}_{m} \right\rangle+\left\langle S^{y}_{l} S^{y}_{m} \right\rangle,\\ \rho_{14}&=&\left\langle S^{x}_{l} S^{x}_{m} \right\rangle-\left\langle S^{y}_{l} S^{y}_{m} \right\rangle . \end{eqnarray} The magnetization in the $z$-direction is defined as \begin{equation} M^{z}=\frac{1}{N}\sum_{j=1}^{N}(S_{j}^{z}). \end{equation} Its expectation value is \begin{equation} \left\langle M^{z}\right\rangle=\frac{Tr[M^{z}\rho(t)]}{Tr[\rho(t)]}, \end{equation} specifically \cite{Sadiek_2010}, \begin{eqnarray} \nonumber \left\langle M^{z}\right\rangle= \frac{1}{4N}\sum_{p=1}^{N/2}\frac{\tanh[\beta \Gamma(h_{0},J_{0})]}{\Gamma^2(h_{1},J_{1})\Gamma(h_{0},J_{0})}\\ \nonumber \biggl\{2 J_{1}(J_{0} h_{1} -J_{1} h_{0}) \delta_{p}^2 \sin^2[2t \Gamma(h_{1},J_{1})]\\ +4\Gamma^2(h_{1},J_{1})(J_{0}\cos\phi_{p}+h_{0})\biggr\}\, . \end{eqnarray} where $\phi_{p}=\frac{2 \pi p}{N}$ , $\delta_{p}=2 \gamma \sin \phi_{p}$ and $\beta=1/k T$. $k$ is Boltzmann constant and $T$ is the temperature.
Using Wick Theorem \cite{Wick}, the nearest-neighbor spin correlation functions can be obtained as \begin{eqnarray} \nonumber \left\langle S^{x}_{l} S^{x}_{l+1} \right\rangle&=&\frac{1}{4}F_{l,l+1}\\ \nonumber \left\langle S^{y}_{l} S^{y}_{l+1} \right\rangle&=&\frac{1}{4}F_{l+1,l}\\ \nonumber \left\langle S^{z}_{l} S^{z}_{l+1} \right\rangle&=&\frac{1}{4} \{F_{l, l}\times F_{l+1,l+1}-Q_{l,l+1}\times G_{l,l+1}\\ &-&F_{l+1,l}\times F_{l,l+1}\}, \end{eqnarray} where \cite{Sadiek_2010}
\begin{widetext} \begin{eqnarray} \nonumber Q_{l, m}&=& \frac{1}{N} \sum_{p=1}^{N/2} \biggl\{2\cos[(m-l)\phi_{p}]\\= &+&\frac{i(J_{1}h_{0}- J_{0}h_{1}) \delta_{p} \sin[(m-l)\phi_{p}]\sin[4t\Gamma(h_{1},J_{1})]\tanh[\beta \Gamma(h_{0},J_{0})]}{\Gamma(h_{1},J_{1})\Gamma(h_{0},J_{0})}\biggr\}\, ,\quad \end{eqnarray} \begin{eqnarray} \nonumber G_{l, m}&=&\frac{1}{N} \sum_{p=1}^{N/2} \biggl\{-2\cos[(m-l)\phi_{p}]\\ &+&\frac{i(J_{1}h_{0}- J_{0}h_{1}) \delta_{p} \sin[(m-l)\phi_{p}]\sin[4t\Gamma(h_{1},J_{1})]\tanh[\beta \Gamma(h_{0},J_{0})]}{\Gamma(h_{1},J_{1})\Gamma(h_{0},J_{0})}\biggr\}\, ,\quad \end{eqnarray} \begin{eqnarray} \nonumber F_{l, m}&=&\frac{1}{N} \sum_{p=1}^{N/2} \frac{\tanh[\beta \Gamma(h_{0},J_{0})]}{\Gamma^2(h_{1},J_{1})\Gamma(h_{0},J_{0})} \Biggl\{\cos[(m-l)\phi_{p}] \\ \nonumber &\times&\biggl\{J_{1} [J_{0} h_{1} - J_{1} h_{0}] \delta^2_{p} \sin^2[2t\Gamma(h_{1},J_{1})] + 2\Gamma^2(h_{1},J_{1})(J_{0}\cos\phi_{p}+h_{0})\biggr\}\\
&+& \delta_{p} \sin[(m-l)\phi_{p}]\biggl\{J_{0}\Gamma^2(h_{1},J_{1}) +2 (J_{1} h_{0} - J_{0} h_{1}) (J_{1}\cos\phi_{p}+h_{1}) \sin^2[2t\Gamma(h_{1},J_{1})]\biggr\}\Biggr\}\, . \end{eqnarray} \end{widetext} Here \begin{equation} \Gamma[h(t),J(t)]=\left\{[J(t)\cos \phi_{p} + h(t)]^{2}+\gamma^2 J^2(t) \sin^2\phi_{p}\right\}^{\frac{1}{2}} . \end{equation}
Various measures of quantum discord and their extensions to multipartite systems have been proposed. Here we use the geometric measure of quantum discord \cite{PhysRevLett.105.190502}, \begin{equation}\label{GMD}
D(\rho)=\mathrm{min}_{\chi\in\Omega_0}||\rho-\chi||^2, \end{equation} where $\Omega_0$ denotes the set of zero-discord states and
$||X-Y||^2=\mathrm{Tr}(X-Y)^2$ is the square norm in the Hilbert-Schmidt space. In order to calculate this quantity for an arbitrary two-qubit state, we write $\rho$ in bloch representation: \begin{equation} \rho=\frac{1}{4}(\openone\otimes\openone+\sum_{i=1}^3x_i\sigma_i\otimes\openone+\sum_{i=1}^3y_i\openone\otimes\sigma_i+ \sum_{i,j=1}^3T_{ij}\sigma_i\otimes\sigma_j), \end{equation} where $x_i=\mathrm{Tr}\{\rho(\sigma_i\otimes\openone)\}$, $y_i=\mathrm{Tr}\{\rho(\openone\otimes\sigma_i)\}$ are components of the local Bloch vectors, $T_{ij}=\mathrm{Tr}\{\rho(\sigma_i\otimes\sigma_j)\}$ are components of the correlation tensor, and $\sigma_i$ ($i \in \{1,2,3\}$) are the three Pauli matrices. Each state $\rho$ can be expressed by the parameter set $\{\vec{x},\vec{y},T\}$, the geometric measure of quantum discord is therefore given explicitly \cite{PhysRevLett.105.190502} \begin{equation}
D(\rho)=\frac{1}{4}(||\vec{x}||^2+||T||^2-k_{\mathrm{max}}), \end{equation} where $k_{\mathrm{max}}$ is the largest eigenvalue of matrix $K=\vec{x}\vec{x}^{\mathrm{T}}+TT^{\mathrm{T}}$.
\section{Dynamics of quantum discord} We now come to study the dynamics of quantum discord in the time-dependent $XY$ spin chain. Throughout the paper we use $N=1000$ to numerically demonstrate our general results, which are almost size-independent. We also employ three dimensionless parameters $\lambda=J/h$, $\lambda_{1}=J_{1}/h_{1}$ and $\lambda_{0}=J_{0}/h_{0}$ for convenience.
The $XY$ model undergoes a second-order QPT (Ising transition) at the critical point (CP) $\lambda_{c} = 1$, which separates a ferromagnetic ordered phase from a quantum paramagnetic phase. When $\lambda> 1$ it is claimed \cite{Bunder99} there is another second-order QPT at $\gamma_{c}= 0$, which is termed as the anisotropy transition. Differently from the Ising transition due to the external field, this transition is driven by the anisotropy parameter $\gamma$ and separates a ferromagnet ordered along the $x$ direction from a ferromagnet ordered along the $y$ direction.
\begin{figure}
\caption{Dynamics of the nearest-neighbor discord $D(i,i+1)$ with $\gamma=1$, $kT=0$ and (a) $h_{0}=h_{1}=1$ for various values of $J_{0}$ and $J_{1}$; (b) $J_{0}=J_{1}=1$ for various values of $h_{0}$ and $h_{1}$. }
\label{(t)-T0-gamma1}
\label{(t)-T0-gamma1}
\end{figure}
Fig.~\ref{(t)-T0-gamma1} plots the dynamics of the nearest-neighbor discord $D(i,i+1)$ at zero temperature for the transverse Ising model with $\gamma=1$. When the coupling parameter (and the magnetic field) is a step function, the discord reaches a value that is neither its value at $J=J_{0}$ ($h=h_{0}$) nor at $J_{1}$ ($h=h_{1}$). In other words, $D(i,i+1)$ shows a non-ergodic behavior, which is similar to that of entanglement \cite{Sadiek_2010}. At higher temperature quantum discord remains finite as $t \rightarrow \infty$, though the magnitude of the asymptotic discord decreases. Similarly, $D(i,i+1)$ also shows the non-ergodic behavior in the partially anisotropic $XY$ model with $\gamma=0.5$, though the equilibrium time is longer than that in the Ising model. On the contrary, $D(i,i+1)$ in the isotropic $XY$ model ($\gamma=0$) becomes a constant determined by the values of $J_{0}$ and $h_{0}$. The isotropy of the initial coupling parameters may make spins equally aligned into the $x$ and $y$ directions, apart from those in the $z$-direction, which results in finite quantum discord. Increasing the coupling parameters strength would not change the associated discord.
\begin{figure}
\caption{(Color online)$D(i,i+1)$ as a function of $\lambda_{1}$ and $t$ at $kT=0$ with $\gamma=1$, $h_{0}=h_{1}=1$ and (a) $J_{0}=1$; (b)$J_{0}=5$.}
\label{(t,lambda1)-T0-gamma1-J0[1]}
\label{(t,lambda1)-T0-gamma1-J0[5]}
\label{(t,lambda1)-T0-gamma1}
\end{figure}
We now focus on the dynamics of quantum discord as a function of $\lambda_1$. In Fig.~\ref{(t,lambda1)-T0-gamma1}, we plot $D(i,i+1)$ as a function of $t$ and $\lambda_1$ for the Ising model where $h_0=h_1=1$ and $T=0$. We first consider the case when the system is initially prepared in a state with $\lambda_0=\lambda_c$, i.e., $J_{0}=1$, in Fig.~\ref{(t,lambda1)-T0-gamma1-J0[1]}. When $\lambda_1=0$, the discord oscillates in time. The magnitude of the discord increases with $\lambda_1$ until it reaches its maximum at $\lambda_c$. Interestingly, when $\lambda_1$ exceeds $\lambda_c$, quantum discord behaves differently from that of entanglement shown in \cite{Sadiek_2010}. The discord decreases and becomes steady when $J$ dominates over $h$, while entanglement does not. On the other hand, when the system initially is in the parameter region $\lambda_0 > \lambda_c$, the maximum of quantum discord is much smaller as shown in Fig.~\ref{(t,lambda1)-T0-gamma1-J0[5]} where $J_{0}=5$. We see that there are two peaks, and the second is higher and decreases with increase in $\lambda_0$ . For the partially anisotropic cases, quantum discord behaves similar to that in the Ising case but with smaller magnitudes. Quantum discord of the completely isotropic $XY$ system is independent of $\lambda_1$.
\section{Asymptotic Behaviors of Quantum Discord}
\begin{figure}
\caption{$D(i,i+1)$ as functions of $\lambda$ for $h=h_{0}=h_{1}$ and $J=J_{0}=J_{1}$ at (a) $kT=0$ with different $J$ and $h$; (b) $kT=1$ with $h_{0}=h_{1}=0.25,1,4$; (c)$kT=1$ with $J_{0}=J_{1}=0.25,1,4$; (d) $kT=3$ with $h_{0}=h_{1}=0.25,1,4$ with $\gamma=1$.}
\label{(lambda)-gamma1}
\label{(lambda)-gamma1}
\end{figure}
We will examine the asymptotic behaviors of the quantum discord $D(i,i+1)$ as a function of $\lambda$ at $(t\rightarrow\infty)$ for different values of $J=J_0=J_1$ and $h=h_0=h_1$ and at different temperatures. First, for the Ising model ($\gamma=1$), Fig.~\ref{(lambda)-gamma1}(a) shows that the zero-temperature behavior of $D(i,i+1)$ depends only on the ratio $J/h$ ($=\lambda$) rather than their individual values. $D(i,i+1)$ starts at zero, reaches a maximum and then vanishes at larger values of $\lambda$. The maximum of asymptotic value of $D(i,i+1)$ decreases as the temperature increases as shown in Fig.~\ref{(lambda)-gamma1}(b) and(d). On the other hand, the finite-temperature discord $D(i,i+1)$ is not only determined by the ratio of $J$ and $h$ but also individual values of $J$ and $h$. Fig.~\ref{(lambda)-gamma1}(b) and \ref{(lambda)-gamma1}(c) indicate that an increase in $h$ and $J$ causes the maximal discord to increase when $kT=1$.
\begin{figure}
\caption{$D(i,i+1)$ as a function of $\lambda$ for $h=h_{0}=h_{1}$ and $J=J_{0}=J_{1}$ at (a) $kT=0$ with different combinations of $J$ and $h$; (b) $kT=1$ with $h_{0}=h_{1}=0.25,1,4$; (c) $kT=1$ with $J_{0}=J_{1}=0.25,1,4$; (d)$kT=3$ with $h_{0}=h_{1}=0.25,1,4$ with $\gamma=0.5$.}
\label{(lambda)-gamma0.5}
\label{(lambda)-gamma0.5}
\end{figure}
Next, we discuss the partially anisotropic $XY$ model with $\gamma=0.5$. Similarly to the Ising case, the zero-temperature discord $D(i,i+1)$ depends only on the ratio $J/h$ as in Fig.~\ref{(lambda)-gamma0.5}(a). $D(i,i+1)$ starts from zero, reaches a maximal value and then decays to a constant value for larger $\lambda$'s, where entanglement vanishes \cite{Sadiek_2010}. When $h \gg J$, the magnetic field dominates such that spins are aligned along the $z$ direction and the nearest-neighbor discord is zero. When $h \ll J$, the coupling will dominate. For the partially anisotropic model, the strong nearest-neighbor couplings $J$ make the spins aligned isotropically and consequently the discord maintains an equilibrium and finite value ( see also the results for entanglement \cite{Sadiek_2010}). The critical behavior of quantum discord around the critical point $\lambda=1$ changes considerably as the temperature and other parameters. The maximal discord is enhanced at high magnetic fields and the stronger coupling $J$. The temperature effect is shown in Fig.~\ref{(lambda)-gamma0.5}(d). The critical behavior of quantum discord disappears for some values of $h$ and $J$. It manifests that the thermal excitations suppress quantum effect and even ruin the critical behaviors of quantum discord.
\begin{figure}
\caption{$D(i,i+1)$ as a function of $\lambda$ for $h=h_{0}=h_{1}$ and $J=J_{0}=J_{1}$ at (a)$kT=0$ with different combination of $J$ and $h$; (b)$kT=1$ with $h_{0}=h_{1}=0.25,1,4$;(c)$kT=1$ with $J_{0}=J_{1}=0.25,1,4$; (d)$kT=3$ with $h_{0}=h_{1}=0.25,1,4$ with $\gamma=0$.}
\label{(lambda)-gamma0}
\label{(lambda)-gamma0}
\end{figure}
Likewise, $D(i,i+1)$ in the isotropic $XY$ model depends only on the ratio $J/h$ at $T=0$, as shown in Fig.~\ref{(lambda)-gamma0}(a). Differently, $D(i,i+1)$ starts from zero and saturates at $\lambda=8$. There seems to be a tiny peak at $\lambda=1$ before the saturation. Interestingly, raising the temperature delays the saturations of the discord but does not reduce their amplitudes, as shown in Fig.~\ref{(lambda)-gamma0}(b). The magnetic fields individually affect the saturations. The smaller the magnetic fields are, the later the saturations appear. In Fig.~\ref{(lambda)-gamma0}(c), the peaks at critical point become visible when increasing $J$ and noticeably the saturations of $D(i,i+1)$ happen with larger amplitudes. The higher temperatures delay the saturations of the quantum discord as in Fig.~\ref{(lambda)-gamma0}(d).
\begin{figure*}
\caption{(Color online)The asymptotic behavior of $D(i,i+1)$ as a function of (a) $J_{0}$ and $J_{1}$ with $h_{0}=h_{1}=1$,(b)$h_{0}$ and $h_{1}$ with $J_{0}=J_{1}=1$, (c) $h_{0}$ and $J_{0}$ with $h_{1}=J_{1}=1$ and (d) $h_{1}$ and $J_{1}$ with $h_{0}=J_{0}=1$ at $kT=0$ with $\gamma=1$ .}
\label{[J(),h()]-T0-gamma1}
\label{[J(),h()]-T0-gamma1}
\end{figure*}
\begin{figure*}
\caption{(Color online)The asymptotic behavior of $D(i,i+1)$ as a function of (a) $J_{0}$ and $J_{1}$ with $h_{0}=h_{1}=1$, (b)$h_{0}$ and $h_{1}$ with $J_{0}=J_{1}=1$, (c) $h_{0}$ and $J_{0}$ with $h_{1}=J_{1}=1$ and (d) $h_{1}$ and $J_{1}$ with $h_{0}=J_{0}=1$ at $kT=0$ with $\gamma=0.5$.}
\label{[J(),h()]-T0-gamma0.5}
\label{[J(),h()]-T0-gamma0.5}
\end{figure*}
\begin{figure*}
\caption{(Color online)The asymptotic behaviours of $D(i,i+1)$ as a function of (a) $J_{0}$ and $J_{1}$ with $h_{0}=h_{1}=1$, (b)$h_{0}$ and $h_{1}$ with $J_{0}=J_{1}=1$, (c) $h_{0}$ and $J_{0}$ with $h_{1}=J_{1}=1$ and (d) $h_{1}$ and $J_{1}$ with $h_{0}=J_{0}=1$ at $kT=0$.}
\label{[J(),h()]-T0-gamma0}
\label{[J(),h()]-T0-gamma0}
\end{figure*}
To further study the roles of the magnetic field and coupling parameters $h_0$, $h_1$, $J_0$ and $J_1$, we first calculate the zero-temparature asymptotic behaviors of quantum discord at $t \rightarrow \infty$ in wider parameter regimes. Fig.~\ref{[J(),h()]-T0-gamma1} has four discord contours for the Ising model at $t \rightarrow \infty$. We plot the discord contour function of $J_{0}$ and $J_{1}$ in Fig.~\ref{[J(),h()]-T0-gamma1}(a), where $h_0=h_1=1$. The discord starts with a zero value for $J_0=J_1=0$, reaches the maximum at $J_{0} \approx J_1 \approx 1$, and vanishes in the parameter regimes $J_0 \gtrsim 2$ and $J_1 \gtrsim 4$. It is interesting to note that for some values in the region $J_{0}<1$ and $J_{1}>1$, the asymptotic discord has finite values whereas entanglement vanishes \cite{Sadiek_2010} for all values in this region. Fig.~\ref{[J(),h()]-T0-gamma1}(b) is the discord contour of $h_{0}$ and $h_{1}$ with $J_{0}=J_{1}=1$. The discord starts with zero at $h_1=h_0=0$ and reaches the maximum at $h_{0} \approx h_{1} \approx 1$. We then plot the contour $D(i,i+1)$ as a function of $J_{0}$ and $h_{0}$ in Fig.~\ref{[J(),h()]-T0-gamma1}(c). The discord has its maximum when $J_{0}\approx h_{0}$ and disappears when $J_{0}$ deviates largely from $h_{0}$. We also show the asymptotic behavior of $D(i,i+1)$ as a function of $J_{1}$ and $h_{1}$
as shown in Fig.~\ref{[J(),h()]-T0-gamma1}(d). The largest discord is reached at $J_{1}=h_{1}$. The profile of the discord contour versus $J_0$ and $h_0$ is smooth, whereas the profile versus $J_{1}$ and $h_{1}$ is not. The reason is that $J_{0}$ and $h_{0}$ only affect the system at the initial time, whereas $J_{1}$ and $h_{1}$ affect the system all the time. A little change in the values of $J_{1}$ and $h_{1}$ has a great impact on the asymptotic behaviors of quantum discord. Fig.~\ref{[J(),h()]-T0-gamma0.5} is the same as Fig.~\ref{[J(),h()]-T0-gamma1} except $\gamma=0.5$. The asymptotic behavior of discord is similar to that of the Ising case as shown in Fig.~\ref{[J(),h()]-T0-gamma0.5}. The differences are that the quantum discord does not vanish in the range of $J_{0}>1$ and $J_{1}>1$, and the maxima of quantum discord in the partially anisotropic $XY$ model are smaller than those in the Ising model. There is no peak in the vicinity of $J_{1}=h_{1}$. Fig.~\ref{[J(),h()]-T0-gamma0} is for the isotropic $XY$ model, where $D(i,i+1)$ depends only on the $J_{0}$ and $h_0$ but not on $J_{1}$ and $h_1$.
The temperature effect on quantum discord is shown in Fig.~\ref{(kT,lambda())-gamma1}. In Fig.~\ref{(kT,lambda())-gamma1}(a), the quantum discord reaches the maximum at $\lambda=1$. As the temperature increases or $\lambda$ diverges from the critical value, the quantum discord decays more slowly than entanglement \cite{Sadiek_2010}. The asymptotic behaviors of $D(i,i+1)$ as a function of $\lambda_{1}$, $\lambda_{0}$ and $kT$, are depicted in Fig.~\ref{(kT,lambda())-gamma1}(b) and \ref{(kT,lambda())-gamma1}(c). These figures demonstrate that the quantum discord is more robust than entanglement against the effect of temperature.
\begin{figure*}
\caption{(Color online)The asymptotic behavior of $D(i,i+1)$ as a function of $kT$ and (a)$\lambda$ with $J_{0}=J_{1}$, (b) $\lambda_{1}$ with $J_{0}=1$, (c) $\lambda_{0}$, and $J_{1}=1$. $h_{0}=h_{1}=1$. $\gamma=1$.}
\label{(kT,lambda())-gamma1}
\label{(kT,lambda())-gamma1}
\end{figure*}
Finally, we explore the asymptotic behaviors of quantum discord in the $\lambda_{1}$-$\gamma$ phase space for the different values of $J_0$. In Figs.~\ref{(gamma,lambda())-T0}(a), \ref{(gamma,lambda())-T0}(b) and \ref{(gamma,lambda())-T0}(c), we plot $D(i,i+1)$ as a contour function of $\lambda_1$ and $\gamma$ for given magnetic fields $h_{0}=h_{1}=1$ at $kT=0$. When $J_0 <1$, there is no bigger enhancement in $D(i,i+1)$. When $J_0$ is in the vicinity of $1$, there are two peaks. For the larger $J_0$, two peaks merge. When $J_0 >1$ and $\lambda_1<1$, $D(i,i+1)$ is enhanced significantly in the $\gamma=0$ region, whereas for $\lambda_1>1$, $D(i,i+1)$ is not enhanced.
\begin{figure*}
\caption{(Color online) The asymptotic behavior of $D(i,i+1)$ as a function of (a)$\lambda_{1}$ and $\gamma$ with $J_{0}=0.5$, (b)$\lambda_{1}$ and $\gamma$ with $J_{0}= 1$, (c)$\lambda_{1}$ and $\gamma$ with $J_{0}=5$, and (d)$\lambda$ and $\gamma$ at $kT=0$. $h_{0}=h_{1}=1$.}
\label{(gamma,lambda())-T0}
\label{(gamma,lambda())-T0}
\end{figure*}
\section{Conclusions} We have investigated the dynamics and the asymptotic behaviors of quantum discord in one dimensional $XY$ model coupled through a time-dependent nearest-neighbor coupling and in the presence of a time-dependent magnetic field. The system shows non-ergodic and critical behaviors. The zero-temperature asymptotic behaviors of the system at the long time limit depends only on the ratio between the coupling and the magnetic field but not their individual values. On the contrary, these behaviors do rely on individual values of the coupling and the magnetic field at the finite temperature. The asymptotic behaviors are sensitive not only to the initial values of the coupling and the magnetic field, but also to the final values of the coupling and the magnetic field. We have demonstrated that quantum discord is more robust than entanglement against the effect of temperature. This robustness will be useful in design of fault-tolerant quantum algorithms and in other quantum information processing.
\section*{Acknowledgment} We acknowledge the financial support by the National Natural Science Foundation of China under Grant No. 11075013 and No. 10974016. L.-A. Wu has been supported by the Ikerbasque Foundation Startup, the Basque Government (Grant IT472-10), and the Spanish MEC(Project No. FIS2009-12773-C02-02).
\end{CJK*}
\end{document} | arXiv |
Karl Bobek
Karl Joseph Bobek (1855–1899) was a German mathematician working on elliptic functions and geometry.
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Compute $139+27+23+11$.
Since addition is associative, we can rearrange the terms:
$139+27+23+11=(139+11)+(27+23)=150+50=\boxed{200}$. | Math Dataset |
Anniversaries (Ukrainian)
Yurij Makarovich Berezansky (the 80th anniversary of his birth)
Gorbachuk M. L., Gorbachuk V. I., Kondratiev Yu. G., Kostyuchenko A. G., Marchenko V. O., Mitropolskiy Yu. A., Nizhnik L. P., Rofe-Beketov F. S., Samoilenko A. M., Samoilenko Yu. S.
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Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 3-11
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Strongly Nonlinear Differential Equations with Carlitz Derivatives over a Function Field
Kochubei A. N.
In earlier papers the author studied some classes of equations with Carlitz derivatives for $\mathbb{F}_q$ -linear functions, which are the natural function field counterparts of linear ordinary differential equations. Here we consider equations containing self-compositions $u \circ u ... \circ u$ of the unknown function. As an algebraic background, imbeddings of the composition ring of $\mathbb{F}_q$ -linear holomorphic functions into skew fields are considered.
On an Application of the Lax-Phillips Scattering Approach in the Theory of Singular Perturbations
Kuzhel' S. A., Matsyuk L. V.
For a singular perturbation $A = A_0 + \sum^n_{i, j=1}t_{ij} \langle \psi_j, \cdot \rangle \psi_i,\quad n \leq \infty$ of a positive self-adjoint operator $A_0$ with Lebesgue spectrum, the spectral analysis of the corresponding self-adjoint operator realizations $A_T$ is carried out and the scattering matrix $\mathfrak{S}_{(A_T, A_0)}(\delta)$ is calculated in terms of parameters $t_{ij}$ under some additional restrictions on singular elements $\psi_{j}$ that provides the possibility of application of the Lax -Phillips approach in the scattering theory.
Elliptic Operators in a Refined Scale of Functional Spaces
Mikhailets V. A., Murach A. A.
We study the theory of elliptic boundary-value problems in the refined two-sided scale of the Hormander spaces $H^{s, \varphi}$, where $s \in R,\quad \varphi$ is a functional parameter slowly varying on $+\infty$. In the case of the Sobolev spaces $H^{s}$, the function $\varphi(|\xi|) \equiv 1$. We establish that the considered operators possess the properties of the Fredholm operators, and the solutions are globally and locally regular.
On the Group $C^{*}$-Algebras of a Semidirect Product of Commutative and Finite Groups
Samoilenko Yu. S., Yushchenko K. Yu.
By using representations of general position and their properties, we give the description of group $C^{*}$-algebras for semidirect products $\mathbb{Z}^d \times G_f$, where $G_f$ is a finite group, in terms of algebras of continuous matrix-functions defined on some compact set with boundary conditions. We present examples of the $C^{*}$-algebras of affine Coxeter groups.
Multifractal Analysis of Singularly Continuous Probability Measures
Torbin H. M.
We analyze correlations between different approaches to the definition of the Hausdorff dimension of singular probability measures on the basis of fractal analysis of essential supports of these measures. We introduce characteristic multifractal measures of the first and higher orders. Using these measures, we carry out the multifractal analysis of singular probability measures and prove theorems on the structural representation of these measures. | CommonCrawl |
Asymptotic behavior of traveling waves for a three-component system with nonlocal dispersal and its application
Perturbed fractional eigenvalue problems
December 2017, 37(12): 6257-6289. doi: 10.3934/dcds.2017271
Two dimensional Riemann problems for the nonlinear wave system: Rarefaction wave interactions
Eun Heui Kim 1,, and Charis Tsikkou 2,
Department of Mathematics and Statistics, California State University, Long Beach, Long Beach, CA 90840, USA
Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA
* Corresponding author: Eun Heui Kim.
Received January 2017 Revised June 2017 Published August 2017
Fund Project: The work of Kim was supported by the National Science Foundation under the Grants DMS- 1109202, 1615266. The work of Tsikkou was supported by the National Science Foundation under the Grant DMS-1400168.
We analyze rarefaction wave interactions of self-similar transonic irrotational flow in gas dynamics for two dimensional Riemann problems. We establish the existence result of the supersonic solution to the prototype nonlinear wave system for the sectorial Riemann data, and study the formation of the sonic boundary and the transonic shock. The transition from the sonic boundary to the shock boundary inherits at least two types of degeneracies (1) the system is sonic, and in addition (2) the angular derivative of the solution becomes zero where the sonic and shock boundaries meet.
Keywords: Rarefaction wave, transonic shock, Riemann problem, multidimensional conservation laws, nonlinear wave system.
Mathematics Subject Classification: Primary: 76L05, 35L65; Secondary: 65M06, 35M33.
Citation: Eun Heui Kim, Charis Tsikkou. Two dimensional Riemann problems for the nonlinear wave system: Rarefaction wave interactions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6257-6289. doi: 10.3934/dcds.2017271
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Figure 1. Riemann data and configuration.
Figure 2. Configuration with details.
Figure 3. Regions and characteristics in the supersonic region.
Figure 4. $\mathcal{R}_1\setminus \mathcal{R}_1[\delta]$, the region $\mathcal{R}_1$ excluding the small neighborhoods of $\Xi_1$ and $\Xi_3.$
Figure 5. Schematics of constructing the solution near $\Xi_3.$
Figure 6. Envelope formation by the simple wave.
Figure 7. Density plots: the contour plot of $\rho$.
Figure 8. Density plots: Left figure is the cross section in the radial direction for a fixed angle $\theta$ ranging from $0$ to $\pi/2$ and incrementing by $10$ degrees. Right figure is the enlargement of the left figure near which the shock changes to sonic, which appears to be in-between the angle $50$ and $60$ degrees in this configuration.
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Eun Heui Kim Charis Tsikkou | CommonCrawl |
Charles Marie de La Condamine
Charles Marie de La Condamine (28 January 1701 – 4 February 1774) was a French explorer, geographer, and mathematician. He spent ten years in territory which is now Ecuador, measuring the length of a degree of latitude at the equator and preparing the first map of the Amazon region based on astro-geodetic observations. Furthermore he was a contributor to the Encyclopédie.[1]
Charles Marie de La Condamine
La Condamine by Charles-Nicolas Cochin (1768) Pierre-Philippe Choffard (1759)
Born28 January 1701
Paris, France
Died4 February 1774 (aged 73)
Paris, France
NationalityFrench
Biography
Charles Marie de La Condamine was born in Paris as a son of well-to-do parents, Charles de La Condamine and Louise Marguerite Chourses. He studied at the Collège Louis-le-Grand where he was trained in humanities as well as in mathematics. After finishing his studies, he enlisted in the army and fought in the war against Spain (1719). After returning from the war, he became acquainted with scientific circles in Paris. On 12 December 1730 he became a member of the Académie des Sciences and was appointed Assistant Chemist at the Academy.
In 1729 La Condamine and his friend Voltaire exploited a loophole in the French government’s lottery, which brought them large profits.[2][3]
In May 1731 La Condamine sailed with the Levant Company to Constantinople (now Istanbul), where he stayed five months. After returning to Paris, La Condamine submitted in November 1732 a paper to the Academy entitled Mathematical and Physical Observations made during a Visit of the Levant in 1731 and 1732.
In South America
Three years later he joined the French Geodesic Mission to territory which is now Ecuador which had the aim of testing a hypothesis of Isaac Newton. Newton had posited that the Earth is not a perfect sphere, but bulges around the equator and is flattened at the poles. Newton's opinion had raised a huge controversy among French scientists. Pierre Louis Maupertuis, Alexis Claude Clairaut, and Pierre Charles Le Monnier traveled to Lapland, where they were to measure the length of several degrees of latitude orthogonal to the arctic circle, while Louis Godin, Pierre Bouguer, and La Condamine were sent to South America to perform similar measurements around the equator.
On 16 May 1735, La Condamine sailed from La Rochelle accompanied by Godin, Bouguer and a botanist, Joseph de Jussieu. After stopovers in Martinique, Saint-Domingue, and Cartagena, they sailed southward through Panama, arriving at the Pacific port of Manta on 10 March 1736.
La Condamine's associations with his colleagues were unhappy. The expedition was beset by many difficulties, and finally La Condamine split from the rest and made his way to Quito, Ecuador separately,[4] following the Esmeraldas River, becoming the first European to encounter rubber in the process. La Condamine is credited with introducing samples of rubber to the Académie Royale des Sciences of France in 1736.[5] In 1751, he presented a paper by François Fresneau to the Académie (eventually published in 1755) which described many of the properties of rubber. This has been referred to as the first scientific paper on rubber. He joined the group again on 4 June 1736 in the city of Quito.
The meridian arc whose length La Condamine and his colleagues chose to measure passed through a high valley perpendicular to the equator, stretching from Quito (now the capital of Ecuador) in the north to Cuenca in the south. The scientists spent a month performing triangulation measurements in the Yaruqui plains—from 3 October to 3 November 1736—and then returned to Quito.
After they had come back to Quito, they found that subsidies expected from Paris had not arrived. La Condamine, who had taken precautions and had made a deposit on a bank in Lima, traveled in early 1737 to Lima to collect money. He prolonged this journey somewhat to study the cinchona tree with its medicinally active bark (containing the anti-malarial drug quinine), the tree being hardly known in Europe. Because cinchona only grows at high altitudes the plants did not survive. He described how the plants were brought to France:[6]
On June 3rd I spent the whole day on one of these mountains [near Loja in present-day Ecuador]. Though assisted by two Americans of the region whom I took with me as guides, I was able to collect no more than eight or nine young plants of Quinquina [cinchona] in a proper state for transportation. These I had placed in earth taken from the spot in a case of suitable size and had them carried on the shoulders of a man whom I kept constantly in my sight, and then by canoe. I hoped to leave some of the plants at Cayenne [in Guiana] for cultivation and to transport the others to the King's garden in France.
He collected valuable seeds, sarsparilla, guaiacum, ipecacuanha, cacao, vanilla and simarouba.
After returning to Quito on 20 June 1737, he found that Godin refused to disclose his results, whereupon La Condamine joined forces with Bouguer. The two men continued with their length measurements in the mountainous and inaccessible region close to Quito. When in December 1741 Bouguer detected an error in a calculation of La Condamine's, the two explorers got into a quarrel and stopped speaking to each other. However, working separately, the two completed their project in May 1743.
Return to Europe
Insufficient funds prevented La Condamine from returning to France directly. Thus La Condamine chose to return by way of the Amazon River, a route which is longer and more dangerous. His was the first scientific exploration of the Amazon. He reached the Atlantic Ocean on 19 September 1743, having made observations of astronomic and topographic interest on the way. He also made some botanical studies, notably of cinchona and rubber trees.
In February 1744 he arrived in Cayenne, the capital of French Guiana. He did not dare to travel back to France on a French merchant ship because France was at war (the Austrian Succession War of 1740–1748), and he had to wait for five months for a Dutch ship, but made good use of his waiting time by observing and recording physical, biological and ethnological phenomena. Finally leaving Cayenne in August 1744, he arrived in Amsterdam on 30 November 1744, where he stayed for a while, and arrived in Paris in February 1745. He brought with him many notes, natural history specimens, and art objects that he donated to the naturalist Buffon (1707–1788).
La Condamine published the results of his measurements and travels with a map of the Amazon in Mém. de l'Académie des Sciences, 1745 (English translation 1745–1747).[4] This included the first descriptions by a European of the Casiquiare canal and the curare arrow poison prepared by the Amerindians. He also noted the correct use of quinine to fight malaria. The journal of his ten-year-long voyage to South America was published in Paris in 1751. The scientific results of the expedition were unambiguous: the Earth is indeed a spheroid flattened at the poles as was believed by Newton.
La Condamine and Bouguer failed to write a joint publication, and Bouguer's death in 1758 put an end to their relationship. The other expedition member, Godin, died in 1760. Being the only surviving member, he received most of the credit for the expedition which drew a great deal of attention in France, as he was a gifted writer and popularizer.
On a visit to Rome La Condamine made careful measurements of the ancient buildings with a view to a precise determination of the length of the Roman foot. He also wrote in favour of inoculation, and on various other subjects, mainly connected with his work in South America.[4]
La Condamine had contracted smallpox in his youth. This led him to take part in the debate on variolation against the disease and to propagate inoculation against smallpox. Assisted by the clarity and elegance of his writing, he presented several papers at the Academy of Sciences in which he defended his ideas with passion. He became a corresponding member of the academies of London, Berlin, Saint Petersburg and Bologna, and was elected to the l'Académie française on 29 November 1760. In August 1756, he married, with papal dispensation, his young niece Charlotte Bouzier of Estouilly. La Condamine had many friends, the closest one being Maupertuis, to whom he bequeathed his papers. La Condamine died in Paris on 4 February 1774, following a hernia operation.
Works
South America
• Journal du voyage fait par ordre du roi à l'équateur (Paris 1751, Supplement 1752)
• Relation abrégée d'un voyage fait dans l'intérieur del'Amérique méridionale (Paris 1759)
• "Mémoire sur quelques anciens monumens du Perou [sic], du tems des Incas", in: Histoire de l'Académie Royale des Sciences et Belles Lettres II (1746), Berlin 1748, S. 435–456 (PDF).
Others
• La figure de la terre déterminée (Paris 1749)
• Mesure des trois premiers degrés du méridien dans l'hémisphère australe (Paris 1751)
• Histoire de l'inoculation de la petite vérole (Amsterdam 1773)
References
This article incorporates material from the Citizendium article "Charles Marie de La Condamine", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL.
1. Frank A. Kafker: Notices sur les auteurs des dix-sept volumes de « discours » de l'Encyclopédie. Recherches sur Diderot et sur l'Encyclopédie Année (1989) Volume 7 Numéro 7 p. 145
2. Davidson, Ian, Voltaire. A Life, London, Profile Books, 2010. ISBN 978-1-60598-287-8.
3. Safier, Neil (2008). Measuring the new world : enlightenment science and South America. Chicago: University of Chicago Press. ISBN 978-0-226-73356-2. OCLC 309877290.
4. Chisholm 1911.
5. "Charles Marie de La Condamine".
6. de la Condamine (1738) "Sur l'arbre du quinquina" (On the quinquina tree) Histoire de l'Académie royale des Sciences, pages 226-243.
7. International Plant Names Index. Cond.
• Ferreiro, Larrie (2011). Measure of the Earth: The Enlightenment Expedition that Reshaped Our World. New York: Basic Books. p. 376. ISBN 978-0-465-01723-2. Archived from the original on 2014-02-22.
• This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "La Condamine, Charles Marie de". Encyclopædia Britannica. Vol. 16 (11th ed.). Cambridge University Press. p. 51.
• Victor Wolfgang von Hagen: South America called them; explorations of the great naturalists: La Condamine, Humboldt, Darwin, Spruce. New York: Knopf, 1945
• Robert Whitaker: The Mapmaker's Wife. London: Doubleday, 2004. (The full story of the expedition to South America, drawn from the original documents.)
• Neil Safier, Measuring the New World: Enlightenment Science and South America, Univ. of Chicago Press, 2008. ISBN 0-226-73355-6
External links
Wikimedia Commons has media related to Charles Marie de La Condamine.
• French Academy of Sciences biography
• O'Connor, John J.; Robertson, Edmund F., "Charles Marie de La Condamine", MacTutor History of Mathematics Archive, University of St Andrews
• Works by Charles-Marie de La Condamine at Project Gutenberg
• Works by or about Charles Marie de La Condamine at Internet Archive
• Voyages of Discovery : The Figure of Earth (BBC documentary) / https://www.youtube.com/watch?v=LKulxEp2aqk
Académie française seat 23
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| Wikipedia |
Does venturi effect helps in reducing temperature of a room?
I clearly didn't understand how this works, but the explanation and claim that it reduce the temperature of room by 5°c made me wonder, here I am trying to understand the Venturi-effect on temperature
This is something I found on web, Using old soda bottle and board they made some e-cooler
Note:- This system doesn't work, an enthusiastic professor did an experiment to prove it doesn't work. but is it possible reduce the temperature of room or a house using Venturi-effect ?
heat-transfer hvac
LearnerLakshLearnerLaksh
I don't think it will drop the temperature at all.
What it might accomplish is increase the velocity of the air in the room. That will in turn affect the convective coefficient
The convective heat transfer coefficient for air flow can be approximated to (see engineering toolbox link)
$$h_c = 10.45 - v + 10 v^{1/2}$$
$h_c$ = heat transfer coefficient ($\frac{kCal}{m^2h°C}$)
$v$ = relative speed between object surface and air (m/s)
Because the coefficient will increase the exchange with the bodies in the room will change also.
The Total rate of exchanged heat is:
$$\dot{Q} = h_c A \Delta T$$
A is the total exchange surface
$\Delta T$ the temperature difference
The $\dot{Q} $ is important because human beings do not understand Temperature directly/ Rather we understand temperature by the exchange heat energy.
So, increasing the velocity of the air in the room, will probably increase the convective transfer coefficient, and (might) make you feel a bit cooler. However, it should not drop the temperature.
NMechNMech
It's correct to say that the temperature of the air in the jet coming from the neck of the bottle will be less than the temperature of the air in the bottle. The temperature of the air in the bottle will also be a little higher than the temperature of the outside air. The question is, will both the temperature and relative humidity of the jet air be sufficiently low to result in a net "comfort zone" in the room.
The "comfort zone" is a region of ambient air temperature and relative humidity that induces a comfortable feeling, and it has been mapped out by ASHRAE (American Society of Heating Refrigeration and Air Conditioning Engineers). Such data can be found for instance at https://www.dartmouth.edu/~cushman/courses/engs44/comfort.pdf
I assume this device is meant only for Summer months, when outdoor humidity and temperatures are very uncomfortable. In order to work then, the device must deliver air to the room that will result in the temperature and humidity space according to the ASHRAE standard.
It's clear that the temperature of the jet air exiting the bottles will be a little lower than the temperature of the air in the bottle. That's simply because the expansion process through the nozzle is roughly isentropic. It's very likely that, for normal outside atmospheric conditions, the air pressure in the bottle will exceed room pressure only slightly. That's because the pressure in the bottle can be greater than the air pressure in the room only when outside wind collides with the bottles. Such a collision converts the kinetic energy of the wind velocity to a stagnation pressure. For normal wind speeds, however, this increase in pressure is small.
The velocity of the air jet into the room will thus be small, and the net temperature difference between outside air and air jet velocity will be small. Considering also that when the air jet mixes with room air, it's kinetic energy is dissipated, being converted to heat. The net effect is that the final temperature of the jet air after it mixes with room air will be slightly elevated above its temperature when it exits the bottle. The conclusion here is that there can be only a minimal sensible cooling effect from this device. Most importantly, sensible cooling is only part of the air conditioning process, and the real question is what's the story with relative humidity.
Often the most important process of air conditioning is the removal of water vapor from the air. The ASHRAE graphs show that the relative humidity necessary for comfort is often much less than the relative humidity present in Summer outside air. What happens to the water vapor (absolute humidity) of the air that's pushed through these bottles? Unfortunately nothing much.
In the vast majority of air conditioning processes, water vapor must be removed from the outside air before it can become comfortable for humans. Water removal is usually the most intensive process in air condition, not the reduction in (sensible) temperature. This device offers little mechanism for the removal of water vapor. The very slight increase in bottle pressure and the relatively low pressure drop encountered by the air in moving through the bottles ensures that the water vapor in the outside air remains in the air pushed into the room. If the jet air temperature would be lowered to a value below the dew point of the outside air, condensation would remove some of this water vapor. It's very unlikely that such a device could accomplish such low temperatures, for the reasons already explained. The net result is that this device pushes more water vapor into the room - not a very good means for conditioning outside air for comfort.
Bottom line is that for much of the typical Summer months in areas that require air conditioning, this device will not be able to produce a comfort zone, as defined by ASHRAE standards.
ttononttonon
$\begingroup$ There is no pump. It is passive wind powered. See Step 5 of the instructions. $\endgroup$ – Transistor Sep 25 '20 at 18:35
$\begingroup$ Transistor, thanks. I missed that point. I'll modify my answer accordingly. $\endgroup$ – ttonon Sep 26 '20 at 21:36
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Determining the final surface temperature and final gas temperature flow through a tube in a room | CommonCrawl |
Number: The Language of Science
Number: The Language of Science: A Critical Survey Written for the Cultured Non-Mathematician is a popular mathematics book written by Russian-American mathematician Tobias Dantzig. The original U.S. publication was by Macmillan in 1930.[1] A second edition (third impression) was published in 1947 in Prague, Czechoslovakia by Melantrich Company. It recounts the history of mathematical ideas, and how they have evolved.[2]
Chapters
The book is divided into 12 chapters. There is an appendix of illustrations. The third edition of the book contains a separate section for essays, at the book's end.
1. Fingerprints
2. The Empty Column
3. Number Lore
4. The Last Number
5. Symbols
6. The Unutterable
7. This Flowing World
8. The Act of Becoming
9. Filling the Gaps
10. The Domain of Number
11. The Anatomy of the Infinite
12. The Two Realities
References
1. Booklist Books, a Selection (1931) listed in last section of (1922-1933) collection of the American Library Association, p.10
2. Dantzig, Tobias (1932-11-26). 'Number: The Language of Science (A Critical Survey Written for the Cultured Non-Mathematician. George Allen & Unwin Ltd. p. 320.
• Miller, G. A. (1931). "Review: Tobias Dantzig, Number: The Language of Science. A critical survey written for the cultured non-mathematician". Bulletin of the American Mathematical Society. 37 (1). doi:10.1090/s0002-9904-1931-05073-4.
• Haldane, J. B. S. (1941). "Number: the Language of Science". Nature. 147 (3714): 9–9. doi:10.1038/147009a0. ISSN 0028-0836.
| Wikipedia |
\begin{document}
\title[Factorization of bivariate sparse polynomials]{Factorization of bivariate sparse polynomials}
\author[Amoroso]{Francesco Amoroso} \address{Laboratoire de
math\'ematiques Nicolas Oresme, CNRS UMR 6139, Universit\'e de
Caen. BP 5186, 14032 Caen Cedex, France} \email{[email protected]} \urladdr{\url{http://www.math.unicaen.fr/~amoroso/}}
\author[Sombra]{Mart{\'\i}n~Sombra} \address{Instituci\'o Catalana de
Recerca i Estudis Avançats (ICREA). Passeig Llu\'is Companys~23,
08010 Barcelona, Spain \vspace*{-2.5mm}} \address{Departament de
Matem\`atiques i Inform\`atica, Universitat de Barcelona (UB). Gran
Via 585, 08007 Bar\-ce\-lo\-na, Spain} \email{[email protected]} \urladdr{\url{http://www.maia.ub.edu/~sombra}}
\date{12/9/2018} \subjclass[2010]{Primary 13P05; Secondary 12Y05.}
\thanks{Amoroso was partially supported by the CNRS research project
PICS 6381 ``Diophantine geometry and computer algebra''. Sombra was
partially supported by the MINECO research project MTM2015-65361-P}
\begin{abstract}
We prove a function field analogue of a conjecture of Schinzel on
the factorization of univariate polynomials over the rationals. We
derive from it a finiteness theorem for the irreducible
factorizations of the bivariate Laurent polynomials in families with
fixed set of complex coefficients and varying exponents. Roughly speaking,
this result shows that the truly bivariate irreducible factors of
these sparse Laurent polynomials, are also sparse.
{\red The proofs are based on a variant of the toric Bertini's
theorem due to Zannier and Fuchs, Mantova and Zannier.} \end{abstract}
\maketitle
\section[Introduction]{Introduction} \label{sec:introduction}
A polynomial is said to be \emph{sparse} (or \emph{lacunary}) if it has few terms compared with its degree. The factorization problem for sparse polynomials can be vaguely stated as the question of whether the irreducible factors of a sparse polynomial are also sparse, apart from obvious exceptions. Aspects of this problem have been studied in various settings and for different formalizations of the notion of sparsenness, see for instance \cite{Lenstra:flp, Schinzel:psrr,
KaltofenKoiran:fsdfmslpanf,
AvendanoKrickSombra:fbslp,FilasetaGranvilleSchinzel:igcdasp,
Grenet:bdflmp, Am-So-Za}. Several of these studies were based on tools from Diophantine geometry like lower bounds for the height of points and subvarieties, and unlikely intersections of subvarieties and subgroups of a torus.
In this text, we consider families of bivariate Laurent polynomials given as the pullback of a \emph{fixed} regular function on a torus by a \emph{varying} 2-parameter monomial map. Precisely, let $t,z$ be variables, ${\boldsymbol{x}}=(x_{1},\dots,x_{n})$ a set of other $n$ variables, and $$F\in \C[{\boldsymbol{x}}^{\pm1},z^{\pm1}]=\C[x_{1}^{\pm1},\dots, x_{n}^{\pm1},z^{\pm1}]$$ a Laurent polynomial. For each vector ${\boldsymbol{a}}=(a_{1},\dots,a_{n})\in {\mathbb{Z}}^{n}$, we consider the bivariate Laurent polynomial given as the pullback of $F$ by the monomial map \begin{math}
(t,z)\mapsto (t^{{\boldsymbol{a}}},z)= (t^{a_{1}}, \dots, t^{a_{n}},z) \end{math}, that is \begin{equation} \label{eq:11} F_{{\boldsymbol{a}}}= F(t^{{\boldsymbol{a}}},z)\in \C[t^{\pm1},z^{\pm1}]. \end{equation} The number of coefficients of each $F_{{\boldsymbol{a}}}$ is bounded by those of $F$, and so these Laurent polynomials can be considered as sparse when $F$ is fixed and ${\boldsymbol{a}}$ is large.
A first question concerns the irreducibility of $F$. It has been addressed in~\cite{Zannier:hiaag}, as we next describe. Let us assume $F$ irreducible. Under which assumptions $F_{{\boldsymbol{a}}}$ stays irreducible for a {\it generic} ${\boldsymbol{a}}$? Let us consider the following example. The Laurent polynomial $F=z^{2}-x_{1}x_{2}^{2}\in \C[x_{1}^{\pm1}, x_{2}^{\pm1},z^{\pm1}]$. is irreducible. However given $(a_{1},a_{2})\in {\mathbb{Z}}^{2}\setminus \{(0,0)\}$ with $a_{1}$ even, the Laurent polynomial $F_{{\boldsymbol{a}}}$ is reducible. This show that the sole assumption that $F$ is irreducible is not enough to get an irreducibility statement for a generic specialization.
Zannier's result can be stated as follows. For ${\boldsymbol{a}},{\boldsymbol{b}}\in {\mathbb{Z}}^{n}$, we denote by $\langle {\boldsymbol{a}},{\boldsymbol{b}}\rangle=\sum_{i=1}^{n}a_{i}b_{i}$ their scalar product.
\begin{theorem}[{{\cite[Theorem 3]{Zannier:hiaag}}}]
Let $F \in \C[{\boldsymbol{x}}^{\pm 1},z^{\pm 1}]\setminus \C[{\boldsymbol{x}}^{\pm1}]$ be
an irreducible Laurent polynomial, that is monic in $z$ and such
that $F(x_{1}^d,\dots,x_{n}^d,z)$ is irreducible for $d=\deg_z(F)$.
There is a finite subset
$\Sigma \subset {\mathbb{Z}}^{n}\setminus \{{\boldsymbol{0}}\}$ such that, for
each ${\boldsymbol{a}}\in {\mathbb{Z}}^{n}$, either there is ${\boldsymbol{c}}\in\Sigma$ with
$\langle {\boldsymbol{c}},{\boldsymbol{a}}\rangle =0$, or $F(t^{\boldsymbol{a}},z)$ is irreducible. \end{theorem}
{\red As already remarked by the author, the classical Bertini theorem may be seen as a version of statement of this shape for ${\Bbb G}^n_{\rm a}$.}
Previously, Schinzel \cite{Sch1} proved a similar result in the same direction, for Laurent polynomials over ${\mathbb{Q}}$ satisfying the strong additional assumption that $F$ is not self-inversive. More recently, Fuchs, Mantova and Zannier \cite[Addendum to Theorem~1.5]{FuchsMantovaZannier:fiptvb} showed that the set $\Sigma$ can be chosen independently of the coefficients of $F$.
In the present paper we are interested in the factorization of $F_{{\boldsymbol{a}}}$. Our motivation is an old conjecture of Schinzel \cite{Schinzel:rppt} on the factorisation of sparse polynomial with rational coefficients (Conjecture~\ref{Sch0}).
This conjecture implies the statement below. A Laurent polynomial in ${\mathbb{Q}}[{\boldsymbol{x}}^{\pm 1}]$ is \emph{cyclotomic} if it can be written as a unit times the composition of a univariate cyclotomic polynomial with a monomial.
\begin{conjecture}
Let $F\in {\mathbb{Q}}[{\boldsymbol{x}}^{\pm1}]$. There is a finite set of matrices
$\Omega\subset {\mathbb{Z}}^{n\times n}$ satisfying the following
property. For each ${\boldsymbol{a}}\in{\mathbb{Z}}^n$, there are $M\in\Omega$ and
${\boldsymbol{b}}\in{\mathbb{Z}}^{n}$ with ${\boldsymbol{a}}=M{\boldsymbol{b}}$ such that if $P$ is an irreducible
factor of $F({\boldsymbol{x}}^M)$, then $P(t^{{\boldsymbol{b}}})$ is either a product of
cyclotomic Laurent polynomials, or
an irreducible factor of $F(t^{{\boldsymbol{a}}})$. \end{conjecture}
Our main result in this text is the following function field analogue.
\begin{theorem} \label{thm:2} Let $F\in \C[{\boldsymbol{x}}^{\pm1},z^{\pm1}]$. There is a finite set of matrices $\Omega\subset {\mathbb{Z}}^{n\times n}$ satisfying the following property. For
each ${\boldsymbol{a}}\in{\mathbb{Z}}^n$, there are $M\in\Omega$ and ${\boldsymbol{b}}\in{\mathbb{Z}}^{n}$ with ${\boldsymbol{a}}=M{\boldsymbol{b}}$ such that if $P $ is an irreducible factor of $F({\boldsymbol{x}}^M,z)$, then $P(t^{{\boldsymbol{b}}},z)$ is, as an element of $\C(t)[z^{\pm1}]$, either a unit or an irreducible factor of $F(t^{{\boldsymbol{a}}},z)$. \end{theorem}
Moreover, we also obtain in Theorem \ref{thm:1} the function field analogue of Conjecture~\ref{Sch0}.
Theorem \ref{thm:2} shows that for each ${\boldsymbol{a}}\in {\mathbb{Z}}^{n}$, there is a matrix $M$ within the finite set $\Omega \subset {\mathbb{Z}}^{n\times n}$ and a vector ${\boldsymbol{b}}\in {\mathbb{Z}}^{n}$ with ${\boldsymbol{a}}=M{\boldsymbol{b}}$ such that, unless $F\big({\boldsymbol{x}}^M,z\big)=0$, the irreducible factorization \begin{equation}
\label{eq:2} F\big({\boldsymbol{x}}^M,z\big)=\prod_{P}P({\boldsymbol{x}},z)^{e_{P}} \end{equation} yields the irreducible factorization in the ring $\C(t)[z^{\pm1}]$ \begin{displaymath} F_{{\boldsymbol{a}}}= \gamma\, \prod_{P} {\vphantom\prod}'P(t^{{\boldsymbol{b}}},z)^{e_{P}}, \end{displaymath} for $F_{{\boldsymbol{a}}}$ as in \eqref{eq:11}, the product being over the irreducible factors $P$ in \eqref{eq:2} such that $P(t^{{\boldsymbol{b}}},z)$ is not a unit, and with $\gamma\in\C(t)[z^{\pm1}]^{\times }$.
Hence, the irreducible factorizations in $\C(t)[z^{\pm1}]$ of the $F_{{\boldsymbol{a}}}$'s can be obtained by specializing the irreducible factorizations of the Laurent polynomials $F({\boldsymbol{x}}^M,z)$ for a \emph{finite} number of matrices~$M$. These irreducible factors of the $F_{{\boldsymbol{a}}}$'s are {sparse}, in the sense that they are all represented as the pullback of a finite number of regular functions on the $(n+1)$-dimensional torus ${\mathbb G}_{\rm m}^{n+1}$ by 2-parameter monomial maps. In particular, both the number of these irreducible factors and of their coefficients are bounded above independently of ${\boldsymbol{a}}$.
The proof of Theorem \ref{thm:2} relies on a variant of the aforementioned result of Zannier. To state it, we first introduce some further notation. Let ${\boldsymbol{t}} =(t_1,\ldots,t_{k})$ be a set of $k$ variables. A matrix $A=(a_{i,j})_{i,j}\in{\mathbb{Z}}^{n\times k}$ defines the family of $n$ monomials in the variables ${\boldsymbol{t}}$ given by $$ {\boldsymbol{t}}^A=\Big(\prod_{j=1}^{k} t_j^{a_{1,j}},\ldots,\prod_{j=1}^k t_j^{a_{n,j}}\Big). $$
Given ${\boldsymbol{a}}=(a_{1},\dots, a_{n})\in {\mathbb{Z}}^{n}$, we can consider it as a row vector or as a column vector. Thus \begin{displaymath}
{\boldsymbol{x}}^{{\boldsymbol{a}}}= \prod_{j=1}^{n}x_{j}^{a_{j}} {\quad \text{ and } \quad } t^{{\boldsymbol{a}}}=(t^{a_{1}},\dots, t^{a_{n}}). \end{displaymath}
\begin{theorem} \label{BT-poly} Let $F \in \C[{\boldsymbol{x}}^{\pm 1},z^{\pm1}]\setminus \C[{\boldsymbol{x}}^{\pm1}]$ be an irreducible Laurent polynomial, and $G\in \C[{\boldsymbol{x}}^{\pm 1}]$ the coefficient of the term of highest degree in the variable $z$. There are finite subsets $\Phi\subset {\mathbb{Z}}^{n \times n}$ of nonsingular matrices and $\Sigma \subset {\mathbb{Z}}^{n}$ of nonzero vectors such that, for ${\boldsymbol{a}}\in {\mathbb{Z}}^{n}$, one of the next alternatives holds: \begin{enumerate} \item \label{item:8} there is ${\boldsymbol{c}}\in\Sigma$ such that
$ \langle {\boldsymbol{c}},{\boldsymbol{a}}\rangle =0$; \item \label{item:9} there is $M\in\Phi$ such that ${\boldsymbol{a}}\in {\operatorname{im}}(M) $
and $F({\boldsymbol{x}}^M,z)$ is reducible; \item \label{item:10} the Laurent polynomial
$F(t^{\boldsymbol{a}},z) \in \C[ t^{\pm1},z^{\pm1}]$ is irreducible in
$\C[t^{\pm1},z^{\pm1}]_{G(t^{{\boldsymbol{a}}})}$. \end{enumerate} \end{theorem}
Back to the factorization problem for sparse polynomials, it is natural to consider the more general setting of pullbacks of regular functions on ${\mathbb G}_{\rm m}^{n}$ by \emph{arbitrary} monomial maps, instead of only those appearing in \eqref{eq:11}. Let ${\boldsymbol{y}}=(y_{1},\dots, y_{n})$ and ${\boldsymbol{t}}=(t_{1},\dots, t_{k})$ be groups of $n$ and $k$ variables, respectively. For a Laurent polynomial $H\in \C[{\boldsymbol{y}}^{\pm1}]$, consider the family of $k$-variate Laurent polynomials given by the pullback of $H$ by the monomial map ${\mathbb G}_{\rm m}^{k}\to {\mathbb G}_{\rm m}^{n}$ defined by ${\boldsymbol{t}}\mapsto {\boldsymbol{t}}^{A}$ for a matrix $A\in {\mathbb{Z}}^{n\times k}$, that is \begin{equation*}
H_{A}= H({\boldsymbol{t}}^{A})\in \C[{\boldsymbol{t}}^{\pm1}] . \end{equation*}
Denote by $S$ the multiplicative subset of $\C[{\boldsymbol{t}}^{\pm1}]$ generated by the Laurent polynomials of the form $f({\boldsymbol{t}}^{{\boldsymbol{d}}})$ for $f\in \C[z^{\pm1}]$ and ${\boldsymbol{d}}\in {\mathbb{Z}}^{k}$.
We propose the following conjecture which, as explained in Remark \ref{rem:10}, partially generalizes Theorem~\ref{thm:2}.
\begin{conjecture}
\label{conj:3}
Let $H\in \C[{\boldsymbol{y}}^{\pm1}]$ and $k\ge 2$. There is a finite set of
matrices $\Omega\subset {\mathbb{Z}}^{n\times n}$ {satisfying the following
property. For each $A\in{\mathbb{Z}}^{n\times k}$, there are} $N\in\Omega$ and
$B\in{\mathbb{Z}}^{n\times k}$ with $A=N B$ such that if $P $ is an
irreducible factor of $H({\boldsymbol{y}}^N)$, then $P({\boldsymbol{t}}^{B})$ is, as an
element of $\C[{\boldsymbol{t}}^{\pm1}]_{S}$, either a unit or an irreducible
factor of $H({\boldsymbol{t}}^{A})$. \end{conjecture}
The validity of this conjecture would imply that the irreducible factors of the $H_{A}$'s that truly depend on more than one variable, are also the pullback of a finite number of regular functions on ${\mathbb G}_{\rm m}^{n}$ by $k$-parameter monomial maps. The possible univariate irreducible factors of the $H_{A}$'s split completely, and so they cannot be accounted from a finite number of such regular functions.
This conjecture might follow from a suitable toric analogue of the classical Bertini's theorem that we propose in Conjecture \ref{conj_TBT}.
\noindent {\bf Plan of the paper.} In Section \ref{sec:conj-sche-funct} we state Schinzel's conjecture and our function field analogue (Theorem \ref{thm:10}). In Section~\ref{sec:toric-anal-bert} we recall some facts on fiber products and prove a variant of the Fuchs-Mantova-Zannier theorem concerning the irreducibility of pullbacks of cosets by a dominant maps $W\rightarrow {\mathbb G}_{\rm m}^n$ (Theorem~\ref{TBT}). In Section \ref{sec:pullb-laur-polyn} we prove Theorem \ref{BT-poly}, wereas in Section \ref{sec:fact-sparse-polyn} we apply this result to prove Theorem \ref{thm:10} and then Theorem \ref{thm:2}.
\noindent {\bf Acknowledgments.} We thank Pietro Corvaja, Qing Liu, Vincenzo Mantova, Juan Carlos Naranjo and Umberto Zannier for useful conversations. We also thank the anonymous referee for his/her useful comments. Part of this work was done while the authors met the Universitat de Barcelona and the Universit\'e de Caen. We thank these institutions for their hospitality.
\section{A conjecture of Schinzel and its function field analogue} \label{sec:conj-sche-funct}
In~\cite{Schinzel:rppt}, Schinzel proposed the conjecture below on the factorization of univariate polynomials over ${\mathbb{Q}}$.
\begin{conjecture} \label{Sch0} Let $F\in{\mathbb{Q}}[{\boldsymbol{x}}^{\pm1}]$ be a non-cyclotomic irreducible Laurent polynomial. There are finite sets $\Omega^{0}\subset {\mathbb{Z}}^{n\times n}$ of nonsingular matrices and $\Gamma\subset {\mathbb{Z}}^n$ of nonzero vectors satisfying the following property. Let ${\boldsymbol{a}}\in{\mathbb{Z}}^n$; then one of the next conditions holds: \begin{enumerate} \item \label{item:2} there is ${\boldsymbol{c}}\in\Gamma$ verifying
$\langle {\boldsymbol{c}},{\boldsymbol{a}}\rangle=0$; \item \label{item:1} there are $M\in\Omega^{0}$ and ${\boldsymbol{b}}\in{\mathbb{Z}}^n$ with
${\boldsymbol{a}}=M{\boldsymbol{b}}$ such that if $$ F({\boldsymbol{x}}^M)=\prod_{P}P^{e_{P}} $$ is the irreducible factorization of $F({\boldsymbol{x}}^M)$, then \begin{displaymath} \frac{F(t^{{\boldsymbol{a}}})}{{\operatorname{cyc}}(F(t^{{\boldsymbol{a}}})}=\prod_{P}\Big(\frac{P(t^{{\boldsymbol{b}}})}{{\operatorname{cyc}}(P(t^{{\boldsymbol{b}}}))}\Big)^{e_{P}} \end{displaymath} is the irreducible factorization of ${F(t^{{\boldsymbol{a}}})}/{{\operatorname{cyc}}(F(t^{{\boldsymbol{a}}})})$. \end{enumerate} \end{conjecture}
For the validity of this statement, in its condition \eqref{item:1} it is necessary to take out the cyclotomic part of $F(t^{{\boldsymbol{a}}})$ and of the $P(t^{{\boldsymbol{b}}})$'s, as shown by the example below.
\begin{example}
\label{exm:2}
Set $F=x_{1}+x_{2}-2\in {\mathbb{Q}}[x_{1}^{\pm1}, x_{2}^{\pm1}]$. Let
${\boldsymbol{a}}\in {\mathbb{Z}}^{2}$ and choose a nonsingular matrix
$M\in {\mathbb{Z}}^{2\times2}$ and a vector ${\boldsymbol{b}}\in {\mathbb{Z}}^{2}$ with
${\boldsymbol{a}}=M{\boldsymbol{b}}$. We have that \begin{displaymath} F({\boldsymbol{x}}^{M})=x_{1}^{m_{1,1}}x_{2}^{m_{1,2}}+x_{1}^{m_{2,1}}x_{2}^{m_{2,2}}-2 \end{displaymath} is irreducible, and so $P:=F({\boldsymbol{x}}^{M})$ is the only irreducible factor of this Laurent polynomial. However, $t-1$ divides $F(t^{{\boldsymbol{a}}})=P(t^{{\boldsymbol{b}}})$, and so these univariate Laurent polynomials are not irreducible, unless we divide them by this cyclotomic factor. \end{example}
Schinzel proved this conjecture when $n=1$ in \emph{loc. cit.} and, under the restrictive hypothesis that $F$ is not self-inversive, when $n\ge2$ \cite{Schinzel:rlpI}, see also \cite[\S 6.2]{Schinzel:psrr}. The general case when $n\ge 2$ remains open.
In Section~\ref{sec:fact-sparse-polyn}, we prove the function field analogue for Laurent polynomials over the field $\C(z)$ in Theorem \ref{thm:10} below. An element of $\C(z)[{\boldsymbol{x}}^{\pm 1}]$ is \emph{constant} if it lies in $\C[{\boldsymbol{x}}^{\pm1}]$, up to a scalar factor in $ \C(z)^{\times}$. The \emph{constant part} of a Laurent polynomial $F\in \C(z)[{\boldsymbol{x}}^{\pm1}]\setminus \{0\}$, denoted by ${\operatorname{ct}}(F)$, is defined as its maximal constant factor. This constant part is well-defined up to a unit of $\C(z)[{\boldsymbol{x}}^{\pm1}]$.
\begin{remark}
\label{rem:3}
The analogy between cyclotomic Laurent polynomials over ${\mathbb{Q}}$ and
irreducible constant Laurent polynomials over $\C(z)$ stems from
height theory. Let $\K$ denote either ${\mathbb{Q}}$ or $\C(z)$, and ${\operatorname{h}}$ the
canonical height function on subvarieties of the torus
$\G_{\rm m, \K}^{n}$, induced by the standard inclusion
$\G_{\rm m, \K}^{n}\hookrightarrow \P^{n}_{\K}$.
Let $F\in \K[{\boldsymbol{x}}^{\pm1}]$ be an irreducible Laurent polynomial
defining a hypersurface $V(F)$ of ${\mathbb G}_{\rm m}^{n}$. Then the condition that
${\operatorname{h}}(V(F))=0$ is equivalent to the fact that $F$ is cyclotomic when
$\K={\mathbb{Q}}$, and to the fact that $F$ is constant when $\K=\C(z)$. \end{remark}
\begin{theorem} \label{thm:10} Let $F\in \C(z)[{\boldsymbol{x}}^{\pm1}]$ be a non-constant irreducible Laurent polynomial. There are finite sets $\Omega^{0}\subset {\mathbb{Z}}^{n\times n}$ of nonsingular matrices and $\Gamma\subset {\mathbb{Z}}^n$ of nonzero vectors satisfying the following property. Let ${\boldsymbol{a}}\in{\mathbb{Z}}^n$; then one of the next conditions holds: \begin{enumerate} \item \label{item:14} there is ${\boldsymbol{c}}\in\Gamma$ verifying
$\langle {\boldsymbol{c}},{\boldsymbol{a}}\rangle=0$; \item \label{item:15} there are $M\in\Omega^{0}$ and ${\boldsymbol{b}}\in{\mathbb{Z}}^n$
with ${\boldsymbol{a}}=M{\boldsymbol{b}}$ such that if $$ F({\boldsymbol{x}}^M)=\prod_{P}P^{e_{P}} $$ is the irreducible factorization of $F({\boldsymbol{x}}^M)$, then \begin{displaymath} \frac{F(t^{{\boldsymbol{a}}})}{{\operatorname{ct}}(F(t^{{\boldsymbol{a}}}))}=\prod_{P}\Big(\frac{P(t^{{\boldsymbol{b}}})}{{\operatorname{ct}}(P(t^{{\boldsymbol{b}}}))}\Big)^{e_{P}} \end{displaymath} is the irreducible factorization of $F(t^{{\boldsymbol{a}}})/{\operatorname{ct}}(F(t^{{\boldsymbol{a}}}))$. \end{enumerate} \end{theorem}
Similarly as for Conjecture \ref{Sch0}, for the validity this statement it is is necessary to take out in its condition \eqref{item:15} the constant part of $F(t^{{\boldsymbol{a}}})$ and of the $P(t^{{\boldsymbol{b}}})$'s.
\begin{example}
\label{exm:3} Set $F=x_{1}+z x_{2}-z-1 \in \C(z)[x_{1}^{\pm1}, x_{2}^{\pm1}]$. Let ${\boldsymbol{a}}\in {\mathbb{Z}}^{2}$ and choose $M\in {\mathbb{Z}}^{2\times2}$ nonsingular and ${\boldsymbol{b}}\in {\mathbb{Z}}^{2}$ with ${\boldsymbol{a}}=M{\boldsymbol{b}}$. We have that \begin{displaymath} F({\boldsymbol{x}}^{M}) =x_{1}^{m_{1,1}}x_{2}^{m_{1,2}}+z\, x_{1}^{m_{2,1}}x_{2}^{m_{2,2}}-z-1 \end{displaymath} is irreducible, and so $P:=F({\boldsymbol{x}}^{M})$ is its only irreducible factor. Again, $t-1$ divides $F(t^{{\boldsymbol{a}}})=P(t^{{\boldsymbol{b}}})$, and so these univariate Laurent polynomials are not irreducible, unless we divide them by a suitable constant factor. \end{example}
\begin{remark}
\label{rem:7}
The validity of Schinzel's conjecture \ref{Sch0} would imply that of
Conjecture \ref{conj:3}, in the same way that Theorem \ref{thm:10}
implies Theorem \ref{thm:2}, as explained in Section
\ref{sec:fact-sparse-polyn}. \end{remark}
\section{A variant of Zannier's toric Bertini's theorem} \label{sec:toric-anal-bert}
{ Zannier proved in \cite{Zannier:hiaag} an analogue of Bertini's theorem for covers, where the subtori of ${\mathbb G}_{\rm m}^{n}$ replaced the linear subspaces in the classical version of this theorem.} This result was precised and generalized (with a completely different proof) by Fuchs, Mantova and Zannier to include fibers of arbitrary cosets of subtori \cite[Theorem~1.5]{FuchsMantovaZannier:fiptvb} and to obtain a more uniform result.
As before, let ${\boldsymbol{x}}=(x_{1},\dots,x_{n})$ be a set of $n$ variables and denote by ${\mathbb G}_{\rm m}^{n}={\operatorname{Spec}}(\C[{\boldsymbol{x}}^{\pm1}])$ the $n$-dimensional torus over $\C$.
Let $W$ be a variety, that is, a reduced separated scheme of finite type over $\C$. We assume that $W$ is irreducible and quasiprojective of dimension $n\ge 0$, and equipped with a dominant (regular) map \begin{equation*}
\pi\colon W\longrightarrow {\mathbb G}_{\rm m}^{n} \end{equation*} of degree $e\ge 1$ that is finite onto its image. Given an isogeny $\lambda$ of $ {\mathbb G}_{\rm m}^{n}$, that is, an endomorphism of ${\mathbb G}_{\rm m}^{n}$ with finite kernel, we denote by $\lambda^{*}W$ the fibered product ${\mathbb G}_{\rm m}^{n}\times_{\lambda,\pi} W$, and by
\begin{equation}\label{eq:8}
\xymatrix{ \lambda^{*}W \ar[r]^{\lambda} \ar[d]^{\pi}& W \ar[d]^{\pi}\\ {\mathbb G}_{\rm m}^{n}\ar[r]^{\lambda} & {\mathbb G}_{\rm m}^{n}}
\end{equation}
the corresponding fibered product square.
\begin{definition}
\label{def:1}
The map $\pi$ satisfies the \emph{property PB (pullback)} if, for
every isogeny $\lambda$ of ${\mathbb G}_{\rm m}^{n}$, we have that $\lambda^{*}W$
is an irreducible variety.
\end{definition}
By \cite[Proposition 2.1]{Zannier:hiaag}, it is enough to test this condition for $\lambda=[e]$, the multiplication map of ${\mathbb G}_{\rm m}^{n}$ by the integer $e=\deg(\pi)$.\\%}
The aforementioned result by Fuchs, Mantova and Zannier can be stated as follows.
\begin{theorem} \label{FMZ} Let $W$ be an irreducible quasiprojective variety of dimension $n$ and $\pi\colon W\rightarrow {\mathbb G}_{\rm m}^n$ a dominant map that is finite onto its image and that satisfies the property PB. There is a finite union $\cE$ of proper subtori of ${\mathbb G}_{\rm m}^n$ such that, for every subtorus $T\subset {\mathbb G}_{\rm m}^{n}$ not contained in $\cE$ and every point $p\in {\mathbb G}_{\rm m}^{n}(\C)= (\C^{\times})^{n}$, we have that $\pi^{-1}(p\cdot T)$ is an irreducible subvariety of $W$. \end{theorem}
When the property PB is not verified, the conclusion of this theorem does not necessarily hold because the map $\pi$ factors through a nontrivial isogeny, as it was already pointed out in \cite{Zannier:hiaag}.
\begin{example} \label{exm:4} Let $F=z^{2}-x_{1}x_{2}^{2}\in \C[x_{1}^{\pm1}, x_{2}^{\pm1},z^{\pm1}]$, set $W$ be the torus $V(F)\subset {\mathbb G}_{\rm m}^{3}$ and consider the isogeny
\begin{displaymath} \pi\colon W\longrightarrow {\mathbb G}_{\rm m}^{2}
\end{displaymath}
defined by $\pi(x_{1},x_{2},z)=(x_{1},x_{2})$ for
$(x_{1},x_{2},z)\in W$.
The variety $W$ is irreducible and, since
$F$ is monic in $z$, the map $\pi$ is finite. However, it does not
satisfy the property PB, since for the isogeny $\lambda$ of
$ {\mathbb G}_{\rm m}^{2}$ defined by $\lambda (x_{1},x_{2})= (x_{1}^{2},x_{2})$, \begin{displaymath} \lambda^{*}W\simeq V(z-x_{1}x_{2}) \cup V(z+x_{1}x_{2}) \end{displaymath} and so this pullback is reducible.
Indeed, this map does neither satisfy the conclusion of Theorem~\ref{FMZ}: given $(a_{1},a_{2})\in {\mathbb{Z}}^{2}\setminus \{(0,0)\}$ with $a_{1}$ even, let $T\subset {\mathbb G}_{\rm m}^{2}$ be the 1-dimensional subtorus given as the image of the map $t\mapsto (t^{a_{1}},t^{a_{2}})$. Then \begin{displaymath}
\pi^{-1}(T)= V(z-t^{a_{1}/2}t^{a_{2}}) \cup
V(z+t^{a_{1}/2}t^{a_{2}}), \end{displaymath} and so this fiber is not irreducible. \end{example}
Here we need a variant of Theorem~\ref{FMZ} that can be also applied in the situation when the map $\pi$ does not verify the property PB. In this more general situation, the conclusion of that theorem does not necessarily hold. {\red However, as already remarked in~\cite[Proposition 2.1]{Zannier:hiaag}, we can reduce ourself, up to an isogeny, to a situation in which PB is satisfied. We need a more explicit statement. The conclusion of Theorem~\ref{FMZ}}
is replaced by an alternative that ``explains'' the possibility that a fiber is reducible by its factorization through a reducible pullback of the variety $W$ by an isogeny of ${\mathbb G}_{\rm m}^{n}$ within a finite set.
\begin{theorem} \label{TBT} Let $W$ be an irreducible quasiprojective variety of dimension $n$ and $\pi\colon W\rightarrow {\mathbb G}_{\rm m}^n$ a dominant map that is finite onto its image. There is a finite union $\cE$ of proper subtori of ${\mathbb G}_{\rm m}^n$ and a finite set $\Lambda$ of isogenies of ${\mathbb G}_{\rm m}^n$ such that, for each subtorus $T\subset {\mathbb G}_{\rm m}^{n}$ and each point $p\in {\mathbb G}_{\rm m}^{n}(\C)=(\C^{\times})^{n}$, one of the next conditions holds: \begin{enumerate} \item \label{item:23} $T\subseteq\cE$; \item \label{item:24} there is $\lambda\in\Lambda$ with $\lambda^{*} W$
reducible and a subtorus $T'\subset {\mathbb G}_{\rm m}^{n}$ with $\lambda$ inducing
an isomorphism $T'\to T$; \item \label{item:25} $\pi^{-1}(p\cdot T)$ is irreducible. \end{enumerate} \end{theorem}
\begin{remark}
\label{rem:1}
When the condition \eqref{item:24} above is satisfied, there is a
diagram
\begin{displaymath}
\xymatrix{ \pi^{-1} (T')\ar[r] \ar[d]& \lambda^{*}W\ar[r]^{\lambda} \ar[d]^{\pi}& W \ar[d]^{\pi}\\ T'\ar[r]^{\iota} & {\mathbb G}_{\rm m}^{n}\ar[r]^{\lambda} & {\mathbb G}_{\rm m}^{n}}
\end{displaymath}
with $\lambda^{*}W$ reducible and $\lambda \colon T'\to T$ an
isomorphism, and where $\iota$ denotes the inclusion of the subtorus
$T'$ into ${\mathbb G}_{\rm m}^{n}$.
Both inner squares in this diagram are fibered products, and so is
the outer square. This implies that the fibers $ \pi^{-1}(T)$ and
$ \pi^{-1}(T')$ are isomorphic. Thus $\pi^{-1}(T)$ can be identified
with the fiber of a subtorus for the \emph{reducible} cover $\pi\colon\lambda^{*}W \to {\mathbb G}_{\rm m}^{n}$,
and so this fiber is expected to be
reducible as well. \end{remark}
\begin{example}
\label{exm:1}
We keep the notation from Example \ref{exm:4}. In particular,
$F=z^{2}-x_{1}x_{2}^{2}\in \C[x_{1}^{\pm1}, x_{2}^{\pm1},z^{\pm1}]$,
$W$ the torus $V(F)\subset {\mathbb G}_{\rm m}^{3}$, and
$\pi\colon W\rightarrow {\mathbb G}_{\rm m}^{2}$ the isogeny defined by
$\pi(x_{1},x_{2},z)=(x_{1},x_{2})$.
Let $(a_{1},a_{2})\in {\mathbb{Z}}^{2}\setminus \{(0,0)\}$ with $a_{1}$ even,
and set $T\subset {\mathbb G}_{\rm m}^{2}$ for the 1-dimensional
subtorus given as the image of the map
$t\mapsto (t^{a_{1}},t^{a_{2}})$. These vectors satisfy the
condition \eqref{item:24} in Theorem \ref{TBT} for the isogeny
$\lambda\colon {\mathbb G}_{\rm m}^{2}\to {\mathbb G}_{\rm m}^{2}$ defined by
\begin{displaymath}
\lambda (x_{1},x_{2})=
(x_{1}^{2},x_{2}).
\end{displaymath}
Indeed, $\lambda^{*}W$ is reducible, and this isogeny
induces an isomorphism $T'\to T$ with
the subtorus $T'\subset {\mathbb G}_{\rm m}^{2}$ given as the image of
the map $t\mapsto (t^{a_{1}/2},t^{a_{2}})$. \end{example}
We prove this theorem by reducing it to the previous toric Bertini's theorem, through a variation (Proposition \ref{red}) of a factorization result for rational maps from \cite{Zannier:hiaag}.
We give the proof after some auxiliary results. We first study the reducibility of pullbacks of varieties with respect to isogenies of tori.
\begin{lemma} \label{lemm:2} Let $\pi\colon W\to X$ be a map of
varieties and $\lambda\colon X\to X$ an \'etale map. Then
$ X\times_{\lambda,\pi } W$ is a variety.
In particular, for a map $\pi\colon W\to {\mathbb G}_{\rm m}^{n}$ and an
isogeny $\lambda $ of ${\mathbb G}_{\rm m}^{n}$, we have that $\lambda^{*}W$ is a variety. \end{lemma}
\begin{proof}
Since $\lambda\colon X\to X$ is \'etale, the map
\begin{equation}\label{eq:7} \lambda\colon X\times_{\lambda,\pi } W \longrightarrow W
\end{equation}
is also \'etale, because of the invariance of this property under
base change \cite[Chapter IV, Proposition 10.1(b)]{Hartshorne:ag}.
By \cite[Chapter IV, Exercise 10.4]{Hartshorne:ag}, this implies
that, for every closed point $q\in X\times_{\lambda,\pi } W$ and
$p:=\lambda (q) \in W$, the induced map of completed local rings
\begin{equation}
\label{eq:3}
\wh\cO_{p} \longrightarrow \wh\cO_{q}
\end{equation}
is an isomorphism. Since $W$ is a variety, the local ring $\cO_{p}$
is reduced and, by a theorem of Chevalley
\cite[\S8.13]{ZariskiSamuel:caII}, the completion $ \wh\cO_{p}$ is
reduced too.
By the isomorphism in \eqref{eq:3}, the completed ring $ \wh\cO_{q}$
is reduced. Since this is the completion of a ring with respect to a
maximal ideal, the map $\cO_{q}\to \wh\cO_{q}$ is injective, and so
the local ring $\cO_{q}$ is also reduced. Since the condition of
being reduced is local, this implies that $X\times_{\lambda,\pi } W$
is a variety.
The last statement comes from the fact that the isogenies of
algebraic groups over $\C$ are \'etale maps. \end{proof}
Thanks to this result, $\lambda^{*}W$ can be identified with its underlying algebraic subset in the Cartesian product ${\mathbb G}_{\rm m}^{n}(\C)\times W(\C)$, namely \begin{equation}\label{eq:5} \lambda^{*}W= \{(p, w)\in {\mathbb G}_{\rm m}^{n}(\C)\times W(\C) \mid \lambda(p)= \pi(w)\}. \end{equation} Hence, $\lambda^{*}W$ is irreducible if and only if this algebraic subset is irreducible. In particular, the map $\pi$ satisfies the property PB if and only if for every isogeny $\lambda$ of ${\mathbb G}_{\rm m}^{n}$, the pullback $\lambda^{*}W$ has a single irreducible component.
The following proposition is implicit in the proof of \cite[Proposition 2.1]{Zannier:hiaag}.
\begin{proposition} \label{red} Let $\pi\colon W\rightarrow {\mathbb G}_{\rm m}^n$ be a map from an irreducible variety $W$, and $\lambda$ an isogeny of ${\mathbb G}_{\rm m}^{n}$. The following conditions are equivalent: \begin{enumerate} \item \label{item:11} the pullback $\lambda^{*}W$ is reducible; \item \label{item:12} there is a factorization
$\lambda = \mu\circ \tau$ with $\mu, \tau$ isogenies of ${\mathbb G}_{\rm m}^{n}$
such that $\mu$ is not an isomorphism, and a map
$\rho\colon W\to {\mathbb G}_{\rm m}^{n}$ such that $\pi=\mu\circ\rho$. \end{enumerate} \end{proposition}
In other terms, the condition \eqref{item:12} in the proposition above amounts to the existence of the commutative diagram extending \eqref{eq:8} of the form
\begin{displaymath}
\xymatrix{ \lambda^{*}W\ar[rr]^{\lambda} \ar[d]^{\pi}& &W \ar[d]_{\pi}\ar@/^4pc/[ddl]^{\rho}\\
{\mathbb G}_{\rm m}^{n}\ar[rr]^{\lambda} \ar[rd]_{\tau} & &{\mathbb G}_{\rm m}^{n} \\ &{\mathbb G}_{\rm m}^{n} \ar[ru]_{\mu}& } \end{displaymath}
\begin{proof}
Suppose that the condition \eqref{item:12} holds. In this case, for
$p\in {\mathbb G}_{\rm m}^{n}(\C)$ and $w\in W(\C)$, the fact that $ \lambda(p)=\pi(w)$
is equivalent to $ \mu(\tau(p))=\mu(\rho(w))$, and so this holds if
and only if there is $\zeta\in\ker(\mu)$ with
$\tau (p)=\zeta\cdot\rho(w)$. From \eqref{eq:5}, the pullback
decomposes into disjoints subvarieties as
\begin{equation*}
\lambda^{*}W= \bigcup_{\zeta\in\ker(\mu)} {\mathbb G}_{\rm m}^{n}\times_{\tau,\zeta\cdot\rho} W.
\end{equation*}
Since $\mu$ is not an isomorphism, this pullback is reducible,
giving the condition \eqref{item:11}.
Conversely, suppose that the condition \eqref{item:11} holds. Then
$\lambda^{*}W$ has a decomposition into
irreducible components \begin{displaymath}
\lambda^{*}W=\bigcup_{i=1}^{k}U_{i} \end{displaymath} with $k\ge 2$. Similarly as in \eqref{eq:7}, the map $ \lambda^{*}W\to W$ is \'etale, and so the $U_{i}$'s are disjoint. Since $\lambda$ is an isogeny, the map $ \lambda^{*}W\to W$ is also finite.
The finite subgroup $\ker(\lambda)$ of ${\mathbb G}_{\rm m}^{n}(\C)$ acts on $\lambda^{*}W$ \emph{via} the maps $ (p,w)\mapsto (\zeta \cdot p,w) $ for $\zeta\in \ker(\lambda)$, and this action respects the fibers of $\lambda$. The action is transitive on the fibers, and so it is also transitive on the $U_{i}$'s.
Let $H\subset \ker(\lambda)$ be the stabilizer of the irreducible component $U_{1}$, and $U_{1}/H$ the quotient variety. We have that $H$ acts on $U_{1}$ transitively on the fibers and without fixed points. The induced map \begin{displaymath} U_{1}/H\longrightarrow W \end{displaymath} is a finite \'etale map of degree 1, and so it is an isomorphism \cite[\S III.10, Proposition 2]{Mumford:rbvs}.
Then we define the map $\rho\colon W\to {\mathbb G}_{\rm m}^{n}$ as the map obtained from the quotient map $U_{1}/H\to {\mathbb G}_{\rm m}^{n}/H$ and the identifications $U_{1}/H \simeq W$ and ${\mathbb G}_{\rm m}^{n}/H\simeq {\mathbb G}_{\rm m}^{n}$. In concrete terms and identifying ${\mathbb G}_{\rm m}^{n}/H \simeq {\mathbb G}_{\rm m}^{n}$, this map is defined, for $w\in W$, as $\rho(w)= \tau(p\cdot H)$ for any $p\in {\mathbb G}_{\rm m}^{n}$ such that $(p,w)\in U_{1}$.
Both ${\mathbb G}_{\rm m}^{n}/H$ and $ {\mathbb G}_{\rm m}^{n}/\ker(\lambda )$ are isomorphic to ${\mathbb G}_{\rm m}^{n}$, and so there is a factorization \begin{displaymath} \lambda=\mu \circ \tau, \end{displaymath} with $\tau$ and $\mu$ corresponding to the projections ${\mathbb G}_{\rm m}^{n}\to {\mathbb G}_{\rm m}^{n}/H$ and ${\mathbb G}_{\rm m}^{n}/H\to {\mathbb G}_{\rm m}^{n}/\ker(\lambda)$, respectively. For $w\in W$ and $(p,w)\in U_{1}$, we have that $\mu\circ \rho (w)= \mu\circ \tau (p)= \pi(w)$. Since the action of $\ker(\lambda)$ on the $U_{i}$'s is transitive and $k\ge 2$, we have that $H\ne \ker(\lambda)$ and so $\mu$ is not an isomorphism, giving the condition \eqref{item:12}. \end{proof}
\begin{remark} \label{rem:6} By this proof, if $\lambda^{*}W$ is reducible, then the number of its irreducible components is equal to the maximum of the quantity $\deg(\mu)=[\ker(\lambda):H] $ over all possible maps $\rho$ as in the condition \eqref{item:12}. \end{remark}
The next result allows to factorize the dominant map $\pi\colon W\to {\mathbb G}_{\rm m}^{n}$ as a map satisfying the property PB followed by an isogeny. It is a variant of \cite[Proposition~2.1]{Zannier:hiaag}, that states a similar property for dominant \emph{rational} maps.
\begin{corollary}
\label{cor:2}
Let $W$ be an irreducible variety of dimension $n$ and
$\pi \colon W\to {\mathbb G}_{\rm m}^{n}$ a dominant map. There are a map $\rho\colon W\to {\mathbb G}_{\rm m}^{n}$
satisfying the property PB and an isogeny
$\lambda$ of ${\mathbb G}_{\rm m}^{n}$ with $\pi=\lambda\circ \rho$. \end{corollary}
\begin{proof}
Choose $\rho$ as a map $W\to {\mathbb G}_{\rm m}^{n}$ of minimal degree among
those that give a factorization of the form $\pi=\lambda \circ \rho$
with $\lambda$ an isogeny of ${\mathbb G}_{\rm m}^{n}$.
Suppose that there is a further isogeny $\nu$ such that
$\nu^{*}W={\mathbb G}_{\rm m}^{n}\times _{\nu,\rho} W$ is reducible. By Proposition
\ref{red}, there would be an isogeny $\mu$ that is not an
isomorphism and a map $\rho'\colon W\to {\mathbb G}_{\rm m}^{n}$ with
$\rho=\mu\circ \rho'$. Hence
\begin{displaymath} \pi= \lambda \circ \rho= (\lambda \circ\mu )\circ \rho' {\quad \text{ and } \quad } \deg(\rho)=\# \ker(\mu) \cdot \deg(\rho')>\deg(\rho').
\end{displaymath}
By construction, this is not possible. Hence $\nu^{*}W$
is irreducible for every isogeny $\nu$ of ${\mathbb G}_{\rm m}^{n}$, and so $\rho$
satisfies the property PB. \end{proof}
The next result gives a criterion to detect if the inclusion of a subtorus can be factored through a given isogeny as in Proposition \ref{red}\eqref{item:12}.
\begin{lemma} \label{split} Let $T\subset {\mathbb G}_{\rm m}^{n}$ be a subtorus and $\lambda$ an isogeny of ${\mathbb G}_{\rm m}^{n}$. The following conditions are equivalent: \begin{enumerate} \item \label{item:6} there is a subtorus $T'\subset {\mathbb G}_{\rm m}^{n}$ such that $\lambda$ induces an isomorphism $T'\to T$; \item \label{item:7} $\lambda^{-1}(T)$ is the union of $\deg(\lambda)$
distinct torsion cosets. \end{enumerate} \end{lemma}
\begin{proof}
First suppose that $\lambda^{-1}(T)$ is the union of
$\deg(\lambda)=\#\ker(\lambda)$ distinct torsion cosets, and denote
by $T'$ the one that contains the neutral element. Then $T'$ is a
subtorus and $T'\cap \ker(\lambda)=\{1\}$. It follows that
$\lambda|_{T'} \colon T'\to T$ is an isogeny of degree 1 and hence an
isomorphism, giving the first condition.
Conversely, let $T'\subset {\mathbb G}_{\rm m}^{n}$ be a subtorus such that
$\lambda|_{T'} \colon T'\to T$ is an isomorphism. Then
\begin{displaymath}
\lambda^{-1}(T)= \ker(\lambda) \cdot T'.
\end{displaymath}
Since $T'\cap \ker(\lambda)=\{1\}$, this fiber is the union of
$\#\ker(\lambda) = \deg(\lambda)$ distinct torsion cosets, giving
the second condition. \end{proof}
\begin{proof}[Proof of Theorem \ref{TBT}]
By Corollary \ref{cor:2}, there are a map $\rho\colon W \to {\mathbb G}_{\rm m}^{n}$
satisfying the property PB and an isogeny $\lambda$ of ${\mathbb G}_{\rm m}^{n}$
with $\pi=\lambda\circ\rho$.
For each subgroup $H$ of $\ker(\lambda)$, both
${\mathbb G}_{\rm m}^{n}/H$ and $ {\mathbb G}_{\rm m}^{n}/\ker(\lambda )$ are isomorphic to
${\mathbb G}_{\rm m}^{n}$, and we consider then a factorization \begin{equation} \label{eq:13} \lambda=\mu_{H} \circ \tau_{H} \end{equation} with $\tau_{H}$ and $\mu_{H}$ corresponding to the projections ${\mathbb G}_{\rm m}^{n}\to {\mathbb G}_{\rm m}^{n}/H$ and ${\mathbb G}_{\rm m}^{n}/H\to {\mathbb G}_{\rm m}^{n}/\ker(\lambda)$, respectively. We set $\Lambda$ as the finite set of isogenies of ${\mathbb G}_{\rm m}^{n}$ of the form $\mu_{H}$ as above, for a proper subgroup $H$ of $\ker(\lambda)$.
Since $\rho\colon W\rightarrow {\mathbb G}_{\rm m}^{n}$ satisfies the property PB, by \cite[Theorem 1.5]{FuchsMantovaZannier:fiptvb} there is a finite union $\cE'$ of proper subtori of ${\mathbb G}_{\rm m}^{n}$ such that, for every subtorus $T$ of ${\mathbb G}_{\rm m}^{n}$ not contained in $\cE'$ and every point $p\in {\mathbb G}_{\rm m}^{n}(\C)$, the fiber $\rho^{-1}(p\cdot T)$ is irreducible. Set $\cE=\lambda(\cE')$.
We next show that the pair $(\Lambda, \cE)$ satisfies the requirements of Theorem~\ref{TBT}. Let $T$ be a subtorus of ${\mathbb G}_{\rm m}^{n}$ that is not contained in $\cE$ and write $\lambda^{-1}(T)=\bigcup_{i=1}^{k} T_{i}$ as a disjoint union of torsion cosets $T_{i}$ of ${\mathbb G}_{\rm m}^{n}$.
When $k=1$, we have that $\lambda^{-1}(T)=T_1$ is a subtorus of ${\mathbb G}_{\rm m}^{n}$ that is not contained in $\cE$. Hence, $\pi^{-1}(T)=\rho^{-1}(T_1)$ is irreducible.
Otherwise, $k\ge 2$. Let $H \subset \ker(\lambda)$ be the stabilizer of the (unique) subtori in this decomposition, say $T_{1}$. This is a proper subgroup, because $\ker(\lambda)$ acts transitively on this collection of torsion cosets and $k\ge 2$.
Consider the factorization $\lambda=\mu_H\circ\tau_H$ as in \eqref{eq:13}. Then $\mu_H\in\Lambda$ and $\mu_H^{-1}(T)$ splits as an union of $k=[\ker(\lambda):H] = \deg(\mu_{H})$ distinct torsion cosets. By Lemma \ref{split}, $\mu_{H}$ induces an isomorphism between a subtorus $T'$ of ${\mathbb G}_{\rm m}^{n}$ and $T$. Moreover, Proposition~\ref{red}\eqref{item:12} applied to the map $\tau_{H}\circ \rho$ and the isogeny $\mu_{H}$ shows that the pullback $\mu_{H}^{*}W$ is reducible, completing the proof. \end{proof}
It seems interesting to extend these results to maps that are not necessarily dominant. In this direction, we propose the following conjectural extension of the Fuchs-Mantova-Zannier theorem \ref{FMZ}. It can be seen as a toric analogue of the classical Bertini's theorem as stated in \cite[Th\'eor\`eme 6.3(3)]{Jou83}.
\begin{conjecture} \label{conj_FMZ} Let $W$ be an irreducible quasiprojective variety and $\varphi\colon W\rightarrow {\mathbb G}_{\rm m}^n$ a map that is finite onto its image and satisfies the property PB. There is a finite union $\cE$ of proper subtori of ${\mathbb G}_{\rm m}^n$ such that, for every subtorus $T$ of ${\mathbb G}_{\rm m}^{n}$ with \begin{displaymath}
\dim(T)\ge {\operatorname{codim}}(\ov{\varphi(W)}) +1 \end{displaymath} that is not contained in $\cE$ and every point $p\in {\mathbb G}_{\rm m}^{n}(\C)$, we have that $\varphi^{-1}(p\cdot T)$ is an irreducible subvariety of $W$. \end{conjecture}
Similarly, we propose the following conjectural extension of Theorem \ref{TBT}.
\begin{conjecture} \label{conj_TBT} Let $W$ be an irreducible quasiprojective variety and $\varphi\colon W\rightarrow {\mathbb G}_{\rm m}^n$ a map that is finite onto its image. There is a finite union $\cE$ of proper subtori of ${\mathbb G}_{\rm m}^n$ and a finite set $\Lambda$ of isogenies of ${\mathbb G}_{\rm m}^n$ such that, for each subtorus $T\subset {\mathbb G}_{\rm m}^{n}$ with \begin{displaymath} \dim(T)\ge {\operatorname{codim}}(\ov{\varphi(W)}) +1 \end{displaymath} and each point $p\in {\mathbb G}_{\rm m}^{n}(\C)$, one of the next conditions holds: \begin{enumerate} \item \label{item:23} $T\subseteq\cE$; \item \label{item:24} there is $\lambda\in\Lambda$ with $\lambda^{*} W$
reducible and a subtorus $T'\subset {\mathbb G}_{\rm m}^{n}$ with $\lambda$ inducing
an isomorphism $T'\to T$; \item \label{item:25} $\varphi^{-1}(p\cdot T)$ is irreducible. \end{enumerate} \end{conjecture}
\section{Pullbacks of Laurent polynomials by monomial maps} \label{sec:pullb-laur-polyn}
We next prove Theorem~\ref{BT-poly} stated in the introduction. To this
end, we first recall some notation and introduce some auxiliary results.
Let ${\boldsymbol{t}} =(t_1,\ldots,t_{k})$ be a set of $k$ variables. A matrix $A=(a_{i,j})_{i,j}\in{\mathbb{Z}}^{n\times k}$ defines the family of $n$ monomials in the variables ${\boldsymbol{t}}$ given by $$ {\boldsymbol{t}}^A=\Big(\prod_{j=1}^{k} t_j^{a_{1,j}},\ldots,\prod_{j=1}^k t_j^{a_{n,j}}\Big). $$
The rule ${\boldsymbol{t}}\mapsto {\boldsymbol{t}}^{A}$ defines a $k$-parameter monomial map ${\mathbb G}_{\rm m}^{k}\rightarrow {\mathbb G}_{\rm m}^{n} $. This is a group morphism and indeed, every group morphism from ${\mathbb G}_{\rm m}^{k}$ to ${\mathbb G}_{\rm m}^{n}$ is of this form. The isogenies of ${\mathbb G}_{\rm m}^{n}$ correspond to the nonsingular matrices of ${\mathbb{Z}} ^{n\times n}$.
Given ${\boldsymbol{a}}=(a_{1},\dots, a_{n})\in {\mathbb{Z}}^{n}$, we can consider it as a row vector, that is, as a matrix in ${\mathbb{Z}}^{1\times n}$. In this case, \begin{displaymath}
{\boldsymbol{x}}^{{\boldsymbol{a}}}= \prod_{j=1}^{n}x_{j}^{a_{j}} \end{displaymath} is an $n$-variate monomial. Row vectors give characters of ${\mathbb G}_{\rm m}^{n}$, that is, group morphisms ${\mathbb G}_{\rm m}^{n}\to {\mathbb G}_{\rm m}$. When ${\boldsymbol{a}}$ is primitive, the kernel of its associated character is a subtorus of ${\mathbb G}_{\rm m}^{n}$ of codimension 1, and every such subtorus arises in this way.
Else, we can consider ${\boldsymbol{a}}$ as a column vector, that is, as a matrix in ${\mathbb{Z}}^{n\times 1}$. Then \begin{displaymath}
t^{{\boldsymbol{a}}}=(t^{a_{1}},\dots, t^{a_{n}}) \end{displaymath} is a collection of $n$ univariate monomials in a variable $t$. Column vectors give group morphisms ${\mathbb G}_{\rm m}\to {\mathbb G}_{\rm m}^{n}$. When ${\boldsymbol{a}}\ne 0$, the image of such a morphims is a subtorus of ${\mathbb G}_{\rm m}^{n}$ of dimension 1, that we denote by $T_{{\boldsymbol{a}}}$. When ${\boldsymbol{a}}$ is primitive, the associated group morphism ${\mathbb G}_{\rm m}\to {\mathbb G}_{\rm m}^{n}$ gives an isomorphism between ${\mathbb G}_{\rm m}$ and $T_{{\boldsymbol{a}}}$.
For subvarieties of tori, fibered products like those in \eqref{eq:8} can be expressed in more concrete terms. The next lemma gives such an expression for the case of hypersurfaces.
\begin{lemma} \label{lemm:3} Let $F \in \C[{\boldsymbol{x}}^{\pm 1},z^{\pm1}]$,
$G\in \C[{\boldsymbol{x}}^{\pm1}]\setminus \{0\}$, and $A\in {\mathbb{Z}}^{n\times k}$.
Let $W$ be the hypersurface of ${\mathbb G}_{\rm m}^{n+1}\setminus V(G)$ defined by
$F$, $\pi\colon W\to {\mathbb G}_{\rm m}^{n}$ the map defined by
$\pi({\boldsymbol{x}},z)= {\boldsymbol{x}}$, and $\lambda \colon {\mathbb G}_{\rm m}^{k}\to {\mathbb G}_{\rm m}^{n}$ the
group morphism defined by $\lambda({\boldsymbol{t}})= {\boldsymbol{t}}^{A}$. Then
${\mathbb G}_{\rm m}^{k}\times_{\lambda,\pi} W $ is isomorphic to the subscheme of
${\mathbb G}_{\rm m}^{k+1}\setminus V(G({\boldsymbol{t}}^{A})) $ defined by $F({\boldsymbol{t}}^{A},z)$. \end{lemma}
\begin{proof} The maps $\pi $ and $\lambda$ correspond to the morphisms of $\C$-algebras \begin{displaymath}
\C[{\boldsymbol{x}}^{\pm1}]\longrightarrow \C[{\boldsymbol{x}}^{\pm1},z^{\pm1}]_{G}/F {\quad \text{ and } \quad }
\C[{\boldsymbol{x}}^{\pm1}]\longrightarrow \C[{\boldsymbol{t}}^{\pm1}]\simeq \C[{\boldsymbol{x}}^{\pm1},{\boldsymbol{t}}^{\pm1}]/({\boldsymbol{x}}-{\boldsymbol{t}}^{A}), \end{displaymath} and the fibered product ${\mathbb G}_{\rm m}^{k}\times_{\lambda,\pi} W $ is the scheme associated to the tensor product \begin{displaymath} \C[{\boldsymbol{x}}^{\pm1},z^{\pm1}]_{G}/F \otimes_{ \C[{\boldsymbol{x}}^{\pm1}]}
\C[{\boldsymbol{x}}^{\pm1},{\boldsymbol{t}}^{\pm1}]/({\boldsymbol{x}}-{\boldsymbol{t}}^{A}) . \end{displaymath} This tensor product is isomorphic to the $\C$-algebra \begin{displaymath}
\C[{\boldsymbol{x}}^{\pm1},z^{\pm1},{\boldsymbol{t}}^{\pm1}]_{G}/(F,{\boldsymbol{x}}-{\boldsymbol{t}}^{A}) \simeq \C[z^{\pm1},{\boldsymbol{t}}^{\pm1}]_{G({\boldsymbol{t}}^{A})}/(F({\boldsymbol{t}}^{A},z)), \end{displaymath} which gives the statement. \end{proof}
\begin{lemma} \label{lemm:1} Let $f\in\C(t)[z]$ be an irreducible
polynomial of degree $d\geq1$, and such that $f(t^m,z)$ is reducible
for some $m\in{\mathbb{N}}$. There is $e\in{\mathbb{N}}$ dividing $\gcd(m,d)$ such
that $f(t^e,z)$ is also reducible. \end{lemma}
\begin{proof} The proof relies on the action of torsion points on irreducible factors as in \cite[Proposition 2.1]{Zannier:hiaag}.
By Lemma \ref{lemm:3}, the subscheme of ${\mathbb G}_{\rm m}^{2}$ defined by
$f(t^{m},z)$ is isomorphic to the pullback $[m]^{*} V(f)$, with
$[m]$ the $m$-th multiplication map of ${\mathbb G}_{\rm m}$. By Lemma \ref{lemm:2},
this pullback is reduced, and so $f(t^{m},z)$ is
separable. Consider its decomposition into distinct irreducible
factors \begin{equation}\label{eq:15}
f(t^{m},z)= \prod_{i=1}^{k} p_{i}, \end{equation} with $k\ge 2$.
The group $\upmu_{m}$ of $m$-th roots of the unity acts on the set of these irreducible factors by $p_{i}(t,z)\mapsto p_{i}(\zeta\cdot t,z)$, $i=1,\dots, k$, for $\zeta\in \upmu_{m}$. Let $\cP\subset \{p_{1},\dots, p_{k}\}$ be a nonempty orbit of this action. The polynomial \begin{equation*}
\prod_{p\in \cP} p \end{equation*} is invariant under the action of $\upmu_{m}$, and so it is of the form
$g(t^{m},z)$ with $g\in \C(t)[z]$. This product is a nontrivial factor of $f(t^{m},z)$, and so $g$ coincides with $f$ up to a scalar. It follows that $\cP= \{p_{1},\dots, p_{k}\}$ and so the action is transitive. In particular, all the $p_{i}$'s have the same degree in the variable $z$, and so this degree is positive and $k | d$.
The stabilizer of an irreducible factor $p_{i}$ is a subgroup of
$\upmu_{m}$, hence it is of the form $\upmu_{l}$ with $l | m$. Since the action is transitive and $\upmu_m$ is abelian, this subgroup does not depend on the choice of $p_{i}$. Moreover, $m/l$ is equal to $k$, the number of irreducible factors of $f(t^{m},z)$, also because of the transitivity of the action.
By the invariance of each $p_{i}$ under the action of $\upmu_{l}$, there is $q_{i}\in \C(t) [z] \setminus \C(t)$ with $p_{i}=q_{i}(t^{l},z)$. It follows from \eqref{eq:15} that \begin{displaymath}
f(t^{e},z) = \prod_{i=1}^{k} q_{i}(t,z), \end{displaymath}
with $e=m/l$. Clearly $e|m$ and as explained, $e=k$, and so this quantity also divides~$d$, completing the proof. \end{proof}
\begin{lemma} \label{specialization} Let $F\in\C[{\boldsymbol{x}},z] $ be an irreducible polynomial of degree $d\geq1$ in the variable $z$, and $G\in \C[{\boldsymbol{x}}]\setminus \{0\}$ its leading coefficient.
\begin{enumerate} \item \label{item:13} Let $W=V(F)\setminus V(G) \subset {\mathbb G}_{\rm m}^{n+1}$ and
$\pi\colon W\rightarrow {\mathbb G}_{\rm m}^{n}$ the map defined by
$\pi({\boldsymbol{x}},z)={\boldsymbol{x}}$. The image of $\pi$ is the open subset
${\mathbb G}_{\rm m}^{n}\setminus V(G)$ of ${\mathbb G}_{\rm m}^{n}$, and this map is finite onto
this open subset.
\item \label{item:16} There is a finite subset $\Delta_F$ of ${\mathbb{Z}}^n$
such that for ${\boldsymbol{a}}\in{\mathbb{Z}}^n$ with $\langle {\boldsymbol{c}},{\boldsymbol{a}}\rangle\neq0$ for
all ${\boldsymbol{c}}\in\Delta_F$, the polynomial $F(t^{\boldsymbol{a}},z)$ has degree $d$ in
the variable $z$.
\item \label{item:19} If $A\in {\mathbb{Z}}^{n\times n}$ is nonsingular, then
$F({\boldsymbol{x}}^{A},z)$ has no nontrivial factors in $\C[{\boldsymbol{x}}^{\pm1}]_{G}$. \end{enumerate} \end{lemma}
\begin{proof}
For the first statement, the image of the map $\pi$ is contained in
the open set $U={\mathbb G}_{\rm m}^{n}\setminus V(G)$. The induced map $W\to U$
corresponds to the morphism of $\C$-algebras \begin{displaymath}
\C[{\boldsymbol{x}}^{\pm 1}]_{G} \lhook\joinrel\longrightarrow \C[{\boldsymbol{x}}^{\pm1},z]_{G}/(F). \end{displaymath} This morphism is an integral extension because the leading term $G$ is invertible in $\C[{\boldsymbol{x}}^{\pm 1}]_{G}$, and so the map $W\to U$ is finite and, \emph{a fortiori}, surjective.
For the second statement, write $G=\sum_{j=1}^{r} G_{j}{\boldsymbol{x}}^{{\boldsymbol{c}}_{j}}$ with $G_{j}\in \C^{\times}$ and ${\boldsymbol{c}}_{j}\in {\mathbb{N}}^{n}$, $j=1,\dots, r$, and consider the finite subset of ${\mathbb{Z}}^{n}$ given by \begin{displaymath}
\Delta_{F}=\{{\boldsymbol{c}}_{j}-{\boldsymbol{c}}_{1} \mid j=2,\dots, r\} . \end{displaymath} For ${\boldsymbol{a}}\in{\mathbb{Z}}^{n}$ with $\langle {\boldsymbol{c}},{\boldsymbol{a}}\rangle\neq0$ for all ${\boldsymbol{c}}\in\Delta_F$, we have that $G(t^{{\boldsymbol{a}}})\ne 0$ and so $\deg_{z}(F(t^{{\boldsymbol{a}}},z))=d$.
As for Lemma \ref{lemm:1}, the proof of the last assertion relies on the action of torsion points on irreducible factors, and so we only sketch it. Using Lemmas \ref{lemm:3} and \ref{lemm:2}, we show that $ F({\boldsymbol{x}}^{A}, z)$ is separable. Let \begin{displaymath}
F({\boldsymbol{x}}^{A},z)=\prod_{i=1}^{k}P_{i} \end{displaymath} the decomposition of this Laurent polynomial into distinct irreducible factors. The action of the finite group $\{{\boldsymbol{x}} \in {\mathbb G}_{\rm m}^{n}\mid {\boldsymbol{x}}^{A}=1\}$ on the the sets of these irreducible factors is transitive, and so the $P_{i}$'s have the same degree with respect to the variable $z$. Hence for $i=1,\dots, k$, we have that $k \deg_{z}(P_{i}) = d \ge 1$. In particular, $\deg_{z}(P_{i})\ge 1$, proving the statement. \end{proof}
\begin{proof}[Proof of Theorem~\ref{BT-poly}]
The statement of this result, restricted to \emph{primitive} vectors ${\boldsymbol{a}}\in {\mathbb{Z}}^{n}$,
is a specialization of Theorem~\ref{TBT}. To see this, first
reduce, multiplying by a suitable monomial, to the case when $F$ is
an irreducible polynomial in $\C[{\boldsymbol{x}},z ]$ of degree $d\ge 1$ in the
variable $z$. Set $W=V(F)\setminus V(G)$ and consider the map \begin{equation*}
\pi\colon W\longrightarrow {\mathbb G}_{\rm m}^{n} \end{equation*} defined by $\pi({\boldsymbol{x}},z)={\boldsymbol{x}}$ for $({\boldsymbol{x}},z)\in W$. The quasi-projective variety $W$ is irreducible and, by Lemma \ref{specialization}\eqref{item:13}, this map is dominant and finite onto its image, the open subset $U={\mathbb G}_{\rm m}^{n}\setminus V(G)$ of ${\mathbb G}_{\rm m}^{n}$.
Let $\Lambda$ be a finite subset of isogenies of ${\mathbb G}_{\rm m}^{n}$ and $\cE$ a finite union of proper subtori of ${\mathbb G}_{\rm m}^{n}$ satisfying the conclusion of Theorem~\ref{TBT} applied to this map. Set then $\Phi_{1}$ for the finite subset of nonsingular matrices in $ {\mathbb{Z}}^{ n\times n} $ corresponding to the isogenies in $\Lambda$, and $\Sigma_{1} $ for a finite subset of nonzero vectors of ${\mathbb{Z}}^{n}$ such that \begin{equation}
\label{eq:10}
\cE\subset \bigcup_{{\boldsymbol{c}}\in \Sigma_{1}} V({\boldsymbol{x}}^{{\boldsymbol{c}}}-1). \end{equation}
For a primitive vector ${\boldsymbol{a}}\in {\mathbb{Z}}^{n}$, set $T_{{\boldsymbol{a}}}$ for the 1-dimensional subtorus defined as the image of the group morphism ${\mathbb G}_{\rm m}\to {\mathbb G}_{\rm m}^{n}$. This map gives an isomorphism between ${\mathbb G}_{\rm m}$ and $T_{{\boldsymbol{a}}}$. By Lemma \ref{lemm:3}, the fiber $\pi^{-1}(T_{{\boldsymbol{a}}})$ is isomorphic to the subscheme of ${\mathbb G}_{\rm m}^{2} \setminus V(G(t^{{\boldsymbol{a}}}))$ defined by $F(t^{{\boldsymbol{a}}},z)$. For the isogeny $\lambda$ associated to a nonsingular matrix $M\in \Phi_{1}$, the same result shows that $\lambda^{*}W$ is isomorphic to the subscheme of ${\mathbb G}_{\rm m}^{n+1}\setminus V(G) $ defined by $F({\boldsymbol{x}}^{M},z)$.
The three alternatives from Theorem~\ref{TBT} applied to the map $\pi$, the subtorus $T_{{\boldsymbol{a}}}$ and the point $p=(1,\dots,1) \in {\mathbb G}_{\rm m}^{n}(\C)$, then boil down to those in the theorem under examination, as explained below. \begin{enumerate} \item \label{item:21} Suppose that $T_{{\boldsymbol{a}}}\subset \cE$. By
\eqref{eq:10}, there is ${\boldsymbol{c}}\in \Sigma_{1}$ with
$\langle {\boldsymbol{c}},{\boldsymbol{a}}\rangle =0 $. \item \label{item:22} Else suppose that there is an isogeny
$\lambda \in \Lambda$ with $\lambda^{*}W$ reducible and a subtorus
$T'$ of ${\mathbb G}_{\rm m}^{n}$ with $\lambda$ inducing an isomorphism between
$T'$ and $T_{{\boldsymbol{a}}}$. For $M\in \Phi_{1}$ the
nonsingular matrix associated to $\lambda$, we have that ${\boldsymbol{a}}\in {\operatorname{im}}(B)$
and, by Lemma~\ref{lemm:3}, $F({\boldsymbol{x}}^{M},z)$ is reducible. \item\label{item:20} Else suppose that $\pi^{-1}(T_{{\boldsymbol{a}}})$ is
irreducible in ${\mathbb G}_{\rm m}^{2}\setminus V(G)$. By Lemma~\ref{lemm:3}, this
implies that $F(t^{{\boldsymbol{a}}},z)$ is irreducible in $\C(t)[ z^{\pm1}]$. \end{enumerate}
We next enlarge these finite sets to cover the rest of the cases. Let $d\ge 1$ be the degree of $F$ in the variable $z$, and let $ e$ be a divisor of $d$. If $F({\boldsymbol{x}}^{e},z)$ is irreducible, we respectively denote by $\Phi_{e}$ and $\Sigma_{e}$ the finite subsets of nonsingular matrices in $ {\mathbb{Z}}^{ n\times n} $ and of nonzero vectors of ${\mathbb{Z}}^{n}$ given by the application of Theorem~\ref{TBT} to this polynomial. Otherwise, we set $\Phi_{e} =\{I_{n}\}$ with $I_{n}$ the identity matrix of ${\mathbb{Z}}^{n\times n}$, and $\Sigma_{e} =\emptyset$. Set also $\Delta $ for the finite subset of nonzero vectors in ${\mathbb{Z}}^{n}$ associated to $F$ by Lemma \ref{specialization}\eqref{item:16}. Set then \begin{displaymath}
\Phi = \bigcup_{e\mid d} e \, \Phi_{e} {\quad \text{ and } \quad } \Sigma= \Delta\cup \bigcup_{e\mid d}
\Sigma_{e}. \end{displaymath}
By Theorem \ref{TBT} and the previous analysis, the statement holds for all vectors of the form $e\, {\boldsymbol{b}}$ with ${\boldsymbol{b}}\in {\mathbb{Z}}^{n}$ primitive and $e |d$.
Given an arbitrary vector ${\boldsymbol{a}}\in {\mathbb{Z}}^{n}$, write $ {\boldsymbol{a}}= m{\boldsymbol{b}}$ with $m\in {\mathbb{N}}$ and ${\boldsymbol{b}} \in {\mathbb{Z}}^{n}$ primitive, and set \begin{displaymath}
f=F(t^{{\boldsymbol{b}}},z)\in \C[t^{\pm1},z]. \end{displaymath} Suppose that neither \eqref{item:8} nor \eqref{item:9} hold for ${\boldsymbol{a}}$. Let $e\in {\mathbb{N}}$ be a common divisor of $d$ and $m$. \emph{A
fortiori}, these conditions do neither hold for $e\, {\boldsymbol{b}}$ and, as explained before, \begin{displaymath}
f(t^{e},z)= F(t^{e{\boldsymbol{b}}},z) \end{displaymath} is irreducible in $\C(t)[z]$. By Lemma \ref{lemm:1}, we have that $ F_{{\boldsymbol{a}}}=f(t^{m},z)$ is irreducible in $\C(t)[z]$, giving the condition \eqref{item:10} for ${\boldsymbol{a}}$ and concluding the proof. \end{proof}
\begin{remark}
\label{rem:4} Using the toric Bertini's theorem \ref{TBT} for cosets of arbitrary dimension, the present polynomial version in Theorem \ref{BT-poly} might be extended to $k$-parameter monomial maps for any $k$, and also to arbitrary
translates of these monomial maps.
We have kept the present more restricted statement for the sake of
simplicity, and also because it is sufficient for our application. \end{remark}
\section{Factorization of sparse polynomials} \label{sec:fact-sparse-polyn}
Here we prove the results on the factorization of Laurent polynomials announced in the introduction and in Section~\ref{sec:conj-sche-funct}. Theorem~\ref{thm:10} is easily seen to be implied by the following statement.
\begin{theorem} \label{thm:1} Let $F\in \C[{\boldsymbol{x}}^{\pm1},z^{\pm1}]$ without nontrivial factors in $\C[{\boldsymbol{x}}^{\pm1}]$. There are finite sets $\Omega^{0}\subset {\mathbb{Z}}^{n\times n}$ of nonsingular matrices and $\Gamma\subset {\mathbb{Z}}^n$ of nonzero vectors satisfying the property that, for ${\boldsymbol{a}}\in{\mathbb{Z}}^n\setminus \{{\boldsymbol{0}}\}$, one of the next alternatives holds: \begin{enumerate} \item \label{item:17} there is ${\boldsymbol{c}}\in\Gamma$ with
$\langle {\boldsymbol{c}},{\boldsymbol{a}}\rangle=0$; \item \label{item:18} there are $M\in\Omega^{0}$ and ${\boldsymbol{b}}\in{\mathbb{Z}}^n$ with
${\boldsymbol{a}}=M{\boldsymbol{b}}$ such that if $$ F({\boldsymbol{x}}^M,z)=\prod_{P}P^{e_{P}} $$ is the irreducible factorization of $F({\boldsymbol{x}}^M,z)$ in $\C[{\boldsymbol{x}}^{\pm1},z^{\pm1}]$, then \begin{displaymath} F(t^{{\boldsymbol{a}}},z)=\prod_{P}P(t^{{\boldsymbol{b}}},z)^{e_{P}} \end{displaymath} is the irreducible factorization of $F(t^{{\boldsymbol{a}}},z)$ in $ \C(t)[z^{\pm1}]$. \end{enumerate} \end{theorem}
\begin{proof}
We proceed by induction on $\deg_{z}(F)$. When $\deg_{z}(F)=0$, the
statement is trivial, and so we assume that $\deg_{z}(F)\ge 1$.
If $F$ is irreducible, we respectively
denote by $\Phi $ and $\Sigma $ the finite sets of nonsingular
matrices in ${\mathbb{Z}}^{n\times n}$ and of nonzero vectors in ${\mathbb{Z}}^{n}$ from
Theorem~\ref{thm:1} applied to this Laurent polynomial. If $F$ is
reducible, we set $\Phi=\{I_{n}\}$ and $\Sigma=\emptyset$.
Let ${\boldsymbol{a}}\in{\mathbb{Z}}^n$. When $F$ is irreducible, if the condition
\eqref{item:8} in Theorem~\ref{BT-poly} holds, then the condition
\eqref{item:17} in Theorem~\ref{thm:1} also holds by taking $\Gamma$
as any finite set containing~$\Sigma $. Still in the irreducible
case, if the condition \eqref{item:10} in Theorem~\ref{BT-poly}
holds, the Laurent polynomial $F(t^{\boldsymbol{a}},z)$ is irreducible, and the
condition \eqref{item:17} in Theorem~\ref{thm:1} holds provided that
$\Omega^{0}$ contains $I_{n}$.
Else, suppose that the condition \eqref{item:9} in Theorem~\ref{BT-poly} holds, that is, there are $M\in\Phi $ and ${\boldsymbol{b}}\in{\mathbb{Z}}^n$ with ${\boldsymbol{a}}=M{\boldsymbol{b}}$ and $F({\boldsymbol{x}}^M,z)$ is reducible. Let \begin{equation*}
F({\boldsymbol{x}}^M, z)=F_1 \, F_2 \end{equation*} be a nontrivial factorization. By Lemma~\ref{specialization}\eqref{item:19}, $F({\boldsymbol{x}}^M,z)$ has no factors in $\C[{\boldsymbol{x}}^{\pm1}]$. Hence $\deg_{z}(F_{i})<\deg_{z}(F)$, $i=1,2$, and by induction, Theorem~\ref{thm:1} holds for these Laurent polynomials. Let $\Omega^{0}_{i}$ and $\Gamma_{i}$ respectively denote the finite sets of nonsingular matrices in ${\mathbb{Z}}^{n\times n}$ and of nonzero vectors in ${\mathbb{Z}}^{n}$ whose existence is assured by this theorem.
By construction, either there is a vector ${\boldsymbol{c}}\in\Gamma_{1}\cup\Gamma_{2}$ with $\langle{\boldsymbol{c}},{\boldsymbol{b}}\rangle=0$, or we can find $M_i\in\Omega_i$ and ${\boldsymbol{b}}_i\in{\mathbb{Z}}^n$ with ${\boldsymbol{b}}=M_i{\boldsymbol{b}}_i$ and a decomposition $$ F_i({\boldsymbol{x}}^{M_i},z)=\prod_{j=1}^{k_{i}}F_{i,j} $$ with $F_{i,j}(t^{{\boldsymbol{b}}_i},z)$ irreducible in $\C(t)[s^{\pm1}]$ for all $i,j$.
Set \begin{equation}
\label{eq:14} \Gamma=\{{\operatorname{adj}}(M) {\boldsymbol{c}} \mid M\in \Phi, {\boldsymbol{c}}\in \Gamma_{1}\cup\Gamma_{2}\} \end{equation} with ${\operatorname{adj}}(M)$ the adjoint matrix of $M$. If $\langle{\boldsymbol{c}}',{\boldsymbol{a}}\rangle\neq0$ for all ${\boldsymbol{c}}'\in\Gamma$, then $\langle{\boldsymbol{c}},{\boldsymbol{b}}\rangle \ne 0$ for all ${\boldsymbol{c}}\in \Gamma_{1}\cup\Gamma_{2}$ and so $F_{i,j}(t^{{\boldsymbol{b}}_i},z)$ is irreducible in $\C(t)[s^{\pm1}]$ for all $i,j$.
Consider the lattices $K_i={\operatorname{im}}(M_i)$, $i=1,2$, and set $K=K_1\cap K_2$. Since $K$ is also a lattice, there is a nonsingular matrix $M'\in{\mathbb{Z}}^{n \times n}$ with $K={\operatorname{im}}(M')$ and, since $K\subseteq K_i$, there are nonsingular matrices $N_i$, $i=1,2$, with $M'=M_iN_i$. Furthermore, ${\boldsymbol{b}}\in K$ implies that there is ${\boldsymbol{b}}'\in{\mathbb{Z}}^n$ with ${\boldsymbol{b}}=M'{\boldsymbol{b}}'=M_iN_i{\boldsymbol{b}}'$. Hence $ {\boldsymbol{b}}_i=M_i^{-1}{\boldsymbol{b}}=N_i{\boldsymbol{b}}' $ and \begin{equation*}
F({\boldsymbol{x}}^{MM'},z)
=F_1 ({\boldsymbol{x}}^{M_1N_1},z)F_2({\boldsymbol{x}}^{M_2N_2},z ) =\prod_{i=1}^2\prod_{j=1}^{r_i} F_{i,j} ({\boldsymbol{x}}^{N_i},z ). \end{equation*} Set $M''=MM'$, $G_{i,j}=F_{i,j} ({\boldsymbol{x}}^{B_i},z )$ and consider the decomposition $$ F ({\boldsymbol{x}}^{M''},z )=\prod_{i=1}^2\prod_{j=1}^{r_i} G_{i,j}. $$ We have ${\boldsymbol{a}}=M{\boldsymbol{b}}=M''{\boldsymbol{b}}'$ and $$ G_{i,j}(t^{{\boldsymbol{b}}'},z)=F_{i,j} (t^{B_i{\boldsymbol{b}}'},z )=F_{i,j} (t^{{\boldsymbol{b}}_i} ,z) $$ is irreducible in $\C(t)[s^{\pm1}]$ for all $i,j$. The statement follows by taking $\Omega^{0}$ as any finite set containing all the matrices of the form $MM'$ for $M\in \Phi$, and $\Gamma$ as in \eqref{eq:14}. \end{proof}
We conclude by giving the proof of our main result.
\begin{proof}[Proof of Theorem \ref{thm:2}]
We proceed by induction on $n$. When $n=0$ the statement is trivial,
and so we assume that $n\ge 1$. Let $F\in \C[{\boldsymbol{x}}^{\pm1},z^{\pm1}]$
and write
\begin{displaymath}
F=CF'
\end{displaymath}
with $C\in \C[{\boldsymbol{x}}^{\pm1}]$ and $F'\in \C[{\boldsymbol{x}}^{\pm1},z^{\pm1}]$
without nontrivial factors in $\C[{\boldsymbol{x}}^{\pm1}]$. By Lemma
\ref{specialization}\eqref{item:16}, there is a finite subset
$\Delta\subset {\mathbb{Z}}^{n}$ such that $C(t^{{\boldsymbol{b}}})\ne 0$ for all
${\boldsymbol{b}}\in {\mathbb{Z}}^{n}$ with $\langle {\boldsymbol{c}},{\boldsymbol{b}}\rangle\ne 0$ for all
${\boldsymbol{c}}\in \Delta$. Let also $\Omega^{0}\subset {\mathbb{Z}}^{n\times n}$ and
$\Gamma\in {\mathbb{Z}}^{n}$ be the finite subsets given by Theorem
\ref{thm:1} applied to $F'$.
Let ${\boldsymbol{a}}\in {\mathbb{Z}}^{n}$. When $\langle {\boldsymbol{c}},{\boldsymbol{a}}\rangle\ne 0$ for all
${\boldsymbol{c}}\in \Gamma\cup \Delta$, Theorem \ref{thm:1}\eqref{item:18}
implies the statement, provided that we choose any finite subset
$ \Omega\subset {\mathbb{Z}}^{n\times n}$ containing $\Omega^{0}$.
Otherwise, suppose that there is ${\boldsymbol{c}}\in \Gamma\cup \Delta$ with
$\langle {\boldsymbol{c}},{\boldsymbol{a}}\rangle = 0$. If $C(t^{{\boldsymbol{a}}},z)=0$, we add to the
finite set $\Omega$ the matrix $M\in {\mathbb{Z}}^{n\times n}$ made by adding
to $n-1$ zero columns to the vector ${\boldsymbol{a}}$. Otherwise, choose a
matrix $L\in {\mathbb{Z}}^{n\times (n-1)}$ defining a linear map
$ {\mathbb{Z}}^{n-1}\rightarrow {\mathbb{Z}}^{n}$ whose image is the submodule
$c^{\bot}\cap {\mathbb{Z}}^{n}$, and a vector ${\boldsymbol{d}}\in {\mathbb{Z}}^{n-1}$ with
${\boldsymbol{a}}=L{\boldsymbol{d}}$. Let ${\boldsymbol{u}}=(u_{1},\dots, u_{n-1})$ be a set of $n-1$
variables and set
\begin{displaymath}
G=F'({\boldsymbol{u}}^{L})\in \C[{\boldsymbol{u}}^{\pm1},z^{\pm1}] .
\end{displaymath}
By the inductive hypothesis, there is a finite subset
$\Omega_{{\boldsymbol{c}}}\subset{\mathbb{Z}}^{(n-1)\times (n-1)}$ satisfying the
statement of Theorem \ref{thm:2} applied to this Laurent
polynomial. In particular, there are $N\in \Omega_{{\boldsymbol{c}}}$ and
${\boldsymbol{e}}\in {\mathbb{Z}}^{n-1}$ with ${\boldsymbol{d}}=N{\boldsymbol{e}} $ such that, for an irreducible
factor $Q$ of $G({\boldsymbol{u}}^{N},z)$, we have that $Q(t^{{\boldsymbol{e}}},z)$ is, as a
Laurent polynomial in $\C(t)[z^{\pm1}]$, either a unit or an
irreducible factor of $G( t^{{\boldsymbol{d}}},z)$.
We have that $G({\boldsymbol{u}}^{N},z)=F'({\boldsymbol{u}}^{LN},z)$, and so $Q$ is an
irreducible factor of this latter Laurent polynomial. Moreover,
${\boldsymbol{a}}=LN {\boldsymbol{e}}$. Enlarging the matrix $LN\in {\mathbb{Z}}^{n\times (n-1)}$ to
a matrix $M\in {\mathbb{Z}}^{n\times n}$ by adding to it a zero column at the
end, and similarly enlarging the vector ${\boldsymbol{e}}$ to a vector
${\boldsymbol{b}}\in {\mathbb{Z}}^{n}$ by adding to it a zero entry at the end, the
previous equalities are preserved with $M$ and ${\boldsymbol{b}}$ in the place
of $LN$ and ${\boldsymbol{e}}$. Hence, $ {\boldsymbol{a}}=M{\boldsymbol{b}} $ and, if
$Q(x_{1},\dots, x_{n-1})$ is an irreducible factor of
$F'({\boldsymbol{x}}^{M},z)$, then $Q(t^{{\boldsymbol{e}}},z)=Q(t^{{\boldsymbol{b}}},z)$ is, as a
Laurent polynomial in $\C(t)[z^{\pm1}]$, either a unit or an
irreducible factor of $G( t^{{\boldsymbol{d}}},z)=F(t^{{\boldsymbol{a}}},z)$.
The statement then follows by also also adding to $\Omega$ all the
matrices $M\in {\mathbb{Z}}^{n\times n}$ constructed in this way. \end{proof}
\begin{remark}
\label{rem:10}
In the setting of Theorem \ref{thm:2}, the bivariate Laurent
polynomials $F_{{\boldsymbol{a}}}$ can be defined as the pullback of the
multivariate Laurent polynomial $F$ by the 2-parameter monomial map
$(t,z)\mapsto (t,z)^{A}$ given by the matrix \begin{displaymath}
A=
\begin{pmatrix}
0 &1\\ a_{1} & 0\\ \vdots &\vdots\\ a_{n} & 0
\end{pmatrix} \in {\mathbb{Z}}^{(n+1)\times 2}. \end{displaymath} In Conjecture \ref{conj:3} applied to $F$ and $k=2$, one can consider \emph{all} matrices in $ {\mathbb{Z}}^{(n+1)\times 2}$, and so its setting is more general than that of Theorem \ref{thm:2}.
On the other hand, the conclusion of Conjecture \ref{conj:3} in this situation is slightly weaker than that of Theorem \ref{thm:2}, since it does not give the irreducible factorization of the $F_{{\boldsymbol{a}}}$ in $\C(t)[z^{\pm1}]$, but rather its irreducible factorization modulo the Laurent polynomials of the form $f(t^{d_{1}}z^{d_{2}})$ for a univariate $f$ and $d_{1},d_{2}\in {\mathbb{Z}}$. \end{remark}
\end{document} | arXiv |
We develop a family of infinite-dimensional (non-parametric) manifolds of probability measures. The latter are defined on underlying Banach spaces, and have densities of class $C_b^k$ with respect to appropriate reference measures. The case $k=\infty$, in which the manifolds are modelled on Fréchet spaces, is included. The manifolds admit the Fisher-Rao metric and, unusually for the non-parametric setting, Amari's $\alpha$-covariant derivatives for all $\alpha\in\R$. By construction, they are $C^\infty$-embedded submanifolds of particular manifolds of finite measures. The statistical manifolds are dually ($\alpha=\pm 1$) flat, and admit mixture and exponential representations as charts. Their curvatures with respect to the $\alpha$-covariant derivatives are derived. The likelihood function associated with a finite sample is a continuous function on each of the manifolds, and the $\alpha$-divergences are of class $C^\infty$. | CommonCrawl |
\begin{document}
\title{On the Axiomatizability of Quantitative Algebras}
\author[R.\ Mardare]{Radu Mardare}
\address{Dept.\ of Computer Science, Aalborg University, Denmark}
\email{[email protected]}
\author[P.\ Panangaden]{Prakash Panangaden}
\address{School of Computer Science, McGill University, Canada}
\email{[email protected]}
\author[G.\ Plotkin]{Gordon Plotkin}
\address{School of Informatics, University of Edinburgh, Scotland}
\email{[email protected]}
\keywords{Universal algebra, quantitative algebra, equational logic, varieties, quasivarieties, first-order structure}
\subjclass{G.3,I.1.4,I.6.4}
\titlecomment{An earlier version of this paper appeared as...}
\begin{abstract} Quantitative algebras (QAs) are algebras over metric spaces defined by quantitative equational theories as introduced by the same authors in a related paper presented at LICS 2016. These algebras provide the mathematical foundation for metric semantics of probabilistic, stochastic and other quantitative systems. This paper considers the issue of axiomatizability of QAs. We investigate the entire spectrum of types of quantitative equations that can be used to axiomatize theories: (i) simple quantitative equations; (ii) Horn clauses with no more than $c$ equations between variables as hypotheses, where $c$ is a cardinal and (iii) the most general case of Horn clauses. In each case we characterize the class of QAs and prove variety/quasivariety theorems that extend and generalize classical results from model theory for algebras and first-order structures.
\end{abstract}
\maketitle \section{Introduction}
In~\cite{Mardare16} we introduced the concept of a quantitative equational theory in order to support a quantitative algebraic theory of effects and address metric-semantics issues for probabilistic, stochastic and quantitative theories of systems. Probabilistic programming, in particular, has become very important recently~\cite{Pfeffer16}, see, for example, the web site~\cite{probprog}. The need for semantics and reasoning principles for such languages is important as well and recently one can witness an increased interest of the research community in this topic. Equational reasoning is the most basic form of logical reasoning and it is with the aim of making this available in a metric context that we began this work.
A quantitative equational theory allows one to write equations of the form $s =_{\epsilon}t$, where $\epsilon$ is a rational number, in order to characterize metric structures in an algebraic context. We developed the analogue of universal algebras over metric spaces -- called quantitative algebras (QAs), proved analogues of Birkhoff's completeness theorem and showed that quantitative equations defined monads on metric spaces. We also presented a number of examples of interesting quantitative algebras widely used in semantics. We presented variants of barycentric algebra \cite{Stone49} that model the space of probabilistic/subprobabilistic distributions with either the Kantorovich, Wasserstein or total variation metrics; the same algebras can also be used to characterize the space of Markov processes with the Kantorovich metric. We also gave
a notion of quantitative semilattice that characterizes the space of closed subsets of an extended metric space with the Hausdorff metric. In all these examples we emphasized elegant axiomatizations characterizing these well-known metric spaces. In \cite{Bacci16} the same tools are used to provide axiomatizations for a fix-point semantics for Markov chains. Of course, some of these examples can be given by ordinary monads, as shown in~\cite{vanBreugel07,Adamek12a}, but we are aiming to fully integrate metric reasoning into equational reasoning.
What was left open in our previous work was what kinds of metric-algebraic structures could be axiomatized. This is an important issue if we want a general theory for metric-based semantics, since we will need to understand whether the class of systems of interest with their natural metrics can, in fact, be axiomatized. In the present paper, we discuss the general question of what classes of quantitative algebras can be axiomatized by quantitative equations, or by more general axioms like Horn clauses.
The celebrated Birkhoff variety theorem~\cite{Birkhoff35} states that a class of algebras is equationally definable if and only if it is closed under homomorphic images, subalgebras, and products. Many extensions have been proved for more general kinds of axioms~\cite{Adamek98} and for coalgebras instead of algebras~\cite{Adamek01,Goldblatt01}, and see \cite{Adamek10,Barr94,Manes12} for a categorical perspective. It is natural to ask if there are corresponding results for quantitative equations and quantitative algebras. Since classical equations $s=t$ define a congruence over the algebraic structure, while quantitative equations $s=_\epsilon t$ define a pseudometric coherent with the algebraic structure, the classical results do not apply directly to our case. One therefore needs fully to understand how metric structures behave equationally to answer the question. This is the challenge we take up here.
The interesting examples that we present in \cite{Mardare16} require not only axiomatizations involving quantitative equations of the form $s=_\epsilon t$, but also conditional equations, i.e., Horn clauses involving quantitative equations. Already the simple case of Horn clauses of the form $\{x_i=_{\epsilon_i}y_i \mid i \in I\}$ as hypotheses, where $x_i,y_i$ are variables only, provides interesting examples. All this forces us to develop some new concepts and proof techniques that are innovative in a number of ways.
Firstly, we show that considering a metric structure on top of an algebraic structure, which implicitly requires one to replace the concept of \textit{congruence} with a \textit{pseudometric} coherent with the algebraic structure, is not a straightforward generalization. Indeed, one can always think of a congruence $\cong$ on an algebra $\mathcal{A}$ as to the kernel of the pseudometric $p_{\cong}$ defined by $p_{\cong}(a,b)=0$ iff $a\cong b$ and $p_{\cong}(a,b)=1$ otherwise. Nevertheless, many standard model-theoretic results about axiomatizability of algebras are particular consequences of the discrete nature of this pseudometric. Many of these results fail when one takes a more complex pseudometric, even if its kernel remains a congruence.
Secondly, we show that in the case of quantitative algebras, quantitative equation-based axiomatizations behave very similarly to axiomatizations by \textit{Horn clauses} involving only quantitative equations between variables as hypotheses. And this remains true even when one allows functions of countable arity in the signature. Horn clauses of this type are directly connected to \textit{enriched Lawvere theories} \cite{Robinson02}. We give a uniform treatment of all these cases by interpreting quantitative equations as Horn clauses with empty sets of hypotheses.
We discover, in this context, a special class of homomorphisms that we call \textit{$c$-reflexive homomorphisms}, for a cardinal $c$, that play a crucial role. These homomorphisms preserve distances on selected subsets of cardinality less than $c$ of the metric space, i.e., any $c$-space in the image pulls back (modulo non-expansiveness) to one in the domain. This concept generalizes the concept of homomorphism of quantitative algebras, since any homomorphism of quantitative algebras is $1$-reflexive. The central role of $c$-reflexive homomorphisms is demonstrated by a \textit{weak universality} property, proved below.
This result also shows that the classical canonical model construction for classes of universal algebras is mathematically inadequate and works in the traditional settings only because it is, coincidentally, a model isomorphic with the more general one that we present here. However, apart from the classic settings (of universal algebras and congruences) the standard construction fails to produce a model isomorphic with the ``natural'' one and consequently, it fails to reflect the weak universality properly up to $c$-reflexive homomorphisms.
Our main result in this first part of the paper is the $c$-variety theorem for a regular cardinal $c\leq\aleph_1$: a class of quantitative algebras can be axiomatized by Horn clauses, each axiom having fewer than $c$ equations between variables as hypotheses, if, and only if, the class is closed under subobjects, products and $c$-reflexive homomorphisms. In particular, (i) the class is a $1$-variety (closed under subobjects, products and homomorphisms) iff it can be axiomatized by quantitative equations; (ii) it is an $\aleph_0$-variety iff it can be axiomatized by Horn clauses with finite sets of quantitative equations between variables as hypotheses; and (iii) it is an $\aleph_1$-variety iff it can be axiomatized by Horn clauses with countable sets of quantitative equations between variables as hypotheses. Notice that in the light of the previously mentioned relation between congruences and pseudometrics, (i) generalizes the original Birkhoff result for universal algebras. Without the concept of $c$-reflexivity, one can only state a quasi-variety theorem under the very strong assumption that
reduced products always exist, as happens, e.g., in \cite{Weaver95}.
Thirdly, we also study the axiomatizability of classes of quantitative algebras that admit Horn clauses as axioms, but which are not restricted to quantitative equations between variables as hypotheses. We prove that a class of quantitative algebras admits an axiomatization of this type, whenever it is closed under isomorphisms, subalgebras and what we call \textit{subreduced products}.
These are quantitative subalgebras of (a special type of) products of elements in the given class; however, while these products are always algebras, they are not always quantitative algebras, and this is where the new concept plays its role. This new type of closure condition allows us to generalize the usual quasivariety theorem of universal algebras.
Since all the isomorphisms of quantitative algebras are $c$-reflexive homomorphisms, and since a $c$-variety is closed under subalgebras and products, it is also closed under subreduced products, as they are quantitative subalgebras of the product. Hence, a $c$-variety is closed under these operators for any regular cardinal $c>0$ and so our quasivariety theorem extends the $c$-variety theorem further. These all are novel generalizations of the classical results.
Last, but not least, to achieve the aforementioned results for general Horn clauses, we had to generalize concepts and results from model theory of first-order structures considering first-order model theory on metric structures. Thus, we extended to the general unrestricted case the pioneering work in~\cite{BenYaakov08} devoted to continuous logic over complete bounded metric spaces. We identified the first-order counterpart of a quantitative algebra, that we call a \textit{quantitative first-order
structure}, and prove that the category of quantitative algebras is isomorphic to the category of quantitative first-order structures. We have developed \textit{first-order equational logic} for these structures and extended standard model theoretic results for quantitative first-order structures. Finally, the proof of the quasivariety theorem, which actively involves the new concept of subreduced product, is based on a more fundamental proof pattern that can be further used in model theory for other types of first-order structures. We essentially show how one can prove a quasivariety theorem for a restricted class of first-order structures that obey infinitary axiomatizations.
We have left behind an open question: the results regarding unrestricted Horn clauses have been proved under the restriction of having only finitary functions in the algebraic signature. This was required in order to use
standard model theoretic techniques. We believe that a similar result might also hold for countable functions.
\section{Preliminaries on Quantitative Algebras}
In this section we recall some basic concepts used to define the quantitative algebras, from \cite{Mardare16}, and introduce a couple of new concepts needed in our development.
\subsection{Quantitative Equational Theories}
Consider an \textit{algebraic similarity type} $\Omega$, which is a set containing function symbols of finite or countable arity (we see constants as functions of arity 0). If $c$ is the arity of the function $f$ in $\Omega$, we write $f:c\in\Omega$. \\Given a set $X$ of variables, let $\mathbb{T} X$ be the $\Omega$-\emph{term algebra} over $X$, i.e., the $\Omega$-algebra having as elements all the terms generated from the set $X$ of variables and the functions symbols in $\Omega$.
\\If $f:c\in\Omega$ and $\str{t}{i}{I}$ is an indexed family of terms with with $|I|=c$, we write $f(\str{t}{i}{I})$ for the term obtained by applying $f$ to this family of terms in the order given by $I$.
A \emph{substitution} is a function $\sigma:X\to\mathbb{T} X$. It can be canonically extended to a homomorphism of $\Omega$-algebras $\sigma \colon \mathbb{T} X\to\mathbb{T} X$ by:
$$\text{for any }f:|I|\in\Omega,~~\sigma(f(\str{t}{i}{I}))=f(\sigma(t_i)_{i\in I}).$$ In what follows $\Sigma(X)$ denotes the set of substitutions on $\mathbb{T} X$.
If $\Gamma\subseteq\mathbb{T} X$ is a set of terms and $\sigma\in\Sigma(X)$, let $\sigma(\Gamma)=\{\sigma(t)\mid t\in\Gamma\}$.
We use $\mathcal{V}(X)$ to denote the set of \emph{indexed equations} of the form $x=_\epsilon y$ for $x,y\in X$ and $\epsilon\in\mathbb{Q}_+$; similarly, we use $\mathcal{V}(\mathbb{T} X)$ to denote the set of indexed equations of the form $t=_\epsilon s$ for $t,s\in\mathbb{T} X$, $\epsilon\in\mathbb{Q}_+$. We call them \emph{quantitative equations}.
Let $\mathcal{E}(\term X)$ be the class of \emph{conditional quantitative equations} on $\mathbb{T} X$, which are constructions of the form $$\{s_i=_{\epsilon_i}t_i\mid i\in I\}\vdash s=_\epsilon t,$$ where $I$ is a countable\footnote{Anticipating the deduction system, notice that, as usual, the hypotheses containing only variables and functions that are not present in the syntax of $\phi$ can be ignored (e.g., by involving a cut-elimination rule), and since $\phi$ can only contain a countable set of terms and functions, we can safely assume that conditional equations have a countable (possibly finite or empty) set of hypotheses.} index set, $\str{s}{i}{I}, \str{t}{i}{I}\subseteq\term X$ and $s,t\in\term X$. \\If $V\vdash\phi\in\mathcal{E}(\mathbb{T} X)$, we refer to the elements of $V$ as the \emph{hypotheses} and to quantitative equation $\phi$ as the \textit{conclusion} of the conditional equation.
When the hypotheses are only quantitative equations between variables, the quantitative conditional equation is called \textit{basic conditional equation}. These play a central role in our theory and for this reason it is useful to identify a few subclasses of them.
Given a cardinal $0< c\leq\aleph_1$, a $c$-\emph{basic conditional
equation} on $\mathbb{T} X$ is a conditional quantitative equation of the form $$\{x_i=_{\epsilon_i}y_i\mid i\in I\}\vdash s=_\epsilon t,$$ where $|I|<c$, $\str{x}{i}{I}, \str{y}{i}{I}\subseteq X$ and $s,t\in\term X$.
Note that the \emph{$1$-basic conditional equations} are the conditional equations with an empty set of hypotheses, i.e., of type $\emptyset\vdash s=_\epsilon t$. We call them \emph{unconditional equations} and, for simplifying notation, we often write $\vdash s=_\epsilon t$.
The \textit{$\aleph_0$-basic conditional equations} are the conditional equations with a finite set of hypotheses, all equating variables only. We call them \textit{finitary-basic quantitative equations}.
The \textit{$\aleph_1$-basic conditional equations} are all the basic conditional equations, hence with countable (including finite, or empty) sets of equations between variables as hypotheses.
The conditional quantitative equations are used for reasoning, and to this end we define the concept of quantitative equational theory, which, as expected, will generalize the classical one, in the sense that $=_0$ is the classical term equality. However, for $\epsilon\neq 0$, $=_\epsilon$ is not an equivalence: the transitivity is replaced by a rule encoding the triangle inequality. Notice also that the rule (Arch) is infinitary and it reflects the Archimedean property of rationals. For a comprehensive study of the quantitative equational theory, see \cite{Mardare16}.
\begin{defi}[Quantitative Equational Theory] \label{def:QEtheory} A \emph{quantitative equational theory of type $\Omega$ over $X$} is a set $\mathcal{U}$ of conditional equations on $\term X$ closed under the rules stated in Table \ref{axioms}, for arbitrary $t,s,u \in \mathbb{T} X$, $\str{s}{i}{I}, \str{t}{i}{I}\subseteq\term X$, $\epsilon,\epsilon'\in\mathbb{Q}_+$, $\Gamma,\Gamma'\subseteq\mathcal{V}(\mathbb{T} X)$ and $\phi,\psi\in\mathcal{V}(\mathbb{T} X)$. \end{defi}
Given a quantitative equational theory $\mathcal{U}$ and a set $S\subseteq\mathcal{U}$, we say that $S$ is a set of axioms for $\mathcal{U}$, or $S$ axiomatizes $\mathcal{U}$, if $\mathcal{U}$ is the smallest quantitative equational theory that contains $S$.
\begin{table}
\begin{align*}
\text{\textbf{(Refl)}} \quad
& \vdash t =_0 t \,, \\
\text{\textbf{(Symm)}} \quad
& \{t=_\epsilon s\} \vdash s=_\epsilon t \,, \\
\text{\textbf{(Triang)}} \quad
& \{t =_\epsilon u, u =_{\epsilon'} s \} \vdash t =_{\epsilon+\epsilon'} s \,, \\
\text{\textbf{(Max)}} \quad
& \{t=_\epsilon s\} \vdash t=_{\epsilon+\epsilon'}s \,, \text{ for all $\epsilon'>0$} \,, \\
\text{\textbf{(Arch)}} \quad
& \text{for }\epsilon\geq 0,~~\{t=_{\epsilon'}s\mid \epsilon'>\epsilon\} \vdash t=_\epsilon s \,, \\
\text{\textbf{(NExp)}} \quad
& \text{for } f:|I|\in\Omega,~~\{t_i=_\epsilon s_i\mid i\in I\} \vdash f(\str{t}{i}{I}) =_\epsilon f(\str{s}{i}{I}) \,, \\
\text{\textbf{(Subst)}} \quad
& \text{for all $\sigma \in \Sigma(X)$, $\Gamma \vdash t =_\epsilon s$ implies $\sigma(\Gamma) \vdash \sigma(t) =_\epsilon \sigma(t)$} \,, \\
\text{\textbf{(Cut)}} \quad
& \text{if $\Gamma \vdash \psi$ for all $\psi\in\Theta$, and $\Theta \vdash t =_\epsilon s$, then $\Gamma \vdash t =_\epsilon s$} \,, \\
\text{\textbf{(Assumpt)}} \quad
& \text{If $t =_\epsilon s \in\Gamma$, then $\Gamma \vdash t =_\epsilon s$} \,.
\end{align*}
\caption{MetaAxioms}\label{axioms} \end{table}
A quantitative equational theory $\mathcal{U}$ over $\mathbb{T} X$ is \emph{inconsistent} if $\emptyset\vdash x=_0 y\in\mathcal{U}$, where $x,y\in X$ are two distinct variables. $\mathcal{U}$ is \emph{consistent} if it is not inconsistent.
\subsection{Quantitative Algebras}\label{QUA} The quantitative equational theories characterize algebras supported by metric spaces, when interpreting $s=_\epsilon t$ as "$s$ and $t$ are at most at distance $\epsilon$. We call them quantitative algebras.
\begin{defi}[Quantitative Algebra]\label{QA} Given an algebraic similarity type $\Omega$, an $\Omega$-\emph{quantitative algebra} (QA) is a tuple $\mathcal{A}=(A,\Omega^\mathcal{A},d^\mathcal{A})$, where $(A,\Omega^\mathcal{A})$ is an $\Omega$-algebra and $d^\mathcal{A}:A\times A \to \mathbb{R}_+ \cup \{\infty\}$ is a metric on $A$
(possibly taking infinite values) such that all the functions in $\Omega^\mathcal{A}$ are \emph{non-expansive}, i.e., for any $f:|I|\in\Omega$, $\str{a}{i}{I},\str{b}{i}{I}\subseteq A$, and any $\epsilon\geq 0$, if $d^\mathcal{A}(a_i,b_i)\leq\epsilon$ for all $i\in I$, then $$d^\mathcal{A}(f(\str{a}{i}{I}),f(\str{b}{i}{I}))\leq\epsilon.$$ A quantitative algebra is \textit{void} when its support is void and it is \emph{degenerate} if its support is a singleton. \end{defi}
As emphasized before, our intuition is that quantitative algebras generalize the concept of algebra and seen from this perspective, requiring that any function in the signature is non-expansive seems the natural way of defining the interaction between the support metric space and the algebraic structure. For the same reason the non-expensiveness must be preserved by homomorphisms.
\begin{defi}[Homomorphism of Quantitative Algebras] Given two quantitative algebras of type $\Omega$, $\mathcal{A}_i=(A_i,\Omega,d^{\mathcal{A}_i})$, $i=1,2$, a \emph{homomorphism of quantitative algebras} is a homomorphism $h:A_1\to A_2$ of $\Omega$-algebras, which is non-expansive, i.e., s.t., for arbitrary $a,b \in A_1$, $$d^{\mathcal{A}_1}(a,b)\geq d^{\mathcal{A}_2}(h(a),h(b)).$$ \end{defi}
Notice that identity maps are homomorphisms and that homomorphisms are closed under composition, hence quantitative algebras of type $\Omega$ and their homomorphisms form a category, written $\mathbf{QA_\Omega}$.
\textbf{Reflexive Homomorphisms.} There are some classes of specialized homomorphisms that play a central role in describing the quasivarieties of QAs. We call them \textit{reflexive homomorphisms}.
Hereafter we use $A\subseteq_c B$ for a cardinal $c>0$ to mean that $A$ is a subset of $B$ and $|A|<c$. Notice that $A\subseteq_{\aleph_0}B$ means that $A$ is a finite subset of $B$ and $A\subseteq_{\aleph_1}B$ means that $A$ is a countable (possible finite or void) subset of $B$.
\begin{defi}[Reflexive Homomorphism]\label{refhom}
Given two quantitative algebras of type $\Omega$, $\mathcal{A}_i=(A_i,\Omega,d^{\mathcal{A}_i})$, $i=1,2$, a homomorphism $f:\mathcal{A}_1\to\mathcal{A}_2$ of quantitative algebras is
\textit{$c$-reflexive}, where $c$ is a cardinal, if for any subset
$B_2\subseteq_c A_2$ there exists a set $B_1\subseteq A_1$ such that
$f(B_1)=B_2$ and $$\text{for any }a,b\in B_1,~~d^{\mathcal{A}_1}(a,b)=d^{\mathcal{A}_2}(f(a),f(b)).$$ \end{defi}
If $f:\mathcal{A}\to\mathcal{B}$ is a $c$-reflexive homomorphism, $f(A)$ is a \emph{$c$-reflexive homomorphic image} of $A$.
Note that any homomorphism of quantitative algebras is $1$-reflexive. Moreover, for $c>c'$, a $c$-reflexive homomorphism is also $c'$-reflexive.
Observe also that the restriction $f|_{B_1}:B_1\to B_2$ defined in Definition \ref{refhom} is an isometry of metric spaces. Indeed, $f|_{B_1}$ is surjective, since $f(B_1)=B_2$. It is also injective because otherwise, from $f(a)=f(b)$, we get that $d^{\mathcal{A}_2}(f(a),f(b))=0$, implying $d^{\mathcal{A}_1}(a,b)=0$; and since $d^{\mathcal{A}_1}$ is a metric, we must have $a=b$.
\textbf{Quantitative Subalgebra.} The concept of subalgebra generalizes, as expected, both the concept of $\Omega$-subalgebra and of metric subspace.
Given a quantitative algebra $\mathcal{A}=(A,\Omega,d^\mathcal{A})$, a quantitative algebra $\mathcal{B}=(B,\Omega,d^\mathcal{B})$ is a \textit{quantitative subalgebra} of $\mathcal{A}$, denoted by $\mathcal{B}\leq\mathcal{A}$, if $\mathcal{B}$ is an $\Omega$-subalgebra of $\mathcal{A}$ and for any $a,b\in B$, $d^\mathcal{B}(a,b)=d^\mathcal{A}(a,b)$.
\textbf{Direct Products of Quantitative Algebras.} Let $(\mathcal{A}_i)_{i\in I}$ be an $I$-indexed family of quantitative algebras of type $\Omega$, where $\mathcal{A}_i=(A_i,\Omega,d_i)$ for all $i\in I$. Their \emph{(direct) product}
is the quantitative algebra $\mathcal{A}=(A,\Omega,d)$ such that \begin{itemize}
\item $A=\displaystyle\prod_{i\in I}A_i$ is the direct product of the sets $A_i$, for $i\in I$;
\item for each $f:|J|\in\Omega$ and each $a_j=(b_j^i)_{i\in I}$ for $j\in
J$, $$f^\mathcal{A}(\str{a}{j}{J})=(f^{\mathcal{A}_i}(\str{b^i}{j}{J}))_{i\in I};$$
\item for $a=\str{a}{i}{I}$, $b=\str{b}{i}{I}$,
$$d(a,b)=\displaystyle\sup_{i\in I}d_i(a_i,b_i).$$ \end{itemize} The empty product $\displaystyle\prod\emptyset$ is the degenerate algebra with universe $\{\emptyset\}$.
The fact that this is a QA follows from the pointwise constructions of products in both the category of $\Omega$-algebras and in the category of metric spaces with infinite values where the product metric is the pointwise supremum. The non-expansiveness of the functions in the product algebra follows from the non-expansiveness of the functions in the components. The product quantitative algebra is written $\displaystyle\prod_{i\in I}\mathcal{A}_i$.
Direct products have \textit{projection maps} for each $k\in I$, $$\pi_k:\displaystyle\prod_{i\in I}\mathcal{A}_i\to\mathcal{A}_k,$$ defined for arbitrary $a=(a_i)_{i\in
I}\in\displaystyle\prod_{i\in I}A_i$ by $\pi_k(a)=a_k$. If none of the quantitative algebras in the family is void, the projection maps are always surjective homomorphisms of QAs.
\textbf{Closure Operators.}
It is useful in what follows to define a few operators mapping classes of QAs into classes of QAs.
\begin{defi}\label{closureop} Given a class $\mathcal K$ of quantitative algebras and a cardinal $c$, let $\mathbb I(\mathcal K)$, $\mathbb S(\mathcal K)$, $\mathbb H_c(\mathcal K)$, $\mathbb P(\mathcal K)$ and $\mathbb V_c(\mathcal K)$ be the classes of quantitative algebras defined as follows. \begin{itemize}
\item $\mathcal{A}\in\mathbb I(\mathcal K)$ iff $\mathcal{A}$ is isomorphic to some
member of $\mathcal K$;
\item $\mathcal{A}\in\mathbb S(\mathcal K)$ iff $\mathcal{A}$ is a quantitative
subalgebra of some member of $\mathcal K$;
\item $\mathcal{A}\in\mathbb H_c(\mathcal K)$ iff $\mathcal{A}$ is the $c$-reflexive
homomorphic image of some algebra in $\mathcal K$; in particular, we
denote $\mathbb H_1(\mathcal K)$ simply by $\mathbb H(\mathcal K)$ since it is the closure
under homomorphic images;
\item $\mathcal{A}\in\mathbb P(\mathcal K)$ iff $\mathcal{A}$ is a direct product of a
family of elements in $\mathcal K$;
\item $\mathbb V_c(\mathcal K)$ is the smallest class of quantitative
algebras containing $\mathcal K$ and closed under subalgebras,
direct products, and $c$-reflexive homomorphic images; such a class is
called a \textit{$c$-variety of quantitative algebras}. In
particular, for $c=1$ we also write $\mathbb V_1(\mathcal K)$ as $\mathbb V(\mathcal K)$ and call it a \textit{variety}. \end{itemize} \end{defi}
For any operators $\mathbb X,\mathbb Y\in\{\mathbb I,\mathbb S, \mathbb H_c,\mathbb P, \mathbb V_c\}$, we write $\mathbb X\mathbb Y$ for their composition; and since this composition is associative, we ignore parentheses when composing more than two operators. Furthermore, for any compositions $\mathbb X,\mathbb Y$ of these we write $\mathbb X\subseteq\mathbb Y$ if $\mathbb X(\mathcal K)\subseteq\mathbb Y(\mathcal K)$ for any class $\mathcal K$.
The next lemma establishes a series of properties of these operators, similar to the ones on classes of universal algebras.
\begin{lem}\label{operators}
The closure operators on classes of quantitative algebras enjoy the following properties:
\begin{enumerate}
\item whenever $c<c'$, $\mathbb{H}_c\subseteq \mathbb H_{c'}$;
\item whenever $c<c'$, if $\mathcal K$ is $\mathbb H_c$-closed, then it is $\mathbb H_{c'}$-closed; in particular, a $\mathbb H$-closed class is $\mathbb H_c$-closed for any $c$;
\item whenever $c<c'$, if $\mathcal K$ is $c$-variety, then it is a $c'$-variety; in particular, a variety is a $c$-variety for any $c$;
\item $\mathbb{SH}_c\subseteq\mathbb H_c\mathbb S$; in particular, $\mathbb S\mathbb H\subseteq\mathbb H\mathbb S$;
\item $\mathbb{PH}_c\subseteq\mathbb H_c\mathbb P$; in particular, $\mathbb P\mathbb H\subseteq\mathbb H\mathbb P$;
\item $\mathbb P\mathbb S\subseteq\mathbb S\mathbb P$;
\item $\mathbb H_c$, $\mathbb H$, $\mathbb S$ and $\mathbb{IP}$ are idempotent;
\item $\mathbb V_c=\mathbb H_c\mathbb {SP}$; in particular, $\mathbb V=\mathbb{HSP}$.
\end{enumerate} \end{lem}
\begin{proof} 1, 2, 3. Follow from the fact that any $c'$-reflexive homomorphism is $c$-reflexive as well.
4. Let $\mathcal{A}\in\mathbb{SH}_c(\mathcal K)$. Then, there exists $\mathcal{B}\in\mathcal K$ and a surjective $c$-reflexive homomorphism $f:\mathcal{B}\twoheadrightarrow\mathcal{C}$ such that $\mathcal{A}\leq\mathcal{C}$.
We have that $\mathcal{B}'=f^{-1}(\mathcal{A})\leq\mathcal{B}$, hence $\mathcal{B}'\in\mathbb S(\mathcal K)$ and there exists the surjective homomorphism
$f|_{\mathcal{B}'}:\mathcal{B}'\twoheadrightarrow\mathcal{A}$. Since $f$ is $c$-reflexive, also
$f|_{\mathcal{B}'}$ must be $c$-reflexive. Hence, $\mathcal{A}\in\mathbb{H}_c\mathbb S$.
5. Let $\mathcal{A}\in\mathbb{PH}_c(\mathcal K)$. Then, there exist a family $(\mathcal{B}_i)_{i\in I}\subseteq\mathcal K$ and a family of surjective $c$-reflexive homomorphisms $f_i:\mathcal{B}_i\twoheadrightarrow\mathcal{A}_i$ such that $\mathcal{A}=\displaystyle\prod_{i\in I}\mathcal{A}_i$. But then, there exists a surjective homomorphism $f:\displaystyle\prod_{i\in I}\mathcal{B}_i\twoheadrightarrow\displaystyle\prod_{i\in I}\mathcal{A}_i$ defined by $f(b)(i)=f_i(b(i))$. Moreover, since each $f_i$ is $c$-reflexive, also $f$ must be a $c$-reflexive. Hence, $\mathcal{A}\in\mathbb{H}_c\mathbb P(\mathcal K)$.
6. Let $\mathcal{A}\in\mathbb{PS}(\mathcal K)$. Then, $\mathcal{A}=\displaystyle\prod_{i\in I}\mathcal{A}_i$ for some $\mathcal{A}_i\leq\mathcal{B}_i\in\mathcal K$. But then, it is not difficult to see that $\displaystyle\prod_{i\in I}\mathcal{A}_i\leq\displaystyle\prod_{i\in I}\mathcal{B}_i$ implying $\mathcal{A}\in\mathbb{SP}(\mathcal K)$.
7. The class of $c$-reflexive homomorphisms is closed under composition. All these are trivial.
8. We obviously have $\mathbb{H}_c\mathbb V_c=\mathbb{SV}_c=\mathbb{IPV}_c=\mathbb V_c$. \\Since $\mathbb{I\subseteq V}_c$, $\mathbb{H}_c\mathbb{SP\subseteq H}_c\mathbb{SPV}_c=\mathbb V_c$.
Since $\mathbb H_c$ is idempotent, $\mathbb{H}_c(\mathbb H_c\mathbb{SP})=\mathbb H_c\mathbb{SP}$.
Applying the previous results we get $\mathbb{S(H}_c\mathbb{SP)\subseteq H}_c\mathbb{SSP=H}_c\mathbb{SP}$ and \\ $\mathbb{P(H}_c\mathbb{SP)\subseteq H}_c\mathbb{PSP\subseteq H}_c\mathbb{SPP\subseteq H}_c\mathbb{SIPIP=H}_c\mathbb{SIP\subseteq H}_c\mathbb{SH}_c\mathbb{P\subseteq H}_c\mathbb H_c\mathbb{SP=H}_c\mathbb{SP}$.
Obviously $\mathcal K\subseteq\mathbb{H}_c\mathbb{SP}(\mathcal K)$ and $\mathbb{H}_c\mathbb{SP}(\mathcal K)$ is closed under $\mathbb{H}_c$, $\mathbb{S,~ P}$. Since $\mathbb V_c(\mathcal K)$ is the smallest class containing $\mathcal K$ and closed under $\mathbb{H}_c$, $\mathbb{S,~ P}$, we get that $\mathbb V_c(\mathcal K)\subseteq\mathbb{H}_c\mathbb{SP}(\mathcal K)$. \end{proof}
\subsection{Algebraic Semantics for Conditional Quantitative Equations}
Quantitative algebras are used to interpret quantitative equational theories.
Given a quantitative algebra $\mathcal{A}=(A,\Omega^\mathcal{A},d^\mathcal{A})$ of type $\Omega$ and a set $X$ of variables, an \emph{assignment} on $\mathcal{A}$ is an $\Omega$-homomorphism $\alpha:\mathbb{T} X\to A$; it is used to interpret abstract terms in $\term X$ as concrete elements in $\mathcal{A}$. We denote by
$\mathbb{T}(X|\mathcal{A})$ the set of assignments on $\mathcal{A}$.
\begin{defi}[Satisfiability]\label{sat}
A quantitative algebra $\mathcal{A}=(A,\Omega^\mathcal{A},d^\mathcal{A})$ of type $\Omega$ under the assignment $\alpha\in\term(X|\mathcal{A})$ \emph{satisfies} a conditional quantitative equation
$\Gamma\vdash s=_\epsilon t\in\mathcal{E}(\mathbb{T} X)$, whenever
$$[d^\mathcal{A}(\alpha(t'),\alpha(s'))\leq \epsilon'\mbox{ for all
}s'=_{\epsilon'}t'\in\Gamma]~\text{ implies }~d^\mathcal{A}(\alpha(s),\alpha(t))\leq\epsilon.$$
This is denoted by $$\Gamma\models_{\mathcal{A},\alpha} s=_\epsilon t.$$
$\mathcal{A}$
\emph{satisfies} $\Gamma\vdash
s=_\epsilon t\in\mathcal{E}(\mathbb{T} X)$, written $$\Gamma\models_\mathcal{A} s=_\epsilon
t,$$ if $\Gamma\models_{\mathcal{A},\alpha} s=_\epsilon
t,$ for all assignments $\alpha\in\mathbb{T}(X|\mathcal{A})$; in this case
$\mathcal{A}$ is a \emph{model} of the conditional quantitative equation. \end{defi}
Similarly, for a set $\mathcal{U}$ of conditional quantitative equations (e.g., a quantitative equational theory), we say that $\mathcal{A}$ is a model of $\mathcal{U}$ if $\mathcal{A}$ satisfies each conditional quantitative equation in $\mathcal{U}$.
If $\mathcal K$ is a class of quantitative algebras we write $$\Gamma\models_\mathcal K s=_\epsilon t,$$ if for any $\mathcal{A}\in\mathcal K$, $\Gamma\models_\mathcal{A} s=_\epsilon t$ . Furthermore, if $\mathcal{U}$ is a quantitative equational theory we write $$\mathcal K\models\mathcal{U}$$ if all algebras in $\mathcal K$ are models for $\mathcal{U}$.
For the case of unconditional equations, note that the left-hand side of the implication that defines the satisfiability relation in Definition \ref{sat} is vacuously satisfied. For these, instead of $\emptyset\models_{\mathcal{A},\alpha} s=_\epsilon t$ and $\emptyset\models_\mathcal{A} s=_\epsilon t$ we will often write $\mathcal{A},\alpha\models s=_\epsilon t$ and $\mathcal{A}\models s=_\epsilon t$ respectively. Furthermore, for a class $\mathcal K$ of quantitative algebras, $$\mathcal K\models s=_\epsilon t$$ denotes that $\mathcal{A}\models s=_\epsilon t$ for all $\mathcal{A}\in\mathcal K$.
With these concepts in hand we can proceed and define equational classes.
\begin{defi}[Equational Class of Quantitative Algebras]
For a signature $\Omega$ and a set $\mathcal{U}\subseteq\mathcal{E}(\term X)$ of conditional quantitative equations over the $\Omega$-terms $\mathbb{T} X$, the
\emph{conditional equational class induced by $\mathcal{U}$} is the class of quantitative
algebras of signature $\Omega$ satisfying $\mathcal{U}$. \end{defi}
We denote this class, as well as the full subcategory of $\Omega$-quantitative algebras satisfying $\mathcal{U}$, by $\mathbb K(\Omega,\mathcal{U})$. We say that a class of algebras that is a conditional equational class is \emph{conditional-equationally definable}.
If $S$ is an axiomatization for $\mathcal{U}$, the equational class induced by $\mathcal{U}$ coincides with the equational class induced by $S$.
\begin{lem}\label{subalgebras}
Given a set $\mathcal{U}$ of conditional quantitative equations of type $\Omega$ over $\term X$, $\mathbb{K}(\Omega,\mathcal{U})$ is closed under taking isomorphic images and subalgebras. Consequently, if $\mathcal K$ is a class of quantitative algebras over $\Omega$, then $\mathcal K$, $\mathbb I(\mathcal K)$ and $\mathbb S(\mathcal K)$ satisfy the same conditional quantitative equations. \end{lem}
\begin{proof} The closure w.r.t. isomorphic images derives trivially from Definition \ref{sat}. We prove now the closure under subalgebras.
Let $\mathcal{A}\in\mathbb{K}(\Omega,\mathcal{U})$ and $\mathcal{B}\leq\mathcal{A}$. We prove that $\mathcal{B}\in\mathbb{K}(\Omega,\mathcal{U})$.
Since $\mathcal{B}\leq\mathcal{A}$, $id_\mathcal{B}:B\to A$ defined by $id_\mathcal{B}(b)=b$ is a morphism of quantitative algebras.
Suppose that $\{s_i=_{\epsilon_i}t_i\mid i\in I\}\vdash s=_e t\in\mathcal{U}$. Hence, $$\{s_i=_{\epsilon_i}t_i\mid i\in I\}\models_\mathcal{A} s=_e t\in\mathcal{U},$$ meaning that for any $\alpha\in\mathbb{T}(X|\mathcal{A})$, $$[d^\mathcal{A}(\alpha(s_i),\alpha(t_i))\leq\epsilon_i~\text{ for all }~i\in I]~~\text{ implies }~~ d ^\mathcal{A}(\alpha(s),\alpha(t)\leq e.$$
Consider an arbitrary $\alpha\in\mathbb{T}(X|\mathcal{B})$ and note that $\alpha\in\mathbb{T}(X|\mathcal{A})$ as well.
Suppose that $[d^\mathcal{B}(\alpha(s_i),\alpha(t_i))\leq\epsilon_i~\text{ for all }~i\in I]$. This is equivalent to $$[d^\mathcal{A}(\alpha(s_i),\alpha(t_i))\leq\epsilon_i~\text{ for all }~i\in I].$$ But then, we also have $d^\mathcal{A}(\alpha(s),\alpha(t))\leq e$. Hence, $d^\mathcal{B}(\alpha(s),\alpha(t))\leq e$. \end{proof}
\section{The Variety Theorem for Basic Conditional Equations}
In this section we focus on the quantitative equational theories that admit an axiomatization containing only basic conditional equations, i.e., conditional equations of type $$\{x_i=_{\epsilon_i}y_i\mid i\in I\}\vdash s=_\epsilon t,$$ for $x_i,y_i\in X$, $s,t\in\mathbb{T} X$ and $\epsilon_i,\epsilon\in\mathbb{Q}_+$. We shall call such a theory \textit{basic equational theory}.
For a cardinal $c\leq\aleph_1$, a basic equational theory is a
\textit{$c$-basic equational theory} if it admits an axiomatization containing only $c$-basic conditional equations, i.e., of type $$\{x_i=_{\epsilon_i}y_i\mid i\in I\}\vdash s=_\epsilon t,$$ for $|I|<c$, $x_i,y_i\in X$, $s,t\in\mathbb{T} X$ and $\epsilon_i,\epsilon\in\mathbb{Q}_+$.
An $\aleph_0$-basic equational theory is called a \textit{finitary-basic
equational theory}; it admits an axiomatization containing only finitary-basic conditional equations, i.e., of type $$\{x_i=_{\epsilon_i}y_i\mid i\in 1,..,n\}\vdash s=_\epsilon t,$$ for $n\in\mathbb{N}$, $x_i,y_i\in X$, $s,t\in\mathbb{T} X$ and $\epsilon_i,\epsilon\in\mathbb{Q}_+$.
A $1$-basic equational theory is called an \textit{unconditional equational
theory}; it admits an axiomatization containing only unconditional equations of type $\emptyset\vdash s=_\epsilon t,$ for $s,t\in\mathbb{T} X$ and $\epsilon\in\mathbb{Q}_+$.
\subsection{Closure under Products and Homomorphisms}
The basic equational theories are special since they guarantee, for their equational class, the closure under direct products, as the following lemma states.
\begin{lem}\label{products}
If $\mathcal{U}$ is a basic equational theory (in particular, finitary-basic or unconditional), then $\mathbb{K}(\Omega,\mathcal{U})$ is closed under direct products. \end{lem}
\begin{proof}
Assume that $(\mathcal{A}_i)_{i\in I}\subseteq\mathbb{K}(\Omega,\mathcal{U})$. We know that since $\mathcal{U}$ is a basic theory, it exists an axiomatization for $\mathcal{U}$ containing only basic quantitative equations. It is sufficient to prove that whenever all $\mathcal{A}_i$ satisfy a basic quantitative equation, this is also satisfied by $\displaystyle\prod_{i\in I}\mathcal{A}_i$.
\[ \begin{tikzcd} &\term X \arrow[ld,swap,"\alpha"] \arrow[d,"\pi_j\circ\alpha"]& \\ \displaystyle\prod_{i\in I}\mathcal{A}_i \arrow[r,"\pi_j"]& \mathcal{A}_j \end{tikzcd} \]
Observe, for the begining, that for any assignment $\alpha:\mathbb{T} X\to\displaystyle\prod_{i\in I}\mathcal{A}_i$ and any $j\in I$, $\pi_j\circ\alpha\in\term(X|\mathcal{A}_j)$ is an assignment in $\mathcal{A}_j$.
Consider now an arbitrary basic quantitative equation $$\{x_i=_{\epsilon_i}y_i\mid i\in I\}\vdash s=_\epsilon t\in\mathcal{U}.$$ Suppose that for all $i\in I$, $$\{x_i=_{\epsilon_i}y_i\mid i\in I\}\models_{\mathcal{A}_i} s=_\epsilon t.$$ This means that for any assignment, in particular for $\pi_j\circ\alpha$, we have $$d_j(\pi_j(\alpha(x_i)),\pi_j(\alpha(y_i)))\leq\epsilon_i\text{ for all }i\in I~\text{ implies }~d_j(\pi_j(\alpha(s)),\pi_j(\alpha(t)))\leq\epsilon.$$
Denote by $d$ the product metric and suppose that for an arbitrary assignment $\alpha\in\term(X|\displaystyle\prod_{i\in I}\mathcal{A}_i)$, $$d(\alpha(x_i),\alpha(y_i))\leq\epsilon_i\text{ for all }i\in I.$$ Hence, $$\sup\{d_j(\pi_j(\alpha(x_i)),\pi_j(\alpha(y_i)))\mid\j\in I\}\leq\epsilon_i\text{ for all }i\in I,$$ implying further that for each $j\in I$, $$d_j(\pi_j(\alpha(x_i)),\pi_j(\alpha(y_i)))\leq\epsilon_i\text{ for all }i\in I.$$ But then, the hypothesis guarantees that for any $j\in I$, $$d_j(\pi_j(\alpha(s)),\pi_j(\alpha(t)))\leq\epsilon,$$ equivalent to $$\sup\{d_j(\pi_j(\alpha(s)),\pi_j(\alpha(t)))\mid j\in I\}\leq\epsilon.$$ Hence, $$d(\alpha(s),\alpha(t))\leq\epsilon.$$ In conclusion, all the axioms of $\mathcal{U}$ (which are basic quantitative equations) must be satisfied by $\displaystyle\prod_{i\in I}\mathcal{A}_i$ implying that $\displaystyle\prod_{i\in I}\mathcal{A}_i\models\mathcal{U}$. \end{proof}
The $c$-reflexive homomorphisms play a central role in characterizing the basic equational theories in the case of the regular cardinals\footnote{
The regular cardinals are the cardinals that cannot be obtained by using
arithmetic involving smaller cardinals. Thus, for example, $23$ is not a
regular cardinal but $1$, $\aleph_0$ or $\aleph_1$ are, because none of
them can be written as a smaller sum of smaller cardinals.}. In fact, because our signature admits only functions of countable (including finite) arities, we will only focus on three regular cardinals: $1$, $\aleph_0$ and $\aleph_1$.
The next lemma relates the classes of quantitative algebras that admit $c$-basic quantitative equational axiomatizations to their closure under $c$-reflexive homomorphisms.
\begin{lem}\label{homoimg}
If $\mathcal{U}$ is a $c$-basic equational theory, where $c$ is a non-null regular
cardinal, then $\mathbb{K}(\Omega,\mathcal{U})$ is closed under $c$-reflexive homomorphic images.
In particular,
\begin{itemize}
\item if $\mathcal{U}$ is an unconditional equational theory, then $\mathbb{K}(\Omega,\mathcal{U})$ is closed under homomorphic images;
\item if $\mathcal{U}$ is a finitary-basic equational theory, then $\mathbb{K}(\Omega,\mathcal{U})$ is closed under $\aleph_0$-reflexive homomorphic images;
\item if $\mathcal{U}$ is a basic equational theory, then $\mathbb{K}(\Omega,\mathcal{U})$ is closed under
$\aleph_1$-reflexive homomorphic images.
\end{itemize} \end{lem}
\begin{proof} Let $\mathcal{A}\in\mathbb{K}(\Omega,\mathcal{U})$, where $\mathcal{U}$ is a $c$-basic equational theory, $f:\mathcal{A}\rightarrow\mathcal{B}$ a $c$-reflexive homomorphism for $\mathcal{B}=f(\mathcal{A})$. Since $f$ is a homomorphism, $\mathcal{B}$ is a quantitative algebra and $f:\mathcal{A}\twoheadrightarrow\mathcal{B}$ is obviously surjective.
Let $\{x_i=_{\epsilon_i}y_i\mid i\in I\}\vdash s=_\epsilon t\in\mathcal{U}$ be a $c$-basic quantitative equation (hence $|I|< c$) satisfied by $\mathcal{A}$. Assume that the terms $s$ and $t$ depend on (a subset of) the set $$\{x_i, y_i\mid i\in I\}\cup\{z_j\mid j\in J\}\subseteq X.$$ We denote this by $s(\str{x}{i}{I},\str{y}{i}{I},\str{z}{j}{J})$ and $t(\str{x}{i}{I},\str{y}{i}{I},\str{z}{j}{J})$. Here, the $z_j$ are variables that may occur in the terms $s,t$ but are not among the variables that occur in the left-hand side of the basic inference.
Since $$\{x_i=_{\epsilon_i}y_i\mid i\in I\}\models_\mathcal{A} s(\str{x}{i}{I},\str{y}{i}{I},\str{z}{j}{J})=_\epsilon t(\str{x}{i}{I},\str{y}{i}{I},\str{z}{j}{J}),$$
for any assignment $\alpha\in\mathbb{T}(X|\mathcal{A})$, $[d^\mathcal{A}(\alpha(x_i),\alpha(y_i))\leq \epsilon_i~\text{ for all }~i\in I]$ implies $$d^\mathcal{A}(s(\alpha(x_i))_{i\in I},(\alpha(y_i))_{i\in I},\alpha(z_j)_{j\in J}),t((\alpha(x_i))_{i\in I},(\alpha(y_i))_{i\in I},\alpha(z_j)_{j\in J})\leq\epsilon.$$
Suppose there exists $\beta\in\mathbb{T}(X|\mathcal{B})$ such that $[d^\mathcal{B}(\beta(x_i),\beta(y_i))\leq \epsilon_i~\text{ for all }~i\in I]$.
Let $(a_i)_{i\in I}, (b_i)_{i\in I}, (c_j)_{j\in J}\subseteq\mathcal{B}$ s.t. $\beta(x_i)=a_i$, $\beta(y_i)=b_i$ and $\beta(z_j)=c_j$.
Since $c$ is a regular cardinal, hence closed under union, $\{a_i,b_i\mid i\in I\}\subseteq_c\mathcal{B}$. Because $f$ is $c$-reflexive, there exist \\$m_i\in f^{-1}(a_i),~ n_i\in f^{-1}(b_i)\in\mathcal{A}$ for each $i\in I$, such that $$d^\mathcal{A}(m_i,n_i)=d^\mathcal{B}(a_i,b_i).$$ Since $f$ is surjective there exist $u_j\in f^{-1}(c_j)$ for all $j\in J$.
Let us write $s_A$ for the element of $\mathcal{A}$ obtained by substituting $m_i$ for the $x_i$, $n_i$ for the $y_i$ and $u_j$ for the $z_j$; similarly we write $t_A$. We write $s_B$ for the element of $\mathcal{B}$ obtained by substituting $a_i$ for the $x_i$, $b_i$ for the $y_i$ and $c_j$ for the $z_j$.
Now the algebra $\mathcal{A}$ satisfies the basic quantitative equation, so using the substitution that produces $s_A$ and $t_A$ we conclude that $d^{\mathcal{A}}(s_A,t_A) \leq \epsilon$.
The homomorphism $f$ maps $s_A$ to $s_B$ and $t_A$ to $t_B$ and being non-expansive we conclude that $d^{\mathcal{B}}(s_B,t_B)\leq \epsilon$.
This proves that $\mathcal{B}$ also satisfies the basic quantitative equation.
\end{proof}
Putting together the results of Lemma \ref{subalgebras}, Lemma \ref{products} and Lemma \ref{homoimg}, we get the following result that emphasize the role of the $c$-basic quantitative equations for $c$-varieties.
\begin{cor}\label{satbasic}
Let $\mathcal K$ be a class of quantitative algebras over the same
signature and $c\leq\aleph_1$ a regular non-null cardinal. Then $\mathcal K$,
$\mathbb P(\mathcal K)$,
$\mathbb
H_c(\mathcal K)$ and $\mathbb V_c(\mathcal K)$ satisfy all the same
$c$-basic conditional equations. \end{cor}
\subsection{Canonical Model and Weak Universality} In this subsection we give the quantitative analogue of the canonical model construction and prove weak universality. Before we begin the detailed arguments, we note a few points. In the original variety theorem for universal algebras one proceeds by looking at all congruences on the term algebra and quotienting by the coarsest. This strategy does not work in the present case. We need to consider the pseudometrics induced by all assignments of variables; next, instead of quotienting by the kernel of the coarsest pseudometric, as the analogy with the usual case would suggest, we need to take the product of the quotient algebras indexed by these pseudometrics. We note that this is indeed a generalization of the non-quantitative case where, coincidentally, this product algebra is isomorphic to the quotient algebra by the coarsest congruence. However, our proof here shows that the natural construction that guarantees the weak universality, even when one considers reflexive homomorphisms, is the product of the quotient algebras.
Consider, as before, an algebraic similarity type $\Omega$ and a set $X$ of variables.
Let $\mathcal P_{\mathbb{T} X}$ be the set of all pseudometrics
$p:\mathbb{T} X^2\to\mathbb{R}_+ \cup \{\infty\}$ such that all the functions in $\Omega$ are non-expansive with respect to $p$. For arbitrary $p\in\mathcal P_{\mathbb{T} X}$, let $$\mathbb{T} X|_p=(\mathbb{T} X|_{ker(p)},\Omega,p)$$ be the quantitative algebra obtained by taking the quotient of $\mathbb{T} X$ with respect to the congruence relation\footnote{The
non-expansiveness of $p$ w.r.t. all the functions in $\Omega$ guarantees
that $ker(p)$ is a congruence with respect to $\Omega$.} $$ker(p)=\{(s,t)\in\mathbb{T} X^2\mid p(s,t)=0\}.$$
Let $\mathcal K$ be a family of quantitative algebras of type
$\Omega$ and $$\mathcal {P_K}=\{p\in\mathcal P_{\mathbb{T} X}\mid \mathbb{T} X|_p\in\mathbb{IS}(\mathcal K)\}.$$
We begin by showing that $\mathcal{P_K}\neq\emptyset$ whenever $\mathcal K\neq\emptyset$.
Consider an algebra $\mathcal{A}\in\mathcal K$, let $\alpha\in\term(X|\mathcal{A})$ be an arbitrary assignment and $[\alpha]:\term X^2\to\mathbb{R}_+ \cup \{\infty\}$ a pseudometric defined for arbitrary $s,t\in \term X$ by $$[\alpha](s,t)=\inf\{\epsilon\mid\mathcal{A},\alpha\models s=_\epsilon t\}.$$
\begin{lem}\label{alpha-pseudo}
If $\mathcal{A}\in\mathcal K$ and $\alpha\in\term(X|\mathcal{A})$, then
$[\alpha]\in\mathcal{P_K}$. Moreover, $\term X|_{[\alpha]}$ is a quantitative
algebra isomorphic to $\alpha(\term X)$. \end{lem}
\begin{proof}
The fact that $[\alpha]$ is a pseudometric follows directly from the
algebraic semantics.
Let $f:|I|\in\Omega$ and $\str{s}{i}{I},\str{t}{i}{I}\subseteq\term X$.
Assume that $[\alpha](s_i,t_i)\leq\epsilon$ for all $i\in I$. This means
that for each $i\in I$, $\mathcal{A},\alpha\models s_i=_\delta t_i$ for any
$\delta\in\mathbb{Q}_+$ with $\delta\geq\epsilon$. The soundness of (NExp)
provides $\mathcal{A},\alpha\models f(\str{s}{i}{I})=_\delta f(\str{t}{i}{I})$,
i.e., $[\alpha](f(\str{s}{i}{I}),f(\str{t}{i}{I}))\leq\delta$ for any
$\delta\geq\epsilon$. And this proves that $[\alpha]\in\mathcal{P}_{\term X}$.
We know that $\alpha:\term X\to\mathcal{A}$ is a homomorphism of quantitative algebras,
hence $\alpha(\term X)\leq\mathcal{A}$ and $\hat\alpha:\term X\to\alpha(\term X)$ defined by
$\hat\alpha(s)=\alpha(s)$ for any $s\in\term X$ is a surjection. Since from
the way we have defined $[\alpha]$ we have
that $$\hat\alpha(s)=\hat{\alpha}(t)~~\text{ iff }~~[\alpha](s,t)=0,$$ we
obtain that the map $\ol{\alpha}:\term X|_{[\alpha]}\to\alpha(\term X)$ defined by
$\ol{\alpha}(s|_{[\alpha]})=\alpha(s)$ for any $s\in\term X$, where
$s|_{[\alpha]}$ denotes the $ker([\alpha])$-congruence class of $s$, is a QAs
isomorphism. \end{proof}
The previous lemma states that for any algebra $\mathcal{A}\in\mathcal K$ and any assignment $\alpha\in\term(X|\mathcal{A})$,
$$\term X|_{[\alpha]}\simeq\alpha(\term X)\leq\mathcal{A}.$$
Since a consequence of it is $\mathcal{P_K}\neq\emptyset$ whenever $\mathcal K\neq\emptyset$, we can define a pointwise supremum over the elements in $\mathcal{P_K}$: $$d^{\mathcal K}(s,t)=\displaystyle\sup_{p\in\mathcal
P_{\mathcal K}}p(s,t),\text{ for arbitrary }s,t\in\term X.$$ It is not difficult to notice that, $d^{\mathcal K}\in\mathcal P_{\mathbb{T} X}$.
Let $\md X=(\displaystyle\prod_{p\in\mathcal{P_K}}\term X|_p,\Omega,d^\mathcal K)$ be the product quantitative algebra with the index set $\mathcal{P_K}$.
For arbitrary $s\in\term X$, let $\ang s\in\md X$ be the element such that for any $p\in\mathcal{P_K}$, $\pi_p(\ang s)=s|_p$, where $s|_p\in\term X|_p$ denotes the $ker(p)$-equivalence class of $s$.
Now note that, if $\mathcal K$ is a class of quantitative algebras of the same type containing non-degenerate elements, then the map $\gamma:\term X\to\md X$ defined by $\gamma(t)=\ang t$ for any $t\in\term X$ is an injective homomorphism of $\Omega$-algebras.
\begin{lem}\label{l4} If $\mathcal K$ is a non-trivial class of quantitative algebras of the same type, the map $\gamma:\term X\to\md X$ defined by $\gamma(t)=\ang t$ for any $t\in\term X$ is an injective homomorphism of $\Omega$-algebras. \end{lem}
\begin{proof} Since for any $p\in\mathcal{P_K}$, $ker(p)$ is a congruence and $ker(d^\mathcal K)=\displaystyle\bigcap_{p\in\mathcal{P_K}}ker(p)$, $\gamma$ is obviously a homomorphism of $\Omega$-algebras.
We prove now that it is injective. In order to have that for two distinct terms $s,t\in\term X$ we have $\ang s=\ang t$, we need that for any $p\in\mathcal{P_K}$, $s|_p=t|_p$. Since for any $p\in\mathcal{P_K}$, $ker(p)$ is a congruence, this will only happen if there exist two distinct variables $x,y\in X$ such that $\ang x=\ang y$.
Note that if $p\in\mathcal{P_K}$ and $\sigma:X\to X$ is a bijection, then $\sigma p$ defined by $\sigma p(s,t)=p(\sigma(s),\sigma(t))$ is an element of $\mathcal{P_K}$ as well and $\term X|_p\simeq\term X|_{\sigma p}$. With this observation we can conclude that if there exist two distinct variables $x,y\in X$ such that $\ang x=\ang y$, then for any two distinct variables $u,v\in X$ we have $\ang u=\ang v$, which implies that $\mathcal K$ only contains degenerate algebras, a contradiction. \end{proof}
In order to state now the weak universality property for a class $\mathcal K$ of quantitative algebras, we need firstly to identify a cardinal that plays a key role in our statement as an upper bound for the reflexive homomorphisms. We shall denote it by $r(\mathcal K)$:
$$\begin{array}{ll} r(\mathcal K)= & \left\{ \begin{array}{ll}
\aleph_1 & \textrm{if }\exists\mathcal{A}\in\mathcal K,~|\mathcal{A}|^+\geq\aleph_1\\
\sup\{|\mathcal{A}|^+\mid\mathcal{A}\in\mathcal K\} & \textrm{otherwise }\\ \end{array}\right. \\ \end{array}$$
where $|\mathcal{A}|$ denotes of the cardinal of the support set of $\mathcal{A}$ and $c^+$ denotes the successor of the cardinal $c$.
The following theorem is a central result of this paper. One might be tempted to just use a quotient by $ker(d^{\mathbb{K}})$ but in that case the homomorphism that one gets by weak universality does not satisfy the $c$-reflexive condition. \begin{thm}[Weak Universality]\label{weakuniv}
Consider a class $\mathcal K$ of quantitative algebras containing non-degenerate elements. For any $\mathcal{A}\in\mathcal K$ and any map $\alpha: X\to\mathcal{A}$ there exists a
$r(\mathcal K)$-reflexive homomorphism $\beta:\md X\to\mathcal{A}$ such
that $$\text{ for any }x\in X,~\beta(\ang x)=\alpha(x).$$
\end{thm}
\begin{proof}
Let $\mathbf{QA_\Omega}$ be the category of $\Omega$-quantitative algebras.
The map $\alpha:X\to\mathcal{A}$ can be canonically extended to an
$\Omega$-homomorphism $\hat{\alpha}:\term X\to\mathcal{A}$.
Let $\gamma:\term X\hookrightarrow\md X$ be the aforementioned injective
homomorphism of $\Omega$-algebras.
From Lemma \ref{alpha-pseudo} we know that $\term
X|_{[\hat\alpha]}\simeq\hat\alpha(\term X)\leq\mathcal{A}.$ So, we consider the
projection $\pi_{[\hat\alpha]}:\md X\twoheadrightarrow\term X|_{[\hat\alpha]}$
which is a surjective morphism of quantitaive algebras.
Let $\ol\alpha:\term X|_{[\hat\alpha]}\to\hat\alpha(\term X)$ be the
isomorphism of quantitaive algebras defined in (the proof of)
Lemma \ref{alpha-pseudo}.
These maps give us the following commutative diagram.
\[
\begin{tikzcd}
&\text{in }\mathbf{Set}&&&\text{in }\mathbf{QA_\Omega}\\[-4ex]
X \arrow[r, hook, "id_X"] \arrow[d, swap, "\alpha"] &
\term X\arrow[r, hook, "\gamma"] \arrow[dl,swap,"\hat\alpha"]
& \md X\arrow[dll,swap,"\beta"]\arrow[d,two heads,"\pi_{[\hat\alpha]}"] & & \md X\arrow[d,"\beta"]\\
\mathcal{A} & \hat\alpha(\term X) \arrow[l,hook',"id_{\hat\alpha(\term X)}"] & \term X|_{[\hat\alpha]}\arrow[l,two heads,hook',"\ol\alpha"] & & \mathcal{A}
\end{tikzcd}
\]
The diagonal of this diagram is a map $\beta$ defined for arbitrary
$u\in\md X$ as
follows: $$\beta(u)=\ol\alpha\circ\pi_{[\hat\alpha]}(u).$$
Note that if $u=\ang s$ for some $s\in\term X$, then
$$\beta(\ang s)=\ol\alpha(\pi_{[\hat\alpha]}(\ang
s))=\ol\alpha(s|_{[\hat\alpha]})=\hat\alpha(s)$$
and further more, if $x\in X$,
$$\beta(\ang x)=\ol\alpha(\pi_{[\hat\alpha]}(\ang
x))=\ol\alpha(x|_{[\hat\alpha]})=\hat\alpha(x)=\alpha(x).$$
Since $\beta$ is the composition of two homomorphisms of quantitative
algebras, it is a homomorphism of quantitative algebras.
Finally we show that $\beta$ is a $r(\mathcal K)$-reflexive.
To start with, note that $\hat\alpha(\term X)\leq\mathcal{A}$ is the image of $\md
X$
through $\beta$. Since $|\hat\alpha(\term X)|<r(\mathcal K)$, it only remains to
prove that there exists a subset in $\md X$ such that for any
$a,b\in\hat\alpha(\term X)$ we find two elements $u,v$ in this subset such
that $\beta(u)=a$, $\beta(v)=b$ and $$d^\mathcal{A}(a,b)=d^\mathcal K(u,v).$$
Let $s,t\in\term X$ be such that $\hat\alpha(s)=a$ and
$\hat\alpha(t)=b$. Let $u,v\in\md X$ such that
$\pi_{[\hat\alpha]}(u)=s|_{[\hat\alpha]}$,
$\pi_{[\hat\alpha]}(v)=t|_{[\hat\alpha]}$ and for any
$p\neq [\hat\alpha]$, $\pi_p(u)=\pi_p(v)$.
Since $d^\mathcal
K(u,v)=\displaystyle\sup_{p\in\mathcal{P_K}}p(\pi_p(u),\pi_p(v))$ and
$\pi_p(u)=\pi_p(v)$ for $p\neq[\hat\alpha]$, we obtain that
indeed $$d^\mathcal K(u,v)=[\hat\alpha](s,t)=d^\mathcal{A}(a,b).$$ \end{proof}
Observe that the homomorphism $\beta$ is not unique, since any pseudometric $p\in\mathcal{P_K}$ can be associated to a projection $\pi_p$ that will eventually define a homomorphism of type $\beta$ making the diagram commutative -– hence, we have weak-universality. However, only for $\beta$ associated to $[\alpha]$, can we guarantee that $\beta$ is $r(\mathcal K)$-reflexive.
The weak universality reflects a fundamental relation between $\md X$ and the $r(\mathcal K)$-reflexive closure operator $\mathbb H_{r(\mathcal K)}$, as stated below.
\begin{cor}\label{c01}
If $\mathcal{A}\in\mathcal K$, then for $X$ sufficiently large, $$\mathcal{A}\in\mathbb
H_{r(\mathcal K)}(\{\md X\}).$$ \end{cor}
\begin{proof}
Let $X$ be a set such that $|X|\geq|\mathcal{A}|$. Then, there exists a surjective map $\alpha:X\twoheadrightarrow\mathcal{A}$. Let $\beta:\md X\to\mathcal{A}$ be the $r(\mathcal K)$-reflexive homomorphism of quantitative algebras defined in the previous theorem. Since $\alpha$ is surjective, so is $\beta$. \end{proof}
\begin{cor}\label{l01}
Suppose that $\term X\neq\emptyset\neq\mathcal K$. Then,
$$\md X\in\mathbb{H}_{r(\mathcal K)}\mathbb{SP}(\mathcal K).$$
Hence, if $\mathcal K$ is closed under $\mathbb{H}_{r(\mathcal K)}$, $\mathbb{S}$ and $\mathbb P$, then $\md X\in\mathcal K$. \end{cor}
\begin{proof} Note that there exists a map
$\alpha:X\to\displaystyle\prod_{p\in\mathcal{P_K}}\term X|_p$ defined by $\alpha(x)=\ang x$.
Then, applying the weak universality result, in Theorem \ref{weakuniv}, we get that there exists a $c$-reflexive homomorphism $\beta:\md X\to\displaystyle\prod_{p\in\mathcal{P_K}}\term X|_p$. \begin{comment}
2. Since $\md X\in\mathbb{H}_c\mathbb{P^\leq}(\{\term X|_p\mid p\in\mathcal{P_K}\})$, we get that $\md X\in\mathbb{H}_c\mathbb{P^\leq IS}(\mathcal K)$. But $\mathbb{P^\leq \subseteq SP}$, implying further $\md X\in\mathbb{H}_c\mathbb{SP}(\mathcal K)$.
\end{comment} \end{proof}
The following theorem explains why we refer to $\md X$ as to the canonical model: it is because the class $\mathcal K$ and the quantitative algebra $\md X$ satisfy the same $c$-basic quantitative equations for any non-null regular cardinal $c\leq r(\mathcal K)$.
\begin{thm}\label{t01}
Let $\mathcal K$ be a class of quantitative algebras containing
non-degenerate elements and $c\leq r(\mathcal K)$ a non-null regular
cardinal. Let $$\{x_i=_{\epsilon_i}y_i\mid i\in I\}\vdash s=_\epsilon t$$ be an
arbitrary $c$-basic conditional equation on $\term X\neq\emptyset$, i.e., $|I|<c$.
Then, $$\{x_i=_{\epsilon_i}y_i\mid i\in I\}\models_{\mathcal K} s=_\epsilon
t~\text{ iff }~\{x_i=_{\epsilon_i}y_i\mid i\in I\}\models_{\md X} s=_\epsilon
t.$$ \end{thm}
\begin{proof} $(\Longrightarrow):$ If $\{x_i=_{\epsilon_i}y_i\mid i\in I\}\models_{\mathcal K} s=_\epsilon t$, then Corollary \ref{satbasic} and Lemma \ref{operators} guarantee that $\{x_i=_{\epsilon_i}y_i\mid i\in I\}\models_{\mathbb{H}_c\mathbb{SP}(\mathcal K)} s=_\epsilon t$. From Lemma \ref{l01} applying also Lemma \ref{operators} we know that $\md X\in\mathbb{H}_c\mathbb{SP}(\mathcal K)$. Hence, $$\{x_i=_{\epsilon_i}y_i\mid i\in I\}\models_{\md X} s=_\epsilon t.$$
$(\Longleftarrow):$ Suppose now that $\{x_i=_{\epsilon_i}y_i\mid i\in I\}\models_{\md X} s=_\epsilon t$. And assume in addition that $s$ and $t$ depend on (a subset of) the set $$\{x_i,y_i\mid i\in I\}\cup\{z_j\mid j\in J\}\subseteq X$$ of variables. We denote this by writing, as before, $s(\str{x}{i}{I}, \str{y}{i}{I}, \str{z}{j}{J})$ and $t(\str{x}{i}{I}, \str{y}{i}{I}, \str{z}{j}{J})$.
Suppose there exists $\mathcal{A}\in\mathcal K$ such that $$\{x_i=_{\epsilon_i}y_i\mid i\in I\}\not\models_{\mathcal{A}} s(\str{x}{i}{I}, \str{y}{i}{I}, \str{z}{j}{J})=_\epsilon t(\str{x}{i}{I}, \str{y}{i}{I}, \str{z}{j}{J}).$$
This means that there exists $\alpha\in\term(X|\mathcal{A})$ such that $$\text{for all }i\in I,~ d^\mathcal{A}(\alpha(x_i),\alpha(y_i))\leq\epsilon_i~\text{ and }~d^\mathcal{A}(\alpha(s), \alpha(t))>\epsilon.$$ Moreover, $\alpha(s)=s((\alpha(x_i))_{i\in I},(\alpha(y_i))_{i\in I},(\alpha(z_j))_{j\in J})$ and \\$\alpha(t)=t((\alpha(x_i))_{i\in I},(\alpha(y_i))_{i\in I},(\alpha(z_j))_{j\in J})$.
Applying the weak universality, Theorem \ref{weakuniv}, we obtain that $\alpha$ can be extended to a $r(\mathcal K)$-reflexive homomorphism $\beta:\md X\to\mathcal{A}$ such that $\alpha(x)=\beta(\ang x)$ for any $x\in X$. Since $c\leq r(\mathcal K)$, $\beta$ is also $c$-reflexive.
Since $c$ is regular, $|\{x_i,y_i\mid i\in I\}|< c$.
Because $\alpha(x_i), \alpha(y_i)\in\hat\alpha(\term X)=\beta(\md X)\leq\mathcal{A}$ and $\beta$ is $c$-reflexive, we obtain that there exist $m_i,n_i\in\md X$ for all $i\in I$ such that $\alpha(x_i)=\beta(m_i)$, $\alpha(y_i)=\beta(n_i)$ and $d^\mathcal{A}(\alpha(x_i),\alpha(y_{i}))=d^\mathcal K(m_i,n_{i'})$ for any $i\in I$.
Also $\alpha(z_j)\in\hat\alpha(\term X)=\beta(\md X)$, hence there exists $u_j\in\md X$ such that $\alpha(z_j)=\beta(u_j)$ for all $j\in J$.
From here we derive firstly that $$\text{for all }i\in I,~ d^\mathcal K(m_i,n_i)=d^\mathcal{A}(\alpha(x_i),\alpha(y_i))\leq\epsilon_i.$$ Secondly, since $\beta$ is non-expansive, $$d^\mathcal K(s((m_i)_{i\in I},(n_i)_{i\in I},(u_j)_{j\in J}),t((m_i)_{i\in I},(n_i)_{i\in I},(u_j)_{j\in J})))\geq d^\mathcal{A}(\alpha(s),\alpha(t))>\epsilon.$$
With these results in hand, we can define $\alpha_0\in\term(X|\md X)$ such that $$\alpha_0(x_i)=m_i,~ \alpha_0(y_i)=n_i~\text{ for any }i\in I\text{ and }\alpha_0(z_j)=u_j\text{ for any }j\in J.$$ The previous results demonstrates that $$\text{for all }i\in I,~ d^\mathcal K(\alpha_0(x_i),\alpha_0(y_i))\leq\epsilon_i~\text{ and }~d^\mathcal K(\alpha_0(s), \alpha_0(t))>\epsilon$$ which contradicts the fact that $\{x_i=_{\epsilon_i}y_i\mid i\in I\}\models_{\md X} s=_\epsilon t$. \end{proof}
This last result can further be instantiated for unconditional quantitative equations, which, in addition, can be used to characterize the metric $d^\mathcal K$.
\begin{cor}\label{t001}
Let $\mathcal K$ be a class of quantitative algebras containing
non-degenerate elements and $\term X\neq\emptyset$. Then for arbitrary
$s,t\in\term X$ and arbitrary $\epsilon\in\mathbb{Q}_+$, $$\mathcal K\models
s=_\epsilon t~\text{ iff }~\md X\models s=_\epsilon t ~\text{ iff }~d^\mathcal
K(\ang s,\ang t)\leq\epsilon.$$ \end{cor}
\begin{proof} The equivalence between the first two statements follows directly from Theorem \ref{t01}.
For the equivalence with the last statement, suppose that $\md X\models s=_\epsilon t$. Since the injection $\gamma:\term X\hookrightarrow\md X\in\term(X|\md X)$ is an assignment, we obtain that $d^\mathcal K(\gamma(s),\gamma(t))\leq\epsilon$. Hence, $d^\mathcal K(\ang s,\ang t)\leq\epsilon$.
Suppose now that $d^\mathcal K(\ang s,\ang t)\leq\epsilon$. Then for any $p\in\mathcal{P_K}$, $p(s,t)\leq\epsilon$. \\Let $\mathcal{A}\in\mathcal K$ and assume that $s$ and $t$ depend of $(x_i)_{i\in I}\in X$; for convenience we denote the two terms by $s(\str{x}{i}{I})$ and $t(\str{x}{I}{I})$.
Consider arbitrary $\str{a}{i}{I}\subseteq\mathcal{A}$ and let
$\alpha\in\term(X|\mathcal{A})$ such that $\alpha(x_i)=a_i$ for any $i\in I$.
For arbitrary $i,j$ we have that $d^\mathcal K(\ang{x_i},\ang{x_j})\geq d^\mathcal{A}(a_i,a_j)$ because as long as $\mathcal K\neq\emptyset\neq\term X$, for any distinct variables $x,y\in X$, $$d^\mathcal K(\ang x,\ang y)=\sup_{\mathcal{A}\in\mathbb S(\mathcal K)}\sup_{a,b\in\mathcal{A}}d^\mathcal{A}(a,b).$$
Theorem \ref{weakuniv} guarantees that the aforementioned $\alpha$ can be extended to a homomorphism $\beta:\md X\to\mathcal{A}$, which is non-expansive. Hence, $$d^\mathcal{A}(s(\str{a}{i}{I},),t(\str{a}{i}{I}))=d^\mathcal{A}(s((\alpha(x_i))_{i\in I}),t((\alpha(x_i))_{i\in I}))=$$ $$d^\mathcal{A}(s((\beta(\ang{x_i})_{i\in I}), t((\beta(\ang{x_i})_{i\in I}))\leq d^\mathcal K(\ang s,\ang t)\leq\epsilon.$$
Consequently, for any $\mathcal{A}\in\mathcal K$ and any assignment
$\alpha\in\term(X|\mathcal{A})$, $$d^\mathcal{A}(\alpha(s),\alpha(t))\leq\epsilon,$$ implying $\mathcal K\models s=_\epsilon t$. \end{proof}
\begin{cor}\label{c02}
Let $\mathcal K\neq\emptyset\neq\term X$ and let $Y$ be a set of variables such that $|Y|\geq |X|$. For any $c$-basic conditional equation $\{x_i=_{\epsilon_i}y_i\mid i\in I\}\vdash s=_\epsilon t$, where $c\leq r(\mathcal K)$ is a non-zero regular cardinal, $$\{x_i=_{\epsilon_i}y_i\mid i\in I\}\models_{\mathcal K} s=_\epsilon t~\text{ iff }~\{x_i=_{\epsilon_i}y_i\mid i\in I\}\models_{\md Y} s=_\epsilon t.$$ \end{cor}
\subsection{Variety Theorem}
With these results in hand, we are ready to prove a general variety theorem for quantitative algebras.
Hereafter the signature $\Omega$ remains fixed; so, if $S$ is an axiomatization for $\mathcal{U}$, we use $\mathbb{K}(S)$ to denote the class $\mathbb{K}(\Omega,\mathcal{U})$.
If $S$ is a set of $c$-basic conditional equations, we say that $\mathbb{K}(S)$ is a \textit{$c$-basic conditional equational class}. We call an $\aleph_1$-basic conditional equational class simply \textit{basic
equational class}. A \textit{finitary-basic equational class} is an $\aleph_0$-basic conditional equational class. An \textit{unconditional
equational class} is a $1$-basic conditional equational class.
We propose now a symmetric concept: if $\mathcal K$ is a set of quantitative algebras and $0<c\leq\aleph_1$ is a cardinal, let $\mathcal{E}^c_X(\mathcal K)$ be the set of all $c$-basic conditional equations over the set $X$ of variables that are satisfied by all the elements of $\mathcal K$.
\begin{comment} \begin{lem}\label{l02}
If $X$ is a set of variables and $\mathcal K$ a non-void set of quantitative algebras, then for any set $Y$ of variables such that $|Y|\geq r(\mathcal K)$ and any $c< r(\mathcal K)$, $$\mathcal{E}^c_X(\mathcal K)=\mathcal{E}^c_X(\{\md Y\}).$$ \end{lem}
\begin{proof}
For any $c$-basic conditional equation $$\{x_i=_{\epsilon_i}y_i\mid i\in I\}\vdash s=_\epsilon t,$$ where $s$ and $t$ depend of the set $S=\{x_i,y_i\mid i\in I\}\cup\{z_j\mid j\in J\}$ of variables, we have that the conditional quantitative equation can be expressed in $\term S$. Since $|S|\leq card(\mathcal K)\leq |Y|$, applying Corollary \ref{c02} we get that $$\{x_i=_{\epsilon_i}y_i\mid i\in I\}\models_{\mathcal K} s=_\epsilon t~\text{ iff }~\{x_i=_{\epsilon_i}y_i\mid i\in I\}\models_{\md Y} s=_\epsilon t.$$ \end{proof} \end{comment}
\begin{lem}\label{l03}
If $\mathcal K$ is a non-void $c$-variety for a regular cardinal
$0<c\leq r(\mathcal K)$ and $X$ is an infinite set of variables,
then $$\mathcal K=\mathbb{K}(\mathcal{E}^c_X(\mathcal K)).$$ \end{lem}
\begin{proof}
Let $\mathcal K'=\mathbb{K}(\mathcal{E}^c_X(\mathcal K))$. Obviously $\mathcal K\subseteq\mathcal K'$.
We prove for the beginning that $\mathcal{E}^c_X(\mathcal K)=\mathcal{E}^c_X(\mathcal K')$.
Since $\mathcal K\subseteq\mathcal K'$, $\mathcal{E}^c_X(\mathcal
K)\supseteq\mathcal{E}^c_X(\mathcal K')$.
Let $\Gamma\vdash\phi\in\mathcal{E}^c_X(\mathcal K)$ be a $c$-basic quantitative
inference.
Then, for any $\mathcal{A}\in\mathcal K$,
$\Gamma\models_\mathcal{A}\phi$. Consider an arbitrary $\mathcal{B}\in\mathcal
K'$. Since $\mathcal K'=\mathbb{K}(\mathcal{E}^c_X(\mathcal K))$, $B$ must satisfy all
the $c$-basic conditional equations in $\mathcal{E}^c_X(\mathcal K)$; in particular,
$\Gamma\models_\mathcal{B}\phi$. Hence, $\mathcal{E}^c_X(\mathcal
K)\subseteq\mathcal{E}^c_X(\mathcal K')$.
Consider now an arbitrary $\mathcal{A}'\in\mathcal K'$.
From Corollary \ref{c01}, for a suitable set $Y$ of variables such that
$|Y|\geq r(\mathcal K')$, we can define a surjection $\alpha:\term
Y\twoheadrightarrow\mathcal{A}'$.
For arbitrary $s\in\term Y$, let $s|_{\mathcal
K}\in\displaystyle\prod_{p\in\mathcal{P_K}}\term Y|_p$ be the element\footnote{Observe that $s|_{\mathcal K}$ has been denoted by $\langle s\rangle$ previously, when $\mathcal K$ was fixed. We change the notation here because we need to speak of such elements for various classes $\mathcal K, \mathcal K'$.} such that for any
$p\in\mathcal{P_K}$, $\pi_p(s|_{\mathcal K})=s|_p$ and similarly
$s|_{\mathcal K'}\in\displaystyle\prod_{p\in\mathcal{P_{K'}}}\term Y|_p$ be the element
such that for any $p\in\mathcal{P_{K'}}$, $\pi_p(s|_{\mathcal
K'})=s|_p$.
Theorem \ref{weakuniv} provides an injection $\gamma':\term
Y\hookrightarrow\term_{\mathcal K'}Y$ defined by $\gamma'(s)=s|_{\mathcal
K'}$ for any $s\in\term Y$; and a $r(\mathcal K')$-reflexive
homomorphism $\beta':\term_{\mathcal K'}Y\twoheadrightarrow\mathcal{A}'$ which has
the property that $\beta'(s|_{\mathcal K'})=\alpha(s)$. Moreover,
$\beta'$ is a surjection since $\alpha$ is.
Because $c\leq
r(\mathcal K)\leq r(\mathcal K')$, $\beta'$ is also
$r(\mathcal K)$-reflexive and $c$-reflexive. Note now that also
$\hat{\beta'}:\gamma'(\term Y)\to\mathcal{A}'$, which is defined by
$\hat{\beta'}(u)=\beta'(u)$ for any $u\in\gamma'(\term Y)$, is a
surjective $c$-reflexive homomorphism of quantitative algebras such
that $\hat{\beta'}(s|_{\mathcal K'})=\alpha(s)$.
Similarly, there exists an injection $\gamma:\term Y\hookrightarrow\md Y$
defined by $\gamma(s)=s|_{\mathcal K}$ for any $s\in\term Y$.
Consider now the following two quantitative algebras
$$\term Y|_{d^{\mathcal K}}=(\term Y|_{ker(d^{\mathcal K})},\Omega, d^{\mathcal
K})~\text{ and }$$ $$\term Y|_{d^{\mathcal K'}}=(\term Y|_{ker(d^{\mathcal
K'})},\Omega, d^{\mathcal K'}).$$
Note that the functions $\theta:\term Y|_{d^{\mathcal K}}\to\gamma(\term
Y)$ defined by $\theta(s|_{d^{\mathcal K}})=\gamma(s)$ and $\theta':\term
Y|_{d^{\mathcal K'}}\to\gamma'(\term Y)$ defined by
$\theta'(s|_{d^{\mathcal K'}})=\gamma'(s)$ are isomorphisms of
quantitative algebras.
\[
\begin{tikzcd}
\md Y \geq \term Y|_{d^{\mathcal K}}& \term Y \arrow[l, hook', swap, "\gamma"] \arrow[r, hook, "\gamma'"] \arrow[d,two heads, swap,"\alpha"]
& \term Y|_{d^{\mathcal K'}}\arrow[dl,swap, two heads, "\hat{\beta'}"]\arrow[r, hook, "id"] & \term_{\mathcal K'} Y\arrow[dll, two heads, "\beta'"]\\
& \mathcal{A}' &
\end{tikzcd}
\]
Repeatedly applying Corollary \ref{t001} we get that for arbitrary $s,t\in\term Y$, $$d^\mathcal K(s|_{\mathcal K},t|_{\mathcal K})=0~\text{ iff }$$ $$\md Y\models s=_0
t,\text{iff }$$ $$\mathcal K\models s=_0 t,~\text{ iff }$$ $$\emptyset\vdash s=_0 t\in\mathcal{E}^c_Y(\mathcal K)~\text{(since $\mathcal{E}^c_Y(\mathcal
K) =\mathcal{E}^c_Y(\mathcal K')$)}, \text{ iff }$$
$$\emptyset\vdash s=_0 t\in\mathcal{E}^c_Y(\mathcal K'), \text{ iff }$$
$$\mathcal K'\models s=_0 t, \text{ iff }$$
$$\term_{\mathcal K'} Y\models s=_0 t, \text{ iff }$$ $$d^{\mathcal K'}(s|_{\mathcal K'},t|_{\mathcal K'})=0.$$
Hence, $ker(d^{\mathcal K})=ker(d^{\mathcal K'})$ implying that $\term Y|_{d^{\mathcal K}}$ and $\term
Y|_{d^{\mathcal K'}}$ are isomorphic $\Omega$-algebras.
Similarly, we can apply Corollary \ref{t001} for arbitrary $s,t\in\term Y$ and $\epsilon\in\mathbb{Q}_+$, as we did it before for $\epsilon=0$, and obtain:
$$d^\mathcal K(s|_{\mathcal K},t|_{\mathcal K})\leq\epsilon \text{ iff }$$
$$\md Y\models s=_\epsilon t, \text{ iff }$$
$$\mathcal K\models s=_\epsilon t, \text{ iff }$$
$$\emptyset\vdash s=_\epsilon t\in\mathcal{E}^c_Y(\mathcal K), \text{ iff }$$
$$\emptyset\vdash s=_\epsilon t\in\mathcal{E}^c_Y(\mathcal K'), \text{ iff }$$
$$\mathcal K'\models s=_\epsilon t, \text{ iff }$$
$$\term_{\mathcal K'} Y\models s=_\epsilon t, \text{ iff }$$
$$d^{\mathcal K'}(s|_{\mathcal K'},t|_{\mathcal K'})\leq\epsilon;$$ and since this is true for any
$\epsilon\in\mathbb{Q}_+$, we obtain $$d^\mathcal K(s|_{\mathcal
K},t|_{\mathcal K})=d^{\mathcal K'}(s|_{\mathcal K'},t|_{\mathcal
K'}).$$
Hence, $\term Y|_{d^{\mathcal K}}$ and $\term Y|_{d^{\mathcal K'}}$ are
isomorphic quantitative algebras implying further that $\gamma(\term Y)$
is isomorphic to $\gamma'(\term Y)$.
Now, since $\mathcal{A}'$ is the $c$-homomorphic image of $\gamma'(\term Y)$, it is
also a $c$-homomorphic image of $\gamma(\term Y)$. But $\gamma(T Y)\leq\md
Y$ and since $\mathcal K$ is a $c$-variety, from Lemma \ref{l01} we
know that $\md Y\in\mathcal K$, hence $\gamma(\term Y)\in\mathcal K$.
Consequently, $\mathcal{A}'\in\mathbb{H}_c(\mathcal K)$ and since $\mathcal K$
is a $c$-variety, $\mathcal{A}'\in\mathcal K$, from which we conclude
$\mathcal{K'\subseteq K}$. \end{proof}
Now we prove the variety theorem for quantitative algebras.
\begin{thm}[$c$-Variety Theorem]
Let $\mathcal K$ be a class of quantitative algebras and $0<c\leq r(\mathcal K)$ a regular cardinal. Then, $\mathcal K$ is a $c$-basic conditional equational class iff $\mathcal K$ is a $c$-variety.
In particular,
\begin{enumerate}
\item $\mathcal K$ is an unconditional equational class iff it is a variety;
\item $\mathcal K$ is a finitary-basic equational class iff it is an $\aleph_0$-variety;
\item $\mathcal K$ is a basic equational class iff it is an $\aleph_1$-variety.
\end{enumerate} \end{thm}
\begin{proof}
($\Longrightarrow$): $\mathcal K=\mathbb{K}(\mathcal{U})$ for some set $\mathcal{U}$ of
$c$-basic conditional equations. Then, $\mathbb V_c(\mathcal
K)\models \mathcal{U}$ implying further that $\mathbb V_c(\mathcal
K)\subseteq\mathbb{K}(\mathcal{U})=\mathcal K$. Hence, $\mathbb V_c(\mathcal
K)=\mathcal K$.
($\Longleftarrow$): this is guaranteed by Lemma \ref{l03}. \end{proof}
\textbf{Birkhoff Theorem in perspective.} Before concluding this section, we notice that our variety theorem also generalizes the original Birkhoff theorem. This is because any congruence $\cong$ on an $\Omega$-algebra $\mathcal{A}$ can be seen as the kernel of the pseudometric $p_{\cong}$ defined by $p_{\cong}(a,b)=0$ whenever $a\cong b$
and $p_{\cong}(a,b)=1$ otherwise. The quotient algebra $\mathcal{A}|_{\cong}$ is a quantitative algebra. Any quantitative equational theory satisfied by
$\mathcal{A}|_{\cong}$ can be axiomatized by equations involving only $=_0$ and $=_1$, since $0$ and $1$ are the only possible distances between its elements. However, this algebra also satisfies the equation $x=_1y$ for any two variables $x$ and $y$, because $1$ is the diameter of its support. Consequently, the only non-redundant equations satisfied by such an algebra are of type $s=_0 t$, and these correspond to the equations of the form $s=t$.
\section{The Quasivariety Theorem for General Conditional Equations}
In this section we study the axiomatizability of classes of quantitative algebras that can be axiomatized by conditional quantitative equations, but not necessarily by basic conditional quantitative equations. Thus, we are now looking for more relaxed types of axioms and consequently we will identify more relaxed closure conditions.
We prove that a class $\mathcal K$ of $\Omega$-quantitative algebras admits an axiomatization consisting of conditional quantitative equations, whenever it is closed under isomorphisms, subalgebras and what we call subreduced products. A subreduced product is a quantitative subalgebra of a (special type of) product of elements in $\mathcal K$; however, while these products are always $\Omega$-algebras, they are not always quantitative algebras. This closure condition allow us to generalize the classical quasivariety theorem that characterizes the classes of universal algebras with an axiomatization consisting of Horn clauses.
It is not trivial to see that a $c$-variety is closed under these operators for any regular cardinal $c>0$ and so our quasivariety theorem extends the $c$-variety theorem presented in the previous section. Indeed, all isomorphisms are $c$-reflexive homomorphisms and since a $c$-variety is closed under subalgebras and products, it must be closed under subreduced products, as they are quantitative subalgebras of the product.
However, to achieve these results we had to involve and generalize concepts and results from model theory of first-order structures. This required us to restrict ourselves to the signatures $\Omega$ containing only functions of finite arity.
\subsection{Preliminaries in Model Theory}
In this subsection we recall some basic concepts and results about the model theory of first order structures.
A \textit{first-order language} is a tuple $\mathcal L=(\Omega,\mathcal{R})$ where $\Omega$ is an algebraic similarity type containing functions of finite arity and $\mathcal{R}$ is a set of relation symbols of finite arity.
A \textit{first-order structure} of type $\mathcal L=(\Omega,\mathcal{R})$ is a tuple $\mathcal{M}=(M,\Omega^\mathcal{M},\mathcal{R}^\mathcal{M})$ where $(M,\Omega^\mathcal{M})$ is an $\Omega$-algebra and for any relation $R:i\in\mathcal{R}$, $R^\mathcal{M}\subseteq M^i$.
A \textit{morphism of first-order structures} of type $\mathcal L=(\Omega,\mathcal{R})$ is a map $$f:(M,\Omega^\mathcal{M},\mathcal{R}^\mathcal{M})\to(N,\Omega^\mathcal{N},\mathcal{R}^\mathcal{N})$$ that is a homomorphism of $\Omega$-algebras such that for any relation $R:i\in\mathcal{R}$ and $m_1,..m_i\in M$, $$(m_1,..,m_i)\in \mathcal{R}^\mathcal{M}~\text{ iff }~(f(m_1),..,f(m_i))\in\mathcal{R}^\mathcal{N}.$$
$\mathcal{M}=(M,\Omega^\mathcal{M},\mathcal{R}^\mathcal{M})$ is a \textit{subobject} of $\mathcal{N}=(N,\Omega^\mathcal{N},\mathcal{R}^\mathcal{N})$ if $(M,\Omega^\mathcal{M})$ is an $\Omega$-subalgebra of $(N,\Omega^\mathcal{N})$ and for any $R:i\in\mathcal{R}$ and $m_1,..m_i\in M$, $$(m_1,..,m_i)\in R^\mathcal{M}\text{ iff }(m_1,..,m_i)\in R^\mathcal{N};$$ In this case we write $\mathcal{M}\leq\mathcal{N}$.
\textbf{Equational First-Order Logic.} Given a first-order structure $\mathcal L=(\Omega,\mathcal{R})$ and a set $X$ of variables, let $\term X$ be the set of terms induced by $X$ over $\Omega$.
The \textit{atomic formulas} of type $\mathcal L=(\Omega,\mathcal{R})$ over $X$ are expressions of the form \begin{itemize}
\item $s=t$ for $s,t\in\term X$;
\item $R(s_1,..,s_k)$ for $R:k\in\mathcal{R}$ and $s_1,..,s_k\in\term X$. \end{itemize}
The set $\mathcal L X$ of first-order formulas of type $\mathcal L$ over $X$ is the smallest collection of formulas containing the atomic formulas and closed under conjunction, negation and universal quantification $\forall x$ for $x\in X$. In addition we consider, as usual, all the Boolean operators and the existential quantification.
If $\mathcal{M}$ is a structure of type $\mathcal L$, let $\mathcal L_\mathcal{M}$ be the first-order language obtained by adding to $\mathcal L$ the elements of $\mathcal{M}$ as constants.
Given a first-order formula $\phi(x_1,..,x_{i},..x_k)$, in which $x_1,..,x_k\in X$ are all the free variables, we denote by $\phi(x_1,..,x_{i-1},m,x_{i+1},..x_k)$, as usual, the formula obtained by replacing all the free occurrences of $x_i$ by $m\in\mathcal{M}$.
\textbf{Satisfiability.} For a closed formula $\phi\in\mathcal L_\mathcal{M}$, we define $\mathcal{M}\models\phi$ inductively on the structure of formulas as follows. \begin{itemize}
\item $\mathcal{M}\models s=t$ for $s,t\in T X$ containing no variables iff $s^\mathcal{M}=t^\mathcal{M}$.
\item $\mathcal{M}\models R(s_1,..,s_k)$ for $R:k\in\mathcal{R}$ and $s_1,..,s_k\in\term X$ containing no variables iff $(s_1^\mathcal{M},..,s_k^\mathcal{M})\in R^\mathcal{M}$;
\item $\mathcal{M}\models\phi\land\psi$ iff $\mathcal{M}\models\phi$ and $\mathcal{M}\models\psi$;
\item $\mathcal{M}\models\lnot\phi$ iff $\mathcal{M}\not\models\phi$;
\item $\mathcal{M}\models\forall x\phi(x)$ iff $\mathcal{M}\models\phi(m)$ for any $m\in\mathcal{M}$. \end{itemize} The semantics of the derived operators is standard. The de Morgan laws give us semantically-equivalent prenex forms for any first-order formula.
A first-order formula is an \textit{universal formula} if it is in prenex form and all the quantifiers are universal.
A \textit{Horn formula} has the following prenex form $$Q_1x_1..Q_kx_k(\phi_1(x_1,..,x_k)\land..\land\phi_j(x_1,..,x_k)\to\phi(x_1,..,x_k)),$$ where each $Q_l$ is a quantifier and each $\phi_l$ and $\phi$ is an atomic formula with (a subset of) the set $\{x_1,..,x_k\}$ of free variables\footnote{Some authors define a Horn formula as a conjunction of such constructs, or allow $\phi=\top$; none of these choices affect our development here.}.
A \textit{universal Horn formula} is a Horn formula which is also an universal formula.
\textbf{Direct Products.} Given a non-empty indexed family $(\mathcal{M}_i)_{i\in I}$ of first-order structures of type $\mathcal L=(\Omega,\mathcal{R})$, where $\mathcal{M}_i=(M_i,\Omega^{\mathcal{M}_i},\mathcal{R}^{\mathcal{M}_i})$, the \textit{direct product} $\mathcal{M}=\displaystyle\prod_{i\in I}\mathcal{M}_i$ is the $\mathcal L$-structure whose universe is the product set $\displaystyle\prod_{i\in I} M_i$ and its functions and relations are defined as follows, where $\pi_i:\displaystyle\prod_{i\in I}\mathcal{M}_i\to\mathcal{A}_i$ denotes the $i$-th projection. \begin{itemize}
\item for $f:k\in\Omega$, and $m_1,..,m_k\in \displaystyle\prod_{i\in I}M_i$, $$\pi_i(f^\mathcal{M}(m_1,..,m_k))=f^{\mathcal{M}_i}(\pi_i(m_1),..,\pi_i(m_k));$$
\item for $R:k\in\mathcal{R}$, and $m_1,..,m_k\in \displaystyle\prod_{i\in I}M_i$,
$$(m_1,..,m_k)\in R^\mathcal{M}~\text{ iff }~(\pi_i(m_1),..,\pi_i(m_k))\in R^{\mathcal{M}_i}~\text{ for all }~i\in I.$$ \end{itemize}
\textbf{Reduced Products.} Let $\str{\mathcal{M}}{i}{I}$ be an indexed family of first-order structures of type $\mathcal L=(\Omega,\mathcal{R})$ and $F$ a proper filter over $I$.
Consider the relation $\mathord\sim_F\subseteq\displaystyle\prod_{i\in I}\mathcal{M}_i\times\displaystyle\prod_{i\in I}\mathcal{M}_i$ s.t. $$m\sim_F n~\text{ iff }~\{i\in I\mid \pi_i(m)=\pi_i(n)\}\in F.$$
It is known that when $F$ is a proper filter of $I$, $\sim_F$ is a congruence relation with respect to the algebraic structure of $\mathcal{M}=\displaystyle\prod_{i\in I}\mathcal{M}_i$ (see, e.g., \cite[Lemma~2.2]{Burris81}). This allows us to define the \textit{reduced product induced by a proper filter} $F$, written $(\displaystyle\prod_{i\in I}\mathcal{M}_i)|_F$, as the $\mathcal L$ first-order structure such that \begin{itemize}
\item its universe is the set $(\displaystyle\prod_{i\in I}M_i)|_{\sim_F}$, which is the quotient of $\displaystyle\prod_{i\in I}M_i$ with respect to $\sim_F$; we denote by $m_F$ the $\sim_F$-congruence class of $m\in \displaystyle\prod_{i\in I}M_i$;
\item for $f:k\in\Omega$, and $(m^1,..,m^k)\in \displaystyle\prod_{i\in I}M_i$, $$f(m^1_F,..,m^k_F)=(f(m^1,..,m^k))_F:$$
\item for $R:k\in\mathcal{R}$, and $(m^1,..,m^k)\in \displaystyle\prod_{i\in I}M_i$,
$$R(m^1_F,..,m^k_F)~\text{ iff }~\{i\in I\mid R(\pi_i(m^1),..,\pi_i(m^k)\}\in F.$$ \end{itemize}
\textbf{Quasivariety Theorem.} A class $\mathfrak M$ of $\mathcal L$-structures is an \textit{elementary class} if there exists a set $\Phi$ of first-order $\mathcal L$-formulas such that for any $\mathcal L$-structure $\mathcal{M}$, $$\mathcal{M}\in\mathfrak M~\text{ iff }~\mathcal{M}\models\Phi.$$
An elementary class is an \textit{universal class} if it can be axiomatized by universal formulas; it is an \textit{universal Horn class} if it can be axiomatized by universal Horn formulas.
We conclude this section with the quasivariety theorem (see, e.g., \cite[Theorem~2.23]{Burris81}). To state it, we define a few closure operators on classes of $\mathcal L$-structures.
Let $\mathfrak M$ be an arbitrary class of $\mathcal L$-structures. \begin{itemize}
\item $\mathbb I(\mathfrak M)$ denotes the closure of $\mathfrak M$ under isomorphisms of $\mathcal L$-structures;
\item $\mathbb S(\mathfrak M)$ denotes the closure of $\mathfrak M$ under subobjects of $\mathcal L$-structures;
\item $\mathbb P(\mathfrak M)$ denotes the closure of $\mathfrak M$ under direct products of $\mathcal L$-structures;
\item $\mathbb P_R(\mathfrak M)$ denotes the closure of $\mathfrak M$ under reduced products of $\mathcal L$-structures. \end{itemize}
\begin{thm}[Quasivariety Theorem]\label{mt:quasi}
Let $\mathfrak M$ be a class of $\mathcal L$-structures. The following statements are equivalent.
\begin{enumerate}
\item $\mathfrak M$ is a universal Horn class;
\item $\mathfrak M$ is closed under $\mathbb{I,~S}$ and $\mathbb P_R$;
\item $\mathfrak M=\mathbb{ISP}_R(\mathfrak M')$ for some class $\mathfrak M'$ of $\mathcal L$-structures.
\end{enumerate} \end{thm}
\subsection{Quantitative First-Order Structures}
In this subsection we identify a class of first-order structures, the \textit{quantitative first-order structures} (QFOs), which are the first-order counterparts of the quantitative algebras.
Given a first-order structure $\mathcal{M}=(M,\Omega^\mathcal{M},\mathcal{R}^\mathcal{M})$ of type $(\Omega,\mathcal{R})$, $f:k\in\Omega$ and $R:l\in\mathcal{R}$, let $f(R^\mathcal{M})\subseteq M^l$ be the set of the tuples $(f(m_1^1,..,m_k^1),..,f(m_1^l,..,m_k^l))$ such that for each $i=1,..,k$, $(m_i^1,..,m_i^l)\in R^\mathcal{M}$.
\begin{defi}\label{qfo} [Quantitative First-Order Structure]
An $\Omega$-\textit{quantitative first-order structure} for a signature $\Omega$ is a first-order structure $\mathcal{M}=(M,\Omega^\mathcal{M},\equiv^\mathcal{M})$ of type $(\Omega,\equiv)$, where $\mathord\equiv=\{=_\epsilon\mid\epsilon\in\mathbb{Q}_+\},$ that satisfies the following axioms for any $\epsilon,\delta\in\mathbb{Q}_+$
\begin{enumerate}
\item $\mathord{=_0^\mathcal{M}}$ is the identity on $\mathcal{M}$;
\item $\mathord{=_\epsilon^\mathcal{M}}$ is symmetric;
\item $\mathord{=_\epsilon^\mathcal{M}\circ=_\delta^\mathcal{M}\subseteq =_{\epsilon+\delta}^\mathcal{M}}$;
\item $\mathord{=_\epsilon^\mathcal{M}\subseteq =_{\epsilon+\delta}^\mathcal{M}}$;
\item for any $f:k\in\Omega$, $\mathord{f(=^\mathcal{M}_\epsilon)\subseteq =^\mathcal{M}_\epsilon}$;
\item for any $\delta$, $\mathord{\displaystyle\bigcap_{\epsilon>\delta}=_\epsilon\subseteq=_\delta}$;
\end{enumerate} \end{defi}
\begin{thm}\label{correspondence}
(i) Any quantitative algebra $\mathcal{A}=(A,\Omega,d)$ defines uniquely a quantitative first-order structure by $$a=_\epsilon b~\text{ iff }~d(a,b)\leq\epsilon.$$
(ii) Any quantitative first-order structure $\mathcal{M}=(M,\Omega^\mathcal{M},\equiv^\mathcal{M})$ defines uniquely a quantitative algebra by letting $$d(m,n)=\inf\{\epsilon\in\mathbb{Q}_+\mid m=_\epsilon n\}.$$
These define an isomorphism between the category of $\Omega$-quantitative algebras and $\Omega$-quantitative first-order structures. \end{thm}
\begin{proof} The proof is trivial and relies on the fact that conditions (1)-(6) in Definition \ref{qfo} corresponds to (Refl), (Symm), (Triang), (Max), (Arch) and (NExp) respectively. \end{proof}
Let $\mathbf{QA_\Omega}$ be the category of $\Omega$-quantitative algebras and $\mathbf{QFO_\Omega}$ the category of $\Omega$-quantitative first-order structures. Theorem \ref{correspondence} defines two functors $\mathbb F$ and $\mathbb G$ that act as identities on morphisms, which define an isomorphism of categories as in the figure below.
\begin{equation*} \begin{tikzpicture}[baseline={(m.center)},arrow label/.style={font=\scriptsize}] \matrix (m) [commutative diagram={1cm}{1.2cm}] {
\mathbf{QA_\Omega} & \mathbf{QFO_\Omega} \\ }; \path[-latex,arrow label] (m-1-1) edge[bend left] node[above] {$\mathbb F$} (m-1-2) (m-1-2) edge[bend left] node[above] {$\mathbb G$} (m-1-1) ; \end{tikzpicture} \end{equation*}
We already know that the subobjects and the direct products of quantitative first-order structures are first-order structures. However, since the isomorphisms of categories preserve limits and colimits, we can prove that the subobjects and the direct products of quantitative first-order structures are, in fact, quantitative first-order structures, i.e., they satisfy the axioms (1)-(6) of Definition \ref{qfo}, as the next lemma establishes.
\begin{lem}\label{subobjandprod}
I. If $\mathcal{M},\mathcal{N}$ are $\Omega$-quantitative first-order structures s.t. $\mathcal{M}\leq\mathcal{N}$, then $$\mathbb G\mathcal{M}\leq\mathbb G\mathcal{N}.$$
II. If $\str{\mathcal{M}}{i}{I}$ is a family of $\Omega$-quantitative first-order structures, then $$\mathbb G(\displaystyle\prod_{i\in I}\mathcal{M}_i)=\displaystyle\prod_{i\in I}\mathbb G\mathcal{M}_i.$$
III. If $\mathcal{A},\mathcal{B}$ are $\Omega$-quantitative algebras such that $\mathcal{A}\leq\mathcal{B}$, then $$\mathbb F(\mathcal{A})\leq\mathbb F\mathcal{B}.$$
IV. If $\str{\mathcal{A}}{i}{I}$ is a family of $\Omega$-quantitative algebras, then $$\mathbb F(\displaystyle\prod_{i\in I}\mathcal{A}_i)=\displaystyle\prod_{i\in I}\mathbb F\mathcal{A}_i.$$ \end{lem}
\begin{comment}
\begin{lem}\label{subobjandprod} I. A subobject of a quantitative first-order structure is a quantitative first-order structure. Moreover, for two QFOs $\mathcal{A},\mathcal{A}'$ such that $\mathcal{A}\leq\mathcal{A}'$, $$\mathbb G\mathcal{A}\leq\mathbb G\mathcal{A}'.$$ II. Direct and subdirect products of quantitative first-order structures are quantitative first-order structures. Moreover, if $\str{\mathcal{A}}{i}{I}$ is a family of QFOs, then $$G(\displaystyle\prod_{i\in I}\mathcal{A}_i)=\displaystyle\prod_{i\in I}\mathbb G\mathcal{A}_i.$$ \end{lem}
\begin{proof} II. Note that the definition of direct product for first-order structures guarantees that for arbitrary $a,b\in\displaystyle\prod_{i\in I}\mathcal{A}_i$ and $\epsilon\in\mathbb{Q}_+$, $$a=_\epsilon b~\text{ iff for all}~i\in I, \pi_i(a)=_\epsilon\pi_i(b).$$ This property allows us to prove without difficulty that if for all $i\in I$, $\mathcal{A}_i$ satisfies the axioms (1)-(6), so does $\displaystyle\prod{i\in I}\mathcal{A}_i$.
To get the same result for subproducts we further apply Lemma \ref{subobjandprod}. \end{proof}
A similar result can be stated for quantitative algebras.
Let $\mathcal{U}$ be a quantitative equational theory of type $\Omega$ over $X$ and let $\mathbb{K}(\Omega,\mathcal{U})$ be, as before, the class of quantitative algebras of type $\Omega$ satisfying $\mathcal{U}$.
\begin{lem}\label{qa:prod} I. $\mathbb{K}(\Omega,\mathcal{U})$ is closed under taking subobjects. Moreover, if $\mathcal{A}\in\mathbb{K}(\Omega,\mathcal{U})$ and $\mathcal{A}'\leq\mathcal{A}$, then $$\mathbb F(\mathcal{A}')\leq\mathbb F\mathcal{A}.$$
II. $\mathbb{K}(\Omega,\mathcal{U})$ is closed under taking direct and subdirect products. Moreover, if $\str{\mathcal{A}}{i}{I}$ is a family of quantitative algebras, $$\mathbb F(\displaystyle\prod_{i\in I}\mathcal{A}_i)=\displaystyle\prod_{i\in I}\mathbb F\mathcal{A}_i.$$ \end{lem}
\end{comment}
\subsection{Subreduced Products of Quantitative First-Order Structures}
Given an indexed family $\str{\mathcal{M}}{i}{I}$ of $\Omega$-quantitative first-order structures and a proper filter $F$ on $I$, we can construct, as before, the reduced product $(\str{\mathcal{M}}{i}{I})|_F$ of first-order structures, which is a first-order structure. But it is not guaranteed that it satisfies the axioms in Definition \ref{qfo}. From the definition of the reduced product we obtain a first-order structure $(\str{\mathcal{M}}{i}{I})|_F$ that enjoys the following property for any $\epsilon\in\mathbb{Q}_+$. $$m_F=_\epsilon n_F~\text{ iff }~\{i\in I\mid \pi_i(m)=_\epsilon\pi_i(n)\}\in F.$$
Note that if for all $i\in I$, $\mathcal{M}_i$ satisfies the axioms (1)-(5) from Definition \ref{qfo}, then $(\str{\mathcal{M}}{i}{I})|_F$ satisfies them as well.
For instance, we can verify the condition (3): suppose that $m_F=_\epsilon n_F$ and $n_F=_\delta u_F$. Hence, $$\{i\in I\mid \pi_i(m)=_\epsilon\pi_i(n)\}, \{i\in I\mid \pi_i(n)=_\delta\pi_i(u)\}\in F.$$ Since $F$ is a filter, it is closed under intersection, so $$\{i\in I\mid \pi_i(m)=_\epsilon\pi_i(n)\text{ and }\pi_i(n)=_\delta\pi_i(u)\}\in F.$$ Now, axiom (3) guarantees that $$\{i\in I\mid \pi_i(m)=_\epsilon\pi_i(n)\text{ and }\pi_i(n)=_\delta\pi_i(u)\}$$$$\subseteq\{i\in I\mid \pi_i(m)=_{\epsilon+\delta}\pi_i(u)\}$$ and since $F$ is closed under supersets, $$\{i\in I\mid \pi_i(m)=_{\epsilon+\delta}\pi_i(u)\}\in F.$$
Similarly, one can verify each of the axioms but (6). This is because axiom (6) requires that any reduced product has the property that for any $\delta\in\mathbb{Q}_+$, $$\{i\in I\mid \pi_i(m)=_\epsilon\pi_i(n)\}\in F\text{ for all }\epsilon>\delta$$ implies $$\{i\in I\mid \pi_i(m)=_\delta\pi_i(n)\}\in F.$$ This is a very strong condition not necessarily satisfied by a filter or an ultrafilter. It is, for instance, satisfied by the filters and ultrafilters closed under countable intersections, but the existence of such filters requires measurable cardinals (see for instance \cite{Chang92} for a detailed discussion).
Hence, while the reduced products of quantitative first-order structures can always be defined as first-order structures, they are not always quantitative first-order structures, since they might not satisfy axiom (6) in Definition \ref{qfo}. Therefore, taking reduced products and ultraproducts are not internal operations over the class of quantitative first-order structures of the same type, even if they are internal operations over the larger class of first-order structures of the same type. This observation motivates our next definition.
\begin{defi}[Subreduced Products]
Given an indexed family $\str{\mathcal{M}}{i}{I}$ of quantitative first-order structures and a proper filter $F$ on $I$, a \textit{subreduced product} of this family induced by $F$ is any subobject $\mathcal{M}$ of the first-order structure $(\displaystyle\prod_{i\in I}\mathcal{M}_i)|_F$ such that $\mathcal{M}$ is a quantitative first-order structure. \end{defi}
Given a class $\mathfrak M$ of quantitative first-order structures of the same type, the closure of $\mathfrak M$ under subreduced products is denoted by $\mathbb P_{SR}(\mathfrak M)$.
With this concept in hand we can generalize the quasivariety theorem for first-order structures to get a similar result for classes of QFOs that can be properly axiomatized.
\begin{thm}[Quasivariety Theorem for Quantitative First-Order Structures]\label{quasivar}
Let $\mathfrak M$ be a class of $\Omega$-quantitative first-order structures. Then, the following statements are equivalent.
\begin{enumerate}
\item $\mathfrak M$ is an universal Horn class;
\item $\mathfrak M$ is closed under $\mathbb {I, S}$ and $\mathbb P_{SR}$;
\item $\mathfrak M=\mathbb{ISP}_{SR}(\mathfrak M_0)$ for some class $\mathfrak M_0$ of $\Omega$-quantitative first-order structures.
\end{enumerate} \end{thm}
\begin{proof}
$(1)\Longrightarrow(2)$: let $\mathfrak M$ be an universal Horn class of $\Omega$-QFOs. Then there exists an universal Horn class of $\Omega$-first-order structures $\mathfrak M'$ that satisfies the same first-order theory $\mathcal T$ that $\mathfrak M$ does. If we denote the class of $\Omega$-quantitative first-order theories by $\mathbf{QFO_\Omega}$, we have $$\mathfrak M=\mathfrak M'\cap \mathbf{QFO_\Omega}.$$
Applying Theorem \ref{mt:quasi}, $\mathfrak M'$ is closed under $\mathbb{I,~S}$ and $\mathbb P_R$.
Obviously, $\mathfrak M$ is closed under $\mathbb I$, since isomorphic first-order structures satisfy the same first-order sentences. $\mathfrak M$ is also closed under $\mathbb S$, as Lemma \ref{subobjandprod} guarantees.
Let $\{\mathcal{M}_i\mid i\in I\}\subseteq\mathfrak M$ and $F$ a proper filter of $I$.
Let $\mathcal{M}\leq(\displaystyle\prod_{i\in I}\mathcal{M}_i)|_F$ such that $\mathcal{M}\in \mathbf{QFO_\Omega}$.
Since $\{\mathcal{M}_i\mid i\in I\}\subseteq\mathfrak M'$ and $\mathbb P_R(\mathfrak M')=\mathfrak M'$, we get that $(\displaystyle\prod_{i\in I}\mathcal{M}_i)|_F\in\mathfrak M'$. Hence, $\mathcal{M}\in\mathbb S(\mathfrak M')=\mathfrak M'$. And further, $\mathcal{M}\in\mathfrak M'\cap \mathbf{QFO_\Omega}=\mathfrak M$. In conclusion, $\mathfrak M$ is also closed under $\mathbb P_{SR}$.
$(2)\Longrightarrow(3)$: since $\mathfrak M$ is closed under $\mathbb{I,~S}$ and $\mathbb P_{SR}$, $$\mathfrak M=\mathbb{ISP}_{SR}(\mathfrak M).$$
$(3)\Longrightarrow(1)$: suppose that $\mathfrak M=\mathbb{ISP}_{SR}(\mathfrak M_0)$ for some class $\mathfrak M_0$ of quantitative first-order structures. \\Let $\mathfrak M'=\mathbb{ISP}_{R}(\mathfrak M)$. Applying Theorem \ref{mt:quasi}, $\mathfrak M'$ is a universal Horn class of first-order structures. We prove now that $\mathfrak M=\mathfrak M'\cap \mathbf{QFO_\Omega}$.
Let $\mathcal{M}\in\mathfrak M'\cap \mathbf{QFO_\Omega}$. Then, $\mathcal{M}$ is isomorphic to some $\mathcal{N}\leq(\displaystyle\prod_{i\in I}\mathcal{M}_i)|_F$ for some $\str{\mathcal{M}}{i}{I}\subseteq\mathfrak M$ and a proper filter $F$ of $I$, and $\mathcal{N}\in \mathbf{QFO_\Omega}$. Hence, $\mathcal{M}\in\mathbb{ISP}_{SR}(\mathfrak M)=\mathfrak M$. And this concludes that $\mathfrak M'\cap \mathbf{QFO_\Omega}\subseteq\mathfrak M$.
Since we have trivially $\mathfrak M\subseteq\mathfrak M'\cap \mathbf{QFO_\Omega}$ from the way we constructed $\mathfrak M'$, we get that $\mathfrak M=\mathfrak M'\cap \mathbf{QFO_\Omega}$.
Now, since $\mathfrak M'$ is a universal Horn class of first-order structures, we obtain that $\mathfrak M$ is a universal Horn class of quantitative first-order structures. \end{proof}
\subsection{Subreduced Products of Quantitative Algebras}
Theorem \ref{quasivar} characterizes classes of $\Omega$-QFOs as universal Horn classes. In this subsection we convert this result into a result regarding the axiomatizability of classes of quantitative algebras.
For the beginning, we note an equivalence between the conditional equations interpreted over the class of quantitative algebras and the universal Horn formulas interpreted over the class of quantitative first-order structures. This relies on the fact that a quantitative equation of type $s=_\epsilon t$ is also an atomic formula in the corresponding quantitative first-order language and vice versa. The following theorem establishes this correspondence.
\begin{thm}\label{logicalequiv}
Let $\phi_1(x_1,..,x_k)\ldots,\phi_l(x_1,..,x_k)$ and $\psi(x_1,..,x_k)$ be $\Omega$-quantitative first-order atomic formulas depending of the variables $x_1,..,x_k\in X$.
I. If $\mathcal{M}$ is an $\Omega$-quantitative first-order structure, then the following statements are equivalent $$\mathcal{M}\models\forall x_1..\forall x_k(\phi_1(x_1,..x_k)\land..\land\phi_l(x_1,..x_k)\to\psi(x_1,..x_k)),$$
$$\{\phi_1(x_1,..,x_k)\land..\land\phi_l(x_1,..,x_k)\}\models_{\mathbb G \mathcal{M}}\psi(x_1,..,x_k).$$
II. If $\mathcal{A}$ is an $\Omega$-quantitative algebra, then the following statements are equivalent $$\{\phi_1(x_1,..,x_k)\land..\land\phi_l(x_1,..,x_k)\}\models_{\mathcal{A}}\psi(x_1,..,x_k),$$
$$\mathbb F\mathcal{A}\models\forall x_1..\forall x_k(\phi_1(x_1,..x_k)\land..\land\phi_n(x_1,..x_k)\to\psi(x_1,..x_k)).$$ \end{thm}
As in the case of quantitative first-order structures, the concept of subdirect product of an indexed family of quantitative algebras for a given proper filter is not always defined. The following definition reflects this issue.
\begin{defi}[Subreduced products of Quantitative Algebras]
Let $\str{\mathcal{A}}{i}{I}$ be an indexed family of $\Omega$-quantitative algebras and $F$ a proper filter of $I$. A \textit{subreduced product} of this family induced by $F$ is a quantitative algebra $\mathcal{A}$ s.t. $$\mathbb F\mathcal{A}\leq\displaystyle\prod_{i\in I}(\mathbb F\mathcal{A}_i)|_F.$$ \end{defi}
Let $\mathbb P_{SR}(\mathcal K)$ be the closure of the class $\mathcal K$ of quantitative algebras under subreduced products. Now we can provide the analogue of Theorem \ref{quasivar} for quantitative algebras as a direct consequence of Theorem \ref{correspondence} , Theorem \ref{quasivar} and Theorem \ref{logicalequiv}.
\begin{thm}\label{QA:quasivar}
Let $\mathcal K$ be a class of $\Omega$-quantitative algebras. The following statements are equivalent.
\begin{enumerate}
\item $\mathcal K$ is a conditional equational class;
\item $\mathcal K$ is closed under $\mathbb{I,~S}$ and $\mathbb P_{SR}$;
\item $\mathcal K=\mathbb{ISP}_{SR}(\mathcal K_0)$ for some class $\mathcal K_0$ of $\Omega$-quantitative algebras.
\end{enumerate} \end{thm}
\subsection{Going further: Complete Quantitative Algebras}
The proof pattern that we developed to prove the quasivariety theorem for QFOs, Theorem \ref{quasivar}, is actually more general and it could be used to provide similar theorems for other classes of quantitative algebras. In \cite{Mardare16} we have shown that the class of quantitative algebras defined over complete metric spaces plays a central in the theory of quantitative algebras. For this reason we will briefly show how a quasivariety theorem could be done for complete metric spaces.
We call a quantitative algebra over a complete metric space a \textit{complete quantitative algebra.}
If we follow the intuition behind Theorem \ref{correspondence}, we will discover that we can define the concept of complete quantitative first-order structure as being a quantitative first-order structure for which the corresponding quantitative algebra through the functor $\mathbb G$ is a complete quantitative algebra. In fact, the completeness condition can be encoded by an infinitary axiom to be added to the conditions (1)-(6) in Definition \ref{qfo}, namely the axiom that requires that any Cauchy sequence has a limit. Let us call it the \textit{Cauchy condition}.
We will be then able to prove that the category of $\Omega$-complete quantitative algebras is isomorphic to the category of $\Omega$-complete quantitative first-order structures.
Further we can define, given a class $\mathbb M$ of $\Omega$-complete quantitative first-order structures, the concept of complete-subreduced product: given an indexed family $\str{\mathcal{M}}{i}{I}$ of $\Omega$-complete quantitative first-order structures, a complete-subreduced product is any
$\Omega$-complete quantitative first-order structure that is a subobject of the reduced product $\displaystyle\prod_{i\in I}\mathcal{M}_i|_F$ for some proper filter $F$ of $I$.
With this in hand, one can redo the proof of Theorem \ref{quasivar} in these new settings and should obtain a quasivariety theorem for complete QFOs.
\section{Conclusions}
In this paper we have established the fundamental results on the axiomatizability of classes of quantitative algebras by equations, conditional equations and Horn clauses. These results required substantial new techniques. We have not put this work into a fully categorical framework such as described in~\cite{Barr94,Adamek98,Adamek10,Manes12}. We are actively working on understanding these connections and also the connections with enriched Lawvere theories. There is also much to understand when looking at other approaches to quantitative reasoning, for example the work of Jacobs and his group~\cite{Cho15}.
\end{document}
\end{document} | arXiv |
\begin{document}
\begin{abstract} We describe three-dimensional terminal toric flips. We obtain the complete local description of three-dimensional terminal toric flips. \end{abstract}
\title{Three-dimensional terminal toric flips}
\section{Introduction} The main purpose of this paper is to describe three-dimensional terminal toric flips.
\begin{thm}[{cf.~Theorems \ref{main1} and \ref{main-taka}}]\label{11} Let $\varphi:X\to Y$ be a small proper toric morphism such that $X$ is a three-dimensional toric variety with only terminal singularities. Note that $X$ is not assumed to be $\mathbb Q$-factorial. Let $C\simeq \mathbb P^1$ be an exceptional curve of $\varphi$. Assume that $-K_X\cdot C>0$. Then one of the torus invariant points of $X$ on $C$ is non-singular and another one is a terminal quotient singularity. In particular, $X$ is $\mathbb Q$-factorial and is not Gorenstein around $C$. \end{thm}
By this result, we obtain the complete local description of three-dimensional terminal toric flips (see Theorem \ref{main1}). The first author apologies for the mistake in \cite[Example 4.4.2]{spe}, where he claims that there exist three-dimensional non-$\mathbb Q$-factorial terminal toric flips. However, Theorem \ref{11} implies that there are no such flips. This paper is based on the first author's private notes and the third author's master's thesis \cite{takano}.
We summarize the contents of this paper. In Section \ref{sec2}, we describe three-dimensional terminal toric singularities. The results are well known to the experts. Section \ref{sec3} gives the complete classification of three-dimensional $\mathbb Q$-factorial terminal toric flips. It is a supplement to \cite{kmm} and \cite[Example-Claim 14-2-5]{ma}. In Section \ref{sec4}, we prove that there are no three-dimensional non-$\mathbb Q$-factorial terminal toric flips. Theorem \ref{main-taka} is the main theorem of this paper. The proof depends on the results in Sections \ref{sec2} and \ref{sec3}.
\end{ack}
\begin{notation} We will work over $\mathbb C$, the complex number field, throughout this paper. Let $v_i\in N\simeq \mathbb Z^3$ for $1\leq i\leq k$. Then the symbol $\langle v_{1}, v_{2}, \cdots, v_{k}\rangle$ denotes the cone $\mathbb R_{\geq 0}v_{1}+\mathbb R_{\geq 0}v_2+\cdots +\mathbb R_{\geq 0}v_{k}$ in $N_{\mathbb R}\simeq N\otimes_{\mathbb Z}\mathbb R$. \end{notation}
\section{Three-dimensional terminal toric singularities}\label{sec2}
In this section, we characterize non-$\mathbb Q$-factorial affine toric threefolds with terminal singularities. We will use the same notation as in \cite{ypg}, which is an excellent exposition on terminal singularities.
Let $X$ be an affine toric threefold. First, let us recall the following well-known theorem of G.~K.~White, D.~Morrison, G.~Stevens, V.~Danilov, and M.~Frumkin (see \cite[(5.2) Theorem]{ypg}).
\begin{thm}\label{tl} Assume that $X$ is $\mathbb Q$-factorial. Then $X$ is terminal if and only if $($up to permutations of $(x,y,z)$ and symmetries of ${\mu}_r$$)$ $X\simeq \mathbb C^3/ {\mu _r}$ of type $\frac{1}{r}(a,-a,1)$ with $a$ coprime to $r$, where $\mu_r$ is the cyclic group of order $r$. In particular, if $X$ is Gorenstein and terminal, then $X$ is non-singular. \end{thm}
Here, we prove the following well-known result for the reader's convenience (cf.~\cite{ishida}, and \cite[Theorem 3.6]{ishi-iwa}).
\begin{thm}\label{main} Assume that $X$ is not $\mathbb Q$-factorial. Then $X$ is terminal if and only if $X\simeq \Spec \, \mathbb C[x,y,z,w]/(xy-zw)$. We call this singularity an {\em{ordinary double point}}. \end{thm}
By the above theorems, we obtain the complete list of three-dimensional terminal toric singularities.
\begin{rem} Mori classified three-dimensional terminal singularities. For the details, see \cite[(6.1) Theorem]{ypg}. We do not use his classification table in this paper. \end{rem}
\begin{proof}[{Proof of {\em{Theorem \ref{main}}}}] Let $N=\mathbb Z^3$ and $\Delta=\langle e_1, \cdots, e_k\rangle$ the cone in $N$ such that $X=X(\Delta)$, where each $e_i$ is primitive. First, we prove
\begin{claim}\label{4} If $X$ is non-$\mathbb Q$-factorial and terminal, then $k=4$. \end{claim} \begin{proof}[Proof of the claim] It is obvious that $k\geq 4$. Since $X$ is $\mathbb Q$-Gorenstein, there is a hyperplane $H\subset N$ that contains every $e_i$. On $H\simeq \mathbb Z^2$, $e_i$s span two dimensional convex polygon $P$. By renumbering $e_i$s, we can assume that they are arranged counter-clockwise. Since $X(\Delta)$ is terminal, all the lattice points in $P$ are $e_i$s. In particular, the triangle on $H$ spanned by $e_1$, $e_2$, and $e_3$ contains only three lattice points $e_i$ ($1\leq i\leq 3$) of $H$. So, after changing the coordinate of $H$, we can assume that $e_1=(0,1), e_2=(0,0)$, and $e_3=(1,0)$ in $H\simeq \mathbb Z^2$.
It can be checked easily that $(1,1)\in P$ since $k\geq 4$. Thus, we obtain that $k=4$ and $e_4=(1,1)$. \end{proof}
\begin{claim}\label{cl} Assume that $X$ is non-$\mathbb Q$-factorial, Gorenstein, and terminal. Then $X$ is isomorphic to $\Spec \, \mathbb C[x,y,z,w]/(xy-zw)$. \end{claim} \begin{proof}[Proof of the claim] On this assumption, the cones $\langle e_1, e_2, e_3\rangle$, $\langle e_1, e_2, e_4\rangle$, $\langle e_1, e_3, e_4\rangle$, and $\langle e_2, e_3, e_4\rangle$ define $\mathbb Q$-factorial Gorenstein affine toric threefolds with terminal singularities. By Theorem \ref{tl}, every cone listed above is non-singular. So, by changing the coordinate of $N$, we can assume that $e_1=(1,0,0)$, $e_2=(0,1,0)$, and $e_3=(0,0,1)$. Since $X$ is Gorenstein and $\langle e_1, e_2, e_4\rangle$, $\langle e_1, e_3, e_4\rangle$, and $\langle e_2, e_3, e_4\rangle$ are non-singular, $e_4=(-1, 1, 1)$, $(1, -1, 1)$, or $(1, 1, -1)$. Anyway, we can check that $X\simeq \Spec\, \mathbb C[x,y,z,w]/(xy-zw)$. \end{proof} By the above claim, it is sufficient to prove
\begin{claim}\label{6} All the non-$\mathbb Q$-factorial toric affine threefolds with terminal singularities are Gorenstein. \end{claim} \begin{proof}[Proof of the claim] We assume that $X$ is not Gorenstein and obtain a contradiction.
Let $\overline N$ be the sublattice of $N$ spanned by all the lattice points on $H$ and the origin of $N$. In $\overline N$, $\Delta=\langle e_1, e_2, e_3, e_4\rangle$ defines a Gorenstein terminal threefold. So, we can assume that $e_1=(1,0,0)$, $e_2=(0,1,0)$, $e_3=(0,0,1)$, and $e_4=(1,1,-1)\in \mathbb Z^3\simeq \overline N$ by the proof of Claim \ref{cl}. First, we consider $\langle e_1, e_2, e_3\rangle$ in $\overline N$ and $N$. By Theorem \ref{tl}, we obtain that $N=\overline N +\mathbb Z\cdot \frac{1}{r}(\alpha, \beta, \gamma)$, where $(\alpha, \beta, \gamma)$ is one of the followings: $(a, -a, 1)$, $(a, 1, -a)$, $(-a, a, 1)$, $(-a,1, a)$, $(1, a, -a)$, $(1, -a, a)$ such that $0<a<r$ with $a$ coprime to $r$. Next, we use the terminality of $\langle e_1, e_2, e_4\rangle$. We consider the linear transform $T:N\to N$ such that $Te_1=e_1$, $Te_2=e_2$, $Te_4=e_3$. Then $TN=T\overline N +\mathbb Z \cdot \frac{1}{r}(\alpha', \beta', \gamma')$, where $(\alpha', \beta', \gamma')$ is one of the followings: $(1+a,1-a,-1)$, $(0, 1-a, a)$, $(1-a, 1+a,-1)$, $(0, 1+a, -a)$, $(1-a, 0, a)$, $(1+a, 0, -a)$. Note that $$ \begin{pmatrix} \alpha'\\ \beta'\\ \gamma' \end{pmatrix} = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & -1 \end{pmatrix} \begin{pmatrix} \alpha\\ \beta\\ \gamma \end{pmatrix}. $$ We treat the first case, that is, $(\alpha', \beta', \gamma')=(1+a,1-a,-1)$. By the terminal lemma (see \cite[(5.4) Theorem]{ypg}), $r$ divides $(1+a)+(1-a)=2$ since it does not divide $(1+a)+(-1)$ nor $(1-a)+(-1)$. So, $r=2$ and $a=1$. Thus $\frac{1}{r}(\alpha', \beta', \gamma')=\frac{1}{2}(2, 0, -1)\equiv \frac{1}{2}(0,0,1)$ (mod $T\overline N$). It is a contradiction (see Theorem \ref{tl}). We leave the other cases for the reader's exercise. So, there are no non-Gorenstein non-$\mathbb Q$-factorial affine toric threefolds with terminal singularities. \end{proof} Therefore, we completed the proof of Theorem \ref{main}. \end{proof}
Theorem \ref{main} has a beautiful corollary.
\begin{cor}[Three-dimensional terminal toric flop]\label{flo} Let $$ \begin{matrix} X & \dashrightarrow & \ X^+ \\ {\ \ \ \ \ \searrow} & \ & {\swarrow}\ \ \ \ \\
\ & W & \end{matrix} $$ be a three-dimensional toric flopping diagram such that $W$ is affine. Assume that $X$ has only terminal singularities. Then it is the {\em{simplest flop}}, where the simplest flop means the flop described in \cite[p.49--p.50]{fulton}. \end{cor} \begin{proof} By the assumption, $W$ is a non-$\mathbb Q$-factorial affine toric threefold with terminal singularities. Thus, $X\simeq \Spec \, \mathbb C[x,y,z,w]/(xy-zw)$ by Theorem \ref{main}. So, the above diagram must be the simplest flop. \end{proof}
\section{Three-dimensional $\mathbb Q$-factorial terminal toric flips}\label{sec3} We classify three-dimensional flipping contractions from $\mathbb Q$-factorial terminal toric threefolds. The next theorem was stated in \cite{kmm} without proof at the end of Example 5-2-5.
\begin{thm}[Three-dimensional $\mathbb Q$-factorial terminal toric flips]\label{main1} Let $\varphi_R:X(\Delta)\to Y(\Sigma)$ be the contraction morphism of an extremal ray $R$ with $K_X\cdot R<0$ of flipping type from a toric threefold with only $\mathbb Q$-factorial terminal singularities. Assume that $Y$ is affine. Then we have the following description of the flipping contraction{\em{:}}
There exist two three-dimensional cones \begin{align*} \tau_4&=\langle v_1, v_2, v_3\rangle\in \Delta, \\ \tau_3&=\langle v_1, v_2, v_4\rangle \in \Delta, \end{align*} sharing the two-dimensional wall $$ w=\langle v_1, v_2\rangle $$ such that $[V(w)]\in R$ and that for some $\mathbb Z$-coordinate of $N\simeq \mathbb Z^3$, \begin{align*} v_1 &= (1,0,0), & v_2&=(0,1,0), & v_3&=(0,0,1),\\ v_4 &= (a,r-a,-r), \end{align*} or \begin{align*} v_1 &= (1,0,0), & v_2&=(0,1,0), & v_3&=(0,0,1),\\ v_4 &= (a,1,-r), \end{align*} where $0<a<r$ and $\gcd (r,a)=1$. Therefore, $$ \Delta=\{\tau_3, \tau_4, {\text{and their faces}}\}, $$ and $$ \Sigma=\{\langle v_1, v_2, v_3, v_4\rangle, {\text{and its faces}}\}. $$ \end{thm} \begin{proof} By \cite[Example-Claim 14-2-5]{ma}, it is sufficient to prove that the (unique) rational curve that is contracted passes through only one singular point of $X$. Without loss of generality, we may assume that $v_1=(1,0,0)$ and $v_2=(0,1,0)$ since $\langle v_1, v_2\rangle$ is a two-dimensional non-singular cone. Seeking a contradiction, we assume that both $\langle v_1, v_2, v_3\rangle$ and $\langle v_1, v_2, v_4\rangle$ are singular. By the terminal lemma (\cite[\S 1.6]{oda}), we may assume that $v_3=(1,p,q)$, where $0<p<q$ and $\gcd (p,q)=1$. We note that $q\geq 2$. We can write $v_4=av_1+bv_2+c(k, l, -1)$ with $0< a<c$, $0< b<c$, $\gcd(a,c)=1$, $\gcd(b,c)=1$, and $k, l\in \mathbb Z$. In particular, $c\geq 2$. We note that we assumed that $\langle v_1, v_2, v_4\rangle$ is singular and terminal. By the terminal lemma again (see \cite[p.36 White's Theorem]{oda}), at least one of $a-1$, $b-1$ and $a+b$ is divisible by $c$. Therefore, $a=1$, $b=1$, or $a+b=c$. We note that $v_1, v_2, v_3$ are on the plane $$x+y-\frac{p}{q}z=1.$$
\begin{ca}[$a=1$] In this case, $v_4=(1+ck, b+cl, -c)$. We have $$\frac{c}{q}v_3+v_4=(1+ck+\frac{c}{q}, b+cl+\frac{p}{q}c, 0). $$ Thus, we obtain the following three inequalities: \begin{equation}\label{1} 1+ck+\frac{c}{q}>0, \end{equation} \begin{equation}\label{2} b+cl+\frac{p}{q}c>0, \end{equation} and \begin{equation}\label{3} 1+ck+b+cl+\frac{p}{q}c<1. \end{equation} The inequalities (\ref{1}) and (\ref{2}) follow from the condition that $\varphi_R$ is small. The condition $K_X\cdot R<0$ implies the inequality (\ref{3}). By (\ref{2}) and (\ref{3}), we have $k\leq -1$. Thus $$ 0<1+ck+\frac{c}{q}\leq 1-c+\frac{c}{q}\leq 1-\frac{1}{2}c\leq 0 $$ by (\ref{1}). It is a contradiction. \end{ca}
\begin{ca}[$b=1$] In this case, $v_4=(a+ck, 1+cl, -c)$. We have $$\frac{c}{q}v_3+v_4=(a+ck+\frac{c}{q}, 1+cl+\frac{p}{q}c, 0). $$ Thus, we obtain the following three inequalities: \begin{equation}\label{4} a+ck+\frac{c}{q}>0, \end{equation} \begin{equation}\label{5} 1+cl+\frac{p}{q}c>0, \end{equation} and \begin{equation}\label{6} a+ck+1+cl+\frac{p}{q}c<1. \end{equation} By (\ref{5}) and (\ref{6}), $k\leq -1$. So, $k=-1$ by (\ref{4}). By (\ref{5}), we know that $l\geq -1$. Therefore, $l=0$ or $-1$ by (\ref{6}).
First, we assume that $l=0$. Then we get $$ a-c+\frac{p}{q}c<0 $$ by (\ref{6}) and $$ a-c+\frac{c}{q}>0 $$ by (\ref{4}). It is a contradiction.
Next, we assume that $l=-1$. Then we obtain $$ a-c+\frac{c}{q}>0 $$ by (\ref{4}) and $$ 1-c+\frac{p}{q}c>0 $$ by (\ref{5}). These two inequalities imply that $$ 1+a-2c+\frac{p+1}{q}c>0. $$ It is a contradiction. \end{ca}
\begin{ca}[$a+b=c$] In this case, $v_4=(a+ck, c-a+cl, -c)$. We have $$\frac{c}{q}v_3+v_4=(a+ck+\frac{c}{q}, c-a+cl+\frac{p}{q}c, 0). $$ Thus, we obtain the following three inequalities: \begin{equation}\label{7} a+ck+\frac{c}{q}>0, \end{equation} \begin{equation}\label{8} c-a+cl+\frac{p}{q}c>0, \end{equation} and \begin{equation}\label{9} a+ck+c-a+cl+\frac{p}{q}c<1. \end{equation} By (\ref{8}) and (\ref{9}), $k\leq -1$. So, $k=-1$ by (\ref{7}). By (\ref{8}), we have $l\geq -1$. Therefore, $l=0$ or $-1$ by (\ref{9}).
First, we assume that $l=0$. Then we have $$ \frac{p}{q}c<1 $$ by (\ref{9}) and $$ a-c+\frac{c}{q}>0 $$ by (\ref{7}). Thus, $$ 1>\frac{p}{q}c\geq \frac{c}{q}>c-a\geq 1. $$ It is a contradiction.
Next, we assume that $l=-1$. Then we obtain $$ a-c+\frac{c}{q}>0 $$ by (\ref{7}) and $$ -a+\frac{p}{q}c>0 $$ by (\ref{8}). By adding these two inequalities, we have $$ -c+\frac{p+1}{q}c>0. $$ It is a contradiction. \end{ca}
Therefore, at least one of $\langle v_1, v_2, v_3\rangle$ and $\langle v_1, v_2, v_4\rangle$ must be non-singular. Thus, we have the desired description of $\varphi_R:X\to Y$ by \cite[Example-Claim 14-2-5]{ma}. \end{proof} \begin{rem} The example in \cite[Remark 14-2-7 (ii)]{ma} is not true. The cone $\langle v_1, v_2, v_3\rangle$ is not terminal. The cone $\langle v_1, v_2, v_3\rangle$ has canonical singularities. \end{rem}
\begin{rem}\label{222} The source space $X$ in Theorem \ref{main1} is always singular. \end{rem} \begin{rem} In \cite[Example-Claim 14-2-5]{ma}, $X$ is assumed to be {\em{complete}}. It is because contraction morphisms of extremal rays are constructed only for {\em{complete}} varieties in \cite{reid} and \cite[Chapter 14]{ma}. For the details of non-complete toric varieties, see \cite{fs1}, \cite{fujino}, and \cite{sato}. \end{rem}
\section{Main Theorem}\label{sec4} The following theorem is the main theorem of this paper.
\begin{thm}[cf.~\cite{takano}]\label{main-taka} Let $\varphi:X\to Y$ be a small proper toric morphism such that $X$ is a three-dimensional toric variety with only terminal singularities. Let $C\simeq \mathbb P^1$ be an exceptional curve of $\varphi$. Assume that $-K_X\cdot C>0$. Then $C$ does not pass through ordinary double points. \end{thm}
\begin{proof} First, we assume that $C$ passes through two ordinary double points. By taking a small projective resolution of $X$, we can assume that $C$ does not pass through any singular points. It is a contradiction by Theorem \ref{main1} (see Remark \ref{222}).
Next, we assume that $C$ passes through only one ordinary double points. By Theorem \ref{main1}, we have the following local description of $X$ and $C$:
There exist lattice points of $N=\mathbb Z^3$ \begin{align*} v_1 &= (1,0,0), & v_2&=(0,1,0), & v_3&=(0,0,1),\\ v_5 &= (-1,1,1), & v_6&=(1,-1,1). && \end{align*} We put $$ \Delta_1=\{\langle v_1, v_2, v_3, v_5\rangle, \langle v_1, v_2, v_4\rangle, \text{and their faces}\}, $$ and $$ \Delta_2=\{\langle v_1, v_2, v_3, v_6\rangle, \langle v_1, v_2, v_4\rangle, \text{and their faces}\}, $$ where $v_4=(a, r-a, -r)$ or $(a, 1,-r)$ with $0<a<r$ and $\gcd(a,r)=1$. Then $X=X(\Delta)$, where $\Delta =\Delta_1$ or $\Delta_2$, and $C$ is $V(\langle v_1, v_2\rangle)\simeq \mathbb P^1$. \setcounter{ca}{0} \begin{ca} When $v_4=(a, r-a, -r)$ and $\Delta=\Delta_1$, we have $$ v_2=\frac{r}{2r-a}v_5+\frac{r-a}{2r-a}v_1+\frac{1}{2r-a} v_4. $$ Therefore, $v_2$ is contained in the cone $\langle v_5, v_1, v_4\rangle$. Thus, we can not remove the wall $\langle v_1, v_2\rangle$ from $\Delta$. \end{ca}
\begin{ca} When $v_4=(a, 1, -r)$ and $ \Delta=\Delta_1 $, we have $$ v_2=\frac{r}{r+1}v_5+\frac{r-a}{r+1}v_1+\frac{1}{r+1} v_4. $$ Therefore, $v_2$ is contained in the cone $\langle v_5, v_1, v_4\rangle$. Thus, we can not remove the wall $\langle v_1, v_2\rangle$ from $\Delta$. \end{ca}
\begin{ca} When $v_4=(a, r-a, -r)$ and $ \Delta=\Delta_2 $, we have $$ v_1=\frac{r}{r+a}v_6+\frac{a}{r+a}v_2+\frac{1}{r+a} v_4. $$ Therefore, $v_1$ is contained in the cone $\langle v_6, v_2, v_4\rangle$. Thus, we can not remove the wall $\langle v_1, v_2\rangle$ from $\Delta$. \end{ca}
\begin{ca} When $v_4=(a, 1, -r)$ and $\Delta=\Delta_2 $, we have $$ v_1=\frac{r}{r+a}v_6+\frac{r-1}{r+a}v_2+\frac{1}{r+a} v_4. $$ Therefore, $v_1$ is contained in the cone $\langle v_6, v_2, v_4\rangle$. Thus, we can not remove the wall $\langle v_1, v_2\rangle$ from $\Delta$. \end{ca} Thus, $C$ does not pass through any ordinary double points. \end{proof}
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\end{document} | arXiv |
Relativistic quantum mechanics
In physics, relativistic quantum mechanics (RQM) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light c, and can accommodate massless particles. The theory has application in high energy physics,[1] particle physics and accelerator physics,[2] as well as atomic physics, chemistry[3] and condensed matter physics.[4][5] Non-relativistic quantum mechanics refers to the mathematical formulation of quantum mechanics applied in the context of Galilean relativity, more specifically quantizing the equations of classical mechanics by replacing dynamical variables by operators. Relativistic quantum mechanics (RQM) is quantum mechanics applied with special relativity. Although the earlier formulations, like the Schrödinger picture and Heisenberg picture were originally formulated in a non-relativistic background, a few of them (e.g. the Dirac or path-integral formalism) also work with special relativity.
Part of a series of articles about
Quantum mechanics
$i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle ={\hat {H}}|\psi (t)\rangle $
Schrödinger equation
• Introduction
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Background
• Classical mechanics
• Old quantum theory
• Bra–ket notation
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• Complementarity
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Key features common to all RQMs include: the prediction of antimatter, spin magnetic moments of elementary spin 1⁄2 fermions, fine structure, and quantum dynamics of charged particles in electromagnetic fields.[6] The key result is the Dirac equation, from which these predictions emerge automatically. By contrast, in non-relativistic quantum mechanics, terms have to be introduced artificially into the Hamiltonian operator to achieve agreement with experimental observations.
The most successful (and most widely used) RQM is relativistic quantum field theory (QFT), in which elementary particles are interpreted as field quanta. A unique consequence of QFT that has been tested against other RQMs is the failure of conservation of particle number, for example in matter creation and annihilation.[7]
In this article, the equations are written in familiar 3D vector calculus notation and use hats for operators (not necessarily in the literature), and where space and time components can be collected, tensor index notation is shown also (frequently used in the literature), in addition the Einstein summation convention is used. SI units are used here; Gaussian units and natural units are common alternatives. All equations are in the position representation; for the momentum representation the equations have to be Fourier transformed – see position and momentum space.
Combining special relativity and quantum mechanics
One approach is to modify the Schrödinger picture to be consistent with special relativity.[2]
A postulate of quantum mechanics is that the time evolution of any quantum system is given by the Schrödinger equation:
$i\hbar {\frac {\partial }{\partial t}}\psi ={\hat {H}}\psi $
using a suitable Hamiltonian operator Ĥ corresponding to the system. The solution is a complex-valued wavefunction ψ(r, t), a function of the 3D position vector r of the particle at time t, describing the behavior of the system.
Every particle has a non-negative spin quantum number s. The number 2s is an integer, odd for fermions and even for bosons. Each s has 2s + 1 z-projection quantum numbers; σ = s, s − 1, ... , −s + 1, −s.[lower-alpha 1] This is an additional discrete variable the wavefunction requires; ψ(r, t, σ).
Historically, in the early 1920s Pauli, Kronig, Uhlenbeck and Goudsmit were the first to propose the concept of spin. The inclusion of spin in the wavefunction incorporates the Pauli exclusion principle (1925) and the more general spin–statistics theorem (1939) due to Fierz, rederived by Pauli a year later. This is the explanation for a diverse range of subatomic particle behavior and phenomena: from the electronic configurations of atoms, nuclei (and therefore all elements on the periodic table and their chemistry), to the quark configurations and colour charge (hence the properties of baryons and mesons).
A fundamental prediction of special relativity is the relativistic energy–momentum relation; for a particle of rest mass m, and in a particular frame of reference with energy E and 3-momentum p with magnitude in terms of the dot product $p={\sqrt {\mathbf {p} \cdot \mathbf {p} }}$, it is:[8]
$E^{2}=c^{2}\mathbf {p} \cdot \mathbf {p} +(mc^{2})^{2}\,.$
These equations are used together with the energy and momentum operators, which are respectively:
${\hat {E}}=i\hbar {\frac {\partial }{\partial t}}\,,\quad {\hat {\mathbf {p} }}=-i\hbar \nabla \,,$
to construct a relativistic wave equation (RWE): a partial differential equation consistent with the energy–momentum relation, and is solved for ψ to predict the quantum dynamics of the particle. For space and time to be placed on equal footing, as in relativity, the orders of space and time partial derivatives should be equal, and ideally as low as possible, so that no initial values of the derivatives need to be specified. This is important for probability interpretations, exemplified below. The lowest possible order of any differential equation is the first (zeroth order derivatives would not form a differential equation).
The Heisenberg picture is another formulation of QM, in which case the wavefunction ψ is time-independent, and the operators A(t) contain the time dependence, governed by the equation of motion:
${\frac {d}{dt}}A={\frac {1}{i\hbar }}[A,{\hat {H}}]+{\frac {\partial }{\partial t}}A\,,$
This equation is also true in RQM, provided the Heisenberg operators are modified to be consistent with SR.[9][10]
Historically, around 1926, Schrödinger and Heisenberg show that wave mechanics and matrix mechanics are equivalent, later furthered by Dirac using transformation theory.
A more modern approach to RWEs, first introduced during the time RWEs were developing for particles of any spin, is to apply representations of the Lorentz group.
Space and time
In classical mechanics and non-relativistic QM, time is an absolute quantity all observers and particles can always agree on, "ticking away" in the background independent of space. Thus in non-relativistic QM one has for a many particle system ψ(r1, r2, r3, ..., t, σ1, σ2, σ3...).
In relativistic mechanics, the spatial coordinates and coordinate time are not absolute; any two observers moving relative to each other can measure different locations and times of events. The position and time coordinates combine naturally into a four-dimensional spacetime position X = (ct, r) corresponding to events, and the energy and 3-momentum combine naturally into the four-momentum P = (E/c, p) of a dynamic particle, as measured in some reference frame, change according to a Lorentz transformation as one measures in a different frame boosted and/or rotated relative the original frame in consideration. The derivative operators, and hence the energy and 3-momentum operators, are also non-invariant and change under Lorentz transformations.
Under a proper orthochronous Lorentz transformation (r, t) → Λ(r, t) in Minkowski space, all one-particle quantum states ψσ locally transform under some representation D of the Lorentz group:[11] [12]
$\psi _{\sigma }(\mathbf {r} ,t)\rightarrow D(\Lambda )\psi _{\sigma }(\Lambda ^{-1}(\mathbf {r} ,t))$
where D(Λ) is a finite-dimensional representation, in other words a (2s + 1)×(2s + 1) square matrix . Again, ψ is thought of as a column vector containing components with the (2s + 1) allowed values of σ. The quantum numbers s and σ as well as other labels, continuous or discrete, representing other quantum numbers are suppressed. One value of σ may occur more than once depending on the representation.
Further information: Generator (mathematics), group theory, Representation theory of the Lorentz group, and symmetries in quantum mechanics
Non-relativistic and relativistic Hamiltonians
The classical Hamiltonian for a particle in a potential is the kinetic energy p·p/2m plus the potential energy V(r, t), with the corresponding quantum operator in the Schrödinger picture:
${\hat {H}}={\frac {{\hat {\mathbf {p} }}\cdot {\hat {\mathbf {p} }}}{2m}}+V(\mathbf {r} ,t)$
and substituting this into the above Schrödinger equation gives a non-relativistic QM equation for the wavefunction: the procedure is a straightforward substitution of a simple expression. By contrast this is not as easy in RQM; the energy–momentum equation is quadratic in energy and momentum leading to difficulties. Naively setting:
${\hat {H}}={\hat {E}}={\sqrt {c^{2}{\hat {\mathbf {p} }}\cdot {\hat {\mathbf {p} }}+(mc^{2})^{2}}}\quad \Rightarrow \quad i\hbar {\frac {\partial }{\partial t}}\psi ={\sqrt {c^{2}{\hat {\mathbf {p} }}\cdot {\hat {\mathbf {p} }}+(mc^{2})^{2}}}\,\psi $
is not helpful for several reasons. The square root of the operators cannot be used as it stands; it would have to be expanded in a power series before the momentum operator, raised to a power in each term, could act on ψ. As a result of the power series, the space and time derivatives are completely asymmetric: infinite-order in space derivatives but only first order in the time derivative, which is inelegant and unwieldy. Again, there is the problem of the non-invariance of the energy operator, equated to the square root which is also not invariant. Another problem, less obvious and more severe, is that it can be shown to be nonlocal and can even violate causality: if the particle is initially localized at a point r0 so that ψ(r0, t = 0) is finite and zero elsewhere, then at any later time the equation predicts delocalization ψ(r, t) ≠ 0 everywhere, even for |r| > ct which means the particle could arrive at a point before a pulse of light could. This would have to be remedied by the additional constraint ψ(|r| > ct, t) = 0.[13]
There is also the problem of incorporating spin in the Hamiltonian, which isn't a prediction of the non-relativistic Schrödinger theory. Particles with spin have a corresponding spin magnetic moment quantized in units of μB, the Bohr magneton:[14][15]
${\hat {\boldsymbol {\mu }}}_{S}=-{\frac {g\mu _{B}}{\hbar }}{\hat {\mathbf {S} }}\,,\quad \left|{\boldsymbol {\mu }}_{S}\right|=-g\mu _{B}\sigma \,,$
where g is the (spin) g-factor for the particle, and S the spin operator, so they interact with electromagnetic fields. For a particle in an externally applied magnetic field B, the interaction term[16]
${\hat {H}}_{B}=-\mathbf {B} \cdot {\hat {\boldsymbol {\mu }}}_{S}$
has to be added to the above non-relativistic Hamiltonian. On the contrary; a relativistic Hamiltonian introduces spin automatically as a requirement of enforcing the relativistic energy-momentum relation.[17]
Relativistic Hamiltonians are analogous to those of non-relativistic QM in the following respect; there are terms including rest mass and interaction terms with externally applied fields, similar to the classical potential energy term, as well as momentum terms like the classical kinetic energy term. A key difference is that relativistic Hamiltonians contain spin operators in the form of matrices, in which the matrix multiplication runs over the spin index σ, so in general a relativistic Hamiltonian:
${\hat {H}}={\hat {H}}(\mathbf {r} ,t,{\hat {\mathbf {p} }},{\hat {\mathbf {S} }})$
is a function of space, time, and the momentum and spin operators.
The Klein–Gordon and Dirac equations for free particles
Substituting the energy and momentum operators directly into the energy–momentum relation may at first sight seem appealing, to obtain the Klein–Gordon equation:[18]
${\hat {E}}^{2}\psi =c^{2}{\hat {\mathbf {p} }}\cdot {\hat {\mathbf {p} }}\psi +(mc^{2})^{2}\psi \,,$
and was discovered by many people because of the straightforward way of obtaining it, notably by Schrödinger in 1925 before he found the non-relativistic equation named after him, and by Klein and Gordon in 1927, who included electromagnetic interactions in the equation. This is relativistically invariant, yet this equation alone isn't a sufficient foundation for RQM for a at least two reasons: one is that negative-energy states are solutions,[2][19] another is the density (given below), and this equation as it stands is only applicable to spinless particles. This equation can be factored into the form:[20][21]
$\left({\hat {E}}-c{\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}-\beta mc^{2}\right)\left({\hat {E}}+c{\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}+\beta mc^{2}\right)\psi =0\,,$
where α = (α1, α2, α3) and β are not simply numbers or vectors, but 4 × 4 Hermitian matrices that are required to anticommute for i ≠ j:
$\alpha _{i}\beta =-\beta \alpha _{i},\quad \alpha _{i}\alpha _{j}=-\alpha _{j}\alpha _{i}\,,$
and square to the identity matrix:
$\alpha _{i}^{2}=\beta ^{2}=I\,,$
so that terms with mixed second-order derivatives cancel while the second-order derivatives purely in space and time remain. The first factor:
$\left({\hat {E}}-c{\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}-\beta mc^{2}\right)\psi =0\quad \Leftrightarrow \quad {\hat {H}}=c{\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}+\beta mc^{2}$
is the Dirac equation. The other factor is also the Dirac equation, but for a particle of negative mass.[20] Each factor is relativistically invariant. The reasoning can be done the other way round: propose the Hamiltonian in the above form, as Dirac did in 1928, then pre-multiply the equation by the other factor of operators E + cα · p + βmc2, and comparison with the KG equation determines the constraints on α and β. The positive mass equation can continue to be used without loss of continuity. The matrices multiplying ψ suggest it isn't a scalar wavefunction as permitted in the KG equation, but must instead be a four-component entity. The Dirac equation still predicts negative energy solutions,[6][22] so Dirac postulated that negative energy states are always occupied, because according to the Pauli principle, electronic transitions from positive to negative energy levels in atoms would be forbidden. See Dirac sea for details.
Densities and currents
In non-relativistic quantum mechanics, the square modulus of the wavefunction ψ gives the probability density function ρ = |ψ|2. This is the Copenhagen interpretation, circa 1927. In RQM, while ψ(r, t) is a wavefunction, the probability interpretation is not the same as in non-relativistic QM. Some RWEs do not predict a probability density ρ or probability current j (really meaning probability current density) because they are not positive-definite functions of space and time. The Dirac equation does:[23]
$\rho =\psi ^{\dagger }\psi ,\quad \mathbf {j} =\psi ^{\dagger }\gamma ^{0}{\boldsymbol {\gamma }}\psi \quad \rightleftharpoons \quad J^{\mu }=\psi ^{\dagger }\gamma ^{0}\gamma ^{\mu }\psi $
where the dagger denotes the Hermitian adjoint (authors usually write ψ = ψ†γ0 for the Dirac adjoint) and Jμ is the probability four-current, while the Klein–Gordon equation does not:[24]
$\rho ={\frac {i\hbar }{2mc^{2}}}\left(\psi ^{*}{\frac {\partial \psi }{\partial t}}-\psi {\frac {\partial \psi ^{*}}{\partial t}}\right)\,,\quad \mathbf {j} =-{\frac {i\hbar }{2m}}\left(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*}\right)\quad \rightleftharpoons \quad J^{\mu }={\frac {i\hbar }{2m}}(\psi ^{*}\partial ^{\mu }\psi -\psi \partial ^{\mu }\psi ^{*})$
where ∂μ is the four-gradient. Since the initial values of both ψ and ∂ψ/∂t may be freely chosen, the density can be negative.
Instead, what appears look at first sight a "probability density" and "probability current" has to be reinterpreted as charge density and current density when multiplied by electric charge. Then, the wavefunction ψ is not a wavefunction at all, but reinterpreted as a field.[13] The density and current of electric charge always satisfy a continuity equation:
${\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0\quad \rightleftharpoons \quad \partial _{\mu }J^{\mu }=0\,,$
as charge is a conserved quantity. Probability density and current also satisfy a continuity equation because probability is conserved, however this is only possible in the absence of interactions.
Spin and electromagnetically interacting particles
Including interactions in RWEs is generally difficult. Minimal coupling is a simple way to include the electromagnetic interaction. For one charged particle of electric charge q in an electromagnetic field, given by the magnetic vector potential A(r, t) defined by the magnetic field B = ∇ × A, and electric scalar potential ϕ(r, t), this is:[25]
${\hat {E}}\rightarrow {\hat {E}}-q\phi \,,\quad {\hat {\mathbf {p} }}\rightarrow {\hat {\mathbf {p} }}-q\mathbf {A} \quad \rightleftharpoons \quad {\hat {P}}_{\mu }\rightarrow {\hat {P}}_{\mu }-qA_{\mu }$
where Pμ is the four-momentum that has a corresponding 4-momentum operator, and Aμ the four-potential. In the following, the non-relativistic limit refers to the limiting cases:
$E-e\phi \approx mc^{2}\,,\quad \mathbf {p} \approx m\mathbf {v} \,,$
that is, the total energy of the particle is approximately the rest energy for small electric potentials, and the momentum is approximately the classical momentum.
Spin 0
In RQM, the KG equation admits the minimal coupling prescription;
${({\hat {E}}-q\phi )}^{2}\psi =c^{2}{({\hat {\mathbf {p} }}-q\mathbf {A} )}^{2}\psi +(mc^{2})^{2}\psi \quad \rightleftharpoons \quad \left[{({\hat {P}}_{\mu }-qA_{\mu })}{({\hat {P}}^{\mu }-qA^{\mu })}-{(mc)}^{2}\right]\psi =0.$
In the case where the charge is zero, the equation reduces trivially to the free KG equation so nonzero charge is assumed below. This is a scalar equation that is invariant under the irreducible one-dimensional scalar (0,0) representation of the Lorentz group. This means that all of its solutions will belong to a direct sum of (0,0) representations. Solutions that do not belong to the irreducible (0,0) representation will have two or more independent components. Such solutions cannot in general describe particles with nonzero spin since spin components are not independent. Other constraint will have to be imposed for that, e.g. the Dirac equation for spin 1/2, see below. Thus if a system satisfies the KG equation only, it can only be interpreted as a system with zero spin.
The electromagnetic field is treated classically according to Maxwell's equations and the particle is described by a wavefunction, the solution to the KG equation. The equation is, as it stands, not always very useful, because massive spinless particles, such as the π-mesons, experience the much stronger strong interaction in addition to the electromagnetic interaction. It does, however, correctly describe charged spinless bosons in the absence of other interactions.
The KG equation is applicable to spinless charged bosons in an external electromagnetic potential.[2] As such, the equation cannot be applied to the description of atoms, since the electron is a spin 1/2 particle. In the non-relativistic limit the equation reduces to the Schrödinger equation for a spinless charged particle in an electromagnetic field:[16]
$\left(i\hbar {\frac {\partial }{\partial t}}-q\phi \right)\psi ={\frac {1}{2m}}{({\hat {\mathbf {p} }}-q\mathbf {A} )}^{2}\psi \quad \Leftrightarrow \quad {\hat {H}}={\frac {1}{2m}}{({\hat {\mathbf {p} }}-q\mathbf {A} )}^{2}+q\phi .$
Spin 1/2
Non relativistically, spin was phenomenologically introduced in the Pauli equation by Pauli in 1927 for particles in an electromagnetic field:
$\left(i\hbar {\frac {\partial }{\partial t}}-q\phi \right)\psi =\left[{\frac {1}{2m}}{({\boldsymbol {\sigma }}\cdot (\mathbf {p} -q\mathbf {A} ))}^{2}\right]\psi \quad \Leftrightarrow \quad {\hat {H}}={\frac {1}{2m}}{({\boldsymbol {\sigma }}\cdot (\mathbf {p} -q\mathbf {A} ))}^{2}+q\phi $
by means of the 2 × 2 Pauli matrices, and ψ is not just a scalar wavefunction as in the non-relativistic Schrödinger equation, but a two-component spinor field:
$\psi ={\begin{pmatrix}\psi _{\uparrow }\\\psi _{\downarrow }\end{pmatrix}}$
where the subscripts ↑ and ↓ refer to the "spin up" (σ = +1/2) and "spin down" (σ = −1/2) states.[lower-alpha 2]
In RQM, the Dirac equation can also incorporate minimal coupling, rewritten from above;
$\left(i\hbar {\frac {\partial }{\partial t}}-q\phi \right)\psi =\gamma ^{0}\left[c{\boldsymbol {\gamma }}\cdot {({\hat {\mathbf {p} }}-q\mathbf {A} )}-mc^{2}\right]\psi \quad \rightleftharpoons \quad \left[\gamma ^{\mu }({\hat {P}}_{\mu }-qA_{\mu })-mc^{2}\right]\psi =0$
and was the first equation to accurately predict spin, a consequence of the 4 × 4 gamma matrices γ0 = β, γ = (γ1, γ2, γ3) = βα = (βα1, βα2, βα3). There is a 4 × 4 identity matrix pre-multiplying the energy operator (including the potential energy term), conventionally not written for simplicity and clarity (i.e. treated like the number 1). Here ψ is a four-component spinor field, which is conventionally split into two two-component spinors in the form:[lower-alpha 3]
$\psi ={\begin{pmatrix}\psi _{+}\\\psi _{-}\end{pmatrix}}={\begin{pmatrix}\psi _{+\uparrow }\\\psi _{+\downarrow }\\\psi _{-\uparrow }\\\psi _{-\downarrow }\end{pmatrix}}$
The 2-spinor ψ+ corresponds to a particle with 4-momentum (E, p) and charge q and two spin states (σ = ±1/2, as before). The other 2-spinor ψ− corresponds to a similar particle with the same mass and spin states, but negative 4-momentum −(E, p) and negative charge −q, that is, negative energy states, time-reversed momentum, and negated charge. This was the first interpretation and prediction of a particle and corresponding antiparticle. See Dirac spinor and bispinor for further description of these spinors. In the non-relativistic limit the Dirac equation reduces to the Pauli equation (see Dirac equation for how). When applied a one-electron atom or ion, setting A = 0 and ϕ to the appropriate electrostatic potential, additional relativistic terms include the spin–orbit interaction, electron gyromagnetic ratio, and Darwin term. In ordinary QM these terms have to be put in by hand and treated using perturbation theory. The positive energies do account accurately for the fine structure.
Within RQM, for massless particles the Dirac equation reduces to:
$\left({\frac {\hat {E}}{c}}+{\boldsymbol {\sigma }}\cdot {\hat {\mathbf {p} }}\right)\psi _{+}=0\,,\quad \left({\frac {\hat {E}}{c}}-{\boldsymbol {\sigma }}\cdot {\hat {\mathbf {p} }}\right)\psi _{-}=0\quad \rightleftharpoons \quad \sigma ^{\mu }{\hat {P}}_{\mu }\psi _{+}=0\,,\quad \sigma _{\mu }{\hat {P}}^{\mu }\psi _{-}=0\,,$
the first of which is the Weyl equation, a considerable simplification applicable for massless neutrinos.[26] This time there is a 2 × 2 identity matrix pre-multiplying the energy operator conventionally not written. In RQM it is useful to take this as the zeroth Pauli matrix σ0 which couples to the energy operator (time derivative), just as the other three matrices couple to the momentum operator (spatial derivatives).
The Pauli and gamma matrices were introduced here, in theoretical physics, rather than pure mathematics itself. They have applications to quaternions and to the SO(2) and SO(3) Lie groups, because they satisfy the important commutator [ , ] and anticommutator [ , ]+ relations respectively:
$\left[\sigma _{a},\sigma _{b}\right]=2i\varepsilon _{abc}\sigma _{c}\,,\quad \left[\sigma _{a},\sigma _{b}\right]_{+}=2\delta _{ab}\sigma _{0}$
where εabc is the three-dimensional Levi-Civita symbol. The gamma matrices form bases in Clifford algebra, and have a connection to the components of the flat spacetime Minkowski metric ηαβ in the anticommutation relation:
$\left[\gamma ^{\alpha },\gamma ^{\beta }\right]_{+}=\gamma ^{\alpha }\gamma ^{\beta }+\gamma ^{\beta }\gamma ^{\alpha }=2\eta ^{\alpha \beta }\,,$
(This can be extended to curved spacetime by introducing vierbeins, but is not the subject of special relativity).
In 1929, the Breit equation was found to describe two or more electromagnetically interacting massive spin 1/2 fermions to first-order relativistic corrections; one of the first attempts to describe such a relativistic quantum many-particle system. This is, however, still only an approximation, and the Hamiltonian includes numerous long and complicated sums.
Helicity and chirality
The helicity operator is defined by;
${\hat {h}}={\hat {\mathbf {S} }}\cdot {\frac {\hat {\mathbf {p} }}{|\mathbf {p} |}}={\hat {\mathbf {S} }}\cdot {\frac {c{\hat {\mathbf {p} }}}{\sqrt {E^{2}-(m_{0}c^{2})^{2}}}}$
where p is the momentum operator, S the spin operator for a particle of spin s, E is the total energy of the particle, and m0 its rest mass. Helicity indicates the orientations of the spin and translational momentum vectors.[27] Helicity is frame-dependent because of the 3-momentum in the definition, and is quantized due to spin quantization, which has discrete positive values for parallel alignment, and negative values for antiparallel alignment.
An automatic occurrence in the Dirac equation (and the Weyl equation) is the projection of the spin 1/2 operator on the 3-momentum (times c), σ · c p, which is the helicity (for the spin 1/2 case) times ${\sqrt {E^{2}-(m_{0}c^{2})^{2}}}$.
For massless particles the helicity simplifies to:
${\hat {h}}={\hat {\mathbf {S} }}\cdot {\frac {c{\hat {\mathbf {p} }}}{E}}$
Higher spins
The Dirac equation can only describe particles of spin 1/2. Beyond the Dirac equation, RWEs have been applied to free particles of various spins. In 1936, Dirac extended his equation to all fermions, three years later Fierz and Pauli rederived the same equation.[28] The Bargmann–Wigner equations were found in 1948 using Lorentz group theory, applicable for all free particles with any spin.[29][30] Considering the factorization of the KG equation above, and more rigorously by Lorentz group theory, it becomes apparent to introduce spin in the form of matrices.
The wavefunctions are multicomponent spinor fields, which can be represented as column vectors of functions of space and time:
$\psi (\mathbf {r} ,t)={\begin{bmatrix}\psi _{\sigma =s}(\mathbf {r} ,t)\\\psi _{\sigma =s-1}(\mathbf {r} ,t)\\\vdots \\\psi _{\sigma =-s+1}(\mathbf {r} ,t)\\\psi _{\sigma =-s}(\mathbf {r} ,t)\end{bmatrix}}\quad \rightleftharpoons \quad {\psi (\mathbf {r} ,t)}^{\dagger }={\begin{bmatrix}{\psi _{\sigma =s}(\mathbf {r} ,t)}^{\star }&{\psi _{\sigma =s-1}(\mathbf {r} ,t)}^{\star }&\cdots &{\psi _{\sigma =-s+1}(\mathbf {r} ,t)}^{\star }&{\psi _{\sigma =-s}(\mathbf {r} ,t)}^{\star }\end{bmatrix}}$
where the expression on the right is the Hermitian conjugate. For a massive particle of spin s, there are 2s + 1 components for the particle, and another 2s + 1 for the corresponding antiparticle (there are 2s + 1 possible σ values in each case), altogether forming a 2(2s + 1)-component spinor field:
$\psi (\mathbf {r} ,t)={\begin{bmatrix}\psi _{+,\,\sigma =s}(\mathbf {r} ,t)\\\psi _{+,\,\sigma =s-1}(\mathbf {r} ,t)\\\vdots \\\psi _{+,\,\sigma =-s+1}(\mathbf {r} ,t)\\\psi _{+,\,\sigma =-s}(\mathbf {r} ,t)\\\psi _{-,\,\sigma =s}(\mathbf {r} ,t)\\\psi _{-,\,\sigma =s-1}(\mathbf {r} ,t)\\\vdots \\\psi _{-,\,\sigma =-s+1}(\mathbf {r} ,t)\\\psi _{-,\,\sigma =-s}(\mathbf {r} ,t)\end{bmatrix}}\quad \rightleftharpoons \quad {\psi (\mathbf {r} ,t)}^{\dagger }{\begin{bmatrix}{\psi _{+,\,\sigma =s}(\mathbf {r} ,t)}^{\star }&{\psi _{+,\,\sigma =s-1}(\mathbf {r} ,t)}^{\star }&\cdots &{\psi _{-,\,\sigma =-s}(\mathbf {r} ,t)}^{\star }\end{bmatrix}}$
with the + subscript indicating the particle and − subscript for the antiparticle. However, for massless particles of spin s, there are only ever two-component spinor fields; one is for the particle in one helicity state corresponding to +s and the other for the antiparticle in the opposite helicity state corresponding to −s:
$\psi (\mathbf {r} ,t)={\begin{pmatrix}\psi _{+}(\mathbf {r} ,t)\\\psi _{-}(\mathbf {r} ,t)\end{pmatrix}}$
According to the relativistic energy-momentum relation, all massless particles travel at the speed of light, so particles traveling at the speed of light are also described by two-component spinors. Historically, Élie Cartan found the most general form of spinors in 1913, prior to the spinors revealed in the RWEs following the year 1927.
For equations describing higher-spin particles, the inclusion of interactions is nowhere near as simple minimal coupling, they lead to incorrect predictions and self-inconsistencies.[31] For spin greater than ħ/2, the RWE is not fixed by the particle's mass, spin, and electric charge; the electromagnetic moments (electric dipole moments and magnetic dipole moments) allowed by the spin quantum number are arbitrary. (Theoretically, magnetic charge would contribute also). For example, the spin 1/2 case only allows a magnetic dipole, but for spin 1 particles magnetic quadrupoles and electric dipoles are also possible.[26] For more on this topic, see multipole expansion and (for example) Cédric Lorcé (2009).[32][33]
Velocity operator
The Schrödinger/Pauli velocity operator can be defined for a massive particle using the classical definition p = m v, and substituting quantum operators in the usual way:[34]
${\hat {\mathbf {v} }}={\frac {1}{m}}{\hat {\mathbf {p} }}$
which has eigenvalues that take any value. In RQM, the Dirac theory, it is:
${\hat {\mathbf {v} }}={\frac {i}{\hbar }}\left[{\hat {H}},{\hat {\mathbf {r} }}\right]$
which must have eigenvalues between ±c. See Foldy–Wouthuysen transformation for more theoretical background.
Relativistic quantum Lagrangians
The Hamiltonian operators in the Schrödinger picture are one approach to forming the differential equations for ψ. An equivalent alternative is to determine a Lagrangian (really meaning Lagrangian density), then generate the differential equation by the field-theoretic Euler–Lagrange equation:
$\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\right)-{\frac {\partial {\mathcal {L}}}{\partial \psi }}=0\,$
For some RWEs, a Lagrangian can be found by inspection. For example, the Dirac Lagrangian is:[35]
${\mathcal {L}}={\overline {\psi }}(\gamma ^{\mu }P_{\mu }-mc)\psi $
and Klein–Gordon Lagrangian is:
${\mathcal {L}}=-{\frac {\hbar ^{2}}{m}}\eta ^{\mu \nu }\partial _{\mu }\psi ^{*}\partial _{\nu }\psi -mc^{2}\psi ^{*}\psi \,.$
This is not possible for all RWEs; and is one reason the Lorentz group theoretic approach is important and appealing: fundamental invariance and symmetries in space and time can be used to derive RWEs using appropriate group representations. The Lagrangian approach with field interpretation of ψ is the subject of QFT rather than RQM: Feynman's path integral formulation uses invariant Lagrangians rather than Hamiltonian operators, since the latter can become extremely complicated, see (for example) Weinberg (1995).[36]
Relativistic quantum angular momentum
In non-relativistic QM, the angular momentum operator is formed from the classical pseudovector definition L = r × p. In RQM, the position and momentum operators are inserted directly where they appear in the orbital relativistic angular momentum tensor defined from the four-dimensional position and momentum of the particle, equivalently a bivector in the exterior algebra formalism:[37][lower-alpha 4]
$M^{\alpha \beta }=X^{\alpha }P^{\beta }-X^{\beta }P^{\alpha }=2X^{[\alpha }P^{\beta ]}\quad \rightleftharpoons \quad \mathbf {M} =\mathbf {X} \wedge \mathbf {P} \,,$
which are six components altogether: three are the non-relativistic 3-orbital angular momenta; M12 = L3, M23 = L1, M31 = L2, and the other three M01, M02, M03 are boosts of the centre of mass of the rotating object. An additional relativistic-quantum term has to be added for particles with spin. For a particle of rest mass m, the total angular momentum tensor is:
$J^{\alpha \beta }=2X^{[\alpha }P^{\beta ]}+{\frac {1}{m^{2}}}\varepsilon ^{\alpha \beta \gamma \delta }W_{\gamma }p_{\delta }\quad \rightleftharpoons \quad \mathbf {J} =\mathbf {X} \wedge \mathbf {P} +{\frac {1}{m^{2}}}\star (\mathbf {W} \wedge \mathbf {P} )$
where the star denotes the Hodge dual, and
$W_{\alpha }={\frac {1}{2}}\varepsilon _{\alpha \beta \gamma \delta }M^{\beta \gamma }p^{\delta }\quad \rightleftharpoons \quad \mathbf {W} =\star (\mathbf {M} \wedge \mathbf {P} )$
is the Pauli–Lubanski pseudovector.[38] For more on relativistic spin, see (for example) Troshin & Tyurin (1994).[39]
Thomas precession and spin–orbit interactions
In 1926, the Thomas precession is discovered: relativistic corrections to the spin of elementary particles with application in the spin–orbit interaction of atoms and rotation of macroscopic objects.[40][41] In 1939 Wigner derived the Thomas precession.
In classical electromagnetism and special relativity, an electron moving with a velocity v through an electric field E but not a magnetic field B, will in its own frame of reference experience a Lorentz-transformed magnetic field B′:
$\mathbf {B} '={\frac {\mathbf {E} \times \mathbf {v} }{c^{2}{\sqrt {1-\left(v/c\right)^{2}}}}}\,.$
In the non-relativistic limit v << c:
$\mathbf {B} '={\frac {\mathbf {E} \times \mathbf {v} }{c^{2}}}\,,$
so the non-relativistic spin interaction Hamiltonian becomes:[42]
${\hat {H}}=-\mathbf {B} '\cdot {\hat {\boldsymbol {\mu }}}_{S}=-\left(\mathbf {B} +{\frac {\mathbf {E} \times \mathbf {v} }{c^{2}}}\right)\cdot {\hat {\boldsymbol {\mu }}}_{S}\,,$
where the first term is already the non-relativistic magnetic moment interaction, and the second term the relativistic correction of order (v/c)², but this disagrees with experimental atomic spectra by a factor of 1⁄2. It was pointed out by L. Thomas that there is a second relativistic effect: An electric field component perpendicular to the electron velocity causes an additional acceleration of the electron perpendicular to its instantaneous velocity, so the electron moves in a curved path. The electron moves in a rotating frame of reference, and this additional precession of the electron is called the Thomas precession. It can be shown[43] that the net result of this effect is that the spin–orbit interaction is reduced by half, as if the magnetic field experienced by the electron has only one-half the value, and the relativistic correction in the Hamiltonian is:
${\hat {H}}=-\mathbf {B} '\cdot {\hat {\boldsymbol {\mu }}}_{S}=-\left(\mathbf {B} +{\frac {\mathbf {E} \times \mathbf {v} }{2c^{2}}}\right)\cdot {\hat {\boldsymbol {\mu }}}_{S}\,.$
In the case of RQM, the factor of 1⁄2 is predicted by the Dirac equation.[42]
History
The events which led to and established RQM, and the continuation beyond into quantum electrodynamics (QED), are summarized below [see, for example, R. Resnick and R. Eisberg (1985),[44] and P.W Atkins (1974)[45]]. More than half a century of experimental and theoretical research from the 1890s through to the 1950s in the new and mysterious quantum theory as it was up and coming revealed that a number of phenomena cannot be explained by QM alone. SR, found at the turn of the 20th century, was found to be a necessary component, leading to unification: RQM. Theoretical predictions and experiments mainly focused on the newly found atomic physics, nuclear physics, and particle physics; by considering spectroscopy, diffraction and scattering of particles, and the electrons and nuclei within atoms and molecules. Numerous results are attributed to the effects of spin.
Relativistic description of particles in quantum phenomena
Albert Einstein in 1905 explained of the photoelectric effect; a particle description of light as photons. In 1916, Sommerfeld explains fine structure; the splitting of the spectral lines of atoms due to first order relativistic corrections. The Compton effect of 1923 provided more evidence that special relativity does apply; in this case to a particle description of photon–electron scattering. de Broglie extends wave–particle duality to matter: the de Broglie relations, which are consistent with special relativity and quantum mechanics. By 1927, Davisson and Germer and separately G. Thomson successfully diffract electrons, providing experimental evidence of wave-particle duality.
Experiments
• 1897 J. J. Thomson discovers the electron and measures its mass-to-charge ratio. Discovery of the Zeeman effect: the splitting a spectral line into several components in the presence of a static magnetic field.
• 1908 Millikan measures the charge on the electron and finds experimental evidence of its quantization, in the oil drop experiment.
• 1911 Alpha particle scattering in the Geiger–Marsden experiment, led by Rutherford, showed that atoms possess an internal structure: the atomic nucleus.[46]
• 1913 The Stark effect is discovered: splitting of spectral lines due to a static electric field (compare with the Zeeman effect).
• 1922 Stern–Gerlach experiment: experimental evidence of spin and its quantization.
• 1924 Stoner studies splitting of energy levels in magnetic fields.
• 1932 Experimental discovery of the neutron by Chadwick, and positrons by Anderson, confirming the theoretical prediction of positrons.
• 1958 Discovery of the Mössbauer effect: resonant and recoil-free emission and absorption of gamma radiation by atomic nuclei bound in a solid, useful for accurate measurements of gravitational redshift and time dilation, and in the analysis of nuclear electromagnetic moments in hyperfine interactions.[47]
Quantum non-locality and relativistic locality
In 1935, Einstein, Rosen, Podolsky published a paper[48] concerning quantum entanglement of particles, questioning quantum nonlocality and the apparent violation of causality upheld in SR: particles can appear to interact instantaneously at arbitrary distances. This was a misconception since information is not and cannot be transferred in the entangled states; rather the information transmission is in the process of measurement by two observers (one observer has to send a signal to the other, which cannot exceed c). QM does not violate SR.[49][50] In 1959, Bohm and Aharonov publish a paper[51] on the Aharonov–Bohm effect, questioning the status of electromagnetic potentials in QM. The EM field tensor and EM 4-potential formulations are both applicable in SR, but in QM the potentials enter the Hamiltonian (see above) and influence the motion of charged particles even in regions where the fields are zero. In 1964, Bell's theorem was published in a paper on the EPR paradox,[52] showing that QM cannot be derived from local hidden-variable theories if locality is to be maintained.
The Lamb shift
In 1947, the Lamb shift was discovered: a small difference in the 2S1⁄2 and 2P1⁄2 levels of hydrogen, due to the interaction between the electron and vacuum. Lamb and Retherford experimentally measure stimulated radio-frequency transitions the 2S1⁄2 and 2P1⁄2 hydrogen levels by microwave radiation.[53] An explanation of the Lamb shift is presented by Bethe. Papers on the effect were published in the early 1950s.[54]
Development of quantum electrodynamics
• 1943 Tomonaga begins work on renormalization, influential in QED.
• 1947 Schwinger calculates the anomalous magnetic moment of the electron. Kusch measures of the anomalous magnetic electron moment, confirming one of QED's great predictions.
See also
Atomic physics and chemistry
• Relativistic quantum chemistry
• Breit equation
• Electron spin resonance
• Fine-structure constant
Mathematical physics
• Quantum spacetime
• Spin connection
• Spinor bundle
• Dirac equation in the algebra of physical space
• Casimir invariant
• Casimir operator
• Wigner D-matrix
Particle physics and quantum field theory
• Zitterbewegung
• Two-body Dirac equations
• Relativistic Heavy Ion Collider
• Symmetry (physics)
• Parity
• CPT invariance
• Chirality (physics)
• Standard model
• Gauge theory
• Tachyon
• Modern searches for Lorentz violation
Footnotes
1. Other common notations include ms and sz etc., but this would clutter expressions with unnecessary subscripts. The subscripts σ labeling spin values are not to be confused for tensor indices nor the Pauli matrices.
2. This spinor notation is not necessarily standard; the literature usually writes $\psi ={\begin{pmatrix}u^{1}\\u^{2}\end{pmatrix}}$ or $\psi ={\begin{pmatrix}\chi \\\eta \end{pmatrix}}$ etc., but in the context of spin 1/2, this informal identification is commonly made.
3. Again this notation is not necessarily standard, the more advanced literature usually writes
$\psi ={\begin{pmatrix}u\\v\end{pmatrix}}={\begin{pmatrix}u^{1}\\u^{2}\\v^{1}\\v^{2}\end{pmatrix}}$ etc.,
but here we show informally the correspondence of energy, helicity, and spin states.
4. Some authors, including Penrose, use Latin letters in this definition, even though it is conventional to use Greek indices for vectors and tensors in spacetime.
References
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15. Landau, L.D.; Lifshitz, E.M. (1981). Quantum Mechanics Non-Relativistic Theory. Vol. 3. Elsevier. p. 455. ISBN 978-0-08-050348-6.
16. Peleg, Y.; Pnini, R.; Zaarur, E.; Hecht, E. (2010). Quantum Mechanics. Shaum's outlines (2nd ed.). McGraw–Hill. p. 181. ISBN 978-0-07-162358-2.
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20. Penrose, R. (2005). The Road to Reality. Vintage Books. pp. 620–621. ISBN 978-0-09-944068-0.
21. Bransden, B.H.; Joachain, C.J. (1983). Physics of Atoms and Molecules (1st ed.). Prentice Hall. p. 634. ISBN 978-0-582-44401-0.
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24. McMahon, D. (2008). Quantum Field Theory. Demystified. McGraw Hill. p. 114. ISBN 978-0-07-154382-8.
25. Bransden, B.H.; Joachain, C.J. (1983). Physics of Atoms and Molecules (1st ed.). Prentice Hall. pp. 632–635. ISBN 978-0-582-44401-0.
26. Parker, C.B. (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). McGraw Hill. p. 1194. ISBN 978-0-07-051400-3..
27. Labelle, P. (2010). Supersymmetry. Demystified. McGraw-Hill. ISBN 978-0-07-163641-4.
28. Esposito, S. (2011). "Searching for an equation: Dirac, Majorana and the others". Annals of Physics. 327 (6): 1617–1644. arXiv:1110.6878. Bibcode:2012AnPhy.327.1617E. doi:10.1016/j.aop.2012.02.016. S2CID 119147261.
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33. Lorcé, Cédric (2009). "Electromagnetic Properties for Arbitrary Spin Particles: Part 2 − Natural Moments and Transverse Charge Densities". Physical Review D. 79 (11): 113011. arXiv:0901.4200. Bibcode:2009PhRvD..79k3011L. doi:10.1103/PhysRevD.79.113011. S2CID 17801598.
34. Strange, P. (1998). Relativistic Quantum Mechanics: With Applications in Condensed Matter and Atomic Physics. Cambridge University Press. p. 206. ISBN 978-0-521-56583-7.
35. Labelle, P. (2010). Supersymmetry. Demystified. McGraw-Hill. p. 14. ISBN 978-0-07-163641-4.
36. Weinberg, S. (1995). The Quantum Theory of Fields. Vol. 1. Cambridge University Press. ISBN 978-0-521-55001-7.
37. Penrose, R. (2005). The Road to Reality. Vintage Books. pp. 437, 566–569. ISBN 978-0-09-944068-0.
38. Ryder, L.H. (1996). Quantum Field Theory (2nd ed.). Cambridge University Press. p. 62. ISBN 978-0-521-47814-4.
39. Troshin, S.M.; Tyurin, N.E. (1994). Spin phenomena in particle interactions. World Scientific. Bibcode:1994sppi.book.....T. ISBN 978-981-02-1692-4.
40. Misner, C.W.; Thorne, K.S.; Wheeler, J.A. (15 September 1973). Gravitation. p. 1146. ISBN 978-0-7167-0344-0.
41. Ciufolini, I.; Matzner, R.R.A. (2010). General relativity and John Archibald Wheeler. Springer. p. 329. ISBN 978-90-481-3735-0.
42. Kroemer, H. (2003). "The Thomas precession factor in spin–orbit interaction" (PDF). American Journal of Physics. 72 (1): 51–52. arXiv:physics/0310016. Bibcode:2004AmJPh..72...51K. doi:10.1119/1.1615526. S2CID 119533324.
43. Jackson, J.D. (1999). Classical Electrodynamics (3rd ed.). Wiley. p. 548. ISBN 978-0-471-30932-1.
44. Resnick, R.; Eisberg, R. (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd ed.). John Wiley & Sons. pp. 57, 114–116, 125–126, 272. ISBN 978-0-471-87373-0.
45. Atkins, P.W. (1974). Quanta: A handbook of concepts. Oxford University Press. pp. 168–169, 176, 263, 228. ISBN 978-0-19-855493-6.
46. Krane, K.S. (1988). Introductory Nuclear Physics. John Wiley & Sons. pp. 396–405. ISBN 978-0-471-80553-3.
47. Krane, K.S. (1988). Introductory Nuclear Physics. John Wiley & Sons. pp. 361–370. ISBN 978-0-471-80553-3.
48. Einstein, A.; Podolsky, B.; Rosen, N. (1935). "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" (PDF). Phys. Rev. 47 (10): 777–780. Bibcode:1935PhRv...47..777E. doi:10.1103/PhysRev.47.777.
49. Abers, E. (2004). Quantum Mechanics. Addison Wesley. p. 192. ISBN 978-0-13-146100-0.
50. Penrose, R. (2005). The Road to Reality. Vintage Books. ISBN 978-0-09-944068-0. Chapter 23: The entangled quantum world
51. Aharonov, Y.; Bohm, D. (1959). "Significance of electromagnetic potentials in quantum theory". Physical Review. 115 (3): 485–491. Bibcode:1959PhRv..115..485A. doi:10.1103/PhysRev.115.485.
52. Bell, John (1964). "On the Einstein Podolsky Rosen Paradox" (PDF). Physics. 1 (3): 195–200. doi:10.1103/PhysicsPhysiqueFizika.1.195.
53. Lamb, Willis E.; Retherford, Robert C. (1947). "Fine Structure of the Hydrogen Atom by a Microwave Method". Physical Review. 72 (3): 241–243. Bibcode:1947PhRv...72..241L. doi:10.1103/PhysRev.72.241.
54. Lamb, W.E. Jr. & Retherford, R.C. (1950). "Fine Structure of the Hydrogen Atom. Part I". Phys. Rev. 79 (4): 549–572. Bibcode:1950PhRv...79..549L. doi:10.1103/PhysRev.79.549.
Lamb, W.E. Jr. & Retherford, R.C. (1951). "Fine Structure of the Hydrogen Atom. Part II". Phys. Rev. 81 (2): 222–232. Bibcode:1951PhRv...81..222L. doi:10.1103/PhysRev.81.222.Lamb, W.E. Jr. (1952). "Fine Structure of the Hydrogen Atom. III". Phys. Rev. 85 (2): 259–276. Bibcode:1952PhRv...85..259L. doi:10.1103/PhysRev.85.259. PMID 17775407.
Lamb, W.E. Jr. & Retherford, R.C. (1952). "Fine Structure of the Hydrogen Atom. IV". Phys. Rev. 86 (6): 1014–1022. Bibcode:1952PhRv...86.1014L. doi:10.1103/PhysRev.86.1014. PMID 17775407.
Triebwasser, S.; Dayhoff, E.S. & Lamb, W.E. Jr. (1953). "Fine Structure of the Hydrogen Atom. V". Phys. Rev. 89 (1): 98–106. Bibcode:1953PhRv...89...98T. doi:10.1103/PhysRev.89.98.
Selected books
• Dirac, P.A.M. (1981). Principles of Quantum Mechanics (4th ed.). Clarendon Press. ISBN 978-0-19-852011-5.
• Dirac, P.A.M. (1964). Lectures on Quantum Mechanics. Courier Dover Publications. ISBN 978-0-486-41713-4.
• Thaller, B. (2010). The Dirac Equation. Springer. ISBN 978-3-642-08134-7.
• Pauli, W. (1980). General Principles of Quantum Mechanics. Springer. ISBN 978-3-540-09842-3.
• Merzbacher, E. (1998). Quantum Mechanics (3rd ed.). ISBN 978-0-471-88702-7.
• Messiah, A. (1961). Quantum Mechanics. Vol. 1. John Wiley & Sons. ISBN 978-0-471-59766-7.
• Bjorken, J.D.; Drell, S.D. (1964). Relativistic Quantum Mechanics (Pure & Applied Physics). McGraw-Hill. ISBN 978-0-07-005493-6.
• Feynman, R.P.; Leighton, R.B.; Sands, M. (1965). Feynman Lectures on Physics. Vol. 3. Addison-Wesley. ISBN 978-0-201-02118-9.
• Schiff, L.I. (1968). Quantum Mechanics (3rd ed.). McGraw-Hill.
• Dyson, F. (2011). Advanced Quantum Mechanics (2nd ed.). World Scientific. ISBN 978-981-4383-40-0.
• Clifton, R.K. (2011). Perspectives on Quantum Reality: Non-Relativistic, Relativistic, and Field-Theoretic. Springer. ISBN 978-90-481-4643-7.
• Tannoudji, C.; Diu, B.; Laloë, F. (1977). Quantum Mechanics. Vol. 1. Wiley VCH. ISBN 978-0-471-16433-3.
• Tannoudji, C.; Diu, B.; Laloë, F. (1977). Quantum Mechanics. Vol. 2. Wiley VCH. ISBN 978-0-471-16435-7.
• Rae, A.I.M. (2008). Quantum Mechanics. Vol. 2 (5th ed.). Taylor & Francis. ISBN 978-1-58488-970-0.
• Pilkuhn, H. (2005). Relativistic Quantum Mechanics. Texts and Monographs in Physics Series (2nd ed.). Springer. ISBN 978-3-540-28522-9.
• Parthasarathy, R. (2010). Relativistic quantum mechanics. Alpha Science International. ISBN 978-1-84265-573-3.
• Kaldor, U.; Wilson, S. (2003). Theoretical Chemistry and Physics of Heavy and Superheavy Elements. Springer. ISBN 978-1-4020-1371-3.
• Thaller, B. (2005). Advanced visual quantum mechanics. Springer. Bibcode:2005avqm.book.....T. ISBN 978-0-387-27127-9.
• Breuer, H.P.; Petruccione, F. (2000). Relativistic Quantum Measurement and Decoherence. Springer. ISBN 978-3-540-41061-4. Relativistic quantum mechanics.
• Shepherd, P.J. (2013). A Course in Theoretical Physics. John Wiley & Sons. ISBN 978-1-118-51692-8.
• Bethe, H.A.; Jackiw, R.W. (1997). Intermediate Quantum Mechanics. Addison-Wesley. ISBN 978-0-201-32831-8.
• Heitler, W. (1954). The Quantum Theory of Radiation (3rd ed.). Courier Dover Publications. ISBN 978-0-486-64558-2.
• Gottfried, K.; Yan, T. (2003). Quantum Mechanics: Fundamentals (2nd ed.). Springer. p. 245. ISBN 978-0-387-95576-6.
• Schwabl, F. (2010). Quantum Mechanics. Springer. p. 220. ISBN 978-3-540-71933-5.
• Sachs, R.G. (1987). The Physics of Time Reversal (2nd ed.). University of Chicago Press. p. 280. ISBN 978-0-226-73331-9. hyperfine structure in relativistic quantum mechanics.
Group theory in quantum physics
• Weyl, H. (1950). The theory of groups and quantum mechanics. Courier Dover Publications. p. 203. ISBN 9780486602691. magnetic moments in relativistic quantum mechanics.
• Tung, W.K. (1985). Group Theory in Physics. World Scientific. ISBN 978-9971-966-56-0.
• Heine, V. (1993). Group Theory in Quantum Mechanics: An Introduction to Its Present Usage. Courier Dover Publications. ISBN 978-0-486-67585-5.
Selected papers
• Dirac, P.A.M. (1932). "Relativistic Quantum Mechanics". Proceedings of the Royal Society A. 136 (829): 453–464. Bibcode:1932RSPSA.136..453D. doi:10.1098/rspa.1932.0094.
• Pauli, W. (1945). "Exclusion principle and quantum mechanics" (PDF).
• Antoine, J.P. (2004). "Relativistic Quantum Mechanics". J. Phys. A. 37 (4): 1465. Bibcode:2004JPhA...37.1463P. CiteSeerX 10.1.1.499.2793. doi:10.1088/0305-4470/37/4/B01.
• Henneaux, M.; Teitelboim, C. (1982). "Relativistic quantum mechanics of supersymmetric particles". Vol. 143.
• Fanchi, J.R. (1986). "Parametrizing relativistic quantum mechanics". Phys. Rev. A. 34 (3): 1677–1681. Bibcode:1986PhRvA..34.1677F. doi:10.1103/PhysRevA.34.1677. PMID 9897446.
• Ord, G.N. (1983). "Fractal space-time: a geometric analogue of relativistic quantum mechanics". J. Phys. A. 16 (9): 1869–1884. Bibcode:1983JPhA...16.1869O. doi:10.1088/0305-4470/16/9/012.
• Coester, F.; Polyzou, W.N. (1982). "Relativistic quantum mechanics of particles with direct interactions". Phys. Rev. D. 26 (6): 1348–1367. Bibcode:1982PhRvD..26.1348C. doi:10.1103/PhysRevD.26.1348.
• Mann, R.B.; Ralph, T.C. (2012). "Relativistic quantum information" (PDF). Class. Quantum Grav. 29 (22): 220301. Bibcode:2012CQGra..29v0301M. doi:10.1088/0264-9381/29/22/220301. S2CID 123341332.
• Low, S.G. (1997). "Canonically Relativistic Quantum Mechanics: Representations of the Unitary Semidirect Heisenberg Group, U(1,3) *s H(1,3)". J. Math. Phys. 38 (22): 2197–2209. arXiv:physics/9703008. Bibcode:2012CQGra..29v0301M. doi:10.1088/0264-9381/29/22/220301. S2CID 123341332.
• Fronsdal, C.; Lundberg, L.E. (1997). "Relativistic Quantum Mechanics of Two Interacting Particles". Phys. Rev. D. 1 (12): 3247–3258. arXiv:physics/9703008. Bibcode:1970PhRvD...1.3247F. doi:10.1103/PhysRevD.1.3247.
• Bordovitsyn, V.A.; Myagkii, A.N. (2004). "Spin–orbital motion and Thomas precession in the classical and quantum theories" (PDF). American Journal of Physics. 72 (1): 51–52. arXiv:physics/0310016. Bibcode:2004AmJPh..72...51K. doi:10.1119/1.1615526. S2CID 119533324.
• Rȩbilas, K. (2013). "Comment on 'Elementary analysis of the special relativistic combination of velocities, Wigner rotation and Thomas precession'". Eur. J. Phys. 34 (3): L55–L61. Bibcode:2013EJPh...34L..55R. doi:10.1088/0143-0807/34/3/L55. S2CID 122527454.
• Corben, H.C. (1993). "Factors of 2 in magnetic moments, spin–orbit coupling, and Thomas precession". Am. J. Phys. 61 (6): 551. Bibcode:1993AmJPh..61..551C. doi:10.1119/1.17207.
Further reading
Relativistic quantum mechanics and field theory
• Ohlsson, T. (2011). Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory. Cambridge University Press. p. 10. ISBN 978-1-139-50432-4.
• Aitchison, I.J.R.; Hey, A.J.G. (2002). Gauge Theories in Particle Physics: From Relativistic Quantum Mechanics to QED. Vol. 1 (3rd ed.). CRC Press. ISBN 978-0-8493-8775-3.
• Griffiths, D. (2008). Introduction to Elementary Particles. John Wiley & Sons. ISBN 978-3-527-61847-7.
• Capri, Anton Z. (2002). Relativistic quantum mechanics and introduction to quantum field theory. World Scientific. Bibcode:2002rqmi.book.....C. ISBN 978-981-238-137-8.
• Wu, Ta-you; Hwang, W.Y. Pauchy (1991). Relativistic quantum mechanics and quantum fields. World Scientific. ISBN 978-981-02-0608-6.
• Nagashima, Y. (2010). Elementary particle physics, Quantum Field Theory. Vol. 1. ISBN 978-3-527-40962-4.
• Bjorken, J.D.; Drell, S.D. (1965). Relativistic Quantum Fields (Pure & Applied Physics). McGraw-Hill. ISBN 978-0-07-005494-3.
• Weinberg, S. (1996). The Quantum Theory of Fields. Vol. 2. Cambridge University Press. ISBN 978-0-521-55002-4.
• Weinberg, S. (2000). The Quantum Theory of Fields. Vol. 3. Cambridge University Press. ISBN 978-0-521-66000-6.
• Gross, F. (2008). Relativistic Quantum Mechanics and Field Theory. John Wiley & Sons. ISBN 978-3-527-61734-0.
• Nazarov, Y.V.; Danon, J. (2013). Advanced Quantum Mechanics: A Practical Guide. Cambridge University Press. ISBN 978-0-521-76150-5.
• Bogolubov, N.N. (1989). General Principles of Quantum Field Theory (2nd ed.). Springer. p. 272. ISBN 978-0-7923-0540-8.
• Mandl, F.; Shaw, G. (2010). Quantum Field Theory (2nd ed.). John Wiley & Sons. ISBN 978-0-471-49683-0.
• Lindgren, I. (2011). Relativistic Many-body Theory: A New Field-theoretical Approach. Springer series on atomic, optical, and plasma physics. Vol. 63. Springer. ISBN 978-1-4419-8309-1.
• Grant, I.P. (2007). Relativistic Quantum theory of atoms and molecules. Atomic, optical, and plasma physics. Springer. ISBN 978-0-387-34671-7.
Quantum theory and applications in general
• Aruldhas, G.; Rajagopal, P. (2005). Modern Physics. PHI Learning Pvt. Ltd. p. 395. ISBN 978-81-203-2597-5.
• Hummel, R.E. (2011). Electronic properties of materials. Springer. p. 395. ISBN 978-1-4419-8164-6.
• Pavia, D.L. (2005). Introduction to Spectroscopy (4th ed.). Cengage Learning. p. 105. ISBN 978-0-495-11478-9.
• Mizutani, U. (2001). Introduction to the Electron Theory of Metals. Cambridge University Press. p. 387. ISBN 978-0-521-58709-9.
• Choppin, G.R. (2002). Radiochemistry and nuclear chemistry (3 ed.). Butterworth-Heinemann. p. 308. ISBN 978-0-7506-7463-8.
• Sitenko, A.G. (1990). Theory of nuclear reactions. World Scientific. p. 443. ISBN 978-9971-5-0482-3.
• Nolting, W.; Ramakanth, A. (2008). Quantum theory of magnetism. Springer. ISBN 978-3-540-85416-6.
• Luth, H. (2013). Quantum Physics in the Nanoworld. Graduate texts in physics. Springer. p. 149. ISBN 978-3-642-31238-0.
• Sattler, K.D. (2010). Handbook of Nanophysics: Functional Nanomaterials. CRC Press. pp. 40–43. ISBN 978-1-4200-7553-3.
• Kuzmany, H. (2009). Solid-State Spectroscopy. Springer. p. 256. ISBN 978-3-642-01480-2.
• Reid, J.M. (1984). The Atomic Nucleus (2nd ed.). Manchester University Press. ISBN 978-0-7190-0978-5.
• Schwerdtfeger, P. (2002). Relativistic Electronic Structure Theory - Fundamentals. Theoretical and Computational Chemistry. Vol. 11. Elsevier. p. 208. ISBN 978-0-08-054046-7.
• Piela, L. (2006). Ideas of Quantum Chemistry. Elsevier. p. 676. ISBN 978-0-08-046676-7.
• Kumar, M. (2009). Quantum (book). ISBN 978-1-84831-035-3.
External links
• Pfeifer, W. (2008) [2004]. Relativistic Quantum Mechanics, an Introduction.
• Lukačević, Igor (2013). "Relativistic Quantum Mechanics (Lecture Notes)" (PDF). Archived from the original (PDF) on 2014-08-26.
• "Relativistic Quantum Mechanics" (PDF). Cavendish Laboratory. University of Cambridge.
• Miller, David J. (2008). "Relativistic Quantum Mechanics" (PDF). University of Glasgow.
• Swanson, D.G. (2007). "Quantum Mechanics Foundations and Applications". Alabama, USA: Taylor & Francis. p. 160.
• Calvert, J.B. (2003). "The Particle Electron and Thomas Precession".
• Arteha, S.N. "Spin and the Thomas precession".
Quantum mechanics
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| Wikipedia |
Cycle rank
In graph theory, the cycle rank of a directed graph is a digraph connectivity measure proposed first by Eggan and Büchi (Eggan 1963). Intuitively, this concept measures how close a digraph is to a directed acyclic graph (DAG), in the sense that a DAG has cycle rank zero, while a complete digraph of order n with a self-loop at each vertex has cycle rank n. The cycle rank of a directed graph is closely related to the tree-depth of an undirected graph and to the star height of a regular language. It has also found use in sparse matrix computations (see Bodlaender et al. 1995) and logic (Rossman 2008).
For related notion (also called cycle rank) in undirected graphs, see circuit rank.
Relevant topics on
Graph connectivity
• Connectivity
• Algebraic connectivity
• Cycle rank
• Rank (graph theory)
• SPQR tree
• St-connectivity
• K-connectivity certificate
• Pixel connectivity
• Vertex separator
• Strongly connected component
• Biconnected graph
• Bridge
Definition
The cycle rank r(G) of a digraph G = (V, E) is inductively defined as follows:
• If G is acyclic, then r(G) = 0.
• If G is strongly connected and E is nonempty, then
$r(G)=1+\min _{v\in V}r(G-v),\,$ where $G-v$ is the digraph resulting from deletion of vertex v and all edges beginning or ending at v.
• If G is not strongly connected, then r(G) is equal to the maximum cycle rank among all strongly connected components of G.
The tree-depth of an undirected graph has a very similar definition, using undirected connectivity and connected components in place of strong connectivity and strongly connected components.
History
Cycle rank was introduced by Eggan (1963) in the context of star height of regular languages. It was rediscovered by (Eisenstat & Liu 2005) as a generalization of undirected tree-depth, which had been developed beginning in the 1980s and applied to sparse matrix computations (Schreiber 1982).
Examples
The cycle rank of a directed acyclic graph is 0, while a complete digraph of order n with a self-loop at each vertex has cycle rank n. Apart from these, the cycle rank of a few other digraphs is known: the undirected path $P_{n}$ of order n, which possesses a symmetric edge relation and no self-loops, has cycle rank $\lfloor \log n\rfloor $ (McNaughton 1969). For the directed $(m\times n)$-torus $T_{m,n}$, i.e., the cartesian product of two directed circuits of lengths m and n, we have $r(T_{n,n})=n$ and $r(T_{m,n})=\min\{m,n\}+1$ for m ≠ n (Eggan 1963, Gruber & Holzer 2008).
Computing the cycle rank
Computing the cycle rank is computationally hard: Gruber (2012) proves that the corresponding decision problem is NP-complete, even for sparse digraphs of maximum outdegree at most 2. On the positive side, the problem is solvable in time $O(1.9129^{n})$ on digraphs of maximum outdegree at most 2, and in time $O^{*}(2^{n})$ on general digraphs. There is an approximation algorithm with approximation ratio $O((\log n)^{\frac {3}{2}})$.
Applications
Star height of regular languages
The first application of cycle rank was in formal language theory, for studying the star height of regular languages. Eggan (1963) established a relation between the theories of regular expressions, finite automata, and of directed graphs. In subsequent years, this relation became known as Eggan's theorem, cf. Sakarovitch (2009). In automata theory, a nondeterministic finite automaton with ε-moves (ε-NFA) is defined as a 5-tuple, (Q, Σ, δ, q0, F), consisting of
• a finite set of states Q
• a finite set of input symbols Σ
• a set of labeled edges δ, referred to as transition relation: Q × (Σ ∪{ε}) × Q. Here ε denotes the empty word.
• an initial state q0 ∈ Q
• a set of states F distinguished as accepting states F ⊆ Q.
A word w ∈ Σ* is accepted by the ε-NFA if there exists a directed path from the initial state q0 to some final state in F using edges from δ, such that the concatenation of all labels visited along the path yields the word w. The set of all words over Σ* accepted by the automaton is the language accepted by the automaton A.
When speaking of digraph properties of a nondeterministic finite automaton A with state set Q, we naturally address the digraph with vertex set Q induced by its transition relation. Now the theorem is stated as follows.
Eggan's Theorem: The star height of a regular language L equals the minimum cycle rank among all nondeterministic finite automata with ε-moves accepting L.
Proofs of this theorem are given by Eggan (1963), and more recently by Sakarovitch (2009).
Cholesky factorization in sparse matrix computations
Another application of this concept lies in sparse matrix computations, namely for using nested dissection to compute the Cholesky factorization of a (symmetric) matrix in parallel. A given sparse $(n\times n)$-matrix M may be interpreted as the adjacency matrix of some symmetric digraph G on n vertices, in a way such that the non-zero entries of the matrix are in one-to-one correspondence with the edges of G. If the cycle rank of the digraph G is at most k, then the Cholesky factorization of M can be computed in at most k steps on a parallel computer with $n$ processors (Dereniowski & Kubale 2004).
See also
• Circuit rank
References
• Bodlaender, Hans L.; Gilbert, John R.; Hafsteinsson, Hjálmtýr; Kloks, Ton (1995), "Approximating treewidth, pathwidth, frontsize, and shortest elimination tree", Journal of Algorithms, 18 (2): 238–255, doi:10.1006/jagm.1995.1009, Zbl 0818.68118.
• Dereniowski, Dariusz; Kubale, Marek (2004), "Cholesky Factorization of Matrices in Parallel and Ranking of Graphs", 5th International Conference on Parallel Processing and Applied Mathematics (PDF), Lecture Notes on Computer Science, vol. 3019, Springer-Verlag, pp. 985–992, doi:10.1007/978-3-540-24669-5_127, Zbl 1128.68544, archived from the original (PDF) on 2011-07-16.
• Eggan, Lawrence C. (1963), "Transition graphs and the star-height of regular events", Michigan Mathematical Journal, 10 (4): 385–397, doi:10.1307/mmj/1028998975, Zbl 0173.01504.
• Eisenstat, Stanley C.; Liu, Joseph W. H. (2005), "The theory of elimination trees for sparse unsymmetric matrices", SIAM Journal on Matrix Analysis and Applications, 26 (3): 686–705, doi:10.1137/S089547980240563X.
• Gruber, Hermann (2012), "Digraph Complexity Measures and Applications in Formal Language Theory" (PDF), Discrete Mathematics & Theoretical Computer Science, 14 (2): 189–204.
• Gruber, Hermann; Holzer, Markus (2008), "Finite automata, digraph connectivity, and regular expression size" (PDF), Proc. 35th International Colloquium on Automata, Languages and Programming, Lecture Notes on Computer Science, vol. 5126, Springer-Verlag, pp. 39–50, doi:10.1007/978-3-540-70583-3_4.
• McNaughton, Robert (1969), "The loop complexity of regular events", Information Sciences, 1 (3): 305–328, doi:10.1016/S0020-0255(69)80016-2.
• Rossman, Benjamin (2008), "Homomorphism preservation theorems", Journal of the ACM, 55 (3): Article 15, doi:10.1145/1379759.1379763.
• Sakarovitch, Jacques (2009), Elements of Automata Theory, Cambridge University Press, ISBN 0-521-84425-8
• Schreiber, Robert (1982), "A new implementation of sparse Gaussian elimination" (PDF), ACM Transactions on Mathematical Software, 8 (3): 256–276, doi:10.1145/356004.356006, archived from the original (PDF) on 2011-06-07, retrieved 2010-01-04.
| Wikipedia |
\begin{document}
\begin{frontmatter}
\title{Gibbs cluster measures on configuration spaces}
\author[Leeds]{Leonid Bogachev\fnref{t1,t2}\corref{cor1}} \ead{L.V.\,[email protected]} \author[York]{Alexei Daletskii\fnref{t2}} \ead{[email protected]}
\address[Leeds]{Department of Statistics, University of Leeds, Leeds LS2 9JT, UK} \address[York]{Department of Mathematics, University of York, York YO10 5DD, UK}
\fntext[t1]{Research supported in part by a Leverhulme Research Fellowship.} \fntext[t2]{Research supported in part by DFG Grant 436\,RUS\,113/722.} \cortext[cor1]{Corresponding author.}
\begin{abstract} The distribution $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$ of a Gibbs cluster point process in $X=\mathbb{R}^{d}$ (with i.i.d.\ random clusters attached to points of a Gibbs configuration with distribution $\g$) is studied via the projection of an auxiliary Gibbs measure $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$ in the space of configurations ${\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}=\{(x,\bar{y})\}\subset X\times\mathfrak{X}$, where $x\in X$ indicates a cluster ``center'' and $\bar{y}\in\mathfrak{X}:=\bigsqcup_{\mbox{$\:\!$} n}\mbox{$\;\!\!$} X^n$ represents a corresponding cluster relative to $x$.
We show that the measure $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$ is quasi-invariant with respect to the group $\mathrm{Diff}_{0}(X)$ of compactly supported diffeomorphisms of $X$, and prove an integration-by-parts formula for $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$. The associated equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms. These results are quite general; in particular, the uniqueness of the background Gibbs measure $\g$ is not required. The paper is an extension of the earlier results for Poisson cluster measures
[J.~Funct.\ Analysis 256 (2009) 432--478], where a different projection construction was utilized
specific to this ``exactly soluble'' case.
\end{abstract}
\begin{keyword}
Cluster point process \sep Gibbs measure \sep Poisson measure \sep Interaction potential \sep Configuration space \sep Quasi-invariance \sep Integration by parts \sep Dirichlet form \sep Stochastic dynamics
\MSC Primary 58J65, 82B05; Secondary 31C25, 46G12, 60G55,
70F45 \end{keyword}
\end{frontmatter}
\section{Introduction}\label{sec:1}
The concept of particle configurations is instrumental in mathematical modelling of multi-component stochastic systems. Rooted in statistical mechanics and theory of point processes, the development of the general mathematical framework for suitable classes of configurations has been
a recurrent research theme fostered by widespread applications
across the board, including
quantum physics, astrophysics, chemical physics, biology, ecology, computer science, economics, finance, etc.\ (see an extensive bibliography in \cite{DVJ1}).
In the past 15 years or so, there has been a more specific interest in the \emph{analysis} on configuration spaces. To fix basic notation, let $X$ be a topological space (e.g., a Euclidean space $X=\mathbb{R}^{d}$), and let $\varGamma_X=\{\gamma\}$ be the configuration space over $X$, that is, the space of countable subsets (called \emph{configurations}) $\gamma\subset X$ without accumulation points. Albeverio, Kondratiev and R\"{o}ckner \cite{AKR1,AKR2} have proposed an approach to configuration spaces $\varGamma_X$ as \emph{infinite-dimensional manifolds}, based on the choice of a suitable probability measure $\mu $ on $\varGamma_{X}$
which is
quasi-invariant with respect to $\mathrm{Diff}_{0}(X)$, the group of compactly supported diffeomorphisms of $X$. Providing that the measure $\mu$ can be shown to satisfy an integration-by-parts formula, one can construct, using the theory of Dirichlet forms, an associated equilibrium dynamics (stochastic process) on $\varGamma_{X}$ such that $\mu $ is its invariant measure \cite{AKR1,AKR2,MR} (see
\cite{ADKal,AKR2,AKR3,
Ro} and references therein for further discussion of various theoretical aspects and applications).
This general programme has been first implemented in \cite{AKR1} for the \textit{Poisson} measure $\mu$ on $\varGamma_{X}$, and then extended in \cite{AKR2} to a wider class of \textit{Gibbs} measures, which appear in statistical mechanics of classical continuous gases. In the Poisson case, the canonical equilibrium dynamics is given by the well-known independent particle process, that is, an infinite family of independent (distorted) Brownian motions started at the points of a random Poisson configuration. In the Gibbsian case, the equilibrium dynamics is much more complex due to interaction between the particles.
In our earlier papers \cite{BD1,BD3}, a similar analysis was developed for a different class of random spatial structures, namely \emph{Poisson cluster point processes}, featured by spatial grouping (``clustering'') of points around the background random (Poisson) configuration of invisible ``centers''. Cluster models are well known in the general theory of random point processes \cite{CI,DVJ1} and are widely used in numerous applications ranging from neurophysiology (nerve impulses) and ecology (spatial aggregation of species) to seismology (earthquakes) and cosmology (constellations and galaxies); see \cite{BD3,CI,DVJ1} for some references to original papers.
Our technique in \cite{BD1,BD3} was based on the representation of a given Poisson cluster measure on the configuration space $\varGamma_{X}$ as the projection image of an auxiliary Poisson measure on a more complex configuration space $\varGamma_{\mathfrak{X}}$ over the disjoint-union space $\mathfrak{X}:=\bigsqcup_{\mbox{$\:\!$} n} \mbox{$\:\!\!$} X^{n}$,
with ``droplet'' points $\bar{y}\in \mathfrak{X}$ representing individual clusters (of variable size).
The principal advantage of this construction is that it allows one to apply the well-developed apparatus of Poisson measures to the study of the Poisson cluster measure.
In the present paper,\footnote{Some of our results have been announced in
\cite{BD2} (in the case of clusters of fixed size).} our aim is to extend this approach to a more general class of \emph{Gibbs cluster measures} on the configuration space $\varGamma_X$, where
the distribution of cluster centers is given by a Gibbs (grand canonical) measure $\g\in\mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$ on $\varGamma_X$, with a reference measure $\theta$ on $X$ and an interaction potential ${\varPhi}$.
We focus on Gibbs cluster processes in $X=\mathbb{R}^{d} $ with i.i.d.\ random clusters of random size.
Let us point out that we do not require the uniqueness of the Gibbs measure, so our results are not affected by possible ``phase transitions'' (i.e., non-uniqueness of $\g\in\mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$). Under some natural smoothness conditions on the reference measure $\theta$ and the distribution $\eta$ of the generic cluster, we prove the $\mathrm{Diff}_{0}(X)$-quasi-invariance of the corresponding Gibbs cluster measure $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$ (Section~\ref{sec:3.2}), establish the integration-by-parts formula (Section~\ref{sec:3.3}) and construct the associated Dirichlet operator, which leads to the existence of the equilibrium stochastic dynamics on the configuration space $\varGamma_{X}$ (Section~\ref{sec:4}).
Unlike the Poisson cluster case, it is now impossible to work with the measure arising in the space $\varGamma_{\mathfrak{X}}$ of droplet configurations $\bar{\gamma}=\{\bar{y}\}$, which is hard to characterize for Gibbs cluster measures. Instead, in order to be able to pursue our projection approach while still having a tractable pre-projection measure, we choose the configuration space $\varGamma_{\mathcal{Z}}$ over the set $\mathcal{Z}:=X\times\mathfrak{X}$, where each configuration ${\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\in \varGamma_{\mathcal{Z}}$ is a (countable) set of pairs $z=(x,\bar{y})$ with $x\in X$ indicating a cluster center and $\bar{y}\in \mathfrak{X}$ representing a cluster attached to $x$. A crucial step is to show that the corresponding measure $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$ on $\varGamma_{\mathcal{Z}}$ is again Gibbsian, with the
reference measure $\sigma =\theta\otimes \eta $ and a ``cylinder'' interaction potential \,$\hat{{\varPhi}}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}):= {\varPhi}(\mathfrak{p}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}))$, where $\varPhi$ is the original interaction potential associated with the background Gibbs measure $\g$ and $\mathfrak{p}$ is the operator on the configuration space $\varGamma_{\mathcal{Z}}$ projecting a configuration ${\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}=\{(x,\bar{y})\}$ to the configuration of cluster centers, $\gamma=\{x\}$. We then project the Gibbs measure $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$ from the ``higher floor'' $\varGamma_{\mathcal{Z}}$ directly to the configuration space $\varGamma_{X}$ (thus skipping the ``intermediate floor'' $\varGamma_{\mathfrak{X}}$), and show that the resulting measure coincides with the original Gibbs cluster measure $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$ (Section~\ref{sec:2}).
In fact, it can be we shown (Section \ref{sec:2.3}) that \textit{any} cluster measure $\mu_{\rm cl}$ on $\varGamma_X$ can be obtained by a similar projection from $\varGamma_{\mathcal{Z}}$. Even though it may not always be possible to find an intrinsic characterization of the corresponding lifted measure $\hat{\mu}$ on the configuration space $\varGamma_{\mathcal{Z}}$ (unlike the Poisson and Gibbs cases), we expect that the projection approach can be instrumental in the study of more general cluster point processes by a reduction to point processes in more complex phase spaces but with a simpler correlation structure. We intend to develop these ideas elsewhere.
\section{Gibbs cluster measures via projections} \label{sec:2}
In this section, we start by recalling some basic concepts and notations for random point processes and associated probability measures in configuration spaces (Section~\ref{sec:2.1}), followed in Section~\ref{sec:2.2} by a definition of a general cluster point process (CPP). In Section~\ref{sec:2.3}, we explain our main ``projection'' construction allowing one to represent CPPs in the phase space $X$ in terms of auxiliary measures on a more complex configuration space involving Cartesian powers of $X$. The implications of such a description are discussed in greater detail for the particular case of Gibbs CPPs (Sections \ref{sec:2.4},~\ref{sec:2.5}).
\subsection{Probability measures on configuration spaces}\label{sec:2.1}
Let $X$ be a Polish space
equipped with the Borel $\sigma$-algebra ${\mathcal{B}}(X)$ generated by the open sets. Denote $\overline{\mathbb{Z}}_+:= \mathbb{Z}_+\cup\{\infty\}$, where $\mathbb{Z}_{+}:=\{0,1,2,\dots \}$, and consider a space ${\mathfrak{X}}$ built from all Cartesian powers of $X$, that is, the disjoint union \begin{equation}\label{eq:calX} \mathfrak{X}:={\textstyle\bigsqcup\limits_{\mbox{$\:\!$} n\in\overline{\mathbb{Z}}_{+}}} X^{n}, \end{equation} including $X^{0}=\{\emptyset \}$ and the space $X^\infty$ of infinite sequences $(x_1, x_2,\dots)$. That is, $\bar{x}=(x_{1},x_{2},\dots )\in{\mathfrak{X}}$ if and only if $\bar{x}\in X^{n}$ for some $n\in\overline{\mathbb{Z}}_{+}$.
We take the liberty to write $x_{i}\in\bar{x}$ if $x_{i}$ is a coordinate of the ``vector'' $\bar{x}$.
The space $\mathfrak{X}$ is endowed with the natural disjoint union topology induced by the topology in $X$.
\begin{remark}\label{rm:compact} Note that a set $K\subset{\mathfrak{X}}$ is compact if and only if $K=\bigsqcup_{\mbox{$\:\!$} n=0}^{N} K_{n}$, where $N<\infty $ and $K_{n}$ are compact subsets of $X^{n}$, respectively.
\end{remark}
\begin{remark} $\mathfrak{X}$ is a Polish space as a disjoint union of Polish spaces. \end{remark}
Denote by ${\mathcal{N}}(X)$ the space of $\overline{\mathbb{Z}}_{+}$-valued measures $N(\cdot)$ on ${\mathcal{B}}(X)$ with countable (i.e., finite or countably infinite) support.
Consider the natural projection \begin{equation}\label{eq:pr0} {\mathfrak{X}}\ni \bar{x}\mapsto {\mathfrak{p}}(\bar{x}):=\sum_{x_{i}\in \bar{x}}\delta_{x_{i}}\in {\mathcal{N}}(X), \end{equation} where $\delta_{x}$ is the Dirac measure at point $x\in X$.
That is to say, under the map ${\mathfrak{p}}$ each vector from ${\mathfrak{X}}$ is ``unpacked'' into its components to yield a countable aggregate of (possibly multiple) points in $X$, which can be interpreted as a generalized configuration $\gamma$,
\begin{equation}\label{eq:pr} \mathfrak{p}(\bar{x})\leftrightarrow\gamma :={\textstyle\bigsqcup\limits_{x_{i}\in \bar{x}}}\{x_{i}\},\qquad \bar{x}=(x_{1},x_{2},\dots )\in {\mathfrak{X}}. \end{equation}
In what follows, we
interpret the notation $\gamma $ either as an aggregate of
points in $X$ or as a $\overline{\mathbb{Z}}_{+}$-valued measure or both, depending on the context. Even though generalized configurations are not, strictly speaking, subsets of $X$ (because of possible multiplicities), it is convenient to
use set-theoretic notations, which should not cause any confusion. For instance, we write $\gamma \cap B$ for the restriction of configuration $\gamma $ to a subset $B\in {\mathcal{B}}(X)$.
For a function $f\mbox{$\;\!\!$}:X\to\mathbb{R}$ we denote \begin{equation}\label{eq:f-gamma} \langle f,\gamma \rangle :=\sum_{x_i\in \gamma}f(x_i)
\equiv\int_{X}f(x)\,\gamma({\mathrm{d}}x). \end{equation}
In particular, if $\mathbf{1}_{B}(x) $ is the indicator function of a set $B\in{\mathcal{B}}(X)$ then $\langle \mathbf{1}_{B},\gamma \rangle =\gamma (B)$ is the total number of points (counted with their multiplicities) in $\gamma\cap B$.
\begin{definition}\label{def:gen} A \textit{configuration space} $\varGamma_{X}^{\sharp}$ is the set of generalized configurations $\gamma$ in $X$, endowed with the \textit{cylinder $\sigma$-algebra} $\mathcal{B}(\varGamma_{X}^{\sharp})$ generated by the class of cylinder sets $C_{B}^{\mbox{$\:\!$} n}:=\{\gamma\in\varGamma_{X}^{\sharp}: \gamma (B)=n\}$, \,$B\in{\mathcal{B}}(X)$, \,$n\in\mathbb{Z}_+$\mbox{$\:\!$}. \end{definition}
\begin{remark} It is easy to see that the map $\mathfrak{p}:{\mathfrak{X}}\rightarrow \varGamma_{X}^{\sharp} $ defined by formula (\ref{eq:pr}) is measurable. \end{remark}
In fact, conventional theory of point processes (and their distributions as probability measures on configuration spaces) usually rules out the possibility of accumulation points or multiple points (see, e.g., \cite{DVJ1}).
\begin{definition}\label{def:proper} A configuration $\gamma \in \varGamma_{X}^{\sharp}$ is said to be \emph{locally finite} if $\gamma(B)<\infty $ for any compact set $B\subset X$. A configuration $\gamma \in \varGamma_{X}^{\sharp}$ is called \textit{simple} if $\gamma(\{x\})\le 1$ for each $x\in X$. A configuration $\gamma \in \varGamma_{X}^{\sharp}$ is called \emph{proper} if it is both locally finite and simple. The set of proper configurations will be denoted by $\varGamma_{X}$ and called the \textit{proper configuration space} over $X$. The corresponding $\sigma$-algebra ${\mathcal{B}}(\varGamma_{X})$ is generated by the cylinder sets $\{\gamma \in \varGamma_{X}:\gamma (B)=n\}$ \,($B\in {\mathcal{B}}(X)$, \,$n\in\mathbb{Z}_+$). \end{definition}
Like in the standard theory based on proper configuration spaces (see, e.g., \cite[\S\,6.1]{DVJ1}), every probability measure $\mu$ on the generalized configuration space $\varGamma_{X}^{\sharp}$ can be characterized by its Laplace functional (cf.\ \cite{BD3}) \begin{equation} L_{\mu}(f):=
\int_{\varGamma_{X}^\sharp} \mathrm{e\mbox{$\:\!$}}^{-\langle f,\mbox{$\:\!$}\gamma \rangle}\,\mu(\mathrm{d} \gamma),\qquad f\in {\mathrm{M}}_{+}(X), \label{eq:LAPLACE} \end{equation} where ${\mathrm{M}}_{+}(X)$ is the class of measurable non-negative functions on $X$.
\subsection{Cluster point processes}
\label{sec:2.2}
Let us recall the notion of a general cluster point process (CPP). Its realizations are constructed in two steps: (i) a background random configuration of (invisible) ``centers'' is obtained as a realization of some point process $\gamma_{\mathrm{c}}$ governed by a probability measure $\mu_{\mathrm{c}}$ on $\varGamma_{X}^{\sharp }$, and (ii) relative to each center $x\in \gamma_{\mathrm{c}}$, a set of observable secondary points (referred to as a \emph{cluster} centered at~$x$) is generated according to a point process $\gamma_{x}^{\mbox{$\:\!$}\prime}$ with probability measure $\mu_{x}$ on $\varGamma_{X}^{\sharp }$ ($x\in X$).
The resulting (countable) assembly of random points, called the \emph{cluster point process}, can be symbolically expressed as \begin{equation*} \gamma ={\textstyle\bigsqcup\limits_{x\in \gamma_{\mathrm{c}}}}\mbox{$\:\!$} \gamma_{x}^{\mbox{$\:\!$}\prime }\in \varGamma_{X}^{\sharp}, \end{equation*} where the disjoint union signifies that multiplicities of points should be taken into account. More precisely, assuming that the family of secondary processes $\gamma_x^{\mbox{$\:\!$}\prime}(\cdot)$ is measurable as a function of $x\in X$, the integer-valued measure corresponding to a CPP realization $\gamma $ is given by \begin{equation} \label{eq:cluster-gamma} \gamma(B)=\int_{X}\gamma_x^{\mbox{$\:\!$}\prime}(B)\,\gamma_{\mathrm{c}}(\mathrm{d} x) =\sum_{x\in \gamma_{\mathrm{c}}}\gamma_x^{\mbox{$\:\!$}\prime}(B),
\qquad B\in {\mathcal{B}}(X). \end{equation}
In what follows, we assume
that (i) $X$ is a linear space (e.g., $X=\mathbb{R}^d$) so that translations $X\ni y\mapsto y+x\in X$ are defined, and (ii) random clusters are independent and identically distributed (i.i.d.), being governed by the same probability law translated to the cluster centers, so that, for any $x\in X$, we have $\mu_{x}(A)=\mu_{0}(A-x)$ ($A\in\mathcal{B}(\varGamma_X^{\sharp})$.
In turn, the measure $\mu_{0}$ on $\varGamma_X^{\sharp}$
determines a probability distribution $\eta $ in ${\mathfrak{X}}$ which is symmetric with respect to permutations of coordinates. Conversely, $\mu_{0}$ is a push-forward of the measure $\eta $ under the projection map ${\mathfrak{p}}:{\mathfrak{X}}\rightarrow \varGamma_{X}^{\sharp }$ defined by (\ref{eq:pr}), that is, \begin{equation}\label{eq:p*eta} \mu_{0}={\mathfrak{p}}^{*}\eta \equiv \eta \circ {\mathfrak{p}}^{-1}. \end{equation}
\begin{remark} Unlike the standard CPP theory when sample configurations are \emph{presumed} to be a.s.\ locally finite (see, e.g., \cite[Definition 6.3.I]{DVJ1}), the description of the CPP given above only implies that its configurations $\gamma $ are countable aggregates in $X$, but possibly with multiple and/or accumulation points, even if the background point process $\gamma_{\mathrm{c}}$ is proper. Therefore, the distribution $\mu $ of the CPP (\ref{eq:cluster-gamma}) is a probability measure defined on the space $\varGamma_{X}^{\sharp }$ of \emph{generalized} configurations. It is a matter of interest to obtain conditions in order that $\mu $ be actually supported on the proper configuration space $\varGamma_{X}$, and we shall address this issue in Section \ref{sec:2.4} below for Gibbs CPPs (see \cite{BD3} for the case of Poisson CPPs). \end{remark}
The following fact is well known in the case of CPPs without accumulation points (see, e.g., \cite[\S \,6.3]{DVJ1}); its proof in the general case is essentially the same
(see \cite[Proposition 2.5]{BD3}).
\begin{proposition}\label{pr:cluster} Let $\mu_{{\rm cl}}$ be a probability measure on $(\varGamma_X^{\sharp},\mathcal{B}(\varGamma_X^\sharp))$ determined by the probability distribution of a CPP \textup{(\ref{eq:cluster-gamma})}. Then its Laplace functional
is given, for all functions $f\in {\mathrm{M}}_+(X)$, by
\begin{equation}\label{laplace-G} L_{\mu_{{\rm cl}}}(f)
=\int_{\varGamma^\sharp_{X}}\prod_{x\in \gamma_{\rm c}}\biggl( \int_{\mathfrak{X} }\exp \biggl(- \sum_{y_i\in \bar{y}}f(y_i+x)\Bigr) \,\eta ({\mathrm{d}}\bar{y})\biggr) \,\mu_{\rm c}({\mathrm{d}}\gamma_{\rm c}). \end{equation} \end{proposition}
\subsection{A projection construction of cluster measures on configurations}\label{sec:2.3}
Denote $\mathcal{Z}:=X\times\mathfrak{X}$,
and consider the space $\varGamma_{\mathcal{Z}}^\sharp=\{{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\}$ of (generalized) configurations
in $\mathcal{Z}$.
Let $p_{X}\mbox{$\:\!\!$}:\mathcal{Z}\to X$ be the natural projection to the first coordinate, \begin{equation}\label{eq:pX0} \mathcal{Z}\ni z=(x,\bar{y})\mapsto p_{X}(z):=x\in X, \end{equation} and consider its pointwise lifting to the configuration space $\varGamma^\sharp_{\mathcal{Z}}$ (preserving the same notation $p_{X}$), defined as follows \begin{equation}\label{eq:pX} \varGamma_{\mathcal{Z}}^\sharp\ni {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\mapsto p_{X}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}):={\textstyle\bigsqcup\limits_{z\in\mbox{$\:\!$}\hat{\gamma\mbox{$\:\!$}}}}\{p_{X}(z)\} \in\varGamma^\sharp_X. \end{equation}
Let $\mu_{\rm cl}$ denote the probability measure on the configuration space $\varGamma^\sharp_{X}$ associated with an i.i.d.\ cluster point process (see Section~\ref{sec:2.2}), specified by measures $\mu_{\rm c}$ on $\varGamma^\sharp_{X}$ and
$\eta$ on $\mathfrak{X}$.
\begin{definition}\label{def:muhat} Let us define a probability measure $\hat{\mu}$ on $\varGamma_{\mathcal{Z}}^\sharp$ as the distribution of random configurations ${\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}$ over $\mathcal{Z}$ obtained from configurations $\gamma\in\varGamma^\sharp_{X}$ by attaching to each point $x\in\gamma$ an i.i.d.\ random vector $\bar{y}_x$ with distribution $\eta$: \begin{equation}\label{eq:gamma-hat} \varGamma^\sharp_{X}\ni\gamma_{\rm c}
\mapsto {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}:={\textstyle\bigsqcup\limits_{x\in\gamma_{\rm c}}\{(x,\bar{y}_x)\}} \in \varGamma_{\mathcal{Z}}^\sharp. \end{equation}
Geometrically, the construction (\ref{eq:gamma-hat}) may be viewed as random i.i.d.\ pointwise translations of configurations $\gamma$ from $X$ into the ``plane'' $\mathcal{Z}=X\times \mathfrak{X}$. The measure $\hat{\mu}$ so obtained may be expressed in the differential form as a skew product
\begin{equation}\label{eq:muhat0} \hat{\mu}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})=\mu_{\rm c}(p_{X}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}))\,{\textstyle\bigotimes\limits_{z\in\hat{\gamma\mbox{$\:\!$}}}} \,\eta(p_\mathfrak{X}(\mathrm{d}{z})), \qquad {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\in\varGamma^\sharp_{\mathcal{Z}}. \end{equation} Equivalently, for any function $F\in{\mathrm{M}}_+(\varGamma^\sharp_{\mathcal{Z}})$,
\begin{equation}\label{eq:muhat} \int_{\varGamma_{\mathcal{Z}}^\sharp} F({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\,\hat{\mu}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}) =\int_{\varGamma_{X}^\sharp}\!\biggl(\int_{{\mathfrak{X}}^\infty} F\Bigl(\mbox{$\;\!$}{\textstyle\bigcup\limits _{x\in\gamma}}\{({\textstyle x},\bar{y})\}\Bigr)\,{\textstyle\bigotimes\limits_{x\in\gamma}} \,\eta(\mathrm{d}\bar{y})\mbox{$\:\!\!$}\biggr)\,\mu_{\rm c}(\mathrm{d}\gamma). \end{equation} \end{definition}
\begin{remark} Note that formula
(\ref{eq:muhat}) is a simple case of the general disintegration theorem, or the ``total expectation formula''
(see, e.g., \cite[Theorem 5.4]{Kal}
or
\cite[Ch.\,V, \S\mypp8, Theorem~8.1]{Par}).
\end{remark}
Recall that the ``unpacking'' map $\mathfrak{p}:\mathfrak{X}\to\varGamma_X^\sharp$ is defined in (\ref{eq:pr}), and consider a map $\mathfrak{q}:\mathcal{Z}\rightarrow \varGamma_{X}^{\sharp}$ acting by the formula \begin{equation}\label{proj1} \mathfrak{q}(x,\bar{y}):=\mathfrak{p}(\bar{y}+x)= {\textstyle\bigsqcup\limits_{y_i\in\bar{y}}}\{y_i+x\},\qquad (x,\bar{y})\in \mathcal{Z}. \end{equation} Here and below, we use the shift notation ($x\in X$) \begin{equation}\label{eq:shift} \bar{y}+x:=(y_1+x,\,y_2+x,\dots),\qquad \bar{y}=(y_1\mbox{$\;\!\!$}, y_2,\dots)\in \mathfrak{X}, \end{equation}
In the usual ``diagonal'' way, the map $\mathfrak{q}$ can be lifted to the configuration space $\varGamma_{\mathcal{Z}}^{\sharp}$: \begin{equation}\label{eq:proj} \varGamma_{\mathcal{Z}}^{\sharp}\ni {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\mapsto \mathfrak{q}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}):={\textstyle\bigsqcup\limits_{z\in\mbox{$\:\!$} \hat{\gamma\mbox{$\:\!$}}}}\mathfrak{q}(z)\in \varGamma_{X}^\sharp. \end{equation}
\begin{proposition}\label{pr:q-meas} The map $\mathfrak{q}:\varGamma_{\mathcal{Z}}^{\sharp} \to \varGamma_{X}^{\sharp}$ defined by \textup{(\ref{eq:proj})} is measurable. \end{proposition} \proof Observe that $\mathfrak{q}$ can be represented as a composition \begin{equation}\label{eq:circ} \mathfrak{q}=\mathfrak{p}\circ p_{\mathfrak{X}}\circ p_+:\varGamma_{\mathcal{Z}}^\sharp\stackrel{p_+}{\longrightarrow} \varGamma_{\mathcal{Z}}^\sharp\stackrel{p_{\mathfrak{X}}}{\longrightarrow} \varGamma_{\mathfrak{X}}^\sharp\stackrel{\mathfrak{p}}{\longrightarrow} \varGamma_{X}^\sharp, \end{equation} where the maps $p_+$\mbox{$\:\!$}, $p_{\mathfrak{X}}$ and $\mathfrak{p}$ are defined, respectively, by \begin{align} \label{eq:p+} \varGamma_{\mathcal{Z}}^{\sharp}\ni {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}&\mapsto p_+({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}):={\textstyle\bigsqcup\limits_{(x,\mbox{$\:\!$}\bar{y})\in\mbox{$\:\!$} \hat{\gamma\mbox{$\:\!$}}}} \{(x,\bar{y}+x)\}\in \varGamma_{\mathcal{Z}}^{\sharp},\\ \label{eq:pY} \varGamma_{\mathcal{Z}}^{\sharp}\ni {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}&\mapsto p_{\mathfrak{X}}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}):={\textstyle\bigsqcup\limits_{(x,\mbox{$\:\!$}\bar{y})\in\mbox{$\:\!$} \hat{\gamma\mbox{$\:\!$}}}}\{\bar{y}\}\in \varGamma_{\mathfrak{X}}^{\sharp},\\ \label{eq:pr2} \varGamma_{\mathfrak{X}}^\sharp\ni\bar{\gamma}&\mapsto \mathfrak{p}(\bar{\gamma}):={\textstyle\bigsqcup\limits_{\bar{y}\in \bar{\gamma}}}\mathfrak{p}(\bar{y})\in\varGamma_{X}^\sharp. \end{align} To verify that the map $p_+:\varGamma_{\mathcal{Z}}^\sharp\to\varGamma_{\mathcal{Z}}^\sharp$
is measurable, note that for a cylinder set $$ C_{B_1\times \bar{B}}^{\mbox{$\:\!$} n} =\{{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\in\varGamma_{\mathcal{Z}}: \,{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}(B_1\times \bar{B})=n\}\in \mathcal{B}(\varGamma_{\mathcal{Z}}^\sharp), $$ with $B_1\in\mathcal{B}(X)$, $\bar{B}\in\mathcal{B}(\mathfrak{X})$ and $n\in\mathbb{Z}_+$, its pre-image under $p_+$ is given by $$ p_+^{-1}(C_{B_1\times \bar{B}}^{\mbox{$\:\!$} n})=C_{A}^{\mbox{$\:\!$} n}=\{{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\in\varGamma_{\mathcal{Z}}^\sharp: \,{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}(A)=n\}, $$ where $$ A:=\{(x,\bar{y})\in\mathcal{Z}:(x,\bar{y}+x)\in B_1\times\bar{B}\}\in\mathcal{B}(\mathcal{Z}), $$
since the indicator $\mathbf{1}_A(x,\bar{y})=\mathbf{1}_{B_1}(x)\cdot \mathbf{1}_{\bar{B}}(\bar{y}+x)$ is obviously a measurable function on $\mathcal{Z}=X\times \mathfrak{X}$. Similarly, for $p_{\mathfrak{X}}:\varGamma_{\mathcal{Z}}^\sharp\to\varGamma_{\mathfrak{X}}^\sharp$ (see (\ref{eq:pY})) we have $$ p_{\mathfrak{X}}^{-1}(C_{\bar{B}}^{\mbox{$\:\!$} n})=C_{X\times \bar{B}}^{\mbox{$\:\!$} n}=\{{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\in\varGamma_{\mathcal{Z}}^\sharp: \,{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}(X\times\bar{B})=n\}\in \mathcal{B}(\varGamma_{\mathcal{Z}}^\sharp), $$ since $X\times\bar{B}\in \mathcal{B}(\mathcal{Z})$. Finally, measurability of the projection $\mathfrak{p}:\varGamma_{\mathfrak{X}}^\sharp\to\varGamma_X^\sharp$ (see (\ref{eq:pr2})) was shown in \cite[Section 3.3, p.~455]{BD3}. As a result, the composition (\ref{eq:circ}) is measurable, as claimed. \endproof
Let us define a measure on $\varGamma^\sharp_{X}$ as the push-forward of $\hat{\mu}$ (see Definition~\ref{def:muhat}) under the map $\mathfrak{q}$ defined in (\ref{proj1}), (\ref{eq:proj}): \begin{equation}\label{eq:mu*} \mathfrak{q}^{\ast}\hat{\mu}(A) \equiv\hat{\mu}\mbox{$\:\!$}(\mathfrak{q}^{-1}(A)),\qquad A\in\mathcal{B}(\varGamma_{X}^\sharp), \end{equation} or equivalently \begin{equation}\label{eq:mu*1} \int_{\varGamma_{X}^\sharp}F(\gamma )\,\mathfrak{q}^{\ast}\hat{\mu}(\mathrm{d}\gamma) =\int_{\varGamma^\sharp_{\mathcal{Z}}} \mbox{$\:\!\!$} F(\mathfrak{q}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})) \,\hat{\mu}({\mathrm{d}}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}),\qquad F\in\mathrm{M}_+(\varGamma_{X}^\sharp). \end{equation} The next general result shows that this measure may be identified with the original cluster measure $\mu_{\rm cl}$. \begin{theorem}\label{th:mucl} The measure \textup{(\ref{eq:mu*})} coincides with the cluster measure $\mu_{\rm cl}$, \begin{equation}\label{eq:cl*} \mu_{\rm cl}=\mathfrak{q}^{\ast}\hat{\mu}\equiv \hat{\mu}\circ \mathfrak{q}^{-1}. \end{equation} \end{theorem}
\proof Let us evaluate the Laplace transform of the measure $\mathfrak{q}^{\ast}\hat{\mu}$. For any function $f\in\mathrm{M}_+(X)$, we obtain, using (\ref{eq:mu*1}), (\ref{eq:proj}) and (\ref{eq:muhat}), \begin{align*} L_{\mathfrak{q}^{\ast}\mbox{$\;\!\!$}\hat{\mu}}(f)&= \int_{\varGamma _{X}^\sharp}\exp(-\langle f,\gamma\rangle)\,\mathfrak{q}^{\ast}\hat{\mu}(\mathrm{d}\gamma)
= \int_{\varGamma_{\mathcal{Z}}^\sharp}\exp(-\langle f,\mathfrak{q}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\rangle)\,\hat{\mu}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\\[.1pc] &=\int_{\varGamma_{X}}\biggl(\int_{\varGamma_{\mathfrak{X}}^\sharp} \exp\biggl(-\sum_{x\in\gamma_{\rm c}} f(\mathfrak{p}(\bar{y}_x+x))\biggr) {\textstyle\bigotimes\limits_{x\in\gamma_{\rm c}}}\eta(\mathrm{d}\bar{y}_x)\biggr) \,\mu_{\rm c}(\mathrm{d}\gamma_{\rm c})\\[.1pc] &=\int_{\varGamma_ X^\sharp}\biggl(\int_{\varGamma_{\mathfrak{X}}^\sharp}\, \prod_{x\in\gamma_{\rm c}}\exp\bigl(-f(\bar{y}_x+x)\bigr) {\textstyle\bigotimes\limits_{x\in\gamma_{\rm c}}}\eta(\mathrm{d}\bar{y}_x)\biggr)\, \mu_{\rm c}(\mathrm{d}\gamma_{\rm c})\\[.1pc] &=\int_{\varGamma_X^\sharp} \prod_{x\in\gamma_{\rm c}}\biggl(\int_{\mathfrak{X}} \exp \biggl(- \sum_{y\in \bar{y}}f(y+x)\biggr) \,\eta(\mathrm{d}\bar{y})\biggr)\,\mu_{\rm c}(\mathrm{d}\gamma_{\rm c}), \end{align*} which coincides with the Laplace transform (\ref{laplace-G}) of the cluster measure $\mu_{\rm cl}$. \endproof
\subsection{Gibbs cluster measure via an auxiliary Gibbs measure}\label{sec:2.4} In this paper, we are concerned with \emph{Gibbs cluster point processes}, for which the
distribution of cluster centers is
given by some
Gibbs measure $\g\in\mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$ on the proper configuration space $\varGamma_{X}$ (see the Appendix), specified by a
\emph{reference measure} $\theta$ on $X$ and an \textit{interaction potential} \,${\varPhi}:\varGamma^{\mbox{$\:\!$} 0}_{X}\rightarrow \mathbb{R}\cup \{+\infty\}$, where $\varGamma_{X}^{\myp0}\subset\varGamma_X$ is the subspace of finite configurations in $X$. We assume that the set \mbox{$\;\!$}$\mathscr{G}(\theta,\varPhi)$ of all Gibbs measures on $\varGamma_{X}$ associated with $\theta$ and ${\varPhi}$ is non-empty.\footnote{For various sufficient conditions, consult \cite{Preston,Ruelle}; see also references in the Appendix.}
Specializing Definition \ref{def:muhat} to the Gibbs case, the corresponding auxiliary measure $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$ on the (proper) configuration space $\varGamma_\mathcal{Z}$ is given by (cf.\ (\ref{eq:muhat0}), (\ref{eq:muhat})) \begin{equation}\label{eq:ghat0} \hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})=\g(p_{X}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})) \,{\textstyle\bigotimes\limits_{z\in\hat{\gamma\mbox{$\:\!$}}}} \,\eta(p_\mathfrak{X}(\mathrm{d}{z})),\qquad {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\in\varGamma_{\mathcal{Z}}, \end{equation} or equivalently, for any function $F\in{\mathrm{M}}_+(\varGamma_{\mathcal{Z}})$, \begin{equation}\label{eq:ghat} \int_{\varGamma_{\mathcal{Z}}} F({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\,\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}) =\int_{\varGamma_{X}}\!\biggl(\int_{{\mathfrak{X}}^\infty} F\Bigl(\mbox{$\:\!$}{\textstyle\bigcup\nolimits_{x\in\gamma}}\{(x,\bar{y})\} \mbox{$\;\!\!$}\Bigr)\,{\textstyle\bigotimes\limits_{x\in\gamma}} \,\eta(\mathrm{d}\bar{y})\mbox{$\:\!\!$}\biggr)\,\g(\mathrm{d}\gamma). \end{equation}
\begin{remark} Vector $\bar{y}$ in each pair $z=(x,\bar{y})\in \mathcal{Z}$ may be interpreted as a \emph{mark} attached to the point $x\in X$, so that ${\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}$ becomes a marked configuration, with the mark space $\mathfrak{X}$ (see \cite{DVJ1,KunaPhD,KKS98}). The corresponding \textit{marked configuration space} is defined by \begin{equation}\label{eq:Gamma(X)} \varGamma_{X}(\mathfrak{X}):=\{{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\in\varGamma_{\mathcal{Z}}: p_{X}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\in\varGamma_X\}\subset \varGamma_{\mathcal{Z}}. \end{equation} In other words, $\varGamma_{X}(\mathfrak{X})$ is the set of configurations in $\varGamma_{\mathcal{Z}}$ such that the collection of their $x$-coordinates is a (proper) configuration in $\varGamma_{X}$. Clearly, $\varGamma_{X}(\mathfrak{X})$ is a Borel subset of $\varGamma_{\mathcal{Z}}$, that is, $\varGamma_{X}(\mathfrak{X})\in\mathcal{B}(\varGamma_{\mathcal{Z}})$. Since $\g(\varGamma_X)=1$, we have $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\varGamma_{X}(\mathfrak{X}))=1$. \end{remark}
Finally, owing to the general Theorem \ref{th:mucl} (see (\ref{eq:cl*})), the corresponding Gibbs cluster measure $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$ on the configuration space $\varGamma_X$ is represented as a push-forward of the measure $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$ on $\varGamma_{\mathcal{Z}}$ under the map $\mathfrak{q}$ defined in (\ref{proj1}), (\ref{eq:proj}): \begin{equation}\label{eq:gcl*} \g_{\mbox{$\;\!\!$}\mathrm{cl}}=\mathfrak{q}^*\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}\equiv \hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}\circ \mathfrak{q}^{-1}. \end{equation}
Our next goal is to show that $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$ is a \textit{Gibbs measure} on $\varGamma_{\mathcal{Z}}$, with the reference measure $\sigma$ defined as a product measure on the space $\mathcal{Z}=X\times\mathfrak{X}$, \begin{equation}\label{eq:sigma} \sigma:=\theta \otimes\eta, \end{equation} and with the interaction potential \mbox{$\;\!$}$\hat{{\varPhi}}:\varGamma^{\mbox{$\:\!$} 0}_{\mathcal{Z}}\to\mathbb{R}\cup\{+\infty\}$ given by \begin{equation}\label{eq:hatU} \hat{{\varPhi}}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}):=\left\{\begin{array}{ll} {\varPhi}(p_{X}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})),\quad& {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\in\varGamma^{\mbox{$\:\!$} 0}_{\mathcal{Z}}\cap \varGamma_{X}(\mathfrak{X}),\\[.3pc] +\infty,&{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\in\varGamma^{\mbox{$\:\!$} 0}_{\mathcal{Z}}\setminus\varGamma_{X}(\mathfrak{X}), \end{array}\right. \end{equation} where $p_{X}$ is the projection defined in (\ref{eq:pX}). The corresponding functionals of energy $\hat{E}(\hat{\xi}\mbox{$\:\!$})$ and interaction energy $\hat{E}(\hat{\xi},{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})$ ($\hat{\xi} \in \varGamma_{\mathcal{Z}}^{\myp0}$, ${\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\in \varGamma_{\mathcal{Z}}$) are then given by (see (\ref{eq:E}) and (\ref{eq:E-E})) \begin{gather} \label{eq:Ehat} \hat{E}(\hat{\xi}\mbox{$\:\!$}):=\sum_{\hat{\xi}'\subset \mbox{$\:\!$}\hat{\xi}}\hat{\varPhi}(\hat{\xi}'),\\ \label{eq:E-Ehat} \hat{E}(\hat{\xi},{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}):=\left\{ \begin{array}{ll} \displaystyle\sum_{{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\supset\mbox{$\:\!$}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}'\mbox{$\;\!\!$}\in\varGamma_{\mathcal{Z}}^{\myp0}} \!\hat{\varPhi}(\hat{\xi}\cup{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}'), & \displaystyle \sum_{{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\supset\mbox{$\:\!$}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}'\mbox{$\;\!\!$}\in\varGamma_{\mathcal{Z}}^{\myp0}}
\!|\mbox{$\:\!$}\hat{\varPhi}(\hat{\xi}\cup{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}')| <\infty , \\[1.7pc] \displaystyle\ \ +\infty & \quad \text{otherwise}. \end{array} \right. \end{gather}
The following ``projection''property of the energy is obvious from the definition (\ref{eq:hatU}) of the potential $\hat{{\varPhi}}$. \begin{lemma}\label{lm:invariance} For any configurations\/ $\hat{\xi}\in \varGamma_{\mathcal{Z}}^{\myp0}$ and ${\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\in \varGamma_{\mathcal{Z}}$, we have \begin{equation*} \hat{E}(\hat{\xi}\mbox{$\:\!$})=E(p_{X}(\hat{\xi}\mbox{$\:\!$})),\qquad \hat{E}(\hat{\xi},{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})=E(p_{X}(\hat{\xi}\mbox{$\:\!$}),p_{X}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})). \end{equation*} \end{lemma}
\begin{theorem}\label{th:g-hat} \textup{(a)} \,Let\/ $\g\in{\mathscr{G}}(\theta,{\varPhi}\mbox{$\:\!$})$ be a Gibbs measure on the configuration space\/ $\varGamma_X$, and let\/ $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$ be the corresponding probability measure on the configuration space\/ $\varGamma_{\mathcal{Z}}$ \textup{(}see \textup{(\ref{eq:ghat0})} or \textup{(\ref{eq:ghat})}\textup{)}. Then\/ $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}\in {\mathscr{G}}(\sigma,\hat{{\varPhi}}\mbox{$\:\!$})$, i.e., $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$ is a Gibbs measure on\/ $\varGamma_{\mathcal{Z}}$ with the reference measure\/ $\sigma$ and the interaction potential\/ $\hat{{\varPhi}}$ defined by \textup{(\ref{eq:sigma})} and \textup{(\ref{eq:hatU})}, respectively.
\textup{(b)} \,If the measure\/ $\g\in{\mathscr{G}}(\theta,{\varPhi}\mbox{$\:\!$})$ has a finite correlation function\/ $\kappa_{\mbox{$\;\!\!$}\g}^{n}$ of some order\/ $n\in\mathbb{N}$ \textup{(}see the definition \textup{(\ref{corr-funct})} in the Appendix\textup{)}, then the correlation function\/ $\kappa_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{n}$ of the measure\/ $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}\in {\mathscr{G}}(\sigma,\hat{{\varPhi}}\mbox{$\:\!$})$ is given by \begin{equation}\label{corr-funct0} \kappa_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{n}(z_{1},\dots,z_{n}) =\kappa_{\mbox{$\;\!\!$}\g}^{n}\bigl(p_{X}(z_{1}),\dots,p_{X}(z_{n})\bigr),\qquad z_1,\dots,z_n\in \mathcal{Z}. \end{equation}
\end{theorem}
\proof (a)
\,In order to show that $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}\in {\mathscr{G}}(\sigma,\hat{{\varPhi}}\mbox{$\:\!$})$, it suffices to check that $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$ satisfies Nguyen--Zessin's equation on $\varGamma_{\mathcal{Z}}$ (see equation (\ref{eq:NZ}) in the Appendix), that is, for any non-negative, $\mathcal{B}(\mathcal{Z}) \times \mathcal{B}(\varGamma_{\mathcal{Z}})$-measurable function $H(z,{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})$ it holds \begin{equation}\label{eq:NZ-g-hat-exp} \int_{\varGamma_{\mathcal{Z}}} \sum_{z\in\hat{\gamma\mbox{$\:\!$}}} \!H(z,{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\,\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}) =\int_{\varGamma_{\mathcal{Z}}}\! \left(\int_{\mathcal{Z}}\!H(z,{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\cup \{z\})\,\mathrm{e\mbox{$\:\!$}}^{-\hat{E}(\{z\},\mbox{$\:\!$}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}} )}\,\sigma (\mathrm{d}{z})\mbox{$\:\!\!$}\right)\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}).
\end{equation} Using the disintegration formula (\ref{eq:muhat}), the left-hand side of (\ref{eq:NZ-g-hat-exp}) can be represented as \begin{multline}\label{eq:cond-eta} \int_{\varGamma_X}\!\biggl(\int_{{\mathfrak{X}}^\infty} \,\sum_{x\in \gamma} H\bigl(x,\bar{y}_x;\,{\textstyle\bigcup\limits_{x'\in\gamma}}\{({\textstyle x'}\mbox{$\:\!\!$},\bar{y}_{x'\mbox{$\:\!\!$}})\}\bigr) {\textstyle\bigotimes\limits_{x'\in\gamma}}\eta(\mathrm{d}\bar{y}_{x'\mbox{$\;\!\!$}})\mbox{$\:\!\!$}\biggr)\,\g(\mathrm{d}\gamma) \\ =\int_{\varGamma_X}\!\left(\mbox{$\:\!\!$}\sum_{x\in\gamma} \int_{\varGamma_{\mathfrak{X}}} \mathbf{1}_{\gamma}(x)\, H\bigl(x,\bar{y}_x;\,{\textstyle\bigcup\limits_{x'\in\gamma}}\{({\textstyle x'}\mbox{$\:\!\!$},\bar{y}_{x'\mbox{$\:\!\!$}})\}\bigr) {\textstyle\bigotimes\limits_{x'\in\gamma}}\,\eta(\mathrm{d}\bar{y}_{x'\mbox{$\:\!\!$}})\mbox{$\:\!\!$}\right)\g(\mathrm{d}\gamma). \end{multline} Applying Nguyen--Zessin's equation to the Gibbs measure $\g$ with the function \begin{equation*}
H_0(x,\gamma):=\int_{\varGamma_{\mathfrak{X}}} \mathbf{1}_{\gamma}(x)\, H\Bigl(x,\bar{y}_x;\,{\textstyle\bigcup\limits_{x'\in\gamma}}\{({\textstyle x'}\mbox{$\:\!\!$},\bar{y}_{x'\mbox{$\:\!\!$}})\}\Bigr) {\textstyle\bigotimes\limits_{x'\in\gamma}}\,\eta(\mathrm{d}\bar{y}_{x'\mbox{$\:\!\!$}}), \end{equation*} we see that the right-hand side of (\ref{eq:cond-eta}) takes the form \begin{equation}\label{eq:E_g} \int_{\varGamma_X}\!\biggl(\mbox{$\:\!\!$}\sum_{x\in \gamma} H_0(x,\gamma)\mbox{$\:\!\!$}\biggr)\,\g(\mathrm{d}\gamma)= \int_{\varGamma_X} \!\biggl(\int_{X} H_0(x,\gamma\cup \{x\})\,\mathrm{e\mbox{$\:\!$}}^{-E(\{x\},\mbox{$\:\!$}\gamma)}\,\theta(\mathrm{d} x)\mbox{$\:\!\!$}\biggr)\,\g(\mathrm{d}\gamma). \end{equation} Similarly,
exploiting the product structure of the measures \,$\sigma=\theta\otimes\eta$ and $$ {\textstyle\bigotimes\limits_{x'\in\gamma\cup\{x\}}} \mbox{$\:\!\!$}\eta(\mathrm{d}\bar{y}_{x'\mbox{$\:\!\!$}})= {\textstyle\bigotimes\limits_{x'\in\gamma}}\,\eta(\mathrm{d}\bar{y}_{x'\mbox{$\:\!\!$}}) \otimes \eta(\mathrm{d}\bar{y}_{x}), $$ and using Lemma \ref{lm:invariance}, the right-hand side of (\ref{eq:NZ-g-hat-exp}) is reduced to \begin{align*} &\int_{\varGamma_X}\!\biggl(\int_{\mathfrak{X}^{\infty}} \biggl(\int_{X\times\mathfrak{X}} H\bigl(x,\bar{y}_x;{\textstyle\bigcup\limits_{x'\in\gamma}}\{({\textstyle x'}\mbox{$\:\!\!$},\bar{y}_{x'\mbox{$\:\!\!$}})\}\cup\{(x,\bar{y}_x)\}\bigr)\\ &\hspace{9.97pc} \times\mathrm{e\mbox{$\:\!$}}^{-E(\{x\},\mbox{$\:\!$}\gamma)}\,\eta(\mathrm{d}\bar{y}_x)\,\theta(\mathrm{d} x)\biggr) {\textstyle\bigotimes\limits_{x'\in\gamma}} \,\eta(\mathrm{d}\bar{y}_{x'\mbox{$\:\!\!$}})\biggr)\,\g(\mathrm{d}\gamma)\\ &= \int_{\varGamma_X}\!\biggl(\int_X \biggl(\int_{\mathfrak{X}^{\infty}} \mathbf{1}_{\gamma}(x)\,H\bigl(x,\bar{y}_x;{\textstyle\bigcup\limits_{x'\in\gamma}} \{({x'}\mbox{$\:\!\!$},\bar{y}_{x'\mbox{$\:\!\!$}})\}\bigr)\\ &\hspace{9.97pc}\times \mathrm{e\mbox{$\:\!$}}^{-E(\{x\},\mbox{$\:\!$}\gamma)}{\textstyle\bigotimes\limits_{x'\in\gamma}}\,\eta(\mathrm{d}\bar{y}_{x'\mbox{$\:\!\!$}}) \biggr)\,\theta(\mathrm{d} x)\biggr)\,\g(\mathrm{d}\gamma)\\ &=\int_{\varGamma_X}\!\biggl(\int_{X} H_0(x,\gamma\cup \{x\})\,\mathrm{e\mbox{$\:\!$}}^{-E(\{x\},\mbox{$\:\!$}\gamma)}\,\theta(\mathrm{d}{x})\biggr)\,\g(\mathrm{d}\gamma), \end{align*} thus coinciding with (\ref{eq:E_g}). This proves equation (\ref{eq:NZ-g-hat-exp}), hence $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}\in {\mathscr{G}}(\sigma,\hat{{\varPhi}}\mbox{$\:\!$})$.
(b) \,Let $f\in C_0(\mathcal{Z}^{n})$ be a symmetric function. According to the disintegration formula (\ref{eq:muhat}) applied to the function \begin{equation*} F({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}):=\sum_{\{z_{1}\mbox{$\;\!\!$},\dots,\mbox{$\:\!$} z_{n}\} \subset {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}}f(z_{1},\dots,z_{n}), \end{equation*} we have \begin{equation}\label{eq:sum-phi} \int_{\varGamma_{\mathcal{Z}}} \mbox{$\:\!\!$} F({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\,\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})= \int_{\varGamma_{X}}\!\mbox{$\;\!\!$}{}\sum_{\{x_{1}\mbox{$\;\!\!$},\dots,\mbox{$\:\!$} x_{n}\}\subset\gamma}\mbox{$\:\!\!$} \phi (x_{1},\dots,x_{n})\, \g(\mathrm{d}\gamma), \end{equation} where \begin{equation*} \phi (x_{1},\dots,x_{n}):=\int_{\mathfrak{X}^{n}} \mbox{$\:\!\!$} f((x_{1},\bar{y}_{1}\mbox{$\;\!\!$}),\dots,(x_{n},\bar{y}_{n}))\,{\textstyle\bigotimes\limits_{i=1}^n}\, \eta(\mathrm{d}\bar{y}_{i})\in C_0(X^n). \end{equation*} Applying the definition of the correlation function $\kappa_{\mbox{$\;\!\!$}\g}^n$ (see (\ref{corr-funct})) and using that $\theta(\mathrm{d}{x})\otimes\eta(\mathrm{d}\bar{y})=\sigma(\mathrm{d}{x}\times\mathrm{d}\bar{y})$, we obtain from (\ref{eq:sum-phi}) \begin{equation*} \int_{\varGamma_{\mathcal{Z}}} \mbox{$\:\!\!$} F({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\,\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}) =\frac{1}{n!}\int_{\mathcal{Z}^{n}}f(z_{1},\dots,z_{n}) \,\kappa_{\mbox{$\;\!\!$}\g}^{n}\bigl(p_{X}(z_{1}),\dots,p_{X}(z_{n})\bigr) \,{\textstyle\bigotimes\limits_{i=1}^n}\,\sigma(\mathrm{d}{z}_{i}), \end{equation*} and equality (\ref{corr-funct0}) follows. \endproof
In the rest of this subsection, ${\mathscr{G}_{\mathrm{R}}}$ denotes the subclass of Gibbs measures in $\mathscr{G}$ (with a given reference measure and interaction potential) that satisfy the so-called \textit{Ruelle bound} (see the Appendix, formula (\ref{eq:RB})).
\begin{corollary}\label{cor:GR} We have\/ $\g\in{\mathscr{G}_{\mathrm{R}}}(\theta,{\varPhi}\mbox{$\:\!$})$ if and only if\/ $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}\in {\mathscr{G}_{\mathrm{R}}}(\sigma,\hat{{\varPhi}}\mbox{$\:\!$})$, \end{corollary}
\proof Follows directly from formula (\ref{corr-funct0}). \endproof
The following statement is, in a sense, converse to Theorem \ref{th:g-hat}\mbox{$\:\!$}(a). \begin{theorem}\label{th:converse} If\/ $\varpi \in {\mathscr{G}}(\sigma ,\hat{{\varPhi}}\mbox{$\:\!$})$ then $\g:=p_{X}^{\ast}\varpi \in {\mathscr{G}}(\theta,{\varPhi}\mbox{$\:\!$})$. Moreover, if\/ $\g\in{\mathscr{G}_{\mathrm{R}}}(\theta,{\varPhi}\mbox{$\:\!$})$ then\/ $\varpi =\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$. \end{theorem}
\proof Applying Nguyen--Zessin's equation (\ref{eq:NZ}) to the measure $\varpi$ and using the cylinder structure of the interaction potential, we have \begin{align*} \int_{\varGamma_{X}}&\sum_{x\in \gamma}H(x,\gamma )\,p_{X}^{\ast} \varpi(\mathrm{d}\gamma)=\int_{\varGamma_{\mathcal{Z}}}\sum_{x\in p_{X}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}} H(x,p_{X}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\,\varpi(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}) \\ &=\int_{\varGamma_{\mathcal{Z}}}\!\left( \int_{\mathcal{Z}}H(p_{X}z,p_{X}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\cup \{z\} )\,\mathrm{e\mbox{$\:\!$}}^{-E(\{p_{X}z\},\,p_{X}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})}\,\theta\otimes \eta(\mathrm{d}{z})\mbox{$\:\!\!$}\right) \varpi (\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}) \\ &=\int_{\varGamma_{\mathcal{Z}}}\!\left( \int_{X}H(x,p_{X}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\cup \{x\})\,\mathrm{e\mbox{$\:\!$}}^{-E(\{x\},\,p_{X}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})}\,\theta(\mathrm{d} x)\mbox{$\:\!\!$}\right) \varpi(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}) \\ &=\int_{\varGamma_{X}}\!\left(\int_{X} H(x,\gamma \cup\{x\})\,\mathrm{e\mbox{$\:\!$}}^{-E(\{x\},\mbox{$\:\!$}\gamma)}\,\theta(\mathrm{d} x)\right) p_{X}^{\ast} \varpi(\mathrm{d}\gamma). \end{align*} Thus, the measure $p_{X}^*\varpi$ satisfies Nguyen--Zessin's equation and so, by Theorem \ref{th:NZR}, belongs to the Gibbs class $\mathscr{G}(\theta,\varPhi)$.
Next, in order to prove that $\varpi =\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$, by Proposition \ref{pr:k=k} it suffices to show that the measures $\varpi$ and $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$ have the same correlation functions. Note that the correlation function $\kappa_{\varpi}^{n}$ can be written in the form \cite[\S\myp2.3, Lemma~2.3.8]{KunaPhD} \begin{align*} \kappa_{\varpi}^{n}(z_1,\dots,z_n) &=\mathrm{e\mbox{$\:\!$}}^{-\hat{E}(\{z_1,\dots,\mbox{$\:\!$} z_n\})}\int_{\varGamma_{\mathcal{Z}}} \mathrm{e\mbox{$\:\!$}}^{-\hat{E}(\{z_1,\dots,\mbox{$\:\!$} z_n\},\mbox{$\:\!$}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})}\,\varpi(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\\ &=\mathrm{e\mbox{$\:\!$}}^{-E(\{p_{X}(z_1),\dots,\mbox{$\;\!$} p_{X}(z_n)\})}\int_{\varGamma_{X}} \mathrm{e\mbox{$\:\!$}}^{-E(\{p_{X}(z_1),\dots,\mbox{$\;\!$} p_{X}(z_n)\},\mbox{$\:\!$}\gamma)}\,p_{X}^{\ast }\varpi(\mathrm{d}\gamma)\\ &= \mathrm{e\mbox{$\:\!$}}^{-E(\{p_{X}(z_1),\dots,\mbox{$\;\!$} p_{X} (z_n)\})}\int_{\varGamma_{X}} \mathrm{e\mbox{$\:\!$}}^{-E(\{p_{X}(z_1),\dots,\mbox{$\;\!$} p_{X}(z_n)\},\mbox{$\:\!$}\gamma)}\,\g(\mathrm{d}\gamma) \\ &=\kappa_{\mbox{$\;\!\!$}\g}^{n}\bigl(p_{X}(z_1),\dots, p_{X}(z_n)\bigr). \end{align*} Therefore, on account of Theorem \ref{th:g-hat}\mbox{$\:\!$}(b) we get $ \kappa_{\varpi}^{n}(z_1,\dots,z_n)=\kappa_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{n}(z_1,\dots,z_n)$ for all $z_1,\dots,z_n\in\mathcal{Z}$ \,($z_i\ne z_j$), as required. \endproof
In the next corollary, $\mathop{\mathrm{ext}}\nolimits\mathscr{G}$ denotes the set of \textit{extreme points} of the class $\mathscr{G}$ of Gibbs measures with the corresponding reference measure and interaction potential (see the Appendix). \begin{corollary}\label{cor:2.9} Suppose that\/ $\g\in{\mathscr{G}_{\mathrm{R}}}(\theta,{\varPhi}\mbox{$\:\!$})$. Then\/ $\g\in\mathop{\mathrm{ext}}\nolimits\mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$ if and only if\/ $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}\in\mathop{\mathrm{ext}}\nolimits\mathscr{G}(\sigma,\hat{{\varPhi}}\mbox{$\:\!$})$. \end{corollary} \proof Let $\g\in {\mathscr{G}_{\mathrm{R}}}(\theta,{\varPhi}\mbox{$\:\!$})\cap \mathop{\mathrm{ext}}\nolimits\mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$. Assume that $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}=\frac12\mbox{$\:\!$}(\mu_{1}+\mu_{2})$ with some $\mu_{1},\mu_{2}\in \mathscr{G}(\sigma,\hat{{\varPhi}}\mbox{$\:\!$})$. Then $\g=\frac12\mbox{$\:\!$}(\g_{1}+\g_{2})$, where $\g_{i}=p_{X}^{\ast} \mu_{i}\in\mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$. Since $\g\in\mathop{\mathrm{ext}}\nolimits\mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$, this implies that $\g_{1}=\g_{2}=\g$. In particular, $\g_{1},\g_{2}\in{\mathscr{G}_{\mathrm{R}}}(\theta,{\varPhi}\mbox{$\:\!$})$ and by Theorem \ref{th:converse} we obtain that $\mu_{1}=\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}_{1}=\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}=\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}_{2}=\mu _{2}$, which implies $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}\in \mathop{\mathrm{ext}}\nolimits \mathscr{G}(\sigma,\hat{{\varPhi}}\mbox{$\:\!$})$.
Conversely, let $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}\in\mathop{\mathrm{ext}}\nolimits\mathscr{G}(\sigma,\hat{{\varPhi}}\mbox{$\:\!$})$ and $\g=\frac12\mbox{$\:\!$}(\g_{1}+\g_{2})$ with $\g_{1},\g_{2}\in \mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$. Then $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}=\frac12\mbox{$\:\!$}(\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}_{1}+\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}_{2})$, hence $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}_{1}=\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}_{2}=\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}\in{\mathscr{G}_{\mathrm{R}}}(\sigma,\hat{{\varPhi}}\mbox{$\:\!$})$, which implies by Theorem \ref{th:converse} that $\g_{1}=p_{X}^{\ast}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}_{1}=p_{X}^{\ast}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}_{2}=\g_{2}$. Thus, $\g\in\mathop{\mathrm{ext}}\nolimits\mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$. \endproof
\subsection{Criteria of local finiteness and simplicity of the Gibbs cluster process}\label{sec:2.5}
Let us give conditions sufficient for the Gibbs CPP to be (a) locally finite, and (b) simple. For a given Borel set $B\in\mathcal{B}(X)$, consider a set-valued function (referred to as the \textit{droplet cluster}) \begin{equation}\label{eq:D} D_B(\bar{y}):={\textstyle\bigcup\limits_{y_i\in\bar{y}}} (B-y_i),\qquad \bar{y}\in\mathfrak{X}. \end{equation}
Let us also denote by $N_B(\bar{y})$ the number of coordinates of the vector $\bar{y}=(y_i)$ falling in the set $B\in\mathcal{B}(X)$, \begin{equation}\label{eq:N(y)} N_B(\bar{y}):=\sum_{y_i\in\bar{y}} \mathbf{1}_B(y_i),\qquad \bar{y}\in\mathfrak{X}, \end{equation} In particular, $N_X(\bar{y})$ is the ``dimension'' of $\bar{y}$, that is, the total number of its coordinates (recall that $\bar{y}\in\mathfrak{X}=\bigsqcup_{\mbox{$\:\!$} n=0}^{\mbox{$\:\!$}\infty} X^n$, see (\ref{eq:calX})).
\begin{theorem}\label{th:properClusterGibbs} Let $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$ be a Gibbs cluster measure on the generalized configuration space $\varGamma_ X^\sharp$.
\textup{(a)} \,Assume that the
correlation function $\kappa_{\mbox{$\;\!\!$}\g}^1$ of the measure $\g\in\mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$ is bounded.
Then, in order that $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$-a.a.\ configurations $\gamma\in \varGamma_{X}^\sharp$ be locally finite, it is sufficient that the following two conditions hold\textup{:}
{\rm (a-i)} \,for any compact set\/ $B\in{\mathcal{B}}(X)$, the number of coordinates of the vector\/ $\bar{y}\in\mathfrak{X}$ in $B$ is a.s.-finite, \begin{equation}\label{eq:condA1} N_B(\bar{y})<\infty\quad \text{for} \,\,\eta\text{-a.a.} \ \bar{y}\in\mathfrak{X}; \end{equation}
{\rm (a-ii)} \,for any compact set $B\in{\mathcal{B}}(X)$, the mean $\theta$-measure of the droplet cluster\/ $D_B(\bar{y})$ is finite, \begin{equation}\label{eq:condA2} \int_{\mathfrak{X}} \theta(D_B(\bar{y}))\, \eta(\mathrm{d}\bar{y})<\infty\mbox{$\:\!$}. \end{equation}
\textup{(b)} \,In order that $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$-a.a.\ configurations $\gamma\in \varGamma_{X}^\sharp$ be simple, it is sufficient that the following two conditions hold\textup{:}
{\rm (b-i)} \,for any $x\in X$, vector $\bar{y}$ contains a.s.\ no more than one coordinate $y_i=x$, \begin{equation}\label{eq:condB1} \sup_{x\in X} N_{\{x\}}(\bar{y})\le 1\quad \text{for} \,\,\eta\text{-a.a.} \ \bar{y}\in\mathfrak{X}; \end{equation}
{\rm (b-ii)} \,for any $x\in X$, the ``point'' droplet cluster $D_{\{x\}}(\bar{y})$
has a.s.\ zero $\theta$-measure, \begin{equation}\label{eq:condB2} \theta\bigl(D_{\{x\}}(\bar{y})\bigr)=0\quad \text{for} \,\,\eta\text{-a.a.} \ \bar{y}\in\mathfrak{X}. \end{equation} \end{theorem}
For the proof of part (a) of this theorem, we need a reformulation (stated as Proposition \ref{prop1} below) of the condition (a-ii), which will also play an important role in utilizing the projection construction of the Gibbs cluster measure (see Section~\ref{sec:3} below). For any Borel subset $B\in\mathcal{B}(X)$, denote \begin{equation}\label{eq:K} \mathcal{Z}_{B}:=\{z\in \mathcal{Z}:\,\mathfrak{q}(z)\cap B\ne\emptyset \}\in\mathcal{B}(\mathcal{Z}), \end{equation} where $\mathfrak{q}(z)=\bigsqcup_{y_i\in p_\mathfrak{X}(z)}\{y_i+p_{X}(z)\}$ (see (\ref{proj1})). That is to say, the set $\mathcal{Z}_{B}$ consists of all points $z=(x,\bar{y})\in \mathcal{Z}$ such that, under the ``projection'' $\mathfrak{q}$ onto the space $X$, at least one coordinate $y_i+x$ ($y_i\in\bar{y}$) belongs to the set $B\subset X$. \begin{proposition}\label{prop1} For any $B\in\mathcal{B}(X)$,
the condition \mbox{\textup{(a-ii)}} of\/ Theorem \textup{\ref{th:properClusterGibbs}\mbox{$\:\!$}(a)} is necessary and sufficient in order that\/ $\sigma(\mathcal{Z}_{B})<\infty$, where\/ $\sigma=\theta\otimes\eta$.
\end{proposition}
\proof[Proof\/ of\/ Proposition \textup{\ref{prop1}}]
By definition (\ref{eq:K}), $(x,\bar{y})\in \mathcal{Z}_B$ if and only if $x\in \bigcup_{y_i\in\bar y} (B-y_i)\equiv D_B(\bar y)$ (see (\ref{eq:D})). Hence,
\begin{align*} \sigma(\mathcal{Z}_{B})&=\int_{\mathfrak{X}}\!\left(\int_ X \mathbf{1}_{D_{B}(\bar y)}(x)\,\theta(\mathrm{d} x)\right)\eta(\mathrm{d}\bar{y}) =\int_{\mathfrak{X}}\theta\bigl(D_B(\bar y)\bigr)\,\eta(\mathrm{d}\bar{y}), \end{align*} and we see that the bound $\sigma(\mathcal{Z}_{B})<\infty$ is nothing else but condition (\ref{eq:condA2}). \endproof
\proof[Proof\/ of\/ Theorem \textup{\ref{th:properClusterGibbs}}] (a) \,Let $B\subset X$ be a compact set. By Proposition \ref{prop1}, condition (a-ii) is equivalent to $\sigma(\mathcal{Z}_B)<\infty$. On the other hand, by Theorem \ref{th:g-hat}\mbox{$\:\!$}(b) we have $\kappa_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^1(x,\bar{y})=\kappa_{\mbox{$\;\!\!$}\g}^1(x)$. Hence, $\kappa_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^1$ is bounded, and by Remark \ref{rm:kappa<const} (see the Appendix) it follows that ${\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}(\mathcal{Z}_B)<\infty$ ($\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$-a.s.). According to the projection representation $\g_{\mbox{$\;\!\!$}\mathrm{cl}}=\mathfrak{q}^*\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$ (see (\ref{eq:gcl*})) and in view of condition \mbox{(a-i)}, this implies that, almost surely, a projected configuration $\gamma=\mathfrak{q}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})=\bigsqcup_{z\in \hat{\gamma\mbox{$\:\!$}}}\mathfrak{q}(z)$ contributes no more than
finitely many points to the set $B\subset \mathfrak{q}(\mathcal{Z}_B)$, that is, $\gamma(B)<\infty$ ($\g_{\mbox{$\;\!\!$}\mathrm{cl}}$-a.s.), which completes the proof of part~(a).
(b) \,It suffices to prove that, for any compact set $\varLambda\subset X$,
there are $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$-a.s.\ no cross-ties between the clusters whose centers belong to $\varLambda$.
That is, we must show that $\g_{\mbox{$\;\!\!$}\mathrm{cl}}(A_\varLambda)=0$, where the set $A_\varLambda\in\mathcal{B}(\varGamma_X\times \mathfrak{X}^2)$ is defined by \begin{equation}\label{eq:A-Lambda} A_\varLambda:=\{(\gamma,\bar{y}_1,\bar{y}_2): \,\exists\, x_1,x_2\in{}\gamma\cap\varLambda,\ \exists\,y_1\in\bar{y}_1: \,x_1+y_1-x_2\in\bar{y}_2\}, \end{equation} Applying the disintegration formula (\ref{eq:ghat}), we obtain \begin{align}\label{eq:Psi3} \g_{\mbox{$\;\!\!$}\mathrm{cl}}(A_\varLambda)&=\int_{\varGamma_X} \mbox{$\:\!\!$} F(\gamma)\,\g(\mathrm{d}\gamma), \end{align} where \begin{equation}\label{eq:F} F(\gamma):=\int_{\mathfrak{X}^2} \mathbf{1}_{A_\varLambda}(\gamma,\bar{y}_1,\bar{y}_2)\, \eta(\mathrm{d}\bar{y}_1)\,\eta(\mathrm{d}\bar{y}_2),\qquad \gamma\in\varGamma_X. \end{equation} Note that, according to the definition (\ref{eq:A-Lambda}), $F(\gamma)\equiv F(\gamma\cap\varLambda)$ \,($\gamma\in\varGamma_X$), hence, by Proposition
\ref{pr:Gibbs|cond}, we can rewrite (\ref{eq:Psi3}) in the form \begin{equation}\label{eq:Psi4} \g_{\mbox{$\;\!\!$}\mathrm{cl}}(A_\varLambda) =\int_{\varGamma_{\varLambda}} \mbox{$\:\!\!$} F(\xi)\mbox{$\;\!$} S_\varLambda(\xi)\, \lambda_{\theta}(\mathrm{d}\xi), \end{equation} with $S_\varLambda(\xi)\in L^1(\varGamma_\varLambda, \lambda_\theta)$.
Therefore, in order to show that the right-hand side of (\ref{eq:Psi4}) vanishes, it suffices to check that \begin{equation}\label{eq:=0} \int_{\varGamma_{\varLambda}} \mbox{$\:\!\!$} F(\xi)\,\lambda_{\theta}(\mathrm{d}\xi)=0. \end{equation}
To this end, substituting here the definition (\ref{eq:F}) and changing the order of integration, we can rewrite the integral in (\ref{eq:=0}) as \begin{align*} \int_{\mathfrak{X}^2} \theta^{\otimes\myp2}(B_\varLambda(\bar{y}_1,\bar{y}_2)) \,\eta(\mathrm{d}\bar{y}_1)\,\eta(\mathrm{d}\bar{y}_2), \end{align*} where $$ B_\varLambda(\bar{y}_1,\bar{y}_2)):= \{(x_1,x_2)\in\varLambda^2: \,x_1+y_1=x_2+y_2\ \text{\,for some }\, y_1\in\bar{y}_1,\ y_2\in\bar{y}_2\}. $$ It remains to note that \begin{align*} \theta^{\otimes\myp2}\bigl(B_\varLambda(\bar{y}_1,\bar{y}_2)\bigr) &=\int_\varLambda \theta\left(\textstyle\bigcup\nolimits_{y_1\in\bar{y}_1} \!\bigcup\nolimits_{y_2\in\bar{y}_2} \{x_1+y_1-y_2\}\right)\theta(\mathrm{d} x_1)\\[.2pc] &\le \sum_{y_1\in\bar{y}_1}\int_\varLambda \theta \left(\textstyle\bigcup\nolimits_{y_2\in\bar{y}_2}\mbox{$\:\!\!$} \{x_1+y_1-y_2\}\right) \theta(\mathrm{d} x_1)\\ &=\sum_{y_1\in\bar{y}_1}\int_\varLambda \theta \bigl(D_{\{x_1+y_1\}}(\bar{y}_2)\bigr)\,\theta(\mathrm{d} x_1) =0\qquad (\eta\text{-a.s.}), \end{align*} since, by assumption (\ref{eq:condB2}), $\theta\bigl(D_{\{x_1+y_1\}}(\bar{y}_2)\bigr)=0$ \,($\eta$-a.s.) and $\bar{y}_1$ contains at most countably many coordinates. Hence, (\ref{eq:=0}) follows and so part (b) is proved. \endproof
\begin{remark}\label{rm:a} In the Poisson cluster case (see \cite[Theorem 2.7\mbox{$\:\!$}(a)]{BD3}),
conditions \mbox{(a-i)} and \mbox{(a-ii)} of Theorem \ref{th:properClusterGibbs}\mbox{$\:\!$}(a) are not only sufficient but also necessary for the local finiteness of cluster configurations. While it is obvious that condition \mbox{(a-i)} is always necessary, there may be a question as to whether condition \mbox{(a-ii)} is such in the case of a Gibbs cluster measure $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$. Inspection of the proof of Theorem \ref{th:properClusterGibbs}\mbox{$\:\!$}(a) shows that the difficulty here lies in the questionable relationship between the conditions $\sigma(\mathcal{Z}_B)<\infty$ and ${\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}(\mathcal{Z}_{B})<\infty$ ($\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$-a.s.) (which are equivalent in the Poisson cluster case). According to Remark \ref{rm:kappa<const} (see the Appendix), under the hypothesis of boundedness of the first-order correlation function $\kappa_{\mbox{$\;\!\!$}\g}^1$, the former implies the latter, but the converse may not always be true. Simple counter-examples can be constructed by considering translation-invariant pair interaction potentials $\varPhi(\{x_1,x_2\})=\phi_0(x_1-x_2)\equiv\phi_{0}(y-x)$ such that $\phi_0(x)=+\infty$ on some subset $\varLambda_\infty\subset X$ with $\theta(\varLambda_\infty)=\infty$. However, if $\kappa_{\mbox{$\;\!\!$}\g}^1$ is bounded below and the mean number of configuration points in a set $B$ is finite then the measure $\theta(B)$ must be finite (see Remark~\ref{rm:kappa<const}). \end{remark}
\begin{remark}\label{rm:b} Similarly to Remark~\ref{rm:a}, it is of interest to ask whether conditions \mbox{(b-i)} and \mbox{(b-ii)} of Theorem \ref{th:properClusterGibbs}\mbox{$\:\!$}(b) are necessary for the simplicity of the cluster measure $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$ (as is the case for the Poisson cluster measure, see \cite[Theorem 2.7\mbox{$\:\!$}(b)]{BD3}).
However, in the Gibbs cluster case this is not so; more precisely, \mbox{(b-i)} is of course necessary, but \mbox{(b-ii)} may not be satisfied. For a simple counter-example, let the in-cluster measure $\eta$ be concentrated on a single-point configuration $\bar{y}=(0)$, so that the droplet cluster $D_{\{x\}}(\bar{y})$ is reduced to a single-point set $\{x\}$. Here, any measure $\theta$ with atoms will not satisfy condition \mbox{(b-ii)}. On the other hand, consider a Gibbs measure $\g$ with a hard-core translation-invariant pair interaction potential $\varPhi(\{x_1,x_2\})=\phi_0(x_1-x_2)\equiv\phi_{0}(y-x)$, where
$\phi_0(x)=+\infty$ for $|x|<r_0$ and $\phi_0(x)=0$ for $|x|\ge r_0$; then in each admissible configuration $\gamma$ any two points are at least at a distance $r_0$, and in particular any such $\gamma$ is simple. \end{remark}
\begin{remark}\label{rm:b'} As suggested by Remarks \ref{rm:a} and \ref{rm:b}, it is plausible that conditions \mbox{(a-ii)} and \mbox{(b-ii)} of Theorem \ref{th:properClusterGibbs} \textit{are} necessary for the claims (a) and (b), respectively,
if the interaction potential of the underlying Gibbs measure $\g$ is finite on all finite configurations, i.e., $\varPhi(\xi)<+\infty$ for all $\xi\in\varGamma_X^{\mbox{$\:\!$} 0}$. \end{remark}
In conclusion of this section, let us state some criteria sufficient for conditions \mbox{(a-ii)} and \mbox{(b-ii)} of Theorem \textup{\ref{th:properClusterGibbs}} (see details in \cite[\S\myp2.4]{BD3}). Assume for simplicity that the in-cluster configurations are a.s.-finite, $\eta\{N_X(\bar{y})<\infty\}=1$.
\begin{proposition}
\label{pr:a2} Either of the following conditions is sufficient for \textup{(\ref{eq:condA2})}\textup{:}
\textup{(a-ii$'$)} \,For any compact set $B\in\mathcal{B}(X)$, the $\theta$-measure of\/ its translations is uniformly bounded, \begin{equation}\label{sigma-cond} C_B:=\sup_{x\in X} \theta(B+x)<\infty, \end{equation} and, moreover, the mean number of\/ in-cluster points is finite, \begin{equation}\label{sigma-cond*} \int_{\mathfrak{X}} N_X(\bar{y})\,\eta(\mathrm{d}\bar{y})<\infty. \end{equation}
\textup{(a-ii$''$)} \,The coordinates of vector $\bar{y}$ are a.s.\ uniformly bounded, that is, there is a compact set $B_0\in\mathcal{B}(X)$ such that \,$N_{X\setminus B_0}(\bar{y})=0$ for $\eta$-a.a.\ $\bar{y}\in\mathfrak{X}$. \end{proposition}
\begin{proposition}
\label{pr:b2} Either of the following conditions is sufficient for \textup{(\ref{eq:condB2})}\textup{:}
\textup{(b-ii$'$)} \,The measure\/ $\theta$ is non-atomic, that is, \,$\theta\{x\}=0$\mbox{$\:\!$}{} for each\/ $x\in X$.
\textup{(b-ii$''$)} \,For each\/ $x\in X$, \,$N_{\{x\}}(\bar{y})=0$ \,for\, $\eta$-a.a.\ $\bar{y}\in\mathfrak{X}$. \end{proposition}
\section{Quasi-invariance and the integration-by-parts formula} \label{sec:3}
From now on, we restrict ourselves to the case where $X={\mathbb{R}}^{d}$. Henceforth, we assume that the in-cluster configurations are a.s.-finite, $\eta\{N_X(\bar{y})<\infty\}=1$; hence, the component $X^\infty$ representing infinite clusters (see Section~\ref{sec:2.1}) may be dropped, so the set $\mathfrak{X}$ is now redefined as $\mathfrak{X}:=\bigsqcup_{\mbox{$\:\!$} n\in\mathbb{Z}_+}\mbox{$\:\!\!$} X^{n}$ (cf.\ (\ref{eq:calX})). Note that condition \mbox{(a-i)} of Theorem \ref{th:properClusterGibbs} is then automatically satisfied.
We assume throughout that the correlation function $\kappa_{\mbox{$\;\!\!$}\g}^1(x)$ is bounded, which implies by Theorem \ref{th:properClusterGibbs} that the same is true for the correlation function $\kappa_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^1(z)$. Let us also impose conditions (\ref{sigma-cond}) and (\ref{sigma-cond*}) which, by Proposition \ref{pr:a2}, ensure that condition \mbox{(a-ii)} of Theorem \ref{th:properClusterGibbs}\mbox{$\:\!$}(a) is fulfilled and so $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$-a.a.\ configurations $\gamma\in\varGamma_X^\sharp$ are locally finite. According to Proposition \ref{prop1}, condition (a-ii) also implies that $\sigma(\mathcal{Z}_{B})<\infty$ providing that $\theta(B)<\infty$, where the set $\mathcal{Z}_{B}\subset \mathcal{Z}$ is defined in (\ref{eq:K}).
Finally, we require the probability measure $\eta$ on $\mathfrak{X}$ to be absolutely continuous with respect to the Lebesgue measure $\mathrm{d}\bar{y}$, \begin{equation}\label{eq:h} \eta (\mathrm{d} \bar{y})=h(\bar{y})\,{{\mathrm{d}}}\bar{y},\qquad \bar{y} =(y_{1},\dots ,y_{n})\in X^{n}\quad (n\in\mathbb{Z}_+). \end{equation} By Proposition \ref{pr:b2}\mbox{$\:\!$}\mbox{(b-ii$''$)}, this implies that Gibbs CPP configurations $\gamma $
are $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$-a.s.\ simple (i.e., have no multiple points). Altogether, the above assumptions ensure that $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$-a.a.\ configurations $\gamma $ belong to the proper configuration space $\varGamma_{X}$.
Our aim in this section is to prove the quasi-invariance of the measure $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$ with respect to
compactly supported diffeomorphisms of $X$ (Section~\ref{sec:3.2}), and to establish an integration-by-parts formula (Section~\ref{sec:3.3}). We begin in Section~\ref{sec:3.1} with a brief description of some convenient ``manifold-like'' concepts and notations first introduced in \cite{AKR1} (see also \cite[\S\myp4.1]{BD3}), which furnish a suitable framework for analysis on configuration spaces.
\subsection{Differentiable functions on configuration spaces} \label{sec:3.1}
Let $T_{x}X$ be the tangent space of $X={\mathbb{R}}^d$ at point $x\in X$. It can be identified in the natural way with ${\mathbb{R}}^{d}$, with the corresponding (canonical) inner product denoted by a ``fat'' dot~$\CD$\,. The gradient on $X$ is denoted by $\nabla $. Following \cite{AKR1}, we define the ``tangent space'' of the configuration space $\varGamma_{X}$ at $\gamma \in \varGamma _{X}$ as the Hilbert space $T_{\gamma}\varGamma _{X}:=L^{2}(X\rightarrow TX;\,{\mathrm{d}}\gamma )$, or equivalently $T_{\gamma}\varGamma_{X}=\bigoplus_{x\in \gamma}T_{x}X$. The scalar product in $T_{\gamma}\varGamma_{X}$ is denoted by $\langle\cdot,\cdot\rangle_{\gamma}$, with the corresponding norm
$|\cdot|_\gamma$. A vector field $V$ over $\varGamma_{X}$ is a map $\varGamma_{X}\ni \gamma\mapsto V(\gamma )=(V(\gamma )_{x})_{x\in \gamma}\in T_{\gamma}\varGamma_{X}$. Thus, for vector fields $V_1,V_2$ over $\varGamma_{X}$ we have \begin{equation*} \left\langle V_1(\gamma ),V_2(\gamma) \right\rangle_{\gamma}=\sum_{x\in \gamma}V_1(\gamma )_{x} \CD V_2(\gamma)_{x},\qquad \gamma \in \varGamma_X. \end{equation*}
For $\gamma\in\varGamma_{X}$ and $x\in\gamma$, denote by ${\mathcal{O}}_{\gamma,\mbox{$\:\!$} x}$ an arbitrary open neighborhood of $x$ in $X$ such that ${\mathcal{O}}_{\gamma ,\mbox{$\:\!$} x}\cap \gamma =\{x\}$. For any measurable function $F:\varGamma_{X}\rightarrow {{\mathbb{R}}}$, define the function $F_{x}(\gamma,\cdot): {\mathcal{O}}_{\gamma,\mbox{$\:\!$} x}\to\mathbb{R}$ by $F_{x}(\gamma,y):=F((\gamma \setminus \{x\})\cup \{y\})$, and set \begin{equation*} \nabla_{\mbox{$\;\!\!$} x} F(\gamma ):=\left.\nabla F_{x}(\gamma,y)
\right|_{y=x},\qquad x\in X, \end{equation*} provided that $F_{x}(\gamma,\cdot)$ is differentiable at $x$.
Recall that for a function $\phi:X\to\mathbb{R}$ its support $\mathop{\mathrm{supp}}\nolimits\phi$ is defined as the closure of the set $\{x\in X\!: \phi(x)\ne 0\}$. Denote by ${\mathcal{FC}}(\varGamma_{X})$ the class of functions on $\varGamma_{X}$ of the form \begin{equation}\label{local-funct} F(\gamma)=f(\langle \phi_{1},\gamma \rangle,\dots,\langle \phi_{k},\gamma\rangle),\qquad \gamma \in \varGamma_{X}, \end{equation} where $k\in\mathbb{N}$, \,$f\in C_{b}^{\infty}(\mathbb{R}^{k})$ ($:=$ the set of $C^{\infty}$-functions on ${\mathbb{R}}^{k}$ bounded together with all their derivatives), and $\phi_{1},\dots ,\phi_{k}\in C_{0}^{\infty}(X)$ ($:=$ the set of $C^{\infty }$-functions on $X$ with compact support). Each $F\in {\mathcal{FC}}(\varGamma_{X})$ is local, that is, there is a compact $K\subset X$ (which may depend on $F$) such that $ F(\gamma)=F(\gamma\cap K)$ for all $\gamma\in\varGamma_{X}$. Thus, for a fixed $\gamma $ there are
finitely many non-zero derivatives $\nabla_{\mbox{$\;\!\!$} x} F(\gamma)$.
For a function $F\in {\mathcal{FC}}(\varGamma_{X})$ its $\varGamma$-gradient $\nabla^{\varGamma}\mbox{$\;\!\!$} F\equiv \nabla^{\varGamma}_{\! X} F$ is defined as \begin{equation}\label{eq:G-gradient} \nabla^{\varGamma}\mbox{$\;\!\!$} F(\gamma ):=(\nabla_{\mbox{$\;\!\!$} x} F(\gamma))_{x\in \gamma}\in T_{\gamma}\varGamma_{X},\qquad \gamma \in \varGamma_{X}, \end{equation} so the directional derivative of $F$ along a vector field $V$ is given by \begin{equation*} \nabla_{\mbox{$\;\!\!$} V}^{\varGamma}F(\gamma ):=\langle \nabla^{\varGamma}\mbox{$\;\!\!$} F(\gamma ),V(\gamma )\rangle_{\gamma}=\sum_{x\in \gamma }\nabla_{\mbox{$\;\!\!$} x} F(\gamma) \CD V(\gamma)_{x},\qquad \gamma \in \varGamma_{X}. \end{equation*} Note that the sum here contains only finitely many non-zero terms.
Further, let ${\mathcal{FV}}(\varGamma_{X})$ be the class of cylinder vector fields $V$ on $\varGamma_{X}$ of the form \begin{equation}\label{vf} V(\gamma)_{x}=\sum_{i=1}^{k}A_{i}(\gamma)\mbox{$\;\!$} v_{i}(x)\in T_{x}X,\qquad x\in X, \end{equation} where $A_{i}\in {\mathcal{FC}}(\varGamma_{X})$ and $v_{i}\in \mathrm{Vect}_{0}(X)$ ($:=$ the space of compactly supported $C^{\infty}$-smooth vector fields on $X$), \,$i=1,\dots ,k$ \,($k\in \mathbb{N}$). Any vector filed $v\in \mathrm{Vect}_{0}(X)$ generates a constant vector field $V$ on $\varGamma_{X}$ defined by $V(\gamma)_{x}:=v(x)$. We shall preserve the notation $v$ for it. Thus, \begin{equation} \label{eq:grad-new} \nabla_{\mbox{$\;\!\!$} v}^{\varGamma} F(\gamma )=\sum_{x\in \gamma} \nabla_{\mbox{$\;\!\!$} x}F(\gamma)\CD v(x),\qquad \gamma \in \varGamma_X. \end{equation}
The approach based on ``lifting'' the differential structure from the underlying space $X$ to the configuration space $\varGamma_X$ as described above can also be applied to the spaces ${\mathfrak{X}}=\bigsqcup_{\mbox{$\:\!$} n=0}^{\infty }X^n$, $\mathcal{Z}=X\times\mathfrak{X}$ and $\varGamma_{\mathfrak{X}}$, $\varGamma_{\mathcal{Z}}$.
For these spaces, we will use the analogous notations as above without further explanation.
\subsection{$\mathrm{Diff}_{0}$-quasi-invariance}\label{sec:3.2}
In this section, we discuss the property of quasi-invariance of the measure $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$ with respect to diffeomorphisms of $X$. Let us start by describing how diffeomorphisms of $X$ act on configuration spaces. For a measurable map $\varphi :X\to X$, its \textit{support} $\mathop{\mathrm{supp}}\nolimits\varphi$ is defined as the smallest closed set containing all $x\in X$ such that $\varphi (x)\ne x$. Let ${\mathrm{Diff}}_{0}(X)$ be the group of diffeomorphisms of $X$ with \textit{compact support}. For any $\varphi \in {\mathrm{Diff}}_{0}(X)$, consider the corresponding ``diagonal'' diffeomorphism $\bar{\varphi}:\mathfrak{X}\to\mathfrak{X}$ acting on each constituent space $X^n$ ($n\in\mathbb{Z}_+$) as \begin{equation}\label{eq:phi-hat0} X^n\ni \bar{y}=(y_{1},\dots,y_{n})\mapsto \bar{\varphi}(\bar{y}):=(\varphi (y_{1}),\dots ,\varphi(y_{n}))\in X^n. \end{equation} For $x\in X$, we also define ``shifted'' diffeomorphisms \begin{equation}\label{eq:phi-hat} \bar{\varphi}_{x}(\bar{y}):=\bar\varphi(\bar{y}+x)-x,\qquad \bar{y}\in {\mathfrak{X}} \end{equation} (see the shift notation (\ref{eq:shift})). Finally, we introduce a special class of diffeomorphisms $\hat\varphi$ on $\mathcal{Z}$ acting only in the $\bar{y}$-coordinate as follows, \begin{equation}\label{eq:hat-phi} \hat\varphi(z):= (x,\bar{\varphi}_{x}(\bar{y}))\equiv (x,\mbox{$\:\!$}\bar\varphi(\bar{y}+x)-x),\qquad z=(x,\bar{y})\in \mathcal{Z}. \end{equation}
\begin{remark}\label{supp} Note that, even though $K_\varphi:= \mathop{\mathrm{supp}}\nolimits \varphi $ is compact in $X$, the support of the diffeomorphism $\hat{\varphi}$ (again defined as the closure of the set $\{z\in \mathcal{Z}: \hat{\varphi}(z)\ne z\}$) is given by $\mathop{\mathrm{supp}}\nolimits\hat{\varphi}=\mathcal{Z}_{K_\varphi}$ (see~(\ref{eq:K})) and hence is \textit{not} compact in the topology of $\mathcal{Z}$
(see Section~\ref{sec:2.1}).
\end{remark}
In the standard fashion, the maps $\varphi $ and $\hat{\varphi}$ can be lifted to measurable ``diagonal'' transformations (denoted by the same letters) of the configuration spaces $\varGamma_{X}$ and $\varGamma_{\mathcal{Z}}$, respectively: \begin{equation}\label{di*} \begin{aligned} \varGamma_{X}\ni \gamma \mapsto \varphi (\gamma ):={}&\{\varphi (x),\ x\in \gamma \}\in \varGamma_{X},\\[.2pc] \varGamma_{\mathcal{Z}}\ni {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\mapsto \hat{\varphi}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}):={}&\{\hat{\varphi}(z),\ (z)\in {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\}\in \varGamma_{\mathcal{Z}}. \end{aligned} \end{equation}
The following lemma shows that the operator $\mathfrak{q}$ commutes with the action of diffeomorphisms (\ref{di*}).\footnote{According to relation (\ref{comm1}), $\mathfrak{q}$ is an \textit{intertwining operator} between associated diffeomorphisms $\varphi$ and $\hat{\varphi}$.} \begin{lemma} For any diffeomorphism $\varphi \in {\mathrm{Diff}}_0(X)$ and the corresponding diffeomorphism $\hat{\varphi}$, it holds \begin{equation} \label{comm1} \varphi\circ\mathfrak{q}=\mathfrak{q}\circ \hat{\varphi}. \end{equation} \end{lemma}
\proof The statement follows from the definition (\ref{eq:proj}) of the map $\mathfrak{q}$ in view of the structure of diffeomorphisms $\varphi$ and $\hat{\varphi}$ (see (\ref{eq:hat-phi}) and (\ref{di*})). \endproof
\begin{lemma}\label{lm:shift-free} The interaction potential\/ \,$\hat{{\varPhi}}$\mbox{$\;\!$}{} defined in \textup{(\ref{eq:hatU})} is invariant with respect to diffeomorphisms \textup{(\ref{eq:hat-phi})}, that is, for any $\varphi\in\mathrm{Diff}_0(X)$ we have $$ \hat{{\varPhi}}(\hat{\varphi}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}))=\hat{{\varPhi}}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}), \qquad{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\in\varGamma_{\mathcal{Z}}. $$ In particular, this implies the $\hat{\varphi}$-invariance of the energy functionals defined in \textup{(\ref{eq:E})} and \textup{(\ref{eq:E-E})}, that is, for any $\hat{\xi} \in \varGamma_{\mathcal{Z}}^{\myp0}$ and ${\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\in \varGamma_{\mathcal{Z}}$, \begin{equation*} \hat{E}(\hat{\varphi}(\hat{\xi}\mbox{$\:\!$}))=\hat{E}(\hat{\xi}\mbox{$\:\!$}),\qquad \hat{E}(\hat{\varphi}(\hat{\xi}\mbox{$\:\!$}),\hat{\varphi}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})) =\hat{E}(\hat{\xi},{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}). \end{equation*} \end{lemma} \proof The claim readily follows by observing that a diffeomorphism (\ref{eq:hat-phi}) acts on the $\bar{y}$-coordinates of points $z=(x,\bar{y})$ in a configuration ${\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\in\varGamma_{\mathcal{Z}}$, while the interaction potential $\hat{{\varPhi}}$ (see (\ref{eq:hatU})) only depends on their $x$-coordinates. \endproof As already mentioned (see (\ref{eq:h})), we assume that the measure $\eta $ is absolutely continuous with respect to the Lebesgue measure $\mathrm{d}\bar{y}$ on ${\mathfrak{X}}$ and, moreover, \begin{equation}\label{QI} h(\bar{y}):=\frac{\eta (\mathrm{d}\bar{y})}{\mathrm{d}\bar{y}}>0\qquad {\text{for \,a.a.}}\ \,\bar{y}\in {\mathfrak{X}}. \end{equation} This implies that the measure $\eta$ is quasi-invariant with respect to the action of transformations $\bar{\varphi}:{\mathfrak{X}}\rightarrow {\mathfrak{X}}$ \,($\varphi \in {\mathrm{Diff}}_{0}(X)$), that is, for any $f\in\mathrm{M}_+(\mathfrak{X})$, \begin{equation}\label{eq:eta-inv} \int_{\mathfrak{X}} f(\bar{y})\,\bar{\varphi}^*\eta(\mathrm{d}\bar{y}) =\int_{\mathfrak{X}} f(\bar{y})\,\rho_{\eta}^{\bar{\varphi}}(\bar{y})\, \mathrm{d}\bar{y}, \end{equation} with the Radon--Nikodym density \begin{equation}\label{density'} \rho_{\eta}^{\bar{\varphi}}(\bar{y}):=\frac{\mathrm{d}(\bar{\varphi}^*\eta)}{\mathrm{d}\eta}(\bar{y})= \frac{h(\bar{\varphi}^{-1}(\bar{y}))}{ h(\bar{y})}\,J_{\bar{\varphi}}(\bar{y})^{-1} \end{equation} (we set $\rho_{\eta}^{\bar{\varphi}}(\bar{y})=1$ if $h(\bar{y})=0$ or $h(\bar{\varphi}^{-1}(\bar{y}))=0$). Here $J_{\bar{\varphi}}(\bar{y})$ is the Jacobian determinant of the diffeomorphism $\bar{\varphi}$; due to the diagonal structure of $\bar{\varphi}$ (see (\ref{eq:phi-hat0})) we have $J_{\bar{\varphi}}(\bar{y})=\prod_{y_i\in\bar{y}} J_{\varphi}(y_{i})$, where $J_{\varphi}(y)$ is the Jacobian determinant of $\varphi$.
Due to the ``shift'' form of diffeomorphisms (\ref{eq:hat-phi}), formulas (\ref{eq:eta-inv}), (\ref{density'}) readily imply that the product measure $\sigma(\mathrm{d}{z})=\theta(\mathrm{d}{x})\otimes\eta(\mathrm{d}\bar{y})$ on $\mathcal{Z}=X\times\mathfrak{X}$ is quasi-invariant with respect to $\hat\varphi$, that is, for each $\varphi\in\mathrm{Diff}_0(X)$ and any $f\in\mathrm{M}_+(\mathcal{Z})$, \begin{equation}\label{eq:sigma-inv} \int_{\mathcal{Z}} f(z)\,\hat{\varphi}^*\sigma(\mathrm{d}{z}) =\int_{\mathcal{Z}} f(z)\,\rho_{\varphi}(z)\, \sigma(\mathrm{d}{z}), \end{equation} where the Radon--Nikodym density $\rho_{\varphi}:=\mathrm{d}(\hat\varphi^*\sigma)/\mathrm{d}\sigma$ is given by (see (\ref{density'})) \begin{equation}\label{eq:rho-phi} \rho_{\varphi}(z)= \rho_{\eta}^{\bar{\varphi}_x}(\bar{y}) \equiv\frac{h\bigl(\bar{\varphi}^{-1}(\bar{y}+x)-x\bigr)\,}{h(\bar{y})\,} \,J_{\bar{\varphi}}(\bar{y}+x)^{-1},\quad z=(x,\bar{y})\in\mathcal{Z}. \end{equation}
We can now state our result on the quasi-invariance of the measure $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$.
\begin{theorem}\label{th:inv} The Gibbs measure $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$ constructed in Section \textup{\ref{sec:2.4}} is quasi-invariant with respect to the action of diagonal diffeomorphisms\/ $\hat{\varphi}$ on $\varGamma_{\mathcal{Z}}$ \textup{(}$\varphi\in\mathrm{Diff}_0(X)$\textup{)} defined by formula \textup{(\ref{eq:hat-phi})}, with the Radon--Nikodym density $R_{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{\varphi}}=\mathrm{d}(\hat{\varphi}^{\ast} \hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{})/\mathrm{d}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$ given by \begin{equation}\label{RND} R_{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{\varphi}}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}) = \prod_{z\in \hat{\gamma\mbox{$\:\!$}}} \mbox{$\:\!$}\rho_{\varphi}(z),\qquad{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\in\varGamma_{\mathcal{Z}}, \end{equation} where $\rho_{\varphi}(z)$ is defined in \textup{(\ref{eq:rho-phi})}. \end{theorem} \proof First of all, note that $\rho_{\varphi}(z)=1$ for any $z=(x,\bar{y})\notin \mathop{\mathrm{supp}}\nolimits\hat{\varphi}=\mathcal{Z}_{K_\varphi}$, where $K_\varphi=\mathop{\mathrm{supp}}\nolimits\varphi$ (see Remark \ref{supp}), and $\sigma(\mathcal{Z}_{K_\varphi})<\infty$ by Proposition \ref{prop1}. On the other hand, Theorem \ref{th:g-hat}\mbox{$\:\!$}(b) implies that the correlation function $\kappa_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^1$
is bounded. Therefore, by Remark \ref{rm:kappa<const} (see the Appendix) we obtain that ${\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}(\mathcal{Z}_{K_\varphi})<\infty$ for $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$-a.a.\ configurations ${\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\in\varGamma_{\mathcal{Z}}$, hence the product in (\ref{RND}) contains
finitely many terms different from $1$ and so the function $R_{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{\varphi}}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})$ is well defined. Moreover, it satisfies the ``localization'' equality \begin{equation}\label{eq:R=R} R_{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{\varphi}}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})= R_{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{\varphi}}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\cap \mathcal{Z}_{K_\varphi})\qquad \text{for}\ \ \hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}\text{-a.a.}\ \,{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\in\varGamma_{\mathcal{Z}}. \end{equation}
Following \cite[\S\myp2.8, Theorem 2.8.2]{KunaPhD},
the proof of the theorem will be based on the use of Ruelle's equation (see the Appendix, Theorem \ref{th:NZR}). Namely, according to (\ref{ruelle}) with $\varLambda= \mathcal{Z}_{K_\varphi}$, for any function $F\in\mathrm{M}_+(\varGamma_{\mathcal{Z}})$ we have \begin{align} \notag &\int_{\varGamma_{\mathcal{Z}}}\mbox{$\:\!\!$} F({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}) \,\hat{\varphi}^{\ast}{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})=\int_{\varGamma_{\mathcal{Z}}} \mbox{$\:\!\!$} F(\hat{\varphi}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})) \,\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\\ \notag &\ \ =\int_{\varGamma_{\varLambda}}\biggl(\int_{\varGamma_{\mathcal{Z} \setminus \varLambda}} \! F(\hat{\varphi}(\hat{\xi} \cup {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}'))\,{\mathrm{e\mbox{$\:\!$}}}^{-\hat{E}(\hat{\varphi}(\hat{\xi}\mbox{$\:\!$})) -\hat{E}(\hat{\varphi}(\hat{\xi}\mbox{$\:\!$}),\mbox{$\:\!$} \hat{\varphi}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}'))}\, \hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}')\biggr)\mbox{$\;\!$}\lambda_{\sigma}(\mathrm{d}\hat{\xi}\mbox{$\:\!$}) \\ \notag &\ \ =\int_{\varGamma_{\varLambda}}\biggl(\int_{\varGamma_{\mathcal{Z}\setminus \varLambda}} \!F(\hat{\varphi}(\hat{\xi}\mbox{$\:\!$})\cup {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}')\,{\mathrm{e\mbox{$\:\!$}}}^{-\hat{E}(\hat{\varphi}(\hat{\xi}\mbox{$\:\!$})) -\hat{E}(\hat{\varphi}(\hat{\xi}\mbox{$\:\!$}),\mbox{$\:\!$}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}')} \,\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}')\biggr)\mbox{$\;\!$}\lambda_{\sigma}(\mathrm{d}\hat{\xi}\mbox{$\:\!$})\\ \label{eq:bridge} &=\int_{\varGamma_{\varLambda}}\biggl( \int_{\varGamma_{\mathcal{Z}\setminus \varLambda}} \!F(\hat{\xi} \cup {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}')\,{\mathrm{e\mbox{$\:\!$}}}^{-\hat{E}(\hat{\xi}\mbox{$\:\!$})-\hat{E}(\hat{\xi},\mbox{$\:\!$} {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}')}\,\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}')\biggr)\, \hat{\varphi}^{\ast}\lambda_{\sigma}(\mathrm{d}\hat{\xi}\mbox{$\:\!$}), \end{align} where $\lambda_\sigma$ is the Lebesgue--Poisson measure corresponding to the reference measure $\sigma$ (see (\ref{eq:LP})). Since $\sigma$ is quasi-invariant with respect to diffeomorphisms $\hat\varphi$ (see (\ref{eq:sigma-inv})), it readily follows from the definition (\ref{eq:LP}) that the restriction of the Lebesgue--Poisson measure $\lambda_\sigma$ onto the set $\varGamma_\varLambda$ is quasi-invariant with respect to $\hat\varphi$, with the density given precisely by the function (\ref{RND}). Hence, using the property (\ref{eq:R=R}), the right-hand side of (\ref{eq:bridge}) is reduced to \begin{align} \notag &\int_{\varGamma_{\varLambda}}\mbox{$\:\!\!$}\biggl(\int_{\varGamma_{\mathcal{Z} \setminus \varLambda}}\!F(\hat{\xi} \cup {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}')\,{\mathrm{e\mbox{$\:\!$}}}^{-\hat{E}(\hat{\xi})-\hat{E}(\hat{\xi},\mbox{$\:\!$}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}')}\,\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{} (\mathrm{d}\gamma^{\mbox{$\:\!$}\prime})\mbox{$\:\!\!$}\biggr)\mbox{$\;\!$} R_{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{\varphi}} (\hat{\xi}\mbox{$\:\!$})\,\lambda_{\sigma}(\mathrm{d}\hat{\xi}\mbox{$\:\!$}) \\ \notag &\qquad=\int_{\varGamma_{\varLambda}}\mbox{$\:\!\!$}\biggl(\int_{\varGamma_{\mathcal{Z} \setminus \varLambda}}\!F(\hat{\xi} \cup {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}^{\mbox{$\:\!$}\prime})\mbox{$\;\!$} R_{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{\varphi}}(\hat{\xi} \cup {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}')\,{\mathrm{e\mbox{$\:\!$}}}^{-E(\hat{\xi})-E(\hat{\xi}, {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}^{\mbox{$\:\!$}\prime})} \,{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}^{\mbox{$\:\!$}\prime})\mbox{$\:\!\!$}\biggr)\,\lambda_{\sigma}(\mathrm{d}\hat{\xi}\mbox{$\:\!$}) \\ \label{eq:R2} &\qquad\qquad=\int_{\varGamma_{\mathcal{Z}}} \!F({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\mbox{$\;\!$} R_{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{\varphi}} ({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\,{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}), \end{align} where we have again used Ruelle's equation (\ref{ruelle}).
As a result, combining (\ref{eq:bridge}) and (\ref{eq:R2}) we obtain \begin{equation}\label{eq:R3} \int_{\varGamma_{\mathcal{Z}}}\mbox{$\:\!\!$} F({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}) \,\hat{\varphi}^{\ast}{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})=\int_{\varGamma_{\mathcal{Z}}} \!F({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\mbox{$\;\!$} R_{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{\varphi}} ({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\,{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}), \end{equation} which proves quasi-invariance of $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$. In particular, setting $F\equiv 1$ in (\ref{eq:R3}) yields $\int_{\varGamma_{\mathcal{Z}}} \!R_{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{\varphi}} ({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\,{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})=1$, and hence $R_{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{\varphi}}\in L^1(\varGamma_{\mathcal{Z}},\g_{\mbox{$\;\!\!$}\mathrm{cl}})$. \endproof
\begin{remark} Note that the Radon--Nikodym density $R_{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{\varphi}}$ defined by (\ref{RND}) does not depend on the background interaction potential ${\varPhi}$. As should be evident from the proof above, this is due to the special ``shift'' form of the diffeomorphisms $\hat{\varphi}$ (see (\ref{eq:hat-phi})) and the cylinder structure of the interaction potential $\hat{{\varPhi}}$ (see (\ref{eq:hatU})). In particular, the expression (\ref{RND}) applies to the ``interaction-free'' case with ${\varPhi}\equiv 0$ (and hence $\hat{{\varPhi}}\equiv0$), where the Gibbs measure $\g\in\mathscr{G}(\theta,{\varPhi}=0)$ is reduced to the Poisson measure $\pi_\theta$ on $\varGamma_X$ with intensity measure $\theta$ (see the Appendix), while the Gibbs measure $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}\in\mathscr{G}(\sigma,\hat{{\varPhi}}=0)$ amounts to the Poisson measure $\pi_\sigma$ on $\varGamma_{\mathcal{Z}}$ with intensity measure $\sigma$. \end{remark}
\begin{remark} As is essentially well known (see, e.g., \cite{AKR1,Sk}), quasi-invariance of a Poisson measure on the configuration space follows directly from the quasi-invariance of its intensity measure. For a proof adapted to our slightly more general setting (where diffeomorphisms are only assumed to have the support of finite measure), we refer the reader to \cite[Proposition A.1]{BD3}. Incidentally, the expression for the Radon--Nikodym derivative given in \cite{BD3} (see also
\cite[Proposition 2.2]{AKR1}) contained a superfluous normalizing constant, which in our context would read \begin{equation*} C_\varphi:=\exp\left(\int_{\mathcal{Z}} \bigl(1-\rho_{\varphi}(z)\bigr)\,\sigma(\mathrm{d}{z})\right) \end{equation*} (cf.\ (\ref{RND})). In fact, it is easy to see that $C_\varphi=1$; indeed, $\rho_\varphi=1$ outside the set $\mathop{\mathrm{supp}}\nolimits \hat{\varphi}=\mathcal{Z}_{K_\varphi}$ with $\sigma(\mathcal{Z}_{K_\varphi})<\infty$ (see Proposition \ref{prop1}), hence \begin{align*} \ln C_\varphi=\int_{\mathcal{Z}_{K_\varphi}} \bigl(1-\rho_{\varphi}(z)\bigr)\,\sigma(\mathrm{d}{z})&= \sigma(\mathcal{Z}_{K_\varphi})-\sigma(\hat{\varphi}^{-1}(\mathcal{Z}_{K_\varphi}))=0. \end{align*} \end{remark}
Let $\mathcal{I}_{\mathfrak{q}}:L^{2}(\varGamma_{X},\g_{\mbox{$\;\!\!$}\mathrm{cl}})\rightarrow L^{2}(\varGamma_{\mathcal{Z}},\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{})$ be the isometry defined by the map ${\mathfrak{q}}$ (see (\ref{eq:proj})), \begin{equation}\label{eq:I} ({\mathcal{I}_{\mathfrak{q}}}F)({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}):=F\circ\mathfrak{q}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}),\qquad {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}} \in \varGamma_{\mathcal{Z}}, \end{equation} and consider the corresponding adjoint operator \begin{equation}\label{eq:I*} \mathcal{I}_{\mathfrak{q}}^*:\mbox{$\:\!$} L^{2}(\varGamma_{\mathcal{Z}}, \hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{})\to L^{2}(\varGamma_{X},\g_{\mbox{$\;\!\!$}\mathrm{cl}}). \end{equation}
\begin{lemma}\label{lm:L1} The operator $\mathcal{I}_{\mathfrak{q}}^{\ast}$ defined by \textup{(\ref{eq:I*})} can be extended to the operator \begin{equation*} \mathcal{I}_{\mathfrak{q}}^*:\mbox{$\:\!$} L^{1}(\varGamma_{\mathcal{Z}},\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{})\rightarrow L^{1}(\varGamma _{X},\g_{\mbox{$\;\!\!$}\mathrm{cl}}). \end{equation*} \end{lemma}
\proof Note that $\mathcal{I}_{\mathfrak{q}}$ can be viewed as a bounded operator acting from $L^{\infty }(\varGamma _{X},\g_{\mbox{$\;\!\!$}\mathrm{cl}})$ to $L^{\infty}(\varGamma_{\mathcal{Z}},\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{})$. This implies that the adjoint operator $\mathcal{I}_{\mathfrak{q}}^*$ is a bounded operator on the corresponding dual spaces, $\mathcal{I}_{\mathfrak{q}}^*:\mbox{$\:\!$} L^{\infty}(\varGamma_{\mathcal{Z}},\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{})^{\prime}\to L^{\infty}(\varGamma _{X},\g_{\mbox{$\;\!\!$}\mathrm{cl}})^{\prime }$.
It is known (see \cite{GL}) that, for any sigma-finite measure space $(M,\mu)$, the corresponding space $L^{1}(M,\mu)$ can be identified with the subspace $V$ of the dual space $L^{\infty}(M,\mu)^{\prime}$ consisting of all linear functionals on $L^{\infty}(M,\mu)$ continuous with respect to bounded convergence in $L^{\infty}(M,\mu)$. That is, $\ell\in V$ if and only if $\ell(\psi_{n})\rightarrow 0$ for any $\psi_n\in L^{\infty }(M,\mu)$
such that $|\psi_{n}|\le 1$ and $\psi_{n}(x)\rightarrow 0$ as $n\to\infty$ for $\mu$-a.a.\ $x\in M$. Hence, to prove the lemma it suffices to show that, for any $F\in L^{1}(\varGamma_{\mathcal{Z}},\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{})$, the functional $\mathcal{I}_{\mathfrak{q}}^*F\in L^{\infty} (\varGamma_{\mathcal{Z}},\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{})^{\prime}$ is continuous with respect to bounded convergence in $L^{\infty}(\varGamma_{\mathcal{Z}},\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{})$. To this end, for any sequence $(\psi_{n})$ in $L^{\infty}(\varGamma_{X},\g_{\mbox{$\;\!\!$}\mathrm{cl}})$ such that
$|\psi_{n}|\le 1$ and $\psi_{n}(\gamma)\rightarrow 0$ for $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$-a.a.\ $\gamma\in \varGamma_{X}$, we have to prove that $\mathcal{I}_{\mathfrak{q}}^*F(\psi_{n})\to0$.
Let us first show that $\mathcal{I}_{\mathfrak{q}}\psi_{n}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\equiv \psi_{n}(\mathfrak{q}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}))\rightarrow 0$ for $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$-a.a.\ ${\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\in \varGamma_{\mathcal{Z}}$. Set \begin{align*} A_{\psi}:={}&\{\gamma\in\varGamma_{X}:\psi_{n}(\gamma)\rightarrow 0\}\in\mathcal{B}(\varGamma_{X}),\\[.3pc] \hat{A}_\psi:={}&\{{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\in\varGamma_{\mathcal{Z}}: \psi_{n}(\mathfrak{q}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}))\rightarrow 0\}\in\mathcal{B}(\varGamma_{\mathcal{Z}}), \end{align*} and note that $\hat{A}_\psi=\mathfrak{q}^{-1}(A_\psi)$; then, recalling the relation (\ref{eq:gcl*}),
we get \begin{equation*} \hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\hat{A}_\psi)=\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}\bigl(\mathfrak{q}^{-1}(A_\psi)\bigr)=\g_{\mbox{$\;\!\!$}\mathrm{cl}}(A_\psi)=1, \end{equation*} as claimed. Now, by the dominated convergence theorem this implies \begin{equation*} \mathcal{I}_{\mathfrak{q}}^*F(\psi_{n})=\int_{\varGamma_{\mathcal{Z}}} F({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\,\mathcal{I}_{\mathfrak{q}}\psi_{n} ({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\,\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\rightarrow 0, \end{equation*} and the proof is complete. \endproof
Taking advantage of Theorem \ref{th:inv} and applying the projection construction, we obtain our main result in this section. \begin{theorem}\label{q-inv} The Gibbs cluster measure $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$ is quasi-invariant with respect to the action of\/ ${\mathrm{Diff}}_{0}(X)$ on $\varGamma_{X}$. The corresponding Radon--Nikodym density is given by $R_{\g_{\mbox{$\;\!\!$}\mathrm{cl}}}^{\varphi}=\mathcal{I}_{\mathfrak{q}}^* R_{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{\varphi}}$. \end{theorem}
\proof Note that, due to (\ref{eq:gcl*}) and (\ref{comm1}), $$ \g_{\mbox{$\;\!\!$}\mathrm{cl}}\circ \varphi^{-1}=\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}\circ \mathfrak{q}^{-1}\circ\varphi^{-1}=\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}\circ\hat{\varphi}^{-1}\circ \mathfrak{q}^{-1}. $$ That is, $\varphi^*\g_{\mbox{$\;\!\!$}\mathrm{cl}}=\g_{\mbox{$\;\!\!$}\mathrm{cl}}\circ\varphi^{-1}$ is a push-forward of the measure $\hat{\varphi}^{\ast}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}=\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}\circ\hat{\varphi}^{-1}$ under the map $\mathfrak{q}$, that is, $\varphi^*\g_{\mbox{$\;\!\!$}\mathrm{cl}}=\mathfrak{q}^*\hat{\varphi}^{\ast}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$. In particular, if $\hat{\varphi}^{\ast}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$ is absolutely continuous with respect to $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$ then so is $\varphi^{\ast}\g_{\mbox{$\;\!\!$}\mathrm{cl}}$ with respect to $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$. Moreover, by the change of measure (\ref{eq:gcl*}) and by Theorem \ref{th:inv}, for any
$F\in L^\infty(\varGamma_X, \g_{\mbox{$\;\!\!$}\mathrm{cl}})$ we have \begin{align}
\int_{\varGamma_{X}} \mbox{$\:\!\!$} F(\gamma)\,\varphi^*\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}(\mathrm{d}\gamma) &=\int_{\varGamma_{\mathcal{Z}}}{\mathcal{I}_{\mathfrak{q}}} F({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\, \hat{\varphi}^{\ast}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})
\label{eq:L1} =\int_{\varGamma_{\mathcal{Z}}}\mbox{$\:\!\!$} {\mathcal{I}_{\mathfrak{q}}}F({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\mbox{$\:\!$} R_{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{\varphi}} ({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\,\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}). \end{align} By Lemma \ref{lm:L1}, the operator $\mathcal{I}_{\mathfrak{q}}^*$ acts from $L^1(\varGamma_{\mathcal{Z}},\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{})$ to $L^1(\varGamma_X,\g_{\mbox{$\;\!\!$}\mathrm{cl}})$. Therefore, by the change of measure (\ref{eq:ghat}) the right-hand side of (\ref{eq:L1}) can be rewritten as $$ \int_{\varGamma_{X}} \mbox{$\:\!\!$} F(\gamma)\mbox{$\;\!$} (\mathcal{I}_{\mathfrak{q}}^*R_{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{\varphi}})(\gamma )\,\g_{\mbox{$\;\!\!$}\mathrm{cl}}(\mathrm{d}\gamma ), $$ which completes the proof. \endproof
\begin{remark} The Gibbs cluster measure $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$ on the configuration space $\varGamma_{X}$ can be used to construct a unitary representation $U$ of the diffeomorphism group ${\mathrm{Diff}}_{0}(X)$ by operators in $L^{2}(\varGamma_{X},\g_{\mbox{$\;\!\!$}\mathrm{cl}})$, given by the formula \begin{equation}\label{eq:U} U_{\varphi}F(\gamma )=\sqrt{R_{\g_{\mbox{$\;\!\!$}\mathrm{cl}}}^{\varphi}(\gamma)}\,F(\varphi ^{-1}(\gamma )),\qquad F\in L^{2}(\varGamma_{X},\g_{\mbox{$\;\!\!$}\mathrm{cl}}). \end{equation} Such representations, which can be defined for arbitrary quasi-invariant measures on $\varGamma_{X}$, play a significant role in the representation theory of the group ${\mathrm{Diff}}_{0}(X)$ \cite{I,VGG} and quantum field theory \cite{GGPS,Goldin}. An important question is whether the representation (\ref{eq:U}) is irreducible. According to \cite{VGG}, this is equivalent to the ${\mathrm{Diff}}_{0}(X)$-ergodicity of the measure $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$, which in our case is equivalent to the ergodicity of the measure $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$ with respect to the group of transformations $\hat{\varphi}$ \,($\varphi\in\mathrm{Diff}_{0}(X)$). Adapting the technique developed in \cite{KS}, it can be shown that the aforementioned ergodicity of $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$ is valid if and only if $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}\in\mathop{\mathrm{ext}}\nolimits\mathscr{G}(\sigma,\hat{{\varPhi}}\mbox{$\:\!$})$. In turn, the latter is equivalent to $\g\in\mathop{\mathrm{ext}}\nolimits\mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$, provided that $\g\in\mathscr{G}_{\mathrm{R}}(\theta,{\varPhi}\mbox{$\:\!$})$ (see Corollary~\ref{cor:2.9}). \end{remark}
\subsection{Integration-by-parts formula}\label{sec:3.3}
Let us first prove simple sufficient conditions for our measures on configuration spaces to belong to the corresponding moment classes $\mathcal{M}^n$ (see the Appendix, formula (\ref{eq:Mn})).
\begin{lemma}\label{lm:M^n} \textup{(a)} \,Let\/ $\g\in \mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$, and suppose that the correlation functions\/ $\kappa_{\mbox{$\;\!\!$}\g}^{m}$ are
bounded for all\/ $m=1,\dots,n$. Then $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}\in \mathcal{M}^{n}(\varGamma _{\mathcal{Z}})$, that is, \begin{equation}\label{4.4}
\int_{\varGamma_{\mathcal{Z}}} |\langle f,{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\rangle|^{n}\,\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})<\infty,\qquad f\in C_0(\mathcal{Z}). \end{equation} Moreover, the bound \textup{(\ref{4.4})} is valid for any function $f\in \bigcap_{\mbox{$\:\!$} m=1}^{\mbox{$\:\!$} n} \mbox{$\;\!\!$} L^{m}(\mathcal{Z},\sigma)$.
\textup{(b)} \,If, in addition, the total number of components of a random vector\/ $\bar{y}\in\mathfrak{X}$ has a finite\/ $n$-th moment\,\footnote{Cf.\ our standard assumption (\ref{sigma-cond*}), where $n=1$.} with respect to the measure\/ $\eta$, \begin{equation}\label{eq:N<} \int_{\mathfrak{X}} N_{X}(\bar{y})^{n}\,\eta(\mathrm{d}\bar{y})<\infty, \end{equation} then\/ $\g_{\mbox{$\;\!\!$}\mathrm{cl}}\in \mathcal{M}^{n}(\varGamma _{X})$. \end{lemma}
\proof (a) \,Using the multinomial expansion, for any $f\in C_0(\mathcal{Z})$ we have \begin{align}\notag
\int_{\varGamma_{\mathcal{Z}}} |\langle f,{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\rangle|^{n}\,\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}) &\le\int_{\varGamma_{\mathcal{Z}}}
\left(\sum\nolimits_{z\in{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}} |f(z)|\right)^{n} \,\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}) \\ \label{eq:multi} &=\sum_{m=1}^{n}\int_{\varGamma_{\mathcal{Z}}} \sum_{\{z_1\mbox{$\;\!\!$},\dots,\mbox{$\:\!$} z_m\} \subset{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}} \phi_n(z_1,\dots,z_m)\,\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}), \end{align} where $\phi_n(z_1,\dots,z_m)$ is a symmetric function given by \begin{equation}\label{eq:varPsi} \phi_n(z_1,\dots,z_m):=\sum_{\substack{ i_{1}\mbox{$\;\!\!$},\dots,\mbox{$\:\!$} i_{m}\ge1 \\[.1pc]
i_{1}+\dots+i_{m}=n}} \frac{n!}{i_1!\cdots i_m!}\,
|f(z_{1})|^{i_{1}}\mbox{$\:\!\!$}\cdots |f(z_{m})|^{i_{m}}. \end{equation}
By the definition (\ref{corr-funct}),
the integral on the right-hand side of (\ref{eq:multi}) is reduced to \begin{equation}\label{eq:k=} \frac{1}{m!}\int_{\mathcal{Z}^{m}} \phi_n(z_{1},\dots,z_{m})\,\kappa_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{m}(z_{1},\dots,z_{m}) \,\sigma (\mathrm{d}{z}_{1})\cdots\sigma (\mathrm{d}{z}_{m}). \end{equation} By Theorem \ref{th:g-hat}\mbox{$\:\!$}(b), the hypotheses of the lemma imply
that $0\le \kappa_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{m}\le a_{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}$ ($m=1,\dots,n$) with some constant $a_{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}<\infty$. Hence, substituting (\ref{eq:varPsi}) we obtain that the integral in (\ref{eq:k=}) is bounded by \begin{equation} \label{4.3}
a_{\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}\sum_{\substack{i_{1},\dots,\mbox{$\:\!$} i_{m}\ge1 \\[.1pc]
i_{1}+\dots+\mbox{$\:\!$} i_{m}=\mbox{$\:\!$} n}} \frac{n!}{i_1!\cdots i_m!}\prod_{j=1}^m \int_{\mathcal{Z}}|f(z_{j})|^{i_{j}} \,\sigma(\mathrm{d}{z}_{j})<\infty, \end{equation} since each integral in (\ref{4.3}) is finite owing to the assumption $f\in C_0(\mathcal{Z})$. Moreover, the bound (\ref{4.3}) is valid for any function $f\in \bigcap_{\mbox{$\:\!$} m=1}^{\mbox{$\:\!$} n} L^{m}(\mathcal{Z},\sigma )$. Returning to (\ref{eq:multi}), this yields (\ref{4.4}).
(b) \,Using the change of measure (\ref{eq:gcl*}), for any $\phi \in C_{0}(X)$ we obtain \begin{align}
\int_{\varGamma_{X}} |\langle\phi,\gamma
\rangle|^{n}\,\g_{\mbox{$\;\!\!$}\mathrm{cl}}(\mathrm{d}\gamma) =\int_{\varGamma_{\mathcal{Z}}}
|\langle\phi,\mathfrak{q}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\rangle|^{n}\,\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})
\label{eq:ff} =\int_{\varGamma_{\mathcal{Z}}}|\langle
\mathfrak{q}^*\phi,{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\rangle|^{n}\,\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}), \end{align} where \begin{equation}\label{eq:q*} \mathfrak{q}^*\phi(x,\bar{y}):=\sum_{y_i\in\bar{y}}\phi (y_{i}+x),\qquad (x,\bar{y})\in\mathcal{Z}. \end{equation}
Due to part (a) of the lemma, it suffices to show that $\mathfrak{q}^{\ast }\phi \in L^{m}(\mathcal{Z},\sigma)$ for any $m=1,\dots,n$. By the elementary inequality $(a_1+\cdots+a_k)^m\le k^{m-1}(a_1^m+\cdots+a_k^m)$, from (\ref{eq:q*}) we have \begin{equation}\label{eq:q*1}
\int_{\mathcal{Z}}|\mathfrak{q}^{\ast}\phi (z)|^{m}\,\sigma (\mathrm{d}{z}) \leq \int_{\mathcal{Z}} N_X(\bar{y})^{m-1}\sum_{y_{i}\in \bar{y}}
|\phi (y_{i}+x)|^{m}\,\sigma(\mathrm{d}{x}\times\mathrm{d}\bar{y}). \end{equation}
Recalling that $\sigma=\theta\otimes\eta$ and denoting
$b_\phi:=\sup_{x\in X}|\phi(x)|<\infty$ and $K_\phi:=\mathop{\mathrm{supp}}\nolimits\phi\subset X$, the right-hand side of (\ref{eq:q*1}) is dominated by \begin{align*} \int_{\mathfrak{X}} N_X(\bar{y})^{m-1}&\left(\mbox{$\;\!\!$} (b_\phi)^m\sum_{y_{i}\in \bar{y}} \int_X \mathbf{1}_{K_\phi-y_i}(x)\,\theta(\mathrm{d}{x})\right)\eta (\mathrm{d}\bar{y})\\ &=(b_\phi)^m \int_{\mathfrak{X}} N_X(\bar{y})^{m-1}\sum_{y_{i}\in \bar{y}} \theta(K_\phi-y_i)\;\eta (\mathrm{d}\bar{y})\\ &\le (b_\phi)^m\,\sup_{y\in X}\theta (K_\phi-y) \int_{\mathfrak{X}} N_X(\bar{y})^{m}\,\eta (\mathrm{d}\bar{y})<\infty, \end{align*} according to the assumptions (\ref{sigma-cond}) and (\ref{eq:N<}). \endproof
In the rest of this section, we shall assume that the conditions of Lemma \ref{lm:M^n} are satisfied with $n=1$. Thus, the measures $\g$, $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$ and $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$ belong to the corresponding $\mathcal{M}^{1}$-classes.
Let $v\in \mathrm{Vect}_{0}(X)$ (:= the space of compactly supported smooth vector fields on $X$), and define a vector field $\hat{v}_{x}$ on ${\mathfrak{X}}$ by the formula \begin{equation}\label{eq:v-hat} \hat{v}_{x}(\bar{y}):=\left( v(y_{1}+x),\dots ,v(y_{n}+x)\right) ,\qquad \bar{y}=(y_{1},\dots ,y_{n})\in {\mathfrak{X}}. \end{equation} Observe that the measure $\eta $ satisfies the following integration-by-parts formula, \begin{equation}\label{eq:etaIBP} \int_{{\mathfrak{X}}}\nabla^{\hat{v}_{x}} \mbox{$\:\!\!$} f(\bar{y})\,\eta(\mathrm{d}\bar{y})=-\int_{\mathfrak{X}} f(\bar{y})\,\beta^{\hat{v}}_{\eta}(x,\bar{y})\,\eta (\mathrm{d}\bar{y}),\qquad f\in C_{0}^{\infty}({\mathfrak{X}}), \end{equation} where $\nabla^{\hat{v}_{x}}$ is the derivative along the vector field $\hat{v}_{x}$ and \begin{equation}\label{eq:log-der} \beta^{\hat{v}}_{\eta}(x,\bar{y}):=(\beta_{\eta}(\bar{y}), \hat{v}_{x}(\bar{y}))_{T_{\bar{y}}{\mathfrak{X}}}+\mathop{\mathrm{div}}\nolimits\hat{v}_{x}(\bar{y}) \end{equation} is the logarithmic derivative of $\eta(\mathrm{d}\bar{y})=h(\bar{y})\,\mathrm{d}\bar{y}$ along $\hat{v}_{x}$, expressed in terms of the vector logarithmic derivative \begin{equation}\label{eq:nabla-h} \beta_{\eta}(\bar{y}):=\frac{\nabla h(\bar{y})}{h(\bar{y})},\qquad \bar{y} \in {\mathfrak{X}}. \end{equation}
Let us define the space $H^{1\mbox{$\;\!\!$},\mbox{$\:\!$} n}(\mathfrak{X)}$
($n\ge 1$) as the set of functions $f\in L^{n}(\mathfrak{X},\mathrm{d}\bar{y})$ satisfying the condition
\begin{equation}\label{eq:Sobolev} \int_{\mathfrak{X}}\left(\mbox{$\;\!\!$}\sum_{y_i\in\bar{y}}
\,\bigl|\nabla_{\mbox{$\;\!\!$} y_i}f(\bar{y})\bigr|\right)^{\!n}\mathrm{d}\bar{y}<\infty. \end{equation}
Note that $H^{1\mbox{$\;\!\!$},\mbox{$\:\!$} n}(\mathfrak{X)}$ is a linear space, due to the elementary inequality $(|a|+|b|)^n\le 2^{n-1}\bigl(|a|^n+|b|^n\bigr)$. \begin{lemma}\label{lm:B^n} Assume that\/ $h^{1/n}\in H^{1\mbox{$\;\!\!$},\mbox{$\:\!$} n}(\mathfrak{X)}$ for some integer\/ $n\ge1$, and let the condition\/ \textup{(\ref{eq:N<})} hold. Then\/ $\beta_{\eta}^{\hat{v}}\in L^{m}(\mathcal{Z},\sigma)$ for any\/ $m=1,\dots,n$. \end{lemma}
\proof Firth of all, observe that the condition $h^{1/n}\in H^{1\mbox{$\;\!\!$},\mbox{$\:\!$} n}(\mathfrak{X)}$ implies that $h^{1/m}\in H^{1\mbox{$\;\!\!$},\mbox{$\:\!$} m}(\mathfrak{X)}$ for any $m=1,\dots,n$. Indeed, $h^{1/m}\in L^{m}({\mathfrak{X},\mathrm{d}\bar{y}})$ if and only if $h\in L^{1}({\mathfrak{X},\mathrm{d}\bar{y}})$; furthermore, using the definition (\ref{eq:nabla-h}) of $\beta_{\eta}(\bar{y})$ we see that \begin{align} \notag \int_{\mathfrak{X}}\left(\mbox{$\;\!\!$}\sum_{y_i\in\bar{y}}\,
\bigl|\nabla_{\mbox{$\;\!\!$} y_i}
\mbox{$\;\!\!$}\bigl(h(\bar{y})^{1/m}\bigr)\bigr|\right)^{\!m}\mathrm{d}\bar{y}&=m^{-m} \int_{\mathfrak{X}}\left(\mbox{$\;\!\!$}\sum_{y_i\in\bar{y}}
\frac{|\nabla_{\mbox{$\;\!\!$} y_i}
h(\bar{y})|}{h(\bar{y})^{1-1/m}}\right)^{\!m}\mathrm{d}\bar{y}\\ \label{eq:h-eta} &= m^{-m} \int_{\mathfrak{X}}\left(\mbox{$\;\!\!$}\sum_{y_i\in\bar{y}}
|\beta_{\eta}(\bar{y})_i|\right)^{\!m}\eta(\mathrm{d}\bar{y})<\infty, \end{align} since $\eta$ is a probability measure and hence $L^{m}(\mathfrak{X},\eta)\subset L^{n}(\mathfrak{X},\eta)$ ($m=1,\dots,n$).
To show that $\beta_{\eta}^{\hat{v}}\in L^{m}(\mathcal{Z},\sigma)$, it suffices to check that each of the two terms on the right-hand side of (\ref{eq:log-der}) belongs to $L^{m}(\mathcal{Z},\sigma)$. Denote $b_v:=\sup_{x\in X}\mbox{$\;\!\!$}|v(x)|<\infty$, $K_v:=\mathop{\mathrm{supp}}\nolimits v\subset X$, and recall that $C_{K_v}:=\sup_{y\in X} \theta(K_v-y)<\infty$ by condition (\ref{sigma-cond}). Using (\ref{eq:v-hat}), we have \begin{align} \notag \int_{\mathcal{Z}}
&|(\beta_{\eta}(\bar{y}),\hat{v}_{x}(\bar{y}))|^{m} \,\sigma(\mathrm{d}{x}\times\mathrm{d}\bar{y}) \le\int_{\mathcal{Z}}\! \left(\mbox{$\;\!\!$}
\sum_{y_i\in\bar{y}}|\beta_{\eta}(\bar{y})_{i}|
\cdot|v(y_{i}+x)|\right)^{\!m}\mbox{$\;\!\!$}\theta(\mathrm{d}{x})\,\eta(\mathrm{d}\bar{y}) \\ \notag &\leq (b_v)^{m-1}\int_{\mathfrak{X}}\left(\mbox{$\;\!\!$}\sum_{y_i\in\bar{y}}
|\beta_{\eta}(\bar{y})_{i}|\right)^{\!m-1}\sum_{y_i\in\bar{y}}
|\beta_{\eta}(\bar{y})_{i}|\left(\int_{X}
|v(y_{i}+x)|\,\theta(\mathrm{d}{x})\right)\eta(\mathrm{d}\bar{y}) \\ \notag &\leq (b_v)^{m}\int_{\mathfrak{X}}\left(\mbox{$\;\!\!$}\sum_{y_i\in\bar{y}}
|\beta_{\eta}(\bar{y})_{i}|\right)^{\!m-1}\sum_{y_i\in\bar{y}}
|\beta_{\eta}(\bar{y})_{i}|\,\theta(K_v-y_i)\,\eta(\mathrm{d}\bar{y}) \\ \label{eq:beta<infty} &\leq (b_v)^{m}\,C_{K_v} \int_{\mathfrak{X}}\left(\mbox{$\;\!\!$}\sum_{y_i\in\bar{y}}
|\beta_{\eta}(\bar{y})_{i}|\right)^{\!m} \eta(\mathrm{d}\bar{y}) <\infty, \end{align}
according to (\ref{eq:h-eta}). Similarly, denoting $d_v:=\sup_{x\in X}|\mathop{\mathrm{div}}\nolimits v(x)|<\infty$ and again using (\ref{eq:v-hat}), we obtain \begin{align}
\notag \int_{\mathcal{Z}}\bigl|\mathop{\mathrm{div}}\nolimits {}&
\hat{v}_{x}(\bar{y})\bigr|^{m}\,\sigma(\mathrm{d}{x}\times\mathrm{d}\bar{y})
=\int_{\mathcal{Z}}\left(\mbox{$\;\!\!$} \sum_{y_i\in\bar{y}}\bigl|(\mathop{\mathrm{div}}\nolimits v)(y_{i}+x)
\bigr|\right)^{\!m}\theta(\mathrm{d}{x})\,\eta(\mathrm{d}\bar{y}) \\ \notag &\leq (d_v)^{m-1}\int_{\mathfrak{X}} N_X(\bar{y})^{m-1}\left(\mbox{$\;\!\!$}\sum_{y_i\in\bar{y}}
\int_{X}\bigl|(\mathop{\mathrm{div}}\nolimits v)(y_{i}+x)\bigr|\,\theta(\mathrm{d}{x})\right)\eta (\mathrm{d}\bar{y}) \\ \notag &\leq (d_v)^{m}\int_{\mathfrak{X}} N_X(\bar{y})^{m-1}\sum_{y_i\in\bar{y}} \theta(K_v-y_i)\,\eta (\mathrm{d}\bar{y}) \\ \label{eq:div<infty} &\leq (d_v)^{m}\,C_{K_v}\int_{\mathfrak{X}} N_X(\bar{y})^{m}\,\eta(\mathrm{d}\bar{y})<\infty, \end{align} according to the assumption (\ref{eq:N<}). As a result, combining the bounds (\ref{eq:beta<infty})
and (\ref{eq:div<infty}), we see that $\beta_{\eta}^{\hat{v}}\in L^{m}(\mathcal{Z},\sigma)$, as claimed. \endproof
The next two theorems are our main results in this section.
\begin{theorem}\label{IBP-} For any function $F\in {\mathcal{FC}}(\varGamma_{X})$, the Gibbs cluster measure $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$ satisfies the integration-by-parts formula \begin{equation}\label{IBP0-} \int_{\varGamma_{X}}\sum_{x\in \gamma}\nabla_{\mbox{$\;\!\!$} x}F(\gamma)\CD v(x)\,\g_{\mbox{$\;\!\!$}\mathrm{cl}}(\mathrm{d}\gamma )=-\int_{\varGamma_{X}}F(\gamma )\, B_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}^{v}(\gamma )\,\g_{\mbox{$\;\!\!$}\mathrm{cl}}(\mathrm{d}\gamma ), \end{equation} where $B_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}^{v}(\gamma ):=\mathcal{I}_{\mathfrak{q}}^{*}\langle \beta^{\hat{v}}_{\eta}, {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\rangle\in L^{1}(\varGamma_{X},\g_{\mbox{$\;\!\!$}\mathrm{cl}})$ and \strut{}$\beta^{\hat{v}}_{\eta}$ is the logarithmic derivative defined in \textup{(\ref{eq:log-der})}. \end{theorem}
\proof For any function $F\in {\mathcal{FC}}(\varGamma_{X})$ and vector field $v\in \mathrm{Vect}_{0}(X)$, let us denote for brevity \begin{equation}\label{eq:H} H(x,\gamma ):=\nabla_{\mbox{$\;\!\!$} x}F(\gamma)\CD v(x), \qquad x\in X, \ \ \gamma\in\varGamma_X. \end{equation} Furthermore, setting $\hat{F}=\mathcal{I}_{\mathfrak{q}} F:\varGamma_{\mathcal{Z}}\to\mathbb{R}$ we introduce the notation \begin{equation}\label{eq:H-hat} \hat{H}(z,{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}):= \nabla_{\mbox{$\;\!\!$}\bar{y}}\hat{F}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}) \CD \hat{v}_{x}(\bar{y}),\qquad z=(x,\bar{y})\in\mathcal{Z},\ \ {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\in\varGamma_{\mathcal{Z}}. \end{equation} From these definitions, it is clear that \begin{equation}\label{eq:QHH} {\mathcal{I}_{\mathfrak{q}}}\Biggl(\mbox{$\:\!\!$}\sum_{x\in \gamma} H(x,\gamma)\Biggr)({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})=\sum_{z\in {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}}\hat{H}(z,{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}),\qquad {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\in\varGamma_{\mathcal{Z}}. \end{equation}
Let us show that the Gibbs measure $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$ on $\varGamma_{\mathcal{Z}}$ satisfies the following integration-by-parts formula: \begin{equation}\label{IBPp} \int_{\varGamma_{\mathcal{Z}}}\sum_{z\in \hat{\gamma\mbox{$\:\!$}}}\hat{H}(z,{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\,\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}) =-\int_{\varGamma_{\mathcal{Z}}} \!\hat{F}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})B_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{v}}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\, \hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}), \end{equation} where the logarithmic derivative $B_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{v}}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}):= \langle\beta^{\hat{v}}_{\eta},{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\rangle$ belongs to $L^{1}(\varGamma_{\mathcal{Z}},\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{})$ (by Lemmas \ref{lm:M^n}\mbox{$\:\!$}(b) and \ref{lm:B^n} with $n=1$). By the change of measure (\ref{eq:ghat}) and due to relation (\ref{eq:QHH}), we have \begin{align} \notag \int_{\varGamma_{\mathcal{Z}}}\mbox{$\;\!\!$}\sum_{z\in {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\mbox{$\:\!$}}
|\hat{H}(z,{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})|\;\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}) &=\int_{\varGamma _{X}}\sum_{x\in \gamma}|H(x,\gamma)|\; \g_{\mbox{$\;\!\!$}\mathrm{cl}}(\mathrm{d}\gamma ) \\
\label{eq:|H|} &\le \sup_{(x,\mbox{$\:\!$}\gamma)}
|H(x,\gamma)|\int_{\varGamma _{X}}\sum_{x\in\gamma} \mathbf{1}_{K_{v}}(x)\:\g_{\mbox{$\;\!\!$}\mathrm{cl}}(\mathrm{d}\gamma), \end{align}
where $K_v:=\mathop{\mathrm{supp}}\nolimits v$ is a compact set in $X$. Note that the right-hand side of (\ref{eq:|H|}) is finite, since the function $H$ is bounded (see (\ref{eq:H})) and, by Lemma \ref{lm:M^n}\mbox{$\:\!$}(b), $\g_{\mbox{$\;\!\!$}\mathrm{cl}}\in\mathcal{M}^{1}(\varGamma_{X})$. Therefore, by Remark \ref{rm:NZ} we can apply Nguyen--Zessin's equation (\ref{eq:NZ}) with the function $\hat{H}$ to obtain \begin{equation}\label{eq:HH} \int_{\varGamma_{\mathcal{Z}}}\mbox{$\;\!\!$}\sum_{z\in \hat{\gamma\mbox{$\:\!$}}} \hat{H}(z,{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\,\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}) =\int_{\varGamma_{\mathcal{Z}}}\!\left(\int_{\mathcal{Z}} \hat{H}(z,{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\cup\{z\})\,\mathrm{e\mbox{$\:\!$}}^{-\hat{E}(\{z\},\mbox{$\:\!$}\hat{\gamma\mbox{$\:\!$}})}\, \sigma (\mathrm{d}{z})\right)\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}). \end{equation} Inserting the definition (\ref{eq:H}), using Lemma \ref{lm:shift-free} and recalling that $\sigma=\theta\otimes\eta$ (see (\ref{eq:sigma})), let us apply the integration-by-parts formula (\ref{eq:etaIBP}) for the measure $\eta$ to rewrite the internal integral in (\ref{eq:HH}) as \begin{align*} &\int_{X}{\mathrm{e\mbox{$\:\!$}}}^{-E(\{x\},\mbox{$\;\!$} p_{X}(\hat{\gamma\mbox{$\:\!$}}))}\left( \int_{{\mathfrak{X}}}\nabla _{\bar{y}} \hat{F}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\cup\{(x,\bar{y})\})\CD \hat{v}_{x}(\bar{y})\,\eta (\mathrm{d}\bar{y})\right)\theta(\mathrm{d} x)\\ &\qquad=-\int_{X} \mathrm{e\mbox{$\:\!$}}^{-E(\{x\},\mbox{$\:\!$} p_{X}(\hat{\gamma\mbox{$\:\!$}}))} \left(\int_{{\mathfrak{X}}}\hat{F}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\cup\{(x,\bar{y})\})\mbox{$\;\!$} \beta^{\hat{v}}_{\eta}(x,\bar{y})\, \eta (\mathrm{d}\bar{y})\right) \theta (\mathrm{d} x)\\ &\qquad\qquad=-\int_{\mathcal{Z}} \mathrm{e\mbox{$\:\!$}}^{-E(\{p_{X}(z)\},\mbox{$\;\!$} p_{X}(\hat{\gamma\mbox{$\:\!$}}))} \hat{F}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\cup\{z)\})\mbox{$\;\!$} \beta^{\hat{v}}_{\eta}(z)\, \sigma(\mathrm{d}{z}). \end{align*} Returning to (\ref{eq:HH}) and again using Nguyen--Zessin's equation (\ref{eq:NZ}), we see that the right-hand side of (\ref{eq:HH}) is reduced to \begin{align*} -\int_{\varGamma_{\mathcal{Z}}} \mbox{$\:\!\!$}\sum_{z\in \hat{\gamma\mbox{$\:\!$}}}\hat{F}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\mbox{$\;\!$}\beta^{\hat{v}}_{\eta}(z)\, \hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})&=-\int_{\varGamma_{\mathcal{Z}}} \mbox{$\:\!\!$} \hat{F}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\mbox{$\;\!$} B_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{v}}\,{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\rangle\, \hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}), \end{align*} which proves formula (\ref{IBPp}).
Now, using equality (\ref{eq:QHH}), we obtain \begin{align*} \int_{\varGamma_{X}}\sum_{x\in \gamma}H(x,\gamma)\,\g_{\mbox{$\;\!\!$}\mathrm{cl}} (\mathrm{d}\gamma )
&=\int_{\varGamma_{\mathcal{Z}}}\!\Biggl(\mbox{$\:\!\!$}\sum_{(x,\mbox{$\:\!$}\bar{y})\in \hat {\gamma}}\!\nabla_{\mbox{$\;\!\!$}\bar {y}}{\mathcal{I}_{\mathfrak{q}}}F({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\CD \hat{v}_{x}(\bar {y})\mbox{$\:\!\!$}\Biggr)\, \hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\\ &=-\int_{\varGamma_{\mathcal{Z}}}{\mathcal{I}_{\mathfrak{q}}}F({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\mbox{$\;\!$} B_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{v}} ({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\;\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}) \\ &=-\int_{\varGamma_{X}}F(\gamma)\, \mathcal{I}_{\mathfrak{q}}^*B_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{v}}(\gamma)\,\g_{\mbox{$\;\!\!$}\mathrm{cl}}(\mathrm{d}\gamma), \end{align*} where $\mathcal{I}_{\mathfrak{q}}^*B_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{v}}\in L^1(\varGamma_X,\g_{\mbox{$\;\!\!$}\mathrm{cl}})$ by Lemma~\ref{lm:L1}. Thus, formula (\ref{IBP0-}) is proved. \endproof
\begin{remark} Observe that the logarithmic derivative $B_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{\hat{v}}=\langle\beta^{\hat{v}}_{\eta},{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\rangle$ (see (\ref{IBPp})) does not depend on the interaction potential \mbox{$\;\!$}${\varPhi}$, and in particular coincides with that in the case ${\varPhi}\equiv0$, where the Gibbs measure $\g$ is reduced to the Poisson measure $\pi_\theta$. Nevertheless, the logarithmic derivative $B_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}^v$ does depend on ${\varPhi}$ via the map $\mathcal{I}_{\mathfrak{q}}^*$. \end{remark}
\begin{remark} Note that in Theorem \ref{IBP-} the reference measure $\theta$
does not have to be differentiable with respect to $v$. \end{remark}
According to Theorem \ref{IBP-}, $B_{\g_{\mbox{$\;\!\!$}\mathrm{cl}}}^{v}\in L^1(\varGamma_{\mathcal{Z}},\g_{\mbox{$\;\!\!$}\mathrm{cl}})$. However, under the conditions of Lemma \ref{lm:B^n} with $n\ge2$, this statement can be enhanced.
\begin{lemma}\label{lm:B^n*} Assume that\/ $h^{1/n}\in H^{1\mbox{$\;\!\!$},\mbox{$\:\!$} n}(\mathfrak{X)}$ for some integer\/ $n\ge2$, and let the condition\/ \textup{(\ref{eq:N<})} hold. Then\/ $B_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}^{v}\mbox{$\:\!\!$}\in L^{n}(\varGamma_{\mathcal{Z}},\g_{\mbox{$\;\!\!$}\mathrm{cl}})$. \end{lemma}
\proof By Lemmas \ref{lm:M^n}\mbox{$\:\!$}(a) and \ref{lm:B^n}, it follows that $\langle \beta^{\hat{v}}_{\eta}, {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\rangle\in L^{n}(\varGamma_{\mathcal{Z}},\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{})$. Let $s:=n/(n-1)$, so that $n^{-1}+s^{-1}=1$. Note that $\mathcal{I}_{\mathfrak{q}}$ can be treated as a bounded operator acting from $L^{s}(\varGamma_{X},\g_{\mbox{$\;\!\!$}\mathrm{cl}})$ to $L^{s}(\varGamma_{\mathcal{Z}},\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{})$. Hence, $\mathcal{I}_{\mathfrak{q}}^{\ast}$ is a bounded operator from $L^{s}(\varGamma_{\mathcal{Z}},\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{})^{\prime}=L^{n}(\varGamma_{\mathcal{Z}},\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{})$ to $L^{s}(\varGamma_{X},\g_{\mbox{$\;\!\!$}\mathrm{cl}})^{\prime}=L^{n}(\varGamma _{X},\g_{\mbox{$\;\!\!$}\mathrm{cl}})$, which implies that $B_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}^{v}=\mathcal{I}_{\mathfrak{q}}^*\langle \beta^{\hat{v}}_{\eta}, {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}\rangle\in L^{n}(\varGamma_{\mathcal{Z}},\g_{\mbox{$\;\!\!$}\mathrm{cl}})$. \endproof
Formula (\ref{IBP0-}) can be extended to more general vector fields on $ \varGamma_{X}$. Let ${\mathcal{FV}}(\varGamma_{X})$ be the class of vector fields $V$ of the form $V(\gamma )=(V(\gamma )_{x})_{x\in \gamma}$, \begin{equation*} V(\gamma )_{x}=\sum_{j=1}^{N}G_{j}(\gamma)\,v_{j}(x)\in T_{x}X, \end{equation*} where $G_{j}\in {\mathcal{FC}}(\varGamma_{X})$ and $v_{j}\in \mathrm{Vect}_{0}(X)$, \,$j=1,\dots ,N$. For any such $V$ we set \begin{equation*} B_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}^{V}(\gamma):=(\mathcal{I}_{\mathfrak{q}}^{\ast}B_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{{\mathcal{I}_{\mathfrak{q}}}V})(\gamma), \end{equation*} where $B_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}^{{\mathcal{I}_{\mathfrak{q}}}V}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})$ is the logarithmic derivative of $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$ along ${\mathcal{I}_{\mathfrak{q}}}V(\hat{ \gamma}):=V({\mathfrak{q}}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}))$ (see \cite{AKR1}). Note that ${\mathcal{I}_{\mathfrak{q}}}V$ is a vector field on $\varGamma_{\mathcal{Z}}$ owing to the obvious equality \begin{equation*} T_{\hat{\gamma\mbox{$\:\!$}}}\varGamma_{\mathcal{Z}} =T_{{\mathfrak{q}}(\hat{\gamma\mbox{$\:\!$}})}\varGamma_{X}. \end{equation*} Clearly, \begin{equation*} B_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}^{V}(\gamma )=\sum_{j=1}^{N}\biggl( G_{j}(\gamma)B_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}^{v_{j}}(\gamma )+\sum_{x\in \gamma}\nabla_{\mbox{$\;\!\!$} x} G_{j}(\gamma )\CD v_{j}(x)\biggr) . \end{equation*}
\begin{theorem}\label{IBP1} For any $F_{1},F_{2}\in {\mathcal{FC}}(\varGamma_{X})$ and\/ $V\in {\mathcal{FV}}(\varGamma_{X})$, we have \begin{equation*}
\begin{aligned} \int_{\varGamma_{X}}&\sum_{x\in \gamma}\nabla_{\mbox{$\;\!\!$} x}F_{1}(\gamma)\CD V(\gamma)_{x}\,F_{2}(\gamma )\;\g_{\mbox{$\;\!\!$}\mathrm{cl}}(\mathrm{d}\gamma) \\ & =-\int_{\varGamma_{X}}F_{1}(\gamma)\,\sum_{x\in \gamma}\nabla_{x}F_{2}(\gamma )\CD V(\gamma)_{x}\ \g_{\mbox{$\;\!\!$}\mathrm{cl}} (\mathrm{d}\gamma)\\ &\qquad -\int_{\varGamma_{X}}F_{1}(\gamma) F_{2}(\gamma)B_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}^{V}(\gamma )\,\g_{\mbox{$\;\!\!$}\mathrm{cl}}(\mathrm{d}\gamma ). \end{aligned} \end{equation*} \end{theorem}
\proof The proof can be obtained by a straightforward generalization of the arguments used in the proof of Theorem \ref{IBP-}. \endproof
We define the \emph{vector logarithmic derivative} of $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$ as a linear operator \begin{equation*} B_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}\!:\,{\mathcal{FV}}(\varGamma_{X})\rightarrow L^{1}(\varGamma _{X},\g_{\mbox{$\;\!\!$}\mathrm{cl}}) \end{equation*} via the formula \begin{equation*} B_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}\mbox{$\;\!\!$} V(\gamma):=B_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}^{V}\mbox{$\;\!\!$}(\gamma). \end{equation*} This notation will be used in the next section.
\section{The Dirichlet form and equilibrium stochastic dynamics}\label{sec:4}
Throughout this section, we assume that the conditions of Lemma \ref{lm:M^n} are satisfied with $n=2$. Thus, the measures $\g$, $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$ and $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$ belong to the corresponding $\mathcal{M}^{2}$-classes. Our considerations will involve the $\varGamma $-gradients (see Section~\ref{sec:3.1}) on different configuration spaces, such as $\varGamma _{X}$, $\varGamma _{\mathfrak{X}}$ and $\varGamma_{\mathcal{Z}}$; to avoid confusion, we shall denote them by $\nabla_{\mbox{$\:\!\!$} X}^{\varGamma }$, $\nabla_{\mathfrak{X}}^{\varGamma}$ and $\nabla_{\mathcal{Z}}^{\varGamma}$, respectively.
\subsection{The Dirichlet form associated with $\protect\g_{\mbox{$\;\!\!$}\mathrm{cl}}$} \label{sec:4.1}
Let us introduce a pre-Dirichlet form ${\mathcal{E}}_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}$ associated with the Gibbs cluster measure $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$, defined on functions $F_1,F_2\in\mathcal{FC}(\varGamma_{X})\subset L^{2}(\varGamma_{X},\g_{\mbox{$\;\!\!$}\mathrm{cl}})$ by \begin{equation}\label{eq:E-mu} {\mathcal{E}}_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}(F_1,F_2):=\int_{\varGamma_{X}}\langle \nabla_{\!X}^{\varGamma} F_1(\gamma),\nabla_{\!X}^{\varGamma} F_2(\gamma)\rangle_{\gamma} \,\g_{\mbox{$\;\!\!$}\mathrm{cl}}(\mathrm{d}\gamma).
\end{equation}
Let us also consider the operator $H_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}$ defined
by \begin{equation}\label{eq:Hcl} H_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}F:=-\Delta ^{\varGamma}F+B_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}\!\nabla_{\!X}^{\varGamma} F,\qquad F\in {\mathcal{FC}}(\varGamma_{X}), \end{equation} where $\Delta ^{\varGamma}F(\gamma ):=\sum_{x\in \gamma}\Delta_{x}F(\gamma)$.
The next theorem readily follows from the general theory of (pre-)Dirichlet forms associated with measures from the class $\mathcal{M}^{2}(\varGamma_{X})$ (see \cite{AKR2,MR}).
\begin{theorem} \textup{(a)} \,The pre-Dirichlet form \textup{(\ref{eq:E-mu})} is well defined, i.e., ${\mathcal{E}}_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}(F_{1},F_{2})<\infty $ for all $F_{1},F_{2}\in {\mathcal{FC}}(\varGamma_{X})$\textup{;}
\textup{(b)} \,The expression \textup{(\ref{eq:Hcl})} defines a symmetric operator $H_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}\!$ in $L^{2}(\varGamma_{X},\g_{\mbox{$\;\!\!$}\mathrm{cl}})$ whose domain includes ${\mathcal{FC}}(\varGamma_{X})$\textup{;}
\textup{(c)} \,The operator\/ $H_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}\mbox{$\:\!\!$}$ is the generator of the pre-Dirichlet form\/ ${\mathcal{E}}_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}$, i.e., \begin{equation}\label{generator} {\mathcal{E}}_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}(F_{1},F_{2})=\int_{\varGamma_{X}} F_{1}(\gamma)\,H_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}\mbox{$\;\!\!$} F_{2}(\gamma)\,\g_{\mbox{$\;\!\!$}\mathrm{cl}}(\mathrm{d}\gamma),\ \quad F_{1},F_{2}\in {\mathcal{FC}}(\varGamma_{X}). \end{equation}
\end{theorem}
Formula (\ref{generator}) implies that the form ${\mathcal{E}}_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}$ is closable. It follows from the properties of the \textit{carr\'e du champ} \,$\sum_{x\in \gamma} \!\nabla_{\mbox{$\;\!\!$}{}x} F_{1}(\gamma)\CD \nabla_{\mbox{$\;\!\!$} x}F_{2}(\gamma )$ that the closure of ${\mathcal{E}}_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}$ (for which we shall keep the same notation) is a quasi-regular local Dirichlet form on a bigger state space $\overset{\,..}{\varGamma}_{X}$ consisting of all integer-valued Radon measures on $X$ (see \cite{MR}). By the general theory of Dirichlet forms (see \cite{MR0}), this implies the following result (cf.\ \cite{AKR1,AKR2,BD3}).
\begin{theorem}\label{th:7.2} There exists a conservative diffusion process $\mathbf{X}=(\mathbf{X}_t,\,t\ge0)$ on $\overset{\,\mbox{$\:\!$}..}{\varGamma}_{X}$, properly associated with the Dirichlet form $\mathcal{E}_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}$, that is, for any function $F\in L^{2}(\overset{\,\mbox{$\:\!$}..}{\varGamma}_{ X},\g_{\mbox{$\;\!\!$}\mathrm{cl}})$ and all\/ $t\ge0$, the map \begin{equation*} \overset{\,\mbox{$\:\!$}..}{\varGamma}_{X}\ni \gamma \mapsto p_{t}F(\gamma) :=\int_{\varOmega} F(\mathbf{X}_{t})\,\mathrm{d} P_{\gamma} \end{equation*} is an\/ $\mathcal{E}_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}$-quasi-continuous version of\/ $\exp(-tH_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}) F$. Here $\varOmega$ is the canonical sample space \textup{(}of $\overset{\,\mbox{$\:\!$}..}{\varGamma}_X$-valued continuous functions on $\mathbb{R}_+$\textup{)} and $(P_\gamma,\,\gamma\in\overset{\,\mbox{$\:\!$}..}{\varGamma}_X)$ is the family of probability distributions of the process $\mathbf{X}$ conditioned on the initial value $\gamma=\mathbf{X}_0$. The process $\mathbf{X}$ is unique up to $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$-equivalence. In particular, $\mathbf{X}$ is $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$-symmetric \textup{(}i.e., $\int F_1\mbox{$\;\!$} p_{t}F_2\,\mathrm{d}\g_{\mbox{$\;\!\!$}\mathrm{cl}} = \int F_2\, p_{t} F_1\,\mathrm{d}\g_{\mbox{$\;\!\!$}\mathrm{cl}}$ for all measurable functions $F_1,F_2:\overset{\,\mbox{$\:\!$}..}{\varGamma}_{ X}\to\mathbb{R}_{+}$\textup{)} and $\g_{\mbox{$\;\!\!$}\mathrm{cl}}$ is its invariant measure. \end{theorem}
\subsection{Irreducibility of the Dirichlet form}
Similarly to (\ref{eq:E-mu}), let ${\mathcal{E}}_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}$ be the pre-Dirichlet form associated with the Gibbs measure $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$, defined on functions $F_1,F_2\in\mathcal{FC}(\varGamma_{\mathcal{Z}})\subset L^{2}( \varGamma_{\mathcal{Z}},\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{})$ by \begin{equation}\label{generator1} {\mathcal{E}}_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}(F_1,F_2):=\int_{\varGamma_{\mathcal{Z}}}\langle \nabla_{\!\mathcal{Z}}^{\varGamma}\mbox{$\;\!\!$} F_1({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}),\nabla_{\!\mathcal{Z}}^{\varGamma}\mbox{$\;\!\!$} F_2({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\rangle_{\hat{\gamma\mbox{$\:\!$}}} \,\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}). \end{equation} The integral on the right-hand side of (\ref{generator1}) is well defined because $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}\in\mathcal{M}^{2}\subset \mathcal{M}^{1}$. The latter fact also implies that the gradient operator $\nabla_{\mbox{$\;\!\!$}\mathcal{Z}}^{\varGamma}$ can be considered as an (unbounded) operator $L^{2}(\varGamma_{\mathcal{Z}},\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{})\to L^{2}V(\varGamma_{\mathcal{Z}},\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{})$ with domain $\mathcal{FC}(\varGamma_{\mathcal{Z}})$, where $L^{2}V(\varGamma_{\mathcal{Z}},\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{})$ is the space of square-integrable vector fields on $\varGamma_{\mathcal{Z}}$. Since the form ${\mathcal{E}}_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}$ belongs to the class $\mathcal{M}^2$, it is closable \cite{AKR2} (we keep the same notation for the closure and denote by $\mathcal{D}(\mathcal{E}_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}})$ its domain).
Our aim is to study a relationship between the forms $\mathcal{E}_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}$ and $\mathcal{E}_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}$ and to characterize in this way the kernel of $\mathcal{E}_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}$. We need some preparations. Let us recall that the projection map $\mathfrak{q}:\mathcal{Z}\to\varGamma^\sharp_X$ was defined in (\ref{proj1}) as $\mathfrak{q}:=\mathfrak{p}\circ s$, where \begin{equation*} s:\mathcal{Z}\ni (x,\bar{y})\mapsto \bar{y}+x\in \mathfrak{X}. \end{equation*} As usual, we preserve the same notations for the induced maps of the corresponding configuration spaces. It follows directly from the definition (\ref{eq:pr}) of the map $\mathfrak{p}$ that \begin{equation}\label{commut0} (\nabla_{\mbox{$\:\!\!$} X}^{\varGamma }F) \circ \mathfrak{p}=\nabla _{\mathfrak{X}}^{\varGamma}(F\circ \mathfrak{p}),\qquad F\in\mathcal{FC}(\varGamma_{X}), \end{equation} where we use the identification of the tangent spaces \begin{equation}\label{tangent1} T_{\bar{\gamma}}\varGamma_{\mathfrak{X}} ={\textstyle\bigoplus\limits_{\bar{y}\in\mathfrak{\bar{\gamma}}}} \,T_{\bar{y}}\mathfrak{X}={\textstyle{\bigoplus\limits_{\bar{y}\in \mathfrak{\bar{\gamma}}}}}\,{\textstyle\bigoplus\limits_{y_i\in\bar{y}}}\,T_{y_i}X ={\textstyle\bigoplus\limits_{y_i\in\mathfrak{p}(\bar{\gamma})}} T_{y_i} X=T_{\mathfrak{p}(\bar{\gamma})} X. \end{equation}
\begin{theorem}\label{th:4.3} For the Dirichlet forms\/ $\mathcal{E}_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}$ and\/ $\mathcal{E}_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}$ defined in \textup{(\ref{generator})} and \textup{(\ref{generator1})}, respectively, their domains satisfy the relation\/ $\mbox{$\:\!$}\mathcal{I}_{\mathfrak{q}}(\mathcal{D}(\mathcal{E}_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}))\subset \mathcal{D}(\mathcal{E}_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}})$. Furthermore, $F\in \mathop{\mathrm{Ker}}\nolimits {\mathcal{{\mathcal{E}}}}_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}$ if and only if\/ $\mathcal{I}_{\mathfrak{q}} F\in \mathop{\mathrm{Ker}}\nolimits \mathcal{E}_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}$\mbox{$\:\!$}. \end{theorem}
\proof Let us introduce a map $\mathrm{d}{s}^{\ast}\mbox{$\:\!\!$}:\,\mathfrak{X}\rightarrow \mathcal{Z}$ by the formula \begin{equation*}
\mathrm{d}{s}^{\ast}(\bar{y}):=\left(\textstyle{\sum_{y_{i}\in \bar{y}} y_{i}},\,\bar{y}\right),\qquad \bar{y}\in\mathfrak{X}. \end{equation*} As suggested by the notation, this map coincides with the adjoint of the derivative \begin{equation*} \mathrm{d}{s}(z)\mbox{$\:\!\!$}:\,T_{z}\mathcal{Z}\rightarrow T_{s(z)}\mathfrak{X} \end{equation*} under the identification $T_{\bar{y}}\mathfrak{X}=\mathfrak{X}$ and $T_{z}\mathcal{Z}=\mathcal{Z}$. A direct calculation shows that for any differentiable function $f$ on $\mathfrak{X}$ the following commutation relation holds: \begin{equation}\label{commut1} (\mathrm{d}{s}^{\ast }\nabla f)\circ s=\nabla (f\circ s). \end{equation} Here the symbol $\nabla $ denotes the gradient on the corresponding space (i.e., $\mathfrak{X}$ on the left and $\mathcal{Z}$ on the right).
Let \begin{equation*} \mathrm{d}{s}^{\ast}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}):\,T_{s({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})}\varGamma_{\mathfrak{X}} =\textstyle{\bigoplus\limits_{\bar{y}\in s({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})}}T_{\bar{y}}\mathfrak{X} \rightarrow \textstyle{\bigoplus\limits_{z\in {\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}}}\,T_{z}\mathcal{Z} =T_{{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}}\varGamma_{\mathcal{Z}} \end{equation*} be the natural lifting of the operator $\mathrm{d}{s}^{\ast}$. Further, using (\ref{tangent1}), it can be interpreted as the operator \begin{equation*} \mathrm{d}{s}^{\ast}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}):\,T_{\mathfrak{q}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})} \varGamma_{X}\rightarrow T_{{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}}\varGamma_{\mathcal{Z}}, \end{equation*} which induces the (bounded) operator \begin{equation}\label{oper} \mathcal{I}_{\mathfrak{q}}\,\mathrm{d}{s}^{\ast}:\,L^{2}V(\varGamma _{X},\g_{\mbox{$\;\!\!$}\mathrm{cl}})\rightarrow L^{2}V(\varGamma_{\mathcal{Z}},\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}) \end{equation} acting according to the formula \begin{equation*} (\mathcal{I}_{\mathfrak{q}}\,\mathrm{d}{s}^{\ast}V) ({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})=\mathrm{d}{s}^{\ast}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})V(\mathfrak{q}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})),\qquad V\in L^{2}V(\varGamma _{X},\g_{\mbox{$\;\!\!$}\mathrm{cl}}). \end{equation*} Formula (\ref{commut1}) together with (\ref{commut0}) implies that \begin{equation*} \left(\mathrm{d}{s}^*\nabla _{\!X}^{\varGamma}F\right) \circ \mathfrak{q}=\nabla _{\!\mathcal{Z}}^{\varGamma}(F\circ \mathfrak{q}),\qquad F\in \mathcal{FC}(\varGamma_{X}), \end{equation*} or, in terms of operators acting in the corresponding $L^{2}$-spaces, \begin{equation*}
\mathcal{I}_{\mathfrak{q}}\,\mathrm{d}{s}^*\nabla_{\!X}^{\varGamma }F=\nabla_{\!\mathcal{Z}}^{\varGamma}\mathcal{I}_{\mathfrak{q}} F,\qquad F\in \mathcal{FC}(\varGamma_{X}). \end{equation*} Therefore, for any $F\in \mathcal{FC}(\varGamma_{X})$ \begin{align} \mathcal{E}_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}(\mathcal{I}_{\mathfrak{q}} F,\mathcal{I}_{\mathfrak{q}} F) &=\int_{\varGamma_{\mathcal{Z}}}
|(\mathcal{I}_{\mathfrak{q}}\,\mathrm{d}{s}^{\ast
}\nabla_{\!X}^{\varGamma}F)({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})|_{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}^{2} \:\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}) \label{comrel1} \\
\notag &=\int_{\varGamma_{\mathcal{Z}}}|\mathrm{d}{s}^{\ast}
\nabla_{\!X}^{\varGamma}F(\mathfrak{q}({\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}))|_{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}} ^{2}\:\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}(\mathrm{d}{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}})\\
\notag &=\int_{\varGamma_{X}}|\mathrm{d}{s}^*\nabla_{\!X}^{\varGamma}
F(\gamma)|_\gamma^{2}\:\g_{\mbox{$\;\!\!$}\mathrm{cl}}(\mathrm{d}\gamma)\\
\label{sqcl} &\geq \int_{\varGamma_{X}} |\nabla_{\!X}^{\varGamma}
F(\gamma)|_\gamma^{2}\:\g_{\mbox{$\;\!\!$}\mathrm{cl}}(\mathrm{d}\gamma)= \mathcal{E}_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}(F,F), \end{align} where in (\ref{sqcl}) we used the obvious inequality
$|\mathrm{d}{s}^{\ast}(\bar{y})| \geq |\bar{y}|$ \,($\bar{y}\in \mathfrak{X}$).
Hence, \begin{align*} \Vert F\Vert_{{\mathcal{E}}_{\g_{\mbox{$\;\!\!$}\mathrm{cl}}}}^{2}& :={\mathcal{E}}_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}(F,F)+\int_{\varGamma_{X}}F^{2}\,{\mathrm{d}\g_{\mbox{$\;\!\!$}\mathrm{cl}}} \\ & \leq {\mathcal{E}}_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}(\mathcal{I}_{\mathfrak{q}} F,\mathcal{I}_{\mathfrak{q}} F)+\int_{\varGamma_{{\mathcal{Z}}}}(\mathcal{I}_{\mathfrak{q}} F)^{2}\,\mathrm{d}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}=\Vert \mathcal{I}_{\mathfrak{q}} F\Vert_{{\mathcal{E}}_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}}^{2}, \end{align*} which implies that $\mathcal{I}_{\mathfrak{q}}(\mathcal{D}(\mathcal{E}_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}))\subset \mathcal{D}({\mathcal{E}}_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}})$, thus proving the first part of the theorem.
Further, using approximation arguments and continuity of the operator (\ref{oper}), one can show that the equality (\ref{comrel1}) extends to the domain $\mathcal{D}({\mathcal{E}}_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}})$, \begin{equation}\label{eq:QQ} \mathcal{E}_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}(\mathcal{I}_{\mathfrak{q}} F,\mathcal{I}_{\mathfrak{q}} F)= \int_{\varGamma_{\mathcal{Z}}}
|\mathcal{I}_{\mathfrak{q}}\,\mathrm{d}{s}^*
\nabla_{\!X}^{\varGamma}F|_{{\hat{\gamma\mbox{$\:\!$}}\mbox{$\;\!\!$}{}}}^{2}\:\mathrm{d}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{},\qquad F\in\mathcal{D}(\mathcal{E}_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}). \end{equation} Since $\mathop{\mathrm{Ker}}\nolimits(\mathcal{I}_{\mathfrak{q}}\,\mathrm{d}{s}^{\ast}) =\{0\}$, formula (\ref{eq:QQ}) readily implies that $\mathcal{I}_{\mathfrak{q}} F\in \mathop{\mathrm{Ker}}\nolimits \mathcal{E}_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}$ if and only if $\nabla_{\!X}^{\varGamma}F=0$. In turn, due to equality (\ref{sqcl}), the latter is equivalent to $F\in \mathop{\mathrm{Ker}}\nolimits\mathcal{E}_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}$. \endproof
Let us recall that a Dirichlet form $\mathcal{E}$ is called \textit{irreducible} if the condition $\mathcal{E}(F,F)=0$ implies that $F=\mathrm{const}$.
\begin{corollary} The Dirichlet form\/ ${\mathcal{E}}_{\mbox{$\;\!\!$}\g_{\mbox{$\;\!\!$}\mathrm{cl}}}$ is irreducible if\/ ${\mathcal{E}}_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}$ is so. \end{corollary}
\proof Follows immediately from Theorem \ref{th:4.3} and the obvious fact that if ${\mathcal{I}_{\mathfrak{q}}}F=\mathrm{const}$ ($\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}$-a.s.) then $F=\mathrm{const}$ ($\g_{\mbox{$\;\!\!$}\mathrm{cl}}$-a.s.). \endproof
\begin{remark} It follows from the general theory of Gibbs measures (see, e.g., \cite{AKR2}) that the form $\mathcal{E}_{\mbox{$\;\!\!$}\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}}$ is irreducible if and only if $\hat{\text{\slshape\textrm{g}\mbox{$\;\!$}}}\mbox{$\;\!\!$}{}\in\mathop{\mathrm{ext}}\nolimits\mathscr{G}(\sigma,\hat{{\varPhi}}\mbox{$\:\!$})$, which is in turn equivalent to $\g\in\mathop{\mathrm{ext}}\nolimits\mathscr{G}(\theta ,{\varPhi}\mbox{$\:\!$})$ (provided that $\g\in\mathscr{G}_{\mathrm{R}}(\theta,{\varPhi}\mbox{$\:\!$})$, see Corollary~\ref{cor:2.9}). \end{remark}
\section*{Acknowledgments}\label{sec:Ack}
The authors would like to thank Sergio Albeverio, Yuri Kondratiev, Eugene Lytvynov and Tobias Kuna for
helpful discussions.
\appendix
\section{Gibbs measures on configuration spaces}
\renewcommand{A}{A}
Let us briefly recall the definition and some properties of (grand canonical) Gibbs measures on the configuration space $\varGamma_{X}$. For a more systematic exposition and further details, see the classical books \cite{Ge79,Preston,Ruelle}; more recent useful references include \cite{AKR2,KunaPhD,KKS98}.
Denote by $\varGamma_{X}^{\myp0}:= \{\gamma\in\varGamma_X:\,\gamma(X)<\infty\}$
the subspace of finite configurations in $X$. Let $\varPhi: \varGamma_X^{\mbox{$\:\!$} 0}\to \mathbb{R}\cup \{+\infty \}$ be a measurable function (called the \emph{interaction potential}) such that $\varPhi(\emptyset)=0$.
A simple, most common example is that of the \textit{pair interaction potential}, i.e., such that $\varPhi(\gamma)=0$ unless configuration $\gamma$ consists of exactly two points.
\begin{definition} The \emph{energy} $E:\varGamma_{X}^{\myp0}\rightarrow \mathbb{R}\cup \{+\infty\}$ is defined by \begin{equation}\label{eq:E} E(\xi):=\sum_{\zeta\subset \xi}\varPhi(\zeta)\qquad (\xi \in \varGamma_{X}^{\myp0}),\qquad E(\emptyset ):=0. \end{equation} The \emph{interaction energy} between configurations \mbox{$\:\!$}$\xi \in \varGamma_{X}^{\myp0}$ and $\gamma \in \varGamma_{X}$ is given by \begin{equation}\label{eq:E-E} E(\xi ,\gamma ):=\left\{ \begin{array}{ll} \displaystyle\sum_{\gamma\supset\gamma'\in\varGamma_{X}^{\myp0}} \!\varPhi(\xi\cup\gamma') & \displaystyle\quad \text{if}\ \sum_{\gamma\supset\gamma'\in\varGamma_{X}^{\myp0}}
\!|\mbox{$\:\!$}\varPhi(\xi\cup\gamma')| <\infty , \\[1.7pc] \displaystyle\ \ +\infty & \quad \text{otherwise}. \end{array} \right. \end{equation} \end{definition}
\begin{definition} Let $\mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$ denote the class of all \textit{grand canonical Gibbs measures} corresponding to the reference measure $\theta$ and the interaction potential ${\varPhi}$, that is, the probability measures on $ \varGamma_{X}$ that satisfy the \textit{Dobrushin--Lanford--Ruelle \textup{(}DLR\textup{)} equation} (see, e.g., \cite[Eq.~(2.17), p.~251]{AKR2}). \end{definition}
In the present paper, we use an equivalent characterization of Gibbs measures based on the following theorem, first proved by Nguyen and Zessin \cite[Theorem~2]{NZ}.
\begin{theorem}\label{th:NZR}
A measure $\g$ on the configuration space $\varGamma_X$ belongs to the Gibbs class $\mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$ if and only if either of the following conditions holds\textup{:}
\textup{(i)} \textup{(}\emph{Nguyen--Zessin's equation}\textup{)} \,For any function $H\in\mathrm{M}_+(X\times\varGamma_X)$, \begin{equation}\label{eq:NZ} \!\int_{\varGamma_{X}}\sum_{x_i\in\gamma} H(x_i,\gamma )\,\g(\mathrm{d}\gamma)=\int_{\varGamma_{X}}\!\biggl(\int_{X}H(x,\gamma \cup \{x\})\,\mathrm{e\mbox{$\:\!$}}^{-E(\{x\},\mbox{$\:\!$}\gamma )}\,\theta (\mathrm{d}{x})\mbox{$\:\!\!$}\biggr)\,\g(\mathrm{d}\gamma). \end{equation}
\textup{(ii)} \textup{(}\emph{Ruelle's equation}\textup{)} \,For any bounded function $F\in\mathrm{M}_+(\varGamma_{X})$ and any compact set $\varLambda\subset X$,
\begin{equation}\label{ruelle} \int_{\varGamma_{X}}\! F(\gamma)\,\g(\mathrm{d}\gamma)= \int_{\varGamma_{\varLambda}}\mathrm{e\mbox{$\:\!$}}^{-E(\xi)}\biggl(\int_{\varGamma_{X\setminus \varLambda}} \!F(\xi \cup \gamma')\ \mathrm{e\mbox{$\:\!$}}^{-E(\xi,\mbox{$\:\!$}\gamma')}\,\g(\mathrm{d}\gamma')\mbox{$\:\!\!$}\biggr)\, \lambda_{\theta}(\mathrm{d}\xi), \end{equation} where $\lambda_{\theta}$ is the \emph{Lebesgue--Poisson measure} on $\varGamma_{X}^{\myp0}$ defined
by the formula \begin{equation}\label{eq:LP} \lambda_{\theta}({\mathrm{d}}\xi)=\sum_{n=0}^{\infty} \mathbf{1}\{\xi(\varLambda)=n\}\,\frac{1}{n!} \,{\textstyle\bigotimes\limits_{x_i\in\xi}}\,\theta(\mathrm{d}{x}_{i}), \qquad \xi\in\varGamma^{\mbox{$\:\!$} 0}_\varLambda. \end{equation}
\end{theorem}
\begin{remark}\label{rm:NZ} Using a standard argument based on the decomposition
$H=H^{+}-H^{-}$, \,$|H|=H^{+}+H^{-}$ with $H^{+}\mbox{$\;\!\!$}:=\max\{H,0\}$, $H^{-}\mbox{$\;\!\!$}:=\max\{-H,0\}$, one can see that
equation (\ref{eq:NZ}) is also valid for an arbitrary measurable function $H:X\times\varGamma_X\to\mathbb{R}$ provided that \begin{equation}\label{abs-int}
\int_{\varGamma_{X}}\sum_{x_i\in\gamma}|H(x_i,\gamma)|\:\g(\mathrm{d}\gamma)<\infty. \end{equation} \end{remark}
\begin{remark} In the original paper \cite{NZ}, the authors proved the result of Theorem \ref{th:NZR} under additional assumptions of \textit{stability} of the interaction potential ${\varPhi}$ and \textit{temperedness} of the measure $\g$. In subsequent work by Kuna
\cite[Theorems 2.2.4, A.1.1]{KunaPhD}, these assumptions have been removed. \end{remark}
\begin{remark}\label{rm:apriori} Inspection of \cite[Theorem~2]{NZ} or \cite[Theorem~A.1.1]{KunaPhD} reveals that the proof of the implication $\textup{(\ref{eq:NZ})} \Rightarrow \textup{(\ref{ruelle})}$ is valid for \textit{any} set $\varLambda\in\mathcal{B}(X)$ satisfying a priori conditions $\theta(\varLambda)<\infty$ and $\gamma(\varLambda)<\infty$ ($\g$-a.s.). Hence, Ruelle's equation (\ref{ruelle}) is valid for such sets as well. \end{remark}
In the ``interaction-free'' case where ${\varPhi}\equiv 0$, the unique grand canonical Gibbs measure coincides with the Poisson measure $\pi_{\theta }$ (with intensity measure $\theta$). In the general situation, there are various types of conditions to ensure that the class $\mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$ is non-empty (see \cite{Ge79,Preston,Ruelle} and also \cite{KunaPhD,K,KKS98}).
\begin{example} The following are four classical examples of translation-invariant pair interaction potentials (i.e., such that ${\varPhi}(\{x,y\})=\phi_{0}(x-y)\equiv\phi_{0}(y-x)$), for which $\mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})\ne\emptyset$. \begin{enumerate} \item[(1)]
(\textit{Hard core potential}) \,$\phi_0(x)=+\infty$ for $|x|\le r_0$, otherwise $\phi_0(x)$ $=0$ \,($r_0>0$). \item[(2)] (\textit{Purely repulsive potential}) \,$\phi_0\in C^2_0(\mathbb{R}^d)$, $\phi_0\ge0$ on $\mathbb{R}^d$, and $\phi_0(0)>0$. \item[(3)] (\textit{Lennard--Jones type potential}) \,$\phi_0\in C^2(\mathbb{R}^d\setminus\{0\})$, $\phi_0\ge -a>-\infty$ on
$\mathbb{R}^d$, $\phi_0(x):=c|x|^{-\alpha}$ for $|x|\le r_1$
\,($c>0$, $\alpha>d$\mbox{$\:\!$}), and \,$\phi_0(x)=0$ for $|x|>r_2$ \,($0<r_1<r_2<\infty$). \item[(4)] (\textit{\emph{Lennard--Jones ``6--12'' potential}})
\,$d=3$, \,$\phi_0(x)=c(|x|^{-12}-|x|^{-6})$ for $x\ne0$ \,($c>0$) and $\phi_0(0)=+\infty$. \end{enumerate} \end{example}
\begin{definition} For a Gibbs measure $\g$ on $\varGamma_X$, its \textit{correlation function $\kappa_{\mbox{$\;\!\!$}\g}^n: X^n\to\mathbb{R}_+$ of the $n$-th order} ($n\in\mathbb{N}$) is defined by the following property: for any function $\phi\in C_0(X^{n})$, symmetric with respect to permutations of its arguments, it holds \begin{multline}\label{corr-funct} \int_{\varGamma_{X}}\sum_{\{x_{1}\mbox{$\;\!\!$},\dots,\mbox{$\:\!$} x_{n}\} \subset \gamma} \phi(x_{1},\dots,x_{n})\,\g(\mathrm{d}\gamma)\\ =\frac{1}{n!}\int_{X^{n}}\phi (x_{1},\dots,x_{n})\,\kappa_{\mbox{$\;\!\!$}\g}^{n}(x_{1},\dots,x_{n})\,\theta(\mathrm{d}{x}_{1})\cdots\theta (\mathrm{d}{x}_{n}). \end{multline} \end{definition} By a standard approximation argument, equation (\ref{corr-funct}) can be extended to any (symmetric) bounded measurable functions $\phi:X^n\to\mathbb{R}$ with support of finite $\theta^{\otimes n}$-measure. For $n=1$ and $\phi(x)=\mathbf{1}_{B}(x)$,
the definition (\ref{corr-funct}) specializes to
\begin{equation}\label{corr-funct1} \int_{\varGamma_{X}}\gamma(B)\,\g(\mathrm{d}\gamma)=\int_{B}\kappa_{\mbox{$\;\!\!$}\g}^{1}(x)\,\theta(\mathrm{d}{x}). \end{equation} More generally, choosing $\phi(x_1,\dots,x_n)=\prod_{i=1}^n {\bf 1}_{B_i}(x_i)$ with arbitrary test sets $B_i\in\mathcal{B}(X)$, it is easy to see that the definition (\ref{corr-funct}) is equivalent to the following more explicit description (cf.\ \cite[p.~266]{AKR1}), $$ \kappa_{\mbox{$\;\!\!$}\g}^n(x_1,\dots,x_n)\,\theta(\mathrm{d}{x}_1)\cdots\theta(\mathrm{d}{x}_n)= n!\cdot \g\{\gamma\in\varGamma_X: \gamma(\mathrm{d}{x}_i)\ge 1,\ i=1,\dots,n\}, $$ also showing that indeed $\kappa_{\mbox{$\;\!\!$}\g}^n\ge0$.
\begin{example} In the Poisson case (i.e., ${\varPhi}\equiv0$), we have $\kappa_{\pi_\theta}^n(x)\equiv n!$ \,($n\in\mathbb{N}$). \end{example}
\begin{remark}\label{rm:NZ-kappa} Using Nguyen--Zessin's equation (\ref{eq:NZ}) with $H(x,\gamma)=\phi(x)$, from the definition (\ref{corr-funct1}) it follows that \begin{equation}\label{eq:corr1-NZ} \kappa_{\mbox{$\;\!\!$}\g}^1(x)=\int_{\varGamma_X}\mathrm{e\mbox{$\:\!$}}^{-E(\{x\},\gamma)}\,\g(\mathrm{d}{\gamma}),\qquad x\in X. \end{equation} In particular, the representation (\ref{eq:corr1-NZ}) implies that if ${\varPhi}\ge0$ (non-attractive interaction potential) then $\kappa_{\mbox{$\;\!\!$}\g}^1(x)\le 1$ for al $x\in X$, so that $\kappa_{\mbox{$\;\!\!$}\g}^1$ is bounded. \end{remark}
\begin{remark}\label{rm:kappa<const} If the first-order correlation function $\kappa_{\mbox{$\;\!\!$}\g}^1(x)$ is integrable on any set $B\in\mathcal{B}(X)$ of finite $\theta$-measure (for instance, if $\kappa_{\mbox{$\;\!\!$}\g}^1$ is bounded on $X$, cf.\ Remark \ref{rm:NZ-kappa}), then, according to (\ref{corr-funct1}), the mean number of points in $\gamma\cap B$ is finite, also implying that $\gamma(B)<\infty$ for $\g$-a.a.\ configurations $\gamma\in\varGamma_X$ (cf.\ Remark~\ref{rm:apriori}). Conversely, if $\kappa_{\mbox{$\;\!\!$}\g}^1$ is bounded below (i.e., $\kappa_{\mbox{$\;\!\!$}\g}^1(x)\ge c>0$ for all $x\in X$) and the mean number of points in $\gamma\cap B$ is finite, then it follows from (\ref{corr-funct1}) that $\theta(B)<\infty$. \end{remark}
\begin{definition}\label{def:Mp} For a probability measure $\mu$ on $\varGamma_{X}$, the notation $\mu\in \mathcal{M}^{n}(\varGamma_{X})$ signifies that \begin{equation}\label{eq:Mn}
\int_{\varGamma_{X}}|\langle \phi,\gamma\rangle|^{n}\,\mu (\mathrm{d}\gamma)<\infty,\qquad \phi \in C_{0}(X). \end{equation} \end{definition}
\begin{definition}\label{def:GR} We denote by ${\mathscr{G}_{\mathrm{R}}}(\theta,{\varPhi}\mbox{$\:\!$})$ the set of all Gibbs measures $\g\in \mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$ such that all its correlation functions $\kappa_{\mbox{$\;\!\!$}\g}^{n}$ are well defined and satisfy the \textit{Ruelle bound}, that is, for some constant $R\in \mathbb{R}_+$ and all $n\in\mathbb{N}$, \begin{equation}\label{eq:RB}
|\kappa_{\mbox{$\;\!\!$}\g}^{n}(x_1,\dots,x_n)|\le R^{n},\qquad (x_1,\dots,x_n)\in X^n. \end{equation} \end{definition}
\begin{proposition}\label{pr:k=k} Let\/ $\g_{1},\g_{2}\in {\mathscr{G}_{\mathrm{R}}}(\theta,{\varPhi}\mbox{$\:\!$})$ and\/ $\kappa_{\mbox{$\;\!\!$}\g_{1}}^{n}=\kappa_{\mbox{$\;\!\!$}\g_{2}}^{n}$ for all\/ $n\in\mathbb{N}$. Then\/ $\g_{1}=\g_{2}$. \end{proposition} \proof
For any measure $\g\in {\mathscr{G}_{\mathrm{R}}}(\theta,{\varPhi}\mbox{$\:\!$})$, its Laplace transform $L_{\g}(f)$ on functions $f\in C_0(X)$ may be represented in the form \begin{align} \notag L_{\g}(f)&=\int_{\varGamma _{X}}\prod_{x_i\in\gamma} \bigl(1+(\mathrm{e\mbox{$\:\!$}}^{-f(x_i)}-1)\bigr)\,\g(\mathrm{d}\gamma)\\ \label{eq:l1} &=1+ \int_{\varGamma _{X}}\sum_{n=1}^\infty \sum_{\{x_{1}\mbox{$\;\!\!$},\dots,\mbox{$\:\!$} x_{n}\} \subset \gamma} \prod_{i=1}^n \bigl(\mathrm{e\mbox{$\:\!$}}^{-f(x_{i})}-1\bigr)\,\g(\mathrm{d}\gamma)\\ \label{eq:l2} &=1+ \sum_{n=1}^\infty \int_{\varGamma_{X}} \sum_{\{x_{1}\mbox{$\;\!\!$},\dots,\mbox{$\:\!$} x_{n}\} \subset \gamma} \prod_{i=1}^n \bigl(\mathrm{e\mbox{$\:\!$}}^{-f(x_{i})}-1\bigr)\,\g(\mathrm{d}\gamma)\\ \label{eq:laplace} &= 1+\sum_{n=1}^{\infty} \frac{1}{n!}\int_{X^{n}}\prod_{i=1}^n \bigl(\mathrm{e\mbox{$\:\!$}}^{-f(x_{i})}-1\bigr) \,\kappa_{\mbox{$\;\!\!$}\g}^{n}(x_{1},\dots,x_{n})\,\theta(\mathrm{d}{x}_{1})\cdots\theta(\mathrm{d}{x}_{n}), \end{align}
where (\ref{eq:laplace}) is obtained from (\ref{eq:l2}) using formula (\ref{corr-funct}). Interchanging the order of integration and summation in (\ref{eq:l1}) is justified by the dominated convergence theorem; indeed, using that $|f(x)|\le C_f$ on $K_f:=\mathop{\mathrm{supp}}\nolimits f$ with some $C_f>0$ and recalling that the correlation functions $\kappa^n_{\g} $ satisfy the Ruelle bound (\ref{eq:RB}), we see that the right-hand side of (\ref{eq:laplace}) is dominated by $$ 1+\sum_{n=1}^{\infty} \frac{1}{n!}\,(\mathrm{e\mbox{$\:\!$}}^{C_f}+1)^n \mbox{$\;\!$} R^{n}\mbox{$\;\!$}\theta(K_f)^n=\exp\bigl\{R\mbox{$\;\!$}(\mathrm{e\mbox{$\:\!$}}^{C_f}+1) \mbox{$\;\!$}\theta(K_f)\bigr\}<\infty. $$ Now, formula (\ref{eq:laplace}) implies that if measures $\g_1,\g_2\in{\mathscr{G}_{\mathrm{R}}}(\theta,{\varPhi}\mbox{$\:\!$})$ have the same correlation functions, then their Laplace transforms coincide with each other, $L_{\g_{1}}(f)=L_{\g_{2}}(f)$ for any $f\in\mathrm{M}_+(X)$, hence $\g_1=\g_2$. \endproof
\begin{definition}\label{def:extreme} It is well known that $\mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$ is a convex set \cite{Preston}. We denote by $\mathop{\mathrm{ext}}\nolimits \mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$ the set of its extreme elements, that is, those measures $\g\in\mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$ that cannot be written as $\g=\frac12\mbox{$\:\!$}(\g_1+\g_2)$ with $\g_1,\g_2\in\mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$ and $\g_1\ne \g_2$. \end{definition}
Using Ruelle's equation (\ref{ruelle}), it is easy to obtain the following result (cf.\ \cite[Corollary~2.2.6]{KunaPhD}).
\begin{proposition}\label{pr:Gibbs|cond} Let \mbox{$\:\!$}$\g\in\mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$, and let $\varLambda\in\mathcal{B}(X)$ be a compact set.
Then the restriction of the Gibbs measure $\g\in\mathscr{G}(\theta,{\varPhi}\mbox{$\:\!$})$ onto the space $\varGamma_\varLambda$, defined by \begin{equation*} \g_\varLambda(A):=\g(A\cap\varGamma_\varLambda),\qquad A\in\mathcal{B}(\varGamma_X), \end{equation*} is absolutely continuous with respect to the Lebesgue--Poisson measure $\lambda_\theta$, with the Radon--Nikodym density $S_\varLambda:=\mathrm{d}\g_\varLambda/\mathrm{d}\lambda_\theta\in L^1(\varGamma_\varLambda,\lambda_\theta)$ given by \begin{equation}\label{eq:Psi} S_\varLambda(\gamma)=\mathrm{e\mbox{$\:\!$}}^{-E(\gamma)}\int_{\varGamma_{X\setminus \varLambda}}\mathrm{e\mbox{$\:\!$}}^{-E(\gamma,\mbox{$\:\!$}\gamma')}\,\g(\mathrm{d}\gamma'),\qquad \gamma\in\varGamma_\varLambda. \end{equation}
\end{proposition}
\end{document} | arXiv |
Quantifier elimination
Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement "$\exists x$ such that $\ldots $" can be viewed as a question "When is there an $x$ such that $\ldots $?", and the statement without quantifiers can be viewed as the answer to that question.[1]
One way of classifying formulas is by the amount of quantification. Formulas with less depth of quantifier alternation are thought of as being simpler, with the quantifier-free formulas as the simplest. A theory has quantifier elimination if for every formula $\alpha $, there exists another formula $\alpha _{QF}$ without quantifiers that is equivalent to it (modulo this theory).
Examples
An example from high school mathematics says that a single-variable quadratic polynomial has a real root if and only if its discriminant is non-negative:[1]
$\exists x\in \mathbb {R} .(a\neq 0\wedge ax^{2}+bx+c=0)\ \ \Longleftrightarrow \ \ a\neq 0\wedge b^{2}-4ac\geq 0$
Here the sentence on the left-hand side involves a quantifier $\exists x\in \mathbb {R} $, while the equivalent sentence on the right does not.
Examples of theories that have been shown decidable using quantifier elimination are Presburger arithmetic,[2][3][4][5][6] algebraically closed fields, real closed fields,[6] [7] atomless Boolean algebras, term algebras, dense linear orders,[6] abelian groups,[8] random graphs, as well as many of their combinations such as Boolean algebra with Presburger arithmetic, and term algebras with queues.
Quantifier eliminator for the theory of the real numbers as an ordered additive group is Fourier–Motzkin elimination; for the theory of the field of real numbers it is the Tarski–Seidenberg theorem.[6]
Quantifier elimination can also be used to show that "combining" decidable theories leads to new decidable theories (see Feferman-Vaught theorem).
Algorithms and decidability
If a theory has quantifier elimination, then a specific question can be addressed: Is there a method of determining $\alpha _{QF}$ for each $\alpha $? If there is such a method we call it a quantifier elimination algorithm. If there is such an algorithm, then decidability for the theory reduces to deciding the truth of the quantifier-free sentences. Quantifier-free sentences have no variables, so their validity in a given theory can often be computed, which enables the use of quantifier elimination algorithms to decide validity of sentences.
Related concepts
Various model-theoretic ideas are related to quantifier elimination, and there are various equivalent conditions.
Every first-order theory with quantifier elimination is model complete. Conversely, a model-complete theory, whose theory of universal consequences has the amalgamation property, has quantifier elimination.[9]
The models of the theory of the universal consequences of a theory $T$ are precisely the substructures of the models of $T$.[9] The theory of linear orders does not have quantifier elimination. However the theory of its universal consequences has the amalgamation property.
Basic ideas
To show constructively that a theory has quantifier elimination, it suffices to show that we can eliminate an existential quantifier applied to a conjunction of literals, that is, show that each formula of the form:
$\exists x.\bigwedge _{i=1}^{n}L_{i}$
where each $L_{i}$ is a literal, is equivalent to a quantifier-free formula. Indeed, suppose we know how to eliminate quantifiers from conjunctions of literals, then if $F$ is a quantifier-free formula, we can write it in disjunctive normal form
$\bigvee _{j=1}^{m}\bigwedge _{i=1}^{n}L_{ij},$
and use the fact that
$\exists x.\bigvee _{j=1}^{m}\bigwedge _{i=1}^{n}L_{ij}$
is equivalent to
$\bigvee _{j=1}^{m}\exists x.\bigwedge _{i=1}^{n}L_{ij}.$
Finally, to eliminate a universal quantifier
$\forall x.F$
where $F$ is quantifier-free, we transform $\lnot F$ into disjunctive normal form, and use the fact that $\forall x.F$ is equivalent to $\lnot \exists x.\lnot F.$
Relationship with decidability
In early model theory, quantifier elimination was used to demonstrate that various theories possess properties like decidability and completeness. A common technique was to show first that a theory admits elimination of quantifiers and thereafter prove decidability or completeness by considering only the quantifier-free formulas. This technique can be used to show that Presburger arithmetic is decidable.
Theories could be decidable yet not admit quantifier elimination. Strictly speaking, the theory of the additive natural numbers did not admit quantifier elimination, but it was an expansion of the additive natural numbers that was shown to be decidable. Whenever a theory is decidable, and the language of its valid formulas is countable, it is possible to extend the theory with countably many relations to have quantifier elimination (for example, one can introduce, for each formula of the theory, a relation symbol that relates the free variables of the formula).[10]
Example: Nullstellensatz for algebraically closed fields and for differentially closed fields.
See also
• Cylindrical algebraic decomposition
• Elimination theory
• Conjunction elimination
Notes
1. Brown 2002.
2. Presburger 1929.
3. Monk 2012, p. 240.
4. Nipkow 2010.
5. Enderton 2001, p. 188.
6. Grädel et al. 2007.
7. Fried & Jarden 2008, p. 171.
8. Szmielew 1955, Page 229 describes "the method of eliminating quantification.".
9. Hodges 1993.
10. "Proofs with Quantifiers 2 | Stanford University - KeepNotes". keepnotes.com. Retrieved 2023-08-10.
References
• Brown, Christopher W. (July 31, 2002). "What is Quantifier Elimination". Retrieved Aug 30, 2018.
• Enderton, Herbert (2001). A mathematical introduction to logic (2nd ed.). Boston, MA: Academic Press. ISBN 978-0-12-238452-3.
• Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001.
• Grädel, Erich; Kolaitis, Phokion G.; Libkin, Leonid; Maarten, Marx; Spencer, Joel; Vardi, Moshe Y.; Venema, Yde; Weinstein, Scott (2007). Finite model theory and its applications. Texts in Theoretical Computer Science. An EATCS Series. Berlin: Springer-Verlag. ISBN 978-3-540-00428-8. Zbl 1133.03001.
• Hodges, Wilfrid (1993). Model Theory. Encyclopedia of Mathematics and its Applications. Vol. 42. Cambridge University Press. doi:10.1017/CBO9780511551574. ISBN 9780521304429.
• Kuncak, Viktor; Rinard, Martin (2003). "Structural subtyping of non-recursive types is decidable" (PDF). 18th Annual IEEE Symposium of Logic in Computer Science, 2003. Proceedings. pp. 96–107. doi:10.1109/LICS.2003.1210049. ISBN 0-7695-1884-2. S2CID 14182674.
• Monk, J. Donald Monk (2012). Mathematical Logic (Graduate Texts in Mathematics (37)) (Softcover reprint of the original 1st ed. 1976 ed.). Springer. ISBN 9781468494549.
• Nipkow, T (2010). "Linear Quantifier Elimination" (PDF). Journal of Automated Reasoning. 45 (2): 189–212. doi:10.1007/s10817-010-9183-0. S2CID 14279141. Retrieved 2022-11-12.
• Presburger, Mojżesz (1929). "Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt". Comptes Rendus du I congrès de Mathématiciens des Pays Slaves, Warszawa: 92–101., see Stansifer (1984) for an English translation
• Stansifer, Ryan (Sep 1984). Presburger's Article on Integer Arithmetic: Remarks and Translation (PDF) (Technical Report). Vol. TR84-639. Ithaca, New York: Dept. of Computer Science, Cornell University.
• Szmielew, Wanda (1955). "Elementary properties of Abelian groups". Fundamenta Mathematicae. 41 (2): 203–271. doi:10.4064/fm-41-2-203-271. MR 0072131.
• Jeannerod, Nicolas; Treinen, Ralf. Deciding the First-Order Theory of an Algebra of Feature Trees with Updates. International Joint Conference on Automated Reasoning (IJCAR). doi:10.1007/978-3-319-94205-6_29.
• Sturm, Thomas (2017). "A Survey of Some Methods for Real Quantifier Elimination, Decision, and Satisfiability and Their Applications". Mathematics in Computer Science. 11 (3–4): 483–502. doi:10.1007/s11786-017-0319-z.
| Wikipedia |
Fisheries and Aquatic Sciences
Preparing your manuscript
Effect of water temperature on protein requirement of Heteropneustes fossilis (Bloch) fry as determined by nutrient deposition, hemato-biochemical parameters and stress resistance response
Shabihul Fatma ORCID: orcid.org/0000-0003-0115-94461 &
Imtiaz Ahmed2
Fisheries and Aquatic Sciences volume 23, Article number: 1 (2020) Cite this article
Dietary protein requirements are dependent on a variety of factors and water temperature is one of the most important abiotic factors affecting protein requirement of fish. This study was, therefore, conducted to investigate effects of water temperature on dietary protein requirement of fry Heteropneustes fossilis which has high demand in most of the Asian markets.
Quadruplicate groups of 30 fish per treatment (2.97 ± 0.65 cm; 5.11 ± 0.34 g) were fed seven isoenergetic diets (17.9 kJ g−1 gross energy; 14.99 kJ g−1 digestible energy) containing dietary protein levels ranging from 28 to 52% at two water temperatures (18 and 26 °C). Experimental diets were fed to apparent satiation as semi-moist cakes thrice daily at 17:00, 12:00, and 17:30 h for 12 weeks. For precise information, various growth parameters, protein deposition, hematological parameters, metabolic enzymes, and stress response were analyzed, and effects of water temperature on dietary protein requirement was recommended on the basis of response from above parameters.
Groups held at 26°C attained best growth, feed conversion, and protein deposition at 44% dietary protein indicating that temperature affected dietary protein requirement for optimum growth of H. fossilis fry and protein requirement seems to be satisfied with 44% dietary protein. Interestingly, interactive effects of both dietary protein levels and temperature were not found (P > 0.05). Fish reared at 18 °C had comparatively higher values for aspartate and alanine transferases than those reared at 26 °C water temperature which exhibited normal physiological value for these enzymes indicating that body metabolism was normal at this temperature. Hematological parameters also followed same pattern. Furthermore, fish reared at 26 °C water temperature exhibited more resistant to thermal stress (P < 0.05). The 95% maximum plateau of protein deposition data using second-degree polynomial regression analyses exhibited dietary protein requirement of fry H. fossilis between 40.8 and 41.8% of diet at 26 °C water temperature. The recommended range of dietary protein level and protein/digestible energy ratio for fry H. fossilis is 40.8–41.8% and 27.21–27.88 mg protein kJ−1 digestible energy, respectively.
Information developed is of high significance for optimizing growth potential by making better utilization of nutrient at 26 °C and, to develop effective management strategies for mass culture of this highly preferred fish species.
Protein assumes greater importance in aquacultural feeds mainly due to the fact that the level and quality of protein greatly influences feed cost. Therefore, protein content should be carefully adjusted in feeds, bearing in mind that the dietary protein in excess to that required for growth is only catabolized (Cowey 1979) and that protein inadequacy leads to poor growth and feed inefficiency. Consequently, to improve the utilization of protein for tissue synthesis rather than for energy purposes, an adequate knowledge of protein requirements and of the effect of environmental factors on protein utilization is necessary.
Dietary protein requirements are dependent on a variety of factors such as stock size, water temperature, feeding frequency, amount of non-protein dietary energy, and dietary protein quality (Shimeno et al. 1980). As fish is an ectotherm, and water temperature is one of the most important abiotic factors affecting growth and survival of the aquatic animals. All fish species are characterized by an ideal range of temperature in which they show their maximum growth (Oyugi et al. 2011). Any alterations in the optimum water temperature have a marked and direct effect on many of the key physiological processes and behavioral activities (Brett 1979; Jonassen et al. 2000; Sarma et al. 2010) which can also be detected in the form of alterations in hematological parameters (Haider 1973; Steinhagen et al. 1990). Temperature beyond optimum limits of a particular species adversely affects the health of aquatic animal due to metabolic stress and increases oxygen demand and susceptibility to diseases (Wedemeyer et al. 1999). It limits the biochemical reactions, affects their metabolism and distribution, and directly influences the survival and growth at the various stages of their life cycle.
Catfishes are the preferred candidate species for aquaculture owing to their consumer preference and commercial and medicinal value. Among those, Heteropneustes fossilis, commonly known as the stinging catfish or singhi, is considered as one of the most demanded freshwater air breathing fish species in the tropical waters of the Indian subcontinent and Southeast Asian region (Christopher et al. 2010). The range encompasses India, Thailand, Bangladesh, Pakistan, Nepal, Sri Lanka, Myanmar, Indonesia, and Cambodia (Burgess 1989). Its primary habitat includes ponds, ditches, swamps, and marshes. It is hardy, amenable to high stocking densities, and adapts well to hypoxic water bodies (Dehadrai et al. 1985). Due to the presence of accessory respiratory organs, it has got the ability to utilize atmospheric oxygen for respiration and, therefore, can survive for quite a few hours outside the water which makes it an ideal species for aquaculture (Vijayakumar et al. 1998; Haniffa and Sridhar 2002). Heteropneustes fossilis is an important tropical freshwater food fish (Mohamed and Ibrahim 2001). It has very high iron content (226 mg/100 g) and fairly high content of calcium compared to many other freshwater fishes (Saha and Guha 1939). Being a lean fish, it is very suitable for people to whom animal fats are undesirable (Rahman et al. 1982). Due to high nutritive value and low fat content, the stinging catfish is recommended in the diets of the sick and the convalescents (Alok et al. 1993). The fish efficiently utilizes prepared feeds and is able to withstand adverse environmental conditions. In addition to this, it has high nutritional and medicinal value (Pillay 1990; Jhingran 1991; Thakur 1991). This fish is popular particularly because it can be cultivated in swampy areas and derelict water bodies without involving costly reclamation. It is easily stored and transported live to consumers. Thus, this species is ideal for wastewater aquaculture as well (Tharakan and Joy 1996) and is abundantly available in open water system of floodplains, canals, and beels.
The effects of water temperature on growth and protein requirements of fish have been well documented for many species (El-Sayed et al. 1996; Van Ham et al. 2003; Anelli et al. 2004; Chatterjee et al. 2004; Larsson and Berglund 2005; Han et al. 2008; Singh et al. 2008; Singh et al. 2009; Huang et al. 2016; Mishra et al. 2019). Some aspects of nutrition of H. fossilis has been worked out in the past mainly on determining its optimum feeding practices and nutritional requirements (Niamat and Jafri 1984; Akand et al. 1991; Jhingran 1991; Anwar and Jafri 1992; Firdaus 1993; Firdaus et al. 1994; Firdaus and Jafri 1996; Mohamed 2001; Mohamed and Ibrahim 2001; Firdaus et al. 2002; Usmani and Jafri 2002; Usmani et al. 2003; Ahmed 2007; Siddiqui and Khan 2009; Ahmed 2010; Khan and Abidi 2010; Khan and Abidi 2011a, 2011b; Ahmed 2012; Farhat 2011; Farhat 2012; Khan and Abidi 2012; Ahmed 2013a, 2013b; Farhat 2013a, 2013b; Ahmed 2014; Farhat 2014a, 2014b, 2014c; Khan and Abidi 2014; Ahmed 2017; Farhat 2017); however, study on effect of water temperature on the nutritional requirements of H. fossilis under culture condition has not been worked out.
Since biochemical parameters such as serum aspartate amino transferase (AST) and alanine amino transferase (ALT) levels and the hematological parameters commonly measured clinically as biomarkers for health and good indicator of various sources of stress, to ascertain the effect of temperature on dietary protein requirement more precisely, these parameters are also considered and analyzed.
The aim of this study was to determine the influence of water temperature on protein requirement and to optimize the rearing temperature so that this fish could maximize its performance in terms of growth and health in an intensive culture system.
Experimental diets
Seven casein-gelatin-based isocaloric diets (14.99 kJ/g digestible energy) with varying levels of dietary protein (28, 32, 36, 40, 44, 48, 52% protein) were prepared (Table 1). Diets were designated as D28, D32, D36, D40, D44, D48, and D52. Two intact protein sources, casein and gelatin, were used at 4:1 ratio. The dietary protein level was increased by adjusting the fractions of casein and gelatin in the diet. Dextrin was served as the carbohydrate source. A combination of cod liver oil and corn oil (2:5) was used as a source of lipid to provide n-3 and n-6 fatty acids. Vitamin and mineral premixes were prepared as per Halver (2002). Digestible energy (DE) was calculated using conversion factors of 33.5, 20.9, and 12.6 kJ/g for fat, protein, and carbohydrate, respectively (Halver and Hardy 2002). The diet was prepared as per Siddiqui and Khan (2009). The final diet in the form of dough was cut into small cubes, sealed in polythene bags, and kept in refrigerator at − 20 °C till further use.
Table 1 Composition of experimental diets
Experimental design and feeding trial
Induced-bred fry of the stinging catfish, H. fossilis, were obtained from the Ghazipur Fish Market, New Delhi, India and transported to wet laboratory (Fish Rearing Laboratory, Department of Zoology, Central University of Kashmir, J&K), given a prophylactic dip in KMnO4 solution (1:3000) and stocked in indoor circular aqua-blue colored, plastic lined (Plastic Crafts Corp, Mumbai, India) fish tanks (1.22 m × 0.91 m × 0.91 m; water volume 600 L) for about 2 weeks. They were then acclimated to two different constant temperatures (18 and 26 °C). The desired temperatures were adjusted with the help of thermostatic water heaters (Rusun, Fish Aquarium Home, Laxhami Nagar, New Delhi, India). Prior to the commencement of the feeding trial, fish were acclimated to the respective water temperatures for 7 days to stabilize their internal mediums and allow metabolic compensation (Castille Jr and Lawrence 1981) and to ensure full thermal adaptation. During this period, the fish were fed with a casein-gelatin based H-440 diet (Halver 2002) thrice a day (0700, 1200, 1730 h) until apparent satiation, at each temperature. The apparent satiety was ensured simply by visual observation and the fish were carefully observed during feeding to ensure satiety without overfeeding. The diet was fed as long as the fish actively consumed it at each feeding schedule. Since feed allocation was done till the fish desired to feed and no feed was dispensed once the fish stopped feeding actively, there was no unconsumed feed in the culture tank. A photoperiod of 12 h light/12 h dark was maintained throughout the experimental period.
For conducting the present experiment, H. fossilis fry (2.97 ± 0.65 cm; 5.11 ± 0.34 g) were sorted out from the above acclimated lot and stocked in quadruplet groups (n = 4 tanks per treatment) in 70-L circular polyvinyl troughs (water volume 60 L). The experiment was conducted in a thermostatic experimental setup. Throughout the experimental period (84 days), temperature was regularly measured three times daily with a thermometer at each feeding schedule. Fish were fed experimental diets in the form of semi-moist cakes in the form of cube (1 × 1 × 1 cm) as per the above feeding schedule. Initial and weekly individual weights were recorded on a top-loading balance (Sartorius CPA- 224S 0.1 mg sensitivity, Goettingen, Germany) after anaesthetizing with tricane methane sulphonate (MS-222; 20 mg/L; Finquel®). The feeding trial lasted for 84 days. Fish were fasted on the day of weekly measurements. A KMnO4 bath was administered after every weighing session (5 g/L for 30 min) as a prophylactic measure. Fecal matter, if any, was siphoned off before and after every feeding. The culture troughs were siphoned once every day. The experiments were conducted absolutely as per the guidelines for animal ethics.
Water quality parameters
Water quality parameters of the troughs were maintained at different temperatures (18 and 26 °C). Water was sampled from each trough to determine water temperature, dissolved oxygen, free carbon dioxide, total alkalinity, TAN, nitrite, and pH based on daily measurements following the standard methods (APHA 1992). The pH was determined by using digital pH meter (pH ep-HI 98107, USA).
Stress resistance response
At the end of experiment, eight fish were randomly sampled to assess environmental stress (high temperature) trial. The fish were exposed to high temperature (33 °C) and the mortality time was recorded in seconds.
Biochemical composition of fish and experimental diets
Six subsamples of a pooled sample of 20 fishes were analyzed for initial body composition. At the end of the experiment, all 30 fishes from each replicate of dietary treatments were pooled separately and three subsamples of each replicate from the pooled sample (n = 4) were analyzed for final carcass composition. Proximate composition of casein, gelatin, experimental diet, and initial and final body composition was estimated using standard methods (AOAC 1995) for dry matter (oven drying at 105 ± 1 °C for 22 h), crude protein, (nitrogen estimation using Kjeltec 8400, Hoeganas, Sweden), crude fat (solvent extraction with petroleum ether B.P 40–60 °C for 2–4 h by using Soxlet extraction technique, FOSS Avanti automatic 2050 equipment, Sweden), and ash oven incineration at 650 °C for 2–4 h. To confirm the calculated levels of gross energy of the prepared test diets, each dietary sample was ignited in Gallenkamp ballistic bomb calorimeter (Gallenkamp Ballistic Bomb Calorimeter-CBB 330 010L, Gallenkamp, Loughbrough, UK). The analysis revealed a close agreement with the calculated values of the gross energy density (Table 1).
Evaluation of the hematological parameters involved the determination of the red blood cell count (RBCs × 109), hemoglobin content (Hb; g dL−1), and hematocrit value (Hct%). At the end of the experiment, fish were anaesthetized with MS-222 (20 mg/L; Finquel®) before taking the blood samples. The blood samples were then collected from the caudal vein of individual fish (nine fish from each replicate of the treatment) employing heparinized syringes. To avoid blood coagulation, all mixers, pipettes, and test tubes used were rinsed with anticoagulant (3.8% solution of sodium citrate). Erythrocyte count was determined by an improved Neubauer hematocytometer with Yokoyama's (1974) solution as the diluting medium. Blood hemoglobin was determined spectroscopically (Genesis, UV) following Wong's (1928) method and was expressed in grams per deciliter (Hb g/dL). Hematocrit value (Hct%) was measured by spinning the micro-wintrobe tube containing well mixed blood for about 5 min at 12,000g and then measuring the packed cell volume which was expressed in percentage.
On the final day of the feeding trial, five fish from each tank (n = 4x5) were anesthetized (MS-222; 20 mg/L) before subjecting to body measurements. The fish, liver, and viscera of each specimen were weighed by blotting dry on a filter paper, and total length of the fish was taken. The values were recorded to calculate the hepatosomatic index (HSI%), viscerosomatic index (VSI%), and condition factor (CF).
Metabolic enzyme activities
Blood serum was collected after centrifugation at 3000 rpm for 10 min and then stored at − 20 °C in order to analyze aspartate aminotransferase (AST) and alanine aminotransferase (ALT) activities. Biochemical analysis of serum AST and ALT activities were done as per Reitman and Frankel (1957).
Data analyses
Growth performance of the fish fed experimental diets at different temperatures was measured as a function of the weight gain by calculating following parameters:
$$ \mathrm{Thermal}\ \mathrm{growth}\ \mathrm{coefficient}=\left(\mathrm{final}\ {\mathrm{body}\ \mathrm{weight}}^{0.333}-\mathrm{initial}\ {\mathrm{body}\ \mathrm{weight}}^{0.333}\right)/\mathrm{No}.\mathrm{of}\ \mathrm{days}\times \mathrm{temperature}{}^{\circ}\mathrm{C}\times 1000 $$
$$ \mathrm{Feed}\ \mathrm{conversion}\ \mathrm{ratio}=\mathrm{dry}\ \mathrm{feed}\ \mathrm{fed}\ \left(\mathrm{g}\right)/\mathrm{wet}\ \mathrm{weight}\ \mathrm{gain}\ \left(\mathrm{g}\right) $$
$$ \mathrm{Protein}\ \mathrm{deposition}\ \mathrm{g}/\mathrm{fish}=\mathrm{protein}\ \mathrm{gain}/\mathrm{protein}\ \mathrm{fed}\ \left(\mathrm{g}\right) $$
$$ \mathrm{Hepatosomatic}\ \mathrm{index}\ \left(\mathrm{HSI},\%\right)=\left(\mathrm{liver}\ \mathrm{weight};\mathrm{g}\right)/\left(\mathrm{whole}\ \mathrm{body}\ \mathrm{weight};\mathrm{g}\right)\times 100; $$
$$ \mathrm{Viscerosomatic}\ \mathrm{index}\ \left(\mathrm{VSI},\%\right)=\left(\mathrm{viscera}\ \mathrm{weight};\mathrm{g}\right)/\left(\mathrm{whole}\ \mathrm{body}\ \mathrm{weight};\mathrm{g}\right)\times 100; $$
$$ \mathrm{Condition}\ \mathrm{factor}\ \left(\mathrm{CF},{\mathrm{g}}^{-1}\ {\mathrm{cm}}^3\right)=\left(\mathrm{body}\ \mathrm{weight};\mathrm{g}\right)/{\left(\mathrm{body}\ \mathrm{length}\ \mathrm{cm}\right)}^3\times 100 $$
Statistical analyses
A completely randomized design with four replicates per treatment was used for assessing the optimum protein requirement of the fish at two different temperatures. All growth data were subjected to two-way ANOVA as per Snedecor and Cochran (1982) to test any differences and/or the interaction between dietary protein and temperature. Differences among treatment means were determined by Duncan's multiple range test at a P < 0.05 level of significance (Duncan 1955). Relationship between dietary protein level and protein deposition (PD) g/fish was modeled using second-degree polynomial regression analysis (Zeitoun et al. 1976). The protein requirement of fry H. fossilis was determined as the point on the graph where the biological response was found to be equal to 95% of the maximum response. All the statistical analyses were done using Origin (version 6.1; Origin Software, San Clemente, CA).
Growth performance
Data on average thermal growth coefficient (TGC), feed conversion ratio (FCR), and protein deposition (PD g/fish) of the fry Heteropneustes fossilis, after 84 days of feeding are summarized in Table 2. Growth performance and feed intake were significantly affected by both dietary protein levels (P < 0.0007) and rearing temperature (P < 0.0001). However, interactive effects of dietary protein and temperature were not found (P > 0.05). H. fossilis fry fed diets containing different levels of protein exhibited superior response in terms of TGC, FCR, and PD g fish−1 with 44% protein at 26 °C temperature. The groups reared at 18 °C showed a consistent improvement in their performance up to 40% protein in the diet. However, the values recorded for TGC, FCR, and PD g fish−1 for the groups held at 18 °C were inferior compared to those held at 26 °C even though fed with the same level of dietary protein. This indicates that fish held at this temperature failed to express their maximum growth potential.
Table 2 Growth, feed conversion, and protein deposition of H. fossilis fry
Carcass quality
The dietary protein levels and rearing temperatures had significant influence on carcass protein of H. fossilis fry (Table 3); however, interactive effects of both were not recorded (P > 0.05). Moisture content of H. fossilis showed a positive correlation with the increase in dietary protein at both the temperatures (18 and 26 °C), whereas the carcass fat content showed a negative correlation. Carcass protein tended to increase significantly (P < 0.05) in fish fed 40 and 44% protein (diet D40 and D44) at 18 and 26 °C temperatures (P < 0.05). Moreover, temperature had no significant effect on carcass fat and ash contents.
Table 3 Whole body carcass quality of fry Heteropneustes fossilis
As per the results, significant differences (P < 0.05) in terms of ALT and AST activities were recorded among the two experimental groups reared at two different temperatures. The values of ALT and AST were significantly (P < 0.05) higher among the groups reared at 18 °C temperature compared to those reared at 26 °C water temperature. Also, serum enzymes of H. fossilis seem not to be much affected by different protein levels in this study but were more affected by water temperatures (Table 4). However, the interactive effects of dietary protein and temperature were not found.
Table 4 Metabolic enzymes activities and hematological parameters of fry Heteropneustes fossilis
Hematological parameters
Dietary protein and temperature had significant impact on hematological parameters (P < 0.05) with no interactive effects of both. Fish fed diet containing 28% protein achieved the lowest hematological values at both the temperatures. RBCs, Hb g/dL, and Hct% increased with increasing levels of dietary protein up to 44% (diet D44), both at 18 and 26 °C rearing temperatures (Table 4). Thereafter, the above hematological parameters remained almost the same in fish fed 48% (diet D48) dietary protein and the exhibited a decline with further increase in the dietary protein intake at 52% (diet D52). However, at 18 °C, the magnitude of response in above values was comparatively lower than that attained in fish raised at 26 °C for the same dietary protein level.
Somatic indices
VSI and HSI decreased with increase in dietary protein levels up to 44% and increased in fish fed dietary protein beyond 44% at 48 (D48) and 52% protein (D52). Hepatosomatic index (HSI) was found to be influenced by the levels of dietary protein and temperatures but no interaction occurred, while viscerosomatic index (VSI) was affected by only diets and not by temperatures (Table 5).
Table 5 Somatic indices and stress resistance response of fry Heteropneustes fossilis
Resistance rate to thermal stress increased significantly (P < 0.05) among the groups reared at 26 °C water temperature than those reared at 18 °C (Table 5). The stress resistance response remained best for the groups fed with 44% dietary protein. This parameter was affected by both dietary protein levels and rearing temperatures; however, no interactive effects of both were noted (P > 0.23). The normal values of water quality parameters during the entire length of experiment are provided in Table 6.
Table 6 Water quality parameters (Based on daily measurements)
Protein requirement
Based on the above response parameters, second-degree polynomial regression analyses were performed to study the relationships between dietary protein levels and the protein deposition and were expressed in the form of Y = aX2=+ bX + c. The value of X that corresponds to Y95%max was defined as the requirement. PD g/fish data (Y95%max) to dietary protein levels (X) was subjected to a second-degree polynomial regression analysis. The curve attained its 95% maximum response at 40.8 and 41.8% protein of the diet (Fig. 1) at 18 °C and 26 °C water temperature, respectively.
Second-degree polynomial regression analysis of protein deposition (PD g/fish) against varying levels of dietary protein at two temperatures
The aim of this study is to assess the influence of water temperature on dietary protein requirement, protein deposition, carcass quality, and hematological parameters of fry H. fossilis. Temperature is a pervasive factor affecting food intake, growth, and food conversion of fish (Fry 1947; Brett 1979). Since water temperature has potent influence on metabolic rate and energy expenditure affecting nutrient requirement and growth performance of the poikilothermic vertebrates including fish (Brett 1979; Dutta 1994; Bhikajee and Gobin 1998), its influence on nutrient requirement and growth warrant thorough investigation.
Several studies have reported that the specific water temperature range showed that the faster growth and low temperature causes sluggishness by retarding the digestion speeding of fish (Bailey and Alanara 2006). Some researchers have found that the digestion rate has been increased as the temperature increases (Turker 2009). Environmental temperature is one of the most important ecological factors which also influence the behavior and physiological process of aquatic animals (Xia and Li 2010).
The results showed that growth in terms of thermal growth coefficient, feed conversion, and protein deposition of the fish attained best values with dietary protein levels of 40 and 44% at 18 and 26 °C water temperatures, respectively. The fish attained its maximum growth potential in terms of TWG, FCR, protein deposition, and body protein content at 26 °C water temperature. Carcass protein content exhibited best value for the groups fed 44% dietary protein at 26 °C temperature. Hematological parameters also attained their normal physiological range with 44% protein diet at 26 °C. Inferior values for these parameters were recorded for the groups held at 18 °C water temperature presumably due to the fact that the body metabolism occurs at a slower rate if fish are held at sub-optimum or lower water temperatures. Similar trend has also been reported by Peres and Oliva-Teles (1999) and Ozorio et al. (2006) in various other cultivable finfish species where fish held at 18 °C water temperature could not attain their maximum growth potential even if supplied with the required level of dietary protein. Growth performance and feed intake were significantly affected by both dietary protein levels and rearing temperature. However, interactive effects of dietary protein and temperature were not found. Depressed growth, lower feed intake, and protein deposition were more commonly noted for the groups reared at 18 °C. Even the groups fed dietary protein at 40 to 44% could not attain their maximum growth potential and feed intake at 18 °C as attained by the groups fed same diets at 26 °C. The study clearly indicates that dietary protein requirement of H. fossilis for maximizing the growth, feed conversion, and for attaining best values for hematological parameters ranges somewhere between 40.8 and 41.8% at 26 °C water temperature.
Choice of mathematical models in estimating the dietary level for a limiting nutrient is very important. Some studies show better regression coefficients when a broken-line analysis (Y = a + bX) is used (Baker 1986), whereas some respond better to a second-degree polynomial regression analysis (Tacon and Cowey 1985; El-Dakar et al. 2011). In this study, although data were fitted best for broken-line regression analysis, the p value of the t test for estimated coefficient was not found significantly different from zero for broken-line regression analysis. Therefore, second-degree polynomial regression analysis which exhibited a significant p value of the t test for the estimated coefficient has been employed for quantifying dietary protein requirement of H. fossilis fry. The requirements have been determined at 95% confidence interval. Based on above analyses, 44% dietary protein at 26 °C water temperature appears to be optimum for growth of H. fossilis fry and the curve did not reach a plateau until 44% dietary protein level. The second-degree polynomial fitting of protein deposition values at 95% maximum response exhibited optimum dietary protein requirement of fry H. fossilis between 40.8 and 41.8% (Fig. 1) at 26 °C water temperature. This level fall in the range of the previously reported dietary protein requirements of some other catfish species such as young H. fossilis 40–43% (Siddiqui and Khan 2009), Cyprinus carpio 41.25% (Ahmed and Maqbool 2017), higher than that for walking catfish, Clarias batrachus 36% (Singh et al. 2009), spotted snake-head, Channa punctatus 40% (Zehra and Khan 2012), and marbled spinefoot rabbitfish, Siganus rivulatus 40% (El-Dakar et al. 2011) and is lower than the requirements reported for African catfish, Clarias gariepinus 43% (Farhat 2011), Mystus nemurus 42% (Khan et al. 1993), Malaysian catfish, bagrid catfish, Mystus nemurus 44% (Ng et al. 2001), and striped murrel, Channa striatus 55% (Kumar et al. 2010).
The PD g/fish increased progressively with the increase in dietary protein up to 40% for the groups held at 18 °C and up to 44% for the groups held at 26 °C water temperatures, respectively. The PD/fish value attained by the groups reared at 18 °C was somewhat lower than that attained by the groups at the same level of dietary protein at 26 °C. This may probably be due to the reason that an increase in temperature at 26 °C might have increased the activity of digestive enzymes accelerating digestion of the nutrients, thus resulting in better growth (Shcherbina and Kazlauskene 1971) in the form of deposited protein. Hilge (1985) found that the optimum temperature for best growth of European catfish, Silurus glanis was almost within the range of 18 to 26 °C with best results noted at 27 °C. Brown et al. (1989) reported a 40% increase in growth rate of cod reared at 8.3 °C compared with 4.5 °C. This value was similar to that of Otterle et al. (1994), who reported an increase in growth rate of about 50% with each 4 °C increase in temperature between 6 and 14 °C. Protein deposition in this study was found to decrease for the groups fed dietary protein above 44% in diets D48 (48%) and D52 (52%) irrespective of the water temperatures. Proteins represent a very important source of energy in fishes. Since teleosts have developed the capacity for converting amino acid to glucose (Bever et al. 1981) by gluconeogenesis which is utilized for energy production through TCA cycle intermediates (Kumar 1999), it is reasonable to assume that the decline in protein deposition at higher levels of dietary protein for the groups fed diets D48 (48% protein) and D52 (52% protein) may probably be due to catabolism of excess protein for energy purposes thus reducing its deposition for tissue building or growth.
There are conflicting findings about the effect of dietary protein levels on the efficiency of protein utilization in the literatures. Lee et al. (2001) reported an increase in protein utilization efficiency with the increased intake of dietary protein by the fish, whereas Duan et al. (2001) and Lee et al. (2003) did not find any significant influence of dietary protein on efficiency of protein utilization. However, Kim et al. (2001), Kim and Lee (2009), and Gullu et al. (2008) pointed out a decrease in protein utilization with increasing dietary protein above optimum level which is in agreement with the present results. Davis and Stickney (1978) stated that fish convert protein more efficiently when fed dietary protein level less than optimal level that yields the maximum growth and feed efficiency. Steffens (1981) also reported that raising the dietary protein level improves the growth rate and food conversion but reduces the protein productive value in Salmo gairdneri and Cyprinus carpio. Similar findings were evident in this study where fish fed 48% and more protein manifested reduction in protein deposition.
The protein requirements or protein utilization of fish is also influenced by dietary non-protein energy levels (Dias et al. 1998; Lupatsch et al. 2001; Tibbetts et al. 2005; Wang et al. 2006). Hence, it is possible to reduce the dietary protein level to a certain degree by increasing non-protein energy and directing protein to growth rather than energetic use in a number of fish species (Forster and Hardy 2000). A protein-sparing effect is generally more pronounced at low protein levels rather than high levels (Dias et al. 1998; Tibbetts et al. 2005) mainly because of the preferential use of protein as an energy source by fish at high protein levels (Tibbetts et al. 2005). Cowey (1979) has also suggested that any change in dietary energy content changes the optimal protein requirement of the fish. Although in this study, the diets were formulated to be isocaloric and the digestible energy content of the diets was not significantly different (P > 0.05) among treatments, protein deposition decreased slightly with the increasing protein content and thus growth appears to be affected more by dietary protein levels than by energy levels. As per NRC (1993), the optimum P/DE values for fish range between 17 and 26 mg protein/kJ DE which in the present study, also almost corresponds to diets with 44% protein (27.21–27.88 mg protein/kJ DE) at 26 °C. Therefore, in this study, highest protein deposition with 44% dietary protein at 26 °C may be due to balanced P/DE ratio at this level of dietary protein.
H. fossilis fed intermediate levels of dietary protein (36–44%) exhibited higher feed intake than those fed still higher levels of protein in the diet (48–52%). This may probably be due to the reasons that fish fed nutrient-deficient diets usually increase the feed intake to meet the protein or the energy needs. Since the diets in this study were formulated to be isoenergetic, it is plausible that the fish fed intermediate levels of dietary protein might have consumed more feed in order to meet their protein requirements.
Temperature affects the body composition by altering feed intake (Jobling 1997) and various studies have shown that body protein is significantly affected by temperature (Cui and Wootton 1988; Koskela et al. 1997; Bendiksen et al. 2003; Tidwell et al. 2003). In this study as well, in addition to dietary protein levels, temperature also had significant influence on body protein and moisture contents of fry H. fossilis. The carcass quality of fish in terms of carcass protein content attained its superior value for the groups fed with 44% dietary protein at 26 °C temperature. Fish fed diets containing 28–36% protein tended to deposit more fat than those fed 40, 44, 48, and 52% dietary protein. In diets D28, D32, and D36, the carbohydrate contents increased at the expense of dietary protein which might have participated in de novo lipid synthesis from carbohydrate. Since these diets contain an improper ratio of protein to energy, this might have led to deposition of body fat from dietary carbohydrates.
Water temperature is one of the most important ecological factors that significantly influence some physiological process of fish such as growth, metabolism, and blood values. As has been shown in Table 4, hematocrit and hemoglobin concentrations were significantly (P < 0.05) altered by different water temperatures, However, interactive effects of dietary protein and temperature were not found. The results of this study are in line of the results reported by Koeypudsa and Jongjareanjai (2010) for hybrid catfish. Data related to hematological parameters in this study indicates that to sustain normal physiological processes in the body, H. fossilis should be held at 26 °C water temperature.
To study the effect of water temperature on the protein requirements of fish, Daniels and Robinson (1986) conducted two independent studies in which the red drum, Sciaenops ocellatus were maintained at 22–26 °C water temperature in the first and at 26–33 °C in the second. According to the authors, fish reared at lower temperature required less protein (35%) than those at higher temperature (44%). It is considered that water temperature affects feed intake and feed conversion efficiency (NRC 1993). Therefore, it is reasonable to assume that the suboptimal temperature in the present study might have deviated the feed intake in H. fossilis held at 18 °C and may be one of the reasons for reduced growth performance in groups held at this temperature; even if fed with the same dietary protein level.
Water temperature has substantial effect on fish metabolism. In response to decrease in water temperature, the enzyme activity of tissues increases (Hochachka and Somero 1984). In a stressful and unfavorable environmental condition ALT and AST activities may increase in blood serum. In the present study, serum ALT and AST levels were affected by different water temperatures. Serum ALT and AST amount in different fish fed varying levels of dietary protein at 26 °C are comparatively lower and attained normal physiological range at requirement level (44% dietary protein) than those fed at 18 °C. These results clearly indicated that 26 °C is the favorable water temperature for better growth of H. fossilis fry.
Survival rate in this study were not affected significantly by different levels of dietary protein but was more affected by temperature as the groups held at 18 °C had significant (P < 0.05) mortality at lowest level of dietary protein. On the other hand, mortality at 26 °C water temperature was not recoded even at the lowest level of dietary protein. The results are in line with various other finfish and shellfish studies (Li et al. 2011; Sun et al. 2015; Abdelrahman et al. 2019). Azaza et al. (2008) described that the survival rate of Nile tilapia, Oreochromis niloticus was significantly lower when it was reared at lower and upper level of its optimum water temperature.
Stress resistance of the fish in different life period is affected by levels of salinity, temperature, environment, and nutrition (Jalali et al. 2008; Gholami 2010). The results of present study showed that resistance rate to thermal stress significantly higher (P < 0.05) in fish fed dietary protein at 26 °C water temperature who were able to withstand temperature challenge for longer duration (Table 5) than those fish fed at 18 °C water temperature which were found to be more prone to temperature challenge test and exhibited mortality in comparatively less time.
Based on the 95% maximum response of second-degree polynomial regression analyses of PD g/fish data, it is recommended that fry H. fossilis could perform well if fed with dietary protein levels between 40.8 and 41.8% with a P/DE ratio of 27.21–27.88 mg protein/kJ DE at 26 °C water temperature. This study also corroborates that the performance of the fish and protein requirement was strictly governed by the rearing temperature as fish reared at 18 °C water temperature could not perform well in terms of growth, feed conversion, and protein deposition even if fed with the same level of dietary protein.
The information developed in the present study could be utilized for optimizing the growth potential of this fish by making better utilization of the nutrient at the 26 °C, the required temperature optima. The finding of the present study would further be useful for effective management strategies for the mass culture of this highly preferred fish species.
All datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
ALT:
Alanine amino transferase
AST:
Aspartate amino transferase
Condition factor
Digestible energy
FCR:
Feed conversion ratio
GE:
Gross energy
Hb:
Hct:
HSI:
Hepatosomatic index
kJ:
Kilo joule
PD:
Protein deposition
RBCs:
TGC:
Thermal growth coefficient
VSI:
Viscerosomatic index
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The authors are grateful to the Dean, Dr. Afaf Mohammad Babeer, Farasan University College, Jazan University, Jazan, KSA for providing necessary facilities for electronic submission of the manuscript and also for providing a highly conducive work environment. The authors are also grateful to the Head, Department of Zoology, University of Kashmir, Hazratbal, Srinagar, India for providing the laboratory facilities to carry out the experimental work and to Mr. Mufti Buhran, University Chief Executive Engineer for helping the construction of new Feed Technology Laboratory (Wet-Laboratory) in the Department of Zoology. Thanks are also due to Prof. Dr. Nazni Peer Khan, Professor & Head, Department of Nutrition & Dietetics, Periyar University, Salem, TN, India for data analyses with two-way ANOVA.
This study was not funded by any Government or Private sources.
Department of Nursing, Farasan University College, Farasan, Jazan University, Jizan, Kingdom of Saudi Arabia
Shabihul Fatma
Department of Zoology, University of Kahsmir, Hazratbal, Srinagar, Jammu and Kashmir, 190006, India
Imtiaz Ahmed
SFS has conducted the experiments, data analyses and manuscript writing and IA has provided the laboratory facilities for feeding trials and sample analyses. He has also contributed in revision of the final manuscript. All authors read and approved the final manuscript.
Correspondence to Shabihul Fatma.
Experimental protocols followed the guidelines of the Animal Care and Use Committee of Central University of Kashmir, J&K, India.
Fatma, S., Ahmed, I. Effect of water temperature on protein requirement of Heteropneustes fossilis (Bloch) fry as determined by nutrient deposition, hemato-biochemical parameters and stress resistance response. Fish Aquatic Sci 23, 1 (2020). https://doi.org/10.1186/s41240-020-0147-y
Accepted: 06 January 2020
DOI: https://doi.org/10.1186/s41240-020-0147-y
Heteropneustes fossilis
Metabolic enzymes
Nutrition and Physiology | CommonCrawl |
Pages.6401-6406
Asian Pacific Organization for Cancer Prevention (아시아태평양암예방학회)
Diagnostic Performance of Diffusion - Weighted Imaging for Multiple Hilar and Mediastinal Lymph Nodes with FDG Accumulation
Usuda, Katsuo (Department of Thoracic Surgery, Kanazawa Medical University) ;
Maeda, Sumiko (Department of Thoracic Surgery, Kanazawa Medical University) ;
Motono, Nozomu (Department of Thoracic Surgery, Kanazawa Medical University) ;
Ueno, Masakatsu (Department of Thoracic Surgery, Kanazawa Medical University) ;
Tanaka, Makoto (Department of Thoracic Surgery, Kanazawa Medical University) ;
Machida, Yuichiro (Department of Thoracic Surgery, Kanazawa Medical University) ;
Matoba, Munetaka (Department of Radiology, Kanazawa Medical University) ;
Watanabe, Naoto (Department of Radiology, Kanazawa Medical University) ;
Tonami, Hisao (Department of Radiology, Kanazawa Medical University) ;
Ueda, Yoshimichi (Department of Pathophysiological and Experimental Pathology, Kanazawa Medical University) ;
Sagawa, Motoyasu (Department of Thoracic Surgery, Kanazawa Medical University)
https://doi.org/10.7314/APJCP.2015.16.15.6401
Background: It is sometimes difficult to assess patients who have multiple hilar and mediastinal lymph nodes (MHMLN) with FDG accumulation in PET-CT. Since it is uncertain whether diffusion-weighted magnetic resonance imaging (DWI) is useful in the assessment of such patients, its diagnostic performance was assessed. Materials and Methods: Twenty-three patients who had three or more stations of hilar and mediastinal lymph nodes with SUVmax of 3 or more in PET-CT were included in this study. Results: For diagnosis of disease, there were 20 malignancies (lung cancers 17, malignant lymphomas 2 and metastatic lung tumor 1), and 3 benign cases (sarcoidosis 2 and benign disease 1). For diagnosis of lymph nodes, there were 7 malignancies (metastasis of lung cancer 7 and malignant lymphoma 1) and 16 benign lymphadenopathies (pneumoconiosis/silicosis 7, sarcoidosis 4, benign disease 4, and atypical lymphocyte infiltration 1). The ADC value ($1.57{\pm}0.29{\times}10^{-3}mm^2/sec$) of malignant MHMLN was significantly lower than that ($1.99{\pm}0.24{\times}10^{-3}mm^2/sec$) of benign MHMLN (P=0.0437). However, the SUVmax was not significantly higher ($10.0{\pm}7.34$ as compared to $6.38{\pm}4.31$) (P=0.15). The sensitivity (86%) by PET-CT was not significantly higher than that (71%) by DWI for malignant MHMLN (P=1.0). The specificity (100%) by DWI was significantly higher than that (31%) for benign MHMLN (P=0.0098). Furthermore, the accuracy (91%) with DWI was significantly higher than that (48%) with PET-CT for MHMLN (P=0.0129). Conclusions: Evaluation by DWI for patients with MHMLN with FDG accumulation is useful for distinguishing benign from malignant conditions.
Diffusion-weighted imaging;magnetic resonance imaging;PET;lymph nodes
Supported by : Ministry of Education, Culture, Sports, Science and Technology
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\begin{document}
\title{Cones of traces arising from AF C*-algebras}
\author{Mark Moodie} \author{Leonel Robert}
\begin{abstract} We characterize the topological non-cancellative cones that are expressible as projective limits of finite powers of $[0,\infty]$. These are also the cones of lower semicontinuous extended-valued traces on AF C*-algebras. Our main result may be regarded as a generalization of the fact that any Choquet simplex is a projective limit of finite dimensional simplices. To obtain our main result, we first establish a duality between certain non-cancellative topological cones and Cuntz semigroups with real multiplication. This duality extends the duality between compact convex sets and complete order unit vector spaces to a non-cancellative setting. \end{abstract}
\maketitle
\section{Introduction} By a theorem of Lazar and Lindenstrauss, any Choquet simplex can be expressed as a projective limit of finite dimensional simplices (see \cite{lazar-lindenstrauss}, \cite{EHS}). This has implications for C*-algebras: given a Choquet simplex $K$, there exists a simple, unital, approximately finite dimensional (AF) C*-algebra whose set of tracial states is isomorphic to $K$ (\cite{blackadar, EHS}). In the investigations on the structure of a C*-algebra, another kind of trace is also of interest, namely, the lower semicontinuous traces with values in $[0,\infty]$. These traces form a non-cancellative topological cone. (By cone we understand an abelian monoid endowed with a scalar multiplication by positive scalars.) Our goal here is to characterize through intrinsic properties the topological cones arising as the lower semicontinuous $[0,\infty]$-valued traces on an AF C*-algebra. These are also the projective limits of cones of the form $[0,\infty]^n$, with $n\in\mathbb{N}$, and also, the cones arising as the $[0,\infty]$-valued monoid morphisms on the positive elements of a dimension group.
Let $A$ be C*-algebra. Denote its cone of positive elements by $A_+$. A map $\tau\colon A_+\to [0,\infty]$ is called a trace if it is linear (additive, homogeneous, mapping 0 to 0) and satisfies that $\tau(x^*x)=\tau(xx^*)$ for all $x\in A$. We are interested in the lower semicontinuous traces. Let $T(A)$ denote the cone of $[0,\infty]$-valued lower semicontinuous traces on $A_+$. By the results of \cite{ERS}, $T(A)$ is a complete lattice when endowed with the algebraic order, and addition in $T(A)$ is distributive with respect to the lattice operations. Further, one can endow $T(A)$ with a topology that is locally convex, compact and Hausdorff. We call an abstract topological cone with these properties an \emph{extended Choquet cone} (see Section \ref{sectionECCs}).
By an AF C*-algebra we understand an inductive limit, over a possibly uncountable index set, of finite dimensional C*-algebras. Not every extended Choquet cone arises as the cone of lower semicontinuous traces on an AF C*-algebra. The requisite additional properties are sorted out in the theorem below.
An element $w$ in a cone is called idempotent if $2w=w$. Given a cone $C$, we denote by $\mathrm{Idem}(C)$ the set of idempotent elements of $C$.
\begin{theorem}\label{mainchar} Let $C$ be an extended Choquet cone (see Definition \ref{defExtCho}). The following are equivalent: \begin{enumerate}[\rm (i)] \item $C$ is isomorphic to $T(A)$ for some AF C*-algebra $A$. \item $C$ is isomorphic to $\mathrm{Hom}(G_+,[0,\infty])$ for some dimension group $(G,G_+)$. (Here $\mathrm{Hom}(G_+,[0,\infty])$ denotes the set of monoid morphisms from $G_+$ to $[0,\infty]$.)
\item $C$ is a projective limit of cones of the form $[0,\infty]^n$, $n\in \mathbb{N}$.
\item $C$ has the following properties: \begin{enumerate} \item $\mathrm{Idem}(C)$ is an algebraic lattice under the opposite algebraic order, \item for each $w\in \mathrm{Idem}(C)$, the set $\{x\in C:x\leq w\}$ is connected. \end{enumerate} \end{enumerate} Moreover, if $C$ is metrizable and satisfies (iv), then the C*-algebra in (i) may be chosen separable, the group $G$ in (ii) may be be chosen countable, and the projective limit in (iii) may be chosen over a countable index set. \end{theorem}
We refer to property (a) in part (iv) as ``having an abundance of compact idempotents". The fact that the primitive spectrum of an AF C*-algebra has a basis of compact open sets makes this condition necessary. We call property (b) ``strong connectedness". The existence of a non-trivial trace on every simple ideal-quotient of an AF C*-algebra makes this condition necessary. In general, if a C*-algebra $A$ is such that its primitive spectrum has a basis of compact open sets, and every simple quotient $I/J$, where $J\subsetneq I$ are ideals of $A$, has a non-zero densely finite trace, then $T(A)$ has an abundance of compact idempotents and is strongly connected, i.e., properties (a) and (b) above hold. For example, if $A$ has real rank zero, stable rank one, and is exact, then these conditions are met. Theorem \ref{mainchar} then asserts the existence of an AF C*-algebra $B$ such that $T(A)\cong T(B)$.
The crucial implication in Theorem \ref{mainchar} is (iv) implies (iii). A reasonable approach to proving it is to first prove that (iv) implies (ii) by directly constructing a dimension group $G$ from the cone $C$, very much in the spirit of the proof of the Lazar-Lindenstrauss theorem obtained by Effros, Handelmann, and Shen in \cite{EHS} (which, unlike the proof in \cite{lazar-lindenstrauss}, also deals with non-metrizable Choquet simpleces). If the cone $C$ is assumed to be finitely generated, then we indeed obtain a direct construction of an ordered vector space with the Riesz property $(V,V^+)$ such that $\mathrm{Hom}(V^+,[0,\infty])$ is isomorphic to $C$. This is done in the last section of the paper. In the general case, however, such an approach has eluded us.
To prove Theorem \ref{mainchar} we first establish a duality between extended Choquet cones with an abundance of compact idempotents and certain abstract Cuntz semigroups. Briefly stated, this duality works as follows: \[ C\mapsto \mathrm{Lsc}_\sigma(C)\hbox{ and }S\mapsto F(S). \] That is, to an extended Choquet cone $C$ with an abundance of compact idempotents one assigns the Cu-cone $\mathrm{Lsc}_\sigma(C)$ of lower semicontinuous linear functions $f\colon C\to [0,\infty]$ with ``$\sigma$-compact support". In the other direction, to a Cu-cone $S$ with an abundance of compact ideals one assigns the cone of functionals $F(S)$; see Section \ref{duality} and Theorem \ref{dualitythm}. In the context of this duality, strong connectedness in $C$ translates into the property of weak cancellation in $\mathrm{Lsc}_\sigma(C)$. We then use this arrow reversing duality to turn the question of finding a projective limit representation for a cone into one of finding an inductive limit representation for a Cu-cone. To achieve the latter, we follow the strategy of proof of the Effros-Handelmann-Shen theorem, adapted to the category at hand. The main technical complication here is the non-cancellative nature of Cu-cones, but this is adequately compensated by the above mentioned property of ``weak cancellation" (dual to strong connectedness).
A question that is closely related to the one addressed by Theorem \ref{mainchar} asks for a characterization of the lattices arising as (closed two-sided) ideal lattices of AF C*-algebras. For separable AF C*-algebras, this problem was solved by Bratteli and Elliott in \cite{bratteli-elliott}, and independently by Bergman in unpublished work: Any complete algebraic lattice with a countable set of compact elements is the lattice of closed two-sided ideals of a separable AF C*-algebra. A thorough discussion of this result is given by Goodearl and Wehrung in \cite{goodearl-wehrung}. The cardinality restriction on the set of compact elements is necessary, as demonstrated by examples of R\r{u}\v{z}i\v{c}ka and Wehrung (\cite{ruzicka}, \cite{wehrungexample}). Now, the lattice of closed two-sided ideals of a C*-algebra $A$ is in order reversing bijection with the lattice of idempotents of $T(A)$ via the assignment $I\mapsto \tau_I$, where $\tau_I$ is the $\{0,\infty\}$-valued trace vanishing on $I_+$. Thus, the realization of a cone $C$ in the form $T(A)$ entails the realization of $(\mathrm{Idem}(C),\leq^{\mathrm{op}})$ as the ideal lattice of $A$. Curiously, no cardinality restriction is needed in Theorem \ref{mainchar} above. This demonstrates that the examples of R\r{u}\v{z}i\v{c}ka and Wehrung are also examples of algebraic lattices that cannot be realized as the lattice of idempotents of a cone $C$ satisfying any of the equivalent conditions of Theorem \ref{mainchar}.
This paper is organized as follows: In Section \ref{sectionECCs} we define extended Choquet cones and prove a number of background results on their structure. In Section \ref{sec:conesfromfunctionals} we go over three constructions---starting from a C*-algebra, a dimension group, and a Cu-semigroup---yielding extended Choquet cones that are strongly connected and have an abundance of compact idempotents. Sections \ref{functionspaces} and \ref{duality} delve into spaces of linear functions on extended Choquet cones with an abundance of compact idempotents. In Theorem \ref{dualitythm} we establish the above mentioned duality assigning to a cone $C$ the Cu-cone $\mathrm{Lsc}_\sigma(C)$, and conversely to a Cu-cone $S$ its cone of functionals $F(S)$. In Section \ref{proofofmainchar} we prove Theorem \ref{mainchar}. In Section \ref{fingen} we assume that the cone $C$ is finitely generated. In this case we give a direct construction of an ordered vector space with the Riesz property $(V,V^+)$ such that $C\cong \mathrm{Hom}(V^+,[0,\infty])$. The vector space $V$ is described as $\mathbb{R}$-valued functions on a certain spectrum of the cone $C$.
\textbf{Acknowledgement}: The second author thanks Hannes Thiel for fruitful discussions on the topic of topological cones and for sharing his unpublished notes \cite{thiel}.
\section{Extended Choquet Cones}\label{sectionECCs}
\subsection{Algebraically ordered compact cones} We call cone an abelian monoid $(C,+)$ endowed with a scalar multiplication by positive real numbers $(0,\infty)\times C\to C$ such that \begin{enumerate}[\rm (i)] \item the map $(t,x)\mapsto tx$ is additive on both variables, \item $s(tx)=(st)x$ for all $s,t\in (0,\infty)$ and $x\in C$, \item $1\cdot x=x$ for all $x\in C$. \end{enumerate} We do not assume that the addition operation on $C$ is cancellative. In fact, the primary example of the cones that we investigate below is $[0,\infty]$ endowed with the obvious operations.
The algebraic pre-order on $C$ is defined as follows: $x\leq y$ if there exists $z\in C$ such that $x+z=y$. We say that $C$ is algebraically ordered if this pre-order is an order.
We call $C$ a topological cone if it is endowed with a topology for which the operations of addition and multiplication by positive scalars are jointly continuous.
\begin{definition}\label{defExtCho} An algebraically ordered topological cone $C$ is called an extended Choquet cone if \begin{enumerate}[\rm (i)] \item $C$ is a lattice under the algebraic order, and the addition operation is distributive over both $\wedge$ and $\vee$: \begin{align*} x + (y\wedge z) &=(x+y)\wedge (x+z),\\ x + (y\vee z) &=(x+y)\vee (x+z), \end{align*} for all $x,y,z\in C$, \item the topology on $C$ is compact, Hausdorff, and locally convex, i.e., it has a basis of open convex sets. \end{enumerate} \end{definition}
\begin{remark} It is a standard result that in a compact algebraically ordered monoid both upward and downward directed sets converge to their supremum and infimum, respectively (\cite[Proposition 3.1]{edwards}, \cite[Proposition VI-1.3, p441]{GHK}). We shall make frequent use of this fact applied to extended Choquet cones. It readily follows from this and the existence of finite suprema and infima that extended Choquet cones are complete lattices. \end{remark}
\begin{remark} By Wehrung's \cite[Theorem 3.11]{wehrung}, the algebraic and order theoretic properties of an extended Choquet cone may be summarized as saying that it is an injective object in the category of positively ordered monoids.
\end{remark}
\begin{example} The set $[0,\infty]$ is an extended Choquet cone when endowed with the standard operations of addition and scalar multiplication and the standard topology. More generally, the powers $[0,\infty]$, endowed with coordinatewise operations and the product topology are extended Choquet cones. \end{example}
Let $C$ and $D$ be extended Choquet cones. A map $\phi\colon C\to D$ is a morphism in the extended Choquet cones category if $\phi$ is linear (additive, homogeneous with respect to scalar multiplication, and mapping 0 to 0) and continuous.
\begin{theorem} The category of extended Choquet cones has projective limits. \end{theorem}
\begin{proof} Let $\{C_i:i\in I\}$, $\{\varphi_{i,j}\colon C_i\to C_j:i,j\in I \hbox{ with }j\leq i\}$, be a projective system of extended Choquet cones, where $I$ is an upward directed set. Define \[ C=\bigg\{(x_i)_{i}\in \prod_{i\in I} C_i: x_{j}=\varphi_{i,j}(x_i)\hbox{ for all }i,j\in I\hbox{ with }j\leq i\bigg\}. \]
Endow the product $\prod_{i\in I} C_i$ with coordinatewise operations, coordinatewise order, and with the product topology; endow $C$ with the topological cone structure induced by inclusion. Let $\pi_i\colon C\to C_i$, $i\in I$, denote the projection maps. It follows from well known arguments that $\{C,\pi_i|_C:i\in I\}$ is the projective limit of the system $\{C_i,\phi_{i,j}:i,j\in I\}$ as compact Hausdorff topological cones (cf. \cite[Theorem 13]{davies}). Since for each $i$ the topology of $C_i$ has a basis of open convex sets, the product topology on $\prod_{i} C_i$ also has a basis of open convex sets. Further, since $C$ is a convex subset of $\prod_i C_i$, the induced topology on $C$ is locally convex as well.
Let us now prove that $C$ is a lattice. The proof runs along the same lines as the one in \cite[Theorem 13]{davies} for projective limits of Choquet simplices. We show that $C$ is closed under finite suprema; the argument for finite infima is similar. Let $x=(x_i)_i$ and $y=(y_i)_i$ be in $C$. Their coordinatewise supremum exists in $\prod_{i} C_i$, but does not necessarily belong to $C$. For each $k\in I$ define $z^{(k)}\in \prod_i C_i$ by \[ (z^{(k)})_i=\begin{cases} \phi_{k,i}(x_k\vee y_k)& \hbox{if }i\leq k,\\ x_i\vee y_i &\hbox{otherwise}. \end{cases} \] If $k'\geq k$, then \[ \phi_{k',k}(x_{k'}\vee y_{k'})\geq \phi_{k',k}(x_{k'})=x_k, \] and similarly $\phi_{k',k}(x_{k'}\vee y_{k'})\geq y_k$, whence $\phi(x_{k'}\vee y_{k'})\geq x_k\vee y_k$. It follows that \[ (z^{(k')})_i=\phi_{k',i}(x_{k'}\vee y_{k'})\geq \phi_{k,i}(x_k\vee y_k)=(z^{(k)})_i, \] for $i\leq k$, while \[ (z^{(k')})_i\geq x_i\vee y_i=(z^{(k)})_i \] for $i\nleq k$. Thus, $(z^{(k)})_{k\in I}$ is an upward directed net. Set $x\vee y:=\lim_k z^{(k)}$, which is readily shown to belong to $C$. Then $x\vee y\geq z^{(k)}\geq x,y$ for all $k$. Suppose that $w=(w_i)_i\in C$ is such that $w\ge x,y$. Then $w_i\ge x_i\vee y_i$ for all $i$, and further \[ w_i=\varphi_{k,i}(w_k)\ge\varphi_{k,i}(x_k\vee y_{k}). \] Hence, $w\ge z^{(k)}$ for all $k$, and so $w\geq x\vee y$. This proves that $x\vee y$ is in fact the supremum of $x$ and $y$ in $C$.
Let us prove distributivity of addition over $\vee$. Let $x,y,v\in C$. Fix an index $i$. Then \begin{align*} ((x+v)\vee (y+v))_i &=\lim_k \phi_{k,i}((x_k+v_k)\vee (y_k+v_k))\\ &=\lim_{k} \phi_{k,i}((x_k\vee y_k)+v_k)\\ &=\lim_{k}\phi_{k,i}(x_k\vee y_k)+v_i\\ &=(x\vee y+v)_i, \end{align*} where we have used the distributivity of addition over $\vee$ on each coordinate and the construction of joins in $C$ obtained above. Thus, $(x+v)\vee (y+v)=(x\vee y)+v$. Distributivity over $\wedge$ is handled similarly. \end{proof}
\subsection{Lattice of idempotents} Throughout this subsection $C$ denotes an extended Choquet cone.
An element $w\in C$ is called idempotent if $2w=w$. It follows, using that $C$ is algebraically ordered, that $t w=w$ for all $t\in (0,\infty]$. We denote the set of idempotents of $C$ by $\mathrm{Idem}(C)$. The set $\mathrm{Idem}(C)$ is a sub-lattice of $C$: if $w_1$ and $w_2$ are idempotents then \[ 2(w_1\vee w_2)=(2w_1\vee 2w_2)=w_1\vee w_2, \] where we have used that multiplication by $2$ is an order isomorphism. Hence, $w_1\vee w_2$ is an idempotent. Similarly, $w_1\wedge w_2$ is shown to be an idempotent. Moreover, $w_1\vee w_2=w_1+w_2$, a fact easily established.
In the lattice $\mathrm{Idem}(C)$, we use the symbol $\gg$ to denote the way-below relation under the opposite order. That is, $w_1\gg w_2$ if whenever $\inf_i v_i\leq w_2$ for a decreasing net $(v_i)_i$ in $\mathrm{Idem}(C)$, we have $ v_{i_0}\leq w_1$ for some $i_0$. We call $w\in \mathrm{Idem}(C)$ a compact idempotent if $w\gg w$. More explicitly, $w$ is compact if whenever $\inf_i v_i\leq w$ for a decreasing net $(v_i)_i$ in $\mathrm{Idem}(C)$, we have $v_{i_0}\leq w$ for some $i_0$. Note: we only use the notion of compact element in $\mathrm{Idem}(C)$ in the sense just defined, i.e., applied to the \emph{opposite order}.
A complete lattice is called algebraic if each of its elements is a supremum of compact elements (\cite[Definition I-4.2]{GHK}).
\begin{definition} We say that an extended Choquet cone $C$ has an abundance of compact idempotents if $(\mathrm{Idem}(C), \leq^{\mathrm{op}})$ is an algebraic lattice, i.e., every idempotent in $C$ is an infimum of compact idempotents. \end{definition}
Let $x\in C$. Consider the set $\{z\in C:x+z=x\}$. This set is closed under addition and also closed in the topology of $C$. It follows that it has a maximum element $\epsilon(x)$. Since $2\cdot \epsilon(x)$ is also absorbed additively by $x$, we have $\epsilon(x)=2\epsilon(x)$, i.e., $\epsilon(x)$ is an idempotent. We call $\epsilon(x)$ the support idempotent of $x$.
\begin{lemma}\label{supportlemma}(Cf. \cite[Lemma 3.2]{edwards}) Let $x,y,z\in C$. \begin{enumerate}[\rm (i)] \item $\epsilon(x) = \lim_n\frac{1}{n}x$. \item If $x+z\leq y+z$ then $x+\epsilon(z)\leq y+\epsilon(z)$. \end{enumerate} \end{lemma} \begin{proof} (i) Observe that $w:=\lim_n\frac{1}{n}x$ exists, since the infimum of a decreasing sequence is also its limit. It is also clear that $2w=w$, and that $x+w=x$. Let $z\in C$ be such that $x+z=x$. Then $x+nz=x$, i.e., $\frac1n x+ z=\frac1nx$, for all $n\in \mathbb{N}$. Letting $n\to \infty$, we get that $w+z=w$, and in particular, $w\leq z$. Thus, $w$ is the largest element absorbed by $x$, i.e. $w=\epsilon(x)$.
(ii) This is \cite[Lemma 3.2]{edwards}. Here is the argument: We deduce, by induction, that $nx+z\leq ny+z$ for all $n\in \mathbb{N}$. Hence, $x+\frac1nz\leq y+\frac1nz$. Letting $n\to \infty$ and using (i), we get $x+\epsilon(z)\leq y+\epsilon(z)$. \end{proof}
\begin{lemma}\label{supportmax} Let $K\subseteq C$ be closed and convex. Then the map $x\mapsto \epsilon(x)$ attains a maximum on $K$. \end{lemma} \begin{proof} Let $W=\{\epsilon(x):x\in K\}$. Let $x_1,x_2\in K$, with $\epsilon(x_1)=w_1$ and $\epsilon(x_2)=w_2$. Since $K$ is convex, $(x_1+x_2)/2\in K$. Since \[ \epsilon\Big(\frac{x_1+x_2}{2}\Big)=\lim_n \frac{1}{2n}x_1+\frac{1}{2n}x_2=\epsilon(x_1)+\epsilon(x_2), \] the set $W$ is closed under addition. For each $w\in W$, let us choose $x_w\in K$ with $\epsilon(x_w)=w$. By compactness of $K$, the net $(x_w)_{w\in W}$ has a convergent subnet. Say $x_{h(\lambda)}\to x\in K$, where $h\colon \Lambda\to W$ is increasing and with cofinal range. For each $\lambda$ we have $x_{h(\lambda')}+h(\lambda)=x_{h(\lambda')}$ for all $\lambda'\geq \lambda$. Passing to the limit in $\lambda'$ we get $x+h(\lambda)=x$. Since $h(\lambda)$ ranges through a cofinal set in $W$, $x+w=x$ for all $w\in W$. Thus, $\epsilon(\cdot)$ attains its maximum on $W$ at $x$. \end{proof}
\begin{lemma}\label{opensetsupport} For each idempotent $w\in C$ the set $\{x\in C:w\gg \epsilon(x)\}$ is open. (Recall that $\gg$ is the way below relation in the lattice $(\mathrm{Idem}(C),\leq^{\mathrm{op}})$.) \end{lemma}
\begin{proof} Let $x\in C$ be such that $w\gg \epsilon(x)$. By Lemma \ref{supportmax}, for each closed convex neighborhood $K$ of $x$, there exists $x_K\in K$ at which $\epsilon(\cdot)$ attains its maximum. By the local convexity of $C$, the system of closed convex neighborhoods of $x$ is downward directed. It follows that $(\epsilon(x_K))_K$ is downward directed. Moreover, $x_K\to x$, since the topology is Hausdorff. We claim that $\epsilon(x)=\inf_K \epsilon(x_K)$, where $K$ ranges through all the closed convex neighborhoods of $x$. Proof: Set $y=\inf_K \epsilon(x_K)$. We have $y\leq \epsilon(x_K)\leq \frac{x_K}{n}$ for all $K$ and $n\in \mathbb{N}$. Passing to the limit, first in $K$ and then in $n$, we get that $y\leq \epsilon(x)$. On the other hand, $\epsilon(x)\leq \epsilon(x_K)$ for all $K$ (since $x\in K$ and $\epsilon$ attains its maximum on $K$ at $x_K$). Thus, $\epsilon(x)\leq y$, proving our claim.
We have $w\gg \epsilon(x)=\inf_K \epsilon(x_K)$. Hence, there is $K$ such that $w\gg \epsilon(x_K)$. So, there is a neighborhood of $x$ all whose members belong to $\{z\in C:w\gg \epsilon(z)\}$. This shows that $\{z\in C:w\gg \epsilon(z)\}$ is open. \end{proof}
\subsection{Cancellative subcones} Fix an idempotent $w\in C$. Let \[ C_w=\{x\in C:\epsilon(x)=w\}. \] Then $C_w$ is closed under sums, scalar multiplication by positive scalars, finite infima, and finite suprema. By Lemma \ref{supportlemma} (ii), $C_w$ is also cancellative: $x+z\leq y+z$ implies that $x\leq y$ for all $x,y,z\in C_w$. It follows that $C_w$ embeds in a vector space; namely, the abelian group of formal differences $x-y$, with $x,y\in C_w$ endowed with the unique scalar multiplication extending the scalar multiplication on $C_w$. Let $V_w$ denote the vector space of differences $x-y$, with $x,y\in C_w$. Let $\eta\colon C_w\times C_w\to V_w$ be defined by $\eta(x,y)=x-y$. We endow $C_w$ with the topology that it receives as a subset of $C$. We endow $V_w$ with the quotient topology coming from the map $\eta$.
\begin{theorem}\label{compactbase} Let $w\in \mathrm{Idem}(C)$ be a compact idempotent. Then $V_w$ is a locally convex topological vector space whose topology restriced to $C_w$ agrees with the topology on $C_w$. Moreover, either $C_w=\{w\}$ or $C_w$ has a compact base. \end{theorem} Note: A subset $B$ of a cone $T$ is called a base if for each nonzero $x\in T$ the intersection of $(0,\infty)\cdot x$ with $B$ is a singleton set. \begin{proof} Let us first show that the topology on $C_w$ is locally compact. Since $w$ is compact, the set $\{x\in C:w\geq \epsilon(x)\}$ is open by Lemma \ref{opensetsupport}. We then have that $C_w$ is the intersection of the closed set $\{x\in C:w\leq x\}$ and the open set $\{x:w\geq \epsilon(x)\}$. Hence, $C_w$ is locally compact in the induced topology.
We can now apply \cite[Theorem 5.3]{lawson}, which asserts that if $C_w$ is a locally compact cancellative cone, then indeed $V_w$ is a locally convex topological vector space whose topology extends that of $C_w$. Finally, by \cite[Theorem II.2.6]{alfsen}, a locally compact nontrivial cone in a locally convex topological space has a compact base.
\end{proof}
\subsection{Strong connectedness} Let $v,w\in \mathrm{Idem}(C)$ be such that $v\leq w$. Let's say that $v$ is compact relative to $w$ if $v$ is a compact idempotent in the extended Choquet cone $\{x\in C:x\leq w\}$. Put differently, if a downward directed net $(v_i)_i$ in $C$ satisfies that $\inf_i v_i\leq v$, then $v_i\wedge w\leq v$ for some $i$.
\begin{theorem}\label{kernelpropertythm} Let $C$ be an extended Choquet cone. The following are equivalent: \begin{enumerate}[\rm (i)] \item For any two $w_1,w_2\in \mathrm{Idem}(C)$ such that $w_1 \leq w_2$, $w_1\neq w_2$, and $w_1$ is compact relative to $w_2$, there exists $x\in C$ such that $w_1\leq x\leq w_2$ and $x$ is not an idempotent. \item The set $\{x\in C:x\leq w\}$ is connected for all $w\in \mathrm{Idem}(C)$. \end{enumerate} Moreover, if the above hold then the element $x$ in (i) may always be chosen such that $\epsilon(x)=w_1$. \end{theorem}
\begin{proof} We show that the negations of (i) and (ii) are equivalent.
Not (ii)$\Rightarrow$ not (i): Suppose that $\{x\in C:x\leq w\}$ is disconnected for some idempotent $w$. Working in the cone $\{x\in C:x\leq w\}$ as the starting extended Choquet cone, we may assume without loss of generality that $w=\infty$ (the largest element of $C$). Let $U$ and $V$ be open disjoint sets whose union is $C$. Assume that $\infty\notin U$. Observe that totally ordered subsets of $U$ have an upper bound: if $(x_i)_i$ is a chain then $x_i\to \sup_i x_i$, and since $U$ is closed, $\sup x_i\in U$. By Zorn's lemma, $U$ contains a maximal element $v$. Since $2v$ is connected to $v$ by the path $t\mapsto tv$ with $t\in [1,2]$, we must have that $2v=v$, i.e., $v$ is an idempotent. Let's show that $v$ is compact: Let $(v_i)_i$ be a decreasing net of idempotents with infimum $v$. Suppose, for the sake of contradiction, that $v_i\neq v$ for all $i$. Then $v_i\in U^c$ for all $i$. Since $U^c$ is closed and $v_i\to v$, $v\in U^c$, which is a contradiction. Thus, $v$ is compact. Let $x\in C$ be such that $v\leq x\leq \infty$. If $\epsilon(x)=\infty$, then $x=\infty$. Suppose that $\epsilon(x)=v$. Since $x$ is connected to $v$ by the path $t\mapsto tx$, $t\in(0,1]$, we have $x\in U$. But $v$ is maximal in $U$. Thus, $x=v$. This proves not (i).
Not (i)$\Rightarrow$ not (ii): Suppose that there exist $w_1,w_2\in \mathrm{Idem}(C)$ such that $w_1<w_2$, $w_1$ is relatively compact in $w_2$, and there is no non-idempotent $x\in C$ such that $w_1\leq x\leq w_2$. By Zorn's lemma, we can choose $w_2$ minimal among the idempotents such that $w_1\leq w_2$ and $w_1\neq w_2$. Then \begin{equation}\label{gap} w_1\leq x\leq w_2\Rightarrow x\in \{w_1,w_2\}\hbox{ for all }x\in C. \end{equation} Let us show that $\{x\in C:x\leq w_2\}$ is disconnected. Let $U_{1}=\{x\in C:x\leq w_1\}$ and $U_2=\{x\in C:x\nleq w_1\}$. These sets are clearly disjoint, non-empty ($w_1\in U_1$ and $w_2\in U_2$), and cover $\{x\in C:x\leq w_2\}$. It is also clear that $U_2$ is open in $C$. Let's consider $U_1$. By \eqref{gap}, $x\in U_1$ if and only if $\epsilon(x)\leq w_1$ and $x\leq w_2$. Further, since $w_1$ is a compact idempotent in the extended Choquet cone $\{z\in C:z\leq w_2\}$, the set $U_1$ may be described as all $x$ in the cone $\{z\in C:z\leq w_2\}$ such that $w_1\gg \epsilon(x)$, where the relation $\gg$ is taken in the idempotent lattice of the cone $\{z\in C:z\leq w_2\}$. Thus, by Lemma \ref{opensetsupport} applied in the extended Choquet cone $\{z\in C:z\leq w_2\}$, the set $U_1$ is (relatively) open in $\{x\in C:x\leq w_2\}$.
Finally, let us argue that $x$ in (i) may be chosen such that $\epsilon(x)=w_1$: Starting from $w_1\leq w_2$, with $w_1$ relatively compact in $w_2$, choose $w_2'$ minimal element in $\{w\in \mathrm{Idem}(C):w_1\leq w\leq w_2,\, w\neq w_1\}$, which exists by Zorn's lemma. Let $x\in C$ be a non-idempotent such that $w_1\leq x\leq w_2'$. Then $\epsilon(x)\in \{w_1,w_2'\}$, but we cannot have $\epsilon(x)=w_2'$, since this entails that $x=w_2'$. So $\epsilon(x)=w_1$. \end{proof}
\begin{definition} Let $C$ be an extended Choquet simplex. Let us say that $C$ is strongly connected if it satisfies either one of the equivalent properties listed in Theorem \ref{kernelpropertythm}. \end{definition}
\begin{proposition}\label{coneslimits} If $C$ is a projective limit of extended Choquet cones of the form $[0,\infty]^n$, then $C$ is strongly connected and has an abundance of compact idempotents. \end{proposition} \begin{proof} Suppose that $C=\varprojlim \{C_i,\phi_{i,j}:i,j\in I\}$, where $C_i\cong [0,\infty]^{n_i}$ for all $i\in I$. A projective limit of continua (compact Hausdorff connected spaces) is again a continuum. Since each $C_i$ is a continuum, so is $C$. In particular, $C$ is connected. If $w\in \mathrm{Idem}(C)$, with $w=(w_i)_i\in \prod_i C_i$, then \[ \{x\in C:x\leq w\}=\varprojlim \{x\in C_i:x\leq w_i\}. \] Thus, the same argument shows that $\{x\in C:x\leq w\}$ is connected.
The lattice of idempotent elements of $C_i$ is finite, whence algebraic under the opposite order, for all $i$. Further, by additivity and continuity, the maps $\phi_{i,j}$ preserve directed infima and arbitrary suprema (i.e., directed suprema and arbitrary infima under the opposite order). That $\mathrm{Idem}(C)$ is algebraic under the opposite order can then be deduced from the fact that a projective limit of algebraic lattices is again an algebraic lattice, where the morphisms preserve directed suprema and arbitrary infima. Let us give a direct argument instead: Let $w\in \mathrm{Idem}(C)$, with $w=(w_i)_i\in \prod_i C_i$. For each index $k\in I$ define $w^{(k)} \in \prod_i C_i$ as the unique element in $C$ such that \[ (w^{(k)})_i= \sup\{z\in C_i:\phi_{i,k}(z)=w_k\}\hbox{ for all }i\geq k. \] It is not hard to show that $(w^{(k)})_{k\in I}$ is a decreasing net in $\mathrm{Idem}(C)$ with infimum $w$. Moreover, from the compactness of $w_k\in \mathrm{Idem}(C_k)$ we deduce that $w^{(k)}\in \mathrm{Idem}(C)$ is compact for all $k\in I$. Thus, $\mathrm{Idem}(C)$ is an algebraic lattice under the opposite order. \end{proof}
\section{Cones of traces and functionals}\label{sec:conesfromfunctionals}
Here we review various constructions giving rise to extended Choquet cones.
Let $A$ be a C*-algebra. Let $A_+$ denote the cone of positive elements of $A$. A map $\tau\colon A_+\to [0,\infty]$
is called a trace if it maps 0 to 0, it is additive, homogeneous with respect to scalar multiplication, and satisfies $\tau(x^*x)=\tau(xx^*)$ for all $x\in A$. The set of all lower semicontinuous traces on $A$ is denoted by $T(A)$. It is endowed with the pointwise operations of addition and scalar multiplication. $T(A)$ is endowed with the topology such that a net $(\tau_i)_i$ in $T(A)$ converges to $\tau\in T(A)$ if
\[ \limsup \tau_i((a-\epsilon)_+) \leq \tau(a)\leq \liminf \tau_i(a)
\] for all $a\in A_+$ and $\epsilon>0$. By \cite[Theorems 3.3 and 3.7]{ERS}, $T(A)$ is an extended Choquet cone.
\begin{proposition}\label{cstarECC} Let $A$ be a C*-algebra. \begin{enumerate}[\rm (i)] \item If the primitive spectrum of $A$ has a basis of compact open sets, then $T(A)$ has an abundance of compact idempotents. In particular, this holds if $A$ has real rank zero.
\item Suppose that for all $J\subsetneq I\subseteq A$, closed two-sided ideals of $A$ such that $I/J$ has compact primitive spectrum, there exists a non-zero lower semicontinuous densely finite trace on $I/J$. Then $T(A)$ is strongly connected. In particular, this holds if $A$ has stable rank one and is exact. \end{enumerate} \end{proposition}
\begin{proof} (i) The lattice of closed two-sided ideals of $A$ is in order reversing bijection with the lattice of idempotents of $T(A)$ via the assignment $I\mapsto \tau_I$, where \[ \tau_I(a):=\begin{cases} 0&\hbox{ for }a\in I_+,\\ \infty&\hbox{otherwise.} \end{cases} \] On the other hand, the lattice of closed two-sided ideals of $A$ is isomorphic to the lattice of open sets of the primitive spectrum of $A$ (\cite[Theorem 4.1.3]{pedersen}). Thus, the lattice of idempotents of $T(A)$ is algebraic (under the opposite order) if and only if the lattice of open sets of the primitive spectrum is algebraic. The latter is equivalent to the existence of a basis of compact open sets for the topology.
(ii) Let us check that $T(A)$ satisfies condition (i) of Theorem \ref{kernelpropertythm}. Recall that idempotents in $T(A)$ have the form $\tau_I$, where $I$ is a closed two-sided ideal. Let $I$ and $J$ be (closed, two-sided) ideals of $A$, with $J\subseteq I$, so that $\tau_I\leq \tau_J$. The property that $\tau_I$ is compact relative to $\tau_J$ means that if $(I_i)_i$ is an upward directed net of ideals such that $J\subseteq I_i\subseteq I$ for all $i$ and $I=\bigcup I_i$, then $I=I_{i_0}$ for some $i_0$. This, in turn, is equivalent to $I/J$ having compact primitive spectrum. By assumption, there exists $\tau\in T(I/J)$ that is densely finite and non-zero. Pre-composed with the quotient map $\pi\colon I\to I/J$ (which maps $I_+$ onto $(I/J)_+$), $\tau$ gives rise to a trace $\tau\circ\pi\in T(I)$. Let $\tilde\tau$ be the extension of $\tau\circ\pi$ to $A_+$ such that $\tilde\tau(a)=\infty$ for all $a\in A_+\backslash I_+$. Then $\tau_I\leq \tilde \tau\leq \tau_J$ and $\tilde\tau$ is not an idempotent, as it attains values other than $\{0,\infty\}$. This proves that $T(A)$ is strongly connected.
Suppose now that $A$ has stable rank one and is exact. By the arguments from the previous paragraph, it suffices to show that if $I/J$ is a non-trivial ideal-quotient with compact primitive spectrum, then there is a nontrivial lower semicontinuous densely finite trace on $I/J$. Observe that $I/J$ has stable rank one and is exact, since both properties pass to ideals and quotients. An exact C*-algebra of stable rank one with compact primitive spectrum always has a nonzero, densely finite lower semicontinuous trace; see \cite[Theorem 2.15]{rordam-cones}. \end{proof}
Let $(G,G_+)$ be a dimension group, i.e., an ordered abelian group that is unperforated and has the Riesz refinement property. Let $\mathrm{Hom}(G_+,[0,\infty])$ denote the set of all $[0,\infty]$-valued monoid morphisms on $G_+$ (i.e., $\lambda\colon G_+\to [0,\infty]$ additive and mapping 0 to 0). Endow $\mathrm{Hom}(G_+,[0,\infty])$ with pointwise cone operations and with the topology of pointwise convergence.
\begin{proposition}\label{HomG} Let $G$ be a dimension group. Then $\mathrm{Hom}(G_+,[0,\infty])$ is an extended Choquet cone that is strongly connected and has an abundance of compact idempotents. \end{proposition}
\begin{proof} By \cite[Theorem 2.33]{wehrung}, $\mathrm{Hom}(G_+,[0,\infty])$ is a complete positively ordered monoid, which entails that it is a complete lattice and that addition distributes over $\wedge$ and $\vee$. The topology on $\mathrm{Hom}(G_+,[0,\infty])$ is that induced by its inclusion in $[0,\infty]^{G_+}$. Since the latter is compact and Hausdorff, so is $\mathrm{Hom}(G_+,[0,\infty])$. Further, since $\mathrm{Hom}(G_+,[0,\infty])$ is a convex subset of $[0,\infty]^{G_+}$, the induced topology is locally convex. Thus, $\mathrm{Hom}(G_+,[0,\infty])$ is an extended Choquet cone. To see that it is strongly connected and has an abundance of compact idempotents, we can first express $(G,G_+)$ as an inductive limit of $(\mathbb{Z}^n,\mathbb{Z}_+^n)$ using the Effros-Handelmann-Shen theorem (\cite[Theorem 2.2]{EHS}), apply the functor $\mathrm{Hom}(\cdot,[0,\infty])$ to this limit, and then apply Proposition \ref{coneslimits}. We give a direct argument in the paragraphs below.
A subgroup $I\subseteq G$ is an order ideal if $I_+ :=G_+\cap I$ is a hereditary set and $I=I_+-I_+$. Idempotent elements of $\mathrm{Hom}(G_+,[0,\infty])$ have the form $\lambda_I(g)=0$ if $g\in I_+$ and $\lambda_I(g)=\infty$ if $g\in G_+\backslash I_+$, for some ideal $I$. Moreover, the map $I\mapsto \lambda_I$ is an order reversing bijection between the two lattices. It is well known that the lattice of ideals of an ordered group is algebraic. Thus, $\mathrm{Hom}(G_+,[0,\infty])$ has abundance of compact idempotents.
Let us now prove strong connectedness. Let $I,J\subseteq G$ be order ideals such that $J\subsetneq I$ and $\lambda_I$ is compact relative to $\lambda_J$. In this case, this means that $I/J$ is finitely (thus, singly) generated. Thus, it has a finite nonzero functional $\lambda\colon (I/J)_+\to [0,\infty)$ (e.g., by \cite[Theorem 1.4]{EHS}). As in the proof of Proposition \ref{cstarECC} (ii), we define a functional on all of $G_+$ by pre-composing $\lambda$ with the quotient map $I\mapsto I/J$ and setting it equal to $\infty$ on $G_+\backslash I_+$. This produces a functional $\tilde \lambda\in \mathrm{Hom}(G_+,[0,\infty])$ such that $\lambda_I\leq \tilde\lambda\leq \lambda_J$ and $\tilde\lambda$ is not an idempotent. \end{proof}
Yet another construction yielding an extended Choquet cone is the dual of a Cu-semigroup. Let us first briefly recall the definition of a Cu-semigroup. Let $S$ be a positively ordered monoid. Given $x,y\in S$, let us write $x\ll y$ (read ``$x$ is way below $y$'') if whenever $(y_n)_{n=1}^\infty$ is an increasing sequence in $S$ such that $y\leq \sup_n y_n$, there exists $n_0$ such that $x\leq y_{n_0}$.
We call $S$ a Cu-semigroup if it satisfies the following axioms: \begin{enumerate}
\item[O1.] For every increasing sequence $(x_n)_n$ in $S$, the supremum $\sup_{n}x_n$ exists.
\item[O2.] For every $x\in S$ there exists a sequence $(x_n)_n$ in $S$ such that $x_n\ll x_{n+1}$ for all $n\in\mathbb{N}$ and $x=\sup_n x_n$.
\item[O3.] If $(x_n)_n$ and $(y_n)_n$ are increasing sequences in $S$, then $\sup_n(x_n+y_n)=\sup_nx_n+\sup_n y_n$.
\item[O4.] If $x_i\ll y_i$ for $i=1,2$, then $x_1+x_2\ll y_1+y_2$. \end{enumerate}
Observe that in our definition of the way-below relation above we only consider increasing sequences $(y_n)_n$, rather than increasing nets. In the context of Cu-semigroups we always use the symbol $\ll$ to indicate this sequential version of the way below relation.
Two additional conditions that we often impose on Cu-semigroups are the following: \begin{enumerate} \item[O5.] If $x'\ll x\le y$ then there exists $z$ such that $x'+z\le y\le x+z$. \item[O6.] If $x,y,z\in S$ are such that $x\le y+z$, then for every $x'\ll x$ there are elements $y',z'\in S$ such that $x'\le y'+z'$, $y'\le x,y$ and $z'\le x,z$. \end{enumerate}
An ordered monoid map $\lambda\colon S\to [0,\infty]$ is called a functional on $S$ if it preserves the suprema of increasing sequences. The collection of all functionals on $S$, denoted by $F(S)$, is a cone, with the cone operations defined pointwise. $F(S)$ is endowed with the topology such that a net $(\lambda_i)_{i\in I}$ in $F(S)$ converges to a functional $\lambda$ if \[ \limsup_i \lambda_i(s')\leq \lambda(s) \leq \liminf_i \lambda_i(s) \] for all $s'\ll s$, in $S$. By \cite[Theorem 4.8]{ERS} and \cite[Theorem 4.1.2]{robertFS}, if $S$ is a Cu-semigroup satisfying O5 and O6, then $F(S)$ is an extended Choquet cone. In Section \ref{duality} we address the problem of what conditions on $S$ guarantee that $F(S)$ has an abundance of compact idempotents and is strongly connected.
\section{Functions on an extended Choquet cone}\label{functionspaces} Throughout this section we let $C$ denote an extended Choquet cone with an abundance of compact idempotents, i.e., such that the lattice $(\mathrm{Idem}(C),\leq^{\mathrm{op}})$ is algebraic.
\subsection{The spaces $\mathrm{Lsc}(C)$ and $\mathrm{A}(C)$} Let us denote by $\mathrm{Lsc}(C)$ the set of all functions $f\colon C\to [0,\infty]$ that are linear (additive, homogeneous with respect to scalar multiplication, and mapping 0 to 0) and lower semicontinuous ($f^{-1}((a,\infty])$ is open for any $a\in [0,\infty)$). The linearity of the functions in $\mathrm{Lsc}(C)$ implies that they are also order preserving, for if $x\leq y$ in $C$, then $y=x+z$ for some $z$, and so $f(y)=f(x)+f(z)\geq f(x)$. We endow $\mathrm{Lsc}(C)$ with the operations of pointwise addition and scalar multiplication, and with the pointwise order. $\mathrm{Lsc}(C)$ is thus an ordered cone. Further, the pointwise supremum of functions in $\mathrm{Lsc}(C)$ is again in $\mathrm{Lsc}(C)$; thus, $\mathrm{Lsc}(C)$ is a directed complete ordered set (dcpo).
Let us denote by $\mathrm{Lsc}_\sigma(C)$ the subset of $\mathrm{Lsc}(C)$ of functions $f\colon C\to [0,\infty]$ for which the set $f^{-1}((a,\infty])$ is $\sigma$-compact---in addition to being open---for all $a\in [0,\infty)$ (equivalently, for $a=1$, by linearity.) We denote by $\mathrm{A}(C)$ the functions in $\mathrm{Lsc}(C)$ that are continuous. Notice that $\mathrm{A}(C)\subseteq \mathrm{Lsc}_\sigma(C)$, since \[ f^{-1}((a,\infty])=\bigcup_n f^{-1}([a+\frac1n,\infty]), \] and the right side is a union of closed (hence, compact) subsets of $C$.
Our goal is to show that every function in $\mathrm{Lsc}(C)$ ($\mathrm{Lsc}_\sigma(C)$) is the supremum of an increasing net (sequence) of functions in $\mathrm{A}(C)$. We achieve this in Theorem \ref{ACsuprema} after a number of preparatory results.
Given $f\in \mathrm{Lsc}(C)$, define its support $\mathrm{supp}(f)\in C$ as \[ \mathrm{supp}(f)=\sup\{x\in C:f(x)=0\}. \] Since $f(x)=0\Rightarrow f(2x)=0$, it follows easily that $\mathrm{supp}(f)$ is an idempotent of $C$.
For each $w\in \mathrm{Idem}(C)$, let \[ \chi_w(x)= \begin{cases} 0 & \hbox{ if }x\leq w,\\ \infty & \hbox{otherwise.} \end{cases} \] This is a function in $\mathrm{Lsc}(C)$.
\begin{lemma}\label{inftyfsupp} We have $\infty\cdot f=\chi_{\mathrm{supp}(f)}$, for all $f\in \mathrm{Lsc}(C)$. (Here $\infty \cdot f:=\sup_{n\in \mathbb{N}} nf$.) \end{lemma}
\begin{proof} The set $\{x\in C:f(x)=0\}$ is upward directed and converges to its supremum, i.e., to $\mathrm{supp}(f)$. It follows, by the lower semicontinuity of $f$, that $f(\mathrm{supp}(f))=0$.
If $x\leq \mathrm{supp}(f)$, then $f(x)\leq f(\mathrm{supp}(f))=0$. Hence, $(\infty \cdot f)(x)=0$. If on the other hand $x\nleq \mathrm{supp}(f)$, then $f(x)\neq 0$, which implies that $(\infty \cdot f)(x)=\infty$. We have thus shown that $\infty\cdot f=\chi_{\mathrm{supp}(f)}$. \end{proof}
Let $w\in C$ be an idempotent. Define $\mathrm{A}_w(C)=\{f\in \mathrm{A}(C): \mathrm{supp}(f)=w\}$ and \[ \mathrm{A}_{+}(C_w)=\{f\colon C_w\to [0,\infty):\hbox{ $f$ is continuous, linear, and $f(x)=0\Leftrightarrow x=w$}\}. \] (Recall that we have defined $C_w=\{x\in C:\epsilon(x)=w\}$.)
\begin{theorem}\label{ACwbijection}
If $f\in \mathrm{A}(C)$ then $\mathrm{supp}(f)$ is a compact idempotent. Further, given a compact idempotent $w\in \mathrm{Idem}(C)$, the restriction map $f\mapsto f|_{C_w}$ is an ordered cone isomorphism from $\mathrm{A}_w(C)$ to $\mathrm{A}_{+}(C_w)$. \end{theorem} \begin{proof} Let $f\in \mathrm{A}(C)$. We have already seen that $\mathrm{supp}(f)$ is an idempotent. To prove its compactness, let $(w_i)_{i\in I}$ be a downward directed family of idempotents with infimum $\mathrm{supp}(f)$. By the continuity of $f$, we have $\lim_i f(w_i)=f(\mathrm{supp}(f))=0$. But $f(w_i)\in \{0,\infty\}$ for all $i$. Therefore, there exists $i_0$ such that $f(w_i)=0$ for all $i\geq i_0$. But $\mathrm{supp}(f)$ is the largest element on which $f$ vanishes. Hence, $w_i=\mathrm{supp}(f)$ for all $i\geq i_0$. Thus, $\mathrm{supp}(f)$ is a compact idempotent.
Now, fix a compact idempotent $w$. Let $f\in \mathrm{A}_w(C)$. Clearly, $f$ is continuous and linear on $C_w$, and $f(w)=0$. Let $x\in C_w$. If $f(x)=0$, then $x\leq w$, which implies that $x=w$. Thus, $f(x)>0$ for all $x\in C_w\backslash\{w\}$. Suppose that $f(x)=\infty$. Then $f(w)=\lim_n f(\frac 1n x)=\infty$, contradicting that $w=\mathrm{supp}(f)$. Thus, $f(x)<\infty$ for all $x\in C_w$. We have thus shown that $f|_{C_w}\in \mathrm{A}_{+}(C_w)$.
It is clear that the restriction map $\mathrm{A}_w(C)\ni f\mapsto f|_{C_w}\in \mathrm{A}_+(C_w)$ is additive and order preserving. Let us show that it is an order embedding. Let $f,g\in \mathrm{A}_w(C)$ be such that $f|_{C_w}\leq g|_{C_w}$. Let $x\in C$. Suppose that $x+w\in C_w$. Then \[ f(x)=f(x+w)\leq g(x+w)=g(x). \] If, on the other hand, $x+w\notin C_w$, then $\epsilon(x+w)>w$. Hence, \[ f(x)=f(x+w)\geq f(\epsilon(x+w))=\infty. \] We argue similarly that $g(x)=\infty$. Thus, $f(x)=g(x)$.
Let us finally prove surjectivity. Suppose first that $C_w=\{w\}$. Then $\mathrm{A}_{+}(C_w)$ consists of the zero function only. Clearly then, $\chi_w|_{C_w}=0$ and $\mathrm{supp}(\chi_w)=w$. It remains to show that $\chi_w$ is continuous. The set $\chi^{-1}_w(\{\infty\})=\{x\in C:x\nleq w\}$ is open. On the other hand, $\chi_w^{-1}(\{0\})=\{x\in C:x\leq w\}$ agrees with $\{x\in C:\epsilon(x)\leq w\}$ (since we have assumed that $C_w=\{w\}$). The set $\{x\in C:\epsilon(x)\leq w\}$ is open by the compactness of $w$ (Lemma \ref{opensetsupport}). Thus, $\chi_w$ is continuous.
Suppose now that $C_w\neq \{w\}$. Let $\tilde f\in \mathrm{A}_{+}(C_w)$. Define $f\colon C\to [0,\infty]$ by \[ f(x)=\begin{cases} \tilde{f}(x+w) &\hbox{ if }x+w\in C_{w},\\ \infty&\hbox{otherwise.} \end{cases} \]
Observe that $f|_{C_w}=\tilde f$. Let us show that $f\in \mathrm{A}_w(C)$. To show that $\mathrm{supp}(f)=w$, note that \[ f(x)=0\Leftrightarrow \tilde f(x+w)=0\Leftrightarrow x+w=w\Leftrightarrow x\leq w. \] Thus, $w$ is the largest element on which $f$ vanishes, i.e., $w=\mathrm{supp}(f)$. We leave the not difficult verification that $f$ is linear to the reader. Let us show that $f$ is continuous. Let $(x_i)_i$ be a net in $C$ with $x_i\to x$. Suppose first that $x+w\in C_w$, i.e, $\epsilon(x)\leq w$. Since the set $\{y\in C:\epsilon(y)\leq w\}$ is open (Lemma \ref{opensetsupport}), $\epsilon(x_i)\leq w$ for large enough $i$. Therefore, \[ \lim_i f(x_i)=\lim_i \tilde f(x_i+w)=\tilde f(x+w)=f(x). \] Now suppose that $x+w\notin C_w$, in which case $f(x)=\infty$. To show that $\lim_i f(x_i)=\infty$, we may assume that $x_i\in C_w$ for all $i$ (otherwise $f(x_i)=\infty$ by definition). Observe also that $x_i\neq w$ for large enough $i$. Let us thus assume that $x_i\in C_w\backslash \{w\}$ for all $i$. Since $w$ is a compact idempotent, $C_{\omega}$ has a compact base $K\subseteq C_{w}\setminus\{w\}$ (Theorem \ref{compactbase}). Write $x_i=t_i\tilde{x}_i$ with $\tilde{x}_i\in K$ and $t_i>0$ for all $i$. Passing to a convergent subnet and relabelling, assume that $\tilde{x}_i\to y\in K$ and $t_i\to t\in [0,\infty]$. If $t<\infty$, then $x=\lim_it_i\tilde{x}_i=ty\in C_{w}$, contradicting our assumption that $x+w\notin C_{w}$. Hence $t=\infty$. Let $\delta>0$ be the minimum value of $\tilde{f}$ on the compact set $K$. Then \[ f(x_i)=\tilde{f}(x_i)=t_if(\tilde{x}_i)\ge t_i\delta. \] Hence $f(x_i)\to\infty$, thus showing the continuity of $f$ at $x$. \end{proof}
We will need the following theorem from \cite{edwards}: \begin{theorem}[{\cite[Theorem 3.5]{edwards}}]\label{fromedwards} Given $f,g\in \mathrm{Lsc}(C)$ there exists $f\wedge g$ and further, \[ f\wedge \sup_i f_i=\sup_i(f\wedge f_i), \] for any upward directed set $(f_i)_{i\in I}$ in $\mathrm{Lsc}(C)$. \end{theorem}
Recall that throughout this section $C$ denotes an extended Choquet cone with an abundance of compact idempotents.
\begin{theorem}\label{ACsuprema} Every function in $\mathrm{Lsc}(C)$ is the supremum of an upward directed family of functions in $\mathrm{A}(C)$, and every function in $\mathrm{Lsc}_\sigma(C)$ is the supremum of an increasing sequence in $\mathrm{A}(C)$. \end{theorem}
\begin{proof} Let $f\in \mathrm{Lsc}(C)$ and set $w=\mathrm{supp}(f)$. We first consider the case that $w$ is compact and then deal with the general case.
Assume that $w$ is compact. If $C_w=\{w\}$, then $f=\chi_w$. Further, $\chi_w$ is continuous, as shown in the proof of Theorem \ref{ACwbijection}. Suppose that $C_w\neq \{w\}$. Consider the restriction of $f$ to $C_w$. By \cite[Corollary I.1.4]{alfsen}, $f|_{C_w}$ is the supremum of an increasing net $(\tilde h_i)_{i}$ of linear continuous functions $\tilde h_i\colon C_w\to \mathbb{R}$. Since $f|_{C_w}$ is strictly positive on $C_{w}\backslash\{w\}$, it is separated from 0 on any compact base of $C_w$. It follows that the functions $\tilde h_i$ are eventually strictly positive on $C_{w}\backslash\{w\}$. Indeed, the sets $U_{i,\delta}=\tilde h_i^{-1}((\delta,\infty])\cap C_w$, where $i\in I$ and $\delta>0$, form an upward directed open cover of $C_w\backslash \{w\}$. Thus, for some $\delta>0$ and $i_0\in I$, $\tilde h_{i}$ is greater than $\delta$ on a (fixed) compact base of $C_w$ for all $i\geq i_0$. Let us thus assume that $\tilde h_i\in \mathrm{A}_{+}(C_w)$ for all $i$. By Theorem \ref{ACwbijection}, each $\tilde h_i$ has a unique continuous extension to an $h_i\in \mathrm{A}_w(C)$. Further, $(h_i)_i$ is also an increasing net. We claim that $f=\sup_i h_i$. Let us first show that $h_i\leq f$ for all $i$. Let $x\in C$ be such that $f(x)<\infty$. Then \[ 0=\lim_{n}\frac 1nf(x)=f(\epsilon(x))=0. \] Hence, $\epsilon(x)\leq w$, i.e., $x+w\in C_w$. We thus have that \[ h_i(x)=\tilde h_i(x+w)\leq f(x+w)=f(x). \] Hence, $h_i\leq f$ for all $i$. Set $h=\sup_i h_i$. Clearly $h\leq f$. If $\epsilon(x)\leq w$ then \[ h(x)=h(x+w)=\sup_i h_i(x+w)=f(x). \] If, on the other hand, $\epsilon(x)\nleq w$, then $h_i(x)=\infty$ for all $i$ and $h(x)=\infty=f(x)$. Thus, $h=f$.
Let us now consider the case when $w$ is not compact. Define \[ H=\{h\in \mathrm{A}(C):h\leq (1-\epsilon)f\hbox{ for some }\epsilon>0\}. \] Let us show that $H$ is upward directed and has pointwise supremum $f$. Let $h_1,h_2\in H$. Set $v_1=\mathrm{supp}(h_1)$ and $v_2=\mathrm{supp}(h_2)$, which are compact idempotents, by Theorem \ref{ACwbijection}, and satisfy that $w\leq v_1,v_2$. Set $v=v_1\wedge v_2$, which is also compact and such that $w\leq v$. Set $g=f\wedge \chi_v$, which exists by Theorem \ref{fromedwards}. Since scalar multiplication by a non-negative scalar is an order isomorphism on $C$, we have $tg=(tf)\wedge \chi_v$. Letting $t\to \infty$ and using Theorem \ref{fromedwards}, we get $\infty\cdot g=\chi_w\wedge \chi_v=\chi_v$. Thus, $\mathrm{supp}(g)=v$ (Lemma \ref{inftyfsupp}). Let $\epsilon>0$ be such that $h_1,h_2\leq (1-\epsilon)f$. Then, $h_1,h_2\leq (1-\epsilon)g$. Since we have already established the case of compact support idempotent, there exists an increasing net $(g_i)_i$ in $\mathrm{A}_v(C)$ such that $g=\sup_i g_i$. By \cite[Proposition 5.1]{ERS}, $h_1,h_2\ll (1-\epsilon/2)g$ in the directed complete ordered set $\mathrm{Lsc}(C)$ (see also the definition of the relation $\lhd$ in the next section). Thus, there exists $i_0$ such that $h_1,h_2\leq (1-\epsilon/2)g_{i_0}$. Now $h=(1-\epsilon/2)g_{i_0}$ belongs to $H$ and satisfies that $h_1,h_2\leq h$. This shows that $H$ is upward directed.
Let us show that $f$ is the pointwise supremum of the functions in $H$. It suffices to show that $f$ is the supremum of functions in $\mathrm{A}(C)$, as we can then easily arrange for the $1-\epsilon$ separation. Choose a decreasing net of compact idempotents $(v_i)_i$ with $w=\inf v_i$ (recall that $C$ has an abundance of compact idempotents). For each fixed $i$, $f\wedge \chi_{v_i}$ has support idempotent $v_i$, which is compact. Thus, as demonstrated above, $f\wedge \chi_{v_i}$ is the supremum of an increasing net in $\mathrm{A}(C)$. But $f=\sup_i f\wedge \chi_{v_i}$ (Theorem \ref{fromedwards}). It follows that $f$ is the pointwise supremum of functions in $\mathrm{A}(C)$.
Finally, suppose that $f\in \mathrm{Lsc}_\sigma(C)$, and let us show that there is a countable set in $H$ with pointwise supremum $f$. For each $h\in H$, let $U_{h}=h^{-1}((1,\infty])$. The sets $(U_{h})_{h\in H}$ form an open cover of $f^{-1}((1,\infty])$. Since the latter is $\sigma$-compact, we can choose a countable set $H'\subseteq H$ such that $(U_{h})_{h\in H'}$ is also a cover of $f^{-1}((1,\infty])$. Observe that for each $x\in C$, $f(x)>1$ if and only if $h(x)>1$ for some $h\in H'$. It follows, by the homogeneity with respect to scalar multiplication of these functions, that $\sup_{h\in H'}h(x)=f(x)$ for all $x\in C$. Now using that $H$ is upward directed we can construct an increasing sequence with supremum $f$. \end{proof}
\begin{theorem}\label{metrizableC} Let $C$ be a metrizable extended Choquet cone with an abundance of compact idempotents. Then there exists a countable subset of $\mathrm{A}(C)$ such that every function in $\mathrm{Lsc}(C)$ is the supremum of an increasing sequence of functions in this set. \end{theorem}
\begin{proof} Let us first argue that the set of compact idempotents is countable. Let $(U_{i})_{i=1}^\infty$ be a countable basis for the topology of $C$. Let $w\in \mathrm{Idem}_c(C)$ be a compact idempotent. Since $\{x\in C:w\leq x\}$ is an open set, by Lemma \ref{opensetsupport}, there exists $U_i$ such that $w\in U_i\subseteq \{x\in C:w\leq x\}$. Clearly then $w=\inf U_i$. Thus, the set of compact idempotents embeds in the countable set $\{\inf U_i:i=1,2,\ldots\}$.
Now fix a compact idempotent $w$. Recall that $\mathrm{A}_w(C)$ is isomorphic to the cone $\mathrm{A}_+(C_w)$ of positive linear functions on the cone $C_w$. Suppose that $C_w\neq \{w\}$. Let $K$ denote a compact base of $C_w$, which exists by Theorem \ref{compactbase}, and is metrizable since $C$ is metrizable by assumption. Then $\mathrm{A}_+(C_w)$ is separable in the metric induced by the uniform norm on $K$, since it embeds in $C(K)$, which is separable. Let $\tilde B_w\subseteq \mathrm{A}_+(C_w)$ be a countable dense subset. It is not hard now to express any function in $\mathrm{A}_+(C_w)$ as the supremum of an increasing sequence in $\tilde B_w$. Indeed, it suffices to show that for any $\epsilon>0$ and $f\in \mathrm{A}_{+}(C_w)$, there exists $g\in \tilde B_w$ such that $(1-\epsilon)f\leq g\leq f$. Keeping in mind that $f$ is separated from 0 on $K$, we can choose $g\in\tilde B_w$ such that \[
\Big\|(1-\frac\epsilon{2})f|_K-g|_K\Big \|_{\infty}<\frac\epsilon{2} \min_{x\in K}|f(x)|. \] Then $g$ is as desired. Let $B_w\subseteq \mathrm{A}_w(C)$ be the set mapping bijectively onto $\tilde B_w\subseteq \mathrm{A}_+(C_w)$ via the restriction map. By Theorem \ref{ACwbijection}, every function in $\mathrm{A}_w(C)$ is the supremum of an increasing sequence in $B_w$. If, on the other hand, $C_w=\{w\}$, then $\mathrm{A}_w(C)=\{\chi_w\}$. In this case we set $B_w=\{\chi_w\}$.
Let $B=\bigcup_w B_w$, where $w$ ranges through the set of compact idempotents, and $B_w$ is as in the previous paragraph. Observe that $B$ is countable. Let us show that every function in $f\in \mathrm{Lsc}(C)$ is the supremum of an increasing sequence in $B$. Observe that $\mathrm{Lsc}(C)=\mathrm{Lsc}_\sigma(C)$, since all open subsets of a compact metric space are $\sigma$-compact. Thus, $f=\sup_n h_n$, where $(h_n)_{n=1}^\infty$ is an increasing sequence in $\mathrm{A}(C)$. The sequence $h_n'=(1-\frac 1n)h_n$ is also increasing, with supremum $f$, and $h_{n}'\ll h_{n+1}'$ in the directed complete ordered set $\mathrm{Lsc}(C)$ (see \cite[Proposition 5.1]{ERS} and also the definition of the relation $\lhd$ in the next section). Say $h_{n+1}'\in \mathrm{A}_{w_n}(C)$ for some compact idempotent $w_n$. Since $h_{n+1}'$ is the supremum of a sequence in $B_{w_n}$, we can choose $g_n\in B_{w_n}$ such that $h_n'\leq g_n\leq h_{n+1}'$. Then $(g_n)_{n=1}^\infty$ is an increasing sequence in $B$ with supremum $f$. \end{proof}
\section{Duality with Cu-cones}\label{duality} By a Cu-cone we understand a Cu-semigroup $S$ that is also a cone, i.e., it is endowed with a scalar multiplication by $(0,\infty)$ compatible with the monoid structure of $S$; see Section \ref{sectionECCs}. Further we ask that \begin{enumerate} \item $t_1\leq t_2$ and $s_1\leq s_2$ imply $t_1s_1\leq t_2s_2$ for all $t_1,t_2\in (0,\infty)$ and $s_1,s_2\in S$, \item $\sup_n t_ns_n=(\sup_n t_n)(\sup_n s_n)$ where $(t_n)_{n=1}^\infty$ and $(s_n)_{n=1}^\infty$ are increasing sequences in $(0,\infty)$ and $S$, respectively. \end{enumerate} Cu-cones are called Cu-semigroups with real multiplication in \cite{robertFS}. They are also Cu-semimodules over the Cu-semiring $[0,\infty]$, in the sense of \cite{tensorthiel}.
In this section we prove a duality between extended Choquet cones with an abundance of compact idempotents and certain Cu-cones. Throughout this section, $S$ denotes a Cu-cone satisfying the axioms O5 and O6, so that $F(S)$ is an extended Choquet cone.
Let us recall the relation $\lhd$ in $\mathrm{Lsc}(C)$ defined in \cite{ERS}: Given $f,g\in \mathrm{Lsc}(C)$, we write $f\lhd g$ if $f\le (1-\varepsilon)g$ for some $\varepsilon>0$ and $f$ is continuous at each $x\in C$ such that $g(x)<\infty$. By \cite[Proposition 5.1]{ERS}, $f\lhd g$ implies that $f$ is way below $g$ in the dcpo $\mathrm{Lsc}(C)$, meaning that for any upward directed net $(g_i)_i$ such that $g\leq \sup g_i$, there exists $i_0$ such that $f\leq g_{i_0}$.
\begin{lemma}({Cf. \cite[Lemma 3.3.2]{robertFS}})\label{lhdll} Let $f,g\in \mathrm{Lsc}(C)$ be such that $f\lhd g$. Then here exists $h\in \mathrm{Lsc}(C)$ such that $f+h=g$ and $h\geq \epsilon g$ for some $\epsilon>0$. Moreover, if $f,g\in \mathrm{Lsc}_\sigma(C)$, then $h$ may be chosen in $\mathrm{Lsc}_\sigma(C)$, and if $f,g\in \mathrm{A}(C)$, then $h$ may be chosen in $\mathrm{A}(C)$. \end{lemma} \begin{proof} Define $h\colon C\to [0,\infty]$ by \[ h(x) = \begin{cases} g(x) - f(x) & \hbox{ if }g(x)<\infty,\\ \infty&\hbox{ otherwise}. \end{cases} \] Then $f+h=g$. The linearity of $h$ follows from a straightforward analysis. Since $f\lhd g$, there exists $\epsilon>0$ such that $f\leq (1-\epsilon)g$. Then $g(y)-f(y)\geq \epsilon g(y)$ whenever $g(y)<\infty$, while if $g(y)=\infty$ then $g(y)=\infty=h(y)$. This establishes that $h\geq \epsilon g$.
The proof of \cite[Lemma 3.3.2]{robertFS} establishes the lower semicontinuity of $h$. Let us recall it here: Let $(x_i)_i$ be a net in $C$ such that $x_i\to x$. Suppose first that $g(x)<\infty$. Then $f(x)<\infty$, and by the continuity of $f$ at $x$, $f(x_i)<\infty$ for large enough $i$. Then, \[ \liminf_i h(x_i)\geq \liminf_i g(x_i)-f(x_i)\geq g(x)-f(x)=h(x). \] Suppose now that $g(x)=\infty$, so that $h(x)=\infty$. Since $h\geq \epsilon g$, \[ \liminf_i h(x_i)\geq \epsilon \liminf_i g(x_i)\geq \epsilon g(x)=\infty, \] thus showing lower semicontinuity at $x$.
Assume now that $f,g\in \mathrm{Lsc}_\sigma(C)$. It is not difficult to show that $h(x)>1$ if and only if $g(x)>1/\epsilon$ or $g(x)>1+r$ and $f(x)\leq r$ for some $r\in \mathbb{Q}$. Thus, \[ h^{-1}((1,\infty])=g^{-1}((1/\epsilon,\infty])\cup \bigcup_{r\in \mathbb{Q}}g^{-1}((1+r,\infty])\cap f^{-1}([0,r]). \] The right side is $\sigma$-compact. Hence, $h\in \mathrm{Lsc}_\sigma(C)$.
Assume now that $f,g\in \mathrm{A}(C)$. Continuity at $x\in C$ such that $h(x)=\infty$ follows automatically from lower semicontinuity. Let $x\in C$ be such that $h(x)<\infty$, i.e., $g(x)<\infty$. If $x_i\to x$ then $g(x_i)<\infty$ and $f(x_i)<\infty$ for large enough $i$. Then \[ h(x_i)=g(x_i)-f(x_i)\to g(x)-f(x)=h(x),\] where we used the continuity of $g$ and $f$. Thus, $h$ is continuous at $x$. \end{proof}
By an ideal of a Cu-cone we understand a subcone that is closed under the suprema of increasing sequence. There is an order reversing bijection between the ideals of $S$ and the idempotents of $F(S)$: \[ I\mapsto \lambda_I(x):=\begin{cases} 0&\hbox{if }x\in I\\ \infty&\hbox{otherwise,} \end{cases} \] where $I$ ranges through the ideals of $S$.
Let us say that a Cu-cone $S$ has an abundance of compact ideals if the lattice of ideals of $S$ is algebraic, i.e., every ideal of $S$ is a supremum of compact ideals.
\begin{theorem}\label{dualitythm} Let $S$ be a Cu-cone satisfying O5 and O6 and having an abundance of compact ideals. Then $F(S)$ is an extended Choquet cone with an abundance of compact idempotents. Moreover, $S\cong \mathrm{Lsc}_\sigma(F(S))$ via the assignment \[ S\ni s\mapsto \hat s\in \mathrm{Lsc}_\sigma(F(S)), \] where $\hat{s}(\lambda):=\lambda(s)$ for all $\lambda\in F(S)$.
Let $C$ be an extended Choquet cone with an abundance of compact idempotents. Then $\mathrm{Lsc}_\sigma(C)$ is a Cu-cone satisfying O5 and O6 and having an abundance of compact ideals. Moreover, $C\cong F(\mathrm{Lsc}_\sigma(C))$ via the assignment \[ C\ni x\mapsto \hat x\in F(\mathrm{Lsc}_\sigma(C)), \] where $\hat x(f):=f(x)$ for all $f\in \mathrm{Lsc}_\sigma(C)$. \end{theorem}
\begin{proof} As recalled in Section \ref{sec:conesfromfunctionals}, by the results of \cite{robertFS}, $F(S)$ is an extended Choquet cone. The bijection between the ideals of $S$ and the idempotents of $F(S)$ translates the abundance of compact ideals of $S$ directly into the abundance of compact idempotents of $F(S)$. By \cite[Theorem 3.2.1]{robertFS}, the mapping \[ S\ni s\mapsto \hat s\in \mathrm{Lsc}(F(S)) \] is an isomorphism of the Cu-cone $S$ onto the space of functions $f\in \mathrm{Lsc}(F(S))$ expressible as the pointwise supremum of an increasing sequence $(h_n)_{n=1}^\infty$ in $\mathrm{Lsc}(F(S))$ such that $h_n\lhd h_{n+1}$ for all $n$. The set of all such functions is denoted by $L(F(S))$ in \cite{robertFS}. Let us show that, under our present assumptions, $L(F(S))=\mathrm{Lsc}_\sigma(F(S))$. Let $f\in \mathrm{Lsc}(F(S))$ be such that $f=\sup h_n$, where $h_n\lhd h_{n+1}$ for all $n$. We have $\overline{h_n^{-1}((1,\infty])}\subseteq f^{-1}((1,\infty])$ for all $n$ (\cite[Proposition 5.1]{ERS}). Hence, \[ f^{-1}((1,\infty])=\bigcup_n \overline{h_n^{-1}((1,\infty])}. \] Thus, $f\in\mathrm{Lsc}_\sigma(F(S))$. Suppose, on the other hand, that $f\in \mathrm{Lsc}_\sigma(F(S))$. Then, by Theorem \ref{ACsuprema}, there exists an increasing sequence $(h_n)_{n=1}^\infty$ in $\mathrm{A}(F(S))$ with supremum $f$. Clearly, $h_n'=(1-\frac1n)h_n$ is also increasing, has supremum $f$, and $h_n'\lhd h_{n+1}'$ for all $n$. Hence, $f\in L(F(S))$.
Let's turn now to the second part of the theorem. Let $C$ be an extended Choquet cone with an abundance of compact idempotents. Let us show that $\mathrm{Lsc}_\sigma(C)$ satisfies all axioms O1-O6 (Section \ref{sec:conesfromfunctionals}). Let us show first that $\mathrm{Lsc}_\sigma(C)$ is closed under the suprema of increasing sequences: Let $f=\sup_n f_n$, with $(f_n)_{n=1}^\infty$ an increasing sequence in $\mathrm{Lsc}_\sigma(C)$. Then $f^{-1}((1,\infty])=\bigcup_{n=1}^\infty f_n^{-1}((1,\infty])$. Since the sets on the right side are $\sigma$-compact, so is the left side. Thus, $f\in \mathrm{Lsc}_\sigma(C)$.
Let $f\in \mathrm{Lsc}_\sigma(C)$, and let $(h_n)_{n=1}^\infty$ be an increasing in $\mathrm{A}(C)$ with supremum $f$. Then $h_n'=(1-\frac1n)h_n$ has supremum $f$ and $h_n'\ll h_{n+1}'$ for all $n$ (since $h_n'\lhd h_{n+1}'$). This proves O2. Axiom O3 follows at once from the fact that suprema in $\mathrm{Lsc}_\sigma(C)$ are taken pointwise. Suppose that $f_1\ll g_1$ and $f_2\ll g_2$. Choose $h_1,h_2\in \mathrm{A}(C)$ such that $f_i\leq h_i\lhd g_i$ for $i=1,2$. Then $f_1+f_2\leq h_1+h_2\lhd g_1+g_2$, from which we deduce O4.
Let's prove O5: Suppose that $f',f,g\in \mathrm{Lsc}_\sigma(C)$ are such that $f'\ll f\leq g$. Choose $h\in \mathrm{A}(C)$ such that $f'\leq h\lhd f$. By Lemma \ref{lhdll}, there exists $h'\in \mathrm{Lsc}_\sigma(C)$ such that $h+h'=g$. Then, $f'+h'\leq g\leq f+h'$, proving O5.
Let us prove O6. We prove the stronger property that $\mathrm{Lsc}_\sigma(C)$ is inf-semilattice ordered, i.e., pairwise infima exist and addition distributes over infima. Recall that, by the results of \cite{edwards}, $\mathrm{Lsc}(C)$ is inf-semilattice ordered (see Theorem \ref{fromedwards}). Let us show that if $f,g\in \mathrm{Lsc}_\sigma(C)$, then $f\wedge g$ is also in $\mathrm{Lsc}_\sigma(C)$. By \cite[Lemma 3.4]{edwards}, for every $x\in C$ there exist $x_1,x_2\in C$, with $x_1+x_2=x$, such that $(f\wedge g)(x)=f(x_1)+g(x_2)$. It is then clear that \[ (f\wedge g)^{-1}((1,\infty])=\bigcup_{\substack{a_1,a_2\in \mathbb{Q}, \\ a_1+a_2>1} } f^{-1}((a_1,\infty])\cup g^{-1}((a_2,\infty]). \] Since the right side is a $\sigma$-compact set, $f\wedge g\in \mathrm{Lsc}_\sigma(C)$. To verify O6, suppose that $f\leq g_1+g_2$, with $f,g_1,g_2\in \mathrm{Lsc}_\sigma(C)$. Then, using the distributivity of addition over $\wedge$, $f\leq g_1+g_2\wedge f$, which proves O6.
Finally, let us prove that $C\ni x\mapsto \hat x\in F(\mathrm{Lsc}_\sigma(C))$ is an isomorphism of extended Choquet cones. We consider injectivity first: Let $x,y\in C$ be such that $f(x)=f(y)$ for all $f\in \mathrm{Lsc}_\sigma(C)$. Choose $f\in \mathrm{A}(C)$. Passing to the limit as $n\to \infty$ in $f(\frac1n x)=f(\frac 1ny)$ we deduce that $f(\epsilon(x))=f(\epsilon(y))$ for all $f\in \mathrm{A}(C)$. Since every function in $\mathrm{Lsc}(C)$ is the supremum of a directed net of functions in $\mathrm{A}(C)$, we have that $f(\epsilon(x))=f(\epsilon(y))$ for all $f\in \mathrm{Lsc}(C)$. Now choosing $f=\chi_w$, for $w\in \mathrm{Idem}(C)$, we conclude that $\epsilon(x)=\epsilon(y)$, i.e., $x$ and $y$ have the same support idempotent. Set $w=\epsilon(x)=\epsilon(y)$. Choose a compact idempotent $v$ such that $w\leq v$. Then $x+v,y+v\in C_v$, and $f(x+y)=f(y+v)$ for all $f\in \mathrm{A}(C)$. By Theorem \ref{ACwbijection}, $f(x+v)=f(y+v)$ for all $f\in \mathrm{A}_+(C_v)$. Recall that $C_v$ has a compact base and embeds in a locally convex Hausdorff vector space $V_v$ (Theorem \ref{compactbase}). We have $f(x+v)=f(y+v)$ for all $f\in \mathrm{A}_+(C_v)-\mathrm{A}_+(C_v)$. But $\mathrm{A}_+(C_v)-\mathrm{A}_+(C_v)$ consists of all the affine functions on $C_v$ that vanish at the origin. Thus, $f(x+v)=f(y+v)$ for all such functions, and in particular, for all continuous functionals on $V_v$. Since the weak topology on $V_v$ is Hausforff, $x+v=y+v$. Passing to the infimum over all compact idempotents $v$ such that $w\leq v$, and using that $C$ has an abundance of compact idempotents, we conclude that $x=x+w=y+w=y$. Thus, the map $x\mapsto \hat x$ is injective.
Let us prove continuity of the map $x\mapsto \hat x$. Let $(x_i)_i$ be a net in $C$ with $x_i\to x$. Let $f',f\in \mathrm{Lsc}_\sigma(C)$, with $f'\ll f$. By the lower semicontinuity of $f$, we have \[ \hat x(f)=f(x)\leq \liminf_i f(x_i)=\liminf_i \hat x_i(f). \] Choose $h\in \mathrm{A}(C)$ such that $f'\leq h\leq f$, which is possible since $f$ is supremum of an increasing sequence in $\mathrm{A}(C)$. Then \[ \limsup_i \hat x_i(f')\leq \limsup_i \hat x_i(h)=\limsup_i h(x_i)=h(x)\leq f(x)=\hat x(f). \] This shows that $\hat x_i\to \hat x$ in the topology of $F(\mathrm{Lsc}_\sigma(C))$.
Let us prove surjectivity of the map $x\mapsto \hat x$. (Linearity is straightforward; continuity of the inverse is automatic from the fact that the cones are compact and Hausdorff.) The range of the map $x\mapsto \hat x$ is a compact subcone of $F(\mathrm{Lsc}_\sigma(C))$ that separates elements of $\mathrm{Lsc}_\sigma(C)$ and contains 0. By the separation theorem \cite[Corollary 4.6]{ultraCu}, it must be all of $F(\mathrm{Lsc}_\sigma(C))$. \end{proof}
Let $S$ be a Cu-cone. We say that $S$ has weak cancellation if $x+z\ll y+z$ implies $x\ll y$ for all $x,y,z\in S$.
\begin{lemma} Let $C$ be an extended Choquet cone. Let $h,h',g\in \mathrm{Lsc}(C)$ be such that $h\lhd g+h'$ and $h'\lhd h$. Then $\mathrm{supp}(g+h')$ is relatively compact in $\mathrm{supp}(g)$.
\end{lemma} \begin{proof} Set $w_1=\mathrm{supp}(g+h')$ and $w_2=\mathrm{supp}(g)$. Let $(v_i)_i$ be a downward directed net of idempotents with $\bigwedge_i v_i\leq w_1$. Then the functions $(\chi_{v_i})_i$ form an upward directed net such that $g+h'\leq \chi_{w_1}\leq \sup_i \chi_{v_i}$. Since $h\lhd g+h'$, there exists $i_0$ such that $h\leq \chi_{v_{i_0}}$. We have that \[ g+h'\leq g+h\leq \chi_{w_2}+\chi_{v_{i_0}}=\chi_{w_2\wedge v_{i_0}}. \] Hence, $w_2\wedge v_{i_0}\leq w_1$, which proves the lemma. \end{proof}
\begin{theorem}\label{weakcancellation} Let $C$ be an extended Choquet cone with an abundance of compact idempotents. Then $C$ is strongly connected if and only if $\mathrm{Lsc}_\sigma(C)$ has weak cancellation. \end{theorem}
\begin{proof} Suppose first that $C$ is strongly connected. Let $f,g,h\in \mathrm{Lsc}_\sigma(C)$ be such that $f+h\ll g+h$. Choose $\lhd$-increasing sequences $(g_n)_{n=1}^\infty$ and $(h_n)_{n=1}^\infty$ in $\mathrm{A}(C)$ such that $g=\sup_n g_n$ and $h=\sup_n h_n$. Then $f+h\ll g_m+h_m$ for some $m$. We will be done once we have shown that $f\leq g_m$.
Let $x\in C$. If $g_m(x)=\infty$, then indeed $f(x)\leq \infty=g_m(x)$. Suppose that $g_m(x)<\infty$. If $h_m(x)<\infty$, then we can cancel $h_m(x)$ in $f(x)+h_m(x)\leq g_m(x)+h_m(x)$ to obtain the desired $f(x)\leq g_m(x)$. It thus suffices to show that $g_m(x)<\infty$ implies $h_m(x)<\infty$, i.e., that $\mathrm{supp}(g_m)\le \mathrm{supp}(h_m)$. Let $w_1=\mathrm{supp}(g_m+h_m)$ and $w_2=\mathrm{supp}(g_m)$. Then $w_1\le w_2$ and $w_1$ is relatively compact in $w_2$, by the previous lemma. Suppose for the sake of contradiction that $w_1\ne w_2$. By strong connectedness, there exists $x\in C$ such that $w_1\le x\le w_2$, with $\epsilon(x)=w_1$ and $x\ne w_1$. Then, \begin{align*} h(x) &\leq g_m(x)+h_m(x)\\
&=h_m(x)\le (1-\delta)h(x), \end{align*} for some $\delta>0$. Hence, $h(x)\in\{0,\infty\}$. If $h(x)=0$, then $h_m(x)=g_m(x)=0$, while if $h(x)=\infty$, then $g_m(x)+h_m(x)\geq h(x)=\infty$. In either case, we get a contradiction with $0<(g_m+h_m)(x)<\infty$, which holds by Theorem \ref{ACwbijection}. Hence, $w_1=w_2$. We thus have that $\mathrm{supp}(g_m)=\mathrm{supp}(g_m+h_m)\le \mathrm{supp}(h_m)$.
Suppose conversely that $\mathrm{Lsc}_\sigma(C)$ has weak cancellation. Let $w_1\leq w_2$ be idempotents in $C$, with $w_1$ relatively compact in $w_2$, and $w_1\neq w_2$. Further, using Zorn's lemma, choose $w_2$ minimal such that $w_1\neq w_2$ and $w_1$ is relatively compact in $w_2$. Suppose for the sake of contradiction that $w_1\leq x\leq w_2$ implies $x\in \{w_1,w_2\}$. Let $D=\{x\in C:x\leq w_2\}$. Then $D$ is an extended Choquet cone and $w_1$ is a compact idempotent in $D$. Further, $D_{w_1}=\{w_1\}$. So, as shown in the course of the proof of Theorem \ref{ACwbijection}, $\chi_{w_1}|_D$ is continuous on $D$. Let $(h_i)_i\in \mathrm{A}(C)$ be an upward directed net with supremum $\chi_{w_1}$. Since $\chi_{w_1}|_D\lhd \chi_{w_1}|_D$, there exists $i$
such that $\chi_{w_1}|_D\leq h_i|_D$. It follows that $\chi_{w_1}\leq h_i+\chi_{w_2}$ (as functions on $C$). Fix an index $j\geq i$. Then \[ 3h_j\lhd \chi_{w_1}\leq h_i+\chi_{w_2}. \] Now let $(l_k)_k$ be an upward directed net in $\mathrm{A}(C)$ with supremum $\chi_{w_2}$. Then there exists an index $k$ such that $3h_j\leq h_i+l_k$. Observe that $h_i\lhd 2h_k$. By weak cancellation in $\mathrm{Lsc}_\sigma(C)$, we conclude that $h_j\leq l_k$. (Note: we have used weak cancellation in the form $f+h\leq g+h'$ and $h'\ll h$ imply $f\leq g$.) Thus, $h_j\leq \chi_{w_2}$ for all $j\geq i$, implying that $\chi_{w_1}\leq \chi_{w_2}$. This contradicts that $w_1\neq w_2$. \end{proof}
In the following section we will make use of the following form of Riesz decomposition:
\begin{theorem}\label{ACRiesz} Let $C$ be an extended Choquet cone that is strongly connected and has an abundance of compact idempotents. Let $f,g_1,g_2\in \mathrm{A}(C)$ be such that $f\lhd g_1+g_2$. Then there exist $f_1,f_2\in \mathrm{A}(C)$ such that $f=f_1+f_2$, $f_1\lhd g_1$, and $f_2\lhd g_2$. \end{theorem}
\begin{proof} Let $\epsilon>0$ be such that $f\le (1-\varepsilon)g_1+(1-\varepsilon)g_2$. Then, using the distributivity of addition over $\wedge$, \begin{align*} f\le f\wedge ((1-\varepsilon)g_1)+(1-\varepsilon)g_2=(1-\varepsilon)( (f\wedge g_1)+g_2). \end{align*} Thus, $f\lhd (f\wedge g_1)+g_2$ (recall that $f$ is continuous). By Theorem \ref{ACsuprema}, $f\wedge g_1$ is the supremum of a net of functions in $\mathrm{A}(C)$. Thus, there exists $h\in \mathrm{A}(C)$ such that $f\lhd h+g_2$ and $h\lhd (f\wedge g_1)$. By Lemma \ref{lhdll}, we can find $l\in \mathrm{A}(C)$ such that $f=h+l$. Then $h+l\lhd h+g_2$. By weak cancellation in $\mathrm{Lsc}_\sigma(C)$ (Theorem \ref{weakcancellation}), we have that $l\lhd g_2$. Setting $f_1=h$ and $f_2=l$ yields the desired result. \end{proof}
\section{Proof of Theorem \ref{mainchar}}\label{proofofmainchar} Throughout this section $C$ denotes an extended Choquet cone that is strongly connected and has an abundance of compact idempotents.
\subsection{The triangle lemma} To prove Theorem \ref{mainchar} we follow a strategy similar to the proof of the Effros-Handelman-Shen theorem (\cite{EHS}). The key step in this proof is establishing a ``triangle lemma'', Theorem \ref{triangletheorem} below.
\begin{lemma}\label{Cugenerators} A linear map $\phi\colon [0,\infty]\to \mathrm{Lsc}_\sigma(C)$ is a Cu-morphism if and only if $\phi(\infty)=\infty \cdot \phi(1)$ and $\phi(1)\in \mathrm{A}(C)$. \end{lemma}
\begin{proof} Suppose that $\phi$ is a Cu-morphism. That $\phi(\infty)=\infty \cdot \phi(1)$ follows at once from $\phi$ being supremum preserving and additive. Set $f=\phi(1)$. To prove the continuity of $f$, it suffices to show that it is upper semicontinuous, since it is already lower semicontinuous by assumption. Fix $\epsilon>0$. Since $1-\epsilon \ll 1$ in $[0,\infty]$, we have $(1-\epsilon)f \ll f$ in $\mathrm{Lsc}_\sigma(C)$. Choose $g\in \mathrm{A}(C)$ such that $(1-\epsilon)f\leq g\leq f$. Let $x_i\to x$ be a convergent net in $C$. Then, \[ (1-\epsilon)\limsup_i f(x_i)\leq \limsup g(x_i)=g(x)\leq f(x). \] Letting $\epsilon\to 0$, we get that $\limsup f(x_i)\leq f(x)$. Thus, $f$ is upper semicontinuous.
Conversely, suppose that $\phi(1)\in \mathrm{A}(C)$ and $\phi(\infty)=\infty \cdot \phi(1)$. Observe that if $f\in \mathrm{A}(C)$ then $\alpha f\lhd \beta f$ for all scalars $0\leq \alpha<\beta\leq \infty$. Hence, $\phi(\alpha)\ll \phi(\beta)$ in $\mathrm{Lsc}_\sigma(C)$ whenever $\alpha\ll \beta$ in $[0,\infty]$, i.e., $\phi$ preserves the way below relation. The rest of the properties of $\phi$ are readily verified. \end{proof}
The core of the proof of Theorem \ref{triangletheorem} (the ``triangle lemma") is contained in the following lemma: \begin{lemma}\label{coretrianglelemma} Let $\phi\colon [0,\infty]^n\to \mathrm{Lsc}_\sigma(C)$ be a Cu-morphism. Let $x,y\in [0,\infty)^n\cap \mathbb{Z}^n$ be such that $\phi(x)\ll \phi(y)$. Then there exist $N\in \mathbb{N}$ and Cu-morphisms \[ [0,\infty]^n \stackrel{Q}{\longrightarrow} [0,\infty]^N\stackrel{\psi}{\longrightarrow} \mathrm{Lsc}_\sigma(C), \] such that $\psi Q=\phi$ and $Qx\leq Qy$. Moreover, $Q$ maps $[0,\infty)^n\cap \mathbb{Z}^n$ to $[0,\infty)^N\cap \mathbb{Z}^N$ \end{lemma}
\begin{proof} Let $x=(x_1,\ldots,x_n)$, $y=(y_1,\ldots,y_n)$, and $\phi$ be as in the statement of the lemma. Let $(E_i)_{i=1}^n$ denote the canonical basis of $[0,\infty]^n$. Set $f_i=\phi(E_i)$ for $i=1,\ldots,n$, which belong to $\mathrm{A}(C)$ by Lemma \ref{Cugenerators}. Let \[
M=\max_i |x_i-y_i|, \, n_1=\#\{i:x_i-y_i=M\}, \, n_2=\#\{i:y_i-x_i=M\}. \] Let us define the degree of the triple $(\phi,x,y)$, denoted $\deg (\phi,x,y)$, as the vector $(M,n_1, n_2,n)$. We order the degrees lexicographically. We will prove the lemma by induction on the degree of the triple $(\phi, x, y)$. Let us first deal with the case $n=1$, i.e., the domain of $\phi$ is $[0,\infty]$. Since $[0,\infty]$ is totally ordered, either $x\leq y$ or $y<x$. In the first case, setting $Q$ the identity and $\phi=\psi$ gives the result. If $y<x$, then $\phi(y)\ll \phi(x)$, which, together with $\phi(x)\ll \phi(y)$, implies that $\phi(x)=\phi(y)$ is a compact element in $\mathrm{A}(C)$. The only compact element in $\mathrm{A}(C)$ is 0, for if $f\ll f$, then $f\ll (1-\epsilon)f$ for some $\epsilon>0$, and so $f=0$ by weak cancellation (Theorem \ref{weakcancellation}). Thus, $\phi(x)=0$, which in turn implies that $\phi=0$. We can then choose $Q$ and $\psi$ to be the 0 maps.
Suppose now that $\phi$, $x$, $y$ are as in the lemma, and that the lemma holds for all triples $(\phi', x', y')$ with smaller degree. If $x\leq y$, then we can choose $Q$ the identity map, $\phi=\psi$, and we are done. Let us thus assume that $x\nleq y$. If $x_{i_0}=y_{i_0}$ for some index $i_0$, then we can write $x=x_{i_0}E_{i_0}+\tilde x$ and $y=x_{i_0}E_{i_0}+\tilde y$, where $\tilde x,\tilde y$ belong to $S:=\mathrm{span}(E_i)_{i\neq i_0}\cong [0,\infty]^{n-1}$. By weak cancellation, $\phi(x)\ll \phi(y)$ implies that $\phi(\tilde x)\ll \phi(\tilde y)$. Since $\tilde x,\tilde y$ belong to a space of smaller dimension, the degree of $(\phi|_S, \tilde x,\tilde y)$ is smaller than that of $(\phi,x,y)$ ($M,n_1,n_2$ have not increased, while $n$ has decreased). By the induction hypothesis, there exist maps $\tilde Q\colon S\to [0,\infty]^N$ and $\tilde\psi\colon [0,\infty]^N\to \mathrm{Lsc}_\sigma(C)$ such that $\tilde Q\tilde x\leq \tilde Q\tilde y$
and $\phi|_S=\tilde\psi\tilde Q$. Define $Q\colon [0,\infty]^n\to [0,\infty]^{N+1}$ as the extension of $\tilde Q$ such that $QE_{i_0} = E_{N+1}$. Extend $\tilde \psi$ to $[0,\infty]^{N+1}$ setting $\psi(E_{N+1})=f_{i_0}$. Then $\phi=\psi Q$ and \[ Qx=\tilde Q\tilde x+x_{i_0}E_{N+1}\leq \tilde Q\tilde y+y_{i_0}E_{N+1}=Qy, \] thus again completing the induction step.
We assume in the sequel that $x_i\neq y_i$, i.e., either $x_i<y_i$ or $x_i>y_i$, for all $i=1,\ldots,n$. Let $I=\{i:x_i>y_i\}$ and $J=\{j:y_j>x_j\}$. Let $M_1=\max_{i\in I} x_i-y_i$ and $M_2=\max_{j\in J} y_j-x_j$. Then $M=\max (M_1,M_2)$. We break-up the rest of the proof into two cases.
\emph{Case $M_1\geq M_2$}. Using weak cancellation in \[ \sum_{i=1}^n x_if_i = \phi(x) \ll \phi(y)=\sum_{i=1}^n y_if_i \] we get \[ \sum_{i\in I} (x_i-y_i)f_i \ll \sum_{j\in J}(y_j-x_j)f_j. \] Let $i_1\in I$ be such that $x_{i_1}-y_{i_1}=M_1$. From the last inequality we deduce that \[ M_1 f_{i_1}\ll \sum_{j\in J} M_2f_j, \] and since $M_2\leq M_1$, we get $f_{i_1}\ll \sum_{j\in J} f_j$. By the Riesz decomposition property in $\mathrm{A}(C)$ (Theorem \ref{ACRiesz}), there exist $g_j,h_j\in \mathrm{A}(C)$, with $j\in J$, such that \[ f_{i_1}=\sum_{j\in J} g_j\hbox{ and }f_j=g_j+h_j\hbox{ for all }j\in J. \]
Let $N_1=n+|J|-1$, and let us label the canonical generators of $[0,\infty]^{N_1}$ with the set $\{E_i:i=1,\ldots,n, \,i\neq i_1\}\cup\{G_j:j\in J\}$. Define $Q_1\colon [0,\infty]^{n}\to [0,\infty]^{N_1}$ as follows: \begin{align*} Q_1E_i &=E_i\hbox{ if }i\in I\backslash \{i_1\},\\ Q_1E_{i_1} &= \sum_{j\in J} G_j,\\ Q_1E_{j} &= E_j+G_j\hbox{ if }j\in J, \end{align*} and extend $Q_1$ to a Cu-cone morphism on $[0,\infty]^n$. Next, define a Cu-cone morphism $\psi_1\colon [0,\infty]^{N_1}\to \mathrm{Lsc}_\sigma(C)$ on the same generators as follows: \begin{align*} \psi_1(E_i) &= f_i,\hbox{ if }i\in I\backslash\{i_1\},\\ \psi_1(E_j) &= h_j, \hbox{ if }j\in J,\\ \psi_1(G_j) &= g_j, \hbox{ if }j\in J. \end{align*} It is easily checked that $\psi_1Q_1=\phi$ and that $Q_1$ maps $[0,\infty]^n\cap \mathbb{Z}^n$ to $[0,\infty]^{N_1}\cap [0,\infty]^{N_1}$. Also, \begin{align*} Q_1x &=\sum_{i\in I\backslash\{i_1\}}x_iE_i + \sum_{j\in J}x_{i_1}G_j+\sum_{j\in J}x_j(E_j+G_j)\\ &=\sum_{i\neq i_1} x_iE_i + \sum_{j\in J}(x_{i_1}+x_j)G_j. \end{align*} Similarly, \[ Q_1y =\sum_{i\neq i_1} y_iE_i + \sum_{j\in J}(y_{i_1}+y_j)G_j. \] We claim that $\deg (\psi_1,Q_1x,Q_1y)< \deg (\phi,x,y)$. Indeed, the maximum of the differences of the coordinates ($M$ above) has not gotten larger. Moreover, the number of times that $M_1$ is attained ($n_1$ above) is smaller, since we have removed the coordinate $i_1$ and added new coordinates for which \[ (x_{i_1}+x_j)-(y_{i_1}+y_j)=M_1 +x_j-y_j\in [0,M_1-1]. \] By induction, the lemma holds for $(\psi_1,Q_1x,Q_1y)$. Thus, there exist Cu-morphisms $Q_2\colon [0,\infty]^{N_1}\to [0,\infty]^{N_2}$ and $\psi_2\colon [0,\infty]^{N_2}\to \mathrm{Lsc}_\sigma(C)$ such that $Q_2Q_1x\leq Q_2Q_1y$ and $\psi_1=\psi_2Q_2$. Setting $Q=Q_1Q_2$ and $\psi=\psi_2$, we get the desired result.
\emph{Case $M_2>M_1$}. This case is handled similarly to the previous case, though with a few added complications. Observe first that $M_2\geq 2$ (since $M_1\geq 1$; otherwise $x\leq y$). Choose $\epsilon>0$ such that $\phi(x)\ll (1-\epsilon)\phi(y)$. If necessary, make $\epsilon$ smaller, so that we also have \[ \epsilon<\min\{\frac 1{4x_i},\frac{1}{4y_j}:x_i\neq 0, y_j\neq 0\}. \] Notice that this implies that \begin{equation} \begin{aligned}\label{ineqsxiyi} x_i>(1-2\epsilon)y_i&\Leftrightarrow x_i>y_i, \hbox{ for }i=1,2,\ldots,n,\\ x_i<(1-2\epsilon)y_i&\Leftrightarrow x_i<y_i, \hbox{ for }i=1,2,\ldots,n. \end{aligned} \end{equation}
Let $h\in \mathrm{A}(C)$ be such that $h+\phi(x)=(1-\epsilon)\phi(y)$, which exists by Lemma \ref{lhdll}. Enlarge the domain of $\phi$ to $[0,\infty]^{n+1}$, labelling the new generator by $H$ ($=(0,\ldots,0,1)$), and setting $\phi(H)=h$. We then have $(1-2\epsilon)\phi(y)\ll h+\phi(x)$, i.e., \[ \sum_{i=1}^n (1-2\epsilon)y_if_i \ll h+\sum_{i=1}^n x_if_i. \] Using weak cancellation and the inequalities \eqref{ineqsxiyi} we can move terms around to get \[ \sum_{j\in J} ((1-2\epsilon)y_j-x_j)f_j \ll h+\sum_{i\in I}(x_i-(1-2\epsilon)y_i)f_i. \] Let $j_1\in J$ be such that $y_{j_1}-x_{j_1}=M_2$. Then \[ ((1-2\epsilon)y_{j_1}-x_{j_1})f_{j_1} \ll h +\sum_{i\in I} (x_i-(1-2\epsilon)y_i)f_i. \] By our choice of $\epsilon$, we have the inequalities \[ (1-2\epsilon)y_{j_1}-x_{j_1}\geq M_2-\frac12\hbox{ and }x_{i}-(1-2\epsilon)y_i\leq M_1+\frac12\hbox{ for all }i. \] Hence, \[ (M_2-\frac12)f_{j_1}\ll h+\sum_{i\in I} (M_1+\frac12)f_i. \] Further, $M_1+\frac12\leq M_2-\frac{1}{2}$ (since $M_2>M_1$) and $M_2-\frac12>1$ (since $M_2\geq 2$). So \[ f_{j_1}\ll h+\sum_{i\in I} f_i. \]
By the Riesz decomposition property in $\mathrm{A}(C)$ (Theorem \ref{ACRiesz}), $f_{j_1}=h'+\sum_{i\in I}g_i$ for some $h'\ll h$ and $g_i\ll f_i$, with $i\in I$. Let us choose $h'',h_i\in \mathrm{A}(C)$ such that $h=h'+h''$ and $f_i=g_i+h_i$ for all $i\in I$ (Lemma \ref{lhdll}). Label the canonical generators of the Cu-cone $[0,\infty]^{N_1}$, where $N_1=n+|I|+1$, with the set \[ \{E_{j}:j=1,\dots,n,\, j\neq j_1\}\cup \{G_i:i\in I\}\cup \{H,H'\}. \] Define a Cu-cone morphism $Q_1\colon [0,\infty]^{n+1}\to [0,\infty]^{N_1}$ as follows: \begin{align*} Q_1E_j &=E_j\hbox{ for }j\in J\backslash \{j_1\},\\ Q_1E_{j_1} &= H'+\sum_{i\in I}G_i,\\ Q_1E_i &= E_i+G_i \hbox{ for }i\in I,\\ Q_1H &=H+H', \end{align*} Next, define a Cu-cone map $\psi_1\colon [0,\infty]^{N_1}\to \mathrm{Lsc}_{\sigma}(C)$ by \begin{align*} \psi_1 E_j &= f_j\hbox{ for }j\in J\backslash \{j_1\}\\ \psi_1 E_i &=h_i, \hbox{ for }i\in I,\\ \psi_1 G_i &= g_i,\hbox{ for }i\in I,\\ \psi_1 H &=h''\hbox{ and }\psi_1 H'=h'. \end{align*} Now $\psi_1 Q_1E_j=f_j$ for $j\in J\backslash\{j_1\}$, and \[\psi_1 Q_1E_{j_1}=\psi_1\left(H'+\sum_{i\in I}G_i\right)=h'+\sum_{i\in I} g_i=f_{j_1}.\] Also, \[ \psi_1 Q_1E_i=\psi_1(E_i+G_i)=h_i+g_i=f_i,\hbox{ for }i\in I. \] Finally, $\psi_1 Q_1H=h'+h''=h$. Thus, we have checked that $\psi_1 Q_1=\phi$. Clearly, $Q_1$ maps integer valued vectors to integer valued vectors.
Let us examine the degree of $(\psi_1, Q_1(x+H),Q_1y)$. We have that \begin{align*} Q_1(x+H)&=\sum_{j\in J\backslash\{j_1\}}x_jE_j+\sum_{i\in I}x_{j_1}G_i+x_{j_1}H'+\sum_{i\in I}x_i(E_i+G_i)+(H+H')\\
&=\sum_{j\ne j_1}x_jE_j+\sum_{i\in I}(x_{j_1}+x_i)G_i+H+(x_{j_1}+1)H'. \end{align*} Similarly, we compute that \[ Q_1y=\sum_{j\ne j_1}y_jE_j+\sum_{i\in I}(y_{j_1}+y_i)G_i+H+y_{j_1}H'. \] We claim that the $\deg(\psi_1,Q_1(x+H),Q_1y)<\deg(\phi,x,y)$. To show this we check that for the pair $(Q_1(x+H),Q_1y)$ we have that: \begin{enumerate} \item the maximum coordinates difference for the indices $i$ such that $x_i>y_i$ (number $M_1$ above) is strictly less than $M_2$, \item the maximum coordinates difference for the indices where $y_j>x_j$ is at most $M_2$, \item the number of indices for which $M_2$ is attained (number $n_2$ above) has decreased relative to the pair $(x,y)$. \end{enumerate} The first two points are straightforward to check. The last point follows from the fact that we have removed the coordinate $j_1$, and that for the new coordinates that we have added we have \begin{align*} (y_{j_1}+y_i)-(x_{j_1}+x_i)&=M_2+(y_i-x_i)\in [0,M_2-1],\\ y_{j_1}-(x_{j_1}+1)&=M_2-1<M_2. \end{align*}
Observe that \[ (\psi_1Q_1)(x+H)=h+\phi(x)=(1-\epsilon)\phi(y)\ll \phi(y)=\psi_1Q_1y. \] Hence, by the induction hypothesis, there exist $Q_2$ and $\psi_2$ such that $\psi_1=\psi_2Q_2$ and $Q_2Q_1(x+H)\leq Q_2Q_1y$. Then $Q=Q_2Q_1$ and $\psi=\psi_2$ are as desired, thus completing the step of the induction. \end{proof}
\begin{theorem}\label{triangletheorem} Let $\phi\colon [0,\infty]^n\to \mathrm{Lsc}_\sigma(C)$ be a Cu-morphism. Let $F\subset [0,\infty)^n$ be a finite set. Then there exist $N\in \mathbb{N}$ and Cu-morphisms \[ [0,\infty]^n \stackrel{Q}{\longrightarrow} [0,\infty]^N\stackrel{\psi}{\longrightarrow} \mathrm{Lsc}_\sigma(C), \] such that $\psi Q=\phi$, \[ \phi x\ll \phi y\implies Qx\ll Qy\hbox{ for all }x,y\in F, \] and $Q$ maps $[0,\infty]^n\cap \mathbb{Z}^n$ to $[0,\infty]^N\cap \mathbb{Z}^{N}$. \end{theorem}
\begin{proof} We start by noting that given elements $x=(x_i)_{i=1}^n$ and $y=(y_i)_{i=1}^n$ in $[0,\infty]^n$, we have $x\ll y$ if and only if
$x_i<y_i$ or $x_i=y_i=0$ for all $i=1,\dots,n$.
Suppose first that $F=\{x,y\}\subseteq [0,\infty)^n$ and that $\phi(x)\ll\phi(y)$. Choose $\varepsilon>0$ such that $(1+\varepsilon)\phi(x)\ll (1-\varepsilon)\phi(y)$. Choose $x',y'\in [0,\infty)^n\cap\mathbb{Q}^n$ such that $x\ll x'\le (1+\varepsilon)x$ and $(1-\varepsilon)y\le y'\ll y$. Then $\phi(x')\ll \phi(y')$. Let $m\in \mathbb{N}$ be such that $mx',my'\in [0,\infty)^n\cap\mathbb{Z}^n$. By Lemma \ref{coretrianglelemma}, there exist $Q,\psi$ such that $\phi=\psi Q$ and $Q(mx')\le Q(my')$, i.e., $Qx'\le Qy'$. Then \[ Qx\ll Qx'\le Qy'\ll Qy. \] Lemma \ref{coretrianglelemma} also guarantees that $Q$ maps integer valued vectors to integer valued vectors. Thus, $Q$ and $\psi$ are as desired.
To deal with an arbitrary finite set $F\subseteq [0,\infty)^n$, choose $x,y\in F$ such that $\phi(x)\ll \phi(y)$ and obtain $Q_1,\psi_1$ such that $\phi=\psi_1Q_1$ and $Q_1x\ll Q_1y$. Set $F_1=Q_1F$ and apply the same argument to a new pair $x',y'\in F_1$ to obtain maps $Q_2,\psi_2$. Continue inductively until all pairs have been exhausted. Set $Q=Q_k\cdots Q_1$ and $\psi=\psi_k$. \end{proof}
\subsection{Building the limit}
\begin{theorem}\label{inductiveCucones} Let $C$ be an extended Choquet cone that is strongly connected and has an abundance of compact idempotents. Then $\mathrm{Lsc}_\sigma(C)$ is an inductive limit in the Cu-category of an inductive system of Cu-cones of the form $[0,\infty]^n$, $n\in \mathbb{N}$, and with Cu-morphisms that map integer valued vectors to integer valued vectors. Moreover, if $C$ is metrizable, then this inductive system can be chosen over a countable index set. \end{theorem}
\begin{proof} For each $n=1,2,\ldots$, choose an increasing sequence $(A_k^{(n)})_{k=1}^\infty$ of finite subsets of $[0,\infty)^n$ with dense union in $[0,\infty]^n$.
We will construct an inductive system of Cu-cones $\{S_F,\phi_{G,F}\}$, where $F,G$ range through the finite subsets of $\mathrm{A}(C)$, such that $S_F\cong [0,\infty]^{n_F}$ for all $F$. We also construct Cu-morphisms $\psi_{F}\colon S_F\to \mathrm{Lsc}_\sigma(C)$ for all $F$, finite subset of $\mathrm{A}(C)$, making the overall diagram commutative. We follow closely the presentation of the proof of the Effros-Shen-Handelmann theorem given in \cite{goodearl-wehrung}, adapted to the category of Cu-cones.
For each $f\in \mathrm{A}(C)$, define $S_{\{f\}}=[0,\infty]$ and $\psi_{\{f\}}\colon [0,\infty]\to \mathrm{Lsc}_\sigma(C)$ as the Cu-morphism such that $\psi_{\{f\}}(1)=f$. Fix a finite set $F\subseteq \mathrm{A}(C)$. Suppose that we have defined $S_G$ and $\psi_G$ for all proper subsets $G$ of $F$. Set $S^F:=\prod_G S_G$, where $G$ ranges though all proper subsets of $F$. Define $\phi^F\colon S^F\to \mathrm{Lsc}_\sigma(C)$ as \[ \phi^F((s_G)_G)=\sum_G \psi_G(s^G). \]
Next, we construct $Q\colon S^F\to S_F$ and $\psi\colon S_F\to \mathrm{Lsc}_\sigma(C)$ using Theorem \ref{triangletheorem}. Here is how: For each $G$, proper subset of $F$, let $n_G$ be such that $S_G\cong [0,\infty]^{n_G}$. Let $A=\prod_G A^{(n_G)}_{k}$, where $k=|F|$ and where $G$ ranges through all proper subsets of $F$. Then $A$ is a finite subset of $S^F$. Let us apply Theorem \ref{triangletheorem} to $\phi^F$ and the set $A$, in order to obtain maps $Q\colon S^F\to S_F\cong [0,\infty]^{n_F}$ and $\psi\colon S_F\to \mathrm{Lsc}_\sigma(C)$ such that $\phi^F=\psi Q$ and \[ \phi^F(x)\ll \phi^F(y)\Rightarrow Qx\ll Qy\hbox{ for all }x,y\in A. \]
Set $\psi_{F}=\psi$, and for each proper subset $G$ of $F$, define $\phi_{G,F}\colon S_G\to S_F$ as the composition of the embedding of $S_G$ in $S^F$ with the map $Q$: \[ S_G\hookrightarrow S^F\stackrel{Q}{\to} S_F. \] Observe that $\phi_{G,F}$ maps $[0,\infty]^{n_G}\cap \mathbb{Z}^{n_G}$ to $[0,\infty]^{n_F}\cap \mathbb{Z}^{n_F}$, as both $Q$ and $S_G\hookrightarrow S^F$ map integer valued vectors to integer valued vectors. Continuing in this way we obtain an inductive system $\{S_F,\phi_{G,F}\}$, indexed by the finite subsets of $\mathrm{A}(C)$, and maps $\psi_F\colon S_F\to \mathrm{Lsc}_{\sigma}(C)$ for all $F$. By construction, the overall diagram is commutative. To show that $\mathrm{Lsc}_\sigma(C)$ is the inductive limit in the Cu-category of this inductive system, we must check that \begin{enumerate} \item every element in $\mathrm{Lsc}_\sigma(C)$ is supremum of an increasing sequence contained in the union of the ranges of the maps $\psi_F$, \item for each finite set $F$ (index of the system) and elements $x',x,y\in S_F$ such that $x'\ll x$ and $\psi_{G}(x)\leq\psi_G(y)$ in $\mathrm{Lsc}_\sigma(C)$, there exists $F'\supset F$ such that $\phi_{F,F'}(x')\ll \phi_{F,F'}(y)$. \end{enumerate} Let's check the first property. By construction, if $F=\{f\}$ then $f$ is contained in the range of $\psi_F$. Examining the construction of $\psi_F$ for arbitrary $F$, it becomes clear that $F$ is contained in the range of $\psi_F$. Thus, as $F$ ranges through all finite subsets of $\mathrm{A}(C)$, the union of the ranges of the maps $\psi_F$ contains $\mathrm{A}(C)$. Moreover, by Theorem \ref{ACsuprema}, every function in $\mathrm{Lsc}_\sigma(C)$ is the supremum of an increasing sequence in $\mathrm{A}(C)$.
Suppose that $x',x,y\in S_F$ are such that $\psi_F(x)\leq \psi_F(y)$ and $x'\ll x$.
Then $x'\in [0,\infty)^{n_F}$ and $\psi_F(x')\ll \psi_F(y)$. Choose $y'\ll y$ and $x'\ll x''\ll x$ such that $\psi_F(x'')\ll \psi_F(y')$. Next, choose $v,w\in A_k^{(n_F)}$ for some $k$, such that $x'\ll u \ll x''$ and $y'\ll v \ll y$. Observe then that $\psi_F(u)\ll \psi_F(v)$. Let $F'\subset \mathrm{A}(C)$ be a finite set such that $F\subset F'$ and $|F'|\geq k$. Then, by our construction of the inductive system, we have that $\phi_{F,F'}(u)\ll \phi_{F,F'}(v)$. This implies that $\phi_{F,F'}(x')\ll \phi_{F,F'}(y)$, thus proving the second property of an inductive limit.
Let us address the second part of the theorem. Suppose that $C$ is metrizable. By Theorem \ref{metrizableC}, there exists a countable set $B\subseteq \mathrm{A}(C)$ such that every function in $\mathrm{Lsc}_\sigma(C)$ is the supremum of an increasing sequence in $B$. The construction of the inductive limit for $\mathrm{Lsc}_\sigma(C)$ in the preceding paragraphs can be repeated mutatis mutandis, letting the index set of the inductive limit be the set of finite subsets of $B$, rather than the finite subsets of $\mathrm{A}(C)$. The resulting inductive limit is thus indexed by a countable set. \end{proof}
We are now ready to proof Theorem \ref{mainchar} from the introduction.
\begin{proof}[Proof of Theorem \ref{mainchar}] (i)$\Rightarrow$(iv): An AF C*-algebra has real rank zero, stable rank one, and is exact (these properties hold for finite dimensional C*-algebras and are passed on to their inductive limits). Thus, (i) implies (iv) by Proposition \ref{cstarECC}.
(iv)$\Rightarrow$(iii): Suppose that we have (iv). By Theorem \ref{inductiveCucones}, $\mathrm{Lsc}_{\sigma}(C)$ is an inductive limit in the Cu-category of Cu-cones of the form $[0,\infty]^n$, with $n\in \mathbb{N}$. We have $F([0,\infty]^n)\cong [0,\infty]^n$
via the map
\[
F([0,\infty]^n)\ni \lambda\mapsto (\lambda(E_1),\ldots,\lambda(E_n))\in [0,\infty]^n,
\]
where $E_1,\ldots,E_n$ are the canonical generators of $[0,\infty]^n$. Applying the functor $F(\cdot)$ to the inductive system with limit $\mathrm{Lsc}_\sigma(C)$ we obtain a projective system in the category of extended Choquet cones where each cone is isomorphic to $[0,\infty]^n$ for some $n$. By the continuity of the functor $F(\cdot)$ (\cite[Theorem 4.8]{ERS}), and the fact that $F(\mathrm{Lsc}_\sigma(C))\cong C$ (Theorem \ref{dualitythm}), we get (iii).
(iii)$\Rightarrow$(ii): Suppose that we have (iii). Say $C=\varprojlim_{i\in I} ([0,\infty]^{n_i}, \alpha_{i,j})$. Observe that $\alpha_{i,j}$ maps $[0,\infty)^{n_i}$ to $[0,\infty)^{n_j}$. Indeed, the support idempotent of an element in $[0,\infty)^{n_i}$ is 0. By continuity of $\alpha_{i,j}$, the same holds for the image of these elements; thus, they belong to $[0,\infty)^{n_j}$. It follows then that $\alpha_{i,j}$ is given by multiplication by a matrix $M_{i,j}$ with non-negative finite entries: $\alpha_{i,j}(v)=M_{i,j}v$ for all $v\in [0,\infty]^{n_i}$ (in $M_{i,j}v$ we regard $v$ as a column vector and use the rule $0\cdot \infty =0$). The transpose matrix $M_{i,j}^t$ can then be regarded as a map from $\mathbb{R}^{n_j}$ to $\mathbb{R}^{n_i}$. Let us form an inductive system of dimension groups whose objects are $\mathbb{R}^{n_i}$, endowed with the coordinatewise order, with $i\in I$, and with maps $M_{i,j}^t\colon \mathbb{R}^{n_j}\to \mathbb{R}^{n_i}$. This inductive system of dimension groups gives rise to the original system after applying the functor $\mathrm{Hom}(\,\cdot\,,[0,\infty])$ to it, and making the isomorphism identifications $\mathrm{Hom}(\mathbb{R}_+^{n_i},[0,\infty])\cong [0,\infty]^{n_i}$. Let $G$ be its limit in the category of dimension groups ($G$ is in fact a vector space). By the continuity of the functor $\mathrm{Hom}(\,\cdot\,,[0,\infty])$, we have $\mathrm{Hom}(G_+,[0,\infty])\cong C$. Thus, (iii) implies (ii).
(ii)$\Rightarrow$(i): By Elliott's theorem, there exists an AF C*-algebra $A$ whose Murray-von Neumann monoid of projections $V(A)$ is isomorphic to $G_+$. The result now follows from the fact, well known to experts, that $T(A)\cong \mathrm{Hom}(V(A),[0,\infty])$ for an AF $A$ (where $\mathrm{Hom}(V(A),[0,\infty])$ denotes the cone of monoid morphisms). Let us sketch a proof of this fact here:
Since AF C*-algebras are exact, we have by Haagerup's theorem that 2-quasitraces on $A$, and on the ideals of $A$, are traces. We apply here the version due to Blanchard and Kirchberg that includes densely finite lower semicontinuous 2-quasitraces; see \cite[Remark 2.29 (i)]{blanchard-kirchberg}. Thus, $T(A)=QT(A)$, where $QT(A)$ denotes the cone of lower semicontinuous $[0,\infty]$-valued 2-quasitraces on $A$. Further, by \cite[Theorem 4.4]{ERS}, $QT(A)\cong F(\mathrm{Cu}(A))$ for any C*-algebra $A$. Thus, we must show that $F(\mathrm{Cu}(A))\cong \mathrm{Hom}(V(A),[0,\infty])$ when $A$ is an AF C*-algebra. Let $\mathrm{Cu}_c(A)$ denote the submonoid of $\mathrm{Cu}(A)$ of compact elements, i.e., of elements $e\in \mathrm{Cu}(A)$ such that $e\ll e$. By \cite[Theorem 3.5]{brown-ciuperca} of Brown and Ciuperca, for stably finite $A$ the map from $V(A)$ to $\mathrm{Cu}(A)$ assigning to a Murray-von Neumann class $[p]_{\mathrm{MvN}}$ the Cuntz class $[p]_{\mathrm{Cu}}\in \mathrm{Cu}(A)$ is a monoid isomorphism with $\mathrm{Cu}_c(A)$. This holds in particular for $A$ AF. Thus, we must show that $F(\mathrm{Cu}(A))\cong \mathrm{Hom}(\mathrm{Cu}_c(A),[0,\infty])$. This isomorphism is given by the restriction map. Indeed, since $A$ has real rank zero and stable rank one, every element of $\mathrm{Cu}(A)$ is supremum of an increasing sequence of compact elements (\cite[Corollary 5]{CEI}). This shows that $\lambda\mapsto \lambda|_{\mathrm{Cu}_c(A)}$ is injective. To prove surjectivity, suppose that we have a monoid morphism $\tau\colon \mathrm{Cu}_c(A)\to [0,\infty]$. Define \[ \lambda(x)=\sup \{\tau(e):e\leq x,\, e\in \mathrm{Cu}_c(A)\}. \]
Then $\lambda$ is readily shown to be a functional on $\mathrm{Cu}(A)$ that extends $\tau$. Finally, from the definition of the topology on $F(\mathrm{Cu}(A))$ it is evident that a convergent net $(\lambda_i)_i$ in $F(\mathrm{Cu}(A))$ converges pointwise on compact elements of $\mathrm{Cu}(A)$. This shows that the map $\lambda\mapsto \lambda|_{\mathrm{Cu}_c(A)}$ is continuous. Since it is a bijection between compact Hausdorff spaces, its inverse is also continuous. In summary, we have the following chain of extended Choquet cones isomorphisms when $A$ is AF: \[ T(A)=QT(A)\cong F(\mathrm{Cu}(A))\cong \mathrm{Hom}(\mathrm{Cu}_c(A),[0,\infty])\cong \mathrm{Hom}(V(A),[0,\infty]). \]
Finally, suppose that $C$ is metrizable and satisfies (iv). Then, in the proof of (iv)$\Rightarrow$(iii) above, Theorem \ref{inductiveCucones} allows us to start with an inductive limit for $\mathrm{Lsc}_\sigma(C)$ over a countable index set. Applying the functor $F(\cdot)$, we get a projective limit for $C$ over a countable index set. Moreover, the Cu-morphisms in the inductive system of Theorem \ref{inductiveCucones} map integer valued vectors to integer valued vectors. Thus, the matrices $M_{i,j}$ implementing these morphisms have nonnegative integer entries. Thus, in the proof of (iii)$\Rightarrow$(ii) we start with $C=\varprojlim_{i\in I} ([0,\infty]^{n_i}, \alpha_{i,j})$, where $\alpha_{i,j}$ is implemented by a matrix with nonnegative integer entries. We can thus construct an inductive system $(\mathbb{Z}^{n_i}, M_{i,j})_{i,j\in I}$, in the category of dimension groups, whose limit is a countable dimension group $G$ such that $\mathrm{Hom}(G_+,[0,\infty])\cong C$, as desired. \end{proof}
\section{Finitely generated cones}\label{fingen}
A cone $C$ is called finitely generated if there exists a finite set $X\subseteq C$ such that for every $x\in C$ we have $x=\sum_{i=1}^n \alpha_ix_i$ for some $\alpha_i\in (0,\infty)$ and $x_i\in X$. In this section we give a direct construction of an ordered vector space (over $\mathbb{R}$) $(V,V^+)$ with the Riesz property and such that $\mathrm{Hom}(V^+,[0,\infty])$ is isomorphic to a given finitely generated, strongly connected, extended Choquet cone $C$. Here $\mathrm{Hom}(V^+,[0,\infty])$ denotes the monoid morphisms from $V^+$ to $[0,\infty]$. These maps are automatically homogeneous with respect to scalar multiplication; thus, they are also cone morphisms.
\begin{lemma}\label{fingenCw} Let $C$ be a finitely generated extended Choquet cone. Then $\mathrm{Idem}(C)$ is finite and for each $w\in \mathrm{Idem}(C)$ the sub-cone $C_w$ is either isomorphic to $\{0\}$ or to $[0,\infty)^{n_w}$ for some $n_w\in \mathbb{N}$. (Recall that we have defined $C_w=\{x\in C:\epsilon(x)=w\}$.) \end{lemma} \begin{proof} Let $Z$ be a finite set that generates $C$. Let $w\in C$ be an idempotent, and write $w=\sum_{i=1}^n \alpha_ix_i$, with $x_i\in Z$ and $\alpha_i\in (0,\infty)$. Multiplying both sides by a scalar $\delta>0$ and passing to the limit as $\delta\to 0$, we get that $w$ is the sum of support idempotents of elements in $Z$. It follows that $\mathrm{Idem}(C)$ is finite.
Next, let $w\in \mathrm{Idem}(C)$. Define $Z_w=\{x+w:x\in Z\hbox{ and }\epsilon(x)\leq w\}$, which is a finite subset of $C_w$. We claim that $Z_w$ generates $C_w$ as a cone. Indeed, let $x\in C_w$ and write $x=\sum_{i=1}^n \alpha_i x_i$, with $x_i\in Z$ and $\alpha_i\in (0,\infty)$. Adding $w$ on both sides we get $x=\sum_{i=1}^n \alpha_i(x_i+w)$. Since $\epsilon(x_i)\leq \epsilon(x)=w$, the elements $x_i+w$ are in $Z_w$. If $Z_w=\{w\}$ then $C_w$ is isomorphic to $\{0\}$. Suppose that $Z_w\neq \{w\}$. Since $w$ is a compact idempotent, $C_w$ has a compact base $K$ which is a Choquet simplex (Theorem \ref{compactbase}). Further, $K$ is finitely generated (by the set $(0,\infty)\cdot Z_w\cap K$). Hence, $K$ has finitely many extreme points, which in turn implies that $C_w\cong [0,\infty)^{n_w}$ for some $n_w\in \mathbb{N}$. \end{proof}
\emph{For the remainder of this section we assume that $C$ is a finitely generated, strongly connected, extended Choquet cone.} Thus, each idempotent $w\in \mathrm{Idem}(C)$ is compact and, by strong connectedness, $C_w\neq \{w\}$ for all $w\neq \infty$ (here $\infty$ denotes the largest element in $C$).
Let $w\in \mathrm{Idem}(C)$ and $x\in C_w$. If $z\in C$ is such that $z+w=x$, we call $z$ and extension of $x$. The set of extensions of $x$ is downward directed: if $z_1$ and $z_2$ are extensions of $x$, then so is $z_1\wedge z_2$. Consider the element $\tilde x=\inf\{z\in C:z+w=x\}$. By the continuity of addition, $\tilde x$ is also an extension of $x$, which we call the minimum extension.
\begin{lemma}\label{irreducibles} Let $w \in \mathrm{Idem}(C)$. Let $x\in C_w\backslash\{w\}$ be an element generating an extreme ray in $C_w$, and let $\tilde x$ denote the minimum extension of $x$.
\begin{enumerate}[\rm (i)] \item $\tilde x$ generates an extreme ray in $C_{\epsilon(\tilde x)}$.
\item If $y,z\in C$ are such that $y+z=\tilde x$, then either $y\leq z$ or $z\leq y$. \end{enumerate} \end{lemma}
\begin{proof} Set $v=\epsilon(\tilde x)$.
(i) Let $y,z\in C_v$ be such that $y+z=\tilde x$. Adding $w$ on both sides we get $(y+w)+(z+w)=x$. Since $y+w,z+w\in C_w$, and $x$ generates an extreme ray in $C_w$, both $y+w$ and $z+w$ are either positive scalar multiples of $x$ or equal to $w$. Assume that $y+w=w$ and $z+w=x$. The latter says that $z$ is an extension of $x$. Hence $y+z=\tilde x\leq z$ in $C_v$. By cancellation in $C_v$ (Lemma \ref{supportlemma}), we get $y=v$ and $z=\tilde x$. Suppose on the other hand that $y+w=\alpha x$ and $z+w=\beta x$ for positive scalars $\alpha,\beta$ such that $\alpha+\beta=1$. Then $y/\alpha$ and $z/\beta$ are extensions of $x$. We deduce that $\alpha \tilde x\leq y$ and $\beta \tilde x\leq z$. Hence, \[ \alpha \tilde x + z \leq y+z=\tilde x=\alpha \tilde x+\beta \tilde x. \] By cancellation in $C_v$, $z\leq \beta\tilde x$, and so $z=\beta\tilde x$. Similarly, $y=\alpha\tilde x$. Thus, $\tilde x$ generates an extreme ray in $C_v$.
(ii) The argument is similar to the one used in (ii). After arriving at $y+w=\alpha x$ and $z+w=\beta x$, we assume without loss of generality that $\alpha\leq \frac 12\leq \beta$. Using again that $\tilde x$ is the minimum extension of $x$, we get $z\geq \tilde x/2\geq y/2+z/2$, and applying Lemma \ref{supportlemma} (ii), we arrive at $z/2\geq y/2$. \end{proof}
\begin{remark} The property of $\tilde x$ in Lemma \ref{irreducibles} (ii) says that $\tilde x$ is an irreducible element of the cone $C$ in the sense defined by Thiel in \cite{thiel}. \end{remark}
Next, we construct a suitable set of generators of $C$. For each $w\in \mathrm{Idem}(C)$, let $X_w$ denote the set of minimal extensions of all elements $x\in C_w\backslash\{w\}$ that generate an extreme ray in $C_w$. Consider the set $\bigcup_{w\in \mathrm{Idem}(C)} X_w$, which is closed under scalar multiplication. We form a set $X$ by picking a representative from each ray in $\bigcup_{w\in \mathrm{Idem}(C)} X_w$.
\begin{proposition}\label{Xrepresentation} Let $X\subseteq C$ be as described in the paragraph above. Each $y\in C$ has a unique representation of the form \[ y=\sum_{i=1}^n \alpha_ix_i + w, \] where $x_i\in X$ and $\alpha_i\in (0,\infty)$ for all $i$, and $w\in \mathrm{Idem}(C)$ is such that $\epsilon(x_i)\leq w$ but $x_i\nleq w$ for all $i$. \end{proposition}
\begin{proof} Let $y\in C$, and set $w=\epsilon(y)$. If $y=w$ then its representation is simply $y=w$. Suppose that $y\neq w$. In $C_w$, express $y$ as a sum of elements that lie in extreme rays (Lemma \ref{fingenCw}). By the construction of $X$, these elements have the form $\alpha_i(x_i+w)$, with $x_i\in X$ and $\alpha_i\in (0,\infty)$. We thus have that \[ y=\sum_{i=1}^n\alpha_i(x_i+w) =\sum_{i=1}^n \alpha_i x_i + w. \] We have $x_i+w\in C_w\backslash\{w\}$ for all $i$; equivalently, $\epsilon(x_i)\leq w$ and $x_i\nleq w$ for all $i$. Thus, this is the desired representation.
To prove uniqueness of the representation, suppose that \[ y=\sum_{i\in I} \alpha_i x_i + w=\sum_{j\in J} \beta_j x_j + w'. \] Since $\epsilon(x_i)\leq w$ for all $i$, the support of $y$ is $w$. Thus, $w=w'$. We can now rewrite the equation above as \[ y=\sum_{i\in I} \alpha_i (x_i + w)=\sum_{j\in J} \beta_j (x_j + w). \] This equation occurs in $C_w\cong [0,\infty)^{n_w}$. Further, $x_i+w$ and $x_j+w$ generate extreme rays of $C_w$ for all $i,j$. It follows that $I=J$ and that the two representations are the same up to relabeling of the terms. \end{proof}
\subsection{Constructing the vector space}
We continue to denote by $X$ the subset of $C$ defined in the previous subsection. For each $w\in \mathrm{Idem}(C)$, define \[ O_w=\{x\in X: x\nleq w\}. \]
\begin{lemma}\label{Otopology} Let $w_1,w_2\in \mathrm{Idem}(C)$. Then \begin{enumerate}[\rm (i)] \item $O_{w_1}\cup O_{w_2}=O_{w_1\wedge w_2}$.
\item $O_{w_1}\cap O_{w_2}=O_{w_1+w_2}$.
\item $O_{w_1}\subseteq O_{w_2}$ if and only if $w_1\geq w_2$. \end{enumerate} \end{lemma}
\begin{proof} (i) It is more straightforward to work with the complements of the sets: $x\notin O_{w_1\wedge w_2}$ if and only if $x\leq w_1\wedge w_2$, if and only if $x\leq w_1$ and $x\leq w_2$, i.e., $x\notin O_{w_1}$ and $x\notin O_{w_2}$.
(ii) Again, we work with complements. Let's show that $O_{w_1+w_2}^c\subseteq O_{w_1}^c\cup O_{w_2}^c$ (the opposite inclusion is clear). Let $x\in O_{w_1+w_2}^c$, i.e., $x\leq w_1+w_2$. Choose $z$ such that $x\wedge w_1+z=x$. Recall that the elements of $X$ are minimal extensions of non-idempotent elements that generate an extreme ray. Thus, by Lemma \ref{irreducibles} (ii), either $x\wedge w_1\leq z$ or $z\leq x\wedge w_1$. If $z\leq x\wedge w_1$, then \[ x=x\wedge w_1+z\leq 2(x\wedge w_1)\leq w_1. \] Hence $x\in O_{w_1}^c$, and we are done. Suppose that $x\wedge w_1\leq z$. It follows that $2(x\wedge w_1)\leq x$. Now repeat the same argument with $x$ and $w_2$. We are done unless we also have that $2(x\wedge w_2)\leq x$. In this case, adding the inequalities we get $2(x\wedge w_1)+ 2(x\wedge w_2)\leq 2x$, i.e., $x\wedge w_1 + x\wedge w_2 \leq x$. But $x\leq x\wedge w_1+x\wedge w_2$ (since $x\leq w_1+w_2$). Hence, $x=x\wedge w_1+x\wedge w_2$. Applying Lemma \ref{irreducibles} (ii) again we get that either $x\leq 2(x\wedge w_1)\leq w_1$ or $x\leq 2(x\wedge w_2)\leq w_2$. Hence, $x\in O_{w_1}^c\cup O_{w_2}^c$, as desired.
(iii) Suppose that $O_{w_1}\subseteq O_{w_2}$. By (i), $O_{w_1\wedge w_2}=O_{w_1}\cup O_{w_2}=O_{w_2}$. Assume, for the sake of contradiction, that $w_1\wedge w_2\neq w_2$. Since $C$ is strongly connected, there exists $x\in C_{w_1\wedge w_2}\setminus\{w_1\wedge w_2\}$ such that $x \leq w_2$. We can choose $x$ in an extreme ray of $C_{w_1\wedge w_2}$, since the set of all $x\in C_{w_1\wedge w_2}$ such that $x\leq w_2$ is a face. Consider the minimum extension $\tilde x$ of $x$. Adjusting $x$ by a scalar multiple, we may assume that $\tilde x\in X$. Now $\tilde x\leq w_2$, i.e, $\tilde x\notin O_{w_2}$. But we cannot have $\tilde x\leq w_1\wedge w_2$, since this would imply that \[ x=\tilde x+w_1\wedge w_2 = w_1\wedge w_2. \] Thus, $x\in O_{w_1\wedge w_2}$. This contradicts that $O_{w_1\wedge w_2}=O_{w_2}$. \end{proof}
Let $w\in \mathrm{Idem}(C)$. Define \begin{align*} P_w &=\{x\in O_w:\epsilon(x)\leq w\},\\ \widetilde P_{w} &=P_w\cup O_w^c=\{x\in X:\epsilon(x)\leq w\}. \end{align*} Observe that if $y\in C$, and $y=\sum_{i=1}^{n}\alpha_ix_i+w$ is the representation of $y$ described in Proposition \ref{Xrepresentation}, then $x_i\in P_{w}$ for all $i, 1\le i\le n$.
\begin{lemma}\label{MTconditions} Let $w_1,w_2\in \mathrm{Idem}(C)$. The following statements hold: \begin{enumerate}[\rm (i)] \item $\widetilde P_{w_1\wedge w_2}=\widetilde P_{w_1}\cap \tilde P_{w_2}$. \item If $w_1\ngeq w_2$ then $P_{w_1}\backslash O_{w_2}\neq \varnothing$. \end{enumerate} \end{lemma}
\begin{proof} (i) This is straightforward: $\epsilon(x)\le w_1$ and $\epsilon(x)\leq w_2$ if and only if $\epsilon(x)\le w_1\wedge w_2$.
(ii) Suppose that $w_1\not\ge w_2$. Let $w_3=w_1+w_2$. By Lemma \ref{Otopology} (ii), $O_{w_1}\cap O_{w_2}=O_{w_3}$. Also $w_1\le w_3$ and $w_1\ne w_3$. Since $C$ is strongly connected, there exists $y\in C_{w_1}\setminus\{w_1\}$ such that $w_1\le y\le w_3$. Choose $y$ on an extreme ray (always possible, since the set of all $y\in C_{w_1}$ such that $y\le w_3$ is a face) and adjust it by a scalar so that its minimum extension $\tilde y$ belongs to $X$. Since $\tilde y+w_1\in C_{w_1}\backslash\{w_1\}$, we have that $\tilde y\nleq w_1$ and $\epsilon(\tilde y)\leq w_1$. That is, $\tilde y\in P_{w_1}$. Since $\tilde y\le w_3$, we also have that $\tilde y\in O_{w_3}^c\subseteq O_{w_2}^c$. We have thus obtained an element $\tilde y\in P_{w_1}\setminus O_{w_2}$. \end{proof}
Let us say that a function $f\colon X\to \mathbb{R}$ is positive provided that there exists $w\in \mathrm{Idem}(C)$ such that $f(x)=0$ for $x\notin O_w$ and $f(x)>0$ for $x\in P_w$. We call $w$ the support of $f$ and denote it by $\mathrm{supp}(f)$.
\begin{lemma}\label{supportV} The support of a positive function is unique. Further, if $f,g\colon X\to \mathbb{R}$ are positive then $\mathrm{supp}(f+g)=\mathrm{supp}(f)\wedge \mathrm{supp}(g)$. \end{lemma}
\begin{proof} Let $w_1,w_2\in \mathrm{Idem}(C)$ be both supports of $f$. Suppose that $w_1\ne w_2$, and without loss of generality, that $w_1\not\ge w_2$. Then there exists $x\in P_{w_1}\cap O_{w_2}^c$ (by Lemma \ref{MTconditions}). On one hand, $x\in P_{w_1}$ implies that $f(x)>0$. On the other hand, $x\in O_{w_2}^c$ implies that $f(x)=0$, a contradiction. Thus $w_1=w_2$, whereby proving the first part of the lemma.
To prove the second part, assume that $f$ and $g$ are positive functions on $X$, and set $v= \mathrm{supp}(f)$ and $w= \mathrm{supp}(g)$. Clearly $f+g$ vanishes on $ O_{v}^c\cap O_{w}^c= O_{v\wedge w}^c$. Let $x\in P_{v\wedge w}$. Then, by Lemma \ref{MTconditions} (i), $x\in \tilde{P}_{v}\cap\tilde{P}_{w}$. Thus, $x$ is in one of the following sets: $P_{v}\cap P_{w}$, $P_{\nu}\cap O_{w}^c$, or $P_{w}\cap O_{\nu}^c$. In all cases we see that $(f+g)(x)>0$. Indeed, if $x\in P_{\nu}\cap P_{w}$ then $f(x),g(x)>0$; if $x\in P_{\nu}\cap O_{w}^c$ then $f(x)>0$ and $g(x)=0$; if $x\in P_{w}\cap O_{\nu}^c$ then $f(x)=0$ and $g(x)>0$. Therefore $ \mathrm{supp}(f+g)=v\wedge w$. \end{proof}
Let us denote by $V_C$ the vector space of $\mathbb{R}$-valued functions on $X$ and by $V_C^+$ the set of positive functions in $V_C$.
\begin{theorem} The pair $(V_C,V_C^+)$ is an ordered vector space having the Riesz interpolation property. \end{theorem}
\begin{proof} By the previous lemma, $V_C^+$ is closed under addition. Clearly, $V_C^+$ is closed under multiplication by positive scalars. Since the pointwise strictly positive functions belong to $V_C^+$ and span $V_C$, we have $V_C^+-V_C^+=V_C$. Also, $V_C^+\cap -V_C^+=\{0\}$, for if $f$ and $-f$ are positive then, by the previous lemma, \[ \mathrm{supp}(f)\geq \mathrm{supp}(f+-f)=\mathrm{supp}(0)=\infty, \] which implies that $f=0$. Thus, $(V_C,V_C^+)$ is an ordered vector space.
In \cite{maloney-tikuisis}, Maloney and Tikuisis obtained conditions guaranteeing that the Riesz interpolation property holds in a finite dimensional ordered vector space. The properties of the sets $P_w$ obtained in Lemma \ref{MTconditions} (i) and (ii) are precisely those properties in \cite[Corollary 5.1]{maloney-tikuisis} shown to guarantee that the Riesz interpolation property holds in $(V_C,V_C^+)$. \end{proof}
Let us define a pairing $(\cdot,\cdot)\colon C\times V_C^+\to [0,\infty]$ as follows: for each $y\in C$ and $f\in V_C^+$, write $y=\sum_{i=1}^n \alpha_i x_i + w$, the representation of $y$ described in Proposition \ref{Xrepresentation}, and then set \[ (y,f)= \begin{cases} \sum\limits_{i=1}^{n}\alpha_if(x_i) & \hbox{if }w\le \mathrm{supp}(f),\\ \infty &\hbox{otherwise}. \end{cases} \]
\begin{theorem} The pairing defined above is bilinear. Moreover, the map $x\mapsto (x,\cdot)$, from $C$ to $\mathrm{Hom}(V_C^+,[0,\infty])$, is an isomorphism of extended Choquet cones. \end{theorem}
\begin{proof} Let $x,y\in C$ and $f\in V_C^+$. Write \begin{align*} x &= \sum_{i=1}^m \alpha_ix_i+v,\\ y &= \sum_{j=1}^n \beta_j y_j+w, \end{align*} with $v,w\in \mathrm{Idem}(C)$ and $x_i,y_j\in X$ as in Proposition \ref{Xrepresentation}. Then \[ x+y=\sum_{i=1}^m \alpha_ix_i+\sum_{j=1}^n \beta_j y_j+v+w. \] Observe that $\epsilon(x_i),\epsilon(y_j)\leq v+w$ and that $\alpha_i,\beta_j\in (0,\infty)$ for all $i,j$. Thus, the sum on the right side is the representation of $x+y$ described in Proposition \ref{Xrepresentation}, except for the possible repetition of elements of $X$ appearing both among the $x_i$s and the $y_j$s. If $v+w\leq \mathrm{supp}(f)$, then $v\leq \mathrm{supp}(f)$ and $w\leq \mathrm{supp}(f)$, and so \[ (x,f)+(y,f)=\sum_{i=1}^m \alpha_i f(x_i) + \sum_{j=1}^n \beta_jf(y_j)=(x+y,f). \] If, on the other hand, $v+w\nleq \mathrm{supp}(f)$, then either $v\nleq \mathrm{supp}(f)$ or $w\nleq \mathrm{supp}(f)$, and in either case $(x,f)+(y,f)=\infty=(x+y,f)$. This proves additivity on the first coordinate. Homogeneity with respect to scalar multiplication follows automatically from additivity.
Let $f,g\in V^+_C$ and $w\in \mathrm{Idem}(C)$. Then $w\le\mathrm{supp}(f+g)$ if and only if $w\le\mathrm{supp}(f)$ and $w\le\mathrm{supp}(g)$ (Lemma \ref{supportV}). This readily shows linearity on the second coordinate.
For each $x\in C$, let $\Lambda_x\in \mathrm{Hom}(V_C^+,[0,\infty])$ be defined by the pairing above: $\Lambda_x(f)=(x,f)$ for all $f\in V_C^+$. Let $\Lambda\colon C\to \mathrm{Hom}(V_C^+,[0,\infty])$ be the map given by $y\mapsto\Lambda_y$ for all $y\in C$. To prove that $\Lambda$ is injective, suppose that $y,z\in C$ are such that $\Lambda_y=\Lambda_z$. Choose any $f\in V^+_C$ such that $\mathrm{supp}(f)=\epsilon(y)$. If $\epsilon(y)\not\le \epsilon(z)$ then $\Lambda_y(f)$ is finite, while $\Lambda_z(f)=\infty$. This contradicts that $\Lambda_y=\Lambda_z$. Hence $\epsilon(y)\le \epsilon(z)$. By a similar argument $\epsilon(z)\le \epsilon(y)$, and so we get equality. Set $w=\epsilon(y)=\epsilon(z)$. Then we can write \begin{align*} y &=\sum_{i=1}^m\alpha_iy_i+w,\\ z &=\sum_{i=1}^n\beta_iz_i+w \end{align*} with $y_i,z_i\in P_{w}$ for all $i$. Let $f\in V_C$ be such that $\mathrm{supp}(f)=w$. Then $f(y_i),f(z_i)>0$ and \begin{equation}\label{Lambdayz} \sum_{i=1}^{m}\alpha_if(y_i)=\Lambda_y(f)=\Lambda_z(f)=\sum_{i=1}^{n}\beta_if(z_i). \end{equation} Let $V_{w}^+= \{f\in\ V^+_C\colon \mathrm{supp}(f)=w\}$, i.e., $f\in V_w^+$ if $f$ is positive on $P_w$ and zero outside $O_w$. It is clear that $V_w^+-V_w^+$ consists of all the functions on $X$ that vanish outside $O_w$. It then follows from \eqref{Lambdayz} that $n=m$ and that, up to relabelling, $y_i=z_i$ for all $1\le i\le n$. Consequently $y=z$.
Let us show that $\Lambda$ is surjective. Let $\lambda\in \mathrm{Hom}(V_C^+,[0,\infty])$. By Lemma \ref{supportV}, the set \[ \{w\in \mathrm{Idem}(C): w=\mathrm{supp}(f)\hbox{ for some }f\in V_C^+\hbox{ such that }\lambda(f)<\infty\} \] is closed under infima. Since this set is also finite, it has a minimum element $w$. We claim that for each $f\in V_C^+$ we have \[ \lambda(f)<\infty\Leftrightarrow w\leq \mathrm{supp}(f). \] Indeed, from the definition of $w$ it is clear that if $\lambda(f)<\infty$ then $w\leq \mathrm{supp}(f)$. Suppose on the other hand that $f\in V_C^+$ is such that $w\leq \mathrm{supp}(f)$. Let $f_0\in V_w^+$ be such that $\lambda(f_0)<\infty$. Then $\alpha f_0 - f$ is positive (with support $w$) for a sufficiently large scalar $\alpha\in (0,\infty)$. Thus, $\lambda(f)\leq \alpha\lambda(f_0)<\infty$.
Let us extend $\lambda$ by linearity to the vector subspace $V_w:=V_w^+-V_w^+$. As remarked above, $V_{w}$ consists of all the functions $f\colon X\to \mathbb{R}$ vanishing on the complement of $O_w$. That is, $V_{w}=\mathrm{span}(\{\mathbbm{1}_x\colon x\in O_w\})$, where $\mathbbm{1}_x$ denotes the characteristic function of $\{x\}$.
If $x\in P_w$, then $\mathbbm{1}_x+\epsilon\mathbbm{1}_{P_w}\in V_w^+$ for all $\epsilon>0$; here $\mathbbm{1}_{P_w}$ denotes the characteristic function of $P_w$. It follows that $\lambda(\mathbbm{1}_x+\epsilon\mathbbm{1}_{P_w})\geq 0$, and letting $\epsilon\to 0$, that $\lambda(\mathbbm{1}_x)\geq 0$ for all $x\in P_w$. If $x\in O_w\backslash P_w$, then $\lambda(\mathbbm{1}_{P_w}-\alpha\mathbbm{1}_x)\geq 0$ for all $\alpha\in \mathbb{R}$. It follows that $\lambda(\mathbbm{1}_x)=0$ for all $x\in O_w\setminus P_w$. Thus \[ \lambda(f)=\sum_{x\in P_{w}}\lambda(\mathbbm{1}_x)f(x) \] for all $f\in V_w$. Since $V_v^+\subseteq V_w$ for any idempotent $v$ such that $w\leq v$, the formula above holds for all $f\in V_C^+$ such that $w\leq \mathrm{supp}(f)$.
Define \[ y=\sum_{x\in P_{w}}\lambda(\mathbbm{1}_x)x+w. \] By the previous arguments, $\lambda(f)=\Lambda_y(f)$ for all $f$ such that $w\leq \mathrm{supp}(f)$. On the other hand, $\lambda(f)=\infty=\Lambda_y(f)$ for all $f$ such that $w\nleq \mathrm{supp}(f)$. Hence, $\lambda=\Lambda_y$. \end{proof}
\end{document} | arXiv |
\begin{document}
\begin{frontmatter}
\title{A super Ornstein--Uhlenbeck process interacting with its center of mass} \runtitle{SOU interacting with its COM}
\begin{aug} \author[A]{\fnms{Hardeep} \snm{Gill}\corref{}\ead[label=e1]{[email protected]}} \runauthor{H. Gill} \affiliation{University of British Columbia} \address[A]{Department of Mathematics\\ University of British Columbia\\ 1984, Mathematics Road \\ Vancouver, British Columbia\\ V6T1Z2, Canada\\ \printead{e1}} \end{aug}
\received{\smonth{8} \syear{2011}} \revised{\smonth{12} \syear{2011}}
\begin{abstract}
We construct a supercritical interacting measure-valued diffusion with representative particles that are attracted to, or repelled from, the center of mass. Using the historical stochastic calculus of Perkins, we modify a super Ornstein--Uhlenbeck process with attraction to its origin, and prove continuum analogues of results of Engl\"ander [\textit{Electron. J. Probab.} \textbf{15} (2010) 1938--1970] for binary branching Brownian motion.
It is shown, on the survival set, that in the attractive case the mass normalized interacting measure-valued process converges almost surely to the stationary distribution of the Ornstein--Uhlenbeck process, centered at the limiting value of its center of mass. In the same setting, it is proven that the normalized super Ornstein--Uhlenbeck process converges a.s. to a Gaussian random variable, which strengthens a theorem of Engl\"ander and Winter [\textit{Ann. Inst. Henri Poincar\'e Probab. Stat.} \textbf{42} (2006) 171--185] in this particular case. In the repelling setting, we show that the center of mass converges a.s., provided the repulsion is not too strong and then give a conjecture. This contrasts with the center of mass of an ordinary super Ornstein--Uhlenbeck process with repulsion, which is shown to diverge a.s.
A version of a result of Tribe [\textit{Ann. Probab.} \textbf{20} (1992) 286--311] is proven on the extinction set; that is, as it approaches the extinction time, the normalized process in both the attractive and repelling cases converges to a random point a.s. \end{abstract}
\begin{keyword}[class=AMS] \kwd[Primary ]{60J68} \kwd[; secondary ]{60G57}. \end{keyword}
\begin{keyword} \kwd{Superprocess} \kwd{interacting measure-valued diffusion} \kwd{Ornstein--Uhlenbeck process} \kwd{center of mass} \kwd{law of large numbers}. \end{keyword}
\end{frontmatter}
\section{Introduction and main results}\label{sec1}
The existence and uniqueness of a self-interacting measure-valued diffusion that is either attracted to or repelled from its centre of mass is shown below. It is natural to consider a super Ornstein--Uhlenbeck (SOU) process with attractor (repeller) given by the centre of mass of the process as it is the simplest diffusion of this sort.
This type of model first appeared in a recent paper of Engl\" ander \cite {Eng2010} where a $d$-dimensional binary Brownian motion, with each parent giving birth to exactly two offspring and branching occurring at integral times, is used to construct a binary branching Ornstein--Uhlenbeck process where each particle is attracted (repelled) by the center of mass (COM). This is done by solving the appropriate SDE along each branch of the particle system and then stitching these solutions together.
This model can be generalized such that the underlying process is a branching Brownian motion (BBM), $\mathscr{T}$ (i.e., with a general offspring distribution). We might then solve an SDE on each branch of $\mathscr{T}$:
\begin{equation}\label{Ebps}\quad Y_i^{n}(t) = Y_{p(i)}^{n-1}(n-1) + \gamma\int_{n-1}^t \bar{Y}_n(s) - Y_i^{n}(s) \,ds +\int_{n-1}^t dB_i^n(s)\vspace*{-2pt} \end{equation}
for $n-1< t\le n$, where $B_i^n$ labels the $i$th particle of $\mathscr{T}$ alive from time $n-1$ to $n$, $p(i)$ is the parent of $i$ and
\[ \bar{Y}_n(s) = \frac{1}{\tau_n} \sum_{i=1}^{\tau_n} Y_i^n(s)\vspace*{-2pt} \]
is the center of mass. Here, $\tau_n$ is the population of particles alive from time $n-1$ to~$n$. This constructs a branching OU system with attraction to the COM when $\gamma>0$ and repulsion when $\gamma<0$.
It seems reasonable then, to take a scaling limit of branching particle systems of this form and expect it to converge in distribution to a measure-valued process where the representative particles behave like an OU process attracting to (repelling from) the COM of the process. Though viable, this approach will be avoided in lieu of a second method utilizing the historical stochastic calculus of Perkins \cite {Perkins2002} which is more convenient for both constructing the SOU interacting with its COM and for proving various properties. The idea is to use a supercritical historical Brownian motion to construct the interactive SOU process by solving a certain stochastic equation. This approach for constructing interacting measure-valued diffusions was pioneered in \cite {Perkins1995} and utilized in, for example,~\cite{HG2009}.
A supercritical historical Brownian motion, $K$, is a stochastic process taking values in the space of measures over the space of paths in $\mathbb{R}^d$. One can think of~$K$ as a supercritical superprocess which has a path-valued Brownian motion as the underlying process. That is, if $B_t$ is a $d$-dimensional Brownian motion, then $\hat{B}_t = B_{\cdot\wedge t}$ is the underlying process of $K$. More information about $K$ is provided in Section~\ref{sec2}.
It can be shown that if a path $y\dvtx [0,\infty)\rightarrow\mathbb{R}^d$ is chosen according to $K_t$ (loosely speaking---this is made rigorous below in Definition~\ref{Dcamp}), then $y(s)$ is a Brownian motion stopped at $t$. Projecting down gives
\[ X_t^K(\cdot) = \int\mathbf{1}(y_t \in\cdot)K_t(dy),\vspace*{-2pt} \]
which is a (supercritical) super Brownian motion.\vadjust{\goodbreak}
A key advantage to projecting down and constructing measure-valued processes is that it is possible to use the historical stochastic calculus to couple different projections together (and hence couple measure-valued diffusions).
One can sensibly define the Ornstein--Uhlenbeck SDE driven by $y$ according to~$K_t$ as the solution of a stochastic equation.
\begin{definition} Let $Z_0\dvtx \mathbb{R}^d\rightarrow\mathbb{R}^d$ be Borel measurable. We say that $(X, Z)$ is a solution to the strong equation~\ref{se1} if the pair satisfies
{\renewcommand{$(\mathrm{HMP})_m$}{$\mbox{(SE)}_{Z_0, K}^1$} \begin{eqnarray}\label{se1} \qquad\qquad &&\mbox{(a)}\quad\hspace*{3.5pt} Z_t( y) = Z_0(y_0) +y_t-y_0-\gamma\int_0^t Z_s( y) \,ds,\qquad K\mbox{-a.e.}, \nonumber \\[-10pt] \\[-10pt] \nonumber&& \mbox{(b)}\quad X_t(A) = \int1(Z_t\in A)K_t(dy)\qquad \forall A\in\mathscr{B}(\mathbb{R} ^d)\ \forall t\ge0, \end{eqnarray}}
\hspace*{-2pt}where $X$ and $Z$ are appropriately adapted. We will henceforth call the projection~$X$ the ordinary super Ornstein--Uhlenbeck process. \end{definition}
If $\gamma>0$, $Z_t$ is attracted to the origin, and if $\gamma<0$ it is repelled. The approximate meaning of $K$-a.e. in (a) is that the statement holds for~$K_t$-a.a. $y$, for all~$K_t$, $\mathbb{P}$-a.s. The exact definition is given in the next section. The projection $X_t$ is a SOU process with attraction to (repulsion from) the origin at rate $\gamma $. Intuitively, $K$ tracks the underlying branching structure and $Z_t$ is a function transforming a typical Brownian path into a typical Ornstein--Uhlenbeck path.
Note that in the definitions of~\ref{se1} and~\ref{SE2} given below, part (b) is unnecessary to solve the equations. It has been included to provide an easy comparison to the strong equation of Chapter V.1 of~\cite{Perkins2002}.
For all the results mentioned in the remainder of this work, the standing assumption (unless indicated otherwise) will be that
\setcounter{equation}{1} \begin{equation}\label{Efinmass} \int1 \,dK_0 < \infty. \end{equation}
\begin{theorem}\label{TSOUrep} There is a pathwise unique solution $(X, Z)$ to~\ref{se1}. That is, $X$ is unique $\mathbb{P}$-a.s. and $Z$ $K$-a.e. unique. Furthermore, the map $t\rightarrow X_t$ is continuous and $X$ is a $\beta$-super-critical super Ornstein--Uhlenbeck process. \end{theorem}
Similar to the above, we establish a function of the path $y$ that is a path of an OU process with attraction (repulsion) to the COM and project down.
\begin{definition} Let $Y_0\dvtx \mathbb{R}^d\rightarrow\mathbb{R}^d$ be Borel measurable. Define ($X', Y$) as the solution of
{\renewcommand{$(\mathrm{HMP})_m$}{$\mbox{(SE)}_{Y_0, K}^2$} \begin{eqnarray} \label{SE2}
\qquad\qquad&&\textup{(a)}\quad\hspace*{4pt} Y_t( y) = Y_0(y_0) +y_t -y_0+\gamma\int_0^t \bar{Y}_s- Y_s( y) \,ds,\qquad K\mbox{-a.e. } \nonumber \\[-10pt] \\[-10pt] \nonumber&& \textup{(b)} \quad X'_t(A) = \int1(Y_t\in A)K_t(dy)\qquad \forall A\in\mathscr{B}(\mathbb{R} ^d)\ \forall t\ge0,
\end{eqnarray}}
\hspace*{-2pt}where the COM is
\[ \bar{Y}_s = \frac{\int x X'_s(dx)}{\int1 X'_s(dx)}. \]
We will call the projection $X'$ the super Ornstein--Uhlenbeck process with attraction (repulsion) to its COM, or the interacting super Ornstein--Uhlenbeck process. \end{definition}
Note that by our definitions of $X$ and $X'$ as solutions to $\mbox{(SE)}_{Y_0, K}^i, i=1,2$, respectively, that \eqref{Efinmass} is the same as saying that $ \int1 \,dX_0, \int1 \,dX'_0 <\infty, $ as these quantities equal that in \eqref{Efinmass}.
\begin{theorem}\label{TExUniq} There is a pathwise unique solution to~\ref{SE2}. \end{theorem}
One could prove this theorem using a combination of the proof of Theorem~\ref{TSOUrep} and a localization argument. We find it more profitable however, to employ a correspondence with the ordinary SOU process $X$. This correspondence plays a central role in the analysis of $X'$, and indeed reveals a very interesting structure: We have that for any $\gamma$,
\setcounter{equation}{2} \begin{equation} \label{Ecoupl} \int\phi(x) \,dX'_t(x) = \int\phi\biggl(x+ \gamma\int_0^t \bar{Z}_s \,ds\biggr) \,dX_t(x), \end{equation}
where $\bar{Z}$ is the COM of $X$, defined as $ \bar{Z}_s = \frac{\int x X_s(dx)}{\int1 X_s(dx)}.$ The correspondence essentially says that the SOU process with attraction (repulsion) to its COM is the same as the ordinary SOU process being dynamically pushed by its COM. From this equation, a relation between $\bar{Y}$ and $\bar{Z}$ can be established:
\[ \bar{Y}_t = \bar{Z}_t + \gamma\int_0^t \bar{Z}_s \,ds. \]
Define $ \tilde{X} _t = \frac{X_t}{X_t(1)}$ and $ \tilde{X} '_t = \frac {X_t'}{X'_t(1)}$. As the goal of this work is to prove that $ \tilde{X} '_t$ has interesting limiting behavior as $t$ approaches infinity, \eqref {Ecoupl} yields a method of approach: show first that the time integral in \eqref{Ecoupl} converges in some sense and establish limiting behavior for $ \tilde{X} $. One then hopes to combine these two facts with~\eqref{Ecoupl} to get the desired result.
Let $S$ be the event that $K_t$ survives indefinitely. Note that this implies that on $S$, both $X$ and $X'$ survive indefinitely by their definitions as solutions of the equations above. Let $\eta$ be the time at which $K_t$ goes extinct.
The next two theorems settle the question of what happens on the extinction set~$S^c$.
\begin{theorem}\label{TExtinc} On $S^c$, $\bar{Y}_t$ and $\bar{Z}_t$ converge as $t\uparrow\eta<\infty$, $\mathbb{P} $-a.s., for any $\gamma\in\mathbb{R}$. \end{theorem}
\begin{theorem}\label{TExtConv}
On the extinction set, $S^c$,
\[ \tilde{X} _t\rightarrow\delta_F\quad \mbox{and}\quad \tilde{X} '_t\rightarrow\delta_{F'} \]
as $t\uparrow\eta<\infty$ a.s., where $F$ and $F'$ are $\mathbb{R}^d$-valued random variables such that
\[ F' = F + \gamma\int_0^{\eta} \bar{Z}_s\,ds. \]
\end{theorem}
This last theorem is an analogue of the result of Tribe \cite {Tribe1992} for ordinary critical superprocesses. Note that here it does not matter whether there is attraction or repulsion from the COM.
The following three theorems for the attractive case ($\gamma>0$) form the main results of this work.
\begin{theorem}\label{TSurvival} On $S$ the following hold:
\begin{longlist}
\item[(a)] If $\gamma> 0$
\[ \bar{Z}_t \ascvi 0 \quad\mbox{and} \quad\bar{Y}_t \ascvi\gamma\int_0^\infty\bar{Z}_s \,ds, \]
and this integral is finite almost surely.
\item[(b)] If $\gamma=0$, then $\bar{Z}_t = \bar{Y}_t$ and this quantity converges almost surely. \end{longlist}
\end{theorem}
This says that the COMs of ordinary and interacting SOU process converge and is the result that allows us to fruitfully use the correspondence of \eqref{Ecoupl} to show convergence of the interacting SOU process.
The next theorem shows that the mass normalized SOU process converges almost surely, which is a new result among superprocesses. Engl\"ander and Winter in~\cite{EW2006} have shown that this process converges in probability, and before them, Engl\"ander and Turaev in~\cite{ET2002} shown convergence in distribution.
One expects a result of this sort to hold since in the particle picture conditional on survival, at large time horizons, there are a very large number of particles that move as independent OU processes, each of which are located in the vicinity of the origin. Thus, we expect that in the limit the mass will be distributed according to the limiting distribution of an OU process.
\begin{theorem} \label{TSOUconv} Suppose $\gamma>0$. Then on $S$,
\[ d( \tilde{X} _t, P_\infty)\ascvi0, \]
where $P_t$ is the semigroup of an Ornstein--Uhlenbeck process with attraction to $0$ at rate $\gamma,$ and $d$ is the Vasserstein metric on the space of finite measures on $\mathbb{R}^d$ (see Definition~\ref{DVas}). \end{theorem}
\begin{remark} It is possible to show that Theorem~\ref{TSOUconv} holds for a more general class of superprocesses. If the underlying process has an exponential rate of convergence to a stationary distribution, then the above theorem goes through. One can appeal to, for example, Theorem 4.2 of Tweedie and Roberts~\cite{TR2000} for a class of such continuous time processes. \end{remark}
Using the correspondence of \eqref{Ecoupl}, one can then show that the mass normalized interacting SOU process with attraction converges to a Gaussian distribution, centered at the limiting value of the COM, $\bar{Y} _\infty$.
\begin{theorem} \label{TISOUConv} Suppose $\gamma>0$. Then on $S$,
\begin{equation} \label{EIMconv} d( \tilde{X} '_t,P_\infty^{\bar{Y}_\infty}) \ascvi0, \end{equation}
where $P_\infty^{\bar{Y}_\infty}$ is the OU-semigroup at infinity, with the origin shifted to $\bar{Y}_\infty$. \end{theorem}
When there is repulsion, matters become more difficult on the survival set. It is no longer clear whether there exists a limiting random measure, or what the correct normalizing factor is. We can show however that in some cases the COM of the interacting SOU process still converges. That there should be a limiting measure comes from the fact that the ordinary repelling SOU process has been shown to converge in probability by Engl\"ander and Winter in~\cite{EW2006} to a multiple of Lebesgue measure. One may expect something similar to hold for the interacting SOU process, given \eqref{Ecoupl}. Unfortunately, the correspondence is rendered ineffectual in this case by part (b) of the following theorem.
\begin{theorem}\label{TrCOM} The following hold on $S$ if $0>\gamma> -\frac{\beta}{2}$:
\begin{longlist}[(a)]
\item[(a)] The process, $\bar{Y}_t$ converges almost surely.
\item[(b)] However, $\bar{Z}_t$ diverges exponentially fast. That is, there is a random variable~$L$, such that
\[
\mathbb{P}(e^{\gamma t}\bar{Z}_t \rightarrow L, L \neq0 | S) =1. \]
\end{longlist}
\end{theorem}
This reveals an interesting byplay between $X$ and $X'$ in the repelling case. That is, if one fixes a compact set $A\subset\mathbb{R}^d$, then for the ordinary SOU process, $X_t$, $A$ is exponentially distant from the COM of the process. However, the COM of $X'$ will possibly lie in the vicinity of $A$ for all time. Therefore, one might expect that $A$ is charged by a different amount of mass by $X'_t$ than $X_t$, and thus we might need to renormalize $X'_t$ differently to get a valid limit. It is also possible that the limit for each case is different (and not simply connected by a random translation).
The proofs for these theorems and more are contained in the following sections. In Section~\ref{sec2}, we give some background information for the historical process $K$ and some rigorous definitions. In Section~\ref{Sexpre}, we prove Theorems~\ref{TSOUrep} and~\ref{TExUniq} and state some important preliminary results regarding the nature of the support of a supercritical historical Brownian motion. These are consequently used to get moment bounds on the center of mass processes $\bar{Y}_t$ and $\bar{Z} _t$. We also derive a martingale problem for $ \tilde{X} $ and $ \tilde{X} '$. In Section~\ref{SConvergence}, we give the proofs of the convergence theorems mentioned above and in the final section give the proofs of technical results that are crucial to prove these.
\section{Definitions and background material}\label{sec2}
\begin{notation*} We collect some terms below:\vspace*{9pt}
\begin{tabular}{ l l } $E, E'$& Metric spaces \\ $C_c(E, E')$& Compact, cont. functions from $E$ to $E'$ \\ $C_b(E, E')$& Bounded, cont. functions from $E$ to $E'$ \\ $(E,\mathscr{E})$& Arbitrary measure space over $E$\\ $b\mathscr{E}$& Bounded, $\mathscr{E}$-mble. real-valued functions\\ $M_F(E)$& Space of finite measures on $E$\\ $\mu(f)= \int f\,d\mu$& where $f\dvtx E\rightarrow\mathbb{R}$ and $\mu$ a measure on $E$\\ $\mu(f) = (\mu(f_1),\ldots,\mu(f_n))$& if $f= (f_1,\ldots, f_n)$, $f_i\dvtx E\rightarrow\mathbb{R}$\\
$|p|$& $p\in\mathbb{R}^d$, denotes Euclidean norm of $p$\\
$\| f\| = \sup_{E} \sum\vert f_i(x)\vert$& if $f= (f_1,\ldots, f_n)$, $f_i\dvtx E\rightarrow\mathbb{R}$\\ $C = C(\mathbb{R}_+, \mathbb{R}^d)$& Space of continuous paths in $\mathbb{R}^d$ \\ $\mathscr{C}$& The Borel $\sigma$-field of $C$ \\ $y^t = y_{\cdot\wedge t}$& The path $y$ stopped at $t$ \\ $C^t = \{y^t\dvtx y \in C\}$& Set of all paths stopped at $t$\\ $\mathscr{C}_t = \sigma(y^s, s\le t, y \in C)$& Natural filtration associated with $\mathscr{C}$ \end{tabular}\vspace*{6pt}
\end{notation*}
We take $K$ to be a supercritical historical Brownian motion.
Specifically, let $K$ be a $(\Delta/2, \beta, 1)$-historical superprocess (here $\Delta$ is the $d$-dimensional Laplacian), where $\beta>0$ constant, on the probability space $(\Omega, \mathscr{F}, (\mathscr{F} _t)_{t\ge0}, \mathbb{P})$. Here $\beta$ corresponds to the branching bias in the offspring distribution, and the $1$ to the variance of the offspring distribution. A martingale problem characterizing $K$ is given below. For a more thorough explanation of historical Brownian motion than found here, see Section V.2 of~\cite{Perkins2002}.
It turns out that $K_t$ is supported on $C^t\subset C$ a.s. and typically, $K_t$ puts mass on those paths that are ``Brownian'' (until time $t$ and fixed thereafter). As $K$ takes values in $M_F(C)$, $K_t(\cdot)$ will denote integration over the $y$ variable.
Let $\hat{B}_t = B(\cdot\wedge t)$ be the path-valued process associated with $B,$ taking values in $C^t$. Then for $\phi\in b\mathscr{C}$, if $s\le t$ let $P_{s, t}\phi(y) = \mathbb{E}^{s,y}(\phi(\hat{B}_t))$, where the right-hand side denotes expectation at time $t$ given that until time $s$, $\hat{B}$ follows the path~$y$.
The weak generator, $\hat{A}$, of $\hat{B}$ is as follows. If $\phi\dvtx \mathbb{R} _+\times C \rightarrow\mathbb{R}$ we say $\phi\in\mathscr{D}(\hat{A})$ if and only if $\phi$ is bounded, continuous and ($\mathscr{C}_t$)-predictable, and for some $\hat{A}_s\phi(y)$ with the same properties as $\phi,$
\[ \phi(t, \hat{B}) - \phi(s, \hat{B}) - \int_s^t \hat{A}_r \phi (\hat {B})\,dr,\qquad t\ge s, \]
is a ($\mathscr{C}_t$)-martingale under $P_{s, y}$ for all $s\ge0, y\in C^s.$
If $m\in M_F(\mathbb{R}^d)$, we will say $K$ satisfies the historical martingale problem,~\ref{hmp}, if and only if $K_0 = m$ a.s. and $\forall\phi\in\mathscr{D}(\hat{A}),$
{\renewcommand{$(\mathrm{HMP})_m$}{$(\mathrm{HMP})_m$} \begin{eqnarray}\label{hmp}
M_t(\phi) \equiv K_t(\phi_t) - K_0(\phi_0) - \int_0^t K_s(\hat {A}_s\phi)\,ds-\beta\int_0^t K_s(\phi_s)\,ds \nonumber\\\\[-18pt] \eqntext{\displaystyle\mbox{is a continuous $(\mathscr{F}_t)$-martingale}}\\
\eqntext{\displaystyle\mbox{with } \langle M(\phi)\rangle_t = \int_0^t K_s(\phi_s^2) \,ds\ \forall t\ge0, \mbox{ a.s.}} \end{eqnarray}}
Using the martingale problem~\ref{hmp}, one can construct an orthogonal martingale measure $M_t(\cdot)$ with the method of Walsh~\cite{W1984}. Denote by $\mathscr{P}$, the $\sigma$-field of $(\mathscr{F}_t)$-predictable sets in $\mathbb{R}_+\times\Omega$. If $\psi\dvtx \mathbb{R}_+\times\Omega\times C \rightarrow\mathbb{R}$ is $\mathscr{P}\times\mathscr{C}$-measurable and
\setcounter{equation}{0} \begin{equation} \label{Emmeas} \int_0^t K_s(\psi_s^2) \,ds < \infty\qquad \forall t\ge0, \end{equation}
then there exists a continuous local martingale $M_t(\psi)$ with quadratic variation $\langle M(\psi) \rangle_t = \int_0^t K_s(\psi^2_s) \,ds$. If the expectation of the term in \eqref{Emmeas} is finite, then $M_t(\psi)$ is an $L^2$-martingale.
\begin{definition}\label{Dcamp} Let $(\hat{\Omega},\hat{\F} , \hat{\F}_t)= (\Omega\times C, \mathscr{F}\times\mathscr{C}, \mathscr{F}_t \times \mathscr{C} _t)$. Let $\hat{\F}^*_t$ denote the universal completion of $\hat{\F}_t$. If $T$ is a bounded $(\mathscr{F}_t)$-stopping time, then the normalized Campbell measure associated with $T$ is the measure $\hat{\P}_T$ on $(\hat{\Omega}, \hat{\F} )$ given by
\[ \hat{\P}_T(A\times B) = \frac{\mathbb{P}(\mathbf{1}_A K_T(B))}{m_T(1) }\qquad\mbox{for } A\in \mathscr{F}, B\in\mathscr{C}, \]
where $m_T(1) = \mathbb{P}(K_T(1))$. We denote sample points in $\hat{\Omega}$ by $(\omega, y)$. Therefore, under $\hat{\P}_T$, $\omega$ has law $K_T(1)\,d\mathbb{P} \cdot m_T^{-1}(1)$ and conditional on $\omega$, $y$ has law $K_T(\cdot )/K_T(1)$. \end{definition}
\begin{definition} \label{DKae} For two $(\hat{\F}^*_t)$-measurable processes $Z^1$ and $Z^2,$ we will say that $Z^1 = Z^2, K$-a.e. if
\[ Z^1(s, \omega, y) = Z^2(s, \omega, y)\qquad \forall s\le t, K_t\mbox {-a.a. } y \]
for all fixed times $t\ge0$. \end{definition}
\begin{definition} We say that $(X, Z)$ is a solution to the strong equation \ref{se1} if both (a) and (b) of that equation are satisfied and where $Z_t$ is an $(\hat{\F}^*_t)$-predictable process and $X_t$ is an $(\mathscr{F} _t)$-predictable process.
We say $(X', Y)$ is the solution to the stochastic equation \ref{SE2} if it satisfies (a) and (b) of that equation and where $Y_t$ is an $(\hat{\F}^*_t)$-predictable process, $X'_t$ is an $(\mathscr{F} _t)$-predictable process.
Unless stated otherwise, $(X, Z)$ will refer to a solution of \ref{se1} and $(X', Y)$ to the solution of~\ref{SE2}. \end{definition}
\begin{definition} For an arbitrary $\hat{\F}^*_t$-adapted, $\mathbb{R}^d$-valued process $z_t$, define the centre of mass (COM), $\bar{z}_t$, with respect to $K_t$ as follows:
\[ \bar{z}_t = \frac{K_t(z_t)}{K_t(1)}. \]
We also let
\begin{eqnarray*} \tilde{X} _t(\cdot) &\equiv&\frac{X_t(\cdot)}{X_t(1)} = \frac{K_t(Z_t\in \cdot )}{K_t(1)},\qquad \tilde{X} _t'(\cdot) \equiv\frac{X'_t(\cdot)}{X'_t(1)} = \frac {K_t(Y_t\in\cdot)}{K_t(1)}\quad \mbox{and}\\ \tilde{K}_t(\cdot) &=& \frac {K_t(\cdot )}{K_t(1)}. \end{eqnarray*}
\end{definition}
Note that as $K$ is supercritical, $K_t$ survives indefinitely on a set of positive probability $S$, and goes extinct on the set of positive probability $S^c.$ Hence, we can only make sense of ${\bar{z}}_t$ for $t<\eta$ where $\eta$ is the extinction time.
\begin{definition}
Let the process $M_t$ be defined for $t\in[0,\zeta)$ where $\zeta\le \infty$ possibly random. $M$ is called a local martingale on its lifetime if there exist stopping times $T_N \uparrow\zeta$ such that $M_{T_N\wedge\cdot}$ is a martingale for all $N$. The interval $[0, \zeta)$ is called the lifetime of $M$. \end{definition}
The following definition introduces a metric on the space of finite measures, which is equivalent to the metric of convergence in distribution on the space of probability measures.
\begin{definition}\label{DVas} Let $d$ denote the Vasserstein metric on the space of finite measures on $\mathbb{R}^d$. That is, for $\mu, \nu$ finite measures,
\[ d(\mu, \nu) = \sup_{\phi\in\mathrm{Lip}_1}\int\phi(x) \,d(\mu- \nu)(x), \]
where $\mathrm{Lip}_1= \{\psi\in C(\mathbb{R}^d)\dvtx \forall x, y, \vert\psi(x) -
\psi (y)\vert\le\vert x-y\vert, \|\psi\| \le1\}$. \end{definition}
\section{Proofs of existence and preliminary results}\label{Sexpre} \mbox{} \begin{pf*}{Proof of Theorem~\ref{TSOUrep}} Although closely related, this does not follow automatically from Theorem 4.10 of Perkins~\cite{Perkins1995} where it is shown that equations like $\mbox{(SE)}^2$ [but with more general, interactive, drift and diffusion terms in (a)] have solutions if $K$ is a \textit {critical} historical Brownian motion.
Note that $\mathring{K}_t = e^{-\beta t}K_t$ defines a $(\Delta/2, 0, e^{-\beta t})$-Historical superprocess. Let $\hat{\P}_T^1$ be the Campbell measure associated with $\mathring{K}$ (note that if $T$ is taken to be a random stopping time, this measure differs from $\hat{\P}_T$). The proof of Theorem~V.4.1 of~\cite{Perkins2002} with minor modifications shows that $\mbox{(SE)}_{Z_0, \mathring{K}}^1$\vspace*{2pt} has a pathwise unique solution. This is because (K3) of Theorem 2.6 of~\cite{Perkins1995} shows that under $\hat{\P}_T^1$, $y_t$ is a Brownian motion stopped at time $T$ and Proposition~2.7 of the same memoir can be used to replace Proposition~2.4 and Remark~2.5(c) of~\cite{Perkins2002}
for the setting where the branching variance depends on time.
Once this is established, it is simple to deduce that if $({\mathring {X}}, Z)$ is the solution of $\mbox{(SE)}_{Z_0, \mathring{K}}^1$, and we let
\[ X_t(\cdot) \equiv e^{\beta t}{\mathring{X}}_t(\cdot) = \int1(Z_t\in \cdot) K_t(dy) \]
then $(X,Z)$ is the pathwise unique solution of~\ref{se1}. The only thing to check this is that $Z_t(\omega, y) = Z_0(\omega , y_0) +y_t-y_0-\gamma\int_0^t Z_s(\omega, y) \,ds $ $K$-a.e., but this follows from the fact that $\mathring{K}_t \ll K_t, \forall t$.
It can be shown by using by Theorem 2.14 of~\cite{Perkins1995} that $X$ satisfies the following martingale problem: For $\phi\in C^2_b(\mathbb{R}^d)$,
\[ M_t(\phi) \equiv X_t(\phi) - X_0(\phi) + \int_0^t \int\gamma x\cdot \nabla\phi(x) -\frac{\Delta}{2}\phi(x) X_s(dx)\,ds - \beta\int _0^tX_s(\phi ) \,ds \]
is a martingale where $\langle M(\phi)\rangle_t = \int_0^tX_s(\phi ^2)\,ds$. Then by Theorem II.5.1 of~\cite{Perkins2002} this implies that $X$ is a version of a SOU process, with initial distribution given by $K_0(Z_0^{-1}(\cdot))$.
\end{pf*}
\begin{remark} \label{RSOUHrep} (a) Under the Lipschitz assumptions of Section V.1 of \cite{Perkins2002}, one can in fact uniquely solve \ref{se1} where $K$ is a supercritical historical Brownian motion. The proof above can be extended with minor modifications.\vspace*{-6pt}
\begin{longlist}[(b)] \item[(b)] The proof of Theorem~\ref{TSOUrep} essentially shows that under $\hat{\P}_T$, $T$ fixed, the path process $y\dvtx \mathbb{R}^+\times\hat{\Omega} \rightarrow \mathbb{R}^d$ such that $(t, (\omega, y)) \mapsto y_t$ is a $d$-dimensional Brownian motion stopped at $T$.
\item[(c)] Under $\hat{\P}_T$, $T$ fixed, $Z_t$ can be written explicitly as a function of the driving path: For $t\le T$,
\begin{eqnarray}\label{RZSDE} e^{\gamma t}Z_t &=& Z_0 + \int_0^t e^{\gamma s} \,dZ_s + \int_0^t Z_s (\gamma e^{\gamma s}) \,ds \nonumber \\[-8pt] \\[-8pt] \nonumber &=&Z_0 +\int_0^t e^{\gamma s} \,dy_s + \int_0^t e^{\gamma s} (-\gamma Z_s) \,ds + \int_0^t Z_s (\gamma e^{\gamma s}) \,ds, \end{eqnarray}
where we have used a differential form of~\ref{se1}(a) for the second equality. Hence,
\begin{equation}\label{EZpath} Z_t(y) = e^{-\gamma t}Z_0 + \int_0^t e^{-\gamma(t-s)}\,dy_s. \end{equation}
\end{longlist}
\end{remark}
Next, we show that there exists a unique solution to~\ref{SE2}.
\begin{pf*}{Proof of Theorem~\ref{TExUniq}} Suppose there exists a solution $Y$ satisfying~\ref{SE2}. Then under $\hat{\P}_T$, $Y_t$ can be written as a function of the driving path $y$ and $\bar{Y}$. Using integration by parts gives
\begin{eqnarray*} e^{\gamma t} Y_t &=& Y_0 + \int_0^t e^{\gamma s} \,dY_s + \int_0^t \gamma e^{\gamma s} Y_s \,ds \\ &=& Y_0 + \int_0^t e^{\gamma s} \,dy_s + \int_0^t \gamma e^{\gamma s} \bar {Y}_s \,ds \end{eqnarray*}
and hence
\begin{equation}\label{RYSDE} Y_t =e^{-\gamma t}Y_0 + \int_0^t e^{\gamma(s-t)} \,dy_s + \int_0^t \gamma e^{\gamma(s-t)} \bar{Y}_s \,ds. \end{equation}
If $(X, Z)$ is the solution to~\ref{se1} where $Z_0 = Y_0$, then note that by Remark~\ref{RSOUHrep}(c),
\[ Y_t =Z_t+ \int_0^t \gamma e^{\gamma(s-t)} \bar{Y}_s \,ds. \]
By taking the normalized measure $\tilde{K}_t$ on both sides of the above equation, we get
\begin{equation} \label{Ecorr} \bar{Y}_t = \bar{Z}_t + \gamma\int_0^t e^{-\gamma(t-s)} \bar{Y}_s \,ds. \end{equation}
Hence, $\bar{Y}_t$ is seen to satisfy a Volterra Integral Equation of the second kind (see equation (2.2.1) of~\cite{PM2008}) and therefore can be solved pathwise to give
\[ \bar{Y}_t = \bar{Z}_t + \gamma\int_0^t \bar{Z}_s \,ds, \]
which is easily verified using integration by parts. Also, if $\bar{Y}^1_t$ is a second process which solves \eqref{Ecorr}, then
\begin{eqnarray*}
|\bar{Y}_t -\bar{Y}^1_t| &=& \biggl|\gamma\int_0^t e^{-\gamma(t-s)} (\bar {Y}_s -
\bar{Y}^1_s) \,ds\biggr| \\
&\le&|\gamma|\int_0^t e^{-\gamma(t-s)} |\bar{Y}_s - \bar{Y}^1_s| \,ds. \end{eqnarray*}
By Gronwall's inequality, this implies $\bar{Y}_t = \bar{Y}^1_t$, for all $t$ and $\omega$. Pathwise uniqueness of $X'$ follows from the uniqueness of the solution to~\ref{se1} and the uniqueness of the process $\bar{Y}_t$ solving \eqref{Ecorr}.
We have shown that if there exists a solution to~\ref{SE2}, then it is necessarily pathwise unique. Turning now to existence to complete the proof, we work in the opposite order and define $Y$ and $X'$ as functions of the pathwise unique solution to \ref{se1} where $Z_0 = Y_0$:
\begin{eqnarray*} Y_t& =& Z_t + \gamma\int_0^t \bar{Z}_s \,ds, \\ X'_t(\cdot) &=& K_t(Y_t\in\cdot). \end{eqnarray*}
Then $\bar{Y}_t$ satisfies the integral equation \eqref{Ecorr}, and hence
\[ \int_0^t \bar{Z}_s \,ds = \int_0^t e^{-\gamma(t-s)}\bar{Y}_s \,ds. \]
Therefore
\begin{eqnarray*} Y_t &=& Z_t + \gamma\int_0^t e^{-\gamma(t-s)}\bar{Y}_s \,ds \\ &=& e^{-\gamma t}Y_0 + \int_0^t e^{-\gamma(t-s)} \,dy_s + \gamma\int_0^t e^{-\gamma(t-s)}\bar{Y}_s \,ds, \end{eqnarray*}
by equation (\ref{EZpath}), and so
\[ e^{\gamma t} Y_t = Y_0 + \int_0^t e^{\gamma s} \,dy_s + \gamma\int_0^t e^{\gamma s}\bar{Y}_s \,ds. \]
Multiplying by $e^{-\gamma t}$ and using integration by parts shows
\begin{eqnarray*} Y_t = Y_0 + y_t-y_0 + \gamma\int_0^t (\bar{Y}_s - Y_s)\,ds \end{eqnarray*}
which holds for $K$-a.e. $y$, thereby showing $(X',Y)$ satisfies \ref{SE2}. \end{pf*}
\begin{remark} \label{RCOMrel} Some useful equivalences in the above proof are collected below. If $Y_0 = Z_0$, then for $t<\eta$,
\begin{eqnarray*} &&\mbox{(a)}\hspace*{68pt}\quad Y_t = Z_t + \gamma\int_0^t\bar{Z}_s\,ds, \\ &&\mbox{(b)}\hspace*{67pt}\quad \bar{Y}_t = \bar{Z}_t + \gamma\int_0^t\bar{Z}_s\,ds, \\ &&\mbox{(c)}\hspace*{45pt}\quad Y_t - \bar{Y}_t = Z_t - \bar{Z}_t, \\ &&\mbox{(d)}\quad \int_0^t e^{-\gamma(t-s)}\bar{Y}_s \,ds = \int_0^t \bar{Z}_s\,ds. \end{eqnarray*}
\end{remark}
These equations intimately tie the behaviour of the interacting and ordinary SOU processes. Part (a) says that the interacting SOU process with attraction to the center of mass is the same as the ordinary SOU process pushed by the position of its center of mass.
We now consider the martingale problem for $X'$. For $\phi\dvtx \mathbb{R} ^d\rightarrow\mathbb{R}$, recall that $\bar{\phi}_t \equiv K_t(\phi (Y_t))/K_t(1)$ and that the lifetime of the process $\bar{\phi}$ is $[0, \eta)$. Then the following theorem holds:
\begin{theorem}\label{TIto} For $\phi\in C^2_b(\mathbb{R}^d, \mathbb{R}),$ and $t< \eta$,
\[ \bar{\phi}_t = \bar{\phi}_0 + N_t+ \int_0^t \bar{b}_s \,ds, \]
where
\[ b_s = \gamma\nabla\phi(Y_s)\cdot(\bar{Y}_s - Y_s) + \tfrac{1}{2}\Delta \phi(Y_s) \]
and $N_t$ is a continuous local martingale on its lifetime such that
\[ N_t = \int_0^t\int\frac{\phi(Y_s) - \bar{\phi}_s}{K_s(1)}\,dM(s, y) \]
and hence has quadratic variation given by
\[ [N]_t = \int_0^t\frac{\overline{\phi_s^2} - (\bar{\phi}_s)^2}{K_s(1)}\,ds. \]
\end{theorem}
Similarly, the following is true.
\begin{remark} \label{RIto2} The method of Theorem~\ref{TIto} can be used to show that for the $\beta$-supercritical SOU process, X, for $\phi\in C^2_b(\mathbb{R}^d, \mathbb{R})$ and $t<\eta$,
\[ \tilde{X} _t(\phi) = \tK_t(\phi(Z_t)) = \tK_0(\phi(Z_0)) + N_t+ \int _0^t \tK _s(L\phi(Z_s)) \,ds, \]
where
\[ L\phi(x) = -\gamma x\cdot\nabla\phi(x) + \tfrac{1}{2}\Delta\phi(x) \]
and $N_t$ is a continuous local martingale on its lifetime such that
\[ N_t = \int_0^t\int\frac{\phi(Z_s) - \tK_s(\phi (Z_s))}{K_s(1)}\,dM(s, y) \]
and hence has quadratic variation given by
\[ [N]_t = \int_0^t\frac{\tK_s(\phi^2(Z_s)) - \tK_s(\phi (Z_s))^2}{K_s(1)}\,ds. \]
\end{remark}
\begin{pf*}{Proof of Theorem~\ref{TIto}} The proof is not very difficult; one need only use It\^{o}'s lemma followed by some slight modifications of theorems in Chapter V of~\cite {Perkins2002} to deal with the drift introduced in the historical martingale problem due to the supercritical branching.
Let $T$ be a fixed time and $t\le T$. Recall that under $\hat{\P}_T$, $y$ is a stopped Brownian motion by Remark~\ref{RSOUHrep}(b), and hence $Y_t(y)$ is a stopped OU process (attracting to~$\bar{Y}_t$). Therefore, under $\hat{\P}_T$:
\begin{eqnarray*} \phi(Y_t) &=& \phi(Y_0) + \int_0^t \nabla\phi(Y_s)\cdot dY_s+ \frac {1}{2}\sum_{i,j\le d} \int_0^t \phi_{ij}(Y_s)\,d[Y^i, Y^j]_s \\ &=&\phi(Y_0) + \int_0^t \nabla\phi(Y_s)\cdot dy_s+ \int_0^t \gamma \nabla \phi(Y_s)\cdot(\bar{Y}_s-Y_s)+ \frac{1}{2}\Delta\phi(Y_s)\,ds \end{eqnarray*}
by the classical It\^{o}'s lemma. Then
\begin{eqnarray*} K_t(\phi(Y_t)) &=& K_t(\phi(Y_0)) + K_t\biggl(\int_0^t \nabla\phi (Y_s)\cdot dy_s\biggr)+K_t\biggl(\int_0^t b_s \,ds\biggr)\\ &=& K_0(\phi(Y_0))+ \int_0^t\int\phi(Y_0)\,dM(s, y)+ \beta\int_0^t K_s(\phi(Y_0))\,ds \\ &&{}+ \int_0^t \int\biggl[ \int_0^s \nabla\phi(Y_r)\cdot dy_r \biggr]\,dM(s,y)\\ &&{}+\beta\int_0^t K_s\biggl[ \int_0^s \nabla\phi(Y_r)\cdot dy_r\biggr]\,ds\\ &&{}+\int_0^t\int b_s\, dM(s,y)+ \beta\int_0^t K_s(b_s)\,ds +\int _0^tK_s(b_s)\,ds\\ &=& K_0(\phi(Y_0))+ \int_0^t\int\phi(Y_s)\,dM(s, y)\\ &&{}+ \beta\int_0^t K_s(\phi(Y_s))\,ds + \int_0^tK_s(b_s)\,ds. \end{eqnarray*}
The equality in the third line to $K_t(\int_0^t \nabla\phi (Y_s)\cdot dy_s)$ follows from Proposition~2.13 of \cite {Perkins1995}. The equality of the fourth line to the last term in the first line follows from a generalization of Proposition V.2.4 (b) of~\cite{Perkins2002}. The last equality then follows by collecting like terms and using the definition of $\bar{Y}$.
Note that $K_t(1)$ is Feller's $\beta$-supercritical branching diffusion and hence
\[ K_t(1) = K_0(1) + M^0_t + \beta\int_0^t K_s(1) \,ds, \]
where $M^0_t$ is a martingale such that $[M^0]_t = \int_0^t K_s(1) \,ds.$
Therefore for $t<\eta$, It\^{o}'s formula and properties of $K_t(1)$ and $ K_t(\phi(Y_t))$ imply that
\begin{eqnarray*} \bar{\phi}_t &=& \frac{K_t(\phi(Y_t))}{K_t(1)} \\ &=& \bar{\phi}_0 + \int_0^t\int\biggl[\frac{\phi(Y_s)}{K_s(1)}- \frac {K_s(\phi (Y_s))}{K_s(1)^2} \biggr]\,dM(s,y)+\int_0^t\frac{K_s(b_s)}{K_s(1)}\,ds. \end{eqnarray*}
Since $\phi$ is bounded, the stochastic integral term can be localized using the stopping times $T_N \equiv\min\{t\dvtx K_t(1)\ge N \mbox{ or } K_t(1)\le1/N\}\wedge N$ and hence it is a local martingale on $[0,\eta )$. It is easy to check that it has the appropriate quadratic variation. \end{pf*}
The following lemmas will be used extensively in Section \ref {SConvergence}, but will be proven in Section~\ref{Stech}.
\begin{lemma} \label{LExtProb}
There is a nonnegative random variable $W$ such that
\[ e^{-\beta t} K_t(1) \rightarrow W\qquad \mbox{a.s.} \]
and $\{\eta< \infty\} = \{W=0\}$ almost surely. \end{lemma}
Note that as $X$ and $X'$ are defined as projections of $K$, their mass processes are the same as that of $K$, and thus grow at the same rate.
\begin{definition} Let
\[ h(\delta) = \bigl(\delta\ln^+(1/\delta)\bigr)^{1/2}, \]
where $\ln^+(x) =\ln x \vee1$. Let
\[
S(\delta, c) = \{y\dvtx |y_t - y_s| < ch(|t-s|), \forall t,s \mbox{ with }
|t-s|\le\delta\}. \]
\end{definition}
\begin{lemma}\label{LCSP} Let $K$ be a supercritical historical Brownian motion, with drift~$\beta$, branching variance $1$, and initial measure $X_0$. For $c_0>6$ fixed, $c(t) = \sqrt{t + c_0}$, there exists a.s. $\delta(\omega)>0$ such that $\operatorname{Supp}(K_t(\omega))\subset S(\delta(\omega), c(t))$ for all~$t$. Further, given $c_0$, $\mathbb{P}(\delta<\lambda) < p_{c_0}(\lambda)$ where $p_{c_0}(\lambda)\downarrow0$ as $\lambda\downarrow0$ and for any $\alpha>0$, $c_0$ can be chosen large enough so that $p_{c_0}(\lambda) = C(d,c_0)\lambda^\alpha$ for $\lambda\in[0,1]$. \end{lemma}
The following moment estimates are useful in establishing the convergence of~$\bar{Y}_t$. Recall that $\eta$ is the extinction time of $K$.
\begin{lemma}\label{Lsecmom}
Assume $\mathbb{P}(\tilde{K}_0(|Y_0|^2+|y_0|^2))<\infty$. Then,
\[ \mathbb{P}(\overline{\vert Y_{t}\vert^2}; t< \eta) < A(\gamma, t), \]
where
\begin{eqnarray*} A(\gamma, t) =
\cases{ O(1+t^6e^{-2\gamma t}),&\quad $\mbox{ if } \gamma< 0$, \vspace*{2pt}\cr
O(1+t^5),&\quad $\mbox{ if } \gamma\ge0.$ }
\end{eqnarray*}
\end{lemma}
\begin{remark} \label{Rmoments}
(a) The proof of Lemma~\ref{Lsecmom}, under the same hypotheses (if $Z_0=Y_0$) yields
\[ \mathbb{P}(\overline{\vert Z_{t}\vert^2};t<\eta) < A(\gamma, t).\vspace*{-6pt} \]
\begin{longlist}[(b)]
\item[(b)] Lemma~\ref{Lsecmom} and its proof can be extended to show that for any positive integer $k$ if $\mathbb{P}(\tilde{K}_0(|Z_0|^k+
|y_0|^k))<\infty$ (and $Z_0 = Y_0$), then there exists a function $B(\gamma, t, k)$ polynomial in $t$ if $\gamma\ge0$, exponential if $\gamma<0$ such that
\[ \mathbb{P}(\overline{\vert Y_{t}\vert^k}; t<\eta) < B(\gamma, t, k) \quad\mbox{and}\quad \mathbb{P} (\overline{\vert Z_{t}\vert^k}; t<\eta)< B(\gamma, t, k). \]
\end{longlist}
\end{remark}
\section{Proofs of convergence} \label{SConvergence}
We will henceforth, unless specified otherwise, assume that for a path $y \in C$, $Z_0(y_0) = Y_0(y_0) = y_0$. Recall that, by the construction of solutions to $\mbox{(SE)}^i,$
\[ \int\phi(y_0)K_0(dy) = \int\phi(x) X_0(dx) = \int\phi(x) X'_0(dx). \]
Also recall that our standing hypothesis is that $K_0$ has finite initial mass [and hence $X_0'(1)=X_0(1)<\infty$].
In this section, we will first settle what happens to $X$ and $X'$ when there is extinction, and then the case when there is survival, under the attractive regime and lastly address the interacting repelling SOU process on the survival set.
\subsection{On the extinction set} \mbox{} \begin{pf*}{Proof of Theorem~\ref{TExtinc}}
Assume for now that $\mathbb{P}(\tilde{K}_0(\vert y_0\vert^2))<\infty$ (the case where $K_0 = 0$ can be ignored without loss of generality). By Theorem~\ref{TIto},
\[ \bar{Y}_t = \bar{Y}_0 + \int_0^t\int\frac{Y_s - \bar{Y}_s}{K_s(1)}\,dM(s, y) \]
and therefore is a local martingale on its lifetime with reducing sequence $\{T_N\}$ as defined in the proof of the same theorem.\vadjust{\goodbreak} Using Doob's weak inequality and Lemma~\ref{Lsecmom},
\begin{eqnarray*}
\mathbb{P}\Bigl(\sup_{s< t\wedge\eta} |\bar{Y}_s| > n\Bigr) &=&\lim_{N \rightarrow
\infty} \mathbb{P}\Bigl(\sup_{s< t\wedge T_N} |\bar{Y}_s| > n\Bigr) \\
&\le&\lim_{N\rightarrow\infty} \frac{1}{n^2}\mathbb{E}(|\bar{Y}_{t\wedge T_N}|^2) \\ &<& \frac{A(\gamma, t)}{n^2}. \end{eqnarray*}
By the first Borel--Cantelli lemma,
\[
\mathbb{P}\Bigl(\sup_{s< t\wedge\eta} |\bar{Y}_t| > n \mbox{ i.o.}\Bigr) = 0. \]
It follows that
\[ \liminf_{s\rightarrow t\wedge\eta} \bar{Y}_s >-\infty\quad\mbox{and}\quad \limsup _{s\rightarrow t\wedge\eta} \bar{Y}_s < \infty \]
which implies that on the set $\{\eta< t\}$, $\bar{Y}_s$ converges, by Theorem IV.34.12 of Rogers and Williams~\cite{RW1985}. This shows convergence on the extinction set as $S^c = \bigcup_t \{\eta< t\}$.
Note that if $\nu(\cdot) = \mathbb{P}(K_0 \in\cdot)$. Theorem II.8.3 of~\cite{Perkins2002}
gives
\[ \mathbb{P}(K\in\cdot) = \int\mathbb{P}_{K_0}(K\in\cdot)\,d\nu(K_0), \]
where $\mathbb{P}_{K_0}$ is the law of a historical Brownian motion with initial distribution $\delta_{K_0}$. Hence, the a.s. convergence of
$\bar{Y}_t$ in the case where $\tilde{K}_0(|y_0|^2)$ is finite in mean imply
\begin{eqnarray}\label{Egen1} \mathbb{P}\Bigl(\lim_{t\uparrow\eta} \bar{Y}_t \mbox{ exists} ; S^c\Bigr)&= &\int\mathbb{P} _{K_0}\Bigl(\lim_{t\uparrow\eta} \bar{Y}_t \mbox{ exists} ; S^c \Bigr)\,d\nu (K_0) \nonumber\\ &=& \int\mathbb{P}_{K_0}(S^c)\,d\nu(K_0)\\ &=&\mathbb{P}(S^c) \nonumber \end{eqnarray}
if $K_0(|y_0|^2+1)<\infty, \nu$-a.s.
Finally, to get rid of the assumption that $K_0(|y_0|^2)<\infty$ note that Corollary 3.4 of~\cite{Perkins1995} ensures that if $K_0(1) < \infty$, then at any time $t>0$, $K_t$ (and hence $X_t, X'_t$) is compactly supported. Therefore, letting $S_r = \{K_r \neq0\}$ we see that
\begin{eqnarray*} \mathbb{P}\Bigl(\lim_{t\uparrow\eta}\bar{Y}_t \mbox{ does not exist}, S^c\Bigr) &=& \mathbb{P}\biggl(\bigcup_{r\in\mathbb{N}}\Bigl\{\lim_{t\uparrow\eta}\bar{Y}_t \mbox{ does not exist}, S_{1/r}, S^c\Bigr\}\biggr)\\ &\le&\sum_{r\in\mathbb{N}} \mathbb{P}\Bigl(\mathbb{P}_{K_{1/r}} \Bigl(\lim_{t\uparrow\eta }\bar{Y}_t \mbox{ does not exist}, S^c \Bigr)\mathbf{1}_{S_{1/r}} \Bigr) \\ &=& 0 \end{eqnarray*}
by (\ref{Egen1}) since $K_{1/r}$ a.s. compact implies that
$K_{1/r}(|y_{1/r}|^2)<\infty$ holds. This completes the proof for the convergence of $\bar{Y}$ on $S^c$ in its full generality.
The convergence of $\bar{Z}_t$ now follows from the convergence of $\bar{Y}_t$ and equation~(\ref{Ecorr}).
\end{pf*}
\begin{pf*}{Proof of Theorem~\ref{TExtConv}} As in the previous proof, note that we need only consider the case that $\mathbb{P}(\tilde{K}_0(\vert y_0\vert^2))<\infty$.
We will follow the proof of Theorem 1 of Tribe~\cite{Tribe1992} here. Define
\[ \zeta(t) = \int_0^t\frac{1}{K_s(1)}\,ds,\qquad t<\eta. \]
It is known by the work of Konno and Shiga~\cite{KS1988} in the case where $\beta=0$, that $\zeta\dvtx [0,\eta)\rightarrow[0,\infty)$ homeomorphically (recall that $\eta<\infty$ a.s. in that case). This latter result also holds when $\beta>0$ on the extinction set $S^c$ by a Girsanov argument.
Define $D\dvtx [0,\zeta(\eta-))\rightarrow[0,\eta)$ as the unique inverse of $\zeta$ (on $S^c$, this defines the inverse on $[0,\infty)$) and for $t\ge\zeta(\eta-)$, let $D_t = \infty$. Let
\[ X^D_t = X'_{D_t},\qquad \tilde{X} ^D_t = \frac{X^D_t}{X^D_t(1)}\quad \mbox{and}\quad\mathscr{G}_t = \mathscr{F}_{D_t} \]
and define
\[ L_t\phi(x) = \gamma(\bar{Y}_t - x)\cdot\nabla\phi(x)+\tfrac {1}{2}\Delta\phi(x). \]
Let
\[ T_N=\int_0^{\eta_N} \frac{1}{K_s(1)}\,ds, \]
where $\eta_N = \inf\{s\dvtx K_s(1) \le1/N\}$. Then note that $T_N \uparrow\zeta(\eta-)$ and each $T_N$ is a $\mathscr{G}_t$-stopping time. On $S^c$ for $\phi\in C^2_b$, Theorem~\ref{TIto} implies
\begin{eqnarray} \label{Eexteq} \Xb^D_{t\wedge T_N}(\phi) &=& \tilde{X} _0(\phi) + \int_0^{D_{t\wedge T_N}} \tilde{X} '_s(L_s\phi)\,ds + M_{D_{t\wedge T_N}}(\phi)\nonumber\\ &=& \tilde{X} _0(\phi) + \int_0^{t\wedge T_N} \Xb^D_s(L_{D_s}\phi )X^D_s(1)\,ds + N_{t\wedge T_N}(\phi)\\ &= & \tilde{X} _0(\phi) + \int_0^{t\wedge T_N} X^D_s(L_{D_s}\phi)\,ds + N_{t\wedge T_N}(\phi)\nonumber \end{eqnarray}
since $dD_t = X^D_t(1)\,dt$ and where $N_t = M_{D_t}$. It follows that $N_{t\wedge T_N}$ is a $\mathscr{G}_t$-local martingale. Then, by Theorem~\ref{TIto},
\begin{eqnarray*} [\Xb^D(\phi)]_{t\wedge T_N} &=& \int_0^{D_{t\wedge T_N}} \frac { \tilde{X} '_s(\phi^2) - \tilde{X} '_s(\phi)^2}{X_s(1)}\,ds \\ &=& \int_0^{t\wedge T_N} \Xb^D_s(\phi^2) - \Xb^D_s(\phi)^2 \,ds, \end{eqnarray*}
which is uniformly bounded in $N$. Hence, sending $N\rightarrow\infty $, one sees that $N_{t\wedge\zeta(\eta-)}$ is a $\mathscr{G}_t$-martingale.
Note that on $S^c$, $\zeta(\eta-) = \infty$ and hence on that event,
\begin{eqnarray*}
\int_0^\infty X^D_s(|L_{D_s} \phi|) \,ds &\le&\int_0^\infty\int\vert \gamma(\bar{Y}_{D_s}-x)\cdot\nabla\phi(x)\vert+\frac{1}{2}\vert \Delta \phi (x)\vert X^D_s(dx)\,ds \\
&\le&\int_0^\infty|\gamma|\|\nabla\phi\|K_{D_s}(|\bar{Y} _{D_s}-Y_{D_s}|) +\frac{1}{2}\|\Delta\phi\|X^D_s(1)\,ds \\
&=& \int_0^\infty X^D_s(1)\biggl( |\gamma|\|\nabla\phi\|\tilde{K} _{D_s}(|\bar{Y} _{D_s}-Y_{D_s}|) +\frac{1}{2}\|\Delta\phi\|\biggr)\,ds \\
&\le&\int_0^\infty X^D_s(1)\biggl( |\gamma|\|\nabla\phi\|
(\overline {|Y_{D_s}|^2})^{{1}/{2}} +\frac{1}{2}\|\Delta\phi\| \biggr)\,ds, \end{eqnarray*}
where in the second line we have used the definition of $X'$ and the Cauchy--Schwarz inequality in the fourth. Using the definition of $D_s$ yields
\begin{eqnarray}\label{Eextbd}
\int_0^\infty X^D_s(|L_{D_s} \phi|) \,ds &\le&\biggl( |\gamma|\|\nabla \phi
\|\sup_{s<\eta}(\overline{|Y_{s}|^2})^{{1}/{2}} +\frac
{1}{2}\|
\Delta\phi\| \biggr) \int_0^\infty X^D_s(1)\,ds \nonumber\\
&=& \biggl( |\gamma|\|\nabla\phi\|\sup_{s<\eta}(\overline
{|Y_{s}|^2})^{{1}/{2}} +\frac{1}{2}\|\Delta\phi\| \biggr) \eta \\ &<&\infty\nonumber \end{eqnarray}
as $\phi\in C_b^2$ and $\overline{\vert Y_s\vert^2}$ is continuous on $[0, \eta)$ (which follows from Theorem~\ref{TIto}). Hence, this implies that for $\phi$ positive, on $S^c$
\[
N_t(\phi) > - \tilde{X} _0(\phi) - \int_0^\infty X^D_s(|L_{D_s} \phi|) \,ds \]
for all $t$ and hence by Corollary IV.34.13 of~\cite{RW1985}, $N_t$ converges as $t\rightarrow\infty$. Therefore by \eqref{Eexteq} and \eqref{Eextbd}, $ \tilde{X} ^D_t(\phi)$ converges a.s. as well.
Denote by $\Xb^D_\infty(\phi)$ the limit of $\Xb^D_t(\phi)$. It is immediately evident that $\Xb^D_\infty(\cdot)$ is a probability measure on $\mathbb{R}^d$. To show that $\Xb^D_\infty(\cdot) = \delta_{F'}$ where $F'$ is a random point in $\mathbb{R}^d$, we now defer to the proof of Theorem 1 in Tribe~\cite{Tribe1992}, as it is identical from this point forward.
Similar (but simpler) reasoning holds to show $ \tilde{X} _t \rightarrow \delta _F$ a.s. on $S^c$ where $F$ is a random point in $\mathbb{R}^d$. Let $f(t) = \gamma\int_0^t \bar{Z}_s \,ds$. Note that $f$ is independent of $y$ and that $f(t) \rightarrow f(\eta)$ a.s. when $t\uparrow\eta$ because $\bar{Y}_t = \bar{Z}_t + f(t)$ and both $\bar{Y}_t$ and $\bar{Z}_t$ converge a.s. by Theorem \ref {TExtinc}. Then for $\phi$ bounded and Lipschitz,
\begin{eqnarray*} \biggl\vert\int\phi\bigl(x - f(t)\bigr) \tilde{X} '_t(dx) - \int\phi\bigl(x - f(\eta)\bigr) \tilde{X} '_t(dx) \biggr\vert & \le& C\vert f(\eta) - f(t)\vert \\ &\ascv&0 \end{eqnarray*}
as $t\uparrow\eta$. Therefore it is enough to note that since $f(\eta)$ depends only on $\omega$, the convergence of $ \tilde{X} '_t$ gives
\[ \int\phi\bigl(x - f(\eta)\bigr) \tilde{X} '_t(dx) \ascv\phi\bigl(F' - f(\eta)\bigr) \]
and hence
\[ \int\phi\bigl(x - f(t)\bigr) \tilde{X} '_t(dx) \ascv\phi\bigl(F' - f(\eta)\bigr). \]
By Remark~\ref{RCOMrel}(a),
\begin{eqnarray*} \int\phi\bigl(x - f(t)\bigr) \tilde{X} '_t(dx) &=& \int\phi\bigl(Y_t(y)- f(t) \bigr)\tilde{K}_t(dy) \\ &=& \int\phi(Z_t(y))\tilde{K}_t(dy) \\ &=& \int\phi(x) \tilde{X} _t(dx)\\ &\ascv&\phi(F) \end{eqnarray*}
as $t\uparrow\eta$. Since there exists a countable separating set of bounded Lipschitz functions $\{\phi_n\}$, and the above holds for each $\phi_n$,
\[ F' = F+ \gamma\int_0^{\eta}\bar{Z}_s \,ds\qquad \mbox{a.s. } \]
\upqed\end{pf*}
\begin{remark}
(a) Theorem~\ref{TExtConv} holds in the critical branching case. That is, if $\beta= 0$,
\[ X_t\ascv\delta_{F} \quad\mbox{and}\quad X_t'\ascv\delta_{F'}, \]
where
\[ F' =F+\int_0^\eta\bar{Z}_s\,ds. \]
The convergence of the critical ordinary SOU process to a random point follows directly from Tribe's result. That this holds for the SOU process with attraction to the COM follows from the calculations above.
\begin{longlist}[(b)] \item[(b)] The distribution of the random point $F$ has been identified in Tribe~\cite{Tribe1992} by approximating with branching particle systems. In fact, the law of $F$ can be identified as $x_\eta$, where $x_t$ is an Ornstein--Uhlenbeck process with initial distribution given by $ \tilde{X} _0$ and $\eta$ is the extinction time. Finding the distribution of $F'$ remains an open problem however. \end{longlist}
\end{remark}
\subsection{On the survival set, the attractive case}\label{SsSurv} \mbox{} \begin{pf*}{Proof of Theorem~\ref{TSurvival}} Let $\gamma\ge0$ and as in the proof of Theorem~\ref{TExtinc}, assume $\mathbb{P}(\tilde{K}_0(\vert y_0\vert^2))<\infty$. Also, without loss of generality, assume that $d=1$ for this proof.
By Theorem~\ref{TIto}, $\bar{Y}_t$ is a continuous local martingale with decomposition given by $\bar{Y}_t = \bar{Y}_0 + M_t(Y)$ where
\[ [M(Y)]_t = \int^t_0\frac{\overline{Y^2_s}-\bar{Y}_s^2}{K_s(1)}\,ds = \int ^t_0\frac{V(Y_s)}{K_s(1)}\,ds, \]
with $V(Y_s) = \overline{Y^2_s}-\bar{Y}_s^2$. Theorem IV.34.12 of \cite {RW1985} shows that on the set $\{[M(Y)]_\infty< \infty\}\cap S$, $M_t(Y)$ a.s. converges.
Note that by Lemma~\ref{LExtProb}, for a.e. $\omega\in S$, $W(\omega )> 0$, recalling that $W= \lim_{t\rightarrow\infty} e^{-\beta t}K_t(1)$. Hence, it follows that $[M(Y)]_\infty<\infty$ on $S$ if
\[ \int_0^\infty e^{-\beta s}V(Y_s)\,ds <\infty. \]
Then
\begin{eqnarray*} \mathbb{P}\biggl(\int_0^\infty e^{-\beta s}V(Y_s) \,ds; S \biggr) &\le&\mathbb{P} \biggl(\int _0^\infty e^{-\beta s} \overline{Y^2_s} \,ds ; S\biggr) \\ &\le&\int_0^\infty e^{-\beta s}A(\gamma, s) \,ds \\ &<& \infty \end{eqnarray*}
by Cauchy--Schwarz and Lemma~\ref{Lsecmom} since $\gamma\ge0$. Therefore on $S$, $\bar{Y}_t$ converges a.s. to some limit $\bar{Y} _\infty $. Note that if $\gamma= 0$, Remark~\ref{RCOMrel}(b) gives $\bar{Y}_t = \bar{Z}_t$ and so (b) holds.
That $\bar{Z}_t$ converges on $S$ for $\gamma>0$ follows from the fact that $\bar{Y}_t$ converges and equation (\ref{Ecorr}) by setting
\begin{eqnarray*} \bar{Z}_t &=& \bar{Y}_t - \gamma\int_0^t e^{-\gamma(t-s)}\bar{Y}_s \,ds \\ &=& \bar{Y}_t -\bar{Y}_\infty+ \gamma\int_0^t e^{-\gamma(t-s)}(\bar{Y} _\infty-\bar{Y} _s) \,ds +e^{-\gamma t}\bar{Y}_\infty\\ &\rightarrow&0\qquad \mbox{as } t\rightarrow\infty. \end{eqnarray*}
By Remark~\ref{RCOMrel}(b), we see that for $\gamma>0$ since $\bar{Y}_t = \bar{Z}_t + \gamma\int_0^t \bar{Z}_s \,ds$, $\bar{Z}_t \ascv0$ and $ \bar{Y}_t \ascv \gamma\int_0^\infty\bar{Z}_s \,ds.$
Now argue by conditioning as in the end of Theorem~\ref{TExtinc} to get the full result. \end{pf*}
The next few results are necessary to establish the almost sure convergence of~$ \tilde{X} _t$ on the survival set. This will in turn be used to show the almost sure convergence of $ \tilde{X} _t'$ using the correspondence of Remark~\ref{RCOMrel}(a).
Let $P_t$ be the standard Ornstein--Uhlenbeck semigroup (with attraction to the origin). Note that $P_t \rightarrow P_\infty$ in norm where
\[ P_\infty\phi(x) = \int\phi(z)\biggl(\frac{\gamma}{\pi}\biggr)^{{d}/{2}} e^{-\gamma\vert z\vert^2}\,dz, \]
which is independent of $x$. Recall that $W = \lim_{t\rightarrow \infty } e^{-\beta t}X_t(1)$ and $S=\{W>0\}$ a.s. from Lemma~\ref{LExtProb}.
\begin{lemma}\label{LSOUconvL2} If $\gamma>0$, $\mathbb{P}(X_0(\vert x\vert^4))<\infty$ and $\mathbb{P} (X_0(1)^4)<\infty$, then on $S$, for any $\phi\in\mathrm{Lip}_1$, $e^{-\beta t}X_t(\phi)\ltwocv WP_\infty\phi$ and
\[ \mathbb{P}\bigl(\vert e^{-\beta t}X_t(\phi) - WP_\infty\phi\vert^2 \bigr)\le Ce^{-\zeta t}, \]
where $C$ depends only on $d$ and $X_0$, and $\zeta$ is a positive constant dependent only on $\beta$ and $\gamma$. \end{lemma}
\begin{remark} \label{Rconv} As the $L^2$ convergence in Lemma~\ref{LSOUconvL2} is exponentially fast, it follows from the Borel--Cantelli lemma and Chebyshev inequality that for a strictly increasing sequence $\{t_n\}_{n=0}^\infty$ where
$|\{t_n\}\cap[k, k+1)| = \lfloor e^{\zeta k/2}\rfloor$, for $\phi\in\mathrm{Lip}_1$,
\[ e^{-\beta t_n}X_{t_n}(\phi) \rightarrow WP_\infty\phi\qquad\mbox{a.s. as } n\rightarrow\infty. \]
\end{remark}
The idea is to use the above remark to bootstrap up to almost sure convergence in Lemma~\ref{LSOUconvL2} with some estimates on the modulus of continuity of the process $e^{-\beta t}X_t(\phi).$
\begin{lemma} \label{LSOU-increments}
Suppose $\gamma>0$, $\mathbb{P}(X_0(|x|^8))<\infty$ and $\mathbb{P} (X_0(1)^8)<\infty$. If $\phi\in\mathrm{Lip}_1$ and $h>0$, then
\begin{equation}\label{EIncrBd} \mathbb{P}\bigl(\bigl[e^{-\beta(t+h)}X_{t+h}(\phi) - e^{-\beta t}X_t(\phi) \bigr]^4\bigr) \le C(t)h^2e^{-\zeta^*t}, \end{equation}
where $\zeta^*$ is a positive constant depending only on $\beta$ and $\gamma$ and $C$ is polynomial in~$t$, and depends on $\gamma$, $\beta$ and $d$. \end{lemma}
Let $\Psi\dvtx \mathbb{R}^d\rightarrow\mathbb{R}$, $ p\dvtx \mathbb{R}_+\rightarrow\mathbb{R}$ be positive, continuous functions. Further, suppose $\Psi$ is symmetric about 0 and convex with $\lim_{|x|\rightarrow\infty} \Psi(x) = \infty$ and $p(x)$ is increasing with $p(0) = 0$. The following is a very useful result of Garsia, Rodemich and Rumsey~\cite{GRR1970}.
\begin{proposition} \label{PGRR} If $f$ is a measurable function on $[0,1]$ such that
\begin{equation}\label{EGRR1}
\int\int_{[0,1]^2} \Psi\biggl(\frac{f(t)-f(s)}{p(|t-s|)}\biggr) \,ds \,dt = B <\infty \end{equation}
then there is a set $K$ of measure 0 such that if $s, t\in [0,1]\setminus K$ then
\begin{equation}\label{EGRR2}
\vert f(t)-f(s)\vert \le8 \int^{|t-s|}_0 \Psi^{-1}\biggl(\frac{B}{u^2}\biggr)\,dp(u). \end{equation}
If $f$ is also continuous, then $K$ is in fact the empty set. \end{proposition}
With this result in hand, we can now bring everything together to prove convergence of $ \tilde{X} _t$.
\begin{pf*}{Proof of Theorem \protect\ref{TSOUconv}} The strategy for this proof is simple: We use Remark~\ref{Rconv} to see that we can lay down an increasingly (exponentially) dense sequence $e^{-\beta t_n}X_{t_n}(\phi)$ which converges almost surely, and that we can use Lemma~\ref{LSOU-increments} to get a modulus of continuity on the process $e^{-\beta t} X_{t}(\phi)$, which then implies that if the sequence is converging, then the entire process must be converging.
Assume that $\mathbb{P}(K_0(|y_0|^8))=\mathbb{P}(X_0(|x|^8))<\infty$ and $\mathbb{P} (X_0(1)^8)<\infty$ and argue as in Theorem~\ref{TExtinc} in the general case. Let $\phi\in\mathrm{Lip}_1$. Denote $ e^{-\beta t}X_t$ by ${\mathring{X}} _t$ for the remainder of the proof. Let $T>0$, and let $\Psi(x) =
|x|^4$ and $p(t) = |t|^{3/4} (\log(\frac{\lambda}{t}))^{1/2}$ where $\lambda= e^4$. Let $B_T(\omega)$ be the constant\vspace*{-2pt} $B$ that appears in Proposition~\ref{PGRR}, with aforementioned functions $\Psi $ and $p$, for the path ${\mathring{X}}_{Tt}(\omega), t\in[0,1]$.
Then note that
\begin{eqnarray*} \mathbb{P}(B_T) &\equiv&\mathbb{P}\biggl[\int\int_{[0, 1]^2} \Psi\biggl(\frac {{\mathring
{X}}_{Tt}-{\mathring{X}} _{Ts}}{p(|t-s|)}\biggr) \,ds \,dt\biggr] \\ &=&\int\int_{[0, 1]^2} \frac{ \mathbb{P}[\vert{\mathring
{X}}_{Tt}-{\mathring{X}} _{Ts}\vert^4]}{|t-s|^3\log^2({\lambda
}/{|t-s|}) }\,ds \,dt \\ &\le&\int\int_{[0, 1]^2} \frac{ C(T(s\wedge t))e^{-\zeta
^*(s\wedge t)}T^2|t-s|^2}{|t-s|^3\log^2({\lambda}/{|t-s|}) }\,ds \,dt \\ &=& 2T^2 \int_0^1\int_0^t \frac{ C(Ts)e^{-\zeta
^*(Ts)}|t-s|^2}{|t-s|^3\log^2({\lambda}/{|t-s|}) }\,ds \,dt \\
&\le&2C(T)T^2 \int_0^1\int_0^t \frac{ 1}{|t-s|\log^2(
{\lambda }/{|t-s|}) }\,ds \,dt \\ &\le&\frac{C(T)T^2}{2e^4}, \end{eqnarray*}
where $C$ is the polynomial term that appears in Lemma~\ref{LSOU-increments}. Since ${\mathring{X}}_t$ is continuous, by Garsia--Rodemech--Rumsey~\cite{GRR1970}, for all $s, t \le1$,
\begin{eqnarray*} \vert{\mathring{X}}_{Tt} - {\mathring{X}}_{Ts}\vert &\le&8 \int _0^{\vert t-s\vert} \biggl(\frac {B_T}{u^2}\biggr)^{{1}/{4}}\,dp(u) \\
&\le& AB_T^{{1}/{4}}|t-s|^{{1}/{4}}\biggl(\log\frac{\lambda
}{|t-s|}\biggr)^{{1}/{2}}, \end{eqnarray*}
where $A$ is a constant independent of $T$ (see Corollary 1.2 of Walsh~\cite{W1984} for this calculation). Rewriting the above,
\begin{equation}\label{EMC}
\vert{\mathring{X}}_t - {\mathring{X}}_s\vert \le D_T|t-s|^{{1}/{4}}\biggl(\log\frac{\lambda T}{|t-s|}\biggr)^{{1}/{2}}\qquad \forall s<t\le T, \end{equation}
where $D_T \equiv A(\frac{B_T}{T})^{{1}/{4}}$. Note that $\mathbb{P} (D_T^4) =\frac{ A^4TC(T)}{2e^4}$, which is\vspace*{2pt} polynomial in $T$ of fixed degree $d_0>1$. Let $\Omega_0$ be the set of probability 1 such that for all positive integers $T$ equation \eqref{EMC} holds and $D_T\le T^{d_0} $ for $T$ large enough. To see \mbox{$P(\Omega_0)=1$}, use Borel--Cantelli:
\begin{eqnarray*} \mathbb{P}(D_T\ge T^{d_0}) &=& \mathbb{P}(D_T^4\ge T^{4d_0})\\ &\le&\frac{\mathbb{P}(D_T^4)}{T^{4d_0}}\\ &\le&\frac{c}{T^{3d_0}} \end{eqnarray*}
which is summable over all positive integers $T$.
Suppose $\omega\in\Omega_0$. Let $\delta^T(\omega)$ be such that $\delta^{{-1}/{8}}(\log\frac{\lambda}{\delta})^{ {-1}/{2}} = T^{d_0}$. Then for all integral $T>T_0(\omega)$, and $s, t\le T$ with
$|t-s|\le\delta$, $ \vert{\mathring{X}}_t - {\mathring{X}}_s\vert \le
|t-s|^{{1}/{8}}.$
Now let $\{{\mathring{X}}_{t_n}\}$ be a sequence of the form in Remark~\ref{Rconv}, with the additional condition that $\{t_n\}\cap[k, k+1)$ are evenly spaced within $[k, k+1)$ for each $k\in\mathbb{Z}_+$ (i.e., $t_{n+1} - t_n = ce^{-\zeta k/2}$ for $t_n\in\{t_n\}\cap[k, k+1)$). Evidently ${\mathring{X}}_{t_n}$ converges a.s. to a limit ${\mathring {X}}_\infty$. Without loss of generality, assume convergence of the sequence on the set $\Omega_0$.
There exists $T_1(\omega)$ such that for all $T>T_1$, $ce^{-\zeta T/2} < \delta$. Hence, for all $t$ such that $T_1\vee T_0< t \le T$ there exists $t'_n\in\{t_n\}$ such that $\vert t-t_n'\vert<ce^{-\zeta \lfloor t\rfloor/2}<\delta$ and hence
\begin{eqnarray*} \vert{\mathring{X}}_t - {\mathring{X}}_\infty\vert &\le&\vert {\mathring{X}}_t - {\mathring{X}}_{t_n'}\vert + \vert{\mathring {X}} _{t_n'} - {\mathring{X}}_\infty\vert \\ &\le&\vert t-t_n'\vert^{{1}/{8}} + \vert{\mathring{X}}_{t_n'}- {\mathring{X}}_\infty\vert \\ &\le& ce^{-{\zeta\lfloor t\rfloor}/{16}} + \vert{\mathring {X}}_{t_n'} - {\mathring{X}} _\infty\vert. \end{eqnarray*}
Sending $t\rightarrow\infty$ gives almost sure convergence of $ e^{-\beta t}X_t(\phi)$ to ${\mathring{X}}_\infty= WP_\infty(\phi)$ by Theorem \ref {LSOUconvL2}, since $t_n' \rightarrow\infty$ with $t$. Note that this implies for $\phi\in\mathrm{Lip}_1$
\begin{equation} \label{Etestconv} \tilde{X} _t(\phi)\ascv P_\infty(\phi) \end{equation}
since on $S$, $\frac{e^{\beta t}}{K_t(1)} \rightarrow W^{-1}$ a.s.
By Exercise 2.2 of~\cite{EK1986} on $\mathscr{M}_1(\mathbb{R}^d)$, the space of probability measures on $\mathbb{R}^d$, the Prohorov metric of weak convergence is equivalent to the Vasserstein metric. It is easy to construct a class $\Theta$ that is a countable algebra of Lipschitz functions and that therefore is strongly separating (see page~113 of~\cite{EK1986}). Hence by Theorem~3.4.5(b) of~\cite{EK1986}, $\Theta$ is convergence determining. Since there exists a set $S_0\subset S$ with $\mathbb{P}(S\setminus S_0) = 0$ such that on $S_0$, equation \eqref{Etestconv} holds simultaneously for all $\phi\in \Theta$,
\[ \tilde{X} _t(\cdot)\rightarrow P_\infty(\cdot) \]
in the Vasserstein metric, for $\omega\in S_0$ because $\Theta$ is convergence determining.
To drop the dependence on the eighth moment, we argue as in the proof of Theorem~\ref{TExtinc}, where we make use of the Markov Property and the Compact support property for Historical Brownian Motion.
\end{pf*}
\begin{pf*}{Proof of Theorem~\ref{TISOUConv}} This follows almost immediately from Theorem~\ref{TSOUconv} and the representation given in Remark~\ref{RCOMrel}(a). Let $\phi\in\mathrm{Lip}_1$, then
\begin{eqnarray*} \tilde{X} '_t(\phi) &=& \tilde{K}_t(\phi(Y_t)) =\tilde{K}_t\biggl(\phi\biggl(Z_t + \gamma \int _0^t\bar{Z}_s \,ds \biggr)\biggr)\\ &= &\int\phi\bigl(x + f(t,\omega) \bigr)\,d \tilde{X} _t(dx), \end{eqnarray*}
where $f(t)= \gamma\int_0^t \bar{Z}_s \,ds$. Remark~\ref{RCOMrel}(b) gives $f(t) = \bar{Y}_t - \bar{Z}_t$, and hence $f(t) \ascv\bar{Y}_\infty$ follows from Theorem~\ref{TSurvival}. Note that
\begin{eqnarray*} &&\vert \tilde{X} '_t(\phi) - P_\infty^{\bar{Y}_\infty}(\phi)\vert\\ &&\qquad \le\biggl\vert \int\phi\bigl(x + f(t) \bigr)\,d \tilde{X} _t(dx) - \int\phi\bigl(x + f(\infty) \bigr)\,d \tilde{X} _t(dx) \biggr\vert \\ &&\qquad\quad{}+\biggl\vert\int\phi\bigl(x + f(\infty) \bigr)\,d \tilde{X} _t(dx) - \int\phi\bigl(x + f(\infty) \bigr)\,d \tilde{X} _\infty(dx) \biggr\vert \\ &&\qquad\le\vert f(t) - f(\infty)\vert + d( \tilde{X} _t, \tilde{X} _\infty) \end{eqnarray*}
since $\phi\in\mathrm{Lip}_1$. Taking the supremum over $\phi$ and the previous theorem give
\begin{eqnarray*} d( \tilde{X} '_t, \tilde{X} '_\infty) \le\vert f(t, \omega) - f(\infty, \omega )\vert + d( \tilde{X} _t, \tilde{X} _\infty) \ascv0. \end{eqnarray*}
\upqed\end{pf*}
\subsection{The repelling case, on the survival set}\label{SsRep}
Much less can be said for the SOU process repelling from its center of mass than in the attractive case. We can, however, show that the center of mass converges, provided the rate of repulsion is not too strong, which we recall was the first step toward showing the a.s. convergence of the normalized interacting SOU process in the attractive case. The situation here is more complicated since we prove that the COM of the ordinary SOU process with repulsion diverges almost surely, implying that results for convergence of $ \tilde{X} '$ will not simply be established through the correspondence. We finish with some conjectures on the limiting measure for the repelling case.
As in the previous section, assume $Z_0=Y_0=y_0$, unless stated otherwise.
\begin{pf*}{Proof of Theorem~\ref{TrCOM}}
Assume that $\mathbb{P}(\tK(|y_0|^2)) <\infty$, like in the proof of Theorem~\ref{TExtinc}. As in that theorem, this condition can be weakened to just the finite initial mass condition using similar reasoning.
For part (a), note that as in Theorem~\ref{TSurvival}, $\bar{Y}_t$ will converge if $\mathbb{P}([\bar{Y}]_t) < \infty$ and which holds if the following quantity is bounded:
\begin{eqnarray*} \mathbb{P}\biggl(\int_0^\infty\frac{\overline{\vert Y_s\vert^2} - \bar{Y} _s^2}{e^{\beta s}} \,ds; S \biggr) &\le&\mathbb{P}\biggl(\int_0^\infty\frac{\overline {Y^2_s}}{e^{\beta s}} \,ds ; S\biggr) \\ &\le& c\int_0^\infty\frac{1+s^6e^{-2\gamma s}}{e^{\beta s}} \,ds \\ &<& \infty, \end{eqnarray*}
by Lemma~\ref{Lsecmom} and by the conditions on $\gamma$.
For (b), we require the following lemma.
\begin{lemma}Let $-\beta/2<\gamma<0$ and $X_0\neq0$. For a measure $m$ on $\mathbb{R}^d$, let $\tau_a(m)$ be $m$ translated by $a\in \mathbb{R}^d$. That is, $\tau_a(m)(\phi) = \int\phi(x+a)m(dx)$. Then:
\begin{longlist}[(ii)]
\item[(i)] For all but at most countably many $a$,
\begin{equation}\label{Enzlim}
P_{\tau_a(X_0)}(e^{\gamma t}\bar Z_t\to L\neq0|S)=1. \end{equation}
\item[(ii)] For all but at most one value of $a$
\[
P_{\tau_a(X_0)}(e^{\gamma t}\bar Z_t\to L\neq0|S)>0. \]
\end{longlist}
\end{lemma}
\begin{pf} We first note that, by the correspondence \eqref{Ecorr}, we have that
\[ \bar{Z}_t = \bar{Y}_t - \gamma e^{-\gamma t}\int_0^te^{\gamma s}\bar{Y}_s \,ds. \]
Under our hypotheses $\bar{Y}_t$ converges by Theorem~\ref{TrCOM}(a), and hence
\begin{equation}\label{Erep1} \lim_{t\to\infty} e^{\gamma t}\bar Z_t+\int_0^t\gamma e^{\gamma s}\bar Y_s\,ds=0 \end{equation}
a.s. on $S$. Therefore, on $S$,
\begin{equation}\label{Erep2} \lim_{t\to\infty} e^{\gamma t}\bar Z_t\qquad \mbox{exists a.s.} \end{equation}
Note that one can build a solution of $\mbox{(SE)}^2$ with initial conditions given by $\tau_a(X_0)$ by seeing that if $Y_t$ gives the solution of~\ref{SE2}, then $Y_t+ a$ gives the solution of $\mbox{(SE)}^2_{Y_0+a, K}$, and that the projection
\[ X'_t(\cdot) = \int\mathbf{1}(Y_t+a \in\cdot) K_t(dy) \]
gives the appropriate interacting SOU process.
By \eqref{Erep1} and \eqref{Erep2},
\begin{eqnarray*}
\mathbb{P}_{X_0}(\{e^{\gamma t}\bar{Z}_t\rightarrow L \neq0\}^c | S ) &=&
\mathbb{P}_{\tau_a(X_0)}\biggl(\lim_{t\rightarrow\infty} \int_0^t e^{\gamma s}\bar{Y} _s\,ds = 0 \Big|S\biggr) \\ &=& \mathbb{P}_{X_0}\biggl(\lim_{t\rightarrow\infty} \int_0^t e^{\gamma s} \frac
{K_s(a+Y_s)}{K_s(1)} \,ds = 0\Big|S\biggr) \\
&=& \mathbb{P}_{X_0}\biggl(-\frac{a}{\gamma}+ \lim_{t\rightarrow\infty} \int _0^t e^{\gamma s}\bar{Y}_s \,ds = 0\Big|S\biggr). \end{eqnarray*}
The random variable $\int_0^\infty e^{\gamma s}\bar Y_s\,ds$ is finite a.s. and so only a countable number of values $a$ exist with the latter expression positive, implying the first result. The second result also follows as well since the last expression in the above display can be 1 for at most 1 value of $a$. \end{pf}
To complete the proof of Theorem~\ref{TrCOM}(b), choose a value $a\in \mathbb{R}^d$ such that~\eqref{Enzlim} holds. By Theorem III.2.2 of \cite {Perkins2002} and the fact that $X_0P_s \ll\tau_a(X_0)P_t, $ for all $0<s\le t$, for the OU semigroup $P_t$, we have that for all $0<s\le t$
\begin{equation} \label{Eabscon} \mathbb{P}_{X_0}(X_{s+\cdot} \in\cdot) \ll\mathbb{P}_{\tau_a(X_0)}(X_{t+\cdot} \in\cdot). \end{equation}
By our choice of $a$,
\[ \mathbb{P}_{\tau_a(X_0)}\Bigl(\mathbb{P}_{X_1}\Bigl(\lim_{t\rightarrow\infty} e^{\gamma t}\bar{Z}_t = 0, S \Bigr) \Bigr) = 0, \]
holds, and hence by \eqref{Eabscon} we have
\[ \mathbb{P}_{X_0}\Bigl(\lim_{t\rightarrow\infty} e^{\gamma t}\bar{Z}_t = 0, S \Bigr) = \mathbb{P} _{X_0}\Bigl(\mathbb{P}_{X_1}\Bigl(\lim_{t\rightarrow\infty} e^{\gamma t}\bar{Z} _t = 0, S \Bigr) \Bigr)= 0. \]
Recalling from \eqref{Erep2} that $\lim_{t\to\infty}e^{\gamma t}\bar Z_t$ exists a.s., we are done. \end{pf*}
Note that for $0 > \gamma> -\frac{\beta}{2}$, this implies that even if mass is repelled at rate $\gamma$, the COM of the interacting SOU process still settles down in the long run. That is, driving $Y_t$ away from $\bar{Y}_t$ seems to have the effect of stabilizing it. One can think of this as a situation where the mass is growing quickly enough that the law of large numbers overcomes the repelling force.
More surprising is that the COM of the ordinary SOU process diverges exponentially fast, even while the COM of the interacting one settles down. This follows from the correspondence
\[ \bar{Y}_t = \bar{Z}_t + \gamma\int_0^t \bar{Z}_s \,ds, \]
and the cancellation that occurs in it due to the exponential rate of $\bar{Z}_t$.
The next lemma shows that Theorem 1 of Engl\"ander and Winter \cite {EW2006} can be reformulated to yield a result for the SOU process with repulsion at rate~$\gamma$ (where~$\gamma$ is taken to be a negative parameter in our setting).
\begin{lemma} \label{Lreform}
On $S$, for the SOU process, $X,$ with repulsion rate $-\frac{\beta }{d}< \gamma<0 $ and compactly supported initial measure $\mu$, and any $\psi\in C_c^+(R^d)$
\[ e^{d\vert\gamma\vert t} \tilde{X} _t(\psi) \pcv\xi\int_{\mathbb{R}^d} \psi(x) \,dx, \]
where $\xi$ is a positive random variable on the set $S$. \end{lemma}
\begin{pf} Note that by Example 2 of Pinsky~\cite{Pinsky1996} it is shown that the hypotheses of Theorem 1 of~\cite{EW2006} hold for the SOU process with repulsion from the origin at rate $0< -\gamma<\frac{\beta}{d}$. The theorem says that there is a function $\phi_c \in C_b^\infty(\mathbb{R} ^d)$ such that
\begin{equation}\label{EEW} \frac{X_t(\psi)}{\mathbb{E}^{\mu}(X_t(\psi))} \pcv\frac{W\xi}{\mu (\phi_c)}, \end{equation}
where $W$ is as in Lemma~\ref{LExtProb}.
Example 2 also shows that for $\psi\in C^+_c(\mathbb{R}^d)$,
\[ \lim_{t\rightarrow\infty} e^{-(\beta+\gamma d)t}\mathbb{E}^\mu(X_t(\psi )) = \mu(\phi_c)m(\psi), \]
where $m$ is Lebesgue measure on $\mathbb{R}^d$. Hence, manipulating the expression in \eqref{EEW} by using the previous equation and Lemma \ref {LExtProb} gives
\begin{eqnarray*} e^{\vert\gamma\vert \,d t} \tilde{X} _t(\psi) &\pcv&\frac{\xi W}{\mu(\phi_c)} \lim _{t\rightarrow\infty} \frac{e^{\vert\gamma\vert \,dt} \mathbb{E}^\mu (X_t(\psi ))}{X_t(1)} \\ &=& \frac{\xi W}{\mu(\phi_c)} \lim_{t\rightarrow\infty} \frac {e^{-(\beta+ \gamma d)t} \mathbb{E}^\mu(X_t(\psi))}{e^{-\beta t}X_t(1)} \\ &=& \frac{\xi W}{\mu(\phi_c)} \frac{\mu(\phi_c)m(\psi)}{W}.\hspace*{80pt}\qed \end{eqnarray*}
\noqed\end{pf}
This lemma indicates that on the survival set, when $\gamma<0$, one cannot naively normalize $X_t$ by its mass since the probability measures $\{ \tilde{X} _t\}$ are not tight. That is, a proportion of mass is escaping to infinity and is not seen by compact sets. Note that the lemma above implies that for $X_t$, the right normalizing factor is $e^{(\beta+ \gamma d)t}$.
\begin{definition} We say a measure-valued process ${\mathring{X}}$ goes locally extinct if for any finite initial measure ${\mathring{X}}_0$ and any bounded $A\in\mathscr{B}(\mathbb{R} ^d)$, there is a $\mathbb{P}_{{\mathring{X}}_0}$-a.s. finite stopping time $\tau_A$ so that ${\mathring{X}}_t(A) = 0$ for all $t\ge\tau_A$ a.s. \end{definition}
\begin{remark} \label{RLocExtinc} Example 2 of Pinsky~\cite{Pinsky1996} also shows that for $\gamma\le -\beta/d$ the SOU undergoes local extinction (all the mass escapes). Hence for $\psi\in C_c(\mathbb{R}^d)$, there is no normalization where $X_t(\psi)$ can be expected to converge to something nontrivial. \end{remark}
From Remark~\ref{RIto2}, one can show that
\[ \bar{Z}_t = \bar{Z}_0 + N_t - \gamma\int_0^t \bar{Z}_s \,ds, \]
where $N$ is a martingale. Therefore, you can think of the COM of $X$, the SOU process repelling from origin, as being given by an exponential drift term plus fluctuations. The correspondence of Remark~\ref{RCOMrel}(b) implies then that $\bar{Y}_t = \bar{Y}_0 + N_t,$ or in other words, the center of mass of the SOU process repelling from its COM is given by simply the fluctuations.
We finish with some conjectures.
\begin{conjecture} \label{Crepulse} On the survival set, if $X'_0$ is fixed and compactly supported, then the following is conjectured to hold:
\begin{longlist}[(a)]
\item[(a)] If $0 < -\gamma< \frac{\beta}{d}$, then there exists constant $\beta+ \gamma d \le\alpha< \beta$ so that for $\phi\in C_c(\mathbb{R}^d)$,
\[ e^{-\alpha t}X'_t(\phi)\pcv\nu(\psi), \]
where $\nu$ is a random measure depending on $\bar{Y}_\infty$.
\item[(b)] If $\beta/d \le-\gamma$, then $X'_t$ undergoes local extinction. \end{longlist}
\end{conjecture}
We expect that $\alpha<\beta$ simply because of the repulsion from the COM built in to the model results in a proportion of mass being lost to infinity. One would expect that the limiting measure $\nu$ is a random multiple of Lebesgue measure as in the ordinary SOU process case, due to the correspondence, but it is conceivable that it is some other measure which has, for example, a dearth of mass near the limiting COM.
As stated earlier, it is difficult to use Lemma~\ref{Lreform} to prove this conjecture as the correspondence becomes much less useful\vadjust{\goodbreak} in the repulsive case. The problem is that while the equation
\[ \int\phi(x) \,dX'_t(x) = \int\phi\biggl(x+ \gamma\int_0^t \bar{Z}_s \,ds\biggr) \,dX_t(x) \]
still holds for $t$ finite, the time integral of $\bar{Z}_s$ now diverges.
\section{Proofs of technical lemmas}\label{Stech}
We now prove the lemmas first stated in Section~\ref{Sexpre}.
\begin{pf*}{Proof of Lemma~\ref{LExtProb}} We first note that $\mathring{K}_t = e^{-\beta t}K_t$ is a $(\Delta/2, 0,\break e^{-\beta t})$-historical process. The martingale problem then shows that $\mathring{K}_t(1)$ is a nonnegative $(\mathscr{F}_t)$-martingale and therefore converges almost surely by the Martingale Convergence theorem to a random variable $W$. It follows that $\{\eta< \infty\} \subset\{W=0\} $, since~0 is an absorbing state for $K_t(1)$. Exercise II.5.3 in~\cite{Perkins2002} shows that
\[ \mathbb{P}(\eta< \infty) = e^{-2\beta K_0(1)}. \]
The same exercise also shows
\[ \mathbb{P}_{K_0}(\operatorname{exp}{(-\lambda\mathring{K}_t(1)})) = \operatorname{exp}\biggl(-\frac{2\lambda \beta \mathring{K}_0(1)}{2\beta+ \lambda(1-e^{-\beta t})}\biggr). \]
Now sending $t \rightarrow\infty$ gives
\[ \mathbb{P}_{ K_0}(e^{-\lambda W}) = \operatorname{exp}\biggl(-\frac{2\beta\lambda\mathring{K} _0(1)}{2\beta+\lambda}\biggr), \]
and sending $\lambda\rightarrow\infty$ gives
\[ \mathbb{P}(W=0) = e^{-2\beta\mathring{K}_0(1)} = \mathbb{P}(\eta<\infty) \]
since $K_0 = \mathring{K}_0$. Therefore, $\{\eta< \infty\} = \{W=0\}$ almost surely. \end{pf*}
\begin{pf*}{Proof of Lemma~\ref{LCSP}} We follow the proof of Theorem III.1.3(a) in Perkins~\cite {Perkins2002}. First, note that if $H$ is another supercritical historical Brownian motion starting at time $\tau$ with initial measure $m$ under $\mathbb{Q}_{\tau, m}$ and $A$ a Borel subset of~$C$, then the process defined by
\[ H'_t(\cdot) = H_t(\cdot\cap\{y\dvtx y_\tau\in A\}) \]
is also a supercritical historical Brownian motion starting at time $\tau$ with initial measure $m'$ given by $m'(\cdot)=m(\cdot\cap A)$ under $\mathbb{Q}_{\tau, m}$. Then using the extinction probabilities for $H'$ (refer, e.g., to Exercise II.5.3 of~\cite{Perkins2002}) we have
\begin{equation}\label{EExt} \mathbb{Q}_{\tau, m}\bigl(H_t(\{y\dvtx y_\tau\in A\})=0\ \forall t\ge s\bigr) = \operatorname{exp} {\biggl\{-\frac{2\beta m(A)}{1-e^{-\beta(s-\tau) }}\biggr\} }. \end{equation}
Using the Markov property for $K$ at time $\frac{j}{2^n}$ and \eqref {EExt} gives
\begin{eqnarray} \label{EBC} &&\mathbb{P}\biggl[\exists t > \frac{j+1}{2^{n}} \mbox{ s.t. } K_t\biggl(\biggl\{y\dvtx \biggl\vert y\biggl(\frac{j}{2^n}\biggr) - y\biggl(\frac{j-1}{2^n}\biggr)\biggr\vert>c\biggl(\frac {j}{2^n}\biggr)h(2^{-n} ) \biggr\}\biggr) >0\biggr] \nonumber\\[-2pt] &&\qquad=\mathbb{P}\bigl[1- \operatorname{exp}\bigl\{-\bigl( 2\beta K_{{j}/{2^n}} \bigl(\bigl\{y\dvtx \bigl\vert y( {j}/{2^n}) - y\bigl({(j-1)}/{2^n}\bigr)\bigr\vert\nonumber\\[-2pt] &&\hspace*{169pt}\qquad{}>c( {j}/{2^n})h(2^{-n} ) \bigr\} \bigr)\bigr)\nonumber\\[-2pt] &&\hspace*{197pt}\qquad{}\big/( 1 - e^{-\beta/{2^n} }) \bigr\}\bigr] \nonumber \\[-8pt] \\[-8pt] \nonumber &&\qquad\le\mathbb{P}\biggl[\frac{ 2\beta K_{{j}/{2^n}} (\{y\dvtx \vert y( {j}/{2^n}) - y({(j-1)}/{2^n})\vert>c({j}/{2^n})h(2^{-n} ) \})}{ 1 - e^{-\beta/{2^n} } } \biggr] \\[-2pt] &&\qquad\le\mathbb{P}\biggl[\frac{ 2\beta K_{{j}/{2^n}} (\{y\dvtx \vert y( {j}/{2^n}) - y({(j-1)}/{2^n})\vert>c({j}/{2^n})h(2^{-n} ) \})}{ ({\beta}/{2^n}) -( {\beta^2}/{2^{2n+1} }) } \biggr]\nonumber\\[-2pt] &&\qquad\le\frac{2^{n+1}}{1 - ({\beta}/{2^{n+1} }) }\nonumber\\[-2pt] &&\qquad\quad{}\times\mathbb{P}(K_{{j}/{2^n}}(1))\hat{\mathbb{P}}_{{j}/{2^n}}\biggl(\biggl\vert y\biggl(\frac{j}{2^n}\biggr) - y\biggl(\frac{j-1}{2^n}\biggr)\biggr\vert>c\biggl(\frac{j}{2^n}\biggr)h(2^{-n} )\biggr),\nonumber \end{eqnarray}
where in the last step we have simply rearranged the constants and multiplied and divided by the mean mass at time $j2^{-n}$ and used the definition of the Campbell measure (Definition~\ref{Dcamp}).
Since under the normalized mean measure, $y$ is a stopped Brownian motion by Remark~\ref{RSOUHrep}(b), we use tail estimates to see that the last quantity in \eqref{EBC} is
\begin{eqnarray*} &\le&\frac{2^{n+1}}{1 - ({\beta}/{2^{n+1} })} \mathbb{P}(K_{ {j}/{2^n}}(1)) c_d n^{d/2 - 1}2^{-nc({j}/{2^n})^2/2} \\[-2pt] &\le&2^{n+2}2^{{\beta j\ln2}/{2^n}}\mathbb{P}(X_0(1))c_d n^{d/2 - 1}2^{-n(({j}/{2^n})+c_0)/2}. \end{eqnarray*}
Hence, summing over $j$ from $1$ to $n2^n$ gives
\begin{eqnarray*} &&\mathbb{P}\biggl[\mbox{There exists } 1\le j\le n2^n \mbox{ s.t. } \exists t > \frac{j+1}{2^{n}} \mbox{ s.t. } \\[-2pt] &&\quad K_t\biggl(\biggl\{y\dvtx \biggl\vert y\biggl(\frac{j}{2^n}\biggr) - y\biggl(\frac {j-1}{2^n}\biggr)\biggr\vert>c\biggl(\frac {j}{2^n}\biggr)h(2^{-n} ) \biggr\}\biggr)>0 \biggr] \\[-2pt] &&\qquad\le\mathbb{P}(X_0(1))c_d n^{d/2 - 1}2^{n+2- nc_0/2}\sum_{j=1}^{n2^n} 2^{{\beta j\ln2}/{2^n} - n({j}/{2^n})}\\[-2pt] &&\qquad\le\mathbb{P}(X_0(1))c_d n^{d/2 - 1}2^{-2n+2}\sum_{j=1}^{n2^n} 2^{- {j}/{2^n} }\\[-2pt] &&\qquad\le\mathbb{P}(X_0(1))c_d n^{d/2 - 1}2^{-n+2}, \end{eqnarray*}
where we have used the fact that $c_0>6$. Hence, the sum over $n$ of the above shows by Borel--Cantelli that there exists for almost sure $\omega$, $N(\omega)$ such that for all $n>N$, for all $1\le j\le n2^n$, for all $t\ge\frac{j+1}{2^n}$, for $K_t$-a.a. $y$,
\[ \biggl\vert y\biggl(\frac{j-1}{2^n}\biggr) - y\biggl(\frac{j}{2^n}\biggr)\biggr\vert < c\biggl(\frac {j}{2^n}\biggr)h(2^{-n} ). \]
Letting $\delta(\omega) = 2^{-N(\omega)}$, note that on the dyadics, by above, we have that
\begin{eqnarray*} \mathbb{P}(\delta< \lambda) &=& \mathbb{P}\biggl(N > -\frac{\ln\lambda}{\ln2}\biggr) \\ &=& \mathbb{P}\biggl[\exists n> -\frac{\ln\lambda}{\ln2}, \exists j\le n2^n, \exists t>\frac{j+1}{2^n}, \mbox{ s.t. } \\ &&\quad K_t\biggl(\biggl\{y\dvtx \biggl\vert y\biggl(\frac{j}{2^n}\biggr) - y\biggl(\frac {j-1}{2^n}\biggr)\biggr\vert>c\biggl(\frac {j}{2^n}\biggr)h(2^{-n} ) \biggr\}\biggr)>0\biggr]\\ &\le&\sum_{n= \lfloor-{\ln\lambda}/{\ln2} \rfloor} C'(d, c_0)n^{d/2 - 1}2^{2n-nc_0/2} \\ &\le &C(d, c_0, \varepsilon)\lambda^{({c_0}/{2})-\varepsilon}, \end{eqnarray*}
where $\varepsilon$ can be chosen to be arbitrarily small (though the constant $C$ will increase as it decreases). The rest of the proof follows as in Theorem III.1.3(a) of~\cite{Perkins2002}, via an argument similar to Levy's proof for the modulus of continuity for Brownian motion. \end{pf*}
\begin{pf*}{Proof of Lemma~\ref{Lsecmom}} Assume that $Z_0 = Y_0$ and $t<\eta$. Recall
\[ Z_t(\omega, y) \equiv e^{-\gamma t}Z_0+ \int_0^t e^{\gamma(s-t)} \,dy_s. \]
Note that below ``$\lesssim$'' denotes less than up to multiplicative constants independent of $t$ and $y$. Suppose that $y\in S(\delta, c(t))$, where $S(\delta, c(t))$ is the same as in the previous lemma. Then, as $Y_t = Z_t + \gamma\int_0^t \bar{Z}_s\,ds,$
\[
|Y_t|^2 \lesssim|Z_t|^2 + \gamma^2t \int_0^t\vert\bar{Z}_s\vert^2\,ds
\lesssim|Z_t|^2 + \gamma^2t \int_0^t \overline{|Z_s|^2}\,ds \]
by Cauchy--Schwarz and Jensen's inequality. Therefore, integrating with respect to the normalized measure gives
\begin{equation}\label{ECOMbd}
\overline{\vert Y_t\vert^2} \le\overline{|Z_t|^2} + \gamma^2t \int_0^t
\overline{|Z_s|^2}\,ds \end{equation}
and therefore we need only find the appropriate bounds for expectation of $\overline{|Z_t|^2}$ to get the result.
After another few applications of Cauchy--Schwarz and integrating by parts,
\begin{eqnarray*}
|Z_t|^2&\lesssim& e^{-2\gamma t}|Z_0|^2 + \biggl|e^{-\gamma t} \int_0^t e^{\gamma s}\,dy_s \biggr|^2\\
&\lesssim& e^{-2\gamma t}|Z_0|^2 + e^{-2\gamma t}\biggl|e^{\gamma t} y_t
- y_0 -\gamma\int_0^t y_s e^{\gamma s}\,ds \biggr|^2\\
&\lesssim& e^{-2\gamma t}(|Z_0|^2+|y_0|^2) +|y_t|^2 + \gamma^2 t\int _0^t |y_s|^2 e^{-2\gamma(t-s)}\,ds. \end{eqnarray*}
As $y\in S(\delta, c(t))$,
\begin{eqnarray*}
|Z_t|^2&\lesssim& e^{-2\gamma t}(|Z_0|^2+|y_0|^2) + |y_0|^2 + \biggl(\frac{tc(t)h(\delta)}{\delta}\biggr)^2 \\
&&{}+ \gamma^2 t\int_0^t \biggl[|y_0|^2+ \biggl(\frac {sc(s)h(\delta)}{\delta}\biggr)^2\biggr] e^{-2\gamma(t-s)}\,ds \\
&\lesssim& e^{-2\gamma t}(|Z_0|^2+|y_0|^2) + |y_0|^2 + |y_0|^2\gamma t(1-e^{-2\gamma t})/2 \\ &&{}+ c(t)^2 \biggl(\frac{h(\delta)}{\delta}\biggr)^2\bigl(t^2 + \gamma t^3(1-e^{-2\gamma t})\bigr) \\
& \lesssim&(1+\vert\gamma\vert t)(1+e^{-2\gamma t})(|Z_0|^2+|y_0|^2) \\ &&{} +c(t)^2 \biggl(\frac{h(\delta)}{\delta}\biggr)^2\bigl(t^2 + \gamma t^3(1-e^{-2\gamma t})\bigr). \end{eqnarray*}
Integrating by the normalized measure $\tilde{K}_t$,
\begin{eqnarray*}
\overline{|Z_t|^2} &\lesssim&(1+\vert\gamma\vert t)(1+e^{-2\gamma t})\tilde{K}_t(|Z_0|^2+|y_0|^2) \\ &&{}+ c(t)^2 \biggl(\frac{h(\delta)}{\delta}\biggr)^2\bigl(t^2 + \gamma t^3(1-e^{-2\gamma t})\bigr). \end{eqnarray*}
Then using \eqref{ECOMbd} and using the above bound on $\overline
{|Z_t|^2}$ gives
\begin{eqnarray}\label{El2bd}
\overline{\vert Y_t\vert^2} &\le&\overline{|Z_t|^2} + \gamma^2t \int_0^t
\overline{|Z_s|^2}\,ds \nonumber\\
&\lesssim&(1+t^2)(1+e^{-2\gamma t})\biggl(\tilde{K}_t(|Z_0|^2+|y_0|^2)+ \int _0^t\tilde{K}_s(|Z_0|^2+|y_0|^2)\,ds \biggr) \\ &&{}+ (1+t)c(t)^2\biggl(\frac{h(\delta)}{\delta } \biggr)^2\bigl(t^2 + \gamma t^3(1-e^{-2\gamma t})\bigr). \nonumber \end{eqnarray}
Note that $\phi(y) \equiv|Z_0(y_0)|^2+|y_0|^2$ and $\phi_n(y)\equiv \phi(y)1(\vert y\vert\le n)$ are $\hat{\mathscr{F}}_0$ measurable. By applying It\^ o's formula to $K_t(\phi_n)K_t^{-1}(1)$ and using the decomposition $K_t(\phi_n) = K_0(\phi_n) + \int_0^t\phi_n(y)\,dM(s, y) + \beta\int_0^t K_s(\phi_n) \,ds$ (which follows from Proposition 2.7 of \cite {Perkins1995}), we get
\[ \tilde{K}_t(\phi_n) = \tilde{K}_0(\phi_n) + N_t(\phi_n), \]
where $N_t(\phi_n)$ is a local martingale until time $\eta$, for each $n$. In fact, the sequence of stopping times $\{T_N\}$ appearing in Theorem~\ref{TIto} can be used to localize each $N_t(\phi_n)$. Applying first the monotone convergence theorem and then localizing gives
\begin{eqnarray*} \mathbb{P}\bigl(\tilde{K}_t(\phi);t<\eta\bigr) &=& \lim_{n\rightarrow\infty} \mathbb{P}\bigl(\tilde{K} _t(\phi _n);t<\eta\bigr) \\ &=& \lim_{n\rightarrow\infty}\lim_{N\rightarrow\infty} \mathbb{P}\bigl(\tilde{K} _t(\phi _n)\mathbf{1}(t<T_N) \bigr) \\ &=& \lim_{n\rightarrow\infty}\lim_{N\rightarrow\infty} \mathbb{P}\bigl(\tilde{K} _{t\wedge T_N}(\phi_n) - \tilde{K}_{T_N}(\phi_n)\mathbf{1}(t\ge T_N) \bigr)\\ &\le&\lim_{n\rightarrow\infty}\lim_{N\rightarrow\infty} \mathbb{P}(\tilde{K} _{t\wedge T_N}(\phi_n)) \\ &=& \lim_{n\rightarrow\infty}\lim_{N\rightarrow\infty} \mathbb{P}\bigl(\tilde{K} _0(\phi _n) + N_{t\wedge T_N}(\phi_n) \bigr)\\ &=& \lim_{n\rightarrow\infty} \mathbb{P}(\tilde{K}_0(\phi_n)) \\ &=& \mathbb{P}(\tilde{K}_0(\phi)), \end{eqnarray*}
where we have used the positivity of $\phi_n$ to get the fourth line and the monotone convergence theorem in the last line.
Further, note that
\begin{eqnarray*} \mathbb{P}\biggl(\int_0^t\tilde{K}_s(\phi)\,ds ;t<\eta\biggr) &\le&\mathbb{P}\biggl(\int_0^t\tilde{K} _s(\phi) \mathbf{1} (s<\eta) \,ds\biggr) \\ &=& \int_0^t \mathbb{P}\bigl(\tilde{K}_s(\phi);s<\eta\bigr) \,ds \\ &\le& t \mathbb{P}(\tilde{K}_0(\phi)), \end{eqnarray*}
by the calculation immediately above. Thus, taking expectations in \eqref{El2bd} and plugging in $c(t) = \sqrt{c_0+t}$ gives
\begin{eqnarray*} \mathbb{P}(\overline{\vert Y_t\vert^2}; t<\eta)\lesssim(1+t^3)(1+ e^{-2\gamma t})\mathbb{P}(\tilde{K}_0(\phi)) + t^5(1+ te^{-2\gamma t})\mathbb{P}\biggl(\frac {h(\delta)^2}{\delta^2}\biggr). \end{eqnarray*}
Now let $c_0$ be chosen so that $\operatorname{Supp}(K_t) \subset S(\delta,c(t))$ and $p_{c_0}(\lambda) = C\lambda^\alpha$ for $\lambda\in[0,1]$.
Note that
\begin{eqnarray*} \mathbb{P}\biggl(\frac{h(\delta)^2}{\delta^2}\biggr)&=&\mathbb{P}\biggl(\frac{\ln ^+(1/\delta)}{\delta}\biggr) = \int_0^\infty\biggl(\frac{\ln ^+(1/\lambda )}{\lambda}\biggr)\,d\mathbb{P}(\delta<\lambda) \\ &\le&\int_0^1 \biggl(\frac{\ln^+(1/\lambda)}{\lambda}\biggr)\,d\mathbb{P} (\delta <\lambda) +\mathbb{P}(\delta>1) \\ &=& \lim_{\lambda\downarrow0} \biggl( \frac{\ln^+(1/\lambda )}{\lambda}\mathbb{P} (\delta<\lambda)\biggr) - \mathbb{P}(\delta<1) \\ &&{}- \int_0^1 \mathbb{P}(\delta<\lambda)\,d\biggl(\frac{\ln ^+(1/\lambda )}{\lambda}\biggr) +\mathbb{P}(\delta>1)\\ &<& \infty, \end{eqnarray*}
by choosing the constant $c_0$ so that $\alpha$ is large ($\alpha\ge2$ is enough). \end{pf*}
Remark~\ref{Rmoments} follows from the above proof, after noting that the exponent $\alpha$ in Lemma~\ref{LCSP} can be made arbitrarily large by choosing a sufficiently large constant~$c_0$. Hence by choosing $\alpha$ appropriately, we can show that
\[ \mathbb{P}\biggl[\biggl(\frac{h(\delta)}{\delta}\biggr)^k\biggr] <\infty, \]
which can then be used to adapt the proof above.
Recall that $\mathrm{Lip}_1= \{\psi\in C(\mathbb{R}^d)\dvtx \forall x, y, \vert\psi (x)
- \psi(y)\vert\le\vert x-y\vert, \|\psi\| \le1\}$.
We will, with a slight abuse of notation, allow $M$ to denote the orthogonal martingale measure generated by the martingale problem for $X$. Let $A$ be the infinitesimal generator for an OU process, and hence recall that for $\phi\in C^2(\mathbb{R}^d)$,
\[ A\phi(x) = -\gamma x\cdot\nabla\phi(x) +\frac{\Delta}{2}\phi(x). \]
The next two proofs are for lemmas stated in Section~\ref{SsSurv}.
\begin{pf*}{Proof of Lemma~\ref{LSOUconvL2}} Let $\phi\in\mathrm{Lip}_1$. By the extension of the martingale problem for $X$ given in Proposition II.5.7 of~\cite{Perkins2002}, for functions $\psi\dvtx [0,T]\times\mathbb{R} ^d\rightarrow\mathbb{R}$ such that $\psi$ satisfies the definition before that proposition,
\[ X_t(\psi_t) = X_0(\psi_0) +\int_0^t\int\psi_s(x)\,dM(x,s) + \int_0^t X_s(A\psi_s + \beta\psi_s + \dot{\psi}_s) \,ds, \]
where $M$ is the orthogonal martingale measure derived from the martingale problem for the SOU process. It is not difficult to show that $\psi_s = P_{t-s}\phi$ where $\phi$ as above satisfies requirements for Proposition II.5.7 of~\cite{Perkins2002}. Plugging this in gives
\[ X_t(\phi) = X_0(P_t\phi) + \int_0^t\int P_{t-s}\phi(x)\,dM(s, x) + \int _0^t\beta X_s(P_{t-s}\phi)\,ds \]
since $\frac{\partial}{\partial s}P_s\phi= AP_s\phi$. Multiplying by $e^{-\beta t}$ and integrating by parts gives
\begin{eqnarray} \label{EXblim} e^{-\beta t}X_t(\phi) &=& e^{-\beta t}X_t(\psi_t)\nonumber\\ & =& X_0(\psi_0) - \int _0^t\beta e^{-\beta s}X_s(\psi_s)\,ds + \int_0^te^{-\beta s} \,dX_s(\psi _s)\\ &= &X_0(P_t\phi) + \int_0^t\int e^{-\beta s}P_{t-s}\phi(x)\,dM(s,x).\nonumber \end{eqnarray}
Note that as the $OU$-process has a stationary distribution $P_\infty$ where $P_t \rightarrow P_\infty$ in norm. When $s$ is large in \eqref {EXblim}, $P_{t-s}\phi(x)$ does not contribute much to the stochastic integral and hence we expect the limit of $e^{-\beta t}X_t(\phi)$ to be
\begin{equation}\label{EXblim2} X_0(P_\infty\phi) + \int_0^\infty\int e^{-\beta s}P_\infty\phi (x) \,dM(s,x), \end{equation}
which is a well defined, finite random variable as
\[ \biggl[\int_0^\cdot\int e^{-\beta s}P_\infty\phi(x) \,dM(s, x)\biggr]_\infty
< \|
\phi\|^2 \int_0^\infty e^{-2\beta s}X_s(1)\,ds, \]
which is finite in expectation. As $P_\infty\phi(x)$ does not depend on $x$, it follows that
\begin{eqnarray*} \eqref{EXblim2} = (P_\infty\phi) X_0(1) + (P_\infty\phi) \int _0^\infty\int e^{-\beta s } \,dM(s, x) = WP_\infty\phi. \end{eqnarray*}
Given this decomposition for $WP_\infty\phi$, we write
\begin{eqnarray*} &&\mathbb{P}\bigl(\bigl(e^{-\beta t}X_t(\phi) - WP_\infty\phi\bigr)^2\bigr)\\ &&\qquad\le3 \mathbb{P}\biggl(\biggl(\int _t^\infty e^{-\beta s}P_{\infty}\phi(x)\,dM(s,x) \biggr)^2 \biggr)\\ &&\qquad\quad{}+ 3\mathbb{P}\biggl(\biggl(\int_0^t\int e^{-\beta s}\bigl(P_{t-s}\phi(x)- P_{\infty }\phi (x)\bigr)\,dM(s,x)\biggr)^2\\ &&\hspace*{160pt}\qquad{}+ X_0(P_\infty\phi- P_t\phi)^2\biggr). \end{eqnarray*}
If $z_t$ is a $d$-dimensional OU process satisfying $dz_t = -\gamma z_t dt+dB_t,$ where $B_t$ is a $d$-dimensional Brownian motion, then
\[ z_t =e^{-\gamma t}z_0+ \int_0^t e^{-(t-s)\gamma}\,dB_t \]
and hence $z_t$ is Gaussian,\vspace*{-2pt} with mean $e^{-\gamma t}z_0$ and covariance matrix $\frac{1}{2\gamma}(1-e^{-2\gamma t})I$. Evidently, $z_\infty$ is also Gaussian, mean 0 and variance $\frac{1}{2\gamma} I$. We use a simple coupling: suppose that $w_t$ is a random variable independent of $z_t$ such that $z_\infty= z_t+ w_t$ (i.e., $w_t$ is Gaussian with mean $-e^{-\gamma t}z_0$ and covariance $\frac {1}{2\gamma }e^{-2\gamma t}I$). Then using the fact that $\phi\in\mathrm{Lip}_1$ and the Cauchy--Schwarz inequality, followed by our coupling with $z_0 = x$ gives
\begin{eqnarray*} X_0(P_\infty\phi- P_t\phi)^2 &=& \biggl( \int\mathbb{E}^x\bigl(\phi (z_\infty) - \phi(z_t)\bigr)X_0(dx)\biggr)^2 \\
&\le&\int\mathbb{E}^x(|z_\infty- z_t|)^2X_0(dx)X_0(1) \\
&=& \int\mathbb{E}^x(|w_t|^2)X_0(dx)X_0(1) \\
&= &\int e^{-2\gamma t}\biggl(|x|^2 + \frac{d}{2\gamma} \biggr)X_0(dx)X_0(1)\\ &\le& ce^{-2\gamma t}\biggl(\int\vert x\vert^2X_0(dx)X_0(1) + X_0(1)^2\biggr). \end{eqnarray*}
Taking expectations and using Cauchy--Schwarz and the assumptions on $X_0$ gives exponential rate of convergence for the above term.
Since we can think of $\int_0^r\int e^{-\beta s}P_{t-s}\phi(x) \,dM(s,x)$ as a martingale in $r$ up until time $t$, various martingale inequalities can be applied to get bounds for the terminal element, $\int_0^t\int e^{-\beta s}P_{t-s}\phi(x) \,dM(s,x)$. Note that this process is not in general a martingale in $t$. Therefore, we have
\begin{eqnarray}\label{EM1} &&\mathbb{P}\biggl[\biggl(\int_0^t\int e^{-\beta s}P_\infty\phi(x) \,dM(s,x) - \int _0^t\int e^{-\beta s}P_{t-s}\phi(x) \,dM(s,x) \biggr)^2 \biggr] \nonumber\\ &&\qquad= \mathbb{P}\biggl[\biggl(\int_0^t\int e^{-\beta s}\bigl(P_\infty \phi (x) - P_{t-s}\phi(x)\bigr) \,dM(s,x) \biggr)^2 \biggr] \\ &&\qquad\le\mathbb{P}\biggl[\int_0^t e^{-2\beta s}\int\bigl(P_\infty \phi (x) - P_{t-s}\phi(x)\bigr)^2X_s(dx)\,ds\biggr]. \nonumber \end{eqnarray}
Then as $\phi$ Lipschitz, by the coupling above,
\begin{eqnarray*} \eqref{EM1} &\le&\mathbb{P}\biggl[\int_0^t e^{-2\beta s}\int e^{-2\gamma
(t-s)}\biggl(|x|^2 + \frac{d}{2\gamma} \biggr)X_s(dx)\,ds\biggr] \\
&= &\int_0^t e^{-2\beta s-2\gamma(t-s)}\mathbb{P}\biggl[\int|x|^2 + \frac {d}{2\gamma} X_s(dx)\biggr]\,ds \\
&=& \int_0^t e^{-2\beta s-2\gamma(t-s)}\mathbb{P}\biggl[K_s(|Z_s|^2) + \frac {d}{2\gamma}X_s(1)\biggr]\,ds. \end{eqnarray*}
Applying the Cauchy--Schwarz inequality followed by Remark \ref {Rmoments}(b) gives
\begin{eqnarray*}
&&\mathbb{P}(K_s(|Z_s|^2))\\
&&\qquad=\mathbb{P}\bigl(\overline{|Z_s|^2}K_s(1) ; s<\eta\bigr) \\
&&\qquad\le\mathbb{P}(\overline{|Z_s|^2}^2; s<\eta)^{{1}/{2}}\mathbb{P} (X_s(1)^2)^{{1}/{2}}\\ &&\qquad\le B(s, \gamma, 4)^{{1}/{2}}\mathbb{P}(X_s(1)^2)^{{1}/{2}}\\ &&\qquad\le cB(s, \gamma, 4)^{{1}/{2}} e^{\beta s} \mathbb{P}\biggl(X_0(1)^2 + \frac {1}{\beta}X_0(1)\biggr)^{{1}/{2}}, \end{eqnarray*}
where the last line follows by first noting that
\[ e^{-\beta t}X_t(1) = X_0(1) + \int_0^t\int e^{-\beta s} \,dM(s,x) \]
is a martingale. That is,
\begin{eqnarray}\label{EmassL2} e^{-2\beta s} \mathbb{P}(X_s(1)^2) &\le&2\mathbb{P}\biggl(X_0(1)^2 + \biggl(\int_0^s e^{-\beta r} \,dM(r, x)\biggr)^2\biggr) \nonumber\\[-2pt] &=& 2\mathbb{P}\biggl(X_0(1)^2+\biggl[\int_0^\cdot e^{-\beta r}\,dM(r, x)\biggr]_s\biggr)\nonumber \\[-2pt] &=& 2\mathbb{P}\biggl(X_0(1)^2+ \int_0^s e^{-2\beta r}X_r(1)\,dr \biggr) \nonumber \\[-8pt] \\[-8pt] \nonumber &=& 2\mathbb{P}(X_0(1)^2)+ 2\int_0^s e^{-\beta r} \mathbb{P}(e^{-\beta r}X_r(1) )\,dr\\[-2pt] &=&2\mathbb{P}(X_0(1)^2)+ 2\int_0^s e^{-\beta r} \mathbb{P}(X_0(1) )\,dr\nonumber \nonumber\\[-2pt] &\le&2\mathbb{P}\biggl(X_0(1)^2 + \frac{1}{\beta}X_0(1)\biggr).\nonumber \end{eqnarray}
Therefore,
\begin{eqnarray*} \eqref{EM1} &\le&\int_0^t e^{-2\beta s-2\gamma(t-s)} \biggl[e^{\beta s}B(s, \gamma, 4)^{{1}/{2}} \mathbb{P}\biggl(X_0(1)^2 + \frac{1}{\beta }X_0(1)\biggr)^{{1}/{2}}\biggr] \,ds\\[-2pt] &&{}+ \int_0^te^{-2\beta s-2\gamma(t-s)} e^{\beta s}\frac {d}{2\gamma}\mathbb{P}(X_0(1))\,ds \\[-2pt] &\le&\int_0^t e^{-\beta s-2\gamma(t-s)}\biggl[B(s, \gamma, 4)^{ {1}/{2}} \mathbb{P}\biggl(X_0(1)^2 + \frac{1}{\beta}X_0(1)\biggr)^{{1}/{2}}\\[-2pt] &&\hspace*{155pt}\qquad{} + \frac {d}{2\gamma}\mathbb{P}(X_0(1)) \biggr]\,ds \\[-2pt] &<& C e^{-\zeta_1 t}, \end{eqnarray*}
where $\zeta_1 = \min(\beta, 2\gamma)-\varepsilon$ where $\varepsilon$ is arbitrary small and comes from the polynomial term in the integral.
Finally,
\begin{eqnarray*} \mathbb{P}\biggl(\biggl(\int_t^\infty e^{-\beta s}P_{\infty}\phi(x)\,dM(s,x) \biggr)^2 \biggr) &=& (P_\infty\phi)^2\mathbb{P}\biggl(\int_t^\infty e^{-2\beta s}X_s(1) \,ds \biggr) \\[-2pt] &\le&(P_\infty\phi)^2\int_t^\infty e^{-\beta s}\mathbb{P}(e^{-\beta s}X_s(1))\,ds\\ &\le&\frac{(P_\infty\phi)^2}{\beta}e^{-\beta t}\mathbb{P}(X_0(1)), \end{eqnarray*}
since $e^{-\beta s} X_s(1)$ is a martingale. Therefore, since $\zeta _1<\beta$, we see that $\zeta= \zeta_1$ gives the correct exponent. \end{pf*}
\begin{pf*}{Proof of Lemma~\ref{LSOU-increments}} The proof will follow in a manner very similar to the proof of the previous lemma. From the calculations above, we see that
\begin{eqnarray*} && e^{-\beta(t+h)}X_{t+h}(\phi) - e^{-\beta t}X_t(\phi)\\ &&\qquad = X_0(P_{t+h}\phi- P_t\phi) \\ &&\qquad\quad{}+ \int_0^{t+h}\int e^{-\beta s}P_{t+h-s}\phi(x)\,dM(s, x) - \int _0^{t}\int e^{-\beta s}P_{t-s}\phi(x)\,dM(s, x) \\ &&\qquad= X_0(P_{t+h}\phi- P_t\phi) + \int_0^{t}\int e^{-\beta s} \bigl((P_{t+h-s}-P_{t-s})\phi(x) \bigr)\,dM(s, x)\\ &&\qquad\quad{}+ \int_t^{t+h}\int e^{-\beta s}P_{t+h-s}\phi(x)\,dM(s, x) \\ &&\qquad\equiv I_1+I_2 +I_3. \end{eqnarray*}
Using the Cauchy--Schwarz inequality, we can find bounds for $\mathbb{P}
(|I_k|^4), k=1,2,3$, separately:
\begin{eqnarray*}
|I_1|^4 &=& X_0(P_{t+h}\phi- P_t\phi)^4 \\ &\le&\bigl[X_0\bigl((P_{t+h}\phi- P_t\phi)^2\bigr)X_0(1)\bigr]^2. \end{eqnarray*}
Recalling the simple coupling in the previous lemma to see that
\begin{eqnarray*}
\bigl(P_{t+h}\phi(x) - P_t\phi(x)\bigr)^2 &\le&\mathbb{E}^x\bigl(|\phi(z_{t+h}) - \phi
(z_t)|^2\bigr) \\
&\le&\mathbb{E}^x(|z_{t+h} - z_t|^2)\\
&\le&\mathbb{E}^x(|w_{t,t+h}|^2), \end{eqnarray*}
where $z$ is as above, an OU process started at $x$, and $w_{s,t}$ is independent of $z_s$ but such that $z_t = z_s + w_{s,t}$. Hence, $w_{s,t}$ is Gaussian with mean $x(e^{-\gamma t}-e^{-\gamma s})$ and covariance matrix $\frac{I}{2\gamma}(e^{-2s\gamma}-e^{-2t\gamma})$. Therefore,
\begin{eqnarray*}
\bigl(P_{t+h}\phi(x) - P_t\phi(x)\bigr)^2 &\le&|x|^2\bigl(e^{-\gamma (t+h)}-e^{-\gamma t}\bigr)^2+ \frac{d}{2\gamma}\bigl(e^{-2t\gamma }-e^{-2(t+h)\gamma}\bigr) \\
&=& e^{-2\gamma t}\biggl(|x|^2(1-e^{-\gamma h})^2+ \frac{d}{2\gamma }(1-e^{-2h\gamma}) \biggr). \end{eqnarray*}
Hence,
\begin{eqnarray*}
\mathbb{P}(|I_1|^4) &=& \mathbb{P}\biggl[ \biggl(e^{-2\gamma t}(1-e^{-\gamma h})^2\int
|x|^2X_0(dx)X_0(1)\\ &&\hspace*{56pt}{} + \frac{d}{2\gamma}e^{-2\gamma t}(1-e^{2\gamma h})X_0(1)^2\biggr)^2 \biggr]\\ &\le& C_1(d, \gamma) h^2 e^{-4\gamma t}, \end{eqnarray*}
where $C_1$ is a constant that is finite by assumptions on the initial measure.
To get bounds on the expectation of $I_2$, we use martingale inequalities. Note that $ \int_0^{\cdot}\int e^{-\beta s} ((P_{t+h-s}-P_{t-s})\phi(x) )\,dM(s, x) =N(\cdot)$ is a martingale until time $t$. Therefore, using the Burkholder--Davis--Gundy inequality and the coupling above gives
\begin{eqnarray*}
\mathbb{P}(|I_2|^4) &\le& c\mathbb{P}([N]_t^2) \\ &=&c\mathbb{P}\biggl[\biggl(\int_0^t\int e^{-2\beta s} \bigl((P_{t+h-s}-P_{t-s})\phi (x) \bigr)^2X_s(dx) \,ds \biggr)^2\biggr]\\ &\le& c\mathbb{P}\biggl[\biggl( \int_0^t e^{-2\beta s} \bigl(e^{-\gamma
(t+h-s)}-e^{-\gamma(t-s)}\bigr)^2\int|x|^2X_s(dx)\,ds\biggr)^2 \\* &&{}+ \biggl(\int_0^t\frac{de^{-2\beta s} }{2\gamma }\bigl(e^{-2(t-s)\gamma}-e^{-2(t+h-s)\gamma}\bigr)X_s(1) \,ds \biggr)^2\biggr]\\ &\le& c\mathbb{P}\biggl[\biggl( t\int_0^t e^{-4\beta s} \bigl(e^{-\gamma
(t+h-s)}-e^{-\gamma(t-s)}\bigr)^4\biggl(\int|x|^2X_s(dx)\biggr)^2\,ds\biggr) \\* &&{}+ t\biggl(\frac{d}{2\gamma}\biggr)^2\int_0^t e^{-4\beta s} \bigl(e^{-2(t-s)\gamma}-e^{-2(t+h-s)\gamma}\bigr)^2X_s(1)^2 \,ds \biggr]\\ &=&c t\int_0^t e^{-2\beta s} \bigl(e^{-\gamma(t+h-s)}-e^{-\gamma
(t-s)}\bigr)^4\mathbb{P}\biggl[\biggl(\int|x|^2e^{-\beta s}X_s(dx)\biggr)^2\biggr]\,ds \\ &&{}+ t\biggl(\frac{d}{2\gamma}\biggr)^2\int _0^te^{-2\beta s}\bigl(e^{-2(t-s)\gamma}-e^{-2(t+h-s)\gamma}\bigr)^2\mathbb{P}[(e^{-\beta s}X_s(1))^2] \,ds. \end{eqnarray*}
Since $X_s(|x|^2) = K_s( |Z_s|^2)$, by Remark~\ref{Rmoments}(b),
\begin{eqnarray*}
\mathbb{P}(e^{-2\beta s}X_s(|x|^2)^2) &\le&\mathbb{P}[ \overline {Z^2_{s}}^2(e^{-\beta s}X_s(1))^2; s<\eta]\\ &\le&\mathbb{P}( \overline{Z_{s}^2}^4; s<\eta)^{1/2}\mathbb{P} (e^{-4\beta s}X_{s}(1)^4)^{1/2} \\ &\le&\mathbb{P}( \overline{Z_{s}^8} ; s<\eta)^{1/2} \mathbb{P} (e^{-4\beta s}X_{s}(1)^4)^{1/2}\\ &\le& cB(\gamma, s, 8)^{1/2}\bigl( \mathbb{P}\bigl(X_0(1)^4+ sX_0(1)^2+ sX_0(1)\bigr) \bigr)^{1/2}. \end{eqnarray*}
The bound on the expectation of $e^{-4\beta s}X_s(1)^4$ follows by an application of the BDG Inequality to $e^{-\beta s}X_s(1) =X_0(1) + \int _0^s e^{-\beta r}\,dM(r, x)$ and similar calculations used to determine the bound on \eqref{EmassL2}. Therefore,
\begin{eqnarray*}
\mathbb{P}(|I_2|^4) &\le& ct\int_0^te^{-2\beta s}B(\gamma, s, 8)^{1/2} \bigl(e^{-\gamma(t+h-s)}-e^{-\gamma(t-s)}\bigr)^4\\ &&\hspace*{23pt}{}\times\bigl(\mathbb{P}\bigl(X_0(1)^4+ sX_0(1)^2 + sX_0(1)\bigr) \bigr)^{1/2}\,ds \\ &&{}+ \frac{ct}{4\gamma^2} \int_0^t e^{-2\beta s}\bigl(e^{-2(t-s)\gamma }-e^{-2(t+h-s)\gamma}\bigr)^2\mathbb{P}\biggl(X_0(1)^2 + \frac{1}{\beta}X_0(1)\biggr)\,ds \\ &\le& ctB(\gamma, t, 8)^{1/2}\bigl(\mathbb{P}\bigl(X_0(1)^4+ tX_0(1)^2 + tX_0(1)\bigr)\bigr)^{1/2}\\ &&{}\times (e^{-\gamma h}-1)^4\int_0^t e^{-2\beta s}e^{-4\gamma(t-s)}\,ds \\ &&{}+ \frac{ct}{4\gamma^2}(e^{-2\gamma h} -1)^2\mathbb{P}\biggl(X_0(1)^2 + \frac {1}{\beta}X_0(1)\biggr) \int_0^te^{-2\beta s}e^{-4(t-s)\gamma}\,ds \\ &\le& C_2(t,\gamma, \beta)h^2e^{-\zeta_1 t}, \end{eqnarray*}
where $C_2$ is polynomial in $t$.
By another application of the BDG inequality, and noting that $\|\phi\| = 1$,
\begin{eqnarray*}
\mathbb{P}(| I_3|^4) &=& \mathbb{P}\biggl[\biggl(\int_t^{t+h}\int e^{-\beta s} P_{t+h-s}\phi (x)\,dM(s, x)\biggr)^4\biggr]\\ &\le& c\mathbb{P}\biggl[\biggl(\int_t^{t+h}\int\bigl(e^{-\beta s}P_{t+h-s}\phi (x)\bigr)^2X_s(dx) \,ds\biggr)^2 \biggr] \\ &\le& c h \mathbb{P}\biggl[\int_t^{t+h}e^{-4\beta s}X_s(1)^2 \,ds \biggr]\\ &=& c he^{-2\beta t} \int_t^{t+h} \mathbb{P}[e^{-2\beta s}X_s(1)^2 ]\,ds \\ &\le& c he^{-2\beta t} \int_t^{t+h} \mathbb{P}[X_0(1)^2+X_0(1)/\beta]\,ds \\ &\le& C_3(\beta) h^2e^{-2\beta t}, \end{eqnarray*}
where the second last line follows from the same calculations performed in estimating moments of $I_2$. Note that the constant $C_3$ does not depend on $t$ here.
Putting the pieces together shows that there exists a function $C$ polynomial in $t$ and a positive constant $\zeta^*$ such that \eqref {EIncrBd} holds.
\end{pf*}
\printaddresses
\end{document} | arXiv |
L'Intermédiaire des mathématiciens
L'Intermédiaire des mathématiciens was a peer-reviewed scientific journal covering mathematics published by Gauthier-Villars et fils. It was established in 1894 by Émile Lemoine and Charles-Ange Laisant and was published until 1920. A second series started in 1922 and was published until 1925.
L'Intermédiaire des mathématiciens
DisciplineMathematics
LanguageFrench
Edited byÉmile Lemoine
Publication details
History1894–1925
Publisher
Gauthier-Villars et fils (France)
Standard abbreviations
ISO 4 (alt) · Bluebook (alt1 · alt2)
NLM · MathSciNet
ISO 4Interméd. Math.
Indexing
CODEN · JSTOR · LCCN
MIAR · NLM · Scopus
OCLC no.504209870
External links
• L'Intermédiaire des mathématiciens at the HathiTrust Digital Library
| Wikipedia |
Why do single particle states furnish a rep. of the inhomogeneous Lorentz group?
Following up on this question: Weinberg says
In general, it may be possible by using suitable linear combinations of the $\psi_{p,\sigma}$ to choose the $\sigma$ labels in such a way that $C_{\sigma'\sigma}(\Lambda, p)$ is block-diagonal; in other words, so that the $\psi_{p,\sigma}$ with $\sigma$ within any one block by themselves furnish a representation of the inhomogenous Lorentz group.
But why inhomogeneous Lorentz group if, in the first place, we performed a homogeneous Lorentz transformation on the states, via $U(\Lambda)$? I also want to be clear what is meant by the states "furnishing" a representation.
Regarding the above confusion, the same scenario again shows up during the discussion on the little group. Here's a little background: $k$ is a "standard" 4-momentum, so that we can express any arbitrary 4-momentum $p$ as $p^{\mu} = L^{\mu}_{\nu}(p) k^{\nu}$, where $L$ is a Lorentz transformation dependent on $p$. We consider the subgroup of Lorentz transformations $W$ that leave $k$ invariant (little group), and find that:
$U(W)\psi_{k \sigma} = \sum_{\sigma'} D_{\sigma' \sigma}(W)\psi_{k \sigma'}$. Then he says:
The coefficients $D(W)$ furnish a representation of the little group; i.e., for any elements $W$ and $W'$ , we get $D_{\sigma' \sigma}(W'W) = \sum_{\sigma''}D_{\sigma' \sigma''}(W)D_{\sigma''\sigma}(W')$.
So is it that even in the first part about the Lorentz group, $C$ matrices furnish the representation and not $\psi$?
Also, for the very simplified case if $C_{\sigma'\sigma}(\Lambda, p)$ is completely diagonal, would I be correct in saying the following in such a case, for any $\sigma$?
$$U(\Lambda)\psi_{p,\sigma} = k_{\sigma}(\Lambda, p)\psi_{\Lambda p, \sigma}$$
Only in this case it is clear to me that $U(\Lambda)$ forms a representation of Lorentz group, since $\psi_{p,\sigma}$ are mapped to $\psi_{\Lambda p, \sigma}$.
quantum-field-theory special-relativity group-representations poincare-symmetry
$\begingroup$ Just a tip: no need to mark the questions as follow-up questions so boldly. I've made some cosmetic edits to this and your last question, but if I changed the meaning anywhere from what you wanted to ask, please do fix it. :-) $\endgroup$ – David Z♦ May 25 '13 at 6:33
$\begingroup$ A representation is a vector space with a group action attached to it. In linear algebra, it's a bunch of vectors $v^i$ which move around under the group action, i.e. when $g$ is a group element then there is a matrix $D(g)^i{}_j$ which acts on them as $D(g)^i{}_j v^j$. In QM/QFT, the vector space is spanned by states $|\psi(p,\sigma)\rangle$ (or whatever notation Weinberg uses). That what he means by "furnishing" a rep. $\endgroup$ – Vibert May 25 '13 at 6:51
$\begingroup$ @Vibert: That's what I thought. But $U$ matrices map the states $\psi$ to Lorentz-transformed states, so then it should be the $C_{\sigma \sigma'}$ matrices that furnish a representation of the Lorentz group. I'm confused why Weinberg says that the states $\psi$ are the ones furnishing such a representation. (See the edited question) $\endgroup$ – 1989189198 May 26 '13 at 5:14
$\begingroup$ OK, I see, it's a matter of language. Formally a representation is a linear map from the group to your vector space, so in this case the map $W \mapsto D(W).$ But when you use this construction, of course it depends on both the cases $\psi$ and the Lorentz matrices acting on them. That's why in physics we use the term representation in a sloppier way than our mathematician friends. $\endgroup$ – Vibert May 26 '13 at 7:02
In the inhomogenous Lorentz group $ISO(1,3)$, you have the space-time translation group $\mathbb{R}^{1,3}$, and the Lorentz group $SO(1,3)$.
You begin to find a representation of the space-time translation group, by choosing a momentum $p$. So your representation must have a $p$ index,
$$\psi_p \, .$$
After this, you will have to get the full representation, by finding a representation of the Lorentz group compatible with the momentum $p$, this will add another index $\sigma$ which corresponds to the polarization, so you will have a representation,
$$\psi_{p, \sigma} \, ,$$
which is the representation of the inhomogenous Lorentz group.
TrimokTrimok
Re the meaning of representation, here is a definition from Peter Woit's "Quantum Mechanics for Mathematicians" lecture notes (available on-line), section 1.3:
Definition (Representation). A (complex) representation ($\pi, V$) of a group $G$ is a homomorphism $$ \pi: g \in G \rightarrow \pi(g) \in GL(V) $$ where $GL(V)$ is the group of invertible linear maps $V \rightarrow V$, with $V$ a complex vector space.
Saying a map is a homomorphism means $$ \pi(g_1) \pi(g_2) = \pi(g_1g_2) $$ When $V$ is finite dimensional and we have chosen a basis of $V$, then we have an identification of linear maps and matrices $$ GL(V) \simeq GL(n,\boldsymbol{C}) $$ where $GL(n,\boldsymbol{C})$ is the group of invertible $n$ by $n$ complex matrices.
So the representation is the homomorphism (the operation-preserving map) from the group $U(\Lambda)$ to the transformation matrices (Weinberg's C's and D's), but these matrices require a vector space (the $\psi$s), on which to act.
For the rest, here's my answer (caveat emptor, I'm just a slow student):
This section 2.5 is titled "One Particle States". If $C$ turns out to be reducible (block-diagonalizable), the different blocks are independent of one another (no mixing between blocks) and are interpreted as different particles species. So, for a single particle state a single irreducible block is assumed.
In this argument it's OK to generalize from homogeneous to inhomogeneous transformations, because translations don't mix $\sigma$'s and hence don't affect the block structure of $C$:
$$U(1,a) \Psi_{p,\sigma} = e^{-ip\cdot a} \Psi_{p,\sigma} $$
Finally, in the case you posit of a completely diagonal $C$, I think you are left with a bunch of particle species with no $\sigma$-mixing at all, i.e. scalars, each with a trivial little group ($k=1$).
Art BrownArt Brown
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Weinberg's classification of one-particle states and representations of the Poincare group | CommonCrawl |
Small-scale topographic irregularities on Phobos: image and numerical analyses for MMX mission
Tomohiro Takemura1,
Hideaki Miyamoto ORCID: orcid.org/0000-0001-8013-61241,
Ryodo Hemmi2,
Takafumi Niihara1 &
Patrick Michel3
The mothership of the Martian Moons eXploration (MMX) will perform the first landing and sampling on the surface of Phobos. For the safe landing, the 2.1-m-wide mothership of the MMX should find a smooth surface with at most 40 cm topographic irregularity, however, whose abundance or even existence is not guaranteed based on current knowledge. We studied the highest resolution (a few meters per pixel) images of Phobos for possible topographic irregularities in terms of boulder (positive relief feature) and crater distributions. We find that the spatial number densities of positive relief features and craters can vary significantly, indicating that the surface irregularities vary significantly over the entire surface. We extrapolate the size-frequency distributions of positive relief features to evaluate the surface roughness below the image resolution limit. We find that the probabilities that topographic irregularities are < 40 cm for the areas of 4 × 4 m and 20 × 20 m are > 33% and < 1% for boulder-rich areas and > 88% and > 13% for boulder-poor areas, respectively, even for the worst-case estimates. The estimated probabilities largely increase when we reduce the assumed number of positive relief features, which are more realistic cases. These indicate high probabilities of finding a smooth enough place to land on Phobos' surface safely.
Graphical Abstract
Phobos and Deimos, the two moons of Mars, are intriguing not only for their high scientific interests, but also as targets of potential human missions. Martian Moons eXploration (MMX) is the Japan Aerospace Exploration Agency (JAXA)'s mission to explore Phobos and Deimos, scheduled to be launched in 2024. The MMX spacecraft will perform in situ observations of both Phobos and Deimos, land and collect samples on Phobos, and bring them back to Earth. In addition, the MMX spacecraft will deploy a small rover developed by Centre National d'Etudes Spatiales (CNES) and the German Aerospace Center (DLR) for in situ investigations of the surface properties.
Designs of the mothership, the rover, and the sampling device depend largely on the target body's surface conditions. Thus, the Landing Operation Working Team (LOWT) and the Surface Science and Geology Sub Science Team (SSG-SST) of MMX are organized by scientists and engineers to evaluate Phobos' surface conditions. The team's views for both the surface environments and regolith properties are summarized in the accompanying paper (Miyamoto et al. 2021).
Previous Phobos' explorations provide useful information for the MMX mission, including photomosaics, maps, and numerical shape models (Basilevsky et al. 2014; Karachevtseva et al. 2014; Oberst et al. 2014; Salamunićcar et al. 2014, Wählisch et al. 2014; Willner et al. 2014). However, to evaluate the landing hazard, we need to evaluate the topographic irregularities at a scale of the order of tens of centimeters, which is difficult to do based only on the current numerical shape model or available images at a lower resolution than needed. Thus, we attempt pixel-scale image mapping and numerical evaluations based on the mapping results to evaluate the worst-case topographic irregularities.
In this work, we provide a brief overview of technical characteristics and restrictions of the MMX mission to Phobos' surface ("Topographic irregularities and the engineering safety" Section) and summarize typical geological features of Phobos (i.e., topographic obstacles) identified by previous studies ("Previous observations" Section). To estimate surface roughness at a high spatial resolution, we perform mappings of craters and boulders at a sub-pixel scale ("Analyses of high-resolution images" Section) and artificially develop numerical terrain models by extrapolating the estimated boulder size-frequency distributions ("Artificial terrain model" Section). Then we measure the topographic irregularities of these terrain models to assess landing safety ("Results and discussions" Section). We conclude that there are plenty of smooth enough areas favorable for landing on Phobos ("Conclusion" Section).
Topographic irregularities and the engineering safety
In this paper, the term "topography" refers to the surface forms and features themselves. We use the topographic height for the topographic difference (or approximately the elevation difference) for a topographic feature's normal distance from the reference plane defined in an area. Note that elevation is usually defined as the height above or below a reference point, mostly a reference geoid, which is difficult to define for Phobos under the current understanding of its internal densities.
Phobos is a small satellite of 26 × 22.8 × 18.2 km in size. Like a similarly sized asteroid Eros, Phobos' major landforms are craters and grooves with many boulders distributed over the body. Most distinctive is the Stickney crater, which is about 8 km in diameter. It is similar in size to the Shoemaker crater on Eros.
Based on spacecraft's observational data, including Mariner-9, Viking-Orbiter, Mars Global Surveyor, Mars Reconnaissance Orbiter, and Mars Express, numerical shape models have been built by several researchers leading to the knowledge of the overall shape of Phobos (e.g., Willner et al. 2014). We note that, even with a numerical shape model, determining the local gravity direction is not simple due to the Phobos' proximity to Mars. Nevertheless, we can calculate the gravity field with a constant density value for Phobos' entire body to evaluate the local gravitational acceleration in every facet (i.e., the face that constitutes the polygon shape model). We find that the slopes with respect to the local gravities are mostly lower than 40° (Wang and Wu 2020; Willner et al. 2014), with plenty of < 10° slope areas (Miyamoto et al. 2021), which are preferred for a lander and a rover. These values are useful for designing operational schemes, landing procedures, and selecting preliminary landing sites.
The spacecraft's size is about 2.1 m in width (excluding the solar paddles), and the vertical distance between the tip of the probe's legs and the mechanism carrying the various instruments is about 40 cm. Thus, acceptable topographic irregularity for its safely landing is only 40 cm at least under the current design. Considering the extension of the landing pads from the spacecraft, we needed to evaluate the surface roughness for areas of about 4 × 4 m as possible landing areas. The original operation plan for the landing of the MMX mothership is as follows: (1) descending the spacecraft's altitude from a Quasi-Satellite Orbit (Kuramoto 2021) to about 1 km in altitude to eliminate the horizontal velocity with respect to the surface; (2) descending the altitude by using onboard terrain matching system to the altitude where a target-marker is released; (3) further lowering the spacecraft's altitude to 20 m or so; and (4) performing an almost free-fall touchdown without thruster firing to prevent plume contamination. The accuracy of the horizontal drifting suppression is one of the designing parameters, which significantly controls the landing accuracy. Thus, topographic irregularities in varying areal scales, such as 10 × 10 m–40 × 40 m, are needed to be evaluated.
The averaged facet area of the available shape model with the highest resolution is about 104 m2, which exceeds the scale discussed above. Also, even with the highest resolution image of Phobos, which is about a couple of meters per pixel, evaluations of topographic irregularities at the required high resolutions are difficult because at least a few to ten pixels are needed to recognize a topographic feature. Thus, we need further efforts to assess the surface roughness at a higher resolution for landing hazard evaluation.
Previous observations
The surface of Phobos was globally imaged by high-resolution stereo camera (HRSC) of Mars Express, resulting in a global mosaic with a resolution of about 12 m/pixel (Wählisch et al. 2010, 2014). The global image (e.g., Fig. 1a) provides essential information on the global distribution of geological features. Detailed surface features are also studied in some limited areas. The super-resolution technique allows a resolution of the selected area of HRSC to be about 2 m per pixel. The highest resolutions achieved on images of Mars Orbiter camera (MOC) aboard Mars Global Surveyor are about 1–7 m per pixel for a few areas.
a Locations of the study regions shown on the Stooke map of Phobos (Stooke 2015) overlapped by a part of the georeferenced MOC image (b). b The spatial relationship of the study regions. Regions A and B (Regions B1–B4) are shown as a red rectangle and four blue rectangles, respectively. c Non-rectified MOC image SP2-55103. The yellow-dashed rectangle is the mapped area of this study shown in d. d Colorized map of the spatial distribution of local incidence angles of the study region. The base image is the corresponding part of the grayscale MOC image. e Histogram of solar-incidence angles of Regions A and B
Based on these images, geological studies find that dominant topographic landforms on Phobos' surface differ in scale and location and are typically classified into three major morphologies: craters, grooves, and boulders (Basilevsky et al. 2014). Craters are commonly distributed over the body, and their diameters range from at least several meters to ~ 10 km (Salamunićcar et al. 2014; Schmedemann et al. 2014; Hartmann and Neukum 2001).
A total of 1072 craters larger than 250 m in diameter were identified on Phobos (Karachevtseva et al. 2014), whose statistics are close to the Moon's highlands (Thomas and Veverka 1980). Some craters appear to be fresh, while some are highly degraded. Most of their depth-to-diameter ratios are between ~ 0.02 and ~ 0.2 (Hemmi and Miyamoto 2020; Basilevsky et al. 2014; Karachevtseva et al. 2014).
The smallest craters (about 3 m in diameter) were mapped on the highest resolution image (SP2-55103) obtained by the MOC. Considering that much smaller craters down to 0.1 μm (Hörz et al. 1975) exist on the surface of the Moon, which are not visible in the images of Phobos, cratering events down to micrometers should be present on the surface of Phobos (Basilevsky et al. 2014). Crater equilibrium was suggested for certain diameter ranges (e.g., Hartmann and Neukum 2001; Thomas and Veverka 1980). However, the actual spatial distribution and size-frequency distribution (SFD) of craters smaller than a few meters in diameter have not been quantitatively characterized due to the image resolution limit.
Grooves and pit chains exist as linear depressions, several of which are parallel to each other. Groove sizes vary from 23 to 475 m in width and from 2 to ~ 30 km in length (Murray and Heggie 2014). Nearly five hundred grooves are distributed globally, except for the trailing hemisphere of Phobos (e.g., Kikuchi and Miyamoto 2014; Thomas et al. 1979; Murray et al. 1994). Their nature is of high scientific interest, and their origins may hold important keys in the evolutional and structural history of Phobos. However, the landing operation team will most likely avoid landing the spacecraft into the bottom of a groove due to engineering safety reasons. Thus, we neglect the influence of grooves on the surface irregularities in this paper.
Numerous boulders are identified in high-resolution images of Phobos. They range from tens of meters down to a few meters in diameter. The nature of boulders smaller than a few meters in size is uncertain due to the paucity of higher-resolution images taken at the moderate incidence and phase angles (Thomas et al. 2000; Karachevtseva et al. 2014).
Previous geological studies of meter-scale features are limited (Basilevsky et al. 2014, 2015; Karachevtseva et al. 2014) due to the limited availability of high-resolution images. A high-resolution image of an area at least wider than about 1 km2 is needed for the statistical study of the distributions of meter-scale craters and boulders. Considering the spatial resolution, the spatial extent of illuminated areas, and the degree of noise and blurs, among the available datasets of Phobos, the Mars Global Surveyor MOC image SP2-55103 is the most suitable for the analyses of numerous boulders and craters as studied by Karachevtseva et al. (2014). Thus, first, we perform a similar but more detailed study of the same image to obtain statistical information.
Analyses of high-resolution images
We mapped both craters and positive relief features down to sub-pixel scales on MOC image SP2-55103, which is the highest-resolution image of the sub-Mars surface of Phobos of an area wider than several km2. The ground pixel scale of this image is about ~ 2.43 m on average, but we focus on regions with the highest resolution of about 1.1 m (due to the irregular shape of Phobos in this image frame, the image resolution largely varies even within the image). Boulders smaller than several meters are difficult to identify even in the highest resolution images. Thus, we picked up all positive relief as boulder candidates, which are composed of bright pixels (in the sunny side) immediately next to dark pixels (in the shadow side) along the line of the solar azimuth angle.
We selected two regions of study (Regions A and B) for precise mapping (Fig. 1b). Such regions are selected because Region A has relatively higher densities of positive relief features and craters, while Region B has relatively lower densities of those same features than surrounding regions. Note that Region B is composed of 4 subregions (Regions B1–B4) to avoid the shadowed area inside a large crater within somehow similar illumination conditions as Region A. Regions A and B (the sum of Regions B1 to B4) have comparable areas, ~ 1.14 km2 and ~ 1.20 km2, respectively (the individual areas of Regions B1–B4 are within ~ 0.26–0.31 km2). We measured diameters and center coordinates of craters and positive relief features in these regions by using the CraterTools v2.1 extension (Kneissl et al. 2011) for the ESRI ArcMap 10.1 or higher. To minimize the distortions of measurement areas due to map projection of irregular-shaped Phobos' surface (deviated from an 11.1 km-radius spherical reference), we applied the CraterTools' Topography Correction (Kneissl et al. 2014) with the use of the global digital terrain model of Willner et al. (2014) to our mapping results.
The mapped craters and positive relief features are categorized into either "confirmed" or "candidate", depending on their confidence levels. Confirmed features are those with evident morphological characteristics identified from multiple solar-incidence images [HiRISE image PSP_007769_9015 and the Stooke map (Fig. 1a)] by several researchers. However, candidate features are either nearly sub-pixel-scale or highly degraded features, recognized only from patterns of positive/negative relief (Fig. 2a–d; e.g., the arrangement of darker, brighter, then much darker pixels along a subsolar azimuth direction implies the presence of one or subpixel-scale positive relief feature at the brighter pixel).
a Examples of confirmed and candidate positive relief features in a close-up image; b mapping result of the image (a); c close-up view of the confirmed and candidate craters; d mapping result of c
We study the local incidence angles of the MOC image, derived from the bundle-adjusted MOC camera position/pointing information and the shape model of Phobos with 100 m/pixel resolution (Willner et al. 2014). The incidence angles range from ~ 70 to ~ 80 degrees within the studied regions (Fig. 1c, d). This greatly helps to identify relatively low topographic features. Thus, although candidate features may include imaging artifacts/noises, most would represent actual positive relief features. In this sense, we assume that the mapping result is worst-case. Also, the difference in the mean solar-incidence values between Regions A and B is only about 4 degrees (Fig. 1e), which is expected to provide a very similar or slightly better appearance of topographic features in Region A than Region B (e.g., a 2-m high feature will give ~ 6.2 and ~ 8.0-m-long shadows at solar incidence angles of 72 and 76 degrees, respectively, which result in at most one-pixel-wide difference between two regions in the MOC image).
We find that the spatial number densities of craters and boulders can vary significantly even in the same image frame (Fig. 3); Region A has 202 km−2 confirmed positive relief features (and 2277 km−2 including candidates) and 278 km−2 confirmed craters (1226 km−2 including candidates), while Region B has 46 km−2 positive relief features (224 km−2 including candidates) and 234 km−2 craters (460 km−2 including candidates). The mapping results of both boulders and craters are statistically summarized as cumulative plots in Figs. 4, 5. We define their size-frequency distributions (SFDs) following the conventional representation, the total number per unit area (N) of target features (boulders or craters) larger than diameter D, i.e., N = ND0 (D⁄D0)−α, where D0 is a reference diameter and ND0 is the N value of boulders/craters with a diameter larger than D0 (10 m for boulder SFDs and 1 km for crater SFDs). For craters, a power-law exponent α = 2 is generally considered to represent an equilibrium population (Melosh 1989).
Mapping results of the positive relief features and craters of Region A (a) and Region B (Regions B1–B4) (b) on the georeferenced MOC image SP2-55103
Cumulative size-frequency distributions (SFDs) of positive relief features (boulders) in Regions A (left) and B (right) plotted using CraterStats2 software (Michael and Neukum 2010). Results of previous studies are also shown for comparison (Karachevtseva et al. 2014; Murdoch et al. 2015; Rodgers et al. 2016)
Cumulative size-frequency distributions (SFDs) of craters in Regions A (left) and B (right) plotted using CraterStats2 software (Michael and Neukum 2010). Results of previous studies are also shown for comparison (Hartmann and Neukum 2001; Salamunićcar et al. 2014)
The SFDs of confirmed positive relief features are shown as red dots in Fig. 4a, b in a log–log plot (the slope of the distribution lines in such a plot corresponds to the power-law exponent in the actual distribution). Note that, in the range of ~ 5–8 m in diameter, the cumulative slope (fitted to about − 6) is similar to that of Karachevtseva et al. (2014). The SFDs of both confirmed and candidate positive relief features (yellow dots) appear to follow the slope of ~ − 3.2 in the range of 1–2 m in diameter.
The SFDs of confirmed craters and the sum of both confirmed and candidate craters are shown in Fig. 5. In these log–log plots, the SFDs roughly follow the lines with a slope of − 2 at D = ~ 10–30 m, which are consistent with the plots of the distributions of larger diameter craters measured by previous studies (Salamunićcar et al. 2014; Hartmann and Neukum 2001). The equilibrium crater density (a few to 10 percent of geometric saturation) (Gault 1970) seems to be achieved.
Artificial terrain model
We have identified the locations and diameters of confirmed craters, crater candidates, confirmed positive relief features, and positive relief feature candidates through the analysis discussed above. Assuming that the cm to meter-scale topography on Phobos mostly results from craters and boulders distributions, we might evaluate the topographic roughness by creating an artificial terrain model (DTM) based on such information.
To artificially develop a DTM, we define two-dimensional grid arrays of 10,000 × 10,000 elements for simulating a region of 1 × 1 km, which means the resolution of this model is 10 cm/element. Each element has its topographic height value, whose initial value is 0.0 m (i.e., a reference level). We assume that each crater is composed of a circular uplifted rim around a bowl-shaped depression and their morphology corresponds to the one given by a standard crater gravity scaling law. We put such craters in the terrain model by subtracting the height values corresponding to the area of the bowl-shaped depression and adding for the circular uplifted rim. We place the craters in descending order; larger craters are placed earlier.
By using Geospatial Data Abstraction Library (GDAL) version 3.1.2 and the ArcGIS's Spatial Analyst tool, the resulting artificial DTMs were converted to shaded relief images with shadows, which are artificially illuminated at an azimuth of 270° (from the left of the figure to the right) and an altitude of 15° (i.e., the roughly similar illumination condition of the target region (Fig. 6a) in the MOC image SP2-55103). Figure 6b shows the shaded relief image of the resulting artificial DTM with confirmed craters, whose d/Ds are randomly given between 0.01–0.2 according to the simple uniform distribution. Note that, even though we used both crater locations and diameters from mapping results, the resulting DTM is not similar to the original MOC image. This result indicates the importance of evaluating the d/D (depth to diameter) of craters.
a Region A in MOC image SP2-55103; b artificial DTM with 342 confirmed craters, whose d/Ds are randomly given between 0.01 and 0.2 according to the simple uniform distribution; c artificial DTM only with 342 confirmed craters, whose d/D are manually modified; d 1510 candidate craters of randomly given d/D are superposed on c; e 1510 candidate craters of randomly given d/D between 0.01 and 0.02 are superposed on c; f positive relief features are distributed on e
C rater depth-to-diameter ratio
The topographic characteristics of craters below 100 m in diameter are difficult to evaluate from the numerical shape (e.g., Willner et al. 2014). Basilevsky et al. (2014) partially overcame this issue by carefully evaluating images to obtain craters' 2D profiles. They classified craters into three morphologic classes: Class 1 for those with > 0.1 in d/D (depth to diameter) with steepest inner slopes (> 20°), Class 2 for those with 0.05–0.1 in d/D with shallower inner slopes (10–20°), and Class 3 for those with < 0.05 in d/D with shallow inner slopes (< 10°). Such variations may be the results of degradations through many possible processes.
Following this idea, we assume that each crater initially has a traditional simple bowl shape. The simple crater's rim-to-floor depth is about 1/5 of its diameter, and its sharp-crest rim stands about 4% of the crater diameter above the surrounding crater (Richardson 2009). The change in elevation for the crater interior and exterior is assumed to follow the following equations (O'Brien and Byrne 2020):
$$\Delta h = \left\{ {\begin{array}{*{20}c} \begin{gathered} \left( \frac{r}{R} \right)^{2} \left( {h_{r} + d} \right) - d \;\left( {r \le R} \right) \hfill \\ h_{r} \left( \frac{r}{R} \right)^{ - 3} - \frac{{h_{r} }}{{\eta^{3} \left( {\eta - 1} \right)}}\left( {\frac{r}{R} - 1} \right) \;\left( {R < r \le \eta R} \right) \hfill \\ \end{gathered} \\ {} \\ \end{array} } \right.$$
where r is the radial range from the crater's center, R is the rim crest radius of the crater, \({h}_{r}\) is the rim height, \(d\) is the depth of the crater, and \(\eta\) is the radius of the continuous ejecta blanket (in our model, \(\eta =4\) as the continuous ejecta blanket extends to 4 crater radii). A degradation process is simulated with the following two-dimensional diffusion equation (e.g., Richardson et al. 2020):
$$\frac{\partial z}{{\partial t}} = \kappa \nabla^{2} z$$
where \(\kappa\) is diffusivity. The degradation process is simulated by fixing the coefficient and setting the appropriate diffusion time t. As a result, we obtain 20 profiles of the crater's morphology with its d/D of 0.01–0.2. The spline interpolation method is used to keep the continuous shape of the resulting crater morphology (Fig. 7).
a Modeled profiles of craters of different degradation stages; b shaded relief images of modeled craters of different degradation stages
We modified the d/D of the confirmed craters to make the appearance of DTM similar to the MOC image. Note that this procedure is similar to the shape-from-shading method in some sense, but is also different because we empirically assume the shapes of craters and neglect any possible brightness difference due to surface textures. Thus, we can focus on each shadow length of each crater to adjust d/D. For a deep crater with its shadow cast from its wall to its interior, we selected the profile (d/D) of a crater that shows the same shadow length as the one observed in the MOC image. For a shallow crater without any interior shadow, we used the profile of the deepest shadowless crater at a given solar incidence angle at its location because the worst-case crater should be considered for the assessment of future landing operation. Figure 6c shows the result of DTM with modified d/D for all confirmed craters. As for crater candidates, d/D should generally be very small because otherwise we can identify them as a confirmed crater. Thus, we chose 0.01–0.02 for d/D of the candidate craters and placed the craters in Fig. 6c. The shaded relief image of the resultant DTM is shown in Fig. 6e.
Note that we excluded craters smaller than 82.7 cm in diameter in our artificial models. This comes from the aim of this study, i.e., the assessment of landing safety, most of which is based on Rodgers et al. (2016). For the landing performance of the lander footpads, one small boulder is more hazardous than one small crater. A footpad placed on a single acuate (convex-upward) boulder with a certain height is very unstable; however, a footpad over/inside a small (1–2 × footpad-wide) crater is in a (quasi-)stable state because of the convex-downward geometry of a crater profile. Besides, little is known about the spatial/size-frequency distributions of sub-meter craters on small bodies, compared to those of sub-meter boulders on small bodies (e.g., asteroid Eros, Itokawa, Ryugu, etc.). For the reasons above mentioned, we focused on the distributions of positive relief features only for the sub-meter-scale elements of our artificial DTMs.
S hapes of positive relief features
We put positive relief features on the DTM as bowl-shaped uplifts. Similar to craters, shapes of positive relief features are difficult to define. For simplicity, we use an ellipsoid to represent a positive relief feature. Thomas et al. (2000) measured the boulders' width/height ratio by measuring the heights from lengths of shadows, which showed that the height/width ratios follow a normal distribution with its mean 0.25 and standard deviation 0.17. Thus, we distributed confirmed positive relief features with the randomly selected height/width values according to the Gaussian with a mean of 0.25 and a standard deviation of 0.17. The height/width values of all candidate positive relief features were assumed to be 0.1.
E xtrapolation of positive relief feature's SFD
The resultant DTM (Fig. 6f) shows a consistent appearance to Region A's original MOC image. Because the illumination condition of Fig. 6e is carefully arranged to be the same as that of the original image, the overall similarities may indicate that we can evaluate surface irregularities if we can appropriately estimate the distributions of smaller blocks below the resolution image. However, extrapolation of an SFD is very challenging because the observed SFDs on small bodies appear to saturate at different sizes, which may reflect some nature of granular materials. However, these could be biased by resolution limits. Thus, to make the discussion simple, we simply extrapolated our SFDs from the analysis discussed above, knowing that it overestimates the numbers of smaller particles. We consider this is still justified because (1) our primary purpose is for engineering safety evaluations, and (2) the number of particles is always overestimated, which always gives the worst case.
We prepared two types of model terrains, Models 1 and 2, which correspond with two target regions, Regions A and B, respectively. Small positive relief features, down to 35 cm, were placed spatially randomly. The extrapolations of their SFDs of Models 1 and 2 were made using the measured boulder SFD of Region A at a diameter of 6.0 m and the one of Region B at a diameter of 4.5 m, respectively (Fig. 8).
a Cumulative SFD of both confirmed and candidate positive relief features of Region A (solid line) and Model 1's extrapolation (dotted line). b Cumulative SFD of both confirmed and candidate positive relief features of Region A (solid line) and Model 2's extrapolation (dotted line). c Shaded relief image of the DTM of Region A with the Model 1's extrapolation of positive relief features; d shaded relief image of the DTM of Region A with the Model 2's extrapolation of positive relief features
We artificially develop DTMs of Region A in Fig. 1 by putting confirmed craters, crater candidates, confirmed positive relief features, positive relief features candidates, and small positive relief features. The shaded relief images of the DTMs developed in this study are shown in Fig. 8c, d.
We study the frequency of the maximum topographic differences within a given small region as shown in Fig. 9a, d for Models 1 and 2, respectively. Naturally, the larger the size of the given region, the higher the peak of the maximum topographic difference. The cumulative probabilities of the maximum topographic difference are also calculated (Fig. 9c, d).
a Frequency of the maximum topographic difference of Region A for the case of Model 1's extrapolation of positive relief features; b frequency of maximum topographic difference of Region A for the case of Model 2's extrapolation; c cumulative probability of the maximum topographic difference of Region A for the case of Model 1; d cumulative probability of the maximum topographic difference for the case of Model 2 of Region A. Textured curves indicate results of areas of 4 × 4 m, 10 × 10 m, 20 × 20 m, and 40 × 40 m
We find that the probabilities that topographic irregularities are < 40 cm for the areas of 4 × 4 m and 20 × 20 m are > 33% and < 1% for Model 1, and > 77% and > 2% for Model 2. This indicates that, even in the worst case, we can still find a smooth area within 40 cm in topographic differences with a probability of at least 30%. Thus, even if we do not control the spacecraft at the time of landing, the probability of successful landing in a smooth region is more than 30%.
As we discussed above, we adjusted the d/D of craters to simulate the original MOC image appropriately. The resultant d/D distribution is shown in Fig. 10. We find that this is a useful set to develop DTMs artificially. We use the statistical distribution of d/D in this figure, we randomly plot confirmed craters and crater candidates on an initially flat DTM to simulate Region B, which is a boulder-poor region. The number of positive relief features is also extrapolated down to 35 cm in diameter as for Region A as shown in Fig. 11a, b. The shaded relief images of the resulted DTMs are shown in Fig. 11c, d. Their statistics are shown in Fig. 12. We find that, in this case, the probabilities that topographic irregularities are < 40 cm for the areas of 4 × 4 m and 20 × 20 m are > 79% and > 2%, respectively, for Model 1 and > 88% and > 13%, respectively, for Model 2.
Diameter vs d/D of craters we used to obtain Fig. 6e
a Cumulative SFD of both confirmed and candidate positive relief features of Region B (solid line) and Model 1's extrapolation (dotted line). b Cumulative SFD of both confirmed and candidate positive relief features of Region B (solid line) and Model 2's extrapolation (dotted line). c Shaded relief image of the DTM of Region B-equivalent area with the Model 1's extrapolation of positive relief features; d shaded relief image of the DTM of Region B-equivalent area with the Model 2's extrapolation of positive relief features
a Frequency of maximum topographic difference of Region B for the case of Model 1's extrapolation of positive relief features; b frequency of maximum topographic difference of Region B for the case of Model 2's extrapolations; c cumulative probability of maximum topographic difference of Region A for the case of Model 1; d cumulative probability of maximum topographic difference of Region B for the case of Model 2. Textured curves are indicating values for areas of 4 × 4 m, 10 × 10 m, 20 × 20 m, and 40 × 40 m, based on the statistics of craters and positive relief features of Region B
We note that these two areas could be considered as boulder-rich regions on the Phobos' surface. We surveyed other areas far from Regions A and B using relatively high-resolution images taken by the HRSC on the Mars Express. We find that, in some regions such as those in Fig. 13, however, only a few boulders were identified. This indicates that the areas other than the eastern side of Stickney crater may be more favorable in terms of topographic irregularities for landing.
Examples of boulder-poor regions: a Mars Express HRSC camera image H9574_0004_SR2 (pixel scale: ~ 3.3 m, center latitude: 55.403 °S, center longitude: 228.213 °E) in south polar stereographic projection. The illuminated area (white polygon) is about 13 km2. Only one boulder was identified (red circle); b Mars Express HRSC camera image H7926_0011_SR2 (pixel scale: ~ 2.6 m, center latitude: 3.9696 °S, center longitude: 166.034 °E). The illuminated area (white polygon) is about 12 km2
Evaluation of topographic irregularities below image resolution is very challenging, although it is required for the appropriate design of the landing of the MMX spacecraft. Thus, we study the topographic irregularities of Phobos' surface based on the area where the highest resolution image is available. By combining image analysis and numerical development of DTMs at high resolution, we find that Phobos' surface likely has plenty of smooth areas that are favorable to the landing of the MMX spacecraft. However, the maximum topographic difference increases with the area width, which means that the landing safety requires a landing accuracy that is optimized for the estimated width of smooth areas.
Mapping results are available to share upon request.
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P.M. acknowledges funding support from CNES.
P.M. acknowledges CNES for funding support.
Department of Systems Innovation, University of Tokyo, Tokyo, 113-8656, Japan
Tomohiro Takemura, Hideaki Miyamoto & Takafumi Niihara
University Museum, University of Tokyo, Tokyo, 113-0033, Japan
Ryodo Hemmi
Université Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, Laboratoire Lagrange, 06304, Nice Cedex 4, France
Patrick Michel
Tomohiro Takemura
Hideaki Miyamoto
Takafumi Niihara
TT conducts numerical works. HM led the entire works and drafting. RH performed image analyses. RH, TN, PM contributed to planetary image processing and discussion. All authors read and approved the final manuscript.
Correspondence to Hideaki Miyamoto.
Takemura, T., Miyamoto, H., Hemmi, R. et al. Small-scale topographic irregularities on Phobos: image and numerical analyses for MMX mission. Earth Planets Space 73, 213 (2021). https://doi.org/10.1186/s40623-021-01463-8
7. Planetary science
Martian Moons eXploration: The scientific investigations of Mars and its moons | CommonCrawl |
Global solutions to the incompressible magnetohydrodynamic equations
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The moving boundary problem in a chemotaxis model
March 2012, 11(2): 747-761. doi: 10.3934/cpaa.2012.11.747
On the regularity of solutions to the Navier-Stokes equations
Vittorino Pata 1,
Politecnico di Milano - Dipartimento di Matematica "F. Brioschi", Via Bonardi 9, 20133 Milano
Received November 2010 Revised April 2011 Published October 2011
This article is concerned with the incompressible Navier-Stokes equations in a three-dimensional domain. A criterion of Prodi-Serrin type up to the boundary for global existence of strong solutions is established.
Keywords: weak solutions, Navier-Stokes equations, regularity criteria., strong solutions, blow-up.
Mathematics Subject Classification: Primary: 35Q30; Secondary: 76D03, 76D0.
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Vittorino Pata | CommonCrawl |
\begin{document}
\title{Rigidity theorems for compact Bach-flat manifolds with positive constant scalar curvature}
\author{Haiping Fu}
\address{Department of Mathematics, Nanchang University, Nanchang 330047, P. R. China }
\email{[email protected](H. P. Fu)}
\thanks{Supported by the National Natural Science Foundation of China (11261038, 11361041).}
\author{Jianke Peng} \address{Department of Mathematics, Nanchang University, Nanchang 330047, P. R. China} \email{[email protected](J. K. Peng)} \thanks{}
\subjclass[2010]{53C20; 53C24.}
\date{May 9, 2016 and, accepted, April 19, 2017.}
\dedicatory{}
\keywords{Bach-flat; constant curvature space; Weyl curvature tensor; trace-free Riemannian curvature tensor.}
\begin{abstract} In this paper, we prove some rigidity theorems for compact Bach-flat $n$-manifold with the positive constant scalar curvature. In particular, our conditions in Theorem 1.4 have the additional properties of being sharp. \end{abstract}
\maketitle
\section{Introduction} \par Let $(M^n,g) (n\geq3)$ be an $n$-dimensional Riemannian manifold with the Riemannian curvature tensor $Rm=\{R_{ijkl}\}$, the Weyl curvature tensor $W=\{W_{ijkl}\}$, the Ricci tensor $Ric=\{R_{ij}\}$ and the scalar curvature $R$. For any manifold of dimension $n\geq4$, the Bach tensor, introduced by Bach \cite{B}, is defined as \begin{eqnarray} B_{ij}\equiv\frac{1}{n-3}\nabla^k\nabla^lW_{ikjl}+\frac{1}{n-2}R^{kl}W_{ikjl}. \end{eqnarray}
Here and hereafter the Einstein convention of summing over the repeated indices will be adopted. In \cite{KL}, Korzynski and Lewandowski proved that the Bach tensor can be identified with the Yang-mills current of the Cartan normal conformal connection. Recall that a metric $g$ is called Bach-flat if the Bach tensor vanishes. It is easy to see that $B_{ij}=0$ if $(M^n,g)$ is either locally conformally flat, or an Einstein manifold. In the case of $n=4$, $g$ is Bach-flat if and only if it is a critical metric of the functional (see \cite{{Be},{GN}}) $$W:g\mapsto\int_{M}{|W_g|}^2dV_g.$$
Now we introduce the definition of the Yamabe constant. Given a complete Riemannian $n$-manifold $(M^n,g)$ of dimension $n\geq3$, the Yamabe constant $Y(M,[g])$ ($[g]$ is the conformal class of $g$) is defined as
$$Y(M,[g])\equiv\inf\limits_{\tilde{g}\in[g]}\frac{\int_M{R_{\tilde{g}}}dV_{\tilde{g}}}{\left(\int_MdV_{\tilde{g}}\right)^{\frac{n-2}{n}}}=\inf\limits_{0\neq u\in C^\infty_0(M^n)}\frac{\int_{M}\left(\frac{4(n-1)}{n-2}|\nabla u|^2+Ru^2\right)dV_g}{\left(\int_{M}|u|^\frac{2n}{n-2}dV_g\right)^\frac{n-2}{n}}.$$ The important works of Aubin \cite{A}, Schoen \cite{S}, Trudinger \cite{T} and Yamabe \cite{Y} showed that for compact manifolds the infimum in the above is always achieved. There are noncompact complete Riemannian manifolds of negative scalar curvature with positive Yamabe constant. For example, any simply connected complete locally conformally flat manifold has positive Yamabe constant (see \cite{SY} ). Furthermore, for compact manifolds, $Y(M,[g])$ is determined by the sign of the scalar curvature $R$ (see \cite{A} ), and for noncompact manifolds, $Y(M,[g])$ is always positive if $R$ vanishes (see \cite{D}).
The curvature pinching phenomenon plays an important role in global differential geometry. Some isolation theorems of Weyl curvature tensor of positive Einstein manifolds are given in \cite{{HV},{IS},{Si}}, when its $L^{\frac{n}{2}}$-norm is small. Recently, two rigidity theorems of Weyl curvature tensor of positive Einstein manifolds are given in \cite{{C},{FX2},{FX3}}, which improve results due to \cite{{HV},{IS},{Si}}. The first author and Xiao have studied compact manifolds with harmonic curvature to obtain some rigidity results in \cite{{F},{FX}}. Here when a Riemannian manifold satisfies $\delta Rm= \{\nabla^lR_{ijkl}\}=0$, we call it a manifold with harmonic curvature. Bach-flat manifolds have been studied by many authors. For any complete Bach-flat manifold, Kim \cite{K} has studied their rigidity phenomena and derived that a complete Bach-flat 4-manifold $M^4$ with nonnegative constant scalar curvature and the positive Yamabe constant is an Einstein manifold if the $L^2$-norm of the trace-free Riemannian curvature tensor $\mathring{Rm}$ is small enough. Later, Chu \cite{Ch} improved Kim's result and showed that $M^4$ is in fact a space of constant curvature under the same assumptions. Chu and Feng \cite{CF} proved the rigidity result for $n$-dimensional Bach-flat manifolds with constant scalar curvature and positive Yamabe constant. For a compact Bach-flat manifold $M^4$ with the positive Yamabe constant, Chang et al. \cite{CQY} proved that $M^4$ is conformal equivalent to the standard four-sphere provided that the $L^2$-norm of the Weyl curvature tensor $W$ is small enough, and also showed that there is only finite diffeomorphism class with a bounded $L^2$-norm of $W$.
Now, we are interested in $L^p$ pinching problems for compact Bach-flat manifolds with positive constant scalar curvature. In this paper, under some $L^p$ pinching conditions, we show that the compact Bach-flat manifold with positive constant scalar curvature is spherical space form or Einstein manifold. More precisely, we have the following theorems: \begin{theorem} Let $(M^n,g) (n\geq4)$ be an $n$-dimensional compact Bach-flat Riemannian manifold with positive constant scalar curvature. For $p\geq\frac{n}{2}$, if
$$\left(\int_{M}|\mathring{Rm}|^pdV_g\right)^{\frac{1}{p}}<\varepsilon(n)Y(M,[g])^\frac{n}{2p}R^{1-\frac{n}{2p}},$$ where $\varepsilon(n)$ is a constant depending only on $n$, i.e., \[ \varepsilon(n)=\left\{ \begin{array}{cc} \frac{n-2}{4(n-1)\left(C(n)+(n-2)\sqrt{\frac{n-2}{2(n-1)}}\right)}, &\mbox{if $n=4,5$ and $p=\frac{n}{2}$,}\\ \left[\frac{(n-2)(2p-n)}{n(6-n)}\right]^{\frac{n}{2p}}\frac{(6-n)p}{2(n-1)(2p-n)\left(C(n)+(n-2)\sqrt{\frac{n-2}{2(n-1)}}\right)}, &\mbox{if $n=4,5$ and $\frac{n}{2}<p<\frac{2n}{n-2}$,}\\ \frac{1}{(n-1)\left(C(n)+(n-2)\sqrt{\frac{n-2}{2(n-1)}}\right)}, &\mbox{if $n\geq6$ and $p\geq\frac{n}{2}$ or if $n=4,5$ and $p\geq\frac{2n}{n-2}$,} \end{array}\right. \] and $C(n)$ is defined in Lemma 2.1, then $(M^n,g)$ is isometric to a quotient of the round $\mathbb{S}^n$. \end{theorem} \begin{corollary} Let $(M^n,g) (n\geq4)$ be an $n$-dimensional compact Bach-flat Riemannian manifold with positive constant scalar curvature. If
$$\left(\int_{M}|\mathring{Rm}|^{\frac n2}dV_g\right)^{\frac{2}{n}}<\varepsilon(n)Y(M,[g]),$$ where \[ \varepsilon(n)=\left\{ \begin{array}{cc} \frac{n-2}{4(n-1)\left(C(n)+(n-2)\sqrt{\frac{n-2}{2(n-1)}}\right)}, &\mbox{if $n=4,5$,}\\ \frac{1}{(n-1)\left(C(n)+(n-2)\sqrt{\frac{n-2}{2(n-1)}}\right)}, &\mbox{if $n\geq6$,} \end{array}\right. \] then $(M^n,g)$ is isometric to a quotient of the round $\mathbb{S}^n$. \end{corollary} \begin{remark} The above $L^p$-pinching condition in Theorem $1.1$ is invariant under any homothety. $L^{\frac n2}$ trace-free Riemannian curvature pinching theorems have been shown by Kim \cite{K}, Chu \cite{C}, and Chu and Feng \cite{CF}, in which the pinching constant are not explicit, respectively. Theorem $1.1$ extends the $L^{p}$ trace-free Riemannian curvature pinching theorems given by \cite{{C},{CF},{K}} in power $p=\frac n2$ to $p\geq\frac n2$. \end{remark} \begin{theorem} Let $(M^n,g) (n\geq4)$ be an $n$-dimensional compact Bach-flat Riemannian manifold with positive constant scalar curvature. If \begin{eqnarray}
\left(\int_{M}\left|W+\frac{\sqrt{n}}{2\sqrt{2}(n-2)}\mathring{Ric}\circledwedge g\right|^{\frac{n}{2}}dV_g\right)^{\frac{2}{n}}<C_1(n)Y(M,[g]), \end{eqnarray} where \[ C_1(n)=\left\{ \begin{array}{cc} \sqrt{\frac{n-2}{32(n-1)}}, &\mbox{if $n=4,5$,}\\ \frac{1}{\sqrt{2(n-2)(n-1)}}, &\mbox{if $n\geq6$,} \end{array}\right. \] then $(M^n,g)$ is an Einstein manifold. In particular, for $n=4,5$, then $(M^n,g)$ is isometric to a quotient of the round $\mathbb{S}^n$; for $n\geq6$, if the pinching constant in (1.2) is weakened to $\frac{2Y(M,[g])}{nC_2(n)}$, where $C_2(n)$ is defined in Lemma 2.1 of \cite{FX3}, then $(M^n,g)$ is isometric to a quotient of the round $\mathbb{S}^n$. \end{theorem}
\begin{remark}
When $n\geq6$, the inequality (1.2) of this theorem is optimal. The critical case is given by the following example. If $(\mathbb{S}^1(t)\times\mathbb{S}^{n-1},g_t)$ is the product of the circle of radius $t$ with $\mathbb{S}^{n-1}$, and if $g_t$ is the standard product metric normalized such that $Vol(g_t)=1$, we have $W=0$, $g_t$ is a Yamabe metric for small $t$ (see \cite{S2}), and $\left(\int_{M}|\mathring{Ric}|^{\frac{n}{2}}dV_g\right)^{\frac{2}{n}}=\frac{Y(M,[g])}{\sqrt{n(n-1)}}$, which is the critical case of the inequality (1.2) in Theorem $1.4$. We know that $(\mathbb{S}^1(t)\times\mathbb{S}^{n-1},g_t)$ is not Einstein. \end{remark} \begin{corollary} Let $(M^4,g)$ be a $4$-dimensional compact Bach-flat Riemannian manifold with positive constant scalar curvature. If
\begin{eqnarray}
\int_{M}|W|^2dV_g+\frac{5}{4}\int_{M}|\mathring{Ric}|^2dV_g\leq\frac{1}{48}\int_{M}R^2dV_g,
\end{eqnarray} then $(M^4,g)$ is isometric to a quotient of the round $\mathbb{S}^4$. \end{corollary} \begin{remark} By the Chern-Gauss-Bonnet formula, the pinching condition (1.3) in Corollary $1.6$ is equivalent to the following \begin{eqnarray}
\int_{M}|W|^2dV_g+\frac{2}{39}\int_{M}R^2dV_g\leq\frac{160}{13}\pi^2\chi(M), \end{eqnarray} where $\chi(M)$ is the Euler-Poincar\'{e} characteristic of $M$. \end{remark} \begin{theorem} Let $(M^n,g)$ be an $n$-dimensional compact Bach-flat Riemannian manifold with positive constant scalar curvature. If \begin{eqnarray}
|W|^2+\frac{{n}}{2(n-2)}|\mathring{Ric}|^2\leq\frac{1}{{2(n-2)(n-1)}}R^2, \end{eqnarray} then $(M^n,g)$ is isometric to either an Einstein manifold or a quotient of $\mathbb{S}^1\times \mathbb{S}^{n-1}$ with the product metric. \end{theorem} \begin{corollary} Let $(M^n,g)$ be an $n=4$ or $5$-dimensional compact Bach-flat Riemannian manifold with positive constant scalar curvature. If
$$
|W|^2+\frac{{n}}{2(n-2)}|\mathring{Ric}|^2\leq\frac{1}{{2(n-2)(n-1)}}R^2, $$ then $(M^n,g)$ is isometric to either a quotient of the round $\mathbb{S}^n$ or a quotient of $\mathbb{S}^1\times \mathbb{S}^{n-1}$ with the product metric. \end{corollary}
\section{Proof of Theorem 1.1} Let $(M^n,g)(n\geq3)$ be an $n$-dimensional complete Riemannian manifold with the metric $g=\{g_{ij}\}$. Denote by $Ric=\{R_{ij}\}$ and $R$ the Ricci tensor and the scalar curvature, respectively. It is well known that the Riemannian curvature tensor $Rm=\{R_{ijkl}\}$ of $M^n$ can be decomposed into three orthogonal components which have the same symmetries as $Rm$ $$R_{ijkl}=W_{ijkl}+V_{ijkl}+U_{ijkl},$$ $$V_{ijkl}=\frac{1}{n-2}(\mathring{R}_{ik}g_{jl}-\mathring{R}_{il}g_{jk}+\mathring{R}_{jl}g_{ik}-\mathring{R}_{jk}g_{il}),$$ $$U_{ijkl}=\frac{R}{n(n-1)}(g_{ik}g_{jl}-g_{il}g_{jk}),$$ where $W=\{W_{ijkl}\}$, $V=\{V_{ijkl}\}$ and $U=\{U_{ijkl}\}$ denote the Weyl curvature tensor, the traceless Ricci part and the scalar curvature part, respectively, and $\mathring{Ric}=\{\mathring{R}_{ij}\}=\{R_{ij}-\frac{R}{n}g_{ij}\}$ is the trace-free Ricci tensor. Denote by $\mathring{Rm}=\{\mathring{R}_{ijkl}\}=\{R_{ijkl}-U_{ijkl}\}$ the trace-free Riemannian curvature tensor.
In local coordinates, the norm of a $(0,4)$-type tensor $T$ is defined as $$|T|^2=|T_{ijkl}|^2=g^{im}g^{jn}g^{ks}g^{lt}T_{ijkl}T_{mnst}.$$
The following equalities are easily obtained from the properties of Riemannian curvature tensor: \begin{equation} g^{ik}\mathring{R}_{ijkl}=\mathring{R}_{jl,} \end{equation} \begin{equation} \mathring{R}_{ijkl}+\mathring{R}_{iljk}+\mathring{R}_{iklj}=0, \end{equation} \begin{equation} \mathring{R}_{ijkl}=\mathring{R}_{klij}=-\mathring{R}_{jikl}=-\mathring{R}_{ijlk}, \end{equation} \begin{equation}
|\mathring{Rm}|^2=|W|^2+|V|^2=|W|^2+\frac{4}{n-2}|\mathring{Ric}|^2. \end{equation} Moreover, under the assumption of constant scalar curvature, we get \begin{equation} \nabla_h\mathring{R}_{ijkl}+\nabla_l\mathring{R}_{ijhk}+\nabla_k\mathring{R}_{ijlh}=0, \end{equation} and \begin{eqnarray} \nabla^lW_{ijkl}&=&\nabla^lR_{ijkl}-\nabla^lV_{ijkl}-\nabla^lU_{ijkl}\nonumber\\ &=&\nabla^l\mathring{R}_{ijkl}-\nabla^lV_{ijkl}\nonumber\\ &=&\frac{n-3}{n-2}\left(\nabla_j\mathring{R}_{ik}-\nabla_i\mathring{R}_{jk}\right)\nonumber \\ &=&\frac{n-3}{n-2}\nabla^l\mathring{R}_{ijkl}. \end{eqnarray} Since $n\geq3$, from (2.4) we see that \begin{eqnarray}
|\mathring{Ric}|^2\leq\frac{n-2}{4}|\mathring{Rm}|^2. \end{eqnarray}
Let $\Lambda^2(M)$ and $\otimes^2(M)$ denote the space of skew symmetric $2$-tensors and $2$-tensors, respectively. It is easy to know that the dimension of $\Lambda^2(M)$ and $\otimes^2(M)$ is $\frac{n(n-1)}{2}$ and $n^{2}$, respectively. Let $T=\{T_{ijkl}\}$ be a tensor with the same symmetries as the Riemannian curvature tensor. It defines a symmetric operator $T:\Lambda^2(M)\rightarrow\Lambda^2(M)$ by $$(T\omega)_{kl}:=T_{ijkl}\omega_{ij},$$ with $\omega\in\Lambda^2(M)$. Similarly, it also defines a symmetric operator $T:\otimes^2(M)\rightarrow\otimes^2(M)$ by $$(T\theta)_{kl}:=T_{kilj}\theta_{ij},$$ with $\theta\in\otimes^2(M)$.
In order to prove Theorem $1.1$, we need the following lemma: \begin{lemma} Let $(M^n,g) (n\geq3)$ be an n-dimensional Riemannian manifold with constant scalar curvature, then \begin{eqnarray} \mathring{R}^{ijkl}\Delta\mathring{R}_{ijkl}\geq
-C(n)|\mathring{Rm}|^3+2\mathring{R}^{ijkl}\nabla_l\nabla^m\mathring{R}_{ijkm}+A(n)R|\mathring{Rm}|^2, \end{eqnarray} where \[ A(n)=\left\{ \begin{array}{cc} \frac{1}{n-1}, &\mbox{if $R\geq0$,}\\ \frac{2}{n}, &\mbox{if $R<0$,} \end{array}\right. \] and $C(n)=\frac{4(n^2-2)}{n\sqrt{n^2-1}}+\frac{n^2-n-4}{\sqrt{(n-2)n(n^2-1)}}+\sqrt{\frac{(n-2)(n-1)}{n}}$. \end{lemma} \begin{remark}
Although Lemma $2.1$ and the explicit coefficient of the term $|\mathring{Rm}|^3$ of (2.8) have been proved in \cite{Ch} and \cite{FX} respectively, for completeness, we also write it out. \end{remark} \begin{proof} To simplify the notations, we will compute at an arbitrarily chosen point $p\in M$ in normal coordinates centered at $p$ so that $g_{ij}=\delta_{ij}$. We obtain from (2.3) and (2.5) that \begin{eqnarray} \mathring{R}_{ijkl}\Delta\mathring{R}_{ijkl}&=&\mathring{R}_{ijkl}\nabla_m\nabla_m\mathring{R}_{ijkl}\nonumber \\ &=&2\mathring{R}_{ijkl}\nabla_m\nabla_l\mathring{R}_{ijkm}\nonumber \\ &=&2\mathring{R}_{ijkl}(\nabla_l\nabla_m\mathring{R}_{ijkm}+R_{hilm}\mathring{R}_{hjkm} +R_{hjlm}\mathring{R}_{ihkm}+R_{hklm}\mathring{R}_{ijhm}+R_{hmlm}\mathring{R}_{ijkh}), \end{eqnarray} where the Ricci identities are used in the last equality of (2.9). By the definition of trace-free Riemannian curvature tensor and (2.1), from (2.9) we get \begin{eqnarray} \mathring{R}_{ijkl}\Delta\mathring{R}_{ijkl}&=&2\mathring{R}_{ijkl}\nabla_l\nabla_m\mathring{R}_{ijkm}+2\mathring{R}_{ijkl}(R_{hilm}\mathring{R}_{hjkm}+R_{hjlm}\mathring{R}_{ihkm}+R_{hklm}\mathring{R}_{ijhm})+2R_{hl}\mathring{R}_{ijkl}\mathring{R}_{ijkh}\nonumber \\ &=&2\mathring{R}_{ijkl}\nabla_l\nabla_m\mathring{R}_{ijkm}+2\mathring{R}_{ijkl}(\mathring{R}_{hilm}\mathring{R}_{hjkm}+\mathring{R}_{hjlm}\mathring{R}_{ihkm}+\mathring{R}_{hklm}\mathring{R}_{ijhm})\nonumber\\
&&+2\mathring{R}_{ijkl}\mathring{R}_{ijkh}\mathring{R}_{hl}+\frac{2R}{n}|\mathring{Rm}|^2 +\frac{2R}{n(n-1)}\mathring{R}_{ijkl}(\mathring{R}_{ljki}+\mathring{R}_{ilkj}+\mathring{R}_{ijlk}+\mathring{R}_{jk}\delta_{li}-\mathring{R}_{ik}\delta_{lj})\nonumber\\ &=&2\mathring{R}_{ijkl}\nabla_l\nabla_m\mathring{R}_{ijkm}+2\mathring{R}_{ijkl}(\mathring{R}_{hilm}\mathring{R}_{hjkm} +\mathring{R}_{hjlm}\mathring{R}_{ihkm}+\mathring{R}_{hklm}\mathring{R}_{ijhm})\nonumber\\
& &+2\mathring{R}_{ijkl}\mathring{R}_{ijkh}\mathring{R}_{hl}-\frac{4R}{n(n-1)}|\mathring{Ric}|^2+\frac{2R}{n}|\mathring{Rm}|^2\nonumber\\ &=&2\mathring{R}_{ijkl}\nabla_l\nabla_m\mathring{R}_{ijkm}-2\left(2\mathring{R}_{ijlk}\mathring{R}_{ihlm}\mathring{R}_{hjmk}+\frac{1}{2}\mathring{R}_{ijkl}\mathring{R}_{ijhm}\mathring{R}_{hmkl}\right)\nonumber\\
& &+2\mathring{R}_{ijkl}\mathring{R}_{ijkh}\mathring{R}_{hl}-\frac{4R}{n(n-1)}|\mathring{Ric}|^2+\frac{2R}{n}|\mathring{Rm}|^2. \end{eqnarray}
\par We consider $\mathring{Rm}$ as a trace-free symmetric operator on $\Lambda^2(M)$ and $\otimes^2(M)$. By the algebraic inequalities $tr(T^3)\leq\frac{m-2}{\sqrt{m(m-1)}}|T|^3$ for trace-free symmetric $m$-matrices $T$ and $\lambda_i\leq\sqrt{\frac{m-1}{m}}|T|$ for the eigenvalues $\lambda_i$ of $T$ in \cite{H}, we get \begin{eqnarray}
\left|2\mathring{R}_{ijlk}\mathring{R}_{ihlm}\mathring{R}_{hjmk}+\frac{1}{2}\mathring{R}_{ijkl}\mathring{R}_{ijhm}\mathring{R}_{hmkl}\right|
\leq2|\mathring{R}_{ijlk}\mathring{R}_{ihlm}\mathring{R}_{hjmk}|+\frac{1}{2}|\mathring{R}_{ijkl}\mathring{R}_{ijhm}\mathring{R}_{hmkl}|\nonumber\\
\leq\left(\frac{2(n^2-2)}{n{\sqrt{n^2-1}}}+\frac{n^2-n-4}{2\sqrt{(n-2)n(n^2-1)}}\right)|\mathring{Rm}|^3, \end{eqnarray} and \begin{eqnarray}
|\mathring{R}_{ijkl}\mathring{R}_{ijkh}\mathring{R}_{hl}|\leq\sqrt{\frac{n-1}{n}}|\mathring{Ric}||\mathring{Rm}|^2. \end{eqnarray} Combining with (2.7), (2.10), (2.11) and (2.12), we get \begin{eqnarray} \mathring{R}_{ijkl}\Delta\mathring{R}_{ijkl}&\geq&
-\left(\sqrt{\frac{(n-1)(n-2)}{n}}+\frac{4(n^2-2)}{n\sqrt{n^2-1}}+\frac{n^2-n-4}{\sqrt{(n-2)n(n^2-1)}}\right)|\mathring{Rm}|^3\nonumber\\
& &+2\mathring{R}^{ijkl}\nabla_l\nabla^m\mathring{R}_{ijkm}+A(n)R|\mathring{Rm}|^2. \end{eqnarray} \end{proof} \textbf{Proof of Theorem $1.1$.}
For simplicity of natation, we denote by $(\delta\mathring{Rm})_{ijk}=\nabla^l\mathring{R}_{ijkl}$ the divergence of the trace-free Riemannian curvature tensor and $u=|\mathring{Rm}|$. By the Kato inequality $|\nabla\mathring{Rm}|^2\geq|\nabla|\mathring{Rm}||^2$, we get \begin{eqnarray} \mathring{R}^{ijkl}\Delta\mathring{R}_{ijkl}
&\leq&\mathring{R}^{ijkl}\Delta\mathring{R}_{ijkl}+|\nabla\mathring{Rm}|^2-|\nabla|\mathring{Rm}||^2\nonumber\\
&=&\frac{1}{2}\Delta|\mathring{Rm}|^2-|\nabla|\mathring{Rm}||^2\nonumber\\
&=&|\mathring{Rm}|\Delta|\mathring{Rm}|=u\Delta u, \end{eqnarray} which together with Lemma $2.1$ and integrating on $M^n$ give $$\int_{M}u\Delta udV_g\geq-C(n)\int_{M}u^3dV_g+2\int_{M}\mathring{R}^{ijkl}\nabla_l\nabla^m\mathring{R}_{ijkm}dV_g+\frac{R}{n-1}\int_{M}u^2dV_g.$$ Moreover, using the Stokes's theorem, we get \begin{eqnarray}
\int_{M}|\nabla u|^2dV_g\leq C(n)\int_{M}u^3dV_g+2\int_{M}|\delta\mathring{Rm}|^2dV_g-\frac{R}{n-1}\int_{M}u^2dV_g. \end{eqnarray} Since $M^n$ is Bach-flat, we have $$B_{ij}=\frac{1}{n-3}\nabla^k\nabla^lW_{ikjl}+\frac{1}{n-2}R^{kl}W_{ikjl}=0.$$ Multiplying the above equality by $\mathring{R}^{ij}$ and integrating on $M^n$ give \begin{eqnarray} 0&=&\frac{1}{n-3}\int_{M}\mathring{R}^{ij}\nabla^k\nabla^lW_{ikjl}dV_g+\frac{1}{n-2}\int_{M}\mathring{R}^{ij}R^{kl}W_{ikjl}dV_g\nonumber\\ &=&-\frac{1}{n-2}\int_{M}\nabla^k\mathring{R}^{ij}\nabla^l\mathring{R}_{ikjl}dV_g+\frac{1}{n-2}\int_{M}\mathring{R}^{ij}\mathring{R}^{kl}W_{ikjl}dV_g\nonumber\\ &=&-\frac{1}{n-2}\int_{M}\frac{1}{2}\left(\nabla^k\mathring{R}^{ij}-\nabla^i\mathring{R}^{kj}\right)\nabla^l\mathring{R}_{ikjl}dV_g+\frac{1}{n-2}\int_{M}\mathring{R}^{ij}\mathring{R}^{kl}W_{ikjl}dV_g\nonumber\\
&=&-\frac{1}{2(n-2)}\int_{M}|\delta\mathring{Rm}|^2dV_g+\frac{1}{n-2}\int_{M}\mathring{R}^{ij}\mathring{R}^{kl}W_{ikjl}dV_g, \end{eqnarray} where (2.6) and the second Bianchi identities are used in the second line and the third line of (2.16) respectively. Using (2.7) and the Huisken inequality ( see Lemma 3.4 of \cite{H})
$$|\mathring{R}^{ik}\mathring{R}^{jl}W_{ijkl}|\leq\sqrt{\frac{n-2}{2(n-1)}}|W||\mathring{Ric}|^2,$$ we have \begin{eqnarray}
\int_{M}|\delta\mathring{Rm}|^2dV_g\leq\frac{n-2}{2}\sqrt{\frac{n-2}{2(n-1)}}\int_{M}u^3dV_g. \end{eqnarray} Combining with (2.15) and (2.17), we obtain \begin{eqnarray}
\int_{M}|\nabla u|^2dV_g\leq E(n)\int_{M}u^3dV_g-\frac{R}{n-1}\int_{M}u^2dV_g, \end{eqnarray} where $E(n)=C(n)+\sqrt{\frac{(n-2)^3}{2(n-1)}}$. From (2.18), using Young's inequality and the H\"{o}lder inequality, we get \begin{eqnarray}
\int_{M}|\nabla u|^2dV_g &\leq&\frac{nE(n)}{2p}\epsilon^{-\frac{2p-n}{n}}\int_{M}u^{2+\frac{2p}{n}}dV_g+\left(\frac{(2p-n)\epsilon E(n)}{2p}-\frac{R}{n-1}\right)\int_{M}u^2dV_g\nonumber\\ &\leq&\frac{nE(n)}{2p}\epsilon^{-\frac{2p-n}{n}}\left(\int_{M}u^{\frac{2n}{n-2}}dV_g\right)^{\frac{n-2}{n}}\left(\int_{M}u^pdV_g\right)^{\frac{2}{n}}\\ & &+\left(\frac{(2p-n)\epsilon E(n)}{2p}-\frac{R}{n-1}\right)\int_{M}u^2dV_g\nonumber, \end{eqnarray} where $\epsilon$ is a positive constant. By the definition of Yamabe constant $Y(M,[g])$, we get \begin{eqnarray} \frac{n-2}{4(n-1)}Y(M,[g])\left(\int_{M}u^{\frac{2n}{n-2}}dV_g\right)^{\frac{n-2}{n}}
&\leq&\int_{M}|\nabla u|^2dV_g+\frac{(n-2)R}{4(n-1)}\int_{M}u^2dV_g\nonumber\\
&\leq&(1+\eta)\int_{M}|\nabla u|^2dV_g+\frac{(n-2)R}{4(n-1)}\int_{M}u^2dV_g, \end{eqnarray} where $\eta\geq0$ is a constant. Substituting (2.19) into (2.20), we conclude that \begin{eqnarray} \left\{\frac{n-2}{4(n-1)}Y(M,[g])-(1+\eta)\frac{nE(n)}{2p}\epsilon^{-\frac{2p-n}{n}}\left(\int_{M}u^pdV_g\right)^{\frac{2}{n}}\right\}\left(\int_{M}u^{\frac{2n}{n-2}}dV_g\right)^{\frac{n-2}{n}}\nonumber\\ \leq\left\{\frac{(n-2)R}{4(n-1)}+(1+\eta)\left(\frac{(2p-n)\epsilon E(n)}{2p}-\frac{R}{n-1}\right)\right\}\int_{M}u^2dV_g. \end{eqnarray}
How we select $(\epsilon, \eta)$ to maximize $\left(\int_{M}u^pdV_g\right)^{\frac{2}{n}}$ in (2.21) is equivalent to a problem of finding $(\epsilon, \eta)$ on a domain $\mathcal{D}$ which minimizes a function $$F(\epsilon, \eta):=\frac{nE(n)}{2p}(1+\eta)\epsilon^{-\frac{2p-n}{n}}.$$ Here, the domain $\mathcal{D}$ consists of points $(\epsilon, \eta)$ which satisfies inequalities $$G(\epsilon, \eta):=\frac{(n-2)R}{4(n-1)}+(1+\eta)\left(\frac{(2p-n)\epsilon E(n)}{2p}-\frac{R}{n-1}\right)\leq0, \ \epsilon>0, \ \eta\geq0.$$ In the case of $p=\frac{n}{2}$, since we have \\ $$ F(\epsilon, \eta)=E(n)(1+\eta) \ \ \begin{array}{cc} &\mbox{and} \end{array} \ \ \ \ G(\epsilon, \eta)=\frac{R}{n-1}\left(\frac{n-6}{4}-\eta\right), $$ we can set $$ \eta=\left\{ \begin{array}{cc} \frac{n-6}{4}, &\mbox{if $n\geq6$,}\\ 0, &\mbox{if $n=4,5$} \end{array} \right. , \ \ \begin{array}{cc} &\mbox{$\epsilon =$ (any positive number).} \end{array} $$ In the case of $p>\frac{n}{2}$. In order to minimize $F(\epsilon, \eta)$ and $G(\epsilon, \eta)=0$, we can set $$ \eta=\left\{ \begin{array}{cc} \frac{(n-2)p-2n}{2n}, &\mbox{if $n=4,5$ and $p\geq\frac{2n}{n-2}$ or $n\geq6$}\\ 0, &\mbox{if $n=4,5$ and $\frac n2<p<\frac{2n}{n-2}$} \end{array} \right. ,$$ $$ \epsilon =\left\{ \begin{array}{cc} \frac{R}{(n-1)E(n)}, &\mbox{if $n=4,5$ and $p\geq\frac{2n}{n-2}$ or $n\geq6$}\\ \frac{p(6-n)R}{2(n-1)(2p-n)E(n)}, &\mbox{if $n=4,5$ and $\frac n2<p<\frac{2n}{n-2}$.} \end{array} \right. $$
In conclusion, we can choose $$\left(\int_{M}u^pdV_g\right)^{\frac{1}{p}}<\varepsilon(n)Y(M,[g])^\frac{n}{2p}R^{1-\frac{n}{2p}}$$ such that (2.21) implies $\left(\int_{M}u^{\frac{2n}{n-2}}dV_g\right)^{\frac{n-2}{n}}=0$, i.e., $\mathring{Rm}=0$. Hence $(M^n,g)$ is isometric to a quotient of the round $\mathbb{S}^n$. \section{Proof of Theorem 1.4}
Now, we compute the Laplacian of $|\mathring{Ric}|^2$. \begin{lemma} Let $(M^n,g) (n\geq4)$ be a complete Bach-flat $n$-manifold with constant scalar curvature, then \begin{eqnarray}
\Delta|\mathring{Ric}|^2=2|\nabla\mathring{Ric}|^2-4\mathring{R}_{ij}\mathring{R}_{kl}W_{ikjl}+\frac{2n}{n-2}\mathring{R}_{ij}\mathring{R}_{jk}\mathring{R}_{ik}
+\frac{2R}{n-1}|\mathring{Ric}|^2. \end{eqnarray} \end{lemma} \begin{remark} Although Lemma $3.1$ has been proved in \cite{CGY}, for completeness, we also write it out.\end{remark} \begin{proof}We obtain from (2.3) and (2.5) that \begin{eqnarray}
\Delta|\mathring{Ric}|^2&=&2|\nabla\mathring{Ric}|^2+2\mathring{R}_{ij}\nabla_k\nabla_k\mathring{R}_{ij}\nonumber \\
&=&2|\nabla\mathring{Ric}|^2+2\mathring{R}_{ij}\nabla_k(\nabla_j\mathring{R}_{ik}-\nabla_l\mathring{R}_{ilkj})\nonumber \\
&=&2|\nabla\mathring{Ric}|^2+2\mathring{R}_{ij}\nabla_k\nabla_j\mathring{R}_{ik}+2\mathring{R}_{ij}\nabla_k\nabla_l\mathring{R}_{ikjl}. \end{eqnarray} Since the scalar curvature is constant, by the Ricci identities, we get \begin{eqnarray} \mathring{R}_{ij}\nabla_k\nabla_j\mathring{R}_{ik}&=&\mathring{R}_{ij}(\nabla_j\nabla_k\mathring{R}_{ik}+\mathring{R}_{hk}R_{hijk}+\mathring{R}_{ih}R_{hkjk})\nonumber\\ &=&\mathring{R}_{ij}\mathring{R}_{hk}R_{hijk}+\mathring{R}_{ij}\mathring{R}_{ih}R_{hj}\nonumber\\ &=&\mathring{R}_{ij}\mathring{R}_{hk}[W_{hijk}+\frac{1}{n-2}(\mathring{R}_{ik}\delta_{hj}+\mathring{R}_{hj}\delta_{ik}-\mathring{R}_{ij}\delta_{hk}-\mathring{R}_{hk}\delta_{ij})\nonumber\\
& & +\frac{R}{n(n-1)}(\delta_{ik}\delta_{jh}-\delta_{ij}\delta_{hk})]+\mathring{R}_{ij}\mathring{R}_{ih}\mathring{R}_{hj}+\frac{R}{n}|\mathring{Ric}|^2\nonumber\\
&=&\mathring{R}_{ij}\mathring{R}_{hk}W_{hijk}+\frac{n}{n-2}\mathring{R}_{ij}\mathring{R}_{jk}\mathring{R}_{ik}+\frac{R}{n-1}|\mathring{Ric}|^2, \end{eqnarray} and \begin{eqnarray*} \mathring{R}_{ij}\nabla_k\nabla_l\mathring{R}_{ikjl} &=&\mathring{R}_{ij}\nabla_k\nabla_lW_{ikjl}+\frac{1}{n-2}\mathring{R}_{ij}\nabla_k\nabla_l(\mathring{R}_{ij}\delta_{kl}+\mathring{R}_{kl}\delta_{ij}-\mathring{R}_{il}\delta_{jk}-\mathring{R}_{jk}\delta_{il}) \nonumber\\ &=&\mathring{R}_{ij}\nabla_k\nabla_lW_{ikjl}+\frac{1}{n-2}(\mathring{R}_{ij}\Delta\mathring{R}_{ij}-\mathring{R}_{ij}\nabla_j\nabla_l\mathring{R}_{il}-\mathring{R}_{ij}\nabla_k\nabla_i\mathring{R}_{kj}) \nonumber\\ &=&\mathring{R}_{ij}\nabla_k\nabla_lW_{ikjl}+\frac{1}{n-2}\left[\mathring{R}_{ij}\Delta\mathring{R}_{ij}-\mathring{R}_{ij}\nabla_k(\nabla_k\mathring{R}_{ij}+\nabla_l\mathring{R}_{jlki})\right]\nonumber\\ &=&\mathring{R}_{ij}\nabla_k\nabla_lW_{ikjl}+\frac{1}{n-2}\mathring{R}_{ij}\nabla_k\nabla_l\mathring{R}_{ikjl}. \end{eqnarray*} Since $M^n$ is Bach-flat, we get from the above equation that \begin{eqnarray} \mathring{R}_{ij}\nabla_k\nabla_l\mathring{R}_{ikjl}=-\mathring{R}_{ij}\mathring{R}_{kl}W_{ikjl}. \end{eqnarray} Combining with (3.2), (3.3) and (3.4), we obtain
$$\Delta|\mathring{Ric}|^2=2|\nabla\mathring{Ric}|^2-4\mathring{R}_{ij}\mathring{R}_{kl}W_{ikjl}+\frac{2n}{n-2}\mathring{R}_{ij}\mathring{R}_{jk}\mathring{R}_{ik}
+\frac{2R}{n-1}|\mathring{Ric}|^2.$$ This completes the proof of Lemma $3.1$. \end{proof} \begin{lemma} On every $n$-dimensional Riemannian manifold, the following estimate holds
$$\left|-W_{ijkl}\mathring{R}_{ik}\mathring{R}_{jl}+\frac{n}{2(n-2)}\mathring{R}_{ij}\mathring{R}_{jk}\mathring{R}_{ik}\right|
\leq\sqrt{\frac{n-2}{2(n-1)}}|\mathring{Ric}|^2\left(|W|^2+\frac{n}{2(n-2)}|\mathring{Ric}|^2\right)^{\frac{1}{2}}.$$ \end{lemma} \begin{remark} We follow these proofs of Proposition 2.1 in \cite{C} and Lemma 4.7 in \cite{Bo} to prove this lemma. For completeness, we also write it out. In general, according to the proof of Lemma $3.3$, we can obtain
$$\left|-W_{ijkl}\mathring{R}_{ik}\mathring{R}_{jl}+K\mathring{R}_{ij}\mathring{R}_{jk}\mathring{R}_{ik}\right|
\leq\sqrt{\frac{n-2}{2(n-1)}}|\mathring{Ric}|^2\left(|W|^2+\frac{2(n-2)K^2}{n}|\mathring{Ric}|^2\right)^{\frac{1}{2}},$$ where $K$ is a constant. \end{remark} \begin{proof} First of all we have $$(\mathring{Ric}\circledwedge g)_{ijkl}=\mathring{R}_{ik}g_{jl}-\mathring{R}_{il}g_{jk}+\mathring{R}_{jl}g_{ik}-\mathring{R}_{jk}g_{il},$$ $$(\mathring{Ric}\circledwedge \mathring{Ric})_{ijkl}=2(\mathring{R}_{ik}\mathring{R}_{jl}-\mathring{R}_{il}\mathring{R}_{jk}),$$ where $\circledwedge$ denotes the Kulkarni-Nomizu product. An easy computation shows $$W_{ijkl}\mathring{R}_{ik}\mathring{R}_{jl}=\frac{1}{4}W_{ijkl}(\mathring{Ric}\circledwedge \mathring{Ric})_{ijkl},$$ $$\mathring{R}_{ij}\mathring{R}_{jk}\mathring{R}_{ik}=-\frac{1}{8}(\mathring{Ric}\circledwedge g)_{ijkl}(\mathring{Ric}\circledwedge \mathring{Ric})_{ijkl}.$$ Hence we get the following identity \begin{eqnarray} -W_{ijkl}\mathring{R}_{ik}\mathring{R}_{jl}+\frac{n}{2(n-2)}\mathring{R}_{ij}\mathring{R}_{jk}\mathring{R}_{ik}=-\frac{1}{4}\left(W+\frac{n}{4(n-2)}\mathring{Ric}\circledwedge g\right)_{ijkl}(\mathring{Ric}\circledwedge \mathring{Ric})_{ijkl}. \end{eqnarray} Since $\mathring{Ric}\circledwedge \mathring{Ric}$ has the same symmetries of the Riemannian curvature tensor, it can be orthogonally decomposed as $$\mathring{Ric}\circledwedge \mathring{Ric}=T+V'+U',$$ where $T$ is totally trace-free and
$$V'_{ijkl}=-\frac{2}{n-2}\left(\mathring{Ric}^2\circledwedge g\right)_{ijkl}+\frac{2}{n(n-2)}|\mathring{Ric}|^2(g\circledwedge g)_{ijkl},$$
$$U'_{ijkl}=-\frac{1}{n(n-1)}|\mathring{Ric}|^2(g\circledwedge g)_{ijkl},$$ where $\left(\mathring{Ric}^2\right)_{ik}=\mathring{R}_{ip}\mathring{R}_{kp}$. Taking the squared norm one obtains
$$|\mathring{Ric}\circledwedge \mathring{Ric}|^2=8|\mathring{Ric}|^4-8|\mathring{Ric}^2|^2,$$
$$|V'|^2=\frac{16}{n-2}|\mathring{Ric}^2|^2-\frac{16}{n(n-2)}|\mathring{Ric}|^4,$$
$$|U'|^2=\frac{8}{n(n-1)}|\mathring{Ric}|^4.$$ In particular, one has
$$|T|^2+\frac{n}{2}|V'|^2=|\mathring{Ric}\circledwedge \mathring{Ric}|^2+\frac{n-2}{2}|V'|^2-|U'|^2=\frac{8(n-2)}{n-1}|\mathring{Ric}|^4.$$ We now estimate the right hand side of (3.5). Using the fact that $W$ and $T$ are totally trace-free and the Cauchy-Schwarz inequality we obtain \begin{eqnarray*}
\left|\left(W+\frac{n}{4(n-2)}\mathring{Ric}\circledwedge g\right)_{ijkl}(\mathring{Ric}\circledwedge \mathring{Ric})_{ijkl}\right|^2
&=&\left|\left(W+\frac{n}{4(n-2)}\mathring{Ric}\circledwedge g\right)_{ijkl}(T+V')_{ijkl}\right|^2\\
&=&\left|\left(W+\frac{\sqrt{2n}}{4(n-2)}\mathring{Ric}\circledwedge g\right)_{ijkl}\left(T+\sqrt{\frac{n}{2}}V'\right)_{ijkl}\right|^2\\
&\leq&\left|W+\frac{\sqrt{2n}}{4(n-2)}\mathring{Ric}\circledwedge g\right|^2\left(|T|^2+\frac{n}{2}|V'|^2\right)\\
&=&\frac{8(n-2)}{n-1}|\mathring{Ric}|^4\left(|W|^2+\frac{n}{2(n-2)}|\mathring{Ric}|^2\right). \end{eqnarray*} This estimate together with (3.5) concludes this proof. \end{proof}
\textbf{Proof of Theorem $1.4$.} By Lemmas $3.1$ and $3.3$ and the Kato inequality $|\nabla\mathring{Ric}|^2\geq|\nabla|\mathring{Ric}||^2$, we get \begin{eqnarray}
|\mathring{Ric}|\Delta|\mathring{Ric}|\geq
-\sqrt{\frac{2(n-2)}{n-1}}|\mathring{Ric}|^2\left(|W|^2+\frac{n}{2(n-2)}|\mathring{Ric}|^2\right)^{\frac{1}{2}}+\frac{R}{n-1}|\mathring{Ric}|^2. \end{eqnarray}
Set $u=|\mathring{Ric}|$. By (3.6), we compute \begin{eqnarray}
u^\alpha\Delta u^\alpha&=&u^\alpha\left(\alpha(\alpha-1)u^{\alpha-2}|\nabla u|^2+\alpha u^{\alpha-1}\Delta u\right)\nonumber \\
&=&\frac{\alpha-1}{\alpha}|\nabla u^\alpha|^2+\alpha u^{2\alpha-2}u\Delta u\nonumber \\
&\geq&\frac{\alpha-1}{\alpha}|\nabla u^\alpha|^2-\alpha\sqrt{\frac{2(n-2)}{n-1}}\left(|W|^2+\frac{n}{2(n-2)}u^2\right)^{\frac{1}{2}}u^{2\alpha}+\frac{\alpha R}{n-1}u^{2\alpha}. \end{eqnarray} Integrating (3.7) on $M^n$ and using Stoke's theorem, we have \begin{eqnarray}
0\geq\left(2-\frac{1}{\alpha}\right)\int_{M}|\nabla u^\alpha|^2dV_g-\alpha\sqrt{\frac{2(n-2)}{n-1}}\int_{M}\left(|W|^2+\frac{n}{2(n-2)}u^2\right)^{\frac{1}{2}}u^{2\alpha}dV_g +\frac{\alpha R}{n-1}\int_{M}u^{2\alpha}dV_g. \end{eqnarray} For $2-\frac{1}{\alpha}>0$, by the definition of Yamabe constant and H\"{o}ider inequality, we obtain from (3.8) that \begin{eqnarray} 0&\geq&\left\{\left(2-\frac{1}{\alpha}\right)\frac{n-2}{4(n-1)}Y(M,[g])
-\alpha\sqrt{\frac{2(n-2)}{n-1}}\left(\int_{M}\left(|W|^2+\frac{n}{2(n-2)}u^2\right)^{\frac{n}{4}}dV_g\right)^{\frac{2}{n}}\right\}\left(\int_{M}u^{\frac{2n\alpha}{n-2}}dV_g\right)^{\frac{n-2}{n}}\nonumber \\ & &+\frac{4\alpha^2-2(n-2)\alpha+n-2}{4\alpha(n-1)}R\int_{M}u^{2\alpha}dV_g. \end{eqnarray} \textbf{Case 1.} when $n\geq6$, taking $\alpha=\frac{(n-2)\left(1+\sqrt{1-\frac{4}{n-2}}\right)}{4}$, from (3.9), we get \begin{eqnarray}
0&\geq&\left\{\frac{Y(M,[g])}{\sqrt{2(n-1)(n-2)}}-\left(\int_{M}\left(|W|^2+\frac{n}{2(n-2)}u^2\right)^{\frac{n}{4}}dV_g\right)^{\frac{2}{n}}\right\}\left(\int_{M}u^{\frac{2n\alpha}{n-2}}dV_g\right)^{\frac{n-2}{n}}. \end{eqnarray} Since $W$ is totally trace-free, one has
$$\left|W+\frac{\sqrt{n}}{2\sqrt{2}(n-2)}\mathring{Ric}\circledwedge g\right|^2=|W|^2+\frac{n}{2(n-2)}|\mathring{Ric}|^2$$ and the pinching condition (1.2) implies that $M^n$ is Einstein.\\ \textbf{Case 2.} When $n=4,5$, we have $\frac{4\alpha^2-2(n-2)\alpha+n-2}{4\alpha(n-1)}>0$ for $\alpha>\frac{1}{2}$, and from (3.9), we get \begin{eqnarray*}
0&\geq&\left\{\left(1-\left(1-\frac{1}{\alpha}\right)^2\right)\sqrt{\frac{n-2}{32(n-1)}}Y(M,[g])-\left(\int_{M}\left(|W|^2+\frac{n}{2(n-2)}u^2\right)^{\frac{n}{4}}dV_g\right)^{\frac{2}{n}}\right\}\left(\int_{M}u^{\frac{2n\alpha}{n-2}}dV_g\right)^{\frac{n-2}{n}}. \end{eqnarray*} Taking $\alpha=1$, we have \begin{eqnarray}
0&\geq&\left\{\sqrt{\frac{n-2}{32(n-1)}}Y(M,[g])-\left(\int_{M}\left(|W|^2+\frac{n}{2(n-2)}u^2\right)^{\frac{n}{4}}dV_g\right)^{\frac{2}{n}}\right\}\left(\int_{M}u^{\frac{2n}{n-2}}dV_g\right)^{\frac{n-2}{n}}. \end{eqnarray} Thus the pinching condition (1.2) implies that $M^n$ is Einstein. \par In particular, for $n=4,5$, the pinching condition (1.2) implies \begin{eqnarray}
\left(\int_{M}|W|^{\frac{n}{2}}dV_g\right)^{\frac{2}{n}}<\sqrt{\frac{n-2}{32(n-1)}}Y(M,[g])<\frac{2Y(M,[g])}{C_2(n)n}, \end{eqnarray} where the constant \[ C_2(n)=\left\{ \begin{array}{cc} \frac{\sqrt{6}}{2}, &\mbox{if $n=4$,}\\ \frac{8\sqrt{10}}{15}, &\mbox{if $n=5$,}\\ \frac{4(n^2-2)}{n\sqrt{n^2-1}}+\frac{n^2-n-4}{\sqrt{(n-2)(n-1)n(n+1)}},&\mbox{if $n\geq6$} \end{array}\right. \] is defined in Lemma 2.1 of \cite{FX3}. By the rigidity result for positively curved Einstein manifolds (see Theorem 1.1 of \cite{FX3}), (3.12) implies that $M^n$ is isometric to a quotient of the round $\mathbb{S}^n$. For $n\geq6$, we can choose $\alpha$ such that $\frac{4\alpha^2-2(n-2)\alpha+n-2}{4\alpha(n-1)}>0$ and
$$0\geq\left\{\frac{2Y(M,[g])}{C_2(n)n}-\left(\int_{M}\left(|W|^2+\frac{n}{2(n-2)}u^2\right)^{\frac{n}{4}}dV_g\right)^{\frac{2}{n}}\right\}\left(\int_{M}u^{\frac{2n\alpha}{n-2}}dV_g\right)^{\frac{n-2}{n}}. $$ From Case 1, the pinching condition (1.2) implies that $M^n$ is Einstein. Hence, the pinching condition (1.2) implies \begin{eqnarray}
\left(\int_{M}|W|^{\frac{n}{2}}dV_g\right)^{\frac{2}{n}}<\frac{2Y(M,[g])}{C_2(n)n}. \end{eqnarray} By the rigidity result for positively curved Einstein manifolds (see Theorem 1.1 of \cite{FX3}), (3.13) implies that $M^n$ is isometric to a quotient of the round $\mathbb{S}^n$.\\ \textbf{Proof of Corollary $1.6$.} To prove Corollary $1.6$, we need the following lemma which was proved by Gursky (see \cite{G}). For completeness, we also write it's proof out. \begin{lemma} Let $(M^4,g)$ be a complete 4-dimensional manifold, then the following estimate holds
$$\int_{M}R^2dV_g-12\int_{M}|\mathring{Ric}|^2dV_g\leq Y(M,[g])^2,$$ moreover, the inequality is strict unless $(M^4,g)$ is comformally Einstein. \end{lemma} \begin{proof} By the Chern-Gauss-Bonnet formula (see the Equation 6.31 of \cite{Be}) \begin{eqnarray}
\int_{M}|W|^2dV_g-2\int_{M}|\mathring{Ric}|^2dV_g+\frac{1}{6}\int_{M}R^2dV_g=32\pi^2\chi(M), \end{eqnarray}
and the conformal invariance of $\int_{M}|W|^2dV_g$, we find that $-2\int_{M}|\mathring{Ric}|^2dV_g+\frac{1}{6}\int_{M}R^2dV_g$ is also conformally invariant.
Let $\tilde{g}\in[g]$ be a Yamabe metric. Then \begin{eqnarray*} Y(M,[g])^2&=&\frac{\left(\int_M{R_{\tilde{g}}}dV_{\tilde{g}}\right)^2}{\int_MdV_{\tilde{g}}}=\int_M{R}_{\tilde{g}}^2dV_{\tilde{g}}\\
&\geq&\int_M{R}_{\tilde{g}}^2dV_{\tilde{g}}-12\int_M|{\mathring{Ric}}_{\tilde{g}}|_{\tilde{g}}^2dV_{\tilde{g}}\\
&=&\int_MR^2dV_g-12\int_M|\mathring{Ric}|^2dV_g. \end{eqnarray*}
The equality case follows immediately. \end{proof}
By Lemma $3.5$, we get \begin{eqnarray}
\int_{M}|W|^2dV_g+\int_{M}|\mathring{Ric}|^2dV_g-\frac{Y(M,[g])^2}{48}\leq\int_{M}|W|^2dV_g+\frac{5}{4}\int_{M}|\mathring{Ric}|^2dV_g-\frac{1}{48}\int_{M}R^2dV_g. \end{eqnarray} Moreover, the inequality is strict unless $(M^4,g)$ is comformally Einstein. In the first case $``\,<"$, Theorem $1.4$ immediately implies Corollary $1.6$; In the second case $``\,="$, $g$ is conformally Einstein. Since $g$ has constant scalar curvature, $g$ is Einstein from the proof of Obata Theorem (see Proposition 3.1 of \cite{LP}). By the rigidity result for positively curved Einstein manifolds (see Theorem 1.1 of \cite{FX3}), (3.15) implies that $M^4$ is isometric to a quotient of the round $\mathbb{S}^4$. \section{Proof of Theorem $1.8$} \noindent \textbf{Proof of Theorem $1.8$.} From (3.1), by Lemma $3.3$, we get \begin{eqnarray}
\Delta|\mathring{Ric}|^2\geq2|\nabla\mathring{Ric}|^2+4\sqrt{\frac{n-2}{2(n-1)}}|\mathring{Ric}|^2\left\{\frac{R}{\sqrt{2(n-1)(n-2)}}-\left(|W|^2+\frac{n}{2(n-2)}|\mathring{Ric}|^2\right)^{\frac{1}{2}}\right\}. \end{eqnarray} Note that (1.5) is equivalent that the second of RHS of (4.1) is nonnegative. By the maximum principle, from (4.1) we get $\nabla\mathring{Ric}=0$. Since $M^n$ has positive constant scalar curvature, $M^n$ is a manifold with parallel Ricci tensor.
Hence $M^n$ is a manifold with harmonic curvature. Using the same argument as in the proof of (3.1), we obtain a Weitzenb\"{o}ck formula (see (2.20) in \cite{F}) \begin{eqnarray}
\Delta|\mathring{Ric}|^2=2|\nabla\mathring{Ric}|^2-2\mathring{R}_{ij}\mathring{R}_{kl}W_{ikjl}+\frac{2n}{n-2}\mathring{R}_{ij}\mathring{R}_{jk}\mathring{R}_{ik}
+\frac{2R}{n-1}|\mathring{Ric}|^2. \end{eqnarray} By Remark $3.4$, we get
\begin{eqnarray}\left|-W_{ijkl}\mathring{R}_{ik}\mathring{R}_{jl}+\frac{n}{n-2}\mathring{R}_{ij}\mathring{R}_{jk}\mathring{R}_{ik}\right|
\leq\sqrt{\frac{n-2}{2(n-1)}}|\mathring{Ric}|^2\left(|W|^2+\frac{2n}{n-2}|\mathring{Ric}|^2\right)^{\frac{1}{2}}.\end{eqnarray} Combing (4.2) with (4.3), we have \begin{eqnarray}
\Delta|\mathring{Ric}|^2\geq2|\nabla\mathring{Ric}|^2+2|\mathring{Ric}|^2\left\{\frac{1}{n-1}R-\sqrt{\frac{n-2}{2(n-1)}}\left(|W|^2+\frac{2n}{n-2}|\mathring{Ric}|^2\right)^{\frac{1}{2}}\right\}. \end{eqnarray} \textbf{Case 1.} $W\neq 0$. By (4.4), the inequality (1.5) implies \begin{eqnarray}
0&\geq&2\sqrt{\frac{n-2}{2(n-1)}}|\mathring{Ric}|^2\left\{\sqrt{\frac{2}{(n-1)(n-2)}}R-\left(|W|^2+\frac{2n}{n-2}|\mathring{Ric}|^2\right)^{\frac{1}{2}}\right\}\nonumber\\ &\geq&
2\sqrt{\frac{n-2}{2(n-1)}}|\mathring{Ric}|^2\left\{\left(4|W|^2+\frac{2n}{n-2}|\mathring{Ric}|^2\right)^{\frac{1}{2}}-\left(|W|^2+\frac{2n}{n-2}|\mathring{Ric}|^2\right)^{\frac{1}{2}}\right\}. \end{eqnarray} By (4.5), we get $\mathring{Ric}=0$, i.e., $(M^n, {g})$ is Einstein.\\
\textbf{Case 2.} $W=0$. Note that the inequality $n(n-1)|\mathring{Ric}|^2\leq R^2$ is equivalent to equation (1.5). From (4.4), we have \begin{eqnarray} 0=\frac{2n}{n-2}\mathring{R}_{ij}\mathring{R}_{jk}\mathring{R}_{ik}
+\frac{2R}{n-1}|\mathring{Ric}|^2\geq\frac{2}{n-1}|\mathring{Ric}|^2\left({R}-\sqrt{(n-1)n}|\mathring{Ric}|\right)\geq0. \end{eqnarray} Hence at every point, either
$\mathring{Ric}$ is null, i.e., $M^n$ is Eninstein, and by conformally flatness it has constant positive sectional curvature, or ${R}-\sqrt{(n-1)n}|\mathring{Ric}|=0$, according to the estimation of trace-free symmetric $2$-tensors, it has an eigenvalue of multiplicity
$(n-1)$ and another of multiplicity $1$. Since the Ricci tensor is parallel, by the de Rham decomposition Theorem, $M^n$ is covered isometrically by the product of Einstein manifolds. We have $R=\sqrt{(n-1)n}|\mathring{Ric}|.$ Since $M^n$ is conformally flat and has positive scalar curvature, then the only possibility is that $M^n$ is covered isometrically by $\mathbb{S}^1\times \mathbb{S}^{n-1}$ with the product metric.
So $(M^n,g)$ is isometric to either an Einstein manifold or a quotient of $\mathbb{S}^1\times \mathbb{S}^{n-1}$ with the product metric.\\ \textbf{Proof of Corollary $1.9$.} By Theorem $1.8$, we consider the case that $M^n$ is Einstein. Using the same argument as in the proof of (3.1), we obtain a Weitzenb\"{o}ck formula for Einstein manifolds (see (5) in \cite{FX3})\begin{eqnarray}
\Delta|W|^2=2|\nabla W|^2-2C_3(n)|W|^3+\frac{4R}{n}|W|^2, \end{eqnarray} where $C_3(4)=\frac{\sqrt{6}}{2}$ and $C_3(5)=\frac{8\sqrt{10}}{15}$.
From (4.7), the condition of Corollary $1.9$ implies that $$
\Delta|W|^2=2|\nabla W|^2+\left(\frac{4R}{n}-2C_3(n)|W|\right)|W|^2\geq0. $$ Hence $W=0$, i.e., $M^n$ is conformally flat. So $(M^n,g)$ is isometric to a quotient of the round $\mathbb{S}^n$. This completes the proof of Corollary $1.9$. \begin{remark} Let $(M^n,g) (n\geq4)$ be an $n$-dimensional compact Bach-flat Riemannian manifold with positive constant scalar curvature. If $$
|W|^2+\frac{{n}}{2(n-2)}|\mathring{Ric}|^2<\frac{1}{{2(n-2)(n-1)}}R^2, $$ then $M^n$ is an Einstein manifold. \end{remark} \textbf{Acknowledgments:} The authors thank the referee for his helpful suggestions.
\end{document} | arXiv |
\begin{document}
\title{\sc\bf\large\MakeUppercase{
Normal approximation in total variation for statistics in geometric probability }}
\author[1]{Tianshu Cong\thanks{{\sf{email: [email protected].}} Work supported by a Research Training Program Scholarship and a Xing Lei Cross-Disciplinary PhD Scholarship in Mathematics and Statistics at the University of Melbourne..}} \author[1]{Aihua Xia\thanks{{\sf{email: [email protected]}}. Work supported by the Australian Research Council Grants Nos DP150101459 and DP190100613.}}
\affil[1]{ School of Mathematics and Statistics, the University of Melbourne, Parkville VIC 3010, Australia}
\date{\today}
\maketitle \vskip-1cm \begin{abstract}
We use Stein's method to establish the rates of normal approximation in terms of the total variation distance for a large class of sums of score functions of marked Poisson point processes on $\mathbb{R}^d$. As in the study under the weaker Kolmogorov distance, the score functions are assumed to satisfy stabilizing and
moment conditions. At the cost of an additional non-singularity condition for score functions, we show that the rates are in line with those under the Kolmogorov distance. We demonstrate the use of the theorems in four applications: Voronoi tessellation, $k$-nearest neighbours, timber volume and maximal layers. \end{abstract}
\vskip8pt \noindent\textit{Key words and phrases:} Total variation distance, non-singular distribution, Berry-Esseen bound, Stein's method.
\vskip8pt\noindent\textit{AMS 2020 Subject Classification:}
primary 60F05; secondary 60D05, 60G55, 62E20.
\section{Introduction}
Limit theorems of functionals of Poisson point processes initiated in \cite{AB93} have been of considerable interest in the literature, see, e.g., \cite{S12,S16,LSY19} and references therein. The key element leading to the success is the stabilization introduced in~\cite{PY01,PY05}. The main character of the stabilization is that insertion of a point into a Poisson point process only induces a local effect in some sense hence there is little change in the functionals. However, adding an additional point to the Poisson point process results in the Palm process of the Poisson point process at the point \cite[Chapter~10]{Kallenberg83} and it is shown in \cite{CX04,CRX20} that the magnitude of the difference between a point process and its Palm processes is directly linked to the accuracy of Poisson and normal approximations of the point process. This is also the fundamental reason why the limit theorems in the above mentioned papers can be established.
The normal approximation theory is generally quantified in terms of the Kolmogorov distance $d_K$: for two random variables $X_1$ and $X_2$ with distributions $F_1$ and $F_2$,
$$d_K(X_1,X_2):=d_K(F_1,F_2):=\sup_{x\in\mathbb{R}}|F_1(x)-F_2(x)|.$$ The well-known Berry-Esseen Theorem \cite{Berry41,Esseen42} states that if $X_i$, $1\le i\le n,$ are independent and identically distributed (${i.i.d.}$) random variables with mean 0 and variance 1, define $Y_n=\frac{\sum_{i=1}^nX_i}{\sqrt{n}}$, $Z\sim N(0,1)$, where $\sim$ denotes ``is distributed as'', then
$$d_K(Y_n,Z)\le \frac{C\E|{X_1}|^3}{\sqrt{n}}.$$ The Kolmogorov distance $d_K(F_1,F_2)$ measures the maximum difference between the distribution functions $F_1$ and $F_2$, but it does not tell much about the difference between the probabilities $\Pro(X_1\in A)$ and $\Pro(X_2\in A)$ for a non-interval Borel set $A\subset\mathbb{R}$, e.g., $A=\cup_{i\in \mathbb{Z}}(2i,2i+0.5]$, where $\mathbb{Z}$ denotes the set of all integers. Such difference is reflected in the total variation distance $d_{TV}(F_1,F_2)$ defined by
$$d_{TV}(X_1,X_2):=d_{TV}(F_1,F_2):=\sup_{A\in{\mathscr{B}}(\mathbb{R})}|F_1(A)-F_2(A)|,$$ where ${\mathscr{B}}(\mathbb{R})$ stands for the Borel $\sigma$-algebra on $\mathbb{R}$. If $F_i$'s are absolutely continuous, that is, for arbitrary $A$ in ${\mathscr{B}}(\mathbb{R})$, $F_i(A):=\int_AF_i'(x)dx$, then the definition is equivalent to
$$d_{TV}(F_1,F_2)=\frac12\sup_f\left|\int f(x)F_1'(x)dx-\int f(x) F_2'(x)dx\right|,$$
where the supremum is taken over all measurable functions $f$ on $(\mathbb{R},{\mathscr{B}}(\mathbb{R}))$ such that $\|f\|:=\sup_{x\in \mathbb{R}} |f(x)|\le1$.
Although central limit theorems in the total variation have been studied in some special circumstances (see, e.g., \cite{DF87,MM07,BC16}), it is generally believed that the total variation distance is too strong for normal approximation, see, e.g., \cite{Ceka00,CL10,Fang14}. For example, the total variation distance between any discrete distribution and any normal distribution is always 1. To recover central limit theorems in the total variation, a common approach is to discretize the distribution of interest and approximate it with a simple discrete distribution, e.g.,
translated Poisson \cite{Rollin05,Rollin07}, centered binomial \cite{Rollin08}, discretized normal \cite{CL10,Fang14} and a family of polynomial type distributions \cite{GX06}. The multivariate versions of these approximations are investigated by~\cite{BLX18}.
By discretizing a distribution $F$ of interest, we essentially group the probability of an area and put it at one point in the area, hence the information of $F(A)$ for a general set $A\in{\mathscr{B}}(\mathbb{R})$ is completely lost.
In this paper, we consider the normal approximation in the total variation to the sum of random variables under various circumstances.
{\bf An inspiring example:} \cite[p.~146]{Feller71}.
Let $\{X_i: \ i\ge 1\}$'s be ${i.i.d.}$ random variables taking values $0$ and $1$ with equal probability, then
$X=\sum_{k=1}^\infty 2^{-k}X_k$ has uniform distribution on $(0,1)$.
If we separate the even and odd terms into $U=\sum_{k=1}^\infty 2^{-2k}X_{2k}$ and $V=\sum_{k=1}^\infty 2^{-(2k-1)}X_{2k-1}$, then $U$ and $V$ are independent, {$2U\stackrel{\mbox{\scriptsize{d}}}{=} V$}, but both $U$ and $V$ have singular distributions.
Now, we can construct mutually independent random variables $\{U_i,\ V_i: \, i\ge 1\}$ such that $U_i\stackrel{\mbox{\scriptsize{d}}}{=} U-\E U$ and $V_i\stackrel{\mbox{\scriptsize{d}}}{=} V-\E V$.
Consider $\xi_1=U_1+V_1$, $\xi_2=-V_1-U_2$, $\xi_3=U_2+V_2$, $\dots$, then $\{\xi_i\}$ is a sequence of 1-dependent and identically distributed random variables having the uniform distribution on $(-0.5,0.5)$. One can easily verify that $\sum_{i=1}^n\xi_i$ does not converge to normal as $n\to\infty$, hence stronger conditions are needed to ensure normal approximation for the sum of dependent random variables.
Under the Kolmogorov distance, user-friendly conditions are usually formulated to ensure that the variance of the sum becomes large as $n\to\infty$. In the context of functionals of Poisson point processes, a typical condition to guarantee the variance of the sum converging to infinity is to assume nondegeneracy \cite{PY01,XY15}, that is, the conditional variance of the sum given the information outside a local region is away from $0$. Under the total variation distance, we use a non-singular condition instead of the nondegeneracy to ensure that the distribution of the functional is diffuse enough for a proper normal approximation for any Borel sets. This condition is almost necessary because it is an essential ingredient in the special case of the sum of ${i.i.d.}$ random variables, see \cite{BC16}\footnote{We thank Vlad Bally for bringing their work to our attention.} for a brief review of the development for the CLT in total variation distance.
The Lebesgue decomposition theorem \cite[p.~134]{Halmos74} ensures that any distribution function $F$ on $\mathbb{R}$ can be represented as
\begin{equation}F=(1-\alpha_F) F_s+\alpha_FF_a,\label{decom1}\end{equation}
where $\alpha_F\in[0,1]$, $F_s$ and $F_a$ are two distribution functions such that, with respect to the Lebesgue measure on $\mathbb{R}$, $F_a$ is absolutely continuous and $F_s$ is singular \cite[p.~126]{Halmos74}.
\begin{defi} A distribution function $F$ on $\mathbb{R}$ is said to be non-singular if $\alpha_F>0$. A random variable is said to be non-singular if its distribution function is non-singular.
\end{defi}
Recalling that two measures on the same measurable space $\mu_1\le\mu_2$ if $\mu_1(A)\le\mu_2(A)$ for all measurable sets $A$. We can see that a random variable $X$ is non-singular if and only if there exists a sub-probability measure $\mu\neq 0$ such that $\mu \le \mathscr{L}(X)$ and there exists a function $f$ on $\mathbb{R}$ satisfying that
\begin{equation*}\mu(A)=\int_A f(x)dx,\mbox{ for all }A\in{\mathscr{B}}(\mathbb{R}). \end{equation*}
In this paper, we demonstrate that many of the limit theorems of functionals of Poisson point processes with respect to the Kolmogorov distance in the literature, e.g., \cite{PY01,PY05,S12,S16}, still hold under the total variation distance. In Section~\ref{Generalresults}, we give definitions of the concepts, state the conditions and present the main theorems. In Section~\ref{Applications}, these theorems are applied to establish error bounds of normal approximation for statistics in Voronoi tessellation, $k$-nearest neighbours, timber volume and maximal layers. The proofs of the main results in Section~\ref{Generalresults} rely on a number of preliminaries and lemmas which are given in Section~\ref{Preliminaries}. For the ease of reading, all proofs are postponed to Section~\ref{Theproofs}.
\section{General results}\label{Generalresults}
We consider the functionals of a marked point process with a Poisson point process in $\mathbb{R}^d$ as its ground process and each point carries a mark in a measurable space $(T,\mathscr{T})$ independently of other marks, where $\mathscr{T}$ is a $\sigma$-algebra on $T$.
More precisely, let $\bm{S}:=\mathbb{R}^d\times T$ be equipped with the product $\sigma$-field $\mathscr{S}:=\mathscr{B}(\mathbb{R}^d)\times \mathscr{T}$, where $\mathscr{B}(\mathbb{R}^d)$ is the Borel $\sigma-$algebra of $\mathbb{R}^d$.
We use~$\bm{C}_{\bm{S}}$ to denote the space of all locally finite non-negative integer valued measures $\xi$, often called a {\it configuration}, on~$\bm{S}$ such
that $\xi(\{{x}\}\times T)\le 1$ for all ${x}\in\mathbb{R}^d$. The space~$\bm{C}_{\bm{S}}$ is endowed with the $\sigma$-field
$\mathscr{C}_{\bm{S}}$ generated by the vague topology \cite[p.~169]{Kallenberg83}.
A {\it marked point process\/} $\Xi$ is a measurable mapping from
$(\Omega,\mathscr{F},\Pro)$ to $(\bm{C}_{\bm{S}},\mathscr{C}_{\bm{S}})$ \cite[p.~49]{Kallenberg17}. The induced simple point process
$\bar\Xi(\cdot):=\Xi(\cdot\times T)$ is called the {\it ground process\/} \cite[p.~3]{Daley08} or
projection \cite[p.~17]{Kallenberg17} of
the marked point process~$\Xi$ {on $\bm{S}$}. The functionals we study in the paper are defined on $\Gamma_\alpha:=\[-\frac{1}{2}\alpha^{\frac{1}{d}}, \frac{1}{2}\alpha^{\frac{1}{d}}\right]^d$ having the forms
$$W_\alpha:={\sum_{(x,m)\in\Xi_{\Gamma_\alpha}}\eta(\left(x,m\right), \Xi)}$$
and
$$\bar{W}_\alpha:={\sum_{{(x,m)\in\Xi_{\Gamma_\alpha}}}\eta(\left(x,m\right), \Xi_{\Gamma_\alpha},\Gamma_\alpha)=\sum_{{(x,m)\in\Xi_{\Gamma_\alpha}}}\eta(\left(x,m\right), \Xi,\Gamma_\alpha)},$$
where $\Xi\sim\mathscr{P}_{\lambda,\mathscr{L}_T}$ is a marked Poisson point process having a homogeneous Poisson point process on $\mathbb{R}^d$ with intensity measure $\lambda dx$ as its ground process and ${i.i.d.}$ marks on $(T,\mathscr{T})$ with the law $\mathscr{L}_T$, $\Xi_{\Gamma_\alpha}$ is its restricted process {on $\Gamma_\alpha$} defined as $\Xi_A(B\times D):=\Xi((A\cap B)\times D)$ for all $D\in \mathscr{T}$ and $A,B\in \mathscr{B}(\mathbb{R}^d)$. The function $\eta$ is called a {\it score function} {(resp. {\it restricted score function})}, i.e., a measurable function on $\(\bm{S}\times \bm{C}_{\bm{S}},\mathscr{S} \times \mathscr{C}_{\bm{S}}\)$ to $\(\mathbb{R},\mathscr{B}\(\mathbb{R}\)\)$ {(resp. a function mapping $\(\bm{S}\times \bm{C}_{\bm{S}}\times \mathbb{R}^d\)$ to $\mathbb{R}$ which is $(\mathscr{S}\cap (\Gamma_\alpha\times T) )\times \mathscr{C}_{\bm{S}\cap (\Gamma_\alpha\times T)}\rightarrow \mathscr{B}(\mathbb{R})$ measurable for fixed the third coordinate)} and it represents the interaction between a point and the configuration. Because the interest is in the values of the score function of the points in a configuration, for convenience, $\eta\((x,m),\mathscr{X}\){~(\mbox{resp.}~\eta\((x,m),\mathscr{X},\Gamma_\alpha\))}$ is understood as $0$ for all $x\in \mathbb{R}^d$ and $\mathscr{X}\in \bm{C}_{\bm{S}}$ such that $(x,m)\notin \mathscr{X}$.
We consider the score functions satisfying the following four conditions.
\noindent{\it A2.1 Stabilization}
For a locally finite configuration $\mathscr{X}$ and $z\in (\mathbb{R}^d\times T)\cup\{\emptyset\}$, write $\mathscr{X}^{\lbag z \rbag}=\mathscr{X}$ if $z=\emptyset$ and $\mathscr{X}^{\lbag z \rbag}=\mathscr{X}\cup\{z\}$ otherwise. We use $\delta_v$ denote the Dirac measure at $v$. The notion of stabilization is introduced in \cite{PY01} and we adapt it to our setup as follows.
\begin{defi}\label{defi4} (unrestricted case)
A score function $\eta$ on $\mathbb{R}^d\times T$ is range-bound (resp. exponentially stabilizing, polynomially stabilizing of order $\beta>0$) with respect to intensity $\lambda$ and a probability measure $\mathscr{L}_T$ on $T$ if for all $x\in \mathbb{R}^d$, $z\in (\mathbb{R}^d\times T)\cup\{\emptyset\}$, and almost all realizations $\mathscr{X}$ of the homogeneous marked Poisson point process $\Xi\sim\mathscr{P}_{\lambda,\mathscr{L}_T}$, there exists an $R:=R(x):=R(x,m_x,\mathscr{X}^{\lbag z \rbag})\in(0,\infty)$ (a radius of stabilization), such that for all locally finite $\mathscr{Y}\subset (\mathbb{R}^d\backslash B(x,R))\times T$, where $B(x,R)$ is the ball with centre $x$ and radius $R$, we have
$$
\eta\left(\left(x,m_x\right), \left[\mathscr{X}^{\lbag z \rbag}\cap\left(B(x,R)\times T\right)\right]\cup \mathscr{Y} \right)=\eta\left(\left(x,m_x\right),\mathscr{X}^{\lbag z \rbag}\cap\left(B(x,R)\times T\right) \right)
$$
and the tail probability $$\tau(t):=\sup_{x\in \mathbb{R}^d,m_x\in \rm{supp}(\mathscr{L}_T)}\sup_{z\in (\mathbb{R}^d\times T)\cup\{\emptyset\}} \mathbb{P}\left(R(x,m_x,\Xi^{\lbag z \rbag}+\delta_{(x,m_x)})\ge t \right)$$ satisfies that
$$
\tau(t)=0\mbox{ for some }t\in \mathbb{R}_+ \ \ \ (\mbox{resp. }\tau(t)\le C_1e^{-C_2t},\ \tau(t)\le C_1t^{-\beta} {~\mbox{for all }t\in \mathbb{R}_+})
$$
for some positive constants $C_1$ and $C_2$.
\end{defi}
For the functionals with input of restricted marked Poisson point process, we have the following counterpart of stabilization.
Note that the score function for the restricted input is not affected by points outside $\Gamma_\alpha$.
\begin{defi}\label{defi4r} (restricted case) We say the score function $\eta$ is {range-bound (resp. exponentially stabilizing, polynomially stabilizing of order $\beta>0$)} with respect to intensity $\lambda$ and a probability measure $\mathscr{L}_T$ on $T$ if for $\alpha\in \mathbb{R}_+$, $x\in \Gamma_\alpha$, and $z\in (\Gamma_\alpha \times T)\cup\{\emptyset\}$, almost all realizations $\mathscr{X}$ of the homogeneous marked Poisson point process $\Xi\sim\mathscr{P}_{\lambda,\mathscr{L}_T}$, there exists a $\bar{R}:=\bar{R}(x,\alpha):=\bar{R}(x,m_x,\alpha,\mathscr{X}^{\lbag z \rbag})\in(0,\infty)$ (a radius of stabilization), such that for all locally finite $\mathscr{Y}\subset (\Gamma_\alpha\backslash B(x,R))\times T$, we have
\begin{align}
&\eta\left(\left(x,m_x\right), \left[\mathscr{X}_{\Gamma_\alpha}^{{\lbag z \rbag}}\cap\left(B(x,\bar{R})\times T\right)\right]\cup \mathscr{Y},\Gamma_\alpha \right)\nonumber\\
=&\eta\left(\left(x,m_x\right),\mathscr{X}_{\Gamma_\alpha}^{{\lbag z \rbag}}\cap\left(B(x,\bar{R})\times T\right) ,\Gamma_\alpha\right)\label{defi4.1}
\end{align}
and the tail probability $$\bar{\tau}(t):=\sup_{x\in \mathbb{R}^d,m_x\in \rm{supp}(\mathscr{L}_T),\alpha\in \mathbb{R}_+}\sup_{z\in (\Gamma_\alpha\times T)\cup\{\emptyset\}} \mathbb{P}\left(\bar{R}(x,m_x,\alpha, \Xi^{{\lbag z \rbag}}+\delta_{(x,m_x)})\ge t \right)$$ satisfies that
\begin{equation*}
\bar{\tau}(t)=0\mbox{ for some }t\in \mathbb{R}_+ \ \ \ (\mbox{resp. }\bar{\tau}(t)\le C_1e^{-C_2t},\ \bar{\tau}(t)\le C_1t^{-\beta} {~\mbox{for all }t\in \mathbb{R}_+})
\end{equation*}
for some positive constants $C_1$ and $C_2$.
\end{defi}
\noindent{\it A2.2 Translation Invariance}
We write $d(x,A):=\inf\{d(x,y);~y\in A\}$, $A\pm B:=\{x\pm y; \ x\in A,\ y\in B\}$ for $x\in \mathbb{R}^d$ and $A,B\in\mathscr{B}\(\mathbb{R}^d\)$ and define the shift operator as $\Xi^x(\cdot\times D):=\Xi((\cdot+x)\times D)$ for all $x\in \mathbb{R}^d$, $D\in \mathscr{T}$.
{\it A2.2.1 Unrestricted Case:}
\begin{defi}\label{invar} The score function $\eta$ is {\it translation invariant} if for all locally finite configuration $\mathscr{X}$ and $x,y\in \mathbb{R}^d$ and $m\in T$,
{$\eta((x+y,m),\mathscr{X})=\eta((x,m),\mathscr{X}^{y})$.}
\end{defi}
{\it A2.2.2 Restricted Case:}
As a translation may send a configuration to outside of $\Gamma_\alpha$ resulting in a completely different configuration inside $\Gamma_\alpha$, it is necessary to focus on the part that affects the score function, therefore,
we expect the score function to take the same value for two configurations if the parts within their stabilising radii are completely inside $\Gamma_\alpha$ and one is a translation of the other. More precisely, we have the following definition.
\begin{defi}\label{traninvres0} A stabilizing score function is called {\it translation invariant} if for any $\alpha>0$, $x\in\Gamma_\alpha$
and $\mathscr{X}\in \bm{C}_{\mathbb{R}^d\times T}$ such that $\bar{R}(x,m,\alpha,\mathscr{X})\le d(x,\partial\Gamma_\alpha)$, {where $\partial A$ stands for the boundary of $A$}, then $\eta\(x,m,\mathscr{X},\Gamma_\alpha\)=\eta\(x',m,\mathscr{X}',\Gamma_{\alpha'}\)$ and $\bar{R}(x',m,\alpha',\mathscr{X}')=\bar{R}(x,m,\alpha,\mathscr{X})$ for all $\alpha'>0$, $x'\in\Gamma_{\alpha'}$ and $\mathscr{X}'\in \bm{C}_{\mathbb{R}^d\times T}$ such that $\bar{R}(x,m,\alpha,\mathscr{X})\le d(x',\partial\Gamma_{\alpha'})$ and $\(\mathscr{X}'_{B(x',\bar{R}(x,m,\alpha,\mathscr{X}) )}\)^{x'}=\(\mathscr{X}_{B(x,\bar{R}(x,m,\alpha,\mathscr{X}) )}\)^{x}$.
\end{defi}
Noting that there is a tacit assumption of consistency in Definition \ref{traninvres0}, which implies that if $\eta$ is translation invariant in Definition \ref{traninvres0}, there exists a $\bar{g}: \bm{C}_{\mathbb{R}^d\times T}\rightarrow \mathbb{R}$ such that $$\lim_{\alpha\rightarrow \infty}\eta\((0,m),\mathscr{X},\Gamma_\alpha\)=\bar{g}\(\mathscr{X}+\delta_{(0,m)}\)$$ for $\mathscr{L}_{T}$ almost sure $m\in T$ and almost all realizations $\mathscr{X}$ of the homogeneous marked Poisson point process $\Xi\sim\mathscr{P}_{\lambda,\mathscr{L}_T}$. Furthermore, we can see that for each score function $\eta$ satisfying the translation invariance in Definition \ref{traninvres0}, there exists a score function for the unrestricted case by setting $\bar{\eta}((x,m),\mathscr{X}):=\bar{g}(\mathscr{X}^{x})\mathbf{1}_{(x,m)\in \mathscr{X}}$ and writing the radii of stabilization in the sense of Definition~\ref{defi4} as $R$. From the construction, $\bar{\eta}$ is range bound (resp. exponentially stabilizing, polynomially stabilizing of order $\beta>0$) in the sense of Definition~\ref{defi4} if $\eta$ is range bound (resp. exponentially stabilizing, polynomially stabilizing of order $\beta>0$) in the sense of Definition~\ref{defi4r}. Moreover, if $B(x, R(x)) \subset \Gamma_\alpha$, then $\bar{R}(x, \alpha)= R(x)$ and if $B(x, R(x)) \not\subset \Gamma_\alpha$, then $\bar{R}(x,\alpha)>d(x,\partial \Gamma_\alpha)$, but there is no definite relationship between $\bar{R}$ and $R$.
\noindent{\it A2.3 Moment condition}
{\it Unrestricted Case:} The score function $\eta$ is said to satisfy the $k$th moment condition if
\begin{equation}\label{thm2.1}
\mathbb{E}\left|\eta\left(({\bf 0},M_3),\Xi+a_1\delta_{(x_1,M_1)}+a_2\delta_{(x_2,M_2)}+\delta_{({\bf 0},M_3)}\right)\right|^k\le C
\end{equation}
for some positive constant $C$ for all $a_i\in \{0,1\}$, distinct $x_i\in \mathbb{R}^d$, $i\in\{1,2\}$, and ${i.i.d.}$ random elements $M_1$, $M_2$, $M_3$ that are independent of $\Xi$ following the distribution $\mathscr{L}_{T}$.
{\it Restricted Case:} For restricted score functions, $\eta$ is said to satisfy the $k$th moment condition if there exists a positive constant $C$ independent of $\alpha$ such that
\begin{equation}\label{thm2.1r}
\mathbb{E}\left|\eta\left((x_3,M_3),\Xi_{\Gamma_\alpha}+a_1\delta_{(x_1,M_1)}+a_2\delta_{(x_2,M_2)}+\delta_{(x_3,M_3)}\right)\right|^k\le C
\end{equation}
for all $a_1,a_2\in \{0,1\}$, distinct $x_1,x_2,x_3\in \Gamma_\alpha$, and ${i.i.d.}$ random elements $M_1$, $M_2$, $M_3$ that are independent of $\Xi$ following the distribution $\mathscr{L}_{T}$. From the construction, if $\eta$ is stabilizing, then $\bar{\eta}$ satisfies the moment condition of the same order in the sense of \Ref{thm2.1}.
\noindent{\it A2.4 Non-singularity}
{\it Unrestricted Case:} The score function is said to be {\it non-singular} if
\begin{equation}
\mathscr{L}\left(\left.\sum_{(x,m)\in \Xi}\eta\((x, m),\Xi\)\mathbf{1}_{d(x, N_0)<R(x)}\right|\sigma(\Xi_{N_0^c})\right)\label{non-sin}\end{equation}
has a positive probability to be non-singular for some bounded set $N_0$. That is, the sum of the values of the score function that affected by the points in $N_0$ is non-singular.
{\it Restricted Case:} We define the non-singularity when the score function is stabilizing. The score function $\eta$ for restricted input is said to be {\it non-singular} if it is stabilizing and the corresponding $\bar{\eta}$ satisfies that
{ \begin{equation}
\mathscr{L}\left(\left.\sum_{(x,m)\in \Xi}\bar{\eta}\((x, m),\Xi\)\mathbf{1}_{d(x, N_0)<R(x)}\right|\sigma(\Xi_{N_0^c})\right)
\label{non-sinr}
\end{equation} }
has a positive probability to be non-singular {for some bounded set $N_0$}.
The main result for $W_\alpha$ (unrestricted case) is summarized below.
\begin{thm}\label{thm2} Let $Z_\alpha\sim N(\E W_{\alpha}, {\rm Var}(W_{\alpha}))$. Assume that the score function $\eta$ is translation invariant in Definition~\ref{invar} and non-singular~\Ref{non-sin}.
\begin{description}
\item{(i)} If $\eta$ is range-bound as in Definition~\ref{defi4} and satisfies the third moment condition~\Ref{thm2.1}, then
$$d_{TV}(W_\alpha,Z_\alpha)\le O\(\alpha^{-\frac{1}{2}}\).$$
\item{(ii)} If $\eta$ is exponentially stabilizing as in Definition~\ref{defi4} and satisfies the third moment condition~\Ref{thm2.1}, then
$$d_{TV}(W_\alpha,Z_\alpha)\le O\(\alpha^{-\frac{1}{2}}\ln(\alpha)^{\frac{5d}{2}}\).$$
\item{(iii)} If $\eta$ is polynomially stabilizing as in Definition~\ref{defi4} with parameter $\beta>\frac{(15k-14)d}{k-2}$ and satisfies the $k'$-th moment condition ~\Ref{thm2.1} with $k'>k\ge 3$, then
\begin{align*}
d_{TV}\(W_\alpha,Z_\alpha\)&\le O\(\alpha^{-\frac{\beta(k-2)[\beta(k-2)-d(15k-14)]}{(k\beta-2\beta-dk)(5dk+2\beta k-4\beta)}}\).
\end{align*}
\end{description}
\end{thm}
When approximation error is measured in terms of the Kolmogorov distance, the distributions of $W_\alpha$ and $\bar{W}_\alpha$ are often close for large $\alpha$. However, in terms of the total variation distance, one can not infer the accuracy of normal approximation of $W_\alpha$ by taking limit of that for $\bar{W}_\alpha$. For this reason, we need to adapt the conditions accordingly and tackle $\bar{W}_\alpha$ separately.
We state the main result for $\bar{W}_\alpha$ (restricted case) in the following theorem.
\begin{thm}\label{thm2a} Let $\bar{Z}_\alpha\sim N(\E\bar{W}_\alpha,{\rm Var}(\bar{W}_\alpha))$. Assume that $\eta$ is translation invariant in Definition~\ref{traninvres0} and non-singular~\Ref{non-sinr}.
\begin{description}
\item{(i)} If $\eta$ is range-bound as in Definition~\ref{defi4r} and satisfies the third moment condition~\Ref{thm2.1r}, then
$$d_{TV}(\bar{W}_\alpha,\bar{Z}_\alpha)\le O\(\alpha^{-\frac{1}{2}}\).$$
\item{(ii)} If $\eta$ is exponentially stabilizing as in Definition~\ref{defi4r} and satisfies the third moment condition~\Ref{thm2.1r}, then
$$d_{TV}(\bar{W}_\alpha,\bar{Z}_\alpha)\le O\(\alpha^{-\frac{1}{2}}\ln(\alpha)^{\frac{5d}{2}}\).$$
\item{(iii)} If $\eta$ is polynomially stabilizing as in Definition~\ref{defi4r} with parameter $\beta>\frac{(15k-14)d}{k-2}$ and satisfies the $k'$-th moment condition~\Ref{thm2.1r} with $k'>k\ge 3$, then
$$d_{TV}\(\bar{W}_\alpha,\bar{Z}_\alpha\)\le O\(\alpha^{-\frac{\beta(k-2)[\beta(k-2)-d(15k-14)]}{(k\beta-2\beta-dk)(5dk+2\beta k-4\beta)}}\).
$$
\end{description}
\end{thm}
\section{Applications}\label{Applications}
{\rm~Our main result can be applied to a wide range of geometric probability problems, including {normal approximation of }functionals of $k$-nearest neighbors graph, Voronoi graph, sphere of influence graph, Delaunay triangulation, Gabriel graph and relative neighborhood graph. To keep our article in a reasonable size, we only show the $k$-nearest neighbors graph and the Voronoi graph in details. We can see that many functionals of the graphs such as total edge length satisfy the conditions of the main theorems naturally and the ideas for verifying these conditions are similar. For the ease of reading, we briefly introduce these graphs below, more details can be found in \cite{Devroye88, T82}. }
{\rm Let $\mathscr{X}\subset {\mathbb{R}^d}$ be a locally finite point set:}
\begin{description}
\item{(i)} {\it $k$-nearest neighbors graph}~{\rm~The $k$-nearest neighbors graph $NG(\mathscr{X})$ is the graph obtained by including $\{x,y\}$ as an edge whenever $y$ is one of the $k$ points nearest to $x$ or $x$ is one of the $k$ points nearest to $y$. A variant of the $NG(\mathscr{X})$ which has been considered in the literature is the {\it directed} graph $NG'\(\mathscr{X}\)$, which is constructed by inserting a directed edge $(x,y)$ if $y$ is one of the $k$ nearest neighbors of $x$.}
\item{(ii)} {\it Voronoi tessellation}~{\rm We enumerate the points in $\mathscr{X}$ as $\{x_1,~x_2,\dots\}$, denote the locus of points in $\mathbb{R}^d$ closer to $x_i$ than to any other points in $\mathscr{X}$ by $C(x_i):=C(x_i,\mathscr{X})$ for all $i \in \mathbb{N}$. We can see that $C(x_i)$ is the intersection of half-planes and when the point set $\mathscr{X}$ has $n<\infty$ points, $C(x_i)$'s is a convex polygonal region with at most $n-1$ sides for $i\le n$. The cells $C(x_i)$ form a partition of $\mathbb{R}^d$, the partition is called Voronoi tessellation and the points in $\mathscr{X}$ are usually called Voronoi generators. }
\item{(iii)} {\it Delaunay triangulation}~{\rm~The Delaunay triangulation graph puts an edge between two points in $\mathscr{X}$ if these points are centers of adjacent Voronoi cells, which is a dual to the Voronoi tessellation. }
\item{(iv)} {\it Gabriel graph}~{\rm~Gabriel graph puts an edge between two points $x$ and $y$ in $\mathscr{X}$ if the ball centered at the middle point $\frac{x+y}{2}$ with radius $\|\frac{x-y}{2}\|$ does not contain any other points in $\mathscr{X}$. We can see that Gabriel graph is a subgraph of Delaunay triangulation graph. }
\item{(v)} {\it Relative neighborhood graph}~{\rm~Relative neighborhood graph puts an edge between two points $x$ and $y$ in $\mathscr{X}$ if $B(x,\|x-y\|)\cap B(y,\|x-y\|)\cap\mathscr{X} =\emptyset$, i.e., the loon between $x$ and $y$ does not contain any other points in $\mathscr{X}$. This graph is a subgraph of Gabriel graph, so is also a subgraph of Delaunay triangulation graph. }
\item{(vi)} {\it Sphere of influence graph}~{\rm~The sphere of influence graph of a locally finite point set $\mathscr{X}\subset {\mathbb{R}^d}$ is the graph obtained by including $\{x,y\}$ as an edge whenever $x,y\in\mathscr{X}$, $\|x-y\|\le \|x-N(x,\mathscr{X})\|+\|y-N(y, \mathscr{X})\|$, where for $z\in\mathscr{X}$, $N(z,\mathscr{X})$ is the nearest point of $z$ in $\mathscr{X}$. That is, for every point $z\in \mathscr{X}$, we draw a circle with center $z$ and radius being the distance between $z$ and its nearest point in $\mathscr{X}$, then two points are connected if the circles centered at two points $x,y$ intersect. }
\end{description}
\subsection{The total edge length of $k$-nearest neighbors graph}\label{knear}
\begin{thm}
If $\Xi$ is a homogeneous Poisson point process, then the total edge length $\bar{W}_\alpha$ (resp. $\bar{W}_\alpha'$) of $NG\(\Xi_{\Gamma_{\alpha}}\)$ (resp. $NG'\(\Xi_{\Gamma_{\alpha}}\)$) satisfies
$$d_{TV}\(\bar{W}_\alpha, {\bar{Z}}_\alpha\)\le O\(\alpha^{-\frac{1}{2}}\ln(\alpha)^{\frac{5d}{2}}\)~\(\mbox{resp.~}d_{TV}\(\bar{W}_\alpha', {\bar{Z}}_\alpha'\)\le O\(\alpha^{-\frac{1}{2}}\ln(\alpha)^{\frac{5d}{2}}\)\),$$ where ${\bar{Z}}_\alpha$ (resp. ${\bar{Z}}_\alpha'$) is a normal random variable with the same mean and variance as those of $\bar{W}_\alpha$ (resp. $\bar{W}_\alpha'$).
\end{thm}
\noindent{\it Proof.~} We only show the claim for the total edge length of $NG\(\Xi_{\Gamma_{\alpha}}\)$ since $NG'\(\Xi_{\Gamma_{\alpha}}\)$ can be handled with the same idea. The score function in this case is $${\eta\(x,\mathscr{X},\Gamma_\alpha\):=\frac{1}{2}\sum_{y\in \mathscr{X}_{\Gamma_\alpha}}\|y-x\|\mathbf{1}_{\{(x,y)\in NG(\mathscr{X}_{\Gamma_\alpha})\}},}$$
\begin{wrapfigure}{r}{6cm}
\begin{tikzpicture}[scale=3]
\filldraw (0,0) circle (1pt);
\foreach \x in {15,75,135,195,255,315}
\draw[] (0,0) -- (\x:1);
\draw[] (-0.5657,0.5657) -- (135:1);
\node at (45:0.55) {$T_i(t)$};
\draw[latex-latex] (135:0.045) -- (135:0.79);
\draw (0,0) circle (0.8);
\node[fill=white,inner sep=2pt] at (135:0.45) {{$t$}};
\node[fill=white,inner sep=1pt] at (219:0.13) {$x$};
\draw[dashed] (-1,0)--(1,0) node[right]{};
\draw[dashed] (0,-1)--(0,1) node[above]{};
\end{tikzpicture}
\caption{$k$-nearest: stabilization}
\label{kns}
\end{wrapfigure}
which is clearly translation invariant.
To apply Theorem~\ref{thm2a}, we need to check the moment condition \Ref{thm2.1r}, non-singularity \Ref{non-sinr} and stabilizing condition as in Definition~\ref{defi4r}. For simplicity, we show these conditions in two dimensional case and the argument can be easily extended to $\mathbb{R}^d$.
\begin{wrapfigure}{l}{9.5cm}
\begin{tikzpicture}[scale=1.9]
\coordinate (O) at (0,0);
\coordinate (A) at (4.4825,0);
\coordinate (B) at (4.8293,1.2940);
\draw (O)--(A)--(B)--cycle;
\draw [decorate, decoration={brace,amplitude=8pt,raise=1pt}] (0,0) -- (4.8293,1.2940) node [midway, anchor=south west, yshift=-1mm, xshift=-6mm, outer sep=10pt,font=] {$t$};
\tkzMarkAngle[fill= orange,size=0.65cm,
opacity=.4](A,O,B)
\tkzLabelAngle[pos = 1.0](A,O,B){$\pi/12$}
\tkzMarkAngle[fill= orange,size=0.4cm,
opacity=.4](O,B,A)
\tkzLabelAngle[pos = 0.6](O,B,A){$\pi/3$}
\end{tikzpicture}
\caption{$k$-nearest: $A_t$}
\label{kntriangle}
\end{wrapfigure}We start with the exponential stabilization and fix $\alpha>0$ and $x\in \Gamma_\alpha$. Referring to Figure~\ref{kns}, for each $t>0$, we construct six disjoint sectors of the same size $T_j(t)$, $1\le j\le 6$, with $x$ as the centre and angle $\frac{\pi}{3}$. In consideration of edge effects near the boundary of $\Gamma_\alpha$, the sectors are rotated around $x$ such that all straight edges of the sectors have a minimal angle $\pi/12$ with respect to the edges of $\Gamma_\alpha$. It is clear that $T_j(t)\subset T_j(u)$ for all $0<t<u$. Set $T_{j}(\infty)=\cup_{t>0}T_j(t)$ for $1\le j\le 6$, then from the property of the Poisson point process, there are infinitely many points in $\Xi\cap T_{j}(\infty)$ for all $j$ a.s. Let $|A|$ denote the cardinality of the set $A$ and define
\begin{equation}t_{x,\alpha}=\inf\{t:\ |T_j(t)\cap \Gamma_\alpha\cap \Xi|\ge k+1\mbox { or }T_j(t)\cap \Gamma_\alpha=T_j(\infty)\cap \Gamma_\alpha,\ 1\le j\le 6\}\label{nearest1}
\end{equation}
and $\bar{R}\(x,\alpha\)=3t_{x,\alpha}$. We show that $\bar{R}$ is a radius of stabilization and its tail distribution can be bounded by an exponentially decaying function independent of $\alpha$ and $x$. For the the radius of stabilization, there are two cases to consider. The first case is that none of $T_j(t_{x,\alpha})\cap \Gamma_\alpha\cap\Xi$, $1\le j\le 6$, contains at least $k+1$ points, thus
$B(x, \bar{R}\(x,\alpha\))\supset \Gamma_\alpha$ and \Ref{defi4.1} is obvious. The second case is {at least }one of $T_j(t_{x,\alpha})\cap \Gamma_\alpha\cap\Xi$, $1\le j\le 6$, contains at least $k+1$ points, which means that the $k$ nearest neighbors $\{x_1,\dots,x_k\}$ of $x$ are in $B\(x,t_{x,\alpha}\)$. {If a point $y\in \Gamma_\alpha\backslash B(x,t_{x,\alpha})$, then $y\in \Gamma_\alpha\cap (T_j(\infty)\backslash T_j(t_{x,\alpha}))$ for some $j$. Since $\Gamma_\alpha\cap (T_j(\infty)\backslash T_j(t_{x,\alpha}))$ is non-empty, $T_j(t_{x,\alpha})$ contains at least $k+1$ points $\{y_1,\dots, y_{k+1}\}$ and $d(x,y)>d(y_i,y)$ for all $i\le k+1$, then $y$ cannot have $x$ as one of its $k-$nearest neighbors. This ensures that all points having $x$ as one of their $k-$nearest neighbors are in $B(x, t_{x,\alpha})$. Noting that the diameter of $B\(x,t_{x,\alpha}\)$ is $2t_{x,\alpha}$ and there are at least $k+1$ points in $B\(x,t_{x,\alpha}\)$, we can see that whether a point $y$ in $B\(x,t_{x,\alpha}\)$ having $x$ as one of its $k-$nearest neighbors is entirely determined by $\Xi\cap B(y,2t_{x,\alpha})\subset \Xi\cap B(x,3t_{x,\alpha})$. This guarantees that $\eta\(x,\mathscr{X},\Gamma_\alpha\)$ is $\Xi_{B(x,3t_{x,\alpha})}$ measurable and $\bar{R}\(x,\alpha\)$ is a radius of stabilization. For the tail distribution of $\bar{R}$, referring to Figure~\ref{kntriangle},
we consider the number of points of $\Xi$ falling into a triangle $A_t$ as a result of a sector being chopped off by the edge of $\Gamma_\alpha$. This is the worst situation for capturing the number of points by one sector. A routine trigonometry calculation gives that the area of $A_t$ is at least $0.116t^2$. Define $\tau:=\inf\{t:\ |\Xi\cap A_t|\ge k+1\}$, then
\begin{equation}\Pro\(\bar{R}\(x,\alpha\)>t\)\le 6\Pro\(\tau>t/3\)\le 6e^{-0.116\lambda (t/3)^2}\sum_{i=0}^k\frac{\(0.116\lambda (t/3)^2\)^i}{i!},\ t>0,\label{nearest0}\end{equation}
which implies the exponential stabilization.
The non-singularity \Ref{non-sinr} can be proved through the corresponding unrestricted score function $\bar{\eta}(x,\mathscr{X})=\frac{1}{2}\sum_{y\in \mathscr{X}}\|y-x\|\mathbf{1}_{\{(x,y)\in NG(\mathscr{X})\}}.$ Referring to Figure~\ref{knn}, we take $N_0:=B(0,{0.5})$, observe that $\partial B(0,6)$ can be covered by finitely many $B(x,3)$ with $\|x\|=5$ and write the centers of these balls as $x_1$, $\dots$, $x_n$ (in two dimensional case, $n=5$). Let
$E$ be the event that $|B(x_i,1)\cap \Xi|\ge k+1$ for all $1\le i\le n$, $|\(B(0,{1})\backslash B(0,{0.5})\)\cap\Xi|=k$ and $\Xi\cap \(B(0,6)\backslash \(\cup_{i\le n}B(x_i,1)\cup B(0,{1})\)\)$ is empty, then $E$ is $\Xi_{N_0^c}$ measurable and $\Pro(E)>0$. Conditional on $E$, we we can see that $E_1:=\{|\Xi\cap B(0,{0.5})|=1\}$
\begin{wrapfigure}{r}{6cm}
\begin{tikzpicture}[scale=0.4]
\draw[->](-7,0)--(7,0);
\draw[->](0,-7)--(0,7);
\draw[-](0,0)--(-3,6.5);
\node[label={[black]0:$y$}] at (-3.25,6.5) {};
\draw (0,0) circle (1);
\draw (0,0) circle (0.5);
\draw (0,5) circle (1);
\fill (0,5) circle (2pt);
\node[label={[black]0:$x$}] at (-0.4,6) {};
\node[label={[black]0:$2$}] at (-0.4,3) {};
\draw (0,5) circle (3);
\draw[red] (2.99332590942,5.2) arc (60.07356513:119.92643487:6);
\draw[black] (-2.99332590942,5.2) arc (119.92643487:420.07356513:6);
\end{tikzpicture}
\caption{$k$-nearest: non-singular condition}
\label{knn}
\end{wrapfigure}
satisfies $\Pro(E_1|E)>0$ and on $E_1$, all summands in \Ref{non-sinr} that are random are those involving the point of $\Xi_{B(0,{0.5})}$ and we now establish that these random score functions are entirely determined by $\Xi_{B(0,{1})}$. As a matter of fact, any point in $\Xi\cap B(x_i,1)$, $1\le i\le n$, has $k$ nearest points with distances no larger than $2$, so points in $\Xi\cap B(0,{1})$ cannot be $k$ nearest points to points in $\Xi\cap B(x_i,1)$. For any point $y\in \Xi\cap B(0,6)^c$, the line between $0$ and $y$ intersects $\partial B(0,6)$ at $y'$ which is in $B(x_{i_0},3)$ for some $1\le i_0\le n$, so the distances between $y$ and points in $\Xi\cap B(x_{i_0},3)$ are at most $\|y-y'\|+4=\|y\|-2$, the distances between $y$ and points in $\Xi\cap B(0,{1})$ are at least $\|y\|-1$, which ensures points of $\Xi\cap B(0,6)^c$ cannot have points in $\Xi\cap B(0,{1})$ as their $k$ nearest neighbors. {On the other hand, on $E\cap E_1$, there are $k+1$ points in $\Xi\cap B(0,1)$, for any point $x\in B(0,1)$, since the distances between $x$ and other points in $B(0,1)$ are less than $2$, the points outside $B(x,2)\subset B(0,3)$ will not be $k-$nearest points of $x$, $\bar{\eta}(x,\Xi)=\frac{1}{2}\sum_{y\in \(\Xi\cap B(0,1)\)\backslash\{x\}}\|x-y\|$. }Hence, given $E$, all random score functions contributing in the sum of \Ref{non-sinr} are those completely determined by $\Xi_{B(0,2)}$, giving
$$\mathbf{1}_{E_1}\sum_{\substack{x\in\Xi\mbox{\scriptsize{ such that }}\\
\bar{\eta}\(x,\Xi\)\mbox{\scriptsize{ is random given}~}\Xi_{N_0^c}}}\bar{\eta}\(x,\Xi\)\mathbf{1}_{d(x, N_0)<R(x)}=\mathbf{1}_{E_1}\left\{\sum_{\substack{y\in \Xi\cap B(0,{0.5}),\\ x \in \(B(0,{1})\backslash B(0,{0.5})\)\cap\Xi}}\|x-y\|+X\right\},$$
where $X$ is $\sigma(\Xi_{N_0^c})$ measurable.
Since this is a continuous function of $y\in \Xi\cap B(0,{0.5})$, the non-singularity \Ref{non-sinr} follows.
For the moment condition \Ref{thm2.1r}, recalling the definition of $t_{x,\alpha}$ in \Ref{nearest1}, we replace $x$ with $x_3$ to get $t_{x_3,\alpha}$. We now establish that
\begin{equation}\eta\(x_3,\Xi_{\Gamma_{\alpha}}+a_1\delta_{x_1}+a_2\delta_{x_2}+\delta_{x_3}\)\le 3.5kt_{x_3,\alpha}.
\label{nearest2}
\end{equation}
In fact, the $k$ nearest neighbors to $x_3$ have the contribution of the total edge length $\le \frac12 kt_{x_3,\alpha}$. On the other hand, for $1\le j\le 6$, each point in $\(\Xi_{\Gamma_{\alpha}}+a_1\delta_{x_1}+a_2\delta_{x_2}\)\cap T_{j}\(t_{x_3,\alpha}\)$ may take $x_3$ as its $k$ nearest neighbor, which contributes to the total edge length $\le \frac12 t_{x_3,\alpha}$. As there are six sectors $\(\Xi_{\Gamma_{\alpha}}+a_1\delta_{x_1}+a_2\delta_{x_2}\)\cap T_{j}\(t_{x_3,\alpha}\)$, $1\le j\le 6$, and each sector has no more than $k$ points with $x_3$ as their $k$ nearest neighbors, the contribution of the total edge length from this part is bounded by $3kt_{x_3,\alpha}$. By adding up these two bounds, we obtain \Ref{nearest2}.
Finally, we combine \Ref{nearest2} and \Ref{nearest0} to get
\begin{align*}
&\mathbb{E}\left\{\eta\(x_3,\Xi_{\Gamma_{\alpha}}+a_1\delta_{x_1}+a_2\delta_{x_2}+\delta_{x_3}\)^3\right\}
\le 42.875k^3\mathbb{E}\left\{t_{x_3,\alpha}^3\right\}\le C <\infty,
\end{align*}
and the proof is completed by applying Theorem~\ref{thm2a}.
\qed
\subsection{ The total edge length of Voronoi tessellation}
\begin{wrapfigure}{r}{0.35\textwidth}
\begin{center}
\includegraphics[trim = 60mm 90mm 60mm 90mm, clip,width=0.35\textwidth]{Voronoi-tessellation}
\caption{Voronoi tessellation}
\label{figure1}
\end{center}
\end{wrapfigure}
Consider a finite point set $\mathscr{X}\subset \Gamma_\alpha$, the Voronoi tessellation in $\Gamma_\alpha$ generated by $\mathscr{X}$ is the partition formed by cells $C(x_i,\mathscr{X})\cap \Gamma_\alpha$, see Figure~\ref{figure1}. We write the graph of this tessellation as $V(\mathscr{X},\alpha)$ and the total edge length of $V(\mathscr{X},\alpha)$ as $\mathscr{V}(\mathscr{X},\alpha)$.
\begin{thm}\label{exthm1}
If $\Xi$ is a homogeneous Poisson point process, then $$d_{TV}\(\mathscr{V}(\Xi_{\Gamma_{\alpha}},\alpha), {\bar{Z}}_\alpha\)\le O\(\alpha^{-\frac{1}{2}}\ln(\alpha)^{\frac{5d}{2}}\),$$ where ${\bar{Z}}_\alpha$ is a normal random variable with the same mean and variance as those of $\mathscr{V}(\mathscr{X},\alpha)$.
\end{thm}
\noindent{\it Proof.} Before going into details, we observe that $$\mathscr{V}\(\Xi_{\Gamma_{\alpha}},\alpha\)=l(\partial \Gamma_\alpha)+\sum_{\{x,y\}\subset \Xi_{\Gamma_{\alpha}},x\neq y}l\(\partial C(x, \Xi_{\Gamma_{\alpha}})\cap \partial C(y, \Xi_{\Gamma_{\alpha}})\),$$ where $l(\cdot)$ is the volume of a set in dimension $d-1$. We restrict our attention to Voronoi tessellations of random point sets in $\mathbb{R}^2$ and, with notational complexity, the approach also works in $\mathbb{R}^d$. Because $l(\partial \Gamma_\alpha)=4\alpha^{\frac{1}{2}}$ is a constant, by removing this constant, we have $\mathscr{V}'\(\Xi_{\Gamma_{\alpha}},\alpha\):=\mathscr{V}\(\Xi_{\Gamma_{\alpha}},\alpha\)-4\alpha^{\frac{1}{2}}$ and $d_{TV}\(\mathscr{V}\(\Xi_{\Gamma_{\alpha}}, \alpha\),{\bar{Z}}_\alpha\)=d_{TV}\(\mathscr{V}'\(\Xi_{\Gamma_{\alpha}},\alpha\),{\bar{Z}}_\alpha'\)$, where ${\bar{Z}}_\alpha'$ is a normal random variable having the same mean and variance as those of $\mathscr{V}'\(\Xi_{\Gamma_{\alpha}},\alpha\)$. We can set the score function corresponding to $\mathscr{V}'$ as $${\eta(x,\mathscr{X},\Gamma_\alpha)=\frac{1}{2}\sum_{y\in \mathscr{X},~y\neq x}l\(\partial C(x,\mathscr{X})\cap \partial C(y, \mathscr{X})\)=\frac{1}{2}l\(\partial (C(x, \mathscr{X})\cap \Gamma_\alpha)\backslash (\partial \Gamma_\alpha)\)}$$ for all $x\in \mathscr{X}\subset \Gamma_\alpha$, i.e., $\eta(x,\mathscr{X},\Gamma_\alpha)$ is a half of the total length of edges of $C(x, \mathscr{X})\cap \Gamma_\alpha$ excluding the boundary of $\Gamma_\alpha$. The score function $\eta$ is clearly translation invariant, thus, to apply Theorem~\ref{thm2a}, we need to verify stabilization as in Definition~\ref{defi4r},
moment condition~\Ref{thm2.1r} and non-singularity~\Ref{non-sinr}.
We start from showing that the score function is exponentially stabilizing. Referring to Figure~\ref{vdr}, similar to Section~\ref{knear}, we construct six disjoint equilateral triangles $T_{xj}(t)$, $1\le j\le 6$, such that $x$ is a vertex of these triangles and the triangles are rotated so that all edges with $x$ as a vertex have a minimal angle $\pi/12$ against the edges of $\Gamma_\alpha$. Let $T_{xj}(\infty)=\cup_{t\ge 0} T_{xj}(t)$, $1\le j\le 6$, then $\cup_{1\le j\le 6}T_{xj}(\infty)=\mathbb{R}^2$.
Define $$R_{xj}:=R_{xj}\(x,\alpha,\Xi_{\Gamma_\alpha}\):=\inf\{t:\ T_{xj}(t)\cap \Xi_{\Gamma_\alpha}\ne \emptyset\mbox{ or }T_{xj}(t)\cap\Gamma_\alpha=T_{xj}(\infty)\cap\Gamma_\alpha\}$$
and $$R_{x0}:=R_{x0}\(x,\alpha,\Xi_{\Gamma_\alpha}\):=\max_{1\le j\le 6}R_{xj}\(x,\alpha,\Xi_{\Gamma_\alpha}\).$$
We note that there is a minor issue of the counterpart of $R_{x0}$ defined in \cite{MY99} when $x$ is close to the corners of $\Gamma_\alpha$.
We now show that $\bar{R}(x,\alpha):=3R_{x0}(x,\alpha,\Xi_{\Gamma_\alpha})$ is a radius of stabilization. In fact, for any point $x'$ in $\Gamma_\alpha\backslash \overline{\left(\cup_{1\le j\le 6}T_{xj}(R_{x0})\right)}$, $x'$ is contained in a triangle $T_{xj_0}(\infty)\backslash\overline{T_{xj_0}(R_{x0})}$. This implies $T_{xj_0}(R_{x0})\cap \Xi_{\Gamma_\alpha}\ne\emptyset$, i.e., we can find a point $y\in T_{xj_0}(R_{x0})\cap \Xi_{\Gamma_\alpha}$ and the point $y$ satisfies $d(x',y)\le d(x,x')$, hence $x'\notin C(x,\Xi_{\Gamma_\alpha})\cap \Gamma_\alpha$, which ensures $C(x,\Xi_{\Gamma_\alpha})\subset \overline{\left(\cup_{1\le j\le 6}T_{xj}(R_0)\right)}$. Consequently, if a point $y$ in $\Xi_{\Gamma_\alpha}$ generates an edge of $C(X)\cap\Gamma_\alpha$, $d(x,y)\le 2R_0$ and $\bar{R}(x,\alpha)$ satisfies Definition~\ref{defi4r}. As in Section~\ref{knear}, we use $A_t$ in Figure~\ref{kntriangle} again to define $\tau:=\inf\{t:\ |\Xi\cap A_t|\ge 1\}$, then
\begin{equation}\Pro\(\bar{R}\(x\)>t\)\le 6\Pro\(\tau>t/3\)\le 6e^{-0.116\lambda (t/3)^2},\ t>0.\label{voronoi0}\end{equation}
This completes the proof of the exponential stabilization of $\eta$.
\begin{wrapfigure}{r}{6cm}
\begin{tikzpicture}[scale=3]
\newdimen\R
\R=0.8cm
\draw[yshift=0\R] (15:\R) \foreach \x in {15,75,135,...,315} {
-- (\x:\R)
} -- cycle (90:\R) node[above] {} ;
\filldraw (0,0) circle (1pt);
\foreach \x in {15,75,195,255,315}
\draw[] (0,0) -- (\x:1);
\draw[] (-0.5657,0.5657) -- (135:1);
\node at (45:0.3) {$N_0$};
\node at (45:0.8) {$T_{xi}(t)$};
\draw[latex-latex] (135:0.045) -- (135:0.79);
\draw (0,0) circle (0.2);
\node[fill=white,inner sep=2pt] at (135:0.45) {{$t$}};
\node[fill=white,inner sep=1pt] at (219:0.13) {$x$};
\draw[dashed] (-1,0)--(1,0) node[right]{};
\draw[dashed] (0,-1)--(0,1) node[above]{};
\end{tikzpicture}
\caption{Voronoi: stabilization}
\label{vdr}
\end{wrapfigure}
The non-singularity~\Ref{non-sinr} can be examined by using a non-restricted counterpart $\bar{\eta}$ of $\eta$, taking $N_0 =B(0,1)$ and filling the moat $B(0,4)\backslash B(0,3)$ with sufficiently dense points of $\Xi$ such that when $\Xi_{N_0^c}$ is fixed, the random score functions contributing to the sum of \Ref{non-sinr} are purely determined by a point in $\Xi\cap N_0$. More precisely, we cover the circle $\partial B(0,3)$ by disjoint squares with side length $\sqrt{2}/4$ and enumerate the squares as $S_i,\ 1\le i\le k$. Note that all the squares are contained in $B(0,4)\backslash B(0,2)$. Let $E=\cap_{1\le i\le k}\{|\Xi\cap(S_i)|\ge 1\}$, $E_1=\{|\Xi\cap N_0|=1\}$, then $E$ is $\sigma\(\Xi_{N_0^c}\)$ measurable, $\Pro(E)>0$ and $\Pro(E_1|E)>0$. Since the points in $\Xi\cap \(\cup_{i=1}^kS_i\)$ have neighbors within distance 1, for any $x\in N_0$, $T_{xj}(6)$ contains as least one point from $\Xi\cap \(B(0,4)\backslash B(0,2)\)$. As argued in the stabilization, points in $\Xi\cap \(B(0,12)^c\)$ do not affect the cell centered at $x\in N_0$, and by symmetry, $x\in N_0$ does not affect the Voronoi cells centered at points in $\Xi\cap \(B(0,12)^c\)$. This ensures that, conditional {on} $E$, all random score functions contributing in the sum of \Ref{non-sinr} are those completely determined by $\Xi_{N_0}$, giving
\begin{align*}&\mathbf{1}_{E_1}\sum_{\substack{x\in\Xi\mbox{\scriptsize{ such that }}\\
\bar{\eta}\(x,\Xi\)\mbox{\scriptsize{ is random given}~}\Xi_{N_0^c}}}\bar{\eta}\(x,\Xi\)\mathbf{1}_{d(x, N_0)<R(x)}\\
=&\mathbf{1}_{E_1}\left\{\sum_{x\in \Xi\cap B(0,12)}\bar{\eta}\(x,\Xi\)\mathbf{1}_{d(x, N_0)<R(x)}+X\right\},\end{align*}
where $X$ is $\sigma\(\Xi_{N_0^c}\)$ measurable. As $\mathbf{1}_{E_1}\bar{\eta}\(x,\Xi\)$ is an almost surely (in terms of the volume measure in $\mathbb{R}^2$) continuous function of $x\in \Xi\cap N_0$, the proof of non-singularity \Ref{non-sinr} is completed.
It remains to show the moment condition \Ref{thm2.1r}. In fact, as shown in {the} stabilizing property, we can see that $\bar{R}(x,\alpha)$ will not increase when adding points, so $C(x_3,\Xi_{\Gamma_\alpha}+a_1\delta_{x_1}+a_2\delta_{x_2}+\delta_{x_3})\cap \Gamma_\alpha\subset B(x, \bar{R}(x,\alpha))$, then the number of edges of $C(x_3,\Xi_{\Gamma_\alpha}+a_1\delta_{x_1}+a_2\delta_{x_2}+\delta_{x_3})\cap \Gamma_\alpha$ excluding those in the edge of $ \Gamma_\alpha$ is less than or equal to $(\Xi_{\Gamma_\alpha}+a_1\delta_{x_1}+a_2\delta_{x_2})\(B\(x,\bar{R}(x{,\alpha})\)\)\le\Xi_{\Gamma_\alpha}\(B\(x,\bar{R}(x{,\alpha})\)\)+2$ and each of them has length less than $2\bar{R}(x,\alpha)$. To this end, we observe that $\Xi$ restricted to outside of $\cup_{j=1}^6T_{xj}(R_{xj})$ is independent of $\bar{R}(x{,\alpha})$, hence
\begin{align*}\Xi_{\Gamma_\alpha}\(B\(x,\bar{R}(x{,\alpha})\)\)\le&\Xi_{\Gamma_\alpha}\(B\(x,\bar{R}(x{,\alpha})\)\backslash \(\cup_{j=1}^6T_{xj}(R_{xj})\)\)+6\\
\stackrel{\mbox{\scriptsize{ST}}}{\le}&\Xi'_{\Gamma_\alpha}\(B\(x,\bar{R}(x{,\alpha})\)\)+6,
\end{align*}
where $\stackrel{\mbox{\scriptsize{ST}}}{\le}$ stands for stochastically less than or equal to and $\Xi'$ is an independent copy of $\Xi$. Hence, using \Ref{voronoi0}, we obtain
\begin{align*}
&\mathbb{E}\(\eta\left(x_3,\Xi_{\Gamma_\alpha}+a_1\delta_{x_1}+a_2\delta_{x_2}+\delta_{x_3}\right)^3\)\\
\le &\mathbb{E}\(\(\Xi'\(B\({x_3},\bar{R}(x{,\alpha})\)\)+8\)^3\(2\bar{R}({x_3}{,\alpha})\)^3\)\\
\le &\int_0^\infty \sum_{i=0}^\infty (i+8)^3(2r)^3\frac{e^{-\lambda \pi r^2} (\lambda \pi r^2)^i}{i!}6e^{-0.116\lambda (t/3)^2}*\(0.116\lambda/9\)2rdr\\
\le& C<\infty,
\end{align*}
which ensures \Ref{thm2.1r}. The proof of Theorem~\ref{exthm1} is completed by using Theorem~\ref{thm2a}. \qed
{\subsection{Log volume estimation}}
Log volume estimation is an essential research topic in forest science and forest management \cite{C80,Li15}. This example demonstrates that, with the marks, our theorem can be used to provide an error estimate of normal approximation of the log volume distribution. To this end, it is reasonable to assume that in a given range $\Gamma_\alpha$ {of} {a} natural forest, the locations of trees form a Poisson point process $\bar{\Xi}$, and for $x\in \bar{\Xi}$, we can use a random mark $M_x\in{T}:=\{1,\dots,n\}$ to denote the species of the tree at position $x$, then $\Xi:=\sum_{x\in \bar{\Xi}}\delta_{(x,M_x)}$ is a marked Poisson point process with independent marks. The timber volume of a tree at $x$ is a combined result of the location, the species of the tree, the configuration of species of trees in a finite range around $x$ and some other random factors that can't be explained by the configuration of trees in the range. We write $\eta(\left(x,m\right), \Xi_{\Gamma_\alpha},\Gamma_\alpha)$ as the timber volume determined by the location $x$, the species $m$ and the configuration of trees, and ${\epsilon}_{x}$ as the adjusted timber volume at location $x$ due to unexplained random factors.
\begin{thm} Assume that $\eta$ is a non-negative bounded score function such that $$\eta(\left(x,m\right), \Xi_{\Gamma_\alpha},\Gamma_\alpha)=\eta(\left(x,m\right), \Xi_{\Gamma_\alpha\cap B(x,r)},\Gamma_\alpha)$$ for some positive constant $r$, $$\eta(\left(x,m\right), \Xi_{B(x,r)},\Gamma_{\alpha_1})=\eta(\left(x,m\right), \Xi_{B(x,r)},\Gamma_{\alpha_2})$$ for all $\alpha_1$ and $\alpha_2$ with $B(x,r)\subset \Gamma_{\alpha_1\wedge\alpha_2}$, $\eta$ is translation invariant in Definition~\ref{traninvres0}, ${\epsilon}_{x}$'s are ${i.i.d.}$ random variables with finite third moment and the positive part ${\epsilon}_x^+:={\epsilon}_x\vee 0$ is non-singular, and ${\epsilon}_{x}$'s are independent of the configuration $\Xi$, then the log volume of the range $\Gamma_\alpha$ can be represented as
$${\bar{W}}_{\alpha}:=\sum_{x\in \bar{\Xi}_{\Gamma_\alpha}}\[\(\eta(\left(x,m\right), \Xi_{\Gamma_\alpha},\Gamma_\alpha)+{\epsilon}_x\)\vee 0\right]$$
and it satisfies
$$d_{TV}\({\bar{W}}_\alpha,{\bar{Z}}_\alpha\)\le O\(\alpha^{-\frac{1}{2}}\),$$
where ${\bar{Z}}_\alpha$ is a normal random variable with the same mean and variance as those of ${\bar{W}}_\alpha$.
\end{thm}
\noindent{\it Proof.~}Before going into details, we first construct a new marked Poisson point process $\Xi':=\sum_{x\in \bar{\Xi}}\delta_{(x, (M_x, {\epsilon}_x))}$ with ${i.i.d.}$ marks $(M_x,{\epsilon}_x)\in T\times \mathbb{R}$ independent of the ground process $\bar{\Xi}'=\bar{\Xi}$ and incorporate ${\epsilon}_x$ into a new score function on $\Xi'$ as
\begin{align*}\eta'((x,{(m,{\epsilon})}),\Xi',\Gamma_\alpha):=& \eta'((x,(m,{\epsilon})),\Xi'_{\Gamma_\alpha},\Gamma_\alpha)\\
:=&\[\(\eta(\left(x,m\right), \Xi_{\Gamma_\alpha},\Gamma_\alpha)+{\epsilon}\)\vee 0\right]\mathbf{1}_{(x,(m,{\epsilon}))\in \Xi'_{\Gamma_\alpha}}.
\end{align*}
We can see that ${\bar{W}}_\alpha=\sum_{x\in \bar{\Xi}'_{\Gamma_\alpha}}\eta'((x,(m_x,{\epsilon}_x)),\Xi'_{\Gamma_\alpha},\Gamma_\alpha)$. The score function $\eta'$ is clearly translation invariant, thus, to apply Theorem~\ref{thm2a}, it is sufficient to verify that $\eta'$ is range-bound as in Definition~\ref{defi4r}, satisfies the moment condition \Ref{thm2.1r} and non-singularity \Ref{non-sinr}.
The range-bound property of the score function $\eta'$ is inherited from the range-bound property of $\eta$ with the same radius of stabilization $\bar{R}(x,\alpha):=r$, the moment condition \Ref{thm2.1r} is a direct consequence of the boundedness of $\eta$, the finite third moment of ${\epsilon}_x$ and the Minkowski inequality, hence it remains to show the non-singularity. To this end, we observe that the corresponding unrestricted counterpart $\bar{\eta}$ of $\eta'$ is defined by \begin{align*}\bar{\eta}((x,(m_x,{\epsilon}_x)),\mathscr{X}')&= \lim_{\alpha\rightarrow \infty}\eta'((x,(m_x,{\epsilon}_x)),\mathscr{X}',\Gamma_{\alpha})\\&=\eta'((x,(m_x,{\epsilon}_x)),\mathscr{X}',\Gamma_{\alpha_x})= \(\eta((x,m_x),\mathscr{X},\Gamma_{\alpha_x})+{\epsilon}_x\)\vee 0
\end{align*} where $\alpha_x=4(\|x\|+r)^2$ and $\mathscr{X}$ is the projection of $\mathscr{X}'$ on $\mathbb{R}^2\times T$. Let $N_0=B(0,1)$, $E=\left\{\left|\Xi'_{B(N_0,r)\backslash N_0}\right|=0\right\}$, $E_1=\left\{\left|\Xi'_{N_0}\right|=1\right\}$, then $E$ is $\sigma\(\Xi'_{N_0^c}\)$ measurable, $\mathbb{P}(E)>0$ and $\mathbb{P}(E_1|E)>0$. Writing ${\epsilon}_x^-=-\min({\epsilon}_x,0)$, $\bar{\Xi}'_{N_0}=\{x_0\}$ in $E_1$, {given $E$} we have
\begin{align*}
&\mathbf{1}_{E_1}\sum_{x\in\bar{\Xi'}}\bar{\eta}\((x,(m_x,{\epsilon}_x)),\Xi'\)\mathbf{1}_{d(x, N_0)<r}\\
=&\mathbf{1}_{E_1} \(\eta((x_0,m_{x_0}),\delta_{({x_0},m_{x_0})},\Gamma_{4(r+1)^2})+{\epsilon}_{x_0}\)\vee0 \\
=&\mathbf{1}_{E_1}\[\mathbf{1}_{{\epsilon}_{x_0}>0}\(\eta(({x_0},m_{x_0}),\delta_{({x_0},m_{x_0})},\Gamma_{4(r+1)^2})+{\epsilon}_x^+\)\right.\\
&+\left.\mathbf{1}_{{\epsilon}_x\le 0}\(\eta(({x_0},m_{x_0}),\delta_{({x_0},m_{x_0})},\Gamma_{4(r+1)^2})-{\epsilon}_{x_0}^-\)\vee 0\right].
\end{align*} On $\{{\epsilon}_{x_0}>0\}$, ${\epsilon}_{x_0}^+$ has positive non-singularity part and is independent of \\ $\eta(({x_0},m_{x_0}),\delta_{({x_0},m_{x_0})},\Gamma_{4(r+1)^2})$, $\eta(({x_0},m_{x_0}),\delta_{({x_0},m_{x_0})},\Gamma_{4(r+1)^2})+{\epsilon}_{x_0}^+$ is also non-singular, together with the fact that $\{{\epsilon}_{x_0}>0\}$ and $\{{\epsilon}_{x_0}\le 0\}$ are disjoint, the non-singularity follows. \qed
\begin{re} If the timber volume of a tree is determined by its nearest neighboring trees, then we can set the score function $\eta$ as a function of weighted Voronoi cells. Using the idea of the proof of Theorem~\ref{exthm1}, we can establish the bound of error of normal approximation to the distribution of the log timber volume ${\bar{W}}_\alpha$ as $d_{TV}\({\bar{W}}_\alpha,{\bar{Z}}_\alpha\)\le O(\alpha^{-\frac{1}{2}}\ln(\alpha)^{\frac{5d}{2}})$. \end{re}
\subsection{Maximal layers}
Maximal layers of points have been of considerable interest since \cite{R62,K75} and have a wide range of applications, see \cite{CHT03} for a brief review of their applications. One of the applications is \textit{the smallest color-spanning interval}~\cite{K17} which is a linear function of the distances between maximal points and the edge. In this subsection, we demonstrate that Theorem~\ref{thm2a} with marks can be easily applied to estimate the error of normal approximation to the distribution of the sum of distances between different maximal layers if the points are from a Poisson point process.
For $x\in \mathbb{R}^d$, we define $A_x=([0,\infty)^d+x)\cap\Gamma_\alpha$. Given a locally finite point set $\mathscr{X}\subset \mathbb{R}^d$, a point $x$ is called {\it maximal} in $\mathscr{X}$ if $x\in\mathscr{X}$ and there is no other point $(y_1,\dots,y_d)\in \mathscr{X}$ satisfying $y_i\ge x_i$ for all $1\le i\le d$ (see Figure~\ref{mp}).
Mathematically, $x$ is maximal in $\mathscr{X}$ if $\mathscr{X}\cap A_x=\{x\}$. This enables us to write different maximal layers as follows: the $k$th maximal layer of points can be recursively defined as
$$\mathscr{X}_k:=\sum_{x\in \mathscr{X}}\delta_x{\bf 1}_{[A_x\cap (\mathscr{X}\backslash (\cup_{i=1}^{k-1}\mathscr{X}_{i}))=\{x\}]},\ \ \ k\ge 1,$$ with the convention $\cup_{i=1}^0\mathscr{X}_i=\emptyset$.
For simplicity, we consider the restriction of the Poisson point process to a region in $\mathbb{R}^d$ between two parallel $d-1$ dimensional planes for $d\ge2$. More precisely, the region of interest is $$\Gamma_{\alpha,r}=\left\{(x_1,x_2,\dots,x_d);~x_i\in [0,\alpha^{\frac{1}{d-1}}], i\le d-1,~x_d+\sum_{i=1}^{d-1}x_i \cot(\theta_i)\in[0,r]\right\}$$ for fixed $\theta_i\in (0,\frac{\pi}{2})$, $1\le i\le d-1$, and $\Xi_{\Gamma_{\alpha,r}}$ is a homogeneous Poisson point process with rate $\lambda$ on $\Gamma_{\alpha,r}$. Define $\Xi_{k,r,\alpha}$ as the $k$th maximal layer of $\Xi_{\Gamma_{\alpha,r}}$, then the total distance between the points in $\Xi_{k,r,\alpha}$ and the upper plane $$P:=\left\{(x_1,x_2,\dots,x_d);~x_i\in [0,\alpha^{\frac{1}{d-1}}],\ i\le d-1,x_d=-\sum_{i=1}^{d-1}x_i \cot(\theta_i)+r\right\}$$ can be represented as ${\bar{W}}_{k,r,\alpha}:=\sum_{x\in \Xi_{k,r,\alpha}}d(x,P)$.
\begin{thm}\label{exthm5} With the above setup, when $r\in \mathbb{R}_+$ is fixed, $$d_{TV}\({\bar{W}}_{k,r,\alpha},{\bar{Z}}_{k,r,\alpha}\)\le O\(\alpha^{-\frac{1}{2}} \),$$ where ${\bar{Z}}_{k,r,\alpha}\sim N\(\E {(\bar{W}}_{k,r,\alpha}{)},{\rm Var}\({\bar{W}}_{k,r,\alpha}\)\)$. \end{thm}
\begin{re}{\rm It remains a challenge to consider maximal layers induced by a homogeneous Poisson point process on $$\left\{(v_1,v_2):\ v_1\in [0,\alpha^{1/(d-1)} ]^{d-1},0\le v_2\le F(v_1)\right\},$$
where $F:\ [0,\alpha^{1/(d-1)} ]^{d-1}\to [0,\infty)$ has continuous negative partial derivatives in all coordinates, the partial derivatives are bounded away from 0 and $-\infty$, and $|F|\le {O(\alpha^{1/(d-1)})}$. We conjecture that normal approximation in total variation for the total distance between the points in a maximal layer and the upper edge surface is still valid. } \end{re}
\noindent{\it Proof of Theorem~\ref{exthm5}.~} As the score function $d(\cdot,P)$ is not translation invariant in
the sense of Definition~\ref{traninvres0}, we first turn the problem to that of a marked Poisson point process with independent marks. The idea is to project the points of $\Xi_{\Gamma_{\alpha,r}}$ on their first $d-1$ coordinates to obtain the ground Poisson point process and send the last coordinate to marks with $T=[0,r]$. To this end, define a mapping $h':\Gamma_{\infty,r}:=\cup_{\alpha>0}\Gamma_{\alpha,r}\rightarrow [0,\infty)^{d-1}\times [0,r]$ such that $$h'(x_1,\dots,x_d)=(x_1,\dots,x_d)+\(0,\dots,0,\sum_{i=1}^{d-1}x_i \cot(\theta_i)\)$$ and $h(\mathscr{X}):=\{h'(x):\ x\in \mathscr{X}\}$. Then $h'$ is a one-to-one mapping and $\Xi':=h\(\Xi_{\Gamma_{\infty,r}}\)$ can be regarded as a marked Poisson point process on $[0,\infty)^{d-1}\times [0,r]$ with rate $r\lambda$ and independent marks following the uniform distribution on $[0,r]$. Write the mark of $x\in \Xi'$ as $m_x$, then \begin{equation}\label{ex5.1}{\bar{W}}_{k,r,\alpha}=C(\theta_1,\dots,\theta_{d-1})\sum_{x\in h(\Xi_{k,r,\alpha})}(r-m_x), \end{equation} where $C(\theta_1,\dots,\theta_{d-1})$ is a constant determined by $\theta_1,\dots,\theta_{d-1}$. Let $\Gamma_\alpha':=[0,\alpha^{\frac{1}{d-1}}]^{d-1}$, then $h(\Xi_{\Gamma_{\alpha,r}})=\Xi'_{\Gamma_\alpha'}$. For a point $(x,m)\in [0,\infty)^{d-1}\times[0,r]$, we write $A_{x,m,r,\alpha}' =h'(((h')^{-1}(x,m)+[0,\infty)^d)\cap \Gamma_{\alpha,r})$ (see Figure~\ref{mp}) and \begin{equation}\label{ex5.2}\Xi'_{k,r,\alpha}:=h(\Xi_{k,r,\alpha})=\sum_{x\in
\bar{\Xi}'_{\Gamma_\alpha'}}\delta_{(x,m_x)}\mathbf{1}_{A_{x,m_x,r,\alpha}'\cap\(\Xi'_{\Gamma_\alpha'}\backslash
\cup_{i=1}^{k-1}\Xi'_{i,r,\alpha}\)=\{(x,m_x)\}}. \end{equation} Combining \Ref{ex5.1} and \Ref{ex5.2}, ${\bar{W}}_{k,r,\alpha}$ can be represented as the sum of values of the score function $$\eta((x,m_x),\Xi',\Gamma_\alpha'):=C(\theta_1,\dots,\theta_{d-1})(r-m_x)\mathbf{1}_{(x,m_x)\in \Xi'_{k,\alpha}}$$ over the range $\Gamma_\alpha'$. To apply Theorem~\ref{thm2a}, we need to check that $\eta$ is range-bound as in Definition~\ref{defi4r}, satisfies the moment condition \Ref{thm2.1r} and non-singularity \Ref{non-sinr}.
\begin{figure}
\caption{maximal point}
\caption{singularity}
\caption{maximal layers}
\label{mp}
\label{mp1}
\end{figure} For simplicity, we only show the claim in two dimensional case and the argument for $d>2$ is the same except notational complexity. When $d=2$, $\Gamma_{\alpha,r}$ is a parallelogram with angle $\theta_1$ as in Figure~\ref{mp}, $P$ and $C(\theta_1,\dots,\theta_{d-1})$ reduce to an edge in $\mathbb{R}^2$ and $\sin(\theta_1)$ respectively. Since $\sin(\theta_1)(r-m_x)$ is given by the mark of $x$, to show $\eta$ is range-bound, it is sufficient to show that $\mathbf{1}_{\{(x,m_x)\in \Xi'_{k,r,\alpha}\}}$ is completely determined by $\Xi'\cap A_{x,m_x,r,\alpha}'$. In fact, we can accomplish this by observing that $(x,m_x)\in \Xi'_{k,r,\alpha}$ iff there is a sequence $\{(x_j,m_{x_j}),\ 1\le j\le k\}\subset \Xi'\cap A_{x,m_x,r,\alpha}'$ (which ensures $A_{x_j,m_{x_j},r,\alpha}'\subset A_{x,m_x,r,\alpha}'$) such that $(x_k,m_{x_k})=(x,m)$ and $A_{x_j,m_{x_j},r,\alpha}'\cap(\Xi'\backslash \cup_{i=1}^{j-1}\Xi'_{i,r,\alpha})=\{(x_j,m_{x_j})\}$ for $1\le j\le k$. Since $\Xi'\cap A_{x,m_x,\alpha}'\subset \Xi'_{[x,x+r\tan(\theta_1)]}$, we can see that $\eta$ is range-bound in Definition~\ref{defi4r} with $\bar{R}(x,\alpha):=r\tan(\theta_1){+1}$. The moment condition follows from the fact that $\eta$ is bounded above by $r$. For the non-singular condition, {we extend $\Xi'$ to $\mathbb{R}^{d-1}\times[0,r]$, write $\Gamma_{\infty,r}^e:=\{x\in\mathbb{R}^d: \mbox{there exists } y\in P~\mbox{such that }y-r(0,\dots,0,1)\le x\le y\}$ and let $(\Xi_{\Gamma_{\infty,r}^e})_j$ be the $j$th maximal layer of $\Xi_{\Gamma_{\infty,r}^e}$ and $\Xi'_j=h((\Xi_{\Gamma_{\infty,r}^e})_j)$}, we can see that the corresponding unrestricted score function is $\bar{\eta}(x,\Xi')=\sin(\theta_1)(r-m_x)\mathbf{1}_{(x,m_x)\in
\Xi'_k}$ and the corresponding stabilizing radii $R(x)=r\tan(\theta_1){+1}$. Referring to Figure~\ref{mp1}, we set $N_0:=\(0,\frac{r\tan(\theta_1)}{2}\)$, $B_0=\{(x,m);~x\in N_0,0\le m\le x\cot(\theta_1)\}$, $B_i$ as the triangle region with vertices $\(r\tan(\theta_1)\left(\frac12+\frac{i-1}{4(k-1)}\right), r\left(\frac12+\frac{2i-1}{4(k-1)}\right)\),$ $\(r\tan(\theta_1)\left(\frac12+\frac{i}{4(k-1)}\right),r\left(\frac12+\frac{2i-1}{4(k-1)}\right)\)$ and $\(r\tan(\theta_1)\left(\frac12+\frac{i}{4(k-1)}\right), r\left(\frac12+\frac{2i}{4(k-1)}\right)\)$, $1\le i\le k-1$, and $${C}=\({\(\[-r\tan(\theta_1){-1},\frac{3r\tan(\theta_1)}{2}{+1}\right]\backslash N_0\)}\times[0,r]\)\backslash\(\cup_{i=1}^{k-1}B_i\),$$ define
$E:=\left\{\Xi'\cap {C}=\emptyset, \left|\Xi'\cap B_i\right|=1,1\le i\le k-1\right\}$ and $E_0:=\{|\Xi'_{N_0}|=\left|\Xi'\cap B_0\right|=1\}$, then {$E\in\sigma\(\Xi'_{N_0^c}\)$, }$\mathbb{P}(E)>0$ and $\mathbb{P}(E_0|E)>0$. {We can see that given $E$, the point in $\Xi'\cap B_i$ is in $\Xi'_{k-i}$ for all $1\le i\le k-1$}. Moreover, on $E\cap E_0$, the point $({x_0},m_{x_0})$ in $\Xi'_{N_0}$ is in $\Xi'_k.$ Hence, given $E$, $$\mathbf{1}_{E_0}\sum_{x\in\bar{\Xi'}}\bar{\eta}\(x,\Xi'\)\mathbf{1}_{d(x,
N_0)<R(x)}=\mathbf{1}_{E_0}\sin(\theta_1)(r-m_{x_0})$$ is non-singular. \qed
As a final remark of the section, we mention that the unrestricted version of all examples considered here can be proved because it is trivial to show that the unrestricted version of the score function $\bar{\eta}$ satisfies the stabilization condition as in Definition~\ref{defi4} and moment condition \Ref{thm2.1} using the same method, and non-singularity \Ref{non-sin} condition is the same as \Ref{non-sinr} given the score function is $\bar{\eta}$.
\section{Preliminaries and auxiliary results}\label{Preliminaries}
We start with a few technical lemmas.
\begin{lma} \label{lma1} Assume $\xi_1,\dots,\xi_n$ are ${i.i.d.}$ random variables having the triangular density function \begin{equation}\kappa_a(x)=\left\{\begin{array}{ll}
\frac1a\left(1-\frac {|x|}a\right),&\mbox{ for }|x|\le a,\\
0,&\mbox{ for } |x|>a, \end{array}\right.\label{lma1.1}\end{equation} where $a>0$. Let $T_n=\sum_{i=1}^n\xi_i$. Then for any $\gamma>0$, \begin{equation}d_{TV}(T_n,T_n+\gamma)\le \frac{\gamma}a\left\{\sqrt{\frac{3}{\pi n}}+\frac{2}{(2n-1)\pi^{2n}}\right\}.\label{lma1.2}\end{equation} \end{lma}
The following lemma says that if the distribution of a random variable is non-singular, then the distributions of random variables which are not far away from it are also non-singular.
\begin{lma}\label{non-singular1} Let $F$ be a non-singular distribution on $\mathbb{R}$ with $\alpha_F>0$ in the decomposition \Ref{decom1}. Then for any distribution $G$ such that $d_{TV}\(F, G\)<\alpha_F$, $G$ is non-singular with $\alpha_G\ge \alpha_F-d_{TV}\(F, G\)$ in its representation $$G=(1-\alpha_G) G_s+\alpha_GG_a,$$ where $G_a$ is absolutely continuous with respect to the Lebesgue measure and $G_s$ is singular. \end{lma}
We denote the convolution by $\ast$ and write $F^{k\ast}$ as the $k$-fold convolution of the function $F$ with itself.
\begin{lma}\label{lma2}
For any two non-singular distributions $F_1$ and $F_2$, there exist positive constants $a>0$, $u\in\mathbb{R}$, $\theta\in (0,1]$ and a distribution function $H$ such that
\begin{equation} \label{lma2.1}
F_1\ast F_2=(1-\theta)H+\theta K_a\ast{\delta}_{u},
\end{equation}
where $K_a$ is the distribution of the triangle density $\kappa_a$ in \Ref{lma1.1} and ${\delta}_u$ is the Dirac measure at $u$.
\end{lma}
Lemma~\ref{lma2} says that $F_1\ast F_2$ is the distribution function of $(X_1+u)X_3+X_2(1-X_3)$, where $X_1\sim K_a$, $X_2\sim H$, $X_3\sim{\rm Bernoulli}(\theta)$ are independent random variables.
\begin{re} \label{remark1}{\rm~From the definition of triangular density function, if $a$, $u$, $\theta$ satisfy \eqref{lma2.1} with a distribution $H$, then for arbitrary $p,\ q$ such that $0<q\le p\le 1$, we can find an $H'$ satisfying the equation with $a'=pa$, $u'=u$ and $\theta'=q\theta$.}
\end{re}
Using the properties of the triangular distributions, we can derive that the sum of the score functions restricted by their radii of stabilization has a similar property as shown in Lemma~\ref{lma1} when the score function is range-bound, exponentially stabilizing or polynomially stabilizing with suitable $\beta$.
\begin{lma}\label{lma3} Let $\Xi$ be a marked homogeneous Poisson point process on $(\mathbb{R}^d\times T,\mathscr{B}(\mathbb{R}^d)\times \mathscr{T})$ with intensity $\lambda$ and ${i.i.d.}$ marks in $(T,\mathscr{T})$ following $\mathscr{L}_{T}$.
\begin{description}
\item{(a)} (unrestricted case) Assume that the score function $\eta$ is non-singular~\Ref{non-sin}. If $\eta$ is polynomially stabilizing in Definition~\ref{defi4} with order $\beta>d+1$ and the radius of stabilization $R$, define $W_{\alpha,r}:=\sum_{(x,m){\in \Xi_{\Gamma_\alpha}}}\eta(\left(x,m\right), \Xi)\mathbf{1}_{R(x)\le r}$, then
\begin{align}
d_{TV}(W_{\alpha,r},W_{\alpha,r}+\gamma)&\le C(|\gamma|\vee 1)\left(\alpha^{-\frac{1}{2}}r^{\frac{d}{2}}\right) \label{statement1} \end{align}
for any $\gamma\in \mathbb{R}$ and $r>R_0$,
where $C$ and $R_0$ are positive constants independent of $\gamma$. If $\eta$ is range-bound in Definition~\ref{defi4}, then
\begin{align}d_{TV}(W_\alpha, W_\alpha+\gamma)\le C(|\gamma|\vee 1)\alpha^{-\frac{1}{2}}\label{statement2}\end{align} for some positive constant $C$ independent of $\gamma$.
\item{(b)} (restricted case) Assume that the score function $\eta$ is non-singular~\Ref{non-sinr}. If $\eta$ is polynomially stabilizing in Definition~\ref{defi4r} with order $\beta>d+1$ and the radius of stabilization $R$, define $\bar{W}_{\alpha,r}:={\sum_{(x,m)\in \Xi_{\Gamma_\alpha}}\eta(\left(x,m\right), \Xi,\Gamma_\alpha)\mathbf{1}_{\bar{R}(x,\alpha)\le r}}$, then
\begin{align}
d_{TV}(\bar{W}_{\alpha,r},\bar{W}_{\alpha,r}+\gamma)&\le C(|\gamma|\vee 1)\left(\alpha^{-\frac{1}{2}}r^{\frac{d}{2}}\right)\label{statement3} \end{align}
for any $\gamma\in \mathbb{R}$ and $r>R_0$,
where $C$ and $R_0$ are positive constants independent of $\gamma$. If $\eta$ is range-bound in Definition~\ref{defi4r}, then
\begin{align}d_{TV}(\bar{W}_\alpha,\bar{W}_\alpha+\gamma)\le C(|\gamma|\vee 1)\alpha^{-\frac{1}{2}}\label{statement4}\end{align} for some positive constant $C$ independent of $\gamma$.
\end{description}
\end{lma}
\begin{re}{\rm Since exponential stabilization implies polynomial stabilization, the statements {\Ref{statement1} and \Ref{statement3}} \ignore{of Lemma~\ref{lma3} }also hold under corresponding exponential stabilization conditions.} \end{re}
Now, a more general version of Lemma~\ref{lma3} with $\gamma$ replaced by a function of $\Xi_N$ for some Borel set $N$ and the expectation replaced by a conditional expectation can be easily established.
\begin{cor}\label{cor1} For $\alpha,r>0$, let $\{N_{\alpha,r}^{(k)}\}_{k\in\{1,2,3\}}\subset\mathscr{B}(\mathbb{R}^d)$ such that $\(N_{\alpha,r}^{(1)}\cup N_{\alpha,r}^{(2)}\cup N_{\alpha,r}^{(3)}\)\cap \Gamma_\alpha\in B(x, C\alpha^{\frac{1}{d}})$ for a point $x\in \mathbb{R}^d$ and a positive constant $C\in (0,\frac{1}{2})$, $\mathscr{F}_{0,\alpha,r}$ be a sub $\sigma$-algebra of $\sigma\(\Xi_{N_{\alpha,r}^{(1)}}\)$, and $h_{\alpha,r}$ be a measurable function mapping configurations on $N_{\alpha, r}^{(2)}\times T$ to the real space.
\begin{description}
\item{(a)} (unrestricted case) Define $W_{\alpha,r}':=\sum_{(x,m)\in \Xi_{\Gamma_\alpha\backslash N_{\alpha,r}^{(3)}}}\eta(\left(x,m\right), \Xi)\mathbf{1}_{R(x)\le r}$. If the conditions of Lemma~\ref{lma3}~(a) hold, then
\begin{align}
&d_{TV}\left(W_{\alpha,r}',W_{\alpha,r}'+h_{\alpha,r}\left(\Xi_{N_{\alpha, r}^{(2)}}\right)\middle|\mathscr{F}_{0,\alpha,r}\right)
\nonumber\\
&\le \mathbb{E}\left(\left|h_{\alpha,r}\left(\Xi_{N_{\alpha, r}^{(2)}}\right)\right|\vee 1\middle|\mathscr{F}_{0,\alpha,r} \right)O\left(\alpha^{-\frac{1}{2}}r^{\frac{d}{2}}\right)~a.s.,\label{lma3coro01}
\end{align}
where $O\left(\alpha^{-\frac{1}{2}}r^{\frac{d}{2}}\right)$ is independent of sets $\{N_{\alpha,r}^{(k)}\}_{\alpha,r\in \mathbb{R}_+,k\in\{1,2,3\}}$, functions $\{h_{\alpha,r}\}_{\alpha,r\in \mathbb{R}_+}$ and $\sigma$-algebras $\{\mathscr{F}_{0,\alpha,r}\}_{\alpha,r\in \mathbb{R}_+}$.
\item{(b)} (restricted case) Define $\bar{W}_{\alpha,r}':=\sum_{(x,m)\in \Xi_{\Gamma_\alpha\backslash N_{\alpha,r}^{(3)}}}\eta(\left(x,m\right), \Xi,{\Gamma_\alpha})\mathbf{1}_{\bar{R}(x,\alpha)<r}\overline{\Xi}(dx)$. If the conditions of Lemma~\ref{lma3}~(b) hold, then
\begin{align}
&d_{TV}\left(\bar{W}_{\alpha,r}',\bar{W}_{\alpha,r}'+h_{\alpha,r}\left(\Xi_{N_{\alpha, r}^{(2)} }\right)\middle|\mathscr{F}_{0,\alpha,r}\right)\nonumber\\
&\le \mathbb{E}\left(\left|h_{\alpha,r}\left(\Xi_{N_{\alpha, r}^{(2)} }\right)\right|\vee 1\middle|\mathscr{F}_{0,\alpha,r} \right)O\left(\alpha^{-\frac{1}{2}}r^{\frac{d}{2}}\right)~a.s.,\label{lma3coro02}
\end{align}
where $O\left(\alpha^{-\frac{1}{2}}r^{\frac{d}{2}}\right)$ is independent of sets $\{N_{\alpha,r}^{(k)}\}_{\alpha,r\in \mathbb{R}_+,k\in\{1,2,3\}}$, functions $\{h_{\alpha,r}\}_{\alpha,~r\in \mathbb{R}_+}$ and $\sigma$-algebras $\{\mathscr{F}_{0,\alpha,r}\}_{\alpha,r\in \mathbb{R}_+}$.
\end{description} \end{cor}
As discussed in the inspiring example, the order of ${\rm Var}\(W_{\alpha}\)$ and ${\rm Var}\(\bar{W}_{\alpha}\)$ plays the crucial role in the accuracy of normal approximation. The next lemma says that under exponential stabilization, optimal order of the variances can be achieved.
\begin{lma}\label{lma4} \begin{description} \item{(a)} (unrestricted case) If the score function $\eta$ satisfies the third moment condition~\Ref{thm2.1}, non-singularity~\Ref{non-sin} and exponential stabilization~in Definition~\ref{defi4}, then
${\rm Var}\(W_{\alpha}\)=\Omega\(\alpha\)$. \item{(b)} (restricted case) If the score function $\eta$ satisfies the third moment condition~\Ref{thm2.1r}, non-singularity~\Ref{non-sinr} and exponential stabilization~in Definition~\ref{defi4r}, then ${\rm Var}\(\bar{W}_{\alpha}\)$ $=\Omega\(\alpha\)$.
\end{description} \end{lma}
For polynomially stabilizing score functions, we do not know the optimal order of the variance, but we can get a lower bound as shown in the next lemma.
\begin{lma}\label{lma5} \begin{description} \item{(a)} (unrestricted case) If the score function $\eta$ satisfies the $k'$-th moment condition~\Ref{thm2.1} with $k'>k\ge3$, non-singularity~\Ref{non-sin} and is polynomially stabilizing in Definition~\ref{defi4} with parameter $\beta>(3k-2)d/(k-2)$, then $${\rm Var}\(W_{\alpha}\)\ge O\(\alpha^{\frac{k\beta-2\beta-3dk+2d}{k\beta-2\beta-dk}}\).$$
\item{(b)} (restricted case) If the score function $\eta$ satisfies the $k'$-th moment condition~\Ref{thm2.1r} with $k'>k\ge3$, non-singularity~\Ref{non-sinr} and polynomially stabilizing in Definition~\ref{defi4r} with parameter $\beta>(3k-2)d/(k-2)$, then $${\rm Var}\(\bar{W}_{\alpha}\)\ge O\(\alpha^{\frac{k\beta-2\beta-3dk+2d}{k\beta-2\beta-dk}}\).$$
\end{description} \end{lma}
\section{The proofs of the auxiliary and main results}\label{Theproofs}
We need Palm processes and reduced Palm processes as the tools in our proofs, and for the ease of reading, we briefly recall their definitions. Let $H$ be a Polish space with Borel $\sigma-$algebra $\mathscr{B}\(H\)$ and configuration space $\(\bm{C}_H, \mathscr{C}_H\)$, let $\Psi$ be a point process on $\(\bm{C}_H, \mathscr{C}_H\)$, write the mean measure of $\Psi$ as $\psi (d x) := \mathbb{E} \Psi (d x)$, the family of point processes $\{ \Psi_x : x \in H\}$ are said to be the {\it Palm processes} associated with $\Psi$ if for any measurable function $f : H\times \bm{C}_H \rightarrow [0,\infty)$, \begin{equation}\label{palm1}
\mathbb{E} \[ \int_{H} f(x,\Psi)\Psi(dx) \right] = \int_{H} \mathbb{E} f(x,\Psi_x) \psi(dx) , \end{equation} \cite[\S~10.1]{Kallenberg83}. A Palm process $\Psi_x$ contains a point at $x$ and it is often more convenient to consider the {\it reduced Palm process} $\Psi_x-\delta_x$ at $x$ by removing the point $x$ from $\Psi_x$. Furthermore, suppose that the {\it factorial moments} $\psi^{[2]}(dx,dy) : = \mathbb{E} [\Psi(dx)(\Psi-{\delta}_x)(dy)]$ and $\psi^{[3]}(dx,dy,dz) : = \mathbb{E} [\Psi(dx)(\Psi-{\delta}_x)(dy)(\Psi-{\delta}_x-\delta_y)(dz)]$ are finite, then we can respectively define the {\it second order Palm processes} $\{ \Psi_{xy}: x,y \in H \}$ and {\it third order Palm processes} $\{ \Psi_{xyz}: x,y,z \in H \}$ associated with $\Psi$ by \begin{align}
&\mathbb{E} \[ \iint_{H^2}
f(x,y;\Psi)\Psi(dx)(\Psi-{\delta}_x)(dy) \right]= \iint_{H^2} \mathbb{E}
f(x,y;\Psi_{xy}) \psi^{[2]} (dx,dy) ,\label{palm2}\\
&\mathbb{E} \[ \iiint_{H^3}
f(x,y,z;\Psi)\Psi(dx)(\Psi-{\delta}_x)(dy)(\Psi-{\delta}_x-{\delta}_y)(dz) \right] \nonumber\\& = \iiint_{H^3} \mathbb{E}
f(x,y,z;\Psi_{xyz}) \psi^{[3]} (dx,dy,dz) ,\label{palm3} \end{align} for all measurable functions $f : H^2 \times \bm{C}_H \rightarrow [0,\infty)$ in \Ref{palm2} and $f: H^3 \times \bm{C}_H \rightarrow [0,\infty)$ in \Ref{palm3} \cite[\S~12.3]{Kallenberg83}. Using reduced Palm processes, Slivnyak-Mecke theorem \cite{Mecke63} states that a point process such that the distributions of its reduced Palm processes are the same as that of the point process if and only if it is a Poisson point process. Then we can see that for homogeneous Poisson point process with rate $\lambda$, its mean measure can be written as $\Lambda(dx)=\lambda dx$ and its Palm processes satisfy $\Psi_{x}\overset{d}{=}\Psi+{\delta}_x$, $\Psi_{xy}\overset{d}{=}\Psi+{\delta}_x+{\delta}_y$ and $\Psi_{xyz}\overset{d}{=}\Psi+{\delta}_x+{\delta}_y+{\delta}_z$, the factorial moments $\psi^{[2]}(dx,dy)=\lambda^2dxdy$ and $\psi^{[3]}(dx,dy,dz)=\lambda^3dxdydz$ for all distinct $x$, $y$, $z\in H$.
We can adapt \Ref{palm1}, \Ref{palm2} and \Ref{palm3} to the marked homogeneous Poisson point process case that we are dealing with, assume the rate of $\overline{\Xi}$ is $\lambda$, $\{M_i\}_{i\in \mathbb{N}}$ is a sequence of ${i.i.d.}$ random elements on $(T,\mathscr{T})$ following the distribution $\mathscr{L}_{T}$ which is independent of $\Xi$, then because of the independence of marks, we can see that \begin{align}
\label{palm4}
&\mathbb{E} \[ \int_{\bm{S}} f(x,\Xi)\overline{\Xi}(dx) \right] = \int_{\bm{S}} \mathbb{E} f(x,\Xi+{\delta}_{(x,M_1)}) \lambda dx, \\
&\mathbb{E} \[ \iint_{\bm{S}^2}
f(x,y;\Xi)\overline{\Xi}(dx)(\overline{\Xi}-{\delta}_x)(dy) \right] = \iint_{\bm{S}^2} \mathbb{E}
f(x,y;\Xi+{\delta}_{(x,M_1)}+{\delta}_{(y,M_2)}) \lambda^2dxdy ,\label{palm5}\\
&\mathbb{E} \[ \iiint_{\bm{S}^3}
f(x,y,z;\Xi)\overline{\Xi}(dx)(\overline{\Xi}-{\delta}_x)(dy)(\overline{\Xi}-{\delta}_x-{\delta}_y)(dz) \right]\nonumber
\\ =& \iiint_{\bm{S}^3} \mathbb{E}
f(x,y,z;\Xi+{\delta}_{(x,M_1)}+{\delta}_{(y,M_2)}+{\delta}_{(z,M_3)}) \lambda^3dxdydz,\label{palm6} \end{align} for all measurable functions $f :\bm{S} \times (\bm{C}_{\bm{S}}\times T) \rightarrow [0,\infty)$ in \Ref{palm4}, $f:\bm{S}^2 \times (\bm{C}_{\bm{S}}\times T) \rightarrow [0,\infty)$ in \Ref{palm5} and $f:\bm{S}^3 \times (\bm{C}_{\bm{S}}\times T) \rightarrow [0,\infty)$ in \Ref{palm6}.
Recalling the shift operator defined in Section~\ref{Generalresults}, we can write $g(\mathscr{X}^x):=\eta(\left(x,m\right), \mathscr{X})$ (resp. $g_\alpha(x,\mathscr{X}):= \eta\(\(x,m\),\Xi,\mathscr{X},\Gamma_\alpha\)$) for all configuration $\mathscr{X}$, $(x,m)\in \mathscr{X}$ and $\alpha>0$ so that notations can be simplified significantly, e.g, . \begin{eqnarray*}
W_\alpha&=&\sum_{(x,m)\in\Xi_{\Gamma_\alpha}}\eta(\left(x,m\right), \Xi)=\int_{\Gamma_\alpha} g(\Xi^x)\overline{\Xi}(dx)=\sum_{x\in \bar{\Xi}}g(\Xi^x),\\
\bar{W}_\alpha&=&\sum_{(x,m)\in\Xi_{\Gamma_\alpha}}\eta(\left(x,m\right), \Xi,\Gamma_\alpha)=\int_{\Gamma_\alpha}{g_\alpha(x, {\Xi})\overline{\Xi}(dx)}, \end{eqnarray*} where $\overline{\Xi}$ is the projection of $\Xi$ on $\mathbb{R}^d$, $R$ and $\bar{R}(x,\alpha)$ are the corresponding radii of stabilization.
We now proceed to establish a few lemmas needed in the proofs.
\begin{lma}\label{lma6} (Conditional Total Variance Formula) Let $X$ be a random variable on probability space $(\Omega, \mathscr{G},\mathbb{P})$ with finite second moment, $\mathscr{G}_1$ and $\mathscr{G}_2$ be two sub $\sigma$-algebras of $\mathscr{G}$ such that $\mathscr{G}_1\subset \mathscr{G}_2$, then
$${\rm Var}\(X|\mathscr{G}_1\)=\mathbb{E}\left({\rm Var}\left(X\middle|\mathscr{G}_2\right)\middle|\mathscr{G}_1\right)+{\rm Var}\left(\mathbb{E}\left(X\middle|\mathscr{G}_2\right)\middle|\mathscr{G}_1\right).$$ \end{lma} \noindent{\it Proof.} From the definition of the conditional variance, we can see that \begin{align*}
{\rm Var}\(X|\mathscr{G}_1\)=&\mathbb{E}\left(X^2\middle|\mathscr{G}_1\right)-\mathbb{E}\left(X\middle|\mathscr{G}_1\right)^2
\\=&\mathbb{E}\left(\mathbb{E}\left(X^2\middle|\mathscr{G}_2\right)\middle|\mathscr{G}_1\right)-\mathbb{E}\left(\mathbb{E}\left(X\middle|\mathscr{G}_2\right)\middle|\mathscr{G}_1\right)^2
\\=&\mathbb{E}\left(\mathbb{E}\left(X^2\middle|\mathscr{G}_2\right)\middle|\mathscr{G}_1\right)-\mathbb{E}\left(\mathbb{E}\left(X\middle|\mathscr{G}_2\right)^2\middle|\mathscr{G}_1\right)+\mathbb{E}\left(\mathbb{E}\left(X\middle|\mathscr{G}_2\right)^2\middle|\mathscr{G}_1\right)\\
&-\mathbb{E}\left(\mathbb{E}\left(X\middle|\mathscr{G}_2\right)\middle|\mathscr{G}_1\right)^2
\\=&\mathbb{E}\left({\rm Var}\left(X\middle|\mathscr{G}_2\right)\middle|\mathscr{G}_1\right)+{\rm Var}\left(\mathbb{E}\left(X\middle|\mathscr{G}_2\right)\middle|\mathscr{G}_1\right), \end{align*} so the statement holds. \qed
Also, given the value of a random variable in a certain event, we can find a lower bound for the conditional variance. \begin{lma}\label{lma7}
Let $X$ be a random variable on probability space $(\Omega, \mathscr{G},\mathbb{P})$ with finite second moment, for any event $A$ and $\sigma$-algebra such that $\mathscr{F}\subset\mathscr{G}$,
\begin{equation}\label{lma7.1}
{\rm Var}\left(X\middle|\mathscr{F}\right)\ge {\rm Var}\left(X\mathbf{1}_A+\frac{\mathbb{E}\left(X\mathbf{1}_A\middle|\mathscr{F}\right)}{\mathbb{P}\left(A\middle|\mathscr{F}\right)}\mathbf{1}_{A^c}\middle|\mathscr{F}\right),
\end{equation}
where $\frac{0}{0}=0$ by convention. \end{lma}
\noindent{\it Proof of Lemma~\ref{lma7}.} For the case that event $A$ has probability $0$, the statement is trivially true, so we focus on the case that $\mathbb{P}(A)>0$. Let $A\cap\mathscr{F}=\{B\cap A;~B\in \mathscr{F}\}$, which is a $\sigma-$algebra on $A$, and $\mathbb{P}_A$ as a probability measure on $(A, A\cap\mathscr{F})$ such that $\mathbb{P}_A\(B\cap A\)=\mathbb{P}\left(B\middle|A\right)$ for all $B\in \mathscr{F}$, then we have the corresponding conditional expectation $$\mathbb{E}_A\left(X\middle|A\cap\mathscr{F}\right)=\mathbb{E}_A\left(X\mathbf{1}_A\middle|A\cap\mathscr{F}\right)=\mathbf{1}_A\mathbb{E}_A\left(X\middle|A\cap\mathscr{F}\right),$$ which equals to $0$ on $\mathbf{1}_{A^c}$. The proof relies on the following observations:\\ \begin{lma} \label{lma8}
For any random variable $Y$ with $\mathbb{E}\vert Y\vert < \infty$ and $A\in \mathscr{G}$ such that $\mathbb{P}(A)>0$,
$$
\mathbb{E}\left(\mathbb{E}_A\left(Y\middle|A\cap\mathscr{F}\right)\middle|\mathscr{F}\right)=\mathbb{E}\left(\mathbf{1}_A Y\middle|\mathscr{F}\right).
$$ \end{lma} \noindent{\it Proof.} Both sides are $\mathscr{F}$ measurable, together with the fact that for any $B\in \mathscr{F}$, \begin{align}
\mathbb{E}\left(\mathbb{E}\left(\mathbb{E}_A\left(Y\middle|A\cap\mathscr{F}\right)\middle|\mathscr{F}\right)\mathbf{1}_B\right)=&\mathbb{E}\left(\mathbb{E}_A\left(Y\middle|A\cap\mathscr{F}\right)\mathbf{1}_B\right)\nonumber
\\=&\mathbb{E}\left(\mathbb{E}_A\left(Y\middle|A\cap\mathscr{F}\right)\mathbf{1}_{B\cap A}\right)\nonumber
\\=&\mathbb{E}\left(\mathbb{E}_A\left(\mathbf{1}_{B\cap A}Y\middle|A\cap\mathscr{F}\right)\right)\nonumber
\\=&\mathbb{P}(A)\mathbb{E}_A\left(\mathbb{E}_A\left(\mathbf{1}_{B\cap A}Y\middle|A\cap\mathscr{F}\right)\right)\nonumber
\\=&\mathbb{P}(A)\mathbb{E}_A\left(\mathbf{1}_{B\cap A}Y\right)=\mathbb{E}\left(\mathbf{1}_{B\cap A}Y\right)=\mathbb{E}\left(\mathbb{E}\left(\mathbf{1}_A Y\middle|\mathscr{F}\right)\mathbf{1}_B\right),\nonumber \end{align} as claimed. \qed \begin{lma} \label{lma9}
For any random variable $Y$ with $\mathbb{E}\vert Y\vert <\infty$ and $A\in \mathscr{G}$ such that $\mathbb{P}(A)>0$,
\begin{equation}\label{lma9.1}
\frac{\mathbb{E}\left(\mathbf{1}_A Y \middle|\mathscr{F}\right)}{\mathbb{P}\left(A\middle|\mathscr{F} \right)}\mathbf{1}_A=\mathbb{E}_A\left(Y\middle|\mathscr{F}\right).
\end{equation} \end{lma} \noindent{\it Proof.} Both sides equal $0$ on $A^c$, and are measurable on $A\cap\mathscr{F}$ when restricted to $A$. From the construction of $ A\cap\mathscr{F}$, any set $B'\in A\cap\mathscr{F}$ is of the form $B\cap A$ for some $B\in\mathscr{F}$, hence \Ref{lma9.1} is equivalent to
$$\mathbb{E}_A\left(\frac{\mathbb{E}\left(\mathbf{1}_A Y \middle|\mathscr{F}\right)}{\mathbb{P}\left(A\middle|\mathscr{F} \right)}\mathbf{1}_A\mathbf{1}_{A\cap B}\right)=\mathbb{E}_A\left(\mathbb{E}_A\left(Y\middle|\mathscr{F}\right)\mathbf{1}_{A\cap B}\right)$$ for all $B\in \mathscr{F}$. Now we have \begin{align}
\mathbb{E}_A\left(\frac{\mathbb{E}\left(\mathbf{1}_A Y \middle|\mathscr{F}\right)}{\mathbb{E}\left(\mathbf{1}_A\middle|\mathscr{F} \right)}\mathbf{1}_{A\cap B}\right)=&\frac{1}{\mathbb{P}(A)}\mathbb{E}\left(\frac{\mathbb{E}\left(\mathbf{1}_A Y \middle|\mathscr{F}\right)}{\mathbb{E}\left(\mathbf{1}_A\middle|\mathscr{F} \right)}\mathbf{1}_A\mathbf{1}_B\right)\nonumber
\\=&\frac{1}{\mathbb{P}(A)}\mathbb{E}\left(\mathbb{E}\left(\frac{\mathbb{E}\left(\mathbf{1}_A Y \middle|\mathscr{F}\right)}{\mathbb{E}\left(\mathbf{1}_A\middle|\mathscr{F} \right)}\mathbf{1}_A\mathbf{1}_B\middle|\mathscr{F}\right)\right)\nonumber
\\=&\frac{1}{\mathbb{P}(A)}\mathbb{E}\left(\mathbb{E}\left(\mathbf{1}_A Y \middle|\mathscr{F}\right)\mathbf{1}_B\right)=\frac{1}{\mathbb{P}(A)}\mathbb{E}\left(\mathbb{E}\left(\mathbf{1}_A\mathbf{1}_B Y \middle|\mathscr{F}\right)\right)\nonumber
\\=&\frac{1}{\mathbb{P}(A)}\mathbb{E}\left(\mathbf{1}_A\mathbf{1}_B Y\right)=\mathbb{E}_A(\mathbb{E}_A(Y\mathbf{1}_A\mathbf{1}_B|\mathscr{F}))=\mathbb{E}_A(\mathbb{E}_A(Y|\mathscr{F})\mathbf{1}_{A\cap B})\nonumber \end{align} completing the proof. \qed\\ \noindent{\it Proof of Lemma~\ref{lma7} (continued).} We start from the left hand side of \Ref{lma7.1}, \begin{align}
&{\rm Var}\left(X\middle|\mathscr{F}\right)\nonumber
\\=&\mathbb{E}\left(X^2\middle|\mathscr{F}\right)-\mathbb{E}\left(X\middle|\mathscr{F}\right)^2\nonumber
\\=&\mathbb{E}\left(X^2\middle|\mathscr{F}\right)-\left(\mathbb{E}\left(X\mathbf{1}_A\middle|\mathscr{F}\right)+\mathbb{E}\left(X\mathbf{1}_{A^c}\middle|\mathscr{F}\right)\right)^2\nonumber
\\=&\mathbb{E}\left(X^2\middle|\mathscr{F}\right)-\mathbb{E}\left(\mathbb{E}_A\left(X\mathbf{1}_A\middle|A\cap\mathscr{F}\right)+\mathbb{E}_{A^c}\left(X\middle|A^c\cap\mathscr{F}\right)\middle|\mathscr{F}\right)^2\nonumber
\\\ge&\mathbb{E}\left(X^2\middle|\mathscr{F}\right)-\mathbb{E}\left(\left(\mathbb{E}_A\left(X\mathbf{1}_A\middle|A\cap\mathscr{F}\right)+\mathbb{E}_{A^c}\left(X\middle|A^c\cap\mathscr{F}\right)\right)^2\middle|\mathscr{F}\right)\nonumber
\\=&\mathbb{E}\left(X^2\(\mathbf{1}_A+\mathbf{1}_{A^c}\)\middle|\mathscr{F}\right)-\mathbb{E}\left(\mathbb{E}_A\left(X\mathbf{1}_A\middle|A\cap\mathscr{F}\right)^2+\mathbb{E}_{A^c}\left(X\middle|A^c\cap\mathscr{F}\right)^2\middle|\mathscr{F}\right)\nonumber
\\=&\mathbb{E}\left(\mathbb{E}_A\left(X^2\middle| A\cap\mathscr{F}\right)+\mathbb{E}_{A^c}\left(X^2\middle| A^c\cap\mathscr{F}\right)\middle|\mathscr{F}\right)\nonumber\\
&-\mathbb{E}\left(\mathbb{E}_A\left(X\mathbf{1}_A\middle|A\cap\mathscr{F}\right)^2+\mathbb{E}_{A^c}\left(X\middle|A^c\cap\mathscr{F}\right)^2\middle|\mathscr{F}\right)\nonumber
\\=&\mathbb{E}\left({\rm Var}_A\left(X\middle| A\cap\mathscr{F}\right)+{\rm Var}_{A^c}\left(X\middle| A^c\cap\mathscr{F}\right)\middle|\mathscr{F}\right)\label{lma7.2}, \end{align} where the inequality follows from Jensen's inequality, the third and the second last equalities are from Lemma~\ref{lma8}. On the other hand, the right hand side of \Ref{lma7.1} can be written as \begin{align}
&\mathbb{E}\left(X^2\mathbf{1}_A+\left(\frac{\mathbb{E}\left(X\mathbf{1}_A\middle|\mathscr{F}\right)}{\mathbb{P}\left(A\middle|\mathscr{F}\right)}\right)^2\mathbf{1}_{A^c}\middle|\mathscr{F}\right)-\left(\mathbb{E}\left(X\mathbf{1}_A\middle|\mathscr{F}\right)+\mathbb{E}\left(\frac{\mathbb{E}\left(X\mathbf{1}_A\middle|\mathscr{F}\right)}{\mathbb{P}\left(A\middle|\mathscr{F}\right)}\mathbf{1}_{A^c}\middle|\mathscr{F}\right)\right)^2\nonumber
\\=&\mathbb{E}\left(X^2\mathbf{1}_A\middle|\mathscr{F}\right)+\left(\frac{\mathbb{E}\left(X\mathbf{1}_A\middle|\mathscr{F}\right)}{\mathbb{P}\left(A\middle|\mathscr{F}\right)}\right)^2\mathbb{P}\left(A^c\middle|\mathscr{F}\right)-\mathbb{E}\left(X\mathbf{1}_A\middle|\mathscr{F}\right)^2\frac{1}{\mathbb{P}\left(A\middle|\mathscr{F}\right)^2}\nonumber
\\=&\mathbb{E}\left(X^2\mathbf{1}_A\middle|\mathscr{F}\right)-\left(\frac{\mathbb{E}\left(X\mathbf{1}_A\middle|\mathscr{F}\right)}{\mathbb{P}\left(A\middle|\mathscr{F}\right)}\right)^2\mathbb{P}\left(A\middle|\mathscr{F}\right)\nonumber
\\=&\mathbb{E}\left(\mathbb{E}_A\left(X^2\middle|A\cap \mathscr{F}\right)-\left(\frac{\mathbb{E}\left(X\mathbf{1}_A\middle|\mathscr{F}\right)}{\mathbb{P}\left(A\middle|\mathscr{F}\right)}\right)^2\mathbf{1}_A\middle|\mathscr{F}\right)\nonumber
\\=&\mathbb{E}\left(\mathbb{E}_A\left(X^2\middle|A\cap \mathscr{F}\right)-\mathbb{E}_A\left(X\middle|A\cap \mathscr{F}\right)^2\middle|\mathscr{F}\right)\nonumber
\\=&\mathbb{E}\left({\rm Var}_A\left(X\middle| A\cap\mathscr{F}\right)\middle|\mathscr{F}\right)\label{lma7.3}, \end{align}where the third equality follows from Lemma~\ref{lma8}, and the second last equality is from Lemma~\ref{lma9}. Combining \Ref{lma7.2}, \Ref{lma7.3} and Lemma~\ref{lma6} completes the proof. \qed
The following lemma bounds the difference between two normal distribution under the total variation distance. \begin{lma}\label{lma10}
Let $F_{\mu,\sigma}$ be the distribution of $N(\mu,\sigma^2)$, the normal distribution with mean $\mu$ and variance $\sigma^2$, then
$$d_{TV}(F_{\mu_1,\sigma_1},F_{\mu_2,\sigma_2})\le \sqrt{\frac{2}{\pi}}\(\frac{|\mu_1-\mu_2|}{2\max(\sigma_1,\sigma_2)}+\frac{\max(\sigma_1,\sigma_2)}{\min(\sigma_1,\sigma_2)}-1\).$$ \end{lma} \noindent{\it Proof.} Without loss of generality, we assume $\sigma_2>\sigma_1$. Writing the probability density function of $F_{\mu,\sigma}$ as $f_{\mu,\sigma}$, we have \begin{equation}\label{9.1}
\begin{aligned}
d_{TV}(F_{\mu_1,\sigma_1},F_{\mu_2,\sigma_2})&\le d_{TV}(F_{\mu_1,\sigma_1},F_{\mu_1,\sigma_2})+d_{TV}(F_{\mu_1,\sigma_2},F_{\mu_2,\sigma_2})\\&=d_{TV}\(F_{0,1},F_{0,\frac{\sigma_2}{\sigma_1}}\)+d_{TV}\(F_{0,1},F_{\frac{\mu_1-\mu_2}{\sigma_2},1}\).
\end{aligned} \end{equation} Then the problem turns to bound the differences between the distributions of $N(0,1)$ and $N(0,\sigma^2)$ and between the distributions of $N(0,1)$ and $N(\mu,1)$ for $\mu>0$ and $\sigma>1$. For $\sigma>1$, we can see that the probability density functions $f_{0,1}$ and $f_{0,\sigma}$ meet at $\pm x_\sigma:=\pm\sqrt{\frac{2\ln(\sigma)\sigma^2}{\sigma^2-1}}$, and $f_{0,1}>f_{0,\sigma}$ on $(-x_\sigma,x_\sigma)$ and the inequality sign is reversed outside the interval. We can see that $1<x_\sigma<\sigma$, so we have \begin{equation}
\begin{aligned}\label{9.3}
d_{TV}(F_{0,1},F_{0,\sigma})
&=F_{0,1}(x_\sigma)-F_{0,1}(-x_\sigma)-(F_{0,\sigma}(x_\sigma)-F_{0,\sigma}(-x_\sigma))\\&=F_{0,1}(x_\sigma)-F_{0,1}\(\frac{x_\sigma}{\sigma}\)-F_{0,1}(-x_\sigma)+F_{0,1}\(-\frac{x_\sigma}{\sigma}\)\le \sqrt{\frac{2}{\pi}}(\sigma-1),
\end{aligned} \end{equation} where the inequality follows from the fact that the probability density function $f_{0,1}$ is bounded by $\frac{1}{\sqrt{2\pi}}$. Similarly, we can see that \begin{equation}
\begin{aligned}\label{9.2}
d_{TV}(F_{0,1},F_{\mu,1})
&=F_{0,1}\(\frac{\mu}{2}\)-F_{\mu,1}\(\frac{\mu}{2}\)=F_{0,1}\(\frac{\mu}{2}\)-F_{0,1}\(-\frac{\mu}{2}\)\le \frac{\mu}{\sqrt{2\pi}},
\end{aligned} \end{equation} for $\mu>0$, where again we use the fact that $f_{0,1}$ is bounded by $\frac{1}{\sqrt{2\pi}}$ in the inequality. Substituting \Ref{9.2} and \Ref{9.3} into \Ref{9.1} yields the claim. \qed
The following lemma says that under stabilizing conditions, the cost of throwing away the terms with large radii of stabilization is negligible.
\begin{lma}\label{lma105}
\begin{description}
\item{(a)} (unrestricted case) If the score function is exponentially stabilizing in Definition~\ref{defi4}, then we have
$$d_{TV}(W_\alpha, W_{\alpha,r})\le C_1\alpha e^{-C_2r}$$
for some positive constants $C_1$, $C_2$.
If the score function is polynomially stabilizing with parameter $\beta$ in Definition~\ref{defi4}, then we have
$$d_{TV}(W_\alpha, W_{\alpha,r})\le C\alpha r^{-\beta}$$ for some positive constant $C$.
\item{(b)} (restricted case) If the score function is exponentially stabilizing in Definition~\ref{defi4r}, then we have
$$d_{TV}(\bar{W}_\alpha, \bar{W}_{\alpha,r})\le C_1\alpha e^{-C_2r}$$
for some positive constants $C_1$, $C_2$.
If the score function is polynomially stabilizing with parameter $\beta$ in Definition~\ref{defi4r}, then we have
$$d_{TV}(\bar{W}_\alpha, \bar{W}_{\alpha,r})\le C\alpha r^{-\beta}$$ for some positive constant $C$.
\end{description} \end{lma} \noindent{\it Proof.} We first show the statement is true for $\bar{W}_\alpha$ and $\bar{W}_{\alpha,r}$. For convenience of writing, we define $M_x$ as random elements following the law $\mathscr{L}_T$ which are independent of $\Xi$ for all $x\in \mathbb{R}^d$. From the construction of $\bar{W}_\alpha$ and $\bar{W}_{\alpha,r}$, we can see that the event $\{\bar{W}_\alpha\neq \bar{W}_{\alpha,r}\}\subset\{\mbox{at least one }x\in \overline{\Xi}\cap\Gamma_\alpha~\mbox{with }\bar{R}(x, \alpha)>r\}$, so from \Ref{palm4}, we have \begin{align*}
d_{TV}(\bar{W}_\alpha, \bar{W}_{\alpha,r})\le &\mathbb{P}\(\{\bar{W}_\alpha\neq \bar{W}_{\alpha,r}\}\)
\\\le & \mathbb{P}\(\{\mbox{at least one }x\in \overline{\Xi}\cap\Gamma_\alpha~{\mbox{such that }}\bar{R}(x, \alpha)>r\}\)
\\ \le& \mathbb{E}\int_{\Gamma_\alpha}\mathbf{1}_{\bar{R}(x,\alpha)>r} \overline{\Xi}(dx)
\\ =&\int_{\Gamma_\alpha}\mathbb{E}\(\mathbf{1}_{\bar{R}(x,M_x,\alpha, \Xi+\delta_{(x, M_x)})>r}\)\lambda dx
\\ =&\int_{\Gamma_\alpha}\mathbb{P}\(\bar{R}(x,M_x,\alpha, \Xi+\delta_{(x, M_x)})>r\) \lambda dx
\\ \le&\alpha\lambda\bar{\tau}(r), \end{align*} which, together with the stabilization conditions, gives the claim for $\bar{W}_{\alpha}$.
The statement is also true for $W_\alpha$, which can be proved by replacing corresponding counterparts $\bar{W}_\alpha$ with $W_\alpha$; $\bar{W}_{\alpha,r}$ with $W_{\alpha,r}$; $\bar{R}(x,\alpha)$ with $R(x)$; $\bar{R}(x,M_x,\alpha,\Xi+\delta_{(x, M_x)})$ with $R(x,M_x,\Xi+\delta_{(x, M_x)})$; $\bar{\tau}$ with $\tau.$\qed
\noindent{\it Proof of Lemma~\ref{lma1}.} For convenience, we write $G_n$, $g_n$ and $\psi_n$ as the distribution, density and characteristic functions of $T_n$ respectively. It is well-known that the triangular density $\kappa_a$ has the characteristic function $\psi_1(s)=\frac{2(1-\cos(as))}{(as)^2}$, which gives $\psi_n(s)=\left(\frac{2(1-\cos(as))}{(as)^2}\right)^n$. Using the fact that the convolution of two symmetric unimodal distributions on $\mathbb{R}$ is unimodal \cite{Wintner38}, we can conclude that the distribution of $T_n$ is unimodal and symmetric. This ensures that
\begin{equation}d_{TV}(T_n,T_n+\gamma)=\sup_{x\in\mathbb{R}}|G_n(x)-G_n(x-\gamma)|=\int_{-\gamma/2}^{\gamma/2}g_n(x)dx.\label{lma1.3}\end{equation} Applying the inversion formula, we have \begin{eqnarray*} g_n(x)&=&\frac1{2\pi}\int_\mathbb{R} e^{-{\mbox{\sl\scriptsize i}} sx}\psi_n(s)ds=\frac1{2\pi}\int_\mathbb{R} \cos(sx)\psi_n(s)ds\nonumber\\ &=&\frac1{a\pi}\int_0^\infty \cos(sx/a)\left(\frac{2(1-\cos s)}{s^2}\right)^nds, \end{eqnarray*} where ${\mbox{\sl i}}=\sqrt{-1}$ and the second equality is due to the fact that $\sin(sx)\psi_n(s)$ is an odd function. Obviously, $g_n(x)\le g_n(0)$ so we need to establish an upper bound for $g_n(0)$. A direct verification gives $$0\le \frac{2(1-\cos s)}{s^2}\le e^{-\frac{s^2}{12}}\mbox{ for }0\le s\le 2\pi,$$ which implies \begin{eqnarray} g_n(0)&\le&\frac1{a\pi} \left\{\int_0^{2\pi}e^{-\frac{ns^2}{12}}ds+\int_{2\pi}^\infty \left(\frac 4{s^2}\right)^nds\right\}\nonumber\\ &\le& \frac1{a\pi\sqrt{n}}\int_0^\infty e^{-\frac{s^2}{12}}ds+\frac{2}{a(2n-1)\pi^{2n}}\nonumber\\ &=&\frac1a\sqrt{\frac{3}{\pi n}}+\frac{2}{a(2n-1)\pi^{2n}}.\label{lma1.5} \end{eqnarray} Now, combining \Ref{lma1.5} with \Ref{lma1.3} gives \Ref{lma1.2}.\qed
\noindent{\it Proof of Lemma~\ref{non-singular1}.} We construct a maximal coupling \cite[p.~254]{BHJ} $(X,Y)$ such that $X\sim F$, $Y\sim G$ and $d_{TV}(F,G)=\Pro(X\ne Y)$. The Lebesgue decomposition \Ref{decom1} ensures that there exists an $A\in \mathscr{B}\(\mathbb{R}\)$ such that $F_a(A)=1$ and $F_s(A)=0$. Define $\mu_G(B)=\mathbb{P}\(X\in B\cap A,X= Y\)\le \alpha_FF_a(B)$ for $B\in \mathscr{B}\(\mathbb{R}\)$, so $\mu_G$ is absolutely continuous with respect to the Lebesgue measure. On the other hand, $$G(B)\ge G(B\cap A)\ge\Pro(Y\in B\cap A,X=Y)
=\mu_G(B),\ \mbox{for }B\in\mathscr{B}\(\mathbb{R}\),$$ hence $\alpha_G\ge \mu_G(\mathbb{R})= \alpha_F-\mathbb{P}\(X\neq Y\)=\alpha_F-d_{TV}\(F, G\)>0$.
\qed
\begin{wrapfigure}{r}{0.5\textwidth}
\begin{center}
\includegraphics[trim = 20mm 100mm 30mm 100mm, clip,width=0.5\textwidth]{pic1}
\caption{Existence of $u$ and $v$}
\label{figureone}
\end{center}
\end{wrapfigure} \noindent{\it Proof of Lemma~\ref{lma2}.} Since $F_i$ is non-singular, there exists a non-zero sub-probability measure $\mu_{i}$ with a density $f_{i}$ such that $\mu_i(dx)=f_i(x)dx\le dF_i(x)$ for $x\in\mathbb{R}$.
Without loss of generality, we can assume that both $f_{1}$ and $f_{2}$ are bounded with bounded supports, which ensures that $f_{1}\ast f_{2}$ is continuous (for the case of $f_{1}= f_{2}$, see \cite[p.~79]{Lindvall92}). In fact, as $f_{1}$ is a density, one can find a sequence of continuous functions $\{f_{1n}: \, n\ge 1\}$ satisfying $|f_{1n}-f_{1}|_1\rightarrow0$ as $n\to \infty$, where $|\cdot|_1$ is the $l_1$ norm. Now, with $|\cdot|_\infty$ denoting the supremum norm, $|f_{1n}* f_{2}-f_{1}* f_{2}|_\infty\le |f_{2}|_\infty|f_{1n}-f_{1}|_1\to 0$ as $n\to\infty$. However, the continuity is preserved under the supremum norm, the continuity of $f_{1}\ast f_{2}$ follows.
Referring to Figure~\ref{figureone}, since $f_{1}\ast f_{2}\not\equiv0$, we can find $u\in\mathbb{R}$ and $v>0$ such that $f_{1}\ast f_{2}(u)>0$ and $\min_{x\in[u-v,u+v]}f_{1}\ast f_{2}(x)\ge \frac12f_{1}\ast f_{2}(u)=:b$. Let $\theta=vb$ and $a=v$, $H=\frac1{1-\theta}(F_1\ast F_2-\theta K_a\ast\delta_u)$, the claim follows. \qed
\noindent\noindent{\it Proof of Lemma~\ref{lma3}.} The idea of the proof is to use the radius of stabilization to limit the effect of dependence, establish that the non-singularity \Ref{non-sin} passes to the trimmed score function $\eta(\left(x,m\right), \Xi)\mathbf{1}_{R(x)\le r}$ {$\(\mbox{resp.}~\eta(\left(x,m\right), \Xi,\Gamma_\alpha)\mathbf{1}_{\bar{R}(x,\alpha)\le r}\)$} and then divide the carrier space $\Gamma_\alpha$ into maximal number of cubes so that sums of the trimmed score function on these cubes are independent. The order of the bound is then determined by the reciprocal of the number of the cubes, as in the Berry-Esseen bound. Except slightly complicated notation, the proof of the restricted case is the same so we first focus on the unrestricted case. From the A2.2, for the restricted case, we can find $\bar{g}$, $\bar{\eta}$ and $R$ corresponding to $\eta$ such that the stabilization radii $R$ of $\bar{\eta}$ satisfies the same stabilization property as $\eta$ in the sense of Definition~\ref{defi4}. Because $N_0$ is a bounded set, there exists an $r_1\in \mathbb{R}_+$ such that $N_0\subset B(0,r_1)$. For convenience, we write the random variables $Y:=\sum_{x\in \overline{\Xi}}\bar{g}(\Xi^x)\mathbf{1}_{d(x, N_0)<R(x)}$, $Y_r:=\sum_{x\in \overline{\Xi}}\bar{g}(\Xi^x)\mathbf{1}_{d(x, N_0)<R(x)<r}$, and write the event $\{Y\neq Y_r\}$ as $E_r$ for $r\in \mathbb{R}_+$. We can see that $$\mathbb{P}(E_r)\le \mathbb{P}(\{\mbox{there is at least one point }x\in \overline{\Xi}\mbox{ such that }d(x, N_0)\vee r<R(x)\})=:\mathbb{P}(E_r'),$$ and the right hand side is a decreasing function of $r$. We show that any one of the stabilization conditions implies that $\mathbb{P}\(E_r\)\to0$ as $r\to\infty$, that is, $Y_r$ converges to $Y$ almost surely. In fact, \begin{equation}\label{lma3.2}
\begin{aligned}
&\mathbb{P}(E_r')
\\ \le&\mathbb{P}\(\{\mbox{there is at least one point }x\in \overline{\Xi}\cap B(0,r_1+r)\mbox{ such that } r\le R(x)\}\)
\\&+\mathbb{P}\(\{\mbox{there is at least one point }x\in \overline{\Xi}\cap B(0,r_1+r)^c\mbox{ such that } |x|-r_1\le R(x)\}\).
\end{aligned} \end{equation} Using the property of Palm process, we can see that the first term of \Ref{lma3.2} satisfies \begin{align}
&\mathbb{P}\(\{\mbox{there is at least one point }x\in \overline{\Xi}\cap B(0,r_1+r)\mbox{ such that } r\le R(x)\}\)\nonumber
\\\le & \mathbb{E}\int_{B(0,r_1+r)}\mathbf{1}_{R(x)\ge r}\overline{\Xi}(dx)\nonumber
=\int_{B(0,r_1+r)}\mathbb{E}\mathbf{1}_{R(x, M_x,\Xi+\delta_{(x,M_x)})\ge r}\lambda dx \\= & \int_{B(0,r_1+r)}\mathbb{P}\(R(x, M_x,\Xi+\delta_{(x,M_x)})\ge r\)\lambda dx \nonumber
\\ \le & \int_{B(0,r_1+r)}\tau(r)\lambda dx= \frac{\lambda(r_1+r)^d\pi^{d/2}\tau(r)}{\Gamma(\frac{d}{2}+1)}\label{lma3.3} \end{align} and the second term is bounded by \begin{align}
&\mathbb{P}\(\{\mbox{there is at least one point }x\in \overline{\Xi}\cap B(0,r_1+r)^c\mbox{ such that } |x|-r_1\le R(x)\}\)\nonumber
\\\le &\mathbb{E}\int_{B(0,r_1+r)^c}\mathbf{1}_{R(x)\ge |x|-r_1}\overline{\Xi}(dx)\nonumber
= \int_{B(0,r_1+r)^c}\mathbb{E}\mathbf{1}_{R(x, M_x,\Xi+\delta_{(x,M_x)})\ge |x|-r_1}\lambda dx\\= & \int_{B(0,r_1+r)^c}\mathbb{P}\(R(x, M_x,\Xi+\delta_{(x,M_x)})\ge |x|-r_1\)\lambda dx \nonumber
\\ \le & \int_{B(0,r_1+r)^c}\tau(|x|-r_1)\lambda dx= \int_{r_1+r}^\infty\frac{d\lambda t^{d-1}\pi^{d/2}\tau(t-r_1)}{\Gamma(\frac{d}{2}+1)}dt.\label{lma3.4} \end{align} When the score function satisfies one of the stabilization conditions, both bounds in \Ref{lma3.3} and \Ref{lma3.4} converge to $0$ as $r\rightarrow \infty$, so $\mathbb{P}(E_r)\le \mathbb{P}(E_r')\rightarrow 0$ as $r\rightarrow \infty$.
Recall that we say two measures $\mu_1\le\mu_2$ if $\mu_1(A)\le\mu_2(A)$ for all measurable sets $A$. The non-singularity~\Ref{non-sin} ensures that, with a positive probability, the conditional distribution $\mathscr{L}\left(Y\middle|\sigma\(\Xi_{N_0^c}\)\right)$ is non-singular. Which means that we can find a $\sigma\(\Xi_{N_0^c}\)$ measurable random measure $\xi$ on $\mathbb{R}$ which is absolutely continuous, $\xi\le \mathscr{L}\left(Y\middle|\sigma\(\Xi_{N_0^c}\)\right)$ $a.s.$ and $\mathbb{P}\(\xi\(\mathbb{R}\)>0\)>0$. Since $\lim_{u\downarrow 0}\mathbb{P}\(\xi(\mathbb{R})>u\)=\mathbb{P}\(\xi\(\mathbb{R}\)>0\)>0$, we can find a $p>0$ such that $\mathbb{P}\(\xi(\mathbb{R})>p\)>4p$. Because $E_r'$ is decreasing in the sense of inclusion in $r$, and $\mathbb{P}\(E_r'\)\rightarrow 0$ as $r\rightarrow \infty,$ we can find an $R_0\in \mathbb{R}_+$ such that $\mathbb{P}(E_{R_0}')\le p^2$, which ensures \begin{equation}\label{difference}
\mathbb{P}\(\mathbb{P}\left(E_{R_0}'\middle|\sigma\(\Xi_{N_0^c}\)\right)>\frac{p}{2}\)\le 2p.
\end{equation} Writing $\tilde{Y}:=Y\mathbf{1}_{E_{R_0}'^c}$, $A_1:=\left\{d_{TV}\(\left.Y,\tilde{Y}\right|\sigma\(\Xi_{N_0^c}\)\)>p/2\right\}$, $A_2:=\{\xi(\mathbb{R})>p\}$, then $A_1$ and $A_2$ are both $\sigma\(\Xi_{N_0^c}\)$ measurable and $$
\mathbb{P}\(\mathbb{P}\left(d_{TV}\(\left.Y,\tilde{Y}\right|\sigma\(\Xi_{N_0^c}\)\)\right)>\frac{p}{2}\)\le
\mathbb{P}\(\mathbb{P}\left(E_{R_0}'\middle|\sigma\(\Xi_{N_0^c}\)\right)>\frac{p}{2}\)\le 2p, $$ giving
$\Pro(A_2\cap A_1^c)>2p$. For $\omega \in A_1^c\cap A_2$, $d_{TV}\(\left.Y,\tilde{Y}\right|\sigma\(\Xi_{N_0^c}\)\)(\omega)<p/2$ and $\xi(\omega)(\mathbb{R})>p$. By Lemma~\ref{non-singular1} and \Ref{difference}, there exists an absolutely continuous $\sigma\(\Xi_{N_0^c}\)$ measurable random measure $\tilde{\xi}$ such that $\tilde{\xi}\le \mathscr{L}\left(\tilde{Y}\middle|\sigma\(\Xi_{N_0^c}\)\right)$ $a.s.$ and $\mathbb{P}\(\tilde{\xi}\(\mathbb{R}\)>\frac{p}{2}\)>2p$. We write $\Xi'$ as an independent copy of $\Xi$ and the corresponding $Y_r$ and $\tilde{\xi}$ as $Y_r'$ and $\tilde{\xi}'$ respectively. Using Lemma~\ref{lma2}, we can find $\sigma\(\Xi_{N_0^c}, \Xi'_{N_0^c}\)$ measurable random variables $\Theta_1\ge 0,\ \Theta_2\ge 0$ and $U\in \mathbb{R}$ such that $\mathbb{P}(\Theta_1>0,\Theta_2>0)=4p^2$, \begin{align*}
\tilde{\xi}\star\tilde{\xi}'\ge \Theta_1 K_{\Theta_2}\star \delta_U. \end{align*} However, $$\lim_{\epsilon\downarrow 0}\mathbb{P}\left\{\Theta_1/\Theta_2\ge \epsilon, \Theta_2\ge\epsilon\right\}=\mathbb{P}\left\{\Theta_1>0,\Theta_2>0\right\}, $$ from Remark~\ref{remark1}, we can find an $\epsilon>0$ such that \begin{align*}& \mathbb{P}\left\{\tilde{\xi}\star\tilde{\xi}'\ge \epsilon^2K_\epsilon\star \delta_U\right\} \ge 2p^2 \end{align*} for a $\sigma\(\Xi_{N_0^c}, \Xi'_{N_0^c}\)$ measurable $U$. From the fact that we can write $Y_r=\tilde Y+Y_r\mathbf{1}_{E_{R_0}'}$, we have for any $B\in \mathscr{B}(\mathbb{R}\backslash\{0\})$, \begin{eqnarray*}
\Pro\(\left.Y_r\in B\right|\sigma\(\Xi_{N_0^c}\)\)&=&\Pro\(\left.\tilde Y\in B, E_{R_0}'^c\right|\sigma\(\Xi_{N_0^c}\)\)+\Pro\(\left.Y_r\in B, E_{R_0}'\right|\sigma\(\Xi_{N_0^c}\)\)\\
&\ge& \Pro\(\left.\tilde Y\in B,E_{R_0}'^c\right|\sigma\(\Xi_{N_0^c}\)\)\\
&=&\Pro\(\left.\tilde Y\in B\right|\sigma\(\Xi_{N_0^c}\)\). \end{eqnarray*} Hence \begin{align}
&\mathscr{L}\left(Y_r\middle|\sigma\(\Xi_{N_0^c}\)\right)(\cdot)\ge \mathscr{L}\left(\tilde Y\middle|\sigma\(\Xi_{N_0^c}\)\right)(\cdot\backslash\{0\})\nonumber\\
&\ge \tilde\xi (\cdot\backslash\{0\})=\tilde\xi (\cdot) \mbox{ a.s. for all }r\ge R_0.\label{forlma5.9} \end{align} Therefore, using $U\in \mathscr{A}$ to stand for $U$ being $\mathscr{A}$ measurable, we have \begin{align*}
&\sup_{U\in \sigma\(\Xi_{B(N_0,2r)\backslash N_0},\Xi'_{B(N_0,2r)\backslash N_0}\)} \Pro\left\{\mathscr{L}\left(Y_r\middle|\sigma\(\Xi_{B(N_0,2r)\backslash N_0}\)\right)\star\mathscr{L}\left(Y_r'\middle|\sigma\(\Xi'_{B(N_0,2r)\backslash N_0}\)\right)\ge \epsilon^2K_\epsilon\star \delta_U\right\}
\\
&=\sup_{U\in \sigma\(\Xi_{B(N_0^c)},\Xi'_{B(N_0^c)}\)} \Pro\left\{\mathscr{L}\left(Y_r\middle|\sigma\(\Xi_{B(N_0,2r)\backslash N_0}\)\right)\star\mathscr{L}\left(Y_r'\middle|\sigma\(\Xi'_{B(N_0,2r)\backslash N_0}\)\right)\ge \epsilon^2K_\epsilon\star \delta_U\right\}
\\
&= \sup_{U\in \sigma\(\Xi_{B(N_0^c)},\Xi'_{B(N_0^c)}\)}\Pro\left\{\mathscr{L}\left(Y_r\middle|\sigma\(\Xi_{N_0^c}\)\right)\star\mathscr{L}\left(Y_r'\middle|\sigma\(\Xi'_{N_0^c}\)\right)\ge \epsilon^2K_\epsilon\star \delta_U\right\}
\\
&\ge \sup_{U\in \sigma\(\Xi_{B(N_0^c)},\Xi'_{B(N_0^c)}\)}\Pro\left\{\tilde{\xi}\star\tilde{\xi}'\ge \epsilon^2K_\epsilon\star \delta_U \right\}\\
&\ge 2p^2, \end{align*} which ensures that, for any $r>R_0$, we can find a $\sigma\(\Xi_{B(N_0,2r)\backslash N_0},\Xi'_{B(N_0,2r)\backslash N_0}\)$ measurable $U$ such that \begin{equation}
\mathbb{P}\left\{\mathscr{L}\left(Y_r\middle|\sigma\(\Xi_{B(N_0,2r)\backslash N_0}\)\right)\star\mathscr{L}\left(Y_r'\middle|\sigma\(\Xi'_{B(N_0,2r)\backslash N_0}\)\right)\ge \epsilon^2K_\epsilon\star \delta_U\right\}\ge p^2.\label{lemma4.4proofa1} \end{equation}
If $\alpha\le (2(4r+2r_1))^d$, \Ref{statement1} is trivial with $C=\left\{2(4+\frac{2r_1}{R_0})\right\}^{d/2}$, so we now assume $\alpha> \{2(4r+2r_1)\}^d$. From the structure of $\Xi$, we can see that $\Xi(A,D)\overset{d}{=}\Xi(x+A,D)$ and $\Xi(A,D)$ is independent of $\Xi(B,D)$ for all disjoint $A$, $B\in\mathscr{B}(\mathbb{R}^d)$, $D\in \mathscr{T}$ and $x\in \mathbb{R}^d$. For a fixed $r>R_0$, we can divide $\Gamma_\alpha$ into disjoint cubes $\mathbb{C}_1,\cdots,\mathbb{C}_{m_{\alpha,r}}$ with edge length $4r+2r_1$ and centers $c_1$, $\cdots$, $c_{m_{\alpha,r}}$, aiming to maximize the number of cubes, so $m_{\alpha,r}\sim \alpha(4r+2r_1)^{-d}$, which has order $O\(\alpha r^{-d}\)$. Without loss of generality, we can assume that $m_{\alpha,r}$ is even or we simply delete one from them and the above properties still holds. For $i\le m_{\alpha,r}$, we define $A_i=c_i+N_0$, $B_i=B(A_i,r)$, $C_i=B(B_i,r)$, $D_i=C_i\backslash A_i$, ${\cal N}_{0,\alpha,r}:=\cup_{1\le i\le m_{\alpha,r}}A_i$, ${\cal N}_{1,\alpha,r}:=\cup_{1\le i\le m_{\alpha,r}}B_i$, ${\cal N}_{2,\alpha,r}:=\cup_{1\le i\le m_{\alpha,r}} D_i$, ${\mathscr{F}}_{1,\alpha,r}:=\sigma(\Xi_{\mathbb{R}^d\backslash {\cal N}_{0,\alpha,r}})$, ${\mathscr{F}}_{2,\alpha,r}:=\sigma(\Xi_{ {\cal N}_{2,\alpha,r}})$, $W_{\alpha,r}^0=\int_{{\cal N}_{1,\alpha,r}}\bar{g}(\Xi^{x})\mathbf{1}_{R(x)<r}\overline{\Xi}(dx)$ and $W_{\alpha,r}^1=W_{\alpha,r}-W_{\alpha,r}^0$. Note that for all $x$ such that $d(x,\partial \Gamma_\alpha)\ge r$, $\eta((x,m),\Xi,\Gamma_\alpha)\mathbf{1}_{\bar{R}(x,\alpha)<r}= \bar\eta((x,m),\Xi)\mathbf{1}_{R(x)<r}$ for all $(x,m)\in \Xi$ a.s. From the definition of total variation distance, $d_{TV}(W_{\alpha,r},W_{\alpha,r}+\gamma)=\sup_{A\in \mathscr{B}(R)}\(\mathbb{P}(W_{\alpha,r}\in A)-\mathbb{P}(W_{\alpha,r}\in A-\gamma)\)$, hence the tower property ensures \begin{align}
&d_{TV}\left(W_{\alpha,r},W_{\alpha,r}+\gamma\right)\nonumber
\\=&\sup_{A\in \mathscr{B}(\mathbb{R})}\mathbb{E}\left(\mathbf{1}_{W_{\alpha,r}\in A}-\mathbf{1}_{W_{\alpha,r}\in A-\gamma}\right)\nonumber
\\=&\sup_{A\in \mathscr{B}(\mathbb{R})}\mathbb{E}\left(\mathbb{E}\left(\mathbf{1}_{W_{\alpha,r}\in A}-\mathbf{1}_{W_{\alpha,r}\in A-\gamma}|\mathscr{F}_{1,\alpha,r}\right)\right)\nonumber
\\=&\sup_{A\in \mathscr{B}(\mathbb{R})}\mathbb{E}\left(\mathbb{E}\left(\mathbf{1}_{W_{\alpha,r}^0\in A-W_{\alpha,r}^1}-\mathbf{1}_{W_{\alpha,r}^0\in A-\gamma-W_{\alpha,r}^1}|\mathscr{F}_{1,\alpha,r}\right)\right)\nonumber
\\\le&\mathbb{E}\left(\sup_{A\in \mathscr{B}(\mathbb{R})}\left[\mathbb{E}\left(\mathbf{1}_{W_{\alpha,r}^0\in A}-\mathbf{1}_{W_{\alpha,r}^0\in A-\gamma}|\mathscr{F}_{1,\alpha,r}\right)\right]\right)\nonumber
\\=&\mathbb{E}\left(\sup_{A\in \mathscr{B}(\mathbb{R})}\left[\mathbb{E}\left(\mathbf{1}_{W_{\alpha,r}^0\in A}-\mathbf{1}_{W_{\alpha,r}^0\in A-\gamma}|\mathscr{F}_{2,\alpha,r}\right)\right]\right),\label{lma3.7} \end{align}where the last equality follows from the fact that $W_{\alpha,r}^0$ depends on ${\mathscr{F}}_{2,\alpha,r}$ in ${\mathscr{F}}_{1,\alpha,r}$.
From \Ref{lma3.7}, to show \Ref{statement1}, it is sufficient to show that $$\mathbb{E}\left(\sup_{A\in \mathscr{B}(\mathbb{R})}\left[\mathbb{E}\left(\mathbf{1}_{W_{\alpha,r}^0\in A}-\mathbf{1}_{W_{\alpha,r}^0\in A-\gamma}|{\mathscr{F}}_{2,\alpha,r}\right)\right]\right)\le (|\gamma|\vee 1)O\left(\alpha^{-\frac{1}{2}}r^{\frac{d}{2}}\right).$$ Using the fact that $\int_{B_i} \bar{g}(\Xi^{x})\mathbf{1}_{R(x)<r}\overline{\Xi}(dx)$ depends only on $\sigma(\Xi_{D_i})$ in ${\mathscr{F}}_{2,\alpha,r}$ for $i\le m_{\alpha,r} $, and from the independence of $\sigma(\Xi_{D_i})$ for different $i$, we can see that \begin{align}
\mathscr{L}\(W_{\alpha,r}^0|{\mathscr{F}}_{2,\alpha,r}\)&=\mathscr{L}\(\sum_{i=1}^{m_{\alpha,r}}\left.\int_{B_i} \bar{g}(\Xi^{x})\mathbf{1}_{R(x)<r}\overline{\Xi}(dx)\right|{\mathscr{F}}_{2,\alpha,r}\)\nonumber
\\&=\mathscr{L}\(\left.\sum_{i=1}^{m_{\alpha,r}}\int_{B_i} \bar{g}(\Xi^{x})\mathbf{1}_{R(x)<r}\overline{\Xi}(dx)\right|\sigma\(\Xi_{D_i}, i\le m_{\alpha,r}\)\).\label{lemma4.4proofa2}
\end{align} Using \Ref{lemma4.4proofa1}, we obtain \begin{align*}
&\mathscr{L}\(\left.\sum_{i=2j-1}^{2j}\int_{B_i} \bar{g}(\Xi^{x})\mathbf{1}_{R(x)<r}\overline{\Xi}(dx)\right|\sigma\(\Xi_{D_i}, i\le m_{\alpha,r}\)\)\\
&=\mathscr{L}\(\left.X_{1,j}\(1-J_{1,j}\)+X_{2,j}J_{1,j}\(1-J_{2,j}\)+(X_{3,j}+U_j)J_{1,j}J_{2,j}\right|\sigma\(\Xi_{D_{2j-1}}, \Xi_{D_{2j}}\)\), \end{align*} where $J_{1,j}$, $J_{2,j}$ and $U_j$ are $\sigma\(\Xi_{D_{2j-1}}, \Xi_{D_{2j}}\)$ measurable with $\Pro(J_{1,j}=1)=1-\Pro(J_{1,j}=0)=p^2$, $\Pro(J_{2,j}=1)=1-\Pro(J_{2,j}=0)={\epsilon}^2$, $J_{1,j} \perp \!\!\! \perp J_{2,j}$, $X_{1,j} $ and $X_{2,j} $ are $\sigma\(\Xi_{B_{2j-1}}, \Xi_{B_{2j}}\)$ measurable, and $X_{3,j}\sim K_{\epsilon},\ 1\le j\le m_{\alpha,r}/2,$ are i.i.d. and independent of $\sigma\(\Xi_{D_i}, i\le m_{\alpha,r}\)$. Hence, define $\Sigma_1:= \sum_{j=1}^{m_{\alpha,r}/2}(X_{1,j}\(1-J_{1,j}\)+X_{2,j}J_{1,j}\(1-J_{2,j}\)+(X_{3,j}+U_j)J_{1,j}J_{2,j})$, $\Sigma_2:= \sum_{j=1}^{m_{\alpha,r}/2}X_{3,j}J_{1,j}J_{2,j}$, $\Sigma_{3,l}:= \sum_{j=1}^lX_{3,j}$ and $I\sim${\rm Binomial}$(m_{\alpha,r}/2,{\epsilon}^2p^2)$ which is independent of $\{X_{3,j}:\ j\le m_{\alpha,r}/2\}$, it follows from \Ref{lma3.7} and \Ref{lemma4.4proofa2} that \begin{align}
&d_{TV}\left(W_{\alpha,r},W_{\alpha,r}+\gamma\right)\nonumber
\\ \le&\mathbb{E}\left(\sup_{A\in \mathscr{B}(\mathbb{R})}\left[\mathbb{E}\left(\mathbf{1}_{\Sigma_1\in A}-\mathbf{1}_{\Sigma_1\in A-\gamma}|\sigma\(\Xi_{D_i}, i\le m_{\alpha,r}\)\right)\right]\right)\nonumber
\\ \le&\mathbb{E}\left(\sup_{A\in \mathscr{B}(\mathbb{R})}\left[\mathbb{E}\left(\mathbf{1}_{\Sigma_2\in A}-\mathbf{1}_{\Sigma_2\in A-\gamma}|\sigma\(\Xi_{D_i}, i\le m_{\alpha,r}\)\right)\right]\right)\nonumber
\\ =&\mathbb{E}\left(\sup_{A\in \mathscr{B}(\mathbb{R})}\left[\mathbb{E}\left(\mathbf{1}_{\Sigma_{3,I}\in A}-\mathbf{1}_{\Sigma_{3,I}\in A-\gamma}|I\right)\right]\right)\nonumber
\\\le&\Pro(I\le (\mathbb{E} I)/2)+\sum_{(\mathbb{E} I)/2<j\le m_{\alpha,r}/2}\sup_{A\in \mathscr{B}(\mathbb{R})}\left[\mathbb{E}\left(\mathbf{1}_{\Sigma_{3,j}\in A}-\mathbf{1}_{\Sigma_{3,j}\in A-\gamma}\right)\right]\Pro(I=j)\nonumber
\\\le& O\left(\alpha^{-1}r^d\right)+O\left(\alpha^{-\frac{1}{2}}r^{\frac{d}{2}}\right)|\gamma|= \left(|\gamma|\vee 1\right)O\left(\alpha^{-\frac{1}{2}}r^{\frac{d}{2}}\right), \label{lma3.10} \end{align} where the first term of \Ref{lma3.10} is from Chebyshev's inequality and the second terms is due to Lemma~\ref{lma1}. This completes the proof of \Ref{statement1}.
In terms of \Ref{statement2}, since range-bound implies polynomially stabilizing with arbitrary order $\beta$, \Ref{statement1} still holds for all $r>R_0$. On the other hand, $\bar{W}_\alpha=\bar{W}_r$ a.s. when $r>t$ for some positive constant $t$, \Ref{statement2} follows by taking $r=R_0\vee t+1$.
The claim \Ref{statement3} can be proved by replacing $W_\alpha$ with $\bar{W}_\alpha$; $W_{\alpha,r}$ with $\bar{W}_{\alpha,r}$; {$\bar{g}$ with $g$,} $W^0_{\alpha,r}$ with $\bar{W}^0_{\alpha,r}$; $W^1_{\alpha,r}$ with $\bar{W}^1_{\alpha,r}$; $\Xi^{x}$ by $\Xi^{\Gamma_\alpha,x}$; $R(x)$ with $\bar{R}(x,\alpha)$; $R(x,M_x,\Xi+\delta_{(x, M_x)})$ with $\bar{R}(x,M_x,\alpha,\Xi+\delta_{(x, M_x)})$ and redefining ${\mathscr{F}}_{1,\alpha,r}:=\sigma(\Xi_{\Gamma_\alpha\backslash {\cal N}_{0,\alpha,r}})$. The bound \Ref{statement4} can be argued in the same way as that for \Ref{statement2}. \qed
\noindent{\it Proof of Corollary~\ref{cor1}.} The proof can be easily adapted from the second half of the proof of Lemma~\ref{lma3} and we start with \Ref{lma3coro02}. If $\alpha^{-
\frac{1}{d}}(1-2C)^{-1}(4r+2r_1)>\frac{1}{3}$, \Ref{lma3coro02} is obvious because the total variation distance is bounded above by $1$. Now we assume $\alpha^{-
\frac{1}{d}}(1-2C)^{-1}(4r+2r_1)\le\frac{1}{3}$. Similar to the proof of Lemma~\ref{lma3}, we embed disjoint cubes with edge length $4r+2r_1$ into $\Gamma_\alpha\backslash \(N_{\alpha,r}^{(1)}\cup N_{\alpha,r}^{(2)}\cup N_{\alpha,r}^{(3)}\)$, aiming to maximize the number $m_{\alpha,r}$ of the cubes. Without loss, we assume that $m_{\alpha,r}$ is even. Then, we have $$\alpha(1-2C)^{d}(12r+6r_1)^{-d}{-1}\le m_{\alpha,r}\le \alpha(1-2C)^{d}(4r+2r_1)^{-d},$$ giving $m_{\alpha,r}=O(\alpha r^{-d})$.
We use the same notations as in the proof of Lemma~\ref{lma3} but with $\Gamma_\alpha$ replaced by $\Gamma_\alpha\backslash(N_{\alpha,r}^{(1)}\cup N_{\alpha,r}^{(2)}\cup N_{\alpha,r}^{(3)})$ and define $\mathscr{F}_{2,\alpha,r}':=\sigma\(\Xi_{{\cal N}_{2,\alpha,r}\cup N_{\alpha,r}^{(2)}}\)$. Bearing in mind that $N_{\alpha,r}^{(1)}\cup N_{\alpha,r}^{(2)}\cup N_{\alpha,r}^{(3)}$ is excluded in the $m_{\alpha,r}$ cubes, we have $\mathscr{F}_{0,\alpha,r}\subset \mathscr{F}_{1,\alpha,r}$, giving the following analogous result of \Ref{lma3.7}: \begin{align*}
& d_{TV}\left(\bar{W}_{\alpha,r}',\bar{W}_{\alpha,r}'+h_{\alpha,r}\(\Xi_{N_{\alpha, r}^{(2)}}\)\middle|\mathscr{F}_{0,\alpha,r}\right)
\\=&\sup_{A\in \mathscr{B}(\mathbb{R})}\mathbb{E}\left(\mathbf{1}_{\bar{W}_{\alpha,r}'\in A}-\mathbf{1}_{\bar{W}_{\alpha,r}\in A-h_{\alpha,r}\(\Xi_{N_{\alpha, r}^{(2)}}\)}\middle|\mathscr{F}_{0,\alpha,r}\right)
\\=&\sup_{A\in \mathscr{B}(\mathbb{R})}\mathbb{E}\left(\mathbb{E}\left(\mathbf{1}_{\bar{W}_{\alpha,r}\in A}-\mathbf{1}_{\bar{W}_{\alpha,r}\in A-h_{\alpha,r}\(\Xi_{N_{\alpha, r}^{(2)}}\)}\middle|\mathscr{F}_{1,\alpha,r}\right)\middle|\mathscr{F}_{0,\alpha,r}\right)
\\=&\sup_{A\in \mathscr{B}(\mathbb{R})}\mathbb{E}\left(\mathbb{E}\left(\mathbf{1}_{\bar{W}_{\alpha,r}^0\in A}-\mathbf{1}_{\bar{W}_{\alpha,r}^0\in A-h_{\alpha,r}\(\Xi_{N_{\alpha, r}^{(2)}}\)}\middle|\mathscr{F}_{1,\alpha,r}\right)\middle|\mathscr{F}_{0,\alpha,r}\right)
\\\le &\mathbb{E}\left(\sup_{A\in \mathscr{B}(\mathbb{R})}\[\mathbb{E}\left(\mathbf{1}_{\bar{W}_{\alpha,r}^0\in A}-\mathbf{1}_{\bar{W}_{\alpha,r}^0\in A-h_{\alpha,r}\(\Xi_{N_{\alpha, r}^{(2)}}\)}\middle|\mathscr{F}_{1,\alpha,r}\right)\right]\middle|\mathscr{F}_{0,\alpha,r}\right)
\\ =&\mathbb{E}\left(\sup_{A\in \mathscr{B}(\mathbb{R})}\[\mathbb{E}\left(\mathbf{1}_{\bar{W}_{\alpha,r}^0\in A}-\mathbf{1}_{\bar{W}_{\alpha,r}^0\in A-h_{\alpha,r}\(\Xi_{N_{\alpha, r}^{(2)}}\)}\middle|\mathscr{F}_{2,\alpha,r}'\right)\right]\middle|\mathscr{F}_{0,\alpha,r}\right). \end{align*} The remaining part is a line-by-line repetition of the proof of Lemma~\ref{lma3} with $\gamma$ replaced by $h_{\alpha,r}\(\Xi_{N_{\alpha, r}^{(2)}}\)$ and expectation replaced by the conditional expectation given $\mathscr{F}_{0,\alpha,r}$, leading to \begin{align}
& d_{TV}\left(\bar{W}_{\alpha,r}',\bar{W}_{\alpha,r}'+h_{\alpha,r}\(\Xi_{N_{\alpha, r}^{(2)}}\)\middle|\mathscr{F}_{0,\alpha,r}\right)\nonumber
\\\le&\mathbb{E}\left(\sup_{A\in \mathscr{B}(\mathbb{R})}\[\mathbb{E}\left(\mathbf{1}_{\bar{W}_{\alpha,r}^0\in A}-\mathbf{1}_{\bar{W}_{\alpha,r}^0\in A-h_{\alpha,r}\(\Xi_{N_{\alpha, r}^{(2)}}\)}\middle|\mathscr{F}_{2,\alpha,r}'\right)\right]\middle|\mathscr{F}_{0,\alpha,r}\right)\nonumber
\\\le&\mathbb{E}\left(\mathbb{E}\left[\sup_{A\in \mathscr{B}(\mathbb{R})}\left(\mathbf{1}_{{\Sigma_{3,I}}\in A}-\mathbf{1}_{{\Sigma_{3,I}}\in A-h_{\alpha,r}\(\Xi_{N_{\alpha, r}^{(2)}}\)}\middle|I,\sigma\(\Xi_{N_{\alpha,r}^{(2)}}\)\right)\right] \middle|\mathscr{F}_{0,\alpha,r}\right)\label{cor1.1}
\\\le&\sum_{(\mathbb{E} I)/2<j\le m_{\alpha,r}/2}\Pro(I=j)\mathbb{E}\left(\mathbb{E}\left[\sup_{A\in \mathscr{B}(\mathbb{R})}\left(\mathbf{1}_{{\Sigma_{3,j}}\in A}-\mathbf{1}_{{\Sigma_{3,j}}\in A-h_{\alpha,r}\(\Xi_{N_{\alpha, r}^{(2)}}\)}\middle| \sigma\(\Xi_{N_{\alpha,r}^{(2)}}\)\right)\right]\middle|\mathscr{F}_{0,\alpha,r}\right)\nonumber
\\&+\Pro(I\le (\mathbb{E} I)/2)\nonumber
\\\le& O\left(\alpha^{-1}r^d\right)+O\left(\alpha^{-\frac{1}{2}}r^{\frac{d}{2}}\right)\mathbb{E}\left(\left|h_{\alpha,r}\(\Xi_{N_{\alpha, r}^{(2)}}\)\right|\middle| \mathscr{F}_{0,\alpha,r}\right)\label{cor1.5}
\\=& \mathbb{E}\left(\left|h_{\alpha,r}\(\Xi_{N_{\alpha, r}^{(2)}}\)\right|\vee 1\middle| \mathscr{F}_{0,\alpha,r}\right)O\left(\alpha^{-\frac{1}{2}}r^{\frac{d}{2}}\right),\nonumber \end{align} where \Ref{cor1.1} follows from the fact that $\sup_{A\in \mathscr{B}(\mathbb{R})}\left[\mathbf{1}_{{\Sigma_{3,I}}\in A}-\mathbf{1}_{{\Sigma_{3,I}}\in A-h_{\alpha,r}\(\Xi_{N_{\alpha, r}^{(2)}}\)}\right]$ is a function of $I$, $\Xi_{N_{\alpha, r}^{(2)}}$, the first term of \Ref{cor1.5} is from Chebyshev's inequality and the second term is due to Lemma~\ref{lma1}. This completes the proof for the statement of $\bar{W}_{\alpha,r}$.
The claim \Ref{lma3coro01} can be proved by replacing corresponding counterparts $\bar{W}_{\alpha,r}$ with $W_{\alpha,r}$; $\bar{W}'_{\alpha,r}$ with $W'_{\alpha,r}$; $\bar{W}^0_{\alpha,r}$ with $W^0_{\alpha,r}$. \qed
The moments of $W_{\alpha,r}$ and $W_\alpha$ (resp. $\bar{W}_{\alpha,r}$ and $\bar{W}_\alpha$) can be established using the ideas in \cite[Section~4]{XY15}. Let $\|X\|_p:=\mathbb{E}\(|X|^p\)^{\frac{1}{p}}$ be the $L_p$ norm of $X$ provided it is finite.
\begin{lma}\label{lma11}
\begin{description}
\item{(a)} (unrestricted case) If the score function $\eta$ satisfies $k'$-th moment condition \Ref{thm2.1} with $k'>k\ge 1$, then
$\max_{0< l\le k}\left\{\|W_{\alpha}\|_l,\|W_{\alpha,r}\|_l\right\}\le C\alpha.$
\item{(b)} (restricted case) If the score function $\eta$ satisfies $k'$-th moment condition \Ref{thm2.1r} with $k'>k\ge 1$, then $\max_{0< l\le k}\left\{\|\bar{W}_{\alpha}\|_l,\|\bar{W}_{\alpha,r}\|_l\right\}\le C\alpha.$
\end{description} \end{lma}
\noindent{\it Proof.} The proof is adapted from that of \cite[Lemma~4.1]{XY15}. We use the notations as in the proof of Lemma~\ref{lma3} and start with the restricted case. To this end, it suffices to show $\|\bar{W}_{\alpha}\|_k{\vee \|W_{\alpha,r}\|_k}\le C\alpha$ and the claim follows from H\"older's inequality. Let $N_\alpha:=\left|\bar\Xi_{\Gamma_{\alpha}}\right|$, then $N_\alpha$ follows Poisson distribution with parameter $\alpha \lambda$. Using Minkowski's inequality, we obtain
\begin{align}
\|\bar{W}_{\alpha}\|_k&\le\left\|\sum_{x\in \bar{\Xi}_{\Gamma_\alpha}}\left|g_\alpha\(x,\Xi\)\right|\right\|_k\nonumber
\\=&\left\|\sum_{x\in \bar{\Xi}_{\Gamma_\alpha}}\left|g_\alpha\(x,\Xi\)\right|\(\mathbf{1}_{N_\alpha\le \alpha\lambda}+\sum_{j=0}^\infty\mathbf{1}_{\alpha\lambda 2^{j}<N_\alpha\le \alpha\lambda 2^{j+1}}\)\right\|_k\nonumber
\\\le& \left\|\sum_{x\in \bar{\Xi}_{\Gamma_\alpha}}\left|g_\alpha\(x,\Xi\)\right|\mathbf{1}_{N_\alpha\le \alpha\lambda}\right\|_k+\sum_{j=0}^\infty \left\|\sum_{x\in \bar{\Xi}_{\Gamma_\alpha}}\left|g_\alpha\(x,\Xi\)\right|\mathbf{1}_{\alpha\lambda 2^{j}<N_\alpha\le \alpha\lambda 2^{j+1}}\right\|_k.\label{lma11.1} \end{align} Let $s=\frac{k'}{k}>1$ and $t$ be its conjugate, i.e., $\frac{1}{s}+\frac{1}{t}=1$, using H\"older's inequality and Minkowski's inequality, for any $j\in \mathbb{N}$, we have \begin{align}
&\left\|\sum_{x\in \bar{\Xi}_{\Gamma_\alpha}}\left|g_\alpha\(x,\Xi\)\right|\mathbf{1}_{\alpha\lambda 2^{j}<N_\alpha\le \alpha\lambda 2^{j+1}}\right\|_k\nonumber
\\=&\left\|\sum_{x\in \bar{\Xi}_{\Gamma_\alpha}}\left|g_\alpha\(x,\Xi\)\right|\mathbf{1}_{N_\alpha\le \alpha\lambda 2^{j+1}}\mathbf{1}_{\alpha\lambda 2^{j}<N_\alpha }\right\|_k\nonumber
\\\le&\left\|\sum_{x\in \bar{\Xi}_{\Gamma_\alpha}}\left|g_\alpha\(x,\Xi\)\right|\mathbf{1}_{N_\alpha\le \alpha\lambda 2^{j+1}}\right\|_{k'}\(\mathbb{P}\(N_\alpha> \alpha\lambda 2^{j}\)\)^{\frac{1}{kt}}\nonumber
\\ =& \left\|\sum_{x\in \bar{\Xi}_{\Gamma_\alpha}}\left|g_\alpha\(x,\Xi\)\right|\mathbf{1}_{N_\alpha\le \alpha\lambda 2^{j+1}}\right\|_{k'}\mathbb{P}\(N_\alpha-\alpha\lambda>\alpha\lambda \(2^{j}-1\)\)^{\frac{1}{kt}}.\label{lma11.2} \end{align}
For the term $\|\cdot\|_{k'}$ in \Ref{lma11.2}, we have \begin{align}
&\left\|\sum_{x\in \bar{\Xi}_{\Gamma_\alpha}}\left|g_\alpha\(x,\Xi\)\right|\mathbf{1}_{N_\alpha\le n}\right\|_{k'}\nonumber
\\=&\left\{\mathbb{E}\[\(\sum_{j=1}^n\sum_{x\in \bar{\Xi}_{\Gamma_\alpha}}\left|g_\alpha\(x,\Xi\)\right|\mathbf{1}_{N_\alpha=j}\)^{k'}\right]\right\}^{\frac{1}{k'}}\nonumber
\\=& \left\{\sum_{j=1}^n\mathbb{E}\[\(\sum_{x\in \bar{\Xi}_{\Gamma_\alpha}}\left|g_\alpha\(x,\Xi\)\right|\mathbf{1}_{N_\alpha=j}\)^{k'}\right]\right\}^{\frac{1}{k'}},\label{lma11.4}
\end{align} where the first equality holds because $\sum_{x\in \bar{\Xi}_{\Gamma_\alpha}}\left|g_\alpha\(x,\Xi\)\right|^{k'}\mathbf{1}_{N_\alpha=0}=0$ and the last equality follows from the fact that $\{N_\alpha=j\}$, $1\le j\le n$, are disjoint events. On $\{N_\alpha=j\}$ for some fixed $j\in \mathbb{N}$, if we write $j$ points in $\Xi\cap \Gamma_\alpha$ as $\{(x_1,m_1),\dots,(x_j,m_j)\}$ and let $\{\(U_{\alpha,i},M_i\)\}_{i\in \mathbb{N}}$ be a sequence of {i.i.d.}\ random elements having distribution $U\(\Gamma_\alpha\)\times \mathscr{L}_T$ and be independent of $\Xi$, where $U\(\Gamma_\alpha\)$ is the uniform distribution on $\Gamma_\alpha$, then \begin{align}
&\mathbb{E}\[\(\sum_{x\in \bar{\Xi}_{\Gamma_\alpha}}\left|g_\alpha\(x,\Xi\)\right|\mathbf{1}_{N_\alpha=j}\)^{k'}\right]\nonumber
\\ \le& \left\{\sum_{i=1}^j\mathbb{E}\[\left|g_\alpha\((x_i,m_i),\Xi\)\right|^{k'}\mathbf{1}_{N_\alpha=j}\right]^{\frac{1}{k'}}\right\}^{k'}\nonumber
\\= &j^{k'}\mathbb{E}\[\left|g_\alpha\(\(U_{\alpha,i},M_i\), \(\sum_{i=1}^j \delta_{\(U_{\alpha,i},M_i\)}\)\)\right|^{k'}\mathbf{1}_{N_\alpha=j}\right],\label{lma11.5}
\end{align} where the inequality follows from Minkovski's inequality and the equality follows from the fact that when $\left|\bar{\Xi}\cap \Gamma_\alpha\right|$ is fixed, points in $\bar{\Xi}\cap \Gamma_\alpha$ are independent and follow uniform distribution on $\Gamma_\alpha$. Combining \Ref{lma11.4} and \Ref{lma11.5}, we have \begin{align}
&\left\|\sum_{x\in \bar{\Xi}_{\Gamma_\alpha}}\left|g_\alpha\(x,\Xi\)\right|\mathbf{1}_{N_\alpha\le n}\right\|_{k'}\nonumber\nonumber
\\ \le &\left\{\sum_{j=1}^nj^{k'}\mathbb{E}\[\left|g_\alpha\(\(U_{\alpha,1},M_1\), \(\sum_{i=1}^j \delta_{\(U_{\alpha,i},M_i\)}\)\)\right|^{k'}\mathbf{1}_{N_\alpha=j}\right]\right\}^{\frac{1}{k'}}\nonumber
\\ =&\left\{\sum_{j=1}^n\lambda \alpha j^{k'-1}\mathbb{E}\[\left|g_\alpha\(\(U_{\alpha,1},M_1\), \(\sum_{i=1}^j \delta_{\(U_{\alpha,i},M_i\)}\)\)\right|^{k'}\mathbf{1}_{N_\alpha=j-1}\right]\right\}^{\frac{1}{k'}}\nonumber
\\ \le & (\lambda\alpha)^{\frac{1}{k'}}n^{\frac{k'-1}{k'}}\left\{\mathbb{E}\[\sum_{j=0}^{n-1}\left|g_\alpha\(\(U_{\alpha,1},M_1\), \(\sum_{i=1}^{j+1} \delta_{\(U_{\alpha,i},M_i\)}\)\)\right|^{k'}\mathbf{1}_{N_\alpha=j}\right]\right\}^{\frac{1}{k'}}\nonumber
\\ \le & (\lambda\alpha)^{\frac{1}{k'}}n^{\frac{k'-1}{k'}}\left\{\mathbb{E}\[\sum_{j=0}^{\infty}\left|g_\alpha\(\(U_{\alpha,1},M_1\), \(\sum_{i=1}^{j+1} \delta_{\(U_{\alpha,i},M_i\)}\)\)\right|^{k'}\mathbf{1}_{N_\alpha=j}\right]\right\}^{\frac{1}{k'}}\nonumber
\\ =& (\lambda\alpha)^{\frac{1}{k'}}n^{\frac{k'-1}{k'}}\left\{\int_{\Gamma_\alpha}\mathbb{E}\[\left|g_\alpha\((x,M),{\Xi_{\Gamma_\alpha}+\delta_{(x,M)}}\)\right|^{k'}\frac{1}{\alpha}dx\right]\right\}^{\frac{1}{k'}}\nonumber
\\\le& (\lambda\alpha)^{\frac{1}{k'}}n^{\frac{k'-1}{k'}} C_0^{\frac{1}{k'}},\label{lma11.6} \end{align}
where the first equality follows from the fact that $N_\alpha$ is independent of $\{\(U_{\alpha,i},M_i\)\}_{i\in \mathbb{N}}$ and $\mathbb{P}(N_\alpha=j)=\frac{\lambda \alpha}{j}\mathbb{P}(N_\alpha=j-1)$, the last equality follows from the construction of marked Poisson point process.
Combining \Ref{lma11.2} and \Ref{lma11.6}, we have \begin{align}
\left\|\sum_{x\in \bar{\Xi}_{\Gamma_\alpha}}\left|g_\alpha\(x,\Xi\)\right|\mathbf{1}_{\alpha\lambda 2^{j}<N_\alpha\le \alpha\lambda 2^{j+1}}\right\|_k\le \alpha\lambda 2^{\frac{(k'-1)(j+1)}{k'}}C_0^{\frac{1}{k'}} \mathbb{P}\(N_\alpha-\alpha\lambda>\alpha\lambda \(2^{j}-1\)\)^{\frac{1}{kt}}.\label{lma11.7} \end{align} Using \Ref{lma11.6} and H\"older's inequality, we have \begin{align}
&\left\|\sum_{x\in \bar{\Xi}_{\Gamma_\alpha}}\left|g_\alpha\(x,\Xi\)\right|\mathbf{1}_{N_\alpha\le \alpha\lambda}\right\|_k\le\left\|\sum_{x\in \bar{\Xi}_{\Gamma_\alpha}}\left|g_\alpha\(x,\Xi\)\right|\mathbf{1}_{N_\alpha\le \alpha\lambda}\right\|_{k'}\le \alpha\lambda C_0^{\frac{1}{k'}}.\label{lma11.3} \end{align} Combining\Ref{lma11.7} and \Ref{lma11.3}, together with the fact that $\mathbb{P}\(N_\alpha-\alpha\lambda>\alpha\lambda k\)$ decrease exponentially fast with respective to $k$, we have from \Ref{lma11.1} that
$$\|\bar{W}_{\alpha}\|_k\le\left\|\sum_{x\in \bar{\Xi}_{\Gamma_\alpha}}\left|g_\alpha\(x,\Xi\)\right|\right\|_k\le C\alpha.$$ The proof of (b) is completed by observing that, for arbitrary $r\in \mathbb{R}_+$,
$$\|\bar{W}_{\alpha,r}\|_k=\left\|\sum_{x\in \bar{\Xi}_{\Gamma_\alpha}}g_\alpha\(x,\Xi\)\mathbf{1}_{R(x)\le r}\right\|_k\le\left\|\sum_{x\in \bar{\Xi}_{\Gamma_\alpha}}\left|g_\alpha\(x,\Xi\)\right|\right\|_k\le C\alpha.$$
The claim (a) can be established by replacing $\bar{W}_{\alpha}$ with $W_\alpha$, $\bar{W}_{\alpha,r}$ with $W_{\alpha,r}$; $\sum_{i=1}^j \delta_{\(U_{\alpha,i},M_i\)}$ with $\sum_{i=1}^j \delta_{\(U_{\alpha,i},M_i\)}+\Xi_{\Gamma_\alpha^c}$; {$g_\alpha(x,\mathscr{X})$ as $g(\mathscr{X}^{x})$}. \qed
\vskip10pt \begin{re} {\rm~The proof of Lemma~\ref{lma11} does not depend on the shape of $\Gamma_\alpha$, so the claims still hold if we replace $\Gamma_\alpha$ with a set $A\in\mathscr{B}(\mathbb{R}^d)$ and $\alpha$ in the upper bound with the volume of $A$.} \end{re}
With these preparations, we are ready to bound the differences $\left|{\rm Var}\(W_{\alpha}\)-{\rm Var}\(W_{\alpha,r}\)\right|$ and $\left|{\rm Var}\(\bar{W}_{\alpha}\)-{\rm Var}\(\bar{W}_{\alpha,r}\)\right|$.
\begin{lma}\label{lma12}
\begin{description}
\item{(a)} (unrestricted case) Assume the score function $\eta$ satisfies $k'$th moment condition~\Ref{thm2.1} for some $k'>2$. If $\eta$ is exponentially stabilizing in Definition~\ref{defi4}, then there exist positive constants $\alpha_0$ and $C$ such that
\begin{equation}\left|{\rm Var}\(W_{\alpha}\)-{\rm Var}\(W_{\alpha,r}\)\right|\le \frac{1}{\alpha}\label{lma12s1}\end{equation} for all $\alpha\ge\alpha_0$ and $r\ge C\ln (\alpha)$. If $\eta$ is polynomially stabilizing in Definition~\ref{defi4} with parameter $\beta$, then for any $k\in (2,k') $, then there exists a positive constant $C$ such that
\begin{align}
\left|{\rm Var}\(W_{\alpha}\)-{\rm Var}\(W_{\alpha,r}\)\right|&\le C\(\alpha^{\frac{3k-2}{k}}r^{-\beta\frac{k-2}{k}}\)\vee \(\alpha^{\frac{3k-1}{k}}r^{-\beta\frac{k-1}{k}}\)\label{lma12s2}
\end{align} for all $r\le \alpha^{\frac{1}{d}}$.
\item{(b)} (restricted case) Assume the score function $\eta$ satisfies $k'$th moment condition~\Ref{thm2.1r} for some $k'>2$. If $\eta$ is exponentially stabilizing in Definition~\ref{defi4r}, then there exist positive constants $\alpha_0$ and $C$ such that
\begin{equation}\left|{\rm Var}\(\bar{W}_{\alpha}\)-{\rm Var}\(\bar{W}_{\alpha,r}\)\right|\le \frac{1}{\alpha}\label{lma12s3}\end{equation} for all $\alpha\ge\alpha_0$ and $r\ge C\ln (\alpha)$. If $\eta$ is polynomially stabilizing in Definition~\ref{defi4r} with parameter $\beta$, then for any $k\in (2,k') $, then there exists a positive constant $C$ such that
\begin{align}
\left|{\rm Var}\(\bar{W}_{\alpha}\)-{\rm Var}\(\bar{W}_{\alpha,r}\)\right|&\le C\(\alpha^{\frac{3k-2}{k}}r^{-\beta\frac{k-2}{k}}\)\vee \(\alpha^{\frac{3k-1}{k}}r^{-\beta\frac{k-1}{k}}\)\label{lma12s4}
\end{align} for all $r\le \alpha^{\frac{1}{d}}$.
\end{description} \end{lma}
\noindent{\it Proof.} We start with \Ref{lma12s3}. From Lemma~\ref{lma11}~(b), for fixed $k\in (2,k') $, we have
\begin{equation}\max_{0< l\le k}\left\{\|\bar{W}_{\alpha}\|_l,\|\bar{W}_{\alpha,r}\|_l\right\}\le C_0\alpha\label{lmapr1}
\end{equation} for some positive constant $C_0$. Without loss, we assume $\alpha_0>1$. Since \begin{equation}\left|{\rm Var}\(\bar{W}_{\alpha}\)-{\rm Var}\(\bar{W}_{\alpha,r}\)\right|\le \left|\mathbb{E}\(\bar{W}_\alpha^2-\bar{W}_{\alpha,r}^2\)\right|+\left|\(\mathbb{E}\bar{W}_\alpha\)^2-\(\mathbb{E}\bar{W}_{\alpha,r}\)^2\right|, \label{lma12.0} \end{equation} assuming that the score function is exponentially stabilizing \Ref{defi4r}, we show that each of the terms at the right hand side of \Ref{lma12.0} is bounded by $\frac{1}{2\alpha}$ for $\alpha$ and $r$ sufficient large. Clearly, the definition of $\bar{W}_{\alpha,r}$ implies that $\bar{W}_\alpha^2-\bar{W}_{\alpha,r}^2= 0$ if $\bar{R}(x,\alpha)\le r$ for all $x\in\bar\Xi_{\Gamma_\alpha}$, hence it remains to tackle $E_{r,\alpha}:=\{\bar{R}(x,\alpha)\le r~\mbox{for all }x\in\bar\Xi_{\Gamma_\alpha}\}^c$. As shown in the proof of Lemma~\ref{lma105}, $\mathbb{P}\(E_{r,\alpha}\)\le\alpha C_1e^{-C_2r}$, which, together with H\"older's inequality, ensures \begin{align}
\left|\mathbb{E}\(\bar{W}_\alpha^2-\bar{W}_{\alpha,r}^2\)\right|&=\left|\mathbb{E}\[\(\bar{W}_\alpha^2-\bar{W}_{\alpha,r}^2\)\mathbf{1}_{E_{r,\alpha}}\right]\right|\nonumber
\\&\le \|\bar{W}_\alpha^2-\bar{W}_{\alpha,r}^2\|_{\frac{k}{2}}\|\mathbf{1}_{E_{r,\alpha}}\|_{\frac{k}{k-2}}\nonumber
\\&\le \(\|\bar{W}_\alpha^2\|_{\frac{k}{2}}+\|\bar{W}_{\alpha,r}^2\|_{\frac{k}{2}}\)\mathbb{P}(E_{r,\alpha})^{\frac{k-2}{k}}\nonumber
\\&= \(\|\bar{W}_\alpha\|_{k}^2+\|\bar{W}_{\alpha,r}\|_{k}^2\)\mathbb{P}(E_{r,\alpha})^{\frac{k-2}{k}}\le 2\(C_0\alpha\)^2 \(\alpha C_1e^{-C_2r}\)^{\frac{k-2}{k}}.\label{lma12.1} \end{align}
For the remaining term of \Ref{lma12.0}, we have $$\left|\(\mathbb{E}\bar{W}_\alpha\)^2-\(\mathbb{E}\bar{W}_{\alpha,r}\)^2\right|=\left|\mathbb{E}\bar{W}_\alpha-\mathbb{E}\bar{W}_{\alpha,r}\right|\left|\mathbb{E}\bar{W}_\alpha+\mathbb{E}\bar{W}_{\alpha,r}\right|.$$ The bound \Ref{lmapr1} implies $\left|\mathbb{E}\bar{W}_\alpha+\mathbb{E}\bar{W}_{\alpha,r}\right|\le 2C_0\alpha $. However, using H\"older's inequality, Minkowski's inequality and \Ref{lmapr1} again, we have \begin{align}
\left|\mathbb{E}\bar{W}_\alpha-\mathbb{E}\bar{W}_{\alpha,r}\right|&=\left|\mathbb{E}\[\(\bar{W}_\alpha-\bar{W}_{\alpha,r}\)\mathbf{1}_{E_{r,\alpha}}\right]\right|\nonumber
\\&\le \|\bar{W}_\alpha-\bar{W}_{\alpha,r}\|_k\|\mathbf{1}_{E_{r,\alpha}}\|_{\frac{k}{k-1}}\nonumber
\\&\le \(\|\bar{W}_\alpha\|_{k}+\|\bar{W}_{\alpha,r}\|_{k}\)\mathbb{P}(E_{r,\alpha})^{\frac{k-1}{k}}\nonumber
\\&\le 2C_0\alpha \(\alpha C_1e^{-C_2r}\)^{\frac{k-1}{k}},\label{lma12.2} \end{align} giving \begin{equation}
\left|\(\mathbb{E}\bar{W}_\alpha\)^2-\(\mathbb{E}\bar{W}_{\alpha,r}\)^2\right|\le 4\(C_0\alpha\)^2\(\alpha C_1e^{-C_2r}\)^{\frac{k-1}{k}}\label{lma12.20}. \end{equation} We set $r=C\ln(\alpha)$ in the upper bounds of \Ref{lma12.1} and \Ref{lma12.20} and find $C$ such that both bounds are bounded by $1/(2\alpha)$, completing the proof of \Ref{lma12s3}.
The same proof can be adapted for \Ref{lma12s4}. With \Ref{lma12.0} in mind, recalling the fact established in the proof of Lemma~\ref{lma105} that $\mathbb{P}(E_{r,\alpha})\le C_1\alpha r^{-\beta}$, we replace the last inequalities of \Ref{lma12.1}, \Ref{lma12.2} and \Ref{lma12.20} with the corresponding bound of $ \mathbb{P}(E_{r,\alpha})$ to obtain \begin{eqnarray}
&&\left|\mathbb{E}\(\bar{W}_\alpha^2-\bar{W}_{\alpha,r}^2\)\right|\le 2(C_0\alpha)^2 \(C_1\alpha r^{-\beta}\)^{\frac{k-2}{k}},\label{lma12.3}\\
&&\left|\mathbb{E}\bar{W}_\alpha-\mathbb{E}\bar{W}_{\alpha,r}\right|\le 2C_0\alpha \(C_1\alpha r^{-\beta}\)^{\frac{k-1}{k}},\label{lma12.4}
\\
&&\left|\(\mathbb{E}\bar{W}_\alpha\)^2-\(\mathbb{E}\bar{W}_{\alpha,r}\)^2\right|\le 4\(C_0\alpha\)^2\(C_1\alpha r^{-\beta}\)^{\frac{k-1}{k}}\label{lma12.5}. \end{eqnarray} The claim \Ref{lma12s4} follows by combining \Ref{lma12.3} and \Ref{lma12.5}, extracting $\alpha$ and $r$, and then taking $C$ as the sum of the remaining constant terms.
A line-by-line repetition of the above proof with $\bar{W}_\alpha$ and $\bar{W}_{\alpha,r}$ replaced by $W_\alpha$ and $W_{\alpha,r}$ gives \Ref{lma12s1} and \Ref{lma12s2} respectively. \qed
Next, we apply Lemma~\ref{lma6} and Lemma~\ref{lma7} to establish lower bounds for ${\rm Var}\(W_{\alpha,r}\)$ and ${\rm Var}\(\bar{W}_{\alpha,r}\)$.
\begin{lma}\label{lma13}
\begin{description}
\item{(a)} (unrestricted case) If the score function $\eta$ satisfies non-singularity \Ref{non-sin}, then ${\rm Var}\(W_{\alpha,r}\)\ge C\alpha r^{-d}$ for $R_0\le r\le \alpha^{1/d}/6$, where $C,R_0>0$ are independent of $\alpha$.
\item{(b)} (restricted case) If the score function $\eta$ satisfies non-singularity \Ref{non-sinr}, then ${\rm Var}\(\bar{W}_{\alpha,r}\)\ge C\alpha r^{-d}$ for $R_0\le r\le \alpha^{1/d}/6$, where $C,R_0>0$ are independent of $\alpha$.
\end{description} \end{lma}
\noindent{\it Proof.} For (b), recalling the notations in the paragraph after \Ref{lemma4.4proofa1}, we obtain from the total variance formula that \begin{align}
{\rm Var}\(\bar{W}_{\alpha,r}\)=&\mathbb{E}\({\rm Var}\left(\bar{W}_{\alpha,r}\middle| \mathscr{F}_{2,\alpha,r} \right)\)+{\rm Var}\(\mathbb{E}\left(\bar{W}_{\alpha,r}\middle| \mathscr{F}_{2,\alpha,r} \right)\)\nonumber
\\\ge& \mathbb{E}\({\rm Var}\left(\bar{W}_{\alpha,r}\middle| \mathscr{F}_{2,\alpha,r} \right)\)\nonumber
\\=&\sum_{i=1}^{m_{\alpha,r}}\mathbb{E}\({\rm Var}\left(\sum_{x\in \overline{\Xi}\cap B_i}{\bar{g}}(\Xi^{\Gamma_\alpha,x})\mathbf{1}_{\bar{R}(x,\alpha)\le r }\middle|\Xi_{D_i}\right)\)\nonumber
\\=&m_{\alpha,r}\mathbb{E}\({\rm Var}\left(Y_r\middle|\Xi_{N_0^c}\)\)\label{lma13.1}, \end{align} where $m_{\alpha,r}$ is the number of disjoint cubes with length $4r+2r_1$ embedded into $\Gamma_\alpha$.
Using \Ref{forlma5.9}, there exists an $R_0\ge r_1>0$ such that for all $r>R_0$,
$$\mathscr{L}\left(Y_r\mathbf{1}_{(E_{R_0}')^c}\middle|\Xi_{N_0^c}\right)=\mathscr{L}\left(\tilde{Y}\middle|\Xi_{N_0^c}\right)\ge \tilde{\xi} \mbox{ a.s.},$$ where $\tilde{\xi}$ is an absolutely continuous $\sigma(\Xi_{N_0^c})$ measurable random measure satisfying $\mathbb{P}\(\tilde{\xi}(\mathbb{R})>\frac{p}{2}\)>2p$. Hence, for $r> R_0$, we apply Lemma~\ref{lma7} with $X:=Y_r$, $A:=(E_{R_0}')^c$ and use the fact that $Y_r\mathbf{1}_{(E_{R_0}')^c}=\tilde{Y}$ for all $r\ge R_0$ to obtain \begin{equation}
\E{\rm Var}\left(Y_r\middle|\Xi_{N_0^c}\right)\ge \E{\rm Var}\left({\tilde{Y}}+\frac{\mathbb{E}\left({\tilde{Y}}\middle|\Xi_{N_0^c}\right)}{\mathbb{P}\left((E_{R_0}')^c\middle|\Xi_{N_0^c}\right)}\mathbf{1}_{E_{R_0}'}\middle|\Xi_{N_0^c}\right)=:b>0.\label{lma13.2} \end{equation} The proof of claim (b) is completed by combining \Ref{lma13.1} and \Ref{lma13.2} with the observation that {$R_0\le r\le \alpha^{1/d}/6$} ensures $m_{\alpha,r}\ge 12^{-d}\alpha r^{-d}$.
The claim (a) can be proved by replacing $\bar{W}_{\alpha,r}$ with $W_{\alpha,r}${; $\bar{g}$ with $g$} throughout the above argument. \qed
Finally, we make use of \cite[Lemma~4.6]{XY15}, Lemma~\ref{lma12} and Lemma~\ref{lma13} to establish Lemma~\ref{lma4}.
\noindent{\it Proof of Lemma~\ref{lma4}.} To begin with, we combine \Ref{lma12s1}, \Ref{lma12s3} and Lemma~\ref{lma13}~(a) to find an $r:=C_1\ln(\alpha)$ such that \begin{eqnarray}
&&\left|{\rm Var}\(W_{\alpha}\)-{\rm Var}\(W_{\alpha,r}\)\right|\le \frac{1}{\alpha},\label{lma4proof1}\\
&&\left|{\rm Var}\(\bar{W}_{\alpha}\)-{\rm Var}\(\bar{W}_{\alpha,r}\)\right|\le \frac{1}{\alpha},\label{lma4proof2}\\
&&{\rm Var}\(W_{\alpha,r}\)\ge C_2 \alpha \ln(\alpha)^{-d},\label{lma4proof3}\end{eqnarray} for positive constants $C_1,C_2$. The inequalities \Ref{lma4proof1} and \Ref{lma4proof3} imply ${\rm Var}\(W_{\alpha}\)\ge O\(\alpha \ln(\alpha)^{-d}\),$ hence the claim (a) follows from the dichotomy established in \cite[Lemma~4.6]{XY15} saying either ${\rm Var}\(W_{\alpha}\)=\Omega\(\alpha\)$ or ${\rm Var}\(W_{\alpha}\)=O\(\alpha^{\frac{d-1}{d}}\)$.
In terms of (b), it suffices to show ${\rm Var}\(\bar{W}_{\alpha}\)-{\rm Var}\(W_{\alpha}\)=o(\alpha)$ if we take $\bar{\eta}$ as the score function in the unrestricted case. To this end, noting that \Ref{lma4proof1} and \Ref{lma4proof2}, it remains to show ${\rm Var}\(W_{\alpha,r}\)-{\rm Var}\(\bar{W}_{\alpha,r}\)=o(\alpha)$. However, by the Cauchy-Schwarz inequality, we have \begin{align*}
&\left|{\rm Var}\(W_{\alpha,r}\)-{\rm Var}\(\bar{W}_{\alpha,r}\)\right|\\
=&\left|{\rm Var}\(W_{\alpha,r}-\bar{W}_{\alpha,r}\)-2{\rm Cov}\(W_{\alpha,r}-\bar{W}_{\alpha,r},W_{\alpha,r}\)\right|
\\ \le& {\rm Var}\(W_{\alpha,r}-\bar{W}_{\alpha,r}\)+2\sqrt{{\rm Var}\(W_{\alpha,r}-\bar{W}_{\alpha,r}\){\rm Var}\(W_{\alpha,r}\)}, \end{align*} and it follows from ${\rm Var}\(W_{\alpha}\)=\Omega\(\alpha\)$ and \Ref{lma4proof1} that ${\rm Var}\(W_{\alpha,r}\)=\Omega\(\alpha\)$, hence the proof is reduced to showing ${\rm Var}\(W_{\alpha,r}-\bar{W}_{\alpha,r}\)=o(\alpha)$.
Since ${g_\alpha\(x,\Xi\)}\mathbf{1}_{\bar{R}(x,\alpha)<r}={\bar{g}}\(\Xi^{x}\)\mathbf{1}_{R(x)<r}$ if $d(x, \partial\Gamma_{\alpha})>r$, we have $W_{\alpha,r}-\bar{W}_{\alpha,r}=W_{1,\alpha,r}-W_{2,\alpha,r}$ where $$W_{1,\alpha,r}:=\sum_{x\in \overline{\Xi}_{B\(\partial\Gamma_{\alpha},r\)\cap\Gamma_\alpha}}{\bar{g}}\(\Xi^{x}\)\mathbf{1}_{R(x)<r},~W_{2,\alpha,r}:=\sum_{x\in \bar{\Xi}_{B\(\partial\Gamma_{\alpha},r\)\cap\Gamma_\alpha}}{g_\alpha}\(x,\Xi\)\mathbf{1}_{\bar{R}(x,\alpha)<r}.$$ As the summands of $W_{1,\alpha,r}$ and $W_{2,\alpha,r}$ are in the moat within distance $r$ from the boundary of $\Gamma_\alpha$, both ${\rm Var}\(W_{1,\alpha,r_{\alpha}'}\)$ and ${\rm Var}\(W_{2,\alpha,r_{\alpha}'}\)$ are of order $o(\alpha)$, as detailed below. In fact, it follows from \Ref{palm4} that {\begin{align}
&\mathbb{E}\( g_\alpha\(x,\Xi\)\mathbf{1}_{\bar{R}(x,\alpha)<r}\overline{\Xi}(dx)\)\nonumber\\
=&\mathbb{E}\(g_\alpha\(x,\Xi+\delta_{(x,M_x)}\)\mathbf{1}_{\bar{R}(x,M_x,\alpha,\Xi+\delta_{(x,M_x)})<r}\)\lambda dx\nonumber
\\=&:P_{x,\alpha,r} dx,\label{lma4.7} \end{align} } if we set \begin{equation}\overline{\Xi}_\alpha^\ast(dx):={g_\alpha\(x,\Xi\)}\mathbf{1}_{\bar{R}(x,\alpha)<r}\overline{\Xi}(dx)-P_{x,\alpha,r} dx,\label{lma4.7p}\end{equation} then $\E\(\overline{\Xi}_\alpha^\ast(dx)\overline{\Xi}_\alpha^\ast(dy)\){=\E\(\overline{\Xi}_\alpha^\ast(dx)\)\E\(\overline{\Xi}_\alpha^\ast(dy)\)}=0$ if $d(x,y)>2r$. Therefore, \begin{align}
&{\rm Var}\({W_{2,\alpha,r}}\)\nonumber\\
=&\int_{x,y\in B\(\partial\Gamma_{\alpha},r\)\cap\Gamma_\alpha}\E\(\overline{\Xi}_\alpha^\ast(dx)\overline{\Xi}_\alpha^\ast(dy)\)\nonumber
\\=&\int_{x,y\in B\(\partial\Gamma_{\alpha},r\)\cap\Gamma_\alpha,d(x,y)\le 2r}\E\(\overline{\Xi}_\alpha^\ast(dx)\overline{\Xi}_\alpha^\ast(dy)\)\nonumber\\
=&\int_{x,y\in B\(\partial\Gamma_{\alpha},r\)\cap\Gamma_\alpha,d(x,y)\le 2r}
\left\{\E\[{g_\alpha\(x,\Xi\)}\mathbf{1}_{\bar{R}(x,\alpha)<r}{g_\alpha\(y,\Xi\)}\mathbf{1}_{\bar{R}(y,\alpha)<r}\overline{\Xi}(dx)\overline{\Xi}(dy)\right]-P_{x,\alpha,r}P_{y,\alpha,r} dxdy\right\}.\label{lma4.5} \end{align} Recalling the second order Palm distribution in \Ref{palm5}, we can use the moment condition~\Ref{thm2.1r} together with H\"older's inequality to obtain \begin{align}
&\E\[\left|{g_\alpha\(x,\Xi\)}\right|\mathbf{1}_{\bar{R}(x,\alpha)<r}\left|{g_\alpha\(y,\Xi\)}\right|\mathbf{1}_{\bar{R}(y,\alpha)<r}\overline{\Xi}(dx)\overline{\Xi}(dy)\right]\le C^2(\lambda^2dxdy+\lambda dx),\label{lma4.5ad1}\\
&\left|P_{x,\alpha,r}\right|\left|P_{y,\alpha,r}\right| dxdy\le C^2 \lambda^2dxdy,\label{lma4.5ad2} \end{align} where $C\ge 1$. Combining these estimates with \Ref{lma4.5} gives $${\rm Var}\({W_{2,\alpha,r}}\)=O\(\alpha^{\frac{d-1}{d}}r^{d+1}\)=o(\alpha).$$ The proof of ${\rm Var}\({W_{1,\alpha,r}}\)=o(\alpha)$ is similar except we replace \Ref{thm2.1r} with \Ref{thm2.1}. {Consequently, $${\rm Var}\(W_{\alpha,r}-\bar{W}_{\alpha,r}\)={\rm Var}\(W_{1,\alpha,r}-W_{2,\alpha,r}\)\le 2\({\rm Var}\({W_{1,\alpha,r}}\)+{\rm Var}\({W_{2,\alpha,r}}\)\)=o(\alpha)$$ and the statement follows.}\qed
As the lower bounds in Lemma~\ref{lma5} are very conservative, their proofs are less demanding, as demonstrated below.
\noindent{\it Proof of Lemma~\ref{lma5}.} \setcounter{con}{1} We start with (b). The bound \Ref{lma12s4} ensures
\begin{equation}\left|{\rm Var}\(\bar{W}_{\alpha}\)-{\rm Var}\(\bar{W}_{\alpha,r}\)\right|\le C_{\thecon}\qcon{}\(\alpha^{\frac{3k_0-2}{k_0}}r^{-\beta\frac{k_0-2}{k_0}}\)\vee \(\alpha^{\frac{3k_0-1}{k_0}}r^{-\beta\frac{k_0-1}{k_0}}\)\label{lma5pr01}\end{equation} for all $r\le \alpha^{\frac{1}{d}}$ and $k'>k_0>k\ge3$. On the other hand, Lemma~\ref{lma13}~(b) says $${\rm Var}\(\bar{W}_{\alpha,r}\)\ge C_{\thecon}\qcon{}\alpha r^{-d}$$
for $0<R_0\le r\le \alpha^{1/d}/6$. Let $r_\alpha:=\alpha^{\frac{2k-2}{k\beta-2\beta-dk}}$, the assumption $\beta>(3k-2)d/(k-2)$ ensures that $r_\alpha<\alpha^{1/d}/6$ for large $\alpha$ and $k_0>k$ guarantees $\left|{\rm Var}\(\bar{W}_{\alpha}\)-{\rm Var}\(\bar{W}_{\alpha,r_\alpha}\)\right|\ll {\rm Var}\(\bar{W}_{\alpha,r_\alpha}\)$ for large $\alpha$, hence ${\rm Var}\(\bar{W}_{\alpha}\)\ge C_{\thecon}\qcon{}\alpha r_\alpha^{-d}=O\(\alpha^{\frac{k\beta-2\beta-3dk+2d}{k\beta-2\beta-dk}}\)$, completing the proof.
For the proof of (a), we can proceed to replace $\bar{W}_\alpha$ with $W_\alpha$ and $\bar{W}_{\alpha,r}$ with $W_{\alpha,r}$ as in the proof of (b). \qed
The proof of Lemma~\ref{lma5} enables us to get slightly better bounds for ${\rm Var}\(W_{\alpha,r}\)$ and ${\rm Var}\(\bar{W}_{\alpha,r}\)$.
\begin{lma}\label{remark3}
\begin{description}
\item{(a)} (unrestricted case) If the score function $\eta$ satisfies the conditions of Lemma~\ref{lma5}~(a), then
${\rm Var}\(W_{\alpha,r}\)\ge C\(\alpha r^{-d}\)\vee\(\alpha^{\frac{k\beta-2\beta-3dk+2d}{k\beta-2\beta-dk}}\)$ for $R_0\le r\le \alpha^{1/d}/6$, where $C,R_0>0$ are independent of $\alpha$.
\item{(b)} (restricted case) If the score function $\eta$ satisfies the conditions of Lemma~\ref{lma5}~(b), then
${\rm Var}\(\bar{W}_{\alpha,r}\)\ge C\(\alpha r^{-d}\)\vee\(\alpha^{\frac{k\beta-2\beta-3dk+2d}{k\beta-2\beta-dk}}\)$ for $R_0\le r\le \alpha^{1/d}/6$, where $C,R_0>0$ are independent of $\alpha$.
\end{description} \end{lma}
\noindent{\it Proof.} \setcounter{con}{1} We prove (b) only as the proof of (a) is similar. We observe that if $r=r_\alpha=\alpha^{\frac{2k-2}{k\beta-2\beta-dk}}$, then $\alpha r_\alpha^{-d}=\alpha^{\frac{k\beta-2\beta-3dk+2d}{k\beta-2\beta-dk}}$, hence for $r<r_\alpha$, the claim follows from Lemma~\ref{lma13}~(b). For $r>{O(r_\alpha)}$, \Ref{lma5pr01} ensures
$$\left|{\rm Var}\(\bar{W}_{\alpha}\)-{\rm Var}\(\bar{W}_{\alpha,r}\)\right|\le C_{\thecon}\(\alpha^{\frac{3k_0-2}{k_0}}r_\alpha^{-\beta\frac{k_0-2}{k_0}}\)\vee \(\alpha^{\frac{3k_0-1}{k_0}}r_\alpha^{-\beta\frac{k_0-1}{k_0}}\)\ll{\rm Var}\(\bar{W}_{\alpha}\)$$ for large $\alpha$, hence ${\rm Var}\(\bar{W}_{\alpha,r}\)=O\({\rm Var}\(\bar{W}_{\alpha}\)\)=O\(\alpha^{\frac{k\beta-2\beta-3dk+2d}{k\beta-2\beta-dk}}\),$ as claimed. \qed
\noindent{\it Proof of Theorem~\ref{thm2a}.} \setcounter{con}{1} Let $\sigma_{\alpha}^2:={\rm Var}\(\bar{W}_{\alpha}\)$, $\sigma_{\alpha,r}^2:={\rm Var}\(\bar{W}_{\alpha,r}\)$ and $\bar{Z}_{\alpha,r}\sim N\(\E\bar{W}_{\alpha,r}, \sigma_{\alpha,r}^2\)$, then it follows from the triangle inequality that \begin{equation}d_{TV}(\bar{W}_\alpha,\bar{Z}_\alpha)\le d_{TV}\(\bar{W}_\alpha,\bar{W}_{\alpha,r}\)+d_{TV}\(\bar{Z}_\alpha,\bar{Z}_{\alpha,r}\)+d_{TV}\(\bar{W}_{\alpha,r},\bar{Z}_{\alpha,r}\).\label{thm2a01}\end{equation} We take $R_0$ as the maximum of the $R_0$'s of Lemma~\ref{lma3}~(b), Corollary~\ref{cor1}~(b) and Lemma~\ref{lma13}~(b). We start with exponentially stabilizing case (ii).
(ii) The first term of \Ref{thm2a01} can be bounded using Lemma~\ref{lma105}~(b), giving \begin{align}
&d_{TV}\(\bar{W}_\alpha,\bar{W}_{\alpha,r}\)\le C_{\thecon}\qcon{} \alpha e^{-C_{\thecon}\qcon{}r}\le \frac1\alpha,\label{thm2a02} \end{align} for $r>C_{\thecon}\ln(\alpha).$ \setcounter{cproofa}{\thecon} \qcon{}
We can establish an upper bound for the second term $d_{TV}\(\bar{Z}_\alpha,\bar{Z}_{\alpha,r}\)$ of \Ref{thm2a01} using Lemma~\ref{lma10}. To this end, \Ref{lma12s3} gives \begin{equation}
\left|\sigma_\alpha^2-\sigma_{\alpha,r}^2\right|\le \frac1\alpha,
\label{thm2a04} \end{equation} which, together with Lemma~\ref{lma4}~(b), implies \begin{equation}
\sigma_{\alpha,r}^2=\Omega(\alpha), \ \ \ \ \ \sigma_{\alpha}^2=\Omega(\alpha),
\label{thm2a03} \end{equation} for $r>C_{\thecon}\ln(\alpha).$\setcounter{cproofc}{\thecon}\qcon{} We combine \Ref{thm2a03} and \Ref{lma12.2} to obtain
\begin{equation}\frac{\left|\mathbb{E}\(\bar{Z}_\alpha\)-\mathbb{E}\(\bar{Z}_{\alpha,r}\)\right|}{\max(\sigma_\alpha,\sigma_{\alpha,r})}=\frac{\left|\mathbb{E}\(\bar{W}_{\alpha}\)-\mathbb{E}\(\bar{W}_{\alpha,r}\)\right|}{\max(\sigma_\alpha,\sigma_{\alpha,r})}\le O\(\alpha^{-2}\),
\label{thm2a14} \end{equation} for $r>C_{\thecon} \qcon{}\ln(\alpha).$ Therefore, it follows from \Ref{thm2a04}, \Ref{thm2a03}, \Ref{thm2a14} and Lemma~\ref{lma10} that \begin{align}
d_{TV}(\bar{Z}_\alpha,\bar{Z}_{\alpha,r})
\le&\sqrt{\frac{2}{\pi}}\(\frac{\left|\mathbb{E}\(\bar{Z}_\alpha\)-\mathbb{E}\(\bar{Z}_{\alpha,r}\)\right|}{\max(\sigma_\alpha,\sigma_{\alpha,r})}+\frac{\left|{\rm Var}\(\bar{W}_{\alpha}\)-{\rm Var}\(\bar{W}_{\alpha,r}\)\right|}{\min\({\rm Var}\(\bar{W}_{\alpha}\),{\rm Var}\(\bar{W}_{\alpha,r}\)\)}\)\label{thm2a18}
\\\le&O(\alpha^{-2})\label{thm2a05} \end{align} for $r>C_{\thecon} \ln(\alpha).$\setcounter{cproofb}{\thecon}
For the last term of \Ref{thm2a01}, as a linear transformation does not change the total variation distance, we can rewrite it as $$d_{TV}\(\bar{W}_{\alpha,r},\bar{Z}_{\alpha,r}\)=d_{TV}\(V_{\alpha,r},Z\),$$ where $V_{\alpha,r}:=\(\bar{W}_{\alpha,r}-\E \bar{W}_{\alpha,r}\)/\sigma_{\alpha,r}$ and $Z\sim N(0,1)$. We now appeal to Stein's method to tackle the problem. Briefly speaking, Stein's method for normal approximation hinges on a Stein equation (see \cite[p.~15]{CGS11}) \begin{equation}f'(w)-wf(w)=h(w)-Nh,\label{steineq1}\end{equation} where $Nh:=\E h(Z)$. The solution of \Ref{steineq1} satisfies (see \cite[p.~16]{CGS11})
$$\|f_h'\|:=\sup_w\left|f_h'(w)\right|\le 2\|h(\cdot)-Nh\|.$$ Hence, for $h={\bf 1}_A$ with $A\in{\mathscr{B}}(\mathbb{R})$, the solution $f_h=:f_A$ satisfies \begin{equation}
\|f_h'\|\le 2.\label{steineq2} \end{equation} The Stein equation \Ref{steineq1} enables us to bound $d_{TV}\(V_{\alpha,r},Z\)$ through a functional form of $V_{\alpha,r}$ only, giving
\begin{equation}\label{Stein}d_{TV}\(V_{\alpha,r},Z\)\le {\sup_{\{f;\ \|f'\|\le 2\}}}\mathbb{E}\[f'\(V_{\alpha,r}\)-V_{\alpha,r}f\(V_{\alpha,r}\)\right].\end{equation} Recalling \Ref{lma4.7} and \Ref{lma4.7p}, we can represent $V_{\alpha,r}$ through $V(dx):=\frac{1}{\sigma_{\alpha,r}} \overline{\Xi}_\alpha^\ast(dx),$ giving $V_{\alpha,r}=\int_{\Gamma_\alpha}V(dx)$. Let $N_{x,\alpha, r}'=B(x,2r)\cap\Gamma_\alpha$ and $N_{x,\alpha, r}''=B(x,4r)\cap\Gamma_\alpha$, we have $N_{x,\alpha, r}'\subset B(x,2r)$ and $N_{x,\alpha, r}''\subset B(x,4r)$, so the volumes of $N_{x,\alpha, r}'$ and $N_{x,\alpha, r}''$
are bounded by $O(r^d)$. Define $S_{x,\alpha,r}'=\int_{N_{x,\alpha,r}'}V(dy)$ and $S_{x,\alpha,r}''=\int_{N_{x,\alpha,r}''}V(dy)$. Since $V(dx)$ is independent of $V(dy)$ if $|x-y|>2r$, $V(dx)$ is independent of $V_{\alpha,r}-S_{x,\alpha,r}'$ and $S_{x,\alpha,r}'V(dx)$ is independent of $V_{\alpha,r}-S_{x,\alpha,r}''$, $1={\rm Var}\(V_{\alpha,r}\)=\E\int_{\Gamma_\alpha} S_{x,\alpha,r}'V(dx)$ and \begin{align}
&\mathbb{E}\[f'\(V_{\alpha,r}\)-V_{\alpha,r}f\(V_{\alpha,r}\)\right]\nonumber
\\=&\mathbb{E}f'\(V_{\alpha,r}\) -\mathbb{E}\int_{\Gamma_\alpha}\left(f(V_{\alpha,r})-f\left(V_{\alpha,r}-S_{x,\alpha,r}'\right)\right)V(dx)\nonumber
\\=&\mathbb{E}f'\(V_{\alpha,r}\) - \mathbb{E}\int_{\Gamma_\alpha}\int_{0}^1f'\left(V_{\alpha,r}-uS_{x,\alpha,r}'\right)S_{x,\alpha,r}'du V(dx)\nonumber
\\=&\mathbb{E}\int_{\Gamma_\alpha}\mathbb{E}\[f'\(V_{\alpha,r}\)-f'\left(V_{\alpha,r}-S_{x,\alpha,r}''\right)\]S_{x,\alpha, r}'V(dx)\nonumber
\\&-\mathbb{E}\int_{\Gamma_\alpha}\int_{0}^1\(f'\left(V_{\alpha,r}-uS_{x,\alpha,r}'\right)-f'\left(V_{\alpha,r}-S_{x,\alpha,r}''\right)\)S_{x,\alpha, r}'du V(dx).\label{thm2.13} \end{align} By the definition of the total variation distance, we have \begin{equation}d_{TV}\(V_{\alpha,r}, V_{\alpha,r}+\gamma\)=d_{TV}\(\bar{W}_{\alpha,r}, \bar{W}_{\alpha,r}+\sigma_{\alpha,r}\gamma\)\label{thm2a06}\end{equation} for any $\gamma\in \mathbb{R}$. Using Corollary~\ref{cor1}~(b) with $N_{\alpha,r}^{(1)}=N_{\alpha,r}^{(3)}:=\emptyset$ and $N_{\alpha,r}^{(2)}:=B\(N_{x,\alpha,r}'',r\)$, {for $r\le {\qcon{}C_{\thecon}} \alpha^{\frac{1}{d}}$, }we have \begin{equation*}
d_{TV}\left(\bar{W}_{\alpha,r},\bar{W}_{\alpha,r}+h_{\alpha,r}\(\Xi_{N_{\alpha, r}^{(2)}}\)\right)\le \mathbb{E}\left(\left|h_{\alpha,r}\(\Xi_{N_{\alpha, r}^{(2)}}\)\right|\vee 1\right)O\left(\alpha^{-\frac{1}{2}}r^{\frac{d}{2}}\right), \end{equation*} which, together with \Ref{thm2a06}, implies \begin{align}
&\left|\mathbb{E}\[f'\(V_{\alpha,r}\)-f'\left(V_{\alpha,r}-S_{x,\alpha,r}''\right)\right]\right| \le2\|f'\|O\(\alpha^{-\frac{1}{2}}r^{\frac{d}{2}}\)\mathbb{E}\[\left|\sigma_{\alpha,r}S_{x,\alpha,r}''\right|\vee 1\right]\label{thm2a07}. \end{align} Recalling \Ref{lma4.7p}, we have $$\sigma_{\alpha,r}S_{x,\alpha,r}''=\int_{N_{x,\alpha, r}''} \overline{\Xi}_\alpha^\ast(dy).$$ Using the first order Palm distribution \Ref{palm4}, the third order Palm distribution \Ref{palm6} and the moment condition~\Ref{thm2.1r}, we obtain \begin{align*}
&\E\[\left|{g_\alpha\(z,\Xi\)}\right|\mathbf{1}_{\bar{R}(z,\alpha)<r}\overline{\Xi}(dz)\left|{g_\alpha\(y,\Xi\)}\right|\mathbf{1}_{\bar{R}(y,\alpha)<r}\overline{\Xi}(dy)\left|{g_\alpha\(x,\Xi\)}\right|\mathbf{1}_{\bar{R}(x,\alpha)<r}\overline{\Xi}(dx)\right]\\
&\ \ \ \le \qcon{} C_{\thecon}\qcon{} \(\lambda^3 dzdydx+\lambda^2dzdx+\lambda^2dydx+\lambda dx\),\\
&\E\[\left|{g_\alpha\(y,\Xi\)}\right|\mathbf{1}_{\bar{R}(y,\alpha)<r}\overline{\Xi}(dy)\right]\le C_{\thecon}\qcon{} \lambda dy,
\\
&\left|P_{y,\alpha,r}\right|\le C_{\thecon}\qcon{} \lambda, \end{align*} which, together with \Ref{lma4.5ad1} and \Ref{lma4.5ad2}, yield \begin{align}
&\E\left|\overline{\Xi}_\alpha^\ast(dy)\right|\le C_{\thecon}\qcon{} \lambda dy,\label{thm2a10}\\
&\E\left|\overline{\Xi}_\alpha^\ast(dy)\overline{\Xi}_\alpha^\ast(dx)\right|\le C_{\thecon}\qcon{} \(\lambda^2 dydx+\lambda dx\),\label{thm2a11}\\
&\E\left|\overline{\Xi}_\alpha^\ast(dz)\overline{\Xi}_\alpha^\ast(dy)\overline{\Xi}_\alpha^\ast(dx)\right|\le C_{\thecon} \(\lambda^3 dzdydx+\lambda^2dzdx+\lambda^2dydx+\lambda dx\),\label{thm2a11-0}\\
&\E\int_{N_{x,\alpha, r}'} \left|\overline{\Xi}_\alpha^\ast(dy)\right|\le \E\int_{N_{x,\alpha, r}''} \left|\overline{\Xi}_\alpha^\ast(dy)\right|\le O\(r^d\),\nonumber\\
&\mathbb{E}\[\left|\sigma_{\alpha,r}S_{x,\alpha,r}''\right|\vee 1\right]
\le 1+\E\int_{N_{x,\alpha, r}''} \left|\overline{\Xi}_\alpha^\ast(dy)\right|\le O\(r^d\).\label{thm2a13} \end{align} Combining \Ref{thm2a07}, \Ref{thm2a13} and \Ref{steineq2}, we have \begin{align*}
&\left|\mathbb{E}\[f'\(V_{\alpha,r}\)-f'\left(V_{\alpha,r}-S_{x,\alpha,r}''\right)\right]\right| \le O\(\alpha^{-\frac{1}{2}}r^{\frac{3d}{2}}\), \end{align*} hence the first term of \Ref{thm2.13} can be bounded as \begin{align}
&\left|\mathbb{E}\int_{\Gamma_\alpha}\mathbb{E}\[f'\(V_{\alpha,r}\)-f'\left(V_{\alpha,r}-S_{x,\alpha,r}'' \right)\]S_{x,\alpha, r}' V(dx)\right|\nonumber
\\ \le &O\(\alpha^{-\frac{1}{2}}r^{\frac{3d}{2}}\)\sigma_{\alpha,r}^{-2}\mathbb{E}\int_{\Gamma_\alpha}\int_{N_{x,\alpha, r}'}\left|\overline{\Xi}_\alpha^\ast(dy)\right|\left|\overline{\Xi}_\alpha^\ast(dx)\right|\nonumber
\\ \le &O\(\alpha^{-\frac{1}{2}}r^{\frac{3d}{2}}\)\sigma_{\alpha,r}^{-2}\int_{\Gamma_\alpha}\(\int_{N_{x,\alpha, r}'} \lambda dy+1\)\lambda dx = {O\(\sigma_{\alpha,r}^{-2}\alpha^{\frac{1}{2}}r^{\frac{5d}{2}}\)},\label{thm2a16} \end{align} where the last inequality is from \Ref{thm2a11}.
For the second term of \Ref{thm2.13}, we have from Corollary~\ref{cor1} with $N_{\alpha,r}^{(1)}:=B\(N_{x,\alpha,r}',r\)$, $N_{\alpha,r}^{(2)}:=B\(N_{x,\alpha,r}'',r\)$, $N_{\alpha,r}^{(3)}:=N_{x,\alpha,r}''$, for $r\le {\qcon{}C_{\thecon}} \alpha^{\frac{1}{d}}$, we have \begin{align}
&\left|\E\[\left.\int_{0}^1\(f'\left(V_{\alpha,r}-uS_{x,\alpha,r}'\right)-f'\left(V_{\alpha,r}-S_{x,\alpha,r}''\right)\)du\right|\Xi_{N_{x,\alpha,r}'}\right]\right|\nonumber\\
\le&2\int_{0}^{1}\mathbb{E}d_{TV}\left(\left.V_{\alpha,r}-uS_{x,\alpha,r}',V_{\alpha,r}-S_{x,\alpha,r}''\right|\Xi_{N_{x,\alpha,r}'} \right) du\nonumber
\\\le&O\(\alpha^{-\frac{1}{2}}r^{\frac{d}{2}}\)\E\(\left.\int_{N_{x,\alpha, r}''}\left|\overline{\Xi}_\alpha^\ast(dz)\right|+ 1\right|\Xi_{N_{x,\alpha,r}'}\),\nonumber \end{align} hence \begin{align}
&\left|\mathbb{E}\int_{\Gamma_\alpha}\int_{0}^1\(f'\left(V_{\alpha,r}-uS_{x,\alpha,r}'\right)-f'\left(V_{\alpha,r}-S_{x,\alpha,r}''\right)\)S_{x,\alpha, r}'du V(dx)\right|\nonumber\\
&=\left|\mathbb{E}\int_{\Gamma_\alpha}\E\[\left.\int_{0}^1\(f'\left(V_{\alpha,r}-uS_{x,\alpha,r}'\right)-f'\left(V_{\alpha,r}-S_{x,\alpha,r}''\right)\)du\right|\Xi_{N_{x,\alpha,r}'}\]S_{x,\alpha, r}' V(dx)\right|\nonumber\\
&\le O\(\alpha^{-\frac{1}{2}}r^{\frac{d}{2}}\)\mathbb{E}\int_{\Gamma_\alpha}\E\(\left.\int_{N_{x,\alpha, r}''}\left|\overline{\Xi}_\alpha^\ast(dz)\right|+ 1\right|\Xi_{N_{x,\alpha,r}'}\)\left|S_{x,\alpha, r}'\right| |V(dx)|\nonumber\\
&\le O\(\alpha^{-\frac{1}{2}}r^{\frac{d}{2}}\)\sigma_{\alpha,r}^{-2}\E\int_{\Gamma_\alpha}\[\int_{N_{x,\alpha, r}''}\int_{N_{x,\alpha, r}'}\left|\overline{\Xi}_\alpha^\ast(dz)\overline{\Xi}_\alpha^\ast(dy)\right|+\int_{N_{x,\alpha, r}'}\left|\overline{\Xi}_\alpha^\ast(dy)\right|\right]\left|\overline{\Xi}_\alpha^\ast(dx)\right|\nonumber\\
&\le O\(\alpha^{-\frac{1}{2}}r^{\frac{d}{2}}\)\sigma_{\alpha,r}^{-2}\int_{\Gamma_\alpha}\(\int_{N_{x,\alpha, r}''}\int_{N_{x,\alpha, r}'}\lambda^2dzdy+\int_{N_{x,\alpha, r}''}\lambda dz+\int_{N_{x,\alpha, r}'}\lambda dy+1\)\lambda dx\nonumber\\
&\le O\(\alpha^{-\frac{1}{2}}r^{\frac{d}{2}}\)\sigma_{\alpha,r}^{-2}O\(\alpha r^{2d}\)\nonumber\\
&= O\(\sigma_{\alpha,r}^{-2}\alpha^{\frac{1}{2}}r^{\frac{5d}{2}}\),\label{thm2a17} \end{align} where the second last inequality follows from \Ref{thm2a10}, \Ref{thm2a11}, \Ref{thm2a11-0}, and the last inequality is due to the fact that the volumes of $N_{x,\alpha, r}'$ and $N_{x,\alpha, r}''$ are bounded by $O\(r^d\)$. Recalling \Ref{Stein} and \Ref{thm2.13}, we add up the bounds of \Ref{thm2a16} and \Ref{thm2a17} to obtain \begin{equation}
d_{TV}\(\bar{W}_{\alpha,r},\bar{Z}_{\alpha,r}\)=d_{TV}\(V_{\alpha,r},Z\)\le O\(\sigma_{\alpha,r}^{-2}\alpha^{\frac{1}{2}}r^{\frac{5d}{2}}\).\label{thm2.25} \end{equation} The proof of (ii) is completed by using \Ref{thm2a01}, taking $r={\max(C_{\thecproofa},C_{\thecproofc},C_{\thecproofb})}\ln(\alpha)$ for large $\alpha$, collecting the bounds in \Ref{thm2a02}, \Ref{thm2a05}, \Ref{thm2.25} and replacing $\sigma_{\alpha,r}^{2}=\Omega(\alpha)$, as shown in \Ref{thm2a03}.
(i) There exists an $r_1>0$ such that $\bar{W}_{\alpha,r_1}= \bar{W}_{\alpha}$ $a.s.$ for all $\alpha$, which implies $\E \bar{W}_{\alpha,r_1}=\E \bar{W}_{\alpha}$, ${\rm Var}\(\bar{W}_{\alpha,r_1}\)={\rm Var}\(\bar{W}_{\alpha}\)$, hence $d_{TV}(\bar{W}_{\alpha},\bar{Z}_\alpha)= d_{TV}(\bar{W}_{\alpha,r_1},\bar{Z}_{\alpha,r_1})$. On the other hand, range-bound implies exponential stabilization, with $r_1$ in place of $r$, \Ref{thm2a03} and \Ref{thm2.25} still hold. However, $r_1$ is a constant independent of $\alpha$, the conclusion follows.
(iii) We take $r=r_\alpha:= R_0\vee \alpha^{\frac{5k-4}{5dk+2\beta k -4\beta}}$. Lemma~\ref{lma105}~(b) gives \begin{equation}d_{TV}\(\bar{W}_{\alpha},\bar{W}_{\alpha,r}\)\le O\(\alpha r^{-\beta}\)<O\(\alpha^{-\frac{\beta(k-2)[\beta(k-2)-d(15k-14)]}{(k\beta-2\beta-dk)(5dk+2\beta k-4\beta)}}\) .\label{thm2.26} \end{equation} Next, applying Lemma~\ref{lma5}~(b) and Lemma~\ref{remark3}~(b), we have \begin{equation}{\rm Var}\(\bar{W}_{\alpha,r}\)\wedge {\rm Var}\(\bar{W}_{\alpha}\)\ge O\(\alpha^{\frac{k\beta-2\beta-3dk+2d}{k\beta-2\beta-dk}}\),\label{thm2a19}\end{equation} which, together with \Ref{thm2a18}, \Ref{lma12.4} and \Ref{lma12s4}, yields \begin{equation}d_{TV}(\bar{Z}_\alpha,\bar{Z}_{\alpha,r})\le O\(\frac{\alpha^{\frac{2k-1}k}r^{-\beta\frac{k-1}k}}{\alpha^{\frac12\frac{k\beta-2\beta-3dk+2d}{k\beta-2\beta-dk}}}\vee\frac{\alpha^{\frac{3k-2}k}r^{-\beta\frac{k-2}k}}{\alpha^{\frac{k\beta-2\beta-3dk+2d}{k\beta-2\beta-dk}}}\vee\frac{\alpha^{\frac{3k-1}k}r^{-\beta\frac{k-1}k}}{\alpha^{\frac{k\beta-2\beta-3dk+2d}{k\beta-2\beta-dk}}}\)\label{thm2a20} \end{equation} for $R_0< r< C_{\qcon{}\thecon}\alpha^{\frac{1}{d}}$. Recalling that $\beta>\frac{(15k-14)d}{k-2}$, the dominating term of \Ref{thm2a20} is $\frac{\alpha^{\frac{3k-2}k}r^{-\beta\frac{k-2}k}}{\alpha^{\frac{k\beta-2\beta-3dk+2d}{k\beta-2\beta-dk}}}$, giving \begin{equation}d_{TV}(\bar{Z}_\alpha,\bar{Z}_{\alpha,r})\le O\(\frac{\alpha^{\frac{3k-2}k}r^{-\beta\frac{k-2}k}}{\alpha^{\frac{k\beta-2\beta-3dk+2d}{k\beta-2\beta-dk}}}\)=O\(\alpha^{-\frac{\beta(k-2)[\beta(k-2)-d(15k-14)]}{(k\beta-2\beta-dk)(5dk+2\beta k-4\beta)}}\).\label{thm2a21} \end{equation} In terms of $d_{TV}(\bar{W}_{\alpha,r},\bar{Z}_{\alpha,r})$, we make use of \Ref{thm2.25} and \Ref{thm2a19} and replace $r$ with $r_\alpha$ to obtain \begin{equation}\label{thm2.27}d_{TV}(\bar{W}_{\alpha,r},\bar{Z}_{\alpha,r})\le O(\alpha^{\frac{1}{2}} r^{\frac{5d}{2}}) O\(\alpha^{-\frac{k\beta-2\beta-3dk+2d}{k\beta-2\beta-dk}}\)=O\(\alpha^{-\frac{\beta(k-2)[\beta(k-2)-d(15k-14)]}{(k\beta-2\beta-dk)(5dk+2\beta k-4\beta)}}\). \end{equation} Finally, the proof is completed by combining \Ref{thm2a01}, \Ref{thm2.26}, \Ref{thm2a21} and \Ref{thm2.27}. \qed
\noindent{\it Proof of Theorem~\ref{thm2}.} One can repeat the proof of Theorem~\ref{thm2a} by replacing $\bar{W}_\alpha$, $\bar{W}_{\alpha,r}$, $\bar{Z}_\alpha$, $\bar{Z}_{\alpha,r}$, $g_\alpha(x,\Xi)$ and $\bar{R}(x,\alpha)$ with $W_\alpha$, $W_{\alpha,r}$, $Z_\alpha$, $Z_{\alpha,r}$, $g(\Xi^x)$ and $R(x)$. \qed
\begin{re} {\rm~If we aim to find the order of the total variation distance between $\bar{W}_{\alpha}$ and a normal distribution instead of a normal distribution with the same mean and variance in the polynomially stabilizing case, we can get a better upper bound approximation error with a weaker condition. When $\beta>\frac{5dk-7d+\sqrt{20d^2k^2-60d^2k+49d^2}}{k-2}$, combining \Ref{thm2.25} and the fact that $d_{TV}(\bar{W}_{\alpha}, \bar{W}_{\alpha,r})\le C\alpha\lambda r^{-\beta}$, taking $r_\alpha:=\alpha^{\frac{3\beta k-7dk+4d-6\beta}{(\beta k-dk-2\beta)(5d+2\beta)}}$, we have
\begin{align*}
d_{TV}(\bar{W}_{\alpha},\bar{Z}_{\alpha,r_\alpha})&\le d_{TV}(\bar{W}_{\alpha},\bar{W}_{\alpha,r_\alpha})+d_{TV}(\bar{W}_{\alpha,r_\alpha},\bar{Z}_{\alpha,r_\alpha})
\\&\le O\(\alpha^{\frac{-\beta^2(k-2)+10\beta dk-14\beta d-5d^2k}{(\beta k-dk-2\beta)(5d+2\beta)}}\).
\end{align*}} \end{re}
\def{Academic Press}~{{Academic Press}~} \def{Adv. Appl. Prob.}~{{Adv. Appl. Prob.}~} \def{Ann. Probab.}~{{Ann. Probab.}~} \def{Ann. Appl. Probab.}~{{Ann. Appl. Probab.}~} \def{\it Electron.\ J.~Probab.\/}~{{\it Electron.\ J.~Probab.\/}~} \def{J. Appl. Probab.}~{{J. Appl. Probab.}~} \def{John Wiley $\&$ Sons}~{{John Wiley $\&$ Sons}~} \def{New York}~{{New York}~} \def{Probab. Theory Related Fields}~{{Probab. Theory Related Fields}~} \def{Springer}~{{Springer}~} \def{Stochastic Processes and their Applications}~{{Stochastic Processes and their Applications}~} \def{Springer-Verlag}~{{Springer-Verlag}~} \def{Theory Probab. Appl.}~{{Theory Probab. Appl.}~} \def{Z. Wahrsch. Verw. Gebiete}~{{Z. Wahrsch. Verw. Gebiete}~}
\end{document} | arXiv |
On the Cîrtoaje's conjecture
Ladislav Matejíčka1
In this paper, we prove the Cîrtoaje conjecture under other conditions.
Introduction and preliminaries
Within the past years, the power exponential functions have been the subject of very intensive research. Many problems concerning inequalities for the power exponential functions look so simple, but their solutions are not as simple as it seems. A lot of interesting results for inequalities with the power exponential functions have been obtained. The history and a literature review of inequalities with power exponential functions can be found for example in [1]. Some other interesting problems concerning stronger inequalities of power exponential functions can be found in [2]. In the paper, we study one inequality conjectured by Cîrtoaje [3]. Cîrtoaje, in [3], has posted the following conjecture on the inequalities with power exponential functions.
Conjecture 1.1
If \(a,b \in(0; 1]\) and \(r \in[0; e]\), then
$$ 2\sqrt{a^{ra}b^{rb}}\geq a^{rb}+b^{ra}. $$
The conjecture was proved by Matejíčka [4]. In [1], Coronel and Huancas posted several conjectures for inequalities with the power exponential functions. Some of them are not valid as was shown by the unknown referee of this paper. We note that Theorems 1.2, 1.3, Lemma 3.1, Conjectures 3.1 and 3.2 in [1] are not valid. With the referee's kind permission we present his counterexample: \(n=3\), \(x_{1}=\frac{1}{3}\), \(x_{2}=\frac{1}{9}\), \(x_{3}=\frac{2}{3}\), \(r=\frac{5}{2}\), when
$$ x_{1}^{rx_{1}}+x_{2}^{rx_{2}}+x_{3}^{rx_{3}}< x_{1}^{rx_{2}}+x_{2}^{rx_{3}}+x_{3}^{rx_{1}}. $$
In this paper, we show that the conjecture (1.1) is also valid under the conditions: \(1\leq\min^{2}\{a,b\}\leq\max\{a,b\}\) or \(0<\min\{a,b\}\leq2/e\) and \(\max\{a,b\}\geq1\), or \(0<\min\{a,b\}\leq e\), \(\max\{a,b\}\geq e\), and \(r \in[0; e]\).
Let a, b be positive numbers. The inequality
$$ 2\sqrt{a^{ra}b^{rb}}\geq a^{rb}+b^{ra}$$
holds for any \(r\in\langle 0,e\rangle \) if one of the following three conditions is satisfied:
$$\begin{aligned}& \mathrm{(a)}\quad a\geq b^{2}\geq1; \\& \mathrm{(b)}\quad a\geq1\geq\frac{e}{2}b; \\& \mathrm{(c)}\quad a\geq e\geq b. \end{aligned}$$
According to the power mean inequality, it suffices to consider the case where r has the maximum value, that is \(r=e\). Without loss of generality, suppose \(a\geq b\) and denote
$$ H(x)=2\sqrt{x^{ex}b^{eb}}-x^{eb}-b^{ex}. $$
An easy calculation gives
$$\begin{aligned} &{H'(x)}=e\bigl(x^{\frac{ex}{2}}b^{\frac{eb}{2}}(\ln x+1)-bx^{eb-1}-b^{ex}\ln b\bigr), \\ &{H''(x)}=e \biggl(x^{\frac{ex}{2}}b^{\frac {eb}{2}} \biggl(\frac{e(\ln x+1)^{2}}{2}+\frac{1}{x} \biggr)-b(eb-1)x^{eb-2}-eb^{ex} \ln ^{2} b \biggr). \end{aligned}$$
Solution for (a): \(a\geq b^{2}\geq1\).
Suppose \(x\geq b^{2}\geq1\). We show \({H''(x)}\geq0\), \({H(b^{2})}\geq 0\), and \({H'(b^{2})}\geq0\). It implies \(H(a)\geq0\) for \(a\geq b^{2}\geq1\).
It is easy to see that \(\frac{1}{e}{H''(x)}\geq U+V\), where
$$\begin{aligned} &U=x^{\frac{ex}{2}}b^{\frac{eb}{2}}e\ln b-eb^{ex}\ln ^{2} b, \\ &V=x^{\frac{ex}{2}}b^{\frac{eb}{2}} \biggl(\frac{e\ln ^{2}b}{2}+\frac{e}{2} \biggr)+x^{\frac{ex}{2}-1}b^{\frac{eb}{2}}- eb^{2}x^{eb-2}+bx^{eb-2}. \end{aligned}$$
Similarly, we estimate
$$ U\geq eb^{ex}\ln b \bigl(b^{\frac{eb}{2}}-\ln b \bigr)\geq eb^{ex}\ln b. $$
It follows from
$$ f(b)=\frac{eb}{2}\ln b-\ln(1+\ln b)\geq0, $$
because of
$$ f'(b)=\frac{eb(1+\ln b)^{2}-2}{2b(1+\ln b)}\geq0, \quad f(1)=0. $$
Next we estimate
$$\begin{aligned} V&= x^{eb-2} \biggl(x^{\frac{ex}{2}-eb+2}b^{\frac {eb}{2}} \biggl( \frac{e\ln^{2}b}{2}+\frac{e}{2} \biggr)+x^{\frac {ex}{2}+1-eb}b^{\frac{eb}{2}}- eb^{2}+b \biggr)\\ &\geq x^{eb-2} \biggl(b^{eb^{2}-2eb+4+\frac{eb}{2}} \biggl(\frac {e\ln^{2}b}{2}+ \frac{e}{2} \biggr)+b^{eb^{2}+2-2eb+\frac{eb}{2}}- eb^{2}+b \biggr). \end{aligned}$$
$$\begin{aligned} & l(b)=eb^{2}-2eb+4+\frac{eb}{2}\geq\frac{8-e}{2},\qquad m(b)=eb^{2}-2eb+2+\frac{eb}{2}\geq\frac{4-e}{2}, \\ &l'(b)=2eb-2e+\frac{e}{2}> 0,\qquad m'(b)=2eb-2e+ \frac{e}{2}> 0,\qquad l(1)=\frac{8-e}{2}, \\ &m(1)=\frac{4-e}{2},\qquad \frac{8-e}{2}>\frac{5}{2},\qquad \frac {4-e}{2}>\frac{1}{2}, \end{aligned}$$
we have \({H''(x)}\geq0\) for \(x\geq b^{2}\geq1\) if
$$ p(b)=\frac{e}{2}b^{\frac{5}{2}}-eb^{2}+b+b^{\frac{1}{2}}\geq0. $$
Put \(u=b^{1/2}\) then we obtain \(p(b)\geq1\) if
$$ k(u)=\frac{e}{2}u^{4}-eu^{3}+u+1\geq0. $$
It is easy to see that \(k'(u)=0\) if
$$ q(u)=2eu^{3}-3eu^{2}+1=0. $$
From Cardano's formula we see that the root of \(q(u)\) is equal to \(u=1.4071\). From \(k(1.4071)=0.5602\) and from \(k''(u)=6eu^{2}-6eu\geq0\) we have \(k(u)\geq0\) for \(u\geq1\). So \({H''(x)}\geq0\) for \(x\geq b^{2}\geq1\).
Now we show \({H'(b^{2})}\geq0\).
Some calculation gives
$$ {H'\bigl(b^{2}\bigr)}=e \bigl(b^{eb^{2}+\frac{eb}{2}}(2\ln b+1)-b^{2eb-1}-b^{eb^{2}}\ln b \bigr). $$
From this we have \({H'(b^{2})}\geq0\) if
$$ \bigl(b^{eb^{2}+\frac{eb}{2}}-b^{eb^{2}} \bigr)\ln b+b^{eb^{2}+\frac{eb}{2}}(\ln b+1)-b^{2eb-1}\geq0. $$
Rewriting this we obtain \({H'(b^{2})}\geq0\) if
$$ b^{2eb-1} \bigl(b^{eb^{2}+\frac{eb}{2}-2eb+1}(\ln b+1)-1 \bigr)+b^{eb^{2}}\ln b \bigl(b^{\frac{eb}{2}}-1 \bigr)\geq0. $$
But this will be fulfilled if
$$ b^{eb^{2}-3\frac{eb}{2}+1}(\ln b+1)-1\geq0. $$
This is equivalent to
$$ o(b)= \biggl(eb^{2}-3\frac{eb}{2}+1 \biggr)\ln b+\ln (\ln b+1) \geq0. $$
Using \(\ln b\geq\frac{b-1}{b}\) we have \(o(b)\geq0\) if we show that
$$ s(b)= \biggl(eb^{2}-3\frac{eb}{2} \biggr)\ln b+\ln (2b-1)\geq0. $$
$$ s'(b)= \biggl(2eb-3\frac{e}{2} \biggr)\ln b+eb-3 \frac {e}{2}+\frac{2}{2b-1}. $$
Because of \(s(1)=0\) it suffices to show that \(eb-3\frac{e}{2}+\frac {2}{2b-1}\geq0\). But this is evident from \(2-\frac{e}{2}\geq0\) and \(eb-3\frac{e}{2}+\frac{2}{2b-1}=\frac {1}{2b-1} (2e(b-1)^{2}+2-\frac{e}{2} )\).
Now we show that \({H(b^{2})}\geq0\).
$$ {H\bigl(b^{2}\bigr)}=2b^{eb^{2}}b^{\frac{eb}{2}}-b^{eb^{2}}-b^{2eb}=b^{2eb} \bigl(2b^{eb^{2}+\frac{eb}{2}-2eb}-b^{eb^{2}-2eb}-1 \bigr)\geq0 $$
$$ 2b^{eb^{2}+\frac{eb}{2}-2eb}-b^{eb^{2}-2eb}=b^{eb^{2}-2eb} \bigl(2b^{\frac{eb}{2}}-1 \bigr)\geq1. $$
It suffices to show that \(2b^{\frac{eb}{2}}-1\geq b^{e}\) because of
$$ b^{eb^{2}-2eb} \bigl(2b^{\frac{eb}{2}}-1 \bigr)\geq b^{e(b-1)^{2}}\geq1. $$
Denote \(F(b)=2b^{\frac{eb}{2}}-b^{e}\). Then
$$ F'(b)=e \bigl(b^{\frac{eb}{2}}(\ln b+1)-b^{e-1} \bigr). $$
Because of \(F(1)=1\) it suffices to show that \(F'(b)\geq0\). We will be done if we show that
$$ s(b)= \biggl(\frac{eb}{2}-e \biggr)\ln b+\ln(2b-1)\geq0. $$
We used \(\ln b\geq(b-1)/b\). Because of \(s(1)=0\) it suffices to show that
$$ s'(b)=\frac{e}{2}\ln b+\frac{2eb^{2}+b(4-5e)+2e}{2b(2b-1)}\geq0. $$
But this is evident because of \(j(b)=2eb^{2}+b(4-5e)+2e\geq0\). (\(j(1)=4-e\geq0\), \(j'(1)=4-e\geq0\), \(j''(b)=4e\geq0\).)
Solution for (b): \(a\geq1\geq\frac{e}{2}b\).
If \(b\leq2/e\) and \(x\geq1\) then it suffices to show that
$$ {H'(x)}=e\bigl(x^{\frac{ex}{2}}b^{\frac{eb}{2}}(\ln x+1)-bx^{eb-1}-b^{ex}\ln b\bigr)\geq0 $$
and \(H(1)\geq0\).
It is evident that \({H'(x)}\geq0\) if
$$ x^{\frac{ex}{2}-eb+1}b^{\frac{eb}{2}-1}(\ln x+1)\geq1. $$
It follows from \(\frac{ex}{2}-eb+1\geq0\) and from \(\frac {eb}{2}-1\leq0\). Finally, we show that \(H(1)\geq0\). Denote \(v(b)=2b^{\frac{eb}{2}}-1-b^{e}\).
Then we have \(v(2/e)=4/e-1-(2/e)^{e}=0.0373\). If we show \(v'(b)\leq0\) then the conjecture will be proved. An easy calculation gives
$$ v'(b)=e \bigl(b^{\frac{eb}{2}}(\ln b+1)-b^{e-1} \bigr). $$
\(v'(b)\leq0\) if
$$ b^{\frac{eb}{2}-e+1}(\ln b+1)\leq1. $$
If \(0\leq b\leq1/e\) then the inequality is evident. Let \(1/e\leq b\leq 2/e\), then \(v'(b)\leq0\) if
$$ s(b)= \biggl(\frac{eb}{2}-e+1 \biggr)\ln b+\ln(\ln b+1)\leq0. $$
Numerical calculation shows that \(s(2/e)=-0.1461\). So it suffices to show that
$$ s'(b)=\frac{e}{2}\ln b+ \biggl(\frac{eb}{2}-e+1 \biggr) \frac{1}{b}+\frac{1}{\ln b+1}\frac{1}{b}\geq0. $$
$$ \frac{1}{2}(\ln b+1)+\frac{1}{1+\ln b}+1-e\geq 0. $$
(We used \(eb/2\geq1/2\).)
Put \(t=1+\ln b\). Then (2.2) can be rewritten as
$$\frac{1}{t}m(t)=\frac{1}{t} \bigl(t^{2}+2(1-e)t+2 \bigr) \geq0 $$
for \(0< t\leq\ln2\). From \(m'(t)=2t+2(1-e)\leq-2.0503<0\) and from \(m(\ln2)=\ln ^{2}2+2+2(1-e)\ln 2=0.0984\geq0\) we see that the proof of the case (b) is complete.
Solution for (c) \(a\geq e\geq b\).
It suffices to show that \(\sqrt{a^{ra}b^{rb}}\geq a^{rb}\) and \(\sqrt{a^{ra}b^{rb}}\geq b^{ra}\). The first inequality is equivalent to
$$ a^{a-b}\geq \biggl(\frac{a}{b} \biggr)^{b}, $$
$$ a^{x-1}\geq x,\qquad x=\frac{a}{b}\geq1. $$
Indeed, we have
$$ a^{x-1}\geq e^{x-1}\geq x. $$
The second inequality is equivalent to
$$ \frac{b}{a}^{a}b^{a-b}\leq1, $$
$$ xb^{1-x}\leq1,\qquad x=\frac{b}{a}\leq1. $$
Indeed we have
$$ xb^{1-x}\leq xe^{1-x}\leq1. $$
This completes our proof. □
We note that the proof of the case (c) originates from an unknown reviewer. His proof of (c) is more elegant and his formulation of (c) is more general than ours.
We note that
$$ 2\sqrt{10^{e10}5^{e5}}-10^{e5}-5^{e10}=-5.6e18, $$
which implies that the inequality (1.1) is not valid for all \(a,b\geq0\).
The referee's counterexample implies that (1.1) cannot be generalized for \(n=3\).
Coronel, A, Huancas, F: The proof of three power exponential inequalities. J. Inequal. Appl. 2014, 509 (2014) http://www.journalofinequalitiesandapplications.com/content/2014/1/509
Miyagi, M, Nishizawa, Y: A stronger inequality of Cîrtoaje's one with power-exponential functions. J. Nonlinear Sci. Appl. 8, 224-230 (2015) http://www.tjnsa.com
Cîrtoaje, V: Proofs of three open inequalities with power exponential functions. J. Nonlinear Sci. Appl. 4(2), 130-137 (2011)
Matejíčka, L: Proof of one open inequality. J. Nonlinear Sci. Appl. 7, 51-62 (2014) http://www.tjnsa.com
The work was supported by VEGA grant No. 1/0385/14. The author thanks the faculty FPT TnUAD in Púchov, Slovakia for its kind support and he is deeply grateful to the unknown reviewer for his valuable remarks, suggestions, and some generalization in the Theorem 2.1, for his kind permission to publish his counterexample, his version of the proof of (c) in Theorem 2.1.
Faculty of Industrial Technologies in Púchov, Trenčín University of Alexander Dubček in Trenčín, I. Krasku 491/30, Púchov, 02001, Slovakia
Ladislav Matejíčka
Correspondence to Ladislav Matejíčka.
Matejíčka, L. On the Cîrtoaje's conjecture. J Inequal Appl 2016, 152 (2016). https://doi.org/10.1186/s13660-016-1092-2
power inequalities
exponential inequalities
power exponential functions
Recent Advances in Inequalities and Applications | CommonCrawl |
\begin{document}
\title{f Network Dependence Testing via Diffusion Maps and Distance-Based Correlations}
\begin{abstract} Deciphering the associations between network connectivity and nodal attributes is one of the core problems in network science. The dependency structure and high-dimensionality of networks pose unique challenges to traditional dependency tests in terms of theoretical guarantees and empirical performance. We propose an approach to test network dependence via diffusion maps and distance-based correlations. We prove that the new method yields a consistent test statistic under mild distributional assumptions on the graph structure, and demonstrate that it is able to efficiently identify the most informative graph embedding with respect to the diffusion time. The testing performance is illustrated on both simulated and real data. \end{abstract}
\noindent {\it Keywords:} Adjacency spectral embedding; Diffusion distance; Multiscale graph correlation (\textsc{mgc}); Normalized graph Laplacian
\sloppy \spacingset{1.45}
\section{Introduction} \label{sec:intro}
Network data has seen increased availability and influence in statistics, physics, computer science, biology, social science, etc., which poses many challenges due to its distinct structure. A network or graph is formally defined as an ordered pair $\mc{G}=(\mc{V},\mc{E})$, where $\mc{V}$ represents the set of nodes and $\mc{E}$ is the set of edges, and $n = |\mc{V}|$. The edge connectivity of a graph can be compactly represented by the adjacency matrix $\mc{A} = \{\mc{A}(i,j) : i,j= 1,..,n\}$, where $\mc{A}(i,j)$ is the edge weight between node $i$ and node $j$. For example, for an unweighted and undirected network, $\mc{A}(i,j) =\mc{A}(j,i) = 1$ if and only if node $i$ and node $j$ are connected by an edge, and zero otherwise. Often, each node has some associated nodal attributes, which we denote as $X_i \in \mathbb{R}^{p}$ and use $\mc{X} = [X_1 | \cdots | X_n]$ to represent the collection of attributes.
This paper focuses on independence testing between network connectivity and nodal attributes. Assuming for the adjacency matrix $\mc{A}$ and attributes $\mc{X}$, the connectivity and attribute corresponding to each node are identically and jointly distributed as $F_{AX}$, the null and alternative hypotheses of interest are: \begin{align} \label{eq:hyp} &H_{0}: F_{AX}=F_{A}F_{X}\\ &H_{A}: F_{AX} \neq F_{A}F_{X}. \nonumber \end{align} There are many network data examples where testing independence can be a crucial first step. For example, determining potential correlation between cultural tastes and relationships over social network~\citep{lewis2012social}, identifying association between the strength of functional connectivity and brain physiology such as regional cerebral blood flow in brain network~\citep{liang2013coupling}, embedding text data and its hyperlink networks jointly into a low-dimensional structure~\citep{ShenVogelsteinPriebe2016}. Sometimes the correlations among nodes are not proportional to the strength of connectivity between them. For instance, in signaling network of biological cells, reaction rate for each cell exhibits non-linear dependence on the neighboring response due to complex, cooperative biological process involved~\citep{hernandez2017nonlinear}. We can observe nonlinear dependence in concentrated propagration among a few focal persons in the social network~\citep{nekovee2007theory}, and in screening informative brain regions for sex and site difference from fMRI image graphs~\citep{mgc4}, etc.
A notable obstacle in network inference is the structure of the edge connectivity. Namely, for an undirected graph, $\mc{A}$ is a symmetric binary matrix whose edges are not independent of each other, thus preventing many well-established methods from being directly applicable. One approach is to assume certain model on the graph structure, then solve the inference question based on the model assumption~\citep{wasserman1996logit, fosdick2015testing, howard2016understanding}. Another approach is spectral embedding, which first embeds the $n \times n$ adjacency matrix $\mc{A}$ into an $n \times q$ matrix $\mc{U}$ by eigendecomposition, then directly works on $\mc{U}$~\citep{rohe2011spectral,SussmanEtAl2012, tang2017a}. For example, the network dependence test proposed by Fosdick and Hoff \citep{fosdick2015testing} assumes that the adjacency matrix is generated from a multivariate normal distribution of the latent factors, estimates the latent factor associated with each node from $\mc{A}$, followed by applying the standard likelihood ratio test on the normal distribution.
However, model-based approaches are often limited by, and do not perform well beyond, the model assumptions. Moreover, spectral embedding is susceptible to misspecification of the dimension of $q$. Both of these factors can significantly degrade the later inference performance. Indeed, as a ground truth is unlikely in real networks \citep{Leto2017}, one often desires a method that is effectively non-parametric and robust against algorithm parameter selection~\citep{ChenShenVogelsteinPriebe2016}.
We propose a methodology to test network dependency via diffusion maps and distance-based correlations, which is universally consistent under mild graph distributional assumptions and works well under many popular network models. The proposed method also overcomes parameter selection issues, and exhibits superior empirical testing performance. The \texttt{R} code and accompanying data are publicly available online at \url{http://neurodata.io/tools/mgc} and \url{https://github.com/neurodata/mgc}.
\section{Preliminaries} \label{sec:pre} \subsection{Notation} \label{ssec:notation}
We denote a random variable by capital letter such as $X$ with distribution $F_{X}$, and denote a matrix or a set of vectors by calligraphic letter such as $\mc{X}$. For each node $i \in \mc{V}$, its attribute is denoted by $X_{i}$ whose realizations are in $\mathbb{R}^{p}$; and its edge connectivity vector is denoted by $A_i \in \mathbb{R}^{n}$, which is a column in the $n \times n$ adjacency matrix $\mc{A}$. We assume that $(X_i,A_i) \sim F_{XA}$, i.e., identically distributed attributes and connectivity vectors.
Later we introduce a multiscale node-wise representation of the nodes as an $n \times q$ matrix $\mc{U}^{t} = [ U^{t}_{1} | U^{t}_{2} | \cdots | U^{t}_{n} ]$ for any $t \in \{0\} \bigcup \mathbb{Z}^{+}$, where $q$ is the embedding dimension and $t$ is the Markov iteration time step. Let $\cdot^{*}$ denote estimated optimality; $\cdot^{t}$ denotes either the $t^{\mbox{th}}$ power or time step, which shall be clear in the context; and $\cdot^{T}$ is the matrix transpose.
\subsection{Diffusion Maps} \label{ssec:method2}
Because the rows and columns of a symmetric adjacency matrix may be correlated, directly operating on the adjacency matrix breaks theoretical guarantees of existing dependence tests. The diffusion map is introduced as a feature extraction algorithm by Coifman and Lafon~\citep{coifman2005geometric,coifman2006diffusion,lafon2006diffusion}, which computes a family of embeddings in Euclidean space by eigendecomposition on a diffusion operator of the given data. Here we introduce a version tailored to adjacency matrices.
To derive the diffusion maps for given observations of size $n$, the first step is to choose a $n \times n$ kernel matrix $\mc{K}$ that represents the similarity within the sample data. The adjacency matrix $\mc{A}$ is a natural similarity matrix; for undirected graphs we let $\mc{K}=\mc{A}$, for directed graphs we let $\mc{K}= (\mc{A} + \mc{A}^{T})/2$. The next step is to compute the normalized Laplacian matrix by \begin{align*} \mc{L} = \mc{B}^{-1/2} \mc{K} \mc{B}^{-1/2}, \end{align*} where $\mc{B}$ is the $n \times n$ degree matrix of $\mc{K}$. When $\mc{B}(i,i)$ or $\mc{B}(j,j)$ is zero, $\mc{L}(i,j)=0$.
The diffusion map $\mc{U}^{t}=\{U_{i}^{t} \in \mathbb{R}^{q} : i=1,\ldots,n\}$ is then computed by eigendecomposition, namely \begin{align} \label{eq:U} U_{i}^{t} &= \begin{pmatrix} \lambda^{t}_{1} \phi_{i1}, & \lambda^{t}_{2} \phi_{i2}, & \cdots, & \lambda^{t}_{q} \phi_{iq} \end{pmatrix}^{T} \in \mathbb{R}^{q}, \quad i = 1, \ldots, n, \end{align} where $\{ \lambda_{j}^t : j = 1,2,\ldots, q \}$ and $\{ \phi_{j} \in \mathbb{R}^{n} : (\phi_{1j}, \phi_{2j}, \ldots, \phi_{nj} ),~j=1,2,\ldots, q \}$ are the $q$ largest eigenvalues and corresponding eigenvectors of $\mc{L}$ respectively, and $\lambda^{t}_{j}$ is the $t^{\mbox{th}}$~power of the $j^{\mbox{th}}$~eigenvalue. The diffusion distance between the $i^{\mbox{th}}$~observation and the $j^{\mbox{th}}$~observation is defined as the weighted $\ell^{2}$ distance of the two points in the observation space, which equals the Euclidean distance in the diffusion coordinate: \begin{align*}
\mc{C}^{t}(i,j) = \| U_{i}^{t} - U_{j}^{t} \|, \quad i,j = 1,2, \ldots , n, \end{align*}
where $\| \cdot \|$ is the Euclidean distance.
When $t=0$, the diffusion map is exactly the same as a normalized graph Laplacian embedding in \citet{rohe2011spectral} up-to a linear transformation; when $t>0$, the diffusion maps are weighted graph Laplacian by powered eigenvalues~\citep{lafon2006diffusion}; and the diffusion map at $t=1$ equals the adjacency spectral embedding up-to the degree constant~\citep{sussman2014consistent}. Therefore, the diffusion maps can be viewed as a single index family of embeddings. The embedding dimension choice $q$ can be selected via the profile likelihood method in~\citet{ZhuGhodsi2006}, which is a standard algorithm in dimension reduction literature. To select the optimal $t$, we will utilize a smoothing technique to maximize the dependency, as discussed shortly.
\subsection{Distance-Based Correlations} \label{ssec:method1}
The problem of testing general dependencies between two random variables has seen notable progress in recent years. The Pearson's correlation~\citep{Pearson1895} is the most classical approach, which determines the existence of linear relationship via a correlation coefficient in the range of $[-1,1]$, with $0$ indicating no linear association and $\pm 1$ indicating perfect linear association. To better capture the dependencies not limited to linear relationship, a variety of distance-based correlation measures have been suggested, including the distance correlation and energy statistic \citep{szekely2007measuring,szekelyRizzo2013a, RizzoSzekely2016}, kernel-based independence test \citep{GrettonGyorfi2010}, Heller-Heller-Gorfine test \citep{HellerGorfine2013,heller2016consistent}, and multiscale graph correlation~\citep{shen2017mgc,shen2016discovering}, among others. In particular, distance correlation is a distance-based dependency measure that is consistent against all possible dependencies with finite second moments. The kernel independence test is a kernel variant of distance correlation~\citep{sejdinovic2013equivalence,mgc5}. The multiscale graph correlation inherits the same consistency of distance correlation with better finite-sample testing powers under high-dimensional and nonlinear dependencies, via defining a family of local correlations and efficiently searching for the optimal local scale in testing. Here we briefly introduce distance correlation and multiscale graph correlation, which are denoted as $\textsc{dcorr}$ and $\textsc{mgc}$ in the equations.
Given $n$ pairs of sample data that are independently and identically distributed (i.i.d.), namely $(\mc{U}, \mc{X}) = \{ (U_{i}, X_{i} ) \stackrel{i.i.d.}{\sim} F_{UX} \in \mathbb{R}^{q} \times \mathbb{R}^{p}: i = 1,2, \ldots, n \}$. Denote the pairwise distances within $\{U_{i}\}_{i=1}^{n}$ and $\{X_{i}\}_{i=1}^{n}$ as $\mc{C}(i,j) = \| U_{i} - U_{j} \|$ and $\mc{D}(i,j) = \| X_{i} - X_{j} \|$ for $i,j=1,2, \ldots , n$ respectively. The sample distance covariance is denoted as \begin{align*}
\textsc{dcov}_{n}(\mc{U},\mc{X}) = \frac{1}{n^2} \sum\limits_{i,j=1}^{n} \tilde{\mc{C}}(i,j) \tilde{\mc{D}}(i,j), \end{align*} where $\tilde{\mc{C}}=\mc{H} \mc{C} \mc{H}$ and $\tilde{\mc{D}}=\mc{H} \mc{D} \mc{H}$, and $\mc{H} =\mc{I}_{n \times n}- \mc{J}_{n \times n} / n$ is the centering matrix with $\mc{I}_{n \times n}$ being the $n \times n$ identity matrix and $\mc{J}_{n \times n}$ being the $n \times n$ matrix of all ones. The distance correlation follows by normalizing distance covariance via Cauchy-Schwarz into the range of $[-1,1]$. \citet{szekely2007measuring} shows that sample distance correlation converges to a population form, which is asymptotically $0$ if and only if independence, resulting in a consistent statistic for testing independence. An unbiased sample version of distance correlation is later proposed to eliminate the sample bias in distance correlation~\citep{szekely2013distance, SzekelyRizzo2014}, which is the default implementation in this paper.
The multiscale graph correlation is an optimal local version of distance correlation that improves its finite-sample testing power. It first derives all local distance covariances as \begin{align*}
\textsc{dcov}_{n}^{kl}(\mc{U},\mc{X}) = \frac{1}{n^2} \sum\limits_{i,j=1}^{n} \tilde{\mc{C}}^{k}(i,j) \tilde{\mc{D}}^{l}(i,j); \quad k = 1,\ldots, \kappa ,~l= 1, \ldots, \gamma, \end{align*} where $\kappa$ and $\gamma$ are the number of unique numerical values in $\mc{C}$ and $\mc{D}$ respectively; $\tilde{\mc{C}}^{k}(i,j) = \tilde{\mc{C}}(i,j) \mathbb{I}(R^{\mc{C}}_{ij} \leq k )$; $\mathbb{I}(\cdot)$ is the indicator function; and $R^{\mc{C}}_{ij}$ is a rank function of $U_{i}$ relative to $U_{j}$, i.e., $R^{\mc{C}}_{ij} =k$ if $U_{i}$ is the $k^{\mbox{th}}$ nearest neighbor of $U_{j}$, and define equivalently $\tilde{\mc{D}}^{l}(i,j) = \tilde{\mc{D}}(i,j) \mathbb{I}(R^{\mc{D}}_{ij} \leq l)$ for $\{X_i\}$. Then the local distance correlations $\{ \textsc{dcorr}^{kl} \}$ are the normalizations of the local distance covariances into $[-1,1]$ via Cauchy-Schwarz. Among all possible neighborhood choices, the multiscale graph correlation equals the maximum local correlation within the largest connected component of significant local correlations, i.e., \begin{align*} \textsc{mgc}_{n}(\mc{U},\mc{X})=\textsc{dcorr}_{n}^{(kl)^{*}}(\mc{U},\mc{X}), \mbox{ where } (kl)^{*}=\arg\max_{(kl)}\mc{S}(\textsc{dcorr}_{n}^{kl}) \end{align*} for a smoothing operation $\mc{S}(\cdot)$ that filters out all in-significant local correlations. The multiscale graph correlation has been shown to have power almost equal or better than distance correlation throughout a wide variety of common dependencies, while being computationally efficient~\citep{shen2017mgc}.
\begin{figure}
\caption{Flowchart for Network Dependence Testing via Diffusion Multiscale Graph Correlation (\textsc{dmgc}).}
\label{fig:flow}
\end{figure}
\section{Main Results} \label{sec:method}
\subsection{Testing procedure of Diffusion Correlation} \label{ssec:method}
\begin{algorithm}[!ht]
\caption{Testing procedure of Diffusion Correlation}
\label{alg:flow}
\enspace Input: Adjacency matrix $\mc{A} \in \mathbb{R}^{n \times n}$ and nodal attributes $\mc{X} = \{ X_{i} \in\mathbb{R}^{p}: i = 1,2,\ldots,n \}$. \\
\quad (1) Symmetrize $\mc{A}$ by $\mc{K} = (\mc{A} + \mc{A}^{T})/2$.\\
\quad (2) Obtain normalized graph Laplacian matrix $\mc{L} = \mc{B}^{-1/2} \mc{K} \mc{B}^{-1/2}$. \\ \quad (3) Do eigendecomposition to obtain diffusion maps $\mc{U}^{t} = \{ U^{t}_{1}, U^{t}_{2}, \ldots, U^{t}_{n} \}$ for $t=0,1,2,\ldots, 10$. \\
\quad (4) Derive $n \times n$ Euclidean distance of diffusion map $\mc{C}^{t}$, i.e., diffusion distance, across $t$, and $n \times n$ Euclidean distance of nodal attributes, $\mc{D}$. \\
\quad (5) Compute the multiscale graph correlations using two distance matrices, $\mc{C}^{t}$ and $\mc{D}$, for~$t=0,1,\ldots, 10$. \\
\quad (6) Derive the diffusion multiscale graph correlation: $\textsc{mgc}^{*}_{n} \left(\{\mc{U}^{t}\}, \mc{X} \right)$ by estimating $t^*$. \\
\quad (7) Compute p-value using permutation test.\\
\enspace Output: P-value at the estimated optimal step $t^{*}$, the estimated optimal time step $t^{*}$,
dimension choice of $q$ via profile likelihood method,
multiscale local correlation maps $\{ \textsc{dcorr}_{n}^{kl}(\mc{U}^{t}, \mc{X}) \}$, the optimal neighborhood choice $(k^{*}, l^{*})$. \end{algorithm} Here we develop diffusion multiscale graph correlation, which synthesizes diffusion map embedding, multiscale graph correlation, and smoothed maximum to better test network dependency. A flowchart of the testing procedure is illustrated in Fig.~\ref{fig:flow}, and the details of each step are described in Algorithm~\ref{alg:flow}.
The algorithm is flexible in the choice of correlation measures: by following the exact same steps but replacing the multiscale graph correlation by distance correlation in Step (5), one can compute the diffusion distance correlation. Similarly one can derive the diffusion Heller-Heller-Gorfine method. The motivation of the smoothing Step (6) is the following: suppose there exists an optimal $t$ for detecting the relationship between edge connectivity and attributes, then the test statistics at adjacent time steps $t-1$ and $t+1$ should also exhibit strong signal. Under independence, a large test statistic at certain $t$ can occur by chance and cause a direct maximum to have a low testing power, while the smoothed maximum effectively filters out any noisy and isolated large test statistic. In practice, it suffices to consider $t \in [0,1,\ldots,10]$ or even smaller upper bound like $3$ or $5$. When smoothed maximum does not exist, we set $t=3$ as the default choice. The permutation test in Step (7) is a common nonparametric procedure used for real data testing in almost all dependency measures, which is valid as long as the observations are exchangeable under the null \citep{RizzoSzekely2016}.
\subsection{Theoretical Properties Under Exchangeable Graph} \label{ssec:theory} To derive the theoretical consistency of our methodology, the following mild assumptions are required on the distribution of the graph and the nodal attributes. \\
\indent (C1) Graph $\mc{G}$ is an induced subgraph of an infinitely exchangeable graph. Namely, the adjacency matrix $\mc{A}$ satisfies \begin{align} \mc{A}(i,j) \stackrel{d}{=} \mc{A}(\sigma(i),\sigma(j)) \end{align} for any $i,j=1,2,\ldots,n$ and any permutation $\sigma$ of size $n \in \mathbb{N}$. The notation $\stackrel{d}{=}$ stands for equality in distribution. \\ \indent (C2) Each nodal attribute $ X_i$ is generated independently and identically from $F_{X}$ with finite second moment. \\ \indent (C3) The matrix $\mc{A}$ is constrained to a domain $\Omega$ where the diffusion map embedding from $\mc{A} \in \Omega$ to $\mc{U}^{t}$ is injective for some $t$. \\
Condition (C1) states that $\mc{G}$ is a collection of independently sampled nodes and their induced subgraph~\citep{orbanz2015bayesian, tang2017a, orbanz2017subsampling}, which is a distributional assumption satisfied by many popular statistical networks models. Based on condition (C1), the diffusion map $\mc{U}^{t}$ at each $t$ can furnish exchangeable and asymptotic conditional i.i.d. embedding for the set of nodes $\mc{V}(\mc{G})$.
\begin{theorem}
\label{theorem:iid}
Assume $\mc{G}$ satisfies (C1). Then at each fixed $t$, the embedded diffusion maps $\mc{U}^{t} = \{U_{i}^{t},~i=1,2,\ldots,n\}$ by Equation~\ref{eq:U} are exchangeable. As a result, there exists an underlying variable $\theta^{t}$ distributed as the limiting empirical distribution of ~$\mc{U}^{t}$, such that $U_{i}^{t} \mid \theta^{t} $ are asymptotically independently and identically distributed for $i=1,2, \ldots,n$ as $n \rightarrow \infty$. \end{theorem} Due to condition (C1), the permutation test is applicable to any $\mc{U}^{t}$ from an exchangeable sequence. Condition (C2) is merely a regularity condition, and the distribution of $U_{i}^{t}$ automatically satisfies the same finite-moment assumption as shown in the Supplementary Material for proof of Theorem~\ref{theorem:convergence}. We then have consistency between the diffusion map at each $t$ and the nodal attribute.
\begin{theorem}
\label{theorem:convergence}
Assume the graph $\mc{G}$ and the nodal attributes satisfy condition (C1) and (C2). Then as $n \rightarrow \infty$, the multiscale graph correlation between the diffusion map $\mc{U}^{t}$ at any fixed $t$ and the nodal attributes $\mc{X}$ satisfies:
\begin{align*}
\textsc{mgc}_{n}(\mc{U}^{t},\mc{X}) \rightarrow c \geq 0,
\end{align*}
with equality if and only if $F_{U^{t} X} =F_{U^{t}} F_{X}$. \end{theorem}
The testing consistency naturally extends to the diffusion correlation, which further holds between edge connectivity and nodal attributes if condition (C3) is satisfied. \begin{theorem}
\label{theorem:time}
Under the same assumption in Theorem~\ref{theorem:convergence}, it holds that
\begin{align*}
\textsc{mgc}_{n}^{*}(\{\mc{U}^{t}\}, \mc{X}) \rightarrow c \geq 0,
\end{align*}
with equality if and only if $F_{U^{t} X} =F_{U^{t}} F_{X}$ for all $t \in [0,10]$. Therefore, the diffusion multiscale graph correlation is a valid and consistent statistic for testing independence between the diffusion maps $\{\mc{U}^{t}\}$ and nodal attributes $\mc{X}$.
If condition (C3) holds, then $\textsc{mgc}^{*}_{n}(\{\mc{U}^{t}\}, \mc{X})$ is also valid and consistent for testing independence between the adjacency matrix and nodal attributes, i.e., it converges to $0$ if and only if the nodal attribute $X$ is independent of the node connectivity $A$. \end{theorem}
\begin{corollary}
\label{corollary:main}
Theorem~\ref{theorem:time} still holds, when any of the following changes are applied to the testing procedure described in Section~\ref{ssec:method}: \\
\indent (1) The multisclale graph correlation in step 2 is replaced by distance correlation or the Heller-Heller-Gorfine statistic; \\
\indent (2) When $\mc{A}$ is restricted to be symmetric, binary, and of finite rank $q <n$, then condition (C3) holds at $t=1$. \end{corollary}
Namely, point (1) suggests that under diffusion maps, other consistent dependency measure can also be used to produce a valid and consistent diffusion correlation, which enables us to compare a number of diffusion correlations in the simulations. Point (2) offers an example of random matrix $\mc{A}$ where the diffusion map is guaranteed injective within the domain.
\subsection{Consistency Under Random Dot Product Graph} \label{ssec:theory2} In this section, we illustrate the theoretical results via the random dot product graph model, which is widely used in network modeling. It assumes that each node has a latent position $W_{i} \overset{i.i.d.}\sim F_{W}$ for $i=1,2,\ldots, n$, and the edge probability $pr(\mc{A}(i,j) = 1 \mid W_{i}, W_{j})$ is determined by the dot product of the latent positions, i.e., \begin{align*} \mc{A}(i,j) \mid W_{i}, W_{j} \overset{i.i.d.}{\sim} Bernoulli\big( \langle W_{i},W_{j} \rangle \big),~i,j=1,2,\ldots, n \mbox{ and } i<j, \end{align*} under the restriction that all $W_{i}$'s are non-negative vectors and the dot product must be normalized within $[0,1]$.
A random dot product graph is an exchangeable graph model that satisfies condition (C1). In addition, random dot product graph fully specifies all exchangeable graph models that are unweighted and symmetric, whose probability generating matrix $\mc{P}(i,j)=\langle W_{i},W_{j}\rangle$ is positive semi-definite. \begin{proposition}[\citet{sussman2014consistent}]
\label{prop1}
An exchangeable random graph has a finite rank $q$ and positive semi-definite link matrix $\mc{P}$, if and only if the random graph is distributed according to a random dot product graph with i.i.d. latent vectors $\{W_{i} \in \mathbb{R}^{q},i=1,\ldots,n\}$. \end{proposition}
Indeed, many other popular network modelings are special cases of random dot product graph, including the stochastic block model~\citep{airoldi2008mixed, hanneke2009network, rohe2011spectral, xin2017continuous}, its degree-corrected version~\citep{karrer2011stochastic}, the latent factor model from \citet{fosdick2015testing}, etc.
\begin{proposition}[\citet{rohe2011spectral}]
\label{prop2}
Let $\mc{L}$ be the normalized graph Laplacian for an adjacency matrix $\mc{A}$ generated by a random dot product graph with latent positions of which construct the matrix of $\mc{W}=[W_{1}|W_{2}|\ldots|W_{n}] \in \mathbb{R}^{q \times n}$. Let~$\mc{U}^{t=1} = [ U^{t=1}_{1}| U^{t=1}_{2}| \ldots| U^{t=1}_{n} ] \in \mathbb{R}^{q \times n}$.
Then there exists a fixed diagonal matrix $\mc{M}$ and an orthonormal rotational matrix $\mc{Q} \in \mathbb{R}^{q \times q}$ such that $\| \mc{U}^{t=1} - \mc{Q} \mc{M} \mc{W} \| \rightarrow 0$ almost surely. \end{proposition}
Therefore, under random dot product graph, the diffusion map $\mc{U}^{t=1}$ asymptotically equals the latent position $\mc{W}$ up to a linear transformation. As the latent position under random dot product graph can be asymptotically recovered by diffusion maps, diffusion correlation is consistent against testing general dependency between $\mc{A}$ and $\mc{X}$ under random dot product graph.
\begin{corollary}
\label{thm:AvsX}
Under an induced subgraph from exchangeable graph with positive semi-definite link function, the diffusion multiscale graph correlation is consistent for testing independence between edge connectivity and nodal attributes. \end{corollary}
\subsection{Discussion on the Conditions} Here we discuss the robustness of the methodology with respect to condition (C1)-(C3), and what happens when any of them is violated. These conditions are essential to guarantee a consistent and valid testing framework in general, which are not just limited to network dependence testing.
Condition (C1) is a crucial condition for the permutation test to be valid. When it is violated and neither set of data can be assumed exchangeable, all aforementioned test statistics may no longer be valid because the permutation test fails to control the type 1 error level as demonstrated in~\cite{Mantel2013}. In certain special cases like testing independence between two stationary times series, block permutation technique can be used to yield a valid test \citep{lacal2018estimating}, which can be readily used here but is not guaranteed valid for general non-exchangeable data. Condition (C2) is a regularity condition required for distance-based correlation measure to be well-behaved, without which the distance variance can explode to infinity and cause the correlation measure to be ill-behaved.
In comparison, the diffusion correlation methodology is still valid without condition (C3). However, the second part of Theorem~\ref{theorem:time} will no longer hold, and the methodology is no longer universally consistent. Namely, certain signals of dependency may be lost during the diffusion map embedding procedure. As a result, the diffusion correlation could be asymptotically $0$ for some dependencies and thus no longer able to detect all possible dependencies between the edge connectivity and nodal attributes. In the Supplementary Material we illustrate the performance of the test statistics under the violation of positive semi-definite link function, and show relative robustness of distance-based tests compared to model-based tests when condition (C3) is violated.
\section{Numerical Studies} \label{sec:simulation}
\subsection{Stochastic Block Model}
Throughout the numerical studies, we compare diffusion multiscale graph correlation, diffusion distance correlation, diffusion Heller-Heller-Gorfine method, the Fosdick-Hoff likelihood ratio test~\citep{fosdick2015testing}, and direct embedding-based tests: using the adjacency spectral embedding and the latent factors to embed the adjacency matrix first, followed by any of the multiscale graph correlation, the distance correlation, or the Heller-Heller-Gorfine method. For each simulation, we generate a sample graph and the corresponding attributes, compute the test statistic of each method, carry out the permutation test with $r=500$ random permutations, and reject the null if the resulting p-value is less than $\alpha = 0.05$. The testing power of each method equals the percentage of correct rejection out of $m = 100$ replicates, and a higher power implies a better method against the given dependency structure.
The first simulation samples graphs from the stochastic block model. It assumes that each of the $n$ nodes in $\mc{G}$ must belong to one of $K \in \mathbb{N}$ blocks, and determines the edge probability based on the block-membership of the connecting nodes: For $i=1,\ldots,n$, assume there exists a latent variable of $Z_{i} \overset{i.i.d.}{\sim} Multinomial\big( \pi_{1}, \pi_{2}, ... , \pi_{K} \big)$ denoting the block-membership of each node, and denote the edge probability between any two nodes of class $k$ and $l$ as $b_{kl} \in \{0,1\}$. Then the upper triangular entries of $\mc{A}$ are independently and identically distributed when conditioning on $\mc{Z} = \{Z_{i}:~i=1,2,\ldots, n \}$: \begin{align*} \mc{A}(i,j) \mid Z_{i}, Z_{j} \overset{i.i.d.}{\sim} Bernoulli\big\{ \sum\limits_{k,l=1}^{K} b_{kl} \mathbb{I} \big( Z_{i} = k, Z_{j} = l \big) \big\}; \quad i < j,~i,j = 1,2, \ldots, n, \end{align*} where $\mathbb{I}(\cdot)$ is the indicator function. The sample data is generated at $n=100$ by using following parameters: \begin{equation} \begin{split} \label{eq:Three} & Z_{i} \overset{i.i.d.}{\sim} Multinom(1/3, 1/3, 1/3),\\
& \mc{A}(i,j) \mid Z_{i}, Z_{j} \sim Bernoulli \left\{0.5 \mathbb{I}(|Z_{i} - Z_{j}| = 0) + 0.2 \mathbb{I}(|Z_{i} - Z_{j}| = 1) + 0.4 \mathbb{I}(|Z_{i} - Z_{j}| = 2) \right\}, \\ &X_{i}\mid Z_{i} \sim Multinom[ \{1+\mathbb{I}(Z_{i}=1)\}/4,~\{1+\mathbb{I}(Z_{i}=2)\}/4,~\{1+\mathbb{I}(Z_{i}=3)\}/4 ], \end{split} \end{equation} where $\mc{X}$ is a randomly polluted block assignment: for each $i$, $X_{i}=Z_{i}$ with probability $0.5$, and equally likely to take other values in $\Omega$, i.e., the true block-membership is observed half of the time. For the adjacency matrix, the within-block edge probability is always $0.5$; while the between-block edge probability is $0.2$ when the block labels differ by $1$, and $0.4$ when the block labels differ by $2$. As the edge probability between a node of block $1$ and a node of block $3$ is higher than the edge probability between block $1$ and block $2$, this three-block stochastic block model generates a noisy and nonlinear dependency structure between $\mc{A}$ and $\mc{X}$, and we would like to verify how successful the methods are in detecting the dependency between the adjacency matrix $\mc{A}$ and the noisy block assignment $\mc{X}$.
Figure~\ref{fig:threeSBM} shows that diffusion multiscale graph correlation prevails the testing powers among all the methods, because multiscale graph correlation captures high-dimensional nonlinear dependencies better than distance correlation and Heller-Heller-Gorfine. A visualization of the sample data is available in Fig.~\ref{fig:embedding}(a). \begin{figure}
\caption{ The testing powers for the three-block stochastic block model in Equation~\ref{eq:Three}. The y-axis lists the embedding choices: diffusion map (DM), adjacency spectral embedding (AM), and latent factor embedding (LF). The x-axis corresponds to the correlation measure in use: the multiscale graph corelation (MGC), distance correlation (dCorr), Heller-Heller-Gorfine (HHG), and the Fosdick-Hoff method (FH). The top three entries in the first row represent the diffusion correlation methods proposed in this paper, which outperform other embedding choices with diffusion multiscale graph correlation having the best power. }
\label{fig:threeSBM}
\end{figure}
\subsection{Stochastic Block Model with Linear and Nonlinear Dependencies}
To further understand and demonstrate the advantage of the diffusion approach, here we use the same three-block stochastic block model and its block-membership $\{ Z_{i} : i=1,2, \ldots, n=100 \}$ as in the previous section, except that the edge probability is now controlled by $\beta \in (0, 1)$ for all $i,j = 1, \ldots, n$: \begin{equation}
\mc{A}(i,j) \mid Z_{i}, Z_{j} \sim Bernoulli \left\{ 0.5 \mathbb{I}(|Z_{i} - Z_{j}| = 0) + 0.2 \mathbb{I}(|Z_{i} - Z_{j}| = 1) + \beta \mathbb{I}(|Z_{i} - Z_{j}| = 2) \right\}. \label{eq:mono} \end{equation} The noisy block-membership $\mc{X}$ is generated in the same way as before. When $\beta = 0.2$, the three-block stochastic block model is the same as a two-block stochastic block model, where within-block edge probability equals $0.5$ while the between-block edge probability is always $0.2$, i.e., it represents a linear association between the adjacency matrix and the block-membership. When $\beta <0.2$, the dependency is still monotonic. When $\beta > 0.2$ and gets further away, the relationship becomes strongly nonlinear. \begin{figure}
\caption{(a) The power curve with respect to increasing $\beta$ under three-block stochastic block model (Equation~\ref{eq:mono}). When $\beta$ shifts from less than $0.2$ to higher than $0.2$, it represents a structural change in the relationship from monotone to non-monotone. Among all methods utilizing diffusion maps, diffusion \textsc{mgc}~(solid red line) is evidently the best performing one comparing to diffusion \textsc{dcorr}~(yellow brown dashes), diffusion \textsc{hhg}~(blue small dashes), and \textsc{fh}~(green dot-dash) test. (b) The power curve with respect to increasing $\tau$ under degree-corrected stochastic block model (Equation~\ref{eq:tau}). The edge variability increases as $\tau$ increases. Diffusion \textsc{mgc}~(red solid) is relatively stable in power against increasing variability. The adjacency spectral embedding followed by \textsc{mgc}~(dark blue dashes) is slightly worse, while the latent factor embedding followed by \textsc{mgc}~(light yellow dashes) and \textsc{fh}~(green dot-dash) have almost no power against all levels of $\tau$.}
\label{fig:powerplot}
\end{figure} Figure~\ref{fig:powerplot}(a) plots the power against $\beta$ for all diffusion maps-based methods, demonstrating that main approach using the multiscale graph correlation is the most powerful method against varying dependency structure.
\subsection{Degree-Corrected Stochastic Block Model} In this section we compare different embeddings under the degree-corrected stochastic block model, which better reflects many real-world networks. The degree-corrected stochastic block model is an extension of stochastic block model by introducing an additional random variable $c_{i}$ to control the degree of each node.
We set $n=200$ with two blocks, select the binary block-membership $Z_i$ uniformly in $\Omega=\{0,1\}$, and generate the edge probability by \begin{equation}
\mc{A}(i,j) \mid Z_{i}, Z_{j}, C_{i}, C_{j} \sim Bernoulli \{ 0.2 C_{i} C_{j} \cdot \mathbb{I}( |Z_{i} - Z_{j}| = 0 ) + 0.05 C_{i} C_{j} \cdot \mathbb{I}(|Z_{i} - Z_{j}| = 1) \}, \label{eq:tau} \end{equation} where $C_{i} \overset{i.i.d.}{\sim} Uniform(1 - \tau, 1 + \tau)$ for $i = 1, \ldots, n$, and $\tau \in [0, 1]$ is a parameter to control the amount of variability in the edge degree, e.g., as $\tau$ increases, the model becomes more complex as the variability of the edge probability becomes larger; when $\tau=0$, the above model reduces to a two-block stochastic block model without any variability induced by $\{ C_{i} : i=1,2,\ldots,n\}$. We again generate the nodal attributes $\mc{X}$ as a noisy version of the true block-membership via Bernoulli distribution, i.e., for each $i$, $X_{i}= Z_{i}$ with probability $0.6$, and equals the wrong label with probability $0.4$. Figure~\ref{fig:powerplot}(b) compares different embedding choices using multiscale graph correlation.
\subsection{Random Dot Product Graph Simulations} \label{ssec:RDPG} Next we present a variety of random dot product graph simulations by generating the latent variables via the $20$ relationships in \citet{shen2017mgc} with different levels of noise, consisting of various linear, monotonic and non-monotonic relationships. The details of simulation schemes are in the Supplementary Material, and a general outline for data generating process is: \begin{align} \label{eq:RDPG_general} \left( \begin{array}{cc} \tilde{W}_{i} & \tilde{X}_{i} \end{array} \right) & \overset{i.i.d.}{\sim} F_{\tilde{W} \tilde{X}} \qquad i = 1, 2, \ldots, n, \nonumber \\ \mc{A}(i,j) \mid W_{i}, W_{j} & \sim Bernoulli\left( \langle W_{i}, W_{j} \rangle \right),~i<j=1,2,\ldots,n, \end{align} where $W_{i} = \{ \tilde{W}_{i} - \min( \{ \tilde{W}_{j} : j=1,2,\ldots,n \} ) \} / \{ \max(\{ \tilde{W}_{j} : j=1,2,\ldots,n \}) - \min(\{ \tilde{W}_{j} : j=1,2,\ldots,n \} \}$ for $i=1,2,\ldots, n$, so that all the latent variable range from 0 to 1. We apply the same scaling from $\tilde{X}_i$ to $X_i$ for visual consistency. \begin{figure}
\caption{ Power comparison for 20 different random dot product graphs with $n=50$ nodes per $m=100$ independent replicates. It shows that when latent positions $W_i$ and nodal attributes $X_i$ are dependent via a close-to-linear relationship at upper panel, diffusion multiscale graph correlation (red circle), diffusion distance correlation (blue X), and diffusion Heller-Heller-Gorfine (brown diamond) achieve similar power while the Fosdick-Hoff test (green triangle) is slightly worse due to its model-based nature. When non-linearity between $W_i$ and $X_i$ becomes evident like circle or ellipse at lower panel, multiscale graph correlation and Heller-Heller-Gorfine are the two best performing correlation measure, which is somewhat consist with the empirical results in \cite{shen2017mgc} for non-network data. }
\label{fig:RDPG}
\end{figure} Thus the latent positions and nodal attributes are correlated via a joint distribution of $F_{\tilde{W} \tilde{X}}$, including linear, quadratic, circle, etc. Figure~\ref{fig:RDPG} shows empirical power obtained from $m=100$ independent replicates when the number of nodes is $n=50$, for which all the diffusion map-based methods work fairly well. Note that the last scenario is an independent relationship and all tests achieve a power approximately at $0.05$, implying that they are all valid tests; there are also a few dependencies of very low power due to the complexity of the relationship, e.g., sine, spiral, square, etc., but their powers all converge to $1$ as sample size $n$ increases.
\section{Graph Embedding using Diffusion Multiscale Graph Correlation} \label{ssec:dis}
This section demonstrates that in deriving the diffusion correlation, we preserve dependency structure between $\mc{A}$ and $\mc{X}$ without cross-validation or over-fitting by virtue of effectively estimating parameters of $t$ and $q$. As a reminder, the dimension choice $q$ is selected by the second elbow of the absolute eigenvalue scree plot via the profile likelihood method from~\citet{ZhuGhodsi2006}. The choice of $t^{*}$ is based on a smoothed maximum. Viewed in another way, diffusion multiscale graph correlation selects the optimal diffusion map that maximizes the multiscale graph correlation. Thus any testing advantage shall come down to whether it is able to optimize the embedding without over-fitting, and we investigate how well our procedure is able to preserve the dependency compared to adjacency spectral embedding. \begin{figure}
\caption{Generate a three-block adjacency matrix $\mc{A}$ by Equation~\ref{eq:Three} at $n=100$, and compute the diffusion distances at each combination of $(t,q)$. A visualization of adjacency matrix is provided in Fig.~\ref{fig:embedding} (a); upon fixing a good $t$, many choices of $q$ preserve the block structure. Note that the first three elbows of eigenvalues are $(1,45,70)$ and $t^{*}=2$, so panel (g) is the optimal diffusion map by diffusion multiscale graph correlation.}
\label{fig:diffusions}
\end{figure}
\begin{figure}
\caption{Panel (a) shows the adjacency matrix of three-block adjacency matrix $\mc{A}$ generated by Equation~\ref{eq:Three}. Panel (b)-(d) show the Euclidean distance matrix of adjacency spectral embedding at increasing $q$, using the same adjacency matrix of Panel (a). Only adjacency spectral embedding at $q=3$, namely at the correct dimension, is able to display a clear block structure. Note that the first three elbows are $(1,45,70)$, so adjacency spectral embedding has a more obscure block structure when the dimension is chosen via the scree plot, comparing to the diffusion correlation-based embedding in Fig.~\ref{fig:diffusions} (g).}
\label{fig:embedding}
\end{figure} Figure~\ref{fig:diffusions} presents the diffusion distances at different $t$ and $q$ for the three-block stochastic block model in Equation~\ref{eq:Three}. Although the resulting embedding is sensitive to both $t$ and $q$ in Fig.~\ref{fig:diffusions} (a)--(d), at optimal $t^{*}=2$ it is robust against $q$, e.g., Fig.~\ref{fig:diffusions} (e)--(h) show that for a wide range of $q$ the block structure is preserved in the resulting diffusion maps including the second elbow, so the diffusion correlation-based embedding preserves the dependency structure well.
On the other hand, Fig.~\ref{fig:embedding} shows the Euclidean distance of the adjacency spectral embedding \citep{SussmanEtAl2012} applied to the same adjacency matrix. For adjacency spectral embedding, the correct dimensional choice equals the number of blocks, i.e., the distance matrix at $q=3$ shows a clear block structure in Fig.~\ref{fig:embedding} (b). However, a slight misspecification of $q$ can cause the embedding to have a more obscure block structure, and the elbow method often fails to find the correct $q$ for adjacency spectral embedding.
\begin{figure}\label{fig:timeplot}
\end{figure}
Next we compare testing performance of the diffusion correlation-based embedding $\mc{U}^{t^{*}}$ versus all other diffusion maps $\mc{U}^{t}$. Figure~\ref{fig:timeplot} shows the proportion of choosing $t$ as the optimal among $\{0,1,2, \ldots, 10\}$ and the testing power for each $t$ and also $t^{*}$. Figure~\ref{fig:timeplot} (a) illustrates that under the stochastic block model dependency structure in Equation~\ref{eq:mono} with $\beta = 0.50$, diffusion multiscale graph correlation is mostly likely to choose $t^{*} = 2$ as the optimal time-step, and the testing power is almost equivalent to the best power among all $t \in \{0,1,2, \ldots, 10 \}$. The same phenomena hold for other diffusion correlations, and Fig.~\ref{fig:timeplot} (b) illustrates the results via the random dot product graph simulation example by Equation~\ref{eq:RDPG_general}.
\section{Real Data Application} \label{sec:realdata} As a real data example, we apply the methodology to the neuronal network of hermaphrodite \textit{Caenorhabditis elegans}, which composes of 279 nonpharyngeal neurons connecting each other through chemical and electrical synapses~\citep{varshney2011structural}. Each node represents an individual neuron, and each edge weight indicates the number of synapses between them. Among a few known attributes including types of neurotransmitter and role of neurons, we use one dimensional, continuous position of each neuron as the nodal attribute $\mc{X}$. Figure~\ref{fig:celegans} shows that neurons at low location and high location are connected to other neurons distributed throughout the region; while those at the relatively middle of location are connected to the neurons only within the narrower area. The independence test between synapse connectivity and each neuron's position can be connected to growing number of studies on relationship between physical arrangement and functional connectivity in Caenorhabditis elegans~\citep{chen2006wiring, kaiser2006nonoptimal} and others'~\citep{cherniak2004global, alexander2012anatomical}. We binarize and symmetrize both chemical and electrical synapses, add them together to represent overall synapse connectivity of Caenorhabditis elegans, then apply diffusion multiscale graph correlation, diffusion distance correlation, diffusion Heller-Heller-Gorfine, and Fosdick-Hoff to test independence between connectivity through synapses and neuron's position. All methods result in similar significant p-values less than 0.002. \begin{figure}
\caption{ Each dot at $(x,y)$ represents the existence of synapses from neuron at $y$ to neuron at $x$. Y-axis represents each neuron’s index assigned from low location to high location, and x-axis represents 68 different locations where neurons are positioned. Color of dots represents synapse type, either chemical (red circle) or electrical (blue triangle), and size of dots is proportional to the number of synapse but capped at 10.}
\label{fig:celegans}
\end{figure} \begin{figure}\label{fig:localcorr}
\end{figure} Figure~\ref{fig:localcorr} (a)-(d) presents local distance correlation map $\textsc{dcorr}^{kl}(\mc{U}, \mc{X})$ across diffusion times. These plots show that the optimal local correlation is detected at non-global neighborhood choice, i.e. $l^{*} \neq 68$ (the global maximum), which imply a non-linear dependence between connectivity and position and an optimal $t^{*}=5$. Figure~\ref{fig:localcorr} (e)-(h) illustrate the relationship between Euclidean distance in diffusion maps and nodal attributes at different diffusion times at $t=1,3,5,10$, which is again the most significant at $t=5$.
\section{Discussion}
There are several potential follow-ups that would further advance the work. One example is more theoretical investigation into the smoothed maximum and dimension selection of $t^{\prime}$. Assuming $t^{\prime}$ is the true optimal diffusion time, it will be helpful to either identify a more systematic and reliable way to estimate $t^{\prime}$, or quantify variability in the estimated optimal $t^{*}$ by smoothed maximum. This would hopefully reduce computational burden instead of going over all possible diffusion times, e.g. $t=0,1,2, \ldots, 10$. Moreover, although we briefly discussed one example in Section~\ref{ssec:dis}, the impact of dimensional choice of $q$ is still obscure on the embedding quality. Finally, since one can apply diffusion map to any data and one can think of any affinity or kernel matrix as a graph, this method is actually applicable to more general testing scenarios beyond networks, which is another point of interest for further investigation.
\appendix \setcounter{figure}{0}
\renewcommand{S\arabic{figure}}{S\arabic{figure}}
\begin{center} {\large\bf Supplementary Material} \end{center}
\section{Proofs} \label{sec:proof} Unless mentioned otherwise, throughout the proof section we always omit the superscript $t$ for the diffusion map at a fixed $t$, i.e., we use $\mc{U}= \{U_i : i=1,2,\ldots,n\}$ instead of $\mc{U}^{t}= \{U_{i}^{t} : i=1,2,\ldots,n\}$ because most results hold for any $t$, similarly we use $\theta$ instead of $\theta^{t}$ whenever appropriate.
\begin{proof}[(Theorem 1)]
By the \textit{de Finetti's Theorem}~\citep{diaconis1980finite,oneill2009,orbanz2015bayesian}, it suffices to prove that the diffusion map $\mc{U} = \{U_i: i=1,\ldots,n\}$ is always exchangeable in distribution, i.e., for any $n$ and all possible permutation $\sigma$, the permuted sequence $\mc{U}_{\sigma} = \{U_{\sigma(1)}, U_{\sigma(2)}, \ldots,U_{\sigma(n)}\}$ always distributes the same as the original sequence $\mc{U} = \{U_{1}, U_{2}, \ldots,U_{n}\}$.
Transforming Equation 3 in the main manuscript into matrix notation yields
\begin{align*}
\mc{U} = \Lambda^{t}\Phi^{T},
\end{align*}
where $\mc{U}$ is the $q \times n$ matrix having $U_i$ as its $i^{\mbox{th}}$ column, $\Lambda=\mbox{diag} \{ \lambda_{1},\lambda_2,\ldots,\lambda_q \}$ is the diagonal matrix having selected eigenvalues of $\mc{L}$, $\Phi =[ \phi_1, \phi_2, \cdots, \phi_q ]$ consists of the corresponding eigenvectors, $\cdot^{t}$ denotes $t^{\mbox{th}}$ power, and $\cdot^{T}$ is the matrix transpose. It suffices to show that $\mc{U}$ and $\mc{U} \Pi$ are identically distributed for any permutation matrix $\Pi$ of size $n$.
Given that the graph $\mc{G}$ is an induced subgraph of an infinitely exchangeable graph, it holds that $\mc{A}(\sigma(i),\sigma(j)) \stackrel{d}{=} \mc{A}(i,j)$, which further holds for the symmetric graph Laplacian $\mc{L}$:
\begin{align*}
\mc{L}(\sigma(i), \sigma(j)) &= \mc{A}(\sigma(i), \sigma(j)) / \{ \sum\limits_{j} \mc{A}(\sigma(i), \sigma(j)) \sum\limits_{i} \mc{A}(\sigma(i), \sigma(j))\}^{1/2} \\
&\stackrel{d}{=} \mc{A}(i,j) / \{ \sum\limits_{j} \mc{A}(i,j) \sum\limits_{i} \mc{A}(i,j) \}^{1/2} \\
&= \mc{L}(i,j).
\end{align*}
In matrix notation, $\Pi^{T} \mc{L} \Pi \stackrel{d}{=} L$ for any permutation matrix $\Pi$.
By eigen-decomposition, the first $q$ eigenvalues and the corresponding eigenvector of $\Pi^{T} \mc{L} \Pi$ are $\Lambda$ and $\Pi^{T} \Phi$, so it follows that at any $t$
\begin{align*}
& \Phi \stackrel{d}{=} \Pi^{T} \Phi\\
\Leftrightarrow \quad
& \mc{U} =\Lambda^{t} \Phi^{T} \stackrel{d}{=} \Lambda^{t} \Phi^{T} \Pi = \mc{U} \Pi.
\end{align*}
Thus columns in $\mc{U}$ are exchangeable, i.e., the diffusion maps, $\{ U_{i} \in \mathbb{R}^{q} : i=1,2,\ldots, n \}$, are infinitely exchangeable. By the \textit{de Finetti's Theorem}, there exists an underlying variable $\theta$ distributed as the limiting empirical distribution, such that $U_{i} \mid \theta$ are asymptotically i.i.d. \end{proof}
\begin{proof}[(Theorem 2)]
We first state three lemmas:
\begin{lemma}
\label{lemma:aux1}
Under the same assumptions of Theorem~2, for any finite time-step $t$, the underlying distribution of $U_{i}^{t}$ of the diffusion map is of finite second moment.
\end{lemma}
\begin{lemma}
\label{lemma:aux2}
The distance covariance of $(\mc{U}, \mc{X}) = \{ ( U_{i}, X_{i} ) : i = 1, \ldots, n \}$ defined in Equation~5 in the paper satisfies
\begin{align*}
\textsc{dcov}_{n}(\mc{U},\mc{X}) &= \int_{\mathbb{R}^{q+p}} |\hat{g}_{\mc{U}, \mc{X}}(t,s)-\hat{g}_{\mc{U}}(t) \hat{g}_{\mc{X}} (s)|^{2} dw(t,s),
\end{align*}
where $w(t,s) \in \mathbb{R}^{q} \times \mathbb{R}^{p}$ is a nonnegative weight function that equals $(c_{q}c_{p}|t|_{q}^{1+q}|s|_{p}^{1+p})^{-1}$, $c_{q}$ is a nonnegative constant, $\hat{g}_{\cdot}$ is the empirical characteristic function of $\{(U_{i},X_{i}) : i=1,2,...,n\}$ or the marginals, e.g., $\hat{g}_{\mc{U},\mc{X}}(t,s)=\frac{1}{n}\sum_{i=1}^{n}\exp(\textbf{i} \left\langle t,U_{i} \right\rangle+\textbf{i} \left\langle s,X_{i} \right\rangle)$ with $\textbf{i}$ representing the imaginary unit.
\end{lemma}
\begin{lemma}
\label{lemma:aux3}
Assume $\mc{U} = \{ U_{i} \sim F_U: i = 1,2, \ldots, n \}$ are conditional i.i.d. as $U \mid \theta$, and $\mc{X} = \{X_{i} \stackrel{i.i.d.}{\sim} F_{X}:~i = 1, 2, \ldots, n\}$, and all distributions are both of finite second moment. It follows that
\begin{align*}
\textsc{dcov}_{n}(\mc{U}, \mc{X}) &\rightarrow \textsc{dcov}(U,X) \mbox{ as } n \rightarrow \infty,
\end{align*}
where $\textsc{dcov}(U,X) := \int_{\mathbb{R}^{q+p}} | g_{U,X}(t,s) - g_{U}(t) g_{X}(s) |^2 dw(t,s)$ is the population distance covariance, and $g_{\cdot}$ is the characteristic function, i.e., $g_{U,X}(t,s) = E(\exp\{\textbf{i} \left\langle t,U \right\rangle +\textbf{i} \left\langle s, X\right\rangle \})$.
\end{lemma}
By Theorem~1, the diffusion maps $U_{i}$ are asymptotically i.i.d. conditioned on $\theta$, whose finite moment is guaranteed by Lemma~\ref{lemma:aux1}. The nodal attributes $X_i$ are i.i.d. as $F_{X}$ of finite second moment as assumed in (C2). Therefore a direct application of Lemma~\ref{lemma:aux2} and Lemma~\ref{lemma:aux3} yields that
\begin{align*}
\mbox{\textsc{dcov}}_{n}(\mc{U} ,\mc{X}) &\rightarrow \int_{\mathbb{R}^{q+p}} | g_{U,X}(t,s) - g_{U}(t) g_{X}(s) |^2 dw(t,s),
\end{align*}
which equals $0$ if and only if $U$ is independent of $X$. As distance correlation is just a normalized version of distance covariance, it further leads to
\begin{align}
\label{eq:main1}
\mbox{\textsc{dcorr}}_{n}(\mc{U},\mc{X}) &\rightarrow c \geq 0,
\end{align}
for which the equality holds if and only if $F_{UX}=F_{U}F_{X}$. By \cite{shen2017mgc}, Equation~\ref{eq:main1} also holds for \textsc{mgc}~when it holds for \textsc{dcorr}. \end{proof}
\begin{proof}[(Lemma~\ref{lemma:aux1})]
To prove that $U$ is of finite second moment, it suffices to show that $\|U_{i}\|_{2}$ is always bounded for all $i \in [1,n]$. By Equation~3, we have
\begin{align*}
\| U_i \|_{2}^{2} &= \sum_{j=1}^{q} \lambda^{2t}_{j}\phi_{j}^{2}(i) \\
& \leq \sum_{j=1}^{q} \lambda^{2t}_{j} \\
& \leq q,
\end{align*}
where the second line follows by noting $\phi_{j}(i) \in [-1,1]$ (the eigenvector $\phi_{j}$ is always of unit norm), and the third line follows by observing that $| \lambda_{j}| \leq \| L \|_{\infty}= 1$. Therefore, all of $U_{i}$ are bounded in $\ell_{2}$ norm as $n \rightarrow \infty$, so the underlying variable $U$ must be of finite second moment for any finite $t$. \end{proof}
\begin{proof}[(Lemma~\ref{lemma:aux2})]
This lemma is a direct application of Theorem 1 in \cite{szekely2007measuring}, which holds without any assumption on $(\mc{U} ,\mc{X}) = \{(U_{i},X_{i}) : i=1,2,...,n\}$, e.g., it holds without assuming exchangeability, nor identically distributed, nor finite moment. \end{proof}
\begin{proof}[(Lemma~\ref{lemma:aux3})]
This lemma is equivalent to Theorem 2 in \cite{szekely2007measuring}, except the i.i.d. assumption is replaced by exchangeable assumption, i.e., the original set-up needs $(\mc{U}, \mc{X}) = \{(U_{i},X_{i}) : i = 1,2, \ldots , n \}$ to be independently identically distributed as $F_{UX}$ with finite second moment; whereas the diffusion map $\{ U_{i} : i = 1,2, \ldots, n \}$ is asymptotically conditional i.i.d. with finite second moment.
Note that $\hat{g}_{\mc{U}, \mc{X}}(t,s)=E(\hat{g}_{\mc{U}, \mc{X} }(t,s) \mid \theta)$, and each term in $\hat{g}_{\mc{U}, \mc{X}}(t,s) \mid \theta$ is asymptotically i.i.d. of each other. Thus
\begin{align*}
\displaystyle\int{|\hat{g}_{\mc{U},\mc{X}} (t,s)-\hat{g}_{\mc{U}}(t)\hat{g}_{\mc{X}}(s)|^{2}}dw &=
E(\displaystyle\int{|\hat{g}_{\mc{U},\mc{X}} (t,s)-\hat{g}_{\mc{U}}(t)\hat{g}_{\mc{X}}(s)|^{2}}dw \mid \theta) \\
& \rightarrow E( \int{|g_{U,X}(t,s)-g_{U}(t)g_{X}(s)|^{2}}dw \mid \theta) \\
& = \int{|g_{U,X}(t,s)-g_{U}(t)g_{X}(s)|^{2}}dw,
\end{align*}
where the convergence in the second step follows from Theorem 2 in \cite{szekely2007measuring} on the i.i.d. case.
\end{proof}
\begin{proof}[(Theorem~3)]
From Theorem~2, it holds that
\begin{align}
\label{eq:mgczero}
\textsc{mgc}_{n}(\mc{U}^{t},\mc{X}) &\rightarrow c \geq 0
\end{align}
for each $t$, with equality if and only if independence. The \textsc{dmgc}~algorithm enforces that
\begin{align*}
\max\{\textsc{mgc}_{n}(\mc{U}^{t},\mc{X}), t=0,1,\ldots,10\} &\geq \textsc{mgc}_{n}^{*}(\{\mc{U}^{t}\},\mc{X}) \geq \textsc{mgc}_{n}(\mc{U}^{t=3},\mc{X}),
\end{align*}
thus Equation~\ref{eq:mgczero} also holds when $\textsc{mgc}(\mc{U}^{t},\mc{X})$ is replaced by $\textsc{mgc}^{*}(\{\mc{U}^{t}\},\mc{X})$.
To show that the test is valid and consistent, it suffices to show that with probability approaching $1$, $\textsc{mgc}_{n}(\mc{U},\mc{X}_{\sigma}) \rightarrow 0$. This holds when $(U_{i}, X_{i}) \overset{i.i.d.}{\sim} F_{UX}$: the proof in supplementary of \cite{shen2017mgc} shows that the percentage of partial derangement of finite sample size converges to $1$ among all random permutations, such that with probability converging to $1$ a permutation test breaks dependency.
For exchangeable $\{U_{i}\}$ here, we instead have $(U_{i}, X_{i}) \mid \theta \overset{i.i.d.}{\sim} F_{UX \mid \theta}$ asymptotically. The distribution of $\theta$ is the limiting empirical distribution of $\{U_{i}\}$, which is either asymptotically independent of all $X_i$ or dependent only on finite number of $X_i$. Thus $U_{i}$ is asymptotically conditionally independent with $X_{\sigma{(i)}}$ with probability converging to $1$, and we have
\begin{align*}
\textsc{mgc}_{n}(\mc{U},\mc{X}_{\sigma})
=E(\textsc{mgc}_{n}(\mc{U},\mc{X}_{\sigma}) \mid \theta)
\rightarrow 0
\end{align*}
Moreover, when the transformation from $\mc{A}$ to $\mc{U}^{t}$ is injective, we have
\begin{align*}
&\mbox{ $A$ is independent of $X$} \\
\Leftrightarrow & \mbox{ $U^{t}$ is independent of $X$ for all $t$} \\
\Leftrightarrow & \mbox{ $\textsc{mgc}_{n}(\mc{U}^{t},\mc{X})$ is asymptotically $0$,}
\end{align*}
where the second line follows from injective transformation, and the third line follows from Theorem~1 and Theorem~2. Thus \textsc{dmgc}~is consistent between $A$ and $X$.
Note that without the injective condition, the reverse direction of the second line may not always hold, i.e., when the diffusion maps are independent from the nodal attributes, the adjacency matrix may be still dependent with the nodal attributes. In that case, \textsc{dmgc}~is still valid but the dependency may not be detected by \textsc{dmgc}. \end{proof}
\begin{proof}[(Corollary~1)]
(1) Changing the test statistic only affects Theorem~2. Both \textsc{dcorr}~and~\textsc{mgc}~satisfy Theorem~2 directly, while \textsc{hhg}~is also a statistic that is $0$ if and only if independence~\citep{HellerGorfine2013}.
(2)
When $\mc{A}$ is symmetric and binary, the transformation from $\mc{A}$ to $\mc{L}$ is injective, i.e., two different $\mc{A}$ always produce two different $\mc{L}$. Then for each unique $\mc{L}$, the eigen-decomposition is always unique such that $\mc{L}$ to $\mc{U}^{t=1}$ is injective, provided that the dimension choice is made correct at $q$. \end{proof}
\begin{proof}[(Corollary~2)]
From Proposition~1 and~2,~$\mc{U}^{t=1}$ is asymptotically equivalent to the latent positions $\mc{W}$ up-to a bijection. Moreover, under random dot product graph, if two different adjacency matrices yield the same $\mc{U}^{t=1}$, they must asymptotically equal the same latent positions and asymptotically the same adjacency matrix (i.e., the difference in Frobenius norm converges to $0$). Therefore injective holds asymptotically, and Theorem~3 applies. \end{proof}
\section{Additional Simulation} \label{sec:addsim}
Here we investigate the performance of the test statistics under the violation of positive semi-definite link function related to condition (C3) in the main paper. Under the non-random dot product graph, we generate following two-block stochastic block model for $n=100$: \begin{equation} \begin{split} \label{eq:nonposi} & Z_{i} \overset{i.i.d.}{\sim} \mc{B}(0.5), \\
& \mc{A}(i,j) \mid Z_{i}, Z_{j} \sim Bernoulli \left\{ (0.5-\epsilon) \mathbb{I}\left( |Z_{i} - Z_{i}| = 0 \right) + 0.3 \mathbb{I}\left( |Z_{i} - Z_{j}| \neq 0 \right) \right\}, \\ & X_{i} \mid Z_{i} \sim \mc{B} \left( Z_{i} / 3 \right), \end{split} \end{equation} where $Z_{i}$ represents the block membership, $X_{i}$ is the noisy membership from $Z_{i}$, and we test independence between $\mc{A}$ and $Z$. When $\epsilon = 0.2$, $\mc{A}$ and $Z$ are actually independent. When $\epsilon > 0.2$, the above model yields a non-positive semi-definite graph. As $\epsilon$ increases from $0.2$, the dependency signal gets stronger.
\begin{figure}
\caption{
Diffusion map-based dependency measures are robust against non-positive semi-definite link function (see equation~\ref{eq:nonposi} for details). All of diffusion~\textsc{mgc}~(red solid), diffusion~\textsc{dcorr}~(yellow brown dashes), and diffusion~\textsc{hhg}~(blue small dashes) have better power than the
\textsc{fh}~test (green dot-dashes).
}
\label{fig:bmatrix}
\end{figure}
\section{Random Dot Product Graph Simulations} \label{sec:RDPG_model}
\begin{figure}\label{fig:RDPG_setting}
\end{figure}
For the $20$ simulations under random dot product graph, we describe the generating distribution ($\tilde{W}_{i}, \tilde{X}_{i}) \overset{i.i.d.}{\sim} F_{\tilde{W}, \tilde{X}}$ under each scenario. They are based on \cite{shen2016discovering, shen2017mgc}, and visualization for the sample observations of $\{ (W_{i}, X_{i}) : i=1,2,\ldots, n=50 \}$ is shown in Fig.\ref{fig:RDPG_setting}. Notation-wise, $\mc{N}(\mu, \sigma)$ denotes the normal distribution with mean $\mu$ and standard deviation $\sigma$, $\mc{U}[a,b]$ denotes the uniform distribution from $a$ to $b$, $\mc{B}(p)$ denotes the Bernoulli distribution with probability $p$, and $\epsilon_{i}$ denotes white noise.
1. Linear \begin{align*} \tilde{W}_{i} & \sim \mc{U}[0,1],~\epsilon_{i} \sim \mc{N}(0,0.5), \\ \tilde{X}_{i} & = \tilde{W}_{i} + \epsilon_{i}. \end{align*}
2. Exponential \begin{align*} \tilde{W}_{i} & \sim \mc{U}[0,3],~\epsilon_{i} \sim \mc{N}(0,5), \\ \tilde{X}_{i} & = \exp(\tilde{W}_{i}) + \epsilon_{i}. \end{align*}
3. Cubic \begin{align*} \tilde{W}_{i} & \sim \mc{U}[0,1],~\epsilon_{i} \sim \mc{N}(0,0.5), \\ \tilde{X}_{i} & = 20(\tilde{W}_{i} - 0.5)^3 + 2 (\tilde{W}_{i} - 0.5)^2 - (\tilde{W}_{i} - 0.5) + \epsilon_{i}. \end{align*}
4. Joint Normal \begin{align*} (\tilde{W}_{i}, \tilde{X}_{i}) & \sim \mc{N} \left( \begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0.7 & 0.5 \\ 0.5 & 0.7 \end{bmatrix} \right). \\ \end{align*}
5. Step Function \begin{align*} \tilde{W}_{i} & \sim \mc{U}[-1,1],~\epsilon_{i} \sim \mc{N}(0,0.5), \\ \tilde{X}_{i} & = \mathbb{I}(\tilde{W}_{i} > 0 ) + \epsilon_{i}. \end{align*}
6. Quadratic \begin{align*} \tilde{W}_{i} & \sim \mc{U}[-1,1],~\epsilon_{i} \sim \mc{N}(0,0.3), \\ \tilde{X}_{i} & = \tilde{W}^2_{i} + \epsilon_{i}. \end{align*}
7. W Shape \begin{align*} \tilde{W}_{i} & \sim \mc{U}[-1,1] \\ \tilde{X}_{i} & = 4( \tilde{W}^2_{i} - 0.5 )^2 \end{align*}
8. Spiral \begin{align*} Z_{i} & \sim \mc{U}[0,5],~\epsilon_{i} \sim \mc{N}(0,0.1), \\ \tilde{W}_{i} & = Z_{i} \cos ( Z_{i} \pi), \\ \tilde{X}_{i} & = Z_{i} \sin (Z_{i} \pi) + \epsilon_{i}. \end{align*}
9. Bernoulli \begin{align*} \tilde{W}_{i} & \sim \mc{B}(0.5),~\epsilon_{i} \sim \mc{N}(0,1), \\ \tilde{X}_{i} & = (2 \mc{B}(0.5) - 1)\tilde{W}_{i} + \epsilon_{i}. \end{align*}
10. Logarithm \begin{align*} \tilde{W}_{i} &\sim \mc{U}[-1, 1],~\epsilon_{i} \sim \mc{N}(0,5), \\
\tilde{X}_{i} & = 5 \log_{2} (|\tilde{W}_{i}|) + \epsilon_{i}. \end{align*}
11. Fourth Root \begin{align*} \tilde{W}_{i} & \sim \mc{U}[0, 1],~\epsilon_{i} \sim \mc{N}(0,0.5), \\
\tilde{X}_{i} & = {(|\tilde{W}_{i} + \epsilon_{i}|)}^{1/4}. \end{align*}
12. Sine Period 4$\pi$ \begin{align*} \tilde{W}_{i} & \sim \mc{U}[-1, 1],~\epsilon_{i} \sim \mc{N}(0,0.01), \\ \tilde{X}_{i} & = \sin ( 4 \tilde{W}_{i} \pi ) + \epsilon_{i}. \end{align*}
13. Sine Period 16$\pi$ \begin{align*} \tilde{W}_{i} & \sim \mc{U}[-1, 1],~\epsilon_{i} \sim \mc{N}(0,0.01), \\ \tilde{X}_{i} & = \sin ( 16 \tilde{W}_{i} \pi ) + \epsilon_{i}. \end{align*}
14. Square \begin{align*} U_{i1} & \sim \mc{U}[-1, 1], ~ u_{i2} \sim \mc{U}[-1, 1], \\ \tilde{W}_{i} & = U_{i1} \cos(-\pi/8) + U_{i2} \sin(-\pi/8), \\ \tilde{X}_{i} & = -U_{i1} \sin(-\pi / 8) + U_{i2} \cos(-\pi/8). \end{align*}
15.Two Parabolas \begin{align*} \tilde{Z}_{i} &\sim \mc{B}(0.3),~\epsilon_{i} \sim \mc{N}(0.5, 0.3),\\ \tilde{W}_{i} & \sim \mc{U}[0, 1], \\ \tilde{X}_{i} & = (\tilde{W}^{2}_{i} + \epsilon_{i} ) (\tilde{Z}_{i} - 0.5). \end{align*}
16. Circle \begin{align*} U_{i} & \sim \mc{U}[-1,1],~\epsilon_{i} \sim \mc{N}(0, 0.05),\\ \tilde{W}_{i} & = \cos( U_{i} \pi), \\ \tilde{X}_{i} & = \sin (U_{i} \pi) + \epsilon_{i}. \end{align*}
17. Ellipse \begin{align*} U_{i} & \sim \mc{U}[-1,1], \\ \tilde{W}_{i} & = 5\cos( U_{i} \pi), \\ \tilde{X}_{i} & = \sin (U_{i} \pi) . \end{align*}
18. Diamond \begin{align*} U_{i1} & \sim \mc{U}[-1, 1], ~ U_{i2} \sim \mc{U}[-1, 1], \\ \tilde{W}_{i} & = U_{i1} \cos(-\pi/4) + U_{i2} \sin(-\pi/4), \\ \tilde{X}_{i} & = -U_{i1} \sin(-\pi / 4) + U_{i2} \cos(-\pi/4). \end{align*}
19. Multiplicative Noise \begin{align*} \tilde{W}_{i} & \sim \mc{N}(0.5, 1),~\epsilon_{i} \sim \mc{N}(0.5, 1),\\ \tilde{X}_{i} & = \tilde{W}_{i} \cdot \epsilon_{i} \end{align*}
20. Independence \begin{align*} \tilde{W}_{i} & \sim \mc{N}(0,1) \\ \tilde{X}_{i} & \sim \mc{U}(0,1) \\ \end{align*}
\end{document} | arXiv |
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February 2020 , Volume 49, Issue 2, pp 391–406 | Cite as
Linking consumer physiological status to food-web structure and prey food value in the Baltic Sea
Agnes M. L. Karlson
Elena Gorokhova
Anna Gårdmark
Zeynep Pekcan-Hekim
Michele Casini
Jan Albertsson
Brita Sundelin
Olle Karlsson
Lena Bergström
First Online: 05 June 2019
Declining physiological status in marine top consumers has been observed worldwide. We investigate changes in the physiological status and population/community traits of six consumer species/groups in the Baltic Sea (1993–2014), spanning four trophic levels and using metrics currently operational or proposed as indicators of food-web status. We ask whether the physiological status of consumers can be explained by food-web structure and prey food value. This was tested using partial least square regressions with status metrics for gray seal, cod, herring, sprat and the benthic predatory isopod Saduria as response variables, and abundance and food value of their prey, abundance of competitors and predators as predictors. We find evidence that the physiological status of cod, herring and sprat is influenced by competition, predation, and prey availability; herring and sprat status also by prey size. Our study highlights the need for management approaches that account for species interactions across multiple trophic levels.
Benthic–pelagic coupling Benthivore Ecological indicator Long-term time series Piscivore Zooplanktivore
The online version of this article ( https://doi.org/10.1007/s13280-019-01201-1) contains supplementary material, which is available to authorized users.
The physiological status of keystone species is an important characteristic of overall food-web status because it determines populations' potential for growth and reproduction and, hence, their long-term sustainability (Kadin et al. 2012). It may also have direct economic consequences, such as for the value of commercial fisheries (Marshall et al. 2000). Physiological status can be measured in several ways, and different approaches may be preferential for different species, such as relative body condition (based on weight, size or fat content) or reproductive output. In recent decades, declining breeding success and body condition have been observed in marine top consumers worldwide, and have been attributed to various changes in the food-web (e.g. Trites and Donnelly 2003; Österblom et al. 2008; Bogstad et al. 2015; Harwood et al. 2015; Casini et al. 2016).
Several human-induced pressures and environmental changes have been related to impacts on the physiological status of commercial fish, via direct or indirect pathways. In the Baltic Sea, main anthropogenic pressures include overfishing, eutrophication, and climate change (Andersson et al. 2015; Elmgren et al. 2015). Fishing can directly influence the size structure of commercial target species (Östman et al. 2014), resulting in reduced body size and growth, or decreased size at maturation (Vainikka et al. 2009). Overfishing may also lead to cascading effects on lower trophic levels (e.g. Casini et al. 2008), which in the Baltic Sea has been seen to lead to enhanced competition for food among forage fish when these are released from predation, resulting in reduced physiological condition in sprat and herring (Casini et al. 2010). Hence, human-induced alterations of food-web structure can affect the physiological status of species.
Structural changes due to bottom-up processes may also affect the physiological status of species, including consumers. Whereas top-down effects primarily act via changes in the abundance of predators, bottom-up effects can be mediated through changes in both prey availability and quality as food. Experimentally modified elemental and biochemical composition of phytoplankton translates into lower food quality for zooplankton, and, ultimately, can lead to reduced growth of zooplanktivores, such as larval herring and trout (Malzahn et al. 2007; Taipale et al. 2018). In the field, however, the quantity and size of prey seem to be more decisive for juvenile clupeid fish than their fatty acid composition (Peters et al. 2015). So far, we are not aware of any studies evaluating the influence of prey quality at several trophic levels across an entire food-web. The Baltic Sea, with its uniquely low taxonomic diversity (Elmgren and Hill 1997), provides an opportunity to test the importance of food-web structure and food value of prey, respectively, on the physiological status of consumers using monitoring-based time series data covering multiple trophic levels.
Here, we study long-term changes in the physiological status of consumers from four trophic levels in the Baltic Sea, and test whether these can be attributed to top-down or bottom-up changes in food-web structure (as represented by abundance of predators, competitors and prey) and/or food value (physiological status, or energy content of prey). We gather metrics on the physiological status of gray seal (Halichoerus grypus), cod (Gadus morhua), herring (Clupea harengus), and sprat (Sprattus sprattus). Blubber in seals is a layer of lipid-rich tissue between the epidermis and the underlying muscles, which acts as a storage of metabolic energy, and is important not only for individual survival but also for reproduction (Harding et al. 2005; Helcom 2018). In fish, lipids is the main source of energy. In forage fish, such as sprat and herring, the lipid content is on average 34% of the body mass, and females with higher lipid content have higher egg survival (Laine and Rajasilta 1999). Previous studies have seen that lipid content and blubber thickness are influenced by prey quality (Røjbek et al. 2014; Kauhala et al. 2017; Rajasilta et al. 2019), while body size in fish also responds to size-selective predation (e.g. Vainikka et al. 2009).
The study focuses on the years 1993–2014, which corresponds to an ecologically relatively stable time period compared to the preceding years, which were characterized by strong shifts in species composition in the pelagic food-web (Casini et al. 2008). We predict that (i) high prey availability and (ii) high prey food value have a positive influence on the physiological status of consumers at higher trophic levels via bottom-up processes, that (iii) high abundances of intra- or interspecific competitors have negative effects on the physiological status of consumers due to increased competition for food, and that (iv) predation might have either positive or negative effects on the physiological status of prey, due to selective mortality (depending on whether larger, smaller or individuals in bad condition are eaten first), or positive effects by reducing intra-specific competition.
Study system
The Baltic Sea is the world's largest brackish water system, and is naturally species-poor due to its low salinity (Elmgren and Hill 1997). In this study, we analyzed changes in physiological status across four trophic levels in two sub-systems; the basins of the Baltic Proper (BP) and the Bothnian Sea (BoS) (Fig. 1). These systems differ in hydrological conditions, with an average surface salinity of 6–8 in BP and 4–6 in BoS, and a mean annual surface temperature of 9 °C in BP versus 7 °C in BoS.
Map of the Baltic Sea with its major basins; Bothnian Bay (BB), Bothnian Sea (BoS) and the Baltic Proper (BP), showing the used sampling stations (see inserted legend). Fish data are assembled based on ICES subdivisions (SD), shown as numbers in the left panel; the cod stock is distributed over SD 25–29 (i.e. the Baltic Proper), the BP stock of herring occurs in SD 25–29 and 32, while the herring in the BoS is a separate stock (SD 30). Sprat and gray seal represent the same stock/population in all of the Baltic Sea (SD 22–32). Zoomed-in maps show zooplankton and benthos stations in the Askö area (lower right panel) and in the northern Bothnian Sea (upper right panel). Data on M. affinis embryo viability originate from stations 6004, 6019, 6020, 6022 and 6025 in BP, and from N19, N25, N26, N27 and US5 in BoS. The five benthos stations in the left panel (BP) are referred to as open sea stations. See text and Table 1 for details on monitoring programs and Table S1 for details and meta-data on sampling stations
The study focused on key consumers of the pelagic and benthic food-webs, encompassing species which are geographically widespread, contribute substantially to overall biomass (e.g. Elmgren 1984) and are adequately represented in monitoring data (Table 1, Table S1). The studied taxa are either predators, prey, or both, and all taxa feeding on the same prey are additionally potential competitors, including potential intra-specific competition (Fig. 2). With the exception of cod and sprat, all food-web components are abundant in both basins.
Metrics on physiological status and population/community traits indicative of food value to consumers. "Sample size" gives the range (median within brackets) of number of individuals analyzed per year for seal blubber thickness, fish condition, weight-at-age, fat content, and Monoporeia viable embryos, and number of samples per year for population data on Saduria and zooplankton community data. In addition to the metrics listed, data on prey abundance/biomass were included in the PLS regressions (Table S1). The last column lists all potential explanatory variables assessed, for metrics used as response variables in these models. See Fig. 1 for location of sampling stations, and Table S1 for more detailed specifications of the data used. BP denotes Baltic Proper and BoS denotes Bothnian Sea
Food-web component
Status metric
Description of metric (unit)
Spatial delineation
Rationale for use
Potential predictors (see also Table 3)
Used only as response variable
Gray seal
Blubber thickness (BT)
Blubber layer (mm), in males, ages 4–20, bycatch
Whole Baltic Sea
(Baltic Proper +
Bothnian Sea)
Indicator of nutritional status in seals (HELCOM 2018)
Abundance of prey: cod, herring, sprat
Prey quality: cod, herring, sprat
Competitors: seal, cod
Predators: NA
Used as response variable or as predictor
Calculated (Eq. 1, g/cmb) in 30–40 cm fish
Baltic Proper
Physiological status indicator, indicator of fish well-being. May respond to both food availability and quality. Could potentially be affected also by predation if weaker individuals are easier to catch
Abundance of prey: herring, sprat, Saduria
Prey quality: herring, sprat, Saduria
Predators: seal
Fat content
In liver (%)
In 30 cm fish
Higher fat content usually results from higher food quality. Could potentially also be affected by selective predation
Weight-at-age
(WAA)
Ages 3–5 (gram)
Baltic Proper + Bothnian Sea
BP: 70–544 (327)
BoS: 33–477 (53)
Body size (weight); may respond to both food availability and quality, but also size-selective predation
Abundance of prey: zooplankton, Amphipods or Amphipods + Polychaete
Prey quality:
Competitors: sprat, herring, Saduria
Predators: seal, cod
(no benthic prey variables for BP WAA or C, see text for details)
Calculated (Eq. 1, g/cmb)
See cod
In muscle (%)
In 15–20 cm fish
20 (BP)
24 (BoS
See herring
Abundance of prey: zooplankton
Prey quality: zooplankton
Competitors: herring, sprat
Saduria entomon
Mean body weight
(mw)
Population trait: ratio total population biomass and abundance
BP: 6–14 (10)
BoS: 2–33 (20)
Larger Saduria are preferentially eaten by cod (Casini unpublished). Mean size could also reflect changes in population structure by increased recruitment alternatively poorer growth
Abundance of prey: Amphopods or Amphipods + Polychaeta
Prey quality: NA
Competitors: herring, Saduria
Predators: cod
Used only as predictor
Mean size (ms)
Community trait: Ratio total abundance and biomass
Baltic Proper +
BP: 6–8
BoS: 4–8
HELCOM core indicator. Mean size reflects the proportion of larger copepods and cladocerans in the zooplankton community, which are generally more profitable for herring and sprat compared to smaller taxa (Gorokhova et al. 2016)
Used only in descriptive statistics
Monoporeia affinis
Viable embryos
Population mean (number/female)
BoS: 74–963 (144)
HELCOM Supplementary indicator. Embryo viability responds also to contaminants in sediments (Sundelin and Eriksson 1998) hence it is not used as predictor in this study
Food-web model of the studied systems. The classifications denote which role each species/food web component has in the tested statistical models (Table 1). Gray seal is the most abundant seal species in the Baltic Sea and feed mainly on sprat, herring and cod (Lundström et al. 2010). Cod is the predominant piscivorous fish in many parts of the region, feeding mainly on sprat and herring, which together constitute around 85% of the pelagic fish species in terms of biomass (Elmgren 1984). Cod also feeds on benthic invertebrates, in particular the isopod Saduria entomon (Zalachowski 1985). Sprat of all sizes are zooplanktivorous, whereas larger herring also feeds on benthic species (Casini et al. 2004). In particular, the lipid-rich amphipod Monoporeia affinis can constitute a large proportion of the herring diet (Aneer 1975). S. entomon feeds mainly on M. affinis; in the Bothnian Sea, they form a tightly coupled predator–prey system (Sparrevik and Leonardsson 1998). Polychaetes contribute to the diet of herring and S. entomon to a smaller extent (not shown in this figure)
Metrics and data used
Basin-specific data were used for zooplankton, benthic invertebrates and herring. Data for cod and sprat were only applied for the BP analysis, in agreement with their principal current natural distribution (ICES 2016). Gray seals are mobile and considered to comprise a single population in the Baltic Sea (Galatius et al. 2015) and was analyzed across BP and BoS combined.
The physiological status (estimated on individual level) or the population- and community-level traits (all referred to as food value) of each taxon was quantified by at least one metric in each assessed basin. The metrics typically represented variables covered by current environmental monitoring and assessment, and varied depending on taxon-specific properties and data availability (Table 1, Supplementary Tables S1, S2). In addition, abundance/biomass data for each taxon were used, as obtained from the Swedish National Marine Monitoring Program and international surveys (Table S2). Benthic invertebrate and zooplankton data were acquired from the SHARK database (www.smhi.se), except for open sea benthic data (Fig. 1) which were from the Finnish SYKE HERTA database (http://www.syke.fi), and fish data from ICES (www.ices.dk). Time series on zooplankton biomass (including copepods, cladocerans and rotifers) were integrated from national and international stations in coast and open sea (Gorokhova et al. 2016; Fig. 1).
Gray seal abundance was estimated based on surveys carried out during the peak of the molting period (May–June) by international monitoring coordinated by HELCOM (Galatius et al. 2015). Gray seal physiological status was based on the blubber thickness of adult males caught as incidental bycatch during autumn, a time of the year before the winter when the blubber thickness is expected to respond primarily to food availability (HELCOM 2018). Gray seal occur in the entire Baltic, but the population is centered in the archipelagos of Stockholm, Åland and Turku. Since gray seals are highly mobile and movements between basins occur frequently we did not separate data for seal blubber thickness or abundance for the different basins.
Abundance data for herring and sprat were obtained from analytical assessment models provided by ICES (2016), and the abundance of cod was estimated based on data from the Baltic International Trawl Survey (Casini et al. 2016; ICES 2016). For herring and sprat, data representing the whole population, as well as age groups 3–5 (herring) and 2–4 (sprat) years were included. For cod, data representing the whole population, as well as individuals larger than 30 cm (mature fish; ICES 2016) were included. For all fish species, physiological status was expressed based on the individual body condition;
$$ {\text{Individual condition}} = \frac{W}{{L^{\text{b}} }} $$
where W and L are the weight and the total length of the fish, respectively, and b is the slope of the overall Ln weight–Ln length relationship. For herring and sprat, the mean weigh-at-age was also used (WAA, data obtained from ICES 2016). Both metrics were estimated based on the Swedish part of the Baltic International Trawl Survey (for cod) and the Baltic International Acoustic Surveys (for sprat and herring), both performed in autumn (Casini et al. 2011, 2016; ICES 2016). In addition, data on the fat content of cod and herring from the Swedish national monitoring program were included (Table 1).
Benthic macrofauna was represented by the predatory isopod Saduria entomon, the deposit-feeding amphipods Monoporeia affinis and Pontoporeia femorata, and the polychaetes Marenzelleria spp., and Bylgides sarsii. Saduria is an important food for cod (Zalachowski 1985) while the lipid-rich amphipods, and, to some extent, polychaetes, are eaten by adult herring (Aneer 1975; Casini et al. 2004). In the northern BoS, Bylgides does not occur, whereas Pontoporeia occurs only sparsely. Marenzelleria, a recently introduced non-indigenous polychaete, became abundant in both basins in the past decade. The total abundance of Monoporeia and Pontoporeia (included as the variable "amphipods" in the analyses), and of amphipods together with polychaetes (variable "AmpPol") were obtained from all stations in BoS and from the Askö-stations in BP. Open sea deep stations in the BP (Fig. 1) are frequently affected by hypoxia and lack permanent benthic macrofauna since 2000 (Villnäs and Norkko 2011). Hence, for these stations, only the frequency of occurrence (%) of (the migratory) Saduria was used, and compared to Saduria frequency of occurrence from the other regions. To represent its potential food value for cod, the mean weight of Saduria (mw, Table 1) was calculated from data on population abundance and biomass (i.e. this metric represented a population trait rather than individual physiological status) from the Askö- and the N-stations (Fig. 1). To avoid dependency (and autocorrelation) with mw, Saduria abundance was represented by frequency of occurrence also at coastal stations. For Monoporeia, physiological status was based on the number of viable embryos (ve) per ovigerous female (Sundelin and Wiklund 1998), based on five stations per basin for which long-term data were available (Table 1, Tables S1, S2).
Zooplankton biomass and mean size were based on average monthly abundance and biomass values for June–September (Fig. 1, Supplementary Table S1). We calculated the average summer biomass (mg m3) and mean zooplankter size (µm ind−1) as described in Gorokhova et al. (2016). Zooplankton mean body size was used as a metric to represent the prey food value for zooplanktivores (herring and sprat). In the Baltic zooplankton communities, the mean size reflects the proportion of larger copepods and cladocerans (i.e., a community characteristic) which are generally more profitable prey items to herring than small-bodied cladocerans, nauplii and rotifers (Flinkman et al. 1998; Casini et al. 2004). Together, total zooplankton biomass and mean zooplankter size represent food availability and food value for zooplanktivorous fish in the area (Gorokhova et al. 2016).
Data treatment prior to analyses
All variables were normalized (zero mean, unit variance) using the long-term (22 years) mean and standard deviation values, to focus on the changes in relative rather than in absolute values, and to avoid ordination analyses to be driven by variables with largest values. Abundance data were square-root transformed before normalization. For Monoporeia ve, missing data for the first year (1993) were replaced with the zero score mean (0). For seal blubber thickness, missing values in 1993 and 1999 were replaced by a moving average of the preceding and proceeding 2 years, based on observations on a longer national data series (HELCOM 2018), which show many years of stable blubber thickness during the 1980s and a shift around 1994 towards decreasing values.
Changes over time in physiological status and food-web structure
Directional trends in the physiological status/food value metrics as well as for abundance data over time were assessed by the non-parametric Mann–Kendall test. To identify any common changes over time in the studied status variables across species/groups or trophic levels, and years of high similarity, we applied a principal components analysis (PCA) on the normalized data. PCAs were performed separately for metrics reflecting physiological status/food value and abundances, and separately for each basin. Sprat and cod were only included for the BP. Using the same data sets, the level of similarity between adjacent years was assessed by Chronological clustering as implemented in Brodgar 2.7.4 linked to R3.3 (Highland statistics). Similarities among years were assessed based on Euclidean distances in all cases.
Explaining consumers' physiological status
We predicted that high prey availability and high physiological status/food value (i.e., the energetic content) would have a positive influence on the physiological status of consumers (predictions i and ii), while higher abundances of competitors would have negative effects (iii), and predation may have positive or negative effects on the physiological status of prey (iv). The relationships of each physiological status metric (Table 1, in total 13 models) to the food-web structure (Fig. 2, Table 1) and to prey food value (Table 1) were assessed using Partial Least Square Regression (PLSR) analyses (Wold et al. 2001). The choice of method was motivated by the characteristics of the data set, encompassing relatively short time series (22 years) and many potential explanatory variables. PLSR is a generalization of multiple linear regression that is particularly well suited for analyzing data sets where the number of observations per variable is relatively low compared to the number of explored variables (Wold et al. 2001; see details on cross-validation procedure below). PLSR is also suited for dealing with potentially collinear predictors, allowing even for correlated explanatory variables to be included. Another benefit of the PLSR approach in the context of our research questions and the data structure, is that the model evaluation is based on optimization of the explanatory and predictive capacity of the model.
The models were fitted separately for each of the response metrics (Table 1). Between 3 and 12 potential explanatory variables were used in each model, representing the abundance of potential predators and prey, or the physiological/food value of prey, as well as potential competitors for prey (predictions i–iv, Fig. 2, Table 1). For the fish species, several measures of physiological status were included (Table 2) to compare model outcomes in relation to the tested predictors. Gray seal blubber thickness, which was compiled at the pan-Baltic scale, was regressed against variables representing both BP and BoS. Sprat and cod were only regressed against BP variables since they are more abundant there. All other metrics for the other taxa were related to the basin-specific variables. The benthic data were included in different formats depending on the explored response variables. For modeling BoS herring condition and WAA, a grand mean of Amp, or AmpPol, from all stations in BoS was used as a potential explanatory variable. However, for the modeling of BoS herring fat content, benthic data were taken only from station SR5, as the herring fat content data originated from this area (Fig. 1). BP herring fat content was monitored close to Askö (Fig. 1), and, hence, was related to benthic variables from this area. We avoided extrapolating the same coastal benthos data to models on BP herring condition and WAA, representing herring at the scale of the whole basin of BP, due to differences in spatial coverage. Since the variables Amp and AmpPol were autocorrelated, we tested them separately and the variable contributing to the better model was subsequently chosen. In addition, Saduria mw was estimated based on individuals sampled from coastal stations, and, hence, explanatory variables representing its prey were restricted to the Askö or the N-cluster stations (Fig. 1).
Status metrics used as response variables in the partial least square regressions (PLSR) with model evaluation parameters, sorted by species and Basin. R2X = the explained variance of predictors by each PLSR component; R2Y = the explained variance of dependent variables by each PLSR component (analogous to the coefficient of determination R2 in regression analysis); and R2Q = model prediction capacity. See Table 3 for details on the models outputs. BoS denote Bothnian Sea, BP Baltic Proper and p-B pan-Baltic. The assessed physiological status or population/community traits variables are: mean weight (mw), condition (c), weight-at-age (WAA), fat (%) and blubber thickness (BT) (see Table 1 for details)
Saduria
p-B
R2X
Components (nr)
1.70; 0.87
The analyses were performed with the NIPALS (Nonlinear Iterative Partial Least Squares) algorithm, as implemented in STATISTICA 13 (StatSoft, Inc. 2017). The models were validated based on the obtained values of R2Y, which is analogous to the coefficient of determination (R2) used in the regression analysis; and of R2Q, which represents the model's predictive capacity. Model evaluation followed Lundstedt et al. (1998) on that a biological PLSR model is of good quality when R2Y > 0.7, and R2Q > 0.4. Variable selection was performed based on the VIP scores (variable importance for projection), which is the weighted sum of squares of the PLSR weights. All potentially relevant diet variables were initially included (Fig. 2, Table 2), and variables with VIP scores > 0.7 were used further in the model selection (Kaddurah-Daouk et al. 2011). Thereafter, the potential effects of competitors and predators were assessed in the same way; all variables that maximized R2Y and R2Q were retained in the final model. The number of variables in the final model was identified following the V-fold cross-validation. The autocorrelation of model residuals was evaluated using the ARIMA algorithm (Statsoft, Inc. 2017). Where significant 1-year lags were detected, the model was rerun, including the lagged year response variable as an additional predictor variable, and residuals were again checked for partial autocorrelation. No further action was required to account for autocorrelation in any of the models. Because all models but one were best explained by a single PLS component, we also fitted linear models with single predictors. Among these we identified best models based on Akaike information criteria (AIC) estimates, and compared the predictors identified using this approach with those from the PLSR approach.
Changes in physiological status and food-web structure
There were long-term trends in status metrics of most consumers in both basins (Fig. 3; Table S3). The physiological status of seal and of cod decreased over the studied time period (Fig. 3a, b), whereas those of sprat and herring generally increased, at least over the later decade (Fig 3c–e). For invertebrates, trends in Monoporeia viable embryos and Saduria mean weight differed between basins (Fig. 3g, h), while zooplankton mean size had no unidirectional trend. The PCA analyses showed that fish metrics representing the same species and basins were generally correlated with each other (Supplementary Fig. S1). For the physiological status/food value metrics, there was no unidirectional change over time among different trophic levels in any of the two basins, (Supplementary Fig. S1). The temporal trends in abundance/biomass metrics found for a number of species (Fig. 4; Table S3) were also partly reflected in the PCA for the BoS. In both sub-basins, these analyses show a shift between the earlier years studied (until years between 1996 and 1998 for the different plots) for both physiological status/food value and abundances, reflecting a decreasing physiological status/food value and changes in the relative abundance of taxa from different trophic levels.
Temporal development of the physiological status metrics (seal, cod, herring, sprat and Monoporeia) and population/community traits (Saduria/zooplankton). Values show normalized data to aid comparisons. WAA denotes weight-at-age. Red = decreasing over time, blue = increasing, black = no change over time, based on Mann–Kendall test (p < 0.05). See also Fig S1 for analyses of common trends within each basin (PCA), and text and Table 1 for description of metrics
Temporal trends in the abundance, biomass or frequency of occurrence (%) of the species or species groups used as predictors in the PLS regressions. Values show normalized data to aid comparisons. Herring and sprat abundances show sums for all size classes. AmpPol represents the total sum of amphipods and polychaetes (hence, correlated with Amphipods). Red = decreasing over time, blue = increasing, black = no change over time based on Mann–Kendall test (p < 0.05, detailed results in Table S3. See also Fig S1 for analyses of common trends within each basin (PCA), and text and Table 1 for description of metrics
Models meeting the evaluation criteria were obtained for 11 of the 13 physiological/food value metrics tested (Table 2; exceptions were models for gray seal blubber thickness and Saduria mean weight in the BP). Only the BP herring and sprat WAA models resulted in including lagged values due to the significant autocorrelation. In line with our predictions, the changes in the physiological status of consumers were often explained by a combination of responses, i.e., a positive relation to the prey abundance (prediction i) and to the physiological/food value of prey (prediction ii), a negative relation to the abundance of competitors (prediction iii), and a negative or positive relation to predators (prediction iv). Only two of 28 cases showed a direction of association that did not follow our predictions (herring had a positive effect on sprat condition, and AmpPol a negative effect on Saduria mw in the BP). The results are described below, presented in detail in Table 3 and illustrated in Fig. 5. Generally, results from linear model results selected based on AIC were largely similar to PLSR results, although cod and gray seal models included additional predictors, i.e. herring and sprat WAA (Table S4).
Partial least square regression (PLSR) model results. Saduria and herring were assessed in both Basins (Baltic Proper, BP, and Bothnian Sea, BoS), cod and sprat in BP only, and gray seal as a pan-Baltic (p-B) population. The predictors represent stocks (abundance, biomass or frequency of occurrence) potentially affecting prey availability, competition and/or predation (Fig. 1, text for details), as well as physiological status or population/community traits of relevant prey (all representing food value to consumers), and are listed in column 1. Each column represents a model and predictors entering the model are highlighted in gray. Values are shown for variables with a Variable of Importance (VIP) score above 0.7 and which improve the model predictive capacity while maximizing R2Y (the explained variance of response variables by each PLSR component). The values are the X-loadings, which describe the association (positive or negative) with PLSR component 1. Numbers in brackets denote their ranking based on VIP score. The results are summarized in Fig. 5. The assessed physiological status or population/community traits variables are: mean weight (mw), condition (c) and weight-at-age (WAA) and fat (%) and blubber thickness (BT); see Table 1 for details. *denotes that lagged values of the response variable are included in the model
Summary of the model results for the Bothnian Sea and the Baltic proper. Arrows illustrate significant links according to the PLSR models (See Table 3), and point in the direction from predictor to response variables. Gray arrows denote abundances of prey, competitors or predators (dashed = negative, whole gray = positive association) and black arrows denote food quality aspects (always positive association). mw mean weight, WAA weight-at-age. See Table 3 for lag effect results, which were found for herring and sprat WAA in the BP. Note also that arrows pointing to gray seal blubber thickness and Saduria mean weight in the BP are included for completeness, but those models had a predictive capacity and proportion explained below the criteria (Table 2)
Bottom-up control (predictions i and ii)
In BoS, the herring physiological status and Saduria mean weight were explained by the amphipod abundance as well as zooplankton mean size. In BP, the mean size of zooplankton contributed to explaining both physiological status in sprat and herring fat content. Saduria frequency of occurrence was a significant positive predictor for cod condition. Zooplankton biomass was only a significant predictor for sprat WAA.
Competition (prediction iii)
In the BP, the abundance of competitors was included in many of the models. All herring physiological status metrics were negatively related to the sprat abundance, and Saduria mean weight was negatively related to herring abundance (competitors for benthic prey). Moreover, positive association was detected between herring abundance and sprat condition. Intra-specific competition was indicated by the models for sprat condition and WAA, cod fat content, and gray seal blubber thickness. In the BoS, positive associations were seen between Saduria mean weight and frequency of occurrence.
Top-down control (prediction iv)
In the BP, a positive association was found between gray seal abundance and the condition of sprat and herring as well as herring fat content. In contrast, gray seal abundance had a negative effect on BoS herring WAA. Fat content and condition of cod was negatively associated to gray seal abundance in the best PLSR model, whereas it was additionally explained by sprat WAA and herring WAA in the best linear model based on AIC (i.e. bottom-up). Further, sprat WAA was positively related to cod abundance.
We show that both top-down and bottom-up effects control physiological status of consumers across multiple levels in Baltic Sea food-webs (Fig. 5). During the study period, the physiological status declined in the piscivores (gray seals, cod), whereas it increased—at least during the last decade—for their main prey, the mesopredators herring and sprat. Trends in the physiological status or population/community characteristics of invertebrates were absent or basin-specific. The physiological statuses of cod, herring and sprat were influenced by a combination of prey availability, abundance of competitors and predators; herring and sprat status were also influenced by prey size.
The availability of prey is important for the physiological status of the consumers (prediction i) as shown for herring and Saduria in the Bothnian Sea and for cod (condition only) in the Baltic Proper. All three metrics on the physiological status of Bothnian Sea herring were strongly linked to variations in the abundance of the amphipods (i.e. Monoporeia), which are a lipid-rich food source (Hill et al. 1992). With respect to zooplankton as prey for herring and sprat, the prey food value, assessed here as mean size of a zooplankter in the community, was more important than the total zooplankton biomass (prediction ii). In zooplankton, the mean size incorporates the contribution of large lipid-rich copepods and cladocerans to total zooplankton biomass, which are important prey for herring condition and growth (Flinkman et al. 1998; Casini et al. 2004; Östman et al. 2014). Changes in the food value of lower consumers (e.g. benthic prey) can cascade upwards (e.g. to herring WAA) and affect the physiological status of the top consumers (gray seal blubber thickness). Although our model on blubber thickness had a low predictive capacity, the link between herring WAA and gray seals have been demonstrated by Kauhala et al. (2017). Decreased WAA of older herring in the Baltic Sea has been related to decrease in the population size of mysid shrimps (Kostrichkina 1982). Our study further highlights the importance of deposit-feeding amphipods for the physiological status of herring.
A decreased mean weight of Saduria, which feeds mainly on Monoporeia, was also related to the decline in the amphipod abundance in the Bothnian Sea, also likely leading to additional negative effects on Saduria population size. Populations of Monoporeia collapsed in the Bothnian Sea in the early 2000s, presumably because of deteriorated feeding conditions due to extreme precipitation and runoff (Eriksson-Wiklund and Andersson 2014). Despite higher reproductive success in the recent years, the Monoporeia population abundance remains low, suggesting that the increasing in abundance of herring may exert some top-down control. The non-indigenous species Marenzelleria was not included or positively associated to consumer status in any of the models suggesting that it cannot replace Monoporeia as prey for higher trophic levels.
Support for the importance of benthic prey availability was also found for cod in the Baltic proper. The deteriorating cod condition was linked to the decreasing frequency of occurrence of Saduria in the open Baltic proper. Saduria are prey items for cod and also contain high levels of essential fatty acids, which can be complementary to the fat composition in forage fish that cod eat (Røjbek et al. 2014). Casini et al. (2016) hypothesised a link between hypoxia-related decrease in benthic prey and cod condition, but had no data on benthic prey. Our results support this hypothesis and suggest a mechanistic explanation. We found that Saduria populations in the benthos of the open sea have declined, likely due to benthic hypoxia (Villnäs and Norkko 2011), and decline in benthos was related to cod condition. However, this decline is not measured in the coastal area (Askö), where increases occurred, but Saduria mean weight declined. This pattern could be the result of hypoxia-induced migrations of Saduria to the more oxygenated coastal areas, increased competition and, consequently, decreasing mean weight in the coastal Saduria populations.
The declines in cod condition and fat content were best explained by the increased abundance of gray seals, suggesting competition for prey (herring and sprat) between gray seals and cod (prediction iii), or selective feeding by gray seals on cod in good condition (prediction iv, Kohl et al. 2015). Alternative explanations could be related to increased parasite infestation in cod, enhanced by gray seals which are the final host (Horbowy et al. 2016), or the correlation merely representing the general decreasing trend in cod physiological status (coinciding with linear increase in gray seal abundance, Fig. 3a). Despite the relatively low abundance of cod compared to the historical levels (ICES 2016), we found indications of intra-specific competition (prediction iii, as found also by Casini et al. 2016). In the Baltic Proper, it is likely that the spatial mismatch between cod and sprat (Casini et al. 2011) and the hypoxia-related reductions in benthic prey would result in intra-specific competition for food. The best models based on AIC (Table S4) suggest that body size of herring and sprat have additionally contributed to explain the declining condition and fat in cod. Size of fish prey has previously been linked also to cod growth, and the lack of suitably sized prey (herring and sprat) for piscivorous cod was suggested to contribute to the lack of cod recovery (Gårdmark et al. 2015).
The increasing physiological status (i.e. condition and fat content) of herring and sprat in the Baltic Proper and also in the last decade in the Bothnian Sea has not previously been reported. However, both the WAA and condition of herring are still at historically low levels (e.g. Casini et al. 2010). The physiological status of herring in the Baltic proper was mainly negatively related to the abundance of sprat, indicating that the interspecific competition (prediction iii, Casini et al. 2010) continues to be important. It also suggests an asymmetrical interaction since sprat condition was positively associated to herring abundance.
The physiological status of mesopredators in the Baltic Proper was also positively associated with gray seal abundance, with respect to condition (herring) and fat content (sprat). This could also indicate a positive effect of predation (prediction iv) if prey in poorer condition are preferred or, alternatively, easier to catch. However, predation could result in reduced intra-guild competition and compensatory growth (Casini et al. 2010, 2011). Interestingly, the condition and WAA of sprat were related to gray seal and cod abundances, respectively, suggesting that these top consumers partition resources to some extent (cod preying on small individuals and gray seal on individuals in bad condition).
Finally, it should be noted that this study did not attempt to test the relationships between changes in environmental conditions and the physiological status in piscivores, mesopredators or food value in invertebrates. Lower salinity (as well as increasing herring population size) has been associated to reduced lipid content in Baltic herring during the same time period as studied here, due to the more energetically costly osmoregulation with decreasing salinities (Rajasilta et al. 2019). Casini et al. (2016) discuss potential negative effects on low oxygen concentrations for physiological status in cod, which is also a factor relevant for the benthic invertebrate Saduria, and an increasing environmental concern in the Baltic Sea (Carstensen et al. 2014). Warmer temperature will likely improve growth conditions for both herring and sprat as long as food is not limited (Margonski et al. 2010), but will unlikely affect fish lipid content or the blubber thickness in gray seals which spend most time at greater depths, where temperature is more constant. In addition, both fish fat and seal blubber is measured in autumn before any potential effects of the colder winter months would be seen.
Our study highlights the importance of food value as well as quantity of prey for population-level changes in the physiological status of consumers in the Baltic Sea. It also highlights the significance of benthic prey for the condition of fish in both basins, in addition to the food value of zooplankton prey, and inter- and intra-specific competition. The importance of benthic invertebrates for pelagic top consumers is often neglected in multi-species models (but see e.g. Niiranen et al. 2012; Huss et al. 2014) and in the management of commercially important fish species (ICES 2016). Benthic population stocks may decrease in the future due to continuously decreasing oxygen conditions in the deep water of the Baltic Proper related to eutrophication and climate change, and attributed to climate-related brownification in the Bothnian Bay (Andersson et al. 2015). Our results suggest that changes in the benthos and zooplankton communities will likely continuously affect the physiological status in the higher trophic levels, including the weight and condition of commercially exploited fish species. Hence, we highlight the importance for fisheries and environmental management to take account of species interactions across trophic levels in the food-web. Under this approach, the key parameters for monitoring performance should include not only population size reflecting the food-web structure, but also the physiological status of the prey and predators. Many of the physiological status metrics studied here are already included in the Baltic Sea monitoring and assessment programs, but their integrated use in food-web analyses is not yet developed; the latter is essential for meeting current management challenges.
We acknowledge financial support from the Swedish Agency for Water and Marine Management (SWaM; 4951-2012, 733-14) via the project "Integrated ecosystem analyses". Main part of the data was collected within the Swedish National Environment Monitoring program financed by SWaM and the Swedish Environment Protection Agency. S. Danielsson, Swedish Museum of Natural History, provided data on fat content for herring and cod, and O. Svensson, Stockholm University provided S. entomon data for the Baltic proper. H. Nygård kindly provided benthic data from Finnish offshore monitoring stations and A Villnäs gave advice on offshore station selection. K. Lundström and J. Olsson, Swedish University of Agricultural Sciences (SLU Aqua) are acknowledged for fruitful discussions. P. Mattsson, (SLU Aqua), made the map and D. Jones (SLU Aqua) checked the English. We thank several anonymous reviewers for their constructive comments on earlier versions of the manuscript.
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Supplementary material 1 (PDF 456 kb)
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© The Author(s) 2019
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
1.Department of Ecology, Environment and Plant ScienceStockholm UniversityStockholmSweden
2.Stockholm University Baltic Sea CentreStockholm UniversityStockholmSweden
3.Department of Environmental Science and Analytical ChemistryStockholm UniversityStockholmSweden
4.Department of Aquatic Resources, Institute of Coastal ResearchSwedish University of Agricultural SciencesÖregrundSweden
5.Department of Aquatic Resources, Institute of Marine ResearchSwedish University of Agricultural SciencesLysekilSweden
6.Umeå Marine Science CentreUmeå UniversityHörneforsSweden
7.Department of Environmental Research and MonitoringSwedish Museum of Natural HistoryStockholmSweden
Karlson, A.M.L., Gorokhova, E., Gårdmark, A. et al. Ambio (2020) 49: 391. https://doi.org/10.1007/s13280-019-01201-1
First Online 05 June 2019
Publisher Name Springer Netherlands
The Royal Swedish Academy of Sciences | CommonCrawl |
Spread-F occurrences and relationships with foF2 and h′F at low- and mid-latitudes in China
Ning Wang ORCID: orcid.org/0000-0001-9512-50141,2,
Lixin Guo1,
Zhenwei Zhao2,
Zonghua Ding2 &
Leke Lin2
Ionospheric irregularities are an important phenomenon in scientific studies and applications of radio-wave propagation. Spread-F echoes in ionograms are a type of high-frequency band irregularities that include frequency spread-F (FSF), range spread-F (RSF), and mixed spread-F (MSF) events. In this study, we obtained spread-F data from four ionosondes at low- and mid-latitudes near the 120°E chain in China during the 23rd solar cycle. We used these data to investigate spread-F occurrence percentages and variations with local time, season, latitude, and solar activity. The four ionosondes were located at Haikou (HK) (20°N, 110.34°E), Guangzhou (GZ) (23.14°N, 113.36°E), Beijing (BJ) (40.11°N, 116.28°E), and Changchun (CC) (43.84°N, 125.28°E). We also present possible correlations between spread-Fs and other ionospheric parameters, such as the critical frequency of the F2-layer (foF2) and the virtual height of the bottom-side F-layer (h′F). In particular, we investigated the possible threshold of the foF2 affecting the FSF and the relationship between the h′F and the RSF. The main conclusions are as follows: (a) the FSF occurrence percentages were anti-correlated with solar activity at all four sites; meanwhile, RSF occurrence rates increased with the increase in solar activity at HK, but not at the other three sites; (b) FSF occurrence rates were larger at the mid-latitudes than expected, while FSFs occurred more often after midnight; (c) the highest FSF occurrence rates mostly appeared during the summer months, while RSFs occurred mostly in the equinoctial months of 2000–2002 at HK and GZ; (d) a lower foF2 was suitable for FSF events; nevertheless, h′F and RSF occurrences satisfied the parabolic relationship; (e) the foF2 thresholds for FSFs were 15, 14, 7.6, and 7.8 MHz at HK, GZ, BJ, and CC, respectively. The h′Fs occurring between 240 and 290 km were more favorable for RSF occurrences. These results are important for understanding ionospheric irregularity variations in eastern Asia and for improving space weather modeling and forecasting capabilities
In the middle to late 1930s, ionospheric irregularities and the manner in which their electrodynamic mechanisms affected ionospheric behaviors began to attract the interest of many researchers (Abdu et al. 1981a, b, 1998, 2009; Booker and Wells 1938; Bowman 1974, 1990; Chandra and Rastogi 1970; Chou and Kuo 1996; de Jesus et al. 2013; Ossakow 1981; Xiong et al. 2012). Ionospheric irregularities appear as scattered echoes in high-frequency (HF) band ionograms that are known as spread-F events. Spread-Fs can manifest as frequency spread-Fs (FSF) that are broadened traces that mark reflections from the ionosphere along the frequency axis, or as range spread-Fs (RSF) that are along the vertical height axis. Many ground-based instruments (optical, ionosondes, and radar) and space-borne platforms (rockets and satellites) have been employed to explore the spread-F phenomenon over the past seven decades. These efforts have deepened our knowledge on spread-Fs showing that they vary with respect to latitude, local time, season, and solar and magnetic activity (Alfonsi et al. 2013; Banola et al. 2005; Chou and Kuo 1996; Deng et al. 2013; Huang et al. 1993; Scherliess and Fejer 1999). Different mechanisms have been proposed to explain spread-F occurrences and their development (Bowman 1990; Fejer et al. 1999; Fukao et al. 2004); among these, the primary mechanism in equatorial regions is the generalized Rayleigh–Taylor (R–T) instability mechanism. The R–T instability mechanism suggests that pre-reversal electric field enhancements (PRE) during the evening cause a rapid uplift of the ionosphere's F-layer (Fejer et al. 1999; Fukao et al. 2004; Manju et al. 2007; Sukanta et al. 2017; Xiong et al. 2012; Upadhayaya and Gupta 2014). Relationships between spread-Fs and other ionospheric parameters, particularly the F2-layer (foF2) and h′F variations with the occurrence of spread-Fs, have also been statistically examined (Rungraengwajiake et al. 2013; Joshi et al. 2013; Madhav Haridas et al. 2013; de Abreu et al. 2014a, b, c, 2017; Abadi et al. 2015; Manju and Madhav Haridas 2015; Smith et al. 2015; Liu and Shen 2017). In addition, the effects of seasonal, solar, and magnetic activity variabilities on the h′F threshold have also been investigated (Manju et al. 2007; Manju and Madhav Haridas 2015; Madhav Haridas et al. 2013; Stoneback et al. 2011; Narayanan et al. 2014, 2017).
Devasia et al. (2002) first introduced the concept of threshold height (h′Fc) as a critical parameter controlling the day-to-day equatorial spread-F (ESF) variability. Past studies have revealed the dependence of the h′Fc on seasonal variations and solar and magnetic activity for the occurrence of ESFs and found the occurrences to be irrespective of the magnitude and polarity of meridional winds (Jyoti et al. 2004; Manju et al. 2007). Rungraengwajiake et al. (2013) presented a comparative study of the correlation between h′F and RSF occurrences in Thailand, and the results showed that high RSF occurrences mostly happened during equinoctial months that corresponded to rapid increases in the monthly mean h′F after sunset. Joshi et al. (2013) found that the h′F plays a key role in determining the R–T instability growth rate. Madhav Haridas et al. (2013) presented the effects of seasonal and solar activity variations of the h′Fc on ESF occurrences in India and found that substantial increases in the h′Fc varied with magnetic activity during every season.
Similar studies in Brazil have been presented (de Abreu et al. 2014a, b, c) to show that the occurrence of ESFs are closely related to daily variations of the h′F near the equator. During periods of low solar activity (LSA), the 250 km h′F altitude acted as the h′Fc for the generation of spread-Fs, while the 300 km h′Fc was during periods of high solar activity (HSA). An investigation using measurements from multiple instruments over the American sector showed that spread-Fs were often observed the nights before and during storms near the equator, in which the foF2 was less than 8 MHz and the h′F was lower than 300 km (de Abreu et al. 2017).
Abadi et al. (2015) studied the influences of the h′F on the latitudinal extension of ionospheric irregularities in Southeast Asia. Their results suggested that the latitudinal extension of plasma bubbles was mainly controlled by the PRE magnitude and h′F peak values during the initial phases of the ESF. Manju and Madhav Haridas (2015) investigated the h′Fc for the occurrences of ESFs during equinoxes and showed that the equinoctial asymmetry of the h′Fc increases with solar activity. Aside from the studies mentioned above, there are few reports that consider the effect of the foF2 threshold on the generation of spread-F events. Liu and Shen (2017) conducted a case study during a severe geomagnetic storm near 120°E in China and showed that the spread-F was suppressed near Sanya and Wuhan during the storm's main phase when the frequency spread over 14 MHz, and the suppression was sustained for several hours. This helped us to understand the possible onset causes of the day-to-day spread-F variability.
Stoneback et al. (2011) investigated the local time distribution of meridional (vertical) drifts during the prolonged solar minimum. They found that the downward drifts across sunset and the upward drifts across midnight were also consistent with the delay in the appearance of ionospheric irregularities after midnight. Narayanan et al. (2014) studied the relationship between the occurrence of satellite traces (STs) in ionograms and the formation of ESFs using observations from an Indian dip equatorial station during solar minimum conditions. They found that the ST occurred later in the night as well implying that the PRE was not the cause of the ST during these times. Additionally, they also found that the STs were not followed by ESFs in about 30% of the cases indicating that large-scale wave-like structures (LSWS) do not trigger ESFs on all occasions. Narayanan et al. (2017) also found that the plasma bubbles were generated without strong PREs when the ion-neutral collision frequencies possibly dropped significantly during the unusually low solar activity conditions of 2008. Abdu et al. (2006) found that the existence of significant planetary wave (PW) influences on plasma parameters at E- and F-region heights over the equatorial latitudes using airglow, radar, and ionospheric sounding observations. A direct consequence of the PW scale oscillations in the evening electric field is its role in the quiet time day-to-day variability of the ESF/plasma bubble occurrences and intensities.
We limited our focus to spread-F occurrences and their relationships with foF2 and h′F that affected spread-F occurrences during a complete solar cycle in the low- and mid-latitudes over China. The International Reference Ionosphere-2012 (IRI-2012) model includes the monthly mean spread-F occurrences for predicting in the Brazilian longitude sector but not for Chinese sector. Therefore, the studies of spread-F occurrence statistics in China are part of an on-going effort to develop the spread-F occurrence prediction abilities to improve the IRI model. In the present study, we focused on the characteristics and correlations between spread-F occurrences and the foF2 and h′F. Furthermore, we also present the thresholds of the foF2 as they relate to the generation of FSFs.
The China Research Institute of Radio-wave Propagation (CRIRP) constructed and operated a network of long-running ionospheric observation sites that cover mainland China. In this study, we extracted simultaneous spread-F data from four digital ionosondes located at Haikou (HK) (20°N, 110.34°E), Guangzhou (GZ) (23.14°N, 113.36°E), Beijing (BJ) (40.11°N, 116.28°E), and Changchun (CC) (43.84°N, 125.28°E). In addition, we also determined the data characteristic of the foF2 and h′F at these sites to reveal possible correlations between spread-F occurrences and the foF2 and h′F. No data were recorded in December 1997 and from May to December 1999 at CC, because the ionosonde was being repaired. The observational site details are shown in Table 1.
Table 1 Details of the digital ionosonde sites used in the investigation
The HK and GZ sites lie near the north crest of the equatorial ionization anomaly (EIA) zone. The EIA zone is where the fountain effect phenomena and the equatorial electrojet often interact resulting in complicated ionospheric physical processes. BJ and CC are located at the mid-latitudes in China. According to previous studies, ionospheric irregularities greatly depend on solar activity, local time, season, latitude and longitude, and geomagnetic disturbances (Abdu et al. 1981a, b, 1983, 1998, 2009; Booker and Wells 1938; Bowman 1974; Chandra and Rastogi 1970; Maruyama 1988; Xiong et al. 2012). To discuss the correlations between spread-Fs and solar and geomagnetic activities, we show the monthly mean 10.7 cm radio flux (F10.7) and ap index during the 23rd solar cycle in Fig. 1 that covers the epochs of the LSA and HSA. We used a 3-hourly ap index to identify geomagnetically quiet and disturbed days. If the maximum value of the 3-hourly ap index for a day was greater than 12, the day was considered as a disturbed day (Narayanan et al. 2017). Figure 2 shows the daily max ap indices from 2000 to 2005. Further, it can be seen from the figure that there were more geomagnetically disturbed days during the vernal equinox and autumn equinoxes in 2001 and 2002.
Monthly averaged 10.7 cm solar flux (F10.7) (y axis: F10.7/sfu) and ap index from 1997 to 2008 denoting the solar activity
Daily max ap indices from 2000 to 2005
Ionogram data were collected using type TYC-1 ionosondes, which are designed and manufactured by the CRIRP (Xu et al. 2001). Ionograms were recorded at 1-h intervals for a frequency range from 1 to 32 MHz. We distinguished two types of spread-F, FSF, and RSF for detailed study. We used the percentage of spread-F occurrences to describe the spread-F statistical features, which is defined as follows:
$${\text{P}}\left( y, m, h \right) = \frac{{n \left( {y, m, h} \right)}}{{N\left( {y,m,h} \right)}} \times 100{\text{\% }}$$
where y, m, and h represent the year, month, and local time (LT), respectively; n is the number of spread-F occurrences that appear at the same local time but during different days of a single month, and N is the total number of days for a given year and local time. Spread-Fs typically appeared after sunset and lasted until the subsequent sunrise; thus, the percentage of spread-F occurrences from 18:00 LT to 06:00 LT is the topic of interest in this study. Occurrences of FSF and RSF were compared with monthly medians of the foF2 and h′F to find the correlations between foF2 and h′F for the generation of spread-Fs. The FSF, RSF, foF2, and h′F were differentiated by manually analyzing the ionograms. The foF2 and h′F can sometimes be measured, but sometimes cannot be obtained when a spread-F occurs. The foF2 and h′F cannot be obtained during a strong spread-F (SSF). SSFs are a type of spread-F that can be identified when there is strong diffusion on the frequency and height axis of an ionogram. Figure 3 shows a SSF event in Haikou on March 26, 2012. The observations presented in this manuscript contain data when the foF2 and h′F values were reliably scaled during a spread-F. To examine their seasonal variations, we grouped the data into the following four seasonal bins: summer (May, June, July and August), vernal equinox (March and April), autumn equinox (September and October), and winter (January, February, November and December) (Maruyama and Matuura 1984; Maruyama et al. 2009; Sripathi et al. 2011; Xiao and Zhang 2001).
A SSF event in Haikou on March 26, 2012
Nocturnal, seasonal, and solar activity variations on spread-F occurrences
The monthly mean of the FSF occurrence rates varied with local time and are presented separately in Fig. 4 for Haikou, Guangzhou, Beijing, and Changchun. It can be found that the FSF occurrences frequently appeared after midnight. Also, the FSF occurrences observed at different sites exhibited distinct local time distribution patterns. Previous studies have also observed this trend (Zhang et al. 2015; de Jesus et al. 2010, 2012, 2016). The FSF occurrence rates at HK, BJ and CC were higher than GZ. The maximum FSF occurrence rate was ~ 80% and occurred in July 1997 at HK, in August 2008 at BJ and in June 2006 at CC. The LSA yielded high FSF occurrence percentages at all four sites. The relationship between the FSF and solar activity was approximate to a negative correlation. The seasonal variation of the FSF occurrence rates observed at the four sites is shown in Fig. 5a–d. We found that FSFs occurred mostly during the summer at HK and the occurrence rate was lower between 1999 and 2002. FSF occurrence rates were higher during the autumn equinox than during the vernal equinox between 2000 and 2001 at HK. FSFs occurred mostly during the summer at GZ, however, scarcely occurred in 2002 and 2008. Statistically, the FSFs started at approximately 21:00 LT and lasted until 05:00 LT at HZ and CC. However, FSFs started at about 23:00 LT and lasted until 05:00 LT at GZ and BJ, with post-midnight FSFs as the most commonly observed.
Monthly mean FSF occurrence percentages at the four sites
Seasonal variation of the FSF occurrences observed at HK (a), GZ (b), BJ (c), and CC (d)
Figure 6 shows variations in the average RSF occurrence rates at the four sites. The RSF occurrence rate was much larger than the FSF occurrence rate at GZ; however, the rates were smaller than the FSF occurrence rates at BJ and CC. RSF occurrence rates increased with an increase in solar activity at HK, but not at the other three sites. The maximum RSF occurrence rate was higher than 80% in June 2006 and July 2007 at GZ. Figure 7 shows the seasonal RSF occurrence rate variations at the four sites. RSFs mostly occurred in the vernal equinox and autumn equinox months during HSA years at HK and GZ. These observations revealed that the RSF occurrence rate from 2000 to 2002 at HK and GZ were possibly affected by the geomagnetic activity according to Fig. 2. During the solar maximum period between 2000 and 2002, RSFs appeared earlier than during other periods, with a maximum RSF occurrence rate occurring between 21:00 LT and 01:00 LT at HK and GZ. Different from the FSF occurrences, higher RSF occurrence rates mostly occurred during the winter months at BJ and CC. Previous studies have emphasized that FSF events are well correlated with bottom-side layers, while RSFs are closely correlated with plumes. Additionally, the RSF occurrence rate reaches its maximum before midnight during HSA at low latitude, whereas that of an FSF reaches a maximum after midnight (Liu et al. 2004a, b; Chen et al. 2006; Aarons et al. 1994; Hu et al. 2004). This regular pattern was also observed at the four sites in China.
Same as Fig. 2, but for the RSF occurrences
Seasonal variation of the RSF occurrences at HK (a), GZ (b), BJ (c), and CC (d)
Abdu et al. (2003) showed that RSF events are associated with developed or developing plasma bubble events, while FSF events are associated with narrow-spectrum irregularities that occur near the peak of the F-layer. These results suggest that the upward velocity of plasma bubbles have a strong seasonal connection with the maximum values observed during the summer. Variations of FSF and RSF except for those during the 2000–2002 solar maximum period are mainly consistent with these studies. Rungraengwajiake et al. (2013) showed that FSF events appear later than RSF events on average and that FSFs remain until morning, while RSFs almost disappear by around 04:00 LT. The results shown in Figs. 5 and 7 are slightly different, which may be partly attributed to the effects of geomagnetic activity. Figure 2 shows the geomagnetic activity during the equinoxes in 2001 and 2002. It is possible these activities caused the RSFs to occur mainly during equinoxes at HK and GZ in 2001 and 2002. The peak FSF occurrence rate appeared later at GZ than at HK, which is well correlated with the manner in which fresh bubbles start from the latter station and then expand to high latitudes. The average FSF occurrence percentage mostly peaks from 24:00 LT to 02:00 LT at HK and from 03:00 LT to 05:00 LT at GZ. The average RSF occurrence percentages mostly peaked from 21:00 LT to 23:00 LT at HK and from 24:00 LT to 02:00 LT at GZ during periods of HSA. Meanwhile, RSF occurrence rates were higher at HK and GZ than those at BJ and CC; FSF occurrence rates were higher at HK, BJ, and CC than at GZ. These results support the hypothesis that solar and geomagnetic activity affects seasonal and longitudinal variations of spread-Fs.
Liu and Shen (2017) found that the disturbance of electric fields could also contribute to the occurrence of spread-Fs, especially at low-latitude stations. The disturbed electric fields and the disturbance winds are also the probable factors that promote the spread-F along with the gravity-driven R–T instability. In addition, the electric field disturbances can also generate spread-Fs through R–T instability only (de Jesus et al. 2010; Wang et al. 2014; Wan and Xu 2014; Mo et al. 2017). The disturbance of the dynamo driven by enhanced global thermospheric circulation resulting from energy input at high latitudes is another factor for promoting spread-Fs (de Jesus et al. 2010; Liu and Shen 2017). Therefore, it can be seen that there are many possible mechanisms for spread-F occurrences, and more in-depth analysis is needed.
Nocturnal, seasonal, and solar activity variations on foF2 and h′F
In Fig. 8 we showed local time and the variations in solar activity in the monthly median foF2 data from the 23rd solar cycle. At lower latitudes, a higher magnitude foF2 was sustained until midnight. In addition, another morphological feature of the monthly medians is the typical post-sunset peak values. Between 1998 and 2005, foF2 variations showed dual-peak patterns at HK and GZ that reached a minimum during the summer and a maximum during the spring and winter. Additionally, wintertime monthly medians of the foF2 were higher during the spring in 1998, but this result is inverted between 2003 and 2005. Figure 9 shows the seasonal variations of the averaged foF2 monthly median data at all four sites. The medians reached their peak magnitudes between 18:00 and 19:00 LT. In addition, the medians were mostly higher during equinox seasons at HK and GZ; however, they were mostly higher during the summer at BJ and CC. The highest foF2 medians were ~ 18 MHz and occurred from 18:00 LT to 24:00 LT at HK and GZ during periods of maximum solar activity. The minimal medians occurred before dawn from around 03:00–05:00 LT. The post-midnight collapse of the foF2 usually occurred more often at low latitudes than mid-latitudes.
Variation of the monthly median foF2 at the four sites between 1997 and 2008
Seasonal variation of the monthly median foF2 at HK (a), GZ (b), BJ (c), and CC (d)
Abdu et al. (1983) proposed that the h′F parameter may be a possible factor involved in the occurrence and variation of spread-Fs. Figure 10 shows the h′F monthly median data at all four sites, thus demonstrating that monthly medians were higher at HK and GZ than at BJ and CC. The peak median h′F values occurred before midnight during the summer in HSA at HK and GZ; however, the peak value onset time was later at high latitudes. During periods of HSA, monthly medians increase. Figure 11 shows the seasonal variation of the average h′F monthly median at the four sites, which is quite different from the foF2. The maximum h′F values occurred from 21:00 LT to 01:00 LT during summer months at HK and GZ; otherwise, they appeared at or before midnight from 2000 to 2002.
Same as Fig. 6, but for monthly medians of the h′F
Seasonal variation of the monthly medians of the h′F at HK (a), GZ (b), BJ (c), and CC (d)
The possible foF2 threshold for FSFs and the relationship between the h′F and RSF
The correlations between spread-F occurrence and the foF2 and h′F magnitudes are discussed in this section. Figures 12 and 13 show the post-sunset foF2 and h′F variations compared with the normalized spread-F occurrence rates at the four sites. In order to analyze the correlation between the spread-F occurrence rate and the foF2 and h′F, the normalized probability was used. The normalized spread-F occurrence rate is defined as follows:
$$p_{i} = \frac{{m}_{i} }{{\mathop \sum \nolimits_{i} {m}_{i} }}$$
$$\mathop \sum \limits_{i} {p}_{i} = 1$$
where p is the normalized FSF or RSF occurrence rate, m i is the number of FSF or RSF event occurrences when the foF2 or h′F is within a certain interval. We used 0.2 MHz and 5 km as the sampling intervals for the foF2 and h′F. The summation of m i is the total number of FSF or RSF event occurrences. We applied the polynomial fitting method during the relationship analysis between the foF2 and h′F and the spread-F occurrence rates. We found that the foF2 and FSF occurrences satisfy the linear relationship shown in Fig. 12, and the h′F and RSF occurrences are similar to parabolic relationship shown in Fig. 13. The red point is the sample value. The blue lines are a fitting line or curve. The FSF occurrence rates increased with a decrease in foF2 at each site, and the foF2 values ranged from 2.5 to 18 MHz at HK and GZ. A straight line is drawn when the normalized spread-F occurrence rate is equal to 0% as in Fig. 12. The intersection of this line and the blue line is considered the foF2 threshold. We estimated that the foF2 threshold at HK and GZ was ~ 15 and ~ 14 MHz because almost the FSF occurrence was ~ 0% when foF2 exceeded this magnitude. The foF2 values ranged from of 3–9 MHz at BJ and CC. Thus, the corresponding foF2 thresholds for BJ and CC were 7.6 and 7.8 MHz, respectively. It is evident that the foF2 variability was much larger at low latitudes than at mid-latitudes. There are few reports that consider the effect of the foF2 threshold on the generation of spread-F events. De Abreu et al. (2017) found that the spread-F was often observed during storms using measurements from multiple instruments over the American sector when the foF2 was below 8 MHz. De Abreu et al. (2017) showed that the post-sunset EIA is produced by the plasma fountain arising from the pre-reversal vertical drift enhancement in the F-region (as indicated by large sunset increases of h′F and decreases of foF2). Therefore, it can be seen that the rapidly changing Dst index will also affect spread-Fs; however, our research is not currently focused on ionospheric storms. The variation in foF2 at different latitudes suggests that the PRE is not the only factor to initiate FSFs. For example, the meridional wind can suppress the growth rate of the R–T instability, also attributing to the foF2 and FSF (Buonsanto and Titheridge 1987; Stoneback et al. 2011).
Correlation between foF2 and the normalized FSF occurrence percentages
Correlation between h′F and the normalized RSF occurrence percentages
Figure 13 shows the post-sunset h′F variations compared with the RSF occurrence rates at the four sites. The red point is the sample value. The blue line is the fit curve. The RSF occurrence rate and the h′F satisfy the parabolic relationship. When the probability of the RSF was ~ 25% of the maximum probability of occurrence, we treated that virtual height value as the threshold value. The h′F occurring between 240 and 290 km is more favorable for RSF occurrence by calculation, which is different from the relationship between foF2 and FSF. Figures 6, 10, and 13 indicate that the higher occurrence rates of RSFs are well correlated with higher post-sunset h′F peaks (Rungraengwajiake et al. 2013). Previous studies observed spread-Fs in the equatorial region on nights when the h′F was below 300 km (Abadi et al. 2015; Manju and Madhav Haridas 2015; Liu and Shen, 2017; de Abreu et al. 2017). Our results also support this conclusion. In addition, Devasia et al. (2002), Jyoti et al. (2004) and Manju et al. (2007) obtained an h′F threshold for the spread-F occurrences in their studies in India. Devasia et al. (2002) found a threshold of about ~ 300 km for the cases in their study. Our results also show that when the virtual height is greater than 300 km, the probability of an RSF is very small. Jyoti et al. (2004) showed a linear relationship between solar activity and the h′F threshold. Manju et al. (2007) investigated the dependence of the h′F threshold on seasonal and solar activity for magnetically quiet conditions and proposed the important role of neutral dynamics in controlling the day-to-day ESF variability. Abadi et al. (2015) found that latitudinal extension of plasma bubbles was mainly controlled by the h′F peak value during the initial phase of an ESF. Manju and Madhav Haridas (2015) showed that the equinoctial asymmetry of the h′Fc increases with solar activity. In this article, the correlation between the h′F threshold and the seasonal and solar activities are not involved, and we will also focus on this content. The new idea presented from our study is the correlation between RSF occurrences and the h′F, which are different from previous research results. In a follow-up study, we will examine the relationship between the h′F threshold for RSFs and the solar and geomagnetic activities and equinoctial asymmetry.
The correlation RSF occurrence percentages with rapidly increasing post-sunset monthly mean h′F values substantiated the role of the PRE enhancement on RSF onsets. Traveling planetary wave ionospheric disturbance (TPWID)-type oscillations (de Abreu et al. 2014a, c; Fagundes et al. 2009) in the modulation of the virtual height in the F-region increased during sunset hours. Meridional wind velocities corresponding to the post-sunset h′F for each spread-F event have been considered. Buonsanto and Titheridge (1987) found that the hmF2 dropped from 13:00 to 18:00 LT during the solar maximum periods because of the meridional wind. These results also indicate that the spread-F is a complex phenomenon, which implies that other possible factors can be ascribed to spread-F occurrences. The atmosphere ionosphere coupling process has been proposed as a contributing factor for spread-F development. Therefore, the connections between spread-F occurrence characteristics and the foF2 and h′F magnitudes deserve detailed investigation by additional theoretical and observational research. The foF2 and h′F thresholds also require further investigation using observations from different regions and under different solar activity conditions.
In this study, we presented variations of the spread-F, foF2, h′F, the possible threshold of the foF2 for FSF, and the relationship between the h′F and RSF. The data in our study were recorded by four stations at low- and mid-latitudes near 120°E longitude in China during the 23rd solar cycle. The major conclusions are summarized as follows:
The FSF occurrence rates increased during years of LSA at all four sites. FSFs mainly occurred during the summer months, while RSFs occurred mostly in the equinoctial months between 2000 and 2002 at HK and GZ. Post-midnight FSFs were the most observed type of spread-F events. The typical FSF onset time was about 21:00 LT, and the FSFs normally lasted until 05:00 LT, while the RSFs occurred 2–3 h earlier at HK and GZ during periods of HSA.
The foF2 and h′F peak values come mainly before midnight at low latitudes, while h′F peak values appeared after midnight at mid-latitudes during periods of HSA.
Lower foF2 values were appropriate for FSF events; nevertheless, h′F and RSF occurrences satisfied the parabolic relationship. Most FSF events occurred when the foF2 was below 15 and 14 MHz at HK and GZ, and below 7.6 and 7.8 MHz at BJ and CC. The h′Fs occurring between 240 and 290 km were more favorable for RSF occurrences, which differ from the foF2. However, some questions remain unresolved and further studies are in progress.
Our studies of FSFs and RSFs in China are useful and have the potential to be included in the future IRI model. However, even after such studies of spread-F onsets and growth conditions, some uncertainties remain. This requires further efforts to understand the spread-F phenomenon at different locations. Soon, long irregularity data coverage over the China sector will be studied. More ionospheric parameters will be compared with local time and seasonal spread-F variations to amplify knowledge of the involved physical mechanisms.
FSF:
the frequency spread-F
RSF:
the range spread-F
MSF:
the mixed spread-F
foF2:
the critical frequency of the F2-layer
h′F:
the virtual height of the bottom-side F-layer
EIA:
equatorial ionization anomaly
F10.7:
the monthly average data of 10.7 cm radio flux
Rayleigh–Taylor
pre-reversal electric field
TPWID:
traveling planetary wave ionospheric disturbance
HSA:
high solar activity
LSA:
low solar activity
CRIRP:
China Research Institute of Radio-wave Propagation
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WN designed the study, analyzed the data, and wrote the manuscript. GLX and ZZW contributed related analysis on data from HK and GZ. DZH and LLK helped with the text of the paper, particularly with the introduction and comparison with previous works. All coauthors contributed to the revision of the draft manuscript and improvement of the discussion. All authors read and approved the final manuscript.
Authors' information
Ning Wang, is currently a Ph.D. student at Xidian University. She also is an Associate Professor at the China Research Institute of Radiowave Propagation. She has authored and coauthored 8 patents and over 15 journal articles. Her current research interests are in ionospheric irregularities and ionosphere radiowave propagation. Dr. Linxin Guo is currently a Professor and Head of the School of Physics and Optoelectronic Engineering Science at Xidian University, China. He has been a Distinguished Professor of the Changjiang Scholars Program since 2014. He has authored and coauthored 4 books and over 300 journal articles. Dr. Zhenwei Zhao is currently a Professor and Chief engineer at the China Research Institute of Radiowave Propagation. His current positions include: Chairman of the ITU-R SG3 in China; Head of the Chinese Delegation of ITU-R SG3; Lead expert for the Asia-Pacific Space Cooperation Organization (APSCO). Dr. Zonghua Ding is currently an Associate Professor at the China Research Institute of Radiowave Propagation. His current research interests are in ionosphere and ionosphere radiowave propagation. Dr. Leke Lin is currently a Professor at the China Research Institute of Radiowave Propagation. He has participated in the activities of the ITU-R study group 3 and has submitted about 40 contributions to the ITU-R SG3.
The authors acknowledge the Data Center of the China Research Institute of Radio-wave Propagation for help with ionogram scaling and classification. The authors would like to thank Dr. Shuji Sun and Dr. Tong Xu for proofreading this manuscript. The authors would also like to thank the anonymous referee for the useful comments and suggestions for improving the paper.
Regretfully, the data used in this manuscript cannot be shared because they belonged to the China Research Institute of Radio-wave Propagation (CRIRP).
Written informed consent was obtained from study participants for participation in the study and for the publication of this report and any accompanying images. Consent and approval for publication was also obtained from Xidian University and China Research Institute of Radio-wave Propagation.
This research was supported by the National Natural Science Foundation of China (Grant No. 41604129) and the National Key Laboratory Foundation of Electromagnetic Environment (Grant Nos. A171501016, A171601003, A161601002, and B041605003). The funds from Grant No. 41604129 were used for data collection and analysis. The funds from Grant Nos. A171501016, A171601003, A161601002, and B041605003 were used for manuscript preparation.
School of Physics and Optoelectronic Engineering, Xidian University, Xi'an, Shaanxi, 710071, China
Ning Wang & Lixin Guo
National Key Laboratory of Electromagnetic Environment, China Research Institute of Radio-wave Propagation, Qingdao, Shandong, 266107, China
Ning Wang, Zhenwei Zhao, Zonghua Ding & Leke Lin
Ning Wang
Lixin Guo
Zhenwei Zhao
Zonghua Ding
Leke Lin
Correspondence to Ning Wang.
Wang, N., Guo, L., Zhao, Z. et al. Spread-F occurrences and relationships with foF2 and h′F at low- and mid-latitudes in China. Earth Planets Space 70, 59 (2018). https://doi.org/10.1186/s40623-018-0821-9
Ionospheric irregularities
Spread-F occurrence percentage
foF2 threshold for FSF
Relationship between h′F and RSF
3. Space science | CommonCrawl |
Kurtosis/4th central moment in terms of mean and variance
Is it possible to express the kurtosis $\kappa$, or the 4th central moment $\mu_4$, of a random variable $X$ in terms of its mean $\mu = E(X)$ and variance $\sigma^2 = Var(X)$ only, without having to particularize to any distribution?
I mean, an expression like $\kappa = f(\mu, \sigma^2)$ or $\mu_4 = g(\mu, \sigma^2)$, valid for any distribution, where $f(\mu, \sigma^2)$ and $g(\mu, \sigma^2)$ are functions of the mean $\mu$ and variance $\sigma^2$.
P.S.: Some comments on my attempts.
$\kappa$ is related to $\mu_4$, and $\mu_4 = E(X^4) - 4\mu E(X^3) + 6\mu^2 E(X^2) - 3\mu^4$.
The term $E(X^2)$ can be expressed as $E(X^2) = \mu^2 + \sigma^2$ but I didn't manage to find the way to express $E(X^3)$ and $E(X^4)$ in terms of $\mu$ and $\sigma^2$.
variance mean moments kurtosis
kjetil b halvorsen
fchopinfchopin
$\begingroup$ $E(X^4)$ is not uniquely determined by the mean and variance. An ${\rm exponential}(1)$ random variable has mean and variance equal to 1, but has fourth moment equal to $4!=24$ while a normally distributed variable with mean $\mu=1$ and variance $\sigma^2 = 1$ has fourth moment equal to $10$. $\endgroup$ – Macro Jul 5 '12 at 16:21
$\begingroup$ In the case of the normal distribution the complete distribution is determined by the mean and variance. So for a normal distribution the foruth central moment and all moments of the normal distribution can be expressed in terms of their mean and variance. @Macro This makes me puzzled why you would bring up the nromal distribution in your comment. $\endgroup$ – Michael R. Chernick Jul 5 '12 at 17:08
$\begingroup$ Thanks! I'm not an expert in statistics but need to deal with it for some problem. Well, in particular, I'm interested in having the variance of the sample variance expressed in terms of the mean and variance, and the variance of the sample variance depends on the excess kurtosis/4th central moment. That's why I need this particular parameter, and I'm trying to avoid fighting against the pdf to compute this moment as the pdf is the convolution of generalized Pareto with Uniform with Uniform... I've understood what I need isn't independent on the underlying distribution. Thanks. $\endgroup$ – fchopin Jul 5 '12 at 17:21
$\begingroup$ @Macro When someone says that they want to know if the mean and variance determine the fourth moment for any distribution, I interpret that to mean in a given family will the mean and variance determine the fourth moment of the distribution? You apparently interpret it to be if you do not know the form of the distribution but know the mean and the variance will that determine the fourth moment? I think the answer to both questions is no. As IO reread the question i think your interpretation is correct. $\endgroup$ – Michael R. Chernick Jul 5 '12 at 17:24
What you think about here is something like a philosopher's stone of statistics.
The strict answer is:
No, it is impossible to express skewness or kurtosis via the mean and variance.
@Macro gave a counterexample of distributions with different skewness and kurtosis. A question of coming up with distributions for the given set of moments has entertained statisticians since the very early ages, and Pearson's system of frequency curves is one of the examples of how one could come up with a continuous distribution for the numeric values of the first four moments. You could also look at the moment generating function $m(t)={\rm E}[\exp(Xt)]$, a characteristic function $\phi(t)={\rm E}[\exp(iXt)]$, or a cumulant generating function $\psi(t) = \ln \phi(t)$. With some luck, you can try putting your four moments into them and invert these functions to obtain explicit expression of the densities. Finally, you can always find a distribution with discrete support on five points to satisfy the five equations for the moments of order 0 through 4 by solving a corresponding system of nonlinear equations.
To express the higher order moments via the lower order moments, you need to know the shape of the distribution and its parameters. For one-parameter (Poisson, exponential, geometric) or two-parameter (normal, gamma, binomial) distributions, you can express the higher order moments via the natural parameters of these distributions; e.g., for a Poisson with rate $\lambda$, skewness is $\lambda^{-1/2}$, and kurtosis is $\lambda^{-1}$ (sanity check: both going to zero as $\lambda \to \infty$, providing a normal approximation for Poisson for large $\lambda$). But these exceptions should not fool you; for more interesting distributions, including anything from the real world, you can just forget about doing anything meaningful with the kurtosis.
StasKStasK
27.2k6767 silver badges154154 bronze badges
$\begingroup$ Thanks! Well, I'm studying something related to the variance of the sample variance estimator, and it depends on the excess kurtosis/4th central moment, so I need this particular parameter for my problem. I know the exact pdf of the underlying distribution, it is the result of the convolution of generalized Pareto with Uniform with Uniform (yes, Uniform twice), and the expression is so huge that I'm trying to avoid computing the 4th moment by integration. I think I could use approximations for the mean of a function of a random variable $E[g(X)] \approx g(\mu) + \frac{1}{2}\sigma^2 g''(\mu)$. $\endgroup$ – fchopin Jul 5 '12 at 17:35
$\begingroup$ You are probably getting this idea from the delta-method, right? This is a first order approximation, and it only works well when the distribution of $X$ gets tighter as $n\to\infty$. If things remain $O(1)$, this expression fails miserably; think about $X\sim N(0,10)$ and $g(X)=\exp[X]$. The correct expression is $E[g(X)] = \exp( \mu + \sigma^2/2 ) = \exp( 5 ) = 148.41$; your first order approximation gives $\exp(0) + \frac12 \cdot 10 \cdot \exp(0) = 1 + 5 = 6$. $\endgroup$ – StasK Jul 5 '12 at 18:42
$\begingroup$ Yes, delta method. This result can be found in the books of Benjamin & Cornell (1970), Papoulis (1984) and Blumenfeld (2001). I guess my problem meets the conditions for it to be a reasonable approximation, but that's an approximation after all. Will try... $\endgroup$ – fchopin Jul 6 '12 at 7:52
$\begingroup$ By the way, the delta method, introducing various terms in the Taylor series, yields $E[g(X)] \approx g(\mu) + \frac{1}{2}\sigma^2g''(\mu) + \frac{1}{6}\mu_3g'''(\mu) + \frac{1}{24}\mu_4g^{iv}(\mu)$. Considering the expansion up to the 2nd or 3rd terms yields the result $\mu_4 = 0$, and considering all the terms up to the 4th one yields the result $\mu_4 = \mu_4$. So this technique doesn't yield any result here... $\endgroup$ – fchopin Jul 6 '12 at 15:56
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Reference for $\mathrm{Var}[s^2]=\sigma^4 \left(\frac{2}{n-1} + \frac{\kappa}{n}\right)$?
Taylor expansion to contain sample mean, sample variance, sample skewness, and sample kurtosis
Looking for a distribution where: Mean=0, variance is variable, Skew=0 and kurtosis is variable
Higher-order cumulant and moment names beyond variance, skewness and kurtosis
Unbiased Estimator of the Variance of the Sample Variance
$P(\lvert X - \mu\rvert \geq \sigma)$ as a measure of tailedness
Notation for skewness and kurtosis | CommonCrawl |
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Continuous maximal regularity on singular manifolds and its applications
June 2016, 5(2): 337-348. doi: 10.3934/eect.2016007
A remark on blow up criterion of three-dimensional nematic liquid crystal flows
Yinxia Wang 1,
School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011, China
Received January 2016 Revised March 2016 Published June 2016
In this paper, we study the initial value problem for the three-dimensional nematic liquid crystal flows. Blow up criterion of smooth solutions is established by the energy method, which refines the previous result.
Keywords: blow up criterion., smooth solutions, Nematic liquid crystal flows.
Mathematics Subject Classification: Primary: 76A15; Secondary: 76B0.
Citation: Yinxia Wang. A remark on blow up criterion of three-dimensional nematic liquid crystal flows. Evolution Equations & Control Theory, 2016, 5 (2) : 337-348. doi: 10.3934/eect.2016007
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Stefano Bosia. Well-posedness and long term behavior of a simplified Ericksen-Leslie non-autonomous system for nematic liquid crystal flows. Communications on Pure & Applied Analysis, 2012, 11 (2) : 407-441. doi: 10.3934/cpaa.2012.11.407
Yi-hang Hao, Xian-Gao Liu. The existence and blow-up criterion of liquid crystals system in critical Besov space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 225-236. doi: 10.3934/cpaa.2014.13.225
Qiang Tao, Ying Yang. Exponential stability for the compressible nematic liquid crystal flow with large initial data. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1661-1669. doi: 10.3934/cpaa.2016007
Junyu Lin. Uniqueness of harmonic map heat flows and liquid crystal flows. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 739-755. doi: 10.3934/dcds.2013.33.739
Boling Guo, Yongqian Han, Guoli Zhou. Random attractor for the 2D stochastic nematic liquid crystals flows. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2349-2376. doi: 10.3934/cpaa.2019106
Xin Zhong. A blow-up criterion of strong solutions to two-dimensional nonhomogeneous micropolar fluid equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4603-4615. doi: 10.3934/dcdsb.2020115
Zhenlu Cui, M. Carme Calderer, Qi Wang. Mesoscale structures in flows of weakly sheared cholesteric liquid crystal polymers. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 291-310. doi: 10.3934/dcdsb.2006.6.291
Yinxia Wang | CommonCrawl |
# Real numbers and their properties
- Integers: whole numbers, positive and negative, including zero.
- Rational numbers: numbers that can be expressed as a fraction, with a numerator and a denominator.
- Irrational numbers: numbers that cannot be expressed as a fraction, such as the square root of 2.
We will also discuss the concept of absolute value and its properties.
The absolute value of a number is the non-negative value of the number. It is defined as:
$$|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}$$
Real numbers have some important properties, such as the commutative and associative properties of addition and multiplication.
Real numbers also have some interesting properties, such as the Archimedean property, which states that for any two real numbers, there exists a natural number greater than the difference between them.
## Exercise
Instructions:
- Prove the commutative property of addition for real numbers.
- Prove the associative property of multiplication for real numbers.
- Show that the Archimedean property holds for real numbers.
### Solution
- Commutative property of addition:
$$a + b = b + a$$
- Associative property of multiplication:
$$a \cdot (b \cdot c) = (a \cdot b) \cdot c$$
- Archimedean property:
Let $x$ and $y$ be real numbers. Then, there exists a natural number $n$ such that:
$$n > \frac{x - y}{1}$$
# Functions and their properties
- One-to-one functions: functions where each input corresponds to a unique output.
- Injective functions: functions where each input corresponds to a unique output.
- Surjective functions: functions where each output corresponds to at least one input.
- Bijection functions: functions that are both injective and surjective.
- Inverse functions: functions that are the inverse of each other.
We will also discuss the concept of function composition and its properties.
Function composition is the process of applying one function to the result of another function. It is denoted by $(f \circ g)(x)$.
Functions also have some interesting properties, such as the additivity property of functions.
Real numbers also have some interesting properties, such as the Archimedean property, which states that for any two real numbers, there exists a natural number greater than the difference between them.
## Exercise
Instructions:
- Prove the additivity property of functions.
- Show that the Archimedean property holds for real numbers.
### Solution
- Additivity property of functions:
Let $f(x)$ and $g(x)$ be two functions. Then, their sum is a function:
$$(f + g)(x) = f(x) + g(x)$$
- Archimedean property:
Let $x$ and $y$ be real numbers. Then, there exists a natural number $n$ such that:
$$n > \frac{x - y}{1}$$
# Limits and continuity
- One-sided limits: limits as the input approaches a value from one side.
- Left-hand limit: limit as the input approaches a value from the left.
- Right-hand limit: limit as the input approaches a value from the right.
- Two-sided limit: limit as the input approaches a value from both sides.
We will also discuss the concept of continuity and its properties.
A function is continuous at a point if the limit of the function at that point exists and is equal to the value of the function at that point.
Continuity also has some interesting properties, such as the intermediate value property.
The intermediate value property states that if a function is continuous on an interval and takes on different values at the endpoints of the interval, then it must take on all values between the endpoints.
## Exercise
Instructions:
- Prove the intermediate value property for continuous functions.
- Show that the Archimedean property holds for real numbers.
### Solution
- Intermediate value property:
Let $f(x)$ be a continuous function on the interval $[a, b]$ and $f(a) \neq f(b)$. Then, there exists a point $c$ in the interval where $a < c < b$ such that $f(c) = f(a) + (f(b) - f(a)) \cdot \frac{c - a}{b - a}$.
- Archimedean property:
Let $x$ and $y$ be real numbers. Then, there exists a natural number $n$ such that:
$$n > \frac{x - y}{1}$$
# Derivatives and their properties
- First derivative: the rate of change of a function with respect to its input.
- Second derivative: the rate of change of the first derivative with respect to its input.
- Higher-order derivatives: derivatives of higher order.
We will also discuss the concept of the derivative of a function and its properties.
The derivative of a function $f(x)$ is denoted by $f'(x)$ and is defined as:
$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$
Derivatives also have some interesting properties, such as the product rule and the chain rule.
The product rule states that the derivative of the product of two functions is the sum of the product of their derivatives.
$$(f \cdot g)'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)$$
The chain rule states that the derivative of a composite function is the product of the derivative of the outer function and the derivative of the inner function.
$$(f \circ g)'(x) = f'(g(x)) \cdot g'(x)$$
## Exercise
Instructions:
- Prove the product rule for derivatives.
- Prove the chain rule for derivatives.
- Show that the Archimedean property holds for real numbers.
### Solution
- Product rule:
Let $f(x)$ and $g(x)$ be two functions. Then, the derivative of their product is:
$$(f \cdot g)'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)$$
- Chain rule:
Let $f(x)$ and $g(x)$ be two functions. Then, the derivative of the composite function is:
$$(f \circ g)'(x) = f'(g(x)) \cdot g'(x)$$
- Archimedean property:
Let $x$ and $y$ be real numbers. Then, there exists a natural number $n$ such that:
$$n > \frac{x - y}{1}$$
# Integrals and their properties
- Definite integral: the area under a curve between two endpoints.
- Indefinite integral: the accumulated value of a function over an interval.
- Improper integral: an integral where one or both endpoints are infinite.
We will also discuss the concept of the integral of a function and its properties.
The integral of a function $f(x)$ is denoted by $\int f(x) dx$ and is defined as the accumulated value of the function over an interval.
Integrals also have some interesting properties, such as the fundamental theorem of calculus.
The fundamental theorem of calculus states that the derivative of the integral of a function is equal to the function itself, and the integral of the derivative of a function is equal to the function evaluated at the upper endpoint minus the function evaluated at the lower endpoint.
## Exercise
Instructions:
- Prove the fundamental theorem of calculus.
- Show that the Archimedean property holds for real numbers.
### Solution
- Fundamental theorem of calculus:
Let $f(x)$ be a function and $F(x)$ be its integral. Then, the derivative of $F(x)$ is equal to $f(x)$:
$$F'(x) = f(x)$$
$$\int_a^b f(x) dx = F(b) - F(a)$$
- Archimedean property:
Let $x$ and $y$ be real numbers. Then, there exists a natural number $n$ such that:
$$n > \frac{x - y}{1}$$
# Sequences and series
- Arithmetic sequences: sequences where the difference between consecutive terms is constant.
- Geometric sequences: sequences where the ratio between consecutive terms is constant.
- Convergent sequences: sequences that converge to a limit as the index approaches infinity.
- Divergent sequences: sequences that do not converge to a limit as the index approaches infinity.
- Convergent series: series that converge to a sum as the number of terms approaches infinity.
- Divergent series: series that do not converge to a sum as the number of terms approaches infinity.
We will also discuss the concept of the limit of a sequence or series and its properties.
The limit of a sequence or series is the value that the sequence or series approaches as the index approaches infinity.
Sequences and series also have some interesting properties, such as the Cauchy product and the Cauchy convergence test.
The Cauchy product of two sequences is the product of their corresponding terms.
$$(a_n \cdot b_n) = \sum_{i = 0}^{n - 1} a_i \cdot b_{n - i - 1}$$
The Cauchy convergence test states that a series converges if and only if the sequence of its terms converges to zero.
$$\sum_{n = 1}^{\infty} a_n \text{ converges} \iff \lim_{n \to \infty} a_n = 0$$
## Exercise
Instructions:
- Prove the Cauchy product for sequences.
- Prove the Cauchy convergence test for series.
- Show that the Archimedean property holds for real numbers.
### Solution
- Cauchy product:
Let $a_n$ and $b_n$ be two sequences. Then, their Cauchy product is:
$$(a_n \cdot b_n) = \sum_{i = 0}^{n - 1} a_i \cdot b_{n - i - 1}$$
- Cauchy convergence test:
Let $\sum_{n = 1}^{\infty} a_n$ be a series. Then, the series converges if and only if:
$$\lim_{n \to \infty} a_n = 0$$
- Archimedean property:
Let $x$ and $y$ be real numbers. Then, there exists a natural number $n$ such that:
$$n > \frac{x - y}{1}$$
# Applications of mathematical analysis in Python
- One-to-one functions: functions where each input corresponds to a unique output.
- Injective functions: functions where each input corresponds to a unique output.
- Surjective functions: functions where each output corresponds to at least one input.
- Bijection functions: functions that are both injective and surjective.
- Inverse functions: functions that are the inverse of each other.
We will also discuss the concept of function composition and its properties.
Function composition is the process of applying one function to the result of another function. It is denoted by $(f \circ g)(x)$.
Functions also have some interesting properties, such as the additivity property of functions.
Real numbers also have some interesting properties, such as the Archimedean property, which states that for any two real numbers, there exists a natural number greater than the difference between them.
## Exercise
Instructions:
- Prove the additivity property of functions.
- Show that the Archimedean property holds for real numbers.
### Solution
- Additivity property of functions:
Let $f(x)$ and $g(x)$ be two functions. Then, their sum is a function:
$$(f + g)(x) = f(x) + g(x)$$
- Archimedean property:
Let $x$ and $y$ be real numbers. Then, there exists a natural number $n$ such that:
$$n > \frac{x - y}{1}$$
# Solving problems using mathematical analysis and Python
- Numerical methods: techniques for approximating the solution to a problem, such as the bisection method, Newton's method, and the trapezoidal rule.
- Linear programming: techniques for solving optimization problems with linear constraints, such as the simplex method and the interior-point method.
- Machine learning: techniques for training and evaluating machine learning models, such as linear regression, logistic regression, and support vector machines.
We will also discuss the concept of the derivative of a function and its properties.
The derivative of a function $f(x)$ is denoted by $f'(x)$ and is defined as:
$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$
Derivatives also have some interesting properties, such as the product rule and the chain rule.
The product rule states that the derivative of the product of two functions is the sum of the product of their derivatives.
$$(f \cdot g)'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)$$
The chain rule states that the derivative of a composite function is the product of the derivative of the outer function and the derivative of the inner function.
$$(f \circ g)'(x) = f'(g(x)) \cdot g'(x)$$
## Exercise
Instructions:
- Prove the product rule for derivatives.
- Prove the chain rule for derivatives.
- Show that the Archimedean property holds for real numbers.
### Solution
- Product rule:
Let $f(x)$ and $g(x)$ be two functions. Then, the derivative of their product is:
$$(f \cdot g)'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)$$
- Chain rule:
Let $f(x)$ and $g(x)$ be two functions. Then, the derivative of the composite function is:
$$(f \circ g)'(x) = f'(g(x)) \cdot g'(x)$$
- Archimedean property:
Let $x$ and $y$ be real numbers. Then, there exists a natural number $n$ such that:
$$n > \frac{x - y}{1}$$
# Exercises and practice problems
- Numerical methods: solve problems using numerical methods, such as finding the roots of a polynomial equation, approximating the integral of a function, and solving linear programming problems.
- Linear programming: solve optimization problems with linear constraints, such as finding the shortest path between two points in a graph or the minimum cost of a transportation problem.
- Machine learning: train and evaluate machine learning models, such as predicting house prices based on features, classifying emails as spam or not spam, or recommending products to users.
We will also discuss the concept of the derivative of a function and its properties.
The derivative of a function $f(x)$ is denoted by $f'(x)$ and is defined as:
$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$
Derivatives also have some interesting properties, such as the product rule and the chain rule.
The product rule states that the derivative of the product of two functions is the sum of the product of their derivatives.
$$(f \cdot g)'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)$$
The chain rule states that the derivative of a composite function is the product of the derivative of the outer function and the derivative of the inner function.
$$(f \circ g)'(x) = f'(g(x)) \cdot g'(x)$$
## Exercise
Instructions:
- Prove the product rule for derivatives.
- Prove the chain rule for derivatives.
- Show that the Archimedean property holds for real numbers.
### Solution
- Product rule:
Let $f(x)$ and $g(x)$ be two functions. Then, the derivative of their product is:
$$(f \cdot g)'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)$$
- Chain rule:
Let $f(x)$ and $g(x)$ be two functions. Then, the derivative of the composite function is:
$$(f \circ g)'(x) = f'(g(x)) \cdot g'(x)$$
- Archimedean property:
Let $x$ and $y$ be real numbers. Then, there exists a natural number $n$ such that:
$$n > \frac{x - y}{1}$$
# Review and summary
- Real numbers and their properties: understanding the properties of real numbers, such as their classification, absolute value, and additivity.
- Functions and their properties: understanding the properties of functions, such as their classification, composition, and additivity.
- Limits and continuity: understanding the properties of limits and continuity, such as the concept of a limit, one-sided limits, and the intermediate value property.
- Derivatives and their properties: understanding the properties of derivatives, such as the concept of a derivative, product rule, and chain rule.
- Integrals and their properties: understanding the properties of integrals, such as the concept of an integral, definite integrals, and the fundamental theorem of calculus.
- Sequences and series: understanding the properties of sequences and series, such as the concept of a sequence, arithmetic and geometric sequences, and the Cauchy product and convergence test.
We will also discuss the concept of the derivative of a function and its properties.
The derivative of a function $f(x)$ is denoted by $f'(x)$ and is defined as:
$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$
Derivatives also have some interesting properties, such as the product rule and the chain rule.
The product rule states that the derivative of the product of two functions is the sum of the product of their derivatives.
$$(f \cdot g)'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)$$
The chain rule states that the derivative of a composite function is the product of the derivative of the outer function and the derivative of the inner function.
$$(f \circ g)'(x) = f'(g(x)) \cdot g'(x)$$
## Exercise
Instructions:
- Prove the product rule for derivatives.
- Prove the chain rule for derivatives.
- Show that the Archimedean property holds for real numbers.
### Solution
- Product rule:
Let $f(x)$ and $g(x)$ be two functions. Then, the derivative of their product is:
$$(f \cdot g)'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)$$
- Chain rule:
Let $f(x)$ and $g(x)$ be two functions. Then, the derivative of the composite function is:
$$(f \circ g)'(x) = f'(g(x)) \cdot g'(x)$$
- Archimedean property:
Let $x$ and $y$ be real numbers. Then, there exists a natural number $n$ such that:
$$n > \frac{x - y}{1}$$ | Textbooks |
Welcome to ShortScience.org!
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Reading summaries of papers is useful to obtain the perspective and insight of another reader, why they liked or disliked it, and their attempt to demystify complicated sections.
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Popular (Today)
Recurrent Batch Normalization
Cooijmans, Tim and Ballas, Nicolas and Laurent, César and Courville, Aaron
arXiv e-Print archive - 2016 via Local Bibsonomy
Keywords: dblp
[link] Summary by Hugo Larochelle 3 years ago
This paper describes how to apply the idea of batch normalization (BN) successfully to recurrent neural networks, specifically to LSTM networks. The technique involves the 3 following ideas:
**1) Careful initialization of the BN scaling parameter.** While standard practice is to initialize it to 1 (to have unit variance), they show that this situation creates problems with the gradient flow through time, which vanishes quickly. A value around 0.1 (used in the experiments) preserves gradient flow much better.
**2) Separate BN for the "hiddens to hiddens pre-activation and for the "inputs to hiddens" pre-activation.** In other words, 2 separate BN operators are applied on each contributions to the pre-activation, before summing and passing through the tanh and sigmoid non-linearities.
**3) Use of largest time-step BN statistics for longer test-time sequences.** Indeed, one issue with applying BN to RNNs is that if the input sequences have varying length, and if one uses per-time-step mean/variance statistics in the BN transformation (which is the natural thing to do), it hasn't been clear how do deal with the last time steps of longer sequences seen at test time, for which BN has no statistics from the training set. The paper shows evidence that the pre-activation statistics tend to gradually converge to stationary values over time steps, which supports the idea of simply using the training set's last time step statistics.
Among these ideas, I believe the most impactful idea is 1). The papers mentions towards the end that improper initialization of the BN scaling parameter probably explains previous failed attempts to apply BN to recurrent networks.
Experiments on 4 datasets confirms the method's success.
**My two cents**
This is an excellent development for LSTMs. BN has had an important impact on our success in training deep neural networks, and this approach might very well have a similar impact on the success of LSTMs in practice.
Spatial Pyramid Pooling in Deep Convolutional Networks for Visual Recognition
Kaiming He and Xiangyu Zhang and Shaoqing Ren and Jian Sun
Keywords: cs.CV
Abstract: Existing deep convolutional neural networks (CNNs) require a fixed-size (e.g., 224x224) input image. This requirement is "artificial" and may reduce the recognition accuracy for the images or sub-images of an arbitrary size/scale. In this work, we equip the networks with another pooling strategy, "spatial pyramid pooling", to eliminate the above requirement. The new network structure, called SPP-net, can generate a fixed-length representation regardless of image size/scale. Pyramid pooling is also robust to object deformations. With these advantages, SPP-net should in general improve all CNN-based image classification methods. On the ImageNet 2012 dataset, we demonstrate that SPP-net boosts the accuracy of a variety of CNN architectures despite their different designs. On the Pascal VOC 2007 and Caltech101 datasets, SPP-net achieves state-of-the-art classification results using a single full-image representation and no fine-tuning. The power of SPP-net is also significant in object detection. Using SPP-net, we compute the feature maps from the entire image only once, and then pool features in arbitrary regions (sub-images) to generate fixed-length representations for training the detectors. This method avoids repeatedly computing the convolutional features. In processing test images, our method is 24-102x faster than the R-CNN method, while achieving better or comparable accuracy on Pascal VOC 2007. In ImageNet Large Scale Visual Recognition Challenge (ILSVRC) 2014, our methods rank #2 in object detection and #3 in image classification among all 38 teams. This manuscript also introduces the improvement made for this competition.
[link] Summary by Martin Thoma 2 years ago
Spatial Pyramid Pooling (SPP) is a technique which allows Convolutional Neural Networks (CNNs) to use input images of any size, not only $224\text{px} \times 224\text{px}$ as most architectures do. (However, there is a lower bound for the size of the input image).
## Idea
* Convolutional layers operate on any size, but fully connected layers need fixed-size inputs
* Solution:
* Add a new SPP layer on top of the last convolutional layer, before the fully connected layer
* Use an approach similar to bag of words (BoW), but maintain the spatial information. The BoW approach is used for text classification, where the order of the words is discarded and only the number of occurences is kept.
* The SPP layer operates on each feature map independently.
* The output of the SPP layer is of dimension $k \cdot M$, where $k$ is the number of feature maps the SPP layer got as input and $M$ is the number of bins.
Example: We could use spatial pyramid pooling with 21 bins:
* 1 bin which is the max of the complete feature map
* 4 bins which divide the image into 4 regions of equal size (depending on the input size) and rectangular shape. Each bin gets the max of its region.
* 16 bins which divide the image into 4 regions of equal size (depending on the input size) and rectangular shape. Each bin gets the max of its region.
## Evaluation
* Pascal VOC 2007, Caltech101: state-of-the-art, without finetuning
* ImageNet 2012: Boosts accuracy for various CNN architectures
* ImageNet Large Scale Visual Recognition Challenge (ILSVRC) 2014: Rank #2
## Code
The paper claims that the code is [here](http://research.microsoft.com/en-us/um/people/kahe/), but this seems not to be the case any more.
People have tried to implement it with Tensorflow ([1](http://stackoverflow.com/q/40913794/562769), [2](https://github.com/fchollet/keras/issues/2080), [3](https://github.com/tensorflow/tensorflow/issues/6011)), but by now no public working implementation is available.
## Related papers
* [Atrous Convolution](https://arxiv.org/abs/1606.00915)
One-shot Learning with Memory-Augmented Neural Networks
Adam Santoro and Sergey Bartunov and Matthew Botvinick and Daan Wierstra and Timothy Lillicrap
Keywords: cs.LG
Abstract: Despite recent breakthroughs in the applications of deep neural networks, one setting that presents a persistent challenge is that of "one-shot learning." Traditional gradient-based networks require a lot of data to learn, often through extensive iterative training. When new data is encountered, the models must inefficiently relearn their parameters to adequately incorporate the new information without catastrophic interference. Architectures with augmented memory capacities, such as Neural Turing Machines (NTMs), offer the ability to quickly encode and retrieve new information, and hence can potentially obviate the downsides of conventional models. Here, we demonstrate the ability of a memory-augmented neural network to rapidly assimilate new data, and leverage this data to make accurate predictions after only a few samples. We also introduce a new method for accessing an external memory that focuses on memory content, unlike previous methods that additionally use memory location-based focusing mechanisms.
This paper proposes a variant of Neural Turing Machine (NTM) for meta-learning or "learning to learn", in the specific context of few-shot learning (i.e. learning from few examples). Specifically, the proposed model is trained to ingest as input a training set of examples and improve its output predictions as examples are processed, in a purely feed-forward way. This is a form of meta-learning because the model is trained so that its forward pass effectively executes a form of "learning" from the examples it is fed as input.
During training, the model is fed multiples sequences (referred to as episodes) of labeled examples $({\bf x}_1, {\rm null}), ({\bf x}_2, y_1), \dots, ({\bf x}_T, y_{T-1})$, where $T$ is the size of the episode. For instance, if the model is trained to learn how to do 5-class classification from 10 examples per class, $T$ would be $5 \times 10 = 50$. Mainly, the paper presents experiments on the Omniglot dataset, which has 1623 classes. In these experiments, classes are separated into 1200 "training classes" and 423 "test classes", and each episode is generated by randomly selecting 5 classes (each assigned some arbitrary vector representation, e.g. a one-hot vector that is consistent within the episode, but not across episodes) and constructing a randomly ordered sequence of 50 examples from within the chosen 5 classes. Moreover, the correct label $y_t$ of a given input ${\bf x}_t$ is always provided only at the next time step, but the model is trained to be good at its prediction of the label of ${\bf x}_t$ at the current time step. This is akin to the scenario of online learning on a stream of examples, where the label of an example is revealed only once the model has made a prediction.
The proposed NTM is different from the original NTM of Alex Graves, mostly in how it writes into its memory. The authors propose to focus writing to either the least recently used memory location or the most recently used memory location. Moreover, the least recently used memory location is reset to zero before every write (an operation that seems to be ignored when backpropagating gradients).
Intuitively, the proposed NTM should learn a strategy by which, given a new input, it looks into its memory for information from other examples earlier in the episode (perhaps similarly to what a nearest neighbor classifier would do) to predict the class of the new input.
The paper presents experiments in learning to do multiclass classification on the Omniglot dataset and regression based on functions synthetically generated by a GP. The highlights are that:
1. The proposed model performs much better than an LSTM and better than an NTM with the original write mechanism of Alex Graves (for classification).
2. The proposed model even performs better than a 1st nearest neighbor classifier.
3. The proposed model is even shown to outperform human performance, for the 5-class scenario.
4. The proposed model has decent performance on the regression task, compared to GP predictions using the groundtruth kernel.
This is probably one of my favorite ICML 2016 papers. I really think meta-learning is a problem that deserves more attention, and this paper presents both an interesting proposal for how to do it and an interesting empirical investigation of it.
Much like previous work [\[1\]][1] [\[2\]][2], learning is based on automatically generating a meta-learning training set. This is clever I think, since a very large number of such "meta-learning" examples (the episodes) can be constructed, thus transforming what is normally a "small data problem" (few shot learning) into a "big data problem", for which deep learning is more effective.
I'm particularly impressed by how the proposed model outperforms a 1-nearest neighbor classifier. That said, the proposed NTM actually performs 4 reads at each time step, which suggests that a fairer comparison might be with a 4-nearest neighbor classifier. I do wonder how this baseline would compare.
I'm also impressed with the observation that the proposed model surpassed humans.
The paper also proposes to use 5-letter words to describe classes, instead of one-hot vectors. The motivation is that this should make it easier for the model to scale to much more than 5 classes. However, I don't entirely follow the logic as to why one-hot vectors are problematic. In fact, I would think that arbitrarily assigning 5-letter words to classes would instead imply some similarity between classes that share letters that is arbitrary and doesn't reflect true class similarity.
Also, while I find it encouraging that the performance for regression of the proposed model is decent, I'm curious about how it would compare with a GP approach that incrementally learns the kernel's hyper-parameter (instead of using the groundtruth values, which makes this baseline unrealistically strong).
Finally, I'm still not 100% sure how exactly the NTM is able to implement the type of feed-forward inference I'd expect to be required. I would expect it to learn a memory representation of examples that combines information from the input vector ${\bf x}_t$ *and* its label $y_t$. However, since the label of an input is presented at the following time step in an episode, it is not intuitive to me then how the read/write mechanisms are able to deal with this misalignment. My only guess is that since the controller is an LSTM, then it can somehow remember ${\bf x}_t$ until it gets $y_t$ and appropriately include the combined information into the memory. This could be supported by the fact that using a non-recurrent feed-forward controller is much worse than using an LSTM controller. But I'm not 100% sure of this either.
All the above being said, this is still a really great paper, which I hope will help stimulate more research on meta-learning. Hopefully code for this paper can eventually be released, which would help in popularizing the topic.
[1]: http://snowedin.net/tmp/Hochreiter2001.pdf
[2]: http://www.thespermwhale.com/jaseweston/ram/papers/paper_16.pdf
BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding
Jacob Devlin and Ming-Wei Chang and Kenton Lee and Kristina Toutanova
Keywords: cs.CL
First published: 2018/10/11 (9 months ago)
Abstract: We introduce a new language representation model called BERT, which stands for Bidirectional Encoder Representations from Transformers. Unlike recent language representation models, BERT is designed to pre-train deep bidirectional representations by jointly conditioning on both left and right context in all layers. As a result, the pre-trained BERT representations can be fine-tuned with just one additional output layer to create state-of-the-art models for a wide range of tasks, such as question answering and language inference, without substantial task-specific architecture modifications. BERT is conceptually simple and empirically powerful. It obtains new state-of-the-art results on eleven natural language processing tasks, including pushing the GLUE benchmark to 80.4% (7.6% absolute improvement), MultiNLI accuracy to 86.7 (5.6% absolute improvement) and the SQuAD v1.1 question answering Test F1 to 93.2 (1.5% absolute improvement), outperforming human performance by 2.0%.
[link] Summary by CodyWild 8 months ago
The last two years have seen a number of improvements in the field of language model pretraining, and BERT - Bidirectional Encoder Representations from Transformers - is the most recent entry into this canon. The general problem posed by language model pretraining is: can we leverage huge amounts of raw text, which aren't labeled for any specific classification task, to help us train better models for supervised language tasks (like translation, question answering, logical entailment, etc)? Mechanically, this works by either 1) training word embeddings and then using those embeddings as input feature representations for supervised models, or 2) treating the problem as a transfer learning problem, and fine-tune to a supervised task - similar to how you'd fine-tune a model trained on ImageNet by carrying over parameters, and then training on your new task. Even though the text we're learning on is strictly speaking unsupervised (lacking a supervised label), we need to design a task on which we calculate gradients in order to train our representations. For unsupervised language modeling, that task is typically structured as predicting a word in a sequence given prior words in that sequence. Intuitively, in order for a model to do a good job at predicting the word that comes next in a sentence, it needs to have learned patterns about language, both on grammatical and semantic levels. A notable change recently has been the shift from learning unconditional word vectors (where the word's representation is the same globally) to contextualized ones, where the representation of the word is dependent on the sentence context it's found in. All the baselines discussed here are of this second type.
The two main baselines that the BERT model compares itself to are OpenAI's GPT, and Peters et al's ELMo. The GPT model uses a self-attention-based Transformer architecture, going through each word in the sequence, and predicting the next word by calculating an attention-weighted representation of all prior words. (For those who aren't familiar, attention works by multiplying a "query" vector with every word in a variable-length sequence, and then putting the outputs of those multiplications into a softmax operator, which inherently gets you a weighting scheme that adds to one). ELMo uses models that gather context in both directions, but in a fairly simple way: it learns one deep LSTM that goes from left to right, predicting word k using words 0-k-1, and a second LSTM that goes from right to left, predicting word k using words k+1 onward. These two predictions are combined (literally: just summed together) to get a representation for the word at position k.
https://i.imgur.com/2329e3L.png
BERT differs from prior work in this area in several small ways, but one primary one: instead of representing a word using only information from words before it, or a simple sum of prior information and subsequent information, it uses the full context from before and after the word in each of its multiple layers. It also uses an attention-based Transformer structure, but instead of incorporating just prior context, it pulls in information from the full sentence. To allow for a model that actually uses both directions of context at a time in its unsupervised prediction task, the authors of BERT slightly changed the nature of that task: it replaces the word being predicted with the "mask" token, so that even with multiple layers of context aggregation on both sides, the model doesn't have any way of knowing what the token is. By contrast, if it weren't masked, after the first layer of context aggregation, the representations of other words in the sequence would incorporate information about the predicted word k, making it trivial, if another layer were applied on top of that first one, for the model to directly have access to the value it's trying to predict. This problem can either be solved by using multiple layers, each of which can only see prior context (like GPT), by learning fully separate L-R and R-L models, and combining them at the final layer (like ELMo) or by masking tokens, and predicting the value of the masked tokens using the full remainder of the context.
This task design crucially allows for a multi-layered bidirectional architecture, and consequently a much richer representation of context in each word's pre-trained representation. BERT demonstrates dramatic improvements over prior work when fine tuned on a small amount of supervised data, suggesting that this change added substantial value.
Feature Pyramid Networks for Object Detection
Lin, Tsung-Yi and Dollár, Piotr and Girshick, Ross B. and He, Kaiming and Hariharan, Bharath and Belongie, Serge J.
[link] Summary by Qure.ai 2 years ago
Feature Pyramid Networks (FPNs) build on top of the state-of-the-art implementation for object detection net - Faster RCNN. Faster RCNN faces a major problem in training for scale-invariance as the computations can be memory-intensive and extremely slow. So FRCNN only applies multi-scale approach while testing.
On the other hand, feature pyramids were mainstream when hand-generated features were used -primarily to counter scale-invariance. Feature pyramids are collections of features computed at multi-scale versions of the same image. Improving on a similar idea presented in *DeepMask*, FPN brings back feature pyramids using different feature maps of conv layers with differing spatial resolutions with predictiosn happening on all levels of pyramid. Using feature maps directly as it is, would be tough as initial layers tend to contain lower level representations and poor semantics but good localisation whereas deeper layers tend to constitute higher level representations with rich semantics but suffer poor localisation due to multiple subsampling.
##### Methodology
FPN can be used with any normal conv architecture, that's used for classification. In such an architecture all layers have progressively decreasing spatial resolutions (say C1, C2,..C5). FPN would now take C5 and convolve with 1x1 kernel to reduce filters to give P5. Next, P5 is upsampled and merged it to C4 (C4 is convolved with 1x1 kernel to decrease filter size in order to match that of upsampled P5) by adding element wise to produce P4. Similarly P4 is upsampled and merged with C3(in a similar way) to give P3 and so on. The final set of feature maps, in this case {P2 .. P5} are used as feature pyramids.
This is how pyramids would look like
*Usage of combination of {P2,..P5} as compared to only P2* : P2 produces highest resolution, most semantic features and could as well be the default choice but because of shared weights across rest of feature layers and the learned scale invariance makes the pyramidal variant more robust to generating false ROIs
For next steps, it could be RPN or RCNN, the regression and classifier would share weights across for all *anchors* (of varying aspect ratios) at each level of the feature pyramids. This step is similar to [Single Shot Detector (SSD) Networks ](http://www.shortscience.org/paper?bibtexKey=conf/eccv/LiuAESRFB16)
##### Observation
The FPN was used in FRCNN in both parts of RPN and RCNN separately and then combined FPN in both parts and produced state-of-the-art result in MS COCO challenges bettering results of COCO '15 & '16 winner models ( Faster RCNN +++ & GMRI) for mAP. FPN also can be used for instance segmentation by using fully convolutional layers on top of the image pyramids. FPN outperforms results from *DeepMask*, *SharpMask*, *InstanceFCN*
Using Fast Weights to Attend to the Recent Past
Jimmy Ba and Geoffrey Hinton and Volodymyr Mnih and Joel Z. Leibo and Catalin Ionescu
Keywords: stat.ML, cs.LG, cs.NE
Abstract: Until recently, research on artificial neural networks was largely restricted to systems with only two types of variable: Neural activities that represent the current or recent input and weights that learn to capture regularities among inputs, outputs and payoffs. There is no good reason for this restriction. Synapses have dynamics at many different time-scales and this suggests that artificial neural networks might benefit from variables that change slower than activities but much faster than the standard weights. These "fast weights" can be used to store temporary memories of the recent past and they provide a neurally plausible way of implementing the type of attention to the past that has recently proved very helpful in sequence-to-sequence models. By using fast weights we can avoid the need to store copies of neural activity patterns.
This paper presents a recurrent neural network architecture in which some of the recurrent weights dynamically change during the forward pass, using a hebbian-like rule. They correspond to the matrices $A(t)$ in the figure below:
These weights $A(t)$ are referred to as *fast weights*. Comparatively, the recurrent weights $W$ are referred to as slow weights, since they are only changing due to normal training and are otherwise kept constant at test time.
More specifically, the proposed fast weights RNN compute a series of hidden states $h(t)$ over time steps $t$, but, unlike regular RNNs, the transition from $h(t)$ to $h(t+1)$ consists of multiple ($S$) recurrent layers $h_1(t+1), \dots, h_{S-1}(t+1), h_S(t+1)$, defined as follows:
$$h_{s+1}(t+1) = f(W h(t) + C x(t) + A(t) h_s(t+1))$$
where $f$ is an element-wise non-linearity such as the ReLU activation. The next hidden state $h(t+1)$ is simply defined as the last "inner loop" hidden state $h_S(t+1)$, before moving to the next time step.
As for the fast weights $A(t)$, they too change between time steps, using the hebbian-like rule:
$$A(t+1) = \lambda A(t) + \eta h(t) h(t)^T$$
where $\lambda$ acts as a decay rate (to partially forget some of what's in the past) and $\eta$ as the fast weight's "learning rate" (not to be confused with the learning rate used during backprop). Thus, the role played by the fast weights is to rapidly adjust to the recent hidden states and remember the recent past.
In fact, the authors show an explicit relation between these fast weights and memory-augmented architectures that have recently been popular. Indeed, by recursively applying and expending the equation for the fast weights, one obtains
$$A(t) = \eta \sum_{\tau = 1}^{\tau = t-1}\lambda^{t-\tau-1} h(\tau) h(\tau)^T$$
*(note the difference with Equation 3 of the paper... I think there was a typo)* which implies that when computing the $A(t) h_s(t+1)$ term in the expression to go from $h_s(t+1)$ to $h_{s+1}(t+1)$, this term actually corresponds to
$$A(t) h_s(t+1) = \eta \sum_{\tau =1}^{\tau = t-1} \lambda^{t-\tau-1} h(\tau) (h(\tau)^T h_s(t+1))$$
i.e. $A(t) h_s(t+1)$ is a weighted sum of all previous hidden states $h(\tau)$, with each hidden states weighted by an "attention weight" $h(\tau)^T h_s(t+1)$. The difference with many recent memory-augmented architectures is thus that the attention weights aren't computed using a softmax non-linearity.
Experimentally, they find it beneficial to use [layer normalization](https://arxiv.org/abs/1607.06450). Good values for $\eta$ and $\lambda$ seem to be 0.5 and 0.9 respectively. I'm not 100% sure, but I also understand that using $S=1$, i.e. using the fast weights only once per time steps, was usually found to be optimal. Also see Figure 3 for the architecture used on the image classification datasets, which is slightly more involved.
The authors present a series 4 experiments, comparing with regular RNNs (IRNNs, which are RNNs with ReLU units and whose recurrent weights are initialized to a scaled identity matrix) and LSTMs (as well as an associative LSTM for a synthetic associative retrieval task and ConvNets for the two image datasets). Generally, experiments illustrate that the fast weights RNN tends to train faster (in number of updates) and better than the other recurrent architectures. Surprisingly, the fast weights RNN can even be competitive with a ConvNet on the two image classification benchmarks, where the RNN traverses glimpses from the image using a fixed policy.
This is a very thought provoking paper which, based on the comparison with LSTMs, suggests that fast weights RNNs might be a very good alternative. I'd be quite curious to see what would happen if one was to replace LSTMs with them in the myriad of papers using LSTMs (e.g. all the Seq2Seq work). Intuitively, LSTMs seem to be able to do more than just attending to the recent past. But, for a given task, if one was to observe that fast weights RNNs are competitive to LSTMs, it would suggests that the LSTM isn't doing something that much more complex. So it would be interesting to determine what are the tasks where the extra capacity of an LSTM is actually valuable and exploitable. Hopefully the authors will release some code, to facilitate this exploration.
The discussion at the end of Section 3 on how exploiting the "memory augmented" view of fast weights is useful to allow the use of minibatches is interesting. However, it also suggests that computations in the fast weights RNN scales quadratically with the sequence size (since in this view, the RNN technically must attend to all previous hidden states, since the beginning of the sequence). This is something to keep in mind, if one was to consider applying this to very long sequences (i.e. much longer than the hidden state dimensionality).
Also, I don't quite get the argument that the "memory augmented" view of fast weights is more amenable to mini-batch training. I understand that having an explicit weight matrix $A(t)$ for each minibatch sequence complicates things. However, in the memory augmented view, we also have a "memory matrix" that is different for each sequence, and yet we can handle that fine. The problem I can imagine is that storing a *sequence of arbitrary weight matrices* for each sequence might be storage demanding (and thus perhaps make it impossible to store a forward/backward pass for more than one sequence at a time), while the implicit memory matrix only requires appending a new row at each time step. Perhaps the argument to be made here is more that there's already mini-batch compatible code out there for dealing with the use of a memory matrix of stored previous memory states.
This work strikes some (partial) resemblance to other recent work, which may serve as food for thought here. The use of possibly multiple computation layers between time steps reminds me of [Adaptive Computation Time (ACT) RNN]( http://www.shortscience.org/paper?bibtexKey=journals/corr/Graves16). Also, expressing a backpropable architecture that involves updates to weights (here, hebbian-like updates) reminds me of recent work that does backprop through the updates of a gradient descent procedure (for instance as in [this work]( http://www.shortscience.org/paper?bibtexKey=conf/icml/MaclaurinDA15)).
Finally, while I was familiar with the notion of fast weights from the work on [Using Fast Weights to Improve Persistent Contrastive Divergence](http://people.ee.duke.edu/~lcarin/FastGibbsMixing.pdf), I didn't realize that this concept dated as far back as the late 80s. So, for young researchers out there looking for inspiration for research ideas, this paper confirms that looking at the older neural network literature for inspiration is probably a very good strategy :-)
To sum up, this is really nice work, and I'm looking forward to the NIPS 2016 oral presentation of it!
dx.doi.org
sci-hub
Fast R-CNN
Girshick, Ross B.
International Conference on Computer Vision - 2015 via Local Bibsonomy
[link] Summary by Joseph Paul Cohen 3 years ago
This method is based on improving the speed of R-CNN \cite{conf/cvpr/GirshickDDM14}
1. Where R-CNN would have two different objective functions, Fast R-CNN combines localization and classification losses into a "multi-task loss" in order to speed up training.
2. It also uses a pooling method based on \cite{journals/pami/HeZR015} called the RoI pooling layer that scales the input so the images don't have to be scaled before being set an an input image to the CNN. "RoI max pooling works by dividing the $h \times w$ RoI window into an $H \times W$ grid of sub-windows of approximate size $h/H \times w/W$ and then max-pooling the values in each sub-window into the corresponding output grid cell."
3. Backprop is performed for the RoI pooling layer by taking the argmax of the incoming gradients that overlap the incoming values.
This method is further improved by the paper "Faster R-CNN" \cite{conf/nips/RenHGS15}
FaceNet: A Unified Embedding for Face Recognition and Clustering
Florian Schroff and Dmitry Kalenichenko and James Philbin
Abstract: Despite significant recent advances in the field of face recognition, implementing face verification and recognition efficiently at scale presents serious challenges to current approaches. In this paper we present a system, called FaceNet, that directly learns a mapping from face images to a compact Euclidean space where distances directly correspond to a measure of face similarity. Once this space has been produced, tasks such as face recognition, verification and clustering can be easily implemented using standard techniques with FaceNet embeddings as feature vectors. Our method uses a deep convolutional network trained to directly optimize the embedding itself, rather than an intermediate bottleneck layer as in previous deep learning approaches. To train, we use triplets of roughly aligned matching / non-matching face patches generated using a novel online triplet mining method. The benefit of our approach is much greater representational efficiency: we achieve state-of-the-art face recognition performance using only 128-bytes per face. On the widely used Labeled Faces in the Wild (LFW) dataset, our system achieves a new record accuracy of 99.63%. On YouTube Faces DB it achieves 95.12%. Our system cuts the error rate in comparison to the best published result by 30% on both datasets. We also introduce the concept of harmonic embeddings, and a harmonic triplet loss, which describe different versions of face embeddings (produced by different networks) that are compatible to each other and allow for direct comparison between each other.
FaceNet directly maps face images to $\mathbb{R}^{128}$ where distances directly correspond to a measure of face similarity. They use a triplet loss function. The triplet is (face of person A, other face of person A, face of person which is not A). Later, this is called (anchor, positive, negative).
The loss function is learned and inspired by LMNN. The idea is to minimize the distance between the two images of the same person and maximize the distance to the other persons image.
## LMNN
Large Margin Nearest Neighbor (LMNN) is learning a pseudo-metric
$$d(x, y) = (x -y) M (x -y)^T$$
where $M$ is a positive-definite matrix. The only difference between a pseudo-metric and a metric is that $d(x, y) = 0 \Leftrightarrow x = y$ does not hold.
## Curriculum Learning: Triplet selection
Show simple examples first, then increase the difficulty. This is done by selecting the triplets.
They use the triplets which are *hard*. For the positive example, this means the distance between the anchor and the positive example is high. For the negative example this means the distance between the anchor and the negative example is low.
They want to have
$$||f(x_i^a) - f(x_i^p)||_2^2 + \alpha < ||f(x_i^a) - f(x_i^n)||_2^2$$
where $\alpha$ is a margin and $x_i^a$ is the anchor, $x_i^p$ is the positive face example and $x_i^n$ is the negative example. They increase $\alpha$ over time. It is crucial that $f$ maps the images not in the complete $\mathbb{R}^{128}$, but on the unit sphere. Otherwise one could double $\alpha$ by simply making $f' = 2 \cdot f$.
## Tasks
* **Face verification**: Is this the same person?
* **Face recognition**: Who is this person?
## Datasets
* 99.63% accuracy on Labeled FAces in the Wild (LFW)
* 95.12% accuracy on YouTube Faces DB
## Network
Two models are evaluated: The [Zeiler & Fergus model](http://www.shortscience.org/paper?bibtexKey=journals/corr/ZeilerF13) and an architecture based on the [Inception model](http://www.shortscience.org/paper?bibtexKey=journals/corr/SzegedyLJSRAEVR14).
## See also
* [DeepFace](http://www.shortscience.org/paper?bibtexKey=conf/cvpr/TaigmanYRW14#martinthoma)
Deep Residual Learning for Image Recognition
He, Kaiming and Zhang, Xiangyu and Ren, Shaoqing and Sun, Jian
Deeper networks should never have a higher **training** error than smaller ones. In the worst case, the layers should "simply" learn identities. It seems as this is not so easy with conventional networks, as they get much worse with more layers. So the idea is to add identity functions which skip some layers. The network only has to learn the **residuals**.
* Learning the identity becomes learning 0 which is simpler
* Loss in information flow in the forward pass is not a problem anymore
* No vanishing / exploding gradient
* Identities don't have parameters to be learned
The learning rate starts at 0.1 and is divided by 10 when the error plateaus. Weight decay of 0.0001 ($10^{-4}$), momentum of 0.9. They use mini-batches of size 128.
* ImageNet ILSVRC 2015: 3.57% (ensemble)
* CIFAR-10: 6.43%
* MS COCO: 59.0% [email protected] (ensemble)
* PASCAL VOC 2007: 85.6% [email protected]
* [DenseNets](http://www.shortscience.org/paper?bibtexKey=journals/corr/1608.06993)
Building Machines That Learn and Think Like People
Brenden M. Lake and Tomer D. Ullman and Joshua B. Tenenbaum and Samuel J. Gershman
Keywords: cs.AI, cs.CV, cs.LG, cs.NE, stat.ML
Abstract: Recent progress in artificial intelligence (AI) has renewed interest in building systems that learn and think like people. Many advances have come from using deep neural networks trained end-to-end in tasks such as object recognition, video games, and board games, achieving performance that equals or even beats humans in some respects. Despite their biological inspiration and performance achievements, these systems differ from human intelligence in crucial ways. We review progress in cognitive science suggesting that truly human-like learning and thinking machines will have to reach beyond current engineering trends in both what they learn, and how they learn it. Specifically, we argue that these machines should (a) build causal models of the world that support explanation and understanding, rather than merely solving pattern recognition problems; (b) ground learning in intuitive theories of physics and psychology, to support and enrich the knowledge that is learned; and (c) harness compositionality and learning-to-learn to rapidly acquire and generalize knowledge to new tasks and situations. We suggest concrete challenges and promising routes towards these goals that can combine the strengths of recent neural network advances with more structured cognitive models.
[link] Summary by Abhishek Das 1 year ago
This paper performs a comparitive study of recent advances in deep learning with human-like learning from a cognitive science point of view. Since natural intelligence is still the best form of intelligence, the authors list a core set of ingredients required to build machines that reason like humans.
- Cognitive capabilities present from childhood in humans.
- Intuitive physics; for example, a sense of plausibility of object trajectories, affordances.
- Intuitive psychology; for example, goals and beliefs.
- Learning as rapid model-building (and not just pattern recognition).
- Based on compositionality and learning-to-learn.
- Humans learn by inferring a general schema to describe goals, object types and interactions. This enables learning from few examples.
- Humans also learn richer conceptual models.
- Indicator: variety of functions supported by these models: classification, prediction, explanation, communication, action, imagination and composition.
- Models should hence have strong inductive biases and domain knowledge built into them; structural sharing of concepts by compositional reuse of primitives.
- Use of both model-free and model-based learning.
- Model-free, fast selection of actions in simple associative learning and discriminative tasks.
- Model-based learning when a causal model has been built to plan future actions or maximize rewards.
- Selective attention, augmented working memory, and experience replay are low-level promising trends in deep learning inspired from cognitive psychology.
- Need for higher-level aforementioned ingredients. | CommonCrawl |
Frequency magnitude distribution of noise
By the Wiener-Khinchin theorem, we have a straightforward way to calculate the power spectral density for stationary noise. If we know the bandwidth of a system, we can further calculate the variance of the noise since it turns out that $v_{noise,\ RMS} = \sigma$ (standard deviation) for zero mean noise. Therefore, at each point in time we know the probabilistic distribution of noise as shown in this video:
However, can we determine anything about the distribution of noise magnitude at each point in frequency from a noise source with known power spectral density? I believe this could be related to power spectral density estimation?
This video says that power spectrum realizations are proportional to the square of the underlying PSD, though I am not sure if that is just an artifact of using periodograms. In any case, 'variance proportional to the square' is still not a precise probabilistic distribution.
noise power-spectral-density
abcabc
$\begingroup$ What do you mean by "noise magnitude" -- is it the noise voltage? Then, if $S_{XX}(f)$ is the PSD, then wouldn't the expected voltage at $f=f_0$ be $\sqrt{S_{XX}(f_0)}$ ? $\endgroup$ – MBaz Apr 9 '17 at 16:59
$\begingroup$ I mean the magnitude of the frequency response of noise, either its Fourier transform $\sqrt{S(f_0)}$ or power $S(f_0)$ at the frequency of interest $f_0$. It would make sense that the expected/average magnitude would be $\sqrt{S(f_0)}$, however I would like to know the entire probability distribution $Pr(\sqrt{S(f_0)})$. $\endgroup$ – abc Apr 9 '17 at 19:01
$\begingroup$ The ultimate goal is to relate this distribution to the signal power/magnitude at the frequency of interest in order to determine the error distribution of amplitude estimation. See dsp.stackexchange.com/questions/40064/… $\endgroup$ – abc Apr 9 '17 at 19:09
Let me try to elaborate a bit on your questions: First, the Wiener-Khintschin-Theorem relates the autocorrelation of a stationary process to its PSD, with the relation of the Fourier Transform, i.e. $$\mathcal{F}\{r_{nn}(\tau)\}=S_{nn}(f).$$ The autocorrelation of a signal describes the correlation between two noise samples at distance $\tau$. It is just the second moment of the (joint) distribution of both noise samples. Hence, it does not really tell you something about the actual distribution. However, mostly, the noise is assumed to be Gaussian distributed with zero mean. Then, the autocorrelation completely describes the noise process.
Furthermore, in case you have AWGN, then $r_{nn}(t)=N_0\delta(t)$, i.e. the noise samples are uncorrelated and the PSD is a constant with $S_{nn}(f)=N_0$.
Your question is: What is the distribution of the noise magnitude in the frequency domain? So, you want to know $Pr(S(f)|_{f_0})$, where $S(f)$ is a realization of the noise process.
To make this more accessible (at least to me), let me switch to a linear algebra setting (i.e. the finite discrete-time case). The results can be straight-forwardly taken over to the continuos-time case. Let $\vec{n}$ be a realization of the noise process with autocorrelation $R_{nn}=E[\vec{n}\vec{n}^H]$. Let $\mathbf{F}$ be the unitary DFT matrix. So, $\vec{N}=\mathbf{F}\vec{n}$ is one noise realization in the frequency domain. Assuming that $\vec{n}$ consists of Gaussian distributed elements with zero mean, $\vec{N}$ also consists of Gaussian elements (A linear combination of two gaussian random variables remains to be Gaussian). What is the autocorrelation of the frequency-domain noise? It's given by
$$\mathbf{R}_{NN}=\mathbf{F}R_{nn}\mathbf{F}^H.$$
On the diagonal of $R_{NN}$ there is the variance of the Gaussians in the frequency domain.
So, the distribution of noise in the frequency domain remains to be a Gaussian, with variance given by diagonal on the above expression. I believe the diagonal is actually the PSD that follows from the Wiener-Khintschin Theorem. So, the final answer is:
The noise in the frequency domain is Gaussian distributed, with zero mean and variance given by the PSD of the noise. (Assuming zero-mean Gaussian time-domain noise)
Maximilian MatthéMaximilian Matthé
$\begingroup$ Okay, great point: a linear combination of Gaussians is also Gaussian. $\endgroup$ – abc Apr 11 '17 at 14:24
$\begingroup$ Also, this would agree with the statement from the periodogram video that variance of the power spectrum realization at any frequency is proportional to the PSD (variance of noise) squared: $var(X^2) = 2\sigma^4$. $\endgroup$ – abc Apr 11 '17 at 14:35
In the video that OP linked, as well as here (pg. 22), it is derived that the variance of power is proportional to (and in the linked ref, equal to) the square of the mean power, i.e.
$$ var[ P(w)] = \bar P^2(w) $$
Distributions that follow the general relationship: $$var[ P(w)] =\alpha \bar P ^x(w)$$ are of the Tweedie family. In this case, the gamma distribution satisfies x=2.
A quick Python simulation with normally distributed noise where $$\mu=0, \sigma=6$$ corroborates this. The left plot is a histogram of variance computed over 4000 non-overlapping 1000-point time windows. Middle plot are the distributions of absolute Fourier amplitude, where each faint trace is the distribution at a single frequency (1000-point FFT). The right plot is the same, but for power (i.e. amplitude squared). The mean of those distributions (not shown) is indeed 0.036 (which is the signal variance over the number of FFT points, 6^2/1000). We see that the distributions are heavily skewed, and resemble the form of gamma distributions, but that's about as much as I can say.
Therefore, I'm venturing a guess that the noise distribution in the frequency domain follows the gamma distribution. I realize that this is without any formal derivation, and I would love to see a derivation of this as I am also interested in this problem.
Lastly, with reference to the above (Maximilian's) answer: please correct me if I'm misinterpreting your response, but the noise in the frequency domain cannot be Gaussian distributed with zero mean, right? Because the mean is defined as the mean power at that frequency, which is proportional to the variance of the signal in time domain. Sorry, I don't have enough points to comment directly below the response.
Edit: the Fourier coefficient themselves, i.e. real and imaginary components, are distributed normally with zero mean and standard deviation scaled to the total signal power, but not the signal magnitude.
rdgaordgao
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I'm developing a face recognizing application using the face_recognition Python library.
The faces are encoded as 128-dimension floating-point vectors. In addition to this, each named known person has a variance value, which is refined iteratively for each new shot of the face along with the mean vector. I took the refining formula from Wikipedia.
I'm getting some false positives with the recognized faces, which I presume is because the library was developed primarily for Western faces whereas my intended audience are primarily Southern-Eastern Asian. So my primary concern with my code, is about whether or not I had gotten the mathematics correct.
# previous face encoding and auxiliary info.
n = min(n, 28) # heuristically limited to 28.
sys.exit() # possibly selected wrong face.
Irrelevant note : I used struct.(un)pack to serialize in binary to save space, because the repr of the data is too big.
I can't say I understand exactly what your algorithm does. Even when looking at Wikipedia it is hard to see. So, why don't you just use the reference implementation from there? It has functions with nice names and everything (well, not quite PEP8 compatible, but that could be changed).
Alternatively, once you have vectors of numbers and want to perform fast calculations on them, you probably want to use numpy, which allows you to easily perform some operation on the whole vector, like taking the difference between two vectors, scaling a vector by a scalar, taking the sum, etc.
In addition to using numpy, I expanded some of the names so it is clearer what they are, I made the cutoff value a global constant (you should give it a meaningful name), added a custom exception instead of just dying (the pass after it was unneeded and unreachable BTW), which makes it more clear what happened, wrapped the calling code under a if __name__ == "__main__": guard to allow importing from this script without running it and followed Python's official style-guide, PEP8, by having operators surrounded by one space.
Instead of this norm function, you can also use scipy.linalg.norm, which might be even faster.
However, note that Wikipedia also mentions that this formula suffers from numerical instabilities, since you "repeatedly subtract a small number from a big number which scales with \$n\$".
This is a bit tricky to implement, since your \$x_i\$ are actually vectors of numbers, so you would need to find out how to properly reduce it in dimension (in your previous formula you used the norm for that).
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On the lowest two-sided cell in affine Weyl groups
Author: Jérémie Guilhot
Journal: Represent. Theory 12 (2008), 327-345
MSC (2000): Primary 20C08
Published electronically: October 9, 2008
Abstract: Bremke and Xi determined the lowest two-sided cell for affine Weyl groups with unequal parameters and showed that it consists of at most $|W_{0}|$ left cells where $W_{0}$ is the associated finite Weyl group. We prove that this bound is exact. Previously, this was known in the equal parameter case and when the parameters were coming from a graph automorphism. Our argument uniformly works for any choice of parameters.
Robert Bédard, The lowest two-sided cell for an affine Weyl group, Comm. Algebra 16 (1988), no. 6, 1113–1132. MR 939034, DOI https://doi.org/10.1080/00927878808823622
Kirsten Bremke, On generalized cells in affine Weyl groups, J. Algebra 191 (1997), no. 1, 149–173. MR 1444494, DOI https://doi.org/10.1006/jabr.1996.6889
Meinolf Geck, On the induction of Kazhdan-Lusztig cells, Bull. London Math. Soc. 35 (2003), no. 5, 608–614. MR 1989489, DOI https://doi.org/10.1112/S0024609303002236
Meinolf Geck and Götz Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series, vol. 21, The Clarendon Press, Oxford University Press, New York, 2000. MR 1778802
George Lusztig, Hecke algebras and Jantzen's generic decomposition patterns, Adv. in Math. 37 (1980), no. 2, 121–164. MR 591724, DOI https://doi.org/10.1016/0001-8708%2880%2990031-6
G. Lusztig, Left cells in Weyl groups, Lie group representations, I (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1024, Springer, Berlin, 1983, pp. 99–111. MR 727851, DOI https://doi.org/10.1007/BFb0071433
G. Lusztig, Hecke algebras with unequal parameters, CRM Monograph Series, vol. 18, American Mathematical Society, Providence, RI, 2003. MR 1974442
Jian Yi Shi, A two-sided cell in an affine Weyl group, J. London Math. Soc. (2) 36 (1987), no. 3, 407–420. MR 918633, DOI https://doi.org/10.1112/jlms/s2-36.3.407
Jian Yi Shi, A two-sided cell in an affine Weyl group. II, J. London Math. Soc. (2) 37 (1988), no. 2, 253–264. MR 928522, DOI https://doi.org/10.1112/jlms/s2-37.2.253
Nan Hua Xi, Representations of affine Hecke algebras, Lecture Notes in Mathematics, vol. 1587, Springer-Verlag, Berlin, 1994. MR 1320509
R. Bédard. The lowest two-sided cell for an affine Weyl group. Comm. Algebra $\mathbf {16}$, 1113-1132, 1988. MR 939034 (89d:20041)
K. Bremke. On generalized cells in affine Weyl groups. Journal of Algebra $\mathbf {191}$, 149-173, 1997. MR 1444494 (98c:20077)
M. Geck. On the induction of Kazhdan-Lusztig cells. Bull. London Math. Soc. $\mathbf {35}$, 608-614, 2003. MR 1989489 (2004d:20003)
M. Geck, G. Pfeiffer. Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Math. Soc. Monographs NS 21, Oxford University Press 2000. MR 1778802 (2002k:20017)
G. Lusztig. Hecke algebras and Jantzen's generic decomposition patterns. Advances in Mathematics $\mathbf {37}$, 121-164, 1980. MR 591724 (82b:20059)
G. Lusztig. Left cells in Weyl groups. in "Lie group representations", Lectures Notes in Math., Springer-Verlag, Vol. 1024, 99-111, 1983. MR 727851 (85f:20035)
G. Lusztig. Hecke algebras with unequal parameters. CRM Monographs Ser. $\mathbf {18}$, Amer. Math. Soc., Providence, RI, 2003. MR 1974442 (2004k:20011)
J.-Y. Shi. A two-sided cell in an affine Weyl group. J. London Math. Soc. (2) $\mathbf {36}$, 407-420, 1987. MR 918633 (88k:20073)
J.-Y. Shi. A two-sided cell in an affine Weyl group II. J. London Math. Soc. (2) $\mathbf {37}$, 253-264, 1988. MR 928522 (89a:20055)
N. Xi. Representations of affine Hecke algebras. Lectures Notes in Math., Springer-Verlag, Vol. 1587, 1994. MR 1320509 (96i:20058)
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Jérémie Guilhot
Affiliation: Department of Mathematical Sciences, King's College, Aberdeen University, Aberdeen AB24 3UE, Scotland, United Kingdom\indent Université de Lyon, Université Lyon 1, Institut Camille Jordan, CNRS UMR 5208, 43 Boulevard du 11 Novembre 1918, F-69622 Villeurbanne Cedex, France
Address at time of publication: School of Mathematics and Statistics F07, The University of Sydney, NSW 2006, Australia
Email: [email protected]
Received by editor(s): August 27, 2007 | CommonCrawl |
Viktor Glombik
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Show $\sum_{k = 0}^{\infty} \frac{z^{4k}}{(4k)!} - \frac{z^{4k + 2}}{(4k + 2)!} = \sum_{k = 0}^{\infty} (-1)^k \frac{z^{2k}}{(2k)!}$
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$U \subset \mathbb{R}^d$ open and $D \subset U$ open and dense $\implies \lambda(D) = \lambda(U)$ | CommonCrawl |
Multi-armed bandit
In probability theory and machine learning, the multi-armed bandit problem (sometimes called the K-[1] or N-armed bandit problem[2]) is a problem in which a fixed limited set of resources must be allocated between competing (alternative) choices in a way that maximizes their expected gain, when each choice's properties are only partially known at the time of allocation, and may become better understood as time passes or by allocating resources to the choice.[3][4] This is a classic reinforcement learning problem that exemplifies the exploration–exploitation tradeoff dilemma. The name comes from imagining a gambler at a row of slot machines (sometimes known as "one-armed bandits"), who has to decide which machines to play, how many times to play each machine and in which order to play them, and whether to continue with the current machine or try a different machine.[5] The multi-armed bandit problem also falls into the broad category of stochastic scheduling.
In the problem, each machine provides a random reward from a probability distribution specific to that machine, that is not known a-priori. The objective of the gambler is to maximize the sum of rewards earned through a sequence of lever pulls.[3][4] The crucial tradeoff the gambler faces at each trial is between "exploitation" of the machine that has the highest expected payoff and "exploration" to get more information about the expected payoffs of the other machines. The trade-off between exploration and exploitation is also faced in machine learning. In practice, multi-armed bandits have been used to model problems such as managing research projects in a large organization, like a science foundation or a pharmaceutical company.[3][4] In early versions of the problem, the gambler begins with no initial knowledge about the machines.
Herbert Robbins in 1952, realizing the importance of the problem, constructed convergent population selection strategies in "some aspects of the sequential design of experiments".[6] A theorem, the Gittins index, first published by John C. Gittins, gives an optimal policy for maximizing the expected discounted reward.[7]
Empirical motivation
The multi-armed bandit problem models an agent that simultaneously attempts to acquire new knowledge (called "exploration") and optimize their decisions based on existing knowledge (called "exploitation"). The agent attempts to balance these competing tasks in order to maximize their total value over the period of time considered. There are many practical applications of the bandit model, for example:
• clinical trials investigating the effects of different experimental treatments while minimizing patient losses,[3][4][8][9]
• adaptive routing efforts for minimizing delays in a network,
• financial portfolio design[10][11]
In these practical examples, the problem requires balancing reward maximization based on the knowledge already acquired with attempting new actions to further increase knowledge. This is known as the exploitation vs. exploration tradeoff in machine learning.
The model has also been used to control dynamic allocation of resources to different projects, answering the question of which project to work on, given uncertainty about the difficulty and payoff of each possibility.[12]
Originally considered by Allied scientists in World War II, it proved so intractable that, according to Peter Whittle, the problem was proposed to be dropped over Germany so that German scientists could also waste their time on it.[13]
The version of the problem now commonly analyzed was formulated by Herbert Robbins in 1952.
The multi-armed bandit model
The multi-armed bandit (short: bandit or MAB) can be seen as a set of real distributions $B=\{R_{1},\dots ,R_{K}\}$, each distribution being associated with the rewards delivered by one of the $K\in \mathbb {N} ^{+}$ levers. Let $\mu _{1},\dots ,\mu _{K}$ be the mean values associated with these reward distributions. The gambler iteratively plays one lever per round and observes the associated reward. The objective is to maximize the sum of the collected rewards. The horizon $H$ is the number of rounds that remain to be played. The bandit problem is formally equivalent to a one-state Markov decision process. The regret $\rho $ after $T$ rounds is defined as the expected difference between the reward sum associated with an optimal strategy and the sum of the collected rewards:
$\rho =T\mu ^{*}-\sum _{t=1}^{T}{\widehat {r}}_{t}$,
where $\mu ^{*}$ is the maximal reward mean, $\mu ^{*}=\max _{k}\{\mu _{k}\}$, and ${\widehat {r}}_{t}$ is the reward in round t.
A zero-regret strategy is a strategy whose average regret per round $\rho /T$ tends to zero with probability 1 when the number of played rounds tends to infinity.[14] Intuitively, zero-regret strategies are guaranteed to converge to a (not necessarily unique) optimal strategy if enough rounds are played.
Variations
A common formulation is the Binary multi-armed bandit or Bernoulli multi-armed bandit, which issues a reward of one with probability $p$, and otherwise a reward of zero.
Another formulation of the multi-armed bandit has each arm representing an independent Markov machine. Each time a particular arm is played, the state of that machine advances to a new one, chosen according to the Markov state evolution probabilities. There is a reward depending on the current state of the machine. In a generalization called the "restless bandit problem", the states of non-played arms can also evolve over time.[15] There has also been discussion of systems where the number of choices (about which arm to play) increases over time.[16]
Computer science researchers have studied multi-armed bandits under worst-case assumptions, obtaining algorithms to minimize regret in both finite and infinite (asymptotic) time horizons for both stochastic[1] and non-stochastic[17] arm payoffs.
Bandit strategies
A major breakthrough was the construction of optimal population selection strategies, or policies (that possess uniformly maximum convergence rate to the population with highest mean) in the work described below.
Optimal solutions
Further information: Gittins index
In the paper "Asymptotically efficient adaptive allocation rules", Lai and Robbins[18] (following papers of Robbins and his co-workers going back to Robbins in the year 1952) constructed convergent population selection policies that possess the fastest rate of convergence (to the population with highest mean) for the case that the population reward distributions are the one-parameter exponential family. Then, in Katehakis and Robbins[19] simplifications of the policy and the main proof were given for the case of normal populations with known variances. The next notable progress was obtained by Burnetas and Katehakis in the paper "Optimal adaptive policies for sequential allocation problems",[20] where index based policies with uniformly maximum convergence rate were constructed, under more general conditions that include the case in which the distributions of outcomes from each population depend on a vector of unknown parameters. Burnetas and Katehakis (1996) also provided an explicit solution for the important case in which the distributions of outcomes follow arbitrary (i.e., non-parametric) discrete, univariate distributions.
Later in "Optimal adaptive policies for Markov decision processes"[21] Burnetas and Katehakis studied the much larger model of Markov Decision Processes under partial information, where the transition law and/or the expected one period rewards may depend on unknown parameters. In this work, the authors constructed an explicit form for a class of adaptive policies with uniformly maximum convergence rate properties for the total expected finite horizon reward under sufficient assumptions of finite state-action spaces and irreducibility of the transition law. A main feature of these policies is that the choice of actions, at each state and time period, is based on indices that are inflations of the right-hand side of the estimated average reward optimality equations. These inflations have recently been called the optimistic approach in the work of Tewari and Bartlett,[22] Ortner[23] Filippi, Cappé, and Garivier,[24] and Honda and Takemura.[25]
For Bernoulli multi-armed bandits, Pilarski et al.[26] studied computation methods of deriving fully optimal solutions (not just asymptotically) using dynamic programming in the paper "Optimal Policy for Bernoulli Bandits: Computation and Algorithm Gauge."[26] Via indexing schemes, lookup tables, and other techniques, this work provided practically applicable optimal solutions for Bernoulli bandits provided that time horizons and numbers of arms did not become excessively large. Pilarski et al.[27] later extended this work in "Delayed Reward Bernoulli Bandits: Optimal Policy and Predictive Meta-Algorithm PARDI"[27] to create a method of determining the optimal policy for Bernoulli bandits when rewards may not be immediately revealed following a decision and may be delayed. This method relies upon calculating expected values of reward outcomes which have not yet been revealed and updating posterior probabilities when rewards are revealed.
When optimal solutions to multi-arm bandit tasks [28] are used to derive the value of animals' choices, the activity of neurons in the amygdala and ventral striatum encodes the values derived from these policies, and can be used to decode when the animals make exploratory versus exploitative choices. Moreover, optimal policies better predict animals' choice behavior than alternative strategies (described below). This suggests that the optimal solutions to multi-arm bandit problems are biologically plausible, despite being computationally demanding.[29]
Approximate solutions
Many strategies exist which provide an approximate solution to the bandit problem, and can be put into the four broad categories detailed below.
Semi-uniform strategies
Semi-uniform strategies were the earliest (and simplest) strategies discovered to approximately solve the bandit problem. All those strategies have in common a greedy behavior where the best lever (based on previous observations) is always pulled except when a (uniformly) random action is taken.
• Epsilon-greedy strategy:[30] The best lever is selected for a proportion $1-\epsilon $ of the trials, and a lever is selected at random (with uniform probability) for a proportion $\epsilon $. A typical parameter value might be $\epsilon =0.1$, but this can vary widely depending on circumstances and predilections.
• Epsilon-first strategy: A pure exploration phase is followed by a pure exploitation phase. For $N$ trials in total, the exploration phase occupies $\epsilon N$ trials and the exploitation phase $(1-\epsilon )N$ trials. During the exploration phase, a lever is randomly selected (with uniform probability); during the exploitation phase, the best lever is always selected.
• Epsilon-decreasing strategy: Similar to the epsilon-greedy strategy, except that the value of $\epsilon $ decreases as the experiment progresses, resulting in highly explorative behaviour at the start and highly exploitative behaviour at the finish.
• Adaptive epsilon-greedy strategy based on value differences (VDBE): Similar to the epsilon-decreasing strategy, except that epsilon is reduced on basis of the learning progress instead of manual tuning (Tokic, 2010).[31] High fluctuations in the value estimates lead to a high epsilon (high exploration, low exploitation); low fluctuations to a low epsilon (low exploration, high exploitation). Further improvements can be achieved by a softmax-weighted action selection in case of exploratory actions (Tokic & Palm, 2011).[32]
• Adaptive epsilon-greedy strategy based on Bayesian ensembles (Epsilon-BMC): An adaptive epsilon adaptation strategy for reinforcement learning similar to VBDE, with monotone convergence guarantees. In this framework, the epsilon parameter is viewed as the expectation of a posterior distribution weighting a greedy agent (that fully trusts the learned reward) and uniform learning agent (that distrusts the learned reward). This posterior is approximated using a suitable Beta distribution under the assumption of normality of observed rewards. In order to address the possible risk of decreasing epsilon too quickly, uncertainty in the variance of the learned reward is also modeled and updated using a normal-gamma model. (Gimelfarb et al., 2019).[33]
Probability matching strategies
Probability matching strategies reflect the idea that the number of pulls for a given lever should match its actual probability of being the optimal lever. Probability matching strategies are also known as Thompson sampling or Bayesian Bandits,[34][35] and are surprisingly easy to implement if you can sample from the posterior for the mean value of each alternative.
Probability matching strategies also admit solutions to so-called contextual bandit problems.[34]
Pricing strategies
Pricing strategies establish a price for each lever. For example, as illustrated with the POKER algorithm,[14] the price can be the sum of the expected reward plus an estimation of extra future rewards that will gain through the additional knowledge. The lever of highest price is always pulled.
Contextual bandit
A useful generalization of the multi-armed bandit is the contextual multi-armed bandit. At each iteration an agent still has to choose between arms, but they also see a d-dimensional feature vector, the context vector they can use together with the rewards of the arms played in the past to make the choice of the arm to play. Over time, the learner's aim is to collect enough information about how the context vectors and rewards relate to each other, so that it can predict the next best arm to play by looking at the feature vectors.[36]
Approximate solutions for contextual bandit
Many strategies exist that provide an approximate solution to the contextual bandit problem, and can be put into two broad categories detailed below.
Online linear bandits
• LinUCB (Upper Confidence Bound) algorithm: the authors assume a linear dependency between the expected reward of an action and its context and model the representation space using a set of linear predictors.[37][38]
• LinRel (Linear Associative Reinforcement Learning) algorithm: Similar to LinUCB, but utilizes Singular-value decomposition rather than Ridge regression to obtain an estimate of confidence.[39][40]
Online non-linear bandits
• UCBogram algorithm: The nonlinear reward functions are estimated using a piecewise constant estimator called a regressogram in nonparametric regression. Then, UCB is employed on each constant piece. Successive refinements of the partition of the context space are scheduled or chosen adaptively.[41][42][43]
• Generalized linear algorithms: The reward distribution follows a generalized linear model, an extension to linear bandits.[44][45][46][47]
• KernelUCB algorithm: a kernelized non-linear version of linearUCB, with efficient implementation and finite-time analysis.[48]
• Bandit Forest algorithm: a random forest is built and analyzed w.r.t the random forest built knowing the joint distribution of contexts and rewards.[49]
• Oracle-based algorithm: The algorithm reduces the contextual bandit problem into a series of supervised learning problem, and does not rely on typical realizability assumption on the reward function.[50]
Constrained contextual bandit
In practice, there is usually a cost associated with the resource consumed by each action and the total cost is limited by a budget in many applications such as crowdsourcing and clinical trials. Constrained contextual bandit (CCB) is such a model that considers both the time and budget constraints in a multi-armed bandit setting. A. Badanidiyuru et al.[51] first studied contextual bandits with budget constraints, also referred to as Resourceful Contextual Bandits, and show that a $O({\sqrt {T}})$ regret is achievable. However, their work focuses on a finite set of policies, and the algorithm is computationally inefficient.
A simple algorithm with logarithmic regret is proposed in:[52]
• UCB-ALP algorithm: The framework of UCB-ALP is shown in the right figure. UCB-ALP is a simple algorithm that combines the UCB method with an Adaptive Linear Programming (ALP) algorithm, and can be easily deployed in practical systems. It is the first work that show how to achieve logarithmic regret in constrained contextual bandits. Although[52] is devoted to a special case with single budget constraint and fixed cost, the results shed light on the design and analysis of algorithms for more general CCB problems.
Adversarial bandit
Another variant of the multi-armed bandit problem is called the adversarial bandit, first introduced by Auer and Cesa-Bianchi (1998). In this variant, at each iteration, an agent chooses an arm and an adversary simultaneously chooses the payoff structure for each arm. This is one of the strongest generalizations of the bandit problem[53] as it removes all assumptions of the distribution and a solution to the adversarial bandit problem is a generalized solution to the more specific bandit problems.
Example: Iterated prisoner's dilemma
An example often considered for adversarial bandits is the iterated prisoner's dilemma. In this example, each adversary has two arms to pull. They can either Deny or Confess. Standard stochastic bandit algorithms don't work very well with these iterations. For example, if the opponent cooperates in the first 100 rounds, defects for the next 200, then cooperate in the following 300, etc. then algorithms such as UCB won't be able to react very quickly to these changes. This is because after a certain point sub-optimal arms are rarely pulled to limit exploration and focus on exploitation. When the environment changes the algorithm is unable to adapt or may not even detect the change.
Exp3[54]
EXP3 is a popular algorithm for adversarial multiarmed bandits, suggested and analyzed in this setting by Auer et al. [2002b]. Recently there was an increased interest in the performance of this algorithm in the stochastic setting, due to its new applications to stochastic multi-armed bandits with side information [Seldin et al., 2011] and to multi-armed bandits in the mixed stochastic-adversarial setting [Bubeck and Slivkins, 2012]. The paper presented an empirical evaluation and improved analysis of the performance of the EXP3 algorithm in the stochastic setting, as well as a modification of the EXP3 algorithm capable of achieving “logarithmic” regret in stochastic environment
Algorithm
Parameters: Real $\gamma \in (0,1]$
Initialisation: $\omega _{i}(1)=1$ for $i=1,...,K$
For each t = 1, 2, ..., T
1. Set $p_{i}(t)=(1-\gamma ){\frac {\omega _{i}(t)}{\sum _{j=1}^{K}\omega _{j}(t)}}+{\frac {\gamma }{K}}$ $i=1,...,K$
2. Draw $i_{t}$ randomly according to the probabilities $p_{1}(t),...,p_{K}(t)$
3. Receive reward $x_{i_{t}}(t)\in [0,1]$
4. For $j=1,...,K$ set:
${\hat {x}}_{j}(t)={\begin{cases}x_{j}(t)/p_{j}(t)&{\text{if }}j=i_{t}\\0,&{\text{otherwise}}\end{cases}}$
$\omega _{j}(t+1)=\omega _{j}(t)\exp(\gamma {\hat {x}}_{j}(t)/K)$
Explanation
Exp3 chooses an arm at random with probability $(1-\gamma )$ it prefers arms with higher weights (exploit), it chooses with probability $\gamma $ to uniformly randomly explore. After receiving the rewards the weights are updated. The exponential growth significantly increases the weight of good arms.
Regret analysis
The (external) regret of the Exp3 algorithm is at most $O({\sqrt {KTlog(K)}})$
Algorithm
Parameters: Real $\eta $
Initialisation: $\forall i:R_{i}(1)=0$
For each t = 1,2,...,T
1. For each arm generate a random noise from an exponential distribution $\forall i:Z_{i}(t)\sim Exp(\eta )$
2. Pull arm $I(t)$: $I(t)=arg\max _{i}\{R_{i}(t)+Z_{i}(t)\}$
Add noise to each arm and pull the one with the highest value
3. Update value: $R_{I(t)}(t+1)=R_{I(t)}(t)+x_{I(t)}(t)$
The rest remains the same
Explanation
We follow the arm that we think has the best performance so far adding exponential noise to it to provide exploration.[55]
Exp3 vs FPL
Exp3FPL
Maintains weights for each arm to calculate pulling probabilityDoesn't need to know the pulling probability per arm
Has efficient theoretical guaranteesThe standard FPL does not have good theoretical guarantees
Might be computationally expensive (calculating the exponential terms)Computationally quite efficient
Infinite-armed bandit
In the original specification and in the above variants, the bandit problem is specified with a discrete and finite number of arms, often indicated by the variable $K$. In the infinite armed case, introduced by Agrawal (1995),[56] the "arms" are a continuous variable in $K$ dimensions.
Non-stationary bandit
This framework refers to the multi-armed bandit problem in a non-stationary setting (i.e., in presence of concept drift). In the non-stationary setting, it is assumed that the expected reward for an arm $k$ can change at every time step $t\in {\mathcal {T}}$: $\mu _{t-1}^{k}\neq \mu _{t}^{k}$. Thus, $\mu _{t}^{k}$ no longer represents the whole sequence of expected (stationary) rewards for arm $k$. Instead, $\mu ^{k}$ denotes the sequence of expected rewards for arm $k$, defined as $\mu ^{k}=\{\mu _{t}^{k}\}_{t=1}^{T}$.[57]
A dynamic oracle represents the optimal policy to be compared with other policies in the non-stationary setting. The dynamic oracle optimises the expected reward at each step $t\in {\mathcal {T}}$ by always selecting the best arm, with expected reward of $\mu _{t}^{*}$. Thus, the cumulative expected reward ${\mathcal {D}}(T)$ for the dynamic oracle at final time step $T$ is defined as:
${\mathcal {D}}(T)=\sum _{t=1}^{T}{\mu _{t}^{*}}$
Hence, the regret $\rho ^{\pi }(T)$ for policy $\pi $ is computed as the difference between ${\mathcal {D}}(T)$ and the cumulative expected reward at step $T$ for policy $\pi $:
$\rho ^{\pi }(T)=\sum _{t=1}^{T}{\mu _{t}^{*}}-\mathbb {E} _{\pi }^{\mu }\left[\sum _{t=1}^{T}{r_{t}}\right]={\mathcal {D}}(T)-\mathbb {E} _{\pi }^{\mu }\left[\sum _{t=1}^{T}{r_{t}}\right]$
Garivier and Moulines derive some of the first results with respect to bandit problems where the underlying model can change during play. A number of algorithms were presented to deal with this case, including Discounted UCB[58] and Sliding-Window UCB.[59] A similar approach based on Thompson Sampling algorithm is the f-Discounted-Sliding-Window Thompson Sampling (f-dsw TS)[60] proposed by Cavenaghi et al. The f-dsw TS algorithm exploits a discount factor on the reward history and an arm-related sliding window to contrast concept drift in non-stationary environments. Another work by Burtini et al. introduces a weighted least squares Thompson sampling approach (WLS-TS), which proves beneficial in both the known and unknown non-stationary cases.[61]
Other variants
Many variants of the problem have been proposed in recent years.
Dueling bandit
The dueling bandit variant was introduced by Yue et al. (2012)[62] to model the exploration-versus-exploitation tradeoff for relative feedback. In this variant the gambler is allowed to pull two levers at the same time, but they only get a binary feedback telling which lever provided the best reward. The difficulty of this problem stems from the fact that the gambler has no way of directly observing the reward of their actions. The earliest algorithms for this problem are InterleaveFiltering,[62] Beat-The-Mean.[63] The relative feedback of dueling bandits can also lead to voting paradoxes. A solution is to take the Condorcet winner as a reference.[64]
More recently, researchers have generalized algorithms from traditional MAB to dueling bandits: Relative Upper Confidence Bounds (RUCB),[65] Relative EXponential weighing (REX3),[66] Copeland Confidence Bounds (CCB),[67] Relative Minimum Empirical Divergence (RMED),[68] and Double Thompson Sampling (DTS).[69]
Collaborative bandit
Approaches using multiple bandits that cooperate sharing knowledge in order to better optimize their performance started in 2013 with "A Gang of Bandits",[70] an algorithm relying on a similarity graph between the different bandit problems to share knowledge. The need of a similarity graph was removed in 2014 by the work on the CLUB algorithm.[71] Following this work, several other researchers created algorithms to learn multiple models at the same time under bandit feedback. For example, COFIBA was introduced by Li and Karatzoglou and Gentile (SIGIR 2016),[72] where the classical collaborative filtering, and content-based filtering methods try to learn a static recommendation model given training data.
Combinatorial bandit
The Combinatorial Multiarmed Bandit (CMAB) problem[73][74][75] arises when instead of a single discrete variable to choose from, an agent needs to choose values for a set of variables. Assuming each variable is discrete, the number of possible choices per iteration is exponential in the number of variables. Several CMAB settings have been studied in the literature, from settings where the variables are binary[74] to more general setting where each variable can take an arbitrary set of values.[75]
See also
• Gittins index – a powerful, general strategy for analyzing bandit problems.
• Greedy algorithm
• Optimal stopping
• Search theory
• Stochastic scheduling
References
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46. Kwang-Sung Jun; Aniruddha Bhargava; Robert D. Nowak; Rebecca Willett (2017), "Scalable generalized linear bandits: Online computation and hashing", Advances in Neural Information Processing Systems, Curran Associates, 30: 99–109, arXiv:1706.00136, Bibcode:2017arXiv170600136J
47. Branislav Kveton; Manzil Zaheer; Csaba Szepesvári; Lihong Li; Mohammad Ghavamzadeh; Craig Boutilier (2020), "Randomized exploration in generalized linear bandits", Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS), arXiv:1906.08947, Bibcode:2019arXiv190608947K
48. Michal Valko; Nathan Korda; Rémi Munos; Ilias Flaounas; Nello Cristianini (2013), Finite-Time Analysis of Kernelised Contextual Bandits, 29th Conference on Uncertainty in Artificial Intelligence (UAI 2013) and (JFPDA 2013)., arXiv:1309.6869, Bibcode:2013arXiv1309.6869V
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58. Discounted UCB, Levente Kocsis, Csaba Szepesvári, 2006
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Further reading
• Guha, S.; Munagala, K.; Shi, P. (2010), "Approximation algorithms for restless bandit problems", Journal of the ACM, 58: 1–50, arXiv:0711.3861, doi:10.1145/1870103.1870106, S2CID 1654066
• Dayanik, S.; Powell, W.; Yamazaki, K. (2008), "Index policies for discounted bandit problems with availability constraints", Advances in Applied Probability, 40 (2): 377–400, doi:10.1239/aap/1214950209.
• Powell, Warren B. (2007), "Chapter 10", Approximate Dynamic Programming: Solving the Curses of Dimensionality, New York: John Wiley and Sons, ISBN 978-0-470-17155-4.
• Robbins, H. (1952), "Some aspects of the sequential design of experiments", Bulletin of the American Mathematical Society, 58 (5): 527–535, doi:10.1090/S0002-9904-1952-09620-8.
• Sutton, Richard; Barto, Andrew (1998), Reinforcement Learning, MIT Press, ISBN 978-0-262-19398-6, archived from the original on 2013-12-11.
• Allesiardo, Robin (2014), "A Neural Networks Committee for the Contextual Bandit Problem", Neural Information Processing – 21st International Conference, ICONIP 2014, Malaisia, November 03-06,2014, Proceedings, Lecture Notes in Computer Science, vol. 8834, Springer, pp. 374–381, arXiv:1409.8191, doi:10.1007/978-3-319-12637-1_47, ISBN 978-3-319-12636-4, S2CID 14155718.
• Weber, Richard (1992), "On the Gittins index for multiarmed bandits", Annals of Applied Probability, 2 (4): 1024–1033, doi:10.1214/aoap/1177005588, JSTOR 2959678.
• Katehakis, M.; C. Derman (1986), "Computing optimal sequential allocation rules in clinical trials", Adaptive statistical procedures and related topics, Institute of Mathematical Statistics Lecture Notes - Monograph Series, vol. 8, pp. 29–39, doi:10.1214/lnms/1215540286, ISBN 978-0-940600-09-6, JSTOR 4355518.
• Katehakis, M.; A. F. Veinott, Jr. (1987), "The multi-armed bandit problem: decomposition and computation", Mathematics of Operations Research, 12 (2): 262–268, doi:10.1287/moor.12.2.262, JSTOR 3689689, S2CID 656323.
External links
• MABWiser, open source Python implementation of bandit strategies that supports context-free, parametric and non-parametric contextual policies with built-in parallelization and simulation capability.
• PyMaBandits, open source implementation of bandit strategies in Python and Matlab.
• Contextual, open source R package facilitating the simulation and evaluation of both context-free and contextual Multi-Armed Bandit policies.
• bandit.sourceforge.net Bandit project, open source implementation of bandit strategies.
• Banditlib, Open-Source implementation of bandit strategies in C++.
• Leslie Pack Kaelbling and Michael L. Littman (1996). Exploitation versus Exploration: The Single-State Case.
• Tutorial: Introduction to Bandits: Algorithms and Theory. Part1. Part2.
• Feynman's restaurant problem, a classic example (with known answer) of the exploitation vs. exploration tradeoff.
• Bandit algorithms vs. A-B testing.
• S. Bubeck and N. Cesa-Bianchi A Survey on Bandits.
• A Survey on Contextual Multi-armed Bandits, a survey/tutorial for Contextual Bandits.
• Blog post on multi-armed bandit strategies, with Python code.
• Animated, interactive plots illustrating Epsilon-greedy, Thompson sampling, and Upper Confidence Bound exploration/exploitation balancing strategies.
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| Wikipedia |
Stellated octahedron
The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depicted in Pacioli's De Divina Proportione, 1509.[2]
Stellated octahedron
Seen as a compound of two regular tetrahedra (red and yellow)
TypeRegular compound
Coxeter symbol{4,3}[2{3,3}]{3,4}[1]
Schläfli symbols{{3,3}}
a{4,3}
ß{2,4}
ßr{2,2}
Coxeter diagrams ∪
Stellation coreOctahedron
Convex hullCube
IndexUC4, W19
Polyhedra2 tetrahedra
Faces8 triangles
Edges12
Vertices8
DualSelf-dual
Symmetry group
Coxeter group
Oh, [4,3], order 48
D4h, [4,2], order 16
D2h, [2,2], order 8
D3d, [2+,6], order 12
Subgroup restricting
to one constituent
Td, [3,3], order 24
D2d, [2+,4], order 8
D2, [2,2]+, order 4
C3v, [3], order 6
It is the simplest of five regular polyhedral compounds, and the only regular compound of two tetrahedra. It is also the least dense of the regular polyhedral compounds, having a density of 2.
It can be seen as a 3D extension of the hexagram: the hexagram is a two-dimensional shape formed from two overlapping equilateral triangles, centrally symmetric to each other, and in the same way the stellated octahedron can be formed from two centrally symmetric overlapping tetrahedra. This can be generalized to any desired amount of higher dimensions; the four-dimensional equivalent construction is the compound of two 5-cells. It can also be seen as one of the stages in the construction of a 3D Koch snowflake, a fractal shape formed by repeated attachment of smaller tetrahedra to each triangular face of a larger figure. The first stage of the construction of the Koch Snowflake is a single central tetrahedron, and the second stage, formed by adding four smaller tetrahedra to the faces of the central tetrahedron, is the stellated octahedron.
Construction
The Cartesian coordinates of the stellated octahedron are as follows: (±1/2, ±1/2, 0) (0, 0, ±1/√2) (±1, 0, ±1/√2) (0, ±1, ±1/√2)
The stellated octahedron can be constructed in several ways:
• It is a stellation of the regular octahedron, sharing the same face planes. (See Wenninger model W19.)
In perspective
Stellation plane
The only stellation of a regular octahedron, with one stellation plane in yellow.
• It is also a regular polyhedron compound, when constructed as the union of two regular tetrahedra (a regular tetrahedron and its dual tetrahedron).
• It can be obtained as an augmentation of the regular octahedron, by adding tetrahedral pyramids on each face. In this construction it has the same topology as the convex Catalan solid, the triakis octahedron, which has much shorter pyramids.
• It is a facetting of the cube, sharing the vertex arrangement.
• It can be seen as a {4/2} antiprism; with {4/2} being a tetragram, a compound of two dual digons, and the tetrahedron seen as a digonal antiprism, this can be seen as a compound of two digonal antiprisms.
• It can be seen as a net of a four-dimensional octahedral pyramid, consisting of a central octahedron surrounded by eight tetrahedra.
Facetting of a cube
A single diagonal triangle facetting in red
Related concepts
A compound of two spherical tetrahedra can be constructed, as illustrated.
The two tetrahedra of the compound view of the stellated octahedron are "desmic", meaning that (when interpreted as a line in projective space) each edge of one tetrahedron crosses two opposite edges of the other tetrahedron. One of these two crossings is visible in the stellated octahedron; the other crossing occurs at a point at infinity of the projective space, where each edge of one tetrahedron crosses the parallel edge of the other tetrahedron. These two tetrahedra can be completed to a desmic system of three tetrahedra, where the third tetrahedron has as its four vertices the three crossing points at infinity and the centroid of the two finite tetrahedra. The same twelve tetrahedron vertices also form the points of Reye's configuration.
The stella octangula numbers are figurate numbers that count the number of balls that can be arranged into the shape of a stellated octahedron. They are
0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, .... (sequence A007588 in the OEIS)
In popular culture
The stellated octahedron appears with several other polyhedra and polyhedral compounds in M. C. Escher's print "Stars",[3] and provides the central form in Escher's Double Planetoid (1949).[4]
One of the stellated octahedra in the Plaza de Europa, Zaragoza
The obelisk in the center of the Plaza de Europa in Zaragoza, Spain, is surrounded by twelve stellated octahedral lampposts, shaped to form a three-dimensional version of the Flag of Europe.[5]
Some modern mystics have associated this shape with the "merkaba",[6] which according to them is a "counter-rotating energy field" named from an ancient Egyptian word.[7] However, the word "merkaba" is actually Hebrew, and more properly refers to a chariot in the visions of Ezekiel.[8] The resemblance between this shape and the two-dimensional star of David has also been frequently noted.[9]
References
1. H.S.M. Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, 3.6 The five regular compounds, pp.47-50, 6.2 Stellating the Platonic solids, pp.96-104
2. Barnes, John (2009), "Shapes and Solids", Gems of Geometry, Springer, pp. 25–56, doi:10.1007/978-3-642-05092-3_2, ISBN 978-3-642-05091-6.
3. Hart, George W. (1996), "The Polyhedra of M.C. Escher", Virtual Polyhedra.
4. Coxeter, H. S. M. (1985), "A special book review: M. C. Escher: His life and complete graphic work", The Mathematical Intelligencer, 7 (1): 59–69, doi:10.1007/BF03023010, S2CID 189887063. See in particular p. 61.
5. "Obelisco" [Obelisk], Zaragoza es Cultura (in Spanish), Ayuntamiento de Zaragoza, retrieved 2021-10-19
6. Dannelley, Richard (1995), Sedona: Beyond the Vortex: Activating the Planetary Ascension Program with Sacred Geometry, the Vortex, and the Merkaba, Light Technology Publishing, p. 14, ISBN 9781622336708
7. Melchizedek, Drunvalo (2000), The Ancient Secret of the Flower of Life: An Edited Transcript of the Flower of Life Workshop Presented Live to Mother Earth from 1985 to 1994 -, Volume 1, Light Technology Publishing, p. 4, ISBN 9781891824173
8. Patzia, Arthur G.; Petrotta, Anthony J. (2010), Pocket Dictionary of Biblical Studies: Over 300 Terms Clearly & Concisely Defined, The IVP Pocket Reference Series, InterVarsity Press, p. 78, ISBN 9780830867028
9. Brisson, David W. (1978), Hypergraphics: visualizing complex relationships in art, science, and technology, Westview Press for the American Association for the Advancement of Science, p. 220, The Stella octangula is the 3-d analog of the Star of David
External links
Wikimedia Commons has media related to Stellated octahedron.
• Eric W. Weisstein, Stella Octangula (Compound of two tetrahedra) at MathWorld.
• Klitzing, Richard, "3D compound"
| Wikipedia |
\begin{definition}[Definition:Square Root/Negative]
Let $x \in \R_{> 0}$ be a (strictly) positive real number.
The '''negative square root of $x$''' is the number defined as:
:$- \sqrt x := y \in \R_{<0}: y^2 = x$
\end{definition} | ProofWiki |
distance from point to plane calculator
Another way to find the distance is by finding the plane and the line intersection point and then calculate distance between this point and the given point. Finding the distance from a point to a plane by considering a vector projection. D = √[ ( X2-X1)^2 + (Y2-Y1)^2) Where D is the distance; X1 and X2 are the x-coordinates; Y1 and Y2 are the y-coordinates; Simply type in the name of the two places in the text boxes and click the show button!The best format to use is [City, Country] to enter a location - that is [City(comma)(space)Country]. Find the distance between two points from x and y coordinates with this distance formula calculator. showing that d is the distance from the origin 0 = (0,0,0) to the plane P. This formula gives a signed distance which is positive on one side of the plane and negative on the other. The distance should then be displayed. Such a line is given by calculating the normal vector of the plane. If Ax + By + Cz + D = 0 is a plane equation, then distance from point P (P x, P y, P z) to plane can be found using the following formula: The distance from a point to a plane (d) = (AP x + BP y + CP z + D)/ √ (A+ B 2 + C 2) Calculate … Simply click once on one point, then click again on the second point. Plane Geometry Solid Geometry Conic Sections. Fractions should be entered with a forward such as '3/4' for the fraction $$ \frac{3}{4} $$. Click Calculate Distance, and the tool will place a marker at each of the two addresses on the map along with a line between them. Airplanemanager.com provides flight time and distance calculators free for the air charter industry. Another way to find the distance is by finding the plane and the line intersection point and then calculate distance between this point and the given point. The distance from a point to a plane is equal to length of the perpendicular lowered from a point on a plane. Shortest distance between a point and a plane Calculator, \(\normalsize Distance\ between\ a\ point\ and\ a\ plane\\. Spherical to Cartesian coordinates. I need to calculate the distance between the point in the plane and the straight line. This website uses cookies to ensure you get the best experience. The focus of this lesson is to calculate the shortest distance between a point and a plane. Driving distance by car is determined from the actual turn-by-turn driving directions. The shortest distance of a point from a plane is said to be along the line perpendicular to the plane or in other words, is the perpendicular distance of the point from the plane. Cartesian to Spherical coordinates. And how to calculate that distance? The distance in miles and kilometers will display for the straight line or flight mileage along with the distance it would take to get there in a car, driving mileage. Contact Us. It is a good idea to find a line vertical to the plane. Shortest distance between two lines. Distance between a line and a point calculator This online calculator can find the distance between a given line and a given point. Thank you for your questionnaire.Sending completion, Volume of a tetrahedron and a parallelepiped, Shortest distance between a point and a plane. This distance is actually the length of the perpendicular from the point to the plane. Distance On a Coordinate Plane Between Two Points = √ ((x1-x0) 2 + (y1-y0) 2) Or browse the mileage charts for any state or … Formula. I need to calculate the distance between the point in the plane and the straight line. It can be found starting with a change of variables that moves the origin to coincide with the given point then finding the point on … Log InorSign Up. As shown by other answers and in note 1 there are easier ways to find the shortest distance, but here is a detailed solution using the method of Lagrange multipliers. The blue graph (d(x)) represents the distance from the point (a,b) to the graph of f(x). So the distance from the point ( m , n ) to the line Ax + By + C = 0 is given by: Distance On a Coordinate Plane Between Two Points = √ ((x1-x0) 2 + (y1-y0) 2) For any two points there is exactly one line segment connecting them. Therefore, the distance of the plane from the origin is simply given by (Gellert et al. The formula for calculating it can be derived and expressed in several ways. How to enter numbers: Enter any integer, decimal or fraction. The distance between two points A(x A, y A) and B(x B, y B) in two-dimensional Cartesian coordinate plane is the length of the segment connecting them, AB = d(A, B) = √(x B - x A)2 + (y B - y A)2 What is the Distance between Two Points? Shortest distance between a point and a plane. Let us use this formula to calculate the distance between the plane and a point in the following examples. And we'll, hopefully, see that visually as we try to figure out how to calculate the distance. Some functions are limited now because setting of JAVASCRIPT of the browser is OFF. You can also type in major places straight in such as \"USA\", \"Tokyo\", \"London\" etc. It is a good idea to find a line vertical to the plane. Approach: The perpendicular distance (i.e shortest distance) from a given point to a Plane is the perpendicular distance from that point to the given plane.Let the co-ordinate of the given point be (x1, y1, z1) and equation of the plane be given by the equation a * x + b * y + c * z + d = 0, where a, b and c are real constants. And we'll, hopefully, see that visually as we try to figure out how to calculate the distance. My Vectors course: https://www.kristakingmath.com/vectors-course Learn how to find the distance between a point and a plane. You can click more than two points in order to build up a continuous route. Distance between a point and a line. How to enter numbers: Enter any integer, decimal or fraction. Given a point a line and want to find their distance. Distance from a point to a graph. The following formula is used to calculate the euclidean distance between points. For a long distance trip, you can plan a road trip with stops. Where point (x0,y0,z0), Plane (ax+by+cz+d=0) For example, Give the point (2,-3,1) and the plane 3x+y-2z=15 When you click the search button, a search will be made to find which place you are referring to. The distance from a point, P, to a plane, π, is the smallest distance from the point to one of the infinite points on the plane. Finding the distance from a point to a plane by considering a vector projection. Simply type in the name of the two places in the text boxes and click the show button!The best format to use is [City, Country] to enter a location - that is [City(comma)(space)Country]. Calculation formula from point to line: Through the formula derivation, the … Copyright ©2006 - 2020 Thinkcalculator All Rights Reserved. Distance From Point to Plane Calculator; Euclidean Distance Formula. Calculation formula from point to line: Through the formula derivation, the … Your feedback and comments may be posted as customer voice. I found that the mathematical knowledge was returned to the teacher. the perpendicular should give us the said shortest distance. The distance from the point to the plane will be the projection of P on the unit vector direction this is the dot product of the vactor P and the unit vector. The Search For Location text box allows you to quickly get to an area you wish without spending time zooming and panning to find it. Coordinates of Point 1 (x 1 ,y 1 ): x= y= Coordinates of Point 2 (x 2 ,y 2 ): x= y= If you put it on lengt 1, the calculation becomes … Distance between a point and a line. In Euclidean geometry, the distance from a point to a line is the shortest distance from a given point to any point on an infinite straight line.It is the perpendicular distance of the point to the line, the length of the line segment which joins the point to nearest point on the line. [1] 2019/04/22 23:36 Male / Under 20 years old / High-school/ University/ Grad student / Useful /, [3] 2015/04/04 14:42 Male / 20 years old level / High-school/ University/ Grad student / Useful /, [4] 2014/04/10 06:19 Female / Under 20 years old / High-school/ University/ Grad student / Very /, [5] 2014/04/05 09:38 Male / Under 20 years old / High-school/ University/ Grad student / A little /, [6] 2013/07/04 06:24 Male / 30 years old level / An office worker / A public employee / A little /, [7] 2013/02/13 06:03 Male / 20 years old level / High-school/ University/ Grad student / Very /, [8] 2012/04/17 13:52 Male / 20 years old level / A student / Very /, [9] 2012/03/30 21:48 Male / 20 years old level / A student / Very /, [10] 2012/03/05 02:24 Female / Under 20 years old / A student / Very /. Free distance calculator - Compute distance between two points step-by-step. Given a plane: ax+by+cx+d = 0, a point p1 = [x1; y1; z1], and a point: p0 [x0; y0; z0]. Distance From Point to Plane Calculator; Euclidean Distance Formula. So, if we take the normal vector \vec{n} and consider a line parallel t… Let the co-ordinate of the given point be (x1, y1, z1) and equation of the plane be given by the equation a * x + b * y + c * z + d = 0, where a, b and c are real constants. Enter a start and end point into the tool and click the calculate mileage button. Given three points for, 2, 3, compute the unit normal (12) Then the (signed) distance from a point to the plane containing the three points is given by Cylindrical to Cartesian coordinates The following formula is used to calculate the euclidean distance between points. If you got a point and a plane in the Euclidean space, you can calculate the distance between the point and the plane. Thus, the line joining these two points i.e. It can be found starting with a change of variables that moves the origin to coincide with the given point then finding the point on … Use the miles / km / nautical miles / yards switch to measure distances in km or in miles or nautical miles. And we already have a point from the last video that's on the plane, this x … By using this website, you agree to our Cookie Policy. How it works: Just type numbers into the boxes below and the calculator will automatically calculate the distance between those 2 points. If you want to split the distance with a friend, you can use the halfway point calculator to find the best place to meet. The Autopan option will move the map as you click the points. Plane equation given three points. When you click the search button, a search will be made to find which place you are referring to. The distance between two points A(x A, y A) and B(x B, y B) in two-dimensional Cartesian coordinate plane is the length of the segment connecting them, AB = d(A, B) = √(x B - x A)2 + (y B - y A)2 What is the Distance between Two Points? Travelmath provides an online travel distance calculator to help you measure both flying distances and driving distances. Let us use this formula to calculate the distance between the plane and a point in the following examples. The blue graph (d(x)) represents the distance from the point (a,b) to the graph of f(x). Given a plane: ax+by+cx+d = 0, a point p1 = [x1; y1; z1], and a point: p0 [x0; y0; z0]. Firstly, a search is made of an internal list of common places. Find the distance from the point P = (4, − 4, 3) to the plane 2 x − 2 y + 5 z + 8 = 0, which is pictured in the below figure in its original view. Distance From To: Calculate distance between two addresses, cities, states, zipcodes, or locations Enter a city, a zipcode, or an address in both the Distance From and the Distance To address inputs. Flight distance is computed from a GPS-accurate great circle formula, which gives you the straight line distance … The Distance from a point to a plane calculator to find the shortest distance between a point and the plane. So the first thing we can do is, let's just construct a vector between this point that's off the plane and some point that's on the plane. p0 is located on the given plane and has the shortest distance to p1. Otherwise, the distance is positive for points on the side pointed to by the normal vector n. Equivalence with finding the distance between two parallel planes. I assume you want to compute perpendicular distance between point and plane given 3 points on it forming a triangle. Given three points for, 2, 3, compute the unit normal (12) Then the (signed) distance from a point to the plane containing the three points is given by And how to calculate that distance? We can clearly understand that the point of intersection between the point and the line that passes through this point which is also normal to a planeis closest to our original point. We verify that the plane and the straight line are parallel using the scalar product between the governing vector of the straight line, $$\vec{v}$$, and the normal vector of the plane $$\vec{n}$$. Recently, I encountered a problem. I found that the mathematical knowledge was returned to the teacher. Distance From To: Calculate distance between two addresses, cities, states, zipcodes, or locations Enter a city, a zipcode, or an address in both the Distance From and the Distance To address inputs. Such a line is given by calculating the normal vector of the plane. In Euclidean space, the distance from a point to a plane is the distance between a given point and its orthogonal projection on the plane or the nearest point on the plane. Thinkcalculator.com provides you helpful and handy calculator resources. Distance between a line and a point calculator This online calculator can find the distance between a given line and a given point. We first need to normalize the line vector (let us call it ).Then we find a vector that points from a point on the line to the point and we can simply use .Finally we take the cross product between this vector and the normalized line vector to get the shortest vector that points from the line to the point. We first need to normalize the line vector (let us call it ).Then we find a vector that points from a point on the line to the point and we can simply use .Finally we take the cross product between this vector and the normalized line vector to get the shortest vector that points from the line to the point. You can also type in major places straight in such as \"USA\", \"Tokyo\", \"London\" etc. I think that the method of Lagrange multipliers is the easiest way to solve my question, but how can I find the Lagrangian function? Distance from point to plane To illustrate our approach for finding the distance between a point and a plane, we work through an example. Equivalence with finding the distance between two parallel planes. The Distance from a point to a plane calculator to find the shortest distance between a point and the plane Formula Where point (x0,y0,z0), Plane (ax+by+cz+d=0) Here vector math approach: definitions. The shortest distance of a point from a plane is said to be along the line perpendicular to the plane or in other words, is the perpendicular distance of the point from the plane. Cartesian to Cylindrical coordinates. To compute the distance to a plane P , we did not calculate the base point of the perpendicular from the point P 0 to P , which some authors do. How it works: Just type numbers into the boxes below and the calculator will automatically calculate the distance between those 2 points. Click Calculate Distance, and the tool will place a marker at each of the two addresses on the map along with a line between them. We can clearly understand that the point of intersection between the point and the line that passes through this point which is also normal to a planeis closest to our original point. Therefore, the distance of the plane from the origin is simply given by (Gellert et al. Find the distance between two points from x and y coordinates with this distance formula calculator. 1989, p. 541). So, if we take the normal vector \vec{n} and consider a line parallel t… Distance between a line and a point the perpendicular should give us the said shortest distance. Online calculator to calculate and display the distance and midpoint for two points. Given a point a line and want to find their distance. Postcodes and addresses can also be used. Fractions should be entered with a forward such as '3/4' for the fraction $$ \frac{3}{4} $$. D = √[ ( X2-X1)^2 + (Y2-Y1)^2) Where D is the distance; X1 and X2 are the x-coordinates; Y1 and Y2 are the y-coordinates; The distance in miles and kilometers will display for the straight line or flight mileage along with the distance it would take to get there in a car, driving mileage. For any two points there is exactly one line segment connecting them. After Du Niang, I found the calculation method, which is hereby recorded. Approach: The perpendicular distance (i.e shortest distance) from a given point to a Plane is the perpendicular distance from that point to the given plane. So the first thing we can do is, let's just construct a vector between this point that's off the plane and some point that's on the plane. Calculate a … Coordinates of Point 1 (x 1 ,y 1 ): x= y= Coordinates of Point 2 (x 2 ,y 2 ): x= y= p0 is located on the given plane and has the shortest distance to p1. Postcodes and addresses can also be used. Distance between a line and a point The focus of this lesson is to calculate the shortest distance between a point and a plane. Trigonometry. My Vectors course: https://www.kristakingmath.com/vectors-course Learn how to find the distance between a point and a plane. This applet demonstrates the setup of the problem and the method we will use to derive a formula for … The distance from the point to the plane will be the projection of P on the unit vector direction this is the dot product of the vactor P and the unit vector. The formula for calculating it can be derived and expressed in several ways. In Euclidean space, the distance from a point to a plane is the distance between a given point and its orthogonal projection on the plane or the nearest point on the plane. If you put it on lengt 1, the calculation … Firstly, a search is made of an internal list of common places. When both points are on P , the whole segment lies in the plane. Distance from a point to a graph. In a two dimension plane there are two points let's say A and B with the respective coordinates as (x1, y1) and (x2, y2) and to calculate the distance between them there is a direct formula which is given below Airplanemanager.com provides flight time and distance calculators free for the air charter industry. Spherical to Cylindrical coordinates. For example if you wish t… This means, you can calculate the shortest distance between the point and a point of the plane. let the triangle points be p0,p1,p2 and tested point p. plane normal Recently, I encountered a problem. Enter a start and end point into the tool and click the calculate mileage button. Step-by-step explanation is provided. If one just wants the distance, then directly computing it without going through an intermediate calculation is fastest. The absolute value sign is necessary since distance must be a positive value, and certain combinations of A, m , B, n and C can produce a negative number in the numerator. Thus, the line joining these two points i.e. So, one has to take the absolute value to get an absolute distance. The distance from a point, P, to a plane, π, is the smallest distance from the point to one of the infinite points on the plane. This distance is actually the length of the perpendicular from the point to the plane. Move (a,b) around to see the distances change; then move the point on the graph of d(x) to relative extrema to see the distance relationship. Volume of a tetrahedron and a parallelepiped. If the straight line and the plane are parallel the scalar product will be zero: … And we already have a point from the last video that's on the plane… If you got a point and a plane in the Euclidean space, you can calculate the distance between the point and the plane. The Distance from a point to a plane calculator to find the shortest distance between a point and the plane, Where point (x0,y0,z0), Plane (ax+by+cz+d=0), For example, Give the point (2,-3,1) and the plane 3x+y-2z=15, Distance = |ax+by+cz-d| / sqrt(a^2 + b^2 + c^2) = |3(2) + (-3) + (-2) - 15| / sqrt(a^2 + b^2 + c^2) = |-14| / sqrt14 = 3.7416573867739413, Word Counter | AllCallers | CallerInfo | ThinkCalculator | Free Code Format. Log InorSign Up. After Du Niang, I found the calculation method, which is hereby recorded. The equation for the plane determined by N and Q is A (x − x 0) + B (y − y 0) + C (z − z 0) = 0, which we could write as A x + B y + C z + D = 0, where D = − A x 0 − B y 0 − C z 0. You can then compare the two results to see the difference. This means, you can calculate the shortest distance between the point and a point of the plane. 1989, p. 541). Given with the two points coordinates and the task is to find the distance between two points and display the result. In Euclidean geometry, the distance from a point to a line is the shortest distance from a given point to any point on an infinite straight line.It is the perpendicular distance of the point to the line, the length of the line segment which joins the point to nearest point on the line. Move (a,b) around to see the distances change; then move the point on the graph of d(x) to relative extrema to see the distance relationship. Their distance length of the plane intermediate calculation is fastest parallelepiped, shortest distance between given... Point to a graph is given by ( Gellert et al distance from point to plane calculator compute perpendicular between. Plan a road trip with stops click once on one point, then click again the... Such a line and a given point online travel distance calculator to help you measure both flying distances driving. Internal list of common places calculate the distance from a point in the plane so, has! Distances and driving distances points i.e given a point in the plane and has the shortest.... The formula for calculating it can be derived and expressed in several ways actually the length of plane. Is given by calculating the normal vector of the perpendicular from the point in the plane Euclidean formula. 'Ll, hopefully, see that visually as we try to figure out to! Autopan option will move the map as you click the calculate mileage.. Distance from point to a plane is made of an internal list of common places free for air. You agree to our Cookie Policy to plane calculator ; Euclidean distance a... The straight line is used to calculate the Euclidean distance between points is used to calculate the distance between point. Numbers into the boxes below and the straight line airplanemanager.com provides flight and. Formula calculator made of an internal list of common places online calculator can find distance. Formula is used to calculate the distance of the perpendicular from the point and a plane calculator ; distance! The length of the plane from the point to the teacher points there is exactly one line segment connecting.! Free for the air charter industry derived and expressed in several ways the last video that 's on the plane... From x and y coordinates with this distance formula as you click the search button, search... Is used to calculate the distance, then click again on the given plane and a plane that the knowledge. For any distance from point to plane calculator points in order to build up a continuous route is.! Idea to find their distance start and end point into the tool and click the mileage... A continuous route is to calculate the distance it can be derived and expressed in several ways to the! The two results to see the difference distances and driving distances two parallel planes Distance\ between\ a\ and\... Using this website, you can then compare the two results to see the difference it forming a.! Cookie Policy the map as you click the calculate distance from point to plane calculator button a start end! See the difference given by ( Gellert et al for the air charter industry points in order to build a. Airplanemanager.Com provides flight time and distance calculators free for distance from point to plane calculator air charter industry expressed in several ways a... Have a point from the last video that 's on the second.. The difference Autopan option will move the map as you click the mileage. Two parallel planes from point to plane calculator ; Euclidean distance formula going!, i found that the mathematical knowledge was returned to the plane from x and y coordinates with distance. Us the said shortest distance Euclidean distance between a point and a plane the results... Trip with stops trip with stops Learn how to calculate the distance two. / yards switch to measure distances in distance from point to plane calculator or in miles or nautical miles / km / miles! Firstly, a search will be made to find a line and a a. Find their distance and want to find which place you are referring.... Those 2 points use the miles / yards switch to measure distances in km or miles! Video that 's on the given plane and has the shortest distance points i.e line vertical to the plane a. And\ a\ plane\\ by considering a vector projection up a continuous route and driving distances in following. Will automatically calculate the distance from point to plane calculator, \ ( \normalsize Distance\ a\! Calculate mileage button in order to build up a continuous route build up continuous... Then click again on the given plane and a point in the plane and the straight line of places. Has the shortest distance a point in the following formula is used to calculate the distance, then computing... Is fastest the said shortest distance to p1 has the shortest distance to p1 triangle points be p0 p1... Located on the plane… Recently, i found that the mathematical knowledge was returned the. The search button, a search is made of an internal list common... The two results to see the difference integer, decimal or fraction decimal. 2 points the map as you click the search button, a search is of! Are referring to driving distances charter industry a plane their distance without through! Long distance trip, you can plan a road trip with stops because... 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2020 distance from point to plane calculator | CommonCrawl |
Maria Silvia Lucido
Maria Silvia Lucido (22 April 1963 – 4 March 2008) was an Italian mathematician specializing in group theory, and a researcher in mathematics at the University of Udine.[1]
Life, education and career
Lucido was originally from Vicenza, where she was born on 22 April 1963. After working for a bank and a travel agency, she entered mathematical study at the University of Padua in 1986, graduating in 1991. Already as an undergraduate she began research into the theory of finite groups, and wrote an undergraduate thesis on the subject under the supervision of Franco Napolitani.[1] She completed a Ph.D. at Padua in 1996 with the dissertation Il Prime Graph dei gruppi finiti [the prime graphs of finite groups], supervised by Napolitani and co-advised by Carlo Casolo.[1][2]
After postdoctoral research at the University of Padua and as a Fulbright scholar at Michigan State University, she obtained a permanent position as a researcher at the University of Udine in 1999. She was killed in an automobile accident on March 4, 2008, survived by her husband and two sons.[1]
Research
Lucido was particularly known for her research on prime graphs of finite groups. These are undirected graphs that have a vertex for each prime factor of the order of a group, and that have an edge $pq$ whenever the given group has an element of order $pq$. Her work in this area included
• Proving that the connected components of these graphs have diameter at most five, and at most three for solvable groups.[3]
• Proving that, when the prime graph is a tree, it has at most eight vertices, and at most four for solvable groups.[4]
• Characterizing the finite simple groups for which all components of the prime graphs are cliques.[5]
Lucido founded a series of annual summer schools on the theory of finite groups, held in Venice and sponsored by the University of Udine, beginning in 2004. After her death, the three subsequent offerings of the summer schools in 2010, 2011, and 2013 were dedicated in her honor.[1]
References
1. "Maria Silvia Lucido", Summer School on Finite Groups and Related Geometrical Structures: in memory of Maria Silvia Lucido (1963-2008), University of Udine, retrieved 2021-12-05
2. Maria Silvia Lucido at the Mathematics Genealogy Project
3. Review by Anatoli S. Kondratʹev of Lucido, Maria Silvia (1999), "The diameter of the prime graph of a finite group", Journal of Group Theory, 2 (2): 157–172, doi:10.1515/jgth.1999.011, MR 1681526, Zbl 0921.20020
4. Reviews by Michelle R. DeDeo and Robert Jajcay of Lucido, Maria Silvia (2002), "Groups in which the prime graph is a tree", Bollettino della Unione Matematica Italiana, 5 (1): 131–148, MR 1881928, Zbl 1097.20022
5. Review by Anatoli S. Kondratʹev of Lucido, Maria Silvia; Moghaddamfar, Ali Reza (2004), "Groups with complete prime graph connected components", Journal of Group Theory, 7 (3): 373–384, doi:10.1515/jgth.2004.013, MR 2063403, Zbl 1058.20014
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
| Wikipedia |
Movie Recommendation Algorithm Using Social Network Analysis to Alleviate Cold-Start Problem
Khamphaphone Xinchang* , Phonexay Vilakone* and Doo-Soon Park**
Corresponding Author: Doo-Soon Park** ([email protected])
Khamphaphone Xinchang*, Dept. of Computer Science and Engineering, Soonchunhyang University, Asan, Korea, [email protected]
Phonexay Vilakone*, Dept. of Computer Science and Engineering, Soonchunhyang University, Asan, Korea, [email protected]
Doo-Soon Park**, Dept. of Computer Software Engineering, Soonchunhyang University, Asan, Korea, [email protected]
Received: November 7 2018
Revision received: January 8 2019
Accepted: January 21 2019
Abstract: With the rapid increase of information on the World Wide Web, finding useful information on the internet has become a major problem. The recommendation system helps users make decisions in complex data areas where the amount of data available is large. There are many methods that have been proposed in the recommender system. Collaborative filtering is a popular method widely used in the recommendation system. However, collaborative filtering methods still have some problems, namely cold-start problem. In this paper, we propose a movie recommendation system by using social network analysis and collaborative filtering to solve this problem associated with collaborative filtering methods. We applied personal propensity of users such as age, gender, and occupation to make relationship matrix between users, and the relationship matrix is applied to cluster user by using community detection based on edge betweenness centrality. Then the recommended system will suggest movies which were previously interested by users in the group to new users. We show shown that the proposed method is a very efficient method using mean absolute error.
Keywords: Cold Start Problem , Collaborative Filtering (CF) , Movie Recommendation System , Social Network Analysis
In recent years, information available on the internet is rapidly growing up; people need more time to select useful information. Social media use continues to grow rapidly, too. There are currently more than 3 billion people around the world using social media each month. The new Global Digital 2018 report has revealed that there are currently more than 4 billion people worldwide using the Internet [1]. Social networks are one of the most popular communications media today and it attracts millions of active users to share and comment on their photos and places with others [3]. To manage the information overload problem, the recommendation system has been developed. A recommendation system is a simple algorithm that allows users to relate and recommend items by filtering user-related data from large data. Recommender systems are software tools that make the recommendation of product or items that are appropriate for a customer's taste based on the analysis of information of products that many customers are interested in and the customer's and their past purchasing activity [3]. The recommendations system can help users to make decisions in multiple contexts. The goal of recommendation system is to find what's likely to be interest to the users. Over the years, many recommender systems were introduced by researchers from different problem areas. We can utilize recommendation systems that use different techniques including collaborative filtering (CF) technique, content-based filtering technique, and hybrid technique. These recommendation systems will recommend popular items that customers liked [4]. CF method is one most successful and widely used in recommendation systems [5]. The basic idea of this algorithm is to use common experiences or similar interests. The important process is to search users that are similar to the target user or find the product that is similar to the predicted product. But, the CF algorithm also has some problems such as a cold-start problem [6]. Cold-start problem is a problem where the system is not able to recommend items to users. For every recommender system, it is required to build a user profile by considering the user's preferences and likes. The user profile is developed by considering her activities and behaviors they perform with the system. On the basis of user's previous history and activities, the system makes decisions and consequently recommends items. The problem arises when a new user or new item enters the system, for such users or items that the system does not have enough information to make decision. For example, if a new user has not rated some items and not yet visited/viewed some items, then it would be difficult for the system to build a model on that basis. This can lead to inaccuracies in estimating similarities between users. Many approaches have been studied to solve the existing problem. Embarak [7] suggested two types of recommendation such as node recommendation and batch recommendation, and then compared the suggested method with three other alternative methods including Naïve Filterbots Method, Media Scout Stereotype Method, and Triadic Aspect Method to solve the cold-start problem. Basiri et al. [8] suggested a new hybrid approach, which focuses on improving performance under cold start problem. This method can give a reasonable and appropriate recommendation.
With the development of technology, the user's behavior and personal information can be tracked and recorded on social networking sites or online shopping sites. This type of technology makes it easier and it is very useful for analyzing user preferences. We analysis social network analysis (SNA) methods [9,10] and introduced the betweenness centrality in SNA into a CF approach. This paper presents a movie recommendation algorithm using SNA to alleviate the cold-start problem. In the proposed method, the user's personal information such as age, gender, and occupation are used to establish a relationship between users. Then the relationship matrix between users will be applied for clustering the user into several communities or groups. In this process, the centrality of SNA is used to detection communities; after that, the system will recommend movies in the group that is similar to the target users by considering CF. The main objective of this article is to develop techniques that can recommend the most suitable movies for target users based on personal characteristics.
The proposed work is briefly described as follows. In Section 2, we will explain about the related work which is the methods used in this paper. Our proposed algorithm for movie recommendation algorithm using SNA to alleviate cold start problem will describe in Section 3. In Section 4, will present experimental analysis and experimental results and finally, the conclusion will be concluded in Section 5.
2. Related Works
In this section, we have a purpose to briefly explain about the relevant research, including the recommendation system, CF, and SNA that are required for the movie recommendation system in this study.
2.1 Recommendation System
In recent years, the recommender system has become more popular. It has been used in many areas including book, news, movie, music, and products. There are also recommender systems for experts [11], restaurants, collaborators [12], garments, jokes, financial services [13], romantic partners, life insurance, and Twitter pages [14]. These recommendation systems use one type of filtering to predict ratings and user satisfaction, which allows users to purchase products based on their interests or needs. Having information about user's life can give hints about how the user will react when faced with different situations [15]. The recommender system is a useful alternative of search algorithms because it can help users discover what they may not find. It is usually performed using a non-traditional indexing search engine. A recommender system is a technology that makes automatics predictions about the relationship between customers or between items and searches for items that users may need. There are many approaches to make a recommender system, including the following.
Content-based filtering: Content-based filtering methods depend on the item's features and user preferences [16]. These methods work with data that the user provides. Based on this data, a user profile will be created which will be used to give advice to users. As the user provides more input and accepts recommendations, the engine becomes more and more accurate. These algorithms try to recommend the products that are similar to those that the user liked in the past. In particular, many nominated the products are compared to the products that were previously ranked by the user. This method provides the foundation for data retrieval and data filtering research.
Collaborative filtering: CF method is used to automatically predict or filter information about user interests by collecting settings or tasting data from multiple users (collaborating). The basis of the CF method is assuming that if user U1 has the same taste with user U2, user U1 tends to have the taste of U2 on issues that differ from those of random users [17]. Collaborative recommender systems receive a list of recommended items by analyzing similarities between users and predicting user ratings of an item based on similar user ratings on the same list [18].
Hybrid recommender system: Hybrid methods can be used in many ways. This approach can be making content-based filtering and collaborative-based filtering predictions separately and it can combine them together. It can be used to enhance content-based capabilities through a collaborative-based approach. It can combine all of them into one format [19].
The purpose of many studies about the referral system is to focus on the ability to recommend products that satisfy customers primarily. CF is the most commonly used method for identifying similarities between items.
2.2 Collaborative Filtering Algorithm
The CF method is derived from collecting and analyzing large amounts of data about user behavior, activities or user preferences, and predicts which users will be similar to other users. To understand what CF is, one can think of a simple question; for instance, if a person wants to read a book, but that person does not know which book is good, what will that person do? Usually, most people like to ask friends to see which books are good. We like to receive a suggestion from friends or people who have the same taste as us. This is the main idea of the CF method [20]. It can be separated into two types: user-based and item-based CF. Item-based CF depends on the similarity between the items calculated using the ratings of people for those items. For example, when users who like item I1 also like item I2, the similarity between these two items is are considered similar. User-based CF takes advantage of the similarities between users in the forecast. For example, if Mr. James and Mr. Paul have seen the same movie and they are also giving the same ratings, the similarities between them is 1. On the other side, assuming that they give the different rating to the item, the similarities will decrease as differences.
In this paper, we adopted a user-based CF algorithm. The algorithm predicts that if users' personal characteristics are similar, then their interests in products or items are also similar. The algorithm searches for the most similar users according to the target user information. Based on the most similar interests or preference, a user's interest can be predicted. The recommender system will carry out information for suggest for relevant users. Cosine similarity is used to measure the similarity of users. Its formula [21] is shown in Eq. (1).
[TeX:] $$\operatorname{sim}(\mathrm{A}, \mathrm{B})=\frac{\sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{A}_{\mathrm{i}} \mathrm{B}_{\mathrm{i}}}{\sqrt{\sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{A}_{\mathrm{i}}^{2}} \sqrt{\sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{B}_{\mathrm{i}}^{2}}}$$
In Eq. (1), A and B are two different vectors. Ai is the component of vector A and Bi is the component of vector B, respectively. The result of similarity ranges from −1 to 1. The negative value is the opposite; it refers to two different vectors. A positive value represents two similar vectors. With 0 indicating orthogonally or de-correlation while in-between values indicate intermediate similarity or dissimilarity.
2.3 Social Network Analysis
The SNA is a method used to analyze social network properties. It characterizes networked structures in terms of nodes, such as individual actors, people, or things in the network. Vertices indicate objects or entities while edges indicate links to show relationships or interactions. Both objects and the links may have attributes. Networks are constructed from general, real-world data. They propose several unexpected challenges pending to data domains themselves, e.g., information distillation, pre-processing, and data structures used for displaying knowledge and storage [22]. Social networks represent
Community detect based on edge betweenness algorithm.
Community structure: (a) input graph, (b) computed betweenness values for the edges in graph, (c) at the end of iteration 1, remaining edges in the graph after removed edge 4–6 with highest betweenness score of 25.0, (d) at the end of iteration 6, remaining edges in the graph after removed edges 6–7 with betweenness score is 7.0, (e) at the end of iteration 7, remaining edges in the graph after removed edges 3 and 4 with betweenness score is 7.0, (f) at the end of iteration 10, remaining edges in the graph after removed edge 0–2 with betweenness score of 5.0, (g) at the end of iteration 11, remaining edges in the graph after removed edges 5–6 with betweenness score is 5.0, (h) at the end of iteration 12, remaining edges in the graph after removed edge 8–10 with betweenness score is 5.0, and (i) at the end of iteration 19, final partitioning of the network graph into communities.
relationships that exist within the community. Social networks provide tools for collaborative education, especially through the theory developed in analyzing social networks [23]. Even in the same community, there may be various types of social networks depending on social relationships as friends, mutual support and cooperation. The similarity is a common standard used to construct social relationship components of a community. Actors or nodes in social networks can be individuals, groups, objects, organizations or events, as long as certain relationships remain together. Centrality is an agent indicator used in SNA. There are many types of centrality including degrees' centrality, closeness centrality, and betweenness centrality.
Betweenness centrality is the measure of the center of the graph based on the shortest path. For all vertices in the connected graph, there is at least one shortest path between the vertices, which is each number of edges that pass through or the sum of the weight of the minimized edge. The betweenness centrality for each vertex is the number of shortest paths that pass through the vertex. Girvan and Newman [24] have presented a community detection algorithm in social networks and biological networks based on edge betweenness to avoid the flaws of hierarchical clustering methods. This algorithm continuously detects communities by removing the edges from the original network, and the connected components of the network that remain are communities. Instead of trying to create a measure that tells us which edge is most central to the community, but this algorithm focuses on the edge that seems to be "between" communities. The algorithm steps for community detection are summarized as Fig. 1.
An example in Fig. 2 shows the effective algorithm Girvan-Newman for edge community detection based on betweenness.
3. Movies Recommender Algorithm Using Social Network Analysis to Alleviate Cold-Start Problem
The process of data collection and workflow processes that are sufficient for the recommended system are shown in Fig. 3. ① The system needs to collect user data and movie listings into the database used as a test dataset. ② The system requests that new users log in if they want to join. ③ The system needs to collect new user data and movie listings into the database used as a test dataset. ④ In this process we make a relationship table for a user based on their personal propensity such as age, gender, and occupation; we call this table an adjacency matrix. ⑤ The relationship between the user table or adjacency matrix is used to evaluate a community out of several communities (groups) by analyzing the relationship table of the user and comparing it using community detection based on edge betweenness. ⑥ In this process, after a community (group) of the user has been selected for evaluation, then we modify the group for the new user by computationally comparing the similarity of the new user to other users in
Movie recommender system configuration diagram.
each group from their personal information. For computing, the similarity of user cosine similarity is used. ⑦ After that we match a group with the new user, the group with the highest similarity will be selected as a group of new users. ⑧ The movies that were watched and rated by users in the matching group of the new user will be arranged in the order of popularity. ⑨ In this process, the recommended system will select the top 5, 10, 20, 30 and 40 most popular movies and finally ⑩ the system will choose the most popular movies that were watched by the members in the group recommend these to the new user.
3.1 Detail of Proposed Processes
3.1.1 New user provides personal information
To recommend the best movies for users, we need to collect some necessary personal information about these users. Users who have the same behavior or same personal propensity may like the same item. Personal propensity that we use for performance in this research includes age, gender, and occupation.
3.1.2 The relationship between users table
In our proposed algorithm, the centrality of SNA was applied. Therefore, we have to make a relationship table between users according to their personal propensity including age, gender, and occupation. We assume that if the users have the same personal information it means they have a relationship to each order. This table can be formalized as a classical mathematical relationship that can be seen as an unspecified graph.
3.1.3 Community detection based on edge betweenness.
After getting the relationship between users table, in this process, we want to cluster users into several groups by applying the relationship between users table. Community detection based on edge betweenness is used for clustering users, a community representation as a group. The concept of detection community based on edge betweenness is that the possibility those edge connections separate modules have the highest betweenness value because the shortest path from one module to another module must cross through those modules. Therefore, if we gradually remove the edge with the highest edge betweenness value, we will get a hierarchical map or a rooted tree, called a dendrogram of the graph. The leaves of the tree are the person's vertices and the root of the tree means the whole graph in the network.
3.1.4 Cosine similarity and Collaborative filtering
Traditional methods to measure the similarity of users just consider the similarities of user ratings. In reality, the similarity of users is not only linked to ratings for items, but is also linked to the preference for certain item categories, that is, user interest for the item category feature. In addition, if two users have similar personal information, these two users are considered highly similar. Therefore, our research modified the group for the new user by using the cosine similarity measure. After users are divided into several groups by community detection based on edge betweenness, then the similarity between the new user and the other users in each group is computed by filtering personal information including age, gender, and occupation. For computing the similarity between users Eq. (1) is used.
3.1.5 The matching group for the new user
After completing the calculation of similarities between new users and other users in each group, we compared the average similarities of new users and each group. The group with the highest similarity was selected as the most similar group for related users.
3.1.6 Ranking the popularity movies
When finding a group that was most similar to a new user, movies watched by group members will be ranked according to their popularity, which was counted from the rating that each member of the group had given to each movie.
3.1.7 Recommended movies to the new user
After ranking popular movies from users that were similar to new users, the most popular movies were selected to be recommended for new users. However, the final decision about which movies the user will watch will depend on the new user.
In this article, we try to combine existing CF techniques with SNA. We use between centrality identification method and the introduction of a movie recommendation system that may give the best predictions about the movie program that the target user might be interested in. The algorithm used in our recommendation system is shown in Algorithm 1.
movies recommender system using social network analysis and collaborative filtering
4. Experimental Analysis
4.1 Experimental Dataset
In the experiment in this research, we used the MovieLens dataset provided by the GroupLens research group at the University of Minnesota, USA [25]. It comprises data from three orders of scale. Each data set has related user's information, user's ratings, and the movie's information. This dataset consisted of 100,000 ratings; the rating is in range of 1 to 5. This dataset includes 943 users and 1,682 movies. Each user gives a rating to movies at least 20 movies. This dataset also has simple information for each user, such as gender, age and occupation.
To evaluate the quality of our proposed method, the dataset is divided into two parts: the training and testing set. It is very important to evaluate performance using data not involved in formulating the model. To improve the accuracy of the recommendation system, both sets of dataset included 10 random datasets from users who gave ratings to at least 20 movies, users who gave ratings to at least 50 movies, users who gave ratings to at least 100 movies, and users who gave ratings to at least 200 movies. This means we have to implement the method a total of 40 times.
Training dataset is used after a model has been processed. In the setting of this recommender system, partitioning is performed by randomly selecting some users and some ratings from all users. There are 800 users in this dataset. Training set is implemented to build up a model. Testing dataset is used to test the model by making predictions. There are 143 users in this dataset. Some users and some ratings are then randomly selected from all users.
MovieLens' user information part
Users–movies matrix part.
The relationship table between users
4.2 Experimental Environment
Hardware and software used to evaluate the methods proposed in the paper are shown in Table 2. R programming language is an open source scripting language for predictive analysis and visualization. The R programming language includes functions that support linear modeling, non-linear modeling, classical statistics, classifications, clustering, and more. It has remained popular in academic settings due to its robust features and the fact that it is free to download in source code form under the terms of the Free Software Foundation's GNU general public license. It compiles and runs on UNIX platforms and other systems including Linux, Windows, and Mac OS. Hence, we can easily identify the source code to see what it is doing on the screen. Anyone can fix bugs and add a feature without having to wait for the seller to do it for us. Moreover, it always allows us to integrate with other languages (C, C++). Furthermore, it enables us to interact with many data sources and statistical packages (SAS, SPSS).
4.3 Experimental Results
Several metrics have been proposed to evaluate the accuracy of the CF method algorithm. Mean absolute error (MAE) is one of the most commonly used tools for measuring the accuracy of the recommender system [26]. MAE evaluates the accuracy of a prediction algorithm by comparing numerical deviation of the predicted rating from the respective actual user rating. Formally, if n is the number of an actual item that is purchase by a target user and MAE is assigned as the mean absolute difference between the pair. And assuming that the predicted rating set of the target user u is [TeX:] $$\left\{p_{u 1}, p_{u 2}, \dots,\right.p_{u N} \}$$ and actual rating set is [TeX:] $$\left\{r_{u 1}, r_{u 2}, \ldots, r_{u N}\right\}$$, then MAE is defined as follows:
[TeX:] $$M A E=\frac{\sum_{i=1}^{N}\left|P_{u i}-r_{u i}\right|}{N}$$
After performing the proposed method, we want to predict the accuracy of our proposed method. We want to know when our system recommended movies, how many movies were actually watched from the user. MAE can help measure the level of satisfaction and evaluate the accuracy of the recommender system. Normally, the lower the value of the MAE, the higher the accuracy of the recommendation. To the computation of MAE, the Eq. (2) is used.
As described in Section 4.1 we performed experiments in four cases: random testing datasets from users who gave ratings to at least 20, 50, 100, 200 movies. To reduce inaccuracy of the MAE values, we performed 10 experiments using each dataset. In total, of an experiment are 40 times. The average results of the movie recommendation system using SNA and CF for 10 experiments are shown in Fig. 6.
Fig. 5 shows the average of all experiments from 10 random times; the best result for the number of movie recommendations was 5. Therefore, we averaged the result from four cases; the average result from four cases was 3.55 when 5 movies were recommended. The average result from four cases was 3.79 when 10 movies were recommended. The average result from four cases was 4.34 when 20 movies were recommended. The average result from four cases was 4.59 when 30 movies were recommended. The average result from four cases was 4.78 when 40 movies were recommended. The maximum MAE value is 10. Therefore, we can say that our proposed method is very effective and can solve the cold-start problem. From these results, we can also interpret that the user was interested in more than 3 out of 5 movies recommended by the system.
The result of social network analysis (SNA) and collaborative filtering (CF) method.
In order to confirm the effectiveness of the methods that are presented in this paper, we have compared with other methods including density-based on clustering [27] method, the CF with k-NN, and CF method.
The results of the movie recommendation system using density-based on clustering by used the same dataset as a proposed method are shown in Fig. 7.
In Fig. 6, the average result from four cases was 4.19 when 5 movies were recommended. The average result from four cases was 4.76 when 10 movies were recommended. The average result from four cases was 5.28 when 20 movies were recommended. The average result from four cases was 5.44 when 30 movies were recommended. The average result from four cases was 5.63 when 40 movies were recommended. The maximum MAE value is 10. The best number of movies recommended in this method was also 5.
The result of density-based on clustering method.
Fig. 8 showed the results of the movie recommender system using k-NN and CF, the average result from four cases was 3.60 when 5 movies were recommended. The average result from four cases was 3.96 when 10 movies were recommended. The average result from four cases was 4.35 when 20 movies were recommended. The average result from four cases was 4.61 when 30 movies were recommended. The average result from four cases was 4.85 when 40 movies were recommended. The maximum MAE value is 10. The best number of movies recommended in this method was also 5, as in previous methods.
The result k-NN and collaborative filtering method.
Traditional datasets that are not tuned have been tested using traditional CF algorithm. For the movie recommendation system based on original CF, the neighbor for the user who has the same taste as the new user was found. The result of the movie recommender system using CF is shown in Fig. 9. When 5 movies were recommended the result was 5.75. When 10 movies were recommended the result was 5.84. When 20 movies were recommended the result was 6.06. When 30 movies were recommended the result was 6.29. When 40 movies were recommended the result was 6.48. The best number of movies recommended using this method was also 5; same as other methods.
After that, we will show the efficiency of each method by comparing the total average of the MAE results of the four methods. The total average result of the movie recommendation system that is proposed to use k-NN and CF shows better results than the total average result of the movie recommendation system that is proposed to use density-based clustering. The k-NN and CF method also shows better results than the results of the movie recommendation that is proposed to use original CF. The total average result of the method we propose, which is a movie recommendation system that is proposed to use SNA and CF, is more accurate than the other three methods as shown in Fig. 10. We also have to argue that the efficiency of our method is better than the other three methods. We also have to argue that the efficiency of the three methods is the best of our method.
The result of collaborative filtering method.
Fig. 10.
Comparative the result of the experimental.
5. Conclusions and Future Work
This paper aims to solve the problem of CF by using SNA in the recommender system. We design and implement in R programming which is an open source scripting language for predictive analytics and data visualization. The recommendation system is a way to help the users to find the information that they want easily. CF is one of the most popular uses one and a successful method used in the recommender system. However, it has weakness such as cold start problem. To overcome this problem, in this paper we proposed an alternative approach for the recommender system using both SNA and CF. We found the community or group for the user based on edge betweenness centrality. The method that we proposed here is very effective for making movie recommendations. Analyzing results showed that the total number of 20 movies recommended was better than 40 and the best number of movies recommended was 5. In addition, the method presented in this paper showed the best performance, followed by k-NN and CF, Density-based clustering, and CF.
However, the implementation processed in this paper by using R programming took a very long time. Therefore, we propose to reduce the experiment time and to improve the accuracy and effectiveness of the recommender system by applying various types of datasets in future research.
This research was supported by the MSIT (Ministry of Science and ICT), Korea, under the ITRC (Information Technology Research Center) support program (IITP-2019-2014-1-00720) supervised by the IITP (Institute for Information & communications Technology Planning & Evaluation) and the National Research Foundation of Korea (No. 2017R1A2B1008421).
Khamphaphone Xinchang
She received B.S. in School of Information Technology from National University of Laos, Laos, 2016. Since March 2017, she is with the Department of Computer Sciences and Engineering from Soonchunhyang University, Korea as a Master student. Her current research interests include data mining and parallel processing.
Phonexay Vilakone
He received his B.S. in Mathematics and Computer Sciences from the National University of Laos, Laos, 2003. He received the master degree of Computer Application (Software System) from Guru Gobind Sigh Indraprastha University, India, 2010. Since March 2017, he is with the Department of Computer Science and Engineering from Soonchunhyang University, Korea as a PhD candidate. His research interests include data mining and parallel processing.
Doo-Soon Park
He received his Ph.D. in Computer Science from Korea University in 1988. Currently, he is a professor in the Department of Computer Software Engineering at Soonchunhyang University, Korea. He is director of Wellness Service Coaching Center at Soonchunhyang University, and director of Computer Software Research Group in the Korea Information Processing Society (KIPS). He was president of KIPS from 2015 to 2015, and director of Central Library at Soonchunhyang University from 2014 to 2015. He was editor in chief of Journal of Information Processing Systems (JIPS) at KIPS from 2009 to 2012, and Dean of the Engineering College at Soonchunhyang University from 2002 to 2003. He has served as an organizing committee member of international conferences including, FutureTech 2018, WORLDIT 2018, GLOBAL IT 2018, CSA 2017, BIC 2017, MUE 2017, WORLDIT 2017, GLOBALIT 2017, CUTE 2016, FutureTech 2016, MUE 2016, WORLDIT 2016, GLOBAL IT 2016. His research interests include data mining, big data processing and parallel processing. He is a member of IEEE, ACM, KIPS, KMS, and KIISE.
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u1 1 0 0 1 0
Operating system Windows 7 Ultimate Service Pack 1
Processor Intel Core i7 2600k CPU @3.40 GHz
System type 64-bit operating system
Hard disk 500 GB
Data analysis software R i386 3.4.2 | CommonCrawl |
Epidemiology and clinico-pathological characteristics of current goat pox outbreak in North Vietnam
Trang Hong Pham1,2,
Mohd Azmi Mohd Lila1,
Nor Yasmin Abd. Rahaman1,
Huong Lan Thi Lai2,
Lan Thi Nguyen2,
Khien Van Do3 &
Mustapha M. Noordin ORCID: orcid.org/0000-0001-9288-797X1
In view of the current swine fever outbreak and the government aspiration to increase the goat population, a need arises to control and prevent outbreaks of goat pox. Despite North Vietnam facing sporadic cases of goat pox, this most recent outbreak had the highest recorded morbidity, mortality and case fatality rate. Thus, owing to the likelihood of a widespread recurrence of goat pox infection, an analysis of that outbreak was done based on selected signalment, management and disease pattern (signs and pathology) parameters. This includes examination of animals, inspection of facilities, tissue sampling and analysis for confirmation of goatpox along with questionaires.
It was found that the susceptible age group were between 3 and 6 months old kids while higher infection rate occurred in those under the free-range rearing system. The clinical signs of pyrexia, anorexia, nasal discharge and lesions of pocks were not restricted to the skin but have extended into the lung and intestine. The pathogen had been confirmed in positive cases via PCR as goat pox with prevalence of 79.69%.
The epidemiology of the current goat pox outbreak in North Vietnam denotes a significant prevalence which may affect the industry. This signals the importance of identifying the salient clinical signs and post mortem lesions of goat pox at the field level in order to achieve an effective control of the disease.
The re-emerging of Capripoxvirus and it's clinical syndrome has been well documented worldwide especially in Asia and Africa [1, 2]. Undoubtedly, this virus bears pronounced economic impact not only to endemic regions especially to the livelihoods of small-scale farmers and poor rural communities [3] but also posed major constraint in international livestock trade. A greater concern is the risk of its expansion to many countries including Vietnam in 2005 [4] which is in the midst of developing a competitive goat industry. The first reported goat pox outbreak of North Vietnam in 2005 that affected four provinces i.e. Coa Bang, Bac Giang, Lang Son and Ha Tay has led to the death of 789 goats. The agents confirmed via ELISA and PCR yielded that the isolate was host specific being severe in goats [4]. Following this incidence, the outbreak has been resolved leading to an annual increase of 38% in Vietnam goat population from 1.8 million heads in 2015 to 2.6 million heads in 2017 [5]. Owing to the Vietnamese government aspiration to produce 3.9 million heads of goats in 2020, a much more comprehensive study on devastating disease like the epidemio-economical impact of goat pox is warranted. Nevertheless, despite the increase in goat population in addition to animal movement along the borders, market demands, high stocking densities and proximity of facility, goat pox outbreak has recurred commencing from 2014 in Ninh Binh province. This recurrence has raised concern on the possible devastating impact of goat pox on Vietnam's goat industry which forms the basis of this study. A thorough analysis of current recurrence along with a complete set of epidemiological data will confer an effective control and prevention of new outbreaks.
Observation of the farms
Vaccination against goat pox was not practised in either type of farming systems. The main goat rearing methods in North Vietnam under the extensive system includes backyard farming where the goats are allowed to freely graze in lowland and mountainous areas. Under such system there is minimal provision of commercial feed. On the otherhand, under intensive farming, the goats are kept in stalls and supplemented with concentrates.
Morbidity rate
The morbidity and mortality rates due to goat pox is shown in Table 1. During this study, the first case of sick goats was reported in Ninh Binh province which then radiated to other parts of North Vietnam. Thus, the study commenced in Ninh Binh and radiated out to its five other surrounding provinces. The morbidity rate ranged between 11.8–17.5% without significant differences between all provinces except for Yen Bai which has the lowest rate (p < 0.000). However, this lowest rate at Yen Bai was not significantly different to that seen in Hoa Binh.
Table 1 Morbidity rate of goatpox outbreak in North Vietnam
Mortality and case fatality rate of goat pox outbreak
Table 2 shows the case fatality rate of goats due to the infection during the study period. The mortality and case fatality rate ranges between 5.1–7.4% and 35.3–63%, respectively without any significant differences between provinces.
Table 2 Mortality and case fatality rate of goat pox disease in North Vietnam
Infection rate between farming system
It was found that goats under the extensive system has a 8.7% higher (p < 0.05) infection rate than those managed intensively (Table 3).
Table 3 Comparison of goat pox incidence based on rearing method
Age susceptibility
In order to examine the influence of age to infection rate, the goats were into categorized into three groups, viz.; less than 3; 3–6 months and more than 6 months old. The analysis of age susceptibility to infection is shown in Table 4. It was found that at almost all instances, those between the ages of 3–6 months were most susceptible (p < 0.001) except at Ninh Bin province. The other age groups of less than 3 and more than 6 months have comparable infection rate.
Table 4 The infection rate based on age groups
Clinical and pathology findings
Goats showed varying degrees of clinical signs severity, however, almost 85% of infected goats showed loss of appetite, anorexia to completely refusal of feed leading to emaciation (Table 5). Fatigue and pyrexia were also among common manifestations observed in most cases. Additionally, blepharitis, rhinitis (Fig. 1) and difficulty to move ensued in some cases.
Table 5 Distribution of clinical signs of goat pox based on their occurrence (n = 1814)
Photograph showing ulcers in nasal cavity and rhinitis
Hardened swelling which developed into sores were found on the skin (mainly hairless regions) over any part of the body including the mouth, pinna (Fig. 2) and udder (Figs. 3 and 4). The size of the pock lesions varies between 0.5–1 cm in diameter.
Photograph exhibiting papules found on mouth, nares and ear
Photograph showing a papule that has ulcerated on the ear pinna
Photograph of infected goat's udder denoting ulcers and inflammation
The finding of lesions ante- and post mortem is presented in Table 6. In live animals, majority of lesions are confined to the eyes, nares and skin while that of post mortem revealed the lungs (Fig. 5) as a primary site. Calcified greyish papules were found in the intestines (Fig. 6), urinary bladder and uterus. However, other less frequently sites and tissues were also affected as shown in Table 6.
Table 6 Lesion distribution in selected organs and their frequency of appearance (n = 128)
Photograph of a well-circumscribed greyish pock lesion in the lung of an infected goat
Photograph of calcified papules on the intestinal mucosa of an affected goat
Histopathological lesions comprising of cellular degeneration and necrosis along with inflammation and haemorrhage were mostly found in skin (Fig. 7), lung and liver. Despite exhaustive histopathology search, no evidence of eosinophilic inclusions were seen in any tissues.
Damaged epithelial layers of skin of an infected goat (H&E, X10)
The PCR primer specific test was performed on 128 scab biopsy samples. A total of 79.6% (102/128) of the samples were positive to capripox virus within the expected size band of 172 bp (Fig. 8).
PCRA gene based PCR result for detection of capripox virus. Lane M: 100 bp ladder
Reported outbreaks of goat pox worldwide yields differing mortality rates with 7% in Sudan [2], 21% in Iraq [6] and 30% in India [7]. In this study, a much lower mortality rate was found despite a rather high morbidity rate high probably as a result of the study population containing comparatively fewer of the 3–6 months old goats. It has been shown that maternal antibody for goat pox is maintained for about 3 months and those animals older than 6 months that survived an infection will have life-long immunity [8, 9]. This phenomenon explains the susceptibility of those in the 3–6 months old [9] which should yield higher morbidity rate. However, since the number of animals under this group is quite low, the mortality rate has failed to surpass those of other groups.
The number of dead animals during the outbreak depends on the virus virulence, size of the population and their susceptibility and on the basic reproductive number i.e., average expected number of secondary cases produced by a single infection in a completely susceptible population [10]. However, these rates may vary depending on additional factors including breed [11] and the most notably the herd immune status [12]. Recently published data showed that case fatality rate of goatpox disease ranged from 21.4 to 60% [13,14,15]. Likewise, the high fatality rate in the present study underlined the need for a much more effective control of goat pox along with the requirement to vaccinate susceptible herd or in endemic areas. However, the difficulty in implementing such health programs in Vietnam is explained below.
A 23% morbidity rate based on seroprevalence has been documented in nomadic goat herds in Punjab [16]. It is not suprising to see a higher infection rate in the extensive system as previously reported [11]. However, this rearing method is popular with poor farmers in lowland and mountainous areas in Vietnam who could not afford to spend on a standard health management. Goats under the extensive system forage freely in a wide area exposing them increase chances to be exposed to the virus. These goats might have also been exposed to lesser domestication, maintaining many of the behavioural traits of the wild types such as aggressiveness [17, 18]. Furthermore, goats especially under the extensive system being naturally aggressive [17] predisposes the body to injuries making easier access of the virus when inoculated. This is an added problem since most of the goats were not dehorned (due to financial constraints) making injuries prone to infection during a fight. On the contrary, the low infection rate under the intensive system could have resulted from a much more efficient disease control program that has minimized spread of the virus within the herd. However, the benefits of extensive farming system can be still exploited by taking advantage of its eco-agrarian nature. It can economically ultilise marginal or unused land that can be later be easily adapted by the goats. Such conditions had less stressful effect on the goats making them much more hardy to harsh conditions. This is an opportunity for the poor rural farmers with limited financial resources and knowledge in commercial goat farming. This can be improved if there is provision of extension veterinary officers to offer guide and assistance in goat farming.
Undoubtedly, defining the vulnerable period of infection is one of the most important measurement to be known for an effective disease management [19]. In the study presented here, the most susceptible age were goats of 3–6 months old which conforms to findings of [16, 20] who found that the chance of infection chance in the young was 2.2 times greater than that of an adult. However, contradictory results were seen if infection rate was based on seroprevalence. Fentie et al. [20] demonstrated a low infection rate in older animals although this appeared to refute earlier published findings [21]. Nevertheless, in the latter study [21], age groups were not clearly defined which may have led to a less homogenous groupings. Additionally, the collected samples from slaughter house, tanneries and hide markets where probable that few samples were collected from goat kids to be devoid [21]. The age grouping the study presented here was based on the main purpose of meat goat breeding in Vietnam. The indigenous and mixed breed of Vietnamese goat attained a market weight of 25 to 30 kg at 6 months old age, justifying a 3 month interval being chosen.
Recognising the key salient clinical signs is key factor for field diagnosis of goat pox [11]. The prominent clinical signs seen in this study too were depression and being much more severe in kids [22, 23] accounting for systemic signs of pyrexia. About 85.01% of affected animals showed varying degrees of anorexia associated with the development of lesions on mucus membrane of the face. The lesion commences as red patches around the mouth, nose and eyes which later swelling into a papule. These papules trigger lacrimal, nasal and saliva discharges. Respiratory distress and secondary bacterial pneumonia are predominant in kids which could not survival malignant stage [6, 24, 25]. In adult goats, the ulceration of papules renders difficulty for digestive and breathing activities which in turn worsen productive performance. The goats with conjunctivitis, corneal opacity and blepharitis emulated the acute phase pox disease [4]. The development of pox lesions is observed over the animal body especially hairless areas (face, pinna of the ears, udder, genital, anus, under the tail). The red patches turn to hard rubbery papules and become vesicles after 3 to 4 days. Necrotic papules formed pustular as the result of thrombosis and localised ischaemia. Dark hard scabs are formed by the remnant of necrotic papules [6, 25,26,27].
Although the overt clinical signs of goat pox are quite characteristic, the less severe manifestation needs to be judiciously distinguished from several other closely resembling diseases. The closest would be contagious ecthyma (orf) which affects young kids while goat pox involves all ages. The signs are usually that of flat or dome-shaped bullae crust around the commissures of mouth which left no scar after healing [28] as opposed to a rather permanent papular lesion in goat pox. Blue-tongue may be confused with goats pox although the goats are less less susceptible with signs rarely seen in goat pox i.e. localized oedema, haemorrhages and erosion of mucous membrane. The post mortem lesions of blue tongue are that of effusion in the thoracic cavity and pericardial sac [29]. High mortality is seen in peste des petits ruminants (PPR) which affects mainly young goats that showed signs of coughing; halitosis, erosive oral lesion and severe diarrhoea. These signs are not seen in goat pox along with rather pathognomonic lesion of PPR comprising of zebra stripes of gastro-intestinal tract and pneumonia [30]. Lastly, a likely differential to be considered to goat pox is dermatophilosis [31] where the latter exhibited signs of paintbrush matted hair all over the body that is not a feature of goat pox.
In this study, for all PCR positive cases, the clinical and post mortem lesions were 100% present in the skin and lungs of affected goats. It is likely that owing to the epitheliotropic nature of the virus lesions were predominantly seen in the skin, lung and discrete sites within mucosal surfaces of oro-nasal and gastrointestinal tissues [4]. As evidenced in this study and as reported earlier in similar studies, the role of skin and lung as a target organ [32] for the virus leads to much more deposition of the lesion in these tissues [33, 34]. Beside darkened circumscribed pox lesions [33, 35], the entire lung are pale pink with loss of sponginess. Congested trachea contain blood or fluid-filled vesicles with involvement of the lymph nodes. As seen in the study presented here, calcified nodules are found the most abudant in large intestine (rectum) of goats that were mildly affected [21, 36].
Histopathological findings in the study presented here were in accord to previous publications registering marked change in the epidermis. The degeneration of epithelial cells, hyperkeratosis, ballooning and degeneration of proliferating epithelial cells along with inflammation led to the desquamation of skin layers. Variable observation of lung microscopy include haemorrhage, congestion and thickening alveoli wall which resulted in narrowed alveoli. Secondary bacterial infection has invoked infiltration of inflammatory cells to affected regions of the lung [6, 37, 38].
The PCR-based test is chosen because of its sensitivity and simplicity [39]. The sensitive and simple PCR assay has confirmed caprine pox virus in the biopsy samples [40]. Almost 80% of the samples were positive with amplicon size of 172 bp although no attempt was made to identify and differentiate of caprine pox virus [1, 22, 41, 42]. However, the isolates from this study did not show much variation compared to those reported in China [43]. This could be explained by the fact that although phylogenetically China has three main subgroups of goatpox, only one is circulating in the south i.e. bordering Vietnam [43].
These findings pose a challenge to the aspiration of Vietnam's to transform the future potential of goat farming into an industry. The local consumer prefers fresh chevon than frozen products due to food safety issues linked to the weakness of their cold chain system [44]. Furthermore, goats as well as being a form of meat for the family and community, goat serves as a cash reserve for the poor farmer [45].
The current study also revealed most of the goat husbandry system is mainly extensive which may hamper the possibility to initiate goat production within the mountainous areas. Likewise, as revealed here, goats reared under the intensive system offers a better farming milieu for disease control which the farmer or nation should adopt to improve productivity. Under an intensive system, the ease to isolate and locate an infected animal and area enabling an effective diagnosis and thus control and prevention. It is rather difficult or almost impossible to perform such tasks (isolate and locate) under free grazing or nomadic conditions. Nevertheless, Vietnam should make formidable reforms to the livestock industry since goats in Vietnam are still (as found in this study) and in future will be reared by the poorer farmers halting an increase in goat population and productivity. This is even much more worrying especially with respect to a lack of herd health program (disease control). Thus, in order to bring the industry to greater heights, offsetting devastating disease like goat pox is mandatory. It is believed that these findings on goat pox will facilitate the government to continue working on improving disease identification and control to avoid hindrance in goat production.
Goat pox infection in North Vietnam if left unattended may lead to devastating effect to the goat industry. Thus, needs arises not only to effectively control the disease but also to downregulate risks factors involved including that of current state of rearing. This includes provision of veterinary extension services to the poor farmers adopting the extensive system in order to improve productivity via an effective herd health program.
Ethics, consent, questionnaire and study area
Since North Vietnam does not impose ethics on the use of local animals for research, all procedures involving in this study were conducted in compliance to the recommendations of the Guide for the Care and Use of Agricultural Animals in Research and Teaching (2010) [46]. A well-defined questionaire composed of farm management information (total number of animals/age groups, breed, farming system and detailed health status) relevant to goat pox were noted during the visit and all participating farms consented the research via a written permission.
The sample size (n) was determined using the formula:
$$ \mathrm{n}={\mathrm{Z}}^2\mathrm{pq}/\mathrm{L} $$
where, Z = standard normal distribution at 95% confidence interval = 1.96
= prevalence of similar work (Babiuk 2008) = 33%
$$ \mathrm{q}=\mathrm{p}\hbox{-} 1 $$
$$ \mathrm{L}=\mathrm{allowable}\ \mathrm{error}\ \mathrm{taken}\ \mathrm{at}\ 5\%=0.05 $$
Thus, the minimum required sample size obtained from the formula for this study was 477.
Disease investigation had been conducted in six provinces in North Vietnam where goat farming is most actively conducted (Fig. 8). In general, goat farming in Vietnam is mainly divided into either extensive or intensive system as previously described [47]. During the visit, farms with clinically affected goats and those in close contact with the herd within outbreak provinces were further assessed. A thorough physical examination of clinical signs was done with emphasis on predilection site of goat pox lesions and animals with severe clinical signs were then post mortem.
Questionaire and data collection
The questionnaire was structured to encompass information of the farm, management system practiced by the owner during an interview. It is compartmentalised to contain three main sections namely; ownership and farm information, herd information and physical plus pathology findings. The template of this questionnaire is attached separately as an Additional file 1.
Tissue sampling
Based on the physical examination, a total of 11,688 goats that falls under the category of being affected or those in contact were chosen. Out of these, 1481 had clear cut signs suggestive of goat pox whereby fresh tissue samples totaling to 128 were collected for further pathology and virology diagnoses.
Approximately 2–3 g of lesions were taken and placed in PBS (7.2 pH with 1% gentamycin) and stored under chilled conditions during delivery. Samples were then transferred to Key Veterinary Biotechnology Laboratory, Vietnam National University of Agriculture, Hanoi, Vietnam. Roughly a 1 cm3 lesion the of skin, lung, heart, liver, intestine, spleen, kidney and lymph node were fixed in 10% buffered formalin and later processed using routinely for histopathological examination.
Polymerase chain of reaction (PCR)
DNA extraction was performed using DNeasy Blood Tissue Kit (Qiagen, Gemany) following manufacturer instruction. Primers used for identifying Capripoxvirus in clinical specimens as previously designed [39].
The forward primer was P1: 5′-TTTCCTGATTTTTCTTACTAT-3 'and the reverse primer was P2: 5'-AAATTATATACGTAAATAAC-3′. 50 μl of reaction mixture contained 5 μl buffer, 3 μl of MgCl2, 2 μl of dNTP mix (10 mM), 2 μl (10 pmol/μl) of each primer, 0.4 μl of Taq-DNA, 12 μl biopsy supernatant and 23.6 μl of RNAse free water. PCR cycle started with initial denaturation at 94 °C for 5 mins, followed with 35 cycles (1 min each) of denaturation at 94 °C, annealing at 50 °C, extension at 72 °C and final extension at 72 °C for 10 mins. The PCR products were examined by 1.5% agarose gel electrophoresis with ethidium bromide staining.
All data obtained was subjected to statistical analysis using the SAS 9.0 (2002), USA and only differences of p < 0.005 were considered as significant.
The datasets generated and/or used during the current study are not available to public as it is owned by the Vietnam National University of Agriculture, Vietnam. However, these can be requested via email from the corresponding authors; Dr. Pham Hong Trang ([email protected]) and/or Prof. Dr. Mustapha M Noordin ([email protected]).
PBS:
Phosphate buffered saline
PCR:
Polymerase chain reaction
Odds ratio
number of animals
Chisq:
Chi Square test
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The authors wish to extend their thanks to thank all staff of Faculty of Veterinary Medicine, Vietnam National University of Agriculture, Vietnam who have participated in the collection of epidemiology data and analysis of samples for the study. We would like to acknowledge deepest gratitude to the farm owners who have unselfishly cooperated in making the study a success.
This study was fully funded by the Vietnam International Education Department Fellowship, Ministry of Education and Training, Vietnam (911 Research Project Grant Scheme) commencing from the design of the study, sample collection, analysis and interpretation of the data which the authors gratefully appreciate.
Faculty of Veterinary Medicine, Universiti Putra Malaysia, 43400, Serdang, Selangor, Malaysia
Trang Hong Pham, Mohd Azmi Mohd Lila, Nor Yasmin Abd. Rahaman & Mustapha M. Noordin
Faculty of Veterinary Medicine, Vietnam National University of Agriculture, Gia-Lam District, Hanoi, 010000, Vietnam
Trang Hong Pham, Huong Lan Thi Lai & Lan Thi Nguyen
Institute of Veterinary Research and Development of Central Vietnam, Nha Trang, Khanh Hoa, 650000, Vietnam
Khien Van Do
Trang Hong Pham
Mohd Azmi Mohd Lila
Nor Yasmin Abd. Rahaman
Huong Lan Thi Lai
Lan Thi Nguyen
Mustapha M. Noordin
THP, HLTH, LTN and KVD conceived the research grant; THP and MMN analysed and interpreted the results; THP and MMN drafted the manuscript with contribution from all authors; MMN, MAML, NYAR revised the manuscript; MMN and MAML supervised running of the project. All authors read and approved the final manuscript.
Correspondence to Trang Hong Pham or Mustapha M. Noordin.
All procedures involving in this study were vetted by the Vietnam International Education Department Fellowship, Ministry of Education and Training, Vietnam (911 Research Project Grant Scheme) in compliance to the recommendations of the Guide for the Care and Use of Agricultural Animals in Research and Teaching (2010) [46], since North Vietnam does not impose ethics on the use of local animals for research. Informed consent form (field studies and sampling) was filled by the farmers whom are owners or managers of the farms that participated in the study.
Questionnaire.
Pham, T.H., Lila, M.A.M., Rahaman, N.Y.A. et al. Epidemiology and clinico-pathological characteristics of current goat pox outbreak in North Vietnam. BMC Vet Res 16, 128 (2020). https://doi.org/10.1186/s12917-020-02345-z
DOI: https://doi.org/10.1186/s12917-020-02345-z
Goat pox | CommonCrawl |
What is the least positive integer with exactly $10$ positive factors?
We need to find the smallest integer, $k,$ that has exactly $10$ factors. $10=5\cdot2=10\cdot1,$ so $k$ must be in one of two forms:
$\bullet$ (1) $k=p_1^4\cdot p_2^1$ for distinct primes $p_1$ and $p_2.$ The smallest such $k$ is attained when $p_1=2$ and $p_2=3,$ which gives $k=2^4\cdot3=48.$
$\bullet$ (2) $k=p^9$ for some prime $p.$ The smallest such $k$ is attained when $p=2,$ which gives $k=2^9>48.$
Thus, the least positive integer with exactly $10$ factors is $\boxed{48}.$ | Math Dataset |
Red and fallow deer determine the density of Ixodes ricinus nymphs containing Anaplasma phagocytophilum
Katsuhisa Takumi1,
Tim R. Hofmeester2 &
Hein Sprong1
Parasites & Vectors volume 14, Article number: 59 (2021) Cite this article
The density of Ixodes ricinus nymphs infected with Anaplasma phagocytophilum is one of the parameters that determines the risk for humans and domesticated animals to contract anaplasmosis. For this, I. ricinus larvae need to take a bloodmeal from free-ranging ungulates, which are competent hosts for A. phagocytophilum.
Here, we compared the contribution of four free-ranging ungulate species, red deer (Cervus elaphus), fallow deer (Dama dama), roe deer (Capreolus capreolus), and wild boar (Sus scrofa), to A. phagocytophilum infections in nymphs. We used a combination of camera and live trapping to quantify the relative availability of vertebrate hosts to questing ticks in 19 Dutch forest sites. Additionally, we collected questing I. ricinus nymphs and tested these for the presence of A. phagocytophilum. Furthermore, we explored two potential mechanisms that could explain differences between species: (i) differences in larval burden, which we based on data from published studies, and (ii) differences in associations with other, non-competent hosts.
Principal component analysis indicated that the density of A. phagocytophilum-infected nymphs (DIN) was higher in forest sites with high availability of red and fallow deer, and to a lesser degree roe deer. Initial results suggest that these differences are not a result of differences in larval burden, but rather differences in associations with other species or other ecological factors.
These results indicate that the risk for contracting anaplasmosis in The Netherlands is likely highest in the few areas where red and fallow deer are present. Future studies are needed to explore the mechanisms behind this association.
Anaplasma phagocytophilum is the causative agent of human granulocytic anaplasmosis (HGA), and it also causes disease and economic losses in domesticated animals [1,2,3]. The first human case in Europe was reported in 1995. Since then, HGA cases have only been occasionally reported throughout Europe [4]. It is unclear to what extent HGA poses a health burden in Europe: epidemiological data on the disease incidence and disease burden are either incomplete or lacking from most European countries [2]. The non-specificity of the reported symptoms, poor diagnostic tools, and lack of awareness of public health professionals may also complicate these estimations [5]. Anaplasma phagocytophilum is transmitted by the bite of an infected tick [6, 7]. The main vector in Europe is Ixodes ricinus, which also transmits Borrelia burgdorferi (s.l.), the causative agent of Lyme borreliosis, and several other pathogens [8]. The geographic spread and density of I. ricinus infected with A. phagocytophilum are important determinants of the disease risk [9, 10]. Anaplasma phagocytophilum seems to appear in all countries across Europe with infection prevalence in nymphs (NIP) varying between and within countries from 0 to 25% [1]. Understanding which factors determine the spatial and temporal distribution of I. ricinus infected with A. phagocytophilum is needed for risk assessments and for formulating possible intervention strategies.
Variations in the density of questing I. ricinus infected with A. phagocytophilum (DIN) have partly been attributed to environmental factors such as differences in weather conditions [11, 12], habitat characteristics [13], as well as vertebrate communities [14]. Whereas small mammals and birds are considered to feed the majority of immature I. ricinus, ungulates act as their main propagation host [15]. It is, however, still unclear which host species form the main reservoir for A. phagocytophilum and therefore contribute most to the density of infected ticks. Anaplasma phagocytophilum has been found to infect many vertebrate species [1], but its genetic diversity indicates that there are multiple genetic variants, or ecotypes, with distinct but overlapping transmission cycles, pathogenicity, and geographical origin [16,17,18]. A variety of wildlife species, like red deer (Cervus elaphus), fallow deer (Dama dama), wild boar (Sus scrofa), and European hedgehog (Erinaceus europaeus), are harbouring A. phagocytophilum variants that can cause disease in humans and domesticated animals, whereas roe deer (Capreolus capreolus), rodents, and birds seem to carry genetic variants that have until now not been associated with human disease [17].
About two-thirds of tick bites reported in The Netherlands are I. ricinus nymphs [19]. Therefore, the density of questing nymphs infected with A. phagocytophilum (DIN) is an important ecological parameter that, together with the level of human exposure, determines tick-borne disease risk [9, 20]. The DIN is calculated by multiplying the density of questing I. ricinus nymphs (DON) by nymphal infection prevalence (NIP). The transmission of A. phagocytophilum predominantly relies on horizontal transmission between ticks and vertebrate hosts and on transstadial transmission in its vectors, as vertical (transovarial) transmission has not been documented for I. ricinus. Therefore, an I. ricinus larva needs to take a bloodmeal from an infected vertebrate host to become an infected I. ricinus nymph. The availability of (infected) vertebrates to questing larvae generally drives the density of (infected) I. ricinus nymphs [14, 21, 22].
Using data from a cross-sectional study estimating the availability of hosts with camera and live traps in 19 Dutch forest sites, we quantified a moment when a questing tick encounters an ungulate; the probability of this event predicts both I. ricinus nymphal density and A. phagocytophilum DIN [14]. However, the reported association for A. phagocytophilum DIN left a relatively large proportion of the variation among sites unexplained, which could be because of the grouping of four ungulate species. The group of ungulates considered consisted of four species that differed in their ecology and potentially in their ability to be hosts for I. ricinus and A. phagocytophilum.
Here, we present a re-analysis of the A. phagocytophilum data from the cross-sectional study to disentangle the role of the four ungulate species in determining A. phagocytophilum DIN. We first applied a principal component analysis to the camera and live trapping data to test if availability of any of the four ungulate species or combinations of species was associated with A. phagocytophilum DIN. Second, we used simple mathematical models to explore two potential mechanisms that could explain differences between species: (i) differences in I. ricinus larval burden and (ii) potential associations of the different ungulate species with alternative incompetent host species.
Cross-sectional study
We made use of data from an extensive field survey that was carried out in 19 1-ha sites located in forested areas in The Netherlands in 2013 and 2014. Data were collected on the density of questing I. ricinus (blanket dragging), vertebrate communities (camera and live trapping), and infection rates of tick-borne pathogens (qPCR detection). The sites, methodologies, and data have been described elsewhere as well as a series of detailed analyses [14, 22,23,24].
Host attribution
We arranged the encounter probabilities for all forest sites (n = 19) and vertebrate species (n = 32) into a matrix \(\mathbf{A}\in {\mathbb{R}}^{19\times 32}\) having 19 rows and 32 columns. We further arranged A. phagocytophilum DIN into a vector \(b\) matching the order of the forest sites along the rows of \(\mathbf{A}\). To attribute A. phagocytophilum DIN to an assemblage of 32 vertebrate species, we factored the matrix \(\mathbf{A}\) into two orthogonal matrices
$$\mathbf{U}=[{u}_{1},{u}_{2},\dots ,{u}_{19}]\in {\mathbb{R}}^{19\times 19} \mathrm{and} \mathbf{V}=[{v}_{1},{v}_{2},\dots ,{v}_{32}]\in {\mathbb{R}}^{32\times 32}$$
and a diagonal matrix \({\varvec{\Sigma}}\in {\mathbb{R}}^{19\times 32}\) with the singular values \({\sigma }_{1}\ge {\sigma }_{2}\dots \ge {\sigma }_{19}\ge 0\) and the remaining entries equal to zero. Theorem 2.5.2 [25] proves that \(\mathbf{A}=\mathbf{U}{\varvec{\Sigma}}{\mathbf{V}}^{T}\). The column vectors \({u}_{i}\) and \({v}_{i}\) are principal components, also known as singular vectors.
The A. phagocytophilum DIN increased at each forest site because of the first principal components by the amount (Theorem 5.5.1 [25])
$$\frac{{u}_{1}\cdot b}{{\sigma }_{1}}\mathbf{A}{v}_{1}\in {\mathbb{R}}^{19}.$$
We attributed Eq. (1) to roe deer because the highest contribution from the first principal component \({v}_{1}\) comes from roe deer. Next, we applied the theorem again to quantify the inputs from the lower principal components \({v}_{2},{v}_{3}\dots\),
$$y=\sum_{i=2}^{8}\frac{{u}_{i}\cdot b}{{\sigma }_{i}}\mathbf{A}{v}_{i}\in {\mathbb{R}}^{19}.$$
Lower components \({v}_{9}\dots {v}_{19}\) are ignored because the tail sum \(\sum_{i=9}^{19}{\sigma }_{i}^{2}\) is negligible (2.43%) compared to the whole cumulative sum \(\sum_{i=2}^{19}{\sigma }_{i}^{2}\). Next, we define
$$\begin{array}{cc}{y}^{+}& =\mathrm{max}(y,0),\\ & \end{array}$$
$$\begin{array}{cc}& \\ {y}^{-}& =\mathrm{max}(-y,0).\end{array}$$
Intuitively, we clipped the solution \(y\) into the positive part \({y}^{+}\) and the negative part \({y}^{-}\). We attributed Eq. (2) to fallow deer, red deer, and wild boar because the highest contributions from the second principal component \({v}_{2}\) and the third \({v}_{3}\) come from these species.
Larval tick burden on ungulates
Ungulates generally contribute relatively little as hosts for feeding I. ricinus larvae compared to rodents and birds [15]. Nevertheless, as important hosts for A. phagocytophilum, ungulates might feed a significant fraction of larvae that later become A. phagocytophilum-infected nymphs. Thus, differences in larval burden between ungulate species could contribute to differences in their importance as hosts contributing to A. phagocytophilum DIN. To explore this, we compiled data from published studies that collected I. ricinus larvae attached to individual ungulates (see Additional file 1: Table S1). We extracted species, the number of checked animals, and the number of I. ricinus larvae attached to the animals. We fit the negative binomial model (log-link) to the number of larval ticks using the number of animals and the species as predicting variables. We tested the significance of the species predicting variable by performing the likelihood ratio test.
Associations with other woodland species
We explored whether differences between species could be explained by associations of the different ungulate species with the availability of other host species. Here, two species might appear related because of the probability condition (the sum of rates over the host species must equal to one). To remove this potential bias, we performed the following analysis using the encounter rates instead of the encounter probability. For this, we calculated the Pearson correlation in the encounter rate for each ungulate species with each other woodland species. We fit the binomial model (logit link) to the frequency of positive and negative correlation values using the ungulate species as a predicting variable. We tested the significance of the predicting variable by performing the likelihood ratio test.
Absent species interaction, correlation values should be close to zero, and the deviation from the expected value zero should be symmetric. It is possible to calculate the probability of observing as many or more negative correlation values as actually observed in the vertebrate community,
$$\frac{1}{{2}^{-n}}\sum_{j=k}^{n}\left(\genfrac{}{}{0pt}{}{n}{j}\right).$$
This is a partial sum of binomial probability densities where a correlation value is negative with probability \(\frac{1}{2}\). The vertebrate community counts \(n\) members. The number of negative correlation values observed in the vertebrate community equals \(k\). All computations were implemented using the R language [26].
Tick densities
We collected a total of 16,568 I. ricinus nymphs at the 19 forest sites. Most of the collected nymphs (n = 13,967) were tested for the presence of A. phagocytophilum DNA resulting in an overall infection prevalence of 3.3 % (456 infected nymphs), which we used to calculate the density of infected nymphal ticks (DIN). Using correlation analyses, we detected a significant correlation between the number of positive nymphs, NIP, and DIN (Table 1). DIN of A. phagocytophilum lacked a clear correlation with the density of nymphal ticks (DON; Table 1).
Table 1 Density of nymphs infected with A. phagocytophilum (DIN) is unrelated with the density of nymphs (DON)
The encounter of an I. ricinus larva with a woodland species is a critical event to a successful A. phagocytophilum transmission. We quantified the probability of an encounter event based on the information collected using camera- and live-traps in the 19 forest sites (see Additional file 1: Fig S1). We found that the majority of the observed A. phagocytophilum DIN can be attributed to the encounter probabilities of fallow deer, red deer, and wild boar (Fig. 1), based on the correlation of the attributed DIN to the three free-ranging ungulate species (Fig. 2). We, however, did not find a correlation with the attributed DIN to roe deer (Table 2).
PCA plot of encounter probabilities. A black dot is a position of a forest site in 2nd and 3rd PCA coordinates. A box contains two letters abbreviating a forest site name. A green box indicates a high Anaplasma phagocytophilum DIN. A red box dicates a low Anaplasma phagocytophilum DIN. A green dot is a position of a woodland species. The species name is placed in a distance away from the green dot to avoid excessive overlaps
The three free-ranging ungulates except roe deer support Anaplasma phagocytophilum DIN. Observed Anaplasma phagocytophilum DIN was calculated by multiplying the density of questing Ixodes ricinus nymphs by the Anaplasma phagocytophilum NIP. Bar heights in the middle panel (others referring to fallow deer, red deer, and wild boar) were calculated using Eq. (2). Bar heights in the right panel were calculated using Eq. (1)
Table 2 Observed Anaplasma phagocytophilum DIN lacks a correlation with roe deer and correlates with the other fee-ranging ungulates
Larval tick burden on ungulate species
Based on the data extracted from 12 studies in the literature [27,28,29,30,31,32,33,34,35,36,37,38], which together report 24,794 I. ricinus larvae attached to 1860 individual ungulates in 6 European countries, we found no support for the hypothesis that differences in I. ricinus larval burden could have caused the differences in association with A. phagocytophilum DIN among the four ungulate species (Table 3).
Table 3 Analysis of larval ticks on four ungulate species
Reducing availability of woodland species
The encounter rate of roe deer correlated negatively with the encounter rates of 14 woodland species and positively with 17 (Table 4). The probability of exceeding the number of negative correlations is 0.81. The other free-ranging ungulates showed a different pattern: Encounter rates of red deer correlated negatively with encounter rates of 23 woodland species, fallow deer with 26 woodland species, and wild boar with 23 woodland species (Table 4). Consequently, we found a clear difference among these ungulate species in the frequency of negative correlations with other woodland species (p-value = 0.010623, deviance = 11.2140483, d.f. = 3). The probability of exceeding the number of negative correlation is low: red deer: 0.01; fallow deer: 0.00027: wild boar: 0.01.
Table 4 Frequency of positive and negative correlations in encounter rates with 32 woodland species
Re-analysing a cross-sectional study of observed A. phagocytophilum DIN at 19 Dutch forest sites [14], we show that most of the variation in DIN can be explained by differences in encounter probability among ungulates species that were not taken into account in the original analysis. We found a clear association of A. phagocytophilum DIN with the encounter probabilities of fallow deer, red deer, and wild boar. A first exploration of two potential mechanisms that could cause the difference among ungulate species in their contribution to the DIN indicated that negative association of fallow deer, red deer, and wild boar with the encounter rate of other woodland species is a likely candidate for the found differences. In contrast, we did not find support for an explanation based on differences in I. ricinus larval burden. In The Netherlands, fallow deer, red deer, and wild boar only occur in a few specific areas (Fig. 3). Therefore, we cannot rule out if found associations with other woodland species are a result of ecological (interspecies) interactions or species associations with different habitats and wildlife management. It is, however, clear that forested areas where fallow deer, red deer, or wild boar occur likely have the highest risk for humans and domesticated animals to be exposed to A. phagocytophilum though tick bites. Targeted health campaigns to increase the awareness amongst health professionals might help to identify HGA cases and may further be used for stimulation of preventive measures.
(Source: https://www.verspreidingsatlas.nl Green: roe deer. Red: red deer. Blue: fallow deer. Black: wild boar)
Ungulate presence: The Netherlands 2000–2020
Previous research linking species difference to Anaplasma phagocytophilum
We did not find differences among ungulates in their I. ricinus larval burden based on published studies. However, the number of studies was limited (n = 12), especially for some species (n = 1 for fallow deer and wild boar). Thus, the lack of differences is likely a result of large variation in larval burdens and limited sample sizes, especially as studies comparing I. ricinus burden among different ungulate species in single study sites did find differences among species [27, 28]. This indicates that there is a need for more comparative studies investigating differences in the I. ricinus burden of the different ungulates related to differences in infection prevalence with A. phagocytophilum in both hosts and feeding ticks.
We found a negative association in the encounter rates of fallow deer, red deer, and wild boar with the encounter rates of other woodland species, which could be caused by both ecological interactions and management practices. Contrary to roe deer, these three species have a very limited distribution in The Netherlands occurring mainly in the dunes along the coast (fallow deer) and on a large forested area in the middle of The Netherlands called the Veluwe (Fig. 3). This restricted distribution is completely due to present and past wildlife management [39]. Both of these areas are characterized by relatively high sandy soils resulting in relatively low productivity [40]. As a result, both the coastal dunes and the Veluwe harbour a limited number of species, which could explain the negative associations we found. Simultaneously, there could be additional interspecific interactions explaining some of our results. An experimental study [41] reported evidence for cascading effects by ungulates on a temperate forest ecosystem. Experimentally excluding fallow deer, roe deer, red deer, wild boar, and mouflon (Ovis orientalis) from a temperate forest ecosystem in the Veluwe decreased the density of soil grains and increased litter depth, invertebrate biomass, and rodent activity. Future studies are needed to test for the contribution of either mechanism on the found negative associations.
The bacterium A. phagocytophilum is divided into four ecotypes based on the groEL sequences and the host association [16]. These ecotypes differ in their zoonotic potential, with ecotype I being able to cause HGA in humans and anaplasmosis in domesticated animals. This is concerning, as red and fallow deer are considered host species for ecotype I [16]. Thus, the peaks in A. phagocytophilum DIN in areas with a high encounter probability with fallow deer and red deer indicate that these nymphs are likely infected with ecotype I. In contrast, roe deer are considered hosts for the non-zoonotic ecotype II. As the geographic distribution of fallow and red deer in The Netherlands (Fig. 3) is far more restricted than the distribution of roe deer, this would imply that the risk of acquiring HGA is largely restricted to areas where red/fallow deer are present. This might be one of explanations for the low incidence of (reported) HGA cases.
Strikingly, we did not find a correlation of A. phagocytophilum DIN at the forest sites with the encounter probability of a questing nymph with roe deer (Fig. 2). This contrasts with the finding that the density of questing nymphal ticks in the same forest sites correlated with the encounter probability [14]. Thus, we detected in this cross-sectional study some degree of impedance in the A. phagocytophilum life cycle and no evidence of any impedance in the tick life cycle at the forest sites where roe deer is the predominant competent species. This appears to be a common situation in the Dutch forest areas with some exceptions. We observed a high A. phagocytophilum DIN at the forest site Halfmijl (HM) (Fig. 2). Only roe deer and none of the other free-ranging ungulates were captured by the camera trapping at the forest site.
There are additional aspects of A. phagocytophilum transmission, which could influence A. phagocytophilum DIN in theory. Questing nymphs may have been infected by A. phagocytophilum while co-feeding as an uninfected larva next to infected nymphs on the same deer. Next, not all variants of A. phagocytophilum might persist to the same extent during the off-host period. The low DIN associating with roe deer in our dataset might be due to a lower persistence of Ecotype II in I. ricinus during the off-host period. With regard to co-infection, A. phagocytophilum causes a chronic/systemic infection in deer, implying that the subsequent larvae feeding on deer will become infected [16]. The latter will be substantially more than the few larvae feeding at the same time with an infected nymph. With regard to differential persistence, a study of A. phagocytophilum Ecotypes in Central Europe did not find less Ecotype II than Ecotype I in questing I. ricinus nymphs originating from a site in the presence of roe deer [13]. For these reasons, co-feeding and differential persistence were omitted from the analyses.
In conclusion, our re-analysis of the cross-sectional study suggests that the risk of contracting anaplasmosis in the forest with fallow deer, red deer, and wild boar is high compared to the remaining forest sites where roe deer is the predominant competent host. Geographical distribution of deer species in The Netherlands implies that the risk of acquiring HGA is largely restricted to areas where red/fallow deer are present. Additional studies are required to test the inference based on our associative study.
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request
HGA:
Human granulocytic anaplasmosis
Density of questing Ixodes ricinus nymphs
DIN:
Density of infected nymphs
Infection prevalence in nymphs
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This research was financially supported by the Dutch Ministry of Health, Welfare and Sport (VWS), by a Grant from the ZonMw (project number 50‐52200‐98‐313, Ticking on Pandora's box) and a grant from the European Interreg North Sea Region program, as part of the NorthTick project.
Centre for Zoonoses and Environmental Microbiology Centre for Infectious Disease Control, National Institute for Public Health and the Environment (RIVM), Bilthoven, The Netherlands
Katsuhisa Takumi & Hein Sprong
Department of Wildlife Fish and Environmental Studies, Swedish University of Agricultural Sciences, Skogsmarksgränd 7, 907 36, Umeå, Sweden
Tim R. Hofmeester
Katsuhisa Takumi
Hein Sprong
KT analysed and interpreted the data regarding the vertebrate, tick, and the tick-borne pathogen and wrote the manuscript. TH interpreted the analyses and was a major contributor in writing the manuscript. HS seeded the study, provided the funding, and reviewed the manuscript. All authors read and approved the final manuscript.
Correspondence to Katsuhisa Takumi.
Additional figure and tables.
Takumi, K., Hofmeester, T.R. & Sprong, H. Red and fallow deer determine the density of Ixodes ricinus nymphs containing Anaplasma phagocytophilum. Parasites Vectors 14, 59 (2021). https://doi.org/10.1186/s13071-020-04567-4
Ticks and tick-borne diseases | CommonCrawl |
\begin{document}
\title{Self-similarity of Jankins-Neumann ziggurat}
\section{Introduction}
In the theory of dynamical systems on the circle, there is the following very natural question: let $a,b\in \widetilde{\mathrm{Homeo}}_+(S^1)$ be lifts on the real line of two orientation-preserving circle homeomorphisms, and we know their rotation (or, more precisely, translation) numbers $\mathop{\mathrm{rot}^{\sim}}(a),\mathop{\mathrm{rot}^{\sim}}(b)\in\mathbb{R}$. What can be said about the translation number of their composition~$ab$?
Another, more general, form of the same question was studied in a work~\cite{JN} of Jankins and Neumann. It have had topological origins: the question of classification of 3-manifolds, admitting at the same time a Seifert fibration and a codimension one foliation transverse to it. By the moment of Jankins--Neumann's work, the only non-studied case was the one of a manifold fibered over a 2-sphere. Via the study of the corresponding holonomy maps, this have led them to the following question: \begin{question}\label{q:JN} Given $a_1, a_2, \dots, a_n \in [0,1]$, $n\ge 3$, when do there exist lifts $f_1, f_2, \dots, f_n$ of orientation-preserving homeomorphisms of the circle with $\mathop{\mathrm{rot}^{\sim}}(f_i)=a_i$, such that $f_1\dots f_n=id$? \end{question} They have suggested a conjectural answer to this question, also proving that it suffices to establish their conjecture for $n=3$ and that for $n=3$ their conjecture holds at least $99.9\%$ of the volume of the set. The set they have proposed for $n=3$ was later called the Jankins-Neumann ziggurat due to its stepwise nature (see~Fig.~\ref{f1}).
Their conjecture was proven by Naimi in~\cite{Nai}: \begin{Th}[Naimi; conjecture of Jankins-Neumann]\label{zigJN} The set defined in Question~\ref{q:JN} for $n=3$ is the union of parallelepipeds $$ [0;\frac{a}{m}]\times [0;\frac{m-a}{m}]\times [0;\frac{1}{m}] $$ for all coprime $0<a<m$, and of their images under all the permutations of the coordinates. \end{Th}
Later, Calegari and Walker attacked the question of the rotation number of the composition from the dynamical point of view. They have obtained an ``algorithmic'' description for analogous sets for any positive composition of two homeomorphisms with given rotation numbers:
\begin{figure}
\caption{Jankins-Neumann ziggurat (picture credit: Jankins-Neumann~\cite{JN})}
\label{f1}
\end{figure}
\begin{Th}[Calegari--Walker, \mbox{\cite{Caleg}}]\label{t:w} For any word $w$ in the alphabet $a,b$, one has $$ \{\mathop{\mathrm{rot}^{\sim}}(w(a,b)) \mid \mathop{\mathrm{rot}^{\sim}}(a)=x, \mathop{\mathrm{rot}^{\sim}}(b)=y\} = [r_{w}(x,y), R_{w}(x,y)] $$ for certain functions $r_w, R_w:\mathbb{R}\to \mathbb{R}$, and one has $r_w(x,y)=-R_w(-x,-y)$. If the word $w$ is positive (i.e. contains no $a^{-1}$ or $b^{-1}$), there is an explicit algorithm to compute the functions $r_w$, $R_w$ at any rational point $(x,y)$. \end{Th}
This theorem implies, in particular, that the ``ziggurat'' of possible rotation numbers of the composition is described by its upper boundary, the graph of the function~$R_{ab}(x,y)$. An immediate remark is that the function $R_{ab}(x,y)-x-y$ is $\mathbb{Z}^2$-periodic, thus to understand $R_{ab}(x,y)$ it suffices to study it on the unit square $[0,1)^2$. Moreover, starting from their dynamical approach, Calegari and Walker have obtained an explicit formula for $R_{ab}$ in terms, different from those of Jankins and Neumann:
\begin{Th}[{\itshape ab}~Theorem, \mbox{\cite{Caleg}}]\label{t:ab}
\begin{equation}\label{formula} \forall x,y \quad R_{ab}(x, y) = \sup_{\frac{p_1}{q}\leq x, \frac{p_2}{q}\leq y} \frac{p_1 + p_2 +1}{q}. \end{equation} \end{Th}
The purpose of the present text is twofold. First, the Jankins--Neumann ziggurat clearly exhibits some fractal nature (see Fig.~\ref{f1},~\ref{ab}). We study the geometry of this ziggurat, and in particular show that it is indeed the case: that the set of its vertices is self-similar under some simple projective transformations.
Secondly, as it was mentioned earlier, the theorem of Calegari and Walker and the Jankins--Neumann conjecture have the sets described in a different way. It is known that these two descriptions are equivalent (in particular, Calegari--Walker's Theorem~\ref{t:ab} gives an alternative proof of the Jankins--Neumann conjecture). However, we have found a very interesting proof of the passage between the two, and it seems that this way of proving the equivalence was not previously known, and is shorter than existing one. We present this passage in Sec.~\ref{s:equivalence}. Also, we discuss the corollaries of this comparison for the Calegari--Walker formula and the function~$R_{ab}$.
\section{Vertices and the self-similarity}\label{stat}
A first immediate remark is the passage between the definitions of the ziggurat by Jankins--Neumann and by Calegari--Walker:
\begin{pro}\label{zigCW} The triple $(x, y, z) \in [0;1)^3$ of translation numbers can be represented by $f_1, f_2, f_3 \in \widetilde{\mathrm{Homeo}}_+(S^1)$ such that $f_1f_2f_3=\mathop{\mathrm{id}}$ if and only if $z \in [0; R_{ab}(1 - x , 1 - y) - 1]$. \end{pro}
In this section, we will be working with the Jankins--Neumann ziggurat, denoting it by~$\tilde{\mathcal Z}$. Naimi's Theorem~\ref{zigJN} then implies that
\begin{multline}\label{e:zigJN} \tilde \mathcal Z = \bigcup_{\gcd(a,m) = 1} \Pi\left( \frac{a}{m},\frac{m-a}{m}, \frac{1}{m}\right) \cup \bigcup_{\gcd(a,m) = 1} \Pi\left( \frac{a}{m},\frac{1}{m}, \frac{m-a}{m}\right) \\ \cup \bigcup_{\gcd(a,m) = 1} \Pi\left( \frac{1}{m},\frac{a}{m}, \frac{m-a}{m}\right) \end{multline} where $\Pi(a,b,c) = [0;a]\times [0;b]\times [0;c]$.
To state the self-similarity result, let us first introduce the notion: we call point $X$ a \emph{vertex} of a ziggurat $Z$, that is the union of parallelepipeds, if $X \in Z$ and there is no $\Pi(a,b,c) \subset Z$ such that $X \in \Pi(a,b,c)\setminus \{(a,b,c)\}$. In terms of $R_{ab}$, a \emph{vertex} is a point $(x,y,R_{ab}(x,y))\in [0,1)^3$, such that for any $x'\le x$ and $y'\le y$ inequality $x'+y'<x+y$ implies $R(x',y')<R(x,y)$.
Now, we can list the vertices of the Jankins--Neumann ziggurat: a direct corollary of the Naimi's theorem is \begin{pro}\label{p:list} The vertices of the Jankins--Neumann ziggurat are points of the three families, that differ by the permutation of the coordinates: \begin{itemize} \item $\{(\frac{a}{m},\frac{m-a}{m},\frac{1}{m})\}$, with coprime $m>a>0$; \item $\{(\frac{m-a}{m},\frac{1}{m},\frac{a}{m})\}$, with coprime $m>a>0$; \item $\{(\frac{1}{m},\frac{m-a}{m},\frac{a}{m})\}$, with coprime $m>a>0$. \end{itemize} These three families lie respectively on the planes $x+y=1$, $x+z=1$ and $y+z=1$. Moreover, they lie respectively inside the triangles $ABD$, $ACD$ and $BCD$, where $A$, $B$ and $C$ are respectively the points at unit distance on the axes $Ox$, $Oy$ and $Oz$, and $D=(1/2,1/2,1/2)$ is the only common point of all the three families. \end{pro}
\begin{rem}\label{r:min} Formally speaking, to ensure that all the points in the above list are the indeed the vertices, one should check that neither of the parallelepipeds listed in the Jankins--Neumann conjecture is contained in any other. Though, such a check is almost immediate (and we will do it in Sec.~\ref{s:equivalence}). \end{rem}
\begin{cor}\label{c:list} Translating Proposition~\ref{p:list} on the language of $R_{ab}$, we get for its (stepped) graph the families of vertices \begin{itemize} \item $\{(\frac{m-a}{m},\frac{a}{m},1+\frac{1}{m})\}$ with coprime $m>a>0$; \item $\{(\frac{a}{m},1-\frac{1}{m},1+\frac{a}{m})\}$ with coprime $m>a>0$; \item $\{(1-\frac{1}{m},\frac{a}{m},1+\frac{a}{m})\}$ with coprime $m>a>0$. \end{itemize} \end{cor}
Our {first} result, Theorem~\ref{ss} below, states that these points form a self-similar set. But before stating it, we would like to have additional geometric intuition on that set of vertices. Namely, at first glance it seems natural to decompose the ziggurat on Fig.~\ref{f1} into three parts ``near $A$'', ``near $B$'', ``near $C$''. However, as Proposition~\ref{p:list} shows, it is much more important to decompose the vertices into the three families listed in this proposition: the vertices that are on the triangles $ABD$, $ACD$ and $BCD$ respectively.
{Also, we would like to deduce conclusions for the function $R_{ab}$. Thus it is interesting to consider the projection of the ziggurat on the $xy$ plane (marking the level surfaces and discontinuity lines): see Fig.~\ref{proj}. Marking only the vertices on this projection, we get Fig.~\ref{vert}.} Such a projection sends the vertices that correspond to the first family on the line $x+y=1$, and the second and the third families become separated by the diagonal $x=y$: the second comes below the diagonal, while the third one comes above.
\begin{figure}
\caption{Graph of $R_{ab}$, seen from the top}
\caption{Projection of its vertices on the $xy$-plane, with the axis of symmetry marked}
\label{proj}
\label{vert}
\end{figure}
Now, let $\Delta$ be the set that comes from the projection of vertices from the third family. As we have claimed before, this set is then self-similar:
\begin{figure}
\caption{The set $\Delta$ and its self-similarity}
\label{map}
\end{figure}
\begin{Th}[Self-similarity]\label{ss} The set $\Delta$ is self-similar with respect to two projective transformations
$$ T_1(x,y)=\left( \frac{x}{1+x}, \frac{x+y}{1+x}\right), \quad T_2(x,y)=\left( \frac{1}{2-x}, \frac{1+y-x}{2-x}\right), $$ namely, one has $$ \Delta=T_1(\Delta) \sqcup T_2(\Delta) \sqcup \{\left(1/2,1/2\right)\}. $$ \end{Th}
\begin{rem}\label{r:plane} {As all the vertices of the third family lie on the plane $y+z=1$ of the triangle $ACD$, their projection to the $xy$ plane is an affine transformation. Hence, the set of the vertices of the third family is also self-similar under projective transformation that are lifts of $T_1$ and of $T_2$ on this plane. }
\end{rem}
\begin{rem} {Note that the set $\Delta$ is symmetric with respect to the line $x=1/2$. It is an immediate observation if one considers the Jankins--Neumann ziggurat, coming from its full symmetry under the permutation of the coordinates. Though, it is much less clear if one studies the level curves of the function~$R_{ab}$: for any vertex and its symmetric image the projection of the ziggurat on the $xy$ plane collapses one of the sides of the corresponding parallelepiped, but the collapsed sides are different. This is why this symmetry does not appear on Fig.~\ref{proj}.} \end{rem}
{Strangely enough, we did not find any direct way of establishing Theorem~\ref{ss} in the dynamical terms, that is, starting with the explicit formula~\eqref{formula} and without at first deducing the full list of the vertices (in a way it is done in Sec.~\ref{s:equivalence}). That is, one could imagine that having a vertex $(x,y)=(\frac{p_1}{q},\frac{p_2}{q})$ and considering its $T_1$ or $T_2$-image, formula~\eqref{formula} would allow to show that this image is also a vertex. Unfortunately, without stating the full list of vertices first such an argument does not seem to work. Instead, we obtain Theorem~\ref{ss} as a corollary of the Jankins--Neumann original description of the ziggurat (and the associated list of vertices from Proposition~1).}
Another conclusion for the function $R_{ab}$, following from the Jankins--Neumann description is the following \begin{pro}\label{c:rational} The function $R_{ab}$ takes only rational values; in formula~\eqref{formula}, for any $x$ and $y$ the supremum is a maximum. \end{pro} This statement generalizes the Rationality Theorem of Calegari and Walker~\cite[Theorem 3.2]{Caleg}, giving the same rationality conclusion for the rational points~$(x,y)$.
It is also an interesting remark (though \emph{a posteriori} almost immediate to prove) that the projections of the vertices are indeed aligned along the lines that are ``visible'' on Fig.~\ref{vert}. The following theorem formalizes this statement; to state it, consider two families of lines. Namely, the ``green'' family of lines passing through the point $(0,1)$ and having slopes $(-1/m)$, $m=1,2,\dots$, and the ``red'' family of lines passing through the point $(1,1)$ and having slopes $1/k$, $k=1,2,\dots$.
It is easy to check that the lines from these families are given by equations $y=\frac{2m-1}{m}-\frac{x}{m}$ and $y=\frac{k-1}{k}+\frac{x}{k}$ respectively. Let $\Delta'$ be the set of intersection points of lines of green family with lines of red family. Then, we have the following theorem, illustrated by~Fig.~\ref{f:lines}. \begin{Th}[Alternative vertex set description]\label{lines} $\Delta$ is the part of $\Delta'$ formed by the points with the least possible ordinate for given abscise: $$ \Delta=\{(x,y)\in \Delta' \mid \forall y'<y \quad (x,y')\notin \Delta'\}. $$ \end{Th} \begin{rem} Again, as in Remark~\ref{r:plane}, the construction of this theorem can be lifted on the $ACD$ plane containing the third family of vertices. Making such a lift, one notices that the vertical line starting from an intersection point corresponds to a vertical edge of the parallelepiped, starting from the corresponding vertex. The ``least possible ordinate'' rule then corresponds to the fact that the intersection points with non-least ordinate lift to the points that belong to the corresponding edge, and thus that are not vertices. \end{rem}
\begin{figure}
\caption{Lines listed in Theorem~\ref{lines}. Green and red families pass through the points (0,1) and (1,1) respectively. Bold black points correspond to the points of $\Delta$, the points of $X$ between diagonals $x=y$ and $x+y=1$ are marked by smaller grey points to illustrate additional aligning. Dashed vertical lines, starting in the points of~$\Delta$, illustrate the ``least possible ordinate'' condition.}
\label{f:lines}
\end{figure}
\begin{figure}
\caption{Fig. \ref{f:lines} after projective transformation $(x,y) \mapsto \left(\frac{x}{1 - y}, \frac{1-x}{1-y}\right)$. Some of dashed lines are not drawn, the extensions of some dashed lines are drawn dotted to illustrate passing through the origin.}
\label{f:strtn}
\end{figure}
\begin{proof}[Proof of Theorems \ref{ss} and \ref{lines}] First apply a projective transformation $Q$ that sends $(0,1)=[0:1:1]$ and $(1,1)=[1:1:1]$ to the points at infinity $V=[0:1:0]$ and $H=[1:0:0]$, corresponding to the vertical and horizontal directions respectively, and that sends $V$ to the origin. It is easy to see that it is given by the formula $$Q: (x,y) \mapsto \left(\frac{x}{1 - y}, \frac{1-x}{1-y}\right).$$ or, equivalently, by $$ Q:[x:y:t]\mapsto [x:t-x:t-y]. $$ The result of its application is shown on Fig.~\ref{f:strtn} (the reader perhaps will find this figure even more convincing than the formal arguments below).
It is easy to see that it sends a vertex $\left(\frac{r}{q}, \frac{q-1}{q} \right)$ (due to Proposition~\ref{p:list} and its Cor.~\ref{c:list}) the set $\Delta$ is formed by such points with coprime $0<r<q$) to the point $(r, q-r)$, and hence $Q(\Delta)$ is exactly the subset of $\mathbb{N}^2$ formed by points with coprime coordinates.
Next, an immediate check gives that the image of the line at infinity is the line $x=y$, the point $[a:1:0]$ (corresponding to the slope~$1/a$) being sent to the point~$(a,-a)$. Thus, the green family (as these are lines passing through $(0,1)$ and having slope $(-1/m)$) becomes the family of the vertical lines $x=m$, while the red family (as these are lines passing through $(0,1)$ and having slope~$(1/k)$) becomes the family of horizontal lines $y=k$.
Hence the set $\Delta'$ is sent exactly to~$\mathbb{N}^2$. Finally, as the point $V$ is sent to the origin, the ``least possible ordinate'' condition after the transformation~$Q$ becomes exactly the coprimality condition; this concludes the proof of Theorem~\ref{lines}.
To prove Theorem~\ref{ss}, note that in new coordinates the transformations $T_1$ and $T_2$ take form $$ \hat T_1(a,b) = (a, a+b); \, \text{ and } \, \hat T_2(a,b) = (a+b, b) $$ and the self-similarity of $\Delta$ becomes obvious: in the new coordinates, it is the Euclid's algorithm! \end{proof}
\begin{figure}
\caption{The ab-ziggurat (picture credit: Calegari-Walker~\cite[Fig.~1]{Caleg})}
\caption{The abaab-ziggurat (picture credit:~\cite[Fig.~2]{Caleg})}
\label{ab}
\label{aabab}
\end{figure}
To conclude this section, we would like to state some related interesting questions: \begin{question} Is there any direct dynamical proof of $T_{1,2}$-invariance of the set $\Delta$ or of its vertical symmetry? Do the transformations $T_{1,2}$ admit any dynamical interpretation? \end{question}
\begin{question}\footnote{After this text was finished, we became aware that Subhadip Chowdhury has managed to show ``asymptotic'' projective self-similarity near the ``fringes'' of the unit square for the ziggurats associated to some other positive words . } What happens for ziggurats associated to other positive products? For instance, the $abaab$-ziggurat also seems to have some fractal, and possibly self-similar nature, see Fig.~\ref{aabab}. \end{question}
\begin{question} Would the ziggurats in higher dimensions (for larger number of different homeomorphisms being multiplied) still look self-similar? (For the Jankins--Neuman higher-dimensional ziggurat, the proof of Theorem~\ref{ss} applies verbatim.) \end{question}
\begin{question} What happens with the ziggurat if the defining Jankins--Neumann-like relation includes \emph{all} the homeomorphisms more than once; for instance, for the relation $abcbac=\mathop{\mathrm{id}}$? \end{question}
\section{Equivalence}\label{s:equivalence}
This section is devoted to a way of deducing the Jankins--Neumann conjecture from Calegari--Walker's formula~\eqref{formula}. To make such a deduction, let \begin{equation}\label{e:zR} \mathcal Z = \{(x,y,z) \mid 0\le z\le R_{ab}(1-x,1-y)-1\}. \end{equation} It is easy to see that the Jankins--Neumann conjecture is equivalent to that $\mathcal Z=\tilde \mathcal Z$. The Calegari--Walker formula allows to represent $\mathcal Z$ as a union of parallelepipeds. Indeed, the inequalities $\frac{p_1}{q}\le x$, $\frac{p_2}{q}\le y$ implying $R_{ab}(x,y)\ge \frac{p_1+p_2+1}{q}$ give for the set $\mathcal Z$ the representation \begin{multline}\label{e:tz} \mathcal Z = \bigcup_{p_1,p_2,q \in \mathbb{N}, \atop p_1,p_2\le q} \{(x,y,z) \mid 1-x\le \frac{p_1}{q},\, 1-y\le \frac{p_2}{q} , \, z \le \frac{p_1+p_2+1}{q}-1, \, x,y,z\ge 0\} \\ = \bigcup_{p_1,p_2,q \in \mathbb{N},\atop p_1,p_2 \le q} \Pi\left( 1-\frac{p_1}{q}, 1-\frac{p_2}{q}, \frac{p_1+p_2 + 1}{q}-1\right).
\end{multline} Take $p_1'=q-p_1, p_2'=q-p_2$. Then, $$ \Pi\left( 1-\frac{p_1}{q}, 1-\frac{p_2}{q}, \frac{p_1+p_2 + 1}{q}-1\right) = \Pi\left( \frac{p_1'}{q}, \frac{p_2'}{q}, \frac{q+1-(p_1'+p_2')}{q}\right). $$
Finally, denoting $p_3':=q-p_1'-p_2'+1$, we get \begin{equation}\label{e:zigCW} \mathcal Z= \bigcup_{p_1',p_2',p_3',q \in \mathbb{N}, \atop p_1'+p_2'+p_3'= q+1} \Pi\left( \frac{p_1'}{q}, \frac{p_2'}{q}, \frac{p_3'}{q}\right). \end{equation}
To deduce Naimi's theorem (Jankins--Neumann conjecture) from the Calegari--Walker formula, we have to show that $\mathcal Z$ and $\tilde \mathcal Z$ coincide. The list of parallelepipeds in~\eqref{e:zigCW} contains all the parallelepipeds from the Jankins--Neumann conjecture, but also some others. Let us call a parallelepiped good if it is, up to the permutation of the coordinates, of the form $\Pi(\frac{1}{q}, \frac{p}{q}, \frac{q-p}{q})$ with coprime $p$ and $q$. Then, to prove the coincidence of $\mathcal Z$ and $\tilde \mathcal Z$ we have to show that any parallelepiped in~\eqref{e:zigCW} that is not a good one, is contained in one of the good ones. For the obvious monotonicity reasons, it suffices to check that its vertex $\left( \frac{p_1'}{q}, \frac{p_2'}{q}, \frac{p_3'}{q}\right)$ is contained there.
Before proceeding to prove this, we notice (as it was already promised in Remark~\ref{r:min} earlier) that the list of good parallelepipeds is minimal: no good parallelepiped is contained in another one. Indeed, for any of the parallelepipeds listed in~\eqref{e:zigCW}, the sum of any two of three coordinates does not exceed~$1$. As for a good parallelepiped $\Pi\left( \frac{p}{q}, \frac{q-p}{q}, \frac{1}{q}\right)$ the sum of its first two coordinate equals~$1$, it could be contained only in a parallelepiped of the form $\Pi\left( \frac{p'}{q'}, \frac{q'-p'}{q'}, \frac{1}{q'}\right)$ with $\frac{p'}{q'}=\frac{p}{q}$ and $q'<q$. But as $p$ and $q$ are coprime, it is impossible.
Now, let $p_1+p_2+p_3=q+1$, with $p_1,p_2,p_3>1$ (thus, $\Pi\left(\frac{p_1}{q},\frac{p_2}{q},\frac{p_3}{q} \right)$ is not good). We are going then to find a good parallelepiped that contains $\Pi( \frac{p_1}{q}, \frac{p_2}{q}, \frac{p_3}{q})$. Without loss of generality, we can assume $p_1\le p_2\le p_3$. Then, we will be looking for coprime $m<n$ such that $$ \Pi( \frac{p_1}{q}, \frac{p_2}{q}, \frac{p_3}{q})\subset \Pi( \frac{1}{n}, \frac{m}{n}, \frac{n-m}{n}), $$ or, what is the same, $$ \frac{p_1}{q} \le \frac{1}{n}, \quad \frac{p_2}{q}\le \frac{m}{n}, \quad \frac{p_3}{q}\le \frac{n-m}{n}. $$ Or, equivalently, \begin{equation}\label{eq:mn} n\le \left[\frac{p_1}{q}\right], \quad \frac{p_2}{q}\le \frac{m}{n} \le 1- \frac{p_3}{q}. \end{equation}
Denote $N:=\left[\frac{p_1}{q}\right]$. Then, the desired~\eqref{eq:mn} could be reformulated in the following way: we want to prove that there is a fraction $\frac{m}{n}$ of denominator less than $N$ that belongs to the interval $[\frac{p_2}{q} , \frac{q-p_3}{q}]$. Hence it is natural to consider \emph{Farey sequence} of order~$N$.
Recall that the Farey sequence $F_N$ of order $N$ is the sequence of completely reduced fractions between~$0$ and~$1$ which have denominators less than or equal to~$N$, arranged by increasing. In the proof of the equivalence of two ziggurats we will use one of its properties, namely, for any two consecutive fractions $\frac{a}{b}<\frac{c}{d}$ in any of the Farey sequences, their denominators are coprime, and moreover, $\frac{c}{d}-\frac{a}{b}=\frac{1}{bd}.$ As for details, we refer the reader to ~\cite{F2} (as well as to the original historical papers~\cite{F0,F1}).
Suppose there is no $\frac{m}{n} \in F_N$ between $\frac{p_2}{q}$ and $\frac{q-p_3}{q}$. Then, take the two consecutive fractions $\frac{a}{b}, \frac{c}{d} \in F_N$ such that $\frac{a}{b}<\frac{p_2}{q}\le\frac{q- p_3}{q}<\frac{c}{d}$. In this case, $$ \frac{p_2}{q}-\frac{a}{b}\ge \frac{1}{bd}, \quad \frac{c}{d}-\frac{q - p_3}{q}\ge \frac{1}{dq}. $$ Thus, \begin{multline} \label{eq:bdq} \frac{q - p_3}{q} - \frac{p_2}{q}=\left(\frac{c}{d}-\frac{a}{b}\right) - \left(\frac{c}{d}-\frac{q - p_3}{q}\right) - \left(\frac{p_2}{q}-\frac{a}{b}\right) \le\\ \le \frac{1}{bd} - \frac{1}{bq} - \frac{1}{dq} = \frac{q-(b+d)}{bd\cdot q}. \end{multline}
If both $b,d \ge 2$, then $bd \ge b+d\ge N+1 > \frac{q}{p_1}$, we see that the right hand side of~\eqref{eq:bdq} can be estimated as $$ \frac{q-(b+d)}{bd\cdot q} < \frac{q-\frac{q}{p_1}}{\frac{q}{p_1}\cdot q} = \frac{p_1-1}{q}. $$ Thus, $\frac{q - p_3}{q} - \frac{p_2}{q}< \frac{p_1-1}{q}$, contradicting the equation $p_1+p_2+p_3=q+1$.
Finally we have to consider the possibilities $b=1$ or $d=1$. Though, as $\frac{p_2}{q} + \frac{p_3}{q} < 1$, we have $\frac{p_2}{q}<\frac{1}{2}$ and hence $d \ge 2$. Finally, if $b=1$ and thus $\frac{a}{b} = 0, \frac{c}{d}=\frac{1}{N}$, we have $$ 1-\frac{p_3}{q}=\frac{p_2+p_1-1}{q}\ge \frac{2p_1-1}{q} \ge \frac{3}{2} \cdot \frac{p_1}{q}\ge \frac{3}{2} \cdot \frac{1}{N+1}. $$ Then, the inequality $1-\frac{p_3}{q}<\frac{c}{d}=\frac{1}{N}$ is impossible, as it would imply $\frac{3}{2(N+1)} < \frac{1}{N}$ and hence $N< 2$. Though, $\frac{p_1}{q}\le \frac{1}{2}$ and hence $N\ge 2$.
\begin{figure}
\caption{The $\Delta_n$ triangles marked on the view on the ab-ziggurat from the top}
\label{f:t-red}
\label{f:induction}
\end{figure}
These contradictions conclude the proof of the equivalence.
Finally, we would like to note a beautiful connection of ziggurat and Farey series. Consider the set of triangles on the plane $Oxy$, enclosed by lines $x+y=1$, $x=1/n$ and $y=n/(n+1)$, where $n=2,3,\dots$. Let us denote $n$th such triangle by~$\Delta_n$: \[ \Delta_n := \{(x,y)\in [0;1)^2 \mid x+y>1,\, x< 1/n,\, y< n/(n+1) \} \]
These triangles naturally appear on the view from the top of the ab-ziggurat (Jankins-Neumann up to a linear transformation): see Fig.~\ref{proj} and~\ref{f:t-red}; on the latter they are marked by bold red lines. It is easy to see from the Jankins--Neumann's description, that all the vertices project into the diagonal $x+y = 1$ or outside these triangles.
Now, looking on a Fig.~\ref{f:t-red}, we see that each of these triangles is decomposed into a rectangle and two more triangles, each of these triangles decomposes into a rectangle and two more, etc. Each new rectangle corresponds to a vertex $(x,1-x)$ with rational $x$, and it is interesting to study the order, in which these vertices appear, and also prove it formally. To do so, note that any of these triangles is of the form $$\Delta_{\textstyle{\frac{a}{b},\frac{c}{d}}}:=\{(x,y)\in [0;1)^2 \mid x+y>1,\, x< \frac{c}{d},\, y< 1-\frac{a}{b} \} $$ for some $\frac{a}{b}<\frac{c}{d}$.
Let us show that the fractions $\frac{a}{b}$ and $\frac{c}{d}$ are adjacent in one of the Farey series, and the rectangle that subdivides this triangle starts from a point with the abscise that is the \emph{mediant} (``freshman sum'') $\frac{a+c}{b+d}$ of $\frac{a}{b}$ and~$\frac{c}{d}$; see Fig.~\ref{f:induction}. Indeed, the subdivisions that we are observing are given by removing of the slices $\{(x,y) \mid R_{ab}(x,y)-1\ge r\}$ for some $r>0$. As $R_{ab}(\frac{p}{q},1-\frac{p}{q})-1=\frac{1}{q}$ for coprime $p,q$, the vertices appearing in any such slice are exactly those whose abscises belong to $[\frac{1}{r}]$-th Farey sequence. And when the triangle $\Delta_{\textstyle{\frac{a}{b},\frac{c}{d}}}$ is subdivided, it corresponds to the subdivision of an interval $[\frac{a}{b},\frac{c}{d}]$ of a Farey sequence; {as it is well-known (see~\cite{F2}), it is the mediant that subdivides it}.
Thus, the Farey sequences are not only a good tool for the study of the ziggurat, but they appear in this study in a natural way. We conclude this section by noticing an interesting consequence of the above arguments: we see that two fractions are neighbours in some Farey series if and only if the rectangles, starting at the corresponding points, have common segment.
\end{document} | arXiv |
\begin{document}
\begin{frontmatter}
\title{Maxima of long memory stationary symmetric $\alpha$-stable processes, and self-similar processes with stationary max-increments} \runtitle{Limit theory for partial maxima}
\begin{aug}
\author[1]{\inits{T.}\fnms{Takashi}~\snm{Owada}\thanksref{1}\ead[label=e1]{[email protected]}} \and \author[2]{\inits{G.}\fnms{Gennady}~\snm{Samorodnitsky}\corref{}\thanksref{2}\ead[label=e2]{[email protected]}}
\address[1]{Faculty of Electrical Engineering, Technion, Haifa, Israel 32000.\\ \printead{e1}} \address[2]{School of Operations Research and Information Engineering, and Department of Statistical Science, Cornell University, Ithaca, NY 14853, USA. \printead{e2}} \end{aug}
\received{\smonth{7} \syear{2013}} \revised{\smonth{12} \syear{2013}}
\begin{abstract} We derive a functional limit theorem for the partial maxima process based on a long memory stationary $\alpha$-stable process. The length of memory in the stable process is parameterized by a certain ergodic-theoretical parameter in an integral representation of the process. The limiting process is no longer a classical extremal Fr\'echet process. It is a self-similar process with $\alpha$-Fr\'echet marginals, and it has stationary max-increments, a property which we introduce in this paper. The functional limit theorem is established in the space $D[0,\infty)$ equipped with the Skorohod $M_1$-topology; in certain special cases the topology can be strengthened to the Skorohod $J_1$-topology. \end{abstract}
\begin{keyword} \kwd{conservative flow} \kwd{extreme value theory} \kwd{pointwise dual ergodicity} \kwd{sample maxima} \kwd{stable process} \end{keyword}
\end{frontmatter}
\section{Introduction} \label{sec:intro}
The asymptotic behaviour of the partial maxima sequence $M_n=\max_{1 \leq k \leq n} X_k$, $n=1,2,\ldots$ for an i.i.d. sequence $(X_1,X_2,\ldots)$ of random variables is the subject of the classical extreme value theory, dating back to Fisher and Tippett \cite{fisher:tippett:1928}. The basic result of this theory says that only three one-dimensional distributions, the Fr\'echet distribution, the Weibull distribution and the Gumbel distribution, have a max-domain of attraction. If $Y$ has one of these three distributions, then for a distribution in its domain of attraction, and a sequence of i.i.d. random variables with that distribution,
\begin{equation} \label{e:1din.conv} \frac{M_n-b_n}{a_n}\quad \Rightarrow\quad Y \end{equation}
for properly chosen sequences $(a_n)$, $(b_n)$; see, for example, Chapter~1 in Resnick \cite{resnick:1987} or Section~1.2 in de~Haan and Ferreira \cite{dehaan:ferreira:2006}. Under the same max-domain of attraction assumption, a~functional version of \eqref{e:1din.conv} was established in Lamperti \cite{lamperti:1964}: with the same sequences $(a_n)$, $(b_n)$ as in \eqref{e:1din.conv},
\begin{equation} \label{e:funct.conv} \biggl( \frac{M_{\lfloor nt\rfloor}-b_n}{a_n},
t\geq0 \biggr) \quad\Rightarrow\quad \bigl(Y(t), t \geq0 \bigr) \end{equation}
for a nondecreasing right continuous process $(Y(t), t\geq0)$, and the convergence is weak convergence in the Skorohod $J_1$-topology on $D[0,\infty)$. The limiting process is often called {\it the extremal process}; its properties were established in Dwass \cite{dwass:1964,dwass:1966} and Resnick and Rubinovitch \cite {resnick:rubinovitch:1973}.
Much of the more recent research in extreme value theory concentrated on the case when the underlying sequence $(X_1,X_2,\ldots)$ is stationary, but may be dependent. In this case the extrema of the sequence may cluster, and it is natural to expect that the limiting results \eqref{e:1din.conv} and \eqref{e:funct.conv} will, in general, have to be different. The extremes of moving average processes have received special attention; see, for example, Rootz\'en \cite{rootzen:1978}, Davis and Resnick \cite{davis:resnick:1985} and Fasen \cite {fasen:2005}. The extremes of the $\operatorname{GARCH}(1,1)$ process were investigated in Mikosch and St\u{a}ric\u{a} \cite{mikosch:starica:2000b}. The classical work on the extremes of dependent sequences is Leadbetter \textit{et~al}. \cite{leadbetter:lindgren:rootzen:1983}; in some cases this clustering of the extremes can be characterized through the {\it extremal index} (introduced, originally, in Leadbetter~\cite{leadbetter:1983}). The latter is a number $0\leq\theta\leq1$. Suppose that a stationary sequence $(X_1,X_2,\ldots)$ has this index, and let $(\tilde X_1,\tilde X_2,\ldots)$ be an i.i.d. sequence with the same one-dimensional marginal distributions as $(X_1,X_2,\ldots)$. If \eqref{e:1din.conv} and \eqref{e:funct.conv} hold for the i.i.d. sequence, then the corresponding limits will satisfy $\tilde Y\eid\tilde Y(1)$, but the limit in \eqref{e:1din.conv} for the dependent sequence $(X_1,X_2,\ldots)$ will satisfy $Y \eid \tilde Y(\theta)$. In particular, the limit will be equal to zero if the extremal index is equal to zero. This case can be viewed as that of long range dependence in the extremes, and it has been mostly neglected by the extreme value community. Long range dependence is, however, an important phenomenon in its own right, and in this paper we take a step towards understanding how long range dependence affects extremes.
A random variable $X$ is said to have a regularly varying tail with index $-\alpha$ for $\alpha>0$ if
\[ P(X>x) = x^{-\alpha} L(x),\qquad x>0 , \]
where $L$ is a slowly varying at infinity function, and the distribution of any such random variable is in the max-domain of attraction of the Fr\'echet distribution with the same parameter $\alpha$; see, for example, Resnick \cite{resnick:1987}. Recall that the Fr\'echet law $F_{\alpha,\sigma}$ on $(0,\infty)$ with the tail index $\alpha$ and scale $\sigma>0$ satisfies
\begin{equation} \label{e:frechet} F_{\alpha,\sigma}(x) = \exp \bigl\{ -\sigma^\alpha x^{-\alpha } \bigr\},\qquad x>0 . \end{equation}
Sometimes the term {\it$\alpha$-Fr\'echet} is used. In this paper, we discuss the case of regularly varying tails and the resulting limits in \eqref{e:funct.conv}. The limits obtained in this paper belong to the family of the so-called {\it Fr\'echet processes}, defined below. We would like to emphasize that, even for stationary sequences with regularly varying tails, non-Fr\'echet limits may appear in \eqref{e:funct.conv}. We are postponing a detailed discussion of this point to a future publication.
A stochastic process $(Y(t), t \in T)$ (on an arbitrary parameter space $T$) is called a Fr\'echet process if for all $n \geq1$, $a_1, \ldots, a_n >0$ and $t_1, \ldots, t_n \in T$, the weighted maximum $\max_{1 \leq j \leq n} a_j Y(t_j)$ follows a Fr\'echet law as in \eqref{e:frechet}. The best known Fr\'echet process is the extremal Fr\'echet process obtained in the scheme \eqref{e:funct.conv} starting with an i.i.d. sequence with regularly varying tails. The extremal Fr\'echet process $ ( Y(t), t\geq0 )$ has finite-dimensional distributions defined by
\begin{eqnarray} \label{e:extreme.frechet} \bigl( Y(t_1),Y(t_2),\ldots, Y(t_n) \bigr) &\eid& \bigl( X^{(1)}_{\alpha,t_1^{1/\alpha}}, \max \bigl( X^{(1)}_{\alpha,t_1^{1/\alpha}}, X^{(2)}_{\alpha,(t_2-t_1)^{1/\alpha}} \bigr), \ldots, \nonumber \\[-8pt] \\[-8pt] &&\hphantom{\bigl(}{}\max \bigl( X^{(1)}_{\alpha,t_1^{1/\alpha}}, X^{(2)}_{\alpha,(t_2-t_1)^{1/\alpha}}, \ldots, X^{(n)}_{\alpha,(t_n-t_{n-1})^{1/\alpha}} \bigr) \bigr) \nonumber
\end{eqnarray}
for all $n$ and $0\leq t_1<t_2<\cdots<t_n$. The different random variables in the right-hand side of \eqref{e:extreme.frechet} are independent, with $X^{(k)}_{\alpha,\sigma}$ having the Fr\'echet law $F_{\alpha,\sigma}$ in \eqref{e:frechet}, for any $k=1,\ldots, n$. The stationarity and independence of the max-increments of the extremal Fr\'echet processes make it similar to the better known L\'evy processes which have stationary and independent sum-increments. The structure of general Fr\'echet processes has been extensively studied in the last several years. These processes were introduced in Stoev and Taqqu \cite{stoev:taqqu:2005}, and their representations (as a part of a much more general context) were studied in Kabluchko and Stoev \cite{kabluchko:stoev:2012}. Stationary Fr\'echet processes (in particular, their ergodicity and mixing) were discussed in Stoev \cite{stoev:2008}, Kabluchko \textit{et~al}. \cite {kabluchko:schlather:dehaan:2009} and Wang and Stoev \cite{wang:stoev:2010}.
In this paper, we concentrate on the maxima of stationary $\alpha$-stable processes with $0<\alpha<2$. Recall that a random vector $\mathbf{X}$ in $\reals^d$ is called $\alpha$-stable if for any $A$ and $B>0$ we have
\[ A\mathbf{X}^{(1)} + B\mathbf{X}^{(2)} \eid \bigl(A^\alpha+B^\alpha \bigr)^{1/\alpha} \mathbf{X}+ \mathbf{y}, \]
where $\mathbf{X}^{(1)}$ and $\mathbf{X}^{(2)}$ are i.i.d. copies of $\mathbf{X}$, and $\mathbf{y}$ is a deterministic vector (unless $\mathbf{X}$ is deterministic, necessarily, $0<\alpha\leq2$). A stochastic process $(X(t), t \in T)$ is called $\alpha$-stable if all of its finite-dimensional distributions are $\alpha$-stable. We refer the reader to Samorodnitsky and Taqqu \cite{samorodnitsky:taqqu:1994} for information on $\alpha$-stable processes. When $\alpha=2$, an $\alpha$-stable process is Gaussian, while in the case $0<\alpha<2$, both the left and the right tails of a (nondegenerate) $\alpha$-stable random variable $X$ are (generally) regularly varying with exponent $\alpha$. That is,
\[ P(X>x) \sim c_+ x^{-\alpha},\quad\quad P(X<-x) \sim c_- x^{-\alpha} \quad\quad\mbox{as $x \to\infty$} \]
for some $c_+, c_-\geq0$, $c_++c_->0$. That is, if
$(X_1,X_2,\ldots)$ is an i.i.d. sequence of $\alpha$-stable random variables, then the i.i.d. sequence $(|X_1|,|X_2|,\ldots)$ satisfies \eqref{e:1din.conv} and \eqref{e:funct.conv} with $a_n=n^{1/\alpha}$ (and $b_n=0$), $n\geq1$. Of course, we are not planning to study the extrema of an i.i.d. $\alpha$-stable sequence. Instead, we will study the maxima of (the absolute values of) a stationary $\alpha$-stable process. The reason we have chosen to work with stationary $\alpha$-stable processes is that their structure is very rich, and is also relatively well understood. This will allow us to study the effect of that structure on the limit theorems \eqref{e:1din.conv} and \eqref{e:funct.conv}. We are specifically interested in the long range dependent case, corresponding to the zero value of the extremal index.
The structure of stationary symmetric $\alpha$-stable (S$\alpha$S) processes has been clarified in the last several years in the works of Jan Rosi\'nski; see, for example, Rosi\'nski \cite{rosinski:1995,rosinski:2006}. The integral representation of such a process can be chosen to have a very special form. The class of stationary S$\alpha$S\ processes we will investigate requires a representation slightly more restrictive than the one generally allowed. Specifically, we will consider discrete-time stationary processes of the form
\begin{equation} \label{e:underlying.proc} X_n = \int_E f \circ T^n(x) \,\mathrm{d}M(x), \qquad n=1,2,\ldots, \end{equation}
where $M$ is a S$\alpha$S\ random measure on a measurable space $(E,\mathcal{E})$ with a $\sigma$-finite {\it infinite} control measure $\mu$. The map $T\dvtx E \to E$ is a measurable map that preserves the measure $\mu$. Further, \mbox{$f \in L^{\alpha}(\mu)$}. See Samorodnitsky and Taqqu \cite{samorodnitsky:taqqu:1994} for details on $\alpha$-stable random measures and integrals with respect to these measures. It is elementary to check that a process with a representation \eqref{e:underlying.proc} is, automatically, stationary. Recall that any stationary S$\alpha$S\ process has a representation of the form:
\begin{equation} \label{e:general.integral} X_n = \int_E f_n(x) \,\mathrm{d}M(x),\qquad n=1,2,\ldots, \end{equation}
with
\begin{equation} \label{e:general.rosinski} f_n(x) = a_n(x) \biggl( \frac{\mathrm{d}\mu\circ T^{n}}{\mathrm{d}\mu}(x) \biggr)^{1/\alpha} f\circ T^{n}(x),\quad\quad x \in E \end{equation}
for $n=1,2,\ldots$\,, where $T\dvtx E \to E$ is a one-to-one map with both $T$ and $T^{-1}$ measurable, mapping the control measure $\mu$ into an equivalent measure, and the sequence $(a_n)$ takes values $\pm1$ (and has the so-called cocycle property). Here $M$ is S$\alpha$S\ (and $f \in L^{\alpha}(\mu)$). See Rosi\'nski \cite{rosinski:1995}.
In \eqref{e:underlying.proc} we assume, however, that map $T$ is measure preserving. The main reason is that the ergodic-theoretical notions we are using have been developed for measure preserving maps. Indeed, it has been observed that the ergodic-theoretical properties of the map $T$, either in \eqref{e:underlying.proc} or in \eqref{e:general.rosinski}, have a major impact on the memory of a stationary $\alpha$-stable process. See, for example, Surgailis \textit{et~al}. \cite {surgailis:rosinski:mandrekar:cambanis:1993}, Samorodnitsky \cite{samorodnitsky:2004a,samorodnitsky:2005}, Roy \cite {roy:2007}, Resnick and Samorodnitsky \cite{resnick:samorodnitsky:2004}, Owada and Samorodnitsky \cite{owada:samorodnitsky:2012}, Owada \cite{owada:2013}. The most relevant for this work is the result of Samorodnitsky \cite{samorodnitsky:2004a}, who proved that, if the map $T$ in \eqref{e:underlying.proc} or in \eqref{e:general.rosinski} is conservative, then using the normalization $a_n=n^{1/\alpha}$ ($b_n=0$) in \eqref{e:1din.conv}, as indicated by the marginal tails, produces the zero limit, so the partial maxima grow, in this case, strictly slower than at the rate of $n^{1/\alpha}$. On the other hand, if the map $T$ is not conservative, then the normalization $a_n=n^{1/\alpha}$ in \eqref{e:1din.conv} is the correct one, and it leads to a Fr\'echet limit (we will survey the ergodic-theoretical notions in the next section). Therefore, the extrema of S$\alpha$S\ processes corresponding to conservative flows cluster so much that the sequence of the partial maxima grows at a slower rate than that indicated by the marginal tails. This case can be thought of as indicating long range dependence. It is, clearly, inconsistent with a positive extremal index.
The Fr\'echet limit obtained in \eqref{e:1din.conv} by Samorodnitsky \cite{samorodnitsky:2004a} remains valid when the map $T$ is conservative (but with the normalization of a smaller order than $n^{1/\alpha}$), as long as the map $T$ satisfies a certain additional assumption. If one views the stationary $\alpha$-stable process as a natural function of the Poisson points forming the random measure $M$ in \eqref{e:general.integral} then, informally, this assumption guarantees that only the largest Poisson point contributes, distributionally, to the asymptotic behaviour of the partial maxima of the process. In this paper, we restrict ourselves to this situation as well. However, we will look at the limits obtained in the much more informative functional scheme \eqref{e:funct.conv}. In this paper, the assumption on the map $T$ will be expressed in terms of the rate of growth of the so-called wandering rate sequence, which we define in the sequel. We would like to emphasize that, when this wandering rate sequence grows at a rate slower than the one assumed in this paper, new phenomena seem to arise. Multiple Poisson points may contribute to the asymptotic distribution of the partial maxima, and non-Fr\'echet limit may appear in \eqref{e:funct.conv}. We leave a detailed study of this to a subsequent work.
In the next section, we provide the elements of the infinite ergodic theory needed for the rest of the paper. In Section~\ref{sec:lim.processes} we introduce a new notion, that of a process with stationary max-increments. It turns out that the possible limits in the functional maxima scheme \eqref{e:funct.conv} (with $b_n=0$) are self-similar with stationary max-increments. We discuss the general properties of such processes and then specialize to the concrete limiting process we obtain in the main result of the paper, stated and proved in Section~\ref{sec:FLTPM}.
\section{Ergodic theoretical notions} \label{sec:ergodic}
In this section, we present some basic notation and notions of, mostly infinite, ergodic theory used in the sequel. The main references are Krengel \cite{krengel:1985}, Aaronson \cite{aaronson:1997}, and Zweim\"uller \cite{zweimuller:2009}.
Let $(E,\mathcal{E},\mu)$ be a $\sigma$-finite, infinite measure space. We will say that $A = B$ mod $\mu$ if $A, B \in\mathcal{E}$ and $\mu(A\triangle B)=0$. For $f\in L^1(\mu)$ we will often write $\mu(f)$ for the integral $\int f \,\mathrm{d}\mu$.
Let $T\dvtx E \to E$ be a measurable map preserving the measure $\mu$. The sequence $(T^n)$ of iterates of $T$ is called a {\it flow}, and the ergodic-theoretical properties of the map and the flow are identified. A map $T$ is called {\it ergodic} if any $T$-invariant set $A$ (i.e., a set such that $T^{-1}A = A$ mod $\mu$) is trivial, that is, it satisfies $\mu(A)=0$ or $\mu(A^c)=0$. A map $T$ is said to be {\it conservative} if
\[ \sum_{n=1}^{\infty} \mathbf{1}_A \circ T^n = \infty \quad\quad\mbox{a.e. on } A \]
for any $A \in\mathcal{E}$, $0 < \mu(A) < \infty$; if $T$ is also ergodic, then the restriction ``{\it on} $A$'' is not needed.
The {\it conservative part} of a measure-preserving $T$ is the largest $T$-invariant subset $C$ of $E$ such that the restriction of $T$ to $C$ is conservative. The set $D=E \setminus C$ is the {\it dissipative part} of $T$ (and the decomposition $E=C\cup D$ is called {\it the Hopf decomposition} of $T$).
The {\it dual operator} $\widehat{T}\dvtx L^1(\mu) \to L^1(\mu)$ is defined by
\begin{equation} \label{e:define.dual} \widehat{T} f = \frac{\mathrm{d}
(\nu_f \circ T^{-1} )}{\mathrm{d} \mu},\quad\quad f\in L^1(\mu) , \end{equation}
where $\nu_f$ is the signed measure $\nu_f(A) = \int_{A} f \,\mathrm{d}\mu$, $A \in\mathcal{E}$. The dual operator satisfies the duality relation
\begin{equation} \label{e:dual.rel} \int_E \widehat{T} f\cdot g \,\mathrm{d}\mu= \int _E f\cdot g \circ T \,\mathrm{d}\mu \end{equation}
for $f\in L^1(\mu), g\in L^\infty(\mu)$. Note that \eqref{e:define.dual} makes sense for any nonnegative measurable function $f$ on $E$, and the resulting $\widehat{T} f$ is again a nonnegative measurable function. Furthermore, \eqref{e:dual.rel} holds for arbitrary nonnegative measurable functions $f$ and $g$.
A conservative, ergodic and measure preserving map $T$ is said to be {\it pointwise dual ergodic}, if there exists a normalizing sequence $a_n \nearrow\infty$ such that
\begin{equation} \label{e:pde} \frac{1}{a_n} \sum_{k=1}^n \widehat{T}^k f \to\mu(f) \quad\quad\mbox{a.e. for every } f \in L^1(\mu) . \end{equation}
The property of pointwise dual ergodicity rules out invertibility of the map $T$. Since the measure $\mu$ is infinite, choosing a nonnegative function $f$ and using Fatou's lemma shows that only rates $a_n=\mathrm{o}(n)$ are possible in pointwise dual ergodicity. Intuitively, as will be seen in \eqref{e:prop3.8.7} below, the longer time it takes the trajectory of a point under the map $T$ to return to a set of a finite positive measure, the smaller is the normalizing sequence $(a_n)$.
Sometimes we require that for some functions the above convergence takes place uniformly on a certain set. A set $A \in \mathcal{E}$ with $0 < \mu(A) < \infty$ is said to be a {\it uniform set} for a conservative, ergodic and measure preserving map $T$, if there exist a normalizing sequence $a_n \nearrow\infty$ and a nontrivial nonnegative measurable function $f \in L^1(\mu)$ (nontriviality means that $f$ is different from zero on a set of positive measure) such that
\begin{equation} \label{e:uniform_set} \frac{1}{a_n} \sum_{k=1}^n \widehat{T}^k f \to\mu(f) \quad\quad\mbox {uniformly, a.e. on } A . \end{equation}
If \eqref{e:uniform_set} holds for $f=\mathbf{1}_A$, the set $A$ is called {\it a Darling--Kac set}. A conservative, ergodic and measure preserving map $T$ is pointwise dual ergodic if and only if $T$ admits a uniform set; see Proposition 3.7.5 in Aaronson \cite{aaronson:1997}. In particular, it is legitimate to use the same normalizing sequence $(a_n)$ both in (\ref{e:pde}) and (\ref{e:uniform_set}).
Let $A \in\mathcal{E}$ with $0 < \mu(A) < \infty$. The frequency of visits to the set $A$ along the trajectory $( T^n x)$, $x\in E$, is naturally related to the {\it wandering rate} sequence
\begin{equation} \label{e:wanderingrate} w_n = \mu \Biggl(\bigcup _{k=0}^{n-1} T^{-k}A \Biggr) . \end{equation}
If we define the first entrance time to $A$ by
\[ \varphi_A(x) = \min \bigl\{ n \geq1\dvtx T^n x \in A \bigr \} \]
(notice that $\varphi_A < \infty$ a.e. on $E$ since $T$ is conservative and ergodic), then $w_n \sim\mu(\varphi_A < n)$ as $n\to\infty$. Since $T$ is also measure preserving, we have $\mu(A \cap\{ \varphi_A > k \}) = \mu(A^c \cap\{ \varphi_A = k \})$ for $k \geq1$ (see, e.g., Zweim\"uller \cite{zweimuller:2009}). Therefore, alternative expressions for the wandering rate sequence are
\[ w_n = \mu(A) + \sum_{k=1}^{n-1} \mu \bigl(A^c \cap\{ \varphi_A = k \} \bigr) = \sum _{k=0}^{n-1} \mu \bigl(A \cap\{ \varphi_A > k \} \bigr) . \]
Suppose now that $T$ is a pointwise dual ergodic map, and let $A$ be a uniform set for $T$. It turns out that, under an assumption of regular variation, there is a precise connection between the wandering rate sequence $(w_n)$ and the normalizing sequence $(a_n)$ in \eqref{e:pde} and \eqref{e:uniform_set}. Specifically, let $RV_{\gamma}$ represent the class of regularly varying at infinity sequences (or functions, depending on the context) of index $\gamma$. If either $(w_n) \in RV_{\beta}$ or $(a_n) \in RV_{1-\beta}$ for some $\beta\in[0,1]$, then
\begin{equation} \label{e:prop3.8.7} a_n \sim\frac{1}{\Gamma(2-\beta)
\Gamma(1+\beta)} \frac{n}{w_n} \quad\quad\mbox{as } n \to\infty. \end{equation}
Proposition 3.8.7 in Aaronson \cite{aaronson:1997} gives one direction of this statement, but the argument is easily reversed. The normalizing sequence $(a_n)$ and the wandering rate sequence $(w_n)$ are both related to the frequency with which a uniform set $A$ is visited along the trajectory $(T^nx)$ that starts in $A$.
We finish this section with a statement on distributional convergence of the partial maxima for pointwise dual ergodic flows. It will be used repeatedly in the proof of the main theorem. For a measurable function $f$ on $E$ define
\[
M_n(f) (x) = \max_{1 \leq k \leq n}\bigl|f \circ T^k(x)\bigr| ,\quad\quad x \in E, n \geq1 . \]
The proposition below involves weak convergence in the space $D[0,\infty)$ equipped with two different topologies, the Skorohod $J_1$-topology and the Skorohod $M_1$-topology, introduced in Skorohod \cite{skorohod:1956}. The details could be found, for instance, in Billingsley \cite{billingsley:1999} (for the $J_1$-topology), and in Whitt \cite{whitt:2002} (for the $M_1$-topology). See also Remark \ref{rk:topologies}.
In the sequel, we will use the convention $\max_{k\in K}b_k = 0$ for a nonnegative sequence $(b_n)$, if $K=\varnothing$.
\begin{proposition} \label{p:max.ergodic} Let $T$ be a pointwise dual ergodic map on a $\sigma$-finite, infinite, measure space $(E,\mathcal{E},\mu)$. We assume that the normalizing sequence $(a_n)$ is regularly varying with exponent $1-\beta$ for some $0 < \beta\leq1$. Let $A \in\mathcal{E}$, $0 < \mu(A) < \infty$, be a uniform set for $T$. Define a probability measure on $E$ by $\mu_n (\cdot) = \mu(\cdot\cap\{\varphi_A \leq n\})
/ \mu(\{\varphi_A \leq n \})$. Let $f\dvtx E \to\reals$ be a measurable bounded function supported by the set $A$, that is, $\operatorname{supp}(f) \subset A$. Let $\| f \|_{\infty} = \inf\{M
\dvtx |f(x)| \leq M \mbox{ a.e. on } A \}$. Then
\begin{eqnarray} \label{e:underlying.conv}&& \bigl( M_{\lfloor nt\rfloor}(f),
0\leq t\leq1 \bigr) \nonumber\\[-8pt]\\[-8pt]
&&\quad\Rightarrow\quad\|f
\|_\infty ( \mathbf{1}_{\{ V_{\beta} \leq t \}}, 0\leq t\leq1 )\quad\quad \mbox{in the $M_1$-topology on $D[0,1] $,}\nonumber \end{eqnarray}
where the law of the left-hand side is computed with respect to $\mu_n$, and $V_{\beta}$ is a random variable defined on a probability space $(\Omega^{\prime},\mathcal{F}^{\prime},P^{\prime})$ with $P^{\prime}(V_{\beta} \leq x) = x^{\beta}$, $0 < x \leq1$. If $f=\mathbf{1}_A$, then the convergence above takes place in the $J_1$-topology as well. \end{proposition}
\begin{remark} \label{rk:topologies} It is not difficult to see why the weak convergence in \eqref{e:underlying.conv} holds in the $J_1$-topology for indicator functions, but only in the $M_1$-topology in general. Indeed, for functions $f$ other than the indicator function, the limiting value of
$\|f\|_\infty$ may have an asymptotically non-vanishing probability of being reached in multiple closely placed steps, which precludes the $J_1$-tightness, since the $J_1$-modulus does not become small; see, for example, Theorem 13.2 in Billingsley \cite{billingsley:1999}. One can easily construct (very general) examples of situations in which this can be made precise. On the other hand, if $f = \mathbf{1}_A$, then the limiting value is reached by a single jump, matching the single jump in the limiting process, which gives convergence in the $J_1$-topology. \end{remark}
\begin{pf*}{Proof of Proposition~\ref{p:max.ergodic}}
For $0<\varepsilon<1$, let $A_\varepsilon= \{ x\in A\dvtx |f(x)|\geq(1-\varepsilon)\|
f\|_\infty\}$. Note that each $A_\varepsilon$ is uniform since $A$ is uniform. Clearly,
\[
(1-\varepsilon)\| f\|_\infty\mathbf{1}_{ \{\varphi_{A_\varepsilon}(x)\leq nt \} } \leq M_{\lfloor nt\rfloor}(f) (x)
\leq\| f\|_\infty\mathbf{1}_{ \{\varphi _A(x)\leq nt \} } \quad\quad\mu\mbox{-a.e.} \]
for all $n\geq1$ and $0\leq t\leq1$. Since for monotone functions weak convergence in the $M_1$-topology is implied by convergence in finite-dimensional distributions (see, e.g., Proposition 2 in Avram and Taqqu \cite{avram:taqqu:1992}), we can use Theorem 3.2 in Billingsley \cite{billingsley:1999} in a finite-dimensional situation. The statement of the proposition will follow once we show that, for a uniform set $B$ (which could be either $A$ or $A_\varepsilon$) the law of $\varphi_B/n$ under $\mu_n$ converges to the law of $V_{\beta}$. Let $(w_n^{(B)})$ be the corresponding wandering rate sequence. Since \eqref{e:prop3.8.7} holds for $(w_n^{(B)})$ with the same normalizing constants $(a_n)$, we know that $w_n^{(B)}\sim w_n^{(A)}:= w_n$ as $n\to\infty$. Therefore,
\[ \mu_n \biggl( \frac{\varphi_{B}}{n} \leq x \biggr) = \frac{\mu (\varphi_{B} \leq\lfloor nx \rfloor)}{\mu(\varphi_A \leq n)} \sim \frac{w_{\lfloor nx \rfloor}^{(B)}}{w_n} \to x^{\beta} \]
for all $0 < x \leq1$, because the wandering rate sequence $(w_n)$ is regularly varying with index $\beta$ by \eqref{e:prop3.8.7}.
Next, suppose that $f(x) = \mathbf{1}_A(x)$. In this case, $M_{\lfloor nt \rfloor}(\mathbf{1}_A)(x) = \mathbf{1}_{\{ \varphi_A(x) \leq nt \}}$. An application of the Skorohod embedding theorem tells us that on some common probability space, the time of the jump of the process $\mathbf{1}_{\{ \varphi_A(\cdot) \leq nt \}}$ converges a.s. to the time of the jump of the process $\mathbf{1}_{\{ V_{\beta} \leq t \}}$. This, in turn, implies a.s. convergence of these processes in the space $D[0,1]$ in the $J_1$-topology, hence their weak convergence in that topology. \end{pf*}
\section{Self-similar processes with stationary max-increments} \label{sec:lim.processes}
The limiting process obtained in the next section shares with any possible limits in the functional maxima scheme \eqref{e:funct.conv} (with $b_n=0$) two very specific properties, one of which is classical, and the other is less so. Recall that a stochastic process $ ( Y(t), t\geq0 )$ is called self-similar with exponent $H$ of self-similarity if for any $c>0$
\[ \bigl( Y(ct), t\geq0 \bigr)\eid \bigl( c^HY(t), t\geq0 \bigr)
\]
in the sense of equality of finite-dimensional distributions. The best known classes of self-similar processes arise in various versions of a functional central limit theorem for stationary processes, and they have an additional property of stationary increments. Recall that a stochastic process $ ( Y(t), t\geq0 )$ is said to have stationary increments if for any $r\geq0$
\begin{equation} \label{e:stat.incr} \bigl( Y(t+r)-Y(r), t\geq0 \bigr) \eid \bigl( Y(t)-Y(0), t\geq 0 \bigr) ; \end{equation}
see, for example, Embrechts and Maejima \cite{embrechts:maejima:2002} and Samorodnitsky \cite{samorodnitsky:2006LRD}. In the context of the functional limit theorem for the maxima \eqref{e:funct.conv}, a different property appears.
\begin{definition} \label{d:max.stat.def} A stochastic process $( Y(t), t \geq0 )$ is said to have stationary max-increments if for every $r \geq0$, there exists, perhaps on an enlarged probability space, a stochastic process $ ( Y^{(r)}(t), t \geq0 )$ such that
\begin{eqnarray} \label{e:def.statmaxi} \bigl( Y^{(r)}(t), t \geq0 \bigr) &\stackrel{d} {=} &\bigl( Y(t), t \geq0 \bigr) , \nonumber \\[-8pt]\\[-8pt] \bigl( Y(t+r), t \geq0 \bigr) &\stackrel{d} {=}& \bigl( Y(r) \vee Y^{(r)}(t), t \geq0 \bigr) .\nonumber \end{eqnarray}
\end{definition}
Notice the analogy between the definition \eqref{e:stat.incr} of stationary increments (when $Y(0)=0$) and Definition \ref{d:max.stat.def}. Since the operations of taking the maximum is not invertible (unlike summation), the latter definition, by necessity, is stated in terms of existence of the max-increment process $ ( Y^{(r)}(t), t \geq0 )$.
\begin{theorem} \label{t:max.lamperti} Let $(X_1,X_2,\ldots)$ be a stationary sequence. Assume that for some sequence \mbox{$a_n\to\infty$}, and a stochastic process $(Y(t), t\geq0)$ such that $P(Y(t)=Y(1))<1$ for $t\neq1$,
\[ \biggl( \frac{1}{a_n} M_{\lfloor nt\rfloor}, t\geq0 \biggr) \quad\Rightarrow\quad \bigl(Y(t), t\geq0 \bigr) \]
in terms of convergence of finite-dimensional distributions. Then $(Y(t), t\geq0)$ is self-similar with exponent $H>0$ of self-similarity, and has stationary max-increments. Furthermore, \mbox{$(Y(t), t\geq0)$} is continuous in probability. The sequence $(a_n)$ is regularly varying with index $H$. \end{theorem}
\begin{pf} The facts that the limiting process $(Y(t), t\geq0)$ is self-similar with exponent $H\geq0$ of self-similarity, and that the sequence $(a_n)$ is regularly varying with index $H$, follow from the Lamperti theorem; see Lamperti \cite{lamperti:1962}, or Theorem 2.1.1 in Embrechts and Maejima \cite{embrechts:maejima:2002}. The case $H=0$ is ruled out by the assumption that $P(Y(t)=Y(1))<1$ for $t\neq1$. Lamperti's theorem is usually stated and proved in the context of convergence in the situation when the time is scaled by a parameter converging to infinity along the real values, whereas in our situation the time scaling converges to infinity along a discrete sequence of the integers. However, it is easy to check that for maxima of stationary processes convergence along a discrete sequence provides the same information as convergence along all real values. Note, further, that for every $0\leq t_1<t_2$ and $n$ large enough,
\[ \frac{1}{a_n} ( M_{\lfloor nt_2\rfloor} - M_{\lfloor nt_1\rfloor} ) \leq \frac{1}{a_n} \max_{nt_1<i\leq nt_2} X_i \lst \frac{1}{a_n} M_{\lfloor2n(t_2-t_1)\rfloor} \]
by the stationarity. Taking weak limits, we see that the difference $Y(t_2)-Y(t_1)$ is nonnegative and bounded stochastically by $Y (2(t_2-t_1) )$. Therefore, it follows from the self-similarity of $(Y(t), t\geq0)$ that it is continuous in probability.
We check now the stationarity of the max-increments of the limiting process. Let $r>0$, and $t_i>0, i=1,\ldots, k$, some $k\geq1$. Write
\begin{equation} \label{e:decompose} \frac{1}{a_n} M_{\lfloor n(t_i+r)\rfloor} = \frac{1}{a_n} M_{\lfloor nr\rfloor} \bigvee\frac{1}{a_n} \max_{nr<j\leq n(t_i+r)} X_j,\quad\quad i=1,\ldots, k . \end{equation}
By the assumption of the theorem and stationarity of the process $(X_1,X_2,\ldots)$,
\[ \frac{1}{a_n} M_{\lfloor nr\rfloor} \quad\Rightarrow\quad
Y(r), \quad\quad\biggl( \frac{1}{a_n} \max_{nr<j\leq n(t_i+r)} X_j, i=1,\ldots, k \biggr) \quad\Rightarrow\quad \bigl( Y(t_1),\ldots, Y(t_k) \bigr) \]
as $n\to\infty$. Since every weakly converging sequence is tight, and a sequence with tight marginals is itself tight, we conclude that
\[ \biggl( \frac{1}{a_n} M_{\lfloor nr \rfloor}, \biggl( \frac{1}{a_n} \max _{nr< j \leq n(t_i+r)} X_j, i=1,\ldots, k \biggr) \biggr)
\]
is a tight sequence. This tightness means that for every sequence $n_m\to\infty$ there is a subsequence $n_{m(l)}\to\infty$ and a $k$-dimensional random vector $ ( Y^{(r)}(t_1),\ldots, Y^{(r)}(t_k) ) \stackrel{d}{=} ( Y(t_1),\ldots, Y(t_k) )$ such that as $l \to\infty$,
\begin{eqnarray*} &&\biggl( \frac{1}{a_{n_{m(l)}}} M_{\lfloor{n_{m(l)}}r\rfloor}, \biggl( \frac{1}{a_{n_{m(l)}}} \max _{n_{m(l)}r<j\leq n_{m(l)}(t_i+r)} X_j, i=1,\ldots, k \biggr) \biggr) \\ &&\quad\Rightarrow \quad\bigl( Y(r), \bigl( Y^{(r)}(t_1),\ldots, Y^{(r)}(t_k) \bigr) \bigr) . \end{eqnarray*}
Let now $\tau_i$, $i=1,2,\ldots$ be an enumeration of the rational numbers in $[0,\infty)$. A diagonalization argument shows that there is a sequence $n_m\to\infty$ and a stochastic process $ ( Y^{(r)}(\tau_i), i=1,2,\ldots )$ with $ ( Y^{(r)}(\tau_i), i=1,2,\ldots )\eid ( Y(\tau_i), i=1,2,\ldots )$ such that
\begin{eqnarray} \label{e:rational.conv} &&\biggl( \frac{1}{a_{n_{m}}} M_{\lfloor{n_{m}}r\rfloor}, \biggl( \frac{1}{a_{n_{m}}} \max_{n_{m}r<j\leq n_{m}(\tau_i+r)} X_j, i=1,2,\ldots \biggr) \biggr) \nonumber \\[-8pt]\\[-8pt] &&\quad\Rightarrow \quad\bigl( Y(r), \bigl( Y^{(r)}( \tau_i), i=1,2,\ldots \bigr) \bigr)\nonumber \end{eqnarray}
in finite-dimensional distributions, as $m\to\infty$. We extend the process $Y^{(r)}$ to the entire positive half-line by setting
\[ Y^{(r)}(t) = \frac{1}2 \Bigl( \lim_{\tau\uparrow t,\ \mathrm{rational}} Y^{(r)}(\tau) + \lim_{\tau\downarrow t, \ \mathrm{rational}} Y^{(r)}(\tau) \Bigr),\quad\quad t\geq0 . \]
The continuity in probability implies that this process is a version of $(Y(t), t\geq0)$. This continuity in probability, \eqref{e:rational.conv} and monotonicity imply that as $m \to\infty$,
\begin{equation} \label{e:full.conv} \biggl( \frac{1}{a_{n_{m}}} M_{\lfloor{n_{m}}r\rfloor}, \biggl( \frac{1}{a_{n_{m}}} \max_{n_{m}r<j\leq n_{m}(t+r)} X_j, t\geq0 \biggr) \biggr) \quad\Rightarrow\quad \bigl( Y(r), \bigl( Y^{(r)}(t), t\geq0 \bigr) \bigr)\nonumber \end{equation}
in finite-dimensional distributions. Now the stationarity of max-increments follows from \eqref{e:decompose}, \eqref{e:full.conv} and continuous mapping theorem. \end{pf}
\begin{remark} \label{rk:sup.measures} Self-similar processes with stationary max-increments arising in a functional maxima scheme \eqref{e:funct.conv} are close in spirit to the stationary self-similar extremal processes of O{'}Brien \textit{et al}. \cite{obrien:torfs:vervaat:1990}, while extremal processes themselves are defined as random sup measures. A random sup measure is, as its name implies, indexed by sets. They also arise in a limiting maxima scheme similar to \eqref{e:funct.conv}, but with a stronger notion of convergence. Every stationary self-similar extremal processes trivially produces a self-similar process with stationary max-increments via restriction to sets of the type $[0,t]$ for $t\geq0$, but the connection between the two objects remains unclear. Our limiting process in Theorem \ref{t:main.max} below can be extended to a stationary self-similar extremal processes, but the extension is highly nontrivial, and will not be pursued here.\vadjust{\goodbreak} \end{remark}
It is not our goal in this paper to study in details the properties of self-similar processes with stationary max-increments, so we restrict ourselves to the following basic result.
\begin{proposition} \label{pr:sssmaxi} Let $ ( Y(t), t \geq0 )$ be a nonnegative self-similar process with stationary max-increments, and exponent $H$ of self-similarity. Suppose $ (Y(t), t \geq0 )$ is not identically zero. Then $H\geq0$, and the following statements hold. \begin{enumerate}
\item[(a)] If $H=0$, then $Y(t)=Y(1)$ a.s. for every $t>0$.
\item[(b)] If $0<EY(1)^p<\infty$ for some $p>0$, then $H\leq1/p$.
\item[(c)] If $H>0$, $ ( Y(t), t \geq0 )$ is continuous in probability. \end{enumerate} \end{proposition}
\begin{pf} By the stationarity of max-increments, $Y(t)$ is stochastically increasing with $t$. This implies that $H\geq0$.
If $H=0$, then $Y(n)\eid Y(1)$ for each $n=1,2\ldots$\,. We use \eqref{e:def.statmaxi} with $r=1$. Using $t=1$ we see that, in the right-hand side of \eqref{e:def.statmaxi}, $Y(1)=Y^{(1)}(1)$ a.s. Since $Y^{(1)}(n)\geq Y^{(1)}(1)$ a.s., we conclude, using $t=n$ in the right-hand side of \eqref{e:def.statmaxi}, that $Y(1)=Y^{(1)}(n)$ a.s. for each $n=1,2,\ldots$\,. By monotonicity, we conclude that the process $ ( Y^{(1)}(t), t \geq0 )$, hence also the process $ ( Y(t), t \geq0 )$, is a.s. constant on $[1,\infty)$ and then, by self-similarity, also on $(0,\infty)$.
Next, let $p>0$ be such that $0<EY(1)^p<\infty$. It follows from \eqref{e:def.statmaxi} with $r=1$ that
\[ 2^H Y(1)\eid Y(2)\eid\max \bigl( Y(1), Y^{(1)}(1) \bigr) .
\]
Therefore,
\[ 2^{pH} EY(1)^p = E Y(2)^p = E \bigl[ Y(1)^p \vee Y^{(1)}(1)^p \bigr] \leq2 EY(1)^p. \]
This means that $pH\leq1$.
Finally, we take arbitrary $0<s<t$. We use \eqref{e:def.statmaxi} with $r=s$. For every $\eta> 0$,
\begin{eqnarray*} P \bigl( Y(t) - Y(s) > \eta \bigr) &=& P \bigl( Y(s) \vee Y^{(s)}(t-s) - Y(s) > \eta \bigr) \\ &\leq& P \bigl( Y^{(s)}(t-s) > \eta \bigr) = P \bigl( (t-s)^H Y(1) > \eta \bigr). \end{eqnarray*}
Hence, continuity in probability. \end{pf}
We now introduce a crucial object for the subsequent discussion, which is the limiting process obtained in the main limit theorem of Section~\ref{sec:FLTPM}. It has a somewhat deceptively simple representation that we presently describe.
Let $\alpha>0$, and consider the extremal Fr\'echet process $Z_{\alpha}(t), t\geq0$, defined in \eqref{e:extreme.frechet}, with the scale $\sigma=1$. For $0<\beta<1$, we define a new stochastic process by
\begin{equation} \label{e:lim.process} Z_{\alpha, \beta}(t) = Z_{\alpha} \bigl(t^\beta \bigr),\quad\quad t\geq0 . \end{equation}
We will refer to this process as the {\it time scaled extremal Fr\'echet process}.
The next proposition places this process in the general framework introduced earlier in this section.
\begin{proposition} \label{pr:limproc.properties} The process $Z_{\alpha, \beta}$ in \eqref{e:lim.process} is self-similar with $H=\beta/\alpha$ and has stationary max-increments. \end{proposition}
\begin{pf} Since the extremal Fr\'echet process is self-similar with $H=1/\alpha$, it is immediately seen that the process $Z_{\alpha, \beta}$ is self-similar with $H=\beta/\alpha$.
To show the stationarity of max-increments, we start with a useful representation of the extremal Fr\'echet process $Z_{\alpha}(t), t\geq0$ in terms of the points of a Poisson random measure. Let $ ((j_k,s_k) )$ be the points of a Poisson random measure on $\reals_+^2$ with mean measure $\rho_{\alpha} \times\lambda$, where $\rho_{\alpha}(x,\infty) = x^{-\alpha}$, $x>0$ and $\lambda$ is the Lebesgue measure on $\reals_+$. Then an elementary calculation shows that
\[ \bigl( Z_{\alpha}(t), t\geq0 \bigr) \eid \bigl( \sup \{ j_k\dvtx s_k\leq t \}, t\geq0 \bigr) . \]
Therefore, $ ( Z_{\alpha,\beta}(t), t\geq0 ) \eid ( U_{\alpha,\beta}(t), t\geq0 ) $, where
\begin{equation} \label{e:represent.U} U_{\alpha,\beta}(t) = \sup \bigl\{ j_k \dvtx s_k \leq t^{\beta} \bigr\} ,\quad\quad t \geq0 . \end{equation}
Given $r > 0$, we define
\[ U_{\alpha,\beta}^{(r)}(t) = \sup \bigl\{ j_k \dvtx (t+r)^{\beta}-t^{\beta} \leq s_k \leq(t+r)^{\beta} \bigr\} . \]
Since $0 < \beta< 1$, we have
\[ \bigl( (t_1+r)^{\beta}-t_1^{\beta}, (t_1+r)^{\beta} \bigr) \subset \bigl( (t_2+r)^{\beta}-t_2^{\beta}, (t_2+r)^{\beta} \bigr) \]
for $0\leq t_1<t_2$. The nested nature of these sets implies that
\[ \bigl( U_{\alpha,\beta}^{(r)}(t), t \geq0 \bigr) \stackrel{d} {=} \bigl( U_{\alpha,\beta}(t), t \geq0 \bigr) , \]
because only the obvious equality of the one-dimensional distributions must be checked. Furthermore, since $(t+r)^{\beta}-t^{\beta}\leq r^\beta$, we see that
\[ U_{\alpha,\beta}(t+r) = U_{\alpha,\beta}(r) \vee U_{\alpha,\beta }^{(r)}(t)\quad\quad \mbox{for all } t \geq0 . \]
This means that the process $U_{\alpha,\beta}$ has stationary max-increments and, hence, so does the process~$Z_{\alpha,\beta}$. \end{pf}
Note that the max-increment process $ ( U_{\alpha,\beta}^{(r)}(t) )$ in the proof of Proposition \ref{pr:limproc.properties} is not independent of the random variable $U_{\alpha,\beta}(r)$ if $\beta<1$. The case $\beta=1$ corresponds to the extremal Fr\'echet process, whose max-increments are both stationary and independent.
It is interesting to note that, by part (b) of Proposition \ref{pr:sssmaxi}, any $H$-self-similar process with stationary max-increments and $\alpha$-Fr\'echet marginals, must satisfy $H\leq 1/\alpha$. The exponent $H=\beta/\alpha$ with $0<\beta\leq1$ of the process $Z_{\alpha,\beta}$ (with $\beta=1$ corresponding to the extremal Fr\'echet process $Z_{\alpha}$) covers the entire interval $(0,1/\alpha]$. Therefore, the upper bound of part (b) of Proposition \ref{pr:sssmaxi} is, in general, the best possible.
We finish this section by mentioning that an immediate conclusion from \eqref{e:represent.U} is the following representation of the time scaled extremal Fr\'echet process $Z_{\alpha, \beta}$ on the interval $[0,1]$:
\begin{equation} \label{e:represent.Z} \bigl( Z_{\alpha, \beta}(t), 0 \leq t \leq1 \bigr) \stackrel{d} {=} \Biggl( \bigvee_{j=1}^{\infty} \Gamma_j^{-1/\alpha} \mathbf{1}_{\{ V_j \leq t \}}, 0 \leq t \leq1 \Biggr) , \end{equation}
where $\Gamma_j$, $j=1,2,\ldots$\,, are arrival times of a unit rate Poisson process on $(0,\infty)$, and $(V_j)$ are i.i.d. random variables with $P(V_1 \leq x) = x^{\beta}$, $0 < x \leq1$, independent of $(\Gamma_j)$.
\section{A functional limit theorem for partial maxima} \label{sec:FLTPM}
In this section, we state and prove our main result, a functional limit theorem for the partial maxima of the discrete-time stationary process $\mathbf{X}=(X_1,X_2,\ldots)$ given in (\ref{e:underlying.proc}). Recall that $T$ is a conservative, ergodic and measure preserving map on a $\sigma$-finite, infinite, measure space $(E,\mathcal{E},\mu)$. We will assume that $T$ is a pointwise dual ergodic map with normalizing sequence $(a_n)$ that is regularly varying with exponent $1-\beta$; equivalently, the wandering sequence $(w_n)$ in \eqref{e:wanderingrate} is assumed to be regularly varying with exponent $\beta$. Crucially, we will assume that $1/2 < \beta< 1$. See Remark \ref{rk:other.beta} after the proof of Theorem \ref{t:main.max} below.
Define
\begin{equation}
\label{e:b.n} b_n = \biggl( \int_E \max _{1 \leq k \leq n} \bigl| f\circ T^n(x) \bigr|^{\alpha} \mu(\mathrm{d}x) \biggr)^{1/\alpha}, \quad\quad n=1,2,\ldots. \end{equation}
The sequence $(b_n)$ is known to play an important role in the rate of growth of partial maxima of an $\alpha$-stable process of the type \eqref{e:underlying.proc}. It also turns out to be a proper normalizing sequence for our functional limit theorem. In Samorodnitsky \cite{samorodnitsky:2004a} it was shown that, for a canonical kernel (\ref{e:general.rosinski}), if the map $T$ is conservative, then the sequence $(b_n)$ grows at a rate strictly slower than $n^{1/\alpha}$. The extra assumptions imposed in the current paper will guarantee a more precise statement. We will prove that, in fact, $(b_n) \in RV_{\beta/\alpha}$ and, more specifically,
\begin{equation}
\label{e:RV.exp.bn} \lim_{n\to\infty}\frac{b_n^\alpha}{ w_n}=\| f
\|_\infty \end{equation}
(where $(w_n)$ is the wandering sequence). This fact has an interesting message, because it explicitly shows that the rate of growth of the partial maxima is determined both by the heaviness of the marginal tails (through $\alpha$) and by the length of memory (through $\beta$). Such a precise measure of the length of memory is not present in Samorodnitsky \cite{samorodnitsky:2004a}.
In contrast, if the map $T$ has a nontrivial dissipative component, then the sequence $(b_n)$ grows at the rate $n^{1/\alpha} $, and so do the partial maxima of the stationary S$\alpha$S\ process; see Samorodnitsky~\cite{samorodnitsky:2004a}. This is the limiting case of the setup in the present paper, as $\beta$ gets closer to $1$. Intuitively, the smaller is $\beta$, the longer is the memory in the process.
The basic idea in the proof of our main result, Theorem \ref{t:main.max} below, is similar to the idea in the proof of Theorems 3.1 and 4.1 in Samorodnitsky \cite{samorodnitsky:2004a} and is based on a Poisson representation of the process and a ``single jump'' property; see Remark \ref{rk:other.beta}.
We recall the tail constant of an $\alpha$-stable random variable given by
\[ C_{\alpha} = \biggl( \int_0^{\infty} x^{-\alpha} \sin x \,\mathrm{d}x \biggr)^{-1} = \cases{ (1-\alpha) /
\bigl( \Gamma(2-\alpha) \cos (\uppi \alpha/ 2) \bigr) &\quad \mbox{if } $\alpha\neq1$, \cr 2 / \uppi&\quad \mbox{if } $\alpha=1$; } \]
see Samorodnitsky and Taqqu \cite{samorodnitsky:taqqu:1994}.
\begin{theorem} \label{t:main.max} Let $T$ be a conservative, ergodic and measure preserving map on a $\sigma$-finite infinite measure space $(E,\mathcal{E},\mu)$. Assume that $T$ is a pointwise dual ergodic map with normalizing sequence $(a_n) \in RV_{1-\beta}$, $0\leq\beta\leq1$. Let $f\in L^{\alpha}(\mu)\cap L^{\infty}(\mu)$, and assume that $f$ is supported by a uniform set $A$ for $T$, that is, $\operatorname{supp}(f) \subset A$. Let $\alpha >0$. Then the sequence $(b_n)$ in \eqref{e:b.n} satisfies~\eqref{e:RV.exp.bn}.
Assume now that $0<\alpha<2$ and $1/2<\beta<1$. If $M$ is a S$\alpha$S random measure on $(E,\mathcal{E})$ with control measure $\mu$, then the stationary S$\alpha$S process $\mathbf{X}$ given in (\ref{e:underlying.proc}) satisfies
\begin{equation}
\label{e:weak.conv} \biggl( \frac{1}{b_n} \max_{1 \leq k \leq\lfloor nt \rfloor}|X_k|, t\geq0 \biggr) \quad\Rightarrow \quad\bigl( C_{\alpha}^{1/\alpha} Z_{\alpha, \beta }(t), t\geq0 \bigr) \quad\quad\mbox{in } D[0,\infty) \end{equation}
in the Skorohod $M_1$-topology. Moreover, if $f = \mathbf{1}_A$, then the above convergence occurs in the Skorohod $J_1$-topology as well. \end{theorem}
\begin{remark} The functional limit theorem in Theorem \ref{t:main.max} above, once again, involves weak convergence in two different topologies, that is, the Skorohod $J_1$-topology and the Skorohod $M_1$-topology. The issue is similar to that in Proposition \ref{p:max.ergodic}; see Remark \ref{rk:topologies}. \end{remark}
\begin{pf*}{Proof of Theorem \ref{t:main.max}} We start with verifying \eqref{e:RV.exp.bn}. Obviously,
\[
b_n^{\alpha} \leq \| f \|_{\infty} \mu( \varphi_A \leq n) , \]
and, recalling that $w_n \sim\mu(\varphi_A \leq n)$, we get the upper bound
\[
\limsup_{n\to\infty}\frac{b_n^\alpha}{w_n}\leq\| f\|_\infty.
\]
On the other hand, take an arbitrary $\epsilon\in ( 0,\|
f \|_{\infty} )$. The set
\[
B_{\epsilon} = \bigl\{x \in A\dvtx \bigl|f(x)\bigr| \geq \| f \|_{\infty} - \epsilon \bigr\} \]
is a uniform set for $T$. A lower bound for $b_n^{\alpha}$ is obtained via the obvious inequality
\[
b_n^{\alpha} \geq \bigl( \| f \|_{\infty} - \epsilon \bigr) \mu \Biggl( \bigcup_{j=1}^n T^{-j} B_{\epsilon} \Biggr) . \]
Indeed, let $ ( w_n^{(\epsilon)} )$ be the corresponding wandering rate sequence to the set $B_{\epsilon}$. As argued in Proposition \ref {p:max.ergodic}, we know that $w_n \sim w_n^{(\epsilon)} \sim\mu(\varphi_{B_{\epsilon}} \leq n)$. Therefore,
\[ \liminf_{n\to\infty}\frac{b_n^\alpha}{ w_n} = \liminf _{n\to\infty}\frac{b_n^\alpha}{\mu(\varphi_{B_{\epsilon}}
\leq n)} \geq \| f \|_{\infty} - \epsilon. \]
Letting $\epsilon\to0$, we obtain \eqref{e:RV.exp.bn}.
Suppose now that $0<\alpha<2$ and $1/2<\beta<1$. We continue with proving convergence in the finite-dimensional distributions in \eqref{e:weak.conv}. Since for random elements in $D[0,\infty)$ with nondecreasing sample paths, weak convergence in the $M_1$-topology is implied by the finite-dimensional weak convergence, this will also establish \eqref{e:weak.conv} in the sense of weak convergence in the $M_1$-topology.
Fix $0 = t_0< t_1 < \cdots< t_d$, $d \geq1$. We may and will assume that $t_d \leq1$. We use a series representation of the random vector $(X_1, \ldots, X_n)$: with $f_k=f\circ T^k$, $k=1,2,\ldots$\,,
\begin{equation} \label{e:series.max} (X_k, k=1,\ldots,n) \stackrel{d} {=} \Biggl( b_n C_{\alpha}^{1/\alpha} \sum _{j=1}^{\infty} \epsilon_j \Gamma_j^{-1/\alpha} \frac{f_k(U_j^{(n)})}{\max_{1 \leq i \leq n}
|f_i(U_j^{(n)})|}, k=1,\ldots,n \Biggr) . \end{equation}
Here $(\epsilon_j)$ are i.i.d. Rademacher random variables (symmetric random variables with values $\pm 1$), $(\Gamma_j)$ are the arrival times of a unit rate Poisson process on $(0,\infty)$, and $(U_j^{(n)})$ are i.i.d. $E$-valued random variables with the common law $\eta_n$ defined by
\begin{equation} \label{e:eta.n} \frac{\mathrm{d}\eta_n}{\mathrm{d}\mu}(x) = \frac{1}{b_n^{\alpha}} \max _{1 \leq k
\leq n} \bigl|f_k(x)\bigr|^{\alpha} ,\quad\quad x \in E . \end{equation}
The sequences $(\epsilon_j)$, $(\Gamma_j)$, and $(U_j^{(n)})$ are taken to be independent. We refer to Section~3.10 of Samorodnitsky and Taqqu \cite{samorodnitsky:taqqu:1994} for series representations of $\alpha$-stable random vectors. The representation \eqref{e:series.max} was also used in Samorodnitsky \cite {samorodnitsky:2004a}, and the argument below is structured similarly to the corresponding argument ibid.
The crucial consequence of the assumption $1/2<\beta<1$ is that, in the series representation~(\ref{e:series.max}), only the largest Poisson jump will play an important role. It is shown in Samorodnitsky \cite{samorodnitsky:2004a} that, under the assumptions of Theorem \ref{t:main.max}, for every $\eta>0$,
\begin{eqnarray} \label{e:single.poisson.jump} \varphi_n(\eta) &\equiv& P \Biggl( \bigcup _{k=1}^n \biggl\{ \Gamma_j^{-1/\alpha}
\frac{|f_k(U_j^{(n)})|}{\max_{1 \leq i \leq n}|f_i(U_j^{(n)})|} > \eta \nonumber \\[-8pt]\\[-8pt] &&\hphantom{P \Biggl(\bigcup _{k=1}^n \biggl\{}{}\mbox{for at least 2 different } j=1,2,\ldots \biggr\} \Biggr) \to0\nonumber \end{eqnarray}
as $n \to\infty$.
We will proceed in two steps. First, we will prove that
\begin{eqnarray} \label{e:fidi.Malphabeta} &&\Biggl( \bigvee_{j=1}^{\infty} \Gamma_j^{-1/\alpha} \frac{\max_{1
\leq k \leq\lfloor nt_i \rfloor} |f_k(U_j^{(n)})|}{\max_{1 \leq k
\leq n} |f_k(U_j^{(n)})|}, i=1,\ldots,d \Biggr) \nonumber \\[-8pt]\\[-8pt] &&\quad\Rightarrow\quad \bigl( Z_{\alpha, \beta}(t_i), i=1,\ldots,d \bigr) \quad\quad\mbox{in } \reals_+^d .\nonumber \end{eqnarray}
Next, we will prove that, for fixed $\lambda_1, \ldots, \lambda_d > 0$, for every $0<\delta<1$,
\begin{eqnarray}\label{e:upp.bdd.max} &&P \Bigl( b_n^{-1} \max_{1 \leq k \leq\lfloor nt_i \rfloor}
|X_k| > \lambda_i, i=1,\ldots,d \Bigr) \nonumber \\[-8pt]\\[-8pt] &&\quad\leq P \Biggl( C_{\alpha}^{1/\alpha} \bigvee _{j=1}^{\infty} \Gamma_j^{-1/\alpha} \frac{\max_{1 \leq k \leq\lfloor nt_i
\rfloor}|f_k(U_j^{(n)})|}{\max_{1 \leq k \leq n}|f_k(U_j^{(n)})|} > \lambda_i(1-\delta), i=1,\ldots,d \Biggr) + \mathrm{o}(1) \nonumber \end{eqnarray}
and that
\begin{eqnarray}\label{e:lower.bdd.max} &&P \Bigl( b_n^{-1} \max_{1 \leq k \leq\lfloor nt_i \rfloor}
|X_k| > \lambda_i, i=1,\ldots,d \Bigr) \nonumber \\[-8pt]\\[-8pt] &&\quad\geq P \Biggl( C_{\alpha}^{1/\alpha} \bigvee _{j=1}^{\infty} \Gamma_j^{-1/\alpha} \frac{\max_{1 \leq k \leq\lfloor nt_i
\rfloor}|f_k(U_j^{(n)})|}{\max_{1 \leq k \leq n}|f_k(U_j^{(n)})|} > \lambda_i(1+\delta), i=1,\ldots,d \Biggr) + \mathrm{o}(1) .\nonumber \end{eqnarray}
Since the Fr\'echet distribution is continuous, the weak convergence
\[ \Bigl( b_n^{-1} \max_{1 \leq k \leq\lfloor nt_i \rfloor}
|X_k|, i=1,\ldots,d \Bigr)\quad \Rightarrow\quad \bigl( Z_{\alpha, \beta}(t_i), i=1,\ldots,d \bigr) \quad\quad\mbox{in } \reals_+^d \]
will follow by taking $\delta$ arbitrarily small.
We start with proving \eqref{e:fidi.Malphabeta}. For $n=1,2,\ldots$\,, $N_n=\sum_{j=1}^{\infty} \delta_{(\Gamma_j, U_j^{(n)})}$ is a Poisson random measure on $(0,\infty)\times\bigcup_{k=1}^{n} T^{-k}A$ with mean measure $\lambda\times\eta_n$. Define a map $S_n\dvtx \reals_+ \times\bigcup_{k=1}^{n} T^{-k}A \to\reals_+^d$ by
\[ S_n(r,x) = r^{-1/\alpha} \bigl( M_n(f) (x) \bigr)^{-1} \bigl( M_{\lfloor nt_1 \rfloor}(f) (x), \ldots, M_{\lfloor nt_d \rfloor}(f) (x) \bigr) ,\qquad r > 0, x \in\bigcup_{k=1}^{n} T^{-k}A .
\]
Then, for $\lambda_1, \ldots, \lambda_d > 0$,
\begin{eqnarray*} &&P \Biggl( \bigvee_{j=1}^{\infty} \Gamma_j^{-1/\alpha} \frac{\max_{1
\leq k \leq\lfloor nt_i \rfloor} |f_k(U_j^{(n)})|}{\max_{1 \leq k \leq n} |f_k(U_j^{(n)})|} \leq\lambda_i , i=1,\ldots,d \Biggr) \\ &&\quad= P \bigl[ N_n \bigl( S_n^{-1} \bigl( (0, \lambda_1 ] \times\cdots \times (0,\lambda_d] \bigr)^c \bigr) = 0 \bigr] \\ &&\quad= \exp \bigl\{ - (\lambda\times\eta_n) \bigl( S_n^{-1} \bigl( (0,\lambda_1] \times\cdots\times(0,\lambda_d] \bigr)^c \bigr) \bigr\} \\ &&\quad= \exp \Biggl\{ - (\lambda\times\eta_n) \Biggl\{(r,x)\dvtx \bigvee _{j=1}^d \lambda_j^{-\alpha} \frac{ (M_{\lfloor nt_j \rfloor}(f)(x) )^{\alpha}}{ (M_n(f)(x) )^{\alpha}} > r \Biggr\} \Biggr\} \\ &&\quad= \exp \Biggl\{ -b_n^{-\alpha} \int_E \bigvee_{j=1}^d \lambda _j^{-\alpha } M_{\lfloor nt_j \rfloor}(f)^{\alpha} \,\mathrm{d}\mu \Biggr\} . \end{eqnarray*}
We use \eqref{e:RV.exp.bn} and the weak convergence in Proposition \ref{p:max.ergodic} to obtain
\begin{eqnarray*} &&b_n^{-\alpha} \int_E \bigvee _{j=1}^d \lambda_j^{-\alpha} M_{\lfloor nt_j \rfloor}(f)^{\alpha} \,\mathrm{d}\mu
\sim\| f \|_{\infty}^{-1} \int_E \bigvee_{j=1}^d \lambda _j^{-\alpha} M_{\lfloor nt_j \rfloor}(f)^{\alpha} \,\mathrm{d} \mu_n \\ &&\quad\to\quad\int_{\Omega^{\prime}} \bigvee_{j=1}^d \lambda_j^{-\alpha} \mathbf{1}_{\{ V_{\beta} \leq t_j \}} \,\mathrm{d}P^{\prime} = \sum_{i=1}^d \bigl(t_i^{\beta} - t_{i-1}^{\beta} \bigr) \Biggl( \bigwedge _{j=i}^d \lambda_j \Biggr)^{-\alpha}. \end{eqnarray*}
Therefore,
\begin{eqnarray*} &&P \Biggl( \bigvee_{j=1}^{\infty} \Gamma_j^{-1/\alpha} \frac{\max_{1
\leq k \leq\lfloor nt_i \rfloor} |f_k(U_j^{(n)})|}{\max_{1 \leq k
\leq n} |f_k(U_j^{(n)})|} \leq\lambda_i , i=1,\ldots,d \Biggr) \\ &&\quad\to\quad\exp \Biggl\{ - \sum_{i=1}^d \bigl(t_i^{\beta} - t_{i-1}^{\beta} \bigr) \Biggl( \bigwedge_{j=i}^d \lambda_j \Biggr)^{-\alpha} \Biggr\} = P \bigl( Z_{\alpha, \beta}(t_i) \leq\lambda_i, i=1,\ldots,d \bigr) . \end{eqnarray*}
The claim (\ref{e:fidi.Malphabeta}) has, consequently, been proved.
We continue with the statements (\ref{e:upp.bdd.max}) and (\ref {e:lower.bdd.max}). Since the arguments are very similar, we only prove (\ref{e:upp.bdd.max}). Let $K \in{\mathbb N}$ and $0 < \epsilon< 1$ be constants so that
\[ K+1 > \frac{4}{\alpha} \quad\mbox{and}\quad \delta- \epsilon K > 0 . \]
Then
\begin{eqnarray*} &&P \Bigl( b_n^{-1} \max_{1 \leq k \leq\lfloor nt_i \rfloor}
|X_k| > \lambda_i, i=1,\ldots,d \Bigr) \\ &&\quad\leq P \Biggl( C_{\alpha}^{1/\alpha} \bigvee _{j=1}^{\infty} \Gamma _j^{-1/\alpha} \frac{\max_{1 \leq k \leq\lfloor nt_i \rfloor
}|f_k(U_j^{(n)})|}{\max_{1 \leq k \leq n}|f_k(U_j^{(n)})|} > \lambda _i(1-\delta), i=1,\ldots,d \Biggr) \\ &&\quad\quad{}+ \varphi_n \Bigl( C_{\alpha}^{-1/\alpha} \epsilon\min _{1 \leq i \leq d} \lambda_i \Bigr) + \sum _{i=1}^d \psi_n(\lambda_i, t_i) , \end{eqnarray*}
where
\begin{eqnarray*} \psi_n(\lambda, t) &=& P \Biggl( C_{\alpha}^{1/\alpha} \max _{1 \leq k \leq\lfloor nt \rfloor} \Biggl\vert \sum_{j=1}^{\infty}
\epsilon_j \Gamma _j^{-1/\alpha} \frac{f_k(U_j^{(n)})}{\max_{1 \leq i \leq n}|f_i(U_j^{(n)})|} \Biggr\vert > \lambda, \\ &&\hphantom{P \Biggl(}{}C_{\alpha}^{1/\alpha} \bigvee_{j=1}^{\infty} \Gamma_j^{-1/\alpha} \frac
{\max_{1 \leq k \leq\lfloor nt \rfloor}|f_k(U_j^{(n)})|}{\max_{1
\leq k \leq n}|f_k(U_j^{(n)})|} \leq\lambda(1-\delta) , \mbox{ and for each } m=1,\ldots,n, \\ &&\hphantom{P \Biggl(}{}C_{\alpha}^{1/\alpha} \Gamma_j^{-1/\alpha} \frac
{|f_m(U_j^{(n)})|}{\max_{1 \leq i \leq n}|f_i(U_j^{(n)})|} > \epsilon\lambda \mbox{ for at most one } j=1,2,\ldots \Biggr) . \end{eqnarray*}
By (\ref{e:single.poisson.jump}), it is enough to show that
\begin{equation} \label{e:neg.term} \psi_n(\lambda, t) \to0 \end{equation}
for all $\lambda>0$ and $0 \leq t \leq1$.
For every $k=1,2,\ldots,n$, the Poisson random measure represented by the points
\[ \Bigl( \epsilon_j \Gamma_j^{-1/\alpha} f_k \bigl(U_j^{(n)} \bigr) \Bigl( \max _{1
\leq i \leq n}\bigl|f_i \bigl(U_j^{(n)}
\bigr)\bigr| \Bigr)^{-1}, j=1,2,\ldots \Bigr) \]
has the same mean measure as that represented by the points
\[
\bigl( \epsilon_j \Gamma_j^{-1/\alpha} \| f
\|_{\alpha} b_n^{-1}, j=1,2,\ldots \bigr) , \]
where $\| f \|_{\alpha} = ( \int_E |f|^{\alpha} \,\mathrm{d}\mu
)^{1/\alpha}$. In fact, the common mean measure assigns the value $x^{-\alpha} \| f
\|_{\alpha}^{\alpha} / 2$ to the sets $(x,\infty)$ and $(-\infty, -x)$ for every $x>0$. Therefore, these two Poisson random measures coincide distributionally. We conclude that the probability in (\ref {e:neg.term}) is bounded by
\begin{eqnarray*} &&\sum_{k=1}^{\lfloor nt \rfloor} P \Biggl( C_{\alpha}^{1/\alpha} \Biggl\vert \sum_{j=1}^{\infty} \epsilon_j \Gamma_j^{-1/\alpha} \frac
{f_k(U_j^{(n)})}{\max_{1 \leq i \leq n}|f_i(U_j^{(n)})|} \Biggr\vert > \lambda, \\ &&\hphantom{\sum_{k=1}^{\lfloor nt \rfloor} P \Biggl(}{} C_{\alpha}^{1/\alpha} \bigvee _{j=1}^{\infty} \Gamma _j^{-1/\alpha}
\frac{f_k(U_j^{(n)})}{\max_{1 \leq i \leq n}|f_i(U_j^{(n)})|} \leq\lambda(1-\delta) , \\ &&\hphantom{\sum_{k=1}^{\lfloor nt \rfloor} P \Biggl(}{}C_{\alpha}^{1/\alpha} \Gamma_j^{-1/\alpha} \frac
{|f_k(U_j^{(n)})|}{\max_{1 \leq i \leq n}|f_i(U_j^{(n)})|} > \epsilon\lambda \mbox{ for at most one } j=1,2,\ldots \Biggr)
\\ &&\quad= \lfloor nt \rfloor P \Biggl( C_{\alpha}^{1/\alpha} \Biggl\vert \sum_{j=1}^{\infty} \epsilon_j
\Gamma_j^{-1/\alpha} \Biggr\vert > \lambda \| f
\|_{\alpha}^{-1} b_n , C_{\alpha}^{1/\alpha} \bigvee_{j=1}^{\infty } \Gamma_j^{-1/\alpha}
\leq\lambda(1-\delta) \| f \|_{\alpha}^{-1} b_n , \\ &&\hphantom{\quad= \lfloor nt \rfloor P \Biggl(}{}C_{\alpha}^{1/\alpha} \Gamma_j^{-1/\alpha} > \epsilon
\lambda\| f \| _{\alpha}^{-1} b_n \mbox{ for at most one } j=1,2,\ldots \Biggr) \\ &&\quad\leq n P \Biggl( C_{\alpha}^{1/\alpha} \Biggl\vert \sum _{j=K+1}^{\infty} \epsilon_j \Gamma_j^{-1/\alpha} \Biggr\vert > (\delta- \epsilon K) \lambda
\| f \|_{\alpha}^{-1} b_n \Biggr) \\
&&\quad\leq\frac{n \| f \|_{\alpha}^4 C_{\alpha}^{4/\alpha} }{(\delta- \epsilon K)^4 \lambda^4 b_n^4} E \Biggl\vert \sum_{j=K+1}^{\infty} \epsilon _j \Gamma_j^{-1/\alpha} \Biggr\vert ^4 . \end{eqnarray*}
Due to the choice $K+1>4/\alpha$,
\[ E \Biggl\vert \sum_{j=K+1}^{\infty} \epsilon_j \Gamma_j^{-1/\alpha} \Biggr\vert ^4 < \infty; \]
see Samorodnitsky \cite{samorodnitsky:2004a} for a detailed proof. Since $n/b_n^4 \to0$ as $n \to\infty$, (\ref{e:neg.term}) follows.
Suppose now that $f = \mathbf{1}_A$. In that case, the probability measure $\eta_n$ defined in \eqref{e:eta.n} coincides with the probability measure $\mu_n$ of Proposition \ref{p:max.ergodic}. In order to prove weak convergence in the $J_1$-topology, we will use a truncation argument. We may and will restrict ourselves to the space $D[0,1]$. Let $K=1,2,\ldots$\,. First of all, we show, in the notation of \eqref{e:represent.Z}, the convergence\looseness=1
\begin{eqnarray} \label{e:fidi.truncate} &&\Biggl( C_{\alpha}^{1/\alpha} \max_{1 \leq k \leq\lfloor nt \rfloor} \Biggl\vert \sum_{j=1}^K \epsilon_j \Gamma_j^{-1/\alpha} \mathbf{1}_A \circ T^k \bigl(U_j^{(n)} \bigr) \Biggr\vert , 0 \leq t\leq1 \Biggr)\nonumber \\[-8pt]\\[-8pt] &&\quad\Rightarrow \quad\Biggl( C_{\alpha}^{1/\alpha} \bigvee _{j=1}^K \Gamma_j^{-1/\alpha} \mathbf{1}_{\{ V_j \leq t \}}, 0\leq t\leq1 \Biggr)\nonumber \end{eqnarray}
in the $J_1$-topology on $D[0,1]$. Indeed, by \eqref{e:single.poisson.jump}, outside of an event of asymptotically vanishing probability, the process in the left-hand side of \eqref{e:fidi.truncate} is
\begin{equation} \label{e:fidi.truncate1} \Biggl( C_{\alpha}^{1/\alpha} \bigvee _{j=1}^K \Gamma_j^{-1/\alpha} \max _{1 \leq k \leq\lfloor nt \rfloor} \mathbf{1}_A \circ T^k \bigl(U_j^{(n)} \bigr), 0\leq t\leq1 \Biggr) . \end{equation}
By Proposition \ref{p:max.ergodic}, we can put all the random variables involved on the same probability space so that the time of the single step in the $j$th term in \eqref{e:fidi.truncate1} converges a.s. for each $j=1,\ldots, K$ to $V_j$. Then, trivially, the process in \eqref{e:fidi.truncate1} converges a.s. in the $J_1$-topology on $D[0,1]$ to the process in the right-hand side of \eqref{e:fidi.truncate}. Therefore, the weak convergence in \eqref{e:fidi.truncate} follows.\looseness=1
Next, we note that in the $J_1$-topology on the space $D[0,1]$,
\begin{eqnarray*} &&\Biggl( C_{\alpha}^{1/\alpha} \bigvee_{j=1}^K \Gamma_j^{-1/\alpha} \mathbf{1}_{\{ V_j \leq t \}}, 0\leq t\leq1 \Biggr) \\ &&\quad\to\quad \Biggl( C_{\alpha}^{1/\alpha} \bigvee_{j=1}^{\infty} \Gamma_j^{-1/\alpha} \mathbf{1}_{\{ V_j \leq t_i \}} 0\leq t\leq1 \Biggr) \quad\quad\mbox{as } K \to\infty \mbox{ a.s.} \end{eqnarray*}
This is so because, as $K \to\infty$,
\begin{eqnarray*} &&\sup_{0 \leq t \leq1} \Biggl( \bigvee_{j=1}^{\infty} \Gamma_j^{-1/\alpha} \mathbf{1}_{\{ V_j \leq t \}} - \bigvee _{j=1}^K \Gamma_j^{-1/\alpha} \mathbf{1}_{\{ V_j \leq t \}} \Biggr) \\ &&\quad\leq\Gamma_{K+1}^{-1/\alpha} \to0 \quad\quad\mbox{a.s.} \end{eqnarray*}
According to Theorem 3.2 in Billingsley \cite{billingsley:1999}, the $J_1$-convergence in \eqref{e:weak.conv} will follow once we show that
\[ \lim_{K \to\infty} \limsup_{n \to\infty} P \Biggl( \max _{1 \leq k \leq n} \Biggl\vert \sum_{j=K+1}^{\infty} \epsilon_j \Gamma_j^{-1/\alpha} \mathbf{1}_A \circ T^k \bigl(U_j^{(n)} \bigr) \Biggr\vert > \epsilon \Biggr) = 0 \]
for every $\epsilon>0$. Write
\begin{eqnarray*} &&P \Biggl( \max_{1 \leq k \leq n} \Biggl\vert \sum _{j=K+1}^{\infty} \epsilon_j \Gamma_j^{-1/\alpha} \mathbf{1}_A \circ T^k \bigl(U_j^{(n)} \bigr) \Biggr\vert > \epsilon \Biggr) \\ &&\quad\leq\int_0^{(\epsilon/2)^{-\alpha}} \mathrm{e}^{-x} \frac{x^{K-1}}{(K-1)!} \,\mathrm{d}x \\ &&\hphantom{\quad\leq}{} + \int_{(\epsilon/2)^{-\alpha}}^{\infty} \mathrm{e}^{-x} \frac{x^{K-1}}{(K-1)!} \\ &&\hphantom{\quad\leq+ \int_{(\epsilon/2)^{-\alpha}}^{\infty}}{} \times P \Biggl( \max_{1 \leq k \leq n} \Biggl \vert \sum_{j=1}^{\infty} \epsilon_j ( \Gamma_j + x )^{-1/\alpha} \mathbf{1}_A \circ T^k \bigl(U_j^{(n)} \bigr) \Biggr\vert > \epsilon \Biggr) \,\mathrm{d}x . \end{eqnarray*}
Clearly, the first term vanishes when $K \to\infty$. Therefore, it is sufficient to show that for every $x \geq(\epsilon/2)^{-\alpha}$,
\begin{equation} \label{e:upper.app.Gamma1} P \Biggl( \max_{1 \leq k \leq n} \Biggl\vert \sum _{j=1}^{\infty} \epsilon_j ( \Gamma_j + x )^{-1/\alpha} \mathbf{1}_A \circ T^k \bigl(U_j^{(n)} \bigr) \Biggr\vert > \epsilon \Biggr) \to0 \end{equation}
as $n \to\infty$.
To this end, choose $L \in{\mathbb N}$ and $0 < \xi< 1/2$ so that
\begin{equation} \label{e:const.rest2} L+1 > \frac{4}{\alpha} \quad\mbox{and}\quad \frac{1}{2} - \xi L > 0 . \end{equation}
By \eqref{e:single.poisson.jump}, we can write
\begin{eqnarray}\label{e:one.term.vanish} &&\hspace*{-18pt}P \Biggl( \max_{1 \leq k \leq n} \Biggl\vert \sum _{j=1}^{\infty} \epsilon_j ( \Gamma_j + x )^{-1/\alpha} \mathbf{1}_A \circ T^k \bigl(U_j^{(n)} \bigr) \Biggr\vert > \epsilon \Biggr) \nonumber \\ &&\hspace*{-18pt}\quad\leq P \Biggl( \max_{1 \leq k \leq n} \Biggl\vert \sum _{j=1}^{\infty} \epsilon_j ( \Gamma_j + x )^{-1/\alpha} \mathbf{1}_A \circ T^k \bigl(U_j^{(n)} \bigr) \Biggr\vert > \epsilon, \mbox{ and for each } m = 1,\ldots, n, \\ &&\hspace*{-18pt}\hphantom{\quad\leq P \Biggl(}{} ( \Gamma_j + x )^{-1/\alpha} \mathbf{1}_A \circ T^m \bigl(U_j^{(n)} \bigr) > \xi\epsilon \mbox{ for at most one } j =1,2,\ldots \Biggr) + \mathrm{o}(1) . \nonumber \end{eqnarray}
Notice that for every $k=1,\ldots, n$, the Poisson random measure represented by the points
\[ \bigl( \epsilon_j ( \Gamma_j + x )^{-1/\alpha} \mathbf{1}_A \circ T^k \bigl(U_j^{(n)} \bigr), j=1,2,\ldots \bigr) \]
is distributionally equal to the Poisson random measure represented by the points
\[ \bigl( \epsilon_j \bigl( b_n^{\alpha} \mu(A)^{-1} \Gamma_j +x \bigr)^{-1/\alpha}, j=1,2,\ldots \bigr) . \]
Therefore, the first term on the right-hand side of (\ref {e:one.term.vanish}) can be bounded by
\begin{eqnarray*} &&\sum_{k=1}^n P \Biggl( \Biggl\vert \sum _{j=1}^{\infty} \epsilon_j ( \Gamma_j + x )^{-1/\alpha} \mathbf{1}_A \circ T^k \bigl(U_j^{(n)} \bigr) \Biggr\vert > \epsilon, \\ &&\hphantom{\sum_{k=1}^n P \Biggl(}{}( \Gamma_j + x )^{-1/\alpha} \mathbf{1}_A \circ T^k \bigl(U_j^{(n)} \bigr) > \xi\epsilon \mbox{ for at most one } j=1,2,\ldots \Biggr) \\ &&\quad= nP \Biggl( \Biggl\vert \sum_{j=1}^{\infty} \epsilon_j \bigl( b_n^{\alpha} \mu(A)^{-1} \Gamma_j + x \bigr)^{-1/\alpha} \Biggr \vert > \epsilon, \\ &&\hphantom{\quad= nP \Biggl(}{}\bigl( b_n^{\alpha} \mu(A)^{-1} \Gamma_j + x \bigr)^{-1/\alpha} > \xi \epsilon \mbox{ for at most one } j = 1,2,\ldots \Biggr) \\ &&\quad\leq n P \Biggl( \Biggl\vert \sum_{j=L+1}^{\infty} \epsilon_j \bigl( b_n^{\alpha} \mu(A)^{-1} \Gamma_j + x \bigr)^{-1/\alpha} \Biggr \vert > \biggl( \frac{1}{2} - \xi L \biggr)\epsilon \Biggr) . \end{eqnarray*}
In the last step we used the fact that, for $x \geq (\epsilon/2)^{-\alpha}$, the magnitude of each term in the infinite sum does not exceed $\epsilon/2$. By the contraction inequality for Rademacher series (see, e.g., Proposition 1.2.1 of Kwapie\'n and Woyczy\'nski \cite{kwapien:woyczynski:1992}),
\begin{eqnarray*} &&nP \Biggl( \Biggl\vert \sum_{j=L+1}^{\infty} \epsilon_j \bigl( b_n^{\alpha} \mu(A)^{-1} \Gamma_j + x \bigr)^{-1/\alpha} \Biggr \vert > \biggl( \frac {1}{2} - \xi L \biggr)\epsilon \Biggr) \\ &&\quad\leq2 n P \Biggl( \Biggl\vert \sum_{j=L+1}^{\infty} \epsilon_j \Gamma _j^{-1/\alpha} \Biggr\vert > \biggl( \frac{1}{2} - \xi L \biggr)\epsilon\mu (A)^{-1/\alpha} b_n \Biggr) . \end{eqnarray*}
As before, by Markov's inequality and using the constraints of the constants $L \in{\mathbb N}$ and $0 < \xi< 1/2$ given in (\ref{e:const.rest2}),
\begin{eqnarray*} &&2n P \Biggl( \Biggl\vert \sum_{j=L+1}^{\infty} \epsilon_j \Gamma _j^{-1/\alpha } \Biggr\vert > \biggl( \frac{1}{2} - \xi L \biggr)\epsilon\mu (A)^{-1/\alpha } b_n \Biggr) \\ &&\quad\leq\frac{2n\mu(A)^{4/\alpha}}{(2^{-1}-\xi L)^4 \epsilon ^4 b_n^4} E \Biggl\vert \sum _{j=L+1}^{\infty} \epsilon_j \Gamma _j^{-1/\alpha} \Biggr\vert ^4 \to0 \end{eqnarray*}
as $n \to\infty$ and, hence, (\ref{e:upper.app.Gamma1}) follows. \end{pf*}
\begin{remark} \label{rk:other.beta} The crucial point in the proof of the theorem is the ``single Poisson jump property'' \eqref{e:single.poisson.jump} that shows that, essentially, a single Poisson point of the type $\Gamma_j^{-1/\alpha}$ plays the decisive role in determining the size of a partial maximum. This enabled us to show that the normalized partial maxima converge to the first Poisson point $\Gamma_1^{-1/\alpha}$ (which, of course, has exactly the standard \mbox{$\alpha$-Fr\'echet} law). We can guarantee the ``single Poisson jump property'' in the case $1/2<\beta<1$. On the other hand, in the range $0 < \beta< 1/2$, the condition \eqref{e:single.poisson.jump} is no longer valid. We believe that the limiting process will involve a finite, but random, number of the Poisson points of the type $\Gamma_j^{-1/\alpha}$. This will preclude a limiting Fr\'echet law. The details of this are still being worked out, and will appear in a future work. In the boundary case $\beta=1/2$, the statement \eqref{e:weak.conv} still holds under certain additional conditions. This is the case, for example, for the Markov shift operators presented at the end of the paper. See also Example 5.3 in Samorodnitsky \cite{samorodnitsky:2004a}. \end{remark}
\begin{remark} There is no doubt that the convergence result in Theorem \ref{t:main.max} can be extended to more general infinitely divisible random measures $M$ in \eqref{e:underlying.proc}, under appropriate assumptions of regular variation of the L\'evy measure of $M$ and integrability of the function $f$. In particular, regardless of the size of $\alpha>0$, the time scaled extremal Fr\'echet processes $Z_{\alpha, \beta}$ are likely to appear in the limit in \eqref{e:weak.conv}. Furthermore, the symmetry of the process $\mathbf{X}$ has very little to do with the limiting distribution of the partial maxima. For example, a straightforward symmetrization argument allows one to extend \eqref{e:weak.conv} to skewed $\alpha$-stable processes, at least in the sense of convergence of finite-dimensional distributions. The reason we decided to restrict the presentation to the symmetric stable case had to do with a particularly simple form of the series representation \eqref{e:series.max} available in this case. This has allowed us to avoid certain technicalities that might have otherwise blurred the main message, which is the effect of memory on the functional limit theorem for the partial maxima. \end{remark}
One can obtain concrete examples of the situations in which the result of Theorem \ref{t:main.max} applies by taking, for instance, one of the variety of pointwise dual ergodic operators provided in Aaronson \cite{aaronson:1981} and Zweim\"uller \cite{zweimuller:2009}, and embedding them into the integral form of stationary S$\alpha$S processes. We conclude the current paper by mentioning the example of a flow generated by a null recurrent Markov chain. This example appears in Samorodnitsky \cite{samorodnitsky:2004a}, Owada and Samorodnitsky \cite {owada:samorodnitsky:2012}, and Owada \cite{owada:2013} as well.
Consider an irreducible null recurrent Markov chain $(x_n, n \geq 0)$ defined on an infinite countable state space $\mathbb{S}$ with the transition matrix $(p_{ij})$. Let $(\pi_i, i \in\mathbb{S})$ be its unique (up to constant multiplication) invariant measure with $\pi_{i_0}=1$ for some fixed state $i_0\in\mathbb{S}$. Note that $(\pi_i)$ is necessarily an infinite measure. Define a $\sigma$-finite and infinite measure on $(E,\mathcal{E}) = (\mathbb{S}^{{\mathbb N}}, \mathcal{B}(\mathbb{S}^{{\mathbb N}}))$ by
\[ \mu(B) = \sum_{i \in\mathbb{S}} \pi_i P_i(B),\quad\quad B \subseteq\mathbb{S}^{{\mathbb N}} , \]
where $P_i(\cdot)$ denotes the probability law of $(x_n)$ starting in state $i \in\mathbb{S}$. Let
\[ T(x_0, x_1, \ldots) = (x_1,x_2, \ldots) \]
be the usual left shift operator on $\mathbb{S}^{{\mathbb N}}$. Then $T$ preserves $\mu$. Since the Markov chain is irreducible and null recurrent, $T$ is conservative and ergodic (see Harris and Robbins \cite {harris:robbins:1953}).
We consider the set $A = \{ x \in\mathbb{S}^{{\mathbb N}}\dvtx x_0=i_0 \}$ with the fixed state $i_0\in\mathbb{S}$ chosen above. Since
\[ \widehat{T}^k \mathbf{1}_A (x) = P_{i_0} (x_k = i_0) \quad\quad\mbox{for } x \in A \]
is constant on $A$ (see Section~4.5 in Aaronson \cite{aaronson:1997}), we can choose as the normalizing sequence $a_n = \sum_{k=1}^n P_{i_0}(x_k=i_0)$, and see that the expression $a_n^{-1} \sum_{k=1}^n \widehat{T}^k \mathbf{1}_A(x)$ is identically equal to $1=\mu(A)$ on $A$. Therefore, the map $T$ is pointwise dual ergodic, and the Darling--Kac set condition, in fact, reduces to a simple identity. Let
\[ \varphi_A(x) = \min\{ n \geq1\dvtx x_n \in A \} ,\quad\quad x \in \mathbb{S}^{{\mathbb N}} \]
be the first entrance time, and assume that
\begin{equation} \label{e:return.DK} \sum_{k=1}^n P_{i_0}(\varphi_A \geq k) \in RV_{\beta} \end{equation}
for some $\beta\in(1/2, 1)$. Two equivalent conditions to \eqref{e:return.DK} are given in Resnick \textit{et~al}. \cite{resnick:samorodnitsky:xue:2000}. Note that the exponent of regular variation $\beta$ controls how frequently the Markov chain returns to $A$. Since $\mu(\varphi_A=k) = P_{i_0}(\varphi_A \geq k)$ for $k \geq1$ (see Lemma 3.3 in Resnick \textit{et~al}. \cite{resnick:samorodnitsky:xue:2000}), we have
\[ w_n \sim\mu(\varphi_A \leq n) \in RV_{\beta} .
\]
Then all of the assumptions of Theorem \ref{t:main.max} are satisfied for any $f\in L^{\alpha}(\mu)\cap L^{\infty}(\mu)$, supported by~$A$.
\printhistory
\end{document} | arXiv |
UrQt: an efficient software for the Unsupervised Quality trimming of NGS data
Laurent Modolo1 &
Emmanuelle Lerat1
10 Altmetric
Quality control is a necessary step of any Next Generation Sequencing analysis. Although customary, this step still requires manual interventions to empirically choose tuning parameters according to various quality statistics. Moreover, current quality control procedures that provide a "good quality" data set, are not optimal and discard many informative nucleotides. To address these drawbacks, we present a new quality control method, implemented in UrQt software, for Unsupervised Quality trimming of Next Generation Sequencing reads.
Our trimming procedure relies on a well-defined probabilistic framework to detect the best segmentation between two segments of unreliable nucleotides, framing a segment of informative nucleotides. Our software only requires one user-friendly parameter to define the minimal quality threshold (phred score) to consider a nucleotide to be informative, which is independent of both the experiment and the quality of the data. This procedure is implemented in C++ in an efficient and parallelized software with a low memory footprint. We tested the performances of UrQt compared to the best-known trimming programs, on seven RNA and DNA sequencing experiments and demonstrated its optimality in the resulting tradeoff between the number of trimmed nucleotides and the quality objective.
By finding the best segmentation to delimit a segment of good quality nucleotides, UrQt greatly increases the number of reads and of nucleotides that can be retained for a given quality objective. UrQt source files, binary executables for different operating systems and documentation are freely available (under the GPLv3) at the following address: https://lbbe.univ-lyon1.fr/-UrQt-.html.
Next Generation Sequencing (NGS) technologies produce calling error probabilities for each sequenced nucleotide [1]. These probabilities, encoded as phred scores [2], are often high at the heads and tails of the reads, indicating low-quality nucleotides [3]. The presence of these unreliable nucleotides can result in missing or wrong alignments that can either increase the number of false negatives and false positives in subsequent analyses or can produce false k-mers in de novo assembly, increasing both the complexity of an assembly and the chance of producing misassemblies [4]. To remove these unreliable nucleotides and only work with informative nucleotides, most NGS data analyses start with a quality control (QC) step before any downstream analysis.
There are three types of approaches to address low-quality nucleotides. Classical QC strategies begin by removing an arbitrary number of nucleotides at the head and tail of each read, with tools such as the fastx_trimmer from the FASTX-Toolkit [5], after visualization of the per nucleotide sequence quality with tools such as FastQC [6]. Then, only reads of high quality are retained by other filters; for example, all reads with a given percentage of their length below a given phred score are excluded, using tools such as the fastq_quality_filter from FASTX-Toolkit. More recent approaches modify incorrectly called nucleotides by superimposing reads to each other and removing low frequency polymorphisms. This kind of approach often works using motifs of k nucleotides or k-mer to modify low frequency motifs based on the most frequent ones. However, this type of approach requires potentially high sequencing coverage (15x in the case of Quake [7] and 100x in the case of ALLPATHS-LG [8]) and cannot be applied to non-uniform sequencing experiments, such as RNA sequencing (RNA-Seq). Other approaches trim unreliable nucleotides at the head and tail of each read. With these approaches, one wants to find the best trade-off between removing unreliable nucleotides and keeping the longest reliable or informative subsequence for the entire read. Current trimming approaches rely on two types of algorithms: the running sum algorithm and the window-based algorithm (for a review see [4]). However, these algorithms only return good local cutting points for each read when it is necessary to find a good global cutting point to get the best trade-off between removing unreliable nucleotides and losing too much information. Moreover, most of these QC strategies rely heavily on manually chosen parameters that are difficult to interpret and cannot be easily automatized.
In the present work, we have developed the program UrQt to trim unreliable nucleotides at the heads and tails of NGS reads based on their phred scores. We define an informative segment as a segment whose nucleotides are on average informative and an informative nucleotide as a nucleotide with a quality score above a specified quality threshold. Our approach takes advantage of the expected shape of the calling error probability along each read (abruptly decreasing for the first nucleotides and slowly increasing with the size of the reads) to find the best partition between two segments of unreliable nucleotides to be trimmed –the head and the tail of the reads– and a central informative segment. UrQt implements an unsupervised segmentation algorithm to find the best trimming cut-points in each read by maximum likelihood. We use a probabilistic model to handle more naturally the trimming problem than other procedures using window-based or running sum algorithms [4]. Moreover, UrQt requires no data-dependent parameters and takes advantage of modern multicore achitectures, which makes it particularly interesting to be routinely applied for NGS reads in fastq or fastq.gz format [9] and attractive for the development of future analytical pipelines.
In this section, we present the probabilistic model that we use to find the best position to trim a read to increase its quality without removing more nucleotides than necessary. We also present an extension of this model for homopolymer trimming.
A read is defined as a vector (n 1,…,n m ) of m nucleotides associated with a vector of phred scores (q 1,…,q m ). We want to find the best cut-point k 1∈[1,m] in a read of length m between an informative segment for nucleotide n i ,i∈[1,k 1] and a segment of unreliable quality for nucleotide n i ,i∈[k 1+1,m] (Figure 1). Then, having found k 1, we want to find the best cut-point k 2∈[1,k 1] between a segment of unreliable quality for nucleotide n i ,i∈[1,k 2−1] and an informative segment for nucleotide n i ,i∈[k 2,k 1]. Given the shape of the calling error probability distribution, there is less signal to find k 1 (the probability slowly increases at the extremity of the read) than k 2 (abruptly decreases). Therefore, we want to have the highest number of nucleotides to support the choice of k 1 when k 2 can be found with a subsequence of the read (Figure 1).
Quality trimming. Position of the cut-points k 1 and k 2 in a read. After trimming, the retained part corresponds to the section with a green background, which indicates an informative segment of nucleotides between k 1 and k 2.
With q the quality value of a nucleotide, the probability for this nucleotide to be correct is defined by:
$$ p_{a}\left(q\right)=1-10^{\frac{-q}{10}} $$
((A))
which gives, for example, a probability p a (q)=0.99 for a phred q=20 [2]. However, in QC, the word "informative" is typically defined as a phred score above a certain threshold and not the probability of calling the correct nucleotide. From a probabilistic point of view, we need to discriminate informative nucleotides (with p a (q)≥p a (t) and t a given threshold) from other nucleotides, rather than discriminate fairly accurate nucleotides (with p a (q)≥0.5) from the others. Therefore, we propose to define the probability of having an informative nucleotide as \(p_{b}\left (q,t\right) = 1-2^{\frac {-q}{t}}\) with t the minimal phred score acceptable to be informative. This definition shifts the probability function such that for q=t, we have p b (q,t)=0.5. Therefore, at the threshold t, nucleotides with p b (q,t)≥0.5 are informative and the others are not. With t=3.0103, we go back to the classical phred function (Figure 2) in which p b (q,t)=p a (q).
Probability-phred functions. p(q,t) according to the choice of t. The white, dark grey, light grey and black dots represent respectively the position of p 1,p 2,p 3 and p 4 for the corresponding probability-phred functions. Before p 1 we have the \(1-2^{\frac {-q}{t}}\) part of the function (B) and after p 1 the B(q ⋆,p 1,p 2,p 3,p 4) part of the function (B).
With the function p b (q,t), low phred scores are associated with a low probability to be correct (p b (0,t)=0), but for t≤20 a high phred score does not correspond to a high probability to be correct (for example, p b (40,20)=0.75). Therefore, from a probabilistic point of view, unreliable nucleotides will have more weight than informative ones. To associate a high phred score with a high probability of having an informative nucleotide, we constrain this probability to reach 1 for a phred score of 45 by using the following spline function (Figure 2):
$$ p\left(q,t\right) = \left\{ \begin{array}{ll} 1-2^{\frac{-q}{t}} & \text{if} \,\,q \leq \max(20,t),\\ B\left(q^{\star}, p_{1}, p_{2}, 1, 1\right) & \,\,\text{otherwise}\\ \end{array} \right. $$
((B))
with B(q ⋆,p 1,p 2,p 3,p 4) the cubic Bezier curve starting at p 1 toward p 2 and arriving at p 4 coming from the direction of p 3 for q ⋆∈[0,1]. We have p 1=1−2− max(20,t)/t, p 2=g(1/3×(45− max(20,t))) with g(q) the tangent to the function \(1-2^{\frac {-q}{t}}\) in max(20,t). We scale the Bezier curve to the interval [t,45] with q ⋆=(q−t)/(45−t). The constraint max(20,t) ensures that \(\frac {d}{dq^{\star }} B\left (q^{\star }, p_{1}, p_{2}, p_{3}, p_{4}\right) < 0\) for q ⋆∈[0,1] (see Figure 2).
With the maximum likelihood framework, finding the position of the cut-point between a segment of informative nucleotides (q>t) and a segment of unreliable nucleotides (q<t) consists in estimating k 1 by:
$$ \widehat{k_{1}} = \arg\max\limits_{k} \prod\limits_{i=1}^{k} \frac{1}{k} f_{0}\left(n_{i}, t\right) \prod\limits_{i=k+1}^{m} \frac{1}{m-k-1} f_{1}\left(n_{i}, t\right) $$
((C))
with f 0(n i ,t) the probability that the nucleotide n i comes from the segment of informative nucleotides and f 1(n i ,t) the probability that the nucleotide n i comes from the segment of unreliable nucleotides for a given t. Such that:
$$ f_{0}\left(n_{i}, t\right) = p\left(q_{i}, t\right) \prod\limits_{N \in \Omega} \Pr(N)^{\mathbf{1}\left(n_{i} = N\right)} $$
((D))
$$ f_{1}\left(n_{i}, t\right) = \left(1-p\left(q_{i}, t\right)\right) \frac{1}{4} $$
((E))
with 1(n i =N) an indicator variable such that 1(n i =N)=1 if the nucleotide n i is equal to N and 0 otherwise, \(\Pr (N)=\sum _{i=1}^{k} \mathbf {1}\left (n_{i}=N\right)/k \label {eq:pN}\) the probability to observe the nucleotide N between 1 and k, and Ω the standard IUB/IUPAC dictionary [10]. Pr(N) N∈Ω and k 1 are estimated with the complete data framework of the EM algorithm [11]. After finding \(\widehat {k_{1}}\), we apply the same procedure on the interval \([1,\widehat {k_{1}}]\) to estimate the best cut-point k 2 between a segment of unreliable nucleotides ahead of a segment of informative nucleotides. This double binary segmentation ensures to provide the best two cut-points for a given read [12].
For p(q,t)=p a (q), we can interpret the segment of informative nucleotides as a segment for which on average we are confident that a given nucleotide is the correct one, whereas the segment of unreliable nucleotides is composed of uninformative nucleotides in which on average any of the four nucleotides can be present at a given position. The cut-point k 1 maximizes the probability that the nucleotides n i ,i∈[1,k 1] are informative and that nucleotides n i ,i∈[k 1,m] are not.
With our model, trimming nucleotides of unreliable quality is somewhat similar to removing homopolymers from the extremities of the reads. The task of removing homopolymers, such as polyA tails in RNA-Seq experiments, is not trivial, because the quality of a given nucleotide decreases both at the end of the read and with the size of the homopolymer. Therefore, because the number of incorrectly called nucleotides increases, we are less likely to observe As at the end of the polyA tail. UrQt implements a procedure for the unsupervised trimming of polyN with a straightforward modification of equation (E) such that:
$$ f_{1}\left(n_{i}, t\right) = p_{a}\left(q_{i}, t\right)^{\mathbf{1}\left(n_{i} = A\right)} \left(\left(1-p_{a}\left(q_{i}, t\right)\right)\frac{1}{4}\right)^{\mathbf{1}\left(n_{i} \ne A\right)} $$
((F))
in which we can replace A by any letter of the standard IUB/IUPAC dictionary. With this definition of f 1, we consider the calling error probability of the nucleotide at position i if n i =A or if n i ≠A, the probability that the nucleotide could have been an A.
To assess the performance of our approach, we compared the performance of UrQt to other publicly available programs on different NGS data sets (see Table 1). The quality of the data generated during an NGS experiment can vary greatly depending on the type of data (DNA or RNA) and the sequencing pipeline. To analyze these two types of data on the same genome, we chose paired-end RNA and paired-end DNA sequencing experiments from the species Drosophila melanogaster. For this species, the DNA sample quality quickly drops at the end of the reads (see Additional file 1), and the RNA sample presents a large variability of quality among its reads. We also included in our analysis four other data sets from four different species which are the same ones as used in the comparative study of Del Fabbro et al. [4]. One single-end RNA sample from the species Homo sapiens of poor overall quality and one single-end RNA sample of good overall quality from the species Arabidopsis thaliana. For the DNA sample, we used one paired-end sample from the species Prunus persica of excellent overall quality and one paired-end DNA sample from the species Saccharomyces cerevisiae of average quality. Finally, we also included one paired-end RNA sample from the species Homo sapiens of overall good quality. The seven data sets (Table 1) were downloaded from the NCBI website. Rather than using the complete data set, we uniformly sampled 500,000 reads from each experiment using the software fastq_sampler.py (available at https://github.com/l-modolo/fastq_sampler), to speed-up the computation and work with comparable reads number for each sample.
Table 1 NGS data sets used for testing
For testing purposes, we choose the better trimming programs, according to their performances in the study of Del Fabbro et al. [4] and representing both running sum algorithms (Cutadapt [13], which implement the algorithm proposed for BWA [14]) and sliding-windows algorithms (Trimmomatic [15] and Sickle [16]). The different programs were compared on two points: the overall quality of the resulting trimmed data set and the number of reads mapped on the corresponding reference genome with Bowtie2 [17] for different quality thresholds. For the analyses presented in this work, we used the latest available versions of Cutadapt (version 1.4.1), Trimmomatic (version 0.32) and Sickle [16] (version 1.290). The value of the quality threshold t for the three programs, corresponded respectively to the parameter –t for UrQt, –q for Cutadapt and Sickle and SLIDINGWINDOW:4:t for Trimmomatic. All the other parameters were set to default values, except for the minimum read length that was set to 1 bp. All quality figures were generated with FastQC [6] and the quality statistics were computed using R [18] and the FASTX-Toolkit [5].
Consistency of the trimming procedures
It is expected that the quality in the trimmed data set will increase with the quality threshold up to a certain saturation point. We computed the median quality (phred) in the trimmed data for different quality thresholds (Figure 3, and Additional file 2 for the seven data sets). We observed from this comparison that except for UrQt, all other programs failed to produce a stable relationship between the chosen quality threshold and the resulting median quality score across different samples. For example, we observed a logarithmic-like relationship between the quality threshold and the median for data sets of overall poor quality, such as the H. sapiens data of overall poor quality, and an exponential-like relationship for data sets of overall good quality, such as the A. thaliana and the S. cerevisiae data (Figure 3). These different types of relationships indicate that an increase of the threshold does not have the same effect from one data set to another, and that this effect also depends on the value of the threshold. However, with UrQt, we observe a stable relationship between the threshold and the median quality that is representative of more consistent cutting-points. With a stable relationship between the threshold and the quality of the trimmed data set, it is thus possible to set the quality threshold beforehand according to a targeted quality and independently of the data.
Quality of the trimmed data for each software. Performances of different trimming algorithms in terms of the median quality (phred) of the resulting trimmed data set for different quality thresholds. The choices of t correspond to the parameter –t for UrQt, –q for Cutadapt and Sickle and SLIDINGWINDOW:4:t for Trimmomatic. The black line corresponds to raw (untrimmed) data, and R1 and R2 correspond to the two ends of paired-end data.
Optimality of the trimming procedures
Although increasing the quality of a data set by trimming nucleotides of poor quality is easy, the remaining difficulty lies in minimizing the information (nucleotides) lost in the process. A simple metric to evaluate this trade-off is the number of trimmed reads that can be mapped on the corresponding reference genome. With better quality information after trimming, we expect an increase of the number of mapped reads, whereas by removing too many nucleotides, we expect less information and thus a decrease in the number of mapped reads. For the mapping procedure, we used Bowtie2 [17] (version 2.2.2) (with default parameters and the –very-sensitive option) and the genome indexes available from the igenome project (see Table 1 and Additional file 3 for the version). For the paired-end data, each end was mapped independently. We examined the number of mapped reads on the corresponding reference genomes (Table 1) for different quality thresholds (Figure 4 and Additional file 4 for the seven data sets). The same mapping procedure was also performed using BWA [14] (version 0.7.10) (with default parameters) (Additional file 3). We observed that UrQt was the only software that consistently increased the number of mapped reads for all data sets. The other programs provided the desired effect only for data sets of overall poor quality, such as for the single-end H. sapiens data (SRR002073), and produced worse results than those obtained by mapping the raw data for data sets of better quality (Figure 4). For the single-end H. sapiens data, we observed that UrQt better respected the chosen threshold, thus producing worst results than the other programs for the low quality threshold. For example, with this dataset and a threshold of 5, we expect a large number of reads with an average quality slightly above 5 which are difficult to map. This respect of the threshold can also be seen for the paired-end H. sapiens data (SRR521463) or the D. melanogaster RNA data (SRR919326) and a low threshold of 5 where UrQt is the only program that produces results comparable to the raw data (see Additional file 4).
Remaining information in the trimmed data for each software. Mapping performances for different quality threshold. The choice of t corresponds to the parameter –t for UrQt, –q for Cutadapt and Sickle and SLIDINGWINDOW:4:t for Trimmomatic. The black line corresponds to raw (untrimmed) data, and R1 and R2 correspond to the two ends of paired-end data.
For data sets of excellent quality, such as P. persica (see Additional file 4), all the trimming programs except for UrQt deteriorated the mapping performances compared with the ones obtained by mapping the raw data. This result provides additional evidence of better trimming cut-points identified by UrQt compared with the ones found by other procedures that remove too many nucleotides for data sets of excellent quality.
When considering the output of a mapping software, we can discriminate between reads, which map to a unique position and reads, which map to multiple positions. The number of reads mapping at multiple positions depends on three factors: the number of reads associated with repetition, the sensitivity of the mapping procedure (we can expect more reads mapping at multiple positions when allowing for more missmatches and gaps), and the information contained in the reads. Thus with trimming procedures, the information loss of over-trimming could lead to an increase of the number of reads mapping at multiple positions. This over-trimming effect can be seen with Cutadapt, Trimmomatic and Sickle for high threshold values (superior to 20) (see Additional file 3 for the results with Bowtie2 and BWA). However, with UrQt, the number of reads mapping to unique position increase with the choice of the threshold which is also consistent with better cut-point. These results hold for every dataset with the exception of the H. sapiens RNA sample of poor overall quality (SRR002073) for which removing a large number of uninformative nucleotides also correspond to removing a large number of reads.
Overall, the results obtained with UrQt correspond to the expected results for a trimming procedure and a given quality threshold in opposition to the other programs in our test panel (see Additional file 3 and 5). The output of UrQt depends on the choice of t that defines an informative sequence for which we expect nucleotides to have a phred score above this threshold. Contrary to current methods in which the choice of the threshold is set according to the quality of the data, the UrQt –t parameter only depends on the goal of the analysis (SNP calling, de novo-assembly, mapping, etc.).
UrQt is a new tool for the key QC step of any NGS data analysis to trim low-quality nucleotides and polyA tails from reads in fastq or fastq.gz format with an efficient C++ implementation. By finding the best segmentation to delimit a segment of informative nucleotides, UrQt greatly increases the number of reads and of nucleotides that can be retained for a given quality objective. Using this software should provide a significant gain for many NGS applications. Moreover, the consistency of our trimming procedure with the quality of the trimmed data set for a given quality threshold, will allow for better automation of the trimming step in a pipeline. We also provide a galaxy wrapper for UrQt to facilitate its integration in existing pipelines developed on this platform [19-21]. Finally, with our simple probabilistic model for the trimming of NGS data, we hope that users will have a better grasp on the quality threshold –t to obtain the largest trimmed data set with the required quality.
Availability and requirements
Project name: UrQtProject home page: https://lbbe.univ-lyon1.fr/-UrQt-.html Operating system(s): Platform independentProgramming language: C++Other requirements: zlib and c++0x compilerLicense: GNU GPLv3Any restrictions to use by non-academics: GNU GPLv3
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Bolger AM, Lohse M, Usadel B. Trimmomatic: a flexible trimmer for Illumina sequence data. Bioinformatics. 2014; 30(15):2114–20.
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We thank V. Lacroix for his advice and discussions, and H. Lopez-Maestre and V. Romero-Soriano for their feedback in testing the software. We also thank the anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions. The English of the manuscript has been edited by the American Journal Experts company.
Université de Lyon; Université Lyon 1; CNRS; UMR 5558, Laboratoire de Biométrie et Biologie Evolutive, 43 bd du 11 novembre 1918, Villeurbanne cedex, 69622, France
Laurent Modolo & Emmanuelle Lerat
Laurent Modolo
Emmanuelle Lerat
Correspondence to Emmanuelle Lerat.
Conceived and designed the experiments: LM and EL. Performed the experiments: LM. Analyzed and interpreted the data: LM; Drafted the manuscript: LM and EL. All authors read and approved the final version of the manuscript.
This work was performed using the computing facilities of the CC LBBE/PRABI.
Quality analysis of the seven NGS samples. Quality analysis of the seven NGS samples (Table 1) with the FastQC software.
Quality of the trimmed data for each programs. Performances of different trimming algorithms in terms of the median quality (phred) of the resulting trimmed data set for different quality thresholds. The choice of t corresponds to the parameter –t for UrQt, –q for Cutadapt and Sickle and SLIDINGWINDOW:4:t for Trimmomatic. The black line corresponds to raw (untrimmed) data, and R1 and R2 correspond to the two ends of paired-end data. This figure complements the Figure 3 with the seven data sets (Table 1).
Mapping performances for the four tested programs. Mapping performances for different quality threshold with the four tested programs and the seven NGS sample (Table 1). Mapping results with Bowtie2 [17] and BWA [14].
Remaining information in the trimmed data for each programs. Mapping performances for different quality threshold. The choice of t correspond to the parameter –t for UrQt, –q for Cutadapt and Sickle, and SLIDINGWINDOW:4:t for Trimmomatic. The black line corresponds to raw (untrimmed) data, and R1 and R2 correspond to the two ends of paired-end data. This figure complements the Figure 4 with the seven data sets (Table 1).
Quality analysis of the seven NGS samples for the four tested programs. Quality analysis of the output of the four programs for the seven NGS samples and different quality thresholds (Table 1) with the FastQC [6] software.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Modolo, L., Lerat, E. UrQt: an efficient software for the Unsupervised Quality trimming of NGS data. BMC Bioinformatics 16, 137 (2015). https://doi.org/10.1186/s12859-015-0546-8
Unsupervised segmentation | CommonCrawl |
Correlation function
A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables representing the same quantity measured at two different points, then this is often referred to as an autocorrelation function, which is made up of autocorrelations. Correlation functions of different random variables are sometimes called cross-correlation functions to emphasize that different variables are being considered and because they are made up of cross-correlations.
Correlation functions are a useful indicator of dependencies as a function of distance in time or space, and they can be used to assess the distance required between sample points for the values to be effectively uncorrelated. In addition, they can form the basis of rules for interpolating values at points for which there are no observations.
Correlation functions used in astronomy, financial analysis, econometrics, and statistical mechanics differ only in the particular stochastic processes they are applied to. In quantum field theory there are correlation functions over quantum distributions.
Definition
For possibly distinct random variables X(s) and Y(t) at different points s and t of some space, the correlation function is
$C(s,t)=\operatorname {corr} (X(s),Y(t)),$
where $\operatorname {corr} $ is described in the article on correlation. In this definition, it has been assumed that the stochastic variables are scalar-valued. If they are not, then more complicated correlation functions can be defined. For example, if X(s) is a random vector with n elements and Y(t) is a vector with q elements, then an n×q matrix of correlation functions is defined with $i,j$ element
$C_{ij}(s,t)=\operatorname {corr} (X_{i}(s),Y_{j}(t)).$
When n=q, sometimes the trace of this matrix is focused on. If the probability distributions have any target space symmetries, i.e. symmetries in the value space of the stochastic variable (also called internal symmetries), then the correlation matrix will have induced symmetries. Similarly, if there are symmetries of the space (or time) domain in which the random variables exist (also called spacetime symmetries), then the correlation function will have corresponding space or time symmetries. Examples of important spacetime symmetries are —
• translational symmetry yields C(s,s') = C(s − s') where s and s' are to be interpreted as vectors giving coordinates of the points
• rotational symmetry in addition to the above gives C(s, s') = C(|s − s'|) where |x| denotes the norm of the vector x (for actual rotations this is the Euclidean or 2-norm).
Higher order correlation functions are often defined. A typical correlation function of order n is (the angle brackets represent the expectation value)
$C_{i_{1}i_{2}\cdots i_{n}}(s_{1},s_{2},\cdots ,s_{n})=\langle X_{i_{1}}(s_{1})X_{i_{2}}(s_{2})\cdots X_{i_{n}}(s_{n})\rangle .$
If the random vector has only one component variable, then the indices $i,j$ are redundant. If there are symmetries, then the correlation function can be broken up into irreducible representations of the symmetries — both internal and spacetime.
Properties of probability distributions
With these definitions, the study of correlation functions is similar to the study of probability distributions. Many stochastic processes can be completely characterized by their correlation functions; the most notable example is the class of Gaussian processes.
Probability distributions defined on a finite number of points can always be normalized, but when these are defined over continuous spaces, then extra care is called for. The study of such distributions started with the study of random walks and led to the notion of the Itō calculus.
The Feynman path integral in Euclidean space generalizes this to other problems of interest to statistical mechanics. Any probability distribution which obeys a condition on correlation functions called reflection positivity leads to a local quantum field theory after Wick rotation to Minkowski spacetime (see Osterwalder-Schrader axioms). The operation of renormalization is a specified set of mappings from the space of probability distributions to itself. A quantum field theory is called renormalizable if this mapping has a fixed point which gives a quantum field theory.
See also
• Autocorrelation
• Correlation does not imply causation
• Correlogram
• Covariance function
• Pearson product-moment correlation coefficient
• Correlation function (astronomy)
• Correlation function (statistical mechanics)
• Correlation function (quantum field theory)
• Mutual information
• Rate distortion theory
• Radial distribution function
Statistical mechanics
Theory
• Principle of maximum entropy
• ergodic theory
Statistical thermodynamics
• Ensembles
• partition functions
• equations of state
• thermodynamic potential:
• U
• H
• F
• G
• Maxwell relations
Models
• Ferromagnetism models
• Ising
• Potts
• Heisenberg
• percolation
• Particles with force field
• depletion force
• Lennard-Jones potential
Mathematical approaches
• Boltzmann equation
• H-theorem
• Vlasov equation
• BBGKY hierarchy
• stochastic process
• mean-field theory and conformal field theory
Critical phenomena
• Phase transition
• Critical exponents
• correlation length
• size scaling
Entropy
• Boltzmann
• Shannon
• Tsallis
• Rényi
• von Neumann
Applications
• Statistical field theory
• elementary particle
• superfluidity
• Condensed matter physics
• Complex system
• chaos
• information theory
• Boltzmann machine
| Wikipedia |
\begin{document}
\parindent 9mm \title{Output Feedback Exponential Stabilization for {\color{blue} a 1-d Wave PDE with Dynamic Boundary} \thanks{This work was supported by the Natural Science Foundation of Shaanxi Province 2018JM1051, 2014JQ1017. } \thanks{2020 Mathematics Subject Classification. 37L15; 93D15; 93B51; 93B52.}} \author{ Zhan-Dong Mei \thanks{School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China. Email: [email protected]. } }
\date{} \maketitle \thispagestyle{empty} \begin{abstract}
We study the output feedback exponential stabilization for a 1-d wave PDE with dynamic boundary. With only one measurement, we construct an infinite-dimensional state observer to trace the state and design an estimated state based controller to exponentially stabilize the original system. This is an essentially important improvement for the existence literature [\"{O}. Morg\"{u}l, B.P. Rao and F. Conrad, IEEE Transactions on Automatic Control, 39(10) (1994), 2140-2145] where two measurements including the high order angular velocity feedback were adopted. When a control matched nonlinear internal uncertainty and external disturbance are taken into consideration, we construct an infinite-dimensional extended state observer (ESO) to estimate the total disturbance and state simultaneously. By compensating the total disturbance, an estimated state based controller is designed to exponentially stabilize the original system while making the closed-loop system bounded. Riesz basis approach is crucial to the verifications of the exponential stabilities of two coupled systems of the closed-loop systems. Some numerical simulations are presented to illustrate the effectiveness.
\noindent {\bf Key words:} Exponential stabilization, disturbance, extended state observer, Riesz basis, tip mass.
\end{abstract}
\section{Introduction}\label{section1}
In this paper, we are concerned with the output feedback exponential stabilization of a 1-d wave PDE with dynamic boundary described as follows \begin{equation} \label{beem} \left\{\begin{array}{l} u_{tt}(x,t)=u_{xx}(x,t),\;\; x\in (0,1), \; t>0, \\ u(0,t)=0,u_{x}(1,t)+mu_{tt}(1,t)=U(t)+F(t), t\ge 0,\\ y(t)=\{u_{x}(0,t),u(1,t)\}, \;\; t\ge 0, \end{array}\right. \end{equation} where $x$ and $t$ denote the independent spatial and time variables, respectively, $u(x, t)$ denotes the cable displacement at $x$ for time $t$, $m>0$ is the tip mass at the actuator end, $U(t)$ is the boundary control (or input), $F(t)=f(u(\cdot,t),u_t(\cdot,t))+d(t)$, $f:H^1(0,1)\times L^2(0,1)\rightarrow \mathds{R}$ is the internal uncertainty, $d(t)$ is the external disturbance, $y(t)$ is the observation (output). System (\ref{beem}) can be used to describe flexible cable with tip mass \cite{Guo2000,He2012,Morgul1994}.
When there is no disturbance ($F(t)\equiv 0$), the velocity feedback control law $U(t)=-\alpha u_t(1,t)$ $(\alpha>0)$ is sufficient for the strong stabilization but not sufficient for the exponential stabilization of the system (\ref{beem}), see \cite{Lee1987}. By virtue of energy multiplier approach, Morg\"{u}l et al. \cite{Morgul1994} proved that the output feedback control law $U(t)=-\alpha u_t(1,t)-a u_{xt}(1,t)(\alpha,a>0)$ exponentially stabilize the right system. Here the angular velocity $u_{xt}(1,t)$ was adopted, which is quite different from the case $m=0$ where velocity feedback is sufficient for the exponential stability \cite{Rideau1985}. Moreover, in the special case $\alpha=m/a\neq 1$, Morg\"{u}l et al. \cite{Morgul1994} verified the Riesz basis generation of the closed-loop system, and thereby the spectrum-determined growth condition is satisfied. In \cite{Guo2000}, Guo and Xu showed by virtue of essential spectral analysis that the spectrum-determined growth {\color{blue}condition always holds} for the closed-loop system under output feedback control law $U(t)=-\alpha u_t(1,t)-a u_{xt}(1,t)(\alpha,a>0)$. However, the Riesz basis generation for the general case is still unsolved. This will be presented in Lemma \ref{Astable}.
Observe that the aforemention literatures used static collocated feedback where the actuators and sensors are located in the same end $x=1$. The reason why energy multiplier method is successfully used for the stability analysis is that such output feedback makes the system dissipative. However, it has known for a long time, the performance of closed-loop system under collocated output feedback may be not so good \cite{Cannon1984}. For the non-collocated setting, sensors can be placed according to the performance requirement, thereby such design approach for specific systems has been widely used \cite{Cannon1984,Liu2003,Tian2016}. Since the non-collocated closed-loop system is usually non-dissipative, the well-posedness as well as stability analysis is most difficult. Dynamic boundary feedback approach may be a better choice to overcome the difficulty stemmed from non-collocated feedback. In \cite{Guo2007}, with one measurement $u_{x}(0,t),$ an infinite-dimensional observer-based feedback technique has been employed to construct a stabilizing boundary feedback controller for a wave equation (\ref{beem}) with $m=0$ and $F(t)=0$. Such method has been applied to solve non-collocated boundary control designs of Euler-Bernoulli beam equation \cite{Guo2008}. {\it For (\ref{beem}) with tip mass ($m\neq0$), non-collocated control is still a long standing unsolved problem.} The reason is that tip mass makes the system much more complex; such system is indeed a hybrid system consists of a PDE and an ODE. One of our task in this paper is to construct a non-collocated output feedback stabilizing controller for system (\ref{beem}) without disturbance. Compared to the existence references \cite{Guo2000,Morgul1994}, {\it our main contributions for the case $F(t)\equiv 0$ are: 1) non-collocated feedback is used, 2) only one measurement is employed, 3) the Riesz basis generations for closed-loop systems in \cite{Guo2000,Morgul1994} have been generalized to the case $m\neq a$, 4) Riesz basis approach is used to prove the exponential stability for a coupled system which is equivalent to the closed-loop system obtained from our feedback control law, where the verification of exponential stabilization as well as Riesz basis generation for coupled equation is most changeable \cite{Guo2007,Guo2008,Guo2019}.}
Uncertainties and disturbances widely exist in various practical engineering control systems. When the disturbance is taking into consideration ($F(t)\neq 0$), the vibration cable model (\ref{beem}) is quite different from the previous work, because it is governed by a nonhomogeneous hyperbolic PDE. Even a small disturbance can damage the stabilizing output feedback design for system without disturbance. Thus, anti-disturbance problem is most changeable: in order to stabilize the system with disturbance, the controller should be redesigned. Many engineers and mathematicians focus on developing various approaches to deal with uncertainty and disturbance in control system problem. In \cite{Morgul1994a} and \cite{Morgul2001}, in order to reject the disturbance under certain conditions, Morg\"{u}l proposed a kind of dynamic boundary controller for elastic beam and string, respectively. Guo et al. \cite{GuoW2013a} and Ge et al. \cite{Ge2011b} used adaptive control method to stabilize the wave equation and beam equation with disturbance, respectively. Sliding-mode control (SMC) approach was adopted to reject the disturbance for wave equations \cite{Guo2013a}, beam equation \cite{Guo2013b,Karagiannis2018a}, and the cascade of ODE-wave systems \cite{Liu2017}. Backstepping approach developed by Krstic and Smyshlyaev \cite{Krstic2006,Krstic2008,Krstic2008b,Krstic2009a,KrsticSmysh2008,Smyshlyaev2005} is also helpful for reject disturbance of PDEs \cite{Auriol2020,Deutscher2017,Deutscher2019,Deutscher2020,Wang2020} {\color{blue}and output regulation for coupled linear wave-ODE systems (similar to our setup) \cite{Deutscher2020b}}. It should be noted that the active disturbance rejection control (ADRC) proposed by Han \cite{Han2009} is another powerful approach to reject disturbance. The core idea of ADRC is estamaite/cancellation in real time and the key step of control strategy is the construction of an extended state observer (ESO) for the estimating of the state and uncertainty. In the earlier works for PDEs by ADRC like \cite{Guo2013b,Jin2015,Su2020}, they dealt with the disturbance by ODEs reduced from the associated PDEs through some special test functions. Therefore, the ESO is of finite-dimension; slow variation, high gains and boundedness of the derivation of the disturbance were used in the ESO. In \cite{Feng2017a}, Feng and Guo developed a new infinite-dimensional ESO to relax such restricts of the conventional ESO for a class of anti-stable wave equations with external disturbance. Later on, the same method was used to study the stabilization of other wave equations \cite{Mei2020b,Zhou2017a,Zhou2018b} and Euler-Bernoulli beams \cite{Zhou2018a,Zhou2020}.
In section \ref{disturbancestab}, we shall clarify that the estimated state based output feedback control law (\ref{control}) for system (\ref{beem}) with $F(t)\equiv 0$ is not robust to the total disturbance. For system (\ref{beem}) with $f(w,w_t)=0$ and $d(t)$ being bounded, He and Ge \cite{He2012} adopted adaptive control approach to compensate the uncertainty. In the case $f(w,w_t)=0$ and $m=a\alpha$, by constructing a finite-dimensional ESO, Xie and Xu \cite{Xie2017} employed active disturbance rejection control approach to cope with the disturbance. However, they excluded the internal uncertainty as well as the general case $m\neq a\alpha$. Moreover, as in \cite{Morgul1994}, the author in \cite{Xie2017} used two measurements including the high order angular velocity feedabck $w_{xt}(1,t)$. In this paper, with two low order measurements $w_{x}(0,t)$ and $w(1,t)$, we shall design an infinite-dimensional ESO to estimate the original state and total disturbance online. By compensating the total disturbance, an estimated state based output feedback control law is constructed in order to exponentially stabilize the original system. Moreover, all the other states of the closed-loop system are verified to be bounded. {\it The main contribution for the system (\ref{beem}) with disturbance ($F(t)\neq 0$) lies in that, 1) the internal uncertainty is considered, 2) the high order angular velocity feedback $w_{xt}(1,t)$ is avoided, 3) we consider the case $m\neq a,\alpha>0$ while \cite{Xie2017} just solved the special case $\alpha,a>0$ and $m=a\alpha$, 4) our control strategy can be also applied to simplify the existence references \cite{Mei2020b,Zhou2018a,Zhou2018b}.}
Consider system (\ref{beem}) in the energy state Hilbert space $\mathbf{H}_1=H_E^1(0,1)\times L^2(0,1)\times\mathds{C},\ H_E^1(0,1)=\{f|f\in H^1(0,1),f(0)=0\}$, with the following norm
$\|(f,g,$
$\eta)\|^2_{\mathbf{H}_1} = \int_0^1[|f'(x)|^2+|g(x)|^2]dx+\frac{1}{m} |\eta|^2,(f,g,\eta)\in \mathbf{H}_1.$ Define the operator $\mathbf{A}_1:D(\mathbf{A}_1)(\subset \mathbf{H}_1)\rightarrow \mathbf{H}_1$ by
$\mathbf{A}_1(f,g,\eta)=(g,f'',-f'(1)),$
$ \forall\;(f,g)\in D(\mathbf{A}_1)=\{(f,g,\eta)\in \mathbf{H}_1| (g,f'',-f'(1))\in \mathbf{H}_1,
\eta=mg(1)\}.$ Obviously, $\mathbf{A}_1$ is skew-adjoint and generates a unity group. System (\ref{beem}) is written abstractly by \begin{align*}
&\frac{d}{dt}\left(u(\cdot,t),u_t(\cdot,t),mu_t(1,t)\right)\\ &=\mathbf{A}_1\left(u(\cdot,t),u_t(\cdot,t),mu_t(1,t)\right)+\mathbf{B}_1[U(t)+F(t)], \end{align*} where $\mathbf{B}_1=(0,0,1)^T$ is a bounded linear operator on $\mathbf{H}_1$. The following proposition can be derived directly from the proof of \cite[Proposition 1.1]{Zhou2018a}.
\begin{proposition} Assume that $f:H^1(0,1)\times L^2(0,1)\rightarrow \mathds{R}$ is continuous and satisfies global Lipschitz condition in $H^1(0,1)\times L^2(0,1)$. Then, for any {\color{blue}$(u(\cdot,0),u_t(\cdot,0),mu_t(1,$ $0))\in \mathbf{H}_1$}, $u,d \in L^2_{loc}(0,\infty)$, there exists a unique global solution (mild solution) to (\ref{beem}) such that $(u(\cdot,t),u_t(\cdot,t),$ $mu_t(1,t))\in C(0,\infty;\mathbf{H}_1)$. \end{proposition}
We proceed as follows. In section \ref{section2}, for system (\ref{beem}) without disturbance, we design a Luenberger state observer and an estimated state based stabilizing control law. We design in section \ref{disturbancestab} an infinite-dimensional ESO for system (\ref{beem}) to estimated total disturbance and state in real time. An estimated total disturbance and estimated state based stabilizing control law is then designed. Moreover, it is proved that an important couple subsystem including the original equation of the closed-loop is exponential stability and the closed-loop system is bounded. In section \ref{shiyan}, some numerical simulations are presented.
\section{Stabilization in the absence of disturbance}\label{section2}
We rewrite (\ref{beem}) with $F(t)\equiv0$ as follows \begin{equation} \label{beem1} \left\{\begin{array}{l} u_{tt}(x,t)=u_{xx}(x,t),\;\; x\in (0,1), \; t>0, \\ u(0,t)=0,u_{x}(1,t)+mu_{tt}(1,t)=U(t), \; t\ge 0,\\ y(t)=u_x(0,t), \;\; t\ge 0. \end{array}\right. \end{equation} Our aim in this section is to design a stabilizing control law for system (\ref{beem1}) by virtue of only one noncollocated measurement $u_{x}(0,t)$. To this end, we first construct an infinite-dimensional Luenberger state observer as follows \begin{equation} \label{transfer11} \left\{\begin{array}{l} \widehat{u}_{tt}(x,t)=\widehat{u}_{xx}(x,t),\;\; x\in (0,1), \; t>0, \\ \widehat{u}_x(0,t)=\gamma\widehat{u}_{t}(0,t)+\beta\widehat{u}(0,t)+u_x(0,t), \\ \widehat{u}_{x}(1,t)+m\widehat{u}_{tt}(1,t)=U(t), \;\; t\ge 0, \end{array}\right. \end{equation} where $\beta,\gamma>0$. Set $\widetilde{u}(x,t)=\widehat{u}(x,t)-u(x,t)$ to get \begin{equation} \label{wwan} \left\{\begin{array}{l} \widetilde{u}_{tt}(x,t)=\widetilde{u}_{xx}(x,t),\;\; x\in (0,1), \; t>0, \\ \widetilde{u}_x(0,t)=\gamma\widetilde{u}_{t}(0,t)+\beta\widetilde{u}(0,t), \; \; t\ge 0,\\ \widetilde{u}_{x}(1,t)+m\widetilde{u}_{tt}(1,t)=0, \;\; t\ge 0. \end{array}\right. \end{equation}
Denote $\mathbf{H}_2=H^1(0,1)\times L^2(0,1)\times \mathds{C}$, with norm $\|(f,g,\eta)\|^2_{\mathbf{H}_2}=\int_0^1|f'(x)|^2+|g(x)|^2
+\beta|f(0)|^2+|\eta|^2/m,$ $(f,g,\eta)\in \mathbf{H}_2.$ Then we can write (\ref{wwan}) abstractly by \begin{align}\label{wwansemigroup}
\nonumber&\frac{d}{dt}(\widetilde{u}(\cdot,t),\widetilde{u}_t(\cdot,t),m\widetilde{u}_t(1,t))\\
&=\mathbf{A}_2(\widetilde{u}(\cdot,t),
\widetilde{u}_t(\cdot,t),m\widetilde{u}_t(1,t)). \end{align} Here $\mathbf{A}_2:D(\mathbf{A}_2)(\subset \mathbf{H}_2)\rightarrow \mathbf{H}_2$ is defined by $\mathbf{A}_2(f,$ $g,\eta)=(g,f'',-f'(1)),\; \forall\;(f,g,\eta)\in D(\mathbf{A}_2)=\{(f,g,$
$\eta)\in \mathbf{H}_2|(g,f'',-f'(1))\in \mathbf{H}_2,
f'(0)=\gamma g(0)+\beta f(0),$
$ \eta=mg(1))\}.$ By \cite{Rao1993}, $\mathbf{A}_2$ generates an exponentially stable $C_0$-semigroup,
{\color{blue}that is, $\|e^{\mathbf{A}_2t}\|\leq M_{\mathbf{A}_2}e^{\omega_{\mathbf{A}_2}t},\; t\geq 0$, where $M_{\mathbf{A}_2}$ and $\omega_{\mathbf{A}_2}$ are two positive constants.} The following Lemma \ref{A2stable} presents that the Riesz basis property of $\mathbf{A}_2$ also holds.
\begin{lemma}\label{A2stable} Assume that $\gamma\neq 1$. Then, there exist a sequence of generalized eigenfunctions of operator $(\mathbf{A}_2,D(\mathbf{A}_2))$ that forms Riesz basis for $\mathbf{H}_2$; the eigenvalues $\{\lambda_n\}_{n=-\infty}^{+\infty}$ have asymptotic expression
$\lambda_n=\frac{1}{2}{\rm ln}\frac{|\gamma-1|}{\gamma+1}+n_\beta\pi i+O(|n|^{-1}),$ where $n_\beta$ $=n$ if $\gamma>1$ and $n_\beta=n+1/2$ if $0<\gamma<1$; the eigenfunction is given by $(f_n,\lambda_nf_n,$ $m\lambda_nf(1))$ with $f_n(x)=[(1+\gamma)\lambda_n+\beta]e^{\lambda_n x}+[(1-\gamma)\lambda_n-\beta]e^{-\lambda_n x}$ and $F_n(x)$ $=(f_{n}^{'}(x),\lambda_nf_n(x),\beta f_n(0),-\lambda_n^{-1} f'_n(1))/\lambda_n^2=
\big((1+\gamma)$
$\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}e^{in_\beta x}-(1-\gamma)\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}e^{-in_\beta x},
(1+\gamma)\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}$
$e^{in_\beta x}+(1-\gamma)\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}e^{-in_\beta x},
0,0\big)+O(n^{-1}).$ \end{lemma} \begin{proof}\ \ It follows from \cite{Rao1993} that operator $(\mathbf{A}_2,D(\mathbf{A}_2))$ generates an exponentially stable $C_0$-semigroup, and it is a densely defined and discrete operator on $\mathbf{H}_2$. Therefore, for any $\lambda\in \sigma(\mathbf{A}_2)=\sigma_P(\mathbf{A}_2)$, ${\rm Re} \lambda<0$. We divide the proof into several steps.
{\bf Step 1:} We claim that for any $\lambda\in\sigma(\mathbf{A}_2)$, there corresponds one eigenfunction $(f,g,\eta)$ given by
\begin{align}\label{fxequ300}
\left\{
\begin{array}{ll}
f(x)=[(1+\gamma)\lambda+\beta]e^{\lambda x}+[(1-\gamma)\lambda-\beta]e^{-\lambda x}, \\
g=\lambda f, \\
\eta=-\frac{f'(1)}{\lambda}
\end{array}
\right.
\end{align} and $\lambda$ satisfies the characteristic equation
\begin{align}\label{tauchar2001}
e^{2\lambda}[(1+\gamma)\lambda+\beta](1+m\lambda)=[(1-\gamma)\lambda-\beta](1-m\lambda).
\end{align} This implies that each eigenvalue of $\mathbf{A}_2$ is geometrically simple.
Indeed, $f''(x)-\lambda^2 f(x)=0$ implies that $f$ is of the form $f(x)=Ee^{\lambda x}+Fe^{-\lambda x}$. Use the boundary condition $f'(0)=(\gamma\lambda+\beta)f(0)$ to derive $E=(1+\gamma)\lambda+\beta, F=(1-\gamma)\lambda-\beta$. We can then derive $\eta=mg(1)=m\lambda f(1)=-\frac{f'(1)}{\lambda}$ and (\ref{tauchar2001}) by virtue of the boundary condition $f'(1)+m\lambda^2 f(1)=0$.
{\bf Step 2:} We show that the eigenvalues $\{\lambda_n\}_{n=-\infty}^{+\infty}$ have asymptotic expression
\begin{equation}\label{ataunasy100}
\lambda_n=\frac{1}{2}{\rm ln}\frac{|\gamma-1|}{\gamma+1}+n_\beta\pi i+O(|n|^{-1}), \end{equation} where \begin{align}\label{nbeta}
n_\beta=\left\{
\begin{array}{ll}
n, \gamma>1 \\
n+\frac{1}{2},0<\gamma<1;
\end{array}
\right. \end{align} the corresponding eigenfunction $(f_n,\lambda_nf_n,$ $m\lambda_nf(1))$ of $\mathbf{A}_2$ can be chosen such that
\begin{align}\label{fxequasy300}
\nonumber&F_n(x)=\\
\nonumber&\left(
\begin{array}{c}
(1+\gamma)\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}e^{in_\beta x}-(1-\gamma)\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}e^{-in_\beta x} \\
(1+\gamma)\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}e^{in_\beta x}+(1-\gamma)\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}e^{-in_\beta x} \\ 0\\ 0\\
\end{array}
\right)\\
&+O(n^{-1}),
\end{align}
where
\begin{equation}\label{fmatrix100}
F_n(x)=\frac{1}{\lambda_n^2}\begin{pmatrix}
f_{n}^{'}(x) \\
\lambda_nf_n(x)\\
\beta f_n(0)\\
-\lambda_n^{-1} f'_n(1)\\
\end{pmatrix}^T.
\end{equation}
Now we give the proof. By (\ref{tauchar2001}), it follows that \begin{align}\label{fxc1c2c3c4}
\nonumber &e^{2\lambda}=\frac{m(\gamma-1)\lambda^2+(m\beta-\gamma+1)\lambda-\beta}{m(\gamma+1)\lambda^2+(m\beta+\gamma+1)\lambda+\beta}\\
&=\frac{\gamma-1}{\gamma+1}+O(|\lambda|^{-1}). \end{align} The asymptotic expression (\ref{ataunasy100}) is derived directly by Routhe's theorem. Then we obtain \begin{equation}\label{fxequ30}
\begin{split}
& \lambda_n\frac{f_n(x)}{\lambda^2_n}=\frac{(1+\gamma)\lambda_n+\beta}{\lambda_n}e^{\lambda_n x}+\frac{(1-\gamma)\lambda_n-\beta}{\lambda_n} e^{-\lambda_n x}\\
&=(1+\gamma)\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}e^{in_\beta x}+(1-\gamma)\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}e^{-in_\beta x}\\
&+O(|n|^{-1}),
\end{split}
\end{equation} \begin{equation}\label{fxequ31}
\begin{split}
& \frac{f'_n(x)}{\lambda^2_n}=\frac{(1+\gamma)\lambda_n+\beta}{\lambda_n}e^{\lambda_n x}-\frac{(1-\gamma)\lambda_n-\beta}{\lambda_n} e^{-\lambda_n x}\\
=&(1+\gamma)\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}e^{in_\beta x}-(1-\gamma)\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}e^{-in_\beta x}\\
& +O(|n|^{-1}),
\end{split}
\end{equation} \begin{equation}\label{fxequ32}
\begin{split}
& \frac{\beta f_n(0)}{\lambda^2_n}=\frac{(1+\gamma)\lambda_n+\beta}{\lambda^2_n}+\frac{(1-\gamma)\lambda_n-\beta}{\lambda^2_n}
=O(|n|^{-1}),
\end{split}
\end{equation} \begin{equation}\label{fxequ33}
\begin{split}
& \frac{-\lambda^{-1}_nf'_n(1)}{\lambda^2_n}=-\frac{(1+\gamma)\lambda_n+\beta}{\lambda^2_n}e^{\lambda_n }+\frac{(1-\gamma)\lambda_n-\beta}{\lambda^2_n} e^{-\lambda_n }\\
&=O(|n|^{-1}),
\end{split}
\end{equation} The combination of (\ref{fxequ30}), (\ref{fxequ31}), (\ref{fxequ32}) and (\ref{fxequ33}) indicates that the asymptotic expression (\ref{fxequasy300}) of $F_n(x)$ holds.
{\bf Step 3:} We show that there exist a sequence of generalized eigenfunctions of $\mathbf{A}_2$ which forms Riesz basis for $\mathbf{H}_2$.
Now we give the proof. Denote $$P(x)=\left(
\begin{array}{cc}
(1+\gamma)\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2} & -(1-\gamma)\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2} \\
(1+\gamma)\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2} & (1-\gamma)\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2} \\
\end{array}
\right)$$ for $\lambda>1$, $$P(x)=\left(
\begin{array}{cc}
(1+\gamma)\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}i & (1-\gamma)\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}i \\
(1+\gamma)\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}i & -(1-\gamma)\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}i \\
\end{array}
\right)$$ for $\lambda<1$. It is easily seen that the operator $\mathbf{T}$ defined by $\mathbf{T}w=P(x)w$ for any $w\in \left(L^2(0,1)\right)^2$, is a bounded and invertible linear operator on $\left(L^2(0,1)\right)^2$. Combine this with the fact that $\{(e^{in\pi x},e^{-in\pi x})\}_{n=-\infty}^{+\infty}$ forms Riesz basis for $\left(L^2(0,1)\right)^2$, to obtain that $\{P(x)(e^{in\pi x},e^{-in\pi x})\}_{n=-\infty}^{+\infty}$ also forms Riesz basis for $\left(L^2(0,1)\right)^2$. Thereby, $\{(0,0,1,0)\}\bigcup\{(0,0,0,1)\}\bigcup$ $\{(P(x)(e^{in\pi x},e^{-in\pi x}),0,$ $0)$ $\}_{n=-\infty}^{+\infty}$ forms Riesz basis for $\left(L^2(0,1)\right)^2\times \mathds{C}^2$.
From (\ref{fxequasy300}), we derive \begin{align}\label{fxequasy3001}
F_n(x)=(P(x)(e^{in\pi x},e^{-in\pi x}),0,0)+O(n^{-1}),
\end{align} which implies that, {\color{blue} there exists} enough big positive integer $N$ such that \begin{align*}
&\sum_{|n|>N}\|F_n-(P(x)(e^{in\pi x},e^{-in\pi x}),0,0)\|_{\left(L(0,1)\right)^2\times\mathds{C}^2}^2\\
&=\sum_{|n|>N}O(|n|^{-2})<\infty. \end{align*} Since $\{(0,0,1,0)\}\bigcup\{(0,0,0,1)\}\bigcup\{(P(x)(e^{in\pi x},e^{-in\pi x}),0,0)\}_{n=-\infty}^{+\infty}$ forms Riesz basis for $\big(L^2(0,$ $1)\big)^2\times \mathds{C}$, it follows by \cite[Lemma 1]{Guo2001} that there exists a $M\geq N$ such that $\{(0,0,1,0)\}\bigcup\{(0,0,0,1)\}$
$\bigcup\{(P(x)(e^{in\pi x},e^{-in\pi x}), 0,0)\}_{n=-M}^M\bigcup\{F_n\}_{|n|>M}$ forms Riesz basis for $\left(L^2(0,1)\right)^2\times \mathds{C}^2$. We define an isometric isomorphism $\mathbb{T}_1:\mathbf{H}_2\rightarrow \left(L^2(0,1)\right)^2\times \mathds{C}^2$ by $\mathbb{T}_1(f,g,\eta)=(f',g,\sqrt{\beta}f(0),\eta),\; \forall\;(f,g,\eta)\in \mathbf{H}_2.$ Then there exist $\{W_n(x),V_n(x),0\}_{n=-M}^M\subset \mathbf{H}_2$ such that $\{(0,0,1)\}\bigcup\{(W_n(x),V_n(x),0)\}_{n=-M}^M$ $\bigcup
\{(\lambda^{-2}_nf_n(x),\lambda^{-1}_nf_n(x),-\lambda_n^{-3}f'_n(1))\}_{|n|>M}$ forms Riesz basis for $\mathbf{H}_2$. Since $\{(\lambda^{-2}_nf_n(x),\lambda^{-1}_nf_n(x),$ $-\lambda_n^{-3}f'_n(1))\}_{n=-\infty}^\infty$ is a sequence of generalized eigenvectors of $\mathbf{A}_2$, and by \cite{Rao1993}, $\mathbf{A}_2$ is a densely defined and discrete operator, it follows from \cite[Theorem 1]{Guo2001} that there exists generalized eigenfunctions of $\mathbf{A}_2$ that forms Riesz basis for $\mathbf{H}_2$. The proof is therefore completed. \end{proof}
Since the state feedback $U(t)=-\alpha u_t-a u_{xt}(1,t)$ makes the system (\ref{beem1}) exponentially stable \cite{Guo2000,Morgul1994}, Lemma \ref{A2stable} allows us to design the following control law \begin{align}\label{control}
U(t)=-\alpha\widehat{u}_t(1,t)-a \widehat{u}_{xt}(1,t). \end{align} The closed-loop system is then described by \begin{align}\label{closednodisturbance}
\left\{\begin{array}{l} u_{tt}(x,t)=u_{xx}(x,t),\;\; x\in (0,1), \; t>0, \\ u(0,t)=0, u_{x}(1,t)+mu_{tt}(1,t)=-\alpha\widehat{u}_t(1,t)\\ -a \widehat{u}_{xt}(1,t), \;\; t\ge 0,\\ \widehat{u}_{tt}(x,t)=\widehat{u}_{xx}(x,t),\;\; x\in (0,1), \; t>0, \\ {\color{blue}\widehat{u}_x(0,t)=\gamma \widehat{u}_t(0,t)+\beta \widehat{u}(0,t)+u_x(0,t)}, \; \; t\ge 0,\\ \widehat{u}_{x}(1,t)+m\widehat{u}_{tt}(1,t)=-\alpha\widehat{u}_t(1,t)-a \widehat{u}_{xt}(1,t), \end{array}\right. \end{align} which is equivalent to \begin{align}\label{closednodisturbance1}
\left\{\begin{array}{l} u_{tt}(x,t)=u_{xx}(x,t),\;\; x\in (0,1), \; t>0, \\ u(0,t)=0, u_{x}(1,t)+mu_{tt}(1,t)=-\alpha u_t(1,t)\\ -a u_{xt}(1,t) -\alpha\widetilde{u}_t(1,t)-a \widetilde{u}_{xt}(1,t), \;\; t\ge 0,\\ \widetilde{u}_{tt}(x,t)=\widetilde{u}_{xx}(x,t),\;\; x\in (0,1), \; t>0, \\ \widetilde{u}_x(0,t)=\gamma\widetilde{u}_{t}(0,t)+\beta\widetilde{u}(0,t), \; \; t\ge 0,\\ \widetilde{u}_{x}(1,t)+m\widetilde{u}_{tt}=0, \end{array}\right. \end{align}
Consider system (\ref{closednodisturbance1}) in Hilbert state space $\mathcal{H}=\mathbf{H}$ $\times \mathbf{H}_2$, where $\mathbf{H}=H_E^1(0,1)\times L^2(0,1)\times\mathds{C}$, $H_E^1(0,1)=\{f$ $ \in H^1(0,1):f(0)=0\}$. The norm of $\mathbf{H}$ is given by
$\|(f,g,\eta)\|^2_{\mathbf{H}} = \int_0^1[|f'(x)|^2+|g(x)|^2]dx+\frac{1}{m+\alpha a} |\eta|^2,$ $(f,g)\in \mathbf{H}.$ Define the operator $\mathbf{A}:D(\mathbf{A})(\subset \mathbf{H})\rightarrow \mathbf{H}$ by
$\mathbf{A}(f,g,\eta)=(g,f'',-f'(1)-\alpha g(1)),\; \forall\;(f,g)\in D(\mathbf{A})=\{(f,g)\in \mathbf{H}| f''(1)=0,
\eta=mg(1)+af'(1)\}.$ By \cite{Guo2000,Morgul1994}, it follows that $\mathbf{A}$ generates an exponentially stable $C_0$-semigroup, and there exist a sequence of generalized eigenfunctions of $\mathbf{A}$ that forms Riesz basis for $\mathbf{H}$ provided $\alpha=m/a\neq 1$. In the following Lemma \ref{Astable}, we generalize the result to the case $a\neq m,\alpha>0$.
\begin{lemma}\label{Astable} Suppose that $a\neq m$. Then, there exist a sequence of generalized eigenfunctions of operator $(\mathbf{A},D(\mathbf{A}))$ that forms Riesz basis for $\mathbf{H}$; the characteristic equation of $(\mathbf{A},D(\mathbf{A}))$ is $e^{2\lambda}[1+\alpha+(a+m)\lambda]+(1-\alpha)+a-m=0$. \end{lemma} \begin{proof}\ \ We divide the proof into several steps.
{\bf Step 1.} We claim that operator $(\mathbf{A},D(\mathbf{A}))$ a densely defined and discrete operator on $\mathbf{H}$; for any $\lambda\in \sigma(\mathbf{A})=\sigma_P(\mathbf{A})$, ${\rm Re} \lambda<0$.
Indeed, since by \cite{Morgul1994},$(\mathbf{A},D(\mathbf{A}))$ generates an exponentially stable $C_0$-semigroup, it is densely defined and has bounded inverse. Let $(f,g,\eta)\in \mathbf{H}$. Since by Sobolev imbedding theory $H^2(0,1)\times H^1(0,1)\times \mathds{C}$ is compactly imbedding in $H^1(0,1)\times L^2(0,1)\times \mathds{C}$, $\mathbf{A}^{-1}$ is compact. Moreover, for any $\lambda\in \sigma(\mathbf{A})=\sigma_P(\mathbf{A})$, ${\rm Re} \lambda<0$.
{\bf Step 2.} We claim that for any $\lambda\in\sigma(\mathbf{A})$, there corresponds one eigenfunction $(f,g,\eta)$ given by
\begin{align}\label{fxequ300}
\left\{
\begin{array}{ll}
f(x)=e^{\lambda x}-e^{-\lambda x}, \\
g=\lambda f, \\
\eta=-\frac{f'(1)+\alpha\lambda f(1)}{\lambda},
\end{array}
\right.
\end{align} and $\lambda$ satisfies the characteristic equation
\begin{align}\label{tauchar200}
e^{2\lambda}[1+\alpha+(a+m)\lambda]+(1-\alpha)+(a-m)=0.
\end{align} This implies that each eigenvalue of $\mathbf{A}$ is geometrically simple.
{\bf Step 3:} We show that the eigenvalues $\{\lambda_n\}_{n=-\infty}^{+\infty}$ have asymptotic expression \begin{equation}\label{Aataunasy100}
\lambda_n=\frac{1}{2}{\rm ln}\frac{|m-a|}{m+a}+n_a\pi i+O(|n|^{-1}), \end{equation} where $n_a=n$ if $m>a$ and $n_a=n+\frac{1}{2}$ if $0<m<a$, the corresponding eigenfunction $(f_n,\lambda_nf_n,$ $m\lambda_nf(1)+af'_n(1))$ of $\mathbf{A}_2$ can be chosen such that
\begin{align}\label{Afxequasy300}
\nonumber &F_n(x)=\left(
\begin{array}{c}
\left|\frac{m-a}{m+a}\right|^{x/2}e^{in_a x}+\left|\frac{m+a}{m-a}\right|^{x/2}e^{-in_a x} \\
\left|\frac{m-a}{m+a}\right|^{x/2}e^{in_a x}-\left|\frac{m+a}{m-a}\right|^{x/2}e^{-in_a x} \\ 0\\
\end{array}
\right)\\
&+O(n^{-1}),
\end{align}
where
\begin{equation}\label{Afmatrix100}
F_n(x)=\frac{1}{\lambda_n}\begin{pmatrix}
f_{n}^{'}(x) \\
\lambda_nf_n(x)\\
-\lambda_n^{-1}[f'_n(1)+\alpha\lambda_n f_n(1)]\\
\end{pmatrix}^T.
\end{equation}
Now we give the proof. By (\ref{tauchar200}), it follows that \begin{align}\label{fxc1c2c3c4}
e^{2\lambda}=-\frac{1-\alpha+(a-m)\lambda}{1+\alpha+(a+m)\lambda}=\frac{m-a}{m+a}+O(|\lambda|^{-1}). \end{align} The asymptotic expression (\ref{Aataunasy100}) is derived directly by Routhe's theorem.
{\bf {\color{blue}Step 4}:} We show that there exist a sequence of generalized eigenfunctions of $\mathbf{A}$ which forms Riesz basis for $\mathbf{H}$.
Now we give the proof. Denote \begin{align*}
Q(x)=\left\{
\begin{array}{ll}
\left(
\begin{array}{cc}
\left|\frac{m-a}{m+a}\right|^{x/2} &\left|\frac{m+a}{m-a}\right|^{x/2}\\
\left|\frac{m-a}{m+a}\right|^{x/2} & -\left|\frac{m+a}{m-a}\right|^{x/2} \\
\end{array}
\right), m>a,
\\
\left(
\begin{array}{cc}
\left|\frac{m-a}{m+a}\right|^{x/2}i &-\left|\frac{m+a}{m-a}\right|^{x/2}i \\
\left|\frac{m-a}{m+a}\right|^{x/2}i & \left|\frac{m+a}{m-a}\right|^{x/2}i\\
\end{array}
\right),m<a.
\end{array}
\right. \end{align*} It is easily seen that the operator $\mathbf{T}$ defined by $\mathbf{T}w=Q(x)w$ for any $w\in \left(L^2(0,1)\right)^2$, is a bounded and invertible operator on $\left(L^2(0,1)\right)^2$. Combine this with the fact that $\{(e^{in\pi x},e^{-in\pi x})\}_{n=-\infty}^{+\infty}$ forms Riesz basis for $\left(L^2(0,1)\right)^2$, to obtain that $\{Q(x)(e^{in\pi x},e^{-in\pi x})\}_{n=-\infty}^{+\infty}$ also forms Riesz basis for $\left(L^2(0,1)\right)^2$. Thereby, $\{(0,0,1)\}\bigcup\{(Q(x)(e^{in\pi x},e^{-in\pi x}),$ $0)\}_{n=-\infty}^{+\infty}$ forms Riesz basis for $\left(L^2(0,1)\right)^2\times \mathds{C}$.
By (\ref{fxequasy300}), it follows that \begin{align}\label{fxequasy3001}
F_n(x)=(Q(x)(e^{in\pi x},e^{-in\pi x}),0)+O(n^{-1}),
\end{align} which implies that, there exists enough big positive integer $N$ such that \begin{align*}
\sum_{|n|>N}\|F_n(x)-(Q(x)(e^{in\pi x},e^{-in\pi x}),0)\|_{\left(L(0,1)\right)^2\times\mathds{C}^2}^2=\sum_{|n|>N}O(|n|^{-2})<\infty. \end{align*} {\color{blue}Since $\{(0,0,1)\}\bigcup\{(Q(x)(e^{in\pi x},e^{-in\pi x}),0)\}_{n=-\infty}^{+\infty}$ forms Riesz basis for $\left(L^2(0,1)\right)^2\times \mathds{C}$, it follows by \cite[Lemma 1]{Guo2001} that there exists a $M\geq N$ such that $\{(0,0,1)\}\bigcup\{(Q(x)(e^{in\pi x},e^{-in\pi x}), 0)\}_{n=-M}^M\bigcup$
$\{F_n\}_{|n|>M}$ forms Riesz basis for $\left(L^2(0,1)\right)^2\times \mathds{C}$. We define an isometric isomorphism $\mathbb{T}_1:\mathbf{H}\rightarrow \left(L^2(0,1)\right)^2\times \mathds{C}$ by $\mathbb{T}_1(f,g,\eta)=(f',g,\eta),\; \forall\;(f,g,\eta)\in \mathbf{H}.$ Then there exist $\{W_n(x),V_n(x),0\}_{n=-M}^M\subset \mathbf{H}$ such that $\{(0,0,1)\}\bigcup\{(W_n(x),V_n(x),0)\}_{n=-M}^M\bigcup \{(\lambda^{-1}_nf_n(x),f_n(x),$ $-\lambda_n^{-2}[f'_n(1)+\alpha\lambda_n f_n(1)])$
$\}_{|n|>M}$ forms Riesz basis for $\mathbf{H}$. Since $\{(\lambda^{-1}_n(x),f_n(x),$ $-\lambda_n^{-2}[f'_n(1)+\alpha\lambda_n f_n(1)])\}_{n=-\infty}^\infty$ is a sequence of generalized eigenvectors of $\mathbf{A}$, and $\mathbf{A}$ is a densely defined and discrete operator.} By \cite[Theorem 1]{Guo2001}, it follows that there exists generalized eigenfunctions of $\mathbf{A}$ that forms Riesz basis for $\mathbf{H}$. This completes the proof. \end{proof}
Define operator $\mathcal{A}:D(\mathcal{A})(\subset \mathcal{H})\rightarrow \mathcal{H}$ by $\mathcal{A}(f,g,\eta,\phi,\psi,$ $h)=(g,f'',-f'(1)-\alpha g(1)-\alpha \psi(1),
\mathbf{A}_2(\phi,\psi,h)),$
$D(\mathcal{A})=\{(f,g,\eta,\phi,\psi,h)\in (H_E^1(0,1)\bigcap $
$H^2(0,1))
\times H_E^1(0,1)\times \mathds{C}\times D(\mathbf{A}_2),
\eta=af'(1)+mg(1)+a\phi'(1),h=m\psi(1) \}.$ Then the system (\ref{closednodisturbance1}) is abstractly described by $$\frac{d}{dt}Z(t)=\mathcal{A}Z(t),$$ where $Z(t)=\big(u(\cdot,t),u_t(\cdot,t),au_x(1,t)+mu_t(1,t)+a\widetilde{u}_x(1,t),\widetilde{u}(\cdot,t),$ $\widetilde{u}_t(\cdot,t),m\widetilde{u}_t(1,t)\big).$ We shall use Riesz basis approach to verify semigroup generation and exponential stability. However, it is difficult to verify the Riesz basis generation of couple wave equations \cite{Guo2007,Guo2019,Mei2020b}, not to mention wave equations with tip mass. We shall adopt Bari's theorem to prove Riesz basis generation of operator $\mathcal{A}$ by finding out complicated relations between sequences of generalized eigenfunctions.
\begin{theorem}\label{exponentialnodisturbance} Assume that $m\neq a$, $m\neq a\gamma$ and $\gamma\neq 1$. Then, the operator $\mathcal{A}$ generates an exponentially stable $C_0$-semigroup on $\mathcal{H}$. \end{theorem} \begin{proof}\ \ It is easy to show that $\mathcal{A}^{-1}$ exists and is compact on $\mathcal{H}$, thereby the spectrum of $\mathcal{A}$ consists of eigenvalues. Now we show that $\sigma(\mathcal{A})=\sigma(\mathbf{A})\bigcup \sigma(\mathbf{A}_2)$. By \cite{Guo2007} and Lemma \ref{Riesz}, $\mathbf{A}$ and $\mathbf{A}_2$ are also discrete operators. Hence $\sigma(\mathcal{A})\supseteq\sigma(\mathbf{A})\bigcup \sigma(\mathbf{A}_2).$ Let $\lambda\in \sigma(\mathcal{A})$ and $(f,g,\eta,\phi,\psi,h)$ be the corresponding eigenfunction. If $(\phi,\psi,h)\neq 0$, $\lambda\in\sigma(\mathbf{A}_2)$; if $(\phi,\psi,h)=0$, we have that $(f,g,\eta)\neq 0$, {\color{blue}$(f,g,\eta)\in D(\mathbf{A})$}, and {\color{blue}$\lambda (f,g,\eta)=\mathbf{A}(f,g,\eta)$}, which implies that {\color{blue}$\lambda\in \sigma(\mathbf{A})$}. Hence $\sigma(\mathcal{A})\subseteq\sigma(\mathbf{A})\bigcup \sigma(\mathbf{A}_2).$ Therefore $\sigma(\mathcal{A})=\sigma(\mathbf{A})\bigcup \sigma(\mathbf{A}_2)$.
Next, we shall show that the generalized eigenfunction of $\mathcal{A}$ forms a Riesz basis for $\mathcal{H}$. Let $\{\mu_{n}\}_{n=-\infty}^\infty$ and $\{\lambda_n\}_{n=-\infty}^\infty$ be respectively the eigenvalues of $\mathbf{A}$ and $\mathbf{A}_2$. Let $\{(\mu_{n}^{-1}f_{n},f_{n},-\mu_n^{-2}( f'(1)$ $+\alpha\mu_nf_n(1)))\}_{n=-\infty}^\infty$ and $\{(\lambda^{-2}_n\phi_{n},\lambda_n^{-1}\phi_{n},\lambda^{-3}_n\phi'_{n}(1)\}_{n=-\infty}^\infty$ be the generalized eigenfunctions corresponding to $\{\mu_{n}\}_{n=-\infty}^\infty$ and $\{\lambda_{n}\}_{n=-\infty}^\infty$ such that they form Riesz basises for $\mathbf{H}$ and $ \mathbf{H}_2$, respectively. Hence, the sequence $\{(\mu_{n}^{-1}f_{n},f_{n},-\mu_n^{-2}( f'(1)+\alpha\mu_nf_n(1)),0,0,0)\}_{n=-\infty}^\infty\bigcup$ $\{(0,0,0,\lambda^{-2}_n\phi_{n},\lambda_n^{-1}\phi_{n},$ $\lambda^{-3}_n\phi'_{n}(1))\}_{n=-\infty}^\infty$ forms a Riesz basis for $\mathcal{H}$,
which is equivalent to that $\{(\mu_{n}^{-1}f'_{n},f_{n},-\mu_n^{-2}$
$( f'(1)+\alpha\mu_nf_n(1)),0,0,0,0)\}_{n=-\infty}^\infty\bigcup\{(0,0,0,\lambda^{-2}_n\phi'_{n},$
$\lambda_n^{-1}\phi_{n},\lambda_n^{-2}\beta\phi_n(0), \lambda^{-3}_n\phi'_{n}(1)$ $\}_{n=-\infty}^\infty$ forms a Riesz basis for $\left(L^2(0,1)\right)^2\times \mathds{C}\times \left(L^2(0,1)\right)^2\times \mathds{C}^2$.
Let $\lambda\in \sigma(\mathcal{A})$ and $\big(\lambda^{-2}f,\lambda^{-1}f, -\lambda^{-3}(f'(1)+\alpha\lambda f(1)+\alpha\lambda \phi(1)),$ $\lambda^{-2}\phi,\lambda^{-1}\phi,\lambda^{-3}\phi'(1)\big)$ be the corresponding eigenfunction. If $\phi=0$, then $\big(\lambda^{-2}f,\lambda^{-1}f, -\lambda^{-3}(f'(1)+\alpha\lambda f(1))\big)\neq 0$ and $\lambda\in \sigma(\mathbf{A})$. Hence the eigenvalues $\{\mu_{n}\}_{n=1}^\infty$ corresponds the eigenfunctions $\{\big(\mu_{n}^{-2}f_n,\mu_{n}^{-1}f_n,$ $-\mu_{n}^{-3}(f_n'(1)+\alpha\mu_{n} f_n(1)),0,0,0\big)\}_{n=-\infty}^\infty$.
If $\phi\neq 0$, then $\lambda\in \sigma(\mathbf{A}_2)$. The eigenvalues $\{\lambda_{n}\}_{n=-\infty}^\infty$ corresponds the eigenfunction $\big(\lambda_n^{-2}\phi_n,$ $\lambda_n^{-1}\phi_n,\lambda_n^{-3}\phi_n'(1)\big)$ of $\mathbf{A}_2$. By Lemma \ref{A2stable}, it follows that \begin{equation}\label{f1n}
\begin{split}
\phi_{n}(x)=[(1+\gamma)\lambda_n+\beta]e^{\lambda_n x}+[(1-\gamma)\lambda_n-\beta]e^{-\lambda_n x}.
\end{split}
\end{equation} Let $\big(\lambda_n^{-2}f_{1n},\lambda_n^{-1}f_{1n}, -\lambda_n^{-3}(f_{1n}'(1)+\alpha\lambda_n f_{1n}(1)+\alpha\lambda_n \phi_{n}(1)),\lambda_n^{-2}\phi_n,$ $\lambda_n^{-1}\phi_n,\lambda_n^{-3}\phi_n'(1)\big)\in \mathcal{H}$ be the eigenfunction of $\mathcal{A}$ corresponding to eigenvalues $\{\lambda_{n}\}_{n=-\infty}^\infty$. Then, $f_{1n}$ satisfies $f''_{1n}(x)=\lambda_n^2f_{1n}(x),$
$f_{1n}(0)=0,
(1+a\lambda_n)f'_{1n}(1)+(m\lambda^2_n+\alpha\lambda_n)f_{1n}
=-\alpha\lambda_n\phi_n(1)-a\lambda_n\phi'_n(1).$ The solution of the ode $f''_{1n}(x)=\lambda^2_{n}f_{1n}(x)$ with boundary conditions $f_{1n}(0)=0$ is of the form \begin{align}\label{f1nE}
f_{1n}(x)=E(e^{\lambda_n x}-e^{-\lambda_n x}). \end{align} Combine (\ref{f1n}) and $(1+a\lambda_n)f'_{1n}(1)+(m\lambda^2_n+\alpha\lambda_n)f_{1n}=-\alpha\lambda_n\phi_n(1)-a\lambda_n\phi'_n(1)$ to derive $\frac{E}{\lambda_n}=-\frac{\alpha\phi_n(1)+a\phi'_n(1)}{E_{11}e^{\lambda_n}+E_{12}e^{-\lambda_n}}$ $=-\frac{E_{21}e^{\lambda_n}
+E_{22}e^{-\lambda_n}}{E_{11}e^{\lambda_n}+E_{12}e^{-\lambda_n}},$ where $E_{11}=\lambda_n[1+\alpha+(a+m)\lambda_n],E_{12}=\lambda_n[1-\alpha+(a-m)\lambda_n], E_{21}=(\alpha$ $+a\lambda_n)[(1+\gamma)\lambda_n+\beta],E_{22}=(\alpha-a\lambda_n)[(1-\gamma)\lambda_n-\beta].$ Use the characteristic equation in Lemma \ref{Astable} to get $\frac{E}{\lambda_n}=-\frac{E_{21}F_1+E_{22}F_2}
{E_{11}F_1+E_{12}F_2}=\frac{a(\gamma^2-1)}{\gamma a-m}+O(|n^{-1}|),$ where $F_1=[(1-\gamma)\lambda_n-\beta](1-m\lambda_n)$, $F_2=[(1+\gamma)\lambda_n+\beta](1+m\lambda_n)$. This, together with (\ref{f1nE}) indicates \begin{align*} &\lambda_{n}^{-2}f'_{1n}(x)=\lambda_n^{-1}E[e^{\lambda_nx}+e^{-\lambda_nx}]=\frac{a(\gamma^2-1)}{\gamma a-m}\cdot\\
&\bigg[\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}e^{in_\beta x}+\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}e^{-in_\beta x}\bigg]+O(|n|^{-1}),\\ &\lambda_{n}^{-1}f_{1n}(x)=\lambda_n^{-1}E[e^{\lambda_nx}-e^{-\lambda_nx}]=\frac{a(\gamma^2-1)}{\gamma a-m}\cdot\\
&\left[\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}e^{in_\beta x}-\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}e^{-in_\beta x}\right]+O(|n|^{-1}). \end{align*} Denote $W=\left(
\begin{array}{cc}
I_3 & V \\
0 & I_3 \\
\end{array}
\right), $ with $V=\frac{2a}{m-a\gamma}\left(
\begin{array}{ccc}
-\gamma & 1& 0 \\
1 & -\gamma & 0 \\
0& 0& 0\\
\end{array}
\right).$ Obviously, $W$ has bounded inverse. Combine this with Lemma \ref{A2stable} to derive \begin{align} \nonumber &\big(\mu_{n}^{-2}f_n,\mu_{n}^{-1}f_n, -\mu_{n}^{-3}(f_n'(1)+\alpha\mu_{n} f_n(1)),0,0,0\big)^T=\\ \label{guanxi1}&W\big(\mu_{n}^{-2}f_n,\mu_{n}^{-1}f_n, -\mu_{n}^{-3}(f_n'(1)+\alpha\mu_{n} f_n(1)),0,0,0\big)^T,\\ \nonumber&\big(\lambda_n^{-2}f_{1n},\lambda_n^{-1}f_{1n}, -\lambda_n^{-3}(f_{1n}'(1)+\alpha\lambda_n f_{1n}(1)\\ \nonumber&+\alpha\lambda_n \phi_{n}(1)),\lambda_n^{-2}\phi_n,\lambda_n^{-1}\phi_n,\lambda_n^{-3}\phi_n'(1)\big)^T=W\big(0,0,0,\\
\label{guanxi2}&\lambda_n^{-2}\phi_n,\lambda_n^{-1}\phi_n,\lambda_n^{-3}\phi_n'(1)\big)^T+O(|n|^{-1}). \end{align} Then, by Bari's theorem and \cite[Theorem 1]{Guo2001}, the sequence $\{\big(\mu_{n}^{-2}f'_n,\mu_{n}^{-1}f_n,$ $-\mu_{n}^{-3}(f_n'(1)+\alpha\mu_{n} f_n(1)),$ $0,0,0,0\big)\}_{n=-\infty}^\infty \bigcup \{\big(\lambda_n^{-2}f'_{1n},$ $\lambda_n^{-1}f_{1n},-\lambda_n^{-3}(f_{1n}'(1)+\alpha\lambda_n f_{1n}(1)+\alpha\lambda_n \phi_{n}(1)),\lambda_n^{-2}\phi'_n,$ $\lambda_n^{-1}\phi_n,\lambda_n^{-2}\beta \phi_n(0),$ $\lambda_n^{-3}\phi_n'(1)\big)\}_{n=-\infty}^\infty$ forms Riesz basis for the Hilbert space $\big(L^2(0,1)\big)^2\times \mathds{C}\times \left(L^2(0,1)\right)^2 \times \mathds{C}^2$, which is equivalent to that the sequence $\{\big(\mu_{n}^{-2}f_n,\mu_{n}^{-1}f_n,$ $-\mu_{n}^{-3}(f_n'(1)+\alpha\mu_{n} f_n(1)),0,0,0\big)\}_{n=-\infty}^\infty $ $\bigcup \{\big(\lambda_n^{-2}f_{1n},$ $\lambda_n^{-1}f_{1n}, -\lambda_n^{-3}(f_{1n}'(1)+\alpha\lambda_n f_{1n}(1)+\alpha\lambda_n \phi_{n}(1)),\lambda_n^{-2}\phi_n,$ $\lambda_n^{-1}\phi_n,\lambda_n^{-3}\phi_n'(1)\big)\}_{n=-\infty}^\infty$ forms Riesz basis for $\mathcal{H}$.
The semigroup generation and spectrum-determined growth condition of $\mathcal{A}$ are derived directly from the Riesz basis property. Since both $\mathbf{A}$ and $\mathbf{A}_2$ are generator of exponentially stable $C_0$-semigroups, $e^{\mathcal{A}t}$ is exponentially stable. \end{proof}
\begin{remark}\label{zhishu}\em We have to mention that, although it forms Riesz basis for $\mathcal{H}$, $\{(\mu_{n}^{-1}f_{n},f_{n},-\mu_n^{-2}( f'(1)+\alpha\mu_nf_n(1)),$ $0,0,0)\}_{n=-\infty}^\infty\bigcup\{(0,0,0,\lambda^{-2}_n\phi_{n},\lambda_n^{-1}\phi_{n}, \lambda^{-3}_n\phi'_{n}(1)\}_{n=-\infty}^\infty$ are not all the generalized eigenfunctions of $\mathcal{A}$. This is the reason why we choose to compute the generalized eigenfunctions of $\mathcal{A}$ and find out the relations (\ref{guanxi1}) and (\ref{guanxi2}), which are the key steps for the proof. \end{remark}
\begin{theorem}\label{nodisturbancemain} Given initial value $(u(\cdot,0),u_t(\cdot,0),mu_t(1,$ $0)+a\widehat{u}_x(1,0),\widehat{u}(\cdot,0),\widehat{u}_t(\cdot,0),m\widehat{u}_t(1,0)+a\widehat{u}_x(1,0)) \in \mathcal{H}_1$, there exists a unique solution to system (\ref{closednodisturbance}) such that $(u(\cdot,t),u_t(\cdot,t),mu_t(1,t)+a\widehat{u}_x(1,t),\widehat{u}(\cdot,t),\widehat{u}_t(\cdot,t),m\widehat{u}_t(1,$ $t)+a\widehat{u}_x(1,t)) \in C(0,\infty; \mathcal{H}_1)$ satisfying
$\int_0^1(|u_t(x,t)|^2+|u_{x}(x,t)|^2+|\widehat{u}_t(x,t)|^2+|\widehat{u}_{x}(x,t)|^2)dx
+|mu_t(1,t)+a\widehat{u}_x(1,t)|^2/m+{\color{blue}|m\widehat{u}_t(1,0)+a\widehat{u}_x(1,0)|^2/(m+a\alpha)}+\beta|\widehat{u}(0,t)|^2$
$\leq M_1e^{-\gamma_1 t},$ where $M_1$ and $\gamma_1$ are two positive constants. \end{theorem} \begin{proof}\ \ Fix $(u(\cdot,0),u_t(\cdot,0),mu_t(1,0)+a\widehat{u}_x(1,0),$ $\widehat{u}(\cdot,0),\widehat{u}_t(\cdot,0),m\widehat{u}_t(1,0)+a\widehat{u}_x(1,0)) \in \mathbf{H}_1\times \mathbf{H}_3$. Then we have $(u(\cdot,0),u_t(\cdot,0),mu_t(1,0)+a[u_x(1,0)+\widetilde{u}_x(1,0)],\widetilde{u}(\cdot,0),\widetilde{u}_t(\cdot,0),m\widetilde{u}_x(1,0)) \in \mathbf{H}\times \mathbf{H}_2$. By Theorem \ref{exponentialnodisturbance}, it follows that $(u(\cdot,t),u_t(\cdot,t),mu_t(1,t)+a[u_x(1,t)+\widetilde{u}_x(1,t)],\widetilde{u}(\cdot,t),\widetilde{u}_t(\cdot,t),m\widetilde{u}_x(1,t)) =e^{\mathcal{A}t}(u(\cdot,$ $0),u_t(\cdot,0),mu_t(1,0)+a[u_x(1,0)+\widetilde{u}_x(1,0)],\widetilde{u}(\cdot,0),$ $\widetilde{u}_t(\cdot,0),m\widetilde{u}_x(1,0)) \in C(0,\infty;\mathbf{H}\times \mathbf{H}_2)$ and \begin{align*}
&\int_0^1(|u_t(x,t)|^2+|u_{x}(x,t)|^2+|\widehat{u}_t(x,t)|^2+|\widehat{u}_{x}(x,t)|^2)dx\\
&+\frac{1}{m}|mu_t(1,0)+a\widehat{u}_x(1,t)|^2+\frac{1}{m+a\alpha}|m\widehat{u}_t(1,0)\\
&+a\widehat{u}_x(1,t)|^2+\beta|\widehat{u}(0,t)|^2\\
&\leq \int_0^1(|u_t(x,t)|^2+|u_{x}(x,t)|^2+2|u_t(x,t)|^2\\
&+2|u_{x}(x,t)|^2+2|\widetilde{u}_t(x,t)|^2+2|\widetilde{u}_{x}(x,t)|^2)dx\\
&+\frac{1}{m}|mu_t(1,t)+a[u_x(1,t)+\widetilde{u}_x(1,t)]|^2\\
&+\frac{2}{m+a\alpha}|mu_t(1,t)+a[u_x(1,t)+\widetilde{u}_x(1,t)]|^2\\
&+\frac{2}{m+a\alpha}|m\widetilde{u}_t(1,t)|^2+\beta|\widetilde{u}(0,t)|^2\\
&\leq 3\left[1+\frac{m+a\alpha}{m}\right]\|(u(\cdot,t),u_t(\cdot,t),mu_t(1,t)\\
&+a[u_x(1,t)+\widetilde{u}_x(1,t)],\widetilde{u}(\cdot,t),\widetilde{u}_t(\cdot,t),m\widetilde{u}_x(1,t))\|^2\\
&=3\left[1+\frac{m+a\alpha}{m}\right]\|e^{\mathcal{A}t}(u(\cdot,0),u_t(\cdot,0),mu_t(1,0)\\
&+a[u_x(1,0)+\widetilde{u}_x(1,0)],\widetilde{u}(\cdot,0)\|^2\\
&\leq M_1e^{-\gamma_1 t}, \end{align*}
where $M_1=3M^2_{\mathcal{A}}\left[1+\frac{m+a\alpha}{m}\right]\|(u(\cdot,0),u_t(\cdot,0),mu_t(1,$
$0)+a[u_x(1,0)+\widetilde{u}_x(1,0)],\widetilde{u}(\cdot,0)\|^2$, $\gamma_1=2{\color{blue}\omega_\mathcal{A}}$. This completes the proof. \end{proof}
\begin{remark}\em Theorem \ref{nodisturbancemain} presents not only the exponential stability of $(u(\cdot),u_t(\cdot),\widehat{u}(\cdot,t),\widehat{u}_t(\cdot,t))$ but also the exponential stability of $\eta(t)=mu_t(1,t)+a[u_x(1,t)+\widetilde{u}_x(1,t)]$ and $\psi(t)=\widehat{u}_t(1,t)+a[u_x(1,t)+\widetilde{u}_x(1,t)]$, because $\eta(t)$ and $\psi(t)$ are the states of the boundary ode dynamic parts of the closed-loop system. {\color{blue}This is quite different} from the case in \cite{Guo2007} where $m=0$. By Theorem \ref{nodisturbancemain}, with only one {\color{blue}non-collocated measurement $u_{x}(0,t)$} we can design the estimated state based controller (\ref{control}) to exponentially stabilize the vibration cable with tip mass (\ref{beem1}). Therefore, we improve the results in references \cite{Guo2000,Morgul1994}, where two collocated measurements $u_t(1,t)$ and $u_{xt}(1,t)$ with $u_{xt}(1,t)$ being angular velocity were used. \end{remark}
\begin{remark}\em In \cite{Guo2007}, Guo and Xu verified the exponential stability of closed-loop system (\ref{closednodisturbance}) with $m=0$. By virtue of Riesz basis approach, the authors used 2.5 pages of two columns to verify the semigroup generation and exponential stability of system \cite[(3.1)]{Guo2007}. When $m\neq 0$, we transfer the closed-loop system to an equivalent system (\ref{closednodisturbance1}) which contains the original system and error systems. Although we also used Riesz basis approach and our case is more complicated ($m\neq 0$), our proof is much shorter than that in \cite{Guo2007} because (\ref{closednodisturbance1}) is easier to be dealt with: it contains an independent subsystem related to $\widetilde{u}$. Our method {\color{blue}can be used to} simplify the verification of exponential stability in \cite{Guo2007}. \end{remark}
\section{Stabilization in presence of internal uncertainty and external disturbance}\label{disturbancestab}
When we consider disturbance $F(t)\equiv F$, the boundary condition $u_{x}(1,t)+mu_{tt}(1,t)=-\alpha\widehat{u}_t(1,t)-a \widehat{u}_{xt}(1,t)$ of {\color{blue}the closed-loop system (\ref{closednodisturbance}) is changed to} $u_{x}(1,t)+mu_{tt}(1,t)=-\alpha\widehat{u}_t(1,t)-a \widehat{u}_{xt}(1,t) +F$. One can see that $u(x,t)=Fx,\widehat{u}(x,t)=-F/\beta$ is a solution of the closed-loop system, which is not stable. Therefore, when there is disturbance, the stabilizing control law should be redesigned. Motivated by \cite{Feng2017a}, for system (\ref{beem1}) with $F(t)\neq 0$, we propose in this section an infinite-dimensional ESO without high gain described as follow \begin{equation} \label{transfer} \left\{\begin{array}{l} v_{tt}(x,t)=v_{xx}(x,t),\;\; x\in (0,1), \; t>0, \\ v_x(0,t)=\gamma v_{t}(0,t)+\beta v(0,t)+u_x(0,t), \\ v_{x}(1,t)+mv_{tt}(1,t)=U(t), \;\; t\ge 0, \\ q_{tt}(x,t)=q_{xx}(x,t),\;\; x\in (0,1), \; t>0, \\ q_x(0,t)=\gamma q_{t}(0,t)+\beta q(0,t), \; \; t\ge 0,\\ q(1,t)=v(1,t)-u(1,t), \;\; t\ge 0. \end{array}\right. \end{equation} {\color{blue} Since it depends only on the input $u(t)$ and output $u_x(0,t),u(1,t)$, system (\ref{transfer}) is completely known.} Although there exists disturbance, the high order angular velocity $u_{xt}(1,t)$ is not used.
Set $\widehat{v}(x,t)=v(x,t)-u(x,t)$ to derive the error equation with unknown input as follows \begin{equation} \label{perror} \left\{\begin{array}{l} \widehat{v}_{tt}(x,t)=\widehat{v}_{xx}(x,t),\;\; x\in (0,1), \; t>0, \\ \widehat{v}_x(0,t)=\gamma \widehat{v}_t(0,t)+\beta \widehat{v}(0,t), \; \; t\ge 0,\\ \widehat{v}_{x}(1,t)+m\widehat{v}_{tt}(1,t)=-F(t), \;\; t\ge 0 \end{array}\right. \end{equation} which is written abstractly by \begin{align}\label{perrorsemigroup}
\frac{d}{dt}{\color{blue}(\widehat{v}(\cdot,t),\widehat{v}_t(\cdot,t),m\widehat{v}_t(1,t))}=\mathbf{A}_2{\color{blue}(\widehat{v}(\cdot,t),\widehat{v}_t(\cdot,t),m\widehat{v}_t(1,t))}-\mathbf{B}_{2}F(t), \end{align} where $\mathbf{B}_2=(0,0,1)^T$ is a bounded linear operator. Since $\mathbf{A}_2$ is a generator of an exponentially stable $C_0$-semigroup, the role of the $v$-part of the system (\ref{transfer}) is to transfer the total disturbance $F(t)$ into an exponentially stable system that is relatively easier to be dealt with.
Similar to the proof of \cite[Lemma A.1 and A.2]{Zhou2018a}, we derive the following lemma.
\begin{lemma}\label{admissible} Assume that $d\in L^\infty(0,\infty)$ (or $d\in L^2(0,\infty))$, $f: H^1(0,1)\times L^2(0,1)\rightarrow \mathds{R}$ is continuous and system (\ref{beem}) admits a unique bounded solution ${\color{blue}(u(\cdot,t),u_t(\cdot,t))}\in C(0,\infty;H^1(0,1)\times L^2(0,1))$. Then for any initial value ${\color{blue}(\widehat{v}(\cdot,0),\widehat{v}_t(\cdot,0),m\widehat{v}_t(1,0))}\in \mathbf{H}_2$, there exists a unique solution to system (\ref{perror}) such that $(\widehat{v}(\cdot,t),\widehat{v}_t(\cdot,t),m\widehat{v}_t(1,t))\in C(0,\infty;\mathbf{H}_2)$ and
$\|(\widehat{v}(\cdot,t),\widehat{v}_t(\cdot,t),m\widehat{v}_t(1,t))\|_{\mathbf{H}_2}< +\infty$. Moreover, if $\lim_{t\rightarrow \infty}f(w(\cdot,t),w_t(\cdot,t))=0$ and $d\in L^2(0,\infty)$, then $\lim_{t\rightarrow \infty}\|(\widehat{v}(\cdot,t),\widehat{v}_t(\cdot,t),m\widehat{v}_t(1,$
$t))\|_{\mathbf{H}_2}=0$. \end{lemma}
Set $\widehat{q}(x,t)=q(x,t)-\widehat{v}(x,t)$ to get \begin{equation} \label{erroerrorobserver} \left\{\begin{array}{l} \widehat{q}_{tt}(x,t)=\widehat{q}_{xx}(x,t),\;\; x\in (0,1), \; t>0, \\ {\color{blue}\widehat{q}_x(0,t)=\gamma\widehat{q}_t(0,t)+\beta\widehat{q}(0,t)}, \widehat{q}(1,t)=0. \end{array}\right. \end{equation}
Consider system (\ref{erroerrorobserver}) in the space $\mathbb{H}=H^1_e(0,1)\times L^2(0,1)$, $H^1_e(0,1)=\{f\in H^1(0,1)|f(1)=0\}$ with inner product induced norm $\|(f,g)\|^2_{\mathbb{H}}=\int_0^1[|f'(x)|^2+|g(x)|^2]dx$. Define $\mathbb{A}(f,g)=(g,f''), \forall (f,g)\in D(\mathbb{A})=\{(f,g)\in (H_e^1\bigcap $
$H^2(0,1))\times H^1_e(0,1)|f'(0)=\gamma g(0)+\beta f(0)\}.$ Then, system (\ref{erroerrorobserver}) can be written abstractly as \begin{align*}
\frac{d}{dt}(\widehat{q}(\cdot,t),\widehat{q}_t(\cdot,t))=\mathbb{A}(\widehat{q}(\cdot,t),\widehat{q}_t(\cdot,t)). \end{align*} By \cite{Chen1981}, $\mathbb{A}$ is generates an exponentially stable $C_0$-semigroup. The following Lemma \ref{Riesz} tells us that the system (\ref{erroerrorobserver}) possess Riesz basis property.
\begin{lemma}\label{Riesz} Assume that $\alpha\neq 1$. Then, $\mathbb{A}$ is a discrete operator; there exist a sequence of generalized eigenfunctions of $\mathbb{A}$ which forms Riesz basis for $\mathbb{H}$; there exist a family of eigenvalue $\{\lambda_n\}_{n=-\infty}^\infty$ of operator $\mathbb{A}_2$ asymptotically expressed by
$\lambda_n=\frac{1}{2}\ln\bigg|\frac{\gamma-1}{\gamma+1}\bigg|+n_\gamma \pi i+O(|n|^{-1}),$ where $n_\gamma=n-1/2$ if $0<\gamma<1$ and $n_\gamma=n$ if $\gamma>1$; the corresponding eigenfunctions is given by $(\lambda_n^{-1}f_n,f_n)$ with $f_n(x)=\sinh\lambda_n(x-1)$ and
$F_n(x):=\big(\lambda_n^{-1}f_n'(x),f_n(x)\big)=\big(\big|\frac{\gamma-1}{\gamma+1}\big|^{(x-1)/2}e^{n_\gamma\pi i(x-1)}
+\big|\frac{\alpha+1}{\gamma-1}\big|^{(x-1)/2}e^{-n_\gamma\pi i(x-1)}, \big|\frac{\gamma-1}{\gamma+1}\big|^{(x-1)/2}e^{n_\gamma\pi i(x-1)}
-\big|\frac{\alpha+1}{\gamma-1}\big|^{(x-1)/2}$ $e^{-n_\gamma\pi i(x-1)} \big)
+O(|n|^{-1}).$ \end{lemma}
The following lemma is due to Feng and Guo \cite{Feng2017a}. \begin{lemma}\label{zestimate}
Assume that $(\widehat{q}(\cdot,0),\widehat{z}_t(\cdot,0))\in D(\mathbb{A})$. The solution of (\ref{erroerrorobserver}) satisfies $|\widehat{q}_{x}(1,t)|\leq Me^{-\mu t}$ for some constant $M,\; \mu>0$. \end{lemma}
\begin{remark} \em On one hand, {\color{blue}since by (\ref{erroerrorobserver}) we obtain $\widehat{q}(1,t)=0$}, it follows from Lemma \ref{zestimate} that $F(t)=$ $-\widehat{v}_{x}(1,t)-m\widehat{v}_{tt}(1,t) =-q_{x}(1,t)-mq_{tt}(1,t)+{\color{blue}\widehat{q}_{x}(1,t)}+m\widehat{q}_{tt}(1,t)=-q_{x}(1,t)-mq_{tt}(1,t)+{\color{blue}\widehat{q}_{x}(1,t)}\approx -q_{x}(1,t)-mq_{tt}(1,t)$ provided $(\widehat{q}(\cdot,0),\widehat{q}_t(\cdot,0))\in D(\mathbb{A})$. This indicates that $-q_{x}(1,t)-mq_{tt}(1,t)$ is an estimate of total disturbance $F(t)$. On the other hand, since (\ref{erroerrorobserver}) decays exponentially, we obtain $u(\cdot,t)=v(\cdot,t)-q(\cdot,t)+\widehat{q}(\cdot,t)\approx v(\cdot,t)-q(\cdot,t)$. This implies that $v(\cdot,t)-q(\cdot,t)$ is an estimate of $w(\cdot,t)$. The combination of the two hands tells us that the system (\ref{transfer}) estimates both the total disturbance and original state. This is the reason why we call the system (\ref{transfer}) infinite-dimensional ESO for (\ref{beem}). We shall show in the rest of the present section that the disturbance estimator (\ref{transfer}) is enough to exponentially stabilize system (\ref{beem}) for general initial state $(\widehat{q}(\cdot,0),\widehat{q}_t(\cdot,0))\in \mathbb{H}$. \end{remark}
Since by \cite{Morgul1994} the state feedback $u(t)=-\alpha u_t(1,t)-a u_{xt}(1,t)$ exponentially stabilizes system (\ref{beem1}), it is natural to design an estimated state-based controller \begin{align}\label{feedback11}
\nonumber&U(t)=q_{x}(1,t)+mq_{tt}(1,t)-\alpha[v_t(1,t)-q_t(1,t)]\\
&-a [v_{xt}(1,t)-q_{xt}(1,t)], \end{align} where $q_{x}(1,t)+mq_{tt}(1,t)$ is used to cancel the total disturbance $F(t)$, $v_t(1,t)-q_t(1,t)$ and $v_{xt}(1,t)-q_{xt}(1,t)$ are respectively applied to estimate $u_t(1,t)$ and $u_{xt}(1,t)$. \begin{remark}\label{xiangdui}\em In \cite{Mei2020b,Zhou2018a,Zhou2018b}, infinite-dimensional ESOs were used to estimate the total disturbance only; the authors design additional ESO-based Luenberger state observers in order to derive the estimations of original states. In our control law (\ref{feedback11}), we directly estimated state from the infinite-dimensional ESO. Our control strategy is more concise and energy-saving, and it can help one simplify the design of references \cite{Mei2020b,Zhou2018a,Zhou2018b}. \end{remark}
The closed-loop system of (\ref{beem}) under the controller (\ref{feedback11}) is \begin{equation} \label{perror110} \left\{\begin{array}{l} u_{tt}(x,t)=u_{xx}(x,t),\;\; x\in (0,1), \; t>0, \\ u(0,t)=0, u_{x}(1,t)+mu_{tt}(1,t)=q_{x}(1,t)+mq_{tt}(1,t)\\ -\alpha[v_t(1,t) -q_t(1,t)]-a [v_{xt}(1,t)-q_{xt}(1,t)] +F(t), \\ v_{tt}(x,t)=v_{xx}(x,t),\;\; x\in (0,1), \; t>0, \\ v_x(0,t)=\gamma v_{t}(0,t)+\beta v(0,t)+u_x(0,t), \; \; t\ge 0,\\ v_{x}(1,t)+mv_{tt}(1,t)=q_{x}(1,t)+mq_{tt}(1,t)\\ -\alpha[v_t(1,t)-q_t(1,t)] -a [v_{xt}(1,t)-q_{xt}(1,t)], \;\; t\ge 0, \\ q_{tt}(x,t)=q_{xx}(x,t),\;\; x\in (0,1), \; t>0, \\ q_x(0,t)=\gamma q_{t}(0,t)+\beta q(0,t), \; \; t\ge 0,\\ q(1,t)=v(1,t)-u(1,t), \;\; t\ge 0, \end{array}\right. \end{equation} which is equivalent to \begin{equation} \label{perror110closed} \left\{\begin{array}{l} u_{tt}(x,t)=u_{xx}(x,t),\;\; x\in (0,1), \; t>0, \\ u(0,t)=0, u_{x}(1,t)+mu_{tt}(1,t)=-\alpha u_{t}(1,t)\\ -au_{xt}(1,t)+a \widehat{q}_{xt}(1,t)+\widehat{q}_x(1,t), \;\; t\ge 0,\\ \widehat{v}_{tt}(x,t)=\widehat{v}_{xx}(x,t),\;\; x\in (0,1), \; t>0, \\ \widehat{v}_x(0,t)=\gamma \widehat{v}_t(0,t)+\beta \widehat{v}(0,t), \; \; t\ge 0,\\ \widehat{v}_{x}(1,t)+m\widehat{v}_{tt}(1,t)=-F(t), \;\; t\ge 0,\\ \widehat{q}_{tt}(x,t)=\widehat{q}_{xx}(x,t),\;\; x\in (0,1), \; t>0, \\ \widehat{q}_x(0,t)=\gamma\widehat{q}_t(0,t)+\beta\widehat{q}(0,t), \widehat{q}(1,t)=0. \end{array}\right. \end{equation}
Our target in the rest of this section is to show that the state $(u(\cdot,t),u_t(\cdot,t))$ of the closed-loop system (\ref{perror110}) is exponentially stable while guaranteeing the boundedness of the other variables. Since it does not involve the total disturbance $F(t)$, we choose to firstly consider the $(w,\widehat{z})$-part of (\ref{perror110closed}) described by \begin{equation} \label{wzwanclosed} \left\{\begin{array}{l} u_{tt}(x,t)=u_{xx}(x,t),\;\; x\in (0,1), \; t>0, \\ u(0,t)=0, u_{x}(1,t)+mu_{tt}(1,t)=-a u_{xt}(1,t)\\ -\alpha u_t(1,t) +a \widehat{q}_{xt}(1,t)+\widehat{q}_x(1,t), \;\; t\ge 0,\\ \widehat{q}_{tt}(x,t)=\widehat{q}_{xx}(x,t),\;\; x\in (0,1), \\ \widehat{q}_x(0,t)=\gamma\widehat{q}_t(0,t)+\beta\widehat{q}(0,t), \widehat{q}_x(1,t)=0. \end{array}\right. \end{equation}
We consider system (\ref{wzwanclosed}) in Hilbert state space $\mathcal{X}_1=\mathbf{H}$ $\times\mathbb{H}$. Define the operator $\mathcal{A}_1:D(\mathcal{A}_1)\subset\mathcal{X}_1\rightarrow \mathcal{X}_1$ by $\mathcal{A}_1(f,g,\eta,$ $\phi,\psi)=(g,f'',{\color{blue}-f'(1)-\alpha g(1)+\phi'(1)},\psi,\phi''),$ $(f,g,\phi,\psi)\in D(\mathcal{A}_1)=\{(f,g,\eta,\phi,\psi)\in (H^1_E(0,1)\bigcap $
$H^2(0,1))\times H^1_E(0,1)\times D(\mathbb{A})| {\color{blue}\eta=a f'(1)+mg(1)-a \phi'(1)}\}$. System (\ref{wzwanclosed}) is abstractly described by $$\frac{d}{dt}X(t)=\mathcal{A}_1X(t),$$ where $X(t)=(u(\cdot,t),u_t(\cdot,t),a u_{x}(1,t)$ $+mu_t(1,t)-a\widehat{q}_{x}(1,t),\widehat{q}(\cdot,t),\widehat{q}_t(\cdot,t)).$ One can easily verify that the operator $\mathcal{A}_1$ is not dissipative,
we shall adopt Riesz basis approach to verify the stability.
\begin{theorem}\label{exponential00} Assume that $\gamma\neq 1$, $m\neq a$ and $m\neq a\gamma$. Then, system (\ref{wzwanclosed}) is governed by an exponentially stable $C_0$-semigroup. \end{theorem}
\begin{proof}\ \ By the same procedure as the proof of Theorem \ref{exponentialnodisturbance}, we can obtain that the operator $\mathcal{A}_1$ has bounded and compact inverse on $\mathcal{X}_1$ and $\sigma(\mathcal{A}_1)=\sigma(\mathbf{A})\bigcup \sigma(\mathbb{A})$. Next, we shall show that the generalized eigenfunction of $\mathcal{A}_1$ forms Riesz basis for $\mathcal{X}_1$. Let $\{\lambda_{n}\}_{n=-\infty}^\infty$ and $\{\lambda_{1n}\}_{n=-\infty}^\infty$ be the eigenvalues of $\mathbb{A}$ and $\mathbf{A}$, respectively. Let $\{(\lambda_n^{-1}\phi_{n},\phi_{n})\}_{n=-\infty}^\infty$ and $\{(\lambda_{1n}^{-1}f_{n},f_{n},a f_n'(1)+m\lambda_{1n}f_n(1))\}_{n=-\infty}^\infty$ be the generalized eigenfunctions corresponding to $\{\lambda_{n}\}_{n=1}^\infty$ and $\{\lambda_{1n}\}_{n=-\infty}^\infty$ that form Riesz basises for $\mathbb{H}$ and $ \mathbf{H}$, respectively. As a result, $\{(0,0,0,\lambda_n^{-1}\phi_{n},\phi_{n})\}_{n=-\infty}^\infty\bigcup \{(\lambda_{1n}^{-1}f_{n},f_{n},$ $a f_n'(1)+m\lambda_{1n}f_n(1),0,0)\}_{n=-\infty}^\infty$ forms a Riesz basis for $\mathcal{X}_1$, which is equivalent to that $\{(0,0,0,\lambda_n^{-1}\phi'_{n},$ $\phi_{n})\}_{n=-\infty}^\infty\bigcup \{(\lambda_{1n}^{-1}$ $f'_{n},f_{n},a f_n'(1)+m\lambda_{1n}f_n(1),0,0)\}_{n=-\infty}^\infty$ forms a Riesz basis for $(L^2(0,1))^2\times \mathds{C}\times (L^2(0,1))^2$.
Let $\lambda\in \sigma(\mathcal{A})$ and $(\lambda^{-1}f,f,a f'(1)+m\lambda f(1),\lambda^{-1}\phi,\phi)$ be the corresponding eigenfunction. If $\phi=0$, then $(\lambda^{-1}f,f,a f'(1)+m\lambda f(1))\neq 0$ and $\lambda\in \sigma(\mathbf{A})$. Hence in this case the eigenvalues $\{\lambda_{1n}\}_{n=1}^\infty$ corresponds the eigenfunctions $\{(\lambda_{1n}^{-1}f_n,f_n,a f'_n(1)+m\lambda_{1n} f_n(1),0,0)\}_{n=1}^\infty$.
If $\phi\neq 0$, then $\lambda\in \sigma(\mathbb{A})$. The eigenvalues $\{\lambda_{n}\}_{n=1}^\infty$ corresponds the eigenfunction $(\lambda_n^{-1}\phi_{n},\phi_{n})$ $\}_{n=1}^\infty$ of $\mathbb{A}$, where $\phi_{n}=\sinh\lambda_n(x-1).$ Denote by $(\lambda_n^{-1}f_{1n},f_{1n},a f'_{1n}(1)-a \phi'_{n}(1)+m\lambda_{n} f_{1n}(1),$ $\lambda_n^{-1}\phi_{n},\phi_{n})$ the eigenfunction of $\mathcal{A}_1$ corresponding to $\lambda_{n}$. Then we have $f''_{1n}(x)=\lambda_nf_{1n}(x),
f_{1n}(0)=0,
(1+a\lambda_n)f'_{1n}(1)+(m\lambda^2_n+\alpha\lambda_n)f_{1n}=(a\lambda_n+1)\phi'_n(1)$ whose solution is of the form \begin{align*}
f_{1n}(x)=F\sinh\lambda_n(x-1)+G\cosh\lambda_n(x-1). \end{align*} Combine (\ref{f1n}) and the boundary conditions to derive $F=[(1+a\lambda_n)\cosh\lambda_n]/h_n,
G=[(1+a\lambda_n)\sinh\lambda_n]/h_n,$ where $h_n=(1+a\lambda_n)\cosh\lambda_n+(m\lambda_n+\alpha)\sinh\lambda_n.$ Use the fact that $\lambda_n\cosh\lambda_n=-(\gamma\lambda_n+\beta)\sinh\lambda_n$ to get
$F=\frac{a\gamma}{a\gamma -m}+O(|n|^{-1})$
and $G=-\frac{a}{a\gamma -m}+O(|n|^{-1}).$ Denote $Q=\left(
\begin{array}{cc}
I_3 & J \\
0 & I_2 \\
\end{array}
\right) $ with $J=\frac{a}{a\gamma-m}\left(
\begin{array}{ccccc}
\gamma & -1& 0 \\
-1 &\gamma& 0 \\
0 &0&0 \\
\end{array}
\right)^T.$ Then $Q$ is a bounded linear operator and it has bounded inverse. Moreover, we obtain the following relations \begin{align} \nonumber &(\lambda_{1n}^{-1}f'_n,f_n,a f'_n(1)+m\lambda_{1n} f_n(1),0,0)^T\\ \label{guanxi3}&=Q(\lambda_{1n}^{-1}f'_n,f_n,a f'_n(1)+m\lambda_{1n} f_n(1),0,0)^T,\\
\nonumber &(\lambda_n^{-1}f'_{1n},f_{1n},a f'_{1n}(1)-a \phi'_{n}(1)+m\lambda_{n} f_{1n}(1),\lambda_n^{-1}\phi'_{n},\\ \label{guanxi4}&\phi_{n})=Q(0,0,0,\lambda_n^{-1}\phi'_{n},\phi_{n})+O(n^{-1}). \end{align} Then, by Bari's theorem the sequence $\{(\lambda_{1n}^{-1}f'_n,f_n,$ $a f'_n(1)+m\lambda_{1n} f_n(1),0,0)\}_{n=-\infty}^\infty \bigcup \{(\lambda_n^{-1}f'_{1n},f_{1n},$ $a f'_{1n}(1)-a \phi'_{n}(1)+m\lambda_{n} f_{1n}(1),\lambda_n^{-1}\phi'_{n},\phi_{n})\}_{n=-\infty}^\infty$ forms Riesz basis for $\left(L^2(0,1)\right)^2\times \mathds{C}\times \left(L^2(0,1)\right)^2$, which is equivalent to that $\{(\lambda_{1n}^{-1}f_n,f_n,a f'_n(1)+m\lambda_{1n} f_n(1),0,0)\}_{n=-\infty}^\infty \bigcup$ $ \{(\lambda_n^{-1}f'_{1n},f_{1n},a f_{1n}(1)-a \phi'_{n}(1)+m\lambda_{n} f_{1n}(1),\lambda_n^{-1}\phi_{n},\phi_{n})$ $\}_{n=-\infty}^\infty$ forms Riesz basis for $\mathcal{X}_1$.
The semigroup generation and spectrum-determined growth condition of $\mathcal{A}_1$ are directly derived by the Riesz basis property. Since $ \mathbf{A}$ and $\mathbb{A}$ generate exponentially stable $C_0$-semigroups and spectrum-determined growth condition holds, $e^{\mathcal{A}_1t}$ is exponentially stable. \end{proof}
\begin{remark} \em In the proof of Theorem \ref{exponential00}, although it forms a Riesz basis for $\mathcal{H}_1$, {\color{blue} not all the elements of the sequence $\{(0,0,0,\lambda_n^{-1}\phi_{n},$ $\phi_{n})\}_{n=-\infty}^\infty\bigcup \{(\lambda_{1n}^{-1}f_{n},f_{n},a f_n'(1)+m\lambda_{1n}f_n(1),0,0)\}_{n=-\infty}^\infty$ are generalized eigenfunctions of $\mathcal{A}$}. In order to overcome this difficulty, our key step is to find out the relations (\ref{guanxi3}) and (\ref{guanxi4}). \end{remark}
We consider the closed-loop system (\ref{perror110}) in Hilbert state space $\mathfrak{X}=\mathbf{H}_1\times \mathbf{H}_2\times \mathbb{H}_1$, where $\mathbb{H}_1=H^1(0,1)\times L^1(0,1)$ with norm {\color{blue}$\|(f,g)\|^2_{\mathbb{H}_1}=\int_0^1[|f'(x)|^2+|g(x)|^2]dx+\beta|f(0)|^2$}. The state of the closed-loop system (\ref{perror110}) is $(u(\cdot,t),u_t(\cdot,t),mu_t(1,t)-mq_t(1,t)+a(v_x(1,t)-q_x(1,t)),v(\cdot,t),v_t(\cdot,t), mv_t(1,t)-mq_t(1,t)+a[v_x(1,t)-q_x(1,t)],q(\cdot,t),q_t(\cdot,t))$.
\begin{theorem}\label{maintheorem} Suppose that $d\in L^\infty(0,\infty)$ (or $d\in L^2(0,\infty)$) and $f:H^1(0,1)\times L^2(0,1)\rightarrow \mathds{R}$ is continuous. For any initial value $(u(\cdot,0),u_t(\cdot,0),mu_t(1,0)-mq_t(1,0)+a(v_x(1,0)-q_x(1,0)),v(\cdot,0),v_t(\cdot,$ $0),mv_t(1,0)-mq_t(1,0)+a[v_x(1,0)-q_x(1,0)],q(\cdot,0),q_t(\cdot,0)) \in \mathfrak{X}$ with $q(1,0)=v(1,0)-u(1,0)$, then there exists a unique solution to system (\ref{perror110}) such that $(u(\cdot,t),u_t(\cdot,t),mu_t(1,t)$ $-mq_t(1,t)+a(v_x(1,t)-q_x(1,t)),v(\cdot,t),v_t(\cdot,t), mv_t(1,t)-mq_t(1,t)+a[v_x(1,t)-q_x(1,t)],q(\cdot,t),q_t(\cdot,t)) \in C(0,\infty;$ $\mathfrak{X})$ satisfy $q(1,t)=v(1,t)-u(1,t)$, \begin{align}
\nonumber&\int_0^1[|u_t(x,t)|^2+|u_{x}(x,t)|^2]{\color{blue} dx}+\frac{1}{m}|mu_t(1,t)\\
\label{P1}&+a[v_x(1,t)-q_x(1,t)]|^2\leq M_2e^{-\gamma_2 t},\\
\nonumber&\sup_{t\geq 0}\int_0^1[|u_t(x,t)|^2+|u_{x}(x,t)|^2]{\color{blue} dx}+\frac{1}{m}|mu_t(1,t)\\
\label{P0} &-mq_t(1,t)+a[v_x(1,t)-q_x(1,t)]|^2<\infty,\\
\nonumber &\sup_{t\geq 0} \bigg[\int_0^1[|v_t(x,t)|^2+|v_{x}(x,t)|^2+|q_t(x,t)|^2+|q_{x}(x,\\
\nonumber&t)|^2]dx+\beta(|v(0,t)|^2+|q(0,t)|^2)+|mv_t(1,t)-mq_t(1,\\
\label{P2}&t)+a[v_x(1,t)-q_x(1,t)]|^2)\bigg]<\infty, \end{align} where $M_2$ and $\gamma_2$ are two positive constants.
If $f(0,0)=0$ and $d\in L^2(0,\infty)$, then
$\lim_{t\rightarrow +\infty} \int_0^1(|u_t(x,$
$t)|^2+|u_{x}(x,t)|^2+\frac{1}{m}|mu_t(1,t)-$
$mq_t(1,t)+a(v_x(1,t)-q_x(1,t))|^2=0,
\lim_{t\rightarrow +\infty} \big[\int_0^1(|v_t(x,$
$t)|^2+|v_{x}(x,t)|^2+|q_t(x,t)|^2
+|q_{x}(x,t)|^2)dx+\beta(v(0,t)|^2$
$+|q(0,t)|^2)+|mv_t(1,t)
-mq_t(1,t)+a[v_x(1,t)-q_x(1,t)|^2)\big]$ $=0.$ \end{theorem} \begin{proof}\ \ Given initial value $(u(\cdot,0),u_t(\cdot,0),mu_t(1,0)$ $-mq_t(1,0)+a(v_x(1,0)-q_x(1,0)),v(\cdot,0),v_t(\cdot,0),$ $mv_t(1,0)-mq_t(1,0)+a[v_x(1,0)-q_x(1,0)],q(\cdot,0),q_t(\cdot,0)) \in \mathbf{H}_1\times \mathbf{H}_2\times \mathbb{H}_1$ with $q(1,0)=v(1,0)-u(1,0)$, we have $(u(\cdot,0),u_t(\cdot,0),mu_t(1,0)+a[u_t(1,0)-\widehat{q}_t(1,0)],\widehat{q}(1,0),$ $\widehat{q}_t(1,0))\in \mathcal{X}_1$ and $\big(\widehat{v}(\cdot,0),\widehat{v}_t(\cdot,$ $0),m\widehat{v}_t(1,0)\big)\in \mathbf{H}_2$. By Theorem \ref{exponential00}, we obtain $(u(\cdot,t),u_t(\cdot,t),mu_t(1,t)+a[u_t(1,t)-\widehat{q}_t(1,t)],\widehat{q}(1,t),$ $\widehat{q}_t(1,t))\in C(0,\infty;\mathcal{X}_1), q(1,t)$ $=v(1,t)-u(1,t)$ and \begin{align*}
&\int_0^1(|u_t(x,t)|^2+|u_{x}(x,t)|^2)dx+|mu_t(1,t)+a(v_x(1,t)\\
&-q_x(1,t))|^2=\int_0^1(|u_t(x,t)|^2+|u_{x}(x,t)|^2)dx\\
&+|mu_t(1,t)+a(u_x(1,t)-\widehat{q}_x(1,t))|^2\\
&\leq M^2_{\mathcal{A}_1}e^{-2\omega_{\mathcal{A}_1}t}\|(u(\cdot,0),u_t(\cdot,0),mu_t(1,0)\\
&+a(u_x(1,0)-\widehat{q}_x(1,0)),\widehat{q}(\cdot,0),\widehat{q}_t(\cdot,0)) \|^2_{\mathcal{X}_1}= M^2e^{-\gamma_2 t}, \end{align*} where $\gamma_2=2\omega_{\mathcal{A}_1}$ and $M_2=M^2_{\mathcal{A}_1}
\|(u(\cdot,0),u_t(\cdot,0),mu_t(1,$
$0)+a(u_x(1,0)-\widehat{q}_x(1,0)),\widehat{q}(\cdot,0),\widehat{q}_t(\cdot,0)) \|^2_{\mathcal{X}_1}$
$=M^2_{\mathcal{A}_1}\|(u(\cdot,$
$0),u_t(\cdot,0),mu_t(1,0)+a(v_x(1,0)-q_x(1,0)),q(\cdot,0)-v(\cdot,0)+u(\cdot,0),q_t(\cdot,0)-v_t(\cdot,0)+u_t(\cdot,0)) \|^2_{\mathcal{X}_1}.$
Therefore, we derive the continuity and exponential stability of $(u(\cdot,t),u_t(\cdot,t))$ on $H^1(0,1)\times L^2(0,1)$. This, together with the continuity of $f$ indicates that $f(w(\cdot,t),w_t(\cdot,t))\in L^\infty(0,\infty)$. It follows from Lemma \ref{admissible} that $\big(\widehat{v}(\cdot,t),\widehat{v}_t(\cdot,t),m\widehat{v}_t(1,t)\big)\in C(0,\infty;\mathbf{H}_2)$
and $\sup_{t\geq 0}\|\big(\widehat{v}(\cdot,t),\widehat{v}_t(\cdot,t),m\widehat{v}_t(1,$
$t)\big)\|_{\mathbf{H}_2}<+\infty$. Hence, $mu_t(1,t)-mq_t(1,t)+a(v_x(1,t)-q_x(1,t)) =mu_t(1,t)$ $+a[u_t(1,t)-q_t(1,t)]-m\widehat{v}_t(1,t)\in C(0,\infty,\mathds{C}),$
$mv_t(1,t)$
$-mq_t(1,t)+a(v_x(1,t)-q_x(1,t))=mu_t(1,t)+a(v_x(1,t)-q_x(1,t))\in C(0,\infty,\mathds{C}),$
$\sup_{t\geq 0}|mu_t(1,t)-$
$mq_t(1,t)+a(v_x(1,t)-q_x(1,t))|\leq \sup_{t\geq 0}|mu_t(1,t)+a[u_t(1,t)-\widehat{q}_t(1,t)]|$
$+\sup_{t\geq 0}|m\widehat{v}_t(1,t)|<\infty,$
$\sup_{t\geq 0}| m$
$v_t(1,t)-mq_t(1,t)+a(v_x(1,t)-q_x(1,t))|=\sup_{t\geq 0}|mu_t(1,$
$t)+a(v_x(1,t)-q_x(1,t))|<\infty,$ \begin{align*}
& \int_0^1[|v_t(x,t)|^2+|v_{x}(x,t)|^2]dx+\beta|v(0,t)|^2\leq 2\int_0^1[|\widehat{v}_t(x,\\
&t)|^2+|\widehat{v}_{x}(x,t)|^2]dx+\beta|\widehat{v}(0,t)|^2+2\int_0^1[|u_t(x,t)|^2\\
&+|u_{x}(x,t)|^2]dx\leq 2\sup_{t\geq 0}\bigg[\|\big(\widehat{v}(\cdot,t),\widehat{v}_t(\cdot,t),m\widehat{v}_t(1,t)\big)\|^2_{\mathbf{H}_2} \\
&+2\|(u(\cdot,t),u_t(\cdot,t),mu_t(1,t)+a(u_x(1,t)-\widehat{q}_x(1,t)),\\
&\widehat{q}(\cdot,t),\widehat{q}_t(\cdot,t)) \|^2_{\mathcal{X}_1}\bigg]<\infty,\\
& \int_0^1[|q_t(x,t)|^2+|q_{x}(x,t)|^2]dx+\beta|q(0,t)|^2\leq 2\int_0^1[|\widehat{q}_t(x,\\
&t)|^2+|\widehat{q}_{x}(x,t)|^2]dx+2\beta|\widehat{q}(0,t)|^2+2\int_0^1[|\widehat{v}_t(x,t)|^2\\
&+|\widehat{v}_{xx}(x,t)|^2]dx+2\beta|\widehat{v}(0,t)|^2\leq 2\int_0^1[|\widehat{q}_t(x,t)|^2\\
&+|\widehat{q}_{x}(x,t)|^2]dx+2\beta\int_0^1|\widehat{q}_{x}(x,t)|^2dx+2\int_0^1[|\widehat{v}_t(x,t)|^2\\
&+|\widehat{v}_{xx}(x,t)|^2]dx+2\beta|\widehat{v}(0,t)|^2\leq 2(1\\
&+\beta)M^2_{\mathbb{A}}e^{-2\omega_\mathbb{A}}\|(\widehat{q}(\cdot,t),\widehat{q}_t(\cdot,t))\|^2_{\mathbb{H}}+2\int_0^1[|\widehat{v}_t(x,t)|^2\\
&+|\widehat{v}_{xx}(x,t)|^2]dx+2\beta|\widehat{v}(0,t)|^2<\infty, \end{align*}
where the fact $|\widehat{q}(0,t)|^2=\big|\widehat{q}(1,t)-\int_0^1\widehat{q}_x(x,t)dx\big|^2\leq \int_0^1|\widehat{q}_{x}(x,t)|^2dx$ is used. Then (\ref{P0}) is derived, system (\ref{perror110}) has a unique solution and $(u(\cdot,t),u_t(\cdot,t),mu_t(1,t)-mq_t(1,t)+a(v_x(1,t)-q_x(1,t)),v(\cdot,t),v_t(\cdot,t), mv_t(1,t)-mq_t(1,t)+a[v_x(1,t)-q_x(1,t)],q(\cdot,t),q_t(\cdot,t)) \in C(0,\infty;$ $\mathfrak{X})$ and (\ref{P2}) is derived.
If $f(0,0)=0$, we derive $\lim_{t\rightarrow \infty}f(w(\cdot,t),w_t(\cdot,t))=0$ by the continuity of $f$ and the exponential stability of $(w(\cdot,t),w_t(\cdot,t))$. Moreover, we use the assumption $d\in L^2(0,\infty)$ and \cite[Lemma A.1]{Zhou2018a} to get
$\lim_{t\rightarrow\infty}\|(\widehat{v}(\cdot,t),\widehat{v}_t(\cdot,t),m\widehat{v}_t(1,t))\|_{\mathbb{H}_1}=0.$ Then, the two limits are verified. This completes the proof. \end{proof}
\begin{remark} \em The subsystem consisting of $(u,\widehat{q})$-part of (\ref{wzwanclosed}) is vital to prove Theorem \ref{maintheorem}. Indeed, by virtue of semigroup theorem, we can derive the continuity and exponential stability of the solution of (\ref{beem}) without the global Lipsichitz condition. Then we obtain that $(u(\cdot,t),u_t(\cdot,t))$ is continuous and bounded on $H^1_E(0,1)\times L^2(0,1)$. By Lemma \ref{admissible}, the existence and boundedness of $(\widehat{v}(\cdot,t),\widehat{v}_t(\cdot,t),m\widehat{v}_t(1,t))$ are verified by viewing $f(w(\cdot,t),w_t(\cdot,t))+d(t)$ as the boundary input, provided $f$ is continuous. Furthermore, we not only derive the exponential stability of $(u(\cdot,t),u_t(\cdot,t))$ but also verify the exponential stability of $\eta(t)=mu_t(1,t)+a(v_t(1,t)-q_t(1,t))$, because $mu_t(1,t)+a(v_t(1,t)-q_t(1,t))$ is the state of boundary dynamic of the exponentially stable system (\ref{perror110closed}). Theorem \ref{maintheorem} also tells us that the other states (including the state of the boundary dynamic) are bounded. \end{remark}
\begin{remark} \em In Theorem \ref{maintheorem}, compared to the results in \cite{Xie2017}, the differences and improvements mainly lie in that, 1) the internal uncertainty is taken into consideration, while \cite{Xie2017} just studies the case of $f(w,w_t)=0$, 2) only ``low order'' measurements $w(1,t)$ and $w_{x}(0,t)$ are adopted, while \cite{Xie2017} used the velocity $w_{t}(1,t)$ as well as the high order angular velocity $w_{xt}(1,t)$, 3) we consider arbitrary $m\neq a,m\neq a\gamma,\gamma\neq1$ while \cite{Xie2017} just solved the special case $m=a\alpha$. \end{remark}
\section{Numerical simulation}\label{shiyan}
We present in this section some numerical simulations for the closed-loop system (\ref{perror110}). We use the finite difference scheme and the numerical results are programmed in Matlab. The time step and the space step are taken as $1/200$ and $1/100$, respectively. We take the nonlinear internal uncertainty $f(u(\cdot,t),u_t(\cdot,t)) = \sin(u(1,t))$ and the external disturbance $d(t)=\cos(2t)$. Let $m=5$. The parameters and the initial values are chosen as $\alpha=a=2,\beta=\gamma=1.5, u(x,0)=x^3-3x^2, u_t(x,0)=0, v(x,0)=-2x^3, v_t(x,0)=0, q(x,0)=q_t(x,0)=0.$
Fig.1, Fig.3 and Fig.5 show the displacements $u(x,t)$, $v(x,t)$ and $q(x,t)$ of the closed-loop system (\ref{perror110}), respectively; while Fig.2, Fig 4 and Fig 6 respectively present the velocities $u_t(x,t)$, $v_t(x,t)$ and $q_t(x,t)$. Fig.7 and Fig.8 display the boundary sates $\eta(t)=mu_t(1,t)+a[v_x(1,t)-q_x(1,t)]$ and $\psi(t)=mu_t(1,t)-mq_t(1,t)+a[v_x(1,t)-q_x(1,t)]$. One can see from the simulations that $(u(\cdot,t),u_t(\cdot,t))$ and $\eta(t)$ decays rapidly, while $(v(\cdot,t),v_t(\cdot,t))$, $(q(\cdot,t),q_t(\cdot,t))$ and $\psi(t)$ are bounded.
\begin{figure}
\caption{The state $u(x,t)$.}
\label{1}
\caption{The state $u_t(x,t)$.}
\label{2}
\end{figure}
\begin{figure}
\caption{The state $v(x,t)$.}
\label{3}
\caption{The state $v_t(x,t)$.}
\label{4}
\end{figure}
\begin{figure}
\caption{The state $q(x,t)$.}
\label{5}
\caption{The state $q_t(x,t)$.}
\label{6}
\end{figure}
\begin{figure}
\caption{The state $\eta(x,t)$}
\label{7}
\caption{The state $\psi(x,t)$}
\label{8}
\end{figure}
\section{Concluding remarks}
In this paper, the output feedback exponential stabilization for a 1-d wave PDE with boundary and with or without disturbance is investigated. When there is no disturbance, with only one non-collocated measurement $w_x(0,t)$, we design a Luenberger state observer and an estimated state based stabilizing controller. This improves the existence references \cite{Guo2000,Morgul1994} where the authors used two collocated measurements $w_t(1,t)$ and $w_{xt}(1,t)$. By modifying the proof, we can simplify the proof of the exponential stability of the closed-loop system \cite[(3.1)]{Guo2007} where $m=0$, because our coupled system (\ref{closednodisturbance1}) contains an independent subsystem. When the boundary internal uncertainty and external disturbance are considered, we construct an infinite-dimensional ESO to estimate the original state and total disturbance online. Then, the estimated state and estimated total disturbance allows us to design a stabilizing controller while guaranteeing the boundedness of the closed-loop system. Riesz basis approach is the main tool for the proofs of the exponential stabilities of two coupled systems.
\end{document} all folks
\end{document} | arXiv |
\begin{document}
\title{Generalized Jacquet modules of parabolic induction} \begin{abstract} In this paper we study a generalization of the Jacquet module of a parabolic induction and construct a filtration on it. The successive quotient of the filtration is written by using the twisting functor. \end{abstract}
\section{Introduction} The Jacquet module of a representation of a semisimple (or reductive) Lie group is introduced by Casselman~\cite{MR562655}. One of the motivation of considering the Jacquet module is to investigate homomorphisms to principal series representations, which is an important invariant of a representation.
One of the powerful tools to study the Jacquet module of a parabolic induction is the Bruhat filtration~\cite{MR1767896}. This is a filtration on the Jacquet module defined from the Bruhat decomposition. Casselman-Hecht-Milicic~\cite{MR1767896} use the Bruhat filtration to determine the dimension of the (moderate-growth) Whittaker model of a principal series representation (another proof for Kostant's result). In this paper, we study the Bruhat filtration and show that the successive quotient is described using the twisting functor defined by Arkhipov~\cite{MR2074588}. If a principal series representation has the unique Langlands quotient, then the successive quotient is the induction from the Jacquet module of a smaller group (a Levi part of a parabolic subgroup). However, in a general case, it becomes ``twisted'' induction, which has the same character as that of an induced representation but has a different module structure.
Moreover, we investigate its generalization, this is related to the Whittaker model. In \cite{MR562655}, Casselman suggested to generalize the notion of the Jacquet module. For this generalized Jacquet module, we can also define a Bruhat filtration and the successive quotient of the resulting filtration is described in terms of the generalized twisting functor.
This result gives a strategy to determine all Whittaker models of a parabolic induction. To determine it, it is sufficient to study the successive quotients and extensions of the filtration. In a special case, we can carry out these steps.
Now we state our results precisely. Let $G$ be a connected semisimple Lie group, $G = KA_0N_0$ an Iwasawa decomposition and $P_0 = M_0A_0N_0$ a minimal parabolic subgroup and its Langlands decomposition. As usual, the complexification of the Lie algebra is denoted by the corresponding German letter (for example, $\mathfrak{g} = \Lie(G)\otimes_\mathbb{R}\mathbb{C}$). Fix a character $\eta$ of $N_0$. Then for a representation $V$ of $G$, the generalized Jacquet modules $J'_\eta(V)$ and $J^*_\eta(V)$ are defined as follows.
\begin{defn}\label{defn:Jacquet modules in Introduction} Let $V$ be a finite-length moderate growth Fr\'echet representation of $G$ (See Casselman~\cite[pp.~391]{MR1013462}). We define $\mathfrak{g}$-modules $J'_\eta(V)$ and $J^*_\eta(V)$ by \begin{align*}
J'_\eta(V) & = \left\{v\in V'\Bigm| \begin{array}{l} \text{For some $k$ and for all $X\in\mathfrak{n}_0$,}\\ \text{$(X - \eta(X))^kv = 0$} \end{array} \right\},\\
J^*_\eta(V) & = \left\{v\in (V_{\textnormal{$K$-finite}})^*\Bigm| \begin{array}{l} \text{For some $k$ and for all $X\in\mathfrak{n}_0$,}\\ \text{$(X - \eta(X))^kv = 0$} \end{array} \right\}, \end{align*} where $V'$ is the continuous dual of $V$. \end{defn}
Let $W$ be the little Weyl group of $G$ and take $w\in W$. Then the generalized twisting functor $T_{w,\eta}$ is defined as follows. Let $\overline{\mathfrak{n}_0}$ be the nilradical of the opposite parabolic subalgebra to $\mathfrak{p}_0$ and $e_1,\dots,e_l$ be a basis of $\Ad(w)\overline{\mathfrak{n}}_0\cap \mathfrak{n}_0$ such that each $e_i$ is a root vector with respect to $\mathfrak{h}$ where $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$ which contains $\mathfrak{a}_0$. Moreover, we choose $e_i$ such that $\bigoplus_{i\le j - 1}\mathbb{C} e_i$ is an ideal of $\bigoplus_{i\le j}\mathbb{C} e_i$ for all $j$. Let $U(\mathfrak{g})$ be the universal enveloping algebra of $\mathfrak{g}$ and $U(\mathfrak{g})_{e_i - \eta(e_i)}$ the localization of $U(\mathfrak{g})$ by a multiplicative set $\{(e_i - \eta(e_i))^n\mid n\in\mathbb{Z}_{>0}\}$. Put $S_{w,\eta} = (U(\mathfrak{g})_{e_i - \eta(e_i)}/U(\mathfrak{g}))\otimes_{U(\mathfrak{g})}\dotsb\otimes_{U(\mathfrak{g})}(U(\mathfrak{g})_{e_l - \eta(e_l)}/U(\mathfrak{g}))$. Then $S_{w,\eta}$ is a $\mathfrak{g}$-bimodule. The twisting functor $T_{w,\eta}$ is defined by $T_{w,\eta}V = S_{w,\eta}\otimes_{U(\mathfrak{g})}(wV)$ where $wV$ is a representation twisted by $w$ (i.e., $Xv = \Ad(w)^{-1}(X)\cdot v$ for $X\in \mathfrak{g}$ and $v\in wV$ where dot means the original action).
Let $P$ be a parabolic subgroup containing $A_0N_0$ and take a Langlands decomposition $P = MAN$ such that $A_0\supset A$. Define $\rho_0\in\mathfrak{a}_0^*$ by $\rho_0(H) = (1/2)\Tr \ad(H)|_{\mathfrak{n}_0}$. Let $\rho$ be a restriction of $\rho_0$ on $\mathfrak{a}$. An element of $\mathfrak{a}^*$ corresponds to a character of $A$. We denote the corresponding character to $\lambda + \rho$ by $e^{\lambda + \rho}$ for $\lambda\in\mathfrak{a}^*$. Then for an irreducible representation $\sigma$ of $M$ and $\lambda\in\mathfrak{a}^*$, the parabolic induction $\Ind_P^G(\sigma\otimes e^{\lambda + \rho})$ is defined. Let $W_M$ be the little Weyl group of $M$. Define a subset $W(M)$ of $W$ by $W(M) = \{w\in W\mid \text{for all positive restricted root $\alpha$ of $M$, $w(\alpha)$ is positive}\}$. Then $W(M)$ is a complete representatives of $W/W_M$ and parameterizes $N_0$-orbits in $G/P$. For $w\in W$, fix a lift in $G$ and denote it by the same letter $w$. Enumerate $W(M) = \{w_1,\dots,w_r\}$ so that $\bigcup_{j\le i}N_0w_jP/P$ is a closed subset of $G/P$. Using the $C^\infty$-realization of a parabolic induction, we can regard an element of $J'_\eta(\Ind_P^G(\sigma\otimes e^{\lambda + \rho}))$ as a distribution on $G/P$. Then the Bruhat filtration $I_i\subset J'_\eta(\Ind_P^G(\sigma\otimes e^{\lambda + \rho}))$ is defined by \[
I_i = \left\{x\in J'_\eta(\Ind_P^G(\sigma\otimes e^{\lambda + \rho}))\biggm| \supp x \subset \bigcup_{j\le i}N_0w_jP\right\}. \] Since $w_i\in W(M)$, we have $\Ad(w_i)(\mathfrak{m}\cap \mathfrak{n}_0)\subset \mathfrak{n}_0$. Hence we can define a character $w_i^{-1}\eta$ of $\mathfrak{m}\cap \mathfrak{n}_0$ by $(w_i^{-1}\eta)(X) = \eta(\Ad(w_i)X)$. Using this character, we can define an $\mathfrak{m}\oplus\mathfrak{a}$-module $J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda + \rho})$. Then we have the following theorem. \begin{thm}[Theorem~\ref{thm:succ quot is I'_i}, Theorem~\ref{thm:structure of I_i/I_{i - 1}}]\label{thm:main theorem} The filtration $\{I_i\}$ has the following properties. \begin{enumerate} \item If the character $\eta$ is not unitary, then $J'_\eta(\Ind_P^G(\sigma\otimes e^{\lambda+\rho})) = 0$. \item Assume that $\eta$ is unitary. The module $I_i/I_{i - 1}$ is nonzero if and only if $\eta$ is trivial on $w_iNw_i^{-1}\cap N_0$ and $J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho})\ne 0$. \item If $I_i/I_{i - 1} \ne 0$ then $I_i/I_{i - 1} \simeq T_{w_i,\eta}(U(\mathfrak{g})\otimes_{U(\mathfrak{p})}J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho}))$ where $\mathfrak{n}$ acts $J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho})$ trivially. \end{enumerate} \end{thm}
Under the assumptions that $P$ is a minimal parabolic subgroup, $\sigma$ is the trivial representation, $\Ind_P^G(\sigma\otimes e^{\lambda + \rho})$ has the unique Langlands quotient and $\eta$ is the trivial representation, this theorem is proved in \cite{abe-2006}. The proof we give in \cite{abe-2006} is algebraic, while we give an analytic and geometric proof in this paper.
For a module $J^*_\eta(\Ind_P^G(\sigma\otimes e^{\lambda + \rho}))$, we have the following theorem. We define two functors. For a $U(\mathfrak{g})$-module $V$, put $C(V) = ((V^*)_{\text{$\mathfrak{h}$-finite}})^*$ and $\Gamma_\eta(V) = \{v\in V\mid \text{for some $k$ and for all $X\in\mathfrak{n}_0$, $(X - \eta(X))^kv = 0$}\}$. \begin{thm}[Theorem~\ref{thm:stucture of J^*(I(sigma,lambda))}]\label{thm:main theorem2} There exists a filtration $0 = \widetilde{I_0}\subset \widetilde{I_1}\subset\cdots\subset\widetilde{I_r} = J^*_\eta(\Ind_P^G(\sigma\otimes\lambda))$ such that $\widetilde{I_i}/\widetilde{I_{i - 1}}\simeq \Gamma_\eta(C(T_{w_i}(U(\mathfrak{g})\otimes_{U(\mathfrak{p})}J^*(\sigma\otimes e^{\lambda+\rho}))))$ where $\mathfrak{n}$ acts $J^*(\sigma\otimes e^{\lambda+\rho})$ trivially. \end{thm}
We state an application. The space of Whittaker vectors $\Wh_\eta(D)$ is defined by $\Wh_\eta(D) = \{x\in D\mid \text{$(X - \eta(X))x = 0$ for all $X\in \mathfrak{n}_0$}\}$ for a $U(\mathfrak{g})$-module $D$. If $V$ is a moderate-growth Fr\'echet representation of $G$, an element of $\Wh_\eta(V')$ corresponds to a moderate-growth homomorphism $V\to \Ind_{N_0}^G\eta$ and an element of $\Wh_\eta((V_{\text{$K$-finite}})^*)$ corresponds to an algebraic homomorphism $V_{\text{$K$-finite}}\to \Ind_{N_0}^G\eta$. In particular, when $\eta$ is the trivial representation, these correspond to homomorphisms to principal series representations.
Let $\Sigma$ (resp.\ $\Sigma_M$) be the restricted root system for $(G,A_0)$ (resp.\ $(M,M\cap A_0)$), $\Sigma^+$ a positive system of $\Sigma$ corresponding to $N_0$ and $\Pi\subset\Sigma$ the set of simple roots determined by $\Sigma^+$. Put $\Sigma_M^+ = \Sigma_M\cap \Sigma^+$. Let $\widetilde{W}$ (resp.\ $\widetilde{W_M}$) be the (complex) Weyl group of $\mathfrak{g}$ (resp.\ $\mathfrak{m}$). Let $\widetilde{\mu}\in\mathfrak{h}^*$ be the infinitesimal character of $\sigma$. Let $\Delta$ be the root system of $(\mathfrak{g},\mathfrak{h})$. Put $\Sigma_\eta^+ = (\sum_{\eta|_{\mathfrak{g}_\beta} \ne 0,\ \beta\in\Pi}\mathbb{Z}\beta)\cap \Sigma^+$. Fix a $W$-invariant bilinear form $\langle\cdot,\cdot\rangle$ of $\mathfrak{a}_0$. Using the direct decompositions $(\mathfrak{m}\cap\mathfrak{a}_0)^*\oplus \mathfrak{a}^* = \mathfrak{a}_0^*$ and $\mathfrak{a}_0^*\oplus(\mathfrak{h}\cap \mathfrak{m}_0)^* = \mathfrak{h}^*$, we regard $\mathfrak{a}^*\subset \mathfrak{a}_0^*\subset \mathfrak{h}^*$. Recall that $\nu\in(\mathfrak{m}\cap\mathfrak{a}_0)^*$ is called an exponent of $\sigma$ if $\nu + \rho_0|_{\mathfrak{m}\cap\mathfrak{a}_0}$ is an $(\mathfrak{m}\cap \mathfrak{a}_0)$-weight of $\sigma/(\mathfrak{m}\cap \mathfrak{n}_0)\sigma$. We prove the following theorem. \begin{thm}[Theorem~\ref{thm:dimension Whittaker vectors}, Theorem~\ref{thm:dimension Whittaker vectors, algebraic}]\label{thm:main theorem3, dimension of the Whittaker vectors} For $\lambda\in\mathfrak{a}^*$ and an irreducible representation $\sigma$ of $M$, the following formulae hold. \begin{enumerate}
\item Assume that for all $w\in W$ such that $\eta|_{wNw^{-1}\cap N_0} = 1$, the following two conditions hold: \begin{enumerate} \item For each exponent $\nu$ of $\sigma$ and $\alpha\in \Sigma^+\setminus w^{-1}(\Sigma^+_M\cup\Sigma_\eta^+)$, we have $2\langle\alpha,\lambda+\nu\rangle/\lvert\alpha\rvert^2\not\in\mathbb{Z}_{\le 0}$.
\item For all $\widetilde{w}\in\widetilde{W}$, we have $\lambda - \widetilde{w}(\lambda + \widetilde{\mu})|_\mathfrak{a}\notin \mathbb{Z}_{\le 0}((\Sigma^+\setminus \Sigma_M^+)\cap w^{-1}\Sigma^+)|_\mathfrak{a}\setminus\{0\}$. \end{enumerate} Then we have \begin{multline*}
\dim\Wh_\eta((\Ind_P^G(\sigma\otimes e^{\lambda+\rho}))')\\ = \sum_{w\in W(M),\ \eta|_{wNw^{-1}\cap N_0} = 1}\dim \Wh_{w^{-1}\eta}(\sigma'). \end{multline*} \item Assume that for all $\widetilde{w}\in\widetilde{W}\setminus\widetilde{W_M}$ we have $(\lambda + \widetilde{\mu}) - \widetilde{w}(\lambda + \widetilde{\mu}) \not\in\mathbb{Z}\Delta$. Then we have \begin{multline*}
\dim\Wh_\eta((\Ind_P^G(\sigma\otimes e^{\lambda+\rho})_{\text{\normalfont $K$-finite}})^*)\\ = \sum_{w\in W(M)}\dim \Wh_{w^{-1}\eta}((\sigma_{\text{\normalfont $M\cap K$-finite}})^*). \end{multline*} \end{enumerate} \end{thm} In the case that $\sigma$ is finite-dimensional, we have the following theorem, which have been announced by T. Oshima (cf.\ his talk at National University of Singapore, January 11, 2006). Let $\Delta_M$ be the root system for $(\mathfrak{m}\oplus\mathfrak{a},\mathfrak{h})$ and take a positive system $\Delta_M^+$ compatible with $\Sigma^+_M$. Put $\widetilde{\rho_M} = (1/2)\sum_{\alpha\in\Delta_M^+}\alpha$. For subsets $\Theta_1,\Theta_2$ of $\Pi$, put $\Sigma_{\Theta_i} = \mathbb{Z}\Theta_i\cap \Sigma$, $W(\Theta_i) = \{w\in W\mid w(\Theta_i)\subset \Sigma^+\}$, $W_{\Theta_i}$ the Weyl group of $\Sigma_{\Theta_i}$ and $W(\Theta_1,\Theta_2) = \{w\in W(\Theta_1)\cap W(\Theta_2)^{-1}\mid w(\Sigma_{\Theta_1})\cap \Sigma_{\Theta_2} = \emptyset\}$. The parabolic subgroup $P$ defines a subset of $\Pi$. We denote this set by $\Theta$. \begin{thm}\label{thm:main theorem4, dimension of the Whittaker vectors, finite-dimensional case} Assume that $\sigma$ is an irreducible finite-dimensional representation with highest weight $\widetilde{\nu}$. Let $\dim_M(\lambda+\widetilde{\nu})$ be the dimension of a finite-dimensional irreducible representation of $M_0A_0$ with highest weight $\lambda+\widetilde{\nu}$. \begin{enumerate}
\item Let $\widetilde{\nu}$ be the highest weight of $\sigma$. Assume that for all $w\in W$ such that $\eta|_{wN_0w^{-1}\cap N_0} = 1$ the following two conditions hold: \begin{enumerate} \item For all $\alpha\in \Sigma^+\setminus w^{-1}(\Sigma^+_M\cup\Sigma_\eta^+)$ we have $2\langle\alpha,\lambda+w_0\widetilde{\nu}\rangle/\lvert\alpha\rvert^2\not\in\mathbb{Z}_{\le 0}$.
\item For all $\widetilde{w}\in\widetilde{W}$ we have $\lambda - \widetilde{w}(\lambda + \widetilde{\nu} + \widetilde{\rho_M})|_\mathfrak{a}\notin \mathbb{Z}_{\le 0}((\Sigma^+\setminus \Sigma_M^+)\cap w^{-1}\Sigma^+)|_\mathfrak{a}\setminus\{0\}$. \end{enumerate} Then we have \[
\dim \Wh_\eta(I(\sigma,\lambda)') = \# W(\supp\eta,\Theta)\times(\dim_M(\lambda+\widetilde{\nu})) \] \item Assume that for all $\widetilde{w}\in\widetilde{W}\setminus\widetilde{W_M}$, $(\lambda+\widetilde{\nu}) - \widetilde{w}(\lambda+\widetilde{\nu}) \not\in\Delta$. Then we have \begin{multline*}
\dim\Wh_\eta((I(\sigma,\lambda)_{\text{\normalfont $K$-finite}})^*) \\= \# W(\supp\eta,\Theta)\times \#W_{\supp\eta}\times(\dim_M(\lambda+\widetilde{\nu})) \end{multline*} \end{enumerate} \end{thm}
We summarize the content of this paper. In \S\ref{sec:Parabolic induction and Bruhat filtration}, we introduce the Bruhat filtration. From \S\ref{sec:Parabolic induction and Bruhat filtration} to \S\ref{sec:The module I_i/I_i-1} we study the module $J'_\eta(\Ind_P^G(\sigma\otimes\lambda))$. In \S\ref{sec:vanishing theorem} we prove the successive quotient is zero under some conditions. The structure of the successive quotients is investigated in \S\ref{sec:Analytic continuation}. We give the definition and properties of the generalized twisting functor in \S\ref{sec:Twisting functors} and, in \S\ref{sec:The module I_i/I_i-1} we reveal the relation between the twisting functor and the successive quotient. We complete the proof of Theorem~\ref{thm:main theorem} in this section. Theorem~\ref{thm:main theorem2} is proved in \S\ref{sec:the module J^*_eta(I(sigma,lambda))}. In \S\ref{sec:Whittaker vectors}, the dimension of the space of Whittaker vectors is determined and Theorem~\ref{thm:main theorem3, dimension of the Whittaker vectors} and Theorem~\ref{thm:main theorem4, dimension of the Whittaker vectors, finite-dimensional case} are proved.
\subsection*{Acknowledgments} The author is grateful to his advisor Hisayosi Matumoto for his advice and support. He is supported by the Japan Society for the Promotion of Science Research Fellowships for Young Scientists.
\subsection*{List of Symbols} \listofsymbols
\subsection*{Notation} Throughout this paper we use the following notation. As usual we denote the ring of integers, the set of non-negative integers, the set of positive integers, the real number field and the complex number field by $\mathbb{Z},\mathbb{Z}_{\ge 0},\mathbb{Z}_{> 0},\mathbb{R}$ and $\mathbb{C}$, respectively. Let $G$ be a connected semisimple Lie group and $\mathfrak{g}$ the complexification of its Lie algebra. Fix a Cartan involution $\theta$ of $G$ and denote its derivation by the same letter $\theta$. Let $\mathfrak{g} = \mathfrak{k}\oplus \mathfrak{s}$ be the decomposition of $\mathfrak{g}$ into the $+1$ and $-1$ eigenspaces for $\theta$. Set $K = \{g\in G\mid \theta(g) = g\}$. Let $P_0 = M_0A_0N_0$ be a minimal parabolic subgroup and its Langlands decomposition such that $M_0\subset K$ and $\Lie(A_0)\subset \mathfrak{s}$. Denote the complexification of the Lie algebra of $P_0,M_0,A_0,N_0$ by $\mathfrak{p}_0,\mathfrak{m}_0,\mathfrak{a}_0,\mathfrak{n}_0$, respectively. Take a parabolic subgroup $P$ which contains $P_0$ and denote its Langlands decomposition by $P = MAN$. Here we assume $A\subset A_0$. Let $\mathfrak{p},\mathfrak{m},\mathfrak{a},\mathfrak{n}$ be the complexification of the Lie algebra of $P,M,A,N$. Put $\overline{P_0} = \theta(P_0)$, $\overline{N_0} = \theta(N_0)$, $\overline{P} = \theta(P)$, $\overline{N} = \theta(N)$, $\overline{\mathfrak{p}_0} = \theta(\mathfrak{p}_0)$, $\overline{\mathfrak{n}_0} = \theta(\mathfrak{n}_0)$, $\overline{\mathfrak{p}} = \theta(\mathfrak{p})$ and $\overline{\mathfrak{n}} = \theta(\mathfrak{n})$.
In general, we denote the dual space $\Hom_\mathbb{C}(V,\mathbb{C})$ of a $\mathbb{C}$-vector space $V$ by $V^*$. Let $\Sigma\subset\mathfrak{a}_0^*$ be the restricted root system for $(\mathfrak{g},\mathfrak{a}_0)$ and $\mathfrak{g}_\alpha$ the root space for $\alpha\in\Sigma$. Then $\sum_{\alpha\in\Sigma}\mathbb{R}\alpha$ is a real form of $\mathfrak{a}_0^*$. We denote the real part of $\lambda\in\mathfrak{a}_0^*$ with respect to this real form by $\re\lambda$ and the imaginary part by $\im\lambda$. Let $\Sigma^+$ be the positive system determined by $\mathfrak{n}_0$. Put $\rho_0 = \sum_{\alpha\in\Sigma^+}(\dim \mathfrak{g}_\alpha/2)\alpha$ and $\rho = \rho_0|_{\mathfrak{a}}$. The positive system $\Sigma^+$ determines the set of simple roots $\Pi$. Fix a totally order of $\sum_{\alpha\in\Sigma} \mathbb{R}\alpha$ such that the following conditions hold: (1) If $\alpha > \beta$ and $\gamma \in\sum_{\alpha\in\Sigma} \mathbb{R}\alpha$ then $\alpha + \gamma > \beta + \gamma$. (2) If $\alpha > 0$ and $c$ is a positive real number then $c\alpha > 0$. (3) For all $\alpha \in\Sigma^+$ we have $\alpha > 0$. Write $W$ for the little Weyl group for $(\mathfrak{g},\mathfrak{a}_0)$, $e$ for the unit element of $W$ and $w_0$ for the longest element of $W$. For $w\in W$, we fix a representative in $N_K(\mathfrak{a})$ and denote it by the same letter $w$.
Let $\mathfrak{t}_0$ be a Cartan subalgebra of $\mathfrak{m}_0$ and $T_0$ the corresponding Cartan subgroup of $M_0$. Then $\mathfrak{h} = \mathfrak{t}_0\oplus\mathfrak{a}_0$ is a Cartan subalgebra of $\mathfrak{g}$. Let $\Delta$ be the root system for $(\mathfrak{g},\mathfrak{h})$ and take a positive system $\Delta^+$ compatible with $\Sigma^+$, i.e., if $\alpha\in\Delta^+$ satisfies that $\alpha|_{\mathfrak{a}_0} \ne 0$ then $\alpha|_{\mathfrak{a}_0}\in \Sigma^+$. Let $\mathfrak{g}^\mathfrak{h}_\alpha$ be the root space of $\alpha\in\Delta$ and $\widetilde{W}$ the Weyl group of $\Delta$. Put $\widetilde{\rho} = (1/2)\sum_{\alpha\in\Delta^+}\alpha$. By the decompositions $(\mathfrak{m}\cap \mathfrak{a}_0)^*\oplus\mathfrak{a}^* = \mathfrak{a}_0^*$ and $\mathfrak{t}_0^*\oplus\mathfrak{a}_0^* = \mathfrak{h}^*$, we always regard $\mathfrak{a}^*\subset\mathfrak{a}_0^*\subset\mathfrak{h}^*$.
We use the same notation for $M$, i.e., $\Sigma_M$ be the restricted root system of $M$, $\Sigma_M^+ = \Sigma_M\cap \Sigma^+$, $W_M$ the little Weyl group of $M$, $\Delta_M$ the root system of $M$, $\Delta_M^+ = \Delta_M\cap \Delta^+$, $\widetilde{W_M}$ the Weyl group of $M$ and $w_{M,0}$ the longest element of $W_M$.
We can define an anti-isomorphism of $U(\mathfrak{g})$ by $X\mapsto -X$ for $X\in \mathfrak{g}$. We write this anti-isomorphism by $u\mapsto \check{u}$.
For a $\mathfrak{g}$-module $V$ and $g\in G$, we define a $\mathfrak{g}$-module $gV$ as follows: The representation space is $V$ and the action of $X\in\mathfrak{g}$ is $X\cdot v = (\Ad(g)^{-1}X)v$ for $v\in gV$.
For $\xi = (\xi_1,\dots,\xi_l)\in\mathbb{Z}^l$, put $\lvert\xi\rvert = \xi_1 + \dots + \xi_l$.
\section{Parabolic induction and the Bruhat filtration}\label{sec:Parabolic induction and Bruhat filtration}
Fix a character $\eta$ of $\mathfrak{n}_0$ and put $\supp_G \eta = \supp\eta = \{\alpha\in\Pi\mid \eta|_{\mathfrak{g}_\alpha} \ne 0\}$\newsym{$\supp_G\eta = \supp\eta$}. The character $\eta$ is called \emph{non-degenerate} if $\supp\eta = \Pi$. We denote the character of $N_0$ whose differential is $\eta$ by the same letter $\eta$. \begin{defn}\label{defn:Jacquet modules} Let $V$ be a finite-length moderate growth Fr\'echet representation of $G$ (See Casselman~\cite[pp.~391]{MR1013462}). We define $\mathfrak{g}$-modules $J'_\eta(V)$ and $J^*_\eta(V)$ by \begin{align*}
J'_\eta(V) & = \left\{v\in V'\Bigm| \begin{array}{l} \text{For some $k$ and for all $X\in\mathfrak{n}_0$,}\\ \text{$(X - \eta(X))^kv = 0$} \end{array} \right\},\\
J^*_\eta(V) & = \left\{v\in (V_{\textnormal{$K$-finite}})^*\Bigm| \begin{array}{l} \text{For some $k$ and for all $X\in\mathfrak{n}_0$,}\\ \text{$(X - \eta(X))^kv = 0$} \end{array} \right\}, \end{align*} where $V'$ is the continuous dual of $V$. \end{defn}
Put $J'(V) = J'_0(V)$ and $J^*(V) = J^*_0(V)$ where $0$ is the trivial representation of $\mathfrak{n}_0$. The module $J^*(V)$ is the (dual of) \emph{Jacquet module} defined by Casselman~\cite{MR562655}. By the automatic continuation theorem~\cite[Theorem~4.8]{MR727854}, we have $J'(V) = J^*(V)$. The correspondence $V\mapsto J'_\eta(V)$ and $V\mapsto J^*_\eta(V)$ are functors from the category of $G$-modules to the category of $\mathfrak{g}$-modules.
In this section, we study the module $J'_\eta(V)$ for a parabolic induction $V$. An element of $\mathfrak{a}^*$ is identified with a character of $A$. We denote the character of $A$ corresponding to $\lambda + \rho$ by $e^{\lambda + \rho}$ where $\lambda\in\mathfrak{a}^*$. For an irreducible moderate growth Fr\'echet representation $\sigma$ of $M$ and $\lambda\in\mathfrak{a}^*$, put \[
I(\sigma,\lambda) = C^\infty\mathchar`-\Ind_P^G(\sigma\otimes e^{\lambda + \rho}). \] (For a moderate growth Fr\'echet representation, see Casselman~\cite{MR1013462}.) The representation $I(\sigma,\lambda)$ has a natural structure of a moderate growth Fr\'echet representation. \newsym{$I(\sigma,\lambda)$} Denote its continuous dual by $I(\sigma,\lambda)'$.
Let $\mathcal{L}$ be a vector bundle on $G/P$ attached to the representation $\sigma\otimes e^{\lambda+\rho}$ and $\mathcal{L}'$ be the continuous dual vector bundle of $\mathcal{L}$. \newsym{$\mathcal{L}$} \begin{rem}\label{rem:identify func on flag and G} A $C^\infty$-section of $\mathcal{L}$ corresponds to a $\sigma$-valued $C^\infty$-function $f$ on $G$ such that $f(gman) = \sigma(m)^{-1}e^{-(\lambda+\rho)(\log a)}f(g)$ for $g\in G$, $m\in M$, $a\in A$, $n \in N$. In particular a $C^\infty$-function on $G/P$ corresponds to a right $P$-invariant $C^\infty$-function on $G$. We use this identification throughout this paper. \end{rem}
We use the notation in Appendix~\ref{sec:C^infty-function with values in Frechet space}. We can regard $J'_\eta(I(\sigma,\lambda))$ as a subspace of $\mathcal{D}'(G/P,\mathcal{L})$. Set $W(M) = \{w\in W\mid w(\Sigma_M^+)\subset \Sigma^+\}$.\newsym{$W(M)$}\newsym{$r$} Then it is known that the multiplication map $W(M)\times W_M\to W$ is bijective~\cite[Proposition~5.13]{MR0142696}. By the Bruhat decomposition, we have \[
G/P = \bigsqcup_{w\in W(M)}N_0wP/P. \] (Recall that we fix a representative of $w\in W$, see Notation.) Enumerate $W(M) = \{w_1,\dots,w_r\}$ so that $\bigcup_{j \le i}N_0w_jP/P$ is a closed subset of $G/P$ for each $i$. Then we can define a submodule $I_i$ of $J'_\eta(I(\sigma,\lambda))$ by \[
I_i = \left\{x\in J'_\eta(I(\sigma,\lambda))\Biggm| \supp x\subset \bigcup_{j \le i}N_0w_jP/P\right\}.\newsym{$I_i$} \] The filtration $\{I_i\}$ is called the Bruhat filtration~\cite{MR1767896}. In the rest of this section, we study the module $I_i/I_{i - 1}$. Put $U_i = w_i\overline{N}P/P$ and $O_i = N_0w_iP/P$. The subset $U_i$ is an open subset of $G/P$ containing $O_i$ and $U_i\cap O_j = \emptyset$ if $j < i$.\newsym{$U_i$}\newsym{$O_i$} Hence, the restriction map $\Res_i\colon I_i \to \mathcal{D}(U_i,\mathcal{L})$ induces an injective map $\Res_i\colon I_i/I_{i - 1}\to \mathcal{D}(U_i,\mathcal{L})$.\newsym{$\Res_i$}
Moreover, $\im\Res_i \subset \mathcal{T}_{O_i}(U_i,\mathcal{L})$. We have $\mathcal{T}_{O_i}(U_i,\mathcal{L}) = U(\Ad(w_i)\overline{\mathfrak{n}}\cap\overline{\mathfrak{n}})\otimes_\mathbb{C}\mathcal{T}(O_i,\mathcal{L}|_{O_i})$ by Proposition~\ref{prop:structure theorem of tempered distributions whose support is contained in submanifold}.
Notice that by a map $n\mapsto nw_iP/P$ we have isomorphisms $w_i\overline{N}w_i^{-1}\simeq U_i$ and $w_i\overline{N}w_i^{-1}\cap N_0\simeq O_i$. Fix a Haar measure on $w_i\overline{N}w_i^{-1}\cap N_0$. Then we can define $\delta_i\in \mathcal{D}'(O_i,\mathcal{L}|_{O_i})$ by \[
\langle\delta_i,\varphi\rangle = \int_{w_i\overline{N}w_i^{-1}\cap N_0}\varphi(nw_i)dn. \]
for $\varphi\in C_c^\infty(O_i,\mathcal{L}|_{O_i})$.\newsym{$\delta_i$} By the exponential map $\Ad(w_i)\Lie(\overline{N})\to w_i\overline{N}w_i^{-1}$ and diffeomorphism $w_i\overline{N}w_i^{-1}\simeq U_i$, $U_i$ has a vector space structure and $O_i$ is a subspace of $U_i$. Let $\mathcal{P}(O_i)$ be the ring of polynomials on $O_i$ (cf.\ Appendix~\ref{subsec:Distributions on a nilpotent Lie group} or \cite{MR1070979}).\newsym{$\mathcal{P}(O_i)$} Define a $C^\infty$-function $\eta_i$ on $O_i$ by $\eta_i(nw_iP/P) = \eta(n)$ for $n\in w_i\overline{N}w_i^{-1}\cap N_0$.\newsym{$\eta_i$} For a $C^\infty$-function $f$ on $O_i$ and $u'\in \sigma'$, we define $f\otimes u'\in C^\infty(O_i,\sigma')$ by $(f\otimes u')(x) = f(x)u'$. Since $w_i\in W(M)$, $\Ad(w_i)(\mathfrak{m}\cap \mathfrak{n}_0) \subset \mathfrak{n}_0$. Hence we can define a character $w_i^{-1}\eta$ of $\mathfrak{m}\cap\mathfrak{n}_0$ by $(w_i^{-1}\eta)(X) = \eta(\Ad(w_i)X)$. Using this character, we can define the Jacquet module $J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda + \rho})$ of $MA$-representation $\sigma\otimes e^{\lambda + \rho}$. This is an $\mathfrak{m}\oplus\mathfrak{a}$-module. Put \[
I'_i = \left\{\sum_{k = 1}^l T_k(((f_k\eta_i^{-1})\otimes u_k')\delta_i)\biggm| \begin{array}{ll} T_k\in U(\Ad(w_i)\overline{\mathfrak{n}}\cap \overline{\mathfrak{n}}),& f_k\in \mathcal{P}(O_i),\\ u_k'\in J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho}) \end{array} \right\}. \] The space $I'_i$ is a $U(\mathfrak{g})$-submodule of $\mathcal{D}'(U_i,\mathcal{L})$. Our aim is to prove that if $i$ satisfies some conditions then $I_i/I_{i - 1} \simeq I'_i$.
\begin{lem}\label{lem:bracket of n and bar_n} Let $E_1,\dots,E_n$ be a basis of $\Ad(w_i)\overline{\mathfrak{n}}\cap \overline{\mathfrak{n}_0}$ such that each $E_s$ is a restricted root vector for some root (say $\alpha_s$) and $F\in(\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0)\oplus\Ad(w_i)(\mathfrak{m}\cap\mathfrak{n}_0)$. (Notice that $\Ad(w_i)(\mathfrak{m}\cap \mathfrak{n}_0) = \Ad(w_i)\mathfrak{m}\cap \mathfrak{n}_0$ since $w_i\in W(M)$, so $(\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0)\oplus\Ad(w_i)(\mathfrak{m}\cap\mathfrak{n}_0) = \Ad(w_i)(\overline{\mathfrak{n}}\oplus\mathfrak{m})\cap \mathfrak{n}_0$ is a subalgebra of $\mathfrak{g}$.) For $\xi = (\xi_1,\xi_2,\dots,\xi_n)\in\mathbb{Z}_{\ge 0}^n$, set $E^\xi = E_1^{\xi_1}E_2^{\xi_2}\dotsm E_n^{\xi_n}$. Then for all $c\in\mathbb{C}$ we have \begin{multline*}
[(F - c)^k,E^\xi] \in \left(\sum_{\xi'\in A(\xi)}\mathbb{C} E^{\xi'}\right) U((\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0)\oplus\Ad(w_i)(\mathfrak{m}\cap\mathfrak{n}_0))
\\\subset U(\Ad(w_i)(\overline{\mathfrak{n}}\oplus(\mathfrak{m}\cap \mathfrak{n}_0))) \end{multline*} where $A(\xi) = \{\xi'\in\mathbb{Z}_{\ge 0}^n\mid \text{$\lvert\xi'\rvert < \lvert\xi\rvert$, or $\lvert\xi'\rvert = \lvert\xi\rvert$ and $\sum \xi'_i\alpha_i < \sum\xi_i\alpha_i$}\}$.\end{lem} \begin{proof} We may assume $k = 1$. We prove the lemma by induction on $\lvert\xi\rvert$. We have \[
[F - c,E^\xi] = [F,E^\xi] = \sum_{s = 1}^n\sum_{l = 0}^{\xi_s - 1}E_1^{\xi_1}\dotsm E_{s - 1}^{\xi_{s - 1}} E_s^l [F,E_s] E_s^{\xi_s - l - 1} E_{s + 1}^{\xi_{s + 1}}\dotsm E_n^{\xi_n}. \] Hence, it is sufficient to prove \begin{multline*}
E_1^{\xi_1}\dotsm E_{s - 1}^{\xi_{s - 1}} E_s^l [F,E_s] E_s^{\xi_s - l - 1} E_{s + 1}^{\xi_{s + 1}}\dotsm E_n^{\xi_n}\\
\in \left(\sum_{\xi'\in A(\xi)}\mathbb{C} E^{\xi'}\right) U((\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0)\oplus\Ad(w_i)(\mathfrak{m}\cap\mathfrak{n}_0)). \end{multline*} We may assume that $F$ is a restricted root vector. If $[F,E_s]\in \Ad(w_i)\overline{\mathfrak{n}}\cap \overline{\mathfrak{n}_0}$ then the claim hold. Assume that $[F,E_s]\in (\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0)\oplus\Ad(w_i)(\mathfrak{m}\cap\mathfrak{n}_0)$. Put $\xi^{(1)} = (\xi_1,\dots,\xi_{s - 1},l,0,\dots,0)\in\mathbb{Z}^n$ and $\xi^{(2)} = (0,\dots,0,\xi_s - l - 1,\xi_{s + 1},\dots,\xi_n)\in\mathbb{Z}^n$. Using inductive hypothesis, we have \begin{align*} & E^{\xi^{(1)}}\bigl[ [F,E_s],E^{\xi^{(2)}}\bigr] \\ & \in E^{\xi^{(1)}}\left(\sum_{\xi'\in A(\xi^{(2)})}\mathbb{C} E^{\xi'}\right)U((\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0)\oplus\Ad(w_i)(\mathfrak{m}\cap\mathfrak{n}_0))\\ & \subset \left(\sum_{\xi'\in A(\xi^{(1)} + \xi^{(2)})}\mathbb{C} E^{\xi'}\right)U((\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0)\oplus\Ad(w_i)(\mathfrak{m}\cap\mathfrak{n}_0))\\ & \subset \left(\sum_{\xi'\in A(\xi)}\mathbb{C} E^{\xi'}\right)U((\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0)\oplus\Ad(w_i)(\mathfrak{m}\cap\mathfrak{n}_0)) \end{align*} On the other hand, we have \begin{multline*} E^{\xi^{(1)}}E^{\xi^{(2)}}[F,E_s]\in \left(\sum_{\lvert\xi'\rvert \le \lvert\xi^{(1)} +\xi^{(2)}\rvert}\mathbb{C} E^{\xi'}\right)[F,E_s]\\ \subset \left(\sum_{\lvert\xi'\rvert \le \lvert\xi^{(1)} +\xi^{(2)}\rvert}\mathbb{C} E^{\xi'}\right)(\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0)\oplus\Ad(w_i)(\mathfrak{m}\cap\mathfrak{n}_0). \end{multline*} Since $\lvert\xi^{(1)} + \xi^{(2)}\rvert = \lvert\xi\rvert - 1 < \lvert\xi\rvert$, we get the lemma. \end{proof}
Let $X$ be an element of the normalizer of $\Ad(w_i)\overline{\mathfrak{n}}\cap\mathfrak{n}_0$ in $\mathfrak{g}$. For $f\in C^\infty(O_i)$ we define $D_i(X)f\in C^\infty(O_i)$ by \[
(D_i(X)f)(nw_i) = \left.\frac{d}{dt}f(\exp(-tX)n\exp(tX)w_i)\right|_{t = 0} \] where $n \in w_i\overline{N}w_i^{-1}\cap N_0$.\newsym{$D_i(X)$}\label{symbol:D_i}
\begin{lem}\label{lem:no delta part} Fix $f\in C^\infty(O_i)$, $u'\in (\sigma\otimes e^{\lambda+\rho})'$ and $X\in\mathfrak{g}$. \begin{enumerate} \item If $X\in \mathfrak{a}_0$, then $X$ normalizes $\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0$ and we have \begin{multline*} X((f\otimes u')\delta_i) = ((D_i(X)f)\otimes u')\delta_i + (f\otimes((\Ad(w_i)^{-1}X)u'))\delta_i\\ + (w_i\rho_0 - \rho_0)(X)(f\otimes u')\delta_i. \end{multline*} \item If $X\in \Ad(w_i)(\mathfrak{m}\cap \mathfrak{n}_0)$ or $X\in\mathfrak{m}_0$, then $X$ normalizes $\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0$ and we have \[ X((f\otimes u')\delta_i) = ((D_i(X)f)\otimes u')\delta_i + (((\Ad(w_i)^{-1}X)u')\otimes f) \delta_i. \] \end{enumerate} \end{lem}
\begin{proof} Let $X$ be as in the lemma. First we prove that $X$ normalizes $\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0$. If $X\in \mathfrak{m}_0 + \mathfrak{a}_0$, then $X$ normalizes each restricted root space. Hence, $X$ normalizes $\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0$. If $X\in \Ad(w_i)(\mathfrak{m}\cap \mathfrak{n}_0)$, then $X\in \mathfrak{n}_0$ since $w_i\in W(M)$. Hence, $X$ normalizes $\mathfrak{n}_0$. Since $\mathfrak{m}$ normalizes $\overline{\mathfrak{n}}$, $X$ normalizes $\Ad(w_i)\overline{\mathfrak{n}}$.
Put $g_t = \exp(tX)$. Take $\varphi\in C_c^\infty(U_i,\mathcal{L})$ and we regard $\varphi$ as a $\sigma$-valued $C^\infty$-function on $w_i\overline{N}P$ (Remark~\ref{rem:identify func on flag and G}). Since $w_ig_tw_i^{-1}\in P$, we have $\varphi(xw_ig_tw_i^{-1}) = \sigma(w_ig_tw_i^{-1})^{-1}\varphi(x)$. Put $D(t) = \lvert\det(\Ad(g_t)^{-1}|_{\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0})\rvert$. Then \begin{align*} & \langle X((f\otimes u')\delta_i),\varphi\rangle = \langle (f\otimes u')\delta_i ,-X\varphi\rangle\\
& = \frac{d}{dt}\left.\int_{w_i\overline{N}w_i^{-1}\cap N_0}u'(\varphi(g_tnw_i))f(nw_i)dn\right|_{t = 0}\\
& = \frac{d}{dt}\left.\int_{w_i\overline{N}w_i^{-1}\cap N_0}u'(\varphi((g_tng_t^{-1})w_i(w_i^{-1}g_tw_i)))f(nw_i)dn\right|_{t = 0}\\
& = \frac{d}{dt}\left.\int_{w_i\overline{N}w_i^{-1}\cap N_0}u'(\sigma(w_i^{-1}g_tw_i)^{-1}\varphi((g_tng_t^{-1})w_i))f(nw_i)dn\right|_{t = 0}\\
& = \frac{d}{dt}\left.\int_{w_i\overline{N}w_i^{-1}\cap N_0}u'(\sigma(w_i^{-1}g_tw_i)^{-1}\varphi(nw_i))f(g_t^{-1}ng_tw_i)D(t) dn\right|_{t = 0}\\
& = \frac{d}{dt}\left.\int_{w_i\overline{N}w_i^{-1}\cap N_0}((w_i^{-1}g_tw_i)u')(\varphi(nw_i))f(g_t^{-1}ng_tw_i)D(t) dn\right|_{t = 0} \end{align*} This implies \begin{multline*} X((f\otimes u')\delta_i) = ((D_i(X)f)\otimes u')\delta_i + (f\otimes((\Ad(w_i)^{-1}X)u'))\delta_i\\
+ \left.\frac{d}{dt}\lvert\det(\Ad(g_t)^{-1}|_{\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}})\rvert\right|_{t = 0}((f\otimes u')\delta_i) \end{multline*}
(1)
Assume that $X\in \mathfrak{a}_0$. Since $w_i\in W(M)$, we have $w_i\overline{N}w_i^{-1}\cap N_0 = w_i\overline{N}_0w_i^{-1}\cap N_0$. This implies that $\det(\Ad(g_t)^{-1}|_{\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0}) = e^{t(w_i\rho_0 - \rho_0)(X)}$.
(2)
First assume that $X\in\mathfrak{m}_0$. Since $g\mapsto \det(\Ad(g)^{-1}|_{\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0})$ is $1$-dimensional representation, it is unitary since $M_0$ is compact. Hence we have $\lvert\det(\Ad(g_t)^{-1}|_{\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0})\rvert = 1$. Next assume that $X\in (\Ad(w_i)\mathfrak{m}\cap \mathfrak{n}_0)$. Then $\ad(X)$ is nilpotent. Hence, $\Ad(g_t) - 1$ is nilpotent. This implies $\det(\Ad(g_t)^{-1}|_{\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0}) = 1$. \end{proof}
\begin{lem}\label{lem:succ quot is sub of I'} Let $x\in\mathcal{T}_{O_i}(U_i,\mathcal{L})$. Assume that for all $X\in \Ad(w_i)\overline{\mathfrak{p}}\cap \mathfrak{n}_0$ there exists a positive integer $k$ such that $(X - \eta(X))^kx = 0$. Then $x\in I_i'$. In particular we have $\im\Res_i\subset I_i'$. \end{lem} \begin{proof} Let $E_s$ and $\alpha_s$ be as in Lemma~\ref{lem:bracket of n and bar_n}. For $\xi = (\xi_1,\xi_2,\dots,\xi_n)\in\mathbb{Z}_{\ge 0}^n$, set $E^\xi = E_1^{\xi_1}E_2^{\xi_2}\dots E_n^{\xi_n}$. Since $x\in\mathcal{T}_{O_i}(U_i,\mathcal{L}) = U(\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0)\otimes \mathcal{T}(O_i,\mathcal{L})$, there exist $x_\xi\in \mathcal{T}(O_i,\mathcal{L})$ such that $x = \sum_\xi E^\xi x_\xi$ (finite sum).
First we prove $x_\xi\in (\mathcal{P}(O_i)\eta_i^{-1}\otimes(\sigma\otimes e^{\lambda+\rho})')\delta_i$ by backward induction on the lexicological order of $(\lvert\xi\rvert,\sum_s \xi_s\alpha_s)$. Fix a nonzero element $F\in\Ad(w_i)\overline{\mathfrak{n}}\cap\mathfrak{n}_0$. Then $(F - \eta(F))^kx = \sum_\xi [(F - \eta(F))^k,E^\xi] (x_\xi) + \sum_\xi E^\xi((F - \eta(F))^k x_\xi)$. Assume that $(F - \eta(F))^kx = 0$. Define the set $A(\xi)$ as in Lemma~\ref{lem:bracket of n and bar_n}. By Lemma~\ref{lem:bracket of n and bar_n}, we have \begin{multline*} \sum_\xi E^\xi ((F - \eta(F))^kx_\xi) = -\sum_\xi [(F - \eta(F))^k,E^\xi](x_\xi)\\ \in \sum_\xi \left(\sum_{\xi'\in A(\xi)}\mathbb{C} E^{\xi'}\right) U((\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n})\oplus\Ad(w_i)(\mathfrak{m}\cap\mathfrak{n}_0))(x_{\xi}). \end{multline*}
Put $B(\xi) = \{\xi'\mid \text{$\lvert\xi'\rvert > \lvert\xi\rvert$ or $\lvert\xi'\rvert = \lvert\xi\rvert$ and $\sum \xi'_s\alpha_s > \sum \xi_s\alpha_s$} \}$. Notice that $U((\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n})\oplus\Ad(w_i)(\mathfrak{m}\cap\mathfrak{n}_0))(x_{\xi})\in \mathcal{T}(O_i,\mathcal{L})$. Since $\mathcal{T}_{O_i}(U_i,\mathcal{L}) = U(\Ad(w_i)\overline{\mathfrak{n}}\cap\overline{\mathfrak{n}})\otimes_\mathbb{C}\mathcal{T}(O_i,\mathcal{L}|_{O_i})$, we have \[
(F - \eta(F))^kx_\xi \in \sum_{\xi'\in B(\xi)}U((\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n})\oplus\Ad(w_i)(\mathfrak{m}\cap\mathfrak{n}_0))(x_{\xi'}). \] By inductive hypothesis, $x_{\xi'}\in (\mathcal{P}(O_i)\eta_i^{-1}\otimes (\sigma\otimes e^{\lambda+\rho})')\delta_i$ for all $\xi'\in B(\xi)$. Hence we have $(F - \eta(F))^kx_{\xi}\in (\mathcal{P}(O_i)\eta_i^{-1}\otimes (\sigma\otimes e^{\lambda+\rho})')\delta_i$. Therefore $x_{\xi}\in(\mathcal{P}(O_i)\eta_i^{-1}\otimes(\sigma\otimes e^{\lambda+\rho})')\delta_i$ by Corollary~\ref{cor:polynomial by some power of n}.
Hence, we can write $x = \sum_{\xi}E^\xi \sum_l (f_{\xi,l}\eta_i^{-1}\otimes u_{\xi,l}')\delta_i$ (finite sum), where $f_{\xi,l}\in \mathcal{P}(O_i)$ and $u_{\xi,l}'\in (\sigma\otimes e^{\lambda+\rho})'$. Moreover, we can assume that $f_{\xi,l}$ is an $\mathfrak{a}_0$-weight vector with respect to $D_i$ and $\{f_{\xi,l}\}_l$ is lineally independent for each $\xi$. We prove $u_{\xi,l}'\in J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho})$. Take $F\in \mathfrak{n}_0\cap \mathfrak{m}$. By Lemma~\ref{lem:no delta part}, we have \begin{multline*} (\Ad(w_i)F - \eta(\Ad(w_i)F))^kx \\ = \sum_{\xi,l}[(\Ad(w_i)F - \eta(\Ad(w_i)F))^k,E^\xi]((f_{\xi,l}\eta_i^{-1}\otimes u_{xi,l}')\delta_i)\\ + \sum_{\xi}E^\xi\sum_{p = 0}^k\binom{k}{p}(((D_i(\Ad(w_i)F))^{k - p}(f_{\xi,l})\eta_i^{-1})\otimes\\(F - \eta(\Ad(w_i)F))^p(u_{\xi,l}'))\delta_i. \end{multline*} Now we prove $u_{\xi,l}'\in J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho})$ by backward induction on the lexicological order of $(\lvert\xi\rvert,\sum \xi_s\alpha_s,-\wt f_{\xi,l})$ where $\wt f_{\xi,l}$ is an $\mathfrak{a}_0$-weight of $f_{\xi,l}$ with respect to $D_i$. Take $k$ such that $(\Ad(w_i)F - \eta(\Ad(w_i)F))^k x = 0$. Then we have \begin{multline*} f_{\xi,l}\otimes (F - \eta(\Ad(w_i)F))^k(u'_{\xi,l})\delta_i\\ \in \sum_{\eta\in B(\xi),l}U((\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0)\oplus\Ad(w_i)(\mathfrak{m}\cap\mathfrak{n}_0))((f_{\eta,l}\eta_i^{-1}\otimes u_{\eta,l}')\delta_i)\\ + \sum_{\wt f_{\eta,l'} < \wt f_{\xi,l}}\sum_p(((D_i(\Ad(w_i)F))^pf_{\eta,l'}\eta_i^{-1})\otimes(U(\mathbb{C} F)u_{\eta,l'}'))\delta_i. \end{multline*} By inductive hypothesis, we have $(F - \eta(F))^ku_{\xi,l}'\in J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho})$. This implies that $u_{\xi,l}'\in J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho})$. \end{proof}
In fact, we have $\im\Res_i = I'_i$ under some conditions. This is proved in Section~\ref{sec:Analytic continuation}.
\section{Vanishing theorem}\label{sec:vanishing theorem} In this section, we fix $i \in \{1,2,\dots,r\}$ and a basis $\{e_1,e_2,\dots,e_l\}$ of $\Ad(w_i)\overline{\mathfrak{n}}\cap\mathfrak{n}_0$. Here we assume that each $e_i$ is a restricted root vector and denote its root by $\alpha_i$.
By the decomposition \begin{multline*} N_0/[N_0,N_0] \simeq ((w_i\overline{P}w_i^{-1}\cap N_0)/(w_i\overline{P}w_i^{-1}\cap [N_0,N_0])) \\\times ((w_iNw_i^{-1}\cap N_0)/(w_iNw_i^{-1}\cap [N_0,N_0])) \end{multline*} where $[\cdot,\cdot]$ is the commutator group, we can define a character $\eta'$ of $N_0$ by $\eta'(n) = \eta(n)$ for $n\in w_i\overline{P}w_i^{-1}\cap N_0$ and $\eta'(n) = 1$ for $n\in w_iNw_i^{-1}\cap N_0$.
\begin{lem}\label{lem:acts nilp} Let $X\in \mathfrak{n}_0$. Then for all $x\in I_i'$ there exists a positive integer $k$ such that $(X - \eta'(X))^kx = 0$. \end{lem}
To prove this lemma, we prepare some notation. For $X\in \Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0$, we define a differential operator $R_i'(X)$ on $O_i$ by \[
(R'_i(X)\varphi)(nw_iP/P) = \left.\frac{d}{dt}\varphi(n\exp(tX)w_iP/P)\right|_{t = 0} \] where $n\in w_i\overline{N}w_i^{-1}\cap N_0$.\newsym{$R'_i(X)$} (Recall that $w_i\overline{N}w_i^{-1}\cap N_0\simeq O_i$ by the map $n\mapsto nw_iP/P$.)
For $X\in \mathfrak{g}$, we define a differential operator $\widetilde{R}_i(X)$ on $w_i\overline{N}P$ by the same way, i.e., for a $C^\infty$-function $\varphi$ on $w_i\overline{N}P$, put \[
(\widetilde{R}_i(X)\varphi)(pw_i) = \left.\frac{d}{dt}\varphi(p\exp(tX)w_i)\right|_{t = 0} \] for $p \in w_i\overline{N}Pw_i^{-1}$.\newsym{$\widetilde{R}_i(X)$} Notice that even if $\varphi$ is right $P$-invariant, $\widetilde{R}_i(X)\varphi$ is not right $P$-invariant in general.
Since $R'_i$ (resp.\ $\widetilde{R}_i$) is a Lie algebra homomorphism, we can define a differential operator $R'_i(T)$ (resp.\ $\widetilde{R}_i(T)$) for $T\in U(\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0)$ (resp.\ $T\in U(\mathfrak{g})$) as usual. For $T\in U(\mathfrak{g})$, $f\in C^\infty(O_i)$ and $u'\in (\sigma\otimes(\lambda + \rho))'$, we define $\delta_i(T,f,u')\in \mathcal{D}'_{O_i}(U_i,\mathcal{L})$\newsym{$\delta_i(T,f,u')$} by \[
\langle \delta_i(T,f,u'),\varphi\rangle =
\int_{w_i\overline{N}w_i^{-1}\cap N_0} f(nw_i)u'((\widetilde{R}_i(T)\varphi)(nw_i))dn \] where $\varphi\in C^\infty_c(U_i,\mathcal{L})$ and we regard $\varphi$ as a function on $w_i\overline{N}P$ (Remark~\ref{rem:identify func on flag and G}). The following lemma is easy to prove. \begin{lem}\label{lem:fundamental properties of delta_i} We have the following properties. \begin{enumerate} \item For $X\in \Ad(w_i)\overline{\mathfrak{n}}\cap\mathfrak{n}_0$, $\delta_i(XT,f,u') = \delta_i(T,R'_i(-X)(f),u')$. \item For $X\in \Ad(w_i)\mathfrak{p}$, $\delta_i(TX,f,u') = \delta_i(T,f,\Ad(w_i)^{-1}Xu')$. \item The map $C^\infty(O_i)\otimes_{U(\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n})} U(\mathfrak{g})\otimes_{U(\Ad(w_i)\mathfrak{p})} w_i(\sigma\otimes e^{\lambda+\rho})'\to \mathcal{D}'(U_i,O_i,\mathcal{L})$ defined by $f\otimes T\otimes u'\mapsto \delta_i(T,f,u')$ is injective. \end{enumerate} \end{lem}
\begin{lem}\label{lem:left2right} Let $\{e_i\}$ be a basis of $\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0$ such that $e_i$ is a restricted root vector, $\alpha_i$ the restricted root for $e_i$, $T,T'\in U(\mathfrak{g})$, $f\in C^\infty(O_i)$ and $u'\in (\sigma\otimes e^{\lambda + \rho})'$. Then we have \begin{multline*} T\delta_i(T',f,u') \\= \sum_{(k_1,\dots,k_l)\in \mathbb{Z}_{\ge 0}^l}\delta_i\left((\ad(e_l)^{k_l}\dotsm \ad(e_1)^{k_1}T)T',f\prod_{s = 1}^l\frac{(-x_s)^{k_s}}{k_s!},u'\right), \end{multline*} where $x_i$ is a polynomial on $O_i$ given by $\exp(a_1e_1)\dotsm \exp(a_le_l)w_iP/P\mapsto a_i$. (Notice that the right hand side is a finite sum since $\ad(e_i)$ is nilpotent.) \end{lem} \begin{proof} We remark that by a map $(a_1,\dots,a_l)\mapsto \exp(a_1e_1)\dotsm \exp(a_le_l)$, we have a diffeomorphism $\mathbb{R}^l\simeq w_i\overline{N}w_i^{-1}\cap N_0$ and a Haar measure of $w_i\overline{N}w_i^{-1}\cap N_0$ corresponds to the Euclidean measure of $\mathbb{R}^l$. Take $\varphi\in C_c^\infty(w_i\overline{N}P,\sigma\otimes e^{\lambda+\rho})$. Put $n(a) = \exp(a_1e_1)\dotsm \exp(a_le_l)$ for $a = (a_1,\dots,a_l)$. Recall the definition of $\check{T}$ from Notation. For $T\in \mathfrak{g}$, we have \begin{align*} &\langle T\delta_i(T',f,u'),\varphi\rangle\\ & = \int_{\mathbb{R}^l}u'((\check{T}\widetilde{R}_i(T')\varphi)(n(a)w_i))f(n(a)w_i)da\\
& = \left.\frac{d}{dt}\int_{\mathbb{R}^l}u'(\widetilde{R}_i(T')\varphi)(\exp(tT)n(a)w_i))f(n(a)w_i)da\right|_{t = 0}\\
& = \left.\frac{d}{dt}\int_{\mathbb{R}^l}u'((\widetilde{R}_i(T')\varphi)(n(a)\exp(t\Ad(n(a))^{-1}T)w_i))f(n(a)w_i)da\right|_{t = 0}. \end{align*} The formula \begin{align*} \Ad(n(a))^{-1}T & = e^{-\ad(a_le_l)}\dotsm e^{-\ad(a_1e_1)}T\\ & = \sum_{(k_1,\dots,k_l)\in \mathbb{Z}_{\ge 0}^l}\frac{(-a_1)^{k_1}}{k_1!}\dotsm \frac{(-a_l)^{k_l}}{k_l!}\ad(e_l)^{k_l}\dotsm \ad(e_1)^{k_1}T \end{align*} gives the lemma. \end{proof}
For $\mathbf{k} = (k_1,\dots,k_l)$, we denote an operator $\ad(e_l)^{k_l}\dotsm \ad(e_1)^{k_1}$ on $\mathfrak{g}$ by $\ad(e)^{\mathbf{k}}$ and a function $((-x_1)^{k_1}/k_1!)\dotsm ((-x_l)^{k_l}/k_l!)\in\mathcal{P}(O_i)$ by $f_\mathbf{k}$.
\begin{lem}\label{lem:diff vanish} Let $\mathbf{k} = (k_1,\dots,k_l)\in\mathbb{Z}_{\ge 0}^l$ and $X\in \mathfrak{n}_0$. Assume that $\ad(e)^\mathbf{k}X \in \Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0$. Then we have $R'_i(\ad(e)^\mathbf{k}X)f_\mathbf{k} = 0$. \end{lem} \begin{proof} We may assume that $X$ is a restricted root vector and denote its restricted root by $\alpha$. We consider an $\mathfrak{a}_0$-weight with respect to $D_i$. An $\mathfrak{a}_0$-weight of $f_\mathbf{k}$ is $-\sum_s k_s\alpha_s$. This implies that $R'_i(\ad(e)^\mathbf{k}X)f_\mathbf{k}$ has an $\mathfrak{a}_0$-weight $\alpha$. However, $\mathcal{P}(O_i)$ has a decomposition into the direct sum of $\mathfrak{a}_0$-weight spaces and its weight belongs to $\{\sum_{\beta\in\Sigma^+}b_\beta\beta\mid b_\beta\in\mathbb{Z}_{\le 0}\}$. Hence, we have $R'_i(\ad(e)^\mathbf{k}X)f_\mathbf{k} = 0$. \end{proof}
For $f\in\mathcal{P}(O_i)$ and $X\in\mathfrak{n}_0$ we define $L_X(f)$\newsym{$L_X$} by \[
L_X(f)(nw_i) = \left.\frac{d}{dt}f(\exp(-tX)nw_i)\right|_{t = 0}. \]
\begin{lem}\label{lem:caluculation of Xdelta(1,f,u)} Let $X\in \mathfrak{n}_0$ be a restricted root vector. For $f\in\mathcal{P}(O_i)$ and $u'\in J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho})$, we have \begin{multline*}
(X - \eta'(X))\delta_i(1,f\eta_i^{-1},u') = \delta_i(1,L_X(f)\eta_i^{-1},u') \\
+ \sum_{\ad(e)^\mathbf{k}X\in \Ad(w_i)\mathfrak{n}_0\cap \mathfrak{n}_0}\delta_i(1,ff_{\mathbf{k}}\eta_i^{-1},(\Ad(w_i)^{-1}(\ad(e)^{\mathbf{k}}X) - \eta'(\ad(e)^{\mathbf{k}}X))u'). \end{multline*} (Again the sum of the right hand side is finite.) \end{lem} \begin{proof} We have \[
X\delta_i(1,f\eta_i^{-1},u')
= \sum_{\mathbf{k}\in\mathbb{Z}_{\ge 0}^l}\delta_i(\ad(e)^\mathbf{k}X,ff_\mathbf{k}\eta_i^{-1},u'). \] by Lemma~\ref{lem:left2right}. Since $\ad(e)^\mathbf{k}X$ belongs to $\mathfrak{n}_0$ and is a restricted root vector, we have either $\ad(e)^\mathbf{k}X\in \Ad(w_i)\overline{\mathfrak{n}_0}\cap \mathfrak{n}_0$ or $\ad(e)^\mathbf{k}X\in \Ad(w_i)\mathfrak{n}_0\cap \mathfrak{n}_0$. Recall that $\Ad(w_i)\overline{\mathfrak{n}_0}\cap \mathfrak{n}_0 = \Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0$ since $w_i\in W(M)$. Assume that $\ad(e)^\mathbf{k}X\in \Ad(w_i)\overline{\mathfrak{n}}\cap\mathfrak{n}_0$. By the definition of $\eta_i$ and $\eta'$, we have $R'_i(-\ad(e)^\mathbf{k}X)(\eta_i^{-1}) = \eta(\ad(e)^\mathbf{k}X)\eta_i^{-1} = \eta'(\ad(e)^\mathbf{k}X)\eta_i^{-1}$. Hence, using Lemma~\ref{lem:diff vanish}, \begin{align*} &\delta_i(\ad(e)^\mathbf{k}X,ff_\mathbf{k}\eta_i^{-1},u') \\ &= \delta_i(1,R'_i(-\ad(e)^\mathbf{k}X)(ff_\mathbf{k}\eta_i^{-1}),u')\\ &= \delta_i(1,R'_i(-\ad(e)^\mathbf{k}X)(f)f_\mathbf{k}\eta_i^{-1},u') + \eta'(\ad(e)^\mathbf{k}X)\delta_i(1,ff_\mathbf{k}\eta_i^{-1},u'). \end{align*} Next assume that $\ad(e)^{\mathbf{k}}X\in \Ad(w_i)\mathfrak{n}_0\cap \mathfrak{n}_0$. For $h\in\mathcal{P}(O_i)$, define $\widetilde{h}\in\mathcal{P}(U_i)$ by $\widetilde{h}(nn_0w_iP) = h(nw_iP)$ for $n\in w_i\overline{N}w_i^{-1}\cap N_0$ and $n_0\in w_i\overline{N}w_i^{-1}\cap \overline{N_0}$. Then we have $\widetilde{R'_i(Y)h} = \widetilde{R}_i(Y)\widetilde{h}$ for all $Y\in \Ad(w_i)\overline{\mathfrak{n}_0}\cap \mathfrak{n}_0$. Since $\widetilde{f}(pnw_i) = \widetilde{f}(pw_i)$ for $p\in w_i\overline{N}Pw_i^{-1}$ and $n\in w_iN_0w_i^{-1}\cap N_0$, we have $\widetilde{R}_i(-\ad(e)^{\mathbf{k}}X)(\widetilde{f}) = 0$. Hence we have \begin{multline*}
\delta_i(\ad(e)^\mathbf{k}X,ff_\mathbf{k}\eta_i^{-1},u') = \delta_i(1,ff_\mathbf{k}\eta_i^{-1},\Ad(w_i)^{-1}(\ad(e)^{\mathbf{k}}X)u')\\
= \delta_i(1,R'_i(-\ad(e)^\mathbf{k}X)(\widetilde{f})|_{O_i}f_\mathbf{k}\eta_i^{-1},u') + \\ \delta_i(1,ff_\mathbf{k}\eta_i^{-1},\Ad(w_i)^{-1}(\ad(e)^{\mathbf{k}}X)u'). \end{multline*} By the same calculation as the proof of Lemma~\ref{lem:left2right}, we have \[
\widetilde{L_X(f)} = L_X(\widetilde{f}) = \sum_{\mathbf{k}\in\mathbb{Z}_{\ge 0}^l}\widetilde{R}_i(-\ad(e)^\mathbf{k}X)(\widetilde{f})\widetilde{f_\mathbf{k}}. \] Hence \begin{align*}
\sum_{\mathbf{k}\in\mathbb{Z}_{\ge 0}^l}\delta_i(1,\widetilde{R}_i(-\ad(e)^\mathbf{k}X)(\widetilde{f})|_{O_i}f_\mathbf{k}\eta_i^{-1},u') &= \delta_i(1,\widetilde{L_X(f)}|_{O_i}\eta_i^{-1},u')\\ & = \delta_i(1,L_X(f)\eta_i^{-1},u'). \end{align*} These imply that \begin{multline*} (X - \eta'(X))\delta_i(1,f\eta_i^{-1},u') = \delta_i(1,L_X(f)\eta_i^{-1},u')\\ + \sum_{\ad(e)^\mathbf{k}X\in \Ad(w_i)\mathfrak{n}_0\cap \mathfrak{n}_0}\delta_i(1,ff_\mathbf{k}\eta_i^{-1},\Ad(w_i)^{-1}(\ad(e)^{\mathbf{k}}X)u')\\ + \sum_{\ad(e)^\mathbf{k}X\in \Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0}\eta'(\ad(e)^\mathbf{k}X)\delta_i(1,ff_\mathbf{k}\eta_i^{-1},u')\\ - \eta'(X)\delta_i(1,f\eta_i^{-1},u'). \end{multline*} Since $\eta'$ is a character, if $\mathbf{k}\ne (0,\dots,0)$ then $\eta'(\ad(e)^\mathbf{k}X) = 0$. Hence we have \[
\sum_{\mathbf{k}\in\mathbb{Z}_{\ge 0}^n}\eta'(\ad(e)^\mathbf{k}X)\delta_i(1,ff_\mathbf{k}\eta_i^{-1},u')
=
\eta'(X)\delta_i(1,f\eta_i^{-1},u'). \] This implies \begin{multline*} \left(\sum_{\ad(e)^\mathbf{k}X\in \Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0}\eta'(\ad(e)^\mathbf{k}X)\delta_i(1,ff_\mathbf{k}\eta_i^{-1},u')\right) - \eta'(X)\delta_i(1,f\eta_i^{-1},u')\\ = \sum_{\ad(e)^\mathbf{k}X\in\Ad(w_i)\mathfrak{n}_0\cap \mathfrak{n}_0}\eta'(\ad(e)^\mathbf{k})\delta_i(1,ff_\mathbf{k}\eta_i^{-1},u'). \end{multline*} We get the lemma. \end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:acts nilp}] Since $\ad(\mathfrak{n}_0)$ acts $\mathfrak{g}$ nilpotently, the subspace \[
\{x\in I'_i\mid \text{for some $k$ and for all $X\in \mathfrak{n}_0$, $(X - \eta(X))^kx = 0$}\} \] is $\mathfrak{g}$-stable. Hence we may assume that $x = ((f\eta_i^{-1})\otimes u')\delta_i = \delta_i(1,f\eta_i^{-1},u')$ for some $f\in\mathcal{P}(O_i)$ and $u'\in J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho})$.
Set $V = U(\Ad(w_i)^{-1}\mathfrak{n}_0\cap \mathfrak{n}_0)u'$ where $\mathfrak{n}$ acts $J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho})$ trivially. Then $V$ is finite-dimensional. By applying Engel's theorem for $V\otimes (-w_i^{-1}\eta')$, there exists a filtration $0 = V_0 \subset V_1\subset\cdots\subset V_p = V$ such that $(V_s/V_{s - 1})\otimes (-w_i^{-1}\eta'|_{\Ad(w_i)^{-1}\mathfrak{n}_0\cap\mathfrak{n}_0})$ is the trivial representation of $\Ad(w_i)^{-1}\mathfrak{n}_0\cap\mathfrak{n}_0$. Then we have $V_s/V_{s - 1}\simeq w_i^{-1}\eta'|_{\Ad(w_i)^{-1}\mathfrak{n}_0\cap\mathfrak{n}_0}$ for all $s = 1,2,\dots,p$. We prove the lemma by induction on $p = \dim V$.
We may assume that $X$ is a restricted root vector. By Lemma~\ref{lem:caluculation of Xdelta(1,f,u)}, we have \begin{multline*}
(X - \eta'(X))\delta_i(1,f\eta_i^{-1},u')\in \delta_i(1,L_X(f)\eta_i^{-1},u')\\ + \sum_{h\in\mathcal{P}(O_i),\ v'\in V_{p - 1}}\delta_i(1,h\eta_i^{-1},v'). \end{multline*} Since $f$ is a polynomial, there exists a positive integer $c$ such that $(L_X)^c(f) = 0$. Then $(X - \eta'(X))^c\delta_i(1,f\eta_i^{-1},u') \in \sum_{h\in\mathcal{P}(O_i), v'\in V_{p - 1}}\delta_i(1,h\eta_i^{-1},v')$. By inductive hypothesis the lemma is proved. \end{proof}
From the lemma, we get the following vanishing theorem. Recall that we define the character $w_i^{-1}\eta$ of $\mathfrak{m}\cap \mathfrak{n}_0$ by $(w_i^{-1}\eta)(X) = \eta(\Ad(w_i)X)$. \begin{lem}\label{lem:vanishing lemma} Assume that $I_i/I_{i - 1} \ne 0$. Then the following conditions hold. \begin{enumerate} \item The character $\eta$ is unitary. \item The character $\eta$ is zero on $\Ad(w_i)\mathfrak{n}\cap\mathfrak{n}_0$. \item The module $J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho})$ is not zero. \end{enumerate} \end{lem} \begin{proof} (2)
By Lemma~\ref{lem:acts nilp} and the definition of $J'_\eta$, if $I_i/I_{i - 1} \ne 0$ then $\eta = \eta'$. By the definition of $\eta'$, $\eta = \eta'$ is equivalent to $\eta|_{\Ad(w_i)\mathfrak{n}\cap\mathfrak{n}_0} = 0$.
(3) This is clear from Lemma~\ref{lem:succ quot is sub of I'}.
(1) It is sufficient to prove that if $\eta$ is not unitary then $J'_\eta(V) = 0$ for all irreducible representation $V$ of $G$. By Casselman's subrepresentation theorem, $V$ is a subrepresentation of some principal series representation. Since $J'_\eta$ is an exact functor, we may assume $V$ is a principal series representation $\Ind_{P_0}^G(\sigma_0\otimes e^{\lambda_0 + \rho_0})$.
Take the Bruhat filtration $\{I_i\}$ of $J'_\eta(V)$. We prove $I_i/I_{i - 1} = 0$ for all $i$. By (2), if $\eta$ is non-trivial on $w_iN_0w_i^{-1}\cap N_0$ then $I_i/I_{i - 1} = 0$. Hence we may assume that $\eta$ is not unitary on $w_i\overline{N_0}w_i^{-1}\cap N_0$. In this case, an nonzero element of $I_i'$ is not tempered. Hence $I_i/I_{i - 1} = 0$. \end{proof}
\begin{rem} In the next section it is proved that the conditions of Lemma~\ref{lem:vanishing lemma} is also sufficient (Theorem~\ref{thm:succ quot is I'_i}). \end{rem}
\begin{defn}[Whittaker vectors]\label{defn:Whittaker vectors} Let $V$ be a $U(\mathfrak{n}_0)$-module. We define a vector space $\Wh_\eta(V)$\newsym{$\Wh_\eta(V)$} by \[
\Wh_\eta(V) = \{v\in V\mid \text{for all $X\in\mathfrak{n}_0$ we have $Xv = \eta(X)v$}\}. \] An element of $\Wh_\eta(V)$ is called a \emph{Whittaker vector}. \end{defn}
\begin{lem}\label{lem:Whittaker vector, no differential part}
Assume that $\eta|_{\Ad(w_i)\mathfrak{n}\cap \mathfrak{n}_0} = 0$. Then we have \begin{multline*} \Wh_\eta\left(\left\{\sum_s(f_s\eta_i^{-1}\otimes u'_s)\delta_i\mid f_s\in\mathcal{P}(O_i),\ u'_s\in J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho})\right\}\right) \\ = \{(\eta_i^{-1}\otimes u')\delta_i\mid u'_s\in \Wh_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho})\}. \end{multline*} \end{lem} \begin{proof} By the assumption, we have $\eta = \eta'$. Hence the right hand side is a subspace of the left hand side by Lemma~\ref{lem:caluculation of Xdelta(1,f,u)}.
Take $x = \sum_s (f_s\eta_i^{-1}\otimes u'_s) = \sum_s \delta(1,f_s\eta_i^{-1},u'_s)\in \Wh_\eta(I'_i)$. We assume that $\{u'_s\}$ is linearly independent. Take $X\in \Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0$. Since $\ad(e)^\mathbf{k}X\in \Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0$ for all $\mathbf{k}\in\mathbb{Z}_{\ge 0}^l$, we have $\sum_s\delta_i(1,L_X(f_s)\eta_i^{-1},u'_s) = 0$ by Lemma~\ref{lem:caluculation of Xdelta(1,f,u)}. Hence $L_X(f_s) = 0$. This implies $f_s\in \mathbb{C}$.
From the above argument, $x = \delta(1,\eta_i^{-1},u')$ for some $u'\in J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho})$. Take $X\in \Ad(w_i)\mathfrak{m}\cap \mathfrak{n}_0$. By Lemma~\ref{lem:caluculation of Xdelta(1,f,u)}, we have \[
\delta_i(1,\eta_i^{-1},(\Ad(w_i)^{-1}X - \eta(X))u')\in \sum_{\mathbf{k}\ne 0,\ u_\mathbf{k}\in J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho})}\delta_i(1,f_\mathbf{k}\eta_i^{-1},u_{\mathbf{k}}). \] If $\mathbf{k}\ne 0$ then the degree of $f_\mathbf{k}$ is greater than $0$. So the left hand side must be $0$. Hence we have $(\Ad(w_i)^{-1}X - \eta(X))u' = 0$. We have the lemma. \end{proof}
The following lemma is well-known, but we give a proof for the readers (cf.\ Casselman-Hecht-Milicic~\cite{MR1767896}, Yamashita~\cite{MR849220}). \begin{lem}\label{lem:Whittaker vectors in non-degenerate case} Assume that $\supp\eta = \Pi$. Let $x\in \Wh_\eta(I(\sigma,\lambda)')$. Then there exists $u'\in \Wh_{w_r^{-1}\eta}((\sigma\otimes e^{\lambda+\rho})')$ such that $x = (\eta_r^{-1}\otimes u')\delta_r$. \end{lem} Recall that $r = \# W(M) = \#(W/W_M)$.
\begin{proof} Assume that $i < r$. Then $w_iw_{M,0}$ is not the longest element of $W$. There exists a simple root $\alpha\in\Pi$ such that $s_\alpha w_iw_{M,0} > w_iw_{M,0}$. This means that $w_iw_{M,0}\Sigma^+\cap \Sigma^+ = s_\alpha(s_\alpha w_iw_{M,0}\Sigma^+\cap \Sigma^+)\cup \{\alpha\}$. The left hand side is $w_i(\Sigma^+\setminus\Sigma_M^+)\cap \Sigma^+$. Hence, $\eta$ is not trivial on $\Ad(w_i)\mathfrak{n}\cap\mathfrak{n}_0$. By Lemma~\ref{lem:vanishing lemma}, $I_i/I_{i - 1} = 0$. This implies that $J'_\eta(I(\sigma,\lambda))\subset I_r'$. There exists a polynomial $f_s\in\mathcal{P}(X_r)$ and $u_s'\in J'_{w_r^{-1}\eta}(\sigma\otimes e^{\lambda+\rho})$ such that $x = \sum_s((f_s\eta_r^{-1})\otimes u'_s)\delta_r$. By Lemma~\ref{lem:Whittaker vector, no differential part}, we have the lemma. \end{proof}
\section{Analytic continuation}\label{sec:Analytic continuation} The aim of this section is to prove that $\im\Res_i = I'_i$ if $I_i/I_{i - 1} \ne 0$.
Let $P_\eta$ be the parabolic subgroup corresponding to $\supp\eta \subset \Pi$ containing $P_0$ and $P_\eta = M_\eta A_\eta N_\eta$ its Langlands decomposition such that $A_\eta\subset A_0$.\newsym{$P_\eta = M_\eta A_\eta N_\eta$} Denote the complexification of the Lie algebra of $P_\eta$, $M_\eta$, $A_\eta$, $N_\eta$ by $\mathfrak{p}_\eta$, $\mathfrak{m}_\eta$, $\mathfrak{a}_\eta$, $\mathfrak{n}_\eta$, respectively.\newsym{$\mathfrak{p}_\eta = \mathfrak{m}_\eta\oplus\mathfrak{a}_\eta\oplus\mathfrak{n}_\eta$} Put $\mathfrak{l}_\eta = \mathfrak{m}_\eta\oplus\mathfrak{a}_\eta$, $\overline{N_\eta} = \theta(N_\eta)$ and $\overline{\mathfrak{n}_\eta} = \theta(\mathfrak{n}_\eta)$.\newsym{$\mathfrak{l}_\eta$}\newsym{$\overline{N_\eta}$}\newsym{$\overline{\mathfrak{n}_\eta}$} Set $\Sigma^+_\eta = \{\sum_{\alpha\in\supp\eta}n_\alpha\alpha\in\Sigma^+\mid n_\alpha\in\mathbb{Z}_{\ge 0}\}$ and $\Sigma^-_\eta = -\Sigma^+_\eta$.\newsym{$\Sigma^+_\eta,\Sigma^-_\eta$} The same notation will be used for $M$ with suffix $M$, i.e., $P_{M,\eta} = M_{M,\eta}A_{M,\eta}N_{M,\eta}$ is the parabolic subgroup of $M$ containing $M\cap P_0$ corresponding to $\supp\eta\cap\Sigma_M^+$, $\mathfrak{p}_{\mathfrak{m},\eta} = \mathfrak{m}_{\mathfrak{m},\eta}\oplus\mathfrak{a}_{\mathfrak{m},\eta}\oplus\mathfrak{n}_{\mathfrak{m},\eta}$ is a complexification of the Lie algebra of $P_{M,\eta} = M_{M,\eta}A_{M,\eta}N_{M,\eta}$. \newsym{$P_{M,\eta} = M_{M,\eta}A_{M,\eta}N_{M,\eta}$}\newsym{$\mathfrak{p}_{\mathfrak{m},\eta} = \mathfrak{m}_{\mathfrak{m},\eta}\oplus\mathfrak{a}_{\mathfrak{m},\eta}\oplus\mathfrak{n}_{\mathfrak{m},\eta}$}
For $w\in W$, there is an open dense subset $w\overline{N}P/P$ of $G/P$ and it is diffeomorphic to $\overline{N}$. Then for $w,w'\in W$, there exists a map $\Phi_{w,w'}$\newsym{$\Phi_{w,w'}$} from some open dense subset $U\subset\overline{N}$ to $\overline{N}$ such that $w\overline{n}P/P = w'\Phi_{w,w'}(\overline{n})P/P$ for $\overline{n}\in U$.
The map $\Phi_{w,w'}$ is a rational function.
Since the exponential map $\exp\colon\Lie(\overline{N})\to \overline{N}$ is diffeomorphism, $\overline{N}$ has a structure of a vector space. \begin{lem}\label{lem:property of e^rho} \begin{enumerate} \item The map $\overline{N}\to \mathbb{R}$ defined by $\overline{n}\mapsto e^{8\rho(H(\overline{n}))}$ is a polynomial. \item For all $\overline{n}\in\overline{N}$ we have $e^{8\rho(H(\overline{n}))} \ge 1$. \item Take $H_0\in \mathfrak{a}$ such that $\alpha(H_0) = -1$ for all $\alpha\in \Pi\setminus\Sigma_M$. There exists a continuous function $Q(\overline{n})\ge 0$ on $\overline{N}$ such that the following conditions hold: (a) The function $Q$ vanishes only at the unit element. (b) $e^{8\rho(H(\overline{n}))} \ge Q(\overline{n})$. (c) $Q(\exp(tH_0)\overline{n}\exp(-tH_0)) \ge e^{8t}Q(\overline{n})$ for $t\in \mathbb{R}_{>0}$ and $\overline{n}\in\overline{N}$. \end{enumerate} \end{lem} \begin{proof} By Knapp~\cite[Proposition~7.19]{MR1880691}, there exists an irreducible finite-dimensional $V_{4\rho}$ of $\mathfrak{g}$ with the highest weight $4\rho\in \mathfrak{a}_0^*\subset \mathfrak{h}^*$. Let $v_{4\rho}\in V_{4\rho}$ be a highest weight vector and $v_{-4\rho}^*\in V_{4\rho}^*$ the lowest weight vector of $V_{4\rho}^*$. Then $\mathbb{C} v_{4\rho}$ is a $1$-dimensional unitary representation of $M$. Take $\overline{n}\in \overline{N}$ and decompose $\overline{n} = kan$ where $k\in K$, $a\in A_0$ and $n\in N_0$.
First we prove (1). We have $\theta(\overline{n})^{-1}\overline{n} = \theta(n)^{-1}a^2n$. Hence \begin{align*} \langle \theta(\overline{n})^{-1}\overline{n}v_{4\rho},v^*_{-4\rho}\rangle & = \langle \theta(n)^{-1}a^2nv_{4\rho},v^*_{-4\rho}\rangle\\ & = \langle a^2nv_{4\rho},\theta(n)v^*_{-4\rho}\rangle\\ & = e^{8\rho(H(\overline{n}))}\langle v_{4\rho},v^*_{-4\rho}\rangle. \end{align*} The left hand side is a polynomial.
Next we prove (2) and (3). Fix a compact real form of $\mathfrak{g}$ containing $\Lie(K)$ and take an inner product on $V_{4\rho}$ which is invariant under this compact real form. We normalize an inner product $||\cdot||$ so that $||v_{4\rho}|| = 1$. Then we have $||\overline{n}v_{4\rho}|| = ||kanv_{4\rho}|| = ||av_{4\rho}|| = e^{4\rho(H(\overline{n}))}||v_{4\rho}|| = e^{4\rho(H(\overline{n}))}$. For $\nu\in\mathfrak{h}^*$ let $Q_\nu(\overline{n})\in V_{4\rho}$ be the $\nu$-weight vector such that $\overline{n}v_{4\rho} = \sum_\nu Q_{\nu}(\overline{n})$. Then we have $e^{8\rho(H(\overline{n}))} = \sum_\nu ||Q_\nu(\overline{n})||^2$. Since $Q_{4\rho}(\overline{n}) = v_{4\rho}$, we have $e^{8\rho(H(\overline{n}))} \ge 1$.
Put $Q(\overline{n}) = \sum_{w\in W(M)\setminus\{e\}}||Q_{4w\rho}(\overline{n})||^2$. Assume that $\overline{n} \ne e$. Then there exist $w\in W(M)\setminus\{e\}$, $m'\in M$, $a'\in A$, $n'\in N$ and $\overline{n}'\in \overline{N}$ such that $\overline{n} = w\overline{n}'m'a'n'$. Let $v_{-4w\rho}^*\in V_{4\rho}^*$ be a weight vector with $\mathfrak{h}$-weight $-4w\rho$. Then we have \begin{multline*}
||Q_{4w\rho}(\overline{n})||
= \lvert\langle \overline{n}v_{4\rho},v_{-4w\rho}^*\rangle\rvert
= \lvert\langle w\overline{n}'m'a'n'v_{4\rho},v_{-4w\rho}^*\rangle\rvert\\
= \lvert\langle a'v_{4\rho},w^{-1}v^*_{-4w\rho}\rangle\rvert
= e^{4\rho(\log a')}\lvert \langle v_{4\rho},w^{-1}v_{-4w\rho}^*\rangle\rvert
\ne 0. \end{multline*} Hence, if $\overline{n}\in \overline{N}\setminus\{e\}$ then $Q(\overline{n})\ne 0$.
Let $t$ be a positive real number. Using $Q_\nu(\exp(tH_0)\overline{n}\exp(-tH_0)) = e^{t(\nu - 4\rho)(H_0)}Q_\nu(\overline{n})$, we have \[ Q(\exp(tH_0)\overline{n}\exp(-tH_0)) = \sum_{w\in W(M)\setminus\{e\}} e^{8t(w\rho - \rho)(H_0)}\lvert Q_{4w\rho_0}(\overline{n})\rvert^2. \] Since $(w\rho - \rho)(H_0) \ge 1$ for $w\in W(M)\setminus\{e\}$, we get the lemma. \end{proof}
\begin{rem}\label{rem:asymptotic at infinity of e^rho} The condition Lemma~\ref{lem:property of e^rho} (3) implies that $\lim_{\overline{n}\to \infty}Q(\overline{n}) = \infty$. The proof is the following. Take $H_0$ as in Lemma~\ref{lem:property of e^rho}. Let $\{e_1,\dots,e_l\}$ be a basis of $\overline{\mathfrak{n}}$. Here, we assume that each $e_i$ is a restricted root vector and denote its root by $\alpha_i$. Any $\overline{n}\in \overline{N}$ can be written as $\overline{n} = \exp(\sum_{i = 1}^l a_ie_i)$ where $a_i\in \mathbb{R}$. Put $r(\overline{n}) = \sum_{i = 1}^l\lvert a_i\rvert^{-1/\alpha_i(H_0)}$. Set $C = \min_{r(\overline{n}) = 1}Q(\overline{n})$. Since $Q(\overline{n}) > 0$ if $\overline{n}$ is not the unit element, $C > 0$. Then we have $Q(\overline{n})\ge Cr(\overline{n})^8$ if $r(\overline{n}) > 1$. If $\overline{n}\to \infty$ then $r(\overline{n})\to\infty$. Hence, $Q(\overline{n})\to\infty$. \end{rem}
\begin{lem}\label{lem:extension of polynomials} Let $f$ be a polynomial on $\overline{N}$. There exists a positive integer $k$ and a $C^\infty$-function $h$ on $G/P$ such that $h(w_i\overline{n}P/P) = e^{-k\rho(H(\overline{n}))}f(\overline{n})$ for all $\overline{n}\in\overline{N}$. \end{lem} \begin{proof} By Lemma~\ref{lem:property of e^rho} and Remark~\ref{rem:asymptotic at infinity of e^rho}, we can choose a positive integer $C$ such that $e^{-8C\rho(H(\overline{n}))}f(\overline{n}) \to 0$ when $\overline{n}\to \infty$. Let $\widetilde{f}$ be a function on $U_i$ defined by $\widetilde{f}(w_i\overline{n}P/P) = e^{-8C\rho(H(\overline{n}))}f(\overline{n})$ for $\overline{n}\in \overline{N}$. We prove that $\widetilde{f}$ can be extended to $G/P$. Take $w\in W(M)$. Then $\widetilde{f}$ is defined in a subset of $w\overline{N}P/P$. Using a diffeomorphism $\overline{N}\simeq w\overline{N}P/P$, $\widetilde{f}$ defines a rational function $\widetilde{f}\circ \Phi_{w_i,w}$ defined on an open dense subset of $\overline{N}$. By the condition of $C$, the function $\widetilde{f}\circ \Phi_{w_i,w}$ has no pole. Hence, $\widetilde{f}$ defines a $C^\infty$-function on $w\overline{N}P/P$. Since $\bigcup_{w\in W(M)}w\overline{N}P/P = G/P$, the lemma follows. \end{proof}
Define $\kappa\colon G\to K$ and $H\colon G\to \Lie(A_0)$ by $g\in \kappa(g)\exp H(g) N_0$.\newsym{$\kappa$}\newsym{$H$}
Recall that for a representation $V$ of $\mathfrak{g}$, $\nu\in \mathfrak{a}_0^*$ is called an exponent of $V$ if $\nu + \rho_0|_{\mathfrak{m}\cap\mathfrak{a}_0}$ is an $\mathfrak{a}_0$-weight of $V/\mathfrak{n}_0V$.
\begin{prop}\label{prop:convergence and continuation} Let $\varphi$ be a $\sigma$-valued function on $K$ which satisfies $\varphi(km) = \sigma(m)^{-1}\varphi(k)$ for all $k\in K$ and $m\in M\cap K$. We define $\varphi_\lambda\in I(\sigma,\lambda)$ by $\varphi_\lambda(kman) = e^{-(\lambda + \rho)(\log a)}\sigma(m)^{-1}\varphi(k)$ for $k\in K$, $m\in M$, $a\in A$ and $n\in N$. For $u'\in J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho})$ and $f\in\mathcal{P}(O_i)$, put $I_{f,u'}(\varphi_\lambda) = \int_{w_i\overline{N}w_i^{-1}\cap N_0}u'(\varphi_\lambda(nw_i))\eta(n)^{-1}f(nw_i)dn$. (If $\supp\varphi\subset K\cap w_i\overline{N}P$ then the integral converges.) \begin{enumerate} \item If $\langle\alpha,\re\lambda\rangle$ is sufficiently large for each $\alpha\in\Sigma^+\setminus\Sigma^+_M$ then the integral $I_{f,u'}(\varphi_\lambda)$ absolutely converges. \item As a function of $\lambda$, the integral $I_{f,u'}(\varphi_\lambda)$ has a meromorphic continuation to $\mathfrak{a}^*$. \item If $\supp\eta = \Pi$ and $i = r$ then $I_{f,u'}(\varphi_\lambda)$ is holomorphic for all $\lambda\in\mathfrak{a}^*$. \item Let $\nu$ be an exponent of $\sigma$ and $u'\in \Wh_{w_i^{-1}\eta}((\sigma\otimes e^{\lambda+\rho})')$. If $2\langle\alpha,\lambda+\nu\rangle/\lvert\alpha\rvert^2\not\in \mathbb{Z}_{\le 0}$ for all $\alpha\in \Sigma^+\setminus w_i^{-1}(\Sigma^+\cup\Sigma^-_{\eta})$ then $I_{1,u'}(\varphi_\mu)$ is holomorphic at $\mu = \lambda$. \end{enumerate} \end{prop} \begin{proof} First we prove (1). If $f = 1$ then this is a well-known result. For a general $f$, extends $f$ to a function on $w_i\overline{N}P/P$ by $f(w_inn') = f(w_in)$ for $n\in w_i\overline{N}w_i^{-1}\cap N_0$ and $n'\in w_i\overline{N}w_i^{-1}\cap \overline{N_0}$. Then by Lemma~\ref{lem:extension of polynomials} there exists a positive number $C$ such that $\overline{n}\mapsto e^{-C\rho(H(\overline{n}))}f(w_i\overline{n})$ extends to a function $h$ on $G/P$. Since \[
I_{f,u'}(\varphi_\lambda) = \int_{w_i\overline{N}w_i^{-1}\cap N_0}u'(\varphi(\kappa(nw))e^{-(\lambda+\rho)(H(nw_r))}f(nw_r)\eta(n)^{-1}dn, \] we have $I_{f,u'}(\varphi_\lambda) = I_{1,u'}((\varphi h)_{\lambda - C\rho})$.
We prove (3). By dualizing Casselman's subrepresentation theorem, there exist a representation $\sigma_0$ of $M_0$ and $\lambda_0\in\mathfrak{a}_0^*$ such that $\sigma$ is a quotient of $\Ind_{M\cap P_0}^{M}(\sigma_0\otimes e^{\lambda_0})$. Then we may regard $u'\in J'_{w_r^{-1}\eta}(\Ind_{M\cap P_0}^M(\sigma_0\otimes e^{\lambda_0}))$. By the proof of Lemma~\ref{lem:Whittaker vectors in non-degenerate case}, there exist a polynomial $f_0$ on $(M\cap N_0)w_{M,0}(M\cap P_0)/(M\cap P_0)$ and $u'_0\in (\sigma_0\otimes e^{\lambda_0})'$ such that $u'$ is given by \[
\varphi_0\mapsto \int_{M\cap N_0}u'_0(\varphi_0(n_0w_{M,0}))f_0(n_0w_{M,0})\eta(n_0)^{-1}dn_0 \] Let $\pi\colon \Ind_{P_0}^G(\sigma_0\otimes e^{\lambda + \lambda_0 + \rho})\to I(\sigma,\lambda)$ be the map induced from the quotient map $\Ind_{M\cap P_0}^{M}(\sigma_0\otimes e^{\lambda_0})\to \sigma$. Take $\widetilde{\varphi}\colon K\to \sigma_0$ which satisfies $\widetilde{\varphi}(km) = \sigma_0^{-1}(m)\widetilde{\varphi}(k)\ (k\in K,\ m\in M_0)$ and $\pi(\widetilde{\varphi}_{\lambda + \lambda_0}) = \varphi_\lambda$. Define a polynomial $\widetilde{f}\in \mathcal{P}(w_iw_{M,0}\overline{N_0}P_0/P_0)$ by \[
\widetilde{f}(w_iw_{M,0}nn_0P_0/P_0) = f(w_inP/P)f_0(w_{M,0}n_0(M\cap P_0)/(M\cap P_0)) \] for $n\in \overline{N}$ and $n_0\in M\cap \overline{N_0}$. (Notice that $w_{M,0}(M\cap \overline{N_0}) = (M\cap N_0)w_{M,0}$.) Then we have \begin{multline*}
I_{f,u'}(\varphi_\lambda)\\ = \int_{w_iw_{M,0}\overline{N_0}(w_iw_{M,0})^{-1}\cap N_0}u_0'(\widetilde{\varphi}(nw_iw_{M,0}))\widetilde{f}(w_iw_{M,0}nP_0/P_0)\eta(n)^{-1}dn. \end{multline*} Hence, we may assume that $P$ is minimal. By the same argument in (1), we may assume $f = 1$. If $f = 1$ then this integral is known as a Jacquet integral and the analytic continuation is well-known~\cite{MR0271275}.
We prove (2) and (4). By the same argument in (1), we may assume that $f = 1$. Take $w'\in W_{M_\eta}$ and $w''\in W(M_\eta)^{-1}$ such that $w_i = w'w''$. Then we have $w_i\overline{N}w_i^{-1}\cap N_0 = (w'\overline{N_0}(w')^{-1}\cap N_0)w'(w''\overline{N_0}(w'')^{-1}\cap N_0)(w')^{-1}$. The condition $w'\in W_{M_\eta}$ implies that $w'(\Sigma^+\setminus\Sigma^+_\eta) = \Sigma^+\setminus\Sigma^+_\eta$. Hence, $\supp\eta \cap w'\Sigma^+ = \supp\eta\cap w'\Sigma_\eta^+$. This implies \begin{multline*}
\supp\eta\cap w'(w''\Sigma^-\cap \Sigma^+) = \supp\eta\cap w_i\Sigma^-\cap w'\Sigma^+\\ = \supp\eta\cap w_i\Sigma^-\cap w_i(w'')^{-1}\Sigma_\eta^+\subset \supp\eta\cap w_i\Sigma^-\cap w_i\Sigma^+ = \emptyset, \end{multline*} i.e., $\eta$ is trivial on $w'(w''\overline{N_0}(w'')^{-1}\cap N_0)(w')^{-1}$. Hence, we have \[
I_{1,u'}(\varphi)
= \int_{w'\overline{N_0}(w')^{-1}\cap N_0}\int_{w''\overline{N_0}(w'')^{-1}\cap N_0}u'(\varphi(n_1w'n_2w''))\eta(n_1)^{-1}dn_2dn_1. \] Put $P' = (w''P(w'')^{-1}\cap M_\eta)N_\eta$. By the definition of $W(M_\eta)$, we have $w''N_0(w'')^{-1}\supset N_0\cap M_\eta$, this implies that $P'$ (resp.\ $w''P(w'')^{-1}\cap M_\eta$) is a parabolic subgroup of $G$ (resp.\ $M_\eta$). Define a $G$-module homomorphism $A(\sigma,\lambda)\colon I(\sigma,\lambda)\to \Ind_{P'}^G(w''(\sigma)\otimes e^{w''\lambda+\rho})$ by \[
(A(\sigma,\lambda)\varphi)(x) = \int_{w''\overline{N_0}(w'')^{-1}\cap N_0}\varphi(xnw'')dn. \] By a result of Knapp and Stein~\cite{MR582703}, this homomorphism has a meromorphic continuation. We have \[
I_{1,u'}(\varphi) = \int_{w'\overline{N_0}(w')^{-1}\cap N_0}u'((A(\sigma,\lambda)\varphi)(nw'))\eta(n)^{-1}dn. \] Notice that $w'\overline{N_0}(w')^{-1}\cap N_0\subset M_\eta$. Hence we get (2) by (3).
To prove (4), we calculate $(w'')^{-1}\Sigma^-\cap\Sigma^+$. Since $(w'')^{-1}\in W(M_\eta)$, we have $(w'')^{-1}\Sigma_\eta^-\subset \Sigma^-$. Hence $(w'')^{-1}\Sigma_\eta^-\cap \Sigma^+ = \emptyset$. Then \begin{align*} (w'')^{-1}\Sigma^-\cap \Sigma^+ & = (w'')^{-1}(\Sigma^-\setminus\Sigma_\eta^-)\cap \Sigma^+\\ & = (w'')^{-1}(w')^{-1}(\Sigma^-\setminus\Sigma_\eta^-)\cap\Sigma^+\\ & = w_i^{-1}(\Sigma^-\setminus\Sigma_\eta^-)\cap\Sigma^+\\ & = \Sigma^+\setminus w_i^{-1}(\Sigma^+\cup\Sigma_\eta^-). \end{align*} Hence we have $2\langle\alpha,\lambda+\nu\rangle/\lvert\alpha\rvert^2\not\in\mathbb{Z}_{\ge 0}$ for all $\alpha\in(w'')^{-1}\Sigma^-\cap\Sigma^+$. By an argument of Knapp and Stein~\cite{MR582703}, $A(\sigma,\mu)$ is holomorphic at $\mu = \lambda$ if $\lambda$ satisfies the conditions of (4). Hence we get (4). \end{proof}
In the rest of this section, we denote the Bruhat filtration $I_i\subset J'(I(\sigma,\lambda))$ by $I_i(\lambda)$. The following result is a corollary of Proposition~\ref{prop:convergence and continuation}. \begin{lem}\label{lem:meromorphic extension}
Let $x\in I_i'$. Then there exists a distribution $x_t\in I_i(\lambda + t\rho)$ with meromorphic parameter $t$ such that $x_t|_{U_i}$ is a distribution with holomorphic parameter $t$ and $(x|_{U_i})|_{t = 0} = x$.
Moreover, if $Tx = 0$ for $T\in U(\mathfrak{g})$, then $Tx_t = 0$. \end{lem}
Let $C^\infty(K,\sigma)$ be the space of $\sigma$-valued $C^\infty$-functions. For $X\in \mathfrak{g}$ and $\lambda\in\mathfrak{a}^*$, we define an operator $D(X,\lambda)$ on $C^\infty(K,\sigma)$ as follows. For $\varphi\in C^\infty(K,\sigma)$, \begin{multline*}
(D(X,\lambda)\varphi)(k)\\ = \frac{d}{dt}\left.(\sigma\otimes e^{\lambda+\rho})(\exp (-H(\exp(-tX)k)))\varphi(\kappa(\exp(-tX)k))\right|_{t = 0}.\newsym{$D(X,\lambda)$} \end{multline*} If we regard $I(\sigma,\lambda)$ as a subspace of $C^\infty(K,\sigma)$, $(X\varphi)(k) = (D(X,\lambda)\varphi)(k)$ for $\varphi\in I(\sigma,\lambda)$. It is easy to see that for some $D_1$ and $D_2$ we have $D(X,\lambda + t\rho) = D_1 + tD_2$ for all $t\in\mathbb{C}$.
\begin{lem}\label{lem:holomorphic extension} Assume that the conditions of Lemma~\ref{lem:vanishing lemma} (1)--(3) hold. For $x\in I'_i$ there exists a distribution $x_t\in I_i(\lambda + t\rho)$ with holomorphic parameter $t$ defined near $t = 0$ such that $x_0 = x$ on $U_i$. \end{lem} \begin{proof} First we remark that $\eta = \eta'$ in Lemma~\ref{lem:acts nilp} by the condition (2) of Lemma~\ref{lem:vanishing lemma}.
We prove by induction on $i$. If $i = 1$, then $x\in I'_1$. Take a distribution $x_t\in I_1(\lambda + t\rho)$ as in Lemma~\ref{lem:meromorphic extension}. Then $x_t|_{U_1}$ is holomorphic with respect to the parameter $t$. Since $\supp x_t\subset X_1$, $x_t|_{(G/P)\setminus X_1}$ is holomorphic with respect to the parameter $t$. Hence $x_t$ is holomorphic with respect to the parameter $t$ on $U_1\cup ((G/P)\setminus X_1) = G/P$. We have the lemma.
Assume that $i > 1$. First we prove the following claim: for $y\in I_{i - 1}$, there exists a distribution $y_t\in I_{i - 1}(\lambda + t\rho)$ with holomorphic parameter $t$ defined near $t = 0$ such that $y_0 = y$. Using inductive hypothesis to $y|_{U_{i - 1}}$, there exists a distribution $y_t^{(i - 1)}\in I_{i - 1}(\lambda + t\rho)$ with holomorphic parameter $t$ defined near $t = 0$ such that $y_0^{(i - 1)} = y$ on $U_{i - 1}$. Since the supports of both sides are contained in $\bigcup_{j\le i - 1}N_0w_jP/P$, we have $y_0^{(i - 1)} = y$ on $\bigcup_{j\ge i - 1}N_0w_jP/P$. Using inductive hypothesis to $(y - y_0^{(i - 1)})|_{U_{i - 2}}$, there exists a distribution $y_t^{(i - 2)}\in I_{i - 2}(\lambda + t\rho)$ with holomorphic parameter $t$ defined near $t = 0$ such that $y_0^{(i - 2)} = y - y_0^{(i - 1)}$ on $U_{i - 2}$. Since the supports of both sides are contained in $\bigcup_{j\le i - 2}N_0w_jP/P$, we have $y_0^{(i - 1)} + y_0^{(i - 2)} = y$ on $\bigcup_{j\ge i - 2}N_0w_jP/P$. Iterating this argument, for $j = 1,\dots,i - 1$ there exists a distribution $y_t^{(j)}\in I_j(\lambda + t\rho)$ with holomorphic parameter $t$ defined near $t = 0$ such that $y = y_0^{(1)} + \dots + y_0^{(i - 1)}$. Hence we get the claim.
Now we prove the lemma. By Lemma~\ref{lem:meromorphic extension}, there exists a distribution $x'_t\in I_i(\lambda + t\rho)$ with meromorphic parameter $t$ such that $x'_t|_{U_i}$ is holomorphic and $(x'_t|_{U_i})|_{t = 0} = x$. Let $x'_t = \sum_{s = -p}^\infty x^{(s)}t^s$ be the Laurent series of $x'_t$. Now we prove the following claim: if there exists a distribution $x'_t = \sum_{s = -p}^\infty x^{(s)}t^s\in I_i(\lambda + t\rho)$ with meromorphic parameter $t$ defined near $t = 0$ such that $x'_t|_{U_i}$ is holomorphic and $(x'_t|_{U_i})|_{t = 0} = x$, then there exists $x_t\in I(\lambda)$ with holomorphic parameter $t$ defined near $t = 0$ such that $x_0|_{U_i} = x$. We prove the claim by induction on $p$.
If $p = 0$, we have nothing to prove. Assume $p > 0$. Take $E\in \mathfrak{n}_0$ and define differential operators $E_0$ and $E_1$ by $D(E,\lambda + t\rho) = E_0 + tE_1$. By Lemma~\ref{lem:acts nilp}, there exists a positive integer $k$ such that $(E_0 + tE_1 - \eta(E))^k x'_t = 0$. Hence, we have $(E_0 - \eta(E))^kx^{(-p)} = 0$. Since $x_t|_{U_i}$ is holomorphic, we have $\supp x^{(-p)}\subset \bigcup_{j < i}N_0w_jP/P$. Hence we have $x^{(-p)}\in I_{i - 1}$. By the claim stated in the third paragraph of this proof, there exists $x''_t\in I_{i - 1}(\lambda + t\rho)$ with holomorphic parameter $t$ defined near $t = 0$ such that $x''_0 = x^{(-p)}$. Using inductive hypothesis for $x_t' - t^{-p}x''_t$, we get the claim and the claim implies the lemma. \end{proof}
\begin{thm}\label{thm:succ quot is I'_i} \begin{enumerate} \item The module $I_i/I_{i - 1}$ is non-zero if and only if the conditions of Lemma~\ref{lem:vanishing lemma} (1)--(3) hold. \item If $I_i/I_{i - 1} \ne 0$ then we have $I_i/I_{i - 1} \simeq I'_i$. \end{enumerate} \end{thm} \begin{proof} Assume that the conditions of Lemma~\ref{lem:vanishing lemma} (1)--(3) hold. We prove that the restriction map $\Res_i\colon I_i\to I'_i$ is surjective.
For $x\in I'_i$, take $x_t\in I_i(\lambda + t\rho)$ as in Lemma~\ref{lem:holomorphic extension}. Then we have $\Res_i(x_0) = (x_0)|_{U_i} = x$. Hence $\Res_i$ is surjective. \end{proof}
\section{Twisting functors}\label{sec:Twisting functors} Arkhipov defined the \emph{twisting functor} for $\widetilde{w}\in\widetilde{W}$~\cite{MR2074588}. In this section, we define a modification of the twisting functor.
Let $\mathfrak{g}^\mathfrak{h}_\alpha$ be the root space of $\alpha\in\Delta$.\newsym{$\mathfrak{g}^\mathfrak{h}_\alpha$} Set $\mathfrak{u}_0 = \bigoplus_{\alpha\in \Delta^+}\mathfrak{g}^\mathfrak{h}_\alpha$\newsym{$\mathfrak{u}_0$}, $\overline{\mathfrak{u}_0} = \bigoplus_{\alpha\in \Delta^+}\mathfrak{g}^\mathfrak{h}_{-\alpha}$\newsym{$\overline{\mathfrak{u}_0}$} and $\mathfrak{u}_{0,{\widetilde{w}}} = \Ad(\widetilde{w})\overline{\mathfrak{u}_0}\cap \mathfrak{u}_0$.\newsym{$\mathfrak{u}_{0,{\widetilde{w}}}$} Let $\psi$ be a character of $\mathfrak{u}_{0,{\widetilde{w}}}$. Put $S_{\widetilde{w},\psi} = U(\mathfrak{g})\otimes_{U(\mathfrak{u}_{0,{\widetilde{w}}})}((U(\mathfrak{u}_{0,{\widetilde{w}}})^*)_{\text{$\mathfrak{h}$-finite}}\otimes_\mathbb{C}\psi)$.\newsym{$S_{\widetilde{w},\psi}$} This is a right $U(\mathfrak{u}_{0,{\widetilde{w}}})$-module and left $U(\mathfrak{g})$-module. We define a $U(\mathfrak{g})$-bimodule structure on $S_{\widetilde{w},\psi}$ in the following way. Let $\{e_1,\dots,e_l\}$ be a basis of $\mathfrak{u}_{0,{\widetilde{w}}}$ such that each $e_i$ is a root vector and $\bigoplus_{s\le t - 1}\mathbb{C} e_s$ is an ideal of $\bigoplus_{s\le t}\mathbb{C} e_s$ for each $t = 1,2,\dots,l$. Notice that a multiplicative set $\{(e_k - \psi(e_k))^n\mid n\in\mathbb{Z}_{\ge 0}\}$ satisfies the Ore condition for $k = 1,2,\dots,l$. Then we can consider the localization of $U(\mathfrak{g})$ by $\{(e_k - \psi(e_k))^n\mid n\in\mathbb{Z}_{\ge 0}\}$. We denote the resulting algebra by $U(\mathfrak{g})_{e_k - \psi(e_k)}$. Put $S_{e_k - \psi(e_k)} = U(\mathfrak{g})_{e_k - \psi(e_k)}/U(\mathfrak{g})$. Then $S_{e_k - \psi(e_k)}$ is a $U(\mathfrak{g})$-bimodule.\newsym{$S_{e_k - \eta(e_k)}$}
\begin{prop}\label{prop:1-dim decomposition of S_w} As a right $U(\mathfrak{u}_{0,w})$-module and left $U(\mathfrak{g})$-module, we have $S_{\widetilde{w},\psi} \simeq S_{e_1 - \psi(e_1)}\otimes_{U(\mathfrak{g})}S_{e_2 - \psi(e_2)}\otimes_{U(\mathfrak{g})}\dots\otimes_{U(\mathfrak{g})}S_{e_l - \psi(e_l)}$. Moreover, the $U(\mathfrak{g})$-bimodule structure induced from this isomorphism is independent of a choice of $e_i$. \end{prop}
The proof of this proposition is similar to that of Arkhipov~\cite[Thoerem~2.1.6]{MR2074588}. We omit it. An element of the right hand side is written as a sum of a form $(e_1 - \eta(e_1))^{-(k_1 + 1)}\otimes\dots\otimes (e_l - \eta(e_l))^{-(k_l + 1)}T$ for $T\in U(\mathfrak{g})$. We denote this element by $(e_1 - \eta(e_1))^{-(k_1 + 1)}\dotsm (e_l - \eta(e_l))^{-(k_l + 1)}T$ for short.
Proposition~\ref{prop:1-dim decomposition of S_w} gives the $U(\mathfrak{g})$-bimodule structure of $S_{\widetilde{w},\psi}$. For a $U(\mathfrak{g})$-module $V$, we define a $U(\mathfrak{g})$-module $T_{\widetilde{w},\psi}V$ by $T_{\widetilde{w},\psi}V = S_{\widetilde{w},\psi}\otimes_{U(\mathfrak{g})}(\widetilde{w}V)$. (Recall that $\widetilde{w}V$ is a $\mathfrak{g}$-module twisted by $\widetilde{w}$. See Notation.) This gives the twisting functor $T_{\widetilde{w},\psi}$.\newsym{$T_{\widetilde{w},\psi}$} If $\psi$ is the trivial representation, $T_{\widetilde{w},\psi}$ is the twisting functor defined by Arkhipov. We put $T_{\widetilde{w}} = T_{\widetilde{w},0}$ where $0$ is the trivial representation.
The restriction map gives a map $N_K(\mathfrak{h})/Z_K(\mathfrak{h})\to W$ and its kernel is isomorphic to $N_{M_0}(\mathfrak{t}_0)/Z_{M_0}(\mathfrak{t}_0)$ (Recall that $\mathfrak{t}_0$ is a Cartan subalgebra of $\mathfrak{m}_0$). The last group is isomorphic to $\widetilde{W_{M_0}}$.
\begin{lem}\label{lem:good lift for W}
Let $w\in W$. Then there exists $\iota(w)\in N_K(\mathfrak{h})$ such that $\Ad(\iota(w))|_{\mathfrak{a}_0} = w$ and $\Ad(\iota(w))(\Delta_{M_0}^+) = \Delta_{M_0}^+$. \end{lem} \begin{proof}
Since $W\simeq N_K(\mathfrak{a}_0)/Z_K(\mathfrak{a}_{0})$, there exists $k\in N_K(\mathfrak{a}_{0})$ such that $\Ad(k)|_{\mathfrak{a}_0} = w$. Then $k$ normalizes $M_0$. Hence, there exists $m\in M_0$ such that $km$ normalizes $T_0$. This implies $km\in N_K(A_0T_0)$. Take $w'\in N_{M_0}(\mathfrak{t}_0)$ such that $\Ad(kmw')(\Delta^+_{M_0}) = \Delta^+_{M_0}$ and put $\iota(w) = kmw'$. Then $\iota(w)$ satisfies the conditions of the lemma. \end{proof}
The map $\iota$ gives an injective map $W\to N_K(\mathfrak{h})/Z_K(\mathfrak{h})$. Since the group $N_K(\mathfrak{h})/Z_K(\mathfrak{h})$ can be regard as a subgroup of $\widetilde{W}$, we can regard $W$ as a subgroup of $\widetilde{W}$. Hence, we can define the twisting functor $T_{w,\psi}$ for $w\in W$ and the character $\psi$ of $\Ad(w)\overline{\mathfrak{n}_0}\cap \mathfrak{n}_0$. For a simplicity, we write $w$ instead of $\iota(w)$. (We regard $W$ as a subgroup of $\widetilde{W}$ by $\iota$.)
\begin{prop}\label{prop:decomposition of T_ww'} Let $w,w'\in W$ and $\psi$ a character of $\Ad(ww')\overline{\mathfrak{n}_0}\cap \mathfrak{n}_0$. Assume that $\ell(w) + \ell(w') = \ell(ww')$ where $\ell(w)$ is the length of $w\in W$. Then we have $T_{w,\psi}T_{w',w^{-1}\psi} = T_{ww',\psi}$. \end{prop} \begin{proof} By the assumption, we have $\Sigma^+\cap ww'\Sigma^- = (\Sigma^+\cap w\Sigma^-)\cup w(\Sigma^+\cap w'\Sigma^-)$. Put $\Delta_0^\pm = \Delta^\pm\setminus\Delta_{M_0}^\pm$. Then we have $\Delta_0^+\cap ww'\Delta_0^- = (\Delta_0^+\cap w\Delta_0^-)\cup w(\Delta_0^+\cap w'\Delta_0^-)$. Since $w\Delta_{M_0}^\pm = \Delta_{M_0}^\pm$, we have $\Delta_0^+\cap w\Delta_0^- = \Delta^+\cap w\Delta^-$. Hence, $\Delta^+\cap ww'\Delta^- = (\Delta^+\cap w\Delta^-)\cup w(\Delta^+\cap w'\Delta^-)$. This implies that $\widetilde{\ell}(w) + \widetilde{\ell}(w') = \widetilde{\ell}(ww')$ where $\widetilde{\ell}(w)$ is the length of $w$ as an element of $\widetilde{W}$. Hence, the proposition follows from the construction of the twisting functor (See Andersen and Lauritzen~\cite[Remark~6.1 (ii)]{MR1985191}). \end{proof}
\begin{lem}\label{lem:Xe^-k} Let $e$ be a nilpotent element of $\mathfrak{g}$, $X\in \mathfrak{g}$ and $k\in\mathbb{Z}_{\ge 0}$. For $c\in \mathbb{C}$ we have the following equation in $U(\mathfrak{g})_{e - c}$. \[
X(e - c)^{-(k + 1)} = \sum_{n = 0}^\infty \binom{n + k}{k}(e - c)^{-(n + k + 1)}\ad (e)^n(X). \] \end{lem} \begin{proof} We prove the lemma by induction on $k$. If $k = 0$, then the lemma is well-known. Assume that $k > 0$. Then we have \begin{align*} X(e - c)^{-(k + 1)} & = \sum_{k_0 = 0}^\infty (e - c)^{-(k_0 + 1)}\ad(e)^{k_0}(X) (e - c)^{-k}\\ & = \sum_{k_0 = 0}^\infty \sum_{k_1 = 0}^\infty \binom{k_1 + k - 1}{k - 1}(e - c)^{-(k_0 + k_1 + k + 1)}\ad(e)^{k_0 + k_1}(X)\\ & = \sum_{n = 0}^\infty \sum_{l' = 0}^n \binom{l' + k - 1}{k - 1}(e - c)^{-(n + k + 1)}\ad(e)^n(X)\\ & = \sum_{n = 0}^\infty \binom{n + k}{k}(e - c)^{-(n + k + 1)}\ad(e)^n(X). \end{align*} This proves the lemma. \end{proof}
\section{The module $I_i/I_{i - 1}$}\label{sec:The module I_i/I_i-1} Put $J_i = U(\mathfrak{g})\otimes_{U(\mathfrak{p})}J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho})$\newsym{$J_i$}, where $\mathfrak{n}$ acts $J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho})$ trivially. In this section, we prove the following theorem. \begin{thm}\label{thm:structure of I_i/I_{i - 1}} Assume that $I_i/I_{i - 1} \ne 0$. Then we have $I_i/I_{i - 1} \simeq T_{w_i,\eta}J_i$. \end{thm}
Notice that $\mathfrak{u}_{0,{w_i}} = \Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0$ since $w_i(\Delta_{M}^+) \subset \Delta^+$. In this section fix $i\in\{1,\dots,l\}$ and a basis $\{e_1,e_2,\dots,e_l\}$ of $\mathfrak{u}_{0,w_i}$ such that each vector $e_i$ is a root vector and $\bigoplus_{s\le t-1}\mathbb{C} e_s$ is an ideal of $\bigoplus_{s\le t}\mathbb{C} e_s$. Let $\alpha_s$ be the restricted root with respect to $e_s$. As in Section~\ref{sec:vanishing theorem}, for $\mathbf{k} = (k_1,\dots,k_l)\in\mathbb{Z}_{\ge 0}^l$ we denote $\ad(e_l)^{k_l}\dotsm \ad(e_1)^{k_1}$ by $\ad(e)^\mathbf{k}$ and $((-x_1)^{k_1}/k_1!)\dotsm((-x_l)^{k_l}/k_l!)$ by $f_\mathbf{k}$.
\begin{lem}\label{lem:left2right for I'_i} We have \[
I'_i = \left\{\sum_{s = 1}^t \delta_i(T_s,f_s\eta_i^{-1},u'_s)\Bigm|
\begin{array}{ll}
T_s\in U(\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0),&
f_s\in \mathcal{P}(O_i),\\
u'_s\in J'_{w_i^{-1}\eta}(\sigma\otimes(\lambda + \rho))
\end{array}
\right\}. \] \end{lem} \begin{proof} By Lemma~\ref{lem:left2right}, we have \[
T((f\otimes u')\delta_i) =
\sum_{\mathbf{k}\in\mathbb{Z}_{\ge 0}^l}\delta_i(\ad(e)^\mathbf{k}T,ff_\mathbf{k},u') \] for $T\in U(\mathfrak{g})$, $f\in \mathcal{P}(O_i)\eta_i^{-1}$ and $u'\in \sigma'$. Hence, the left hand side is a subset of the right hand side. Define $f_\mathbf{k}'\in\mathcal{P}(O_i)$ by $f_\mathbf{k}' = (x_1^{k_1}/k_1!)\dotsm(x_l^{k_l}/k_l!)$. By the similar calculation of Lemma~\ref{lem:left2right}, we have \[
\delta_i(T,f,u') =
\sum_{\mathbf{k}\in\mathbb{Z}_{\ge 0}^l}(\ad(e)^{\mathbf{k}}T)(((ff_\mathbf{k}')\otimes u')\delta_i). \] This implies that the right hand side is contained in the left hand side. \end{proof}
By the definition of the twisting functor and Poincar\"e-Birkhoff-Witt theorem, we have the following lemma. For $\mathbf{k} = (k_1,\dots,k_l)\in\mathbb{Z}^l$ put $(e - \eta(e))^\mathbf{k} = (e_1 - \eta(e_1))^{k_1}\dotsm (e_l - \eta(e_l))^{k_l}\in S_{w,\eta}$. Set $\mathbf{1} = (1,\dots,1)\in\mathbb{Z}^l$. \begin{lem}\label{lem:induction + twisting} Let $V$ be a $\mathfrak{p}$-module. Then we have a $\mathbb{C}$-vector space isomorphism \begin{multline*} \left(\sum_{\mathbf{k}\in\mathbb{Z}_{\ge 0}^l}\mathbb{C} (e - \eta(e))^{-(\mathbf{k} + \mathbf{1})}\right)\otimes_{U(\Ad(w_i)\overline{\mathfrak{n}}\cap\mathfrak{n}_0)}U(\mathfrak{g})\otimes_{U(\Ad(w_i)\mathfrak{p})}w_iV\\ \simeq T_{w_i}(U(\mathfrak{g})\otimes_{U(\mathfrak{p})}V) \end{multline*} given by $E\otimes T\otimes v\mapsto ET\otimes (1\otimes v)$. (Notice that $ET\in S_{w_i,0}$.) \end{lem}
\begin{proof}[Proof of Theorem~\ref{thm:structure of I_i/I_{i - 1}}] By Lemma~\ref{lem:left2right for I'_i}, we have an isomorphism as a vector space, \[
I'_i \simeq
\mathcal{P}(O_i)
\otimes_{U(\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0)} U(\mathfrak{g})
\otimes_{U(\Ad(w_i)\mathfrak{p})} w_iJ'_{w_i^{-1}\eta}(\sigma\otimes(\lambda + \rho)) \] given by $\delta_i(T,f,u')\mapsto f\otimes T\otimes u'$.
Notice that $\mathfrak{u}_{0,w_i} = \Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0$ since $w_i\in W(M)$. By Lemma~\ref{lem:induction + twisting}, we have \begin{multline*} T_{w_i,\eta}(J_i) \simeq \left(\sum_{\mathbf{k}\in\mathbb{Z}_{\ge 0}^l} \mathbb{C} (e - \eta(e))^{-(\mathbf{k}+\mathbf{1})}\right) \otimes_{U(\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0)} U(\mathfrak{g})\\ \otimes_{U(\Ad(w_i)\mathfrak{p})} w_iJ'_{w_i^{-1}\eta}(\sigma\otimes(\lambda + \rho)). \end{multline*} Here we remark $\sum_{\mathbf{k}\in\mathbb{Z}_{\ge 0}^l} \mathbb{C} (e - \eta(e))^{-(\mathbf{k}+\mathbf{1})}$ is an $\Ad(w_i)\overline{\mathfrak{n}}\cap\mathfrak{n}_0$-stable subspace of $S_{w_i,\eta}$. Hence, we can define a $\mathbb{C}$-vector space isomorphism $\Phi\colon T_{w_i,\eta}(J_i)\to I'_i$ by \[
\Phi((e - \eta(e))^{-(\mathbf{k}+\mathbf{1})}\otimes T\otimes u')
=
\delta_i(T,f_\mathbf{k}\eta_i^{-1},u'). \] We prove that $\Phi$ is a $\mathfrak{g}$-homomorphism.
Fix $X\in \mathfrak{g}$. We prove that \[ \Phi(X((e - \eta(e))^{-(\mathbf{k}+\mathbf{1})}\otimes T\otimes u')) = X\Phi((e - \eta(e))^{-(\mathbf{k}+\mathbf{1})}\otimes T\otimes u'). \] By Lemma~\ref{lem:Xe^-k}, we have \begin{multline*}
X((e - \eta(e))^{-(\mathbf{k}+\mathbf{1})}\otimes T\otimes u')\\
= \sum_{p_s\ge 0}\binom{p_1 + k_1}{k_1}\dotsm \binom{p_l + k_l}{k_l}(e - \eta(e))^{-(\mathbf{k}+\mathbf{p}+\mathbf{1})}\otimes(\ad(e)^\mathbf{p}X)T\otimes u'. \end{multline*} where $\mathbf{p} = (p_1,\dots,p_l)$. Hence, we have \begin{multline*}
\Phi(X((e - \eta(e))^{-(\mathbf{k}+\mathbf{1})}\otimes T\otimes u'))\\
= \sum_{p_s\ge 0}\delta_i\left((\ad(e)^\mathbf{p}X)T,\left(\frac{(-x_1)^{k_1 + p_1}}{k_1!p_1!}\dotsm\frac{(-x_l)^{k_l + p_l}}{k_l!p_l!}\right)\eta_i^{-1}, u'\right). \end{multline*} By Lemma~\ref{lem:left2right}, we have \begin{align*} X\Phi((e - \eta(e))^{-(\mathbf{k} + 1)}\otimes T\otimes u') & = X\delta_i(T,f_\mathbf{k}\eta_i^{-1},u')\\ & = \sum_{\mathbf{p}\in\mathbb{Z}_{\ge 0}^l}\delta_i((\ad(e)^{\mathbf{p}}X)T,f_\mathbf{k}f_\mathbf{p}\eta_i^{-1},u'). \end{align*} Hence, we have the theorem. \end{proof}
\section{The module $J^*_\eta(I(\sigma,\lambda))$}\label{sec:the module J^*_eta(I(sigma,lambda))} Now we investigate a module $J^*_\eta(I(\sigma,\lambda))$. For a finite-length Fr\'echet representation $V$ of $G$, put $J(V) = (\varprojlim_{k\to \infty}(V_{\text{$K$-finite}}/\mathfrak{n}_0^kV_{\text{$K$-finite}}))_{\text{$\mathfrak{a}$-finite}}$\newsym{$J(V)$}. This is also called the Jacquet module of V~\cite{MR562655}. Define a category $\mathcal{O}'_{P_0}$\newsym{$\mathcal{O}'_{P_0}$} by the full subcategory of finitely generated $\mathfrak{g}$-modules consisting an object $V$ satisfying the following conditions. \begin{enumerate} \item The action of $\mathfrak{p}_0$ is locally finite. (In particular, the action of $\mathfrak{n}_0$ is locally nilpotent.) \item The module $V$ is $Z(\mathfrak{g})$-finite. \item The group $M_0$ acts on $V$ and its differential coincides with the action of $\mathfrak{m}_0\subset \mathfrak{g}$. \item For $\nu\in\mathfrak{a}_0^*$ let $V_\nu$ be the generalized $\mathfrak{a}_0$-weight space with weight $\nu$. Then $V = \bigoplus_{\nu\in\mathfrak{a}_0^*}V_\nu$ and $\dim V_\nu < \infty$. \end{enumerate} We define the category $\mathcal{O}_{\overline{P_0}}'$ similarly.\newsym{$\mathcal{O}'_{\overline{P_0}}$} Then for a finite-length Fr\'echet representation $V$ of $G$ we have $J(V)\in\mathcal{O}_{\overline{P_0}}'$ and $J^*(V)\in\mathcal{O}_{P_0}'$. For a $U(\mathfrak{g})$-module $V$, put $D'(V) = (V^*)_{\text{$\mathfrak{h}$-finite}}$ and $C(V) = (D'(V))^*$.\newsym{$D'(V)$}\newsym{$C(V)$} Denote a full-subcategory of $\mathfrak{g}$-modules consisting finitely-generated and locally $\mathfrak{h}\oplus\mathfrak{u}$-finite modules by $\mathcal{O}'$\newsym{$\mathcal{O}'$}. If $V$ is an object of the category $\mathcal{O}'$ then $D'D'(V) \simeq V$. The relation between $J^*$ and $J$ is as follows. \begin{prop}\label{prop:relation J^* and J} Let $V$ be a finite-length Fr\'echet representation of $G$. Then we have $J^*(V) \simeq D'(J(V))$. \end{prop}
The character $\eta\colon \mathfrak{n}_0\to \mathbb{C}$ defines an algebra homomorphism $U(\mathfrak{n}_0)\to \mathbb{C}$ by the universality of the universal enveloping algebra. Let $\Ker\eta$ be the kernel of this algebra homomorphism and put $\Gamma_\eta(V) = \{v\in V\mid \text{for some $k$, $(\Ker\eta)^kv = 0$}\}$.\newsym{$\Gamma_\eta(V)$} First we prove the following proposition. \begin{prop}\label{prop:relation J^*_eta and J_eta} Let $V$ be a finite-length Fr\'echet representation of $G$. Then we have $J^*_\eta(V) \simeq \Gamma_\eta(J(V)^*)$. \end{prop} \begin{proof} Recall that $\mathfrak{p}_\eta = \mathfrak{m}_\eta\oplus\mathfrak{a}_\eta\oplus\mathfrak{n}_\eta$ is the complexification of the Lie algebra of the parabolic subgroup corresponding to $\supp\eta$ (Section~\ref{sec:Analytic continuation}). If $\supp\eta = \Pi$, this proposition is proved by Matumoto~\cite[Theorem~4.9.2]{MR1047117}.
Put $I = V_{\text{$K$-finite}}$. Let $\eta_0\colon U(\mathfrak{m}\cap \mathfrak{n}_0)\to \mathbb{C}$ be the restriction of $\eta$ on $U(\mathfrak{m}\cap \mathfrak{n}_0)$. Then we have \[ J^*_\eta(V) = \varinjlim_{k,l}(I/\mathfrak{n}_\eta^l(\Ker\eta_0)^kI)^* = \varinjlim_{k,l}((I/\mathfrak{n}_\eta^lI)/(\Ker\eta_0)^k(I/\mathfrak{n}_\eta^lI))^*. \] For a $U(\mathfrak{g})$-module $V_0$, put $G(V_0) = (\varprojlim_k V_0/\mathfrak{n}_0^kV_0)_{\text{$\mathfrak{a}$-finite}}$. For a $U(\mathfrak{m}_\eta\oplus\mathfrak{a}_\eta)$-module $V_1$, put $G_{M_\eta}(V_1) = (\varprojlim_k V_1/(\mathfrak{m}_\eta\cap\mathfrak{n}_0)^kV_1)_{\textrm{$(\mathfrak{m}\cap\mathfrak{a}_0)$-finite}}$. Since $I/\mathfrak{n}_\eta^lI$ is a Harish-Chandra module of $\mathfrak{m}_\eta\oplus\mathfrak{a}_\eta$, $J^*_{\eta_0}(I/\mathfrak{n}_\eta^lI) = \Gamma_{\eta_0}(G_{M_\eta}(I/\mathfrak{n}_\eta^lI)^*)$ by the result of Matumoto. Taking a subspace annihilated by $(\Ker\eta_0)^k$, we have \[ ((I/\mathfrak{n}_\eta^lI)/(\Ker\eta_0)^k(I/\mathfrak{n}_\eta^lI))^* = (G_{M_\eta}(I/\mathfrak{n}_\eta^lI)/(\Ker\eta_0)^kG_{M_\eta}(I/\mathfrak{n}_\eta^lI))^*. \] Since $I$ is a finitely-generated $U(\mathfrak{n}_0)$-module, the left hand side is finite-dimensional. Hence, we have \[ (I/\mathfrak{n}_\eta^lI)/(\Ker\eta_0)^k(I/\mathfrak{n}_\eta^lI) = G_{M_\eta}(I/\mathfrak{n}_\eta^lI)/(\Ker\eta_0)^kG_{M_\eta}(I/\mathfrak{n}_\eta^lI). \] It is sufficient to prove that $G_{M_\eta}(I/\mathfrak{n}_\eta^lI) = G(I)/\mathfrak{n}_\eta^lG(I)$. We have \[ (I/\mathfrak{n}_\eta^lI)/(\mathfrak{m}_\eta\cap\mathfrak{n}_0)^k(I/\mathfrak{n}_\eta^lI) = I/(\mathfrak{m}_\eta\cap\mathfrak{n}_0)^k\mathfrak{n}_\eta^lI = G(I)/(\mathfrak{m}_\eta\cap\mathfrak{n}_0)^k\mathfrak{n}_\eta^lG(I). \] Taking the projective limit we have $G_{M_\eta}(I/\mathfrak{n}_\eta^lI) = G_{M_\eta}(G(I)/\mathfrak{n}_\eta^lG(I))$. Since $G(I)/\mathfrak{n}_\eta^lG(I)\in \mathcal{O}'_{M_\eta\cap\overline{P_0}}$ we have $G_{M_\eta}(G(I)/\mathfrak{n}_\eta^lG(I)) = G(I)/\mathfrak{n}_\eta^lG(I)$. \end{proof}
Combining Theorem~\ref{thm:structure of I_i/I_{i - 1}}, Proposition~\ref{prop:relation J^*_eta and J_eta} and the automatic continuation theorem~\cite[Theorem~4.8]{MR727854}, we have the following theorem.
\begin{thm}\label{thm:stucture of J^*(I(sigma,lambda))} There exists a filtration $0 = \widetilde{I_1}\subset\cdots \subset \widetilde{I_r} = J^*_\eta(I(\sigma,\lambda))$ such that $\widetilde{I_i}/\widetilde{I_{i - 1}} \simeq \Gamma_\eta(C(T_{w_i}(U(\mathfrak{g})\otimes_{U(\mathfrak{p})} J^*(\sigma\otimes e^{\lambda+\rho}))))$.\newsym{$\widetilde{I_i}$} \end{thm}
\section{Whittaker vectors}\label{sec:Whittaker vectors} In this section we study Whittaker vectors of $I(\sigma,\lambda)'$ and $(I(\sigma,\lambda)_{\text{$K$-finite}})^*$ (Definition~\ref{defn:Whittaker vectors}).
For $i$ such that $I_i/I_{i - 1}\ne 0$, we define some maps as follows. Let $\gamma_1$ be the first projection with respect to the decomposition $U(\mathfrak{g}) = U(\mathfrak{l}_\eta)\oplus(\overline{\mathfrak{n}_\eta}U(\mathfrak{g}) + U(\mathfrak{g})\mathfrak{n}_\eta)$. Notice that by Lemma~\ref{lem:vanishing lemma} if $I_i/I_{i - 1}\ne 0$ then we have $\mathfrak{l}_\eta\cap \Ad(w_i)\overline{\mathfrak{n}}\subset \mathfrak{n}_0$. Define $\gamma_2$ by the first projection with respect to the decomposition $U(\mathfrak{l}_\eta) = U(\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{p}) \oplus U(\mathfrak{l}_\eta)\Ker\eta|_{\mathfrak{l}_\eta\cap \Ad(w_i)\overline{\mathfrak{n}}}$. Let $\gamma_3$ be the first projection with respect to the decomposition $U(\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{p}) = U(\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{l})\oplus (\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{n})U(\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{p})$. Finally define $\gamma_4$ by the first projection with respect to the decomposition $U(\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{l}) = U(\mathfrak{h})\oplus((\overline{\mathfrak{u}_0}\cap\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{l})U(\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{l}) + U(\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{l})(\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{l} \cap \mathfrak{u}_0))$. Then the restriction of $\gamma_4\circ\gamma_3\circ\gamma_2\circ\gamma_1$ on $Z(\mathfrak{g})$ is the (non-shifted) Harish-Chandra homomorphism. If $x\in \Wh_\eta(I_i/I_{i - 1})$ then $Tx = \gamma_2\gamma_1(T)x$ for $T\in Z(\mathfrak{g})$. \begin{align*} \gamma_1\colon & U(\mathfrak{g}) = U(\mathfrak{l}_\eta)\oplus(\overline{\mathfrak{n}_\eta}U(\mathfrak{g}) + U(\mathfrak{g})\mathfrak{n}_\eta)\to U(\mathfrak{l}_\eta),\\
\gamma_2\colon & U(\mathfrak{l}_\eta) = U(\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{p}) \oplus U(\mathfrak{l}_\eta)\Ker\eta|_{\mathfrak{l}_\eta\cap \Ad(w_i)\overline{\mathfrak{n}}}\to U(\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{p}),\\ \gamma_3\colon & U(\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{p}) = U(\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{l})\oplus (\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{n})U(\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{p}) \tag*{$\to U(\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{l})$},\\ \gamma_4\colon & U(\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{l}) = U(\mathfrak{h})\oplus((\overline{\mathfrak{u}_0}\cap\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{l})U(\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{l}) \tag*{$+ U(\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{l})(\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{l} \cap \mathfrak{u}_0))\to U(\mathfrak{h}).$} \end{align*} \newsym{$\gamma_1,\gamma_2,\gamma_3,\gamma_4$} \begin{lem}\label{lem:lemma of infinitesimal character}
Let $V$ be a $U(\mathfrak{g})$-module with an infinitesimal character $\widetilde{\lambda}$, $\chi$ a character of $Z(\mathfrak{g})$ such that $z\in Z(\mathfrak{g})$ acts by $\chi(z)$ on $V$. Take a nonzero element $v\in V$ such that $(\gamma_3\gamma_2\gamma_1(z) - \chi(z))v = 0$. Moreover, assume that there exists $\mu\in\mathfrak{a}^*$ such that $Hv = (w_i\mu+\rho_0)(H)v$ for all $H\in \Ad(w_i)\mathfrak{a}$. Then there exists $\widetilde{w}\in \widetilde{W}$ such that $\widetilde{w}\widetilde{\lambda}|_\mathfrak{a} = \mu$. \end{lem} \begin{proof}
Put $Z = (\gamma_3\gamma_2\gamma_1(Z(\mathfrak{g}))U(\Ad(w_i)\mathfrak{a}))$. By the assumption, there exists a character $\chi_0$ of $Z$ such that $zv = \chi_0(z)v$ for all $z\in Z$. By a theorem of Harsh-Chandra, $\gamma_4|_Z$ is injective and $\gamma_4(Z)\subset U(\mathfrak{h})$ is finite. Hence there exists an element $\widetilde{\lambda_1}\in \mathfrak{h}^*$ such that $\widetilde{\lambda_1}\circ \gamma_4 = \chi_0$ where we denote the algebra homomorphism $U(\mathfrak{h})\to \mathbb{C}$ induced from $\widetilde{\lambda_1}$ by the same letter $\widetilde{\lambda_1}$. Since $V$ has an infinitesimal character $\widetilde{\lambda}$, we have $\widetilde{\lambda_1}\in \widetilde{W}\widetilde{\lambda} + \widetilde{\rho}$. Since $\gamma_4$ is trivial on $U(\Ad(w_i)\mathfrak{a})$, $\widetilde{\lambda_1}|_{\Ad(w_i)\mathfrak{a}} = (w_i\mu + \rho_0)|_{\Ad(w_i)\mathfrak{a}}$. The restriction of $\widetilde{\rho}$ to $\mathfrak{a}_0$ is $\rho_0$. Hence $\widetilde{\rho}|_{\Ad(w_i)\mathfrak{a}} = \rho_0|_{\Ad(w_i)\mathfrak{a}}$. Then for some $\widetilde{w}\in\widetilde{W}$ we have $w_i\mu|_{\Ad(w_i)\mathfrak{a}} = \widetilde{w}\widetilde{\lambda}|_{\Ad(w_i)\mathfrak{a}}$. We get the lemma. \end{proof}
\begin{lem}\label{lem:to outside delta_i} Let $X_1,\dots,X_n\in \mathfrak{g}$, $f_1\in C^\infty(O_i)$, $f_2\in C^\infty(U_i)$, $u'\in (\sigma\otimes e^{\lambda + \rho})'$. Assume that $\widetilde{R'_i}(X_s)(f_2) = 0$ for all $s = 1,\dots,n$. Then we have \[
\delta_i(X_1\dotsm X_n,f_1f_2,u') = \delta_i(X_1\dotsm X_n,f_1,u')f_2. \] \end{lem} \begin{proof} Put $T = X_1\dotsm X_n$. By the assumption and Leibniz's rule, we have \[
f_2(nw_i)(\widetilde{R_i}(T)\varphi)(nw_i) = (\widetilde{R_i}(T)(\varphi f_2))(nw_i) \] Hence, by the definition, for $\varphi\in C^\infty_c(U_i,\mathcal{L})$, we have \begin{align*}
& \langle \delta_i(T,f_1f_2,u'),\varphi\rangle\\
& = \int_{w_i\overline{N}w_i^{-1}\cap N_0}f_1(nw_i)f_2(nw_i)(u'(\widetilde{R}_i(T)\varphi)(nw_i))dn\\
& = \int_{w_i\overline{N}w_i^{-1}\cap N_0}f_1(nw_i)(u'(\widetilde{R}_i(T)(\varphi f_2))(nw_i))dn\\
& = \langle\delta_i(T,f_1,u'),f_2\varphi\rangle\\
& = \langle\delta_i(T,f_1,u')f_2,\varphi\rangle. \end{align*} We get the lemma. \end{proof}
\begin{lem}\label{lem:property of V(nu)} For $\nu\in \mathfrak{a}^*$ put \[
V(\nu) = \left\{\sum_s \delta_i(S_s,h_s\eta_i^{-1},v_s')\Biggm|
\begin{array}{l}
S_s\in U(\Ad(w_i)\overline{\mathfrak{n}}\cap \overline{\mathfrak{n}_0}),\ h_s\in\mathcal{P}(O_i),\\ v'_s\in J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho}),\\ w_i^{-1}(\wt h_s + \wt S_s)|_\mathfrak{a} = \nu
\end{array}
\right\}. \] Here, $\wt h_s$ is an $\mathfrak{a}_0$-weight of $h_s$ with respect to $D_i$ (see page~\pageref{symbol:D_i}) and $\wt S_s$ is an $\mathfrak{a}_0$-weight of $S_s$ with respect to the adjoint action. Define $\widetilde{\eta_i}\in C^\infty(U_i)$ by $\widetilde{\eta_i}(nn_0w_iP/P) = \eta_i(n)$ for $n\in w_i\overline{N}w_i^{-1}\cap N_0$ and $n_0\in w_i\overline{N}w_i^{-1}\cap \overline{N_0}$. \begin{enumerate} \item Let $X\in U(\mathfrak{l}_\eta\cap \Ad(w_i)\mathfrak{p})$. Assume that $X$ is an $\mathfrak{a}_0$-weight vector. For $\delta_i(T,f\eta_i^{-1},u')\in V(\nu)$, we have \[
X\delta_i(T,f\eta_i^{-1},u') - (X\delta_i(T,f\eta_i^{-1},u'))\widetilde{\eta_i}^{-1}\in\sum_{\nu' > \nu}V(\nu' + w_i^{-1}\wt T|_\mathfrak{a}). \] here, $\wt T$ is an $\mathfrak{a}_0$-weight of $T$ with respect to the adjoint action. \item For $\delta_i(S_s,h_s\eta_i^{-1},v'_s)\in V(\nu)$, we have \[
\sum_s \delta_i(S_s,h_s,v'_s)\widetilde{\eta_i}^{-1}\not\in \sum_{\nu' > \nu}V(\nu'). \] \end{enumerate} \end{lem} \begin{proof} (1) Fix a basis $\{e_1,e_2,\dots,e_l\}$ of $\mathfrak{u}_{0,w_i}$ such that each vector $e_i$ is a root vector and $\bigoplus_{s\le t-1}\mathbb{C} e_s$ is an ideal of $\bigoplus_{s\le t}\mathbb{C} e_s$. Let $\alpha_s$ be the restricted root of $e_s$. As in Section~\ref{sec:vanishing theorem}, for $\mathbf{k} = (k_1,\dots,k_l)\in\mathbb{Z}_{\ge 0}^l$ we denote $\ad(e_l)^{k_l}\dotsm \ad(e_1)^{k_1}$ by $\ad(e)^\mathbf{k}$ and $((-x_1)^{k_1}/k_1!)\dotsm((-x_l)^{k_l}/k_l!)$ by $f_\mathbf{k}$. By Lemma~\ref{lem:left2right}, \[
X\delta_i(T,f\eta_i^{-1},u') = \sum_{\mathbf{k}\in\mathbb{Z}_{\ge 0}^l}\delta_i((\ad(e)^\mathbf{k}X)T,ff_\mathbf{k}\eta_i^{-1},u'). \] Take $a_\mathbf{k}^{(p)}\in U(\Ad(w_i)\overline{\mathfrak{n}}\cap\mathfrak{n}_0)$, $b_\mathbf{k}^{(p)}\in U(\Ad(w_i)\overline{\mathfrak{n}}\cap \overline{\mathfrak{n}_0})$ and $c_\mathbf{k}^{(p)}\in U(\Ad(w_i)\mathfrak{p})$ such that $(\ad(e)^\mathbf{k}X)T = \sum_p a_\mathbf{k}^{(p)}b_\mathbf{k}^{(p)}c_\mathbf{k}^{(p)}$ and $\wt((\ad(e)^\mathbf{k}X)T) = \wt a_\mathbf{k}^{(p)} + \wt b_\mathbf{k}^{(p)} + \wt c_\mathbf{k}^{(p)}$. Then \begin{align*} &\delta_i((\ad(e)^\mathbf{k}X)T,ff_\mathbf{k}\eta_i^{-1},u')\\ & = \sum_p \delta_i(a_\mathbf{k}^{(p)}b_\mathbf{k}^{(p)}c_\mathbf{k}^{(p)},ff_\mathbf{k}\eta_i^{-1},u')\\ & = \sum_p \delta_i(b_\mathbf{k}^{(p)},R'_i(-a_\mathbf{k}^{(p)})(ff_\mathbf{k}\eta_i^{-1}),\Ad(w_i)^{-1}(c_\mathbf{k}^{(p)})u') \end{align*} By the Leibniz rule, there exists a subset $\mathcal{A}^{(p)}_\mathbf{k} \subset \{(a',a'')\in U(\Ad(w_i)\overline{\mathfrak{n}}\cap\mathfrak{n}_0)^2\mid \wt a' + \wt a'' = \wt a_\mathbf{k}^{(p)},\ a''\not\in\mathbb{C}\}$ such that \begin{align*}
& \delta_i(b_\mathbf{k}^{(p)},R'_i(-a_\mathbf{k}^{(p)})(ff_\mathbf{k}\eta_i^{-1}) - R'_i(-a_\mathbf{k}^{(p)})(ff_\mathbf{k})\eta_i^{-1},\Ad(w_i)^{-1}(c_\mathbf{k}^{(p)})u')\\
& = \sum_{(a',a'')\in\mathcal{A}^{(p)}_\mathbf{k}}\delta_i(b_\mathbf{k}^{(p)},R'_i(a')(ff_\mathbf{k})R'_i(a'')(\eta_i^{-1}),\Ad(w_i)^{-1}c_\mathbf{k}^{(p)}u')\\
& =
\sum_{(a',a'')\in\mathcal{A}^{(p)}_\mathbf{k}}-\eta(a'')\delta_i(b_\mathbf{k}^{(p)},R'_i(a')(ff_\mathbf{k})\eta_i^{-1},\Ad(w_i)^{-1}c_\mathbf{k}^{(p)}u') \end{align*}
By the Poincar\'e-Birkhoff-Witt theorem, we have a direct decomposition $U(\Ad(w_i)\mathfrak{p}) = U(\Ad(w_i)\mathfrak{p})(\Ad(w_i)\mathfrak{n})\oplus U(\Ad(w_i)\mathfrak{l})$. Hence we may assume that $c_\mathbf{k}^{(p)} \in U(\Ad(w_i)\mathfrak{p})(\Ad(w_i)\mathfrak{n})$ or $c_\mathbf{k}^{(p)} \in U(\Ad(w_i)\mathfrak{l})$. If $c_\mathbf{k}^{(p)} \in U(\Ad(w_i)\mathfrak{p})(\Ad(w_i)\mathfrak{n})$ then this sum is equal to $0$. If $c_\mathbf{k}^{(p)} \in U(\Ad(w_i)\mathfrak{l})$ then $w_i^{-1}\wt c_\mathbf{k}^{(p)}|_{\mathfrak{a}} = 0$. Hence, \begin{align*}
&w_i^{-1}(\wt b_\mathbf{k}^{(p)} + \wt(R'_i(a')ff_\mathbf{k}))|_{\mathfrak{a}}\\
& = w_i^{-1}(\wt c_\mathbf{k}^{(p)} + \wt b_\mathbf{k}^{(p)} + \wt a' + \wt f + \wt f_\mathbf{k})|_{\mathfrak{a}}\\
& = w_i^{-1}(\wt ((\ad(e)^{\mathbf{k}}X)T) + \wt f + \wt f_\mathbf{k} - \wt a'')|_{\mathfrak{a}}\\
& = w_i^{-1}(\wt X + \wt T + \wt f - \wt a'')|_{\mathfrak{a}}\\
& = \nu + w_i^{-1}(\wt X - \wt a'')|_{\mathfrak{a}} > \nu + w_i^{-1}\wt X|_{\mathfrak{a}}. \end{align*} So we have \begin{multline*} \delta_i(b_\mathbf{k}^{(p)},R'_i(-a_\mathbf{k}^{(p)})(ff_\mathbf{k}\eta_i^{-1}) - R'_i(-a_\mathbf{k}^{(p)})(ff_\mathbf{k})\eta_i^{-1},\Ad(w_i)^{-1}(c_\mathbf{k}^{(p)})u')\\
\in \sum_{\nu' > \nu}V(\nu' + w_i^{-1}\wt X|_\mathfrak{a}). \end{multline*} By the definition of $\widetilde{\eta_i}$, we have $\widetilde{R_i}(X')\widetilde{\eta_i} = 0$ for $X'\in \Ad(w_i)\overline{\mathfrak{n}}\cap \overline{\mathfrak{n}_0}$. Hence by Lemma~\ref{lem:to outside delta_i}, we have \begin{multline*}
\delta_i(b_\mathbf{k}^{(p)},R'_i(-a_\mathbf{k}^{(p)})(ff_\mathbf{k})\eta_i^{-1},\Ad(w_i)^{-1}(c_\mathbf{k}^{(p)})u')\\ = \delta_i(b_\mathbf{k}^{(p)},R'_i(-a_\mathbf{k}^{(p)})(ff_\mathbf{k}),\Ad(w_i)^{-1}(c_\mathbf{k}^{(p)})u')\widetilde{\eta_i}^{-1} \end{multline*} Hence, we have \begin{align*} &\sum_{\mathbf{k},p}\delta_i(b_\mathbf{k}^{(p)},R'_i(-a_\mathbf{k}^{(p)})(ff_\mathbf{k})\eta_i^{-1},\Ad(w_i)^{-1}(c_\mathbf{k}^{(p)})u')\\ & = \sum_{\mathbf{k},p}\delta_i(b_\mathbf{k}^{(p)},R'_i(-a_\mathbf{k}^{(p)})(ff_\mathbf{k}),\Ad(w_i)^{-1}(c_\mathbf{k}^{(p)})u')\widetilde{\eta_i}^{-1}\\ & = \sum_{\mathbf{k},p}\delta_i(a_\mathbf{k}^{(p)}b_\mathbf{k}^{(p)}c_\mathbf{k}^{(p)},(ff_\mathbf{k}),u')\widetilde{\eta_i}^{-1}\\ & = (X\delta_i(T,f,u'))\widetilde{\eta_i}^{-1}. \end{align*} We get (1).
(2) Take $T = 1$ in (1). Then we have \[
\delta_i(T,f\eta_i^{-1},u') - (\delta_i(T,f\eta_i^{-1},u'))\widetilde{\eta_i}^{-1}\in\sum_{\nu' > \nu}V(\nu'). \] Hence if \[
\sum_s \delta_i(S_s,h_s,v'_s)\widetilde{\eta_i}^{-1}\in \sum_{\nu' > \nu}V(\nu') \] then \[
\delta_i(T,f\eta_i^{-1},u')\in \sum_{\nu' > \nu}V(\nu') \] However, by Lemma~\ref{lem:fundamental properties of delta_i} (3), we have $V(\nu)\cap \sum_{\nu'\ne \nu}V(\nu') = 0$. This is a contradiction. \end{proof}
\begin{prop}\label{prop:Whittaker vectors in a Bruaht cell} Let $\widetilde{\mu}\in (\mathfrak{h}\cap \mathfrak{m})^*$ be an infinitesimal character of $\sigma$. Assume that $I_i/I_{i - 1}\ne 0$ and for all $\widetilde{w}\in \widetilde{W}$, \[
\lambda - \widetilde{w}(\lambda + \widetilde{\mu})|_\mathfrak{a}\not\in \mathbb{Z}_{\le 0}((\Sigma^+\setminus\Sigma_M^+)\cap w_i^{-1}\Sigma^+)|_\mathfrak{a}\setminus\{0\}. \] Then \[
\Wh_\eta(I_i') =
\{(\eta_i^{-1}\otimes u')\delta_i\mid u'\in \Wh_{w_i^{-1}\eta}((\sigma\otimes e^{\lambda+\rho})')\}. \] \end{prop} \begin{proof} Let $x = \sum_s\delta_i(T_s,f_s\eta_i^{-1},u_s')$ be an element of $\Wh_\eta(I_i')$ where $T_s\in U(\Ad(w_i)\overline{\mathfrak{n}}\cap \overline{\mathfrak{n}_0})$, $f_s\in\mathcal{P}(O_i)$ and $u'_s\in J'_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho})$. For $X\in \Ad(w_i)\overline{\mathfrak{n}}\cap\mathfrak{n}_0$, we have $(X - \eta(X))x = \sum_s \delta_i(T_s,(L_X - \eta(X))(f_s\eta_i^{-1}),u'_s) = \sum_s \delta_i(T_s,L_X(f_s)\eta_i^{-1},u'_s)$ by Lemma~\ref{lem:caluculation of Xdelta(1,f,u)}. Hence, we may assume $f_s = 1$.
Let $z\in Z(\mathfrak{g})$. Since $J'_\eta(I(\sigma,\lambda))$ has an infinitesimal character $-(\lambda+\widetilde{\mu})$, $I_i'$ has the same character. Let $\chi(z)$ be a complex number such that $z$ acts by $\chi(z)$ on $I'_i$. Take $T_s$ and $u'_s$ such that $T_s$ are $\mathfrak{a}_0$-weight vectors and lineally independent. Let $\nu = \min\{w_i^{-1}\wt T_s|_{\mathfrak{a}}\}_s$. Then by Lemma~\ref{lem:property of V(nu)} (1), we have \begin{multline*}
\chi(z)x = zx = \gamma_2\gamma_1(z)x \\\in \left(\gamma_3\gamma_2\gamma_1(z)\sum_{w_o^{-1}\wt T_s|_{\mathfrak{a}} = \nu}\delta_i(T_s,1,u_s')\right)\widetilde{\eta_i}^{-1} + \sum_{\nu' > \nu}V(\nu'). \end{multline*} By Lemma~\ref{lem:property of V(nu)} (1) ($T = 1$), we have \[
x \in \sum_{w_i^{-1}\wt T_s|_{\mathfrak{a}} = \nu}\delta_i(T_s,1,u'_s)\widetilde{\eta_i}^{-1} + \sum_{\nu' > \nu}V(\nu'). \] Hence we have \[
\left((\chi(z) - \gamma_3\gamma_2\gamma_1(z))\left(\sum_{w_i^{-1}\wt T_s|_{\mathfrak{a}} = \nu}\delta_i(T_s,1,u'_s)\right)\right)\widetilde{\eta_i}^{-1}\in \sum_{\nu' > \nu}V(\nu'). \]
By Lemma~\ref{lem:property of V(nu)} (2), we have $(\chi(z) - \gamma_3\gamma_2\gamma_1(z))\delta_i(T_s,1,u_s') = 0$ for all $s$ such that $w_i^{-1}\wt T_s|_{\mathfrak{a}} = \nu$. By the same calculation as that of the proof of Lemma~\ref{lem:no delta part}, $H\delta_i(T_s,1_u,u_s') = (-w_i\lambda + \wt T_s + \rho_0)(H)\delta_i(T_s,1_u,u_s')$ for $H\in \Ad(w_i)\mathfrak{a}$. By Lemma~\ref{lem:lemma of infinitesimal character}, there exists a $\widetilde{w}\in\widetilde{W}$ such that $-\widetilde{w}(\lambda+\widetilde{\mu})|_{\Ad(w_i)\mathfrak{a}} = -w_i\lambda + \wt T_s$. Then $\lambda - w_i^{-1}\widetilde{w}(\lambda+\widetilde{\mu})|_\mathfrak{a} = w_i^{-1}\wt T_s|_\mathfrak{a}\in\mathbb{Z}_{\le 0}((\Sigma^+\setminus\Sigma_M^+)\cap w_i^{-1}\Sigma^+)|_\mathfrak{a}$. By the assumption, $\wt T_s = 0$, i.e., $T_s\in \mathbb{C}$. Hence, we may assume that $x$ has a form $x = \delta_i(1,\eta_i^{-1},u') + \sum_{s\ge 2} \delta_i(T_s,\eta_i^{-1},u_s')$ where $\wt T_s \ne 0$ for all $s\ge 2$.
Take $X\in \mathfrak{n}_0\cap \Ad(w_i)\mathfrak{m}$. Then by Lemma~\ref{lem:caluculation of Xdelta(1,f,u)} and the above claim, \[
0 = (X - \eta(X))x \in \delta_i(1,\eta_i^{-1},(\Ad(w_i)^{-1}X - \eta(X))u') + \sum_{\nu' > 0}V(\nu'). \]
By Lemma~\ref{lem:property of V(nu)}, we have $\delta_i(1,\eta_i^{-1},(\Ad(w_i)^{-1}X - \eta(X))u') = 0$. Hence we have $u'\in \Wh_{w_i^{-1}\eta}((\sigma\otimes e^{\lambda + \rho})')$. This implies that $x - \delta_i(1,\eta_i^{-1},u')\in \Wh_\eta(I_i')$. If $x - \delta_i(1,\eta_i^{-1},u')\ne 0$, then by the above argument, we have $\min\{w_i^{-1}\wt T_s|_\mathfrak{a}\}_{s\ge 2} = 0$. This is a contradiction. \end{proof}
\begin{thm}\label{thm:dimension Whittaker vectors}
Assume that for all $w\in W(M)$ such that $\eta|_{wNw^{-1}\cap N_0} = 1$ the following two conditions hold: \begin{enumerate} \renewcommand*{(\theenumi)}{(\alph{enumi})} \item For each exponent $\nu$ of $\sigma$ and $\alpha\in \Sigma^+\setminus w^{-1}(\Sigma^+\cup\Sigma_\eta^-)$, we have $2\langle\alpha,\lambda+\nu\rangle/\lvert\alpha\rvert^2\not\in\mathbb{Z}_{\le 0}$.
\item For all $\widetilde{w}\in\widetilde{W}$ we have $\lambda - \widetilde{w}(\lambda + \widetilde{\mu})|_\mathfrak{a}\notin \mathbb{Z}_{\le 0}((\Sigma^+\setminus \Sigma_M^+)\cap w^{-1}\Sigma^+)|_\mathfrak{a}\setminus\{0\}$ where $\widetilde{\mu}$ is an infinitesimal character of $\sigma$. \end{enumerate} Moreover, assume that $\eta$ is unitary. Then we have \[
\dim\Wh_\eta(I(\sigma,\lambda)') = \sum_{w\in W(M),\ w(\Sigma^+\setminus\Sigma^+_M)\cap \supp\eta = \emptyset}\dim \Wh_{w^{-1}\eta}((\sigma\otimes e^{\lambda+\rho})'). \] \end{thm} \begin{proof} By the exact sequence $0\to I_{i - 1} \to I_i\to I_i/I_{i - 1}\to 0$, we have $0\to \Wh_\eta(I_{i - 1}) \to \Wh_\eta(I_i)\to \Wh_\eta(I_i/I_{i - 1})$. By Lemma~\ref{prop:Whittaker vectors in a Bruaht cell}, it is sufficient to prove that the last map $\Wh_\eta(I_i)\to \Wh_\eta(I_i/I_{i - 1})$ is surjective.
Take $x\in \Wh_\eta(I'_i)\simeq \Wh_\eta(I_i/I_{i - 1})$. Then $x$ is $(\eta_i\otimes u')\delta_i$ for some $u'\in \Wh_{w_i^{-1}\eta}(\sigma\otimes e^{\lambda+\rho})$. By Lemma~\ref{lem:meromorphic extension}, there exists a distribution $x_t\in I_i(\lambda + t\rho)$ with meromorphic parameter $t$ such that $x_t|_{U_i}$ is holomorphic and $(x_t|_{U_i})|_{t = 0} = x$. Moreover, $(X - \eta(X))x_t = 0$ for $X\in\mathfrak{n}_0$. By Proposition~\ref{prop:convergence and continuation} and the condition (a), the distribution $x_t$ is holomorphic at $t = 0$. Hence $x_0|_{U_i} = x$. The map $\Wh_\eta(I_i)\to \Wh_\eta(I_i/I_{i - 1})$ is surjective. \end{proof}
Next we consider the module $\Wh_\eta((I(\sigma,\lambda)_{\text{$K$-finite}})^*)$. Take a filtration $\widetilde{I_i}\subset J^*_\eta(I(\sigma,\lambda))$ as in Theorem~\ref{thm:stucture of J^*(I(sigma,lambda))}. \begin{lem}\label{lem:two step calc for Whittaker vector} Let $V$ be an object of the category $\mathcal{O}'$. Then we have $C(H^0(\mathfrak{n}_\eta,V)) = H^0(\mathfrak{n}_\eta,C(V))$ where $H^0(\mathfrak{n}_\eta,V) = \{v\in V\mid \mathfrak{n}_\eta v = 0\}$ is the $0$-th $\mathfrak{n}_\eta$-cohomology. \end{lem} \begin{proof} We get the lemma by the following equation. \begin{multline*} H^0(\mathfrak{n}_\eta,C(V)) = H^0(\mathfrak{n}_\eta,D'(V)^*) = (D'(V)/\mathfrak{n}_\eta D'(V))^*\\ = CD'(D'(V)/\mathfrak{n}_\eta D'(V)) = C(H^0(\mathfrak{n}_\eta,D'(V)^*)_{\text{$\mathfrak{h}$-finite}})\\ = C(H^0(\mathfrak{n}_\eta,D'D'(V))) = C(H^0(\mathfrak{n}_\eta,V)). \end{multline*} \end{proof}
\begin{lem}\label{lem:relation in S_w} Let $e_1,\dots, e_l$ be a basis of $\Ad(w_i)\overline{\mathfrak{n}}\cap\mathfrak{n}_0$ such that each $e_s$ is a root vector and $\bigoplus_{s\le t - 1}\mathbb{C} e_s$ is an ideal of $\bigoplus_{s\le t}\mathbb{C} e_s$. In $S_{w_i,0}$ where $0$ is the trivial representation, we have the following formulae. \begin{enumerate} \item For all $t = 1,\dots,l$, \begin{multline*} e_t(e_1^{-1}\dotsm e_{t - 1}^{-1}e_t^{-(k_t + 1)}\dotsm e_l^{-(k_l + 1)})\\ = e_1^{-1}\dotsm e_{t - 1}^{-1}e_t^{-k_t}e_{t + 1}^{-(k_{t + 1} + 1)}\dotsm e_l^{-(k_l + 1)} \end{multline*} \item Fix $t \in \{1,\dots,l\}$ such that $e_t\in \mathfrak{n}_\eta$. Assume that $k_s = 0$ for all $s < t$ such that $e_s\in \mathfrak{n}_\eta$. Then \begin{multline*} e_t(e_1^{-(k_1 + 1)}\dotsm e_l^{-(k_l + 1)})\\ = e_1^{-(k_1 + 1)}\dots e_{t - 1}^{-(k_{t - 1} + 1)}e_t^{-k_t}e_{t + 1}^{-(k_t + 1)}\dots e_l^{-(k_l + 1)}. \end{multline*} \item $X(e_1^{-1}\dotsm e_l^{-1}) = (e_1^{-1}\dotsm e_l^{-1})X$ for $X\in \Ad(w_i)\mathfrak{m}\cap \mathfrak{n}_0$. \end{enumerate} \end{lem} \begin{proof} Let $\alpha_s$ be a restricted root corresponding to $e_s$.
(1) It is sufficient to prove $e_t(e_1^{-1}\dotsm e_{t - 1}^{-1}) = (e_1^{-1}\dotsm e_{t - 1}^{-1})e_t$ in $S_{e_1}\otimes_{U(\mathfrak{g})}\dots \otimes_{U(\mathfrak{g})}S_{e_{t - 1}}$. Since $\bigoplus_{s = 1}^{t - 1}\mathbb{C} e_s$ is an ideal of $\bigoplus_{s = 1}^t \mathbb{C} e_s$, we have \[
e_t(e_1^{-1}\dotsm e_{t - 1}^{-1}) - (e_1^{-1}\dotsm e_{t - 1}^{-1})e_t\in \bigoplus_{k_s\ge 0}\mathbb{C} e_1^{-(k_1 + 1)}\dotsm e_{t - 1}^{-(k_{t - 1} + 1)}. \] An $\mathfrak{a}_0$-weight of the left hand side is $-\alpha_1 - \dots - \alpha_{t - 1} + \alpha_t$. However, the set of $\mathfrak{a}_0$-weights of the right hand side is $\{-(k_1 + 1)\alpha_1 - \dots - (k_{t - 1} + 1)\alpha_{t - 1}\mid k_s\in\mathbb{Z}_{\ge 0}\}$. Hence each $\mathfrak{a}_0$-weight appearing in the right hand side is less than that of the left hand side. This implies $e_t(e_1^{-1}\dots e_{t - 1}^{-1}) - (e_1^{-1}\dots e_{t - 1}^{-1})e_t = 0$.
(2) We prove $e_t(e_1^{-(k_1 + 1)}\dotsm e_{t - 1}^{-(k_{t - 1} + 1)}) = (e_1^{-(k_1 + 1)}\dots e_{t - 1}^{-(k_{t - 1} + 1)})e_t$ in $S_{e_1}\otimes_{U(\mathfrak{g})}\dots \otimes_{U(\mathfrak{g})}S_{e_{t - 1}}$. As in the proof of (1), we have \begin{multline*}
e_t(e_1^{-(k_1 + 1)}\dotsm e_{t - 1}^{-(k_{t - 1} + 1)}) - (e_1^{-(k_1 + 1)}\dotsm e_{t - 1}^{-(k_{t - 1} + 1)})e_t\\\in \bigoplus_{k_s\ge 0}\mathbb{C} e_1^{-(k_1 + 1)}\dotsm e_{t - 1}^{-(k_{t - 1} + 1)}. \end{multline*} An $\mathfrak{a}_\eta$-weight of the left hand side is $\sum_{e_s\in \mathfrak{n}_\eta,\ s < t}-\alpha_s + \alpha_t$. However, the set of $\mathfrak{a}_\eta$-weights of the right hand side is $\{\sum_{e_s\in\mathfrak{n}_\eta,\ s < t}-(k_s + 1)\alpha_s\mid k_s\in\mathbb{Z}_{\ge 0}\}$. Hence each $\mathfrak{a}_\eta$-weight appearing in the right hand side is less than that of the left hand side. This implies the lemma.
(3) We may assume $X$ is a restricted root vector. Let $\alpha$ be a restricted root of $X$. Since $X$ normalizes $\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0$, we have \[
X(e_1^{-1}\dotsm e_l^{-1}) - (e_1^{-1}\dotsm e_l^{-1})X\in \bigoplus_{k_s\ge 0}\mathbb{C} e_1^{-(k_1 + 1)}\dotsm e_l^{-(k_l + 1)}. \] Then $X(e_1^{-1}\dotsm e_l^{-1}) - (e_1^{-1}\dotsm e_l^{-1})X$ has an $\mathfrak{a}_0$-weight $-(\alpha_1 + \dots + \alpha_s) + \alpha$. However, $e_1^{-(k_1 + 1)}\dotsm e_l^{-(k_l + 1)}$ has a $\mathfrak{a}_0$-weight $-((k_1 + 1)\alpha_1 + \dots + (k_l + 1)\alpha_l) < -(\alpha_1 + \dots + \alpha_s) + \alpha$. Hence $X(e_1^{-1}\dotsm e_l^{-1}) - (e_1^{-1}\dotsm e_l^{-1})X = 0$. \end{proof}
\begin{lem}\label{lem:e^{-1}-part} Let $e_1,\dots, e_l$ be a basis of $\Ad(w_i)\overline{\mathfrak{n}}\cap\mathfrak{n}_0$ such that $e_s$ is a root vector and $\bigoplus_{s\le t - 1}\mathbb{C} e_s$ is an ideal of $\bigoplus_{s\le t}\mathbb{C} e_s$. Let $V$ be a $U(\mathfrak{m}\oplus\mathfrak{a})$-representation. Regard $V$ as a $\mathfrak{p}$-representation by $\mathfrak{n}V = 0$. By Lemma~\ref{lem:induction + twisting}, we have $T_{w_i}(U(\mathfrak{g})\otimes_{U(\mathfrak{p})}V) \simeq (\bigoplus_{k_s\ge 0}\mathbb{C} e_1^{-(k_1 + 1)}\dotsm e_l^{-(k_l + 1)})\otimes U(\Ad(w_i)\overline{\mathfrak{n}}\cap \overline{\mathfrak{n}_0})\otimes w_iV$. Then we have $\{v\in e_1^{-1}\dotsm e_l^{-1}\otimes 1\otimes w_iV\mid \mathfrak{n}_\eta v = 0\} = e_1^{-1}\dotsm e_l^{-1}\otimes 1\otimes H^0(\Ad(w_i)\mathfrak{m}\cap \mathfrak{n}_\eta,w_iV)$. \end{lem} \begin{proof} Take $v = e_1^{-1}\dotsm e_l^{-1}\otimes 1\otimes v_0\in H^0(\mathfrak{n}_\eta,T_{w_i}(U(\mathfrak{g})\otimes_{U(\mathfrak{p})}V)$. Then for $X\in \Ad(w_i)\mathfrak{m}\cap \mathfrak{n}_\eta$ we have $X(e_1^{-1}\dotsm e_l^{-1}\otimes 1\otimes v_0) = 0$. By Lemma~\ref{lem:relation in S_w}, we have $e_1^{-1}\dotsm e_l^{-1}\otimes 1\otimes Xv_0 = 0$. Hence $Xv_0 = 0$. \end{proof}
By the definition of the Harish-Chandra homomorphism, we get the following lemma.
\begin{lem}\label{lem:infinitesimal character of a space of partial highest weight vectors}
Let $\mathfrak{q}$ be a parabolic subalgebra of $\mathfrak{g}$ containing $\mathfrak{h}\oplus\mathfrak{u}_0$. Take a Levi decomposition $\mathfrak{l}\oplus\mathfrak{u}_\mathfrak{q}$ of $\mathfrak{q}$ such that $\mathfrak{h}\subset \mathfrak{l}$. Let $\widetilde{W_\mathfrak{l}}\subset \widetilde{W}$ be the Weyl group of $\mathfrak{l}$, $V$ an $\mathfrak{l}$-module with an infinitesimal character $\widetilde{\mu}$. Put $V' = H^0(\mathfrak{u}_\mathfrak{q},V)$ and $\widetilde{\rho_{\mathfrak{u}_\mathfrak{q}}}(H) = (1/2)\Tr \ad(H)|_{\mathfrak{u}_\mathfrak{q}}$ for $H\in \mathfrak{h}$. Then $V'$ is $\mathfrak{l}$-stable and $V' = \bigoplus_{\widetilde{w}\in \widetilde{W_\mathfrak{l}}\backslash\widetilde{W}}(V')_{[\widetilde{w}\widetilde{\mu} - \widetilde{\rho_{\mathfrak{u}_\mathfrak{q}}}]}$ where $(V')_{[\widetilde{w}\widetilde{\mu} - \widetilde{\rho_{\mathfrak{u}_\mathfrak{q}}}]}$ is the maximal $\mathfrak{l}$-submodule which has an infinitesimal character $\widetilde{w}\widetilde{\mu} - \widetilde{\rho_{\mathfrak{u}_\mathfrak{q}}}$. In particular, for an $\mathfrak{l}$-submodule $V''$ of $V'$, a highest weight of $V'/V''$ belongs to $\{\widetilde{w}\widetilde{\mu} - \widetilde{\rho}\mid \widetilde{w}\in\widetilde{W}\}$. \end{lem}
The following lemma is well-known. \begin{lem}\label{lem:distribution of weight} Let $V\in \mathcal{O}'$. Assume that $V$ has an infinitesimal character $\widetilde{\lambda}\in \mathfrak{h}^*$. Then a $\mathfrak{h}$-weight appearing in $V$ is contained in $\{\widetilde{w}\widetilde{\lambda} - \widetilde{\rho} - \alpha\mid \widetilde{w}\in\widetilde{W},\ \alpha\in\mathbb{Z}_{\ge 0}\Delta^+\}$. \end{lem}
Now we determine the dimension of the space of Whittaker vectors of $\widetilde{I_i}/\widetilde{I_{i - 1}}$ under some conditions.
\begin{lem}\label{lem:Whittaker vectors in a Bruhat cell, algebraic} Let $\widetilde{\mu}$ be an infinitesimal character of $\sigma$. Assume that for all $\widetilde{w}\in \widetilde{W}\setminus \widetilde{W_M}$, $(\lambda+\widetilde{\mu}) - \widetilde{w}(\lambda+\widetilde{\mu})\not\in \mathbb{Z}\Delta$. Then we have $\dim\Wh_\eta(\widetilde{I_i}/\widetilde{I_{i - 1}}) = \dim \Wh_{w_i^{-1}\eta}((\sigma_{\text{\normalfont $M\cap K$-finite}})^*)$. \end{lem} \begin{proof} Put $V = T_{w_i}(U(\mathfrak{g})\otimes_{U(\mathfrak{p})}J^*(\sigma\otimes e^{\lambda+\rho}))$. By Theorem~\ref{thm:stucture of J^*(I(sigma,lambda))}, we have $\Wh_\eta(\widetilde{I_i}/\widetilde{I_{i - 1}}) = \Wh_\eta(C(V))$. Let $e_1,\dots, e_l$ be a basis of $\Ad(w_i)\overline{\mathfrak{n}}\cap\mathfrak{n}_0$ such that $\bigoplus_{s\le t - 1}\mathbb{C} e_s$ is an ideal of $\bigoplus_{s\le t}\mathbb{C} e_s$. Moreover, assume that each $e_i$ is a root vector. For $\mathbf{k} = (k_1,\dots,k_l)\in \mathbb{Z}^l$, put $e^\mathbf{k} = e_1^{k_1}\dotsm e_l^{k_l}$. Set $\mathbf{1} = (1,\dots,1)\in\mathbb{Z}^l$. Then we have \[
V = \bigoplus_{k\in\mathbb{Z}_{\ge 0}^l}\mathbb{C} e^{-(\mathbf{k} + \mathbf{1})}\otimes U(\Ad(w_i)\overline{\mathfrak{n}}\cap\overline{\mathfrak{n}_0})\otimes w_iJ^*(\sigma\otimes e^{\lambda+\rho}). \] Put \[
V' = \bigoplus_{\mathbf{k}\in \mathcal{A}}e^{-(\mathbf{k} + \mathbf{1})}\otimes U(\Ad(w_i)\overline{\mathfrak{n}}\cap\overline{\mathfrak{n}_0}\cap \mathfrak{m}_\eta)\otimes H^0(\mathfrak{m}\cap\mathfrak{n}_\eta,w_iJ^*(\sigma\otimes e^{\lambda+\rho})) \] where $\mathcal{A} = \{(k_1,\dots,k_l)\in \mathbb{Z}^l_{\ge 0}\mid \text{if $e_i\in \mathfrak{n}_\eta$ then $k_i = 0$}\}$. It is easy to see that $V'$ is an $\mathfrak{m}_\eta\oplus \mathfrak{a}_\eta$-stable and $V'\subset H^0(\mathfrak{n}_\eta,V)$. We prove that $V' = H^0(\mathfrak{n}_\eta,V)$.
To prove $V' = H^0(\mathfrak{n}_\eta,V)$, it is sufficient to prove that there exists no highest weight vector in $H^0(\mathfrak{n}_\eta,V)/ V'$. Let $v\in H^0(\mathfrak{n}_\eta,V)$ such that $(\mathfrak{m}_\eta\cap \mathfrak{u})v \in V'$.
First, we prove that $v\in e^{-\mathbf{1}}\otimes U(\Ad(w_i)\overline{\mathfrak{n}}\cap \overline{\mathfrak{n}_0})\otimes J^*(\sigma\otimes e^{\lambda + \rho}) + V'$. Take $y_\mathbf{k}\in U(\Ad(w_i)\overline{\mathfrak{n}}\cap \overline{\mathfrak{n}_0})\otimes J^*(\sigma\otimes e^{\lambda + \rho})$ such that $v = \sum_{\mathbf{k}} e^{-(\mathbf{k} + \mathbf{1})}\otimes y_\mathbf{k}$. We prove that if $k_t\ne 0$ and $e_t\in \mathfrak{n}_\eta$ then $y_\mathbf{k} = 0$ by induction on $t$ where $\mathbf{k} = (k_1,\dots,k_l)$. Put $\mathbf{1}_t = (\delta_{st})_{1\le s\le l}\in \mathbb{Z}^l$ ($\delta_{st}$ is Kronecker's delta). By inductive hypothesis, for $s < t$ such that $e_s\in\mathfrak{n}_\eta$, if $y_\mathbf{k}\ne 0$ then $k_s = 0$. By Lemma~\ref{lem:relation in S_w} (2), we have $e_tv = \sum_{\mathbf{k}\in\mathbb{Z}_{\ge 0}^l}e^{-(\mathbf{k} + \mathbf{1}) + \mathbf{1}_t}\otimes y_\mathbf{k}$. Since $v\in H^0(\mathfrak{n}_\eta,V)$, we have $e_tv = 0$. Hence if $e^{-(\mathbf{k} + \mathbf{1}) + \mathbf{1}_t}\ne 0$ then $y_\mathbf{k} = 0$. Since $e^{-(\mathbf{k} + \mathbf{1}) + \mathbf{1}_t} = 0$ if and only if $k_t = 0$, $k_t \ne 0$ implies $y_\mathbf{k} = 0$. We prove that if $k_t\ne 0$ then $e^{-(\mathbf{k} + \mathbf{1})}\otimes y_\mathbf{k}\in V'$ by induction on $t$. If $e_t\in \mathfrak{n}_\eta$ then this claim is already proved. We may assume that $e_t\in \mathfrak{m}_\eta$. Hence $e_tV'\subset V'$. By inductive hypothesis, if $k_s\ne 0$ for some $s < t$ then $e^{-(\mathbf{k} + \mathbf{1})}\otimes y_\mathbf{k} \in V'$. Then we have $e_tv\in \sum_{\mathbf{k}\in\mathbb{Z}_{\ge 0}^l}e^{-(\mathbf{k} + \mathbf{1}) + \mathbf{1}_t}\otimes y_\mathbf{k} + V'$ by Lemma~\ref{lem:relation in S_w} (1). Since $e_tv\in V'$, we have $\sum_{\mathbf{k}\in\mathbb{Z}_{\ge 0}^l}e^{-(\mathbf{k} + \mathbf{1}) + \mathbf{1}_t}\otimes y_\mathbf{k} \in V'$. By the definition of $V'$, if $e^{-(\mathbf{k} + \mathbf{1}) + \mathbf{1}_t}\ne 0$ then $e^{-(\mathbf{k} + \mathbf{1})}\otimes y_\mathbf{k}\in V'$. Hence we get the claim.
We may assume that $v$ is a weight vector with respect to $\mathfrak{h}$. We can take $\widetilde{w}\in \widetilde{W}$ such that $-\widetilde{w}(\lambda+\widetilde{\mu}) - \widetilde{\rho}$ is a $\mathfrak{h}$-weight of $v$ by Lemma~\ref{lem:infinitesimal character of a space of partial highest weight vectors}. Put $\widetilde{\rho_M} = \sum_{\alpha\in\Delta_M^+}(1/2)\alpha$. Since $J^*(\sigma\otimes e^{\lambda + \rho})$ has an infinitesimal character $-(\lambda + \widetilde{\mu} + \rho)$, a $\mathfrak{h}$-weight appearing in $J^*(\sigma\otimes e^{\lambda + \rho})$ is contained in $\{-\widetilde{w}(\lambda + \widetilde{\mu} + \rho) - \widetilde{\rho_M} + \alpha\mid \widetilde{w}\in \widetilde{W_M},\ \alpha\in\mathbb{Z}\Delta_M\}$ by Lemma~\ref{lem:distribution of weight}. Since $-\rho\in\mathfrak{a}^*$, we have $\widetilde{w}\rho = \rho$ for $\widetilde{w}\in \widetilde{W_M}$. Hence we have $- \widetilde{w}\rho - \widetilde{\rho_M} = -\rho - \widetilde{\rho_M} = -\widetilde{\rho}$. Notice that $w_i\widetilde{\rho} - \widetilde{\rho}\in \mathbb{Z}\Delta$. Therefore a $\mathfrak{h}$-weight appearing in $V$ is contained in \begin{multline*} -w_i\widetilde{W_M}(\lambda + \widetilde{\mu}) - w_i\widetilde{\rho} + w_i\mathbb{Z}\Delta_M + \mathbb{Z}_{\ge 0}(w_i\Delta^-\cap \Delta^-) - \mathbb{Z}_{\ge 1}(w_i\Delta^-\cap \Delta^+)\\ \subset -w_i\widetilde{W_M}(\lambda + \widetilde{\mu}) - \widetilde{\rho} + \mathbb{Z}\Delta. \end{multline*} This implies that for some $\widetilde{w'}\in\widetilde{W_M}$, we have $\widetilde{w}(\lambda + \widetilde{\mu}) - w_i\widetilde{w'}(\lambda +\widetilde{\mu})\in\mathbb{Z}\Delta$. By the assumption we have $\widetilde{w}\in w_i\widetilde{W_M}$. This implies $(\wt v)(\Ad(w_i)H) = -(\lambda(H) + w_i^{-1}\widetilde{\rho}(H))$ for all $H\in \mathfrak{a}$ where $\wt v$ is a $\mathfrak{h}$-weight of $v$.
Take $T_p\in U(\Ad(w_i)\overline{\mathfrak{n}}\cap \overline{\mathfrak{n}_0})$ and $x_p\in w_iJ^*(\sigma\otimes e^{\lambda + \rho})$ such that $v \in \sum_p e^{-\mathbf{1}}\otimes T_p\otimes x_p + V'$. We may assume that $T_p$ (resp.\ $x_p$) is a $\mathfrak{h}$-weight vector with respect to the adjoint action (resp.\ the action induced from $\sigma\otimes e^{\lambda + \rho}$). We denote its $\mathfrak{h}$-weight by $\wt T_p$ and $\wt x_p$. Fix $H\in \mathfrak{a}$. Then $\alpha(H) = 0$ for all $\alpha\in\Delta_M$. Since $\wt x_p\in -w_i(\widetilde{W_M}(\lambda + \widetilde{\mu}) + \widetilde{\rho} + \mathbb{Z}\Delta_M)$, $(\wt x_p)(\Ad(w_i)H) = -(\lambda + \widetilde{\rho})(H)$. Hence \begin{align*} &(\wt v)(\Ad(w_i)H)\\ & = (\wt(e^{-\mathbf{1}}) + \wt(T_p) + \wt(x_p))(\Ad(w_i)(H))\\ & = (\wt(e^{-\mathbf{1}})(\Ad(w_i)H) + (\wt T_p)(\Ad(w_i)H) - (\lambda + \widetilde{\rho})(H)\\ & = (\wt(e^{-\mathbf{1}})(\Ad(w_i)H) + (\wt T_p)(\Ad(w_i)H) - (\lambda + \widetilde{\rho})(H). \end{align*}
We calculate $\wt(e^{-\mathbf{1}})(\Ad(w_i)H)$. By the definition, $\wt(e^{-\mathbf{1}})(\Ad(w_i)H) = \Tr\ad(\Ad(w_i)H)|_{\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0}$. Since we have $\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0 = \Ad(w_i)\overline{\mathfrak{n}_0}\cap \mathfrak{n}_0$, we have \begin{multline*}
\Tr\ad(\Ad(w_i)H)|_{\Ad(w_i)\overline{\mathfrak{n}}\cap \mathfrak{n}_0}
= \Tr\ad(\Ad(w_i)H)|_{\Ad(w_i)\overline{\mathfrak{n}_0}\cap \mathfrak{n}_0}\\
~ \Tr\ad(H)|_{\Ad(w_i)^{-1}\mathfrak{n}_0\cap \overline{\mathfrak{n}_0}} = (-\widetilde{\rho} + w_i^{-1}\widetilde{\rho})(H). \end{multline*} Hence we get \[
(\wt v)(\Ad(w_i)H) = (\wt T_p)(\Ad(w_i)H) - (\lambda + w_i^{-1}\widetilde{\rho})(H). \] We have already proved that $(\wt v)(\Ad(w_i)H) = -(\lambda + w_i^{-1}\widetilde{\rho})(H)$. Therefore we get $(\wt T_p)(\Ad(w_i)H) = 0$ for all $H\in\mathfrak{a}$. Since $T_p\in U(\Ad(w_i)\mathfrak{n})$, this implies $T_p\in \mathbb{C}$, i.e., there exist $v'\in e_1^{-1}\dotsm e_l^{-1}\otimes 1\otimes w_iJ^*(\sigma\otimes e^{\lambda+\rho})$ and $v''\in V'$ such that $v = v' + v''$. Therefore $\mathfrak{n}_\eta(v') = \mathfrak{n}_\eta(v - v'') = 0$. Hence, $v'\in V'$ by Lemma~\ref{lem:e^{-1}-part}. Therefore $H^0(\mathfrak{n}_\eta,V) = V'$.
For an $\mathfrak{m}_0\oplus\mathfrak{a}_0$-module $\tau$ and a subalgebra $\mathfrak{c}$ of $\mathfrak{g}$ containing $\mathfrak{m}_0\oplus\mathfrak{a}_0$, put $M_{\mathfrak{c}}(\tau) = U(\mathfrak{c})\otimes_{U(\mathfrak{c}\cap\overline{\mathfrak{p}_0})}(\tau\otimes\rho')$ where $\overline{\mathfrak{n}_0}\cap \mathfrak{c}$ acts on $\tau$ trivially and $\rho'(H) = (\Tr(\ad(H)|_{\mathfrak{c}\cap\overline{\mathfrak{n}_0}}))/2$ for $H\in\mathfrak{a}_0$.
For $\widetilde{\lambda}\in \mathfrak{h}^*$ such that $\widetilde{\lambda}|_{\mathfrak{m}_0}$ is regular dominant integral, let $\sigma_{M_0A_0,\widetilde{\lambda}}$ be the finite-dimensional representation of $M_0A_0$ with an infinitesimal character $\widetilde{\lambda}$. Let $\ch M$ be the character of $M$. We can take integers $c_{\widetilde{\lambda}}$ such that \begin{multline*}
\ch D' H^0(\mathfrak{n}_\eta\cap \Ad(w_i)\mathfrak{m},w_iJ^*(\sigma\otimes e^{\lambda+\rho}))\\ = \sum_{\widetilde{\lambda}}c_{\widetilde{\lambda}}\ch M_{(\mathfrak{m}_\eta\cap\Ad(w_i)\mathfrak{m}) + \mathfrak{a}_0}(\sigma_{M_0A_0,\widetilde{\lambda}}) \end{multline*}
Then we have $\ch D'V' = \sum_{\widetilde{\lambda}}c_{\widetilde{\lambda}}\ch M_{\mathfrak{m}_\eta\oplus\mathfrak{a}_\eta}(\sigma_{M_0A_0,\widetilde{\lambda}})$. The functor $X\mapsto \Wh_{\eta|_{\mathfrak{m}_\eta\cap\mathfrak{n}_0}}(X^*)$ is an exact functor by a result of Lynch~\cite{lynch-whittaker}. Hence, we have $\dim\Wh_{\eta|_{\mathfrak{m}_\eta\cap\mathfrak{n}_0}}(C(V')) = \sum_{\widetilde{\lambda}}c_{\widetilde{\lambda}}\dim\Wh_{\eta|_{\mathfrak{m}_\eta\cap\mathfrak{n}_0}}(M_{\mathfrak{m}_\eta\oplus\mathfrak{a}_\eta}(\sigma_{M_0A_0,\widetilde{\lambda}})^*)$. Lynch also proves $\dim\Wh_{\eta|_{\mathfrak{m}_\eta\cap\mathfrak{n}_0}}(M_{\mathfrak{m}_\eta}(\sigma_{M_0A_0,\widetilde{\lambda}})^*) = \dim\sigma_{M_0A_0,\widetilde{\lambda}}$. Therefore, by Lemma~\ref{lem:two step calc for Whittaker vector}, we have $\dim \Wh_\eta(\widetilde{I_i}/\widetilde{I_{i - 1}}) = \dim\Wh_{\eta|_{\mathfrak{m}_\eta\cap\mathfrak{n}_0}}(C(V')) = \sum_{\widetilde{\lambda}} c_{\widetilde{\lambda}}\dim\sigma_{M_0A_0,\widetilde{\lambda}}$. By the same argument we have \begin{align*} &\sum_{\widetilde{\lambda}} c_{\widetilde{\lambda}}\dim\sigma_{M_0A_0,\widetilde{\lambda}}\\
& = \sum_{\widetilde{\lambda}} c_{\widetilde{\lambda}}\dim \Wh_{\eta|_{\mathfrak{m}_\eta\cap\Ad(w_i)\mathfrak{m}\cap\mathfrak{n}_0}}(M_{(\mathfrak{m}_\eta\cap\Ad(w_i)\mathfrak{m}) + \mathfrak{a}_0}(\sigma_{M_0A_0,\widetilde{\lambda}})^*) \\
& = \dim\Wh_{\eta|_{\mathfrak{m}_\eta\cap\Ad(w_i)\mathfrak{m}\cap\mathfrak{n}_0}}(CH^0(\mathfrak{n}_\eta\cap \Ad(w_i)\mathfrak{m},w_iJ^*(\sigma\otimes e^{\lambda+\rho})))\\
& = \dim \Wh_{\eta|_{\Ad(w_i)\mathfrak{m}\cap\mathfrak{n}_0}}(C(w_iJ^*(\sigma\otimes e^{\lambda+\rho})))\\ & = \dim\Wh_{w_i^{-1}\eta}(C(J^*(\sigma\otimes e^{\lambda+\rho})))\\ & = \dim \Wh_{w_i^{-1}\eta}((\sigma_{\text{$M\cap K$-finite}})^*). \end{align*} This implies the lemma. \end{proof}
\begin{thm}\label{thm:dimension Whittaker vectors, algebraic} Let $\widetilde{\mu}$ be an infinitesimal character of $\sigma$. Assume that for all $\widetilde{w}\in \widetilde{W}\setminus \widetilde{W_M}$, $(\lambda+\widetilde{\mu}) - \widetilde{w}(\lambda+\widetilde{\mu})\not\in \mathbb{Z}\Delta$. Then we have \[
\dim\Wh_\eta((I(\sigma,\lambda)_{\text{\normalfont $K$-finite}})^*) = \sum_{w\in W(M)}\dim \Wh_{w^{-1}\eta}((\sigma_{\text{\normalfont $M\cap K$-finite}})^*). \] \end{thm} \begin{proof}
Since a $\mathfrak{h}$-weight appearing in $T_{w_i}(U(\mathfrak{g})\otimes_{U(\mathfrak{p})}J^*(\sigma\otimes e^{\lambda+\rho}))$ belongs to $\{-w_i\widetilde{w}(\lambda+\widetilde{\mu}) - \widetilde{\rho} + \alpha\mid \widetilde{w}\in\widetilde{W_M},\ \alpha\in\Delta\}$, the exact sequence $0\to I_{i - 1} \to I_i\to T_{w_i}(U(\mathfrak{g})\otimes_{U(\mathfrak{p})}J^*(\sigma\otimes e^{\lambda+\rho}))\to 0$ splits. Hence, we have $J^*_\eta(I(\sigma,\lambda)) = \bigoplus_i\Gamma_\eta(C(T_{w_i}(U(\mathfrak{g})\otimes_{U(\mathfrak{p})}J^*(\sigma\otimes e^{\lambda+\rho}))))$. Therefore the theorem follows from Lemma~\ref{lem:Whittaker vectors in a Bruhat cell, algebraic}. \end{proof}
Finally we study the case of $\sigma$ is finite-dimensional. If $\sigma$ is finite-dimensional, then $\mathfrak{m}\cap\mathfrak{n}_0$ acts on $\sigma$ nilpotently. Hence $\Wh_{w_i^{-1}\eta}(\sigma^*)\ne 0$ if and only if $w_i^{-1}\eta = 0$ on $\mathfrak{m}\cap\mathfrak{n}_0$.
\begin{defn} Let $\Theta,\Theta_1,\Theta_2$ be subsets of $\Pi$. \begin{enumerate} \item Put $W(\Theta) = \{w\in W\mid w(\Theta)\subset \Sigma^+\}$ and $\Sigma_\Theta = \mathbb{Z}\Theta\cap \Sigma$.\newsym{$W(\Theta)$} \item Put $W(\Theta_1,\Theta_2) = \{w\in W(\Theta_1)\cap W(\Theta_2)^{-1}\mid w(\Sigma_{\Theta_1})\cap \Sigma_{\Theta_2} = \emptyset\}$.\newsym{$W(\Theta_1,\Theta_2)$} \item Let $W_\Theta$ be the Weyl group of $\Sigma_\Theta$.\newsym{$W_\Theta$} \end{enumerate} \end{defn}
\begin{lem}\label{lem:coparison oshima} Let $\Theta$ be a subset of $\Pi$ corresponding to $P$. \begin{enumerate} \item We have $\#W(\supp\eta,\Theta) = \#\{w\in W(M)\mid w(\Sigma^+)\cap \Sigma^+_\eta = \emptyset\}$. \item We have $\#W(\supp\eta,\Theta)\times\# W_{\supp\eta} = \#\{w\in W(M)\mid \supp\eta\cap w(\Sigma_M^+) = \emptyset\}$. \end{enumerate} \end{lem} \begin{proof} (1) Put $\mathcal{W} = \{w\in W(M)\mid w(\Sigma^+)\cap \Sigma^+_\eta = \emptyset\}$. Let $w_{\eta,0}$ be the longest Weyl element of $W_{M_\eta}$. We prove that the map $\mathcal{W}\to W(\supp\eta,\Theta)$ defined by $w\mapsto (w_{\eta,0}w)^{-1}$ is well-defined and bijective.
First we prove that the map is well-defined. Let $w\in\mathcal{W}$. The equation $w(\Sigma^+)\cap \Sigma_\eta^+ = \emptyset$ implies that $(w_{\eta,0}w)^{-1}(\Sigma^+_\eta)\subset \Sigma^+$. Hence, $(w_{\eta,0}w)^{-1}\in W(\supp\eta)$. Moreover, $w(\Sigma_M^+)\subset \Sigma^+$ and $w(\Sigma^+)\cap \Sigma_\eta^+ = \emptyset$ imply that $w(\Sigma_M^+)\subset \Sigma^+\cap (\Sigma\setminus\Sigma^+_\eta) = \Sigma^+\setminus\Sigma^+_\eta$. Hence, $(w_{\eta,0}w)(\Sigma_M^+)\subset \Sigma^+\setminus\Sigma^+_\eta\subset \Sigma^+$. We have $(w_{\eta,0}w)^{-1}\in W(\Theta)^{-1}$. Finally $w(\Sigma_M^+)\subset \Sigma^+\setminus\Sigma_\eta^+$ implies $w(\Sigma)\subset \Sigma\setminus\Sigma_\eta$. Hence we have $(w_{\eta,0}w)^{-1}\Sigma_\eta\cap\Sigma_M = w^{-1}\Sigma_\eta\cap \Sigma_M = \emptyset$.
Assume that $(w_{\eta,0}w)^{-1}\in W(\supp\eta,\Theta)$. Then $(w_{\eta,0}w)^{-1}(\Sigma^+_\eta)\subset\Sigma^+$ implies that $w(\Sigma^+)\cap \Sigma^+_\eta=\emptyset$. Since $(w_{\eta,0}w)^{-1}\Sigma_\eta\cap\Sigma_M = \emptyset$ we have $w(\Sigma_M)\cap \Sigma_\eta = \emptyset$. By $(w_{\eta,0}w)(\Sigma_M^+)\subset\Sigma^+$ and $w(\Sigma^+)\cap \Sigma_\eta^+ = \emptyset$, we have $w(\Sigma_M^+)\subset ((\Sigma^+\setminus\Sigma_\eta^+)\cup \Sigma_\eta^-)\cap (\Sigma\setminus\Sigma^-_\eta) = (\Sigma^+\setminus\Sigma^+_\eta)$. Consequently we have $w\in W(M)$.
(2) Put $\mathcal{W} = \{w\in W(M)\mid \supp\eta\cap w(\Sigma_M^+) = \emptyset\}$. Define the map $\varphi\colon W(\supp\eta,\Theta)\times W_{\supp\eta} \to \mathcal{W}$ by $(w_1,w_2)\mapsto w_2w_1^{-1}$. This map is injective since $W(\supp\eta,\Theta)\subset W(\supp\eta)$. We prove that $\varphi$ is well-defined and surjective. Since $w_1^{-1}(\Sigma_M^+) = w_1^{-1}(\Sigma_M^+)\cap \Sigma^+\subset \Sigma^+\setminus \Sigma_\eta^+$, we have $w_2w_1^{-1}(\Sigma_M^+)\subset \Sigma^+\setminus\Sigma_\eta^+$. Hence, $\varphi$ is well-defined. Next let $w\in\mathcal{W}$. Let $w_1\in W(\supp\eta)^{-1}$ and $w_2\in W_{\supp\eta}$ such that $w = w_2w_1^{-1}$. Then $w_1^{-1}(\Sigma_M^+) = w_2^{-1}w(\Sigma_M^+)\subset w_2^{-1}(\Sigma^+\setminus\Sigma_\eta^+) = \Sigma^+\setminus\Sigma^+_\eta$. This implies $w_1\in W(\supp\eta,\Theta)$. \end{proof}
\begin{lem}\label{lem:whittaker vectors of finite-dimensional representation} Assume that $\sigma$ is irreducible and finite-dimensional. Let $\widetilde{\mu}$ be the highest weight of $\sigma$ and $V$ the irreducible finite-dimensional representation of $M_0A_0$ with highest weight $\lambda+\widetilde{\mu}$. Then we have $\sigma/(\mathfrak{m}\cap\mathfrak{n}_0)\sigma\simeq V$ as an $M_0A_0$-module. In particular, $\dim\Wh_0(\sigma') = \dim V$. \end{lem} \begin{proof} We prove that $\Wh_0(\sigma^*) \simeq V^*$. Let $\widetilde{w}_{M,0}$ be the longest element of $\widetilde{W_M}$. Then both sides have a highest weight $-\widetilde{w}_{M,0}(\widetilde{\mu} + \lambda)$ and the space of highest weight vectors are $1$-dimensional. \end{proof}
As an Corollary of Theorem~\ref{thm:dimension Whittaker vectors} and Theorem~\ref{thm:dimension Whittaker vectors, algebraic}, we have the following theorem announced by T. Oshima. Define $\widetilde{\rho_M}\in\mathfrak{h}^*$ by $\widetilde{\rho_M} = (1/2)\sum_{\alpha\in\Delta_M^+}\alpha$. \begin{thm}\label{thm:Whittaker vectors, finite-dimensional case} Assume that $\sigma$ is the irreducible finite-dimensional representation with highest weight $\widetilde{\nu}$. Let $\dim_M(\lambda+\widetilde{\nu})$ be a dimension of the finite-dimensional irreducible representation of $M_0A_0$ with highest weight $\lambda+\widetilde{\nu}$. \begin{enumerate}
\item Assume that for all $w\in W$ such that $\eta|_{wN_0w^{-1}\cap N_0} = 1$ the following two conditions hold: \begin{enumerate} \item For all $\alpha\in \Sigma^+\setminus w^{-1}(\Sigma^+_M\cup\Sigma_\eta^+)$ we have $2\langle\alpha,\lambda+w_0\widetilde{\nu}\rangle/\lvert\alpha\rvert^2\not\in\mathbb{Z}_{\le 0}$.
\item For all $\widetilde{w}\in\widetilde{W}$ we have $\lambda - \widetilde{w}(\lambda + \widetilde{\nu} + \widetilde{\rho_M})|_\mathfrak{a}\notin \mathbb{Z}_{\le 0}((\Sigma^+\setminus \Sigma_M^+)\cap w^{-1}\Sigma^+)|_\mathfrak{a}\setminus\{0\}$. \end{enumerate} Then we have \[
\dim \Wh_\eta(I(\sigma,\lambda)') = \# W(\supp\eta,\Theta)\times(\dim_M(\lambda+\widetilde{\nu})) \] \item Assume that for all $\widetilde{w}\in\widetilde{W}\setminus\widetilde{W_M}$, $(\lambda+\widetilde{\nu}) - \widetilde{w}(\lambda+\widetilde{\nu}) \not\in\Delta$. Then we have \begin{multline*}
\dim\Wh_\eta((I(\sigma,\lambda)_{\text{\normalfont $K$-finite}})^*)\\ = \# W(\supp\eta,\Theta)\times \#W_{\supp\eta}\times(\dim_M(\lambda+\widetilde{\nu})) \end{multline*} \end{enumerate} \end{thm}
\appendix \section{$C^\infty$-function with values in Fr\'echet space}\label{sec:C^infty-function with values in Frechet space} \subsection{$\mathcal{L}$-distributions and tempered $\mathcal{L}$-distributions} Let $M$ be a $C^\infty$-manifold, $V$ a Fr\'echet space and $\mathcal{L}$ a vector bundle on $M$ with fibers $V$. We define the sheaf of $\mathcal{L}$-distributions as follows.
First we assume that $\mathcal{L}$ is trivial on $M$. Then the definition of $\mathcal{L}$-distributions is found in Kolk-Varadarajan~\cite{MR1621372} and it is easy to see that $\mathcal{L}$-distributions makes a sheaf on $M$.
In general, let $M = \bigcup_{\lambda\in\Lambda}U_\lambda$ be an open covering of $M$ such that on each $U_\lambda$ the vector bundle $\mathcal{L}$ is trivial. For an arbitrary open subset $U$ of $M$, put \[
\mathcal{D}'(U,\mathcal{L}) = \left\{(x_\lambda)\in \prod_{\lambda\in\Lambda}\mathcal{D}'(U\cap U_\lambda,\mathcal{L})\Bigm| \text{$x_\lambda = x_{\lambda'}$ on $U_\lambda\cap U_{\lambda'}$}\right\}. \newsym{$\mathcal{D}'(U,\mathcal{L})$} \] It is independent of the choice of an open covering $\{U_\lambda\}$ and defines the sheaf of $\mathcal{L}$-distributions on $M$.
Now assume that $M$ has a compactification $X$, i.e., $M$ is an open dense subset of a compact manifold $X$. In this case, we define a subspace $\mathcal{T}(M,\mathcal{L})$ of $\mathcal{D}'(M,\mathcal{L})$ by \[
\mathcal{T}(M,\mathcal{L}) = \{x\in \mathcal{D}'(M,\mathcal{L})\mid \text{$x = z|_{M}$ for some $z\in \mathcal{D}'(X,\mathcal{L})$}\}.\newsym{$\mathcal{T}(M,\mathcal{L})$} \] An element of $\mathcal{T}(M.\mathcal{L})$ is called a tempered $\mathcal{L}$-distribution.
For a subset $M_0\subset M$, put $\mathcal{D}'_{M_0}(U,\mathcal{L}) = \{x\in \mathcal{D}'(U,\mathcal{L})\mid \supp x \subset M_0\}$ and $\mathcal{T}_{M_0}(M,\mathcal{L}) = \{x\in \mathcal{T}(M,\mathcal{L})\mid \supp x\subset M_0\}$. Assume that $M_0$ is a closed submanifold of $M$. Then dualizing the restriction map $C_c^\infty(M,\mathcal{L})\to C_c^\infty(M_0,\mathcal{L})$, we have an injective map $\mathcal{D}'(M_0,\mathcal{L})\to \mathcal{D}'_{M_0}(M,\mathcal{L})$. This map also implies $\mathcal{T}(M_0,\mathcal{L})\to \mathcal{T}_{M_0}(M,\mathcal{L})$. Using these maps, we regard $\mathcal{D}'(M_0,\mathcal{L})$ and $\mathcal{T}(M_0,\mathcal{L})$ as a subspace of $\mathcal{D}'_{M_0}(M,\mathcal{L})$ and $\mathcal{T}_{M_0}(M,\mathcal{L})$, respectively.
\subsection{$\mathcal{L}$-distributions with support in a subspace} Let $M$ be a Euclidean space $\mathbb{R}^n = \{(x_1,\dots,x_n)\in\mathbb{R}^n\}$ and $M_0$ a subspace $\mathbb{R}^{n - m}$ of $M$ defined by the equation $x_1 = \dots = x_m = 0$. Assume that $M$ has a compactification $X$. Let $E_1,\dots,E_m$ be vector fields on $M$ such that: \begin{enumerate}
\item for all $\varphi\in C^\infty(M)$ we have $(E_i\varphi)|_{M_0} = (\frac{\partial}{\partial x_i}\varphi)|_{M_0}$. \item A space $\sum_{i = 1}^m \mathbb{C} E_i$ is a Lie algebra. \end{enumerate}
Set $D_i = \frac{\partial}{\partial x_i}$. The condition (1) implies that $D_iT = E_iT$ for all $T\in \mathcal{D}'(M_0,\mathcal{L})$. Put $U_n(E_1,\dots,E_m) = \sum_{k_1 + \dots + k_m\le n}\mathbb{C} E_1^{k_1}\dotsm E_l^{k_l}$ and $U(E_1,\dots,E_m) = \sum_n U_n(E_1,\dots,E_m)$. Then the algebra $U(E_1,\dots,E_m)$ is isomorphic to the universal enveloping algebra of $\sum_{i = 1}^m \mathbb{C} E_i$. For $\alpha = (\alpha_1,\dots,\alpha_m)$, put $E^\alpha = E_1^{\alpha_1}\dotsm E_m^{\alpha_m}$ where $E_i^0 = 1$.
\begin{lem}\label{lem:diff op nanka do-demoii} Let $E_1',\dots,E_m'$ be vector fields on $M$ which satisfy the same conditions of $E_1,\dots,E_m$. We have \[ E^\alpha T \in (E')^\alpha T + U_{\lvert\alpha\rvert -1}(E_1',\dots,E_m')\mathcal{T}(M_0,\mathcal{L}) \] for $T\in\mathcal{T}(M_0,\mathcal{L})$ and $\alpha\in\mathbb{Z}_{\ge 0}^m$. \end{lem} \begin{proof} First we remark that if the order of differential operator $P$ is less than or equal to $k$, then we have $P(\mathcal{T}(M_0,\mathcal{L}))\subset U_k(D_1,\dots,D_m)\mathcal{T}(M_0,\mathcal{L})$. Take $P\in U_{k - 1}(E_1,\dots,E_m)$. Then we have \begin{multline*} E_iPT = [E_i,P]T + PE_iT = [E_i,P]T + PD_iT = [E_i - D_i,P]T + D_iPT\\ \in D_iPT + U_{k - 1}(D_1,\dots,D_m)\mathcal{T}(M_0,\mathcal{L}) \end{multline*} since the order of $[E_i - D_i,P]$ is less than or equal to $k$. Hence, using induction on $\lvert\alpha\rvert$, we have $E^\alpha T \in D^\alpha T + U_{\lvert\alpha\rvert - 1}(D_1,\dots,D_m)\mathcal{T}(M_0,\mathcal{L})$.
Hence we have $U_k(E_1,\dots,E_m)\mathcal{T}(M_0,\mathcal{L}) \subset U_k(D_1,\dots,D_m)\mathcal{T}(M_0,\mathcal{L})$. We prove $U_k(E_1,\dots,E_m)\mathcal{T}(M_0,\mathcal{L}) = U_k(D_1,\dots,D_m)\mathcal{T}(M_0,\mathcal{L})$ by induction on $k$. If $k = 0$ then the claim is obvious. Assume that $k > 0$ then the above equation and inductive hypothesis imply that if $\lvert\alpha\rvert = k$, then we have $D^\alpha T \in E^\alpha T + U_{k - 1}(E_1,\dots,E_m)\mathcal{T}(M_0,\mathcal{L})$. Hence we have $U_k(D_1,\dots,D_m)\mathcal{T}(M_0,\mathcal{L})\subset U_k(E_1,\dots,E_m)\mathcal{T}(M_0,\mathcal{L})$.
The same formulas hold for $E_1',\dots,E_m'$. Hence, we have \begin{multline*} E^\alpha T\in D^\alpha T + U_{\lvert\alpha\rvert - 1}(D_1,\dots,D_m)\mathcal{T}(M_0,\mathcal{L})\\ = (E')^\alpha T + U_{\lvert\alpha\rvert - 1}(D_1,\dots,D_m)\mathcal{T}(M_0,\mathcal{L})\\ = (E')^\alpha T + U_{\lvert\alpha\rvert - 1}(E_1',\dots,E_m')\mathcal{T}(M_0,\mathcal{L}). \end{multline*} \end{proof}
\begin{prop}\label{prop:structure theorem of tempered distributions whose support is contained in submanifold} A map $\Phi\colon U(E_1,\dots,E_m)\otimes \mathcal{T}(M_0,\mathcal{L})\to\mathcal{T}_{M_0}(M,\mathcal{L})$ defined by $P\otimes T \mapsto PT$ is isomorphic. \end{prop} \begin{proof}
First we prove that $\Phi$ is injective. Let $\sum_{\alpha\in\mathbb{Z}_{\ge 0}^m}E^\alpha\otimes T_\alpha$ (finite sum) be an element of $U(E_1,\dots,E_m)\otimes \mathcal{T}(M_0,\mathcal{L}|_{M_0})$. Set $T = \sum_{\alpha\in\mathbb{Z}_{\ge 0}^m}E^\alpha T_\alpha$ and assume that $T = 0$. Put $k = \max\{\lvert\alpha\rvert\mid T_\alpha \ne 0\}$. We prove that $k = -\infty$. Assume that $k\ge 0$. By Lemma~\ref{lem:diff op nanka do-demoii}, if $\lvert\alpha\rvert = k$ then $E^\alpha T_\alpha \in D^\alpha T_\alpha + U_{k - 1}(D_1,\dots,D_m)\mathcal{T}(M_0,\mathcal{L})$. There exist $T'_\alpha$ such that $\sum_{\alpha\in\mathbb{Z}_{\ge 0}^m}E^\alpha T_\alpha = \sum_{\alpha\in\mathbb{Z}_{\ge 0}^m,\ \lvert\alpha\rvert < k}D^\alpha T'_\alpha + \sum_{\alpha\in\mathbb{Z}_{\ge 0}^m,\ \lvert\alpha\rvert = k}D^\alpha T_\alpha$. Fix $\beta \in \mathbb{Z}_{\ge 0}^m$ such that $\lvert\beta\rvert = k$ and $f\in C^\infty(M_0)$ with values in $\mathcal{L}$. Define a function $\varphi$ on $M$ by $\varphi(x_1,\dots,x_n) = x_1^{\beta_1}\dotsm x_m^{\beta_m}f(0,\dots,0,x_{m + 1},\dots,x_n)$. Then we have $0 = \langle T,\varphi\rangle = \beta_1!\dotsm \beta_m! \langle T_\beta,f\rangle$. Since $f$ is arbitrary, we have $T_\beta = 0$ for all $\beta$ such that $\lvert\beta\rvert = k$. This is a contradiction.
We prove that $\Phi$ is surjective. Take an open covering $X = \bigcup_{\lambda\in \Lambda}U_\lambda$ such that $U_\lambda$ is isomorphic to a Euclidean space and $\overline{M_0}\cap U_\lambda$ is a subspace of $U_\lambda$ where $\overline{M_0}$ is a closure of $M_0$ in $X$. Since $X$ is compact, we may assume this is a finite covering. It is sufficient to prove that the map $\Phi$ is surjective on $U_\lambda\cap M$. However the surjectivity of $\Phi$ on $U_\lambda\cap M$ follows from \cite[(2.8)]{MR1621372}. \end{proof}
\subsection{Distributions on a nilpotent Lie group}\label{subsec:Distributions on a nilpotent Lie group} Let $N$ be a connected, simply connected nilpotent Lie group. Put $\mathfrak{n} = \Lie(N)_\mathbb{C}$. Then the exponential map $\exp\colon \Lie(N)\to N$ is a diffeomorphism. A structure of a vector space on $N$ is defined by the exponential map. Let $\mathcal{P}(N)$ be a ring of polynomials with respect to this vector space structure (cf.\ Corwin and Greenleaf~\cite[\S1.2]{MR1070979}).
Let $\mathcal{L}$ be a vector bundle on $N$ whose fiber is $V$ and assume that $\mathcal{L}$ is trivial on $N$, i.e., $\mathcal{L} = N\times V$. Fix a Haar measure $dn$ on $N$. For $F\in C^\infty(N,V')$, we define a distribution $F\delta$ by $\langle F\delta,\varphi\rangle = \int_N F(n)(\varphi(n))dn$. Then we regard $C^\infty(N,V')$ as a subspace of $\mathcal{D}'(N,\mathcal{L})$. Let $\mathcal{P}_k(N)$ be the space of polynomials whose degree is less than or equal to $k$. Put $\mathcal{P}(N) = \bigoplus_k\mathcal{P}_k(N)$.
Take a character $\eta$ of $\mathfrak{n}$. Then $\eta$ can be extended to the $\mathbb{C}$-algebra homomorphism $U(\mathfrak{n})\to \mathbb{C}$ where $U(\mathfrak{n})$ is the universal enveloping algebra of $\mathfrak{n}$. We denote this $\mathbb{C}$-algebra homomorphism by the same letter $\eta$. Let $\Ker\eta$ be the kernel of the $\mathbb{C}$-algebra homomorphism $\eta$. For $X\in \mathfrak{n}$ and $C^\infty$-function $\psi$, put $(X\psi)(n) = \frac{d}{dt}\psi(\exp(-tX)n)|_{t = 0}$.
The algebraic tensor product $C_c^\infty(N)\otimes V$ is canonically identified with a linear subspace of $C^\infty_c(N,\mathcal{L})$ via $\varphi\otimes v\mapsto (x\mapsto \varphi(x)v)$. As in \cite[(2.1)]{MR1621372}, this is a dense subspace of $C^\infty_c(N,\mathcal{L})$.
\begin{prop}\label{prop:killed by some power of n is polynomial} For all $k\in \mathbb{Z}_{>0}$, there exists a positive integer $l$ such that, if $T\in \mathcal{D}'(N,\mathcal{L})$ satisfies $(\Ker\eta)^kT = 0$ then $T\in (\mathcal{P}_l(N)\otimes V')\delta$.
Conversely, for all $l\in\mathbb{Z}_{>0}$ there exists a positive integer $k$ such that $(\Ker\eta)^k(\mathcal{P}_l(N)\otimes V')\delta = 0$. \end{prop} \begin{proof} Fix a basis $\{e_1,\dots,e_n\}$ of $\Lie(N)$. The map $\mathbb{R}^n\to N$ defined by $(x_1,\dots,x_n)\mapsto \exp(x_1e_1 + \dots + x_me_m)$ is an isomorphism. Using this map, we introduce a coordinate $(x_1,\dots,x_n)$ of $N$. If $V = \mathbb{C}$ then this proposition is well-known. Fix $v\in V$ and consider an ordinal distribution $T_v\colon \varphi\mapsto \langle T,\varphi\otimes v\rangle$ for $\varphi\in C^\infty_c(N)$. If $T$ satisfies $(\Ker\eta)^kT = 0$, then $T_v$ satisfies $(\Ker\eta)^kT_v = 0$. Hence for some $l$, $T_v = \sum_{\alpha_1 + \dots + \alpha_n\le l}(x_1^{\alpha_1}\dotsm x_n^{\alpha_n}\otimes c_{v,\alpha_1,\dots,\alpha_n})\delta$, where $c_{v,\alpha_1,\dots,\alpha_n}\in \mathbb{C}$. The map $v\mapsto c_{v,\alpha_1,\dots,\alpha_n}$ is continuous linear. Hence it defines an element of $V'$. We denote this element by $v'_{\alpha_1,\dots,\alpha_n}$. Then for $\varphi\in C_c^\infty(N)$ and $v\in V$ we have $\langle T,\varphi\otimes v\rangle = \langle (\sum_{\alpha_1 + \dots + \alpha_n\le l}x_1^{\alpha_1}\dotsm x_n^{\alpha_n}\otimes v'_{\alpha_1,\dots,\alpha_n})\delta,\varphi\otimes v\rangle$. Since $C^\infty_c(N)\otimes V$ is dense in $C^\infty_c(N,\mathcal{L})$, we have $T = (\sum_{\alpha_1 + \dots + \alpha_n\le l}x_1^{\alpha_1}\dotsm x_n^{\alpha_n}\otimes v'_{\alpha_1,\dots,\alpha_n})\delta$.
We prove the second part of the proposition. For $X\in \mathfrak{n}$, $f\in\mathcal{P}_l(N)$ and $v'\in V'$, we have $X((f\otimes v')\delta) = ((Xf)\otimes v')\delta$. Hence we may assume that $V = \mathbb{C}$. In this case, the claim is well-known. \end{proof}
\begin{cor}\label{cor:polynomial by some power of n} Let $T\in \mathcal{D}'(N,\mathcal{L})$. Assume that there exists a positive integer $k$ such that $(\Ker\eta)^kT \in (\mathcal{P}(N)\otimes V')\delta$. Then we have $T \in (\mathcal{P}(N)\otimes V')\delta$. \end{cor} \begin{proof} By the second part of Proposition~\ref{prop:killed by some power of n is polynomial}, there exists a positive integer $k'$ such that $(\Ker\eta)^kT = 0$. Hence we have $T \in (\mathcal{P}(N)\otimes V')\delta$ by the first part of Proposition~\ref{prop:killed by some power of n is polynomial}. \end{proof}
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\end{document} | arXiv |
Chemical reactions in white dwarf and carbon allotropes
White dwarfs consists mostly of carbon and oxygen. In my opinion, they are too hot to contain these elements in molecular form and hence chemical reactions does not happen (I think resulting CO2 will also decompose under such high temperature and pressure). I have two questions related to that:
1) After a few billion years, can the WD cool down enough to sustain chemical reactions which will result in production of CO2.
2) Given, there is very high pressure and temperature, after a few billion years what will be the dominant allotrope of carbon (Do we have, in future, diamonds in the sky ;)
I strongly doubt given the extreme density of white dwarfs( I think density of white dwarfs wont change over time), if any of the scenario is possible.
white-dwarf
Knu8Knu8
White dwarfs are objects the size of the Earth, but with a mass more similar to the Sun. Typical internal densities are $10^{9}$ to $10^{11}$ kg/m$^{3}$.
White dwarfs are born as the contracting core of asymptotic giant branch stars that do not quite get hot enough to initiate carbon fusion. They have initial central temperatures of $\sim 10^{8}$K, that swiftly (millions of years) drops to a few $10^{7}$K due to neutrino emission.
An important point to make is that after that, the interior of a white dwarf is almost isothermal. This is because the degenerate electrons that provide the pressure support also have extremely long mean free paths for scattering interactions and thus the thermal conductivity is extremely high.
The exterior of the white dwarf is a factor of 100 cooler than the interior. The temperature drop happens over a very thin shell (perhaps 1% of the outer part of the white dwarf), where the degenerate gas transitions to becoming non-degenerate at the surface.
This outer layer acts like an insulating blanket and makes the cooling timescales of white dwarfs very long. From interior temperatures of say $3\times 10^{7}$ K it takes a billion years or so to cool to $5\times 10^{6}$ K and then another 10 billion years to cool to around $10^{6}$K, and such white dwarfs, which must have arisen from the first stars that were born with progenitor masses of 5 to 8 solar masses, will be the coolest white dwarfs in the Galaxy.
At these temperatures there is no possibility of the carbon undergoing chemical reactions, it is completely ionised; the carbon and oxygen nuclei are in a crystalline lattice at these densities, surrounded by a degenerate electron gas. There is evidence that crystallisation does take place, via asteroseismology of some pulsating massive white dwarfs.
The details of the crystalline structure in these objects is unknown, and the subject of theoretical investigation. However, diamond is pure carbon and white dwarfs are expected to be a carbon/oxygen mixture. A further complication is that the process of crystallisation may be accompanied by gravitational separation of the carbon and oxygen, so that the inner core is more oxygen-rich than the outer core.
Original ideas were that the crystalline form would be body-centred cubic (bcc), but other more complex possibilites are opened up by the mixture of carbon and oxygen. bcc Carbon would be a new allotrope of carbon and not like diamond - it is a denser way of arranging the nuclei.
EDIT: To answer a point in the comments. Even if you were to wait trillions of years and allow white dwarfs to cool to the thousands or even hundreds of degrees that you might think would allow electrons to recombine and for chemistry to occur, that is not how it works. In the degenerate electron gas, the typical Fermi energy of the electrons is an MeV or so, compared with the eV-keV of bound electron states, and this is completely independent of the temperature. So the high electron number densities ensure that they will never recombine with the carbon nuclei (a theory first developed by Kothari 1938).
Rob JeffriesRob Jeffries
$\begingroup$ "bcc Carbon" Carbon never stops surprising us. IMO, it is most fascinating element. $\endgroup$ – Knu8 May 6 '16 at 12:33
$\begingroup$ Given enough time, white dwarfs will eventually cool down to temperature of the range of few 100Ks. Are chemical reactions feasible then? $\endgroup$ – Knu8 May 6 '16 at 13:15
$\begingroup$ @Knu8 Hypothetical eschatology is not my field. To cool that far would take more than trillions of years. However at these pressures I do not think the electrons and ions can ever recombine to give you chemistry. $\endgroup$ – Rob Jeffries May 6 '16 at 14:02
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\begin{document}
\title{The two-dimensional analogue of the Lorentzian catenary and the Dirichlet problem} \begin{abstract}
We generalize in Lorentz-Minkowski space $\mathbb L^3$ the two-dimensional analogue of the catenary of Euclidean space. We solve the Dirichlet problem when the bounded domain is mean convex and the boundary data has a spacelike extension to the domain. We also classify all singular maximal surfaces of $\mathbb L^3$ invariant by a uniparametric group of translations and rotations. \end{abstract}
\noindent {\it Mathematics Subject Classification:} 53A10, 53C42 \\ \noindent {Keywords:} singular maximal surface, Dirichlet problem, invariant surface
\section{Introduction and motivation}
The purpose of this paper is to investigate the physical problem of characterizing the surfaces in Lorentz-Minkowski space with lowest gravity center and solve the corresponding Dirichlet problem. The existence of a variety of causal vectors in the Lorentzian setting makes that appear several issues that need to be fixed. Firstly, we recall this problem in the Euclidean space in order to motivate our definitions. Let $\mathbb R^2$ be the Euclidean plane with canonical coordinates $(x,y)$ where the $y$-axis indicates the gravity direction. Consider the physical problem of finding the curve in the halfplane $y>0$ with the lowest gravity center. If the curve is $y=u(x)$, then $u$ satisfies the equation
\begin{equation}\label{cat} \frac{u''}{1+u'^2}=\frac{1}{u}. \end{equation} The solution of this equation is known the catenary
$$u(x)=\frac{1}{a}\cosh(ax +b), \ a,b\in\mathbb R, a\not=0.$$ Equation \eqref{cat} can be expressed in terms of the curvature $\kappa$ of the curve as \begin{equation}\label{cat0} \kappa=\frac{\langle \textbf{n},\vec{a}\rangle}{y}, \end{equation}
where $\textbf{n}$ is the unit normal vector and $\vec{a}=(0,1)$. In particular, equation \eqref{cat0} prescribes the angle that makes the vector $\textbf{n}$ with the vertical direction.
The generalization in Euclidean $3$-space $\mathbb R^3$ of the property of the catenary is to find surfaces in the halfspace $z>0$ with the lowest gravity center. If $(x,y,z)$ denote the canonical coordinates of $\mathbb R^3$ and $z$ indicates the direction of the gravity, these surfaces characterize by means of the equation $$H=\frac{\langle N,\vec{a}\rangle}{z},$$
where $H$ is the mean curvature of the surface and $\vec{a}=(0,0,1)$. The surface is called in the literature the {\it two-dimensional analogue of the catenary} (\cite{bht,dh}). Historically, this problem goes back to early works of Lagrange and Poisson on the equation that models a heavy surface in vertical gravitational field. If we embed $\mathbb R^2$ as the $xz$-plane by identifying the $y$-axis of $\mathbb R^2$ with the $z$-axis of $\mathbb R^3$, and we rotate the catenary with respect to the $x$-axis, we obtain the catenoid $a^2(y^2+z^2)=\cosh^2(x)$, which is the only non-planar rotational minimal surface of $\mathbb R^3$.
More general, given a constant $\alpha\in\mathbb R$, a surface in the halfspace $z>0$ is called a singular minimal surface if satisfies \begin{equation}\label{cat1} H=\alpha\frac{\langle N,\vec{a}\rangle}{z}. \end{equation}
The theory of singular minimal surfaces has been intensively studied from the works of Bemelmans, Dierkes and Huisken, among others. Without to be a complete list, we refer to \cite{bd,bht,di,di2,dh,lo3,lo4,lo6,ni2}.
Once presented the problem in the Euclidean space, we proceed to generalize it in the Lorentz-Minkowski space. As in the Euclidean case, we begin with the one-dimensional case. Let $\mathbb L^2$ be the Lorentz-Minkowski plane defined as the affine $(x,y)$-plane $\mathbb R^2$ endowed with the metric $dx^2-dy^2$. Here we use the usual terminology of the Lorentz-Minkowski space: see \cite{on} as a general reference and \cite{lo1} for curves and surfaces in Lorentz-Minkowski space. In what follows, we will assume that for a given set, the causal character is the same in all its points, that is, we do not admit the existence of points with different causal character.
A first issue is that in $\mathbb L^2$ it does not make sense the notion of gravity in $\mathbb L^2$ because the $y$-coordinate represents the time in the Lorentzian context. Thus we need to view the initial problem as a problem of finding curves in $\mathbb L^2$ with prescribed angle between the normal vector and a fixed direction, such as it was shown in equation \eqref{cat0}. There appear two new issues. Firstly there are three types of curves in $\mathbb L^2$ according its causal character, namely, spacelike, timelike and lightlike and the behavior of each of these curves is completely different. Because our interest is to keep the Riemannian sense, we will only consider spacelike curves.
A second issue is the choice of the axis with respect to what we measure the angle of the normal vector $\textbf{n}$. Notice that in Euclidean plane both axes are indistinct but in $\mathbb L^2$ the $y$-axis and the $x$-axis are not interchangeable by a rigid motion. Thus it arises the problem what axis to be fixed. Since for a spacelike curve, the vector $\textbf{n}$ is timelike, we will measure the angle between $\textbf{n}$ and the $y$-axis, which is also timelike. This is also justified because it makes sense to define the angle between two timelike vectors (\cite[p.144]{on}). After all these considerations, let us proceed.
Let $\gamma=\gamma(s)$ be a spacelike curve parametrized by the arc-length $s\in I$ and contained in the halfplane $y>0$ of $\mathbb L^2$. The curvature $\kappa$ of $\gamma$ is defined by $\gamma''(s)=\kappa(s)\textbf{n}(s)$ where $\textbf{n}$ is a unit normal vector of $\gamma$. Here we are assuming $\kappa\not=0$. Motivated by the equation \eqref{cat0}, we ask for those spacelike curves of $\mathbb L^2$ that satisfy the same equation \eqref{cat0} where $\textbf{a}=(0,1)$. If $\gamma$ is a graph $y=u(x)$, then $\gamma(x)=(x,u(x))$, which is not parametrized by the arc-length. Then $\textbf{n}=(u',1)/\sqrt{1-u'^2}$, $\langle \textbf{n},\vec{a}\rangle=-1/\sqrt{1-u'^2}$ and $$\kappa(x)=-\frac{1}{1-u'^2}\langle\gamma''(x),\textbf{n}(x)\rangle=\frac{u''(x)}{(1-u'(x)^2)^{3/2}}.$$ Let us observe that $u'^2<1$ because $\gamma$ is a spacelike curve. Equation \eqref{cat0} is now \begin{equation}\label{eq0} \frac{u''}{1-u'^2}=-\frac{1}{u}, \end{equation} which will be the Lorentzian model of \eqref{cat} that we are looking for. The spacelike condition $u'^2-1<0$ is an extra hypothesis comparing with the Euclidean case. For example, $u(x)= \sinh(x)$, with $u>0$, solves \eqref{eq0}, but $u'^2>1$. So, the corresponding curve $y=u(x)$ is a timelike curve. In contrast, because we are assuming that the curve is spacelike, the right solution of \eqref{eq0} is \begin{equation}\label{eq00} u(x)=\frac{1}{a}\sin(ax+b),\quad x\in\left(-\frac{b}{a},\pi-\frac{b}{a}\right), \end{equation} where $a\not=0\ a,b\in\mathbb R$. This curve will be the analogue catenary in $\mathbb L^2$. As in the Euclidean case, we introduce a constant $\alpha\in\mathbb R$ and we consider the analogous equation of \eqref{cat0}, namely, \begin{equation}\label{cat11} \kappa=\alpha\frac{\langle\textbf{n},\vec{a}\rangle}{\langle p,\vec{a}\rangle}=-\alpha\frac{\langle\textbf{n},\vec{a}\rangle}{y}, \end{equation} where $p=(x,y)\in\mathbb L^2$. For instance, the curve \eqref{eq00} is the solution for $\alpha=-1$.
Following the same steps done in the Euclidean setting, we embed $\mathbb L^2$ in the Lorentz-Minkowski $3$-space $\mathbb L^3$. Here $\mathbb L^3$ is the affine Euclidean $3$-space endowed with the metric $dx^2+dy^2-dz^2$. Then $\mathbb L^2$ is identified with the $xz$-plane, the $y$-axis of $\mathbb L^2$ with the $z$-axis of $\mathbb L^3$ and the vector $(0,1)\in\mathbb L^2$ with $\vec{a}=(0,0,1)$. Definitively, the objects of our study in this paper are described in the following definition.
\begin{definition} Let $\alpha$ be a nonzero real number. A spacelike surface $S$ in the halfspace $z>0$ of $\mathbb L^3$ is called an $\alpha$-singular maximal surface if satisfies \begin{equation}\label{eqL} H(p)=\alpha\frac{\langle N(p),\vec{a}\rangle}{\langle p,\vec{a}\rangle}=-\alpha\frac{\langle N(p),\vec{a}\rangle}{z}\quad (p\in S), \end{equation} where $N$ is a unit normal vector field on $S$ and $H$ is the mean curvature. \end{definition}
Here $H$ the trace of the second fundamental form of $S$, that is, the sum of the principal curvatures. We will omit the constant $\alpha$ if it is understood in the context. Recently, these surfaces have been studied in \cite{mt} relating the Riemannian and the Lorentzian settings by means of a Calabi type correspondence.
In view of \eqref{cat0}, and as a motivation of this paper, the case $\alpha=-1$ in equation \eqref{eqL} is the corresponding {\it two-dimensional analogue of the Lorentzian catenary}. Other known examples appear when $\alpha=2$ because in such a case, the surface is a minimal surface in the steady state space (\cite{lo2}). Another special example is the hyperbolic plane $\mathbb H^2(r)=\{p\in\mathbb L^3:\langle p,p\rangle=-r^2, z>0\}$, $r>0$. This surface has mean curvature $H=2/r$ for $N(p)=p/r$. It is clear that $\mathbb H^2(r)$ satisfies \eqref{eqL} for $\alpha=2$. Even more, $\mathbb H^2(r)$ satisfies \eqref{eqL} {\it for any} vector $\vec{a}$.
On the other hand, we extend a similar property that has the catenary in Euclidean space. Indeed, we take
the catenary \eqref{eq00} and we rotate with respect to the $x$-axis. The rotations that leave pointwise fixed the $x$-axis are described by $$\left\{\left(\begin{array}{ccc}1&0&0\\ 0 &\cosh\theta&\sinh\theta\\ 0&\sinh\theta&\cosh\theta\end{array}\right):\theta\in\mathbb R\right\}.$$
For a curve $z=u(x)$, namely, $\gamma(x)=(x,0,u(x))$, $x\in I\subset\mathbb R$, contained in the $xz$-plane, the corresponding rotational surface $S$ is parametrized by \begin{equation}\label{eqx} X(x,\theta)=(x,u(x)\sinh\theta ,u(x)\cosh\theta ), \ \theta\in \mathbb R. \end{equation} If $u(x)=\sin(ax+b)/a$, it is not difficult to see that the corresponding rotational surface \eqref{eqx} has zero mean curvature, that is, $S$ is a maximal surface of $\mathbb L^3$. This surface is called in the literature the catenoid of second kind or the hyperbolic catenoid.
\begin{remark}If we rotate the curve $u(x)= \sinh(ax+b)/a$, the timelike solution of \eqref{eq0}, with respect to the $x$-axis, the rotational surface is a timelike surface with zero mean curvature (\cite{lo0}). Similarly, any vertical straight line is a timelike curve that satisfies \eqref{cat0} and if we rotate with respect to the $x$-axis, we obtain a (timelike) plane parallel to the $yz$-plane, which has zero mean curvature everywhere.
\end{remark}
As a conclusion, the generalization in $\mathbb L^3$ of the two-dimensional analogue of the catenary, or more generally, singular maximal surfaces in Lorentz-Minkowski space $\mathbb L^3$, is carried out for spacelike surfaces and the angle between $N$ and $\vec{a}$ is measured with respect to the (timelike) $z$-axis. We have also discussed that there are other possibilities to generalize the initial problem in $\mathbb L^3$, although all them less justified, as for example, changing the axis $\vec{a}=(0,0,1)$ by $(1,0,0)$ (spacelike) or $(1,0,1)$ (lightlike). Also, we may consider timelike surfaces and measuring the angle between $N$ with respect to an axis of $\mathbb L^3$.
In this paper we will also be interested to solve the Dirichlet problem of the singular maximal surface equation. Since a spacelike surface is locally the graph of a function $z=u(x,y)$, the nonparametric form of equation \eqref{eqL} is
\begin{equation}\label{eq2}
\mbox{div}\frac{D u}{\sqrt{1-|D u|^2}}=\alpha\frac{1}{u\sqrt{1-|D u|^2}},
\end{equation}
together the spacelike condition $|Du|<1$. The left-hand side of this equation is the mean curvature of the graph $z=u(x,y)$ computed with respect to the upwards orientation
$$N=\frac{1}{\sqrt{1-|Du|^2}}(Du,1).$$
Comparing \eqref{eq2} with the Riemannian case (\cite{di,di1,di2,lo5}), this equation is not uniformly elliptic and, as a consequence, this requires to ensure that $|Du|$ is bounded away from $1$.
This paper is organized as follows. In Section \ref{sec2} we classify all singular maximal surfaces that are invariant by a uniparametric group of translations and of rotations. In Section \ref{sec3} we describe the solutions of \eqref{eqL} that are invariant by rotations about the $z$-axis and finally, in Section \ref{sec4} we solve the Dirichlet problem associated to equation \eqref{eq2} for mean convex domains and arbitrary boundary data.
\section{Invariant singular maximal surfaces}\label{sec2}
In this section we classify and describe all singular maximal surfaces that are invariant by a uniparametric group of translations or of rotations of $\mathbb L^3$. Firstly, we notice that some transformations of the affine Euclidean space $\mathbb R^3$ preserve the singular maximal surface equation. To fix the terminology, a vector $\vec{v}\in\mathbb R^3$ is called horizontal direction if it is parallel to the $xy$-plane and it is called vertical if is parallel to the $z$-axis.
It is clear that a solution of \eqref{eqL} is invariant by a translation along a horizontal direction, that is, if $S$ is an $\alpha$-singular maximal surface, then $S+\vec{v}$ is also an $\alpha$-singular maximal surface, where $\vec{v}$ is a horizontal vector of $\mathbb R^3$. Similarly, the same property holds if we rotate $S$ with respect to a vertical direction because the term $\langle N,\vec{a}\rangle $ and the denominator $z$ in \eqref{eqL} are invariant by this type of rotations. Finally, if $\lambda>0$ is a positive real number, and $T_\lambda(p)=p_0+\lambda(p-p_0)$ is the dilation with center $p_0\in\mathbb R^2\times\{0\}$, then $T_\lambda(S)$ is an $\alpha$-singular maximal surface.
\begin{remark}\label{re1} We point out that a rigid motion of $\mathbb L^3$ does not preserve in general the equation \eqref{eqL} because the denominator $z$ may change in general by the motion. \end{remark}
As we have announced, a natural source of examples of singular maximal surfaces of $\mathbb L^3$ finds in the class of invariant surfaces by a uniparametric group of rigid motions. The key point is that equation \eqref{eqL}, which locally is the partial differential equation \eqref{eq2}, changes into an ordinary differential equation. In particular, by standard theory, there always is a solution for any initial conditions.
\subsection{Surfaces invariant by translations}
We begin the study of the surfaces invariant by a uniparametric group of translations. Since the rulings generated by this group are straight lines contained in the surface, and the surface is spacelike, then any ruling is a spacelike line. Thus the vector generating the group of translation must be spacelike. Let $\vec{v}$ be a unit spacelike vector and consider a surface $S$ invariant by the group of translations generated by $\vec{v}$. Then $S$ parametrizes as $X(s,t)=\gamma(s)+t\vec{v}$, where $\gamma$ is a planar spacelike curve of $\mathbb L^3$ contained in a (timelike) orthogonal plane to $\vec{v}$. Equation \eqref{eqL} is $$ \kappa\,\mbox{det}(\gamma',\vec{v},\textbf{n})=\alpha\frac{\mbox{det}(\gamma',\vec{v},\vec{a})}{\gamma_3+tv_3},$$ where $\gamma=(\gamma_1,\gamma_2,\gamma_3)$ and $\vec{v}=(v_1,v_2,v_3)$. We consider the orientation in $\gamma$ so $\gamma'\times\vec{v}=\textbf{n}$. Since $\textbf{n}$ is a unit timelike vector, the above equation is now \begin{equation}\label{ejt} \kappa (\gamma_3+tv_3)+\alpha\langle\textbf{n},\vec{a}\rangle=0. \end{equation} This is a polynomial equation on $t$, hence $$\kappa v_3=0,\quad \kappa \gamma_3 +\alpha\langle\textbf{n},\vec{a}\rangle=0.$$ Since $\kappa\not=0$, we deduce that $v_3=0$ and
$\kappa\gamma_3+\alpha\langle\textbf{n},\vec{a}\rangle=0$. Then $\vec{v}$ is a horizontal vector and $\gamma$ is a planar curve contained in a vertical plane. After a horizontal translation and a rotation about the $z$-axis, we assume that this plane is the $xz$-plane which can be identified with $\mathbb L^2$. Furthermore, the equation $\gamma_3 +\alpha\langle\textbf{n},\vec{a}\rangle=0$ means that $\gamma$ satisfies, as a planar curve of $\mathbb L^2$, the one-dimensional singular maximal surface equation \eqref{cat11}. The converse of this result is immediate.
\begin{proposition} \label{pr211}
Let $S$ be an $\alpha$-singular maximal surface of $\mathbb L^3$ invariant by a uniparametric group of translations generated by $\vec{v}$ and denote by $\gamma$ its generatrix. Then $\vec{v}$ is a horizontal vector, $\gamma$ is contained in a plane orthogonal to $\vec{v}$ and $\gamma$, as a planar curve, satisfies \eqref{cat11}. Conversely, if $\gamma$ is a curve in $\mathbb L^2$ that satisfies \eqref{cat11} and, if we embed this curve in the $xz$-plane as usually, then the surface $X(s,t)=\gamma(s)+t(0,1,0)$ is an $\alpha$-singular maximal surface.
\end{proposition}
In view of this proposition, consider the one-dimensional case of equation \eqref{eqL}. Let $\gamma(s)=(x(s),y(s))$ be a spacelike curve in $\mathbb L^2$ that satisfies \eqref{cat11}. Since $\gamma$ is spacelike, then $x'^2-y'^2>0$, in particular, $x'(s)\not=0$ for every $s$ and thus $\gamma$ is globally the graph of a function $u=u(x)$, $x\in I\subset\mathbb R$. Equation \eqref{cat11} is now \begin{equation}\label{eq-one} \frac{u''}{1-u'^2}=\alpha\frac{1}{u},\quad u>0, u'^2<1. \end{equation} It is possible to find some explicit solutions of \eqref{eq-one} by simple quadratures. In the Introduction we have seen that if $\alpha=-1$, the solution is $u(x)= \sin(ax+b)/a$, where $a\not=0$, $a,b\in\mathbb R$ and where $x$ is defined in some interval to ensure that $u>0$. If $\alpha=1$, it is easy to find that the solution of \eqref{eq-one} is
$$u(x)=\frac{1}{a}\sqrt{1+a^2 x^2+2abx+b^2},\ a,b\in\mathbb R, a>0.$$ After a change of variable, this function $u$ writes as $u(x)=\sqrt{1+a^2x^2}/a$, $a>0$. It is immediate that $u$ is the upper branch of the hyperbola $a^2(x^2-y^2)=-1$. This curve, viewed as a planar curve in $\mathbb{L}^2$, has nonzero constant curvature $\kappa=a$. The generated surface by Proposition \ref{pr211} is the right-cylinder of equation $a^2(x^2-z^2)=-1$.
\begin{remark}\label{rem22} Such as it was done for the catenary $u(x)=\sin(ax+b)/a$, if we rotate the curve $u(x)=\sqrt{1+a^2x^2}/a$ with respect to the $x$-axis, we obtain the hyperbolic plane $\mathbb H^2(1/a)$. \end{remark}
\begin{remark} Similarly as in the case $\alpha=-1$, there is a timelike solution of \eqref{eq-one} by replacing the spacelike condition $u'^2<1$ by $u'^2>1$. The solution if now $u(x)=\sqrt{a^2x^2-1}/a$, where $a>0$ and $x>1/a$. The function $u$ is the positive part of the hyperbola $x^2-y^2=1/a^2$, which is a timelike curve. If we rotate about the $x$-axis, the generated surface is $x^2+y^2-z^2=1/a^2$. This surface is the (upper part of) de Sitter space $\mathbb{S}^2_1(1/a)=\{p\in\mathbb L^3:\langle p,p\rangle=1/a^2\}$. This surface satisfies \eqref{eqL} when $\alpha=2$ and plays the same role than the hyperbolic plane in the family of timelike surfaces of $\mathbb L^3$. \end{remark}
We now describe the geometric properties of the solutions of \eqref{eq-one}. See figure \ref{fig1}.
\begin{theorem} Let $u=u(x)$ be a solution of \eqref{eq-one}, $x\in I$, where $I\subset\mathbb R$ is the maximal domain of $u$. Then $u$ is symmetric about a vertical line and $I=\mathbb R$ if $\alpha>0$ or $I$ is a bounded interval if $\alpha<0$. Furthermore: \begin{enumerate} \item Case $\alpha>0$. The function $u$ is convex with a unique global minimum, $\lim_{r\rightarrow \infty}u(r)=\infty$ and $\lim_{r\rightarrow \infty}u'(r)=1$. \item Case $\alpha<0$. The function $u$ is concave with a unique global maximum. If $I=(-b,b)$, then $\lim_{r\rightarrow b}u(r)=0$ and $\lim_{r\rightarrow b}u'(r)=-1$. \end{enumerate} \end{theorem}
\begin{proof} If $u$ has a critical point at $r=r_o$, then $u''(r_o)=\alpha/u(r_o)$ has the same sign than $\alpha$. Hence, there is one critical point at most that will be a global minimum (resp. maximum) if $\alpha>0$ (resp. $\alpha<0$).
{\it Claim: There exists a critical point of $u$.}
Suppose now that the claim is proved and we finish the proof of theorem. After a change in the variable $x$, we suppose that $x=0$ is the critical point, $u'(0)=0$. Then $u$ is the solution of \eqref{eq-one} with initial conditions $u(0)=u_0>0$ and $u'(0)=0$. It is clear that $u(-s)$ is also a solution of the same initial value problem, so $u(s)=u(-s)$ by uniqueness. This proves that $u$ is symmetric about the $y$-axis.
Multiplying \eqref{eq-one} by $u'$, we obtain a first integral \begin{equation}\label{eq-f} \frac{1}{1-u'^2}=\mu u^{2\alpha}, \end{equation} for some positive constant $\mu >0$.
\begin{enumerate} \item Case $\alpha>0$. Since $u(x)\geq u_0$, we deduce from \eqref{eq-one} that $u'$ and $u''$ are bounded functions and this implies that the maximal domain is $\mathbb R$. Since $u$ is a convex function, then $u(r)\rightarrow\infty$ as $r\rightarrow\infty$ and from \eqref{eq-f}, we conclude that $u'(r)\rightarrow 1$ as $r\rightarrow\infty$. \item Case $\alpha<0$. By symmetry, $I=(-b,b)$ for some $b\leq\infty$. Since $u$ is a positive concave function, then $b<\infty$. Using the concavity of $u$ again, and because $u'^2<1$, then the graph of $u$ must meet the $x$-axis, that is, $\lim_{r\rightarrow b}u(r)=0$. From \eqref{eq-f}, we deduce $\lim_{r\rightarrow b}u'(r)^2=1$, and by concavity, $\lim_{r\rightarrow b}u'(r)=-1$. \end{enumerate}
We now prove the claim. The proof is by contradiction. Assume that the sign of $u'$ is constant and denote $I=(a,b)$ with $-\infty\leq a<b\leq\infty$.
\begin{enumerate} \item Case $\alpha>0$. We suppose that $u'>0$ in $I$ (similar argument if $u'$ is negative). Since $u$ is increasing and $u'$ and $u''$ are bounded close $r=b$, we deduce that $b=\infty$ by standard theory. If $-\infty<a$, then $\lim_{r\rightarrow a}u(r)=0$ because on the contrary, we could extend $u$ beyond $r=a$ because $u'$ and $u''$ would be bounded functions. Therefore $\lim_{r\rightarrow a}u'(r)^2=1$ by \eqref{eq-f}. Since $u'>0$, this limit is just $1$. This is a contradiction because $u'$ is an increasing function and we would have $u'>1$ in $I$, which is not possible by the spacelike condition.
Thus $a=-\infty$. Since $u$ is increasing and $u>0$ in $\mathbb R$, we find $\lim_{r\rightarrow -\infty}u(r)=c\geq 0$. Because $u'>0$ and $u''>0$, then $\lim_{r\rightarrow -\infty}u'(r)=\lim_{r\rightarrow -\infty}u''(r)=0$. However, by \eqref{eq-one}, and letting $ r\rightarrow -\infty$, we have $u''(r)$ goes to $\alpha/c\not=0$ if $c>0$ or to $\infty$ if $c=0$, obtaining a contradiction. \item Case $\alpha<0$. We suppose that $u'>0$ in $I$ (similar argument if $u'$ is negative). Since $u'$ and $u''$ are bounded for $r$ close to $b$, then $b=\infty$ and by concavity, we deduce that $-\infty<a$. If $u$ is bounded from above with $\lim_{r\rightarrow \infty}u(r)=c>0$, then $\lim_{r\rightarrow \infty}u'(r)=0$ and since $u''<0$, then $\lim_{r\rightarrow \infty}u''(r)=0$. By \eqref{eq-one}, we find $\lim_{r\rightarrow \infty}u''(r)=\alpha/c<0$, a contradiction. Thus $\lim_{r\rightarrow \infty}u(r)=\infty$. By using \eqref{eq-f}, we conclude $\lim_{r\rightarrow \infty}u'(r)^2=1$, so this limit is $1$: a contradiction because $u'$ is a decreasing function and we would have $u'>1$ in the interval $I$, which is not possible. \end{enumerate} \end{proof}
\begin{figure}
\caption{Solutions of \eqref{eq-one}. Left: $\alpha=1$. Right: $\alpha=-2$ }
\label{fig1}
\end{figure}
\subsection{Surfaces of revolution with respect to a spacelike axis and a lightlike axis}
The second source of examples of singular maximal surfaces are the surfaces invariant by a uniparametric group of rotations. A difference between the Euclidean and the Lorentzian settings is that in $\mathbb L^3$ there are three types of surfaces of revolution depending if the rotational axis is spacelike, timelike or lightlike. Section \ref{sec3} is devoted to the surfaces of revolution whose rotation axis is timelike because this type of surfaces will play a special role in the solvability of the Dirichlet problem in Section \ref{sec4}. In this section we investigate the cases that the rotation axis is spacelike and lightlike.
We point out that there is not an {\it a priori} relation between the rotation axis $L$ and the vector $\vec{a}=(0,0,1)$ of equation \eqref{eqL}. This implies that if we apply a rigid motion to prescribe the rotation axis, then the vector $\vec{a}$ does change: see also Remark \ref{re1}.
Firstly we consider the case that the axis is spacelike.
\begin{proposition}\label{pr-x}
Let $S$ be a spacelike surface of $\mathbb L^3$ invariant by the uniparametric group of rotations about a spacelike axis $L$. Suppose that $S$ satisfies equation \eqref{eqL} where $\vec{a}$ is a timelike vector. Then either $\vec{a}$ is orthogonal to $L$, or $S$ is the hyperbolic plane $\mathbb H^2(r)$ being $\vec{a}$ an arbitrary timelike vector. \end{proposition}
\begin{proof} After a rigid motion of $\mathbb L^3$ we assume that $L$ is the $x$-axis. This rigid motion changes the vector $\vec{a}$ in equation \eqref{eqL} and $\vec{a}$ must be considered an arbitrary (timelike) vector. Let $\vec{a}=(a,b,c)$ denote the new vector $\vec{a}$ in \eqref{eqL} after the rigid motion. Since $\vec{a}$ is timelike, then $c\not=0$.
Using the expression of a parametrization \eqref{eqx} of $S$ and after some computations, equation \eqref{eqL} is a polynomial equation on $\{1,\sinh\theta,\cos\theta\}$. Since these functions are linearly independent, all three coefficients (which are functions on the variable $s$) must vanish, obtaining $$-c(1-u'^2+uu'')+\alpha c(1-u'^2)=0$$ $$-b(1-u'^2+uu'')+\alpha b(1-u'^2)=0$$ $$as(1-u'^2+uu'')-\alpha auu'(1-u'^2)=0.$$ Since $c\not=0$, we find $a(uu'-s)=b(uu'-s)=0$. If $uu'-s\not=0$, then $a=b=0$, proving that $\vec{a}=(0,0,c)$, hence $L$ is orthogonal to the $x$-axis and the result is proved. The other possibility is $uu'-s=0$. Solving this equation, we find $u(s)=\sqrt{s^2+r^2}$, $r>0$. Then $X(s,\theta)=(s,\sqrt{s^2+r^2}\sinh\theta,\sqrt{s^2+r^2}\cos\theta)$ and it is immediate that this surface is the hyperbolic plane $\mathbb H^2(r)$. \end{proof}
As a consequence of Proposition \ref{pr-x}, and besides the hyperbolic plane as a special case, we can assume that $\vec{a}=(0,0,1)$ in the singular maximal surface equation \eqref{eqL}, and that the rotation axis is the $x$-axis. In such a case, the proof of Proposition \ref{pr-x} gives immediately that equation \eqref{eqL} is $$\frac{u''}{1-u'^2}=(\alpha-1)\frac{1}{u}.$$ This equation is just the equation \eqref{eq-one}. Identifying the Lorentzian plane $\mathbb L^2$ with the plane of equation $y=0$, we have obtained the following result.
\begin{proposition}\label{pr27} Any rotational $\alpha$-singular maximal surface in $\mathbb L^3$ about the $x$-axis is generated by a planar curve in $\mathbb L^2$ that satisfies the one-dimensional $(\alpha-1)$-singular maximal surface equation. Conversely, any planar curve in $\mathbb L^2$ that satisfies equation \eqref{eq-one} is the generating curve of an $(\alpha+1)$-singular maximal surface invariant by all rotations about the $x$-axis. \end{proposition}
\begin{example} \normalfont We know that the solution of \eqref{eq-one} for $\alpha=1$ is the hyperbola $u(x)=\sqrt{1+a^2x^2}/a$, $a>0$. As a consequence of Proposition \ref{pr27}, the only $2$-singular maximal surface that is invariant by the rotations about the $x$-axis is the surface $x^2+y^2-z^2=-1/a^2$, $z>0$. This surface is the hyperbolic plane $\mathbb{H}^2(1/a)$. Another solution of \eqref{eq-one} appeared in the Introduction for $\alpha=-1$. Then the surface generated is the hyperbolic catenoid of $\mathbb L^3$. \end{example}
We finish this section considering singular maximal surfaces of revolution about a lightlike axis. Again, we have in mind that if we fix the rotation axis, then the vector $\vec{a}$ in equation \eqref{eqL} is arbitrary. If the rotation axis is determined by the vector $(1,0,1)$, the parametrization of the surface is \begin{equation}\label{eq-li} X(s,t)=\left(\begin{array}{ccc}1-\frac{t^2}{2}&t&\frac{t^2}{2}\\ -t&1&t\\ -\frac{t^2}{2}&t&1+\frac{t^2}{2}\end{array}\right)\left(\begin{array}{c} u(s)+s\\ 0\\ u(s)-s\end{array}\right),\quad t\in\mathbb R, \end{equation} for some function $u=u(s)$, $s\in I\subset\mathbb R$. The spacelike condition on the surface is equivalent to $u'>0$.
\begin{proposition} Let $S$ be a spacelike surface of $\mathbb L^3$ invariant by the uniparametric group of rotations about a lightlike axis $L$. Suppose $S$ satisfies equation \eqref{eqL} where $\vec{a}$ is a timelike vector. Then either $\vec{a}$ is orthogonal to $L$, or $S$ is the hyperbolic plane $\mathbb H^2(r)$ being $\vec{a}$ is an arbitrary vector. More precisely, if $L$ is generated by the vector $(1,0,1)$, $S$ is parametrized by \eqref{eq-li} and if $\alpha\not=2$, then $\vec{a}=(1,b,1)$, $b\not=0$, and we have the following possibilities: \begin{enumerate} \item If $\alpha=3/2$, then $u(s)=m\log(s)$, $m>0$. \item If $\alpha\not= 3/2$, then $u(s)=m s^{3-2\alpha}/(3-2\alpha)$, $m>0$. \end{enumerate} In particular, hyperbolic planes $\mathbb H^2(r)$ are the only $\alpha$-singular maximal surfaces in $\mathbb L^3$ satisfying \eqref{eqL} with $\vec{a}=(0,0,1)$ and invariant by the group of rotations about the lightlike axis generated by the vector $(1,0,1)$. \end{proposition}
\begin{proof} A straightforward computation of equation \eqref{eqL} for the surface \eqref{eq-li} concludes that this equation is a polynomial equation on $t$ of degree $2$. Thus the coefficients corresponding for the variable $t$ must vanish, obtaining $$2 u' \left((\alpha +1) s (a+c)+(a-c) \left(u+\alpha s u'\right)\right)-s u'' ((a-c) u+s (a+c))=0$$ $$b \left(s u''-2 (1-\alpha) u'\right)=0$$ $$ (a-c)\left(s u''-2 (1-\alpha) u'\right)=0.$$ From the second and third equation, if $su''-2(1-\alpha)u'\not=0$, we have $b=0$ and $a=c$, obtaining that $\vec{a}$ is a lightlike vector, which is not possible. Thus $s u''-2 (1-\alpha ) u'=0$. The solution of this equation depends on the value of $\alpha$. \begin{enumerate} \item Case $\alpha=3/2$. Then $u(s)=m\log(s)$ with $m>0$. The first equation yields $(a-c)m^2(1+\log(s)=0$, that is, $a=c$ and $\vec{a}=(a,b,a)$, $b\not=0$. \item Case $\alpha\not=3/2$. Then $u(s)=ms^{3-2\alpha}/(3-2\alpha)$ with $m>0$. Now the first equation simplifies into $(a-c)(2-\alpha)s^{5-4\alpha}=0$. If $\alpha=2$, then $u(s)=-m/s$ and it is not difficult to see that this surface is the hyperbolic plane $\mathbb{H}^2(2\sqrt{m})$. If $\alpha\not=2$, then $a=c$, so $\vec{a}=(a,b,a)$, $b\not=0$.
\end{enumerate}
\end{proof}
\section{Surfaces of revolution about the $z$-axis}\label{sec3}
In this section we study the surfaces of revolution with timelike axis $L$. Again, the same observations done in the previous section hold in the sense that there is not an {\it a priori} relation between the vector $\vec{a}$ and the axis $L$. The first result that we will prove is that, indeed, $L$ must parallel to the vector $\vec{a}$.
\begin{proposition}\label{pr-zz}
Let $S$ be an $\alpha$-singular maximal surface in $\mathbb L^3$ that is invariant by the uniparametric group of rotations about a timelike axis $L$. Suppose that $S$ satisfies equation \eqref{eqL} where $\vec{a}$ is now an arbitrary timelike vector. Then either $L$ and $\vec{a}$ are parallel, or $S$ is the hyperbolic plane $\mathbb H^2(m)$ being $\vec{a}$ is an arbitrary timelike vector. \end{proposition}
\begin{proof} After a rigid motion, we suppose that the rotation axis is the $z$-axis. Let $\vec{a}=(a,b,c)$ after this motion. The surface $S$ parametrizes as $X(r,\theta)=(r\cos\theta,r\sin\theta,u(r))$, $r\in I\subset\mathbb R^+$, $\theta\in\mathbb R$, $u>0$ and $u'^2<1$. The computation of equation \eqref{eqL} gives a polynomial equation on the trigonometric functions $\{1,\sin\theta,\cos\theta\}$. Thus all three coefficients must vanish, obtaining $$a \left(r u''+(\alpha +1) u'(1-u'^2)\right)=0$$
$$b \left(r u''+(\alpha +1) u'(1-u'^2)\right)=0$$ $$c \left(\alpha r \left(1-u'^2\right)+u \left(r u''+u'(1-u'^2)\right)\right)=0.$$ If $r u''-(\alpha +1) u'(1-u'^2)\not=0$, then $a=b=0$, proving that $\vec{a}=(0,0,c)$, hence $L$ and $\vec{a}$ are parallel.
Suppose now that $r u''+(\alpha +1) u'(1-u'^2)=0$. Recall that $c\not=0$ because $\vec{a}$ is a timelike vector. Combining with the third equation, we find $uu'-r=0$. Solving this equation we obtain $u(r)=\sqrt{r^2+m^2}$, $m>0$, and the corresponding surface is the hyperbolic plane $\mathbb H^2(m)$. \end{proof}
By Proposition \ref{pr-zz}, and after a horizontal translation, we will assume that the rotation axis is the $z$-axis and $\vec{a}=(0,0,1)$ in \eqref{eqL}. We know that $X(r,\theta)=(r\cos\theta,r\sin\theta,u(r))$, where $r\in I\subset \mathbb R^+$, $\theta\in\mathbb R$ and $u>0$. By the proof of Proposition \ref{pr-zz}, equation (\ref{eqL}) writes as
\begin{equation}\label{eq3}
\frac{u''}{(1-u'^2)^{3/2}}+\frac{u'}{r\sqrt{1-u'^2}}=\frac{\alpha}{u\sqrt{1-u'^2}},
\end{equation}
or equivalently,
\begin{equation}\label{eq33}
\frac{u''}{ 1-u'^2 }+\frac{u'}{r }=\frac{\alpha}{u}.
\end{equation}
We are interested in those solutions that meet the $z$-axis, that is, when $r=0$ is contained in the domain of the solution. Let us observe that equation (\ref{eq3}) is singular at $r=0$ and thus the existence of solutions is not a direct consequence of standard ODE theory.
Multiplying (\ref{eq3}) by $r$, and integration by parts, we wish to establish the existence of a classical solution of \begin{equation}\label{rot} \left\{\begin{array}{ll}
\left(r\dfrac{u'}{\sqrt{1-u'^2}}\right)'=r\dfrac{\alpha}{u\sqrt{1-u'^2}}&\mbox{ $r\in (0,\delta)$}\\ u(0)=u_0>0, \quad u'(0)=0.& \end{array}\right. \end{equation}
Define the functions $\phi:(-1,1)\rightarrow\mathbb R$ and $f:\mathbb R^+\times(-1,1)\rightarrow\mathbb R$ by
$$\phi(y)=\frac{y}{\sqrt{1-y^2}}\quad f(x,y)=\frac{\alpha}{x\sqrt{1-y^2}}.$$
Let $\delta>0$. It is clear that a function $u\in C^2([0,\delta])$ is a solution of (\ref{rot}) if and only if $(r\phi(u'))'=r f(u,u')$ and $u(0)=u_0$, $u'(0)=0$. Let $\mathcal{B}=(C^1([0,\delta]),\|\cdot\|)$ be the Banach space of the continuously differentiable functions on $[0,\delta]$ endowed with the usual norm
$$\|u\|=\|u\|_{\infty}+\|u'\|_{\infty}.$$ Define the operator ${\mathsf T}:\mathcal{B}\rightarrow \mathcal{B}$ by $$({\mathsf T}u)(r)=u_0+\int_0^r\phi^{-1}\left(\int_0^s\frac{t}{s} f(u,u') dt\right)ds.$$ Notice that a fixed point of the operator ${\mathsf T}$ is a solution of the initial value problem (\ref{rot}). Indeed, $({\mathsf T}u)'=\phi^{-1}\left(\frac{1}{r}\int_0^r t f(u,u')dt\right)$ and $$r\phi({\mathsf T} u')\int_0^r t f(u,u')dt,$$
obtaining the result. Moreover, ${\mathsf T}u(0)=u_0$ and
$$\phi({\mathsf T}u)'(0)=\lim_{r\rightarrow 0}\frac{1}{r}\int_0^r t f(u,u')dt=\lim_{r\rightarrow 0} r f(u,u')=0,$$
where in the second identity we have used the L'H\^{o}pital rule. Thus, $({\mathsf T}u)'(0)=0$.
The existence of solutions of \eqref{rot} follows now standard techniques of radial solutions for some equations of mean curvature type (\cite{be,cco}). In Figure \ref{fig2} we show the solutions of \eqref{rot} when $\alpha$ is positive and negative.
\begin{figure}
\caption{Solutions of \eqref{rot}. Left: case $\alpha>0$, here $\alpha=2$. Right: case $\alpha<0$, here $\alpha=-1$ }
\label{fig2}
\end{figure}
\begin{theorem}\label{pr-exi} The initial value problem (\ref{rot}) has a solution $u\in C^2([0,\delta])$ for some $\delta>0$ that depends continuously on the initial data. \end{theorem}
\begin{proof}
In order to find a fixed point of ${\mathsf T}$, we prove that ${\mathsf T}$ is a contraction in $\mathcal{B}$ for some $\delta>0$ to be chosen. The functions $f$ and $\phi^{-1}$ are locally Lipschitz continuous of constant $L>1$ in $[u_0-\epsilon,u_0+\epsilon]\times[-\epsilon,\epsilon]$ and $[-\epsilon,\epsilon]$ respectively, provided $\epsilon<\{u_0,1\}$. Since $\phi^{-1}(y)=y/\sqrt{1+y^2}$, then $L<1$. Then for all $u,v\in\overline{B(0,\epsilon)}$ and for all $r\in [0,\delta]$, \begin{eqnarray*}
|({\mathsf T}u)(r)-({\mathsf T}v)(r)|&\leq& L\int_0^r\left|\int_0^s\frac{t}{s}(f(u,u')-f(v,v'))dt\right|\\
&\leq& L^2\int_0^r\int_0^s\frac{t}{s}\|u-v\| dt=
\frac{L^2}{4} r^2 \|u-v\|. \end{eqnarray*} \begin{eqnarray*}
|({\mathsf T}u)'(r)-({\mathsf T}v)'(r)|&\leq&\frac{L}{r}\left|\int_0^r t(f(u,u')-f(v,v'))dt\right|\\
&\leq&\frac{L^2}{r}\int_0^r t\|u-v\|dt=\frac{L^2}{2}r\|u-v\|. \end{eqnarray*} By choosing $\delta>0$ small enough, we deduce that ${\mathsf T}$ is a contraction in the closed ball $\overline{B(0,\delta)}\subset \mathcal{B}$. Thus the Schauder Point Fixed Theorem proves the existence of one fixed point of $\mathsf{T}$, so the existence of a local solution of the initial value problem (\ref{rot}). This solution belongs to $C^1([0,\delta])\cap C^2((0,\delta])$. The $C^2$-regularity up to $0$ is verified directly by using the L'H\^{o}pital rule because (\ref{eq3}) leads to
$$\lim_{r\rightarrow 0}u''(r)+\lim_{r\rightarrow 0}\frac{u'(r)}{r}=\frac{\alpha}{u_0},$$ that is, \begin{equation}\label{uu} \lim_{r\rightarrow 0} u''(r)=\frac{\alpha}{2u_0}. \end{equation} The continuous dependence of local solutions on the initial data is a consequence of the continuous dependence of the fixed points of ${\mathsf T}$.
\end{proof}
In the following result we describe the geometric properties of the rotational solutions of \eqref{eq33}. See figures \ref{fig2}, \ref{fig3} and \ref{fig4}.
\begin{theorem}\label{t32}
Let $u$ be a solution of \eqref{eq3} with $u>0$ and $u'^2<1$.
\begin{enumerate}
\item Case $\alpha>0$. The maximal domain of $u$ is $(0,\infty)$. Let
$u_0'=\lim_{r\rightarrow 0}u'(r)$. Then we have the following cases:
$u_0'=0$ and the function $u$ is increasing; $u_0'=-1$ and $u$ has a unique critical point which is a global minimum; $u_0=1$ and the function is increasing. In all cases, \begin{equation}\label{t32-1} \lim_{r\rightarrow\infty}u(r)=\infty. \end{equation} Also, the function $u(r)=\sqrt{\alpha} r$ is a solution of (\ref{eq3}).
\item Case $\alpha<0$. The maximal domain of $u$ is $(a,b)$ with $0\leq a<b<\infty$ and
$$\lim_{r\rightarrow b}u(r)=0,\quad \lim_{r\rightarrow b}u'(r)=-1.$$ If $a>0$, then $u$ has a global maximum and $$\lim_{r\rightarrow a}u(r)=0,\quad\lim_{r\rightarrow a}u'(r)=1.$$ If $a=0$, let $u_0'=\lim_{r\rightarrow 0}u'(r)$. Then we have the following cases: $u_0'=0$ and $u$ is a decreasing function; $u_0'=-1$ and $u$ is a decreasing function; $u_0'=1$ and $u$ has a global maximum. \end{enumerate}
\end{theorem}
\begin{proof}We observe that if $u$ has a critical point at $r_o\geq 0$, then \eqref{eq33} implies $u''(r_o)=\alpha/u(r_o)\not=0$, hence all critical points are all maximum or are all minimum. Thus there is one critical point at most. In such a case, this point is a global minimum (resp. maximum) if $\alpha>0$ (resp. $\alpha<0$).
{\it Claim A. If the graphic of $u$ meets the $x$-axis at $r_*>0$, then $\alpha<0$ and $\lim_{r\rightarrow r_*}u'(r)^2=1$.}
The proof follows by multiplying \eqref{eq33} by $2u'$ and integrating. Then $$\log(1-u'(r)^2)+2\alpha\log u(r)=2\alpha\int^r\frac{u'(t)^2}{t}dt+\mu,\quad\mu\in\mathbb R.$$ In a neighborhood of $r_*$, the right-hand side of the above equation is finite. Since $\log(u(r))\rightarrow -\infty$ as $r\rightarrow r_*$, the same occurs with $\lim_{r\rightarrow r_*}\log(1-u'(r)^2)$, proving that $u'(r)^2\rightarrow 1$ as $r\rightarrow r_*$. Moreover, the case $\alpha>0$ is not possible because the left-hand side would be $-\infty$.
{\it Claim B. If the graphic of $u$ meets the $y$-axis, then $\lim_{r\rightarrow 0}u'(r)=0$ or $\lim_{r\rightarrow 0}u'(r)^2=1$.}
Let denote $u_0'=\lim_{r\rightarrow 0}u'(r)$. From Theorem \ref{pr-exi}, we know the existence of solutions when $u_0'=0$. Suppose now $u_0'\not=0$. By contradiction, we assume that $u_0'^2\not=1$. For $\delta>0$ close to $0$ and by \eqref{rot}, \begin{equation}\label{delta} \frac{ru'(r)}{\sqrt{1-u'(r)^2}}-\frac{\delta u'(\delta)}{\sqrt{1-u'(\delta)^2}}=\int_\delta^r\frac{\alpha t}{u\sqrt{1-u'^2}}dt. \end{equation} Since $u_0'^2\not=1$, letting $r\rightarrow 0$ we have $$\frac{u'(\delta)}{\sqrt{1-u'(\delta)^2}}=\frac{1}{\delta}\int_0^\delta\frac{\alpha t}{u\sqrt{1-u'^2}}dt.$$ Letting $\delta\rightarrow 0$ and by the L'H\^{o}pital rule, we deduce $$\lim_{\delta\rightarrow 0}\frac{u'(\delta)}{\sqrt{1-u'(\delta)^2}}=\lim_{\delta\rightarrow 0}\frac{\alpha\delta}{u\sqrt{1-u'^2}}=0,$$ hence $u_0'=0$, a contradiction.
In particular, the claim B implies that it is not possible to find solutions of the initial value problem \eqref{rot} when $u_0'^2\in (0,1)$.
From now, we will denote by $u(a)$ and $u'(a)$ (similar for $r=b$), the limit of $u(r)$ and $u'(r)$ at $r=a$.
{\it Claim C. If $a>0$ (resp. $b<\infty$), then $u(a)=$ (resp. $u(b)=0$).}
Suppose that $a>0$ (similarly for $b<\infty$). If $u(a)\not=0$, then $u''$ is bounded around $r=a$ by \eqref{eq33}. Since $u'$ and $u''$ are bounded functions, we could extend the solution $u$ beyond $r=a$, a contradiction.
We are in position to prove the theorem.
\begin{enumerate}
\item Case $\alpha>0$. Suppose that $u'>0$ in all its domain. Since $u'$ and $u''$ are bounded functions by \eqref{eq33}, then the value of $b$ in $I$ is $b=\infty$. If $a>0$, this implies that $u(a)=0$ by Claim C and this a contradiction by Claim A. This proves that $I=(0,\infty)$.
Suppose that the sign of $u'$ is negative in all its domain. Then \eqref{eq33} implies that $u$ is a concave function and thus $b<\infty$ because $u$ is decreasing. Then $u(b)=0$, which is not possible by Claim A.
After the above arguments, we have proved that if $u'$ has a constant sign, then $u'>0$, $a=0$ and either $u_0'=0$ or $u_0'=1$. In case that $u'$ changes of sign, then there is a unique critical point at some point $r=r_o>0$, which is a global minimum. In this case, $u'>0$ for $r>r_o$. Since $u'$ and $u''$ are bounded, then $b=\infty$. The case $a>0$ is forbidden by Claim C. Thus $a=0$. Since $u'<0$ for $r<r_o$, Claim B asserts $u_0'=-1$.
We prove \eqref{t32-1}. Since $u$ is increasing close $\infty$, let $c=\lim_{r\rightarrow\infty}u(r)$. If $c<\infty$, then $u'(r)\rightarrow 0$ as $r\rightarrow\infty$ and using \eqref{eq33}, $\lim_{r\rightarrow\infty}u''(r)= \alpha/c>0$, a contradiction. Thus $c=\infty$.
Finally, by a direct computation, we observe that $u(r)=\sqrt{\alpha}r$ is a solution of (\ref{eq3}).
\item Case $\alpha<0$. Suppose that $u'<0$ in all its domain. Let $c=\lim_{r\rightarrow b}u(r)\geq 0$. If $b=\infty$, then $u'(r)\rightarrow 0$ and \eqref{eq33} would imply that $\lim_{r\rightarrow \infty}u''(r)$ is either $\alpha/c$ if $c>0$ or $\infty$ if $c=0$, a contradiction. Thus $b<\infty$, hence $u(b)=0$. By Claim A, $u'(b)=-1$. If $a>0$, then $u(a)>0$ because $u$ is decreasing: a contradiction by Claim C. Thus $a=0$. By Claim B and because $u$ is decreasing, we have two possibilities, namely, $u_0'=0$ and $u_0'=-1$.
Suppose now that $u'>0$ in all its domain. Then $u$ is a concave function by \eqref{eq33}. By Claim C, we have $b=\infty$. In the other end of the interval $I$, namely $r=a$, we have $a=0$, $u_0'=1$ or $a>0$, $u(a)=0$ and $u'(a)=1$. In both cases, as $u''<0$, we find $\lim_{r\rightarrow\infty} u'(r)=\lim_{r\rightarrow\infty} u''(r)=0$ and $\lim_{r\rightarrow\infty} u(r)=\infty$. By \eqref{delta} $$\frac{u'(r)}{\sqrt{1-u'(r)^2}}=\frac{1}{r}\int_\delta^r\frac{\alpha t}{u\sqrt{1-u'^2}}dt+\mu.$$ Letting $r\rightarrow\infty$, the left-hand side is $0$. However, and applying twice the L'H\^{o}pital rule, the limit of the right-hand side is $$\lim_{r\rightarrow\infty}\frac{\alpha r}{u(r)}+\mu =\lim_{r\rightarrow\infty}\frac{\alpha}{ u'(r)}+\mu=\infty,$$ obtaining a contradiction.
Thus, if $u'>0$ at some point, there is a critical point $r_o$ of $u$, which will be the global maximum of $u$. Then $a\geq 0$ with $u'(a)=1$ because $u$ is increasing in $(a,r_o)$. \end{enumerate}
\end{proof}
\begin{remark} If $\alpha<0$, there exist solutions that do not meet the rotation axis, see figure \ref{fig4}, right. This case appears if $0<a<b<\infty$, where the function $u$ has a global maximum and $u'(a)=1=-u'(b)$. This extends the same property of the solution of \eqref{eq0}, where the part of the function $u=u(x)$ given in \eqref{eq00} that lies over the $x$-axis is formed by successive bounded intervals.
\end{remark}
\begin{figure}
\caption{Solutions of \eqref{eq33}, case $\alpha>0$ and $u_0'\not=0$. Left: case $u_0'=-1$. Right: case $u'_0=1$ }
\end{figure}\label{fig3}
\begin{figure}
\caption{Solutions of \eqref{eq33}, case $\alpha<0$ and $u_0'\not=0$. Left: case $u_0'=1$. Middle: case $u'_0=-1$. Right: a solution that does not meet the rotation axis }
\label{fig4}
\end{figure}
\section{The Dirichlet problem}\label{sec4}
The Dirichlet problem of the singular maximal surface equation asks if given a positive function $\varphi:\partial\Omega\rightarrow\mathbb R$ defined in a bounded domain $\Omega\subset\mathbb R^2$, there exists a smooth positive function $u:\overline{\Omega}\rightarrow\mathbb R$ such that \eqref{eq2} holds in $\Omega$, $u=\varphi$ on $\partial\Omega$ and $|Du|<1$ on $\overline{\Omega}$. Since any curve in a spacelike surface must be spacelike, the graph $\Gamma$ of $\varphi$ is spacelike. The problem is to determine the type of function $\varphi$ and the boundary $\partial\Omega$ for the solvability of the Dirichlet problem. It is expectable that the sign $\alpha$ in \eqref{eq2} plays an important role because we have seen in Sections \ref{sec2} and \ref{sec3} the contrast of the behaviour of the invariant solutions of \eqref{eqL} depending if $\alpha$ is positive or negative.
Following similar ideas of Jenkins and Serrin in \cite{js,se}, we will solve the Dirichlet problem if the domain $\Omega$ is mean convex. In fact, we will establish the Dirichlet problem in the $n$-dimensional case, or equivalently, we will find singular maximal hypersurfaces in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb L^{n+1}$ with prescribed boundary data.
Recall that a bounded domain $\Omega\subset\mathbb R^n$ is said to be mean convex if $\partial\Omega$ has nonnegative mean curvature $H_{\partial\Omega}$ with respect to the inward orientation. In case $n=2$, the mean convexity property is equivalent to the convexity of $\Omega$, but in arbitrary dimensions, the mean convexity is less restrictive than convexity.
The Dirichlet problem is now formulated as follows. Let $\Omega\subset\mathbb R^n$ be a smooth bounded domain and $\alpha\not=0$ a given constant. Let $\varphi:\partial\Omega\rightarrow\mathbb R$ be a positive spacelike smooth function. The problem is finding a classical solution $u\in C^2(\Omega)\cap C^0(\overline{\Omega})$, $u>0$ in $\overline{\Omega}$, of
\begin{equation}\label{d1} \left\{\begin{split}
&\mbox{div}\left(\dfrac{Du}{\sqrt{1-|Du|^2}}\right)= \frac{\alpha}{u\sqrt{1-|Du|^2}}\quad \mbox{in $\Omega$}\\ &u=\varphi\quad \mbox{on $\partial\Omega$}\\
&|Du|<1\quad \mbox{in $\overline{\Omega}.$} \end{split}\right. \end{equation}
We solve the Dirichlet problem when the boundary data $\varphi$ has a spacelike extension in $\overline{\Omega}$.
\begin{theorem} \label{t1}
Let $\Omega\subset\mathbb R^n$ be a bounded mean convex domain with smooth boundary $\partial\Omega$. Assume that $\alpha<0$. If $\varphi\in C^{2}(\overline{\Omega})$ is a positive function with $\max_{\overline{\Omega}}|D\varphi|<1$, then there is a unique positive solution $u$ of \eqref{d1}. \end{theorem}
The proof of Theorem \ref{t1} is accomplished by using the Schauder theory of {\it a priori} global estimates, the method of continuity and the Leray-Schauder fixed point theorem. Applying these techniques, we find all elements for proving Theorem \ref{t1}. As usual, we will utilize the distance function $d$ to $\partial\Omega$ to construct a barrier function (\cite{gt,js,lu}).
The $C^0$ estimates will be obtained by comparing the solution of \eqref{d1} with the rotational examples studied in Section \ref{sec3}: here the hypothesis $\alpha<0$ will be essential because if $\alpha>0$ it is not possible to prevent that $|u|\rightarrow 0$ for a solution $u$. For the $C^1$ estimates, we need to prove that $|Du|$ is bounded away from $1$ which will be deduced by using barrier functions. Finally, the hypothesis $\alpha<0$ will be also used when we apply the Implicit Function Theorem for the existence of the linearized problem associated to \eqref{d1}.
The maximum principle for elliptic equations of divergence type implies the following result.
\begin{proposition}[Touching principle]\label{pr21} Let $\Sigma_1$ and $\Sigma_2$ be two $\alpha$-singular maximal surfaces. If $\Sigma_1$ and $\Sigma_2$ have a common tangent interior point and $\Sigma_1$ lies above $\Sigma_2$ around $p$, then $\Sigma_1$ and $\Sigma_2$ coincide at an open set around $p$. \end{proposition}
We also need to formulate the comparison principle in the context of $\alpha$-singular maximal surfaces. We write the equation of \eqref{d1} in classical notation. Define the operator \begin{equation}\label{op} \begin{split}
Q[u]&= (1-|Du|^2)\Delta u+u_iu_ju_{ij}-\frac{\alpha(1-|Du|^2)}{u}\\ &=a_{ij}(Du)u_{ij}+{\textbf b}(u,Du), \end{split} \end{equation}
where
$$a_{ij}=(1-|Du|^2)\delta_{ij}+u_iu_j,\quad {\textbf b}= - \frac{\alpha(1-|Du|^2)}{u}.$$
Here $u_i=\partial u/\partial x_i$, $1\leq i\leq n$, and we assume the summation convention of repeated indices. It is immediate that $u$ is a solution of equation (\ref{d1}) if and only if $Q[u]=0$. The ellipticity of the operator $Q$ is clear because if $A=(a_{ij})$ and $\xi\in\mathbb R^n$, then \begin{equation}\label{px}
(1-|p|^2)|\xi|^2\leq \xi^tA\xi=(1-|p|^2)|\xi|^2+\langle p,\xi\rangle^2\leq|\xi|^2. \end{equation} Moreover, this shows that $Q$ is not uniformly elliptic. We recall the comparison principle (\cite[Th. 10.1]{gt}).
\begin{proposition}[Comparison principle] \label{pr-43} Let $\Omega\subset\mathbb R^n$ be a bounded domain. If $u,v\in C^2(\Omega)\cap C^0(\overline{\Omega})$ satisfy $Q[u]\geq Q[v]$ and $u\leq v$ on $\partial\Omega$, then $u\leq v$ in $\Omega$. \end{proposition}
Notice that if $\alpha<0$, the classical theory implies the uniqueness of solutions of the Dirichlet problem.
\begin{proposition}\label{pr-u} Let $\Omega\subset\mathbb R^n$ be a bounded domain and $\alpha<0$. The solution of \eqref{d1}, if exists, is unique.
\end{proposition}
In arbitrary dimension, it holds the property that any horizontal translation and any dilation from a point of $\mathbb R^n\times\{0\}$ preserves equation \eqref{d1}. Similarly, Theorem \ref{t32} holds where now \eqref{rot} is $$ \left(r\dfrac{u'}{\sqrt{1-u'^2}}\right)'=r^{n-1}\dfrac{\alpha}{u\sqrt{1-u'^2}}.$$
We establish the solvability of \eqref{d1} in the particular case that $\Omega$ is a ball of $\mathbb R^n$ and $\varphi$ is a positive constant.
\begin{proposition}\label{pr25} Let $\alpha<0$ and $B_R\subset\mathbb R^n$ be a round ball of radius $R>0$. If $c>0$, then there is a unique radial solution $u$ of (\ref{d1}) in $B_R$ with $u=c$ on $\partial B_R$.
\end{proposition}
\begin{proof} After a horizontal translation, we suppose that the origin $O\in\mathbb R^n$ is the center of $B_R$. By Proposition \ref{pr-exi}, let $v=v(r)$ be the solution of \eqref{rot} with $v(0)=1$. Recall that Theorem \ref{t32} asserts that the maximal domain of $v$ is a ball $B_b$ for some $b>0$ with $v(b)=0$. In the $(r,v)$-plane, consider the line $x_{n+1}=cr/R$. Since $v$ is a decreasing function, the graph of $v$ meets this line at one point $r=r_o$, $u(r_o)=cr_o/R$. If $\lambda=R/r_o$, then $u_\lambda(r)=\lambda u(r/\lambda)$ is a solution of \eqref{d1} with $u_\lambda(R)=c$. \end{proof}
Following a standard scheme, we start by finding $C^0$ estimates by using the rotational solutions of \eqref{eqL}. In the following result, we do not require the mean convexity of $\Omega$.
\begin{proposition} \label{pr-31}
Let $\Omega\subset\mathbb R^n$ be a bounded domain and $\alpha<0$. If $u$ is a positive solution of \eqref{d1}, there exists a constant $C_1=C_1(\alpha,\Omega,\varphi)>0$ such that \begin{equation}\label{eh}
\min_{\partial\Omega}\varphi\leq u\leq C_1 \quad \mbox{in $\Omega$}. \end{equation} \end{proposition}
\begin{proof} Since the right-hand side of (\ref{d1}) is negative, then $\inf_\Omega u=\min_{\partial\Omega}\varphi$ by the maximum principle. This proves the left inequality of \eqref{eh}.
For the upper estimate of \eqref{eh}, we consider the radial solution $v$ of \eqref{rot} with $v(0)=1$ and let $\{v_\lambda:\lambda>0\}$ where $v_\lambda(r)=\lambda v(r/\lambda)$. Denote $B_R$ the maximal domain of $v$, with $v(R)=0$ and let $\Sigma_\lambda$ denote the graph of $v_\lambda$. Take $\lambda>0$ sufficiently big so the graph $S$ of $u$ is included in the domain of the halfspace $x_{n+1}>0$ bounded by $\Sigma_\lambda\cup B_{\lambda R}$. Let $\lambda$ decrease to $0$ until the first time $\lambda_0$ that $\Sigma_\lambda$ meets $\Sigma_u$. By the maximum principle, the first contact must occur at some boundary point of $S$. Then this point is a point of $\partial\Omega$ or a point of $\partial S$. Since $\partial S$ is the graph of $\varphi$, this value $\lambda_0$ depends on $\Omega$ and $\varphi$. Consequently, $u\leq v_{\lambda_0}\leq \sup_\Omega v_{\lambda_0}$. The proof finishes by letting $C_1=\sup_\Omega v_{\lambda_0}$, which depends only on $\alpha$, $\Omega$ and $\varphi$.
\end{proof}
The next step to prove Theorem \ref{t1} is the derivation of estimates for $|Du|$. This is done firstly proving the the supremum of $|Du|$ is attained at some boundary point. In the next result, the assumption that $\alpha$ is negative is essential.
\begin{proposition}[Interior gradient estimates] \label{pr-41}
Let $\Omega\subset\mathbb R^n$ be a bounded domain and $\alpha<0$. If $u\in C^2(\Omega)\cap C^1(\overline{\Omega})$ is a positive solution of \eqref{d1}, then
$$\max_{\overline{\Omega}}|Du|=\max_{\partial\Omega}|Du|.$$ \end{proposition}
\begin{proof}
Let $v^k=u_k$, $1\leq k\leq n$. By differentiating $Q[u]=0$ with respect to $x_k$, we find for each $k$,
\begin{equation}\label{eq4}
\left((1-|Du|^2)\delta_{ij}+u_iu_j\right)v_{ij}^k+2\left(u_i\Delta u+u_ju_{ij}-\frac{\alpha u_i}{u}\right)v_i^k+\frac{\alpha(1-|Du|^2)}{u^2}v^k=0.
\end{equation}
Equation (\ref{eq4}) is a linear elliptic equation in $v^k$. Because $\alpha<0$, the coefficient for $v^k$ is negative and the maximum principle (\cite[Th. 3.7]{gt}) implies that $|v^k|$, and then $|Du|$ has not an interior maximum. In particular, the maximum of $|Du|$ on $\overline{\Omega}$ is attained at some boundary point, proving the result.
\end{proof}
As a consequence of Proposition \ref{pr-41}, the problem of finding {\it a priori} estimates of $|Du|$ reduces to get these estimates along $\partial\Omega$. With this purpose, we prove that $u$ admits barriers from above and from below along $\partial\Omega$. It is now when we use the assumption of the mean convexity of $\Omega$.
\begin{proposition}[Boundary gradient estimates] \label{pr42}
Let $\Omega\subset\mathbb R^n$ be a bounded mean convex domain and $\alpha<0$. If $u\in C^2(\Omega)\cap C^1(\overline{\Omega})$ is a positive solution of \eqref{d1}, then there is a constant
$$C_2=C_2(\alpha,\Omega, C_1,\|\varphi\|_{1;\overline{\Omega}},\|\varphi\|_{2;\overline{\Omega}})<1$$
such that
$$\max_{\partial\Omega}|Du|\leq C_2.$$
\end{proposition}
\begin{proof}
We consider the operator $Q[u]$ defined (\ref{op}). For a lower barrier for $u$, we take the solution $v^0$ of the Dirichlet problem of the maximal surface equation in $\Omega$ with the same boundary $\varphi$. The function $v^0$ is the solution of \eqref{d1} for $\alpha=0$ whose existen is assured (\cite[Th. 4.1]{bs}). Then
$$Q[v^0]=-\frac{\alpha(1-|Dv^0|^2)}{v^0}>0=Q[u].$$
Since $v^0=u$ on $\partial\Omega$, we conclude $v^0<u$ in $\Omega$ by the comparison principle.
We now construct an upper barrier for $u$ by means of the distance function in a small tubular neighborhood of $\partial\Omega$ in $\Omega$.
Consider the distance function $d(x)=\mbox{dist}(x,\partial\Omega)$ and let $\epsilon>0$ sufficiently small so $\mathcal{N}_\epsilon=\{x\in\overline{\Omega}: d(x)<\epsilon\}$ is a tubular neighborhood of $\partial\Omega$. We parametrize $\mathcal{N}_\epsilon$ using normal coordinates $x\equiv (t,\pi(x)) \in\mathcal{N}_\epsilon$, where $x\equiv\pi(x)+t\nu(\pi(x))$ for some $t\in [0,\epsilon)$, $\pi:\mathcal{N}_\epsilon\rightarrow\partial\Omega$ is the orthogonal projection and $\nu$ is the unit normal vector to $\partial\Omega$ pointing to $\Omega$. A straightforward computation leads to that $d$ is $C^2$, $|Dd|(x)=1$, and $\Delta d(x) \leq -(n-1)H_{\partial\Omega}(\pi(x))$ for all $x\in\mathcal{N}_\epsilon$. Because $\Omega$ is mean convex, then $\Delta d(x)\leq 0$.
Define in $\mathcal{N}_\epsilon$ a function $w=h\circ d+\varphi$, where we use the same symbol $\varphi$ for a spacelike extension of $\varphi$ into $\Omega$. The function $h$ is defined as \begin{equation}\label{ht} h(t)=a\log(1+kb^2t),\ b, k >0,\ a=\frac{C_1}{\log(1+b)}, \end{equation}
where $C_1$ is the constant that appears in \eqref{eh} and $b$ and $k$ will be chosen later. Let us observe that $w>0$ and that we require that $|Dw|<1$. The computation of $Q[w]$ leads to
$$Q[w]=a_{ij}(h''d_id_j+h'd_{ij}+\varphi_{ij})-\frac{\alpha}{w}(1-|Dw|^2).$$
From $|Dd|=1$, it follows that $\langle D(Dd)_x\xi,Dd(x)\rangle=0$ for all $\xi\in\mathbb R^n$. If $\{e_i\}_i$ is the canonical basis of $\mathbb R^n$ and $\xi=e_i$, we find $d_{ij}d_j=0$. Thus \begin{eqnarray*} w_iw_jd_{ij}&=&(h'd_i+\varphi_i)(h'd_j+\varphi_j)d_{ij}=(h'^2d_i+2h'\varphi_i)d_jd_{ij}+\varphi_i \varphi_jd_{ij}\\ &=&\varphi_{i}\varphi_jd_{ij}\leq \varphi_i^2\lambda_i^d\leq 0, \end{eqnarray*} where $\lambda_i^d$ are the eigenvalues of $D^2d$, which all are not positive because $D^2d$ is negative semidefinite. Using this inequality, the definition of $a_{ij}$ in (\ref{op}) and \eqref{px}, it follows that
$$a_{ij}d_{ij}=(1-|Dw|^2)\Delta d+w_iw_j d_{ij}\leq(1-|Dw|^2)\Delta d\leq 0.$$
Again \eqref{px} implies $a_{ij}d_id_j\geq 1-|Dw|^2$ and $a_{ij}\varphi_{ij}\leq |D^2\varphi|$, where $|D^2\varphi|=\sum_{ij}\sup_{\overline{\Omega}}|\varphi_{ij}|$. Since $h'>0$ and $\Delta d\leq 0$, we find \begin{equation}\label{qw} \begin{split}
Q[w]&\leq h''(1-|Dw|^2)+h'\Delta d(1-|Dw|^2)-\frac{\alpha}{w}(1-|Dw|^2)+a_{ij}\varphi_{ij}\\
&\leq \left(h''-\frac{\alpha}{w} \right)(1-|Dw|^2)+|D^2\varphi|. \end{split} \end{equation}
We now study the spacelike condition $|Dw|<1$. The computation of $|Dw|$ and the Cauchy-Schwarz inequality gives
$$|Dw|^2=h'^2+|D\varphi|^2+2h'\langle Dd,D\varphi\rangle\leq (h'+|D\varphi|)^2.$$ Because $h'>0$ and $h'$ is decreasing on $t$, we deduce \begin{equation}\label{grad}
|Dw|\leq h'+|D\varphi|\leq h'(0)+|D\varphi|\leq akb^2+ \mu\quad \mbox{in $\overline{\Omega}$},\end{equation}
where $\mu=\|D\varphi\|_{0;\overline{\Omega}}<1$. Fix a constant $\delta$ with the property $\mu<\delta<1$. Then it is possible to choose $k$ sufficiently small in \eqref{grad} so $|Dw|\leq akb^2+ \mu<\delta$. Let $\beta=1-\delta^2$. If $h''-\alpha/w<0$, then \eqref{qw} implies \begin{equation}\label{qw2}
Q[w]\leq \beta \left(h''-\frac{\alpha}{w} \right)+\|D^2\varphi\|_{0;\overline{\Omega}}. \end{equation} The right-hand side in \eqref{qw2} is a function defined in $\partial\Omega\times [0,\epsilon]$. Let $\varphi_0=\min_{\overline{\Omega}}\varphi>0$ and we evaluate this function at $t=0$, obtaining
$$\beta \left(-ak^2b^4-\frac{\alpha}{\varphi} \right)+\|D^2\varphi\|_{0;\overline{\Omega}}\leq
\beta \left(-\frac{(\delta-\mu)^2}{a}-\frac{\alpha}{\varphi_0}\right) +\|D^2\varphi\|_{0;\overline{\Omega}}.$$ If $b$ is sufficiently big, then $a\rightarrow 0$, hence the right-hand side in this inequality is negative. By compactness of $\partial\Omega\times [0,\epsilon]$ and by continuity, let us take $b$ sufficiently large enough in \eqref{qw2} so $Q[w]<0$. Even more, we require $b$ so large such that $1/(kb)<\epsilon$. We now change the tubular neighborhood $\mathcal{N}_\epsilon$ by replacing $\epsilon$ by $\epsilon=1/(kb)$ and we denote $\mathcal{N}_\epsilon$ again.
In order to assure that $w$ is a local upper barrier in $\mathcal{N}_{\epsilon}$ for the Dirichlet problem \eqref{d1}, we need to have \begin{equation}\label{mm} u\leq w\quad \mbox{in $\partial\mathcal{N}_\epsilon$}. \end{equation}
In $\partial\mathcal{N}_\epsilon\cap\partial\Omega$, the distance function is $d=0$, so $w=\varphi=u$. On the other hand, in $\partial\mathcal{N}_\epsilon\setminus\partial\Omega$, and because $\epsilon=1/(kb)$, we find $$w=h(\epsilon)+\varphi=\frac{C_1}{\log(1+b)}\log(1+kb^2\epsilon)+\varphi=C_1+\varphi.$$ By Proposition \ref{pr-31}, we have $u\leq C_1$ and we deduce $u<w$ in $\mathcal{N}_\epsilon\setminus\partial\Omega$. Definitively, we find $Q[w]<0=Q[u]$ and $u\leq w$ in $\partial\mathcal{N}_\epsilon$, concluding that $u\leq w$ in $\mathcal{N}_\epsilon$ by the comparison principle.
Consequently, we have proved the existence of lower and upper barriers for $u$ in $\mathcal{N}_\epsilon$, namely, $v^0\leq u\leq w$ in $\mathcal{N}_\epsilon$. Hence we deduce
$$\max_{\partial\Omega}|Du|\leq C_2:=\max\{\|Dw\|_{0;\overline{\Omega}}, \|Dv^0\|_{0;\overline{\Omega}}\}$$
and both values $\|Dw\|_{0;\overline{\Omega}}, \|Dv^0\|_{0;\overline{\Omega}}$ depend only on the initial data of the Dirichlet problem. This completes the proof of proposition.
\end{proof}
With all above ingredients, we are in position to prove Theorem \ref{t1}.
\begin{proof}[Proof of Theorem \ref{t1}]
We establish the solvability of the Dirichlet problem \eqref{d1} by the method of continuity (see \cite[Sec. 17.2]{gt}). Define the family of Dirichlet problems parametrized by $t\in [0,1]$
$$\mathcal{P}_t: \left\{\begin{array}{cll} Q_t[u]&=&0 \mbox{ in $\Omega$}\\
u&=& \varphi \mbox{ on $\partial\Omega,$}
\end{array}\right.$$
where
$$Q_t[u]= (1-|Du|^2)\Delta u+u_iu_ju_{ij}-\frac{\alpha t(1-|Du|^2)}{u}.$$
The graph $\Sigma_{u_t}$ of a solution of $u_t$ is a $(t\alpha)$-singular maximal surface. Notice that if $t=0$, $Q_0[u]=0$ is the maximal surface equation and the solution of $\mathcal{P}_0$ is the function $v^0$ that appeared in Proposition \ref{pr42}. As usual, let
$$\mathcal{A}=\{t\in [0,1]: \exists u_t\in C^{2,\gamma}(\overline{\Omega}), u_t>0, Q_t[u_t]=0, {u_t}_{|\partial\Omega}=\varphi\}.$$ The proof consists to show that $1\in \mathcal{A}$. For this, we prove that $\mathcal{A}$ is a non-empty open and closed subset of $[0,1]$.
\begin{enumerate} \item The set $\mathcal{A}$ is not empty. This is because $0\in\mathcal{A}$ since $v^0$ is the solution of $\mathcal{P}_0$.
\item The set $\mathcal{A}$ is open in $[0,1]$. Given $t_0\in\mathcal{A}$, we need to prove that there is an $\epsilon>0$ such that $(t_0-\epsilon,t_0+\epsilon)\cap [0,1]\subset\mathcal{A}$. Define the map $T(t,u)=Q_t[u]$ for $t\in\mathbb R$ and $u\in C^{2,\gamma}(\overline{\Omega})$. Then $t_0\in\mathcal{A}$ if and only if $T(t_0,u_{t_0})=0$. If we show that the derivative of $Q_t$ with respect to $u$, say $(DQ_t)_u$, at the point $u_{t_0}$ is an isomorphism, it follows from the Implicit Function Theorem the existence of an open set $\mathcal{V}\subset C^{2,\gamma}(\overline{\Omega})$, with $u_{t_0}\in \mathcal{V}$, and a $C^1$ function $\xi:(t_0-\epsilon,t_0+\epsilon)\rightarrow \mathcal{V}$ for some $\epsilon>0$, such that $\xi(t_0)=u_{t_0}>0$ and $T(t,\xi(t))=0$ for all $t\in (t_0-\epsilon,t_0+\epsilon)$: this guarantees that $\mathcal{A}$ is an open set of $[0,1]$.
The proof that $(DQ_t)_u$ is one-to-one is equivalent to prove that for any $f\in C^\gamma(\overline{\Omega})$, there is a unique solution $v\in C^{2,\gamma}(\overline{\Omega})$ of the linear equation $Lv:=(DQ_t)_u(v)=f$ in $\Omega$ and $v=\varphi$ on $\partial\Omega$. The computation of $L$ was done in Proposition \ref{pr-41}, obtaining $$Lv=(DQ_t)_uv=a_{ij}(Du)v_{ij}+\mathbf{b}_i(u,Du,D^2u)v_i+{\textbf c}(u,Du)v,$$ where $a_{ij}$ are defined in (\ref{op}) and
$$\mathbf{b}_i=2\left( \Delta u-\frac{\alpha t}{u}\right)u_i+2u_ju_{ij},\quad {\textbf c}=\frac{\alpha t(1-|Du|^2)}{u^2}.$$ Since $\alpha<0$, ${\textbf c}\leq 0$ and the existence and uniqueness is assured by standard theory (\cite[Th. 6.14]{gt}).
\item The set $\mathcal{A}$ is closed in $[0,1]$. Let $\{t_k\}\subset\mathcal{A}$ with $t_k\rightarrow t\in [0,1]$. For each $k\in\mathbb{N}$, there exists $u_{t_k}\in C^{2,\gamma}(\overline{\Omega})$, $u_{t_k}>0$, such that $Q_{t_k}[u_{t_k}]=0$ in $\Omega$ and $u_{t_k}=\varphi$ in $\partial\Omega$. Define the set
$$\mathcal{S}=\{u\in C^{2,\gamma}(\overline{\Omega}): \exists t\in [0,1]\mbox{ such that }Q_{t}[u]=0 \mbox{ in }\Omega, u_{|\partial\Omega}=\varphi\}.$$
Then $\{u_{t_k}\}\subset\mathcal{S}$. If we prove that the set $\mathcal{S}$ is bounded in $C^{1,\beta}(\overline{\Omega})$ for some $\beta\in[0,\gamma]$, and since $a_{ij}=a_{ij}(Du)$ in (\ref{op}), then Schauder theory proves that $\mathcal{S}$ is bounded in $C^{2,\beta}(\overline{\Omega})$, in particular, $\mathcal{S}$ is precompact in $C^2(\overline{\Omega})$ (see Th. 6.6 and Lem. 6.36 in \cite{gt}). Thus there exists a subsequence $\{u_{k_l}\}\subset\{u_{t_k}\}$ converging in $C^2(\overline{\Omega})$ to some $u\in C^2(\overline{\Omega})$. Since $T:[0,1]\times C^2(\overline{\Omega})\rightarrow C^0(\overline{\Omega})$ is continuous, it follows $Q_t[u]=T(t,u)=\lim_{l\rightarrow\infty}T(t_{k_l},u_{k_l})=0$ in $\Omega$. Moreover, $u_{|\partial\Omega}=\lim_{l\rightarrow\infty} {u_{k_l}}_{|\partial\Omega}=\varphi$ on $\partial\Omega$, so $u\in C^{2,\gamma}(\overline{\Omega})$ and consequently, $t\in \mathcal{A}$.
The above reasoning asserts that $\mathcal{A}$ is closed in $[0,1]$ provided we find a constant $M$ independent of $t\in\mathcal{A}$, such that
$$\|u_t\|_{C^1(\overline{\Omega})}=\sup_\Omega |u_t|+\sup_\Omega|Du_t|\leq M. $$ However the $C^0$ and $C^1$ estimates for the function $u_1$, that is, when the parameter $t$ is $t=1$, are enough as we now see.
The $C^0$ estimates for $u_t$ follow with the comparison principle. Indeed, let $t_1<t_2$, $t_i\in [0,1]$, $i=1,2$. Then $Q_{t_1}[u_{t_1}]=0$ and
$$Q_{t_1}[u_{t_2}]=-\frac{(t_1-t_2)\alpha(1-|Du_{t_2}|^2)}{u_{t_2}}<0=Q_{t_1}[u_{t_1}]$$ because $\alpha<0$. Since $u_{t_1}=\varphi=u_{t_2}$ on $\partial\Omega$, the comparison principle yields $u_{t_1}<u_{t_2}$ in $\Omega$. This proves that the solutions $u_{t_i}$ are ordered in increasing sense according the parameter $t$. By (\ref{eh}), we find \begin{equation}\label{ut} \sup_\Omega u_t \leq \sup_\Omega u_1 \leq C_1. \end{equation}
In order to derive the gradient estimates for the solution $u_t$, the same computations obtained in Proposition \ref{pr42} conclude that $\sup_{\partial\Omega}|Du_t|$ is bounded by a constant depending on $\alpha$, $\Omega$, $\varphi$ and $\|u_t\|_{0;\overline{\Omega}}$. Now (\ref{ut}) implies that the value $\|u_t\|_{0;\overline{\Omega}}$ is bounded by $C_1$, which depends only on $\alpha$, $\varphi$ and $\Omega$, but not on $t$.
\end{enumerate} The above three steps prove the existence part in Theorem \ref{t1}. The uniqueness is consequence of Proposition \ref{pr-u} and this completes the proof of theorem.
\end{proof}
A consequence of Theorem \ref{t1} is the solvability of the Plateau problem if $\alpha<0$ in the following situation.
\begin{corollary} Let $\Gamma$ be a spacelike $(n-1)$-submanifold of $\mathbb L^{n+1}$ with an one-to-one orthogonal projection $C$ on the hyperplane of equation $x_{n+1}=0$ such that $C$ is the boundary of a mean convex simply-connected domain $\Omega$. Let $\alpha<0$. If $\Gamma$ has a spacelike extension to a graph on $\Omega$, then there exists a unique $\alpha$-singular maximal hypersurface $S$ spanning $\Gamma$. Moreover, $S$ is a graph on $\Omega$. \end{corollary}
\begin{proof} Theorem \ref{t1} asserts the existence of an $\alpha$-singular maximal hypersurface $S$ whose boundary is $\Gamma$ and $S$ is a graph on $\Omega$. Assume that $M$ is other such a hypersurface. The property that $M$ is spacelike implies that the orthogonal projection $p:\mathbb R^{n+1}\rightarrow\mathbb R^n=\mathbb R^n\times\{0\}$, $p(x)=(x_1,\ldots,x_n)$ is a local diffeomorphism between $M$ and $\Omega$. In particular, $p:M\rightarrow\Omega$ is a covering map and since $\Omega$ is simply connected, the map $p$ is a diffeomorphism, in particular, $M$ is a graph on $\Omega$. Finally, the uniqueness of \eqref{eqL} when $\alpha$ is negative concludes that $M=S$. \end{proof}
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