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\begin{document}
\setcounter{page}{1}
\title[local derivations on Witt algebras]{local derivations on Witt algebras}
\author{Yang Chen}
\address{Mathematics Postdoctoral Research Center,
Hebei Normal University, Shijiazhuang 050016, Hebei, China}
\email{chenyang1729@hotmail.com}
\author{Kaiming Zhao}
\address{Department of Mathematics, Wilfrid
Laurier University, Waterloo, ON, Canada N2L 3C5, and College of
Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050016, Hebei, China}
\email{kzhao@wlu.ca}
\author{Yueqiang Zhao}
\address{School of
Mathematical Sciences, Hebei Normal University, Shijiazhuang 050016, Hebei, China}
\email{yueqiangzhao@163.com}
\date{}
\maketitle
\begin{abstract} In this paper, we prove that every local derivation on Witt algebras $W_n, W_n^+$ or $W_n^{++} $ is a derivation for any $n\in\mathbb{N}$. As a consequence we obtain that every local derivation on a centerless generalized Virasoro algebra of higher rank is a derivation.
\
{\it Keywords:} Lie algebra, Witt algebra, generalized Virasoro algebra, derivation, local derivation.
{\it AMS Subject Classification:} 17B05, 17B40, 17B66.
\end{abstract}
\section{Introduction}
Let $A$ a Banach (or associative) algebra, $X$ be an
$A$-bimodule. A linear mapping
$\Delta:A\to X$ is said to be a \textit{local derivation} if for
every $x$ in $A$ there exists a derivation $D_x :A\to X$,
depending on $x$, satisfying $\Delta(x) = D_x (x).$ When $X$ is taken to be $A$, such a local derivation
is called a local derivation on $A$. The concept of
local derivation for Banach (or associative) algebras was introduced by Kadison \cite{Kad90}, Larson and Sourour \cite{LarSou} in 1990.
Since then there have been a lot of studies on local derivations on various algebras. See for example the recent papers
\cite{ AyuKud, AyuKudPer, AyuKudRak, LiuZh} and the references therein.
Local derivations on various algebras are some kind of local properties for the algebras,
which turn out to be very interesting.
Kadison actually proved in \cite{Kad90} that each continuous local derivation
of a von Neumann algebra $M$ into a dual Banach $M$-bimodule is a
derivation, he established the existence of local derivations on the algebra
$\mathbb{C}(x)$ of rational functions which are not derivations and
showed that any local derivation of the polynomial ring $\mathbb{C}[x_1, \cdots, x_n]$
is a derivation.
Recently, several papers have been devoted to studying local derivations for Lie (super)algebras.
In \cite{AyuKudRak}, Ayupov and Kudaybergenov proved that every local derivation on a finite dimensional semisimple Lie algebra over an algebraically closed field
of characteristic 0 is automatically a derivation, and gave examples of nilpotent Lie
algebra (so-call filiform Lie algebras) which admit local derivations which are not derivations. In \cite{Yus} and \cite{AY} Ayupov and Yusupov studied 2-local derivations on univariate Witt algebras. In \cite{Zhao}, we study 2-local derivations on
multivariate Witt algebras. In the present paper we study local derivations on Witt algebras.
Let $n\in\mathbb{N}$. The Witt algebra $W_n$ of vector fields on an $n$-dimensional torus is the derivation Lie algebra of the Laurent
polynomial algebra $A_n=\mathbb{C}[t_1^{\pm1},t_2^{\pm1},\cdots,
t_n^{\pm1}]$. Witt algebras were one of the four classes of Cartan type Lie algebras originally introduced in 1909 by Cartan \cite{C} when he studied
infinite dimensional simple Lie algebras.
Over the last two decades, the representation theory of Witt algebras was extensively
studied by many mathematicians and physicists; see for example \cite{BF, BMZ, GLLZ}. Very recently, Billig and Futorny
obtained the classification for all simple Harish-Chandra
$W_n$-modules in their remarkable paper \cite{BF}.
The present paper is arranged as follows. In Section 2 we recall some known results and establish some related properties concerning Witt algebras. In Section 3 we prove that every local derivation on Witt algebras $W_n$ is a derivation. As a consequence we obtain that every local derivation on a centerless generalized Virasoro algebra of higher rank is a derivation. The methods used in \cite{AyuKudRak} for finite dimensional semisimple Lie algebras no longer work for Witt algebras since Witt algebras have very different algebraic structure with finite dimensional semisimple Lie algebras. We have to establish new methods in the proofs of this section.
Finally, in Section 4 we show that the above methods and conclusions are applicable for Witt algebras $W_n^+$ and $W_n^{++}$.
Throughout this paper, we denote by $\mathbb{Z}$, $\mathbb{N}$, $\mathbb{Z}_+$ and $\mathbb{C}$ the sets of all integers, positive integers, non-negative integers and complex numbers respectively. All algebras are over $\mathbb{C}$.
\section{The Witt algebras}
In this section we recall definitions, symbols and some known
results for later use in this paper.
A derivation on a Lie algebra $L$ is a linear map
$D: L\rightarrow L$ which satisfies the Leibniz
law
$$
D([x,y])=[D(x),y]+[x, D(y)],\
\forall x,y\in L.$$ The set of all derivations of
$L$ is a Lie algebra and
is denoted by $\text{Der}(L)$. Clearly derivations on $L$ are local derivations, but the converse may not be true in general. For any $a\in L$,
the map
$$\text{ad}(a): L\to L, \ \text{ad}(a)x=[a,x],\ \forall x\in L$$ is a derivation and derivations of this form are called
\textit{inner derivations}. The set of all inner derivations of
$L$, denoted by $\text{Inn}(L),$ is a Lie ideal of
$\text{Der}(L)$.
For $n\in\mathbb{N}$, let $A_n= \mathbb{C}[t_1^{\pm1},t_2^{\pm1},\cdots,t_n^{\pm1}]$ be the Laurent polynomial algebra and $W_n= \text{Der}(A_n)$ be the Witt algebra of vector fields on an $n$-dimensional torus. Thus $W_n$ has a natural structure of a left $A$-module, which is free of rank $n$. Denote $d_1= t_1\frac{\partial}{\partial{t_1}}, \ldots, d_n= t_n\frac{\partial}{\partial{t_n}}$, which form a basis of this $A$-module:
$$W_n=\bigoplus_{i=1}^n A_n d_i.$$
Denote $t^\alpha=t_1^{\alpha_1}\cdots t_n^{\alpha_n}$ for $\alpha= (\alpha_1, \ldots, \alpha_n)\in \mathbb{Z}^n$ and let $\{\epsilon_1, \ldots, \epsilon_n\}$ be the standard basis of $\mathbb{Z}^n$. Then we can write the Lie bracket in $W_n$ as follows:
$$[t^\alpha d_i, t^\beta d_j]= \beta_i t^{\alpha+ \beta} d_j- \alpha_j t^{\alpha+ \beta} d_i,\ i, j= 1, \ldots, n;\ \alpha, \beta\in \mathbb{Z}^n.$$
The subspace $\mathfrak{h}$ spanned by $d_1, \ldots, d_n$ is the Cartan subalgebra of $W_n$. We may write any nonzero element in $W_n$ as $\sum_{\alpha\in S} t^\alpha d_\alpha$, where $S$ is the finite subset consisting of all $\alpha\in \mathbb{Z}^n$ with $d_\alpha\in \mathfrak{h}\setminus\{0\}$. For $d_\alpha= c_1 d_1+ \cdots+ c_n d_n\in \mathfrak{h}$ and $\beta=(\beta_1,\beta_2,\cdots, \beta_n)\in \mathbb{Z}^n$, define $$(d_\alpha, \beta)=c_1 \beta_1+ \cdots+ c_n \beta_n.$$ Then we get the following formula
$$[t^\alpha d_\alpha, t^\beta d_\beta]= t^{\alpha+ \beta}((d_\alpha, \beta)d_\beta- (d_\beta, \alpha)d_\alpha),\,\,\forall \alpha, \beta\in\mathbb{Z}^n, d_\beta d_\alpha\in \mathfrak{h}.$$
For convenience when we write
\begin{equation}X=\sum_{\alpha\in \mathbb{Z}^n} \sum_{i=1}^nc_{\alpha, i} t^\alpha d_i\in W_n,\end{equation}
where $c_{\alpha, i}\in\mathbb{C}$, the coefficient $c_{\alpha, i}$ will be denoted by $(X)_{t^\alpha d_i}.$
We make the convention that when $X\in W_n$ is written as in (2.1) we always assume that the sum is finite, i.e., there are only finitely many $c_{\alpha, i}$ nonzero.
\begin{definition}\label{def} \textit{We call a vector $\mu\in \mathbb{C}^n$ generic if $\mu\cdot \alpha\neq 0$ for all $\alpha\in \mathbb{Z}^n \setminus \{0\}$ , where $\mu\cdot \alpha$ is the standard inner product on $\mathbb{C}^n$.}
\end{definition}
For a generic vector $\mu=(\mu_1,\mu_2, \cdots, \mu_n) \in \mathbb{C}^n$ let $d_\mu= \mu_1 d_1+ \cdots+ \mu_n d_n$. Then we have the Lie subalgebra of $W_n$:
$$W_n(\mu)= A_n d_\mu,$$
which is called (centerless) generalized Virasoro algebra of rank $n$, see \cite{PZ}.
From Proposition 4.1 and Theorem 4.3 in \cite{DZ} we know that
any derivation on $W_n$ is inner. Then for the Witt algebra $W_n$, the above definition of the local derivation can be reformulated as follows. A linear map $\Delta$ on $W_n$ is a local derivation on $W_n$ if
for every elements $x\in W_n$ there exists an element
$a_x \in W_n$ such that $\Delta(x) = [a_x, x]$.
\section{Local derivations on $W_n$}
In this section we shall mainly prove the following result concerning local derivations on $W_n$ for $n\in\mathbb{N}$.
\begin{theorem}\label{thm31} \textit{Every local derivation
on the Witt algebra $W_n$ is a derivation.}
\end{theorem}
Since the proof of this theorem is long, we first setup six lemmas as preparations. Let $\mu=(\mu_1,\mu_2, \cdots, \mu_n) $ be a fixed generic vector in $\mathbb{C}^n$ and $d_\mu= \mu_1 d_1+ \cdots+ \mu_n d_n$.
For a given $\alpha\in \mathbb{Z}^n$, we define an equivalence relation $\stackrel{\alpha}{\sim}$ on $\mathbb{Z}^n$ {for } $\beta, \gamma\in \mathbb{Z}^n$:
$$ \beta\stackrel{\alpha}{\sim} \gamma\ \text{if and only if}\ \gamma- \beta= k\alpha,\ \text{for some }\ k\in \mathbb{Z}.$$
Let $[\gamma]:= \{\beta\in \mathbb{Z}^n\mid \gamma\stackrel{\alpha}{\sim} \beta\}$ denote the equivalence class containing $\gamma$. The set of equivalence classes of $\mathbb{Z}^n$ defined by $\alpha$ is denoted by
$\mathbb{Z}^n/ \alpha$.
Let $\Delta$ be a local derivation on $W_n$ with $\Delta(d_\mu)= 0$. For $t^\alpha d_\mu$ with $\alpha\ne0$, since $\Delta$ is a local derivation there is an element $a= \sum_{\beta\in \mathbb{Z}^n} t^\beta d''_\beta\in W_n$, where $d''_{\beta}\in \mathfrak{h} $, such that
\begin{equation}
\Delta(t^\alpha d_\mu)= [a, t^\alpha d_\mu]= \sum_{[\gamma]\in F}\sum_{k= p_\gamma}^{q_\gamma}t^{\gamma+ k\alpha} d_{\gamma+ k\alpha},
\end{equation}
where $F$ is a finite subset of $\mathbb{Z}^n/\alpha$ and $p_\gamma\leq q_\gamma\in \mathbb{Z}$.
It is clear that
\begin{equation}
d_\alpha= (d''_0, \alpha)d_\mu.
\end{equation}
For $t^\alpha d_\mu+ xd_\mu$ where $x\in\mathbb{C}^*$, since $\Delta$ is a local derivation there is an element
$$\sum_{[\gamma]\in F}\sum_{k= p'_{\gamma}}^{q'_{\gamma}}t^{\gamma+ k\alpha} d'_{\gamma+ k\alpha}\in W_n,$$
where $d'_{\gamma+ k\alpha}\in \mathfrak{h} $ and $p'_\gamma\leq q'_\gamma\in \mathbb{Z}$, such that
\begin{equation}\aligned &\Delta(t^\alpha d_\mu)= \Delta(t^\alpha d_\mu+ xd_\mu)= [\sum_{[\gamma]\in F}\sum_{k= p'_{\gamma}}^{q'_{\gamma}}t^{\gamma+ k\alpha} d'_{\gamma+ k\alpha}, t^\alpha d_\mu+ xd_\mu]\\
= &\sum_{[\gamma]\in F}\sum_{k= p'_{\gamma}}^{q'_{\gamma}+ 1} t^{\gamma+ k\alpha}((d'_{\gamma+ (k-1)\alpha}, \alpha)d_\mu- (d_\mu, \gamma+ (k-1)\alpha)d'_{\gamma+ (k-1)\alpha}\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -x(d_\mu, \gamma+ k\alpha)d'_{\gamma+ k\alpha}),\endaligned\end{equation}
where we have assigned $d'_{\gamma+ (p'_{\gamma}- 1)\alpha}= d'_{\gamma+ (q'_{\gamma}+ 1)\alpha}= 0$. Note that we have the same $F$ in (3.1) and (3.3).
\begin{lemma}\label{lem32'} \textit{Let $\Delta$ be a local derivation
on $W_n$ such that $\Delta(d_\mu)= 0$.
Then $F=\{[0]\}$ in (3.1) and (3.3).}
\end{lemma}
\begin{proof} We have assumed that $\alpha\ne0$.
Suppose that $d_{\gamma+ p_\gamma \alpha}\neq 0$ and $d_{\gamma+ q_\gamma \alpha}\neq 0$, $d'_{\gamma+ p'_\gamma \alpha}\neq 0$ and $d'_{\gamma+ q'_\gamma \alpha}\neq 0$ for some $[\gamma]\neq [0]$. Comparing the right hand sides of (3.1) and (3.3) we see that $p_\gamma=p'_\gamma$ and $q_\gamma\le q'_\gamma+1$. If $q_\gamma<q'_\gamma+1$, from (3.1) and (3.3) we deduce that
$$(d'_{\gamma+ q'_\gamma \alpha}, \alpha)d_\mu- (d_\mu, \gamma+ q'_\gamma \alpha)d'_{\gamma+ q'_\gamma \alpha}= 0.$$
Since $(d_\mu, \gamma+ q'_\gamma \alpha)\neq 0$, we see that $d'_{\gamma+ q'_\gamma \alpha}= c d_\mu$ for some $c\in \mathbb{C}^*$, and furthermore
$$(c d_\mu, \alpha)d_\mu- (d_\mu, \gamma+ q'_\gamma \alpha)c d_\mu= c(d_\mu, -\gamma- (q'_\gamma-1) \alpha)d_\mu= 0.$$
Then $ \gamma+(q'_\gamma-1) \alpha=0$, i.e., $[\gamma]= [0]$, a contradiction. Thus $q_\gamma=q'_\gamma+1$, and $p_\gamma<q_\gamma$.
Comparing (3.1) and (3.3) we deduce that (at least two equations)
\begin{equation}
\aligned &-x(d_\mu, \gamma+ p_\gamma \alpha)d'_{\gamma+ p_\gamma \alpha}= d_{\gamma+ p_\gamma \alpha};\\
&(d'_{\gamma+ p_\gamma\alpha}, \alpha)d_\mu- (d_\mu, \gamma+ p_\gamma\alpha)d'_{\gamma+ p_\gamma\alpha}- x(d_\mu, \gamma+ (p_\gamma+ 1)\alpha)d'_{\gamma+ (p_\gamma+ 1)\alpha}= d_{\gamma+ (p_\gamma+1)\alpha};\\
& \cdots\cdots \cdots\cdots;\\
&(d'_{\gamma+ (q_\gamma-2)\alpha}, \alpha)d_\mu- (d_\mu, \gamma+ (q_\gamma-2)\alpha)d'_{\gamma+ (q_\gamma-2)\alpha}- x(d_\mu, \gamma+ (q_\gamma-1)\alpha)d'_{\gamma+ (q_\gamma-1)\alpha}\\
&\hskip 3cm = d_{\gamma+ (q_\gamma-1)\alpha};\\
&(d'_{\gamma+ (q_\gamma-1)\alpha}, \alpha)d_\mu- (d_\mu, \gamma+ (q_\gamma-1)\alpha)d'_{\gamma+ (q_\gamma-1)\alpha}=
d_{\gamma+ q_\gamma\alpha}.\endaligned
\end{equation}
Since $(d_\mu, \gamma+ k\alpha)\neq 0$ for $k\in\mathbb{Z}$, eliminating $d'_{\gamma+ p_\gamma\alpha}, \ldots, d'_{\gamma+ (q_\gamma- 1)\alpha}$ in this order by substitution we see that
\begin{equation}d_{\gamma+ q_\gamma\alpha}+ *x^{-1}+ \cdots+ *x^{-q_\gamma+p_\gamma}= 0,\end{equation}
where $*\in \mathfrak{h}$ are independent of $x$. We always find some $x\in \mathbb{C}^*$ not satisfying (3.5), which is a contradiction.
The lemma follows.
\end{proof}
Now (3.1) and (3.3) become
\begin{equation}
\aligned &\sum_{k= p}^{q}t^{k\alpha} d_{ k\alpha}= \sum_{k= p'}^{q'+ 1} t^{ k\alpha}((d'_{ (k-1)\alpha}, \alpha)d_\mu- (d_\mu, (k-1)\alpha)d'_{ (k-1)\alpha} -x(d_\mu,k\alpha)d'_{ k\alpha}),\endaligned\end{equation}
where $p=p_0, q=q_0, p'=p'_0, q'=q'_0$ and we have assigned $d'_{(p'- 1)\alpha}= d'_{(q'+ 1)\alpha}= 0$.
We may assume that $d_{p \alpha}\neq 0$ and $d_{ q\alpha}\neq 0$, $d'_{ p' \alpha}\neq 0$ and $d'_{q' \alpha}\neq 0$. Clearly $p'\le p\le q\le q'+1$, and $p'=p$ if $p'\ne 0$. Our destination is to prove that $p=q=1$.
\begin{lemma}\label{lem32''} \textit{Let $\Delta$ be a local derivation
on $W_n$ such that $\Delta(d_\mu)= 0$.
Then $p'\ge0$ and $p\ge 1$ in (3.6).}
\end{lemma}
\begin{proof} To the contrary we assume that $p'< 0$. Then $p'= p$.
If further $q'\geq -1$, from (3.6) we obtain a set of (at least two) equations
\begin{equation}
\aligned &-x(d_\mu, p \alpha)d'_{p \alpha}= d_{p \alpha};\\
&(d'_{p\alpha}, \alpha)d_\mu- (d_\mu, p\alpha)d'_{p\alpha}- x(d_\mu, (p+ 1)\alpha)d'_{(p+ 1)\alpha}= d_{(p+1)\alpha};\\
&\cdots\cdots;\\
&(d'_{-\alpha}, \alpha)d_\mu- (d_\mu, -\alpha)d'_{-\alpha}= d_0.\endaligned
\end{equation}
If $d_0\ne 0$, using the same arguments as for (3.4), the equations (3.7) makes contradictions.
So we consider the case that $d_0=0$. From the last equation in (3.7) we see that $ d'_{-\alpha}=0$. We continue upwards in (3.7) in this manner to some step. We get
$$d_0= d_{-\alpha}= \cdots= d_{l\alpha}= 0,\,\,\,
d_{(l-1)\alpha}\neq 0, \ \text{and}\ d'_{-\alpha}= \cdots= d'_{(l-1)\alpha}= 0.$$
If $p+1< l (\leq 0),$
then (3.7) becomes
\begin{equation}
\aligned &-x(d_\mu, p \alpha)d'_{p \alpha}= d_{p \alpha};\\
&(d'_{p\alpha}, \alpha)d_\mu- (d_\mu, p\alpha)d'_{p\alpha}- x(d_\mu, (p+ 1)\alpha)d'_{(p+ 1)\alpha}= d_{(p+1)\alpha};\\
&\cdots\cdots;\\
&(d'_{(l-2)\alpha}, \alpha)d_\mu- (d_\mu, (l-2)\alpha)d'_{(l-2)\alpha}= d_{(l-1)\alpha}.\endaligned
\end{equation}
Using the same arguments as for (3.4), the equations (3.8) makes contradictions. We need only to consider the case that $p+1=l$, i.e., $l-1=p$.
In this case we have that $0=d'_{(l-1)\alpha}=d'_{p'\alpha}\ne0$, again a contradiction.
Therefore $q'< -1$.
If $q<q'+1$, from (3.6) we see that
$$(d'_{q' \alpha}, \alpha)d_\mu- (d_\mu, q' \alpha)d'_{q' \alpha}= 0.$$
Since $(d_\mu, q' \alpha)\neq 0$, we see that $d'_{ q' \alpha}= c d_\mu$ for some $c\in \mathbb{C}^*$, and furthermore
$$(c d_\mu, \alpha)d_\mu- (d_\mu, q' \alpha)c d_\mu= c(d_\mu, - (q' -1) \alpha)d_\mu= 0.$$
Then $ (q'-1) \alpha=0$, i.e., $q'=1$, a contradiction. So
$q=q'+1$ and $p<q$. We obtain a set of (at least two) equations from (3.6)
\begin{equation}
\aligned &-x(d_\mu, p \alpha)d'_{p \alpha}= d_{p \alpha};\\
&(d'_{p\alpha}, \alpha)d_\mu- (d_\mu, p\alpha)d'_{p\alpha}- x(d_\mu, (p+ 1)\alpha)d'_{(p+ 1)\alpha}= d_{(p+1)\alpha};\\
&\cdots\cdots;\\
&(d'_{(q-1)\alpha}, \alpha)d_\mu- (d_\mu, (q-1)\alpha)d'_{(q-1)\alpha}= d_{q\alpha}.\endaligned
\end{equation}
Using the same arguments as for (3.4), the equations (3.9) makes contradictions. Hence
$p'\geq 0.$
If $p'\ge1$, then $p= p'\ge 1$.
If $p'= 0$, then $d_0= (d'_{-\alpha}, \alpha)d_\mu- (d_\mu, -\alpha)d'_{-\alpha}= 0$ by (3.6). So
$p\geq 1$ also. \end{proof}
\begin{lemma}\label{lem32} \textit{Let $\Delta$ be a local derivation
on $W_n$ such that $\Delta(d_\mu)= 0$.
Then $$\Delta(t^\alpha d_\mu)\in\mathbb{C}t^\alpha d_\mu,\,\,\,\forall \alpha\in \mathbb{Z}^n.$$}
\end{lemma}
\begin{proof} From Lemma \ref{lem32''}, we know that $q\ge p\ge 1$. We need only to prove that $q=1$ in (3.6). Otherwise we assume that $q> 1$, and then $q'>0$.
{\bf Case 1:} $q'>1$.
In this case we can show that $q=q'+ 1$ as in the above arguments.
If $p'\geq 1$, we see that $p=p'$ and $p<q$. From (3.6) we obtain a set of (at least two) equations
\begin{equation}
\aligned &-x(d_\mu, p \alpha)d'_{p \alpha}= d_{p \alpha};\\
&(d'_{p\alpha}, \alpha)d_\mu- (d_\mu, p\alpha)d'_{p\alpha}- x(d_\mu, (p+ 1)\alpha)d'_{(p+ 1)\alpha}= d_{(p+1)\alpha};\\
&\cdots\cdots;\\
&(d'_{(q-1)\alpha}, \alpha)d_\mu- (d_\mu, (q-1)\alpha)d'_{(q-1)\alpha}= d_{q\alpha}.\endaligned
\end{equation}
Using the same arguments as for (3.4), the equation (3.10) makes contradictions.
So $p'=0.$ Now we have $$p'=0, p\ge 1, q=q'+1>2.$$ By (3.6) and (3.2) we have
\begin{equation}
(d'_0, \alpha)d_\mu- x(d_\mu, \alpha)d'_\alpha= d_\alpha= (d''_0, \alpha)d_\mu;
\end{equation}
\begin{equation}
(d'_\alpha, \alpha)d_\mu- (d_\mu, \alpha)d'_\alpha- x(d_\mu, 2\alpha)d'_{2\alpha}= d_{2\alpha}.
\end{equation}
Equation (3.11) implies that $d'_\alpha= c d_\mu$, $c\in \mathbb{C}$. Then Equation (3.12) can be simplified to $-x(d_\mu,2\alpha)d'_{2\alpha}= d_{2\alpha}$. Again from (3.6) we obtain a set of (at least two) equations
\begin{equation}
\aligned &-x(d_\mu,2\alpha)d'_{2\alpha}= d_{2\alpha}\\
&(d'_{2\alpha}, \alpha)d_\mu- (d_\mu, 2\alpha)d'_{2\alpha}- x(d_\mu, 3\alpha)d'_{3\alpha}= d_{3\alpha}\\
&\cdots\cdots\\
&(d'_{(q-1)\alpha}, \alpha)d_\mu- (d_\mu, (q-1)\alpha)d'_{(q-1)\alpha}= d_{q\alpha}.\endaligned
\end{equation}
Using the same arguments again, (3.13) makes contradictions. So $q'=1$. Now we have
{\bf Case 2:} $q'=1$.
We need only consider the case that $q=2$. In this case we still have Equations (3.11) and (3.12) with $d'_{2\alpha}=0$. Equation (3.11) implies that $d'_\alpha= c d_\mu$, $c\in \mathbb{C}$. Then Equation (3.12) implies that $d_{2\alpha}=0$ which is a contradiction.
Therefore $p= q= 1$ and $\Delta(t^\alpha d_\mu)= t^\alpha d_\alpha= (d''_0, \alpha)t^\alpha d_\mu$ by (3.2).
The lemma follows.
\end{proof}
\begin{lemma}\label{lem33}
\textit{Let $\Delta$ be a local derivation on $W_n$ such that $\Delta(d_\mu)= \Delta(t_i d_\mu)= 0$ for a given $1\leq i\leq n$.
Then $\Delta(t_i^m d_\mu)= 0$ for any $m\in \mathbb{Z}$.}
\end{lemma}
\begin{proof}
We may assume that $m\neq 0, 1$. By Lemmas \ref{lem32}, there is $c\in \mathbb{C}$ and
$$\sum_{[\gamma]\in F}\sum_{k= p_{\gamma}}^{q_{\gamma}}t^{\gamma+ k(m-1)\epsilon_i} d_{\gamma+ k(m-1)\epsilon_i}\in W_n,$$
where $F$ is a finite subset of $\mathbb{Z}^n/(m-1)\epsilon_i$ and $p_\gamma\leq q_\gamma\in \mathbb{Z}$, $d_{\gamma+ k(m-1)\epsilon_i}\in \mathfrak{h}$, such that
\begin{equation}
\aligned ct_i^m d_\mu=& \Delta(t_i^m d_\mu)= \Delta(t_i^m d_\mu+ t_i d_\mu)=
[\sum_{[\gamma]\in F}\sum_{k= p_{\gamma}}^{q_{\gamma}}t^{\gamma+ k(m-1)\epsilon_i} d_{\gamma+ k(m-1)\epsilon_i}, t_i^m d_\mu+ t_i d_\mu]\\
=& [\sum_{k= p_0}^{q_0} t_i^{k(m-1)} d_{k(m-1)\epsilon_i}+ \sum_{[0]\neq [\gamma]\in F}\sum_{k= p_{\gamma}}^{q_{\gamma}}t^{\gamma+ k(m-1)\epsilon_i} d_{\gamma+ k(m-1)\epsilon_i}, t_i^m d_\mu+ t_i d_\mu]\\
=& [\sum_{k= p_0}^{q_0} t_i^{k(m-1)} d_{k(m-1)\epsilon_i}, t_i^m d_\mu+ t_i d_\mu]\\
=& \sum_{k= p_0}^{q_0+1}t_i^{k(m-1)+1}((d_{(k-1)(m-1)\epsilon_i}, m\epsilon_i)d_\mu- (d_\mu, (k-1)(m-1)\epsilon_i)d_{(k-1)(m-1)\epsilon_i})\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + ((d_{k(m-1)\epsilon_i}, \epsilon_i)d_\mu- (d_\mu, k(m-1)\epsilon_i)d_{k(m-1)\epsilon_i}).\endaligned
\end{equation}
where we have assigned $d_{(p_0-1)(m-1)\epsilon_i}= d_{(q_0+1)(m-1)\epsilon_i}= 0$, and we have used the fact that
$$[\sum_{[0]\neq [\gamma]\in F}\sum_{k= p_{\gamma}}^{q_{\gamma}}t^{\gamma+ k(m-1)\epsilon_i} d_{\gamma+ k(m-1)\epsilon_i}, t_i^m d_\mu+ t_i d_\mu]= 0$$
since it cannot contain elements from $t_i^m \mathfrak{h}$ or eliminate any term in
$$[\sum_{k= p_0}^{q_0} t_i^{k(m-1)} d_{k(m-1)\epsilon_i}, t_i^m d_\mu+ t_i d_\mu].$$
We may assume that $d_{p_0(m-1)\epsilon_i}\neq 0$
and $d_{q_0(m-1)\epsilon_i}\neq 0$.
Our destination is to prove that $c=0$, To the contrary we assume that $c\ne0$. Then
$$p_0\le 1, q_0\ge0,\text{ and }p_0\le q_0.$$
{\bf Claim 1.} $p_0= 0$ or $1$.
Suppose that $p_0<0$, by (3.14) we deduce that
$$(d_{p_0(m-1)\epsilon_i}, \epsilon_i)d_\mu- (d_\mu, p_0(m-1)\epsilon_i)d_{p_0(m-1)\epsilon_i}= 0.$$
Since $(d_\mu, p_0(m-1)\epsilon_i)\neq 0$, we have $d_{p_0(m-1)\epsilon_i}
= c'd_\mu$ for some $c'\in \mathbb{C}^*$ and furthermore
$$\aligned &(d_{p_0(m-1)\epsilon_i}, \epsilon_i)d_\mu- (d_\mu, p_0(m-1)\epsilon_i)d_{p_0(m-1)\epsilon_i}\\
= &(c'd_\mu, \epsilon_i)d_\mu- (d_\mu, p_0(m-1)\epsilon_i)c'd_\mu\\
= &c'(d_\mu, (1- p_0(m-1))\epsilon_i)d_\mu= 0\endaligned.$$
It follows that $1- p_0(m-1)= 0$, and thus $p_0=-1, m=0$, a contradiction. Hence $p_0= 0$ or $1$.
{\bf Claim 2.} $q_0= 0$.
Suppose that $q_0> 0$, by (3.14) we deduce that
$$(d_{q_0(m-1)\epsilon_i}, m\epsilon_i)d_\mu- (d_\mu, q_0(m-1)\epsilon_i)d_{q_0(m-1)\epsilon_i}= 0.$$
Since $(d_\mu, q_0(m-1)\epsilon_i)\neq 0$, we have $d_{q_0(m-1)\epsilon_i}
= c'd_\mu$ for some $c'\in \mathbb{C}^*$ and furthermore
$$\aligned &(d_{q(m-1)\epsilon_i}, m\epsilon_i)d_\mu- (d_\mu, q(m-1)\epsilon_i)d_{q(m-1)\epsilon_i}\\
= &(c'd_\mu, m\epsilon_i)d_\mu- (d_\mu, q_0(m-1)\epsilon_i)c'd_\mu\\
= &c'(d_\mu, (m- q_0(m-1))\epsilon_i)d_\mu= 0\endaligned.$$
It follows that $m- q_0(m-1)= 0$, and thus $m= 2$ and $q_0= 2$. Now from (3.14) we deduce that
\begin{align}
&(d_0, \epsilon_i)d_\mu= 0;\\
&(d_0, 2\epsilon_i)d_\mu+ ((d_{\epsilon_i}, \epsilon_i)d_\mu- (d_\mu, \epsilon_i)d_{\epsilon_i})= cd_\mu;\\
&((d_{\epsilon_i}, 2\epsilon_i)d_\mu- (d_\mu, \epsilon_i)d_{\epsilon_i})+ ((d_{2\epsilon_i}, \epsilon_i)d_\mu- (d_\mu, 2\epsilon_i)d_{2\epsilon_i})= 0;\\
&(d_{2\epsilon_i}, 2\epsilon_i)d_\mu- (d_\mu, 2\epsilon_i)d_{2\epsilon_i}= 0.
\end{align}
We see that $d_{\epsilon_i}= c''d_\mu$, $c''\in \mathbb{C}$ by (3.18) and (3.17). Substituting into (3.16)
it follows that $c= 0$, a contradiction. Claim 2 follows.
From Claims 1 and 2 we have $p_0= q_0= 0$. By (3.14) we have $(d_0, \epsilon_i)d_\mu= 0$ and still $cd_\mu= (d_0, m\epsilon_i)d_\mu= 0$. The proof is completed.
\end{proof}
\begin{lemma}\label{lem34}
\textit{Let $\Delta$ be a local derivation on $W_n$ such that $\Delta(d_\mu)= \Delta(t_i d_\mu)= 0$ for all $1\leq i\leq n$. Then $\Delta(t^\alpha d_\mu)= 0$ for all $\alpha\in\mathbb{Z}^n$.}
\end{lemma}
\begin{proof}
From Lemma \ref{lem33}, we have
$$\Delta(t^{m\epsilon_i} d_\mu)= 0, \forall 1\le i\le n, m\in \mathbb{Z}.$$
We may assume that $n>1$ and $\alpha\in \mathbb{Z}^n\setminus \cup_{1\le i\le n} \mathbb{Z}\epsilon_i$.
Take an fixed integer $m> |\alpha_i|$ for any $1\leq i\leq n$. Define
$$I=\{i:\alpha_i\geq 0\}\ \text{and}\ I'=\{i: \alpha_i< 0\}.$$
Then by Lemma \ref{lem32} and Lemma \ref{lem33} there is $c\in \mathbb{C}$ and an element $ x=\sum_{\beta\in \mathbb{Z}^n} t^\beta d_\beta\in W_n$ such that
\begin{equation}\aligned ct^\alpha d_\mu&= \Delta(t^\alpha d_\mu)= \Delta(t^\alpha d_\mu+ \sum_{i\in I}t_i^m d_\mu+ \sum_{i\in I'}t_i^{-m} d_\mu)\\
&= [\sum_{\beta\in \mathbb{Z}^n} t^\beta d_\beta, t^\alpha d_\mu+ \sum_{i\in I}t_i^m d_\mu+ \sum_{i\in I'}t_i^{-m} d_\mu].\endaligned\end{equation}
{\bf Claim 1.} If $[t^\gamma d_\gamma, t_i^k d_\mu]= 0$ for some $1\leq i\leq n$, where $\gamma\in \mathbb{Z}^n\setminus\{0\}$, $d_\gamma\in \mathfrak{h}\setminus\{0\}$ and $k\in \mathbb{Z}$,
then $t^\gamma d_\gamma= c_it_i^k d_\mu$ for some $c_i\in\mathbb{C}^*$.
Since
$$[t^\gamma d_\gamma, t_i^k d_\mu]= t^{\gamma+ k\epsilon_i}((d_\gamma, k\epsilon_i)d_\mu- (d_\mu, \gamma)d_\gamma)= 0.$$
and $(d_\mu, \gamma)\neq 0$, we have $d_\gamma= c_i d_\mu$ for some $c_i\in \mathbb{C}^*$ and furthermore
$$(d_\gamma, k\epsilon_i)d_\mu- (d_\mu, \gamma)d_\gamma= (c_i d_\mu, k\epsilon_i)d_\mu- (d_\mu, \gamma)c_i d_\mu= c_i(d_\mu, k\epsilon_i- \gamma)d_\mu= 0.$$
It implies that $\gamma= k\epsilon_i$. Claim 1 follows.
Take a nonzero term $t^\gamma d_\gamma$ in the expression of $x$ with maximal degree with respect to $t_i$ for some $i\in I$.
{\bf Claim 2.} If $\gamma_i\ge 0$ and $\gamma\ne0$, then $t^\gamma d_\gamma=c_it_i^m d_\mu$ for some $c_i\in\mathbb{C}^*$.
The term $[t^\gamma d_\gamma, t_i^m d_\mu]$ is of maximal degree with respect to $t_i$ in the expression of
$$[\sum_{\beta\in \mathbb{Z}^n} t^\beta d_\beta, t^\alpha d_\mu+ \sum_{i\in I}t_i^m d_\mu+ \sum_{i\in I'}t_i^{-m} d_\mu].$$
Since $\gamma+m\epsilon_i\ne\alpha$, it follows that $[t^\gamma d_\gamma, t_i^m d_\mu]= 0$. We see that $t^\gamma d_\gamma= c_it_i^m d_\mu$ for some $c_i\in\mathbb{C}^*$ by Claim 1.
Similarly, a nonzero term $t^\gamma d_\gamma$ in the expression of $x$ with minimal degree with respect to $t_i$ for $i\in I'$ is $c_it_i^{-m} d_\mu$ for some $c_i\in\mathbb{C}^*$ if $\gamma_i\le 0$ and $\gamma\ne0$.
Moreover, $c_it_i^m d_\mu$ (resp. $c_it_i^{-m} d_\mu$) is the only possible term of non-negative maximal degree (resp. non-positive minimal degree) with respect to $t_i$, $i\in I$ (resp. $i\in I'$) in $x$. If there is such a term, we consider $c_it_i^m d_\mu$, $i\in I$ without loss of generality. Now, to delete the term $[c_it_i^m d_\mu, t^\alpha d_\mu]= c_i(d_\mu, \alpha- m\epsilon_i)t^{\alpha+ m\epsilon_i} d_\mu\neq 0$ in
$$[\sum_{\beta\in \mathbb{Z}^n} t^\beta d_\beta, t^\alpha d_\mu+ \sum_{i\in I}t_i^m d_\mu+ \sum_{i\in I'}t_i^{-m} d_\mu],$$
there must be $c_it^\alpha d_\mu$ in $\sum_{\beta\in \mathbb{Z}^n} t^\beta d_\beta$. To delete the term $[c_it^\alpha d_\mu, t_j^m d_\mu]\neq 0$, $i\neq j\in I$ (resp. $[c_it^\alpha d_\mu, t_j^{-m} d_\mu]$, $j\in I'$) in
$$[\sum_{\beta\in \mathbb{Z}^n} t^\beta d_\beta, t^\alpha d_\mu+ \sum_{i\in I}t_i^m d_\mu+ \sum_{i\in I'}t_i^{-m} d_\mu],$$
there must be $c_it_j^m d_\mu$, $i\neq j\in I$ (resp. $c_it_j^{-m} d_\mu$, $j\in I'$).
Thus $c_i=c_1$ for all $1\le i\le n$.
Let $x'= x- c_1(t^\alpha d_\mu+ \sum_{i\in I}t_i^m d_\mu+ \sum_{i\in I'}t_i^{-m} d_\mu)$. Then
\begin{equation}
ct^\alpha d_\mu= [x', t^\alpha d_\mu+ \sum_{i\in I}t_i^m d_\mu+ \sum_{i\in I'}t_i^{-m} d_\mu].
\end{equation}
Note that, if $t^\gamma d_\gamma$ is a nonzero term in the expression of $x'$, then $\gamma=0$ or \begin{equation}
\left\{\begin{matrix}
\gamma_i <0 &\text{ if }i\in I, \\
\gamma_i >0& \text{ if }i\in I'.
\end{matrix}\right.
\end{equation}
{\bf Claim 3.} In the expression of $x'$, the term $t^0d_0=0$.
If $d_0\ne0$,
considering the highest (resp. lowest) degree term with respect to $t_i$ for $i\in I$ (resp. $i\in I'$) in (3.20), we deduce that $(d_0, \epsilon_i)=0$ for all $1\le i\le n$. Thus $d_0=0$. Claim 3 follows.
Suppose that there exists a nonzero term $t^\gamma d_\gamma$ in the expression of $x'$ with maximal degree with respect to some $t_i$ for $i\in I$. Then $\gamma\neq 0$, and the term $[t^\gamma d_\gamma, t_i^m d_\mu]$ is of maximal degree with respect to $t_i$ in the expression of
$$[x', t^\alpha d_\mu+ \sum_{i\in I}t_i^m d_\mu+ \sum_{i\in I'}t_i^{-m} d_\mu].$$
It is only possible that $0\neq [t^\gamma d_\gamma, t_i^m d_\mu]\in t^\alpha\mathfrak{h}$ by Claim 1.
Then $\gamma+ m\epsilon_i= \alpha$. We have $0> \gamma_j= \alpha_j\geq 0$, $i\neq j\in I$ and $0< \gamma_j= \alpha_j< 0$, $i\neq j\in I'$ by (3.21), a contradiction. So $x'= 0$. Therefore $\Delta(t^\alpha d_\mu)= 0$.\end{proof}
\begin{lemma}\label{lem35} \textit{Let $\Delta$ be a local derivation
on $W_n$ such that $\Delta(d_\mu)=0$.
Then $\Delta\mid_\mathfrak{h}= 0$}.
\end{lemma}
\begin{proof} This is trivial for $n=1$. Next we assume that $n>1$.
For a given $i\in \{1,\dots,n\}$, there is an element $\sum_{\alpha\in \mathbb{Z}^n} t^\alpha d_\alpha^{(i)}\in W_n$, where $d_\alpha^{(i)}\in \mathfrak{h}$ such that
$$\Delta(d_i)= [\sum_{\alpha\in \mathbb{Z}^n} t^\alpha d_\alpha^{(i)}, d_i]= -\sum_{\alpha\in \mathbb{Z}^n} \alpha_i t^\alpha d_\alpha^{(i)}.$$
So we may suppose that
$$\Delta(d_i)= \sum_{\alpha\in \mathbb{Z}^n\setminus\{0\}}\sum_{1\leq j\leq n} c^{(i)}_{\alpha,j}t^\alpha d_j,$$
where $ c^{(i)}_{\alpha,j}\in\mathbb{C}$.
For a given $\alpha\in \mathbb{Z}^n\setminus\{0\}$, we have $\alpha_k\neq 0$ for some $1\leq k \leq n$. Let $c_{\alpha, j}= c^{(k)}_{\alpha,j}/\alpha_k$.
For $i\neq k$, let $d= \alpha_k d_i- \alpha_i d_k$. We have
$$(\Delta(d))_{t^\alpha d_j}= \alpha_k c^{(i)}_{\alpha,j}- \alpha_i c^{(k)}_{\alpha,j}.$$
On the other hand, there is an element $\sum_{\alpha\in \mathbb{Z}^n}\sum_{1\leq j\leq n} b_{\alpha,j}t^\alpha d_j\in W_n$ where $b_{\alpha,j}\in\mathbb{C}$
such that
$$(\Delta(d))_{t^\alpha d_j}= ([\sum_{\alpha\in \mathbb{Z}^n}\sum_{1\leq j\leq n} b_{\alpha,j}t^\alpha d_j, \alpha_k d_i- \alpha_i d_k])_{t^\alpha d_j}
= -b_{\alpha,j}\alpha_k\alpha_i+ b_{\alpha,j}\alpha_i\alpha_k= 0.$$
Then $c^{(i)}_{\alpha,j}= \alpha_i(c^{(k)}_{\alpha,j}/\alpha_k)= \alpha_i c_{\alpha,j}$, yielding that
$$0=( \Delta(d_\mu))_{t^\alpha d_j}= \sum_{i=1}^n\mu_i c^{(i)}_{\alpha,j}= c_{\alpha,j}\sum_{i=1}^n\mu_i\alpha_i.$$
We deduce that $c_{\alpha,j}= 0$. So $$c^{(i)}_{\alpha,j}= 0,\,\, \forall 1\leq i,j\leq n, \alpha\in\mathbb{Z}^n\setminus\{0\},$$ that is, $\Delta(d_i)=0$ for $1\leq i\leq n$. Therefore $\Delta\mid_\mathfrak{h}= 0$.
\end{proof}
Now we are in the position to prove Theorem \ref{thm31}.
\
\textit{Proof of Theorem} \ref{thm31}. Let $\Delta$ be a local derivation on $W_n$. We fix an arbitrary generic $\mu\in\mathbb{C}^n$. There is an element $a\in W_n$ such that $\Delta(d_\mu)=[a, d_\mu]$.
Set $\Delta_1= \Delta- \text{ad}(a)$. Then $\Delta_1$ is a local derivation such that $\Delta_1(d_\mu)= 0$. From Lemma \ref{lem35}, we know that $\Delta_1(\mathfrak{h})=0$.
By Lemma \ref{lem32}, there are $c_i\in \mathbb{C}$ such that
$$\Delta_1(t_i d_\mu)= c_i t_i d_\mu,\,\,\forall 1\leq i\leq n.$$
Set $\Delta_2= \Delta_1- \sum_{i=1}^n c_i \text{ad}(d_i)$. Then $\Delta_2$ is a local derivation such that $$\Delta_2(\mathfrak{h})=0, \text{ and }\Delta_2(t_i d_\mu)= 0,\,\,\forall 1\leq i\leq n.$$
By Lemma \ref{lem34}, for any generic $\mu\in\mathbb{C}^n$
we have $$\Delta_2(t^\alpha d_\mu)= 0,\,\,\forall \alpha\in\mathbb{Z}^n.$$
For any generic $\lambda\in\mathbb{C}^n$ that is not a multiple of $\mu$ since $\Delta_2(d_{\lambda})=0$, from Lemma \ref{lem32}
there is $c_\alpha\in \mathbb{C}$ such that $\Delta_2(t^\alpha d_{\lambda})= c_\alpha t^\alpha d_{\lambda}$.
There is $\sum_{\beta\in \mathbb{Z}^n}t^\beta d_\beta$, where $d_\beta\in\mathfrak{h}$, such that
$$\aligned c_\alpha t^\alpha d_{\lambda}=& \Delta_2(t^\alpha d_\alpha)= \Delta_2(t^\alpha d_\mu+ t^\alpha d_{\lambda})
=[\sum_{\beta\in \mathbb{Z}^n}t^\beta d_\beta, t^\alpha d_\mu+ t^\alpha d_{\lambda}]\\
=&\sum_{\beta\in \mathbb{Z}^n} t^{\alpha+\beta}((d_\beta, \alpha)(d_\mu+ d_{\lambda})-(d_\mu+ d_{\lambda},\beta)d_\beta).\endaligned$$
We see that $c_\alpha t^\alpha d_{\lambda}=(d_0, \alpha)(d_\mu+ d_{\lambda})$, yielding that
$c_\alpha= 0$. Thus for any generic
vector $\lambda$,
$$\Delta_2(t^\alpha d_{\lambda})= 0,\,\,\forall \alpha\in\mathbb{Z}^n.$$
Since the set $\{d_{\lambda}:\lambda\ \text{is generic}\}$ can span $\mathfrak{h}$ we must have $\Delta_2= 0$.
Hence $\Delta= \text{ad}(a)+ \sum_{i=1}^n c_i \text{ad}(d_i)$ is a derivation. The proof is completed.
\hfill$\Box$
By Theorem 3.4 in \cite{DZ1} any derivation on the generalized Virasoro algebra $W_n(\mu)$ can be seen as the restriction
of a inner derivation on $W_n$. All the proofs in this section with minor modifications are valid
for the generalized Virasoro algebra $W_n(\mu)$. Therefore
we obtain the following consequence.
\begin{corollary}\label{lem36} \textit{
Let $n\in\mathbb{N}$, and let $\mu\in\mathbb{C}^n$ be generic. Then any local derivation
on the generalized Virasoro algebra $W_n(\mu)$ is a derivation.
}
\end{corollary}
\section{Local derivations on $W_n^+$ and $W_n^{++}$}
For $n\in\mathbb{N}$, we have the Witt algebra $W_n^+=\text{Der}(\mathbb{C}[t_1 ,t_2 ,\cdots, t_n])$ which is a subalgebra of $W_n$. We use $ \mathfrak{h}$ to denote the Cartan subalgebra of $W_n$ which is also a Cartan subalgebra (not unique) of $W_n^+$. We know that
$$W_n^+= \sum_{\alpha\in \mathbb{Z}_+^n} t^\alpha \mathfrak{h}+ \sum_{i=1}^n \mathbb{C} t_i^{-1} d_i.$$
Furthermore $W_n^+$ has a subalgebra
$$W_n^{++}= \sum_{\alpha\in \mathbb{Z}_+^n} t^\alpha \mathfrak{h}.$$
It is well-known that $W_n^+$ is a simple Lie algebra, but $W_n^{++}$ is not.
From Proposition 4.1 and Theorem 4.3 in \cite{DZ} we know that any derivation on $W_n^+$ is inner .
Using same arguments as the proof of Proposition 3.3 in \cite{DZ1} we can show that
any derivation on $W_n^{++}$ is inner.
Hence, the proofs and conclusions with slight modifications in Section 3 are applicable to $W_n^+$ and $W_n^{++}$. It is routine to verify this. We omit the details and directly state the following theorem.
\begin{theorem}\label{thm51} \textit{Every local derivation
on Witt algebras $W_n^+$ or $\ W_n^{++}$ is a derivation.}
\end{theorem}
\vspace{2mm}
\noindent
{\bf Acknowledgements. } This research is partially supported by NSFC (11871190) and NSERC (311907-2015). | {"config": "arxiv", "file": "1911.05015.tex"} |
TITLE: Signal loss in non-reflected light through a tube proportional to square of the length?
QUESTION [0 upvotes]: Reading a patent I came across the claim: "...a portion of light intersecting the inner metal surface is not reflected, resulting in a loss in signal intensity... the signal loss is proportional to the square of the length of the light channel." (0015)
I can see it being proportional to the square of the length if it was making no contact with the tube because of free-space path loss, but that wouldn't be applicable inside the tube because the light could not spread spherically. I'm fairly certain it means that some of the light is absorbed, resulting in less intensity. But they provide no justification for why it is proportional to the square of the length. It seems to be like it would be more dependent on the material that the tube is made of as opposed to the length of the tube. Why is the length the main concern?
REPLY [0 votes]: If the tube was 100% reflective - like the sun tubes in some homes - then there would be no loss. In this case the tube is absorbing a portion of the light on each reflection, so if a given ray starts with intensity R, and looses a fraction x on each bounce, after one reflection the intensity would be R(1-x); after n bounces it would be R(1-x)^n.
Now consider a large number of rays: if they all entered the tube parallel to the axis, there would be no loss at all. But if instead the light is randomly directed, as from a diffuse source, each ray will, on average, take n bounces to make it through the tube, where n increases with the length of the tube.
If you know the angular distribution of the rays you could calculate the number of bounces for each angular range; but for a longer, narrower tube the number will be approximately proportional to the tube length. | {"set_name": "stack_exchange", "score": 0, "question_id": 233823} |
TITLE: Outer measure, Caratheodory measurability
QUESTION [2 upvotes]: Let X be a nonempty set.
If $ m^* : \mathcal{P}(X) \rightarrow [0, + \infty] $ is an outer measure, we say that $ B \subseteq X $ is $m^*$-measurable if:
$$ m^*(A) = m^*(A \cap B) + m^*(A \cap B^c), \forall A \in \mathcal{P}(X) $$
I can't think of an outer measure and a set where this property fails. Can you show me an example? (We defined outer measure as a monotonous, $\sigma$-subadditive function $m^* : \mathcal{P}(X) \rightarrow [0, + \infty]$ which satisfies $m^*(\emptyset) = 0 $)
REPLY [5 votes]: An example I put in my book ... (p. 155 of 2nd edition, Measure, Topology, and Fractal Geometry Springer 2008)
For subsets $A$ of $\mathbb R$, define
$$
m^*(A) = \inf \sum_{k=1}^\infty |I_k|^{1/2}
$$
where the infimum is over all countable covers of $A$ by open intervals $I_k$. Here, $|I_k|$ denotes the length of the interval. This is an outer measure.
Then we can show $m^*([0,1]) = m^*((1,2]) = 1$ and $m^*([0,2]) = \sqrt{2}$ so that $[0,1]$ is not measurable. | {"set_name": "stack_exchange", "score": 2, "question_id": 314048} |
TITLE: Random walk : probability of reaching value $i$ without passing by negative value $j$
QUESTION [2 upvotes]: This is just some question that popped out of nowhere while starting studying random walks, and I don't really know how to approach this.
Say I have a random walk that starts at zero, and goes up or down by one at each step with equal probability.
For some integer $i$, we stop the walk whenever it reaches either $i$ or $-i$.
Suppose we are given that the walk stopped by reaching $i$.
I'm interested in the minimum value the walk passed through.
In other words, for some $0 \geq j > -i$, what it the probability that the walk took value $j$ at some point, but not $j - 1$ ?
REPLY [1 votes]: To compute $P(j\cap i)$, the probability of reaching $i$ having reached $j$ as minimum, consider playing two games after each other:
$G$ Is the game that is won by starting from zero, ending at $j$ without having reached $i$
$H$ is the game that is won by starting from $j$, ending at $i$ without having reached $j-1$
For the first one we have
$$
P(G)=\frac{i}{i+|j|}
$$
for the second we have
$$
P(H)=\frac{1}{1+i+|j|}
$$
and so
$$
P(j\cap i)=P(H)P(G)=\frac{i}{(i+|j|)(1+i+|j|)}
$$
and as I understood it, you asked for the conditional probability $P(j\mid i)$ of having reached $j$ as a minimum given that we end at $i$. This should then be
$$
P(j\mid i)=\frac{P(j\cap i)}{P(i)}=\frac{2i}{(i+|j|)(1+i+|j|)}
$$
since $P(i)=\frac12$ if I am not mistaken. This formula should work for $-i<j\leq 0$ then. I calculated a table of probabilities for the example $i=10$ using Wolfram|Alpha. It looks plausible. For one thing, the probabilities sum to $1$, and of course they decrease with $j$ as they should - which is evident from the formula. | {"set_name": "stack_exchange", "score": 2, "question_id": 1217475} |
TITLE: Two parts of first ratio and one part of second ratio
QUESTION [0 upvotes]: I was preparing for Quantitative aptitude exams and I came across this question of ratios
An alloy contains copper and zinc in ratio 5:2 and another alloy contains zinc and tin in the ratio 3:2. If 2 parts of 1st alloy and one part of second alloy are melted together to form a new alloy of copper, zinc and tin, the ration of the metals will be?
What I've understood is that 2 parts of 1st ratio(lets call it A) means 2*A=10:4 and 1 part of the second ratio(lets call it B) means 1*B=3:2.
Now when these parts are melted together the ratios will be copper:zince:tin=10:4+3:2
Did I get this right or not?
REPLY [0 votes]: In the first alloy, copper : zinc = 5:2
in the 2nd , zinc : tin = 3:2
Let the 70x of first alloy is mixed with 35x of 2nd alloy ( 35 being LCM of (5+2 = 7 and 3+2 =5))
so amount of zinc and copper in the mixture , 70x = 50x -copper , 20x -zinc
35 x of 2nd will contain 21x - zinc and 14x tin
so required ratio of copper : zinc : tin = 50x :(20x+21x):14x = 50:41 : 14
for more concepts-https://www.handakafunda.com/ratio-and-proportion-concepts-properties-and-cat-questions/ | {"set_name": "stack_exchange", "score": 0, "question_id": 3193393} |
TITLE: find all continuous functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying $f(x+y)+f(x-y)=2f(x)+2f(y)$ for all $x,y \in \mathbb{R}^n$
QUESTION [6 upvotes]: find all continuous functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying
\begin{equation*}
f(x+y)+f(x-y)=2f(x)+2f(y)~\forall x,y \in \mathbb{R}^n.
\end{equation*}
My attempt: I manage to show that for any $q \in \mathbb{Q}$, $f(qx)=q^2f(x)$ for all $x \in \mathbb{R}^n$. I have a feeling that the answer is $f(x)=A\| x \|^2$, but I'm unable to prove it.
Can anyone give some hint?
REPLY [3 votes]: YES, $f$ must be a quadratic homogeneous polynomial (QHP in abbreviated
notation) as you expected.
We have a continuous function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying
$$
f(x+y)+f(x-y)=2f(x)+2f(y)~\forall x,y \in \mathbb{R}^n \tag{1}
$$
Let $a,b\in{\mathbb R}^n$. For $k\in{\mathbb N}$, define $\phi_{a,b}(k)=f(a+kb)$.
Taking $x=a+(k+1)b$ and $y=b$ in (1) above, we see that $\phi_{a,b}$ satisfies a second-order linear recurrence formula :
$$
\phi_{a,b}(k+2)-2\phi_{a,b}(k+1)+\phi_{a,b}(k)=2f(b) \tag{2}
$$
It follows easily from (2), by induction on $k\in{\mathbb N}$ that
$$
\phi_{a,b}(k)=k\phi_{a,b}(1)-(k-1)\phi_{a,b}(0)+k(k-1)f(b) \tag{3}
$$
Note that (3) can be rewritten as (putting $\mu=k$)
$$
f(a+\mu b)=f(a)+\mu(f(a+b)-f(a)-f(b))+\mu^2f(b) \ (\mu\in{\mathbb N}) \tag{3'}
$$
Now, let $\lambda,\mu\in{\mathbb N}$. We have
\begin{equation}
\begin{split}
f(\lambda a+\mu b) &= f(\lambda a)+\mu\Bigg(f(\lambda a+b)-f(\lambda a)-f(b)\Bigg)+\mu^2f(b) \\
&= \lambda ^2 f(a)+\mu\Bigg(\Bigg[f(b+\lambda a)\Bigg]-\lambda^2 f(a)-f(b)\Bigg)+\mu^2f(b) \\
&= \lambda ^2 f(a)+\mu\Bigg(\Bigg[f(b)+\lambda(f(a+b)-f(a)-f(b))+\lambda^2 f(a)\Bigg] -\lambda^2 f(a)-f(b)\Bigg)+\mu^2f(b) \\
&= \lambda ^2 f(a)+\lambda\mu\Bigg(f(a+b)-f(a)-f(b)\Bigg)+\mu^2f(b) \ \ \ \ \ \ \ \ \ \ \ \ \ (4) \\
\end{split}
\end{equation}
Replacing $a$ or $b$ with their opposites, we see that (4) still holds
when $\lambda$ or $\mu$ is negative. By homogeneity, (4) still holds when $\lambda,\mu\in{\mathbb Q}$.
By continuity, (4) still holds when $\lambda,\mu\in{\mathbb R}$. So the restriction of $f$
to any two-dimensional subspace is a QHP.
As a special case of (4), we have for $x,a\in{\mathbb R}^n$ and $\lambda \in {\mathbb R}$,
$$
f(x+\lambda a)=f(x)+\lambda (f(x+a)-f(x)-f(a)) +\lambda^2 f(a) \tag{5}
$$
Viewing $x+\lambda a+\lambda a'$ as $(x+\lambda a)+\lambda' a'$ and using (5) repeatedly, we
obtain for $x,a,a'\in{\mathbb R}^n$ and $\lambda,\lambda' \in {\mathbb R}$
$$
\begin{array}{lcl}
f(x+\lambda a+\lambda' a') &=&
f(x)+(\lambda+\lambda')(f(x+a)-f(x)-f(a))+\\
& & \lambda \lambda'
(f(x+a+a')-f(x+a)-f(x+a')+f(x)) +\\
& & \lambda^2f(a)+{\lambda'}^2f(a') \tag{6}
\end{array}
$$
Replacing $x$ with $\lambda''a''$ in (6) above and using (5) repeatedly one more time, we
obtain for $a,a',a''\in{\mathbb R}^n$ and $\lambda,\lambda',\lambda'' \in {\mathbb R}$,
$$
\begin{array}{lcl}
f(\lambda a+\lambda' a'+\lambda''a'')&=&
\lambda \lambda' (f(a+a')-f(a)-f(a')) + \\
& &\lambda \lambda'' (f(a+a'')-f(a)-f(a'')) + \\
& & \lambda' \lambda'' (f(a'+a'')-f(a')-f(a'')) + \\
& & \lambda^2f(a)+{\lambda'}^2f(a')+{\lambda''}^2f(a'') + \\
& & \lambda\lambda'\lambda''(f(a+a'+a'')-f(a+a')-f(a+a'')-f(a'+a'')+f(a)+f(a')+f(a'')) \\\tag{7}
\end{array}
$$
Note that when $\lambda=\lambda'=\lambda''=t$, the LHS reduces to $t^2f(a+a'+a'')$ which has no
$t^3$ term. So in (7) above the $\lambda\lambda'\lambda''$-coefficient must be zero. Then (7)
shows that the restriction of $f$ to any three-dimensional subspace is a QHP. If we put
$B(a,b)=f(a+b)-f(a)-f(b)$, it follows that for any $a$, $B(a,.)$ is linear on any two-dimensional
subspace, and hence linear everywhere. So $B$ is bilinear, which concludes the proof. | {"set_name": "stack_exchange", "score": 6, "question_id": 1297689} |
TITLE: Example of a Young function which does not satisfy $\Delta_{2}$ condition
QUESTION [0 upvotes]: Let me define first
(1) A convex function $\Phi \colon \mathbb{R}\to \mathbb{R}^+$ which satisfies the conditions,
(a) $\Phi(0)=0$
(b)$\Phi(-x)=\Phi(x)$
(c) $\lim_{x \to \infty}\Phi(x)=+\infty$, is called the Young function.
(2) A Young function $\Phi \colon \mathbb{R}\to \mathbb{R}^+$ is said to satisfy the $\Delta_{2}$ condition if, $\Phi(2x)\leq K\Phi(x)$ for $x\geq x_{0}\geq 0$ for some absolute constant $K>0$
Can someone give some example that does not satisfy $\Delta_{2}$ condition?Thanks
REPLY [1 votes]: Consider the function $\Phi \colon \mathbb R \to {\mathbb R}^+, \, \Phi(x) = \mathrm e^{x^2} - 1.$ Then we have:
$\Phi(0) = \mathrm e^0 - 1 = 0$.
$\Phi(-x) = \mathrm e^{(-x)^2} - 1 = \mathrm e^{x^2} - 1 = \Phi(x)$ for all $x \in \mathbb R$.
$\lim_{x \to \infty} \Phi(x) = \lim_{x \to \infty} \mathrm e^{x^2} - 1 = \infty$ since $\mathrm e^x \to \infty$ and $x^2 \to \infty$ if $x \to \infty$.
Finally, $\Phi$ is convex as $\Phi''(x) = (x) = 2 \mathrm e^{x^2} (1 + 2 x^2) > 0$ for all $x \in \mathbb R$.
Now suppose there exist $K > 0$ and $x_0 \geq 0$ such that
$\Phi(2x) \leq K \Phi(x)$ for all $x \geq x_0$. Then $\mathrm e^{(2x)^2} - 1 = \Phi(2 x) \leq K \Phi(x) = K \mathrm e^{x^2} - K$. However, if $x \to \infty$ the above estimate does not hold since $\mathrm e^{4x^2}$ grows way faster than $\mathrm e^{x^2}$ as $x \to \infty$. | {"set_name": "stack_exchange", "score": 0, "question_id": 4507826} |
\begin{document}
\title[On period polynomials of degree $2^m$]{On period polynomials of degree $\bm{2^m}$\\ for finite fields}
\author{Ioulia N. Baoulina}
\address{Department of Mathematics, Moscow State Pedagogical University, Krasnoprudnaya str. 14, Moscow 107140, Russia}
\email{jbaulina@mail.ru}
\date{}
\maketitle
\begin{abstract}
We obtain explicit factorizations of reduced period polynomials of degree $2^m$, $m\ge 4$, for finite fields of characteristic $p\equiv 3\text{\;or\;}5\pmod{8}$. This extends the results of G.~Myerson, who considered the cases $m=1$ and $m=2$, and S.~Gurak, who studied the case $m=3$.
\end{abstract}
\keywords{{\it Keywords}: Period polynomial; cyclotomic period; $f$-nomial period; reduced period polynomial; Gauss sum; Jacobi sum; factorization.}
\subjclass{2010 Mathematics Subject Classification: 11L05, 11T22, 11T24}
\thispagestyle{empty}
\section{Introduction}
Let $\mathbb F_q$ be a finite field of characteristic~$p$ with $q=p^s$ elements, $\mathbb F_q^*=\mathbb F_q^{}\setminus\{0\}$, and let $\gamma$ be a fixed generator of the cyclic group $\mathbb F_q^*$ . By ${\mathop{\rm Tr}\nolimits}:\mathbb F_q\rightarrow\mathbb F_p$ we denote the trace mapping, that is, ${\mathop{\rm
Tr}\nolimits}(x)=x+x^p+x^{p^2}+\dots+x^{p^{s-1}}$ for $x\in\mathbb F_q$. Let $e$ and $f$ be positive integers such that $q=ef+1$. Denote by $\mathcal{H}$ the subgroup of $e$-th powers in $\mathbb F_q^*$. For any positive integer $n$, write $\zeta_n=\exp(2\pi i/n)$.
The cyclotomic (or $f$-nomial) periods of order $e$ for $\mathbb F_q$ with respect to $\gamma$ are defined by
$$
\eta_k=\sum_{x\in\gamma^k\mathcal{H}}\zeta_p^{{\mathop{\rm Tr}\nolimits}(x)}=\sum_{h=0}^{f-1}\zeta_p^{{\mathop{\rm Tr}\nolimits}(\gamma^{eh+k})},\quad k=0,1,\dots,e-1.
$$
The period polynomial of degree $e$ for $\mathbb F_q$ is the polynomial
$$
P_e(X)=\prod_{k=0}^{e-1}(X-\eta_k).
$$
The reduced cyclotomic (or reduced $f$-nomial) periods of order $e$ for $\mathbb F_q$ with respect to $\gamma$ are defined by
$$
\eta_k^*=\sum_{x\in\mathbb F_q}\zeta_p^{{\mathop{\rm Tr}\nolimits}(\gamma^k x^e)}=1+e\eta_k,\quad k=0,1,\dots,e-1,
$$
and the reduced period polynomial of degree $e$ for $\mathbb F_q$ is
$$
P_e^*(X)=\prod_{k=0}^{e-1}(X-\eta_k^*).
$$
The polynomials $P_e(X)$ and $P_e^*(X)$ have integer coefficients and are independent of the choice of generator~$\gamma$. They are irreducible over the rationals when $s=1,$ but not necessarily irreducible when $s>1$. More precisely, $P_e(X)$ and $P_e^*(X)$ split over the rationals into $\delta=\gcd(e,(q-1)/(p-1))$ factors of degree~$e/\delta$ (not necessarily distinct), and each of these factors is irreducible or a power of an irreducible polynomial. Furthermore, the polynomials $P_e(X)$ and $P_e^*(X)$ are irreducible over the rationals if and only if $\gcd(e,(q-1)/(p-1))=1$. For proofs of these facts, see~\cite{M}.
In the case $s=1$, the period polynomials were determined explicitly by Gauss for ${e\in\{2, 3, 4\}}$ and by many others for certain small values of~$e$. In the general case, Myerson~\cite{M} derived the explicit formulas for $P_e(X)$ and $P_e^*(X)$ when $e\in\{2,3,4\}$, and also found their factorizations into irreducible polynomials over the rationals. Gurak~\cite{G3} obtained similar results for $e\in\{6,8,12,24\}$; see also \cite{G2} for the case $s=2$, $e\in\{6,8,12\}$. Note that if $-1$ is a power of $p$ modulo $e$, then the period polynomials can also be easily obtained. Indeed, if $e>2$ and $e\mid(p^{\ell}+1)$, with $\ell$ chosen minimal, then $2\ell\mid s$, and \cite[Proposition~20]{M} yields
$$
P_e^*(X)=(X+(-1)^{s/2\ell}(e-1)q^{1/2})(X-(-1)^{s/2\ell}q^{1/2})^{e-1}.
$$
Baumert and Mykkeltveit~\cite{BM} found the values of cyclotomic periods in the case when $e>3$ is a prime, $e\equiv 3\pmod{4}$ and $p$ generates the quadratic residues modulo~$e$; see also \cite[Proposition~21]{M}.
It is seen immediately from the definitions that $P_e(X)=e^{-e}P_e^*(eX+1)$, and so it suffices to factorize only $P_e^*(X)$.
The aim of this paper is to obtain the explicit factorizations of the reduced period polynomials of degree $2^m$ with $m\ge 4$ in the case that $p\equiv 3\text{\;or\;}5\pmod{8}$. Notice that in this case $\mathop{\rm ord}_2(q-1)=\mathop{\rm ord}_2(p^s-1)=\mathop{\rm ord}_2 s+2$. Hence, for $p\equiv 3\pmod{8}$,
$$
\gcd(2^m,(q-1)/(p-1))=\begin{cases}
2^m&\text{if $2^{m-1}\mid s$,}\\
2^{m-1}&\text{if $2^{m-2}\parallel s$.}
\end{cases}
$$
Appealing to \cite[Theorem~4]{M}, we conclude that in the case when $2^{m-1}\mid s$, $P_{2^m}^*(X)$ splits over the rationals into linear factors. If $2^{m-2}\parallel s$, then $P_{2^m}^*(X)$ splits into irreducible polynomials of degrees at most 2. Similarly, for $p\equiv 5\pmod{8}$,
$$
\gcd(2^m,(q-1)/(p-1))=\begin{cases}
2^m&\text{if $2^m\mid s$,}\\
2^{m-1}&\text{if $2^{m-1}\parallel s$,}\\
2^{m-2}&\text{if $2^{m-2}\parallel s$.}
\end{cases}
$$
Using \cite[Theorem~4]{M} again, we see that $P_{2^m}^*(X)$ splits over the rationals into linear factors if $2^m\mid s$, splits into linear and quadratic irreducible factors if $2^{m-1}\parallel s$, and splits into linear, quadratic and biquadratic irreducible factors if $2^{m-2}\parallel s$. Our main results are Theorems~\ref{t1} and \ref{t2}, which give the explicit factorizations of $P_{2^m}^*(X)$ in the cases $p\equiv 3\pmod{8}$ and $p\equiv 5\pmod{8}$, respectively. All the evaluations in Sections~\ref{s3} and \ref{s4} are effected in terms of parameters occuring in quadratic partitions of some powers of~$p$.
\section{Preliminary Lemmas}
\label{s2}
In the remainder of the paper, we assume that $p$ is an odd prime. Let $\psi$ be a nontrivial character on $\mathbb F_q$. We extend $\psi$ to all of $\mathbb F_q$ by setting $\psi(0)=0$. The Gauss sum
$G(\psi)$ over $\mathbb F_q$ is defined by
$$
G(\psi)=\sum_{x\in\mathbb F_q}\psi(x)\zeta_p^{{\mathop{\rm
Tr}\nolimits}(x)}.
$$
Gauss sums occur in the Fourier expansion of a reduced cyclotomic period.
\begin{lemma}
\label{l1}
Let $\psi$ be a character of order $e>1$ on $\mathbb F_q$ such that $\psi(\gamma)=\zeta_e$. Then for $k=0,1,\dots, e-1$,
$$
\eta_k^*=\sum_{j=1}^{e-1} G(\psi^j)\zeta_e^{-jk}.
$$
\end{lemma}
\begin{proof}
It follows from \cite[Theorem~1.1.3 and Equation~(1.1.4)]{BEW}.
\end{proof}
In the next three lemmas, we record some properties of Gauss sums which will be used throughout this paper. By $\rho$ we denote the quadratic character on $\mathbb F_q$ ($\rho(x)=+1, -1, 0$ according as $x$ is a square, a non-square or zero in $\mathbb F_q$).
\begin{lemma}
\label{l2}
Let $\psi$ be a nontrivial character on $\mathbb F_q$
with $\psi\ne\rho$. Then
\begin{itemize}
\item[\textup{(a)}]
$G(\psi)G(\bar\psi)=\psi(-1)q$;
\item[\textup{(b)}]
$G(\psi)=G(\psi^p)$;
\item[\textup{(c)}]
$G(\psi)G(\psi\rho)=\bar\psi(4)G(\psi^2)G(\rho)$.
\end{itemize}
\end{lemma}
\begin{proof}
See \cite[Theorems~1.1.4(a, d) and 11.3.5]{BEW} or \cite[Theorem~5.12(iv, v) and Corollary~5.29]{LN}.
\end{proof}
\begin{lemma}
\label{l3}
We have
$$
G(\rho)=
\begin{cases}
(-1)^{s-1}q^{1/2}&\text{if\,\, $p\equiv 1\pmod{4}$,}\\
(-1)^{s-1}i^s q^{1/2}&\text{if\,\, $p\equiv 3\pmod 4$.}
\end{cases}
$$
\end{lemma}
\begin{proof}
See \cite[Theorem~11.5.4]{BEW} or \cite[Theorem~5.15]{LN}.
\end{proof}
\begin{lemma}
\label{l4}
Let $p\equiv 3\pmod{8}$, $2\mid s$ and $\psi$ be a biquadratic character on $\mathbb F_q$. Then $G(\psi)=-q^{1/2}$.
\end{lemma}
\begin{proof}
It is a special case of \cite[Theorem~11.6.3]{BEW}.
\end{proof}
Let $\psi$ be a nontrivial character on $\mathbb F_q$. The Jacobi sum $J(\psi)$ over $\mathbb F_q$ is defined by
$$
J(\psi)=\sum_{x\in\mathbb F_q}\psi(x)\psi(1-x).
$$
The following lemma gives a relationship between Gauss sums and Jacobi sums.
\begin{lemma}
\label{l5}
Let $\psi$ be a nontrivial character on $\mathbb F_q$
with $\psi\ne\rho$. Then
$$
G(\psi)^2=G(\psi^2)J(\psi).
$$
\end{lemma}
\begin{proof}
See \cite[Theorem~2.1.3(a)]{BEW} or \cite[Theorem~5.21]{LN}.
\end{proof}
Let $\psi$ be a character on $\mathbb F_q$. The lift $\psi'$ of
the character $\psi$ from $\mathbb F_{q^{\vphantom{r}}}$ to the
extension field $\mathbb F_{q^r}$ is given by
$$
\psi'(x)=\psi({\mathop{\rm N}}_{\mathbb F_{q^r}/\mathbb
F_{q^{\vphantom{r}}}}(x)), \qquad x\in\mathbb F_{q^r},
$$
where ${\mathop{\rm N}}_{\mathbb F_{q^r}/\mathbb
F_{q^{\vphantom{r}}}}(x)=x\cdot x^q\cdot x^{q^2}\cdots
x^{q^{r-1}}=x^{(q^r-1)/(q-1)}$ is the norm of $x$ from
$\mathbb F_{q^r}$ to $\mathbb F_{q^{\vphantom{r}}}$.
\begin{lemma}
\label{l6}
Let $\psi$ be a character on $\mathbb
F_{q^{\vphantom{r}}}$ and let $\psi'$ denote the lift of $\psi$
from $\mathbb F_{q^{\vphantom{r}}}$ to $\mathbb F_{q^r}$. Then
\begin{itemize}
\item[\textup{(a)}]
$\psi'$ is a character on $\mathbb F_{q^r}$;
\item[\textup{(b)}]
a character $\lambda$ on $\mathbb F_{q^r}$ equals the lift $\psi'$ of some character $\psi$ on $\mathbb F_q$ if and only if the order of $\lambda$ divides $q-1$;
\item[\textup{(c)}]
$\psi'$ and $\psi$ have the same order.
\end{itemize}
\end{lemma}
\begin{proof}
See \cite[Theorem~11.4.4(a, c, e)]{BEW}.
\end{proof}
The following lemma, which is due to Davenport and Hasse, connects a Gauss sum and its lift.
\begin{lemma}
\label{l7}
Let $\psi$ be a nontrivial character on $\mathbb F_q$
and let $\psi'$ denote the lift of $\psi$ from $\mathbb F_{q^{}}$
to $\mathbb F_{q^r}$. Then
$$
G(\psi')=(-1)^{r-1}G(\psi)^r.
$$
\end{lemma}
\begin{proof}
See \cite[Theorem~11.5.2]{BEW} or \cite[Theorem~5.14]{LN}.
\end{proof}
Now we turn to the case $p\equiv 3\text{\;or\;}5\pmod{8}$. We recall a few facts which were established in our earlier paper~\cite{B2} in more general settings.
\begin{lemma}
\label{l8}
Let $p\equiv 3\text{\;or\;}5\pmod{8}$ and
$\psi$ be a character of order~$2^r$ on $\mathbb F_q$, where
$$
r\ge\begin{cases}
4&\text{if $p\equiv 3\pmod{8}$,}\\
3&\text{if $p\equiv 5\pmod{8}$.}
\end{cases}
$$
Then $G(\psi)=G(\psi\rho)$.
\end{lemma}
\begin{proof}
See \cite[Lemma 2.13]{B2}.
\end{proof}
\begin{lemma}
\label{l9}
Let $p\equiv 3\text{\;or\;}5\pmod{8}$, $r\ge 3$, and
$\psi$ be a character of order~$2^r$ on $\mathbb F_q$. Then
$$
\psi(4)=
\begin{cases}
1 & \text{if $p\equiv 3\pmod{8}$,}\\
(-1)^{s/2^{r-2}}&\text{if $p\equiv 5\pmod{8}$.}
\end{cases}
$$
\end{lemma}
\begin{proof}
See \cite[Lemma 2.16]{B2}.
\end{proof}
\begin{lemma}
\label{l10}
Let $p\equiv 3\text{\;or\;}5\pmod{8}$, $n\ge 1$ and $r\ge 3$ be integers, $r\ge n$. Then
$$
\sum_{v=0}^{2^{r-2}-1}\zeta_{2^n}^{p^v}=\begin{cases}
-2^{r-2}&\text{if $n=1$,}\\
2^{r-2}i&\text{if $n=2$ and $p\equiv 5\pmod{8}$,}\\
2^{r-3}i\sqrt{2}&\text{if $n=3$ and $p\equiv 3\pmod{8}$,}\\
0&\text{otherwise.}
\end{cases}
$$
\end{lemma}
\begin{proof}
It is an immediate consequence of \cite[Lemma 2.2]{B2}.
\end{proof}
The next lemma relates Gauss sums over $\mathbb F_q$ to Jacobi sums over a subfield of $\mathbb F_q$.
\begin{lemma}
\label{l11}
Let $p\equiv 3\text{\;or\;}5\pmod{8}$, and $\psi$ be a character of order $2^r$ on $\mathbb F_q$, where
$$
r\ge n=\begin{cases}
3&\text{if $p\equiv 3\pmod{8}$,}\\
2&\text{if $p\equiv 5\pmod{8}$,}
\end{cases}
$$
Assume that $2^{r-1}\mid s$. Then $\psi^{2^{r-n}}$ is equal to the lift of some character $\chi$ of order~$2^n$ on $\mathbb F_{p^{s/2^{r-n+1}}}$. Moreover,
$$
G(\psi)=q^{(2^{r-n+1}-1)/2^{r-n+2}}J(\chi)\cdot\begin{cases}
1&\text{if $p\equiv 3\pmod{8}$,}\\
(-1)^{s(r-1)/2^{r-1}}&\text{if $p\equiv 5\pmod{8}$.}
\end{cases}
$$
\end{lemma}
\begin{proof}
We prove the assertion of the lemma by induction on $r$, for $r\ge n$. Let $2^{n-1}\mid s$ and $\psi$ be a character of order $2^n$ on $\mathbb F_q$. As $2^n\mid(p^{s/2}-1)$, Lemma~\ref{l6} shows that $\psi$ is equal to the lift of some character $\chi$ of order $2^n$ on $\mathbb F_{p^{s/2}}$, that is, $\chi'=\psi$. Lemmas~\ref{l5} and \ref{l7} yield $G(\psi)=G(\chi')=-G(\chi)^2=-G(\chi^2)J(\chi)$. Note that $\chi^2$ has order~$2^{n-1}$. Thus, by Lemmas~\ref{l3} and \ref{l4},
$$
G(\chi^2)=\begin{cases}
-q^{1/4}&\text{if $p\equiv 3\pmod{8}$,}\\
(-1)^{(s/2)-1}q^{1/4}&\text{if $p\equiv 5\pmod{8}$,}
\end{cases}
$$
and so
$$
G(\psi)=q^{1/4}J(\chi)\cdot\begin{cases}
1&\text{if $p\equiv 3\pmod{8}$,}\\
(-1)^{s/2}&\text{if $p\equiv 5\pmod{8}$.}
\end{cases}
$$
This completes the proof for the case $r=n$.
Suppose now that $r\ge n+1$, and assume that the result is true when $r$ is replaced by $r-1$. Let $2^{r-1}\mid s$ and $\psi$ be a character of order $2^r$ on $\mathbb F_q$. Then $2^{r-2}\mid\frac s2$, and so $2^r\mid(p^{s/2}-1)$. By Lemma~\ref{l6}, $\psi$ is equal to the lift of some character $\phi$ of order $2^r$ on $\mathbb F_{p^{s/2}}$, that is $\phi'=\psi$. Applying Lemmas \ref{l2}(c), \ref{l3}, \ref{l7}, \ref{l8} and using the fact that $2^n\mid s$, we deduce
\begin{equation}
\label{eq1}
G(\psi)=-G(\phi)^2=-G(\phi)G(\phi\rho_0)=-\bar\phi(4)G(\phi^2)G(\rho_0)=\bar\phi(4)q^{1/4}G(\phi^2),
\end{equation}
where $\rho_0$ denotes the quadratic character on $\mathbb F_{p^{s/2}}$. Note that $\phi^2$ has order $2^{r-1}$ and $2^{r-2}\mid\frac s2$. Hence, by inductive hypothesis, $(\phi^2)^{2^{r-1-n}}=\phi^{2^{r-n}}$ is equal to the lift of some character $\chi$ of order $2^n$ on $\mathbb F_{p^{(s/2)/2^{r-n}}}=\mathbb F_{p^{s/2^{r-n+1}}}$ and
$$
G(\phi^2)=(p^{s/2})^{(2^{r-n}-1)/2^{r-n+1}}J(\chi)\cdot\begin{cases}
1&\text{if $p\equiv 3\pmod{8}$,}\\
(-1)^{(s/2)(r-2)/2^{r-2}}&\text{if $p\equiv 5\pmod{8}$,}
\end{cases}
$$
that is,
$$
G(\phi^2)=q^{(2^{r-n}-1)/2^{r-n+2}}J(\chi)\cdot\begin{cases}
1&\text{if $p\equiv 3\pmod{8}$,}\\
(-1)^{s(r-2)/2^{r-1}}&\text{if $p\equiv 5\pmod{8}$.}
\end{cases}
$$
Substituting this expression for $G(\phi^2)$ into \eqref{eq1} and using Lemma~\ref{l9}, we obtain
$$
G(\psi)=q^{(2^{r-n+1}-1)/2^{r-n+2}}J(\chi)\cdot\begin{cases}
1&\text{if $p\equiv 3\pmod{8}$,}\\
(-1)^{s(r-1)/2^{r-1}}&\text{if $p\equiv 5\pmod{8}$.}
\end{cases}
$$
It remains to show that $\psi^{2^{r-n}}$ is equal to the lift of $\chi$. Indeed, for any $x\in\mathbb F_q$ we have
\begin{align*}
\chi({\mathop{\rm N}}_{\mathbb F_q/\mathbb F_{p^{s/2^{r-n+1}}}}(x))&=\chi(x^{(p^s-1)/(p^{s/2^{r-n+1}}-1)})\\
&=\chi((x^{(p^s-1)/(p^{s/2}-1)})^{(p^{s/2}-1)/(p^{s/2^{r-n+1}}-1)})\\
&=\chi({\mathop{\rm N}}_{\mathbb F_{p^{s/2}}/\mathbb F_{p^{s/2^{r-n+1}}}}(x^{(p^s-1)/(p^{s/2}-1)}))=\phi^{2^{r-n}}(x^{(p^s-1)/(p^{s/2}-1)})\\
&=\left(\phi({\mathop{\rm N}}_{\mathbb F_{p^s}/\mathbb F_{p^{s/2}}}(x))\right)^{2^{r-n}}=\psi^{2^{r-n}}(x).
\end{align*}
Therefore $\chi'=\psi^{2^{r-n}}$, and the result now follows by the principle of mathematical induction.
\end{proof}
For an arbitrary integer $k$, it is convenient to set $\eta_k^*=\eta_{\ell}^*$, where $k\equiv {\ell}\pmod{e}$, $0\le\ell\le e-1$.
\begin{lemma}
\label{l12}
For any integer $k$, $\eta_{kp}^*=\eta_k^*$.
\end{lemma}
\begin{proof}
It is a straightforward consequence of \cite[Proposition~1]{G1}.
\end{proof}
From now on we shall assume that $p\equiv 3\text{\;or\;}5\pmod{8}$, $e=2^m$ with $m\ge 3$, and $\lambda$ is a character of order $2^m$ on $\mathbb F_q$ such that $\lambda(\gamma)=\zeta_{2^m}$. We observe that $2^{m-2}\mid s$.
\begin{lemma}
\label{l13}
We have
$$
P_{2^m}^*(X)=(X-\eta_0^*)(X-\eta_{2^{m-1}}^*)\prod_{t=0}^{m-2}(X-\eta_{2^t}^*)^{2^{m-t-2}}(X-\eta_{-2^t}^*)^{2^{m-t-2}}.
$$
\end{lemma}
\begin{proof}
Write
\begin{align*}
P_{2^m}^*(X)&=(X-\eta_0^*)(X-\eta_{2^{m-1}}^*)\prod_{t=0}^{m-2}\,\prod_{\substack{k=1\\ 2^t\parallel k}}^{2^m-1}(X-\eta_k^*)\\
&=(X-\eta_0^*)(X-\eta_{2^{m-1}}^*)\prod_{t=0}^{m-2}\,\prod_{\substack{k_0=1\\ 2\nmid k_0}}^{2^{m-t}-1}(X-\eta_{2^tk_0}^*).
\end{align*}
Since $p\equiv 3\text{\;or\;}5\pmod{8}$, \,$\pm p^0,\pm p^1,\dots, \pm p^{2^{m-t-2}-1}$ is a reduced residue system modulo $2^{m-t}$ for each $0\le t\le m-2$. Thus
$$
P_{2^m}^*(X)=(X-\eta_0^*)(X-\eta_{2^{m-1}}^*)\prod_{t=0}^{m-2}\,\prod_{j=0}^{2^{m-t-2}-1}(X-\eta_{2^t p^j}^*)(X-\eta_{-2^t p^j}^*).
$$
The result now follows from Lemma~\ref{l12}.
\end{proof}
\begin{lemma}
\label{l14}
We have
\begin{align*}
\eta_0^*=\,&G(\rho)+\sum_{r=2}^m 2^{r-2}\left(G(\lambda^{2^{m-r}})+G(\bar\lambda^{2^{m-r}})\right),\\
\eta_{2^{m-1}}^*=\,&G(\rho)+\sum_{r=2}^{m-1} 2^{r-2}\left(G(\lambda^{2^{m-r}})+G(\bar\lambda^{2^{m-r}})\right)-2^{m-2}\left(G(\lambda)+G(\bar\lambda)\right),
\end{align*}
and, for $0\le t\le m-2$,
\begin{align*}
\eta_{\pm 2^t}^*=\,&\sum_{r=2}^t 2^{r-2}\left(G(\lambda^{2^{m-r}})+G(\bar\lambda^{2^{m-r}})\right),\\
&+\begin{cases}
-G(\rho)&\text{if $t=0$,}\\
G(\rho)-2^{t-1}\left(G(\lambda^{2^{m-t-1}})+G(\bar\lambda^{2^{m-t-1}})\right)&\text{if $t>0$,}
\end{cases}\\
&\mp\begin{cases}
0&\text{if $p\equiv 3\pmod{8}$,}\\
2^ti\,\left(G(\lambda^{2^{m-t-2}})-G(\bar\lambda^{2^{m-t-2}})\right)&\text{if $p\equiv 5\pmod{8}$,}
\end{cases}\\
&\mp\begin{cases}
2^ti\sqrt{2}\,\left(G(\lambda^{2^{m-t-3}})-G(\bar\lambda^{2^{m-t-3}})\right)&\text{if $p\equiv 3\pmod{8}$ and $t\le m-3$,}\\
0&\text{otherwise.}
\end{cases}
\end{align*}
\end{lemma}
\begin{proof}
From Lemma~\ref{l1} we deduce that
$$
\eta_k^*=\sum_{j=1}^{2^m-1}G(\lambda^j)\zeta_{2^m}^{-jk}=\sum_{r=1}^m \sum_{\substack{j=1\\ 2^{m-r}\parallel j}}^{2^m-1}G(\lambda^j)\zeta_{2^m}^{-jk}
=\sum_{r=1}^m \sum_{\substack{j_0=1\\ 2\nmid j_0}}^{2^r-1}G(\lambda^{2^{m-r}j_0})\zeta_{2^r}^{-j_0k}.
$$
Since $\lambda^{2^{m-r}}$ has order $2^r$ and, for $r\ge 2$, $\pm p^0,\pm p^1,\dots,\pm p^{2^{r-2}-1}$ is a reduced residue system modulo $2^r$, we conclude that
$$
\eta_k^*=(-1)^k G(\rho)+\sum_{r=2}^m\, \sum_{u\in\{\pm 1\}}\sum_{v=0}^{2^{r-2}-1}G(\lambda^{2^{m-r}up^v})\zeta_{2^r}^{-kup^v},
$$
or, in view of Lemma~\ref{l2}(b),
\begin{equation}
\label{eq2}
\eta_k^*=(-1)^k G(\rho)+\sum_{r=2}^m\left[G(\lambda^{2^{m-r}})\sum_{v=0}^{2^{r-2}-1}\zeta_{2^r}^{-kp^v}+G(\bar\lambda^{2^{m-r}})\sum_{v=0}^{2^{r-2}-1}\zeta_{2^r}^{kp^v}\right].
\end{equation}
The expressions for $\eta_0^*$ and $\eta_{2^{m-1}}^*$ follow immediately from \eqref{eq2}. Next we assume that $0\le t\le m-2$. If $r>t+3$, then, by Lemma~\ref{l10},
$$
\sum_{v=0}^{2^{r-2}-1}\zeta_{2^r}^{2^t p^v}=\sum_{v=0}^{2^{r-2}-1}\zeta_{2^r}^{-2^t p^v}=0,
$$
and so \eqref{eq2} yields
\begin{align*}
\eta_{2^t}^*=\,&\sum_{r=2}^t 2^{r-2}\left(G(\lambda^{2^{m-r}})+G(\bar\lambda^{2^{m-r}})\right),\\
&+\begin{cases}
-G(\rho)&\text{if $t=0$,}\\
G(\rho)-2^{t-1}\left(G(\lambda^{2^{m-t-1}})+G(\bar\lambda^{2^{m-t-1}})\right)&\text{if $t>0$,}
\end{cases}\\
&+G(\lambda^{2^{m-t-2}})\sum_{v=0}^{2^t-1}i^{-p^v}+G(\bar\lambda^{2^{m-t-2}})\sum_{v=0}^{2^t-1}i^{p^v}\\
&+\begin{cases}
G(\lambda^{2^{m-t-3}})\sum_{v=0}^{2^{t+1}-1}\zeta_8^{-p^v}+G(\bar\lambda^{2^{m-t-3}})\sum_{v=0}^{2^{t+1}-1}\zeta_8^{p^v}&\text{if $t\le m-3$,}\\
0&\text{if $t=m-2$.}
\end{cases}
\end{align*}
The asserted result now follows from Lemmas~\ref{l4} and \ref{l10}. The expression for $\eta_{-2^t}^*$ can be obtained in a similar manner.
\end{proof}
\section{The Case $p\equiv 3\pmod{8}$}
\label{s3}
In this section, $p\equiv 3\pmod{8}$. As before, $2^m\mid(q-1)$ and $\lambda$ is a character of order~$2^m$ on $\mathbb F_q$ with $\lambda(\gamma)=\zeta_{2^m}$.
For $3\le r\le m$, define the integers $A_r$ and $B_r$ by
\begin{gather}
p^{s/2^{r-2}}=A_r^2+2B_r^2,\qquad A_r\equiv -1\pmod{4},\qquad p\nmid A_r,\label{eq3}\\
2B_r\equiv A_r(\gamma^{(q-1)/8}+\gamma^{3(q-1)/8})\pmod{p}.\label{eq4}
\end{gather}
It is well known that for each fixed $r$, the conditions \eqref{eq3} and \eqref{eq4} determine $A_r$ and $B_r$ uniquely.
\begin{lemma}
\label{l15}
Let $r$ be an integer with $2^{r-1}\mid s$ and $3\le r\le m$. Then
$$
G(\lambda^{2^{m-r}})+G(\bar\lambda^{2^{m-r}})=2A_r q^{(2^{r-2}-1)/2^{r-1}}
$$
and
$$
G(\lambda^{2^{m-r}})-G(\bar\lambda^{2^{m-r}})=2B_r q^{(2^{r-2}-1)/2^{r-1}}i\sqrt{2}.
$$
\end{lemma}
\begin{proof}
We observe that $\lambda^{2^{m-r}}$ has order $2^r$. By Lemma~\ref{l11}, $(\lambda^{2^{m-r}})^{2^{r-3}}=\lambda^{2^{m-3}}$ is equal to the lift of some octic character $\chi$ on $\mathbb F_{p^{s/2^{r-2}}}$ and
$$
G(\lambda^{2^{m-r}})\pm G(\bar\lambda^{2^{m-r}})=q^{(2^{r-2}-1)/2^{r-1}}(J(\chi)\pm J(\bar\chi)).
$$
Note that $\gamma^{(q-1)/(p^{s/2^{r-2}-1})}$ is a generator of the cyclic group $\mathbb F_{p^{s/2^{r-2}}}^*$ and, by the defition of the lift, $\chi(\gamma^{(q-1)/(p^{s/2^{r-2}-1})})=\chi({\mathop{\rm N}}_{\mathbb F_q/\mathbb F_{p^{s/2^{r-2}}}}(\gamma))=\lambda^{2^{m-3}}(\gamma)=\zeta_8$. By \cite[Lemma~17]{B1}, $J(\chi)=A_r+B_ri\sqrt{2}$, and the result follows.
\end{proof}
We are now in a position to prove the main result of this section.
\begin{theorem}
\label{t1}
Let $p\equiv 3\pmod{8}$ and $m\ge 4$. Then $P_{2^m}^*(X)$ has a unique decomposition into irreducible polynomials over the rationals as follows:
\begin{itemize}
\item[\rm (a)]
if $2^{m-1}\mid s$, then
\begin{align*}
P_{2^m}^*(X)=\,& (X-q^{\frac 12}+4B_3 q^{\frac 14})^{2^{m-2}} (X-q^{\frac 12}-4B_3 q^{\frac 14})^{2^{m-2}}\\
&\times (X-q^{\frac 12}+8B_4 q^{\frac 38})^{2^{m-3}} (X-q^{\frac 12}-8B_4 q^{\frac 38})^{2^{m-3}}\\
&\times \Bigl(X+3q^{\frac 12}-\sum_{r=3}^{m-2} 2^{r-1}A_r q^{\frac{2^{r-2}-1}{2^{r-1}}}+2^{m-2}A_{m-1} q^{\frac{2^{m-3}-1}{2^{m-2}}}\Bigr)^2\\
&\times \Bigl(X+3q^{\frac 12}-\sum_{r=3}^{m-1} 2^{r-1}A_r q^{\frac{2^{r-2}-1}{2^{r-1}}}+2^{m-1}A_m q^{\frac{2^{m-2}-1}{2^{m-1}}}\Bigr)\\
&\times \Bigl(X+3q^{\frac 12}-\sum_{r=3}^m 2^{r-1}A_r q^{\frac{2^{r-2}-1}{2^{r-1}}}\Bigr)\prod_{t=2}^{m-3}Q_t(X)^{2^{m-t-2}};
\end{align*}
\item[\rm (b)]
if $2^{m-2}\parallel s$ and $m\ge 5$, then
\begin{align*}
P_{2^m}^*(X)=\,& (X-q^{\frac 12}+4B_3 q^{\frac 14})^{2^{m-2}} (X-q^{\frac 12}-4B_3 q^{\frac 14})^{2^{m-2}}\\
&\times (X-q^{\frac 12}+8B_4 q^{\frac 38})^{2^{m-3}} (X-q^{\frac 12}-8B_4 q^{\frac 38})^{2^{m-3}}\\
&\times \Bigl(X+3q^{\frac 12}-\sum_{r=3}^{m-2} 2^{r-1}A_r q^{\frac{2^{r-2}-1}{2^{r-1}}}+2^{m-2}A_{m-1} q^{\frac{2^{m-3}-1}{2^{m-2}}}\Bigr)^2\\
&\times \left(\Bigl(X+3q^{\frac 12}-\sum_{r=3}^{m-1} 2^{r-1}A_r q^{\frac{2^{r-2}-1}{2^{r-1}}}\Bigr)^2+2^{2(m-1)}A_m^2 q^{\frac{2^{m-2}-1}{2^{m-2}}}\right)\\
&\times \left(\Bigl(X+3q^{\frac 12}-\sum_{r=3}^{m-3} 2^{r-1}A_r q^{\frac{2^{r-2}-1}{2^{r-1}}}+2^{m-3}A_{m-2} q^{\frac{2^{m-4}-1}{2^{m-3}}}\Bigr)^2\right.\\
&\hskip42pt+\Biggl.2^{2(m-1)}B_m^2 q^{\frac{2^{m-2}-1}{2^{m-2}}}\Biggr)^2\, \prod_{t=2}^{m-4}Q_t(X)^{2^{m-t-2}};
\end{align*}
\item[\rm (c)]
if $4\parallel s$, then
\begin{align*}
P_{16}^*(X)=\,&(X+3q^{\frac 12}+4A_3 q^{\frac 14})^2 (X-q^{\frac 12}+4B_3 q^{\frac 14})^4 (X-q^{\frac 12}-4B_3 q^{\frac 14})^4\\
&\times\left((X+3q^{\frac 12}-4A_3 q^{\frac 14})^2+64A_4^2 q^{\frac 34}\right)\left((X-q^{\frac 12})^2+64B_4^2 q^{\frac 34}\right)^2.
\end{align*}
\end{itemize}
The integers $A_r$ and $|B_r|$ are uniquely determined by~\eqref{eq3}, and
\begin{align*}
Q_t(X)=\,&\Bigl(X+3q^{\frac 12}-\sum_{r=3}^t 2^{r-1}A_r q^{\frac{2^{r-2}-1}{2^{r-1}}}+2^t A_{t+1} q^{\frac{2^{t-1}-1}{2^t}}+2^{t+2}B_{t+3}q^{\frac{2^{t+1}-1}{2^{t+2}}}\Bigr)\\
&\times\Bigl(X+3q^{\frac 12}-\sum_{r=3}^t 2^{r-1}A_r q^{\frac{2^{r-2}-1}{2^{r-1}}}+2^t A_{t+1} q^{\frac{2^{t-1}-1}{2^t}}-2^{t+2}B_{t+3}q^{\frac{2^{t+1}-1}{2^{t+2}}}\Bigr).
\end{align*}
\end{theorem}
\begin{proof}
Since $4\mid s$, Lemmas~\ref{l3} and \ref{l4} yield $G(\rho)=G(\lambda^{2^{m-2}})=G(\bar\lambda^{2^{m-2}})=-q^{1/2}$. Appealing to Lemmas \ref{l14} and \ref{l15}, we deduce that
\begin{align}
\eta_0^*&=-3q^{\frac 12}+\sum\limits_{r=3}^{m-1} 2^{r-1}A_r q^{\frac{2^{r-2}-1}{2^{r-1}}}+2^{m-2}\left(G(\lambda)+G(\bar\lambda)\right),\label{eq5}\\
\eta_{2^{m-1}}^*&=-3q^{\frac 12}+\sum\limits_{r=3}^{m-1} 2^{r-1}A_r q^{\frac{2^{r-2}-1}{2^{r-1}}}-2^{m-2}\left(G(\lambda)+G(\bar\lambda)\right),\label{eq6}\\
\eta_{\pm 2^{m-2}}^*&=-3q^{\frac 12}+\sum\limits_{r=3}^{m-2} 2^{r-1}A_r q^{\frac{2^{r-2}-1}{2^{r-1}}}-2^{m-2}A_{m-1}q^{\frac{2^{m-3}-1}{2^{m-2}}},\label{eq7}\\
\eta_{\pm 2^{m-3}}^*&=\begin{cases}
q^{\frac 12}\mp 2i\sqrt{2}\left(G(\lambda)-G(\bar\lambda)\right)&\text{if $m=4$,}\\
-3q^{\frac 12}+\sum\limits_{r=3}^{m-3} 2^{r-1}A_r q^{\frac{2^{r-2}-1}{2^{r-1}}}&\\
-2^{m-3}A_{m-2}q^{\frac{2^{m-4}-1}{2^{m-3}}}\mp 2^{m-3}i\sqrt{2}\left(G(\lambda)-G(\bar\lambda)\right)&\text{if $m\ge 5$,}
\end{cases}\label{eq8}\\
\eta_{\pm 1}^*&= q^{\frac 12}\pm 4B_3 q^{\frac 14}.\label{eq9}
\end{align}
Moreover, if $m\ge 5$, then
\begin{equation}
\label{eq10}
\eta_{\pm 2}^*=q^{\frac 12}\pm 8B_4 q^{\frac 38}
\end{equation}
and, for $2\le t\le m-4$,
\begin{equation}
\eta_{\pm 2^t}^*=-3q^{\frac 12}+\sum_{r=3}^t 2^{r-1}A_r q^{\frac{2^{r-2}-1}{2^{r-1}}}-2^t A_{t+1} q^{\frac{2^{t-1}-1}{2^t}}\pm 2^{t+2}B_{t+3}q^{\frac{2^{t+1}-1}{2^{t+2}}}.\label{eq11}
\end{equation}
Assume that $2^{m-1}\mid s$. Combining \eqref{eq5}~--~\eqref{eq11} with Lemma~\ref{l15}, we obtain the values of the cyclotomic periods, which are all integers. Part~(a) now follows from Lemma~\ref{l13}.
Next assume that $2^{m-2}\parallel s$. We have $2^m\parallel(q-1)$, and so $\lambda(-1)=-1$. Hence, by
Lemma~\ref{l2}(a),
$$
\left(G(\lambda)\pm G(\bar\lambda)\right)^2=G(\lambda)^2+G(\bar\lambda)^2\pm 2\lambda(-1)q=G(\lambda)^2+G(\bar\lambda)^2\mp 2q.
$$
Lemmas~\ref{l2}(c), \ref{l3}, \ref{l8}, \ref{l9} and \ref{l15} yield
\begin{align*}
G(\lambda)^2+G(\bar\lambda)^2&=G(\lambda)G(\lambda\rho)+G(\bar\lambda)G(\bar\lambda\rho)=\bar\lambda(4)G(\lambda^2)G(\rho)+\lambda(4)G(\bar\lambda^2)G(\rho)\\
&=-q^{1/2}(G(\lambda^2)+G(\bar\lambda^2))=-2A_{m-1}q^{(2^{m-2}-1)/2^{m-2}},
\end{align*}
and thus
\begin{equation}
\label{eq12}
\left(G(\lambda)\pm G(\bar\lambda)\right)^2=-2q^{(2^{m-2}-1)/2^{m-2}}(A_{m-1}\pm p^{s/2^{m-2}}).
\end{equation}
Note that
$$
A_{m-1}^2+2B_{m-1}^2=p^{s/2^{m-3}}=(p^{s/2^{m-2}})^2=(A_m^2+2B_m^2)^2
=(A_m^2-2B_m^2)^2+2\cdot(2A_mB_m)^2.
$$
Hence $A_{m-1}=\pm(A_m^2-2B_m^2)$. Since $p^{s/2^{m-2}}=A_m^2+2B_m^2\equiv 3\pmod{8}$, $B_m$ is odd, and so $A_{m-1}=A_m^2-2B_m^2$. Substituting the expressions
for $p^{s/2^{m-2}}$ and $A_{m-1}$ into \eqref{eq12}, we find that
\begin{align*}
\left(G(\lambda)+G(\bar\lambda)\right)^2&=-4A_m^2q^{(2^{m-2}-1)/2^{m-2}},\\
\left(G(\lambda)-G(\bar\lambda)\right)^2&=8B_m^2q^{(2^{m-2}-1)/2^{m-2}}.
\end{align*}
The last two equalities together with \eqref{eq5}, \eqref{eq6} and \eqref{eq9} imply
\begin{align}
(X-\eta_0^*)(X-\eta_{2^{m-1}}^*)=\,&
\Bigl(X+3q^{\frac 12}-\sum\limits_{r=3}^{m-1} 2^{r-1}A_r q^{\frac{2^{r-2}-1}{2^{r-1}}}\Bigr)^2\notag\\
&+2^{2(m-1)}A_m^2q^{\frac{2^{m-2}-1}{2^{m-2}}},\label{eq13}\\
(X-\eta_{2^{m-3}}^*)(X-\eta_{-2^{m-3}}^*)=\,&\Bigl(X+3q^{\frac 12}-\sum\limits_{r=3}^{m-3} 2^{r-1}A_r q^{\frac{2^{r-2}-1}{2^{r-1}}}+2^{m-3}A_{m-2}q^{\frac{2^{m-4}-1}{2^{m-3}}}\Bigr)^2\notag\\
&+2^{2(m-1)}B_m^2q^{\frac{2^{m-2}-1}{2^{m-2}}}\qquad\text{if $m\ge 5$,}\label{eq14}\\
(X-\eta_{2^{m-3}}^*)(X-\eta_{-2^{m-3}}^*)=\,&(X-q^{\frac 12})^2+64B_4^2 q^{\frac 34}\qquad\text{if $m=4$.}\label{eq15}
\end{align}
Clearly, the quadratic polynomials on the right sides of \eqref{eq13}~--~\eqref{eq15} are irreducible over the rationals.
Putting \eqref{eq7}, \eqref{eq9}~--~\eqref{eq11}, \eqref{eq13}~--~\eqref{eq15} together and appealing to Lemma~\ref{l13}, we deduce parts~(b) and (c). This completes the proof.
\end{proof}
\begin{remark}
\label{r}
{\rm The result of Gurak~\cite[Proposition~3.3(iii)]{G3} can be reformulated in terms of $A_3$ and $B_3$. Namely, $P_8^*(X)$ has the following factorization into irreducible polynomials over the rationals:
\begin{align*}
P_8^*(X)=\,& (X-q^{1/2})^2 (X-q^{1/2}+4B_3 q^{1/4})^2 (X-q^{1/2}-4B_3 q^{1/4})^2 &\\
&\times (X+3q^{1/2}+4A_3 q^{1/4})(X+3q^{1/2}-4A_3 q^{1/4})&\text{if $4\mid s$,}\\
P_8^*(X)=\,&(X-3q^{1/2})^2 &\\
&\times\left((X+q^{1/2})^2+16A_3^2 q^{1/2}\right)\left((X+q^{1/2})^2+16B_3^2 q^{1/2}\right)^2 &\text{if $2\parallel s$.}
\end{align*}
We see that Theorem~\ref{t1} is not valid for $m=3$.}
\end{remark}
\section{The Case $p\equiv 5\pmod{8}$}
\label{s4}
In this section, $p\equiv 5\pmod{8}$. As in the previous sections, $2^m\mid(q-1)$ and $\lambda$ denotes a character of order~$2^m$ on $\mathbb F_q$ such that $\lambda(\gamma)=\zeta_{2^m}$.
For $2\le r\le m-1$, define the integers $C_r$ and $D_r$ by
\begin{gather}
p^{s/2^{r-1}}=C_r^2+D_r^2,\qquad C_r\equiv 1\pmod{4},\qquad p\nmid C_r,\label{eq16}\\
D_r \gamma^{(q-1)/4}\equiv C_r\pmod{p}.\label{eq17}
\end{gather}
If $2^{m-1}\mid s$, we extend this notation to $r=m$. It is well known that for each fixed $r$, the conditions~\eqref{eq16} and \eqref{eq17} determine $C_r$ and $D_r$ uniquely.
\begin{lemma}
\label{l16}
Let $r$ be an integer with $2^{r-1}\mid s$ and $2\le r\le m$. Then
$$
G(\lambda^{2^{m-r}})+G(\bar\lambda^{2^{m-r}})=\begin{cases}
-2C_r q^{(2^{r-1}-1)/2^r}&\text{if $2^r\mid s$,}\\
(-1)^r\cdot 2C_r q^{(2^{r-1}-1)/2^r}&\text{if $2^{r-1}\parallel s$,}
\end{cases}
$$
and
$$
G(\lambda^{2^{m-r}})-G(\bar\lambda^{2^{m-r}})=\begin{cases}
2D_r q^{(2^{r-1}-1)/2^r}i&\text{if $2^r\mid s$,}\\
(-1)^{r-1}\cdot 2D_r q^{(2^{r-1}-1)/2^r}i&\text{if $2^{r-1}\parallel s$.}
\end{cases}
$$
\end{lemma}
\begin{proof}
The proof proceeds exactly as for Lemma~\ref{l15}, except that at the end, \cite[Proposition~3]{KR} is invoked instead of \cite[Lemma~17]{B1}.
\end{proof}
We are now ready to establish our second main result.
\begin{theorem}
\label{t2}
Let $p\equiv 5\pmod{8}$ and $m\ge 4$. Then $P_{2^m}^*(X)$ has a unique decomposition into irreducible polynomials over the rationals as follows:
\begin{itemize}
\item[\rm (a)]
if $2^m\mid s$, then
\begin{align*}
P_{2^m}^*(X)=\,& (X-q^{\frac 12}+2D_2 q^{\frac 14})^{2^{m-2}} (X-q^{\frac 12}-2D_2 q^{\frac 14})^{2^{m-2}}\\
&\times\Bigl(X+q^{\frac 12}+\sum_{r=2}^{m-1}2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}-2^{m-1}C_m q^{\frac{2^{m-1}-1}{2^m}}\Bigr)\\
&\times\Bigl(X+q^{\frac 12}+\sum_{r=2}^m 2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}\Bigr)\prod_{t=1}^{m-2}R_t(X)^{2^{m-t-2}};
\end{align*}
\item[\rm (b)]
if $2^{m-1}\parallel s$, then
\begin{align*}
P_{2^m}^*(X)=\,& (X-q^{\frac 12}+2D_2 q^{\frac 14})^{2^{m-2}} (X-q^{\frac 12}-2D_2 q^{\frac 14})^{2^{m-2}}\\
&\times\left(\Bigl(X+q^{\frac 12}+\sum_{r=2}^{m-1}2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}\Bigl)^2 -2^{2(m-1)}C_m^2 q^{\frac{2^{m-1}-1}{2^{m-1}}}\right)\\
&\times\Biggl(\Bigl(X+q^{\frac 12}+\sum_{r=2}^{m-2}2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}-2^{m-2}C_{m-1}q^{\frac{2^{m-2}-1}{2^{m-1}}}\Bigr)^2\Biggr.\\
&\hskip20pt -\Biggl.2^{2(m-1)}D_m^2 q^{\frac{2^{m-1}-1}{2^{m-1}}}\Biggr)\prod_{t=1}^{m-3}R_t(X)^{2^{m-t-2}};
\end{align*}
\item[\rm (c)]
if $2^{m-2}\parallel s$, then
\begin{align*}
P_{2^m}^*(X)=\,& (X-q^{\frac 12}+2D_2 q^{\frac 14})^{2^{m-2}} (X-q^{\frac 12}-2D_2 q^{\frac 14})^{2^{m-2}}\\
&\times\biggl(\Bigl(X+q^{\frac 12}+\sum_{r=2}^{m-3}2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}-2^{m-3}C_{m-2}q^{\frac{2^{m-3}-1}{2^{m-2}}}\Bigr)^2\biggr.\\
&\hskip26pt-\biggl.2^{2(m-2)}D_{m-1}^2 q^{\frac{2^{m-2}-1}{2^{m-2}}}\biggr)^2
\end{align*}
\begin{align*}
&\times\Biggl(\biggl(\Bigl(X+q^{\frac 12}+\sum_{r=2}^{m-2}2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}\Bigr)^2+2^{2(m-2)}C_{m-1}^2 q^{\frac{2^{m-2}-1}{2^{m-2}}}+2^{2m-3}q\biggr)^2\Biggr.\\
&\hskip26pt -2^{2(m-1)}C_{m-1}^2 q^{\frac{2^{m-2}-1}{2^{m-2}}}\Biggl.\Bigl(X+(2^{m-2}+1)q^{\frac 12}+\sum_{r=2}^{m-2}2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}\Bigr)^2\Biggr)\\
&\times\prod_{t=1}^{m-4}R_t(X)^{2^{m-t-2}}.
\end{align*}
\end{itemize}
The integers $C_r$ and $|D_r|$ are uniquely determined by~\eqref{eq16}, and
\begin{align*}
R_t(X)=\,&\Bigl(X+q^{\frac 12}+\sum_{r=2}^t 2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}-2^t C_{t+1} q^{\frac{2^t-1}{2^{t+1}}}+2^{t+1} D_{t+2} q^{\frac{2^{t+1}-1}{2^{t+2}}}\Bigr)\\
&\times\Bigl(X+q^{\frac 12}+\sum_{r=2}^t 2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}-2^t C_{t+1} q^{\frac{2^t-1}{2^{t+1}}}-2^{t+1} D_{t+2} q^{\frac{2^{t+1}-1}{2^{t+2}}}\Bigr).
\end{align*}
\end{theorem}
\begin{proof}
As $s$ is even, Lemma~\ref{l3} yields $G(\rho)=-q^{1/2}$. Then, applying Lemmas~\ref{l14} and \ref{l16}, we obtain
\begin{align}
\eta_0^*=\,&-q^{\frac 12}-\sum_{r=2}^{m-2}2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}+2^{m-3}\left(G(\lambda^2)+G(\bar\lambda^2)\right)\notag\\
&+2^{m-2}\left(G(\lambda)+G(\bar\lambda)\right),\label{eq18}\\
\eta_{2^{m-1}}^*=\,&-q^{\frac 12}-\sum_{r=2}^{m-2}2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}+2^{m-3}\left(G(\lambda^2)+G(\bar\lambda^2)\right)\notag\\
&-2^{m-2}\left(G(\lambda)+G(\bar\lambda)\right),\label{eq19}\\
\eta_{\pm 2^{m-2}}^*=\,&-q^{\frac 12}-\sum_{r=2}^{m-2}2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}-2^{m-3}\left(G(\lambda^2)+G(\bar\lambda^2)\right)\notag\\
&\mp 2^{m-2}i\left(G(\lambda)-G(\bar\lambda)\right),\label{eq20}\\
\eta_{\pm 2^{m-3}}^*=\,&-q^{\frac 12}-\sum_{r=2}^{m-3}2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}+2^{m-3}C_{m-2}q^{\frac{2^{m-3}-1}{2^{m-2}}}\notag\\
&\mp 2^{m-3}i\left(G(\lambda^2)-G(\bar\lambda^2)\right),\label{eq21}\\
\eta_{\pm 1}^*=\,&q^{\frac 12}\pm 2D_2 q^{\frac 14},\label{eq22}
\end{align}
and, for $1\le t\le m-4$,
\begin{equation}
\label{eq23}
\eta_{\pm 2^t}^*=-q^{\frac 12}-\sum_{r=2}^t 2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}+2^t C_{t+1} q^{\frac{2^t-1}{2^{t+1}}}\pm 2^{t+1} D_{t+2} q^{\frac{2^{t+1}-1}{2^{t+2}}}.
\end{equation}
First suppose that $2^m\mid s$. By combining \eqref{eq18}~--~\eqref{eq23} with Lemma~\ref{l16}, we find the values of the cyclotomic periods, which are all integers. Now part~(a) follows from Lemma~\ref{l13}.
Next suppose that $2^{m-1}\parallel s$. Using \eqref{eq18}~--~\eqref{eq23} and Lemma~\ref{l16} again, we find the values of the cyclotomic periods. We observe that $\eta_0^*$ and $\eta_{2^{m-1}}^*$ as well as $\eta_{2^{m-2}}^*$ and $\eta_{-2^{m-2}}^*$ are algebraic conjugates of degree~2 over the rationals, and the remaining cyclotomic periods are integers. Therefore the polynomials
$$
(X-\eta_0^*)(X-\eta_{2^{m-1}}^*)=\Bigl(X+q^{\frac 12}+\sum_{r=2}^{m-1}2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}\Bigr)^2-2^{2(m-1)}C_m^2 q^{\frac{2^{m-1}-1}{2^{m-1}}}
$$
and
\begin{align*}
(X-\eta_{2^{m-2}}^*)(X-\eta_{-2^{m-2}}^*)=\,&\Bigl(X+q^{\frac 12}+\sum_{r=2}^{m-2}2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}-2^{m-2}C_{m-1}q^{\frac{2^{m-2}-1}{2^{m-1}}}\Bigr)^2\\
&-2^{2(m-1)}D_m^2 q^{\frac{2^{m-1}-1}{2^{m-1}}}
\end{align*}
are irreducible over the rationals. Part~(b) now follows in view of Lemma~\ref{l13}.
Finally, suppose that $2^{m-2}\parallel s$. Making use of \eqref{eq18}~--~\eqref{eq20} and Lemma~\ref{l16}, we obtain
\begin{align}
\eta_0^*=\,&-q^{\frac 12}-\sum_{r=2}^{m-2}2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}-(-1)^m\cdot 2^{m-2}C_{m-1}q^{\frac{2^{m-2}-1}{2^{m-1}}}\notag\\
&+2^{m-2}\left(G(\lambda)+G(\bar\lambda)\right),\label{eq24}\\
\eta_{2^{m-1}}^*=\,&-q^{\frac 12}-\sum_{r=2}^{m-2}2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}-(-1)^m\cdot 2^{m-2}C_{m-1}q^{\frac{2^{m-2}-1}{2^{m-1}}}\notag\\
&-2^{m-2}\left(G(\lambda)+G(\bar\lambda)\right),\label{eq25}\\
\eta_{\pm 2^{m-2}}^*=\,&-q^{\frac 12}-\sum_{r=2}^{m-2}2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}+(-1)^m\cdot 2^{m-2}C_{m-1}q^{\frac{2^{m-2}-1}{2^{m-1}}}\notag\\
&\mp 2^{m-2}i\left(G(\lambda)-G(\bar\lambda)\right).\label{eq26}
\end{align}
By employing the same type of argument as in the proof of Theorem~\ref{t1}, we see that
$$
\left(G(\lambda)\pm G(\bar\lambda)\right)^2=\mp\, 2q^{(2^{m-1}-1)/2^{m-1}}\left(q^{1/2^{m-1}}\pm (-1)^m\cdot C_{m-1}\right).
$$
Combining this with \eqref{eq24}~--~\eqref{eq26}, we conclude that
\begin{align*}
(X-\eta_0^*)(X-&\eta_{2^{m-1}}^*)\\
=\,&\Bigl(X+q^{\frac 12}+\sum_{r=2}^{m-2}2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}+(-1)^m\cdot 2^{m-2}C_{m-1}q^{\frac{2^{m-2}-1}{2^{m-1}}}\Bigr)^2\\
&+2^{2m-3}q^{\frac{2^{m-1}-1}{2^{m-1}}}\left(q^{\frac 1{2^{m-1}}}+(-1)^m\cdot C_{m-1}\right)
\end{align*}
and
\begin{align*}
(X-\eta_{2^{m-2}}^*)(X-&\eta_{-2^{m-2}}^*)\\
=\,&\Bigl(X+q^{\frac 12}+\sum_{r=2}^{m-2}2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}-(-1)^m\cdot 2^{m-2}C_{m-1}q^{\frac{2^{m-2}-1}{2^{m-1}}}\Bigr)^2\\
&+2^{2m-3}q^{\frac{2^{m-1}-1}{2^{m-1}}}\left(q^{\frac 1{2^{m-1}}}-(-1)^m\cdot C_{m-1}\right).
\end{align*}
Since $q^{1/2^{m-2}}=p^{s/2^{m-2}}=C_{m-1}^2+D_{m-1}^2$, we have $q^{1/2^{m-1}}>|C_{m-1}|$. This means that the polynomials $(X-\eta_0^*)(X-\eta_{2^{m-1}}^*)$ and $(X-\eta_{2^{m-2}}^*)(X-\eta_{-2^{m-2}}^*)$ are irreducible over the reals. Furthermore, since $2^{m-2}\parallel s$, the polynomials $(X-\eta_0^*)(X-\eta_{2^{m-1}}^*)$ and $(X-\eta_{2^{m-2}}^*)(X-\eta_{-2^{m-2}}^*)$ belong to $\mathbb R[X]\setminus\mathbb Q[X]$. Since $\mathbb R[X]$ is a unique factorization domain, it follows that the polynomial
\begin{align*}
(X-&\eta_0^*)(X-\eta_{2^{m-1}}^*)(X-\eta_{2^{m-2}}^*)(X-\eta_{-2^{m-2}}^*)\\
=\,&\biggl(\Bigl(X+q^{\frac 12}+\sum_{r=2}^{m-2}2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}\Bigr)^2+2^{2(m-2)}C_{m-1}^2 q^{\frac{2^{m-2}-1}{2^{m-2}}}+2^{2m-3}q\biggr)^2\\
&-2^{2(m-1)}C_{m-1}^2 q^{\frac{2^{m-2}-1}{2^{m-2}}}\Bigl(X+(2^{m-2}+1)q^{\frac 12}+\sum_{r=2}^{m-2}2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}\Bigr)^2
\end{align*}
is irreducible over the rationals. Further, by Lemma~\ref{l16} and \eqref{eq21},
$$
\eta_{\pm 2^{m-3}}^*=-q^{\frac 12}-\sum_{r=2}^{m-3}2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}+ 2^{m-3}C_{m-2}q^{\frac{2^{m-3}-1}{2^{m-2}}}\pm (-1)^m\cdot 2^{m-2}D_{m-1}q^{\frac{2^{m-2}-1}{2^{m-1}}},
$$
and so $\eta_{2^{m-3}}^*$ and $\eta_{-2^{m-3}}^*$ are algebraic conjugates of degree~2 over the rationals. Hence, the polynomial
\begin{align*}
(X-\eta_{2^{m-3}}^*)(X-\eta_{-2^{m-3}}^*)=\,&\Bigl(X+q^{\frac 12}+\sum_{r=2}^{m-3}2^{r-1}C_r q^{\frac{2^{r-1}-1}{2^r}}-2^{m-3}C_{m-2}q^{\frac{2^{m-3}-1}{2^{m-2}}}\Bigr)^2\\
&-2^{2(m-2)}D_{m-1}^2 q^{\frac{2^{m-2}-1}{2^{m-2}}}
\end{align*}
is irreducible over the rationals. The remaining cyclotomic periods $\eta_{\pm 2^t}^*$, $0\le t\le m-4$, are integers, in view of \eqref{eq22} and \eqref{eq23}. Now Part~(c) follows by appealing to Lemma~\ref{l13}. This concludes the proof.
\end{proof}
\begin{remark}
{\rm Myerson has shown \cite[Theorem~17]{M} that $P_4^*(X)$ is irreducible if $2\nmid s$,
\begin{align*}
P_4^*(X)=\,& (X+q^{1/2}+2C_2 q^{1/4})(X+q^{1/2}-2C_2 q^{1/4})&\\
&\times(X-q^{1/2}+2D_2 q^{1/4}) (X-q^{1/2}-2D_2 q^{1/4})&\text{if $4\mid s$,}\\
\intertext{and, with a slight modification,}
P_4^*(X)=\,&\left((X+q^{1/2})^2-4C_2^2q^{1/2}\right)\left((X-q^{1/2})^2-4D_2^2q^{1/2}\right)&\text{if $2\parallel s$,}
\end{align*}
where in the latter case the quadratic polynomials are irreducible over the rationals. Furthermore, the result of Gurak~\cite[Proposition~3.3(ii)]{G3} can be reformulated in terms of $C_2$, $D_2$, $C_3$ and $D_3$. Namely, $P_8^*(X)$ has the following factorization into irreducible polynomials over the rationals:
\begin{align*}
P_8^*(X)=\,& (X-q^{1/2}+2D_2 q^{1/4})^2 (X-q^{1/2}-2D_2 q^{1/4})^2\\
&\times(X+q^{1/2}+2C_2 q^{1/4}+4C_3 q^{3/8})(X+q^{1/2}+2C_2 q^{1/4}-4C_3 q^{3/8})&\\
&\times(X+q^{1/2}-2C_2 q^{1/4}+4D_3 q^{3/8})(X+q^{1/2}-2C_2 q^{1/4}-4D_3 q^{3/8})&\!\text{if $8\mid s$,}\\
P_8^*(X)=\,& (X-q^{1/2}+2D_2 q^{1/4})^2 (X-q^{1/2}-2D_2 q^{1/4})^2&\\
&\times\left((X+q^{1/2}+2C_2 q^{1/4})^2-16C_3^2 q^{3/4}\right)&\\
&\times\left((X+q^{1/2}-2C_2 q^{1/4})^2-16D_3^2 q^{3/4}\right)&\!\text{if $4\parallel s$,}\\
P_8^*(X)=\,&\left((X-q^{1/2})^2-4D_2^2 q^{1/2}\right)^2&\\
&\times\left(\left((X+q^{1/2})^2+4C_2^2 q^{1/2}+8q\right)^2-16C_2^2 q^{1/2}(X+3q^{1/2})^2\right)&\!\text{if $2\parallel s$.}
\end{align*}
Thus part~(a) of Theorem~\ref{t2} remains valid for $m=2$ and $m=3$. Moreover, for $m=3$, part~(b) of Theorem~\ref{t2} is still valid (cf. Remark~\ref{r}).}
\end{remark} | {"config": "arxiv", "file": "1604.01007.tex"} |
TITLE: Spherical transformation in Peskin Book
QUESTION [2 upvotes]: In the page 27 the authors calculate the propagator
$$D(x-y)=\int{\frac{\mathrm d^3p}{(2\pi)^3}\frac{1}{2E_{\textbf p}}e^{i\textbf{p}\cdot\textbf{r}}}$$
and it transforms the integral from Cartesian to spherical coordinates for $p$ and they found that
$$D(x-y)=\frac{2\pi}{(2\pi)^3}\int_{0}^{\infty}{~\mathrm dp~\frac{p^2}{2E_{\textbf p}}\frac{ e^{i pr}- e^{-i pr}}{ipr}}$$
I can not understand how $e^{i\textbf{p}\cdot\textbf{r}}$ became $\frac{ e^{i px}- e^{-i pr}}{2ipr}$.
How is this is possible?
REPLY [5 votes]: Without loss of generality, we can create spherical coordinates $(p, \theta, \phi)$ such that $\mathbf{r}$ is along the $\theta=0$ axis. This may seem a little backwards, since we usually have $\mathbf{r}$ as the position vector, which naturally varies. But here we leave $\mathbf{r}$ constant, and vary in $\mathbf{p}$-space, since that's what we're integrating over. In particular, variations in $\theta$ and $\phi$ correspond to variations in $\mathbf{p}$ rather than $\mathbf{r}$. Then, we can transform the integral into those coordinates, perform the trivial integral over $\phi$, then change coordinates again using $\sin\theta\, d\theta = -d\cos\theta$, and finally perform that integral too, leaving only the integral over $p$.
\begin{align}
D(x-y) &=
\int_0^\infty \int_0^\pi \int_0^{2\pi} \frac{1}{(2\pi)^3}\,
\frac{1}{2E_{\mathbf{p}}} e^{i\, p\, r\, \cos\theta}\, p^2\, \sin\theta\,
d\phi\, d\theta\, dp \\
&=
\frac{1}{(2\pi)^2}\,
\int_0^\infty \frac{p^2}{2E_{\mathbf{p}}} \left[\int_0^\pi
e^{i\, p\, r\, \cos\theta}\, \sin\theta\,
d\theta\right]\, dp \\
&=
\frac{1}{(2\pi)^2}\,
\int_0^\infty \frac{p^2}{2E_{\mathbf{p}}} \left[\int_1^{-1}
-e^{i\, p\, r\, x}\, dx\right]\, dp \\
&=
\frac{1}{(2\pi)^2}\,
\int_0^\infty \frac{p^2}{2E_{\mathbf{p}}}
\frac{-e^{-i\, p\, r} - -e^{i\, p\, r}}{i\, p\, r}\, dp.
\end{align}
Note that this assumes $E_{\mathbf{p}}$ is independent of $\theta$ and $\phi$, but that's fine because we usually have $E_{\mathbf{p}} = \sqrt{ \lvert \mathbf{p} \rvert^2 + m^2} = \sqrt{ p^2 + m^2}$. | {"set_name": "stack_exchange", "score": 2, "question_id": 276673} |
\begin{document}
\maketitle
\begin{abstract}
In this article the author presents results on the selection for the space of convex hulls of $n$ points and compacts of the complete convex metric space of Busemann nonpositve curvature. Namely, we determine Lipschitz and H\"older properties of barycenter and cirumcenter mappings.
\end{abstract}
\section{Introduction}
In this article the author presents results on the selection for the space of convex hulls of $n$ points and compacts of the complete metric space. This problem seems to be of interest for some time and has the abundant set of solutions in Hadamard spaces \cite{LPS, SB}, in Banach spaces \cite{Shvarts} and in special metric spaces \cite{Khamsi1}. The same can be said about properties of weighted points of the finite sets \cite{Vesely, LPS}. Note also that there exists the description of the Chebyshev center behaviour \cite{SV, BI} in special Banach spaces. Note that here we present the result generalising Theorem 1 of the latter paper to nonlinear spaces. Moreover we show that the behaviour of the distance function $d(\mathrm{cheb}(A), \mathrm{cheb}(B))$, especially non-Lipschitz side of it, is controlled only by convexity properties of the unit ball of metric space.
\subsection{Notation} Here we mostly study sets of two spaces, namely $\Sigma_n(H)$, being the space of all $n$-nets of $H$ and $K(H)$ --- space of all compact convex subsets of $H$.
Also we denote by $\mathrm{cheb}(M)$ the set of Chebyshev centers of $M \subset H$ and by $\mathrm{diam}(M)$ diameter of the set $M$. Also $\mathrm{Hd}(V, W)$ equals Hausdorff distance between $V, W \subset H$.
Note also that we usually assume that space $H$ is such that any two points can be connected with unique geodesic line.
\subsection{Main results} In the paper we present inductive construction of the desired point. There are two approaches using 1) mean point (in some sense generalising barycenter) or 2) Chebyshev center of the set. At first we investigate properties of these mappings in non-linear spaces. The first of them is correctness of the definition of the mean point.
{\bf Statement 1.}{\it
There exists a common limit point $\mathrm{mp}(\sigma)$ for all sequences of points $(x^k_i)_{k=1}^{\infty}$, $i =1, \ldots, n$.}
Then we must consider continuity properties of this mapping.
Denote by $x_1, x_2, \ldots , x_n \in H$ elements of $\sigma \in \Sigma_n(N)$. Then the following holds true.
{\bf Statement 2.}{\it
The mapping $\mathrm{mp}: (\Sigma_n(H), \mathrm{Hd}) \to H$ is $1$-Lipshitz in $\min\{d(x_i, x_j)| i, j =1, \ldots, n, i\neq j \}/2$-neighbourhood of $\sigma$ .}
Moreover the following holds true:
{\bf Statement 7.}{\it
The mean point exists for any $V \in K(H)$.}
We note also that $\mp(V)$ coincides with usual barycenter of $V$ in case $H$ is Hadamard space.
Then we apply methods of \cite{IS, IS2} to prove the following statement:
{\bf Statement 9.}{\it
The Chebyshev selection map $\mathrm{cheb}: K(H) \to H$, $V \mapsto \mathrm{cheb}(V)$ is generalised H\"older with power constant $1/2$ \cite{BI, SV} in the Hadamard space $H$.}
{\bf Statement 10.}{\it
Let $H$ be strictly convex space of Busemann nonpositive curvature.
Then the upper bound for the power of the generalised H\"older map $\mathrm{cheb}: B(H) \to H$ is $1/2$.}
Finally, we construct Lipschitz selection points for the convex hulls of no more than $n$ points of $H$. Naturally, Lipschitz constants depends on the number of points, since, for instance, there can be no Lipschitz selection in this case even in Banach spaces \cite{PY, Lang}.
\section{Mean point.}
\subsection{Mean point of $n$-net.}
Let $H$ be a Busemann nonpositively curved space.
Let $\sigma \in \Sigma_n(H)$, $\sigma=\{x_1, \ldots, x_n\}$ such that $x_i\neq x_j$ for $i \neq j$.
Consider the inductive construction of the mean point of $\sigma$. For $n=2$ put $\mathrm{mp}(\{x_1, x_2\})=m(x_1, x_2)$. Assume now that we have correctly defined mapping $\mathrm{mp}: \Sigma_{n-1}(H) \to H$. Then for $\sigma=\{x_1, \ldots, x_n\}$ one may consider the set $\sigma^1=\{x^1_1, \ldots, x_n^1\}$, here $x_i^1= \mathrm{mp}\{x_1, \ldots, \widehat{x_i}, \ldots, x_n\}$ and $\widehat{x_i}$ means that this point is excluded from $\sigma$. Then repeat the procedure with $\sigma^1$ to get $\sigma^2$.
\begin{statement}
There exists a common limit point $\mathrm{mp}(\sigma)$ for all sequences of points $(x^k_i)_{k=1}^{\infty}$, $i =1, \ldots, n$.
\end{statement}
\begin{proof}
It suffices to prove that $d(x^k_i, x^k_j) \leq 1/2 d(x^{k-1}_i, x^{k-1}_j)$.
The proof is by induction on $n$ (the number of points).
The base of the induction is the first consistent case $n=3$. The statement follows from the definition of the nonpositive curvature.
Now assume that the statement holds true for all $l \leq n-1$ and prove it for $n$. Let us prove it for $\sigma$ and $\sigma^1$ because the rest will easily follow from it. Fix any pair of points $x_i, x_j \in \sigma$.
Recall the construction of $x^1_i, x^1_j$ and note that we start from the mean points of the sets $\{x_i, x_1, \ldots,\widehat{x_i}, \ldots, \widehat{x_i}, \ldots, \widehat{x_l}, x_n\}$ and $\{x_j, x_1, \ldots,\widehat{x_i}, \ldots, \widehat{x_j}, \ldots, \widehat{x_k}, x_n\}$. For each pair of points $\mathrm{mp}(\{x_i, x_1, \ldots,\widehat{x_i}, \ldots, \widehat{x_i}, \ldots, \widehat{x_l}, x_n\})$ and $\mathrm{mp}(\{x_j, x_1, \ldots,\widehat{x_i}, \ldots, \widehat{x_j}, \ldots, \widehat{x_k}, x_n\})$ from the described sets we have the desired inequality for the triple $x_i, x_j, x_k$ and since by induction hypothesis points of $\{x_i, x_1, \ldots,\widehat{x_i}, \ldots, \widehat{x_i}, \ldots, \widehat{x_l}, x_n\}$ and $\{x_j, x_1, \ldots,\widehat{x_i}, \ldots, \widehat{x_j}, \ldots, \widehat{x_k}, x_n\}$ converge to the limit points $x^1_i, x^1_j$ respectively, we get the inequality.
\end{proof}
\begin{statement}
The mapping $\mathrm{mp}: (\Sigma_n(H), \mathrm{Hd}) \to H$ is $1$-Lipshitz in $\min\limits_{x_i \neq x_j \in \sigma}{d(x_i, x_j)}/2$-neighbourhood of each $\sigma \in \Sigma_n(H)$.
\end{statement}
\begin{proof}
The proof is again by induction.
The base of the induction is $\Sigma_2(H)$ for which the inequality follows from the convexity of the distance function between two segments.
The rest of the proof repeats corresponding part of the proof of Statement 1.
\end{proof}
Let $\sigma \in \Sigma_n(H)$, $\sigma=\{x_1, \ldots, x_n\}$.
Recall now the definition of the barycenter $\mathrm{bc}(\sigma)$ as the point providing minimum to the function $f_{\sigma}(x)=\sum\limits_{i=1}^{n} d^2(x, x_i)$.
Since in Busemann space the relation $d^2(z, \gamma(t)) \leq (1-t) d^2(z, \gamma(0))+ t d^2(z, \gamma(1))- t(1-t)d^2(\gamma(0), \gamma(1))$ does not necessarily take place, we can not apply arguments of \cite{LPS}; nevertheless we get the following proposition:
\begin{statement}
The mean point $\mathrm{mp}(\sigma)$ minimizes function which is majorized by barycenter one for any $\sigma \in \Sigma_n(H)$ in Hadamard space $H$.
\end{statement}
\begin{proof}
The proof is again by induction on $n$. The base of the induction is the trivial case of $n=2$. Then $\mathrm{bc}(\{x_1, x_2\})=m(x_1, x_2)=\mathrm{mp}(\{x_1, x_2\})$.
It suffices to prove that $\forall x \in \mathrm{co}(\sigma) \setminus \sigma^1$, $f_{\sigma}(x) \geq \sum\limits_{i=1}^{n} d^2(x_i, \mathrm{co}(\sigma^1))$ since then the first approximation of the point $\mathrm{mp}(\sigma)$ is also the first approximation of the barycenter.
In order to prove it one must consider the auxiliary number $\frac{1}{n-1} \sum\limits_{i=1}^{n}\sum\limits_{j\neq i} d^2(x_i, x^1_j)$. Then convexity of both $\mathrm{co}(\sigma^1)$ and distance function implies that $\sum\limits_{i=1}^{n} d^2(x_i, \mathrm{co}(\sigma^1)) \leq \frac{1}{n-1} \sum\limits_{i=1}^{n}\sum\limits_{j\neq i} d^2(x_i, x^1_j)$. At the same time for any $x \in H$ we have the relation $\sum\limits_{i=1}^{n} d^2(x, x_i) \geq \frac{1}{n-1} \sum\limits_{i=1}^{n}\sum\limits_{j\neq i} d^2(x_i, x^1_j)$. Combination of these two inequalities completes the proof for the set $\sigma^1$.
Since the space $H$ is Hadamard the "parallelogramm inequality" $(n-1)d^2(x, x^1_i) \leq \sum\limits_{j=1, j \neq i}^{n}d^2(x, x_j)-\sum\limits_{j=1, j \neq i}^{n}d^2(x^1_i, x_j)$ holds true. Note than in Euclidean space it becomes an equality. Thus $(n-1)\sum\limits_{i=1}^{n}d^2(x, x^1_i)+\sum\limits_{i=1}^{n}\sum\limits_{j=1, j \neq i}^{n}d^2(x^1_i, x_j) \leq \sum\limits_{i=1}^{n}\sum\limits_{j=1, j \neq i}^{n}d^2(x, x_j)=(n-1) \sum\limits_{j=1}^{n}d^2(x, x_j)$. Here the second summond $\sum\limits_{i=1}^{n}\sum\limits_{j=1, j \neq i}^{n}d^2(x^1_i, x_j)$ is minimised in the first step of the proof and the first summond $\sum\limits_{i=1}^{n}d^2(x, x^1_i)$ can be analysed similarly. This completes the proof.
\end{proof}
Note that there is no such relation between mean point and barycenter of the set in the space of Busemann nonpositive curvature.
\begin{example}
If $H =(\mathbb{R}^n, \|\cdot\|_p)$ or $L^p(\mathbb{R})$ for $p\geq2$ then for any $\sigma \in \Sigma_k(H)$ the point $\mathrm{mp}(\sigma)$ minimises the function $f: H \to \mathbb{R}^+$, $f(x)=\sum\limits_{j=1}^{k} \|x - x_j\|_p^p$.
Let us first consider the simplest nontrivial case of $(\mathrm{R}^n, \|\cdot\|_p)$ and $\Sigma_3(\mathrm{R}^n)$. Then the statement follows from the inequality $1+t^p \leq (1+t)^p + (1-t)^p$ for $t \in (0,1)$.
Now consider the case of $\Sigma_{k+1}(\mathrm{R}^n)$. Then for arbitrary point $x \in H$
$$\sum\limits_{i=1}^{k} \|x-x_i\|^p=\sum\limits_{i=1}^{k} \|pr(x)-x_i\|^p+k \|x-pr(x)\|^p=
$$
$$
\sum\limits_{i=1}^{k} \|x^1_{k+1} + (pr(x)-x^1_{k+1})-x_i\|^p+k \|x-pr(x)\|^p \geq
$$
$$
\geq \sum\limits_{i=1}^{k} \|x^1_{k+1}-x_i\|^p + k \|x-pr(x)\|^p+
p \sum\limits_{i=1}^{k}\sum\limits_{j=1}^{n}|(x_i)_j|^{p-1} (pr(x)-x^1_{k+1})_j+
$$
$$
+\frac{p (p-1)}{2}\sum\limits_{i=1}^{k}\sum\limits_{j=1}^{n}|(x_i)_j|^{p-2} |(pr(x)-x^1_{k+1})_j|^2 + \ldots =
$$
$$
=\sum\limits_{i=1}^{k} \|x^1_{k+1}-x_i\|^p + k \|x-pr(x)\|^p + k\| pr(x)-x^1_{k+1} \|^p - k\| pr(x)-x^1_{k+1} \|^p +
$$
$$
+p \sum\limits_{i=1}^{k}\sum\limits_{j=1}^{n}|(x_i)_j|^{p-1} (pr(x)-x^1_{k+1})_j+ \frac{p (p-1)}{2}\sum\limits_{i=1}^{k}\sum\limits_{j=1}^{n}|(x_i)_j|^{p-2} |(pr(x)-x^1_{k+1})_j|^2 + \ldots.
$$
Hence it suffices to prove that $- k\| pr(x)-x^1_{k+1} \|^p + p \sum\limits_{i=1}^{k}\sum\limits_{j=1}^{n}|(x_i)_j|^{p-1} (pr(x)-x^1_{k+1})_j+ \frac{p (p-1)}{2}\sum\limits_{i=1}^{k}\sum\limits_{j=1}^{n}|(x_i)_j|^{p-2} |(pr(x)-x^1_{k+1})_j|^2 \geq 0$. Since $x^1_{k+1}$ is the minimum point of $\sum\limits_{i=1}^{k}\|x -x_i\|^p$ $\sum\limits_{i=1}^{k}\sum\limits_{j=1}^{n}|(x_i)_j|^{p-1} (pr(x)-x^1_{k+1})_j=0$ and since $l \in \mathrm{co}\{x_1, \ldots, n\}$ $\sum\limits_{j=1}^{n}|(x_i)_j|^{p-2}>\|l_i\|^{p-2}$ for any $i=1, \ldots n$. This completes the proof.
\end{example}
\subsection{Mean point construction for the space of nonpositive Busemann curvature.}
Let us define a mean point of the arbitrary set $V \subset H$. Consider the sequence $\sigma_n \in \Sigma_n(H)$, $\sigma_n \subset V$ as in the previous section. Then call the limit point of the sequence (in case such a point exists) $\mathrm{mp} (\sigma_n)$ the mean point of $V$.
\begin{statement}
The mean point exists for any compact infinite set $V =\{x_1, x_2, \ldots| x_i \in H \}$ such that $\sum\limits_{i=1}^{\infty}d(x_i, x_{i+1})$ converges.
\end{statement}
\begin{proof}
Construction of the mean point implies that for any $\sigma \subset V$ $d(\mathrm{mp}(\sigma), \mathrm{mp}(\sigma \bigcup\limits_{j=1}^{k}\{x_j\})) \leq \max\{d(x_j, \mathrm{co}(\sigma))| j= 1, \ldots k\}$.
First let us show that for $\sigma \in \Sigma_n(H)$ $\sigma=\{x_1, \ldots, x_n\}$ $d(\mathrm{mp}(\sigma), \mathrm{mp}(\sigma \bigcup\{x\})) \leq \max\limits_{i=1, \ldots, n}\{d(x, x_i)\}$. The proof is by induction on $n$. The base of the induction is as usual $n=1$, in which case the proof is trivial.
So assume that for $k \leq n-1$ the claim holds true and prove it for $n$. So we must estimate $d(\mathrm{mp}(\sigma), \mathrm{mp}(\sigma \bigcup\{x\}))$. Recall the construction of $\mathrm{mp}(\sigma)$. Then any $n$-subnet of $\sigma \bigcup\{x\}$ is either $\sigma$ or $\delta_{n-1} \bigcup \{x\}$, here $\delta_{n-1}$ is some $n-1$-subnet of $\sigma$. By induction hypothesis the distances $d(\mathrm{mp}(\delta_{n-1} \bigcup \{x\}), \mathrm{mp}(\delta_{n-1}))$ satisfy the desired relation. Hence the result follows from the convexity of the distance function between sets.
Now consider the general case of $\sigma$ and $\sigma\bigcup\limits_{j=1}^{k}\{x_j\}$.
\end{proof}
\begin{statement}
Let $\sigma \in \Sigma_{n-1}$, $\sigma=\{x_1, \ldots, x_{n-1}\}$ and $x \in H$ then $d(\mathrm{mp}(\sigma_n), \mathrm{mp}(\sigma \bigcup \{x\})) \leq \max\limits_{i=1, \ldots, n-1}\{d(x, x_i)\}/n$.
\end{statement}
\begin{proof}
The proof is again by induction on $n$. Assume that the statement holds true for $\Sigma_{n-2}$. Then recall the construction from the proof of Statement ??. Thus $d(\mathrm{mp}(\sigma), \mathrm{mp}(\sigma \bigcup \{x\})) \leq (\frac{1}{n-1}-\frac{1}{n-1}(\frac{1}{n-1})+\frac{1}{n-1} (\frac{1}{n-1}(\frac{1}{n-1}))+ \ldots ) \max\limits_{i=1, \ldots, n}\{d(x, x_i)\}=(\frac{1}{n-1}-\frac{1}{(n-1)^2}+\frac{1}{(n-1)^3}-\ldots ) \max\limits_{i=1, \ldots, n}\{d(x, x_i)\}=\frac{\max\limits_{i=1, \ldots, n}\{d(x, x_i)\}}{n}$.
\end{proof}
Similar considerations enable us to get
\begin{statement}
Let $\sigma \in \Sigma_n$, $\sigma=\{x_1, \ldots, x_n\}$ and $\sigma' \in \Sigma_k$, $\sigma'=\{y_1, \ldots, y_k\}$ then $d(\mathrm{mp}(\sigma_n), \mathrm{mp}(\sigma_n \bigcup \sigma')) \leq \max\limits_{i=1, \ldots, n, j=1, \ldots, k}\{d(x_i, y_j)\} \frac{k}{n+k}$.
\end{statement}
\begin{proof}
The proof is by induction on $k+n$. As in the previous statement we get an estimate $d(\mathrm{mp}(\sigma), \mathrm{mp}(\sigma \bigcup \sigma')) \leq (\frac{k}{n+k-1}-\frac{1}{n+k-1}(\frac{k}{n+k-1})+\frac{1}{n+k-1} (\frac{1}{n+k-1}(\frac{k}{n+k-1}))+ \ldots ) \max\limits_{i=1, \ldots, n}\{d(x, x_i)\}=(\frac{k}{n+k-1}-\frac{k}{(n+k-1)^2}+\frac{k}{(n+k-1)^3}-\ldots ) \max\limits_{i=1, \ldots , n}\{d(x, x_i)\}=\max\limits_{i=1, \ldots, n, j=1, \ldots, k}\{d( x_i, y_j)\}\frac{k }{n+k}$.
\end{proof}
Assume now that we vary masses of the points so that $\sigma= \bigcup\limits_{i=1}^{n} \{x_i, m_i\}$ and $\sigma'=\bigcup\limits_{i=1}^{n} \{x_i, m^{'}_i\}$, here $m_i, m^{'}_i \geq 0$, $i= 1, \ldots , n$.
\begin{corollary}
$d(\mathrm{mp}(\sigma), \mathrm{mp}(\sigma')) \leq \mathrm{diam}(\sigma)\sum\limits_{i=1}^{n}|m_i -m^{'}_{i}|$.
\end{corollary}
\begin{proof}
Assume first that $m_i, m^{'}_i \in \mathbb{Q}^+$. Hence $m_i=\frac{p_i}{q_i}$ and $m^{'}_i=\frac{p^{'}_i}{q^{'}_i}$. Consider $q=\mathrm{LCM}\{q_i, q^{'}_i | i=1, \ldots, n\}$.
Now we must represent $x_i \in \sigma$ and $x^{'}_{i} \in \sigma'$ as the union of $p_i q/q_i$ and $p^{'}_i q/q^{'}_i$ of points of the same mass $1/q$, respectively.
Then we apply statement 6 to get the result.
The general case easily follows.
\end{proof}
\begin{statement}
The mean point exists for any $V \in K(H)$.
\end{statement}
\begin{proof}
Fix $\varepsilon>0$.
Consider an $\varepsilon$-net $\sigma_1$ of $V$ such that the measure of the set $\sigma_1^{'}=\{x \in V| \exists i, j, i \neq j, d(x, x_i) <\varepsilon, d(x, x_j) <\varepsilon\} < \delta \mu(V)$. Then for any other uniformly distributed $\varepsilon$-net $\sigma_2$ the set of points $\sigma_2^{'}$ for which we have no one-to-one correspondence between $\sigma_1$ and $\sigma_2$ consists of no more than $\delta |\sigma_2|$ points if cardinalities of $\sigma_1$ and $\sigma_2$ coincide.
Thus statement 6 and its corollary imply that $d(\mathrm{mp}(\sigma_1), \mathrm{mp}(\sigma_2)) \leq d(\mathrm{mp}(\sigma_1), \mathrm{mp}(\sigma_1 \setminus \sigma_1^{'}))+d(\mathrm{mp}(\sigma_1\setminus \sigma_1^{'}), \mathrm{mp}(\sigma_2\setminus \sigma_2^{'}))+ d(\mathrm{mp}(\sigma_2), \mathrm{mp}(\sigma_1 \setminus \sigma_2^{'})) < 2 \delta + 2 \varepsilon$. This completes the proof.
\end{proof}
Evidently we arrive to the statement analogous to one of \cite{LPS}.
\begin{corollary}
Let $\mu_1$ and $\mu_2$ be two probability measures with supports $V_1$ and $V_2$, absolutely continuous with respect to the Hausdorff measure $\mu_H$ with Radon-Nikodym derivatives $\theta_1$ and $\theta_2$. Then $d(\mathrm{mp}(V_1), \mathrm{mp}(V_2)) \leq \int\limits_{V_1 \bigcup V_2} |\theta_1 -\theta_2| d\mu_H$.
\end{corollary}
\begin{proof}
It suffices to prove the statement for the finite sets $\sigma_1$ and $\sigma_2$ consisting of points of equal mass. Consider $\sigma=\sigma_1 \bigcap \sigma_2$. Then triangle inequality combined with statement 7 implies that
$d(\mathrm{mp}(\sigma_1), \mathrm{mp}(\sigma_2)) \leq d(\mathrm{mp}(\sigma_1), \mathrm{mp}(\sigma))+ d(\mathrm{mp}(\sigma_2), \mathrm{mp}(\sigma)) \leq \mathrm{diam}(\sigma_1 \bigcup \sigma_2) (\frac{|\sigma_1|-|\sigma|}{|\sigma_1|}+ \frac{|\sigma_2|-|\sigma|}{|\sigma_2|})=\mathrm{diam}(V_1 \bigcup V_2)(\int\limits_{\sigma_1 \setminus \sigma_2} |\theta_1 -\theta_2| d \mu_H+ \int\limits_{\sigma_2 \setminus \sigma_1} |\theta_1 -\theta_2| d \mu_H) \leq \mathrm{diam}(V_1 \bigcup V_2)\int\limits_{\sigma_1 \bigcup \sigma_2} |\theta_1 -\theta_2| d \mu_H$. This completes the proof.
\end{proof}
Moreover as an easy corollary from statements 3 and 7 we get the proposition
\begin{statement}
The mean point $\mathrm{mp}(V)$ minimizes function which is majorized by barycenter one for any compact $V$ in Hadamard space $H$.
\end{statement}
\section{Behaviour of the Chebyshev center}
\subsection{Reduction to disjoint set of points}
Let us consider properties of the mapping $\mathrm{cheb}: K(H) \to H$.
\begin{statement}
In any finite-dimensional strictly convex metric space $H$ Chebyshev center and radius of the set $V$ can be determined by finite number of points from $V$.
\end{statement}
\begin{proof}
Let us first consider $n=\mathrm{dim} H$ points $x_1, \ldots, x_{n}$ in the intersection $I=S_{r_c(V)} (\mathrm{ch}(V)) \bigcap V$, here $r_c, \mathrm{ch}(V)$ are Chebyshev radius and center of the set $V$. Since $H$ is strictly convex, the set $T=S_{r_c(V)}(x_1) \bigcap \ldots \bigcap S_{r_c(V)}(x_{n})$ is discrete and finite.
Let us denote elements of $T$ by $y_i$, $i=1, \ldots, k$. Then for any $y_i \in T$, $y_i \neq \mathrm{ch}(V)$ there exists a point $z_i \in V$ such that $z_i \not\in B_{r_c(V)}(y_i)$. This holds true since Chebyshev center is unique for any subset of convex metric space. Let us add $z_i$ to the set $I$ and repeat the procedure for any other point of $T$. Thus there exist no more than $2 n -1$ points of $V$ that determine its Chebyshev radius and center.
\end{proof}
\begin{statement}
In any strictly convex infinite-dimensional metric space $H$ both Chebyshev center $\mathrm{cheb}(V)$ (in case such a point exists) and radius $r(V)$ of the set $V$ can be determined by the number of points from $V$ of cardinality less than equal to dimension of $H$.
\end{statement}
\begin{proof}
Consider $V \subset H$.
Since by assumption there exists $\mathrm{cheb}(V)$, we can consider also the ball $B_{\mathrm{cheb}(V)}(r(V))$. Now we must construct $r(V)$-net $V_{r(V)}$ of the set $S_{\mathrm{cheb}(V)}(r(V)) \bigcap V$ and apply statement from \cite{GK}.
Now this net may not be sufficient to determine Chebyshev center of $V$. Then the intersection $I=\bigcap\limits_{x \in V_{r(V)}} B_{x}(r(V))$ contains more than one point. Consider the projection $\pi_i(I)$ of $I$ onto some coordinate line of $H$. Suppose $\pi_i(I)$ consists of more than one point. Then considerations similar to that of the previous statement provide us with the sequence of points $(x^i_k)_{k=1}^{\infty}$ of $V$, such that $\pi_i(I \bigcap B_{x_k^i}(r(V))) \to \pi_i(\mathrm{cheb}(V))$, $k \to \infty$. Since $V$ is compact this sequence possess a converging subsequence $(x^i_{k_l})_{l=1}^{\infty}$, $x^i_{k_l} \to x^i$, $l \to \infty$. Now we simply add this limit point to the set $V_{r(V)}$. The same procedure must be applied to each coordinate line of $H$.
The cardinality of the set of added points is clearly less than equal to the dimension of $H$. This completes the proof.
\end{proof}
\subsection{Behaviour of Chebyshev center}
\begin{statement}
The Chebyshev selection map $\mathrm{cheb}: K(H) \to H$, $V \mapsto \mathrm{cheb}(V)$ is generalised H\"older with power constant $1/2$ \cite{BI, SV} in the Hadamard space $H$.
\end{statement}
\begin{proof}
The proof is done using ideas of \cite{IS, IS2}.
First recall from statement 9 that for any set $V \in K(H)$ its Chebyshev center is uniquely determined by the finite set of points $\sigma(V) \subset \Sigma_{\dim(H)+1}$.
Let us show now that any variation of some point of $\sigma(V)$ can be represented as combination of the deformations along edges of $co(\sigma(V))$.
The next step is to prove that supremum of the relation $d(\mathrm{cheb}(\sigma), \mathrm{cheb}(\sigma'))/\mathrm{Hd}(\sigma, \sigma')$ is achieved for $\sigma$ and $\sigma'$ being $3$-nets described as follows: $\sigma=\{x, y, z\}$, $\sigma'=\{x, y, z'\}$, here $z \in [x, z']$, $x, y, z \in S_{d(x, y)/2}(m(x, y))$, $x, y, z' \in S_{d(x, z')/2}(m(x, z'))$.
In order to prove this one must consider first the modulus of convexity of the space of non-positive curvature. It is known \cite{Khamsi} that it is a quadratic function. Thus limit of the relation $d(\mathrm{cheb}(\sigma), \mathrm{cheb}(\sigma'))/\mathrm{Hd}(\sigma, \sigma')$ equals $\infty$ for $z \to y$.
Let us show now that the relation $d(\mathrm{cheb}(\{x, y, z_1\}), \mathrm{cheb}(\{x, y, z_2\}))/\mathrm{Hd}(\{x, y, z_1\}, \{x, y, z_2\})$ is bounded from above.
Consider $x, y \in H$ and a geodesic ray $\gamma$ starting at $x$. It suffices to show that there exists a constant $L(\gamma, x, y) \in \mathbb{R}^+$ such that for any two points $z, z'$ between $z_1$ and $z_2$. $d(\mathrm{cheb}(\{x, y, z\}), \mathrm{cheb}(\{x, y, z'\})) \leq L d(z, z')$. The only problem here is to determine the behaviour of the function $f(z, z')=d(\mathrm{cheb}(\{x, y, z\}), \mathrm{cheb}(\{x, y, z'\}))/d(z, z')$ for $d(z, z') \to 0$. Assume that for some sequences $z_n$ and $z{'}_n$ $f(z_n, z^{'}_n) \to \infty$. Then there exists a limit point $\gamma(t_0)$ on the geodesic ray $\gamma$.
Note now that since Chebyshev radius is a Lipschitz function it suffices to estimate $Hd(S_{\mathrm{cheb}(\{x, y, z\})}(r), S_{\mathrm{cheb}(\{x, y, z'\})}(r'))$. Thus $Hd(S_{\mathrm{cheb}(\{x, y, z\})}(r), S_{\mathrm{cheb}(\{x, y, z'\})}(r'))/ d(z, z') \to \infty$. Hence the chord $[y, z]$ of the sphere $S_{\mathrm{cheb}(\{x, y, z\})}(r)$ is tangent to the it. Hence there exists an infinite number of geodesic lines tangent to this chord at point $z$. This is the contradiction with convexity of $H$.
These two facts combined prove that there exist $w, w' \in H$ sufficiently close to $y$ for any $(z_1, z_2) \in [z, z'] \times [z, z']$ such that both of the following claims hold true
1) $w \in [x, w']$, $x, y, w \in S_{d(x, y)/2}(m(x, y))$, $x, y, w' \in S_{d(x, w')/2}(m(x, w'))$;
2) $d(\mathrm{cheb}(\{x, y, z_1\}), \mathrm{cheb}(\{x, y, z_2\}))/\mathrm{Hd}(\{x, y, z_1\}, \{x, y, z_2\}) \leq$\\ $\leq d(\mathrm{cheb}(\{x, y, w\}), \mathrm{cheb}(\{x, y, w'\}))/\mathrm{Hd}(\{x, y, w\}, \{x, y, w'\})$.
The last estimate then can be derived from consideration of \cite{Khamsi} on convexity modulus of non-positively curved spaces.
The general case of $n$-nets can be analysed similarly.
\end{proof}
\begin{example}
The behaviour of the Chebyshev center for the spaces of Busemann nonpositive curvature is not as easily described as in the case of Hadamard manifolds and surely does not possess the same nice properties.
1. $H=(\mathbb{R}^2, d =\|\cdot\|_p)$, $p>2$. Then for the point $(1, 0)$ of the unit sphere $S_0(1)$ we get the convexity modulus equal to $\varepsilon^p$. Thus the mapping $\mathrm{cheb}: (\Sigma_3(\mathbb{R}^2), \mathrm{Hd}) \to (\mathbb{R}^2, \|\cdot\|_p)$ is H\"older in the neighbourhood of $\sigma \in \Sigma_2$, $\sigma=\{(-1, 0), (1, 0)\}$ with the power coefficient equal to $1/p$.
2. $H=(\bigotimes\limits_{i=1}^{\infty}\mathbb{R}^2, \|\cdot\|)$, here $\|x\|=(\sum\limits_{i=1}^{\infty}(|x_{2i}|^{i+2}+|x_{2i -1}|^{i+2})^{\frac{2}{i+2}})^2$. Note first that this space is uniform convex in any direction, so Chebyshev center is unique \cite{Gar}. Then the convexity of this space in any direction is not bounded from below by $\varepsilon^p$ for any $p \in \mathbb{N}$. Thus Chebyshev selection map is not even H\"older one.
\end{example}
The only thing we can be sure at is that the behaviour of the Chebyshev center is the best in $CAT(0)$ of all convex metric spaces.
\begin{statement}
Let $H$ be strictly convex space of Busemann nonpositive curvature.
Then the upper bound for the power of the generalised H\"older map $\mathrm{cheb}: B(H) \to H$ is $1/2$.
\end{statement}
\begin{proof}
Consider the construction involving $\sigma=\{x, y, z\}$ and $\sigma'=\{x, y, z'\}$ from the previous statement.
This is the direct consequence of the convexity properties of spheres from $H$.
The only obstacle is estimation of the distance not between points on the spheres but the centers of them. Assume that $d(m(x, z'), m(x, y)) = o(d(z, z'))$. Then there exist two geodesic lines passing through $x$ in the same direction. This contradicts convexity of $H$. Hence for any $y \in S_{1}(x)$ there exists $L_x>0$ such that $d(m(x, z'), m(x, y))/ d(z, z') \geq L_x$.
\end{proof}
Note that uniform convexsity of the space is crucial for H\"older behaviour of the Chebyshev center.
\begin{example}
Let us describe behaviour of Chebyshev center for $(\mathbb{R}^n, \|\cdot \|_1)$.
First note that to prove that the mapping $\mathrm{cheb}$ is Lipschitz it suffices to find upper estimate for the distance between angle points of the hyperplane subset $\mathrm{cheb}\{x, y\}$ for arbitrary pair $x, y \in \mathbb{R}^n$. In order to do this consider a pair of points $x=(x_1, \ldots, x_n)$ and $y=(y_1, \ldots, y_n)$. Let us denote by $t=(t_1, \ldots, t_n)$ points of the set $\mathrm{cheb}(\{x, y\})$. Then the angle points of $\mathrm{cheb}(\{x, y\})$ have coordinates $t_1, \ldots, t_i, \ldots, t_n$, here $t_j$ equals either $t_{j, m}=\min \{x_j, y_j\}$ or $t_{j, M}=\max \{x_j, y_j\}$ and $|t_i-x_i|=\frac{\sum\limits_{j=1}^{n}|x_j -y_j|-2\sum\limits_{j\neq i}|t_j -t_{j, m}|}{2}$. Consider $I=\{j\in\{1, \ldots, n\}| t_j =t_{j, m}\}$. Assume without loss of generality that $x_i \leq y_i$. Now let $x'$ and $y'$ of $\mathbb{R}^n$ be such that both $\|x-x'\|_1$ and $\|y-y'\|_1$ are less or equal to $\varepsilon>0$.
Then
$$
\|t-t'\|_1=\sum\limits_{j=1}^{n}|t_j - t_j^{'}|=\sum\limits_{j\neq i}|t_j-t_j^{'}|+|t_i -t_i^{'}|
\leq
2\varepsilon+
$$
$$
+
\frac{|\sum\limits_{j=1}^{n}|x_j -y_j|-2\sum\limits_{j\neq i}|t_j -t_{j, m}|-\sum\limits_{j=1}^{n}|x^{'}_j -y^{'}_j|-2\sum\limits_{j\neq i}|t^{'}_j -t^{'}_{j, m}|+2(x_i -x_i^{'})|}{2}
\leq
$$
$$
\leq
2 \varepsilon + \frac{|\sum\limits_{j\not\in I}|x_j -y_j| -\sum\limits_{j \in I}|x_j -y_j|-\sum\limits_{j\not\in I}|x^{'}_j -y^{'}_j|+ \sum\limits_{j\in I}|x^{'}_j -y^{'}_j|+2(x_i -x_i^{'})|}{2}
\leq
4\varepsilon.
$$
Hence since Chebyshev center of the subset of $(\mathbb{R}^n, \|\cdot \|_1)$ is determined by at most $n$ pairs of points, $L \leq 4n$.
Next let us show that there exists also a lower estimate for Lipschitz constant of $\mathrm{cheb}$. In order to show this consider Chebyshev center of the sets
$$
\sigma_1=\{(0, \ldots, 0), (1, \ldots, 1), (0, 1, \ldots, 1), (1, 0, \ldots, 0), \ldots, (1, \ldots, 1, 0, 1), (0, \ldots, 0, 1, 0)\}
$$
and
$$
\sigma_2=\{(\varepsilon/n, \ldots, \varepsilon/n), (1+\varepsilon/n, \ldots, 1+\varepsilon/n), (-\varepsilon/n, 1+\varepsilon/n, \ldots, 1+\varepsilon/n),
$$
$$
(1-\varepsilon/n, \varepsilon/n, \ldots, \varepsilon/n),
\ldots,
$$
$$
(1+\varepsilon/n, \ldots, 1+\varepsilon/n, -\varepsilon/n , 1+\varepsilon/n), (\varepsilon/n, \ldots, \varepsilon/n, 1-\varepsilon/n, \varepsilon/n)\}.
$$
Then $d(\mathrm{cheb}(\sigma_1), \mathrm{cheb}(\sigma_2)) \geq (n-1) \varepsilon$. Thus $L \geq n-1$.
\end{example}
The second similar example ---$l^{\infty}$-space --- was presented in \cite{IS2}.
\section{Lipschitz selection for $\Sigma_n(H)$.}
In this section we construct Lipschitz selections for the convex hulls of no more than $n$ points of metric space $H$.
\subsection{Construction based on the mapping $\mathrm{mp}$}
Let us now construct the mapping $c$ from the set of all $n$-nets $\Sigma_n \subset \mathbb{R}^n$ to $\mathbb{R}^n$, such that $c$ is Lipschitz with constant $4$. This means that $d(c(\sigma), c(\sigma'))\leq \alpha(\sigma, \sigma')$, here $\alpha$ is Hausdorff metric.
Let us first consider this problem locally.
The construction is inductive.
It is clear that $c(\sigma)=x$ for any $\sigma =\{x\} \in \Sigma_1$. This map obviously is Lipschitz.
To make our construction correct we must assume also that
\begin{equation}
\forall \sigma, \sigma' \in \Sigma_n, \sigma \setminus\{x_n\}=\sigma'\setminus\{x'_n\},\;
d(x_n, x'_n) \leq \varepsilon \Rightarrow d(c(\sigma), c(\sigma')) \leq \varepsilon/2.
\end{equation}
This obviously holds true for $\Sigma_2$ and $c(\{x_1, x_2\})=m(x_1, x_2)$.
Assume that the construction is valid for any $\Sigma_l$, here $1\leq l\leq n-1$.
Consider $\sigma \in \Sigma_n$.
We put $c(\sigma)=b(\sigma)$ if for any point $x_i \in \sigma$, $d(x_i, co(\sigma\setminus\{x_i\}))\geq \mathrm{diam}(\sigma)/2$.
Assume now that there exist points $x_1, \ldots, x_k$, $k\leq n$, such that the distances $d(x_i, co(\sigma\setminus\{x_i\}))$ are less than $\mathrm{diam}(\sigma)/2$.
Let us construct the set $C$ consisting of points $c_i=c(\sigma \setminus x_i)$, $i=\overline{1, k}$ and
consider the barycenter $b(C)$.
Consider now the segment $[b(\sigma), b(C)]$.
We define $c(\sigma) \in [b(\sigma), b(C)]$ to be the point which divides $[b(\sigma), b(C)]$ in relation $d(b(C), c) : d(b(\sigma), b(C)) = \mathrm{min}\{d(x_i, co(\sigma\setminus \{x_i\}))\}_{i=\overline{1,k}}/\mathrm{diam}(\sigma) : 1$.
Now we must verify condition (1) for the set $\Sigma_n$.
By assumption points $c_i=c(\sigma \setminus \{x_i\})$ depend on $x_i$ in Lipschitz way with the constant $L_{n-1}$.
The mapping $c$ is continuous by construction.
The only thing we must show is that its Lipschitz constant exists. Let us estimate this constant as one of the combination of the mappings, namely, shifts of the points of $C$ and path along the segment itself.
Then the constant $L_n=L_{n-1}$(shift of the point $m(C)$) $+1$ (path along the segment; it does not depend on the number of shifted point but only on the minimal shift) $ + 1$ (shift of the point $m(\sigma)$). Thus $L_n=2+L_{n-1}$, hence the constant exists.
Note now that Hausdorff metric locally coincides with Fedorchuk one \cite{Fed}. Recall also that the latter metric is inner, thus the estimate $L_n$ for Lipschitz constant of $c: C(X) \to X$ is globally true for the space of $n$-nets endowed with Fedorchuk metric.
\subsection{Selection based on the mapping $\mathrm{cheb}$}
Now assume as in the previous section that there exists a Lipshitz selection $l: \Sigma_{n-1}(H) \to H$ with Lipschitz constant $L_{n-1}$. Let us expand this mapping to $\Sigma_n(H)$. Consider Chebyshev center $\mathrm{cheb}(\sigma)$ of the set $\sigma \in \Sigma_n(H)$. It is known that the selection $\mathrm{cheb}:\Sigma_n(H) \to H$ is generalised H\"older one with H\"older constant $H_n \mathrm{diam}^{1/2}(\sigma)$ and power coefficient $1/2$. Thus as in the first section we consider the set $\{\sigma_1, \ldots, \sigma_n\}$ of $n-1$ subnets of $\sigma$. Then there exists a set $\sigma_l=\{l(\sigma_1), \ldots, l(\sigma_n)\}$. Consider barycenter $m(\sigma_l)$ and connect it with Chebyshev center of $\sigma$ by segment. We define $l(\sigma) \in [m(\sigma_l), \mathrm{cheb}(\sigma)]$ as the point which divides $[m(\sigma_l), \mathrm{cheb}(\sigma)]$ in relation $d(l(\sigma), m(\sigma_l)) : d(\mathrm{cheb}(\sigma), l(\sigma)) = \mathrm{min}\{d^{1/2}(x_i, co(\sigma\setminus \{x_i\}))\}_{i=\overline{1,k}}/(\mathrm{diam}(\sigma) \max\{1, H_n\}): 1 - \mathrm{min}\{d^{1/2}(x_i, co(\sigma\setminus \{x_i\}))\}_{i=\overline{1,k}}/\mathrm{diam}(\sigma) \max\{1, H_n\})$. This construction provides us with the desired selection $l: \Sigma_n(H) \to H$ with Lipschitz constant equal to $1+L_{n-1}+3/2 L_{n-1}$.
It seems that the last construction can be easily extended to the space $K(H)$. That is, one may take into consideration $n$-nets defining Chebyshev center of the set $V \subset K(H)$ together with all their possible $n-1$-subnets. Nevertheless, there exists a natural obstruction, namely, a decent measure on the so-called perimeter or boundary of the set $V$. The author still does not know the solution of this problem.
\subsection{Construction of the Lipschitz selection for the subsets of convex hulls of finite sets}
The most natural generalisation of the construction given in previous paragraph is as follows: | {"config": "arxiv", "file": "math0702361.tex"} |
TITLE: Elementary proof of $\gcd(a, b) = 1 \wedge a\ |\ b\ c \Rightarrow a \ | \ c $
QUESTION [1 upvotes]: How does on prove $\gcd(a, b) = 1 \wedge a\ |\ b\ c \Rightarrow a \ | \ c $ with as elementary steps as possible (i.e. not using the fundamental theorem of arithmetic (unique prime factorization))?
EDIT: I saw that this theorem is called Gauss Theorem and is proved formally for integers $\mathbb Z$ in Coq, https://coq.inria.fr/library/Coq.ZArith.Znumtheory.html#Gauss
EDIT: Clarification: I forgot to tell that I want to prove this for the natural numbers $\mathbb N \geq 0$. Is Bezout's lemma applicable for the natural numbers, or is some other method needed?
REPLY [0 votes]: I am not sure if Bezout's theorem is allowed or not.
If $\gcd(a,b)=1$, then by Bezouts's theorem there exist $m,n \in \mathbb{Z}$ such that $ma+nb=1$. It follows that $mac+nbc=c$. Therefore since $a$ divides $mac$ and $nbc$ it must divide $c=mac+nbc$. | {"set_name": "stack_exchange", "score": 1, "question_id": 1538789} |
TITLE: Sufficient condition for symmetric matrices to be an orthogonal projection matrix
QUESTION [0 upvotes]: I am trying to prove a version of Cochran's theorem(Theorem 5.14) given here
The problem:-
Let $A_1, A_2, \cdots , A_m$ be $n\times n$ symmetric matrices and $A = \sum_i A_i $ with $\text{rank}(A_j)$$=n_j$. Then,
If
$A_jA_k=0,\forall j\ne k$
$\sum n_j= \text{rank}(A)$ ( should $\text{rank}(A)$ be further constrained to be equal to $n$?? )
Then,
$A_j$ is an orthogonal projection for all $j$.
What i have tried:-
$$\text{Since, each } A_i\text{ is symmetric }\implies N(A_i) = (C(A_i))^{\perp} \tag{a}$$
$$\because A_jA_k=0,\forall j\ne k\implies C(A_i)\cap C(A_j)=\{0\} \tag{b}$$
$$\because (A = \sum A_i)\text{ and (b)} \implies Ax = A_i x, \forall x\in C(A_i) \tag{c}$$
$$\because A_jA_k=0,\forall j\ne k\implies A^2 = \sum A_i^2\text{ and } A^2x = A_i^2 x, \forall x\in C(A_i) \tag{d}$$
$$\because (A = \sum A_i) \implies C(A) \subset C(A_1)+C(A_2)+\cdots\cdots+C(A_m) \tag{e}$$
$$\because \sum n_j= \text{rank}(A) \implies \dim C(A) = \dim C(A_1) + \dim C(A_2)+\cdots\cdots+ \dim C(A_m) \tag{f}$$
$$\because \sum n_j= \text{rank}(A) \text{ and (e) and (f) } \implies C(A_i)\cap\big(\bigcup_{j\ne i}C(A_j)\big)={0} \tag{g}$$
$(e)+(f)+(g) \implies C(A) = C(A_1)\oplus C(A_2)\oplus \cdots \cdots \oplus C(A_m) \tag{h}$
$C(A) = $column space of $A$; $N(A) = $null space of $A$
REPLY [2 votes]: For $n=4$ let $$A_1=\begin{pmatrix} 2 & 1& 0& 0\\
1 & 1 & 0 & 0\\
0 & 0& 0 & 0\\
0 & 0& 0 & 0
\end{pmatrix} \quad A_2=\begin{pmatrix}0 & 0& 0 & 0\\
0 & 0& 0 & 0 \\ 0 & 0 & 2 & 1\\
0 & 0 & 1 & 1
\end{pmatrix}$$
and $A=A_1+A_2.$
Then $A_1A_2=A_2A_1=0,$ $\operatorname{rank}(A_1)=\operatorname{rank}(A_2)=2$ and $\operatorname{rank}(A)=4.$
None of the matrices is a projection or a multiple of a projection. | {"set_name": "stack_exchange", "score": 0, "question_id": 4558528} |
\begin{document}
\title[Coactions of Hopf $C^*$-algebras on Cuntz-Pimsner algebras]{Coactions of Hopf $C^*$-algebras on Cuntz-Pimsner algebras}
\author{Dong-woon Kim}
\subjclass[2010]{46L08, 46L55, 47L65}
\keywords{$C^*$-correspondence, Cuntz-Pimsner algebra, multiplier correspondence, Hopf $C^*$-algebra, coaction, reduced crossed product}
\address{Dong-woon Kim: Department of Mathematical Sciences and Research Institute of Mathematics,
Seoul National University, Seoul 151-747, Korea}\email{dwkim0962@gmail.com}
\begin{abstract}
Unifying two notions of an action and coaction of a locally compact group on a $C^*$-cor\-re\-spond\-ence we introduce a coaction $(\sigma,\delta)$ of a Hopf $C^*$-algebra $S$ on a $C^*$-cor\-re\-spond\-ence $(X,A)$. We show that this coaction naturally induces a coaction $\zeta$ of $S$ on the associated Cuntz-Pimsner algebra $\mathcal{O}_X$ under the weak $\delta$-invariancy for the ideal $J_X$. When the Hopf $C^*$-algebra $S$ is defined by a well-behaved multiplicative unitary, we construct a $C^*$-cor\-re\-spond\-ence $(X\rtimes_\sigma\widehat{S},A\rtimes_\delta\widehat{S})$ from $(\sigma,\delta)$ and show that it has a representation on the reduced crossed product $\mathcal{O}_X\rtimes_\zeta\widehat{S}$ by the induced coaction $\zeta$. This representation is used to prove an isomorphism between the $C^*$-algebra $\mathcal{O}_X\rtimes_\zeta\widehat{S}$ and the Cuntz-Pimsner algebra $\mathcal{O}_{X\rtimes_\sigma\widehat{S}}$ under the covariance assumption which is guaranteed in particular if the ideal $J_{X\rtimes_\sigma\widehat{S}}$ of $A\rtimes_\delta\widehat{S}$ is generated by the canonical image of $J_X$ in $M(A\rtimes_\delta\widehat{S})$ or the left action on $X$ by $A$ is injective. Under this covariance assumption, our results extend the isomorphism result known for actions of amenable groups to arbitrary locally compact groups. The Cuntz-Pimsner covariance condition which was assumed for the same isomorphism result concerning group coactions is shown to be redundant.
\end{abstract}
\maketitle
\section{Introduction}
In this paper, we introduce coactions of Hopf $C^*$-algebras on $C^*$-cor\-re\-spond\-ences, and study the \emph{induced} coactions on the associated Cuntz-Pimsner algebras and their crossed products.
A $C^*$-cor\-re\-spond\-ence $(X,A)$ is a (right) Hilbert $A$-module $X$ equipped with a left action $\varphi_A:A\rightarrow\mathcal{L}(X)$. For $(X,A)$ with injective $\varphi_A$, a $C^*$-algebra $\mathcal{O}_X$ was constructed in \cite{Pim} generalizing crossed product by $\mathbb{Z}$ and Cuntz-Krieger algebra \cite{CK}. The construction was extended in \cite{Kat} to arbitrary $C^*$-cor\-re\-spond\-ences $(X,A)$ by considering an ideal $J_X$ of $A$ --- the largest ideal that is mapped injectively into $\mathcal{K}(X)$ by $\varphi_A$ --- and requiring that a covariance-like relation should hold on $J_X$. The $C^*$-algebra $\mathcal{O}_X$, called the Cuntz-Pimsner algebra associated to $(X,A)$, is generated by $k_X(X)$ and $k_A(A)$ for the universal covariant representation $(k_X,k_A)$. The class of Cuntz-Pimsner algebras is known to be large enough and include in particular graph $C^*$-algebras. In addition, there have been significant results concerning Cuntz-Pimsner algebras such as gauge invariant uniqueness theorem, criteria on nuclearity or exactness, six-term exact sequence, and description of ideal structure (\cite{Pim,Kat,Kat2}). Thus the Cuntz-Pimsner algebras can be viewed as a well-understood class of $C^*$-algebras, and in view of this, it would be advantageous to know that a given $C^*$-algebra is a Cuntz-Pimsner algebra.
Our work was inspired by \cite{HaoNg} and \cite{KQRo2} in which group actions and coactions on $C^*$-cor\-re\-spond\-ences are shown to induce actions and coactions on the associated Cuntz-Pimsner algebras, and the crossed products by the induced actions or coactions are proved to be realized as Cuntz-Pimsner algebras. (We refer to \cite{EKQR} for the definition of actions and coactions of locally compact groups on $C^*$-cor\-re\-spond\-ences.) More precisely, if $(\gamma,\alpha)$ is an action of a locally compact group $G$ on a $C^*$-cor\-re\-spond\-ence $(X,A)$, one can form two constructions: an action $\beta$ of $G$ on $\mathcal{O}_X$ induced by $(\gamma,\alpha)$ \cite[Lemma~2.6.(b)]{HaoNg} on the one hand, and the crossed product correspondence $(X\rtimes_{\gamma,r}G,A\rtimes_{\alpha,r}G)$ of $(X,A)$ by $(\gamma,\alpha)$ (\cite[Proposition~3.2]{EKQR} or \cite{HaoNg}) on the other. It was shown in \cite{HaoNg} that if $G$ is amenable, then the crossed product by the action $\beta$ is isomorphic to the Cuntz-Pimsner algebra associated to $(X\rtimes_{\gamma}G,A\rtimes_{\alpha}G)$:
\begin{equation}\label{Intro.1}
\mathcal{O}_X\rtimes_\beta G\cong\mathcal{O}_{X\rtimes_\gamma G}.
\end{equation}
Similarly, it was shown in \cite{KQRo2} that a nondegenerate coaction $(\sigma,\delta)$ of a locally compact group $G$ on $(X,A)$ satisfying an invariance condition induces a coaction $\zeta$ of $G$ on $\mathcal{O}_X$ \cite[Proposition~3.1]{KQRo2}, and under the hypothesis of Cuntz-Pimsner covariance, the crossed product by $\zeta$ is again a Cuntz-Pimsner algebra \cite[Theorem~4.4]{KQRo2}:
\begin{equation}\label{Intro.2}
\mathcal{O}_X\rtimes_\zeta G\cong\mathcal{O}_{X\rtimes_\sigma G},
\end{equation}
where $X\rtimes_\sigma G$ is the $C^*$-cor\-re\-spond\-ence over $A\rtimes_\delta G$ arising from the coaction $(\sigma,\delta)$ \cite[Proposition 3.9]{EKQR}.
The study in \cite{BS} proposed the framework of reduced Hopf $C^*$-algebras arising from multiplicative unitaries including both Kac algebras \cite{ES} and compact quantum groups \cite{Wo1,Wo2} (of course locally compact groups as well). The study also established the reduced crossed products of $C^*$-algebras by reduced Hopf $C^*$-algebra coactions, which are shown to be a natural generalization of crossed products by group actions and coactions. To each multiplicative unitary $V$, two reduced Hopf $C^*$-algebras $S_V$ and $\widehat{S}_V$ are associated in \cite{BS} under the regularity condition which was modified later in \cite{Wo4,SW} with manageability or modularity; in particular, the multiplicative unitaries of locally compact quantum groups \cite{KV} are known to be manageable. Thus the reduced Hopf $C^*$-algebras arising from multiplicative unitaries are a vast generalization of groups and their dual structures.
The goal of this paper is to show that essentially the same results can be obtained if group actions or coactions studied in \cite{HaoNg,KQRo2} are replaced by Hopf $C^*$-algebra coactions. To this end, we first need a concept of a coaction of a Hopf $C^*$-algebra on a $C^*$-cor\-re\-spond\-ence. In \cite{BS0}, coaction of a Hopf $C^*$-algebra $S$ on a Hilbert $A$-module $X$ was introduced as a pair $(\sigma,\delta)$ of a linear map $\sigma:X\rightarrow M(X\otimes S)$ and a homomorphism $\delta:A\rightarrow M(A\otimes S)$ which are required to be, among other things, compatible with the Hilbert module structure of $X$. This notion was originally aimed to define equivariant KK-groups and generalize the Kasparov product in the setting of Hopf $C^*$-algebras. Since then, the notion of coactions on Hilbert modules has been extensively dealt with in various situations: for example \cite{Bui,Buss,BM,dCY,GZ,KQ0,TdC,Tom,Verg}. In this paper, we propose a definition of coaction of a Hopf $C^*$-algebra $S$ on a $C^*$-cor\-re\-spond\-ence $(X,A)$ as a coaction $(\sigma,\delta)$ of $S$ on the Hilbert $A$-module $X$ which is also compatible with the left action $\varphi_A$ (see Definition~\ref{DefofCoactions} for the precise definition), and show that this definition unifies the separate notions of group actions and nondegenerate group coactions on $(X,A)$ (Remark~\ref{Unify.act.coact.}). We then proceed to show that the passage from a group action or coaction on $(X,A)$ to an action or coaction on $\mathcal{O}_X$ can be generalized nicely in the Hopf $C^*$-algebra framework (Theorem~\ref{induced coactions on O_X}). When the Hopf $C^*$-algebra under consideration is a reduced one defined by a well-behaved multiplicative unitary in the sense of \cite{Timm}, we construct the reduced crossed product correspondences (Theorem~\ref{crossed product correspondences}), and prove an isomorphism result analogous to \eqref{Intro.1} and \eqref{Intro.2} under a suitable condition (Theorem~\ref{Main.Theorem.}). Applying our results we improve and extend the main results of \cite{HaoNg} and \cite{KQRo2} (Remark~\ref{Sec.5.Improve.KQRo2} and Corollary~\ref{Cor1.to.Main.Thm}).
There have been plenty of works concerning ``natural'' coactions of compact quantum groups on the Cuntz algebra $\mathcal{O}_n$ with the focus on their fixed point algebras: for example, see \cite{Ga,KNW,M,Pao} among others. These coactions are the ones induced by coactions on the finite dimensional $C^*$-cor\-re\-spond\-ences $(\mathbb{C}^n,\mathbb{C})$, and actually, can be considered within a more general context of graph $C^*$-algebras \cite{KPRR,KPR,FLR}. In fact, we will extend in our upcoming paper the notion of labeling of a graph given in \cite{KQRa} to the setting of compact quantum groups. We will also consider coactions of compact quantum groups on any finite graphs. Natural coactions on $\mathcal{O}_n$ then can be viewed as the ones arising from labelings of the graph consisting of one vertex and $n$ edges, or alternatively, the ones arising from coactions on such a graph. Moreover, the crossed products by those natural coactions can be realized as Cuntz-Pimsner algebras. In light of these facts, it is natural and desirable to extend the works of \cite{HaoNg,KQRo2} from the point of view of Hopf $C^*$-algebra coactions.
This paper is organized as follows.
In Section~\ref{Sec.2}, we review basic facts from \cite[Chapter 1]{EKQR} and \cite[Appendix~A]{DKQ} on multiplier correspondences. We also collect from \cite{Kat,BS} definitions and facts on Cuntz-Pimsner algebras and reduced crossed products by Hopf $C^*$-algebra coactions on $C^*$-algebras. Note that in \cite{EKQR}, Hilbert $A$-$B$ bimodules were considered while we are concerned only with Hilbert $A$-$A$ bimodules, namely nondegenerate $C^*$-cor\-re\-spond\-ences $(X,A)$.
In Section~\ref{Sec.3}, we define a coaction $(\sigma,\delta)$ of a Hopf $C^*$-algebra $S$ on a $C^*$-cor\-re\-spond\-ence $(X,A)$ (Definition~\ref{DefofCoactions}), and prove in Theorem~\ref{induced coactions on O_X} that if $(\sigma,\delta)$ is a coaction of $S$ on $(X,A)$ such that the ideal $J_X$ is weakly $\delta$-invariant (Definition~\ref{Sec.3.delta.invariant}), then $(\sigma,\delta)$ induces a coaction $\zeta$ of $S$ on the associated Cuntz-Pimsner algebra $\mathcal{O}_X$.
Section~\ref{Sec.4} is devoted to constructing the reduced crossed product correspondence $(X\rtimes_\sigma\widehat{S},A\rtimes_\delta\widehat{S})$ from a coaction $(\sigma,\delta)$ of a reduced Hopf $C^*$-algebra $S$ defined by a well-behaved multiplicative unitary.
In Section~\ref{Sec.5}, we prove an isomorphism analogous to \eqref{Intro.1} and \eqref{Intro.2} in the reduced Hopf $C^*$-algebra setting. Along the way we answer the question posed in \cite[Remark~4.5]{KQRo2}; specifically, we prove that Theorem 4.4 of \cite{KQRo2} still holds without the hypothesis of the Cuntz-Pimsner covariance for the canonical embedding of $(X,A)$ into the crossed product correspondence (see Remark~\ref{Sec.5.Improve.KQRo2}). The $C^*$-cor\-re\-spond\-ence $(X\rtimes_\sigma\widehat{S},A\rtimes_\delta\widehat{S})$ is shown to have a representation $(k_X\rtimes_\sigma{\rm id},k_A\rtimes_\delta{\rm id})$ on the reduced crossed product $\mathcal{O}_X\rtimes_\zeta\widehat{S}$ by the induced coaction $\zeta$ (Proposition~\ref{embedding XxShat to OxShat is a Toep.rep.}). We prove in Theorem \ref{Main.Theorem.} that
\[\mathcal{O}_X\rtimes_\zeta\widehat{S}\cong\mathcal{O}_{X\rtimes_\sigma\widehat{S}}\]
under the assumption that $(k_X\rtimes_\sigma{\rm id},k_A\rtimes_\delta{\rm id})$ is covariant. By applying this to group actions, we extend Theorem 2.10 of \cite{HaoNg} (see Corollary \ref{Cor1.to.Main.Thm}) to any locally compact groups. It is however, not so easy to determine whether the representation $(k_X\rtimes_\sigma{\rm id},k_A\rtimes_\delta{\rm id})$ is covariant or not without understanding the ideal $J_{X\rtimes_\sigma\widehat{S}}$ of $A\rtimes_\delta\widehat{S}$. Actually, $J_{X\rtimes_\sigma\widehat{S}}$ is not known even for the commutative case with some exceptions. For an action $(\gamma,\alpha)$ of a locally compact group $G$, it was shown that $J_{X\rtimes_\gamma G}=J_X\rtimes_\alpha G$ if $G$ is amenable (\cite[Proposition~2.7]{HaoNg}), which was the most difficult part in proving the main result of \cite{HaoNg} as was mentioned in the introductory section there. Recently, the same has been shown for a discrete group $G$ if $G$ is exact or if the action $\alpha$ has Exel's Approximation Property (\cite[Theorem~5.5]{BKQR}). However, we only know in general that $J_{X\rtimes_\sigma\widehat{S}}$ contains the ideal of $A\rtimes_\delta\widehat{S}$ generated by the image $\delta_\iota(J_X)$ (Proposition~\ref{Ext.Main.Thm.of.KQR2}). We bypass the difficulty regarding the ideal $J_{X\rtimes_\sigma\widehat{S}}$ by focusing our attention on the $(A\otimes\mathscr{K})$-multiplier correspondence $(M_{A\otimes\mathscr{K}}(X\otimes\mathscr{K}),M(A\otimes\mathscr{K}))$ in which the $C^*$-cor\-re\-spond\-ence $(X\rtimes_\sigma\widehat{S},A\rtimes_\delta\widehat{S})$ lies. This leads us to two equivalent conditions that the representation $(k_X\rtimes_\sigma{\rm id},k_A\rtimes_\delta{\rm id})$ is covariant (Theorem~\ref{Equiv.of.cov.}). From these equivalent conditions, we see that $(k_X\rtimes_\sigma{\rm id},k_A\rtimes_\delta{\rm id})$ is covariant if, in particular, $J_{X\rtimes_\sigma\widehat{S}}$ is generated by $\delta_\iota(J_X)$ or the left action $\varphi_A$ is injective (Corollary~\ref{Sec.5.Cor.varphi.inj.}), even though we do not know the ideal $J_{X\rtimes_\sigma\widehat{S}}$ explicitly.
In Appendix~\ref{App.A}, we generalize \cite[Corollary~3.4]{APT} to our $C^*$-cor\-re\-spond\-ence setting, and then show that there exists a one-to-one correspondence between actions of a locally compact group $G$ and coactions of the commutative Hopf $C^*$-algebra $C_0(G)$ on a $C^*$-cor\-re\-spond\-ence. In Appendix~\ref{App.B}, we prove a $C^*$-cor\-re\-spond\-ence analogue to the well-known fact that $\mathcal{L}_A(A\otimes\mathcal{H})=M(A\otimes\mathcal{K}(\mathcal{H}))$ for a $C^*$-algebra $A$ and a Hilbert space $\mathcal{H}$. Using this, we recover from our construction of $(X\rtimes\widehat{S},A\rtimes\widehat{S})$ the crossed product correspondences $(X\rtimes_rG,A\rtimes_rG)$ for actions of locally compact groups $G$ given in \cite{EKQR}.
\section{Preliminaries}\label{Sec.2}
In this section, we review some definitions and properties related to multiplier correspondences, Cuntz-Pimsner algebras, and reduced crossed products by reduced Hopf $C^*$-algebra coactions. Our references include \cite{EKQR,DKQ,Kat,BS}.
\subsection{$C^*$-correspondences}
Let $A$ be a $C^*$-algebra. For two (right) Hilbert $A$-modules $X$ and $Y$, we denote by $\mathcal{L}(X,Y)=\mathcal{L}_A(X,Y)$ the Banach space of all adjointable operators from $X$ to $Y$, and by $\mathcal{K}(X,Y)=\mathcal{K}_A(X,Y)$ the closed subspace of $\mathcal{L}(X,Y)$ generated by the operators $\theta_{\xi,\eta}$:
\[\theta_{\xi,\eta}(\eta')=\xi\cdot\langle\eta,\eta'\rangle_A\quad(\xi\in Y,\ \eta,\eta'\in X).\]
We simply write $\mathcal{L}(X)$ and $\mathcal{K}(X)$ when $X=Y$; in this case $\mathcal{L}(X)$ becomes a maximal unital $C^*$-algebra containing $\mathcal{K}(X)$ as an essential ideal.
A \emph{$C^*$-correspondence} over a $C^*$-algebra $A$ is a Hilbert $A$-module $X$ equipped with a homomorphism $\varphi_A:A\rightarrow\mathcal{L}(X)$, called the \emph{left action}. We use the notation $(X,A)$ of \cite{KQRo2} to refer to a $C^*$-correspondence $X$ over $A$. We say that $(X,A)$ is \emph{nondegenerate} if $\varphi_A$ is nondegenerate, that is, $\overline{\varphi_A(A)X}=X$.
Every $C^*$-algebra $A$ has the natural structure of a nondegenerate $C^*$-cor\-re\-spond\-ence over itself, called the \emph{identity correspondence}, with the left action of identifying $A$ with $\mathcal{K}(A)$ (p.\ 368 of \cite{Kat}). When we regard a $C^*$-algebra as a $C^*$-cor\-re\-spond\-ence, we always mean this $C^*$-cor\-re\-spond\-ence. We also regard a Hilbert space $\mathcal{H}$ as a Hilbert $\mathbb{C}$-module such that the inner product is conjugate linear in the first variable; it is a $C^*$-cor\-re\-spond\-ence with $\varphi_{\mathbb{C}}(1)=1_{\mathcal{L}(\mathcal{H})}$.
\subsection{Multiplier correspondences}
Throughout the paper, we restrict ourselves to nondegenerate $C^*$-cor\-re\-spond\-ences, which in particular allows us to consider their multiplier correspondences that are a generalization of multiplier $C^*$-algebras.
Let $(X,A)$ be a $C^*$-correspondence, and let $M(X):=\mathcal{L}(A,X)$. The \emph{multiplier correspondence} of $X$ is the $C^*$-cor\-re\-spond\-ence $M(X)$ over the multiplier algebra $M(A)$ with the Hilbert $M(A)$-module operations
\begin{equation}\label{Prel.module.ops}
m\cdot a:=m a,\quad\langle m,n\rangle_{M(A)}:=m^*n
\end{equation}
and the left action
\begin{equation}\label{Prel.module.ops.2}
\varphi_{M(A)}(a)m:=\overline{\varphi_A}(a)m
\end{equation}
for $m,n\in M(X)$ and $a\in M(A)$, where $\overline{\varphi_A}$ is the strict extension of the nondegenerate homomorphism $\varphi_A$ and $ma$, $m^*n$, and $\varphi_{M(A)}(a)m$ mean the compositions $m\circ a$, $m^*\circ n$, and $\varphi_{M(A)}(a)\circ m$, respectively. The identification of $X$ with $\mathcal{K}(A,X)$, in which each $\xi\in X$ is regarded as the operator $A\ni a\mapsto\xi\cdot a\in X$, gives an embedding of $X$ into $M(X)$, and we will always regard $X$ as a subspace of $M(X)$ through this embedding. Note that $\mathcal{K}(M(X))\subseteq M(\mathcal{K}(X))$ nondegenerately.
The \emph{strict topology} on $M(X)$ is the locally convex topology such that a net $\{m_i\}$ in $M(X)$ converges strictly to 0 if and only if for $T\in\mathcal{K}(X)$ and $a\in A$, the nets $\{Tm_i\}$ and $\{m_i\cdot a\}$ both converge in norm to 0. It can be shown that $M(X)$ is the strict completion of $X$.
Let $(X,A)$ and $(Y,B)$ be $C^*$-correspondences. A pair
\[(\psi,\pi):(X,A)\rightarrow(M(Y),M(B))\]
of a linear map $\psi:X\rightarrow M(Y)$ and a homomorphism $\pi:A\rightarrow M(B)$ is called a \emph{correspondence homomorphism} if
\begin{itemize}
\item[\rm(i)] $\psi(\varphi_A(a)\xi)=\varphi_{M(B)}(\pi(a))\,\psi(\xi)$ for $a\in A$ and $\xi\in X$;
\item[\rm(ii)] $\pi(\langle\xi,\eta\rangle_A)=\langle\psi(\xi),\psi(\eta)\rangle_{M(B)}$ for $\xi,\eta\in X$.
\end{itemize}
It is automatic that $\psi(\xi\cdot a)=\psi(\xi)\cdot\pi(a)$ (see the comment below \cite[Definition~2.3]{Kat2}). We say that $(\psi,\pi)$ is \emph{injective} if $\pi$ is injective; if so $\psi$ is isometric. We also say that $(\psi,\pi)$ is \emph{nondegenerate} if
$\overline{\psi(X)\cdot B}=Y$ and $\overline{\pi(A)B}=B$.
In this case, $(\psi,\pi)$ extends uniquely to a strictly continuous correspondence homomorphism
\[(\overline{\psi},\overline{\pi}):(M(X),M(A))\rightarrow(M(Y),M(B))\]
(\cite[Theorem 1.30]{EKQR}). Note that if $(\psi,\pi)$ is injective, then so is $(\overline{\psi},\overline{\pi})$.
A correspondence homomorphism $(\psi,\pi):(X,A)\rightarrow(M(Y),M(B))$ determines a (unique) homomorphism
$\psi^{(1)}:\mathcal{K}(X)\rightarrow\mathcal{K}(M(Y))\subseteq M(\mathcal{K}(Y))$
such that
\[\psi^{(1)}(\theta_{\xi,\eta})=\psi(\xi)\psi(\eta)^*\quad(\xi,\eta\in X)\]
(see for example \cite[Definition 2.4]{Kat2} and the comment below it). If $(\psi,\pi)$ is nondegenerate, then so is $\psi^{(1)}$; it is straightforward to verify that
\begin{equation}\label{Pre.psi.1}
\psi(T\xi)=\overline{\psi^{(1)}}(T)\psi(\xi),\quad\overline{\psi^{(1)}}(mn^*)=\overline{\psi}(m)\overline{\psi}(n)^*
\end{equation}
for $T\in\mathcal{L}(X)$, $\xi\in X$, and $m,n\in M(X)$. The first relation of \eqref{Pre.psi.1} shows that $\overline{\psi^{(1)}}$ is injective whenever $\psi$ is injective.
\subsection{Tensor product correspondences}
In this paper, the tensor products of Hilbert modules always mean the exterior ones (\cite[pp.\ 34--35]{Lance}). The tensor products of $C^*$-algebras are the minimal ones.
Let $(X_1,A_1)$ and $(X_2,A_2)$ be $C^*$-cor\-re\-spon\-dences. We will freely use the following identification
\[\mathcal{K}(X_1\otimes X_2)=\mathcal{K}(X_1)\otimes\mathcal{K}(X_2)\]
via $\theta_{\xi_1\otimes\xi_2,\,\eta_1\otimes\eta_2}=\theta_{\xi_1,\,\eta_1}\otimes\theta_{\xi_2,\,\eta_2}$. Equipped with the left action
$\varphi_{A_1\otimes A_2}=\varphi_{A_1}\otimes\varphi_{A_2},$
the tensor product $X_1\otimes X_2$ then becomes a $C^*$-cor\-re\-spon\-dence over $A_1\otimes A_2$, called the \emph{tensor product correspondence}.
Let $(\psi_i,\pi_i):(X_i,A_i)\rightarrow(M(Y_i),M(B_i))$ $(i=1,2)$ be correspondence homomorphisms. Then there exists a unique correspondence homomorphism
\[(\psi_1\otimes\psi_2,\pi_1\otimes\pi_2):(X_1\otimes X_2,A_1\otimes A_2)\rightarrow(M(Y_1\otimes Y_2),M(B_1\otimes B_2))\]
such that $(\psi_1\otimes\psi_2)(\xi_1\otimes\xi_2)=\psi_1(\xi_1)\otimes\psi_2(\xi_2)$. If both $(\psi_i,\pi_i)$ are nondegenerate then $(\psi_1\otimes\psi_2,\pi_1\otimes\pi_2)$ is also nondegenerate (\cite[Proposition 1.38]{EKQR}).
\subsection{Cuntz-Pimsner algebras}
Let $(X,A)$ be a $C^*$-cor\-re\-spond\-ence, and let
\[J_X:=\varphi_A^{-1}(\mathcal{K}(X))\cap\{a\in A:ab=0\ {\rm for}\ b\in\ker\varphi_A\}.\]
Then $J_X$ is characterized as the largest ideal of $A$ which is mapped injectively into $\mathcal{K}(X)$ by $\varphi_A$. A correspondence homomorphism $(\psi,\pi):(X,A)\rightarrow(B,B)$ into an identity correspondence $(B,B)$ is called a \emph{representation} of $(X,A)$ on $B$ and denoted simply by $(\psi,\pi):(X,A)\rightarrow B$. We say that $(\psi,\pi)$ is \emph{covariant} if
\[\psi^{(1)}(\varphi_A(a))=\pi(a)\quad(a\in J_X)\]
(\cite[Definition 3.4]{Kat}). We denote by $(k_X,k_A)$ the universal covariant representation of $(X,A)$ which is known to be injective (\cite[Proposition~4.9]{Kat}). The \emph{Cuntz-Pimsner algebra} $\mathcal{O}_X$ is the $C^*$-algebra generated by $k_X(X)$ and $k_A(A)$. Note that the embedding $k_A:A\hookrightarrow\mathcal{O}_X$ is nondegenerate by our standing assumption that $(X,A)$ is nondegenerate. From the universality of $(k_X,k_A)$, if $(\psi,\pi)$ is a covariant representation of $(X,A)$ on $B$, there exists a unique homomorphism $\psi\times\pi:\mathcal{O}_X\rightarrow B$ called the \emph{integrated form} of $(\psi,\pi)$ such that $\psi=(\psi\times\pi)\circ k_X$ and $\pi=(\psi\times\pi)\circ k_A$.
A representation $(\psi,\pi)$ of $(X,A)$ is said to \emph{admit a gauge action} if there exists an action $\beta$ of the unit circle $\mathbb{T}$ on the $C^*$-subalgebra generated by $\psi(X)$ and $\pi(A)$ such that $\beta_z(\psi(\xi))=z\psi(\xi)$ and $\beta_z(\pi(a))=\pi(a)$ for $z\in\mathbb{T}$, $\xi\in X$, and $a\in A$. The universal covariant representation $(k_X,k_A)$ clearly admits a gauge action. The \emph{gauge invariant uniqueness theorem} \cite[Theorem 6.4]{Kat} asserts that an injective covariant representation $(\psi,\pi)$ admits a gauge action if and only if $\psi\times\pi$ is injective.
\subsection{$C$-multiplier correspondences}
We recall from \cite[Appendix~A]{DKQ} basic definitions and facts on $C$-multiplier correspondences. We also fix some notations and provide results that will be used in the subsequent sections.
Let $(X,A)$ be a $C^*$-correspondence, $C$ be a $C^*$-algebra, and $\kappa:C\rightarrow M(A)$ be a nondegenerate homomorphism. The \emph{$C$-multiplier correspondence} $M_C(X)$ of $X$ and the \emph{$C$-multiplier algebra} $M_C(A)$ of $A$ are defined by
\[M_C(X):=\{m\in M(X):\varphi_{M(A)}(\kappa(C))m\cup m\cdot\kappa(C)\subseteq X\},\]
\[M_C(A):=\{a\in M(A):\kappa(C)a\cup a\kappa(C)\subseteq A\}.\]
Under the restriction of the operations \eqref{Prel.module.ops} and \eqref{Prel.module.ops.2}, $(M_C(X),M_C(A))$ becomes a $C^*$-cor\-re\-spond\-ence (\cite[Lemma A.9.(2)]{DKQ}).
\begin{notation}\rm
We mean by $M_A(X)$ the $A$-multiplier correspondence
\[M_A(X)=\big\{m\in M(X):\varphi_A(A)m\subseteq X\big\}\]
determined by $\kappa={\rm id}_A$, and by $M_A(\mathcal{K}(X))$ the $A$-multiplier algebra
\[M_A(\mathcal{K}(X))=\{m\in M(\mathcal{K}(X)):\varphi_A(A)m\cup m\varphi_A(A)\subseteq\mathcal{K}(X)\}\]
determined by the left action $\varphi_A$.
\end{notation}
Note that $\mathcal{K}(M_A(X))\subseteq M_A(\mathcal{K}(X))$ (\cite[Lemma A.9.(3)]{DKQ}).
The \emph{$C$-strict topology} on $M_C(X)$ is the locally convex topology whose neighborhood system at 0 is generated by the family $\{m:\|\varphi_{M(A)}(\kappa(c))m\|\leq\epsilon\}$ and $\{m:\|m\cdot\kappa(c)\|\leq\epsilon\}$ ($c\in C$, $\epsilon>0$). The $C$-strict topology is stronger than the relative strict topology on $M_C(X)$, and $M_C(X)$ is the $C$-strict completion of $X$. Likewise, the \emph{$C$-strict topology} on $M_C(A)$ is the locally convex topology defined by the family of seminorms $\|\kappa(c)\cdot\|+\|\cdot\kappa(c)\|$ ($c\in C$).
\begin{rmk}\label{Prel.C-strict.top.}\rm
Let $(X,A)$ be a $C^*$-correspondence and $M_{C_i}(X)$ be the $C_i$-multiplier correspondence determined by a nondegenerate homomorphism $\kappa_i:C_i\rightarrow M(A)$ ($i=1,2$). It is clear that if $\kappa_1(C_1)$ is nondegenerately contained in $M(\kappa_2(C_2))(\subseteq M(A))$, then $M_{C_1}(X)\subseteq M_{C_2}(X)$ and the $C_1$-strict topology on $M_{C_1}(X)$ is stronger than the relative $C_2$-strict topology. In particular, $M_C(X)\subseteq M_A(X)$ and the $C$-strict topology is stronger than the relative $A$-strict topology.
\end{rmk}
For a not necessarily nondegenerate correspondence homomorphism, we still have an extension by \cite[Proposition A.11]{DKQ}. Let $(\psi,\pi):(X,A)\rightarrow(M_D(Y),M_D(B))$ be a correspondence homomorphism, where $(M_D(Y),M_D(B))$ is a $D$-mul\-ti\-pli\-er correspondence determined by a nondegenerate homomorphism $\kappa_D:D\rightarrow M(B)$. Assume that $\kappa_C:C\rightarrow M(A)$ and $\lambda:C\rightarrow M(\kappa_D(D))(\subseteq M(B))$ are nondegenerate homomorphisms such that $\pi(\kappa_C(c)a)=\lambda(c)\pi(a)$ for $c\in C$ and $a\in A$. Then $(\psi,\pi)$ extends uniquely to a $C$-strict to $D$-strictly continuous correspondence homomorphism
$(\overline{\psi},\overline{\pi}):(M_C(X),M_C(A))\rightarrow (M_D(Y),M_D(B)),$
where $(M_C(X),M_C(A))$ is the $C$-multiplier correspondence determined by $\kappa_C$.
\begin{rmks}\label{Prel.Str.Ext.}\rm
(1) If $(\psi,\pi)$ is nondegenerate, then every $C$-strict to $D$-strictly continuous extension of $(\psi,\pi)$ coincides with the restriction of its usual strict extension.
(2) Suppose that $\overline{\psi}_i:M_{C_i}(X)\rightarrow M_{D_i}(Y)$ are $C_i$-strict to $D_i$-strictly continuous extensions $(i=1,2)$. If $M_{C_1}(X)\subseteq M_{C_2}(X)$ and $M_{D_1}(Y)\subseteq M_{D_2}(Y)$ and if the $C_1$-strict and $D_1$-strict topologies are stronger than the relative $C_2$-strict and $D_2$-strict topologies, respectively, then $\overline{\psi}_1=\overline{\psi}_2|_{M_{C_1}(X)}$.
\end{rmks}
We will frequently need the following special form of \cite[Proposition A.11]{DKQ}.
\begin{thm}[{\cite[Corollary A.14]{DKQ}}]\label{DKQ,Cor.A.14}
Let $(\psi,\pi):(X,A)\rightarrow B$ be a representation with $\pi$ nondegenerate. Then
{\rm(i)} $(\psi,\pi)$ extends uniquely to an $A$-strictly continuous correspondence homomorphism
\[(\overline{\psi},\overline{\pi}):(M_A(X),M(A))\rightarrow M_A(B),\]
where $M_A(B)$ is the $A$-multiplier algebra determined by $\pi$.
{\rm(ii)} $\psi^{(1)}:\mathcal{K}(X)\rightarrow B$ extends uniquely to an $A$-strictly continuous homomorphism $\overline{\psi^{(1)}}:M_A(\mathcal{K}(X))\rightarrow M_A(B)$; moreover, \[\overline{\psi^{(1)}}=\overline{\psi}^{\,(1)}\] on $\mathcal{K}(M_A(X))$, that is, $\overline{\psi^{(1)}}(mn^*)=\overline{\psi}(m)\overline{\psi}(n)^*$ for $m,n\in M_A(X)$.
\end{thm}
\begin{notation}\rm
Let $(X,A)$ be a $C^*$-correspondence and $C$ be a $C^*$-algebra. Consider the representation $(k_X\otimes{\rm id}_C,k_A\otimes{\rm id}_C):(X\otimes C,A\otimes C)\rightarrow\mathcal{O}_X\otimes C$. Since $k_A\otimes{\rm id}_C$ is nondegenerate, $k_X\otimes{\rm id}_C$ extends to the $(A\otimes C)$-strictly continuous map
\begin{equation*}\label{Prel.Str.Ext.k.times.id}
\overline{k_X\otimes{\rm id}_C}:M_{A\otimes C}(X\otimes C)\rightarrow M_{A\otimes C}(\mathcal{O}_X\otimes C)
\end{equation*}
by Theorem \ref{DKQ,Cor.A.14}.(i). Throughout the paper, we mean by $\overline{k_X\otimes{\rm id}_C}$ this extension, and by $M_{A\otimes C}(\mathcal{O}_X\otimes C)$ the $(A\otimes C)$-multiplier algebra determined by $k_A\otimes{\rm id}_C$. On the other hand, $(M_C(X\otimes C),M_C(A\otimes C))$ is the $C$-multiplier correspondence determined by the embedding $C\hookrightarrow M(A\otimes C)$ onto the last factor.
\end{notation}
For an ideal $I$ of a $C^*$-algebra $B$, let
\[M(B;I):=\{m\in M(B): mB\cup Bm\subseteq I\}.\]
By \cite[Lemma 2.4.(i)]{KQRo1}, $M(B;I)$ is the strict closure of $I$ in $M(B)$.
\begin{lem}\label{Lemma.for.Ext.Main.Thm.of.KQR2 0}
Let $(X,A)$ be a $C^*$-correspondence. Then the ideal $J_{M_A(X)}$ is contained in the strict closure of $J_X$, that is,
\[J_{M_A(X)}\subseteq M(A;J_X).\]
\end{lem}
\begin{proof}
We need to show that the ideal $AJ_{M_A(X)}$ is contained in $J_X$. By definition, we have
\[\varphi_A(AJ_{M_A(X)})\subseteq\varphi_A(A)\mathcal{K}(M_A(X))\subseteq\varphi_A(A)M_A(\mathcal{K}(X))\subseteq\mathcal{K}(X).\]
We also have
\[AJ_{M_A(X)}\ker\varphi_A\subseteq J_{M_A(X)}\ker\varphi_{M(A)}=0.\]
Consequently, $AJ_{M_A(X)}\subseteq J_X$.
\end{proof}
The next lemma, contained in the proof of \cite[Lemma 2.5]{KQRo2}, will be useful in proving Theorem \ref{induced coactions on O_X}, Proposition \ref{Ext.Main.Thm.of.KQR2}, and Theorem \ref{Equiv.of.cov.}.
\begin{lem}\label{Lemma.for.Ext.Main.Thm.of.KQR2}
Let $(X,A)$ be a $C^*$-correspondence and $C$ be a $C^*$-algebra. Then
\begin{equation}\label{Covariancy.of.the.Strict.Ext.1}
\overline{(k_X\otimes{\rm id}_C)^{(1)}}\circ\varphi_{M(A\otimes C)}=\overline{k_A\otimes{\rm id}_C}
\end{equation}
holds on $M(A\otimes C;J_X\otimes C)$, that is, the diagram
\[ \xymatrix{ M_{A\otimes C}(\mathcal{K}(X\otimes C)) \ar[drr]^-{\qquad\overline{(k_X\otimes{\rm id}_C)^{(1)}}} && \\
M(A\otimes C;J_X\otimes C) \ar[u]^-{\varphi_{M(A\otimes C)}} \ar[rr]_-{\overline{k_A\otimes{\rm id}_C}} &&
M_{A\otimes C}(\mathcal{O}_X\otimes C)
}\]
commutes.
\end{lem}
\begin{proof}
By definition, the vertical map makes sense and is $(A\otimes C)$-strictly continuous. Also, Theorem \ref{DKQ,Cor.A.14}.(ii) says that $(k_X\otimes{\rm id}_C)^{(1)}$ extends $(A\otimes C)$-strictly to the homomorphism $\overline{(k_X\otimes{\rm id}_C)^{(1)}}$ indicated by the lower right arrow. Hence the composition on the left side of \eqref{Covariancy.of.the.Strict.Ext.1} is well-defined on $M(A\otimes C;J_X\otimes C)$ and $(A\otimes C)$-strictly continuous. On the other hand, the horizontal map is the restriction of the usual strict extension $\overline{k_A\otimes{\rm id}_C}$ and $(A\otimes C)$-strictly continuous. Since \eqref{Covariancy.of.the.Strict.Ext.1} is valid on $J_X\otimes C$, the conclusion now follows by $(A\otimes C)$-strict continuity and the fact that $J_X\otimes C$ is $(A\otimes C)$-strictly dense in $M(A\otimes C;J_X\otimes C)$.
\end{proof}
Recall from \cite[Definition 12.4.3]{BO} the following terminology. Let $A$ and $C$ be $C^*$-algebras and $J$ be a closed subspace of $A$. The triple $(J,A,C)$ is said to \emph{satisfy the slice map property} if the space
\[ F(J,A,C)=\{x\in A\otimes C: ({\rm id}\otimes\omega)(x)\in J\ {\rm for}\ \omega\in C^*\} \]
equals the norm closure $J\otimes C$ of the algebraic tensor product $J\odot C$ in $A\otimes C$.
\begin{rmks}\label{S.M.P.}\rm
(1) If $J$ is an ideal of $A$, then $(J,A,C)$ satisfies the slice map property if and only if the sequence
\[ 0\longrightarrow J\otimes C\longrightarrow A\otimes C\longrightarrow (A/J)\otimes C \longrightarrow0 \]
is exact; this is the case if $A$ is locally reflexive or $C$ is exact (see below \cite[Definition~12.4.3]{BO}).
(2) Let $\mathcal{H}$ be a Hilbert space. If $C$ is a $C^*$-subalgebra of $\mathcal{L}(\mathcal{H})$, then $F(J,A,C)$ equals the norm closure of the following space
\[ \{x\in A\otimes C: ({\rm id}\otimes\omega)(x)\in J\ {\rm for}\ \omega\in\mathcal{L}(\mathcal{H})_*\}. \]
\end{rmks}
\begin{cor}\label{Sec.3.prop.for.thm.in.Sec.5}
Let $(X,A)$ be a $C^*$-correspondence and $C$ be a $C^*$-algebra. Suppose that $(J_X,A,C)$ satisfies the slice map property. Then
\[J_{X\otimes C}=J_X\otimes C.\]
Furthermore,
\[J_{M_{A\otimes C}(X\otimes C)}\subseteq M(A\otimes C;J_X\otimes C)\]
and the injective representation
\[(\overline{k_X\otimes{\rm id}_C},\overline{k_A\otimes{\rm id}_C}):(M_{A\otimes C}(X\otimes C),M(A\otimes C))\rightarrow M_{A\otimes C}(\mathcal{O}_X\otimes C)\]
is covariant.
\end{cor}
\begin{proof}
We always have $J_{X\otimes C}\supseteq J_X\otimes C$ as shown in the first part of the proof of \cite[Lemma 2.6]{KQRo2}. We thus only need to show the converse $J_{X\otimes C}\subseteq F(J_X,A,C)=J_X\otimes C$. But, this can be done in the same way as the second part of the proof of \cite[Lemma 2.6]{KQRo2}, and then the first assertion of the corollary follows. Lemma~\ref{Lemma.for.Ext.Main.Thm.of.KQR2 0} then verifies the second assertion on the inclusion. Finally, since $\varphi_{M(A\otimes C)}$ maps $J_{M_{A\otimes C}(X\otimes C)}$ into $\mathcal{K}(M_{A\otimes C}(X\otimes C))$ on which
\[\overline{(k_X\otimes{\rm id})^{(1)}}=\overline{k_X\otimes{\rm id}_C}^{\,(1)}\]
by Theorem \ref{DKQ,Cor.A.14}.(ii), the representation is covariant by Lemma \ref{Lemma.for.Ext.Main.Thm.of.KQR2}.
\end{proof}
\begin{cor}\label{O_(XotimesB)=O_XotimesB}
Under the same hypothesis of Corollary \ref{Sec.3.prop.for.thm.in.Sec.5}, the injective representation
\[(k_X\otimes{\rm id}_C,k_A\otimes{\rm id}_C):(X\otimes C,A\otimes C)\rightarrow\mathcal{O}_X\otimes C\]
is covariant and the integrated form $(k_X\otimes{\rm id}_C)\times(k_A\otimes{\rm id}_C):\mathcal{O}_{X\otimes C}\rightarrow\mathcal{O}_X\otimes C$ is a surjective isomorphism.
\end{cor}
\begin{proof}
Generally we have $J_Y\subseteq J_{M_B(Y)}$ for a $C^*$-cor\-re\-spond\-ence $(Y,B)$ since $J_Y$ is an ideal of $M(B)$ and is mapped injectively into $\mathcal{K}(Y)\subseteq\mathcal{K}(M_B(Y))$ by $\varphi_{M(B)}$. Hence $J_{X\otimes C}\subseteq J_{M_{A\otimes C}(X\otimes C)}$, and therefore $(k_X\otimes{\rm id}_C,k_A\otimes{\rm id}_C)$ is covariant by Corollary \ref{Sec.3.prop.for.thm.in.Sec.5}. The integrated form is clearly surjective. Since $(k_X,k_A)$ admits a gauge action, and hence so does $(k_X\otimes{\rm id}_C,k_A\otimes{\rm id}_C)$, the integrated form must be injective by \cite[Theorem 6.4]{Kat}.
\end{proof}
\subsection{Reduced and dual reduced Hopf $C^*$-algebras}
By a \emph{Hopf $C^*$-algebra} we always mean a bisimplifiable Hopf $C^*$-algebra in the sense of \cite{BS}, that is, a pair $(S,\Delta)$ of a $C^*$-algebra $S$ and a nondegenerate homomorphism $\Delta:S\rightarrow M(S\otimes S)$ called the \emph{comultiplication} of $S$ satisfying
\begin{itemize}
\item[(i)] $\overline{\Delta\otimes{\rm id}}\circ\Delta=\overline{{\rm id}\otimes\Delta}\circ\Delta$;
\item[(ii)] $\overline{\Delta(S)(1_{M(S)}\otimes S)}=S\otimes S=\overline{\Delta(S)(S\otimes1_{M(S)})}$.
\end{itemize}
Let $G$ be a locally compact group. Then $(C_0(G),\Delta_G)$ is a Hopf $C^*$-algebra with the comultiplication $\Delta_G(f)(r,s)=f(rs)$ for $f\in C_0(G)$ and $r,s\in G$. The full group $C^*$-algebra $C^*(G)$ equipped with the comultiplication given by $r\mapsto r\otimes r$ for $r\in G$ is also a Hopf $C^*$-algebra. The same is true for the reduced group $C^*$-algebra $C^*_r(G)$ such that the canonical surjection $\lambda:C^*(G)\rightarrow C^*_r(G)$ is a morphism in the sense of \cite{BS} (also see \cite[Example 4.2.2]{Timm}).
Let $\mathcal{H}$ be a Hilbert space. A unitary operator $V$ acting on $\mathcal{H}\otimes\mathcal{H}$ is said to be \emph{multiplicative} if it satisfies the pentagonal relation $V_{12}V_{13}V_{23}=V_{23}V_{12}$, where we use the leg-numbering notations $V_{ij}$ such that $V_{12}\in\mathcal{L}(\mathcal{H}\otimes\mathcal{H}\otimes\mathcal{H})$ denotes the unitary $V\otimes1$ for example (see \cite[p.\ 428]{BS}). For each functional $\omega\in\mathcal{L}(\mathcal{H})_*$, define the operators $L(\omega)$ and $\rho(\omega)$ in $\mathcal{L}(\mathcal{H})$ by
\[L(\omega)=\overline{\omega\otimes{\rm id}}(V),\quad\rho(\omega)=\overline{{\rm id}\otimes\omega}(V),\]
where the maps $\overline{\omega\otimes{\rm id}}$ and $\overline{{\rm id}\otimes\omega}$ denote the usual strict extension to the multiplier algebra $M(\mathcal{K}(\mathcal{H})\otimes\mathcal{K}(\mathcal{H}))(=\mathcal{L}(\mathcal{H}\otimes\mathcal{H}))$. The \emph{reduced algebra} $S_V$ and the \emph{dual reduced algebra} $\widehat{S}_V$ are defined as the following norm closed subspaces of $\mathcal{L}(\mathcal{H})$:
\[S_V=\overline{\{L(\omega):\omega\in\mathcal{L}(\mathcal{H})_*\}},\quad
\widehat{S}_V=\overline{\{\rho(\omega):\omega\in\mathcal{L}(\mathcal{H})_*\}}.\]
They are known to be nondegenerate subalgebras of $\mathcal{L}(\mathcal{H})$ (\cite[Prop\-o\-si\-tion~1.4]{BS}).
A multiplicative unitary $V$ acting on $\mathcal{H}\otimes\mathcal{H}$ is said to be \emph{well-behaved} if both $S_V$ and $\widehat{S}_V$ are Hopf $C^*$-algebras with the comultiplications $\Delta_V(s)=V(s\otimes1)V^*$ and $\widehat{\Delta}_V(x)=V^*(1\otimes x)V$
for $s\in S$ and $x\in\widehat{S}$, and $V\in M(\widehat{S}\otimes S)$ (\cite[Definition~7.2.6.i)]{Timm}).
\begin{rmk}\rm
When we consider a well-behaved multiplicative unitary $V$, we will not need the last property $V\in M(\widehat{S}\otimes S)$. In fact, we only need the property that $V$ gives rise to two Hopf $C^*$-algebras $S_V$ and $\widehat{S}_V$.
It should be stressed though that many important and significant Hopf $C^*$-algebras come from well-behaved multiplicative unitaries the class of which includes those with regularity \cite{BS}, manageability \cite{Wo4} and modularity \cite{SW}. In particular, locally compact quantum groups \cite{KV} are the Hopf $C^*$-algebras arising from well-behaved multiplicative unitaries.
\end{rmk}
For a locally compact group $G$, let $W_G$ and $\widehat{W}_G$ be the regular multiplicative unitaries acting on $L^2(G)\otimes L^2(G)$ by
\[(W_G\xi)(r,s)=\xi(r,r^{-1}s),\quad(\widehat{W}_G\xi)(r,s)=\xi(sr,s)\]
for $\xi\in C_c(G\times G)$ and $r,s\in G$. It can be shown that $S_{W_G}=C^*_r(G)=\widehat{S}_{\widehat{W}_G}$ as Hopf $C^*$-algebras. Let $\mu_G$ and $\check{\mu}_G$ be the nondegenerate embeddings $C_0(G)\hookrightarrow\mathcal{L}(L^2(G))$ given by
\begin{equation}\label{Prel.pi.and.U}
\big(\mu_G(f)h\big)(r)=f(r)h(r),\quad\big(\check{\mu}_G(f)h\big)(r)=f(r^{-1})h(r)
\end{equation}
for $h\in C_c(G)$. Then $\mu_G$ and $\check{\mu}_G$ are isomorphisms from the Hopf $C^*$-algebra $C_0(G)$ onto $\widehat{S}_{W_G}$ and $S_{\widehat{W}_G}$, respectively (see for example \cite[Example 9.3.11]{Timm}).
\subsection{Reduced crossed products}
By a \emph{coaction} of a Hopf $C^*$-algebra $(S,\Delta)$ on a $C^*$-algebra $A$ we always mean a nondegenerate homomorphism $\delta:A\rightarrow M(A\otimes S)$ such that
\begin{itemize}
\item[(i)] $\delta$ satisfies the \emph{coaction identity} $\overline{\delta\otimes{\rm id}}\circ\delta=\overline{{\rm id}\otimes\Delta}\circ\delta$;
\item[(ii)] $\delta$ satisfies the \emph{coaction nondegeneracy} $\overline{\delta(A)(1_{M(A)}\otimes S)}=A\otimes S$.
\end{itemize}
Let $V$ be a well-behaved multiplicative unitary acting on $\mathcal{H}\otimes\mathcal{H}$. Let $\delta$ be a coaction of the reduced Hopf $C^*$-algebra $S_V$ on $A$, and $\iota_{S_V}:S_V\hookrightarrow M(\mathcal{K}(\mathcal{H}))$ be the inclusion map. We denote by $\delta_\iota$ the following composition
\begin{equation}\label{Prel.delta.iota}
\delta_\iota:=\overline{{\rm id}_A\otimes\iota_{S_V}}\circ\delta:A\rightarrow M(A\otimes\mathcal{K}(\mathcal{H})).
\end{equation}
The \emph{reduced crossed product} $A\rtimes_\delta\widehat{S}_V$ of $A$ by the coaction $\delta$ of $S_V$ is defined to be the following norm closed subspace of $M(A\otimes\mathcal{K}(\mathcal{H}))$
\[A\rtimes_\delta\widehat{S}_V=\overline{\delta_\iota(A)(1_{M(A)}\otimes\widehat{S}_V)},\]
where $1_{M(A)}\otimes\widehat{S}_V$ denotes the image of the canonical embedding $\widehat{S}_V\hookrightarrow M(A\otimes\mathcal{K}(\mathcal{H}))$. By \cite[Lemma 7.2]{BS}, $A\rtimes_\delta\widehat{S}_V$ is a $C^*$-algebra.
\begin{rmk}\rm
In the literature, the reduced crossed product $A\rtimes_\delta\widehat{S}_V$ is usually defined as a subalgebra of $\mathcal{L}_A(A\otimes\mathcal{H})$ which can be identified with $M(A\otimes\mathcal{K}(\mathcal{H}))$. For the arguments concerning multiplier correspondences and the relevant strict topologies, it seems to be more convenient to work with $M(A\otimes\mathcal{K}(\mathcal{H}))$ rather than $\mathcal{L}_A(A\otimes\mathcal{H})$. This leads us to regard $A\rtimes_\delta\widehat{S}_V$ as a subalgebra of $M(A\otimes\mathcal{K}(\mathcal{H}))$.
\end{rmk}
Let $G$ be a locally compact group and $A$ be a $C^*$-algebra. It is well-known that there exists a one-to-one correspondence between actions of $G$ on $A$ and coactions of $C_0(G)$ on $A$ such that each action $\alpha$ determines a coaction $\delta^\alpha$ given by $\delta^\alpha(a)(r)=\alpha_r(a)$ for $a\in A$ and $r\in G$. Moreover, if $\alpha:G\rightarrow Aut(A)$ is an action then the reduced crossed product $A\rtimes_{\alpha,r}G$ coincides with the crossed product $A\rtimes_{\delta^\alpha_G}\widehat{S}_{\widehat{W}_G}$ by the coaction
\[\delta^\alpha_G=\overline{{\rm id}_A\otimes\check{\mu}_G}\circ\delta^\alpha:A\rightarrow M(A\otimes S_{\widehat{W}_G})\]
when viewed as subalgebras of $M(A\otimes\mathcal{K}(L^2(G)))$ (see for example \cite[Chapter~9]{Timm}).
A \emph{nondegenerate coaction} of $G$ on a $C^*$-algebra $A$ is an injective coaction $\delta$ of the Hopf $C^*$-algebra $C^*(G)$ on $A$ (\cite[Definition A.21]{EKQR}). Let
\begin{equation}\label{Prel.coaction.of.G}
\delta_\lambda:=\overline{{\rm id}_A\otimes\lambda}\circ\delta:A\rightarrow M(A\otimes C^*_r(G))=M(A\otimes S_{W_G}).
\end{equation}
The crossed product $A\rtimes_\delta G$ by $\delta$ is defined to be the reduced crossed crossed product $A\rtimes_{\delta_\lambda}\widehat{S}_{W_G}$ by $\delta_\lambda$ (\cite[Definition~A.39]{EKQR}).
\section{Coactions of Hopf $C^*$-algebras on $C^*$-correspondences}\label{Sec.3}
In this section, we define a coaction $(\sigma,\delta)$ of a Hopf $C^*$-algebra on a $C^*$-cor\-re\-spond\-ence $(X,A)$, and prove that $(\sigma,\delta)$ induces a coaction on the associated Cuntz-Pimsner algebra $\mathcal{O}_X$ under a certain invariance condition (Theorem~\ref{induced coactions on O_X}). Recall that the $C^*$-cor\-re\-spon\-dences considered in this paper are always nondegenerate.
\begin{defn}\label{DefofCoactions}\rm
A \emph{coaction} of a Hopf $C^*$-algebra $(S,\Delta)$ on a $C^*$-cor\-re\-spond\-ence $(X,A)$ is a nondegenerate correspondence homomorphism
\[(\sigma,\delta):(X,A)\rightarrow(M(X\otimes S),M(A\otimes S))\]
such that
\begin{itemize}
\item[\rm(i)] $\delta$ is a coaction of $S$ on the $C^*$-algebra $A$;
\item[\rm(ii)] $\sigma$ satisfies the \emph{coaction identity} $\overline{\sigma\otimes{\rm id}_S}\circ\sigma=\overline{{\rm id}_X\otimes\Delta}\circ\sigma$;
\item[\rm(iii)] $\sigma$ satisfies the \emph{coaction nondegeneracy}
\[ \overline{\varphi_{M(A\otimes S)}(1_{M(A)}\otimes S)\,\sigma(X)}=X\otimes S.\]
\end{itemize}
\end{defn}
Note that the strict extensions $\overline{\sigma\otimes{\rm id}_S}$ and $\overline{{\rm id}_X\otimes\Delta}$ in (ii) are well-defined because the tensor product of two nondegenerate correspondence homomorphisms is also nondegenerate (\cite[Proposition 1.38]{EKQR}).
\begin{rmks}\rm
(1) It should be noted that
\[\overline{\sigma(X)\cdot(1_{M(A)}\otimes S)}=X\otimes S.\]
This follows by the same argument as Remark 2.11.(1) and 2.11.(2) of \cite{EKQR}. We then have $\sigma(X)\subseteq M_S(X\otimes S)\subseteq M_{A\otimes S}(X\otimes S)$.
(2) A coaction on $(X,A)$ in our sense is a coaction on the Hilbert $A$-module $X$ in the sense of \cite[Definition~2.2]{BS}.
\end{rmks}
\begin{rmk}\label{Unify.act.coact.}\rm
Let $G$ be a locally compact group and $(X,A)$ be a $C^*$-cor\-re\-spond\-ence. Theorem \ref{actions=coactions} says that every action of $G$ on $(X,A)$ in the sense of \cite[Definition 2.5]{EKQR} determines a coaction of the Hopf $C^*$-algebra $C_0(G)$ on $(X,A)$, and one can define in this way a one-to-one correspondence between actions of $G$ on $(X,A)$ and coactions of $C_0(G)$ on $(X,A)$. On the other hand, a nondegenerate coaction of $G$ \cite[Definition 2.10]{EKQR} is by definition a coaction $(\sigma,\delta)$ of the Hopf $C^*$-algebra $C^*(G)$ on $(X,A)$ such that $\delta$ is injective. Definition \ref{DefofCoactions} thus unifies the notions of actions and nondegenerate coactions of locally compact groups on $C^*$-cor\-re\-spond\-ences.
\end{rmk}
By Proposition 2.27 (Proposition 2.30, respectively) of \cite{EKQR}, an action (nondegenrate coaction, respectively) of a locally compact group $G$ on $(X,A)$ determines an action (coaction, respectively) of $G$ on $\mathcal{K}(X)$, and the left action $\varphi_A$ satisfies an equivariance condition. The next proposition generalizes this in the Hopf $C^*$-algebra setting. Recall that we identify $\mathcal{K}(X_1\otimes X_2)=\mathcal{K}(X_1)\otimes\mathcal{K}(X_2)$ for two Hilbert modules $X_1$ and $X_2$. In particular, if $(X,A)$ is a $C^*$-cor\-re\-spond\-ence and $C$ is a $C^*$-algebra then $\mathcal{K}(X\otimes C)=\mathcal{K}(X)\otimes C$.
\begin{prop}\label{coaction on KX}
Let $(\sigma,\delta)$ be a coaction of a Hopf $C^*$-algebra $S$ on a $C^*$-cor\-re\-spond\-ence $(X,A)$. Then the nondegenerate homomorphism
\[\sigma^{(1)}:\mathcal{K}(X)\rightarrow M(\mathcal{K}(X\otimes S))= M(\mathcal{K}(X)\otimes S)\]
is a coaction of $S$ on $\mathcal{K}(X)$ and the left action $\varphi_A$ is $\delta$-$\sigma^{(1)}$ equivariant, that is, $\overline{\varphi_A\otimes{\rm id}_S}\circ\delta=\overline{\sigma^{(1)}}\circ\varphi_A$. If $\delta$ is injective then so is $\sigma^{(1)}$.
\end{prop}
\begin{proof}
As noted in \cite[Remark~2.8.(a)]{BS0}, $\sigma^{(1)}$ satisfies the coaction identity. We also have
\begin{equation}\label{Sec.3.Eqn.for.Coact.KX}
\begin{aligned}
\overline{\sigma^{(1)}(\mathcal{K}(X))(1_{M(\mathcal{K}(X))}\otimes S)}
&= \overline{\sigma(X)\sigma(X)^*\varphi_{M(A\otimes S)}(1_{M(A)}\otimes S)} \\
&= \overline{\sigma(X)\big(\varphi_{M(A\otimes S)}(1_{M(A)}\otimes S)\,\sigma(X)\big)^*} \\
&= \overline{\sigma(X)\big((\sigma(X)\cdot(1_{M(A)}\otimes S))\cdot(1_{M(A)}\otimes S)\big)^*} \\
&= \overline{\big(\sigma(X)\cdot(1_{M(A)}\otimes S)\big)\big(\sigma(X)\cdot(1_{M(A)}\otimes S)\big)^*} \\
&= \overline{(X\otimes S)(X\otimes S)^*} = \mathcal{K}(X\otimes S),
\end{aligned}
\end{equation}
in the third and fifth step of which we use the coaction nondegeneracy of $\sigma$. This shows that $\sigma^{(1)}$ satisfies the coaction nondegeneracy, and thus $\sigma^{(1)}$ is a coaction.
The first relation of \eqref{Pre.psi.1} and the fact that $(\sigma,\delta)$ is a correspondence homomorphism yield
\[\overline{\sigma^{(1)}}(\varphi_A(a))\,\sigma(\xi)=\sigma(\varphi_A(a)\xi)=\varphi_{M(A\otimes S)}(\delta(a))\,\sigma(\xi).\]
for $a\in A$ and $\xi\in X$. Multiplying by $1_{M(A)}\otimes s$ on both end sides from the right gives
\[\overline{\sigma^{(1)}}(\varphi_A(a))\big(\sigma(\xi)\cdot(1_{M(A)}\otimes s)\big)=\varphi_{M(A\otimes S)}(\delta(a))\big(\sigma(\xi)\cdot(1_{M(A)}\otimes s)\big)\]
which leads to $\overline{\sigma^{(1)}}(\varphi_A(a))=\varphi_{M(A\otimes S)}(\delta(a))$ by the coaction nondegeneracy of $\sigma$. But $\varphi_{M(A\otimes S)}=\overline{\varphi_A\otimes{\rm id}_S}$ by definition, and then the $\delta$-$\sigma^{(1)}$ equivariancy of $\varphi_A$ follows. For the last assertion, see the comment below \cite[Lemma~2.4]{Kat}.
\end{proof}
\begin{defn}\label{Sec.3.delta.invariant}\rm
Let $(\sigma,\delta)$ be a coaction of a Hopf $C^*$-algebra $S$ on a $C^*$-cor\-re\-spond\-ence $(X,A)$. We say that the ideal $J_X$ is \emph{weakly $\delta$-invariant} if
\[\delta(J_X)(1_{M(A)}\otimes S)\subseteq J_X\otimes S.\]
\end{defn}
\begin{rmk}\label{WI.for.delta}\rm
The coaction nondegeneracy of $\delta$ implies that $J_X$ is weakly $\delta$-invariant if and only if $\delta(J_X)(A\otimes S)\subseteq J_X\otimes S$, namely
\[\delta(J_X)\subseteq M(A\otimes S;J_X\otimes S).\]
\end{rmk}
Under the assumption of the last inclusion in Remark \ref{WI.for.delta} with $S=C^*(G)$, it was proved in \cite[Proposition~3.1]{KQRo2} that every coaction of a locally compact group $G$ on $(X,A)$ induces a coaction of $G$ on the associated Cuntz-Pimsner algebra $\mathcal{O}_X$. Modifying the proof of \cite[Proposition 3.1]{KQRo2} we now prove the next theorem.
\begin{thm}\label{induced coactions on O_X}
Let $(\sigma,\delta)$ be a coaction of a Hopf $C^*$-algebra $S$ on a $C^*$-cor\-re\-spond\-ence $(X,A)$ such that the ideal $J_X$ is weakly $\delta$-invariant. Then the representation
\[
(\overline{k_X\otimes{\rm id}_S}\circ\sigma,\overline{k_A\otimes{\rm id}_S}\circ\delta):(X,A)\rightarrow M_{A\otimes S}(\mathcal{O}_X\otimes S)
\]
is covariant, and its integrated form $\zeta:=(\overline{k_X\otimes{\rm id}_S}\circ\sigma)\times(\overline{k_A\otimes{\rm id}_S}\circ\delta)$ is a coaction of $S$ on $\mathcal{O}_X$ such that the diagram
\begin{equation}\label{Sec.3.diagram}
\begin{gathered}
\xymatrix{(X,A) \ar[rr]^-{(\sigma,\delta)} \ar[d]_-{(k_X,k_A)}
&& (M_{A\otimes S}(X\otimes S),M(A\otimes S)) \ar[d]^-{(\overline{k_X\otimes{\rm id}_S},\overline{k_A\otimes{\rm id}_S})} \\
\mathcal{O}_X \ar[rr]_-{\zeta} && M_{A\otimes S}(\mathcal{O}_X\otimes S)}
\end{gathered}
\end{equation}
commutes. If $\delta$ is injective then so is $\zeta$.
\end{thm}
\begin{proof}
Let us first prove that $(\overline{k_X\otimes{\rm id}_S}\circ\sigma,\overline{k_A\otimes{\rm id}_S}\circ\delta)$ is covariant, that is,
\begin{equation*}\label{Sec.3.Lemma1.Pf.1}
\big(\overline{k_X\otimes{\rm id}_S}\circ\sigma\big)^{(1)}\circ\varphi_A=\overline{k_A\otimes{\rm id}_S}\circ\delta
\end{equation*}
on $J_X$. Since $\sigma(X)\subseteq M_{A\otimes S}(X\otimes S)$ and thus $\sigma^{(1)}(\mathcal{K}(X))\subseteq\mathcal{K}(M_{A\otimes S}(X\otimes S))$, we have
\[\big(\overline{k_X\otimes{\rm id}_S}\circ\sigma\big)^{(1)}=\overline{(k_X\otimes{\rm id}_S)^{(1)}}\circ\sigma^{(1)}\]
on $\mathcal{K}(X)$ by Theorem \ref{DKQ,Cor.A.14}.(ii). We then have
\begin{align*}
\big(\overline{k_X\otimes{\rm id}_S}\circ\sigma\big)^{(1)}\circ\varphi_A
&=\overline{(k_X\otimes{\rm id}_S)^{(1)}}\circ\sigma^{(1)}\circ\varphi_A \\
&=\overline{(k_X\otimes{\rm id}_S)^{(1)}}\circ\overline{\varphi_{A\otimes S}}\circ\delta
\end{align*}
on $J_X$ since $\overline{\sigma^{(1)}}\circ\varphi_A=\overline{\varphi_{A\otimes S}}\circ\delta$ by \cite[Lemma~3.3]{KQRo1}. Hence, the requirement that $(\overline{k_X\otimes{\rm id}_S}\circ\sigma,\overline{k_A\otimes{\rm id}_S}\circ\delta)$ be covariant amounts to that
\begin{equation*}\label{Sec.3.Lemma1.Pf.4}
\overline{(k_X\otimes{\rm id}_S)^{(1)}}\circ\overline{\varphi_{A\otimes S}}\circ\delta=\overline{k_A\otimes{\rm id}_S}\circ\delta
\end{equation*}
holds on $J_X$. By Remark \ref{WI.for.delta}, this equality will follow if we show that
\[\overline{(k_X\otimes{\rm id}_S)^{(1)}}\circ\overline{\varphi_{A\otimes S}}=\overline{k_A\otimes{\rm id}_S}\]
on $M(A\otimes S;J_X\otimes S)$. But, this is the content of Lemma \ref{Lemma.for.Ext.Main.Thm.of.KQR2}, and therefore the representation $(\overline{k_X\otimes{\rm id}_S}\circ\sigma,\overline{k_A\otimes{\rm id}_S}\circ\delta)$ is covariant.
We now show that $\zeta$ is a coaction of $(S,\Delta)$ on $\mathcal{O}_X$. Since
\begin{align*}
\overline{(1_{M(\mathcal{O}_X)}\otimes S)\zeta(k_X(X))}
&= \overline{\overline{k_A\otimes{\rm id}_S}(1_{M(A)}\otimes S)\overline{k_X\otimes{\rm id}_S}(\sigma(X))} \\
&= \overline{k_X\otimes{\rm id}_S\big(\varphi_{M(A\otimes S)}(1_{M(A)}\otimes S)\,\sigma(X)\big)} \\
&= \overline{k_X(X)\odot S},
\end{align*}
we have
\[
\overline{\zeta(k_X(X)^*)(1_{M(\mathcal{O}_X)}\otimes S)}
= \big(\,\overline{(1_{M(\mathcal{O}_X)}\otimes S)\zeta(k_X(X))}\,\big)^*
= \overline{k_X(X)^*\odot S}.
\]
We also have $\overline{\zeta(k_X(X))(1_{M(\mathcal{O}_X)}\otimes S)}=\overline{k_X(X)\odot S}$. From these and the coaction nondegeneracy of $\delta$, we can deduce that $\zeta$ satisfies the coaction nondegeneracy.
The coaction nondegeneracy of $\zeta$ implies $\zeta(\mathcal{O}_X)\subseteq M_{A\otimes S}(\mathcal{O}_X\otimes S)$, and then we have the commutative diagram \eqref{Sec.3.diagram}.
We can easily see that $\overline{\zeta\otimes{\rm id}_S}\circ\zeta\circ k_A=\overline{{\rm id}_{\mathcal{O}_X}\otimes\Delta}\circ\zeta\circ k_A$ by \eqref{Sec.3.diagram}, strict continuity, and the coaction identity of $\delta$. To prove the corresponding equality for $k_X$, we first note the followings. Let $x\in A\otimes S$ and $m\in M_{A\otimes S}(\mathcal{O}_X\otimes S)$. Then
\begin{align*}
(\zeta\otimes{\rm id}_S)\big((k_A\otimes{\rm id}_S)(x)\,m\big)
&=\overline{k_A\otimes{\rm id}_S\otimes{\rm id}_S}\big((\delta\otimes{\rm id}_S)(x)\big)\,\overline{\zeta\otimes{\rm id}_S}(m), \\
({\rm id}_{\mathcal{O}_X}\otimes\Delta)\big((k_A\otimes{\rm id}_S)(x)\,m\big)
&=\overline{k_A\otimes{\rm id}_S\otimes{\rm id}_S}\big(({\rm id}_{A}\otimes\Delta)(x)\big)\,\overline{{\rm id}_{\mathcal{O}_X}\otimes\Delta}(m),
\end{align*}
and similarly for $(\zeta\otimes{\rm id}_S)\big(m\,(k_A\otimes{\rm id}_S)(x)\big)$ and $({\rm id}_{\mathcal{O}_X}\otimes\Delta)\big(m\,(k_A\otimes{\rm id}_S)(x)\big)$. From these relations and also the nondegeneracy of $\delta\otimes{\rm id}_S$ and ${\rm id}_A\otimes\Delta$, we deduce that the restrictions
\[\overline{\zeta\otimes{\rm id}_S},\ \overline{{\rm id}_{\mathcal{O}_X}\otimes\Delta}:M_{A\otimes S}(\mathcal{O}_X\otimes S)\rightarrow M_{A\otimes S\otimes S}(\mathcal{O}_X\otimes S\otimes S)\]
are $(A\otimes S)$-strict to $(A\otimes S\otimes S)$-strictly continuous (cf.\ \cite[Lemma~A.5]{DKQ}). Therefore the following compositions
\begin{multline}\label{Sec.3.Pf.Thm.mult.1}
\overline{\zeta\otimes{\rm id}_S}\circ\overline{k_X\otimes{\rm id}_S},\ \overline{{\rm id}_{\mathcal{O}_X}\otimes\Delta}\circ\overline{k_X\otimes{\rm id}_S}:\\
M_{A\otimes S}(X\otimes S)\rightarrow M_{A\otimes S\otimes S}(\mathcal{O}_X\otimes S\otimes S)
\end{multline}
are $(A\otimes S)$-strict to $(A\otimes S\otimes S)$-strictly continuous. Similarly, both maps
\[\overline{\sigma\otimes{\rm id}_S},\ \overline{{\rm id}_X\otimes\Delta}:M_{A\otimes S}(X\otimes S)\rightarrow M_{A\otimes S\otimes S}(X\otimes S\otimes S)\]
are $(A\otimes S)$-strict to $(A\otimes S\otimes S)$-strictly continuous, and hence so are the maps
\begin{multline}\label{Sec.3.Pf.Thm.mult.2}
\overline{k_X\otimes{\rm id}_S\otimes{\rm id}_S}\circ\overline{\sigma\otimes{\rm id}_S},\ \overline{k_X\otimes{\rm id}_S\otimes{\rm id}_S}\circ\overline{{\rm id}_X\otimes\Delta}:\\
M_{A\otimes S}(X\otimes S)\rightarrow M_{A\otimes S\otimes S}(\mathcal{O}_X\otimes S\otimes S).
\end{multline}
Since the equalities
\begin{align*}
\overline{\zeta\otimes{\rm id}_S}\circ\overline{k_X\otimes{\rm id}_S}
&=\overline{k_X\otimes{\rm id}_S\otimes{\rm id}_S}\circ\overline{\sigma\otimes{\rm id}_S}, \\
\overline{k_X\otimes{\rm id}_S\otimes{\rm id}_S}\circ\overline{{\rm id}_X\otimes\Delta}
&=\overline{{\rm id}_{\mathcal{O}_X}\otimes\Delta}\circ\overline{k_X\otimes{\rm id}_S}
\end{align*}
hold on $X\odot S$ which is $(A\otimes S)$-strictly dense in $M_{A\otimes S}(X\otimes S)$ and since $\sigma(X)\subseteq M_{A\otimes S}(X\otimes S)$, we now have
\begin{align*}
\overline{\zeta\otimes{\rm id}_S}\circ\zeta\circ k_X
&=\overline{\zeta\otimes{\rm id}_S}\circ\overline{k_X\otimes{\rm id}_S}\circ\sigma \\
&=\overline{k_X\otimes{\rm id}_S\otimes{\rm id}_S}\circ\overline{\sigma\otimes{\rm id}_S}\circ\sigma \\
&=\overline{k_X\otimes{\rm id}_S\otimes{\rm id}_S}\circ\overline{{\rm id}_X\otimes\Delta}\circ\sigma \\
&=\overline{{\rm id}_{\mathcal{O}_X}\otimes\Delta}\circ\overline{k_X\otimes{\rm id}_S}\circ\sigma
=\overline{{\rm id}_{\mathcal{O}_X}\otimes\Delta}\circ\zeta\circ k_X
\end{align*}
by the $(A\otimes S)$-strict to $(A\otimes S\otimes S)$-strict continuity of the maps of \eqref{Sec.3.Pf.Thm.mult.1} and \eqref{Sec.3.Pf.Thm.mult.2} and also by the coaction identity of $\sigma$. Thus $\zeta$ satisfies the coaction identity.
For the last assertion of the theorem, assume that $\delta$ is injective. We only need to show by \cite[Theorem~6.4]{Kat} that the injective covariant representation $(\overline{k_X\otimes{\rm id}_S}\circ\sigma,\overline{k_A\otimes{\rm id}_S}\circ\delta)$ admits a gauge action. Let $\beta:\mathbb{T}\rightarrow Aut(\mathcal{O}_X)$ be the gauge action. Note that for each $z\in\mathbb{T}$, the strict extension $\overline{\beta_z\otimes{\rm id}_S}$ on $M(\mathcal{O}_X\otimes S)$ maps $M_{A\otimes S}(\mathcal{O}_X\otimes S)$ onto itself. Then the composition
\begin{multline}\label{Sec.3.Comp.Str.Maps}
(\overline{\beta_z\otimes{\rm id}_S}\circ\overline{k_X\otimes{\rm id}_S},\,\overline{\beta_z\otimes{\rm id}_S}\circ\overline{k_A\otimes{\rm id}_S}): \\
(M_{A\otimes S}(X\otimes S),M(A\otimes S))\rightarrow M_{A\otimes S}(\mathcal{O}_X\otimes S)
\end{multline}
gives a representation which is clearly $(A\otimes S)$-strictly continuous. Since the equalities
\begin{align*}
\overline{\beta_z\otimes{\rm id}_S}\circ\overline{k_X\otimes{\rm id}_S}(m)
&=z\,\overline{k_X\otimes{\rm id}_S}(m), \\
\quad\overline{\beta_z\otimes{\rm id}_S}\circ\overline{k_A\otimes{\rm id}_S}(n)
&=\overline{k_A\otimes{\rm id}_S}(n)
\end{align*}
are valid for $m\in X\odot S$ and $n\in A\odot S$, and the representation \eqref{Sec.3.Comp.Str.Maps} is $(A\otimes S)$-strictly continuous, the above equalities still hold for $m\in M_{A\otimes S}(X\otimes S)$ and $n\in M(A\otimes S)$. Since $\sigma(X)\subseteq M_{A\otimes S}(X\otimes S)$, it thus follows that
\[\overline{\beta_z\otimes{\rm id}_S}\circ\overline{k_X\otimes{\rm id}_S}\circ\sigma=z\,\overline{k_X\otimes{\rm id}_S}\circ\sigma,\]
and similarly that $\overline{\beta_z\otimes{\rm id}_S}\circ\overline{k_A\otimes{\rm id}_S}\circ\delta=\overline{k_A\otimes{\rm id}_S}\circ\delta$. This proves that the restrictions of $\overline{\beta_z\otimes{\rm id}_S}$ to $\zeta(\mathcal{O}_X)$ $(z\in\mathbb{T})$ define a gauge action of $\mathbb{T}$ on $\zeta(\mathcal{O}_X)$, which establishes the theorem.
\end{proof}
\begin{defn}\rm
We call $\zeta$ in Theorem \ref{induced coactions on O_X} the coaction \emph{induced} by $(\sigma,\delta)$.
\end{defn}
\begin{rmks}\label{Sec.3.Rmk.to.Thm}\rm
(1) Let $G$ be a locally compact group. If $(\sigma,\delta)$ is a coaction of $C_0(G)$ on $(X,A)$, then $\overline{\delta(J_X)(1_{M(A)}\otimes S)}=J_X\otimes S$ by \cite[Lemma 2.6.(a)]{HaoNg} and Theorem \ref{actions=coactions}. Hence, $J_X$ is automatically weakly $\delta$-invariant in this case.
(2) Replacing in the diagram \eqref{Sec.3.diagram} the $(A\otimes S)$-multiplier correspondence and $(A\otimes S)$-multiplier algebra by $(M_S(X\otimes S),M_S(A\otimes S))$ and $M_S(\mathcal{O}_X\otimes S)$, respectively, we can regard $(\overline{k_X\otimes{\rm id}_S},\overline{k_A\otimes{\rm id}_S})$ as the $S$-strict extension by Remarks \ref{Prel.Str.Ext.}.(2).
\end{rmks}
\section{Reduced crossed product correspondences}\label{Sec.4}
In this section and the next, we restrict our attention to coactions of reduced Hopf $C^*$-algebras defined by well-behaved multiplicative unitaries.
\begin{notation}\rm
To simplify the notations, we often write $S$ and $\widehat{S}$, respectively, for the ``reduced'' and ``dual reduced'' Hopf $C^*$-algebras $S_V$ and $\widehat{S}_V$ defined by a well-behaved multiplicative unitary $V$. We also write $\mathscr{K}$ for $\mathcal{K}(\mathcal{H})$.
Let $(\sigma,\delta)$ be a coaction of $S$ on $(X,A)$ and $\iota_S:S\hookrightarrow M(\mathscr{K})$ be the inclusion map. As $\delta_\iota$ in \eqref{Prel.delta.iota}, we denote by $\sigma_\iota$ the composition
\[\sigma_\iota=\overline{{\rm id}_X\otimes\iota_{S}}\circ\sigma,\]
where $\overline{{\rm id}_X\otimes\iota_{S}}$ is the strict extension. Evidently, $(\sigma_\iota,\delta_\iota)$ is a nondegenerate correspondence homomorphism:
\[ \xymatrix @C=0pc {(X,A) \ar[rr]^-{(\sigma_\iota,\delta_\iota)} \ar[dr]_-{(\sigma,\delta)}
&& (M(X\otimes\mathscr{K}),M(A\otimes\mathscr{K})) \\
& (M(X\otimes S),M(A\otimes S)) \ar[ur]_-{\qquad(\overline{{\rm id}_X\otimes\iota_S},\,\overline{{\rm id}_A\otimes\iota_S})} &
}\]
\end{notation}
For a coaction $(\sigma,\delta)$ of $S$ on $(X,A)$, we have $\sigma_\iota(X)\cdot(1_{M(A)}\otimes\widehat{S})\subseteq M(X\otimes\mathscr{K})$ and similarly for $\varphi_{M(A\otimes\mathscr{K})}(1_{M(A)}\otimes\widehat{S})\,\sigma_\iota(X)$ since $(M(X\otimes\mathscr{K}),M(A\otimes\mathscr{K}))$ is a $C^*$-cor\-re\-spond\-ence.
\begin{lem}[{\cite[Proposition~1.3]{Bui}}]\label{Lemma similar to BS's}
The norm closures in $M(X\otimes\mathscr{K})$ of the subspaces $\sigma_\iota(X)\cdot(1_{M(A)}\otimes\widehat{S})$ and $\varphi_{M(A\otimes\mathscr{K})}(1_{M(A)}\otimes\widehat{S})\,\sigma_\iota(X)$ coincide.
\end{lem}
Although \cite[Proposition~1.3]{Bui} assumes the regularity condition \cite{BS} for multiplicative unitaries, the proof uses only the pentagonal relation, and hence works the same for well-behaved multiplicative unitaries.
We denote by $X\rtimes_\sigma\widehat{S}$ the norm closure of the subspaces considered in Lemma~\ref{Lemma similar to BS's}:
\[X\rtimes_\sigma\widehat{S}:=\overline{\sigma_\iota(X)\cdot(1_{M(A)}\otimes\widehat{S})}
=\overline{\varphi_{M(A\otimes\mathscr{K})}(1_{M(A)}\otimes\widehat{S})\,\sigma_\iota(X)}.\]
It is obvious that $X\rtimes_\sigma\widehat{S}$ is a Hilbert $(A\rtimes_\delta\widehat{S})$-module with respect to the restriction of the operations on $(M(X\otimes\mathscr{K}),M(A\otimes\mathscr{K}))$.
\begin{thm}\label{crossed product correspondences}
Let $(\sigma,\delta)$ be a coaction of a reduced Hopf $C^*$-algebra $S$ on a $C^*$-cor\-re\-spon\-dence $(X,A)$. Then $(X\rtimes_\sigma\widehat{S},A\rtimes_\delta\widehat{S})$ is a nondegenerate $C^*$-cor\-re\-spond\-ence such that the inclusion
\begin{equation*}\label{Sec.4.Inc.in.Thm}
(X\rtimes_\sigma\widehat{S},A\rtimes_\delta\widehat{S})\hookrightarrow
(M(X\otimes\mathscr{K}),M(A\otimes\mathscr{K}))
\end{equation*}
is a nondegenerate correspondence homomorphism. The left action $\varphi_{A\rtimes_\delta\widehat{S}}$ is injective if $\varphi_A$ is injective. Also,
\begin{equation*}\label{Sec.4.Cpt.Eq}
\mathcal{K}(X\rtimes_\sigma\widehat{S})=\mathcal{K}(X)\rtimes_{\sigma^{(1)}}\widehat{S},
\end{equation*}
where $\sigma^{(1)}$ is the coaction in Proposition \ref{coaction on KX}, and
\begin{equation*}\label{PhiDelta=DeltaMu}
\varphi_{A\rtimes_\delta\widehat{S}}\big(\delta_\iota(a)(1_{M(A)}\otimes x)\big)=\overline{\sigma^{(1)}_\iota}(\varphi_A(a))(1_{M(\mathcal{K}(X))}\otimes x)
\end{equation*}
for $a\in A$ and $x\in\widehat{S}$.
\end{thm}
\begin{proof}
The first assertion amounts to saying that
\begin{itemize}
\item[(i)] the Hilbert $(A\rtimes_\delta\widehat{S})$-module $X\rtimes_\sigma\widehat{S}$ is a nondegenerate $C^*$-cor\-re\-spond\-ence such that $\varphi_{A\rtimes_\delta\widehat{S}}=\varphi_{M(A\otimes\mathscr{K})}|_{A\rtimes_\delta\widehat{S}}$, namely
\[\overline{\varphi_{M(A\otimes\mathscr{K})}\big(\delta_\iota(A)(1_{M(A)}\otimes\widehat{S})\big)\,
\sigma_\iota(X)\cdot(1_{M(A)}\otimes\widehat{S})}=X\rtimes_\sigma\widehat{S};\]
\item[(ii)] the inclusion $(X\rtimes_\sigma\widehat{S},A\rtimes_\delta\widehat{S})\hookrightarrow
(M(X\otimes\mathscr{K}),M(A\otimes\mathscr{K}))$ is a nondegenerate correspondence homomorphism, namely
\[\overline{(X\rtimes_\sigma\widehat{S})\cdot(A\otimes\mathscr{K})}=X\otimes\mathscr{K},\quad
\overline{(A\rtimes_\delta\widehat{S})(A\otimes\mathscr{K})}=A\otimes\mathscr{K}.\]
\end{itemize}
Lemma \ref{Lemma similar to BS's} shows that
\[\overline{\varphi_{M(A\otimes\mathscr{K})}(1_{M(A)}\otimes\widehat{S})\,
\sigma_\iota(X)\cdot(1_{M(A)}\otimes\widehat{S})}=\overline{\sigma_\iota(X)\cdot(1_{M(A)}\otimes\widehat{S})}.\]
Since $\varphi_A$ is nondegenerate, this equality combined with the following
\[\varphi_{M(A\otimes\mathscr{K})}(\delta_\iota(A))\,\sigma_\iota(X)=\sigma_\iota(\varphi_A(A)X)\]
gives (i). Since $S$ and $\widehat{S}$ are both nondegenerate subalgebras of $M(\mathscr{K})$, we have
\begin{equation}\label{Sec.4.in.the.Pf.thm}
\begin{aligned}
\overline{(X\rtimes_\sigma\widehat{S})\cdot(A\otimes\mathscr{K})}
&=\overline{\sigma_\iota(X)\cdot(A\otimes \widehat{S}\mathscr{K})} \\
&=\overline{\sigma_\iota(X)\cdot(1_{M(A)}\otimes S)\cdot(A\otimes \widehat{S}\mathscr{K})} \\
&=\overline{(X\otimes S)\cdot(A\otimes\mathscr{K})} = X\otimes\mathscr{K}
\end{aligned}
\end{equation}
and similarly $\overline{(A\rtimes_\delta\widehat{S})(A\otimes\mathscr{K})}=A\otimes\mathscr{K}$. This verifies (ii), and the first assertion of the theorem is established. Since $\varphi_{A\rtimes_\delta\widehat{S}}$ is the restriction of $\overline{\varphi_A\otimes{\rm id}_{\mathscr{K}}}$ which is injective if $\varphi_A$ is, the assertion on the injectivity of $\varphi_{A\rtimes_\delta\widehat{S}}$ follows.
As in the computation \eqref{Sec.3.Eqn.for.Coact.KX}, but using Lemma \ref{Lemma similar to BS's} instead of coaction nondegeneracy, we can deduce the equality $\mathcal{K}(X\rtimes_\sigma\widehat{S})=\mathcal{K}(X)\rtimes_{\sigma^{(1)}}\widehat{S}$. Finally,
\begin{align*}
\varphi_{A\rtimes_\delta\widehat{S}}\big(\delta_\iota(a)(1_{M(A)}\otimes x)\big)
&= \overline{\varphi_A\otimes{\rm id}_{\mathscr{K}}}\circ\overline{{\rm id}_A\otimes\iota_S}(\delta(a))\,(1_{M(\mathcal{K}(X))}\otimes x) \\
&= \overline{{\rm id}_{\mathcal{K}(X)}\otimes\iota_{S}}\circ\overline{\varphi_A\otimes{\rm id}_S}(\delta(a))\,(1_{M(\mathcal{K}(X))}\otimes x) \\
&= \overline{{\rm id}_{\mathcal{K}(X)}\otimes\iota_{S}}\circ\overline{\sigma^{(1)}}(\varphi_A(a))\,(1_{M(\mathcal{K}(X))}\otimes x) \\
&= \overline{\sigma^{(1)}_\iota}(\varphi_A(a))(1_{M(\mathcal{K}(X))}\otimes x),
\end{align*}
in the third step of which we use the $\delta$-$\sigma^{(1)}$ equivariancy of $\varphi_A$ obtained in Proposition \ref{coaction on KX}. This completes the proof.
\end{proof}
\begin{defn}\label{Sec.4.Def}\rm
We call the $C^*$-correspondence $(X\rtimes_\sigma\widehat{S},A\rtimes_\delta\widehat{S})$ in Theorem~\ref{crossed product correspondences} the \emph{reduced crossed product correspondence} of $(X,A)$ by the coaction $(\sigma,\delta)$ of $S$.
\end{defn}
\begin{rmk}\label{sigmaL,deltaL define a non-deg.corr.hom.}\rm
We require no universal property of the crossed product $A\rtimes_\delta\widehat{S}$ to define the left action $\varphi_{A\rtimes_\delta\widehat{S}}:A\rtimes_\delta\widehat{S}\rightarrow\mathcal{L}(X\rtimes_\sigma\widehat{S})$. It is just the restriction of $\varphi_{M(A\otimes\mathscr{K})}$.
\end{rmk}
\begin{rmk}\label{Sec.4.Rmk.for.Justi.}\rm
For an action $(\gamma,\alpha)$ of a locally compact group $G$ on $(X,A)$, one can form the crossed product correspondence $(X\rtimes_{\gamma,r}G,A\rtimes_{\alpha,r}G)$ by \cite[Proposition~3.2]{EKQR}. We will see in Corollary \ref{Appendix.B.Cor.1} that it is isomorphic to the reduced crossed product correspondence $(X\rtimes_{\sigma^\gamma_G}\widehat{S}_{\widehat{W}_G},A\rtimes_{\delta^\alpha_G}\widehat{S}_{\widehat{W}_G})$, where $(\sigma^\gamma_G,\delta^\alpha_G)$ is the coaction of the Hopf $C^*$-algebra $S_{\widehat{W}_G}$ given in \eqref{Appendix.B.Cor.1.Eqn}. On the other hand, if $(\sigma,\delta)$ is a nondegenerate coaction of $G$ on $(X,A)$ (\cite[Definition 2.10]{EKQR}) and if $\sigma_\lambda:=\overline{{\rm id}_X\otimes\lambda}\circ\sigma$ as \eqref{Prel.coaction.of.G}, then the crossed product correspondence by $(\sigma,\delta)$ in the sense of \cite[Proposition~3.9]{EKQR} is just the reduced crossed product correspondence by the coaction $(\sigma_\lambda,\delta_\lambda)$ of the Hopf $C^*$-algebra $S_{W_G}$. Construction in Theorem~\ref{crossed product correspondences} thus extends both of the crossed product correspondences by actions and nondegenerate coactions of locally compact groups on $C^*$-cor\-re\-spond\-ences.
\end{rmk}
As in \cite[Remark 2.7]{KQRo2}, we have the following corollary, the proof of which is routine.
\begin{cor}\label{Sec.4.jxja}
Let $(\sigma,\delta)$ be a coaction of $S$ on $(X,A)$. Then the map
\[(j_X^\sigma,j_A^\delta):(X,A)\rightarrow (M(X\rtimes_\sigma\widehat{S}),M(A\rtimes_\delta\widehat{S}))\]
defined by
\[j_X^\sigma(\xi)\cdot c:=\sigma_\iota(\xi)\cdot c,\quad j_A^\delta(a)c:=\delta_\iota(a)c\]
for $\xi\in X$, $a\in A$, and $c\in A\rtimes_\delta\widehat{S}$ is a nondegenerate correspondence homomorphism such that $j_X^\sigma(X)\subseteq M_{A\rtimes_\delta\widehat{S}}(X\rtimes_\sigma\widehat{S})$.
\end{cor}
\begin{rmk}\label{Sec.4.Rmk.for.Pf.Main.Thm}\rm
Since $(X\rtimes_\sigma\widehat{S},A\rtimes_\delta\widehat{S})$ is by Theorem~\ref{crossed product correspondences} a nondegenerate $C^*$-cor\-re\-spond\-ence, we have by Theorem~\ref{DKQ,Cor.A.14}.(i) the $(A\rtimes_\delta\widehat{S})$-strict extension
\[(\overline{k_{X\rtimes_\sigma\widehat{S}}},\overline{k_{A\rtimes_\delta\widehat{S}}}):
(M_{A\rtimes_\delta\widehat{S}}(X\rtimes_\sigma\widehat{S}),M(A\rtimes_\delta\widehat{S}))\hookrightarrow M_{A\rtimes_\delta\widehat{S}}(\mathcal{O}_{X\rtimes_\sigma\widehat{S}})\]
for the universal covariant representation $(k_{X\rtimes_\sigma\widehat{S}},k_{A\rtimes_\delta\widehat{S}})$. The composition
\[(\overline{k_{X\rtimes_\sigma\widehat{S}}}\circ j_X^\sigma,\overline{k_{A\rtimes_\delta\widehat{S}}}\circ j_A^\delta):(X,A)\rightarrow M_{A\rtimes_\delta\widehat{S}}(\mathcal{O}_{X\rtimes_\sigma\widehat{S}}).\]
then gives a representation of $(X,A)$. It can be shown that this representation is covariant although we do not need this fact in the sequel.
\end{rmk}
\section{Reduced crossed products}\label{Sec.5}
In this section, we first show that the $C^*$-cor\-re\-spond\-ence $(X\rtimes_\sigma\widehat{S},A\rtimes_\delta\widehat{S})$ has a representation $(k_X\rtimes_\sigma{\rm id},k_A\rtimes_\delta{\rm id})$ on the reduced crossed product $\mathcal{O}_X\rtimes_\zeta\widehat{S}$. We then provide a couple of equivalent conditions that this representation is covariant, which is readily seen to be the case if the ideal $J_{X\rtimes_\sigma\widehat{S}}$ of $A\rtimes_\delta\widehat{S}$ is generated by $\delta_\iota(J_X)$ or the left action $\varphi_A$ is injective. Under this covariance condition, the integrated form of the representation $(X\rtimes_\sigma\widehat{S},A\rtimes_\delta\widehat{S})$ gives an isomorphism between the $C^*$-algebra $\mathcal{O}_X\rtimes_\zeta\widehat{S}$ and the Cuntz-Pimsner algebra $\mathcal{O}_{X\rtimes_\sigma\widehat{S}}$. The representation
\begin{equation*}\label{Sec.5.Begin}
(\overline{k_X\otimes{\rm id}_{\mathscr{K}}},\overline{k_A\otimes{\rm id}_{\mathscr{K}}}):
(M_{A\otimes\mathscr{K}}(X\otimes\mathscr{K}),M(A\otimes\mathscr{K}))\rightarrow M_{A\otimes\mathscr{K}}(\mathcal{O}_X\otimes\mathscr{K})
\end{equation*}
will play an important role in our analysis.
Recall that $\overline{k_X\otimes\id_C}$ denotes the $(A\otimes C)$-strict extension to $M_{A\otimes C}(X\otimes C)$.
\begin{lem}\label{the diagram}
Let $(X,A)$ be a $C^*$-correspondence. Let $S$ be a reduced Hopf $C^*$-algebra and $\iota_{S}:S\hookrightarrow M(\mathscr{K})$ be the inclusion. Then the following diagram commutes:
\begin{equation}\label{Sec.5.diagram}
\begin{gathered}
\xymatrix{(M_{A\otimes S}(X\otimes S),M(A\otimes S))
\ar[rr]^-{(\overline{{\rm id}_X\otimes\iota_{S}},\overline{{\rm id}_A\otimes\iota_{S}})}
\ar[d]_-{(\overline{k_X\otimes{\rm id}_{S}},\overline{k_A\otimes{\rm id}_{S}})}
&& (M_{A\otimes\mathscr{K}}(X\otimes\mathscr{K}),M(A\otimes\mathscr{K}))
\ar[d]^-{(\overline{k_X\otimes{\rm id}_{\mathscr{K}}},\overline{k_A\otimes{\rm id}_{\mathscr{K}}})} \\
M_{A\otimes S}(\mathcal{O}_X\otimes S)
\ar[rr]_-{\overline{{\rm id}_{\mathcal{O}_X}\otimes\iota_{S}}}
&& M_{A\otimes\mathscr{K}}(\mathcal{O}_X\otimes\mathscr{K}) }
\end{gathered}
\end{equation}
\end{lem}
\begin{proof}
By \cite[Proposition A.11]{DKQ}, we see that the upper and lower horizontal maps are $(A\otimes S)$-strict to $(A\otimes\mathscr{K})$-strictly continuous. Hence the two compositions in \eqref{Sec.5.diagram} are $(A\otimes S)$-strict to $(A\otimes\mathscr{K})$-strictly continuous. Since the diagram commutes on $(X\odot S,A\odot S)$, the conclusion follows by strict continuity.
\end{proof}
\begin{cor}\label{Sec.5.cor.to.diagram}
Let $(\sigma,\delta)$ be a coaction of $S$ on $(X,A)$ such that $J_X$ is weakly $\delta$-invariant. Then $\sigma_\iota(X)\subseteq M_{A\otimes\mathscr{K}}(X\otimes\mathscr{K})$ and
\begin{equation*}\label{Sec.5.upper.zeta.iota.2}
X\rtimes_\sigma\widehat{S}\subseteq M_{A\otimes\mathscr{K}}(X\otimes\mathscr{K}).
\end{equation*}
Also,
\begin{equation}\label{Sec.5.zeta.iota}
\overline{k_X\otimes{\rm id}_{\mathscr{K}}}(\sigma_\iota(\xi))=\zeta_\iota(k_X(\xi)), \quad
\overline{k_A\otimes{\rm id}_{\mathscr{K}}}(\delta_\iota(a))=\zeta_\iota(k_A(a))
\end{equation}
for $\xi\in X$ and $a\in A$.
\end{cor}
\begin{proof}
By Theorem \ref{induced coactions on O_X}, we can consider the induced coaction $\zeta$ on $\mathcal{O}_X$ making the diagram \eqref{Sec.3.diagram} commute. Combining \eqref{Sec.3.diagram} and \eqref{Sec.5.diagram} we see that
\[\sigma_\iota(X)=\overline{{\rm id}_X\otimes\iota_S}(\sigma(X))\subseteq M_{A\otimes\mathscr{K}}(X\otimes\mathscr{K}),\]
and thus
\begin{align*}
X\rtimes_\sigma\widehat{S}&=\overline{\sigma_\iota(X)\cdot(1_{M(A)}\otimes\widehat{S})}\\
&\subseteq M_{A\otimes\mathscr{K}}(X\otimes\mathscr{K})\cdot M(A\otimes\mathscr{K})=M_{A\otimes\mathscr{K}}(X\otimes\mathscr{K}).
\end{align*}
The equalities of \eqref{Sec.5.zeta.iota} are also immediate from \eqref{Sec.3.diagram} and \eqref{Sec.5.diagram}.
\end{proof}
\begin{rmk}\label{Sec.5.Rmk.to.Cor.to.Diag}\rm
From Corollary \ref{Sec.5.cor.to.diagram} (and also from Theorem \ref{crossed product correspondences}), we have an injective correspondence homomorphism
\[(X\rtimes_\sigma\widehat{S},A\rtimes_\delta\widehat{S})\hookrightarrow
(M_{A\otimes\mathscr{K}}(X\otimes\mathscr{K}),M(A\otimes\mathscr{K})).\]
We then have
\[\mathcal{K}(X\rtimes_\sigma\widehat{S})\subseteq \mathcal{K}\big(M_{A\otimes\mathscr{K}}(X\otimes\mathscr{K})\big) \subseteq M_{A\otimes\mathscr{K}}\big(\mathcal{K}(X\otimes\mathscr{K})\big).\]
\end{rmk}
\begin{prop}\label{embedding XxShat to OxShat is a Toep.rep.}
Let $(\sigma,\delta)$ be a coaction of $S$ on $(X,A)$ such that $J_X$ is weakly $\delta$-invariant. Then, the restriction of $(\overline{k_X\otimes{\rm id}_{\mathscr{K}}},\overline{k_A\otimes{\rm id}_{\mathscr{K}}})$ to $(X\rtimes_\sigma\widehat{S},A\rtimes_\delta\widehat{S})$ defines an injective representation
\begin{equation*}\label{Sec.5.Prop.1}
(k_X\rtimes_\sigma{\rm id}_{\widehat{S}},k_A\rtimes_\delta{\rm id}_{\widehat{S}}):(X\rtimes_\sigma\widehat{S},A\rtimes_\delta\widehat{S})\rightarrow\mathcal{O}_X\rtimes_{\zeta}\widehat{S}
\end{equation*}
such that
\begin{equation}\label{Sec.5.Rep.1}
\begin{aligned}
k_X\rtimes_\sigma{\rm id}_{\widehat{S}}\big(\sigma_\iota(\xi)\cdot(1_{M(A)}\otimes x)\big) &=\zeta_\iota(k_X(\xi))(1_{M(\mathcal{O}_X)}\otimes x), \\
k_A\rtimes_\delta{\rm id}_{\widehat{S}}\big(\delta_\iota(a)(1_{M(A)}\otimes x)\big)
&=\zeta_\iota(k_A(a))(1_{M(\mathcal{O}_X)}\otimes x), \\
k_X\rtimes_\sigma{\rm id}_{\widehat{S}}\big(\varphi_{M(A\otimes\mathscr{K})}(1_{M(A)}\otimes x)\sigma_\iota(\xi)\big)
&= (1_{M(\mathcal{O}_X)}\otimes x)\zeta_\iota(k_X(\xi))
\end{aligned}
\end{equation}
for $\xi\in X$, $x\in\widehat{S}$, and $a\in A$.
\end{prop}
\begin{proof}
Since $X\rtimes_\sigma\widehat{S}\subseteq M_{A\otimes\mathscr{K}}(X\otimes\mathscr{K})$, the restriction
\[(k_X\rtimes_\sigma{\rm id}_{\widehat{S}},k_A\rtimes_\delta{\rm id}_{\widehat{S}}):=(\overline{k_X\otimes{\rm id}_{\mathscr{K}}}|_{X\rtimes_\sigma\widehat{S}},\,\overline{k_A\otimes{\rm id}_{\mathscr{K}}}|_{A\rtimes_\delta\widehat{S}})\]
makes sense and is an injective representation of $(X\rtimes_\sigma\widehat{S},A\rtimes_\delta\widehat{S})$ on $M_{A\otimes \mathscr{K}}(\mathcal{O}_X\otimes\mathscr{K})$. Using the equalities \eqref{Sec.5.zeta.iota}, we have
\begin{align*}
k_X\rtimes_\sigma{\rm id}_{\widehat{S}}\big(\sigma_\iota(\xi)\cdot(1_{M(A)}\otimes x)\big)
&=\overline{k_X\otimes{\rm id}_{\mathscr{K}}}\big(\sigma_\iota(\xi)\cdot(1_{M(A)}\otimes x)\big) \\
&=\overline{k_X\otimes{\rm id}_{\mathscr{K}}}(\sigma_\iota(\xi))\,\overline{k_A\otimes{\rm id}_{\mathscr{K}}}(1_{M(A)}\otimes x) \\
&=\zeta_\iota(k_X(\xi))(1_{M(\mathcal{O}_X)}\otimes x)
\end{align*}
for $\xi\in X$ and $x\in\widehat{S}$, and similarly for $k_A\rtimes_\delta{\rm id}_{\widehat{S}}$. This proves the first two equalities of \eqref{Sec.5.Rep.1}, and hence $(k_X\rtimes_\sigma{\rm id}_{\widehat{S}},k_A\rtimes_\delta{\rm id}_{\widehat{S}})$ is a representation on $\mathcal{O}_X\rtimes_\zeta\widehat{S}$. The last of \eqref{Sec.5.Rep.1} can be seen similarly.
\end{proof}
For an action $(\gamma,\alpha)$ of a locally compact group group $G$ on $(X,A)$, the ideal $J_{X\rtimes_{\gamma,r}G}$ for the crossed product correspondence $(X\rtimes_{\gamma,r}G,A\rtimes_{\delta,r}G)$ is known to be equal to the crossed product $J_X\rtimes_{\alpha,r}G$ if $G$ is amenable (\cite[Proposition~2.7]{HaoNg}) or if $G$ is discrete such that it is exact or $\alpha$ has Exel's Approximation Property (\cite[Theorem 5.5]{BKQR}). We now give a partial analogue of this fact in the Hopf $C^*$-algebra setting.
\begin{prop}\label{Ext.Main.Thm.of.KQR2}
Let $(\sigma,\delta)$ be a coaction of $S$ on $(X,A)$ such that $J_X$ is weakly $\delta$-invariant. Then
\begin{equation}\label{Sec.5.Drop.CP.KQR2}
\delta_\iota(J_X)(1_{M(A)}\otimes\widehat{S})\subseteq J_{X\rtimes_\sigma\widehat{S}}.
\end{equation}
In particular, if $J_X=A$ then $J_{X\rtimes_\sigma\widehat{S}}=A\rtimes_\delta\widehat{S}$.
\end{prop}
\begin{proof}
The last assertion of the proposition is an immediate consequence of the first. Hence we only need to prove \eqref{Sec.5.Drop.CP.KQR2}, which will follow by \cite[Proposition~3.3]{Kat2} if we show that
\[k_A\rtimes_\delta{\rm id}_{\widehat{S}}\big(\delta_\iota(J_X)(1_{M(A)}\otimes\widehat{S})\big)\subseteq (k_X\rtimes_\sigma{\rm id}_{\widehat{S}})^{(1)}\big(\mathcal{K}(X\rtimes_\sigma\widehat{S})\big)\]
since the representation $(k_X\rtimes_\sigma{\rm id}_{\widehat{S}},k_A\rtimes_\delta{\rm id}_{\widehat{S}})$ is injective. Let us first note the following. By Theorem~\ref{DKQ,Cor.A.14}.(ii), we have
\[\overline{(k_X\otimes{\rm id}_{\mathscr{K}})^{(1)}}=\overline{k_X\otimes{\rm id}_{\mathscr{K}}}^{\,(1)}\]
on $\mathcal{K}(M_{A\otimes\mathscr{K}}(X\otimes\mathscr{K}))$. Hence
\begin{align*}
\overline{(k_X\otimes{\rm id}_{\mathscr{K}})^{(1)}}\big(\mathcal{K}(X\rtimes_\sigma\widehat{S})\big)
&=\overline{k_X\otimes{\rm id}_{\mathscr{K}}}^{\,(1)}\big(\mathcal{K}(X\rtimes_\sigma\widehat{S})\big) \\
&=(k_X\rtimes_\sigma{\rm id}_{\widehat{S}})^{(1)}\big(\mathcal{K}(X\rtimes_\sigma\widehat{S})\big)
\end{align*}
by Remark~\ref{Sec.5.Rmk.to.Cor.to.Diag} and Proposition~\ref{embedding XxShat to OxShat is a Toep.rep.}.
In much the same way as the calculation \eqref{Sec.4.in.the.Pf.thm} in the proof of Theorem~\ref{crossed product correspondences}, we see that
\[\delta_\iota(J_X)(1_{M(A)}\otimes\widehat{S})\subseteq M(A\otimes\mathscr{K};J_X\otimes\mathscr{K})\]
since $J_X$ is weakly $\delta$-invariant. It therefore follows by Proposition \ref{embedding XxShat to OxShat is a Toep.rep.}, Lemma~\ref{Lemma.for.Ext.Main.Thm.of.KQR2}, and the above equality that
\begin{multline*}
k_A\rtimes_\delta{\rm id}_{\widehat{S}}\big(\delta_\iota(J_X)(1_{M(A)}\otimes\widehat{S})\big) \\
\begin{aligned}
&= \overline{k_A\otimes{\rm id}_{\mathscr{K}}}\big(\delta_\iota(J_X)(1_{M(A)}\otimes\widehat{S})\big) \\
&= \overline{(k_X\otimes{\rm id}_{\mathscr{K}})^{(1)}}\circ\varphi_{M(A\otimes\mathscr{K})}\big(\delta_\iota(J_X)(1_{M(A)}\otimes\widehat{S})\big) \\
&= \overline{(k_X\otimes{\rm id}_{\mathscr{K}})^{(1)}}\big(\sigma^{(1)}_\iota(\varphi_A(J_X))(1_{M(\mathcal{K}(X))}\otimes\widehat{S})\big) \\
&\subseteq\overline{(k_X\otimes{\rm id}_{\mathscr{K}})^{(1)}}\big(\mathcal{K}(X)\rtimes_{\sigma^{(1)}}\widehat{S}\big) \\
&=(k_X\rtimes_\sigma{\rm id}_{\widehat{S}})^{(1)}\big(\mathcal{K}(X\rtimes_\sigma\widehat{S})\big),
\end{aligned}
\end{multline*}
where the third and last step come from the $\delta$-$\sigma^{(1)}$ equivariancy of $\varphi_A$ and equality $\mathcal{K}(X)\rtimes_{\sigma^{(1)}}\widehat{S}=\mathcal{K}(X\rtimes_\sigma\widehat{S})$, respectively. This establishes the proposition.
\end{proof}
\begin{rmk}\label{Sec.5.Improve.KQRo2}\rm
Recall from Definition 3.1 and Lemma 3.2 of \cite{KQRo1} that a nondegenerate correspondence homomorphism $(\psi,\pi):(X,A)\rightarrow(M(Y),M(B))$ is \emph{Cuntz-Pimsner covariant} if $\psi(X)\subseteq M_B(Y)$ and $\pi(J_X)\subseteq M(B;J_Y)$. Corollary~\ref{Sec.4.jxja} and Proposition \ref{Ext.Main.Thm.of.KQR2} then assure us that the representation $(j_X^\sigma,j_A^\delta)$ is always Cuntz-Pimsner covariant since \eqref{Sec.5.Drop.CP.KQR2} is obviously equivalent to
\[j_A^\delta(J_X)\subseteq M(A\rtimes_\delta\widehat{S};J_{X\rtimes_\sigma\widehat{S}})\]
which was a hypothesis of \cite[Theorem 4.4]{KQRo2} for $S=C^*(G)$. Therefore, Theorem~4.4 of \cite{KQRo2} can be improved as follows: if $(\sigma,\delta)$ is a nondegenerate coaction of a locally compact group $G$ on $(X,A)$ such that $\delta(J_X)\subseteq M(A\otimes C^*(G);J_X\otimes C^*(G))$, then we always have $\mathcal{O}_X\rtimes_\zeta G\cong \mathcal{O}_{X\rtimes_\sigma G}$.
\end{rmk}
\begin{thm}\label{Equiv.of.cov.}
Let $(\sigma,\delta)$ be a coaction of a reduced Hopf $C^*$-algebra $S$ on a $C^*$-cor\-re\-spon\-dence $(X,A)$ such that $J_X$ is weakly $\delta$-invariant. Then the following conditions are equivalent:
\begin{itemize}
\item[\rm (i)] The representation $(k_X\rtimes_\sigma{\rm id}_{\widehat{S}},k_A\rtimes_\delta{\rm id}_{\widehat{S}}):(X\rtimes_\sigma\widehat{S},A\rtimes_\delta\widehat{S})\rightarrow\mathcal{O}_X\rtimes_\zeta\widehat{S}$ is covariant.
\item[\rm (ii)] The ideal $J_{X\rtimes_\sigma\widehat{S}}$ is contained in $M(A\otimes\mathscr{K};J_X\otimes\mathscr{K})$.
\item[\rm (iii)] The product $J_{X\rtimes_\sigma\widehat{S}}\,(\ker\varphi_A\otimes\mathscr{K})$ is zero.
\end{itemize}
\end{thm}
\begin{proof}
${\rm (i)}\Leftrightarrow{\rm (ii)}$: Suppose (i). Since $(k_X\rtimes_\sigma{\rm id}_{\widehat{S}},k_A\rtimes_\delta{\rm id}_{\widehat{S}})$ is injective, we have
\[J_{X\rtimes_\sigma\widehat{S}}=(k_A\rtimes_\sigma{\rm id}_{\widehat{S}})^{-1}\big((k_X\rtimes_\sigma{\rm id}_{\widehat{S}})^{(1)}(\mathcal{K}(X\rtimes_\sigma\widehat{S}))\big)\]
by the comment below \cite[Proposition 5.14]{Kat2}. The same reason shows
\[J_{M_{A\otimes\mathscr{K}}(X\otimes\mathscr{K})}=(\overline{k_A\otimes{\rm id}_{\mathscr{K}}})^{-1}\big(\overline{k_X\otimes{\rm id}_{\mathscr{K}}}^{\,(1)}\big(\mathcal{K}(M_{A\otimes\mathscr{K}}(X\otimes\mathscr{K}))\big)\big)\]
since $\mathscr{K}$ is nuclear and then $(\overline{k_X\otimes{\rm id}_{\mathscr{K}}},\overline{k_A\otimes{\rm id}_{\mathscr{K}}})$ is covariant by Corollary~\ref{Sec.3.prop.for.thm.in.Sec.5}. It thus follows that $J_{X\rtimes_\sigma\widehat{S}}\subseteq J_{M_{A\otimes\mathscr{K}}(X\otimes\mathscr{K})}$ by Remark~\ref{Sec.5.Rmk.to.Cor.to.Diag} and Proposition \ref{embedding XxShat to OxShat is a Toep.rep.}. But, the latter is contained in $M(A\otimes\mathscr{K};J_X\otimes\mathcal{K})$ again by Corollary \ref{Sec.3.prop.for.thm.in.Sec.5}. This proves ${\rm (i)}\Rightarrow{\rm (ii)}$. Conversely, suppose (ii). Restricting the equality \eqref{Covariancy.of.the.Strict.Ext.1} of Lemma~\ref{Lemma.for.Ext.Main.Thm.of.KQR2} to the subalgebra $J_{X\rtimes_{\sigma}\widehat{S}}$, we can write
\[(k_X\rtimes_\sigma{\rm id}_{\widehat{S}})^{(1)}\circ\varphi_{A\rtimes_\delta\widehat{S}}=k_A\rtimes_\delta{\rm id}_{\widehat{S}},\]
which verifies ${\rm (ii)}\Rightarrow{\rm (i)}$.
${\rm(ii)}\Leftrightarrow{\rm(iii)}$: Assuming (ii) we have
\begin{align*}
J_{X\rtimes_{\sigma}\widehat{S}}(\ker\varphi_A\otimes\mathscr{K})
&=J_{X\rtimes_{\sigma}\widehat{S}}(A\otimes\mathscr{K})(\ker\varphi_A\otimes\mathscr{K}) \\
&\subseteq(J_X\otimes\mathscr{K})(\ker\varphi_A\otimes\mathscr{K})=0,
\end{align*}
and hence we get (iii). Finally, we always have
\[\varphi_{A\otimes\mathscr{K}}\big((A\otimes\mathscr{K})J_{X\rtimes_\sigma\widehat{S}}\big)
\subseteq\varphi_{A\otimes\mathscr{K}}(A\otimes\mathscr{K})\mathcal{K}(X\rtimes_\sigma\widehat{S})
\subseteq \mathcal{K}(X\otimes\mathscr{K})\]
by Remark \ref{Sec.5.Rmk.to.Cor.to.Diag}. Since $\ker\varphi_{A\otimes\mathscr{K}}=\ker(\varphi_A\otimes{\rm id}_{\mathscr{K}})=\ker\varphi_A\otimes\mathscr{K}$ by the exactness of $\mathscr{K}$, (iii) implies
\[\big((A\otimes\mathscr{K})J_{X\rtimes_\sigma\widehat{S}}\big)\ker\varphi_{A\otimes\mathscr{K}}
=(A\otimes\mathscr{K})\big(J_{X\rtimes_\sigma\widehat{S}}(\ker\varphi_A\otimes\mathscr{K})\big)=0.\]
Therefore $(A\otimes\mathscr{K})J_{X\rtimes_\sigma\widehat{S}}\subseteq J_{X\otimes\mathscr{K}}$. But $J_{X\otimes\mathscr{K}}=J_X\otimes\mathscr{K}$ by Corollary~\ref{Sec.3.prop.for.thm.in.Sec.5}, which proves ${\rm(iii)}\Rightarrow{\rm(ii)}$.
\end{proof}
\begin{cor}\label{Sec.5.Cor.varphi.inj.}
Let $(\sigma,\delta)$ be a coaction of $S$ on $(X,A)$ such that $J_X$ is weakly $\delta$-invariant. Assume that either (i) the ideal $J_{X\rtimes_\sigma\widehat{S}}$ of $A\rtimes_\delta\widehat{S}$ is generated by $\delta_\iota(J_X)$
or (ii) $\varphi_A$ is injective. Then $(k_X\rtimes_\sigma{\rm id}_{\widehat{S}},k_A\rtimes_\delta{\rm id}_{\widehat{S}})$ is covariant.
\end{cor}
\begin{proof}
Assume (i), that is, $J_{X\rtimes_\sigma\widehat{S}}=\overline{(1_{M(A)}\otimes\widehat{S})\delta_\iota(J_X)(1_{M(A)}\otimes\widehat{S})}$. The nondegeneracy of $S$ and $\widehat{S}$ shows that
\begin{align*}
J_{X\rtimes_\sigma\widehat{S}}\,(A\otimes\mathscr{K})
&=\overline{(1_{M(A)}\otimes\widehat{S})\delta_\iota(J_X)(1_{M(A)}\otimes\widehat{S})\,(A\otimes\mathscr{K})} \\
&=\overline{(1_{M(A)}\otimes\widehat{S})\delta_\iota(J_X)\,(A\otimes\mathscr{K})} \\
&=\overline{(1_{M(A)}\otimes\widehat{S})\delta_\iota(J_X)(1_{M(A)}\otimes S)\,(A\otimes\mathscr{K})} \\
&\subseteq\overline{(1_{M(A)}\otimes\widehat{S})(J_X\otimes S)(A\otimes\mathscr{K})} =J_X\otimes\mathscr{K},
\end{align*}
in which the last inclusion follows from the weak $\delta$-invariancy of $J_X$. Hence we get the equivalent condition (ii) in Theorem~\ref{Equiv.of.cov.}. Evidently, assumption (ii) implies (iii) in Theorem~\ref{Equiv.of.cov.}.
\end{proof}
\begin{cor}\label{Sec.5.Cor.delta.trivial}
Let $(\sigma,\delta)$ be a coaction of $S$ on $(X,A)$ such that $\delta$ is trivial, that is, $\delta(a)=a\otimes1_{M(S)}$ for $a\in A$. If the triple $(J_X,A,\widehat{S})$ satisfies the slice map property, then $(k_X\rtimes_\sigma{\rm id}_{\widehat{S}},k_A\rtimes_\delta{\rm id}_{\widehat{S}})$ is covariant. Moreover, $J_{X\rtimes_\sigma\widehat{S}}=J_X\otimes\widehat{S}$.
\end{cor}
\begin{proof}
Since $\delta$ is trivial, $J_X$ is evidently weakly $\delta$-invariant. Hence the representation $(k_X\rtimes_\sigma{\rm id}_{\widehat{S}},k_A\rtimes_\delta{\rm id}_{\widehat{S}})$ on $\mathcal{O}_X\rtimes_\zeta\widehat{S}$ makes sense by Proposition~\ref{embedding XxShat to OxShat is a Toep.rep.}. To show that it is covariant, we check the equivalent condition (iii) in Theorem~\ref{Equiv.of.cov.}. First note that $\varphi_{A\rtimes_\delta\widehat{S}}=\varphi_{A\otimes\widehat{S}}=\varphi_A\otimes{\rm id}_{\widehat{S}}$. Then
\begin{equation*}\label{Ker.for.Trivial}
\ker\varphi_{A\rtimes_\delta\widehat{S}}=\ker(\varphi_A\otimes{\rm id}_{\widehat{S}})=\ker\varphi_A\otimes\widehat{S}
\end{equation*}
by Remarks \ref{S.M.P.}.(1). Since $\widehat{S}$ is a nondegenerate subalgebra of $\mathcal{L}(\mathcal{H})$, it follows that
\[J_{X\rtimes_\sigma\widehat{S}}(\ker\varphi_A\otimes\mathscr{K})
=\big(J_{X\rtimes_\sigma\widehat{S}}(\ker\varphi_A\otimes\widehat{S})\big)(1_{M(A)}\otimes\mathscr{K})=0,\]
and therefore $(k_X\rtimes_\sigma{\rm id}_{\widehat{S}},k_A\rtimes_\delta{\rm id}_{\widehat{S}})$ is covariant.
Let $\omega\in\mathcal{L}(\mathcal{H})_*$ and $T\in\mathscr{K}$. Applying the slice map ${\rm id}_A\otimes(\omega T)$ to $J_{X\rtimes_\sigma\widehat{S}}$ and then multiplying $a\in A$ yields
\[a({\rm id}_A\otimes(\omega T))(J_{X\rtimes_\sigma\widehat{S}})=({\rm id}_A\otimes\omega)\big((a\otimes T)J_{X\rtimes_\sigma\widehat{S}}\big)\subseteq J_X,\]
in which the last inclusion is due to the equivalent condition (ii) of Theorem~\ref{Equiv.of.cov.}. We thus have $({\rm id}_A\otimes\omega)(J_{X\rtimes_\sigma\widehat{S}})\subseteq J_X$ for $\omega\in\mathcal{L}(\mathcal{H})_*$, and conclude by Remarks~\ref{S.M.P.}.(2) that $J_{X\rtimes_\sigma\widehat{S}}\subseteq F(J_X,A,\widehat{S})=J_X\otimes\widehat{S}$. The converse follows from Proposition~\ref{Ext.Main.Thm.of.KQR2}.
\end{proof}
We now state and prove our main theorem.
\begin{thm}\label{Main.Theorem.}
Let $(\sigma,\delta)$ be a coaction of a reduced Hopf $C^*$-algebra $S$ on a $C^*$-cor\-re\-spon\-dence $(X,A)$ such that $J_X$ is weakly $\delta$-invariant. Suppose that the representation $(k_X\rtimes_\sigma{\rm id}_{\widehat{S}},k_A\rtimes_\delta{\rm id}_{\widehat{S}})$ is covariant. Then the integrated form
\[(k_X\rtimes_\sigma{\rm id}_{\widehat{S}})\times(k_A\rtimes_\delta{\rm id}_{\widehat{S}}):\mathcal{O}_{X\rtimes_\sigma\widehat{S}}\rightarrow\mathcal{O}_X\rtimes_\zeta\widehat{S}\]
is a surjective isomorphism.
\end{thm}
\begin{proof}
Set $\Psi=(k_X\rtimes_\sigma{\rm id}_{\widehat{S}})\times(k_A\rtimes_\delta{\rm id}_{\widehat{S}})$. Note that the embedding $k_A\rtimes_\delta{\rm id}_{\widehat{S}}$ is clearly nondegenerate, and hence $\Psi$ is also nondegenerate.
We claim that $\Psi(\mathcal{O}_{X\rtimes_\sigma\widehat{S}})$ contains all the elements of the form
\[(1_{M(\mathcal{O}_X)}\otimes x)\big(\zeta_\iota\big(k_X(\xi_1)\cdots k_X(\xi_n)k_X(\eta_m)^*\cdots k_X(\eta_1)^*\big)\big)(1_{M(\mathcal{O}_X)}\otimes y)\]
for nonnegative integers $m$ and $n$, vectors $\xi_1,\ldots,\xi_n,\eta_1,\ldots,\eta_m\in X$, and $x,y\in\widehat{S}$. This will prove that $\Psi$ is surjective (\cite[Proposition 2.7]{Kat}). Since
\[\zeta_\iota(k_A(A))(1_{M(\mathcal{O}_X)}\otimes\widehat{S})\subseteq\Psi(\mathcal{O}_{X\rtimes_\sigma\widehat{S}})\]
by \eqref{Sec.5.Rep.1} of Proposition \ref{embedding XxShat to OxShat is a Toep.rep.}, we only show by considering adjoints that
\begin{equation}\label{Sec.5.Pf.Main.Thm}
(1_{M(\mathcal{O}_X)}\otimes x)\zeta_\iota(k_X(\xi_1)\cdots k_X(\xi_n))\in\Psi(\mathcal{O}_{X\rtimes_\sigma\widehat{S}})
\end{equation}
for positive integers $n$, vectors $\xi_1,\ldots,\xi_n\in X$, and $x\in\widehat{S}$. We now proceed by induction on $n$. For $n=1$, \eqref{Sec.5.Pf.Main.Thm} follows from the last equality of \eqref{Sec.5.Rep.1}. Suppose that \eqref{Sec.5.Pf.Main.Thm} is true for an $n$. Let $\xi,\xi_1,\ldots,\xi_n$ be $n+1$ vectors in $X$ and $x\in\widehat{S}$. Take an element $C\in\mathcal{O}_{X\rtimes_\sigma\widehat{S}}$ such that
\[\Psi(C)=(1_{M(\mathcal{O}_X)}\otimes x)\zeta_\iota\big(k_X(\xi_1)k_X(\xi_2)\cdots k_X(\xi_n)\big).\]
By Remarks \ref{Sec.4.Rmk.for.Pf.Main.Thm}.(2), we have
\[\overline{k_{X\rtimes_\sigma\widehat{S}}}(j_X^\sigma(\xi))\in M_{A\rtimes_\delta\widehat{S}}(\mathcal{O}_{X\rtimes_\sigma\widehat{S}}).\]
We claim that
\begin{equation}\label{Sec.5.Pf.Main.Thm-2}
\overline{\Psi}\big(\overline{k_{X\rtimes_\sigma\widehat{S}}}(j_X^\sigma(\xi))\big)=j_{\mathcal{O}_X}^\zeta(k_X(\xi)),
\end{equation}
where $j_{\mathcal{O}_X}^\zeta:\mathcal{O}_X\rightarrow M(\mathcal{O}_X\rtimes_\zeta\widehat{S})$ is the canonical homomorphism such that $j_{\mathcal{O}_X}^\zeta(c)D=\zeta_\iota(c)D$ for $c\in\mathcal{O}_X$ and $D\in\mathcal{O}_X\rtimes_\zeta\widehat{S}$. In fact, for
\[v=\Psi\big(k_{A\rtimes_\delta\widehat{S}}\big(\delta_\iota(a)(1_{M(A)}\otimes x)\big)\big)=\zeta_\iota(k_A(a))(1_{M(\mathcal{O}_X)}\otimes x),\]
we have
\begin{align*}
\overline{\Psi}\big(\overline{k_{X\rtimes_\sigma\widehat{S}}}(j_X^\sigma(\xi))\big)\,v
&=\Psi\big(\overline{k_{X\rtimes_\sigma\widehat{S}}}(j_X^\sigma(\xi))\,k_{A\rtimes_\delta\widehat{S}}\big(\delta_\iota(a)(1_{M(A)}\otimes x)\big)\big) \\
&=\Psi\big(k_{X\rtimes_\sigma\widehat{S}}\big(j_X^\sigma(\xi)\cdot(\delta_\iota(a)(1_{M(A)}\otimes x))\big)\big) \\
&=\Psi\big(k_{X\rtimes_\sigma\widehat{S}}\big(\sigma_\iota(\xi\cdot a)\cdot(1_{M(A)}\otimes x)\big)\big) \\
&=\zeta_\iota\big(k_X(\xi\cdot a)\big)(1_{M(\mathcal{O}_X)}\otimes x) \\
&=j_{\mathcal{O}_X}^\zeta(k_X(\xi))\,\zeta_\iota(k_A(a))(1_{M(\mathcal{O}_X)}\otimes x)
=j_{\mathcal{O}_X}^\zeta(k_X(\xi))\,v,
\end{align*}
which verifies the equality \eqref{Sec.5.Pf.Main.Thm-2} since $k_A\rtimes_\delta{\rm id}_{\widehat{S}}$ is nondegenerate. It is now obvious that for the product $C\,\overline{k_{X\rtimes_\sigma\widehat{S}}}(j_X^\sigma(\xi))\in\mathcal{O}_{X\rtimes_\sigma\widehat{S}}$ we have
\[\Psi\big(C\,\overline{k_{X\rtimes_\sigma\widehat{S}}}(j_X^\sigma(\xi))\big)
=(1_{M(\mathcal{O}_X)}\otimes x)\zeta_\iota\big(k_X(\xi_1)\cdots k_X(\xi_n)k_X(\xi)\big).\]
Consequently, the statement \eqref{Sec.5.Pf.Main.Thm} is shown to be true for all positive integer $n$, and hence $\Psi$ is surjective.
Let $\beta:\mathbb{T}\rightarrow Aut(\mathcal{O}_X)$ be the gauge action. Then the strict extensions $\overline{\beta_z\otimes{\rm id}_{\mathscr{K}}}$ are automorphisms on $M(\mathcal{O}_X\otimes\mathscr{K})$. We have
\begin{equation*}\label{EqualityforGIUT1}
\begin{aligned}
\overline{\beta_z\otimes{\rm id}_{\mathscr{K}}}\big(\zeta_\iota(k_X(\xi))\big)&=z\zeta_\iota(k_X(\xi)), \\
\overline{\beta_z\otimes{\rm id}_{\mathscr{K}}}\big(\zeta_\iota(k_A(a))\big)&=\zeta_\iota(k_A(a))
\end{aligned}
\end{equation*}
in the same way as the last part of the proof of Theorem \ref{induced coactions on O_X}. Therefore,
\begin{multline*}
\overline{\beta_z\otimes{\rm id}_{\mathscr{K}}}\big((k_X\rtimes_\sigma{\rm id}_{\widehat{S}})\big(\sigma_\iota(\xi)\cdot(1_{M(A)}\otimes x)\big)\big) \\
\begin{aligned}
&=\overline{\beta_z\otimes{\rm id}_{\mathscr{K}}}\big(\zeta_\iota(k_X(\xi))(1_{M(\mathcal{O}_X)}\otimes x)\big) \\
&=z\,\zeta_\iota(k_X(\xi))(1_{M(\mathcal{O}_X)}\otimes x) \\
&=z\,(k_X\rtimes_\sigma{\rm id}_{\widehat{S}})\big(\sigma_\iota(\xi)\cdot(1_{M(A)}\otimes x)\big)
\end{aligned}
\end{multline*}
and similarly
\[\overline{\beta_z\otimes{\rm id}_{\mathscr{K}}}\big((k_A\rtimes_\delta{\rm id}_{\widehat{S}})\big(\delta_\iota(a)(1_{M(A)}\otimes x)\big)\big)=(k_A\rtimes_\delta{\rm id}_{\widehat{S}})\big(\delta_\iota(a)(1_{M(A)}\otimes x)\big).\]
This proves that the restriction $\overline{\beta_z\otimes{\rm id}_{\mathscr{K}}}|_{\mathcal{O}_X\rtimes_\zeta\widehat{S}}$ defines an automorphism on $\mathcal{O}_X\rtimes_\zeta\widehat{S}=\Psi(\mathcal{O}_{X\rtimes_\sigma\widehat{S}})$, and the injective covariant representation $(k_X\rtimes_\sigma{\rm id}_{\widehat{S}},k_A\rtimes_\delta{\rm id}_{\widehat{S}})$ admits a gauge action. We thus conclude by \cite[Theorem~6.4]{Kat} that $\Psi$ is injective as well, which completes the proof.
\end{proof}
Applying Theorem \ref{Main.Theorem.} to group actions we can extend Theorem 2.10 of \cite{HaoNg} as Corollary~\ref{Cor1.to.Main.Thm} states below, the proof of which will be given in Appendix B. Let $(\gamma,\alpha)$ be an action of a locally compact group $G$ on $(X,A)$. By Theorem \ref{actions=coactions}, $(\gamma,\alpha)$ defines a coaction $(\sigma^\gamma,\delta^\alpha)$ of $C_0(G)$ on $(X,A)$, which induces a coaction $\zeta$ of $C_0(G)$ on $\mathcal{O}_X$ by Theorem \ref{induced coactions on O_X} and Remarks~\ref{Sec.3.Rmk.to.Thm}.(1). Let $\beta^\zeta$ be the action of $G$ on $\mathcal{O}_X$ corresponding to the coaction $\zeta$. In a similar way to \cite[Corollary~2.9]{HaoNg}, we define a representation
\[(k_X\rtimes_\gamma G,k_A\rtimes_\alpha G):(X\rtimes_{\gamma,r}G,A\rtimes_{\alpha,r}G)\rightarrow\mathcal{O}_X\rtimes_{\beta^\zeta,r}G\]
by
\[(k_X\rtimes_\gamma G)(f)(r)=k_X(f(r)),\quad(k_A\rtimes_\alpha G)(g)(r)=k_A(g(r))\]
for $f\in C_c(G,X)$, $g\in C_c(G,A)$, and $r\in G$.
\begin{cor}\label{Cor1.to.Main.Thm}
Let $(\gamma,\alpha)$ be an action of a locally compact group $G$ on $(X,A)$. If the representation $(k_X\rtimes_\gamma G,k_A\rtimes_\alpha G)$ is covariant, then its integrated form $(k_X\rtimes_\gamma G)\times(k_A\rtimes_\alpha G):\mathcal{O}_{X\rtimes_{\gamma,r}G}\rightarrow\mathcal{O}_X\rtimes_{\beta^\zeta,r}G$ is a surjective isomorphism.
\end{cor}
For the amenability in the next theorem, we refer to \cite{BS}. See also \cite{Ng}.
\begin{thm}\label{amen.V}
Let $(\sigma,\delta)$ be a coaction on $(X,A)$ of a reduced Hopf $C^*$-algebra $S$ defined by an amenable regular multiplicative unitary such that $J_X$ is weakly $\delta$-invariant. If $A$ is nuclear (or exact, respectively), then the same is true for $\mathcal{O}_X\rtimes_{\zeta}\widehat{S}$.
\end{thm}
\begin{proof}
If $A$ is nuclear (or exact, respectively), then so is $A\rtimes_\delta\widehat{S}$ by \cite[Theorem~3,4]{Ng} (or by \cite[Theorem 3.13]{Ng}, respectively). Hence, the Toeplitz algebra $\mathcal{T}_{X\rtimes_{\sigma}\widehat{S}}$ is nuclear by \cite[Corollary 7.2]{Kat} (or exact by \cite[Theorem 7.1]{Kat}, respectively). Since nuclearity or exactness passes to quotients, it suffices to show that the representation $(k_X\rtimes_\sigma{\rm id}_{\widehat{S}},k_A\rtimes_\delta{\rm id}_{\widehat{S}})$ gives rise to a surjection from $\mathcal{T}_{X\rtimes_\sigma\widehat{S}}$ onto $\mathcal{O}_X\rtimes_\zeta\widehat{S}$. The proof of this then goes parallel to the one given in the proof of Theorem \ref{Main.Theorem.} using the embedding $(i_{X\rtimes_\sigma\widehat{S}},i_{A\rtimes_\delta\widehat{S}})$ of $(X\rtimes_\sigma\widehat{S},A\rtimes_\delta\widehat{S})$ into $\mathcal{T}_{X\rtimes_\sigma\widehat{S}}$ instead of $(k_{X\rtimes_\sigma\widehat{S}},k_{A\rtimes_\delta\widehat{S}})$ used in there.
\end{proof}
Applying the previous results we consider coactions on crossed products by $\mathbb{Z}$ which form an important example of Cuntz-Pimsner algebras.
Let $\varphi$ be an automorphism on a $C^*$-algebra $A$. Equipped with the left action $\varphi_A(a)b=\varphi(a)b$ for $a,b\in A$, the Hilbert $A$-module $A$ then becomes a $C^*$-cor\-re\-spond\-ence (\cite[Examples~(3)]{Pim}), which we call a \emph{$\varphi$-cor\-re\-spond\-ence} and denote by $A(\varphi)$. For a $\varphi$-cor\-re\-spond\-ence $A(\varphi)$, it is clear that the multiplier correspondence $M(A(\varphi))$ coincides with the $\overline{\varphi}$-cor\-re\-spond\-ence $M(A)(\overline{\varphi})$ and the strict topology on $M(A(\varphi))$ is the usual one on the multiplier algebra $M(A)$.
We want to consider a coaction of a Hopf $C^*$-algebra on a $\varphi$-cor\-re\-spond\-ence $(A(\varphi),A)$ and its induced coaction on $\mathcal{O}_{A(\varphi)}$. Before that, let us observe the following.
\begin{lem}\label{Corr.Hom.bet'nIdentityCorr's}
Let $\varphi$ and $\varphi'$ be automorphisms on $C^*$-algebras $A$ and $B$, respectively, and $\pi:A\rightarrow M(B)$ be a nondegenerate homomorphism. Let $v\in M(B)$ be a unitary such that
\[v\,\pi(\varphi(a))=\overline{\varphi'}(\pi(a))\,v\quad(a\in A).\]
Define
\[\psi(a):=v\pi(a)\quad(a\in A).\]
Then $(\psi,\pi):(A(\varphi),A)\rightarrow(M(B(\varphi')),M(B))$ is a nondegenerate correspondence homomorphism. Moreover, every nondegenerate correspondence homomorphism from $(A(\varphi),A)$ into $(M(B(\varphi')),M(B))$ is of this form.
\end{lem}
\begin{proof}
For $a,a'\in A$,
\[ \psi(\varphi(a)a')
= v\pi(\varphi(a)a')=v\pi(\varphi(a))\,\pi(a')
=\overline{\varphi'}(\pi(a))\,v\pi(a')=\overline{\varphi'}(\pi(a))\,\psi(a') \]
and $\langle\psi(a),\psi(a')\rangle_{M(B)}=\pi(a)^*v^*v\pi(a')=\pi(\langle a,a'\rangle_A)$. Hence $(\psi,\pi)$ is a correspondence homomorphism, and obviously nondegenerate.
For the converse, let $(\psi,\pi):(A(\varphi),A)\rightarrow(M(B(\varphi')),M(B))$ be a nondegenerate correspondence homomorphism, and consider its strict extension $(\overline{\psi},\overline{\pi})$. Let $v=\overline{\psi}(1_{M(A)})$. Since $(\overline{\psi},\overline{\pi})$ is a correspondence homomorphism, we have
\[v^*v=\langle v,v\rangle_{M(B)}=\overline{\pi}(\langle1_{M(A)},1_{M(A)}\rangle_{M(A)})=1_{M(B)}.\]
We also have
\[ vv^*(\psi(a) b)
=\overline{\psi}(1_{M(A)})\langle\overline{\psi}(1_{M(A)}),\psi(a) b\rangle_{M(B)}
=\overline{\psi}(1_{M(A)})\pi(a)b=\overline{\psi}(a) b \]
for $a\in A$ and $b\in B$ so that $vv^*=1_{M(B)}$. Hence $v$ is a unitary in $M(B)$. Finally,
\[v\pi(\varphi(a))=\psi(\varphi(a))=\psi\big(\varphi(a)1_{M(A)}\big)=\overline{\varphi'}(\pi(a))v.\]
This completes the proof.
\end{proof}
Let $\delta$ be a coaction of a Hopf $C^*$-algebra $(S,\Delta)$ on a $C^*$-algebra $A$ and $\varphi\in Aut(A)$. Let $v$ be a cocycle for the coaction $\delta$, that is, a unitary $v\in M(A\otimes S)$ satisfying
\[v_{12}\,\overline{\delta\otimes{\rm id}_S}(v)=\overline{{\rm id}_A\otimes\Delta}(v)\]
(\cite[Definition 0.4]{BS}), and suppose that
\begin{equation}\label{EquivarianceforA,Aw.varphi}
v\,\delta(\varphi(a))=\overline{\varphi\otimes{\rm id}_S}(\delta(a))\,v\quad(a\in A).
\end{equation}
Define $\sigma:A(\varphi)\rightarrow M(A\otimes S(\varphi\otimes{\rm id}_S))=M(A(\varphi)\otimes S)$ by
\[\sigma(a):=v\delta(a)\quad(a\in A).\]
Then $(\sigma,\delta)$ is a coaction of $S$ on the $\varphi$-cor\-re\-spond\-ence $(A(\varphi),A)$. Indeed, it is a nondegenerate correspondence homomorphism by Lemma \ref{Corr.Hom.bet'nIdentityCorr's}. Also, the computation
\begin{align*}
\overline{\sigma\otimes{\rm id}_S}(\sigma(a))
&=v_{12}\,\overline{\delta\otimes{\rm id}_S}(v\delta(a)) \\
&=v_{12}\,\overline{\delta\otimes{\rm id}_S}(v)\,\overline{\delta\otimes{\rm id}_S}(\delta(a)) \\
&=\overline{{\rm id}_A\otimes\Delta}(v)\,\overline{{\rm id}_A\otimes\Delta}(\delta(a)) = \overline{{\rm id}_A\otimes\Delta}(\sigma(a))
\end{align*}
verifies the coaction identity of $\sigma$. The coaction nondegeneracy of $\delta$ gives
\begin{align*}
\overline{(1_{M(A)}\otimes S)\sigma(A)}
&=\overline{(1_{M(A)}\otimes S)\,v\delta(A)}=\overline{(1_{M(A)}\otimes S)\,v\delta(\varphi(A))} \\
&=\overline{(1_{M(A)}\otimes S)\,\big(\overline{\varphi\otimes{\rm id}_S}\,\delta(A)\big)\,v} \\
&=\overline{\overline{\varphi\otimes{\rm id}_S}\big((1_{M(A)}\otimes S)\,\delta(A)\big)}\,v =(A\otimes S)v = A\otimes S
\end{align*}
so that $\sigma$ satisfies coaction nondegeneracy. Hence $(\sigma,\delta)$ is a coaction.
The Cuntz-Pimsner algebra $\mathcal{O}_{A(\varphi)}$ is isomorphic to the crossed product $A\rtimes_\varphi\mathbb{Z}$ and an isomorphism $\mathcal{O}_{A(\varphi)}\cong A\rtimes_\varphi\mathbb{Z}$ can be given as follows. Let $(\pi,u)$ be the canonical covariant representation of the $C^*$-dynamical system $(A,\mathbb{Z},\varphi)$ on $M(A\rtimes_\varphi\mathbb{Z})$. Define $\psi:A(\varphi)\rightarrow A\rtimes_\varphi\mathbb{Z}$ by
\[\psi(a)=u^{*}\pi(a)\quad(a\in A(\varphi)).\]
It can be easily checked that $(\psi,\pi)$ is a covariant representation of $(A(\varphi),A)$ on $A\rtimes_\varphi\mathbb{Z}$. Furthermore, the integrated form $\psi\times\pi:\mathcal{O}_{A(\varphi)}\rightarrow A\rtimes_\varphi\mathbb{Z}$ gives a surjective isomorphism. We will identify in this way the universal covariant representations $(k_X,k_A)=(\psi,\pi)$ as well as the $C^*$-algebras $\mathcal{O}_{A(\varphi)}=A\rtimes_\varphi\mathbb{Z}$.
Since $J_{A(\varphi)}=A$ is evidently weakly $\delta$-invariant, it follows by Theorem \ref{induced coactions on O_X} that $(\sigma,\delta)$ induces a coaction $\zeta$ of $S$ on $\mathcal{O}_{A(\varphi)}=A\rtimes_\varphi\mathbb{Z}$ which can be described explicitly on the canonical generators of $A\rtimes_\varphi\mathbb{Z}$ as follows. Theorem \ref{induced coactions on O_X} says that
\[\zeta(\pi(a))=\overline{\pi\otimes{\rm id}_S}(\delta(a)),\]
\[
\zeta(u^{*}\pi(a))
=\zeta(\psi(a))=\overline{\psi\otimes{\rm id}_S}(\sigma(a))
=(u^{*}\otimes1_{M(S)})\,\overline{\pi\otimes{\rm id}_S}(v\delta(a))
\]
for $a\in A$. Note that $\overline{\zeta}(u^*)=(u^{*}\otimes1_{M(S)})\,\overline{\pi\otimes{\rm id}_S}(v)$. Hence,
\begin{equation*}\label{Sec.3.Example1}
\zeta(\pi(a)u^n)=\overline{\pi\otimes{\rm id}_S}(\delta(a))\big((u^{*}\otimes1_{M(S)})\,\overline{\pi\otimes{\rm id}_S}(v)\big)^{-n}
\end{equation*}
for $a\in A$ and $n\in\mathbb{Z}$.
Assume now that the Hopf $C^*$-algebra $S$ is reduced. Then we can form the reduced crossed product correspondence $(A(\varphi)\rtimes_\sigma\widehat{S},A\rtimes_\delta\widehat{S})$ by Theorem~\ref{crossed product correspondences}. Let $v_\iota=\overline{{\rm id}_A\otimes\iota_S}(v)$. Since
the multiplication by $v_\iota$ from the left gives a Hilbert module isomorphism from $A\rtimes_\delta\widehat{S}$ onto $A(\varphi)\rtimes_\sigma\widehat{S}$, we may --- and do --- regard the $C^*$-cor\-re\-spond\-ence $A(\varphi)\rtimes_\sigma\widehat{S}$ as the Hilbert module $A\rtimes_\delta\widehat{S}$ with the left action
\begin{equation}\label{Sec.4.Example1}
\varphi_{A\rtimes_\delta\widehat{S}}(c)\,d=v_\iota^*\overline{\varphi\otimes{\rm id}_{\mathcal{K}(\mathcal{H})}}(c)v_\iota\,d
\end{equation}
for an element $c$ in the $C^*$-algebra $A\rtimes_\delta\widehat{S}$ and a vector $d$ in the Hilbert module $A\rtimes_\delta\widehat{S}$. Note that $\varphi_{A\rtimes_\delta\widehat{S}}$ is injective. Since $\varphi_A$ is injective, $\mathcal{O}_{A(\varphi)}\rtimes_\zeta\widehat{S}$ is the Cuntz-Pimsner algebra $\mathcal{O}_{A(\varphi)\rtimes_\sigma\widehat{S}}$ by Corollary \ref{Sec.5.Cor.varphi.inj.} and Theorem \ref{Main.Theorem.}.
We can summerize what we have seen so far as follows.
\begin{prop}\label{Sec.6.Prop.1}
Let $\varphi$ be an automorphism on a $C^*$-algebra $A$ and $\delta$ be a coaction of a Hopf $C^*$-algebra $S$ on $A$. Let $v$ be a cocyle for $\delta$ satisfying \eqref{EquivarianceforA,Aw.varphi}. Define $\sigma:A(\varphi)\rightarrow M(A(\varphi)\otimes S)$ by $\sigma(a)=v\delta(a)$. Then the following hold.
{\rm(i)} $(\sigma,\delta)$ is a coaction of $S$ on the $\varphi$-cor\-re\-spond\-ence $(A(\varphi),A)$.
{\rm(ii)} Let $(\pi,u)$ be the canonical covariant representation of $(A,\mathbb{Z},\varphi)$ on $M(A\rtimes_\varphi\mathbb{Z})$. Then, the homomorphism \[\overline{\pi\otimes{\rm id}_S}\circ\delta:A\rightarrow M((A\rtimes_\varphi\mathbb{Z})\otimes S)\] and the unitary $\overline{\pi\otimes{\rm id}_S}(v^*)(u\otimes1_{M(S)})\in M((A\rtimes_\varphi\mathbb{Z})\otimes S)$ form a covariant representation of $(A,\mathbb{Z},\varphi)$ on $M((A\rtimes_\varphi\mathbb{Z})\otimes S)$ such that the integrated form gives a coaction $\zeta$ of $S$ on $A\rtimes_\varphi\mathbb{Z}$ and coincides with the coaction induced by $(\sigma,\delta)$.
{\rm(iii)} If $S$ is reduced then $A(\varphi)\rtimes_\sigma\widehat{S}=A\rtimes_\delta\widehat{S}$ as Hilbert $(A\rtimes_\delta\widehat{S})$-modules and the left action is given by \eqref{Sec.4.Example1}. The reduced crossed product $(A\rtimes_\varphi\mathbb{Z})\rtimes_\zeta\widehat{S}=\mathcal{O}_{A(\varphi)}\rtimes_\zeta\widehat{S}$ is the Cuntz-Pimsner algebra $\mathcal{O}_{A(\varphi)\rtimes_\sigma\widehat{S}}$.
\end{prop}
We can say further if we take the cocycle $v$ in Proposition \ref{Sec.6.Prop.1} to be the identity. Let $v=1_{M(A\otimes S)}$. Then \eqref{EquivarianceforA,Aw.varphi} reduces to
\[\delta\circ\varphi=\overline{\varphi\otimes{\rm id}_S}\circ\delta,\]
and then $\varphi_{A\rtimes_\delta\widehat{S}}$ maps $A\rtimes_\delta\widehat{S}$ onto itself:
\[
\varphi_{A\rtimes_\delta\widehat{S}}\big(\delta_\iota(a)(1_{M(A)}\otimes x)\big)
=\overline{\varphi\otimes{\rm id}_{\mathcal{K}(\mathcal{H})}}\big(\delta_\iota(a)(1_{M(A)}\otimes x)\big)
=\delta_\iota(\varphi(a))(1_{M(A)}\otimes x)
\]
for $a\in A$ and $x\in\widehat{S}$. Hence $\varphi_{A\rtimes_\delta\widehat{S}}$ defines an automorphism $\varphi\rtimes{\rm id}$ on $A\rtimes_\delta\widehat{S}$ such that
\[(\varphi\rtimes{\rm id})\big(\delta_\iota(a)(1_{M(A)}\otimes x)\big)=\delta_\iota(\varphi(a))(1_{M(A)}\otimes x)\]
for $a\in A$ and $x\in\widehat{S}$. We thus see that $A(\varphi)\rtimes_\sigma\widehat{S}$ is the $(\varphi\rtimes{\rm id})$-cor\-re\-spond\-ence $A\rtimes_\delta\widehat{S}(\varphi\rtimes{\rm id})$. We have the equality
\begin{equation}\label{Sec.5.Example1}
\mathcal{O}_{A(\varphi)\rtimes_\sigma\widehat{S}}=\mathcal{O}_{A\rtimes_\delta\widehat{S}(\varphi\rtimes{\rm id})}=(A\rtimes_\delta\widehat{S})\rtimes_{\varphi\rtimes{\rm id}}\mathbb{Z}
\end{equation}
as well as
\begin{equation}\label{Sec.5.Example2}
\mathcal{O}_{A(\varphi)}\rtimes_\zeta\widehat{S}=(A\rtimes_\varphi\mathbb{Z})\rtimes_\zeta\widehat{S},
\end{equation}
and then have a surjective isomorphism
\[\Psi:(A\rtimes_\delta\widehat{S})\rtimes_{\varphi\rtimes{\rm id}}\mathbb{Z}=\mathcal{O}_{A(\varphi)\rtimes_\sigma\widehat{S}}
\rightarrow\mathcal{O}_{A(\varphi)}\rtimes_\zeta\widehat{S}=(A\rtimes_\varphi\mathbb{Z})\rtimes_\zeta\widehat{S}.\]
Let us describe $\Psi$ on the canonical generators of the iterated crossed products $(A\rtimes_\delta\widehat{S})\rtimes_{\varphi\rtimes{\rm id}}\mathbb{Z}$ and $(A\rtimes_\varphi\mathbb{Z})\rtimes_\zeta\widehat{S}$. As $(\pi,u)$ in Proposition~\ref{Sec.6.Prop.1}, let $(\widetilde{\pi},\widetilde{u})$ be the canonical covariant representation of the $C^*$-dynamical system $(A\rtimes_\delta\widehat{S},\,\mathbb{Z},\,\varphi\rtimes{\rm id})$ on $M((A\rtimes_\delta\widehat{S})\rtimes_{\varphi\rtimes{\rm id}}\mathbb{Z})$. Let
\begin{align*}
d_1 &= k_{A(\varphi)\rtimes_\delta\widehat{S}}\big(\delta_\iota(a)\cdot(1_{M(A)}\otimes x)\big)\in \mathcal{O}_{A(\varphi)\rtimes_\sigma\widehat{S}}, \\
d_2 &= \widetilde{u}^{*}\widetilde{\pi}\big(\delta_\iota(a)(1_{M(A)}\otimes x)\big)\in (A\rtimes_\delta\widehat{S})\rtimes_{\varphi\rtimes{\rm id}}\mathbb{Z}, \\
d_3 &= \zeta_\iota(k_{A(\varphi)}(a))(1_{M(\mathcal{O}_{A(\varphi)})}\otimes x) \in \mathcal{O}_{A(\varphi)}\rtimes_\zeta\widehat{S}, \\
d_4 &= \zeta_\iota(u^{*}\pi(a))(1_{M(A\rtimes_\varphi\mathbb{Z})}\otimes x)\in (A\rtimes_\varphi\mathbb{Z})\rtimes_\zeta\widehat{S}.
\end{align*}
We then have $d_1=d_2$ in \eqref{Sec.5.Example1}, and $d_3=d_4$ in \eqref{Sec.5.Example2}. Since $\Psi(d_1)=d_3$, we may write $\Psi(d_2)=d_4$. Note that $\overline{\Psi}(\widetilde{u})=\overline{\zeta_\iota}(u)$. Therefore
\[\Psi\big(\widetilde{u}^n\,\widetilde{\pi}\big(\delta_\iota(a)(1_{M(A)}\otimes x)\big)\big)=\zeta_\iota(u^n\pi(a))(1_{M(A\rtimes_\varphi\mathbb{Z})}\otimes x),\]
or equivalently, by the fact that $(\pi,u)$ and $(\widetilde{\pi},\widetilde{u})$ are covariant representations,
\begin{equation}\label{Sec.6.Cor.1}
\Psi\big(\widetilde{\pi}\big(\delta_\iota(a)(1_{M(A)}\otimes x)\big)\,\widetilde{u}^n\big)
=\zeta_\iota(\pi(a)u^n)(1_{M(A\rtimes_\varphi\mathbb{Z})}\otimes x)
\end{equation}
for $a\in A$, $x\in\widehat{S}$, and $n\in\mathbb{Z}$. We summarize this in the next corollary.
\begin{cor}
Under the hypothesis and notation in Proposition \ref{Sec.6.Prop.1} with $v$ replaced by $1_{M(A\otimes S)}$, the formula
\[\zeta(\pi(a)u^n)=\overline{\pi\otimes{\rm id}_S}(\delta(a))(u^n\otimes1_{M(S)})\quad(a\in A,\ n\in\mathbb{Z})\]
defines a coaction $\zeta$ of $S$ on $A\rtimes_\varphi\mathbb{Z}$. Moreover, if $S$ is reduced then there exists a surjective isomorphism
\[\Psi:(A\rtimes_\delta\widehat{S})\rtimes_{\varphi\rtimes{\rm id}}\mathbb{Z}
\rightarrow(A\rtimes_\varphi\mathbb{Z})\rtimes_\zeta\widehat{S}\]
between the iterated crossed products such that \eqref{Sec.6.Cor.1} holds.
\end{cor}
\appendix
\section{Coactions of $C_0(G)$ on $(X,A)$}\label{App.A}
{\renewcommand*{\thesection}{\Alph{section}}
The goal of this section is to show that there exists a one-to-one correspondence between actions of a locally compact group $G$ on $(X,A)$ in the sense of \cite{EKQR} and coactions of the commutative Hopf $C^*$-algebra $C_0(G)$ on $(X,A)$ (Theorem~\ref{actions=coactions}).
Let us fix some notations. Let $(X,A)$ be a nondegenerate $C^*$-cor\-re\-spond\-ence as before, and $G$ be a locally compact Hausdorff space. By $M(X)_s$ we mean the multiplier correspondence $M(X)$ endowed with the strict topology. We denote by $C_b(G,M(X)_s)$ the Banach space of all bounded continuous functions from $G$ to $M(X)_s$ with the sup-norm, and by $C_0(G,X)$ the closed subspace of $C_b(G,M(X)_s)$ consisting of functions with values in $X$ which are also norm continuous and vanishes at infinity.
For an identity correspondence $(X,A)=(A,A)$, the Banach space $C_b(G,M(X)_s)$ becomes a $C^*$-algebra under the usual point-wise operations. In this case,
\[C_b(G,M(X)_s)=M(X\otimes C_0(G))\]
(\cite[Corollary 3.4]{APT}). We first generalize this in Theorem \ref{Cbstr=M} to nondegenerate $C^*$-cor\-re\-spond\-ences, which will enable one to prove the bijective correspondence between $G$-actions and $C_0(G)$-coactions on $(X,A)$.
\begin{prop}\label{CbG,MXs}
The Banach space $C_b(G,M(X)_s)$ is a $C^*$-cor\-re\-spond\-ence over $C_b(G,M(A)_s)$ with the following point-wise operations
\begin{equation}\label{Appendix.A.BimoduleOperation}
\begin{aligned}
(m\cdot l)(r) &= m(r)\cdot l(r), \\
\langle m,n\rangle_{C_b(G,M(A)_s)}(r) &= \langle m(r),n(r)\rangle_{M(A)}, \\
\big(\varphi_{C_b(G,M(A)_s)}(l)\,m\big)(r) &= \varphi_{M(A)}(l(r))\,m(r)
\end{aligned}
\end{equation}
for $m,n\in C_b(G,M(X)_s)$, $l\in C_b(G,M(A)_s)$, and $r\in G$.
\end{prop}
\begin{proof}
Write $\varphi=\varphi_{C_b(G,M(A)_s)}$. The only part requiring proof is that the functions on \eqref{Appendix.A.BimoduleOperation} are strictly continuous. We prove this only for the function $\varphi(l)\,m$. The others can be handled in the same way. Let $\{r_i\}$ be a net in $G$ converging to an $r\in G$, $a\in A$, and $T\in\mathcal{K}(X)$. Evidently, $(\varphi(l)m)(r_i)\cdot a-(\varphi(l)m)(r)\cdot a$ converges to 0. Factor $T=T'\varphi_A(a')$ for some $T'\in\mathcal{K}(X)$ and $a'\in A$, which is possible by the Hewitt-Cohen factorization theorem (see for example \cite[Proposition 2.33]{RaWi}) since the left action $\varphi_A$ is nondegenerate. Then the difference
\begin{align*}
T(\varphi(l)m)(r_i)-T(\varphi(l)m)(r)
&=\big(T'\varphi_A(a'l(r_i))\,m(r_i)-T'\varphi_A(a'l(r))\,m(r_i)\big) \\
&\quad + \big(T\varphi_{M(A)}(l(r))\,m(r_i)-T\varphi_{M(A)}(l(r))\,m(r)\big)
\end{align*}
converges to 0 by the strict continuity of both $l$ and $m$ and also by the boundedness of $m$. Hence $\varphi(l)m$ is strictly continuous.
\end{proof}
It is clear that $(C_0(G,X),C_0(G,A))$ is also a $C^*$-cor\-re\-spon\-dence with respect to the restriction of operations \eqref{Appendix.A.BimoduleOperation}.
We call a correspondence homomorphism $(\psi,\pi):(X,A)\rightarrow(Y,B)$ an \emph{isomorphism} if both $\psi$ and $\pi$ are bijective. In this case, $(X,A)$ and $(Y,B)$ are said to be \emph{isomorphic}. The next corollary is an easy consequence of Corollary \ref{O_(XotimesB)=O_XotimesB}.
\begin{cor}\label{C0GX=XotimesC0X}
The $C^*$-correspondence $(C_0(G,X),C_0(G,A))$ and the tensor product correspondence $(X\otimes C_0(G),A\otimes C_0(G))$ are isomorphic.
\end{cor}
\begin{lem}\label{Preparation.for.Applying.EKQR}
With respect to the operations \eqref{Appendix.A.BimoduleOperation}, the following hold.
\begin{itemize}
\item[\rm(i)] $\varphi_{C_b(G,M(A)_s)}\big(C_b(G,M(A)_s)\big)\,C_0(G,X)= C_0(G,X)$,
\item[\rm(ii)] $C_0(G,X)\cdot C_b(G,M(A)_s)= C_0(G,X)$,
\item[\rm(iii)] $C_b(G,M(X)_s)\cdot C_0(G,A)= C_0(G,X)$.
\end{itemize}
\end{lem}
\begin{proof}
It is obvious that the inclusion $\supseteq$ holds on each of (i) and (ii). The same is true for (iii) by the Hewitt-Cohen factorization theorem since $(C_0(G,X),C_0(G,A))$ is isomorphic to the nondegenerate $C^*$-cor\-re\-spond\-ence $(X\otimes C_0(G),A\otimes C_0(G))$. For the inclusion $\subseteq$ in (i), let $l\in C_b(G,M(A)_s)$ and $x\in C_0(G,X)$, and write $x=\varphi_{C_0(G,A)}(f)y$ for some $f\in C_0(G,A)$ and $y\in C_0(G,X)$. Then
\[\varphi_{C_b(G,M(A)_s)}(l)\,x=\varphi_{C_0(G,A)}(lf)\,y\in C_0(G,X),\]
which proves (i). Similarly we have the inclusion $\subseteq$ in (ii). Finally, the triangle inequality verifies that the functions in the left-hand side space of (iii) are continuous, which gives $\subseteq$ in (iii).
\end{proof}
Henceforth, we identify $C_0(G,X)=X\otimes C_0(G)$ as well as $C_b(G,M(A)_s)=M(A\otimes C_0(G))$. The next theorem generalizes \cite[Corollary 3.4]{APT}.
\begin{thm}\label{Cbstr=M}
The map
\[(\psi,{\rm id}):(C_b(G,M(X)_s),C_b(G,M(A)_s))\rightarrow(M(X\otimes C_0(G)),M(A\otimes C_0(G)))\]
given by
\begin{equation*}\label{Appendix.A.Cbstr=M}
\psi(m)\cdot f=m\cdot f
\end{equation*}
for $m\in C_b(G,M(X)_s)$ and $f\in A\otimes C_0(G)$ is an isomorphism.
\end{thm}
\begin{proof}
By Lemma \ref{Preparation.for.Applying.EKQR}, we can apply \cite[Proposition 1.28]{EKQR} to see that $(\psi,{\rm id})$ is an injective correspondence homomorphism. It thus remains to show that $\psi$ is surjective. Let $n\in M(X\otimes C_0(G))$. For each $r\in G$, define $m_n(r):A\rightarrow X$ and $m_n^*(r):X\rightarrow A$ by
\begin{equation*}\label{Appendix.A.1}
m_n(r)(a):=\big(n\cdot(a\otimes\phi_r)\big)(r),\quad m_n^*(r)(\xi):=\big(n^*(\xi\otimes\phi_r)\big)(r),
\end{equation*}
where $\phi_r\in C_c(G)$ such that $\phi_r\equiv1$ on a neighborhood of $r$. It is immaterial which $\phi_r$ we take to define $m_n(r)$ and $m_n^*(r)$ as long as $\phi_r\equiv1$ near $r$. Since
\[\langle n\cdot(a\otimes\phi_r),\xi\otimes\phi_r\rangle_{A\otimes C_0(G)}=\langle a\otimes\phi_r,n^*(\xi\otimes\phi_r)\rangle_{A\otimes C_0(G)},\]
we have $\langle m_n(r)\cdot a,\xi\rangle_A=\langle a,m_n^*(r)\xi\rangle_A$ by evaluating at $r$, and thus obtain a function $m_n:G\rightarrow M(X)$ with $m_n(r)^*=m_n^*(r)$. By definition, $\|m_n(r)\|\leq\|n\|$ for $r\in G$, and hence $m_n$ is bounded. To see that $m_n$ is strictly continuous, let $\{r_i\}$ be a net in $G$ converging to an $r\in G$, $a\in A$, and $\xi,\eta\in X$. Evidently, $\{m_n(r_i)\cdot a\}$ converges to $m_n(r)\cdot a$. The same is true for the net $\{m_n(r_i)^*\xi\}$, and hence $\{\theta_{\eta,\xi}m_n(r_i)\}=\{\theta_{\eta,m_n(r_i)^*\xi}\}$ converges to $\theta_{\eta,m_n(r)^*\xi}=\theta_{\eta,\xi}m_n(r)$, and consequently $\{Tm_n(r_i)\}$ converges to $Tm_n(r)$ for $T\in\mathcal{K}(X)$. Therefore $m_n\in C_b(G,M(X)_s)$. Finally, we have
\[\big(n\cdot(a\otimes\phi_r)\big)(r)=m_n(r)\cdot a = \big(m_n\cdot(a\otimes\phi_r)\big)(r)\]
for $a\in A$ and $r\in G$, which shows $\psi(m_n)=n$.
\end{proof}
Let $Aut(X,A)$ be the group of isomorphisms from $(X,A)$ onto itself. Recall from \cite[Definition 2.5]{EKQR} that an \emph{action} of a locally compact group $G$ on $(X,A)$ is a homomorphism $(\gamma,\alpha):G\rightarrow Aut(X,A)$ such that for each $\xi\in X$ and $a\in A$, the maps
\[G\ni r\mapsto\gamma_r(\xi)\in X,\quad G\ni r\mapsto\alpha_r(a)\in A\]
are both continuous. The formulas
\begin{equation}\label{actions=coactions.formulars}
\sigma^\gamma(\xi)(r)=\gamma_r(\xi),\quad\delta^\alpha(a)(r)=\alpha_r(a)
\end{equation}
then define elements $\sigma^\gamma(\xi)\in M(X\otimes C_0(G))$ and $\delta^\alpha(a)\in M(A\otimes C_0(G))$ for $\xi\in X$ and $a\in A$ by Theorem~\ref{Cbstr=M}. Note that $(\sigma^\gamma,\delta^\alpha)$ is by definition a correspondence homomorphism. The proof of the next theorem is omitted since the $C^*$-algebra counterpart is well-known (for example, \cite[Theorem~9.2.4]{Timm}) and readily transferred to the $C^*$-cor\-re\-spond\-ence setting with an aid of Theorem~\ref{Cbstr=M}.
\begin{thm}\label{actions=coactions}
The formulas \eqref{actions=coactions.formulars} define a one-to-one correspondence between actions of $G$ on $(X,A)$ and coactions of $C_0(G)$ on $(X,A)$.
\end{thm}
}
\section{$C^*$-correspondences $X\rtimes_\sigma\widehat{S}_{\widehat{W}_G}$}\label{App.B}
{\renewcommand*{\thesection}{\Alph{section}}
It is well-known that $\mathcal{L}_A(A\otimes\mathcal{H})=M(A\otimes\mathcal{K}(\mathcal{H}))$ for a $C^*$-algebra $A$ and a Hilbert space $\mathcal{H}$. We generalize this in Proposition \ref{Appendix.B.Prop.1} to a nondegenerate $C^*$-cor\-re\-spond\-ence:
\[\mathcal{L}_A(A\otimes\mathcal{H},X\otimes\mathcal{H})=M(X\otimes\mathcal{K}(\mathcal{H})).\]
Using this, we show in Corollary \ref{Appendix.B.Cor.1} that the construction of Theorem~\ref{crossed product correspondences} recovers the crossed product correspondence $(X\rtimes_{\gamma,r}G,A\rtimes_{\alpha,r}G)$ in \cite{EKQR} for an action $(\gamma,\alpha)$ of $G$ on $(X,A)$.
We first clarify the $C^*$-cor\-re\-spon\-dence $(\mathcal{L}_A(A\otimes\mathcal{H},X\otimes\mathcal{H}),\mathcal{L}_A(A\otimes\mathcal{H}))$ in the next lemma whose proof is trivial, and so we omit it.
\begin{lem}\label{Appendix.B.Lemma0}
The Banach space $\mathcal{L}_A(A\otimes\mathcal{H},X\otimes\mathcal{H})$ is a $C^*$-cor\-re\-spond\-ence over $\mathcal{L}_A(A\otimes\mathcal{H})$ with the following operations
\begin{equation}\label{Appendix.B.Bimod.Operations}
m\cdot l=m\circ l,\quad
\langle m,n\rangle_{\mathcal{L}_A(A\otimes\mathcal{H})}=m^*\circ n,\quad
\varphi_{\mathcal{L}_A(A\otimes\mathcal{H})}=\varphi_{M(A\otimes\mathcal{K}(\mathcal{H}))}
\end{equation}
for $m,n\in\mathcal{L}_A(A\otimes\mathcal{H},X\otimes\mathcal{H})$ and $l\in\mathcal{L}_A(A\otimes\mathcal{H})$.
\end{lem}
Note that $(\mathcal{K}_A(A\otimes\mathcal{H},X\otimes\mathcal{H}),\mathcal{K}_A(A\otimes\mathcal{H}))$ is also a $C^*$-cor\-re\-spond\-ence with the restriction of the operations given in \eqref{Appendix.B.Bimod.Operations}.
\begin{lem}
There exists an isomorphism
\[(\psi_0,{\rm id}):(\mathcal{K}_A(A\otimes\mathcal{H},X\otimes\mathcal{H}),\mathcal{K}_A(A\otimes\mathcal{H}))
\rightarrow(X\otimes\mathcal{K}(\mathcal{H}),A\otimes\mathcal{K}(\mathcal{H}))\]
such that $\psi_0(\theta_{\xi\otimes h,\,a\otimes k})=\xi\cdot a^*\otimes\theta_{h,k}$ for $\xi\in X$, $a\in A$, and $h,k\in\mathcal{H}$.
\end{lem}
\begin{proof}
Let $\xi_i\in X$, $a_i\in A$, and $h_i,k_i\in\mathcal{H}$ for $i=1,\ldots,n$. We claim that that the norm of the operator $\sum_{i=1}^n\theta_{\xi_i\otimes h_i,\,a_i\otimes k_i}$ agrees with that of $\sum_{i=1}^n\xi_i\cdot a_i^*\otimes\theta_{h_i,k_i}$, which proves that $\psi_0$ is well-defined and isometric. For this, we may assume that the vectors $h_i$ are mutually orthonormal and similarly for $k_i$. Then
\begin{align*}
\big\|\sum_{i=1}^n\theta_{\xi_i\otimes h_i,\,a_i\otimes k_i}\big\|^2
&= \big\|\sum_{i,j=1}^n\theta_{a_i\otimes k_i,\,\xi_i\otimes h_i}\theta_{\xi_j\otimes h_j,\,a_j\otimes k_j}\big\| \\
&= \big\|\sum_{i,j=1}^n\theta_{(a_i\otimes k_i)\cdot\langle\xi_i\otimes h_i,\xi_j\otimes h_j\rangle_A,\,a_j\otimes k_j}\big\| \\
&= \big\|\sum_{i=1}^n\theta_{a_i\langle\xi_i,\xi_i\rangle_A\otimes k_i,\,a_i\otimes k_i}\big\|.
\end{align*}
By \cite[Lemma 2.1]{KPWa}, the last of the above equalities coincides with the norm of the following product of two positive $n\times n$ matrices
\[\Big(\big\langle a_i\langle\xi_i,\xi_i\rangle_A\otimes k_i,\,a_j\langle\xi_j,\xi_j\rangle_A\otimes k_j\big\rangle_A\Big)^{1/2}
\Big(\langle a_i\otimes k_i,a_j\otimes k_j\rangle_A\Big)^{1/2}\]
which is diagonal by orthogonality. Let
\[b_i=\langle\xi_i\cdot a_i^*,\,\xi_i\rangle_A\quad(i=1,\ldots,n).\]
Then
\begin{align*}\label{Appendix.B.Eqn1.in.Lem}
\big\|\sum_{i=1}^n\theta_{\xi_i\otimes h_i,\,a_i\otimes k_i}\big\|^2
&= \max_{i=1,\ldots,n}\big\|\big(\langle\xi_i\cdot a_i^*,\,\xi_i\rangle_A^*\langle\xi_i\cdot a_i^*,\,\xi_i\rangle_A\big)^{1/2}(a_i^*a_i)^{1/2}\big\| \\
&=\max_{i=1,\ldots,n}\|(b_i^*b_i)^{1/2}(a_i^*a_i)^{1/2}\|.
\end{align*}
On the other hand,
\begin{align*}
\big\|\sum_{i=1}^n\xi_i\cdot a_i^*\otimes\theta_{h_i,k_i}\big\|^2
&= \big\|\sum_{i,j=1}^n\big\langle\xi_i\cdot a_i^*\otimes\theta_{h_i,k_i},\,\xi_j\cdot a_j^*\otimes\theta_{h_j,k_j}\big\rangle_{A\otimes\mathcal{K}(\mathcal{H})}\big\| \\
&= \big\|\sum_{i,j=1}^n\langle\xi_i\cdot a_i^*,\,\xi_j\cdot a_j^*\rangle_A\otimes\theta_{k_i\langle h_i,h_j\rangle,\,k_j}\big\| \\
&= \max_{i=1,\ldots,n}\|\langle\xi_i\cdot a_i^*,\,\xi_i\rangle_A\,a_i^*\|=\max_{i=1,\ldots,n}\|b_ia_i^*\|
\end{align*}
again by orthonormality. Our claim then follows since
\begin{align*}
\|(b_i^*b_i)^{1/2}(a_i^*a_i)^{1/2}\|^2
&= \|(a_i^*a_i)^{1/2}b_i^*b_i(a_i^*a_i)^{1/2}\| \\
&= \|b_i(a_i^*a_i)^{1/2}\|^2 = \|b_ia_i^*a_ib_i^*\| = \|b_ia_i^*\|^2.
\end{align*}
The remaining parts of the lemma are now easily verified.
\end{proof}
In the next proposition, we identify $\mathcal{K}_A(A\otimes\mathcal{H},X\otimes\mathcal{H})=X\otimes\mathcal{K}(\mathcal{H})$. Note that for $m\in\mathcal{L}_A(A\otimes\mathcal{H},X\otimes\mathcal{H})$ and $f\in A\otimes\mathcal{K}(\mathcal{H})(=\mathcal{K}_A(A\otimes\mathcal{H}))$, the product $m\cdot f$ defines an element of $X\otimes\mathcal{K}(\mathcal{H})$.
\begin{prop}\label{Appendix.B.Prop.1}
There exists an isomorphism
\[(\psi,{\rm id}):(\mathcal{L}_A(A\otimes\mathcal{H},X\otimes\mathcal{H}),\mathcal{L}_A(A\otimes\mathcal{H}))
\rightarrow(M(X\otimes\mathcal{K}(\mathcal{H})),M(A\otimes\mathcal{K}(\mathcal{H})))\]
such that
\begin{equation*}\label{Appendix.B.Define.psi}
\psi(m)\cdot f=m\cdot f
\end{equation*}
for $m\in\mathcal{L}_A(A\otimes\mathcal{H},X\otimes\mathcal{H})$ and $f\in A\otimes\mathcal{K}(\mathcal{H})$.
\end{prop}
\begin{proof}
For the operations on \eqref{Appendix.B.Bimod.Operations}, the following can be easily seen to hold:
\begin{itemize}
\item[(i)] $\mathcal{K}_A(A\otimes\mathcal{H},X\otimes\mathcal{H})\cdot\mathcal{L}_A(A\otimes\mathcal{H})=
\mathcal{K}_A(A\otimes\mathcal{H},X\otimes\mathcal{H})$;
\item[(ii)] $\varphi_{\mathcal{L}_A(A\otimes\mathcal{H})}\big(\mathcal{L}_A(A\otimes\mathcal{H})\big)\,
\mathcal{K}_A(A\otimes\mathcal{H},X\otimes\mathcal{H}) =\mathcal{K}_A(A\otimes\mathcal{H},X\otimes\mathcal{H})$;
\item[(iii)] $\mathcal{L}_A(A\otimes\mathcal{H},X\otimes\mathcal{H})\cdot\mathcal{K}_A(A\otimes\mathcal{H})=
\mathcal{K}_A(A\otimes\mathcal{H},X\otimes\mathcal{H})$.
\end{itemize}
Thus $(\psi,{\rm id})$ is an injective correspondence homomorphism by \cite[Proposition 1.28]{EKQR}. To see that $\psi$ is surjective, let $n\in M(X\otimes\mathcal{K}(\mathcal{H}))$. Take a net $\{x_i\}$ in $X\otimes\mathcal{K}(\mathcal{H})$ strictly converging to $n$. Then the limits $\lim_ix_ih$ and $\lim_ix_i^*k$ clearly exist for $h\in A\otimes\mathcal{H}$ and $k\in X\otimes\mathcal{H}$. Define $m_n:A\otimes\mathcal{H}\rightarrow X\otimes\mathcal{H}$ and $m_n^*:X\otimes\mathcal{H}\rightarrow A\otimes\mathcal{H}$ by
\[m_nh=\lim_ix_ih,\quad m_n^*k=\lim_ix_i^*k.\]
We see from
\[\langle m_nh,k\rangle_A=\lim_i\langle x_ih,k\rangle_A=\lim_i\langle h,x_i^*k\rangle_A=\langle h,m_n^*k\rangle_A\]
that $m_n\in\mathcal{L}_A(A\otimes\mathcal{H},X\otimes\mathcal{H})$ with the adjoint $m_n^*$. Obviously $\psi(m_n)=n$.
\end{proof}
From now on, we identify $\mathcal{L}_A(A\otimes\mathcal{H},X\otimes\mathcal{H})=M(X\otimes\mathcal{K}(\mathcal{H}))$.
\begin{rmk}\label{Appendix.B.Rmk.EmbedA}\rm
Let $\mu_G:C_0(G)\hookrightarrow\mathcal{L}(L^2(G))$ be the embedding in \eqref{Prel.pi.and.U}. The strict extension $\overline{{\rm id}_X\otimes\mu_G}$ then embeds $M(X\otimes C_0(G))$ into $M(X\otimes\mathcal{K}(L^2(G)))$ such that if $m\in C_b(G,M(X)_s)$ and $h\in C_c(G,A)\subseteq A\otimes L^2(G)$, then
\begin{equation*}\label{Appendix.B.Rmk.1}
(\overline{{\rm id}\otimes\mu_G}(m)h)(r)=m(r)\cdot h(r)\quad (r\in G)
\end{equation*}
by strict continuity.
\end{rmk}
Let $(\gamma,\alpha)$ be an action of a locally compact group $G$ on $(X,A)$. The \emph{crossed product correspondence} $(X\rtimes_{\gamma,r}G,A\rtimes_{\alpha,r}G)$ is the completion of the $C_c(G,A)$-bimodule $C_c(G,X)$ such that
\begin{align*}
(x\cdot f)(r) &=\int_Gx(s)\cdot\alpha_s(f(s^{-1}r))\,ds, \\
\langle x,y\rangle_{A\rtimes_{\alpha,r}G}(r) &=\int_G\alpha_s^{-1}(\langle x(s),y(sr)\rangle_A)\,ds, \\
\big(\varphi_{A\rtimes_{\alpha,r}G}(f)\,x\big)(r) &=\int_G\varphi_A(f(s))\,\gamma_s(x(s^{-1}r))\,ds
\end{align*}
for $x,y\in C_c(G,X)$, $f\in C_c(G,A)$, and $r\in G$ (\cite[Proposition 3.2]{EKQR}).
\begin{rmk}\label{Appendix.B.Cc.Density}\rm
The algebraic tensor product $X\odot C_c(G)$ is dense in $X\rtimes_{\gamma,r}G$. This is because $X\odot C_c(G)$ is $L^1$-norm dense in $C_c(G,X)$ and the crossed product norm on $C_c(G,A)$ is dominated by its $L^1$-norm.
\end{rmk}
The proof of the next theorem is only sketched.
\begin{thm}\label{Appendix.B.Thm.}
Let $(\gamma,\alpha)$ be an action of a locally compact group $G$ on a $C^*$-cor\-re\-spond\-ence $(X,A)$. Then, there exists an injective correspondence homomorphism
\[(\psi_\gamma,\pi_\alpha):(X\rtimes_{\gamma,r}G,A\rtimes_{\alpha,r}G)
\rightarrow(\mathcal{L}_A(A\otimes L^2(G),X\otimes L^2(G)),\mathcal{L}_A(A\otimes L^2(G)))\]
such that
\begin{align*}\label{Appendix.B.Int.Form}
\big(\psi_\gamma(x)h\big)(r) &=\int_G\gamma_{r}^{-1}(x(s))\cdot h(s^{-1}r)\,ds, \\
\big(\pi_\alpha(f)h\big)(r) &=\int_G\alpha_r^{-1}(f(s))\,h(s^{-1}r)\,ds
\end{align*}
for $x\in C_c(G,X)$, $f\in C_c(G,A)$, $h\in C_c(G,A)$, and $r\in G$.
\end{thm}
\begin{proof}
It is well-known that $\pi_\alpha$ gives a nondegenerate embedding.
For each $x\in C_c(G,X)\subseteq X\rtimes_{\gamma,r}G$, define $\rho_\gamma(x):C_c(G,X)\rightarrow C_c(G,A)$ by
\[(\rho_\gamma(x)k)(r)=\int_G\Delta(r^{-1})\langle\gamma_{s}^{-1}(x(sr^{-1})),k(s)\rangle_A\,ds\]
for $k\in C_c(G,X)$ and $r\in G$, where $\Delta$ is the modular function of $G$. A routine computation yields
\begin{equation}\label{Appendix.B.Adj.Formula}
\langle\psi_\gamma(x)h,k\rangle_A=\langle h,\rho_\gamma(x)k\rangle_A,\quad
\rho_\gamma(x)\psi_\gamma(y)=\pi_\alpha(\langle x,y\rangle_{A\rtimes_{\alpha,r}G})
\end{equation}
for $h\in C_c(G,A)\subseteq A\otimes L^2(G)$, $k\in C_c(G,X)\subseteq X\otimes L^2(G)$, and $x,y\in C_c(G,X)$. This shows that $\psi_\gamma$ and $\rho_\gamma$ both extend continuously to all of $X\rtimes_{\gamma,r}G$, and $\psi_\gamma(x)\in\mathcal{L}_A(A\otimes L^2(G),X\otimes L^2(G))$ for $x\in X\rtimes_{\gamma,r}G$ with the adjoint $\psi_\gamma(x)^*=\rho_\gamma(x)$.
The second relation in \eqref{Appendix.B.Adj.Formula} gives
$\langle\psi_\gamma(x),\psi_\gamma(y)\rangle_{\mathcal{L}_A(A\otimes L^2(G))}=\pi_\alpha(\langle x,y\rangle_{A\rtimes_{\alpha,r}G})$
for $x,y\in X\rtimes_{\gamma,r}G$. Let $a\in A$ and $\phi\in C_c(G)\subseteq C^*_r(G)$. Since the strict extension $\overline{\pi_\alpha}$ embeds $A$ into $\mathcal{L}_A(A\otimes L^2(G))$ such that $\big(\overline{\pi_\alpha}(a)h\big)(r)=\alpha_r^{-1}(a)h(r)$, we deduce that
\begin{equation*}\label{Appendix.B.Cov.Rep.1}
\big(\varphi_{\mathcal{L}_A(A\otimes L^2(G))}(\overline{\pi_\alpha}(a))\,k\big)(r)=\varphi_A(\alpha_r^{-1}(a))k(r)
\end{equation*}
for $k\in C_c(G,X)$ and $r\in G$. Similarly,
\begin{equation*}\label{Appendix.B.Cov.Rep.2}
\big(\varphi_{\mathcal{L}_A(A\otimes L^2(G))}\big(\overline{\pi_\alpha}(\phi)\big)k\big)(r)=\int_G\phi(s)k(s^{-1}r)\,ds
\end{equation*}
for $k\in C_c(G,X)$ and $r\in G$. An easy computation then verifies
\[\big(\psi_\gamma\big(\varphi_{A\rtimes_{\alpha,r}G}(a\otimes\phi)x\big)h\big)(r)=\big(\varphi_{\mathcal{L}_A(A\otimes L^2(G))}\big(\pi_\alpha(a\otimes\phi)\big)\psi_\gamma(x)h\big)(r)\]
for $x\in C_c(G,X)$, $h\in C_c(G,A)$, and $r\in G$, which gives
\begin{equation*}\label{Appendix.B.Eqn.7}
\psi_\gamma\big(\varphi_{A\rtimes_{\alpha,r}G}(a\otimes\phi)\,x\big)=\varphi_{\mathcal{L}_A(A\otimes L^2(G))}\big(\pi_\alpha(a\otimes\phi)\big)\psi_\gamma(x).
\end{equation*}
The same equality is now true for $f\in A\rtimes_{\alpha,r}G$ in place of $a\otimes\phi$ and for $x\in X\rtimes_{\gamma,r}G$, and hence $(\psi_\gamma,\pi_\alpha)$ is a correspondence homomorphism. Finally, it is injective since $\pi_\alpha$ is injective.
\end{proof}
Let $(\gamma,\alpha)$ be an action of $G$ on $(X,A)$, and $(\sigma^\gamma,\delta^\alpha)$ be the corresponding coaction. Define
\begin{equation}\label{Appendix.B.Cor.1.Eqn}
\sigma^\gamma_G=\overline{{\rm id}_X\otimes\check{\mu}_G}\circ\sigma^\gamma,\quad
\delta^\alpha_G=\overline{{\rm id}_X\otimes\check{\mu}_G}\circ\delta^\alpha,
\end{equation}
where $\check{\mu}_G:C_0(G)\rightarrow S_{\widehat{W}_G}$ is the Hopf $C^*$-algebra isomorphism given in \eqref{Prel.pi.and.U}. Then $(\sigma^\gamma_G,\delta^\alpha_G)$ is a coaction of $S_{\widehat{W}_G}$ on $(X,A)$. In the next corollary, we regard $\sigma^\gamma_{G\iota}(X)=\overline{{\rm id}_X\otimes\iota_{S_{\widehat{W}_G}}}(\sigma^\gamma_G(X))$ as a subspace of $\mathcal{L}_A(A\otimes L^2(G),X\otimes L^2(G))$.
\begin{cor}\label{Appendix.B.Cor.1}
Let $(\gamma,\alpha)$ be an action of a locally compact group $G$ on $(X,A)$. Then $(\psi_\gamma,\pi_\alpha)$ in Theorem \ref{Appendix.B.Thm.} gives an isomorphism from $(X\rtimes_{\gamma,r}G,A\rtimes_{\alpha,r}G)$ onto $(X\rtimes_{\sigma^\gamma_G}\widehat{S}_{\widehat{W}_G},A\rtimes_{\delta^\alpha_G}\widehat{S}_{\widehat{W}_G})$ such that
\begin{equation}\label{Appendix.B.Cor.1.F}
\psi_\gamma(\xi\otimes\phi)=\sigma^\gamma_{G\iota}(\xi)\cdot(1_{M(A)}\otimes\phi),\quad
\pi_\alpha(a\otimes\phi)=\delta^\alpha_{G\iota}(a)(1_{M(A)}\otimes\phi)
\end{equation}
for $\xi\in X$, $a\in A$, and $\phi\in C_c(G)$.
\end{cor}
\begin{proof}
We only need to prove that $\psi_\gamma$ satisfies the first equality in \eqref{Appendix.B.Cor.1.F} and gives a surjection onto $X\rtimes_{\sigma^\gamma_G}\widehat{S}_{\widehat{W}_G}$. Let $\xi\in X$ and $\phi\in C_c(G)$. We see from Remark \ref{Appendix.B.Rmk.EmbedA} that
\[(\sigma^\gamma_{G\iota}(\xi)\,h)(r)=\gamma_r^{-1}(\xi)\cdot h(r)\]
for $h\in C_c(G,A)$ and $r\in G$. Hence
\[\big(\psi_\gamma(\xi\otimes\phi)h\big)(r) = \gamma_r^{-1}(\xi)\cdot\int_G\phi(s)h(s^{-1}r)\,ds =
\big(\sigma^\gamma_{G\iota}(\xi)\big((1_{M(A)}\otimes\phi)h\big)\big)(r),\]
which shows the first equality in \eqref{Appendix.B.Cor.1.F}. Since $X\odot C_c(G)$ is dense in $X\rtimes_{\gamma,r}G$ by Remark \ref{Appendix.B.Cc.Density} and $\psi_\gamma$ is isometric, we must have $\psi_\gamma(X\rtimes_{\gamma,r}G)=X\rtimes_{\sigma^\gamma_G}\widehat{S}_{\widehat{W}_G}$.
\end{proof}
We now provide a proof of Corollary \ref{Cor1.to.Main.Thm}.
\begin{proof}[Proof of Corollary \ref{Cor1.to.Main.Thm}]
Let
\[\zeta_G=\overline{{\rm id}_{\mathcal{O}_X}\otimes\check{\mu}_G}\circ\zeta.\]
Clearly, $\zeta_G$ is the coaction of $S_{\widehat{W}_G}$ on $\mathcal{O}_X$ induced by $(\sigma^\gamma_G,\delta^\alpha_G)$. Define a representation
\[(k_X\rtimes_\gamma G,k_A\rtimes_\gamma G):(X\rtimes_{\gamma,r}G,A\rtimes_{\alpha,r}G)\rightarrow\mathcal{O}_X\rtimes_{\beta^\zeta,r}G\]
to be the composition as indicated in the following diagram:
\[ \xymatrix{(X\rtimes_{\gamma,r}G,A\rtimes_{\alpha,r}G) \ar[rr]^-{(\psi_\gamma,\pi_\alpha)} \ar @{-->}[d]_-{(k_X\rtimes_\gamma G,k_A\rtimes_\alpha G)}
&& (X\rtimes_{\sigma^\gamma_G}\widehat{S}_{\widehat{W}_G},A\rtimes_{\delta^\alpha_G}\widehat{S}_{\widehat{W}_G})
\ar[d]^-{(k_X\rtimes{\rm id}_{\widehat{S}_{\widehat{W}_G}},k_A\rtimes{\rm id}_{\widehat{S}_{\widehat{W}_G}})} \\
\mathcal{O}_X\rtimes_{\beta^\zeta,r}G && \mathcal{O}_X\rtimes_{\zeta_G}\widehat{S}_{\widehat{W}_G}. \ar @{=}[ll] } \]
By definition (\eqref{Sec.5.Rep.1} and \eqref{Appendix.B.Cor.1.F}), we have
\[(k_X\rtimes_\gamma G)(f)(r)=k_X(f(r)),\quad(k_A\rtimes_\alpha G)(g)(r)=k_A(g(r))\]
for $f\in C_c(G,X)$, $g\in C_c(G,A)$, and $r\in G$. The conclusion then follows by Theorem \ref{Main.Theorem.}.
\end{proof}
} | {"config": "arxiv", "file": "1407.6106.tex"} |
TITLE: Significance of exchange operator commuting with Hamiltonian
QUESTION [4 upvotes]: In an Introduction to Quantum Mechanics by Griffiths (pg. 180), he claims that
"P and H are compatible observables, and hence we can find a complete set of functions that are simultaneous eigenstates of both. That is to say, we can find solutions to the Schrodinger equation that are either symmetric (eigenvalue +1) or antisymmetric (eigenvalue -1) under exchange"
I understand why commuting (or as he calls them, compatible) observables share a common eigenbasis. What I don't see is why P, the exchange operator, and H, the Hamiltonian, need to commute for the second sentence to be true. If P is an observable, then supposedly it is a Hermitian operator whose eigenstates span the L2 Hilbert space. Isn't that a sufficient condition for us to say that we can construct any solution to the Schrodinger equation via a linear combination of those eigenstates, and that the eigenstates themselves are solutions to the Schrodinger equation? What is he trying to show by stating the P and H commute?
REPLY [1 votes]: $[H,P]=0$ means the Hamiltonian does not mix symmetric and antisymmetric states, i.e. for any $|\psi_\mathrm{S}\rangle$ and $|\psi_\mathrm{A}\rangle$ we have
$$
\langle\psi_\mathrm{S}|H|\psi_\mathrm{A}\rangle=0.
$$
There are two consequences:
1) If we try to find the eigenstate of $H$ as a mixture of symmetric and antisymmetric states,
$$
H(\alpha|\psi_\mathrm{S}\rangle+\beta|\psi_\mathrm{A}\rangle)=E(\alpha|\psi_\mathrm{S}\rangle+\beta|\psi_\mathrm{A}\rangle),
$$
then, by multiplying it by $\langle\psi_\mathrm{S}|$ or $\langle\psi_\mathrm{A}|$ from the left, we get:
$$
\alpha(E-\langle\psi_\mathrm{S}|H|\psi_\mathrm{S}\rangle)=0,\qquad\beta(E-\langle\psi_\mathrm{A}|H|\psi_\mathrm{A}\rangle)=0.
$$
So $\alpha$ and $\beta$ cannot be simultaneously nonzero if the energies of symmetric and antisymmetric states are different: $\langle\psi_\mathrm{S}|H|\psi_\mathrm{S}\rangle\neq\langle\psi_\mathrm{A}|H|\psi_\mathrm{A}\rangle$. For any realistic system of interacting particles it is the case.
Thus the eigenstates of interacting many-particle system are either symmetric or antisymmetric.
2) If $|\psi(t)\rangle$ is symmetric/antisymmetric at $t=0$, then, after the Schrodinger time evolution
$$
i\hbar\frac{\partial|\psi\rangle}{\partial t}=H|\psi\rangle,
$$
$|\psi(t)\rangle$ will remain symmetric/antisymmetric at $t>0$.
So we need to choose, depending on physical reasons, which sector of the Hilbert space (symmetric or antisymmetric) we are working in with the system of particles of given kind. There will be no physically realistic problem (both stationary $-$ when you find the Hamiltonian eigenstates, or nonstationary $-$ when you calculate time evolution) where you will need to mix symmetric and antisymmetric states.
Update: Symmetric and antisymmetric states have different energies, at least, by two reasons. First, the fermions need to fill the single-particle states with higher and higher energies up to the Fermi energy because of the Pauli exclusion principle, while bosons can condense in a single lowest-energy state. Second, the exchange energy of many-particle system, appearing in the Hartree-Fock approximation, is positive for bosons and negative for fermions. So it is reasonable to assume that the energies of ground as well as excite states of a Hamiltonian are different in symmetric and antisymmetric sectors (except some rare random coincidences), although I don't know the proof of this statement. Therefore the combined spectrum $\{E_n,P_n=\pm1\}$ of $H$ and $P$ is nondegenerate, and each eigenstate of $H$ is automatically an eigenstate of $P$ and vise versa. | {"set_name": "stack_exchange", "score": 4, "question_id": 533546} |
TITLE: Incorrect modulus character computation in Casselman's notes?
QUESTION [0 upvotes]: In Proposition 6.3.3 in Casselman's notes on representation theory of $p$-adic groups, I believe there is an error in the statement of the Proposition.
Let $G$ be the points of a connected, reductive group over a $p$-adic field $k$, $A_0$ a maximal split torus with $P_0 = M_0N_0$ a minimal Levi and $P_{\theta} = M_{\theta}N_{\theta}$ a standard parabolic subgroup for $\theta \subset \Delta$.
If $\Sigma$ is the set of reduced roots of $A_0$ in $G$, then the modulus character $\delta_{P_{\theta}}$ for $P_{\theta}$ is given by the sum of the roots of $A_0$ in $N_{\theta}$, written multiplicatively as
$$\prod\limits_{\alpha \in \Sigma^+ \backslash \Sigma_{\theta}^+} \alpha^{m_{\alpha}}$$
where $m_{\alpha}$ is an associated multiplicity. For $\Omega \subset \Delta$, one considers an element $w$ in the Weyl group $W$ such that $w$ is the element of smallest length $W_{\theta}wW_{\Omega}$, and a representative $x \in N(A_0)$ of $w$. One considers the modulus character $\gamma$ of $N_{\Omega}/(N_{\Omega} \cap x^{-1}P_{\theta}x)$, and the character $(w^{-1}\delta_P^{\frac{1}{2}})\gamma$.
The first statement "The character $(w^{-1} \delta_{\theta}^{\frac{1}{2}}) \gamma$ is thus the square root of the norm of the rational character equal to..." looks good. But then "This in turn is equal to..." doesn't look right. In fact, none of these three products are equal to another, as far as I can tell.
For example, let me try to show directly the equality of the first and the third terms:
$$\prod\limits_{\alpha \in \Sigma^+ \setminus \Sigma_{\theta}^+} w^{-1}\alpha^{m(\alpha)} \prod\limits_{\substack{\alpha \in \Sigma^+ \setminus \Sigma_{\Omega}^+ \\ w\alpha \in \Sigma^- \setminus \Sigma_{\theta}^-}} \alpha^{2m(\alpha)} = \prod\limits_{\substack{\alpha \in \Sigma^+ \setminus \Sigma_{\Omega}^+\\ w\alpha \in \Sigma^- - \Sigma_{\theta}^-}} \alpha^{m(\alpha)} \prod\limits_{\substack{\alpha \not\in \Sigma^+ \setminus \Sigma_{\Omega}^+ \\ w\alpha \in \Sigma^- \setminus \Sigma_{\theta}^-}} \alpha^{-m(\alpha)}$$
This is equivalent to
$$\prod\limits_{\alpha \in \Sigma^+ \setminus \Sigma_{\theta}^+} w^{-1}\alpha^{m(\alpha)} \prod\limits_{\substack{\alpha \in \Sigma^+ \setminus \Sigma_{\Omega}^+ \\ w\alpha \in \Sigma^- \setminus \Sigma_{\theta}^-}} \alpha^{m(\alpha)} = \prod\limits_{\substack{\alpha \not\in \Sigma^+ \setminus \Sigma_{\Omega}^+ \\ w\alpha \in \Sigma^- \setminus \Sigma_{\theta}^-}} \alpha^{-m(\alpha)}$$
which is equivalent to
$$\prod\limits_{\alpha \in \Sigma^+ \setminus \Sigma_{\theta}^+} w^{-1}\alpha^{m(\alpha)} = \prod\limits_{ w\alpha \in \Sigma^- \setminus \Sigma_{\theta}^-} \alpha^{-m(\alpha)}$$
As far as I can tell, this last equality is not true unless $w = w_l w_l^{\theta}$, which is to say that $w$ is the canonical coset representative in $W_{\theta} \backslash W$ of largest length. In general, $w^{-1}(\theta) > 0$, but some of the roots in $w^{-1}. \Sigma^+ - \Sigma_{\theta}$ may remain positive.
REPLY [1 votes]: Oh wait nevermind it is true..the last product I wrote, is equal to $$\prod\limits_{w \alpha \in \Sigma^+ \setminus \Sigma^+_{\theta}} \alpha^{m(\alpha)}$$ which is definitely equal to $$\prod\limits_{ \alpha \in \Sigma^+ \setminus \Sigma^+_{\theta}} w^{-1} \alpha^{m(\alpha)}$$
So this proposition in Casselman's notes is correct after all. | {"set_name": "stack_exchange", "score": 0, "question_id": 2994793} |
\begin{document}
\begin{abstract}
We establish a non-formal link between the structure of the group of Fourier integral operators
$Cl^{0,*}_{odd}(S^1,V)\rtimes Diff_+(S^1)$ and the solutions
of the Kadomtsev-Petviashvili hierarchy, using infinite-dimensional groups of series of non-formal
pseudo-differential operators.
\end{abstract}
\maketitle
\textit{Keywords:} Kadomtsev-Petviashvili hierarchy, Birkhoff-Mulase factorization, infinite jets,
Fourier-integral operators, odd class pseudo-differential operators.
\smallskip
\smallskip
\textit{MSC(2010):} 35Q51; 37K10; 37K25; 37K30; 58J40 Secondary: 58B25; 47N20
\section{Introduction}
The Kadomtsev-Petviashvili hierarchy (KP hierarchy for short)
is an integrable system on an infinite number of dependent variables
which is related to several soliton equations \cite{D,MJD2000}, and which also appears in quantum field theory
and algebraic geometry among other fields, see for instance \cite{Kaz,Mick,M2} and references therein.
In the 1980's M. Mulase published fundamental papers on the
algebraic structure and formal integrability properties of the KP hierarchy, see \cite{M1,M2,M3}. A common
theme in these papers was the use of a powerful theorem on the
factorization of a group of formal pseudo-differential operators
of infinite order which integrates the algebra of formal
pseudo-differential operators: this factorization ---a delicate
algebraic generalization of the Birkhoff decomposition of loop
groups appearing for example in \cite{PS}--- allowed him to
{\em solve} the Cauchy problem for the KP hierarchy in an
algebraic context. These results have been re-interpreted and extended in the context of (generalized)
differential geometry on diffeological and Fr\"olicher spaces and used
to prove well-posedness of the KP hierarchy, see \cite{ER2013,ERMR,Ma2013,MR2016}. Hereafter we make no distinction
between the KP hierarchy of PDEs and the non-linear equation on (formal) pseudo-differential operators which induces it.
In these papers the operators under consideration are formal pseudo-differential operators.
They are called ``formal" because they cannot be understood as operators
acting on smooth maps or smooth sections of vector bundles. They differ from non-formal pseudo-differential operators
by an (unknown) smooth kernel operator, a so-called smoothing operator. As is well-known, any classical non-formal
pseudo-differential operator $A$ generates a formal operator (the one obtained from the asymptotic expansion of the
symbol of $A$, see \cite{Gil}), but there is no canonical way to recover a non-formal operator from a formal one.
The aim of this paper is to show how the non-linear equation on formal
pseudo-differential operators which gives rise to the KP hierarchy of PDEs, see \cite{D}, can be posed on groups of
(non-formal) Fourier integral operators called {\em groups of $Diff(S^1)-$pseudo-dif\-fer\-en\-tial operators}, in
which $Diff(S^1)$ is the group
of diffeomorphisms of $S^1$. These groups arise as central extensions of $Diff(S^1)$ by a group of bounded
(non-formal) classical
pseudo-differential operators in a specific class called the odd-class. This class was first described by Kontsevich
and Vishik in \cite{KV1,KV2} (in order to deal with spectral functions and renormalized determinants) while, to the best
of our knowledge, groups of $Diff(S^1)-$pseudo-differential operators were first described (with $S^1$ replaced by
a compact Riemannian manifold $M$) in \cite{Ma2016}, in the context of differential geometry of non-parametrized,
non-linear grassmannians.
In this paper we specialize some of the results in
\cite{Ma2016} to the group of Fourier integral operators $FCl^{0,*}_{Diff_+,odd}(S^1,V),$ which is described as the central
extension of the group of orientation-preserving diffeomorphisms $Diff_+(S^1)$ by the group of all odd-class, bounded and
invertible classical pseudo-differential operators $Cl^{0,*}_{odd}(S^1,V)$ which act on smooth sections of a trivial (finite
rank) vector bundle $S^1 \times V$, see (\ref{aste}) below. Working with this group we can prove an analogue of the
Birkhoff-Mulase decomposition. In turn, this decomposition motivates us to introduce a parameter-dependent KP hierarchy
and it allows us to solve its corresponding Cauchy problem.
Our decomposition and parameter-dependent hierarchy also involves two other groups:
$\bullet$ The group of {\em integral operators} $Cl^{-1,*}_{odd}(S^1,V),$ which is the kernel of the principal symbol map
(as a morphism of groups) defined in $Cl^{0,*}_{odd}(S^1,V)$;
$\bullet$ The group $Cl_{h,odd}(S^1,V),$ defined along the lines of \cite{Ma2013}. Its elements are formal series
$\sum A_n h^n$ in a parameter $h$, of classical odd class pseudo-differential
operators whose constant term $A_0$ is invertible and bounded, and such that the order of a monomial
(as a classical pseudo-differential operator) is controlled by the order in $h.$ The groups $Cl^{-1,*}_{odd}(S^1,V)$ and
$Cl_{h,odd}(S^1,V)$ are {\em regular} (in a
sense to be explained in Section 2) if endowed with some classical topologies, as we comment in Section 3.
\smallskip
The group $FCl^{0,*}_{Diff_+,odd}(S^1,V)$ already mentioned
is defined through the short exact sequence
\begin{equation} \label{aste}
0 \rightarrow Cl^{0,*}_{odd}(S^1,V) \rightarrow FCl^{0,*}_{Diff_+,odd}(S^1,V) \rightarrow Diff_+(S^1) \rightarrow 0 \; ,
\end{equation}
in the spirit of \cite{Ma2016}.
One of our main observations is that this sequence splits under a {\em Birkhoff-Mulase sequence}
$$
0 \rightarrow Cl^{-1,*}_{odd}(S^1,V) \rightarrow FCl^{0,*}_{Diff_+,odd}(S^1,V) \rightarrow
DO^{0,*}(S^1,V)\rtimes Diff_+(S^1) \rightarrow 0
$$
in which $DO^{0,*}(S^1,V)$ is the loop group of $GL(V)$, see (\ref{astast}) and section \ref{Birkhoff} below.
In agreement with the remarks made at the beginning of this section, we choose our
terminology so as to take into account Mulase's generalization \cite{M1,M2,M3} of
the classical Birkhoff decomposition \cite{PS}. Motivated by Mulase's work and this splitting,
we introduce our non-formal KP hierarchy and study its Cauchy problem in Section \ref{KP}.
Our KP hierarchy is an $h-$deformed KP-hierarchy constructed with the help of the group $Cl_{h,odd}(S^1,V)$ and a scaling
introduced in \cite{Ma2013} in order to solve the Cauchy problem for a (classical) KP hierarchy.
We note that our scaling differs from the one used
in \cite{hKP1,hKP2}: in these papers the authors apply scaling so as to obtain a deformation of the {\em dispersionless}
KP hierarchy.
In Section 5 we also highlight a non-formal operator $U_h \in Cl_{h,odd}(S^1,V)$ which depends on the initial
condition of our $h-$deformed KP hierarchy; this operator generates its solutions by using the Birkhoff-Mulase
decomposition proved in Section 4.
{Finally, in section \ref{S1} we
show how to
recover the operator $U_h$ by analysing the Taylor expansion of functions in the image of the twisted operator
$A : f \in C^\infty(S^1; V ) \mapsto S_0^{-1}(f) \circ g$, in which $g \in Diff_+(S^1)$ and $S_0$ is our version of the
dressing operator (see \cite{D}) for the initial value of the $h-$deformed KP hierarchy.}
\section{Preliminaries on categories of regular Fr\"olicher Lie groups} \label{regular}
In this section we recall briefly the formal setting which allows us to work rigorously
with spaces of pseudo-differential operators and exponential mappings. We begin with the notion of a diffeological space:
\begin{Definition} Let $X$ be a set.
\noindent $\bullet$ A \textbf{p-parametrization} of dimension $p$
on $X$ is a map from an open subset $O$ of $\R^{p}$ to $X$.
\noindent $\bullet$ A \textbf{diffeology} on $X$ is a set $\p$
of parametrizations on $X$ such that:
- For each $p\in\N$, any constant map $\R^{p}\rightarrow X$ is in $\p$;
- For each arbitrary set of indexes $I$ and family $\{f_{i}:O_{i}\rightarrow X\}_{i\in I}$
of compatible maps that extend to a map $f:\bigcup_{i\in I}O_{i}\rightarrow X$,
if $\{f_{i}:O_{i}\rightarrow X\}_{i\in I}\subset\p$, then $f\in\p$.
- For each $f\in\p$, $f : O\subset\R^{p} \rightarrow X$, and $g : O' \subset \R^{q} \rightarrow O$, in which $g$ is
a smooth map (in the usual sense) from an open set $O' \subset \R^{q}$ to $O$, we have $f\circ g\in\p$.
\vskip 6pt If $\p$ is a diffeology on $X$, then $(X,\p)$ is
called a \textbf{diffeological space} and, if $(X,\p)$ and $(X',\p')$ are two diffeological spaces,
a map $f:X\rightarrow X'$ is \textbf{smooth} if and only if $f\circ\p\subset\p'$.
\end{Definition}
The notion of a diffeological space is due to J.M. Souriau, see \cite{Sou}; see also \cite{Igdiff} and \cite{Chen}
for related constructions. Of particular interest to us is the following subcategory of the category of diffeological
spaces.
\begin{Definition}
A \textbf{Fr\"olicher} space is a triple $(X,\F,\C)$ such that
- $\C$ is a set of paths $\R\rightarrow X$,
- $\F$ is the set of functions from $X$ to $\R$, such that a function
$f:X\rightarrow\R$ is in $\F$ if and only if for any
$c\in\C$, $f\circ c\in C^{\infty}(\R,\R)$;
- A path $c:\R\rightarrow X$ is in $\C$ (i.e. is a \textbf{contour})
if and only if for any $f\in\F$, $f\circ c\in C^{\infty}(\R,\R)$.
\vskip 5pt If $(X,\F,\C)$ and $(X',\F',\C ')$ are two
Fr\"olicher spaces, a map $f:X\rightarrow X'$ is \textbf{smooth}
if and only if $\F'\circ f\circ\C\subset C^{\infty}(\R,\R)$.
\end{Definition}
This definition first appeared in \cite{FK}; we use terminology appearing in Kriegl and Michor's book \cite{KM}.
A short comparison of the notions of diffeological and Fr\"olicher spaces is in \cite{Ma2006-3}; the reader can also
see \cite{Ma2013,Ma2018-2,MR2016,Wa} for extended expositions.
Any family of maps $\F_{g}$ from $X$ to $\R$ generates a Fr\"olicher
structure $(X,\F,\C)$ by setting, after \cite{KM}:
- $\C=\{c:\R\rightarrow X\hbox{ such that }\F_{g}\circ c\subset C^{\infty}(\R,\R)\}$
- $\F=\{f:X\rightarrow\R\hbox{ such that }f\circ\C\subset C^{\infty}(\R,\R)\}.$
We call $\F_g$ a \textbf{generating set of functions}
for the Fr\"olicher structure $(X,\F,\C)$. One easily see that
$\F_{g}\subset\F$. {This notion will be useful for us, see for instance
Proposition \ref{fd} below.} A Fr\"olicher space $(X,\F,\C)$
carries a natural topology, the pull-back topology of
$\R$ via $\F$. In the case of a finite dimensional
differentiable manifold $X$ we can take $\F$ as the set of all smooth
maps from $X$ to $\R$, and $\C$ the set of all smooth paths from
$\R$ to $X.$ Then, the underlying topology of the
Fr\"olicher structure is the same as the manifold topology
\cite{KM}.
We also remark that if $(X,\F, \C)$ is a Fr\"olicher space, we can
define a natural diffeology on $X$ by using the following family
of maps $f$ defined on open domains $D(f)$ of Euclidean spaces, see \cite{Ma2006-3}:
$$
\p_\infty(\F)=
\coprod_{p\in\N}\{\, f: D(f) \rightarrow X; \, \F \circ f \in C^\infty(D(f),\R) \quad \hbox{(in
the usual sense)}\}.$$
If $X$ is a finite-dimensional differentiable manifold, this diffeology has been
called the { \em n\'ebuleuse diffeology}. by J.-M. Souriau, see \cite{Sou}.
\begin{Proposition} \label{fd} \cite{Ma2006-3}
Let$(X,\F,\C)$
and $(X',\F',\C')$ be two Fr\"olicher spaces. A map $f:X\rightarrow X'$
is smooth in the sense of Fr\"olicher if and only if it is smooth for
the underlying diffeologies $\p_\infty(\F)$ and $\p_\infty(\F').$
\end{Proposition}
\begin{Proposition} \label{prod1} Let $(X,\p)$ and $(X',\p')$
be two diffeological spaces. There exists a diffeology $\p\times\p'$ on
$X\times X'$ made of plots $g:O\rightarrow X\times X'$
that decompose as $g=f\times f'$, where $f:O\rightarrow X\in\p$
and $f':O\rightarrow X'\in\p'$. We call it the \textbf{product diffeology},
and this construction extends to an infinite product.
\end{Proposition}
We apply this result to the case of Fr\"olicher spaces and we derive very easily, (compare with e.g. \cite{KM})
the following:
\begin{Proposition} \label{prod2} Let $(X,\F,\C)$
and $(X',\F',\C')$ be two Fr\"olicher spaces equipped with their natural
diffeologies $\p$ and $\p'$ . There is a natural structure of Fr\"olicher space
on $X\times X'$ which contours $\C\times\C'$ are the 1-plots of
$\p\times\p'$.
\end{Proposition}
We can also state the above result for infinite products;
we simply take cartesian products of the plots or of the contours.
Now we remark that given an algebraic structure, we can define a
corresponding compatible diffeological structure, see for instance \cite{Les}. For example, a
$\R-$vector space equipped with a diffeology is called a
diffeological vector space if addition and scalar multiplication
are smooth (with respect to the canonical diffeology on $\R$). An
analogous definition holds for Fr\"olicher vector spaces. The example of
Fr\"olicher Lie groups will arise in Section 3 below.
\begin{remark} \label{comp}
Fr\"olicher and Gateaux smoothness are the same notion
if we restrict to a Fr\'echet context, see \cite[Theorem 4.11]{KM}.
Indeed, for a smooth map $f : (F, \p_1(F)) \rightarrow \R$ defined
on a Fr\'echet space with its 1-dimensional diffeology, we have
that $\forall (x,h) \in F^2,$ the map $t \mapsto f(x + th)$ is
smooth as a classical map in $C^\infty(\R,\R).$ And hence, it is
Gateaux smooth. The converse is obvious.
\end{remark}
We follow \cite{Sou,Igdiff}: Let $(X,\p)$ be a diffeological space,
and let $X'$ be a set. Let $f:X\rightarrow X'$ be a map.
We define the \textbf{push-forward diffeology} as the coarsest (i.e. the smallest for inclusion) among the diffologies
on $X'$, which contains $f \circ \p.$
\begin{Proposition} \label{quotient} let $(X,\p)$ b a diffeological
space and $\rel$ an equivalence relation on $X$. Then, there is
a natural diffeology on $X/\rel$, noted by $\p/\rel$, defined as
the push-forward diffeology on $X/\rel$ by the quotient projection
$X\rightarrow X/\rel$. \end{Proposition}
Given a subset $X_{0}\subset X$, where $X$ is a Fr\"olicher space
or a diffeological space, we equip $X_{0}$ with structures
induced by $X$ as follows:
\begin{enumerate}
\item If $X$ is equipped with a diffeology $\p$, we define
a diffeology $\p_{0}$ on $X_{0}$ called the \textbf{subset or trace
diffeology}, see \cite{Sou,Igdiff}, by setting
\[
\p_{0}=\lbrace p\in\p \hbox{ such that the image of }p\hbox{ is a subset of
}X_{0}\rbrace\; .
\]
\item If $(X,\F,\C)$ is a Fr\"olicher space, we take as a generating
set of maps $\F_{g}$ on $X_{0}$ the restrictions of the maps $f\in\F$.
In this case, the contours (resp. the induced diffeology) on $X_{0}$
are the contours (resp. the plots) on $X$ whose images are a subset
of $X_{0}$.
\end{enumerate}
Let $(X,\p)$ and $(X',\p')$ be diffeological spaces. Let $M \subset C^\infty(X,X')$ be a set of smooth maps.
The \textbf{functional diffeology} on $S$ is the diffeology $\p_S$
made of plots
$$ \rho : D(\rho) \subset \R^k \rightarrow S$$
such that, for each $p \in \p, $
the maps $\Phi_{\rho, p}: (x,y) \in D(p)\times D(\rho) \mapsto \rho(y)(x) \in X'$ are plots of $\p'.$
This definition allows us to prove the following classical properties, see \cite{Igdiff}:
\begin{Proposition} \label{cvar}
Let $X,Y,Z$ be diffeological spaces. Then,
$$C^\infty(X\times Y,Z) = C^\infty(X,C^\infty(Y,Z)) = C^\infty(Y,C^\infty(X,Z))$$
as diffeological spaces equipped with functional diffeologies.
\end{Proposition}
\smallskip
Since we are interested in infinite-dimensional analogues of Lie groups, we need to consider tangent spaces of
diffeological spaces, as we have to deal with Lie algebras and exponential maps. We state the following general
definition after \cite{DN2007-1} and \cite{CW}:
\begin{enumerate}
\item[(i)] the \textbf{internal tangent cone} . For each $x\in X,$ we consider
$$C_{x}=\{c \in C^\infty(\R,X)| c(0) = x\}$$ and take the equivalence relation $\mathcal{R}$ given by
$$c\mathcal{R}c' \Leftrightarrow \forall f \in C^\infty(X,\R), \partial_t(f \circ c)|_{t = 0} =
\partial_t(f \circ c')|_{t = 0}.$$ Equivalence classes of
$\mathcal{R}$ are called {\bf germs} and are denoted by $\partial_t c(0)$ or $\partial_tc(t)|_{t=0}$.
The {\bf internal tangent cone} at $x$ is the quotient $^iT_xX = C_x / \mathcal{R}.$ If
$X = \partial_tc(t)|_{t=0} \in {}^iT_X, $ we define the derivation $Df(X) = \partial_t(f \circ c)|_{t = 0}\, .$
\item[(ii)] The \textbf{internal tangent space} at $x \in X$ is the vector space generated by the internal tangent
cone.
\end{enumerate}
{ The reader may compare this definition to the one appearing in \cite{KM} for manifolds in the ``convenient"
$c^\infty-$setting}.
{We remark that the internal tangent cone at a point $x$ is not a vector space in many examples;
this motivates item (ii) above, see \cite{CW,DN2007-1}.
Fortunately, the internal tangent cone at $x\in X$ {\em is} a vector space for the objects under consideration
in this work, see Proposition \ref{leslie} below; it will be called, simply, the tangent space at $x \in X$.}
\begin{Definition}
Let $G$ be a group equipped with a diffeology $\p.$ We call it a \textbf{diffeological group} if both multiplication
and inversion are smooth.
\end{Definition}
Analogous definitions hold for Fr\"olicher groups.
Following Iglesias-Zemmour, see \cite{Igdiff}, we do not assert that arbitrary diffeological groups have associated
Lie algebras; however, the following holds, see \cite[Proposition 1.6.]{Les} and also \cite{MR2016}.
\begin{Proposition} \label{leslie}
Let $G$ be a diffeological group. Then the tangent cone at the neutral element $T_eG$ is a diffeological vector space.
\end{Proposition}
\begin{Definition}
The diffeological group $G$ is a \textbf{diffeological Lie group} if and only if
the derivative of the Adjoint action of $G$ on the diffeological vector space $^iT_eG$ defines a Lie bracket.
In this case, we call $^iT_eG$ the Lie algebra of $G$ and we denote it by $\mathfrak{g}.$
\end{Definition}
Let us concentrate on Fr\"olicher Lie groups. If $G$ is a Fr\"olicher Lie group then, after (i) and (ii) above we
have (see \cite{Ma2013} and \cite{Les}):
$$
\mathfrak{g} = \{ \partial_t c(0) ; c \in \C \hbox{ and } c(0)=e_G \}
$$
is the space of germs of paths at $e_G.$ Moreover:
\begin{itemize}
\item Let $(X,Y) \in \mathfrak{g}^2,$ $X+Y = \partial_t(c.d)(0)$ where $c,d \in \C ^2,$ $c(0) = d(0) =e_G ,$
$X = \partial_t c(0)$ and $Y = \partial_t d(0).$
\item Let $(X,g) \in \mathfrak{g}\times G,$ $Ad_g(X) = \partial_t(g c g^{-1})(0)$ where $c \in \C ,$ $c(0) =e_G ,$
and $X = \partial_t c(0).$
\item Let $(X,Y) \in \mathfrak{g}^2,$ $[X,Y] = \partial_t( Ad_{c(t)}Y)$ where $c \in \C ,$ $c(0) =e_G ,$
$X = \partial_t c(0).$
\end{itemize}
All these operations are smooth and thus well-defined as operations on Fr\"olicher spaces, see
\cite{Les,Ma2013,Ma2018-2,MR2016}.
The basic properties of adjoint, coadjoint actions, and of Lie brackets, remain globally the same
as in the case of finite-dimensional Lie groups, and the proofs are similar: see \cite{Les}
and \cite{DN2007-1} for details.
\begin{Definition} \label{reg1} \cite{Les} A Fr\"olicher Lie group $G$ with Lie algebra $\mathfrak{g}$
is called \textbf{regular} if and only if there is a smooth map
\[
Exp:C^{\infty}([0;1],\mathfrak{g})\rightarrow C^{\infty}([0,1],G)
\]
such that $g(t)=Exp(v(t))$ is the unique solution
of the differential equation \begin{equation}
\label{loga}
\left\{ \begin{array}{l}
g(0)=e\\
\frac{dg(t)}{dt}g(t)^{-1}=v(t)\end{array}\right.\end{equation}
We define the exponential function as follows:
\begin{eqnarray*}
exp:\mathfrak{g} & \rightarrow & G\\
v & \mapsto & exp(v)=g(1) \; ,
\end{eqnarray*}
where $g$ is the image by $Exp$ of the constant path $v.$
\end{Definition}
When the Lie group $G$ is a vector space $V$, the notion of regular Lie group specialize to what is called
{\em regular vector space} in \cite{Ma2013} and {\em integral vector space} in \cite{Les}; we follow the latter
terminology.
\begin{Definition} \label{reg2} \cite{Les}
Let $(V,\p)$ be a Fr\"olicher vector space.
The space $(V,\p)$ is \textbf{integral} if there is a smooth map
$$ \int_0^{(.)} : C^\infty([0;1];V) \rightarrow C^\infty([0;1],V)$$ such that $\int_0^{(.)}v = u$ if and only if $u$
is the unique solution of the differential equation
\[
\left\{ \begin{array}{l}
u(0)=0\\
u'(t)=v(t)\end{array}\right. .\]
\end{Definition}
This definition applies, for instance, if $V$ is a complete locally convex topological vector space equipped with its
natural Fr\"olicher structure given by the Fr\"olicher completion of its n\'ebuleuse diffeology, see
\cite{Igdiff,Ma2006-3,Ma2013}.
\begin{Definition}
Let $G$ be a Fr\"olicher Lie group with Lie algebra $\mathfrak{g}.$ Then, $G$ is \textbf{regular with integral
Lie algebra} if
$\mathfrak{g}$ is integral and $G$ is regular in the sense of Definitions $\ref{reg1}$ and $\ref{reg2}$.
\end{Definition}
The properties of the specific groups which we will use in the following sections are consequences of the following two
structural results which we quote for completeness:
\begin{Theorem} \label{regulardeformation} \cite{Ma2013}
Let $(A_n)_{n \in \mathbb{N}^*} $ be a sequence of integral (Fr\"olicher) vector spaces
equipped with a graded smooth multiplication operation
on $ \bigoplus_{n \in \mathbb{N}^*} A_n ,$ i.e. a multiplication such that for each $n,m
\in \mathbb{N}^*$,
$A_n .A_m \subset A_{n+m}$ is smooth with respect to the corresponding Fr\"olicher structures.
Let us define the (non unital) algebra of formal series:
$$\mathcal{A}= \left\{ \sum_{n \in \mathbb{N}^*} a_n | \forall n \in \mathbb{N}^* , a_n \in A_n \right\},$$
equipped with the Fr\"olicher structure of the infinite product.
Then, the set
$$1 + \mathcal{A} = \left\{ 1 + \sum_{n \in \mathbb{N}^*} a_n | \forall n \in \mathbb{N}^* , a_n \in A_n \right\} $$
is a regular Fr\"olicher Lie group with
integral Fr\"olicher Lie algebra $\mathcal{A}.$
Moreover, the exponential map defines a smooth bijection $\mathcal{A} \rightarrow 1+\mathcal{A}.$
\end{Theorem}
\noindent
\textbf{Notation:} for each $u \in \mathcal{A},$ we write $[u]_n$ for the $A_n$-component of $u.$
\begin{Theorem}\label{exactsequence} \cite{Ma2013}
Let
$$ 1 \longrightarrow K \stackrel{i}{\longrightarrow} G \stackrel{p}{\longrightarrow} H \longrightarrow 1 $$
be an exact sequence of Fr\"olicher Lie groups, such that there is a smooth section $s : H \rightarrow G,$ and such that
the trace diffeology from $G$ on $i(K)$ coincides with the push-forward diffeology from $K$ to $i(K).$
We consider also the corresponding sequence of Lie algebras
$$ 0 \longrightarrow \mathfrak{k} \stackrel{i'}{\longrightarrow} \mathfrak{g} \stackrel{p}{\longrightarrow}
\mathfrak{h} \longrightarrow 0 . $$
Then,
\begin{itemize}
\item The Lie algebras $\mathfrak{k}$ and $\mathfrak{h}$ are integral
if and only if the
Lie algebra $\mathfrak{g}$ is integral
\item The Fr\"olicher Lie groups $K$ and $H$ are regular if and only if the Fr\"olicher Lie group $G$ is regular.
\end{itemize}
\end{Theorem}
Similar results, as in theorem \ref{exactsequence}, are valid for Fr\'echet Lie groups,
see \cite{KM}.
\section{Preliminaries on Fourier integral and pseudo-differential operators} \label{PDO}
We introduce the groups and algebras
of (non-formal!) pseudo-differential operators needed to set up a KP hierarchy and to prove a non-formal
Birkhoff-Mulase decomposition. In this section $E$ is a real or complex finite-dimensional vector bundle over $S^1$;
below we will specialize our considerations to the case $E = S^1 \times V$ in which $V$ is a finite-dimensional vector space.
The following definition appears in \cite[Section 2.1]{BGV}.
\begin{Definition}
The graded algebra of differential operators acting on the space of smooth sections $C^\infty(S^1,E)$ is the
algebra $DO(E)$ generated by:
$\bullet$ Elements of $End(E),$ the group of smooth maps $E \rightarrow E$ leaving each fibre globally invariant
and which restrict to linear maps on each fibre. This group acts on sections of $E$ via (matrix) multiplication;
$\bullet$ The differentiation operators
$$\nabla_X : g \in C^\infty(S^1,E) \mapsto \nabla_X g$$ where $\nabla$
is a connection on $E$ and $X$ is a vector field on $S^1$.
\end{Definition}
Multiplication operators are operators of order $0$; differentiation operators and vector fields are operators
of order 1. In local coordinates, a differential operator of order $k$ has the form
$ P(u)(x) = \sum p_{i_1 \cdots i_r} \nabla_{x_{i_1}} \cdots \nabla_{x_{i_r}} u(x) \; , \quad r \leq k \; ,$
in which the coefficients $p_{i_1 \cdots i_r}$ can be matrix-valued.
We note by $DO^k(S^1)$,$k \geq 0$, the differential operators of order less or equal than $k$.
The algebra $DO(E)$ is graded by order. It is a subalgebra of the algebra of classical pseudo-differential operators
$Cl(S^1,E),$ an algebra that contains, for example, the square root of the Laplacian, its inverse, and all
trace-class operators on $L^2(S^1,E).$ Basic facts on pseudo-differential operators defined
on a vector bundle $E \rightarrow S^1$ can be found in \cite{Gil}.
A global symbolic calculus for pseudo-differential operators has been defined independently
by J. Bokobza-Haggiag, see \cite{BK} and H. Widom, see \cite{Wid}. In these papers is shown how the
geometry of the base manifold $M$ furnishes an obstruction to generalizing
local formulas of composition and inversion of symbols; we do not recall these
formulas here since they are not involved in our computations.
Following \cite{Ma2016}, we assume henceforth that $S^1$ is equipped with charts such that the changes
of coordinates are translations.
\vskip 6pt
\noindent
\textbf{Notations.}
We note by $ PDO (S^1, E) $
(resp. $ PDO^o (S^1, E)$, resp. $Cl(S^1,E)$) the space of pseudo-differential operators
(resp. pseudo-differential operators of order $o$, resp. classical pseudo-differential operators) acting on smooth
sections of $E$, and by $Cl^o(S^1,E)= PDO^o(S^1,E) \cap Cl(S^1,E)$ the space of classical pseudo-differential operators
of order $o$. We also denote by $Cl^{o,\ast}(S^1,E)$ the group of units of $Cl^o(S^1,E)$ and by ${\mathcal F}Cl^{0,*}(S^1,E)$
the group of units of the algebra ${\mathcal F}Cl^{0}(S^1,E)$. If the vector bundle $E$ is
trivial, i.e. $E = S^1\times V$ or $E = S^1 \times \K^p$ with $\K = \R$ or $\mathbb{C},$
we use the notation $Cl(S^1,V)$ or $Cl(S^1,\K^p)$ instead of $Cl(S^1,E).$
\vskip 6pt
A topology on spaces of classical pseudo differential operators has been described by Kontsevich and Vishik in
\cite{KV1}; see also \cite{CDMP,PayBook,Scott} for other descriptions.
We use all along this work the Kontsevich-Vishik topology. This is a Fr\'echet topology such that each space
$Cl^o(S^1,E)$ is closed in $Cl(S^1,E).$
We set
$$ PDO^{-\infty}(S^1,E) = \bigcap_{o \in \Z} PDO^o(S^1,E) \; .$$
It is well-known that $PDO^{-\infty}(S^1,E)$ is a two-sided ideal of $PDO(S^1,E)$, see e.g.
\cite{Gil,Scott}.
Therefore, we can define the quotients
$$\mathcal{F}PDO(S^1,E) = PDO(S^1,E) / PDO^{-\infty}(S^1,E),$$
$$\F Cl(S^1,E) = Cl(S^1,E) / PDO^{-\infty}(S^1,E),$$
$$ \quad \F Cl^o(S^1,E) = Cl^o(S^1,E) / PDO^{-\infty}(S^1,E)\; .$$
{The script font $\F$ stands for} {\it formal pseudo-differential operators}.
The quotient $\mathcal{F}PDO(S^1,E)$ is an algebra isomorphic to the set of formal symbols, see \cite{BK},
and the identification is a morphism of $\mathbb{C}$-algebras for the usual multiplication on formal symbols
(see e.g. \cite{Gil}).
\begin{Theorem}
The groups $Cl^{0,*}(S^1,E),$ $Diff_+(S^1)$ and $\mathcal{F}Cl^{0,*}(S^1,E)$ are regular Fr\"olicher Lie groups.
\end{Theorem}
Even more, it has been noticed in \cite{Ma2006}, by applying the results of \cite{Gl2002}, that the group
$Cl^{0,*}(S^1,V)$ (resp.$\F Cl^{0,*}(S^1,V)$ ) is open in $Cl^0(S^1,V)$ (resp. $\F Cl^{0}(S^1,V)$) and also
that it is a regular {\em Fr\'echet} Lie group. It follows from \cite{Ee,Om} that $Diff_+{S^1}$ is open in the
Fr\'echet manifold $C^\infty(S^1,S^1)$. This fact makes it a Fr\'echet
manifold and, following \cite{Om}, a regular Fr\'echet Lie group.
\begin{Definition} \label{d7}
A classical pseudo-differential operator $A$ on $S^1$ is called odd class
if and only if for all $n \in \Z$ and all $(x,\xi) \in T^*S^1$ we have:
$$ \sigma_n(A) (x,-\xi) = (-1)^n \sigma_n(A) (x,\xi)\; ,$$
in which $\sigma_n$ is the symbol of $A$.
\end{Definition}
This {particular} class of pseudo-differential operators has been introduced in \cite{KV1,KV2}.
Odd class operators are also called ``even-even class'' operators, see \cite{Scott}. We choose to follow the terminology
of the first two references. Hereafter, the notation $Cl_{odd}$ will refer to odd class classical
pseudo-differential operators.
\begin{Proposition}
The algebra $Cl_{odd}^0(S^1,V)$ is a closed subalgebra of $Cl^0(S^1,V)$. Moreover, $Cl_{odd}^{0,*}(S^1,V)$ is
\begin{itemize}
\item an open subset of $Cl^0(S^1,V)$ and,
\item a regular Fr\'echet Lie group.
\end{itemize}
\end{Proposition}
\begin{proof}
We note by $\sigma(A)(x,\xi)$ the total formal symbol of $A \in Cl^0(S^1,V).$ Let
$\phi: Cl^0(S^1,V)\rightarrow \mathcal{F}Cl^{0}(S^1,V)$ defined by
$$\phi(A) = \sum_{n \in \mathbb{N}}\sigma_{-n}(x,\xi) - (-1)^n\sigma_{-n}(x,-\xi).$$
This map is smooth, and $$Cl^{0}_{odd}(S^1,V)= Ker(\phi),$$ which shows that $Cl_{odd}^0(S^1,V)$ is a
closed subalgebra of $Cl^0(S^1,V).$
Moreover, if $H = L^2(S^1,V),$ $$Cl^{0,*}_{odd}(S^1,V) = Cl^{0}_{odd}(S^1,V)\cap GL(H),$$
which proves that $Cl_{odd}^{0,*}(S^1,V)$ is open in the Fr\'echet algebra $Cl^0(S^1,V),$ and it follows
that it is a regular Fr\'echet Lie group by arguing along the lines of \cite{Gl2002,Neeb2007}.
\end{proof}
In addition to groups of pseudo-differential operators, we also need {a restricted class of} groups of
Fourier integral operators which we will call $Diff_+(S^1)-$pseudo-dif\-fer\-en\-tial operators following \cite{Ma2016}.
These groups appear as central extensions of $Diff_+(S^1)$ by groups of (often bounded) pseudo-differential operators.
We do not state the basic facts on Fourier integral operators here (they can be found in the classical paper \cite{Horm}),
but we recall the following theorem, which was stated in \cite{Ma2016} for a general base manifold $M$.
\begin{Theorem} \label{DiffPDO} \cite[Theorem 4]{Ma2016}
Let $H$ be a regular Lie group of pseudo-differential operators acting on smooth sections of a trivial bundle
$E \sim V \times S^1 \rightarrow S^1.$
The group $Diff(S^1)$ acts smoothly on $C^\infty(S^1,V),$ and it is assumed to act smoothly on $H$ by adjoint action.
If $H$ is stable under the $Diff(S^1)-$adjoint action, then there exists a regular Lie group $G$ of Fourier integral
operators defined through the exact sequence:
$$ 0 \rightarrow H \rightarrow G \rightarrow Diff(S^1) \rightarrow 0.$$ If $H$ is a Fr\"olicher Lie group, then $G$
is a Fr\"olicher Lie group.
\end{Theorem}
This result follows from Theorem \ref{exactsequence}.
The pseudo-differential operators considered in this theorem can be classical, odd class, or anything else. Applying
the formulas of ``changes of coordinates'' (which can be understood as adjoint
actions of diffeomorphisms) of e.g. \cite{Gil}, we obtain that odd class pseudo-differential operators are stable
under the adjoint action of $Diff(S^1).$ Thus, we can define the following group:
\begin{Definition}
The group $FCl_{Diff(S^1),odd}^{0,*}(S^1,V)$ is the regular Fr\"olicher Lie group $G$ obtained in Theorem
$\ref{DiffPDO}$ with $H=Cl^{0,*}_{odd}(S^1,V).$
\end{Definition}
Following \cite{Ma2016}, we remark that operators $A$ in this group can be understood as operators in
$Cl^{0,*}_{odd}(S^1,V)$ twisted by diffeomorphisms, this is,
\begin{equation} \label{aux}
A = B \circ g \; ,
\end{equation}
where $g \in Diff(S^1)$ and $B \in Cl^{0,*}_{odd}(S^1,V).$ We note that the diffeomorphism $g$ is
the phase of the operators, but here the phase (and hence the decomposition (\ref{aux})) is unique, which
is not the case for general Fourier integral operators, see e.g. \cite{Horm}.
\begin{remark} \label{Baaj}
This construction of phase functions of $Diff(M)-$pseudo-differential operators
differs from the one described by Omori \cite{Om}
and Adams, Ratiu and Schmid \cite{ARS2} for the groups of Fourier integral operators; the exact relation among these
constructions still needs to be investigated.
\end{remark}
We finish our constructions by using again Theorem \ref{DiffPDO}: we note that
the group $Diff(S^1)$ decomposes into two connected components
$Diff(S^1) = Diff_+(S^1) \cup Diff_-(S^1)\; ,$
where the connected component of the identity, $Diff_+(S^1)$, is the group of orientation preserving diffeomorphisms
of $S^1$. We make the following definition:
\begin{Definition}
The group $FCl_{Diff_+(S^1),odd}^{0,*}(S^1,V)$ is the regular Fr\"olicher Lie group of all operators
in $FCl_{Diff(S^1),odd}^{0,*}(S^1,V)$ whose phase diffeomorphisms
lie in $Diff_+(S^1).$
\end{Definition}
We observe that the central extension $FCl_{Diff(S^1),odd}^{0,*}(S^1,V)$ contains as a subgroup the group
\begin{equation} \label{astast}
DO^{0,*}(S^1,V) \rtimes Diff_+(S^1)\; ,
\end{equation}
where $DO^{0,*}(S^1,V) = C^\infty(S^1,GL(V))$ is the loop group on $GL(V).$
\smallskip
Let us now assume that $V$ is a complex vector space.
By the symmetry property stated in Definition \ref{d7}, an odd class pseudo-differential operator $A$ has a
partial symbol of non-negative order $n$ that reads
\begin{equation} \label{alfa}
\sigma_{n}(A)(x,\xi) = \gamma_n(x) (i\xi)^n \, ,
\end{equation}
where $\gamma_n \in C^\infty(S^1,L(V))$. {{\em This consequence of Definition $\ref{d7}$}} allows
us to check the following direct sum decomposition:
\begin{Proposition}\label{SD}
$$Cl_{odd}(S^1,V) = Cl_{odd}^{-1}(S^1,V) \oplus DO(S^1,V)\; .$$
\end{Proposition}
The second summand of this expression will be specialized in the sequel to differential operators having symbols of order 1.
Because of (\ref{alfa}), we can understand these symbols as elements of $Vect(S^1)\otimes Id_V.$
\section{$FCl^{0,*}_{Diff_+,odd}(S^1,V)$ and the Birkhoff-Mulase decomposition} \label{Birkhoff}
Mulase's factorization theorem (see \cite{M3}) alluded to in Section 1, tells us that
an appropriate group of formal pseudo-differential operators of infinite order decomposes uniquely as the product of a group
of differential operators of infinite order and a group of the form $1 + \mathcal{I}$, in which $\mathcal{I}$ is the
algebra of formal pseudo-differential operators of order at most $-1$. In our context, we consider the Fr\'echet
Lie group $Cl^{-1,*}_{odd}(S^1,V)$ of invertible operators of the form $A = Id + A_{-1}\; ,$
where $A_{-1} \in Cl^{-1}_{odd}(S^1,V).$
We have the following
{non-formal version of the Birkhoff-Mulase decomposition:}
\begin{Theorem}\label{SY}
Let $U \in FCl^{0,*}_{Diff_+,odd}(S^1,V).$ There exists an unique pair
$$(S,Y) \in Cl^{-1,*}_{odd}(S^1,V)\times \left(DO^{0,*}(S^1,V) \rtimes Diff_+(S^1)\right)$$
such that $$U = S\,Y\;.$$
Moreover, the map $U \mapsto (S,Y)$ is smooth and, there is a short exact sequence
of Lie groups:
$$0 \rightarrow Cl^{-1,*}_{odd}(S^1,V) \rightarrow FCl_{Diff_+,odd}^{0,*}(S^1,V) \rightarrow
DO^0(S^1,V)\rtimes Diff_+(S^1) \rightarrow 0$$
for which the $Y$-part defines a smooth global section, and which is a morphism of groups (canonical inclusion).
\end{Theorem}
\begin{proof}
We already know that $U$ splits in an unique way as
$ U = A_0\,.\,g\; ,$
in which $g \in Diff_+(S^1)$ and $A_0 \in Cl^{0,*}_{odd}(S^1,V).$
By Proposition \ref{SD}, the pseudo-differential operator $A_0$ can be written uniquely as a sum, $A = A_I + A_D$,
in which $A_D \in DO^0(S^1,V) \subset Cl_{odd}(S^1,V)$.
Since $A_0$ is invertible, $\sigma_0(A_0) \in C^\infty(S^1,GL(V))$ and hence $A_D \in DO^{0,*}(S^1,V).$
In this way, we have
$$U = A_0. A_D^{-1}.A_D.g.$$
We get $Y=A_D.g \in DO^{0,*}(S^1,V)\rtimes Diff_{+}(S^1)$ and $S = A_0. A_D^{-1} \in Cl^{0,*}_{odd}(S^1,V)$
(the inverse of an odd class operator is an odd class operator).
Let us compute the principal symbol $\sigma_0(S)$:
$$\sigma_0(S)= \sigma_0(A_0)\sigma_0(A_D^{-1})= \sigma_0(A_0)\sigma_0(A_0)^{-1} = Id_V.$$
Thus, $S \in Cl^{-1,*}_{odd}(S^1,V).$ Moreover, the maps $U \mapsto g$ and
$A_0 \mapsto A_D$ are smooth, which ends the proof.
\end{proof}
We have already stated that the Lie groups $Cl^{-1,*}_{odd}(S^1,V),$ $ FCl_{Diff_+,odd}^{0,*}(S^1,V)$ and
$DO^0(S^1,V)\rtimes Diff_+(S^1)$ are regular. Let us take advantage of the existence of exponential mappings.
Let us consider a curve
$$
L(t) \in C^\infty([0;1],Cl^1_{odd}(S^1,V)) \;;
$$
we compare the exponential $\exp(L)(t) \in C^\infty([0;1],FCl^{0,*}_{Diff_+}(S^1,V))$ with
$$
\exp(L_D)(t) \in C^\infty \left( [0;1],DO^{0,*}(S^1,V))\rtimes Diff_+(S^1) \right)
$$
and $$
\exp(L_S)(t) \in C^\infty([0;1],Cl^{-1,*}_{odd}(S^1,V))\; .
$$
On one hand, we can write
$$
\exp(L)(t) = S(t)Y(t)
$$
according to Theorem \ref{SY}, and we know that the paths $t\mapsto S(t)$
and $t \mapsto Y(t)$ are smooth. On the other hand, using the definition of the left exponential map, we get
$$
\frac{d}{dt}\exp(L)(t) = exp(L)(t). L(t) \; .
$$
Thus, gathering the last two expressions we obtain
\begin{eqnarray*}
\frac{d}{dt}\exp(L)(t) & = & \frac{d}{dt}\left( S(t) Y(t)\right) \\
& = & \left(\frac{d}{dt}S(t)\right)S^{-1}(t)S(t) Y(t)+ S(t)Y(t)Y^{-1}(t)\left(\frac{d}{dt}Y(t)\right)\\
& = & \left(\frac{d}{dt}S(t)S^{-1}(t)\right) \exp(L)(t)+ \exp(L)(t)Y^{-1}(t)\left(\frac{d}{dt}Y(t)\right) \\
& = & \exp(L)(t)\left(Ad_{\exp(L)(t)^{-1}}\left( \left(\frac{d}{dt}S(t)S^{-1}(t)\right)\right) +
Y^{-1}(t)\left(\frac{d}{dt}Y(t)\right) \right)\; .
\end{eqnarray*}
Now, $Y^{-1}(t) \frac{d}{dt}Y(t)$ is a smooth path on the space of differential operators of order 1, and
we have
$$Ad_{\exp(L)(t)^{-1}}\left( \left(\frac{d}{dt}S(t)S^{-1}(t)\right)\right) \in Cl^{-1}_{odd}(S^1,V) \; .$$
In this way we obtain:
\begin{Proposition}
Let us assume that $L(t) \in C^\infty([0;1],Cl^1_{odd}(S^1,V))$,
$L = L_S + L_D$ with $L_S \in Cl^{-1}_{odd}(S^1,V)$ and
$L_D \in DO^1(S^1,V)$, and that $\exp(L)(t) = S(t) Y(t)$. Then,
$$Y(t) = \exp(L_D)(t)$$
and $$ S(t) = \exp\left(Ad_{\exp(L)(t)}\left(L_S\right)\right)(t).$$
\end{Proposition}
\begin{proof}
We have obtained that
$$L_D = Y(t)^{-1}\frac{d}{dt}Y(t)$$
and that
$$L_S =Ad_{\exp(L)(t)^{-1}}\left( \left(\frac{d}{dt}S(t)S^{-1}(t)\right)\right)$$
by the uniqueness of the decomposition $$L=L_S + L_D\; .$$
We obtain the result by passing to the exponential maps on the groups $Cl^{-1,*}_{odd}(S^1,V)$ and
$DO^{0,*}(S^1,V)\rtimes Diff_+(S^1).$
\end{proof}
\section{The h-KP hierarchy with non formal odd-class operators} \label{KP}
We make the following definition, along the lines of the theory developed in \cite{Ma2013} for formal
pseudo-differential operators:
\begin{Definition} \label{dfs}
Let $h$ be a formal parameter. The set of odd class $h-$pseudo-differential operators is the set of formal series
\begin{equation} \label{alpha}
Cl_{h,odd}(S^1,V)=\left\{ \sum_{n \in \mathbb{N}} a_n h^n \, | \, a_n \in Cl_{odd}^{n}(S^1,V) \right\}\; .
\end{equation}
\end{Definition}
We state the following result on the structure of $Cl_{h,odd}(S^1,V)$:
\begin{Theorem} \label{fs}
The set $Cl_{h,odd}(S^1,V)$ is a Fr\'echet algebra, and its group of units
given by
\begin{equation} \label{beta}
Cl_{h,odd}^*(S^1,V)=
\left\{ \sum_{n \in \mathbb{N}} a_n h^n \, | \, a_n \in Cl_{odd}^{n}(S^1,V), a_0\in Cl_{odd}^{0,*}(S^1,V) \right\} \; ,
\end{equation}
is a regular Fr\'echet Lie group.
\end{Theorem}
\begin{proof}
{ This result is mostly an application of Theorem \ref{regulardeformation}, hence the growth conditions on the
coefficients $a_n$ appearing in (\ref{alpha}) and (\ref{beta}).}
From the work \cite{Gl2002} by Gl\"ockner, we know that $Cl^{0,*}_{odd}(S^1,V)$ is a regular Fr\'echet Lie group
since it is open in $Cl^0_{odd}(S^1,V)$. According to classical properties of composition of pseudo-differential
operators \cite{Scott}, see also \cite{KV1}, the natural multiplication on $Cl^{0,*}_{odd}(S^1,V)$ is smooth for the
product topology inherited from the classical topology on
classical pseudo-differential operators, and inversion is smooth using the classical formulas of inversion of series.
In this way we conclude that $Cl_{h,odd}(S^1,V)$ is a Fr\'echet algebra.
Moreover, the series $\sum_{n \in \mathbb{N}} a_nh^n \in Cl_{h,odd}(S^1,V)$ is invertible if and only if
$a_0 \in Cl_{odd}^{0,*}(S^1,V),$ which shows that $Cl_{h,odd}^*(S^1,V)$
is open in $Cl_{h,odd}(S^1,V)$. The same result as before, from \cite{Gl2002}, ends the proof.
\end{proof}
{
\begin{remark}
The assumption $a_n \in Cl^n_{odd}$ in Definition $\ref{dfs}$ and Theorem $\ref{fs}$ can be relaxed to the condition
$$a_0 \in Cl^{0,*}_{odd} \hbox{ and } \forall n \in \N^*, a_n \in Cl_{odd}\; ;$$
this is sufficient for having a regular Lie group. However, the Birkhoff-Mulase decomposition seems to fail in this context.
Following \cite{MR2016}, we find that the growth conditions imposed in {\rm (\ref{alpha})} and {\rm (\ref{beta})} will
ensure both regularity {\em and} existence of a Birkhoff-Mulase decomposition.
\end{remark}
}
The decomposition $L = L_S + L_D$, $L_S \in Cl^{-1}_{odd}(S^1,V)$,
$L_D \in DO^1(S^1,V)$, which is valid on $Cl_{odd}(S^1,V)$, see Proposition \ref{SD}, extends straightforwardly to
the algebra $Cl_{h,odd}(S^1,V)$. We now introduce the $h-$KP hierarchy with non-formal pseudo-differential operators.
Let us assume that $t_1, t_2, \cdots, t_n , \cdots,$ are an infinite number of different formal variables. Then, again
adapting work carried out in \cite{Ma2013}, we make the following definition:
\begin{Definition}
Let $S_0 \in Cl^{-1,*}_{odd}(S^1,V)$ and let $L_0 = S_0 (h\frac{d}{dx})S_0^{-1}.$
We say that an operator $$ L(t_1,t_2,\cdots) \in Cl_{h,odd}(S^1,V)[[ht_1,...,h^nt_n...]]$$ satisfies the
$h-$deformed KP hierarchy if and only if
\begin{equation} \label{jph}
\left\{\begin{array}{cl}
L(0) = & L_0 \\
\frac{d}{dt_n}L =& \left[(L^n)_D, L\right] \; .
\end{array}
\right.
\end{equation}
\end{Definition}
We recall from \cite{Ma2013} that the $h-$KP hierarchy is obtained from the classical KP hierarchy by means of the
$h-$scaling
$$\left\{ \begin{array}{ccc}t_n& \mapsto & h^nt_n \\
\frac{d}{dx} & \mapsto &h\frac{d}{dx} \end{array}\right. \; ,$$
and we also recall that formal series in $t_1, \cdots , t_n, \cdots$ can be also understood as smooth functions on the
algebraic sum
$$T= \bigoplus_{n \in \mathbb{N}^*}(\mathbb{R}t_n)$$ for the product topology and product Fr\"olicher structure,
see Proposition \ref{prod2} and \cite{Ma2013}. Now we solve the initial value problem for (\ref{jph}).
\begin{Theorem} \label{hKP}
{ Let $U_h(t_1,...,t_n,...) = \exp\left(\sum_{n \in N^*} h^nt_n (L_0)^n\right) \in Cl_{h,odd}(S^1,V).$ }
Then:
\begin{itemize}
\item There exists a unique pair $(S,Y)$ such that
\begin{enumerate}
\item $U_h = S^{-1}Y,$
\item $Y \in Cl_{h,odd}^*(S^1,V)_D$
\item $S \in Cl_{h,odd}^*(S^1,V)$ and $S - 1 \in Cl_{h,odd}(S^1,V)_S.$
\end{enumerate}
Moreover, the map
$$(S_0,t_1,...,t_n,...)\in Cl^{0,*}_{odd}(S^1,V)\times T \mapsto (U_h,Y)\in (Cl_{h,odd}^*(S^1,V))^2$$ is smooth.
\item The operator $L \in Cl_{h,odd}(S^1,V)[[ht_1,...,h^nt_n...]]$ given by $L = S L_0 S^{-1} = Y L_0 Y^{-1}$,
is the unique solution to the hierarchy of equations
{\begin{equation} \label{formalKP}
\left\{\begin{array}{ccl}
\frac{d}{dt_n}L &=& \left[(L^n)_D(t), L(t)\right] = -\left[(L^n)_S(t), L(t)\right]\\
L(0) & = & L_0 \\
\end{array}\right. \; ,
\end{equation}}
in which the operators in this infinite system are understood as formal operators.
\item The operator $L \in Cl_{h,odd}(S^1,V)[[ht_1,...,h^nt_n...]]$ given by $L = S L_0 S^{-1} = Y L_0 Y^{-1}$
is the unique solution of the hierarchy of equations
\begin{equation}
\label{trueKP} \left\{\begin{array}{ccl}
\frac{d}{dt_n}L &=& \left[(L^n)_D(t), L(t)\right] = -\left[(L^n)_S(t), L(t)\right]\\
L(0) & = & L_0 \\
\end{array}\right.
\end{equation}
in which the operators in this infinite system are understood as odd class, non-formal operators.
\end{itemize}
\end{Theorem}
\begin{proof}
The part of the theorem on formal operators is proved in \cite{Ma2013} or, it can be derived from the results of
\cite{MR2016} by the $h-$scaling defined before.
We now have to pass from decompositions and equations for formal pseudo-differential operators to the same properties
for non-formal, odd class operators. For this, we denote by $\sigma(A)$ the formal operator (or equivalently the
asymptotic expansion of the symbol) corresponding to the operator $A.$
Following \cite{Ma2013}, existence and uniqueness of the decomposition holds on formal operators, that is,
there exist non-formal odd class operators $Y$ and $W$ defined up to smoothing operators such that
$$\sigma(U_h) = \sigma(W)^{-1}\sigma(Y)\; .$$
Now, $\sigma(Y)$ is a formal series in $h, t_1, \cdots t_n,\cdots$ of symbols of differential operators, which are
in one-to-one correspondence with a series of (non-formal)
differential operators. Thus, the operator $Y$ is uniquely defined, not up to a smoothing operator; it depends
smoothly on $U_h$, and so does $W = Y U_h^{-1}.$ This ends the proof of the first point.
The second point on the $h-$deformed KP hierarchy is proven along the lines of \cite{Ma2013}.
The proof of third point is similar to the proof of the first point. We have that $L= Y L_0 Y^{-1}$ is well-defined
and, following classical computations which can be found in e.g. \cite{ER2013}, see also \cite{MR2016}, we have:
\begin{enumerate} \item
$L^k = Y L_0^{\;k} Y^{-1}$
\item $U_h\,L_0^{\;k} U_h^{-1}
= L_0^{\;k}$ since $L_0$ commutes with $U_h = \exp (\sum_k h^kt_k\,L_0^{\; k}).$
\end{enumerate}
It follows that $L^k = Y L_0^{\;k} Y^{-1} = W W^{-1}
Y L_0^{\;k} Y^{-1} W W^{-1} = W L_0^{\;k} W^{-1}$.
We take $t_k$-derivative of $U$ for each $k \geq 1$. We get the equation
\[
\frac{d U_h}{dt^k} = - W^{-1}\frac{dW}{dt_k} W^{-1} Y + S^{-1} \frac{dY}{dt_k}
\]
and so, using $U_h = S^{-1}\,Y$, we obtain the decomposition
\[
W L_0^{\; k} W^{-1} = - \frac{dW}{dt_k} W^{-1} + \frac{dY}{dt_k} Y^{-1} \; .
\]
Since $\frac{dW}{dt_k} W^{-1} \in Cl_{h,odd}(S^1,V)_S$ and $\frac{dY}{dt_k} Y^{-1} \in
Cl_{h,odd}(S^1,V)_D$, we conclude that
$$(L^k)_D = \frac{dY}{dt_k} Y^{-1} \; \; \mbox{ and } \; \; (L^k)_S = - \frac{dW}{dt_k} W^{-1}.$$ Now we take
$t_k$-derivative of $L$:
\begin{eqnarray*}
\frac{d L}{d t_k} & = & \frac{dY}{dt_k} L_0 Y^{-1} - Y L_0 Y^{-1} \frac{dY}{dt_k}
Y^{-1} \\
& = & \frac{dY}{dt_k} Y^{-1} Y L_0 Y^{-1} - Y L_0 Y^{-1}\frac{dY}{dt_k}
Y^{-1} \\
& = &{ (L^k)_D\, L - L\, (L^k)_D } \\
& = &{ [ (L^k)_D , L ] \; . }
\end{eqnarray*}
It remains to check the initial condition: We have $L(0) = Y(0)
L_0 Y(0)^{-1}$, but $Y(0) = 1$ by the definition of $U_h.$
Smoothness with respect to the variables $(S_0, t_1,...,t_n,...)$ is already proved by construction,
and we have established smoothness of the map $L_0 \mapsto Y$ at the beginning of the proof. Thus, the map
$$L_0 \mapsto L(t)= Y(t) L_0 Y^{-1}(t)$$ is smooth. The corresponding equation
$$\frac{d}{dt_k}L = -\left[(L^k)_S,L\right]$$ is obtained the same way.
Let us finish by checking that the announced solution is the unique solution to the non-formal hierarchy
(\ref{trueKP}).
This is still true at the formal level, but two solutions which differ by a smoothing operator may appear at
this step of the proof.
Let $(L+K)(t_1,...)$ be another solution, in which $K$ is a smoothing operator depending on the variables
$t_1,...$, and $L$ is the solution derived from $U_h.$ Then, for each $n \in \N^*$ we have
$$(L+K)^n_D = L^n_D\; ,$$
which implies that
$K$ satisfies the {\em linear} equation
$$ \frac{d K}{dt_n} = [L^n_D,K]$$
with initial conditions
$K|_{t=0} = 0.$ We can construct the unique solution $K$ by induction on $n$, beginning with $n=1$.
Let $g_n$ be such that $$(g_n^{-1} dg_n)(t_n) = L^n_D(t_1,...t_{n-1},t_n,0,...) \; .$$ Then we get that
$$K(t_1,...t_n,0....) = Ad_{g_n(t_n)}\left(K(t_1,t_{n-1},0...)\right) \; , $$
and hence, by induction,
$$K(0)=0 \Rightarrow K(t_1,0...)=0 \Rightarrow \cdots \Rightarrow K(t_1,...t_n,0....) =0 \Rightarrow \cdots \; ,$$
which implies that $K=0.$
\end{proof}
\section{KP equations and $Diff_{+}(S^1)$} \label{S1}
Let $A_0 \in Cl_{odd}^{-1}(S^1,V)$, and set $S_0 = \exp(A_0)$. The operator $S_0 \in Cl_{odd}^{-1,\ast}(S^1,V)$ is our
version of the dressing operator of standard KP theory, see for instance \cite[Chapter 6]{D}. We define the operator $L_0$
by
$$
f \mapsto L_0(f) = h \left( S_0 \circ \frac{d}{dx} \circ S_0^{-1} \right) (f)
$$
for $f \in C^\infty(S^1,V)$. We note that $L_0^k(f) = h^k S_0 \frac{d^k}{dx^k}(S_0^{-1}(f))$, a formula which we will use
presently. Our aim is to connect the operator
$$
U_h = \exp\left(\sum_{n \in \N*} h^n t_n L_0^n\right) \; ,
$$
which generates the solutions of the $h-$deformed KP hierarchy described in Theorem \ref{hKP}, with the Taylor expansion
of functions in the image of the twisted operator
$$
A : f \in C^\infty(S^1,V) \mapsto S_0^{-1}(f) \circ g \; ,
$$
in which $g \in Diff_+(S^1)$. We remark that $A \in FCl^{0,*}_{Diff_+,odd}(S^1,V)$ for each $g \in Diff_+(S^1)$,
{and that it is smooth with respect to $g$ due to our Birkhoff-Mulase decomposition theorem}.
For convenience, we identify $S^1$ with $[0;2\pi[\sim \mathbb{R}/2\pi\mathbb{Z},$ assuming implicitly that all the
values under consideration are up to terms of the form $2 k \pi,$ for $k \in \mathbb{Z}.$ Set $c = S_0^{-1}(f)\circ g
\in C^\infty(S^1,V)$. We compute:
\begin{eqnarray*}
c(x_0+h) & = & \left( S^{-1}_0(f) \circ g \right)(x_0 + h) \\
& \sim_{x_0} & \left( S^{-1}_0(f) \circ g \right)(x_0) + \sum_{n \in \mathbb{N}^*}
\left[ \frac{h^n}{n !}\, \frac{d^n}{dx^n} \left( S^{-1}_0(f) \circ g \right) \right](x_0) \\
& = & \left( S^{-1}_0(f) \circ g \right)(x_0) + \\
& & \sum_{n \in \mathbb{N}^*}
\left[\frac{h^n}{n!}\sum_{k = 1}^n B_{n,k}(u_1(x_0),...,u_{n-k+1}(x_0))\frac{d^k}{dx^k}\left(S_0^{-1}(f)\circ g\right)(x_0)\right] \; ,
\end{eqnarray*}
in which we have used the classical Fa\'a de Bruno formula for the higher chain rule in terms of Bell's polynomials
$B_{n,k}$, and $u_i(x_0) = g^{(i)}(x_0)$ for $i = 1, \cdots n-k+1$. We can rearrange the last sum and write
\begin{eqnarray*}
c(x_0+h) & \sim_{x_0} & \left( S^{-1}_0(f) \circ g \right)(x_0) + \\
& & \sum_{k \in \mathbb{N}^*} \sum_{n \geq k} \left[ \frac{h^n}{n !}\, B_{n,k}(u_1(x_0),...,u_{n-k+1}(x_0)) \frac{d^k}{dx^k}\left(S_0^{-1}(f)\right)\right] (g(x_0))
\end{eqnarray*}
or,
\begin{eqnarray} \label{t1}
c(x_0+h) & \sim_{x_0} & \sum_{k \in \mathbb{N}} \left[ a_k h^k \frac{d^k}{dx^k}\left(S_0^{-1}(f)\right)\right] (g(x_0))
\end{eqnarray}
in which $a_0=1$ and
$$ a_k = \sum_{n \geq k} \frac{h^{n-k}}{n!} B_{n,k}(u_1(x_0),...,u_{n-k+1}(x_0)) $$
for $k \geq 1$. In terms of the operator $L_0$, Equation (\ref{t1}) means that
\begin{eqnarray} \label{t2}
c(x_0+h) & \sim_{x_0} & S_0^{-1} \sum_{k \in \mathbb{N}} \left[ a_k\, L_0^k(f)\right] (g(x_0)) \; .
\end{eqnarray}
We now define the sequence $(t_n)_{n \in \N^*}$ by the formula
\begin{equation} \label{t3}
\log\left( \sum_{k \in \N} a_kX^k \right) = \sum_{n \in \N^*} t_n X^n \; ,
\end{equation}
so that both, $a_k$ and $t_n$, are series in the variable $h$. We obtain
\begin{eqnarray*}
c(x_0+h) & \sim_{x_0} & S_0^{-1} \exp \left( \sum_{n \in \mathbb{N}^*} \frac{t_n}{h^n}\, L_0^k(f) \right) (g(x_0)) \; .
\end{eqnarray*}
We state the following theorem:
\begin{Theorem}
Let $f \in C^\infty(S^1,V)$ and set $c = S_0^{-1}(f) \circ g \in C^\infty(S^1,V).$
The Taylor series at $x_0$ of the function $c$ is given by
$$c(x_0+h) \sim_{x_0} S_0^{-1} \left( U_h(t_1/h,t_2/h^2,...)(f) \right)(g(x_0)) \; ,$$
in which the times $t_i$ are related to the derivatives of $g$ via Equation $(\ref{t3})$.
\end{Theorem}
The coefficients of the series $a_k$ and $t_n$ appearing in (\ref{t3}) depend smoothly on $g \in Diff_+(S^1)$ and
$x_0 \in S^1$. Indeed, the map
$$
(x,g) \in S^1 \times Diff_+(S^1) \mapsto \left( g(x), (u_n(x))_{n \in \N^*} \right) \in S^1 \times \R^{\N^*}
$$
is smooth due to Proposition \ref{prod2} (more precisely, due to the generalization of Proposition \ref{prod2} to
infinite products); smoothness $a_k$ then follows, while smoothness of $t_n$ is consequence of
Equation (\ref{t3}).
\medskip
\begin{remark}
As a by-product of the foregoing computations, we notice the following relation. If $f \in C^\infty(S^1,V),$
we can write
$$ f(x_0+h) \sim_{x_0} f(x_0)+ \sum_{n \in \mathbb{N}^*}\left(\frac{h^n}{n!}\left(\frac{d}{dx}\right)^n f\right) (x_0) =
\left(\exp\left(h\frac{d}{dx}\right)f\right)(x_0) \in J^\infty(S^1,V)$$
for $x_0 \in S^1.$
Thus, the operator $\exp\left(h\frac{d}{dx}\right)$ belongs to the space $Cl_h(S^1,V).$
\end{remark}
\vskip 12pt
\paragraph{\bf Acknowledgements:} Both authors have been partially
supported by CONICYT (Chile) via the {\em Fondo Nacional de Desarrollo
Cient\'{\i}fico y Tecnol\'{o}gico} operating grant \# 1161691. The authors would like to thank Saad Baaj for comments
leading to Remark \ref{Baaj}. | {"config": "arxiv", "file": "1808.03791.tex"} |
\section{Cohomology}
\label{sec:cohom}
In this section we review several cohomology theories of pattern spaces.
\subsection{Pattern equivariant functions and Cohomology \cite{KellendonkPEF}}
\label{subsec:PEC}
Let $\Lambda \subset \mathbb{R}^d$ be a Delone set and $\Omega_\Lambda$ its associated pattern space. We say that $f:\mathbb{R}^d\rightarrow\mathbb{C}$ is (strongly) $\Lambda$-equivariant if for some $r > 0$
$$B_{r} \cap \varphi_x(\Lambda) = B_{r} \cap \varphi_y(\Lambda) \Rightarrow f(x) = f(y).$$
\begin{remark}
There is a notion of \emph{weakly} $\Lambda$-equivariant functions defined in \cite{KellendonkPEF}. Since we will only use strongly pattern equivariant functions, we will omit the adverb ``strongly'' and just write ``$\Lambda$-equivariant''.
\end{remark}
We denote by $C^k_{\Lambda}(\mathbb{R}^d,V)$ the space of $\Lambda$-equivariant $C^k$ functions on $\mathbb{R}^d$ with values in the vector space $V$. Smooth $\Lambda$-equivariant forms can be then identified with elements of $C^\infty_{\Lambda}(\mathbb{R}^d,\bigwedge(\mathbb{R}^{d})^*)$. We will denote those spaces by $\Delta_{\Lambda}^*$ and by $\star$ the Hodge-$\star$ operator which induces an isomorphism $\star:\Delta_\Lambda^k\rightarrow\Delta_\Lambda^{d-k}$ for any $k\in\{0,\dots, d\}$. We will denote by $(\star 1)$ the canonical Lebesgue volume form in $\mathbb{R}^d$.
The spaces $\Delta_\Lambda^k$ form a subcomplex of the de Rham complex. As such, we can define the pattern equivariant cohomology
\begin{equation}
\label{eqn:PEcoh}
H^k(\Omega_\Lambda;\mathbb{C}) = \frac{\ker \{d:\Delta_{\Lambda}^k\rightarrow \Delta_{\Lambda}^{k+1} \}}{\mathrm{Im}\{ d:\Delta_{\Lambda}^{k-1}\rightarrow \Delta_{\Lambda}^{k}\}}.
\end{equation}
\begin{definition}
A function $f: \Omega_\Lambda\rightarrow \mathbb{C}$ is \emph{transversally locally constant} if for every $\Lambda'\in\Omega_\Lambda$ there is a $\epsilon>0$ such that $f$ is constant on $\mathcal{C}_{\Lambda',\epsilon,\{\bar{0}\}}$.
\end{definition}
We denote by $C^\infty_{tlc}(\Omega_\Lambda, \bigwedge \mathbb{C}^d)$ the transversally locally constant functions in $C^\infty(\Omega_\Lambda, \bigwedge \mathbb{C}^d)$ which are smooth along the leaves of the foliation.
\begin{theorem}[\cite{Kellendonk-Putnam:RS}]
\label{thm:tlc}
Every smooth $\Lambda $-equivariant function $g:\mathbb{R}^d\rightarrow \mathbb{C}$ defines a unique smooth, transversally locally constant function $\bar{f}_g:\Omega_\Lambda\rightarrow\mathbb{C}$ and we have that $\varphi^*_t \bar{f}_g(\Lambda) = g(t)$. This induces an algebra isomorphism between transversally locally constant functions which are smooth along leaves, and smooth $\Lambda$-equivariant functions.
\end{theorem}
We shall denote by
\begin{equation}
\label{eqn:AlgIso}
i_\Lambda: C^\infty_{tlc}(\Omega_\Lambda)\rightarrow \Delta^0_{\Lambda}
\end{equation}
the isomorphism given by above theorem.
\subsection{\v{C}ech cohomology}
\label{subsec:Cech}
For clarity, this subsection will use the setting of tilings instead of Delone sets. We will make some mild assumptions on our tilings: that our tiling is built out of polyhedra for which there are only finitely many tile types (up to translation) with tiles meeting full face to full face. This assumption gives the tiling finite local complexity. Note that any such tiling may be associated with a Delone set satisfying finite local complexity; see Remark \ref{rmk:backandforth}.
Let $T$ be a tiling of $\mathbb{R}^{d}$ satisfying the conditions above. We let $\check{H}^{*}(\Omega_T;\mathbb{C})$ denote the \v{C}ech cohomology of $\Omega_T$ with coefficients in $\mathbb{C}$. It is now well known that it is possible to compute $\check{H}^{*}(\Omega_\Lambda,\mathbb{C})$ using the inverse limit structure of $\Omega_{T}$, described below. While there are many presentations of $\Omega_T$ as an inverse limit, we use the method initially outlined by Anderson-Putnam and generalized by G\"{a}hler. More information can be found in \cite{sadun:inverse}, which our presentation follows.
Suppose $T$ is our given tiling, and let $P$ be a patch in $T$. The first corona of $P$ is defined to be all tiles $t$ in $T$ for which $t \cap P \ne \varnothing$. The first corona is thus a new patch $P_{1}$ in $T$, and we may consider the second corona $P_{2}$ of $P$ to be the patch obtained as the first corona of the patch $P_{1}$. In this way, given the $n$-th corona $P_{n}$ of $P$, we define the $n+1$st corona of $P$ to be the patch $P_{n+1}$ defined as the first corona of $P_{n}$.
Let $n \in \mathbb{N}$. We say two tiles $t_1, t_2$ in $T$ are $n$-equivalent if a patch of $T$ consisting of $t_1$ and it's $n$-th corona is equal, up to translation, to a similar patch around $t_2$. Because we assume $T$ has finite local complexity, there are only finitely many $n$-equivalence classes of tiles, and we refer to such a class as an $n$-collared tile. Thus an $n$-collared tile consists of a tile, along with all the information of its $n$-th corona. Let $\{p_{i}^{(n)}\}_{i=1}^{m(n)}$ denote the collection of $n$-collared tiles in $T$. Consider now the disjoint union $P^{(n)} = \bigsqcup_{i=1}^{m(n)}p_{i}^{(n)}$, and consider the relation on $P^{(n)}$ obtained by identifying two boundary faces of $n$-collared tiles $p_{j}^{(n)}$ and $p_{k}^{(n)}$ in $P^{(n)}$ if there is a patch in $T$ for which those boundary faces meet. We let $\Gamma_{n}$ denote the quotient of $P^{(n)}$ under this relation, and refer to it as the $n$-th Anderson-Putnam complex (for $T$). The spaces $\Gamma_{n}$ also come equipped with a branched manifold structure, allowing the notion of smooth functions, and smooth $k$-forms on $\Gamma_{n}$ to be defined (see \cite[\S 5.1]{sadun:book}). We refer to the collection of smooth $k$-forms on $\Gamma_{n}$ as $\Delta^{k}(\Gamma_{n})$. \\
\indent There is a map $f_{n}:\Gamma_{n+1} \to \Gamma_{n}$ given by 'forgetting' the $n+1$-st corona; that is, for $x \in \Gamma_{n+1}$ lying in an $n+1$-st collared tile, we send $x$ to the corresponding $n$-collared tile it lies in. This map extends naturally to branch points in $\Gamma_{n}$, and we get an inverse system $\{\Gamma_{n},f_{n}\}$. \\
\indent The following, due to Anderson and Putnam in the case of substitutions and later G{\"a}hler in more generality, shows that this construction can be used to present the tiling space $\Omega_{T}$ as an inverse limit. See \cite[Ch. 2]{sadun:book}.
\begin{theorem}
\label{APGTheorem}
$\Omega_{T}$ is homeomorphic to $\varprojlim\{\Gamma_{n},f_{n}\}$.
\end{theorem}
Given $n \in \mathbb{N}$, we may use the tiling $T$ of $\mathbb{R}^{d}$ to construct a map $\pi_{n}:\mathbb{R}^{d} \to \Gamma_{n}$: for $x \in \mathbb{R}^{d}$, $\pi_{n}(x) \in \Gamma_{n}$ is obtained by inspecting the $n$-th corona of $x$ in $T$. A key fact, which is not hard to show, is that a function $f$ on $\mathbb{R}^{d}$ is pattern-equivariant if and only if there is some $n$ for which there exists some smooth function $g$ on $\Gamma_{n}$ such that $f = \pi_{n}^{*}(g) = g\circ\pi_{n}$. The following, found in \cite[Theorem 5.4]{sadun:book}, follows immediately from this.
\begin{proposition}
\label{prop:PEformsPullBack}
The collection of pattern-equivariant $k$-forms $\Delta_{T}^{k}$ is naturally identified with $\bigcup_{n \in \mathbb{N}}\pi_{n}^{*}(\Delta^{k}(\Gamma_{n}))$.
\end{proposition}
This can be used to show the following, originally proved by Kellendonk and Putnam \cite{Kellendonk-Putnam:RS}.
\begin{theorem}
\label{IsoCohomology}
For a tiling $T$ of finite local complexity, the pattern-equivariant cohomology $H^{*}(\Omega_T;\mathbb{C})$ is naturally isomorphic to the \v{C}ech cohomology $\check{H}^{*}(\Omega_{T};\mathbb{C})$.
\end{theorem}
\subsection{Cohomology for cut and project sets}
\label{subsec:CAPScohom}
In this section we summarize the results of \cite{FHK:topological, GHK:cohomology} which are relevant to our work. Namely, we review sufficient conditions under which the cohomology spaces defined in (\ref{eqn:PEcoh}) are finite dimensional for cut and project constructions.
Recall from \S \ref{subsec:CAPS} that our CAPS is given by an Euclidean space $E$ of dimension $n$ containing some lattice $\Gamma$ such that $E = \mathbb{R}^d \oplus \mathbb{R}^{n-d}$ with associated projections $\pi_\parallel:E\rightarrow \mathbb{R}^d$ and $\pi_\perp:E\rightarrow \mathbb{R}^{n-d}$. It is assumed that $\mathbb{R}^d$ and $\mathbb{R}^{n-d}$ are in \emph{total irrational position} with respect to $\Gamma$, meaning that $\pi_\parallel$ and $\pi_\perp$ are one to one and with dense image on the lattice. Let $K\subset \mathbb{R}^{n-d}$ be the window which we will take to be a finite union of compact non-degenerate polyhedra in $\mathbb{R}^{n-d}$ with boundary $\partial K$ made up of faces of dimension $n-d-1$. Denote by $\Gamma^\parallel = \pi_\parallel(\Gamma)$ and $\Gamma^\perp = \pi_\perp(\Gamma)$ the rank $n$ subgroups of $\mathbb{R}^d$ and $\mathbb{R}^{n-d}$, respectively.
We denote by
$$\Lambda(\Gamma,K) = \{\pi_\parallel(z):z\in\Gamma\mbox{ and }\pi_\perp(z)\in K\}.$$
For each point $x\in \mathbb{R}^n$ denote
\begin{equation}
\label{eqn:LambdaX}
\Lambda_x(\Gamma,K) = \{\pi_\parallel(z):z\in\Gamma\mbox{ and }\pi_\perp(z+x)\in K\}.
\end{equation}
the point in $\Omega_{\Lambda(K,\Gamma)}$ corresponding to $x$.
The set of \emph{singular points} in $E$ is the set
$$\mathcal{S}(\Gamma,K) = \{ x\in E: \pi_\perp(x)\in \partial K + \Gamma^\perp\}$$
and its complement $N\mathcal{S}$ is the set of \emph{nonsingular points}.
\begin{definition}
We call the CAPS $\Lambda(\Gamma,K)$ \emph{almost canonical} if for each face $f_i$ of the window $K$, the set $f_i + \Gamma^\perp$ contains the affine space spanned by $f_i$.
\end{definition}
This is equivalent \cite{GHK:cohomology} to having a finite family of $(n-d-1)$-dimensional affine subspaces
$$\mathcal{W} = \{W_\alpha\subset \mathbb{R}^{n-d}\}_{\alpha\in I_{n-d-1}}$$
indexed by some finite set $I_{n-d-1}$ such that
\begin{equation}
\label{eqn:Singular}
\mathcal{S}(\Gamma,K) = \mathbb{R}^d + \Gamma^\perp + \bigcup_{\alpha\in I_{n-d-1}} W_\alpha.
\end{equation}
Suppose we have an almost canonical CAPS and we have chosen such a family of subspaces $\mathcal{W}$. An affine subspace $W_\alpha + \gamma \subset \mathbb{R}^{n-d}$ for any $\alpha\in I_{n-d-1}$ and $\gamma\in\Gamma^\perp$ is called a \emph{singular space}. The set of singular spaces is independent of the generating set $\mathcal{W}$ chosen.
Two singular spaces with non-trivial intersection give rise to singular spaces of lower dimensions. As such, $\Gamma$ acts on the set of all singular spaces as follows. If $W$ is a singular space of dimension $r$ and $\gamma\in\Gamma$, then so is $\gamma\cdot W = W + \pi_\perp(\gamma)$. The \emph{stabilizer} $\Gamma^W$ of a singular space is the subgroup $\{\gamma\in\Gamma: \gamma\cdot W = W\}$.
\begin{definition}
\begin{enumerate}
\item For each $0\leq r < n-d$, let $\mathcal{P}_r$ be the set of all singular $r$-spaces. Their orbit space under the translation action by $\Gamma$ is denoted by $I_r = \mathcal{P}_r/\Gamma$.
\item Since $\Gamma$ is abelian, the stabilizer $\Gamma^W$ of a singular $r$-space $W$ depends only on the orbit class $\Theta\in I_r$ of $W$. We denote the stabilizer of an orbit class $\Theta$ by $\Gamma^\Theta$.
\item For $r<k<n-d$ and $W\in \mathcal{P}_k$ of orbit class $\Theta\in I_k$, let $\mathcal{P}^W_r$ be the set $\{U\in\mathcal{P}_r:U\subset W\}$, the set of singular $r$-spaces lying in $W$. Then $\Gamma^\Theta$ acts on $\mathcal{P}^W_r$ and we write $I^\Theta_r = \mathcal{P}^W_r/\Gamma^\Theta$, a set which depends only on the class $\Theta$ of $W$. Therefore $I^\Theta_r\subset I_r$ consists of those orbits of singular $r$-spaces which have a representative which lies in a singular $k$-space of class $\Theta$.
\item We denote the cardinalities of these sets by $L_r = |I_r|$ and $L^\Theta_r = |I_r^\Theta|$.
\end{enumerate}
\end{definition}
Given the collection $\mathcal{W}$ of affine subspaces as defined above, let $\mathcal{N}(\mathcal{W}) = \{u_{W} \mid W \in \mathcal{W}\}$ be a collection of unit vectors for which $u_{W}$ is normal to the subspace $W \in \mathcal{W}$, and such that $\mathcal{N}(\mathcal{W})$ spans $\mathbb{R}^{n-d}$. Following \cite{FHK:topological}, we say $\mathcal{N}(\mathcal{W})$ is decomposable if there exists a partition $\mathcal{N}(\mathcal{W}) = \mathcal{N}_{1} \cup \mathcal{N}_{2}$ such that $\textnormal{span}\mathcal{N}_{1} \cap \textnormal{span}\mathcal{N}_{2} = 0$; if there is no such partition, we call $\mathcal{N}(\mathcal{W})$ indecomposable. Finally, call $\mathcal{W}$ indecomposable if there exists an indecomposable collection $\mathcal{N}(\mathcal{W})$ for $\mathcal{W}$.\\
\indent With all this, we have the following important result \cite{FHK:topological, GHK:cohomology}.
\begin{theorem}
\label{thm:L0andcohomology}
\begin{enumerate}
\item $L_0$ is finite if and only if $H^*(\Omega_\Lambda;\mathbb{R})$ is finite dimensional.
\item If $L_0$ is finite then all the $L_r$ and $L_r^\Theta$ are finite as well, and $\nu = n/(n-d)$ is an integer. Moreover, if $\mathcal{W}$ is indecomposable, then $\mathrm{rank}\,\Gamma^U = \nu\cdot\mathrm{dim}(U)$ for any singular space if and only if $L_0$ is finite.
\end{enumerate}
\end{theorem}
We now go over concrete geometrical conditions which guarantee that the cohomological spaces are finitely generated.
\begin{definition}
A \emph{rational subspace} of $E$ is a subspace spanned by vectors from $\mathbb{Q}\Gamma$. A \emph{rational affine subspace} is a translate of a rational subspace.
\end{definition}
\begin{definition}
\label{def:rational}
A \emph{rational projection method pattern} is any Delone set arising from an almost canonical CAPS satisfying the following rationality conditions:
\begin{enumerate}
\item The number $\nu = n/(n-d)$ is an integer.
\item There is a finite set $\mathcal{D}$ of rational affine subspaces of $E$ in one to one correspondence under $\pi_\perp$ with the set $\mathcal{W}$, i.e., each $W\in\mathcal{W}$ is of the form $W = \pi_\perp(D)$ for some unique $D\in\mathcal{D}$.
\item The elements of $\mathcal{D}$ are $\nu(n-d-1)$-dimensional, and any intersection of finitely many members of $\mathcal{D}$ or their translates is either empty or a rational affine subspace $R$ of dimension $\nu\cdot\mathrm{dim}\,\pi_\perp(R)$.
\end{enumerate}
\end{definition}
For any affine subspace $R$ in $E$, denote by $\Gamma^R$ the stabilizer subgroup of $\Gamma$ under its translation action on $E$.
\begin{theorem}[\cite{FHK:topological, GHK:cohomology}]
\label{thm:finiteCohom}
Suppose $\Lambda(\Gamma,K)$ is a rational projection method pattern and $\mathbb{T}^n = \mathbb{R}^n/\mathbb{Z}^n$. Then
\begin{enumerate}
\item There are isomomorphisms $H^s(\mathbb{T}^n;\mathbb{C}) \cong H^s(\Omega_\Lambda;\mathbb{C})$ for $s< \nu - 1$.
\item For any commutative ring $S$, the cohomology $H^*(\Omega_\Lambda;S)$ is finitely generated over $S$.
\end{enumerate}
\end{theorem} | {"config": "arxiv", "file": "1511.07557/cohomology.tex"} |
TITLE: What's the difference between semiconductor and insulator (besides band gap)?
QUESTION [2 upvotes]: The typical classification of electronic materials is metal-semiconductor-insulator. Is there any actual difference between a semiconductor and an insulator, besides the size of the bandgap?
REPLY [1 votes]: Basically they are the same from a physics point of view, if you only look at crystalline materials. A semiconductor is defined to be insulating at $ 0 K $, while conducting at room temperature, although I don't recall, what level of conductivity is required to count as semiconductor.
Technically, insulators are a more general group of materials, since they could also be amorphous, while semiconductor materials are in the best case monocrystalline with very few defects or at least polycrystalline with a certain grain size in order to show proper semiconducting behavior due to the bandstructure. If there are to many defects, grain boundaries, impurities, the real bandstructure can differ from the ideal one by a lot. | {"set_name": "stack_exchange", "score": 2, "question_id": 203577} |
TITLE: Why is this condition required? (Schwarz's lemma)
QUESTION [2 upvotes]: Good afternoon!
I was having a look at some complex analysis, and encountered the following lemma (Schwarz's):
If $f$ is holomorphic in the unit disc, and $f(0) = 0$, then $|f(z)| \leq |z|$ for any $z$ in the unit disc $\mathbb{D}$.
The proof is quite straightforward:
Let $g$ be the function defined by $g(z) = f(z)/z$ if $z \ne 0$, and $g(0) = f'(0)$. Then, a simple application of the maximum modulus principle implies $|g(z)| \leq 1$, which finishes the proof.
However, I do not quite understand why we would need the condition $f(0) = 0$. Wikipedia says this enforces differentiability of $g$ at $z = 0$. But why is that so? It looks like some application of L'Hopital's rule but this is leaving me quite puzzled.
And even so, why couldn't we just define $g$ on the unit disc without $0$? The maximum modulus principle only requires the domain to be open and connected, which $\mathbb{D}\backslash\{0\}$ is. So why would this not work?
Thanks for your time :)
REPLY [1 votes]: If $f$ has a zero, say $f(a)=0$, we can reduce it to the use of the Schwarz's lemma. Consider the function $\varphi_a\colon\Delta\rightarrow\Delta$ given by
$$
\varphi_a(z)=\frac{z-a}{1-\bar{a}z}
.
$$
(Can you prove that, indeed, $|\varphi_a(z)|<1$ for every $|z|<1$?) Here $\Delta$ denotes the unit disk.
Then $h=f\circ\varphi_a$ is an analytic function from $\Delta$ to itself which satisfies $h(0)=0$. We apply Schwarz's lemma to get
$$
|h(z)|\leq|z| \quad\Rightarrow\quad |f(\varphi_a(z))|\leq|z|\quad\forall\:|z|\leq1
.
$$
Now, these $\varphi_a$'s are invertible:
$\varphi_{-a}\circ\varphi_a=\varphi_a\circ\varphi_{-a}=\mathrm{id}_\Delta$ (I leave the computations to you!). Then, from the inequality above we obtain
$$
|f(z)|\leq|\varphi_{-a}(z)|=\left|\frac{z+a}{1+\bar{a}z}\right|
.
$$
Maybe not a satisfactory inequality since for $z=a$ we have the upper bound $2|a|/(1+|a|^2)$ for zero... But at least this ilustrates some use of the $\varphi_a$'s. | {"set_name": "stack_exchange", "score": 2, "question_id": 3559968} |
\DOC CHEAT_TAC
\TYPE {CHEAT_TAC : tactic}
\SYNOPSIS
Proves goal by asserting it as an axiom.
\DESCRIBE
Given any goal {A ?- p}, the tactic {CHEAT_TAC} solves it by using {mk_thm},
which in turn involves essentially asserting the goal as a new axiom.
\FAILURE
Never fails.
\USES
Temporarily plugging boring parts of a proof to deal with the interesting
parts.
\COMMENTS
Needless to say, this should be used with caution since once new axioms are
asserted there is no guarantee that logical consistency is preserved.
\SEEALSO
new_axiom, mk_thm.
\ENDDOC
| {"subset_name": "curated", "file": "formal/hol/Help/CHEAT_TAC.doc"} |
TITLE: Motivation for symmetric and antisymmetric atoms in QM
QUESTION [1 upvotes]: For a two particle system in QM, if we have identical particles, which are fermions then we require that the overall wave function (position and spin) is antisymmetric and if the the particles are identical bosons then we require that overall wave function is symmetric, i.e.
$$\psi(r_{1},r_{2}) = A[\psi_{a}(r_{1})\psi_{b}(r_{2})\pm\psi_{b}(r_{1})\psi_{a}(r_{2}) ].$$
Question:
But what are the rules when considering atoms. For example, helium or hydrogen seem to follow the fermion rules regarding when a wave function is symmetric or antisymmetric, that is the overall wave function should be antisymmetric. Do we follow the rules related to fermions for atoms as well since we have electrons involved? What is the basic motivation for these rules? I am just starting to learn about spin in QM, hence am using an introductory text.
Thanks.
REPLY [2 votes]: Indeed certain atoms are bosons and other atoms are fermions. Just to provide you with an example, Bose-Einstein condensates are often realized with ultracold bosonic atoms, such as $^4\mathrm{He}$ or Rubidium. Since these kinds of atoms are bosons, if you cool them down below a critical temperature, the wavefunctions of each single atom overlap and you can start talking about a macroscopic wavefunction: that's condensation!
The basic motivation is that every object is either a boson or a fermion, no matter if we are talking about a single electron, an hydrogen atom, a photon or an atom. Once you know what kind of particle you are dealing with, you automatically know which rule you have to follow. | {"set_name": "stack_exchange", "score": 1, "question_id": 307353} |
\begin{document}
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\title{Self-similar dilatation structures and automata}
\author{Marius Buliga}
\address{"Simion Stoilow"
Institute of Mathematics, Romanian Academy,
P.O. BOX 1-764, RO 014700,
Bucure\c sti, Romania}
\email{Marius.Buliga@imar.ro}
\subjclass[2000]{22A30; 05C12; 68Q45}
\date{14.09.2007}
\begin{abstract}
We show that on the boundary of the dyadic tree, any self-similar
dilatation structure induces a web of interacting automata.
\end{abstract}
\maketitle
\section*{Introduction}
In this paper we continue the study of dilatation structures, introduced in
\cite{buligadil1}.
A dilatation structure $(X,d,\delta)$ describes the approximate
self-similarity of the
metric space $(X,d)$. Metric spaces which admit strong dilatation structures
(definition
\ref{defweakstrong}) have metric tangent spaces at any point (theorem
7 \cite{buligadil1}). By theorems 8, 10 \cite{buligadil1},
any such metric tangent space has an algebraic structure of a conical group.
Particular examples of conical groups are Carnot groups, that is simply connected Lie
groups whose Lie algebra admits a positive graduation.
Here we are concerned with dilatation structures on ultrametric spaces.
The special case considered is the boundary of the infinite dyadic tree,
topologically the same as the middle-thirds Cantor set. This is also
the space of infinite words over the alphabet $X = \left\{ 0, 1\right\}$.
Self-similar dilatation structures are introduced and
studied on this space.
We show that on the boundary of the dyadic tree, any self-similar
dilatation structure is described by a web of interacting automata. This
is achieved in theorems \ref{tstruc} and \ref{th2}. These theorems are
analytical in nature, but they admit an easy interpretation in terms
of automata by using classical results as theorem \ref{tcont} and
proposition \ref{psync}. Due to the limitations in length of the paper,
we leave this straightforward interpretation, as well as examples, for
a further paper (but see also the slow-paced introduction into the subject
\cite{buligadil2}).
The subject is relevant for applications to the hot topic of
self-similar groups of isometries of the dyadic tree (for an introduction
into self-similar groups see \cite{bargrinek}).
\section{Words and the Cantor middle-thirds set}
Let $X$ be a finite, non empty set. The elements of $X$ are called letters.
The collection of words of finite length in the alphabet $X$ is denoted by $X^{*}$. The empty word
$\emptyset$ is an element of $X^{*}$.
The length of any word $w\in X^{*}$,
$\displaystyle w = a_{1} ... a_{m}$, $\displaystyle a_{k} \in X$
for all $k=1, ... , m $,
is denoted by $\mid w \mid = m$.
The set of words which are infinite at right is denoted by
$$X^{\omega} = \left\{ f \ \mid \ \ f: \mathbb{N}^{*}\rightarrow X \right\} = X^{\mathbb{N}^{*}} \quad . $$
Concatenation of words is naturally defined. If $\displaystyle q_{1},q_{2} \in X^{*}$ and
$w \in X^{\omega}$ then $\displaystyle q_{1}q_{2} \in X^{*}$ and $q_{1}w \in X^{\omega}$.
The shift map $\displaystyle s : X^{\omega} \rightarrow X^{\omega}$ is defined
by $\displaystyle w = w_{1} \, s(w) $,
for any word $\displaystyle w \in X^{\omega}$. For any $\displaystyle k \in \mathbb{N}^{*}$ we define
$\displaystyle [w]_{k} \in X^{k} \subset X^{*}$, $\displaystyle \left\{ w \right\}_{k} \in X^{\omega}$ by
$$ w = [w]_{k} \,Ês^{k}(w) \quad , \quad \left\{ w \right\}_{k} = s^{k}(w) \quad . $$
The topology on $\displaystyle X^{\omega}$ is generated by cylindrical sets $qX^{\omega}$,
for all $q\in X^{*}$. The topological space $\displaystyle X^{\omega}$ is compact.
To any $q\in X^{*}$ is associated a continuous injective transformation
$\hat{q}:X^{\omega}\rightarrow X^{\omega}$, $\hat{q}(w) = qw$. The semigroup
$X^{*}$ (with respect to concatenation) can be identified with the semigroup
(with respect to function composition) of these transformations. This
semigroup is obviously generated by $X$. The empty word $\emptyset$
corresponds to the identity function.
The dyadic tree $\mathcal{T}$ is the infinite rooted planar binary tree.
Any node has two descendants. The nodes are coded by elements of
$\displaystyle X^{*}$, $X = \left\{ 0,1\right\}$. The root is coded by the
empty word and if a node is coded by $x\in X^{*}$ then its left hand side
descendant has the code $x0$ and its
right hand side descendant has the code $x1$. We shall therefore identify the
dyadic tree with $\displaystyle X^{*}$ and we put on the dyadic tree the
natural (ultrametric) distance on $\displaystyle X^{*}$. The boundary (or
the set of ends) of the dyadic tree is then the same as the compact
ultrametric space $\displaystyle X^{\omega}$.
\section{Automata}
In this section we use the same notations as \cite{grigorchuk3}.
\begin{definition}
An (asynchronous) automaton is an oriented set $\displaystyle (X_{I}, X_{O},
Q, \pi, \lambda)$, with:
\begin{enumerate}
\item[(a)] $\displaystyle X_{I}, X_{O}$ are finite sets, called the input and
output alphabets,
\item[(b)] $Q$ is a set of internal states of the automaton,
\item[(c)] $\pi$ is the transition function, $\displaystyle \pi: X_{I} \times
Q \rightarrow Q$,
\item[(d)] $\lambda$ is the output function, $\displaystyle
\lambda: X_{I} \times Q \rightarrow X_{O}^{*}$.
\end{enumerate}
If $\lambda$ takes values in $\displaystyle X_{O}$ then the automaton is
called synchronous.
\end{definition}
The functions $\lambda$ and $\pi$ can be continued to the set $\displaystyle
X_{I}^{*} \times Q$ by: $\pi(\emptyset, q)\, = \, q$, $\lambda(\emptyset, q) \,
= \, \emptyset$,
$$\pi(xw, q) \, = \, \pi(w, \pi(x,q)) \quad , \quad \lambda(xw, q) \, =
\, \lambda(x,q) \lambda(w, \pi(x,q))$$
for any $\displaystyle x \in X_{I}, q \in Q$ and any $\displaystyle
w \in X_{I}^{*}$.
An automaton is nondegenerate if the functions $\lambda$ and $\pi$ can be
uniquely extended by the previous formul\ae to
$\displaystyle X_{I}^{\omega} \times Q$.
To any nondegenerated automaton $\displaystyle (X_{I}, X_{O},
Q, \pi, \lambda)$ and any $q \in Q$ is associated the function
$\displaystyle \lambda(\cdot , q) : X_{I}^{\omega} \rightarrow
X_{O}^{\omega}$. The following is theorem 2.4 \cite{grigorchuk3}.
\begin{theorem}
The mapping $\displaystyle f: X_{I}^{\omega} \rightarrow
X_{O}^{\omega}$ is continuous if and only if it is defined by a certain
nondegenerate asynchronous automaton.
\label{tcont}
\end{theorem}
The proof given in \cite{grigorchuk3} is interesting to read because it
provides a construction of an automaton which defines the continuous function
$f$.
\section{Isometries of the dyadic tree}
An isomorphism of $\mathcal{T}$ is just an invertible transformation which preserves the structure of the
tree. It is well known that isometries of $\displaystyle (X^{\omega}, d)$ are the same as isometries of
$\mathcal{T}$.
Let $\displaystyle A \in Isom(X^{\omega}, d)$ be such an isometry. For any finite word $\displaystyle
q\in X^{*}$ we may define $\displaystyle A_{q} \in Isom(X^{\omega}, d)$ by
$$A(qw) = A(q) \, A_{q}(w)$$
for any $\displaystyle w \in X^{\omega}$. Note that in the previous relation $A(q)$ makes sense because
$A$ is also an isometry of $\mathcal{T}$.
The following description of isometries of the dyadic tree in terms of automata
can be deduced from an equivalent formulation of proposition 3.1
\cite{grigorchuk3} (see also proposition 2.18 \cite{grigorchuk3}).
\begin{proposition}
A function $\displaystyle X^{\omega} \rightarrow X^{\omega}$ is an isometry
of the dyadic tree if and only if it is generated by a synchronous automaton
with $\displaystyle X_{I} = X_{O} = X$.
\label{psync}
\end{proposition}
\section{Motivation: linear structure in terms of dilatations}
For the normed, real, finite dimensional vector space $\mathbb{V}$, the dilatation based
at $x$, of coefficient $\varepsilon>0$, is the function
$$\delta^{x}_{\varepsilon}: \mathbb{V} \rightarrow \mathbb{V} \quad , \quad
\delta^{x}_{\varepsilon} y = x + \varepsilon (-x+y) \quad . $$
For fixed $x$ the dilatations based at $x$ form a one parameter group which
contracts any bounded neighbourhood of $x$ to a point, uniformly with respect
to $x$.
The algebraic structure of $\displaystyle \mathbb{V}$ is encoded in
dilatations. Indeed, using dilatations we can recover the operation of addition and multiplication by scalars.
For $\displaystyle x,u,v \in \mathbb{V}$ and $\varepsilon>0$
define
$$\Delta_{\varepsilon}^{x}(u,v) = \delta_{\varepsilon^{-1}}^{\delta_{\varepsilon}^{x} u}
\delta^{x}_{\varepsilon} v \quad , \quad
\Sigma_{\varepsilon}^{x}(u,v) = \delta_{\varepsilon^{-1}}^{x} \delta_{\varepsilon}^{\delta_{\varepsilon}^{x} u}
(v) \quad , \quad inv^{x}_{\varepsilon}(u) = \delta_{\varepsilon^{-1}}^{\delta_{\varepsilon}^{x} u} x \quad . $$
The meaning of this functions becomes clear if we take the limit as
$\varepsilon \rightarrow 0$ of these expressions:
\begin{equation}
\lim_{\varepsilon\rightarrow 0} \Delta_{\varepsilon}^{x}(u,v) =
\Delta^{x}(u,v) = x+(-u+v) \quad
\label{exad}
\end{equation}
$$\lim_{\varepsilon\rightarrow 0} \Sigma_{\varepsilon}^{x}(u,v) = \Sigma^{x}(u,v) = u+(-x+v) \quad ,$$
$$\lim_{\varepsilon\rightarrow 0} inv^{x}_{\varepsilon}(u) = inv^{x}(u) = x-u+x \quad , $$
uniform with respect to $x,u,v$ in bounded sets.
The function $\displaystyle \Sigma^{x}(\cdot,\cdot)$ is a group operation, namely the addition operation
translated such that the neutral element is $x$. Thus, for $x=0$, we recover the group operation. The function
$\displaystyle inv^{x}(\cdot)$ is the inverse function, and $\displaystyle \Delta^{x}(\cdot,\cdot)$ is the difference function.
Dilatations behave well with respect to the distance $d$ induced by the norm, in the following sense:
for any $\displaystyle x,u,v\in \mathbb{V}$ and any $\varepsilon>0$ we have
\begin{equation}
\frac{1}{\varepsilon} d( \delta_{\varepsilon}^{x} u ,
\delta^{x}_{\varepsilon} v ) = d(u,v)
\label{coned}
\end{equation}
This shows that from the metric point of view the space $\displaystyle
(\mathbb{V}, d)$ is a metric cone, that is $\displaystyle
(\mathbb{V}, d)$ looks the same at all scales.
Affine continuous transformations $A:\mathbb{V} \rightarrow \mathbb{V}$ admit the following
description in terms of dilatations. (We could dispense of continuity hypothesis in this situation, but
we want to illustrate a general point of view, described further in the paper).
\begin{proposition}
A continuous transformation $A:\mathbb{V} \rightarrow \mathbb{V}$ is affine if and only if for any
$\varepsilon \in (0,1)$, $x,y \in \mathbb{V}$ we have
\begin{equation}
A \delta_{\varepsilon}^{x} y \ = \ \delta_{\varepsilon}^{Ax} Ay \quad .
\label{eq1proplin}
\end{equation}
\label{1proplin}
\end{proposition}
The proof is a straightforward consequence of representation formul{\ae} for the addition, difference and
inverse operations in terms of dilatations.
Further on we shall take the dilatations as basic data associated to an
ultrametric space. In order to understand our aim we describe it as follows:
we shall study a particular ultrametric space (the infinite dyadic tree)
as if we study the vector space $\mathbb{V}$ by using
only the distance $d$ and the dilatations $\displaystyle
\delta^{x}_{\varepsilon}$ for all $x \in X$ and $\varepsilon > 0$.
We shall call a triple $(X,d, \delta)$
a dilatation structure (see further definition \ref{defweakstrong}), where
$(X,d)$ is a locally compact metric space and $\delta$ is a collection of
dilatations of the metric space $(X,d)$.
We shall add
some compatibility relations between the distance $d$ and dilatations
$\delta$, which will prescribe:
\begin{enumerate}
\item[-] the behaviour of the distance
with respect to dilatations, for example some form of relation
(\ref{coned}),
\item[-] the interaction between dilatations, for example the existence of
the limit from the left hand side of relation (\ref{exad}).
\end{enumerate}
\section{Dilatation structures}
This section contains the axioms of a dilatation structure, introduced in
Buliga \cite{buligadil1}.
\subsection{Notations}
Let $\Gamma$ be a topological separated commutative group endowed with a continuous group morphism
$$\nu : \Gamma \rightarrow (0,+\infty)$$ with $\displaystyle \inf \nu(\Gamma) = 0$. Here $(0,+\infty)$ is
taken as a group with multiplication. The neutral element of $\Gamma$ is denoted by $1$. We use the multiplicative notation for the operation in $\Gamma$.
The morphism $\nu$ defines an invariant topological filter on $\Gamma$ (equivalently, an end). Indeed,
this is the filter generated by the open sets $\displaystyle \nu^{-1}(0,a)$, $a>0$. From now on
we shall name this topological filter (end) by "0" and we shall write $\varepsilon \in \Gamma \rightarrow
0$ for $\nu(\varepsilon)\in (0,+\infty) \rightarrow 0$.
The set $\displaystyle \Gamma_{1} = \nu^{-1}(0,1] $ is a semigroup. We note $\displaystyle
\bar{\Gamma}_{1}= \Gamma_{1} \cup \left\{ 0\right\}$
On the set $\displaystyle
\bar{\Gamma}= \Gamma \cup \left\{ 0\right\}$ we extend the operation on $\Gamma$ by adding the rules
$00=0$ and $\varepsilon 0 = 0$ for any $\varepsilon \in \Gamma$. This is in agreement with the invariance
of the end $0$ with respect to translations in $\Gamma$.
We shall use the following convenient notation: by $\mathcal{O}(\varepsilon)$ we mean a positive function defined on $\Gamma$ such that $\displaystyle \lim_{\varepsilon \rightarrow 0} \mathcal{O}(\nu(\varepsilon)) \ = \ 0$.
\subsection{The axioms}
The first axiom is a preparation for the next axioms. That is why we
counted it as axiom 0.
\begin{enumerate}
\item[{\bf A0.}] The dilatations $$ \delta_{\varepsilon}^{x}: U(x)
\rightarrow V_{\varepsilon}(x)$$ are defined for any
$\displaystyle \varepsilon \in \Gamma, \nu(\varepsilon)\leq 1$. The sets
$\displaystyle U(x), V_{\varepsilon}(x)$ are open neighbourhoods of $x$.
All dilatations are homeomorphisms (invertible, continuous, with
continuous inverse).
We suppose that there is a number $1<A$ such that for any $x \in X$ we have
$$\bar{B}_{d}(x,A) \subset U(x) \ .$$
We suppose that for all $\varepsilon \in \Gamma$, $\nu(\varepsilon) \in
(0,1)$, we have
$$ B_{d}(x,\nu(\varepsilon)) \subset \delta_{\varepsilon}^{x} B_{d}(x,A)
\subset V_{\varepsilon}(x) \subset U(x) \ .$$
There is a number $B \in (1,A]$ such that for
any $\varepsilon \in \Gamma$ with $\nu(\varepsilon) \in (1,+\infty)$ the
associated dilatation
$$\delta^{x}_{\varepsilon} : W_{\varepsilon}(x) \rightarrow B_{d}(x,B) \ , $$
is injective, invertible on the image. We shall suppose that
$\displaystyle W_{\varepsilon}(x) \in \mathcal{V}(x)$, that
$\displaystyle V_{\varepsilon^{-1}}(x) \subset W_{\varepsilon}(x) $
and that for all $\displaystyle \varepsilon \in \Gamma_{1}$ and
$\displaystyle u \in U(x)$ we have
$$\delta_{\varepsilon^{-1}}^{x} \ \delta^{x}_{\varepsilon} u \ = \ u \ .$$
\end{enumerate}
We have therefore the following string of inclusions, for any $\varepsilon \in \Gamma$, $\nu(\varepsilon) \leq 1$, and any $x \in X$:
$$ B_{d}(x,\nu(\varepsilon)) \subset \delta^{x}_{\varepsilon} B_{d}(x, A)
\subset V_{\varepsilon}(x) \subset
W_{\varepsilon^{-1}}(x) \subset \delta_{\varepsilon}^{x} B_{d}(x, B) \quad . $$
A further technical condition on the sets $\displaystyle V_{\varepsilon}(x)$ and $\displaystyle W_{\varepsilon}(x)$ will be given just before the axiom A4. (This condition will be counted as part of
axiom A0.)
\begin{enumerate}
\item[{\bf A1.}] We have
$\displaystyle \delta^{x}_{\varepsilon} x = x $ for any point $x$. We also have $\displaystyle \delta^{x}_{1} = id$ for any $x \in X$.
Let us define the topological space
$$ dom \, \delta = \left\{ (\varepsilon, x, y) \in \Gamma \times X \times X
\mbox{ : } \quad \mbox{ if } \nu(\varepsilon) \leq 1 \mbox{ then } y \in U(x) \,
\, ,
\right.$$
$$\left. \mbox{ else } y \in W_{\varepsilon}(x) \right\} $$
with the topology inherited from the product topology on
$\Gamma \times X \times X$. Consider also $\displaystyle Cl(dom \, \delta)$,
the closure of $dom \, \delta$ in $\displaystyle \bar{\Gamma} \times X \times X$ with product topology.
The function $\displaystyle \delta : dom \, \delta \rightarrow X$ defined by
$\displaystyle \delta (\varepsilon, x, y) = \delta^{x}_{\varepsilon} y$ is continuous. Moreover, it can be continuously extended to $\displaystyle Cl(dom \, \delta)$ and we have
$$\lim_{\varepsilon\rightarrow 0} \delta_{\varepsilon}^{x} y \, = \, x \quad . $$
\item[{\bf A2.}] For any $x, \in K$, $\displaystyle \varepsilon, \mu \in \Gamma_{1}$ and $\displaystyle u \in
\bar{B}_{d}(x,A)$ we have:
$$ \delta_{\varepsilon}^{x} \delta_{\mu}^{x} u = \delta_{\varepsilon \mu}^{x} u \ .$$
\item[{\bf A3.}] For any $x$ there is a function $\displaystyle (u,v) \mapsto d^{x}(u,v)$, defined for any $u,v$ in the closed ball (in distance d) $\displaystyle
\bar{B}(x,A)$, such that
$$\lim_{\varepsilon \rightarrow 0} \quad \sup \left\{ \mid \frac{1}{\varepsilon} d(\delta^{x}_{\varepsilon} u, \delta^{x}_{\varepsilon} v) \ - \ d^{x}(u,v) \mid \mbox{ : } u,v \in \bar{B}_{d}(x,A)\right\} \ = \ 0$$
uniformly with respect to $x$ in compact set.
\end{enumerate}
\begin{remark}
The "distance" $d^{x}$ can be degenerated: there might exist
$\displaystyle v,w \in U(x)$ such that $\displaystyle d^{x}(v,w) = 0$.
\label{imprk}
\end{remark}
For the following axiom to make sense we impose a technical condition on the co-domains $\displaystyle V_{\varepsilon}(x)$: for any compact set $K \subset X$ there are $R=R(K) > 0$ and
$\displaystyle \varepsilon_{0}= \varepsilon(K) \in (0,1)$ such that
for all $\displaystyle u,v \in \bar{B}_{d}(x,R)$ and all $\displaystyle \varepsilon \in \Gamma$, $\displaystyle \nu(\varepsilon) \in (0,\varepsilon_{0})$, we have
$$\delta_{\varepsilon}^{x} v \in W_{\varepsilon^{-1}}( \delta^{x}_{\varepsilon}u) \ .$$
With this assumption the following notation makes sense:
$$\Delta^{x}_{\varepsilon}(u,v) = \delta_{\varepsilon^{-1}}^{\delta^{x}_{\varepsilon} u} \delta^{x}_{\varepsilon} v . $$
The next axiom can now be stated:
\begin{enumerate}
\item[{\bf A4.}] We have the limit
$$\lim_{\varepsilon \rightarrow 0} \Delta^{x}_{\varepsilon}(u,v) = \Delta^{x}(u, v) $$
uniformly with respect to $x, u, v$ in compact set.
\end{enumerate}
\begin{definition}
A triple $(X,d,\delta)$ which satisfies A0, A1, A2, A3, but $\displaystyle d^{x}$ is degenerate for some
$x\in X$, is called degenerate dilatation structure.
If the triple $(X,d,\delta)$ satisfies A0, A1, A2, A3 and
$\displaystyle d^{x}$ is non-degenerate for any $x\in X$, then we call it a
dilatation structure.
If a dilatation structure satisfies A4 then we call it strong dilatation
structure.
\label{defweakstrong}
\end{definition}
\section{Dilatation structures on the boundary of the dyadic tree}
Dilatation structures on the boundary of the dyadic tree will have a simpler
form than general, mainly because the distance is ultrametric.
We shall take the group $\Gamma$ to be the set of integer powers of $2$, seen as a subset of
dyadic numbers. Thus for any $p \in \mathbb{Z}$ the element $\displaystyle 2^{p} \in \mathbb{Q}_{2}$
belongs to $\Gamma$. The operation is the multiplication of dyadic numbers and the morphism
$\nu : \Gamma \rightarrow (0,+\infty)$ is defined by
$$\nu(2^{p}) = d(0, 2^{p}) = \frac{1}{2^{p}} \in (0,+\infty) \quad . $$
\paragraph{Axiom A0.} This axiom states that for any $p\in \mathbb{N}$ and any $x \in X^{\omega}$ the dilatation
$$\delta^{x}_{2^{p}} : U(x) \rightarrow V_{2^{p}}(x) $$
is a homeomorphism, the sets $U(x)$ and $\displaystyle V_{2^{p}}(x) $ are open and
there is $A>1$ such that the ball centered in $x$ and radius $A$ is contained in $U(x)$. But this means
that $\displaystyle U(x) = X^{\omega}$, because $\displaystyle X^{\omega} = B(x,1)$.
Further, for any $p \in \mathbb{N}$ we have the inclusions:
\begin{equation}
B(x, \frac{1}{2^{p}}) \subset \delta^{x}_{2^{p}} X^{\omega} \subset V_{2^{p}}(x) \quad .
\label{a01}
\end{equation}
For any $\displaystyle p \in \mathbb{N}^{*}$ the associated dilatation
$\displaystyle \delta^{x}_{2^{-p}} : W_{2^{-p}}(x) \rightarrow B(x,B) =
X^{\omega}$ ,
is injective, invertible on the image. We suppose that
$\displaystyle W_{2^{-p}}(x)$ is open, that
\begin{equation}
V_{2^{p}}(x) \subset W_{2^{-p}}(x)
\label{a02}
\end{equation}
and that for all $\displaystyle p \in \mathbb{N}^{*}$ and $\displaystyle u \in X^{\omega}$ we have
$\displaystyle \delta_{2^{-p}}^{x} \ \delta^{x}_{2^{p}} u \ = \ u$ .
We leave aside for the moment the interpretation of the technical condition before axiom A4.
\paragraph{Axioms A1 and A2.} Nothing simplifies.
\paragraph{Axiom A3.} Because $d$ is an ultrametric distance and $\displaystyle X^{\omega}$ is compact, this axiom has very strong consequences, for a non degenerate dilatation structure.
In this case the axiom A3 states that there is a non degenerate distance function
$\displaystyle d^{x}$ on
$\displaystyle X^{\omega}$ such that we have the limit
\begin{equation}
\lim_{p \rightarrow \infty} 2^{p} d( \delta^{x}_{2^{p}} u, \delta^{x}_{2^{p}} v) = d^{x}(u,v)
\label{a30}
\end{equation}
uniformly with respect to $\displaystyle x,u,v \in X^{\omega}$.
We continue further with first properties of dilatation structures.
\begin{lemma}
There exists $\displaystyle p_{0} \in \mathbb{N}$ such that for any $\displaystyle x, u,
v \in X^{\omega}$ and for any $\displaystyle p \in \mathbb{N}$, $\displaystyle p \geq p_{0}$, we have
$$ 2^{p} d( \delta^{x}_{2^{p}} u, \delta^{x}_{2^{p}} v) = d^{x}(u,v) \quad . $$
\label{l1}
\end{lemma}
\begin{proof}
From the limit (\ref{a30}) and the non degeneracy of the distances $\displaystyle d^{x}$ we deduce that
$$\lim_{p \rightarrow \infty} \log_{2} \left( 2^{p} d( \delta^{x}_{2^{P}} u, \delta^{x}_{2^{P}} v)\right) =
\log_{2} d^{x}(u,v) \quad , $$
uniformly with respect to $\displaystyle x,u,v \in X^{\omega}$, $u \not = v$. The right hand side term is finite and the sequence from the limit at the left hand side is included in $\mathbb{Z}$. Use this and
the uniformity of the convergence to get the desired result.
\end{proof}
In the sequel $\displaystyle p_{0}$ is the smallest natural number satisfying lemma \ref{l1}.
\begin{lemma}
For any $\displaystyle x \in X^{\omega}$ and for any
$\displaystyle p \in \mathbb{N}$, $\displaystyle p \geq p_{0}$, we have
$\displaystyle \delta^{x}_{2^{p}} X^{\omega} = [x]_{p} X^{\omega}$.
Otherwise stated, for any $\displaystyle x, y \in X^{\omega}$, any $\displaystyle q \in X^{*}$,
$\displaystyle \mid q \mid \geq p_{0}$ there exists $\displaystyle w \in X^{\omega}$ such that
$$ \delta^{qx}_{2^{\mid q \mid}} w = qy $$
and for any $\displaystyle z \in X^{\omega}$ there is
$\displaystyle y \in X^{\omega}$ such that
$\displaystyle \delta^{qx}_{2^{\mid q \mid}} z = qy$ .
Moreover, for any $\displaystyle x \in X^{\omega}$ and for any $\displaystyle p \in \mathbb{N}$, $\displaystyle p \geq p_{0}$ the inclusions from (\ref{a01}), (\ref{a02}) are equalities.
\label{l2}
\end{lemma}
\begin{proof}
From the last inclusion in (\ref{a01}) we get that for any
$\displaystyle x, y \in X^{\omega}$, any $\displaystyle q \in X^{*}$,
$\displaystyle \mid q \mid \geq p_{0}$ there exists $\displaystyle w \in X^{\omega}$ such that
$\displaystyle \delta^{qx}_{2^{\mid q \mid}} w = qy$ .
For the second part of the conclusion we use lemma \ref{l1} and axiom A1. From there we see that
for any $\displaystyle p \geq p_{0}$ we have
$$2^{p} d( \delta^{x}_{2^{p}} x, \delta^{x}_{2^{p}} u) = 2^{p} d( x, \delta^{x}_{2^{p}} u) = d^{x}(x,u)
\leq 1\quad . $$
Therefore $\displaystyle 2^{p} d( x, \delta^{x}_{2^{p}} u) \leq 1$, which is equivalent with the second part of the lemma.
Finally, the last part of the lemma has a similar proof, only that we have to use also the last part of axiom A0.
\end{proof}
The technical condition before the axiom A4 turns out to be trivial. Indeed, from lemma \ref{l2} it follows that for any $\displaystyle p \geq p_{0}$, $p \in \mathbb{N}$, and any $\displaystyle x, u, v \in
X^{\omega}$ we have $\displaystyle \delta^{x}_{2^{p}} u = [x]_{p} w$, $\displaystyle w \in X^{\omega}$. It follows that
$$\delta^{x}_{2^{p}} v \in [x]_{p} X^{\omega} = W_{2^{-p}} (x) = W_{2^{-p}} ( \delta^{x}_{2^{p}} u) \quad . $$
\begin{lemma}
For any $\displaystyle x, u,
v \in X^{\omega}$ such that $\displaystyle 2^{p_{0}} d(x,u) \leq 1$,
$\displaystyle 2^{p_{0}} d(x,v) \leq 1$ we have
$\displaystyle d^{x}(u,v) = d(u,v)$ .
Moreover, under the same hypothesis, for any $\displaystyle p \in \mathbb{N}$ we have
$$ 2^{p} d( \delta^{x}_{2^{p}} u, \delta^{x}_{2^{p}} v) = d(u,v) \quad . $$
\label{l3}
\end{lemma}
\begin{proof}
By lemma \ref{l1}, lemma \ref{l2} and axiom A2. Indeed, from lemma \ref{l1} and axiom A2, for any $\displaystyle p \in \mathbb{N}$ and any
$\displaystyle x, u', v' \in X^{\omega}$ we have
$$d^{x}(u',v') = 2^{p_{0}+p} d( \delta^{x}_{2^{p+p_{0}}} u' , \delta^{x}_{2^{p+p_{0}}} v' ) = $$
$$ = 2^{p} \, 2^{p_{0}} d( \delta^{x}_{2^{p_{0}}} \delta^{x}_{2^{p}}u', \delta^{x}_{2^{p_{0}}} \delta^{x}_{2^{p}}v') = 2^{p} d^{x}( \delta^{x}_{2^{p}}u' , \delta^{x}_{2^{p}}v') \quad . $$
This is just the cone property for $\displaystyle d^{x}$. From here we deduce
that for any $\displaystyle p \in \mathbb{Z}$ we have
$\displaystyle d^{x}(u', v') = 2^{p} d^{x}(\delta^{x}_{2^{p}}u' ,
\delta^{x}_{2^{p}}v')$ .
If $\displaystyle 2^{p_{0}} d(x,u) \leq 1$, $\displaystyle 2^{p_{0}} d(x,v)
\leq 1$ then write $x = qx'$, $\displaystyle \mid q \mid = p_{0}$, and use
lemma \ref{l2} to get the existence of $\displaystyle u', v' \in X^{\omega}$
such that $\displaystyle \delta^{x}_{2^{p_{0}}} u' = u$ , $\displaystyle
\delta^{x}_{2^{p_{0}}} v' = v$ .
Therefore, by lemma \ref{l1}, we have
$$ d(u,v) = 2^{-p_{0}} d^{x}( u', v') = d^{x}( \delta^{x}_{2^{-p_{0}}} u' , \delta^{x}_{2^{-p_{0}}} v' ) =
d^{x}(u,v) \quad . $$
The first part of the lemma is proven. For the proof of the second part write again
$$2^{p} d( \delta^{x}_{2^{p}} u, \delta^{x}_{2^{p}} v) = 2^{p} d^{x}( \delta^{x}_{2^{p}} u, \delta^{x}_{2^{p}} v) = d^{x}(u,v) = d(u,v)
$$
which finishes the proof.
\end{proof}
The space $\displaystyle X^{\omega}$ decomposes into a disjoint union of
$\displaystyle 2^{p_{0}}$ balls which are isometric. There is no connection
between the dilatation structures on these balls, therefore we shall
suppose further that $\displaystyle p_{0} = 0$.
Our purpose is to find the general form of a
dilatation structure on $\displaystyle X^{\omega}$, with
$\displaystyle p_{0} = 0$.
\begin{definition}
A function $\displaystyle W: \mathbb{N}^{*} \times X^{\omega} \rightarrow
Isom(X^{\omega})$ is smooth if for any $\varepsilon > 0$ there exists $\mu(\varepsilon) > 0$ such that
for any $\displaystyle x, x' \in X^{\omega}$ such that $d(x,x')< \mu(\varepsilon)$ and for any $\displaystyle y \in X^{\omega}$ we have
$$ \frac{1}{2^{k}} \, d( W^{x}_{k} (y) , W^{x'}_{k} (y) ) \leq \varepsilon \quad , $$
for an $k$ such that $\displaystyle d(x,x') < 1 / 2^{k} $.
\label{defwsmooth}
\end{definition}
\begin{theorem}
Let $\displaystyle (X^{\omega}, d, \delta)$ be a dilatation structure on $\displaystyle (X^{\omega}, d)$, where $d$ is the standard distance on $\displaystyle X^{\omega}$, such that $\displaystyle p_{0} = 0$. Then there exists a smooth (according to definition \ref{defwsmooth}) function
$$W: \mathbb{N}^{*} \times X^{\omega} \rightarrow Isom(X^{\omega}) \quad , \quad W(n,x) = W^{x}_{n} $$
such that for any
$\displaystyle q \in X^{*}$, $\alpha \in X$, $x, y \in X^{\omega}$ we have
\begin{equation}
\delta_{2}^{q \alpha x} q \bar{\alpha} y = q \alpha \bar{x_{1}} W^{q \alpha x}_{\mid q \mid + 1}
(y) \quad .
\label{eqtstruc}
\end{equation}
Conversely, to any smooth function $\displaystyle W: \mathbb{N}^{*} \times X^{\omega} \rightarrow
Isom(X^{\omega})$ is associated a dilatation structure $\displaystyle (X^{\omega}, d, \delta)$,
with $\displaystyle p_{0} = 0$, induced by functions $\displaystyle \delta_{2}^{x}$, defined by
$\displaystyle \delta_{2}^{x} x = x$ and otherwise by relation (\ref{eqtstruc}).
\label{tstruc}
\end{theorem}
\begin{proof}
Let $\displaystyle (X^{\omega}, d, \delta)$ be a dilatation structure on
$\displaystyle (X^{\omega}, d)$, such that $\displaystyle p_{0} = 0$.
Any two different elements of $\displaystyle X^{\omega}$ can be written in the
form $q \alpha x$ and $q \bar{\alpha} y$, with $\displaystyle q \in X^{*}$,
$\alpha \in X$, $x, y \in X^{\omega}$. We also have
$\displaystyle d(q \alpha x , q \bar{\alpha} y) = 2^{- \mid q \mid}$ .
From the following computation (using $\displaystyle p_{0} = 0$ and axiom A1):
$$2^{-\mid q \mid -1} = \frac{1}{2} d(q \alpha x , q \bar{\alpha} y) = d( q
\alpha x , \delta_{2}^{q \alpha x} q \bar{\alpha} y) \quad , $$
we find that there exists $\displaystyle w^{q \alpha x}_{\mid q \mid + 1}(y)
\in X^{\omega}$ such that
$\displaystyle \delta_{2}^{q \alpha x} q \bar{\alpha} y = q \alpha
w^{q \alpha x}_{\mid q \mid + 1}(y)$ .
Further on, we compute:
$$ \frac{1}{2} d(q \bar{\alpha} x , q \bar{\alpha} y) = d( \delta_{2}^{q \alpha x} q \bar{\alpha} x , \delta_{2}^{q \alpha x} q \bar{\alpha} y) = d( q \alpha w^{q \alpha x}_{\mid q \mid + 1}(x) , q \alpha w^{q \alpha x}_{\mid q \mid + 1}(y)) \quad . $$
From this equality we find that
$\displaystyle 1 > \frac{1}{2} d(x,y) = d( w^{q \alpha x}_{\mid q \mid + 1}(x)
, w^{q \alpha x}_{\mid q \mid + 1}(y))$ ,
which means that the first letter of the word $\displaystyle w^{q \alpha x}_{\mid q \mid + 1}(y)$
does not depend on $y$, and is equal to the first letter of the word $\displaystyle w^{q \alpha x}_{\mid q \mid + 1}(x)$. Let us denote this letter by $\beta$ (which depends only on $q$, $\alpha$, $x$). Therefore we may write:
$$w^{q \alpha x}_{\mid q \mid + 1}(y) = \beta W^{q \alpha x}_{\mid q \mid + 1}(y) \quad , $$
where the properties of the function $\displaystyle y \mapsto W^{q \alpha x}_{\mid q \mid + 1}(y)$ remain to be determined later.
We go back to the first computation in this proof:
$$2^{-\mid q \mid -1} = d( q \alpha x , \delta_{2}^{q \alpha x} q \bar{\alpha} y) = d( q \alpha x , q \alpha
\beta W^{q \alpha x}_{\mid q \mid + 1}(y)) \quad . $$
This shows that $\displaystyle \bar{\beta}$ is the first letter of the word $x$. We proved the relation
(\ref{eqtstruc}), excepting the fact that the function $\displaystyle y \mapsto W^{q \alpha x}_{\mid q \mid + 1}(y)$ is an isometry. But this is true. Indeed, for any $\displaystyle u,v \in X^{\omega}$ we have
$$\frac{1}{2} d(q \bar{\alpha} u , q \bar{\alpha} v) = d( \delta_{2}^{q \alpha x} q \bar{\alpha} u ,
\delta_{2}^{q \alpha x} q \bar{\alpha} v) = d( q \alpha \bar{x_{1}} W^{q \alpha x}_{\mid q \mid + 1}(x) , q \alpha \bar{x_{1}} W^{q \alpha x}_{\mid q \mid + 1}(y)) \quad . $$
This proves the isometry property.
The dilatations of coefficient $2$ induce all dilatations (by axiom A2).
In order to satisfy the continuity
assumptions from axiom A1, the function
$\displaystyle W: \mathbb{N}^{*} \times X^{\omega} \rightarrow
Isom(X^{\omega})$ has to be smooth in the sense of definition
\ref{defwsmooth}. Indeed, axiom A1 is equivalent to the fact that
$\displaystyle \delta^{x'}_{2}(y')$ converges uniformly to $
\displaystyle \delta^{x}_{2}(y)$, as $d(x,x'), d(y,y')$ go to zero.
There are two cases to study.
Case 1: $d(x,x') \leq d(x,y)$, $d(y,y') \leq d(x,y)$. It means that $x = q \alpha q' \beta X$, $y = q \bar{\alpha} q" \gamma Y$, $x' = q \alpha q' \bar{\beta} X'$, $y' = q \bar{\alpha} q" \bar{\gamma} Y'$,
with $d(x,y) = 1 / 2^{k}$, $k = \mid q \mid$ .
Suppose that $q' \not = \emptyset$. We compute then:
$\displaystyle \delta_{2}^{x}(y) = q \alpha \bar{q'_{1}} W_{k+1}^{x}( q"
\gamma Y)$ , $\displaystyle \delta_{2}^{x'}(y') =
q \alpha \bar{q'_{1}} W_{k+1}^{x'}(q" \bar{\gamma} Y')$.
All the functions denoted by a capitalized "W" are isometries, therefore we get the estimation:
$$d( \delta_{2}^{x}(y) , \delta_{2}^{x'}(y')) = \frac{1}{2^{k+2}} \, d( W_{k+1}^{x}( q" \gamma Y) , W_{k+1}^{x'}(q" \bar{\gamma} Y')) \leq $$
$$ \leq \frac{1}{2^{k+2}} \, d( q" \gamma Y , q" \bar{\gamma} Y') +
\, \frac{1}{2^{k+2}} \, d( W_{k+1}^{x}(q" \gamma Y) , W_{k+1}^{x'}(q" \gamma Y)) = $$
$$ = \frac{1}{2} d(y,y') +
\, \frac{1}{2^{k+2}} \, d( W_{k+1}^{x}(q" \gamma Y) , W_{k+1}^{x'}(q" \gamma Y)) \quad . $$
We see that if $W$ is smooth in the sense of definition \ref{defwsmooth} then
the structure $\delta$ satisfies the uniform continuity assumptions for this
case. Conversely, if $\delta$ satisfies A1 then
$W$ has to be smooth.
If $q' = \emptyset$ then a similar computation leads to the same conclusion.
Case 2: $d(x,x') > d(x,y) > d(y,y')$. It means that $x = q \alpha q' \beta X$, $x' = q \bar{\alpha} X'$,
$y = q \alpha q' \bar{\beta} q" \bar{\gamma} Y$, $y' = q \alpha q' \bar{\beta} q" \gamma Y'$, with
$d(x,x') = 1 \ 2^{k}$, $k = \mid q \mid$ .
We compute then:
$\displaystyle \delta_{2}^{x}(y) = q \alpha q ' \beta \bar{X_{1}}
W_{k+2+ \mid q' \mid}^{x}( q" \bar{\gamma} Y)$,
$$\delta_{2}^{x'}(y') =
q \bar{\alpha} \bar{X'_{1}} W_{k+1}^{x'}(q' \bar{\beta} q" \gamma Y') \leq
\frac{1}{2^{k}} = d(x, x') \quad . $$
Therefore in his case the continuity is satisfied, without any supplementary constraints on the function $W$.
The first part of the theorem is proven.
For the proof of the second part of the theorem we start from the function
$\displaystyle W: \mathbb{N}^{*} \times X^{\omega} \rightarrow Isom(X^{\omega})$. It is sufficient to prove for any $\displaystyle
x, y, z \in X^{\omega}$ the equality
$$\frac{1}{2} d(y,z) = d(\delta_{2}^{x} y, \delta_{2}^{x} z) \quad .$$
Indeed, then we can construct the all dilatations from the dilatations of
coefficient $2$ (thus we satisfy A2). All axioms, excepting A1, are satisfied.
But A1 is equivalent with the smoothness of the function
$W$, as we proved earlier.
Let us prove now the before mentioned equality.
If $y = z$ there is nothing to prove. Suppose that $y \not = z$. The distance $d$ is ultrametric, therefore the proof splits in two cases.
Case 1: $d(x,y) = d(x,z) > d(y,z)$. This is equivalent to $x = q \bar{\alpha} x'$, $y = q \alpha q' \beta y'$,
$z = q \alpha q' \bar{\beta} z'$, with $\displaystyle q, q' \in X^{*}$, $\alpha, \beta \in X$,
$\displaystyle x', y', z' \in X^{\omega}$. We compute:
$$d( \delta_{2}^{x} y, \delta_{2}^{x} z) = d(\delta_{2}^{q \bar{\alpha} x'} q \alpha q' \beta y', \delta_{2}^{q \bar{\alpha} x'} q \alpha q' \bar{\beta} z') = $$
$$= d(q \bar{\alpha} \bar{x'_{1}} W^{x}_{\mid q \mid +1} ( q' \beta y') , q \bar{\alpha} \bar{x'_{1}} W^{x}_{\mid q \mid +1} ( q' \bar{\beta} z')) = 2^{-\mid q \mid - 1} d( W^{x}_{\mid q \mid +1} ( q' \beta y'),
W^{x}_{\mid q \mid +1} ( q' \bar{\beta} z')) = $$
$$ = 2^{-\mid q \mid - 1} d(q' \beta y', q' \bar{\beta} z')) = \frac{1}{2} d(q \alpha q' \beta y' , q \alpha q' \bar{\beta} z') = \frac{1}{2} d(y,z) \quad . $$
Case 2: $d(x,y) = d(y,z) > d(x,z)$. If $x = z$ then we write $x = q \alpha u$, $y = q \bar{\alpha} v$ and we have
$$ d(\delta_{2}^{x} y, \delta_{2}^{x} z) = d( q \alpha \bar{u_{1}} W^{x}_{\mid q \mid +1} (v), q \alpha u)
= 2^{-\mid q \mid + 1} = \frac{1}{2} d(y,z) \quad . $$
If $x \not = z$ then we can write $z = q \bar{\alpha} z'$, $y = q \alpha q' \beta y'$,
$x = q \alpha q' \bar{\beta} x'$, with $\displaystyle q, q' \in X^{*}$, $\alpha, \beta \in X$,
$\displaystyle x', y', z' \in X^{\omega}$. We compute:
$$d( \delta_{2}^{x} y, \delta_{2}^{x} z) = d(\delta_{2}^{q \alpha q' \bar{\beta} x'} q \alpha q' \beta y' ,
\delta_{2}^{q \alpha q' \bar{\beta} x'} q \bar{\alpha} z') = $$
$$ = d(q \alpha q' \bar{\beta} \bar{x'_{1}} W^{x}_{\mid q \mid + \mid q' \mid +2}(y'), q \alpha \gamma
W^{x}_{\mid q \mid + 1} (z')) \quad , $$
with $\gamma \in X$, $\displaystyle \bar{\gamma} = q'_{1}$ if $q' \not = \emptyset$, otherwise
$\gamma = \beta$. In both situations we have
$\displaystyle d( \delta_{2}^{x} y, \delta_{2}^{x} z) =
2^{- \mid q \mid - 1} = \frac{1}{2} d(y,z)$ .
The proof is done.
\end{proof}
\subsection{Self-similar dilatation structures}
Let $\displaystyle (X^{\omega}, d, \delta)$ be a dilatation structure. There are induced
dilatations structures on $\displaystyle 0X^{\omega}$ and $\displaystyle 1X^{\omega}$.
\begin{definition}
For any $\alpha \in X$ and $\displaystyle x, y \in X^{\omega}$ we define $\displaystyle \delta_{2}^{\alpha, x} y$ by the relation
$$\delta_{2}^{\alpha x} \alpha y = \alpha \, \delta_{2}^{\alpha, x} y \quad . $$
\end{definition}
The following proposition has a straightforward proof, therefore we skip it.
\begin{proposition}
If $\displaystyle (X^{\omega}, d, \delta)$ is a dilatation structure and $\alpha \in X$ then
$\displaystyle (X^{\omega}, d, \delta^{\alpha})$ is a dilatation structure.
If $\displaystyle (X^{\omega}, d, \delta')$ and $\displaystyle (X^{\omega}, d, \delta")$ are dilatation structures then $\displaystyle (X^{\omega}, d, \delta)$ is a dilatation structure, where
$\delta$ is uniquely defined by $\displaystyle \delta^{0} = \delta'$, $\displaystyle \delta^{1} = \delta"$.
\end{proposition}
\begin{definition}
A dilatation structure $\displaystyle (X^{\omega}, d, \delta)$ is self-similar if for any $\alpha \in X$ and $\displaystyle x,y \in X^{\omega}$ we have
$$\delta_{2}^{\alpha x} \, \alpha y = \alpha \, \delta_{2}^{x} y \quad . $$
\label{defself}
\end{definition}
Self-similarity is thus related to linearity. Indeed, let us compare
self-similarity with the following definition of linearity.
\begin{definition}
For a given dilatation structure $\displaystyle (X^{\omega}, d, \delta)$, a
continuous transformation $\displaystyle A: X^{\omega} \rightarrow X^{\omega}$ is linear (with respect
to the dilatation structure) if for any $\displaystyle x , y \in X^{\omega}$ we
have
$$A \, \delta^{x}_{2} y \, = \, \delta^{A x}_{2} A y$$
\label{defline}
\end{definition}
The previous definition provides a true generalization of linearity for
dilatation structures. This can be seen by comparison with the characterisation
of linear (in fact affine) transformations in vector spaces from the
proposition \ref{1proplin}.
The definition of self-similarity \ref{defself} is related to linearity in
the sense of definition \ref{defline}. To see this, let us consider the
functions $\displaystyle \hat{\alpha} : X^{\omega} \rightarrow X^{\omega}$,
$\displaystyle \hat{\alpha} x \, = \, \alpha x$, for $\alpha \in X$. With this
notations, the definition \ref{defself} simply states that a dilatation
structure is self-similar if these two functions, $\displaystyle \hat{0}$ and
$\displaystyle \hat{1}$, are linear in the sense of definition \ref{defline}.
The description of self-similar dilatation structures on the boundary of the
dyadic tree is given in the next theorem.
\begin{theorem}
Let $\displaystyle (X^{\omega}, d, \delta)$ be a self-similar dilatation structure and
$\displaystyle W: \mathbb{N}^{*} \times X^{\omega} \rightarrow Isom(X^{\omega})$ the function
associated to it, according to theorem \ref{tstruc}. Then there exists a
function $W: X^{\omega} \rightarrow Isom(X^{\omega})$ such that:
\begin{enumerate}
\item[(a)] for any $\displaystyle q \in X^{*}$ and any $\displaystyle x \in X^{\omega}$ we have
$\displaystyle W_{\mid q \mid + 1}^{q x} = W^{x}$ ,
\item[(b)] there exists $C>0$ such that for any $\displaystyle x, x', y \in X^{\omega}$ and for any
$\lambda > 0$, if $d(x,x') \leq \lambda$ then
$\displaystyle d(W^{x}(y) , W^{x'}(y)) \leq C \lambda$ .
\end{enumerate}
\label{th2}
\end{theorem}
\begin{proof}
We define $\displaystyle W^{x} = W^{x}_{1}$ for any $\displaystyle x \in X^{\omega}$ .
We want to prove that this function satisfies (a), (b).
(a) Let $\displaystyle \beta \in X$ and any $\displaystyle x, y \in X^{\omega}$,
$x = q \alpha u$, $y = q \bar{\alpha} v$. By self-similarity we obtain:
$\displaystyle \beta q \alpha \bar{u_{1}} W_{\mid q \mid + 2}^{\beta x} (v) =
\delta_{2}^{\beta x} \beta y = \beta \delta_{2}^{x} y =
\beta q \alpha \bar{u_{1}} W_{\mid q \mid + 1}^{x} (v)$ .
We proved that
$$ W_{\mid q \mid + 2}^{\beta x} (v) = W_{\mid q \mid + 1}^{x} (v) $$
for any $\displaystyle x, v \in X^{\omega}$ and $\beta \in X$ This implies (a).
(b) This is a consequence of smoothness, in the sense of definition \ref{defwsmooth}, of the function
$\displaystyle W: \mathbb{N}^{*} \times X^{\omega} \rightarrow Isom(X^{\omega})$. Indeed,
$\displaystyle (X^{\omega}, d, \delta)$ is a dilatation structure, therefore by theorem \ref{tstruc}
the previous mentioned function is smooth.
By (a) the smoothness condition becomes: for any $\varepsilon > 0$ there is $\mu(\varepsilon) > 0$ such that for any $\displaystyle y \in X^{\omega}$, any $k \in \mathbb{N}$ and any $x, x' \in X^{\omega}$,
if $d(x,x') \leq 2^{k} \mu(\varepsilon)$ then
$$d( W^{x}(y), W^{x'} (y)) \leq 2^{k} \varepsilon \quad . $$
Define then the modulus of continuity: for any $\varepsilon > 0$ let $\bar{\mu}(\varepsilon)$ be given by
$$\bar{\mu}(\varepsilon) = \sup \left\{ \mu \, \mbox{ : } \forall x, x', y \in X^{\omega} \, \, d(x,x')\leq \mu \Longrightarrow d( W^{x}(y), W^{x'} (y)) \leq \varepsilon \right\} \quad .$$
We see that the modulus of continuity $\bar{\mu}$ has the property
$$\bar{\mu}(2^{k} \varepsilon) = 2^{k} \bar{\mu}(\varepsilon) $$
for any $k \in \mathbb{N}$. Therefore there exists $C> 0$ such that
$\displaystyle \bar{\mu}( \varepsilon) = C^{-1} \varepsilon$ for any $\displaystyle \varepsilon = 1/2^{p}$, $p \in \mathbb{N}$. The point (b) follows immediately.
\end{proof} | {"config": "arxiv", "file": "0709.2224.tex"} |
TITLE: The probability that $3$ organizers win
QUESTION [1 upvotes]: A local fraternity is conducting a raffle where $55$ tickets are to be sold - one per customer. There are $3$ prizes to be awarded. If the $4$ organizers of the raffle each buy one ticket, what are the follow probabilities?
(a) The probability that the $4$ organizers win all of the prizes?
For part (a), I was told the answer was $\dfrac{ \binom{3}{3} \binom{52}{1}}{\binom{55}{4}}$
But why wouldn't it be $\dfrac{ \binom{4}{3} \binom{51}{1}}{\binom{55}{4}}$ considering there are $4$ organizers?
(b) The probability that the four organizers win exactly two of the prizes?
For part (b) I need to understand how to do part (a) correctly. So I'm guessing it would either be $$\dfrac{ \binom{3}{2} \binom{52}{2}}{\binom{55}{4}} \text{ or } \dfrac{ \binom{4}{2} \binom{51}{2}}{\binom{55}{4}}$$
REPLY [2 votes]: The denominator in (a) is total number of unordered 4-plets - all combinations of tickets organizers can get if we don't care which organizer got what. So numerator should also be number of unordered 4-plets consisting of 3 winning tickets and 1 losing. And number of such 4-plets is equal to number of losing tickets. | {"set_name": "stack_exchange", "score": 1, "question_id": 3187901} |
TITLE: What is a Lie Group in layman's terms?
QUESTION [28 upvotes]: I'm having trouble getting my head arround the concept.
Can someone explain it to me?
REPLY [49 votes]: I think that understanding comes through examples. The most fundamental example I believe to be the rotation group. Consider the sphere $S^2\subset \mathbb{R}^3$. The sphere has rotational symmetries. If we rotate the sphere by any angle, the sphere doesn't change.
The collection of all rotations forms a Lie group. The group property basically means that
if we rotate the sphere over any angle $\alpha$, after this over an angle $\beta$, it is the same if we would have rotated it in one go (over some different angle). Also any rotation has an inverse (rotating it over the opposite angle). This makes the rotations a group. The "Lie" in Lie group means that these rotations can be done arbitrary small. Many small rotations makes for a big rotation.
Lie groups capture the concept of "continuous symmetries".
REPLY [16 votes]: Consider the set of $(n\times n)$ matrices that have non-zero determinant. Such a matrix corresponds to a system of linear equations ($n$ equations in $n$ unknowns) that has a unique solution. You can think of the solution as the unique point of intersection between the graph of a function and a horizontal hyperplane. Here it is helpful to think of $n=1$. In other words, the coefficients of the system correspond to a transformation of space: the variables $x_1, \ldots x_n$ are transformed to $\sum a_{ij} x_i$. The set of such transformations form a group: the matrices can be multiplied, each has an inverse, the multiplication is associative, and the identity transformation fixes each point of space.
Intuitively, it is easy to see which transformations are close to one another. They are close if they move points that are nearby to points that are nearby. Arithmetically, if the entries in the matrix are close, then the transformations are close: thus $0.14x + .33y$, is a reasonable approximation to $x/7+y/3$.
Thus the set of invertible $(n\times n)$ matrices is a space of invertible $(n\times n)$ matrices. What is not easy for a layman to see is that its spacial characteristics are defined via the determinant since as a set, the $(n\times n)$ matrices are a subset of $n^2$-space. The non-singular matrices are the pull-back of a regular value of the determinant function. [There is a small lie here: this is true for for matrices of determinant 1, but all non-zero determinant matrices deform onto that smaller space].
One important spacial characteristic is that these matrices form a smooth manifold. This is something that is analogous to the surface of a sphere (which is NOT a lie group), the surface of a torus (which is) or the $3$-dimensional sphere that consists of the set of $(x,y,z,w)$ such that $x^2+y^2+z^2+w^2=1$ (which also happens to be a Lie group).
From these examples, we abstract the idea of a Lie group which is a group (that can be thought of as a set of transformations or symmetries) that has the structure of a smooth manifold --- at small scales it is indistinguishable from ordinary Euclidean space. The multiplication and inversion maps are a differentiable functions. And these multiplications occur between pairs of symmetries --- they should not be confused with the action of the matrices on the vector space which is where I started the discussion.
Examples include the real line, the non-zero real numbers, the circle, the torus, the $3$-sphere, the set of rotations of 3-dimensional space, and the special unitary groups representations of which determine particles in physics.
There are some small problems with the definition that I gave. A smooth manifold is a topological space which is paracompact and Hausdorff (neither definition will play a role in the layman's understanding), and that is covered by coordinate charts with specific properties. I imagine that Wikipedia has the relevant definitions articulated carefully. | {"set_name": "stack_exchange", "score": 28, "question_id": 22967} |
TITLE: When should a supervisor be a co-author?
QUESTION [162 upvotes]: What are people's views on this? To be specific: suppose a PhD student has produced a piece of original mathematical research. Suppose that student's supervisor suggested the problem, and gave a few helpful comments, but otherwise did not contribute to the work. Should that supervisor still be named as a co-author, or would an acknowledgment suffice?
I am interested in two aspects of this. Firstly the moral/etiquette aspect: do you consider it bad form for a student not to name their supervisor? Or does it depend on that supervisor's input? And secondly, the practical, career-advancing aspect: which is better, for a student to have a well-known name on his or her paper (and hence more chance of it being noticed/published), or to have a sole-authored piece of work under their belt to hopefully increase their chances of being offered a good post-doc position?
[To clarify: original question asked by MrB ]
REPLY [0 votes]: I am just a self-taught amateur matematician that has published most of his works alone, but my personal point of view is that a supervisor of a PhD dissertation is worth to be added as a coauthor of a paper related to the aforementioned work, if and only if he has given a substantial contribute to the ideas/concept/proofs of that original research.
Now, some PhD candidates are unable to write a preprint with a proper form (a fact), but this is not sufficient (IMHO) to let a supervisor take the coauthorship of their original work if his contribute has merely been helping the candidate to arrange its own results in a more strict form, in order to let it pass easier the peer-review process.
Just my two cents. | {"set_name": "stack_exchange", "score": 162, "question_id": 57337} |
TITLE: Systems of Diophantine Equations
QUESTION [1 upvotes]: Find all ordered 4-tuples of integers $(a,b,c,d)$ that satisfy:
$$a^n+b^n=c^n+d^n$$
for ALL positive integers $n$.
Trivial solutions are $(k,p,k,p)$ and $(k,p,p,k)$ for any integers $k$ and $p$. But does there exist any non-trivial solutions?
REPLY [1 votes]: Hint:-
$a+b=c+d \implies (a-c)=(d-b)\tag{1}$
$\begin{align}a^2+b^2=c^2+d^2 & \implies \left(a^2-c^2\right) \left(d^2-b^2\right)\\&\implies (a+c)(a-c)=(d-b)(d+b)\\&\implies (a+c)=(b+d)\\&\implies(a-b)=(d-c)\tag{2}\end{align}$ | {"set_name": "stack_exchange", "score": 1, "question_id": 1061975} |
TITLE: What causes the evasion of the Goldstone theorem here?
QUESTION [2 upvotes]: For simplicity, I'll consider perhaps the simplest possible example of a gauge theory.
Consider a spontaneously broken ${\rm U(1)}$ gauge theory of a charged scalar field coupled to the electromagnetic field $$\mathscr{L}=(D_\mu\phi)^*(D^\mu\phi)-\mu^2\phi^*\phi-\lambda(\phi^*\phi)^2-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\tag{1}$$ with $\lambda>0$ and $\mu^2<0$. When the field $\phi$ with the polar parametrization $$\phi(x)=\frac{1}{\sqrt{2}}\big(v+h(x)\big)\exp{[i\zeta(x)/v]}\tag{2}$$ plugged into Eq.$(1)$, the field $\zeta$ disappeaears from the theory upon making a suitable gauge transformation. Therefore, there is no Goldstone mode.
Question What causes the Goldstone theorem not to be applicable here? I mean, is there a crucial assumption used in the derivation of Goldstone theorem fails here?
REPLY [0 votes]: This is exactly the Higgs mechanism, which is what happens you have spontaneous symmetry breaking of a gauge symmetry. You are seeing the Goldstone mode get "eaten" by photon. You gauge-transform away the $\xi(x)$ field but you are left with a mass term for the photon, $\frac{1}{2}v^2 A_\mu(x) A^\mu(x)$. | {"set_name": "stack_exchange", "score": 2, "question_id": 540761} |
\begin{document}
\selectlanguage{english}
\maketitle
\begin{abstract}
We study convergent sequences of Baumslag-Solitar groups in the space of marked groups. We prove that
$BS(\m,\n) \to \F_2$ for $|\m|,|\n| \to \infty$ and $BS(1,\n) \to \Z\wr\Z$ for $|\n| \to \infty$. For $\m$ fixed,
$|\m| \geqslant 2$, we show that the sequence $(BS(\m,\n))_{\n}$ is not convergent and characterize many
convergent subsequences. Moreover if $X_\m$ is the set of $BS(\m,\n)$'s for $\n$ relatively prime to $\m$ and $|\n|
\geqslant 2$, then the map $BS(\m,\n) \mapsto \n$ extends continuously on $\overline{X_\m}$ to a surjection onto
invertible $\m$-adic integers.
\end{abstract}
\section{Introduction}\label{Intro}
Let ${\cal G}_2$ be the space of finitely generated marked groups on two generators (see Section \ref{Def} for
definition) and let $\F_2 = \Pres{a,b}{\varnothing}$ be the free group on two generators. Baumslag-Solitar groups
are defined by the presentations
$$
BS(\m,\n)= \Pres{a,b}{ab^{\m}a^{-1} = b^\n}
$$
for $\m,\n \in \Z^* = \Z \setminus \{0\}$. The purpose of the present paper is to understand how Baumslag-Solitar
groups are distributed in ${\cal G}_2$. More precisely, we determine convergent sequences and in some cases we
are able to give the limit group. In the following results, we mark $\F_2$ and $BS(\m,\n)$ by $\{a,b\}$.
\begin{Thm}\label{ToFree}
For $|\m|,|\n| \to \infty$, we have $BS(\m,\n) \to \F_2$.
\end{Thm}
In particular, the property of being Hopfian is not open in ${\cal G}_2$ since $BS(2^k,3^k)$ is known to be
non Hopfian for all $k\geqslant 1$ (see \cite{LynSch}, Chapter IV, Theorem 4.9.) while $\F_2$ is Hopfian.
Theorem \ref{ToFree} is not so surprising because the length of the relator appearing in the presentation of
$BS(\m,\n)$ tends to $\infty$ when $|\m|,|\n| \to \infty$. However, this relator is not the shortest relation in
the group for many values of $(\m,\n)$. To prove Theorem \ref{ToFree}, we give a lower bound for the length of
shortest relations in the more general setting of HNN-extensions (see Section \ref{LowBd}).
We now fix the parameter $\m$. In the case $\m = \pm 1$, we show that the sequence $(BS(\pm 1,\n))_{\n\in\Z}$
is convergent and we can identify the limit.
\begin{Thm}\label{ToWreath}
Let the wreath product $\Z \wr \Z = \Z \ltimes \oplus_{i\in\Z} \Z$ be marked by the elements $(1,0)$ and
$(0,e_0)$ where $e_0 \in \oplus_{i\in\Z} \Z$ is the Dirac mass at $0$. Then $BS(\pm 1,\n) \to \Z \wr \Z$ when
$|\n| \to \infty$.
\end{Thm}
This illustrates the fact that a limit of metabelian groups is metabelian, see \cite{ChaGui}, Section 2.6.
In the case $|\m| \geqslant 2$, we show that the sequence $(BS(\m,\n))_{\n\in\Z}$ is not convergent in ${\cal
G}_2$. As ${\cal G}_2$ is compact, it has convergent subsequences. The next result we state in this
introduction, among subsequences, characterizes many convergent ones. However we don't actually know what the
limits are. Notice that the result also holds for $\m= \pm 1$, even if it is in this case weaker than Theorem
\ref{ToWreath}.
\begin{Thm}\label{ConvSub}
Let $\m\in\Z^*$ and let $(\k_n)_n$ be a sequence of integers relatively prime to $\m$. The sequence $(BS(\m,\k_n))_n$ is
convergent in ${\cal G}_2$ if and only if one (and only one) of the following assertions holds:
\begin{enumerate}
\item[(a)] $(\k_n)_n$ is eventually constant;
\item[(b)] $|\k_n| \to \infty$ and $(\k_n)_n$ is eventually constant
modulo $\m^h$ for all $h\geqslant 1$.
\end{enumerate}
\end{Thm}
Note that for $|\m| \geqslant 2$, condition (b) precisely means that $|\k_n| \to \infty$ and $(\k_n)_n$ is
convergent in $\Z_\m$, the ring of $\m$-adic integers. The link between Baumslag-Solitar groups and $\m$-adic
integers can be made more precise. We define $X_\m$ to be the set of $BS(\m,\n)$'s, for $\n$ relatively prime to
$\m$ and $|\n| \geqslant 2$ and we denote by $\Z_\m^{\times}$ the set of invertible elements of $\Z_\m$.
\begin{Thm}\label{UnifCont}
For $|\m| \geqslant 2$, the map $\Psi : X_\m \to \Z_\m^\times \ ; \ BS(\m,\n) \mapsto \n$ extends
continuously to $\overline{X_\m}$. The extension is surjective, but not injective.
\end{Thm}
An immediate corollary of Theorem \ref{ConvSub} or Theorem \ref{UnifCont} is that (for $|\m|\geqslant 2$) the
sequence $(BS(\m,\n))_\n$ admits uncountably many accumulation points, namely at least one for each invertible
$\m$-adic integer.
We end this introduction by a remark on markings of Baumslag-Solitar groups. In this paper we always mark the
group $BS(\m,\n)$ by the generators coming from its canonical presentation given above. Nevertheless, it is
also an interesting approach to consider different markings on $BS(\m,\n)$. For instance, take $\m$ and $\n$
greater than $2$ and relatively prime, so that $\Gamma = BS(\m,\n)$ is non-Hopfian, the epimorphism
$\phi:\Gamma \to \Gamma$ given by $a \mapsto a$ and $b \mapsto b^\m$ being non-injective (see again
\cite{LynSch}, Chapter IV, Theorem 4.9.). In \cite{ABLRSV}, the authors consider the sequence of groups
$\Gamma_n = \Gamma/\ker(\phi^n)$ (marked by $a$ and $b$). They show that the sequence $(\Gamma_n)_n$
converges to an amenable group while, being all isomorphic to $\Gamma$ as groups, the $\Gamma_n$'s are not
amenable. This allow them to prove that $\Gamma$ is non-amenable, but not uniformly (cf. Proposition 13.3)
and also shows that the property of being amenable is not open in ${\cal G}_2$.
\paragraph{Structure of the article.} In Section \ref{Def}, we give the necessary preliminaries. In Section
\ref{LowBd}, we estimate the length of the shortest relations in a HNN-extension and prove Theorem
\ref{ToFree}. Section \ref{m1} is devoted to the case $\m =1$, namely the proof of Theorem \ref{ToWreath},
and Section \ref{1pBS} to other one-parameter families of Baumslag-Solitar groups, i.e. the proof of Theorem
\ref{ConvSub}. Finally, we prove Theorem \ref{UnifCont} in Section \ref{Topol}, linking Baumslag-Solitar
groups and $\m$-adic integers.
\paragraph{Acknowledgements.} I would like to thank Luc Guyot and Alain Valette for their useful comments and hints,
and Indira Chatterji, Pierre de la Harpe, Nicolas Monod and Alain Robert for comments on previous versions.
\section{Preliminaries}\label{Def}
In this Section, we collect some definitions and material which are needed in the rest of the paper. The
reader who is familiar with the notions of $\m$-adic integers, HNN-extensions and
topology on the space of marked groups can skip this Section.
\paragraph{The ring of $\m$-adic integers.}
For $\m \in \Z$, $|\m| \geqslant 2$ we define $\Z_\m$ to be the completion of $\Z$
with respect to the ultrametric distance given by the following "absolute value":
$$
|a \cdot \m^{k}|_\m := \left( \frac{1}{|\m|} \right)^{k} \ \text{ for } a \text{ not a multiple of } \m \text
{ and } k\geqslant 0 \ .
$$
(Note that it is not multiplicative in general, but one has $|-x|_\m = |x|_\m$ which is sufficient to induce a distance.)
The space $\Z_\m$ has a ring structure obtained by continuous extensions of the ring laws for $\Z$ and we call it
the \emph{ring of $\m$-adic integers}. The symbol $\Z_\m^\times$ denotes the invertible elements of $\Z_\m$. The
distance induced by $|.|_\m$ is called the \emph{$\m$-adic ultrametric distance}, since it satisfies the ultrametric inequality
$$
|x-z|_\m \leqslant \text{max} \big( |x-y|_\m,|y-z|_\m \big) \ .
$$
As a topological ring, $\Z_\m$ is the projective limit of the system
$$
\ldots \to \Z/\m^h\Z \to \Z/\m^{h-1}\Z \to \ldots \to \Z/\m^2\Z \to \Z/\m\Z
$$
where the arrows are the canonical (surjective) homomorphisms. This shows that $\Z_\m$ is compact and it
is coherent with this characterization to set $\Z_{\m} = \{0\}$ for $\m = \pm 1$. It becomes also
nearly obvious that the group of invertible elements of $\Z_\m$ is given by
$$
\Z_\m^\times = \Z_\m \setminus (p_1 \Z_\m \cup \ldots \cup p_k \Z_\m)
$$
where $p_1, \ldots, p_k$ are the prime factors of $\m$.
To conclude this short summary about $\m$-adic integers, let us notice that, for $|\m| >1$
and $\m$ not prime, the ring $\Z_\m$ has zero divisors.
\paragraph{Marked groups and their topology.} Introductory expositions of these topics can be found in
\cite{Cha} or \cite{ChaGui}. We only recall some basics and what we need in following sections.
The free group on $k$ generators will be denoted $\F_k$, or $F_S$ (with $S = (s_1, \ldots, s_k)$)
if we want to precise the names of (canonical) generating elements.
A \emph{marked group on $k$ generators} is a pair $(\Gamma,S)$ where $\Gamma$ is a group and
$S = (s_1, \ldots, s_k)$ is a family which generates $\Gamma$. A marked group $(\Gamma,S)$ comes always
with a canonical epimorphism $\phi: \F_S \to \Gamma$, giving an isomorphism of marked groups between
$\F_S/ \ker \phi$ and $\Gamma$. Hence a class of marked groups can always be represented by a quotient of
$\F_S$. In particular if a group is given by a presentation, this defines a marking on it. The nontrivial elements of
${\cal R} := \ker \phi$ are called \emph{relations} of $(\Gamma,S)$.
Let $w = x_1^{\varepsilon_1} \cdots x_n^{\varepsilon_n}$ be a reduced word in $\F_S$ (with $x_i \in S$ and
$\varepsilon_i \in \{ \pm 1 \}$). The integer $n$ is called the \emph{length} of $w$ and denoted $\ell(w)$.
The length of the shortest relation(s) of $\Gamma$ will be denoted $g_\Gamma$, for we observe it is the girth
of the Cayley graph of $\Gamma$ (with respect to the generating set $S$). In case ${\cal R} = \varnothing$,
we set $g_\Gamma = +\infty$.
If $(\Gamma,S)$ is a marked group on $k$ generators, and $\gamma \in \Gamma$ the \emph{length} of $\gamma$ is
\begin{eqnarray*}
\ell_\Gamma(\gamma) & := & \min\{ n: \gamma = s_1 \cdots s_n \text{ with } s_i \in S \sqcup S^{-1} \} \\
& = & \min\{ \ell(w) : w \in \F_S, \ \phi(w) = \gamma \} \ .
\end{eqnarray*}
Let ${\cal G}_k$ be the set of marked groups on $k$ generators (up to marked isomorphism). Let us recall that
the topology on ${\cal G}_k$ comes from the following ultrametric: for $(\Gamma_1, S_1) \neq (\Gamma_2, S_2)
\in {\cal G}_k$ we set $d \big( (\Gamma_1, S_1), (\Gamma_2, S_2) \big) := e^{-\lambda}$ where $\lambda$ is
the length of a shortest element of $\F_k$ which vanishes in one group and not in the other one. But what the
reader has to keep in mind is the following characterization of convergent sequences.
\begin{Prop}
Let $(\Gamma_n, S_n)$ be a sequence of marked groups (on $k$ generators). The following are equivalent:
\begin{enumerate}
\item[(i)] $(\Gamma_n, S_n)$ is convergent in ${\cal G}_k$;
\item[(ii)] for all $w \in \F_k$ we have either $w = 1$ in $\Gamma_n$ for $n$ large enough,
or $w \neq 1$ in $\Gamma_n$ for $n$ large enough.
\end{enumerate}
\end{Prop}
\textbf{Proof.} (i)$\Rightarrow$(ii): Set $(\Gamma,S) = \underset{n\to \infty}{\lim} (\Gamma_n,S_n)$ and take
$w \in \F_k$. For $n$ sufficiently large we have $d\big( (\Gamma,S),(\Gamma_n,S_n) \big) < e^{- \ell(w)}$,
which implies that we have $w=1$ in $\Gamma_n$ if and only if $w=1$ in $\Gamma$.
(ii)$\Rightarrow$(i): Set $N = \{ w \in \F_k : w=1 \text{ in } \Gamma_n \text{ for } n \text{ large enough }
\}$, $\Gamma = \F_k/N$, and fix $r \geqslant 1$. For $n$ large enough, $\Gamma_n$ and $\Gamma$ have the same
relations up to length $r$ (for the balls in $\F_k$ are finite) and hence $d(\Gamma,\Gamma_n) < e^{- r}$ (we
drop the markings since they are obvious). This implies $\Gamma_n \underset{n \to \infty}{\to} \Gamma$. \qed
\paragraph{HNN-extensions and Baumslag-Solitar groups.} Suppose now that $(H,S)$ is a marked group on $k$
generators, and that $\phi: A \to B$ is an isomorphism between subgroups of $H$. The \emph{HNN extension}
of $H$ with respect to $A$, $B$ and $\phi$ is given by
$$
HNN(H,A,B,\phi) := \frac{H * \langle t \rangle}{\cal N} \ .
$$
where ${\cal N}$ is the normal subgroup generated by the $t^{-1}at \phi(a)^{-1}$ for $ a \in A$.
Unless specified otherwise, we always mark a HNN-extension by $S$ and $t$. An element $\gamma \in HNN(H,A,B,\phi)$
can always be written
\begin{equation}\label{decHNN}
\gamma = h_0 t^{\varepsilon_1} h_1 \cdots t^{\varepsilon_n} h_n \text{ with } n \geqslant 0, \
\varepsilon_i \in \{\pm 1\}, \ h_i \in H\ .
\end{equation}
The decomposition of $\gamma$ in (\ref{decHNN}) is called \emph{reduced} if no subword of type
$t^{-1}at$ (with $a \in A$) or $tbt^{-1}$ (with $b \in B$) appears. We recall the following result, which
is called \emph{Britton's Lemma}.
\begin{Lem}\label{Britton}
(\cite{LynSch}, Chapter IV.2.) Let $\gamma \in HNN(H,A,B,\phi)$ and write as in (\ref{decHNN}) $\gamma =
h_0 t^{\varepsilon_1} h_1 \cdots t^{\varepsilon_n} h_n$. If $n \geqslant 1$ and if the decomposition is
reduced, then $\gamma \neq 1$ in $HNN(H,A,B,\phi)$.
\end{Lem}
This shows in particular that the integer $n$ appearing in a reduced decomposition is uniquely determined by $\gamma$.
Let us finally recall that \emph{Baumslag-Solitar groups} are defined by the presentations $BS(\m,\n)= \Pres{a,b}{ab^{\m}a^{-1} =
b^\n}$ for $\m,\n \in \Z^*$. Setting $\phi(\n k) = \m k$, we have $BS(\m,\n) = HNN(\Z, \n\Z, \m\Z, \phi)$.
\section{Shortest relations in a HNN-extension and convergence of Baumslag-Solitar groups}\label{LowBd}
Let $(H,S)$ be a marked group and $\Gamma = HNN(H,A,B,\phi)$. In this Section we give a lower estimate for
$g_\Gamma$. As a higher estimate, we obviously get $g_\Gamma \leqslant g_H$, because a shortest relation
in $H$ is also a relation in $\Gamma$. Let us define:
\begin{eqnarray*}
\alpha & := & \min\big\{ \ell_H(a) \ : \ a \in A \setminus \{1\} \big\} \ ; \\
\beta & := & \min\big\{ \ell_H(b) \ : \ b \in B \setminus \{1\} \big\} \ .
\end{eqnarray*}
\begin{Thm}\label{Low}
Let $(H,S)$, $\Gamma$, $\alpha$ and $\beta$ be defined as above. Then
we have
$$
\min\{ g_H, \ \alpha+\beta+2, \ 2\alpha+6, \ 2\beta+6 \} \leqslant g_\Gamma \leqslant g_H \ .
$$
\end{Thm}
As the case of Baumslag-Solitar groups (treated below) will show, the lower bound given in
Theorem \ref{Low} is in fact sharp. This sharpness is the principal interest of this Theorem, because the
estimate $\min\{ g_H, \ \alpha, \ \beta \} \leqslant g_\Gamma$, which follows easily from Lemma \ref{rel}
and Britton's Lemma, suffices to prove Theorem \ref{ToFree} (replace Proposition \ref{BSgirth} by the estimate
$g_{BS(\m,\n)} \geqslant \min(\m,\n)$). I would like to thank the referee for having pointed
this fact to me.
Before proving Theorem \ref{Low}, let us begin with a simple observation.
\begin{Lem}\label{rel}
Let $(H,S)$ and $\Gamma$ be defined as above and let $r$ be a relation of $\Gamma$ contained in $\F_S$.
Then $r$ is a relation of $H$. In particular, $\ell(r) \geqslant g_H$.
\end{Lem}
\textbf{Proof.} Since $r=1$ in $\Gamma$ and since the canonical map $H \to \Gamma$ is injective, we get $r=1$ in $H$.
Hence the first assertion. The second one follows by definition of $g_H$. \qed
\smallskip
\textbf{Proof of Theorem \ref{Low}.} The second inequality has already been discussed. To establish the first one, let us take a
relation $r$ of $\Gamma$ and show that $\ell(r) \geqslant m$, where we set $m := \min\{ g_H, \ \alpha+\beta+2,
\ 2\alpha+6, \ 2\beta+6 \}$.
Write $r = h_0 t^{\varepsilon_1} h_1 \cdots t^{\varepsilon_n} h_n$ with $\varepsilon_i \in \{\pm 1\}$, $h_i
\in \F_S$ and $h_i \neq 1$ if $\varepsilon_i = - \varepsilon_{i+1}$. Up to replacement by a (shorter)
conjugate, we may assume that $r$ is cyclically reduced. If $n \neq 0$, we may also assume that $h_0 = 1$.
Since $r=1$ in $\Gamma$, one clearly has $\sum_{i=1}^{n} \varepsilon_i = 0$. In particular, $n$ is even. Let
us distinguish several cases and show $\ell(r) \geqslant m$ in each one:
\textbf{Case $n=0$:} We get $r = h_0 \in \F_S$. Thus $\ell(r) \geqslant g_H \geqslant m$ by Lemma \ref{rel}.
\textbf{Case $n=2$:} One gets $r = t^{\varepsilon} h_1 t^{-\varepsilon} h_2$. If we look at $r$ in $\Gamma$, we have
$r=1$ and thus $h_1 \in A$ (if $\varepsilon = -1$) or $h_1 \in B$ (if $\varepsilon =1$) by Britton's Lemma. Suppose
$\varepsilon = -1$ (in case $\varepsilon = 1$ the proof is similar and left to the reader). Looking at $h_1$ in $\F_S$,
there are two possibilities (remember that we assumed $h_1 \neq 1$).
\begin{itemize}
\item If $h_1 = 1$ in $\Gamma$, Lemma \ref{rel} implies $\ell(r) \geqslant \ell(h_1) \geqslant g_H \geqslant m$.
\item If $h_1 \neq 1$ in $\Gamma$, then $\ell(h_1) \geqslant \alpha$. On the other hand $h_2^{-1} = t^{-1} h_1 t \in
B$; thus $\ell(h_2) \geqslant \beta$ and $\ell(r) \geqslant \alpha + \beta + 2 \geqslant m$.
\end{itemize}
\textbf{Case $n \geqslant 4$:} We have $r = 1$ in $\Gamma$. By Britton's Lemma, there is an index $i$ such
that either $\varepsilon_i = -1$, $\varepsilon_{i+1} = 1$ and $h_i \in A$, or $\varepsilon_i = 1$,
$\varepsilon_{i+1} = -1$ and $h_i \in B$. Since cyclic conjugations preserve length, we may assume $i=1$, so that
$r = t^{\varepsilon_1} h_1 t^{-\varepsilon_1} h_2 t^{\varepsilon_3} h_3 \cdots t^{\varepsilon_n} h_n$. Let us
moreover assume $\varepsilon_1 = -1$ (again the case $\varepsilon_1 = 1$ is similar and left to the reader).
Set $r' = w h_2 t^{\varepsilon_3} h_3 \cdots t^{\varepsilon_n} h_n$ where $w \in F_X$ is such that $w =
t^{-1}h_1t$ in $\Gamma$ (in fact this element is in $B$). Applying Britton's Lemma to $r'$, one sees there
exists an index $j \geqslant 3$ such that either $\varepsilon_{j} = -1 = -\varepsilon_{j+1}$ and $h_j \in A$,
or $\varepsilon_j = 1 = -\varepsilon_{j+1}$ and $h_j \in B$. There are three possibilities:
\begin{itemize}
\item If $h_1 = 1$ or $h_j = 1$ in $\Gamma$, Lemma \ref{rel} implies $\ell(r) \geqslant g_H \geqslant m$ as
above.
\item If $h_1 \neq 1$ in $\Gamma$, $h_j \neq 1$ in $\Gamma$ and $\varepsilon_j = 1$, then $\ell(h_1) \geqslant \alpha$
and $\ell(h_j) \geqslant \beta$. Thus $\ell(r) \geqslant \alpha + \beta + 4 > m$.
\item If $h_1 \neq 1$ in $\Gamma$, $h_j \neq 1$ in $\Gamma$ and $\varepsilon_j = -1$, then we write $r =
t^{-1}h_1t w_1 t^{-1}h_jt w_2$ with $\ell(h_1) \geqslant \alpha$ and $\ell(h_j) \geqslant \alpha$. The
subwords $w_1, w_2$ are not empty because $r$ is cyclically reduced. Thus $\ell(r) \geqslant 2 \alpha + 6
\geqslant m$ (Remark that $2\alpha + 6$ would be replaced by $2 \beta +6$ in the case $\varepsilon_1 = 1,
\varepsilon_j = 1$).
\end{itemize}
The proof is complete. \qed
\smallskip
We now turn to prove that for Baumslag-Solitar groups, the lower bound coming
from Theorem \ref{Low} is in fact the length of the shortest relation. More precisely we have the following
statement:
\begin{Prop}\label{BSgirth}
Let $\m,\n \in \Z^*$. We have
$$
g_{BS(\m,\n)} = \min\big\{ |\m|+|\n|+2, \ 2|\m|+6, \ 2|\n|+6 \big\} \ .
$$
\end{Prop}
\textbf{Proof.} Set $m := \min\{ |\m|+|\n|+2, \ 2|\m|+6, \ 2|\n|+6 \}$ and $\Gamma = BS(\m,\n)$. We have $g_\Z =
+\infty$, $\alpha = |\n|$ and $\beta = |\m|$. Thus, Theorem \ref{Low} implies $g_\Gamma \geqslant m$. To prove
that $g_\Gamma \leqslant m$, we produce relations of length $|\m|+|\n|+2$, $2|\m|+6$, and $2|\n|+6$. Namely:
\begin{itemize}
\item $ab^\m a^{-1}b^{-\n}$ has length $|\m|+|\n|+2$;
\item $ab^\m a^{-1}b ab^{-\m}a^{-1}b^{-1}$ has length $2|\m|+6$;
\item $a^{-1}b^\n ab a^{-1}b^{-\n}ab^{-1}$ has length $2|\n|+6$. \qed
\end{itemize}
Theorem \ref{ToFree} of introduction is now a consequence of Proposition \ref{BSgirth}, since
a sequence of groups $\Gamma_n = \Pres{a,b}{{\cal R}_n}$ converges to the free group on two
generators (marked by its canonical basis) if and only if $g_{\Gamma_n}$ tends to $\infty$.
\section{Limit of Solvable Baumslag-Solitar groups}\label{m1}
This section is entirely devoted to the proof of Theorem \ref{ToWreath}. We notice first that
$BS(1,\n) = BS(-1,-\n)$ as marked groups. Thus we
may assume $\m = 1$. Hence, we let $\Gamma_\n = BS(1,\n)$ and $\Gamma = \Z \wr \Z$. In $\Z \wr \Z$, let us set
$a= (1,0)$ and $b = (0,e_0)$. We have to show that for all $w \in \F_2$:
\begin{enumerate}
\item[(1)] if $w=1$ in $\Z \wr \Z$, then $w=1$ in $BS(1,\n)$ for $|\n|$ large enough;
\item[(2)] if $w\neq 1$ in $\Z \wr \Z$, then $w\neq 1$ in $BS(1,\n)$ for $|\n|$ large enough.
\end{enumerate}
Let $w \in \F_2$. One can write $w = a^\alpha a^{\alpha_1} b^{\beta_1} a^{-\alpha_1} \cdots
a^{\alpha_k} b^{\beta_k} a^{-\alpha_k}$. The image of $w$ in $\Gamma$ is $(\alpha, \sum_{i=1}^k \beta_i
e_{\alpha_i})$, where $e_j \in \oplus_{h \in \Z} \Z$ is the Dirac mass at $j$.
Let $m = \min_{1 \leqslant i \leqslant k} \alpha_i$. In $\Gamma_\n = BS(1,\n)$, we have
\begin{eqnarray*}
w & = & a^\alpha a^m a^{\alpha_1-m} b^{\beta_1} a^{m-\alpha_1} \cdots a^{\alpha_k-m} b^{\beta_k} a^{m-\alpha_k} a^{-m} \\
& = & a^\alpha a^m b^{\beta_1 \n^{\alpha_1-m}} \cdots b^{\beta_k \n^{\alpha_k-m}} a^{-m} \\
& = & a^\alpha a^m b^{\sum_{h \in \Z} \left( \sum_{\alpha_i = h} \beta_i \right) \n^{h-m}} a^{-m}
\end{eqnarray*}
(1) As $w \underset{\Gamma}{=} 1$, we have $\alpha = 0$ and $\forall h \in \Z$, $\sum_{\alpha_i = h}
\beta_i = 0$. Hence
$$
w \underset{\Gamma_\n}{=} a^0 a^m b^{\sum_{h \in \Z} 0 \cdot \n^{h-m}} a^{-m} = 1 \ \forall \n\in \Z^* \ .
$$
(2) As $w \underset{\Gamma}{\neq} 1$, either $\alpha \neq 0$ or $\exists h \in \Z$ such that $\sum_{\alpha_i = h}
\beta_i \neq 0$. The image of $w$ by the morphism $\Gamma_\n \to \Z$ given by $a \mapsto
1, b \mapsto 0$ is $\alpha$. Hence, if $\alpha \neq 0$, then $w \underset{\Gamma_\n}{\neq} 1 \ \forall \n$. If
$\alpha = 0$, we set $h_0$ to be the maximal value of $h$ such that $\sum_{\alpha_i = h} \beta_i \neq 0$. For
$|\n|$ large enough, we have
$$
\left| \sum\limits_{\alpha_i = h_0} \beta_i \n^{h_0 - m} \right| > \left| \sum\limits_{h<h_0} \sum\limits_{\alpha_i = h}
\beta_i \n^{h-m} \right| \ .
$$
For those values of $\n$, we get
$$
w \underset{\Gamma_\n}{=} a^m b^{\left( \sum_{\alpha_i = h_0} \beta_i \right) \n^{h_0 - m} +
\sum_{h < h_0} \left( \sum_{\alpha_i = h} \beta_i \right) \n^{h-m}} a^{-m} \underset{\Gamma_\n}{\neq} 1 \ .
$$
The proof is complete. \qed
\section{General one-parameter families of Baumslag-Solitar groups}\label{1pBS}
We now treat the case $|\m| \geqslant 2$. More precisely, we begin the proof of Theorem \ref{ConvSub}. We also
have $BS(\m,\n) = BS(-\m,-\n)$ as marked groups. This equality will allow us to assume $\m > 0$ in following
proofs. We begin with a Lemma which already shows that the sequence
$(BS(\m,\n))_\n$ is not itself convergent.
\begin{Lem}\label{Congruence}
Let $\m, \n \in \Z^*$, $d= \text{gcd}(\m, \n)$. We write $\m = d \m_1$, $\n = d \n_1$. Let $k\in \Z$, $h\geqslant 1$ and
$$
w = a^{h+1} b^{\m} a^{-1} b^{-k} a^{-h} b a^{h+1} b^{-\m} a^{-1} b^k a^{-h} b^{-1} \ .
$$
If $|\n| \geqslant 2$, we have $w = 1$ in $BS(\m, \n)$ if and only if $\n \equiv k \
(\text{mod } \m_1^h d)$.
\end{Lem}
The congruence modulo $\m_1^h d$ (instead of $\m^h$) is the reason for the hypothesis "$\n$ relatively prime
to $\m$" appearing in Theorem \ref{ConvSub}.
\smallskip
\textbf{Proof.} Let $\Gamma_{\n} = BS(\m, \n)$. We have
$$
w \underset{\Gamma_{\n}}{=} a^h b^{\n - k} a^{-h} b a^{h} b^{k-\n} a^{-h} b^{-1} \ .
$$
Let us now distinguish three cases:
\textbf{Case 1:} $\n \not\equiv k \ (\text{mod } \m)$. \\
We have $w \underset{\Gamma_{\n}}{\neq} 1$ by Britton's Lemma, since $|\n| \geqslant 2$.
\textbf{Case 2:} $\n \not\equiv k \ (\text{mod } \m_1^h d)$, but $\n \equiv k \ (\text{mod } \m)$. \\
We write $\n - k = \ell \m_1^g d$ with $g<h$ and $\ell$ not a multiple of $\m_1$. Hence $\ell \n_1^g d$ is not a multiple
of $\m = \m_1 d$, for $\m_1$ is relatively prime to $\n_1$. We have
$$
w \underset{\Gamma_{\n}}{=} a^{h-g} b^{\ell \n_1^g d} a^{g-h} b a^{h-g} b^{-\ell \n_1^g d} a^{g-h} b^{-1}
\underset{\Gamma_{\n}}{\neq} 1
$$
by Britton's Lemma (again because $|\n| \geqslant 2$).
\textbf{Case 3:} $\n \equiv k \ (\text{mod } \m_1^h d)$. \\
Let us write $\n - k = \ell \m_1^h d$. Then $w \underset{\Gamma_{\n}}{=} b^{\ell \n_1^h d} b b^{-\ell \n_1^h d}
b^{-1} \underset{\Gamma_{\n}}{=} 1$. \qed
\smallskip
\textbf{Proof of Theorem \ref{ConvSub}.} The "if" is a particular case of Theorem \ref{ConvSubOnlyIf}
below.
We prove now the "only if" part. Let $\Gamma_n = BS(\m,\k_n)$. We assume the sequence
$(\Gamma_n)_n$ to converge and condition (a) not to hold. We have to show that condition (b) holds.
Fix $h \geqslant 1$. For $k \in \Z$ we set
$$
w_k = a^{h+1} b^\m a^{-1} b^{-k} a^{-h} b a^{h+1} b^\m a^{-1} b^k a^{-h} b^{-1} \ .
$$
As $(\Gamma_n)_n$ converges, we have (for each $k \in \Z$) either $w_k \underset{\Gamma_n}{=} 1$ for $n$ large enough,
or $w_k \underset{\Gamma_n}{\neq} 1$ for $n$ large enough. As $\k_n$ is relatively prime to $\m$ for all $n$, Lemma
\ref{Congruence} ensures that (for each $k \in \Z$) either $\k_n \equiv k \ (\text{mod } \m^h)$ for $n$ large enough,
or $\k_n \not\equiv k \ (\text{mod } \m^h)$ for $n$ large enough. This implies that $\k_n$ is eventually constant
modulo $\m^h$ ($\forall h \geqslant 1$).
It remains to show that $|\k_n| \to \infty$. Assume by contradiction that there exists some $\ell \in \Z$ such
that $\k_n = \ell$ for infinitely many $n$. As (a) does not hold (i.e $(\k_n)_n$ is not eventually constant),
it is sufficient to treat the two following cases:
\textbf{Case 1:} $\exists \ell' \neq \ell$ such that $\k_n = \ell'$ for infinitely many $n$. \\
Take $h$ large enough so that $\m^h > |\ell - \ell'|$. The sequence $\k_n$ cannot be eventually constant
modulo $\m^h$, in contradiction with the first part of the proof.
\textbf{Case 2:} $\exists$ a subsequence $(\k_{n_j})_j$ of $(\k_n)_n$ such that $|\k_{n_j}| \to \infty$. \\
We set $w = ab^\m a^{-1}b^{-\ell}$. For infinitely many $n$ (those values for which $\k_n = \ell$), we have
$w \underset{\Gamma_n}{=} 1$. On the other hand $|\k_{n_j}| > \ell$ for $j$ large enough. For these values
of $j$, we have
$$
w \underset{\Gamma_{n_j}}{=} b^{\k_{n_j} - \ell} \underset{\Gamma_{n_j}}{\neq} 1 \ .
$$
This contradicts the assumption on the sequence $(\Gamma_n)_n$ to converge. \qed
\smallskip
What remains now to do is to prove the following Theorem, which is a little bit more general than the
"if" part of Theorem \ref{ConvSub}. The proof will need some preliminary lemmas.
\begin{Thm}\label{ConvSubOnlyIf}
Let $\m \in \Z^*$ and let $(\k_n)_n$ be a sequence in $\Z^*$. If $|\k_n| \to \infty$ and if $\forall h
\geqslant 1$ the sequence $(\k_n)_n$ is eventually constant modulo $\m^h$, then the sequence $(BS(\m,\k_n))_n$ is
convergent in ${\cal G}_2$.
\end{Thm}
\begin{Lem}\label{Prelim1}
Let $\m,\n,\n' \in \Z^*$ and $h \geqslant 1$. If $\n \equiv \n'$ $(\text{mod } \m^h)$, there exists
$s_0, \ldots, s_h$; $s'_0, \ldots, s'_h$; $r_1, \ldots, r_h$, which are unique, such that:
\begin{enumerate}
\item[(i)] $0 \leqslant r_i < \m \ \forall i$; $s_0 = 1 = s'_0$;
\item[(ii)] $s_{i-1} \n = s_i \m + r_i $ and $s'_{i-1} \n' = s'_i \m + r_i$ $\forall \, 1\leqslant i \leqslant h$;
\item[(iii)] $s_i \equiv s'_i \ (\text{mod } \m^{h-i}) \ \forall \, 0 \leqslant i \leqslant h$.
\end{enumerate}
\end{Lem}
\textbf{Proof.} Given the congruence $\n \equiv \n'$ $(\text{mod } \m^h)$, we obtain (by Euclidean division)
$s_{0} \n = \n = s_1 \m + r_1 $ and $s'_{0} \n' = \n' = s'_1 \m + r_1$ with $0 \leqslant r_1 \leqslant \m$
and $s_1 \equiv s'_1$ $(\text{mod } \m^{h-1})$. Hence we have $s_1 \n \equiv s'_1 \n'$ $(\text{mod } \m^{h-1})$.
(Let us emphasize that we do not necessary have $s_1 \n \equiv s'_1 \n'$ $(\text{mod } \m^h)$.)
Now, it just remains to iterate the above and uniqueness follows from construction. \qed
\smallskip
Given a word $w$ in $\F_2$, we may use Britton's Lemma to reduce it in $BS(\m,\n)$ or $BS(\m,\n')$. But $w$
could be reducible in one of these groups and not in the other one. Even if it is reducible in both groups
the result is not the same word in general. The purpose of next statement is, under some assumptions, to
control the parallel process of reduction in both groups. This will be useful to ensure that $w$ is a
relation in $BS(\m,\n)$ if and only if it is one in $BS(\m,\n')$ (under some assumptions).
\begin{Lem}\label{Prelim2}
Let $\m,\n,\n' \in \Z^*$ and $h \geqslant m \geqslant 1$. Assume that $\n \equiv \n'$ $(\text{mod } \m^h)$
and let
\begin{eqnarray*}
\alpha & = & k_0 + k_1 \n + k_2 s_1 \n + \ldots + k_m s_{m-1} \n \\
\alpha' & = & k_0 + k_1 \n' + k_2 s'_1 \n' + \ldots + k_m s'_{m-1} \n'
\end{eqnarray*}
where $s_0, \ldots, s_h$; $s'_0, \ldots, s'_h$; $r_1, \ldots, r_h$ are given by Lemma \ref{Prelim1}. We
assume moreover that we have $|k_0| < \min(|\n|,|\n'|)$.
\begin{enumerate}
\item[(i)] We have $\alpha \equiv 0$ $(\text{mod } \m)$ if and only if $\alpha' \equiv 0$ $(\text{mod } \m)$.
If this happens we get $ab^\alpha a^{-1} \underset{BS(\m,\n)}{=} b^\beta$ and $ab^{\alpha'} a^{-1}
\underset{BS(\m,\n')}{=} b^{\beta'}$ with
\begin{eqnarray*}
\beta & = & \ell_1 \n + \ell_2 s_1 \n + \ldots + \ell_{m+1} s_{m} \n \\
\beta' & = & \ell_1 \n' + \ell_2 s'_1 \n' + \ldots + \ell_{m+1} s'_{m} \n' \ .
\end{eqnarray*}
\item[(ii)] We have $\alpha \equiv 0$ $(\text{mod } \n)$ if and only if $\alpha' \equiv 0$ $(\text{mod } \n')$.
If this happens we get $a^{-1}b^\alpha a \underset{BS(\m,\n)}{=} b^\beta$ and $a^{-1} b^{\alpha'} a
\underset{BS(\m,\n')}{=} b^{\beta'}$ with
\begin{eqnarray*}
\beta & = & \ell_0 + \ell_1 \n + \ell_2 s_1 \n + \ldots + \ell_{m-1} s_{m-2} \n \\
\beta' & = & \ell_0 + \ell_1 \n' + \ell_2 s'_1 \n' + \ldots + \ell_{m-1} s'_{m-2} \n' \ .
\end{eqnarray*}
\end{enumerate}
\end{Lem}
\textbf{Proof.} (i) We have $\alpha \equiv \alpha' \ (\text{mod } \m)$ by construction. Assume now that
$\alpha \equiv 0$ and $ \alpha' \equiv 0 \ (\text{mod } \m)$. We have
$$
\alpha = k_0 + k_1 r_1 + \ldots + k_m r_m + k_1 s_1 \m + \ldots + k_m s_m \m \ .
$$
As $\alpha \equiv 0 \ (\text{mod } \m)$, we obtain $ab^\alpha a^{-1} \underset{BS(\m,\n)}{=} b^\beta$ with
$$
\beta = \frac{\n}{\m}(k_0 + k_1 r_1 + \ldots + k_m r_m) + k_1 s_1 \n + \ldots + k_m s_m \n
$$
Thus we set $\ell_1 = \frac{1}{\m}(k_0 + k_1 r_1 + \ldots + k_m r_m)$ and $\ell_i = k_{i-1}$ for $2 \leqslant
i \leqslant m+1$, and doing the same calculation with $\alpha'$ in $BS(\m,\n')$, we obtain also
$$
\beta' = \ell_1 \n' + \ell_2 s'_1 \n' + \ldots + \ell_{m+1} s'_{m} \n' \ .
$$
(ii) As $|\n|> |k_0|$ and $|\n'| > |k_0|$, we have $\alpha \equiv 0 \ (\text{mod } \n)$ if and only if $k_0 = 0$
if and only if $\alpha' \equiv 0 \ (\text{mod } \n')$. Suppose now that it is the case. We have $ab^\alpha
a^{-1} = b^\beta$ in $BS(\m,\n)$ with
\begin{eqnarray*}
\beta & = & k_1 \m + k_2 s_1 \m + \ldots + k_m s_{m-1} \m \\
& = & k_1 \m - k_2 r_1 - \ldots - k_m r_{m-1} + k_2 \n + k_3 s_1 \n + \ldots + k_m s_{m-2} \n \ .
\end{eqnarray*}
Hence we set $\ell_0 = k_1 \m - k_2 r_1 - \ldots - k_m r_{m-1}$ and $\ell_i = k_{i+1}$ for $1 \leqslant
i\leqslant m-1$. Again, doing the same calculation with $\alpha'$ in $BS(\m,\n')$, we obtain also
$$
\beta' = \ell_0 + \ell_1 \n' + \ell_2 s'_1 \n' + \ldots + \ell_{m-1} s'_{m-2} \n' \ .
$$
This completes the proof. \qed
\begin{Lem}\label{Prelim3}
Let $\m \in \Z^*$, let $(\k_n)_n$ be a sequence in $\Z^*$ such that $|\k_n| \to \infty$ and $\forall h
\geqslant 1$ $(\k_n)_n$ is eventually constant modulo $\m^h$ and let $w \in \F_2$. Then, the following alternative
holds:
\begin{enumerate}
\item[(a)] either $w = b^{\lambda_n}$ in $BS(\m,\k_n)$ for $n$ large enough;
\item[(b)] or $w$ is in $BS(\m,\k_n) \setminus \langle b \rangle$ for $n$ large enough.
\end{enumerate}
\end{Lem}
\textbf{Proof.} We define $\Gamma_n = BS(\m,\k_n)$. Let us write $w = b^{\alpha_0} a^{\varepsilon_1} b^{\alpha_1}
\cdots a^{\varepsilon_m} b^{\alpha_m}$ with $\varepsilon_i = \pm 1$ and $\alpha_i \in \Z$, reduced in
the sense that $\alpha_i = 0$ implies $\varepsilon_{i+1} = \varepsilon_{i}$ for all $i \in \{1, \ldots, m-1 \}$.
We assume (b) not to hold, i.e. $w= b^{\lambda_n}$ in $\Gamma_n$ for infinitely many $n$. Then the sum
$\varepsilon_1 + \ldots + \varepsilon_m$ has clearly to be zero (in particular $m$ is even). We have to show
that $w= b^{\lambda_n}$ for $n$ large enough.
For $n$ large enough, we may assume that, $|\k_n| > |\alpha_j|$ for all $1 \leqslant j \leqslant m$
and the $\k_n$'s are all congruent modulo $\m^m$. We take a value of $n$ such that moreover $w= b^{\lambda_n}$ in
$\Gamma_n$ (there are infinitely many ones) and apply Britton's Lemma. This ensures the existence of
an index $j$ such that $\varepsilon_j = 1 = -\varepsilon_{j+1}$ and $\alpha_j \equiv 0 \ (\text{mod } \m)$
(since $|\k_n| > |\alpha_j|$ for all $j$). By Lemma \ref{Prelim2}, for all $n$ large enough
$$
w \underset{\Gamma_n}{=} b^{\alpha_0} \cdots a^{\varepsilon_{j-1}} b^{\alpha_{j-1} + \beta_j + \alpha_{j+1}}
a^{\varepsilon_{j+2}} \cdots b^{\alpha_m}
$$
with $\beta_j = \ell_1 \k_n$ (depending on $n$). Hence we are allowed to write
$$
w \underset{\Gamma_n}{=} b^{\alpha'_{0,n}} a^{\varepsilon'_1} b^{\alpha'_{1,n}}
\cdots a^{\varepsilon'_{m-2,n}} b^{\alpha'_{m-2,n}}
$$
for $n$ large enough, with $\varepsilon'_i = \pm 1$ and $\alpha'_{j,n} = k'_{0,j} + k'_{1,j} \k_n$, where the
$k'_{i,j}$'s do not depend on $n$.
Now, for $n$ large enough, we may assume that, $|\k_n| > |k'_{0,j}|$ for all $1 \leqslant j \leqslant m-1$
(and the $\k_n$'s are all congruent modulo $\m^m$). Again we take a value of $n$ such that moreover $w=b^{\lambda_n}$ in
$\Gamma_n$ and apply Britton's Lemma. This ensures the existence of an index $j$ such that either
$\varepsilon'_j = 1 = -\varepsilon'_{j+1}$ and $\alpha'_{j,n} \equiv 0 \ (\text{mod } \m)$, or
$\varepsilon'_j = -1 = -\varepsilon'_{j+1}$ and $\alpha'_{j,n} \equiv 0 \ (\text{mod } \k_n)$. In both cases,
while applying Lemma \ref{Prelim2}, we obtain
$$
w \underset{\Gamma_n}{=} b^{\alpha''_{0,n}} a^{\varepsilon''_1} b^{\alpha''_{1,n}}
\cdots a^{\varepsilon''_{m-4,n}} b^{\alpha''_{m-4,n}}
$$
for $n$ large enough, with $\varepsilon''_i = \pm 1$ and $\alpha''_{j,n} = k''_{0,j} + k''_{1,j} \k_n + k''_{2,j} s_{1,n} \k_n$,
where the $k''_{i,j}$'s do not depend on $n$.
And so on, and so forth, setting $m' = \frac{m}{2}$, we get finally $w = b^{\alpha^{(m')}_{0,n}}$ in $\Gamma_n$
for $n$ large enough, with
$$
\alpha^{(m')}_{0,n} = k^{(m')}_{0,0} + k^{(m')}_{1,0} \k_n + k^{(m')}_{2,0} s_{1,n} \k_n +
\ldots + k^{(m')}_{m',0} s_{m'-1,n} \k_n
$$
where the $k^{m'}_{i,0}$'s do not depend on $n$. It only remains to set $\lambda_n = \alpha^{(m')}_{0,n}$.
\qed
\smallskip
Let us now introduce the homomorphisms $\psi_\n : BS(\m,\n) \to \text{Aff}(\R)$ (for $\n \in \Z^*$) given by
$\psi_\n (a)(x) = \frac{\n}{\m} x$ and $\psi_\n (b)(x) = x+1$.
\begin{Lem}\label{Prelim4}
Let $w \in \F_2$. We have either $\psi_\n (w) = 1$ for $|\n|$ large enough or $\psi_\n (w) \neq 1$
for $|\n|$ large enough.
\end{Lem}
\textbf{Proof.} Let us write $w = b^{\alpha_0} a^{\varepsilon_1} b^{\alpha_1} \ldots a^{\varepsilon_k}
b^{\alpha_k}$ with $\varepsilon_i = \pm 1$ and $\alpha_i \in \Z$. Set next $\sigma_0 = 0$, $\sigma_i =
\varepsilon_1 + \ldots + \varepsilon_i$ for $1 \leqslant i \leqslant k$ and $m = \min_{0 \leqslant i
\leqslant k} \sigma_i$. We get by calculation that
$$
\psi_\n (w)(x) = \left( \frac{\n}{\m} \right)^{\sigma_k} x + \left( \frac{\n}{\m} \right)^m P_w \left(\frac{\n}{\m} \right)
$$
where $P_w$ is the polynomial defined by $P_w (y) = \sum_{i=0}^k \alpha_i y^{\sigma_i - m}$. Let us assume the
second term of alternative not to hold, i.e. $\psi_\n(w) = 1$ for infinitely many values of $\n$. Hence we have
$\sigma_k = 0$ and for all those values of $\n$, $P_w(\frac{\n}{\m}) = 0$. As $P_w$ is a polynomial with infinitely many
roots, it is the zero polynomial. This shows that $\psi_\n(w) = 1$ for all $\n$. \qed
\smallskip
\textbf{Proof of Theorem \ref{ConvSubOnlyIf}.} It is easy to show that a word $w$ is equal to $1$ in
$BS(\m,\n)$ if and only if it is in the subgroup generated by $b$ and $\psi_\n(w) = 1$. It is also
a consequence of (the proof of) Theorem 1 in \cite{GalJan}.
Let $w\in \F_2$. Lemmas \ref{Prelim3} and \ref{Prelim4} immediately imply that either $w=1$ in $BS(\m,\k_n)$
for $n$ large enough or $w \neq 1$ in $BS(\m,\k_n)$ for $n$ large enough. \qed
Theorem \ref{ConvSub} is now completely established.
\section{Baumslag-Solitar groups and $\m$-adic integers}\label{Topol}
This last section is entirely devoted to the proof of Theorem \ref{UnifCont}. We show first that the map
$\Psi$ is uniformly continuous. In view of the distances we put on ${\cal G}_2$ and $\Z_\m^\times$, it is equivalent
to show that for any $h \geqslant 1$ there exists $r \geqslant 1$ such that we have $\n \equiv \n' \ (\text{mod } \m^h)$
whenever $BS(\m,\n)$ and $BS(\m,\n')$ (taken in $X_\m$) have the same relations up to length $r$.
Fix $h\geqslant 1$. Our candidate is $r = 2\m^h + 4h + 2\m + 4$. Assume that $BS(\m,\n)$ and $BS(\m,\n')$ have the same
relations up to length $r$. For $0\leqslant k \leqslant \m^h-1$ let
$$
w_k = a^{h+1} b^\m a^{-1} b^{-k} a^{-h} b a^{h+1} b^\m a^{-1} b^k a^{-h} b^{-1} \ .
$$
(Remark that these words are exactly those which appear in the proof of the "only if" part of theorem \ref{ConvSub}.
We are in fact improving this proof in order to get the uniform continuity.) We have $\ell(w_k) \leqslant r$
for all $k$. Having by assumption $w = 1$ in $BS(\m,\n)$ if and only if $w = 1$ in $BS(\m,\n')$, Lemma
\ref{Congruence} implies $\n \equiv \n' \ (\text{mod } \m^h)$.
The space $\Z_\m^\times$ being complete and the uniform continuity of $\Psi$ being now proved, the existence
of a (unique) uniformly continuous extension $\overline{\Psi}$ on $\overline{X_\m}$ is a standard fact
(see \cite{Dug}, Chapter XIV, Theorem 5.2. for instance).
Let us now show that $\overline{\Psi}$ is surjective. The space $\overline{X_\m}$ being compact,
$\text{im}(\overline{\Psi})$ is closed in $\Z_\m^\times$. Moreover it is dense since it contains the set of
$\n$'s relatively prime to $\m$ and such that $|\n| \geqslant 2$.
Finally, we consider the sequence $(BS(\m,1+\m+\m^n))_n$, which is convergent by theorem \ref{ConvSub} and we
call the limit $\Gamma$. We have $\overline{\Psi}(\Gamma) = 1+\m = \overline{\Psi}(BS(\m,1+\m))$. On the other
hand, we have $\Gamma \neq BS(1+\m)$, since $ab^\m a^{-1}b^{-(\m+1)} = 1$ in $BS(\m,1+\m)$ while
$ab^\m a^{-1}b^{-(\m+1)} \neq 1$ in $BS(\m,1+\m+\m^n)$ for all $n$. This is the non-injectivity of
$\overline{\Psi}$ and completes the proof. \qed
\bibliographystyle{alpha} | {"config": "arxiv", "file": "math0403241.tex"} |
\begin{document}
\title{Stepanov-like Weighted Pseudo-Almost Automorphic Solutions on Time Scales for a Novel High-order BAM Neural Network with Mixed
Time-varying Delays in the Leakage Terms}
\author{Adn\`ene Arbi}
\institute{
Adn\`ene Arbi\\
adnen.arbi@enseignant.edunet.tn, arbiadnene@yahoo.fr\\
Laboratory of Engineering Mathematics (LR01ES13), Tunisia
Polytechnic School, University of Carthage, El Khawarizmi Street, Carthage 2078, Tunisia.\\
} \maketitle
\begin{abstract}
We first propose the concept of Stepanov-like weighted pseudo almost automorphic on time-space scales and we apply this type of oscillation
to high-order BAM neural networks with mixed delays. Then, we study the existence and exponential stability of Sp-weighted pseudo-almost automorphic
on time-space scales solutions for the suggested system. Some criteria assuring the convergence are proposed. Our method is mainly based on the Banachs
fixed point theorem, the theory of calculus on time scales and the Lyapunov- Krasovskii functional method. Moreover, a numerical example is given to show
the effectiveness of the main results.
\keywords{Time scales; High-order BAM neural networks;
Stepanov-like weighted pseudo-almost automorphic solution; Global
exponential stability; Leakage delays.}
\end{abstract}
\vspace{-6pt}
\section{Introduction}
The concept of weighted pseudo almost automorphic on time-space
scales functions for the nabla and delta derivative is recently
introduced (see \cite{tomorphic2}, \cite{chang},
\cite{adnene+ahmed}). It is a natural generalization of almost
automorphic on time-space scales functions introduced in
\cite{carlos}. In 2010, the concept of Stepanov-like weighted pseudo
almost automorphy which is a natural generalization of the almost
automorphy is presented (see \cite{diagana}). Moreover, there is no
definition of the notion of Stepanov-like almost automorphy and
Stepanov-like weighted pseudo almost periodic on time-space scales
in the previous work.
On the other hand, many researchers have been devoting the dynamics
of various class of neural networks due to its wide application in
pattern recognition, associative memory, image, and signal
processing (see \cite{adnene+ahmed}, \cite{add1}, \cite{add2}, \cite{adn0}, \cite{adn00}, \cite{adn01}, \cite{adn1},
\cite{aaa}, \cite{adnene+jinde}, \cite{maas}, \cite{kren},
\cite{zhang1}, \cite{zhang2}, \cite{zhang3}). Furthermore, BAM
neural networks as an extension of the unidirectional autoassociator
of Hopfield neural network (see \cite{adn0},
\cite{adn01}, \cite{adn1}, \cite{aaa}), was firstly introduced by
Kosko \cite{kosko}. In recent years, many scholars pay much
attention to the dynamical behavior of bidirectional associative
memory (BAM) neural networks. Considering that time delays are
unavoidable because of the finite switching of amplifiers in
practical implementation of neural networks, and the time delay may
result in oscillation and instability; many authors focus on the
dynamical properties of BAM neural networks with time delays (see
\cite{maas}, \cite{bam0}, \cite{bam1}, \cite{bam2}, \cite{bam3},
\cite{bam4}, \cite{bam5}, \cite{bam6}). In real application, when
robot move, the joints are properly described by almost periodic
solutions of a dynamic neural network. For this reason, it is very
important to study almost periodic solutions of neural networks
models. In \cite{bam0}, the authors investigated the almost periodic
solution of
\begin{equation*} \left\{
\begin{array}{ll}
\dot{x}_{i}(t)=-\alpha_{i}(t)x_{i}(t)+\sum\limits_{j=1}^{m}b_{ij}(t)(t)f_{j}(y_{j}(t-\tau))+I_{i}(t), \\
\dot{y}_{j}(t)=-c_{j}(t)y_{j}(t)+\sum\limits_{j=1}^{n}\bar{b}_{ij}(t)(t)f_{j}(x_{j}(t-\sigma))+J_{j}(t),\,\
t\geq 0\\
x_{i}(t)=\phi_{i}(t), \,\ y_{j}(t)=\psi_{j}(t), \,\
t\in[-\tau^{*},0], \\ i=1,...,n \,\ j=1,...,m, \ \ n,m\in
\mathbb{Z}_{+}. \text{}
\end{array}\right.
\end{equation*}
The generalized high-order BAM neural network with mixed delays,
defined as follows, has faster convergence rate, higher fault
tolerance, and stronger approximation property. The problem of
existence and global exponential stability of periodic solution for
high-order discrete-time BAM neural networks has been studied in
\cite{highbam1}. In fact, the study of the existence periodic
solutions, as well as its numerous generalizations to almost
periodic solutions, pseudo almost periodic solutions, weighted
pseudo almost periodic solutions, and so forth, is one of the most
attracting topics in the qualitative theory of differential
equations due both to its mathematical interest as well as to their
applications in various areas of applied science. The authors in
\cite{highbam} propose some several sufficient conditions for
ensuring existence, global attractivity and global asymptotic
stability of the periodic solution for the higher-order
bidirectional associative memory neural networks with periodic
coefficients and delays by using the continuation theorem of
Mawhin's coincidence degree theory, the Lyapunov functional and the
non-singular $M$-matrix:
\begin{equation*} \left\{
\begin{array}{ll}
\dot{x}_{i}(t)=-\alpha_{i}x_{i}(t)+\sum\limits_{j=1}^{m}b_{ij}(t)f_{j}(y_{j}(t-\tau))
\\+\sum\limits_{j=1}^{m}\sum\limits_{l=1}^{m}c_{ijl}(t)f_{j}(y_{j}(t-\tau))f_{l}(y_{l}(t-\tau))
+I_{i}(t),
\\
\dot{y}_{j}(t)=-c_{j}y_{j}(t)+\sum\limits_{j=1}^{n}\bar{b}_{ij}(t)f_{j}(x_{j}(t-\sigma))
\\+\sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}\bar{c}_{ijl}(t)g_{i}(x_{i}(t-\sigma))g_{l}(x_{l}(t-\sigma))
+J_{j}(t),\,\ t\geq 0.
\end{array}\right.
\end{equation*}
With initial condition
\begin{equation*} \left\{
\begin{array}{ll}
x_{i}(t)=\phi_{i}(t), \,\ y_{j}(t)=\psi_{j}(t), \,\
t\in[-\tau^{*},0],\\ i=1,...,n \,\ j=1,...,m, \ \ n,m\in
\mathbb{Z}_{+}. \text{}
\end{array}\right.
\end{equation*}
Furthermore, it has been reported that if the parameters and time
delays are appropriately chosen, the delayed high-order BAM neural
network can exhibit complicated behaviors even with strange chaotic
attractors. Based on the aforementioned arguments, the study of the
high-order BAM neural network with mixed delays and its analogous
equations have attracted worldwide interest (see \cite{bam6},
\cite{ada1}, \cite{ada2}, \cite{ada3}, \cite{ada4}, \cite{highbam1},
\cite{highbam}, \cite{leak1}, \cite{leak2}, \cite{leak3}). In fact,
it is important that systems contain some information about the
derivative of the past state to further describe the dynamics for
such complex neural reactions. In real world, the mixed time-varying
delays and leakage delay should be taken into account when modeling
realistic neural networks (see \cite{adnene+ahmed},
\cite{adnene+jinde}, \cite{maas}).
As a continuation of our previous published results, we shall
consider a high-order BAM neural network with mixed delays:
\begin{equation*}
\left\{
\begin{array}{ccc}
x^{\Delta}_{i}\left( t\right) =
-\alpha_{i}(t)x_{i}\left(t-\eta_{i}(t)\right)
+\sum\limits_{j=1}^{m}D_{ij}\left( t\right) f_{j}\left(x_{j}\left(
t\right) \right) \\+\sum\limits_{j=1}^{m}D_{ij}^{\tau}\left(
t\right) f_{j}\left(x_{j}\left( t -\tau_{ij}(t)\right) \right)
+\sum\limits_{j=1}^{m}\overline{D}_{ij}\left(
t\right)\int_{t-\sigma_{ij}(t)} ^{t}f_{j}\left(x_{j}\left( s\right)
\right)\Delta s \\+\sum\limits_{j=1}^{m}\widetilde{D}_{ij}\left(
t\right)\int_{t-\xi_{ij}(t)} ^{t}f_{j}\left(x_{j}^{\Delta}\left(
s\right) \right)\Delta s+I_{i}\left( t\right)\\
+\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{m}T_{ijk}(t)f_{k}(x_{k}(t-\chi_{k}(t)))f_{j}(x_{j}(t-\chi_{j}(t))),\,\ i=1,...,n, \\
y^{\Delta}_{j}\left( t\right) =
-c_{j}(t)y_{j}\left(t-\varsigma_{j}(t)\right)
+\sum\limits_{i=1}^{n}E_{ij}\left( t\right) f_{j}\left(x_{j}\left(
t\right) \right) \\+\sum\limits_{i=1}^{n}E_{ij}^{\tau}\left(
t\right) f_{j}\left(x_{j}\left( t -\tau_{ij}(t)\right) \right)
+\sum\limits_{i=1}^{n}\overline{E}_{ij}\left(
t\right)\int_{t-\sigma_{ij}(t)} ^{t}f_{j}\left(x_{j}\left( s\right)
\right)\Delta s \\+\sum\limits_{i=1}^{n}\widetilde{E}_{ij}\left(
t\right)\int_{t-\xi_{ij}(t)}^{t}f_{j}\left(x_{j}^{\Delta}\left(
s\right) \right)\Delta s+J_{j}\left( t\right)\\
+\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}\overline{T}_{ijk}(t)f_{k}(x_{k}(t-\chi_{k}(t)))f_{j}(x_{j}(t-\chi_{j}(t)))
,\,\ t\in\mathbb{T}, \,\ j=1,...,m.
\end{array}
\right.
\end{equation*}
To the best of our knowledge, the existence of Stepanov-like
weighted pseudo-almost automorphic solution on time-space scales to
BAM neural networks (BAMs) and high-order BAM neural network
(HOBAMs) with variable coefficients, mixed delays and leakage delays
have not been studied yet. It has been reported that if the
parameters and time delays are appropriately chosen, the delayed
neural networks in leakage term can exhibit complicated behaviors
even with strange chaotic attractors (see \cite{adnene+ahmed},
\cite{adnene+jinde}, \cite{maas}, \cite{leak1}, \cite{leak2},
\cite{leak3}, \cite{leak0}). In addition, the theory of time scales,
which has recently received much attention, was introduced by Hilger
in his PhD thesis in 1988 to unify continuous and discrete analysis
\cite{hilger}.
Our main purpose of this paper is to introduce the Stepanov-like
weighted pseudo almost automorphic functions on time-space scales,
study some of their basic properties and establish the existence,
uniqueness, stability and convergence of Stepanov-like weighted
pseudo almost automorphic solutions of HOBAMs on time-space scales.
we prove new composition theorems for Stepanov-like weighted pseudo
almost automorphic functions on time-space scales.
The remainder of this paper is organized as follows. In Section 2,
we will present the model of HOBAMs. In section 3, we will introduce
some necessary notations, definitions and fundamental properties of
the weighted pseudo-almost automorphic on time-space scales
environment, which will be used in the paper. In Section 4, some
sufficient conditions will be derived ensuring the existence of the
Stepanov-like weighted pseudo-almost automorphic solution on
time-space scales. Section 5 will be devoted to the exponential
stability of the Stepanov-like weighted pseudo-almost automorphic
solution on time-space scales of a HOBAMs model, and the convergence
of all solutions to its unique Stepanov-like weighted pseudo-almost
automorphic solution. At last, one illustrative numerical example
will be given.
\section{Preliminaries and function spaces}
In the following, we introduce some definitions and state some
preliminary results.
\subsection{Time-space scales and delta derivative}
\begin{definition}(\cite{adnene+ahmed})
Let $\mathbb{T}$ be a nonempty closed subset (time scale) of
$\mathbb{R}$. The forward and backward jump operators $\sigma,\rho:
\mathbb{T}\longrightarrow \mathbb{T}$ and the graininess
$\nu:\mathbb{T}\longrightarrow\mathbb{R}_{+}$ are defined,
respectively, by $\sigma(t)=\inf\{s\in\mathbb{T}:s>t\}$,
$\rho(t)=\sup\{s\in\mathbb{T}:s<t\}$ and $\nu(t)=\sigma(t)-t$.
\end{definition}
\begin{lemma}(\cite{adnene+ahmed}, \cite{advance})
Considering that $f,g$ be delta differentiable functions on
$\mathbb{T}$, then:
\begin{description}
\item[(i)] $(\lambda_{1}f+\lambda_{2}g)^{\Delta}=\lambda_{1}f^{\Delta}+\lambda_{2}g^{\Delta}$, for any constants $\lambda_{1},\lambda_{2}$;
\item[(ii)] $(fg)^{\Delta}(t)=f^{\Delta}(t)g(t)+f(\sigma(t))g^{\Delta}(t)=f(t)g^{\Delta}(t)+f^{\Delta}(t)g(\sigma(t))$;
\item[(iii)] If $f$ and $f^{\Delta}$ are continuous, then $\left(\int_{a}^{t}f(t,s)\Delta s\right)^{\Delta}=f(\sigma(t),t)+\int_{a}^{t}f(t,s)\Delta s$.
\end{description}
\end{lemma}
\begin{lemma}(\cite{adnene+ahmed}, \cite{advance}) Assume that $p,q:\mathbb{T}\longrightarrow\mathbb{R}$ are two regressive functions, then
\begin{description}
\item[(i)] $e_{0}(t,s)\equiv1$ and $e_{p}(t,t)\equiv1$;
\item[(ii)] $e_{p}(t,s)=\frac{1}{e_{p}(s,t)}=e_{\ominus p}(s,t)$;
\item[(iii)] $e_{p}(t,s)e_{p}(s,r)=e_{p}(t,r)$;
\item[(iv)] $\left(e_{p}(t,s)\right)^{\Delta}=p(t)e_{p}(t,s)$.
\end{description}
\end{lemma}
\begin{lemma}(\cite{adnene+ahmed}, \cite{advance})
Assume that $p(t)\geq0$ for $t\geq s$, then $e_{p}(t,s)\geq1$.
\end{lemma}
\begin{definition}(\cite{adnene+ahmed})
A function $p:\mathbb{T}\longrightarrow\mathbb{R}$ is called
regressive provided $1+\mu(t)p(t)\neq0$ for all
$t\in\mathbb{T}^{k}$; $p:\mathbb{T}\longrightarrow\mathbb{R}$ is
called positively provided $1+\mu(t)p(t)>0$ for all
$t\in\mathbb{T}^{k}$. The set of all regressive and rd-continuous
functions $p:\mathbb{T}\longrightarrow\mathbb{R}$ will be denoted by
$\mathcal{R}=\mathcal{R}(\mathbb{T},\mathbb{R})$ and the set of all
positively regressive functions and rd-continuous functions will be
denoted $\mathcal{R}^{+}=\mathcal{R}^{+}(\mathbb{T},\mathbb{R})$.
\end{definition}
\begin{lemma} (\cite{adnene+ahmed}, \cite{advance})
Suppose that $p\in\mathcal{R}^{+}$, then:
\begin{description}
\item[(i)] $e_{p}(t,s)>0$, for all $t,s\in\mathbb{T}$;
\item[(ii)] if $p(t)\leq q(t)$ for all $t\geq s$, $t,s\in\mathbb{T}$, then $e_{p}(t,s)\leq e_{q}(t,s)$ for all $t\geq s$.
\end{description}
\end{lemma}
\begin{lemma} (\cite{adnene+ahmed}, \cite{advance})
If $p\in\mathcal{R}$ and $a,b,c\in \mathbb{T}$, then $[e_{p}(c,.)]^{\Delta}=-p[e_{p}(c,.)]^{\sigma}$,\\
and $\int_{a}^{b}p(t)e_{p}(c,\sigma(t))\Delta
t=e_{p}(c,a)-e_{p}(c,b)$.
\end{lemma}
\begin{lemma} (\cite{adnene+ahmed}, \cite{advance})
Let $a\in\mathbb{T}^{k}$, $b\in\mathbb{T}$ and assume that
$f:\mathbb{T}\times\mathbb{T}^{k}\longrightarrow\mathbb{R}$ is
continuous at $(t,t)$, where $t\in\mathbb{T}^{k}$ with $t>a$.
Additionally assume that $f^{\Delta}(t,.)$ is rd-continuous on
$[a,\sigma(t)]$. Suppose that for each $\epsilon>0$, there exists a
neighborhood $U$ of $\tau\in[a,\sigma(t)]$ such that
\begin{equation*}
|f(\sigma(t),\tau)-f(s,\tau)-f^{\Delta}(t,\tau)(\sigma(t)-s)|\leq\epsilon|\sigma(t)-s|,
\,\ \forall s\in U,
\end{equation*}
where $f^{\Delta}$ denotes the derivative of $f$ with respect to the
first variable. Then
\begin{description}
\item[(i)] $g(t):=\int_{0}^{t}f(t,\tau)\Delta\tau$ implies $g^{\Delta}(t):=\int_{a}^{t}f^{\Delta}(t,\tau)\Delta\tau+f(\sigma(t),t)$;
\item[(ii)] $h(t):=\int_{t}^{b}f(t,\tau)\Delta\tau$ implies $h^{\Delta}(t):=\int_{t}^{b}f^{\Delta}(t,\tau)\Delta\tau-f(\sigma(t),t)$.
\end{description}
\end{lemma}
For more details of time scales and $\Delta$-measurability, one is
referred to read the excellent books (\cite{advance},
\cite{guseinov}).
\subsection{Stepanov-like weighted pseudo-almost automorphic functions on time-space scales}
In the following, we recall some definitions of Stepanov almost
automorphic functions and Stepanov-like weighted pseudo-almost
automorphic functions on time-space scales.
\begin{definition}(\cite{tomorphic2})
A time scale $\mathbb{T}$ is called an almost periodic time scale if
\begin{equation*}
\Pi:=\left\{\tau\in\mathbb{R}: t\pm\tau\in\mathbb{T}, \forall
t\in\mathbb{T}\right\}\neq0.
\end{equation*}
\end{definition}
\begin{definition}
A function $f:\mathbb{T}\longrightarrow\mathbb{R}$ is Bochner
integrable, or integrable for short, if there is a sequence of
functions such that $f_{n}(t)\longrightarrow f(t)$ pointwise a.e. in
$\mathbb{T}$ and
\begin{equation*}
\lim\limits_{n\longrightarrow+\infty}\int_{\mathbb{T}}\|f(s)-f_{n}(s)\|\Delta
s=0,
\end{equation*}
and the integral of $f$ is defined by
\begin{equation*}
\int_{\mathbb{T}}f(s)\Delta
s=\lim\limits_{n\longrightarrow+\infty}f_{n}(s)\Delta s,
\end{equation*}
where the limit exists strongly in $\mathbb{R}$.
\end{definition}
\begin{definition}
Let $E\subset\mathbb{T}$ and $f:\mathbb{T}\longrightarrow\mathbb{R}$
be a strongly $\Delta$-measurable function. If, for a given $p$,
$1\leq p<\infty$, $f$ satisfies
\begin{equation*}
\int_{K}\|f(s)\|^{p}\Delta s<+\infty,
\end{equation*}
where $K$ is a compact subset of $E$, then $f$ is called
$p$-integrable in the Bochner sense. The set of all such functions
is denoted by $L_{loc}^{p}(\mathbb{T},\mathbb{R})$.
\end{definition}
From now on, for $a,b\in\mathbb{R}$ and $a\leq b$, we denote
$a^{*}=\inf\{s\in\mathbb{T}, s\geq a\}$,
$b^{*}=\inf\{s\in\mathbb{T}, s\geq b\}$ and for integrable function
$f$, we denote $\int_{a}^{b}f(s)\Delta
s=\int_{a^{*}}^{b^{*}}f(s)\Delta s$. Obviously,
$a^{*},b^{*}\in\mathbb{T}$. If $a\in\mathbb{T}$, then $a^{*}=a$; If
$b\in\mathbb{T}$, then $b^{*}=b$.
Throughout the rest of paper we fix $p$, $1\leq p<\infty$. We say
that a function $f\in L_{loc}^{p}(\mathbb{T},\mathbb{R})$ is
$p$-Stepanov bounded ($S_{l}^{p}$-bounded) if
\begin{equation*}
\|f\|_{S_{l}^{p}}=\sup\limits_{t\in\mathbb{T}}\left(\frac{1}{l}\int_{t}^{t+l}\|f(s)\|^{p}\Delta
s\right)^{\frac{1}{p}}<+\infty,
\end{equation*}
where $l>0$ is a constant.
We denote by $L_{S}^{p}$ the set of all $S_{l}^{p}$-bounded
functions from $\mathbb{T}$ into $\mathbb{R}$.
\begin{definition}
Let $\mathbb{T}$ be an almost periodic time scale. A function
$f(t):\mathbb{T}\longrightarrow \mathbb{R}^{n}$ is said to be
$S^{p}$-almost automorphic, if for any sequence
$\{s_{n}\}_{n=1}^{\infty}\subset\Pi$, there is a subsequence
$\{\tau_{n}\}_{n=1}^{\infty}\subset\{s_{n}\}_{n=1}^{\infty}$ such
that
\begin{equation*}
\|g(t)-f(t+\tau_{n})\|_{S_{l}^{p}}\longrightarrow0, \,\ \text{as}
\,\ n\longrightarrow+\infty,
\end{equation*}
is well defined for each $t\in\Pi$ and
\begin{equation*}
\|g(t-\tau_{n})-f(t)\|_{S_{l}^{p}}\longrightarrow0, \,\ \text{as}
\,\ n\longrightarrow+\infty,
\end{equation*}
for each $t\in\Pi$. Denote by $S^{p}AA(\mathbb{T},\mathbb{R}^{n})$
the set of all such functions.
\end{definition}
Let
\begin{equation*}
S^{p}AA(\mathbb{T})=\left\{f\in
S^{p}C_{rd}(\mathbb{T},\mathbb{R}^{n}): f \,\ \text{is Stepanov
almost automorphic}\right\}
\end{equation*}
and $S^{p}BC(\mathbb{T},\mathbb{R}^{n})$ denote the space of all
bounded continuous functions, in the Stepanov sens, from
$\mathbb{T}$ to $\mathbb{R}^{n}$.
Let $\mathcal{U}$ be the set of all functions
$\nu:\mathbb{T}\longrightarrow(0,+\infty)$ which are positive and
locally $\Delta$-integrable over $\mathbb{T}$. For given
$r\in(0,+\infty)\cap \Pi$, set \begin{equation}\label{poid}
m(r,\nu,t_{0}):=\int_{Q_{r}}\nu(s)\Delta s, \,\ \text{for each} \,\
\nu\in\mathcal{U},
\end{equation}
where $Q_{r}:=[t_{0}-r,t_{0}+r]_{\mathbb{T}}$
($t_{0}=\min\{[0,\infty)_{\mathbb{T}}\}$). If $\nu(s)=1$ for each
$\nu\in\mathbb{T}$, then
$\lim\limits_{t\longrightarrow\infty}\nu(Q_{r})=\infty$.
Consequently, we define the space of weights $\mathcal{U}_{\infty}$
by
\begin{equation*}
\mathcal{U}_{\infty}:=\left\{\nu\in\mathcal{U}: \,\
\lim\limits_{r\longrightarrow+\infty}m(r,\nu,t_{0})=+\infty\right\}.
\end{equation*}
In addition to the aforesaid section, we define the set of weights
$\mathcal{U}_{B}$ by
\begin{equation*}
\mathcal{U}_{B}:=\left\{\nu\in\mathcal{U}_{\infty}: \,\ \nu \,\
\text{is bounded in the Stepanov sens and }\,\
\inf\limits_{s\in\mathbb{T}}\nu(s)>0\right\}.
\end{equation*}
It is clear that $\mathcal{U}_{B}\subset
\mathcal{U}_{\infty}\subset\mathcal{U}$. Let
$S^{p}BCU^{(0)}(\mathbb{T},\mathbb{R}^{n})$ denote the space of all
bounded uniformly continuous functions from $\mathbb{T}$ to
$\mathbb{R}^{n}$,
\begin{multline*}
\phantom{+++++}S^{p}AA^{(0)}(\mathbb{T})=S^{p}AA^{(0)}(\mathbb{T},\mathbb{R}^{n})=\left\{f\in
S^{p}BCU(\mathbb{T},\mathbb{R}^{n}):\right.\\\left. f \,\ \text{is
Stepanov almost automorphic}\right\}\phantom{+++++}
\end{multline*}
and define for $t_{0}\in\mathbb{T}, r\in\Pi$, the class of functions
$WPAA_{0}(\mathbb{T},\nu,t_{0})$ as follows:
\begin{multline*}
WPAA_{0}(\mathbb{T},\nu,t_{0})=\left\{f\in S^{p}BCU(\mathbb{T},\mathbb{R}^{n}): f \,\ \text{is delta measurable such that}\right.\\
\left.\lim\limits_{r\longrightarrow+\infty}\frac{1}{m(r,\nu,t_{0})}\int_{t_{0}-r}^{t_{0}+r}|f(s)|\nu(s)\Delta
s=0 \right\}.
\end{multline*}
We are now ready to introduce the sets $S^{p}WPAA(\mathbb{T},\nu)$
of Stepanov-like weighted pseudo-almost automorphic on time-space
scales functions:
\begin{definition}\label{defwpap}
A function $f\in S^{p}C_{rd}(\mathbb{T},\mathbb{R}^{n})$ is called
Stepanov-like weighted pseudo almost automorphic on time-space
scales if $f=g+\phi$, where $g\in S^{p}AA(\mathbb{T})$ and $\phi\in
WPAP_{0}(\mathbb{T},\nu)$. Denote by $S^{p}WPAA(\mathbb{T})$, the
set of Stepanov-like weighted pseudo-almost automorphic on
time-space scales functions.
\end{definition}
\subsection{Results on composition theorems}
By Definition \ref{defwpap}, one can easily show that
\begin{lemma}
Let $\phi\in S^{p}BC_{rd}(\mathbb{T},\mathbb{R}^{n})$, then $\phi\in
WPAA_{0}(\mathbb{T},\nu)$, where $\nu\in\mathcal{U}_{B}$ if and only
if, for every $\epsilon>0$,
\begin{equation*}
\lim\limits_{r\longrightarrow+\infty}\frac{1}{m(r,\nu,t_{0})}\nu_{\Delta}(M_{r,\epsilon,t_{0}}(\phi))=0,
\end{equation*}
where $r\in\Pi$ and
$M_{r,\epsilon,t_{0}}(\phi):=\left\{t\in[t_{0}-r,t_{0}+r]_{\mathbb{T}}:
\,\ \|\phi(t)\|\geq\epsilon\right\}$.
\end{lemma}
Proof. The demonstration is similar to the proof of Lemma 3.2 in
\cite{tomorphic2}.
\begin{lemma}
$WPAA_{0}(\mathbb{T},\nu)$ is a translation invariant set of
$S^{p}BC_{rd}(\mathbb{T},\mathbb{R}^{n})$ with respect to $\Pi$ if
$\nu\in\mathcal{U}_{B}$, i.e. for any $s\in\Pi$, one has
\begin{equation*}
\phi(t+s):=\theta_{s}\phi\in WPAA_{0}(\mathbb{T},\nu)\,\ \text{if}
\,\ \nu\in\mathcal{U}_{B}.
\end{equation*}
\end{lemma}
Proof. Similar to proof of Lemma 3.3 in \cite{tomorphic2}.
\begin{lemma}
Let $\phi\in S^{p}AA(\mathbb{T},\mathbb{R}^{n})$, then the range of
$\phi$, $\phi(\mathbb{T})$ is a relatively compact subset
$\mathbb{R}^{n}$.
\end{lemma}
Proof. Similar to proof of Lemma 3.3 in \cite{tomorphic2}.
\begin{lemma}\label{lem5}
If $f=g+\phi$ with $g\in S^{p}AA(\mathbb{T},\mathbb{R}^{n})$ and
$\phi\in WPAA_{0}(\mathbb{T},\nu)$, where $\nu\in\mathcal{U}_{B}$,
then $g(\mathbb{T})\subset \overline{f(\mathbb{T})}$.
\end{lemma}
Proof. The demonstration is similar to the proof of Lemma 3.5 in
\cite{tomorphic2}.
\begin{lemma}
The decomposition of a Stepanov-like weighted pseudo-almost
automorphic on time-space scales function according to
$S^{p}AA\oplus WPAA_{0}$ is unique for any $\nu\in\mathcal{U}_{B}$.
\end{lemma}
Proof. Assume that $f_{1}=g_{1}+\phi_{1}$ and
$f_{2}=g_{2}+\phi_{2}$. Then $(g_{1}-g_{2})+(\phi_{1}-\phi_{2})=0$.
Since $g_{1}-g_{2}\in S^{p}AA(\mathbb{T},\mathbb{R}^{n})$ , and
$\phi_{1}-\phi_{2}\in WPAA_{0}$ in view of Lemma \ref{lem5}, we
deduce that $g_{1}-g_{2}=0$. Consequently $\phi_{1}-\phi_{2}=0$,
i.e. $\phi_{1}=\phi_{2}$. This completes the proof.
\begin{lemma}
For $\nu\in\mathcal{U}_{B}$,
$(S^{p}WPAA(\mathbb{T},\nu),\|.\|_{S_{l}^{p}})$ is a Banach space.
\end{lemma}
Proof. Assume that $\{f_{n}\}_{n\in\mathbb{N}}$ is a Cauchy sequence
in $S^{p}WPAA(\mathbb{T},\nu)$. We can write uniquely
$f_{n}=g_{n}+\phi_{n}$. Using Lemma \ref{lem5}, we see that
$\|g_{p}-g_{q}\|\leq\|f_{p}-f_{q}\|_{\infty}$, from which we deduce
that $\{g_{n}\}_{n\in\mathbb{N}}$ is a Cauchy sequence in
$AA(\mathbb{T},\mathbb{R}^{n})$. Hence, $\phi_{n}=f_{n}-g_{n}$ is a
Cauchy sequence in $WPAA_{0}(\mathbb{T},\nu)$. We deduce that
$g_{n}\longrightarrow g\in AA(\mathbb{T},\mathbb{R}^{n})$,
$\phi_{n}\longrightarrow \phi\in WPAA(\mathbb{T},\nu)$ and finally
$f_{n}\longrightarrow g+\phi\in S^{p}WPAA(\mathbb{T},\nu)$. This
complete the proof.
\begin{definition}(\cite{tomorphic2})
Let $\nu_{1},\nu_{2}\in\mathcal{U}_{\infty}$. One says that
$\nu_{1}$ is equivalent to $\nu_{2}$, written $\nu_{1}\sim \nu_{2}$
if $\nu_{1}/\nu_{2}\in \mathcal{U}_{B}$.
\end{definition}
\begin{lemma}
Let $\nu_{1},\nu_{2}\in\mathcal{U}_{\infty}$. If $\nu_{1}\sim
\nu_{2}$, then
$S^{p}WPAA(\mathbb{T},\nu_{1})=S^{p}WPAA(\mathbb{T},\nu_{2})$.
\end{lemma}
Proof. The demonstration is similar to the proof of Theorem 3.8 in
\cite{tomorphic2}.
\begin{lemma}
Let $f=g+\phi\in S^{p}WPAA(\mathbb{T},\nu)$, where
$\nu\in\mathcal{U}_{B}$. Assume that $f$ and $g$ are Lipshitzian in
$x\in\mathbb{R}^{n}$ uniformly in $t\in\mathbb{T}$, then
$f(.,h(.))\in S^{p}WPAA(\mathbb{T},\nu)$ if $h\in
S^{p}WPAA(\mathbb{T},\nu)$.
\end{lemma}
Proof. Similar to proof of Theorem 3.10 in \cite{tomorphic2}.
\begin{lemma}\label{uniform} (\cite{diaganna})
If $f(t)$ is almost automorphic, $F(.)$ is uniformly continuous on
the value field of $f(t)$, then $F\circ f$ is almost automorphic.
\end{lemma}
\begin{lemma}
If $f\in S^{p}C(\mathbb{R},\mathbb{R})$ satisfies the Lipschitz
condition (with $L$ is a constant of Lipschitz), $\varphi\in
S^{p}WPAA(\mathbb{T},\nu)$, $\theta\in
S^{p}C_{rd}^{1}(\mathbb{T},\Pi)$ and
$\eta:=\inf\limits_{t\in\mathbb{T}}\left(1-\theta^{\Delta}(t)\right)>0$,
then $f(\varphi(t-\theta(t)))\in S^{p}WPAA(\mathbb{T},\nu)$.
\end{lemma}
Proof.\\
From Definition \ref{defwpap}, we have
$\varphi=\varphi_{1}+\varphi_{2}$, where $\varphi_{1}\in
AP(\mathbb{T})$ and $\varphi_{2}\in WPAP_{0}(\mathbb{T},\nu,t_{0})$.
Set
\begin{eqnarray*}
E(t)&=& f(\varphi(t-\theta(t)))=f(\varphi_{1}(t-\theta(t)))+\left[f(\varphi_{1}(t-\theta(t))-\varphi_{2}(t-\theta(t)))\right.\\
&-&\left.f(t-\varphi_{1}(t-\theta(t)))\right]=E_{1}(t)+E_{2}(t).
\end{eqnarray*}
Firstly, it follows from Lemma \ref{uniform} that $E_{1}\in
S^{p}AP(\mathbb{T})$. Next, we show that $E_{2}\in
WPAA_{0}(\mathbb{T},\nu,t_{0})$. Since
\begin{eqnarray*}
&&\lim\limits_{r\longrightarrow+\infty}\frac{1}{m(r,\nu,t_{0})}\int_{t_{0}-r}^{t_{0}+r}\left\vert E_{2}(s)\right\vert \nu(s)\Delta s\\
&=& \lim\limits_{r\longrightarrow+\infty}\frac{1}{m(r,\nu,t_{0})}\int_{t_{0}-r}^{t_{0}+r}\left\vert f(\varphi_{1}(t-\theta(t))-\varphi_{2}(t-\theta(t)))\right.\\
&-&\left.f(t-\varphi_{1}(t-\theta(t)))\right\vert\nu(s)\Delta s\\
&\leq&
\lim\limits_{r\longrightarrow+\infty}\frac{L}{m(r,\nu,t_{0})}\int_{t_{0}-r}^{t_{0}+r}\left\vert\varphi_{2}(t-\theta(t)))\right\vert
\nu(s)\Delta s
\end{eqnarray*}
and
\begin{eqnarray*}
0&\leq&\frac{L}{m(r,\nu,t_{0})}\int_{t_{0}-r}^{t_{0}+r}\left\vert\varphi_{2}(t-\theta(t)))\right\vert \nu(s)\Delta s\\
&=& \frac{L}{m(r,\nu,t_{0})}\int_{t_{0}-r-\theta(t_{0}-r)}^{t_{0}+r-\theta(t_{0}+r)}\frac{1}{1-\theta^{\Delta}(s)}\left\vert\varphi_{2}(u))\right\vert \nu(u)\Delta u\\
&\leq&
\frac{1}{\eta}\frac{r+\theta^{+}}{r}\frac{L}{m(r,\nu,t_{0})}\int_{t_{0}-(r+\theta^{+})}^{t_{0}+r+\theta^{+}}\left\vert\varphi_{2}(u))\right\vert
\nu(u)\Delta u=0,
\end{eqnarray*}
$E_{2}\in WPAA_{0}(\mathbb{T},\nu,t_{0})$. Thus $E\in
S^{p}WPAA(\mathbb{T},\nu)$. The proof is achieved.
\section{Model description and hypotheses}
In this paper, we consider a class of $n$-neuron high-order BAM
neural networks (HOBAMs) with mixed time-varying delays and leakage
delays on time-space scales which are defined in the following
lines:
\begin{equation}\label{eq1}
\left\{
\begin{array}{ccc}
x^{\Delta}_{i}\left( t\right) =
-\alpha_{i}(t)x_{i}\left(t-\eta_{i}(t)\right)
+\sum\limits_{j=1}^{m}D_{ij}\left( t\right) f_{j}\left(y_{j}\left(
t\right) \right) +\sum\limits_{j=1}^{m}D_{ij}^{\tau}\left( t\right)
f_{j}\left(x_{j}\left( t -\tau_{ij}(t)\right) \right)\\
+\sum\limits_{j=1}^{m}\overline{D}_{ij}\left(
t\right)\int_{t-\sigma_{ij}(t)} ^{t}f_{j}\left(y_{j}\left( s\right)
\right)\Delta s +\sum\limits_{j=1}^{m}\widetilde{D}_{ij}\left(
t\right)\int_{t-\xi_{ij}(t)} ^{t}f_{j}\left(y_{j}^{\Delta}\left(
s\right) \right)\Delta s\\
+\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{m}T_{ijk}(t)f_{k}(y_{k}(t-\chi_{k}(t)))f_{j}(y_{j}(t-\chi_{j}(t)))
+I_{i}\left( t\right),\\
y^{\Delta}_{j}\left( t\right) =
-c_{j}(t)y_{j}\left(t-\eta_{j}(t)\right)
+\sum\limits_{i=1}^{n}E_{ij}\left( t\right) f_{j}\left(x_{j}\left(
t\right) \right) +\sum\limits_{i=1}^{n}E_{ij}^{\tau}\left( t\right)
f_{j}\left(x_{j}\left( t -\tau_{ij}(t)\right) \right)\\
+\sum\limits_{i=1}^{n}\overline{E}_{ij}\left(
t\right)\int_{t-\sigma_{ij}(t)} ^{t}f_{j}\left(x_{j}\left( s\right)
\right)\Delta s +\sum\limits_{i=1}^{n}\widetilde{E}_{ij}\left(
t\right)\int_{t-\xi_{ij}(t)}^{t}f_{j}\left(x_{j}^{\Delta}\left(
s\right) \right)\Delta s\\
+\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}\overline{T}_{ijk}(t)f_{k}(x_{k}(t-\chi_{k}(t)))f_{j}(x_{j}(t-\chi_{j}(t)))
+J_{j}\left( t\right),\,\ t\in\mathbb{T},
\end{array}
\right.
\end{equation}
where $i=1,\cdots ,n$ and $j=1,\cdots ,m$; $\mathbb{T}$ is an almost
periodic time scale; $x_{i}(t)$ and $y_{i}(t)$ are the neuron
current activity level of ith neuron in the first layer and the jth
neuron in the second layer respectively at time $t$ ($i=1,\cdots ,n,
\,\ j=1,\cdots ,m$); $\alpha_{i}(t), c_{j}(t)$ are the time variable
of the neuron $i$ in the first layer and the neuron $j$ in the
second neuron respectively; $f_{i}(x_{i}(t))$ and $f_{j}(x_{j}(t))$
are the output of neurons; $I_{i}(t)$ and $J_{j}(t)$ denote the
external inputs on the ith neuron at time $t$ for the first layer
and the jth neuron at the second layer at time $t$;
\begin{multline*} \left.
\begin{array}{ll}
t\longmapsto D_{ij}(t)\\
t\longmapsto D_{ij}^{\tau}(t)\\
t\longmapsto \overline{D}_{ij}(t)\\
t\longmapsto \widetilde{D}_{ij}(t)\\
t\longmapsto E_{ij}(t)\\
t\longmapsto E_{ij}^{\tau}(t)\\
t\longmapsto \overline{E}_{ij}(t)\\
t\longmapsto \widetilde{E}_{ij}(t)
\end{array}\right\} \text{represent the connection weights and the synaptic weights }\\
\text{of delayed feedback between the $i$th neuron and the $j$th
neuron respectively};
\end{multline*}
for all $i,k=1,...n, \,\ j=1,...,m \,\ t\longmapsto T_{ijk}(t)$ and $t\longmapsto\overline{T}_{ijk}(t)$ are the second-order connection weights of delayed feedback;\\
$t\longmapsto I_{i}(t)$, $t\longmapsto J_{i}(t)$ denote the external
inputs on the $i$th neuron at time $t$; $t\longmapsto \eta_{i}(t)$
and $t\longmapsto \varsigma_{i}(t)$ are leakage delays and satisfy
$t-\eta_{i}(t)\in\mathbb{T}$, $t-\varsigma_{i}(t)\in\mathbb{T}$ for
$t\in\mathbb{T}$; $\tau_{ij}(t)$, $\sigma_{ij}(t)$, $\chi_{k}(t)$
and $\xi_{ij}(t)$ are transmission delays and satisfy
$t-\tau_{ij}(t)\in\mathbb{T}$, $t-\sigma_{ij}(t)\in\mathbb{T}$,
$t-\chi_{k}(t)\in\mathbb{T}$ and $t-\xi_{ij}(t)\in\mathbb{T}$ for
$t\in\mathbb{T}$.
For convenience, we introduce the following notations:
\begin{multline*}
\alpha_{i}^{+}=\sup\limits_{t\in\mathbb{T}}|\alpha_{i}(t)|, \,\ \alpha_{i}^{-}=\inf\limits_{t\in\mathbb{T}}|\alpha_{i}(t)|>0,\,\ c_{i}^{+}=\sup\limits_{t\in\mathbb{T}}|c_{i}(t)|,\,\ c_{i}^{-}=\inf\limits_{t\in\mathbb{T}}|c_{i}(t)|>0,\\
\eta_{i}^{+}=\sup\limits_{t\in\mathbb{T}}|\eta_{i}(t)|, \,\
\varsigma_{i}^{+}=\sup\limits_{t\in\mathbb{T}}|\varsigma_{i}(t)|,\,\
D_{i}^{+}=\sup\limits_{t\in\mathbb{T}}|D_{i}(t)|,\,\
(D_{i}^{\tau})^{+}=\sup\limits_{t\in\mathbb{T}}|D_{i}^{\tau}(t)|,\\
D_{ij}^{+}=\sup\limits_{t\in\mathbb{T}}|D_{ij}(t)|,\,\
(D_{ij}^{\tau})^{+}=\sup\limits_{t\in\mathbb{T}}|D_{ij}^{\tau}(t)|,\,\
\overline{D}_{ij}^{+}=\sup\limits_{t\in\mathbb{T}}|\overline{D}_{ij}(t)|,\\
(\widetilde{D}_{ij})^{+}=\sup\limits_{t\in\mathbb{T}}|\widetilde{D}_{ij}(t)|,\,\
E_{i}^{+}=\sup\limits_{t\in\mathbb{T}}|E_{i}(t)|,\,\
(E_{i}^{\tau})^{+}=\sup\limits_{t\in\mathbb{T}}|E_{i}^{\tau}(t)|,\,\
E_{ij}^{+}=\sup\limits_{t\in\mathbb{T}}|E_{ij}(t)|,\\
(E_{ij}^{\tau})^{+}=\sup\limits_{t\in\mathbb{T}}|E_{ij}^{\tau}(t)|,\,\
\overline{E}_{ij}^{+}=\sup\limits_{t\in\mathbb{T}}|\overline{E}_{ij}(t)|,\,\
(\widetilde{E}_{ij})^{+}=\sup\limits_{t\in\mathbb{T}}|\widetilde{E}_{ij}(t)|,\\
T_{ijk}^{+}=\sup\limits_{t\in\mathbb{T}}|T_{ijk}(t)|,\,\
\overline{T}_{ijk}^{+}=\sup\limits_{t\in\mathbb{T}}|\overline{T}_{ijk}(t)|,\,\
\tau_{ij}^{+}=\sup\limits_{t\in\mathbb{T}}|\tau_{ij}(t)|,\\
\sigma_{ij}^{+}=\sup\limits_{t\in\mathbb{T}}|\sigma_{ij}(t)|,\,\
\xi_{ij}^{+}=\sup\limits_{t\in\mathbb{T}}|\xi_{ij}(t)|,\,\
\chi_{j}^{+}=\sup\limits_{t\in\mathbb{T}}|\chi_{j}(t)|,\,\
i,=1,...,n, \,\ j=1,...,m.
\end{multline*}
We denote that $[a,b]_{\mathbb{T}}=\{t, t\in[a,b]\cap \mathbb{T}\}$.
The initial conditions associated with system (\ref{eq1}), are of
the form:
\begin{equation*}
x_{i}\left( s\right) = \varphi_{i}\left(s\right),\,\
y_{j}\left(s\right)= \phi_{j}\left( s\right),\,\ s\in [-\theta
,0]_{\mathbb{T}} ,1\leq i\leq n,\,\ 1\leq j\leq m,
\end{equation*}
where $\varphi_{i}(.)$ and $\phi_{i}(.)$ are the real-valued bounded
$\Delta$-differentiable functions defined on
$[-\theta,0]_{\mathbb{T}}$,
\begin{multline*}
\theta=\max\{\eta,\tau,\chi,\sigma,\xi,\varsigma\}, \,\
\eta=\max\limits_{1\leq i\leq n}\eta_{i}^{+}, \,\
\tau=\max\limits_{1\leq i\leq n, 1\leq j\leq m}\tau_{ij}^{+}, \,\
\chi=\max\limits_{1\leq j\leq m}\chi_{j}^{+},\\
\sigma=\max\limits_{1\leq i\leq n, 1\leq j\leq m}\sigma_{ij}^{+},\,\
\xi=\max\limits_{1\leq i\leq n, 1\leq j\leq m}\xi_{ij}^{+}\,\
\text{and} \,\ \varsigma=\max\limits_{1\leq i\leq n, 1\leq j\leq
m}\varsigma_{ij}^{+}.
\end{multline*}
\begin{remark}
This is the first time to study the Stepanov-like weighted
pseudo-almost automorphic solutions of system (\ref{eq1}) for the
both cases: continuous and discrete. Furthermore, there is no result
about automorphic, Stepanov almost automorphic and Stepanov-like
weighted pseudo-almost automorphic solutions of networks
(\ref{eq1}).
\end{remark}
Let us list some assumptions that will be used throughout the rest
of this paper.
\begin{description}
\item[($H_{1}$)]
For all $1\leq i,j\leq n,$, the functions $\alpha_{i}\left(\cdot
\right), c_{j}\left(\cdot \right)\in\mathcal{R}_{\nu}^{+}$ and
$D_{ij}(.)$, $D^{\tau}_{ij}(.)$ $\overline{D}_{ij}(.)$,
$\widetilde{D}_{ij}(.)$, $T_{ijk}(.)$, $E_{ij}(.)$,
$E^{\tau}_{ij}(.)$ $\overline{E}_{ij}(.)$, $\widetilde{E}_{ij}(.)$,
$\overline{T}_{ijk}(.)$, $\eta_{i}(.)$, $\varsigma_{j}(.)$,
$\tau_{ij}(.)$, $\chi_{j}(.)$, $\sigma_{ij}(.)$, $\xi_{ij}(.)$,
$I_{i}(.)$, $J_{j}(.)$ are $ld$-continuous Stepanov-like weighted
pseudo-almost automorphic
functions for $i=1,...,n$, $j=1,...,m$.\\
\item[($H_{2}$)]
The functions $f_{j}\left(\cdot\right)$ are $\Delta$-differential
and satisfy the Lipschitz condition, i.e., there are constants
$L_{j}>0$ such that for all $x,y\in\mathbb{R}$, and for all $1\leq
j\leq \max\{n,m\},$ one has $ \left\vert f_{j}\left( x\right)
-f_{j}\left( y\right) \right\vert \leq L_{j}\left\vert
x-y\right\vert. $
\item[($H_{3}$)]
\begin{multline*}
\phantom{++++++}\max\limits_{1\leq i\leq
n}\left\{\frac{M_{i}}{\alpha_{i}^{-}},\left(1+\frac{\alpha_{i}^{+}}{\alpha_{i}^{-}}\right)M_{i},\frac{N_{i}}{c_{i}^{-}},
\left(1+\frac{c_{i}^{+}}{c_{i}^{-}}\right)N_{i}\right\}\leq r \\
\text{and} \,\ \max\limits_{1\leq i\leq
n}\left\{\frac{\overline{M}_{i}}{\alpha_{i}^{-}},\left(1+\frac{\alpha_{i}^{+}}{\alpha_{i}^{-}}\right)\overline{M}_{i},
\frac{\overline{N}_{i}}{c_{i}^{-}},\left(1+\frac{c_{i}^{+}}{c_{i}^{-}}\right)\overline{N}_{i}\right\}\leq
1,\phantom{++}
\end{multline*}
where $r$ is a constant, for $i=1,...,n$ and $j=1,...,m$,
\begin{eqnarray*}
M_{i}&=&\alpha_{i}^{+}\eta_{i}^{+}r+\sum\limits_{j=1}^{m}\left(D_{ij}^{+}+(D_{ij}^{\tau})^{+}+\overline{D}_{ij}^{+}\sigma_{ij}^{+}
+\widetilde{D}_{ij}^{+}\xi_{ij}^{+}\right)(L_{j}r+|f_{j}(0)|)\\
&+&\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{m}T_{ijk}^{+}\left(L_{k}r+\left\vert f_{k}(0)\right\vert\right)\left(L_{j}r+\left\vert f_{j}(0)\right\vert\right)+I_{i}^{+},\\
\overline{M}_{i}&=&\alpha_{i}^{+}\eta_{i}^{+}+\sum\limits_{j=1}^{m}\left(D_{ij}^{+}+(D_{ij}^{\tau})^{+}
+\overline{D}_{ij}^{+}\sigma_{ij}^{+}+\widetilde{D}_{ij}^{+}\xi_{ij}^{+}\right)L_{j}\\
&+&\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{m}(T_{ijk}^{+}+T_{ikj}^{+})(L_{k}r+|f_{k}(0)|),\\
N_{j}&=&c_{j}^{+}\varsigma_{j}^{+}r+\sum\limits_{i=1}^{n}\left(E_{ij}^{+}+(E_{ij}^{\tau})^{+}+\overline{E}_{ij}^{+}\sigma_{ij}^{+}
+\widetilde{E}_{ij}^{+}\xi_{ij}^{+}\right)(L_{i}r+|f_{i}(0)|)\\
&+&\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}\overline{T}_{ijk}^{+}\left(L_{k}r+\left\vert f_{k}(0)\right\vert\right)\left(L_{i}r+\left\vert f_{i}(0)\right\vert\right)+J_{j}^{+},\\
\overline{N}_{j}&=&
c_{j}^{+}\varsigma_{j}^{+}+\sum\limits_{i=1}^{n}\left(E_{ij}^{+}+(E_{ij}^{\tau})^{+}
+\overline{E}_{ij}^{+}\sigma_{ij}^{+}+\widetilde{E}_{ij}^{+}\xi_{ij}^{+}\right)L_{i}\\
&+&\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}(\overline{T}_{ijk}^{+}+\overline{T}_{ikj}^{+})(L_{k}r+|f_{k}(0)|).
\end{eqnarray*}
\item[($H_{4}$)]
$\inf\limits_{t\in\mathbb{T}}\left(1-\sigma_{ij}^{\nabla}(t)\right)>0$,
$\inf\limits_{t\in\mathbb{T}}\left(1-\xi_{ij}^{\nabla}(t)\right)>0$,
and for all $s\in\Pi$,
\begin{equation*}
\limsup\limits_{|t|\longrightarrow+\infty}\frac{\nu(t+s)}{\nu(t)}<\infty.
\end{equation*}
\end{description}
\begin{remark}
The bidirectional associative memory (BAM) neural networks with
mixed time-varying delays and leakage time-varying delays on
time-space scales is investigated in \cite{maas}. Some sufficient
conditions are given for the existence, convergence and the global
exponential stability of the weighted pseudo almost-periodic
solution. However, Theorem 4.1, Theorem 5.1 and Theorem 6.1 proposed
in \cite{maas} are not applicable for the HOBAMs with mixed
time-varying delays in the leakage terms.
\end{remark}
\section{The existence of Stepanov-like weighted pseudo-almost automorphic on time-space scales solutions}
In this section, based on Banach's fixed point theorem and the
theory of calculus on time-space scales, we will present a new
condition for the existence and uniqueness of weighted pseudo-almost
automorphic on time-space scales solutions of (\ref{eq1}).
Additionally, we will show a result about the delta derivative of
the only Stepanov-like weighted pseudo-almost automorphic on
time-space scales solution of system (\ref{eq1}).
Let
\begin{equation*}
\mathbb{B}=\{(\varphi_{1},\varphi_{2},...,\varphi_{n},\phi_{1},\phi_{2},...,\phi_{m})^{T}:\varphi_{i},\phi_{j}\in
C^{1}(\mathbb{T},\mathbb{R}),\,\ i=1,...,n, \,\ j=1,...,m\}.
\end{equation*}
with the norm $
\|\psi\|_{\mathbb{B}}=\sup\limits_{t\in\mathbb{T}}\max\limits_{i=1,...,n\,\
j=1,...,m}\{|\varphi_{i}(t)|,|\phi_{j}(t)|,|\varphi^{\Delta}_{i}(t)|,|\phi^{\Delta}_{j}(t)|\},
$
then $(\mathbb{B},\|\psi\|_{\mathbb{B}})$ is a Banach space.\\
For every
$\psi=(\varphi_{1},...\varphi_{n},\phi_{1},...,\phi_{m})\in\mathbb{B}$,
we consider the following system
\begin{equation}
x_{i}^{\Delta}(t)=-\alpha_{i}(t)x_{i}(t)+F_{i}(t,\varphi_{i}),\,\
y_{j}^{\Delta}(t)=-c_{j}(t)y_{j}(t)+G_{j}(t,\phi_{i}), \,\
t\in\mathbb{T},
\end{equation}
where, for $i=1,...,n$ and $j=1,...,m$
\begin{eqnarray*}
F_{i}(t,\varphi_{i}(t))&=&\alpha_{i}(t)\int_{t-\eta_{i}(t)}^{t}\varphi_{i}^{\Delta}(s)\Delta s +\sum\limits_{j=1}^{m}D_{ij}\left( t\right) f_{j}\left(\phi_{j}\left( t\right) \right)\\
&+&\sum\limits_{j=1}^{m}D_{ij}^{\tau}\left(t\right)
f_{j}\left(\phi_{j}\left( t -\tau_{ij}(t)\right) \right)
+\sum\limits_{j=1}^{m}\overline{D}_{ij}\left(t\right)\int_{t-\sigma_{ij}(t)}^{t}f_{j}\left(\phi_{j}\left( s\right) \right)\Delta s\\
&+&\sum\limits_{j=1}^{m}\widetilde{D}_{ij}\left( t\right)\int_{t-\xi_{ij}(t)} ^{t}f_{j}\left(\phi_{j}^{\Delta}\left( s\right) \right)\Delta s\\
&+&\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{n}T_{ijk}(t)f_{k}(\phi_{k}(t-\chi_{k}(t)))f_{j}(\phi_{j}(t-\chi_{j}(t)))+I_{i}(t),\\
G_{j}(t,\phi_{j}(t))&=&c_{j}(t)\int_{t-\varsigma_{j}(t)}^{t}\phi_{j}^{\Delta}(s)\Delta
s+\sum\limits_{i=1}^{n}E_{ij}\left( t\right) f_{i}\left(\varphi_{i}\left( t\right) \right)\\
&+&\sum\limits_{i=1}^{n}E_{ij}^{\tau}\left(t\right)
f_{i}\left(\varphi_{i}\left( t -\tau_{ij}(t)\right) \right)
+\sum\limits_{i=1}^{n}\overline{E}_{ij}\left(t\right)\int_{t-\sigma_{ij}(t)}^{t}f_{i}\left(\varphi_{i}\left( s\right) \right)\Delta s\\
&+&\sum\limits_{i=1}^{n}\widetilde{E}_{ij}\left( t\right)\int_{t-\xi_{ij}(t)} ^{t}f_{i}\left(\varphi_{i}^{\Delta}\left( s\right) \right)\Delta s\\
&+&\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}\overline{T}_{ijk}(t)f_{k}(\varphi_{k}(t-\chi_{k}(t)))f_{i}(\varphi_{i}(t-\chi_{i}(t)))+J_{j}(t).
\end{eqnarray*}
Let
$y_{\psi}(t)=\left(x_{\varphi_{1}}(t),...,x_{\varphi_{n}}(t),y_{\phi_{1}}(t),...,y_{\phi_{m}}(t)\right)^{T}$,
where:
\begin{eqnarray*}
x_{\varphi_{i}}(t)&=&\int_{-\infty}^{t}\hat{e}_{-\alpha_{i}}(t,\sigma(s))F_{i}(t,\varphi_{i}(s))\Delta s,\\
y_{\phi_{j}}(t)&=&\int_{-\infty}^{t}\hat{e}_{-c_{j}}(t,\sigma(s))G_{j}(t,\phi_{j}(s))\Delta
s,
\end{eqnarray*}
\begin{lemma}
Suppose that assumptions $(H_{1})-(H_{4})$ hold. Define the
nonlinear operator $\Gamma:\mathbb{F}\longrightarrow \mathbb{F}$ by
for each $\psi\in WPAA(\mathbb{T},\nu)$
\begin{equation*}
(\Gamma\psi)(t)=y_{\psi}(t), \,\ \psi\in\mathbb{F}.
\end{equation*}
Then $\Gamma$ maps $WPAA(\mathbb{T},\nu)$ into itself.
\end{lemma}
Proof.\\
We show that for any $\psi\in\mathbb{F}$, $\Gamma\psi\in\mathbb{F}$.
\begin{eqnarray*}
\left\vert F_{i}(s,\varphi_{i}(s))\right\vert
&\leq&\alpha_{i}^{+}\eta_{i}^{+}r
+\sum\limits_{j=1}^{n}D_{ij}^{+}\left(L_{j}\left\vert\varphi_{j}\left(s\right)\right\vert+\left\vert f_{j}(0)\right\vert\right)\\
&+&\sum\limits_{j=1}^{n}(D_{ij}^{\tau})^{+}\left(L_{j}\left\vert\varphi_{j}\left(s-\tau_{ij}(s)\right)\right\vert+\left\vert f_{j}(0)\right\vert\right)\\
&+&\sum\limits_{j=1}^{n}\overline{D}_{ij}^{+}\sigma_{ij}^{+}\left(L_{j}r+\left\vert
f_{j}(0)\right\vert\right)+
\sum\limits_{j=1}^{n}\widetilde{D}_{ij}^{+}\xi_{ij}^{+}\left(L_{j}r+\left\vert f_{j}(0)\right\vert\right)\\
&+&
\sum\limits_{j=1}^{n}\sum\limits_{k=1}^{n}T_{ijk}^{+}\left(L_{k}r+\left\vert
f_{k}(0)\right\vert\right)\left(L_{j}r+\left\vert f_{j}(0)\right\vert\right)+I_{i}^{+}\\
&\leq& M_{i}.
\end{eqnarray*}
In a similar way, we have
\begin{equation*}
\left\vert G_{j}(s,\phi_{j}(s))\right\vert\leq N_{j}.
\end{equation*}
Which leads to, for $i=1,...,n$, $j=1,...,m$.
\begin{eqnarray*}
\sup\limits_{t\in\mathbb{T}}\left\vert x_{\varphi_{i}}(t)\right\vert&=&\sup\limits_{t\in\mathbb{T}}\left\vert\int_{-\infty}^{t}\hat{e}_{-\alpha_{i}^{-}}(t,\sigma(s))F_{i}(t,\varphi_{i}(s))\Delta s\right\vert\\
&\leq& \sup\limits_{t\in\mathbb{T}}\int_{-\infty}^{t}\hat{e}_{-\alpha_{i}^{-}}(t,\sigma(s))\left\vert F_{i}(s,\varphi_{i}(s))\right\vert \Delta s\\
&\leq& \frac{M_{i}}{\alpha_{i}^{-}},
\end{eqnarray*}
and
\begin{equation*}
\sup\limits_{t\in\mathbb{T}}\left\vert
y_{\phi_{j}}(t)\right\vert=\sup\limits_{t\in\mathbb{T}}\left\vert\int_{-\infty}^{t}\hat{e}_{-c_{i}}(t,\sigma(s))G_{i}(t,\phi_{i}(s))\Delta
s\right\vert\leq \frac{N_{j}}{c_{j}^{-}}.
\end{equation*}
Otherwise, for $i=1,...,n$, we have
\begin{eqnarray*}
\sup\limits_{t\in\mathbb{T}}\left\vert
x_{\varphi_{i}}^{\Delta}(t)\right\vert
&=&\sup\limits_{t\in\mathbb{T}}\left\vert\left(\int_{-\infty}^{t}\hat{e}_{-\alpha_{i}}(t,\sigma(s))F_{i}(t,\varphi_{i}(s))\Delta s\right)^{\Delta} \right\vert\\
&=&\sup\limits_{t\in\mathbb{T}}\left\vert F_{i}(t,\varphi_{i}(t))-\alpha_{i}(t)\int_{-\infty}^{t}\hat{e}_{-\alpha_{i}}(t,\sigma(s))F_{i}(t,\varphi_{i}(s))\Delta s\right\vert\\
&\leq&\alpha_{i}^{+}\eta_{i}^{+}r+\sum\limits_{j=1}^{m}\left(D_{ij}^{+}+(D_{ij}^{\tau})^{+}+(\overline{D}_{ij})^{+}\sigma_{ij}^{+}
+(\widetilde{D}_{ij})^{+}\xi_{ij}^{+}\right)(L_{j}r+\left\vert f_{j}(0)\right\vert)\\
&+&\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{n}T_{ijk}^{+}\left(L_{k}r+\left\vert f_{k}(0)\right\vert\right)\left(L_{j}r+\left\vert f_{j}(0)\right\vert\right)\\
&+&I_{i}^{+}+\frac{\alpha_{i}^{+}}{\alpha_{i}^{-}}\left(\alpha_{i}^{+}\eta_{i}^{+}r
+\sum\limits_{j=1}^{m}\left(D_{ij}^{+}+(D_{ij}^{\tau})^{+}+(\overline{D}_{ij})^{+}\sigma_{ij}^{+}\right.\right.\\
&+&\left.(\widetilde{D}_{ij})^{+}\xi_{ij}^{+}\right)(L_{j}r+\left\vert
f_{j}(0)\right\vert)+\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{n}T_{ijk}^{+}\left(L_{k}r+\left\vert
f_{k}(0)\right\vert\right)\\
&\times&\left(L_{j}r+\left\vert f_{j}(0)\right\vert\right)
+\left. I_{i}^{+}\right)\\
&=&\left(1+\frac{\alpha_{i}^{+}}{\alpha_{i}^{-}}\right)M_{i}.
\end{eqnarray*}
Similarly,
\begin{equation*}
\sup\limits_{t\in\mathbb{T}}\left\vert
y_{\phi_{j}}^{\Delta}(t)\right\vert\leq
\left(1+\frac{c_{j}^{+}}{c_{j}^{-}}\right)N_{j}.
\end{equation*}
From hypothesis $(H_{3})$, we can obtain
\begin{equation*}
\|\Gamma \psi\|_{\mathbb{B}}\leq r,
\end{equation*}
which implies that operator $\Gamma$ is a self-mapping from $\mathbb{F}$ to $\mathbb{F}$.\\
\begin{theorem}\label{th0}
Let $(H_{1})-(H_{4})$ hold. The system (\ref{eq1}) has a unique
Stepanov-like weighted pseudo-almost automorphic solution in the
region $\mathbb{F}=\{\psi\in\mathbb{B}: \|\psi\|_{\mathbb{B}}\leq
r\}.$
\end{theorem}
Proof.\\
First, for
$\psi=\left(\varphi_{1},...,\varphi_{n},\phi_{1},...,\phi_{m}\right)^{T}$,
$\Omega=\left(u_{1},...,u_{n},
v_{1},...,v_{m}\right)^{T}\in\mathbb{F}$, we have
\begin{eqnarray*}
\sup\limits_{s\in\mathbb{T}}\left\Vert
x_{\varphi_{i}}(s)-x_{u_{i}}(s) \right\Vert
&\leq&\frac{1}{\alpha_{i}^{-}}\left(\alpha_{i}^{+}\eta_{i}^{+}+\sum\limits_{j=1}^{m}\left(D_{ij}^{+}+(D_{ij}^{\tau})^{+}
+(\overline{D}_{ij})^{+}\sigma_{ij}^{+}+(\widetilde{D}_{ij})^{+}\xi_{ij}^{+}\right)L_{j}\right.\\
&+&\left.\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{m}(T_{ijk}^{+}+T_{ikj}^{+})(L_{k}r+|f_{k}(0)|)\right)\|\psi-\Omega\|_{\mathbb{B}}\\
&=&
\frac{\overline{M}_{i}}{\alpha_{i}^{-}}\|\psi-\Omega\|_{\mathbb{B}}.
\end{eqnarray*}
Besides,
\begin{eqnarray*}
\sup\limits_{s\in\mathbb{T}}\left\Vert
x^{\Delta}_{\varphi_{i}}(s)-x^{\Delta}_{u_{i}}(s) \right\Vert
&\leq&\left(\alpha_{i}^{+}\eta_{i}^{+}+\sum\limits_{j=1}^{m}\left(D_{ij}^{+}+(D_{ij}^{\tau})^{+}
+(\overline{D}_{ij})^{+}\sigma_{ij}^{+}+(\widetilde{D}_{ij})^{+}\xi_{ij}^{+}\right)L_{j}\right.\\
&+&\left.\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{m}(T_{ijk}^{+}+T_{ikj}^{+})(L_{k}r+|f_{k}(0)|)\right)\|\psi-\Omega\|_{\mathbb{B}}\\
&+&\frac{\alpha_{i}^{+}}{\alpha_{i}^{-}}\left(\alpha_{i}^{+}\eta_{i}^{+}\left\Vert
\phi_{i}^{\Delta}(s)-u_{i}^{\Delta}(s)\right\Vert+\sum\limits_{j=1}^{m}
D_{ij}^{+}\left\Vert \phi_{i}(t)-u_{i}(t)\right\Vert\right.\\
&+&\sum\limits_{j=1}^{m}(D_{ij}^{\tau})^{+}L_{j}\left\Vert \phi_{j}(s-\tau_{ij}(s))-u_{j}(s-\tau_{ij}(s))\right\Vert\\
&+&\sum\limits_{j=1}^{m}(\overline{D}_{ij})^{+}L_{j}\sigma_{ij}^{+}\left\Vert \phi_{j}(s)-u_{j}(s)\right\Vert\\
&+&\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{m}(T_{ijk}^{+}+T_{ikj}^{+})(L_{k}r+|f_{k}(0)|)\left\Vert\phi_{j}(s)-u_{j}(s)\right\Vert\\
&+&\left.\sum\limits_{j=1}^{m}(\widetilde{D}_{ij})^{+}L_{j}\xi_{ij}^{+}\left\Vert
\phi_{j}^{\Delta}(s)-u^{\Delta}_{j}(s)\right\Vert\right)\\
&\leq&
\left(1+\frac{\alpha_{i}^{+}}{\alpha_{i}^{-}}\right)\overline{M}_{i}\|\psi-\Omega\|_{\mathbb{B}}.
\end{eqnarray*}
Similarly,
\begin{equation*}
\sup\limits_{s\in\mathbb{T}}\left\Vert y_{\phi_{j}}(s)-y_{v_{j}}(s)
\right\Vert \leq
\left(\frac{\overline{N}_{j}}{c_{j}^{-}}\right)\|\psi-\Omega\|_{\mathbb{B}},
\end{equation*}
and
\begin{equation*}
\sup\limits_{s\in\mathbb{T}}\left\Vert
y^{\Delta}_{\phi_{j}}(s)-y^{\Delta}_{v_{j}}(s) \right\Vert \leq
\left(1+\frac{c_{j}^{+}}{c_{j}^{-}}\right)\overline{N}_{j}\|\psi-\Omega\|_{\mathbb{B}},
\end{equation*}
therefore,
\begin{equation*}
\|\Gamma\psi-\Gamma\Omega\| \leq \kappa\|\psi-\Omega\|_{\mathbb{B}},
\,\ \text{where}\,\ \kappa<1.
\end{equation*}
According to the well-known contraction principle there exists a
unique fixed point $ h^{\ast }\left( \cdot \right) $ such that
$\Gamma h^{\ast }=h^{\ast }$. So, $h^{\ast }$ is a weighted
pseudo-almost automorphic on time-space scales solution of the model
(\ref{eq1}) in $\mathbb{F}=\{\psi\in\mathbb{B}: \,\
\|\psi\|_{\mathbb{B}}\leq r\}$. This completes the proof.
\begin{remark}
To the best of our knowledge, there have been no results focused on
the automorphic solutions, pseudo-almost automorphic ones and
Stepanov-like weighted pseudo-almost automorphic solutions on
time-space scales for high-order BAM neural networks with time
varying coefficients, mixed delays and leakage until now. Hence, the
obtained results are essentially new and the investigation methods
used in this paper can also be applied to study the Stepanov-like
weighted pseudo-almost automorphic solutions on time-space scales
for some other models of dynamical neural networks, such as
Cohen-Grossberg neural networks.
\end{remark}
\begin{theorem}\label{th1}
Let $(H_{1})-(H_{4})$ hold. The delta derivative of the only
Stepanov-like weighted pseudo-almost automorphic on time-space
scales solution of system (\ref{eq1}) is also Stepanov-like weighted
pseudo-almost automorphic on time-space scales (i.e. the unique
solution of (\ref{eq1}) is delta differentiable Stepanov-like
weighted pseudo-almost automorphic on time-space scales).
\end{theorem}
Proof. From system (\ref{eq1}), the expression of delta derivative
of the only Stepanov-like weighted pseudo-almost automorphic on
time-space scales solution is
\begin{eqnarray*}
x^{\Delta}_{i}\left( t\right)
&=&-\alpha_{i}(t)x_{i}\left(t-\eta_{i}(t)\right)
+\sum\limits_{j=1}^{m}D_{ij}\left( t\right) f_{j}\left(y_{j}\left( t\right) \right)\\
&+&\sum\limits_{j=1}^{m}D_{ij}^{\tau}\left( t\right)
f_{j}\left(y_{j}\left( t -\tau_{ij}(t)\right) \right)
+\sum\limits_{j=1}^{m}\overline{D}_{ij}\left( t\right)\int_{t-\sigma_{ij}(t)} ^{t}f_{j}\left(y_{j}\left( s\right) \right)\Delta s\\
&+&\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{m}T_{ijk}(t)f_{k}(y_{k}(t-\chi_{k}(t)))f_{j}(y_{j}(t-\chi_{j}(t)))\\
&+&\sum\limits_{j=1}^{m}\widetilde{D}_{ij}\left(
t\right)\int_{t-\xi_{ij}(t)} ^{t}f_{j}\left(y_{j}^{\Delta}\left(
s\right) \right)\Delta s+I_{i}\left( t\right),
\end{eqnarray*}
and
\begin{eqnarray*}
y^{\Delta}_{j}\left( t\right)
&=&-c_{j}(t)y_{j}\left(t-\eta_{j}(t)\right)
+\sum\limits_{i=1}^{n}E_{ij}\left( t\right) f_{j}\left(x_{j}\left( t\right) \right)\\
&+&\sum\limits_{i=1}^{n}E_{ij}^{\tau}\left( t\right)
f_{j}\left(x_{j}\left( t -\tau_{ij}(t)\right) \right)
+\sum\limits_{i=1}^{n}\overline{E}_{ij}\left( t\right)\int_{t-\sigma_{ij}(t)}^{t}f_{j}\left(x_{j}\left( s\right) \right)\Delta s\\
&+&\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}\overline{T}_{ijk}(t)f_{k}(x_{k}(t-\chi_{k}(t)))f_{j}(x_{j}(t-\chi_{j}(t)))\\
&+&\sum\limits_{i=1}^{n}\widetilde{D}_{ij}\left(
t\right)\int_{t-\xi_{ij}(t)}^{t}f_{j}\left(x_{j}^{\Delta}\left(
s\right) \right)\Delta s+J_{j}\left( t\right),\,\ t\in\mathbb{T}.
\end{eqnarray*}
Since all coefficients of the system (\ref{eq1}) are Stepanov-like
weighted pseudo-almost automorphic on time-space scales, derivative
of solution of system (\ref{eq1}) is Stepanov-like weighted
pseudo-almost automorphic on time-space scales.
\begin{remark}
In practice, time delays, leakage delay and parameter perturbations
are unavoidably encountered in the implementation of high-order BAM
neural networks, and they may destroy the stability of Stepanov-like
weighted pseudo-almost automorphic solution of HOBAMs, so it is
necessary and vital to study the dynamic behaviors of Stepanov-like
weighted pseudo-almost automorphic on time-space scales solution of
HOBAMs with time delays, leakage term and parameter perturbations.
\end{remark}
\section{Global exponential stability and convergence of Stepanov-like weighted pseudo-almost automorphic on time-space scales solution}
\subsection{Global exponential stability}
\begin{definition}
Let $Z^{\ast }\left(t\right) =\left(x_{1}^{\ast }\left(t\right),
x_{2}^{\ast }\left( t\right), \cdots ,x_{n}^{\ast
}\left(t\right),y_{1}^{\ast}\left(t\right),y_{2}^{\ast}\left(t\right),\cdots,
y_{m}^{\ast}\left(t\right) \right)^{T}$ be the Stepanov-like
weighted pseudo-almost automorphic solution on time-space scales of
system (\ref{eq1}) with initial value
$\psi^{\ast}\left(t\right)=\left(\varphi_{1}^{\ast}\left(t\right),\varphi_{2}^{\ast}\left(t\right),\cdots,\varphi_{n}^{\ast}\left(t\right),\right.$\\$\left.\phi_{1}^{\ast}\left(t\right),\phi_{2}^{\ast}\left(t\right),\cdots,
\phi_{m}^{\ast}\left(t\right)\right)^{T}$.
$Z^{\ast}\left(\cdot\right)$ is said to be globally exponential
stable if there exist constants $\gamma>0$,
$\ominus_{\nu}\gamma\in\mathcal{R}_{+}$ and $M>1$ such that for
every solution
\begin{equation*}
Z\left(t\right) =\left(x_{1}\left(t\right),x_{2}^{\ast
}\left(t\right),
\cdots,x_{n}\left(t\right),y_{1}\left(t\right),y_{2}\left(t\right),\cdots,y_{n}\left(t\right)
\right)^{T}
\end{equation*}
of system (\ref{eq1}) with any initial value
\begin{equation*}
\psi\left(t\right)=\left(\varphi_{1}\left(t\right),\varphi_{2}\left(t\right),\cdots,\varphi_{n}\left(t\right),\phi_{1}\left(t\right),\phi_{2}\left(t\right),\cdots,
\phi_{m}\left(t\right)\right)^{T}, \,\ \forall
t\in(0,+\infty)_{\mathbb{T}}, t\geq t_{0},
\end{equation*}
\begin{eqnarray*}
\left\Vert Z\left(t\right)-Z^{\ast}\left(t\right)\right\Vert_{0}
&=&\max \left\{\left\Vert
x\left(t\right)-x^{\ast}\left(t\right)\right\Vert_{\infty},\Vert
x^{\Delta}\left(t\right)
-x^{\ast\Delta}\left(t\right)\Vert_{\infty},\right.\\
&&\left.\left\Vert y\left(t\right)-y^{\ast
}\left(t\right)\right\Vert_{\infty},\Vert y^{\Delta}\left(t\right)
-y^{\ast\Delta }\left(t\right) \Vert_{\infty}\right\}\\
&\leq & M e_{\ominus\gamma}(t,t_{0})\|\psi\|_{1} \\
&=&M
e_{\ominus\gamma}(t,t_{0})\sup\limits_{t\in[-\theta,0]_{\mathbb{T}}}\max
\left\{\left\Vert\varphi\left(t\right)-\varphi^{\ast}\left(t\right)\right\Vert_{\infty},\left\Vert\varphi^{\Delta}\left(
t\right)-\varphi^{\ast\Delta}\left(t\right)
\right\Vert_{\infty},\right.\\
&&\left.\left\Vert\phi\left(t\right)-\phi^{\ast}\left(t\right)\right\Vert_{\infty},\left\Vert\phi^{\Delta}\left(
t\right)-\phi^{\ast\Delta}\left(t\right)
\right\Vert_{\infty}\right\},
\end{eqnarray*}
where $t_{0}=\max\{[-\theta,0]_{\mathbb{T}}\}$.
\end{definition}
\begin{theorem}\label{th2}
Let $(H_{1})-(H_{4})$ hold. The unique Stepanov-like weighted
pseudo-almost automorphic on time-space scales solution of system
(\ref{eq1}) is globally exponentially stable.
\end{theorem}
Proof. From Theorem \ref{th1} the system (\ref{eq1}) has one and
only one weighted pseudo-almost automorphic on time-space scales
solution on time scales
\begin{equation*}
Z^{*}(t)=(x^{*}_{1}(t),...x^{*}_{n}(t),y^{*}_{1}(t),...,y^{*}_{m}(t))^{T}\in\mathbb{R}^{n\times
m},
\end{equation*}
with the initial condition
\begin{equation*}
\psi^{*}(t)=(\varphi^{*}_{1}(t),...,\varphi^{*}_{n}(t),\phi^{*}_{1}(t),...,\phi^{*}_{m}(t))^{T}.
\end{equation*}
Let $Z(t)=(x_{1}(t),...,x_{n}(t),y_{1}(t),...,y_{m}(t))$ one arbitrary solution of (\ref{eq1}) with initial condition $\psi(t)=(\varphi_{1}(t),...,\varphi_{n}(t),\phi_{1}(t),...,\phi_{m}(t))^{T}$.\\
From system (\ref{eq1}), for $t\in\mathbb{T}$, we obtain:
\begin{equation}\label{eq4}
\left\{
\begin{array}{ccc}
u^{\Delta}_{i}\left( t\right) = -\alpha_{i}(t)u_{i}\left(t\right)
+\alpha_{i}(t)\int_{t-\eta_{i}(t)}^{t}u^{\Delta}_{i}(s)\Delta s
+\sum\limits_{j=1}^{m}D_{ij}\left( t\right)p_{j}\left(v_{j}(t)\right)\\
+\sum\limits_{j=1}^{m}D^{\tau}_{ij}\left(
t\right)p_{j}(v_{j}\left(\left(t-\tau_{ij}(t)\right)\right)
+\sum\limits_{j=1}^{m}\overline{D}_{ij}\left( t\right)\int_{t-\sigma_{ij}(t)} ^{t}p_{j}\left(v_{j}\left( s\right) \right)\Delta s\\
+\sum\limits_{j=1}^{m}\widetilde{D}_{ij}\left(
t\right)\int_{t-\xi_{ij}(t)}^{t}h_{j}\left(v_{j}^{\Delta}\left(
s\right) \right)\Delta s
+\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{m}T_{ijk}(t)q_{j,k}(v_{j}(t-\chi_{j}(t)),v_{k}(t-\chi_{j}(t))),\\
v_{i}^{\Delta}(t)=-c_{i}(t)v_{i}(t)+c_{i}(t)\int_{t-\varsigma_{i}(t)}^{t}v_{i}^{\Delta}(s)\Delta
s+\sum\limits_{i=1}^{n}E_{ij}\left( t\right)p_{j}\left(u_{j}(t)\right)\\
+\sum\limits_{i=1}^{n}E^{\tau}_{ij}\left(
t\right)p_{j}(u_{j}\left(\left(t-\tau_{ij}(t)\right)\right)
+\sum\limits_{i=1}^{n}\overline{E}_{ij}\left( t\right)\int_{t-\sigma_{ij}(t)}^{t}p_{j}\left(u_{j}\left( s\right) \right)\Delta s\\
+\sum\limits_{i=1}^{n}\widetilde{E}_{ij}\left(
t\right)\int_{t-\xi_{ij}(t)}^{t}h_{j}\left(u_{j}^{\Delta}\left(
s\right)\right)\Delta s
+\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}\overline{T}_{ijk}(t)q_{j,k}(u_{j}(t-\chi_{j}(t)),u_{k}(t-\chi_{j}(t))),\,\
\end{array}
\right.
\end{equation}
where
\begin{multline*} u_{i}(t)=x_{i}(t)-x^{*}_{i}(t),\,\ v_{i}(t)=y_{i}(t)-y_{i}^{*}(t),\,\
p_{j}(u_{j}(t))=f_{j}(x_{j}(t))-f_{j}(x_{j}^{*}(t)),\\ h_{j}(u_{j}^{\Delta}(t))=f_{j}(x_{j}^{\Delta}(t))-f_{j}({x_{j}^{*}}^{\Delta}(t)),\\
p_{j}(v_{j}(t))=f_{j}(y_{j}(t))-f_{j}(y_{j}^{*}(t)),\\ h_{j}(v_{j}^{\Delta}(t))=f_{j}(y_{j}^{\Delta}(t))-f_{j}({y_{j}^{*}}^{\Delta}(t)),\\
q_{j,k}(u_{j}(t),u_{k}(t))=f_{k}(x_{k}(t))f_{j}(x_{j}(t))-f_{k}(x^{*}_{k}(t))f_{j}(x^{*}_{j}(t))\\
q_{j,k}(v_{j}(t),v_{k}(t))=f_{k}(y_{k}(t))f_{j}(y_{j}(t))-f_{k}(y^{*}_{k}(t))f_{j}(y^{*}_{j}(t)).\\
\end{multline*}
For $i=1,...,n$ and $j=1,...,m$, the initial condition of
(\ref{eq4}) is
\begin{equation*}
u_{i}(s)=\varphi_{i}(s)-\varphi_{i}^{*}(s), \,\
v_{j}(s)=\phi_{j}(s)-\phi_{j}^{*}(s), \,\ s\in
[-\theta,0]_{\mathbb{T}}.
\end{equation*}
Multiplying the first equation in system (\ref{eq4}) by
$\hat{e}_{-\alpha_{i}}(t_{0},\sigma(s))$ and the second equation by
$\hat{e}_{-c_{j}}(t_{0},\sigma(s))$, and integrating over
$[t_{0},t]_{\mathbb{T}}$, where $t_{0}\in[-\theta,0]_{\mathbb{T}}$,
we obtain
\begin{equation}\label{eq5}
\left\{
\begin{array}{ccc}
u_{i}\left( t\right)=
u_{i}\left(t_{0}\right)\hat{e}_{-\alpha_{i}}(t,t_{0})+\int_{t_{0}}^{t}\hat{e}_{-\alpha_{i}}(t,\sigma(s))
\left(\alpha_{i}(s)\int_{s-\eta_{i}(s)}^{t}u^{\Delta}_{i}(u)\Delta u\right.\\
+\sum\limits_{j=1}^{m}D_{ij}\left(s\right)p_{j}\left(v_{j}(s)\right)
+\sum\limits_{j=1}^{m}D^{\tau}_{ij}\left(s\right)p_{j}(v_{j}\left(\left(s-\tau_{ij}(s)\right)\right)\\
+\sum\limits_{j=1}^{m}\overline{D}_{ij}\left(
s\right)\int_{s-\sigma_{ij}(s)}^{s}p_{j}\left(v_{j}\left( u\right)
\right)\Delta
u+\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{m}T_{ijk}(s)q_{j,k}(v_{j}(s-\chi_{j}(s)),v_{k}(s-\chi_{k}(s)))
\\ \left.+\sum\limits_{j=1}^{m}\widetilde{D}_{ij}\left(
s\right)\int_{s-\xi_{ij}(s)}^{t}h_{j}\left(v_{j}^{\Delta}\left(
u\right) \right)\Delta u
\right)\Delta s,\\
v_{j}(t)=v_{j}(t_{0})\hat{e}_{-c_{j}}(t,t_{0})
+\int_{t_{0}}^{t}\hat{e}_{-c_{j}}(t,\sigma(s))\left(c_{j}(s)\int_{s-\varsigma_{j}(s)}^{s}v_{j}^{\Delta}(u)\Delta
u\right.\\
+\sum\limits_{i=1}^{n}E_{ij}\left(s\right)p_{i}\left(u_{i}(s)\right)
+\sum\limits_{i=1}^{n}E^{\tau}_{ij}\left(s\right)p_{i}(u_{i}\left(\left(s-\tau_{ij}(s)\right)\right)\\
+\sum\limits_{i=1}^{n}\overline{E}_{ij}\left(
s\right)\int_{s-\sigma_{ij}(s)}^{s}p_{i}\left(u_{i}\left( u\right)
\right)\Delta u
+\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}\overline{T}_{ijk}(s)q_{j,k}(u_{i}(s-\chi_{i}(s)),u_{k}(s-\chi_{k}(s)))\\
\left.\left.+\sum\limits_{i=1}^{n}\widetilde{E}_{ij}\left(
s\right)\int_{s-\xi_{ij}(s)}^{t}h_{i}\left(u_{i}^{\Delta}\left(
u\right) \right)\Delta u \right)\Delta s\right)\Delta s,\,\
t\in\mathbb{T},
\end{array}
\right.
\end{equation}
Now, we define $G_{i}$, $\overline{G}_{j}$, $H_{i}$ and $\overline{H}_{j}$ as follows:\\
\begin{multline*}
G_{i}(w)=\alpha_{i}^{-}-w-\left(\exp\left(w
\sup\limits_{s\in\mathbb{T}}\nu(s)\right)\left(\alpha_{i}^{+}\eta_{i}^{+}
\exp\left(w\eta_{i}^{+}\right)+\sum\limits_{j=1}^{m}D_{ij}^{+}L_{j}\right.\right.\\
+\sum\limits_{j=1}^{m}(D_{ij}^{\tau})^{+}L_{j}\exp\left(w\tau_{ij}^{+}\right)
+\sum\limits_{j=1}^{m}\overline{D}_{ij}^{+}L_{j}\sigma_{ij}^{+}\exp\left(w\sigma_{ij}^{+}\right)
+\sum\limits_{j=1}^{m}\widetilde{D}_{ij}^{+}L_{j}\xi_{ij}^{+}\exp\left(w\xi_{ij}^{+}\right)\\
\left.+\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{m}T_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{j}r+|f_{j}(0)|)\right),\phantom{++++++}\\
\overline{G}_{j}(w)=c_{j}^{-}-w-\left(\exp\left(w
\sup\limits_{s\in\mathbb{T}}\nu(s)\right)\left(c_{j}^{+}\varsigma_{j}^{+}
\exp\left(w\varsigma_{j}^{+}\right)+\sum\limits_{i=1}^{n}E_{ij}^{+}L_{i}\right.\right.\\
+\sum\limits_{i=1}^{n}(E_{ij}^{\tau})^{+}L_{i}\exp\left(w\tau_{ij}^{+}\right)
+\sum\limits_{i=1}^{n}\overline{E}_{ij}^{+}L_{i}\sigma_{ij}^{+}\exp\left(w\sigma_{ij}^{+}\right)
+\sum\limits_{i=1}^{n}\widetilde{E}_{ij}^{+}L_{i}\xi_{ij}^{+}\exp\left(w\xi_{ij}^{+}\right)\\
\left.+\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}\overline{T}_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{i}r+|f_{i}(0)|)\right),\phantom{++++++}\\
H_{i}(w)=\alpha_{i}^{-}-w-\left(\alpha_{i}^{+}\exp\left(w
\sup\limits_{s\in\mathbb{T}}\nu(s)+\alpha_{i}^{-}-\beta\right)\left(\alpha_{i}^{+}\eta_{i}^{+}
\exp\left(w\eta_{i}^{+}\right)\right.\right.\\
+\sum\limits_{j=1}^{m}D_{ij}^{+}L_{j}+\sum\limits_{j=1}^{m}(D_{ij}^{\tau})^{+}L_{j}\exp\left(w\tau_{ij}^{+}\right)
+\sum\limits_{j=1}^{m}\overline{D}_{ij}^{+}L_{j}\sigma_{ij}^{+}\exp\left(w\sigma_{ij}^{+}\right)
+\sum\limits_{j=1}^{m}\widetilde{D}_{ij}^{+}L_{j}\xi_{ij}^{+}\exp\left(w\xi_{ij}^{+}\right)
\end{multline*}
\begin{multline*}
\left.+\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{m}T_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{j}r+|f_{j}(0)|)\right),\phantom{++++++}\\
\overline{H}_{j}(w)=c_{j}^{-}-w-\left(c_{j}^{+}\exp\left(w
\sup\limits_{s\in\mathbb{T}}\nu(s)+c_{j}^{-}-\beta\right)\left(c_{j}^{+}\varsigma_{j}^{+}
\exp\left(w\varsigma_{j}^{+}\right)\right.\right.\\
+\sum\limits_{i=1}^{n}E_{ij}^{+}L_{i}+\sum\limits_{i=1}^{n}(E_{ij}^{\tau})^{+}L_{i}\exp\left(w\tau_{ij}^{+}\right)
+\sum\limits_{i=1}^{n}\overline{E}_{ij}^{+}L_{i}\sigma_{ij}^{+}\exp\left(w\sigma_{ij}^{+}\right)
+\sum\limits_{i=1}^{n}\widetilde{E}_{ij}^{+}L_{i}\xi_{ij}^{+}\exp\left(w\xi_{ij}^{+}\right)\\
\left.+\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}T_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{i}r+|f_{i}(0)|)\right),\phantom{++++++}\\
\end{multline*}
where $i=1,...,n$, $j=1,...,m$, $w\in(0,+\infty)$.\\
From $\left( H_{3} \right)$, we have\\
\begin{multline*}G_{i}(0)=\alpha_{i}^{-}-\left(\alpha_{i}^{+}\eta_{i}^{+}
+\sum\limits_{j=1}^{m}D_{ij}^{+}L_{j}+\sum\limits_{j=1}^{m}(D_{ij}^{\tau})^{+}L_{j}
+\sum\limits_{j=1}^{m}\overline{D}_{ij}^{+}L_{j}\sigma_{ij}^{+}\right.\\
\left.+\sum\limits_{j=1}^{m}\widetilde{D}_{ij}^{+}L_{j}\xi_{ij}^{+}
+\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{m}T_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{j}r+|f_{j}(0)|)\right),\\
\overline{G}_{j}(0)=c_{j}^{-}+\sum\limits_{i=1}^{n}E_{ij}^{+}L_{i}
+\sum\limits_{i=1}^{n}(E_{ij}^{\tau})^{+}L_{i}
+\sum\limits_{i=1}^{n}\overline{E}_{ij}^{+}L_{i}\sigma_{ij}^{+}\\
\left.+\sum\limits_{i=1}^{n}\widetilde{E}_{ij}^{+}L_{i}\xi_{ij}^{+}
+\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}\overline{T}_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{i}r+|f_{i}(0)|)\right),\\
H_{i}(0)=\alpha_{i}^{-}-\alpha_{i}^{+}\exp\left(\alpha_{i}^{-}-\beta\right)\left(\alpha_{i}^{+}\eta_{i}^{+}
+\sum\limits_{j=1}^{m}D_{ij}^{+}L_{j}+\sum\limits_{j=1}^{m}(D_{ij}^{\tau})^{+}L_{j}
+\sum\limits_{j=1}^{m}\overline{D}_{ij}^{+}L_{j}\sigma_{ij}^{+}\right.\\
\left.+\sum\limits_{j=1}^{m}\widetilde{D}_{ij}^{+}L_{j}\xi_{ij}^{+}
+\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{m}T_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{j}r+|f_{j}(0)|)\right),
\end{multline*}
\begin{multline*}
\overline{H}_{j}(0)=c_{j}^{-}-c_{j}^{+}\exp\left(c_{j}^{-}-\beta\right)\left(c_{j}^{+}\varsigma_{j}^{+}
+\sum\limits_{i=1}^{n}E_{ij}^{+}L_{i}+\sum\limits_{i=1}^{n}(E_{ij}^{\tau})^{+}L_{i}
+\sum\limits_{i=1}^{n}\overline{E}_{ij}^{+}L_{i}\sigma_{ij}^{+}\right.\\
\left.+\sum\limits_{i=1}^{n}\widetilde{E}_{ij}^{+}L_{i}\xi_{ij}^{+}
+\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}\overline{T}_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{i}r+|f_{i}(0)|)\right),
\end{multline*}
Since the functions $G_{i}(.)$, $\overline{G}_{j}(.)$, $H_{i}(.)$
and $\overline{H}_{j}(.)$ are continuous on $[0,+\infty)$ and
$G_{i}(w)$, $\overline{G}_{j}(w)$, $H_{i}(w)$,
$\overline{H}_{j}(w)\longrightarrow-\infty$ when
$w\longrightarrow+\infty$, it exist $\eta_{i}, \bar{\eta}_{j},
\epsilon_{i}, \bar{\epsilon}_{j}>0$ such as
\begin{equation*}
H_{i}(\eta_{i})=\overline{H}_{j}(\bar{\eta}_{j})=G_{i}(\epsilon_{i})=\overline{G}_{j}(\bar{\epsilon}_{j})=0
\end{equation*}
and
\begin{eqnarray*}
&&G_{i}(w)>0\,\ \text{for} \,\ w\in(0,\eta_{i}), \,\ \overline{G}_{j}(w)>0 \,\ \text{for} \,\ w\in(0,\bar{\eta}_{j}),\\
&&H_{i}(w)>0\,\ \text{for} \,\ w\in(0,\epsilon_{i}), \,\
\overline{H}_{j}(w)>0\,\ \text{for} \,\ w\in(0,\bar{\epsilon}_{j}).
\end{eqnarray*}
Let $a=\min\limits_{1\leq i\leq n}\left\{\eta_{i}, \bar{\eta}_{j},
\epsilon_{i}, \bar{\epsilon}_{j}\right\}$, we obtain
\begin{equation*}
H_{i}(a)\geq0,\,\ \overline{H}_{j}(a)\geq0, \,\ G_{i}(a)\geq0,\,\
\text{and} \,\ \overline{G}_{j}(a)\geq0,\,\ i=1,...,n, j=1,...,m.
\end{equation*}
So, we can choose the positive constant
$0<\gamma<\min\limits_{1\leq i \leq n, 1\leq j\leq m}\{a,\alpha_{i}^{-},c_{j}^{-}\},$\\
$\text{such that}\,\ H_{i}(\gamma)>0,\,\ \overline{H}_{j}(\gamma)>0,
\,\ G_{i}(\gamma)>0\,\ \text{and} \,\ \overline{G}_{j}(\gamma)>0,
\,\ i=1,...,n, j=1,...,m.$ which imply that, for $i=1,...,n$ and
$j=1,...,m$
\begin{multline*}
\frac{1}{\alpha_{i}^{-}-\gamma}\sum\limits_{j=1}^{m}\left(\exp(\gamma\sup\limits_{s\in\mathbb{T}}\nu(s))\left(\alpha_{i}^{+}\eta_{i}^{+}+\sum\limits_{j=1}^{m}
\left(D_{ij}^{+}+(D_{ij}^{\tau})^{+}+\overline{D}_{ij}^{+}\sigma_{ij}^{+}
+\widetilde{D}_{ij}^{+}\xi_{ij}^{+}\right)L_{j}\right.\right.\\
\left.\left.+\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{n}T_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{j}r+|f_{j}(0)|)\right)\right)<1,\\
\frac{1}{c_{j}^{-}-\gamma}\sum\limits_{i=1}^{n}\left(\exp(\gamma\sup\limits_{s\in\mathbb{T}}\nu(s))\left(c_{j}^{+}\varsigma_{j}^{+}+\sum\limits_{i=1}^{n}
\left(E_{ij}^{+}+(E_{ij}^{\tau})^{+}+\overline{E}_{ij}^{+}\sigma_{ij}^{+}
+\widetilde{E}_{ij}^{+}\xi_{ij}^{+}\right)L_{i}\right.\right.\\
\left.\left.+\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}\overline{T}_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{i}r+|f_{i}(0)|)\right)\right)<1,\\
\left(1+\frac{\alpha_{i}^{+}\exp(\gamma\sup\limits_{s\in\mathbb{T}}\nu(s))}{\alpha_{i}^{-}-\gamma}\right)
\sum\limits_{j=1}^{m}\left(\exp(\gamma\sup\limits_{s\in\mathbb{T}}\nu(s))\left(\alpha_{i}^{+}\eta_{i}^{+}
+\sum\limits_{j=1}^{m}\left(D_{ij}^{+}+(D_{ij}^{\tau})^{+}\right.\right.\right.\\
\left.\left.\left.+\overline{D}_{ij}^{+}\sigma_{ij}^{+}+\widetilde{D}_{ij}^{+}\xi_{ij}^{+}\right)L_{j}
+\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{m}T_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{j}r+|f_{j}(0)|)\right)\right)<1,\\
\left(1+\frac{c_{j}^{+}\exp(\gamma\sup\limits_{s\in\mathbb{T}}\nu(s))}{c_{j}^{-}-\gamma}\right)
\sum\limits_{i=1}^{n}\left(\exp(\gamma\sup\limits_{s\in\mathbb{T}}\nu(s))\left(c_{j}^{+}\varsigma_{j}^{+}
+\sum\limits_{i=1}^{n}\left(E_{ij}^{+}+(E_{ij}^{\tau})^{+}\right.\right.\right.\\
\left.\left.\left.+\overline{E}_{ij}^{+}\sigma_{ij}^{+}+\widetilde{E}_{ij}^{+}\xi_{ij}^{+}\right)L_{i}
+\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}\overline{T}_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{i}r+|f_{i}(0)|)\right)\right)<1,\\
\end{multline*}
Let
\begin{equation}\label{abcd5}
K=\max\limits_{1\leq i\leq n, 1\leq j\leq
m}\left\{\frac{\alpha_{i}^{-}}{K^{*}},
\frac{c_{j}^{-}}{P^{*}}\right\},
\end{equation}
where
\begin{multline*}
\phantom{++++++}K^{*}=\alpha_{i}^{+}\eta_{i}^{+}+\sum\limits_{j=1}^{m}\left(D_{ij}^{+}+(D_{ij}^{\tau})^{+}+\overline{D}_{ij}^{+}\sigma_{ij}^{+}
+\widetilde{D}_{ij}^{+}\xi_{ij}^{+}\right)L_{j}\\
+\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{m}T_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{j}r+|f_{j}(0)|),\phantom{+++}
\end{multline*}
and
\begin{multline*}
\phantom{++++++}P^{*}=c_{j}^{+}\varsigma_{j}^{+}+\sum\limits_{i=1}^{n}\left(E_{ij}^{+}+(E_{ij}^{\tau})^{+}+\overline{E}_{ij}^{+}\sigma_{ij}^{+}
+\widetilde{E}_{ij}^{+}\xi_{ij}^{+}\right)L_{i}\\
+\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}\overline{T}_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{i}r+|f_{i}(0)|).\phantom{+++}
\end{multline*}
By hypothesis ($H_{3}$), we have $K>1$, therefore,
\begin{equation}\label{abcd6}
\|Z(t)-Z^{*}(t)\|\leq K \hat{e}_{\ominus_{\nu}
\gamma}(t,t_{0})\|\psi\|_{0},\,\ \forall t\in[t_{0},0]_{\mathbb{T}},
\end{equation}
where $\ominus_{\nu}\gamma\in \mathcal{R}_{\nu}^{+}$. We claim that
\begin{equation}\label{abcd71}
\|Z(t)-Z^{*}(t)\|\leq K \hat{e}_{\ominus_{\nu}
\gamma}(t,t_{0})\|\psi\|_{0},\,\ \forall
t\in[t_{0},+\infty)_{\mathbb{T}}.
\end{equation}
To prove (\ref{abcd71}), we show that for any $\varpi>1$, the
following inequality holds:
\begin{equation}\label{abcd72}
\|Z(t)-Z^{*}(t)\|\leq \varpi K \hat{e}_{\ominus_{\nu}
\gamma}(t,t_{0})\|\psi\|_{0},\,\ \forall
t\in[t_{0},+\infty)_{\mathbb{T}}.
\end{equation}
If (\ref{abcd72}) is not true, then there must be some
$t_{1}\in(0,+\infty)_{\mathbb{T}}$, $d\geq1$ such that
\begin{equation}\label{abcd73}
\|Z(t_{1})-Z^{*}(t_{1})\|=d \varpi K \hat{e}_{\ominus_{\nu}
\gamma}(t_{1},t_{0})\|\psi\|_{0},
\end{equation}
and
\begin{equation}\label{abcd74}
\|Z(t)-Z^{*}(t)\|\leq d \varpi K \hat{e}_{\ominus_{\nu}
\gamma}(t,t_{0})\|\psi\|_{0}, \,\ t\in[t_{0},t_{1}]_{\mathbb{T}}.
\end{equation}
By (\ref{eq5}), (\ref{abcd73}), (\ref{abcd74}) and
$(H_{1})-(H_{3})$, we have for $i=1,...,n$
\begin{eqnarray*}
|u_{i}(t_{1})|&\leq&
\hat{e}_{-\alpha_{i}}(t_{1},t_{0})\|\psi\|_{0}+d \varpi K
\hat{e}_{\ominus_{\nu}\gamma}(t_{1},t_{0})\|\psi\|_{0}\int_{t_{0}}^{t_{1}}
\hat{e}_{-\alpha_{i}}(t_{1},\sigma(s))\hat{e}_{\gamma}(t_{1},\sigma(s))\nonumber\\
&\times&\left(\alpha_{i}^{+}\int_{s-\eta_{i}(s)}^{s}\hat{e}_{\gamma}(\sigma(u),u)\Delta
u+\sum\limits_{j=1}^{n}
D_{ij}^{+}L_{j}\hat{e}_{\gamma}(\sigma(s),s)\right.\nonumber\\
&+&\sum\limits_{j=1}^{n}\sum\limits_{k=1}^{n}T_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{j}r+|f_{j}(0)|)\hat{e}_{\gamma}(\sigma(s),s-\chi_{j}(s))\hat{e}_{\gamma}(\sigma(s),
s-\chi_{k}(s))\nonumber\\
&+&(D_{ij}^{\tau})^{+}L_{j}\hat{e}_{\gamma}(\sigma(s),s-\tau_{ij}(s))+\sum\limits_{j=1}^{n}
\overline{D}_{ij}^{+}L_{j}\int_{s-\sigma_{ij}(s)}^{s}\hat{e}_{\gamma}(\sigma(u),u)\Delta u\nonumber\\
&+&\left. \sum\limits_{j=1}^{n}\widetilde{D}_{ij}^{+}L_{j}\int_{s-\xi_{ij}(s)}^{s}\hat{e}_{\gamma}(\sigma(u),u)\Delta u\right)\Delta s\nonumber\\
&\leq& \hat{e}_{-\alpha_{i}}(t_{1},t_{0})\|\psi\|_{0}+d \varpi K
\hat{e}_{\ominus_{\nu}\gamma}(t_{1},t_{0})\|\psi\|_{0}\int_{t_{0}}^{t_{1}}
\hat{e}_{-\alpha_{i}}(t_{1},\sigma(s))\hat{e}_{\gamma}(t_{1},\sigma(s))\nonumber\\
&\times&\left(\alpha_{i}^{+}\hat{e}_{\gamma}(\sigma(s),s-\eta_{i}(s))+\sum\limits_{j=1}^{n}
D_{ij}^{+}L_{j}\hat{e}_{\gamma}(\sigma(s),s)\right.\nonumber\\
&+&\sum\limits_{j=1}^{n}\sum\limits_{k=1}^{n}T_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{j}r+|f_{j}(0)|)\hat{e}_{\gamma}(\sigma(s),s-\chi_{j}(s))\hat{e}_{\gamma}(\sigma(s)
,s-\chi_{k}(s))\nonumber\\
&+&\sum\limits_{j=1}^{n}(D_{ij}^{\tau})^{+}L_{j}\hat{e}_{\gamma}(\sigma(s),s-\tau_{ij}(s))+\sum\limits_{j=1}^{n}
\overline{D}_{ij}^{+}L_{j}\int_{s-\sigma_{ij}(s)}^{s}\hat{e}_{\gamma}(\sigma(u),u)\Delta
u \nonumber\\&+&\left. \sum\limits_{j=1}^{n}
\widetilde{D}_{ij}^{+}L_{j}\int_{s-\xi_{ij}(s)}^{s}\hat{e}_{\gamma}(\sigma(u),u)\Delta u\right)\Delta s\nonumber\\
&\leq& \hat{e}_{-\alpha_{i}}(t_{1},t_{0})\|\psi\|_{0}+d \varpi K
\hat{e}_{\ominus_{\nu}\gamma}(t_{1},t_{0})\|\psi\|_{0}\int_{t_{0}}^{t_{1}}
\hat{e}_{-\alpha_{i}}(t_{1},\sigma(s))\hat{e}_{\gamma}(t_{1},\sigma(s))\nonumber\\
&\times&\left(\alpha_{i}^{+}\eta_{i}^{+}\exp\left[\gamma\left(\eta_{i}^{+}+\sup\limits_{s\in\mathbb{T}}\nu(s)\right)\right]
+\sum\limits_{j=1}^{n}D_{ij}^{+}L_{j}\exp\left(\gamma\sup\limits_{s\in\mathbb{T}}\nu(s)\right)\right.\nonumber\\
&+&(D_{ij}^{\tau})^{+}L_{j}\exp\left[\gamma\left(\tau_{ij}^{+}+\sup\limits_{s\in\mathbb{T}}\nu(s)\right)\right]
+\sum\limits_{j=1}^{n}\overline{D}_{ij}^{+}L_{j}\exp\left[\gamma\left(\sigma_{ij}^{+}+\sup\limits_{s\in\mathbb{T}}\nu(s)\right)\right]\\
&+&\sum\limits_{j=1}^{n}\widetilde{D}_{ij}^{+}L_{j}\exp\left[\gamma\left(\xi_{ij}^{+}+\sup\limits_{s\in\mathbb{T}}\nu(s)\right)\right]
+\sum\limits_{j=1}^{n}\sum\limits_{k=1}^{n}T_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{j}r+|f_{j}(0)|)\nonumber\\
&\times&\left.\exp\left[\gamma\left(\chi_{j}^{+}+\sup\limits_{s\in\mathbb{T}}\nu(s)\right)\right]\exp\left[\gamma\left(\chi_{k}^{+}
+\sup\limits_{s\in\mathbb{T}}\nu(s)\right)\right] \right)\Delta s
\end{eqnarray*}
\begin{eqnarray}\label{from}
&\leq& d\varpi K \hat{e}_{\ominus_{\nu}
\gamma}(t_{1},t_{0})\|\psi\|_{0}\left\{\frac{1}{K}\hat{e}_{-\alpha_{i}\oplus_{\nu}\gamma}(t_{1},t_{0})+
\left[\exp\left(\gamma\sup\limits_{s\in\mathbb{T}}\nu(s)\right)\right.\right.\nonumber\\
&\times& \left(\alpha_{i}^{+}\eta_{i}^{+}\exp(\gamma\eta_{i}^{+})+
\sum\limits_{j=1}^{n}D_{ij}^{+}L_{j}+\sum\limits_{j=1}^{n}(D_{ij}^{\tau})^{+}L_{j}\exp(\gamma\tau_{ij}^{+})
+\sum\limits_{j=1}^{n}\overline{D}_{ij}^{+}\sigma_{ij}^{+}L_{j}\exp(\gamma\sigma_{ij}^{+})\right.\nonumber\\
&+&\sum\limits_{j=1}^{n}\widetilde{D}_{ij}^{+}L_{j}\xi_{ij}^{+}\exp(\gamma\xi_{ij}^{+})
+\sum\limits_{j=1}^{n}\sum\limits_{k=1}^{n}T_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{j}r+|f_{j}(0)|)\nonumber\\
&\times&\left.\left.\left.\exp(\gamma \chi_{j}^{+})\exp(\gamma
\chi_{k}^{+})\right)\right]\frac{1-\hat{e}_{-\alpha_{i}\oplus_{\nu}\gamma}(t_{1},t_{0})}{\alpha_{i}^{-}-\gamma}\right\}\nonumber\\
&\leq& d\varpi K \hat{e}_{\ominus_{\nu}
\gamma}(t_{1},t_{0})\|\psi\|_{0}\left\{\frac{1}{K}-\frac{1}{\alpha_{i}^{-}-\gamma}\left(
\exp\left(\gamma\sup\limits_{s\in\mathbb{T}}\nu(s)\right) \left(\alpha_{i}^{+}\eta_{i}^{+}\exp(\gamma\eta_{i}^{+})\right.\right.\right.\nonumber\\
&+&\sum\limits_{j=1}^{n}D_{ij}^{+}L_{j}+\sum\limits_{j=1}^{n}(D_{ij}^{\tau})^{+}L_{j}\exp(\gamma\tau_{ij}^{+})
+\sum\limits_{j=1}^{n}\overline{D}_{ij}^{+}\sigma_{ij}^{+}L_{j}\exp(\gamma\sigma_{ij}^{+})\nonumber\\
&+&\sum\limits_{j=1}^{n}\widetilde{D}_{ij}^{+}L_{j}\xi_{ij}^{+}\exp(\gamma\xi_{ij}^{+})
+\sum\limits_{j=1}^{n}\sum\limits_{k=1}^{n}T_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{j}r+|f_{j}(0)|)\nonumber\\
&\times&\left.\left.\left.\exp(\gamma \chi_{j}^{+})\exp(\gamma
\chi_{k}^{+})\right)\right) +
B_{i}^{+}\right]\hat{e}_{-\alpha_{i}\oplus_{\nu}\gamma}(t_{1},t_{0})
+\frac{1}{\alpha_{i}^{-}-\gamma}\left(\exp\left(\gamma\sup\limits_{s\in\mathbb{T}}\nu(s)
\left(\alpha_{i}^{+}\eta_{i}^{+}\exp(\gamma\eta_{i}^{+})\right.\right.\right.\nonumber\\
&+&\sum\limits_{j=1}^{n}D_{ij}^{+}L_{j}+\sum\limits_{j=1}^{n}(D_{ij}^{\tau})^{+}L_{j}\exp(\gamma\tau_{ij}^{+})
+\sum\limits_{j=1}^{n}\overline{D}_{ij}^{+}\sigma_{ij}^{+}L_{j}\exp(\gamma\sigma_{ij}^{+})\nonumber\\
&+&\sum\limits_{j=1}^{n}\widetilde{D}_{ij}^{+}L_{j}\xi_{ij}^{+}\exp(\gamma\xi_{ij}^{+})
+\sum\limits_{j=1}^{n}\sum\limits_{k=1}^{n}T_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{j}r+|f_{j}(0)|)\nonumber\\
&\times&\left.\left.\left.\left.\exp(\gamma \chi_{j}^{+})\exp(\gamma \chi_{k}^{+})\right)\right)\right)\right\}\nonumber\\
&\leq& d\varpi K \hat{e}_{\ominus_{\nu}
\gamma}(t_{1},t_{0})\|\psi\|_{0}.
\end{eqnarray}
In addition, we have
\begin{eqnarray*}
|v_{j}(t_{1})| &\leq& \hat{e}_{-c_{j}}(t_{1},t_{0})\|\psi\|_{0}+d
\varpi K
\hat{e}_{\ominus_{\nu}\gamma}(t_{1},t_{0})\|\psi\|_{0}\int_{t_{0}}^{t_{1}}
\hat{e}_{-c_{j}}(t_{1},\sigma(s))\hat{e}_{\gamma}(t_{1},\sigma(s))\nonumber\\
&\times&\left(c_{j}^{+}\varsigma_{j}^{+}\exp\left[\gamma\left(\varsigma_{j}^{+}+\sup\limits_{s\in\mathbb{T}}\nu(s)\right)\right]
+\sum\limits_{i=1}^{n}E_{ij}^{+}L_{i}\exp\left(\gamma\sup\limits_{s\in\mathbb{T}}\nu(s)\right)\right.\nonumber\\
&+&(E_{ij}^{\tau})^{+}L_{i}\exp\left[\gamma\left(\tau_{ij}^{+}+\sup\limits_{s\in\mathbb{T}}\nu(s)\right)\right]
+\sum\limits_{i=1}^{n}\overline{E}_{ij}^{+}L_{i}\exp\left[\gamma\left(\sigma_{ij}^{+}+\sup\limits_{s\in\mathbb{T}}\nu(s)\right)\right]
\\&+&\sum\limits_{i=1}^{n}\widetilde{E}_{ij}^{+}L_{i}\exp\left[\gamma\left(\xi_{ij}^{+}+\sup\limits_{s\in\mathbb{T}}\nu(s)\right)\right]
+\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}\overline{T}_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{i}r+|f_{i}(0)|)\nonumber\\
&\times&\left.\exp\left[\gamma\left(\chi_{i}^{+}+\sup\limits_{s\in\mathbb{T}}\nu(s)\right)\right]\exp\left[\gamma\left(\chi_{k}^{+}
+\sup\limits_{s\in\mathbb{T}}\nu(s)\right)\right]
\right)\Delta s\nonumber\\
\end{eqnarray*}
\begin{eqnarray*}
&\leq& d\varpi K \hat{e}_{\ominus_{\nu}
\gamma}(t_{1},t_{0})\|\psi\|_{0}\left\{\frac{1}{K}\hat{e}_{-c_{j}\oplus_{\nu}\gamma}(t_{1},t_{0})+
\left[\exp\left(\gamma\sup\limits_{s\in\mathbb{T}}\nu(s)\right)\right.\right.\nonumber\\
&\times&
\left(c_{j}^{+}\varsigma_{j}^{+}\exp(\gamma\varsigma_{j}^{+})+
\sum\limits_{i=1}^{n}E_{ij}^{+}L_{i}+\sum\limits_{i=1}^{n}(E_{ij}^{\tau})^{+}L_{i}\exp(\gamma\tau_{ij}^{+})
+\sum\limits_{i=1}^{n}\overline{E}_{ij}^{+}\sigma_{ij}^{+}L_{i}\exp(\gamma\sigma_{ij}^{+})\right.\nonumber\\
&+&\sum\limits_{i=1}^{n}\widetilde{E}_{ij}^{+}L_{i}\xi_{ij}^{+}\exp(\gamma\xi_{ij}^{+})
+\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}\overline{T}_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{i}r+|f_{i}(0)|)\nonumber\\
&\times&\left.\left.\left.\exp(\gamma \chi_{i}^{+})\exp(\gamma
\chi_{k}^{+})\right)\right]\frac{1-\hat{e}_{-c_{j}\oplus_{\nu}\gamma}(t_{1},t_{0})}{c_{j}^{-}-\gamma}\right\}\nonumber\\
&\leq& d\varpi K \hat{e}_{\ominus_{\nu}
\gamma}(t_{1},t_{0})\|\psi\|_{0}\left\{\frac{1}{K}-\frac{1}{c_{j}^{-}-\gamma}\left(
\exp\left(\gamma\sup\limits_{s\in\mathbb{T}}\nu(s)\right) \left(c_{j}^{+}\varsigma_{j}^{+}\exp(\gamma\varsigma_{j}^{+})\right.\right.\right.\nonumber\\
&+&\sum\limits_{i=1}^{n}E_{ij}^{+}L_{i}+\sum\limits_{i=1}^{n}(E_{ij}^{\tau})^{+}L_{i}\exp(\gamma\tau_{ij}^{+})
+\sum\limits_{i=1}^{n}\overline{E}_{ij}^{+}\sigma_{ij}^{+}L_{i}\exp(\gamma\sigma_{ij}^{+})\nonumber\\
&+&\sum\limits_{i=1}^{n}\widetilde{E}_{ij}^{+}L_{i}\xi_{ij}^{+}\exp(\gamma\xi_{ij}^{+})
+\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}\overline{T}_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{i}r+|f_{i}(0)|)\nonumber\\
&\times&\left.\left.\left.\exp(\gamma \chi_{i}^{+})\exp(\gamma
\chi_{k}^{+})\right)\right)\right]\hat{e}_{-c_{j}\oplus_{\nu}\gamma}(t_{1},t_{0})
+\frac{1}{c_{j}^{-}-\gamma}\left(\exp\left(\gamma\sup\limits_{s\in\mathbb{T}}\nu(s)
\left(c_{j}^{+}\varsigma_{j}^{+}\exp(\gamma\varsigma_{j}^{+})\right.\right.\right.\nonumber\\
&+&\sum\limits_{i=1}^{n}E_{ij}^{+}L_{i}+\sum\limits_{i=1}^{n}(E_{ij}^{\tau})^{+}L_{i}\exp(\gamma\tau_{ij}^{+})
+\sum\limits_{i=1}^{n}\overline{E}_{ij}^{+}\sigma_{ij}^{+}L_{i}\exp(\gamma\sigma_{ij}^{+})\nonumber\\
&+&\sum\limits_{i=1}^{n}\widetilde{E}_{ij}^{+}L_{i}\xi_{ij}^{+}\exp(\gamma\xi_{ij}^{+})
+\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}\overline{T}_{ijk}^{+}(L_{k}r+|f_{k}(0)|)(L_{i}r+|f_{i}(0)|)\nonumber\\
&\times&\left.\left.\left.\left.\exp(\gamma \chi_{i}^{+})\exp(\gamma \chi_{k}^{+})\right)\right)\right)\right\}\nonumber\\
&\leq& d\varpi K \hat{e}_{\ominus_{\nu}
\gamma}(t_{1},t_{0})\|\psi\|_{0}.
\end{eqnarray*}
We can easily obtain some upper bound of the derivative
$|u_{i}^{\Delta}(t_{1})|$ and $|u_{i}^{\Delta}(t_{1})|$ as follow:
\begin{equation}
|u_{i}^{\Delta}(t_{1})|\leq d\varpi K \hat{e}_{\ominus_{\nu}
\gamma}(t_{1},t_{0})\|\psi\|_{0},
\end{equation}
and
\begin{equation}\label{from1}
|v_{j}^{\Delta}(t_{1})|\leq d\varpi K \hat{e}_{\ominus_{\nu}
\gamma}(t_{1},t_{0})\|\psi\|_{0}.
\end{equation}
From (\ref{from})-(\ref{from1}), we obtain
\begin{equation}
\|Z(t_{1})-Z^{*}(t_{1})\|< d\varpi K \hat{e}_{\ominus_{\nu}
\gamma}(t_{1},t_{0})\|\psi\|_{0}.
\end{equation}
which contradicts (\ref{abcd73}), therefore (\ref{abcd72}) holds.
Letting $\varpi\longrightarrow1$, then (\ref{abcd71}) holds. Which
implies that only Stepanov-like weighted pseudo-almost automorphic
on time-space scales solution of system (\ref{eq1}) is globally
exponentially stable.
\subsection{Convergence}
\begin{definition} (\cite{adnene+ahmed}) For each $t\in\mathbb{T}$, let $N$ be a neighborhood of $t$.
Then, we define the generalized derivative (or Dini derivative on
time-space scales) $D^{+}V^{\Delta}(t)$, to mean that, given
$\epsilon>0$, there exists a right neighborhood $N_{\epsilon}\subset
N$ of $t$ such
\begin{equation*}
D^{+}V^{\Delta}(t)=D^{+}V^{\Delta}(t,x(t))=\frac{V(\sigma(t)),x(\sigma(t))-V(t,x(t)}{\nu(t)}.
\end{equation*}
\end{definition}
\begin{theorem}\label{th3}
Suppose that assumptions ($H_{1}$)-($H_{4}$) hold.\\
Let $h^{\ast}\left( \cdot\right)=\left(x_{1}^{\ast
}\left(\cdot\right),\cdots
,x_{n}^{\ast}\left(\cdot\right),y_{1}^{\ast}\left(\cdot\right),...,y_{m}^{\ast}\left(\cdot\right)\right)^{T}$
be a Stepanov-like weighted pseudo-almost automorphic on time-space
scales solution of system (\ref{eq1}). If
\begin{multline*}
\alpha_{i}^{-}-\sum\limits_{j=1}^{m}\left[L_{j}\left(D_{ij}^{+}+(D_{ij}^{\tau})^{+}
+(\overline{D}_{ij})^{+}\sigma_{ij}^{+}+(\widetilde{D}_{ij})^{+}\xi_{ij}^{+}
+\sum\limits_{k=1}^{m}T_{ijk}^{+}(L_{k}r+|f_{k}(0)|)\right)\right.\\
+\left.(L_{j}r+|f_{j}(0)|)\sum\limits_{k=1}^{m}T_{ijk}^{+}L_{k}\right]>0,
\end{multline*}
and
\begin{multline*}
c_{j}^{-}-\sum\limits_{i=1}^{n}\left[L_{i}\left(E_{ij}^{+}+(E_{ij}^{\tau})^{+}
+(\overline{E}_{ij})^{+}\sigma_{ij}^{+}+(\widetilde{E}_{ij})^{+}\xi_{ij}^{+}
+\sum\limits_{k=1}^{n}T_{ijk}^{+}(L_{k}r+|f_{k}(0)|)\right)\right.\\
+\left.(L_{i}r+|f_{i}(0)|)\sum\limits_{k=1}^{n}T_{ijk}^{+}L_{k}\right]>0,
\end{multline*}
then all solutions
$\psi=(\varphi_{1},...,\varphi_{n},\phi_{1},...,\phi_{n})$ of
(\ref{eq1}) satisfying
\begin{equation*}
x_{i}^{\ast}\left(0\right)=\varphi_{i}\left(0\right),\,\
y_{j}^{\ast}\left(0\right)=\phi_{j}\left(0\right),1\leq i\leq n, \,\
1\leq j\leq m
\end{equation*}
converge to its unique Stepanov-like weighted pseudo-almost
automorphic on time-space scales solution $h^{\ast}$.
\end{theorem}
Proof. Let
$h^{\ast}\left(\cdot\right)=\left(x_{1}^{*}\left(\cdot\right),...,x_{n}^{*}\left(\cdot\right),
y_{1}^{*}\left(\cdot\right),...,y_{m}^{*}\left(\cdot \right)\right)$
be a solution of (\ref{eq1}) and
$\psi\left(\cdot\right)=\left(\varphi_{1}\left(\cdot
\right),...,\varphi_{n}\left(\cdot\right),\phi_{1}\left(\cdot\right),...,\phi_{m}\left(\cdot\right)\right)$
be a Stepanov-like weighted pseudo almost automorphic on time-space
scales solution of (\ref{eq1}). First, one verifies without
difficulty that
\begin{eqnarray*}
&&\left(x_{i}^{\ast}\left(t\right)-\alpha_{i}(t)\int_{t-\eta_{i}(t)}^{t}x_{i}(u)\Delta
u\right)^{\Delta}
-\left(\varphi_{i}^{\ast}\left(t\right)-\alpha_{i}(t)\int_{t-\eta_{i}(t)}^{t}\varphi_{i}(u)\Delta u\right)^{\Delta}\\
&=&-\alpha_{i}(t)\left(x_{i}^{\ast}\left(t-\eta_{i}(t)\right)-\varphi_{i}\left(t-\eta_{i}(t)\right)
\right)+\sum\limits_{j=1}^{m}D_{ij}\left(t\right)\left[f_{j}(x_{j}^{\ast}\left(
t\right))-f_{j}(\varphi_{j}\left(t\right))\right]\\
&+&\sum\limits_{j=1}^{m}D_{ij}^{\tau}\left(t\right)\left[f_{j}(x_{j}^{\ast}\left(t-\tau_{ij}(t)\right))-f_{j}(\varphi_{j}\left(t-\tau_{ij}(t)\right))\right]
\\&+&\sum\limits_{j=1}^{m}\overline{D}_{ij}\left(t\right)\int\limits_{t-\sigma_{ij}(t)}^{t}\left[
f_{j}(x_{j}^{\ast}\left(t-u\right))
-f_{j}(\varphi_{j}\left( t-u\right))\right]\Delta u\\
&+&\sum\limits_{j=1}^{m}\widetilde{D}_{ij}\left(t\right)\int\limits_{t-\xi_{ij}(t)}^{t}\left(f_{j}\left((x_{j}^{\ast})^{\Delta}\left(u\right)\right)-f_{j}\left(\varphi_{j}^{\Delta}\left(u\right)\right)\right)\Delta
u\\&+&\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{m}T_{ijk}(t)
\left\vert
f_{k}(x_{k}(t-\chi_{k}(t)))f_{j}(x_{j}(t-\chi_{j}(t)))-f_{k}(\varphi_{k}(t-\chi_{k}(t)))f_{j}(\varphi_{j}(t-\chi_{j}(t)))\right\vert,
\end{eqnarray*}
and
\begin{eqnarray*}
&&\left(y_{j}^{\ast}\left(t\right)-c_{j}(t)\int_{t-\varsigma_{j}(t)}^{t}y_{j}(u)\Delta
u\right)^{\Delta}
-\left(\phi_{j}^{\ast}\left(t\right)-c_{j}(t)\int_{t-\varsigma_{j}(t)}^{t}\phi_{j}(u)\Delta u\right)^{\Delta}\\
&=&-c_{j}(t)\left(y_{j}^{\ast}\left(t-\varsigma_{j}(t)\right)-\phi_{j}\left(t-\varsigma_{j}(t)\right)
\right)
+\sum\limits_{j=1}^{n}E_{ij}\left(t\right)\left[f_{j}(x_{j}^{\ast}\left(
t\right))-f_{j}(\varphi_{j}\left(t\right))\right]\\
&+&\sum\limits_{j=1}^{n}E_{ij}^{\tau}\left(t\right)\left[f_{j}(x_{j}^{\ast}\left(t-\tau_{ij}(t)\right))-f_{j}(\varphi_{j}\left(t-\tau_{ij}(t)\right))\right]
\\&+&\sum\limits_{j=1}^{n}\overline{E}_{ij}\left(t\right)\int\limits_{t-\sigma_{ij}(t)}^{t}\left[
f_{j}(x_{j}^{\ast}\left(t-u\right))
-f_{j}(\varphi_{j}\left( t-u\right))\right]\Delta u\\
&+&\sum\limits_{j=1}^{n}\widetilde{E}_{ij}\left(t\right)\int\limits_{t-\xi_{ij}(t)}^{t}\left(f_{j}\left((x_{j}^{\ast})^{\Delta}\left(u\right)\right)-f_{j}\left(\varphi_{j}^{\Delta}\left(u\right)\right)\right)\Delta u\\
&+&\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\sum\limits_{k=1}^{n}\overline{T}_{ijk}(t)
\left\vert
f_{k}(x_{k}(t-\chi_{k}(t)))f_{j}(x_{j}(t-\chi_{j}(t)))-f_{k}(\varphi_{k}(t-\chi_{k}(t)))f_{j}(\varphi_{j}(t-\chi_{j}(t)))\right\vert,
\end{eqnarray*}
Now, consider the following (ad-hoc) Lyapunov-Krasovskii functional
\begin{equation*}
\begin{array}{cccc}
V: & \mathbb{R} & \longrightarrow & S^{p}WPAP\left(\mathbb{T},\mathbb{R}^{n}\right) \\
& t & \longmapsto & V_{1}(t)+V_{2}(t)+V_{3}(t)+V_{4}(t)+V_{5}(t),
\end{array}
\end{equation*}
where
\begin{eqnarray*}
V_{1}(t)&=&\sum\limits_{i=1}^{n}\left\vert \left(x_{i}^{\ast }\left(
t\right)-\alpha_{i}(t)\int_{t-\eta_{i}(t)}^{t}x_{i}(u)\Delta
u\right)
-\left(\varphi_{i}^{\ast }\left( t\right)-\alpha_{i}(t)\int_{t-\eta_{i}(t)}^{t}\varphi_{i}(u)\Delta u\right)\right\vert,\\
V_{2}\left(t\right)&=&\sum\limits_{j=1}^{n}\sum\limits_{i=1}^{n}\int\limits_{t-\tau_{j}(t)
}^{t}L_{j}\left(D_{ij}^{+}+(D_{ij}^{\tau})^{+}\right)\left\vert
x_{i}^{\ast}\left( s\right)
-\varphi_{i}\left( s\right)\right\vert\Delta s,\\
V_{3}\left(t\right)
&=&\sum\limits_{j=1}^{n}\sum\limits_{i=1}^{n}\int\limits_{t-\sigma_{ij}(t)}^{t
}\int\limits_{t-s}^{t}L_{j}\overline{D}_{ij}^{+}\left\vert
x_{i}^{\ast
}\left(u\right)-\varphi_{i}\left( u\right)\right\vert \Delta u \Delta s,\\
V_{4}\left( t\right)
&=&\sum\limits_{j=1}^{n}\sum\limits_{i=1}^{n}\int\limits_{t-\xi_{ij}(t)}^{t}\int\limits_{s}^{0}L_{j}\widetilde{D}_{ij}^{+}\left\vert
(x_{i}^{\ast})^{\Delta}\left(
u\right)-(\varphi_{i})^{\Delta}\left(u\right)\right\vert\Delta
u\Delta s,
\end{eqnarray*}
and
\begin{eqnarray*}
V_{5}(t)&=&\sum\limits_{j=1}^{n}\left\vert \left(y_{j}^{\ast }\left(
t\right)-c_{j}(t)\int_{t-\varsigma_{j}(t)}^{t}y_{j}(u)\Delta
u\right)
-\left(\phi_{j}^{\ast}\left(t\right)-c_{j}(t)\int_{t-\varsigma_{j}(t)}^{t}\phi_{j}(u)\Delta
u\right)\right\vert.
\end{eqnarray*}
Let us calculate the upper right Dini derivative on time-space
scales $D^{+}V^{\Delta}\left(t\right)$ of $V$ along the trajectory
of the solution of the equation above. Then one has
\begin{eqnarray*}
D^{+}V_{1}^{\Delta}(t) &\leq
&-\sum\limits_{i=1}^{m}\alpha_{i}^{-}\left\vert
x_{i}^{\ast}\left(t\right)-\varphi_{i}\left(t\right)\right\vert
+\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{m}D_{ij}^{+} L_{j}\left\vert x_{j}^{\ast}\left(t\right)-\varphi _{j}\left( t \right) \right\vert\\
&+&\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{m}(D_{ij}^{\tau})^{+}L_{j}\left\vert
x_{j}^{\ast}\left(t-\tau_{j}(t)\right)
-\varphi_{j}\left(t-\tau_{j}(t)\right)\right\vert \\
&+&\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{m}\overline{D}_{ij}^{+}\int\limits_{t-\sigma_{ij}(t)}^{t}L_{j}\left\vert(x_{j}^{\ast}\left(
t-u\right)
-\varphi_{j}\left( t-u\right) \right\vert \Delta u\\
&+&\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{m}
\widetilde{D}_{ij}^{+}\int\limits_{t-\xi_{ij}(t)}^{t}
L_{j}\left\vert(x_{j}^{\ast})^{\Delta}
\left( u\right)-\varphi_{j}^{\Delta }\left( u\right)\right\vert \Delta u\\
&+&\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{m}T_{ijk}^{+} (L_{k}r+|f_{k}(0)|)L_{j}\left\vert x_{j}(t-\chi_{j}(t))-\varphi_{j}(t-\chi_{j}(t))\right\vert\\
&+&\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{m}\sum\limits_{k=1}^{m}T_{ijk}^{+}(L_{j}r+|f_{j}(0)|)L_{k}\left\vert
x_{k}(t-\chi_{k}(t))-\varphi_{k}(t-\chi_{k}(t))\right\vert
\end{eqnarray*}
Obviously,
\begin{eqnarray*}
D^{+}V_{2}^{\Delta}(t) &\leq
&\sum\limits_{j=1}^{m}\sum\limits_{i=1}^{m}L_{j}(D_{ij}^{+}+(D_{ij}^{\tau})^{+})\left[\left\vert
x_{i}^{\ast }\left(t\right)-\varphi_{i}\left(t\right)
\right\vert-\left\vert x_{i}^{\ast }\left(t-\tau_{ij}(t) \right)-\varphi_{i}\left(t-\tau_{ij}(t)\right) \right\vert \right]\\
&\leq&\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{m}L_{j}(D_{ij}^{+}
+(D_{ij}^{\tau})^{+})\left\vert x_{i}^{\ast}\left( t\right)-\varphi_{i}\left( t\right) \right\vert\\
&-&\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{m}L_{j}(D_{ij}^{+}+(D_{ij}^{\tau})^{+})\left\vert
x_{i}^{\ast}\left(t-\tau_{j}(t)\right)-\varphi_{i}\left(t-\tau_{j}(t)\right)\right\vert
\end{eqnarray*}
and
\begin{eqnarray*}
D^{+}V_{3}^{\Delta}(t)&\leq&\sum\limits_{j=1}^{m}\sum\limits_{i=1}^{m}L_{j}\overline{D}_{ij}^{+}\int_{t-\sigma_{ij}(t)}^{t}
\left[\left\vert x_{i}^{\ast }\left(t\right)-\varphi_{i}\left(
t\right) \right\vert
-\left\vert x_{i}^{\ast}\left(t-s\right)-\varphi_{i}\left(t-s\right)\right\vert\right]\Delta s \\
&\leq&\sum\limits_{j=1}^{m}\sum\limits_{i=1}^{m}L_{j}\overline{D}_{ij}^{+}\int_{t-\sigma_{ij}(t)}^{t}\left\vert
x_{i}^{\ast }\left( t\right)-\varphi_{i}\left( t\right) \right\vert \Delta s\\
&-&\sum\limits_{j=1}^{m}\sum\limits_{i=1}^{m}L_{j}
\overline{D}_{ij}^{+}\int_{t-\sigma_{ij}(t)}^{t}\left\vert
x_{i}^{\ast }\left(t-s\right)-\varphi_{i}\left(t-s\right)\right\vert
\Delta s.
\end{eqnarray*}
Reasoning in a similar way, we can obtain the following estimation
\begin{eqnarray*}
D^{+}V_{4}^{\Delta}\left(t\right)&=&\sum\limits_{j=1}^{m}\sum\limits_{i=1}^{m}\int_{t-\xi_{ij}(t)}^{t}\int\limits_{s}^{0}L_{j}\widetilde{D}_{ij}^{+}\left\vert
(x_{i}^{\ast})^{\Delta}\left(u\right)-\varphi_{i}^{\Delta}\left(u\right)
\right\vert \Delta u\,\ \Delta s \\
&\leq&-\sum\limits_{j=1}^{m}\sum\limits_{i=1}^{m}L_{j}\widetilde{D}_{ij}^{+}\int_{t-\xi_{ij}(t)}^{t}\left\vert
(x_{i}^{\ast})^{\Delta}\left(s\right)-\varphi_{i}^{\Delta}\left(s\right)\right\vert
\Delta s,
\end{eqnarray*}
and,
\begin{eqnarray*}
D^{+}V_{5}^{\Delta}\left(t\right)&\leq
&-\sum\limits_{j=1}^{n}c_{j}^{-}\left\vert
y_{j}^{\ast}\left(t\right)-\phi_{j}\left(t\right)\right\vert
+\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}E_{ij}^{+} L_{j}\left\vert y_{j}^{\ast}\left(t\right)-\phi _{j}\left( t \right) \right\vert\\
&+&\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}(E_{ij}^{\tau})^{+}L_{j}\left\vert
y_{j}^{\ast}\left(t-\tau_{j}(t)\right)
-\phi_{j}\left(t-\tau_{j}(t)\right)\right\vert \\
&+&\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\overline{E}_{ij}^{+}\int\limits_{t-\sigma_{ij}(t)}^{t}L_{j}\left\vert(y_{j}^{\ast}\left(
t-u\right)
-\phi_{j}\left( t-u\right) \right\vert \Delta u\\
&+&\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}
\widetilde{E}_{ij}^{+}\int\limits_{t-\xi_{ij}(t)}^{t}
L_{j}\left\vert(y_{j}^{\ast})^{\Delta} \left(
u\right)-\phi_{j}^{\Delta }\left( u\right)\right\vert \Delta
u+\left\vert y_{j}(t)-\phi_{j}(t)\right\vert
\end{eqnarray*}
\begin{eqnarray*}
&+&\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\sum\limits_{k=1}^{n}\overline{T}_{ijk}^{+} (L_{k}r+|f_{k}(0)|)L_{j}\left\vert y_{j}(t-\chi_{j}(t))-\phi_{j}(t-\chi_{j}(t))\right\vert\\
&+&\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\sum\limits_{k=1}^{n}\overline{T}_{ijk}^{+}(L_{j}r+|f_{j}(0)|)L_{k}\left\vert
y_{k}(t-\chi_{k}(t))-\varphi_{k}(t-\chi_{k}(t))\right\vert
\end{eqnarray*}
By using the inequality of the Dini derivative on time-space scales
\begin{equation*}
D^{+}\left(F_{1}^{\Delta}+F_{2}^{\Delta}\right)\leq
D^{+}\left(F_{1}^{\Delta}\right) +D^{+}\left(F_{2}^{\Delta}\right),
\end{equation*}
we get
\begin{eqnarray*}
D^{+}\left( V^{\Delta}(t)\right) &\leq
&D^{+}V_{1}^{\Delta}(t)+D^{+}V_{2}^{\Delta}(t)+D^{+}V_{3}^{\Delta}(t)+D^{+}V_{4}^{\Delta}(t) \\
&\leq &-\sum\limits_{i=1}^{m}
\sum\limits_{j=1}^{n}\min\left\{\alpha_{i}^{+}-D_{ij}^{+}L_{j}-(D_{ij}^{\tau})^{+}L_{j}-\overline{D}_{ij}^{+}\sigma_{ij}^{+}L_{j}\right.\\
&-&\widetilde{D}_{ij}^{+}\xi_{ij}^{+}L_{j}+L_{j}\sum\limits_{k=1}^{n}T_{ijk}^{+}(L_{k}r+|f_{k}(0)|)
+(L_{j}r+|f_{j}(0)|)\sum\limits_{k=1}^{n}T_{ijk}^{+}L_{k},\\
&&
c_{j}^{+}-E_{ij}^{+}L_{j}-(E_{ij}^{\tau})^{+}L_{j}-\overline{E}_{ij}^{+}\sigma_{ij}^{+}L_{j}
-\widetilde{E}_{ij}^{+}\xi_{ij}^{+}L_{j}+L_{j}\sum\limits_{k=1}^{m}\overline{T}_{ijk}^{+}(L_{k}r+|f_{k}(0)|)\\
&+&\left.(L_{j}r+|f_{j}(0)|)\sum\limits_{k=1}^{m}\overline{T}_{ijk}^{+}L_{k}\right\} \| h^{\ast}\left(t\right) -\psi\left( t\right) \| \\
&=&-\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}\min\{\beta
_{i},\beta_{j}\}\| h^{\ast }\left(t\right)-\psi\left(t\right)\| <0.
\end{eqnarray*}
By integrating the above inequality from $t_{0}$ to $t$, we get
\begin{equation*}
V(t)+\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}\min\{\beta
_{i},\beta_{j}\}\int\limits_{t_{0}}^{t}\|
y^{\ast}\left(t\right)-\psi\left(t\right)\|\Delta
s<V(t_{0})<+\infty.
\end{equation*}
Now, we remark that $V(t)>0.$ It follows that
\begin{equation*}
\lim\limits_{t\rightarrow+\infty}\sup\int\limits_{t_{0}}^{t}\min\{\beta
_{i},\beta_{j}\}\| h^{\ast}\left(s\right) -\psi\left( s\right)\|
\Delta s<V(t_{0})<+\infty.
\end{equation*}
Note that $h^{\ast }\left(\cdot\right)$ is bounded on
$\mathbb{T}^{+}$. Therefore
\begin{equation*}
\lim\limits_{t\rightarrow+\infty}\|
h^{\ast}\left(t\right)-\psi\left(t\right)\|=0.
\end{equation*}
The proof of this theorem is now completed.
\begin{remark}
Theorem \ref{th0}, Theorem \ref{th1}, Theorem \ref{th2} and Theorem
\ref{th3} are new even for the both cases of differential equations
($\mathbb{T}=\mathbb{R}$) and difference equations
($\mathbb{T}=\mathbb{Z}$).
\end{remark}
\section{Numerical example}
In system (\ref{eq1}), let $n=3,\,\ m=2$, and take coefficients as
follows:
\begin{multline*}
\left(
\begin{array}{ccc}
D_{11}(t) & D_{12}(t) \\
D_{21}(t) & D_{22}(t) \\
D_{31}(t) & D_{32}(t) \\
\end{array}
\right)=\left(
\begin{array}{ccc}
D^{\tau}_{11}(t) & D^{\tau}_{12}(t) \\
D^{\tau}_{21}(t) & D^{\tau}_{22}(t) \\
D^{\tau}_{31}(t) & D^{\tau}_{32}(t) \\
\end{array}
\right)=\left(
\begin{array}{ccc}
\tilde{D}_{11}(t) & \tilde{D}_{12}(t) \\
\tilde{D}_{21}(t) & \tilde{D}_{22}(t) \\
\tilde{D}_{31}(t) & \tilde{D}_{32}(t) \\
\end{array}
\right)\\=\left(
\begin{array}{ccc}
\bar{D}_{11}(t) & \bar{D}_{12}(t) \\
\bar{D}_{21}(t) & \bar{D}_{22}(t) \\
\bar{D}_{31}(t) & \bar{D}_{32}(t) \\
\end{array}
\right)=\left(
\begin{array}{ccc}
\frac{\sin(t)}{20} & \frac{\sin(t)}{20} \\
\frac{\cos(t)}{20} & \frac{\sin(t)}{20} \\
\frac{\cos(t)}{20} & \frac{\sin(t)}{20} \\
\end{array}
\right);
\\
\left(
\begin{array}{ccc}
E_{11}(t) & E_{12}(t) \\
E_{21}(t) & E_{22}(t) \\
E_{31}(t) & E_{32}(t) \\
\end{array}
\right)=\left(
\begin{array}{ccc}
E^{\tau}_{11}(t) & E^{\tau}_{12}(t) \\
E^{\tau}_{21}(t) & E^{\tau}_{22}(t) \\
E^{\tau}_{31}(t) & E^{\tau}_{32}(t) \\
\end{array}
\right)=\left(
\begin{array}{ccc}
\tilde{E}_{11}(t) & \tilde{E}_{12}(t) \\
\tilde{E}_{21}(t) & \tilde{E}_{22}(t) \\
\tilde{E}_{31}(t) & \tilde{E}_{32}(t) \\
\end{array}
\right)
\\=\left(
\begin{array}{ccc}
\bar{E}_{11}(t) & \bar{E}_{12}(t) \\
\bar{E}_{21}(t) & \bar{E}_{22}(t) \\
\bar{E}_{31}(t) & \bar{E}_{32}(t) \\
\end{array}
\right)=\left(
\begin{array}{ccc}
\frac{\sin(t)}{15} & \frac{\sin(t)}{15} \\
\frac{\cos(t)}{15} & \frac{\sin(t)}{15} \\
\frac{\cos(t)}{15} & \frac{\sin(t)}{15} \\
\end{array}
\right);\\
\alpha_{1}(t)=\alpha_{2}(t)=0.73+0.02\sin\frac{1}{2+\cos(t)+\cos(\sqrt{2}t)};\\
f_{1}(x)=f_{2}(x)=0.1\arctan x;\,\ \nu(t)=\hat{e}(t,\sigma(t)) \,\ \text{for all} \,\ t\in[0,\infty)_{\mathbb{T}}; \\ \nu(t)=1 \,\ \text{for all}\,\ t\in(-\infty,0)_{\mathbb{T}};\\
c_{1}(t)=0.54-0.02\sin\frac{1}{2+\cos(t)+\cos(\sqrt{2}t)};\\
c_{2}(t)=0.54+0.02\sin\frac{1}{2+\cos(t)+\cos(\sqrt{2}t)};\\
I_{1}(t)=0.01\sin\frac{1}{2+\sin(t)+\sin(\sqrt{2}t)}+0.01e^{-t^{4}\cos^{2}(t)};\\
I_{2}(t)=0.02\sin\frac{1}{2+\sin(t)+\sin(\sqrt{2}t)}+0.02e^{-t^{4}\cos^{2}(t)};\phantom{+++++}\\
J_{1}(t)=0.02\sin\frac{1}{2+\cos(t)+\cos(\sqrt{2}t)}+0.02e^{-t^{2}\sin^{4}(t)};\\
J_{2}(t)=0.1\sin\frac{1}{2+\sin(t)+\sin(\sqrt{2}t)};\\
\eta_{1}(t)=0.04\sin\frac{1}{2+\sin(t)+\sin(\sqrt{2}t)}+0.03e^{-t^{4}\cos^{2}(t)};\\
\eta_{2}(t)=0.01\sin\frac{1}{2+\cos(t)+\cos(\sqrt{2}t)}+0.01e^{-t^{4}\cos^{2}(t)};\\
\varsigma_{1}(t)=0.01\sin\frac{1}{2+\cos(t)+\cos(\sqrt{2}t)}+0.01e^{-t^{2}\sin^{4}(t)};\\
\varsigma_{2}(t)=0.01\sin\frac{1}{2+\cos(t)+\cos(\sqrt{2}t)}+0.01e^{-t^{2}\sin^{4}(t)};\\
\tau_{11}(t)=0.02\sin(\sqrt{2}t)+e^{-t^{2}\sin^{4}(t)};\\
\tau_{12}(t)=0.02\sin\frac{1}{2+\sin(t)+\sin(\sqrt{2}t)}+0.01e^{-t^{2}\sin^{4}(t)};\\
\tau_{21}(t)=0.01\sin\frac{1}{2+\cos(t)+\cos(\sqrt{2}t)}+0.01e^{-t^{4}\cos^{2}(t)};\\
\tau_{22}(t)=0.02\sin\frac{1}{2+\sin(t)+\sin(\sqrt{2}t)}+0.01e^{-t^{4}\cos^{2}(t)};\phantom{++++}
\end{multline*}
\begin{eqnarray*}
\sigma_{11}(t)&=&0.02\sin\frac{1}{2+\cos(t)+\cos(\sqrt{2}t)}+0.02e^{-t^{2}\sin^{4}(t)};\\
\sigma_{12}(t)&=&0.01\sin\frac{1}{2+\cos(t)+\cos(\sqrt{2}t)}+0.01e^{-t^{2}\sin^{4}(t)};\\
\sigma_{21}(t)&=&0.02\sin\frac{1}{2+\sin(t)+\sin(\sqrt{2}t)}+0.01e^{-t^{4}\cos^{2}(t)};\\
\sigma_{12}(t)&=&0.02\sin\frac{1}{2+\cos(t)+\cos(\sqrt{2}t)}+e^{-t^{2}\sin^{4}(t)};\\
\xi_{11}(t)&=&0.02\sin\frac{1}{2+\cos(t)+\cos(\sqrt{2}t)}+0.01e^{-t^{4}\cos^{2}(t)};\\
\xi_{12}(t)&=&0.03\sin\frac{1}{2+\sin(t)+\sin(\sqrt{2}t)}+0.02e^{-t^{2}\sin^{4}(t)};\\
\xi_{21}(t)&=&0.01\sin\frac{1}{2+\cos(t)+\cos(\sqrt{2}t)}+0.01e^{-t^{4}\cos^{2}(t)};\\
\xi_{22}(t)&=&0.02\sin\frac{1}{2+\sin(t)+\sin(\sqrt{2}t)}+0.02e^{-t^{2}\sin^{4}(t)}.
\end{eqnarray*}
By a simple calculation, we have
\begin{multline*}
\left(
\begin{array}{ccc}
D_{11}^{+} & D_{12}^{+} \\
D_{21}^{+} & D_{22}^{+} \\
D_{31}^{+} & D_{32}^{+} \\
\end{array}
\right)=\left(
\begin{array}{ccc}
(D^{\tau}_{11})^{+} & (D^{\tau}_{12})^{+} \\
(D^{\tau}_{21})^{+} & (D^{\tau}_{22})^{+} \\
(D^{\tau}_{31})^{+} & (D^{\tau}_{32})^{+} \\
\end{array}
\right)=\left(
\begin{array}{ccc}
\tilde{D}_{11}^{+} & \tilde{D}_{12}^{+}\\
\tilde{D}_{21}^{+} & \tilde{D}_{22}^{+} \\
\tilde{D}_{31}^{+} & \tilde{D}_{32}^{+}\\
\end{array}
\right)=\left(
\begin{array}{ccc}
\bar{D}_{11}^{+} & \bar{D}_{12}^{+} \\
\bar{D}_{21}^{+} & \bar{D}_{22}^{+} \\
\bar{D}_{31}^{+} & \bar{D}_{32}^{+} \\
\end{array}
\right)=\left(
\begin{array}{ccc}
\frac{1}{20} & \frac{1}{20} \\
\frac{1}{20} & \frac{1}{20} \\
\frac{1}{20} & \frac{1}{20} \\
\end{array}
\right);
\\
\left(
\begin{array}{ccc}
E_{11}^{+} & E_{12}^{+} \\
E_{21}^{+} & E_{22}^{+} \\
E_{31}^{+} & E_{32}^{+} \\
\end{array}
\right)=\left(
\begin{array}{ccc}
(E^{\tau}_{11})^{+} & (E^{\tau}_{12})^{+} \\
(E^{\tau}_{21})^{+} & (E^{\tau}_{22})^{+} \\
(E^{\tau}_{31})^{+} & (E^{\tau}_{32})^{+} \\
\end{array}
\right)=\left(
\begin{array}{ccc}
\tilde{E}_{11}^{+} & \tilde{E}_{12}^{+}\\
\tilde{E}_{21}^{+} & \tilde{E}_{22}^{+} \\
\tilde{E}_{31}^{+} & \tilde{E}_{32}^{+}\\
\end{array}
\right)=\left(
\begin{array}{ccc}
\bar{E}_{11}^{+} & \bar{E}_{12}^{+} \\
\bar{E}_{21}^{+} & \bar{E}_{22}^{+} \\
\bar{E}_{31}^{+} & \bar{E}_{32}^{+} \\
\end{array}
\right)=\left(
\begin{array}{ccc}
\frac{1}{15} & \frac{1}{15} \\
\frac{1}{15} & \frac{1}{15} \\
\frac{1}{15} & \frac{1}{15} \\
\end{array}
\right);\\
I_{1}^{+}=0.02; \,\ I_{2}^{+}=0.04;\,\ I_{3}^{+}=0.05;\,\ J_{1}^{+}=0.01;\,\ J_{2}^{+}=-0.1;\\
\eta_{1}^{+}=0.07; \,\ \eta_{2}^{+}=0.02;\,\ \eta_{3}^{+}= \,\ \varsigma_{1}^{+}=0.02; \varsigma_{2}^{+}=0.02;\\
(\sigma_{ij}^{+})_{1\leq i\leq 3;1\leq j\leq 2}=\left(
\begin{array}{ccc}
1.04 & 1.02 \\
1.03 & 1.02 \\
1.03 & 1.02 \\
\end{array}
\right); (\xi_{ij}^{+})_{1\leq i\leq 3;1\leq j\leq 2}=\left(
\begin{array}{ccc}
1.03 & 1.05 \\
1.02 & 1.04 \\
1.02 & 1.04 \\
\end{array}
\right);\phantom{+++++++++++++}\\
(T_{1jk})_{2\times2}=\left[\begin{array}{ccc}
0.01\sin \sqrt{2}t & 0.01\cos \sqrt{2}t \\
0.01\cos \sqrt{2}t & 0.01\frac{\cos \sqrt{2}t}{2} \\
\end{array}
\right];\,\ (T_{2jk})_{2\times2}=\left[
\begin{array}{ccc}
0.01\sin \sqrt{2}t & 0.01\frac{\cos \sqrt{2}t}{2} \\
0.01\sin \sqrt{2}t & 0.01\sin \sqrt{2}t \\
\end{array}
\right];\\ (T_{3jk})_{2\times2}=\left[
\begin{array}{ccc}
0.01\sin \sqrt{2}t & 0.01\frac{\cos \sqrt{2}t}{2} \\
0.01\sin \sqrt{2}t & 0.01\sin \sqrt{2}t \\
\end{array}
\right]; (\overline{T}_{1jk})_{2\times2}=\left[\begin{array}{ccc}
0.01\sin \sqrt{2}t & 0.01\cos \sqrt{2}t1 \\
0.01\cos \sqrt{2}t & 0.01\frac{\cos \sqrt{2}t}{2} \\
\end{array}
\right];\\ (\overline{T}_{2jk})_{2\times2}=\left[
\begin{array}{ccc}
0.01\sin \sqrt{2}t & 0.01\frac{\cos \sqrt{2}t}{2} \\
0.01\sin \sqrt{2}t & 0.01\sin \sqrt{2}t \\
\end{array}
\right];\,\ (\overline{T}_{3jk})_{2\times2}=\left[
\begin{array}{ccc}
0.01\sin \sqrt{2}t & 0.01\frac{\cos \sqrt{2}t}{2} \\
0.01\sin \sqrt{2}t & 0.01\sin \sqrt{2}t \\
\end{array}
\right];
\end{multline*}
We can take $L_{1}=L_{2}=0.1$, $r=0.43$ and we have
\begin{eqnarray*}
M_{1}&=&\alpha_{1}^{+}\eta_{1}^{+}r+\sum\limits_{j=1}^{2}\left(D_{1j}^{+}+(D_{1j}^{\tau})^{+}+\overline{D}_{1j}^{+}\sigma_{1j}^{+}\right.
+\left.\widetilde{D}_{1j}^{+}\xi_{1j}^{+}\right)(L_{j}r+|f_{j}(0)|)\\
&+&\sum\limits_{j=1}^{2}\sum\limits_{k=1}^{2}T_{1jk}^{+}(L_{k}r+|f_{k}(0)|)(L_{j}r+|f_{j}(0)|)+I_{1}^{+},\\
&=& 0.75\times0.07\times0.43+\left(\frac{1}{15}+\frac{1}{15}+\frac{1}{15}\times 1.04+\frac{1}{15}\times 1.03\right)\left(0.1\times0.43+0.1\right)\\
&+& \left(\frac{1}{15}+\frac{1}{15}+\frac{1}{15}\times 1.02+\frac{1}{15}\times 1.05\right)\left(0.1\times0.43+0.1\right)\\
&+&0.04\left(0.1\times0.43+0.1\right)^{2}+0.02\\
&=& 0.119\\
\overline{M}_{1}&=&\alpha_{1}^{+}\eta_{1}^{+}+\sum\limits_{j=1}^{2}\left(D_{1j}^{+}+(D_{1j}^{\tau})^{+}+\overline{D}_{1j}^{+}\sigma_{ij}^{+}
+\widetilde{D}_{1j}^{+}\xi_{1j}^{+}\right)L_{j}\\
&+& \sum\limits_{j=1}^{2}\sum\limits_{k=1}^{2}(T_{1jk}^{+}+T_{1kj}^{+})(L_{k}r+|f_{k}(0)|)\\
&=& 0.75\times0.07+\left(\frac{1}{15}+\frac{1}{15}+\frac{1}{15}\times 1.03+\frac{1}{15}\times 1.02\right)\left(0.1\times0.43+0.1\right)\\
&+&\left(\frac{1}{15}+\frac{1}{15}+\frac{1}{15}\times 1.02+\frac{1}{15}\times 1.04\right)\left(0.1\times0.43+0.1\right)+0.08\left(0.1\times0.43+0.1\right)\\
&=& 0.139\\
N_{1}&=&c_{1}^{+}\varsigma_{1}^{+}r+\sum\limits_{i=1}^{3}\left(E_{i1}^{+}+(E_{i1}^{\tau})^{+}+\overline{E}_{i1}^{+}\sigma_{i1}^{+}
+\widetilde{E}_{i1}^{+}\xi_{i1}^{+}\right)(L_{i}r+|f_{i}(0)|)\\
&+&\sum\limits_{i=1}^{3}\sum\limits_{k=1}^{3}\overline{T}_{i1k}^{+}\left(L_{k}r+\left\vert f_{k}(0)\right\vert\right)\left(L_{i}r+\left\vert f_{i}(0)\right\vert\right)+J_{1}^{+}\\
&=& 0.56\times1.02\times0.43+\left(\frac{1}{20}+\frac{1}{20}+\frac{1}{20}\times 1.04+\frac{1}{20}\times 1.03\right)\left(0.1\times0.43+0.1\right)\\
&+& \left(\frac{1}{20}+\frac{1}{20}+\frac{1}{20}\times 1.03+\frac{1}{20}\times 1.02\right)\left(0.1\times0.43+0.1\right)+0.04\left(0.1\times0.43+0.1\right)^{2}+0.04\\
&=&0.343\\
\overline{N}_{1}&=&
c_{1}^{+}\varsigma_{1}^{+}+\sum\limits_{i=1}^{3}\left(E_{i1}^{+}+(E_{i1}^{\tau})^{+}
+\overline{E}_{i1}^{+}\sigma_{i1}^{+}+\widetilde{E}_{i1}^{+}\xi_{i1}^{+}\right)L_{i}\\
&+&\sum\limits_{i=1}^{3}\sum\limits_{k=1}^{3}(\overline{T}_{i1k}^{+}+\overline{T}_{ik1}^{+})(L_{k}r+|f_{k}(0)|)\\
&=&0.56\times 1.02+\left(\frac{6}{20}+1.04\times\frac{1}{20}+3.09\times\frac{1}{20}+2.04\times\frac{1}{20}\right)\times 0.1+0.018\\
&=&0.65 \\
M_{2}&=&\alpha_{2}^{+}\eta_{2}^{+}r+\sum\limits_{j=1}^{2}\left(D_{2j}^{+}+(D_{2j}^{\tau})^{+}+\overline{D}_{2j}^{+}\sigma_{2j}^{+}
+\widetilde{D}_{2j}^{+}\xi_{2j}^{+}\right)(L_{j}r+|f_{j}(0)|)\\
&+& \sum\limits_{j=1}^{2}\sum\limits_{k=1}^{2}(T_{2jk}^{+}+T_{2kj}^{+})(L_{k}r+|f_{k}(0)|)+I_{2}^{+}\\
&=&0.75\times0.02\times0.43+\left(\frac{1}{15}+\frac{1}{15}+\frac{1}{15}\times 1.03+\frac{1}{15}\times 1.02\right)\left(0.1\times0.43+0.1\right)\\
&+& \left(\frac{1}{15}+\frac{1}{15}+\frac{1}{15}\times 1.02+\frac{1}{15}\times 1.04\right)\left(0.1\times0.43+0.1\right)\\
&+&0.04+0.08\left(0.1\times0.43+0.1\right)\\
&=& 0.52
\end{eqnarray*}
\begin{eqnarray*}
\overline{M}_{2}&=&\alpha_{2}^{+}\eta_{2}^{+}+\sum\limits_{j=1}^{2}\left(D_{2j}^{+}+(D_{2j}^{\tau})^{+}+\overline{D}_{2j}^{+}\sigma_{2j}^{+}
+\widetilde{D}_{2j}^{+}\xi_{2j}^{+}\right)L_{j}\\
&+& \sum\limits_{j=1}^{2}\sum\limits_{k=1}^{2}T_{2jk}^{+}(L_{k}r+|f_{k}(0)|)(L_{j}r+|f_{j}(0)|)\\
&=& 0.75\times0.02+\left(\frac{1}{15}+\frac{1}{15}+\frac{1}{15}\times 1.03+\frac{1}{15}\times 1.02\right)\times0.1\\
&+& \left(\frac{1}{15}+\frac{1}{15}+\frac{1}{15}\times 1.02+\frac{1}{15}\times 1.04\right)\times0.1+0.04\left(0.1\times0.43+0.1\right)^{2}\\
&=& 0.069\\
N_{2}&=&c_{2}^{+}\varsigma_{2}^{+}r+\sum\limits_{i=1}^{3}\left(E_{i2}^{+}+(E_{i2}^{\tau})^{+}+\overline{E}_{i2}^{+}\sigma_{i2}^{+}
+\widetilde{E}_{i2}^{+}\xi_{i2}^{+}\right)(L_{i}r+|f_{i}(0)|)\\
&+&\sum\limits_{i=1}^{3}\sum\limits_{k=1}^{3}\overline{T}_{i2k}^{+}\left(L_{k}r+\left\vert f_{k}(0)\right\vert\right)\left(L_{i}r+\left\vert f_{i}(0)\right\vert\right)+J_{2}^{+}\\
&=& 0.213\\
\overline{N}_{2}&=&
c_{2}^{+}\varsigma_{2}^{+}+\sum\limits_{i=1}^{3}\left(E_{i2}^{+}+(E_{i2}^{\tau})^{+}
+\overline{E}_{i2}^{+}\sigma_{i2}^{+}+\widetilde{E}_{i2}^{+}\xi_{i2}^{+}\right)L_{i}\\
&+&\sum\limits_{i=1}^{3}\sum\limits_{k=1}^{3}(\overline{T}_{i2k}^{+}+\overline{T}_{ik2}^{+})(L_{k}r+|f_{k}(0)|)\\
&=&0.343\\
M_{3}&=&\alpha_{3}^{+}\eta_{3}^{+}r+\sum\limits_{j=1}^{2}\left(D_{3j}^{+}+(D_{3j}^{\tau})^{+}+\overline{D}_{3j}^{+}\sigma_{3j}^{+}\right.
+\left.\widetilde{D}_{3j}^{+}\xi_{3j}^{+}\right)(L_{j}r+|f_{j}(0)|)\\
&+&\sum\limits_{j=1}^{2}\sum\limits_{k=1}^{2}T_{3jk}^{+}(L_{k}r+|f_{k}(0)|)(L_{j}r+|f_{j}(0)|)+I_{3}^{+},\\
&=& 0.75\times0.07\times0.43+\left(\frac{1}{15}+\frac{1}{15}+\frac{1}{15}\times 0.04+\frac{1}{15}\times 0.03\right)\left(0.1\times0.43+0.1\right)\\
&+& \left(\frac{1}{15}+\frac{1}{15}+\frac{1}{15}\times 0.02+\frac{1}{15}\times 0.05\right)\left(0.1\times0.43+0.1\right)\\
&+&0.04\left(0.1\times0.43+0.1\right)^{2}+0.02\\
&=& 0.12\\
\overline{M}_{3}&=&\alpha_{3}^{+}\eta_{3}^{+}+\sum\limits_{j=1}^{2}\left(D_{3j}^{+}+(D_{3j}^{\tau})^{+}+\overline{D}_{3j}^{+}\sigma_{ij}^{+}
+\widetilde{D}_{3j}^{+}\xi_{3j}^{+}\right)L_{j}\\
&+& \sum\limits_{j=1}^{2}\sum\limits_{k=1}^{2}(T_{3jk}^{+}+T_{3kj}^{+})(L_{k}r+|f_{k}(0)|)\\
&=& 0.75\times0.07+\left(\frac{1}{15}+\frac{1}{15}+\frac{1}{15}\times 0.03+\frac{1}{15}\times 0.02\right)\left(0.1\times0.43+0.1\right)\\
&+&\left(\frac{1}{15}+\frac{1}{15}+\frac{1}{15}\times 0.02+\frac{1}{15}\times 0.04\right)\left(0.1\times0.43+0.1\right)+0.08\left(0.1\times0.43+0.1\right)\\
&=& 0.0136
\end{eqnarray*}
The conditions $(H_{1})$, $(H_{2})$ and $(H_{4})$ are satisfied and
it is easy to verify
\begin{multline*}
\max\left\{\frac{M_{1}}{\alpha_{1}^{-}},\frac{M_{2}}{\alpha_{2}^{-}},\frac{M_{3}}{\alpha_{3}^{-}},
\left(1+\frac{\alpha_{1}^{+}}{\alpha_{1}^{-}}\right)M_{1},
\left(1+\frac{\alpha_{2}^{+}}{\alpha_{2}^{-}}\right)M_{2},\left(1+\frac{\alpha_{3}^{+}}{\alpha_{3}^{-}}\right)M_{3},\right.\\
\left.\frac{N_{1}}{c_{1}^{-}},\left(1+\frac{c_{1}^{+}}{c_{1}^{-}}\right)N_{1},
\frac{N_{2}}{c_{2}^{-}},\left(1+\frac{c_{2}^{+}}{c_{2}^{-}}\right)N_{2}\right\}\leq
0.43,
\end{multline*}
and
\begin{multline*}
\max\left\{\frac{\overline{M}_{1}}{\alpha_{1}^{-}},\frac{\overline{M}_{2}}{\alpha_{2}^{-}},\frac{\overline{M}_{3}}{\alpha_{3}^{-}},\left(1+\frac{\alpha_{1}^{+}}{\alpha_{1}^{-}}\right)\overline{M}_{1},
\left(1+\frac{\alpha_{2}^{+}}{\alpha_{2}^{-}}\right)\overline{M}_{2},\left(1+\frac{\alpha_{3}^{+}}{\alpha_{3}^{-}}\right)\overline{M}_{3},\right.\\
\left.\frac{\overline{N}_{1}}{c_{1}^{-}},\frac{\overline{N}_{2}}{c_{2}^{-}},
\left(1+\frac{c_{1}^{+}}{c_{1}^{-}}\right)\overline{N}_{1},\left(1+\frac{c_{2}^{+}}{c_{2}^{-}}\right)\overline{N}_{2}\right\}\leq
1.
\end{multline*}
So, condition $(H_{3})$ holds. Therefore, using Theorem \ref{th0}, \ref{th1} and Theorem \ref{th2}, we conclude that the HOBAMs (\ref{eq1}) with the coefficients and parameters defined above has one and only one Stepanov-like weighted pseudo-almost periodic solution. Besides, this unique solution is globally exponentially stable.\\
Let $h^{\ast}\left( \cdot\right)=\left(x_{1}^{\ast
}\left(\cdot\right),x_{1}^{\ast
}\left(\cdot\right),x_{3}^{\ast}\left(\cdot\right),y_{1}^{\ast}\left(\cdot\right),y_{2}^{\ast}\left(\cdot\right)\right)^{T}$
be a Stepanov-like weighted pseudo-almost periodic on time-space
scales solution of system (\ref{eq1}). We have clearly for $i=1,2,3$
and $j=1,2$
\begin{multline*}
\alpha_{i}^{-}-\sum\limits_{j=1}^{2}\left[L_{j}\left(D_{ij}^{+}+(D_{ij}^{\tau})^{+}
+(\overline{D}_{ij})^{+}\sigma_{ij}^{+}+(\widetilde{D}_{ij})^{+}\xi_{ij}^{+}
+\sum\limits_{k=1}^{2}T_{ijk}^{+}(L_{k}r+|f_{k}(0)|)\right)\right.\\
+\left.(L_{j}r+|f_{j}(0)|)\sum\limits_{k=1}^{2}T_{ijk}^{+}L_{k}\right]>0,
\end{multline*}
and
\begin{multline*}
c_{j}^{-}-\sum\limits_{i=1}^{3}\left[L_{i}\left(E_{ij}^{+}+(E_{ij}^{\tau})^{+}
+(\overline{E}_{ij})^{+}\sigma_{ij}^{+}+(\widetilde{E}_{ij})^{+}\xi_{ij}^{+}
+\sum\limits_{k=1}^{3}T_{ijk}^{+}(L_{k}r+|f_{k}(0)|)\right)\right.\\
+\left.(L_{i}r+|f_{i}(0)|)\sum\limits_{k=1}^{3}T_{ijk}^{+}L_{k}\right]>0,
\end{multline*}
Then from Theorem \ref{th3} all solutions
$\psi=(\varphi_{1},\varphi_{2},\varphi_{3},\phi_{1},\phi_{2})$ of
(\ref{eq1}) satisfying
\begin{equation*}
x_{i}^{\ast }\left( 0\right) =\varphi _{i}\left( 0\right), \,\
y_{j}^{\ast }\left( 0\right) =\phi _{j}\left( 0\right), 1\leq i\leq
3,\,\ 1\leq j\leq 2,
\end{equation*}
converge to its unique Stepanov-like weighted pseudo-almost periodic
on time-space scales solution $h^{\ast}$.
\begin{remark}
The model studied in \cite{maas} is without the second-order
connection weights of delayed feedback. Moreover, the results
published in \cite{maas} can not be applicable for our example in
this brief. Hence, the results presented here are more general than
that outcomes published in \cite{maas}, \cite{bam0} and
\cite{leak3}. Consequently, this analysis of dynamics behavior of
the Stepanov weighted pseudo almost automorphic on time-space scales
solutions for HOBAMs model with Stepanov-like weighted pseudo almost
automorphic (SWPAA) coefficients and mixed delays improve the
previous study in \cite{maas}, \cite{bam0} and \cite{leak3}.
\end{remark}
\section{Conclusion and open problem}
In this paper, the Stepanov-like weighted pseudo-almost automorphic
on time-space scales are concerned for HOBAMs with mixed delays and
leakage time-varying delays by using Banachs fixed point theorem,
the theory of calculus on time scales and the Lyapunov-Krasovskii
functional method method. The results obtained in this paper are
completely new and complement the previously known works of (Ref.
\cite{maas,bam0,leak3}). Finally, numerical example was given to
demonstrate the effectiveness of our theory. As an interesting
problem, other almost automorphic on time scales type of solutions
for neural networks were deserved to be studied by applying some
suitable methods, such as the study of Stepanov-like weighted
pseudo-almost automorphic on time-space scales for Cohen Grossberg
BAM neural networks. The corresponding results will appear in the
near future. | {"config": "arxiv", "file": "1803.01039.tex"} |
TITLE: Numerically finding roots of function - converges?
QUESTION [2 upvotes]: Well this question was in my homework, I have difficulty to "proof" it (or more correctly: seeing how I would solve it).
Consider a floating point system ($s \cdot b^e$ where $1\leq s \leq 10 - 1 \cdot 10^{-15}$ $b = 10$, $-300 \leq e \leq 300$ with special symbols for minus sign, 0 and infinity added). (coincidentally this is almost the computer's floating point system)
Newton's method of approximating a function's root is applied in this system to the function $f(x) = x^2 + \tfrac{1}{x}$ which has a unique root at $x=-1$
a zero derivative at $ \hat{x} = \sqrt[3]{\tfrac{1}{2}}$ and a singularity at $x = 0$.
Does the iteration converge for $x_0 = \hat{x}$?
the answer should be "yes"
When ignoring the number system and simply solving it, a zeo derivative means the tangent is a horizontal line. So the tangent never crosses the x axis and $x_1 = \infty$. However $\hat{x}$ can't be described in the floating point system, hence it will be approximated and rounding will cause the derivative to become non 0.
However this still doesn't prove newton's method will converge, it may also stay around this "local minimum", right? Or is this enough to say it converges?
REPLY [0 votes]: Interval Newton method is perhaps more appropriate in place of the usual Newton's method when calculating with intervals (accounting for finite floating point representation), and particularly over intervals where the derivative contains zero, http://en.wikipedia.org/wiki/Interval_arithmetic#Interval_Newton_method | {"set_name": "stack_exchange", "score": 2, "question_id": 680763} |
\begin{document}
\maketitle
\begin{abstract}
We provide a rigorous derivation of Einstein's formula for the effective viscosity of dilute suspensions of $n$ rigid balls, $n \gg 1$, set in a volume of size $1$. So far, most justifications were carried under a strong assumption on the minimal distance between the balls: $d_{min} \ge c n^{-\frac{1}{3}}$, $c > 0$. We relax this assumption into a set of two much weaker conditions: one expresses essentially that the balls do not overlap, while the other one gives a control of the number of balls that are close to one another. In particular, our analysis covers the case of suspensions modelled by standard Poisson processes with almost minimal hardcore condition.
\end{abstract}
\section{Introduction}
Mixtures of particles and fluids, called {\em suspensions}, are involved in many natural phenomena and industrial processes. The understanding of their rheology, notably the so-called {\em effective viscosity} $\mu_{eff}$ induced by the particles, is therefore crucial. Many experiments or simulations have been carried out to determine $\mu_{eff}$ \cite{Guaz}. For $\lambda$ large enough, they seem to exhibit some generic behaviour, in terms of the ratio between the solid volume fraction $\lambda$ and the maximal flowable solid volume fraction $\lambda_c$, {\it cf.} \cite{Guaz}. Still, a theoretical derivation of the relation $\mu_{eff} = \mu_{eff}(\lambda/\lambda_c)$ observed experimentally is missing, due to the complex interactions involved: hydrodynamic interactions, direct contacts, \dots Mathematical works related to the analysis of suspensions are mostly limited to the {\em dilute regime}, that is when $\lambda$ is small.
\mspace
In these mathematical works, the typical model under consideration is as follows. One considers $n$ rigid balls $B_i = \overline{B(x_i, r_n)}$, $1 \le i \le n$, in a fixed compact subset of $\R^3$, surrounded by a viscous fluid.
The inertia of the fluid is neglected, leading to the Stokes equations
\begin{equation}
\label{Sto}
\left\{
\begin{aligned}
-\mu \Delta u_n + \na p_n & = f_n, \quad x \in \Omega_n = \R^3 \setminus \cup B_i , \\
\div u_n & = 0, \quad x \in \Omega_n , \\
u_n\vert_{B_i} & = u_{n,i} + \omega_{n,i} \times (x-x_i).
\end{aligned}
\right.
\end{equation}
The last condition expresses a no-slip condition at the rigid spheres, where the velocity is given by some translation velocities $u_{n,i}$ and some rotation vectors $\omega_{n,i}$, $1 \le i \le n$. We neglect the inertia of the balls: the $2n$ vectors $u_{n,i}, \omega_{n,i}$ can then be seen as Lagrange multipliers for the $2n$ conditions
\begin{equation}
\label{Sto2}
\begin{aligned}
\int_{\pa B_i} \sigma_\mu(u,p) \nu & = - \int_{B_i} f_n , \quad \int_{\pa B_i} \sigma_\mu(u,p) \nu \times (x-x_i) = - \int_{B_i} (x-x_i) \times f_n
\end{aligned}
\end{equation}
where $\sigma_\mu = 2\mu D(u) \nu - p \nu$ is the usual Newtonian tensor, and $\nu$ the normal vector pointing outward $B_i$.
\mspace
The general belief is that one should be able to replace \eqref{Sto}-\eqref{Sto2} by an effective Stokes model, with a modified viscosity taking into account the average effect of the particles:
\begin{equation}
\label{Stoeff}
\left\{
\begin{aligned}
-\div (2 \mu_{eff} D u_{eff} ) + \na p_{eff} & = f, \quad x \in \R^3, \\
\div u_{eff} & = 0, \quad x \in \R^3,
\end{aligned}
\right.
\end{equation}
with $D = \frac{1}{2}(\na + \na^t)$ the symmetric gradient. Of course, such average model can only be obtained asymptotically, namely when the number of particles $n$ gets very large. Moreover, for averaging to hold, it is very natural to impose some averaging on the distribution of the balls itself. Our basic hypothesis will therefore be the existence of a limit density, through
\begin{equation} \label{A0} \tag{A0}
\frac{1}{n} \sum_i \delta_{x_i} \xrightarrow[n \rightarrow +\infty]{} \rho(x) dx \quad \text{weakly in the sense of measures}
\end{equation}
where $\rho \in L^\infty(\R^3)$ is assumed to be zero outside a smooth open bounded set $\mO$. After playing on the length scale, we can always assume that $|\mO| = 1$. Of course, we expect $\mu_{eff}$ to be different from $\mu$ only in this region $\mO$ where the particles are present.
\mspace
The volume fraction of the balls is then given by $\lambda = \frac{4\pi}{3} n r_n^3$. We shall consider the case where $\lambda$ is small (dilute suspension), but independent of $n$ so as to derive a non-trivial effect as $n \rightarrow +\infty$. The mathematical questions that follow are:
\begin{itemize}
\item Q1 : Can we approximate system \eqref{Sto}-\eqref{Sto2} by a system of the form \eqref{Stoeff} for large $n$?
\item Q2 : If so, can we provide a formula for $\mu_{eff}$ inside $\mO$? In particular, for small $\lambda$, can we derive an expansion
$$ \mu_{eff} = \mu + \lambda \mu_1 + \dots \quad ? $$
\end{itemize}
Regarding Q1, the only work we are aware of is the recent paper \cite{DuerinckxGloria19}. It shows that $u_n$ converges to the solution $u_{eff}$ of an effective model of the type \eqref{Sto2}, under two natural conditions:
\begin{enumerate}
\item[i)] the balls satisfy the separation condition $\inf_{i \neq j} |x_i - x_j| \ge M \ r_n$, $M > 2$. Note that this is a slight reinforcement of the natural constraint that the balls do not overlap.
\item[ii)] the centers of the balls are obtained from a stationary ergodic point process.
\end{enumerate}
We refer to \cite{DuerinckxGloria19} for all details. Note that in the scalar case, with the Laplacian instead of the Stokes operator, similar results were known since the work of Kozlov, see \cite[paragraph 8.6]{MR1329546}.
\mspace
Q2, and more broadly quantitative aspects of dilute suspensions, have been studied for long. The pioneering work is due to Einstein \cite{Ein}. By {\em neglecting the interaction between the particles}, he computed a first order approximation of the effective viscosity of homogeneous suspensions:
$$ \mu_{eff} = (1 + \frac{5}{2} \lambda) \mu \quad \text{ in } \mO.$$
This celebrated formula was confirmed experimentally afterwards. It was later extended to the inhomogenous case, with formula
\begin{equation} \label{Almog-Brenner}
\mu_{eff} = (1 + \frac{5}{2} \lambda \rho) \mu,
\end{equation}
see \cite[page 16]{AlBr}. Further works investigated the $o(\lambda^2)$ approximation of the effective viscosity, {\it cf.} \cite{BaGr1}
and the recent analysis \cite{DGV_MH}.
\mspace
Our concern in the present paper is the justification of Einstein's formula. To our knowledge, the first rigorous studies on this topic are \cite{MR813656} and \cite{MR813657}: they rely on homogenization techniques, and are restricted to suspensions that are periodically distributed in a bounded domain. A more complete justification, still in the periodic setting but based on variational principles, can be found in \cite{MR2982744}. Recently, the periodicity assumption was relaxed in \cite{HiWu}, \cite{NiSc}, and replaced by an assumption on the minimal distance:
\begin{equation} \label{A1} \tag{A1}
\text{There exists an absolute constant $c$, such that } \quad \forall n, \forall 1 \le i \neq j \le n, \quad |x_i - x_j| \ge c n^{-\frac{1}{3}}.
\end{equation}
For instance, introducing the solution $u_{E}$ of the Einstein's approximate model
\begin{equation} \label{Sto_E}
-\div (2 \mu_E Du_E) + \na p_E = f, \quad \div u = 0 \quad \text{ in } \: \R^3
\end{equation}
with $\mu_E = (1 + \frac{5}{2} \lambda \rho) \mu$, it is shown in \cite{HiWu} that for all $ 1 \le p < \frac{3}{2}$,
$$ \limsup_{n \to \infty} ||u_n - u_E||_{L^p_{loc}(\R^3)} = O(\lambda^{1+\theta}), \quad \theta = \frac{1}{p} - \frac{2}{3}. $$
We refer to \cite{HiWu} for refined statements, including quantitative convergence in $n$ and treatment of polydisperse suspensions.
\mspace
Although it is a substantial gain over the periodicity assumption, hypothesis \eqref{A1} on the minimal distance is still strong. In particular, it is much more stringent that the condition that the rigid balls can not overlap. Indeed, this latter condition reads: $\forall i \neq j$, $|x_i - x_j| \ge 2 r_n$, or equivalently $|x_i - x_j| \ge c \, \lambda^{1/3} n^{-\frac{1}{3}}$, with $c = 2 (\frac{3\pi}{4})^{1/3}$. It follows from \eqref{A1} at small $\lambda$. On the other hand, one could argue that a simple non-overlapping condition is not enough to ensure the validity of Einstein's formula. Indeed, it is based on neglecting interaction between particles, which is incompatible with too much clustering in the suspension. Still, one can hope that if the balls are not too close from one another {\em on average}, the formula still holds.
\mspace
This is the kind of result that we prove here. Namely, we shall replace \eqref{A1} by a set of two relaxed conditions:
\begin{align}
\label{B1} \tag{B1}
& \text{There exists $M >2$, such that} \quad \forall n, \: \forall 1 \le i \neq j \le n, \quad |x_i - x_j| \ge M r_n. \\
\label{B2} \tag{B2}
& \text{There exists $C > 0$, such that} \quad \forall \eta > 0, \quad \#\{i, \: \exists j, \: |x_i - x_j| \le \eta n^{-\frac13}\} \le C \eta^3 n
\end{align}
Note that \eqref{B1} is slightly stronger than the non-overlapping condition, and was already present in the work \cite{DuerinckxGloria19} to ensure the existence of an effective model. As regards \eqref{B2}, one can show that it is satisfied almost surely in the case when the particle positions
are generated by a stationary ergodic point process if the process does not favor too much close pairs of points. In particular, it is satisfied by
a (hard-core) Poisson point process. We postpone further discussion to Section \ref{sec:prob}.
\mspace
Under these general assumptions, we obtain:
\begin{theorem} \label{main}
Let $\lambda > 0$, $f \in L^1(\R^3) \cap L^\infty(\R^3)$. For all $n$, let $r_n$ such that $\displaystyle \lambda = \frac{4\pi}{3} n r_n^3$, let $f_n \in L^{\frac65}(\R^3)$, and $u_n$ in $\displaystyle \dot{H}^1(\R^3)$ the solution of \eqref{Sto}-\eqref{Sto2}. Assume \eqref{A0}-\eqref{B1}-\eqref{B2}, and that $f_n \rightarrow f$ in $L^{\frac65}(\R^3)$. Then, there exists $p_{min} > 1$ such that for any $p < p_{min}$, any $q < \frac{3 p_{min}}{3 - p_{min}}$, one can find $\delta > 0$ with the estimate
$$ ||\na (u - u_E)||_{L^p} + \limsup_{n \rightarrow +\infty} ||u_n - u_E||_{L^q_{loc}} = O(\lambda^{1+\delta}) \quad \text{as } \: \lambda \rightarrow 0, $$
where $u$ is any weak accumulation point of $u_n$ in $\displaystyle \dot{H}^1(\R^3)$ and
$u_E$ satisfies Einstein's approximate model \eqref{Sto_E}.
\end{theorem}
\noindent Here, we use the notation $\dot H^1(\R^3)$ for the homogeneous Sobolev space
$\dot H^1(\R^3) = \{ w \in L^6(\R^3) : \nabla w \in L^2(\R^3)\}$ equipped with the $L^2$ norm of the gradient.
\mspace
The rest of the paper is dedicated to the proof of this theorem.
\section{Main steps of proof}
To prove Theorem \ref{main}, we shall rely on an enhancement of the general strategy explained in \cite{DGV}, to justify various effective models for conducting and fluid media. Let us point out that one of the examples considered in \cite{DGV} is the scalar version of \eqref{Sto}-\eqref{Sto2}. It leads to a proof of a scalar analogue of Einstein's formula, under assumptions \eqref{A0}, \eqref{B1}, plus an abstract assumption intermediate between \eqref{A1} and \eqref{B2}. We refer to the discussion at the end of \cite{DGV} for more details. Nevertheless, to justify the effective fluid model \eqref{Sto_E} under the mild assumption \eqref{B2} will require several new steps. The main difficulty will be to handle particles that are close to one another, and will involve sharp $W^{1,q}$ estimates obtained in \cite{Hof2}.
\mspace
Concretely, let $\varphi$ be a smooth and compactly supported divergence-free vector field. For each $n$, we introduce the solution $\phi_n \in \dot{H}^1(\R^3)$ of
\begin{equation} \label{Sto_phi}
\begin{aligned}
- \div(2\mu D \phi_n) + \na q_n & = \div (5 \lambda \mu \rho D \varphi) \: \text{ in } \: \Omega_n, \\
\div \phi_n & = 0 \: \text{ in } \: \Omega_n, \\
\phi_n & = \varphi + \phi_{n,i} + w_{n,i} \times (x-x_i) \: \text{ in } \: B_i, \: 1 \le i \le n
\end{aligned}
\end{equation}
where the constant vectors $\phi_{n,i}$, $w_{n,i}$ are associated to the constraints
\begin{equation} \label{Sto2_phi}
\begin{aligned}
\int_{\pa B_i} \sigma_\mu(\phi_n,q_n) \nu & = - \int_{\pa B_i} 5 \lambda \mu \rho D \varphi \nu, \\
\int_{\pa B_i}(x-x_i) \times \sigma_\mu(\phi_n,q_n) \nu & = - \int_{\pa B_i}(x-x_i) \times 5 \lambda \mu \rho D \varphi \nu.
\end{aligned}
\end{equation}
Testing \eqref{Sto} with $\varphi - \phi_n$, we find after a few integration by parts that
$$
\int_{\R^3} 2\mu_E Du_n : D \varphi = \int_{\R^3} f_n \cdot \varphi - \int_{\R^3} f_n \cdot \phi_n.
$$
Testing \eqref{Sto_E} with $\varphi$, we find
$$
\int_{\R^3} 2\mu_E Du_E : D \varphi = \int_{\R^3} f \cdot \varphi.
$$
Combining both, we end up with
\begin{equation} \label{weak_estimate}
\int_{\R^3} 2\mu_E D(u_n - u_E) : D \varphi = \int_{\R^3} (f_n - f) \cdot \varphi - \int_{\R^3} f_n \cdot \phi_n.
\end{equation}
We remind that vector fields $u_n, u_E, \phi_n$ depend implicitly on $\lambda$.
\mspace
The main point will be to show
\begin{proposition} \label{main_prop}
There exists $p_{min} > 1$ such that for all $p < p_{min}$, there exists $\delta > 0$ and $C > 0$, independent of $\varphi$, such that
\begin{equation} \label{estimateR}
\limsup_{n \to \infty} \big| \int_{\R^3} f_n \cdot \phi_n \big| \le C \lambda^{1+\delta} ||\na \varphi||_{L^{p'}}, \quad p' = \frac{p}{p-1}.
\end{equation}
\end{proposition}
\noindent
Let us show how the theorem follows from the proposition. First, by standard energy estimates, we find that $u_n$ is bounded in $\dot{H}^1(\R^3)$ uniformly in $n$. Let $u = \lim u_{n_k}$ be a weak accumulation point of $u_n$ in this space. Taking the limit in \eqref{weak_estimate}, we get
$$ \int_{\R^3} 2\mu_E D(u - u_E) : D \varphi = \langle R, \varphi \rangle $$
where $\langle R , \varphi \rangle = \lim_{k \rightarrow +\infty} \int_{\R^3} f_{n_k} \cdot \phi_{n_k}$.
Recall that $\varphi$ is an arbitrary smooth and compactly supported divergence-free vector field and that such functions are dense in the homogeneous Sobolev space of divergence-free functions $\dot{W}^{1,p}_\sigma$.
Thus, Proposition \ref{main_prop} implies that $R$ is an element of $\dot{W}_\sigma^{-1,p}$ with $||R||_{\dot{W}_\sigma^{-1,p}} = O(\lambda^{1+\delta})$. Moreover, the previous identity is the weak formulation of
$$ - \div(2 \mu_E D(u - u_E)) + \na q = R, \quad \div (u - u_E) = 0 \quad \text{ in } \: \R^3. $$
Writing these Stokes equations with non-constant viscosity as
$$ -\mu \Delta (u - u_E) + \na q = R + \div (5 \lambda \mu \rho D(u-u_E)), \quad \div (u - u_E) = 0 \quad \text{ in } \: \R^3. $$
and using standard estimates for this system, we get
$$ ||\na (u-u_E)||_{L^p} \le C \left(||R||_{\dot{W}^{-1,p}_\sigma} + \lambda ||\na (u-u_E)||_{L^p} \right). $$
For $\lambda$ small enough, the last term is absorbed by the left-hand side, and finally
$$ ||\na (u-u_E)||_{L^p(\R^3)} \le C\lambda^{1+\delta}$$
which implies the first estimate of the theorem. Then, by Sobolev imbedding, for any $q \le \frac{3p}{3-p}$, and any compact $K$,
\begin{equation} \label{Sob_imbed}
||u-u_E||_{L^q(K)} \le C_{K,q} \, \lambda^{1+\delta}.
\end{equation}
We claim that
$ \limsup_{n \to \infty} ||u_n-u_E||_{L^q(K)} \le C_{K,q} \, \lambda^{1+\delta}$. Otherwise, there exists a subsequence $u_{n_k}$ and $\eps > 0$ such that $\displaystyle ||u_{n_k} - u_E||_{L^q(K)} \ge C_{K,q} \, \lambda^{1+\delta} + \eps$ for all $k$. Denoting by $u$ a (weak) accumulation point of $u_{n_k}$ in $\dot{H}^1$, Rellich's theorem implies that, for a subsequence still denoted $u_{n_k}$, $||u_{n_k} - u||_{L^q(K)} \rightarrow 0$, because $q < 6$ (for $p_{min}$ taken small enough). Combining this with \eqref{Sob_imbed}, we reach a contradiction. As $p$ is arbitrary in $(1, p_{min})$, $q \leq \frac{3p}{3-p}$ is arbitrary in $(1, \frac{3 p_{min}}{3 - p_{min}})$. The last estimate of the theorem is proved.
\mspace
It remains to prove Proposition \ref{main_prop}.
Therefore, we need a better understanding of the solution $\phi_n$ of \eqref{Sto_phi}-\eqref{Sto2_phi}. Neglecting any interaction between the balls, a natural attempt is to approximate $\phi_n$ by
\begin{equation} \label{approx_phi}
\phi_n \approx \phi_{\R^3} + \sum_i \phi_{i,n}
\end{equation}
where $\phi_{\R^3}$ is the solution of
\begin{equation} \label{eq_phi_R3}
- \mu \Delta \phi_{\R^3} + \na p_{\R^3} = \div(5 \lambda \mu \rho D\varphi), \quad \div \phi_{\R^3} = 0 \quad \text{in } \: \R^3
\end{equation}
and $\phi_{i,n}$ solves
\begin{equation} \label{eq_phi_i_n}
- \mu \Delta \phi_{i,n} + \na p_{i,n} = 0, \quad \div\phi_{i,n} = 0 \quad \text{ outside } \: B_i, \quad \phi_{i,n}\vert_{B_i}(x) = D \varphi(x_i) \, (x-x_i)
\end{equation}
Roughly, the idea of approximation \eqref{approx_phi} is that $\phi_{\R^3}$ adjusts to the source term in \eqref{Sto_phi}, while for all $i$, $\phi_{i,n}$ adjusts to the boundary condition at the ball $B_i$. Indeed, using a Taylor expansion of $\varphi$ at $x_i$, and splitting $\na \varphi(x_i)$ between its symmetric and skew-symmetric part, we find
$$ \phi_n\vert_{B_i}(x) \approx D\varphi(x_i) \, (x- x_i) \: + \: \text{\em rigid vector field} = \phi_{i,n}\vert_{B_i}(x) \: + \: \text{\em rigid vector field}. $$
Moreover, $\phi_{i,n}$ can be shown to generate no force and torque, so that the extra rigid vector fields (whose role is to ensure the no-force and no-torque conditions), should be small.
\mspace
Still, approximation \eqref{approx_phi} may be too crude : the vector fields $\phi_{j,n}$, $j \neq i$, have a non-trivial contribution at $B_i$, and for the balls $B_j$ close to $B_i$, which are not excluded by our relaxed assumption \eqref{B1}, these contributions may be relatively big. We shall therefore modify the approximation, restricting the sum in \eqref{approx_phi} to balls far enough from the others.
\mspace
Therefore, for $\eta > 0$, we introduce a {\em good} and a {\em bad} set of indices:
\begin{equation} \label{good_bad_sets}
\mG_\eta = \{ 1 \le i \le n, \: \forall j \neq i, |x_i - x_j| \ge \eta n^{-\frac{1}{3}} \}, \quad \mB_\eta = \{1, \dots n\} \setminus \mG_\eta.
\end{equation}
The good set $\mG_\eta$ corresponds to balls that are at least $\eta n^{-\frac{1}{3}}$ away from all the others. The parameter $\eta > 0$ will be specified later: we shall consider $\eta = \lambda^\theta$ for some appropriate power $0 < \theta < 1/3$. We set
\begin{equation} \label{def_phi_app}
\phi_{app,n} = \phi_{\R^3} + \sum_{i \in \mG_\eta} \phi_{i,n}
\end{equation}
Note that $\phi_{\R^3}$ and $\phi_{i,n}$ are explicit:
$$ \phi_{\R^3} = \mU \star \div(5 \lambda \rho D\varphi), \quad \mU(x) = \frac{1}{8\pi} \left( \frac{I}{|x|} + \frac{x \otimes x}{|x|^3} \right)$$
and
\begin{equation} \label{def_phi_in}
\phi_{i,n} = r_n V[D \varphi(x_i)]\left(\frac{x-x_i}{r_n}\right)
\end{equation}
where for all trace-free symmetric matrix $S$, $V[S]$ solves
$$ -\Delta V[S] + \na P[S] = 0, \: \div V[S] = 0 \quad \text{outside } \: B(0,1), \quad V[S](x) = Sx, \: x \in B(0,1). $$
with expressions
$$ V[S] = \frac{5}{2} S : (x \otimes x) \frac{x}{|x|^5} + Sx \frac{1}{|x|^5} - \frac{5}{2} (S : x \otimes x) \frac{x}{|x|^7}, \quad P[S] = 5 \frac{S : x \otimes x}{|x|^5}. $$
Eventually, we denote
$$ \psi_n = \phi_n - \phi_{app,n}. $$
Tedious but straightforward calculations show that
$$ - \div (\sigma_\mu(V[S], P[S])) = 5 \mu S x s^1 = - \div (5 \mu S 1_{B(0,1)}) \quad \text{in } \: \R^3 $$
where $s^1$ denotes the surface measure at the unit sphere. It follows that
\begin{equation}
- \mu \Delta \phi_{app,n} + \na p_{app,n} = \div \Big( 5 \lambda \mu \rho D\varphi - \sum_{i \in \mG_\eta} 5 \mu D \varphi(x_i) 1_{B_i} \Big), \quad \div \phi_{app,n} = 0 \quad \text{in} \: \R^3,
\end{equation}
Moreover, for all $1 \le i \le n$,
\begin{align*}
\int_{\pa B_i} \sigma_\mu(\phi_{app,n}, p_{app,n}) \nu & = - \int_{\pa B_i} 5 \lambda \mu \rho D\varphi \nu, \\
\int_{\pa B_i} (x-x_i) \times \sigma_\mu(\phi_{app,n}, p_{app,n}) \nu & = - \int_{\pa B_i} (x-x_i) \times 5 \lambda \mu \rho D\varphi \nu.
\end{align*}
Hence, the remainder $\psi_n$ satisfies
\begin{equation} \label{Sto_psi}
\begin{aligned}
- \mu \Delta \psi_n + \na q_n & = 0 \: \text{ in } \: \Omega_n, \\
\div \psi_n & = 0 \: \text{ in } \: \Omega_n, \\
\psi_n & = \varphi - \phi_{app,n} + \psi_{n,i} + w_{n,i} \times (x-x_i) \: \text{ in } \: B_i, \: 1 \le i \le n
\end{aligned}
\end{equation}
where the constant vectors $\psi_{n,i}$, $w_{n,i}$ are associated to the constraints
\begin{equation} \label{Sto2_psi}
\begin{aligned}
\int_{\pa B_i} \sigma_\mu(\psi_n,q_n) \nu & = 0, \\
\int_{\pa B_i}(x-x_i) \times \sigma_\mu(\psi_n,q_n) \nu & = 0.
\end{aligned}
\end{equation}
Estimates on $\phi_{app,n}$ and $\psi_n$ will be postponed to sections \ref{sec_app} and \ref{sec_rem} respectively. Regarding $\phi_{app,n}$, we shall prove
\begin{proposition} \label{prop_phi_app}
For all $p \ge 1$,
\begin{equation} \label{estimate_phi_app}
\limsup_{n \to \infty} \big|\int_{\R^3} f_n \cdot \phi_{app,n} \big| \le C_{p,f} (\lambda \eta^3)^{\frac{1}{p}} ||\na \varphi||_{L^{p'}}.
\end{equation}
\end{proposition}
\noindent
Regarding the remainder $\psi_n$, we shall prove
\begin{proposition} \label{prop_psi}
One can find $p_0 \in (1,2)$, $\theta_0 > 0$, $p_{min} > 1$ satisfying: for all $1 < p < p_{min}$, for all $1 < q < p'$, there exists $c,C > 0$ such that for all $\eta \ge c \lambda^{1/3}$,
$$ \limsup_{n \to \infty} ||\na\psi_n||_{L^{p_0}(\R^3) + L^2(\R^3) + L^q(\R^3)} \le C \Big(\lambda^{1+\theta_0} + \lambda^{1+ \frac{2-p}{2p}} \, \eta^{-\frac{3}{p}} + \big( \eta^3 \lambda \big)^{\frac{1}{r}} \Big) ||\na \varphi||_{L^{p'}}.$$
with $\frac{1}{r} + \frac{1}{p'} = \frac{1}{q}$.
\end{proposition}
\mspace
Let us explain how to deduce Proposition \ref{main_prop} from these two propositions. Let $1 < p < p_{min}$, with $p_{min}$ as in Proposition \ref{prop_psi}. By standard estimates, we see that $\phi_n$ and $\phi_{app,n}$ are bounded uniformly in $n$ in $\dot{H}^1$, and so the same holds for $\psi_n$. It follows that
\begin{align*}
\limsup_{n \to \infty} \Big| \int_{\R^3} f_n \cdot \phi_n \Big| = \limsup_{n \to \infty} \Big| \int_{\R^3} f \cdot \phi_n \Big| & \le \limsup_{n \to \infty} \Big| \int_{\R^3} f \cdot \phi_{app,n} \Big| + \limsup_{n \to \infty} \Big| \int_{\R^3} f \cdot \psi_n \Big|
\\
& \le C_{p,f} (\lambda \eta^3)^{\frac{1}{p}} ||\na \varphi||_{L^{p'}} + \Big| \int_{\R^3} f \cdot \psi \Big|
\end{align*}
with $\psi$ a weak accumulation point of $\psi_n$. Note that we used Proposition \ref{prop_phi_app} to go from the first to the second inequality.
By Proposition \ref{prop_psi}, for all $1 < q < p'$, for all $\eta \ge c \lambda^{1/3}$,
$$ ||\na\psi||_{L^{p_0}(\R^3) + L^2(\R^3) + L^q(\R^3)} \le C \Big(\lambda^{1+\theta_0} + \lambda^{1+ \frac{2-p}{2p}} \, \eta^{-\frac{3}{p}} + \big( \eta^3 \lambda \big)^{\frac{1}{r}} \Big) ||\na \varphi||_{L^{p'}} $$
with $\frac{1}{r} + \frac{1}{p'} = \frac{1}{q}$. As $f \in L^1 \cap L^\infty$, it belongs to the dual of $\dot{W}^{1,p_0}(\R^3) + \dot{W}^{1,2}(\R^3) + \dot{W}^{1,q}(\R^3)$ so that eventually
\begin{align*}
\limsup_{n \to \infty} \Big| \int_{\R^3} f_n \cdot \phi_n \Big| & \le C \Big( \lambda^{1+\theta_0} + \lambda^{1+ \frac{2-p}{2p}} \, \eta^{-\frac{3}{p}} + \big( \eta^3 \lambda \big)^{\frac{1}{r}} \Big) ||\na \varphi||_{L^{p'}}
\end{align*}
To conclude, we adjust properly the various parameters. We look for $\eta$ in the form $\eta = \lambda^{\theta}$, with $\theta = \theta(p) < \frac{1}{3}$, so that the lower bound on $\eta$ needed in Proposition \ref{prop_psi} will be satisfied for small enough $\lambda$. First, we notice that for any $p < 2$, if $\theta$ is such that $2 - p - 6 \theta > 0$, then $\lambda^{1+ \frac{2-p}{2p}} \, \eta^{-\frac{3}{p}} = \lambda^{1+\delta_1}$ for some $\delta_1 > 0$. Such a choice of $\theta$ being made, we can choose $r= r(p)$ close enough to $1$ in a way that
$\big( \eta^3 \lambda \big)^{\frac{1}{r}} = \lambda^{1+\delta_2}$ for some $\delta_2 > 0$. Upon taking smaller $p_{min}$, we can further ensure that $\frac{1}{r} + \frac{1}{p'} < 1$ so that we can define $q$ by $\frac{1}{r} + \frac{1}{p'} = \frac{1}{q}$, and use the previous estimates. Eventually, for any $1 < p < p_{min}$, for $\lambda$ small enough,
\begin{align*}
\limsup_{n \to \infty} \Big| \int_{\R^3} f_n \cdot \phi_n \Big| & \le C \lambda^{1+\delta} ||\na \varphi||_{L^{p'}(\R^3)}
\end{align*}
where $\delta = \min(\theta_0, \delta_1, \delta_2)$.
\section{Bound on the approximation} \label{sec_app}
This section is devoted to the proof of Proposition \ref{prop_phi_app}. We decompose
$$\phi_{app,n} = \phi_{app,n}^1 + \phi_{app,n}^2 + \phi_{app,n}^3$$
where
\begin{align*}
& - \mu \Delta \phi^1_{app,n} + \na p^1_{app,n} = \div \Big( 5 \lambda \mu \rho D\varphi - \sum_{1 \le i \le n} 5 \mu D \varphi(x_i) 1_{B_i} \Big), \quad \div \phi^1_{app,n} = 0 \quad \text{in} \: \R^3, \\
& - \mu \Delta \phi^2_{app,n} + \na p^2_{app,n} = \div \Big( \sum_{i \in \mB_\eta} 5 \mu D \varphi(x) 1_{B_i} \Big), \quad \div \phi^1_{app,n} = 0 \quad \text{in} \: \R^3, \\
& - \mu \Delta \phi^3_{app,n} + \na p^3_{app,n} = \div \Big( \sum_{i \in \mB_\eta} 5 \mu (D \varphi(x_i) - D\varphi(x)) 1_{B_i} \Big), \quad \div \phi^1_{app,n} = 0 \quad \text{in} \: \R^3.
\end{align*}
By standard energy estimates, $\phi^k_{app,n}$ is seen to be bounded in $n$ in $\dot{H^1}$, for all $1 \le k \le 3$. We shall prove next that
$\phi^1_{app,n}$ and $\phi^3_{app,n}$ converge in the sense of distributions to zero, while for any $f$ with $\displaystyle D ( \Delta)^{-1} \mathbb{P} f \in L^\infty$ ($\mathbb{P}$ denoting the standard Helmholtz projection), for any $p \ge 1$,
\begin{equation} \label{estimate_phi_app_2}
\Big|\int_{\R^3} f \cdot \phi^2_{app,n} \Big| \le C_{f,p} (\lambda \eta^3)^{\frac{1}{p}} ||\na \varphi||_{L^{p'}}, \quad p' = \frac{p}{p-1}.
\end{equation}
Proposition \ref{prop_phi_app} follows easily from those properties.
\mspace
We start with
\begin{lemma}
Under assumption \eqref{A0}, $\: \sum_{1 \le i \le n} D\varphi(x_i) \mathbf{1}_{B_i} \: \rightharpoonup \: \lambda \rho D\varphi $ weakly* in $L^\infty$.
\end{lemma}
\noindent
{\em Proof}. As the balls are disjoint, $|\sum_{1 \le i \le n} D\varphi(x_i) \mathbf{1}_{B_i}| \le ||D\varphi||_{L^\infty}$. Let $g \in C_c(\R^3)$, and denote $\delta_n = \frac{1}{n} \sum_{i} \delta_{x_i}$ the empirical measure. We write
\begin{align*}
\int_{\R^3} \sum_{1 \le i \le n} D\varphi(x_i) \mathbf{1}_{B_i}(y) g(y) dy & = \sum_{1 \le i \le n} D\varphi(x_i) \int_{B(0,r_n)} g(x_i+y) dy \\
& = n \int_{\R^3} D\varphi(x) \int_{B(0,r_n)} g(x+y) dy d\delta_n(x) \\
& = n r_n^3 \int_{\R^3} \int_{B(0,1)} g(x+r_nz) dz d\delta_n(x).
\end{align*}
The sequence of bounded continuous functions $x \rightarrow \int_{B(0,1)} g(x+r_n z) dz$ converges uniformly to the function $x \rightarrow \frac{4\pi}{3} g(x)$ as $n \rightarrow +\infty$. We deduce:
$$ \lim_{n \to \infty} \int_{\R^3} \sum_{1 \le i \le n} D\varphi(x_i) \mathbf{1}_{B_i}(y) g(y) dy = \lim_{n \to \infty} \lambda \int_{\R^3} D\varphi(x) g(x) d\delta_n(x) = \lambda \int_{\R^3} D\varphi(x) g(x) \rho(x) dx $$
where the last equality comes from \eqref{A0}. The lemma follows by density of $C_c$ in $L^1$.
\mspace
Let now $h \in C^\infty_c(\R^3)$ and $v = (\Delta)^{-1} \mathbb{P} h$. We find
\begin{align*}
\langle \phi_{app,n}^1 , h \rangle & = \langle \phi_{app,n}^1 , \Delta v \rangle = \langle \Delta \phi_{app,n}^1 , v \rangle \\
& = \int_{\R^3} \big( 5 \lambda \mu \rho D\varphi - \sum_{1 \le i \le n} 5 \mu D \varphi(x_i) 1_{B_i} \big) \cdot Dv \: \rightarrow 0 \quad \text{ as } \: n \rightarrow +\infty
\end{align*}
where we used the previous lemma and the fact that $Dv$ belongs to $L^1_{loc}$ and $\varphi$ has compact support. Hence, $\phi_{app,n}^1$ converges to zero in the sense of distributions. As regards $\phi^3_{app,n}$, we notice that
\begin{align*}
||\sum_{i \in \mB_\eta} 5 \mu (D \varphi(x) - D\varphi(x_i)) 1_{B_i}||_{L^1} & \le ||\na^2 \varphi||_{L^\infty} \sum_{1 \le i \le n} \int_{B_i} |x-x_i| dx \\
& \le ||\na^2 \varphi||_{L^\infty} \lambda r_n \rightarrow 0 \quad \text{ as } \: n \rightarrow +\infty
\end{align*}
Using the same duality argument as for $\phi^1_{app, n}$ (see also below), we get that $\phi^3_{app,n}$ converges to zero in the sense of distributions.
\mspace
It remains to show \eqref{estimate_phi_app_2}. We use a simple H\"older estimate, and write for all $p \ge 1$:
\begin{align*}
||\sum_{i \in \mB_\eta} 5 \mu D \varphi 1_{B_i}||_{L^1} & \le 5 \mu || \sum_{i \in \mB_\eta} 1_{B_i}||_{L^p} ||D\varphi||_{L^{p'}} = 5 \mu \big( \text{card} \ \mB_\eta \, \frac{4\pi}{3} r_n^3 \big)^{\frac{1}{p}} ||D\varphi||_{L^{p'}} \\
& \le C (\eta^3 \lambda)^{\frac{1}{p}} ||D\varphi||_{L^{p'}}
\end{align*}
where the last inequality follows from \eqref{B2}. Denoting $v = ( \Delta)^{-1} \mathbb{P} f$, we have this time
\begin{align*}
\int_{\R^3} f \cdot \phi_{app,n}^2 & = \int_{\R^3} D v \cdot \sum_{i \in \mB_\eta} 5 \mu D \varphi 1_{B_i} \le C ||Dv||_{L^\infty} (\eta^3 \lambda)^{\frac{1}{p}} ||D\varphi||_{L^{p'}}
\end{align*}
which implies \eqref{estimate_phi_app_2}.
\section{Bound on the remainder} \label{sec_rem}
We focus here on estimates for the remainder $\psi_n = \phi_n - \phi_{app,n}$, which satisfies \eqref{Sto_psi}-\eqref{Sto2_psi}. We will use as a black box sharp $W^{1,q}$ estimates derived recently in \cite{Hof2}, which notably apply to systems of the form
\begin{equation} \label{Sto_Psi}
-\mu \Delta \psi + \na p = 0, \quad \div \psi = 0 \quad \text{ in } \: \Omega_n, \quad
D \psi = D \tilde{\psi} \quad \text{ in } \: B_i, \quad 1 \le i \le n
\end{equation}
together with the constraints
\begin{equation} \label{Sto2_Psi}
\int_{\pa B_i} \sigma_\mu(\psi, p)\nu = \int_{\pa B_i} (x-x_i) \times \sigma_\mu(\psi, p)\nu = 0, \quad 1 \le i \le n.
\end{equation}
We first recall that, {\em under assumption \eqref{B1}}, one has the somehow standard estimate
\begin{equation} \label{L2_estimate_Psi}
||\na \psi||_{L^2(\R^3)} \le C \| D \tilde{\psi} \|_{L^2(\cup B_i)}
\end{equation}
for a constant $C$ independent of $n$. A sketch of proof is as follows. By a classical variational characterization of $\psi$, we have
$$
||\na \psi||_{L^2(\R^3)}^2 = 2 ||D \psi||_{L^2(\R^3)}^2 = \inf \big\{ 2 ||D U||_{L^2(\R^3)}^2, \: D U = D \tilde{\psi} \: \text{ on } \cup_i B_i\big\}.
$$
Hence it is enough to build one vector field $U$ satisfying the same condition on $\cup_{i} B_i$ and fulfilling the estimate. By adding an appropriate rigid vector field to $\tilde{\psi}$ on each $B_i$, so that $D \tilde{\psi}$ is unchanged, one can always assume a Poincar\'e and a Korn inequality: $ r_n^{-2} ||\tilde{\psi}||^2_{L^2(B_i)} \le C ||\na \tilde{\psi}||^2_{L^2(B_i)} \le C' ||D \tilde{\psi}||^2_{L^2(B_i)}$, where the factor $r_n^{-2}$ comes from scaling considerations. Hence, after these simplifications, it is enough to find a $U$ satisfying $U = \tilde{\psi}$ on each $B_i$, and such that $\displaystyle \|\na U\|^2_{L^2(\R^3)} \le C \left(r_n^{-2} ||\psi||^2_{L^2(\cup B_i)} + \|\na \tilde{\psi}\|^2_{L^2(\cup B_i)} \right)$. This can then be done under assumption \eqref{B1}, using standard considerations on the Bogovskii operator: one can even choose $U$ to be supported in a vicinity of $\cup B_i$. We refer to \cite[Lemma 5]{DGV} for details.
\mspace
However, the derivation of an $L^p$ analogue of \eqref{L2_estimate_Psi} is much more unclear, as no variational characterization is available. This problem was tackled recently by the second author in \cite{Hof2}, through a careful analysis of the so-called {\em method of reflections}, introduced by Smoluchowski. We shall use the following
\begin{theorem} {\bf \cite[Proposition 5.2 and Remark 5.6]{Hof2}} \label{theo_Lp}
\noindent Assume that $\inf_{i \neq j} |x_i - x_j| \ge \, M \, r_n$. Then,
\begin{description}
\item[i)] for all $M > 2$, there exists $\eps_M > 0$ such that for all $q \in [2-\eps_M , 2+\eps_M]$, and $\tilde{\psi} \in L^q(\cup_i B_i)$, there exists a unique solution $\psi \in \dot{W}^{1,q}(\R^3)$ of \eqref{Sto_Psi}-\eqref{Sto2_Psi} such that
$$ ||\na \psi||_{L^q(\R^3)} \le C_M \| D \tilde{\psi} \|_{L^q(\cup B_i)} $$
\item[ii)] for all $q > 1$, there exists $M_q$ such that for all $M \ge M_q$, and $\tilde{\psi} \in L^q(\cup_i B_i)$, there exists a unique solutions $\psi \in \dot{W}^{1,q}(\R^3)$ of \eqref{Sto_Psi}-\eqref{Sto2_Psi} such that
$$ ||\na \psi||_{L^q(\R^3)} \le C_q \| D \tilde{\psi} \|_{L^q(\cup B_i)} $$
\end{description}
\end{theorem}
\mspace
We now come back to the analysis of $\psi_n$. We decompose the field $\varphi - \phi_{app,n}$ that appears in the third line of \eqref{Sto_psi}: we write
$$\varphi - \phi_{app,n} = \tilde{\psi}^1_n + \tilde{\psi}^2_n + \tilde{\psi}^3_n $$
where
\begin{align*}
& \forall 1 \le i \le n, \: \forall x \in B_i, \quad \tilde{\psi}^1_n(x) = -\phi_{\R^3}(x) - \sum_{\substack{j \neq i, \\ j \in \mG_\eta}} \phi_{j,n}(x)
\end{align*}
and
\begin{align*}
& \forall i \in \mG_\eta, \: \forall x \in B_i, \quad \tilde{\psi}^2_n(x) = \varphi(x) - \varphi(x_i) - \na\varphi(x_i) (x-x_i) + \Bigl(\varphi(x_i) + \frac 1 2 \curl \varphi(x_i) \times (x-x_i)\Bigr), \\
& \forall i \in \mB_\eta, \: \forall x \in B_i, \quad \tilde{\psi}^2_n(x) = 0, \\
& \forall i \in \mG_\eta, \: \forall x \in B_i, \quad \tilde{\psi}^3_n(x) = 0, \\
& \forall i \in \mB_\eta, \: \forall x \in B_i, \quad \tilde{\psi}^3_n(x) = \varphi(x).
\end{align*}
We remind that the sum in \eqref{def_phi_app} is restricted to indices $i \in \mG_\eta$ and that $\phi_{i,n}(x) = D\varphi(x_i) (x-x_i)$ for $x$ in $B_i$. This explains the distinction between $\tilde{\psi}^2_n$ and $\tilde{\psi}^3_n$.
\mspace
Clearly, $\psi_n = \sum_{k=1}^3 \psi^k_n$, where $\psi^k_n$ is the solution of \eqref{Sto_Psi}-\eqref{Sto2_Psi} associated to data $\tilde \psi^k_n$. The control of $\psi^2_n$ is the simplest to obtain: we apply estimate \eqref{L2_estimate_Psi}, valid under the sole assumption \eqref{B1}. We find
\begin{align*}
||\na \psi^2_n||_{L^2(\R^3)} & \le C \|D \tilde \psi^2_n\|_{L^2(\cup B_i)} \le C ||D^2 \varphi||_{L^\infty} \Big( \sum_{i \in \mG_\eta} \int_{B_i} |x-x_i|^2 dx \Bigr)^{1/2} \le C' \lambda^{1/2} r_n.
\end{align*}
Hence,
\begin{equation} \label{lim_psi_2n}
\lim_{n \rightarrow +\infty} || \na \psi^2_n||_{L^2(\R^3)} = 0.
\end{equation}
As regards $\psi^1_n$, we apply again \eqref{L2_estimate_Psi} and find
\begin{align} \label{bound_psi_2n}
||\na \psi^1_n||_{L^2(\R^3)} & \le C \Big( \|D \phi_{\R^3}\|_{L^2(\cup B_i)} + \Big(\sum_i \int_{B_i} \big| \sum_{\substack{j \neq i, \\ j \in \mG_\eta}} D\phi_{j,n}\big|^2 \Big)^{1/2} \Big)
\end{align}
For any $r,s < +\infty$ with $\frac{1}{r} + \frac{1}{s} = \frac{1}{2}$, we obtain
\begin{equation} \label{bound_phi_R3}
\|D \phi_{\R^3}\|_{L^2(\cup B_i)} \: \le \: ||1_{\cup B_i}||_{L^r(\R^3)} ||D \phi_{\R^3}||_{L^s(\R^3)} \: \le C ||1_{\cup B_i}||_{L^r(\R^3)} ||\lambda \rho D\varphi||_{L^s(\R^3)}
\end{equation}
using standard $L^s$ estimate for system \eqref{eq_phi_R3}. Hence,
$$ \|D \phi_{\R^3}\|_{L^2(\cup B_i)} \le C' \lambda^{1+ \frac{1}{r}} ||D\varphi||_{L^s(\R^3)}. $$
Note that we can choose any $s >2$, this lower bound coming from the requirement $\frac{1}{r} + \frac{1}{s} = \frac{1}{2}$. Introducing $p$ such that $s= p'$, we find that for any $p < 2$,
\begin{equation} \label{bound_Dphi_R3}
\|D \phi_{\R^3}\|_{L^2(\cup B_i)} \le C' \lambda^{\frac{1}{2} + \frac{1}{p}} ||D\varphi||_{L^{p'}(\R^3)}.
\end{equation}
The treatment of the second term at the r.h.s. of \eqref{bound_psi_2n} is more delicate. We write, see \eqref{def_phi_in}:
\begin{align} \label{decompo_phi_jn}
D\phi_{j,n}(x) & = DV[D\varphi(x_j)]\Big(\frac{x-x_j}{r_n}\Big) = \mV[D\varphi(x_j)]\Big(\frac{x-x_j}{r_n}\Big) \: + \: \mW[D\varphi(x_j)]\Big(\frac{x-x_j}{r_n}\Big)
\end{align}
where $\: \mV[S] = D\Big( \frac{5}{2} S : (x \otimes x) \frac{x}{|x|^5} \Big)$, $\: \mW[S] = D \Big( \frac{Sx}{|x|^5} - \frac{5}{2} (S : x \otimes x) \frac{x}{|x|^7} \Big)$.
\mspace
We have:
\begin{align*}
& \sum_i \int_{B_i} \Big| \sum_{\substack{j \neq i, \\ j \in \mG_\eta}} \mW[D\varphi(x_j)]\Big(\frac{x-x_j}{r_n}\Big)\Big|^2 dx
\: \le \: C \, r_n^{10} \, \sum_i \int_{B_i} \Big( \sum_{\substack{j \neq i, \\ j \in \mG_\eta}} |D\varphi(x_j)| \, |x-x_j|^{-5} \Big)^2 dx
\end{align*}
For all $i$, for all $j \in \mG_\eta$ with $j \neq i$, and all $(x,y) \in B_i \times B(x_j, \frac{\eta}{4} n^{-\frac{1}{3}})$, we have for some absolute constants $c,c' > 0$:
$$|x-x_j| \: \ge \: c \, |x - y| \ge c' \, \eta n^{-\frac{1}{3}}. $$
Denoting $B_j^* = B(x_j,\frac{\eta}{4} n^{-\frac{1}{3}})$ We deduce
\begin{align*}
& \sum_i \int_{B_i} \Big| \sum_{\substack{j \neq i, \\ j \in \mG_\eta}} \mW[D\varphi(x_j)]\Big(\frac{x-x_j}{r_n}\Big)\Big|^2 dx \\
& \le C \, r_n^{10} \sum_i \int_{B_i}
\Big( \sum_{\substack{j \neq i, \\ j \in \mG_\eta}} \frac{1}{|B_j^*|} \int_{B_j^*} |x - y|^{-5} 1_{\{|x-y| > c' \eta n^{-\frac{1}{3}}\}}(x-y) |D\varphi(x_j)| dy \Big)^2 dx\\
& \le C' \, n^2 \frac{r_n^{10}}{\eta^6} \int_{\R^3} 1_{\cup B_i}(x) \Big( \int_{\R^3} |x - y|^{-5} 1_{\{|x-y| > c' \eta n^{-\frac{1}{3}}\}}(x-y) \sum_{1 \le j \le n} |D\varphi(x_j)| 1_{B_j^*}(y) dy \Big)^2 dx
\end{align*}
Using H\"older and Young's convolution inequalities, we find that for all $r,s$ with $\frac{1}{r} + \frac{1}{s} = 1$,
\begin{align*}
& \int_{\R^3} 1_{\cup B_i}(x) \Big( \int_{\R^3} |x - y|^{-5} 1_{\{|x-y| > c' \eta n^{-\frac{1}{3}}\}}(x-y) \sum_{1 \le j \le n} |D\varphi(x_j)| 1_{B_j^*}(y) dy \Big)^2 dx \\
& \le ||1_{\cup B_i}||_{L^r} \, || \big(|x|^{-5} 1_{\{|x| > c' \eta n^{-\frac{1}{3}}\}}\big) \star \sum_{1 \le j \le n} |D\varphi(x_j)| 1_{B_j^*} ||_{L^{2s}}^2 \\
& \le ||1_{\cup B_i}||_{L^r} \, |||x|^{-5} 1_{\{|x| > c' \eta n^{-\frac{1}{3}}\}}||_{L^1}^2 \, ||\sum_{1 \le j \le n} |D\varphi(x_j)| 1_{B_j^*} ||_{L^{2s}}^2 \\
& \le C \lambda^{\frac{1}{r}} \, (\eta n^{-\frac{1}{3}})^{-4} \, \Big( \sum_j |D\varphi(x_j)|^{2s} \eta^3 n^{-1} \Big)^{\frac{1}{s}}
\end{align*}
Note that, by \eqref{A0}, $\frac{1}{n} \sum_j |D\varphi(x_j)|^t \rightarrow \int_{\R^3} |D\varphi|^t \rho$ as $n \rightarrow +\infty$. We end up with
\begin{align*}
& \limsup_{n \to \infty} \, \sum_i \int_{B_i} \Big| \sum_{\substack{j \neq i, \\ j \in \mG_\eta}} \mW[D\varphi(x_j)]\Big(\frac{x-x_j}{r_n}\Big)\Big|^2 dx
\le C \, \lambda^{\frac{10}{3} + \frac{1}{r}} \, \eta^{-10+\frac{3}{s}} ||D\varphi||^2_{L^{2s}(\mO)}.
\end{align*}
We can take any $s > 1$, which yields by setting $p$ such that $p'=2s$: for any $p < 2$
\begin{equation} \label{bound_mW}
\limsup_{n \to \infty} \, \sum_i \int_{B_i} \Big| \sum_{\substack{j \neq i, \\ j \in \mG_\eta}} \mW[D\varphi(x_j)]\Big(\frac{x-x_j}{r_n}\Big)\Big|^2 dx
\le C \, \lambda^{\frac{10}{3} + \frac{2-p}{p}} \, \eta^{-4-\frac{6}{p}} ||D\varphi||^2_{L^{2s}(\mO)}.
\end{equation}
To treat the first term in the decomposition \eqref{decompo_phi_jn}, we write
$$\mV[D\varphi(x_j)]\Big(\frac{x-x_j}{r_n}\Big) = r_n^3 \, \mM(x-x_j) \, D\varphi(x_j) $$
for $\mM$ a matrix-valued Calderon-Zygmund operator.
We use that for all $i$ and all $j \neq i$, $j \in \mG_\eta$ we have for all $(x,y) \in B_i \times B_j^\ast$
\begin{align*}
|\mM(x-x_j) - \mM(x-y)| \leq C \eta n^{-1/3} |x- y|^{-4}
\end{align*}
Thus, by similar manipulations as before
\begin{align*}
& \sum_i \int_{B_i} \Big| \sum_{\substack{j \neq i, \\ j \in \mG_\eta}} \mV[D\varphi(x_j)]\Big(\frac{x-x_j}{r_n}\Big)\Big|^2 dx \\
& \leq C r_n^6 \sum_i \int_{B_i}
\Big( \sum_{\substack{j \neq i, \\ j \in \mG_\eta}} \frac{1}{|B_j^*|} \int_{B_j^*} \mM(x-y) 1_{\{|x-y| > c \eta n^{-\frac{1}{3}}\}}(x-y) |D\varphi(x_j)| dy \Big)^2 dx\\
& + C \frac{\eta^2}{n^{2/3}} r_n^6 \sum_i \int_{B_i}
\Big( \sum_{\substack{j \neq i, \\ j \in \mG_\eta}} \frac{1}{|B_j^*|} \int_{B_j^*} |x- y|^{-4} 1_{\{|x-y| > c \eta n^{-\frac{1}{3}}\}}(x-y) |D\varphi(x_j)| dy \Big)^2 dx \\
& \leq C n^2 \frac{r_n^6}{\eta^6} \, ||1_{\cup B_i}||_{L^r} \, || \big(\mM(x) 1_{\{|x| > \eta n^{-\frac{1}{3}}\}}\big) \star \sum_{1 \le j \le n} \, |D\varphi(x_j)| 1_{B_j^*} ||_{L^{2s}}^2 \\
&+C n^2 \frac{\eta^2}{n^{2/3}} \frac{r_n^6}{\eta^6} ||1_{\cup B_i}||_{L^r} \, |||x|^{-4} 1_{\{|x| > c \eta n^{-\frac{1}{3}}\}}||_{L^1}^2 \, ||\sum_{1 \le j \le n} |D\varphi(x_j)| 1_{B_j^*} ||_{L^{2s}}^2 \\
\end{align*}
As seen in \cite[Lemma 2.4]{DGV_MH}, the kernel $\mM(x) 1_{\{|x| > c \eta n^{-\frac{1}{3}}\}}$ defines a singular integral that is continuous over $L^t$ for any $1 < t < \infty$, with operator norm bounded independently of the value $\eta n^{-\frac{1}{3}}$ (by scaling considerations). Applying this continuity property with $t=2s$, writing as before $p'=2s$, we get for all $p < 2$,
\begin{align*}
& \limsup_{n \to \infty} \sum_i \int_{B_i} \Big| \sum_{\substack{j \neq i, \\ j \in \mG_\eta}} \mV[D\varphi(x_j)]\Big(\frac{x-x_j}{r_n}\Big)\Big|^2 dx \le C \lambda^{2+ \frac{2-p}{p}} \eta^{-\frac{6}{p}} ||D\varphi||^2_{L^{2s}(\mO)}
\end{align*}
Combining this last inequality with \eqref{decompo_phi_jn} and \eqref{bound_mW}, we finally get: for all $p < 2$,
\begin{align} \label{bound_sum_phi_jn}
& \limsup_{n \to \infty} \Big(\sum_i \int_{B_i} \big| \sum_{\substack{j \neq i, \\ j \in \mG_\eta}} D\phi_{j,n}\big|^2 \Big)^{1/2} \le C' \lambda^{1+ \frac{2-p}{2p}} \, \eta^{-\frac{3}{p}} ||D\varphi||_{L^{p'}(\mO)}
\end{align}
Finally, if we inject \eqref{bound_Dphi_R3} and \eqref{bound_sum_phi_jn} in \eqref{bound_psi_2n}, we obtain that for any $p < 2$,
\begin{align} \label{bound_psi_1n_final}
\limsup_{n \to \infty} ||\na \psi^1_n||_{L^2(\R^3)} & \le C \big( \lambda^{\frac{1}{2}+\frac{1}{p}} + \lambda^{1+ \frac{2-p}{2p}} \, \eta^{-\frac{3}{p}} \big) ||D\varphi||_{L^{p'}(\R^3)}
\end{align}
\bspace
The final step in the proof of Proposition \ref{prop_psi} is to establish bounds on $\psi^3_n$. This term expresses the effect of the balls that are close to one another, and to control it will require the sharp estimates of Theorem \ref{theo_Lp}. Let $M$ the constant in assumption \eqref{B1}, and $\eps_M > 0$ given in point i) of Theorem \ref{theo_Lp}. We fix once for all $p_0 \in (2-\eps_M, 2)$ and $t_0 \in (p_0, 2)$. Set $q_0$ such that $\frac{1}{t_0} + \frac{1}{q_0} = \frac{1}{p_0}$. Let $1 < p < (q_0)'$, $q < p'$, and $M_q$ as given in point ii) of Theorem \ref{theo_Lp}. We set $\eta_q = \max(M_{q_0},M_q) (\frac{3}{4\pi} \lambda)^{\frac13} $, so that $\eta_q n^{-\frac13} = \max(M_{q_0}, M_q) r_n$. We restrict here to $\eta \ge \eta_q$, a condition that will be guaranteed by our choice of $\eta$. We further divide
$$ \mB_\eta \: = \: \Big( \mB_\eta \cap \mG_{\eta_q} \Big) \: \cup \: \Big( \mB_\eta \cap \mB_{\eta_q} \Big) = \mB' \cup \mB''. $$
and correspondingly: $\psi^3_n = \psi' + \psi''$, where
\begin{align*}
-\mu \Delta \psi' + \na p' & = 0, \quad \div \psi' = 0, \quad x \in \R^3\setminus \cup_{i \in \mG_\eta \cup \mB'} \, B_i
\end{align*}
and
\begin{align*}
D \psi'\vert_{B_i} & = 0, \: \forall i \in \mG_\eta, \quad D \psi'\vert_{B_i} = D\varphi\vert_{B_i}, \: i \in \mB' \\
\end{align*}
plus no torque and no force at balls $B_i$, $i \in \mG_\eta \cup \mB'$. We insist that balls $B_i$ with $i \in \mB''$ are completely left aside in the definition of $\psi'$. The point is that for all $\eta \ge \eta_q$, for all $i \neq j \in \mG_{\eta} \cup \mB'$, $|x_i - x_j| \ge M_q r_n$. We can apply point ii) of Theorem \ref{theo_Lp} to deduce
\begin{align*}
||\na \psi'||_{L^q(\R^3)} & \le C \|D\varphi\|_{L^q(\cup_{i \in \mB'} B_i)} \le C \|1_{\cup_{i \in \mB'} B_i}\|_{L^r(\R^3)} \|D\varphi\|_{L^s(\cup_{i \in \mB'} B_i)} \\
& \le C' \Bigl( \text{card} \mB' \, r_n^3 \Big)^{\frac{1}{r}} \|D\varphi\|_{L^s(\cup_{i \in \mB'} B_i)}
\end{align*}
for all $r,s$ with $\frac{1}{r} + \frac{1}{s} = \frac{1}{q}$. Moreover, by assumption \eqref{B2}, $\text{card}\mB' \le \text{card}\mB_\eta \le C \eta^3 n$.
Finally, taking $s = p'$, we find that for all $p > 1$, for all $q < p'$, and all $\eta > \eta_q$:
\begin{align} \label{bound_psi'}
||\na \psi'||_{L^q(\R^3)} & \le C \big( \eta^3 \lambda \big)^{\frac{1}{r}} \|D\varphi\|_{L^{p'}(\R^3)}, \quad \frac{1}{r} + \frac{1}{p'} = \frac{1}{q}.
\end{align}
A similar reasoning holds with $q_0$ instead of $q$. As $p < (q_0)'$, we find that for $r_0$ such that $\frac{1}{r_0} + \frac{1}{p'} = \frac{1}{q_0}$,
\begin{align} \label{bound_psi'_0}
||\na \psi'||_{L^{q_0}(\R^3)} & \le C \big( \eta^3 \lambda \big)^{\frac{1}{r_0}} \|D\varphi\|_{L^{p'}(\R^3)}.
\end{align}
Eventually, the last term $\psi''$ satisfies
\begin{align*}
-\mu \Delta \psi'' + \na p'' & = 0, \quad \div \psi'' = 0, \quad x \in \Omega_n
\end{align*}
and
\begin{align*}
D \psi''\vert_{B_i} & = 0, \: \forall i \in \mG_\eta \cup \mB', \quad D \psi''\vert_{B_i} = D\varphi\vert_{B_i} - D\psi'\vert_{B_i}, \: i \in \mB''
\end{align*}
plus no force and no torque at all balls. Here, we can only rely on assumption \eqref{B1} and point i) of Theorem \ref{theo_Lp}: we find
\begin{align*}
||\na \psi''||_{L^{p_0}(\R^3)} & \le C \Big( \|D\varphi\|_{L^{p_0}(\cup_{i \in \mB''} B_i)} + \|D\psi'\|_{L^{p_0}(\cup_{i \in \mB''} B_i)} \Big) \\
& \le C \|1_{\cup_{i \in \mB''} B_i}\|_{L^{t_0}(\R^3)} \left( \|D\varphi\|_{L^{q_0}(\cup_{i \in \mB'} B_i)} + \|D\psi'\|_{L^{q_0}(\R^3)} \right)
\end{align*}
as $\frac{1}{t_0} + \frac{1}{q_0} = \frac{1}{p_0}$. By assumption \eqref{B2}, $\text{card} \mB'' \le C \eta_q^3 n \le C'_q \lambda n$. Hence,
\begin{align*}
||\na \psi''||_{L^{p_0}(\R^3)} & \le C (\lambda^2)^{\frac{1}{t_0}} \, \left( \|D\varphi\|_{L^{q_0}(\cup_{i \in \mB'} B_i)} + \|D\psi'\|_{L^{q_0}(\R^3)} \right)
\end{align*}
Remember that $t_0 < 2$. Combining with \eqref{bound_psi'_0}, we see that there exists $\theta_0 > 0$ such that for $p < (q_0)'$,
$$ ||\na \psi''||_{L^{p_0}(\R^3)} \le \lambda^{1+\theta_0} ||D \varphi||_{L^{p'}(\R^3)} $$
Combining with \eqref{bound_psi'}, we arrive at: for all $1 < p < (q_0)'$, for all $q < p'$, and $r$ such that $\frac{1}{r} + \frac{1}{p'} = \frac{1}{q}$,
\begin{equation} \label{bound_psi_3n}
||\na \psi^3_n||_{L^{p_0}(\R^3) + L^q(\R^3)} \le C \Big( \big( \eta^3 \lambda \big)^{\frac{1}{r}} + \lambda^{1+\theta_0} \Big) ||D \varphi||_{L^{p'}(\R^3)}
\end{equation}
\noindent
Proposition \ref{prop_psi} follows from collecting \eqref{lim_psi_2n}, \eqref{bound_psi_1n_final} and \eqref{bound_psi_3n}.
\section{Discussion of assumption \texorpdfstring{\eqref{B2}}{(B2)}} \label{sec:prob}
Let $\Phi^\delta = \{y_i\}_i \subset \R^3$ be a stationary ergodic point process on $\R^3$ with intensity $\delta$ and hard-core radius $R$, i.e., $|y_i - y_j| \geq R$ for all $i \neq j$. An example of such a process is a hard-core Poisson point process, which is obtained from a Poisson point process upon deleting all points with a neighboring point closer than $R$. We refer to \cite{MR1950431, MR2371524} for the construction and properties of such processes.
\mspace
Assume that $\mO$ is convex and contains the origin. For $\eps > 0$, we consider the set
\begin{align*}
\eps \Phi^\delta \cap \mO =: \{ x^\eps_i, i =1, \dots, n_\eps\}.
\end{align*}
Let $r < R/2$ and denote $r_\eps = \eps r$ and consider $B_i = \overline{ B(x_i,r_\eps)}$.
The volume fraction of the particles depends on $\eps$ in this case.
However, it is not difficult to generalize our result to the case when the volume fraction converges to $\lambda$ and
this holds in the setting under consideration since
\begin{align*}
\frac{4 \pi}{3} n_\eps r_\eps^3 \to \frac{4 \pi}{3} \delta r^3 =: \lambda(r,\delta) \quad \text{almost surely as } \eps \to 0.
\end{align*}
Clearly, $\lambda(r,\delta) \to 0$, both if $r \to 0$ and if $\delta \to 0$.
However, the process behaves fundamentally different in those cases. Indeed, if we take $r \to 0$ (for $\delta$ and $R$ fixed), we find that condition \eqref{A1}, which implies \eqref{B2}, is satisfied
almost surely for $\eps$ sufficiently small as
\begin{align*}
n_\eps^{1/3} |x_i^\eps - x_j^\eps| \geq n_\eps^{1/3} \eps R \to \delta^{1/3} R.
\end{align*}
\mspace
In the case when we fix $r$ and consider $\delta \to 0$ (e.g. by randomly deleting points from a process), \eqref{A1} is in general not satisfied.
We want to characterize processes for which \eqref{B2} is still fulfilled almost surely as $\eps \to 0$.
Indeed, using again the relation between $\eps$ and $n_\eps$, it suffices to show
\begin{align} \label{eq:B2.prob}
\forall \eta > 0, \quad \#\{i, \: \exists j, \: |x_i - x_j| \le \eta \eps \} \le C \eta^3 \delta^2 \eps^{-3}.
\end{align}
Let $\Phi^\delta_\eta$ be the process obtained from $\Phi^\delta$ by deleting those points $y$ with $B(y,\eta) \cap \Phi^\delta = \{y\}$.
Then, the process $\Phi^\delta_\eta$ is again stationary ergodic (since deleting those points commutes with translations\footnote{In detail: let $\mathcal E_\eta$ be the operator that erases all points without a neighboring point closer than $\eta$, and let $T_x$ denote a translation by $x$. Now, let $\mu$ be the measure for the original process $\Phi^\delta$. Then the measure for $\Phi^\delta_\eta$
is given by $\mu_\eta = \mu \circ \mathcal E_\eta^{-1}$. Since $\mathcal E_\eta T_x = T_x \mathcal E_\eta$ (for all $x$, in particular for $T_{-x} = T^{-1}_x$),
we have for any measurable set $A$ that $T_x \mathcal E_\eta^{-1} A = \mathcal E_\eta^{-1} T_x A$.
This immediately implies that the new process adopts stationarity and ergodicity.}), so that almost surely as $\eps \to 0$
\begin{align*}
\eps^3 \#\{i, \: \exists j, \: |x_i - x_j| \le \eta \eps \} \to \E[\# \Phi^\delta_\eta \cap Q],
\end{align*}
where $Q = [0,1]^3$. Clearly,
$$ \E[\# \Phi^\delta_\eta \cap Q] \le \E \sum_{y \in \Phi^\delta \cap Q} \sum_{y' \neq y \in \Phi^\delta} 1_{B(0,\eta)}(y' - y). $$
We can express this expectation in terms of the 2-point correlation function $\rho^\delta_2(y,y')$ of $\Phi^\delta$ yielding
$$ \E[\# \Phi^\delta_\eta \cap Q] \le \int_{\R^6} 1_Q(y) 1_{B(0,\eta)}(y'-y) \rho^\delta_2(y,y') \dd y \dd y'. $$
Hence, \eqref{eq:B2.prob} and therefore also \eqref{B2} is in particular satisfied if $\rho^\delta_2 \leq C \delta^2$
which is the case for a (hard-core) Poisson point process.
\section*{Acknowledgement}
The first author acknowledges the support of the Institut Universitaire de France, and of the SingFlows project, grant ANR-18-CE40-0027 of the French National Research Agency (ANR)
The second author has been supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
through the collaborative research center ``The Mathematics of Emerging Effects'' (CRC 1060, Projekt-ID 211504053)
and the Hausdorff Center for Mathematics (GZ 2047/1, Projekt-ID 390685813). | {"config": "arxiv", "file": "2002.04846.tex"} |
TITLE: Knowing that $a+b\equiv 1 \pmod{7^{n+1}}$ show that $a^7+b^7\equiv 1 \pmod{7^{n+2}}$
QUESTION [1 upvotes]: Knowing that $a,b$ are prime integers and $a+b\equiv 1 \pmod{7^{n+1}}$ show that $a^7+b^7\equiv 1 \pmod{7^{n+2}}$
I used $a^7+b^7=(a+b)(a^6-a^5b+a^4b^2-a^3b^3+a^2b^4-ab^5+b^6)$ and tried to show that $(a^6-a^5b+a^4b^2-a^3b^3+a^2b^4-ab^5+b^6)\equiv 1 \pmod 7$ but after some trial and error I figured that it might not help as much as I thought. I also tried to show that $ a^7+b^7-1\equiv 0 \pmod{7^{n+2}}$, but I got stuck.
How should I solve this?
REPLY [3 votes]: It's not true. Take $a=7, b=43$. Then $a+b\equiv 1\bmod 49$, but $a^7+b^7\equiv 295\bmod 343$.
In fact, I would think that the primality or otherwise of $a$ and $b$ is irrelevant, given that we are only interested in their values mod $7^{n+1}$; but perhaps if you require that they be co-prime to $7$, your result might hold.
Edited to add: No, that doesn't work either. Take $a=53,b=193$ (both prime, and both co-prime to $7$). Then $a+b\equiv 1\bmod 49$, but $a^7+b^7\equiv 134\bmod 343$. So it seems your claim is simply false. | {"set_name": "stack_exchange", "score": 1, "question_id": 3228479} |
TITLE: How can 0-dimensional particles or 1-dimensional strings be 3D matter?
QUESTION [1 upvotes]: According to the latest information we got
String theory is a theoretical framework in which the point-like
particles of particle physics are replaced by one-dimensional objects
called strings.
We know that anything we've ever seen on a microscopic level or a gigantic level that matter has thickness? let it's thickness be 1.6 x 10-35 m (Plank length) or let it be 1 m , It has some thickness. Right? So everything we've seen is 3-dimensional right? no matter how much we zoom , we get protons that have thickness.So if matter has thickness then how can we say that Something is "1-dimensional".
Anything that's 3D is made up of something 3D. If we add 0 thickness we get 0 thickness but if we add some thickness like atoms and molecules we get 3D matter as the constituents have thickness.
Then how particles of particle physics are replaced by one-dimensional objects called strings?
REPLY [1 votes]: Consider an electron. An electron is an elementary particle whose observed size is small enough that if it was exactly zero there would be no difference in our observations. (Except (I think?) that the implication of being point-like would mean that electrons are naked ring singularities, but as GR and QM don't play nicely together and string theory is an effort to resolve that failure, that's something I'm going to merely acknowledge and not deal with).
Despite being zero-dimensional, electrons still:
Create a 3-dimensional electric force field around themselves, pulling them towards positively charged objects like protons.
Are subject to Heisenburg uncertainty, putting them into a superposition of many possible positions. An important part of this point is that from the way they behave, electrons look like a superposition of point-like particles and not like a particle with non-zero radius.
Are subject to certain constraints on their location, so that when they 'orbit' a proton in a certain way, the wave function describing the probability of being in any given position looks like a shell. Note that this is different from saying "the electron itself is shaped like a shell", as this is just the probability that any given observation will detect the electron in any given position. Because of the nature of these constraints, atoms can bind in interesting ways with complicated and stiff 3-dimensional structures.
For non-elementary particles such as protons and neutrons, a similar argument applies to the smaller fundamental particles that they are made from. | {"set_name": "stack_exchange", "score": 1, "question_id": 431333} |
TITLE: Does $\operatorname EX_n\to0$ as $n\to\infty$?
QUESTION [1 upvotes]: Suppose that $X_1,X_2,\ldots$ are non-negative random variables defined on a probability space $(\Omega,\mathcal F,P)$ with $\operatorname E|X_n|^p<\infty$ with some $p>2$ for each $n\ge1$. Suppose that there exists $\Omega_0\subset\Omega$ such that $P(\Omega_0)=1$ and for each $\omega\in\Omega_0$ there exists $N(\omega)\ge1$ such that $X_n(\omega)=0$ when $n\ge N(\omega)$. In other words, the sequence $\{X_n(\omega):n\ge1\}$ is eventually equal to $0$ for each $\omega\in\Omega_0$.
Can we deduce that $\operatorname EX_n\to0$ as $n\to\infty$?
The first thing that comes to my mind is the monotone convergence theorem. For each $\omega\in\Omega_0$, $0=X_n(\omega)\le X_{n+1}(\omega)=0$ when $n\ge N(\omega)$. If we want to apply the dominated convergence theorem, this inequality has to hold for all $n\ge1$ and all $\omega\in\Omega_0$. We could drop the first $N(\omega)-1$ terms for each $\omega\in\Omega_0$, but $N(\omega)$ depends on $\omega$ so it is not necessarily possible to have $X_n(\omega)\le X_{n+1}(\omega)$ for each sufficiently large $n$.
Any help is much appreciated!
REPLY [2 votes]: No. Example: $X_n =nI_{(0,1/n)}$ on the space $(0,1)$ with Lebesgue measure. | {"set_name": "stack_exchange", "score": 1, "question_id": 2771940} |
TITLE: Why is electric flux through a cube the same as electric flux through a spherical shell?
QUESTION [11 upvotes]: If a point charge $q$ is placed inside a cube (at the center), the electric flux comes out to be $q/\varepsilon_0$, which is same as that if the charge $q$ was placed at the center of a spherical shell.
The area vector for each infinitesimal area of the shell is parallel to the electric field vector, arising from the point charge, which makes the cosine of the dot product unity, which is understandable. But for the cube, the electric field vector is parallel to the area vector (of one face) at one point only, i.e., as we move away from centre of the face, the angle between area vector and electric field vector changes, i.e., they are no more parallel, still the flux remains the same?
To be precise, I guess, I am having some doubt about the angles between the electric field vector and the area vector for the cube.
REPLY [9 votes]: The net flux is the same, but this doesn’t mean the flux is uniform.
Think of a similar situation where you place a lightbulb inside a closed lampshade. The net flux is the total amount of light passing through the lampshade. This depends only on the amount of light produced by the lightbulb, not by the position of the lightbulb.
In other words, if you can take a 60W lightbulb and move it anywhere inside your (closed) lampshade, and this will not change the total amount of light that goes through the lampshade. Of course unless you place the light bulb exactly at the center of a spherical lampshade, the amount of light will be NOT be uniform on every surface of your lampshade, but that not the net flux, which is the sum total of light of all the light on the entire lampshade.
Note I didn’t discuss the shape of the lampshade or its size. The net flux is determined by the strength of the source, not by the surface through which the light passes. | {"set_name": "stack_exchange", "score": 11, "question_id": 550820} |
TITLE: Assume that there is a test for cancer that is 90% accurate for both those who do and do not have cancer.
QUESTION [0 upvotes]: Assume that there is a test for cancer that is 90% accurate for both those who do and do not have cancer. Also assume that 5% of the population has cancer. If an individual takes this test and tests positive for cancer, find the probability that they really do have cancer?
For this problem, I did a tree diagram with population on top then split it into 5% cancer and 95% no cancer. Then I split the 5% into 90% positive test and 10% negative test. Finally I split the 95% into 10% positive test and 90% negative. However, I feel like I'm off somewhere and I do not know how to continue from here. Any help would be much appreciated.
REPLY [0 votes]: I prefer to use a contingency table rather than a tree diagram.
$$\begin{array}{|c|c|c|c|}
\hline
& \text{Test Positive} & \text{Test Negative} & \\
\hline
\text{Has Cancer} & a & b & a + b \\
\hline
\text{No Cancer} & c & d & c + d \\
\hline
& a+c & b+d & a+b+c+d \\
\hline
\end{array}$$
We are told that $95\%$ of the population does not have cancer, and $5\%$ does. So let's pick $a+b+c+d = 1000$ people in our population; then $a+b = (0.05)(1000) = 50$, and $c+d = 950$. Of those with cancer, the test is positive $90\%$ of the time, thus $a = (0.90)(50) = 45$, and $b = 5$. Similarly, of those who do not have cancer, the test is negative $90\%$ of the time, so $d = (0.9)(950) = 855$, and $c = 950 - 855 = 95$. So the table looks like this:
$$\begin{array}{|c|c|c|c|}
\hline
& \text{Test Positive} & \text{Test Negative} & \\
\hline
\text{Has Cancer} & 45 & 5 & 50 \\
\hline
\text{No Cancer} & 95 & 855 & 950 \\
\hline
& 140 & 860 & 1000 \\
\hline
\end{array}$$
Now, how do we use this to answer the question? If an individual tests positive, they are among the $140$ people in the table. Of these, how many actually have cancer? $45$. | {"set_name": "stack_exchange", "score": 0, "question_id": 3840443} |
\section{Approaching separatrices and passing through separatrices} \label{s:decompose}
In this section we state a technical theorem on crossing a small neighborhood of separatrices and reduce Theorem~\ref{t:main} to this technical theorem. Then we split the technical theorem into three lemmas on approaching separatrices, crossing separatrices, and moving away from separatrices.
\begin{itemize}
\item For given $X_{init} \in \mathcal A_3$, denote $\lambda_{init} = \lambda(X_{init})$ and let $X(\lambda)$ be the solution of the perturbed system~\eqref{e:perturbed-pq} with initial data $X(\lambda_{init}) = X_{init}$.
\item Given $\overline X_0 \in \mathbb R^{n+1}_{h, z}$ and $\lambda_0$, denote by $\overline X(\lambda)$ the solution of the averaged system (one needs to specify in which domain $\mathcal B_i$ or in which union of these domains) with initial data $\overline X(\lambda_0) = \overline X_0$.
\end{itemize}
\begin{theorem} \label{t:sep-pass}
Given any $z_* \in \mathcal Z_B$, for any small enough $c_z, c_h > 0$ for any $C_0, \Lambda > 0$ there exists $C > 0$ such that for any small enough $\varepsilon$ there exists $\mathcal E \subset \mathcal A_3$ with
\[
m(\mathcal E) \le C \sqrt{\varepsilon} |\ln^5 \varepsilon|
\]
such that the following holds for any $X_{init} \in \mathcal A_3 \setminus \mathcal E$.
Suppose at some time
\[
\lambda_0 \in [\lambda_{init}, \lambda_{init} + \varepsilon^{-1} \Lambda]
\]
the point $X_0 = X(\lambda_0)$ satisfies
\[
X_0 \in \mathcal A_3, \qquad
\norm{z(X_0) - z_*} \le c_z, \qquad
h(X_0) = c_h.
\]
Then there exists $i=1,2$ and $\lambda_1 > \lambda_0$ such that
\[
X(\lambda_1) \in \mathcal A_i, \qquad
h(X(\lambda_1)) = -c_h.
\]
Take any $\overline X_0 \in \mathbb R^{n+1}_{h, z}$ with
\[
\norm{\overline X_0 - (h(X_0), z(X_0))} < C_0 \sqrt{\varepsilon} |\ln \varepsilon|
\]
and consider the solution $\overline X(\lambda)$ of averaged system corresponding to capture from $\mathcal B_3$ to $\mathcal B_i$ with initial data $\overline X(\lambda_0) = \overline X_0$.
Then for any $\lambda \in [\lambda_0, \lambda_1]$ we have
\begin{equation} \label{e:thm-est}
|h(X(\lambda)) - h(\overline X(\lambda))| < C \sqrt{\varepsilon} |\ln \varepsilon|, \qquad
\norm{z(X(\lambda)) - z(\overline X(\lambda))} < C \sqrt{\varepsilon} |\ln \varepsilon|.
\end{equation}
\end{theorem}
Let us now prove the main theorem using the technical theorem to cover passage near separatrices and~\cite[Theorem 1 and Corollary 3.1]{neishtadt2014} far from separatrices. Let us now state this result from~\cite{neishtadt2014} using our notation.
\begin{theorem}[{\cite{neishtadt2014}}]
Pick $\hat{\overline X}_0$ and $\Lambda > 0$. Suppose that solution of the averaged system $\hat{\overline X}$ with initial data $\hat{\overline X}(0) = \hat{\overline X}_0$ stays far from the separatrices for $\lambda \in [0, \varepsilon^{-1} \Lambda]$ and satisfies certain conditions (discussed right after the statement of theorem).
Then for small enough $r > 0$ for any small enough $\varepsilon > 0$ there exists $\mathcal E \subset A_r(\hat{\overline X}_0)$ with $m(\mathcal E) = O(\sqrt{\varepsilon})$ such that for any
$X_0 \in A_r(\hat{\overline X}_0) \setminus \mathcal E$ we have
\[
\abs{ h(X(\lambda)) - h(\overline X(\lambda)) } = O(\sqrt \varepsilon |\ln \varepsilon|),
\qquad
\norm{ z(X(\lambda)) - z(\overline X(\lambda)) } = O(\sqrt \varepsilon |\ln \varepsilon|)
\]
for $\lambda \in [\lambda_0, \lambda_0 + \varepsilon^{-1} \Lambda]$,
where $\lambda_0 = \lambda(X_0)$, $X(\lambda)$ is the solution of perturbed system with initial data $X(\lambda_0) = X_0$ and
$\overline X(\lambda)$ is the solution of averaged system with initial data $\overline X(\lambda_0) = (h(X_0), z(X_0))$.
\end{theorem}
\noindent For a full statement of the conditions, we refer the reader to~\cite[Section 2]{neishtadt2014}. When we apply this theorem below, these conditions are satisfied, the conditions in Section~\ref{s:main-theorem} are written for this purpose.
We will also need the lemma below, it is proved in Appendix~\ref{a:proof-aux}.
\begin{lemma} \label{l:volume}
For any $\Lambda > 0$ there exists $C > 1$ such that the flow $g^\lambda$ of~\eqref{e:perturbed-pq} satisfies the following.
Take open
$A \subset \mathbb R^{2+n}_{p, q, z} \times [0, 2\pi]_\lambda$
with $m(A) < \infty$. Then for any $\lambda \in [-\varepsilon^{-1} \Lambda, \varepsilon^{-1} \Lambda]$ we have
\[
m(g^\lambda(A)) \le C m(A).
\]
\end{lemma}
\begin{proof}[Proof of Theorem~\ref{t:main}]
Take $c_h, c_z > 0$ such that we can apply Theorem~\ref{t:sep-pass} with these constants.
Recall that
$\hat{\overline X}_i(\lambda) = (\hat{\overline h}_i(\lambda), \hat{\overline z}_i(\lambda))$
denotes the solution of averaged system describing capture in $\mathcal B_i$ with $\hat{\overline X}_i(0) = \hat{\overline X}_0$.
Define $\lambda_+$ by $\hat{\overline h}_1(\lambda_+) = \hat{\overline h}_2(\lambda_+) = 2c_h/3$ and $\lambda_{-, i}$ by $\hat{\overline h}_i(\lambda_{-,i}) = -2c_h/3$, $i=1,2$.
The number $\Lambda_1$ from the conditions for the main theorem is such that if $\lambda$ is $\varepsilon^{-1} \Lambda_1$-close to $\lambda_*$, we have $|\hat{\overline h}_i(\lambda)| < c_h/2$ and $\norm{\hat{\overline z}_i(\lambda) - z_*} < c_z/2$ for $i=1,2$.
Thus
\[
\lambda_+ < \lambda_* - \varepsilon^{-1} \Lambda_1 < \lambda_* + \varepsilon^{-1} \Lambda_1 < \lambda_{-, i}.
\]
This means condition $B$ from~\cite[Section 2]{neishtadt2014} is satisfied for
\[
\hat{\overline X}_i(\lambda), \lambda \in [0, \lambda_+]
\qquad \text{and} \qquad
\hat{\overline X}_i(\lambda), \lambda \in [\lambda_{-, i}, \varepsilon^{-1} \Lambda].
\]
For small enough $r$ solutions $\overline X'(\lambda) = (\overline h'(\lambda), \overline z'(\lambda))$ of averaged system with any initial condition
$\overline X'(0) \in B_r(\hat{\overline X}_0)$ satisfy
\begin{equation} \label{e:loc-5925}
\overline h'(\lambda_+) \in [c_h/2, 3c_h/4],
\qquad
\overline h_i'(\lambda_{-, i}) \in [-3c_h/4, -c_h/2].
\end{equation}
By~\cite[Corollary 3.1]{neishtadt2014} we have~\eqref{e:mt-close} for $\lambda \in [0, \lambda_+]$, given that $X_0$ is not in some set $\mathcal E_1$ of measure $\lesssim \sqrt{\varepsilon}$.
Together with~\eqref{e:loc-5925} for small $\varepsilon$ this implies $h(X(\lambda_0 + \lambda_+)) < 4c_h/5$.
By continuity we have $h(X(\lambda_0 + \lambda'_+)) = c_h$ for some $\lambda'_+ \in [0, \lambda_+]$.
Thus we can apply Theorem~\ref{t:sep-pass} (with $\lambda_0$ in this theorem equal to $\lambda_0 + \lambda'_+$). This theorem gives (possibly after reducing $r$) a set $\mathcal E_2$ of measure $\lesssim \sqrt{\varepsilon} |\ln^5 \varepsilon|$ such that if
$X_0 \not \in \mathcal E_1 \cup \mathcal E_2$,
there is $i=1,2$ and $\lambda'_{-, i}$ such that $X(\lambda_0 + \lambda'_{-, i}) \in \mathcal A_i$, with $h(X(\lambda_0 + \lambda'_{-, i})) = -c_h$ and~\eqref{e:mt-close} holds for
$\lambda \in [\lambda_0 + \lambda_+, \lambda_0 + \lambda'_{-, i}]$.
We have $\overline h_i(\lambda_0 + \lambda'_{-, i}) < -4c_h/5$,
by~\eqref{e:loc-5925} this implies $\lambda'_{-, i} > \lambda_{-, i}$ and so~\eqref{e:mt-close} holds for $\lambda = \lambda_0 + \lambda_{-, i}$.
Denote
\[
\hat{\overline X}_{-,1} = \hat{\overline X}_1(\lambda_{-, 1}), \qquad
\hat{\overline X}_{-,2} = \hat{\overline X}_2(\lambda_{-, 2}).
\]
By~\cite[Corollary 3.1]{neishtadt2014} there exist $r_-$ and $\mathcal E'_{-, i}, i=1,2$ with $m(\mathcal E'_{-, i}) = O(\sqrt{\varepsilon})$ such that for $i=1,2$ solutions starting in $A_{r_-}(\hat{\overline X}_{-,i}) \setminus \mathcal E'_{-, i}$ are approximated by solutions of the averaged system (with the same initial $h, z$) with error
$O(\sqrt{\varepsilon} \ln \varepsilon)$.
Let $\mathcal E_{-, i}$ be the preimage of $\mathcal E'_{-, i}$ under the flow of perturbed system over time $\lambda_{-, i}$, we have $m(\mathcal E_{-, i}) = O(\sqrt{\varepsilon})$ by Lemma~\ref{l:volume}.
We can now write the exceptional set in the current theorem:
$\mathcal E = \mathcal E_1 \cup \mathcal E_2 \cup \mathcal E_{-, 1} \cup \mathcal E_{-, 2}$.
Reducing $r$ if needed, we may assume that solutions of averaged system describing capture in $\mathcal B_i$ starting in $A_r(\hat{\overline X}_0)$ at $\lambda=\lambda_0$ are in $A_{r_-/2}(\hat{\overline X}_{-, i})$ at $\lambda=\lambda_0 + \lambda_{-,i}$ for $i=1,2$. Thus $X(\lambda)$ is $O(\sqrt{\varepsilon}|\ln \varepsilon|)$-close to the solution of averaged system with initial data
\[
\overline X(\lambda_0 + \lambda_{-, i}) = (h(X(\lambda_0 + \lambda_{-, i})), z(X(\lambda_0 + \lambda_{-, i})))
\]
for
$\lambda \in [\lambda_0 + \lambda_{-, i}, \lambda_0 + \varepsilon^{-1} \Lambda]$.
The difference between this solution of averaged system and $\overline X_i(\lambda)$ at the moment $\lambda=\lambda_0 + \lambda_{-, i}$ is $O(\sqrt \varepsilon |\ln \varepsilon|)$, it stays of the same order, as the dynamics in slow time takes time $O(1)$ and the averaged system is smooth far from separatrices. Thus we have~\eqref{e:mt-close} for $\lambda \in [\lambda_0 + \lambda_{-, i}, \lambda_0 + \varepsilon^{-1} \Lambda]$. Now we have proved~\eqref{e:mt-close} for all
$\lambda \in [\lambda_0, \lambda_0 + \varepsilon^{-1} \Lambda]$,
as required.
\end{proof}
Let us now split Theorem~\ref{t:sep-pass} into a lemma on approaching the separatrices, lemma on crossing immediate neighborhood of separatrices, and lemma on moving away from the separatrices.
\begin{itemize}
\item Take $\rho = 5$, suppose we are given $C_\rho > 0$. Denote $h_* = C_\rho \varepsilon |\ln^\rho \varepsilon|$.
\end{itemize}
The immediate neighborhood of separatrices is given by $|h| < h_*(\varepsilon)$.
\begin{lemma}[On approaching separatrices] \label{l:approach-sep}
Take any $z_* \in \mathcal Z_B$, any small enough $c_h, c_z > 0$, any large enough $C_\rho > 0$.
Then for any $C_0, \Lambda > 0$ there exists $C > 0$ such that for any small enough $\varepsilon$ there exists $\mathcal E \subset \mathcal A_3$ with
\[
m(\mathcal E) < C \sqrt{\varepsilon}
\]
such that for any $X_{init} \in \mathcal A_3 \setminus \mathcal E$ the following holds.
Suppose that at some time
\[
\lambda_0 \in [\lambda_{init}, \lambda_{init} + \varepsilon^{-1} \Lambda]
\]
the point $X_0 = X(\lambda_0)$ satisfies
\[
X_0 \in \mathcal A_3, \qquad
h(X_0) = c_h, \qquad
\norm{z(X_0) - z_*} \le c_z.
\]
Then at some time $\lambda_1 > \lambda_0$ we have
\[
X(\lambda_1) \in \mathcal A_3, \qquad h(X(\lambda_1)) = h_*(\varepsilon).
\]
Take any $\overline X_0 \in \mathbb R^{n+1}_{h, z}$ with
\[
\norm{\overline X_0 - (h(X_0), z(X_0)} < C_0 \sqrt{\varepsilon} |\ln \varepsilon|.
\]
and consider the solution $\overline X(\lambda)$ of averaged system in $\mathcal B_3$ with initial data $\overline X(\lambda_0) = \overline X_0$.
Then for any $\lambda \in [\lambda_0, \lambda_1]$ we have
\begin{equation}
|h(X(\lambda)) - h(\overline X(\lambda))| < C \sqrt{\varepsilon} |\ln \varepsilon|, \qquad
\norm{z(X(\lambda)) - z(\overline X(\lambda))} < C \sqrt{\varepsilon} |\ln \varepsilon|.
\end{equation}
Moreover,
\begin{equation}
|h(X(\lambda_1)) - h(\overline X(\lambda_1))| < C \sqrt{\varepsilon}.
\end{equation}
\end{lemma}
\noindent This lemma will be proved in Section~\ref{s:approach-proof}.
\begin{lemma}[On moving away from separatrices] \label{l:away-sep}
Take any $z_* \in \mathcal Z_B$, any small enough $c_h, c_z > 0$, any large enough $C_\rho > 0$.
Then for any $C_0, \Lambda > 0$ there exists $C > 0$ such that for any small enough $\varepsilon$ there exists $\mathcal E \subset \mathcal A_3$ with
\[
m(\mathcal E) < C \sqrt{\varepsilon}
\]
such that for any $X_{init} \in \mathcal A_3 \setminus \mathcal E$ the following holds.
Suppose that for $i=1$ or $i=2$ at some time
\[
\lambda_0 \in [\lambda_{init}, \lambda_{init} + \varepsilon^{-1} \Lambda]
\]
the point $X_0 = X(\lambda_0)$ satisfies
\[
X_0 \in \mathcal A_i, \qquad
h(X_0) = -h_*(\varepsilon), \qquad
\norm{z(X_0) - z_*} \le c_z.
\]
Take any $\overline X_0 = (\overline h_0, \overline z_0) \in \mathbb R^{n+1}_{h, z}$ with
\[
|\overline h_0 - h(X_0)| < C_0 \sqrt{\varepsilon}, \qquad
\norm{\overline z_0 - z(X_0)} < C_0 \sqrt{\varepsilon} |\ln \varepsilon|
\]
and consider the solution $\overline X(\lambda)$ of averaged system in $\mathcal B_i$ with initial data $\overline X(\lambda_0) = \overline X_0$.
Then for any $\lambda \in [\lambda_0, \lambda_{init} + \varepsilon^{-1} \Lambda]$ we have
\begin{equation}
|h(X(\lambda)) - h(\overline X(\lambda))| < C \sqrt{\varepsilon} |\ln \varepsilon|, \qquad
\norm{z(X(\lambda)) - z(\overline X(\lambda))} < C \sqrt{\varepsilon} |\ln \varepsilon|.
\end{equation}
\end{lemma}
\noindent This lemma is proved similarly to the previous one, thus we omit the proof. In the proofs of these two lemmas we estimate the difference between solutions of perturbed and averaged system in the chart $w = (I, z)$, and this distance is $O(\sqrt{\varepsilon}|\ln \varepsilon|)$.
This and the estimate $\norm{\pdv{h}{w}} = O(\ln^{-1} h)$ in Lemma~\ref{l:d-f-0-I} below explains why near separatrices (when $\ln h \sim \ln \varepsilon$) the distance in $h$ is $O(\sqrt{\varepsilon})$, while far from separatrices this distance is $O(\sqrt{\varepsilon}|\ln \varepsilon|)$.
\begin{lemma}[On passing separatrices] \label{l:near-sep-local}
Take any $z_* \in \mathcal Z$, any small enough $c_z > 0$, any large enough $C_\rho > 0$. Then for any $\Lambda > 0$ and $\gamma \in \mathbb R$ there exists $C > 0$ such that for any small enough $\varepsilon > 0$
there exists a set $\mathcal E \subset \mathcal A_3$
with
\[
m(\mathcal E) \le C \sqrt{\varepsilon} |\ln^{\rho - \gamma + 1} \varepsilon|
\]
such that for any
$X_{init} \in \mathcal A_3 \setminus \mathcal E$
the following holds.
Suppose that at some time
\[
\lambda_0 \in [\lambda_{init}, \lambda_{init} + \varepsilon^{-1} \Lambda]
\]
the point $X_0 = X(\lambda_0)$ satisfies
\begin{equation}
X_0 \in \mathcal A_3, \qquad
\norm{z(X_0) - z_*} \le c_z, \qquad
h(X_0) = h_*(\varepsilon).
\end{equation}
Then at some time $\lambda_1 > \lambda_0$ with
\[
\varepsilon(\lambda_1 - \lambda_0) \le \sqrt{\varepsilon} |\ln^\gamma \varepsilon|.
\]
we have
\[
X(\lambda_1) \in \mathcal A_1 \cup \mathcal A_2, \qquad
h(X(\lambda_1)) = -h_1.
\]
\end{lemma}
\noindent This lemma will be proved in Section~\ref{s:pass-sep}.
\begin{proof}[Proof of Theorem~\ref{t:sep-pass}]
Suppose that $c_z^{(0)}$ is small enough for all three lemmas, take in the theorem $c_z = c_z^{(0)} / 3$. Take $c_h$ in the theorem such that
\begin{itemize}
\item it is small enough for all three lemmas;
\item while any solution of averaged system describing passage from $\mathcal B_3$ to $\mathcal B_1$ or $\mathcal B_2$ passes from $h = c_h$ to $h = -2 c_h$ the total variation of $z$ is at most $c_z/3$.
This can be done, as $\dv{z}{h} = O(\ln h)$ for solutions of averaged system.
\end{itemize}
Take $C_\rho$ large enough for all three lemmas.
Take $\Lambda^{(0)}$ such that solutions of averaged systems describing capture in $\mathcal B_1$ and $\mathcal B_2$ starting with $h=c_h$ reach $h=-2c_h$ after time less than $\varepsilon^{-1} \Lambda^{(0)}$ passes. We will apply the three lemmas with $\Lambda$ greater than given in the theorem by $\Lambda^{(0)}$.
Given $C_0$ and $\Lambda$, Lemma~\ref{l:approach-sep} gives us $C, \mathcal E$ that we denote by $C^{(1)}, \mathcal E^{(1)}$.
For $X_{init} \in \mathcal A_3 \setminus \mathcal E^{(1)}$ the solution $X(\lambda)$ reaches $h = h_*(\varepsilon)$ at some moment that we denote $\lambda^{(1)}$. For $\lambda \in [\lambda_0, \lambda^{(1)}]$ we have~\eqref{e:thm-est} with $C = C^{(1)}$.
Note that $\norm{z(X(\lambda^{(1)})) - z_*} < c_z^{(0)}$ for small enough $\varepsilon$ due to~\eqref{e:thm-est} and $\norm{z(\overline X(\lambda^{(1)})) - z_*} < c_z/3$ (this holds by our choice of $c_h$).
Apply Lemma~\ref{l:near-sep-local} with $\gamma=1$, $c_z = c_z^{(0)}$, $C_0 = C^{(1)}$, it gives us $C, \mathcal E$ that we denote by $C^{(2)}, \mathcal E^{(2)}$.
For $X_{init} \in \mathcal A_3 \setminus (\mathcal E^{(1)} \cup \mathcal E^{(2)})$ the solution $X(\lambda)$ reaches $h = -h_*(\varepsilon)$ at some moment that we denote $\lambda^{(2)}$.
We have $\varepsilon(\lambda^{(2)} - \lambda^{(1)}) \le \sqrt{\varepsilon} |\ln \varepsilon|$.
The change of $h, z, \overline h, \overline z$ during this time is bounded by
$C^{(2)} \sqrt{\varepsilon} |\ln \varepsilon|$ for some $C^{(2)} > 0$.
Then for $\lambda \in [\lambda^{(1)}, \lambda^{(2)}]$ we have~\eqref{e:thm-est} with $C = C^{(1)} + 2C^{(2)}$. As above, we get $\norm{z(X(\lambda^{(2)})) - z_*} < c_z^{(0)}$.
Finally, apply Lemma~\ref{l:away-sep} with $c_z = c_z^{(0)}$ and $C_0 = C^{(1)} + 2 C^{(1)}$, it gives us $C, \mathcal E$ that we denote by $C^{(3)}, \mathcal E^{(3)}$. Set $\mathcal E$ in the theorem equal to
$\mathcal E^{(1)} \cup \mathcal E^{(2)} \cup \mathcal E^{(3)}$.
For $X_{init} \in \mathcal A_3 \setminus \mathcal E$
the solution $X(\lambda)$ reaches $h = -c_h$ at some moment that we denote $\lambda^{(3)}$.
For $\lambda \in [\lambda^{(2)}, \lambda^{(3)}]$ we have~\eqref{e:thm-est} with $C = C^{(3)}$.
Take in the theorem $C = C^{(3)} + 2C^{(1)} + 2 C^{(2)}$.
\end{proof} | {"config": "arxiv", "file": "2108.08540/decompose.tex"} |
TITLE: An inequality relating the ratio of the areas of two triangles
QUESTION [6 upvotes]: The conjecture below is a modified version of this question:
Prove that, given a triangle with sides $a,b,c$, there exists a triangle with sides $a+2b,b+2c,c+2a$ that has an area three times the original
Conjecture:
If $u$ is the area of a triangle with sides $a,b,c$, and $v$ is the area of a triangle with sides $a+2b,b+2c,c+2a$, then ${\large{\frac{v}{u}}}\ge 9$.
Remarks:
For the equilateral case ($a=b=c$), we get ${\large{\frac{v}{u}}}=9$.
Limited data testing seems to support the truth of the conjecture.
Trying to prove the claim via Heron's formula appears to be a disaster.
Question:$\;$Is the conjecture true?
REPLY [3 votes]: Let $x=b+c-a\geq 0$, $y=a+c-b\geq 0$, $z=a+b-c\geq 0$.
Let $p=\frac{1}{2}(a+b+c)$ be the semiperimeter of the triangle with sides $a$, $b$, $c$. Similarly let $P=\frac{1}{2}(A+B+C)=3p$ be the semiperimeter of the triangle with sides $A=b+2c$, $B=c+2a$, $C=a+2b$. Then, by Heron's formula, the inequality is equivalent to
$$
\begin{align}0&\leq \frac{v^2}{u^2}-9^2=\frac{P(P-A)(P-B)(P-C)}{p(p-a)(p-b)(p-c)}-9^2\\&=
\frac{6((x^2y+y^2z+z^2x)+2(xy^2+yz^2+zx^2)-9xyz)}{xyz}
\end{align}$$
which holds because by the AGM inequality
$$x^2y+y^2z+z^2x\geq 3xyz\quad\mbox{and}\quad xy^2+yz^2+zx^2\geq 3xyz.$$ | {"set_name": "stack_exchange", "score": 6, "question_id": 2845620} |
TITLE: Prove convexity of split function (linear and quadratic)
QUESTION [1 upvotes]: Prove that
$$
f(x)=\begin{cases}
\frac{2}{3}x^2+\frac{2}{3}x-\frac{1}{12} &\quad x<-0.5 \\
-0.25 &\quad x\geq -0.5 \\
\end{cases}
$$
is convex over $\mathbb{R}$.
As far as I understand I cannot use second derivative because the function is non-differentiable (at $x=-0.5$).
If both $x,y \geq0.25$ or $x,y < 0.25$ this is easy (both cases are convex). But I couldn't find an algebric approach to prove the case where $x<-0.5$ and $y \geq -0.5$.
Please advise.
Thank you.
REPLY [2 votes]: Thanks to @dxiv.
I didn't notice that the function $f(x)$ actually is differentiable.
$$
f'(x)=\begin{cases}
\frac{4}{3}x+\frac{2}{3} &\quad x<-0.5 \\
0 &\quad x\geq -0.5 \\
\end{cases}
\quad
f''(x)=\begin{cases}
4/3 &\quad x<-0.5 \\
0 &\quad x\geq -0.5 \\
\end{cases}
$$
One can see that $f'_+(-0.5)=0=f'_-(-0.5)$ (left and right derivatives are equal), so $f$ is continuously differentiable over $\mathbb{R}$.
From Wikipedia:
differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval
A twice differentiable function of one variable is convex on an interval if and only if its second derivative is non-negative there | {"set_name": "stack_exchange", "score": 1, "question_id": 4157629} |
TITLE: Local time for conditioned simple random walk
QUESTION [2 upvotes]: Consider simple random walk started at $1$ on the path $[0,M]$. Let $\tau_x$ be the time to reach $x$ and condition on the event $\tau_M < \tau_0$ (the walk hits $M$ before reaching $0$). Conditioned on this event, let $V_k$ be the number of visits to $k$ up to time $\tau_M$. What is $\mathbb E V_k$?
All we need is a bound independent of $M$. Something like there exists $C>0$ such that
$$\sup_{M >0 }\Bigl[ \max_{0< k < M} \mathbb E V_k \Bigr]< C.$$
REPLY [3 votes]: To second Anthony Quas's answer, here's a simple way to calculate $\mathbb{E}V_k$. Notice that conditional on $\tau_M<\tau_0$ your walk will visit $k$ at least once. Once we are at $k$, the number of consecutive visits is a positive geometric random variable, since the walk is a Markov process. The parameter is $1-\mathbb{P}(V_k=1 | \tau_M<\tau_0)$, and therefore its expectation value is
$$\mathbb{E}[V_k]=\frac{1}{\mathbb{P}(V_k=1 | \tau_M<\tau_0)} = \frac{\mathbb{P}(\tau_M < \tau_0)}{\mathbb{P}(\tau_k<\tau_0)\frac{1}{2}\mathbb{P}(\tau_{M-k}<\tau_0)},$$
since the walk has to visit $k$ before $0$, step to $k+1$, and then visit $M$ before returning to $k$. Since $\mathbb{P}(\tau_M < \tau_0) = 1/M$ (ballot problem) we find $$\mathbb{E}[V_k]= 2 k (1-k/M)$$. | {"set_name": "stack_exchange", "score": 2, "question_id": 258151} |
\label{app:statmech_finite}
In this Appendix, we collect some facts about rooted spanning forests, killed random walks and their link to
network random walks and spanning trees, that are useful for Section \ref{sec:statmech}.
Suppose for the moment that $\Gs$ is a finite connected
(not necessarily isoradial) graph, with a massive
Laplacian $\Delta^m$. Equivalently, by Equation~\eqref{equ:operator_general},
$\Gs$ is endowed with positive conductances $(\rho(e))_{e\in\Es}$
and positive masses $(m^2(x))_{x\in\Vs}$.
Consider the graph $\Gs^\rs=(\Vs^\rs,\Es^\rs)$ obtained from $\Gs$
by adding a root vertex $\rs$ and joining every vertex of $\Gs$ to $\rs$, as in Section \ref{subsec:rsf_ust}. The graph $\Gs^\rs$ is weighted by the function
$\rho^m$, see~\eqref{equ:defrhom}.
\subsection{Massive harmonicity on $\Gs$ and harmonicity on $\Gs^\rs$}
\label{subsec:massive_harmo}
There is a natural (non-massive) Laplacian $\Delta_\rs$ on $\Gs^\rs$,
acting on functions $f$ defined on vertices of $\Gs^\rs$:
\begin{equation*}
\forall\ x\in\Vs^\rs,\quad \Delta_\rs f(x) =
\sum_{xy\in\Es^\rs} \rho^m(xy)[f(x)-f(y)].
\end{equation*}
Then the restriction $\Delta^{(\rs)}_{\rs}$ of the matrix of $\Delta_\rs$
to vertices of $\Gs$, obtained by removing the row and column corresponding to
$\rs$, is exactly the matrix $\Delta^m$.
Functions on vertices of $\Gs$ are in bijection with functions on vertices of $\Gs^\rs$ taking
value $0$ on $\rs$ (by extension/restriction). This bijection is compatible with
the Laplacians on $\Gs$ and $\Gs^\rs$: if $f$ is a function on $\Gs$ and
$\widetilde{f}$ is its extension to $\Gs^\rs$ such that $\widetilde{f}(\rs)=0$, then
$\Delta^m f = \Delta_\rs \widetilde{f}$ on
$\Gs$.
The operator $\Delta^m$ is invertible, and its inverse is
$G^m$, the massive Green function of $\Gs$.
The matrix $\Delta_\rs$ is not invertible: its kernel is exactly the space of
constant functions on $\Gs^\rs$, but its restriction to functions on
$\Gs^{\rs}$ vanishing at $\rs$ is invertible, and its negated inverse is exactly
$\widetilde{G}^m$, the extension of $G^m$ to $\Gs^\rs$, taking the value 0 at $\rs$:
\begin{equation*}
\forall\ x,y\in\Vs^{\rs},\quad \widetilde{G}^m(x,y)=\widetilde{G}^{m}(y,x)=
\begin{cases}
G^m(x,y) & \text{if $x$ and $y$ are vertices of $\Gs$,} \\
0 & \text{if $x$ or $y$ is equal to $\rs$.}
\end{cases}
\end{equation*}
\subsection{Random walks}
\label{subsec:rw_app}
The \emph{network random walk} $(Y_j)_{j\geq 0}$ on $\Gsr$ with initial state $x_0$ is defined by $Y_0=x_0$ and
jumps
\begin{equation*}
\forall\,x,y\in \Vs^\rs,\quad
P_{x,y}=\PP_{x_0}[Y_{j+1}=y|Y_j=x]=
\begin{cases}
\displaystyle\frac{\rho^m(xy)}{\rho^m(x)}&\text{ if $y\sim x$,}\\
0&\text{ otherwise},
\end{cases}
\end{equation*}
where
\begin{equation}
\label{eq:def_rho^m}
\rho^m(x)=
\sum_{y\in\Vs^\rs:y\sim x}\rho^m(xy)=
\begin{cases}
\sum\limits_{y\in\Vs:y\sim x}\rho(xy)+m^2(x)& \text{if $x\neq \rs$,} \\
\sum\limits_{y\in\Vs} m^2(y) & \text{if $x=\rs$.}
\end{cases}
\end{equation}
The Markov matrix $P=(P_{x,y})$ is related to the Laplacian $\Delta_\rs$ as follows: if
$A_{\rs}$ denotes the diagonal matrix whose entries are the diagonal entries of the Laplacian $\Delta_\rs$, then
\begin{equation}
\label{eq:link_matrix_P_Laplacian}
P = I+ (A_\rs)^{-1}\Delta_\rs.
\end{equation}
This random walk is positive recurrent. The \emph{potential} $V_\rs(x,y)$ of this
random walk is defined as the
difference in expectation of the number of visits at $y$ starting from $x$ and
from $y$:
\begin{equation*}
V_{\rs}(x,y)=
\mathbb{E}_x\Biggl[\sum_{j=0}^{\infty}\mathbb{I}_{\{Y_j=y\}}\Biggr] -
\mathbb{E}_y\Biggl[\sum_{j=0}^{\infty}\mathbb{I}_{\{Y_j=y\}}\Biggr].
\end{equation*}
Although both sums separately are infinite, the difference makes sense and is
finite, as can be seen by computing $V_{\rs}(x,y)$ with
a coupling of the random walks starting from $x$ and $y$, where they evolve
independently until they meet (in finite time a.s.), and stay
together afterward.
Because $(Y_j)$ is (positive) recurrent, the time $\tau_\rs$ for $(Y_j)$ to
hit $\rs$ is finite a.s.
We can define the killed random walk
$(X_j)=(Y_{j\wedge(\tau_r-1)})$, absorbed at the
root $\rs$. The process $(X_j)$ visits only a finite number of vertices of
$\Gs$ before being absorbed: every vertex is thus transient. If $x$ and
$y$ are two vertices of $\Gs$, then we can define the potential of $(X_j)$,
$V^{m}(x,y)$, as the expected number of visits at $y$ of $(X_j)$
starting from $x$, before it gets absorbed. $V^m$ and $V_\rs$ are linked by the
formula below, which directly follows from the strong Markov property:
\begin{equation}
\label{eq:massive_potential}
\forall\ x,y\in\Gs,\quad
V^{m}(x,y) = V_{\rs}(x,y)-V_{\rs}(\rs,y).
\end{equation}
As a matrix, $V^m$ is equal to $(I-Q^m)^{-1}$ where $Q^m$ is the substochastic
transition matrix for the killed process $(X_j)$. Given that
$Q^m=I-(A^m)^{-1}\cdot\Delta^m$, where $A^m$ is the diagonal
matrix extracted from $\Delta^m$, $V^m$ is related to the Green function by the
following formula:
\begin{equation}
\label{eq:potential_vs_green}
V^m(x,y) = \frac{1}{A^m_{x,x}}(\Delta^m)^{-1}_{x,y} =
\frac{G^m(x,y)}{\rho^m(x)}.
\end{equation}
Another quantity related to the potential is the \emph{transfer impedance
matrix} $\Hs$, whose rows and columns are indexed by oriented edges of the graph.
If $e=(x,y)$ and $e'=(x',y')$ are two directed edges of $\Gs^\rs$,
the coefficient $\Hs(e,e')$ is the expected number of times that this
random walk $(Y_j)$, started at $x$ and stopped the first time it hits $x$,
crosses the edge $(x',y')$ minus the expected number of times that
it crosses the edge $(y',x')$:
\begin{equation*}
\Hs(e,e')=
[V_{\rs}(x,x')-V_{\rs}(y,x')] P_{x',y'} -
[V_{\rs}(x,y')-V_{\rs}(y,y')] P_{y',x'}.
\end{equation*}
The quantity $\Hs(e,e')/\rho(e')$ is symmetric in $e$ and $e'$, and is changed to
its opposite if the orientation of one edge is reversed.
When $e$ and $e'$ are in fact edges of $\Gs$,
by~\eqref{eq:massive_potential} and the definition of the
transition probabilities for the processes $(Y_j)$ and $(X_j)$,
$V_{\rs}(x,x')-V_{\rs}(y,x')= V^{m}(x,x')-V^{m}(y,x')$ and
$P_{x',y'}=Q_{x',y'}=\rho(x'y')/\rho^m(x)$
(and similarly when exchanging the roles of $x'$ and $y'$). Therefore,
\begin{align}
\label{eq:tranfer_impedance}
\Hs(e,e')&=
[V^m(x,x')-V^m(y,x')] Q_{x',y'} -
[V^m(x,y')-V^m(y,y')] Q_{y',x'} \nonumber \\
&=\rho(x'y')[G^m(x,x')-G^m(y,x')-G^m(x,y')-G^m(y,y')].
\end{align}
If one of the vertices of $e$ or $e'$ is $\rs$, then the same formula holds if
we replace $G^m$ by $\widetilde{G}^{m}$, \emph{i.e.}, if we put to $0$ all the terms
involving the root $\rs$.
\subsection{Spanning forests on $\Gs$ and spanning trees on $\Gs^{\rs}$}\label{sec:app_derivation_classical}
Recall the definition of rooted spanning forests on $\Gs$ and spanning trees of
$\Gs^{\rs}$ from Section~\ref{subsec:rsf_ust}.
Kirchhoff's matrix-tree theorem~\cite{Kirchhoff} states that
spanning trees of $\Gsr$ are counted by the
determinant of $\Delta_{\rs}^{(\rs)}$, obtained from
$\Delta_{\rs}$ by deleting the row and column corresponding to
$\rs$:
\begin{thm}[\cite{Kirchhoff}]\label{thm:matrix_tree}
The spanning forest partition function of the graph $\Gs$ is equal to:
\begin{equation*}
\Zforest(\Gs,\rho,m)=\det \Delta_{\rs}^{(\rs)}.
\end{equation*}
\end{thm}
Using the fact stated in Section~\ref{subsec:massive_harmo} that
$\Delta_r^{(r)}=\Delta^m$, we exactly obtain
Theorem~\ref{thm:matrix_forest}.
The explicit expression for the Boltzmann measure of spanning trees is due to
Burton and Pemantle~\cite{BurtonPemantle}.
Fix an arbitrary orientation of the edges of $\Gsr$.
\begin{thm}[\cite{BurtonPemantle}]
For any distinct edges $e_1,\dotsc,e_k$ of $\Gsr$, the probability that these
edges belong to a spanning tree of $\Gsr$ is:
$$
\PPtree(e_1,\dotsc,e_k)=\det (\Hs(e_i,e_j))_{1\leq i,j\leq k}.
$$
\end{thm}
Using the correspondence between edges (connected to $\rs$, or not) in the spanning tree
of $\Gs^\rs$ and edges and roots for the corresponding rooted spanning forest of
$\Gs$, together with the expression of the transfer impedance matrix $\Hs$ in
terms of the massive Green function on $\Gs$ from
Equation~\eqref{eq:tranfer_impedance}, one exactly gets the statement of
Theorem~\ref{thm:transimp_forest}.
Due to the bijection between spanning trees on $\Gs^\rs$
and rooted spanning forests on $\Gs$, the latter can be generated by Wilson's
algorithm~\cite{Wilson} from the killed random walk $(X_j)$. Indeed, if we take $\rs$ as starting point of the spanning tree,
and construct its branches by loop erasing the random walk $(Y_j)$, the obtained
trajectories are exactly loop erasures of $(X_j)$.
\subsection{Killed random walk on infinite graphs and convergence of the Green functions along exhaustions}
\label{sec:rw}
In this section we define the killed random walk on an infinite graph $\Gs$, as well as its
associated potential and Green function. We then prove (Lemma
\ref{lem:convergence_Green}) that the Green functions associated to an exhaustion
$(\Gs_n)_{n\geq 1}$ of $\Gs$ converge pointwise to the Green function of $\Gs$.
Lemma~\ref{lem:convergence_Green} is an important preliminary result to Theorem
\ref{thm:infinite_vol_meas}.
In the case where $\Gs$ is infinite, it is not possible to consider the network
random walk $(Y_j)$ on $\Gs^\rs$, the graph obtained from $\Gs$ by adding the root $\rs$
connected to the other vertices, because the degree of $\rs$ is infinite and the
conductances associated to edges connected to $\rs$ are bounded from below by a
positive quantity, and are thus not summable. However, it is
possible to directly
define the walk $(X_j)$, killed when it reaches
$\rs$. Its transition
probabilities are:
\begin{equation}
\label{eq:P_Green}
Q^m_{x,y}=\PP(X_{j+1}=y|X_j=x)=
\left\{\begin{array}{cl}
\displaystyle\frac{\rho(xy)}{\sum_{z\sim x} \rho(xz) + m^2(x)} &\text{if $y$
and $x$ are neighbors,} \\
0 & \text{otherwise,}
\end{array}\right.
\end{equation}
and the probability of being absorbed at $x$ is
$\overline{Q^m_x}=\PP(X_{j+1}=\rs|X_j=x)=1-\sum_{xy\in\Es}Q^m_{x,y}$.
Under the condition that the conductances and masses are uniformly bounded away
from 0 and
infinity (which is the case on isoradial graphs, as soon as $k>0$ and the angles
of the rhombi are bounded away from 0 and $\frac{\pi}{2}$), the probability of
being absorbed at any given site
is bounded from below by some uniform positive quantity. The process $(X_j)$ is
thus absorbed in finite time, and vertices of $\Gs$ are transient. We will
assume that this condition is fulfilled.
There is the same link~\eqref{eq:link_matrix_P_Laplacian} as in Section~\ref{subsec:rw_app}
between the substochastic matrix $Q^m=(Q^m_{x,y})$ and the Laplacian $\Delta^m$.
The {potential} $V^{m}$ of the discrete random walk $(X_j)$ is a
function on $\Gs\times\Gs$ defined at $(x,y)$ as the expected time spent at
vertex $y$ by the discrete random walk $(X_j)$ started at $x$ before being
absorbed (below, $\tau_\rs$ is defined as the first hitting time of $\rs$, as in Section \ref{subsec:rw_app}):
\begin{equation}
\label{eq:def_pot_krw}
V^{m}(x,y)=\mathbb{E}_x\Biggl[\sum_{j=0}^{ \tau_\rs-1}\mathbb{I}_{\{y\}}(X_{j})\Biggr].
\end{equation}
In Section
\ref{subsec:cont} we give the standard interpretation of the Green function in terms of continuous time
random processes.
We now come to the convergence of the Green functions along an exhaustion of the graph.
Let $(\Gs_n)_{n\geq 1}$ be an exhaustion of the infinite graph $\Gs$. Let $(Y^n_j)$ be the
network random walk of $\Gs_n$ and $(X_j)$ be the killed random walk of $\Gs$. We introduce
$\tau_\rs^n=\inf\{j>0:Y_j^n=\rs\}$ and $(X^n_j)=(Y^n_{j\wedge (\tau_\rs-1)})$,
the random walk on $\Gs_n$, killed
at the vertex $\rs$. It is absorbed in finite time by $\rs$. Finally,
$\tau_{\partial \Gs_n}=\inf\{j>0:X_j^n\notin \Gs_n\}=\inf\{j>0:X_j\notin \Gs_n\}$
(if the starting point belongs to $\Gs_n$) is the first exit time from the domain $\Gs_n$.
\begin{lem}
\label{lem:convergence_Green}
For any $x,y\in\Vs$, one has $\lim_{n\to\infty} G_n^m(x,y)=G^m(x,y)$.
\end{lem}
\begin{proof}
To use an interpretation with random walks,
we prove Lemma \ref{lem:convergence_Green} for the potential instead of the Green function; this is equivalent
by~\eqref{eq:potential_vs_green}.
The potential function for the killed walk $(X^n_{j})$ is
\begin{equation*}
V_n^m(x,y)=\mathbb{E}_x\Biggl[\sum_{j=0}^{\infty}\mathbb{I}_{\{y\}}(X^n_{j})\Biggr]
=\mathbb{E}_x\Biggl[\sum_{j=0}^{ \tau_\rs^n-1}\mathbb{I}_{\{y\}}(Y^n_{j})\Biggr].
\end{equation*}
The potential $V^m(x,y)$ for $(X_{j})$ is the same as
above without the subscript $n$, see \eqref{eq:def_pot_krw}.
One has
\begin{equation*}
V_n^m(x,y)=\mathbb{E}_x\Biggl[\sum_{j=0}^{ \tau_\rs^n-1}\mathbb{I}_{\{y\}}(X^n_{j});\tau_\rs^n<\tau_{\partial \Gs_n}\Biggr]
+\mathbb{E}_x\Biggl[\sum_{j=0}^{ \tau_\rs^n-1}\mathbb{I}_{\{y\}}(X^n_{j});\tau_\rs^n>\tau_{\partial \Gs_n}\Biggr].
\end{equation*}
In the first term we replace $X^n_{j}$ by $X_{j}$ (as $x\in\Gs_n$), $\tau_\rs^n$ by
$\tau_\rs$, and we use the monotone convergence theorem (as $n\to \infty$,
$\tau_{\partial \Gs_n}\to\infty$ monotonously). The first term goes to
$V^m(x,y)$. We now prove that the second term goes to $0$ as $n\to\infty$. It is
less than $\mathbb{E}_x[\tau_\rs^n;\tau_\rs^n>\tau_{\partial \Gs_n}]$.
Conductances and masses are bounded away from 0 and
$\infty$, so $\tau_\rs^n$ is integrable and dominated by a geometric random
variable not depending on $n$.
We conclude since $\tau_{\partial \Gs_n}\to\infty$.
\end{proof}
\subsection{Laplacian operators and continuous time random processes}
\label{subsec:cont}
In this section we briefly recall the probabilistic interpretation of the Laplacian $\Delta^m$ on
the infinite graph $\Gs$
(introduced in \eqref{equ:operator_general} of Section \ref{sec:defLap}). A similar interpretation
holds for Laplacian operators on other graphs (like on the finite graphs of Section \ref{subsec:rw_app}).
The Laplacian $\Delta^m$
is the generator of a continuous time Markov process $(X_t)$ on $\Gs$, augmented with an absorbing
state (the \emph{root} $\rs$): when at $x$ at time $t$, the process waits an exponential time (with
parameter equal to the diagonal coefficient $d_x$), and then jumps to a neighbor of $x$
with probability \eqref{eq:P_Green}. For the same reasons as for $(X_j)$ and under the same hypotheses,
the random process $(X_t)$ will be absorbed by the vertex $\rs$ in finite time.
The matrix $Q^m$ in \eqref{eq:P_Green} is a substochastic matrix, corresponding
to the discrete time counterpart $(X_j)$ of $(X_t)$, just tracking the jumps.
The Green function
$G^m(x,y)$ represents the total time spent at $y$ by the process $(X_t)$ started at $x$ at time $t=0$ before being absorbed. | {"config": "arxiv", "file": "1504.00792/section_statmech_finite.inv_v2.tex"} |
TITLE: Are X and Y independent random variables?
QUESTION [0 upvotes]: $\bullet$ Let $Z$ be uniformly distributed on $[-1,1]$.
$\bullet$ $X$ is a random variable such that $X=1$ when $Z>0$ and $X=-1$ otherwise.
$\bullet$ $Y$ is a random variable such that $Y=ZX$
Are $X$ and $Y$ independent??
From reading the question my first conclusion was that they are not independent but I am having difficulty finding a counterexample such that $$\mathbb{P}(A \cap B) \neq \mathbb{P}(A)\mathbb{P}(B)$$
Any help would be appreciated
REPLY [0 votes]: Consider two borelian $>0$ functions $f,g$.
$$
\begin{align}
E(f(X)g(Y)) &=
\frac 12\left(\int _{-1}^0 f(-1) g(-x) dx + \int_0^1 f(1)g(x)dx\right)
\\&=
\frac 12\left(\int _{0}^1 f(-1) g(x) dx + \int_0^1 f(1)g(x)dx\right)
\\&=
\frac 12\left(f(-1) + f(1)\right)
\int _{0}^1 g(x)dx
\\&=
\left[\frac 12\left( \int_{-1}^0 f(-1) dx + \int_0^1 f(1) dx \right)\right]
\left[\frac 12\left( \int _{-1}^0 g(-x) dx + \int_0^1 g(x) dx \right)\right]
\\&= E(f(X))\times E(g(Y))
\end{align}
$$
hence $X,Y $ are independant. | {"set_name": "stack_exchange", "score": 0, "question_id": 971702} |
TITLE: Factorization, simple expression!
QUESTION [0 upvotes]: I must factor out this expression:
(I don't speak English, I don't know the right mathematical terms you use in English, sorry if I make a mistake).
$a^{3}(\frac{a^{3}-2b^{3}}{a^{3}+b^{3}})^{3}+b^{3}(\frac{2a^{3}-b^{3}}{a^{3}+b^{3}})^{3}$
The answer is: $a^{3}-b^{3}$
This is what I found:
$\frac{[a(a^{3}-2b^{3})]^{3}}{(a^{3}+b^{3})^{3}}+\frac{[b(2a^{3}-b^{3})]^{3}}{(a^{3}+b^{3})^{3}}$
$\frac{1}{(a^{3}+b^{3})^{3}}([a(a^{3}-2b^{3})]^{3}+[b(2a^{3}-b^{3})]^{3})$
Now I don't know what to do, principally with this two terms: $(a^{3}-2b^{3})$ and $(2a^{3}-b^{3})$
I hope you help me, thanks :)
REPLY [1 votes]: Let $x=a(a^3-2b^3)$ and $y=b(2a^3-b^3)$.
Use the identity $x^3+y^3=(x+y)(x^2-xy+y^2)$ and you will get (after some manipulations):
$x+y = (a-b)(a+b)^3$
$x^2-xy+y^2 = (a^2-ab+b^2)^3(a^2+ab+b^2)$.
Now put everything together and you have $a^3-b^3$. | {"set_name": "stack_exchange", "score": 0, "question_id": 2293848} |
TITLE: To keep ratios unchanged, How much white colour do we have to add?
QUESTION [0 upvotes]: The ratio of the colours given below.
$\dfrac{R}{B} = \dfrac {3}{4} \space , \dfrac{R}{W} = \dfrac{12}{5}$
Where $B$ is blue colour, $R$ is red colour and $W$ is white colour. $48$ gr of the blue colour is added into the mixture that is made by using the ratio of the colours. To keep the ratios unchanged, How much white colour do we have to add?
Let's recall $R = 12k$, $B = 16k$ and $W = 15k$ then we have
$$12k + 16k + 15k = 43k $$
However, there will be no solution from here. Could you take a look at it?
Regards
REPLY [0 votes]: ==== new answer: easier ======
Maybe easier:
If you are adding $48$ g of blue, you must add $r$ g of red in proportion to $\frac 34$. So we need $\frac r{48} = \frac 34$. So what is $r$?
And if we are adding $r$ g of red, you must add $w$ g of white in proportion to $\frac {12}{5}$. So we need $\frac rw = \frac {12}{5}$. So what is $w$?
It doesn't actually matter what we started with. To keep proportions, we must add in proportion.
===== old answer: slightly harder, more detail but more thorough =====
$12k + 16k + 15k = 43k$
That is not correct. It should be $12k + 16k + 5k = 33k$.
$\frac RB = \frac 34$ so if you have $3n$ grams of red for some unknown quantity of $n$, you must have $4n$ grams of blue. So that the proportions are $\frac 34$. That is $\frac {3n}{4n} = \frac 34$.
$\frac RW = \frac {12}{5}$ so if you have $12k$ grams of red for some unknown quantity of $k$, you must have $5k$ grams of blue. $\frac {12k}{5k} = \frac {12}{5}$.
Now you have $3n$ grams of red and that is the same as $12k$ grams of red. So $n = 4k$. So you have $4n = 4*4k = 16k$ of blue.
So you have $12k$ of red; $16k$ of blue, and $5k$ of white.
So you have $12k + 16k + 5k = 33k $ grams total. But that's completely irrelevant.
Now you add $48$ grams of blue so you know have $16k + 48$ grams of blue.
But $\frac {12k}{16k + 48} \ne \frac 34$. We must add some red to keep the proportions correct.
So we must add $r$ grams of red to get:
$$\frac {12k + r}{16k + 48} = \frac 34$$
But now $\frac {12k + r}{5k} \ne \frac {12}{5}$! We must add some white to keep the proportions.
So we must add $w$ grams of white to get:
$$\frac {12k + r}{5k + w} = \frac {12}{5}$$
Now, we do have the equation $(12k + r) + (16k+48)+(5k + w) = 33k + r + 48 + w$ but... that is completely irrelevant.
So if $\frac {12k + r}{16k + 48} = \frac 34$ then
$4(12k + r) = 3(16k + 48)$.
and if $\frac {12k + r}{5k + w} = \frac {12}{5}$ then
$5(12k + r) = 12(5k + w)$.
Solve for $w$.
Note: You will never know what $k$ is. That is not the point. | {"set_name": "stack_exchange", "score": 0, "question_id": 2794472} |
\begin{document}
\title{On the roots of truncated hypergeometric series over prime fields}
\author{ Amit Ghosh and Kenneth Ward}
\date{}
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\begin{abstract}
We consider canonical polynomial truncations of hypergeometric functions over the finite field $\mathbb{F}_{p}$, for primes $p \to \infty$. For these truncations, we obtain bounds for the number of roots of various modulo $p$ congruences, which represent rational point counts where methods from classical algebraic geometry fail. Via a correspondence to families of elliptic curves, we obtain sharp bounds in some cases. We show that these truncations are also associated with certain surfaces, which we prove are K3. By a modification of methods from transcendence theory, we obtain a power saving for a large natural class of the parameter values within an algebraic closure of $\mathbb{F}_p$ for the Kummer hypergeometrics. Some computations are included to illustrate and supplement our results.
\end{abstract}
\maketitle
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\fancyhead[CO]{\small On the roots of truncated hypergeometric series over prime fields}
\fancyhead[CE]{\small Amit Ghosh and Kenneth Ward}
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\section{Introduction}
In \cite{GhWa}, we considered the following problem in modular arithmetic: Given a prime number $p>3$ and a power series $F(x) = \sum_{n=0}^{\infty} a_{n}x^{n}$ with rational coefficients, suppose that there is an integer $N$ as large as possible, but not exceeding $p-1$, such that the polynomial $F_{N}(x)=\sum_{n=0}^{N} a_{n}x^{n}$ is well-defined over the prime field $\mathbb{F}_{p}$. If $N$ grows with $p$, then for any fixed $m$, what is the number of solutions to the congruence $F(x) \equiv m \pmod{p}$ ?
The reason for our consideration of this problem originally derived from the following failure of the Weil bound: for a polynomial $f$ of degree $d$, the number of roots to the congruence $f \equiv m \pmod{p}$ is bounded by $d\sqrt{p}$, which is trivial if $d$ exceeds $\sqrt{p}$. If $F(x)$ satisfies certain linear differential equations of order $k$ with polynomial coefficients, then for $k=1$, particularly for the truncations of the power series for the exponential and logarithm functions $$E(x) = 1 + x + \frac{x^2}{2!} + \cdots + \frac{x^{p-1}}{(p-1)!} \quad \text{ and }\quad L(x) = x + \frac{x^2}{2} + \cdots + \frac{x^{p-1}}{p-1},$$ Mit'kin ($E(x)$ and $L(x)$) and independently, Heath-Brown ($L(x)$) showed that the number of roots in $\mathbb{F}_p$ is bounded by $O(p^{2/3})$ as $p \rightarrow \infty$ . For each $k\geq 2$, examples of $F$ are given in \cite{GhWa}, for which the congruences have $o(p)$ solutions, for any given $m$, and in some cases a power saving is attained (see \cite{GhWa} for a more complete discussion and references).
In this paper, we explore this question by considering the hypergeometric functions and their truncations. In much of what follows, we shall consider hypergeometric functions with rational parameters, with the exception of Section 8 and Theorem \ref{KummerE} below, where it is natural to take parameter values in $\overline{\mathbb{F}}_p$ for the modular analogue to classical transcendence theory. The bulk of the paper considers the functions $_{2}^{}F^{}_{1}$ and $_{3}^{}F^{}_{2}$ defined as follows: For $d\geq 2$,
\begin{equation}\label{definitionofhypergeometric}
_{d}^{}F_{d-1}^{}\left(\alpha_{1}, \ldots, \alpha_{d};\beta_{1}, \ldots, \beta_{d-1};x\right) = \sum_{n=0}^{\infty}\frac{(\alpha_{1})_{n} \ldots (\alpha_{d})_{n}}{n!(\beta_{1})_{n} \ldots (\beta_{d-1})_{n}}x^{n},
\end{equation}
where $(a)_{n}= a(a +1)\cdots(a +n-1)$ is the Pochammer product. The Kummer hypergeometric function $_{1}^{}F^{}_{1}$ is defined similarly. We will evaluate these functions with parameter values arising from geometry: For the functions $_{2}^{}F^{}_{1}$ and $_{3}^{}F^{}_{2}$, these parameters give rise to non-cocompact arithmetic groups as the associated monodromy groups, and for $_{1}^{}F^{}_{1}$, the parameters will be precisely those analogous to {\it non-integers} over $\mathbb{F}_p$.
In the definition \eqref{definitionofhypergeometric} of the hypergeometric function, if the $\alpha_{i}$'s and $\beta_{j}$'s are rational numbers between zero and one, then there is a natural truncation of the given hypergeometric function $F$ for any prime integer $p > 2$, which we will denote by $F^{(p)}$ (see Section 3 for details). These truncations are polynomials with degree $N$ where $N \leq p-1$ and $\frac{N}{p} \sim r$ as $p$ grows, with $r$ a positive rational number that depends explicitly on the parameters of the hypergeometric function, via the residue classes of $p$ modulo these parameters. A classical example is the Hasse polynomial $H_p(x)$, which is obtained by truncating $_{2}^{}F^{}_{1}\left(\half,\half;1;x\right)$ at $N=\frac{p-1}{2}$ for $p>2$. It is known, for example, that the function $H_p(x)$ divides $x^{(p^2-1)/8}-1$, i.e., all roots of $H$ are $8$th powers of elements of $\mathbb{F}_{p^2}$, and that the number of roots of $H_p(x)$ in $\mathbb{F}_p$ is equal to precisely $$N_p(H_p) =\begin{cases}0, & \text{if }p \equiv 1 \mod 4 \\3h(-p), & \text{if }p \equiv 3 \mod 4,\end{cases}$$ where $h(-p)$ is the class number of $\mathbb{Q}(\sqrt{-p})$ \cite{BrillhartMorton}. Since this example forms the basis of the geometric arguments which follow, we give a detailed explanation here.
Consider the Legendre family of elliptic curves $y^{2}=x(x+1)(x+\lambda)$ over $\mathbb{F}_{p}$ with $\lambda \neq 0, -1 \pmod{p}$. For each curve in this family determined by a $\lambda$, we let $n_{p}(\lambda)$ denote the number of $\mathbb{F}_{p}$-rational points, so that $a_{p}(\lambda):= n_{p}(\lambda)-p-1$ satisfies the Hasse estimate $|a_{p}(\lambda)|\leq 2\sqrt{p}$. It was shown by Igusa (as well as Manin and Dwork) that $a_p(\lambda) \equiv (-1)^{(p-1)/2}\ _{2}^{}F^{(p)}_{1}\left(\half,\half;1,\lambda\right) \pmod{p}$. Thus, for a fixed integer $m$, the number of solutions to the polynomial congruence $_{2}^{}F^{(p)}_{1}\left(\half,\half;1,\lambda\right) \equiv m \pmod{p}$ is the same as the number of $\mathbb{F}_p$-isomorphism classes of Legendre elliptic curves satisfying the condition $a_p(\lambda)\equiv (-1)^{(p-1)/2} m \pmod{p}$. The case $m=0$ counts the number of supersingular such curves. The Hasse estimates imply that there exist no such $\lambda$ if $|m|> 2\sqrt{p}$. On the other hand, if $|m| < 2\sqrt{p}$, Deuring \cite{Deuring} showed that the number of such isomorphism classes for the family $y^{2} = x^{3} + ax +b$ with parameters $a$ and $b$ is essentially equal to the Hurwitz-Kronecker class number $H(m^2 -4p)$ of an imaginary quadratic field (the classes counted up to a weight of $1/|\text{Aut}(E)|$). From this we deduce an upper bound for the number of such isomorphism classes restricted to our 1-parameter family (see Section 7) so that the number of solutions to the congruence $_{2}^{}F^{(p)}_{1}\left(\half,\half;1,\lambda\right) \equiv m \pmod{p}$ is $O\left(H(m^2 -4p)\right)$. Then using Proposition 1.9 of Lenstra \cite{Lenstra} we conclude that the number of roots in $\mathbb{F}_p$ of the congruence is at most $O \left( \sqrt{p}\log p (\log \log p)^{2} \right)$ for $|m|< 2\sqrt{p}$ and is zero otherwise. Since the $m$'s are concentrated in a narrow range, it is clear that there are congruence classes $m$ for which the root count exceeds $\varepsilon \sqrt{p}$ for $p$ large enough, so that the upper bound is quite sharp.
The goal of this paper is a study of examples of truncated functions, arising naturally from geometry, for which one can obtain (sometimes sharp) bounds for the number of roots of the associated congruence. In Section 5, we consider three other families of elliptic curves apart from the Legendre family, which give rise to truncations of $_{2}^{}F^{}_{1}\left(\frac{a}{b},1-\frac{a}{b};1,\lambda\right)$ with $b = 3, 4 $ and $6$. Various classical transformation formulae are used in Section 4 to find analogous formulae over $\mathbb{F}_{p}$, which give relations between truncations of hypergeometric functions. The classical Clausen formula yields truncations of certain $_{3}F_{2}$ hypergeometrics, which are then associated to four families of K3 surfaces. We apply the same argument using Deuring's theorem to the three elliptic families to deduce, using the transformations formulae, bounds for the number of roots for congruences of the type $F(x) \equiv m \pmod{p}$. We note that due to the nature of the classical transformations, it is sometimes necessary to restrict the variables $x$ to certain subsequences, and thus in some cases one may only conclude a bound for $m=0$. We obtain non-trivial results for truncations of thirteen $_{2}F_{1}$ and four $_{3}F_{2}$ hypergeometric functions. The following Theorem and Corollary give these.
\begin{theorem}\label{onehalf} Let $\alpha \in \left\{ \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{6}\right\} $ and put $\alpha=\frac{1}{b}$. Let $Q$ denote any of the polynomials $_{2}^{}F^{(p)}_{1}\left( \alpha,1-\alpha;1,x\right)$\ and let $X_{p}$ denote the quantity $\sqrt{p}\log p (\log \log p)^{2}$. Then for any $m$ satisfying $|m|<2\sqrt{p}$, the congruence $Q \equiv m \pmod{p}$ has $O\left( X_{{p}}\right ) $ solutions for all $p$ sufficiently large. Otherwise, the congruence has no solutions. Moreover, there are $m$'s such that the corresponding congruences have more than $(\frac{1}{4}-\varepsilon) \sqrt{p}$ solutions for sufficiently small $\varepsilon >0$.
\end{theorem}
\begin{corollary}\label{onehalfone} For $\alpha$ and $p$ as above we have the following: :
\begin{enumerate}[(I)]
\item Let $Q$ denote the polynomials $ _{3}^{}F^{(p)}_{2}\left( \alpha,1-\alpha,\half;1,1,4x(1-x)\right)$. Then the congruence $Q \equiv m \pmod{p}$ has $O\left( X_{{p}}\right ) $ solutions for all $m$ satisfying $m \equiv a^2 \pmod{p}$ with $|a|<2\sqrt{p}$. Otherwise, the congruence has no solutions.
\item Let $Q$ be any of the polynomials $_{2}^{}F^{(p)}_{1}\left( \alpha,\alpha;1,x\right)$ and $_{2}^{}F^{(p)}_{1}\left( 1-\alpha,1-\alpha;1,x\right)$ . Then, the congruence $Q \equiv 0 \pmod{p}$ has $O\left( X_{{p}}\right ) $ solutions. If $m\neq 0$, the same conclusion holds for the congruence $Q \equiv m \pmod{p}$ for those large $p$ satisfying $p \equiv 1 \pmod{b}$.
\item Let $Q=\ _{2}^{}F^{(p)}_{1}\left( \frac{\alpha}{2},\half-\frac{\alpha}{2};1,4x(1-x)\right) $. If $p \equiv 1$ or $b-1 \pmod{2b}$, then the same conclusion holds for the congruence $Q \equiv m \pmod{p}$ as in the Theorem.
\item Let $Q$ denote the polynomials $\ _{2}^{}F^{(p)}_{1}\left( \alpha,\half;1,1-x^2\right) $. If $p \equiv 1 \pmod{2b}$, then the congruence $Q \equiv m \pmod{p}$ has $O\left(X_{{p}}\right ) $ solutions. For general $p$, the congruence $Q \equiv 0 \ (\text{mod} p)$ has at most the same number of solutions.
\end{enumerate}
For all the cases there are $m$'s such that the corresponding congruences have more than $\varepsilon \sqrt{p}$ solutions for sufficiently small $\varepsilon >0$.
\end{corollary}
\begin{remark}
In \cite{Ono}, a pairing of an elliptic family and a $K3$ family is considered to determine when elements of the $K3$ family are modular. A transformation formula was determined independently of the classical formulae. In the Appendix, we provide a separate proof of such a transformation formula over $\mathbb{F}_{p}$. The analysis in \cite{Ono} uses character sums, while our exposition for congruences is simpler, using binomial coefficients. A consequence is that the classical Clausen formula follows from that of the prime field case (see Remark \ref{rem1}). We expect that such equivalences are true for all of the formulae considered in Lemma \ref{trans}.
\end{remark}
We give some sample plots for the distribution of $\mathcal{N}_{p}(m)$, the number of roots of $F \equiv_{p} m$ against $m$, with $m$ in appropriate ranges. Our computations have $p$ relatively small due to the time it takes to complete them (as the degrees of the polynomials are quite large), primarily on a PC, and so we do not make any extravagant suggestions.
We see a rather singular behavior shown by Theorem 1.1 for the first four hypergeometric functions where the value distribution modulo $p$ mimics the value distribution of the Hurwitz-Kronecker class numbers, as illustrated in Figs. 1 and 2. In Fig. 1 we plot $\mathcal{N}_{p}(m)$ against $m$ together with the histogram for the distribution of $\mathcal{N}_{p}(m)$, whilst in Fig. 2 we plot $H(m^2 -4p)$ against $m$ and its histogram.
\begin{figure}[ht!]
\centering
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{A312619F.pdf}
\caption{ \smaller $ F \equiv_{p} m$, \ $0\leq |m|\leq 2\sqrt{p}$}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{A312619Fhisto.pdf}
\caption{ {\smaller Histograms for value distribution.}}
\label{fig:subA2}
\end{subfigure}
\caption{\smaller $\ _{2}^{}F^{(p)}_{1}\left(\frac{1}{6},\frac{5}{6},1;x\right)$;\quad $p=312619$}
\end{figure}
\begin{figure}[ht!]
\centering
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{ClassNbr312619.pdf}
\caption{ \smaller $ H(m^{2 } -4p)$, \ $0\leq |m|\leq 2\sqrt{p}$}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{ClassNbrHisto312619.pdf}
\caption{ {\smaller Histograms for value distribution.}}
\label{fig:subA2}
\end{subfigure}
\caption{\smaller Class number:\quad $p=312619$}
\end{figure}
\begin{figure}[ht!]
\centering
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{B312619F.pdf}
\caption{ {\smaller $F \equiv_{p} m$,\ \ $ 0\leq |m| \leq \frac{p-1}{2}$}}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{B312619Fzoom.pdf}
\caption{ {\smaller $- 5\sqrt{p} \leq m \leq 5\sqrt{p}$}}
\end{subfigure}
\caption{{\smaller $\ _{2}^{}F^{(p)}_{1}\left(\frac{1}{2},\frac{1}{3},1;1-x^2\right)$;\quad $p=312619$}}
\end{figure}
The monodromy groups associated with the $ _{2}F_{1}$ hypergeometric functions appearing in Theorem 1.1 and the Corollary are equivalent to the nine non-cocompact arithmetic triangle groups (see \cite{Takeuchi}). In Fig. 3, we see an example where the distribution is more spread out but with a concentration again in a short range.
\begin{figure}[ht!]
\centering
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{C350381F.pdf}
\caption{{\smaller $ F \equiv_{p} m\ ,\ 0\leq m\leq p-1$ }}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{C350381Fpt2.pdf}
\caption{ {\smaller $F \equiv_{p} m^2\ ,\ 0\leq |m|\leq 2\sqrt{p}$ }}
\end{subfigure}
\vspace{10pt}
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{C350381Fhisto.pdf}
\caption{ {\smaller Histogram: $F \equiv_{p} m^2\ ,\ 0\leq |m|\leq 2\sqrt{p}$. }}
\label{fig:subB2}
\end{subfigure}
\caption{{\smaller $\ _{3}^{}F^{(p)}_{2}\left(\frac{1}{2},\frac{1}{2},\frac{1}{2};1,1;4x(1-x)\right)$;\quad $p=350381$.}}
\end{figure}
By \cite{FMS}, there are six (orthogonal, quasiunipotent) arithmetic hypergeometric groups, of which four fix isotropic forms while two fix anisotropic forms (see the Table 2 and the Appendix of \cite{FMS}). The four $_{3}F_{2}$'s in Corollary 1.2 have monodromy groups equivalent to these four hypergeometric groups, and each have an associated fundamental domain that is non-compact. The value distribution of these truncated functions is illustrated in Fig. 4; computationally, we find that the value distribution of the truncated hypergeometric functions associated with the two anisotropic forms have a different behavior, akin to the pictures in Figs. 5 and 6 (the associated fundamental domains here are compact).
\begin{figure}[ht!]
\centering
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{D104773F.pdf}
\caption{ {\smaller $F \equiv_{p} m$,\ \ $|m|\leq \frac{p-1}{2}$}}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{D104773Fhisto.pdf}
\caption{ {\smaller Histogram: Poisson mean 1}}
\end{subfigure}
\caption{{\smaller Compact monodromy: $\ _{3}^{}F^{(p)}_{2}\left(\frac{1}{3},\frac{2}{3},\frac{1}{2};\frac{1}{6},\frac{5}{6};x\right)$,\quad $p=104773$}}
\end{figure}
It appears that Fig. 5 represents the generic situation in many computations we have done. One might venture to guess, by looking at the graph in Fig. 5 that in these cases the maximum value for $\mathcal{N}_{p}(m)$ is bounded (independently of $p$), but we suspect that it is more likely growing but perhaps at a rate much slower than $p^{\varepsilon}$ (for algebraic hypergeometrics, it might well be bounded). For instance, if one assumes the Poisson-like behavior for $\mathcal{N}_{p}(m)$ of Fig. 5, then one might expect that the maximum value for $\mathcal{N}_{p}(m)$ will have size about $\frac{\log{p}}{\log\log{p}}$.
\begin{figure}[ht!]
\centering
\includegraphics[width=.5\linewidth]{4F3halfhisto162601.pdf}
\caption{{\smaller $\ _{4}^{}F^{(p)}_{3}\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};1,1,1;x\right)$,\quad $p=162601$}}
\end{figure}
We find this Poisson-like behavior for thirteen of the fourteen $ _{4}F_{3}$ hypergeometrics associated with Calabi-Yau 3-folds listed in \cite{ShV}; these fourteen have sympletic monodromy groups and half are thin (\cite{BrTh}) and the other half arithmetic (\cite{ShV}, \cite{Singh}). The histograms for the associated $\mathcal{N}_{p}(m)$ all appear to be Poisson with mean 1 (as illustrated in Fig. 5), with the lone exception of $\ _{4}^{}F^{(p)}_{3}\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};1,1,1;x\right)$ whose histogram (Fig. 6) appears to be exponential in nature (so that we expect the maximal value for $\mathcal{N}_{p}(m)$ to be $\log p$ in this case).
In Figures 7 and 8, for $F = \ _{2}^{}F^{(p)}_{1}\left(\frac{1}{2},\frac{1}{2},1;x\right)$, we let $M_{p}$ denote the maximum value of $\mathcal{N}_{p}(m)$. Then, we let $H_{p}$ denote the maximum of the Hurwitz-Kronecker class numbers $H(m^{2}-4p)$ over the same range of $m$'s. The graphs denote a comparision of the plots of the points $(p,M_{p})$ and $(p,H_{p})$ as the primes $p$ range over an interval. By Lemma 7.1, we know that $M_{p} \leq 6H_{p}$, while the graphs show that perhaps more is true. In Figure 6, we plot the ratio $M_{p}/H_{p}$ with $p$ varying in the same range, but distinguishing the congruence classes modulo 4.
\begin{figure}[ht]
\centering
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{Comparision7933to8821OneTwo.pdf}
\caption{ {\smaller $p \equiv 1 \pmod{4}$}}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{Comparision7933to8821ThreeTwo.pdf}
\caption{ {\smaller $p \equiv 3 \pmod{4}$}}
\end{subfigure}
\caption{{\smaller Maximal root counts (black) for $ \ _{2}^{}F^{(p)}_{1}\left(\frac{1}{2},\frac{1}{2},1;x\right)$ vs maximal class number (gray); primes $7919 \leq p \leq 8821$}}
\label{fig:EE}
\end{figure}
\begin{figure}[ht]
\centering
\begin{minipage}{.5\linewidth}
\includegraphics[width=\linewidth]{Ratio7933to8821Two.pdf}
\end{minipage}
\caption{{\small Ratio of maximal values relative to maximal class number (black: $p \equiv 1 \pmod{4}$).}}
\end{figure}
\vspace{15pt}
We now turn to hypergeometric functions for which the geometric method above is not available so that for these we use auxiliary polynomials as in transcendence theory (Stepanov's method). This approach(see \cite{GhWa} for example) has two distinct parts: \begin{enumerate} \item the construction of auxiliary polynomials in many variables (of not too high degrees); and \item after specialisation, the non-vanishing of such polynomials, for which we need a form of algebraic independence of the truncated hypergeometrics and their derivatives over $\mathbb{F}_{p}$. \end{enumerate}
This latter problem is quite difficult, and the bounds obtained are not very sharp. However its advantage is it generality, and in some cases allows us to save a small power of $p$, as we obtained in \cite{GhWa} for the truncation of the Bessel function. In this vein, we show that a power saving is also possible for Kummer hypergeometric function $_{1}F_{1}\left(\alpha;\beta;z\right)=\sum_{n}^{\infty}\frac{(\alpha)_{n}}{(\beta)_{n}}\frac{z^{n}}{n!}$, for which we do not have an analog of the method using elliptic curves.
\begin{theorem}\label{KummerE} Let $p$ be a sufficiently large prime number, let $\alpha \in \mathbb{F}_q$ ($q = p^m$) such that $\alpha \notin \mathbb{F}_p$, and let $\beta$ be a fixed rational number. Then there is a bounded integer $k\geq 1$, determined by $\alpha$, $\beta$ and $p$, and an effectively computable constant $\kappa>0$ such that the number of solutions in $\mathbb{F}_p$ to $_{1}^{}F^{(p)}_{1}\left(\alpha;\beta;x^{k}\right) \equiv 0 \pmod{p}$ is $O(p^{1-\kappa}) $. If instead $\beta \in \mathbb{F}_q$ and $\alpha, \alpha -\beta \notin \mathbb{F}_p$, then we may take $k=1$.
\end{theorem}
\begin{example} Let $\alpha$ be a generator of $\mathbb{F}_q^*$ $(q = p^m, m \geq 2)$ and $\beta = \frac{1}{3}$. Then $_{1}^{}F^{(p)}_{1}\left(\alpha;\beta;x\right)$ has degree $\frac{p-1}{3}$ or $\frac{2p-1}{3}$ (depending on $p \equiv \pm 1 \pmod{6}$) and we take $k=3$.
\end{example}
\begin{example} Let $\alpha,\beta \in \mathbb{F}_q$ such that the sets $\{\alpha + a \;|\; a \in \mathbb{F}_p\}$ and $\{\beta + a \;|\; a \in \mathbb{F}_p\}$ are disjoint. Then $_{1}^{}F^{(p)}_{1}\left(\alpha;\beta;x\right)$ has degree $p-1$ and so we take $k=1$.
\end{example}
The proof differs from that given for the Bessel function in \cite{GhWa}, where an adaptation of methods from transcendence theory was used to count roots of the truncated series. There the non-vanishing of the auxiliary polynomial for the Bessel function relied upon a modular version of the residue theorem, whereas here the argument for the Kummer hypergeometric relies upon a non-vanishing of certain coefficients of large degree {\it less than} $p$. We believe that these methods carry sufficiently general features that similar results are possible for other such hypergeometric functions.
\begin{remark} The proof in Section 8 allows us to take $\kappa=\frac{1}{7}$. Unlike in Theorem \ref{onehalf}, the element $\alpha$ (and $\beta$, if $\beta \in \mathbb{F}_q$) in Theorem \ref{KummerE} depend on $p$. The definition of the Pochhammer product in $\mathbb{F}_q$ is just the same as the usual one: If $a \in \mathbb{F}_q$, then $$(a)_n = a(a+1)\cdots(a+n-1).$$ The notation $\equiv \pmod{p}$ in the statement of Theorem \ref{KummerE} is abusive, as the function $_{1}^{}F^{(p)}_{1}\left(\alpha;\beta;x^{k}\right)$ is {\it a priori} reduced modulo $p$ by taking $\alpha, \beta \in \mathbb{F}_q$. If $\alpha,\beta \in \mathbb{F}_q \backslash \mathbb{F}_p$, then the natural truncation occurs at $p-1$, and otherwise the truncation will have degree of the type $\frac{p-u}{v}$. In either case, $_{1}^{}F^{(p)}_{1}\left(\alpha;\beta;x^{k}\right)$ will satisfy a differential equation with polynomial coefficients. If $v\neq 1$, there will be a coefficient whose degree will grow with $p$, which makes the construction of good auxiliary polynomials impossible. To overcome this, we impose the most natural change of variable to $x^k$, so that the resulting differential equation has coefficients with bounded degree.
\end{remark}
\begin{remark}
As mentioned before, the algebraic independence over $\mathbb{F}_{p}$ is the difficult part of the proof. Our considerations rely on known proofs for the algebraic independence of functions and their derivatives over the complex numbers. This is a difficult problem in transcendence theory. There are some results known for Siegel $E$-functions due to Siegel \cite{Siegel}, Shidlovski \cite{Shidlovskii}, Mahler \cite{Mahler}, Oleinikov \cite{Oleinikov} and others. In characteristic $p > 0$, the derivative of $x^p$ vanishes, and our argument relies upon an analysis of terms of highest degree less than $p$ in the expression of a rational solution to the Riccati differential equation $$y' + \frac{\beta - z}{z} y + y^2 \equiv\ \frac{\alpha}{z} \pmod{p}\ .$$
\end{remark}
\vskip 0.2in
In Section 2, we make some observations about truncations of {\it rational} functions $G$, for which we see that the value distribution of a truncation $G^{(p)}$ is quite different from those of the polynomials we have considered above. There is usually a bounded number of roots in each congruence class, except possibly one class, which may account for almost all roots. The same kind of phenomenon is seen for algebraic hypergeometric functions (that is, all of those with finite monodromy group). This suggests that the value distribution modulo $p$ of polynomials derived from algebraic power series is fundamentally different from those which are not algebraic.
\begin{remark}
In everything that follows, we use the notation $a \equiv_{q} b$ to mean the congruence $a \equiv b$ modulo $q$. We allow $a$ and $b$ to be rational numbers whose denominators are coprime to $q$.
\end{remark}
\vskip 0.2in
\noindent{\small {\bf Acknowledgments.}}\\
AG thanks Enrico Bombieri and Nick Katz for some conversations related to this work, and he thanks Peter Sarnak for some long discussions. He also thanks the Institute for Advanced Study (IAS) for their hospitality. He gratefully acknowledges financial support from the IAS, the College of A\&S and the Department of Mathematics of his home university, and the Simons Foundation for a Collaboration Grant.
\noindent KW thanks the NYU-ECNU Institute of Mathematical Sciences and the American University Mellon Research Fund for their financial support.
\noindent Computations were done using Mathematica$^{\copyright}10.2$ on a Linux PC, and also the Zorro High-Performance Computing System at American University.
\vskip 0.2in
\section{Rational functions}
Suppose that $F(x)=c\frac{A(x)}{B(x)}$ ($A, B \in \mathbb{Z}[x]$) is a nonzero rational function regular at $x=0$, with the Maclaurin series expansion $\sum_{n=0}^{\infty} f_{n}x^{n}$, so that $f_{n}\in\mathbb{Q}$ for all $n$. Then for all primes $p$ large enough, the coefficients $f_{n}$ have denominators not divisible by $p$. Given such a function $F$, we consider only those $p$ sufficiently large that $f_{p-k} \not\equiv_p 0$ for some bounded $k\geq 1$ (here and in what follows, ``bounded" means, ``bounded relative to $p$"). Let $\it{S}_{F,p}$ denote those $x\in \mathbb{F}_{p}$ such that $B(x) \equiv_p 0$ if $k=1$, and if $k\geq 2$, we also include $x=0$. Let $F_{p}(x)=\sum_{n=0}^{p-k} f_{n}x^{n}$, and put $F-F_{p} = x^{p-k+1}G_{p}(x)$. Then if $$Q_{p}(x):=B(x)F_{p}(x) - A(x),$$ we have $Q_{p}(x)= -x^{p-k+1}G_{p}B(x)$, so that $Q_{p}(x)$ has a zero of order at least $p-k+1$ at $x=0$. We write $Q_{p}(x)= x^{p-k+1}Q^{*}_{p}(x)$ for a polynomial $Q^{*}_{p}(x)$ of bounded degree. It follows from this that
\[ x^{k-1}(B(x)F_{p}(x) - A(x)) \equiv xQ^{*}_{p}(x) \mod (x^{p}-x), \]
so that $R_{p}(x):= x^{k-1}(B(x)F_{p}(x) - A(x)) - xQ^{*}_{p}(x)$ vanishes for all $x$ modulo $p$.
Fix an integer $m$, and suppose that $F_{p}(x_{0})\equiv_p m$ for some $x_{0} \in \mathbb{Z} \text{ mod }p$. Define $$R_{p,m}(x):= x^{k-1}(B(x)m - A(x)) - xQ^{*}_{p}(x),$$ which is of bounded degree and has $x_{0}$ as a root modulo $p$. If $R_{p,m}(x)$ is not identically zero in $\mathbb{F}_{p}$, then since it is of bounded degree, it possesses only a bounded number of roots, so that there are also only a bounded number of distinct roots to the congruence $F_{p}(x_{0})\equiv_p m$. On the other hand, if $R_{p,m}(x)$ vanishes identically modulo $p$, we will have $$R_{p}(x)\equiv_p x^{k-1}B(x)(F_{p}(x) - m),$$ so that $F_{p}(x)\equiv_p m$ for all $x \not\in \it{S}_{F,p}$. It is also clear that this latter equivalence can occur for at most one congruence class $m$ modulo $p$.
Next, suppose that $x_{0}$ is a root of multiplicity $l$ of the polynomial $F_{p}(x)-m$ modulo $p$ such that $l$ grows with $p$. From the above, we have by definition that
\[
x^{k-1}B(x)(F_{p}(x)-m) + x^{k-1}(mB(x)-A(x)) =x^{p}Q^{*}_p(x),
\]
so that differentiating $l$ times with respect to $x$ yields
\[
\frac{d^l}{dx^l}\left(x^{k-1}B(x)(F_{p}(x)-m)\right) \equiv_p 0 \mod (F_{p}(x)-m),
\]
in $\mathbb{F}_{p}$, due to the bounded degrees of the other polynomials. Hence $x_0^{k-1}B(x_{0})\equiv_p 0$, so that $x_{0} \in \it{S}_{F,p}$. We then have the following:
\begin{proposition}\label{lem2} Let $F$ be a rational function as above. For a fixed $k\geq 1$, let $p$ be a sufficiently large prime such that $f_{p-k}\not\equiv_p 0$. Then the congruence $$F_{p}(x) \equiv_p m$$ has at most a bounded number of solutions for each $m$, with at most one possible exceptional value $m=m_{0}$. For the exceptional $m_{0}$ (if it exists), all except a bounded number of elements $x \in \mathbb{F}_{p}$ are solutions, upon which the congruence $F_{p}(x) \equiv_p m$ has no solutions, except for a bounded number of values of $m$.
\end{proposition}
If we put $F(x)=\frac{1}{1-x}$, so that for $k=1$ we have $F_{p}(x)=\sum_{n=0}^{p-1} x^{n}$, it then is clear that the Proposition holds with the exceptional $m=1$. There are cases where one can prove the analogue of Proposition \ref{lem2} for truncations of algebraic power series.
\begin{example}
For $d\geq 2$, $ v \geq 1$ and $1\leq u \leq v$, consider the algebraic generalised hypergeometric functions
\[
F(x):=\ _{_{d}}F_{_{d-1}}\left(\frac{u}{vd},\frac{u+v}{vd}, \ldots ,\frac{u+(d-1)v}{vd};\frac{1}{d}, \frac{2}{d}, \ldots ,\frac{d-1}{d}; x\right).
\]
It is easily checked that the $n$th coefficient of the Taylor series about $x=0$ is equal to
\[
a(n)= \frac{1}{(dn)!}\prod_{j=0}^{dn -1}\left(\frac{u}{v}+j\right).
\]
Let $p$ be any prime number such that $p\equiv 1 \ ({\rm mod}\ dv)$. Then for $0\leq n\leq E$ with $E=\frac{p-1}{vd}u$, we have $a(n) \equiv_{p}\ (-1)^{dn}\binom{(p-1)\frac{u}{v}}{dn}$. Truncation of the hypergeometric function at $E$ and denoting by $F^{(p)}(x)$ the resulting polynomial gives
\[
F^{(p)}((-x)^{d}) \equiv_{p} \ \sum_{n=0}^{E}\binom{(p-1)\frac{u}{v}}{dn}x^{dn} = \sum_{\substack{n=0\\d|n}}^{dE} \binom{dE}{n}x^{n} \equiv_{p}\ \frac{1}{d}\sum_{g\in \Sigma(d)}\left(1+gx\right)^{dE},
\]
where $\Sigma(d)=\left\{g : g^{d} \equiv_{p} 1 \right\}$. For any $m$, suppose that there is a value $x_{0}$ such that $F^{(p)}(x_{0}^{d})\equiv_p m$. It is then not difficult to show that there is a polynomial $G$, of degree $v^{d}$ and with coefficients independent of $x_{0}$, such that $G(m)\equiv_{p} 0$. Thus, for all but a bounded number of congruence classes $m$, the congruence $F^{(p)}(x^{d})\equiv_p m$ has no solutions. It also follows that there is a congruence class $m_{0}$ that has a positive proportion of roots.
For example, if $d=2$, let $u, v$ be coprime positive integers such that $0<\frac{u}{v}\leq 1$, and consider the hypergeometric function $_{2}F_{1}(\frac{u}{2v},\frac{u+v}{2v},\frac{1}{2};x)$. It is algebraic with a dihedral monodromy group. Then for primes $p$ such that $p \equiv 1\ ({\rm mod} \ 2v)$ and $E=\frac{p-1}{2v}u$, the truncated function $_{2}F_{1}^{(p)}(\frac{u}{2v},\frac{u+v}{2v},\frac{1}{2};x)$ satisfies
\[
_{2}F_{1}^{(p)}\left(\frac{u}{2v},\frac{u+v}{2v},\frac{1}{2};x^2\right) \equiv_p\ \frac{1}{2}\left[(1-x)^{2E} + (1+x)^{2E}\right]
\]
Choosing, say, $u=2$ and $v=3$, we have $E=\frac{p-1}{3}$ with $p\equiv_{3} 1$, and it follows from the above that $m$ is a root of the congruence $(4m^{3}-1)^{3}-27m^{3}\equiv_p0$. This equation has at most $7$ distinct roots modulo $p$, so that for all except at most $7$ congruence classes $m$ modulo $p$, the equation $_{2}^{}F_{1}^{(p)}\left(\frac{1}{3},\frac{5}{6},\frac{1}{2};x^{2}\right) \equiv_p m$ has no solutions.
\end{example}
\section{Preliminary lemmas and notation}
Given any power series $\sum_{m}^{\infty} a_{m}x^{m}$, we wish to determine a natural truncation modulo prime integers $p$. For hypergeometric functions, there is such a truncation when the parameters are rational numbers. We shall consider only hypergeometric functions of the type $$_{d}F_{d-1}(\alpha_{1}, \ldots, \alpha_{d};\alpha_{i+1}, \ldots, \alpha_{2d-1};x),\;\;\;\;\alpha_{i}=\frac{a_{i}}{b_{i}},\;\;\;\;a_i,b_i \in \mathbb{N},\;\;\;\;(a_{i},b_{i})=1, \;\;\;\;a_i \leq b_i.$$ For any odd prime $p$ sufficiently large so that $p$ does not divide $a_{i}b_{i}$, for all $i$, we observe that $(\alpha_{i})_{m} \equiv_{p} 0$ implies the same for all $n\geq m$. Thus for the numerator and denominator values $\alpha_{i}$, we seek the largest $N\leq p-1$ such that $(\alpha_{i})_{N}$ is not divisible by $p$. This is equivalent to determining the smallest $n_{i}$ such that $a_{i}+n_{i}b_{i} \equiv_{p} 0$. If $b_{i}=1$, we take $n_{i}=p-a_{i}$. Otherwise, for $b_{i}>1$, let $u_{i}$ be the smallest positive residue of $a_{i}\overline{p}$ modulo $b_{i}$. Then the requisite $n_{i}$ is given by $\frac{u_{i}p-a_{i}}{b_{i}}$ (note that $1\leq n_{i}\leq p-1$). Therefore, the natural truncation must occur with $m\leq N$ where $N = \min_{i} n_{i}$, the minimum of all of the values of $n_{i}$ determined by all of the parameters. It follows that there exists a parameter $\frac{a}{b}$ and an integer $ 1\leq \omega \leq b-1$, determined by the prime $p$, such that the natural truncation occurs at $N = \frac{\omega p -a}{b}$. In particular, we have that $\frac{N}{p}$ is asymptotically a positive rational number less than one. For the example $_{2}F_{1}\left(\frac{1}{3},\frac{5}{6},\frac{1}{2};x\right)$ considered in Section 2, if $p\equiv_{6}1$, then the values of $n$ are $\left\{\frac{p-1}{3},\frac{5(p-1)}{6},\frac{p-1}{2}\right\}$, so that $N=\frac{p-1}{3}$, but if on the other hand $p\equiv_{6}5$, then the values of $n$ are $\left\{\frac{2p-1}{3},\frac{p-5}{6},\frac{p-1}{2}\right\}$, so that now one must truncate at $N=\frac{p-5}{6}$.
In everything we consider henceforth, we will use this natural truncation. For a hypergeometric function $F$ and a (sufficiently large) prime number $p$, we will denote the natural truncation by $F^{(p)}$, where the {\it degree} $N$ as determined above is implicit.
\vspace{10pt}
We now state some results that will be used in later sections. Here and in what follows, $p$ is any odd prime, and $D=\frac{p-1}{2}$.
\begin{lemma}\label{pre1} Suppose that $a$ and $b$ are non-negative integers satisfying $0 \leq a + b \leq 2D$. Then
\[
\binom{2D-a}{b} \equiv_p (-1)^b \binom{a+b}{a} \equiv_p (-1)^{a+b} \binom{2D-b}{a}.
\]
\end{lemma}
\begin{proof} This follows from the identities
\[
\binom{2D-a}{b} = \frac{1}{b!}\prod_{j=0}^{b-1}(2D-a-j) \equiv_p \frac{(-1)^b}{b!}\prod_{j=0}^{b-1} (1+a+j) = (-1)^b \binom{a+b}{b}.
\]
\end{proof} Taking $a=0$, we obtain:
\begin{corollary}\label{-1} For each integer $s$ with $0 \leq s \leq 2D$, we have
\[
\binom{2D}{s} \equiv_p (-1)^s.
\]
\end{corollary}
\begin{corollary}\label{4} Suppose that $0 \leq a \leq D$. Then
\[
\binom{2D-a}{a} \equiv_p 4^a \binom{D}{a}.
\]
\end{corollary}
\begin{proof} By Lemma \ref{pre1}, we have $\binom{2D-a}{a} \equiv_p (-1)^a \binom{2a}{a}$. Also, by definition, we have
\[
\binom{D}{a} 2^a a! \equiv_p \prod_{j=0}^{a-1} (-1-2j) = (-1)^a \frac{(2a)!}{2^a (a!)}.
\]
The result follows.
\end{proof}
If $a$ and $b$ are integers with $a,b \geq 0$, we define
\begin{equation}\label{Sfunction}
S(a,b):= \sum_{x \in \mathbb{F}_p} x^a (1+x)^b.
\end{equation}
The following characterisation of $S(a,b)$ will be important for rational point counts.
\begin{lemma}\label{pre2} Suppose that $a$ and $b$ are integers with $a,b \geq 0$.
\begin{enumerate}[(I)]
\item If $a>0$ and if $2D \nmid b$, then
\[
S(a,b) \equiv_p - \sum_{\substack{j=0\\2D| a+j}}^b \binom{b}{j};
\]
\item If additionally $a+b < 4D$, then
\[
S(a,b) \equiv_p -\binom{b}{2D-a}.
\]
\end{enumerate} \end{lemma}
\begin{proof} By definition, we have
\[
S(a,b) = \sum_{j=0}^b \binom{b}{j} \sum_{x \in \mathbb{F}_p} x^{a+j}.
\]
If $\alpha$ is a non-negative integer so that $2D \nmid \alpha$, then $\sum_{x \in \mathbb{F}_p} x^\alpha \equiv_p 0$. If on the other hand $2D | \alpha$ and $\alpha \geq 1$, then $\sum_{x \in \mathbb{F}_p} x^\alpha \equiv_p -1$. Also trivially if $\alpha = 0$, then $\sum_{x \in \mathbb{F}_p} x^\alpha \equiv_p 0$. As $2D \nmid a$, it follows that $a+j \geq 1$. This proves (i).
For (II), if $a+b < 4D$, then $2D | (a+j)$ in the sum implies that $a+j$ is equal to $0$ or $2D$. The sum is zero at $j=0$, so we only need consider the case $a+j = 2D$. The result then follows from the previous arguments. \end{proof}
We state some classical transformation formulae for $_{2}F_{1}$ and $_{3}F_{2}$ hypergeometric functions as follows.
\begin{lemma}\label{trans}\quad
\begin{enumerate}[(I)]
\item \text{Euler}: \[ _{2}F_{1}(a,b,c;x) = (1-x)^{c-a-b}\ _{2}F_{1}(c-a,c-b,c;x)\ . \]
\item \text{Pfaff}: \[ _{2}F_{1}(a,b,c;x) = (1-x)^{-a}\ _{2}F_{1}\left(a,c-b,c;\frac{-x}{1-x}\right). \]
\item \text{Quadratic}:
\begin{enumerate}[(i)] \item \begin{align*} _{2}F_{1}(a,1-a,c;x) &= (1-x)^{c-1}\ _{2}F_{1}\left(\half a,\half -\half a,c;4x(1-x)\right)\ ,\\
&= (1-x)^{c-1}(1-2x)^{2c -1}\ _{2}F_{1}\left(c- \half a,c- \half +\half a,c;4x(1-x)\right)\ ;
\end{align*}
\item \begin{align*} _{2}F_{1}(a,b,2b;x) = (1-x)^{-a/2}\ {_{2}F_{1}}\left(\frac{1}{2}a,b-\frac{1}{2}a,b + \frac{1}{2};\frac{x^2}{4x-4}\right)\ .\end{align*}
\end{enumerate}
\item \text{Clausen}:
\[
_{2}F_{1}\left(a,b,a+b+\frac{1}{2};x\right)^{2} = \ _{3}F_{2}\left(2a,2b,a+b;a+b+\frac{1}{2},2a+2b;x\right).
\]
\end{enumerate}
\end{lemma}
The natural $\mathbb{F}_p$ analogues of these formulae will be proven in Section 4.
\section{Transformation formulae}
\subsection{Euler and Pfaff transformations}\
We consider here transformation formulae modulo $p$ for truncations of hypergeometric functions of the form $_{2}^{} F^{}_1\left(\frac{a}{b},\frac{a}{b},1;x\right)$ and $_{2}^{}F^{}_1\left(\frac{a}{b},1-\frac{a}{b},1;x\right)$, where as before, $1\leq a < b $ with $a$ and $b$ coprime. We first determine the degrees of the truncations. Let $1\leq \omega\leq b-1$ (see Section 3) satisfy $p\omega \equiv a$ modulo $b$, and let $E := \frac{\omega p -a}{b}$ denote the degree of $_{2}^{} F^{(p)}_1\left(\frac{a}{b},\frac{a}{b},1;x\right)$. Then the corresponding $\omega'$ for $_{2}^{} F^{(p)}_1\left(1-\frac{a}{b},1-\frac{a}{b},1;x\right)$ is $\omega'=b-\omega$, so that the degree of the truncation is equal to $2D-E$. Finally, for $_{2}^{} F^{(p)}_1\left(\frac{a}{b},1-\frac{a}{b},1;x\right)$, the corresponding values of $\omega$ (resp. $\omega'$) are $\omega$ and $b-\omega$, so that the degree of the truncation is equal to $E^{*}:= \min(E,2D-E)$. We have the following:
\begin{proposition}\label{10b}Let $1\leq a < b$ be coprime integers. Then with $E$ and $E^{*}$ as above and for any (odd) prime $p>b$, we have:
\noindent(I) If $E < \frac{p}{2}$ then
\[
_{2}^{} F^{(p)}_1\left(\frac{a}{b},1-\frac{a}{b},1;x\right) \equiv_{p}\ (1-x)^{E^{*}}\ _{2}^{} F^{(p)}_1\left(\frac{a}{b},\frac{a}{b},1;\frac{-x}{1-x}\right) \ .
\]
\noindent(II) If $E > \frac{p}{2}$ then
\[
_{2}^{} F^{(p)}_1\left(\frac{a}{b},1-\frac{a}{b},1;x\right) \equiv_{p}\ (1-x)^{E^{*}}\ _{2}^{} F^{(p)}_1\left(1-\frac{a}{b},1-\frac{a}{b},1;\frac{-x}{1-x}\right) \ .
\]
\end{proposition}
\begin{proof}
For the first case, $E^{*}=E$, and we use $m!\binom{E}{m}\equiv_{p} (-1)^{m}(\frac{a}{b})_{m}$. Then the right-hand side is congruent modulo $p$ to
\[
\sum_{n}\binom{E}{n}^{2}(-x)^{n}(1-x)^{E-n} = \sum_{m}\theta_{m}(-x)^{m},
\]
where
\[
\theta_{m} = \sum_{l}\binom{E-m+l}{l}\binom{E}{m-l}^{2}= \binom{E}{m}\sum_{l}\binom{m}{l}\binom{E}{m-l}=\binom{E}{m}\binom{E+m}{m}.
\]
Using $m!\binom{E+m}{m}\equiv_{p} (\frac{b-a}{b})_{m}$ then gives the result.
The proof of the second case is identical with $E^{*}= 2D-E$.
\end{proof}
We now consider the case of those $b$ with $\phi(b)=2$, so that $b=3, 4$ or $6$. Then $\frac{a}{b} = \frac{1}{b}$ or $1-\frac{1}{b}$. In the former case, $\omega = \overline {p}$, and in the latter, $\omega = b-\overline {p}$. Moreover, since $(p,b)=1$, it follows that $p\equiv_{b} \pm 1$. Hence if $p\equiv_{b} 1$, then $E=\frac{p-1}{b} < \frac{p}{2}$ when $\frac{a}{b}=\frac{1}{b}$, so that we may apply Proposition \ref{10b}(I). Similarly, if $p\equiv_{b} -1$, then $E=p - \frac{p+1}{b} > \frac{p}{2}$, so we apply Proposition \ref{10b}(II) to obtain:
\begin{corollary}\label{10c} Let $b=3, 4$ or $6$ and $p>b$, and let $K=\left\lfloor \frac{p-1}{b}\right\rfloor$. Then we have
\noindent(I) If $p \equiv 1$ {\text mod}\ $b$, then
\[
_{2}^{} F^{(p)}_1\left(\frac{1}{b},1-\frac{1}{b},1;x\right) \equiv_{p}\ (1-x)^{K}\ _{2}^{} F^{(p)}_1\left(\frac{1}{b},\frac{1}{b},1;\frac{-x}{1-x}\right) \ .
\]
\noindent(II) If $p \equiv -1$ {\text mod}\ $b$, then
\[
_{2}^{} F^{(p)}_1\left(\frac{1}{b},1-\frac{1}{b},1;x\right) \equiv_{p}\ (1-x)^{K}\ _{2}^{} F^{(p)}_1\left(1-\frac{1}{b},1-\frac{1}{b},1;\frac{-x}{1-x}\right) \ .
\]
The degree of the polynomials on both sides of each equation is equal to $K$.
\end{corollary}
The case of $\frac{a}{b}=1-\frac{1}{b}$ is of course identical to this.
\begin{proposition}\label{10d} Let $1\leq a < b$ be coprime integers. Then with $E$ as above, for any odd prime $p$, we have
\noindent(I) If $E < \frac{p}{2}$, then
\[
(1-x)^{2D-2E} \ _{2}^{} F^{(p)}_1\left(\frac{a}{b},\frac{a}{b},1;x\right) \equiv_{p}\ _{2}^{} F^{(p)}_1\left(1-\frac{a}{b},1-\frac{a}{b},1;x\right) \ .
\]
\noindent(II) If $E > \frac{p}{2}$, then
\[
_{2}^{} F^{(p)}_1\left(\frac{a}{b},\frac{a}{b},1;x\right) \equiv_{p}\ (1-x)^{2E-2D}\ _{2}^{} F^{(p)}_1\left(1-\frac{a}{b},1-\frac{a}{b},1;x\right) \ .
\]
\end{proposition}
\begin{proof} For case (I), we again use $m!\binom{E}{m}\equiv_{p} (-1)^{m}(\frac{a}{b})_{m}$, so that $$_{2}^{}F^{(p)}_1\left(\frac{a}{b},\frac{a}{b},1;x\right) \equiv_{p} \sum_{m=0}^{E}\binom{E}{m}^{2}x^{m}.$$ Expanding the left-hand side as a polynomial in $x$, we write it as $\sum_{m=0}^{2D-E} \alpha_m x^m$, where $\alpha_m = \sum_u (-1)^u \binom{2D-2E}{u} \binom{E}{m-u}^2$. Via
\[
\binom{2D-2E}{u} \binom{E}{m-u} = \binom{2D-E}{E}^{-1} \binom{2D-E}{m} \binom{2D-E-m}{2D-2E-u}\binom{m}{u},
\]
we write $\alpha_m = \binom{2D-E}{E}^{-1} \binom{2D-E}{m} \beta_m$, so that after the substitution $u \to m-u$, we obtain
\[
\beta_m = (-1)^m \sum_u (-1)^u \binom{m}{u} \binom{2D-E-m}{E-u} \binom{E}{u}.
\]
The last two binomial coefficients in the expression above equal the coefficient of $x^{E-u}y^u$ in $(1+x)^{2D-E-m}(y+1)^{E}$. The substitution $y \rightarrow -x/t$ then yields equality to the coefficient of $x^{E} t^{E}$ in
\[
(-1)^u (1+x)^{2D-E-m} t^u [(1+t)-(1+x)]^{E}.
\]
Summing over $u$ shows that $\beta_m$ is equal, up to multiplication by $(-1)^m$, to the coefficient of $x^{E} t^{E}$ in
\[
(1+x)^{2D-E-m}(1+t)^m[(1+t)-(1+x)]^{E}.
\]
Expanding this expression yields
\[
\beta_m = (-1)^m \sum_s (-1)^s \binom{E}{s}\binom{E+m-s}{E} \binom{2D-E-m+s}{E}.
\]
By Lemma \ref{pre1}, we have
\[
\binom{2D-E-m+s}{E} = \binom{2D-(E+m-s)}{E} \equiv_p (-1)^{m-s} \binom{2D-E}{E+m-s}.
\]
Thus
\[
\binom{E}{s}\binom{E+m-s}{E} \binom{2D-E-m+s}{E} \equiv_p (-1)^{m-s}\binom{2D-E}{m}\binom{2D-E-m}{E-s} \binom{m}{s},
\]
whence it follows that
\[
\beta_m \equiv_p \binom{2D-E}{m} \sum_s \binom{2D-E-m}{E-s}\binom{m}{s} = \binom{2D-E}{m}\binom{2D-E}{E}.
\]
Substituting this expression for $\beta_m$ into $\alpha_m$ and using $m!\binom{2D-E}{m}\equiv_{p} (-1)^{m}(\frac{b-a}{b})_{m}$ yields the result.
For case (II), we replace $\frac{a}{b}$ in part (I) with $\frac{b-a}{b}$, which has the effect of replacing $E$ with $2D-E < \frac{p}{2}$. Application of part (I) to the resulting expression gives the formula in (II), completing the proof.
\end{proof}
From this it follows:
\begin{corollary}\label{10f} Let $b=3, 4$ or $6$, $p>b$, and $K=\left\lfloor \frac{p-1}{b}\right\rfloor$. Then we have
\noindent(I) If $p \equiv 1$ {\text mod}\ $b$, then
\[
(1-x)^{2D-2K} \ _{2}^{} F^{(p)}_1\left(\frac{a}{b},\frac{a}{b},1;x\right) \equiv_{p}\ _{2}^{} F^{(p)}_1\left(1-\frac{a}{b},1-\frac{a}{b},1;x\right) \ .
\]
\noindent(II) If $p \equiv -1$ {\text mod}\ $b$, then
\[
_{2}^{} F^{(p)}_1\left(\frac{a}{b},\frac{a}{b},1;x\right) \equiv_{p}\ (1-x)^{2D-2K}\ _{2}^{} F^{(p)}_1\left(1-\frac{a}{b},1-\frac{a}{b},1;x\right) \ .
\]
\end{corollary}
\subsection{Clausen transformations}\
Consider the truncated hypergeometric function $_{3}^{}F^{(p)}_{2}\left(\alpha,1-\alpha,\frac{1}{2};1,1;x\right)$, where $\alpha=\frac{a}{b}$ and $1\leq a < b$, with $a$ and $b$ coprime. We compute the degree of this truncation. Let $1 \leq \Omega < b$ with $p\Omega \equiv_{b} a$, where $p > b$ is a prime number, and put $\hat{N}=\frac{p\Omega -a}{b}$. Since $1-\alpha =\frac{b-a}{b}$ with $(b-a,b)=1$, the corresponding $\omega$ must satisfy $p\omega \equiv_{b}\ b-a$ with $0<\omega<b$, so that $\omega =b-\Omega$. The corresponding $N$ (see Section 3) is $2D - \hat{N}$ and the degree of the truncation is $N^{*} = \min\left(\hat{N}, 2D - \hat{N}, D\right)$.
It is easily checked that if $p > b$, then $\alpha = \half$ if, and only if, $\hat{N}=D$ and $N^{*}=D$. Moreover, if $\alpha \neq \half$, then $\hat{N}< D$ if, and only if, $2\Omega < b$. Thus, in this case, $N^{*}= \hat{N}$ if $2\Omega < b$, and $N^{*}= 2D-\hat{N}$ if $2\Omega > b$. We apply this to the values $b = 2, 3, 4$ and $6$, those values of $b$ with at most two reduced residue classes, to obtain:
\begin{lemma}\label{Clausen1} \
Let $N^{*}$ denote the degree of $_{3}^{}F^{(p)}_{2}\left(\frac{1}{b},1- \frac{1}{b},\frac{1}{2};1,1;x\right)$, where $b= 2, 3, 4$ or $6$. Then if $p > b$, we have
\[
N^{*} = \begin{cases}\frac{p-1}{b}, & \mbox{if}\quad p \equiv _{b}\ 1 \\
\frac{p+1}{b}-1, & \mbox{if}\quad p \equiv _{b} -1\ .
\end{cases}
\]
\end{lemma} The degrees of $_{2}^{}F^{(p)}_{1}\left(\frac{\alpha}{2},\half - \frac{\alpha}{2};1;x\right)$ and $_{2}^{}F^{(p)}_{1}\left(1-\frac{\alpha}{2},\half + \frac{\alpha}{2};1;x\right)$ are computed next, with $\alpha = \frac{a}{b}$ as above.
\subsubsection{} For $_{2}^{}F^{(p)}_{1}\left(\frac{\alpha}{2},\half - \frac{\alpha}{2};1;x\right)$, we have several cases: (I) $a$ even, $b$ odd; (II) $a$ and $b$ both odd; and (III) $a$ odd, $b$ even.
\begin{enumerate}[(I)]
\item Put $a=2A$ and $b=B$ with $B$ odd, so that $\frac{\alpha}{2}=\frac{A}{B}$ with $(A,B)=1$. Define $0 < \omega < B$ such that $p\omega \equiv_{B} \ A$, and put $E= \frac{\omega p -A}{B}$. Then $\half - \frac{\alpha}{2}= \frac{B-2A}{2B}$ with $(B-2A,2B)=1$. Thus the corresponding $\omega$, denoted by $\omega'$, satisfies $p\omega' \equiv_{2B} \ B-2A$ with $0<\omega'<2B$. It follows that if $p > b$, then $\omega'=B-2\omega$ or $3B-2\omega$ depending on whether $\omega <\frac{B}{2}$ or $\omega >\frac{B}{2}$, respectively. The corresponding $E$ (denoted by $E'$) is then $D-E$ and $p-(E-D)$, respectively. It should be noted that if $p>b$, then $\omega <\frac{B}{2}$ if, and only if, $E<D$. By definition, the degree $N^{*}$ of $_{2}^{}F^{(p)}_{1}\left(\frac{\alpha}{2},\half - \frac{\alpha}{2};1;x\right)$ is equal to $\min(E,E')$, so that we have:
\vspace{4pt}
\noindent(i)\quad If \ $\omega <\frac{B}{2}$\ , then
\[
N^{*} = \begin{cases}E, & \mbox{if}\quad E\leq \half D \\
D-E, & \mbox{if}\quad \half D\leq E<D \ .
\end{cases}
\]
\noindent(ii)\quad If\ $\omega >\frac{B}{2}$\ , then
\[
N^{*} = \begin{cases}E, & \mbox{if}\quad D<E\leq \half(p+D) \\
p+D-E, & \mbox{if}\quad E\geq \half(p+D)\ .
\end{cases}
\]
When $b= 2, 3, 4$ or $6$, this case occurs only when $a=2$ and $b=3$. Then if $N^{*}$ is the degree of $_{2}^{}F^{(p)}_{1}\left(\frac{1}{3},\frac{1}{6};1;x\right)$, we have for $p>3$:
\begin{equation}\label{Clausen2}
N^{*} = \begin{cases}\frac{1}{3}(p-1), & \mbox{if}\quad p \equiv _{6}\ 1 \\
\frac{1}{3}(2p-1), & \mbox{if}\quad p \equiv _{6}\ 5\ .
\end{cases}
\end{equation}
\item Since $a$ and $b$ are both odd and the hypergeometric is unchanged if $a$ is replaced with $b-a$, we have the case above, so that $N^{*}$ is the same.
\item If $a$ is odd and $b$ is even, then $\omega$ satisfies $p\omega \equiv_{2b} \ a$, with $1\leq \omega <2b$ and $E= \frac{\omega p -a}{2b}$. Since $(b-a,2b)=1$, it follows that $0< \omega' <2b$, with $p\omega \equiv_{2b} \ b-a$ and $E'= \frac{\omega' p -(b-a)}{2b}$. It then follows that $\omega'=b-\omega$ if $\omega < b$ and $\omega'=3b-\omega$ if $\omega > b$. The corresponding $E'$ are then $E'=D-E$ and $E'=p-(E-D)$, respectively (as in the previous cases). For the choices of $b$ as in Proposition \ref{Clausen1}, we then have $b= 2, 4$ or $6$, so that
\begin{enumerate}[(i)]
\item If $b=2$, $a=1$, then for $p\geq 3$
\[
N^{*} = \begin{cases}\frac{1}{4}(p-1), & \mbox{if}\quad p \equiv _{4}\ 1 \\
\frac{1}{4}(3p-5), & \mbox{if}\quad p \equiv _{4}\ 3\ .
\end{cases}
\]
\item If $b=4$, $a=1$ or $3$, then for $p\geq 5$
\[
N^{*} = \begin{cases}\frac{1}{8}(p-1), & \mbox{if}\quad p \equiv _{8}\ 1 \\
\frac{1}{8}(p-3), & \mbox{if}\quad p \equiv _{8}\ 3\\
\frac{1}{8}(5p-1), & \mbox{if}\quad p \equiv _{8}\ 5\\
\frac{1}{8}(5p-3), & \mbox{if}\quad p \equiv _{8}\ 7 \ .
\end{cases}
\]
\item If $b=6$, $a=1$ or $5$, then for $p\geq 7$
\[
N^{*} = \begin{cases}\frac{1}{12}(p-1), & \mbox{if}\quad p \equiv _{12}\ 1 \\
\frac{1}{12}(p-5), & \mbox{if}\quad p \equiv _{12}\ 5\\
\frac{1}{12}(7p-1), & \mbox{if}\quad p \equiv _{12}\ 7\\
\frac{1}{12}(7p-5), & \mbox{if}\quad p \equiv _{12}\ 11 \ .
\end{cases}
\]
\end{enumerate}
\end{enumerate}
\subsubsection{} For $_{2}^{}F^{(p)}_{1}\left(1-\frac{a}{2b},\half + \frac{a}{2b};1;x\right)$ the analysis is similar:
\begin{enumerate}[(I)]
\item If $a$ is even and $b$ is odd, we write $1-\frac{a}{2b}=\frac{A}{B}$ and $\half + \frac{a}{2b}=\frac{3B-2A}{2B}$, with $B$ odd and all corresponding numerators and denominators coprime. Then we have $0<\omega <B$ satisfying $p\omega \equiv_{B}\ A$ and $E=\frac{p\omega -A}{B}$, and $0<\omega' <2B$ satisfying $p\omega \equiv_{2B}\ 3B-2A$ and $E'=\frac{p\omega' -(3B-2A)}{2B}$. It follows that $B| (2\omega +\omega')$, and since $\omega'$ is odd, that necessarily $\omega'=B-2\omega$ if $\omega < \frac{B}{2}$ or $\omega' = 3B-2\omega$ otherwise. Hence $E'=D-1-E$ or $E'= p+D-1-E$ in the two instances, respectively (note that if $p > b$, then $E<D$ if, and only if, $\omega < \frac{B}{2}$). We thus have
\vspace{4pt}
\noindent(i)\quad If\ $\omega <\frac{B}{2}$\ , then
\[
N^{*} = \begin{cases}E, & \mbox{if}\quad E\leq \frac{1}{2}(D-1) \\
D-1-E, & \mbox{if}\quad \frac{1}{2}(D-1)\leq E<D \ .
\end{cases}
\]
\noindent(ii) \quad If\ $\omega >\frac{B}{2}$\ , then
\[
N^{*} = \begin{cases}E, & \mbox{if}\quad D< E\leq \frac{3}{2}D \\
p+D-1-E, & \mbox{if}\quad E\geq \frac{3}{2}D\ .
\end{cases}
\]
Applying this to the case $b=3$ and $a=2$, gives for $p>3$ that
\begin{equation}\label{Clausen4}
N^{*} = \begin{cases}\frac{2}{3}(p-1), & \mbox{if}\quad p \equiv _{6}\ 1 \\
\frac{1}{6}(p-5), & \mbox{if}\quad p \equiv _{6}\ 5\ .
\end{cases}
\end{equation}
\item If $a$ and $b$ are both odd, (so that when $b=3$, $a=1$), the result is the same as in case (I) above.
\item If $a$ is odd and $b$ is even, we write $1- \frac{a}{2b} = \frac{A}{B}$ with $A=2b-a$ and $B=2b$, so that $(A,B)=1$. We also write $\half + \frac{a}{2b} = \frac{U}{V}$ with $U=\frac{3}{2}B -A$ and $V=B$, so that $(U,V)=1$. Then $0< \omega <B$ with $p\omega \equiv_{B}\ A$ and $E=\frac{\omega p-A}{B}$. Also, $\omega'$ ($0<\omega'<B$) satisfies $p\omega' \equiv_{B}\ \frac{3}{2}B -A$. It follows that $B|2(\omega + \omega')$, so that if $p>b$, we have $\omega' = \half B- \omega$ if $\omega <\half B$, and $\omega' = \frac{3}{2}B-\omega$ otherwise. Hence $E'=D-E$ or $E'=p+D-E$ . We then have the cases
\vspace{4pt}
\begin{enumerate}[(i)]
\item If $b=2$, $a=1$, then for $p\geq 3$,
\[
N^{*} = \begin{cases}\frac{1}{4}(3p+1), & \mbox{if}\quad p \equiv _{4}\ 1 \\
\frac{1}{4}(p-3), & \mbox{if}\quad p \equiv _{4}\ 3\ .
\end{cases}
\]
\item If $b=4$, $a=1$ or $3$, then for $p\geq 5$
\[
N^{*} = \begin{cases}\frac{1}{8}(5p+3), & \mbox{if}\quad p \equiv _{8}\ 1 \\
\frac{1}{8}(5p-7), & \mbox{if}\quad p \equiv _{8}\ 3\\
\frac{1}{8}(p-5), & \mbox{if}\quad p \equiv _{8}\ 5\\
\frac{1}{8}(p-7), & \mbox{if}\quad p \equiv _{8}\ 7 \ .
\end{cases}
\]
\item If $b=6$, $a=1$ or $5$, then for $p\geq 7$
\[
N^{*} = \begin{cases}\frac{1}{12}(7p+5), & \mbox{if}\quad p \equiv _{12}\ 1 \\
\frac{1}{12}(7p-11), & \mbox{if}\quad p \equiv _{12}\ 5\\
\frac{1}{12}(p-7), & \mbox{if}\quad p \equiv _{12}\ 7\\
\frac{1}{12}(p-11), & \mbox{if}\quad p \equiv _{12}\ 11 \ .
\end{cases}
\]
\end{enumerate}
\end{enumerate}
\begin{proposition}\label{K3} Let $1\leq a < b$ be coprime integers with $b = 3,\ 4 $ or $6$ as above. For any odd prime $p>b$
\[
\ _{3}^{}F^{(p)}_{2}\left(\frac{a}{b},1-\frac{a}{b},\frac{1}{2};1,1;x\right) \equiv_{p}\ \begin{cases}_{2}^{}F^{(p)}_{1}\left(\frac{a}{2b},\frac{1}{2} - \frac{a}{2b};1;x\right)^2, & \mbox{if}\quad p \equiv _{2b} 1, \ b-1 \\
(1-x) \ _{2}^{}F^{(p)}_{1}\left(1-\frac{a}{2b},\half +\frac{a}{2b};1;x\right)^2, & \mbox{} {\rm otherwise}\ ;
\end{cases}
\]
where for $b=3$ one discards the congruence class $p \equiv _{6}\ 2$.
\end{proposition}
\begin{proof}
The polynomials on both sides of the equations have the same degree modulo $p$. The results follow from an application of the classical transformation formulae. We provide the details for the first case, as the proof is the same as that for all of the other cases.
First, if $p \equiv_{4} 1$, then the degree of the $_{3}^{}F^{(p)}_{2}$ in question is equal to $D$, while that of the $_{2}^{}F^{(p)}_{1}$ is equal to $E=\frac{D}{2}$. Thus, comparing coefficients, we need to show that for $0\leq m\leq D$,
\begin{equation}\label{5a2}
\left(\frac{\left(\frac{1}{2}\right)_m}{m!}\right)^3 \equiv_p\ \sum_{\substack{0 \leq u,v, \leq E\\ u + v = m}} \left(\frac{\left(\frac{1}{4}\right)_u \left(\frac{1}{4}\right)_v}{u!v!}\right)^2 .
\end{equation}
However, by the definition of {\it degree}, the Pochhammer symbols on the right vanish modulo $p$ if $u$ or $v$ exceed $E$. We may then remove the restrictions on $u$ and $v$ entirely (provided that $m\leq D$), so that \eqref{5a2} holds with equality using Clausen's formula in Lemma \ref{trans}.
For $p \equiv_{4} 3$, the degree of the $_{3}^{}F^{(p)}_{2}$ is equal to $D$, while the degree of the $_{2}^{}F^{(p)}_{1}$ is equal to $E= \frac{p-3}{4}$. The proof by considering coefficients follows in exactly the same manner as above, by instead using a combination of Euler's and Clausen's formula as in Lemma \ref{trans} of the form
\[
\ _{3}F_{2}\left(2a,2b,a+b;a+b+\frac{1}{2},2a+2b;\ x\right) = (1-x)\ _{2}F_{1}\left(a+\half,b+\half,a+b+\half;\ x\right)^{2}.
\]
\end{proof}
\subsection{Quadratic transformations}\
We now turn to identities which relate these formulae to elliptic curves and K3 surfaces.
\begin{proposition}\label{10e}\
Let $1\leq a < b$ be coprime integers with $b = 3,\ 4 $ or $6$ as above. For any odd prime $p>b$ we have
\[
_{2}^{} F^{(p)}_1\left(\frac{a}{b},\frac{b-a}{b},1;x\right) \equiv_{p}\ \begin{cases} _{2}^{} F^{(p)}_1\left(\frac{a}{2b},\half -\frac{a}{2b},1;\ 4x(1-x)\right) , & \mbox{} p \equiv _{2b} 1, \ b-1 \\
(1-2x)\ _{2}^{} F^{(p)}_1\left(1-\frac{a}{2b},\half+\frac{a}{2b},1;\ 4x(1-x)\right), & \mbox{} {\rm otherwise}\ ;
\end{cases}\
\]
where for $b=3$ one discards the congruence class $p \equiv _{6}\ 2$.
\end{proposition}
\begin{proof}
Since $\phi(b)=2$, we may choose $a=1$, as all of the formulae are symmetric with $a\rightarrow b-a$. For $b$ even, the degree of $_{2}^{} F^{(p)}_1\left(\frac{1}{b},1-\frac{1}{b},1;x\right)$ is equal to $E^{*} = \frac{p-1}{b}$ or $\frac{p-(b-1)}{b}$, depending on the two congruence classes $p \equiv_{b} \pm 1$ respectively. It is then easily checked that the degree of $_{2}^{} F^{(p)}_1\left(\frac{1}{2b},\half -\frac{1}{2b},1;\ x\right)$ is equal to $\half E^{*}$ for the two congruence classes $p \equiv _{2b}\ 1 \ {\text or}\ b-1 $. Similarly, one sees that the degree of $_{2}^{} F^{(p)}_1\left(1-\frac{1}{2b},\half+\frac{1}{2b},1;\ x\right)$ is equal to $\half(E^{*}-1)$ for the other two cases. Thus both sides of the equations have precisely the same degrees. The result now follows by using the classical quadratic transformations as in Lemma \ref{trans} and comparing coefficients in exactly the same manner as described above.
The case $b=3$ is similar.
\end{proof}
\begin{lemma}\label{degree}Let $b = 3,\ 4$ or $6$. \quad Suppose $E^*$ denotes the degree of $_{2}^{} F^{(p)}_1 \left(\frac{1}{2},\frac{1}{b},1;x\right)$, $N^*$ the degree of $_{2}^{} F^{(p)}_{1} \left(\frac{1}{2b}, \frac{1}{2} - \frac{1}{2b},1; x \right)$ and $N^{**}$ the degree of ${_{2}^{} F^{(p)}_1} \left(\frac{1}{2}+\frac{1}{2b}, 1 - \frac{1}{2b},1; x \right)$.
\begin{enumerate}[(I)] \item If\ $p \equiv_{2b}\ 1$ or $p \equiv_{2b}\ b-1$, then the values of $E^*$ and $N^*$ are as follows: \begin{enumerate}[(a)] \item If $b=3$, then for $p > 3$,\quad $(E^*, N^{*}) = \left(\frac{1}{3}(p-1),\frac{1}{6}(p-1)\right)$,\quad if\quad $p \equiv _{6}\ 1$.
\item If $b=4$, then for $p\geq 5$,
\[
(E^*, N^{*}) = \begin{cases}\left(\frac{1}{4}(p-1),\frac{1}{8}(p-1)\right), & \mbox{if}\quad p \equiv _{8}\ 1 \\ \left(\frac{1}{2}(p-1),\frac{1}{8}(p-3)\right), & \mbox{if}\quad p \equiv _{8}\ 3.
\end{cases}
\] \item If $b=6$, then for $p\geq 7$,
\[
(E^*,N^{*}) = \begin{cases}\left(\frac{1}{6}(p-1),\frac{1}{12}(p-1)\right), & \mbox{if}\quad p \equiv _{12}\ 1 \\
\left(\frac{1}{2}(p-1),\frac{1}{12}(p-5)\right), & \mbox{if}\quad p \equiv _{12}\ 5.
\end{cases}
\]
\end{enumerate}
\item If $p \equiv_{2b}\ b+1$ or $p \equiv_{2b}\ 2b-1$, then the values of $E^{*}$ and $N^{**}$ are as follows: \begin{enumerate}[(a)] \item If $b=3$, then for $p > 3$,\quad $(E^*, N^{**}) = \left(\frac{1}{2}(p-1),\frac{1}{6}(p-5)\right)$\quad if\quad $p \equiv _{6}\ 5\ $.
\item If $b=4$, then for $p\geq 5$,
\[
(E^*, N^{**}) = \begin{cases}\left(\frac{1}{4}(p-1),\frac{1}{8}(p-5)\right), & \mbox{if}\quad p \equiv _{8}\ 5\\
\left(\frac{1}{2}(p-1),\frac{1}{8}(p-7)\right), & \mbox{if}\quad p \equiv _{8}\ 7 \ .
\end{cases}
\] \item If $b=6$, then for $p\geq 7$,
\[
(E^*,N^{**}) = \begin{cases}\left(\frac{1}{6}(p-1),\frac{1}{12}(p-7)\right), & \mbox{if}\quad p \equiv _{12}\ 7\\
\left(\frac{1}{2}(p-1),\frac{1}{12}(p-11)\right), & \mbox{if}\quad p \equiv _{12}\ 11 \ .
\end{cases}
\]
\end{enumerate}
\end{enumerate}
\end{lemma}
\begin{proof} (I) Let $E'$, $N'$, and $N''$ denote the natural truncations associated with $(\frac{1}{b})_n$, $(\frac{1}{2b})_n$, and $(\frac{1}{2}-\frac{1}{2b})_n$ respectively, then by definition (see Section 3) we find that
\begin{enumerate}[(a)] \item If $b=3$, then for $p > 3$, $(E', E^*, N',N'', N^{*}) = (\frac{p-1}{3},\frac{p-1}{3},\frac{p-1}{6},\frac{p-1}{3},\frac{p-1}{6})$ if $p \equiv _{6}\ 1$.
\item If $b=4$, then for $p\geq 5$,
\[
(E', E^*, N',N'', N^{*}) = \begin{cases}(\frac{p-1}{4},\frac{p-1}{4},\frac{p-1}{8}, \frac{3(p-1)}{8},\frac{p-1}{8}), & \mbox{if}\quad p \equiv _{8}\ 1 \\
(\frac{3p-1}{4},\frac{p-1}{2},\frac{3p-1}{8},\frac{p-3}{8},\frac{p-3}{8}), & \mbox{if}\quad p \equiv _{8}\ 3.
\end{cases}
\] \item If $b=6$, then for $p\geq 7$,
\[
(E', E^*, N',N'', N^{*}) = \begin{cases}(\frac{p-1}{6},\frac{p-1}{6},\frac{p-1}{12},\frac{5(p-1)}{12},\frac{p-1}{12}), & \mbox{if}\quad p \equiv _{12}\ 1 \\
(\frac{5p-1}{6},\frac{p-1}{2},\frac{5p-1}{12},\frac{p-5}{12},\frac{p-5}{12}), & \mbox{if}\quad p \equiv _{12}\ 5.
\end{cases}
\]
\end{enumerate}
The claim then follows from the definition of truncation.
The cases in (II) are obtained by a similar argument.
\end{proof}
\begin{proposition}\label{10k1}\
Let $b = 3,\ 4 $ or $6$. For any odd prime $p>b$ we have: \begin{enumerate}[(I)] \item If
\[K = \begin{cases}
\frac{p-1}{2b} & \text{if } p \equiv_{2b}\ 1\\
\frac{(b-1)p-1}{2b} & \text{if } p \equiv_{2b}\ b-1,
\end{cases}\]
then \[{_{2}^{} F^{(p)}_1} \left(\frac{1}{2},\frac{1}{b},1; x \right) {\equiv_{p}}\ (1-x)^{K}{_{2}^{} F^{(p)}_1}\left(\frac{1}{2b}, \frac{1}{2} - \frac{1}{2b},1; \frac{x^2}{4x-4}\right)\ .\]
\item If
\[K = \begin{cases}
\frac{p-(b+1)}{2b} & \text{if }p \equiv_{2b}\ b+1\\
\frac{(b-1)p-(b+1)}{2b} & \text{if }p \equiv_{2b}\ 2b-1,
\end{cases}\]
then
\[{_{2}^{} F^{(p)}_1} \left(\frac{1}{2},\frac{1}{b},1; x \right) {\equiv_{p}}\ \frac{1}{2} (2-x)(1-x)^{K}{_{2}^{} F^{(p)}_1}\left(\frac{1}{2}+\frac{1}{2b}, 1 - \frac{1}{2b},1; \frac{x^2}{4x-4}\right).\] \end{enumerate}
\end{proposition}
\begin{proof} In characteristic zero, by Lemma \ref{trans}(III)(ii) (a quadratic identity) with $a = \frac{1}{b}$ we have $${_{2}F_1}\left(\frac{1}{b},\frac{1}{2},1;x\right) = (1-x)^{-\frac{1}{2b}} {_{2}F_1} \left(\frac{1}{2b},\frac{1}{2}-\frac{1}{2b},1; \frac{x^2}{4x-4}\right).$$
Then, by Lemma \ref{trans}(I) (Euler's identity), we have \begin{align*}{_{2} F_1} \left(\frac{1}{2b},\frac{1}{2}-\frac{1}{2b},1; \frac{x^2}{4x-4}\right) &= \left(1 - \frac{x^2}{4x-4} \right)^{\frac{1}{2}} {_{2} F_1}\left(1 - \frac{1}{2b},\frac{1}{2} + \frac{1}{2b},1;\frac{x^2}{4x-4}\right), \\& = \frac{1}{2} \left(\frac{(x-2)^2}{1-x}\right)^{\frac{1}{2}}\ _{2} F_{1}\left(1 - \frac{1}{2b},\frac{1}{2} + \frac{1}{2b},1;\frac{x^2}{4x-4}\right), \\& = \frac{1}{2}(1-x)^{-\frac{1}{2}} (2-x)\ {_{2} F_1}\left(1 - \frac{1}{2b},\frac{1}{2} + \frac{1}{2b},1;\frac{x^2}{4x-4}\right),\end{align*} where the last equality follows as ${_{2}^{} F_1}(\cdot,\cdot,\cdot;0) = 1$. Combining the above two formulae gives us
\begin{equation}\label{X1}
{_{2}F_1}\left(\frac{1}{b},\frac{1}{2},1;x\right) = \frac{1}{2} (1-x)^{-\frac{1}{2}-\frac{1}{2b}}\ (2-x)\ {_{2} F_1}\left(1 - \frac{1}{2b},\frac{1}{2} + \frac{1}{2b},1;\frac{x^2}{4x-4}\right).
\end{equation}
We first prove case (II). By Lemma \ref{degree}, if $p \equiv_{2b}\ b+1$, then $$E^{*} = \frac{p-1}{b} \quad\quad \text{and} \quad\quad N^{**} = \frac{p-(b+1)}{2b},$$ and if $p \equiv_{2b}\ 2b-1$, then $$E^{*} = \frac{p-1}{2} \quad\quad \text{and} \quad\quad N^{**} = \frac{p-(2b-1)}{2b}\ .$$ To ensure that the degrees of both sides of our identities are equal and that any denominators containing powers of $4x-4$ are removed, we require that $K \geq N^{**}$ and $E^{*} = K+1+N^{**}$, so that we need $E^{*} \geq 2 N^{**}+1$.
By Lemma \ref{degree}, we have that if $p \equiv_{2b}\ b+1$, then $$N^{**} = \frac{p-(b+1)}{2b} = \frac{p-1}{b} -\frac{p-(b+1)}{2b}-1 \ = \ K,$$ and if $p \equiv_{2b} 2b-1$, then
\[
N^{**} = \frac{p - (2b-1)}{2b} < \frac{(b-1)p - (b+1)}{2b} = \frac{p-1}{2} - \frac{p-(2b-1)}{2b} - 1 = K.
\]
We write ${_{2} F_1} \left(\frac{1}{2} + \frac{1}{2b},1-\frac{1}{2b},1; x\right) = \sum_n \beta_{n} x^n$.
With this notation,
\begin{align*} \frac{1}{2} (2-x)(1-x)^{K}\ {_{2}^{} F^{(p)}_{1}} &\left(\frac{1}{2}+\frac{1}{2b}, 1 - \frac{1}{2b},1; \frac{x^2}{4x-4}\right) \\& = \frac{1}{2} (2-x) \sum_{n \leq N^{**}} \beta_{n} x^{2n} \left(\frac{-1}{4}\right)^{n} (1-x)^{K-n}, \\& =\frac{1}{2} \sum_{n \leq N^{**}} \beta_{n} x^{2n} \left(\frac{-1}{4}\right)^{n} \left[ (1-x)^{K+1-n} + (1-x)^{K-n}\right],\\& = \frac{1}{2} \sum_{n \leq N^{**}} \sum_{j\geq 0} \beta_n \left(\frac{-1}{4}\right)^n \left[\binom{K+1-n}{j} + \binom{K-n}{j}\right]x^{2n+j}.\end{align*}
Putting $m = j+ 2n$; using the fact that necessarily $j \leq K+1$ and using $E^{*} = K+1+N^{**}$, we see that $0 \leq m \leq E^{*}$. Hence, the sum above is
$$\frac{1}{2}\ \sum_{0 \leq m \leq E^{{*}}} x^m \left(\sum_{\substack{n \leq N^{**}, \;j \geq 0 \\ j+2n = m}} \beta_n \left(\frac{-1}{4}\right)^{n} \left\{ \binom{K+1-n}{j} + \binom{K-n}{j}\right\}\right).$$
For $m$ in this range, the $\beta_{n}$'s vanish (modulo $p$) if $n > N^{**}$, so that we may drop the restriction on $n$. Next, by elementary considerations, if $0 \leq j \leq p-1$ and $p \equiv_{2b}\ b+1$, then for $s \in \mathbb{Z}$ $$\binom{K+s}{j} = \binom{\frac{p-(b+1)}{2b} + s}{j} \equiv_p\ \binom{\frac{-(b+1)}{2b} + s}{j} = \binom{-\frac{1}{2} - \frac{1}{2b} + s}{j}\ ;$$ and if $p \equiv_{2b}\ 2b-1$, then also $$\binom{K+s}{j} = \binom{\frac{(b-1)p-(b+1)}{2b} + s}{j} \equiv_p\ \binom{\frac{-(b+1)}{2b} + s}{j} = \binom{-\frac{1}{2} - \frac{1}{2b} + s}{j}.$$
Moreover since $j \leq m < p$, we see that there are no restrictions on $j$ beyond that imposed by $m = 2n + j$. It follows that the coefficient of $x^m$ in the previous sum is equal modulo $p$ to the respective coefficient in characteristic zero, whence the identity then follows from the classical case in \eqref{X1}.
In case (I), we have if either $p \equiv_{2b} 1$ or $p \equiv_{2b} b-1$ that $K = N^*$. The argument then follows as in case (II), where it is only necessary to consider the coefficient $\binom{K-n}{j} \equiv_p\ \binom{-\frac{1}{2b}-n}{j}$.
\end{proof}
\section{Elliptic families}
Those $_2 F_1(a,b,c;z)$ hypergeometric functions with integer coefficients and $c=1$ take the form $_2 F_1(a,1-a,c;z)$, where $a=\frac{1}{2},\frac{1}{3},\frac{1}{4},$ or $\frac{1}{6}$. We now determine the families of elliptic curves associated with the mod $p$ truncations of these functions.
\begin{enumerate}[(I)]
\item If $b=2$, we have
\begin{align*}
_{2}^{}F_1^{(p)}\left(\frac{1}{2},\frac{1}{2};1;\lambda \right) &= \sum_{n=0}^{D}\frac{\left(\half\right)_{n}^{2}}{(n!)^{2}}\lambda^{n} \\&\equiv_{p} \sum_{n=0}^{D}\binom{D}{n}^{2}\lambda^{n}\\ &\equiv_{p}\ -\sum_{n=0}^{D}\binom{D}{n}\sum_{x\in \mathbb{F}_{p}} x^{2D-n}(x+1)^{D}\lambda^{n} \\&\equiv_{p} \sum_{x\in \mathbb{F}_{p}} \big[x(x+1)(x+\lambda)\big]^{D}.
\end{align*}
This is associated with the Legendre family
\[
E_{\lambda,2} : \qquad\quad z^{2} = x(x+1)(x+\lambda)\ .
\]
\vspace{2pt}
\item If $b=3$, we first observe that $$\frac{1}{(n!)^{2}}\left(\frac{1}{3}\right)_{n}\left(\frac{2}{3}\right)_{n}\left(\frac{27}{4}\right)^{n} \equiv_{p}\ \binom{D}{n}\binom{2D-2n}{n}.$$ By Lemma \ref{pre2}, we obtain
\begin{align*}
_{2}^{}F_1^{(p)}\left(\frac{1}{3},\frac{2}{3};1;\frac{27}{4}\lambda \right) &\equiv_{p} \sum_{n=0}^{D}\binom{D}{n}\binom{2D-2n}{n}\lambda^{n} \\&\equiv_{p}\ 1-\sum_{n=1}^{D}\binom{D}{n}\left(\sum_{x\in \mathbb{F}_{p}} x^{2D-n}(x+1)^{2D-2n}\right)\lambda^{n}\\ &\equiv_{p} -1 - \sum_{x\in \mathbb{F}_{p}} \big[x\left(x(x+1)^{2}+ \lambda\right)\big]^{D} \\&\equiv_{p}\ \sum_{x\in \mathbb{F}_{p}} \big[x^{3} + (x+\lambda)^{2}\big]^{D}.
\end{align*}
Note that we have used a change of variable $x\rightarrow \frac{\lambda}{x}$ if $\lambda \neq 0$. The associated elliptic family is
\[
E_{\lambda,3} : \qquad\quad z^{2} = x^{3} + (x+\lambda)^{2}\ .
\]
\vspace{2pt}
\item If $b=4$, we then note (similarly to the previous case) that $$\frac{1}{(n!)^{2}}\left(\frac{1}{4}\right)_{n}\left(\frac{3}{4}\right)_{n}\left(\frac{1}{4}\right)^{n} \equiv_{p}\ \binom{D}{n}\binom{D-n}{n}.$$ Hence by Lemma \ref{pre2}, we find
\begin{align*}
_{2}^{}F_1^{(p)}\left(\frac{1}{4},\frac{3}{4};1;\frac{1}{4}\lambda \right) &\equiv_{p} \sum_{n=0}^{D}\binom{D}{n}\binom{D-n}{n}\lambda^{n} \\&\equiv_{p}\ -\sum_{n=0}^{D}\binom{D}{n}\left(\sum_{x\in \mathbb{F}_{p}} x^{2D-n}(x+1)^{D-n}\right)\lambda^{n}\\&\equiv_{p} - \sum_{x\in \mathbb{F}_{p}} \big[x\big(x(x+1)+ \lambda\big)\big]^{D}.
\end{align*}
The associated elliptic family is given by
\[
E_{\lambda,4} : \qquad\quad z^{2} = x\big(x(x+1)+\lambda\big)\ .
\]
\vspace{2pt}
\item If $b=6$, then using $$\frac{1}{(n!)^{2}}\left(\frac{1}{6}\right)_{n}\left(\frac{5}{6}\right)_{n}\left(\frac{27}{4}\right)^{n} \equiv_{p}\ \binom{D}{n}\binom{D-n}{2n},$$ we obtain
\begin{align*}
_{2}^{}F_1^{(p)}\left(\frac{1}{6},\frac{5}{6};1;\frac{27}{4}\lambda \right) &\equiv_{p} \sum_{n=0}^{D}\binom{D}{n}\binom{D-n}{2n}\lambda^{n} \\&\equiv_{p}\ -\sum_{n=0}^{D}\binom{D}{n}\left(\sum_{x\in \mathbb{F}_{p}} x^{2D-2n}(x+1)^{D-n}\right)\lambda^{n}\\ &\equiv_{p} - \sum_{x\in \mathbb{F}_{p}} \big[\big(x^{2}(x+1)+ \lambda\big)\big]^{D} .
\end{align*}
This yields the elliptic family
\[
E_{\lambda,6} : \qquad\quad z^{2} = x^{2}(x+1)+\lambda\ .
\]
\end{enumerate}
\begin{remark} The methods of this section, notably the determination that the coefficients of a given $_2 F_1$ hypergeometric function are integers, employ the following:
\begin{lemma}\label{binomial} If $a$, $b$, and $n$ are positive integers, then $$\frac{b^n}{\gcd(a,b)^{2n}} \cdot \frac{\prod_{j=0}^{n-1} (a + bj)}{n!}$$ is also an integer. \end{lemma} The case $a=1$ was shown in \cite{StMoAm}. We give the full proof of Lemma \ref{binomial} in the Appendix. \end{remark}
\subsection{A note on {\boldmath $_{2}^{}F_1\left(\frac{1}{4},\frac{3}{4};1;*\right)$}}\
Let $\{E_\lambda\}$ be the family of elliptic curves defined as
\begin{equation}\label{Elambda}
E_\lambda: y^2 = (x-1)\left(x^2 - \frac{1}{\lambda + 1}\right), \;\;\;\;\;\;\;\;\; \lambda \in \mathbb{F}_p \backslash \{0,-1\}.
\end{equation} The family $\{E_\lambda\}$ is just the same as that considered in \cite{Ono}. We express the number of $\mathbb{F}_p$-rational points of members of the family $\{E_\lambda\}$ in terms of the natural truncation of a hypergeometric function. As before, we set $D = \frac{p-1}{2}$. The counting function $I_\lambda$ for the elliptic family $\{E_\lambda\}$ is defined as
\begin{equation}\label{ilambda}
I_\lambda: = \sum_{x \in \mathbb{F}_p} \left[(x-1)\left(x^2 - \frac{1}{\lambda + 1}\right)\right]^D.
\end{equation}
\begin{lemma}\label{4a} Let $E=\left\lfloor\frac{D}{2}\right\rfloor$. The function $I_\lambda$ satisfies
\[
I_\lambda \equiv_p - 2^{D}\ _{2}^{}F_1^{(p)}\left(\frac{1}{4},\frac{3}{4};1;\frac{\lambda}{1+\lambda}\right) \ ,
\] and the truncation of $_{2}^{}F_1^{(p)}$ occurs at $E$.
\end{lemma}
\begin{proof}
We perform the change of variable $x \to 2x +1$ in \eqref{ilambda} to obtain
\begin{align}\label{4eq1}
\nonumber I_\lambda & = 8^{D}\sum_{x \in \mathbb{F}_p} x^D\left(x(x+1)+\frac{\lambda}{4(\lambda+1)}\right)^D\\&= 8^{D}\sum_{v}\binom{D}{v} \left(\frac{\lambda}{4(\lambda+1)}\right)^{v}\sum_{x \in \mathbb{F}_p}x^{2D-v}(x+1)^{D-v}\\& \nonumber \equiv_{p}\ - 2^D \sum_{0 \leq 2v \leq D} \binom{D}{v}\binom{D-v}{v} \left(\frac{\lambda}{4(1+\lambda)}\right)^v\ ,
\end{align}
by Lemma \ref{pre2}.
By definition, we have for all $0 \leq 2v \leq D$ that
\begin{align*}
\binom{D-v}{v} &= \frac{1}{v!} \prod_{j=0}^{v-1} (D-v-j)\\& \equiv_p \frac{(-1)^v}{v!} \prod_{j=v}^{2v-1} \left(\frac{1}{2}+j\right) \\& = \frac{(-1)^v}{v!} \left(\frac{1}{2}\right)^{-1}_v \prod_{j=0}^{v-1}\left( \frac{1}{2}+2j\right) \prod_{k=0}^{v-1} \left(\frac{1}{2} + 2k+1\right) \\& = \frac{(-1)^v}{v!} \left(\frac{1}{2}\right)^{-1}_v 4^v \left(\frac{1}{4}\right)_v \left(\frac{3}{4}\right)_v.
\end{align*}
From the equivalence
\[
\binom{D}{v} = \frac{1}{v!} \prod_{j=0}^{v-1}(D-j) \equiv_p \frac{(-1)^v}{v!}\left(\frac{1}{2}\right)_v,
\]
we obtain
\[
I_\lambda \equiv_p - 2^D \sum_{0 \leq 2v \leq D} \frac{\left(\frac{1}{4}\right)_v \left(\frac{3}{4}\right)_v}{(1)_v v!} \left(\frac{\lambda}{1+\lambda}\right)^v \ .
\]
The result follows.
\end{proof}
\section{Families of K3 surfaces and {\boldmath$_3F_2\left(\frac{1}{b},\frac{b-1}{b},\frac{1}{2};1,1;*\right)$.} }
We now consider $4$ families of K3 surfaces with parameter $\lambda$. These are associated with the hypergeometrics $_3F_2\left(\frac{1}{b},\frac{b-1}{b},\frac{1}{2};1,1;*\right)$ with $b= 2,\ 3,\ 4$ and $6$.
\vspace{3pt}
\begin{enumerate}[(I)]
\item If $b=2$, then we see that
\begin{equation}\label{5a1}
{}_{3}^{}F_2^{(p)}\left(\frac{1}{2},\frac{1}{2},\frac{1}{2};1,1;-\lambda\right) \equiv_p \sum_{n=0}^D \frac{(-1)^n \left(\frac{1}{2}\right)_n^3}{(n!)^3} \lambda^n \equiv_p\ \sum_{n=0}^D \binom{D}{n}^3 \lambda^n .
\end{equation}
By Lemma \ref{pre2}, we have
\[
\binom{D}{n} \equiv_p -\left(\sum_{\substack{j=0 \\ 2D|2D-n+j}}^D \binom{D}{j}\right) \equiv_p - \sum_{x \in \mathbb{F}_p} x^{2D-n}(x+1)^D\
\]
and
\[
\binom{D}{n} = \binom{D}{D-n} \equiv_p \left(\sum_{\substack{j=0 \\ 2D|D+n+j}}^D \binom{D}{j}\right) \equiv_p -\sum_{y \in \mathbb{F}_p} y^{D+n} (y+1)^D .
\]
Hence
\begin{align*}
{}_{3}^{}F_2^{(p)}\left(\frac{1}{2},\frac{1}{2},\frac{1}{2};1,1;-\lambda\right)& \equiv_p \sum_{n=0}^D \binom{D}{n} \left(\sum_{x \in \mathbb{F}_p} x^{2D-n} (x+1)^D \right) \left( \sum_{y \in \mathbb{F}_p} y^{D+n} (y+1)^D \right) \lambda^n\\&\equiv_p \sum_{x,y \in \mathbb{F}_p} \left[x(x+1)y(y+1)(x+\lambda y)\right]^D \ .
\end{align*}
This is the counting function for the K3 family \begin{equation}\label{Xlambda2}X_{\lambda,2}:\quad\quad z^2 = x(x+1)y(y+1)(x+\lambda y), \;\;\;\;\;\;\;\;\; \lambda \in \mathbb{F}_p \backslash \{0,-1\},\end{equation} which is associated to the family $\{E_\lambda\}$ of elliptic curves (see \eqref{Elambda}). The families $\{X_{\lambda,2}\}$ and $\{E_\lambda\}$ were studied together in \cite{Ono} in order to determine the values of $\lambda$ for which $X_{\lambda,2}$ is modular.
\vspace{3pt}
\item If $b=3$, we then use the identity
\[
\left(-\frac{27}{4}\right)^{n}\frac{1}{(n!)^{3}}\left(\frac{1}{3}\right)_{n}\left(\frac{2}{3}\right)_{n}\left(\frac{1}{2}\right)_{n} \equiv_{p}\ \binom{D}{n}^{2}\binom{2D-2n}{n}\ ,
\]
this being valid {\it a priori} for all $n$ satisfying $3n\leq 2D$, and then by extension to all $0\leq n\leq D$, when both sides are congruent to zero.
For $1\leq n\leq D$, we apply Lemma \ref{pre2} to obtain
\[
\binom{2D-2n}{n} \equiv_{p} -\sum_{y\in \mathbb{F}_p}y^{2D-n}(y+1)^{2D-2n}\ .
\]
Hence
\begin{align*}
{}_{3}^{}F_2^{(p)}\left(\frac{1}{3},\frac{2}{3},\frac{1}{2};1,1;-\frac{27}{4}\lambda\right)& \equiv_p -1+ \sum_{n=0}^D \binom{D}{n} \left(\sum_{x \in \mathbb{F}_p} x^{2D-n} (x+1)^D \right) \left( \sum_{y \in \mathbb{F}_p} y^{2D-n} (y+1)^{2D-2n} \right) \lambda^n\\&\equiv_p \sum_{x,y \in \mathbb{F}_p} \left[x(x+1)y(xy+\lambda(y+1)^{2} )\right]^D \ .
\end{align*} This is the counting function for the family of surfaces
\begin{equation}\label{Xlambda3}
X_{\lambda,3}:\quad\quad z^2 = x(x+1)y\left(xy +\lambda(y+1)^{2}\right)\ .
\end{equation}
\vspace{3pt}
\item If $b=4$, then we have for $0\leq n\leq D$ the identity
\[
\left(-4\right)^{n}\frac{1}{(n!)^{3}}\left(\frac{1}{4}\right)_{n}\left(\frac{3}{4}\right)_{n}\left(\frac{1}{2}\right)_{n} \equiv_{p}\ \binom{D}{n}^{2}\binom{D-n}{n}\ ,
\]
so that by Lemma \ref{pre2}, we obtain
\begin{align*}
{}_{3}^{}F_2^{(p)}\left(\frac{1}{4},\frac{3}{4},\frac{1}{2};1,1;-4\lambda\right)& \equiv_p \sum_{n=0}^D \binom{D}{n} \left(\sum_{x \in \mathbb{F}_p} x^{2D-n} (x+1)^D \right) \left( \sum_{y \in \mathbb{F}_p} y^{2D-n} (y+1)^{D-n} \right) \lambda^n\\&\equiv_p \sum_{x,y \in \mathbb{F}_p} \left[x(x+1)y(xy(y+1)+\lambda )\right]^D \ .
\end{align*}
The transformation $x \to \frac{x}{y}$ then gives us the counting function of the family of surfaces
\begin{equation}\label{Xlambda4}
X_{\lambda,4}:\quad\quad z^2 = x(x+y)y\Big{(}x(y+1) +\lambda\Big{)}\ .
\end{equation}
\vspace{3pt}
\item If $b=6$, then we have for $0\leq n\leq D$ that
\[
\left(-27\right)^{n}\frac{1}{(n!)^{3}}\left(\frac{1}{6}\right)_{n}\left(\frac{5}{6}\right)_{n}\left(\frac{1}{2}\right)_{n} \equiv_{p}\ \binom{D}{n}\binom{D-n}{n}\binom{D-2n}{n}\ ,
\]
so that Lemma \ref{pre2} gives
\begin{align*}
{}_{3}^{}F_2^{(p)}\left(\frac{1}{6},\frac{5}{6},\frac{1}{2};1,1;-27\lambda\right)& \equiv_p \sum_{n=0}^D \binom{D}{n} \left(\sum_{x \in \mathbb{F}_p} x^{2D-n} (x+1)^{D-n} \right) \left( \sideset{}{^*}\sum_{y \in \mathbb{F}_p} y^{2D-n} (y+1)^{D-2n} \right) \lambda^n\\&\equiv_p \sideset{}{^*}\sum_{x,y \in \mathbb{F}_p} \left[xy\left(xy(x+1)(y+1)+\frac{\lambda}{y+1} \right)\right]^D \ ,
\end{align*} where the $\sideset{}{^*}\sum$ denotes the sum over all $y\neq -1$. After some transformations similar to the case where $b=4$, this is reduced to the counting function of the family of surfaces \begin{equation}\label{Xlambda6}
X_{\lambda,6}:\quad\quad z^2 = xy\Big{(}x(x+y+1) +\lambda y^{3}\Big{)}\ .
\end{equation}
\end{enumerate}
There is an algebraic criterion given in \cite[\S 4]{StBeuk} for determining that a surface $\mathcal{K}$ is K3. For the affine Weierstrass equation $$\mathcal{K}: \quad\quad z^2 + a_1 xz + a_3 z = x^3 + a_2 x^2 + a_4 x + a_6,$$ where each $a_i = a_i(y) \in \mathbb{F}_p[y]$, the {\it Weierstrass $g$-invariants} are defined by the identities \begin{enumerate}[(a)] \item $g_2 := \frac{1}{12}\left[(a_1^2 + 4a_2)^2 - 24(a_1 a_3 + 2 a_4)\right]$ \item $g_3 := \frac{1}{216}\left[-(a_1^2 + 4a_2)^3+ 36(a_1^2 + 4a_2)(a_1 a_3 + 2a_4) - 216(a_3^2 + 4a_6)\right]$. \end{enumerate} Precisely, the condition that $\mathcal{K}$ is a K3 surface amounts to satisfying all of the following criteria:
\begin{enumerate}[(1)] \item The discriminant $\Delta:= \Delta(y) = g_2^3 - 27 g_3^2$ is not a constant in $\mathbb{F}_p[y]$. \item $\deg a_i(y) \leq Ni$, and $N=2$ is the smallest such integer such that this inequality is satisfied for all $i=1,\ldots,6$. \item Neither $\gcd(g_2(y)^3,g_3(y)^2)$ nor $\gcd(y^{12N} g_2(y^{-1})^3,y^{12N}g_3(y^{-1})^2)$ is divisible by a $12$th power of a non-constant polynomial in $\mathbb{F}_p[y]$. \end{enumerate}
For each of the families $\{X_{\lambda,b}\}$ $(b=3,4,6)$, the value $\lambda = 0$ will be implicitly excluded. As mentioned in \cite{Ono}, it is already known that the family $\{X_{\lambda,2}\}$ is K3. We also have the following:
\begin{proposition} Each of the families $\{X_{\lambda,b}\}$ $(b=3,4,6)$ is K3. \end{proposition}
\begin{proof} Each of these families is in simplified Weierstrass form, hence the $g$-invariants may be written in simplified form as \begin{enumerate}[(a')] \item $g_2 = \frac{1}{12}\left[(4a_2)^2 - 24(2 a_4)\right] = \frac{4}{3}a_2^2 - 4a_4$ \item $g_3 = \frac{1}{216}\left[-(4a_2)^3+ 36(4a_2)(2a_4) - 216(4a_6)\right] = - \frac{8}{27} a_2^3 + \frac{4}{3} a_2 a_4 - 4 a_6$. \end{enumerate}
\begin{enumerate}[(I)] \item If $b=3$, then $$X_{\lambda,3} : z^2 = x(x+1)y(xy+\lambda(y+1)^2) = y^2 x^3 + (y^2 + \lambda y(y+1)^2) x^2 + \lambda y(y+1)^2 x .$$ We perform the transformation $z \rightarrow zy$ to obtain $$z^2 = x^3 + \left(1 + \lambda \frac{y(y+1)^2}{y^2}\right)x^2 + \lambda \frac{y(y+1)^2}{y^2} x= x^3 + \left(1 + \lambda \frac{(y+1)^2}{y}\right) x^2 + \lambda \frac{(y+1)^2}{y}x .$$ We now transform $z \rightarrow y^{-3}z$, $x \rightarrow y^{-2}x$. This yields $$y^{-6} z^2 = y^{-6} x^3 + y^{-4} \left(1 + \lambda \frac{(y+1)^2}{y}\right)x^2 + y^{-2}\lambda \frac{(y+1)^2}{y}x,$$ whence $$z^2 =x^3 + \left(y^2 + \lambda {y(y+1)^2}\right) x^2 + \lambda y^3{(y+1)^2}x.$$ Therefore $a_2 = y^2 + \lambda {y(y+1)^2}$, $a_4 = \lambda y^3{(y+1)^2}$, and $a_6 = 0$. Thus the degree condition (2) is satisfied when $\lambda \neq 0$. The $g$-invariants are \begin{enumerate}[(i)] \item $g_2 = \frac{4}{3}a_2^2 - 4a_4 = \frac{4}{3}\lambda^2 y^6 + \frac{4}{3}(4\lambda^2 -\lambda) y^5 + 4(2\lambda^2 - \frac{2}{3}\lambda + \frac{1}{3})y^4 + \frac{4}{3}(4\lambda^2-\lambda)y^3 + \frac{4}{3}\lambda^2 y^2.$ \item $g_3 = - \frac{8}{27} a_2^3 + \frac{4}{3} a_2 a_4 = -\frac{8}{27}\lambda^3 y^9 +\frac{4}{9}(-4\lambda^3 + \lambda^2) y^8 + \frac{4}{9}(-10\lambda^3+4\lambda^2 + \lambda) y^7 + \frac{8}{3}(-\frac{20}{9}\lambda^3 + \lambda^2 + \frac{1}{3}\lambda -\frac{1}{9})y^6 + \frac{4}{9}(-10\lambda^3 + 4\lambda^2 + \lambda) y^5 + \frac{4}{9}(-4\lambda^3 + \lambda^2) y^4 - \frac{8}{27} \lambda^3 y^3$.
\end{enumerate} If follows that $$\Delta(y) = 16 \lambda^2 y^8 (y+1)^4 (\lambda y^2 + (2\lambda-1)y +\lambda)^2,$$ which is non-constant in $\mathbb{F}_p[y]$ if $p > 2$. Hence condition (1) is satisfied. We may write $$g_2^3 = \frac{64}{27} y^6 \left[z^2 + y^4 z^2 + y z (-1 + 4 z) + y^3 z (-1 + 4 z) + y^2 (1 - 2 z + 6 z^2)\right]^3$$ and \begin{align*} g_3^2 &= \frac{16}{729} y^6 \Big[2 z^3 + 2 y^6 z^3 + 3 y z^2 (-1 + 4 z) + 3 y^5 z^2 (-1 + 4 z) + \\& 3 y^2 z (-1 - 4 z + 10 z^2) + 3 y^4 z (-1 - 4 z + 10 z^2) + 2 y^3 (1 - 3 z - 9 z^2 + 20 z^3)\Big]^2.\end{align*} As $y^6$ is a common factor of $g_2$ and $g_3$, inspection of the terms of lowest and highest degree in $y$ yields (respectively) the first and second parts of condition (3) when $\lambda \neq 0$.
\item If $b=4$, then $$X_{\lambda,4} : z^2 = x(x+y)y(x(y+1)+\lambda) = y(y+1) x^3 + \left(y^2(y+1)+ \lambda y\right) x^2 + \lambda y^2 x.$$ The transformation $x \rightarrow x(y(y+1))^{-1}$ and $z \rightarrow z(y(y+1))^{-1}$ yields $$(y(y+1))^{-2} z^2= (y(y+1))^{-2} x^3 + (y(y+1))^{-2} \left(y^2(y+1)+ \lambda y\right) x^2+ (y(y+1))^{-1} \lambda y^2 x.$$ Thus $$z^2 = x^3 + \left(y^2(y+1)+ \lambda y\right) x^2 + \lambda y^3 (y+1)x.$$ Therefore $a_2 = y^2(y+1)+ \lambda y$, $a_4 = \lambda y^3 (y+1)$, and $a_6 = 0$. In particular, the degree condition (2) is satisfied for {\it any} $\lambda \in \mathbb{F}_p$. The $g$-invariants are \begin{enumerate}[(i)] \item $g_2 = \frac{4}{3}a_2^2 - 4a_4 = \frac{4}{3} y^6 + \frac{8}{3} y^5 + \frac{4}{3}(1-\lambda)y^4 - \frac{4}{3}\lambda y^3 + \frac{4}{3} \lambda^2 y^2$ \item $g_3 = - \frac{8}{27} a_2^3 + \frac{4}{3} a_2 a_4 - 4 a_6 = -\frac{8}{27} y^9 -\frac{8}{9} y^8 +\frac{4}{9} (\lambda - 2) y^7 + \frac{8}{27} (3\lambda-1) y^6 + \frac{4}{9}\lambda(\lambda+1) y^5 + \frac{4}{9} \lambda^2 y^4 - \frac{8}{27} \lambda^3 y^3$. \end{enumerate} It follows that $$\Delta(y) = 16 \lambda^2 y^8 (y+1)^2 (y^2+y - \lambda)^2,$$ which is non-constant in $\mathbb{F}_p[y]$ if $p > 2$. This is condition (1). As with $b=4$, we may write $$g_2^3 = \frac{64}{27} y^6 (2 y^3 + y^4 - y^2 (-1 + z) - y z + z^2)^3$$ and $$g_3^2 = \frac{16}{729} y^6 \Big[6 y^5 + 2 y^6 + y^3 (2 - 6 z) - 3 y^4 (-2 + z) -
3 y z^2 + 2 z^3 - 3 y^2 z (1 + z)\Big]^2.$$ The analysis is as in the case $b=3$ for $\lambda \neq 0$.
\item If $b=6$, then $$X_{\lambda,6}: z^2 = xy(xy(x+y+1) + \lambda) = y^2 x^3 + y^2(y+1) x^2 + y \lambda x.$$ We transform $z \rightarrow zy$ to obtain $$z^2 = x^3 + (y+1) x^2 + \lambda \frac{1}{y} x .$$ We now perform the same transformation as in $X_{\lambda,3}$ to obtain $$z^2 = x^3 + y^2(y+1) x^2 + \lambda y^3 x .$$ \end{enumerate} This yields $a_2 = y^2(y+1)$, $a_4 = \lambda y^3$, and $a_6 = 0$, whence the degree condition is again satisfied for any $\lambda \in \mathbb{F}_p$. The $g$-invariants are \begin{enumerate}[(i)] \item $g_2 = \frac{4}{3}a_2^2 - 4a_4 = \frac{4}{3} y^6 + \frac{8}{3} y^5 + \frac{4}{3} y^4 - 4\lambda y^3$ \item $g_3 = - \frac{8}{27} a_2^3 + \frac{4}{3} a_2 a_4 = -\frac{8}{27} y^9- \frac{8}{9} y^8 -\frac{8}{9} y^7 + \frac{4}{3}(\lambda - \frac{2}{9}) y^6 + \frac{4}{3}\lambda y^5 $. \end{enumerate} It follows that $$\Delta(y) = 16 \lambda^2 y^9 (y^3 + 2y^2 + y - 4\lambda),$$ which is non-constant in $\mathbb{F}_p[y]$ if $p > 2$, which yields condition (1). We write $$g_2^3 = \frac{64}{27} y^9 (y + 2 y^2 + y^3 - 3 z)^3$$ and $$g_3^2 = \frac{16}{729} y^{10} (1 + y)^2 (2 y + 4 y^2 + 2 y^3 - 9 z)^2.$$ As in both of the previous cases, condition (3) is satisfies provided that $\lambda \neq 0$.
\end{proof}
We now denote by $J_{\lambda,2}$ the counting function for members of the K3 family $\{X_{\lambda,2}\}$. A suitable relationship between $J_{\lambda,2}$ and the counting function $I_\lambda$ for members of the family $\{E_\lambda\}$ \eqref{Elambda} will allow us to count the number of rational points of $\{X_{\lambda,2}\}$ in Section 7. Indeed, we have the following:
\begin{corollary}\label{Clausen} If $\lambda \in \mathbb{F}_p\backslash\{0,-1\}$, then the following relation holds:
\[
J_{\lambda,2} \equiv_p (1+\lambda)^D I_\lambda^2.
\]
\end{corollary}
\begin{proof}
If $p \equiv_{4} 3$, then $E=\frac{p-3}{4}$. By Proposition \ref{10b} and Lemma \ref{4a}, we obtain
\[
(1+\lambda)^{D-1} I_\lambda^2 \equiv_{p} \ {}_{2}^{}F^{(p)}_1\left(\frac{3}{4},\frac{3}{4},1;-\lambda\right)^2\ .
\]
Therefore, we must show that
\[
_{3}^{}F_2^{(p)}\left(\frac{1}{2},\frac{1}{2},\frac{1}{2};1,1;\lambda\right) \equiv_{p}\ (1+\lambda)\ {}_{2}^{}F^{(p)}_1\left(\frac{3}{4},\frac{3}{4},1;\lambda\right)^2.
\] This is analogous to Clausen's formula combined with Euler's formula (see Section 4).
The case $p \equiv_4 1$ follows in a similar way. \end{proof}
\begin{remark}\label{rem1}
We have shown here that Corollary \ref{Clausen} is a consequence of Clausen's formula. In fact, it is equivalent to it in the following sense: If the Proposition is true for all primes $p\geq 3$, then formula \eqref{5a2} holds for $0 \leq m \leq D$, and Clausen's formula follows. To prove Clausen's formula, it suffices to show that for any $m\geq 0$
\begin{equation}\label{5a3}
\left(\frac{\left(\frac{1}{2}\right)_m}{m!}\right)^3 = \sum_{\substack{u,v, \geq 0\\ u + v = m}} \left(\frac{\left(\frac{1}{4}\right)_u \left(\frac{1}{4}\right)_v}{u!v!}\right)^2 .
\end{equation}
For any fixed $m\geq 2$, let $p$ be a prime number exceeding $2m$, so that for this choice of $m$, \eqref{5a2} holds. Then as discussed in Section 4.6 (see \eqref{5a2}), we may remove the condition $u,v \leq E$, and we conclude that $p$ divides the difference in \eqref{5a3} for all primes exceeding $2m$, so that \eqref{5a3} holds.
Since Corollary \ref{Clausen} is equivalent to Clausen's formula, it is worthwhile to give a proof of this which is independent of Clausen's formula. This was done in \cite{Ono} using character sums. We provide another independent proof of this in the Appendix.
\end{remark}
\section{Elliptic families and Deuring's theorem}
In Section 1, we discussed the Hasse polynomial $$H_p(\lambda):= _{2}^{}F^{(p)}_{1}\left(\half,\half;1,\lambda\right) = \sum_{n=0}^{D} \binom{D}{n}^2 \lambda^n.$$ In his study of splitting, Deuring \cite{Deuring} determined the isogeny class of an elliptic curve over $\mathbb{F}_p$ via its endomorphism ring (which was generalised by Waterhouse to abelian varieties). Via the $j$-invariant, the number of elliptic curves in a given isomorphism class was determined according to formulae originally obtained by Eichler for Hurwitz-Kronecker class numbers. The value sets of $H_p(\lambda)$ correspond to isomorphism classes of elliptic curves over $\mathbb{F}_p$.
We first estimate the number of curves belonging to a single $\mathbb{F}_p$-isomorphism class in the family $\{E_\lambda\}$ of \eqref{Elambda}, which is associated with the K3 family $\{X_{\lambda, 2}\}$. We then do the same for each of the families $\{E_{\lambda,b}\}$ ($b=2,3,4,6$) of Section 5. In particular, this allows us to obtain estimates for the number of $\mathbb{F}_p$-rational points for the families $\{X_{\lambda,b}\}$, for $b=2,3,4,6$, via estimates of Lenstra \cite{Lenstra} for the number of curves with a given number of $\mathbb{F}_p$-rational points.
\begin{enumerate}[(I)] \item $\{E_\lambda\}$: If $\lambda_1, \lambda_2 \in \mathbb{F}_p \backslash \{0,-1\}$, then we perform the variable change $(x,y) \rightarrow (x+1,y)$ to obtain the form
\[
E_{\lambda_i}: y^2 = x\left(x^2 + 2x + \frac{\lambda_i}{\lambda_i+1}\right)\;\;\;\;\;\;\;\;\;\;\; (i=1,2).
\]
This yields the Weierstrass form
\[
E_{\lambda_i}: y^2 = x^3 + 2x^2 + \frac{\lambda_i}{\lambda_i+1}x \;\;\;\;\;\;\;\;\;\;\; (i=1,2).
\]
The elliptic curves $E_{\lambda_1}$ and $E_{\lambda_2}$ are isomorphic over $\mathbb{F}_p$ precisely when there exist $a,b,c,d \in \mathbb{F}_p$ with $a \neq 0$, so that the change of variables
\[
(x,y) \rightarrow (a^2 x + b,a^3 y + a^2 c x + d)
\]
transforms the equation of $E_{\lambda_1}$ into that of $E_{\lambda_2}$ \cite{Silverman}. As $E_{\lambda_1}$ and $E_{\lambda_2}$ lack any linear terms in $y$ (i.e., the curves are in simplified Weierstrass form), this criterion reduces to finding $a,b \in \mathbb{F}_p$, again with $a \neq 0$, which satisfy the three identities
\begin{equation}\label{list1}
2 a^2 = 2 +3b,
\end{equation}
\begin{equation}\label{list2}
\frac{a^4 \lambda_2}{\lambda_2+1} = \frac{\lambda_1}{\lambda_1+1} + 4b + 3b^2
\end{equation}
\begin{equation}\label{list3}
0 = \frac{ b \lambda_1}{\lambda_1+1} + 2 b^2 + b^3.
\end{equation}
In particular, equation \eqref{list3} implies that $b=0$ or $b = - 1 \pm \sqrt{1 - \frac{\lambda_1}{\lambda_1 + 1}}$. To simplify notation, we denote these values of $b$ as $0,\alpha,\beta$. \begin{enumerate}[(I)] \item If $b=0$, then by equation \eqref{list1}, we find $a = \pm 1$, and hence by equation \eqref{list2}, we obtain $\frac{\lambda_2}{\lambda_2 + 1} = \frac{\lambda_1}{\lambda_1 + 1}$, so that $\lambda_1 = \lambda_2$.
\item If $b \in \{\alpha,\beta\}$, then equation \eqref{list1} uniquely determines $a^2$ in terms of $b$. Application of equation \eqref{list1} to \eqref{list2} yields
\begin{align*}
\frac{1}{4}\left[4 - \frac{\lambda_1}{\lambda_1+1} + 2b\right]\lambda_2 & = \frac{1}{4}\left[4 + 4b + b^2 \right]\lambda_2 \\ & = \left[1 + b + \frac{1}{4}b^2 \right] \lambda_2 \\ & = \left[\left(1+ \frac{3b}{2}\right)^2 - 2(b+b^2)\right] \lambda_2 \\ & = \left[\left(1 + \frac{3b}{2}\right)^2 - \left(\frac{\lambda_1}{\lambda_1+1} + 4b + 3b^2\right)\right] \lambda_2 \\& = \frac{\lambda_1}{\lambda_1+1} + 4b + 3b^2.
\end{align*}
This gives a unique value for $\lambda_2$ in terms of $\lambda_1$ and the value of $b$, except if $4 - \frac{\lambda_1}{\lambda_1+1} + 2b = 0$. As $b \in \{\alpha,\beta\}$, we obtain $$4 - \frac{\lambda_1}{\lambda_1+1} + 2\left(-1 \pm \sqrt{1 - \frac{\lambda_1}{\lambda_1 + 1}}\right) = 0.$$ This implies that $( \frac{\lambda_1}{\lambda_1+1})^2 = 0$, and hence that $\lambda_1 = 0$. This is impossible, as by definition, $\lambda \in \mathbb{F}_p \backslash \{0,1\}$ for each member of the family $\{E_\lambda\}$. Thus $\lambda_2$ is uniquely determined by $\lambda_1$ and the value of $b$. \end{enumerate} It follows that each $E_{\lambda_1} \in \{E_\lambda\}$ is $\mathbb{F}_p$-isomorphic to at most two other curves in the family $\{E_\lambda\}$.
\item $\{E_{\lambda,b}\}$, $b=2$: This is precisely the family of Legendre elliptic curves. Thus according to the $j$-invariant, there exist at most $6$ elements of this family in a given $\mathbb{F}_p$-isomorphism class. See also \cite{FengWu}, which calculates precisely the number of classes of $\mathbb{F}_p$-isomorphic curves for this family (their calculation is also valid over $\mathbb{F}_q$).
\item $\{E_{\lambda,b}\}$, $b=3$: This family is given by $$E_{\lambda,3}: y^2 = x^3 + (x+\lambda)^2 = x^3 + x^2 + 2\lambda x + \lambda^2.$$ As with the family $\{E_\lambda\}$, the members of the family $\{E_{\lambda,3}\}$ are already in simplified Weierstrass form, whence $E_{\lambda_1,3}$ is isomorphic to $E_{\lambda_2,3}$ if, and only if, there exist $a,b \in \mathbb{F}_p$ such that $a \neq 0$,\begin{equation}\label{first}a^2 = 1 + 3b,\end{equation}
\begin{equation}\label{second} 2a^4\lambda_2 = 2\lambda_1 +2b + 3b^2,\end{equation} and
\begin{equation}\label{third}a^6 \lambda_2^2 = \lambda_1^2 + 2b\lambda_1 + b^2 + b^3.\end{equation} By equation \eqref{first}, $a = \pm \sqrt{1+3b}$, and the equations \eqref{second} and \eqref{third} may therefore be read as \begin{equation}\label{quadratic1} (2\lambda_2 - 3)b^2 + (12\lambda_2-2)b+2(\lambda_2 - \lambda_1)=0\end{equation} and
\begin{equation}\label{cubic1} (27\lambda_2^2 - 1)b^3 +(27\lambda_2^2 - 1)b^2 +(9\lambda_2^2 - 2\lambda_1) b +(\lambda_2^2 - \lambda_1^2) = 0.\end{equation} The intersection of the quadratic \eqref{quadratic1} and cubic \eqref{cubic1} imply that $b$ is either zero (whence $\lambda_2 = \lambda_1$) or any root of the sextic polynomial \begin{align*} f(z) =& 239 z^6+ 515 z^5+ (316 \lambda_1+ 313) z^4 -(4 \lambda_1^2 - 444 \lambda_1- 25 ) z^3 \\& + (60 \lambda_1^2+ 20 \lambda_1-4) z^2 - ( 44 \lambda_1^2+12 \lambda_1) z -12 \lambda_1^2,\end{align*} and that $\lambda_2$ may be expressed uniquely in terms of $b$ and $\lambda_1$ according to
\begin{align*}\lambda_2 = &\frac{1}{128 (4 \lambda_1^2+ 180 \lambda_1 -23)} \\& \qquad\qquad\qquad\times\Bigg\{ 239 ( 34 \lambda_1+1533) b^5+ 2 (8038 \lambda_1+363319 ) b^4 \\& \;\;\;\;\qquad\qquad\qquad+ (10744 \lambda_1^2+ 492458 \lambda_1 +355139 ) b^3 \\& \;\;\;\;\qquad\qquad\qquad - 4 (34 \lambda_1^3- 1767 \lambda_1^2 - 149028 \lambda_1+5623 ) b^2 \\& \;\;\;\;\qquad\qquad\qquad + 4 (516 \lambda_1^3+ 22920 \lambda_1^2 - 16629 \lambda_1-736) b\\&\;\;\;\;\qquad\qquad\qquad -4 \lambda_1 (466 \lambda_1^2+ 21045 \lambda_1+736 )\Bigg\}.\end{align*} It follows that there exist at most 13 curves in this family belonging to the isomorphism class of a particular $E_{\lambda_1,3}$ over $\mathbb{F}_p$: the curve $E_{\lambda_1,3}$ and its transformations, corresponding to the values of $b$ given by the above sextic and $a = \pm \sqrt{1+3b}$.
\item $\{E_{\lambda,b}\}$, $b=4$: This family is given by \[
E_{\lambda,4} : \qquad\quad z^{2} = x\big(x(x+1)+\lambda\big) = x^3 + x^2 + \lambda x.\
\] As when $b=2,3$, members of the family $\{E_{\lambda,4}\}$ are in simplified Weierstrass form. Two curves $E_{\lambda_1,4}$ and $E_{\lambda_2,4}$ ($\lambda_1,\lambda_2\neq 0$) are $\mathbb{F}_p$-isomorphic if, and only if, there exist $a,b \in \mathbb{F}_p$ with $a \neq 0$ so that \begin{equation}\label{first2} a^2 = 1 + 3b,\end{equation} \begin{equation}\label{second2} a^4 \lambda_2 = \lambda_1 + 2b + 3b^2, \end{equation} and \begin{equation}\label{third2}0=\lambda_1 b + b^2 + b^3.\end{equation} Together, equations \eqref{first2} and \eqref{second2} imply that \begin{equation}\label{quadratic2} (1 + 3b)^2 \lambda_2 = \lambda_1 + 2b + 3b^2.\end{equation} The intersection of the cubic \eqref{third2} and quadratic \eqref{quadratic2} yield solutions $\lambda_2 = \lambda_1$ and $$\lambda_2 = \frac{1 -13\lambda_1 + 36\lambda_1^2 \pm \sqrt{1 - 4\lambda_1}(1 - 3\lambda_1)}{2(2-9\lambda_1)^2}.$$ It follows that there are at most 3 curves in this family belonging to the isomorphism class of a particular $E_{\lambda_1,4}$ over $\mathbb{F}_p$.
\item $\{E_{\lambda,b}\}$, $b=6$: This family is given by \[
E_{\lambda,6} : \qquad\quad z^{2} = x^{2}(x+1)+\lambda\ = x^3 + x^2 + \lambda.
\] As in previous cases, two members $E_{\lambda_1,6}$ and $E_{\lambda_2,6}$ of this family are isomorphic over $\mathbb{F}_p$ if, and only if, there exist $a,b \in \mathbb{F}_p$ with $a \neq 0$, so that \begin{equation*}a^2 = 1 +3b,\;\;\; 0=2b + 3b^2,\;\;\; a^6 \lambda_2 = \lambda_1 +b^2 + b^3.\end{equation*} Together, these imply that $\lambda_2 = \lambda_1$ or $\lambda_2 = -\frac{4}{27}-\lambda_1$. It follows that there exist at most two members of the family $\{E_{\lambda,6}\}$ in a given $\mathbb{F}_p$-isomorphism class.
\end{enumerate}
For a family $\mathcal{F}$ of elliptic curves, we denote by $C_{\mathbb{F}_p}(\mathcal{F})$ the number of elements of $\mathcal{F}$ in a given $\mathbb{F}_p$-isomorphism class. By the previous arguments, we have obtained the following:
\begin{lemma}\label{isomorphismclasses} \begin{enumerate}[(I)] \item The elliptic family $\{E_\lambda\}$ of \eqref{Elambda} satisfies $C_{\mathbb{F}_p}(\{E_\lambda\}) \leq 3$. \item The elliptic families $\{E_{\lambda,b}\}$ $(b=2,3,4,6)$ satisfy \begin{enumerate}[(i)] \item $C_{\mathbb{F}_p}(\{E_{\lambda,2}\}) \leq 6$, \item $C_{\mathbb{F}_p}(\{E_{\lambda,3}\}) \leq 13$, \item $C_{\mathbb{F}_p}(\{E_{\lambda,4}\}) \leq 3$, and \item $C_{\mathbb{F}_p}(\{E_{\lambda,6}\}) \leq 2$. \end{enumerate} \end{enumerate} \end{lemma}
As noted in Section 1, one might be able to prove even sharper bounds than those given in Lemma \ref{isomorphismclasses}. We now give the proof of Theorem \ref{onehalf}.
\begin{proof}[Proof of Theorem \ref{onehalf}] By Proposition 1.9 of \cite{Lenstra}, following Deuring \cite{Deuring}, if $a \in \mathbb{Z}$ satisfies $|a - (p+1)| \leq 2\sqrt{p}$, then
\[
\bigg| \left\{ \text{Elliptic curves}\ E\ \text{such that}\ E(\mathbb{F}_p) = a \right\} / \sim \mathbb{F}_p \bigg| \leq c \sqrt{p}\log{p} (\log\log{p})^2,
\]
for an effective constant $c$. Let $\alpha = \frac{1}{b}$, with $b=2,3,4,6$. By the arguments of section 5, we have, up to multiplication by $-1$ and a constant multiplier on $\lambda$ depending only on $b$, that $_{2}^{}F_1^{(p)}\left(\alpha,1-\alpha;1;\lambda \right)$ is equivalent modulo $p$ to the counting function for $E_{\lambda,b}$. Hence the result of Deuring and Lenstra may be applied to the value sets of $_{2}^{}F_1^{(p)}\left(\alpha,1-\alpha;1;\lambda \right)$. By Lemma \ref{isomorphismclasses}, the number of elements of $\{E_{\lambda,b}\}$ in any $\mathbb{F}_p$-isomorphism class is bounded by an absolute (and effective) constant. Thus the value sets of $_{2}^{}F_1^{(p)}\left(\alpha,1-\alpha;1;\lambda \right)$ are bounded by $c' \sqrt{p}\log{p} (\log\log{p})^2$, for an effective constant $c'$ depending at most on $b$. \end{proof}
We may now give a proof of Corollary \ref{onehalfone}.
\begin{proof}[Proof of Corollary \ref{onehalfone}]\
\begin{enumerate}[(I)]
\item \quad Using Proposition \ref{K3}, with $x$ replaced with $4x(1-x)$ in Proposition \ref{10e}, we find that
\[
_{3}^{}F^{(p)}_{2}\left(\frac{a}{b},1-\frac{a}{b},\frac{1}{2};1,1;4x(1-x)\right) \equiv_{p}\ {_{2}^{} F^{(p)}_1} \left(\frac{a}{b},1-\frac{a}{b},1;4x(1-x)\right)^{{2}}.
\] Then the conclusions follow directly from Theorem \ref{onehalf}.
\vspace{5pt}
\item \quad By combining Corollaries \ref{10c} and \ref{10f}, the conclusion for the congruence $Q \equiv_{p}\ 0$ follows from the Theorem (using the fact that $\frac{x}{1-x}$ is invertible if $x \neq 0,\ 1$ modulo $p$, which we assume henceforth). For the congruence $Q(\alpha) \equiv_{p}\ m$ we have that $K=\frac{p-1}{b}$ if $p \equiv_{b}\ 1$, so that raising everything to the $b$th power in the Corollaries give us
\[ {_{2}^{} F^{(p)}_1} \left(\frac{1}{b},1- \frac{1}{b},1;-\frac{\alpha}{1-\alpha}\right)^{b} \equiv_{p}\ Q\left(\alpha\right)^{b} \equiv_{p}\ m^{b}\ .
\]
The result follows from Theorem \ref{onehalf} since $b$ is bounded.
\vspace{5pt}
\item \quad This follows directly from Proposition \ref{10e} and the Theorem.
\vspace{5pt}
\item \quad If $p \equiv_{2b}\ 1$, we have $K=\frac{p-1}{2b}$ in Proposition \ref{10k1}. Replacing $x$ by $1- x^{2}$ in Proposition \ref{10k1} and $x$ by $\frac{x+1}{2x}$ in Proposition \ref{10e} gives us
\[{}
{_{2}^{} F^{(p)}_1} \left(\frac{1}{b},\frac{1}{2},1;1-x^{2}\right) \equiv_{p}\ x^{2K}\ {_{2}^{} F^{(p)}_1} \left(\frac{1}{b},1-\frac{1}{b},1;\frac{x+1}{2x}\right) \ .
\]
Raising to the $b$th power and using the argument of part (II) gives the result for $Q \equiv_{p}\ m$, with $m\neq 0$. When $m=0$, the result follows from combining Propositions \ref{10k1} and \ref{10e} without raising to powers.
\end{enumerate}
\end{proof}
\begin{remark}
It is also possible to interpret $J_{\lambda ,2}$, the counting function for the $K3$ family $X_{\lambda,2}$ (discussed in Section 6) in terms of $I_\lambda$ via zeta functions. For example, in Theorem 4.2 of \cite{Ono}, it is shown that
\[
J_{\lambda,2} = 1 + p + 19p + \left( \frac{\lambda+1}{p}\right)_L (\pi^2 + \overline{\pi}^2 + p) \;\;\;\;\text{and}\;\;\;\;I_\lambda = (p+1)-(\pi + \overline{\pi}),
\]
where $\pi^{-1}$ is a root of the zeta function $Z(E_\lambda/\mathbb{F}_p,T)$, and that each member of $\{E_\lambda\}$ has good reduction at $p$. As $\pi \overline{\pi} = p$, it follows that $\pi^2 + \overline{\pi}^2 \equiv_p\ (\pi + \overline{\pi})^2 \equiv_p (I_\lambda-1)^2 $. (Here, $\left(\frac{\lambda+1}{p}\right)_L$ denotes the Legendre symbol). Therefore
\[
J_{\lambda,2} \equiv_p 1 + \left(\frac{\lambda+1}{p}\right)_L (I_\lambda-1)^2.
\]
This implies that $J_{\lambda,2} \equiv_p 1 $ precisely when $I_\lambda \equiv_p 1 $. Thus for any $a \in \mathbb{F}_p$, it follows that $J_{\lambda,2} \equiv_p a $ if, and only if,
\[
\left(\frac{\lambda+1}{p}\right)_L (I_\lambda-1)^2 \equiv_p a-1 .
\]
The latter equivalence implies that $(I_\lambda-1)^4 \equiv_p (a-1)^2 $. This implies that for any such $a$, there exist at most 4 numbers $a_1,\ldots,a_4 \in \mathbb{F}_p$ so that $I_\lambda \equiv_p a_i $. It is therefore natural to ask the when the surfaces $X_{\lambda,b}$ are modular for $b=3,4,6$, and also whether the rank of the N\'{e}ron-Severi group can be explicitly calculated. We do not pursue this question here.
\end{remark}
\section{Hypergeometric $E$-functions}
A relative of this argument may be performed for hypergeometric E-functions. For example, the classical Kummer hypergeometric function is defined as $$K_{\nu,\lambda}(z) = \sum_{n=0}^\infty \frac{(\nu)_n}{(\lambda)_n} \frac{z^n}{n!} \quad\quad(\lambda \neq 0,-1,-2,...),$$ which satisfies the second-order differential equation $zy'' + (\lambda - z)y' - \nu y = 0$. For an element $\alpha \in \mathbb{F}_q$ ($q = p^m$), the Pochhammer product is defined as $$(\alpha)_n = \alpha(\alpha+1) \cdots (\alpha + n-1).$$ Let $\nu \in \mathbb{F}_q$, and let $\lambda$ be a rational number or an element of $\mathbb{F}_q$. The degree $N^*_{\nu,\lambda}$ of the truncation $K_{\nu,\lambda}^{(p)}(x)$ depends upon the values of $\lambda$ and $\nu$, and takes the form $$N^*_{\nu,\lambda} = \frac{\omega_{\nu,\lambda} p - a_{\nu,\lambda}}{b_{\nu,\lambda}}.$$ We let $\eta(z) = K_{\nu,\lambda}(z^{b_{\nu,\lambda}})$. Then ${\eta^{(p)}}(x) = K_{\nu,\lambda}^{(p)}(x^{b_{\nu,\lambda}})$ has degree equal to $\omega_{\nu,\lambda} p - a_{\nu,\lambda}$. From the differential equation for the function $\eta(z)$ and inspection of degrees, one obtains that ${\eta^{(p)}}(x)$ is a solution to the differential equation \begin{equation}\label{KummerDE} \frac{x}{b_{\nu,\lambda}} y'' + \left[(\lambda - x^b_{\nu,\lambda}) - \frac{b_{\nu,\lambda}-1}{b_{\nu,\lambda}}\right] y' - b_{\nu,\lambda} x^{b_{\nu,\lambda}-1} \nu y \equiv_p 0 \mod x^{p - a_{\nu,\lambda}-1}. \end{equation}
\begin{proof}[Proof of Theorem \ref{KummerE}] The function ${\eta^{(p)}}(x)$ is a solution to equation \eqref{KummerDE} multiplied by $x^{a_{\nu,\lambda}+1}$ (a small degree relative to $p$) modulo $x^p$. By Lemma 5.1 of \cite{GhWa}, if $(m+1)n^2 < p$, then the function ${\eta^{(p)}}(x)$ does not satisfy an equation $$a_n(x) T^n + \cdots + a_1(x) T + a_0(x) \equiv_p 0 \mod x^p,$$ where $a_i(x) \in \overline{\mathbb{F}}_p[x]$ and $\deg(a_i(x)) \leq m$ for all $i=1,\ldots,n$ \cite[Lemma 5.1]{GhWa}. By Siegel's argument (see \S II.10 of \cite{Siegel}, Theorem 1.3 of \cite{GhWa}, or Lemma 2 of \cite{Shidlovskii}), it is not hard to show that if a polynomial $P$ in two variables over $\overline{\mathbb{F}}_p[x]$ is of sufficiently small degree such that $P({\eta^{(p)}}'(x),{\eta^{(p)}}(x)) \equiv_p 0 \text{ mod } x^p$, then there exists a solution $y^*$ to the reduced Kummer equation whose logarithmic derivative is algebraic over $\overline{\mathbb{F}}_p(x)$, also of small degree. Precisely, this solution is found by aggregating the (homogeneous) terms of highest total degree in $P$ as a function $H$, and for two linearly independent solutions $y_1, y_2$ to Kummer's differential equation, choosing $y^* = c_1 y_1 + c_2 y_2$ so that $H( c_1 y_1' + c_2 y_2', c_1 y_1 + c_2 y_2) \equiv_p 0$.
Expanding the Riccati equation $$u' + \frac{\lambda - z}{z} u + u^2 \equiv_p \frac{\nu}{z}$$ for $u = {y^*}'/y^*$ around $x = \infty$, which exists as a Puiseux series as we have tamed ramification, gives us that the expression $$u = \sum_{k=0}^\infty c_k x^{r_k}$$ with $r_0 > r_1 > \cdots$ rational powers are in fact integer powers \cite[Lemma 8]{Shidlovskii}. By the Riccati equation, the only possible branch points of $u$ are at $x=0,\infty$, whence $u$ must be a rational function in $\overline{\mathbb{F}}_p(x)$. Let $u = P_0/P_1$, where $P_0,P_1 \in \overline{\mathbb{F}}_p[x]$, $P_1 \not\equiv_p 0$. By construction, the form $R = P_1 w' + P_0 w$ satisfies $R(y^*) \equiv_p 0$ and $x \frac{d}{dx}R(y^*) \equiv_p 0$ at the solution $w = y^*$. By comparison of degrees in $x$, there exist $a,b \in \overline{\mathbb{F}}_p$ such that $$x \frac{d}{dx}R \equiv_p (ax+b)R.$$ This yields two differential equations in terms of $P_0$ and $P_1$: $$P_1' - \left(\frac{\lambda+b}{x}+a-1\right)P_1 + P_0 \equiv_p 0,$$ and $$P_0' - \left(\frac{b}{x}+a\right) P_0 + \frac{\nu}{x}P_1 \equiv_p 0.$$ There are four possible cases: $a\equiv_p 0$ or $a \equiv_p 1$ and $\lambda + b \equiv_p 0$ or $\lambda + b \not\equiv_p 0$. We note that if $a\equiv_p 0$, then $\deg(P_0) = \deg(P_1)$. We let $a_i$ denote the coefficient of $x^i$ for $P_0$, and $b_i$ the same for $P_1$. This pair of differential equations yields the following result.
\begin{enumerate}[(I)] \item If $a \equiv_p 0$ and $\lambda + b \equiv_p 0$, then by the first differential equation, the largest $m<p$ for which the coefficient of $x^m$ in $P_1$ is nonzero is the same as that for $P_0$, and for this $m$, $a_m + b_m \equiv_p 0$. By the second differential equation, we have $\nu -\lambda \equiv_p m$.
\item If $a \equiv_p 0$ and $\lambda + b \not\equiv_p 0$, then $b \equiv_p 0$, as $P_0$ and $P_1$ are relatively prime. With $m$ the same as the previous case, it follows that $\nu \equiv_p m$.
\item If $a \equiv_p 1$ and $\lambda + b \equiv_p 0$, then again for the largest $m < p$ such that the coefficient of $x^m$ in $P_1$, is nonzero, $a_{m-1} + m b_m \equiv_p 0$. As the largest such nonzero coefficient for $P_0$ is equal to $m-1$ by the first differential equation, we have $- a_{m-1} + \nu b_m \equiv_p 0$. Hence $\nu \equiv_p -m$.
\item If $a \equiv_p 1$ and $\lambda + b \not\equiv_p 0$, then by the first differential equation, we have that $x| P_1$. As $P_0$ and $P_1$ are relatively prime, we have as in case (2) that $b \equiv_p 0$. Also by the first differential equation, we have $m b_m - \lambda b_m + a_{m-1} \equiv_p 0$. Also, $-\lambda b_p + a_{p-1} \equiv_p 0$ and, by the second differential equation, $-a_{p-1} + \nu b_p \equiv_p 0$. If $a_{p-1} \not\equiv_p 0$, then $-\lambda \equiv_p -\nu$, whence $\lambda = \nu$. If $a_{p-1} \equiv_p 0$, then as $\lambda,\nu \not\equiv_p 0$, $b_p \equiv_p 0$. It follows that the largest $m < p$ for which the coefficient of $x^m$ in $P_1$ is nonzero is again one more than that for $P_0$, and thus for this $m$ that, by the second differential equation, $- a_{m-1} + \nu b_m \equiv_p 0$. Thus $\lambda b_m - m b_m \equiv_p a_{m-1} \equiv_p \nu b_m$, whence $\lambda - m \equiv_p \nu$. Thus $\lambda - \nu \equiv_p m$.
\end{enumerate} In any case, we have that $\nu \in \mathbb{F}_p$ from (II) and (III) or $\lambda - \nu \in \mathbb{F}_p$ from (I) and (IV). If $\lambda$ and $\nu$ are chosen so that $\nu, \lambda - \nu \notin \mathbb{F}_p$, we obtain a contradiction. We let $A$ denote the maximal degree of the coefficients of $P$ and $B$ the total degree of $P$ in its two variables. Via an auxiliary polynomial $\Phi(x,{\eta^{(p)}}(x),x^p,{\eta^{(p)}}'(x))$, we must choose $C$ and $D$ simultaneously so that $$[2(A+1)B + 1](2B^2)^2 < p \text{ \;\;\;and\;\;\; } D(A+C+2D) < \frac{AC(B+2)}{2}.$$ We let $A = \delta \lfloor p^{2/7} \rfloor$, $B=C=\delta \lfloor p^{1/7} \rfloor$, $D=\delta^3 \lfloor p^{2/7} \rfloor$, and $\varepsilon = \frac{1}{7}$. (See \cite{Heathbrown} or \cite{GhWa}; the denominator of 7 is a general feature of second-order differential equations of this kind.) Hence the number of solutions $\lambda \in \mathbb{F}_p$ to the congruence ${\eta^{(p)}}(\lambda) \equiv_p 0$ is bounded by $(A+(p-1)B+pC)/D \ll_\delta p^{1-\varepsilon}$. \end{proof}
Suppose that $m \geq 2$ and $y_0$ is a solution of minimal degree to an algebraic differential equation of the form \begin{equation}\label{m} y^{(m)} + Q_{m-1}y^{(m-1)} + \cdots + Q_1 y' + Q_0 y = R\end{equation} with $R,Q_0,\ldots,Q_{m-1} \in \mathcal{L}$, a field of analytic functions (over $\mathbb{C}$) which is closed under differentiation. As in \cite[Lemma 11, \S 6]{Shidlovskii}, if $P \in \mathcal{L}[x_1,\ldots,x_m]$ and $P(y_0,y_0',\ldots,y_0^{(m-1)}) = 0$, then there exists a solution $y^*$ to \eqref{m} such that the terms $P_s$ of highest total degree in $P$ satisfy $P_s(y^*,{y^*}',\ldots,{y^*}^{(m-1)})=0$. Theorem \ref{KummerE} is the case $m=2$ over $\mathbb{F}_p$. If $m=3$, then the solution $y^*$ satisfies a Riccati differential equation in terms of $P$. It is very interesting to consider whether results of this type hold for differential equations of order greater than two.
\newpage
\section{Appendix}
Here, we give a proof of Corollary \ref{Clausen} which does not use Clausen's formula. This proof has some features in common with the proof of Theorem 4.2 of \cite{Ono}; however, we dispense with character sums and provide some simplifications.
\begin{proof}[Proof of Corollary \ref{Clausen}] By definition of $J_{\lambda,2}$, we write
\[
J_{\lambda,2} = \sum_{l=0}^D \binom{D}{l} S(2D-l,D)S(D+l,D) \lambda^l.
\]
For all of the terms in this sum with $l \neq 0$, Lemma \ref{pre2} applies, so that we may write
\[
J_{\lambda,2} \equiv_p S(2D,D)S(D,D) + \sum_{1 \leq l \leq D} \binom{D}{l}^3 \lambda^l.
\]
As $S(2D,D) \equiv_p S(D,D) \equiv_p -1$, it follows that $J_{\lambda,2} \equiv_p \sum_{l=0}^D \binom{D}{l}^3 \lambda^l$. We perform the change of variables $\mu = -\frac{\lambda}{1+\lambda}$ ($\mu \neq 0,-1,\infty$, $\lambda \neq 0,-1,\infty$), so that
\[
J_{\lambda,2} \equiv_p (1+\lambda)^D \sum_{l=0}^D \binom{D}{l}^3 (\mu+1)^{D-l}(-\mu)^l.
\]
Let us write $\binom{D}{l}^2 = \binom{D}{l}\binom{D}{D-l}$, and we recognise this product as the coefficient of $x^l y^{D-l}$ of $(1+x)^D (1+y)^D$, and hence the coefficient of $x^D y^D$ of $x^{D-l}(1+x)^D y^l (1+y)^D$. Therefore $J_{\lambda,2} \equiv_p (1+\lambda)^D J_{\lambda,2}^*$, where $J_{\lambda,2}^*$ is the coefficient of $x^D y^D$ of
\[
(1+x)^D (1+y)^D \sum_{l=0}^D \binom{D}{l}(x(\mu+1))^{D-l}(-\mu y)^l = (1+x)^D (1+y)^D [x+\mu(x-y)]^D.
\]
We write $x-y = (1+x)-(1+y)$, and we expand the previous sum in terms of powers of $\mu$ to obtain
\[
\sum_{m=0}^D \binom{D}{m} \mu^m \sum_{r=0}^m \binom{m}{r} (1+x)^{D+r} x^{m-r} (1+y)^{D+m-r} (-1)^{m-r}.
\]
Isolating the coefficient of $x^D y^D$ yields via Corollaries \ref{-1} and \ref{4} that
\[
J_{\lambda,2}^* \equiv_p \sum_{m=0}^D \left(\frac{-\mu}{4}\right)^m \binom{D-m}{m} \sum_{r=0}^m \binom{m}{r}\binom{D+r}{m}\binom{D+m-r}{D}(-1)^r.
\]
By definition, $\binom{m}{r}\binom{D+r}{m} = \binom{D+r}{D}\binom{D}{m-r}$. Replacement of $r$ with $m-s$ yields
\[
J_{\lambda,2}^* \equiv_p \sum_{m=0}^D \left(\frac{\mu}{4}\right)^m \sum_{s=0}^m (-1)^s \binom{D+s}{s}\binom{D}{s}\binom{D+m-s}{D}\binom{2D-m}{m}.
\]
We have
\[
\binom{D}{s}\binom{D+m-s}{D}\binom{2D-m}{m} = \binom{D+m-s}{m}\binom{2D-m-s}{m-s}\binom{2D-m}{s}.
\]
By Lemma \ref{pre1}, we also have
\[
\binom{D+s}{s} = \binom{2D - (D-s)}{s} \equiv_p (-1)^s \binom{D}{s}
\]
and
\[
\binom{D+m-s}{m} \equiv_p (-1)^{D+s} \binom{2D-m}{D-s}.
\]
Therefore
\begin{equation}\label{appendixeq1}
J_{\lambda,2}^* \equiv_p (-1)^D \sum_{m=0}^D \left(\frac{\mu}{4}\right)^m \sum_{s=0}^m \binom{2D-m}{s}(-1)^s \alpha_{m,s},
\end{equation}
where $\alpha_{m,s} = \binom{2D-m-s}{m-s}\binom{2D-m}{D-s}\binom{D}{s}$ is the coefficient of $x^{m-s}y^{D-s}z^s$ in $(1+x)^{2D-m-s}(1+y)^{2D-m}(1+z)^D$, and thus the $x^{2D}y^D z^D$ coefficient of
\[
[x(1+x)(1+y)]^{2D-m}(z(1+z))^D \left(\frac{xy}{(1+x)z}\right)^s.
\]
In the sum over $s$, if $s > m$, then the power of $x$ exceeds $x^{2D}$ and thus contributes nothing. Therefore, we may extend the sum, which gives us the $x^{2D} y^D z^D$ coefficient of
\[
[x(1+x)(1+y)]^{2D-m} (z(1+z))^D \left(1 - \frac{xy}{(1+x)z}\right)^{2D-m}
\]
for the sum over $s$. The substitution $z \rightarrow \frac{1}{z}$ and multiplication by $z^{2D}$ yields the $x^{2D} y^D z^D$ coefficient of
\[
\left[x(1+x)(1+y)\right]^{2D-m}(1+z)^D\left(1 - \frac{xyz}{1+x}\right)^{2D-m}.
\]
We now make the following sequence of changes of variables:
\begin{enumerate}[(i)]
\item $z \rightarrow \frac{z}{1+y}$, multiply by $(1+y)^D$; the $x^{2D}y^D z^D$ coefficient of
\[[x(1+x)]^{2D-m}(1+y+z)^D\left[1+y - \frac{xyz}{1+x}\right]^{2D-m}.\]
\item $y \rightarrow \frac{1+z}{y}$, multiply by $(1+z)^{-D} y^{2D}$; the $x^{2D}y^D z^D$ coefficient of
\[[x(1+x)]^{2D-m}[y(1+y)]^D\left[1+ \frac{1+z}{y}\left(1 - \frac{xz}{x+1}\right)\right]^{2D-m}.\]
\item $z \rightarrow -\frac{z}{x}$; the $x^D y^D z^D$ coefficient of \[(-1)^D [x(1+x)]^{2D-m}[y(1+y)]^D \left[1 + \frac{1}{y} \left(1 - \frac{z(z+1)}{x(x+1)} \right)\right]^{2D-m}.\]
\end{enumerate}
Computing the $y^D$ coefficient, then the $z^D$ coefficient, and finally the $x^D$ coefficient, we find that the sum over $s$ is equivalent modulo $p$ to
\[
(-1)^D \sum_{r=0}^D \binom{2D-m}{r} \binom{D}{r} \sum_{u=0}^r \binom{r}{u}\binom{u}{D-u}\binom{2D-m-u}{m+u - D} (-1)^u.
\]
We now let $v = D-u$ and interchange the sums, which yields
\begin{equation}\label{appendixeq2}
\sum_{v=0}^D (-1)^v \binom{D-v}{v}\binom{D-(m-v)}{m-v} \sum_{r=D-v}^D \binom{D}{r} \binom{2D-m}{r}\binom{r}{D-v}.
\end{equation}
As $\binom{D}{r}\binom{r}{D-v} = \binom{D}{v}\binom{v}{D-r}$, the sum over $r$ is equal to
\[
\binom{D}{v}\sum_{r=D-v}^D \binom{2D-m}{r}\binom{v}{D-r} = \binom{D}{v} \binom{2D-m+v}{D},
\]
which by Lemma \ref{pre1} is equivalent modulo $p$ to $(-1)^{m+v+D}\binom{D}{v}\binom{D}{m-v}$. Substitution into \eqref{appendixeq2} and \eqref{appendixeq1} then yields
\begin{align*}
J_{\lambda,2} &\equiv_p (1+\lambda)^D \sum_{m=0}^D \left(\frac{\lambda}{4(1+\lambda)}\right)^m \left[\sum_{v=0}^m \binom{D-v}{v}\binom{D-(m-v)}{m-v}\binom{D}{v}\binom{D}{m-v}\right]\\& = (1+\lambda)^D \left[\sum_{v=0}^D \binom{D}{v}\binom{D-v}{v} \left(\frac{\lambda}{4(1+\lambda)}\right)^v \right]^2.
\end{align*}
The result follows using \eqref{4eq1}. \end{proof}
We now give a proof of the claim in Lemma \ref{binomial}.
\begin{proof}[Proof of Lemma \ref{binomial}] It suffices to show that $$b^n \cdot \frac{\prod_{j=0}^{n-1} (a + bj)}{n!}$$ is an integer when $\gcd(a,b)=1$, for all $n \in \mathbb{N}$. Via Legendre, we have at a prime integer $p$, $$v_p(n!)= \sum_{k=1}^{\lfloor \log_p(n) \rfloor} \left\lfloor \frac{n}{p^k} \right\rfloor.$$ In particular, let $p| b$ and $v_p(b) = m$. Thus by Legendre we also have $$v_p(n!) = \frac{n-s_p(n)}{p-1} \leq n \leq mn = v_p(b^n),$$ where $s_p(n)$ denotes the sum of the base-$p$ digits of $n$. As the product $\prod_{j=0}^{n-1} (a + bj)$ is relatively prime to $b$, it follows that $$v_p\left(b^n \cdot \frac{\prod_{j=0}^{n-1} (a + bj)}{n!}\right) \geq 0.$$ Let us now consider the case where $p| a$. We then have $$v_p\left(b^n \cdot \frac{\prod_{j=0}^{n-1} (a + bj)}{n!}\right) = v_p\left(\frac{\prod_{j=0}^{n-1} (a + bj)}{n!}\right).$$ Furthermore, \begin{align*} v_p\left(\frac{\prod_{j=0}^{n-1} (a + bj)}{n!}\right) & = v_p\left(\prod_{j=0}^{n-1} (a + bj)\right) - v_p(n!).\end{align*} By Legendre's formula, we again have $$v_p(n!)= \sum_{k=1}^{\lfloor \log_p(n) \rfloor} \left\lfloor \frac{n}{p^k} \right\rfloor.$$ We find similarly for Pochhammer's factorial $\prod_{j=0}^{n-1} (a + bj)$. We have $a + bj \equiv bj \text{ mod } p$, and as $\gcd(p,b)=\gcd(a,b) = 1$, the element $a+bj$ is divisible by $p$ precisely when $j| p$. It follows that \begin{align*}\bigg|\bigg\{a+bj \;\bigg|\; j=0,\ldots,n-1, \; p | a+bj\bigg\}\bigg| = \left\lfloor \frac{n-1}{p} \right\rfloor+1\geq \left\lfloor \frac{n}{p} \right\rfloor.\end{align*} Similarly, if $k \geq 2$, then $p^k | a+bj$ precisely when $j$ is of the form $$j = \frac{\lambda p^k - a}{b} \in \mathbb{Z},$$ and $j$ is of this form precisely when $p^k | a+b(j+p^k)$, so that every $p^k$th element of $a+bj$, $j=0,\ldots,n-1$ is divisible by $p^k$. It follows that $$\bigg|\bigg\{a+bj \;\bigg|\; j=0,\ldots,n-1, \; p^k | a+bj\bigg\}\bigg| \geq \left\lfloor \frac{n}{p^k} \right\rfloor.$$ As in the proof of Legendre's theorem, every additional power of $p$ dividing $a + bj$ contributes once to the $p$-adic valuation of the Pochhammer product. We thus have $$v_p\left(\prod_{j=0}^{n-1} (a + bj)\right) \geq \sum_{k=1}^{\lfloor \log_p(a+b(n-1)) \rfloor} \left\lfloor \frac{n}{p^k} \right\rfloor \geq \sum_{k=1}^{\lfloor \log_p(n) \rfloor} \left\lfloor \frac{n}{p^k} \right\rfloor = v_p(n!).$$ Let us now consider a prime $p \nmid a, b$. As with $p | a$, we have $$v_p\left(b^n \cdot \frac{\prod_{j=0}^{n-1} (a + bj)}{n!}\right) = v_p\left(\frac{\prod_{j=0}^{n-1} (a + bj)}{n!}\right).$$ We note that the valuations decompose as \begin{align*} v_p\left(\frac{\prod_{j=0}^{n-1} (a + bj)}{n!}\right) & = v_p\left(\prod_{j=0}^{n-1} (a + bj)\right) - v_p(n!) \\& = v_p\left(\prod_{j=0}^{n-1} (a + bj)\right) - v_p\left(\prod_{j=0}^{n-1} (1+j) \right). \end{align*} In this case, $p^k$ divides $j+1$ if, and only if, $j = lp^k-1$, and $p^k$ divides $bj+a$ if, and only if, $j = lp^k - \alpha_k$, $\alpha_k \in \{1,\ldots,p^k-1\}$, $\alpha_k b \equiv a \text{ mod } p^k$. It follows that \begin{align*} v_p\left(b^n \cdot \frac{\prod_{j=0}^{n-1} (a + bj)}{n!}\right) &= \sum_{j=0}^{n-1} \left[v_p(a+bj) - v_p(1+j)\right] \\& = \sum_k \left\lfloor\frac{n-1+\alpha_k}{p^k}\right\rfloor - \left\lfloor\frac{n}{p^k}\right\rfloor \geq 0. \end{align*} We have thus proven for all prime integers $p$ that $$v_p\left(b^n \cdot \frac{\prod_{j=0}^{n-1} (a + bj)}{n!}\right) \geq 0,$$ and therefore that $$b^n \cdot \frac{\prod_{j=0}^{n-1} (a + bj)}{n!}$$ is an integer, as desired. \end{proof}
\bibliographystyle{amsalpha}
\raggedright
\bibliography{references}
{\small
\vspace{30pt}
\noindent AMIT GHOSH, Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA \hfill {\itshape E-mail address}: ghosh@okstate.edu
\vspace{12pt}
\noindent KENNETH WARD, Department of Mathematics \& Statistics, American University, Washington, DC 20016, USA \hfill {\itshape E-mail address}: kward@american.edu
\vspace{12pt}
}
\end{document} | {"config": "arxiv", "file": "1601.06765/HyperFinalRev18.tex"} |
\section{Final Remarks}\label{s:conclusion}
The main purpose of this paper has been to introduce the ideas of signed graph colorings and trace diagrams. A secondary purpose has been to provide a lexicon for their translation into linear algebra. The advantage in this approach to linear algebra lies in the ability to \emph{generalize} results, as was done in Section \ref{s:generalization}.
There is much more to be said about trace diagrams. The case $n=2$ was the starting point of the theory \cite{Lev56} and has been studied extensively, most notably providing the basis for spin networks \cite{CFS95,Kau91} and the Kauffman bracket skein module \cite{BFK96}. In the general case, the coefficients of the characteristic equation of a matrix can be understood as the $n+1$ ``simplest'' closed trace diagrams \cite{Pet08}.
The diagrammatic language also proves to be extremely useful in invariant theory. It allows for easy expression of the ``linearization'' of the characteristic equation \cite{Pet08}, from which several classical results of invariant theory are derived \cite{Dre07}. Diagrams have already given new insights in the theory of character varieties and invariant theory \cite{Bu97,LP09,Sik01}, and it is likely that more will follow. | {"config": "arxiv", "file": "0903.1373/conclusion.tex"} |
TITLE: Existence of continuous functions with finitely many prescribed values
QUESTION [2 upvotes]: This question seems very basic but I have no clue how to show this statement nor have I been able to find some references for it.
Let $X$ and $Y$ be two uniform Hausdorff spaces (i.e. completely regular topological Hausdorff spaces). Consider a finite collection of pairs $(x_i, y_i)$,
$i=1,\dots,N$ where $x_i \in X$ and $y_i \in Y$ with $x_i \neq x_j$ whenever $i \neq j$.
Does there always exist a continuous (or even uniformly continuous) function $f: X \to Y$ with $f(x_i) = y_i$ for all $i=1,\dots,N$?
When $Y$ is a (real) vector space it seems straight forward to construct such a function by interpolation. But how to proceed in the general case of uniform/topological spaces?
I would be grateful for any hint how to proof such a statement as well as any pointers to relevant literature. On the other hand, if there is a counterexample, under which assumptions does the statement hold?
REPLY [2 votes]: Find pairwise disjoint $U_i\ni x_i$ and $f_i\colon X\to [0,1]$ with $f_i(x_i)=1$ and $=0$ outside $U_i$. We obtain a map to a star graph space $Z$ consisting of $n$ copies of $[0,1]$ glued together at their $0$-ends, namely by mapping $x\in X$ to the point $f_i(x)$ on the $i$th copy of $[0,1]$ if $x\in U_i$, and map to the star centre otherwise. This is possible for all allowed $X$, so the actual question is: For what $Y$ can we find a continuous map $Z\to Y$ such that the ends are mapped to given points $y_i$?
Necessarily, the image of $Z$ is path connected. On the other hand, if $Y$ is path connected, then we can pick $z\in Y$ and paths from $z$ to the $y_i$ and combine these to obtain the desired $Z\to Y$. | {"set_name": "stack_exchange", "score": 2, "question_id": 4571113} |
TITLE: What is a mass moment?
QUESTION [1 upvotes]: I am currently reading through a document Finding Moments of Inertia from MIT, page 4, and I am a little confused as to one of the concepts that they use.
In this document, there is mention of a mass moment. Could someone possibly define this for me please? I can't find anything too clear on the Internet.
Is this synonymous with the first moment of mass?
REPLY [0 votes]: Mass moment is slightly different from moment of inertia. It is moment of inertia x total Mass | {"set_name": "stack_exchange", "score": 1, "question_id": 491915} |
\begin{document}
\title{\vspace*{0cm}Hopf-Rinow Theorem in the Sato Grassmannian\footnote{2000 MSC. Primary 22E65; Secondary 58E50,
58B20.}}
\date{}
\author{Esteban Andruchow and Gabriel Larotonda}
\maketitle
\abstract{\footnotesize{\noindent Let $U_2({\cal H})$ be the Banach-Lie group of unitary operators in the Hilbert space ${\cal H}$ which are Hilbert-Schmidt perturbations of the identity $1$. In this paper we study the geometry of the unitary orbit
$$
\{ upu^*: u\in U_2({\cal H})\},
$$
of an infinite projection $p$ in ${\cal H}$. This orbit coincides with the connected component of $p$ in the Hilbert-Schmidt restricted Grassmannian $Gr_{res}(p)$ (also known in the literature as the Sato Grassmannian) corresponding to the polarization ${\cal H}=p({\cal H})\oplus p({\cal H})^\perp$. It is known that the components of $Gr_{res}(p)$ are differentiable manifolds. Here we give a simple proof of the fact that $Gr_{res}^0(p)$ is a smooth submanifold of the affine Hilbert space $p+{\cal B}_2({\cal H})$, where ${\cal B}_2({\cal H})$ denotes the space of Hilbert-Schmidt operators of ${\cal H}$. Also we show that $Gr_{res}^0(p)$ is a homogeneous reductive space. We introduce a natural metric,
which consists in endowing every tangent space with the trace inner product, and consider its Levi-Civita connection. This connection has been considered before, for instance its sectional curvature has been computed. We show that the Levi-Civita connection coincides with a linear connection induced by the reductive structure, a fact which allows for the easy computation of the geodesic curves. We prove that the geodesics of the connection, which are of the
form $\gamma(t)=e^{tz}pe^{-tz}$, for $z$ a $p$-codiagonal anti-hermitic element of ${\cal B}_2({\cal H})$, have minimal length provided that $\|z\|\le \pi/2$. Note that the condition is given in terms of the usual operator norm, a fact which implies that there exist minimal geodesics of arbitrary length. Also we show that any two points $p_1,p_2\in Gr_{res}^0(p)$ are joined by a minimal geodesic. If moreover $\|p_1-p_2\|<1$, the minimal geodesic is unique. Finally, we
replace the $2$-norm by the $k$-Schatten norm ($k>2$), and prove that the geodesics are also minimal for these norms, up to a critical value of $t$, which is estimated also in terms of the usual operator norm. In the process, minimality results in the $k$-norms are also obtained for the group $U_2({\cal H})$.
}\footnote{{\bf Keywords and
phrases:} Sato Grassmannian, Infinite projections, Hilbert-Schmidt operators}}
\setlength{\parindent}{0cm}
\section{Introduction}
Let $\h$ be an infinite dimensional Hilbert space and ${\cal B}(\h)$ the space
of bounded linear operators acting in $\h$. Denote by $\gl$ and $U(\h)$ the
groups of invertible and unitary operators in $\h$, and by $\be$ the space of
Hilbert-Scmidt operators, that is
$$
\be=\{ a\in {\cal B}(\h): Tr(a^*a)<\infty\},
$$
where $Tr$ is the usual trace in ${\cal B}(\h)$.
This space is a two sided ideal of ${\cal B}(\h)$, and a Hilbert space with the
inner product
$$
<a,b>=Tr(b^*a).
$$
The norm induced by this inner product is called the $2$-norm, and denoted by
$$
\|a\|_2=Tr(a^*a)^{1/2}.
$$
Throughout this paper, $\| \ \|$ denotes the usual operator norm.
Consider the following groups of operators:
$$
\gldos=\{g\in \gl: g-1\in \be\},
$$
and
$$
\u2=\{u\in U(\h): u-1\in \be\},
$$
here $1\in {\cal B}(\h)$ denotes the identity operator. These groups are
examples of what in the literature \cite{harpe} is called a {\it classical}
Banach-Lie group. They have differentiable structure when endowed with the
metric $\|g_1-g_2\|_2$ (note that $g_1-g_2\in\be$).
Fix a selfadjoint infinite projection $p\in{\cal B}(\h)$. The aim of this paper is the
geometric study of the set
$$
\o2=\{ upu^*: u\in\u2\},
$$
the connected component of $p$ in the (Hilbert-Schmidt) restricted Grassmannian corresponding
to the polarization $\h=R(p)\oplus R(p)^\perp$ \cite{segalwilson}.
Since both $R(p)$ and
$R(p)^\perp$ are infinite dimensional, the $\u2$-orbit of $p$ lies inside $Gr_{res}^0(p)$ ,
\cite{pressleysegal}. The fact that the group $\u2$ acts transitively on $\o2$ was
proved by Stratila and Voiculescu in \cite{stravoic} (see also \cite{carey}).
The Hopf-Rinow Theorem is not valid in infinite dimensional complete manifolds: two
points in a Hilbert-Riemann complete manifold may not be joined by a minimal
geodesic \cite{grossman}, \cite{mcalpin}, or even a geodesic \cite{atkin}. The main
results in this paper establish the validity of the Hopf Rinow Theorem for $\o2$
(\ref{hr1}, \ref{hr2}). In the process we prove minimality results for $\u2$, which
are perhaps well known but for which we could find no references. We also prove
minimality results, both in $\u2$ and $\o2$, for the Finsler metrics given by the
Schatten $k$-norms ($k\ge 2$).
If one chooses unitaries $\omega_n$,
$n\in{\mathbb Z}$, in the different components of ${\cal U}_{res}(\h)$, then the
connected components of $Gr_{res}(p)$ are
$$
Gr_{res}^n(p)=\omega_n\o2 \omega_n^*=\{\omega_n upu^*\omega_n^*: u\in\u2 \},
$$
so that the results described above are valid also in the other components of $Gr_{res}(p)$.
The restricted Grasamannian is related to several areas of mathematics and physics:
loop groops \cite{segalwilson}, \cite{pressleysegal}, integrable systems
\cite{sato}, \cite{segalwilson}, \cite{mulase}, \cite{zelikin}, group representation
theory \cite{stravoic}, \cite{carey}, \cite{segal}, string theory
\cite{alvarezgomez}, \cite{bowickrajeev}.
Unitary orbits of operators, and in particular projections, have been studied before in
(\cite{cprprojections}, \cite{pr}, \cite{belrattum}, \cite{bona}, \cite{larotonda},
\cite{andsto}). In this particular framework, restricting the action to
classical groups, certain results can be found in \cite{carey}, \cite{belrattum} and
\cite{bona}. In the latter paper, the author considered the orbit of a finite
rank projection.
If $p$ has infinite rank and corank, then $p$ and $\be$ are linearly
independent. We shall denote by
$$
p+\be=\{p+a: a\in \be\}.
$$
Note that every element $x$ in $p+\be$ has a unique decomposition $x=p+a$, $a\in
\be$. We shall endow this affine space with the metric induced by the $2$-norm:
if $x=p+a$ and $y=p+b$, $\|x-y\|_2:=\|a-b\|_2$. Apparently, $p+\be$ is a Hilbert
space.
The orbit $\o2$ sits inside $p+\be$:
\begin{eqnarray}
q & = & upu^*=(1+(u-1))p(1-(u-1)^*)\nonumber\\
& = &p+(u-1)p+p(u-1)^*+(u-1)p(u-1)^*\in p+\be.\nonumber
\end{eqnarray}
Therefore we shall consider $\o2$ with the topology induced by the $2$-metric of
$p+\be_2$.
Throughout this paper, $L_2$ denotes the length functional for piecewise smooth
curves, either in $\u2$ or $\o2$, measured with the $2$-norm:
$$
L_2(\alpha)=\int_{t_0}^{t_1} \|\dot{\alpha}(t)\|_2\, d t.
$$
We use the subscript $h$ (resp. $ah$) to denote the sets of hermitic (resp.
anti-hermitic.) operators, e.g. $\beah=\{x\in\be: x^*=-x\}$.
Let us describe the contents and main results of the paper.
In section 2 we prove (Theorem 2.4) that $\o2$ is a smooth submanifold of the
affine Hilbert space $p+\be$, and that the map $\pi_p:\u2\to \o2$,
$\pi_p(u)=upu^*$ is a submersion.
In section 3 we introduce a linear connection, which is the Levi-Civita
connection of the trace inner product in $\o2$. This connection was considered
in \cite{pekonen}, where its sectional curvature was computed, and shown to be
non negative. It is presented here as the connection obtained from the reductive
structure for the action of $\u2$:
$$
\beah=\{y\in\beah: py=yp\} \oplus \{z\in\beah: pzp=(1-p)z(1-p)=0\},
$$
regarded as the decomposition of the Lie algebra of $\u2$ (equal to $\beah$),
the first subspace is the Lie algebra of the isotropy group, and the second
subspace is its orthogonal complement with respect to the trace. Therefore the geodesics can be
explicitely computed.
In section 4 we prove the main results (Theorems \ref{inicial}, \ref{hr1} and
\ref{hr2}) on minimality of
geodesics with given initial (respectively boundary) data. These
results show that any pair of points in $\o2$ can be joined by a minimal geodesic, and that there are mimimal geodesics which have arbitrary
length.
In section 5 we consider the minimality problem, when the length of a curve is
measured with (the Finsler metric given by) the Schatten $k$-norms, for
$2<k<\infty$. Here we obtain (Theorem 5.5) that for these metrics, the
geodesics of the connection have minimal length up to a critical value of $t$
(which depends on the usual operator norm of the initial
data). In both settings, $k=2$ and $k>2$, the minimality results are proved
first in $\u2$, and then derived for $\o2$ via a natural inmersion of
projections as unitaries (more specifically, symmetries).
\section{Differentiable structure of $\o2$}
As said above, it is known that $\o2$, being a connected component of
$Gr_{res}(p)$ (the component of virtual dimension $0$ \cite{segalwilson}), is a differentiable
manifold.
Here we show that $\o2\subset p+\be_{h}$ is a differentiable (real analytic)
submanifold.
The action of $\u2$ induces the map
$$
\pi_p:\u2\to \o2 \ , \ \ \pi_p(u)=upu^*.
$$
This map, regarded as a map on $p+\be$, is real analytic. Its differential at
the identity is the linear map
$$
\delta_p: \beah \to \beh ,\ \delta_p(x)=xp-px.
$$
Here we have identified the Banach-Lie algebra of $\u2$ with the space $\beah$ of
anti-hermitic elements in $\be$, and used the fact that $\pi_p$ takes hermitic
values, i.e. in the set $\beh$ of hermitic elements of $\be$.
\begin{lem}\label{proyecciondelta}
The map $\delta_p^2$ defines an idempotent operator acting on $\beh$. Moreover,
it is symmetric for the trace inner product in $\beh$.
\end{lem}
\begin{proof}
Straightforward computations show that if one regards $\delta_p$ as a linear map
from $\b(\h)$ to $\b(\h)$, then it verifies $\delta_p^3=\delta_p$. Therefore
$\delta_p^2$ is an idempotent, whose range and kernel coincide with the range
and kernel of $\delta_p$. Note that
$$
\delta_p^2(x)=xp-2pxp+px.
$$
Clearly $\delta_p^2$ maps $\beh$ into $\beh$ and $\beah$ into $\beah$, so that
in particular it defines an idempotent (real) linear operator acting in $\beh$.
Finally, if $x,y\in \beh$,
\begin{eqnarray}
<\delta_p^2(x),y> &=&Tr(y(xp-2pxp+px)=Tr(pyx)-2Tr(pypx)+Tr(ypx)\nonumber\\
&=&Tr((py-2pyp+yp)x)=<x,\delta_p^2(y)>.\nonumber
\end{eqnarray}
\end{proof}
Next let us show that the map $\pi_p$ is a fibration:
\begin{prop}
The map
$$
\pi_p:\u2\to \o2\subset p+\beh
$$
has continuous local cross sections. In particular, it is a locally trivial
fibre bundle.
\end{prop}
\begin{proof}
It is well known that if $p,q$ are projections such that $\|p-q\|<1$, then
the element
$s=qp+(1-q)(1-p)$ is invertible. If $q\in \o2$, then $s\in \gldos$. Indeed,
there exists $u\in \u2$ (i.e. a unitary such that $u_0=u-1\in\be$) such that
$q=upu^*$.
Then
\begin{eqnarray}
s & =& p+u_0p+pu_0^*p+u_0pu_0^*p +1-p +u_0(1-p)+(1-p)u_0^*(1-p)\nonumber\\
&& +u_0(1-p)u_0^*(1-p)\in1+\be.\nonumber
\end{eqnarray}
Morever, $s$ verifies $sp=qp=qs$. Let $s=w|s|$ be the polar decomposition of
$s$. Note that $sp=qs$ implies that $ps^*=s^*q$, and then $s^*s$ commutes with
$p$. Therefore
$$
wpw^*=s|s|^{-1}p|s|^{-1}s^*=s(s^*s)^{-1/2}p(s^*s)^{-1/2}s^*=sp(s^*s)^{-1}s^*=sps^{-1}=q.
$$
We claim that $w\in\u2$. Indeed, $\mathbb{C} 1+\be$ is a *-Banach algebra (it is the
unitization of $\be$) with the $2$-norm: $\|\lambda
1+a\|_2^2=|\lambda|^2+\|a\|_2^2$. Since $s$ lies in $\gldos$, in particular it
is an invertible element of this Banach algebra, and therefore by the Riesz functional
calculus, $w=s|s|^{-1}\in \mathbb{C}+\be$, so that $w=\mu 1+w_0$ with $w_0\in\be$.
On the other hand, note that $s^*s$, it is a positive operator which lies
in the C$^*$-algebra $\mathbb{C} 1+{\cal K}(\h)$, the unitization of the ideal
${\cal K}(\h)$ of compact operators. Therefore its square root is of the form $r
1+b$, with $r\ge 0$ and $b$ compact. Then $s^*s=(r 1+b)^2=r^2 1+b'$, with $b'\in
{\cal K}(\h)$. One has that $s^*s\in\gldos$, so that it is of the form $1$ plus
a compact operator, and since $\mathbb{C} 1$ and ${\cal K}(\h)$ are linearly
independent, it follows that $r=1$. Then $w=s|s|^{-1}$ is of the form $1$ plus
compact. By the same argument as above, this implies that $\mu=1$.
The map sending an arbitrary invertible element $g\in \gldos$ to its unitary
part $u\in\u2$ is a continuous map between these groups. In fact, it is real
analytic, by the regularity properties of the Riesz functional calculus.
Summarizing, consider the open set $\{q\in \o2: \|q-p\|_2<1\}$ in $\o2$. If
$q$ lies in this open set, then in particular $\|q-p\|\le \|q-p\|_2<1$, so
that $s$ defined above lies in $\gldos$, and therefore its unitary part
$u\in\u2$ verifies $upu^*=q$. Denote by $u=\sigma_p(q)$. Clearly $\sigma_p$ is
continuous, being the composition of continuous maps. Thus it is a continuous
local cross section for $\pi_p$ on a neighbourhood of $p$. One obtains cross
sections defined on neighbourhoods of other points of $\o2$ by translating this
one with the action of $\u2$ in a standard fashion.
\end{proof}
Let us transcribe the following result contained in the appendix of the paper
\cite{rae} by I. Raeburn, which is a consequence of the implicit function
theorem in Banach spaces.
\begin{lem}
Let $G$ be a Banach-Lie group acting smoothly on a Banach space $X$. For a fixed
$x_0\in X$, denote by $\pi_{x_0}:G\to X$ the smooth map $\pi_{x_0}(g)=g\cdot
x_0$. Suppose that
\begin{enumerate}
\item
$\pi_{x_0}$ is an open mapping, when regarded as a map from $G$ onto the orbit
$\{g\cdot x_0: g\in G\}$ of $x_0$ (with the relative topology of $X$).
\item
The differential $d(\pi_{x_0})_1:(TG)_1\to X$ splits: its kernel and range are
closed complemented subspaces.
\end{enumerate}
Then the orbit $\{g\cdot x_0: g\in G\}$ is a smooth submanifold of $X$, and the
map
$\pi_{x_0}:G\to \{g\cdot x_0: g\in G\}$ is a smooth submersion.
\end{lem}
This lemma applies to our situation:
\begin{teo}
The orbit $\o2$ is a real analytic submanifold of $p+\be$ and the map
$$
\pi_p: \u2\to \o2 \ , \ \ \pi_p(u)=upu^*
$$
is a real analytic submersion.
\end{teo}
\begin{proof}
In our case, $G=\u2$, $X=p+\be$, $x_0=p$. The above proposition implies that
$\pi_p$ is open. The differential $d(\pi_p)_1$ is $\delta_p$, its kernel is
complemented because it is a closed subspace of the real Hilbert space
$\beah=(T\u2)_1$. The range of $\delta_p$ equals $\delta_p^2(\beh)$, and therefore it is closed and complemented in $\be$, by Lemma \ref{proyecciondelta}. In
our context, smooth means real analytic (the group and the action are real
analytic).
\end{proof}
\section{Linear connections in $\o2$ and $\u2$}
The tangent space of $\o2 $ at $q$ is
$$
(T\o2)_q=\{xq-qx: x\in\beah\},
$$
or equivalently, the range of $\delta_q$, the differential of $\pi_p$ at $q$. As noted
above, it is a closed linear subspace of $\beh$, and the operator $\delta_q^2$
is the orthogonal projection onto $(T\o2)_q$. It is natural to consider the
Hilbert-Riemann metric in $\o2$ which consists of endowing each tangent space
with the trace inner product. Therefore the Levi-Civita connection of this
metric is given by differentiating in the ambient space $\beh$ and projecting
onto $T\o2$. That is, if $X$ is a tangent vector field along a curve $\gamma$
in $\o2$, then
$$
\frac{D X}{d t}=\delta_\gamma^2(\dot{X}).
$$
This same connection can be obtained by other means, it is the connection
induced by the action of $\u2$ on $\o2$ and the decomposition of the Banach-Lie algebra $\beah$ of
$\u2$:
$$
\beah=\{y\in \beah: yp=py\}\oplus \{z\in \beah: pzp=(1-p)z(1-p)=0\},
$$
or, if one regards operators as $2\times 2$ matrices in terms of $p$, the
decomposition of $\beah$ in diagonal plus codiagonal matrices. This type of
decomposition, where the first subspace is the Lie algebra of the isotropy group
of the action (at $p$), and the second subspace is invariant under the inner
action of the isotropy group, is what in differential geometry is called a
reductive structure of the homogeneous space \cite{sharpe}. We do not perform
this construction here, it can be read in \cite{cprprojections}, where it is
done in a different context, but with computations that are formally identical.
This alternative description of the Levi-Civita connection of $\o2$ allows for
the easy computation of the geodesics curves of the connection. The unique
geodesic $\delta$ of $\o2$ satisfying
$$
\delta(0)=q \ \hbox{ and } \ \dot{\delta}(0)=xq-qx
$$
is given by
$$
\delta(t)=e^{tz}qe^{-tz}
$$
where $z$ is the unique codiagonal element in $\beah$ ($qzq=(1-q)z(1-q)=0$) such
that
$$
zq-qz=xq-qx.
$$
Equivalently, $z=\delta_q(xp-px)$ is the projection of $x$ in the decomposition
$$
x=y+z\in \{y\in \beah: yp=py\}\oplus \{z\in \beah: pzp=(1-p)z(1-p)=0\}.
$$
Although our main interest in this paper are projections, it will be useful to
take a brief look at the natural Riemannian geometry of the group $\u2$. Namely,
the metric given by considering real part of the trace inner product, and
therefore, the $2$-norm at each tangent space. The tangent spaces of $\u2$
identify with
$$
(T\u2)_u=u \beah= \beah u.
$$
As with $\o2$, the covariant derivative consists of differentiating in the
ambient space, and projecting onto $T\u2$. Geodesics of the Levi-Civita
connection are curves of the form
$$
\mu(t)=u e^{tx},
$$
for $u\in \u2$ and $x\in\beah$.
The exponential mapping of this connection is the map
$$
exp: \beah\to \u2 , \ \ exp(x)=e^x.
$$
\begin{rem}\label{remarko}
\noindent
\begin{enumerate}
\item
The exponential map
$$
exp: \beah\to \u2
$$
is surjective. This fact is certainly well known. Here is a simple proof. If
$u\in\u2$, then it has a spectral decomposition $u=p_0+\sum_{k\ge
1}(1+\alpha_k)p_k$, where $\alpha_k$ are the non zero eigenvalues of
$u-1\in\beah$. There exist $t_k\in{\mathbb R}$ with $|t_k|\le \pi$ such
that $e^{it_k}=1+\alpha_k$. One has the elementary estimate
$$
|t_k|^2(1-\frac{|t_k|^2}{12})\le |e^{it_k}-1|^2=|\alpha_k|^2,
$$
which implies that the sequence $(t_k)$ is square summable. Let $z=\sum_{k\ge 1}
it_kp_k$ (note that $p_k$ are finite rank pairwise orthogonal projections). Thus
$z\in \beah$ and clearly $e^z=u$.
\item
The exponential map is a bijection between the sets
$$
\beah\supset \{z\in\beah: \|z\|<\pi\}\to \{u\in\u2: \|1-u\|<2\}.
$$
Clearly if $z\in\beah$ with $\|z\|<\pi$, then $e^z\in\u2$ and
$\|e^z-1\|<2$.
Suppose that $u\in\u2$ with $\|u-1\|<2$. Then there exist $x\in{\cal B}(\h)$,
$x^*=-x$ and $\|x\|<\pi$, and $z\in\beah$, such that $e^x=e^z=u$. Since
$\|x\|<\pi$, $x$ equals a power series in $u=e^z$, which implies that $z$
commutes with $x$. Then $e^{z-x}=1$ and thus (note that $z-x$ is anti-hermitic)
$z-x=\sum_{k\ge 1}2k\pi i p_k$ for certain projections $p_k$. Let $z=\sum_{j\ge 1}
\lambda_j q_j$ be the spectral decomposition of $z$. Note that $e^z=\sum_{j\ge 1}
e^{\lambda_j} q_j$, and since $x$ commutes with $e^z$, this implies that $x$ commutes
with $q_j$, and also with $z$. Also it is clear that $q_j$ and $p_k$ also commute,
and that $q_j$ have finite ranks. Then
$$
x=\sum_{j\ge 1} \lambda_j q_j+\sum_{|k|\ge 1}2k\pi i p_k.
$$
The fact that $\|x\|<\pi$ implies that the terms $2k\pi i
p_k$ are cancelled by some of the $\lambda_j q_j$, in order that none of the
remaining $\lambda_j$ verify $|\lambda_j|\ge \pi$. It follows that the $p_k$
have finite ranks, and that there are finitely many. Thus we can define $z'$
adding the remaining $\lambda_j q_j$. Clearly $z'$ verifies $\|z'\|<\pi$ and
$e^{z'}=e^z=u$.
\item
The argument above in fact shows that when one considers the exponential
$$
exp:\{z\in \beah: \|z\|\le \pi\}\to \u2,
$$
then it is surjective.
\end{enumerate}
\end{rem}
If $x\in{\cal B}(\h)$ with $\|x\|<\pi$ then it is well known that
\begin{equation}\label{arcoseno}
\|e^x-1\|=2\sin(\frac{\|x\|}{2}).
\end{equation}
There is a natural way to imbedd projections in the unitary group by means of
the map $q\mapsto \epsilon_q=2q-1$. The unitary $\epsilon_q=2q-1$ is a symmetry,
i.e. a selfadjoint unitary: $\epsilon^*=\epsilon$, $\epsilon^2=1$. See for
instance \cite{pr}, where this simple trick was used to characterize the
minimality of geodesics in the Grassmann manifold of a C$^*$-algebra. However,
if $q\in \o2$, $\epsilon_q$ does not belong to $\u2$ (recall that $q$ has
infinite rank and corank). One can slightly modify this imbedding in order that
it takes values in $\u2$. Consider:
$$
S:\o2\to \u2, \ \ S(q)=(2q-1)(2p-1)=\epsilon_q\epsilon_p.
$$
Clearly it takes unitary values. Let us show that these unitaries belong to
$\u2$. Note that if $q=upu^*$ with $u\in\u2$, then
$$
\epsilon_q=u\epsilon_pu^*=\epsilon_p+(u-1)\epsilon_p+\epsilon_p(u^*-1)+(u-1)\epsilon_p(u^*-1)\in
\epsilon_p+\be,
$$
so that
$$
\epsilon_q\epsilon_p\in (\epsilon_p+\be)\epsilon_p=1+\be.
$$
\begin{prop}\label{mapaS}
The map $S$ preserves geodesics, and its differential is $2$ times an isometry.
\end{prop}
\begin{proof}
Let $\delta$ be a geodesic in $\o2$, $\delta(t)=e^{tz}qe^{-tz}$ with $z\in\beah$
and $z$ codiagonal in terms of $q$. This latter condition is equivalent to $z$
anticommuting with $\epsilon_q$: $z\epsilon_q=-\epsilon_q z$. Which implies, as
remarked in \cite{cprprojections}, that $\epsilon_qe^{-tz}=e^{tz}\epsilon_q$,
and thus $e^{tz}\epsilon_q e^{-tz}=e^{2tz}\epsilon_q$.
Therefore
$$
S(\delta(t))=e^{2tz}\epsilon_q\epsilon_p,
$$
which is a geodesic in $\u2$.
The differential of $S$ at $q$ is given by
$$
dS_q(v)=2v\epsilon_p, \ \ v\in (T\o2)_q.
$$
Right multiplication by a fixed unitary operator is isometric in $\be$,
therefore this map is $2$ times an isometry.
\end{proof}
\section{Minimality of geodesics}
In this section we prove that the geodesics of the linear connection have
minimal length up to a certain critical value of $t$. This could be derived from
the general theory of Hilbert-Riemann manifolds.
We shall prove it here, and in the process obtain a uniform lower bound for the
geodesic
radius, i.e. the radius of normal neighbourhoods.
First we need minimality results in the group $\u2$. These results are perhaps
well known. We include proofs here for we could not find
references for them, and they are central to our argument on $\o2$.
\begin{lem}
Suppose that $x\in \beah$ has finite spectrum and $\|x\|\le\pi$, and let
$u\in\u2$. Then the (geodesic) curve $\mu(t)=ue^{tx}$, $t\in[0,1]$, has minimal
length among all piecewise smooth curves in $\u2$ joining the same endpoints.
\end{lem}
\begin{proof}
Since the action of left multiplication by $u$ is an isometric isomorphism of
$\u2$, it suffices to consider the case $u=1$. Let
$\sigma(x)=\{\lambda_0=0,\lambda_1,\dots ,\lambda_n\}$ be the spectrum of $x$.
Then $x=\sum_{i=1}^n \lambda_i p_i$ for $p_i$ finite rank projections, and
denote by $p_0$ the projection onto the kernel of $x$. Note that
$e^{tx}=p_0+\sum_{i=1}^n e^{t\lambda_i}p_i$. Let $r_i^2=Tr(p_i)$, $i=1,\dots, n$,
and denote by ${\cal S}_i$ the sphere in $\be$ of radius $r_i$,
$$
{\cal S}_i=\{a\in \be: Tr(a^*a)=r_i^2\},
$$
with its natural Hilbert-Riemann metric induced by the (trace) inner product in
the Hilbert space $\be$.
Consider the following smooth map
$$
\Phi: \u2\to {\cal S}_1\times \dots \times {\cal S}_n , \ \ \Phi(u)=(p_1u,\dots,
p_nu).
$$
Here the product of spheres is considered with the product metric.
Apparently $\Phi$ is well defined and smooth. Note that the curve $\Phi(\mu(t))$
is a minimal geodesic of the manifold ${\cal S}_1\times \dots \times {\cal
S}_n$. Indeed,
$$
\Phi(\mu(t))=(e^{t\lambda_1}p_1,\dots, e^{t\lambda_n}p_2),
$$
where each coordinate $e^{t\lambda_i}p_i$ is a geodesic of the corresponding
sphere ${\cal S}_i$, with length equal to $|\lambda_i|r_i\le \|x\|r_i\le\pi
r_i$, and therefore it is minimal. Then $\Phi(\mu(t))$ is minimal, being the
cartesian product of $n$ minimal geodesics in the factors.
Next, note that the length of $\Phi(\mu)$ equals the length of $\mu$:
$$
L_2(\Phi(\mu))=\int_0^1 \|(\lambda_1 e^{t\lambda_1}p_1,\dots,\lambda_n
e^{t\lambda_n}p_n)\| d t=\{\sum_{i=1}^n
|\lambda_i|^2r_i^2\}^{1/2}=\|x\|_2=L_2(\mu).
$$
If $\nu(t)$, $t\in[0,1]$ is any other smooth curve in $\u2$, we claim that
$L_2(\Phi(\nu))\le L_2(\nu)$. Clearly this would prove the lemma. Indeed, since
$\Phi(\mu)$ is minimal in ${\cal S}_1\times \dots \times {\cal S}_n$, one has
$L_2(\Phi(\mu))\le L_2(\Phi(\nu))$, and therefore
$$
L_2(\nu)\ge L_2(\Phi(\nu))\ge L_2(\Phi(\mu))=L_2(\mu).
$$
Note that
$$
L_2(\Phi(\nu))=\int_0^1\{\sum_{i=1}^n\|\dot{\nu}p_i\|_2^2\}^{1/2} dt .
$$
Since $\sum_{i=1}^np_i=1-p_0$ and $\dot{\nu}^*(1-p_0)\dot{\nu}\le
\dot{\nu}^*\dot{\nu}$, one has that
$$
\sum_{i=1}^n\|\dot{\nu}p_i\|_2^2=\sum_{i=1}^nTr(\dot{\nu}^*p_i\dot{\nu})=
Tr(\dot{\nu}^*(1-p_0)\dot{\nu})\le Tr(\dot{\nu}^*\dot{\nu})=\|\dot{\nu}\|_2^2.
$$
Therefore
$$
L_2(\Phi(\nu))\le\int_0^1 \|\dot{\nu}\|_2 d t =L_2(\nu).
$$
\end{proof}
\begin{teo}\label{minimalidadunitariainicial}
Let $u\in\u2$ and $x\in\beah$ with $\|x\|\le \pi$. Then the curve
$\mu(t)=ue^{tx}$, $t\in[0,1]$ is shorter than any other pieceise smooth curve in
$\u2$ joining the same endpoints. Moreover, if $\|x\|<\pi$, then $\mu$ is unique with
this property.
\end{teo}
\begin{proof}
Again, by the same argument as in the previous lemma, we may suppose $u=1$.
Assume that $\mu$ is not minimal. Let
$\gamma(t)$, $t\in[0,1]$ be a piecewise smooth curve in $\u2$ with
$L_2(\gamma)+\delta=\|x\|_2=L_2(\mu)$, for some $\delta>0$. Let $z\in \beah$
be a finite rank operator close enough to $x$ in the $2$-norm in order that
$$
y=log(e^{-x}e^z) \hbox{ verifies } \|y\|_2<\delta/4,
$$
$$
|\|x\|_2-\|z\|_2|<\delta/4,
$$
and
$$
\|z\|<\pi.
$$
Let $\rho(t)=e^xe^{ty}$, and consider $\gamma\#\rho$ the curve $\gamma$ followed
by $\rho$, which joins $1$ to $e^{z}$.
Then
$$
L_2(\gamma\#\rho)=L_2(\gamma)+L_2(\rho)=L_2(\gamma)+\|y\|_2<L_2(\gamma)+\delta/4=\|x\|_2-3\delta/4<\|z\|_2-\delta/2,
$$
which contradicts the minimality of the curve $e^{tz}$ proved in the previous
lemma, because $\|z\|\le\pi$.
Suppose now that $\|x\|<\pi$.
By the general theory of Hilbert-Riemann manifolds \cite{lang}, any minimal
curve starting at $u$ is a geodesic of the linear connection, i.e. a curve of
the form $ue^{tw}$. If it joins the same endpoints as $\mu$, then it must be
$e^w=e^{x}$. Since $\|x\|<\pi$, $x$ is a power series in terms of
$e^x$, and therefore $w$ commutes with $x$. Then $e^{w-x}=1$. Suppose that $w\ne
x$, then
$$
w-x=\sum_{k=1}^m 2k\pi i p_i,
$$
for certain pairwise orthogonal (non nil) projectors $p_i\in\be$.
Then,
$$
\|w-x\|_2^2=\sum_{k=1}^m 4\pi^2 Tr(p_i)^2\ge 4\pi^2.
$$
Since $\|x\|<\pi$, this inequality clearly impies that $\|w\|\ge \pi$,
therefore leading to a contradiction.
\end{proof}
\begin{rem}
The proof in Theorem \ref{minimalidadunitariainicial} shows that if
$x\in \beah$, the curve $e^{tx}$ remains minimal as long as $t\|x\|\le \pi$. One has
coincidence $\|x\|=\|x\|_2$ only for rank one operators. In general, the number
$C_x=\|x\|_2/\|x\|$ can be arbitrarily large. Therefore, for a specific
$x\in\beah$, in terms of the $2$-norm, $e^{tx}$ will remain minimal as long as
$$
t\|x\|_2\le C_x \pi.
$$
\end{rem}
\begin{coro}
There are in $\u2$ minimal geodesics of arbitrary length. Thus the Riemannian
diameter of $\u2$ is infinite.
\end{coro}
\begin{teo}\label{minimalidadunitariaborde}
Let $u_0, u_1 \in\u2$. Then there exists a minimal geodesic curve joining them.
If $\|u_0-u_1\|<2$, then this geodesic is unique.
\end{teo}
\begin{proof}
Again, using the isometric property of the left action of
$\u2$ on itself, we may suppose $u_0=1$.
The first assertion follows from the surjectivity of the exponential map
$exp:\{x\in\beah: \|x\|\le \pi\}\to\o2$ in Remark \ref{remarko}, and Theorem
\ref{minimalidadunitariainicial}. The uniqueness assertion also follows from Remark
\ref{remarko}.
\end{proof}
Denote by $d_2$ the geodesic distance, i.e. the metric induced by the $2$-norm on
the tangent spaces, both in $\u2$ and $\o2$.
\begin{prop}
If $u,v\in\u2$ then
$$
\sqrt{ 1-\frac{\pi^2}{12} } \; d_2(u,v) \le \|u-v\|_2\le d_2(u,v).
$$
In particular the metric space $(\u2, d_2)$ is complete.
\end{prop}
\begin{proof}
Since left multiplication by $v^*$ is an isometry for both metrics, we may assume
that $v=1$. As in Remark \ref{remarko}, we may assume that $u=p_0 +\sum_{k\ge
1}e^{it_k}p_k$, with $p_i$ mutually orthogonal projections and $\mid t_k\mid \le
\pi$. Then
$$
\|u-1\|_2^2=\|\sum_{k\ge 1} (e^{it_k}-1)p_k\|_2^2=\sum_{k\ge 1}\mid
e^{it_k}-1\mid^2 r_k^2=\sum_{k\ge 1} 2(1-\cos(t_k)) r_k^2,
$$
where $r_k=Tr(p_k)$.
Now
$$\mid t \mid ^2 \ge 2(1-\cos(t))\ge \mid t\mid^2 (1-\frac{\mid t\mid^2}{12} )\ge
\mid t\mid^2 (1-\frac{ \pi^2}{12} )
$$
for any $t\in [-\pi,\pi]$. Let $z=\sum_{k\ge 1}i t_k p_k$; clearly $z\in \beah$ by
the inequality above and $e^z=u$. If $\gamma(t)=e^{tz}$, then $\gamma$ is a minimal
geodesic in $\u2$ joining $1$ to $u$ because $\|z\|\le \pi$. Then
$d_2(u,1)=L_2(\gamma)=\|z\|_2$, and from the two inequalities above we obtain
$\sqrt{ 1-\frac{\pi^2}{12} } \; \|z\|_2 \le \|u-v\|_2\le \|z\|_2$, which proves the
assertion of the proposition.
Therefore, $\u2$ is complete with the geodesic distance, because $(\u2,\| \ \|_2)$
is complete. This fact is certainly well known. We include a short proof. Suppose
that $u_n$ is a Cauchy sequence in $\u2$ for the $2$-norm. Since the $2$-norm bounds
the operator norm, it follows that there exists a unitary operator $u$ such that
$\|u_n-u\|\to 0$. On the other hand $u_n-1$ is a Cauchy sequence in $\be$, and
therefore it converges to some operator in $\be$. Thus $u\in\u2$.
\end{proof}
\begin{rem}
If $x,y$ are anti-hermitic operators with $\|x\|,\|y\|<\pi$, then $e^x=e^y$ implies
$x=y$. If $\|x\|=\|y\|=\pi$, from Theorem \ref{minimalidadunitariainicial}, it
follows that $e^x=e^y$ implies $\|x\|_2=\|y\|_2$, because the curves $e^{tx},
e^{ty}$ are both minimal geodesics joining the same endpoints, hence they have the
same length.
\end{rem}
Now our main results on minimal geodesics of $\o2$ follow:
\begin{teo}\label{inicial}
Let $z\in\beah$ which is codiagonal with respect to $q\in \o2$, and such that
$\|z\|\le\pi/2$. Then the geodesic $\alpha(t)=e^{tz}qe^{-tz}$, $t\in[0,1]$ has
minimal length among all piecewise smooth curves in $\o2$ joining the same
endpoints. Moreover, if $\|z\|<\pi/2$, then $\alpha$ is unique having this property.
\end{teo}
\begin{proof}
Let $\beta$ be any other piecewise smooth curve in $\o2$ having the same
endpoints as $\alpha$. Consider $S(\alpha)$ and $S(\beta)$ in $\u2$. Note that
$S(\alpha)(t)=e^{2tz}\epsilon_q\epsilon_p$, with $2\|z\|\le \pi$. Therefore
$$
L_2(\alpha)=\frac12 L_2(S(\alpha))\le \frac12 (S(\beta))=L_2(\beta).
$$
The uniqueness part is an easy consequence.
\end{proof}
\begin{rem}
Again, as remarked after \ref{minimalidadunitariainicial}, for specific $z$ (of
rank greater than one), the geodesic $e^{tz}qe^{tz}$ will remain minimal as
long as
$$
t\|z\|_2\le \frac{\pi}{2} C_z,
$$
where again $C_z=\frac{\|z\|_2}{\|z\|}$ can be arbitrarily large.
\end{rem}
Analogously as for $\u2$, one has
\begin{coro}
There are in $\o2$ minimal geodesics of arbitrary length, thus $\o2$ has infinite
Riemannian diameter.
\end{coro}
\begin{teo}\label{hr1}
Let $q_0,q_1\in \o2$ such that $\|q_0-q_1\|<1$.
Then there exists a unique geodesic joining them, which has minimal length.
\end{teo}
\begin{proof}
The action of $\u2$ on $\o2$ is isometric, therefore we may suppose without loss
of generality that $q_0=p$.
Then \cite{pr}, \cite{cprprojections} there exists $z\in{\cal B}(\h)$, $z^*=-z$,
$\|z\|<\pi/2$, $z$ $p$-codiagonal such that $e^zpe^{-z}=q_1$. Therefore
$$
\epsilon_{q_1}=e^z\epsilon_pe^{-z}=e^{2z}\epsilon_p,
$$
and thus $z=\frac12 log(\epsilon_{q_1}\epsilon_p)$, where $log$ is well defined
because
$\|1-\epsilon_{q_1}\epsilon_p\|=\|\epsilon_{q_1}-\epsilon_p\|=2\|p-q_1\|<2$.
On the other hand $\epsilon_{q_1}\epsilon_p\in\u2$, therefore by Remark
\ref{remarko}, $z\in\beah$. Moreover,
$$
2\|z\|\le \pi,
$$
and therefore the curve $\mu(t)=e^{2tz}\epsilon_p$ is a minimal geodesic in
$\u2$. Again, as in the previous theorem, this implies that the geodesic curve
$\delta(t)=e^{tz}pe^{-tz}$, which joins $p$ and $q_1$, is minimal.
\end{proof}
Next let us consider the case when $\|q_1-q_2\|=1$. The problem of existence of minimal curves in
this case, in the context of abstract $C^*$-algebras, and measuring with the operator
norm, has been studied by Brown in \cite{brown}.
\begin{teo}\label{hr2}
Let $q_0,q_1\in\o2$ with $\|q_0-q_1\|=1$. Then there exists a minimal geodesic
joining them.
\end{teo}
\begin{proof}
Again, without loss of generality, we may suppose $q_0=p$.
Consider the following subspaces:
$$
H_{00}=\ker p\cap \ker q_1 \ , \ \ H_{01}=\ker p\cap R(q_1) \ , \ \ H_{10}=R(p)\cap
\ker q_1\ , \ \ H_{11}=R(p)\cap R(q_1) \ ,
$$
and
$$
H_0=(H_{00}\oplus H_{01} \oplus H_{10} \oplus H_{11})^\perp.
$$
These are the usual subspaces to regard when considering the unitary equivalence of
two projections \cite{dixmier}. The space $H_0$ is usually called the generic part
of $H$. It is invariant both for $p$ and $q_1$. Also it is
clear that $H_{00}$ and $H_{11}$ are invariant for $p$ and $q_1$, and that $p$ and
$q_1$ coincide here. Thus in order to find a unitary operator $e^z$ conjugating $p$
and $q_1$, with $z\in \beah$, which is codiagonal with respect to $p$, and such that
$\|z\|\le \pi/2$, one needs to focus on the subspaces $H_0$ and $H_{01}\oplus H_{10}$.
Let us treat first $H_0$, denote by $p'$ and $q'_1$ the projections $p$ and $q_1$ reduced to $H_0$. These projections are in what in the literature is called generic position. In \cite{halmos} Halmos showed that two projections in generic position are unitarily equivalent, more specifically, he showed that there exists a unitary operator $w:H_0\to K\times K$ such that
$$
wp'w*=p''=\left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right) \ \ \hbox{ and } \ \
wq'_1w*=q''_1=\left( \begin{array}{cc} c^2 & cs \\ cs & s^2 \end{array} \right),
$$
where $c,s$ are positive commuting contractions acting in $K$ and satisfying $c^2+s^2=1$. We claim that there exists an anti-hermitic operator $y$ acting on $K\times K$, which is a co-diagonal matrix, and such that $e^yp''e^{-y}=q''_1$. In that case, the element $z_0=w^*yw$ is an anti-hermitic operator in $H_0$, which verifies $e^{z_0}p'e^{-z_0}=q'_1$, and is co-diagonal with respect to $p'$. Moreover. we claim that $y$ is a Hilbert-Schmidt operator in $K\times K$ with $\|y\|\le \pi/2$, so that $z_0$ is also a Hilbert-Schmidt operator in $H_0$ with $\|z_0\|\le\pi/2$. Let us prove these claims. By a functional calculus argument, there exists a positive element $x$ in the C$^*$ algebra generated by $c$, with $\|x\|\le \pi/2$, such that $c=cos(x)$ and $s=sin(x)$. Since $q''_1$ lies in the Hilbert-Schmidt Grassmannian of $p''$, in particular one has that $q''_1|_{R(p'')}$ is a Hilbert-Schmidt operator. That is, the operator $cos(x)\ sin(x)+sin(x)^2$ is Hilbert-Schmidt in $K$. By a strightforward functional calculus argument, it follows that $x$ is a Hilbert-Schmidt operator. Consider the operator
$$
y=\left( \begin{array}{cc} 0 & -x \\ x & 0 \end{array} \right)
$$
Clearly $y^*=-y$, $\|y\|\le \pi/2$. A straightforward computation shows that
$$
e^yp''e^{-y}=p''_1,
$$
and our claims follow.
Let us consider now the space $H_{01}\oplus H_{10}$. Recall
\cite{segalwilson} that an alternative definition of $\o2$ states that if $q_1\in \o2$ then
$$
pq_1|_{R(q_1)}:R(q_1)\to R(p)
$$
is a Fredholm operator of index $0$. Note that $H_{01}=\ker(pq_1|_{R(q_1)})$. Thus
in particular $\dim H_{01}<\infty$.
On the other hand, it is also apparent that $H_{10}\subset R(pq_1)^\perp \cap R(p)$,
and therefore also $\dim H_{10}<\infty$. Therefore, the fact that $pq_1|_{R(q_1)}$
has zero index implies that
$$
\dim H_{01}\le \dim H_{10}.
$$
The fact that $q_1$ lies in the connected component of $p$ in the Sato Grassmannian
corresponding to the polarization given by $p$, implies that, reciprocally, $p$ lies
in the component of $q_1$, in the Grassmannian corresponding to the polarization
given by $q_1$. Thus, by symmetry,
$$
\dim H_{01}= \dim H_{10}.
$$
Let $v:H_{10}\to H_{01}$ be a surjective isometry, and consider
$$
w: H_{01}\oplus H_{10}\to H_{01}\oplus H_{10} \ , w(\xi'+\xi'')=v^*\xi'+v\xi''.
$$
In matrix form (in terms of the decomposition $H_{01}\oplus H_{10}$),
$$
w=
\left(
\begin{array}{ll}
0 & v \\ v^* & 0
\end{array}
\right).
$$
Apparently, $w p|_{H_{01}\oplus H_{10}} w^*=q_1|_{H_{01}\oplus H_{10}}$. Let
$$
z_{2}= \pi/2
\left(
\begin{array}{ll}
0 & v \\ -v^* & 0
\end{array}
\right).
$$
Note that $z_2$ is an anti-hermitic operator in $H_{01}\oplus H_{10}$, with norm
equal to $\pi/2$.
A straightforward matrix computation shows that $e^{z_2}=w$. Consider now
$$
z=z_0+z_1+z_2,
$$
where $z_1=0$ in $H_{00}\oplus H_{11}$, and $z_0$ is the anti-hermitic operator in
the generic part $H_0$ of $H$ found above. Then it is clear that $z$ is
anti-hermitic, Hilbert-Schmidt ($\dim(H_{01}\oplus H_{10})<\infty$), $p$-codiagonal,
$\|z\|=\pi/2$, and $e^zpe^{-z}=q_1$.
\end{proof}
Also completeness of the geodesic metric follows:
\begin{coro}
The metric space $(\o2, d_2)$ is complete.
\end{coro}
\begin{proof}
Let $q_n$ be a Cauchy sequence in $\o2$. Since the map $S:\o2\to \u2$ of Proposition
\ref{mapaS} is $2$ times an isometry, then $S(q_n)$ is a Cauchy sequence in $\u2$,
and therefore converges to an element $u$ of $\u2$ in the metric $d_2$. Moreover,
$S(\o2)$ is closed in $\u2$ and then exists $q\in \o2$ such that $S(q)=u$. Clearly
$d_2(q_n,q)\to 0$.
\end{proof}
\section{$k$-norms}
In this section we study the minimality problem of geodesics in $\o2$ measured
in
the $k$-norms, for $k\in{\mathbb R}$, $k>2$. To do this, as with the case
$k=2$, we study first short curves in $\u2$ with these norms. Minimality of
geodesics in $\o2$ will follow with
arguments
similar as in the previous section. We shall endow now the tangent spaces of
$\u2$ and $\o2$ with the Schatten $k$-norm:
$$
\|x\|_k=Tr(|x|^k)^{1/k}=Tr((x^*x)^{k/2})^{1/k}.
$$
Note that since the tangent spaces live inside $\be$, and $k>2$, the
$k$-norm of $x$ is finite. We shall denote by $L_k$ the functional which measures
the length of a curve (either in $\u2$ or $\o2$) in the $k$-norm:
$$
L_k(\alpha)=\int_{t_o}^{t_1} \|\dot{\alpha}(t)\|_k d t .
$$
We are now then in the realm of (infinite dimensional) Finsler geometry.
To prove our results, two inequalities proved by Hansen and Pedersen in
\cite{pedersenhansen} will play a fundamental role. Let us transcribe these
inequalities, called Jensen's inequalities.
\noindent
\begin{enumerate}
\item
The first is the version for C$^*$-algebras (\cite{pedersenhansen}, Th. 2.7):
if $f(t)$ is a convex continuous
real function, defined on an interval $I$ and and $A$ is a C$^*$-algebra with
finite unital trace $tr$, then the inequality
\begin{equation}
tr\bigl( f(\sum_{i=1}^n b_i^*a_ib_i)\bigr)\le tr\bigl( \sum_{i=1}^n
b_i^*f(a_i)b_i\bigr)
\end{equation}
is valid for every $n$-tuple $(a_1,\dots,a_n)$ of selfadjoint elements in $A$
with spectra contained in $I$ and every $n$-tuple $(b_1,\dots,b_n)$ in $A$ with
$\sum_{i=1}^n b_i^*b_i=1$.
We shall use it in a simpler form:
if $a$ is a selfadjoint element in a $C^*$-algebra with trace $tr$, then
\begin{equation}\label{jensen}
tr(f(a))\le f(tr(a))
\end{equation}
for every convex continuous real function defined in the
spectrum of $a$.
\item
The second inequality is valid for finite matrices (\cite{pedersenhansen}, Th.
2.4): let $f$ be a convex continuous function defined on $I$ and let $m$ and $n$
be natural numbers, then
\begin{equation}\label{jensenmatrices}
Tr(f(\sum_{i=1}^n a_i^*x_ia_i))\le Tr(\sum_{i=1}^n a_i^*f(x_i) a_i)
\end{equation}
for every $n$-tuple $(x_1,\dots,x_n)$ of selfadjoint $m\times m$ matrices with
spectra contained in $I$ and every $n$-tuple $(a_1,\dots,a_n)$ of $m\times m$
matrices with $\sum_{i=1}^n a_i^*a_i=1$.
We shall need a simpler version, namely if $r\in{\mathbb R}$, $r\ge 1$,
then
\begin{equation}\label{jensenmatricessimplificada}
Tr(a^r)=Tr( (\sum_{j=0}^n p_ja^rp_j)\ge Tr( (\sum_{j=0}^n
p_jap_j)^r)=Tr(\sum_{j=0}^n (p_jap_j)^r),
\end{equation}
for $p_0,p_1,\dots,p_n$ projections with $\sum_{j=0}^n p_j=1$ and $p_1,\dots,
p_n$ of finite rank, and $a$ a positive trace class operator. A simple
aproximation argument shows that one can obtain (\ref{jensenmatricessimplificada})
from (\ref{jensenmatrices}). Indeed, let $\{\xi_1^j,\dots,\xi_{k_j}^j\}$ be an
orthonormal basis for the range of $p_j$, $j=1,\dots, n$ and
$\{\varphi_i,\varphi_2,\dots \}$ be an orthonormal basis for the range of
$p_0$. For any integer $N\ge 1$, let $e_N$ denote the orthogonal projection onto
the subspace generated by $\{\xi_i^j, j=1,\dots, n, \ i=1,\dots
k_j\}\cup\{\varphi_k, k=1,\dots, N\}$. Clearly $e_N$ is a finite rank projection
such that $p_j\le e_N$ , for $j=1,\dots,n$ and such that $e_Np_0e_N=p_{0,N}$ is
also a projection. Let $a_N=e_Nae_N$. Then the following facts are apparent:
\begin{enumerate}
\item
$p_ja_Np_j=p_jap_j$ for $j=1,\dots, n$.
\item
$a_N\to a$ and $p_{0,N}a_Np_{0,N}\to p_0ap_0$ in $\| \ \|_1$, and therefore
$p_ja_N^rp_j\to p_ja^rp_j$ for $j=0,1,\dots,n$ and $r\ge 1$ in $\| \ \|_1$.
\end{enumerate}
It follows that one can reduce to prove (\ref{jensenmatricessimplificada}) for the
operator $a_N$ and the projections $p_{0,N},p_1,\dots, p_n$, all of which are operators
in the range of $e_N$, which is finite dimensional.
\end{enumerate}
Let us first state the following lemma which is a simple consequence of
(\ref{jensen}).
\begin{lem}\label{lemaresucitado}
Let $a\in \b(\h)$ be a positive operator and $p$ a finite rank projection.
Then, if $r\in{\mathbb R}$, $r\ge 1$
$$
Tr(pap)^r\le Tr(p)^{r-1}Tr((pap)^r).
$$
\end{lem}
\begin{proof}
If $p=0$ the result is trivial. Suppose $Tr(p)\ne 0$.
Consider the finite C$^*$-algebra $p\b(\h) p$, with unit $p$ and normalized
finite trace
$tr(pxp)=\frac{Tr(pxp)}{Tr(p)}$.
Then by Jensen's trace inequality for the map $f(t)=t^r$,
$$
\frac{Tr(pap)^r}{Tr(p)^r}\le \frac{Tr((pap)^r)}{Tr(p)},
$$
which is the desired inequality.
\end{proof}
Denote by ${\cal S}_R^k$ the unit sphere of $\b_k(\h)$:
$$
{\cal S}_R^k=\{x\in\b_k(\h): \|x\|_k=R\}.
$$
If $\mu(t)$ is a curve of unitaries in $\u2$, and $p$ is finite rank projection
with $Tr(p)=R^k$, then $\mu(t)p$ is a curve in ${\cal S}_R^k$: $\|\mu
p\|_k=Tr((p\mu^*\mu p)^{k/2})^{1/k}=R$.
\begin{lem}
Let $p$ be a finite rank projection with $Tr(p)=R^k$ and $\mu(t)$ be a smooth
curve in $\u2$, such that $\mu(0)p=p$ and
$\mu(1)p=e^{\alpha}p$ with $-\pi\le\alpha \le \pi$. Then the curve $\mu p$
of
${\cal S}_R^k$, measured with the $k$-norm, is longer than the curve
$\epsilon(t)=e^{it\alpha}p$.
\end{lem}
\begin{proof}
The length of $\mu p$ is (in the $k$-norm) measured by
$$
\int_0^1 \|\dot{\mu}(t) p\|_k dt=\int_0^1
Tr((p\dot{\mu}(t)^*\dot{\mu}(t)p)^{k/2})^{1/k} dt.
$$
by the inequality in the above lemma,
$$
L_k(\mu p)\ge Tr(p)^{\frac{1-k/2}{k}} \int_0^1
Tr(p\dot{\mu}(t)^*\dot{\mu}(t)p)^{1/2}dt.
$$
This last integral measures the length of the curve $\mu p$ in the $2$-sphere
${\cal S}_{R^{k/2}}^2$ of radius $R^{k/2}$ in the Hilbert space $\be$. The curves
$\epsilon(t)=e^{t\alpha}p$ are minimizing geodesics of these
spheres,
provided that $|\alpha| R^{k/2}\le\pi R^{k/2}$, which holds because
$|\alpha|\le\pi$. It follows that
$$
\int_0^1 Tr(p\dot{\mu}(t)^*\dot{\mu}(t)p)^{1/2}\ge
L_2(\epsilon)=|\alpha|Tr(p)^{1/2}.
$$
Then
$$
L_k(\mu p)\ge |\alpha|Tr(p)^{\frac{1-k/2}{k}}
Tr(p)^{1/2}=|\alpha|R=L_k(\epsilon).
$$
\end{proof}
\begin{lem}
Let $x\in\beah$ with finite spectrum, $x=\sum_{i=1}^n\alpha_i p_i$ with
$\sum_{i=1}^n p_i=1-p_0$ ($p_0$ the kernel projection of $x$) and
$-\pi \le\alpha_i\le\pi$ (i.e. $\|x\|\le\pi$). Then the curve
$\delta(t)=e^{itx}$, $t\in[0,1]$ is the shortest curve in $\u2$ joining its
endpoints, when measured with the $k$-norm.
\end{lem}
\begin{proof}
Let $Tr(p_i)=R_i^k$, $i=1,\dots,n$. Note that the kernel projection $p_0$ has
infinite rank.
The length $L_k(\mu)$ of $\mu$ is measured by
$\int_0^1 \|\dot{\mu}(t)\|_k dt$.
Then, by inequality (\ref{jensenmatricessimplificada}), with
$a=\dot{\mu}(t)^*\dot{\mu}(t)\ge 0$ in $\b_1(\h)$, one has
\begin{equation}\label{desigualdad}
\|\dot{\mu}(t)\|_k\ge \{\sum_{j=0}^n
Tr\bigl((p_i\dot{\mu}(t)^*\dot{\mu}(t)p_i)^{k/2}\bigr)\}^{1/k}=
\{\sum_{i=0}^n\|\dot{\mu}(t)p_i\|_k^k\}^{1/k}.
\end{equation}
On the other hand, note that
$$
\|\dot{\delta}\|_k=\{\sum_{i=1}^n
|\alpha_i|^k R_i^k)\}^{1/k}.
$$
Trivially, $\{\sum_{j=0}^n
Tr\bigl((p_i\dot{\mu}(t)^*\dot{\mu}(t)p_i)^{k/2}\bigr)\}^{1/k}\ge \{\sum_{j=1}^n
Tr\bigl((p_i\dot{\mu}(t)^*\dot{\mu}(t)p_i)^{k/2}\bigr)\}^{1/k}$, (i.e. we omit
the term corresponding to the projection $p_0$, which has infinite trace).
We finish the proof by establishing that
$$
\{\sum_{j=1}^n
Tr\bigl((p_i\dot{\mu}(t)^*\dot{\mu}(t)p_i)^{k/2}\bigr)\}^{1/k}
\ge \{\sum_{i=1}^n
|\alpha_i|^k R_i^k)\}^{1/k}=L_k(\delta).
$$
There is a classic Minkowski type inequality (see
inequality {\bf 201} of \cite{hardylittlewoodpolya}) which
states that if $f_1,\dots,f_n$ are non negative functions, then
$$
\int_0^1 \{\sum_{i=1}^n f_i^k(t)\}^{1/k} dt \ge \bigl(\sum_{i=1}^n\{\int_0^1
f_i(t)\}^k\bigr)^{1/k}.
$$
In our case $f_i(t)=\|\dot{\mu}(t)p_i\|_k$:
$$
\int_0^1 \{\sum_{i=1}^n \|\dot{\mu}(t)p_i\|_k^k)\}^{1/k} dt \ge
\bigl(\sum_{i=1}^n\{\int_0^1 \|\dot{\mu}(t)p_i\|_k dt\}^k\bigr)^{1/k}\ge
\{\sum_{i=1}^n |\alpha_i|^k R_i^k\}^{1/k},
$$
where in the last inequality we use the previous lemma: $\int_0^1
\|\dot{\mu}(t)\|_k dt\ge |\alpha_i| R_i$ for
$i=1,\dots ,n$.
\end{proof}
\begin{teo}\label{minimalidadk}
Let $x\in \beah$ with $\|x\|\le\pi$, and $v\in \u2$. Then
the curve
$\delta(t)=ve^{tx}$ has minimal length among piecewise smooth curves in $\u2$
joining the same endpoints, measured with the $k$-norm.
\end{teo}
\begin{proof}
There is no loss of generality if we suppose $v=1$. Indeed, for any curve $\mu$
of unitaries, $L_k(\mu)=L_k(v^*\mu)$.
Suppose that there exists a piecewise $C^1$ curve of unitaries $\mu$ which is
strictly shorter than $\delta$,
$L_k(\mu)<L_k(\delta)-\epsilon=\|x\|_k-\epsilon$. The element $x$ can be
approximated in the $k$-norm topology of $\b_k(\h)$ by anti-hermitic elements
$z\in\b_k(\h)$, with finite spectrum and the following conditions:
\begin{enumerate}
\item
$\|z\|\le \|x\|\le\pi$.
\item
$\|x\|_k- \epsilon/2<\|z\|_k\le \|x\|_k$.
\item
There exists a $C^\infty$ curve of unitaries joining $e^{x}$ and $e^{z}$ of
$k$-length $L_k$ less than $\epsilon/2$.
\end{enumerate}
The first two are clear. The third condition can be obtained as follows. By
the third condition $e^{-x}e^{z}=e^{y}$, with $y\in \beah$. Moreover $z$ can
be adjusted so as to obtain $y$ of arbitrarily small $k$-norm. Then the curve of
unitaries $\gamma(t)=e^{x}e^{ty}$ is $C^\infty$, joins $e^{x}$ and $e^{z}$,
with $k$-length $\|y\|_k<\epsilon/2$.
Consider now the curve $\mu'$, which is the curve $\mu$ followed by the curve
$e^{x}e^{ty}$ above. Then clearly
$$
L_k(\mu')\le L_k(\mu)+\|y\|_k<L_k(\mu)+\epsilon/2 .
$$
Therefore $L_k(\mu')< \|x\|_k-\epsilon/2$. On the other hand, since $\mu'$
joins $1$ and $e^{z}$, by the lemma above, it must have length greater than or
equal to $\|z\|_k$. It follows that
$$
\|z\|_k\le \|x\|_k-\epsilon/2 ,
$$
a contradiction.
\end{proof}
One obtains minimality of geodesics in $\o2$ for the $k$-norm analogously as in
the previous section:
\begin{teo}
Let $z\in\beah$, codiagonal with respect to $q\in\o2$, with $\|z\|\le \pi/2$. Then
the geodesic
$\alpha(t)=e^{tz}qe^{-tz}$, $t\in[0,1]$, has minimal length for the $k$-norm
among all piecewise smooth curves in $\o2$ having the same endpoints. If
$\|z\|<\pi/2$, this curve $\alpha$ is unique with this property.
\end{teo}
\begin{proof}
The proof follows as in the analogous result for the $2$ norm in the previous
section, noting that the map $S$ is also isometric for the $k$-norms.
\end{proof}
\begin{teo}
Let $q_1,q_2\in\o2$, then there exists a geodesic joining them, which has minimal
length fot the $k$-norm.
\end{teo}
\begin{proof} The proof follows as in the above result, the geodesic
$\alpha(t)=e^{tz}q_1e^{-tz}$ with $\|z\|\le
\pi/2$ exists by virtue of (\ref{hr1}) and (\ref{hr2}).
\end{proof} | {"config": "arxiv", "file": "0808.2525.tex"} |
\begin{document}
\title{Kac-Moody groups as discrete groups}
\author{Bertrand R\'emy}
\maketitle
\pec
{\footnotesize
{\sc Abstract}.---~
This survey paper presents the discrete group viewpoint on Kac-Moody groups.
Over finite fields, the latter groups are finitely generated; they act on new
buildings enjoying remarkable negative curvature properties.
The study of these groups is shared between proving results supporting the analogy with
some $S$-arithmetic groups, and exhibiting properties showing that they are new groups.}
\pec
{\footnotesize
{\bf Keywords:} Kac-Moody group, Tits system, pro-$p$ group, lattice, arithmetic groups,
algebraic group, Bruhat-Tits building, hyperbolic building, commensurator superrigidity,
linearity.
\pec
{\bf Mathematics Subject Classification (2000):}
22F50,
22E20,
51E24,
53C24,
22E40,
17B67.
}
\vskip 10mm
\section*{Introduction}
Kac-Moody groups were initially designed to generalize algebraic groups \cite{Tit87}.
They share many properties with the latter groups, mainly of combinatorial nature.
For instance, they admit $BN$-pair structures \cite[IV.2]{Bou81}, and actually a much
finer combinatorial structure -- called {\it twin root datum~} -- formalizing the existence
of root subgroups permuted by a (possibly infinite) Coxeter group \cite{Tit92} (survey papers
on this are for instance \cite{Tit89} and \cite{RemBie}).
Moreover Kac-Moody groups and some twisted versions of them are expected to be the group side
in the classification of a reasonable class of buildings, the so-called Moufang 2-spherical
twin buildings.
Our intention is not to go into detail about this; nevertheless, we shortly recalled these
facts in order to emphasize that the goal of this paper is to present a significant change of
viewpoint on Kac-Moody groups.
\pec
The geometric counterpart to the group combinatorics of a $BN$-pair is the existence
of an action on a remarkable geometry: a building.
To any Coxeter system $(W,S)$ is attached a simplicial complex $\Sigma$ on the
maximal simplexes of which the group $W$ acts simply transitively \cite[\S 2]{RonLec}.
A {\it building~} is a simplicial complex, covered by subcomplexes all isomorphic to the
same $\Sigma$ -- called {\it apartments}, and satisfying remarkable incidence
properties: any two {\it facets~} (i.e. simplices) are contained in an apartment, and given
any two apartments $A, A' \simeq \Sigma$, there is a simplicial isomorphism $A \simeq A'$
fixing $A \cap A'$ -- see \cite[p. 77]{BroBuildings}, and also \cite[\S 3]{RonLec} for the
chamber system approach.
\pec
In order to define a general Kac-Moody group $\Lambda$, we need a integral matrix $A$
satisfying properties much weaker than those defining Cartan matrices in the classical sense
of complex semisimple Lie algebras \cite[Introduction]{Tit87}.
We also need to choose a ground field (which will always be a finite field ${\bf F}_q$ in
what follows).
Writing down a presentation of $\Lambda$ would require a lot of combinatorial and Lie
algebra material we wouldn't use later -- see \cite[Subsect. 3.6]{Tit87} and
\cite[Sect. 9]{RemAst} for details.
It is a basic result of the theory that to a Kac-Moody group is naturally attached a pair of
twin buildings (via $BN$-pairs) on the product of which it acts diagonally
(Fact \ref{fact - buildings} of the present paper).
The standard example of such a group is $\Lambda={\bf G}\bigl( {\bf K}[t,t^{-1}] \bigr)$ for
${\bf G}$ a semisimple group over a field ${\bf K}$.
Now, if we choose the ground field to be ${\bf F}_q$, the latter example is
an arithmetic group in positive characteristic, and in this case the buildings alluded to
above are the Bruhat-Tits buildings of the non Archimedean semisimple Lie groups
${\bf G}\bigl({\bf F}_q(\!(t)\!)\bigr)$ and ${\bf G}\bigl({\bf F}_q(\!(t^{-1})\!)\bigr)$.
\pec
The latter examples are a very special case of Kac-Moody groups, called of {\it affine
type}, but many other cases are available.
On the one hand, this suggests to see Kac-Moody groups as generalized arithmetic groups
over function fields, which leads to natural questions, e.g. asking whether some classical
properties of discrete subgroups of Lie groups are relavant or true.
On the other hand, it can be shown that some new buildings can be produced thanks to
Kac-Moody groups, and this leads to asking whether the groups attached to exotic buildings
are themselves new.
Note that \og new\fg in this context means that the buildings are neither of spherical nor
of affine type, i.e. don't come from the classical Borel-Tits (resp. Bruhat-Tits) theory on
algebraic groups over arbitrary (resp. local) fields.
\pec
From the point of view of metric spaces, Kac-Moody theory is an algebraic way to construct
spaces with non-positively curved, often hyperbolic, distances and admitting highly
transitive isometry groups.
The algebraic origin of these groups enables to obtain interesting structure results for
various isometry groups (discrete or much bigger).
Therefore, in the case of hyperbolic Kac-Moody buildings, techniques from group
combinatorics such as $BN$-pairs and from hyperbolic spaces \og \`a la Gromov\fg can be
combined.
This leads us to say that the general trend to study finitely generated Kac-Moody groups is
from algebraic and combinatorial methods to geometric and dynamical ones.
In this paper, we explain for instance how the theory of finitely generated Kac-Moody
groups, i.e. Kac-Moody groups over finite fields, naturally leads to studying uncountable
totally disconnected groups generalizing semsimple groups over local fields of positive
characteristic, groups which we call {\it topological Kac-Moody groups}.
We are especially interested in proving that the groups we obtain are new in general, by
proving that they cannot be linear over any field.
In short, the study of finitely generated Kac-Moody groups is shared between proving
classical properties by comparing them to (linear) lattices of non Archimedean Lie groups,
and finally standing by a difference with the classical situation to disprove linearities.
\pec
This appraoch is not new, since for lattices of products of trees M. Burger and Sh. Mozes
managed to prove many classical properties of lattices (among which the normal subgroup
theorem) but proved that an important difference is the possibility to obtain non residually
finite groups, which is impossible for finitely generated linear groups \cite{BurMozCras},
\cite{BurMozTrees}, \cite{BurMozProd}.
The main application is the construction of the first finitely generated torsion free simple
groups.
We note also that Y. Shalom's work \cite{Shalom} shows that thanks to representation
theory, many properties can be proved for irreducible lattices of products of general
locally compact groups.
Since trees are one-dimensional special cases of buildings, the previous
references encourage us to think that finitely generated Kac-Moody groups will produce
interesting examples of groups which are not linear but handable via their diagonal
actions on products of buildings.
\pec
This paper is organized as follows.
In Sect. \ref{s - arithmetic}, it is shown why Kac-Moody groups over finite fields should
be seen as generalizations of some $S$-arithmetic groups in positive characteristic.
It is also explained how they provide new buildings and why Kac-Moody groups should be
expected to be new groups.
In Sect. \ref{s - generalized KM}, we are interested in a specific class of hyperbolic
buildings.
In this context, we can produce non-isomorphic Kac-Moody groups with the same buildings, and
discrete groups which are close to Kac-Moody groups, but with several ground fields: they
have strong non-linearity properties.
In Sect. \ref{s - topological}, we are interested in totally disconnected groups generalizing
semisimple groups over local fields, arising as closures of non-discrete actions of Kac-Moody
groups on buildings.
We quote the existence of a nice combinatorial structure, as well as a
topological simplicity result for them.
In Sect. \ref{s - hard NL}, we sketch the proof of complete non-linearity of some Kac-Moody
groups.
This is where we use the topological groups of the previous section.
We actually mention that there are some Kac-Moody groups all of whose linear images are
finite, whatever the target field.
In Sect. \ref{s - conjectures}, we ask some questions about the various groups previously
defined in the paper.
We conjecture the non-linearity of a wide class of finitely generated Kac-Moody groups, and
the non-amenablity as well as the abstract simplicity of topological Kac-Moody groups.
\pec
The author expresses his deep gratitude to the organizers of the conference \og
Geometric Group Theory\fg (Guwahati, Assam -- India), organized by the Indian Institute of
Technology Guwahati and the Indian Statistical Institute, and supported by the National Board
of Higher Mathematics.
Meenaxi Bhattacharjee managed to make a tremendous human and scientific event out of a
mathematical conference.
\gec
\section{Generalized arithmetic groups acting on new buildings}
\label{s - arithmetic}
This section is mainly dedicated to quoting results supporting the analogy between Kac-Moody
groups over finite fields and $\{ 0;\infty \}$-arithmetic groups over function fields
\cite[Sect. 2-3]{RemNewton}.
The arguments are: the existence of a discrete diagonal action on a product
of two buildings (\ref{ss - lattice}), cohomological finiteness properties
(\ref{ss - finiteness}) and continuous cohomology vanishings (\ref{ss - T}).
In the last two subsections (\ref{ss - hyperbolic} and \ref{ss - easy NL}), we provide
arguments showing that Kac-Moody theory does provide interesting new group-theoretic/geometric
situations.
\subsection{Discrete actions on buildings and finite covolume}
\label{ss - lattice}
A Kac-Moody group is generated by those of its root groups which are indexed by simple
roots and their opposites (in finite number), and by a suitable maximal torus normalizing
them \cite[\S 3.6]{Tit87}.
Since over ${\bf F}_q$ all these groups are finite, we obtain:
\begin{fact}
\label{fact - finitely generated}
Any Kac-Moody group $\Lambda$ over any finite field is finitely generated.
\end{fact}
From now on, $\Lambda$ is a Kac-Moody group defined over ${\bf F}_q$.
Recall that a group action on a building is {\it strongly transitive~}if it is transitive on
the inclusions of a chamber in an apartment.
Combining \cite[Subsect. 5.8, Proposition 4]{Tit87} and \cite[Theorem 5.2]{RonLec}, we
obtain:
\begin{fact}
\label{fact - buildings}
To $\Lambda$ are attached two isomorphic, locally finite buildings $X_\pm$, each of them
admitting a strongly transitive $\Lambda$-action.
\end{fact}
This fact is fundamental to understand Kac-Moody groups: the geometry of the buildings is the
basic substitute for a natural structure on
$\Lambda$ arising from infinite-dimensional algebraic geometry.
In the specific case of an $S$-arithmetic group $\Lambda={\bf G}\bigl( {\bf F}_q[t,t^{-1}]
\bigr)$ (for $S=\{0;\infty\}$), the buildings $X_\pm$ are the Bruhat-Tits buildings of ${\bf
G}\bigl( {\bf F}_q (\!( t )\!) \bigr)$ and ${\bf G}\bigl( {\bf F}_q (\!( t^{-1} )\!) \bigr)$.
In general, it is still true that for an arbitrary Kac-Moody group, the diagonal action on
the product of buildings $X_- \times X_+$ is discrete \cite{RemCras}.
Moreover the action has a nice convex fundamental domain contained in a single apartment and
defined as an intersection of roots (seen as half-apartments)
\cite[\S 3, Corollary 1]{Abr97}.
The next step in the analogy with arithmetic groups consists in asking whether the finitely
generated group $\Lambda$ is a {\it lattice~} of $X \times X_-$, meaning that the locally
compact group ${\rm Aut}(X_-) \times {\rm Aut}(X_+)$ moded out by the image of
$\Lambda$ carries a finite invariant measure \cite[0.40]{Margulis}.
This is the main result of \cite{RemCras}:
\begin{theorem}
\label{th - lattice}
Assume the Weyl group $W$ of $\Lambda$ is infinite and denote by
$W(t):=\sum_{w \in W}t^{\ell(w)}$ its growth series.
Assume that $W({1\over q}) < \infty$.
Then $\Lambda$ is a lattice of $X \times X_-$ for its diagonal action,
and for any point $x_- \! \in \! X_-$ the stabilizer $\Lambda(x_-)$ is a lattice of $X$.
These lattices are never cocompact.
\end{theorem}
In the arithmetic case of ${\bf G}(\mathbf{F}_q[t,t^{-1}])$, the Weyl group has polynomial
growth (it is virtually abelian because it is an affine reflection group), so the condition
$W({1 \over q}) < \infty$ is empty and ${\bf G}(\mathbf{F}_q[t,t^{-1}])$ is always a
lattice of ${\bf G}\bigl( {\bf F}_q (\!( t )\!) \bigr) \times
{\bf G}\bigl( {\bf F}_q (\!( t^{-1} )\!) \bigr)$, whatever the value of the prime power $q$.
This is a particular case of a well-known result in reduction theory in positive
characteristic \cite{BehrRed}, \cite{Harder}.
\subsection{Finiteness properties}
\label{ss - finiteness}
Cohomological finiteness properties is a very hard problem for arithmetic groups in positive
characteristic \cite{Behr}, which makes a sharp difference with the number field case.
Nevertheless, it is natural to expect that some results, similar to those which are known
for arithmetic groups in the function field case, should hold for Kac-Moody groups.
This is indeed the case, up to taking into account more carefully the submatrices in the
generalized Cartan matrix defining the group -- this is the theme of P.
Abramenko's book \cite{Abr97}.
The following theorem sums up Theorems 1 and 2 from \cite{AbrBie} in a slightly
different language.
For instance, we introduce the parabolic subgroups (defined by means of $BN$-pairs in
[loc. cit.]) via the group action on the buildings: such a subgroup is a facet fixator.
Note also that the below quoted results are still valid in the more general context of
abstract {\it twin $BN$-pairs~} \cite[\S 1]{Abr97}.
\begin{theorem}
\label{th - finiteness for parbolics}
Let $\Lambda$ be a Kac-Moody group over ${\bf F}_q$, and let $\Gamma$ be a facet fixator.
Let $\Sigma$ be the Coxeter complex of the Weyl group $W$, i.e. the model for any
apartment in the building $X_\pm$ of $\Lambda$.
Let $R$ be a chamber of $\Sigma$ and let $\Pi$ be the set of reflections in the codimension
one faces of $R$.
\begin{enumerate}
\item[(i)] If any two reflections of $\Pi$ generate a finite group and if $q > 3$, then
$\Gamma$ is finitely generated; but it is not of cohomological type $FP_2$, in particular
not finitely presented, whenever some set of three reflections in $\Pi$ generates an infinite
subgroup of $W$.
\item[(ii)] If any three reflections of $\Pi$ generate a finite group and if $q > 6$, then
$\Gamma$ is finitely presentable; but it is not of cohomological type $FP_3$ whenever some
set of four reflections in $\Pi$ generates an infinite subgroup of $W$.
\end{enumerate}
\end{theorem}
When any two reflections of $\Pi$ generate a finite group, the Kac-Moody group is called
{\it $2$-spherical}.
Along with the Moufang property \cite[6.4]{RonLec} and twinnings \cite[Definition 3]{Abr97},
this notion plays a major role in the classification of buildings with infinite Weyl
groups.
In \cite{Abr97} some further results are available; they deal with the higher
cohomological finiteness properties $FP_n$ and $F_n$ \cite[VIII.5]{BroCohomology}.
According to Fact \ref{fact - finitely generated}, finite presentability is the
first finiteness property to be considered for the group $\Lambda$ itself.
A result due to P. Abramenko and B. M\"uhlherr \cite{AbrMue} is available:
\begin{theorem}
\label{th - finiteness for KM}
With the same notation as above, if any two reflections of $\Pi$ generate a finite group and
if $q > 3$, then $\Lambda$ is finitely presentable.
\end{theorem}
For instance, finite presentability holds for Kac-Moody groups whose buildings have chambers
isomorphic to hyperbolic regular triangles of angle ${\pi \over 4}$ and $q>3$, but never holds
for those whose buildings are covered by chambers isomorphic to a regular right-angled
$r$-gon, $r \geq 5$ (we will see in \ref{ss - hyperbolic} that such Kac-Moody groups exist
in both cases).
\pec
\subsection{Continuous cohomology and Kazhdan's property (T)}
\label{ss - T}
By results of J. Dymara and T. Januszkiewicz, being 2-spherical also implies useful
continuous cohomology vanishings for automorphism groups of buildings. The result below is
a special case of \cite[Theorem E]{DJ02}.
\begin{theorem}
\label{th - Kazhdan}
Let $\Lambda$ be a Kac-Moody group over $\mathbf{F}_q$, defined by a generalized
Cartan matrix $A$ of size $n \times n$.
Let $m<n$ be an integer such that all the principal submatrices of size $m \times m$ of $A$
are Cartan matrices (i.e. of finite type).
Then for $1 \leq k \leq m-1$ and $q >\!\!> 1$, the continuous cohomology groups
$H^k_{\rm ct}\bigl( {\rm Aut}(X_\pm),\rho \bigr)$ vanish for any unitary representation
$\rho$.
\end{theorem}
The first cohomology case is extremely useful since vanishing of $H^1_{\rm ct}(G,\rho)$ for
any unitary representation $\rho$ is equivalent to property (T) \cite[Chap. 4]
{HarVal}.
Therefore, when $\Lambda$ is 2-spherical, Theorem \ref{th - Kazhdan} implies property (T)
for the full automorphism groups
${\rm Aut}(X_\pm)$ with $q >\!\! >1$, hence for
their product, and finally for any lattice in this product, by S.P. Wang's Theorem
\cite[III, Theorem 2.12]{Margulis}.
The above result says in particular that many Kac-Moody groups have property (T), a
fundamental property for lattices of higher-rank simple algebraic groups
\cite[III]{Margulis}.
\subsection{Hyperbolic examples}
\label{ss - hyperbolic}
The existence of buildings with prescribed shapes of apartements and links around
vertices is well-known in many cases \cite{RonFree}, \cite{BouMostow}.
Some examples lead to interesting full automorphism groups and lattices \cite{BouPaj}, and
some other examples have surprisingly few automorphisms, though they are quite familiar
since they are Euclidean and tiled by regular triangles \cite{Bar}.
The result below \cite[Proposition 2.3]{RemGD} shows that some of these buildings are both
relevant to Kac-Moody theory and to (generalized) hyperbolic geometry \cite[III.H]{BriHae}.
This enables to mix arguments of algebraic and geometric nature in the study of the
corresponding Kac-Moody groups.
\pec
\begin{prop}
\label{prop - hyperbolics exist}
Let $P$ be a polyhedron in the hyperbolic space ${\Bbb H}^n$, with dihedral angles equal to
$0$ or to ${\pi \over m}$ with $m=2$, $3$, $4$ or $6$.
For any prime power $q$, there is a Kac-Moody group whose building has constant
thickness equal to $q+1$ and where the apartments are all tilings of ${\Bbb H}^n$ by $P$;
this building is a complete geodesic ${\rm CAT}(-1)$-metric space.
\end{prop}
Recall that the {\it thickness~} at a codimension 1 cell $C$ of a building is the number of
maximal cells containing $C$.
Note that according to G. Moussong \cite{Moussong}, any Coxeter
group acts discretely and cocompactly on a CAT(0)-space, which provides a good metric
realization of the associated Coxeter complex \cite[Chap. 2]{RonLec} (in which only
spherical facets are represented). Moreover, the metric space under consideration is
CAT(-1) if and only if the Coxeter group is Gromov-hyperbolic (which is a weaker property in
general \cite[Chap. 3]{GhyHar}). Using G. Moussong's non-positively curved complex as a
model for apartments, M. Davis
\cite{Davis} proved the existence of a CAT(0)-metric realization for any building, which is
CAT(-1) whenever the Weyl group of the building is Gromov-hyperbolic.
In the latter case, the metric complex is usually not as nice as a hyperbolic tiling as in
Proposition \ref{prop - hyperbolics exist}.
\subsection{Easy non-linearities}
\label{ss - easy NL}
On the one hand, in \ref{ss - lattice}-\ref{ss - T} we have quoted arguments
supporting the analogy between arithmetic groups over function fields and Kac-Moody groups
over finite fields.
On the other hand, \ref{ss - hyperbolic} shows that the
geometries are certainly new since hyperbolic buildings can be obtained.
Still, it is not proved at this stage that the groups are not well-known groups, only with
new actions: what about arguments showing that the groups are new?
This question leads to rigidity problems, i.e. proving that groups acting naturally on some
geometries cannot act via a big quotient on another geometry.
A way to attack the problem is to prove that the groups are not linear over any field.
Easy non-linearities are summed up in the following \cite[Theorem 4.6]{RemNewton}:
\begin{theorem}
\label{th - easy non-linearity}
Let $\Lambda$ be a Kac-Moody group over $\mathbf{F}_q$ with infinite Weyl group, and let
$\Gamma$ be a facet fixator in $\Lambda$.
Then $\Gamma$ always contains an infinite group of exponent $p$.
Therefore $\Gamma$ cannot be linear over any field of characteristic $\neq p$.
\end{theorem}
Exhibiting an infinite group of exponent $p$ is made possible thanks to arguments on
Kac-Moody root systems and pairs of parallel walls in infinite Coxeter complexes.
Then, elementary Zariski closure and algebraic group arguments imply that an
infinite group of exponent $p$ cannot be linear over any field of characteristic different
from $p$ \cite[Lemma VIII.3.7]{Margulis}.
\gec
\section{Generalized Kac-Moody groups. Non-isomorphic groups with the same building}
\label{s - generalized KM}
In this section, we concentrate on buildings whose apartments are tilings of the
hyperbolic plane.
Since we are interested in understanding new groups and geometries, it is natural to
consider the case of Fuchsian Weyl groups, corresponding to the simplest exotic Kac-Moody
groups.
The results are mainly of algebraic and combinatorial flavour; they emphasize the
role of conditions previously introduced to classify buildings.
Abstract isomorphisms between Kac-Moody groups with the same hyperbolic building can be
naturally factorized (\ref{ss - automorphisms}).
This is an analogy with the spherical building case, but a difference comes
when exhibiting some non-isomorphic groups with the same building (\ref{ss -
non-isomorphic}).
Moreover the local structure of right-angled Fuchsian buildings is
particularly simple: this enables to construct groups which are very close to Kac-Moody
groups, but so to speak defined over several ground fields (\ref{ss - mixing}).
Back to geometric group theory, this provides lattices of buildings of arbitrary large rank
satisfying a strong non-linearity property (\ref{ss - infinite kernel}).
\subsection{Factorization of abstract automorphisms}
\label{ss - automorphisms}
We say that a building is {\it Fuchsian~}if its Weyl group is the reflection
group of a tiling of the hyperbolic plane (in which case the latter tiling is a pleasant
model for apartments).
According to Poincar\'e's Theorem \cite[4.H]{Maskit}, Fuchsian tilings are nice metric
realizations of Coxeter complexes (the most familiar ones after Euclidean tilings).
Elementary facts from hyperbolic geometry enables to prove the following
\cite[Theorem 3.1]{RemGD}:
\begin{theorem}
\label{th - devissage}
Let $G$ and $G'$ be two Kac-Moody defined over the same finite field ${\hbox {\bf F}_q}$ of
cardinality $q \geq 4$.
Assume that the associated buildings are all isomorphic $(\star)$ either to the same locally
finite tree, $(\star\star)$ or to the same Fuchsian building with regular chambers.
Then, up to conjugacy in $G$, any abstract isomorphism from $G$ to $G'$ is the composition
of a permutation of the simple roots and possibly a global opposition of the sign of all
roots.
\end{theorem}
This factorisation result is close to R. Steinberg's classical result on finite groups of Lie
type, saying that an abstract automorphism of such a group is the product of a Dynkin diagram
automorphism, a ground field automorphism and an inner automorphism \cite[Theorem
12.5.1]{Carter}.
Other factorizations of isomorphisms have recently been obtained by P.-E. Caprace; the latter work is complementary since it deals with algebraically closed
fields \cite[Theorem 6]{Caprace}.
\subsection{Several isomorphism classes}
\label{ss - non-isomorphic}
Let $R$ be a right-angled $r$-gon of the hyperbolic plane ${\Bbb H}^2$.
For any integer $q \geq 2$ there is a unique building $I_{r,q+1}$ whose apartments are
Poincar\'e tilings of ${\Bbb H}^2$ by $R$, and such that the link at each vertex is the
complete bipartite graph of parameters $(q+1,q+1)$ \cite[2.2.1]{BouMostow}.
Here the {\it link~} of a vertex is a small ball around it, seen as a graph.
We call such a building the {\it right-angled Fuchsian building~} of parameters $r$ and
$q+1$.
These buildings are interesting because they locally look like products of trees, making
them simple combinatorially, but globally their Weyl group is irreducible and
Fuchsian, leading to remarkable rigidity properties \cite{BouMostow}, \cite{BouPaj}.
By uniqueness and Proposition \ref{prop - hyperbolics exist}, for each $r \geq 5$
the building $I_{r,q+1}$ comes from a Kac-Moody group whenever $q$ is a prime power.
Using Theorem \ref{th - devissage}, it can be proved that there are abstractly
non-isomorphic Kac-Moody groups with the same building \cite[\S 4, Proposition]{RemGD}:
\begin{cor}
\label{cor - non-isomorphic}
Let $q \geq 4$ be a prime power and let $r \geq 5$ be an integer.
Then there are several abstract isomorphism classes of Kac-Moody groups whose associated
buildings are the same $I_{r,q+1}$.
\end{cor}
This result is in contrast with the spherical case, where according to J. Tits'
classification, a spherical building of rank $\geq 3$ uniquely determines a field and
an algebraic group over this field \cite{TitsSpherical}.
Here the rank $r$ of the building is an arbitrary integer $\geq 5$.
This is an argument explaining why in the classification of Moufang
twinnings \cite{MuhRon}, \cite{Muhlherr}, the buildings must be assumed 2-spherical:
Kac-Moody groups with $I_{r,q+1}$ buildings obviously do not satisfy this property.
\subsection{Mixing ground fields}
\label{ss - mixing}
There is an even stronger argument showing that being 2-spherical is a necessary condition
for a Moufang twin building to be part of a reasonable classification (where the group side
would be given by Kac-Moody groups and their twisted versions).
Indeed, some generalizations of Kac-Moody groups with several ground fields can be
constructed, provided the buildings have apartments tiled by regular hyperbolic
right-angled $r$-gons.
Here \og generalization\fg means that the groups satisfy the same
combinatorial axioms as Kac-Moody groups (namely, those of twin root data \cite{Tit92}).
We have \cite[Theorem 3.E]{RemRon}:
\begin{theorem}
\label{th - exotic Fuchsian}
A right-angled Fuchsian building belongs to a Moufang twinning whenever its thicknesses
at panels are cardinalities of projective lines.
\end{theorem}
In the case of trees, a more theoretical construction is sketched by J. Tits in his Notes de
Cours au Coll\`ege de France \cite[\S 9]{TitsCDF}.
Still, the rank of the infinite dihedral Weyl group there is always equal to 2, whereas in
the above result it can be any integer $\geq 5$.
\subsection{Stronger non-linearities}
\label{ss - infinite kernel}
Besides its combinatorial interest, the above construction can be seen as a way to produce
lattices of hyperbolic buildings with remarkable non-linearity properties.
Recall that according to Theorem \ref{th - easy non-linearity} the characteristic $p$ of the
ground field of a Kac-Moody group prevents this group from being linear over any field of
characteristic $\neq p$.
Therefore, mixing fields of different characteristics in Theorem \ref{ss - mixing} should
enable to produce groups which are not linear over any field.
This is indeed the case \cite[Theorem 4.A]{RemRon}:
\begin{theorem}
\label{th - infinite kernel}
Let $r\geq 5$ be an integer and $\{ {\bf K}_i \}_{i \in {\bf Z}/r}$ be a family of
fields, among which are two fields with different positive characteristics.
Let $\Lambda$ be a group defined as in Theorem \ref{th - exotic Fuchsian} from these fields
and let $\Gamma$ be a chamber fixator.
Then, any group homomorphism $\rho: \Gamma \to \prod_{\alpha \in A} {\bf G}_\alpha({\bf
F}_\alpha)$ has infinite kernel, whenever the index set $A$ is finite and ${\bf G}_\alpha$
is a linear algebraic group defined over the field ${\bf F}_\alpha$ for each $\alpha \! \in
\! A$.
\end{theorem}
Note that the result is stronger than a plain non-linearity since mixing ground fields is
also allowed at the (linear) right hand-side of the representation.
Moreover the kernel is not only non-trivial, but always infinite.
From the purely Kac-Moody viewpoint, this result says that the truly difficult case of
non-linearity is when the characteristic of the target algebraic group is the characteristic
of the ground field of the Kac-Moofy group.
This case is reviewed in Sect. \ref{s - hard NL}.
\gec
\section{Generalized algebraic groups over local fields}
\label{s - topological}
A Kac-Moody group $\Lambda$ acts discretely on the product of its buildings, but its action
on a single factor is no longer discrete.
Therefore it makes sense to take the closure $\overline\Lambda < {\rm Aut}(X_\pm)$ of the
image of a Kac-Moody group in such a non-discrete action.
The result is called a {\it topological Kac-Moody group~} (the kernel of the
$\Lambda$-action on $X_\pm$ is the finite center of $\Lambda$).
In the classical case $\Lambda={\rm SL}_n({\bf F}_q[t,t^{-1}])$, $X_\pm$ is the
Bruhat-Tits building of ${\rm SL}_n \bigl( {\bf F}_q(\!(t)\!) \bigr)$ or
${\rm SL}_n \bigl( {\bf F}_q(\!(t^{-1})\!) \bigr)$, respectively.
If $\mu_n({\bf F}_q)$ denotes the $n$-th roots of unity in ${\bf F}_q$, the image
$\Lambda/Z(\Lambda)$ of ${\rm SL}_n({\bf F}_q[t,t^{-1}])$ under the action on $X_\pm$ is
${\rm SL}_n({\bf F}_q[t,t^{-1}])/\mu_n({\bf F}_q)$ and the
completions $\overline\Lambda$ are ${\rm PSL}_n \bigl( {\bf F}_q(\!(t)\!) \bigr)$ and
${\rm PSL}_n \bigl( {\bf F}_q(\!(t^{-1})\!) \bigr)$, respectively.
In fact, there are many arguments to
compare a topological Kac-Moody group with a semisimple group over a local field:
existence of a combinatorial structure refining Tits systems (\ref{ss - RTS}),
group-theoretic characterization of chamber-fixators in terms of pro-$p$ subgroups
(\ref{ss - Iwahori}), topological simplicity (\ref{ss - top simple}).
Still, as for discrete groups we have phenomena suggesting that some so-obtained
totally disconnected groups are new (\ref{ss - F boundaries}).
We finally quote a result exhibiting the coexistence of non-linear non-uniform lattices and
uniform lattices embedding convex-cocompactly in hyperbolic spaces (\ref{ss - coexistence}).
\subsection{Refined Tits systems}
\label{ss - RTS}
The structure of a {\it refined Tits system~} is due to V. Kac and D. Peterson
\cite{KacPet}; it is a generalization of a split $BN$-pair, a notion from the theory of finite
groups of Lie type. The difference with a plain Tits system is the formalization in the
axioms of the existence of an abstract unipotent subgroup in the Borel subgroup.
We have \cite[Theorem 1.C]{RemRon}:
\begin{theorem}
\label{th - refined TS}
Let $\Lambda$ be a Kac-Moody group over the finite field ${\bf F}_q$ of characteristic $p$.
The associated topological Kac-Moody group $\overline\Lambda$ admits a refined Tits system,
and the Tits system gives rise to a building in which any facet-fixator is, up to finite index,
a pro-$p$ group.
\end{theorem}
It is well-known that a group acting strongly transitively on a building
admits a natural Tits system \cite[Theorem 5.2]{RonLec}, so the point in the first
assertion lies in the difference between a Tits system and a refined Tits
system.
Moreover standard properties of double cosets in Bruhat decompositions imply
that topological Kac-Moody group are compactly generated \cite[Corollary 1.B.1]{RemLin}.
It is explained in \cite[1.B.1]{RemLin} why refined Tits systems for $\overline\Lambda$
imply the existence of a lot of torsion in facet fixators -- called {\it parahoric
subgroups}.
This explains why the analogy with semisimple groups over local fields is relevant
only when the local field has the same characteristic $p$ as the finite ground field of
$\Lambda$ (parahoric subgroups of semisimple groups over finite extensions of ${\bf Q}_l$
are virtually torsion free).
In analogy with the classical case of Bruhat-Tits theory \cite{BruhatTits1},
\cite{BruhatTits2}, we call {\it Iwahori subgroup~} a chamber fixator in
$\overline\Lambda$. Ê
\subsection{Iwahori subgroups}
\label{ss - Iwahori}
Back to the case $\Lambda={\rm SL}_n({\bf F}_q[t,t^{-1}])$, let $v$ be a vertex in the
Bruhat-Tits building of ${\rm SL}_n \bigl( {\bf F}_q(\!(t)\!) \bigr)$.
Then its fixator is isomorphic to ${\rm SL}_n({\bf F}_q[[t]])$, and for some chamber
containing $v$ the corresponding Iwahori subgroup is the group of matrices in
${\rm SL}_n({\bf F}_q[[t]])$ reducing to upper triangular matrices modulo $t$.
Moreover the first congruence subgroup of ${\rm SL}_n({\bf F}_q[[t]])$, i.e. the matrices
reducing to the identity modulo $t$, is a maximal pro-$p$ subgroup whose normalizer is the
above Iwahori subgroup.
In the general Kac-Moody setting, the result below \cite[Proposition 1.B.2]{RemLin} is a
generalization of \cite[Theorem 3.10]{PR94}.
\begin{prop}
\label{prop - Iwahori}
The Iwahori subgroups are group-theoretically characterized as the normalizers of
the pro-$p$ Sylow subgroups.
\end{prop}
In general, the analogue of the first congruence subgroup of a vertex fixator must be
defined as the (pointwise) fixator of the star around the vertex under consideration.
\subsection{Topological simplicity}
\label{ss - top simple}
In view of the simplicity of adjoint algebraic simple groups over large enough fields, the
theorem below \cite[Theorem 2.A.1]{RemLin} is not surprising:
\begin{theorem}
\label{th - top simple}
For $q \geq 4$, a topological Kac-Moody group over a finite field is the direct product of
finitely many topologically simple groups, with one factor for each connected component of
its Dynkin diagram.
\end{theorem}
This result is extremely useful when extending abstract representations of finitely
generated Kac-Moody to continuous representations of topological Kac-Moody groups
(Theorem \ref{th - embedding}).
The arguments of the proof are basically: a normal subgroup in an irreducible Tits system
either is chamber-transitive or acts trivially on the associated building
\cite[IV.2]{Bou81}, Iwahori subgroups are virtually pro-$p$, and a generating set for a
Kac-Moody group can be chosen in a finite collection of finite subgroups of Lie type (which
are perfect whenever $q \geq 4$).
It is reasonable to think that topological Kac-Moody groups are in fact abstractly simple
(Question \ref{quest - abstractly simple}).
\subsection{Non-homogeneous Furstenberg boundaries}
\label{ss - F boundaries}
When the building $X_\pm$ has hyperbolic apartments, the existence of many hyperbolic
translations leads to an interesting connection with topological dynamics.
Let $Y$ be a compact metrizable space and let us denote by $M^1(Y)$ the space
of probability measures on $Y$.
Recall that $M^1(Y)$ is compact and metrizable for the weak-$*$ topology.
If $Y$ admits a continuous action by a topological group $G$, we say that $Y$ is a {\it
Furstenberg boundary~} for $G$ if it is $G$-{\it minimal~} and $G$-{\it strongly proximal~}
\cite[VI.1.5]{Margulis}.
The first condition says that any $G$-orbit in $Y$ is dense and the second one says that any
$G$-orbit closure in $M^1(Y)$ contains a Dirac mass.
The arguments of \cite[Lemma 4.B.1]{RemLin} give:
\pec
\begin{lemma}
\label{lemma - Furstenberg}
Let $\Lambda$ be a Kac-Moody group whose buildings have apartments isomorphic to a
hyperbolic tiling.
Then the asymptotic boundary $\partial_\infty X_\pm$ is a Furstenberg boundary for any closed
automorphism group of $X_\pm$ containing $\overline\Lambda$.
\end{lemma}
This existence of non-homogeneous boundaries is new with respect to semisimple Lie groups
(archimedean or not), since in the latter case any Furstenberg boundary is equivariantly
isomorphic to a flag variety of the group \cite[9.37]{GJT}.
\subsection{Coexistence of two kinds of lattices}
\label{ss - coexistence}
We close this section by stating a result which says that some topological Kac-Moody
groups contain lattices of surprisingly different nature \cite[Proposition 4.B]{RemRon}.
\begin{prop}
\label{prop - two lattices}
There exist topological Kac-Moody groups $\overline\Lambda$ over ${\bf F}_q$ which contain
both non-uniform lattices which cannot be linear over any field of
characteristic prime to $q$,
and uniform lattices which have convex-cocompact embeddings into real
hyperbolic spaces.
The limit sets of the latter embeddings often have Hausdorff dimension $>2$.
\end{prop}
It follows from \cite[Corollary 0.5]{BurMoz96} that if a Kac-Moody group $\Lambda$ is
$S$-arithmetic and such that $\overline\Lambda$ is a higher-rank simple Lie group, then any
lattice of $\overline\Lambda$ fixes a point in each of its actions on proper
CAT(-1)-spaces. Therefore the above phenomenon is excluded in the classical algebraic
situation, unless the building of $\overline\Lambda$ is a tree, meaning that the Weyl group
$\Lambda$ is infinite dihedral (of rank 2 as a Coxeter group).
The rank $r$ of a Fuchsian building $I_{r,q+1}$ may be chosen arbitrarily large $\geq 5$.
\gec
\section{Non-linearity in equal characteristics}
\label{s - hard NL}
In view of the easy non-linearity results (Theorem \ref{th - easy non-linearity}), the
remaining linearity to disprove is for Kac-Moody groups over finite fields of characteristic
$p$ into linear groups of characteristic $p$.
But some Kac-Moody $S$-arithmetic groups such as ${\rm SL}_n \bigl( {\bf F}_q[t,t^{-1}]
\bigr)$ are linear in equal characteristic.
Therefore the question is rather, for any finite field of characteristic $p$, to find
examples of Kac-Moody groups that cannot be linear over any field of characteristic $p$
either.
The main idea is to use some superrigidity property (\ref{ss - commensurator
superrigidity}) to show that the existence of a faithful abstract homomorphism from a
finitely generated Kac-Moody group into an algebraic group implies the existence of
an embedding of a topological Kac-Moody group into a non-Archimedean simple group
(\ref{ss - embedding}).
The existence of such an embedding is expected to be simpler to disprove because it comes
with an embedding of the vertices of the (possibly exotic) Kac-Moody building to a Euclidean
one, which enables to take advantage of the incompatibility between hyperbolic and Euclidean
geometries. For some Kac-Moody groups with Fuchsian buildings this is indeed the case
(\ref{ss - NL Fuchsian}).
The same circle of ideas allows to show a complementary result: a non-faithful representation
from a finitely generated Kac-Moody group most of the time has a virtually solvable image
(\ref{ss - v solvable}).
Combining the latter results leads to groups all of whose linear images are
finite (\ref{ss - finite linear images}).
\pec
\subsection{Commensurator superrigidity}
\label{ss - commensurator superrigidity}
Let us first recall that the {\it commensurator~} of a group inclusion $\Delta<G$ is the
group:
\pec
\centerline{
${\rm Comm}_{G}(\Delta):= \{ g \! \in \! G \mid \Delta \cap
g\Delta g^{-1} \hbox{\rm ~has finite
index in both } \Delta \hbox{\rm ~and } g \Delta g^{-1}\}$.}
\pec
The next theorem is basically due to G.A. Margulis \cite[VII.5.4]{Margulis}, and the
formulation below can be found in \cite[Theorem 1]{Bonvin}.
G.A. Margulis proved it when $G$ is a semisimple group over a local field.
A certainly non-exhaustive list of later contributions is the following:
in \cite{AcaBur}, the semisimple group $G$ is replaced by a group containing an amenable
subgroup $P<G$ similar to a minimal parabolic subgroup;
in \cite{Burger} the existence of $P$ is replaced by the existence of a substitute for a
Furstenberg boundary;
in \cite{BurMon} such boundaries are constructed for compactly generated groups $G$;
and the double ergodicity Theorem in \cite{Kaim} shows that suitable boundaries are
available for any locally compact second countable group, via Poisson boundary theory.
Different approaches lead to similar results: \cite{Marg94} by means of equivariant
generalized harmonic mappings and \cite{Shalom} by means of representation theory.
\begin{theorem}
\label{th - superrigidity}
Let $G$ be a locally compact second countable topological group, $\Gamma < G$ be a lattice
and $\Lambda$ be a subgroup of $G$ with $\Gamma < \Lambda < {\rm Comm}_G (\Gamma)$.
Let $k$ be a local field and $H$ be a connected almost $k$-simple algebraic group.
Assume $\pi: \Lambda \to H_k$ is a homomorphism such that $\pi(\Lambda)$ is dense in the
Zariski topology on $H$ and $\pi(\Gamma)$ is unbounded in the Hausdorff topology on $H_k$.
Then $\pi$ extends to a continuous homomorphism $\overline{\Lambda} \to H_k / Z(H_k)$,
where $Z(H_k)$ is the center of $H_k$.
\end{theorem}
A more difficult superrigidity consists in extending representations of lattices
(instead of their commensurators) to continuous representations of the ambient
topological group (to algebraic representations when so is the ambient group).
This is a more difficult result which requires stronger assumptions (e.g. higher rank
for algebraic ambient groups), and again the main ideas are due to G.A. Margulis.
The first results in positive characteristic were proved by T.N. Venkataramana
\cite{Venka}.
\pec
\subsection{Embedding theorem}
\label{ss - embedding}
For the following theorem, we were inspired by a paper due to A. Lubotzky, Sh. Mozes and
R.J. Zimmer \cite{LMZ94}, where superrigidity is used to disprove
linearities for commensurators of tree lattices.
Trees are one-dimensional buildings, and Kac-Moody theory naturally leads to lattices for
buildings of arbitrary dimension.
We have \cite[1.B Lemma 2]{RemRon}:
\begin{lemma}
\label{lemma - commensurator}
Let $\Lambda$ be any Kac-Moody group over ${\bf F}_q$ and let $\Gamma$ be any negative facet
fixator in $\Lambda$.
Then: $\Lambda < {\rm Comm}_{\overline\Lambda}(\Gamma)$.
\end{lemma}
Therefore it makes sense to use commensurator superrigidity because when $q >\!\!> 1$,
$\Gamma$ is a lattice of $\overline\Lambda$ by Theorem \ref{th - lattice}.
This enables to obtain \cite[\S 3, Theorem]{RemLin}:
\begin{theorem}
\label{th - embedding}
Let $\Lambda$ be a Kac-Moody group over the finite
field ${\bf F}_q$ of
characteristic $p$ with $q > 4$ elements, with infinite Weyl group
$W$ and buildings $X_+$ and
$X_-$.
Let $\overline\Lambda$ be the corresponding Kac-Moody topological group.
We make the following assumptions:
\pec
\hskip 4mm {\rm (TS)~} the group $\overline\Lambda$ is topologically simple;
\hskip 4mm {\rm (NA)~} the group $\overline\Lambda$ is not amenable;
\hskip 4mm {\rm (LT)~} the group $\Lambda$ is a lattice of $X_+ \times X_-$ for its diagonal
action.
\pec
Then, if $\Lambda$ is linear over a field of characteristic $p$, there exists:
-- a local field $k$ of characteristic $p$ and a connected adjoint $k$-simple
group ${\bf G}$,
-- a topological embedding $\mu: \overline\Lambda \to {\bf G}(k)$
with Hausdorff unbounded
and Zariski dense image,
-- and a $\mu$-equivariant embedding $\iota: V_{X_+} \to V_\Delta$
from the set of vertices of the Kac-Moody building $X_+$ of $\Lambda$
to the set of vertices of
the Bruhat-Tits building $\Delta$ of ${\bf G}(k)$.
\end{theorem}
Conditions (TS) and (LT) are satisfied whenever the Weyl group $W$ of
$\Lambda$ is irreducible and $q >\!\!> 1$.
Condition (NA) is discussed in \ref{s - conjectures}, where it is conjectured that
it is empty (Problem \ref{pb - NA}).
Note that the conclusion of Theorem \ref{th - superrigidity} provides a
continuous extension but it is not clear that this topological homomorphism is a closed map.
Indeed, there is still work to do \cite[Lemma 3.C]{RemLin}
and, besides the topological simplicity of
$\overline\Lambda$, the arguments used for that are of combinatorial nature: basically
the Bruhat decomposition of $\overline\Lambda$ with respect to an Iwahori subgroup.
\subsection{Non-linear Fuchsian Kac-Moody groups}
\label{ss - NL Fuchsian}
In the case of some Kac-Moody groups with Fuchsian buildings, it can be proved that the
associated topological groups cannot be closed subgroups of any simple non-Archimedean Lie
group, thus implying the non-linearity of the involved finitely generated groups
\cite[Theorem 4.C.1]{RemLin}.
\begin{theorem}
\label{th - Fuchsian NL}
Let $\Lambda$ be a countable Kac-Moody group over ${\bf F}_q$ with right-angled
Fuchsian associated buildings.
Assume that any prenilpotent pair of roots not contained in a spherical root system leads
to a trivial commutation of the corresponding root groups.
Then for $q >\!\!> 1$, the group $\Lambda$ is not linear over any field.
\end{theorem}
The condition on commutation of root groups is technical: prenilpotency of pairs
of roots is relevant to abstract root systems of Coxeter groups \cite{Tit87},
\cite[1.4.1]{RemAst}.
It is not very restrictive and actually a weaker assumption may be
required -- see the remark after \cite[Lemma 4.A.2]{RemLin}.
Seeing roots as half-apartments, a pair of roots is prenilpotent if and only if the
walls of the roots intersect or if a root contains the other.
\pec
The main idea at this stage is to use a dynamical characterization of parabolic subgroups
proved by G. Prasad in a paper on strong approximation \cite[Lemma 2.4]{Prasad} in the
algebraic case, e.g. the case of the target ${\bf G}(k)$ of the continuous extension
$\mu : \overline\Lambda \to {\bf G}(k)$.
In the case of hyperbolic buildings, one must first say that a parabolic subgroup is by
definition a boundary point fixator (in analogy with the symmetric space or Bruhat-Tits
building case), and then show that the dynamical characterization holds too \cite[Lemma
4.B.2]{RemLin}.
We now have a dynamical roundabout to the non-existence of an algebraic structure on
$\overline\Lambda$, and this enables to show that under the continuous homomorphism $\mu$,
parabolic subgroups go to parabolic subgroups.
The contradiction comes when one notes that, thanks to the strong dynamics of a Fuchsian
group on the asymptotic boundary of the hyperbolic plane, the dynamical analogues of
unipotent radicals on the left hand-side are not normalized by the parabolic subgroups
containing them, whereas it is the case by definition on the right
hand-side of $\mu$.
\pec
\subsection{Virtual solvability of non-faithful linear images}
\label{ss - v solvable}
In \ref{ss - embedding}, the starting point is a faithful abstract representation from a
finitely generated Kac-Moody group.
Starting from non-faithful representations is interesting too, since it can be shown that in
this case the images are solvable up to finite index.
In other words, when the kernel is non-trivial, it is big \cite[Theorem 11]{RemImage}:
\begin{theorem}
\label{th - virtually solvabe}
Let $\Lambda$ be a Kac-Moody group over the finite field ${\bf F}_q$ of characteristic $p$,
with connected Dynkin diagram and $q >\!\!> 1$.
Let $\rho: \Lambda \to {\rm GL}_n({\bf F})$ be a linear representation.
If $\rho$ is not faithful, the group $\rho(\Lambda)$ is virtually solvable.
In particular if $\Lambda$ is Kazhdan, $\rho(\Lambda)$ is finite.
\end{theorem}
Recall that by Theorem \ref{th - Kazhdan}, many Kac-Moody groups over ${\bf F}_q$ are Kazhdan
whenever $q >\!\!> 1$.
The above theorem is proved thanks to the same kind of arguments as for Theorem \ref{th -
embedding}.
In order to be in position to apply Theorem \ref{th - superrigidity}, we have to take the
Zariski closure of the image $\rho(\Lambda)$ and to mod out by the radical
$R\bigl(\overline{\rho(\Lambda)}^Z\bigr)$ of the latter algebraic group.
It then suffices to show that the image of $\Lambda$ in
$\overline{\rho(\Lambda)}^Z/R\bigl(\overline{\rho(\Lambda)}^Z\bigr)$ is finite,
which can be done thanks to Burnside's theorem \cite[4.5, Exercise 8]{Jacobson}.
A way to fulfill the unboundedness assumption on the image of $\Lambda$ (in the Hausdorff
topology) is to use tricks from Tits' paper on the existence of free groups in linear groups
\cite{Tit72}.
\pec
\subsection{Groups all of whose linear images are finite}
\label{ss - finite linear images}
From the previous result, it can be shown that some Kac-Moody groups with hyperbolic
buildings don't have any infinite linear image \cite[Theorem 16]{RemImage}.
\begin{theorem}
\label{th - finite linear images}
There is an integer $N$ such that for any Kac-Moody group $\Lambda$ over ${\bf F}_q$ whose
buildings have apartments isomorphic to the tesselation of the hyperbolic plane by regular
triangles of angle ${\pi \over 4}$, if $q \geq N$ then any linear image of $\Lambda$ is
finite, whatever the target field.
\end{theorem}
The proof roughly goes as follows.
By Theorem \ref{th - virtually solvabe}, it is enough to show that some Kac-Moody groups
enjoying Kazhdan's property (T) are not linear over any field.
According to Theorem \ref{th - Kazhdan}, the groups as in the theorem are
Kazhdan, and by twin root datum arguments it can be proved that such groups contain
Kac-Moody-like subgroups to which the proof of Theorem \ref{th - Fuchsian NL} applies.
\gec
\section{Conjectures and questions}
\label{s - conjectures}
In this final section, we propose a few questions about finitely generated and
totally disconnected Kac-Moody groups.
Here, the understatement in {\it totally disconnected~}Êis that the group under
consideration is uncountable and not endowed with the discrete topology.
\subsection{Finitely generated groups}
\label{ss - q discrete}
For discrete groups, we think that the class of non-linear Kac-Moody groups
is much wider than the one for which the property has been proved so far.
\begin{conj}
\label{conj - hyperbolic implies non-linear}
If the Weyl group $W$ is Gromov-hyperbolic and if $q >\!\!> 1$, then the
Kac-Moody group $\Lambda$ is not linear.
\end{conj}
Recall that according to G. Moussong, in the class of Coxeter groups hyperbolicity is
equivalent to acting discretely and cocompactly on a ${\rm CAT}(-1)$-space \cite{Moussong}.
The non-linearity proof of Sect. \ref{s - hard NL} deals with groups $\Lambda$ whose Weyl
groups have pleasant Coxeter complexes since the latter complexes are Fuchsian tilings.
In general, Moussong's complex is defined by isometric gluings of cells and is less easy to
understand, so one may have to use arguments of combinatorial nature on Kac-Moody
root systems at some points.
Note also that one should get rid of the technical condition on prenilpotent pairs of roots
in the statement of Theorem \ref{th - Fuchsian NL}.
\pec
Another natural question is:
\begin{quest}
\label{quest - on q}
Can the assumption \og $q >\!\!> 1$\fg be removed in non-linearity results?
\end{quest}
In other words: is there a generalized Cartan matrix $A$ and a prime number $p$ such that the
corresponding group is linear over ${\bf Z}/p$ but no longer linear over
${\bf F}_{p^r}$ for large enough $r$?
Note that for $\Lambda$ to be a lattice of the product of its buildings -- which is a
crucial argument in our proof, the value of $q$ is important (Theorem \ref{th - lattice}).
So if the answer to the above problem is yes, the non-linearity proof should require new
ideas.
Moreover a counter-example due to P. Abramenko shows that for buildings whose chambers are
regular hyperbolic triangles of angles ${\pi \over 4}$, facet fixators in some groups over
${\bf F}_2$ or ${\bf F}_3$ are not finitely generated \cite[Counter-example 1, Remark
2]{AbrBie}, hence cannot have property (T) \cite[Theorem III.2.7]{Margulis}.
But according to Theorem \ref{th - Kazhdan}, for $q >\!\!> 1$ the groups do enjoy Kazhdan's
property (T).
\pec
The most general question on non-linearity for Kac-Moody groups is:
\begin{pb}
\label{pb - GCM}
Find necessary and sufficient conditions for the non-linearity of a finitely generated
Kac-Moody group, only in terms of the generalized Cartan matrix defining the group.
\end{pb}
Of course, if the generalized Cartan matrix $A$ is of affine type, meaning that the Weyl
group is a Euclidean reflection group, then the corresponding Kac-Moody groups are
$S$-arithmetic groups, hence are linear.
A restatement is: are there other linear examples than the affine ones?
\pec
At last, there is another problem which comes from the analogy with the situation of
lattices, namely the problem of arithmeticity.
Of course when algebraic structures are available, the definition is well-known
\cite[Definitions 6.1.1 and 10.1.11]{Zimmer}.
In the general context of an inclusion $\Delta < G$ of a discrete subgroup $\Delta$ in a
locally compact group $G$, it has now become classical to say that $\Delta$ is {\it
arithmetic~}Êin $G$ if by definition ${\rm Comm}_G(\Delta)$ is dense in $G$
(\ref{ss - commensurator superrigidity}).
This is reasonable since by Margulis' commensurator criterion, a lattice in a non-compact
simple Lie group is arithmetic in the classical sense if and only if so it is in the
above sense \cite[Theorem 6.2.5]{Zimmer}.
This definition makes sense in the case where the topological group $G$ is the isometry group
of some metric space, e.g. of some building with enough autmorphisms.
Since Kac-Moody buildings are close with many respects to Bruhat-Tits buildings, it is
natural to ask:
\begin{quest}
\label{quest - arithmetic}
Let $X_+$ be the positive building of some Kac-Moody group $\Lambda$ over ${\bf F}_q$.
Is a negative facet fixator in $\Lambda$ arithmetic in ${\rm Aut}(X_+)$?
More generally, which lattices, not necessarily relevant to Kac-Moody groups, are arithmetic
in ${\rm Aut}(X_+)$?
\end{quest}
Note that by definition of a topological Kac-Moody group (Sect. \ref{s - topological},
introduction), any negative facet fixator $\Gamma$ is arithmetic in $\overline\Lambda$ since
by Lemma \ref{lemma - commensurator}, ${\rm Comm}_{\overline\Lambda}(\Gamma)$ contains
$\Lambda$.
Here is a list of known arithmeticity results:
\pec
\begin{enumerate}
\item[(i)] Cocompact lattices in a locally finite tree $T$ are all arithmetic in ${\rm
Aut}(T)$ \cite{Liu}.
\item[(ii)] The Nagao lattice ${\rm SL}_2 \bigl( {\bf F}_q [t^{-1}] \bigr)$
is a non-uniform arithmetic lattice in the full automorphism group of the Bruhat-Tits tree of
${\rm SL}_2 \bigl( {\bf F}_q (\!( t )\!) \bigr)$ \cite{MozNagao}.
This was later extended to some Moufang twin trees, which enables to deal with non-linear
non-uniform tree lattices \cite{AbrRem}.
\item[(iii)] Uniform lattices in some hyperbolic buildings, not necessarily from Kac-Moody
theory \cite{HagComm}.
\end{enumerate}
See the end of \cite{AbrRem} for more precise questions.
\subsection{Totally disconnected groups}
\label{ss - q td}
Theorem \ref{th - embedding} shows that non-linearity of some finitely generated groups
roughly amounts to non-linearity of much bigger topological groups.
The study of topological Kac-Moody groups in Sect. \ref{s - topological} made appear an
interesting class of pro-$p$ subgroups.
A way to disprove some linearities would be to answer the following:
\begin{quest}
\label{quest - non-linear pro-p}
Which topological Kac-Moody groups contain non-linear pro-$p$ subgroups?
\end{quest}
Forgetting discrete groups, the question is interesting in its own: for instance, the
linearity of free pro-$p$ subgroups is a question so far admitting only partial answers
(E. Zelmanov).
\pec
In the classical case, topological Kac-Moody groups correspond to groups of rational points
of adjoint semisimple groups, so it makes sense to ask the following:
\begin{quest}
\label{quest - abstractly simple}
Are topological Kac-Moody groups direct products of abstractly simple groups?
\end{quest}
Theorem \ref{th - top simple} only says that the latter groups are direct products
of topologically simple groups.
F. Haglund and F. Paulin, elaborating on J. Tits' proof for trees \cite{TitsTree},
showed the abstract simplicity of full automorphism groups of many hyperbolic buildings
\cite[Theorems 1.1 and 1.2]{HagPau}.
\pec
At last, in two important cases topological Kac-Moody groups $\overline\Lambda$ are not
amenable.
(Recall that a simple Lie group, when non-compact, is not amenable.)
\begin{pb}
\label{pb - NA}
Show that topological Kac-Moody groups are never amenable.
\end{pb}
Solving the problem, even for $q>\!\!>1$, would enable to remove assumption (NA) in Theorem
\ref{th - embedding}.
It is solved when the Weyl group $W$ of $\Lambda$ is Gromov-hyperbolic, and when property
(T) holds for $\overline\Lambda$.
The latter case is clear because amenability and property (T) imply compactness
\cite[Corollary 7.1.9]{Zimmer}, which then implies to have a fixed point in the building
$X_+$ by non-positive curvature \cite[VI.4]{BroBuildings}: contradiction with the
chamber-transitivity of the $\Lambda$-action on $X_+$.
The former case follows from a Furstenberg lemma for CAT(-1)-spaces \cite[Lemma
2.3]{BurMoz96} -- see \cite[Sect. 3, Introduction]{RemLin} for details.
\bibliographystyle{amsalpha}
\bibliography{DiscreteSurvey}
\vspace{1cm}
\addtolength{\parindent}{-1.6pt}
\gec
Institut Fourier -- UMR CNRS 5582\\
Universit\'e de Grenoble 1 -- Joseph Fourier\\
100, rue des maths -- BP 74\\
38402 Saint-Martin-d'H\`eres Cedex -- France\\
{\tt bremy@fourier.ujf-grenoble.fr}
\end{document} | {"config": "arxiv", "file": "math0402300/DiscreteSurvey.tex"} |
TITLE: Convergence with the $5$-adic metric.
QUESTION [3 upvotes]: I'm struggling to get any intuition for the following example:
Consider the sequence $\{a_k\} = \{3, \ 33, \ 333, \ 3333, \ \ldots\}$
It's easy to show, using the geometric series formula, that $a_k = \frac{1}{3}(10^k-1)$.
It follows that $3a_k = 10^k-1$ and hence $3a_k+1=10^k$, and so $3a_k+1 \equiv 0 \bmod 5^k$.
By the definition of the $5$-adic metric, we have $0 \le |3a_k+1|_5 \le 5^{-k}$.
As $k \to \infty$, $5^{-k} \to 0$ and so $|3a_k+1|_5 \to 0$. We conclude that $a_k \to -\frac{1}{3}$ ($5$-adically).
How can a sequence of positive integers tend to a negative fraction?
It seems that $|a_k|_5 = 1$ for all $k$, but then $a_k \to -\frac{1}{3}$?
REPLY [6 votes]: Promoting some of the comments to an answer with a view of A) removing this from the list of unanswered questions, B) covering some of the features of the $p$-adic metric that surprise learners at first, and C) also aiming to draw an analogy with the ring of formal power series.
1. The $p$-adic "size" of rational numbers does not match the intuition based on the archimedean valuation better known as the absolute value. $2$-adically a high power of $2$ is tiny in comparison to a meager $-1$. Therefore, for example, the sequences $2^n\to0$ and $2^n-1\to-1$. In other words, in a series like
$$1+2+4+8+16+32+\cdots$$ the first term $1$ is the dominant one. This is not atypical of converging series in all domains :-)
2. The formula for the sum of a geometric series is your friend. Whenever it converges the sum of the series is surely (by the usual argument)
$$
a+aq+aq^2+aq^3+\cdots=\frac a{1-q}.\qquad(*)
$$
Here it converges if and only if the ratio $q$ has $p$-adic size $<1$ (hardly a surprise!). In other words, the formula $(*)$ holds iff $|q|_p<1$. Therefore, as Fly by Night observed
$$
...33333=3+30+300+3000+30000+\cdots=\frac3{1-10}
$$
whenever $|10|_p<1$. This inequality holds when $p=5$ or $p=2$, and in both cases the sum of this series is $-1/3$.
3. The mildly surprising fact about this limit is that a sequence of positive rationals has a negative limit. This will cease to amaze a learner whenever they recall that $2$-adically $1024$ is very close to $-1024$, but relatively far away from $1025$. A more formal way of phrasing this is that the $p$-adic fields do not have a total ordering - a relation that would allow us to, among other things, partition the $p$-adic numbers into negative and positive numbers. One rigorous argument for that parallels the reasoning why we don't have a total ordering in $\Bbb{C}$ either. Remember that if $i$ were either positive or negative, then its square should be positive, which it ain't. Similarly in all $p$-adic fields some negative integers have square roots. There is a $5$-adic $\sqrt{-1}$ (see here for a crude description of the process of finding a sequence of integers converging $5$-adically to a number with square $=-1$. Exactly which integers have $p$-adic square roots is number-theoretic in nature. For example when $p=2$ it turns out that $\sqrt{m}$ of an odd integer $m$ exists inside $\Bbb{Q}_2$, iff $m\equiv1\pmod8$, so $-7$ has a $2$-adic square root.
4. The same themes recur, if we want to define a $p$-adic exponential function with the usual power series
$$
\exp(x):=\sum_{n=0}^\infty\frac{x^n}{n!}.
$$
The problem is with convergence. Contrary to expectations from real analysis this series will not converge for all $x\in\Bbb{Q}_p$. The reason is the denominators. For large $n$, the factorial will be divisible by higher and higher powers of $p$. Therefore we are dividing by a sequence of small numbers tending towards zero, and the numerator $x^n$ needs to compensate for that. A more careful analysis of the situation reveals, that this series converges, iff
$|x|_p<p^{-1/(p-1)}$.
5. Apropos series. An analogue of the $p$-adic metric you may be familiar from courses on analysis is the $x$-adic topology (can turn it into a metric if so desired) on (formal) power series (with coefficients in, say, $\Bbb{R}$!). Two power series are close to each other $x$-adically, iff their difference is divisible by a high power of $x$. Therefore we can say that $e^x$ and $1+x+x^2/2$ are already quite close to each other, but $\sin x$ and $x-x^3/6+x^5/120$ are closer still. A common feature of all non-archimedean metrics ($p$-adic, $x$-adic,...) is that adding "small" numbers together will never make a large number, no matter how many of them you add together. So w.r.t. the $2$-adic metric no matter how many numbers divisible by four you add together you will never get a large number, say an odd number. Similarly, if you add together several formal power series divisible by $x$ you never create a non-zero constant term. This has an impact on some things. For example, when defining integrals, we want to approximate something by a sum of small things. In the $p$-adic world we need to... | {"set_name": "stack_exchange", "score": 3, "question_id": 1867813} |
\begin{document}
\title{Weierstrass filtration on Teichm\"{u}ller curves and Lyapunov exponents: Upper bounds}
\author{Fei Yu}
\author{Kang Zuo}
\address{Universit{\"a}t Mainz, Fachbereich 17, Mathematik, 55099 Mainz, Germany}
\email{yuf@uni-mainz.de,vvyufei@gmail.com}
\address{Universit{\"a}t Mainz, Fachbereich 17, Mathematik, 55099 Mainz, Germany}
\email{zuok@uni-mainz.de}
\date{\today. \\This work is supported by the SFB/TR 45 ¡®Periods, Moduli Spaces and Arithmetic of Algebraic
Varieties¡¯ of the DFG}
\maketitle
\begin{abstract}We get an
upper bound of the slope of each graded quotient for the
Harder-Narasimhan filtration of the Hodge bundle of a
Teichm\"{u}ller curve. As an application, we show that the sum of
Lyapunov exponents of a Teichm\"{u}ller curve does not exceed
${(g+1)}/{2}$, with equality reached if and only if the curve lies
in the hyperelliptic locus induced from
$\mathcal{Q}(2k_1,...,2k_n,-1^{2g+2})$ or it is some special
Teichm\"{u}ller curve in $\Omega\mathcal{M}_g(1^{2g-2})$. Under some
additional assumptions, we also get an upper bound of individual
Lyapunov exponents; in particular we get Lyapunov exponents in
hyperelliptic loci and low genus non-varying strata.
\end{abstract}
\tableofcontents
\section{Introduction}
Let $\mathcal{M}_g$ be the moduli space of Riemann surfaces of genus
$g$, and $\Omega\mathcal{M}_g\rightarrow \mathcal{M}_g$ the bundle
of pairs $(X,\omega)$, where $\omega\neq 0$ is a holomorphic 1-form
on $X\in \mathcal{M}_g$. Denote
$\Omega\mathcal{M}_g(m_1,...m_k)\hookrightarrow \Omega\mathcal{M}_g$
the stratum of pairs $(X,\omega)$, where $\omega(\neq 0)$ have $k$
distinct zeros of order $m_1,...,m_k$ respectively.
There is a nature action of $GL_2^+(\mathbb{R})$ on
$\Omega\mathcal{M}_g(m_1,...m_k)$, whose orbits project to complex
geodesics in $\mathcal{M}_g$. The projection of an orbit is almost
always dense. However, if the stabilizer $SL(X,\omega)\subset
SL_2(\mathbb{R})$ of a given form is a lattice, then the
projection of its orbit gives a closed, algebraic Teichm\"{u}ller
curve $C$.
After suitable base change and compacfication, we can get a
universal family $f:S\rightarrow C$, which is a relative minimal
semistable model with disjoint sections $D_1,..,D_k$; here $D_i|_X$
is a zero of $\omega$ when restrict to each fiber $X$.
The relative canonical bundle formula \eqref{canonical} of the
Teichm\"{u}ller curve is(\cite{CM11}\cite{EKZ}):
$$\omega_{S/C}\simeq f^*\mathcal{L}\otimes \mathcal{O}(\sum_i m_i
D_i)$$ Here $\mathcal{L}\subset f_*{\omega_{S/C}}$ be the line
bundle whose fiber over the point corresponding to $X$ is
$\mathbb{C}\omega$, the generating differential of Teichm\"{u}ller
curves.
There are many nature vector subbundles of the Hodge bundle
$f_*(\omega_{S/C})$:
$$\mathcal{L}\otimes f_*\mathcal{O}(\sum d_i D_i)\subset
\mathcal{L}\otimes f_*\mathcal{O}(\sum m_i D_i)=f_*(\omega_{S/C})$$
One can construct many filtration by using these subbundles. In
particular, using properties of Weierstrass semigroups, we have
constructed the Harder-Narasimhan filtration of $f_*(\omega_{S/C})$
for Teichm\"{u}ller curves in hyperelliptic loci and some low genus
nonvarying strata \cite{YZ}. In this article, we will get an upper
bound of the slope of each graded quotient for the Harder-Narasimhan
filtration of $f_*(\omega_{S/C})$ of Teichm\"{u}ller curves in each
stratum.
For a vector bundle $V$, define $\mu_i(V)=\mu(gr^{HN}_j)$ if
$rk(HN_{j-1}(V))<i\leq rk(HN_j(V))$. Write $w_i$ for
$\mu_i(f_*(\omega_{S/C}))/deg(\mathcal{L})$.
\begin{lemma}(Lemma \ref{ubhn}) For a Teichm\"{u}ller
curve which lies in $\Omega\mathcal{M}_g(m_1,...m_k)$, we have
inequalities:
$$w_i\leq 1+a_{H_i(P)}$$ Here $a_i$ is the $i$-th largest number in
$\{-\frac{j}{m_i+1}|1\leq j\leq m_i,1\leq i\leq k\}$, $P$ is the
special permutation \eqref{arrange} and $H_i(P)\geq 2i-2$.
\end{lemma}
Fix an $SL_2(\mathbb{R})$-invariant, ergodic measure $\mu$ on
$\Omega\mathcal{M}_g$. The Lyapunov exponents for the
Teichm\"{u}ller geodesic flow on $\Omega\mathcal{M}_g$ measure the
logarithm of the growth rate of the Hodge norm of cohomology classes
under the parallel transport along the geodesic flow.
In general, it is difficult to compute the Lyapunov exponents.
There are some algebraic attempts to compute the sum of certain
Lyapunov exponents, all of which are based on the following fact:
the sum of these Lyapunov exponents is related with the degree of
certain vector bundles (cf. Theorem \ref{sumly}). In particular, the
sum of Lyapunov exponents of a Teichm\"{u}ller curve equals
$deg(f_*(\omega_{S/C}))/deg(\mathcal{L})$. This algebraic
interpretation combined with information about the
Harder-Narasimhan filtration gives the following estimate:
\begin{theorem} (Theorem \ref{main}) The sum of Lyapunov exponents of a Teichm\"{u}ller curve in $\Omega\mathcal{M}_g(m_1,...m_k)$ satisfies the inequality
$$L(C)\leq \frac{g+1}{2}$$
Furthermore, equality occurs if and only if it lies in the
hyperelliptic locus induced from
$\mathcal{Q}(2k_1,...,2k_n,-1^{2g+2})$ or it is some special
Teichm\"{u}ller curve in $\Omega\mathcal{M}_g(1^{2g-2})$.
\end{theorem}
D.W. Chen and M. M\"{o}ller have obtained many interesting upper
bounds in \cite{CM11}\cite{Mo12}.
The Harder-Narasimhan filtration also gives rise to an upper bound
of the degrees of any vector subbundles, especially those related
to the sum of certain Lyapunov exponents (cf. Proposition \ref{pa}).
For individual Lyapunov exponents, due to the lack of algebraic
interpretation, we will make the following assumption:
\begin{assumption}\label{assumption}$f_*(\omega_{S/C})$ equals $(\overset{k}{\underset{i=1}{\oplus}} L_i) \oplus W$, here $L_i$ are line bundles
such that the $i$-th Lyapunov exponent satisfies the equality:
$$\lambda_i=\{
\begin{array}{cc}
deg(L_i)/deg(\mathcal{L}) &1\leq i\leq k \\
0 &k< i\leq g
\end{array}
$$
\end{assumption}
There are many examples satisfying this assumption: triangle groups
\cite{BM10}, square tiled cyclic covers \cite{EKZ11}\cite{FMZ11a},
square tiled abelian covers \cite{Wr1}, some wind-tree models
\cite{DHL}, and algebraic primitives.
Our estimate on the slopes of the Harder-Narasimhan filtration will
give the following upper bound for individual Lyapunov exponents:
\begin{proposition}(Proposition \ref{single}) For a Teichm\"{u}ller curve which
satisfies the assumption \ref{assumption} and lies in
$\Omega\mathcal{M}_g(m_1,...m_k)$, the $i$-th Lyapunov exponent
satisfies the inequality:
$$\lambda_i \leq 1+a_{H_i(P)}$$
Here $a_i$ is the $i$-th largest number in $\{-\frac{j}{m_i+1}|1\leq
j\leq m_i,1\leq i\leq k\}$, $P$ is the special permutation
\eqref{arrange} and $H_i(P)\geq 2i-2$.
\end{proposition}
The equality can be reached for an algebraic primitive
Teichm\"{u}ller curve lying in the hyperelliptic locus induced from
$\mathcal{Q}(2k_1,...,2k_n,-1^{2g+2})$.
For Teichm\"{u}ller curves lying in hyperelliptic loci
and some low genus nonvarying strata, the following proposition is
obvious because we have constructed the Harder-Narasimhan filtration
in \cite{YZ}.
\begin{proposition}(Proposition \ref{nonva}) For a Teichm\"{u}ller curve which
satisfies the assumption \ref{assumption} and lies in hyperelliptic
loci or one of the following strata:\\
$\overline{\Omega\mathcal{M}}_3(4),\overline{\Omega\mathcal{M}}_3(3,1),\overline{\Omega\mathcal{M}}^{odd}_3(2,2),\overline{\Omega\mathcal{M}}_3(2,1,1)$\\
$\overline{\Omega\mathcal{M}}_4(6),\overline{\Omega\mathcal{M}}_4(5,1),\overline{\Omega\mathcal{M}}^{odd}_4(4,2),\overline{\Omega\mathcal{M}}^{non-hyp}_4(3,3),\overline{\Omega\mathcal{M}}^{odd}_4(2,2,2),\overline{\Omega\mathcal{M}}_4(3,2,1)$\\
$\overline{\Omega\mathcal{M}}_5(8),\overline{\Omega\mathcal{M}}_5(5,3),\overline{\Omega\mathcal{M}}^{odd}_5(6,2)$\\
The $i$-th Lyapunov exponent $\lambda_i$ equals the $w_i$ which is computed in the theorem \ref{YZ}.
\end{proposition}
\paragraph{\textbf{Acknowledgement}}We thank Ke Chen for a careful
reading and his many suggestions.
\section{Harder-Narasimhan
filtration}
The readers are referred to \cite{HL} for details about sheaves on
algebraic varieties. Let $C$ be a smooth projective curve, $V$ a
vector bundle over $C$ of slope $\mu(V):=\frac{deg(V)}{rk(V)}$. We
call $V$ semistable (resp.stable) if $\mu(W)\leq\mu(V)$
(resp.$\mu(W)<\mu(V)$) for any subbundle $W\subset V$. If $V_1,V_2$
are semistable such that $\mu(V_1)>\mu(V_2)$, then any map
$V_1\rightarrow V_2$ is zero.
A Harder-Narasimhan filtration for $V$ is an increasing filtration:
$$0=HN_0(V)\subset HN_1(V)\subset ...\subset HN_k(V)$$
such that the graded quotients $gr^{HN}_i=HN_i(V)/HN_{i-1}(V)$ for
$i=1,...,k$ are semistable vector bundles and
$$\mu(gr^{HN}_1)>\mu(gr^{HN}_2)>...>\mu(gr^{HN}_k)$$
The Harder-Narasimhan filtration is unique.
A Jordan-H\"{o}lder filtration for semistable vector bundle $V$ is a
filtration:
$$0=V_0\subset V_1\subset ...\subset V_k=V$$
such that the graded quotients $gr^{V}_i=V_i/V_{i-1}$ are stable of
the same slope.
Jordan-H\"{o}lder filtration always exist. The graded objects
$gr^{V}_i=\oplus gr^{V}_i$ do not depend on the choice of the
Jordan-H\"{o}lder filtration.
For a vector bundle $V$, define $\mu_i(V)=\mu(gr^{HN}_j)$ if
$rk(HN_{j-1}(V))<i\leq rk(HN_j(V))$. Obviously we have $\mu_1(V)\geq
...\geq \mu_k(V)$.
\begin{lemma}\label{control} Let $V$ and $U$ be two vector bundles of rank $n$ over $C$, with increasing filtration
$$0=V_0\subset V_1\subset ...\subset V_n=V$$
$$0=U_0\subset U_1\subset ...\subset U_n=U$$
such that $V_i/V_{i-1},U_i/U_{i-1}$ are line bundles ,
$V_i/V_{i-1}\subset U_i/U_{i-1}$ and the degrees $deg(U_i/U_{i-1})$
decrease in $i$ ($1\leq i\leq n$). Then $\mu_i(V)\leq
deg(U_i/U_{i-1})$.
\end{lemma}
\begin{proof}If there is some $\mu_i(V)$ bigger than $
deg(U_i/U_{i-1})$, where $\mu_i(V)=\mu(gr^{HN(V)}_j)$, then
$\mu_i(V)>deg(U_i/U_{i-1})\geq deg(U_l/U_{l-1})\geq
deg(V_l/V_{l-1})$, for $l\geq i$.
We will show that the canonical morphism $HN_j(V)\hookrightarrow
V\rightarrow V/V_{i-1}$ is zero, namely $HN_j(V)\hookrightarrow
V_{i-1}$, which is a contradiction because $rk(HN_j(V))\geq
i>rk(V_{i-1})$.
For $m\leq j,l\geq i$, the quotients $gr^{HN(V)}_m,V_l/V_{l-1}$ are
semistable and $\mu(gr^{HN(V)}_m)\geq\mu_i(V)>deg(V_l/V_{l-1})$, so
any map $gr^{HN(V)}_m\rightarrow V_l/V_{l-1}$ is zero. Thus any map
$gr^{HN(V)}_m\rightarrow V/V_{i-1}$ is zero by induction on $l$, and
any map $HN_j(V) \rightarrow V/V_{i-1}$ is zero by induction on $m$.
\end{proof}
Let $grad(HN(V))$ denote the direct sum of the graded quotients of
the Harder-Narasimhan filtration: $grad(HN(V))=\oplus gr^{HN(V)}_i$.
\begin{lemma}\label{directsum}Given vector bundles $V_1,...,V_n$, we have: $$grad(HN(V_1\oplus...\oplus V_n))=grad(HN(V_1))\oplus...\oplus grad(HN(V_n))$$
and $\mu_i(V_j)$ equals $\mu_k(V_1\oplus...\oplus V_n)$ for some
$k$.
\end{lemma}
\begin{proof}By induction, we only need to show the case $n=2$. Let
$$0=HN_0(V_1)\subset HN_1(V_1)\subset ...\subset HN_{k_1}(V_1)$$
$$0=HN_0(V_2)\subset HN_1(V_2)\subset ...\subset HN_{k_2}(V_2)$$
be the Harder-Narasimhan filtration of $V_1,V_2$ respectively.
Set $0=HN_0(V_1\oplus V_2)=HN_0(V_1)\oplus HN_0(V_2)$. Assume we
have set $HN_i(V_1\oplus V_2)=HN_{i_1}(V_1)\oplus HN_{i_2}(V_2)$. We
will get $HN_{i+1}(V_1\oplus V_2)$ by the following rule:
\begin{itemize}
\item If $\mu(HN_{i_1+1}(V_1)/HN_{i_1}(V_1))>\mu(HN_{i_2+1}(V_2)/HN_{i_2}(V_2))$
then let $HN_{i+1}(V_1\oplus V_2)=HN_{i_1+1}(V_1)\oplus
HN_{i_2}(V_2)$.
\item If $\mu(HN_{i_1+1}(V_1)/HN_{i_1}(V_1))=\mu(HN_{i_2+1}(V_2)/HN_{i_2}(V_2))$
then let $HN_{i+1}(V_1\oplus V_2)=HN_{i_1+1}(V_1)\oplus
HN_{i_2+1}(V_2)$.
\item If $\mu(HN_{i_1+1}(V_1)/HN_{i_1}(V_1))<\mu(HN_{i_2+1}(V_2)/HN_{i_2}(V_2))$
then let $HN_{i+1}(V_1\oplus V_2)=HN_{i_1}(V_1)\oplus
HN_{i_2+1}(V_2)$.
\end{itemize}
It is easy to check that the vector bundle $gr^{HN(V_1\oplus V_2)}_i=HN_{i+1}(V_1\oplus V_2)/HN_i(V_1\oplus V_2)$ is semistable of
slope $$max\{\mu(gr^{HN(V_1)}_{i_1+1}),\mu(gr^{HN(V_2)}_{i_2+1})\}$$
and the slope is strictly decreasing in $i$.
We have thus constructed the
Harder-Narasimhan filtration of $V_1\oplus V_2$. From the
construction, we also have
$$grad(HN(V_1\oplus V_2))=grad(HN(V_1))\oplus grad(HN(V_2))$$
and $\mu_i(V_1)=\mu(gr^{HN(V_1)}_j)$ always equals $\mu_k(V_1\oplus
V_2)$ for some $k$.
\end{proof}
\section{Filtration of the Hodge bundle} Let
$\Omega\mathcal{M}_g(m_1,...,m_k)$ be the stratum parameterizing
$(X,\omega)$ where $X$ is a curve of genus $g$ and $\omega$ is an
Abelian differentials (i.e.a holomorphic one-form) on $X$ that have
$k$ distinct zeros of order $m_1,...,m_k$. Denote by
$\overline{\Omega\mathcal{M}}_g(m_1,...,m_k)$ the Deligne-Mumford
compactification of $\Omega\mathcal{M}_g(m_1,...,m_k)$. Denote by
$\Omega\mathcal{M}^{hyp}_g(m_1,...,m_k)$( resp. odd, resp. even) the
hyperelliptic (resp. odd theta character, resp. even theta
character) connected component. (\cite{KZ03})
Let $\mathcal{Q}(d_1,...,d_n)$ be the stratum parameterizing $(X,q)$
where $X$ is a curve of genus $g$ and $q$ is a meromorphic quadratic
differentials with at most simple zeros on $X$ that have $k$
distinct zeros of order $d_1,...,d_n$ respectively.
If the quadratic differential is not a global square of a one-form,
there is a canonical double covering $\pi:Y\rightarrow X$ such that
$\pi^*q=\omega^2$. This covering is ramified precisely at the zeros
of odd order of $q$ and at the poles. It give a map
$$\phi:\mathcal{Q}(d_1,...,d_n)\rightarrow\Omega\mathcal{M}_g(m_1,...,m_k)$$
A singularity of order $d_i$ of $q$ give rise to two zeros of degree
$m=d_i/2$ when $d_i$ is even, single zero of degree $m=d+1$ when $d$
is odd. Especially, the hyperelliptic locus in a stratum
$\Omega\mathcal{M}_g(m_1,...,m_k)$ induces from a stratum
$\mathcal{Q}(d_1,...,d_k)$ satisfying $d_1+...+d_n=-4$.
There is a nature action of $GL_2^+(\mathbb{R})$ on
$\Omega\mathcal{M}_g(m_1,...,m_k)$, whose orbits project to complex
geodesics in $\mathcal{M}_g$. The projection of an orbit is almost
always dense. If the stabilizer $SL(X,\omega)\subset
SL_2(\mathbb{R})$ of given form is a lattice, however, then the
projection of its orbit gives a closed, algebraic Teichm\"{u}ller
curve $C$.
The Teichm\"{u}ller curve $C$ is an
algebraic curve in $\overline{\Omega\mathcal{M}}_g$ that is totally
geodesic with respect to the Teichm\"{u}ller metric.
After suitable base change, we can get a universal family
$f:S\rightarrow C$, which is a relatively minimal semistable model
with disjoint sections $D_1,..,D_k$; here $D_i|_X$ is a zero of
$\omega$ when restrict to each fiber $X$. (\cite{CM11})
Let $\mathcal{L}\subset f_*{\omega_{S/C}}$ be the line bundle
whose fiber over the point corresponding to $X$ is
$\mathbb{C}\omega$, the generating differential of Teichm\"{u}ller
curves; it is also known as the "maximal Higgs" line bundle. Let
$\Delta\subset \overline{B}$ be the set of points with singular
fibers, then the property of being ''maximal Higgs'' says by
definition that $\mathcal{L}\cong
\mathcal{L}^{-1}\otimes\omega_C(log{\Delta})$ and
$$deg(\mathcal{L})=(2g(C)-2+|\Delta|)/2,$$
together with an identification (relative canonical bundle
formula)(\cite{CM11}\cite{EKZ}):
\begin{equation}\label{canonical}
\omega_{S/C}\simeq f^*\mathcal{L}\otimes \mathcal{O}(\Sigma m_i D_i)
\end{equation}
By the adjunction formula we get
$$D^2_i=-\omega_{S/C}D_i=-m_iD^2_i-deg{\mathcal{L}}$$
and thus
\begin{equation}\label{intersection}
D^2_i=-\frac{1}{m_i+1}deg \mathcal{L}
\end{equation}
For a line bundle $\mathcal{L}$ of degree $d$ on $X$, denote by
$h^0(\mathcal{L})$ the dimension of $dim(H^0(X,\mathcal{L}))$. From
the exact sequence
$$0\rightarrow f_*\mathcal{O}(d_1D_1+...+d_kD_k)\rightarrow f_*\mathcal{O}(m_1D_1+...+m_kD_k)=f_*(\omega_{S/C})\otimes\mathcal{L}^{-1}$$
and the fact that all subsheaves of a locally free sheaf on a curve
are locally free, we deduce that $f_*\mathcal{O}(d_1D_1+...+d_kD_k)$
is a vector bundle of rank $h^0(d_1p_1+...+d_kp_k)$, here
$p_i=D_i|_F$, $F$ is a generic fiber. We have constructed many
filtration of the Hodge bundle by using those vector bundles in
\cite{YZ}.
A fundament exact sequence for those filtration is the following:
\begin{equation}\label{basic}
0\rightarrow f_*\mathcal{O}(\sum (d_i-a_i)D_i)\rightarrow
f_*\mathcal{O}(\sum d_iD_i)\rightarrow f_*\mathcal{O}_{\sum
a_iD_i}(\sum d_iD_i)\overset{\delta}{\rightarrow}
\end{equation}
$$R^1f_*\mathcal{O}(\sum (d_i-a_i)D_i)\rightarrow
R^1f_*\mathcal{O}(\sum d_iD_i)\rightarrow 0$$
There are many properties of these filtration:
\begin{lemma}(\cite{YZ})\label{EE}
If $h^0(\sum d_ip_i)=h^0(\sum (d_i-a_i)p_i)$ holds in a general
fiber, then we have the equality $f_*\mathcal{O}(\sum
d_iD_i)=f_*\mathcal{O}(\sum (d_i-a_i)D_i)$.
\end{lemma}
\begin{lemma}(\cite{YZ})\label{SP}
If $h^0(\sum d_ip_i)=h^0(\sum (d_i-a_i)p_i)+\sum a_i$ is
non-varying, then
$$f_*\mathcal{O}(\sum d_iD_i)/f_*\mathcal{O}(\sum (d_i-a_i)D_i)=f_*\mathcal{O}_{\sum a_iD_i}(\sum d_iD_i)=\oplus f_*\mathcal{O}_{a_iD_i}(d_iD_i)$$
\end{lemma}
\begin{lemma}(\cite{YZ})\label{HN}
The Harder-Narasimhan filtration of $f_*\mathcal{O}_{aD}(dD)$ is
$$0\subset f_*\mathcal{O}_{D}((d-a+1)D)...\subset ...f_*\mathcal{O}_{(a-1)D}((d-1)D)\subset f_*\mathcal{O}_{aD}(dD)$$
and the direct sum of the graded quotient of this filtration is
$$grad(HN(f_*\mathcal{O}_{aD}(dD)))=\overset{a-1}{\underset{i=0}{\oplus}}
\mathcal{O}_{D}((d-i)D)$$
\end{lemma}
\begin{lemma}(\cite{YZ})\label{UP}The degree $deg(f_*\mathcal{O}(\sum d_iD_i)/f_*\mathcal{O}(\sum (d_i-a_i)D_i))$
is smaller than the maximal sums of $h^0(\sum d_ip_i)-h^0(\sum
(d_i-a_i)p_i)$ line bundles in
$$\underset{i}{\bigcup}\overset{a_i-1}{\underset{j=0}{\bigcup}}\mathcal{O}_{D_i}((d_i-j)D_i)$$
(here $p_i=D_i|_F$, $F$ being a general fiber).
\end{lemma}
For a Teichm\"{u}ller curves lying in hyperelliptic loci and low
genus non-varying strata, we have constructed the Harder-Narasimhan
filtration.
Write $w_i$ for $\mu_i(f_*(\omega_{S/C}))/deg(\mathcal{L})$.
\begin{theorem}\cite{YZ} \label{YZ}Let $C$ be a Teichm\"{u}ller curve in the hyperelliptic locus of
a stratum $\overline{\Omega\mathcal{M}}_g(m_1,...,m_k)$, and denote
by $(d_1,...d_n)$ the orders of singularities of underlying
quadratic differentials. Then the $w_i$'s for $C$ are
$$1,\{1-\frac{2k}{d_j+2}\}_{0<2k\leq d_j+1}$$
For a Teichm\"{u}ller curve lying in some low genus non varying
strata, the $w_i$'s are computed in Table 1,Table 2,Table 3.
\end{theorem}
\begin{table}
\caption{genus 3}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
zeros & component & $w_2$& $w_3$ & $\sum w_i$ \\
\hline
(4)& hyp & 3/5 & 1/5 & 9/5 \\
\hline
(4) & odd & 2/5 & 1/5 & 8/5 \\
\hline
(3,1) & & 2/4 & 1/4 & 7/4 \\
\hline
(2,2) & hyp & 2/3 & 1/3 & 2 \\
\hline
(2,2) & odd & 1/3 & 1/3 & 5/3 \\
\hline
(2,1,1) & & 1/2 & 1/3 & 11/6 \\
\hline
(1,1,1,1) & & & & $\leq 2$ \\
\hline
\end{tabular}
\end{table}
\begin{table}
\caption{genus 4}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
zeros & component & $w_2$& $w_3$ &$w_4$ & $\sum w_i$ \\
\hline
(6)& hyp & 5/7 & 3/7 & 1/7& 16/7 \\
\hline
(6) & even& 4/7 & 2/7 & 1/7 &14/7 \\
\hline
(6) & odd & 3/7 & 2/7 & 1/7 &13/7 \\
\hline
(5,1) & & 1/2 & 2/6 & 1/6 & 2 \\
\hline
(3,3) & hyp & 3/4 & 2/4 & 1/4 & 5/2 \\
\hline
(3,3) & non-hyp & 2/4 & 1/4 & 1/4 & 2 \\
\hline
(4,2) & even & 3/5 & 1/3 & 1/5 &32/15 \\
\hline
(4,2) & odd & 2/5 & 1/3 & 1/5 &29/15 \\
\hline
(2,2,2) & & 1/3 & 1/3 & 1/3 &2 \\
\hline
(3,2,1) & & 1/2 & 1/3 & 1/4 &25/12 \\
\hline
\end{tabular}
\end{table}
\begin{table}
\caption{genus 5}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
zeros & component & $w_2$& $w_3$ &$w_4$ &$w_5$ & $\sum w_i$ \\
\hline
(8)& hyp & 7/9 & 5/9 & 3/9 &1/9& 25/9 \\
\hline
(8) & even& 5/9 & 3/9 & 2/9 &1/9&20/9 \\
\hline
(8) & odd & 4/9 & 3/9 & 2/9 &1/9&19/9 \\
\hline
(5,3) & & 1/2 & 1/3 & 1/4 &1/6& 9/4 \\
\hline
(6,2) & odd & 3/7 & 1/3 & 2/7 &1/7& 46/21 \\
\hline
(4,4) & hyp & 4/5 & 3/5 & 2/5 &1/5& 3 \\
\hline
\end{tabular}
\end{table}
\section{Lyapunov exponents}
A good introduction to Lyapunov exponents with a lot of motivating
examples is the survey by Zorich (\cite{Zo}).
Fix an $SL_2(\mathbb{R})$-invariant, ergodic measure $\mu$ on $\Omega\mathcal{M}_g$. Let V be the
restriction of the real Hodge bundle (i.e. the bundle with fibers
$H^1(X,\mathbb{R})$) to the support $M$ of $\mu$. Let $S_t$ be the
lift of the geodesic flow to $V$ via the Gauss-Manin connection.
Then Oseledec's multiplicative ergodic theorem guarantees the
existence of a filtration
$$0\subset V_{\lambda_g}\subset ...\subset V_{\lambda_1}=V$$
by measurable vector subbundles with the property that, for almost
all $m\in M$ and all $v\in V_m\backslash\{0\}$ one has
$$||S_t(v)||=exp(\lambda_it+o(t))$$
where $i$ is the maximal index such that $v$ is in the fiber of
$V_i$ over $m$ i.e.$v\in(V_i)_m$. The numbers $\lambda_i$ for
$i=1,...,k\leq rank(V)$ are called the \emph{Lyapunov exponents} of
$S_t$. Since $V$ is symplectic, the spectrum is symmetric in the
sense that $\lambda_{g+k}=-\lambda_{g-k+1}$. Moreover, from
elementary geometric arguments it follows that one always has
$\lambda_1=1$.
There is an algebraic interpretation of the sum of certain Lyapunov
exponents:
\begin{theorem}(\cite{KZ97}\cite{Fo02}\cite{BM10})\label{sumly}If the Variation of Hodge structure (VHS) over the Teichm\"uller curve $C$ contains a sub-VHS $\mathbb{W}$
of rank $2k$, then the sum of the $k$ corresponding to non-negative
Lyapunov exponents equals
$$\overset{k}{\underset{i=1}{\sum}}\lambda^{\mathbb{W}}_i=\frac{2deg \mathbb{W}^{(1,0)}}{2g(C)-2+|\Delta|}$$
where $\mathbb{W}^{(1,0)}$ is the $(1,0)$-part of the Hodge
filtration of the vector bundle associated with $\mathbb{W}$. In
particular, we have
$$\overset{g}{\underset{i=1}{\sum}}\lambda_i=\frac{2degf_*\omega_{S/C}}{2g(C)-2+|\Delta|}$$
\end{theorem}
Let $L(C)=\overset{g}{\underset{i=1}{\sum}}\lambda_i$ be the sum of
Lyapunov exponents, and put
$k_{\mu}=\frac{1}{12}\overset{k}{\underset{i=1}{\sum}}\frac{m_i(m_i+2)}{m_i+1}$.
Eskin, Kontsevich and Zorich obtain a formula to compute $L(C)$ (for
the Teichm\"{u}ller geodesic flow):
\begin{theorem}(\cite{EKZ})For the VHS over the Teichm\"uller curve $C$, we have
$$L(C)=k_{\mu}+\frac{\pi^2}{3}c_{area}(C)$$
where $c_{area}(C)$ is the Siegel-Veech constant corresponding to
$C$.
\end{theorem}
Because the Siegel-Veech constant is non-negative, there is a lower
bound $L(C)\geq k_{\mu}$.
\section{Upper bounds}
Denote by $|\mathcal{L}|$ the projective space of one-dimensional
subspaces of $H^0(X,\mathcal{L})$. For a (projective) $r$-dimension
linear subspace $V$ of $|\mathcal{L}|$, we call $(\mathcal{L},V)$ a
linear series of type $g^r_d$.
\begin{theorem}[Clifford's theorem \cite{Ha}]\label{clifford}
Let $\mathcal{L}$ be an effective special divisor (i.e.
$h^1(\mathcal{L})\neq 0$) on the curve $X$. Then
$$h^0(\mathcal{L})\leq 1+\frac{1}{2}deg(\mathcal{L})$$
Furthermore, equality occurs if and only if either $\mathcal{L}=0$
or $\mathcal{L}=K$ or $X$ is hyperelliptic and $\mathcal{L}$ is a
multiple of the unique linear series of type $g^1_2$ on $X$.
\end{theorem}
Let $C$ be a Teichm\"{u}ller curve lying in
$\Omega\mathcal{M}_g(m_1,...m_k)$. Let $P=(p'_1,...,p'_{2g-2})$ be
a permutation of $2g-2$ points
\[
\overbrace{
\underbrace{p_1,...,p_1}_\text{$m_1$},...,
\underbrace{p_k,...,p_k}_\text{$m_k$}
}^\text{$2g-2$}
\]
The point $p_i$ is the intersection of the section $D_i$ with the
generic fiber $F$.
For $j=1,...,g$, denote
$H_j(P)=i$ if $h^0(p'_1+...+p'_{i-1})=j-1$ and
$h^0(p'_1+...+p'_i)=j$.
First by Clifford's Theorem $h^0(p'_1+...+p'_i)\leq
1+\frac{deg(p'_1+...+p'_i)}{2}$, we have $H_j(P)\geq 2j-2$. When
$j<g$, if the equality holds then $C$ lies in the hyperelliptic
locus.
Next by using vector bundles $f_*\mathcal{O}(D'_1+...+D'_i),(1\leq
i\leq 2g-2)$, we construct a filtration
$$0\subset V'_1\subset V'_2... \subset V'_g=f_*\mathcal{O}(m_1D_1+...+m_kD_k)$$
where $V'_j$ is a rank $j$ vector bundle and
$V'_j=f_*\mathcal{O}(D'_1+...+D'_{H_j(P)})=...=f_*\mathcal{O}(D'_1+...+D'_{H_{j+1}(P)-1})$
by lemma \ref{EE}.
From the exact sequence
$$0\rightarrow f_*\mathcal{O}(D'_1+...+D'_{H_j(P)-1})\rightarrow f_*\mathcal{O}(D'_1+...+D'_{H_j(P)})\rightarrow \mathcal{O}_{D'_{H_j(P)}}(D'_1+...+D'_{H_j(P)})$$
we see that the graded quotients $V'_j/V'_{j-1}$ has an upper bound
$\mathcal{O}_{D'_{H_j(P)}}(D'_1+...+D'_{H_j(P)})$ by lemma \ref{UP}.
\begin{theorem} \label{main} The sum of Lyapunov exponents of a Teichm\"{u}ller curve in $\Omega\mathcal{M}_g(m_1,...m_k)$ satisfies the inequality
$$L(C)\leq \frac{g+1}{2}$$
Furthermore, equality occurs if and only if it lies in the
hyperelliptic locus induced from
$\mathcal{Q}(2k_1,...,2k_n,-1^{2g+2})$ or it is some special
Teichm\"{u}ller curve in $\Omega\mathcal{M}_g(1^{2g-2})$.
\end{theorem}
\begin{proof} In $\Omega\mathcal{M}_g(m_1,...m_k)$, there is a
direct sum decomposition (\cite{Mo11}):
$$f_*\omega_{S/C}=\mathcal{L}\otimes(\mathcal{O}_C\oplus f_*\mathcal{O}(m_1D_1+...+m_kD_k)/\mathcal{O}_C)$$
We want to estimate the maximal degree of the rank $g-1$ subbundle
$f_*\mathcal{O}(m_1D_1+...+m_kD_k)/\mathcal{O}_C)$ because we want
to obtain an upper bound of
$L(C)=deg(f_*\omega_{S/C})/deg(\mathcal{L})$.
By exact sequence \eqref{basic}, we have
$$ f_*\mathcal{O}(m_1D_1+...+m_kD_k)/\mathcal{O}_C\subset f_*\mathcal{O}_{\sum m_iD_i}(\sum m_iD_i)=\underset{i}{\oplus} f_*\mathcal{O}_{m_iD_i}(m_iD_i)$$
The last equality follows from the fact that the $D_i$'s are
disjoint. By lemma \ref{HN}, we have
$grad(HN(\mathcal{O}_{m_iD_i}(m_iD_i))=\overset{m_i}{\underset{j=1}{\oplus}}\mathcal{O}_{D_i}(jD_i)$
. By lemma \ref{directsum}, the direct sum of the graded quotients
of $HN(\underset{i}{\oplus} f_*\mathcal{O}_{m_iD_i}(m_iD_i))$ is
$$grad(HN(\underset{i}{\oplus} f_*\mathcal{O}_{m_iD_i}(m_iD_i)))=\underset{i}{\oplus}\overset{m_i}{\underset{j=1}{\oplus}}\mathcal{O}_{D_i}(jD_i)$$
Consider the degree of each summand, we can easily construct a
filtration of
$\underset{i}{\oplus}\overset{m_i}{\underset{j=1}{\oplus}}\mathcal{O}_{D_i}(jD_i)$:
$$0\subset V_1\subset V_2... \subset V_{2g-2}=\underset{i}{\oplus}\overset{m_i}{\underset{j=1}{\oplus}}\mathcal{O}_{D_i}(jD_i)$$
satisfying:
1). $V_i/V_{i-1}$ is a line bundle,
2). $deg(V_i/V_{i-1})$ decreases in $i$.
We rearrange the $2g-2$ points $m_1p_1,...,m_kp_k$ of generic fiber.
If $V_i/V_{i-1}=\mathcal{O}_{D_j}(dD_j)$, then let $p'_i=p_j$. Thus
we get a special permutation
\begin{equation}\label{arrange}
P=(p'_1,p'_2,...,p'_{2g-2})
\end{equation}
Because $D^2_j< 0$, and
$deg(\mathcal{O}_{D_j}(D_j)>...>deg(\mathcal{O}_{D_j}((d-1)D_j)>deg(\mathcal{O}_{D_j}(dD_j)$,
there are only $d-1$ $p_j$'s appearing before $p'_i$
$$D'_i=D_j, D'_1+...+D'_i=dD_j+(\texttt{not contain } D_i\texttt{ part})$$
So
$$grad(HN(\mathcal{O}_{\overset{i}{\underset{k=1}{\sum}}D'_k}(\overset{i}{\underset{k=1}{\sum}}D'_k)))/grad(HN(\mathcal{O}_{\overset{i-1}{\underset{k=1}{\sum}}D'_k}(\overset{i-1}{\underset{k=1}{\sum}}D'_k)))=\mathcal{O}_{D_j}(dD_j)$$
By induction we get:
\begin{equation}
V_i=grad(HN(\mathcal{O}_{D'_1+...+D'_i}(D'_1+...+D'_i)))
\end{equation}
Use vector bundles $f_*\mathcal{O}(D'_1+...+D'_i)$, we also
construct a filtration
\begin{equation}\label{filtration}
0\subset V'_1\subset V'_2... \subset
V'_g=f_*\mathcal{O}(m_1D_1+...+m_kD_k)
\end{equation}
where the equalities
$V'_j=f_*\mathcal{O}(D'_1+...+D'_{H_j(P)})=...=f_*\mathcal{O}(D'_1+...+D'_{H_{j+1}(P)-1})$,
by lemma \ref{EE}.
We get the following exact sequence by using \eqref{basic}:
$$0\rightarrow f_*\mathcal{O}(D'_1+...+D'_{H_j(P)-1})\rightarrow f_*\mathcal{O}(D'_1+...+D'_{H_j(P)})\rightarrow V_{H_j(P)}/V_{H_j(P)-1}$$
The lemma \ref{UP} and the Clifford theorem give us:
$$deg(V'_j/V'_{j-1})\leq deg(V_{H_j(P)}/V_{H_j(P)-1})\leq deg(V_{2j-2}/V_{2j-3})$$
We set
$a_i:=deg(V_i/V_{i-1})/deg(\mathcal{L}),b_i:=deg(V'_i/V'_{i-1})/deg(\mathcal{L})$.
By definition $b_1=0$ and $a_i$ is the $i$-th largest number of
$\{-\frac{j}{m_i+1}|1\leq j\leq m_i,1\leq i\leq k\}$.
Hence
$$b_j=deg(V'_j/V'_{j-1})/deg(\mathcal{L})\leq deg(V_{2j-2}/V_{2j-3})/deg(\mathcal{L})=a_{2j-2}$$
After some element computations:
\begin{align*}
\overset{g}{\underset{j=2}{\sum}} b_j &\leq \overset{g-1}{\underset{j=1}{\sum}} a_{2j}\leq \overset{g-1}{\underset{j=1}{\sum}} (a_{2j-1}+a_{2j})/2=\frac{1}{2}\overset{2g-2}{\underset{j=1}{\sum}} a_{j}\\
&=\frac{1}{2}deg(\underset{i}{\oplus}\overset{m_i}{\underset{j=1}{\oplus}}\mathcal{O}_{D_i}(jD_i))/deg(\mathcal{L})=\frac{1}{2}\overset{k}{\underset{l=1}{\sum}} \overset{m_k}{\underset{i=1}{\sum}} (-\frac{i}{m_l+1})\\
&=\frac{1}{4}\overset{k}{\underset{l=1}{\sum}} (-m_l)=-\frac{g-1}{2}
\end{align*}
We get
$$L(C)=g+\frac{deg(f_*\mathcal{O}(m_1D_1+...+m_kD_k))}{deg(\mathcal{L})}\leq g+\overset{g-1}{\underset{j=1}{\sum}} b_j=g+\overset{g-1}{\underset{j=2}{\sum}}b_j=\frac{g+1}{2}$$
When the inequality becomes equal, we have
$a_{2j-1}=a_{2j}=b_{j+1}$. If
$b_{k+1}=a_1=a_2=...=a_{2k}>a_{2k+1}=a_{2k+2}=b_{k+2}$, then the
exact sequence
$$0\rightarrow f_*\mathcal{O}(D'_1+...+D'_{2k})\rightarrow f_*\mathcal{O}(D'_1+...+D'_{2k+1})\rightarrow V_{2k+1}/V_{2k}$$
give us $h^0(p'_1+...+p'_{2k})\geq k+1$, otherwise the inequality
$a_{2k}=b_{k+1}\leq a_{2k+1}$ leads to a contradiction. Thus by
Clifford's theorem $k+1\leq h^0(p'_1+...+p'_{2k})\leq
1+\frac{2k}{2}$, its generic fibers is hyperelliptics unless
$a_1=a_2=...=a_{2g-2}$ which means $m_1=...=m_{2g-2}=1$.
The hyperelliptic locus in a stratum
$\Omega\mathcal{M}_g(m_1,...,m_k)$ induces from a stratum
$\mathcal{Q}(d_1,...,d_k)$ satisfying $d_1+...+d_n=-4$. A
singularity of order $d_i$ of $q$ give rise to two zeros of degree
$m=d_i/2$ when $d_i$ is even, single zero of degree $m=d+1$ when $d$
is odd.
$$\underset{d_j\texttt{odd}}{\sum} (d_j+1)+\underset{d_j\texttt{even}}{\sum}d_j=2g-2$$
By the formula of sums for the hyperelliptic locus in \cite{EKZ},
$$L(C)=\frac{1}{4}\underset{d_j
\texttt{odd}}{\sum}\frac{1}{d_j+2}\leq \frac{1}{4}\underset{d_j
\texttt{odd}}{\sum} 1=\frac{g+1}{2}$$ a Teichm\"{u}ller curve in the
hyperelliptic locus satisfies $L(C)=\frac{g+1}{2}$ if and only if it
is induced from $\mathcal{Q}(2k_1,...,2k_n,-1^{2g+2})$.
\end{proof}
\begin{remark}D.W. Chen and M.M\"{o}ller (\cite{CM11}) have constructed a Teichm\"{u}ller curve
$C\in \Omega\mathcal{M}_3(1,1,1,1)$ with $L(C)=2$, but it is not
hyperelliptic: the square tiled surface given by the permutations
$$(\pi_r=(1234)(5)(6789),\pi_{\mu}=(1)(2563)(4897))$$
They also have obtained
a bound by using Cornalba-Harris-Xiao's slope inequality
(\cite{Mo12}):
$$L(C) \leq \frac{3g}{(g-1)}\kappa_{\mu}=\frac{g}{4(g-1)}\overset{k}{\underset{i=1}{\sum}}\frac{m_i(m_i+2)}{m_i+1}$$
\end{remark}
In fact we have obtained an upper bound of the slope of each graded
quotient of the Harder-Narasimhan filtration of $f_*(\omega_{S/C})$
for Teichm\"{u}ller curves:
\begin{lemma}\label{ubhn}For a Teichm\"{u}ller
curve which lies in $\Omega\mathcal{M}_g(m_1,...m_k)$, we have
inequalities: $$w_i\leq 1+a_{H_i(P)}$$ Here $a_i$ is the $i$-th
largest number in $\{-\frac{j}{m_i+1}|1\leq j\leq m_i,1\leq i\leq
k\}$, $P$ is the special permutation \eqref{arrange} and $H_i(P)\geq
2i-2$.
\end{lemma}
\begin{proof}For the vector bundle
$f_*\mathcal{O}(m_1D_1+...+m_kD_k)$, the filtration
\eqref{filtration} gives
$$0\subset V'_1\subset V'_2... \subset V'_g=f_*\mathcal{O}(m_1D_1+...+m_kD_k)$$
It is controlled by the following filtration:
$$0\subset \mathcal{O}\subset\mathcal{O}\oplus V_{H_2(P)}/V_{H_2(P)-1}\subset... \subset
\mathcal{O}\oplus\overset{g}{\underset{j=2}{\oplus}}V_{H_j(P)}/V_{H_j(P)-1}$$
By lemma \ref{control},
$\mu_i(f_*\mathcal{O}(m_1D_1+...+m_kD_k))\leq
deg(V_{H_i(P)}/V_{H_i(P)-1})$. So we get
$$w_i=\mu_i(f_*(\omega_{S/C}))/deg(\mathcal{L})=1+\mu_i(f_*\mathcal{O}(m_1D_1+...+m_kD_k))/deg(\mathcal{L})\leq
1+a_{H_i(P)}$$
\end{proof}
The Harder-Narasimhan filtration always give an upper bound of
degrees of any sub vector bundles, especially those related to the
sum of certain Lyapunov exponents.
\begin{proposition}\label{pa}If the VHS over the Teichm\"uller curve $C$ contains a sub-VHS $\mathbb{W}$
of rank $2k$, then the sum of the $k$ corresponding non-negative
Lyapunov exponents is the sum of $w_{i_1},...,w_{i_k}$ (where $i_j$
are different to each other) and satisfies
$$\overset{k}{\underset{i=1}{\sum}}\lambda^{\mathbb{W}}_i\leq\sum^k_{i=1}(1+a_{H_i(P)})$$
\end{proposition}
\begin{proof}$\mathbb{W}^{(1,0)}$ is summand of $f_*(\omega_{S/C})$ by Deligne's semisimplicity theorem. The slope $\mu_j(\mathbb{W}^{(1,0)})$ is equal to $\mu_{i_j}(f_*(\omega_{S/C}))$ for some
$j$ by lemma \ref{directsum}, here we can choose $i_j$ such that
each other is different.
$$\overset{k}{\underset{i=1}{\sum}}\lambda^{\mathbb{W}}_i=\frac{2deg \mathbb{W}^{(1,0)}}{2g(C)-2+|\Delta|}= \frac{\overset{k}{\underset{j=1}{\sum}}\mu_j(\mathbb{W}^{(1,0)})}{deg(\mathcal{L})}=\overset{k}{\underset{j=1}{\sum}}\mu_{i_j}(f_*(\omega_{S/C}))/deg(\mathcal{L})=\overset{k}{\underset{j=1}{\sum}}w_{i_j}$$
By lemma \ref{ubhn} and $a_i$ decrease in $i$,
$$\overset{k}{\underset{i=1}{\sum}}\lambda^{\mathbb{W}}_i=\overset{k}{\underset{j=1}{\sum}}w_{i_j}\leq\sum^k_{i=1}(1+a_{H_{i_j}(P)})\leq\sum^k_{i=1}(1+a_{H_i(P)})$$
\end{proof}
We only present an example to explain the general principle on how
to improve the upper bound when we know more information about
Weierstrass semigroups of general fibers.
\begin{corollary}
A Teichm\"{u}ller curve which lies in the non
hyperelliptic locus of $\mathcal{M}_4(2,2,1,1)$ satisfies
$$L(C)\leq 13/6$$
\end{corollary}
\begin{proof} $a_i$ equal:
$-1/3,-1/3,-1/2,-1/2,-2/3,-2/3$. By Clifford theorem,
$H_2(P)\geq 3,H_3(P)=5,H_4(P)=6$, so we choose the third (or the fourth), the fifth, the sixth
element of $a_i:-1/2,-1/3,-1/3$. Finally we have
$$L(C)\leq\sum^k_{i=1}(1+a_{H_i(P)})=13/6$$
\end{proof}
This result has appeared in \cite{CM11}.
\begin{proposition}\label{single}For a Teichm\"{u}ller curve which
satisfies the assumption \ref{assumption} and lies in
$\Omega\mathcal{M}_g(m_1,...m_k)$, the $i$-th Lyapunov exponent
satisfies the inequality::
$$\lambda_i \leq 1+a_{H_i(P)}$$
Here $a_i$ is the $i$-th largest number in $\{-\frac{j}{m_i+1}|1\leq
j\leq m_i,1\leq i\leq k\}$, $P$ is the special permutation
\eqref{arrange} and $H_i(P)\geq 2i-2$.
\end{proposition}
\begin{proof}The assumption \ref{assumption} and the lemma \ref{directsum} give us $$grad(HN(f_*(\omega_{S/C})))=(\overset{k}{\underset{i=1}{\oplus}} L_i) \oplus grad(HN(W))$$
so there are different $j_i$ such that $\lambda_1
=w_{j_1}\geq\lambda_2 =w_{j_2}\geq...\geq\lambda_k=w_{j_k}$. By
lemma \ref{ubhn}, we have
$$\lambda_i=w_{j_i}\leq w_i\leq 1+a_{H_i(P)}$$
\end{proof}
The equality can be reached for an algebraic primitive
Teichm\"{u}ller curve lying in the hyperelliptic locus induced from
$\mathcal{Q}(2k_1,...,2k_n,-1^{2g+2})$.
\section{Assumptions}
\paragraph{\textbf{Abelian covers}}
The Lyapunov spectrum has been computed for triangle groups (\cite{BM10}), square tiled cyclic covers (\cite{EKZ11} \cite{FMZ11a}) and
square tiled abelian covers (\cite{Wr1}). They all satisfy the assumption
\ref{assumption}. Here we give the description of square tiled
cyclic covers:
Consider an integer $N\geq 1$ and a quadruple of integers
$(a_1,a_2,a_3,a_4)$ satisfying the following conditions:
$$0<a_i\leq N;\quad gcd(N,a_1,...,a_4)=1;\quad\sum^4_{i=1}a_i\equiv 0 \quad(mod\quad N)$$
Let $z_1,z_2,z_3,z_4\in \mathbb{C}$ be four distinct points. By
$M_N(a_1,a_2,a_3,a_4)$ we denote the closed connected nonsingular
Riemann surface obtained by normalization of the one defined by the
equation
$$w^N=(z-z_1)^{a_1}(z-z_2)^{a_2}(z-z_3)^{a_3}(z-z_4)^{a_4}$$ Varying the cross-ratio $(z_1,z_2,z_3,z_4)$ we
obtain the moduli curve $\mathcal{M}_{(a_i),N}$. As an abstract
curve it is isomorphic to $\mathcal{M}_{0,4}\simeq
\mathbb{P}^1-\{0,1,\infty\}$; more strictly speaking, it should be
considered as a stack. The canonical generator $T$ of the group of
deck transformations induces a linear map $T^*:H^{1,0}(X)\rightarrow
H^{1,0}(X)$. $H^{1,0}(X)$ admits a splitting into a direct sum of
eigenspaces $V^{1,0}(k)$ of $T^*$ and satisfies the assumption
\ref{assumption}. (cf. Theorem $2$ in \cite{EKZ11})
For even $N$, $M_N(N-1,1,N-1,1)$ has Lyapunov spectrum
(\cite{EKZ11}):
$$\{\frac{2}{N},\frac{2}{N},\frac{4}{N},\frac{4}{N},...,\frac{N-2}{N},\frac{N-2}{N},1\}$$
\begin{remark} By the Theorem \ref{main} and the genus formular
$g=N+1-\frac{1}{2}\sum^4_{i=1}gcd(a_i,N)$, $M_N(N-1,1,N-1,1)$ lies
in the hyperelliptic locus which induced from $\mathcal
{Q}(N-2,N-2,-1^{2N})$, because $L(C)$ equal $\frac{g+1}{2}$.
\end{remark}
\paragraph{\textbf{Algebraic primitives}} The variation of Hodge structures
over a Teichm\"{u}ller curve decomposes into sub-VHS
\begin{equation}\label{VHS}
R^1f_*\mathbb{C}=(\oplus^r_{i=1} \mathbb{L}_i)\oplus \mathbb{M}
\end{equation}
Here $\mathbb{L}_i$ are rank-2 subsystems, maximal Higgs
$\mathbb{L}^{1,0}_1\simeq \mathcal{L}$ for $i=1$, non-unitary but
not maximal Higgs for $i\neq 1$ (\cite{Mo11}). It is obvious that
the Teichm\"{u}ller curve satisfies the assumption \ref{assumption}
if $r\geq g-1$.
If $r=g$, it is called algebraic primitive Teichm\"{u}ller curves.
We know there are only finite algebraic primitive Teichm\"{u}ller
curves
in the stratum $\Omega\mathcal{M}_3(3,1)$ by M\"{o}ller and
Bainbridge in \cite{BM09}, and they conjecture that the algebraic
primitive Teichm\"{u}ller curves in each stratum is finite
(\cite{Mo12}).
\begin{remark}Algebraic
primitive Teichm\"{u}ller
curves
in the stratum $\Omega\mathcal{M}_3(3,1)$ has Lyapunov spectrum
$\{1,\frac{2}{4},\frac{1}{4}\}$ by proposition \ref{nonva}.
\end{remark}
\paragraph{\textbf{Wind-tree models}}A wind-tree model or the infinite
billiard table is defined as:
$$T(a,b):=\mathbb{R}^2\setminus\underset{m,n\in\mathbb{Z}}{\bigcup}[m,m+a]\times[n,n+b]$$
with $0<a,b<1$. Denote by $\phi^{\theta}_t:T(a,b)\rightarrow T(a,b)$
the billiard flow: for a point $p\in T(a,b)$, the point
$\phi^{\theta}_t$ is the position of a particle after time $t$
starting from position $p$ in direction $\theta$.
\begin{theorem} (\cite{DHL})Let $d(.,.)$ be the Euclidean distance on $\mathbb{R}^2$.\begin{itemize}
\item (Case 1) If
$a$ and $b$ are rational numbers or can be written as
$1/(1-a)=x+y\sqrt{D},1/(1-b)=(1-x)+y\sqrt{D}$ with $x,y\in
\mathbb{Q}$ and $D$ a positive square-free integer then for Lebesgue
almost all $\theta$ and every point $p$ in $T(a,b)$.
\item (Case 2) For Lebesgue-almost all $(a,b)\in (0,1)^2$, Lebesgue-almost
all $\theta$ and every point $p$ in $T(a,b)$ (with an infinite
forward orbit):
\begin{equation}\label{wind}
\underset{T\rightarrow +\infty}{lim sup}\frac{log
d(p,\phi^{\theta}_T(p))}{logT}=\frac{2}{3}
\end{equation}\end{itemize}
\end{theorem}
We are interested in the case $1$ because it is related to
Teichm\"{u}ller curves. By the Katok-Zemliakov construction, the
billiard flow can be replaced by a linear flow on a(non
compact)translation surface which is made of four copies of $T(a,b)$
that we denote $X_{\infty}(a,b)$. The surface $X_{\infty}(a,b)$ is
$\mathbb{Z}^2$ -periodic and we denote by $X(a,b)$ the quotient of
$X_{\infty}(a,b)$ under the $\mathbb{Z}^2$ action.
The surface $X(a,b)$ is a covering (with Deck group
$\mathbb{Z}/2\times \mathbb{Z}/2$) of the genus $2$ surface
$L(a,b)\in \Omega\mathcal{M}_2(2)$ which is called L-shaped surface
(\cite{Ca} \cite{Mc03}). The orbit of $X(a,b)$ for the
Teichm\"{u}ller flow belongs to the moduli space
$\Omega\mathcal{M}_5(2,2,2,2)$.
The Teichm\"{u}ller curve generated by the surface $X(a,b)$
satisfies the assumption \ref{assumption} because there is an
$SL_2(\mathbb{R})$-equivalent splitting of the Hodge bundle. Its
Lyapunov spectrum is
$\{1,\frac{2}{3},\frac{2}{3},\frac{1}{3},\frac{1}{3}\}$, the
equation \eqref{wind} is equivalence to say that
$\lambda_2=\frac{2}{3}$ (\cite{DHL}).
\begin{remark}
In fact, a Teichm\"{u}ller curve which satisfies the assumption
\ref{assumption} and lies in $\Omega\mathcal{M}_5(2,2,2,2)$
satisfies $\lambda_2\leq\frac{2}{3}$ by the proposition
\ref{single}. By the Theorem \ref{main}, $X(a,b)$ is lies in the
hyperelliptic locus which induced from $\mathcal {Q}(4,4,-1^{12})$,
because $L(C)$ equal $\frac{g+1}{2}$.
\end{remark}
\section{Non varying strata}
Recently, there are many progresses about the phenomenon that the
sum of Lyapunov exponents is non varying in some strata
(\cite{CM11}\cite{CM12}\cite{YZ}). The following proposition is a an
immediate corollary of the theorem \ref{YZ}.
\begin{proposition}\label{nonva}For a Teichm\"{u}ller curve which
satisfies the assumption \ref{assumption} and lies in hyperelliptic
loci or one of the following strata:\\
$\overline{\Omega\mathcal{M}}_3(4),\overline{\Omega\mathcal{M}}_3(3,1),\overline{\Omega\mathcal{M}}^{odd}_3(2,2),\overline{\Omega\mathcal{M}}_3(2,1,1)$\\
$\overline{\Omega\mathcal{M}}_4(6),\overline{\Omega\mathcal{M}}_4(5,1),\overline{\Omega\mathcal{M}}^{odd}_4(4,2),\overline{\Omega\mathcal{M}}^{non-hyp}_4(3,3),\overline{\Omega\mathcal{M}}^{odd}_4(2,2,2),\overline{\Omega\mathcal{M}}_4(3,2,1)$\\
$\overline{\Omega\mathcal{M}}_5(8),\overline{\Omega\mathcal{M}}_5(5,3),\overline{\Omega\mathcal{M}}^{odd}_5(6,2)$\\
The $i$-th Lyapunov exponent $\lambda_i$ equals the $w_i$ which is
computed in the theorem \ref{YZ}.
\end{proposition}
\begin{proof}The assumption \ref{assumption} and the lemma \ref{directsum} give us
$$grad(HN(f_*(\omega_{S/C})))=(\overset{k}{\underset{i=1}{\oplus}} L_i) \oplus grad(HN(W))$$
We have constructed the Harder-Narasimhan filtration with $w_i>0$ in
\cite{YZ}. If $k<g$, then $deg(W)=0$ by the assumption
\ref{assumption}. Using lemma \ref{directsum}, we get
$$0=\frac{deg(W)}{deg(\mathcal{L})}=\frac{\overset{g-k}{\underset{i=1}{\sum}}
\mu_i(W)}{deg(\mathcal{L})}=\frac{\overset{g-k}{\underset{i=1}{\sum}}
\mu_{j_i}(f_*(\omega_{S/C}))}{deg(\mathcal{L})}=\overset{g-k}{\underset{i=1}{\sum}}
w_{j_i}>0$$ It is contradiction! Thus we have
$grad(HN(f_*(\omega_{S/C})))=\overset{g}{\underset{i=1}{\oplus}}
L_i$ and $\lambda_i=w_i$.
\end{proof}
\paragraph{\textbf{Hyperelliptic loci}}It has been shown in \cite{EKZ11} that the ''stairs''
square tiled surface $S(N)$ satisfies the assumption
\ref{assumption} and belongs to the hyperelliptic
connected component $\overline{\Omega\mathcal{M}}^{hyp}_g(2g-2)$, for $N=2g-1$
or
$\overline{\Omega\mathcal{M}}^{hyp}_g(g-1,g-1)$, for $N=2g$ .
\begin{remark}
The Proposition \ref{nonva} also implies that
the Lyapunov spetrum of the Hodge bundles over the corresponding
arithmetic Teichm\"{u}ller curves is
$$\Lambda Spec=\{
\begin{array}{cc}
\frac{1}{N},\frac{3}{N},\frac{5}{N},...,\frac{N}{N}& N=2g-1 \\
\frac{2}{N},\frac{4}{N},\frac{6}{N},...,\frac{N}{N}& N=2g
\end{array}
$$
Which has been shown in \cite{EKZ11} by using the fact $S(N)$ is
quotient of $M_N(N-1,1,N-1,1)$ (resp. $M_{2N}(2N-1,1,N,N)$) for $N$
is even (resp. odd).
\end{remark}
\paragraph{\textbf{Prym varieties}} McMullen, use Prym eigenforms, has constructed infinitely many
primitive Teichm\"{u}ller curves for $g=2,3$ and $4$ (\cite{Mc06a}).
Let $W_{D}(6)$ be the Prym Teichm\"{u}ller curves in
$\Omega\mathcal{M}_4$. It has VHS decomposition:
$$R^1f_*\mathbb{C}=(\mathbb{L}_1\oplus \mathbb{L}_2)\oplus \mathbb{M}$$
So it map to curves $W^X_D$ in the Hilbert modular surface
$X_D=\mathbb{H}^2/SL(\mathcal{O}_D\oplus\mathcal{O}_D^{\vee})$.
\begin{remark}
The proposition \ref{pa} tells us that the number
$deg(\mathbb{L}^{1,0}_2)/deg(\mathcal{L})$ equals one of the numbers
$\{\frac{4}{7},\frac{2}{7},\frac{1}{7}\}$. In fact it has been shown
that $W^X_D$ is the vanishing locus of a modular form of weight
$(2,14)$, so $deg(\mathbb{L}^{1,0}_2)/deg(\mathcal{L})$ is
$\frac{1}{7}$. (\cite{Mo12}\cite{Wei12})
\end{remark} | {"config": "arxiv", "file": "1209.2733.tex"} |
TITLE: An inequality equivalent to Hörmander's condition $\sup_{y\in\mathbb R^n}\int_{\{x: |x|>2|y|\}}|K(x-y)-K(x)|\,dx<\infty$
QUESTION [3 upvotes]: This problem has been asked in MSE, but got no answers. I guess that this exam problem may be a small lemma in some research papers, so I post it here on MathOverflow.
Let $K\in L_{\text{loc}}^1(\mathbb R^n\setminus\{0\})$. Prove that
$$\sup_{y\in\mathbb R^n}\int_{\{x: |x|>2|y|\}}|K(x-y)-K(x)|\,dx<\infty\label{1}\tag{1}$$
if and only if
$$\sup_{r>0}\frac1{r^n}\int_{B(0,r)}\int_{\{x: |x|>2r\}}|K(x-y)-K(x)|\,dx\,dy<\infty.\label{2}\tag{2}$$
This is an old exam problem on Harmonic Analysis. Formula \eqref{1} is called the Hörmander's condition for singular integrals. The proof of \eqref{1}$\Rightarrow$\eqref{2} is quite easy: assume
$$\int_{\{x: |x|>2|y|\}}|K(x-y)-K(x)|\,dx\leq M,\qquad \forall y\in\mathbb R^n,$$
then for $r>0$ and $y\in B(0,r)$ we have $\{x: |x|>2r\}\subset \{x: |x|>2|y|\}$, so
$$\int_{\{x: |x|>2r\}}|K(x-y)-K(x)|\,dx\leq \int_{\{x: |x|>2|y|\}}|K(x-y)-K(x)|\,dx\leq M,$$
hence
$$\frac1{r^n}\int_{B(0,r)}\int_{\{x: |x|>2r\}}|K(x-y)-K(x)|\,dx\,dy\leq \frac1{r^n}\int_{B(0,r)}M\,dy=M|B(0,1)|,\ \ \ \forall r>0.$$
This completes the proof of \eqref{1}$\Rightarrow$\eqref{2}.
However, for \eqref{2}$\Rightarrow$\eqref{1}, I don't know how to start.
Any help would be appreciated!
REPLY [3 votes]: Condition \eqref{2} implies \eqref{1} with $5|y|$ instead of $2|y|$. Fix $y$, let $r=|y|$ and $I=\int_{|x| >5|y| }|K(x-y)-K(x)|\, dx$. Then
$$I \leq \int_{|x| >5r }|K(x-y)-K(x-z)|\, dx+\int_{|x| >5r }|K(x-z)-K(x)|\, dx:=I_1(z)+I_2(z)
$$
for every $|z| \leq r$. If $K$ is the supremum in \eqref{2}, then $r^{-n} \int_{B(0,r)} I_2(z)\, dz \leq K$. In $I_1$ we set $\xi=x-z$ so that $|\xi| \geq 4r$ and
$$I_1(z) \leq \int_{|\xi| \geq 4r} |K(\xi-(y-z))-K(\xi)|\, d\xi.$$ Since $|y-z| \leq 2r$, then
$$r^{-n}\int_{B(0,r)} I_1(z)\, dz \leq r^{-n} \int_{B(0,2r)}|K(\xi-w)-K(\xi)|\, dw
\leq 2^n K.$$ The estimate of $I$ in terms of $K$ now follows by averaging the inequality $I \leq I_1(z)+I_2(z)$ over $B(0,r)$. | {"set_name": "stack_exchange", "score": 3, "question_id": 432195} |
TITLE: Gradient-like vector fields
QUESTION [14 upvotes]: Let $M$ be a compact manifold (without boundary) and let $f:M\to \mathbb{R}$ be a fixed Morse-function. My goal is to better understand gradient-like vector fields for $f$.
Question: Do any two gradient-like vector fields necessarily coincide on a sufficiently small neighborhood of any critical point of $f$ ?
This seems to be a natural question to ask.
So I'm quite surprised that it's not addressed in any book about Morse theory that I've seen so far.
To eliminate any misunderstanding I will recall the definition of gradient-like vector fields:
Definition: A vector field $X\in$Vect$(M)$ is called gradient-like for $f$ if
1. $\forall q\in M\setminus Crit(f):\; df(q)X(q)<0$.
2. For each critical point $p\in Crit(f)$ of $f$, $\exists$ Morse coordinate chart $\phi:U\to\mathbb{R^n}$ (i.e. $\phi(p)=0$ and $f\circ\phi^{-1}(x)=f(p)-\sum_{i=1}^k x_i^2+\sum_{i=k+1}^n x_i^2$ for $x\in\phi(U)$, $k=index(p)$) such that $$\phi_*X(x)=d\phi(\phi^{-1}(x))X(\phi^{-1}(x))=(2x_1,\ldots,2x_k,-2x_{k+1},\ldots,-2x_n)^T$$ for $x\in\phi(U)$ (i.e. in this Morse-chart $X$ coincides with the negative gradient of $f$ w.r.t. the euclidean metric on $\mathbb{R^n}$).
Some motivation: I'm interested whether the space of all gradient-like vector fields for $f$ is contractible. If the answer to my question is affirmative then clearly any convex combination of a gradient-like vector field is still gradient-like and hence the space of gradient-like vector fields is contractible.
Any relevant references are also much appreciated.
Thanks for any contribution.
Edit: I've posted a related question on MathOverflow.
REPLY [3 votes]: No. Since your question is about a neighborhood of a critical point, we can work over $\mathbb{R}^n$ instead of the compact manifold $M$.
Consider $\mathbb{R}^2$ with the following two coordinate charts in a neighborhood of 0. First we have the standard $x,y$ coordinates. Next we have the coordinates
$$ z = x \cos r^2 + y \sin r^2 \qquad w = y \cos r^2 - x \sin r^2 $$
where $r^2 = x^2 + y^2$. We easily verify that $z^2 + w^2 = x^2 + y^2 = r^2$. So that both $(x,y)$ and $(z,w)$ are Morse charts for $f = r^2$.
Let the vector field $X$ be $- x\partial_x - y\partial_y$ in the $(x,y)$ coordinates, and $X'$ be $- z\partial_z - w\partial_w$ in the $(z,w)$ coordinates. You can compute the change of variables explicitly and see that $X \neq X'$ except at the origin.
(It may be easier to see in standard polar coordinates, where $X = r\partial_r$ and $X' = r\partial_r + 2r^2\partial_\theta$. With this you also see that by adding a cut-off at finite $r$ for the perturbation, we can also directly extend this example to any two dimensional manifold. Higher dimensional analogues are also immediate.) | {"set_name": "stack_exchange", "score": 14, "question_id": 322330} |
TITLE: Definition of exterior derivative from a connection
QUESTION [0 upvotes]: I fail to see what is the meaning of the symbol $d_{\nabla}$ in (1.2) of
http://arxiv.org/pdf/hep-th/9712042v2.pdf
I know the meaning of that symbol in the context of forms taking values on some vector bundle with connection $\nabla$, but this is different since it is a boundary operator of the standard de Rham complex which at degree zero acts as $\nabla$.
Thanks.
REPLY [1 votes]: I will assume that you know about connections on the tangent bundle.
These connections induce connections on the tensor bundle $\mathcal{T}^{r,s}M$ of $(r,s)$ tensor fields.
Given $∇ : Γ(TM) \to Γ(TM \otimes T^{*}M)$ there is a unique connection $d_∇ : Γ(\mathcal{T}^{r,s}M) \to Γ(\mathcal{T}^{r,s}M \otimes T^{*}M)$ satisfying
$d_∇ = ∇$ on $TM$,
$(d_∇X)f = X(f)$ for functions $f ∈ C^{∞}(M)$,
Product rule: $(d_∇X)(F \otimes G) = d_∇X F \otimes G + F \otimes d_∇X G$,
Trace invariance: $d_∇X(\mathrm{tr}(F)) = \mathrm{tr}(d_∇X F)$.
The almost complex structure $J ∈ Γ(TM \otimes T^{*}M)$ is a $(1,1)$ tensor.
Similarly, this works for forms. | {"set_name": "stack_exchange", "score": 0, "question_id": 1208756} |
TITLE: Prove $\lim\limits_{n \to \infty} \sqrt[k]{X_n} = \sqrt[k]{\lim\limits_{n \to \infty} X_n}$, where $\{X_n\}_{n=1}^\infty$ converges
QUESTION [0 upvotes]: Let $\{X_n\}_{n=1}^\infty$ be a convergence sequence such that $X_n \geq 0$ and $k \in \mathbb{N}$. Then
$$
\lim_{n \to \infty} \sqrt[k]{X_n} = \sqrt[k]{\lim_{n \to \infty} X_n}.
$$
Can someone help me figure out how to prove this?
REPLY [1 votes]: Hint:
$$
a^p - b^p = (a-b)(a^{p-1}b^0 + \cdots + a^0b^{p-1} )
$$
details:
with $a = x_n^{1/p}, b = (\lim x_n)^{1/p}$ then
$$
x_n - \lim x_n = ( x_n^{1/p}-(\lim x_n)^{1/p})
( x_n^{(p-1)/p}(\lim x_n)^{0/p} + \cdots
+ x_n^{0/p} (\lim x_n)^{(p-1)/p} )
$$
The term $x_n^{(p-1)/p}(\lim x_n)^{0/p} + \cdots
+ x_n^{0/p} (\lim x_n)^{(p-1)/p} $
is $\ge$ $p\times \lim x_n/2$ when $n$ is big, because for such an $n$
$$
x_n \ge \lim x_n/2.
$$
Hence
$$
|x_n - \lim x_n| = |x_n^{1/p}-(\lim x_n)^{1/p}|
( x_n^{(p-1)/p}(\lim x_n)^{0/p} + \cdots
+ x_n^{0/p} (\lim x_n)^{(p-1)/p} ) \\
|x_n - \lim x_n| \ge p\times \lim x_n/2| x_n^{1/p}-(\lim x_n)^{1/p}|
$$
When $n\to\infty$, LHS oges to zero, and so does
$$
| x_n^{1/p}-(\lim x_n)^{1/p}|
$$
REPLY [0 votes]: Assume uniqueness of solutions to $r = a^k$ when $k > 0$ and $a \geq 0$ and $r \geq 0$. Then you just need to prove that $(\lim_n a_n)^k = \lim_n a_n^k$. For this show that $f(x) = x^k$ is continuous, and the rest will follow. | {"set_name": "stack_exchange", "score": 0, "question_id": 761024} |
TITLE: If $\sum\limits_{n=1}^{\infty} a_n^p$ converges, for some $p>1$ then $\sum\limits_{n=1}^{\infty} \frac{a_n}{n}$ converges.
QUESTION [4 upvotes]: If $\{ a_n \}_{n=1}^{\infty}$ is a sequence of positive real numbers and $\sum\limits_{n=1}^{\infty} a_n^p$ converges, for some $p>1$ then $\sum\limits_{n=1}^{\infty} \frac{a_n}{n}$ converges.
Above is the full question. I have found this question in other ways, but not with a general $p$ power. I have looked at other answers that use the inequality that $|ab| \leq \frac{1}{2} (a^2 + b^2)$. I was just wondering if there was another way without this inequality?
REPLY [4 votes]: Let $q$ be the conjugated exponent of $p$, i.e. $q=\frac{p}{p-1}$, such that $\frac{1}{p}+\frac{1}{q}=1$.
By Holder's inequality
$$\left|\sum_{n=1}^{N}\frac{a_n}{n}\right| \leq\left(\sum_{n=1}^{N}a_n^p\right)^{1/p}\left(\sum_{n=1}^{N}\frac{1}{n^q}\right)^{1/q} $$
and by letting $N\to +\infty$ we get $\left|\sum_{n\geq 1}\frac{a_n}{n}\right|\leq \zeta(q)^{1/q} \sum_{n\geq 1}a_n^p.$ | {"set_name": "stack_exchange", "score": 4, "question_id": 2711738} |
theory Discipline_Function
imports
Arities
begin
(**********************************************************)
paragraph\<open>Discipline for \<^term>\<open>fst\<close>\<close>
(* ftype(p) \<equiv> THE a. \<exists>b. p = \<langle>a, b\<rangle> *)
arity_theorem for "empty_fm"
arity_theorem for "upair_fm"
arity_theorem for "pair_fm"
definition
is_fst :: "(i\<Rightarrow>o)\<Rightarrow>i\<Rightarrow>i\<Rightarrow>o" where
"is_fst(M,x,t) \<equiv> (\<exists>z[M]. pair(M,t,z,x)) \<or>
(\<not>(\<exists>z[M]. \<exists>w[M]. pair(M,w,z,x)) \<and> empty(M,t))"
synthesize "fst" from_definition "is_fst"
notation fst_fm (\<open>\<cdot>fst'(_') is _\<cdot>\<close>)
arity_theorem for "fst_fm"
definition fst_rel :: "[i\<Rightarrow>o,i] \<Rightarrow> i" where
"fst_rel(M,p) \<equiv> THE d. M(d) \<and> is_fst(M,p,d)"
reldb_add relational "fst" "is_fst"
reldb_add functional "fst" "fst_rel"
definition
is_snd :: "(i\<Rightarrow>o)\<Rightarrow>i\<Rightarrow>i\<Rightarrow>o" where
"is_snd(M,x,t) \<equiv> (\<exists>z[M]. pair(M,z,t,x)) \<or>
(\<not>(\<exists>z[M]. \<exists>w[M]. pair(M,z,w,x)) \<and> empty(M,t))"
synthesize "snd" from_definition "is_snd"
notation snd_fm (\<open>\<cdot>snd'(_') is _\<cdot>\<close>)
arity_theorem for "snd_fm"
definition snd_rel :: "[i\<Rightarrow>o,i] \<Rightarrow> i" where
"snd_rel(M,p) \<equiv> THE d. M(d) \<and> is_snd(M,p,d)"
reldb_add relational "snd" "is_snd"
reldb_add functional "snd" "snd_rel"
context M_trans
begin
lemma fst_snd_closed:
assumes "M(p)"
shows "M(fst(p)) \<and> M(snd(p))"
unfolding fst_def snd_def using assms
by (cases "\<exists>a. \<exists>b. p = \<langle>a, b\<rangle>";auto)
lemma fst_closed[intro,simp]: "M(x) \<Longrightarrow> M(fst(x))"
using fst_snd_closed by auto
lemma snd_closed[intro,simp]: "M(x) \<Longrightarrow> M(snd(x))"
using fst_snd_closed by auto
lemma fst_abs [absolut]:
"\<lbrakk>M(p); M(x) \<rbrakk> \<Longrightarrow> is_fst(M,p,x) \<longleftrightarrow> x = fst(p)"
unfolding is_fst_def fst_def
by (cases "\<exists>a. \<exists>b. p = \<langle>a, b\<rangle>";auto)
lemma snd_abs [absolut]:
"\<lbrakk>M(p); M(y) \<rbrakk> \<Longrightarrow> is_snd(M,p,y) \<longleftrightarrow> y = snd(p)"
unfolding is_snd_def snd_def
by (cases "\<exists>a. \<exists>b. p = \<langle>a, b\<rangle>";auto)
lemma empty_rel_abs : "M(x) \<Longrightarrow> M(0) \<Longrightarrow> x = 0 \<longleftrightarrow> x = (THE d. M(d) \<and> empty(M, d))"
unfolding the_def
using transM
by auto
lemma fst_rel_abs:
assumes "M(p)"
shows "fst(p) = fst_rel(M,p)"
using fst_abs assms
unfolding fst_def fst_rel_def
by (cases "\<exists>a. \<exists>b. p = \<langle>a, b\<rangle>";auto;rule_tac the_equality[symmetric],simp_all)
lemma snd_rel_abs:
assumes "M(p)"
shows "snd(p) = snd_rel(M,p)"
using snd_abs assms
unfolding snd_def snd_rel_def
by (cases "\<exists>a. \<exists>b. p = \<langle>a, b\<rangle>";auto;rule_tac the_equality[symmetric],simp_all)
end \<comment> \<open>\<^locale>\<open>M_trans\<close>\<close>
relativize functional "first" "first_rel" external
relativize functional "minimum" "minimum_rel" external
context M_trans
begin
lemma minimum_closed[simp,intro]:
assumes "M(A)"
shows "M(minimum(r,A))"
using first_is_elem the_equality_if transM[OF _ \<open>M(A)\<close>]
by(cases "\<exists>x . first(x,A,r)",auto simp:minimum_def)
lemma first_abs :
assumes "M(B)"
shows "first(z,B,r) \<longleftrightarrow> first_rel(M,z,B,r)"
unfolding first_def first_rel_def using assms by auto
(* TODO: find a naming convention for absoluteness results like this.
See notes/TODO.txt
*)
lemma minimum_abs:
assumes "M(B)"
shows "minimum(r,B) = minimum_rel(M,r,B)"
proof -
from assms
have "first(b, B, r) \<longleftrightarrow> M(b) \<and> first_rel(M,b,B,r)" for b
using first_abs
proof (auto)
fix b
assume "first_rel(M,b,B,r)"
with \<open>M(B)\<close>
have "b\<in>B" using first_abs first_is_elem by simp
with \<open>M(B)\<close>
show "M(b)" using transM[OF \<open>b\<in>B\<close>] by simp
qed
with assms
show ?thesis unfolding minimum_rel_def minimum_def
by simp
qed
end \<comment> \<open>\<^locale>\<open>M_trans\<close>\<close>
subsection\<open>Discipline for \<^term>\<open>function_space\<close>\<close>
definition
is_function_space :: "[i\<Rightarrow>o,i,i,i] \<Rightarrow> o" where
"is_function_space(M,A,B,fs) \<equiv> M(fs) \<and> is_funspace(M,A,B,fs)"
definition
function_space_rel :: "[i\<Rightarrow>o,i,i] \<Rightarrow> i" where
"function_space_rel(M,A,B) \<equiv> THE d. is_function_space(M,A,B,d)"
reldb_rem absolute "Pi"
reldb_add relational "Pi" "is_function_space"
reldb_add functional "Pi" "function_space_rel"
abbreviation
function_space_r :: "[i,i\<Rightarrow>o,i] \<Rightarrow> i" (\<open>_ \<rightarrow>\<^bsup>_\<^esup> _\<close> [61,1,61] 60) where
"A \<rightarrow>\<^bsup>M\<^esup> B \<equiv> function_space_rel(M,A,B)"
abbreviation
function_space_r_set :: "[i,i,i] \<Rightarrow> i" (\<open>_ \<rightarrow>\<^bsup>_\<^esup> _\<close> [61,1,61] 60) where
"function_space_r_set(A,M) \<equiv> function_space_rel(##M,A)"
context M_Pi
begin
lemma is_function_space_uniqueness:
assumes
"M(r)" "M(B)"
"is_function_space(M,r,B,d)" "is_function_space(M,r,B,d')"
shows
"d=d'"
using assms extensionality_trans
unfolding is_function_space_def is_funspace_def
by simp
lemma is_function_space_witness:
assumes "M(A)" "M(B)"
shows "\<exists>d[M]. is_function_space(M,A,B,d)"
proof -
from assms
interpret M_Pi_assumptions M A "\<lambda>_. B"
using Pi_replacement Pi_separation
by unfold_locales (auto dest:transM simp add:Sigfun_def)
have "\<forall>f[M]. f \<in> Pi_rel(M,A, \<lambda>_. B) \<longleftrightarrow> f \<in> A \<rightarrow> B"
using Pi_rel_char by simp
with assms
show ?thesis unfolding is_funspace_def is_function_space_def by auto
qed
lemma is_function_space_closed :
"is_function_space(M,A,B,d) \<Longrightarrow> M(d)"
unfolding is_function_space_def by simp
\<comment> \<open>adding closure to simpset and claset\<close>
lemma function_space_rel_closed[intro,simp]:
assumes "M(x)" "M(y)"
shows "M(function_space_rel(M,x,y))"
proof -
have "is_function_space(M, x, y, THE xa. is_function_space(M, x, y, xa))"
using assms
theI[OF ex1I[of "is_function_space(M,x,y)"], OF _ is_function_space_uniqueness[of x y]]
is_function_space_witness
by auto
then show ?thesis
using assms is_function_space_closed
unfolding function_space_rel_def
by blast
qed
lemmas trans_function_space_rel_closed[trans_closed] = transM[OF _ function_space_rel_closed]
lemma is_function_space_iff:
assumes "M(x)" "M(y)" "M(d)"
shows "is_function_space(M,x,y,d) \<longleftrightarrow> d = function_space_rel(M,x,y)"
proof (intro iffI)
assume "d = function_space_rel(M,x,y)"
moreover
note assms
moreover from this
obtain e where "M(e)" "is_function_space(M,x,y,e)"
using is_function_space_witness by blast
ultimately
show "is_function_space(M, x, y, d)"
using is_function_space_uniqueness[of x y] is_function_space_witness
theI[OF ex1I[of "is_function_space(M,x,y)"], OF _ is_function_space_uniqueness[of x y], of e]
unfolding function_space_rel_def
by auto
next
assume "is_function_space(M, x, y, d)"
with assms
show "d = function_space_rel(M,x,y)"
using is_function_space_uniqueness unfolding function_space_rel_def
by (blast del:the_equality intro:the_equality[symmetric])
qed
lemma def_function_space_rel:
assumes "M(A)" "M(y)"
shows "function_space_rel(M,A,y) = Pi_rel(M,A,\<lambda>_. y)"
proof -
from assms
interpret M_Pi_assumptions M A "\<lambda>_. y"
using Pi_replacement Pi_separation
by unfold_locales (auto dest:transM simp add:Sigfun_def)
from assms
have "x\<in>function_space_rel(M,A,y) \<longleftrightarrow> x\<in>Pi_rel(M,A,\<lambda>_. y)" if "M(x)" for x
using that
is_function_space_iff[of A y, OF _ _ function_space_rel_closed, of A y]
def_Pi_rel Pi_rel_char mbnr.Pow_rel_char
unfolding is_function_space_def is_funspace_def by (simp add:Pi_def)
with assms
show ?thesis \<comment> \<open>At this point, quoting "trans\_rules" doesn't work\<close>
using transM[OF _ function_space_rel_closed, OF _ \<open>M(A)\<close> \<open>M(y)\<close>]
transM[OF _ Pi_rel_closed] by blast
qed
lemma function_space_rel_char:
assumes "M(A)" "M(y)"
shows "function_space_rel(M,A,y) = {f \<in> A \<rightarrow> y. M(f)}"
proof -
from assms
interpret M_Pi_assumptions M A "\<lambda>_. y"
using Pi_replacement Pi_separation
by unfold_locales (auto dest:transM simp add:Sigfun_def)
show ?thesis
using assms def_function_space_rel Pi_rel_char
by simp
qed
lemma mem_function_space_rel_abs:
assumes "M(A)" "M(y)" "M(f)"
shows "f \<in> function_space_rel(M,A,y) \<longleftrightarrow> f \<in> A \<rightarrow> y"
using assms function_space_rel_char by simp
end \<comment> \<open>\<^locale>\<open>M_Pi\<close>\<close>
locale M_N_Pi = M:M_Pi + N:M_Pi N for N +
assumes
M_imp_N:"M(x) \<Longrightarrow> N(x)"
begin
lemma function_space_rel_transfer: "M(A) \<Longrightarrow> M(B) \<Longrightarrow>
function_space_rel(M,A,B) \<subseteq> function_space_rel(N,A,B)"
using M.function_space_rel_char N.function_space_rel_char
by (auto dest!:M_imp_N)
end \<comment> \<open>\<^locale>\<open>M_N_Pi\<close>\<close>
(***************** end Discipline ***********************)
abbreviation
"is_apply \<equiv> fun_apply"
\<comment> \<open>It is not necessary to perform the Discipline for \<^term>\<open>is_apply\<close>
since it is absolute in this context\<close>
subsection\<open>Discipline for \<^term>\<open>Collect\<close> terms.\<close>
text\<open>We have to isolate the predicate involved and apply the
Discipline to it.\<close>
(*************** Discipline for injP ******************)
definition (* completely relational *)
injP_rel:: "[i\<Rightarrow>o,i,i]\<Rightarrow>o" where
"injP_rel(M,A,f) \<equiv> \<forall>w[M]. \<forall>x[M]. \<forall>fw[M]. \<forall>fx[M]. w\<in>A \<and> x\<in>A \<and>
is_apply(M,f,w,fw) \<and> is_apply(M,f,x,fx) \<and> fw=fx\<longrightarrow> w=x"
synthesize "injP_rel" from_definition assuming "nonempty"
arity_theorem for "injP_rel_fm"
context M_basic
begin
\<comment> \<open>I'm undecided on keeping the relative quantifiers here.
Same with \<^term>\<open>surjP\<close> below. It might relieve from changing
@{thm exI allI} to @{thm rexI rallI} in some proofs.
I wonder if this escalates well. Assuming that all terms
appearing in the "def\_" theorem are in \<^term>\<open>M\<close> and using
@{thm transM}, it might do.\<close>
lemma def_injP_rel:
assumes
"M(A)" "M(f)"
shows
"injP_rel(M,A,f) \<longleftrightarrow> (\<forall>w[M]. \<forall>x[M]. w\<in>A \<and> x\<in>A \<and> f`w=f`x \<longrightarrow> w=x)"
using assms unfolding injP_rel_def by simp
end \<comment> \<open>\<^locale>\<open>M_basic\<close>\<close>
(****************** end Discipline **********************)
(**********************************************************)
subsection\<open>Discipline for \<^term>\<open>inj\<close>\<close>
definition (* completely relational *)
is_inj :: "[i\<Rightarrow>o,i,i,i]\<Rightarrow>o" where
"is_inj(M,A,B,I) \<equiv> M(I) \<and> (\<exists>F[M]. is_function_space(M,A,B,F) \<and>
is_Collect(M,F,injP_rel(M,A),I))"
declare typed_function_iff_sats Collect_iff_sats [iff_sats]
synthesize "is_funspace" from_definition assuming "nonempty"
arity_theorem for "is_funspace_fm"
synthesize "is_function_space" from_definition assuming "nonempty"
notation is_function_space_fm (\<open>\<cdot>_ \<rightarrow> _ is _\<cdot>\<close>)
arity_theorem for "is_function_space_fm"
synthesize "is_inj" from_definition assuming "nonempty"
notation is_inj_fm (\<open>\<cdot>inj'(_,_') is _\<cdot>\<close>)
arity_theorem intermediate for "is_inj_fm"
lemma arity_is_inj_fm[arity]:
"A \<in> nat \<Longrightarrow>
B \<in> nat \<Longrightarrow> I \<in> nat \<Longrightarrow> arity(is_inj_fm(A, B, I)) = succ(A) \<union> succ(B) \<union> succ(I)"
using arity_is_inj_fm' by (auto simp:pred_Un_distrib arity)
definition
inj_rel :: "[i\<Rightarrow>o,i,i] \<Rightarrow> i" (\<open>inj\<^bsup>_\<^esup>'(_,_')\<close>) where
"inj_rel(M,A,B) \<equiv> THE d. is_inj(M,A,B,d)"
abbreviation
inj_r_set :: "[i,i,i] \<Rightarrow> i" (\<open>inj\<^bsup>_\<^esup>'(_,_')\<close>) where
"inj_r_set(M) \<equiv> inj_rel(##M)"
locale M_inj = M_Pi +
assumes
injP_separation: "M(r) \<Longrightarrow> separation(M,injP_rel(M, r))"
begin
lemma is_inj_uniqueness:
assumes
"M(r)" "M(B)"
"is_inj(M,r,B,d)" "is_inj(M,r,B,d')"
shows
"d=d'"
using assms is_function_space_iff extensionality_trans
unfolding is_inj_def by simp
lemma is_inj_witness: "M(r) \<Longrightarrow> M(B)\<Longrightarrow> \<exists>d[M]. is_inj(M,r,B,d)"
using injP_separation is_function_space_iff
unfolding is_inj_def by simp
lemma is_inj_closed :
"is_inj(M,x,y,d) \<Longrightarrow> M(d)"
unfolding is_inj_def by simp
lemma inj_rel_closed[intro,simp]:
assumes "M(x)" "M(y)"
shows "M(inj_rel(M,x,y))"
proof -
have "is_inj(M, x, y, THE xa. is_inj(M, x, y, xa))"
using assms
theI[OF ex1I[of "is_inj(M,x,y)"], OF _ is_inj_uniqueness[of x y]]
is_inj_witness
by auto
then show ?thesis
using assms is_inj_closed
unfolding inj_rel_def
by blast
qed
lemmas trans_inj_rel_closed[trans_closed] = transM[OF _ inj_rel_closed]
lemma inj_rel_iff:
assumes "M(x)" "M(y)" "M(d)"
shows "is_inj(M,x,y,d) \<longleftrightarrow> d = inj_rel(M,x,y)"
proof (intro iffI)
assume "d = inj_rel(M,x,y)"
moreover
note assms
moreover from this
obtain e where "M(e)" "is_inj(M,x,y,e)"
using is_inj_witness by blast
ultimately
show "is_inj(M, x, y, d)"
using is_inj_uniqueness[of x y] is_inj_witness
theI[OF ex1I[of "is_inj(M,x,y)"], OF _ is_inj_uniqueness[of x y], of e]
unfolding inj_rel_def
by auto
next
assume "is_inj(M, x, y, d)"
with assms
show "d = inj_rel(M,x,y)"
using is_inj_uniqueness unfolding inj_rel_def
by (blast del:the_equality intro:the_equality[symmetric])
qed
lemma def_inj_rel:
assumes "M(A)" "M(B)"
shows "inj_rel(M,A,B) =
{f \<in> function_space_rel(M,A,B). \<forall>w[M]. \<forall>x[M]. w\<in>A \<and> x\<in>A \<and> f`w = f`x \<longrightarrow> w=x}"
(is "_ = Collect(_,?P)")
proof -
from assms
have "inj_rel(M,A,B) \<subseteq> function_space_rel(M,A,B)"
using inj_rel_iff[of A B "inj_rel(M,A,B)"] is_function_space_iff
unfolding is_inj_def by auto
moreover from assms
have "f \<in> inj_rel(M,A,B) \<Longrightarrow> ?P(f)" for f
using inj_rel_iff[of A B "inj_rel(M,A,B)"] is_function_space_iff
def_injP_rel transM[OF _ function_space_rel_closed, OF _ \<open>M(A)\<close> \<open>M(B)\<close>]
unfolding is_inj_def by auto
moreover from assms
have "f \<in> function_space_rel(M,A,B) \<Longrightarrow> ?P(f) \<Longrightarrow> f \<in> inj_rel(M,A,B)" for f
using inj_rel_iff[of A B "inj_rel(M,A,B)"] is_function_space_iff
def_injP_rel transM[OF _ function_space_rel_closed, OF _ \<open>M(A)\<close> \<open>M(B)\<close>]
unfolding is_inj_def by auto
ultimately
show ?thesis by force
qed
lemma inj_rel_char:
assumes "M(A)" "M(B)"
shows "inj_rel(M,A,B) = {f \<in> inj(A,B). M(f)}"
proof -
from assms
interpret M_Pi_assumptions M A "\<lambda>_. B"
using Pi_replacement Pi_separation
by unfold_locales (auto dest:transM simp add:Sigfun_def)
from assms
show ?thesis
using def_inj_rel[OF assms] def_function_space_rel[OF assms]
transM[OF _ \<open>M(A)\<close>] Pi_rel_char
unfolding inj_def
by auto
qed
end \<comment> \<open>\<^locale>\<open>M_inj\<close>\<close>
locale M_N_inj = M:M_inj + N:M_inj N for N +
assumes
M_imp_N:"M(x) \<Longrightarrow> N(x)"
begin
lemma inj_rel_transfer: "M(A) \<Longrightarrow> M(B) \<Longrightarrow> inj_rel(M,A,B) \<subseteq> inj_rel(N,A,B)"
using M.inj_rel_char N.inj_rel_char
by (auto dest!:M_imp_N)
end \<comment> \<open>\<^locale>\<open>M_N_inj\<close>\<close>
(*************** end Discipline *********************)
(*************** Discipline for surjP ******************)
definition
surjP_rel:: "[i\<Rightarrow>o,i,i,i]\<Rightarrow>o" where
"surjP_rel(M,A,B,f) \<equiv>
\<forall>y[M]. \<exists>x[M]. \<exists>fx[M]. y\<in>B \<longrightarrow> x\<in>A \<and> is_apply(M,f,x,fx) \<and> fx=y"
synthesize "surjP_rel" from_definition assuming "nonempty"
context M_basic
begin
lemma def_surjP_rel:
assumes
"M(A)" "M(B)" "M(f)"
shows
"surjP_rel(M,A,B,f) \<longleftrightarrow> (\<forall>y[M]. \<exists>x[M]. y\<in>B \<longrightarrow> x\<in>A \<and> f`x=y)"
using assms unfolding surjP_rel_def by auto
end \<comment> \<open>\<^locale>\<open>M_basic\<close>\<close>
(****************** end Discipline **********************)
(**********************************************************)
subsection\<open>Discipline for \<^term>\<open>surj\<close>\<close>
definition (* completely relational *)
is_surj :: "[i\<Rightarrow>o,i,i,i]\<Rightarrow>o" where
"is_surj(M,A,B,I) \<equiv> M(I) \<and> (\<exists>F[M]. is_function_space(M,A,B,F) \<and>
is_Collect(M,F,surjP_rel(M,A,B),I))"
synthesize "is_surj" from_definition assuming "nonempty"
notation is_surj_fm (\<open>\<cdot>surj'(_,_') is _\<cdot>\<close>)
definition
surj_rel :: "[i\<Rightarrow>o,i,i] \<Rightarrow> i" (\<open>surj\<^bsup>_\<^esup>'(_,_')\<close>) where
"surj_rel(M,A,B) \<equiv> THE d. is_surj(M,A,B,d)"
abbreviation
surj_r_set :: "[i,i,i] \<Rightarrow> i" (\<open>surj\<^bsup>_\<^esup>'(_,_')\<close>) where
"surj_r_set(M) \<equiv> surj_rel(##M)"
locale M_surj = M_Pi +
assumes
surjP_separation: "M(A)\<Longrightarrow>M(B)\<Longrightarrow>separation(M,\<lambda>x. surjP_rel(M,A,B,x))"
begin
lemma is_surj_uniqueness:
assumes
"M(r)" "M(B)"
"is_surj(M,r,B,d)" "is_surj(M,r,B,d')"
shows
"d=d'"
using assms is_function_space_iff extensionality_trans
unfolding is_surj_def by simp
lemma is_surj_witness: "M(r) \<Longrightarrow> M(B)\<Longrightarrow> \<exists>d[M]. is_surj(M,r,B,d)"
using surjP_separation is_function_space_iff
unfolding is_surj_def by simp
lemma is_surj_closed :
"is_surj(M,x,y,d) \<Longrightarrow> M(d)"
unfolding is_surj_def by simp
lemma surj_rel_closed[intro,simp]:
assumes "M(x)" "M(y)"
shows "M(surj_rel(M,x,y))"
proof -
have "is_surj(M, x, y, THE xa. is_surj(M, x, y, xa))"
using assms
theI[OF ex1I[of "is_surj(M,x,y)"], OF _ is_surj_uniqueness[of x y]]
is_surj_witness
by auto
then show ?thesis
using assms is_surj_closed
unfolding surj_rel_def
by blast
qed
lemmas trans_surj_rel_closed[trans_closed] = transM[OF _ surj_rel_closed]
lemma surj_rel_iff:
assumes "M(x)" "M(y)" "M(d)"
shows "is_surj(M,x,y,d) \<longleftrightarrow> d = surj_rel(M,x,y)"
proof (intro iffI)
assume "d = surj_rel(M,x,y)"
moreover
note assms
moreover from this
obtain e where "M(e)" "is_surj(M,x,y,e)"
using is_surj_witness by blast
ultimately
show "is_surj(M, x, y, d)"
using is_surj_uniqueness[of x y] is_surj_witness
theI[OF ex1I[of "is_surj(M,x,y)"], OF _ is_surj_uniqueness[of x y], of e]
unfolding surj_rel_def
by auto
next
assume "is_surj(M, x, y, d)"
with assms
show "d = surj_rel(M,x,y)"
using is_surj_uniqueness unfolding surj_rel_def
by (blast del:the_equality intro:the_equality[symmetric])
qed
lemma def_surj_rel:
assumes "M(A)" "M(B)"
shows "surj_rel(M,A,B) =
{f \<in> function_space_rel(M,A,B). \<forall>y[M]. \<exists>x[M]. y\<in>B \<longrightarrow> x\<in>A \<and> f`x=y }"
(is "_ = Collect(_,?P)")
proof -
from assms
have "surj_rel(M,A,B) \<subseteq> function_space_rel(M,A,B)"
using surj_rel_iff[of A B "surj_rel(M,A,B)"] is_function_space_iff
unfolding is_surj_def by auto
moreover from assms
have "f \<in> surj_rel(M,A,B) \<Longrightarrow> ?P(f)" for f
using surj_rel_iff[of A B "surj_rel(M,A,B)"] is_function_space_iff
def_surjP_rel transM[OF _ function_space_rel_closed, OF _ \<open>M(A)\<close> \<open>M(B)\<close>]
unfolding is_surj_def by auto
moreover from assms
have "f \<in> function_space_rel(M,A,B) \<Longrightarrow> ?P(f) \<Longrightarrow> f \<in> surj_rel(M,A,B)" for f
using surj_rel_iff[of A B "surj_rel(M,A,B)"] is_function_space_iff
def_surjP_rel transM[OF _ function_space_rel_closed, OF _ \<open>M(A)\<close> \<open>M(B)\<close>]
unfolding is_surj_def by auto
ultimately
show ?thesis by force
qed
lemma surj_rel_char:
assumes "M(A)" "M(B)"
shows "surj_rel(M,A,B) = {f \<in> surj(A,B). M(f)}"
proof -
from assms
interpret M_Pi_assumptions M A "\<lambda>_. B"
using Pi_replacement Pi_separation
by unfold_locales (auto dest:transM simp add:Sigfun_def)
from assms
show ?thesis
using def_surj_rel[OF assms] def_function_space_rel[OF assms]
transM[OF _ \<open>M(A)\<close>] transM[OF _ \<open>M(B)\<close>] Pi_rel_char
unfolding surj_def
by auto
qed
end \<comment> \<open>\<^locale>\<open>M_surj\<close>\<close>
locale M_N_surj = M:M_surj + N:M_surj N for N +
assumes
M_imp_N:"M(x) \<Longrightarrow> N(x)"
begin
lemma surj_rel_transfer: "M(A) \<Longrightarrow> M(B) \<Longrightarrow> surj_rel(M,A,B) \<subseteq> surj_rel(N,A,B)"
using M.surj_rel_char N.surj_rel_char
by (auto dest!:M_imp_N)
end \<comment> \<open>\<^locale>\<open>M_N_surj\<close>\<close>
(*************** end Discipline *********************)
definition
is_Int :: "[i\<Rightarrow>o,i,i,i]\<Rightarrow>o" where
"is_Int(M,A,B,I) \<equiv> M(I) \<and> (\<forall>x[M]. x \<in> I \<longleftrightarrow> x \<in> A \<and> x \<in> B)"
reldb_rem relational "inter"
reldb_add absolute relational "ZF_Base.Int" "is_Int"
synthesize "is_Int" from_definition assuming "nonempty"
notation is_Int_fm (\<open>_ \<inter> _ is _\<close>)
context M_basic
begin
lemma is_Int_closed :
"is_Int(M,A,B,I) \<Longrightarrow> M(I)"
unfolding is_Int_def by simp
lemma is_Int_abs:
assumes
"M(A)" "M(B)" "M(I)"
shows
"is_Int(M,A,B,I) \<longleftrightarrow> I = A \<inter> B"
using assms transM[OF _ \<open>M(B)\<close>] transM[OF _ \<open>M(I)\<close>]
unfolding is_Int_def by blast
lemma is_Int_uniqueness:
assumes
"M(r)" "M(B)"
"is_Int(M,r,B,d)" "is_Int(M,r,B,d')"
shows
"d=d'"
proof -
have "M(d)" and "M(d')"
using assms is_Int_closed by simp+
then show ?thesis
using assms is_Int_abs by simp
qed
text\<open>Note: @{thm Int_closed} already in \<^theory>\<open>ZF-Constructible.Relative\<close>.\<close>
end \<comment> \<open>\<^locale>\<open>M_basic\<close>\<close>
(**********************************************************)
subsection\<open>Discipline for \<^term>\<open>bij\<close>\<close>
reldb_add functional "inj" "inj_rel"
reldb_add functional relational "inj_rel" "is_inj"
reldb_add functional "surj" "surj_rel"
reldb_add functional relational "surj_rel" "is_surj"
relativize functional "bij" "bij_rel" external
relationalize "bij_rel" "is_bij"
(* definition (* completely relational *)
is_bij :: "[i\<Rightarrow>o,i,i,i]\<Rightarrow>o" where
"is_bij(M,A,B,bj) \<equiv> M(bj) \<and> is_hcomp2_2(M,is_Int,is_inj,is_surj,A,B,bj)"
definition
bij_rel :: "[i\<Rightarrow>o,i,i] \<Rightarrow> i" (\<open>bij\<^bsup>_\<^esup>'(_,_')\<close>) where
"bij_rel(M,A,B) \<equiv> THE d. is_bij(M,A,B,d)" *)
synthesize "is_bij" from_definition assuming "nonempty"
notation is_bij_fm (\<open>\<cdot>bij'(_,_') is _\<cdot>\<close>)
abbreviation
bij_r_class :: "[i\<Rightarrow>o,i,i] \<Rightarrow> i" (\<open>bij\<^bsup>_\<^esup>'(_,_')\<close>) where
"bij_r_class \<equiv> bij_rel"
abbreviation
bij_r_set :: "[i,i,i] \<Rightarrow> i" (\<open>bij\<^bsup>_\<^esup>'(_,_')\<close>) where
"bij_r_set(M) \<equiv> bij_rel(##M)"
locale M_Perm = M_Pi + M_inj + M_surj
begin
lemma is_bij_closed : "is_bij(M,f,y,d) \<Longrightarrow> M(d)"
unfolding is_bij_def using is_Int_closed is_inj_witness is_surj_witness by auto
lemma bij_rel_closed[intro,simp]:
assumes "M(x)" "M(y)"
shows "M(bij_rel(M,x,y))"
unfolding bij_rel_def
using assms Int_closed surj_rel_closed inj_rel_closed
by auto
lemmas trans_bij_rel_closed[trans_closed] = transM[OF _ bij_rel_closed]
lemma bij_rel_iff:
assumes "M(x)" "M(y)" "M(d)"
shows "is_bij(M,x,y,d) \<longleftrightarrow> d = bij_rel(M,x,y)"
unfolding is_bij_def bij_rel_def
using assms surj_rel_iff inj_rel_iff is_Int_abs
by auto
lemma def_bij_rel:
assumes "M(A)" "M(B)"
shows "bij_rel(M,A,B) = inj_rel(M,A,B) \<inter> surj_rel(M,A,B)"
using assms bij_rel_iff inj_rel_iff surj_rel_iff
is_Int_abs\<comment> \<open>For absolute terms, "\_abs" replaces "\_iff".
Also, in this case "\_closed" is in the simpset.\<close>
unfolding is_bij_def by simp
lemma bij_rel_char:
assumes "M(A)" "M(B)"
shows "bij_rel(M,A,B) = {f \<in> bij(A,B). M(f)}"
using assms def_bij_rel inj_rel_char surj_rel_char
unfolding bij_def\<comment> \<open>Unfolding this might be a pattern already\<close>
by auto
end \<comment> \<open>\<^locale>\<open>M_Perm\<close>\<close>
locale M_N_Perm = M_N_Pi + M_N_inj + M_N_surj + M:M_Perm + N:M_Perm N
begin
lemma bij_rel_transfer: "M(A) \<Longrightarrow> M(B) \<Longrightarrow> bij_rel(M,A,B) \<subseteq> bij_rel(N,A,B)"
using M.bij_rel_char N.bij_rel_char
by (auto dest!:M_imp_N)
end \<comment> \<open>\<^locale>\<open>M_N_Perm\<close>\<close>
(*************** end Discipline *********************)
(******************************************************)
subsection\<open>Discipline for \<^term>\<open>eqpoll\<close>\<close>
relativize functional "eqpoll" "eqpoll_rel" external
relationalize "eqpoll_rel" "is_eqpoll"
synthesize "is_eqpoll" from_definition assuming "nonempty"
arity_theorem for "is_eqpoll_fm"
notation is_eqpoll_fm (\<open>\<cdot>_ \<approx> _\<cdot>\<close>)
context M_Perm begin
is_iff_rel for "eqpoll"
using bij_rel_iff unfolding is_eqpoll_def eqpoll_rel_def by simp
end \<comment> \<open>\<^locale>\<open>M_Perm\<close>\<close>
abbreviation
eqpoll_r :: "[i,i\<Rightarrow>o,i] => o" (\<open>_ \<approx>\<^bsup>_\<^esup> _\<close> [51,1,51] 50) where
"A \<approx>\<^bsup>M\<^esup> B \<equiv> eqpoll_rel(M,A,B)"
abbreviation
eqpoll_r_set :: "[i,i,i] \<Rightarrow> o" (\<open>_ \<approx>\<^bsup>_\<^esup> _\<close> [51,1,51] 50) where
"eqpoll_r_set(A,M) \<equiv> eqpoll_rel(##M,A)"
context M_Perm
begin
lemma def_eqpoll_rel:
assumes
"M(A)" "M(B)"
shows
"eqpoll_rel(M,A,B) \<longleftrightarrow> (\<exists>f[M]. f \<in> bij_rel(M,A,B))"
using assms bij_rel_iff
unfolding eqpoll_rel_def by simp
end \<comment> \<open>\<^locale>\<open>M_Perm\<close>\<close>
context M_N_Perm
begin
(* the next lemma is not part of the discipline *)
lemma eqpoll_rel_transfer: assumes "A \<approx>\<^bsup>M\<^esup> B" "M(A)" "M(B)"
shows "A \<approx>\<^bsup>N\<^esup> B"
proof -
note assms
moreover from this
obtain f where "f \<in> bij\<^bsup>M\<^esup>(A,B)" "N(f)"
using M.def_eqpoll_rel by (auto dest!:M_imp_N)
moreover from calculation
have "f \<in> bij\<^bsup>N\<^esup>(A,B)"
using bij_rel_transfer by (auto)
ultimately
show ?thesis
using N.def_eqpoll_rel by (blast dest!:M_imp_N)
qed
end \<comment> \<open>\<^locale>\<open>M_N_Perm\<close>\<close>
(****************** end Discipline ******************)
(******************************************************)
subsection\<open>Discipline for \<^term>\<open>lepoll\<close>\<close>
relativize functional "lepoll" "lepoll_rel" external
relationalize "lepoll_rel" "is_lepoll"
synthesize "is_lepoll" from_definition assuming "nonempty"
notation is_lepoll_fm (\<open>\<cdot>_ \<lesssim> _\<cdot>\<close>)
arity_theorem for "is_lepoll_fm"
context M_inj begin
is_iff_rel for "lepoll"
using inj_rel_iff unfolding is_lepoll_def lepoll_rel_def by simp
end \<comment> \<open>\<^locale>\<open>M_inj\<close>\<close>
abbreviation
lepoll_r :: "[i,i\<Rightarrow>o,i] => o" (\<open>_ \<lesssim>\<^bsup>_\<^esup> _\<close> [51,1,51] 50) where
"A \<lesssim>\<^bsup>M\<^esup> B \<equiv> lepoll_rel(M,A,B)"
abbreviation
lepoll_r_set :: "[i,i,i] \<Rightarrow> o" (\<open>_ \<lesssim>\<^bsup>_\<^esup> _\<close> [51,1,51] 50) where
"lepoll_r_set(A,M) \<equiv> lepoll_rel(##M,A)"
context M_Perm
begin
lemma def_lepoll_rel:
assumes
"M(A)" "M(B)"
shows
"lepoll_rel(M,A,B) \<longleftrightarrow> (\<exists>f[M]. f \<in> inj_rel(M,A,B))"
using assms inj_rel_iff
unfolding lepoll_rel_def by simp
end \<comment> \<open>\<^locale>\<open>M_Perm\<close>\<close>
context M_N_Perm
begin
(* the next lemma is not part of the discipline *)
lemma lepoll_rel_transfer: assumes "A \<lesssim>\<^bsup>M\<^esup> B" "M(A)" "M(B)"
shows "A \<lesssim>\<^bsup>N\<^esup> B"
proof -
note assms
moreover from this
obtain f where "f \<in> inj\<^bsup>M\<^esup>(A,B)" "N(f)"
using M.def_lepoll_rel by (auto dest!:M_imp_N)
moreover from calculation
have "f \<in> inj\<^bsup>N\<^esup>(A,B)"
using inj_rel_transfer by (auto)
ultimately
show ?thesis
using N.def_lepoll_rel by (blast dest!:M_imp_N)
qed
end \<comment> \<open>\<^locale>\<open>M_N_Perm\<close>\<close>
(****************** end Discipline ******************)
(******************************************************)
subsection\<open>Discipline for \<^term>\<open>lesspoll\<close>\<close>
relativize functional "lesspoll" "lesspoll_rel" external
relationalize "lesspoll_rel" "is_lesspoll"
synthesize "is_lesspoll" from_definition assuming "nonempty"
notation is_lesspoll_fm (\<open>\<cdot>_ \<prec> _\<cdot>\<close>)
arity_theorem for "is_lesspoll_fm"
context M_Perm begin
is_iff_rel for "lesspoll"
using is_lepoll_iff is_eqpoll_iff
unfolding is_lesspoll_def lesspoll_rel_def by simp
end \<comment> \<open>\<^locale>\<open>M_Perm\<close>\<close>
abbreviation
lesspoll_r :: "[i,i\<Rightarrow>o,i] => o" (\<open>_ \<prec>\<^bsup>_\<^esup> _\<close> [51,1,51] 50) where
"A \<prec>\<^bsup>M\<^esup> B \<equiv> lesspoll_rel(M,A,B)"
abbreviation
lesspoll_r_set :: "[i,i,i] \<Rightarrow> o" (\<open>_ \<prec>\<^bsup>_\<^esup> _\<close> [51,1,51] 50) where
"lesspoll_r_set(A,M) \<equiv> lesspoll_rel(##M,A)"
text\<open>Since \<^term>\<open>lesspoll_rel\<close> is defined as a propositional
combination of older terms, there is no need for a separate ``def''
theorem for it.\<close>
text\<open>Note that \<^term>\<open>lesspoll_rel\<close> is neither $\Sigma_1^{\mathit{ZF}}$ nor
$\Pi_1^{\mathit{ZF}}$, so there is no ``transfer'' theorem for it.\<close>
end | {"subset_name": "curated", "file": "formal/afp/Transitive_Models/Discipline_Function.thy"} |
\begin{document}
\maketitle
\begin{abstract}
We prove existence and uniqueness of solutions to a class of
stochastic semilinear evolution equations with a monotone nonlinear
drift term and multiplicative noise, considerably extending
corresponding results obtained in previous work of ours. In
particular, we assume the initial datum to be only measurable and we
allow the diffusion coefficient to be locally
Lipschitz-continuous. Moreover, we show, in a quantitative fashion,
how the finiteness of the $p$-th moment of solutions depends on the
integrability of the initial datum, in the whole range
$p \in ]0,\infty[$. Lipschitz continuity of the solution map in
$p$-th moment is established, under a Lipschitz continuity
assumption on the diffusion coefficient, in the even larger range
$p \in [0,\infty[$. A key role is played by an It\^o formula for
the square of the norm, in the variational setting, for processes
satisfying minimal integrability conditions, which yields pathwise
continuity of solutions. Finally, we show how the regularity of the
initial datum and of the diffusion coefficient improves the
regularity of the solution and, if applicable, of the invariant
measures.
\medskip\par\noindent
\emph{AMS Subject Classification:} Primary: 60H15, 47H06,
37A25. Secondary: 46N30.
\medskip\par\noindent
\emph{Key words and phrases:} stochastic evolution equations,
singular drift, variational approach, monotonicity methods,
invariant measures.
\end{abstract}
\section{Introduction}
\label{sec:intro}
We consider semilinear stochastic partial differential equations on a
smooth bounded domain $D \subseteq \erre^d$ of the form
\begin{equation}
\label{eq:0}
dX_t + AX_t\,dt + \beta(X_t)\,dt \ni B(t,X_t)\,dW_t,
\qquad X(0)=X_0,
\end{equation}
where $A$ is a coercive maximal monotone operator on (a subspace of)
$H:=L^2(D)$, $\beta$ is a maximal monotone graph in $\erre \times \erre$
defined everywhere, $W$ is a cylindrical Wiener process on a separable
Hilbert space $U$, and $B$ is a process taking values in the space of
Hilbert-Schmidt operators from $U$ to $L^2(D)$ satisfying a (local)
Lipschitz continuity condition. Precise assumptions on the data of the
problem are given in \S\ref{sec:ass} below.
Assuming that the initial datum $X_0$ has finite second moment and the
diffusion coefficient $B$ is globally Lipschitz continuous, we proved
in \cite{cm:luca} that equation \eqref{eq:0} admits a unique solution,
in a generalized variational sense, whose trajectories are weakly
continuous in $H$. The contribution of this work is to extend these
results in several directions. As a first step we show that the
solution $X$ is pathwise strongly continuous in $H$, rather than just
weakly continuous. This is possible thanks to an It\^o-type formula,
interesting in its own right, for the square of the $H$-norm of
processes satisfying minimal integrability conditions, in a
variational setting extending the classical one by Pardoux
\cite{Pard}. The strong pathwise continuity allows us to prove that
existence and uniqueness of solutions to \eqref{eq:0} continues to
hold under much weaker assumptions on the initial datum and on the
diffusion coefficient. In particular, $X_0$ needs only be measurable
and $B$ can be locally Lipschitz-continuous with linear
growth. Denoting by $\Omega$ the underlying probability space, the
solution map $X_0 \mapsto X$ is thus defined on $L^0(\Omega;H)$, with
codomain contained in $L^0(\Omega;E)$, where $E$ is a suitable path
space. By the results of \cite{cm:luca} we also have that the solution
map restricted to $L^2(\Omega;H)$ has codomain contained in
$L^2(\Omega;E)$. As a further result, we extrapolate these mapping
properties to the whole range of exponents $p \in [0,\infty[$, that
is, we show that if $X_0 \in L^p(\Omega;H)$ then
$X \in L^p(\Omega;E)$ for every positive finite $p$, and we provide an
explicit upper bound on the $L^p(\Omega;E)$-norm of the solution in
terms of the $L^p(\Omega;H)$-norm of the initial datum. If, in
addition, $B$ is Lipschitz-continuous, we show that the solution map
is Lipschitz-continuous from $L^p(\Omega;H)$ to $L^p(\Omega;E)$ for
all $p \in [0,\infty[$. In the particular case $p=0$, this implies
that solutions converge uniformly on $[0,T]$ in probability if the
corresponding initial data converge in probability. Finally, we show
how the smoothness of the solution improves (as well as of invariant
measures, if they exist) if the initial datum and the diffusion
coefficient are smoother, without any further regularity assumption on
the (possibly singular) monotone drift term $\beta$. For example, if
$A$ (better said, the part of $A$ in $H$) is self-adjoint, the
solution has paths belonging to the domain of $A$ in $H$ if $X_0$ and
$B$, roughly speaking, take values in the domain of $A^{1/2}$. This
implies that $X$ is a strong solution in the classical sense, not just
in the variational one.
In the classical variational theory of SPDEs, existence and uniqueness
of solutions under a local Lipschitz condition on $B$ and
measurability of $X_0$ were obtained by Pardoux in \cite{Pard}. Our
results do not follow from his, however, as equation \eqref{eq:0}
cannot be cast in the usual variational setting. Stochastic equations
where \emph{all} nonlinear terms are locally Lipschitz-continuous have
been considered in the semigroup approach (see, e.g., \cite{KvN2} and
references therein), but our existence results are not covered, as
$\beta$ can be discontinuous and have arbitrary growth. Moreover, the
properties of the solution map between $L^p(\Omega;H)$ and
$L^p(\Omega;E)$ do not seem to have been addressed even in the
classical variational setting. On the other hand, the continuity of
the solution map in the case $p=0$ for ordinary SDEs in $\erre^n$ with
Lipschitz coefficients has been studied, also with very general
semimartingale noise (see, e.g., \cite{Eme:stab}).
The text is organized as follows. In \S\ref{sec:ass} we state the main
assumptions and we recall the well-posedness result for \eqref{eq:0}
obtained in \cite{cm:luca}.
In \S\ref{sec:cont} we prove a generalized It\^o formula for the
square of the norm, as well as the strong pathwise continuity of
solutions.
In \S\ref{sec:X0} we prove existence and uniqueness of strong
variational solutions to \eqref{eq:0} assuming first that $B$ is
locally Lipschitz-continuous with linear growth and that $X_0$ is
square integrable, hence removing the latter assumption in a second
step, allowing $X_0$ to be merely measurable. While in the former case
solutions have finite second moment, in the latter case one needs to
work with processes that are just measurable (in $\omega$), so that
uniqueness has to be proved in a much larger space. This is achieved
by a suitable application of the It\^o formula of {\S}\ref{sec:cont}
and stopping arguments.
In \S\ref{sec:mom} we show that $X_0$ having finite $p$-th moment
implies that the solution belongs to a space of processes with finite
$p$-moment as well, with explicit control of its norm. The Lipschitz
continuity of the solution map is then established in a particular case.
Further regularity of the solution and of invariant measures is
obtained in the last section, under additional regularity assumptions
on $X_0$ and $B$.
\section{Assumptions and preliminaries}
\label{sec:ass}
\subsection{Notation and terminology}
Given a Banach space $E$, its (topological) dual will be denoted by
$E'$. Given a further Banach space $F$, the (Banach) space of linear
bounded operators from $E$ to $F$ will be denoted by $\cL(E,F)$. If
$E$ and $F$ are Hilbert spaces, $\cL^2(E,F)$ stands for the space of
Hilbert-Schmidt operators from $E$ to $F$. We recall that
Hilbert-Schmidt operators form a two-sided ideal on linear bounded
operators.
A graph $\gamma$ in $E$ is a subset of $E\times E$ and the domain of
$\gamma$ is defined as
$\dom(\gamma) := \{ x\in E: \exists\,y\in E: (x,y) \in \gamma \}$. We
shall identify linear unbounded operators between Banach spaces with
their graphs, as usual. If $E$ is a Hilbert space, $\gamma$ is
monotone if $(x_1,y_1)$, $(x_2,y_2) \in \gamma$ implies
$\ip{y_2-y_1}{x_2-x_1}_E \geq 0$, where $\ip{\cdot}{\cdot}_E$ is the
scalar product in $E$. The notion of maximal monotone graph is
immediate once graphs are ordered by inclusion.
We shall use the standard notation of stochastic calculus (see, e.g.,
\cite{Met}). In particular, given a c\`adl\`ag process $Y$ with values
in a separable Banach space $E$, the process $Y^*$ is defined as
$Y^*(t):=\sup_{s \in [0,t]} \norm{Y(s)}$. For notational convenience,
we shall also denote the time index as a subscript rather than within
parentheses. Moreover, a process $Y$ stopped at a stopping time $S$ is
denoted by $Y^S$, and the stochastic integral of $K$ with respect to a
local martingale $M$ is denoted by $K \cdot M$.
\subsection{Assumptions}
\label{ssec:ass}
Let $D$ be a bounded domain in $\erre^d$ with smooth boundary, and $V$
a real separable Hilbert space densely, continuously, and compactly
embedded in $H:=L^2(D)$. The scalar product and the norm of $H$ will
be denoted by $\ip{\cdot}{\cdot}$ and $\norm{\cdot}$,
respectively. Identifying $H$ with its dual $H'$, the triple
$(V,H,V')$ is a so-called Gelfand triple: the duality form between $V$
and $V'$ extends the scalar product of $H$,
i.e. $\ip{v}{w}={}_V\ip{v}{w}_{V'}$ for any $v$,
$w \in H$. For this reason, we shall simply denote the duality form of
$V$ and $V'$ by the same symbol used for the scalar product in $H$.
The following assumptions on the linear operator $A \in \cL(V,V')$
will be tacitly assumed to hold throughout the whole text:
\begin{itemize}
\item[(i)] there exists $C>0$ such that $\ip{Av}{v} \geq C\norm{v}_V^2$
for every $v\in V$;
\item[(ii)] the part of $A$ in $H$ can be extended to an $m$-accretive
operator $A_1$ on $L^1(D)$;
\item[(iii)] for every $\delta>0$, the resolvent $(I+\delta A_1)^{-1}$
is sub-Markovian, i.e. for every $f\in L^1(D)$ such that
$0 \leq f \leq 1$ a.e. on $D$, one has
$0 \leq (I+\delta A)^{-1}f \leq 1$ a.e. on $D$;
\item[(iv)] there exists $m \in \enne$ such that
$(I+\delta A_1)^{-m} \in \cL(L^1(D), L^\infty(D))$.
\end{itemize}
We shall occasionally refer to hypothesis (i) as coercivity of $A$,
and to hypothesis (iv) as ultracontractivity of the resolvent of
$A_1$.
\smallskip
Let us now state the assumptions on the nonlinear part of the drift:
$\beta \subset \erre \times \erre$ is a maximal monotone graph such
that $0 \in \beta(0)$ and $\dom(\beta)=\erre$. Let
$j:\erre \to [0,+\infty)$ be the unique convex lower-semicontinuous
function such that $j(0)=0$ and $\beta=\partial j$, where $\partial$
stands for the subdifferential in the sense of convex analysis. We
assume that
\[
\limsup_{|r|\to\infty} \frac{j(r)}{j(-r)} < \infty.
\]
Denoting the Moreau-Fenchel conjugate of $j$ by $j^*$, the fact that
$\dom(\beta)=\erre$ is equivalent to the superlinearity of $j^*$ at
infinity, i.e. to
\[
\lim_{|r|\to\infty}\frac{j^*(r)}{|r|} = +\infty.
\]
For a comprehensive treatment of maximal monotone operators and their
connection with convex analysis we refer to, e.g.,
\cite{Barbu:type}. Here we limit ourselves to recalling that, for any
maximal monotone graph $\gamma$ on a Hilbert space $E$, its resolvent
and Yosida approximation of $\gamma$ are defined as
$(I+\lambda\gamma)^{-1}$ and
\[
\gamma_\lambda := \frac{1}{\lambda}
\bigl( I - (I+\lambda\gamma)^{-1} \bigr),
\]
respectively, that both are continuous operators on $E$, and that the
former is a contraction, while the latter is Lipschitz-continuous with
Lipschitz constant bounded by $1/\lambda$.
\smallskip
Let $(\Omega,\cF,\P)$ be a probability space, endowed with a
right-continuous and completed filtration $(\cF_t)_{t \in [0,T]}$, on
which a cylindrical Wiener process $W$ on a real separable Hilbert
space $U$ is defined. The diffusion coefficient
\[
B:\Omega \times[0,T] \times H \to \cL^2(U,H)
\]
is assumed to be such that $B(\cdot,\cdot,x)$ is progressively
measurable for every $x\in H$, and to grow at most linearly in its
third argument, uniformly with respect to the others. That is, we
assume that there exists a constant $N$ such that
\[
\norm[\big]{B(t,\omega,x)}_{\cL^2(U,H)} \leq
N \bigl( 1+\norm{x} \bigr)
\]
for all $(\omega,t,x) \in \Omega \times [0,T] \times H$. In addition
to this, we shall consider two different assumptions, namely
\begin{itemize}
\item[(B1)] $B$ is Lipschitz continuous in its third argument,
uniformly with respect to the others, i.e.
\[
\norm[\big]{B(\omega,t,x) - B(\omega,t,y)}_{\cL^2(U,H)}
\leq N \norm{x-y}
\]
for all $(\omega,t) \in \Omega \times [0,T]$ and $x$, $y \in H$.
\item[(B2)] $B$ is locally Lipschitz continuous in its third argument,
uniformly with respect to the others, i.e. there exists a function
$R \mapsto N_R: \erre_+ \to \erre_+$ such that
\[
\norm[\big]{B(\omega,t,x) - B(\omega,t,y)}_{\cL^2(U,H)}
\leq N_R \norm{x-y}
\]
for all $(\omega,t) \in \Omega \times [0,T]$ and $x$, $y \in H$ with
$\norm{x}$, $\norm{y} \leq R$.
\end{itemize}
\smallskip
Finally, $X_0$ is assumed to be an $H$-valued $\cF_0$-measurable
random variable.
\smallskip
Let us now define the concept of solution to equation \eqref{eq:0}.
\begin{defi}
\label{def:sol}
A strong solution to \eqref{eq:0} is a pair $(X,\xi)$, where $X$ is
a $V$-valued adapted process and $\xi$ is an $L^1(D)$-valued
predictable process, such that, $\P$-almost surely,
\begin{gather*}
X \in L^\infty(0,T; H) \cap L^2(0,T; V), \qquad
\xi \in L^1(0,T; L^1(D)),\\
\xi \in \beta(X) \quad\text{a.e.~in } (0,T)\times D,
\end{gather*}
and
\[
X(t) + \int_0^t AX(s)\,ds + \int_0^t\xi(s)\,ds
= X_0 + \int_0^tB(s,X(s))\,dW(s)
\]
in $V'\cap L^1(D)$ for all $t \in [0,T]$.
\end{defi}
It is convenient to introduce the family of sets $(\cJ_p)_{p\geq 0}$
as follows:
\[
\cJ_p \subset \Bigl( L^p(\Omega; C([0,T];H))
\cap L^p(\Omega; L^2(0,T;V)) \Bigr)
\times L^{p/2}(\Omega; L^1( (0,T) \times D)
\]
formed by processes $(\phi,\psi)$ such that $\phi$ is adapted with
values in $V$, $\psi$ is predictable with values in $L^1(D)$,
$\psi \in \beta(\phi)$ a.e. in $\Omega \times (0,T) \times D$, and
$j(\phi)+j^*(\psi) \in L^{p/2}(\Omega; L^1((0,T) \times D)$.
The following well-posedness result has been proved in
\cite{cm:luca}. Just for the purposes of this statement, we shall
denote the space $\cJ_2$ with $L^\infty(0,T;H)$ in place of
$C([0,T];H)$ by $\tilde{\cJ}_2$.
\begin{thm}
\label{thm:WP}
If $X_0 \in L^2(\Omega,\cF_0;H)$ and $B$ satisfies the global Lipschitz
condition \emph{(B1)}, then there exists a unique strong solution
$(X,\xi)$ to \eqref{eq:0} belonging to $\tilde{\cJ}_2$. Furthermore, the
trajectories of $X$ are weakly continuous in $H$ and the solution
map
\begin{align*}
L^2(\Omega;H) &\longto L^2(\Omega; L^\infty(0,T; H)) \cap
L^2(\Omega; L^2(0,T; V))\\
X_0 &\longmapsto X
\end{align*}
is Lipschitz-continuous.
\end{thm}
Our main result is the following far-reaching extension of
Theorem~\ref{thm:WP}: under the more general local Lipschitz
continuity assumption (B2), for any $X_0\in L^p(\Omega,\cF_0,\P;H)$,
$p \in [0,\infty[$, there exists a strong solution $(X,\xi)$ belonging
to $\cJ_p$, which is unique in $\cJ_0$. In particular, the
trajectories of $X$ are strongly continuous in $H$. Precise statements
and proofs are given in {\S}\ref{sec:X0}.
\section{Pathwise continuity via a generalized It\^o formula}
\label{sec:cont}
In this section we prove that, under the assumptions of
Theorem~\ref{thm:WP}, the unique strong solution $(X,\xi)$ in $\cJ_2$
to \eqref{eq:0} is such that $X$ admits a modification with strongly
continuous trajectories in $H$, rather than just weakly continuous. To
this purpose, we need a generalized It\^o's formula for the square of
the norm under minimal integrability assumptions, that will play a
fundamental role throughout.
We first need some preparations.
Let us recall that the part of $A$ in $H$ is the linear (unbounded)
operator on $H$ defined by $A_2 := A \cap (V \times H)$. In particular,
\[
\dom(A_2) = \bigl\{ u \in V:\, Au \in H \bigr\}
\qquad \text{and} \qquad
A_2u = Au \quad \forall u \in \dom(A_2).
\]
It is well known (see, e.g., \cite{ISEM18}) that $A_2$ is closed and that
$\dom(A_2)$ is a Banach space with respect to the graph norm
\[
\norm{u}_{\dom(A_2)}^2:= \norm{u}^2 + \norm{Au}^2.
\]
Moreover, $\dom(A_2)$ is continuously and densely embedded in $V$.
\begin{lemma}
\label{lm:cvV}
Let $v \in V$ and $v_\lambda := (I+\lambda A_1)^{-1}v$. Then
$v_\lambda \to v$ in $V$ as $\lambda \to 0$.
\end{lemma}
\begin{proof}
Let $v\in V$ and $\varepsilon>0$: since $\dom(A_2)$ is densely
embedded in $V$, we can choose $u \in \dom(A_2)$ such that
$\norm{v-u}_V < \varepsilon$. Setting
$u_\lambda:=(I+\lambda A_1)^{-1}u$, we have
\[
\norm{v-v_\lambda}_V \leq \norm{v-u}_V + \norm{u-u_\lambda}_V
+ \norm{u_\lambda - v_\lambda}_V.
\]
Since $u,v\in V$, we have $u_\lambda-v_\lambda=(I+\lambda A_2)^{-1}(u-v)$, and
recalling that $A_2$ is the part of $A$ in $H$ we have
\[
(u_\lambda-v_\lambda) + \lambda A(u_\lambda-v_\lambda)=u-v,
\]
where the identity holds in $V$ as well. Taking the duality product
with $A(u_\lambda-v_\lambda) \in V'$, by coercivity and boundedness
of $A$ it follows that
\[
\prescript{}{V'}{\ip[\big]{A(u_\lambda-v_\lambda)}{u_\lambda-v_\lambda}_V}
+ \lambda \prescript{}{V'}{
\ip[\big]{A(u_\lambda-v_\lambda)}{A(u_\lambda-v_\lambda)}_V} \geq
C\norm[\big]{u_\lambda-v_\lambda}_V^2
+ \lambda\norm[\big]{A(u_\lambda-v_\lambda)}^2
\]
and
\[
\prescript{}{V'}{\ip{A(u_\lambda-v_\lambda)}{u}}_V \leq
\norm{A}_{\cL(V,V')}\norm{u_\lambda-v_\lambda}_V\norm{u}_V,
\]
hence
\[
C\norm{u_\lambda-v_\lambda}_V^2 + \lambda\norm{A(u_\lambda-v_\lambda)}^2 \leq
\norm{A}_{\cL(V,V')}\norm{u_\lambda-v_\lambda}_V\norm{u}_V,
\]
which implies that there exists a constant $N>0$, independent of
$\lambda$, such that
\[
\norm{u_\lambda-v_\lambda}_V\leq N\norm{u-v}_V,
\]
or, equivalently, that $(I+\lambda A_1)^{-1}$ is uniformly bounded
in $V$ with respect to $\lambda$. This implies that
\[
\norm{u_\lambda - v_\lambda}_V\leq N\norm{u-v}_V\leq N\varepsilon.
\]
It remains to estimate the term $\norm{u-u_\lambda}_V$. Since
$u \in \dom(A_2)$ and
\[
u_\lambda := (I+\lambda A_1)^{-1}u = (I+\lambda A_2)^{-1}u,
\]
one has $u_\lambda \in \dom(A_2^2)$, hence, recalling that $A_2$ is
the part of $A$ in $H$,
\[
Au_\lambda + \lambda A(Au_\lambda) = Au
\]
in $H \embed V'$. Taking the duality pairing with $Au_\lambda \in
\dom(A_2) \embed V$, one has
\[
\prescript{}{V'}{\ip[\big]{Au_\lambda}{Au_\lambda}_V}
+ \lambda \prescript{}{V'}{\ip[\big]{A(Au_\lambda)}{Au_\lambda}_V}
= \prescript{}{V'}{\ip[\big]{Au}{Au_\lambda}_V},
\]
where
\begin{gather*}
\prescript{}{V'}{\ip[\big]{Au_\lambda}{Au_\lambda}_V} =
\norm[\big]{Au_\lambda}^2, \qquad
\prescript{}{V'}{\ip[\big]{A(Au_\lambda)}{Au_\lambda}_V}
\geq C \norm[\big]{Au_\lambda}_V^2,\\
\prescript{}{V'}{\ip[\big]{Au}{Au_\lambda}_V} =
\ip[\big]{Au}{Au_\lambda} \leq \frac12 \norm[\big]{Au}^2 + \frac12
\norm[\big]{Au_\lambda}^2,
\end{gather*}
hence
\[
\norm[\big]{Au_\lambda}^2 + \lambda C \norm[\big]{Au_\lambda}_V^2
\leq \frac12 \norm[\big]{Au}^2 + \frac12
\norm[\big]{Au_\lambda}^2,
\]
which implies that
$\sqrt{\lambda} \norm[\big]{Au_\lambda}_V \leq N \norm[\big]{Au}$,
with a constant $N$ independent of $\lambda$. Therefore, since
$u \in \dom(A_2)$,
\[
\norm[\big]{u_\lambda - u}_V = \lambda \norm[\big]{Au_\lambda}_V
\leq N \sqrt{\lambda} \norm[\big]{Au}.
\]
Choosing $\lambda$ such that
$N \sqrt{\lambda} \norm[\big]{Au} < \varepsilon$, one has then
\[
\norm{v_\lambda-v}_V < (2+N)\varepsilon,
\]
from which the conclusion follows by arbitrariness of $\epsilon$.
\end{proof}
We recall that (see, e.g., \cite{KPS}) if two Banach spaces $F$ and
$G$ are continuously embedded in a separated topological vector space
$E$, their sum $F+G$ is defined as the subspace of $E$
\[
F+G := \bigl\{ u \in E:\, \exists f \in F, \, g \in G: \, u = f+g \bigr\}.
\]
Endowed with the norm
\[
\norm[\big]{u}_{F+G} := \inf_{u=f+g}\bigl( \norm{f}_F + \norm{g}_G \bigr),
\]
$F+G$ is a Banach space. Similarly, the intersection $F \cap G$ is
also a Banach space if endowed with the norm
\[
\norm[\big]{u}_{F \cap G} := \norm{u}_F + \norm{u}_G.
\]
Moreover, if $F \cap G$ is dense in both $F$ and $G$, then $F'$ and
$G'$ are continuously embedded in $(F \cap G)'$, and
$(F+G)'=F' \cap G'$.
In the following we shall deal with $F:=L^1(0,T; H)$ and
$G:=L^2(0,T; V')$, so that as ambient space $E$ one can simply take
$L^1(0,T; V')$. In this case $F \cap G$ is dense in both $F$ and $G$,
hence, by reflexivity of $V$,
\[
\bigl( L^1(0,T;H) + L^2(0,T;V') \bigr)' = L^\infty(0,T;H) \cap L^2(0,T;V).
\]
\begin{thm}
\label{thm:Ito}
Let $Y$, $v$ and $g$ be adapted processes such that
\begin{gather*}
Y \in L^0(\Omega; L^\infty(0,T; H)\cap L^2(0,T; V)), \\\
v \in L^0(\Omega; L^1(0,T; H) + L^2(0,T; V')), \\
g \in L^0(\Omega; L^1(0,T; L^1(D))), \\
\exists\,\alpha>0:\quad j(\alpha Y) + j^*(\alpha g) \in
L^0(\Omega;L^1((0,T) \times D)).
\end{gather*}
Moreover, let $Y_0 \in L^0(\Omega,\cF_0; H)$ and $G$ be a
progressive $\cL^2(U,H)$-valued process such that
\[
G \in L^0(\Omega; L^2(0,T;\cL^2(U,H))).
\]
If
\[
Y(t) + \int_0^tv(s)\,ds + \int_0^t g(s)\,ds = Y_0 +
\int_0^tG(s)\,dW(s) \qquad \forall t \in [0,T] \quad \P\text{-a.s.}
\]
in $V' \cap L^1(D)$, then
\begin{align*}
&\frac12 \norm{Y(t)}^2 + \int_0^t\ip[\big]{v(s)}{Y(s)}\,ds
+ \int_0^t\!\!\int_D g(s,x) Y(s,x)\,dx\,ds \\
&\hspace{3em} = \frac12 \norm{Y_0}^2
+ \frac12\int_0^t \norm{G(s)}^2_{\cL^2(U,H)}\,ds
+ \int_0^t Y(s)G(s)\,dW(s)
\end{align*}
for all $t \in [0,T]$ with probability one.
\end{thm}
\begin{proof}
Since the resolvent of $A_1$ is
ultracontractive by assumption, there exists $m\in\enne$ such that
\[
(I+\delta A_1)^{-m}: L^1(D) \to H \qquad\forall \delta>0.
\]
Using a superscript $\delta$ to denote the action of
$(I+\delta A_1)^{-m}$, we have
\[
Y^\delta(t) + \int_0^tv^\delta(s)\,ds + \int_0^t g^\delta(s)\,ds
= Y^\delta_0 + \int_0^tG^\delta(s)\,dW(s)
\]
where $g^\delta \in L^1(0,T; H)$, hence the classical It\^o's
formula yields, for every $\delta>0$,
\begin{align*}
&\frac12 \norm{Y^\delta(t)}^2
+ \int_0^t\ip[\big]{v^\delta(s)}{Y^\delta(s)}\,ds
+ \int_0^t\!\!\int_D g^\delta(s,x) Y^\delta(s,x)\,dx\,ds \\
&\hspace{3em} = \frac12 \norm{Y^\delta_0}^2
+ \frac12\int_0^t \norm{G^\delta(s)}^2_{\cL^2(U,H)}\,ds
+ \int_0^t Y^\delta(s)G^\delta(s)\,dW(s).
\end{align*}
Let us pass to the limit as $\delta \to 0$. Since the resolvent of
$A_1$ coincides on $H$ with the resolvent of $A_2$, which converges
to the identity in $\cL(H)$ in the strong operator topology, we
immediately infer that
\begin{alignat*}{3}
Y^\delta(t) &\longto Y(t) & &\quad\text{in } H \quad\forall t\in[0,T],\\
g^\delta &\longto g & &\quad\text{in } L^1(0,T; L^1(D)),\\
Y_0^\delta &\longto Y_0 & &\quad\text{in } H,\\
G^\delta &\longto G & &\quad\text{in } L^2(0,T; \cL^2(U,H))
\end{alignat*}
where the last statement, which follows by well-known continuity properties
of Hilbert-Schmidt operators, also implies
\[
\int_0^t \norm{G^\delta(s)}^2_{\cL^2(U,H)}\,ds \longto
\int_0^t \norm{G(s)}^2_{\cL^2(U,H)}\,ds.
\]
Moreover, by the previous lemma we have
\[
Y^\delta \longto Y \quad\text{in } L^2(0,T; V),
\]
and $Y \in L^\infty(0,T;H)$ and the contractivity in $H$ of the
resolvent of $A_1$ immediately imply, by the dominated convergence
theorem, that $Y^\delta \to Y$ weakly* in $L^\infty(0,T;H)$.
Therefore, by reflexivity of $V$,
\[
Y^\delta \longto Y \quad\text{weakly* in }
L^\infty(0,T;H) \cap L^2(0,T;V).
\]
Since $v \in L^1(0,T; H) + L^2(0,T; V')$, we have that $v=v_1+v_2$,
with $v_1 \in L^1(0,T; H)$ and $v_2 \in L^2(0,T; V')$. In this
case $v^\delta$ has to be interpreted as
\[
v^\delta:=(I+\delta A_1)^{-m}v_1 + (I+\delta A)^{-m}v_2.
\]
Note that this is very natural since $A_1$ and $A$ coincide on
$\dom(A_1) \cap V$. By the properties of the resolvent it easily
follows that
\[
v_1^\delta \longto v_1 \quad \text{ in } L^1(0,T; H).
\]
Moreover, since $A^{-1}v_2 \in L^2(0,T; V)$ and
$A^{-1}v_2^\delta = (I+\delta A_2)^{-m}A^{-1}v_2$, by
Lemma~\ref{lm:cvV} we have that $A^{-1}v_2^\delta \to A^{-1}v_2$ in
$L^2(0,T; V)$, hence also, by continuity of $A$,
\[
v_2^\delta \longto v_2 \quad \text{ in } L^2(0,T; V').
\]
The convergences of $v^\delta$ and $Y^\delta$ just proved thus imply
\[
\int_0^t\ip[\big]{v^\delta(s)}{Y^\delta(s)}\,ds \longto
\int_0^t\ip[\big]{v(s)}{Y(s)}\,ds
\]
for all $t \in [0,T]$.
We are now going to prove that
$\bigl( (Y^\delta G^\delta) \cdot W - (YG) \cdot W \bigr)^*_T \to 0$
in probability. Setting $M_\delta := (Y^\delta G^\delta) \cdot W$
and $M := (YG) \cdot W$, it is well known that it suffices to show
that the quadratic variation of $M_\delta-M$ converges to $0$ in
probability. One has
\begin{align*}
\bigl[ M_\delta-M,M_\delta-M \bigr]
&= \norm[\big]{Y^\delta G^\delta - YG}^2_{L^2(0,T;\cL^2(U,\erre))}\\
&\leq \norm[\big]{Y^\delta G^\delta - Y^\delta G}^2_{L^2(0,T;\cL^2(U,\erre))}
+ \norm[\big]{Y^\delta G - YG}^2_{L^2(0,T; \cL^2(U,\erre))}\\
&\leq\norm{Y}^2_{L^\infty(0,T; H)}\norm{G^\delta-G}^2_{L^2(0,T; \cL^2(U,H))}+
\norm{Y^\delta G - YG}^2_{L^2(0,T; \cL^2(U,\erre))},
\end{align*}
where the convergence to zero of the first term in the last
expression has already been proved, and
\[
\norm{Y^\delta G - YG}^2_{L^2(0,T; \cL^2(U,\erre))}
\leq \int_0^T \norm[\big]{Y^\delta(s)-Y(s)}^2
\norm[\big]{G(s)}^2_{\cL^2(U,H)}\,ds \longto 0,
\]
by the dominated convergence theorem, because $Y^\delta \to Y$
pointwise in $H$ and
$\norm{Y^\delta - Y} \leq 2 \norm{Y} \in L^\infty(0,T)$. We have
thus shown that
\[
\int_0^\cdot Y^\delta(s) G^\delta(s)\,dW(s) \longto
\int_0^\cdot Y(s) G(s)\,dW(s)
\]
in probability, hence $\P$-a.s. along a subsequence of $\delta$.
Finally, it is clear that $Y^\delta g^\delta \to Yg$ in measure in
$(0,T) \times D$, and that, thanks to the assumptions on $j$,
\[
\pm \alpha^2 Y^\delta g^\delta \leq
j(\pm\alpha Y^\delta) + j^*(\alpha g^\delta)
\lesssim 1 + j(\alpha Y^\delta) + j^*(\alpha g^\delta),
\]
where the second inequality follows from the fact that, thanks to
the assumption on the growth of $j$ at $\infty$, there exists a
constant $M>0$ such that
\[
j(r) \leq M \bigl( 1+ j(-r) \bigr) \qquad \forall r \in \erre.
\]
Jensen's inequality for sub-Markovian operators
(see, e.g., \cite{Haa07}) thus yields
\[
j(\alpha Y^\delta) + j^*(\alpha g^\delta) \leq
(I+\delta A_1)^{-m} \bigl( j(\alpha Y) + j^*(\alpha g) \bigr),
\]
so that
\[
\alpha^2 \abs[\big]{Y^\delta g^\delta} \lesssim
1 + (I+\delta A_1)^{-m} \bigl( j(\alpha Y) + j^*(\alpha g) \bigr).
\]
Since $j(\alpha Y) + j^*(\alpha g) \in L^1((0,T) \times D)$ by
assumption, the contractivity of the resolvent in $L^1(D)$ and the
dominated convergence theorem imply that the right-hand side in the
last inequality is convergent in $L^1((0,T) \times D)$. Hence
$(Y^\delta g^\delta)_\delta$ is uniformly integrable and, by Vitali's
theorem,
\[
\int_0^t\!\!\int_D g^\delta(s,x) Y^\delta(s,x)\,dx\,ds \longto
\int_0^t\!\!\int_D g(s,x) Y(s,x)\,dx\,ds
\]
for all $t \in [0,T]$. The proof is thus completed.
\end{proof}
As a first important consequence of the generalized It\^o formula we
show that (the first component of) strong solutions are pathwise
strongly continuous in $H$.
\begin{thm}
\label{thm:cont}
Let $(X,\xi)$ be the unique strong solution to \eqref{eq:0}
belonging to $\cJ_2$. Then $X$ has strongly continuous paths in $H$,
i.e.~there exists $\Omega'\in\cF$ with $\P(\Omega')=1$ such that
\[
X(\omega) \in C([0,T]; H) \qquad\forall \omega \in \Omega'.
\]
\end{thm}
\begin{proof}
Let $r \in [0,T]$. We have to prove that $X(t) \to X(r)$ in $H$ as
$t \to r$, $t \in [0,T]$. It follows from Theorem~\ref{thm:Ito}
that for every $t\in[0,T]$ there exists $\Omega' \in \cF_0$ with
$\P(\Omega')=1$ such that
\begin{align*}
\frac12\norm{X(t)}^2 - \frac12\norm{X(r)}^2
&=
-\int_r^t \ip{AX(s)}{X(s)}\,ds - \int_r^t\!\!\int_D\xi(s)X(s)\,ds\\
&+ \frac12 \int_r^t \norm[\big]{B(s)}^2_{\cL^2(U,H)}
+ \int_r^tX(s)B(s,X(s))\,dW(s)
\end{align*}
everywhere on $\Omega'$. By the definition of strong solution, we
can assume that $X \in L^\infty(0,T; H)$, $AX\in L^2(0,T; V')$ and
$j(X)+j^*(\xi) \in L^1((0,T)\times D)$, as well as that
$B(\cdot, X) \in L^2(0,T;\cL^2(U,H))$, everywhere on $\Omega'$.
Since $X\xi = j(X)+j^*(\xi)$, it
follows that the process
\[
[0,T] \ni s \longmapsto \psi(s) := -\ip{AX(s)}{X(s)}
- \int_D\xi(s)X(s) + \frac12 \norm[\big]{B(s,X(s))}^2_{\cL^2(U,H)}
\]
belongs to $L^1(0,T)$ everywhere on $\Omega'$. Therefore, writing
\[
\frac12\norm{X(t)}^2 - \frac12\norm{X(r)}^2 = \int_r^t\phi(s)\,ds
+ \int_r^t X(s) B(s,X(s))\,dW(s),
\]
since $\psi \in L^1(0,T)$ and the stochastic integral has continuous
trajectories, we have, as $t \to r$,
\[
\norm{X(t)}^2 - \norm{X(r)}^2 \to 0,
\]
so that $\norm{X(t)}\to\norm{X(r)}$. Furthermore, $X(t) \to X(r)$
weakly in $H$ as $t \to r$ by Theorem~\ref{thm:WP}, hence, since $H$
is uniformly convex, we conclude that $X(t) \to X(r)$ in $H$ (cf.,
e.g., \cite[Proposition~3.32]{Bre-FA}).
\end{proof}
\section{Existence and uniqueness}
\label{sec:X0}
We begin with a simple estimate that will be used several times.
\begin{lemma}
\label{lm:BDGY}
Let $F$ and $G$ be progressive process with values in $H$ and
$\cL^2(U,H)$, respectively, such that $FG$ is integrable with
respect to $W$. For any numbers $p$, $\varepsilon>0$ and any
stopping time $S$ one has
\[
\norm[\big]{\bigl( (FG) \cdot W \bigr)^*_S}_{L^p(\Omega)}
\lesssim \varepsilon \norm[\big]{F_S^*}^2_{L^{2p}(\Omega)}
+ \frac{1}{\varepsilon} \norm[\big]{
G\ind{[\![0,S]\!]}}^2_{L^{2p}(\Omega;L^2(0,T;\cL^2(U,H)))}
\]
\end{lemma}
\begin{proof}
The BDG inequality asserts that
\[
\norm[\big]{\bigl( (FG) \cdot W \bigr)^*_S}_{L^p(\Omega)}
\eqsim \norm[\big]{[(FG) \cdot W,(FG) \cdot W]_S^{1/2}}_{L^p(\Omega)},
\]
where, by the ideal property of Hilbert-Schmidt operators and
Young's inequality,
\begin{align*}
[(FG) \cdot W,(FG) \cdot W]_S^{1/2}
&= \biggl( \int_0^S \norm[\big]{F(t)G(t)}_{\cL^2(U,\erre)}^2\,dt
\biggr)^{1/2}\\
&\leq \biggl( \int_0^S \norm[\big]{F(t)}^2
\norm[\big]{G(t)}^2_{\cL^2(U,H)}\,dt \biggr)^{1/2}\\
&\leq F_S^* \biggl( \int_0^S \norm[\big]{G(t)}^2_{\cL^2(U,H)}\,dt
\biggr)^{1/2}\\
&\leq \varepsilon F_S^{*2} + \frac{1}{\varepsilon}
\int_0^S \norm[\big]{G(t)}^2_{\cL^2(U,H)}\,dt.
\end{align*}
Therefore, taking the $L^p(\Omega)$-(quasi)norm on both sides,
\[
\norm[\big]{\bigl( (FG) \cdot W \bigr)^*_S}_{L^p(\Omega)}
\lesssim \varepsilon \norm[\big]{F_S^*}^2_{L^{2p}(\Omega)}
+ \frac{1}{\varepsilon} \norm[\big]{
G\ind{[\![0,S]\!]}}^2_{L^{2p}(\Omega;L^2(0,T;\cL^2(U,H)))}
\qedhere
\]
\end{proof}
\medskip
Let $(X,\xi)$ and $(Y,\eta) \in \cJ_0$ be strong solutions, in the
sense of Definition~\ref{def:sol}, to the equation
\[
dX + AX\,dt + \beta(X)\,dt \ni B(\cdot,X)\,dW
\]
with initial conditions $X_0$ and $Y_0$, both elements of
$L^0(\Omega,\cF_0,\P;H)$, respectively. Here and throughout this
section we assume that $B$ is locally Lipschitz-continuous in the
sense of assumption (B2).
Let us also introduce the sequence of stopping times
$(T_n)_{n\in\enne}$ defined as
\[
T_n := \inf \bigl\{ t \geq 0: \norm{X_\Gamma(t)} \geq n \; \text{ or }
\; \norm{Y_\Gamma(t)} \geq n \bigr\} \wedge T.
\]
Here and in the following, for any $\Gamma \in \cF_0$, we shall denote
multiplication by $\ind{\Gamma}$ by a subscript $\Gamma$. Even though
the stopping times $T_n$ depend on $\Gamma$, we shall not indicate
this explicitly to avoid making the notation too cumbersome.
The stopping times $T_n$ are well defined because, by definition of
$\cJ_0$, $X$ and $Y$ have continuous paths with values in
$H$. Moreover, $T_n \neq 0$ for sufficiently large $n$.
\medskip
The estimate in the following lemma is an essential tool, from which,
for instance, uniqueness and a local property of solutions will follow
as easy corollaries.
\begin{lemma}
\label{lm:gm}
Let $\Gamma \in \cF_0$ be such that $X_{0\Gamma},\, Y_{0\Gamma} \in
L^2(\Omega,\cF_0,\P;H)$. One has, for every $n \in \enne$,
\[
\E\bigl( X_\Gamma- Y_\Gamma \bigr)^{*2}_{T_n} \lesssim
\E\norm[\big]{\pG{X_0}-\pG{Y_0}}^2,
\]
with implicit constant depending on $T$ and on the Lipschitz
constant of $B$ in the ball in $H$ of radius $n$.
\end{lemma}
\begin{proof}
One has
\[
(X-Y) + \int_0^t A(X-Y)\,ds + \int_0^t (\xi-\eta)\,ds = X_0-Y_0
+ \int_0^t (B(X)-B(Y))\,dW.
\]
We recall that, for any $\cF_0$-measurable random variable $\zeta$
and any stochastically integrable process $K$, one has
$\zeta(K \cdot W) = (\zeta K)\cdot W$. Therefore
\[
\pG{(X-Y)} + \int_0^t A\pG{(X-Y)}\,ds
+ \int_0^t \pG{(\xi-\eta)}\,ds = \pG{(X_0-Y_0)}
+ \int_0^t \pG{(B(X)-B(Y))}\,dW.
\]
The It\^o formula of Theorem~\ref{thm:Ito} yields
\begin{align*}
&\norm[\big]{\pG{X}-\pG{Y}}^2(t \wedge T_n)
+ 2\int_0^{t \wedge T_n}
\ip[\big]{A(\pG{X}-\pG{Y})}{\pG{X}-\pG{Y})}\,ds
+ 2\int_0^{t \wedge T_n}\!\!\int_D \pG{((X-Y)(\xi-\eta))}\,ds\\
&\qquad = \norm[\big]{\pG{X_0} - \pG{Y_0}}^2
+ \int_0^{t \wedge T_n} \norm[\big]{\pG{(B(X)-B(Y))}}^2_{\cL^2(U,H)}\,ds\\
&\qquad\quad + 2\int_0^{t \wedge T_n} \pG{(X-Y)} \pG{(B(X)-B(Y))}\,dW,
\end{align*}
where (a) the second and term terms on the left-hand side are
positive by monotonicity of $A$ and $\beta$, and by the assumption
that $\xi \in \beta(X)$, $\eta \in \beta(Y)$ a.e. in
$\Omega \times (0,T) \times D$; (b) one has
\[
\pG{(B(X)-B(Y))} = \ind{\Gamma} \bigl( B(\pG{X}) - B(\pG{Y}) \bigr),
\]
hence
\[
\ind{[\![0,T_n]\!]} \norm[\big]{\pG{(B(X)-B(Y))}}^2_{\cL^2(U,H)}
\lesssim_n \ind{[\![0,T_n]\!]} \, \ind{\Gamma}
\norm[\big]{\pG{X}-\pG{Y}}.
\]
Taking supremum in time and expectation,
\begin{align*}
\E \bigl( \pG{X}^{T_n} - \pG{Y}^{T_n} \bigr)^{*2}_{t}
&\lesssim \E\norm[\big]{\pG{X_0}-\pG{Y_0}}^2
+ \int_0^t \E \bigl( \pG{X}^{T_n} - \pG{Y}^{T_n} \bigr)^{*2}_{s}\,ds\\
&\quad + \E\sup_{s \leq t} \int_0^{s \wedge T_n}
\pG{(X-Y)} \pG{(B(X)-B(Y))}\,dW,
\end{align*}
where, by Lemma~\ref{lm:BDGY}, the last term on the right-hand side
is bounded by
\begin{align*}
&\varepsilon \E \bigl( \pG{X}^{T_n} - \pG{Y}^{T_n} \bigr)^{*2}_{t}
+ N(\varepsilon) \E\int_0^{t \wedge T_n}
\norm[\big]{\pG{(B(X)-B(Y))}}^2_{\cL^2(U,H)}\,ds\\
&\qquad \leq \varepsilon \E\bigl( \pG{X}^{T_n}-\pG{Y}^{T_n} \bigr)^{*2}_{t}
+ N(\varepsilon,n) \int_0^t \E
\bigl( \pG{X}^{T_n} - \pG{Y}^{T_n} \bigr)^{*2}_{s}\,ds.
\end{align*}
Choosing $\varepsilon$ small enough, it follows by Gronwall's
inequality that
\[
\E\bigl( X_\Gamma - Y_\Gamma \bigr)^{*2}_{T_n} =
\E\bigl( \pG{X}^{T_n}-\pG{Y}^{T_n} \bigr)^{*2}_{T} \lesssim
\E\norm[\big]{\pG{X_0}-\pG{Y_0}}^2,
\]
with an implicit constant that depends on $T$ and on the Lipschitz
constant of $B$ on the ball in $H$ of radius $n$.
\end{proof}
\begin{coroll}
\label{cor:uniq}
Uniqueness of strong solutions in $\cJ_0$ holds for \eqref{eq:0}.
\end{coroll}
\begin{proof}
Let $(X,\xi)$, $(Y,\eta) \in \cJ_0$ be strong solutions to
\eqref{eq:0}. For any $\Gamma \in \cF_0$ such that
$X_{0\Gamma} \in L^2(\Omega;H)$ the previous lemma yields
$X^{T_n}_\Gamma = Y^{T_n}_\Gamma$ for all $n \in \enne$, hence
$X_\Gamma = Y_\Gamma$. Writing
\[
\Omega = \bigcup_{k\in\enne} \Omega_k, \qquad
\Omega_k := \bigl\{ \omega \in \Omega: \norm{X_0(\omega)} \leq k \bigr\},
\]
and choosing $\Gamma$ as $\Omega_k$, it follows that
$X\ind{\Omega_k} = Y\ind{\Omega_k}$ for all $k$, hence $X=Y$. By
comparison, $\xi = \eta$ a.e. in $\Omega \times (0,T) \times D$.
\end{proof}
\begin{rmk}
To prove the corollary, by inspection of the proof of
Lemma~\ref{lm:gm} it is evident that one may directly take
$\Gamma=\Omega$, as in this case $X_0-Y_0=0$, whose second moment is
obviously finite. This immediately implies $X^{T_n} = Y^{T_n}$ for
all $n \in \enne$, hence $X=Y$.
\end{rmk}
\begin{coroll}
\label{cor:loc}
Let $\Gamma \in \cF_0$. If $X_{0\Gamma}=Y_{0\Gamma}$, then
$X_\Gamma=Y_\Gamma$, and $\xi_\Gamma = \eta_\Gamma$ a.e. in $\Omega
\times (0,T) \times D$.
\end{coroll}
\begin{proof}
Write $\Omega = \bigcup_{k\in\enne} \Omega_k$, where
\[
\Omega_k := \bigl\{ \omega \in \Omega: \norm{X_0(\omega)} \leq k \bigr\}
\cap \bigl\{ \omega \in \Omega: \norm{Y_0(\omega)} \leq k \bigr\}.
\]
Then $X_0\ind{\Gamma \cap \Omega_k}$, $Y_0\ind{\Gamma \cap \Omega_k}
\in L^2(\Omega;H)$, and Lemma~\ref{lm:gm} implies that
$X_{\Gamma\cap\Omega_k} = Y_{\Gamma\cap\Omega_k}$ for all $k \in
\enne$, hence $X_\Gamma = Y_\Gamma$, as well as, again by
comparison, $\xi_\Gamma = \eta_\Gamma$ a.e. in $\Omega \times (0,T)
\times D$.
\end{proof}
Now that uniqueness is cleared, we turn to the question of existence
of strong solutions. For this we need some preparations. For $R>0$,
let us consider the truncation operator $\sigma_R:H \to H$ defined as
\[
\sigma_R: x \longmapsto
\begin{cases}
x, & \norm{x} \leq R,\\
R x / \norm{x}, & \norm{x} > R.
\end{cases}
\]
We shall then define
\begin{align*}
B_R: \Omega \times[0,T] \times H &\longto \cL^2(U,H)\\
(\omega,t,x) &\longmapsto B(\omega,t,\sigma_R(x)).
\end{align*}
Let us check that $B_R$ is Lipschitz-continuous for every $R>0$. The
progressive measurability of $B_R$ follows from the one of $B$ and the
fact that $\sigma_R:H\to H$ is (Lipschitz) continuous. Moreover, since
$\sigma_R$ is $1$-Lipschitz continuous, thanks to the local Lipschitz
continuity and the linear growth of $B$, for every $\omega\in\Omega$,
$t\in[0,T]$ and $x,y\in H$ one has
\[
\norm{B_R(\omega,t,x)-B_R(\omega,t,y)}_{\cL^2(U,H)}\leq
N_R\norm{\sigma_R(x)-\sigma_R(y)}\leq N_R\norm{x-y}
\]
as well as
\[
\norm{B_R(\omega, t, x)}_{\cL^2(U,H)}\leq N(1+\norm{\sigma_R(x)})\leq N(1+\norm{x}).
\]
Thanks to Theorems~\ref{thm:WP} and \ref{thm:cont}, as well as
Lemma~\ref{lm:gm}, the equation
\begin{equation}
\label{eq:n}
dX_n + AX_n\,dt + \beta(X_n)\,dt = B_n(X_n)\,dW, \qquad X_n(0)=X_0,
\end{equation}
admits a strong solution $(X_n,\xi_n)$, which belongs to $\cJ_2$ and
is unique in $\cJ_0$, for every $n \in \enne$.\footnote{Note that
Theorem~\ref{thm:WP} only shows that $(X_n,\xi_n)$ is unique in
$\cJ_2$, while Lemma~\ref{lm:gm} yields uniqueness in the larger
space $\cJ_0$.} Moreover, by the strong continuity of the paths of
$X_n$, one can define the increasing sequence of stopping times
$(\tau_n)_{n\in\enne}$ by
\[
\tau_n := \inf \bigl\{ t \in [0,T]: \norm{X_n(t)} \geq n \bigr\},
\]
as well as the stopping time
\[
\tau := \lim_{n\to\infty} \tau_n = \sup_{n\in\enne} \tau_n.
\]
As first step we show that the sequence of processes $(X_n,\xi_n)$ satisfies a
sort of consistency condition.
\begin{lemma}
\label{lm:ind}
One has $X_{n+1}^{\tau_n} = X_n^{\tau_n}$ for all $n \in \enne$, as
well as $\xi_n\ind{[\![0,\tau_n]\!]} = \xi_{n+1}\ind{[\![0,\tau_n]\!]}$ in
$L^0(\Omega \times (0,T) \times D)$.
\end{lemma}
\begin{proof}
It\^o's formula yields, in view of the monotonicity of $A$ and
$\beta$,
\begin{align*}
\norm[\big]{X_{n+1}-X_n}^2(t \wedge \tau_n)
&\lesssim \int_0^{t \wedge \tau_n} (X_{n+1}-X_n)
\bigl( B_{n+1}(X_{n+1})-B_n(X_n) \bigr)(s)\,dW(s)\\
&\quad
+ \int_0^{t \wedge \tau_n}
\norm[\big]{B_{n+1}(X_{n+1}(s))-B_n(X_n(s))}^2_{\cL^2(U,H)}\,ds.
\end{align*}
Note that $B_{n+1}=B_n$ on the ball of radius $n$ in $H$, hence
$B_n(X_n)=B_{n+1}(X_n)$ on $[\![0,\tau_n]\!]$. Therefore, since
$B_{n+1}$ is Lipschitz continuous,
\begin{align*}
\E \bigl( X^{\tau_n}_{n+1} - X^{\tau_n}_n \bigr)^{*2}_t
&\lesssim
\E\bigl(
((X_{n+1}-X_n) (B_{n+1}(X_{n+1}) - B_{n+1}(X_n))) \cdot W \bigr)^*_{t \wedge \tau_n}\\
&\quad + \int_0^t \E\bigl( X^{\tau_n}_{n+1} - X^{\tau_n}_n \bigr)^{*2}_s\,ds,
\end{align*}
where the first term on the right-hand side can be estimated, thanks
to the BDG inequality and the ideal property of Hilbert-Schmidt
operators, by
\begin{align*}
&\E\biggl( \int_0^{t \wedge \tau_n}
\norm[\big]{X_{n+1}-X_n}^2(s)
\norm[\big]{B_{n+1}(X_{n+1}(s))-B_{n+1}(X_n(s))}^2_{\cL^2(U,H)}\,ds
\biggr)^{1/2}\\
&\hspace{3em} \lesssim_n \E\bigl(
X_{n+1}^{\tau_n} - X_n^{\tau_n} \bigr)^*_t \biggl( \int_0^t
\norm[\big]{X_{n+1}^{\tau_n}-X_n^{\tau_n}}^2(s) \biggr)^{1/2}\\
&\hspace{3em} \lesssim_n
\varepsilon \E \bigl( X_{n+1}^{\tau_n} - X_n^{\tau_n} \bigr)^{*2}_t
+ \frac{1}{\varepsilon} \int_0^t
\E \bigl( X_{n+1}^{\tau_n} - X_n^{\tau_n} \bigr)^{*2}_s\,ds.
\end{align*}
Choosing $\varepsilon$ small enough, Gronwall's inequality implies
\[
\E \bigl( X_{n+1}^{\tau_n} - X_n^{\tau_n} \bigr)^{*2}_t = 0
\]
for all $t \leq T$, hence $X_{n+1}^{\tau_n} =
X_n^{\tau_n}$. The first claim is thus proved.
In order to prove the second claim, note that it holds
\begin{align*}
X_{n+1}^{\tau_n}(t) + \int_0^{t \wedge \tau_n} AX_{n+1}\,ds
+ \int_0^{t \wedge \tau_n} \xi_{n+1}\,ds
&= X_0 + \int_0^{t \wedge \tau_n} B_{n+1}(X_{n+1})\,dW,\\
X_{n}^{\tau_n}(t) + \int_0^{t \wedge \tau_n} AX_{n}\,ds
+ \int_0^{t \wedge \tau_n} \xi_{n}\,ds
&= X_0 + \int_0^{t \wedge \tau_n} B_{n}(X_{n})\,dW,
\end{align*}
where $B_n(X_n)$ on the right-hand side of the second identity can
be replaced by $B_{n+1}(X_{n+1})$ because the paths of
$X^{\tau_n}_{n+1}$ remain within a ball of radius $n$ in $H$ and
$X_{n+1}^{\tau_n}=X_n^{\tau_n}$. This identity also yields, by
comparison,
\[
\int_0^t \xi_{n+1}\ind{[\![0,\tau_n]\!]}\,ds
= \int_0^{t \wedge \tau_n} \xi_{n+1}\,ds
= \int_0^{t \wedge \tau_n} \xi_{n}\,ds
= \int_0^t \xi_{n}\ind{[\![0,\tau_n]\!]}\,ds,
\]
which implies the second claim.\footnote{The argument in fact proves
the following slightly stronger statement: setting
$\Xi_n:=\int_0^\cdot \xi_n\,ds$, the processes
$\Xi_{n+1}^{\tau_n}$ and $\Xi_n^{\tau_n}$ are indistinguishable
for all $n$.}
\end{proof}
The lemma implies that one can define processes $X$ and $\xi$ on
$[\![0,\tau]\!]$ by the prescriptions $X := X_n$ and $\xi := \xi_n$ on
$[\![0,\tau_n]\!]$ for all $n \in \enne$, or equivalently (but perhaps
less tellingly), as $X=\lim_{n\to\infty} X_n$ and
$\xi=\lim_{n\to\infty} \xi_n$.
\medskip
We are now going to show that the linear growth assumption on $B$
implies that $\tau=T$. We shall first establish a priori estimates for
the solution to equation \eqref{eq:n}.
\begin{lemma}
\label{lm:ap}
There exists a constant $N>0$, independent of $n$, such that
\[
\E\norm[\big]{X_n}^2_{C([0,T];H)}
+ \E\norm[\big]{X_n}^2_{L^2(0,T;V)}
+ \E\norm[\big]{\xi_nX_n}_{L^1((0,T) \times D)}
< N \bigl( 1 + \E\norm{X_0}^2 \bigr).
\]
\end{lemma}
\begin{proof}
The It\^o formula of Theorem~\ref{thm:Ito} yields
\begin{align*}
&\norm{X_n(t)}^2
+ 2\int_0^{t} \ip{AX_n(s)}{X_n(s)}\,ds
+ 2\int_0^{t} \!\!\int_D\xi_n(s)X_n(s)\,dx\,ds\\
&\hspace{3em} = \norm{X_0}^2
+ \int_0^{t} \norm[\big]{B_n(s,X_n(s))}^2_{\cL^2(U,H)}\,ds
+ 2\int_0^{t} X_n(s)B_n(s,X_n(s))\,dW(s),
\end{align*}
where, recalling that $B_n=B(\cdot,\cdot,\sigma_n(\cdot))$ and
$\sigma_n$ is a contraction in $H$, and that $B$ grows at most
linearly,
\[
\int_0^t \norm[\big]{B_n(s,X_n(s))}^2_{\cL^2(U,H)}
\lesssim T + \int_0^T \norm{X_n(s)}^2\,ds.
\]
Denoting the stochastic integral on the right-hand side by $M_n$,
taking supremum in time and expectation we get, by the coercivity of
$A$,
\begin{align*}
&\E\norm[\big]{X_n}^2_{C([0,T];H)}
+ \E\norm[\big]{X_n}^2_{L^2(0,T;V)}
+ \E\norm[\big]{\xi_nX_n}_{L^1((0,T) \times D)}\\
&\hspace{3em} \lesssim 1 + \E\norm{X_0}^2
+ \E\int_0^T \norm{X_n(s)}^2\,ds
+ \E M_T^{*2},
\end{align*}
where the implicit constant depends on $T$. By Lemma~\ref{lm:BDGY}
we have, for any $\varepsilon>0$,
\[
\E M_T^{*2} \lesssim \varepsilon \E\norm[\big]{X_n}^2_{C([0,T];H)}
+ N(\varepsilon) \E \int_0^T \norm{X_n(s)}^2\,ds,
\]
therefore, choosing $\varepsilon$ sufficiently small,
\begin{align*}
&\E\norm[\big]{X_n}^2_{C([0,T];H)}
+ \E\norm[\big]{X_n}^2_{L^2(0,T;V)}
+ \E\norm[\big]{\xi_nX_n}_{L^1((0,T) \times D)}\\
&\hspace{3em} \lesssim 1 + \E\norm{X_0}^2
+ \E\int_0^T \norm{X_n(s)}^2\,ds.
\end{align*}
Since this inequality holds also with $T$ replaced by any
$t \in ]0,T]$, we also have
\[
\E\norm[\big]{X_n}^2_{C([0,t];H)} \lesssim
1 + \E\norm{X_0}^2 + \int_0^T \E\norm[\big]{X_n}^2_{C([0,s];H)}\,ds,
\]
hence, by Gronwall's inequality,
\[
\E\norm[\big]{X_n}^2_{C([0,T];H)} \lesssim 1 + \E\norm{X_0}^2,
\]
with implicit constant depending on $T$. Since $C([0,T];H)$ is
continuously embedded in $L^2(0,T;H)$, one easily deduces
\[
\E\norm[\big]{X_n}^2_{C([0,T];H)} +
\E\norm[\big]{X_n}^2_{L^2(0,T;V)} +
\E\norm[\big]{\xi_nX_n}_{L^1((0,T) \times D)} \lesssim 1 +
\E\norm{X_0}^2.
\qedhere
\]
\end{proof}
\begin{lemma}
\label{lm:tau}
One has
\[
\P\Bigl( \limsup_{n \to \infty} \{\tau_n \leq T\} \Bigr) = 0.
\]
In particular, $\tau=T$.
\end{lemma}
\begin{proof}
By Markov's inequality and the previous lemma,
\[
\P\bigl( \norm{X_n}_{C([0,T];H)} \geq n \bigr)
\leq \frac{1}{n^2} \E\norm{X_n}^2_{C([0,T]; H)}
\lesssim \frac{1}{n^2} \bigl( 1 + \E\norm{X_0}^2).
\]
Since the event $\{ \norm{X_n}_{C([0,T];H)} \geq n \}$ coincides
with $\{\tau_n \leq T\}$, one has
\[
\sum_{n=1}^\infty \P\bigl( \tau_n \leq T \bigr) < \infty,
\]
thus also, by the Borel-Cantelli lemma,
\[
\P\left(\bigcap_{n\in\enne}\bigcup_{k\geq n}\{\tau_k\leq T\}\right) = 0.
\]
In other words, the sequence $(\tau_n)$ is ultimately constant: for
each $\omega$ in a subset of $\Omega$ of $\P$-measure one, there
exists $m=m(\omega)$ such that $\tau_n(\omega)=T$ for all $n>m$. In
particular, $\tau=T$ $\P$-almost surely.
\end{proof}
This lemma implies that the processes $X$ and $\xi$ defined
immediately after the proof of Lemma~\ref{lm:ind} are indeed defined
on the whole interval $[0,T]$.
We can now prove the first existence result.
\begin{thm}
\label{thm:int}
Assume that $X_0 \in L^2(\Omega,\cF_0,\P;H)$. Then equation
\eqref{eq:0} admits a unique strong solution, which belongs to
$\mathscr{J}_2$.
\end{thm}
\begin{proof}
Uniqueness of strong solutions is proved, in more generality, by
Corollary~\ref{cor:uniq}. Let us prove existence. By stopping at
$\tau_n$, one has
\[
X_n^{\tau_n}(t) + \int_0^{t \wedge \tau_n} AX_n(s)\,ds
+ \int_0^{t \wedge \tau_n} \xi_n(s)\,ds
= X_0 + \int_0^{t \wedge \tau_n} B_n(X_n(s))\,ds,
\]
where, by definition of $X$, $X_n^{\tau_n}=X^{\tau_n}$, as well as,
by definition of $B_n$,
\[
\int_0^{t \wedge \tau_n} B_n(X_n(s))\,ds
= \int_0^{t \wedge \tau_n} B(X(s))\,ds.
\]
Similarly, by definition of $\xi$ it follows that
\[
\int_0^{t \wedge \tau_n} \xi_n(s)\,ds
= \int_0^{t \wedge \tau_n} \xi(s)\,ds,
\]
hence that
\[
X^{\tau_n}(t) + \int_0^{t \wedge \tau_n} AX(s)\,ds
+ \int_0^{t \wedge \tau_n} \xi(s)\,ds
= X_0 + \int_0^{t \wedge \tau_n} B(X(s))\,ds.
\]
Since this identity holds for all $n \in \enne$ and $\tau_n \to T$
as $n \to \infty$, we infer that
\[
X(t) + \int_0^t AX(s)\,ds
+ \int_0^t \xi(s)\,ds
= X_0 + \int_0^t B(X(s))\,ds
\]
for all $t \in [0,T]$ $\P$-a.s.. Moreover, for every $n \in \enne$,
$\xi_n \in \beta(X_n)$ a.e. in $\Omega \times (0,T) \times D$, hence
$\xi_n\ind{[\![0,\tau_n]\!]} \in \beta(X_n)\ind{[\![0,\tau_n]\!]}$,
thus also
$\xi\ind{[\![0,\tau_n]\!]} \in \beta(X)\ind{[\![0,\tau_n]\!]}$
a.e. in $\Omega \times (0,T) \times D$. Recalling that
$\tau_n \to T$ as $n \to \infty$, this in turn implies
$\xi \in \beta(X)$ a.e. in $\Omega \times (0,T) \times D$.
Moreover, since $(X,\xi)$ is the almost sure limit of $(X_n,\xi_n)$,
we immediately infer that $X$ and $\xi$ are predictable $H$-valued
and $L^1(D)$-valued processes, respectively. The a priori estimates
of Lemma~\ref{lm:ap} and Fatou's lemma then yield
\[
X \in L^2(\Omega;C([0,T];H)) \cap L^2(\Omega;L^2(0,T;V)),
\qquad \xi \in L^1(\Omega \times (0,T) \times D).
\]
Similarly, $\xi_n \in \beta(X_n)$ implies
$X_n\xi_n = j(X_n)+j^*(\xi_n)$, hence
\[
\E\int_0^T\!\!\int_D \bigl( j(X_n)+j^*(\xi_n) \bigr) \lesssim 1 +
\E\norm{X_0}^2
\]
for all $n \in \enne$, and again by Fatou's lemma, as well as by the
lower-semicontinuity of convex integrals, one obtains
\[
j(X) + j^*(\xi) \in L^1(\Omega;L^1(0,T;L^1(D))).
\]
We have thus proved that $(X,\xi) \in \cJ_2$, so the proof is
completed.
\end{proof}
The second existence result, which allows $X_0$ to be merely
$\cF_0$-measurable, follows by a further ``gluing'' procedure.
\begin{thm}
Assume that $X_0 \in L^0(\Omega,\cF_0,\P;H)$. Then equation
\eqref{eq:0} admits a unique strong solution.
\end{thm}
\begin{proof}
Uniqueness of strong solutions has already been proved in
Corollary~\ref{cor:uniq}. It is hence enough to prove existence. Let
us define the sequence $(\Gamma_n)_{n\in\enne}$ of elements of
$\cF_0$ as
\[
\Gamma_n := \bigl\{ \omega \in \Omega: \, \norm{X_0} \leq n \bigr\}.
\]
It is evident that $(\Gamma_n)$ is a sequence increasing to
$\Omega$, and that
$X_{0\Gamma_n} = X_0\ind{\Gamma_n} \in L^2(\Omega;H)$. Therefore, by
the previous theorem, for each $n \in \enne$ there exists a unique
strong solution $(X_n,\xi_n)$ to \eqref{eq:0} with initial condition
$X_{0\Gamma_n}$. By the local property of solutions established in
Corollary~\ref{cor:loc}, we have that $X_{n+1}\ind{\Gamma_n}$ and
$X_n\ind{\Gamma_n}$ are indistinguishable, and
$\xi_{n+1}\ind{\Gamma_n}=\xi_n\ind{\Gamma_n}$ a.e. in
$\Omega \times (0,T) \times D$. Since $(\Gamma_n)$ is increasing, it
makes sense to define the processes $X$ and $\xi$ by
\[
X\ind{\Gamma_n} = X_n\ind{\Gamma_n}, \quad
\xi\ind{\Gamma_n} = \xi_n\ind{\Gamma_n}
\]
for all $n \in \enne$. This amounts to saying that $X$ and $\xi$
are the $\P$-a.s. limits of $X_n$ and $\xi_n$, respectively,
which
immediately implies that $X$ and $\xi$ are predictable processes
with values in $H$ and $L^1(D)$, respectively. Moreover, by
construction, we also have
\[
X \in L^0(\Omega;C([0,T];H) \cap L^2(0,T;V)), \qquad
\xi \in L^0(\Omega;L^1(0,T;L^1(D)))
\]
In fact, writing $E:=C([0,T];H) \cap L^2(0,T;V)$ for compactness of
notation, by the previous theorem we have $X_n \in L^2(\Omega;E)$
and $\xi \in L^1(\Omega \times (0,T) \times D)$, and for any
arbitrary but fixed $\omega$ in a subset of $\Omega$ of probability
one, there exists $n=n(\omega)$ such that
$(X(\omega),\xi(\omega)) =(X_n(\omega),\xi_n(\omega)) \in E \times
L^1((0,T) \times D)$. Furthermore, since $\xi_n \in \beta(X_n)$
a.e. for all $n \in \enne$, it is easy to see that
\[
\xi\ind{\Gamma_n} = \xi_n\ind{\Gamma_n} \in \beta(X_n)\ind{\Gamma_n}
= \beta(X_n\ind{\Gamma_n}) \ind{\Gamma_n}
= \beta(X) \ind{\Gamma_n}
\]
for all $n \in \enne$, so that $\xi \in \beta(X)$ a.e. because
$\Gamma_n \uparrow \Omega$. Similarly,
\[
j(X_n) \ind{\Gamma_n} = j(\ind{\Gamma_n}X_n) \ind{\Gamma_n}
= j(X) \ind{\Gamma_n}
\]
as well as, by the same reasoning,
$j^*(\xi_n) \ind{\Gamma_n} = j^*(\xi) \ind{\Gamma_n}$. Since, by the
previous theorem,
$j(X_n) + j^*(\xi_n) \in L^1(\Omega;L^1((0,T) \times D)$ for all
$n \in \enne$, it follows that
\[
\bigl( j(X) + j^*(\xi) \bigr) \ind{\Gamma_n} \in
L^1(\Omega;L^1((0,T) \times D) \qquad \forall n \in \enne,
\]
hence $j(X) + j^*(\xi) \in L^0(\Omega;L^1((0,T) \times D)$.
\end{proof}
\section{Moment estimates and dependence on the initial datum}
\label{sec:mom}
We are now going to show that the integrability of the solution is
determined by the integrability of the initial condition.
\begin{thm}
Let $p \geq 0$. If $X_0 \in L^p(\Omega,\cF_0,\P;H)$, then the unique
strong solution to equation \eqref{eq:0} belongs to
$\mathscr{J}_p$.
\end{thm}
\begin{proof}
It\^o's formula yields
\begin{align*}
&\norm{X(t)}^2
+ 2\int_0^{t} \ip{AX(s)}{X(s)}\,ds
+ 2\int_0^{t} \!\!\int_D\xi(s)X(s)\,dx\,ds\\
&\hspace{3em} = \norm{X_0}^2
+ \int_0^{t} \norm[\big]{B(s,X(s))}^2_{\cL^2(U,H)}\,ds
+ 2\int_0^{t} X(s)B(s,X(s))\,dW(s).
\end{align*}
For any $\alpha>0$, it follows by the integration-by-parts formula
that
\begin{align*}
&e^{-2\alpha t} \norm{X(t)}^2
+ 2\alpha \int_0^t e^{-2\alpha s} \norm{X(s)}^2\,ds
+ 2\int_0^t e^{-2\alpha s} \ip{AX(s)}{X(s)}\,ds\\
&\quad + 2\int_0^t\!\!\int_D e^{-2\alpha s} \xi(s)X(s)\,dx\,ds\\
&\quad \hspace{3em} = \norm{X_0}^2
+ \int_0^t e^{-2\alpha s} \norm[\big]{B(s,X(s))}^2_{\cL^2(U,H)}\,ds
+ 2\int_0^t e^{-2\alpha s} X(s)B(s, X(s))\,dW(s).
\end{align*}
Let $M$ denote the stochastic integral on the right-hand side, and
$Y(t):=e^{-\alpha t}X(t)$. Since $X$ has continuous paths in $H$,
one can introduce the sequence of stopping times
$(T_n)_{n\in\enne}$, increasing to $T$, as
\[
T_n := \inf \bigl\{ t \geq 0: \, \norm{X(t)} \geq n \bigr\} \wedge T.
\]
It follows by the local Lipschitz-continuity property of $B$ that
\begin{align*}
&\norm{Y^{T_n}(t)}^2
+ 2\alpha \int_0^{t \wedge T_n} \norm{Y(s)}^2\,ds
+ 2C \int_0^{t \wedge T_n} \norm{Y(s)}^2_V\,ds
+2\int_0^{t \wedge T_n}\!\!\int_D e^{-2\alpha s} \xi(s)X(s)\,dx\,ds\\
&\hspace{3em} \leq \norm{X_0}^2
+ \int_0^{t \wedge T_n} e^{-2\alpha s}
\norm[\big]{B_n(s,X(s))}^2_{\cL^2(U,H)}\,ds
+ 2M^{T_n}(t).
\end{align*}
Recalling that $B_n=B(\cdot,\cdot,\sigma_n(\cdot))$ and
$\sigma_n$ is a contraction in $H$, and that $B$ grows at most
linearly, one has
\[
e^{-2\alpha s} \norm[\big]{B_n(s,X(s))}^2_{\cL^2(U,H)}
\lesssim e^{-2\alpha s} + \norm{Y(s)}^2,
\]
hence
\begin{equation}
\label{eq:alfetta}
\int_0^{t \wedge T_n} e^{-2\alpha s}
\norm[\big]{B_n(s,X(s))}^2_{\cL^2(U,H)}\,ds \lesssim
\frac{1}{2\alpha} + \int_0^{t \wedge T_n} \norm{Y(s)}^2\,ds.
\end{equation}
Taking supremum in time and the $L^{p/2}(\Omega)$-(quasi)norm,
recalling the BDG inequality and the fact that
$e^{-\alpha t}\xi_nX_n\geq e^{-\alpha T}\xi_nX_n$, we are left with
\begin{align*}
&\norm[\big]{Y_{T_n}^*}^2_{L^p(\Omega)}
+ \alpha \norm[\big]{Y\ind{[\![0,T_n]\!]}}^2_{L^p(\Omega;L^2(0,T;H))}
+ \norm[\big]{Y\ind{[\![0,T_n]\!]}}^2_{L^p(\Omega;L^2(0,T;V))}\\
&\qquad +e^{-\alpha T}
\norm[\big]{\xi X\ind{[\![0,T_n]\!]}}_{L^{p/2}(\Omega; L^1((0,T)\times D))}\\
&\qquad\qquad \lesssim \norm[\big]{X_0}^2_{L^p(\Omega;H)}
+ \frac{1}{2\alpha}
+ \norm[\big]{Y\ind{[\![0,T_n]\!]}}^2_{L^p(\Omega;L^2(0,T;H))}
+ \norm[\big]{[M,M]_{T_n}^{1/2}}_{L^{p/2}(\Omega)}.
\end{align*}
Lemma~\ref{lm:BDGY} and \eqref{eq:alfetta} yield
\[
[M,M]^{1/2}_{T_n}
\lesssim \varepsilon Y_{T_n}^{*2}
+ \frac{1}{\varepsilon} \biggl( \frac{1}{2\alpha}
+ \norm[\big]{Y\ind{[\![0,T_n]\!]}}^2_{L^2(0,T;H)} \biggr),
\]
hence
\[
\norm[\big]{[M,M]_{T_n}^{1/2}}_{L^{p/2}(\Omega)}
\lesssim \varepsilon \norm[\big]{Y_{T_n}^*}^2_{L^p(\Omega)}
+ \frac{1}{\varepsilon}
\norm[\big]{Y\ind{[\![0,T_n]\!]}}^2_{L^p(\Omega;L^2(0,T;H))}
+ \frac{1}{2\alpha\varepsilon},
\]
where the implicit constant is independent of $\alpha$ and of an
arbitrary $\varepsilon>0$ to be chosen later. We thus have
\begin{align*}
&\norm[\big]{Y_{T_n}^*}^2_{L^p(\Omega)}
+ \alpha \norm[\big]{Y\ind{[\![0,T_n]\!]}}^2_{L^p(\Omega;L^2(0,T;H))}
+ \norm[\big]{Y\ind{[\![0,T_n]\!]}}^2_{L^p(\Omega;L^2(0,T;V))}\\
&\qquad +e^{-\alpha T}
\norm[\big]{\xi X\ind{[\![0,T_n]\!]}}_{L^{p/2}(\Omega; L^1((0,T)\times D))}\\
&\qquad\qquad \lesssim \norm[\big]{X_0}^2_{L^p(\Omega;H)}
+ \varepsilon \norm[\big]{(Y^{T_n})^*_T}^2_{L^p(\Omega)}
+ (1+1/\varepsilon) \norm[\big]{Y\ind{[\![0,T_n]\!]}}^2_{L^p(\Omega;L^2(0,T;H))}
+ \frac{1}{2\alpha} (1+1/\varepsilon).
\end{align*}
Since the implicit constant is independent of $\alpha$ and
$\varepsilon$, one can take $\varepsilon$ small enough and $\alpha$
large enough so that
\[
\norm[\big]{Y_{T_n}^*}^2_{L^p(\Omega)}
+ \norm[\big]{Y\ind{[\![0,T_n]\!]}}^2_{L^p(\Omega;L^2(0,T;V))}
+\norm[\big]{\xi X\ind{[\![0,T_n]\!]}}_{L^{p/2}(\Omega; L^1((0,T)\times D))}
\lesssim 1 + \norm[\big]{X_0}^2_{L^p(\Omega;H)}.
\]
As the implicit constant is independent of $n$ and $T_n$ increases to $T$,
we get
\[
\norm[\big]{Y}^2_{L^p(\Omega;C([0,T];H))}
+ \norm[\big]{Y}^2_{L^p(\Omega;L^2(0,T;V))}
+\norm[\big]{\xi X}_{L^{p/2}(\Omega; L^1((0,T)\times D))}
\lesssim 1 + \norm[\big]{X_0}^2_{L^p(\Omega;H)}.
\]
The proof is completed noting that, for
$E:=C([0,T];H) \cap L^2(0,T;V)$,
\[
\norm{X}_E \leq e^{\alpha T} \norm{Y}_E.
\qedhere
\]
\end{proof}
If $B$ is Lipschitz-continuous, related arguments show that the
solution map is Lipschitz-continuous between spaces with finite $p$-th
moment in the whole range $p \in [0,\infty[$. We consider the cases
$p>0$ and $p=0$ separately.
\begin{prop}
Let $p>0$. If $B$ is Lipschitz-continuous in the sense of assumption
\emph{(B1)}, then the solution map
\begin{align*}
L^p(\Omega;H) &\longto L^p(\Omega;C([0,T];H)) \cap
L^p(\Omega;L^2(0,T;V))\\
X_0 &\longmapsto X
\end{align*}
is Lipschitz-continuous.
\end{prop}
\begin{proof}
Let $X_0$, $Y_0 \in L^p(\Omega;H)$. The previous theorem asserts
that the (unique) strong solutions $(X,\xi)$ and $(Y,\eta)$ to
\eqref{eq:0} with initial condition $X_0$ and $Y_0$, respectively,
belong to $L^p(\Omega;E)$, where, as before, $E$ stands for
$C([0,T];H) \cap L^2(0,T;V)$. By It\^o's formula,
\begin{align*}
&\norm{X-Y}^2
+ 2\int_0^t \ip{A(X-Y)}{X-Y}\,ds
+ 2\int_0^{t} \!\!\int_D(\xi-\eta)(X-Y)\,ds\\
&\hspace{3em} = \norm{X_0-Y_0}^2
+ \int_0^{t} \norm[\big]{B(X)-B(Y)}^2_{\cL^2(U,H)}\,ds\\
&\hspace{3em}\quad
+ 2\int_0^{t} (X-Y)(B(X)-B(Y))\,dW,
\end{align*}
where the third term on the left-hand side is positive by
monotonicity of $\beta$. Let $\alpha>0$ be a constant to be chosen
later, and set $X_\alpha:= X e^{-\alpha \cdot}$,
$Y_\alpha:= Y e^{-\alpha \cdot}$. It follows by the
integration-by-parts formula, in complete analogy to the proof of
the previous theorem, by the Lipschitz continuity of $B$, and by the
coercivity of $A$, that
\begin{align*}
&\norm{X_\alpha - Y_\alpha}^2
+ \alpha \int_0^t \norm{X_\alpha-Y_\alpha}^2\,ds
+ \int_0^t \norm{X_\alpha-Y_\alpha}^2_V\,ds\\
&\hspace{3em} \lesssim \norm{X_0 - Y_0}^2
+ \int_0^t \norm[\big]{X_\alpha-Y_\alpha}^2\,ds
+ M,
\end{align*}
where $M:= \bigl( e^{-2\alpha \cdot}(X-Y)(B(X)-B(Y)) \bigr) \cdot W$.
Taking supremum in time and the $L^{p/2}(\Omega)$-(quasi)norm yields
\begin{align*}
&\norm[\big]{X_\alpha-Y_\alpha}^2_{L^p(\Omega;C([0,T];H))}
+ \alpha \norm[\big]{X_\alpha-Y_\alpha}^2_{L^p(\Omega;L^2(0,T;H))}
+ \norm[\big]{X_\alpha-Y_\alpha}^2_{L^p(\Omega;L^2(0,T;V))}\\
&\hspace{3em} \lesssim \norm[\big]{X_0-Y_0}^2_{L^p(\Omega;H)}
+ \norm[\big]{X_\alpha - Y_\alpha}^2_{L^p(\Omega;L^2(0,T;H))}
+ \norm[\big]{M^*_{T}}_{L^{p/2}(\Omega)},
\end{align*}
where, by Lemma~\ref{lm:BDGY},
\[
\norm[\big]{M^*_{T}}_{L^{p/2}(\Omega)} \lesssim
\varepsilon \norm[\big]{X_\alpha-Y_\alpha}^2_{L^p(\Omega;C([0,T];H))}
+ N(\varepsilon) \norm[\big]{X_\alpha - Y_\alpha}^2_{L^p(\Omega;L^2(0,T;H))}
\]
for any $\varepsilon>0$. Choosing first $\varepsilon$ small enough,
then $\alpha$ sufficiently large, we obtain
\[
\norm[\big]{X_\alpha-Y_\alpha}^2_{L^p(\Omega;C([0,T];H))}
+ \norm[\big]{X_\alpha-Y_\alpha}^2_{L^p(\Omega;L^2(0,T;V))}
\lesssim \norm[\big]{X_0-Y_0}^2_{L^p(\Omega;H)},
\]
which completes the proof noting that
$\norm{X-Y}_E \leq e^{\alpha T} \norm{X_\alpha-Y_\alpha}_E$.
\end{proof}
Lipschitz continuity of the solution map can also be obtained in the
case $p=0$. As already seen, the space
$E:=C([0,T];H) \cap L^2(0,T;V)$, equipped with the norm
\[
\norm[\big]{u}_E := \norm[\big]{u}_{C([0,T];H)}
+ \norm[\big]{u}_{L^2(0,T;V)},
\]
is a Banach space. Then $L^0(\Omega;E)$, endowed with the topology of
convergence in probability, is a complete metrizable topological
vector space. In particular, the distance
\[
d(f,g) := \E\bigl( \norm{f-g}_E \wedge 1 \bigr)
\]
generates its topology.
\begin{prop}
If $B$ is Lipschitz-continuous in the sense of assumption
\emph{(B1)}, then the solution map
\begin{align*}
L^0(\Omega;H) &\longto L^0(\Omega;E)\\
X_0 &\longmapsto X
\end{align*}
is Lipschitz-continuous.
\end{prop}
\begin{proof}
Let $X_0$, $Y_0 \in L^0(\Omega,\cF_0,\P;H)$, and $(X,\xi)$,
$(Y,\eta)$ the unique solutions in $\cJ_0$ to equation \eqref{eq:0}
with initial datum $X_0$ and $Y_0$, respectively. The stopping time
\[
T_1 := \inf\Bigl\{ t \geq 0: \, (X-Y)_t^*
+ \biggl( \int_0^t \norm{X(s)-Y(s)}_V^2\,ds \biggr)^{1/2} \geq 1
\Bigr\} \wedge T.
\]
is well defined thanks to the pathwise continuity of $X$ and $Y$.
For every $\alpha>0$, using the same notation as in the previous proof,
Theorem~\ref{thm:Ito} yields, by monotonicity of $\beta$ and
coercivity of $A$,
\begin{align*}
&\bigl( X_\alpha - Y_\alpha \bigr)_{t}^{*2} + \int_0^{t}
\norm{X_\alpha(s)-Y_\alpha(s)}_V^2\,ds
+ \alpha \int_0^{t} \norm{X_\alpha(s)-Y_\alpha(s)}^2\,ds\\
&\hspace{3em} \lesssim \norm{X_0-Y_0}^2
+ \int_0^t \norm[\big]{(B(X(s))-B(Y(s)))_\alpha}^2_{\cL^2(U,H)}\,ds\\
&\hspace{3em} \quad + \bigl( (X_\alpha-Y_\alpha)(B(X)-B(Y))_\alpha \cdot W \bigr)^*_t
\end{align*}
Raising to the power $1/2$, stopping at $T_1$, and taking
expectation, we get, by the Lipschitz continuity of $B$,
\begin{align*}
&\E\bigl( X_\alpha - Y_\alpha \bigr)_{T_1}^*
+ \E\biggl( \int_0^{T_1} \norm{X_\alpha(s)-Y_\alpha(s)}_V^2\,ds \biggr)^{1/2}
+ \sqrt{\alpha} \E\biggl( \int_0^{T_1} \norm{X_\alpha(s)-Y_\alpha(s)}^2\,ds \biggr)^{1/2}\\
&\hspace{3em} \lesssim \E\ind{[\![0,T_1]\!]}\norm{X_0-Y_0}
+ \E\biggl( \int_0^{T_1} \norm{X_\alpha(s)-Y_\alpha(s)}^2\,ds \biggr)^{1/2}\\
&\hspace{3em} \quad + \E\bigl( (X_\alpha-Y_\alpha)(B(X)-B(Y))_\alpha \cdot W
\bigr)^{*1/2}_{T_1},
\end{align*}
where, by Lemma~\ref{lm:BDGY} and Lipschitz continuity of $B$, the
last term on the right-hand side is bounded by
\[
\varepsilon \E\bigl( X_\alpha - Y_\alpha \bigr)_{T_1}^*
+ N(\varepsilon) \E\biggl( \int_0^{T_1} \norm{X_\alpha(s)-Y_\alpha(s)}^2\,ds \biggr)^{1/2}
\]
for every $\varepsilon>0$. Therefore, choosing $\varepsilon$ small
enough and $\alpha$ large enough, we are left with
\[
\E\bigl( X - Y \bigr)_{T_1}^* + \E\biggl( \int_0^{T_1}
\norm{X(s)-Y(s)}_V^2\,ds \biggr)^{1/2} \lesssim
\E\ind{[\![0,T_1]\!]}\norm{X_0-Y_0}.
\]
The proof is concluded noting that, by definition of $T_1$,
\[
\bigl( X - Y \bigr)_{T_1}^* + \biggl( \int_0^{T_1}
\norm{X(s)-Y(s)}_V^2\,ds \biggr)^{1/2} =
\norm[\big]{X-Y}_{C([0,T];H) \cap L^2(0,T;V)} \wedge 1,
\]
and $\norm{X_0-Y_0} \leq 1$ on $[\![0,T_1]\!]$, hence
\[
\E\ind{[\![0,T_1]\!]}\norm{X_0-Y_0}
= \E\ind{[\![0,T_1]\!]} \bigl( \norm{X_0-Y_0} \wedge 1 \bigr)
\leq \E\bigl( \norm{X_0-Y_0} \wedge 1 \bigr).
\qedhere
\]
\end{proof}
\section{A regularity result}
\label{sec:reg}
We are going to show that the regularity of the solution to equation
\eqref{eq:0} improves, if the initial datum and the diffusion
coefficient are smoother, irrespective of the (possible) singularity
of the drift coefficient $\beta$. In particular, we provide sufficient
conditions implying that the variational solution to \eqref{eq:0} is
also an analytically strong solution, in the sense that it takes
values in the domain of the part of $A$ in $H$ (see
{\S}\ref{sec:cont}). If the solution to \eqref{eq:0} generates a
Markovian semigroup on $C_b(H)$ admitting an invariant measure, we
also show that improved regularity of the solution carries over to
further regularity of the invariant measure, in the sense that its
support is made of smoother functions.
\begin{thm}
\label{thm:reg1}
Assume that the hypotheses of {\S}\ref{ssec:ass} are satisfied and
that
\begin{equation}
\label{hyp_reg}
X_0\in L^2(\Omega,\cF_0,\P; V), \qquad
B(\cdot,X) \in L^2(\Omega; L^2(0,T; \cL^2(U,V))).
\end{equation}
Then the unique solution $(X,\xi)$ to the equation \eqref{eq:0} satisfies
\[
X \in L^2(\Omega;C([0,T];H))\cap L^2(\Omega; L^\infty(0,T; V)) \cap L^2(\Omega; L^2(0,T; \dom(A_2))).
\]
\end{thm}
For the proof we need the following positivity result.
\begin{lemma}\label{lm:pos}
Let $A_\lambda$ and $\beta_\lambda$ be the Yosida approximations of
$A_2$ and $\beta$, respectively. One has
\[
\ip[\big]{A_\lambda u}{\beta_\lambda(u)} \geq 0 \qquad \forall u \in H.
\]
\end{lemma}
\begin{proof}
Let $j_\lambda:\erre \to \erre$ be the positive convex function
defined as $j_\lambda(x):=\int_0^x \beta_\lambda(y)\,dy$. Then, for
any $u$, $v \in L^2(D)$,
\[
j_\lambda(v)-j_\lambda(u) \geq j_\lambda'(u)(v-u)
\]
a.e. in $D$, hence, integrating over $D$,
\[
\int_D j_\lambda(v) - \int j_\lambda(u)
\geq \ip[\big]{\beta_\lambda(u)}{v-u}.
\]
Choosing $v=(I+\lambda A_2)^{-1}u$, one has
$u-v = \lambda A_\lambda u$, thus also
\[
\lambda \ip[\big]{\beta_\lambda(u)}{A_\lambda u}
\geq \int j_\lambda(u) - \int j_\lambda((I+\lambda A_2)^{-1}u).
\]
Since $A_1$ is an extension of $A_2$ and $u \in L^1(D)$, Jensen's
inequality for sub-Markovian operators and accretivity of $A_1$ in
$L^1(D)$ imply
\[
\int j_\lambda((I+\lambda A_2)^{-1}u)
\leq \int (I+\lambda A_2)^{-1} j_\lambda(u)
\leq \int j_\lambda(u).
\qedhere
\]
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:reg1}]
For any $\lambda>0$, let $J_\lambda$ and $A_\lambda$ be the
resolvent and the Yosida approximations of $A_2$, the part of $A$ in
$H$, as defined in {\S}\ref{sec:cont}. That is,
\[
J_\lambda := (I+\lambda A_2)^{-1}, \qquad
A_\lambda := \frac{1}{\lambda}(I-J_\lambda).
\]
We recall that $J_\lambda$ and $A_\lambda$ are bounded linear
operators on $H$, that $J_\lambda$ is a contraction, and that
$A_\lambda=AJ_\lambda$.
Setting $G:=B(\cdot,X)$, let us consider the equation
\[
dX_\lambda(t) + A_\lambda X_\lambda(t)\,dt + \beta_\lambda(X_\lambda(t))\,dt =
G(t)\, dW(t), \qquad X_\lambda(0)=X_0.
\]
Since $A_\lambda$ is bounded and $\beta_\lambda$ is
Lipschitz-continuous, it admits a unique strong solution
\[
X_\lambda \in L^2(\Omega;C([0,T];H)),
\]
for which It\^o's formula for the square of the $H$-norm yields
\begin{align*}
&\frac12 \norm{X_\lambda(t)}^2
+ \int_0^t\ip[\big]{A_\lambda X_\lambda(s)}{X_\lambda(s)}\,ds
+\int_0^t\!\!\int_D\beta_\lambda(X_\lambda(s))X_\lambda(s)\,ds\\
&\hspace{3em} = \frac12\norm{X_0}^2
+ \frac12 \int_0^t \norm{G(s)}^2_{\cL^2(U,H)}\,ds
+ \int_0^t X_\lambda(s)G(s)\,dW(s)
\end{align*}
for all $t \in [0,T]$ $\P$-almost surely. Writing
\[
X_\lambda = J_\lambda X_\lambda + X_\lambda - J_\lambda X_\lambda
= J_\lambda X_\lambda + \lambda A_\lambda X_\lambda
\]
and recalling that $A_\lambda = AJ_\lambda$ and that $A$ is
coercive, we have, after taking supremum in time and expectation,
\begin{align*}
&\frac12 \E \norm[\big]{X_\lambda}^2_{C([0,T];H)}
+ C \E\int_0^T \norm[\big]{J_\lambda X_\lambda(s)}_V^2\,ds\\
&\hspace{3em} \qquad + \lambda \E\int_0^T \norm{A_\lambda X_\lambda(s)}^2\,ds
+ \E\int_0^T\!\!\int_D \beta_\lambda(X_\lambda(s))X_\lambda(s)\,ds\\
&\hspace{3em} \leq \frac12 \E\norm{X_0}^2
+ \frac12\E\int_0^T \norm[\big]{G(s)}^2_{\cL^2(U,H)}\,ds
+\E\sup_{t\in[0,T]} \abs[\bigg]{\int_0^t X_\lambda(s)G(s)\,dW(s)},
\end{align*}
where, by Lemma~\ref{lm:BDGY}, the last term on the right-hand side
is bounded by
\[
\varepsilon \E \norm{X_\lambda}^2_{C([0,T];H)}
+ C_\varepsilon \E\int_0^T \norm[\big]{G(s)}^2_{\cL^2(U,H)}\,ds,
\]
so that, rearranging terms and choosing $\varepsilon$ small enough,
we deduce that there exists a constant $N>0$ independent of
$\lambda$ such that
\begin{equation}
\label{estimates1}
\norm[\big]{X_\lambda}_{L^2(\Omega;C([0,T];H))}^2
+ \norm[\big]{J_\lambda X_\lambda}^2_{L^2(\Omega;L^2(0,T;V))}
+\norm[\big]{
\beta_\lambda(X_\lambda)X_\lambda}_{L^1(\Omega\times(0,T)\times D)}
< N.
\end{equation}
Moreover, let us introduce the function
\begin{align*}
\varphi_\lambda : H &\longto [0,+\infty[,\\
u &\longmapsto \frac12 \ip{A_\lambda u}{u}.
\end{align*}
The linearity and the boundedness of $A_\lambda$ immediately implies
that $\varphi_\lambda \in C^2(H)$ with
$D\varphi_\lambda(u) = A_\lambda$, and, by linearity,
$D^2\varphi_\lambda(u)=A_\lambda$, for all $u \in H$ (in the latter
statement $A_\lambda$ has to be considered as an element of
$\cL_2(H)$, the space of bounded bilinear maps on $H$). It\^o's
formula applied to $\varphi_\lambda(X_\lambda)$ then yields
\begin{align*}
&\varphi_\lambda(X_\lambda(t))
+ \int_0^t\norm[\big]{A_\lambda X_\lambda(s)}^2\,ds +
\int_0^t \ip[\big]{A_\lambda X_\lambda(s)}{\beta_\lambda(X_\lambda(s))}\,ds\\
&\hspace{3em} = \varphi_\lambda(X_0)
+ \frac12 \int_0^t \operatorname{Tr} \bigl(
G^*(s) D^2\varphi_\lambda(X_\lambda(s)) G(s) \bigr)\,ds
+ \int_0^t A_\lambda X_\lambda(s) G(s)\,dW(s)
\end{align*}
for every $t\in[0,T]$ $\P$-almost surely. Writing, as before,
$X_\lambda = J_\lambda X_\lambda + \lambda A_\lambda X_\lambda$, the
coercivity of $A$ implies that
\[
\varphi_\lambda(X_\lambda) = \frac12 \ip[\big]{A_\lambda X_\lambda}{X_\lambda}
\geq \frac{C}{2} \norm[\big]{J_\lambda X_\lambda}_V^2
+ \frac12 \lambda \norm[\big]{A_\lambda X_\lambda}^2
\gtrsim \norm[\big]{J_\lambda X_\lambda}_V^2.
\]
The continuity of $J_\lambda$ on $V$ (see Lemma~\ref{lm:cvV}) instead implies
\[
\varphi_\lambda(X_0) = \ip[\big]{AJ_\lambda X_0}{X_0}
\leq \norm[\big]{A}_{\cL(V,V')} \norm[\big]{J_\lambda X_0}_V
\norm[\big]{X_0}_V
\lesssim \norm[\big]{A}_{\cL(V,V')} \norm[\big]{X_0}_V^2.
\]
Denoting a complete orthonormal basis of $U$ by $(u_k)_k$,
we have, recalling again the continuity of $J_\lambda$ on $V$ and
that $D^2\varphi_\lambda(u) = A_\lambda$ for all
$u \in H$,
\begin{align*}
\operatorname{Tr} \bigl( G^* D^2\varphi_\lambda(X_\lambda) G \bigr)
&= \sum_{k=0}^\infty \ip[\big]{
G^* D^2\varphi_\lambda(X_\lambda) Gu_k}{u_k}_U
= \sum_{k=0}^\infty \ip[\big]{
A_\lambda G u_k}{Gu_k}\\
&= \sum_{k=0}^\infty \ip[\big]{AJ_\lambda G u_k}{Gu_k}
\leq\norm[\big]{A}_{\cL(V,V')} \sum_{k=0}^\infty
\norm[\big]{J_\lambda Gu_k}_V \norm[\big]{Gu_k}_V\\
&\lesssim \norm[\big]{A}_{\cL(V,V')} \sum_{k=0}^\infty \norm[\big]{Gu_k}_V^2
=\norm[\big]{A}_{\cL(V,V')} \norm[\big]{G}_{\cL^2(U,V)}^2.
\end{align*}
Moreover, by Lemma~\ref{lm:BDGY},
\begin{align*}
\E \bigl( (A_\lambda X_\lambda G) \cdot W \bigr)^*_T
&\lesssim \varepsilon \E \sup_{t\in[0,T]}
\norm[\big]{A_\lambda X_\lambda(t)}^2_{V'}
+ N(\varepsilon) \E\int_0^T \norm[\big]{G(s)}^2_{\cL^2(U,V)}\,ds\\
&\leq \varepsilon \norm[\big]{A}^2_{\cL(V,V')}
\E \sup_{t\in[0,T]} \norm[\big]{J_\lambda X_\lambda(t)}^2_V
+ N(\varepsilon) \E \int_0^T \norm[\big]{G(s)}^2_{\cL^2(U,V)}\,ds
\end{align*}
for every $\varepsilon >0$.
Taking supremum in time and expectations in the It\^o formula for
$\varphi_\lambda(X_\lambda)$, choosing $\varepsilon$ small enough we
obtain, thanks to the previous lemma and hypothesis \eqref{hyp_reg},
that there exists a constant $N>0$ independent of $\lambda$, such
that
\begin{equation}\label{estimates2}
\norm[\big]{J_\lambda X_\lambda}^2_{L^2(\Omega; L^\infty(0,T; V))} +
\norm[\big]{A_\lambda X_\lambda}^2_{L^2(\Omega; L^2(0,T; H))} < N.
\end{equation}
Reasoning as in \cite{cm:luca}, it follows by
\eqref{estimates1} that
\begin{alignat*}{2}
X_\lambda &\longto X
&&\quad \text{ weakly in } L^2(\Omega; L^2(0,T; H)),\\
J_\lambda X_\lambda &\longto X
&&\quad \text{ weakly in } L^2(\Omega;L^2(0,T; V)),\\
\beta_\lambda(X_\lambda) &\longto \xi
&&\quad \text{ weakly in } L^1(\Omega\times(0,T)\times D),
\end{alignat*}
where $(X,\xi)$ is the unique solution to \eqref{eq:0}. Moreover,
by \eqref{estimates2} we have
\[
J_\lambda X_\lambda - X_\lambda = \lambda A_\lambda X_\lambda
\longto 0 \qquad\text{ in } L^2(\Omega;L^2(0,T; H)),
\]
hence, $\P$-almost surely and for almost every $t \in (0,T)$,
$J_\lambda X_\lambda(t)$ converges to $X(t)$ in $H$. Since the
function $\norm{\cdot}_V^2$ is lower semicontinuous on $H$, we infer
that
\[
\norm{X(t)}^2_V\leq \liminf_{\lambda\to 0}
\norm[\big]{J_\lambda X_\lambda(t)}_V^2
\]
for almost every $t$. Hence, taking supremum in time and
expectation, we deduce that
\[
X \in L^2(\Omega;L^\infty(0,T;V)).
\]
Moreover, by \eqref{estimates2} we
also have
\[
A_\lambda X_\lambda \longto \eta \quad
\text{ weakly in } L^2(\Omega; L^2(0,T; H)),
\]
hence, since $J_\lambda X_\lambda \to X$ weakly in
$L^2(\Omega; L^2(0,T; V))$, by the continuity and the linearity of
$A$ we necessarily have $\eta=AX$. In particular,
$X \in L^2(\Omega;L^2(0,T;\dom(A_2)))$.
\end{proof}
As last result we show that if the solution to \eqref{eq:0} generates
a Markovian semigroup $P=(P_t)_{t\geq 0}$ on $C_b(H)$ admitting an
invariant measure, then the improved regularity of solutions given by
Theorem~\ref{thm:reg1} implies better integrability properties also
for the invariant measures. Existence and uniqueness of invariant
measures, ergodicity, and the Kolmogorov equation associated to
\eqref{eq:0} were studied in \cite{cm:inv}. In particular, we recall
the following result (see \cite[{\S}5]{cm:inv}). The set of invariant
measures of $P$ will be denoted by $\mathscr{M}$.
\begin{prop}
Assume that the the hypotheses of \S\ref{ssec:ass} are satisfied,
that $X_0\in H$ is non-random, and that $B$ is non-random and
time-independent. Then the solution $X$ to \eqref{eq:0} is Markovian
and its associated transition semigroup $P$ admits an
ergodic invariant measure. Moreover, there exists a positive
constant $N$ such that
\[
\int_H \norm{u}_V^2\,\mu(du)
+ \int_H\!\int_D j(u)\,\mu(du)
+ \int_H\!\int_Dj^* (\beta^0(u))\,\mu(du) < N
\qquad \forall \mu \in \mathscr{M}.
\]
If $\beta$ is superlinear, there exists a unique invariant measure
for $P$, which is strongly mixing as well.
\end{prop}
\begin{thm}
Assume that the hypotheses of \S\ref{ssec:ass} are satisfied and that
\[
B:H\to\cL^2(U,V), \qquad
\norm{B(v)}_{\cL^2(U,V)} \lesssim 1 + \norm{v}_V.
\]
Then there exists a positive constant $N$ such that
\[
\int_H\norm{Au}^2\,\mu(du) < N \qquad \forall \mu \in \mathscr{M}.
\]
In particular, every invariant measure $\mu$ is concentrated on
$\dom(A_2)$, that is $\mu(\dom(A_2))=1$.
\end{thm}
\begin{proof}
For every $x\in V$, let $(X^x,\xi^x)$ be the unique strong solution
to \eqref{eq:0} with initial datum $x$. Setting $G:=B(X^x)$,
It\^o's formula for $\varphi_\lambda(X_\lambda)$ as in the proof of
the previous theorem yields
\begin{align*}
&\varphi_\lambda(X_\lambda(t))
+ \int_0^t\norm[\big]{A_\lambda X_\lambda(s)}^2\,ds +
\int_0^t \ip[\big]{A_\lambda X_\lambda(s)}{\beta_\lambda(X_\lambda(s))}\,ds\\
&\hspace{3em} = \varphi_\lambda(x)
+ \frac12 \int_0^t \operatorname{Tr} \bigl(
G^*(s) D^2\varphi_\lambda(X_\lambda(s)) G(s) \bigr)\,ds
+ \int_0^t A_\lambda X_\lambda(s) G(s)\,dW(s).
\end{align*}
Since $A_\lambda X_\lambda \in L^2(\Omega; L^\infty(0,T; H))$ and
$G\in L^2(\Omega; L^2(0,T; \cL^2(U,H)))$, the last term on the right
hand side is a martingale; hence, taking expectations and recalling
that $\varphi_\lambda(x)\lesssim \norm{x}_V^2$, it follows by
Lemma~\ref{lm:pos} and by the estimates obtained in the proof of the
previous theorem that
\[
\E\int_0^t\norm[\big]{A_\lambda X_\lambda(s)}^2\,ds
\lesssim \norm{x}_V^2 + \E\norm{G}^2_{L^2(0,t; \cL^2(U,V))}.
\]
Since this holds for every $\lambda>0$, letting $\lambda \to 0$ and
recalling that, as in the proof of the previous theorem,
$A_\lambda X_\lambda$ converges to $AX$ weakly in
$L^2(\Omega;L^2(0,T;H))$, a weak lower semicontinuity argument and
the linear growth assumption on $B$ yield
\begin{equation}
\label{eq:im}
\E \int_0^t \norm[\big]{AX^x(s)}^2\,ds
\lesssim 1 + \norm{x}_V^2
\end{equation}
for every $t\in[0,T]$ and $x\in V$.
Let us introduce the function $F:H \to [0,+\infty]$ defined as
\[
F(u) := \begin{cases}
\norm{Au}^2 \quad&\text{if } u\in\dom(A_2),\\
+\infty \quad&\text{if } u\in H\setminus\dom(A_2),
\end{cases}
\]
and the sequence of functions $(F_n)_{n\in\enne}$, $F_n: H \to [0,+\infty)$,
defined as
\[
F_n(u) := \norm[\big]{A_{1/n}u}^2 \wedge n^2.
\]
It is easily seen that $F_n \in C_b(H)$ for all $n \in \enne$ and
that $F_n$ converges pointwise to $F$ from below. Therefore, for any
invariant measure $\mu$, it follows by Fubini's theorem that
\begin{align*}
\int_H F_n(x)\,\mu(dx)
&= \int_0^1\!\!\int_H F_n(x)\,\mu(dx)\,ds
= \int_0^1\!\!\int_HP_sF_n(x)\,\mu(dx)\,ds\\
&=\int_H\!\int_0^1 \E \bigl( \norm[\big]{A_{1/n}X^x(s)}^2 \wedge n^2 \bigr)
\,ds\,\mu(dx)\\
&\leq \int_H\!\E\int_0^1 \norm[\big]{A_{1/n}X^x(s)}^2 \,ds\,\mu(dx)
\end{align*}
Recalling that $\norm{A_\lambda u} \leq \norm{Au}$ for all
$u \in H$, it follows by \eqref{eq:im} that
\[
\int_H F_n(x)\,\mu(dx) \lesssim 1
+ \int_H \norm{x}_V^2\,\mu(dx).
\]
Since $\norm{\cdot}_V^2\in L^1(H,\mu)$ by
\cite[Theorem~5.3]{cm:inv}, we get
\[
\int_H F_n(x)\,\mu(dx) \lesssim N
\]
for a positive constant $N$, independent of $n$ and $\mu$. Letting
$n\to\infty$, by the monotone convergence theorem we deduce that
$F \in L^1(H,\mu)$, hence $F$ is finite $\mu$-almost everywhere in
$H$, and in particular $\mu(\dom(A_2))=1$.
\end{proof}
\bibliographystyle{amsplain}
\bibliography{ref}
\end{document} | {"config": "arxiv", "file": "1711.11091/arXiv1.tex"} |
TITLE: Solve for y in sin(y) = cos(y) using a fixed point procedure
QUESTION [0 upvotes]: I'm reading an programming book that uses a lot of math equations and formulas as coding examples. In one lesson, it demonstrates finding the fixed point for $\sin(x) + \cos(x)$ by repeatedly calling the same function on the function's result until a tolerable answer is found.
$$
f(x) = \cos(x) + \sin(x)
$$
$$
f(f(f(...f(1)))) \iff 1.2587282 \iff \cos(1.2587282) + \sin(1.2587282)
$$
When I plot $y = \sin(x) + \cos(x)$ that is correct.
Another beautiful example was given to demonstrate that the golden ratio
$φ$ is a fixed point of the translation
$$
f(x) = 1 + 1/x
$$
$$
f(f(f(...f(x)))) \iff 1.618033988 \iff φ
$$
My brain was wandering and I was curious to write my own equation and solve it, but I'm not particularly good at math beyond basic geometry/trigonometry so I got stuck.
I thought, "Ok, let's try to use the fixed point procedure to find the solution for $\sin(x) = \cos(x)$...". I ran into trouble when I tried creating the function for it.
Can $\sin(x) = \cos(x)$ be represented as a function that can be solved using the fixed point procedure above?
$$
g(x) = ... ???
$$
$$
g(g(g(...g(1)))) \iff ? \iff \sin(x) \iff \cos(x)
$$
(Yes, I realize that there are an infinite number of fixed points for $\sin(x) = \cos(x)$ but I'm okay with just getting the pair one closest to zero.)
REPLY [1 votes]: A fixed point $x$ of a function $f$ is one such that $x=f(x)$.
If you want $\sin x = \cos x$, you could try $g_1(x)=\arcsin(\cos x)$ or $g_2(x)=\arccos(\sin x)$. This way, when you solve $x =\arcsin(\cos x)$ you end up with $\sin x = \cos x$ (similarly for the other).
Edit: Actually, having plotted $g_1$ and $g_2$, I realized that neither of them would converge under a fixed-point method (the quick/handwaving explanation is that "near" the fixed point, both $g_1$ and $g_2$ are perpendicular to the identity function).
Solving $\sin x=\cos x$ then would be better approached by a (perhaps more general) root-finding procedure on $g(x)=\sin x - \cos x$ instead of the fixed-point approach. | {"set_name": "stack_exchange", "score": 0, "question_id": 1243037} |
TITLE: Can someone just check this for me please
QUESTION [1 upvotes]: I have an exam and this has stumped me. I think it is a typo error. So this is part of the question.
a curve of the form $y = Ax^3$, where A is a constant, joining the two endpoints $(1,1)$ and $(2,4)$.
The answer that has been given is $y=\frac{1}{2}x^3$.
Obviously it is just a simple case of inputting the endpoints to get an equation for A. But it leads to two contradicting values of $A$, which leads me to believe that $(1,1)$ should be $(1,1/2)$.
For anyone that is interested in what the question is about, http://www.maths.liv.ac.uk/Past_Exams/PDF_FILES/MATH323-jan08-exam.pdf It is question two.
The answer is supplied here, http://www.maths.liv.ac.uk/Past_Exams/PDF_FILES/MATH323-jan08-soln.pdf
REPLY [2 votes]: Yes, you are indeed correct that the given endpoints cannot both be correct:
We have $y = Ax^3$
$(1, 1):\quad 1 = A$
$(2, 4): \quad 4 = 8A\iff A = 1/2$
Contradiction
But given the points $(1, \frac 12), (2, 4)$ as endpoints, then indeed, $A = \frac 12$. | {"set_name": "stack_exchange", "score": 1, "question_id": 636875} |
\begin{document}
\title[mode=title]{A Riemann Difference Scheme for Shock Capturing in Discontinuous Finite Element Methods}
\shorttitle{A Riemann Difference Scheme for Shock Capturing in Discontinuous Finite Element Methods}
\shortauthors{T. Dzanic~\etal}
\author[1]{T. Dzanic}[orcid=0000-0003-3791-1134]
\cormark[1]
\cortext[cor1]{Corresponding author}
\ead{tdzanic@tamu.edu}
\author[1]{W. Trojak}[orcid=0000-0002-4407-8956]
\author[1]{F. D. Witherden}[orcid=0000-0003-2343-412X]
\address[1]{Department of Ocean Engineering, Texas A\&M University, College Station, TX 77843}
\begin{abstract}
We present a novel structure-preserving numerical scheme for discontinuous finite element approximations of nonlinear hyperbolic systems. The method can be understood as a generalization of the Lax--Friedrichs flux to a high-order staggered grid and does not depend on any tunable parameters. Under a presented set of conditions, we show that the method is conservative and invariant domain preserving. Numerical experiments on the Euler equations show the ability of the scheme to resolve discontinuities without introducing excessive spurious oscillations or dissipation.
\end{abstract}
\begin{highlights}
\item A novel staggered grid spectral element method is presented for conservation laws
\item The method is proved to be invariant domain preserving under set conditions
\item Numerical test cases with discontinuities show the high resolution and low dissipation of the method
\end{highlights}
\begin{keywords}
High order \sep Hyperbolic systems \sep Finite element methods \sep Spectral difference \sep Shock capturing
\end{keywords}
\maketitle
\input{introduction}
\input{prelim}
\input{scheme}
\input{numerical}
\input{results}
\input{conclusions}
\section*{Acknowledgements}
\label{sec:ack}
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
\bibliographystyle{unsrtnat}
\bibliography{reference}
\clearpage
\begin{appendices}
\end{appendices}
\end{document} | {"config": "arxiv", "file": "2011.06418/rd_shock_main.tex"} |
TITLE: Weight distributions
QUESTION [2 upvotes]: If a man is standing on two weighing machines (scales), with one foot on each, Will both machines show equal weight or his weight will be distributed in two machines?
REPLY [2 votes]: If the person in concern is standing perfectly, such that his weight distribution is equal over both legs, and if both the weights are calibrated perfectly, then yes, both the weights will show equal readings. (The readings each being half the weight of the person.)
In a realistic case (without 100% perfection), however, the weight would be unequally distributed over both the machines. The sum of these two readings would equal the weight of the person within a tiny error margin.
REPLY [1 votes]: $N_1$ the reading of force on the first weighing machine
$N_2$ the reading of force on the second weighing machine
$X_1$ the horizontal displacement of first leg from COM
$X_2$ the horizontal displacement of second leg from COM
we know that the man is in equilibrium. so
the weights shown on both meters will have their sum equal to the force of gravity on the man.
$$N_1 + N_2 = W$$
then the forces must satisfy that the total torque on the man is zero, that is
$$ N_1X_1 = N_2X_2 $$
by solving $$ N_1 = WX_2/(X_1+X_2) $$ $$ N_2 = WX_1/(X_1+X_2)$$
the thing is : it depends. it depends on the distances that your legs spread horizontally with the center of mass.if both distances are same then the weights are equal. | {"set_name": "stack_exchange", "score": 2, "question_id": 183324} |
TITLE: On whether a formula of KP is $\Pi_3$
QUESTION [1 upvotes]: In the context of KP, is the formula $\forall w(w\in x \leftrightarrow\forall y\exists z F(w,y,z))$ $\Pi_3$ when $F(w,y,x)$ is $\Delta_0$?
REPLY [3 votes]: It seems so:
Unwinding the expression, we get $$G(x)\equiv\forall w [(w\in x\implies \forall y\exists z(F(w, y, z)))\wedge (\forall y\exists z(F(w, y, z))\implies w\in x)]$$ $$\iff \forall w[(\forall y\exists z(F(w, y, z))\vee w\not\in x)\wedge (\exists y\forall z(F(w, y, z))\vee w\in x)]$$ $$\iff [\forall w\forall y\exists z(F(w, y, z)\vee w\not\in x)]\wedge[\forall w\exists y\forall z(F(w, y, z)\vee w\in x)];$$ this last statement is of the form $\Pi^0_2\wedge \Pi^0_3$ (using $F\in \Sigma^0_1$ for the left conjunct, and $F\in \Pi^0_1$ for the right conjunct), so is $\Pi^0_3$.
(I'm using, e.g., "$\Pi^0_3$" instead of "$\Pi_3$" because I sometimes quantify over subsets of an admissible set, so I like keeping track of the relative type I'm talking about.) | {"set_name": "stack_exchange", "score": 1, "question_id": 194689} |
TITLE: what is the right notation for this operation on two vectors?
QUESTION [2 upvotes]: I am trying to find the right notation for this operation on two vectors of size N and M. I do not believe it is the dot product, because there is no sum for the multiplication of each element. Instead, the result is a vector that is of size N * M:
v1 = [a,b,c]
v2 = [d,e]
result: [ad,ae,bd,be,cd,ce]
What is the notation for the operation I just performed between v1 and v2?
REPLY [4 votes]: In a certain ordering of the basis, that's a tensor product, denoted $v_1 \otimes v_2$.
The tensor product $V \otimes W$ of vector spaces has a formal definition in terms of a universal property, but one way you can concretely construct it: if you're given a basis $\{e_1, \dots, e_n\}$ of $V$ and a basis $\{f_1, \dots, f_m\}$ of $W$, then $V \otimes W$
is an $mn$-dimensional vector space whose basis you denote by $\{e_i \otimes f_j\}$ for $i = 1, \dots, n$ and $j = 1, \dots, m$.
Given any vectors $v \in V$ and $w \in W$, you can form a corresponding vector $v \otimes w$ in the tensor product space as follows: Write $v = \sum_{i=1}^n c_i e_i$
and $w = \sum_{j=1}^m d_j f_j$; then define the vector $v \otimes w$ to be
$$
\sum_{i=1}^n \sum_{j=1}^m c_i d_j (e_i \otimes f_j).
$$
This assignment is bilinear, meaning that if you fix $v$, then $v \otimes w$ varies linearly with $w$, and if you fix $w$ then $v \otimes w$ varies linearly with $v$. | {"set_name": "stack_exchange", "score": 2, "question_id": 112567} |
\begin{document}
\title{Green's Function of the Screened Poisson's Equation on the Sphere}
\author{Ramy Tanios}
\address{American University of Beirut, Beirut, Lebanon}
\curraddr{xxx}
\email{rgt09@mail.aub.edu}
\author{Samah El Mohtar}
\address{American University of Beirut, Beirut, Lebanon}
\curraddr{King Abdullah University of Science and Technology, Thuwal 23955, Saudi Arabia}
\email{Samah.Mohtar@kaust.edu.sa}
\author{Omar Knio}
\address{Duke University, Durham, NC 27708, USA}
\curraddr{King Abdullah University of Science and Technology, Thuwal 23955, Saudi Arabia}
\email{Omar.Knio@kaust.edu.sa}
\author{Issam Lakkis}
\address{American University of Beirut, Beirut, Lebanon}
\email{issam.lakkis@aub.du.lb}
\date{\today}
\keywords{Screened Poisson equation; shallow-water equations; sphere}
\begin{abstract}
In geophysical fluid dynamics, the screened Poisson equation appears in the shallow-water, quasi geostrophic equations. Recently, many attempts have been made to solve those equations on the sphere using different numerical methods. These include vortex methods, which solve a Poisson equation to compute the streamfunction from the (relative) vorticity. Alternatively, the streamfunction can be computed directly from potential vorticity (PV), which would offer the possibility of constructing more attractive vortex methods because PV is conserved along material trajectories in the inviscid case. On the spherical shell, however, the screened Poisson equation does not admit a known Green's function, which limits the extension of such approaches to the case of a sphere. In this paper, we derive an expression of Green's function for the screened Poisson equation on the spherical shell in series form and in integral form. A proof of convergence of the series representation is then given. As the series is slowly convergent, a robust and efficient approximation is obtained using a split form which isolates the singular behavior. The solutions are illustrated and analyzed for different values of the screening constant.
\end{abstract}
\maketitle
\section{Introduction}
\label{sec:intro}
In geophysical fluid dynamics, the screened Poisson equation arises in the shallow-water, quasi-geostrophic, potential vorticity equation~\cite{Pedlosky1987,Vallis2006}, for a finite Rossby radius of deformation (baroclinic case). The equation relates the
streamfunction of the geostrophic flow to the potential vorticity, where the ``screening'' is the inverse of the Rossby radius of deformation.
For an infinite Rossby radius of deformation (barotropic case), the screened Poisson equation reduces to the Poisson equation, whose Green's function is known on the spherical shell~\cite{Bogomolov1977,KimuraOkamoto1987}. Recently, many attempts using Lagrangian methods have been made to (numerically) solve the shallow-water quasi-geostrophic potential vorticity equation on the spherical shell.
For instance, Bosler et al.~\cite{Bosler2014} solved the barotropic vorticity (BVE) equation (infinite Rossby radius of deformation) using a Lagrangian particle/panel method. The flow field was computed from the position of particles carrying (relative) vorticity, and advecting with
a velocity expressed in terms of the Biot-Savat law. However, the method did not take advantage of the conservation of potential vorticity, i.e., (relative) vorticity carried by each particle had to be updated at the new particle positions, thus requiring an additional computational cost.
In \cite{MohammadianMarshall2010}, Mohammadian \& Marshall used a vortex-in-cell (VIC) method, in which particles carried (relative) vorticity. The flow field was obtained from the streamfunction, which was computed from the vorticity by inverting a Poisson equation on an underlying Eulerian grid.
Allowing particles to carry potential vorticity enables taking advantage of the fact that potential vorticity is materially conserved in the
inviscid limit. Consequently, in this case advecting/transporting particles along flow trajectories would avoid the need to integrate an
evolution equation for their strengths, provided that the flow field can be immediately computed from the particle distribution. To this end,
we focus on this work on deriving expressions of Green's function for the screened Poisson equation on the spherical shell.
Specifically, in section~\ref{sec:gf-der}, we apply a spectral decomposition of the Laplace-Beltrami operator to arrive at a series
representation of the Green's function. The convergence properties of this representation are then analyzed in section~\ref{sec:gf-conv},
and a computational strategy for evaluating the series is outlined in section~\ref{sec:numer}. In section~\ref{sec:int-form}, an alternative,
integral form of the Green's function is constructed. Implementation of the series and integral solutions is then illustrated in
section~\ref{sec:res}, in light of results obtained for representative test cases. Concluding remarks are given in section~\ref{sec:conc}.
\section{Derivation of Green's function}
\label{sec:gf-der}
Let $\Omega = \{(\rho,\theta,\varphi) \in \mathbb{R}^+\times[0,2\pi]\times[0,\pi] \ / \ \rho = R\}$.
Consider the screened Poisson's equation on $\Omega$:
\begin{equation}
\nabla^2_s \psi(\theta,\varphi) - \frac{1}{L_d^2}\psi(\theta,\varphi)= f(\theta,\varphi)
\label{eq:SP}
\end{equation}
where $L_d \in \mathbb{R}^{+}$ is the Rossby radius of deformation and $\nabla^2_s$ is the \textbf{Laplace-Beltrami} operator on the sphere of radius $R$,
\begin{equation}
\nabla^2_s = \frac{1}{R^2}\frac{1}{\sin^2 \theta }\pdv[2]{}{\varphi} + \frac{1}{R^2}\frac{1}{\sin\theta}\pdv[]{}{\theta} (\sin\theta \pdv[]{}{\theta}).
\end{equation}
The spherical harmonics $Y_{l,m}(\theta,\varphi)$ form a complete basis set of the Hilbert space of all square-integrable functions,
${\mathcal H} = \left\{ f: \Omega \to \mathbb{R} / \int_{\Omega} f^2 < \infty \right\}$. Thus every function in ${\mathcal H}$ can be decomposed
in terms of the mean-square convergent sum:
\begin{equation}
f(\theta,\varphi) = \sum_{l=0}^{\infty} \sum_{m=-l}^l f_{lm}(r)Y_{l,m}(\theta,\varphi),
\label{eq:f-harmonic}
\end{equation}
and the solution, $\psi$, of (\ref{eq:SP}) can be expressed as:
\begin{equation}
\psi(\theta,\varphi) = \sum_{l=0}^{\infty} \sum_{m=-l}^l u_{lm}(r)Y_{l,m}(\theta,\varphi).
\end{equation}
Using the $L_2$ inner product:
\begin{equation}
<h(\theta,\varphi),k(\theta,\varphi)> \equiv \int\limits_{\Omega} h(\theta,\varphi)k(\theta,\varphi) dS \quad (h,k) \in {\mathcal H}^2
\end{equation}
the coefficients in (\ref{eq:f-harmonic}) are given by:
\begin{equation}
f_{lm} = \frac{1}{R^2}\int\limits_{\Omega} Y_{l,m}(\theta,\varphi) f(\theta,\varphi) dS = \int_{\theta=0}^{\pi} \int_{\varphi=0}^{2\pi}Y_{l,m}(\theta,\varphi)f(\theta,\varphi) \sin\theta \,d\theta\,d\varphi.
\end{equation}
Because the spherical harmonics are eigenfunctions of $\nabla^2_s|_{R=1}$~\cite{SphericalFunctions}, that is
\begin{equation}
\left. \nabla^2_s \right| _{R=1} Y_{l,m} = -l(l+1) Y_{l,m}
\end{equation}
we have
\begin{equation}
\nabla^2_s Y_{l,m} =\frac{1}{R^2} \left. \nabla^2_{s} \right| _{R=1} Y_{l,m}=\frac{-l(l+1)}{R^2}Y_{l,m}
\end{equation}
Now we write (\ref{eq:SP}) as:
\begin{equation}
\sum_{l=0}^{\infty} \sum_{m=-l}^l \Big[ u_{lm} \frac{(-l)(l+1)}{R^2} Y_{l,m}(\theta,\varphi)-\frac{1}{L_d^2}u_{lm}(r)Y_{l,m}(\theta,\varphi)\Big] = \sum_{l=0}^{\infty} \sum_{m=-l}^l f_{lm}Y_{l,m}(\theta,\varphi) .
\end{equation}
From the orthogonality of the basis, we obtain
\begin{equation}
u_{lm} = \frac{ f_{lm}}{ \frac{(-l)(l+1)}{R^2}-\frac{1}{L_d^2}}= \frac{ \int_{\theta'=0}^{\pi} \int_{\varphi'=0}^{2\pi}Y_{l,m}(\theta',\varphi')f(\theta',\varphi') \sin\theta' \,d\theta'\,d\varphi'
}{ \frac{(-l)(l+1)}{R^2}-\frac{1}{L_d^2}}
\end{equation}
and
\begin{align}
\psi(\theta,\varphi) &= \sum_{l=0}^{\infty} \sum_{m=-l}^l \frac{ f_{lm}}{ \frac{(-l)(l+1)}{R^2}-\frac{1}{L_d^2}}Y_{l,m}(\theta,\varphi) \\&=\sum_{l=0}^{\infty} \sum_{m=-l}^l \frac{ \int_{\theta'=0}^{\pi} \int_{\varphi'=0}^{2\pi}Y_{l,m}(\theta',\varphi')f(\theta',\varphi') \sin\theta' \,d\theta'\,d\varphi'
}{ \frac{(-l)(l+1)}{R^2}-\frac{1}{L_d^2}}Y_{l,m}(\theta,\varphi) \\ &=
\int_{\theta'=0}^{\pi} \int_{\varphi'=0}^{2\pi} \sum_{l=0}^{\infty} \sum_{m=-l}^l \frac{ Y_{l,m}(\theta,\varphi)Y_{l,m}(\theta',\varphi')}{(-l)(l+1)- \frac{R^2}{L_d^2}}f(\theta',\varphi') R^2 \sin\theta' \,d\theta'\,d\varphi'
\end{align}
Because $\psi = G * f$, we have:
\begin{align}
G((R,\theta,\varphi),(R,\theta',\varphi')) & =\sum_{l=0}^{\infty} \sum_{m=-l}^l \frac{ Y_{l,m}(\theta',\varphi')Y_{l,m}(\theta,\varphi)}{(-l)(l+1)-\frac{R^2}{L_d^2}} \\
& = \sum_{l=0}^{\infty}\frac{1}{(-l)(l+1)-\frac{R^2}{L_d^2}}\sum_{m=-l}^l Y_{l,m}(\theta',\varphi')Y_{l,m}(\theta,\varphi)
\end{align}
Using (i) the spherical harmonics addition theorem~\cite{SphericalFunctions}:
\begin{equation}
\forall (R,\theta,\varphi),(R,\theta',\varphi') \in \Omega, \ \frac{4\pi}{2l+1} \sum_{m=-l}^{l}Y_{l,m}(\theta,\varphi)Y_{l,m}(\theta',\varphi') = P_l(\cos\gamma)
\end{equation}
where $\gamma$ is the angle at the center between $(R,\theta,\varphi)$ and $(R,\theta',\varphi')$, and (ii) the identity $\cos\gamma =\cos\theta\cos\theta'+\sin\theta\sin\theta'\cos(\varphi-\varphi')$, we obtain
\begin{align}
G((R,\theta,\varphi),(R,\theta',\varphi')) &= \sum_{l=0}^{\infty}\frac{1}{(-l)(l+1)-\frac{R^2}{L_d^2}} \frac{2l+1}{4\pi}P_l(\cos(\gamma))\\ &= \boxed{\frac{-1}{4\pi}\sum_{l=0}^{\infty}\frac{2l+1}{l(l+1)+\frac{R^2}{L_d^2}}P_l(\cos(\gamma))}
\label{eq:G-series}
\end{align}
\section{Convergence of the series representation}
\label{sec:gf-conv}
In this section, we briefly examine properties of the Green's function series representation.
To this end, we study the behavior of:
\begin{equation}
f\left(w,\gamma\right)=\sum_{l\geq0}\frac{\left(2l+1\right)P_{l}\left(\cos\left(\gamma\right)\right)}{l\left(l+1\right)+w},\,w\in\mathbb{R}^{+}.
\end{equation}
For $\cos(\gamma)=1$, we have $P_{l} \left(\cos\left(\gamma\right)\right)=1,\ \forall l \in\mathbb{N}$, consequently
\begin{equation}
\sum_{l\geq0}\frac{2l+1}{l\left(l+1\right)+w}\sim\sum_{l\geq0}\frac{2}{l+w} ,
\end{equation}
and so the series diverges like the harmonic series.
For $\cos(\gamma) \neq 1$, we have
\begin{equation}
f\left(w,\gamma\right)=2\sum_{l\geq0}\frac{lP_{l}\left(\cos\left(\gamma\right)\right)}{l\left(l+1\right)+w}+\sum_{l\geq0}\frac{P_{l}\left(\cos\left(\gamma\right)\right)}{l\left(l+1\right)+w}.
\label{eq:f2series}
\end{equation}
The second series is absolutely convergent because
\begin{equation}
\sum_{l\geq0}\left|\frac{P_{l}\left(\cos\left(\gamma\right)\right)}{l\left(l+1\right)+w}\right|\leq\sum_{l\geq0}\frac{1}{l^{2}+w}<\infty,\,\forall w\in\mathbb{R}^{+}
\end{equation}
Now consider the first series,
let $A_l = \frac{l}{l(l+1)+w}$ and $B_l = P_l(\cos(\gamma))$. Clearly, $A_l \geq 0$ and $\lim_{l\to\infty} A_l = 0$.
Hence, there exists $ l^* \in \mathbb{N}$ such that $A_{l+1}\leq A_l$ for all $l\ \geq l^*$.
We now rewrite the first series on the right hand side of (\ref{eq:f2series}) as:
\begin{equation}
\underbrace{\sum_{l=0}^{l^*}\frac{lP_{l}\left(\cos\left(\gamma\right)\right)}{l\left(l+1\right)+w}}_{\text{finite sum}} + \sum_{k=0}^{\infty}\frac{(l^*+k+1)P_{(l^*+k+1)}\left(\cos\left(\gamma\right)\right)}{(l^*+k+1)\left((l^*+k+2)\right)+w}.
\end{equation}
Let us show that the second member is convergent using the Dirichlet Test. We have already shown that $A_l \to 0$ and that it is monotonically decreasing for $l \geq l^*$.
It remains to show that there exists $M \geq 0$ such that $ \left| \sum_{l=0}^N P_l(\cos(\gamma)) \right| \leq M$ for all $N$.
Note that if $\cos(\gamma)=-1.0$, the $P_l\left( \cos(\gamma) \right) = (-1)^l P_l(1) = (-1)^l$, therefore $\sum_{l=0}^{N} P_l \left( \cos(\gamma) \right) = \pm 1$, which is bounded.
We may thus assume that $ \left| \cos(\gamma) \right| < 1$. We make use of (i) the Legendre generating function:
\begin{equation}
\sum_{l=0}^{\infty} u^l P_l(y) = \frac{1}{\sqrt{u^2 - 2yu+1}},
\end{equation}
with $y=\cos(\gamma)$, and (ii) the binomial series expansion $(1+x)^{1/2}$ with $x = u^2 - 2y$.
For $0<\cos(\gamma)<1$, we let $u=1$, so we have $x=1-2\cos(\gamma) <1$. Consequently, the binomial series converges and the
Legendre sum is bounded. For $-1<\cos(\gamma)<0$, we use the fact that $P_l(-\cos(\gamma)) = (-1)^l P_l(\cos(\gamma))$, let $u = -1$ and $x = 1-2|\cos(\gamma)|.$ Following the same argument as before, we conclude that the sum is bounded. Finally, if $\cos(\gamma)=0$, we may set
$u \pm 1$, which leads $x = 1$, and the same conclusion as before.
Consequently, by the Dirichlet test, the series $\sum_{l=0}^{\infty}\frac{lP_{l}\left(\cos\left(\gamma\right)\right)}{l\left(l+1\right)+w}$ converges when
$\cos(\gamma) < 1$.
\section{Numerical approximation of the series representation}
\label{sec:numer}
In the tests below, we assess two approaches for estimating the Green's function based on its
series representation. The first is a straightforward approach based on truncating (\ref{eq:G-series}) at
a suitably large index, $l'$, namely through:
\begin{equation}
G((R,\theta,\varphi),(R,\theta',\varphi')) \approx \frac{-1}{4\pi}\sum_{l=0}^{l'}\frac{2l+1}{(l)(l+1)+\frac{R^2}{L_d^2}}P_l(\cos(\gamma)) .
\label{eq:G-trunc}
\end{equation}
We refer to (\ref{eq:G-trunc}) as the truncated approximation.
A second, alternative approach is developed based on first splitting (\ref{eq:G-series}) according to:
\begin{equation}
G((R,\theta,\varphi),(R,\theta',\varphi')) = \frac{-1}{4\pi R^2}\sum_{l=0}^{l'-1}\frac{2l+1}{\frac{l(l+1)}{R^2}+\frac{1}{L_d^2}}P_l(\cos(\gamma)) +
\frac{-1}{4\pi R^2}\sum_{l=l'}^{\infty}\frac{2l+1}{\frac{l(l+1)}{R^2}+\frac{1}{L_d^2}}P_l(\cos(\gamma)) .
\label{eq:G-alt1}
\end{equation}
Selecting $l'$ such that $\frac{l'(l'+1)}{R^2} >> \frac{1}{L_d^2}$, we may approximate $G$ according to:
\begin{equation}
G((R,\theta,\varphi),(R,\theta',\varphi')) \approx \frac{-1}{4\pi R^2}\sum_{l=0}^{l'-1}\frac{2l+1}{\frac{l(l+1)}{R^2}+\frac{1}{L_d^2}}P_l(\cos(\gamma)) +
\frac{-1}{4\pi R^2}\sum_{l=l'}^{\infty}\frac{2l+1}{\frac{l(l+1)}{R^2}}P_l(\cos(\gamma)) .
\label{eq:G-alt2}
\end{equation}
Let
\begin{equation}
G^*((\theta,\varphi),(\theta',\varphi')) = \frac{1}{4\pi}\log(\frac{\text{e}}{2}(1-\cos(\gamma)))
\end{equation}
denote the Green's function of the Poisson equation on the sphere,
\begin{equation}
\nabla_s^2\psi(\theta,\varphi) = f(\theta,\varphi).
\end{equation}
$G^*$ can be expressed in series form as:
\begin{equation}
G^*((\theta,\varphi),(\theta',\varphi')) = \frac{-1}{4\pi}\sum_{l=1}^{\infty}\frac{2l+1}{(l)(l+1)}P_l(\cos(\gamma))
\label{eq:Gstar}
\end{equation}
Inserting (\ref{eq:Gstar}) into (\ref{eq:G-alt2}) and rearranging we finally obtain:
\begin{align}
G((R,\theta,\varphi),(R,\theta',\varphi')) &\approx -\frac{L_d^2}{4\pi R^2}-\frac{1}{4\pi R^2}\sum_{l=1}^{l'-1} \left[\frac{2l+1}{\frac{(l)(l+1)}{R^2} + \frac{1}{L_d^2}} - \frac{2l+1}{\frac{(l)(l+1)}{R^2}}
\right] P_l(\cos(\gamma)) \nonumber \\&+ \frac{1}{4\pi} \log(\frac{e}{2}(1-\cos(\gamma))) ,
\label{eq:G-split}
\end{align}
which we refer to as the split approximation.
Because the Rossby radius, $L_d$, defines a distance on the circumference of the sphere, and the distance between the source and target on the sphere is $R \gamma$, we introduce the characteristic
angle of the problem $\gamma^* \equiv L_d/R$. The Green's function and its split sum approximation are then expressed in terms of $\gamma^*$ as:
\begin{align}
G(\gamma,\gamma^*) \approx G_{l'}(\gamma,\gamma^*) & = -\frac{\gamma^{*2}}{4\pi }-\frac{1}{4\pi}\sum_{l=1}^{l'-1} \left[\frac{2l+1}{(l)(l+1)+ \frac{1}{\gamma^{*2}} } - \frac{2l+1}{(l)(l+1)} \right] P_l(\cos(\gamma)) \nonumber \\&+ \frac{1}{4\pi} \log(\frac{e}{2}(1-\cos(\gamma))) .
\label{eq:G-split1}
\end{align}
As further discussed below, with the same truncation index, $l'$, the split approximation leads to estimates exhibiting
appreciably smaller relative errors than straightforward truncation. It also exhibits faster and more robust convergence
as $l'$ increases.
\section{Integral form}
\label{sec:int-form}
In this section, we exploit the series solution (\ref{eq:G-series}) to derive an alternative, integral form of the Green's function.
To this end, we make use of the following identity,
\begin{equation}
\frac{1}{l+R} = \int_{0}^{+\infty} e^{-z(l+R)} \,dz, \quad {\rm Re}\left( l+R \right) > 0 ,
\label{eq:identity}
\end{equation}
and factor term $l(l+1) + \frac{R^2}{L_d^2}$ as $(l-S_1)(l-S_2)$ where $S_1,S_2 \in \mathbb{C}$.
We then perform the partial fraction expansion,
$$\frac{2l+1}{l(l+1) + \frac{R^2}{L_d^2}} = \frac{s_1}{l-S_1} + \frac{s_2}{l-S_2},$$
where $s_1,s_2 \in \mathbb{C}$. Note that $S_1$ and $S_2$ are complex conjugates that are independent of $l$, and
so are $s_1$ and $s_2$.
Next, we apply (\ref{eq:identity}) to re-express the fractions $s_1/(l-S_1)$ and $s_2/(l-S_2)$ respectively according to:
$$\frac{s_1}{l-S_1} = s_1 \int_{0}^{+\infty} e^{-z(l-S_1)} \,dz ,$$
and
$$\frac{s_2}{l-S_2} = s_2 \int_{0}^{+\infty} e^{-z(l-S_2)} \,dz.$$
Substituting these representations into (\ref{eq:G-series}), we get:
\begin{align}
G(R,\theta,\phi) &= \frac{-1}{4\pi}\sum_{l=0}^{\infty} (\int_{0}^{+\infty} s_1e^{-z(l-S_1)} + s_2 e^{-z(l-S_2)} \,dz)P_l(\cos(\gamma)) \\
&= \frac{-1}{4\pi}\sum_{l=0}^{\infty} (\int_{0}^{+\infty} e^{-zl}[s_1e^{zS_1} + s_2 e^{zS_2}] \,dz)P_l(\cos(\gamma)) \\
&= \frac{-1}{4\pi}\int_{0}^{+\infty} (s_1e^{zS_1} + s_2 e^{zS_2}) \sum_{l=0}^{\infty} (e^{-z})^l P_l(\cos(\gamma)) \,dz . \label{eq:leg-sum}
\end{align}
Finally, we use the Legendre generating function to re-express the summation in (\ref{eq:leg-sum}), which results in:
\begin{equation}
G(R,\theta,\phi) = \frac{-1}{4\pi} \int_{0}^{+\infty} \frac{(s_1e^{zS_1} + s_2 e^{zS_2})}{\sqrt{e^{-2z} - 2e^{-z} \cos(\gamma) + 1}} dz .
\label{eq:G-int1}
\end{equation}
A more convenient form of (\ref{eq:G-int1}) can be obtained by substituting the values of $S_1$, $S_2$, $s_1$ and $s_2$, namely
$s_1 = s_2 = 1$, and
$$
S_1 = -\frac{1}{2} + i \beta , \quad S_1 = -\frac{1}{2} - i \beta ,
$$
where
$$
\beta \equiv \sqrt{ \frac{R^2}{L_d^2} - \frac{1}{4} } .
$$
Performing the substitution and rearranging, we obtain:
\begin{equation}
G(R,\theta,\phi) = \frac{-1}{2\pi} \int_{0}^{+\infty} \frac{ e^{-z/2} \cos(\beta z)}{\sqrt{e^{-2z} - 2e^{-z} \cos(\gamma) + 1}} dz .
\label{eq:G-int-form}
\end{equation}
Below, we use a quadrature approximation of (\ref{eq:G-int-form}) to verify results obtained using the series representation.
Note that for particular values of $\cos(\gamma)$, the integral in (\ref{eq:G-int-form}) can be evaluated analytically with the
result expressed in terms of elementary functions. Specifically, for $\cos(\gamma) = -1$, we have (see~\cite{GR}, article 3.981):
\begin{equation}
\left. G \right|_{\cos(\gamma) = -1} = \frac{-1}{4 \cosh(\pi\beta)}
\label{eqCOSM1}
\end{equation}
whereas for $\cos(\gamma) = 0$ we obtain (see~\cite{GR}, article 3.985):
\begin{equation}
\left. G \right|_{\cos(\gamma) = 0} = \frac{-1}{8 \pi^{3/2}} \Gamma \left( \frac{1}{4} + i \frac{\beta}{2} \right) \Gamma \left( \frac{1}{4} - i \frac{\beta}{2} \right)
\label{eqCOS0}
\end{equation}
where $\Gamma$ denotes the gamma function. In particular, we use the results to verify both our quadrature and series approximations.
Also note the integral representation in (\ref{eq:G-int-form}) can alternatively be expressed in terms of associated Legendre function,
namely according to:
\begin{equation}
G(R,\theta,\phi) = - \frac{1}{4 \cosh(\pi \beta)} P_{-\frac{1}{2} + i \beta}\left( \cos(\pi - \gamma) \right)
\label{eq:alf}
\end{equation}
Because the zeros of $P_{-\frac{1}{2} + i \beta}( z )$ are all real and greater than unity (see~\cite{GR}, article 8.784), we conclude
that $G(R,\theta,\phi)$ is negative in the entire range $0 < \gamma \leq \pi$ whereas it diverges to $-\infty$ as $\gamma \to 0$.
\section{Results}
\label{sec:res}
In this section, we first compare the split sum to the direct sum in terms of their behavior as a function of the number of terms, for the case $L_d = 1000 \ \text{km}$. The computations are carried out in Fortran using quadruple precision for real numbers and double precision for integers.
Second, we show the absolute error versus the number of terms for the split sum approximation, for $L_d=100 \ \text{km}$ and $L_d=1000 \ \text{km}.$ These two values
are selected to represent the low and high values of the Rossby radius of deformation in the ocean and the atmosphere\cite{Gill82,chelton98}.
Third, we compare, for $L_d = 1000 \ \text{km}$, the values of the Green's function computed using the split sum approximation to those computed using the high precision Numerical Integration Polyalgorithm of Maple$^\text{TM}$.
Fourth, we present plots, for different values of $L_d$, of the Green's function computed using the split sum versus the angle. Finally, tables of the Green's function versus the angle for different values of $L_d$ are also presented in the appendix.
\begin{figure}[h]
\centering
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{cos-1.pdf}
\caption{$\cos(\gamma)=-1$}
\label{fig:sub1}
\end{subfigure}
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{cos0dot9.pdf}
\caption{$\cos(\gamma)=0.9$}
\label{fig:sub2}
\end{subfigure}
\caption{$G(\theta,\varphi)$ versus the number of terms for (a) $\cos(\gamma)=-1$, (b) $\cos(\gamma)=0.9$. Curves are generated for $L_d = 1000 \ \text{km}$ using the direct and split sum as indicated.}
\label{fig:test}
\end{figure}
Figure~\ref{fig:test} depicts the estimates obtained using the split sum and the direct sum as a function of the number of terms retained in the corresponding expansions.
Two cases are considered, namely $\cos(\gamma)=-1.0$ and $\cos(\gamma) = 0.9$. The first case corresponds to a maximum separation between the target and the source. In the second case, the angle separating the target from the source is small. In both cases $L_d = 1000 \ \text{km}$. We observe from Figures \ref{fig:sub1} and \ref{fig:sub2} that the split sum converges faster than the direct sum. One can also observe that the rate of convergence is slower for $\cos(\gamma)=-1.0$. In fact, for a given value of $L_d$, the rate of convergence of the split sum is the slowest when $\cos(\gamma)=-1.0$. This is because the Legendre polynomial in the split sum approximation of equation (\ref{eq:G-split}) switches sign every term, i.e. $P_l(-1)P_{l+1}(-1)<0$.
\begin{figure}[h]
\centering
\includegraphics[width=0.8\textwidth]{Error1.pdf}
\caption{The absolute error (Eq.~\ref{eq:abs-error}) versus the number of terms for $\gamma=\gamma^*$. Curves are generated using the split sum, for $L_d = 1000 \ \text{km}$ and $100$~km as indicated.}
\label{figure:AbsoluteError}
\end{figure}
Figure \ref{figure:AbsoluteError} shows the absolute error of the split sum estimate versus the number of terms at $\frac{\gamma}{\gamma^*}=1$, for $L_d=1000 \ \text{km}$ and $L_d=100 \ \text{km}$. The absolute error is computed as
\begin{equation}
E(l) = \abs{\hat{G}(\gamma,\gamma^*)-G_{l}(\gamma,\gamma^*)},
\label{eq:abs-error}
\end{equation}
where the ``converged'' solution $\hat{G}$ is the value of $G$ obtained after a sufficiently large number of terms, $l'$, is used in the summation, such that the coefficient multiplying the Legendre polynomial in the $l'$th term of the split sum is within quadruple machine precision. (Actually, it may be shown that for a desired cutoff value of this coefficient, $\epsilon$, the number of terms needed is $l' \simeq \sqrt[3]{ \frac{2}{\epsilon \gamma^{*2}}}$.) It can be observed that the asymptotic rate of convergence for both values of $L_d$ appears to be similar ($\sim -3.5$ on the log-log plot), though evidently a larger number of terms must be included as $L_d$ decreases.
Note that $\hat{G}(\gamma,\gamma^*)$ is in close agreement with the value computed using the Numerical Integration Polyalgorithm of Maple$^\text{TM}$ to within 16 decimal points, which is the precision of the Maple integration, as can be seen in Table \ref{Table:splitSumVsMaple}.
Table \ref{Table:splitSumVsMaple} shows that the split sum approximation matches the numerical integration using the Numerical Integration Polyalgorithm of Maple$^\text{TM}$ over the range $0.001 \leq \frac{\gamma}{\gamma^*} \leq 10$ for $L_d = 1000 \ \text{km}$ ($\gamma^* = 0.15696123$). The agreement improves from 8 significant digits at $\frac{\gamma}{\gamma^*} =0.001$ to 16 significant digits at $\frac{\gamma}{\gamma^*} =10$. For the cases $\cos(\gamma)=0$ and $\cos(\gamma)=-1$, the split sum approximation matches the closed form solutions (\ref{eqCOS0}) and (\ref{eqCOSM1}), as shown in Table \ref{Table:splitSumVsEQNS} for different values of $L_d$.
\begin{table}[]
\centering
\caption{The integral approximation of the Green's function using the Numerical Integration Polyalgorithm of Maple$^\text{TM}$ and the split sum approximation for $L_d = 1000 \ \text{km}$ ($\gamma^* = 0.15696123$). The stopping criterion used for the split sum is when the absolute value of the factor multiplying the Legendre polynomial reaches quadruple machine precision. Except for the first two entries, the Maple integral was calculated to 16 significant digits. For $\gamma/\gamma^*=0.001$, the maximum number of digits attained was $12$ whereas for $\gamma/\gamma^*=0.005$, it was $15$ digits.}
\label{Table:splitSumVsMaple}
\begin{tabular}{|l|l|l|}
\hline
$\gamma/\gamma^*$ & \textbf{Maple} & \textbf{Split Sum} \\ \hline
0.001 & -1.11851154768 & -1.1185115466790030 \\ \hline
0.005 & -0.862367611582155 & -0.86236761566019138 \\ \hline
0.01 & -0.7520661831497065 & -0.75206618670609104 \\ \hline
0.02 & -0.6418055952187355 & -0.64180559877292642 \\ \hline
0.03 & -0.5773592542889705 & -0.57735925783975817 \\ \hline
0.04 & -0.5316835690653312 & -0.53168357261181198 \\ \hline
0.05 & -0.4963020973589550 & -0.49630209499805700 \\ \hline
0.06 & -0.4674384556430355 & -0.46743845917847821 \\ \hline
0.07 & -0.4430777083432368 & -0.44307770767107485 \\ \hline
0.08 & -0.4220167638214708 & -0.42201676734310684 \\ \hline
0.09 & -0.4034793378064005 & -0.40347933155959809 \\ \hline
0.1 & -0.3869351828355970 & -0.38693518049861692 \\ \hline
0.15 & -0.3237421092103450 & -0.32374210306526552 \\ \hline
0.2 & -0.2796019048871758 & -0.27960190262165774 \\ \hline
0.25 & -0.2459853828209132 & -0.24598538282091331 \\ \hline
0.3 & -0.2190766246998643 & -0.21907661890074209 \\ \hline
0.4 & -0.1780154655314450 & -0.17801546345860769 \\ \hline
0.5 & -0.1477460718583630 & -0.14774607185836305 \\ \hline
0.6 & -0.1243523124660145 & -0.12435230751870802 \\ \hline
0.7 & -0.1057148171283462 & -0.10571481912301701 \\ \hline
0.8 & -9.055025072697948E-002 & -9.0550249089805523E-002 \\ \hline
0.9 & -7.801957852946700E-002 & -7.8019581252887993E-002 \\ \hline
1 & -6.754292534703262E-002 & -6.7542925347032642E-002 \\ \hline
2 & -1.846304821245086E-002 & -1.8463048212450855E-002 \\ \hline
3 & -5.714300085292792E-003 & -5.7143000852927931E-003 \\ \hline
4 & -1.870693766774068E-003 & -1.8706937667740684E-003 \\ \hline
5 & -6.333928838453482E-004 & -6.3339288384534867E-004 \\ \hline
6 & -2.195637338314424E-004 & -2.1956373383144240E-004 \\ \hline
7 & -7.750693244142898E-005 & -7.7506932441429000E-005 \\ \hline
8 & -2.777791606780832E-005 & -2.7777916067808478E-005 \\ \hline
9 & -1.009014887236524E-005 & -1.0090148872365168E-005 \\ \hline
10 & -3.711685976274890E-006 & -3.7116859762750061E-006 \\ \hline
\end{tabular}
\end{table}
\begin{table}[]
\centering
\caption{The split sum approximation for $\gamma=\pi/2$ and $\pi$ compared to Equations (\ref{eqCOS0}) and (\ref{eqCOSM1}).}
\label{Table:splitSumVsEQNS}
\begin{tabular}{|l|l|l|}
\hline
$L_d$ (km) & \textbf{Split Sum ($\cos(\gamma)=0$)} & \textbf{Equation (\ref{eqCOS0})} \\ \hline
300 & -1.4225814713795086E-016 & -1.4225594675545431E-0016 \\ \hline
400 & -6.8995961107067088E-013 & -6.8995960864883886E-0013 \\ \hline
500 & -1.1530544611076218E-010 & -1.1530544610815334E-0010 \\ \hline
600 & -3.5617983195546507E-009 & -3.5617983195574219E-009 \\ \hline
700 & -4.1828668580602192E-008 & -4.1828668580605125E-008 \\ \hline
800 & -2.6799479475630453E-007 & -2.6799479475630768E-007 \\ \hline
900 & -1.1452992927108748E-006 & -1.1452992927108717E-006 \\ \hline
1000 & -3.6839641135260536E-006 & -3.6839641135260568E-006 \\ \hline
1100 & -9.6329029580597044E-006 & -9.6329029580597082E-006 \\ \hline
1200 & -2.1556389233382265E-005 & -2.1556389233382263E-005 \\ \hline
1300 & -4.2781705365952404E-005 & -4.2781705365952408E-005 \\ \hline
1400 & -7.7247341254381796E-005 & -7.7247341254381793E-005 \\ \hline
1500 & -1.2929790801598909E-004 & -1.2929790801598909E-004 \\ \hline
1600 & -2.0346827424618137E-004 & -2.0346827424618136E-004 \\ \hline
1700 & -3.0428682816304554E-004 & -3.0428682816304552E-004 \\ \hline
1800 & -4.3611425145919584E-004 & -4.3611425145919584E-004 \\ \hline
1900 & -6.0302349401310045E-004 & -6.0302349401310041E-004 \\ \hline
2000 & -8.0871962518105109E-004 & -8.0871962518105106E-004 \\ \hline \hline
$L_d$ (km) & \textbf{Split Sum ($\cos(\gamma)=-1$)} & \textbf{Equation (\ref{eqCOSM1})} \\ \hline
600 & -1.6930096452253692E-015 & -1.6890307829549300E-015 \\ \hline
700 & -1.9949869771970800E-013 & -1.9950267658105915E-013 \\ \hline
800 & -7.1590372634355031E-012 & -7.1590332845768866E-012 \\ \hline
900 & -1.1609776217297870E-010 & -1.1609776615183131E-010 \\ \hline
1000 & -1.0797889438705467E-009 & -1.0797889398916860E-009 \\ \hline
1100 & -6.7027265321956257E-009 & -6.7027265361744720E-009 \\ \hline
1200 & -3.0722971586707354E-008 & -3.0722971582728503E-008 \\ \hline
1300 & -1.1152376939660639E-007 & -1.1152376939262755E-007 \\ \hline
1400 & -3.3703407951899458E-007 & -3.3703407952297341E-007 \\ \hline
1500 & -8.7963567819438533E-007 & -8.7963567819836426E-007 \\ \hline
1600 & -2.0379565833152491E-006 & -2.0379565833112706E-006 \\ \hline
1700 & -4.2803676572173898E-006 & -4.2803676572213691E-006 \\ \hline
1800 & -8.2844410488393729E-006 & -8.2844410488353935E-006 \\ \hline
1900 & -1.4967463004072891E-005 & -1.4967463004068912E-005 \\ \hline
2000 & -2.5504778524850583E-005 & -2.5504778524854560E-005 \\ \hline
\end{tabular}
\end{table}
Equation (\ref{eq:G-split1}) can be expressed as
\begin{equation}
G_{l'}(\gamma,\gamma^*) = -\frac{\gamma^{*2}}{4\pi }-\frac{1}{4\pi}\sum_{l=1}^{l'-1} \left[\frac{2l+1}{(l)(l+1)+ \frac{1}{\gamma^{*2}} } - \frac{2l+1}{(l)(l+1)} \right] P_l(\cos(\gamma)) + G^*(\gamma) ,
\label{eq:G-split2}
\end{equation}
where $G^*(\gamma) = \frac{1}{4\pi} \log(\frac{e}{2}(1-\cos(\gamma)))$ is the Green's function of Poisson's equation on the sphere, without the screening term. To explore the departure of $G$ from $G^*$, plots of $(G-G^*)(\gamma, \gamma^*)$ and $-G^*(\gamma, \gamma^*)$ versus $\gamma/\gamma^*$ are presented in Figure \ref{fig:GG*} for the range $50 \text{ km} \le L_d \le 1000 \text{ km}$.
\begin{figure}[h]
\centering
\includegraphics[width=0.8\textwidth]{GGSTARandGSTARA.pdf}
\caption{$(G-G^*)$ and $-G^*$ vs. $\gamma/\gamma^*$. Curves are generated using the split sum for $L_d= 50,100,200,300,400,500,600,700,800,900 \ \text{and} \ 1000 \ \text{km}$.}
\label{fig:GG*}
\end{figure}
It can be observed from figure \ref{fig:GG*} that for sufficiently large $\gamma/\gamma^*$, the $-G^*$ curve collapses onto the $G-G^*$ curve, indicating that the Green's function of the screened Poisson equation decays at a much faster rate than that of the Poisson equation. This is attributed to the role of the screening term $\psi/L_d^2$ in localizing the solution to a neighborhood of the order $\gamma^*$. This can also be seen in Figures \ref{fig:sub1} and \ref{fig:sub2}, where $G$, $G^*$, and $G-G^*$ are plotted versus$\gamma/\gamma^*$ for $L_d=50$ km and $L_d=1000$ km, respectively. It can be observed that $G$ becomes increasingly localized on the sphere as $L_d$ decreases. As such, a compact approximation of $G$ for small values of $L_d$ may prove suitable.
\begin{figure}[hh]
\centering
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{All50kA.pdf}
\caption{$L_d = 50 \ \text{km}$}
\label{fig:sub1}
\end{subfigure}
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{All1000kA.pdf}
\caption{$L_d = 1000 \ \text{km}$}
\label{fig:sub2}
\end{subfigure}
\caption{$(G-G^*)$, $G^*$ and $G$ vs. $\gamma/\gamma^*$ for (a) $L_d = 50 \ \text{km}$, (b) $L_d = 100 \ \text{km}$. Curves are generated using the split sum.}
\label{fig:Gdecomp}
\end{figure}
\section{Conclusions}
\label{sec:conc}
In this paper, analytical expressions are derived of the Green's function of the screened Poisson's equation on the sphere,
namely in the form of an integral representation and of a series solution involving Legendre polynomials. A robust and efficient numerical approximation of the series representation is then developed. This approximation is based
on a splitting of the series representation that is tailored to isolate the singular behavior. Efficiency and robustness of the split
series approximation was established by showing the rapid decay of the truncation error with the number of terms, and by comparing
estimates with results obtained using high precision numerical integration. The solutions presented, in both graph and tabular forms,
for different values of the normalized screening constant versus the normalized angle provide an effective means for accurate evaluation
of the Green's function.
\section{Acknowledgments}
\label{sec:ackn}
This work is supported by the University Research Board of the American University of Beirut. The authors would like to acknowledge Professor Leila Issa of the Lebanese American University-Beirut for her insightful feedback on the mathematical derivation of the convergence of the Green's Function.
\section{Appendix}
\label{sec:appendix}
In this section, we present tables of the values of the Green's function versus $\gamma/\gamma^*$, for values of the screening constant $L_d$ of $50,100,200,400,800,1000$ km.
\begin{longtable}[c]{|l|l|l|l|}
\caption{$G(\theta,\varphi)$ function of $\gamma/\gamma^*$ for $L_d=50,100,200 \ \text{km}$.The values presented are computed using the split sum. The stopping critera used is when the term contribution of $G(l)$ to the split sum drops down below 1E-20.}
\label{my-label}\\
\hline
$10\gamma/\gamma^*$ & $G(\theta,\varphi)$, $L_d = 50 \ \text{km}$ & $G(\theta,\varphi)$, $L_d = 100 \ \text{km}$ & $G(\theta,\varphi)$, $L_d = 200 \ \text{km}$ \\ \hline
\endfirsthead
\multicolumn{4}{c}
{{\bfseries Table \thetable\ continued from previous page}} \\
\hline
$10\gamma/\gamma^*$ & $G(\theta,\varphi)$, $L_d = 50 \ \text{km}$ & $G(\theta,\varphi)$, $L_d = 100 \ \text{km}$ & $G(\theta,\varphi)$, $L_d = 200 \ \text{km}$ \\ \hline
\endhead
1 & -0.386281669 & -0.386286557 & -0.386306107 \\ \hline
2 & -0.278953105 & -0.278957963 & -0.278977364 \\ \hline
3 & -0.218435407 & -0.218440190 & -0.218459383 \\ \hline
4 & -0.177384391 & -0.177389115 & -0.177407995 \\ \hline
5 & -0.147127405 & -0.147132024 & -0.147150531 \\ \hline
6 & -0.123747990 & -0.123752505 & -0.123770580 \\ \hline
7 & -0.105126463 & -0.105130859 & -0.105148457 \\ \hline
8 & -8.99792090E-02 & -8.99834707E-02 & -9.00005400E-02 \\ \hline
9 & -7.74669051E-02 & -7.74710327E-02 & -7.74875507E-02 \\ \hline
10 & -6.70094490E-02 & -6.70134276E-02 & -6.70293644E-02 \\ \hline
11 & -5.81887029E-02 & -5.81925362E-02 & -5.82078733E-02 \\ \hline
12 & -5.06933816E-02 & -5.06970622E-02 & -5.07117920E-02 \\ \hline
13 & -4.42856625E-02 & -4.42891903E-02 & -4.43033017E-02 \\ \hline
14 & -3.87800299E-02 & -3.87834013E-02 & -3.87968980E-02 \\ \hline
15 & -3.40292826E-02 & -3.40325013E-02 & -3.40453833E-02 \\ \hline
16 & -2.99149491E-02 & -2.99180150E-02 & -2.99302880E-02 \\ \hline
17 & -2.63405293E-02 & -2.63434462E-02 & -2.63551176E-02 \\ \hline
18 & -2.32266262E-02 & -2.32293960E-02 & -2.32404824E-02 \\ \hline
19 & -2.05073487E-02 & -2.05099769E-02 & -2.05204897E-02 \\ \hline
20 & -1.81276016E-02 & -1.81300901E-02 & -1.81400459E-02 \\ \hline
21 & -1.60410143E-02 & -1.60433669E-02 & -1.60527844E-02 \\ \hline
22 & -1.42083438E-02 & -1.42105669E-02 & -1.42194629E-02 \\ \hline
23 & -1.25962095E-02 & -1.25983069E-02 & -1.26067009E-02 \\ \hline
24 & -1.11760953E-02 & -1.11780716E-02 & -1.11859832E-02 \\ \hline
25 & -9.92354099E-03 & -9.92540177E-03 & -9.93285049E-03 \\ \hline
26 & -8.81749671E-03 & -8.81924666E-03 & -8.82625207E-03 \\ \hline
27 & -7.83978775E-03 & -7.84143247E-03 & -7.84801599E-03 \\ \hline
28 & -6.97468081E-03 & -6.97622448E-03 & -6.98240474E-03 \\ \hline
29 & -6.20851712E-03 & -6.20996533E-03 & -6.21576235E-03 \\ \hline
30 & -5.52941579E-03 & -5.53077273E-03 & -5.53620607E-03 \\ \hline
31 & -4.92701819E-03 & -4.92828898E-03 & -4.93337726E-03 \\ \hline
32 & -4.39227652E-03 & -4.39346535E-03 & -4.39822720E-03 \\ \hline
33 & -3.91727407E-03 & -3.91838606E-03 & -3.92283872E-03 \\ \hline
34 & -3.49507318E-03 & -3.49611230E-03 & -3.50027299E-03 \\ \hline
35 & -3.11958441E-03 & -3.12055484E-03 & -3.12444032E-03 \\ \hline
36 & -2.78545590E-03 & -2.78636138E-03 & -2.78998748E-03 \\ \hline
37 & -2.48797797E-03 & -2.48882244E-03 & -2.49220454E-03 \\ \hline
38 & -2.22300179E-03 & -2.22378876E-03 & -2.22694129E-03 \\ \hline
39 & -1.98686752E-03 & -1.98760070E-03 & -1.99053762E-03 \\ \hline
40 & -1.77634496E-03 & -1.77702762E-03 & -1.77976210E-03 \\ \hline
41 & -1.58857903E-03 & -1.58921431E-03 & -1.59175904E-03 \\ \hline
42 & -1.42104481E-03 & -1.42163562E-03 & -1.42400258E-03 \\ \hline
43 & -1.27150724E-03 & -1.27205648E-03 & -1.27425697E-03 \\ \hline
44 & -1.13798620E-03 & -1.13849656E-03 & -1.14054140E-03 \\ \hline
45 & -1.01872673E-03 & -1.01920078E-03 & -1.02109998E-03 \\ \hline
46 & -9.12171672E-04 & -9.12611722E-04 & -9.14375007E-04 \\ \hline
47 & -8.16938817E-04 & -8.17347143E-04 & -8.18983535E-04 \\ \hline
48 & -7.31800625E-04 & -7.32179382E-04 & -7.33697321E-04 \\ \hline
49 & -6.55665994E-04 & -6.56017161E-04 & -6.57424738E-04 \\ \hline
50 & -5.87564951E-04 & -5.87890449E-04 & -5.89195115E-04 \\ \hline
51 & -5.26634336E-04 & -5.26935910E-04 & -5.28144767E-04 \\ \hline
52 & -4.72106069E-04 & -4.72385378E-04 & -4.73505061E-04 \\ \hline
53 & -4.23296093E-04 & -4.23554680E-04 & -4.24591417E-04 \\ \hline
54 & -3.79595032E-04 & -3.79834353E-04 & -3.80793936E-04 \\ \hline
55 & -3.40459781E-04 & -3.40681203E-04 & -3.41569073E-04 \\ \hline
56 & -3.05406109E-04 & -3.05610913E-04 & -3.06432194E-04 \\ \hline
57 & -2.74002145E-04 & -2.74191494E-04 & -2.74950959E-04 \\ \hline
58 & -2.45862553E-04 & -2.46037584E-04 & -2.46739626E-04 \\ \hline
59 & -2.20643400E-04 & -2.20805145E-04 & -2.21453927E-04 \\ \hline
60 & -1.98037596E-04 & -1.98187015E-04 & -1.98786423E-04 \\ \hline
61 & -1.77770882E-04 & -1.77908878E-04 & -1.78462506E-04 \\ \hline
62 & -1.59598261E-04 & -1.59725663E-04 & -1.60236872E-04 \\ \hline
63 & -1.43300756E-04 & -1.43418350E-04 & -1.43890255E-04 \\ \hline
64 & -1.28682659E-04 & -1.28791173E-04 & -1.29226697E-04 \\ \hline
65 & -1.15568961E-04 & -1.15669085E-04 & -1.16070922E-04 \\ \hline
66 & -1.03803170E-04 & -1.03895516E-04 & -1.04266182E-04 \\ \hline
67 & -9.32452676E-05 & -9.33304182E-05 & -9.36722572E-05 \\ \hline
68 & -8.37699772E-05 & -8.38484848E-05 & -8.41636574E-05 \\ \hline
69 & -7.52651904E-05 & -7.53375498E-05 & -7.56280788E-05 \\ \hline
70 & -6.76305353E-05 & -6.76972195E-05 & -6.79649602E-05 \\ \hline
71 & -6.07761576E-05 & -6.08375885E-05 & -6.10842908E-05 \\ \hline
72 & -5.46215779E-05 & -5.46781630E-05 & -5.49054203E-05 \\ \hline
73 & -4.90947168E-05 & -4.91468236E-05 & -4.93561311E-05 \\ \hline
74 & -4.41309930E-05 & -4.41789707E-05 & -4.43717036E-05 \\ \hline
75 & -3.96725482E-05 & -3.97167169E-05 & -3.98941556E-05 \\ \hline
76 & -3.56675264E-05 & -3.57081772E-05 & -3.58715042E-05 \\ \hline
77 & -3.20694453E-05 & -3.21068510E-05 & -3.22571577E-05 \\ \hline
78 & -2.88366336E-05 & -2.88710471E-05 & -2.90093503E-05 \\ \hline
79 & -2.59317294E-05 & -2.59633835E-05 & -2.60906163E-05 \\ \hline
80 & -2.33212249E-05 & -2.33503379E-05 & -2.34673662E-05 \\ \hline
81 & -2.09750706E-05 & -2.10018407E-05 & -2.11094648E-05 \\ \hline
82 & -1.88663071E-05 & -1.88909180E-05 & -1.89898783E-05 \\ \hline
83 & -1.69707491E-05 & -1.69933737E-05 & -1.70843505E-05 \\ \hline
84 & -1.52666980E-05 & -1.52874909E-05 & -1.53711153E-05 \\ \hline
85 & -1.37346760E-05 & -1.37537836E-05 & -1.38306377E-05 \\ \hline
86 & -1.23572072E-05 & -1.23747632E-05 & -1.24453845E-05 \\ \hline
87 & -1.11186018E-05 & -1.11347299E-05 & -1.11996142E-05 \\ \hline
88 & -1.00047755E-05 & -1.00195894E-05 & -1.00791931E-05 \\ \hline
89 & -9.00308260E-06 & -9.01668682E-06 & -9.07143294E-06 \\ \hline
90 & -8.10216807E-06 & -8.11465998E-06 & -8.16493684E-06 \\ \hline
91 & -7.29183557E-06 & -7.30330430E-06 & -7.34947025E-06 \\ \hline
92 & -6.56292559E-06 & -6.57345345E-06 & -6.61583863E-06 \\ \hline
93 & -5.90721220E-06 & -5.91687558E-06 & -5.95578467E-06 \\ \hline
94 & -5.31730620E-06 & -5.32617469E-06 & -5.36188782E-06 \\ \hline
95 & -4.78656784E-06 & -4.79470600E-06 & -4.82748146E-06 \\ \hline
96 & -4.30903310E-06 & -4.31649960E-06 & -4.34657522E-06 \\ \hline
97 & -3.87934051E-06 & -3.88619037E-06 & -3.91378535E-06 \\ \hline
98 & -3.49267475E-06 & -3.49895777E-06 & -3.52427310E-06 \\ \hline
99 & -3.14470549E-06 & -3.15046805E-06 & -3.17368972E-06 \\ \hline
100 & -2.83154236E-06 & -2.83682698E-06 & -2.85812553E-06 \\ \hline
101 & -2.54968745E-06 & -2.55453324E-06 & -2.57406555E-06 \\ \hline
102 & -2.29599664E-06 & -2.30043929E-06 & -2.31834997E-06 \\ \hline
103 & -2.06764298E-06 & -2.07171570E-06 & -2.08813753E-06 \\ \hline
104 & -1.86208513E-06 & -1.86581826E-06 & -1.88087324E-06 \\ \hline
105 & -1.67703740E-06 & -1.68045904E-06 & -1.69425948E-06 \\ \hline
106 & -1.51044492E-06 & -1.51358063E-06 & -1.52622977E-06 \\ \hline
107 & -1.36045946E-06 & -1.36333290E-06 & -1.37492543E-06 \\ \hline
108 & -1.22541883E-06 & -1.22805159E-06 & -1.23867483E-06 \\ \hline
109 & -1.10382803E-06 & -1.10623989E-06 & -1.11597399E-06 \\ \hline
110 & -9.94342145E-07 & -9.96551535E-07 & -1.00546993E-06 \\ \hline
111 & -8.95751498E-07 & -8.97775294E-07 & -9.05945456E-07 \\ \hline
112 & -8.06967819E-07 & -8.08821312E-07 & -8.16305430E-07 \\ \hline
113 & -7.27011923E-07 & -7.28709381E-07 & -7.35564299E-07 \\ \hline
114 & -6.55002964E-07 & -6.56557290E-07 & -6.62835419E-07 \\ \hline
115 & -5.90148204E-07 & -5.91571393E-07 & -5.97320707E-07 \\ \hline
116 & -5.31734372E-07 & -5.33037337E-07 & -5.38301890E-07 \\ \hline
117 & -4.79119592E-07 & -4.80312451E-07 & -4.85132716E-07 \\ \hline
118 & -4.31726249E-07 & -4.32818126E-07 & -4.37231193E-07 \\ \hline
119 & -3.89034426E-07 & -3.90033819E-07 & -3.94073737E-07 \\ \hline
120 & -3.50576187E-07 & -3.51490826E-07 & -3.55188860E-07 \\ \hline
121 & -3.15930350E-07 & -3.16767370E-07 & -3.20152168E-07 \\ \hline
122 & -2.84717800E-07 & -2.85483736E-07 & -2.88581560E-07 \\ \hline
123 & -2.56597247E-07 & -2.57298069E-07 & -2.60133021E-07 \\ \hline
124 & -2.31261438E-07 & -2.31902618E-07 & -2.34496824E-07 \\ \hline
125 & -2.08433789E-07 & -2.09020371E-07 & -2.11394052E-07 \\ \hline
126 & -1.87865254E-07 & -1.88401842E-07 & -1.90573601E-07 \\ \hline
127 & -1.69331628E-07 & -1.69822428E-07 & -1.71809290E-07 \\ \hline
128 & -1.52630975E-07 & -1.53079881E-07 & -1.54897435E-07 \\ \hline
129 & -1.37581523E-07 & -1.37992075E-07 & -1.39654645E-07 \\ \hline
130 & -1.24019579E-07 & -1.24395001E-07 & -1.25915690E-07 \\ \hline
131 & -1.11797668E-07 & -1.12140981E-07 & -1.13531790E-07 \\ \hline
132 & -1.00783062E-07 & -1.01096965E-07 & -1.02368901E-07 \\ \hline
133 & -9.08561688E-08 & -9.11431712E-08 & -9.23063155E-08 \\ \hline
134 & -8.19092989E-08 & -8.21716881E-08 & -8.32352640E-08 \\ \hline
135 & -7.38454560E-08 & -7.40853068E-08 & -7.50577840E-08 \\ \hline
136 & -6.65772504E-08 & -6.67965026E-08 & -6.76856118E-08 \\ \hline
137 & -6.00259824E-08 & -6.02263910E-08 & -6.10392377E-08 \\ \hline
138 & -5.41207683E-08 & -5.43039427E-08 & -5.50470141E-08 \\ \hline
139 & -4.87977516E-08 & -4.89651484E-08 & -4.96443917E-08 \\ \hline
140 & -4.39993748E-08 & -4.41523511E-08 & -4.47732091E-08 \\ \hline
141 & -3.96738145E-08 & -3.98136066E-08 & -4.03810603E-08 \\ \hline
142 & -3.57743737E-08 & -3.59021080E-08 & -3.64207224E-08 \\ \hline
143 & -3.22589777E-08 & -3.23756879E-08 & -3.28496341E-08 \\ \hline
144 & -2.90897102E-08 & -2.91963431E-08 & -2.96294473E-08 \\ \hline
145 & -2.62324260E-08 & -2.63298432E-08 & -2.67255995E-08 \\ \hline
146 & -2.36563462E-08 & -2.37453328E-08 & -2.41069404E-08 \\ \hline
147 & -2.13337188E-08 & -2.14150138E-08 & -2.17453984E-08 \\ \hline
148 & -1.92395699E-08 & -1.93138252E-08 & -1.96156691E-08 \\ \hline
149 & -1.73513719E-08 & -1.74191932E-08 & -1.76949460E-08 \\ \hline
150 & -1.56488333E-08 & -1.57107714E-08 & -1.59626712E-08 \\ \hline
151 & -1.41136454E-08 & -1.41702143E-08 & -1.44003156E-08 \\ \hline
152 & -1.27293438E-08 & -1.27809985E-08 & -1.29911744E-08 \\ \hline
153 & -1.14810517E-08 & -1.15282264E-08 & -1.17201910E-08 \\ \hline
154 & -1.03553974E-08 & -1.03984670E-08 & -1.05737916E-08 \\ \hline
155 & -9.34028943E-09 & -9.37961975E-09 & -9.53973700E-09 \\ \hline
156 & -8.42486525E-09 & -8.46077342E-09 & -8.60699600E-09 \\ \hline
157 & -7.59931229E-09 & -7.63209584E-09 & -7.76562104E-09 \\ \hline
158 & -6.85479051E-09 & -6.88471902E-09 & -7.00664504E-09 \\ \hline
159 & -6.18333251E-09 & -6.21065332E-09 & -6.32198160E-09 \\ \hline
160 & -5.57775426E-09 & -5.60269475E-09 & -5.70434011E-09 \\ \hline
161 & -5.03158537E-09 & -5.05434494E-09 & -5.14714760E-09 \\ \hline
162 & -4.53897409E-09 & -4.55975213E-09 & -4.64447547E-09 \\ \hline
163 & -4.09467127E-09 & -4.11363610E-09 & -4.19097956E-09 \\ \hline
164 & -3.69393183E-09 & -3.71123487E-09 & -3.78183973E-09 \\ \hline
165 & -3.33246919E-09 & -3.34826056E-09 & -3.41270923E-09 \\ \hline
166 & -3.00642644E-09 & -3.02084047E-09 & -3.07966852E-09 \\ \hline
167 & -2.71233902E-09 & -2.72548850E-09 & -2.77918244E-09 \\ \hline
168 & -2.44705856E-09 & -2.45905718E-09 & -2.50806331E-09 \\ \hline
169 & -2.20776197E-09 & -2.21870988E-09 & -2.26343566E-09 \\ \hline
170 & -1.99189643E-09 & -2.00188843E-09 & -2.04270645E-09 \\ \hline
171 & -1.79717496E-09 & -1.80628723E-09 & -1.84353699E-09 \\ \hline
172 & -1.62151270E-09 & -1.62982572E-09 & -1.66381753E-09 \\ \hline
173 & -1.46304457E-09 & -1.47062817E-09 & -1.50164581E-09 \\ \hline
174 & -1.32008116E-09 & -1.32700273E-09 & -1.35530509E-09 \\ \hline
175 & -1.19111354E-09 & -1.19742449E-09 & -1.22324773E-09 \\ \hline
176 & -1.07475628E-09 & -1.08051634E-09 & -1.10407716E-09 \\ \hline
177 & -9.69787584E-10 & -9.75037384E-10 & -9.96533633E-10 \\ \hline
178 & -8.75078177E-10 & -8.79870343E-10 & -8.99481212E-10 \\ \hline
179 & -7.89633137E-10 & -7.94003419E-10 & -8.11894552E-10 \\ \hline
180 & -7.12542081E-10 & -7.16527948E-10 & -7.32849004E-10 \\ \hline
181 & -6.42990494E-10 & -6.46622145E-10 & -6.61510458E-10 \\ \hline
182 & -5.80230419E-10 & -5.83545989E-10 & -5.97126126E-10 \\ \hline
183 & -5.23609933E-10 & -5.26629684E-10 & -5.39017220E-10 \\ \hline
184 & -4.72515582E-10 & -4.75272321E-10 & -4.86571061E-10 \\ \hline
185 & -4.26419122E-10 & -4.28929697E-10 & -4.39235009E-10 \\ \hline
186 & -3.84819399E-10 & -3.87111510E-10 & -3.96510380E-10 \\ \hline
187 & -3.47288559E-10 & -3.49375445E-10 & -3.57947338E-10 \\ \hline
188 & -3.13416904E-10 & -3.15322934E-10 & -3.23139737E-10 \\ \hline
189 & -2.82855184E-10 & -2.84593071E-10 & -2.91721508E-10 \\ \hline
190 & -2.55280103E-10 & -2.56861588E-10 & -2.63362082E-10 \\ \hline
191 & -2.30390998E-10 & -2.31835107E-10 & -2.37763254E-10 \\ \hline
192 & -2.07933740E-10 & -2.09250714E-10 & -2.14655807E-10 \\ \hline
193 & -1.87667978E-10 & -1.88868629E-10 & -1.93797103E-10 \\ \hline
194 & -1.69382397E-10 & -1.70473691E-10 & -1.74967846E-10 \\ \hline
195 & -1.52878210E-10 & -1.53873220E-10 & -1.57970317E-10 \\ \hline
196 & -1.37981140E-10 & -1.38890732E-10 & -1.42626216E-10 \\ \hline
197 & -1.24539587E-10 & -1.25368757E-10 & -1.28774297E-10 \\ \hline
198 & -1.12408867E-10 & -1.13164270E-10 & -1.16269397E-10 \\ \hline
199 & -1.01460999E-10 & -1.02149539E-10 & -1.04980309E-10 \\ \hline
200 & -9.15805209E-11 & -9.22080745E-11 & -9.47886630E-11 \\ \hline
\end{longtable}
\newpage
\begin{longtable}[c]{|l|l|l|l|}
\caption{$G(\theta,\varphi)$ function of $\gamma/\gamma^*$ for $L_d=400,800,1000 \ \text{km}$.The values presented are computed using the split sum. The stopping critera used is when the term contribution of $G(l)$ to the split sum drops down below 1E-20.}
\label{my-label}\\
\hline
$10\gamma/\gamma^*$ & $G(\theta,\varphi)$, $L_d = 400 \ \text{km}$ & $G(\theta,\varphi)$, $L_d = 800 \ \text{km}$ & $G(\theta,\varphi)$, $L_d = 1000 \ \text{km}$ \\ \hline
\endfirsthead
\multicolumn{4}{c}
{{\bfseries Table \thetable\ continued from previous page}} \\
\hline
$10\gamma/\gamma^*$ & $G(\theta,\varphi)$, $L_d = 400 \ \text{km}$ & $G(\theta,\varphi)$, $L_d = 800 \ \text{km}$ & $G(\theta,\varphi)$, $L_d = 1000 \ \text{km}$ \\ \hline
\endhead
1 & -0.386384428 & -0.386698574 & -0.386935174 \\ \hline
2 & -0.279055119 & -0.279367000 & -0.279601902 \\ \hline
3 & -0.218536198 & -0.218844444 & -0.219076619 \\ \hline
4 & -0.177483588 & -0.177786931 & -0.178015471 \\ \hline
5 & -0.147224635 & -0.147522002 & -0.147746071 \\ \hline
6 & -0.123842940 & -0.124133401 & -0.124352314 \\ \hline
7 & -0.105218887 & -0.105501644 & -0.105714813 \\ \hline
8 & -9.00688842E-02 & -9.03433040E-02 & -9.05502513E-02 \\ \hline
9 & -7.75536671E-02 & -7.78192431E-02 & -7.80195743E-02 \\ \hline
10 & -6.70931637E-02 & -6.73494935E-02 & -6.75429255E-02 \\ \hline
11 & -5.82692884E-02 & -5.85160889E-02 & -5.87023981E-02 \\ \hline
12 & -5.07707670E-02 & -5.10078520E-02 & -5.11869080E-02 \\ \hline
13 & -4.43598181E-02 & -4.45870832E-02 & -4.47588041E-02 \\ \hline
14 & -3.88509482E-02 & -3.90683748E-02 & -3.92327383E-02 \\ \hline
15 & -3.40969786E-02 & -3.43046039E-02 & -3.44616435E-02 \\ \hline
16 & -2.99794525E-02 & -3.01773753E-02 & -3.03271618E-02 \\ \hline
17 & -2.64018904E-02 & -2.65902579E-02 & -2.67328992E-02 \\ \hline
18 & -2.32849084E-02 & -2.34639104E-02 & -2.35995445E-02 \\ \hline
19 & -2.05626264E-02 & -2.07324829E-02 & -2.08612736E-02 \\ \hline
20 & -1.81799550E-02 & -1.83409173E-02 & -1.84630472E-02 \\ \hline
21 & -1.60905365E-02 & -1.62428729E-02 & -1.63585451E-02 \\ \hline
22 & -1.42551288E-02 & -1.43991308E-02 & -1.45085575E-02 \\ \hline
23 & -1.26403589E-02 & -1.27763264E-02 & -1.28797302E-02 \\ \hline
24 & -1.12177096E-02 & -1.13459527E-02 & -1.14435637E-02 \\ \hline
25 & -9.96272545E-03 & -1.00835599E-02 & -1.01756109E-02 \\ \hline
26 & -8.85435659E-03 & -8.96809902E-03 & -9.05482657E-03 \\ \hline
27 & -7.87442829E-03 & -7.98139721E-03 & -8.06303602E-03 \\ \hline
28 & -7.00720632E-03 & -7.10771605E-03 & -7.18449941E-03 \\ \hline
29 & -6.23903004E-03 & -6.33339258E-03 & -6.40555192E-03 \\ \hline
30 & -5.55801764E-03 & -5.64653799E-03 & -5.71429962E-03 \\ \hline
31 & -4.95380722E-03 & -5.03678387E-03 & -5.10037038E-03 \\ \hline
32 & -4.41734865E-03 & -4.49507311E-03 & -4.55470011E-03 \\ \hline
33 & -3.94072337E-03 & -4.01347689E-03 & -4.06935439E-03 \\ \hline
34 & -3.51698906E-03 & -3.58504499E-03 & -3.63737578E-03 \\ \hline
35 & -3.14005348E-03 & -3.20367469E-03 & -3.25265480E-03 \\ \hline
36 & -2.80456175E-03 & -2.86400085E-03 & -2.90981843E-03 \\ \hline
37 & -2.50580069E-03 & -2.56130029E-03 & -2.60413601E-03 \\ \hline
38 & -2.23961752E-03 & -2.29141000E-03 & -2.33143684E-03 \\ \hline
39 & -2.00234959E-03 & -2.05065659E-03 & -2.08804011E-03 \\ \hline
40 & -1.79076288E-03 & -1.83579559E-03 & -1.87069364E-03 \\ \hline
41 & -1.60199881E-03 & -1.64395850E-03 & -1.67652150E-03 \\ \hline
42 & -1.43352943E-03 & -1.47260714E-03 & -1.50297780E-03 \\ \hline
43 & -1.28311617E-03 & -1.31949317E-03 & -1.34780735E-03 \\ \hline
44 & -1.14877592E-03 & -1.18262402E-03 & -1.20901014E-03 \\ \hline
45 & -1.02875044E-03 & -1.06023205E-03 & -1.08481187E-03 \\ \hline
46 & -9.21479717E-04 & -9.50748275E-04 & -9.73636983E-04 \\ \hline
47 & -8.25578638E-04 & -8.52779020E-04 & -8.74085352E-04 \\ \hline
48 & -7.39816925E-04 & -7.65085628E-04 & -7.84912205E-04 \\ \hline
49 & -6.63100916E-04 & -6.86566520E-04 & -7.05009967E-04 \\ \hline
50 & -5.94457961E-04 & -6.16241363E-04 & -6.33392832E-04 \\ \hline
51 & -5.33022627E-04 & -5.53237507E-04 & -5.69182623E-04 \\ \hline
52 & -4.78024449E-04 & -4.96777589E-04 & -5.11596852E-04 \\ \hline
53 & -4.28777217E-04 & -4.46168735E-04 & -4.59937815E-04 \\ \hline
54 & -3.84669518E-04 & -4.00793273E-04 & -4.13583126E-04 \\ \hline
55 & -3.45156266E-04 & -3.60100210E-04 & -3.71977425E-04 \\ \hline
56 & -3.09751369E-04 & -3.23597807E-04 & -3.34624754E-04 \\ \hline
57 & -2.78021209E-04 & -2.90847180E-04 & -3.01082298E-04 \\ \hline
58 & -2.49578821E-04 & -2.61456298E-04 & -2.70954217E-04 \\ \hline
59 & -2.24078656E-04 & -2.35074913E-04 & -2.43886810E-04 \\ \hline
60 & -2.01212213E-04 & -2.11390012E-04 & -2.19563706E-04 \\ \hline
61 & -1.80703821E-04 & -1.90121791E-04 & -1.97701956E-04 \\ \hline
62 & -1.62307202E-04 & -1.71019958E-04 & -1.78048329E-04 \\ \hline
63 & -1.45802158E-04 & -1.53860659E-04 & -1.60376163E-04 \\ \hline
64 & -1.30991830E-04 & -1.38443531E-04 & -1.44482517E-04 \\ \hline
65 & -1.17700161E-04 & -1.24589249E-04 & -1.30185595E-04 \\ \hline
66 & -1.05769635E-04 & -1.12137255E-04 & -1.17322532E-04 \\ \hline
67 & -9.50593094E-05 & -1.00943726E-04 & -1.05747371E-04 \\ \hline
68 & -8.54430400E-05 & -9.08798320E-05 & -9.53292547E-05 \\ \hline
69 & -7.68078753E-05 & -8.18301341E-05 & -8.59508509E-05 \\ \hline
70 & -6.90527013E-05 & -7.36911607E-05 & -7.75069275E-05 \\ \hline
71 & -6.20869396E-05 & -6.63701576E-05 & -6.99030570E-05 \\ \hline
72 & -5.58294523E-05 & -5.97839353E-05 & -6.30545183E-05 \\ \hline
73 & -5.02075345E-05 & -5.38578897E-05 & -5.68852702E-05 \\ \hline
74 & -4.51560409E-05 & -4.85250930E-05 & -5.13270279E-05 \\ \hline
75 & -4.06165636E-05 & -4.37254967E-05 & -4.63184842E-05 \\ \hline
76 & -3.65367523E-05 & -3.94051967E-05 & -4.18045638E-05 \\ \hline
77 & -3.28696624E-05 & -3.55158154E-05 & -3.77358010E-05 \\ \hline
78 & -2.95731879E-05 & -3.20139188E-05 & -3.40677325E-05 \\ \hline
79 & -2.66095667E-05 & -2.88605006E-05 & -3.07604096E-05 \\ \hline
80 & -2.39449200E-05 & -2.60205252E-05 & -2.77779127E-05 \\ \hline
81 & -2.15488508E-05 & -2.34625259E-05 & -2.50879511E-05 \\ \hline
82 & -1.93940832E-05 & -2.11582283E-05 & -2.26614848E-05 \\ \hline
83 & -1.74561319E-05 & -1.90822248E-05 & -2.04723947E-05 \\ \hline
84 & -1.57130198E-05 & -1.72116779E-05 & -1.84971850E-05 \\ \hline
85 & -1.41450209E-05 & -1.55260641E-05 & -1.67147118E-05 \\ \hline
86 & -1.27344174E-05 & -1.40069278E-05 & -1.51059503E-05 \\ \hline
87 & -1.14653030E-05 & -1.26376763E-05 & -1.36537765E-05 \\ \hline
88 & -1.03233888E-05 & -1.14033865E-05 & -1.23427744E-05 \\ \hline
89 & -9.29584166E-06 & -1.02906370E-05 & -1.11590643E-05 \\ \hline
90 & -8.37113475E-06 & -9.28735335E-06 & -1.00901480E-05 \\ \hline
91 & -7.53890936E-06 & -8.38267351E-06 & -9.12476662E-06 \\ \hline
92 & -6.78985862E-06 & -7.56682175E-06 & -8.25278130E-06 \\ \hline
93 & -6.11561973E-06 & -6.83100325E-06 & -7.46505657E-06 \\ \hline
94 & -5.50867571E-06 & -6.16729949E-06 & -6.75335968E-06 \\ \hline
95 & -4.96227085E-06 & -5.56858458E-06 & -6.11027281E-06 \\ \hline
96 & -4.47033199E-06 & -5.02844159E-06 & -5.52910888E-06 \\ \hline
97 & -4.02739897E-06 & -4.54109431E-06 & -5.00384112E-06 \\ \hline
98 & -3.62856304E-06 & -4.10134044E-06 & -4.52903441E-06 \\ \hline
99 & -3.26941017E-06 & -3.70449447E-06 & -4.09978884E-06 \\ \hline
100 & -2.94597044E-06 & -3.34633683E-06 & -3.71168539E-06 \\ \hline
101 & -2.65467452E-06 & -3.02306603E-06 & -3.36073822E-06 \\ \hline
102 & -2.39231076E-06 & -2.73125738E-06 & -3.04335163E-06 \\ \hline
103 & -2.15599152E-06 & -2.46782497E-06 & -2.75628190E-06 \\ \hline
104 & -1.94311815E-06 & -2.22998847E-06 & -2.49660184E-06 \\ \hline
105 & -1.75135312E-06 & -2.01524199E-06 & -2.26167026E-06 \\ \hline
106 & -1.57859324E-06 & -1.82132658E-06 & -2.04910316E-06 \\ \hline
107 & -1.42294596E-06 & -1.64620656E-06 & -1.85674855E-06 \\ \hline
108 & -1.28270835E-06 & -1.48804668E-06 & -1.68266388E-06 \\ \hline
109 & -1.15634771E-06 & -1.34519212E-06 & -1.52509517E-06 \\ \hline
110 & -1.04248466E-06 & -1.21615074E-06 & -1.38245855E-06 \\ \hline
111 & -9.39877509E-07 & -1.09957716E-06 & -1.25332360E-06 \\ \hline
112 & -8.47408728E-07 & -9.94258130E-07 & -1.13639840E-06 \\ \hline
113 & -7.64072240E-07 & -8.99099007E-07 & -1.03051593E-06 \\ \hline
114 & -6.88962245E-07 & -8.13112649E-07 & -9.34621937E-07 \\ \hline
115 & -6.21263268E-07 & -7.35408548E-07 & -8.47763715E-07 \\ \hline
116 & -5.60241062E-07 & -6.65183222E-07 & -7.69080543E-07 \\ \hline
117 & -5.05234368E-07 & -6.01711747E-07 & -6.97794405E-07 \\ \hline
118 & -4.55647807E-07 & -5.44339969E-07 & -6.33202205E-07 \\ \hline
119 & -4.10945205E-07 & -4.92477454E-07 & -5.74668263E-07 \\ \hline
120 & -3.70643619E-07 & -4.45591553E-07 & -5.21617949E-07 \\ \hline
121 & -3.34308112E-07 & -4.03201284E-07 & -4.73531713E-07 \\ \hline
122 & -3.01546834E-07 & -3.64872591E-07 & -4.29939774E-07 \\ \hline
123 & -2.72006929E-07 & -3.30213538E-07 & -3.90417199E-07 \\ \hline
124 & -2.45370416E-07 & -2.98870390E-07 & -3.54579697E-07 \\ \hline
125 & -2.21350916E-07 & -2.70523657E-07 & -3.22079671E-07 \\ \hline
126 & -1.99690348E-07 & -2.44884887E-07 & -2.92602635E-07 \\ \hline
127 & -1.80156249E-07 & -2.21693597E-07 & -2.65864031E-07 \\ \hline
128 & -1.62539081E-07 & -2.00714510E-07 & -2.41606443E-07 \\ \hline
129 & -1.46650109E-07 & -1.81735132E-07 & -2.19596799E-07 \\ \hline
130 & -1.32319187E-07 & -1.64563517E-07 & -1.99624267E-07 \\ \hline
131 & -1.19393036E-07 & -1.49026292E-07 & -1.81497953E-07 \\ \hline
132 & -1.07733470E-07 & -1.34966811E-07 & -1.65045066E-07 \\ \hline
133 & -9.72159810E-08 & -1.22243534E-07 & -1.50109145E-07 \\ \hline
134 & -8.77283313E-08 & -1.10728593E-07 & -1.36548522E-07 \\ \hline
135 & -7.91693537E-08 & -1.00306416E-07 & -1.24234930E-07 \\ \hline
136 & -7.14478503E-08 & -9.08725823E-08 & -1.13052174E-07 \\ \hline
137 & -6.44816183E-08 & -8.23327042E-08 & -1.02895029E-07 \\ \hline
138 & -5.81965480E-08 & -7.46014734E-08 & -9.36681559E-08 \\ \hline
139 & -5.25258272E-08 & -6.76017820E-08 & -8.52851940E-08 \\ \hline
140 & -4.74092232E-08 & -6.12639113E-08 & -7.76678917E-08 \\ \hline
141 & -4.27924221E-08 & -5.55248469E-08 & -7.07453296E-08 \\ \hline
142 & -3.86264531E-08 & -5.03276070E-08 & -6.44532392E-08 \\ \hline
143 & -3.48671669E-08 & -4.56206770E-08 & -5.87333808E-08 \\ \hline
144 & -3.14747375E-08 & -4.13574632E-08 & -5.35329470E-08 \\ \hline
145 & -2.84132664E-08 & -3.74958411E-08 & -4.88040683E-08 \\ \hline
146 & -2.56503689E-08 & -3.39976971E-08 & -4.45033486E-08 \\ \hline
147 & -2.31568436E-08 & -3.08285770E-08 & -4.05914271E-08 \\ \hline
148 & -2.09063540E-08 & -2.79573058E-08 & -3.70326099E-08 \\ \hline
149 & -1.88751450E-08 & -2.53556820E-08 & -3.37945316E-08 \\ \hline
150 & -1.70417902E-08 & -2.29981953E-08 & -3.08478256E-08 \\ \hline
151 & -1.53869646E-08 & -2.08617656E-08 & -2.81658483E-08 \\ \hline
152 & -1.38932359E-08 & -1.89255083E-08 & -2.57244359E-08 \\ \hline
153 & -1.25448789E-08 & -1.71705263E-08 & -2.35016540E-08 \\ \hline
154 & -1.13277068E-08 & -1.55797206E-08 & -2.14776001E-08 \\ \hline
155 & -1.02289244E-08 & -1.41376129E-08 & -1.96342125E-08 \\ \hline
156 & -9.23698362E-09 & -1.28301973E-08 & -1.79550899E-08 \\ \hline
157 & -8.34146974E-09 & -1.16447962E-08 & -1.64253517E-08 \\ \hline
158 & -7.53298401E-09 & -1.05699325E-08 & -1.50314801E-08 \\ \hline
159 & -6.80304790E-09 & -9.59521707E-09 & -1.37612064E-08 \\ \hline
160 & -6.14400975E-09 & -8.71124506E-09 & -1.26033859E-08 \\ \hline
161 & -5.54896618E-09 & -7.90949883E-09 & -1.15478977E-08 \\ \hline
162 & -5.01168707E-09 & -7.18226900E-09 & -1.05855493E-08 \\ \hline
163 & -4.52655158E-09 & -6.52257182E-09 & -9.70799086E-09 \\ \hline
164 & -4.08848599E-09 & -5.92408433E-09 & -8.90763641E-09 \\ \hline
165 & -3.69291242E-09 & -5.38107958E-09 & -8.17759194E-09 \\ \hline
166 & -3.33569927E-09 & -4.88837015E-09 & -7.51159668E-09 \\ \hline
167 & -3.01311731E-09 & -4.44125758E-09 & -6.90395785E-09 \\ \hline
168 & -2.72180123E-09 & -4.03548572E-09 & -6.34950403E-09 \\ \hline
169 & -2.45871346E-09 & -3.66719832E-09 & -5.84353499E-09 \\ \hline
170 & -2.22111218E-09 & -3.33290218E-09 & -5.38177947E-09 \\ \hline
171 & -2.00652250E-09 & -3.02943159E-09 & -4.96035346E-09 \\ \hline
172 & -1.81271054E-09 & -2.75391820E-09 & -4.57572913E-09 \\ \hline
173 & -1.63766012E-09 & -2.50376275E-09 & -4.22469837E-09 \\ \hline
174 & -1.47955059E-09 & -2.27660957E-09 & -3.90434662E-09 \\ \hline
175 & -1.33673883E-09 & -2.07032347E-09 & -3.61202490E-09 \\ \hline
176 & -1.20774146E-09 & -1.88296911E-09 & -3.34532624E-09 \\ \hline
177 & -1.09121912E-09 & -1.71279202E-09 & -3.10206394E-09 \\ \hline
178 & -9.85962534E-10 & -1.55820190E-09 & -2.88025204E-09 \\ \hline
179 & -8.90880258E-10 & -1.41775669E-09 & -2.67808775E-09 \\ \hline
180 & -8.04986577E-10 & -1.29014888E-09 & -2.49393417E-09 \\ \hline
181 & -7.27391591E-10 & -1.17419319E-09 & -2.32630715E-09 \\ \hline
182 & -6.57291721E-10 & -1.06881426E-09 & -2.17386131E-09 \\ \hline
183 & -5.93961436E-10 & -9.73037095E-10 & -2.03537787E-09 \\ \hline
184 & -5.36745648E-10 & -8.85977292E-10 & -1.90975413E-09 \\ \hline
185 & -4.85052665E-10 & -8.06832823E-10 & -1.79599446E-09 \\ \hline
186 & -4.38348247E-10 & -7.34875882E-10 & -1.69320002E-09 \\ \hline
187 & -3.96150029E-10 & -6.69446276E-10 & -1.60056179E-09 \\ \hline
188 & -3.58022251E-10 & -6.09945150E-10 & -1.51735369E-09 \\ \hline
189 & -3.23571503E-10 & -5.55828938E-10 & -1.44292556E-09 \\ \hline
190 & -2.92442404E-10 & -5.06604481E-10 & -1.37669776E-09 \\ \hline
191 & -2.64314043E-10 & -4.61824246E-10 & -1.31815625E-09 \\ \hline
192 & -2.38896541E-10 & -4.21081975E-10 & -1.26684785E-09 \\ \hline
193 & -2.15928109E-10 & -3.84008964E-10 & -1.22237653E-09 \\ \hline
194 & -1.95172295E-10 & -3.50270563E-10 & -1.18439991E-09 \\ \hline
195 & -1.76415521E-10 & -3.19562904E-10 & -1.15262611E-09 \\ \hline
196 & -1.59464858E-10 & -2.91610125E-10 & -1.12681164E-09 \\ \hline
197 & -1.44146042E-10 & -2.66161732E-10 & -1.10675868E-09 \\ \hline
198 & -1.30301672E-10 & -2.42990295E-10 & -1.09231402E-09 \\ \hline
199 & -1.17789528E-10 & -2.21889243E-10 & -1.08336740E-09 \\ \hline
200 & -1.06481164E-10 & -2.02670977E-10 & -1.07985043E-09 \\ \hline
\end{longtable}
\bibliographystyle{amsplain}
\bibliography{bibfile.bib}
\end{document} | {"config": "arxiv", "file": "1911.10944/paperARXIV.tex"} |
TITLE: Elementary geometrical interpretation of $2\langle a,x\rangle a -x$
QUESTION [1 upvotes]: Let $V$ be an euclidean vector space with $a \in V$ of length 1.
I showed that $$f: V \rightarrow V, x \rightarrow 2\langle a,x\rangle a -x $$ is an orthogonal transformation and hence either a rotation, reflection or a combination of both.
How excatly am I supposed to figure out, what kind of transformation is taking place?
REPLY [0 votes]: $\newcommand{\Brak}[1]{\left\langle #1\right\rangle}$A natural approach (whether you know the answer or not) is to try to compute
$$
f(x) = 2\Brak{a, x} a - x
$$
for as many vectors as you can. Here, the primary impediment is our (lack of) ability to evaluate $\Brak{a, x}$.
Since $\|a\| = 1$, we have $\Brak{a, a} = \|a\|^{2} = 1$. Consequently,
$$
f(a) = 2\Brak{a, a} a - a = a.
$$
Success!
If $\Brak{a, x} = 0$, i.e., $x$ is orthogonal to $a$, then
$$
f(x) = -x.
$$
Again, success!
But this answers the question: $a$ and the set of vectors orthogonal to $a$ span $V$, so we've found the eigenspace decomposition of $f$. | {"set_name": "stack_exchange", "score": 1, "question_id": 2316060} |
\begin{document}
\title{Chaos representations for Marked Point Processes.}
\author{Samuel N. Cohen\\ samuel.cohen@maths.ox.ac.uk\\University of Oxford}
\date{\today}
\maketitle
\begin{abstract}
We show that for a large class of marked point processes there exists a random measure $m$ with the predictable representation property such that iterated integrals with respect to $m$ span the space of square integrable random variables.
\end{abstract}
\section{Introduction}
A fundamental result in the stochastic analysis on Wiener spaces is the Wiener-It\=o chaos representation theorem \cite{Wiener1938}. This theorem allows the representation of any square integrable random variable as the sum of iterated stochastic integrals with respect to the underlying Wiener process, and provides an approach to the Malliavin calculus of variations. Such a representation is often termed a chaos representation, and is closely linked to representations in terms of Hermite polynomials.
For processes with jumps, it is also possible to construct a theory of chaos expansions. This has been studied in the context of Markov chains in Kroeker \cite{Kroeker1980} (see also Biane \cite{Biane1990}) and for the Binomial process in Privault and Schoutens \cite{Privault2002}. In these works, the approach is based on the principle of finding an analogue to the Hermite polynomials appropriate to these spaces. Emery \cite{Emery1989} studies chaos representations in terms of iterated integrals, assuming that the underlying martingale satisfies a certain structure condition, related to the Az\'ema martingale. Many authors have since expanded on these ideas.
In this paper we give a general approach to chaos decompositions for an arbitrary marked point process, where we simply assume that the compensating measure for the underlying process is absolutely continuous (in both time and space) with respect to some (locally finite in time) deterministic measure. Instead of searching for a polynomial chaos interpretation, we focus on the representation in terms of iterated stochastic integrals with respect to fundamental martingales. For this, we use the fundamental martingales constructed in Elliott \cite{Elliott1976} and Davis \cite{Davis1976}, which make no assumptions about independence of the increments. In such a setting, we shall show that the iterated stochastic integrals span the entirety of $L^2(\F)$.
This result complements the construction of the Malliavin calculus for Marked Point Processes as in Decreusefond \cite[Section 4]{Decreusefond1998}. In \cite{Decreusefond1998} it is simply assumed that the space under consideration admits a chaos decomposition. The contribution of this paper is to show general conditions under which this is the case.
\section{Martingales for Marked Point Processes}
We begin by constructing an explicit martingale representation result. We do this mainly for copleteness, in Section \ref{sec:Chaos} we shall simply assume that some martingale representation is given, which may or may not be the one constructed here.
The setting for this analysis is taken from Elliott \cite{Elliott1976} (see also Davis \cite{Davis1976}, and Elliott \cite{Elliott1977}), we shall state relevant results without proof or further reference. For a simpler and more gentle introduction to this style of analysis for marked point processes, see \cite[Ch. 17]{Elliott1982}.
\subsection{The jump setting}\label{subsec:setting}
Let $(E, \E)$ be a Blackwell space. Consider a right-constant jump process $X$ taking values in $E$, which is initially in the fixed position $X_0 = \xi_0\in E$.
At a random time $T_1$, $X$ jumps to a random location $\xi_1\neq \xi_0$, at which it stays until a random time $T_2$, when it jumps to a random location $\xi_2\neq \xi_1$, etc... As $X$ is right-constant, we know that for each path, the jumps $T_i$ are well ordered, and there are at most countably many jumps. For simplicity, in this paper we shall assume that there are at most finitely many jumps on any compact, that is, $\lim_{n\to\infty} T_n = \infty$ for (almost) all paths.
We then have a measurable space $(\Omega, \F)$, where $\F = \sigma\{X_s, s<\infty\}$, and $\Omega \subset ([0,\infty]\times E)^\bN$ is a list of all the jump times and locations of $X$, with the property that $X$ can only jump once at each time, and must jump to a new location. We suppose a probability measure $\bP$ is given on this space. We denote by $\F_t$ the $\bP$-completed $\sigma$-algebra generated by $X$ up to time $t$, that is $\F_t = \sigma\{X_s; s\leq t\}\vee\{\text{null sets}\}$. This space will be kept fixed throughout the paper.
\subsection{Fundamental martingales}
Suppose $T_\alpha$, $\alpha\in\bN$ is a jump time. The distribution of the pair $(T_{\alpha}, \xi_{\alpha})$ given $\F_{T_{\alpha-1}}$ is described by a random measure (that is, a regular family of conditional probability distributions) $\mu^{\alpha}(\omega; \cdot)$ on $[0,\infty]\times E$. Properties of $\mu^\alpha$ are given in \cite{Elliott1977}. Define
\[F^\alpha_t(\omega; A) = \mu^\alpha(\omega; ]t,\infty]\times A)\]
so that, omitting $\omega$ for notational convenience, $F_t^\alpha(A)$ is the conditional probability that $T_{\alpha} >t$ and $\xi_{\alpha} \in A$ given $\F_{T_{\alpha-1}}$. For convenience $F^\alpha_t:=F^\alpha_t(E)$ and we write
\[\lambda^\alpha(t, A)= \left.\frac{dF^\alpha_\cdot(A)}{dF^\alpha_\cdot(E)}\right|_t,\]
the rate at which the $\alpha$th jump is into $A$ at time $t$. We can then define the stochastic processes
\[\begin{split}
p^\alpha(t, A) &:= I_{t\geq T_{\alpha}} I_{\xi_{\alpha} \in A}\\
\tilde p^\alpha(t, A) &:= -\int_{]0, t\wedge T_{\alpha}]} (F_{u-}^{\alpha})^{-1} dF_u^{\alpha} (A) = -\int_{]0,t\wedge T_\alpha]}\lambda^\alpha(s,A) (F^\alpha_{s-})^{-1}dF_s^\alpha\\
q^\alpha(t, A) &:= p^\alpha(t, A) - \tilde p^\alpha(t, A)
\end{split}
\]
so that $q^\alpha(t, A)$ is an $\F_t$-martingale with predictable quadratic variation
\[\langle q^\alpha(t, A)\rangle = \tilde p^\alpha(t, A) - \sum_{0< u\leq t\wedge T_{\alpha}} \frac{\lambda^\alpha(u, A)^2 (\Delta F^{\alpha}_u)^2}{(F^{\alpha}_{u-})^2}.\]
We shall see that these martingales provide a basis from which we can obtain a martingale representation theorem in these spaces. Note that $\tilde p^\alpha$ is simply the compensator of the finite variation process $p^\alpha$, and $q^\alpha$ is then the martingale part of $p^\alpha$. Note also that if $\tilde p^\alpha$ is continuous in $t$, then $\Lambda^\alpha(t, A)\Delta F^\alpha \equiv 0$, and $\langle q^\alpha(t, A)\rangle = \tilde p^\alpha(t,A)$.
Write $G^\alpha$ for the set of measurable functions $\{g^\alpha:\Omega\times[0,\infty]\times E\to \bR\}$ such that for each $(t,x)\in [0,T]\times E$, $g^\alpha$ is $\F_{T_{\alpha-1}}$-measurable. As for fixed $\alpha, t$ and $\omega$ we know $p^\alpha(t, A)$ and $\tilde p^\alpha(t,A)$ are both countably additive in $A$, for suitable $g^\alpha \in G^\alpha$ we have
\[\begin{split}
\int_{\Omega} g^\alpha(s, x) p^\alpha(ds, dx) &= g^\alpha(T_{\alpha}, x_{\alpha})\\
\int_{\Omega} g^\alpha(s, x) \tilde p^\alpha(ds, dx) &= - \int_{]0, T_{\alpha+1}]} \int_{E} g^\alpha(s,x) \lambda^\alpha(\omega; s, dx) \frac{ dF^\alpha_s}{F^\alpha_{s-}}
\end{split}
\]
\begin{lemma}\label{lem:Msumofdeltas}
For any square-integrable martingale $M$, we define
\[\Delta M^\alpha := M_{T_\alpha} - M_{T_{\alpha-1}}.\]
Then for every $\alpha\in \bN$, we have
\[M_{T_\alpha} = \sum_{\beta\leq \alpha} \Delta M^\beta.\]
\end{lemma}
This leads to a precursor to the martingale representation result in this context.
\begin{theorem}\label{thm:martdiffrepMPP}
Suppose $M$ is a square-integrable martingale; write
\[N^\alpha_t = M_{T_{\alpha}\wedge t} - M_{T_{\alpha-1} \wedge t}\]
Then for each $\alpha\in\bN$, there exists a function $g^\alpha\in G^\alpha$ such that
\[N^\alpha_t = \int_{]0,t]\times E} g^\alpha(s, x) q^\alpha(ds, dx) \quad a.s.\]
We shall say that $g^\alpha$ represents $N^\alpha$. Furthermore, $g^\alpha(T_\alpha, x_\alpha) = \Delta N^\alpha_{T_\alpha}$, up to the addition of a $\F_{T_{\alpha-1}}$-measurable random variable.
\end{theorem}
\section{Martingale representation theorem}
We now depart from the presentation of the martingale representation theorem in \cite{Elliott1977}, to present a slight variant which more naturally leads to the chaos representation, and is of a more familiar form. Our presentation depends on the following lemma and associated definition.
\begin{lemma}
For each $\omega$, any $t$, any $A\in\E$, $p^\alpha(\omega;t, A)$ and $\tilde p^\alpha(\omega;t, A)$ vary in $t$ only on the set $t\in]T_{\alpha-1}, T_{\alpha}]$. In particular, the measures $\{dp^\alpha\}_{\alpha\in\bJ}$ on $[0,\infty]\times E$ have disjoint supports, and similarly for $\{d\tilde p^\alpha\}_{\alpha\in\bJ}$.
Therefore, we can define the disjoint sum
\[p(\omega; dt, dx) = \sum_{\alpha\in\bN} p^\alpha(\omega; dt, dx),\]
and similarly for $\tilde p$ and hence for $q=p-\tilde p$.
\end{lemma}
\begin{corollary}\label{cor:martdiffrepMPPzeroed}
Let $g^\alpha$ be as in Theorem \ref{thm:martdiffrepMPP}. Let $\tilde g^\alpha$ be defined as
\[\tilde g^\alpha(t,x) = I_{\{t\in]T_{\alpha-1}, T_\alpha]\}} g^\alpha(t,x)\]
Then $\tilde g^\alpha$ also represents $N^\alpha$
\end{corollary}
\begin{proof}
This follows as we have only modified $g^\alpha$ off the support of $q^\alpha$.
\end{proof}
We can now state our first martingale representation theorem.
\begin{theorem}
Let $M$ be a square-integrable $\{\F_t\}$-martingale. Then there exists an $\{\F_t\}$-predictable process $g(t, x)$ such that
\[M_t = M_0 + \int_{]0,t]\times E} g(t,x) q(dt, dx).\]
\end{theorem}
\begin{proof}
Let $\tilde g^\alpha$ be as in Corollary \ref{cor:martdiffrepMPPzeroed}. By Lemma \ref{lem:Msumofdeltas} and the fact $\tilde g^\alpha$ represents $N^\alpha$, we have
\[M_t-M_0 = \sum_{\alpha} N^\alpha_t = \sum_{\alpha}\int_{]0,t]\times E} \tilde g^\alpha(s,x) q(ds, dx).\]
We then define $g(s, x) := \sum_{\alpha\leq\gamma} \tilde g^\alpha(s, x)$, this again being a disjoint sum. As there are almost surely finitely many jumps up to time $t$, and $\tilde g^\alpha$ is zero for $\alpha$ greater than the index of the next jump, for almost all $\omega$ this is a finite sum, and so we can exchange the order of integration and summation.
\end{proof}
This martingale representation theorem has a simple interpretation, as it is based purely on the compensated indicator functions of the state of the underlying process $X$. However, it has a significant flaw for our purposes, as iterated integrals are not necessarily orthogonal. For this reason, we need to rescale $q$, for which we need the following assumption. This assumption poses the only restriction on the processes we shall consider.
\begin{assumption}\label{Ass1}
For all $\alpha$, there exists a \emph{deterministic} measure $\zeta^\alpha$ on $\bR^+\times E$ such that $\tilde p^\alpha(\omega, \cdot, \cdot)$ is almost surely equivalent to $\zeta^\alpha$, and such that $\zeta^\alpha([0,t]\times E)<\infty$ for all $t<\infty$. For simplicity, we shall assume that $\zeta^\alpha$ is continuous with respect to $t$.
\end{assumption}
\begin{lemma}
There exists a predictable function $\psi:\Omega \times \bR^+\times E \to ]0,1]$ such that for all measurable functions $f$, for all $\alpha\in \bJ$,
\[\begin{split}
&\bE\left[\int_{]T_\alpha,T_{\alpha+1}]\times E} f(\omega, s,x) \psi(\omega, s,x) \tilde p(\omega, ds, dx)\right]\\
&= \int_{\bR^+\times E}\bE[I_{s\in]T_\alpha,T_{\alpha+1}]}f(\omega, s, x)] \zeta^\alpha(du,dx)
\end{split}
\]
\end{lemma}
\begin{proof}
Simply take
\[\psi(\omega,t,x)=\sum_\alpha I_{t\in]T_{\alpha-1}, T_{\alpha}]}\left(\left.\frac{d\zeta^\alpha}{d\tilde p^\alpha(\omega, \cdot, \cdot)}\right|_{(t,x)}\right).\]
\end{proof}
\begin{definition}
We shall denote by $q_\psi$ the signed measure $q$ rescaled by $\psi$, that is,
\[q_\psi(t, A) := \int_{]0,t]\times A} \psi(\omega, s,x)q(\omega, ds,dx).\]
For simplicity, we may write $\psi_{t,x}$ for $\psi(\omega, t,x)$.
\end{definition}
\begin{lemma}
If $f$ is $q$-integrable, then $f \cdot \psi^{-1}$ is $q_\psi$-integrable and the two integrals agree. If $f$ is predictable, then so is $f \cdot \psi^{-1}$.
\end{lemma}
\begin{proof}
This is clear as $\psi$ is predictable and for each $\omega$ equals the Radon-Nikodym derivative $dq_\psi/dq$.
\end{proof}
Using the previous lemma, we immediately see that our martingale representation theorem can be equivalently stated in terms of $q_\psi$, rather than $q$. This will be preferable, as $q_\psi$ has significantly better orthogonality properties than $q$, and so we shall hereafter focus on $q_\psi$.
We now seek to understand the space of integrands which yield square integrable martingales, when integrated with respect to $q_\psi$. As our martingale representation is not given by an orthonormal set of martingales, but rather by a random measure $q_\psi$ with $q_\psi(t,A)$ and $q_\psi(t, B)$ correlated, we need to be careful in our definition of the appropriate space of integrands.
\begin{lemma}
For all $f, g$ such that $\int_{]0,t]\times E} f(t,x)q_\psi(dt, dx)$ is square integrable (and similarly for $g$), we have the isometry
\[\begin{split}
(f,g)_{q_\psi}&:=\bE\left[\left(\int_{\bR^+\times E} f(t,x) q_\psi(dt, dx)\right)\left(\int_{\bR^+\times E} g(t,x) q_\psi(dt, dx)\right)\right] \\
&= \bE\left[\int_{\bR^+\times E}f(t,x)g(t, x) \psi_{t,x}^2 \tilde p(\omega, dt, dx)\right]\\
&=\sum_\alpha \int_{\bR^+\times E}\bE\left[I_{t\in]T_\alpha,T_{\alpha+1}]}f(t,x)g(t, x)\right] \zeta^\alpha(dt, dx)
\end{split}
\]
We shall write $\|f\|_{q_\psi}^2= (f,f)_{q_\psi}$.
\end{lemma}
\begin{proof}
From \cite{Elliott1977}, we know that the quadratic variation of $q$ is given by $\tilde p$, as we have assumed that $\zeta^\alpha$, and hence $\tilde p^\alpha$, is continuous in $t$. As $q_\psi$ is simply a rescaled version of $q$, this quickly establishes the first isometry. The second then follows by breaking up the integral into the intervals $]T_\alpha, T_{\alpha+1}]$, and extracting the sum.
\end{proof}
From this lemma, we can see that our use of the martingale random measure $q_\psi$ is a slight generalisation of constructing a martingale representation using `normal' martingales, that is, martingales with predictable quadratic variation given by Lebesgue measure (see, for example, Emery \cite{Emery1989}). Here we replace Lebesgue measure with an arbitrary deterministic measure $\zeta^\alpha$, which can vary in $\alpha$, and we retain the presence of the jump space $E$.
\section{Chaos representation property}\label{sec:Chaos}
From this point onwards, we will not restrict ourselves to this particular choice of martingale representation. In fact, there may be cases where an alternative martingale representation is available and more convenient. We shall simply make the following assumption.
\begin{assumption}
We are in the setting described in Section \ref{subsec:setting}, and there exists a random measure $m$ such that
\begin{itemize}
\item $\int_{]0,t]\times E} f(t,x) m(dt, dx)$ is a martingale for all predictable, sufficiently bounded functions $f$,
\item every square integrable martingale has a representation $\int_{]0,t]\times E} f(t,x) m(dt, dx)$ for some predictable function $f$,
\item for all sufficiently integrable predictable $f$ and $g$,
\[\begin{split}
(f,g)_m&:=\bE\left[\left(\int_{\bR^+\times E} f(t,x) m(dt, dx)\right)\left(\int_{\bR^+\times E} g(t,x) m(dt, dx)\right)\right] \\
&=\sum_\alpha \int_{\bR^+\times E}\bE\left[I_{t\in]T_\alpha,T_{\alpha+1}]}f(t,x)g(t, x)\right] \zeta^\alpha(dt, dx).
\end{split}
\]
for some family of deterministic measures $\zeta^\alpha$. As before $\|f\|_m^2 := (f,f)_m$.
\end{itemize}
\end{assumption}
Under Assumption \ref{Ass1}, $m=q_\psi$ satisfies these requirements. However, it may be convenient to take an alternative representation, particularly in cases when Assumption \ref{Ass1} does not hold. A simple example of this is when $E$ posesses a group structure (e.g. when $E$ is a vector space). If $E$ is discrete, for example, when we consider a countable-state Markov chain, then the representation based on $p$ will often not satisfy Assumption \ref{Ass1}, as the previous state $\xi_{\alpha-1}$ is a null set of the measure $p^\alpha$, however is stochastic, which often contradicts the equivalence with the deterministic measure $\zeta^\alpha$. On the other hand, we could use a representation based on the fundamental processes $\pi^\alpha(t, A) = I_{t\geq T_\alpha} I_{\xi_\alpha-\xi_{\alpha-1} \in A}$ (in the place of $p^\alpha$), that is, we use the indicator functions of the jumps themselves, rather than the indicator of the location after the jump. This representation (appropriately rescaled) will satisfy our assumption as soon as the set of possible values (occuring with rate $>0$) for the $\alpha$th jump is deterministic.
Using the martingale $m$, we now prove the existence of the Chaos representation of a random variable.
\begin{definition}
For two (stopping) times $T, T'\leq \infty$, we shall write
\[\F_{T\curlywedge T'} = \F_T\cap \F_{T'-}\]
and
\[\int_0^{T\curlywedge T'} (\cdot) m(dt, dx) := \int_{(]0, T]\cap ]0,T'[)\times E} (\cdot) m(dt, dx).\]
For simplicity, in place of $m(dt, dx)$ we may write $dm$, similarly $dm_1$ for $m(dt_1, dx_1)$, $dm_2$ for $m(dt_2, dx_2)$, etc. and also $d\zeta^\alpha$ for $\zeta^\alpha(dt,dx)$, $d\zeta^\alpha_1$ for $\zeta^\alpha(dt_1,dx_1)$, etc.
\end{definition}
Note that if $\tau=\infty$, then $L^2(\F_{T\curlywedge\tau})=L^2(\F_{T})$, as we have assumed $\F_{\infty-} = \F_\infty$.
\begin{definition}
Let $k<\infty$, $\tau\leq\infty$. Let $\{g_i\}$ be a family of measurable functions $g_i:\Omega\times(\bR^+\times E)^i\to \bR$. Then we define the $k$-fold iterated integral operator via the recursion
\[\I^{k}_\tau(\{g_i\}) = g_0 + \int_0^{T_{k}\curlywedge \tau} \I^{k-1}_t(\{g_{i-1}(t,x, \cdots)\}_{i=1}^{k}) dm,\]
with initial value $\I^0_\tau(g_0)=g_0$. For simplicity, we write $\I^k(\{g_i\}):=\I^k_\infty(\{g_i\})$.
\end{definition}
With this definition, the first few terms of our integral operator are
\[\begin{split}
\I^\emptyset_\tau(\{g_0\}) &= g_0\\
\I^{1}_\tau(\{g_0, g_1\}) &= g_0+\int_0^{T_1\curlywedge \tau} g_1(t,x)m(dt, dx)\\
\I^{2}_\tau(\{g_i\}_{i=0}^2)& =g_0+\int_0^{T_2\curlywedge \tau}\left(g_1(t_1,x_1)+ \int_0^{T_1\curlywedge t_1} g_2(t_1,x_1,t_2,x_2) dm_2\right) dm_1\\
\I^{3}_\tau(\{g_i\}_{i=0}^3)& =g_0+\int_0^{T_3\curlywedge \tau}\left(g_1+ \int_0^{T_2\curlywedge t_1} \left(g_2 +\int_0^{T_1\curlywedge t_2} g_3\, dm_3\right)dm_2\right) dm_1\\
\end{split}
\]
The important point to notice is that the `internal' integrals are taken only up to the preceding jump times in our sequence.
We can now state a precursor to the chaos representation theorem, using the iterated integrals $\I$.
\begin{theorem}\label{thm:chaosfiniteprecursor}
Let $Y\in L^2(\F_{T_k\curlywedge \tau})$ for $k<\infty$, and deterministic $\tau\leq\infty$. Then there exists a sequence of deterministic functions $\{g_i\}_{i=1}^k$ such that
\[Y=\I^{k}_\tau(\{g_i\}).\]
\end{theorem}
\begin{proof}
First assume $\tau<\infty$. We shall use induction, iterating in $\alpha\leq k$ over the cases where $Y\in L^2(\F_{T_\alpha\curlywedge \tau})$. For the initial case, suppose $Y\in L^2(\F_0)$. Then $Y$ is a constant, so $Y= \I^0(g^0) = g^0$ for some constant $g^0$.
Suppose $Y\in L^2(\F_{T_\alpha\curlywedge \tau})$ and that the result holds for all $Y'\in L^2(\F_{T_{\alpha-1}\curlywedge t})$ for $t\leq \tau$. By the martingale representation theorem, $Y$ has a representation of the form
\[Y = E[Y] + \int_0^{T_\alpha\curlywedge \tau} \tilde g(\omega, t, x) dm\]
for some predictable function $\tilde g$ with $\|\tilde g\|_m<\infty$. As $\tilde g$ is predictable, for every $(t,x)$ the random variable $\tilde g(\cdot,t,x)$ is $(\F_{T_{\alpha-1}\curlywedge t})$-measurable. As we have
\[\begin{split}
\infty&> \|\tilde g\|_m^2= \bE\left[\left(\int_0^{T_\alpha\curlywedge \tau} \tilde g dm\right)^2\right]\\
&=\sum_{\beta\leq \alpha}\int_0^{\tau-} \bE[I_{t\in ]T_{\beta-1}, T_\beta]} \tilde g^2(t,x)] d\zeta^\beta
\end{split}
\]
we know $\bE[I_{t\in ]T_{\beta-1}, T_\beta]} \tilde g(t,x)^2]<\infty$ $\zeta^\beta$-a.s. for all $\beta\leq \alpha$. Taking the sum over $\beta\leq \alpha$, we see $\bE[\tilde g(t,x)^2]<\infty$. Therefore $\tilde g(t,x) \in L^2(\F_{T_{\alpha-1}\curlywedge t})$. As we have supposed that the result holds on $L^2(\F_{T_{\alpha-1}\curlywedge t})$, we can find deterministic $\{g^{(t,x)}_i(\cdots)\}_{i=0}^{\alpha-1}$ such that
\[\tilde g_n(\omega,t,x)= \I^{\alpha-1}_t(\{g_i^{(t,x)}(\cdots)\})\]
from which we define
\[g_i(t,x, \cdots) = g^{(t,x)}_{i-1}(\cdots) \text{ for }i>1; \quad g_0 = E[Y].\]
This yields the representation of $Y$,
\[Y = \I^{\alpha}_\tau(\{g_i\}) = E[Y] + \int_0^{T_\alpha\curlywedge \tau} \I^{\alpha-1}_t(\{g_i^{(t,x)}\}) dm.\]
By induction, the result is proven for $Y\in L^2(\F_{T_{k}\curlywedge \tau})$ for all $k<\infty$. We now seek to let $\tau\to\infty$. This is easily done by the convergence of square-integrable martingales. For $Y\in L^2(\F_{T_k})$, let $Y_\tau :=\bE[Y|\F_{\tau}]$, so that $Y_{\tau-}\in L^2(\F_{T_k\curlywedge \tau})$. Therefore $Y_{\tau-} = \I^k_\tau(\{g_i^\tau\})$ for some collection of functions $\{g_i^\tau\}$. It is easy to verify that these functions are consistent, that is, $g_i^\tau = g_i^{\tau'}$ on $[0, \tau\wedge \tau'[\times E$, and hence $\{g_i^\tau\}$ can be taken to be independent of $\tau$. Therefore, by martingale convergence,
\[Y\leftarrow \bE[Y|\F_{\tau-}]=Y_{\tau-} = \I^k_\tau(\{g_i\}) \to \I^k_\infty(\{g_i\})\quad a.s.\]
from which we see $Y=\I^k_\infty(\{g_i\})$.
\end{proof}
\begin{remark}
Intuitively, this representation in terms of $\I^k$ has a simple interpretation. From the martingale representation theorem, we know we can write any $\F_{T_k\curlywedge \tau}=\F_{T_k}\cap\F_{\tau-}$-measurable random variable in terms of the stochastic integral on $[0,T_k]\cap[0,\tau[$ of a predictable process $\tilde g_t$. However, up to time $T_k$, a predictable process $\tilde g_t$ is $\F_{T_{k-1}\curlywedge t}$-measurable for each $t$, and so by induction can itself be written as an integral on $[0,T_{k-1}]\cap [0,t[$. Hence any $T_k$-measurable random variable can be written as the iterated stochastic integral, where each integral is at most up to an earlier jump time.
\end{remark}
We can now construct the chaos representation.
\begin{definition}
For $T\leq\infty$ a stopping time, we shall write
\[\bS^n_T:= \{(s_1, s_2,...,s_n): 0\leq s_n<s_{n-1}<...<s_1\leq T\}\subset [0,T]^{n}\]
For $T$ a stopping time, we define the $n$-fold iterated integral
\[\begin{split}
J^n_T(g) &= \int_{\bS^n_T} g(\{(s_k, x_k)\}) \bigotimes^n_{k=1} m(ds_k,dx_k)\\
&=\int_0^T\int_0^{s_1-}\int_0^{s_{2}-}...\int_0^{s_n-} g(...) dm_{n-1}\,...\, dm_2 \, dm_1.
\end{split}
\]
For convenience, $J^0_T(g) := g$ for all constants $g$.
\end{definition}
\begin{definition}
For $T$ a stopping time, let \[H_T^m:=\overline{\mathrm{span}}\{J^n_T(g): \bE[(J^n_T(g))^2]<\infty, n\leq m\},\] the $L^2(\bP)$-closure of the span of the square integrable iterated stochastic integrals of order at most $m$ up to $T$. This is a Hilbert space, with the same inner product as $L^2(\bP)$.
\end{definition}
We can now prove the the Chaos representation theorem for random variables known after finitely many jumps.
\begin{theorem}\label{thm:chaosfinitejump}
For any $k<\infty$, we have $H_{T_k}^k = L^2(\F_{T_k})$.
\end{theorem}
\begin{proof}
Clearly $H_{T_k}^k \subseteq L^2(\F_{T_k})$, and $H_{T_k}$ is a Hilbert subspace. Therefore, either $H_{T_k}^k=L^2(\F_{T_k})$ or there exists a nonzero random variable $Y\in L^2(\F_{T_k})$ which is orthogonal to every element of $H_{T_k}^k$.
By Theorem \ref{thm:chaosfiniteprecursor}, the space spanned by the iterated integrals $\I^k$ is $L^2(\F_{T_k})$. Hence $Y$ has a representation of the form $Y=\I^k(\{g_i\})$, for some functions $\{g_i\}$. We seek to show that $g_i=0$ for all $i$ on the relevant range of integration. We shall do this using induction, however due to notational complexity, we shall simply write out the first three steps, the rest follow in the same manner.
\textbf{For $g_0$,} note that we must have
\[\begin{split}
0&= \bE[Y J_{T_k}^0(g_0)] \\
&= \bE\left[\left(g_0+ \int_0^{T_k} \I^{k-1}_t (\{g_{i-1}(t,x,\cdots)\})dm\right) (g_0)\right]\\
&=g_0^2 + g_0 \bE\left[\int_0^{T_k} \I^{k-1}_t (\{g_{i-1}(t,x,\cdots)\})dm\right]\\
&=g_0^2
\end{split}
\]
and we see $g_0\equiv 0$.
\textbf{For $g_1$,} note that as $g_0\equiv 0$, we know
\[Y = \int_0^{T_k}\I^{k-1}_{t_1}(\{g_{i-1}(\cdots)\}) dm_1 = \int_0^{T_k}(g_1(t_1,x_1) + \xi(t_1,x_1)) dm_1 .\]
where $\xi(t_1,x_1) = \int_0^{T_{k-1}\curlywedge t_1} \I^{k-2}_{t_2}(\{g_{i-2}(t_1,x_1,\cdots)\})dm_2$. Note that $\bE[\xi(t,x)] \equiv 0$. Then
\[\begin{split}
0&= \bE[Y J^1_{T_k}(g_1)]\\
&=\bE\left[\left(\int_0^{T_k}(g_1(t_1,x_1) + \xi(t_1,x_1)) dm_1 \right)\left(\int_0^{T_1} g_1 dm_1\right)\right]\\
&=\|I_{t_1\leq T_k}g_1(t_1,x_1)\|_m^2 + \sum_\alpha \int_0^\tau \bE\left[\xi(t_1,x_1))\right] g_1(t_1, x_1) d\zeta^\alpha\\
&=\|I_{t_1\leq T_k}g_1(t_1,x_1)\|_m^2
\end{split}
\]
and so $g_1(t_1,x_1)\equiv 0$ on $[0,T_k]$ (up to a set of $\zeta^\alpha$-measure zero for all relevant $\alpha\leq k$).
\textbf{For $g_2$,} note that as $g_0=g_1=0$, we know
\[\begin{split}
Y&= \int_0^{T_k}\int_0^{T_{k-1}\curlywedge t_1}\I^{k-2}_{t_2}(\{g_{i-2}(\cdots)\}) dm_2 dm_1 \\
&= \int_0^{T_k}\int_0^{T_{k-1}\curlywedge t_1}(g_2(t_1,x_1,t_2, x_2) + \xi(t_1,x_1,t_2, x_2)) dm_2 dm_1.
\end{split}\]
where $\bE[\xi(...)]\equiv 0$. Hence, expanding in the same way as above
\[\begin{split}
0&= \bE[Y J^2_{T_k}(g_2)]\\
&=\sum_{\alpha\leq k}\int_0^\infty \bE\left[I_{t_1\in ]T_{\alpha-1}, T_\alpha]} \left(\int_0^{T_{k-1}\curlywedge t_1} g_2(t_1,x_1 t_2,x_2)dm_2\right)^2\right]d\zeta^\alpha_1\\
&=\sum_{\alpha\leq k}\sum_{\beta\leq \alpha}\int_0^\infty\int_0^{t_1} \bE[I_{t_1\in ]T_{\alpha-1}, T_\alpha]}I_{t_2\in ]T_{\beta-1}, T_\beta]}] (g_2(t_1, x_1, t_2,x_2))^2 d\zeta^\beta_2 d\zeta^\alpha_1\\
\end{split}
\]
and so $g_2\equiv 0$ up to a set of measure zero, on its relevant domain.
\textbf{Continuing the induction}, we see that $g_i \equiv 0$ for all $i\leq k$. Therefore $Y\equiv 0$, and there is no element of $L^2(\F_{T_k})$ orthogonal to all of $H_{T_k}$. Therefore the spaces coincide.
\end{proof}
Finally, we can expand our Chaos representation theorem to all of $L^2(\F)$.
\begin{theorem}
Any square integrable random variable can be arbitrarily well approximated in $L^2$ by a sum of iterated integrals, or equivalently,
\[L^2(\F) = \overline{\left\{\cup_k H^k_{T_k}\right\}}.\]
\end{theorem}
\begin{proof}
We shall again use the convergence of square integrable martingales. For any $Y\in L^2(\F)$, let $Y_k = \bE[Y|\F_{T_k}]$. This is a martingale in the discrete filtration $\G_k=\F_{T_k}$, and so $Y_k \to Y$ in $L^2$. Hence for any $\epsilon>0$ there exists a $k$ such that $\bE[(Y-Y_k)^2]<\epsilon/4$. By Theorem \ref{thm:chaosfinitejump} there is also a sequence $\{g_n\}_{n=1}^k$ such that $\bE\left[\left(Y_k - \sum_{n=0}^k J^n_{T_k}(g_n)\right)^2\right]<\epsilon/4$. By the triangle inequality,
\[\bE\left[\left(Y- \sum_{i=1}^k J^n_{T_k}(g_n)\right)^2\right]<\epsilon.\]
\end{proof}
\section{Conclusions}
We have shown general conditions such that an arbitrary marked point process generates a martingale random measure $m$ for which a martingale representation theorem holds, and such that $L^2(\F)$ admits a chaos decomposition.
A key motivating application of this result is to allow a general development of Malliavin calculus for marked point processes. This development is done in \cite{Decreusefond1998}, under the assumption that a chaos decomposition exists.
The only assumptions that we have made on the processes in question is that the compensating measure $\tilde p^\alpha$ is equivalent to some deterministic measure $\zeta^\alpha$, which is continuous in time and finite for finite times. It seems reasonable that some relaxation of this assumption is possible, for example, to only assuming that $\tilde p^\alpha$ is absolutely continuous with respect to $\zeta^\alpha$. The difficulty in doing this arises as we cannot then write the quadratic variation of a martingale in terms of a product measure $\bP\times \zeta$, and therefore cannot directly show that iterated integrals of different orders are orthogonal.
It also seems reasonable that a relaxation of the assumption that there are finitely many jumps should be possible. In \cite{Elliott1976} no such assumption is made, however this leads to the need for transfinite induction in the proof of the martingale representation theorem. Having convergent sequences of jumps also requires a relaxation of the continuity of $\tilde p$ in time (which we assume through the equivalence of $\tilde p^\alpha$ and $\zeta^\alpha$, coupled with the continuity of $\zeta^\alpha$). This may also be possible, however it leads to a more complex quadratic variation for stochastic integrals, as it allows the possibility of accessible jump times.
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\end{document} | {"config": "arxiv", "file": "1110.0635/chaos.arxiv.tex"} |
TITLE: Find direction of the vector
QUESTION [0 upvotes]: I have two points A(1, 2) and B(3, 4) and vector AB between them. How can I find the direction of a vector?
I do not know if direction is appropriate word here. By direction I mean following: if I will have point C(2, 1) then I would have to create a vector CD of length, say 2, with the same direction as vector AB and find coordinates of point D.
REPLY [2 votes]: The direction of the vector from point $\;A\;$ to point $\;B\;$ is defined to be
$$\vec{AB}:=B-A=(2,2)$$
Sometimes is useful to take direction vectors of length 1, so you may want to normalize the above:
$$\overline u:=\frac{\vec{AB}}{||\vec{AB}||}=\frac1{2\sqrt2}(2,2)=\left(\,\frac1{\sqrt2},\,\frac1{\sqrt2}\,\right)$$
so you can talk of the direction $\;\vec{AB}\;$ or the direction $\;\overline u\;$ , it just is the same. But read carefully your definitions. | {"set_name": "stack_exchange", "score": 0, "question_id": 1658533} |
TITLE: Integral of a nonnegative Lebesgue-measurable function on $ [0,1] $.
QUESTION [1 upvotes]: Let $ f $ be a nonnegative Lebesgue-measurable function on $ [0,1] $. Suppose that $ f $ is bounded above by $ 1 $ and that $ \displaystyle \int_{[0,1]} f = 1 $.
Problem. Show that $ f(x) = 1 $ almost everywhere on $ [0,1] $.
I don’t know how to start. Would it help to represent $ f $ as the pointwise limit of a sequence $ (f_{n})_{n \in \Bbb{N}} $ of simple functions on $ [0,1] $?
REPLY [2 votes]: Let $X=[0,1]$. Consider $1-f$. If $f\ne 1$ a.e. then $1-f\ne 0 $ a.e. and indeed there is a positive measure set, $E$ so that $1-f>0$ on $E$. But then
$$0=1-1=\int_X 1-\int_X f=\int_X(1-f)=\int_E (1-f)>0.$$ | {"set_name": "stack_exchange", "score": 1, "question_id": 1338392} |
TITLE: Easy way to find out limit of $a_n = \left (1+\frac{1}{n^2} \right )^n$ for $n \rightarrow \infty$?
QUESTION [3 upvotes]: What's an easy way to find out the limit of $a_n = \left (1+\frac{1}{n^2} \right )^n$ for $n \rightarrow \infty$?
I don't think binomial expansion like with $\left (1-\frac{1}{n^2} \right )^n = \left (1+\frac{1}{n} \right )^n \cdot \left (1-\frac{1}{n} \right )^n$ is possible.
And Bernoulli's inequality only shows $\left (1+\frac{1}{n^2} \right )^n \geq 1 + n \cdot \frac{1}{n^2} = 1 + \frac{1}{n} \geq 1$ which doesn't seem to help as well.
REPLY [4 votes]: Here is a purely algebraic approach. First we note that
$$\left(1+\frac1{n^2}\right)\left(1-\frac1{n^2}\right)\le 1$$
From $(1)$ it is easy to see that
$$\left(1+\frac1{n^2}\right)^n\le \frac1{\left(1-\frac{1}{n^2} \right)^n}\tag 2$$
Applying Bernoulli's Inequality to the term on the left-hand side of $(2)$ reveals
$$1\le \left(1+\frac1{n^2}\right)^n\le \frac1{1-\frac1n}$$
whereupon applying the sqeeze theorem yields the covetes limit
$$ \lim_{n\to\infty} \left(1+\frac1{n^2}\right)^n=1$$ | {"set_name": "stack_exchange", "score": 3, "question_id": 2284250} |
TITLE: Equivalence/Similarity of matrices over domains
QUESTION [3 upvotes]: Let $K$ be a field, $R$ some domain with $R\subsetneq K$. Take some square-matrix $A \in \operatorname{Mat}_n(R)$ with $n>1$, and suppose the matrix $B \in \operatorname{Mat}_n(R)$ is equivalent to $A$ over $K$, i.e. there are $S,T \in \operatorname{GL}_n(K)$ such that $A = SBT$. My question is which conditions need to be imposed on the ring to gurantee that $A = S'BT'$ with unimodular $S',T' \in \operatorname{Mat}_n(R)$. What happens if we replace equivalence by similarity?
I'm under the impression that I once read that a PID suffices regarding the similarity..
REPLY [0 votes]: I think that the two following remarks shed some light on your question.
Notation: if $R$ is a ring, $\mathrm{GL}_n(R)$ is the group of invertible matrices over $R$, i.e. the group made up of matrices $A$ such that $\mathrm{det}(A)$ is invertible in $R$. If $R$ is a ring and $A$ is a matrix, $I_k(A)$ is the ideal of $R$ generated by the $k \times k$ minors of $A$.
If $K$ is a field and $A, B \in \mathrm{Mat}_{m,n}(K)$, then there exist $M \in \mathrm{GL}_m(K)$, $N \in \mathrm{GL}_n(K)$ such that $B = MAN$ if and only if $\mathrm{rank} A = \mathrm{rank} B$.
(Smith normal form) If $R$ is a PID and $A, B \in \mathrm{Mat}_{m,n}(R)$, then there exist $M \in \mathrm{GL}_m(R)$, $N \in \mathrm{GL}_n(R)$ such that $B = MAN$ if and only if $I_k(A) = I_k(B)$ for all $k \geq 1$.
For example, if $L/K$ is a field extension and $A, B \in \mathrm{Mat}_{m,n}(K)$ are two matrices that are equivalent over $L$, then they are equivalent over $K$. The same is true for similarity. | {"set_name": "stack_exchange", "score": 3, "question_id": 439108} |
TITLE: Solving the recurrence $T(n) = 2T(\frac{n}{4}) + (n^{0.5})$ with the Master theorem
QUESTION [0 upvotes]: Would the result will be $\theta(^{0.5})$? This is what I found by using the second option of the master theorem.
REPLY [0 votes]: By using the Master theorem we have that:
$a=2,b=4$
$log_ba=0.5$ so $f(n)$ (which in your case is $n^{0.5}$) is $\Theta(n^{log_ba})$
Thus by falling in the second case of the Master Theorem:
$T(n)=\Theta(n^{log_ba}*log_bn)=\Theta(n^{0.5}*logn)=\Theta(\sqrt{n}*logn)$ | {"set_name": "stack_exchange", "score": 0, "question_id": 3383195} |
TITLE: Calculus 3: Lagrange Multipliers
QUESTION [1 upvotes]: Find the minimum and maximum of $f(x,y,z) = x^2+y^2+z^2$ subject to two constraints, $x+2y+z=8$ and $x−y=5$.
Looking at the equation, it's clear that there is no maximum.
After working this problem out, I found:
$x = 41/11$ , $y = -14/11$ , and $z = 9/11$
After plugging this into the original equation, I found the minimum to be $178/11$
However, my online homework is saying my answer is incorrect. Did I do something wrong?
Thank you in advance to anyone who can help me out with this.
REPLY [0 votes]: I solved equation, and got an answer of $x=\dfrac{23}{3}$ ,$y=\dfrac{8}{3}$,$z=-5$. Is it the correct answer, by your online homework? If it is, I will share my solution with you. By the way, your answer doesn't fit the first constraint, you can check it by plugging in values. | {"set_name": "stack_exchange", "score": 1, "question_id": 2996327} |
TITLE: Show that every row of matrix $S$ is a linear combination of its bottom row and the row (1 1 1 1 1 1 )
QUESTION [1 upvotes]: Couldn't solve the following three questions.
$$S=\begin{pmatrix}
36 & 35 & 34 &33&32&31 \\
25 & 26 & 27&28&29&30 \\
24&23&22&21&20&19 \\
13&14&15&16&17&18 \\
12&11&10&9&8&7 \\
1&2&3&4&5&6 \\
\end{pmatrix}$$
(a) Show that every row of matrix $S$ is a linear combination of its bottom row and the row${(1 1 1 1 1 1 )}$
(b) Deduce that the rank of $S$ is at most 2
(c) Show that the rank of $S$ can't be 0 or 1, therefore, the rank of $S$ is 2
REPLY [2 votes]: These hints are probably best read after you've tried what david has to say, so don't read them until you do.
(a) Let $R_n$ be the $n$th row of $S$ as a row vector and let $A$ be the row vector of all ones. What you should do for (a) is take $\pm R_6 + kA$ with the appropriate choice of $k$ to get the row you want.
(b) simply express A in terms of $R_6$ and $R_5$. Then we know from (a) that any row is a linear combination of the two rows, and so we see that $S$ has rank 2. Do you see this?
(c) David already spells it out for you in his answer. I recommend reviewing the definition of rank if you're having trouble.
EDIT: I think $R_5$ is slightly easier to work with than $R_1$, which David used, but it's up to preference. I just like small numbers. | {"set_name": "stack_exchange", "score": 1, "question_id": 911686} |
\begin{document}
\title[]{The Landis conjecture on exponential decay}
\subjclass{Primary 35J15; Secondary 30C62}
\keywords{Landis' conjecture, Schr\"odinger equation, quasiconformal mappings, vanishing order.}
\author{A. Logunov}
\address{Alexander Logunov: Department of Mathematics, Princeton University, Princeton, NJ, USA}
\email{log239@yandex.ru}
\author{E. Malinnikova}
\address{Eugenia Malinnikova: Department of Mathematics, Stanford University, Stanford, CA, USA}
\email{eugeniam@stanford.edu}
\author{N. Nadirashvili}
\address{Nicolai Nadirashvili: \phantom{n}CNRS, \phantom{n}Institut de Math\' ematique de Marseille, Marseille, France
}
\email{nikolay.nadirashvili@univ-amu.fr}
\author{F. Nazarov}
\address{Fedor Nazarov: Department of Mathematics, Kent State University, Kent, OH, USA}
\email{nazarov@math.kent.edu}
\begin{abstract}
Consider a solution $u$ to $\Delta u +Vu=0$ on $\mathbb{R}^2$, where $V$ is real-valued, measurable and $|V|\leq 1$.
If $|u(x)| \leq \exp(-C |x| \log^{1/2}|x|)$, $|x|>2$,
where $C$ is a sufficiently large absolute constant, then
$u\equiv 0$.
\end{abstract}
\maketitle
\section{The main result.}
Let $u$ be a solution to
\begin{equation} \label{eq:schr}
\Delta u + V u =0
\end{equation}
in $\mathbb{R}^n$, where $V$ is
a measurable function with $|V | \leq 1 $ in the whole space. According to \cite{C15},\cite{KL88}, in the late 1960s Landis conjectured that if
$$ |u(x)| \leq \exp(-C|x|), $$
where $C>0$ is a sufficiently large constant, then $u\equiv0$.
The weaker statement, which was also conjectured by Landis according to \cite{C15}, states that
if $|u(x)|$ tends to $0$ faster than exponentially at $\infty$, i.e., $$|u(x)| \leq \exp(-|x|^{1+\varepsilon}),\varepsilon >0,$$ then $u\equiv 0$.
There are two versions of Landis' conjectures: real and complex.
Meshkov \cite{M92} constructed a counter-example to the complex version of Landis' conjecture. He showed that there is a complex-valued potential $V$ with $|V|\leq 1$ and a non-zero solution $u$ to \eqref{eq:schr} on $\mathbb{R}^2$ such that $|u(x)| \leq \exp(-c |x|^{4/3}).$
Meshkov also showed (in any dimension $n$) that if
$$ \sup_{\mathbb{R}^n} |u(x)| e^{-\tau |x|^{ 4/3}}< \infty \text{ for all } \tau>0,$$
then $u\equiv 0$.
The question whether the Landis conjecture is true for real-valued $V$ is open.
The main result of this article confirms the weak version of the Landis conjecture in dimension two.
\begin{theorem} \label{main}
Suppose that $\Delta u +Vu=0$ on $\mathbb{R}^2$, where $u$ and $V$ are real-valued and $|V|\leq 1$.
If $|u(x)| \leq \exp(-C |x| \log^{1/2}|x|)$, $|x|>2$,
where $C$ is a sufficiently large absolute constant, then
$u\equiv 0$.
\end{theorem}
A similar striking difference between the decay estimates for real and complex solutions has also been observed in \cite{C15}, where a closely related equation $\Delta u+W\cdot\nabla u=0$ with a bounded vector field $W:\mathbb{R}^2 \to \mathbb{R}^2$ was studied.
There is a simple example of a solution to \eqref{eq:schr} with bounded $V$ that
decays exponentially. Define $u=e^{-|x|}$ in $\{ |x| > 1\}$ and extend it to a $C^2$ smooth positive function on the plane. Then $|\Delta u| \leq C|u|$ and by taking $u(\frac{1}{\sqrt C} \cdot)$ in place of $u$ one can make $|V|\leq 1$ in this example.
The assumption that $u$ is real-valued is redundant because in the case of real-valued $V$
the real and imaginary parts of $u$ also satisfy \eqref{eq:schr}. But in the proof we will use that $u$ is real-valued.
The proof of Theorem \ref{main} combines the technique of quasiconformal mappings with two tricks. The tricks involve nodal sets (zero sets) of $u$ and holes that are made in nodal domains (connected components of the complement of the zero set). We describe the idea in Section \ref{Idea}. Some two-dimensional tools are used in the proof and the Landis conjecture in higher dimensions is still open.
Our second result is a local version of Landis' conjecture.
\begin{theorem} \label{local 1} Let $u$ be a real solution to $\Delta u+ Vu=0$ in $B(0,2R)\subset\mathbb{R}^2$, where $V$ is real-valued and $|V|\leq 1$.
Suppose that $|u(0)|=\sup\limits_{B(0,2R)} |u|= 1 $. Then
for any $x_0$ with $|x_0|=R/2>2$, we have
$$ \sup\limits_{B(x_0,1)}|u| \geq \exp(-C R \log^{3/2}R)$$
with some absolute constant $C>0$.
\end{theorem}
The previous best known bound $\sup\limits_{B(x_0,1)}|u| \geq \exp(-CR^{4/3}\log R)$, was obtained in any dimension by Bourgain and Kenig \cite{BK05} in their proof of Anderson localization for the Bernoulli model, see also \cite{C05}.
Theorem \ref{local 1} follows from the main local Theorem \ref{local main}, where we don't assume that $|u(0)|=\sup\limits_{B(0,2R)} |u|= 1 $, and prove a version of the three balls inequality.
Landis' conjecture was a subject to an extensive study. Under additional assumptions on $V$, some versions of Landis' conjecture are known, see \cite{B12},\cite{D19}, \cite{DKW19},\cite{EKPV},\cite{C05},\cite{C15},\cite{K98},\cite{Z16} and references therein. A related problem in a cylinder was studied in \cite{G81}.
\textbf{Notation.} By $c,C,C',... >0$ we denote various constants. Typically small constants are denoted by small letters and we use capital letters for large constants. If a constant $C$ depends on a domain (or some other parameter), we say it. Sometimes we state theorems without reminding that the functions are assumed to be real-valued and $u$ is a solution to \eqref{eq:schr} on $\mathbb{R}^2$.
A ball with center at $x$ of radius $r$ is denoted by $B(x,r)$ and the two-dimensional Lebesgue measure is denoted by $m_2$.
\textbf{Acknowledgements.} The authors
are grateful to Misha Sodin, Alexandru Ionescu, Charles Fefferman and Carlos Kenig
for fruitful discussions.
This work was completed during the time A.L. served as a Clay Research Fellow and Packard Fellow.
E.M. was partially supported by NSF grant DMS-1956294 and by Research Council of Norway, Project 275113.
F.N. was partially supported by NSF grant DMS-1900008.
\section{Strategy of the proof and local versions.} \label{Idea}
The proof consists of three acts. First, we will explain the main ideas of each of them.
\textbf{Description of Act I.}
We will use the following well-known fact about nodal sets, which is proved in the Appendix (Lemma \ref{le: diameter}) for reader's convenience. There is an absolute constant $r_0>0$ such that if $u$ is a solution to $\Delta u+ Vu$ in a neighborhood of a closed ball $\overline{B(z_0,r)}$ with $|V|\leq 1$, $u(z_0)=0$ and $0<r<r_0$, then the circle
$ C(z_0,r)=\{ z:|z-z_0|=r\}$
is intersecting the zero set of $u$.
It is also true that the singular set $$S=\{ x: u(x)=0 \textup{ and } \nabla u(x)= 0\}$$
consists of isolated points and the nodal set
$$F_0=\{x:u(x)=0\}$$ is a union of smooth curves, see \cite{CF85}.
However the proof will not use it, but this structural result about nodal sets makes it easier to think about them.
Now, assume that $u$ is a solution to \eqref{eq:schr} in $B(0,R)$, $R>1$. Take $\varepsilon>0$ (a small parameter to be chosen later) and add finitely many $C\varepsilon$ -- separated closed disks of radius $\varepsilon$ to $F_0$ so that the distance from each disk to $F_0$ is $\geq C\varepsilon$
and
$$F_0\cup \textup{ union of the disks } \cup \{ z:|z|\geq R\}$$
is a $3C\varepsilon$ -- net on the plane (assume $C>2$). Let us denote by $F_1$ the union of the closed disks, see Figure \ref{puncture fig}.
\begin{figure}[h!]
\includegraphics[width=0.8\textwidth]{puncture.png}
\caption{Puncturing nodal domains} \label{puncture fig}
\end{figure}
It can be shown that
$$\Omega=\{z:|z| < R, z \notin F_0\cup F_1 \}$$
is an open (possibly disconnected) set with the Poincare constant $\leq C'\varepsilon^2$, i.e., for every $u \in W_0^{1,2}(\Omega)$,
we have
$$ \int_{\Omega} u^2 \leq C'\varepsilon^2 \int_{\Omega} |\nabla u|^2.$$
\newpage
It allows one to construct a function $\varphi$ in $B(0,R)$ such that
\begin{itemize}
\item $\Delta \varphi + V \varphi=0$ in $\Omega$,
\item $\varphi - 1 \in W_0^{1,2}(\Omega)$,
\item $\| \varphi-1\|_{\infty} \leq C''\varepsilon^2$.
\end{itemize}
The details are given in Section \ref{sec:Poincare}.
\textbf{Description of Act II.}
Consider $f= \frac{u}{\varphi}$. Then $f$ satisfies
$$ {\rm{div}}(\varphi^2 \nabla f)=0$$
in $\Omega$. The set $\Omega$ is usually not connected and the functions $\varphi$ and $f$ may be not smooth across $F_0$. However due to the fact that $F_0$ is the zero set of $u$ it appears (after some work) that the equation ${\rm{div}}(\varphi^2 \nabla f)=0$ holds through $F_0$ in the whole $B(0,R)\setminus F_1$.
Here the theory of quasiconformal mappings joins the game.
After noticing that $f \in W_{loc}^{1,2}$, we may use the Stoilow factorization theorem to make a $K$-- quasiconformal change of variables $g$ mapping $0$ to $0$ and
$B(0,R)$ onto $B(0,R)$ such that
$$ f =h \circ g$$
where $h $ is a harmonic function in $B(0,R)\setminus g(F_1)$.
Moreover, $K$ is very close to $1$ when $\| \varphi - 1 \|_\infty$ is small:
$$ K \leq \frac{1+\left\|\frac{1-\varphi^2}{1+\varphi^2} \right\|_\infty}{1-\left \|\frac{1-\varphi^2}{1+\varphi^2} \right \|_\infty} \leq 1+C \varepsilon^2.$$
Mori's theorem tells us how much the distances are distorted depending on $K$:
\begin{equation}
\frac{1}{16}\left|\frac{z_1 - z_2}{R}\right|^{K} \leq \frac{|g(z_1) - g(z_2)|}{R}
\leq 16\left|\frac{z_1 - z_2}{R}\right|^{1/K}.
\end{equation}
We choose $$ \varepsilon \sim \frac{1}{\sqrt{\log R}}$$
so that the distortion on scales from $
\frac{1}{R}$ to $R$ is bounded and, moreover, the images of the disks in $F_1$ have size comparable to $\varepsilon$.
Then we get a harmonic function $h$ in $B(0,R)\setminus g(F_1)$, where $g(F_1)$ is the union of sets of diameter $\sim \varepsilon$ and each set
(the image of a single disk) is surrounded by an annulus of
width $\sim C\varepsilon$ in which $h$ does not change sign.
\textbf{Description of Act III.}
By rescaling we get the following question:
Let $h$ be harmonic in a punctured domain $B(0,R') \setminus \cup_j D_j$ where $R' \sim \frac{R}{\varepsilon}\sim R \sqrt{\log R}$ and $D_j$ are $1000$-- separated unit disks. Assume also that $h$ does not change sign in $5D_j\setminus D_j$. What can be said about the decay of $|h|$?
\begin{theorem} \label{exercise 1} Under the above assumptions, we have
$$ \sup_{B(0,R') \setminus \cup_j 3D_j}|h| \leq \exp(CR')
\sup_{ \{ z: R'/8<|z|<R' \} \setminus \cup_j 3D_j} |h| \quad \textup{for } R' > 2000$$
with some absolute constant $C>0$.
\end{theorem}
Theorem \ref{exercise 1} is an immediate consequence of a more general Theorem \ref{local 3}.
The outcome is that $|u|$ cannot decay faster than $\exp(-CR\sqrt{\log R})$.
A different proof of the estimate for harmonic functions in a punctured domain (with a slightly worse bound)
is given in the Appendix. The second proof works in higher dimensions and uses the Carleman inequality with log linear weight.
\textbf{Local versions.}
Local versions of Theorem \ref{main} (on the two dimensional plane) are also true. Here is the main local Theorem \ref{local main}.
\begin{theorem} \label{local main}
If $u$ is a solution to $\Delta u + V u =0$ in $B(0, R)$, $R>2$, $V$ is real-valued,
$|V| \leq 1$, and
$$\frac{\sup_{B(0,R)}|u|}{\sup_{B(0,R/2)}|u|} \leq e^N, $$
then
\begin{equation} \label{eq: Br}
\sup_{B(0,r)}|u| \geq (r/R)^{C (R \log^{1/2}R +N)}\sup_{B(0,R)}|u|
\end{equation}
for any $r<R/4$, where $C$ is an absolute positive constant.
\end{theorem}
Theorem \ref{local main} implies Theorems \ref{main} and \ref{local 1}.
In order to deduce Theorem \ref{main}, we may assume that $|u|$ attains its global maximum at some point on the plane, otherwise $|u|$ does not tend to $0$ near infinity.
Let $$|u(z_{max})|= \max_{\mathbb{R}^2} |u|=1.$$ Then for any $ R > 6|z_{max}|$ and any $x$ with $|x|=R/3$, we have $$\sup_{B(x,R)}|u| = \sup_{B(x,R/2)}|u|=1$$ and if additionally $R>2$, then by Theorem \ref{local main} applied to $u(\cdot + x)$, we have $$\sup_{B(x,R/4)}|u| \geq e^{-C R \log^{1/2} R}$$ and therefore $$\sup_{|z|>R/12}|u| \geq e^{-C R \log^{1/2} R}.$$
In order to deduce Theorem \ref{local 1} note that $$\sup_{B(x,R/2)}|u| = \sup_{B(x,R)}|u|=1$$
for any $x$ with $|x|=R/2$ because $|u(0)|=\max_{B(0,2R)} |u|$. Applying Theorem \ref{local main} to $u(\cdot + x)$ we get $$\sup\limits_{B(x,1)}|u| \geq R^{-C R\log^{1/2} R }= e^{-C R\log^{3/2}R}.$$
\begin{corollary} \label{local 2}
Let $A>4$. If $u$ is a solution to $\Delta u + V u =0$ in $B(0, 1)$, $V$ is real-valued,
$|V| \leq A$, and
$$\frac{\sup_{B(0,1)}|u|}{\sup_{B(0,1/2)}|u|} \leq \exp(N), $$
then
\begin{equation} \label{eq:B_r}
\sup_{B(0,r)}|u| \geq r^{C( \sqrt{A \log A}+N)}\sup_{B(0,1)}|u| \quad \text{ for } r\leq 1/4,
\end{equation}
where $C$ is an absolute positive constant.
\end{corollary}
For the proof, consider $u(\frac{1}{\sqrt A}\cdot)$ in place of $u$. We obtain a solution to $\Delta u+ Vu=0$
in $B(0,\sqrt A)$ with $|V| \leq 1$ and
$$\frac{\sup_{B(0,\sqrt A)}|u|}{\sup_{B(0,\sqrt A /2)}|u|} \leq e^{N}$$
and we can apply Theorem \ref{local main} to the new $u$ and $R=\sqrt A$.
\begin{remark}
Inequality \eqref{eq:B_r} implies that the vanishing order of $u$ at $0$ is bounded by $C (\sqrt{A \log A}+N)$.
This question was previously studied in \cite{B12},\cite{K98},\cite{Z16}.
\end{remark}
On any smooth two dimensional Riemannian manifold $(M,g)$ every equation $\Delta_g u+ Vu=0$ can
be simplified in local isothermal coordinates to $\Delta u + V'u=0$ (with ordinary Euclidean Laplacian $\Delta$). Corollary \ref{local 2} gives information on the distribution of solutions to Schrodinger equations on compact manifolds of dimension 2.
\begin{corollary}\label{global} Let $(M,g)$ be a smooth closed (compact and without boundary) Riemannian manifold of dimension $2$. Then for any function $u$ satisfying $\Delta_g u + V u =0$ on $M$ with $|V| \leq \lambda$, $\lambda >2$, we have
$$\sup\limits_{B_r}|u| \geq r^{C \sqrt{\lambda \log \lambda}}\sup_{M}|u|$$
for any ball $B_r$ of radius $r<1/2$. The constant $ C$ depends on the manifold.
\end{corollary}
This result follows from Corollary \ref{local 2} by iterations (see the argument in \cite{DF}, page 162, after formula (1.5)).
\begin{remark}
A slightly better bound was obtained in \cite{DF} by Donnelly and Fefferman for Laplace eigenfunctions on closed Riemannian manifolds of any dimension. If $\Delta_g u+ \lambda u=0$ on $(M,g)$, then
$$ \sup_{B_r} |u| \geq c r^{C \sqrt \lambda } \sup_M |u|, \quad r \leq \frac 1 2.$$
So the vanishing order at any point is at most $C\sqrt \lambda$.
\end{remark}
In Act I and Act II we will reduce (with a logarithmic loss) the main local Theorem \ref{local main}
to a general Theorem \ref{local 3}, which is a local statement about two dimensional harmonic functions.
\section{Act I} \label{sec:Poincare}
\subsection{Poincare constant for porous domains.}
\begin{lemma} \label{Poincare}
Let $F$ be a closed set in $B(0,R)$, $R>1$, such that
\begin{enumerate}
\item[a)] For every $z_0 \in F$, $r\in (0,1]$,
the circle
$ C(z_0,r)=\{ z:|z-z_0|=r\}$
intersects $F\cup \partial B(0,R)$.
\item[b)] $ F \cup \partial B(0,R)$ is $C-$dense in $B(0,R)$, $C>1$.
\end{enumerate}
Then the Poincare constant of $\Omega= B(0,R)\setminus F$
is bounded by some constant $\widetilde C$ that depends only on $C$.
\end{lemma}
\begin{proof}
Let $f \in C_0^\infty(\Omega)$. Extend $f$ by zero outside $\Omega$.
First, we will show that if $z\in F\cup \partial B(0,R)$, then
$$ \int_{B(z,3C)} |f|^2 \lesssim \int_{B(z,3C)} |\nabla f|^2 .$$
Every circle $C_r=\partial B(z,r)$, $r\in(0,1)$, has a zero of $f$, whence
$$\max\limits_{C_r}|f| \leq \int\limits_{C_r }|\nabla f|$$
and
$$\int\limits_{B(z,1)}|f|^2 = \int_0^1 \left(\,\int\limits_{C_r}|f|^2\right) dr \leq \int_0^1 |C_r|\max\limits_{C_r}|f|^2dr \leq \int_0^1 |C_r|\left(\,\int\limits_{C_r }|\nabla f|\right)^2 dr \leq$$
$$ \leq \int_0^1 |C_r|^2\left(\,\int\limits_{C_r }|\nabla f|^2\right) dr \leq (2\pi)^2 \int_0^1 \left(\,\int\limits_{C_r }|\nabla f|^2\right)dr = (2\pi)^2 \int\limits_{B(z,1)} |\nabla f|^2. $$
We therefore can find $r\in (1/2,1)$ such that
$$ \int\limits_{C_r }|f|^2 \leq C_1 \int\limits_{B(z,1)}|f|^2 \leq C_2 \int\limits_{B(z,1)} |\nabla f|^2.$$
Let $\Gamma_\psi$, $\psi \in[0,2\pi)$, be a segment starting at the point $$x_\psi:= z + re^{i\psi}$$
and ending at the point $z+3Ce^{i\psi}$. Note that
$$ \max_{\Gamma_\psi} |f|^2 \leq \left(|f(x_\psi)| + \int_{\Gamma_\psi}|\nabla f|\right)^2 \leq 2|f(x_\psi)|^2+2\left(\int_{\Gamma_\psi}|\nabla f|\right)^2 \leq$$
$$ \leq 2 |f(x_\psi)|^2 + 2|\Gamma_\psi|\int_{\Gamma_\psi}|\nabla f|^2 \leq 2 |f(x_\psi)|^2 + 6C \int_{\Gamma_\psi}|\nabla f|^2$$
and therefore
$$ \int\limits_{ B(z,3C)\setminus B(z,1)} f^2 \leq (3C)^2 \int_0^{2\pi} \max_{\Gamma_\psi} |f|^2 d\psi \leq $$ $$ \leq C_1(C) \left[ \quad \int\limits_{C_r }|f|^2 + \int_0^{2\pi} \left(\int_{\Gamma_\psi}|\nabla f|^2\right) d\psi \right] \leq C_2(C) \int_{B(z,3C)} |\nabla f|^2. $$
Thus $$ \int_{B(z,3C)} |f|^2 \leq C_3(C) \int_{B(z,3C)} |\nabla f|^2 .$$
We can choose a finite collection $Z_*$ of points $z$ in $F \cup \partial B(0,R)$ such that the balls $B(z,3C)$ cover $B(0,R)$ and
each point is covered a bounded number of times.
Finally, we have
$$ \int\limits_{B(0,R)} f^2 \leq \sum\limits_{z\in Z_*} \int_{B(z,3C)} |f|^2 \leq C_3(C) \sum\limits_{z\in Z_*} \int_{B(z,3C)} |\nabla f|^2 \leq $$
$$\leq C_4(C) \int\limits_{B(0,R)} |\nabla f|^2.$$
\end{proof}
We start proving Theorem \ref{local main}.
Recall that $\Delta u + Vu =0$ in the ball $B(0,R)$ (we may think that $R$ is a large number) and $F_0$ is the zero set of $u$.
We will use the fact that $u\in C^1(B(0,R))$, which is proved in the Appendix, see Fact \ref{fact5}.
Now, consider the following setting:
\begin{figure}[h!] \label{contours}
\includegraphics[width=0.8\textwidth]{holes.png}
\caption{Puncturing nodal domains}
\end{figure}
\noindent Take $\varepsilon>0$ (a small parameter to be chosen later). Choose finitely many $C\varepsilon$ -- separated closed disks of radius $\varepsilon$, whose union will be denoted by $F_1$, so that the distance from each disk to $F_0$ and $\partial B(0,R)$ is $\geq C\varepsilon$
and $$F_0\cup F_1\cup \partial B(0,R)$$
is a $3C\varepsilon$ -- net in $B(0,R)$ (we assume $C>2$). \\
For instance, one can get $F_1$ by
considering the maximal number of open non-intersecting disks of radius $(C+1)\varepsilon$ in $B(0,R)\setminus F_0 $. The centers of the disks are $(2C+2)\varepsilon$ -- separated. There is no point $x$ in $B(0,R)\setminus F_0$ that is $(2C+2)\varepsilon$ far from the centers of the disks and from $F_0 \cup \partial B(0,R)$, otherwise we could add one more disk of radius $(C+1)\varepsilon$ with center at this point. So we may choose the disks of radius $\varepsilon>0$ with the same centers, they will be $C\varepsilon$ -- separated and $F_0\cup F_1\cup \partial B(0,R)$ will be a $2(C+1)\varepsilon$ -- net.
\noindent \textbf{Two points to avoid.} Now, let us remove from $F_1$ the disks that are $C\varepsilon$ close to $0$ or to the point $z_{\max} \in \overline{B(0,R/2)}$ such that $$|u(z_{\max})|= \sup_{B(0,R/2)}|u|.$$ The set $F_0\cup F_1\cup \partial B(0,R) $ will still be a $10C\varepsilon$ -- net, but now all disks from $F_1$ are also $C\varepsilon$- separated from $0$ and $z_{\max}$.
The detail about avoiding those two points will be used only in the end of Act II.
Recall that $F_0$ has the property that for any $z_0 \in F_0$, every circle $C(z_0,r)$ with $r<r_0$ intersects
$F_0$ or $\partial B_R$. Taking $u(\varepsilon \cdot)$ in place of $u$ (so the assumptions of Lemma \ref{Poincare} hold for $\varepsilon< r_0$) and applying Lemma \ref{Poincare} we arrive to the following conclusion. \\
\noindent\textbf{Outcome.}
The domain $$\Omega= B(0,R)\setminus (F_0 \cup F_1)$$
has Poincare constant $\leq C'\varepsilon^2$ and $B(0,R)\setminus F_1$ contains $0$ and $z_{\max}$.
\subsection{Solving $\Delta \varphi+ V\varphi=0$.}
The goal of this section is to construct an auxiliary solution to \eqref{eq:schr} in a domain with a small Poincare constant, so that the solution has boundary values $1$ and is uniformly close to $1$.
\begin{lemma} \label{solving Schrodinger}
Let $\Omega$ be a bounded open set with the Poincare constant $k^2$. Let $V\in L^\infty(\Omega)$. Assume that
$$k^2 \|V\|_\infty \ll 1.$$
Then there exists $\varphi=1+\tilde \varphi$ with
$$\tilde \varphi \in W_{0}^{1,2}(\Omega), \|\tilde \varphi\|_\infty \leq Ck^2\|V\|_\infty$$
such that $\varphi$ is a weak solution to $\Delta \varphi + V\varphi=0$ in $\Omega$, where $C$ is an absolute positive constant.
\end{lemma}
\begin{proof}
We will use the following fact, which is proved in the Appendix, Lemma \ref{lem: solving} and Lemma \ref{lem: solving2}.
\noindent \textbf{Fact.} When the Poincare constant of $\Omega$ is $ 1$, $v\in L^{\infty}(\Omega)$, there is a solution $\varphi$ to $\Delta \varphi = v$ in $W_{0}^{1,2}(\Omega)$ with $$\| \varphi\|_\infty \leq C\| v\|_\infty$$
and
$$ \| \varphi \|_{W_0^{1,2}} \leq C \|v\|_2.$$
\noindent \textbf{Corollary} (follows by rescaling). If $\Omega$ has Poincare constant $k^2$, then we can find a solution $\varphi$ to $\Delta \varphi = v$ with
$$\| \varphi \|_\infty \leq Ck^2\| v\|_\infty$$
and
$$ \| \varphi \|_{W_0^{1,2}} \leq C_1(k) \|v\|_2. $$
Now, let $\varphi_1$ solve $\Delta \varphi_1= -V$ and for $n\geq 2$
let $\varphi_n$ solve
$$\Delta \varphi_n= -V\varphi_{n-1}.$$
Note that this sequence is well defined since on each step the right-hand side is in $L^\infty$.
We have $$\|\varphi_n\|_\infty\leq Ck^2\|V\|_\infty\|\varphi_{n-1}\|_\infty,\quad n\geq 2,$$
and $\|\varphi_1\|_\infty \leq Ck^2 \|V\|_\infty.$
We are assuming that $Ck^2 \|V\|_\infty \leq 1/2$.
Hence $\|\varphi_n\|_\infty \leq 2^{-n+1} Ck^2\|V\|_\infty $
and
$$\|\varphi_n\|_{W_0^{1,2}} \leq C_1(k) \|\varphi_{n-1}\|_2 \leq C_2(k) \|\varphi_{n-1}\|_{\infty} \leq C_3(k) 2^{-n}.$$
Thus the series $$\tilde \varphi = \varphi_1+\varphi_2+\dots$$
converges both in $L^{\infty}$ and in $W_0^{1,2}(\Omega)$
with $$\| \tilde \varphi\|_{\infty} \leq C'k^2 \|V\|_\infty.$$
Also for any $h \in W_0^{1,2}(\Omega)$, we have $\int \nabla \varphi_n \nabla h = \int V \varphi_{n-1}h$ for $n \geq 2$ and $\int \nabla \varphi_1 \nabla h = \int Vh$.
Thus $\Delta \tilde \varphi= - V(1+\tilde \varphi)$ and
$$\Delta(1+\tilde \varphi) + V(1+\tilde \varphi) =0 \quad \text{ in } \Omega $$ as required.
\end{proof}
\noindent \textbf{Outcome.} Since the Poincare constant of $\Omega= B(0,R) \setminus (F_0 \cup F_1)$ is $\leq \widetilde C\varepsilon^2$, using Lemma \ref{solving Schrodinger}, we can find $\varphi$ such that
\begin{itemize}
\item $\Delta \varphi + V \varphi=0$ in $\Omega$,
\item $\varphi - 1 \in W_0^{1,2}(\Omega)$,
\item $\| \varphi-1\|_{\infty} \leq C'\varepsilon^2$.
\end{itemize}
\section{Act II.}
\subsection{Reduction to a divergence type equation in a domain with holes.}
Recall that $u$ is a solution to $\Delta u+ Vu=0$ in $B(0,R)$ and $F_0$ is the zero set of $u$.
Extend the function $\varphi$ by 1 outside $$\Omega=B(0,R) \setminus (F_0\cup F_1).$$
\begin{lemma} \label{div equation}
The function $\frac{u}{\varphi} \in W^{1,2}_{loc}(B(0,R))$ and it is a solution to $$\textup{ div}(\varphi^2 \nabla (\frac{u}{\varphi})) = 0$$ in
$B(0,R) \setminus F_1$ in the weak sense.
\end{lemma}
\noindent \textbf{Remark.} The lemma takes care of all ``continuations through nodal lines" of $u$.
\begin{proof}
First, we would like to notice that the extended functions
$\frac{1}{\varphi}, \varphi \in W^{1,2}_{loc}(\mathbb{R}^2)$ and
\begin{equation} \label{grad 1/phi}
\nabla \frac{1}{\varphi}= -\mathbbm{1}_{\Omega} \frac{\nabla \varphi}{\varphi^2} \quad \text{and} \quad \nabla \varphi = \mathbbm{1}_\Omega \nabla \varphi
\end{equation}
in $\mathbb{R}^2$ in the sense of distributions:
$$ \int_{\mathbb{R}^2} \frac{1}{\varphi} \nabla \xi= \int_{\Omega}\frac{\nabla \varphi}{\varphi^2} \xi \quad \text{and} \quad \int_{\mathbb{R}^2} \varphi \nabla \xi= -\int_{\Omega}\nabla \varphi \xi$$
for any $\xi
\in C^\infty_0(\mathbb{R}^2) $. The formal check is performed in Fact \ref{fact6} in the Appendix.
Now, we would like to verify that $ \frac{u}{\varphi} \in W^{1,2}_{loc}(B(0,R))$ and
$$ \nabla \frac{u}{\varphi} = \frac{\varphi\nabla u}{\varphi^2} - \frac{u \nabla \varphi }{\varphi^2} \mathbbm{1}_\Omega.$$
\begin{fact} \label{product}
Let $u,v \in W^{1,2}_{loc}(B(0,R))\cap L^{{}^{\scriptsize \infty}}_{loc}(B(0,R))$. Then $uv \in W^{1,2}_{loc}(
B(0,R)) $ and $\nabla(uv) = u \nabla v + v \nabla u$.
\end{fact}
\noindent Fact \ref{product} is proved in the Appendix.
Recall that $\varphi$ is extended by $1$ outside $\Omega$, $\frac{1}{\varphi} \in W^{1,2}_{loc}(\mathbb{R}^2)$
and $u$ is $C^1$-smooth in $B(0,R)$ by Fact \ref{fact5}.
By Fact \ref{product} we know that $ \frac{u}{\varphi} \in W^{1,2}_{loc}(B(0,R))$ and, as expected,
$$ \nabla \frac{u}{\varphi} = \frac{\varphi \nabla u}{\varphi^2} - \frac{u \nabla \varphi}{\varphi^2} \mathbbm{1}_\Omega$$
in $B(0,R)$ in the sense of distributions. To establish the divergence-type equation for $\nabla \frac{u}{\varphi}$
we want to show that for every test function $h \in C_0^\infty(B(0,R)\setminus F_1)$, we have
$$ \int_{B(0,R)\setminus F_1} \varphi^2 \nabla (\frac{u}{\varphi}) \nabla h =0.$$
So we need to prove that
\begin{equation} \label{need}
\int_{B(0,R)\setminus F_1} (\varphi \nabla u - u \nabla \varphi \mathbbm{1}_\Omega) \cdot \nabla h =0.
\end{equation}
Since $u$ is a solution to $\nabla u +Vu =0$ in $B(0,R)$,
we have
\begin{equation} \label{have1}
\int_{B(0,R)\setminus F_1} \nabla u \cdot (\varphi \nabla h + h \nabla \varphi) = \int_{B(0,R)\setminus F_1} V\varphi u h
\end{equation}
(we know the last equality under the assumption that $
\varphi$ is smooth, but it is also true for $\varphi \in W^{1,2}_{loc}(\mathbb{R}^2)$ by taking the norm limit).
Consider a function $\xi \in C_0^\infty(B(0,R) \setminus F_0)$ that descends from $1$ to $0$ in
the $\varepsilon$ -- neighborhood of $F_0\cup \partial B(0,R)$ with $|\nabla \xi| < C/\varepsilon$.
Since $\Delta \varphi + V \varphi =0$ in $$\Omega= B(0,R) \setminus (F_0 \cup F_1)$$ and $uh\xi \in C_0^1(\Omega)$, we have
\begin{equation} \label{have2}
\int_\Omega \nabla \varphi \cdot(h \nabla u \xi +u\nabla h \xi + uh \nabla \xi)= \int_\Omega V\varphi uh\xi.
\end{equation}
Note that $\int_\Omega V\varphi uh\xi$ tends to $\int_\Omega V\varphi u h$ as $\varepsilon \to 0$ (the functions $V,\varphi,uh,\xi$ are uniformly bounded and
the convergence holds pointwise in $\Omega$ because $\xi \to 1 $ in $B(0,R)\setminus F_0$).
Note that
$$h \nabla u \xi \to h\nabla u \quad \text{pointwise in } \Omega$$
and
$$u\nabla h \xi \to u\nabla h \quad \text{pointwise in } \Omega $$
because $\xi \to 1$ in $\Omega$.
Hence
\begin{equation} \label{have3} \int_\Omega \nabla \varphi \cdot(h \nabla u \xi +u\nabla h \xi) \to \int_\Omega \nabla \varphi (h\nabla u + u\nabla h) \quad \text {as } \varepsilon \to 0
\end{equation}
by the Lebesgue dominated convergence theorem with the majorant $|\nabla \varphi| (|h||\nabla u| + |u||\nabla h|)$.
In order to prove \eqref{need} we will show
that
$$ \int_{\Omega} \nabla \varphi \cdot (uh\nabla \xi) \to 0.$$
And here is the main place where we use that $F_0$ is the zero set of $u$!
Note that $uh \in C^1_0(B(0,R))$ and vanishes on $F_0$, so $|uh| \leq C_1(u,h)\varepsilon$ in the $\varepsilon$-- neighborhood of the zero set of $u$. Thus $|uh\nabla \xi|$ is bounded by some constant $C(u,h)$ in $B(0,R)$. Also $m_2(\textup{supp} \nabla \xi)$ goes to 0. Hence
$$ \int_\Omega \nabla \varphi \cdot (hu\nabla \xi) \leq C(u,h) \sqrt{m_2(\textup{supp} \nabla \xi)} \sqrt{\int_\Omega |\nabla \varphi|^2} \to 0.$$
By \eqref{have2},\eqref{have3} we obtain
$$\int_{\Omega} \nabla \varphi \cdot(h \nabla u +u\nabla h) = \int_{\Omega} V\varphi uh=\int_{B(0,R)\setminus F_1} V\varphi uh$$
(the second equality is due to the fact that $u=0$ on $F_0$).
Using $\nabla \varphi = \nabla \varphi \mathbbm{1}_\Omega$ in the sense of distributions, we have
\begin{equation} \label{have4}
\int_\Omega \nabla \varphi (h\nabla u + u\nabla h) = \int_{B(0,R)\setminus F_1} \nabla \varphi (h\nabla u + u\nabla h).
\end{equation}
Thus
$$\int_{B(0,R)\setminus F_1} \nabla \varphi \cdot(h \nabla u +u\nabla h) = \int_{B(0,R)\setminus F_1} V\varphi uh$$
and, subtracting \eqref{have1}, we finish the proof of \eqref{need}.
\end{proof}
\subsection{Quasiconformal change of variables.}
We briefly describe some facts from the theory of quasiconformal mappings, which are used in the study of the solutions to equations in divergence form on the plane, and explain why the solutions behave like ordinary harmonic functions. We partially follow the exposition from \cite{NPS}, where the quasiconformal mappings are applied to quasi-symmetry of Laplace eigenfunctions.
Let $B$ be a disk on the plane. Consider a real-valued function $f \in W^{1,2}_{loc}(B)$ satisfying
\begin{equation} \label{eq:div}
\textup{div}(\varphi^2 \nabla f) = 0
\end{equation}
and assume that $ 0< c < \varphi(x) < C < +\infty$ in $B$.
One can find a function $\tilde f \in W^{1,2}_{loc}(B)$ such that
$$ \varphi^2 f_x = \tilde f_y \textup{ and } \varphi^2 f_y = -\tilde f_x$$
(see Section \ref{sec: divergence}) and $f$ appears to be the real part of $w=f+i\tilde f$. A direct computation shows that $w$ is a solution to the Beltrami equation:
\begin{equation} \label{eq:Beltrami}
\frac{\partial w}{\partial \overline z} = \mu \frac{\partial w}{\partial z}
\end{equation}
with the Beltrami coefficient
\begin{equation} \label{eq:mu}
\mu= \frac{1-\varphi^2}{1+\varphi^2}\cdot \frac{f_x+i f_y}{f_x-if_y}.
\end{equation}
When $\nabla f=0$, we put $\mu=0$.
We are going to apply the theory of quasiconformal mappings in a situation when $f=\frac{u}{\varphi} $ and the domain $$\Omega_1:= B(0,R)\setminus F_1$$ is not simply connected. In this case $w$ and $\tilde f$ can be defined only locally, but not in the whole $\Omega_1$. However the Beltrami coefficient $\mu$ is well defined by \eqref{eq:mu} in $\Omega_1$ and
$$|\mu| \leq \frac{1-\varphi^2}{1+\varphi^2} \leq C\varepsilon^2.$$
Let us extend $\mu$ by zero outside $\Omega_1$ to the whole complex plane. Now $\mu$ has a compact support.
The existence Theorem 5.3.2 \cite{AIM09} claims that there is a $K$-quasiconformal homeomorphism $\psi$ of the complex plane such that
\begin{itemize}
\item $\psi \in W^{1,2}_{loc}$,
\item
$ \frac{\partial \psi}{\partial \overline z} = \mu \frac{\partial \psi}{\partial z}$,
\item
$ K \leq \frac{1+\sup |\mu|}{1 - \sup |\mu|}.$
\end{itemize}
In our case $$K \leq 1+ C' \varepsilon^2.$$
\textbf{Claim.} The function $f \circ \psi^{-1}$ is harmonic in $\psi(\Omega_1)$.
Indeed, for any ball $B \subset \Omega_1$, we can define $w\in W^{1,2}_{loc}(B)$ such that $f=\Re w$ and $w$,$\psi$ solve the same Beltrami equation. Stoilow factorization theorem (\cite{AIM09}, p.179, Theorem 5.5.1)
claims that
there is a holomorphic function $W$ such that
$$ w = W(\psi(z))$$
and therefore the harmonic function $\Re W$ satisfies
$$ f(z)= \Re W(\psi(z)).$$
Clearly, the local observation shows that $f\circ \psi^{-1}$ is a harmonic function in $\psi(\Omega_1)$.
Note that $\psi(B(0,R))$ is a simply connected domain (and not the whole plane).
Using the Riemann uniformisation theorem we can find a conformal map that sends $\psi(B(0,R))$ back to $B(0,R)$ and $\psi(0)$ to $0$. The composition of this conformal map and the $K$-quasiconformal homeomorphism $\psi$ will be a $K$-quasiconformal homeomorphism
$g$ of $B(0,R)$ onto itself with $g(0)=0$.
Then the function $h=f \circ g^{-1}$ is harmonic in $g(\Omega_1)$.
\textbf{Distortion of quasiconformal mappings.}
Mori's theorem (\cite{Ahl66}, Chapter III, Section C)
tells us that distances are changed by $g$ in a controlled way:
\begin{equation} \label{eq:distortion}
\frac{1}{16}\left|\frac{z_1 - z_2}{R}\right|^{K} \leq \frac{|g(z_1) - g(z_2)|}{R} \leq 16\left|\frac{z_1 - z_2}{R}\right|^{1/K}.
\end{equation}
We choose $$ \varepsilon = \frac{c}{\sqrt{\log R}} $$
so that $$K\in [1,1+C c^2/ \log R), \quad R^K\asymp R \asymp R^{1/K}$$ and the distortion on scales from $
\frac{1}{R}$ to $R$ is bounded. Namely, we may choose $c$ so small that if $\frac{1}{R}\leq|z_1-z_2| \leq2R$, then
$$\frac{1}{32}|z_1-z_2| \leq|g(z_1)-g(z_2)| \leq 32|z_1-z_2|.$$
Note that in the statement of Theorem \ref{local main} one can safely assume that $R$ is sufficiently large ($R\gg 1$) by rescaling, which makes $\|V\|_\infty$ only smaller. It is needed to make $\varepsilon \geq 1/R$. Then we get a harmonic function $h$ in $B(0,R)\setminus g(F_1)$, where $g(F_1)$ is the union of sets of diameter $\sim \varepsilon$. The image of a single disk of radius $\varepsilon$ will be contained in a disk of radius $32\varepsilon$. Let us denote these disks of radius $32\varepsilon$ by $D_j$. The images of disks from $F_1$ are $ \frac{C}{32}\varepsilon$ -- separated from each other and from the zero set of $h$. Hence $D_j$ are $ (\frac{C}{32}\varepsilon - 128\varepsilon)=C_132\varepsilon$ -- separated from each other and from the zero set of $h$, and $h$ does not change sign in $C_1D_j \setminus D_j$. We have $$C_1 = \frac{C}{32^2} - 4 > 100$$
if $C= 10^6$.
We specifically asked that $0$ and $z_{\max}$ (the point where $\sup_{B(0,R/2)}|u|$ is attained) are $C\varepsilon$ -- separated from the disks. Recall that $g(0)=0$, so the disks $C_1D_j$ do not contain $0$ and $g(z_{\max})$. The distortion estimate implies that
$g(z_{\max}) \in \overline{B(0,R-R/64)}$. Since we had
$$\frac{\sup_{B(0,R)}|u|}{\sup_{B(0,R/2)}|u|} \leq e^N,$$
we conclude that
$$ \frac{\sup_{B(0,R) \setminus \cup 3D_j}|h|}{\sup_{B(0,R-R/64)\setminus \cup 3D_j}|h|} \leq e^N.$$
If we make the rescaling by a factor of $32 \varepsilon$, then the disks $D_j$ become 100-separated unit disks and $R$ becomes $$R'=R\cdot32\varepsilon\sim R\sqrt{\log R}.$$
The goal of Theorem \ref{local main} is to estimate $\sup_{B(0,r)}|u|$ from below.
If $r<1/R$, the image of $B(0,r)$ may have radius significantly smaller than $r$.
However $g(B(0,r))$ contains a disk with center at $0$ of radius $$\frac{R}{16}\left(\frac{r}{R}\right)^{K} \geq \frac{R}{16} \left(\frac{r}{R}\right)^2.$$
Let $\tilde g = \frac{1}{32\varepsilon} g$. Then $\tilde g(B(0,r))$ contains a ball $B(0,r')$, where $$r' \geq
\frac{R'}{16} \left(\frac{r}{R}\right)^2.$$
So $$\frac{R'}{r'} \leq 16 \frac{R^2}{r^2}.$$
In order to prove estimate \eqref{eq: Br}, it is enough to show that
$$ \sup_{B(0,r')\setminus \cup 3D_j}|h| \geq c (r'/R')^{C (R' +N)}\sup_{B(0,R')\setminus \cup 3D_j}|h|. $$
It will be proved in Theorem \ref{local 3}.
\section{Act III} \label{sec: toy}
Before we formulate and prove the promised local Theorem \ref{local 3} we will explain the main idea in the global case.
\begin{theorem}[Toy problem] \label{thm: toy}
Let $\{D_j\}$ be a collection of $100$-separated disks with unit radius on the complex plane
$\mathbb{C}$. Suppose that $u$ is a harmonic function in $\mathbb{C} \setminus \cup_j D_j$ which
preserves sign in each annulus $5D_j\setminus D_j$. If $|u(z)| \leq e^{-L|z|}$ for all $z \in \mathbb{C} \setminus \cup_j D_j$ and $L$ is sufficiently large, then $u \equiv 0$.
\end{theorem}
\begin{proof} We start with a simple observation.
\begin{claim}
Let $m_j= \min\limits_{\partial 3D_j}|u|$. Then for some absolute
constant $A>0$, we have
\begin{enumerate}
\item $\max\limits_{\partial 3D_j}|u| \leq Am_j$,
\item $\max\limits_{\partial 3D_j}|\nabla u| \leq Am_j$.
\end{enumerate}
\end{claim}
\begin{proof}
By the Harnack inequality there exists a constant $A>0$ such that
$$ \sup\limits_{4D_j\setminus 2 D_j}|u| \leq A\inf\limits_{4D_j\setminus 2 D_j}|u| \leq Am_j,$$
which proves the first part of the claim. The second part follows from the Cauchy inequality.
\end{proof}
\newpage
Let $k\in (0,L)$ and consider the numbers $m_j e^{k \Re z_j}$, where $z_j$ is the rightmost point of $3D_j$.
\includegraphics[width=0.5\textwidth]{rightmost.png}
\noindent Since $$m_j \leq |u(z_j)| \leq e^{-L|z_j|},$$
there is $j_0$ such that
$$ m_{j_0} e^{k \Re z_{j_0}}= \max\limits_j m_je^{k \Re z_j}.$$
Now, consider the analytic in $\mathbb{C} \setminus \cup (3D_j)$ function $f=(u_x - i u_y) e^{kz}$.
If $|u(z)| \leq e^{-L|z|}$ in $\mathbb{C} \setminus \cup (D_j)$, then
$$|\nabla u(z)| \leq C \sup_{B(z,1)}| u| \leq Ce^{-L(|z|-1)} \quad \text{ for } z \in \mathbb{C} \setminus \cup (2D_j)$$
and
$f(z) \to 0$ as $z \to \infty$, $z \in \mathbb{C} \setminus \cup (2D_j)$.
So, by the maximum principle, there exists $j_1$ such that
$$ \max\limits_{\mathbb{C} \setminus \cup (3D_j)}|f|= \max\limits_{\partial 3D_{j_1}} |f| \leq Am_{j_1} e^{k\Re z_{j_1}} \leq Am_{j_0} e^{k\Re z_{j_0}},$$
whence $|\nabla u| \leq Am_{j_0} e^{-k\Re(z-z_{j_0})}$ in $\mathbb{C} \setminus \cup (3D_j)$. We may assume that
$m_{j_0}\neq 0$, otherwise $u$ is constant and therefore zero.
Now, consider the ray $\{z_{j_0} + y: y \in (0,+\infty) \}$.
There are two possibilities:
\begin{itemize}
\item[(i)] The ray goes to $\infty$ without hitting any other disks $(3D_j)$.
Then for any $y>0$,
$$|u(z_{j_0}+y) - u(z_{j_0})| \leq \int_{0}^{\infty}|\nabla u(z_{j_0}+t)| dt \leq \int_{0}^{\infty} Am_{j_0} e^{-kt}dt = \frac A k m_{j_0}. $$
Since $|u(z_{j_0})| \geq m_{j_0}$, wee see that $|u|$ stays bounded from below by
$(1- \frac A k)m_{j_0}$ on the ray. If $k>A$, this contradicts the decay assumption.
\newpage
\item[(ii)] The ray hits another disk $3D_j$
\includegraphics[width=0.9\textwidth]{ray6}
\noindent at some point $z_{j}'= z_{j_0}+y$.
Then we still have $|u(z_j')|\geq (1-\frac A k )m_{j_0}$
and, due to the fact that the disks are separated,
$$ \Re (z_{j}' - z_{j_0} )=|z_{j}' - z_{j_0}| \geq 1.$$
Hence $$m_j e^{k\Re z_j } \geq m_j e^{k \Re z_j'} \geq \frac{|u(z_j')|}{A} e^{k(\Re z_{j_0} +1)} \geq$$
$$\geq \frac{1}{A}(1 - \frac{A}{k})e^k m_{j_0} e^{\Re z_{j_0}} > m_{j_0} e^{\Re z_{j_0}} $$
as soon as $k>2A$, which contradicts the choice of $j_0$.
\end{itemize}
\noindent This proves the theorem with any $L>2A$.
\end{proof}
Now we formulate and prove the harmonic counterpart of the main local theorem.
\begin{theorem} \label{local 3}
Let $D_j$ be a collection of 100 -- separated unit disks on $\mathbb{R}^2=\mathbb{C}$ such that
$0 \notin \cup 3D_j$. Let $R> 10^4$, $0<r\leq R/4$.
Consider any harmonic function $u$ in $B(0,R) \setminus \cup D_j$ such that $u$ does not change sign in
$(5D_j\setminus D_j) \cap B(0,R)$ for every $j$. Assume that
$$\sup \limits _{B(0,R-R/64) \setminus
\cup 3D_j}|u| \geq e^{- N} \sup \limits_{B(0,R) \setminus
\cup 3D_j}|u|.$$
Then
\begin{equation}\label{eq:*}
\sup \limits _{B(0,r) \setminus
\cup 3D_j}|u| \geq \left(\frac r R \right)^{ C (R + N) } \sup \limits_{B(0,R) \setminus
\cup 3D_j}|u|,
\end{equation}
with some absolute constant $C>0$.
\end{theorem}
\begin{proof}
WLOG, $\sup \limits _{B(0,R-R/64) \setminus
\cup 3D_j}|u|=1$. Fix $k=[C(N+R)]$ with sufficiently large $C>0$ and assume that
$$ \sup \limits _{B(0,r) \setminus
\cup 3D_j}|u| \leq \left(\frac r R \right)^{3k}.$$
Consider the
domain $$\Omega:=\{r/2<|z|< R - 1 \} \setminus \cup (3D_j).$$
Let $W_1$ be the connected component of $\partial \Omega$ that intersects $\partial B(0,r/2)$.
Note that each point of $W_1$ is either on $\partial B(0,r/2)$ or lies on some $\partial 3D_j$ that intersects $\partial B(0,r/2)$.\\
\textbf{Estimate on $W_1$.}
Recall that if $5D_j \subset B(0,R)$, we have
\begin{enumerate}
\item $\max\limits_{\partial3 D_j }|u| \leq A \min\limits_{\partial 3D_j}|u|$,
\item $\max\limits_{\partial 3D_j}|\nabla u| \leq A \min\limits_{\partial 3D_j}|u|$.
\end{enumerate}
Hence on $W_1 \setminus \partial B(0,r/2) $, we have
$$ |u|,|\nabla u| \leq A \sup_{B(0,r)\setminus \cup (3D_j)} |u| \leq A \left(\frac r R \right)^{3k}. $$
If $x \in \partial B(0,r/2) \setminus \cup (3D_j)$, then either $x \in 4D_j$ for some $j$ or $B(x, min(1,r/2)) \subset B(0,r) \setminus \cup (3D_j) $. In the first case $u$ does not change sign in $B(x,1)$ and
$$|\nabla u(x)| \leq A|u(x)| \leq A\sup_{B(0,r)\setminus \cup (3D_j)} |u|\leq A\left(\frac r R \right)^{3k}.$$
In the second case, we have
$$ |\nabla u(x)| \leq\frac{A}{\min(1,r/2)}\sup_{B(0,r)\setminus \cup (3D_j)}|u| \leq \frac{A}{\min(1,r/2)} \left(\frac r R \right)^{3k}. $$
Thus in all cases, if $C$ in the definition of $k$ is large enough, we have
$$ \max_{W_1}|u|, \max_{W_1}|\nabla u| \leq \frac{A}{\min(1,r/2)} \left(\frac r R \right)^{3k} \leq \left(\frac r R \right)^{2k} \quad $$
because $$\left( \frac R r \right)^{k} \geq 4^k > A$$
and
$$\left( \frac R r \right)^{k} \geq 4^{k-1} \frac R r \geq \frac{2A}{r}.$$
Let $W_2$ be the connected component of $\partial \Omega$ that intersects $\partial B(0,R-1)$.
Note that each point of $W_2$ is either on $\partial B(0,R-1)$ or lies on some $\partial 3D_j$ that intersects $\partial B(0,R-1)$.\\
\textbf{Estimate on $W_2$.} Any point $x\in \overline{B(0,R-1)}\setminus \cup (3D_j)$ is either in $4D_j$ for some $j$
or $x\in \overline{B(0,R-1)}\setminus \cup (4D_j)$. In the first case $u$ does not change sign in $B(x,1)$ and therefore
$$|\nabla u(x)|\leq A|u(x)| \leq A e^N.$$
In the second case $B(x,1) \subset B(0,R)\setminus \cup (3D_j)$ and $|\nabla u(x)| \leq A e^N$.
Thus
$$ \max_{W_2}|u|, \max_{W_2}|\nabla u| \leq A e^N.$$
Note also that
$$ W_2 \subset \overline{B(0,R-1)}\setminus B(0,R-7).$$
Now, consider the analytic in $\Omega$ function
$$f(z)=\frac{u_x-iu_y}{z^k}, \quad |f(z)|=\frac{|\nabla u(z)|}{|z|^k}.$$
Since $$\sup\limits_{B(0,R-R/64)\setminus \cup (3D_j)}|u|=1> \sup_{B(0,r/2)\setminus \cup 3D_j}|u|,$$ $$\max\limits_{W_1} |u| \leq \left( \frac r R \right)^{2k} < \frac{1}{2},$$
and since any point in $$\Omega_1=B(0,R-R/64)\setminus \cup (3D_j)$$ can be connected with $W_1$ by a curve of length at most $4R$ within $\Omega_1$, we must
have $$\sup_\Omega|\nabla u| \geq \sup_{\Omega_1}|\nabla u| \geq \frac{1}{8R}$$ and
$$ \sup_{\Omega_1} |f|\geq \left( R-R/64\right)^{-k} \sup_{\Omega_1}|\nabla u| \geq \frac{1}{8R} \left( R-R/64 \right)^{-k}.$$
However
$$ \max_{W_1}|f|\leq \max_{W_1}|\nabla u |\left(\frac{2}{r} \right)^k \leq \left(\frac{r}{R} \right)^{2k}\left(\frac{2}{r} \right)^k= \left(\frac{2r}{R} \right)^{k} R^{-k} \leq $$ $$ \leq 2^{-k} R^{-k} < \frac{1}{8R} R^{-k} < \sup_{\Omega_1} |f|\leq \sup_{\Omega} |f| $$
and
$$ \max_{W_2}|f|\leq A e^N \frac{1}{(R-7)^k} = A e^N \left( \frac{R-R/64}{R-7}\right)^k \left( R-R/64\right)^{-k}\leq
$$ $$\leq A e^N \left( \frac{126}{127}\right)^k \left( R-R/64\right)^{-k} < \frac{1}{8R} \left( R-R/64\right)^{-k} \leq \sup_{\Omega} |f| $$
if $R-7> \frac{127}{128}R $ and $C$ in the definition of $k$ is large enough.
By the maximum principle for holomorphic functions $\sup_\Omega |f|$ is achieved on $\partial 3D_j$ for some
$3D_j\subset B(0,R-1)\setminus \overline{B(0,r/2)}$.
For every disk $D_j$ with $3D_j\subset B(0,R-1)\setminus \overline{B(0,r/2)}$, consider the point $z_j$ on $\partial3D_j$ closest to the origin. All $3D_j$ that are not in the annulus $B(0,R-1)\setminus \overline{B(0,r/2)}$ will not be considered further. Put $m_j=\min_{\partial3D_j}|u|$. Let $j_0$ be the index such that $$\frac{m_{j_0}}{|z_{j_0}|^k} = \max_j \frac{m_{j}}{|z_{j}|^k}.$$
If $\sup_\Omega|f|$ is achieved on $\partial3D_{j_1}$, then for $x\in \Omega$,
$$ \frac{|\nabla u(x)|}{|x|^k}\leq \frac{1}{|z_{j_1}|^k} \max_{\partial3D_{j_1}}|\nabla u|\leq A \frac{m_{j_1}}{|z_{j_1}|^k} \leq A \frac{m_{j_0}}{|z_{j_0}|^k}.$$
So we conclude that
$$|\nabla u(x)| \leq \left( \frac{|x|}{|z_{j_0}|} \right)^k A m_{j_0}.$$
Recalling that $\frac{1}{8R} \leq \sup_\Omega|\nabla u|$, we get
$$ \frac{1}{8R} \leq \left(\frac{2R}{r}\right)^k Am_{j_0},$$
so
$$ |u(z_{j_0})| \geq m_{j_0} \geq \frac{1}{8AR} \left(\frac{r}{2R}\right)^k.$$
Now, let $(sz_{j_0},z_{j_0})$ be the longest subinterval of the radius $[0,z_{j_0}]$ starting at $z_{j_0}$ that is contained in $\Omega$. We have
$$|u(sz_{j_0})| \geq |u(z_{j_0})| - |z_{j_0}| \int_{s}^{1}|\nabla u(t z_{j_0})| dt \geq |u(z_{j_0})| - |z_{j_0}| \int_{0}^{1} Am_{j_0} t^k dt \geq $$
$$ \geq m_{j_0}(1-\frac{AR}{k+1}) \geq \frac{m_{j_0}}{2}$$
if the constant $C$ in the definition of $k$ is large enough.
Note that $$\left(\frac{R}{r}\right)^{k} \geq 4^{k} > 16 AR \cdot 2^k.$$
Hence $$\frac{m_{j_0}}{2} \geq \frac{1}{16AR} \left(\frac{r}{2R}\right)^k > \left(\frac{r}{R}\right)^{2k}$$
and the point $sz_{j_0}$ cannot belong to $W_1$, whence it belongs to some $\partial 3D_j$ with $3D_j \subset B(0,R-1)\setminus \overline{B(0,r/2)}$.
Then
$$ \frac{m_j}{|z_j|^k} \geq \frac{|u(sz_{j_0})|}{A|sz_{j_0}|^k} \geq \frac{1}{2As^k} \frac{m_{j_0}}{|z_{j_0}|^k}.$$
It remains to notice that, since the distance from $3D_j$ to $3D_{j_0}$ is at least $96$, we have
$$ s^k \leq (1-96/R)^k< \frac{1}{2A} $$
if the constant $C$ in the definition of $k$ is large enough.
But then $\frac{m_j}{|z_j|^k} > \frac{m_{j_0}}{|z_{j_0}|^k}$, which contradicts the choice of $j_0$.
\end{proof}
\section{Appendix. }
\subsection{The toy problem for harmonic functions in higher dimensions: a proof with extra logarithm.}
Here we present another proof of a slightly worse bound for the toy problem for harmonic functions in a punctured domain. However this proof works in higher dimensions.
We will denote by $B_R$ the ball in $\mathbb{R}^n$ with center at $0$ and of radius $R$.\\
\textbf{Toy problem with extra logarithm.}
Let $D_j$ be a collection of unit, $100$ -- separated balls on the plane and let $R>100$.
Then for any harmonic function $h$ in $B_R \setminus \cup D_j$ such that $h$ does not change sign in each $B_R\cap 5D_j\setminus D_j$,
we have
$$ \int\limits_{B_{R}\setminus(B_{R/2} \bigcup (\cup 3D_j))} h^2 \geq \exp(-CR\log R) \int\limits_{B_{R/2}\setminus \cup 3D_j} h^2,$$
where $C$ is an absolute positive constant.
\noindent This inequality implies that Theorem \ref{thm: toy} holds in higher dimensions if we assume that
$|u(z)| \leq e^{-L|z|\log|z|}$ for sufficiently large $L$.
\begin{proof}
The proof is based on the Carleman inequality with log linear weight.
Most of Carleman inequalities require strict log convexity-type properties of the weight.
The next inequality is an exception:
\begin{equation} \label{eq: Carleman}
\int_{B_R} |{\Delta} u|^2 e^{kx_1} \geq \frac{ck^2}{R^2} \int_{B_R} u^2 e^{kx_1}
\end{equation}
for any $u\in C^2_0(B_R)$. The inequality is not difficult to prove.
Let $v=ue^{kx_1/2}$, then
$$ e^{kx_1/2} \Delta u = \Delta v - k v_{x_1} + \frac{k^2}{4} v$$
and
$$\int_{B_R} |{\Delta} u|^2 e^{kx_1}= \int_{B_R} |\Delta v + \frac{k^2}{4} v|^2 + \int_{B_R} |k v_{x_1}|^2 - \int_{B_R} 2 (\Delta v + \frac{k^2}{4} v) kv_{x_1}. $$
Note that
$$\int_{B_R} 2 v v_{x_1} =\int_{B_R} \frac{\partial}{\partial x_1}v^2=0.$$
Integrating by parts, we see that
$$-\int_{B_R} \Delta v_{x_1} v=\int_{B_R} \Delta v v_{x_1} = \int_{B_R} v \Delta v_{x_1}$$
and therefore $$\int_{B_R} \Delta v v_{x_1} = 0.$$
Hence
$$\int_{B_R} |{\Delta} u|^2 e^{kx_1}= \int_{B_R} |\Delta v + \frac{k^2}{4} v|^2 + \int_{B_R} |k v_{x_1}|^2 \geq \int_{B_R} |k v_{x_1}|^2 \geq $$
(by Poincare's inequailty)
$$ \geq \frac{\pi^2}{4}\frac{k^2}{ R^2} \int_{B_R} v^2= c\frac{k^2}{ R^2} \int_{B_R} u^2 e^{kx_1}.$$
So we proved \eqref{eq: Carleman} and would like to apply it for the harmonic function $h$. However $h$ is not in
$C^2_0(B_{R})$ and inequality \eqref{eq: Carleman} should be applied to $$u=h\eta,$$
where $\eta$ is a positive, $C^2$-smooth cut-off function:
\begin{itemize}
\item $\eta=0$ in each $2D_j$ and in $\{x: |x|>R-11\}$,
\item $\eta=1$ in $B_{\frac{3}{4}R \setminus \cup 3D_j}$,
\item the function $\eta$, as well as its first and second derivatives are bounded by a numerical constant.
\end{itemize}
We will choose the parameter $k$ later. For now we have
$$\int_{B_{\frac{3}{4}R}\setminus\cup 3D_j} |{\Delta} h|^2 e^{kx_1} + \textup{``cut-off integrals"} \geq \frac{ck^2}{R^2} \int_{B_{\frac{3}{4}R}\setminus \cup 3D_j} h^2 e^{kx_1} =: \textup{RHS}.$$
It is good that $\Delta h=0$, so only the cut-off integrals are left on the left-hand side.
There are two kinds of cut-off integrals:
$$\textup{I}= \sum\limits_{5D_j \subset B_{\frac{3R}{4}}} \int_{3D_j\setminus 2D_j} \textup{``cut-off terms"}$$
and
$$ \textup{II}= \int_{B_{R-11}\setminus B_{(\frac{3R}{4}-10)}} \textup{``cut-off terms"}$$
where
$$ | \textup{``cut-off terms"}| \lesssim (h^2+|\nabla h|^2)e^{kx_1}$$
(recall that $\Delta h =0$).
Note that $$\int_{3D_j\setminus 2D_j} e^{kx_1} \lesssim e^{-k/2}\int_{4D_j\setminus 3D_j} e^{kx_1}$$
because $4D_j\setminus 3D_j$ contains an open disk of radius $\frac{1}{4}$, where the function $e^{kx_1}$
is pointwise bigger than $e^{k/2} \cdot \sup_{3D_j\setminus 2D_j} e^{kx_1}$.
Now, assuming $5D_j \subset B_R$ we will use the sign condition in $5D_j\setminus D_j$. By the Harnack inequality and the Cauchy estimate
we know that there is a constant $a_j\geq 0$ such that
$$ |h| \asymp a_j \textup{ and } |\nabla h| \lesssim a_j \textup{ in } 4D_j\setminus 2D_j.$$
So
$$\int_{3D_j\setminus 2D_j} \textup{``cut-off terms"} \lesssim
a_j^2 \int_{3D_j\setminus 2D_j} e^{kx_1} \lesssim a_j^2 e^{-k/2} \int_{4D_j\setminus 3D_j} e^{kx_1} \lesssim $$
$$ \lesssim e^{-k/2} \int_{4D_j\setminus 3D_j} h^2 e^{kx_1}.$$
Hence $$ \textup{I} \lesssim e^{-k/2}\int_{B_{\frac{3}{4}R}\setminus \cup 3D_j} h^2 e^{kx_1}.$$
Note that
$$\textup{RHS}= \frac{ck^2}{R^2} \int_{B_{\frac{3}{4}R}\setminus \cup 3D_j} h^2 e^{kx_1} > 2 \textup{I}$$
if $$ k^2/{R^2} \gg e^{-k/2}.$$
We make the choice $$ k = C \log R$$
and it yields
$$ \sup_{B_{R}} e^{kx_1} \leq e^{CR\log R}.$$
Since $$\textup{I}+ \textup{II} \geq \textup{RHS} \quad \textup{and} \quad I\leq \frac{1}{2}\textup{RHS},$$
we have
$$ \int_{B_{R-11}\setminus B_{(\frac{3R}{4}-10)}} \textup{``cut-off terms"} = \textup{II} \geq \frac{1}{2}\textup{RHS} \asymp k^2/{R^2} \int_{B_{\frac{3}{4}R}\setminus \cup 3D_j} h^2 e^{kx_1} \geq $$
$$ \geq \exp(-CR\log R)\int_{B_{\frac{1}{2}R}\setminus \cup 3D_j} h^2.$$
If $5D_j \subset B_R$, then $\int_{3D_j\setminus 2D_j} h^2 \asymp \int_{4D_j\setminus 3D_j} h^2$, whence
$$ \int\limits_{B_{R}\setminus(B_{R/2} \bigcup (\cup 3D_j))} h^2
\geq c_1 \int\limits_{B_{R-10}\setminus(B_{R/2} \bigcup(\cup 2D_j))} h^2 \geq $$
(by Cauchy estimate)
$$\geq c_2 \int\limits_{B_{R-11}\setminus(B_{R/2} \bigcup(\cup 3D_j))} \! \! \! (h^2 +|\nabla h|^2)$$
and therefore
$$ \sup_{B_{R}} e^{kx_1} \int\limits_{B_{R}\setminus(B_{R/2} \bigcup (\cup 3D_j))} h^2 \geq c_2 \int\limits_{B_{R-11}\setminus(B_{R/2} \bigcup(\cup 3D_j))} (h^2 +|\nabla h|^2) e^{kx_1} \geq c_3 \textup{II}. $$
Thus
$$ \int\limits_{B_{R}\setminus(B_{R/2} \bigcup( \cup 3D_j))} h^2 \geq \exp(-C'R\log R) \int\limits_{B_{R/2}\setminus \cup 3D_j} h^2.$$
\end{proof}
\noindent \textbf{Deduction of Theorem \ref{exercise 1} from Theorem \ref{local 3}}.
We may assume that $$\sup\limits_{B(0,R')\setminus \cup 3D_j}|h|=\sup\limits_{B(0,R'/8)\setminus \cup 3D_j}|h|,$$ otherwise the statement is trivial.
Consider any point $x$ on $\partial B(0,R'/4)\setminus \cup 3D_j$.
Note that $ B(0,R'/8) \subset B(x,3R'/8)$ and $B(x,3R'/4) \subset B(0,R')$. Hence
$$ \sup\limits_{B(x,3R'/8)\setminus \cup 3D_j}|h|=\sup\limits_{B(x,3R'/4)\setminus \cup 3D_j}|h|.$$
Applying Theorem \ref{local 3} for the disk with center at $x$ (in place of $0$) of radius $R=3R'/4$ and $N=0$, we obtain the bound
$$\sup\limits_{\{R'/8<|z|< R' \}\setminus \cup 3D_j}|h|\geq \sup\limits_{B(x,R'/8)\setminus \cup 3D_j}|h| \geq e^{-CR'}\sup\limits_{B(0,R')\setminus \cup 3D_j}|h|.$$
\subsection{Sketches of general elliptic theory.}
\begin{fact} \label{fact1}
Denote by $E(z)=
\frac{1}{2\pi} \log|z|$ the fundamental solution of the Laplace operator on the plane
in the sense that for every $C^\infty$ compactly supported function $h$, we have
$$ h = E * \Delta h.$$
\end{fact}
\begin{fact} \label{fact2}
Let $\Omega$ be a bounded open set and $g \in L^1(\Omega)$. Put $f= E* g$.
Then
\begin{enumerate}
\item $f\in L^p(\Omega)$ for all $p \geq 1$.
\item $\Delta f =g $ in the sense that for every $h \in C_0^\infty(\Omega)$, we have
$$\int_{\Omega} f \Delta h = \int_{\Omega} gh.$$
\end{enumerate}
\end{fact}
\noindent \textbf{Agreement}. Writing $E*g$ we assume that $g$ is extended to $\mathbb{R}^2\setminus \Omega$ by zero.
\begin{proof} \quad
\begin{enumerate}
\item
Let $D=\textup{diam }\Omega$. Let $E_D= \mathbbm{1}_{B(0,D)} E$. Then in $\Omega$ we have $f=g* E_D$.
Since $E_D \in L^p(\mathbb{R}^2)$ for all $p\geq 1$ and $g\in L^1$, the result follows from Young's convolution inequality.
\item
We have
$$\int f\Delta h = \int (E*g)\Delta h =\int \limits_{\Omega \times \Omega} E(z-\zeta)g(\zeta) \Delta h(z)dm_2(z)dm_2(\zeta)=$$
$$= \int\limits_{\Omega} g(\zeta) \left[ \int\limits_{\Omega} E(z-\zeta)\Delta h(z) dm_2(z) \right] dm_2(\zeta)= \int\limits_\Omega gh. $$
\end{enumerate}
\end{proof}
\begin{fact} \label{fact3}
Let $V \in L^\infty(\Omega)$, $u\in L^1_{loc}(\Omega)$ and $\Delta u+ Vu=0$ in $\Omega$
in the sense that for every $h\in C^{\infty}_{0}(\Omega)$ we have
$$\int\limits_{\Omega}u\Delta h + \int\limits_{\Omega}Vuh=0.$$
Then $u \in L^p_{loc}(\Omega)$ for every $ p\geq 1$.
\end{fact}
\begin{proof}
Passing to a smaller bounded domain $\Omega'$, we may assume that $u\in L^1(\Omega)$, $\Omega$ is bounded.
Consider $f= E * (Vu)$. By Fact \ref{fact2}, $f \in L^p(\Omega)$ for all $p \geq 1$. Note that $u-f \in L^1(\Omega)$ and $\Delta(u-f)=0$ in the sense of distributions. Hence, by Weyl's lemma, $u-f$ is harmonic in $\Omega$,
so $u-f \in L^p_{loc}(\Omega)$ and therefore $u \in L^p_{loc}(\Omega) $ too.
\end{proof}
\begin{fact} \label{fact4}
Let $g \in L^p(\Omega)$ with $p>2$ and
let $\Omega$ be bounded. Then $g * E \in C^1(\Omega)$.
\end{fact}
\begin{proof}
$$E(z+t)-E(z)= \frac{1}{2\pi}(\log|z+t|-\log|z|)= \frac{1}{2\pi} \log\left|1+\frac{t}{z}\right| =$$
$$= \frac{1}{2\pi}
\Re \frac{t}{z} + O\left(
\begin{cases}
\frac{|t|^2}{|z|^2}, \quad \frac{1}{2} \geq \frac{|t|}{|z|} \\
\frac{|t|}{|z|}+|\log \bigl|1+\frac{t}{z}|\bigr|,\quad \frac{1}{2} \leq \frac{|t|}{|z|}
\end{cases}
\right).
$$
Define $$ W(\zeta) =\begin{cases}
\frac{1}{|\zeta|^2}, \quad \zeta > 2 \\
\frac{1}{|\zeta|}+\bigl|\log|1+\frac{1}{\zeta}|\bigr|,\quad \zeta \leq 2.
\end{cases}$$
Taking the convolution and applying Holder's inequality, we have
$$ (g*E)(z+t)- (g*E)(z) = \left[ g* \frac{1}{2\pi}
\Re \frac{t}{\cdot} \right](z) + O\left(\|g\|_p \|W(\cdot/t)\|_q\right) = $$
$$= \Re\left(t \left[g*\frac{1}{2\pi \cdot} \right] (z)\right) + O\left(\|W(\cdot/t)\|_q\right), \quad \text{where } 1/p+1/q=1.$$
Since $g*\frac{1}{2\pi \cdot} \in C(\Omega)$, the first term (as a function of $t \in \mathbb{C}=\mathbb{R}^2$) is a linear operator
from $\mathbb{R}^2$ to $\mathbb{R}$, which depends continuously on $z$.
It is enough to show that $\|W\|_q< \infty$ because $$\|W(\cdot/t)\|_q = \|W\|_q |t|^{2/q} = \|W\|_q \,o(t) \quad \text{ as } t \to 0$$
($1<q<2$ if $p>2$).
Indeed,
$$\int|W|^q \lesssim \int_{|\zeta|>2} \frac{1}{|\zeta|^{2q}} + \int_{|\zeta|\leq 2} \left( \frac{1}{|\zeta|^q}+ \left|\log\Bigl|1+\frac{1}{\zeta}\Bigr|\right|^{q} \right)< \infty.$$
\end{proof}
\begin{fact} \label{fact5}
Let $V \in L^\infty(\Omega)$. If $\Delta u + Vu=0$ in $\Omega$ in the sense of distributions and
$u \in L^1_{loc}(\Omega)$, then $u \in C^1(\Omega)$.
\end{fact}
\begin{proof}
By Fact \ref{fact3}, $u \in L^p_{loc}(\Omega)$ with $p > 2$.
Again passing to a subdomain, if necessary, we may assume that $\Omega$ is bounded and
$u \in L^p(\Omega)$. Consider $f= E*(Vu)$. Since $Vu \in L^p(\Omega)$, $f \in C^1(\Omega)$.
However $u-f$ is harmonic. Hence $u \in C^1(\Omega)$.
\end{proof}
\begin{lemma} \label{lem: solving}
Let $\Omega$ be a bounded domain with Poincare constant smaller than 1.
Then for any $v\in L^\infty(\Omega)$, we can find a solution $u \in W^{1,2}_0(\Omega)$ to
$\Delta u = v$ in the sense of distributions
such that
$$\|u\|_{W^{1,2}_0(\Omega)} \leq 4 \|v\|_2.$$
\end{lemma}
\begin{remark} Note that if $u \in W^{1,2}_{loc}(\Omega)$ and $h \in C^\infty_0(\Omega)$, then
$$ \int_\Omega \nabla u \nabla h = -\int_\Omega u \Delta h.$$
So $u \in W^{1,2}_{loc}(\Omega)$ is a solution to $\Delta u = v$ in the sense of distributions if and only if
$$ \int_\Omega \nabla u \nabla h = -\int_\Omega v h $$
for any $h \in C^\infty_0(\Omega)$.
\end{remark}
\begin{proof}
Consider the functional
$$ \Phi(u)= \int |\nabla u|^2 + \int vu$$
for $u \in W_{0}^{1,2}(\Omega)$. Integrals in the next few lines will be over the domain $\Omega$. Notice that by Poincare's inequality
$$\Phi(u) \geq \frac{1}{2} \int |\nabla u|^2 + \frac{1}{2} \int |u|^2 - \|v\|_2\|u\|_2 \geq$$
$$\geq \frac{1}{2} \|u\|^2_{W_{0}^{1,2}} - \|v\|_2\|u \|_{W_{0}^{1,2}} \geq -\frac{1}{2}\|v\|_2. $$
Thus $\Phi(u)$ is bounded from below. Note that $\Phi(u)>0=\Phi(0)$
as soon as $\|u\|_{W_{0}^{1,2}} > 2\|v\|_2.$
Let now $u_k\in C_0^{\infty}(\Omega)$ be any minimizing sequence for $\Phi$. Note that
$$ \frac{\Phi(u')+ \Phi(u'')}{2} - \Phi\left(\frac{u'+u''}{2}\right) = \frac{1}{4} \int |\nabla(u'-u'')|^2 \geq \frac{1}{8} \|u'- u''\|^2_{W_0^{1,2}}.$$
Hence $u_k$ is a Cauchy sequence in $W_0^{1,2}(\Omega)$, so the limit $u=\lim u_k$ exists and minimizes $\Phi$.
Now, take any test function $h \in C_0^\infty(\Omega)$
and consider
$$ \Phi(u+th)= \Phi(u) + t \left(2\int \nabla u \nabla h + \int vh \right)
+t^2 \int |\nabla h|^2.$$
Since $u$ is a minimizer, we must have
$$2\int \nabla u \nabla h + \int vh=0,$$ i.e.,
$\Delta u = v/2$ in the sense of distributions.
Taking $2u$ in place of $u$ we get a solution to $\Delta u = v$ with
$$\|u\|_{W_0^{1,2} } \leq 4 \|v\|_2.$$
\end{proof}
The next step is to
show that $$\| u\|_\infty \leq C\| v\|_\infty$$
with some absolute constant $C>0$.
WLOG, we will assume $|v|\leq 1$.
\subsection{Uniform bound via Di Giorgi method}
Let $\Omega$ be any bounded open set in $\mathbb{R}^2$ with
Poincare constant $k^{2} \leq k_{0}^{2},$ i.e.,
\[
\int u^{2} \leq k^{2} \int|\nabla u|^{2} \text { for all } u \in C_{0}^{\infty}(\Omega).
\]
\textbf{Claim I}: Let $k_0$ be sufficiently small and consider any smooth
$\varepsilon-$minimizer of
\[
\Phi(u)=\int_{\Omega}|\nabla u|^{2}+\int_{\Omega} vu \quad\left(\|v\|_{\infty} \leq 1\right)
, \text{ i.e.}, \]
$u\in C^\infty_0(\Omega)$ and for any $\tilde u\in C^\infty_0(\Omega)$, $\Phi(\tilde u) \geq \Phi(u) -\varepsilon$.
Then $u$ satisfies $$\int_{B\cap \Omega} u^{2} \leq Ck^2(k^2+\varepsilon)$$
for every unit ball $B \subset \mathbb{R}^{2}$.
\begin{proof}
Let $\varphi(x)$ be a smooth positive radial function such that
\begin{itemize}
\item $\varphi(x)=1$ in $B(0,1)$,
\item $\varphi(x) \in (0,1]$,
\item
$\varphi(x) \asymp e^{-|x|}$,
\item $|\nabla \varphi| \leq \varphi$.
\end{itemize}
Let $ \psi=\varphi^{2}, \text { so }|\nabla \psi| \leqslant 2|\psi|$.
Applying the Poincare inequality to $\varphi u$, we get
$$
\int \varphi^{2} u^{2}\leq k^{2} \int|\varphi \nabla u+u \nabla \varphi|^{2}
\leq 2 k^{2}\left(\int \varphi^{2}|\nabla u|^{2}+\int u^{2}|\nabla \varphi|^{2}\right) \leq $$
$$\leq 2k^{2} \int \varphi^{2}|\nabla u|^{2}+2 k^{2} \int u^{2} \varphi^{2}, \text { whence } $$
$$ \int \varphi^2 u^2 \leq \frac{2 k^{2}}{1-2 k^{2}} \int \varphi^2|\nabla u|^{2}, \text{ i.e.}, $$
\begin{equation} \label{eq: *}
\int u^{2} \psi \leq \frac{2 k^{2}}{1-2 k^{2}} \int |\nabla u|^{2} \psi \leq 4 k^{2} \int |\nabla u|^{2} \psi \quad \text{ if } \quad k_{0} \leq \frac{1}{2}.
\end{equation}
Now, consider the competitor $\tilde{u}=(1-\psi) u$.
We have
\[
\quad \Phi(\tilde{u})=\int|(1-\psi) \nabla u-u \nabla \psi|^{2}+\int v(1-\psi) u\leq \]
\[\leq \int(1-\psi)^{2}|\nabla u|^{2}+2 \int|\nabla u| | u||\nabla \psi|+\int u^{2}|\nabla \psi|^{2}+\int v(1-\psi) u
\]
$$\leq \int(\nabla u)^{2}+\int v u-\int |\nabla u|^{2}\psi+4 \int |\nabla u| | u| \psi +4 \int u^{2} \psi-\int v \psi u$$
(we used the inequalities $(1-\psi)^{2} \leq 1-\psi, |\nabla \psi| \leqslant 2 \psi, \psi^{2} \leq \psi$).
Since $\Phi(\tilde{u}) \geqslant \Phi(u)-\varepsilon,$ we must have
\[
\int|\nabla u|^{2} \psi \leq 4\left(\int|\nabla u| |u| \psi+\int u^{2} \psi\right)+\varepsilon+\int | v \psi u |.
\]
However
\[
\int u^{2} \psi \leqslant 4 k^{2} \int |\nabla u|^{2} \psi \quad \text { by \eqref{eq: *}, }
\]
and
$$\int|\nabla u| |u| \psi \leq \sqrt{\int u^{2} \psi} \sqrt{\int | \nabla u|^{2} \psi} \leqslant 2 k \int|\nabla u|^{2} \psi.$$
So for sufficiently small $k_0$,
$$ 4\left(\int|\nabla u| |u| \psi+\int u^{2} \psi\right) \leq \frac{1}{2} \int|\nabla u|^{2} \psi.$$
Hence
\[
\int|\nabla u|^{2} \psi\leq 2\varepsilon+ 2\int|v u \psi |
\]
\[
\leq 2\left(\varepsilon+\sqrt{\int \psi} \sqrt{\int u^{2} \psi} \, \, \right) \leqslant C\left(\varepsilon+ k \cdot \sqrt{\int |\nabla u|^{2} \psi} \, \,\right).
\]
If the first term dominates, then
$\int|\nabla u|^{2} \psi \leqslant C \varepsilon $. Otherwise $ \int|\nabla u|^{2} \psi \leqslant C k^{2}$.
By \eqref{eq: *} it follows that $$\int u^{2} \psi \leq C k^{2}\left(k^{2}+\varepsilon\right).$$
\end{proof}
Note that we did not care in Claim I where the Poincare constant came from and what was special about the geometry of $\Omega$ that made it small. The next lemma gives a simple bound for the Poincare constant of ``thin" domains.
\begin{lemma}\label{le: thin}
Assume $\Omega$ is open and $m_2(\Omega \cap Q)\leq c < 1$ for all unit squares $Q\subset \mathbb{R}^2$. Then the Poincare constant of $\Omega$ is at most $2 + \frac{2}{1-c}$.
\end{lemma}
\begin{proof}
Let $f \in C_0^{\infty}(\Omega)$.
Extend $f$ by zero outside $\Omega$. It is sufficient to show that
if $Q$ is a unit square, then
$$ \int_Q | f|^2 \leq \left(2 + \frac{2}{1-c}\right)\int_Q | \nabla f|^2.$$
By tiling the plane with unit squares, it implies
$$ \int_\Omega | f|^2 \leq \left(2 + \frac{2}{1-c}\right) \int_\Omega | \nabla f|^2.$$
Let $Q=[0,1]^2$,
$$I_x=((x,y): y\in [0,1]) \quad \text{ and } \quad I^y=((x,y): x\in [0,1]).$$
Let $X_0$ be the set of $x\in [0,1]$ such that $I_x$ contains a zero of $f$. Then
for every $x\in X_0$, we have
$$\max_{I_x} |f| \leq \int_{I_x} |\nabla f|$$
and
$$\int_{I_x} f^2 \leq \max_{I_x} f^2 \leq \left(\int_{I_x} |\nabla f|\right)^2 \leq \int_{I_x} |\nabla f|^2.$$
Hence
$$\int_{X_0 \times [0,1]} f^2 \leq \int_Q |\nabla f|^2.$$
The set $X_0$ has Lebesgue measure at least $1-c$, whence
there is $x_0 \in X_0$ such that
$$ \int_{I_{x_0}} f^2 \leq \frac{1}{1-c} \int_Q |\nabla f|^2. $$
\textbf{Claim.} Let $I$ be a unit interval and let $z$ be any point in $I$.
Then $$\int_I f^2 \leq 2|f(z)|^2 + 2 \int_I |\nabla f|^2.$$
Indeed,
$$\int_I f^2 \leq \max_I f^2 \leq \left(|f(z)| + \int_I |\nabla f|\right)^2 \leq \left(|f(z)| + \sqrt{\int_I |\nabla f|^2}\,\,\right)^2 \leq $$
$$\leq 2|f(z)|^2 + 2\int_I |\nabla f|^2.$$
For every $y \in [0,1]$, it yields
$$\int_{I^y} f^2 \leq 2|f(x_0,y)|^2 + 2 \int_{I^y} |\nabla f|^2.$$
Thus
$$\int_Q f^2 = \int_0^1 \left(\int_{I^y} f^2dx\right)dy \leq 2\int_{I_{x_0}} f^2+2 \int_{Q} |\nabla f|^2 \leq \left(\frac{2}{1-c}+2\right)\int_{Q} |\nabla f|^2.$$
\end{proof}
\begin{corollary}\label{cor: thin}
If $$m_2(\Omega \cap Q)\leq k^2 \ll 1$$
for any unit square $Q$, then
the Poincare consant of $\Omega$ is smaller than $Ck^2$.
\end{corollary}
\begin{proof}
For every square $Q_{2k}$ of size $2k$, we have
$$m_2(\Omega \cap Q_{2k})\leq \frac{1}{4} m_2( Q_{2k}).$$
By $2k$ rescaling we reduce the problem to Lemma \ref{le: thin}.
\end{proof}
\pagebreak
Now we are almost ready to run the Di Georgi scheme. The only remaining preparatory part is smooth surgery.
Let $u\in C_0^\infty(\Omega)$. Fix $t>0$ (level) and $\delta>0 $ (extremely small number).
Let $\Theta$ be a $C^\infty$-smooth function on $\mathbb{R}$ described by Figure \ref{Theta}.
\begin{figure}[h!]
\includegraphics[width=0.7\textwidth]{theta.pdf}
\caption{ $y=\Theta(x)$} \label{Theta}
\end{figure}
The function $\Theta$ has the following properties:
\begin{itemize}
\item $ 0 \leq \Theta' \leq 1,$
\item $ \Theta(0)= 0,$
\item $ x-4\delta\leq \Theta(x) \leq x,$
\item $\Theta(x) = t-2\delta \text{ on } (t -\delta, t+\delta).$
\end{itemize}
Let $\tilde u = \Theta \circ u$. Then $|u-\tilde u| \leq 4\delta$ and
$|\nabla \tilde u| \leq |\nabla u|$ pointwise. Thus, if $u$ was an $\varepsilon$-minimizer,
then $\tilde u$ is an $\varepsilon+ A\delta$-minimizer ($\delta$ is purely qualitative and $A=4\int_\Omega |v|$).
Define $$\Theta_-(x)=
\begin{cases}
\Theta(x), \quad x\leq t+\delta\\
t-2\delta, \quad x\geq t+\delta
\end{cases} \quad \text{and} \quad \Theta_+=\Theta-\Theta_-. $$
The function $\tilde u$ naturally splits into two smooth compactly supported terms:
$$ \tilde u= \tilde u_-+ \tilde u_+, \text{ where } \tilde u_\pm = \Theta_\pm \circ u. $$
The function $\tilde u_+$ is compactly supported in $\{u>t\}$ ($\text{supp } \tilde u_+ \subset\{u \geq t+\delta \}$)
and $\nabla \tilde u_+$ and $\nabla \tilde u_-$ have disjoint supports.
We may then try to replace $\tilde u_+$ by some smooth competitor $w\in C_0^\infty(\{u>t\})$ and see if the functional
can drop. Note that
$$ \Phi(\tilde u_-+w)= \int_\Omega|\nabla \tilde u_-|^2 + \int_{\{u>t\}}|\nabla w|^2 + \int_\Omega v\tilde u_- + \int_{\{u>t\}} v w,$$
so we just need to compare
$$\int_\Omega|\nabla \tilde u_+|^2 + \int_\Omega v\tilde u_+ \quad \text{with} \quad \int_{\{u>t\}}|\nabla w|^2 + \int_{\{u>t\}} v w.$$
Hence $ \tilde u_+$ is an $(\varepsilon+A\delta)$-minimizer in the new domain $\{u> t \}$.
We shall now fix the initial Poincare constant to be $k_0$ from Claim I. If $u$ is an $\varepsilon$-minimizer, then
$ \int_B u^2 \leq Ck_0^2(k_0^2+\varepsilon)$ for every unit ball $B$, so
$$m_2(\{u>t_0\} \cap B)\leq \frac{Ck_0^2(k_0^2+\varepsilon)}{t_0^2}.$$
Choose $t_{0}=C' \sqrt{k_{0}}$ with sufficiently large absolute constant $C'$. Then the domain
$\Omega_{1}=\left\{u>t_{0}\right\}$ satisfies $m_2(\Omega_1\cap B) \leq \frac{C}{C'} k_{0}\left(k_{0}^{2}+\varepsilon\right)$ for any unit ball $B$ and, by Corollary \ref{cor: thin},
the Poincare constant of $\Omega_1$ is at most $k^{2}_1:=k_{0} \frac{k_{0}^{2}+\varepsilon}{2}$.
Also $u_{1}=\tilde{u}_{+} \in C_{0}^{\infty}\left(\Omega_{1}\right)$ will be an $\varepsilon_{1}=\varepsilon+A\delta_{0}$-minimizer of $\Phi$
in $C_{0}^{\infty}\left(\Omega_{1}\right)$ where $\delta_{0}>0$ can be chosen arbitrary small. Finally, note
that $u \leq t_{0}+u_{1}+4\delta_{0}$ everywhere in $\Omega$. We can now repeat this construction with $ u_{1}, \Omega_{1}, \varepsilon_1$ instead of $u, \Omega, \varepsilon$ to
get $u_{2}, \Omega_{2}, \varepsilon_{2}$ and so on. We shall get a sequence of domains $\Omega_j$, functions
$u_j \in C_0^{\infty}(\Omega_j)$ and numbers $k_j,t_j,\varepsilon_j, \delta_j>0$ such that
$$ \Omega_1 \supset \Omega_2 \supset ... \quad, \quad k_j^2= k_{j-1}\frac{k_{j-1}^2+\varepsilon_{j-1}}{2},$$
$$ \varepsilon_j= \varepsilon_{j-1}+ A\delta_{j-1}, \quad t_j=C'\sqrt{k_j}, $$
$$m_2(\Omega_j\cap B) \leq \frac{C}{C'}k_{j-1}(k_{j-1}^2 +\varepsilon_{j-1}) \quad \text{ for any unit ball } B, $$
and
$$ u \leq t_0+t_1+...+t_{l}+u_{l+1}+4\delta_0+4\delta_1+...+4\delta_{l} \quad \text{for any } l\geq 0.$$
In this construction, we can choose $\delta_j>0$ as small as we want, so putting $\delta_j=\frac{c\varepsilon}{2^j}$ with sufficiently small absolute constant $c>0$, we can guarantee that all $\varepsilon_j \leq 2 \varepsilon$.
Let $l$ be the first index for which $k_l^2<2\varepsilon$. Then (provided that $C' > 8C$, $\varepsilon <1/2$), we also have
$m_2(\Omega_{l+1} \cap B)<\frac{\varepsilon}{2}$ for every unit ball $B$. For all $j\leq l$, we have
$$k_j^2 \leq k_{j-1}^3,$$ so, if $k_0$ was chosen less than $\frac{1}{4}$, it implies that $k_j^2 \leq 2^{-j-2}$
for $j=0,1,...,l$, whence $t_j \leq C'2^{-\frac{j}{2}-1}$ and
$$ u \leq t_0+t_1+...+t_{l}+4\delta_0+4\delta_1+...+4\delta_{l} \leq C' \sum_{j\geq 0} 2^{-\frac{j}{2}-1}+\sum_{j\geq 0} 4\delta_j \leq $$
$$\leq 2C' +\varepsilon \leq 2C'+1=C_0$$
in $\Omega\setminus \Omega_{l+1}$ because $u_{l+1}=0$ in $\Omega\setminus \Omega_{l+1}$.
Thus
$$m_2(\{ u>C_0\}\cap B)<\varepsilon/2.$$
Considering $-u$ instead of $u$, we conclude that also $m_2(\{ u<-C_0\}\cap B)<\varepsilon/2$ and
therefore
$$m_2(\{ |u|>C_0\}\cap B)<\varepsilon$$
for every $\varepsilon$-minimizer $u$. Since the true minimizer $u$ is the limit
of $\varepsilon$-minimizers in $L^2(\Omega)$, we get
$$ m_2(\{|u|>C_0\}\cap B)=0 \quad \text{ for any unit ball } B,$$
so $m_2(\{|u|>C_0\})=0$.
\noindent \textbf{Conclusion.} If $|v|\leq 1$ and the Poincare constant of $\Omega$ is not greater than $k_0^2\ll 1$, then the minimizer
of $\int_\Omega |\nabla u|^2 + \int_\Omega vu$ in $W_0^{1,2}(\Omega)$ satisfies
\begin{equation}
\|u\|_\infty\leq C_0.
\end{equation}
If the Poincare constant of $\Omega$ is $k$, we put $\widetilde \Omega = \frac{k_0}{k}\Omega$, $\widetilde u = \frac{k_0^2}{k^2}u(\frac{k}{k_0}\cdot)$, $\widetilde v = v(\frac{k}{k_0}\cdot)$, so
$\widetilde \Phi(\widetilde u)= \int_{\widetilde \Omega} |\nabla \widetilde u |^2+ \int_{\widetilde \Omega} \widetilde v \widetilde u = \frac{k_0^4}{k^4}\Phi(u)$. Applying the result that
was just obtained, we get the final observation.
\begin{lemma} \label{lem: solving2}
If the Poincare constant of $\Omega$ is $k>0$,
then the minimizer of $\Phi(u)=\int_\Omega |\nabla u|^2 + \int_\Omega vu$
in $W_0^{1,2}(\Omega)$ (i.e., the solution to $\Delta u = v/2$) satisfies:
\begin{equation} \label{eq: uniform}
\|u\|_\infty\leq Ck^2\|v\|_\infty,
\end{equation}
where $C$ is an absolute positive constant.
\end{lemma}
\subsection{Other standard facts used in the proof.}
\begin{fact} \label{trivialities 1}
Let $\Omega$ be a bounded open set, let $u \in W^{1,2}_{0}(\Omega) $ satisfy $\| u\|_\infty \leq 1$. Then there exists a sequence $u_k
\in C^\infty_0(\Omega)$ with $\|u_k\|_\infty \leq 2$ such that $ u_k \to u, \nabla u_k \to \nabla u $ in $L^2(\Omega)$ and almost everywhere in $\Omega$.
\end{fact}
\begin{proof}
By the definition of $W^{1,2}_0(\Omega)$, we can find $\tilde u_k
\in C^\infty_0(\Omega) $ with $ \tilde u_k \to u, \nabla \tilde u_k \to \nabla u $ in $L^2$.
Let $\Theta $ be defined by Figure \ref{Theta2}.
\begin{figure}[h!]
\includegraphics[width=0.7\textwidth]{theta1.pdf}
\caption{The graph of $\Theta$} \label{Theta2}
\end{figure}
\pagebreak
The function $\Theta$ has the following properties.
\begin{itemize}
\item $\Theta(x)=x$ for $x\in [-1.5,1.5]$,
\item $ \Theta $ is $C^\infty$-smooth and $|\Theta| \leq 1.75 $,
\item $|\Theta'|\leq 1$ and $|\Theta(x)| \leq |x|$.
\end{itemize}
Put $u_k = \Theta(\tilde u_k)$. Note that $u_k= \tilde u_k$ and $\nabla u_k = \nabla \tilde u_k$ if $|\tilde u_k|\leq 1.5$ and we always have $|u_k|\leq |\tilde u_k|$, $|\nabla u_k|\leq |\nabla \tilde u_k|$.
We need to show that $\|u_k-u\|_2$, $\|\nabla u_k- \nabla u\|_2 \to 0$.
Since $\tilde u_k$ converge in $W^{1,2}_0(\Omega)$, the functions $|\tilde u_k|^2$ and $|\nabla \tilde u_k|^2$ are uniformly integrable,
i.e., for any $\varepsilon>0$, there is $\delta>0$ such that if $m_2(E)<\delta$, then
$$\int_E |\tilde u_k|^2, \int_E |\nabla \tilde u_k|^2 <
\varepsilon.$$
In the following computation $\int$ will denote the integral over $\Omega$:
$$ \int |u_k-u|^2 \leq 2 \left[ \int |u_k-\tilde u_k|^2 +\int |\tilde u_k - u|^2 \right]\leq$$
$$ \leq 4 \!\!\!\!\!\!\int\limits_{\{|\tilde u_k|>1.5\}} (|u_k|^2+|\tilde u_k|^2) + 2\int |\tilde u_k - u|^2 \leq$$
$$\leq 8\!\!\!\!\!\!\int\limits_{\{|\tilde u_k|>1.5\}} |\tilde u_k|^2 + 2 \int|\tilde u_k - u|^2.$$
The second term tends to zero by the choice of $\tilde u_k$.
Note that $$m_2(\{|\tilde u_k|>1.5\}) \leq m_2(|u-\tilde u_k|\geq 0.5) \leq 4 \|u-\tilde u_k\|_2^2 \to 0.$$ So the first term tends to zero by the uniform integrability property. In a similar way one can show that
$\int|\nabla u_k - \nabla u|^2 \to 0 $.
Now we would like to choose a subsequence such that
$ u_k \to u, \nabla u_k \to \nabla u $ almost everywhere in $\Omega$.
It can be done by choosing any subsequence $u_{k_j}$ with $$\|u_{k_j}-u\|_2 \leq \frac{1}{4^j}, \quad
\|\nabla u_{k_j}- \nabla u\|_2 \leq \frac{1}{4^j}.$$
Let $E_j$ be the set of points $x\in \Omega$ such that $|u_{k_j}(x)-u(x)| \geq \frac{1}{2^{j}}$.
Then $$ \sqrt{m_2(E_j) \frac{1}{4^{j}} } \leq \|u_{k_j}-u\|_2 \leq \frac{1}{4^{j}}, \quad {\text{so}} \quad m_2(E_j) \leq \frac{1}{4^j}.$$ Note that if $x \notin \cup_{j=n}^\infty E_j$, then $u_{k_j}(x)$ converge to $u(x)$.
However $$m_2(\cup_{j=n}^\infty E_j) \leq \sum_{j=n}^\infty m_2( E_j) \leq \frac{1}{2^{n}}.$$
Thus $u_{k_j}$ converge to $u$ almost everywhere in $\Omega$. In a similar way one can show that $\nabla u_{k_j}$ also converge to $\nabla u$ almost everywhere.
\end{proof}
\noindent \textbf{Fact \ref{product}.}
Let $\Omega$ be an open set. Assume that $u,v \in W^{1,2}_{loc}(
\Omega)\cap L^{{}^{\scriptsize \infty}}_{loc}(\Omega)$. then $uv \in W^{1,2}_{loc}(
\Omega) $ and $\nabla(uv) = u \nabla v + v \nabla u$.
\begin{proof}
Clearly, the fact is local. So we may assume $\Omega= B(0,r)$ and $u,v \in W^{1,2}(B(0,r)
)\cap L^\infty(B(0,r))$.
Let us fix a small $\delta>0$ and
let $$K_\varepsilon(z)=\frac{1}{\varepsilon^2}\kappa(|z|/\varepsilon)$$ be a $C^\infty$-approximation to identity with $\text{supp} K_\varepsilon(z)\subset B(0,\varepsilon)$, $\varepsilon < \delta$.
Then $K_\varepsilon* u$ and $K_\varepsilon*v$
converge to $u$ and $v$ in $L^2(B(0,r-\delta))$ and a.e. in $B(0,r-\delta)$ as $\varepsilon \to 0$. Consider any test function $\eta \in C_0^\infty(B(0,r-\delta))$ and extend it by $0$ outside $B(0,r-\delta)$. Then $\eta*K_\varepsilon \in C_0^\infty(B(0,r))$ and $K_\varepsilon * \nabla \eta = \nabla (K_\varepsilon*\eta) $.
By Fubini's theorem we have
$$ \int_{B(0,r-\delta)} (K_\varepsilon* u) \nabla \eta =\int_{B(0,r)} u (K_\varepsilon* \nabla \eta)=$$
$$=\int_{B(0,r)} u \nabla(K_\varepsilon* \eta)= -\int_{B(0,r)} \nabla u (K_\varepsilon* \eta)= $$
$$=- \int_{B(0,r)}\left(\int_{B(0,r-\delta)} \nabla u(x) K_\varepsilon(x-y)\eta(y) dy \right) dx=$$ $$= - \int_{B(0,r-\delta)}\left(\int_{B(0,r)} \nabla u(x) K_\varepsilon(y-x)\eta(y) dx \right) dy = -\int_{B(0,r-\delta)} (\nabla u *K_\varepsilon) \eta.$$
So $$u_\varepsilon:=K_\varepsilon* u \quad \textup{and} \quad v_\varepsilon:=\quad K_\varepsilon* v$$ are in $W^{1,2}(B(0,r-\delta))\cap C^\infty(B(0,r-\delta))$ with
\begin{enumerate}
\item $\nabla u_\varepsilon = \nabla u * K_\varepsilon$,$\nabla v_\varepsilon = \nabla v * K_\varepsilon$,
\item
$\nabla u * K_\varepsilon \to \nabla u$, $\nabla v * K_\varepsilon \to \nabla v$ in $L^2(B(0,r-\delta))$ and a.e. in $B(0,r-\delta)$ as $\varepsilon \to 0$,
\item By Young's inequality for convolutions, we have $|u_\varepsilon|< \|u\|_{L^\infty(B(0,r))}$, $|v_\varepsilon|< \|v\|_{L^\infty(B(0,r))}$ in $B(0,r-\delta)$.
\end{enumerate}
We know that the convergence of $u_\varepsilon v_\varepsilon$ holds a.e. in $B(0,r-\delta)$ and
$|u_\varepsilon v_\varepsilon|$ are bounded by $\|u\|_\infty\|v\|_\infty$ a.e. in $B(0,r-\delta)$ if $\varepsilon< \delta$. So by the Lebesgue dominated convergence theorem $u_\varepsilon v_\varepsilon \to uv$ in $L^2(B(0,r-\delta))$.
We want to show that
$$\nabla(uv) = u \nabla v + v \nabla u$$
in the sense of $W^{1,2}(B(0,r-\delta))$. It is clear that $u \nabla v + v \nabla u$ is in $L^2(B(0,r-\delta))$ because $u,v$ are bounded and their gradients are in $L^2(B(0,r-\delta))$.
Consider again a test function $\eta \in C_0^\infty(B(0,r-\delta))$. We have
$$ \int uv \nabla \eta = \lim_{\varepsilon \to 0} \int u_\varepsilon v_\varepsilon \nabla \eta = - \lim_{\varepsilon \to 0} \int (\nabla u_\varepsilon v_\varepsilon + u_\varepsilon \nabla v_\varepsilon ) \eta=$$
$$ =- \lim_{\varepsilon \to 0} \left[ \int \left((\nabla u_\varepsilon - \nabla u) v_\varepsilon + u_\varepsilon (\nabla v_\varepsilon - \nabla v) \right) \eta + \int ( \nabla u v_\varepsilon + u_\varepsilon \nabla v ) \eta \right].$$
Note that $\int (\nabla u_\varepsilon - \nabla u) v_\varepsilon \eta \to 0$ because $\nabla u_\varepsilon\to \nabla u$ in $L^2(B(0,r-\delta))$ and $|v_\varepsilon \eta | < \|v\|_\infty\| \eta\|_\infty$ in $B(0,r-\delta)$. Similarly, $ \int u_\varepsilon (\nabla v_\varepsilon - \nabla v) \eta \to 0$.
Finally, by the Lebesgue dominated convergence theorem $$ \int ( \nabla u v_\varepsilon + u_\varepsilon \nabla v ) \eta \to \int ( \nabla u v + u \nabla v ) \eta$$
because the convergence of the functions holds a.e. in $B(0,r-\delta)$ and there is the integrable majorant
$(|\nabla u| \|v\|_\infty + \|u\|_\infty |\nabla v |) \|\eta\|_\infty$.
Thus
$\int uv \nabla \eta = - \int ( \nabla u v + u \nabla v ) \eta$.
\end{proof}
\begin{fact} \label{fact6}
Let $\Omega$ be a bounded open set. Let $\varphi=1+\psi$, where $\psi \in W^{1,2}_0(\Omega)$,
$\|\psi\|_\infty \leq \frac{1}{3}$. Then the functions
$$ \tilde \varphi =
\begin{cases}
\varphi \textup{ in } \Omega \\
1 \textup{ outside } \Omega
\end{cases}
\quad \textup{ and } \quad
\eta=
\begin{cases}
\frac 1 \varphi \textup{ in } \Omega \\
1 \textup{ outside } \Omega
\end{cases}
$$
are in $W^{1,2}_{loc}(\mathbb{R}^2)$ and
$$\nabla \tilde \varphi= \nabla \varphi \mathbbm{1}_\Omega, \quad \nabla \eta = - \frac{\nabla \varphi}{\varphi^2}\mathbbm{1}_\Omega.$$
\end{fact}
\begin{proof}
Consider a sequence of functions $\psi_k \in C_0^\infty(\Omega)$ such that
$\| \psi_k\|_\infty \leq \frac 2 3 $ and $\psi_k \to \psi$, $\nabla \psi_k \to \nabla \psi$ in $L^2(\Omega)$
and a.e. in $\Omega$. We can extend $\psi_k$ by zero outside $\Omega$ and get a sequence of $C_0^\infty(\mathbb{R}^2)$ functions, which we will still denote by $\psi_k$, such that $
\psi_k=0, \nabla \psi_k=0$ in $\mathbb{R}^2\setminus \Omega$ while $\psi_k \to \psi \mathbbm{1}_\Omega$ in $L^2(\mathbb{R}^2)$ and a.e., $\nabla \psi_k \to \nabla \psi \mathbbm{1}_\Omega$ in $L^2(\mathbb{R}^2)$. This immediately implies that $1+\psi_k \to \tilde \varphi$, $\nabla (1+\psi_k)=\nabla \psi_k \to \nabla \psi \mathbbm{1}_\Omega$ in $L^2_{loc}(\mathbb{R}^2)$,
so $\tilde \varphi \in W^{1,2}_{loc}(\mathbb{R}^2)$ and $\nabla \tilde \varphi= \tilde \varphi \mathbbm{1}_\Omega.$
Note that
$$ \left| \frac{1}{1+\psi_k}- \frac{1}{1+\psi}\right|= \left| \frac{\psi_k-\psi}{(1+\psi_k)(1+\psi)}\right| \leq 9|\psi_k - \psi|$$
and
$$ \left| \nabla \frac{1}{1+\psi_k}+ \frac{\nabla \psi}{(1+\psi)^2}\right|= \left| \frac{\nabla \psi_k}{(1+\psi_k)^2} - \frac{\nabla \psi}{(1+\psi)^2}\right|\leq$$
$$\leq |\nabla \psi_k -\nabla \psi| \frac{1}{(1+\psi_k)^2}+|\nabla \psi|\left| \frac{1}{(1+\psi_k)^2} - \frac{1}{(1+\psi)^2}\right| \quad \textup{ in } \Omega.$$
Also $\frac{1}{1+\psi_k}=1$, $\nabla\frac{1}{1+\psi_k}=0$ in $\mathbb{R}^2\setminus \Omega$.
Hence $\frac{1}{1+\psi_k} \to \eta$ in $L^2(\mathbb{R}^2)$ and we would like to show that
$$\nabla\frac{1}{1+\psi_k} \to - \frac{\nabla \psi}{(1+\psi)^2} \mathbbm{1}_\Omega= - \frac{\nabla \varphi}{\varphi^2}\mathbbm{1}_\Omega \quad \textup{ in } L^2(\mathbb{R}^2).$$
To see the latter, note that $\frac{1}{(1+\psi_k)^2} \leq 9$, so
$$\int_\Omega |\nabla \psi_k - \nabla \psi|^2 \frac{1}{(1+\psi_k)^4} \leq 81 \int_\Omega |\nabla \psi_k - \nabla \psi|^2 \to 0,$$
and the functions $|\nabla \psi|^2 \left[ \frac{1}{(1+\psi_k)^2} - \frac{1}{(1+\psi)^2} \right]^2$
have the integrable majorant $81|\nabla \psi|^2$ and tend to $0$ almost everywhere in $\Omega$.
Thus $\eta \in W^{1,2}_{loc}(\mathbb{R}^2)$ and $\nabla \eta = - \frac{\nabla \varphi}{\varphi^2}\mathbbm{1}_\Omega$
as required.
\end{proof}
\begin{lemma} \label{le: W0} Let $\Omega$ be a bounded open set and let a function $f\in C^1(\overline{\Omega})$ be zero on $\partial \Omega$. Then $f \in W^{1,2}_{0}(\Omega)$.
\end{lemma}
\begin{proof}
Let $\varepsilon>0$, denote by $\Omega_\varepsilon$ the set of points $x$ in $\Omega$ with distance to the boundary of $\Omega$ at least $\varepsilon$. Let $\eta$ be a function in $C_0^\infty(\Omega)$ with the following properties:
\begin{itemize}
\item $\eta(x)= 1$, if $x \in \Omega_\varepsilon$.
\item $0 \leq \eta \leq 1$ and $|\nabla \eta| \leq \frac{C}{\varepsilon}$ in $\Omega$.
\end{itemize}
The function $f\eta$ is in $C_0^1(\Omega) \subset W_0^{1,2}(\Omega) $.
We want to show that $f\eta$ converge to $f$ in $W_0^{1,2}(\Omega)$ norm as $\varepsilon \to 0$. Observe that $|\nabla f|$ is uniformly bounded in $\Omega$ by some constant $A=A(f)$, so $|f(x)| \leq A\varepsilon $ if the distance from $x$ to $\partial \Omega$ is smaller than $\varepsilon$ and therefore $$|\nabla(f\eta)|\leq |\nabla f||\eta|+ | f|| \nabla \eta| \leq A + AC \quad \text{ in } \Omega\setminus \Omega_\varepsilon.$$
Then $$\int\limits_\Omega |f -f \eta|^2 = \int\limits_{\Omega \setminus \Omega_\varepsilon } |f -f \eta|^2 \leq \int\limits_{\Omega \setminus \Omega_\varepsilon } |f|^2 \to 0$$
and
$$\int\limits_\Omega |\nabla f - \nabla (f \eta)|^2 = \int\limits_{\Omega \setminus \Omega_\varepsilon } |\nabla f -\nabla(f \eta)|^2 \leq 2 \int\limits_{\Omega \setminus \Omega_\varepsilon } (|\nabla f|^2 +|\nabla(f \eta)|^2) \leq$$
$$ \leq m_2(\Omega \setminus \Omega_\varepsilon) C_1 A^2. $$
Since $m_2(\Omega \setminus \Omega_\varepsilon) \to 0$ as $\varepsilon \to 0$, we have verified that $f \in W_0^{1,2}(\Omega)$.
\end{proof}
\begin{lemma} \label{le: diameter}
Let $u$ be a solution to $\Delta u + Vu=0$, $|V|\leq 1$, in a ball $B(x,r)$,
where $r<r_0$ and $r_0$ is a sufficiently small universal constant.
If $u$ is continuous up to $\partial B(x,r)$ and $u>0$ on $\partial B(x,r)$, then
$u>0$ in $B(x,r)$.
\end{lemma}
\begin{proof}
We may assume that $u$ is larger than a positive constant $\delta$ on $\partial B(x,r)$.
Consider the set $\Omega=\{ x \in B(x,r): u(x)< \frac \delta 2 \}$.
This is an open set strictly inside $B(x,r)$ and if $u$ is not positive in $B(x,r)$, then $\Omega$ is not empty.
Since $u \in C^1(\overline{\Omega})$ by Fact \ref{fact5} and $u=\frac \delta 2$ on $\partial \Omega$, we know by Lemma \ref{le: W0} that $(u-\frac \delta 2) \in W_0^{1,2}(\Omega)$.
Note that $\overline{\Omega} \subset B(0,r) $, so $\Omega$ has a Poincare constant smaller than $Cr^2$.
By Lemma \ref{solving Schrodinger}, if $r$ is sufficiently small, we can find $\varphi=1+\tilde \varphi$ with
$\tilde \varphi \in W^{1,2}_0(\Omega), \|\tilde \varphi\|_\infty<\frac{1}{2}$
such that $\varphi$ is a solution to $\Delta \varphi + V \varphi=0$ in $\Omega$.
Then $ (\frac{\delta}{2}\varphi - \frac{\delta}{2}) \in W^{1,2}_0(\Omega)$ and therefore the function
$g=(\frac{\delta}{2}\varphi - u) \in W^{1,2}_0(\Omega) $. The function $g$ is also a solution to $\Delta g+ Vg=0$. For any $\eta\in C_0^\infty(\Omega)$, we have $\int_\Omega \nabla g \nabla \eta= \int_\Omega Vg\eta$
and taking the limit in $W^{1,2}_0(\Omega)$, we get
$$ \int_\Omega |\nabla g|^2 = \int_\Omega Vg^2 \leq \int_\Omega g^2.$$
However Poincare's inequality implies
$$ \int_\Omega g^2 \leq Cr^2 \int_\Omega |\nabla g|^2. $$
If $r$ is sufficiently small, this could happen only if $g=0$ in $\Omega$.
So $u= \frac{\delta}{2}\varphi$ in $\Omega$, but $\varphi> \frac 1 2$ in $\Omega$.
So $u>\frac \delta 4$ in $\Omega$ and in $B(x,r)$.
\end{proof}
\subsection{Divergence free vector fields on the plane} \label{sec: divergence}
If $F=(F_1,F_2): B \to \mathbb{R}^2$ is a $C^1$- smooth vector field in a disk $B$ on the plane such
that $F$ is divergence free: $\text{div} F=0$ in $B$, then
there is a smooth function $u$ such that $$(F_1,F_2)= \nabla \times u := (u_{x_2},-u_{x_1}).$$
Sometimes people refer to the statement above as to Poincare's lemma or the fundamental theorem of calculus, or the inverse gradient theorem. Here is the sketch of the standard proof. WLOG, $B=B(0,1)$. Consider any point $Q\in B$ and
the rectangle $R \subset B$ with opposite vertices $0$ and $Q$, and sides parallel to $x_1$ and $x_2$ axes.
\begin{figure}[h!]
\includegraphics[width=0.5\textwidth]{rectangle2}
\end{figure}
Note that
that the contour integral $$ \int_{\partial R} (-F_2,F_1)\cdot dx= \int_{\partial R} F\cdot n(x) |dx| = \int_R \text{div } F$$
is zero. There are two simple paths that start at $0$, go along the sides of $R$ and end at $Q$. The integrals $\int(-F_2,F_1)dx$ over those two paths are the same and we define $u(Q)$ to be equal to both of them. The differentiation of $u(Q)$ in the horizontal and vertical directions shows
that $(F_1,F_2)= (u_{x_2},-u_{x_1})$.
We need the version with less regularity assumptions on the divergence free vector field $F$:
if $F\in L_{\text{loc}}^p(B(0,1))$, $1\leq p<\infty$, and $\int_{B(0,1)}F\nabla h=0$ for any $h \in C^\infty_0(B(0,1))$ (the divergence free condition), then there is a function $u\in W^{1,p}_{loc}(B(0,1))$ such that $(F_1,F_2)= \nabla \times u := (u_{x_2},-u_{x_1}).$
Indeed, let $$K_\varepsilon(z)=\frac{1}{\varepsilon^2}\kappa(|z|/\varepsilon)$$ be a $C^\infty$-approximation to identity with $\text{supp} K_\varepsilon(z)\subset B(0,\varepsilon)$.
Define $$F_\varepsilon= F*K_\varepsilon= (F_1*K_\varepsilon, F_2*K_\varepsilon)$$ in the smaller ball $B(0,1-\varepsilon)$. Then
$F_\varepsilon$ is divergence free in $B(0,1-\varepsilon)$.
Indeed, if $f\in C_0^\infty(B(0,1-\varepsilon)$, then by Fubini's theorem
$$\int (F*K_\varepsilon) \nabla f= \int F (K_\varepsilon * \nabla f)=\int F \nabla (K_\varepsilon * f)=0.$$
So there is a $C^\infty$ function $u_\varepsilon$ such that $F_\varepsilon=\nabla \times u_\varepsilon$ in $B(0,1-\varepsilon)$. Fix $\delta \in (0,1)$.
By the Lebesgue theory $F*K_\varepsilon$ converge to $F$ in $L^p(B(0,1-\delta))$.
Thus $\nabla u_\varepsilon$ is a Cauchy sequence in $L^p(B(0,1-\delta))$. Let us add a constant to $u_\varepsilon$ so that $\int_{B(0,1-\delta)}u_\varepsilon=0$. By the Poincaré--Wirtinger inequality (see p.275, Theorem 1 in \cite{P}) $u_\varepsilon$ is a Cauchy sequence in $L^p(B(0,1-\delta))$.
Thus we can find a function $\tilde u_\delta$ such that $u_\varepsilon$ converge to $\tilde u_\delta$ in $W^{1,p}(B(0,1-\delta))$
and $\nabla \times \tilde u_\delta = \lim_{\varepsilon \to 0} \nabla \times u_\varepsilon=(F_1,F_2)$.
For any $\delta_1,\delta_2 \in (0,1)$, the gradients of $\tilde u_{\delta_1}$ and $\tilde u_{\delta_2}$ are
the same in $B(0,1-\max(\delta_1,\delta_2))$ and therefore $\tilde u_{\delta_1} - \tilde u_{\delta_2}$ is constant almost everywhere in $B(0,1-\max(\delta_1,\delta_2))$.
Finally, let us modify $\tilde u_\delta $ by subtracting a constant so that $ \int_{B(0,1/2)} \tilde u_\delta = 0 $
for all $\delta <1/2$. Then $u$ is well-defined by $u=\tilde u_\delta$ in $B(0,1-\delta)$. | {"config": "arxiv", "file": "2007.07034/Landis_4.tex"} |
\chapter*{Introduction}\label{sec:intro}
Let $L$ be a link in the 3-sphere~$S^3$. Consider a closed 3-ball~$B^3\subset S^3$ whose boundary intersects~$L$ transversely. Then $L\cap B^3$ is essentially what we call a tangle, the main protagonist of this thesis. We define a tangle invariant $\HFT$, a Heegaard Floer homology for tangles, and study its properties.\\
Heegaard Floer homology theories were first defined by Ozsváth and Szabó in~2001 \cite{OSHF3mfds}. With an oriented, closed 3-dimensional manifold~$M$, they associated a family of homological invariants, the simplest of which is denoted by~$\widehat{\text{HF}}(M)$. Given an oriented knot in~$M$, Ozsváth and Szabó, and independently Rasmussen, then defined filtrations on the chain complexes which give rise to the respective flavours of knot Floer homology \cite{OSHFK,Jake}. This was later generalised to oriented links in~$S^3$~\cite{OSHFL}. Our tangle Floer homology should be understood as a generalisation of the hat version of link Floer homology $\HFL$ to oriented tangles.
\subsection*{$\HFT$ and an Alexander polynomial for tangles. } Like $\HFL$, our tangle Floer homology is a finitely generated Abelian group which comes with two gradings: a relative homological $\mathbb{Z}$-grading and an Alexander grading, which is an additional relative $\mathbb{Z}$-grading for each component of the tangle. However, unlike $\HFL$, our tangle Floer homology depends on some extra data, a site, associated with a tangle, see definition~\sref{def:site}. For a tangle $T$ with $n$ open strands, there are $\binom{2n}{n-1}$ such sites $s$, and for each of them, we define a bigraded chain complex
$$\CFT(T,s)=\bigoplus_{\substack{h\in\mathbb{Z}~\leftarrow \text{homological grading}\hspace{-2.96cm}\\ a\in\mathbb{Z}^{\vert T\vert}~\leftarrow \text{Alexander grading}\hspace{-2.7cm}}}\CFT_h(T,s,a),$$
where $\vert T\vert$ denotes the number of components of $T$.
\begin{theorem}[\sref{thm:HFTiswelldefandinvariant}]
Given a tangle \(T\) and a site \(s\) for \(T\), the bigraded chain homotopy type of \(\CFT(T,s)\) is an invariant of \(T\). We denote its homology by \(\HFT(T,s)\) and call it the \textbf{tangle Floer homology} of \(T\).
\end{theorem}
In link Floer homology, the ``Alexander'' in ``Alexander grading'' comes from the fact that, given a link $L$ in $S^3$, the graded Euler characteristic of $\HFL(L)$ recovers the Alexander polynomial of $L$, a classical polynomial link invariant, named after its discoverer~\cite{Alexander}. We say link Floer homology \textit{categorifies} the Alexander polynomial. Similarly, for tangles, we obtain polynomial invariants
$$
\chi(\HFT(T,s))=\sum_{h, a} (-1)^h\operatorname{rk}(\HFT_h(T,s,a))\cdot t_{1\phantom{\vert}}^{a_{1\phantom{\vert}}}\!\!\cdots t_{\vert T\vert}^{a_{\vert T\vert}}\in\mathbb{Z}[t_1^{\pm1},\dots,t_{\vert T\vert}^{\pm1}],
$$
which are well-defined up to multiplication by a unit. In chapter~\ref{chapter:polynomial}, we give a purely combinatorial definition of a normalised version $\nabla_T^s$ of these polynomial invariants in terms of Kauffman states and Alexander codes and study their properties.
\subsection*{Mutation. }
A new invariant can already be interesting because of its simplicity or its aesthetic appeal. But its true value should be determined by its capacity to answer questions about existing theory. So the primary purpose of any tangle Floer homology should be to learn more about ``the local nature'' of knot and link Floer homology and, ultimately, of the geometric objects themselves. A prime example of an open question one might hope to address is how link Floer homology behaves under mutation.
\begin{definition}\label{def:INTROmutation}
Let $L$ be a link. Construct a new link $L'$ by cutting out a 4-ended tangle $\Gamma$ and glueing it back in after a half-rotation, as illustrated below:
$$
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\rput(0,0){$\Gamma$}
}
\rput(0,0){$\longrightarrow$}
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$$
We say $L'$ is obtained from $L$ by \textbf{Conway mutation}. We call~$\Gamma$ the \textbf{mutating tangle} and $L'$ a \textbf{mutant} of~$L$. If~$L$ is oriented, we define an orientation on $L'$ such that it agrees with the one on~$L$ outside~$\Gamma$. If this means that we need to reverse the orientation of the two open components of~$\Gamma$, we also reverse the orientation of any closed components of~$\Gamma$; otherwise, we do not change the orientation on~$\Gamma$.
\end{definition}
We know from~\cite{OSmutation} that knot and link Floer homology is, in general, not invariant under mutation. However, we have the following conjecture from \cite[conjecture~1.5]{BaldwinLevine}.
\begin{conjecture}\label{conj:MutInvHFL}
Let \(L\) be a link and let \(L'\) be obtained from \(L\) by Conway mutation. Then \(\HFL(L)\) and \(\HFL(L')\) agree after collapsing the bigrading to a single \(\mathbb{Z}\)-grading, known as the \(\delta\)-grading. In short: \(\delta\)-graded link Floer homology is mutation invariant.
\end{conjecture}
Let us see what the new invariants tell us on the decategorified level, i.\,e.\ on the level of the Alexander polynomial.
\begin{theorem}[\sref{thm:mutation}]\label{thm:IntroMutationDECAT}
The multivariate Alexander polynomial is invariant under Conway mutation after identifying the variables corresponding to the two open strands of the mutating tangle.
\end{theorem}
This result has long been known for the single-variate Alexander polynomial, see for example~\cite[proposition~11]{LickorishMillett}, but I have been unable to find a result for the multivariate polynomial in the literature. The proof of theorem~\ref{thm:IntroMutationDECAT} relies on certain symmetry relations between the invariants $\nabla_T^s$ for 4-ended tangles $T$ and varying sites $s$, see proposition~\sref{prop:fourended}.
For $\HFT$, we can prove similar symmetry relations which categorify those for $\nabla_T^s$. As conjecture~\ref{conj:MutInvHFL} suggests, in general, they only hold for $\delta$-graded tangle Floer homology, see proposition~\sref{prop:fourendedHFT} and example~\sref{exa:pretzeltangle}. However, these symmetry relations are not sufficient to prove the conjecture. This is because, unlike $\nabla_T^s$, $\HFT$ \emph{alone} is insufficient to state a glueing formula.
\subsection*{Glueing tangle Floer homologies. }
The main tool for glueing 3-manifolds with boundary in Heegaard Floer homology is bordered Heegaard Floer homology, developed by Lipshitz, Ozsváth and Thurston \cite{LOT}. In \cite{Zarev}, Zarev generalised this theory to sutured manifolds. We interpret our tangle Floer homology $\HFT$ in terms of Zarev's theory to add a glueing structure to $\HFT$. This glueing structure essentially takes the form of extra differentials between the tangle Floer chain complexes $\CFT(T,s)$ for different sites $s$. As an accompaniment to this thesis, we provide the Mathematica package~\cite{BSFH.m} which allows us to compute the bordered sutured invariants for any bordered sutured manifold from nice diagrams, see appendix~\ref{app:manualBSFH} for a documentation of \cite{BSFH.m}. In particular, it allows us to confirm conjecture~\ref{conj:MutInvHFL} for mutation about a particular non-trivial tangle.
\begin{theorem}[\sref{thm:2m3pt}, \sref{exa:HFTdpretzeltangle}]\label{thm:23pretzelintro}
Consider the following \((2,-3)\)-pretzel tangle:
\begin{center}
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Two knots or links that are related by mutation of this tangle have the same bigraded knot or link Floer homologies after identifying the Alexander gradings corresponding to the two open strands. If the orientation of one of those two strands is reversed, then their \(\delta\)-graded knot or link Floer homologies agree.
\end{theorem}
We offer two independent proofs of this result, both of which, however, rely on calculations using the program~\cite{BSFH.m}. The first proof follows from computing the glueing structure for the $(2,-3)$-pretzel tangle in two different ways corresponding to mutation and observing that the two results are homotopy equivalent, see theorem~\sref{thm:2m3pt}. However, this proof does not really tell us \textit{why} they are homotopic. The second proof answers this question more satisfactorily, replacing the previous ad hoc construction by a conceptually more refined approach, see example~\sref{exa:HFTdpretzeltangle} in conjunction with theorem~\sref{thm:CFTdGeneralGlueing}. To explain this, let us discuss bordered invariants in a little more detail.
\subsection*{Bordered theory and Fukaya categories. }
In general, the invariants and the glueing theorems in bordered and also bordered sutured Heegaard Floer homology look rather complicated. With a closed 3-manifold~$M$ split along a (parametrised) closed surface~$F$ into two components~$M_1$ and~$M_2$, one associates a differential graded algebra~$\mathcal{A}(F)$, a so-called type D module~$\CFD(M_1)$ and a type A module~$\CFA(M_2)$ over~$\mathcal{A}(F)$. Then~$\HF(M)$ can be computed as the homology of a special tensor product~$\boxtimes$ of $\CFD(M_1)$ and $\CFA(M_2)$ over $\mathcal{A}(F)$. If one wants to split $M$ into more than two pieces, there are also bimodule invariants of type AA, AD, DA and DD, depending on whether we treat the glueing surfaces of the components as type A or type D sides.\pagebreak[3]\\
Especially the algebra $\mathcal{A}(F)$ can be quite complicated, and in general, it will be so for our tangle Floer homology.
In \cite{Auroux}, Auroux gave an interpretation of the bordered algebra $\mathcal{A}(F)$ in terms of some partially wrapped Fukaya category and outlined a conjectural reformulation of bordered Heegaard Floer theory in this framework. In~\cite{LOTMor}, Lipshitz, Ozsváth and Thurston gave further evidence for this conjectural relationship by restating their glueing theorem purely in terms of the homology of the space of morphisms between the two type D modules of $M_1$ and $M_2$. Moreover, Rasmussen, Hanselman and Watson very recently gave an elegant reformulation of the glueing theorem for a surprisingly large class of manifolds with torus boundary, which they called loop-type \cite{HRW}. For such manifolds, the type D structure can be interpreted as an object in the Fukaya category of immersed curves on a (punctured) torus. In particular, glueing corresponds to taking Lagrangian intersection homology on the torus. \\
For 4-ended tangles, a similar story seems to be true, which we explore in chapter~\ref{chapter:HFTd}.
\subsection*{A glueing structure for 4-ended tangles. }
\begin{theorem}[\sref{defthm:CFTd}]
Given a 4-ended tangle~\(T\), we can endow the tangle Floer homology \(\HFT\) with an additional structure of a curved complex, which we denote by \(\CFTd(T)\). (Roughly speaking, a curved complex is a type~D module for which the differential does not square to 0, see definition~\ref{exa:HighBrowDefcurvedTypeD}.) \(\CFTd(T)\) is a tangle invariant. We call it the \textbf{peculiar module} of \(T\).
\end{theorem}
This enables us to explicitly calculate objects for 4-ended tangles in the (triangulated enlargement of the) fully wrapped Fukaya category $\TwFuk(S^2,4)$ of the 4-punctured sphere, via the $A_\infty$-functor $\mathcal{L}$ in the following theorem.
\begin{theorem}[\sref{thm:TwFukpqModEquivalent}]\label{thm:INTROTwFukpqModEquivalent}
Let \(\pqMod\) be the category of peculiar modules. There exist two non-trivial \(A_\infty\)-functors \vspace*{-0.4cm}
$$\begin{tikzcd}[column sep=2cm]
\pqMod
\arrow[in=175,out=5]{r}{\mathcal{L}}
&
\TwFuk(S^2,4).\nmathphantom{\TwFuk(S^2,4)}\phantom{\pqMod}
\arrow[in=-5,out=185]{l}{\mathcal{M}}
\end{tikzcd}$$
\end{theorem}
This allows us to interpret the sites of 4-ended tangles explicitly in terms of generators of the Fukaya category
$\TwFuk(S^2,4)$ via the following proposition.
\begin{proposition}[\sref{prop:CFTsiteFromFukCFTd}]\label{prop:INTROCFTsiteFromFukCFTd}
Let \(T\) be a 4-ended tangle and \(s\) a site of~\(T\). Then, there exists a generator \(L_s\) of \(\TwFuk(S^2,4)\) such that \(\CFT(T,s)\) is bigraded chain homotopic to the Lagrangian intersection chain complex
$$\left(\Mor\left(L_s,\mathcal{L}\left(\CFTd(T)\right)\right),\mutw_1\right),$$
where \(\mutw_1\) denotes the differential on morphism spaces in \(\TwFuk(S^2,4)\).
\end{proposition}
We expect that one can extend the proof of the result above to show the following.
\begin{conjecture}[\sref{conj:TwFukpqModEquivalent}]
The functors \(\mathcal{M}\) and \(\mathcal{L}\) from theorem~\ref{thm:INTROTwFukpqModEquivalent} define an equivalence of \(A_\infty\)-categories.
\end{conjecture}
\begin{figure}[t]
\psset{unit=1.4}
\begin{subfigure}[b]{0.3\textwidth}\centering
\begin{pspicture}(-1.5,-1.5)(1.5,1.5)
\psecurve(1.2,1)(0,1.4)(-1.2,1)(1.4,1)(0,0.6)(-1.4,1)(1.2,1)(0,1.4)(-1.2,1)
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\psline[linestyle=dashed](1,-1)(-1,-1)
\psline[linestyle=dashed](-1,-1)(-1,1)
\psline[linestyle=dashed](-1,1)(1,1)
\psset{dotsize=5pt}
\psdot[linecolor=red](-1,0.8)
\psdot[linecolor=red](-1,0.663)
\psdot[linecolor=gold](0.125,1)
\psdot[linecolor=gold](-0.125,1)
\psdot[linecolor=darkgreen](1,0.8)
\psdot[linecolor=darkgreen](1,0.663)
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\pscircle[fillstyle=solid, fillcolor=white](-1,1){0.08}
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\pscircle[fillstyle=solid, fillcolor=white](-1,-1){0.08}
\end{pspicture}
\caption{}\label{fig:INTROloopbottom}
\end{subfigure}
\quad
\begin{subfigure}[b]{0.3\textwidth}\centering
\begin{pspicture}(-1.5,-1.5)(1.5,1.5)
\psecurve(-0.8,-1.2)(-1.2,-1.2)(1.2,1.1)(-1.2,0.9)(0,1.4)(1.4,1.2)(-0.8,-1.2)(-1.2,-1.2)(1.2,1.1)
\psline[linestyle=dashed](1,1)(1,-1)
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\psdot[linecolor=red](-1,-0.635)
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\psdot[linecolor=darkgreen](1,-0.03)
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\pscircle[fillstyle=solid, fillcolor=white](-1,1){0.08}
\pscircle[fillstyle=solid, fillcolor=white](1,-1){0.08}
\pscircle[fillstyle=solid, fillcolor=white](-1,-1){0.08}
\end{pspicture}
\caption{}\label{fig:INTROloopmiddle}
\end{subfigure}
\quad
\begin{subfigure}[b]{0.3\textwidth}\centering
\begin{pspicture}(-1.5,-1.5)(1.5,1.5)
\psrotate(0,0){180}{
\psecurve(1.2,1)(0,1.4)(-1.2,1)(1.4,1)(0,0.6)(-1.4,1)(1.2,1)(0,1.4)(-1.2,1)
\psline[linestyle=dashed](1,1)(1,-1)
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\psdot[linecolor=darkgreen](-1,0.8)
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\psdot[linecolor=blue](-0.125,1)
\psdot[linecolor=red](1,0.8)
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\pscircle[fillstyle=solid, fillcolor=white](1,1){0.08}
\pscircle[fillstyle=solid, fillcolor=white](-1,1){0.08}
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\pscircle[fillstyle=solid, fillcolor=white](-1,-1){0.08}
}
\end{pspicture}
\caption{}\label{fig:INTROlooptop}
\end{subfigure}
\caption{The three loops on the 4-punctured sphere respresenting $\CFTd$ for the $(2,-3)$-pretzel tangle of theorem~\ref{thm:23pretzelintro}. The intersection points of the loops with the dashed arcs connecting the four punctures calculate the underlying non-glueable tangle Floer homology $\HFT$. The splitting of $\HFT$ according to the four different sites is indicated by the colouring of the intersection points.}\label{fig:INTROmutationexamplefinalresult}
\end{figure}
For rational tangles, but also for more complicated ones like the $(2,-3)$-pretzel tangle, the peculiar modules are loop-type in a similar sense to \cite{HRW}: the tangle Floer homology of these tangles is represented by a collection of loops in the 4-punctured 2-sphere, see figure~\ref{fig:INTROmutationexamplefinalresult} and examples~\sref{exa:CFTdRatTang} and~\sref{exa:HFTdpretzeltangle}.
Furthermore, computations suggest that we can indeed compute link Floer homology as the Lagrangian intersection homology of such immersed curves. In fact, we have the following glueing result.
\begin{theorem}[\sref{thm:CFTdGlueingTrivial} and \sref{thm:CFTdGeneralGlueing}]\label{thm:INTROglueing}
Let \(T_1\) and \(T_2\) be two 4-ended tangles and \(L\) the link obtained by glueing them together according to the following picture.
$$
\psset{unit=0.6}
\begin{pspicture}(-2.5,-1.6)(2.5,1.6)
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\pscircle*[linecolor=white](1.5,0){1}
\pscircle(1.5,0){1}
\rput[c](1.5,0){$T_2$}
\end{pspicture}
$$
Then there exists a certain type~\(AA\) bimodule \(\mathcal{P}\) such that \(\CFL(L)\) is equal to
$$
\CFTd(T_1)\boxtimes\mathcal{P}\boxtimes \CFTd(T_2)
$$
up to at most three stabilisations, i.\,e.\ tensoring with a certain 2-dimensional vector space. Furthermore, for loop-type~\(\CFTd(T_1)\) and trivial~\(T_2\), \(\CFL(L)\) agrees with the Lagrangian intersection Floer homology of the loop of~\(T_1\) with that of the trivial tangle, up to at most a single stabilisation.
\end{theorem}
The computation of the type AA bimodule~$\mathcal{P}$ is done using~\cite{BSFH.m}. The second statement follows from computing a type A structure, which can be done by hand, or alternatively, by simplifying $\mathcal{P}\boxtimes \CFTd(T_2)$. We expect the second statement to generalise to pairings of arbitrary (loop-type) tangles. Unfortunately, however, the bimodule~$\mathcal{P}$ looks rather complicated and not like the one to be expected from the Fukaya category. So there remains work to be done, see conjectures~\sref{conj:CFTdBetterGlueing} and~\sref{conj:GlueingCFTdFUK}.\pagebreak[3]\\
Nonetheless, the mere existence of a glueing theorem for $\CFTd$ allows us to infer properties of link Floer homology. For example, the peculiar invariant of the $(2,-3)$-pretzel tangle has an intrinsic symmetry (see figure~\ref{fig:INTROmutationexamplefinalresult}), which gives us the second proof of theorem~\ref{thm:23pretzelintro}. We expect other 2-stranded pretzel tangles to have the same kind of symmetry; however, a slightly more careful analysis of holomorphic curves is needed to determine the structure maps.
\begin{conjecture}
Theorem~\ref{thm:23pretzelintro} generalises to any 2-stranded pretzel-tangle.
\end{conjecture}
\pagebreak
For general 4-ended tangles, we are only able to prove slightly weaker symmetry relations for $\CFTd$. They support the mutation conjecture, see propositions \sref{prop:CFTdAll4sites} and~\sref{prop:CFTdPeculiarRanks}, but they do not seem to be sufficient to prove it. \\
As another application of the existence of a glueing theorem, we show that peculiar modules detect rational tangles:
\begin{theorem}[\sref{thm:CFTdDetectsRatTan}]
A 4-ended tangle \(T\) is rational iff \(\CFTd(T)\) is homotopic to a single loop that corresponds to an embedded loop on the 4-punctured sphere.
\end{theorem}
Furthermore, we can easily reprove the existence of an unoriented skein exact sequence \cite{Manolescu}, see theorem~\sref{thm:ResolutionExactTriangle}. Similarly, we obtain the following slight generalisation of Ozsváth and Szabó's oriented skein exact sequence \cite{OSHFK}.
\begin{theorem}[\sref{thm:nTwistSkeinRelation}, see also \sref{rem:nTwistSkeinRelation}]\label{thm:INTROnTwistSkeinRelation}
Let \(T_n\) be the positive \(n\)-twist tangle, \(T_{-n}\) the negative \(n\)-twist tangle and \(T_0\) the trivial tangle, see figure~\ref{fig:INTROOrientedSkeinRelation}. Then there is an exact triangle:
$$\begin{tikzcd}[row sep=0.9cm, column sep=-0.5cm]
\CFTd(T_{-n})
\arrow{rr}
& &
\CFTd(T_{n})
\arrow{dl}
\\
&
\CFTd(T_{0})\otimes V
\arrow{lu}
\end{tikzcd}$$
where \(V\) is some 2-dimensional vector space. If the tangles are oriented and coloured consistently, one obtains (bi)graded versions of this triangle.
Furthermore, it gives rise to an exact triangle relating the (appropriately stabilised) link Floer homologies of links that differ in these three tangles.
\end{theorem}
\begin{figure}[t]
\centering
\psset{unit=0.35}
\begin{subfigure}[b]{0.25\textwidth}\centering
$n\left\{\raisebox{-0.85cm}{
\begin{pspicture}(-1.1,-3.1)(1.1,3.1)
\psecurve(1,5)(-1,3)(1,1)(-1,-1)
\pscircle*[linecolor=white](0,2){0.3}
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\rput(0,0.3){$\vdots$}
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\pscircle*[linecolor=white](0,-2){0.3}
\psecurve(1,-5)(-1,-3)(1,-1)(-1,1)
\end{pspicture}}\right.\quad
$
\caption{$T_{n}$}\label{fig:INTROOrientedSkeinRelationTn}
\end{subfigure}
\quad
\begin{subfigure}[b]{0.25\textwidth}\centering
$n\left\{\raisebox{-0.85cm}{
\begin{pspicture}(-1.1,-3.1)(1.1,3.1)
\psecurve(-1,5)(1,3)(-1,1)(1,-1)
\pscircle*[linecolor=white](0,2){0.3}
\psecurve(1,5)(-1,3)(1,1)(-1,-1)
\rput(0,0.3){$\vdots$}
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\pscircle*[linecolor=white](0,-2){0.3}
\psecurve(-1,-5)(1,-3)(-1,-1)(1,1)
\end{pspicture}}\right.\quad
$
\caption{$T_{-n}$}\label{fig:INTROOrientedSkeinRelationTmn}
\end{subfigure}
\quad
\begin{subfigure}[b]{0.25\textwidth}\centering
\begin{pspicture}(-5,-3.5)(5,3.5)
\psecurve(-2,6)(-1,3)(-1,-3)(-2,-6)
\psecurve(2,6)(1,3)(1,-3)(2,-6)
\end{pspicture}
\caption{$T_0$}\label{fig:INTROOrientedSkeinRelationT0}
\end{subfigure}
\caption{The basic tangles for the skein exact sequence from theorem~\ref{thm:INTROnTwistSkeinRelation}}\label{fig:INTROOrientedSkeinRelation}
\end{figure}
\subsection*{Parallels to Khovanov homology. }
Our peculiar invariants $\CFTd$ are also interesting from another, perhaps more philosophical perspective, namely the relationship between link Floer homology and Khovanov homology. Khovanov homology is another homology theory for knots and links, first defined by Khovanov in 1999 \cite{Khovanov}. It categorifies the Jones polynomial in the same way that $\HFL$ categorifies the Alexander polynomial. Although the two theories are defined and computed in very different ways, they look quite similar from a formal point of view, see for example \cite{JakeComparisonKhHFK}.
\begin{figure}[t]
\centering
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\psset{xunit=1.5,nodesep=2pt,nrot=:U}
\begin{pspicture}(-5,-3.4)(5,3.6)
\rput(1,1){\rnode{A}{Khovanov homology}}
\rput(-3,1){\rnode{B}{knot Floer homology}}
\rput(-3,-3){\rnode{C}{Alexander polynomial}}
\rput(1,-3){\rnode{D}{Jones polynomial}}
\rput(3,3){\rnode{E}{\Centerstack{Bar-Natan's Khovanov\\ homology for tangles}}}
\rput(-1,3){\rnode{F}{?}}
\rput(-1,-1){\rnode{G}{?}}
\rput(3,-1){\rnode{H}{\Centerstack{Jones polynomial \\ for tangles}}}
\ncline[linestyle=dotted,dotsep=1pt]{F}{G}
\ncline[linestyle=dotted,dotsep=1pt]{<->}{G}{H}
\ncline[linestyle=dotted,dotsep=1pt]{->}{C}{G}
\ncline{E}{H}\ncput*{categorifies}
\ncline{->}{D}{H}
\ncline{<->}{A}{B}
\ncline{B}{C}\ncput*{categorifies}
\ncline{<->}{C}{D}
\ncline{A}{D}\ncput*{categorifies}
\ncline{->}{A}{E}
\ncline{->}{B}{F}
\ncline{<->}{E}{F}
\end{pspicture}
\caption{Another perspective on $\nabla_T^s$, $\HFT$ and especially $\CFTd$}\label{fig:introcontext}
\end{figure}
\noindent
In \cite{BarNatanKhT}, Bar-Natan gave an elegant generalisation of Khovanov homology to tangles. With a tangle diagram, he associated an (up to homotopy) invariant chain complex over a certain category which essentially (that is, up to grading) consists of only finitely many objects and morphisms. For example, for 4-ended tangles, there are just two objects and at most four morphisms in each hom-set. For $\CFTd$, we similarly get two candidates for such basic objects, and we sketch how to write the peculiar module of any tangle as a chain complex in these basic objects, up to a large tensor factor, see remark~\sref{rem:singularcrossings}. For the $(2,-3)$-pretzel tangle, this tensor factor can be removed.
\begin{questions}
Given any oriented 4-ended tangle \(T\), can we write \(\CFTd(T)\) as a chain complex in two basic objects? If so, can we describe the chain maps?
\end{questions}
An affirmative answer to these two questions would not only be aesthetically pleasing. The basic objects are symmetric under mutation, so if the chain maps between the basic complexes also have this symmetry, one might be able to prove the mutation conjecture. In Khovanov homology, such an approach has been successful: Khovanov homology with $\mathbb{Z}/2$-coefficients is known to be mutation invariant, see \cite{WehrliKhMutInv}.
\subsection*{Similar work by other people. }
There are several other groups of people working on similar ideas to those described in this thesis. \\
In 2014, Petkova and Vértesi defined a combinatorial tangle Floer homology using grid diagrams and ideas from bordered Floer homology \cite{cHFT}. They use a more general definition of tangles, namely those with a ``top'' and a ``bottom'', i.\,e.\ braids with caps and cups. In \cite{DecatCTFH}, they and Ellis show that the decategorification of their invariant agrees with Sartori's generalisation of the Alexander polynomials to top-bottom-tangles via representations of $U_q(\mathfrak{gl}(1\vert 1))$ \cite{Sartori14}. Thus, Petkova and Vértesi's theory fits very nicely into $\mathfrak{sl}_n$-homology theories arising from Khovanov homology. \\
Very recently, Ozsváth and Szabó developed a completely algebraically defined knot homology theory, which they conjecture to be equivalent to knot Floer homology \cite{OSKauffmanStates}. Like Petkova and Vértesi, they cut up a knot diagram into elementary pieces, so they automatically obtain tangle invariants, too. This theory seems to be frightfully powerful: from a computational point of view, since they can compute their homology from diagrams with over 50 crossings; but also from a more theoretical point of view, since their theory includes the hat- as well as the more sophisticated ``$-$''-version of knot Floer homology without reference to holomorphic curves or grid diagrams. Interestingly, the generators in their theory correspond to Kauffman states like in ours. \pagebreak\\
In \cite{HHK13,HHK15}, Hedden, Herald and Kirk study the Lagrangian intersection homology of immersed curves on a 4-punctured sphere (the ``pillowcase'') in the context of instanton knot Floer homology, a computationally more difficult knot Floer homology due to Kronheimer and Mrowka, which also categorifies the Alexander polynomial and is conjecturally closely related to $\HFK$ \cite{KM}. The curve they associate with the trivial tangle does not agree with ours, but it looks very similar to the curve we associate with a singular crossing, see proposition~\sref{prop:singularcrossing} and remark~\sref{rem:singularcrossings}.\\
Finally, I want to mention recent work of Lambert-Cole on conjecture~\ref{conj:MutInvHFL}. In \cite{LambertCole1}, he shows that the bigraded knot Floer homologies of a mutant knot pair obtained by introducing a sufficiently large number of twists into a given (positive) mutant knot pair agree. This follows from a certain stabilisation property of knot Floer homology with respect to twisting that he proves in the same paper. In \cite{LambertCole2}, he investigates conjecture~\ref{conj:MutInvHFL} from a more axiomatic point of view, using basepoint maps and Manolescu's unoriented skein exact triangle as basic ingredients. He is able to confirm conjecture~\ref{conj:MutInvHFL} for mutating tangles that can be closed to an unlink by a rational tangle. In particular, this result encompasses the $\delta$-graded part of theorem~\ref{thm:23pretzelintro}.
\subsection*{Outline. }
The thesis is split into three chapters, following not only a logical, but incidentally also a roughly chronological order.\\
The first chapter is purely concerned with the decategorified story, the combinatorial definition of the polynomial tangle invariants~$\nabla_T^s$ and their properties. The chapter can be seen as a playground for testing ideas for chapters~\ref{chapter:categorification} and~\ref{chapter:HFTd}. In fact, some properties of $\nabla_{T}^s$ are immediate consequences of their categorified counterparts. However, other questions, most prominently the one concerning mutation invariance, are still unresolved in the categorified setting, whilst being relatively easy to answer for the decategorified invariants. On a first read, one can skip all sections of this chapter except the very first.\\
In chapter~\ref{chapter:categorification}, we define the tangle Floer homology~$\HFT$; first, via sutured Floer homology, then in more detail via Heegaard diagrams for tangles, imitating the definition of link Floer homology. We then interpret our invariant in terms of Zarev's bordered sutured theory.\\
In the third and final chapter, we specialise to 4-ended tangles and repackage the glueing structure in this special case into the peculiar invariant~$\CFTd$. We investigate some of its properties and discuss several applications and open questions.\\
There are four appendices. In appendix~\ref{appendix:AlgStructFromGDCats}, we describe the algebraic structures appearing in this thesis from an abstract category-theoretic point of view and derive useful tools for working with them, which form the basis of all our computations. In the second appendix, we give a proof of the generalised clock theorem, which is essential for studying the polynomial invariants~$\nabla_T^s$ in chapter~\ref{chapter:polynomial}.
The last two appendices are documentations for the programs~\cite{APT.m} and~\cite{BSFH.m}.
\subsection*{Acknowledgements}
First and foremost, I would like to thank my supervisor Jake Rasmussen for his generous support throughout the entire time of my PhD and before, during Part III. I consider myself very fortunate to have been his student.\pagebreak\\
My PhD was funded by an EPSRC scholarship covering tuition fees and a DPMMS grant for maintenance, for which I thank the then Head of Department Martin Hyland. I also gratefully acknowledge a research studentship from the Cambridge Philosophical Society for Michaelmas Term 2016.\\
I thank my examiners Ivan Smith and András Juhász for many valuable comments on and corrections to the soft-bound version of this thesis that I prepared for my viva. I also thank Mohammed Abouzaid, Guillem Cazassus, Celeste Damiani, Artem Kotelskiy, Peter Lambert-Cole, Adam Levine, Ina Petkova, Vera Vértesi and my brother Marcus Zibrowius for helpful conversations. My special thanks go to Liam Watson for his interest in my work, his support and the opportunity to speak in Glasgow twice.\\
I am very grateful to my PhD brothers Tom Brown, Tom Gillespie and Paul Wedrich, and my fellow PhD students Nina Friedrich and Christian Lund for their company and friendship. I would also like to thank Senja Barthel, Fyodor Gainullin, Tom Hockenhull and Marco Marengon for organising yearly student conferences at Imperial College London, all of which were terrific.\\
I thank Johnny Nicholson for helpful comments on an earlier draft of chapter~\ref{chapter:polynomial} and for sharing his computations with me during an undergraduate summer research project in 2016.\\
I am indebted to Tom Brown, Nina Friedrich, Paul Wedrich, Marcus Zibrowius and, especially, my father for their proof-reading services.
\bigskip\\
No line of this thesis would have been written without the love and support of my brother and parents. This thesis is dedicated to them.
\begin{center}
\vspace{35pt}
\includegraphics[width=6cm]{knotkobold.pdf}\\
\end{center}
\vspace*{\fill} | {"config": "arxiv", "file": "1610.07494/sections/0_Intro.tex"} |
TITLE: Convergence to the non-wandering set (for a compact dynamical system)
QUESTION [3 upvotes]: Let $X$ be a compact metric space and let $T\colon X \to X$ be continuous and injective. A point $x$ is said to be wandering if there exists an open neighborhood $V \ni x$ and a time $N \in \mathbb{N}^*$ such that, for all $n \geq N$,
$$
T^n(V) \cap V = \emptyset.
$$
A point is said to be non-wandering, well, if it is not wandering. Denote by $W$ the set of wandering points and $M$ its complement. As a matter of fact, $W$ is open and positively invariant ($T(W) \subset W$), while $M$ is closed (thus compact) and invariant ($T(M) = M$).
The question is whether or not $\bigcap_{n \in \mathbb{N}} T^n(W) = \emptyset$, or in other words is it true that for any $x \in W$, $d(T^n(x),M) \to_n 0$.
REPLY [2 votes]: There is a nice proof in Birkhoff's Dynamical Systems (1927), page 192. With more modern notation it gives the following.
Let $\epsilon > 0$ and define the following compact set,
$$
K = \{ x \in X, ~d(x,M) \geq \epsilon\} \subset W.
$$
For all $x \in K$, there exists an open neighbourhood $V_x$ of $x$ and $n_x > 0$ such that for all $n \geq n_x$,
$$
\varphi^n(V_x) \cap V_x = \emptyset.
$$
By compactness of $K$, we can extract a finite cover of $K$ from $(V_x)_{x \in K}$, say given by $E = \{x_1, \dots, x_m \}$. We note $N = \max_{x \in E} n_x$. If $x \in K$, then $x$ belongs to some $V_y$ with $y \in E$. However $x$ can only stay for at most $N$ times, and never comes back. It might reach another $V_{y'}$ with $y' \in E$ different than $y$, and again can only stay at most $N$ times and never visit again $V_{y'}$. Eventually, all $V_z$ with $z \in E$ are exhausted, so that $x$ reaches and remains in $X \setminus K$.
If $x \in X \setminus K$, then either it remains in $X \setminus K$, or it leaves and the previous reasoning brings $x$ in $X \setminus K$ forever. As a result, any initial condition tends toward $M$. | {"set_name": "stack_exchange", "score": 3, "question_id": 3303442} |
:: General theory and tools for proving algorithms in nominative data systems
:: by Adrian Jaszczak
environ
vocabularies NOMIN_1, NUMBERS, SUBSET_1, XBOOLE_0, RELAT_1, FUNCT_1, FINSEQ_1,
NAT_1, ARYTM_3, PARTFUN1, XBOOLEAN, TARSKI, NOMIN_3, NOMIN_4, XCMPLX_0,
NOMIN_2, PARTPR_1, PARTPR_2, CARD_1, CAT_6, NOMIN_7, XXREAL_0, ARYTM_1,
NOMIN_8, MARGREL1, FUNCOP_1;
notations TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, MARGREL1, RELAT_1,
FUNCT_1, RELSET_1, PARTFUN1, FUNCOP_1, CARD_1, FINSEQ_1, XXREAL_0,
XCMPLX_0, NOMIN_1, PARTPR_1, PARTPR_2, NOMIN_2, NOMIN_3, NOMIN_4,
NOMIN_5, NOMIN_6, NOMIN_7;
constructors RELSET_1, NOMIN_2, NOMIN_3, NOMIN_4, NOMIN_5, NOMIN_6, NOMIN_7;
registrations XBOOLE_0, RELAT_1, FUNCT_1, ORDINAL1, FUNCOP_1, RELSET_1, INT_1,
NOMIN_1, MARGREL1, NOMIN_2, NAT_1, XXREAL_0, XREAL_0, CARD_1, FINSEQ_1,
PARTFUN1, NOMIN_7, PARTPR_2;
requirements NUMERALS, BOOLE, SUBSET, REAL, ARITHM;
definitions TARSKI, RELAT_1, NOMIN_7;
equalities NOMIN_1, PARTPR_2, NOMIN_2, NOMIN_5, NOMIN_6, NOMIN_7, FINSEQ_1;
expansions TARSKI, PARTFUN1, NOMIN_7;
theorems XBOOLE_0, NOMIN_1, NOMIN_2, NOMIN_3, NOMIN_4, FUNCT_1, PARTFUN1,
TARSKI, XBOOLE_1, INT_1, XREAL_1, XXREAL_0, FINSEQ_3, NAT_1, FINSEQ_1,
RELAT_1, NOMIN_7, CARD_1;
schemes PARTPR_2, RECDEF_1, NAT_1, FINSEQ_1;
begin
registration
let D be non empty set;
cluster non empty D-valued for FinSequence;
existence
proof
take <*the Element of D*>;
thus thesis;
end;
end;
registration
let D be non empty set, n be Nat;
cluster D-valued n-element for FinSequence;
existence
proof
set p = Seg n --> the Element of D;
dom p = Seg n;
then reconsider p as FinSequence by FINSEQ_1:def 2;
take p;
thus rng p c= D by RELAT_1:def 19;
Seg len p = dom p by FINSEQ_1:def 3;
hence thesis by CARD_1:def 7,FINSEQ_1:6;
end;
end;
reserve D for non empty set;
reserve f1,f2,f3,f4,f5,f6,f7,f8,f9,f10 for BinominativeFunction of D;
reserve p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11 for PartialPredicate of D;
reserve q1,q2,q3,q4,q5,q6,q7,q8,q9,q10 for total PartialPredicate of D;
reserve n,m,N for Nat;
reserve fD for PFuncs(D,D)-valued FinSequence;
reserve fB for PFuncs(D,BOOLEAN)-valued FinSequence;
reserve V,A for set;
reserve val for Function;
reserve loc for V-valued Function;
reserve d1 for NonatomicND of V,A;
reserve p for SCPartialNominativePredicate of V,A;
reserve d,v for object;
reserve size for non zero Nat;
reserve inp,pos for FinSequence;
reserve prg for non empty FPrg(ND(V,A))-valued FinSequence;
definition
let D,f1,f2,f3,f4,f5,f6,f7;
func PP_composition(f1,f2,f3,f4,f5,f6,f7) -> BinominativeFunction of D equals
PP_composition(PP_composition(f1,f2,f3,f4,f5,f6),f7);
coherence;
end;
::$N Unconditional composition rule for 7 programs
theorem Th1:
<*p1,f1,p2*> is SFHT of D & <*p2,f2,p3*> is SFHT of D &
<*p3,f3,p4*> is SFHT of D & <*p4,f4,p5*> is SFHT of D &
<*p5,f5,p6*> is SFHT of D & <*p6,f6,p7*> is SFHT of D &
<*p7,f7,p8*> is SFHT of D &
<*PP_inversion(p2),f2,p3*> is SFHT of D &
<*PP_inversion(p3),f3,p4*> is SFHT of D &
<*PP_inversion(p4),f4,p5*> is SFHT of D &
<*PP_inversion(p5),f5,p6*> is SFHT of D &
<*PP_inversion(p6),f6,p7*> is SFHT of D &
<*PP_inversion(p7),f7,p8*> is SFHT of D
implies <*p1,PP_composition(f1,f2,f3,f4,f5,f6,f7),p8*> is SFHT of D
proof
assume that
A1: <*p1,f1,p2*> is SFHT of D and
A2: <*p2,f2,p3*> is SFHT of D and
A3: <*p3,f3,p4*> is SFHT of D and
A4: <*p4,f4,p5*> is SFHT of D and
A5: <*p5,f5,p6*> is SFHT of D and
A6: <*p6,f6,p7*> is SFHT of D and
A7: <*p7,f7,p8*> is SFHT of D and
A8: <*PP_inversion(p2),f2,p3*> is SFHT of D and
A9: <*PP_inversion(p3),f3,p4*> is SFHT of D and
A10: <*PP_inversion(p4),f4,p5*> is SFHT of D and
A11: <*PP_inversion(p5),f5,p6*> is SFHT of D and
A12: <*PP_inversion(p6),f6,p7*> is SFHT of D and
A13: <*PP_inversion(p7),f7,p8*> is SFHT of D;
<*p1,PP_composition(f1,f2,f3,f4,f5,f6),p7*>
is SFHT of D by A1,A2,A3,A4,A5,A6,A8,A9,A10,A11,A12,NOMIN_7:3;
hence thesis by A7,A13,NOMIN_3:25;
end;
definition
let D,f1,f2,f3,f4,f5,f6,f7,f8;
func PP_composition(f1,f2,f3,f4,f5,f6,f7,f8) ->
BinominativeFunction of D equals
PP_composition(PP_composition(f1,f2,f3,f4,f5,f6,f7),f8);
coherence;
end;
::$N Unconditional composition rule for 8 programs
theorem Th2:
<*p1,f1,p2*> is SFHT of D & <*p2,f2,p3*> is SFHT of D &
<*p3,f3,p4*> is SFHT of D & <*p4,f4,p5*> is SFHT of D &
<*p5,f5,p6*> is SFHT of D & <*p6,f6,p7*> is SFHT of D &
<*p7,f7,p8*> is SFHT of D & <*p8,f8,p9*> is SFHT of D &
<*PP_inversion(p2),f2,p3*> is SFHT of D &
<*PP_inversion(p3),f3,p4*> is SFHT of D &
<*PP_inversion(p4),f4,p5*> is SFHT of D &
<*PP_inversion(p5),f5,p6*> is SFHT of D &
<*PP_inversion(p6),f6,p7*> is SFHT of D &
<*PP_inversion(p7),f7,p8*> is SFHT of D &
<*PP_inversion(p8),f8,p9*> is SFHT of D
implies <*p1,PP_composition(f1,f2,f3,f4,f5,f6,f7,f8),p9*> is SFHT of D
proof
assume that
A1: <*p1,f1,p2*> is SFHT of D and
A2: <*p2,f2,p3*> is SFHT of D and
A3: <*p3,f3,p4*> is SFHT of D and
A4: <*p4,f4,p5*> is SFHT of D and
A5: <*p5,f5,p6*> is SFHT of D and
A6: <*p6,f6,p7*> is SFHT of D and
A7: <*p7,f7,p8*> is SFHT of D and
A8: <*p8,f8,p9*> is SFHT of D and
A9: <*PP_inversion(p2),f2,p3*> is SFHT of D and
A10: <*PP_inversion(p3),f3,p4*> is SFHT of D and
A11: <*PP_inversion(p4),f4,p5*> is SFHT of D and
A12: <*PP_inversion(p5),f5,p6*> is SFHT of D and
A13: <*PP_inversion(p6),f6,p7*> is SFHT of D and
A14: <*PP_inversion(p7),f7,p8*> is SFHT of D and
A15: <*PP_inversion(p8),f8,p9*> is SFHT of D;
<*p1,PP_composition(f1,f2,f3,f4,f5,f6,f7),p8*>
is SFHT of D by A1,A2,A3,A4,A5,A6,A7,A9,A10,A11,A12,A13,A14,Th1;
hence thesis by A8,A15,NOMIN_3:25;
end;
definition
let D,f1,f2,f3,f4,f5,f6,f7,f8,f9;
func PP_composition(f1,f2,f3,f4,f5,f6,f7,f8,f9) ->
BinominativeFunction of D equals
PP_composition(PP_composition(f1,f2,f3,f4,f5,f6,f7,f8),f9);
coherence;
end;
::$N Unconditional composition rule for 9 programs
theorem Th3:
<*p1,f1,p2*> is SFHT of D & <*p2,f2,p3*> is SFHT of D &
<*p3,f3,p4*> is SFHT of D & <*p4,f4,p5*> is SFHT of D &
<*p5,f5,p6*> is SFHT of D & <*p6,f6,p7*> is SFHT of D &
<*p7,f7,p8*> is SFHT of D & <*p8,f8,p9*> is SFHT of D &
<*p9,f9,p10*> is SFHT of D &
<*PP_inversion(p2),f2,p3*> is SFHT of D &
<*PP_inversion(p3),f3,p4*> is SFHT of D &
<*PP_inversion(p4),f4,p5*> is SFHT of D &
<*PP_inversion(p5),f5,p6*> is SFHT of D &
<*PP_inversion(p6),f6,p7*> is SFHT of D &
<*PP_inversion(p7),f7,p8*> is SFHT of D &
<*PP_inversion(p8),f8,p9*> is SFHT of D &
<*PP_inversion(p9),f9,p10*> is SFHT of D
implies <*p1,PP_composition(f1,f2,f3,f4,f5,f6,f7,f8,f9),p10*> is SFHT of D
proof
assume that
A1: <*p1,f1,p2*> is SFHT of D and
A2: <*p2,f2,p3*> is SFHT of D and
A3: <*p3,f3,p4*> is SFHT of D and
A4: <*p4,f4,p5*> is SFHT of D and
A5: <*p5,f5,p6*> is SFHT of D and
A6: <*p6,f6,p7*> is SFHT of D and
A7: <*p7,f7,p8*> is SFHT of D and
A8: <*p8,f8,p9*> is SFHT of D and
A9: <*p9,f9,p10*> is SFHT of D and
A10: <*PP_inversion(p2),f2,p3*> is SFHT of D and
A11: <*PP_inversion(p3),f3,p4*> is SFHT of D and
A12: <*PP_inversion(p4),f4,p5*> is SFHT of D and
A13: <*PP_inversion(p5),f5,p6*> is SFHT of D and
A14: <*PP_inversion(p6),f6,p7*> is SFHT of D and
A15: <*PP_inversion(p7),f7,p8*> is SFHT of D and
A16: <*PP_inversion(p8),f8,p9*> is SFHT of D and
A17: <*PP_inversion(p9),f9,p10*> is SFHT of D;
<*p1,PP_composition(f1,f2,f3,f4,f5,f6,f7,f8),p9*>
is SFHT of D by A1,A2,A3,A4,A5,A6,A7,A10,A11,A12,A13,A14,A15,A8,A16,Th2;
hence thesis by A9,A17,NOMIN_3:25;
end;
definition
let D,f1,f2,f3,f4,f5,f6,f7,f8,f9,f10;
func PP_composition(f1,f2,f3,f4,f5,f6,f7,f8,f9,f10) ->
BinominativeFunction of D equals
PP_composition(PP_composition(f1,f2,f3,f4,f5,f6,f7,f8,f9),f10);
coherence;
end;
::$N Unconditional composition rule for 10 programs
theorem
<*p1,f1,p2*> is SFHT of D & <*p2,f2,p3*> is SFHT of D &
<*p3,f3,p4*> is SFHT of D & <*p4,f4,p5*> is SFHT of D &
<*p5,f5,p6*> is SFHT of D & <*p6,f6,p7*> is SFHT of D &
<*p7,f7,p8*> is SFHT of D & <*p8,f8,p9*> is SFHT of D &
<*p9,f9,p10*> is SFHT of D & <*p10,f10,p11*> is SFHT of D &
<*PP_inversion(p2),f2,p3*> is SFHT of D &
<*PP_inversion(p3),f3,p4*> is SFHT of D &
<*PP_inversion(p4),f4,p5*> is SFHT of D &
<*PP_inversion(p5),f5,p6*> is SFHT of D &
<*PP_inversion(p6),f6,p7*> is SFHT of D &
<*PP_inversion(p7),f7,p8*> is SFHT of D &
<*PP_inversion(p8),f8,p9*> is SFHT of D &
<*PP_inversion(p9),f9,p10*> is SFHT of D &
<*PP_inversion(p10),f10,p11*> is SFHT of D implies
<*p1,PP_composition(f1,f2,f3,f4,f5,f6,f7,f8,f9,f10),p11*> is SFHT of D
proof
assume that
A1: <*p1,f1,p2*> is SFHT of D and
A2: <*p2,f2,p3*> is SFHT of D and
A3: <*p3,f3,p4*> is SFHT of D and
A4: <*p4,f4,p5*> is SFHT of D and
A5: <*p5,f5,p6*> is SFHT of D and
A6: <*p6,f6,p7*> is SFHT of D and
A7: <*p7,f7,p8*> is SFHT of D and
A8: <*p8,f8,p9*> is SFHT of D and
A9: <*p9,f9,p10*> is SFHT of D and
A10: <*p10,f10,p11*> is SFHT of D and
A11: <*PP_inversion(p2),f2,p3*> is SFHT of D and
A12: <*PP_inversion(p3),f3,p4*> is SFHT of D and
A13: <*PP_inversion(p4),f4,p5*> is SFHT of D and
A14: <*PP_inversion(p5),f5,p6*> is SFHT of D and
A15: <*PP_inversion(p6),f6,p7*> is SFHT of D and
A16: <*PP_inversion(p7),f7,p8*> is SFHT of D and
A17: <*PP_inversion(p8),f8,p9*> is SFHT of D and
A18: <*PP_inversion(p9),f9,p10*> is SFHT of D and
A19: <*PP_inversion(p10),f10,p11*> is SFHT of D;
<*p1,PP_composition(f1,f2,f3,f4,f5,f6,f7,f8,f9),p10*> is SFHT of D
by A1,A2,A3,A4,A5,A6,A7,A11,A12,A13,A14,A15,A16,A8,A17,A9,A18,Th3;
hence thesis by A10,A19,NOMIN_3:25;
end;
theorem Th5:
<*p1,f1,q1*> is SFHT of D & <*q1,f2,p2*> is SFHT of D implies
<*p1,PP_composition(f1,f2),p2*> is SFHT of D
proof
<*PP_inversion(q1),f2,p2*> is SFHT of D by NOMIN_3:19;
hence thesis by NOMIN_3:25;
end;
theorem Th6:
<*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D &
<*q2,f3,p2*> is SFHT of D implies
<*p1,PP_composition(f1,f2,f3),p2*> is SFHT of D
proof
assume that
A1: <*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D and
A2: <*q2,f3,p2*> is SFHT of D;
A3: <*PP_inversion(q2),f3,p2*> is SFHT of D by NOMIN_3:19;
<*p1,PP_composition(f1,f2),q2*> is SFHT of D by A1,Th5;
hence thesis by A2,A3,NOMIN_3:25;
end;
theorem Th7:
<*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D &
<*q2,f3,q3*> is SFHT of D & <*q3,f4,p2*> is SFHT of D implies
<*p1,PP_composition(f1,f2,f3,f4),p2*> is SFHT of D
proof
assume that
A1: <*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D &
<*q2,f3,q3*> is SFHT of D and
A2: <*q3,f4,p2*> is SFHT of D;
A3: <*PP_inversion(q3),f4,p2*> is SFHT of D by NOMIN_3:19;
<*p1,PP_composition(f1,f2,f3),q3*> is SFHT of D by A1,Th6;
hence thesis by A2,A3,NOMIN_3:25;
end;
theorem Th8:
<*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D &
<*q2,f3,q3*> is SFHT of D & <*q3,f4,q4*> is SFHT of D &
<*q4,f5,p2*> is SFHT of D implies
<*p1,PP_composition(f1,f2,f3,f4,f5),p2*> is SFHT of D
proof
assume that
A1: <*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D &
<*q2,f3,q3*> is SFHT of D & <*q3,f4,q4*> is SFHT of D and
A2: <*q4,f5,p2*> is SFHT of D;
A3: <*PP_inversion(q4),f5,p2*> is SFHT of D by NOMIN_3:19;
<*p1,PP_composition(f1,f2,f3,f4),q4*> is SFHT of D by A1,Th7;
hence thesis by A2,A3,NOMIN_3:25;
end;
theorem Th9:
<*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D &
<*q2,f3,q3*> is SFHT of D & <*q3,f4,q4*> is SFHT of D &
<*q4,f5,q5*> is SFHT of D & <*q5,f6,p2*> is SFHT of D implies
<*p1,PP_composition(f1,f2,f3,f4,f5,f6),p2*> is SFHT of D
proof
assume that
A1: <*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D &
<*q2,f3,q3*> is SFHT of D & <*q3,f4,q4*> is SFHT of D &
<*q4,f5,q5*> is SFHT of D and
A2: <*q5,f6,p2*> is SFHT of D;
A3: <*PP_inversion(q5),f6,p2*> is SFHT of D by NOMIN_3:19;
<*p1,PP_composition(f1,f2,f3,f4,f5),q5*> is SFHT of D by A1,Th8;
hence thesis by A2,A3,NOMIN_3:25;
end;
theorem Th10:
<*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D &
<*q2,f3,q3*> is SFHT of D & <*q3,f4,q4*> is SFHT of D &
<*q4,f5,q5*> is SFHT of D & <*q5,f6,q6*> is SFHT of D &
<*q6,f7,p2*> is SFHT of D implies
<*p1,PP_composition(f1,f2,f3,f4,f5,f6,f7),p2*> is SFHT of D
proof
assume that
A1: <*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D &
<*q2,f3,q3*> is SFHT of D & <*q3,f4,q4*> is SFHT of D &
<*q4,f5,q5*> is SFHT of D & <*q5,f6,q6*> is SFHT of D and
A2: <*q6,f7,p2*> is SFHT of D;
A3: <*PP_inversion(q6),f7,p2*> is SFHT of D by NOMIN_3:19;
<*p1,PP_composition(f1,f2,f3,f4,f5,f6),q6*> is SFHT of D by A1,Th9;
hence thesis by A2,A3,NOMIN_3:25;
end;
theorem Th11:
<*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D &
<*q2,f3,q3*> is SFHT of D & <*q3,f4,q4*> is SFHT of D &
<*q4,f5,q5*> is SFHT of D & <*q5,f6,q6*> is SFHT of D &
<*q6,f7,q7*> is SFHT of D & <*q7,f8,p2*> is SFHT of D implies
<*p1,PP_composition(f1,f2,f3,f4,f5,f6,f7,f8),p2*> is SFHT of D
proof
assume that
A1: <*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D &
<*q2,f3,q3*> is SFHT of D & <*q3,f4,q4*> is SFHT of D &
<*q4,f5,q5*> is SFHT of D & <*q5,f6,q6*> is SFHT of D &
<*q6,f7,q7*> is SFHT of D and
A2: <*q7,f8,p2*> is SFHT of D;
A3: <*PP_inversion(q7),f8,p2*> is SFHT of D by NOMIN_3:19;
<*p1,PP_composition(f1,f2,f3,f4,f5,f6,f7),q7*> is SFHT of D by A1,Th10;
hence thesis by A2,A3,NOMIN_3:25;
end;
theorem Th12:
<*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D &
<*q2,f3,q3*> is SFHT of D & <*q3,f4,q4*> is SFHT of D &
<*q4,f5,q5*> is SFHT of D & <*q5,f6,q6*> is SFHT of D &
<*q6,f7,q7*> is SFHT of D & <*q7,f8,q8*> is SFHT of D &
<*q8,f9,p2*> is SFHT of D implies
<*p1,PP_composition(f1,f2,f3,f4,f5,f6,f7,f8,f9),p2*> is SFHT of D
proof
assume that
A1: <*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D &
<*q2,f3,q3*> is SFHT of D & <*q3,f4,q4*> is SFHT of D &
<*q4,f5,q5*> is SFHT of D & <*q5,f6,q6*> is SFHT of D &
<*q6,f7,q7*> is SFHT of D & <*q7,f8,q8*> is SFHT of D and
A2: <*q8,f9,p2*> is SFHT of D;
A3: <*PP_inversion(q8),f9,p2*> is SFHT of D by NOMIN_3:19;
<*p1,PP_composition(f1,f2,f3,f4,f5,f6,f7,f8),q8*> is SFHT of D by A1,Th11;
hence thesis by A2,A3,NOMIN_3:25;
end;
theorem
<*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D &
<*q2,f3,q3*> is SFHT of D & <*q3,f4,q4*> is SFHT of D &
<*q4,f5,q5*> is SFHT of D & <*q5,f6,q6*> is SFHT of D &
<*q6,f7,q7*> is SFHT of D & <*q7,f8,q8*> is SFHT of D &
<*q8,f9,q9*> is SFHT of D & <*q9,f10,p2*> is SFHT of D implies
<*p1,PP_composition(f1,f2,f3,f4,f5,f6,f7,f8,f9,f10),p2*> is SFHT of D
proof
assume that
A1: <*p1,f1,q1*> is SFHT of D & <*q1,f2,q2*> is SFHT of D &
<*q2,f3,q3*> is SFHT of D & <*q3,f4,q4*> is SFHT of D &
<*q4,f5,q5*> is SFHT of D & <*q5,f6,q6*> is SFHT of D &
<*q6,f7,q7*> is SFHT of D & <*q7,f8,q8*> is SFHT of D &
<*q8,f9,q9*> is SFHT of D and
A2: <*q9,f10,p2*> is SFHT of D;
A3: <*PP_inversion(q9),f10,p2*> is SFHT of D by NOMIN_3:19;
<*p1,PP_composition(f1,f2,f3,f4,f5,f6,f7,f8,f9),q9*> is SFHT of D
by A1,Th12;
hence thesis by A2,A3,NOMIN_3:25;
end;
definition
let D,fD such that
A1: 0 < len fD;
reconsider X = fD.1 as Element of PFuncs(D,D) by PARTFUN1:45;
defpred P[Nat,object,object] means
ex g being PartFunc of D,D st g = $2 & $3 = PP_composition(g,fD.($1+1));
func PP_compositionSeq(fD) -> FinSequence of PFuncs(D,D) means :Def5:
len it = len fD & it.1 = fD.1 &
for n being Nat st 1 <= n < len fD holds
it.(n+1) = PP_composition(it.n,fD.(n+1));
existence
proof
A2: for n st 1 <= n & n < len fD
for x being Element of PFuncs(D,D)
ex y being Element of PFuncs(D,D) st P[n,x,y]
proof
let n;
assume 1 <= n & n < len fD;
let x be Element of PFuncs(D,D);
reconsider g = x as PartFunc of D,D by PARTFUN1:46;
reconsider y = PP_composition(g,fD.(n+1)) as Element of PFuncs(D,D)
by PARTFUN1:45;
take y;
thus thesis;
end;
consider F being FinSequence of PFuncs(D,D) such that
A3: len F = len fD and
A4: (F.1 = X or len fD = 0) and
A5: for n st 1 <= n & n < len fD holds P[n,F.n,F.(n+1)]
from RECDEF_1:sch 4(A2);
take F;
thus len F = len fD by A3;
thus F.1 = fD.1 by A1,A4;
let n be Nat;
assume 1 <= n < len fD;
then P[n,F.n,F.(n+1)] by A5;
hence thesis;
end;
uniqueness
proof
let F1,F2 being FinSequence of PFuncs(D,D) such that
A6: len F1 = len fD and
A7: F1.1 = fD.1 and
A8: for n being Nat st 1 <= n < len fD holds
F1.(n+1) = PP_composition(F1.n,fD.(n+1)) and
A9: len F2 = len fD and
A10: F2.1 = fD.1 and
A11: for n being Nat st 1 <= n < len fD holds
F2.(n+1) = PP_composition(F2.n,fD.(n+1));
A12: for n st 1 <= n & n < len fD
for x,y1,y2 being Element of PFuncs(D,D) st P[n,x,y1] & P[n,x,y2]
holds y1 = y2;
A13: len F1 = len fD & (F1.1 = X or len fD = 0) &
for n st 1 <= n & n < len fD holds P[n,F1.n,F1.(n+1)]
proof
thus len F1 = len fD by A6;
thus (F1.1 = X or len fD = 0) by A7;
let n;
assume 1 <= n & n < len fD;
then F1.(n+1) =
PP_composition(F1.n,fD.(n+1)) by A8;
hence P[n,F1.n,F1.(n+1)];
end;
A14: len F2 = len fD & (F2.1 = X or len fD = 0) &
for n st 1 <= n & n < len fD holds P[n,F2.n,F2.(n+1)]
proof
thus len F2 = len fD by A9;
thus (F2.1 = X or len fD = 0) by A10;
let n;
assume 1 <= n & n < len fD;
then F2.(n+1) =
PP_composition(F2.n,fD.(n+1)) by A11;
hence P[n,F2.n,F2.(n+1)];
end;
thus F1 = F2 from RECDEF_1:sch 8(A12,A13,A14);
end;
end;
definition
let D,fD;
func PP_composition(fD) -> BinominativeFunction of D equals
PP_compositionSeq(fD).len PP_compositionSeq(fD);
coherence;
end;
definition
let D,fD,fB;
pred fD,fB are_composable means
1 <= len fD & len fB = len fD + 1 &
(for n st 1 <= n <= len fD holds <*fB.n,fD.n,fB.(n+1)*> is SFHT of D) &
(for n st 2 <= n <= len fD holds <*PP_inversion(fB.n),fD.n,fB.(n+1)*>
is SFHT of D);
end;
::$N Composition rule for sequences of programs
theorem
fD,fB are_composable implies
<*fB.1,PP_composition(fD),fB.(len fB)*> is SFHT of D
proof
assume that
A1: 1 <= len fD and
A2: len fB = len fD + 1 and
A3: for n st 1 <= n <= len fD holds <*fB.n,fD.n,fB.(n+1)*> is SFHT of D and
A4: for n st 2 <= n <= len fD holds <*PP_inversion(fB.n),fD.n,fB.(n+1)*>
is SFHT of D;
set G = PP_compositionSeq(fD);
defpred P[Nat] means
1 <= $1 <= len fD implies <*fB.1,G.$1,fB.($1+1)*> is SFHT of D;
A5: P[0];
A6: for k being Nat st P[k] holds P[k+1]
proof
let k be Nat;
assume that
A7: P[k] and
A8: 1 <= k+1 and
A9: k+1 <= len fD;
per cases;
suppose
A10: k = 0;
G.1 = fD.1 by A1,Def5;
hence <*fB.1,G.(k+1),fB.(k+1+1)*> is SFHT of D by A1,A3,A10;
end;
suppose k > 0;
then
A11: 0+1 <= k by NAT_1:13;
A12: k < len fD by A9,NAT_1:13;
A13: k <= k+1 by NAT_1:11;
A14: <*fB.(k+1),fD.(k+1),fB.(k+1+1)*> is SFHT of D by A3,A8,A9;
<*PP_inversion(fB.(k+1)),fD.(k+1),fB.(k+1+1)*> is SFHT of D
by A4,A9,A11,XREAL_1:6;
then <*fB.1,PP_composition(G.k,fD.(k+1)),PP_or(fB.(k+1+1),fB.(k+1+1))*>
is SFHT of D by A7,A9,A11,A13,A14,XXREAL_0:2,NOMIN_3:24;
hence thesis by A11,A12,Def5;
end;
end;
A15: for k being Nat holds P[k] from NAT_1:sch 2(A5,A6);
len G = len fD by A1,Def5;
hence thesis by A1,A2,A15;
end;
definition
let V,A;
let val be FinSequence;
set size = len val;
set D = PFuncs(ND(V,A),ND(V,A));
defpred P[Nat,object] means $2 = denaming(V,A,val.$1);
func denamingSeq(V,A,val) -> FinSequence of PFuncs(ND(V,A),ND(V,A)) means
len it = len val &
for n being Nat st 1 <= n <= len it holds it.n = denaming(V,A,val.n);
existence
proof
A1: for n being Nat st n in Seg size ex x being Element of D st P[n,x]
proof
let n be Nat;
assume n in Seg size;
reconsider x = denaming(V,A,val.n) as Element of D by PARTFUN1:45;
take x;
thus thesis;
end;
consider p being FinSequence of D such that
A2: dom p = Seg size and
A3: for n being Nat st n in Seg size holds P[n,p.n] from FINSEQ_1:sch 5(A1);
take p;
thus len p = size by A2,FINSEQ_1:def 3;
thus thesis by A2,A3,FINSEQ_3:25;
end;
uniqueness
proof
let F1,F2 being FinSequence of D such that
A4: len F1 = size and
A5: for n being Nat st 1 <= n <= len F1 holds F1.n = denaming(V,A,val.n) and
A6: len F2 = size and
A7: for n being Nat st 1 <= n <= len F2 holds F2.n = denaming(V,A,val.n);
for n st 1 <= n <= size holds F1.n = F2.n
proof
let n be Nat;
assume
A8: 1 <= n <= size;
hence F1.n = denaming(V,A,val.n) by A4,A5
.= F2.n by A6,A7,A8;
end;
hence thesis by A4,A6,FINSEQ_1:def 17;
end;
end;
definition
let V,A,loc;
let val be FinSequence such that
A1: len val > 0;
let p be SCPartialNominativePredicate of V,A;
set D = PFuncs(ND(V,A),BOOLEAN);
set size = len val;
set X = SC_Psuperpos(p,denaming(V,A,val.len val),loc/.len val);
defpred P[Nat,object,object] means
ex f being SCPartialNominativePredicate of V,A st f = $2 &
$3 = SC_Psuperpos(f,denaming(V,A,val.(len val-$1)),loc/.(len val-$1));
func SC_Psuperpos_Seq(loc,val,p) -> FinSequence of PFuncs(ND(V,A),BOOLEAN)
means :Def9:
len it = len val &
it.1 = SC_Psuperpos(p,denaming(V,A,val.len val),loc/.len val) &
for n being Nat st 1 <= n < len it holds
it.(n+1) =
SC_Psuperpos(it.n,denaming(V,A,val.(len val-n)),loc/.(len val-n));
existence
proof
A2: for n being Nat st 1 <= n & n < size
for x being Element of D
ex y being Element of D st P[n,x,y]
proof
let n be Nat;
assume 1 <= n & n < size;
let x be Element of D;
reconsider f = x as SCPartialNominativePredicate of V,A by PARTFUN1:47;
set y = SC_Psuperpos(f,denaming(V,A,val.(len val-n)),loc/.(len val-n));
reconsider y as Element of D by PARTFUN1:45;
take y;
thus P[n,x,y];
end;
reconsider X as Element of D by PARTFUN1:45;
consider F being FinSequence of D such that
A3: len F = size & (F.1 = X or size = 0) and
A4: for n st 1 <= n & n < size holds P[n,F.n,F.(n+1)]
from RECDEF_1:sch 4(A2);
take F;
thus len F = len val &
F.1 = SC_Psuperpos(p,denaming(V,A,val.len val),loc/.len val) by A1,A3;
let n be Nat;
assume 1 <= n < len F;
then P[n,F.n,F.(n+1)] by A3,A4;
hence thesis;
end;
uniqueness
proof
let F1,F2 being FinSequence of D such that
A5: len F1 = size and
A6: F1.1 = X and
A7: for n being Nat st 1 <= n < len F1 holds F1.(n+1) =
SC_Psuperpos(F1.n,denaming(V,A,val.(len val-n)),loc/.(len val-n)) and
A8: len F2 = size and
A9: F2.1 = X and
A10: for n being Nat st 1 <= n < len F2 holds F2.(n+1) =
SC_Psuperpos(F2.n,denaming(V,A,val.(len val-n)),loc/.(len val-n));
reconsider X as Element of D by PARTFUN1:45;
A11: for n st 1 <= n & n < size
for x,y1,y2 being Element of D st
P[n,x,y1] & P[n,x,y2] holds y1 = y2;
A12: len F1 = size & (F1.1 = X or size = 0) &
for n st 1 <= n & n < size holds P[n,F1.n,F1.(n+1)]
proof
thus len F1 = size by A5;
thus F1.1 = X or size = 0 by A6;
let n;
assume 1 <= n & n < size;
then F1.(n+1) =
SC_Psuperpos(F1.n,denaming(V,A,val.(len val-n)),loc/.(len val-n))
by A5,A7;
hence P[n,F1.n,F1.(n+1)];
end;
A13: len F2 = size & (F2.1 = X or size = 0) &
for n st 1 <= n & n < size holds P[n,F2.n,F2.(n+1)]
proof
thus len F2 = size by A8;
thus F2.1 = X or size = 0 by A9;
let n;
assume 1 <= n & n < size;
then F2.(n+1) =
SC_Psuperpos(F2.n,denaming(V,A,val.(len val-n)),loc/.(len val-n))
by A8,A10;
hence P[n,F2.n,F2.(n+1)];
end;
thus F1 = F2 from RECDEF_1:sch 8(A11,A12,A13);
end;
end;
theorem Th15:
for size being non zero Nat
for val being size-element FinSequence holds
loc,val,size are_correct_wrt d1 &
1 <= n & n <= len LocalOverlapSeq(A,loc,val,d1,size) &
1 <= m & m <= len LocalOverlapSeq(A,loc,val,d1,size) implies
LocalOverlapSeq(A,loc,val,d1,size).n in dom denaming(V,A,val.m)
proof
let size be non zero Nat;
let val be size-element FinSequence;
set F = LocalOverlapSeq(A,loc,val,d1,size);
set v = val.m;
assume that
A1: loc,val,size are_correct_wrt d1 and
A2: 1 <= n & n <= len F;
A3: len F = size by NOMIN_7:def 4;
A4: dom denaming(V,A,v) = {d where d is NonatomicND of V,A: v in dom d}
by NOMIN_1:def 18;
A5: F.n is NonatomicND of V,A by A2,NOMIN_7:def 6;
assume 1 <= m <= len F;
then v in dom(F.n) by A1,A2,A3,NOMIN_7:10;
hence thesis by A4,A5;
end;
definition
let V,A,inp,d;
let val be FinSequence;
pred inp is_valid_input V,A,val,d means
ex d1 being NonatomicND of V,A st d = d1 & val is_valid_wrt d1 &
for n being Nat st 1 <= n <= len inp holds d1.(val.n) = inp.n;
end;
definition
let V,A,inp;
let val be FinSequence;
defpred P[object] means inp is_valid_input V,A,val,$1;
func valid_input(V,A,val,inp) -> SCPartialNominativePredicate of V,A
means :Def11:
dom it = ND(V,A) &
for d being object st d in dom it holds
(inp is_valid_input V,A,val,d implies it.d = TRUE) &
(not inp is_valid_input V,A,val,d implies it.d = FALSE);
existence
proof
A1: ND(V,A) c= ND(V,A);
consider p being SCPartialNominativePredicate of V,A such that
A2: dom p = ND(V,A) & (for d being object st d in dom p holds
(P[d] implies p.d = TRUE) & (not P[d] implies p.d = FALSE))
from PARTPR_2:sch 1(A1);
take p;
thus thesis by A2;
end;
uniqueness
proof
for p,q being SCPartialNominativePredicate of V,A st
(dom p = ND(V,A) & (for d being object st d in dom p holds
(P[d] implies p.d = TRUE) & (not P[d] implies p.d = FALSE))) &
(dom q = ND(V,A) & (for d being object st d in dom q holds
(P[d] implies q.d = TRUE) & (not P[d] implies q.d = FALSE)))
holds p = q from PARTPR_2:sch 2;
hence thesis;
end;
end;
registration
let V,A,inp;
let val be FinSequence;
cluster valid_input(V,A,val,inp) -> total;
coherence by Def11;
end;
definition
let V,A,d;
let Z,res be FinSequence;
pred res is_valid_output V,A,Z,d means
ex d1 being NonatomicND of V,A st d = d1 & Z is_valid_wrt d1 &
for n being Nat st 1 <= n <= len Z holds d1.(Z.n) = res.n;
end;
definition
let V,A;
let Z,res be object;
set D = {d where d is TypeSCNominativeData of V,A: d in dom denaming(V,A,Z)};
defpred P[object] means <*res*> is_valid_output V,A,<*Z*>,$1;
func valid_output(V,A,Z,res) -> SCPartialNominativePredicate of V,A
means
dom it = {d where d is TypeSCNominativeData of V,A: d in dom denaming(V,A,Z)}
& for d being object st d in dom it holds
(<*res*> is_valid_output V,A,<*Z*>,d implies it.d = TRUE) &
(not <*res*> is_valid_output V,A,<*Z*>,d implies it.d = FALSE);
existence
proof
A1: D c= ND(V,A)
proof
let v;
assume v in D;
then ex d being TypeSCNominativeData of V,A st v = d &
d in dom denaming(V,A,Z);
hence thesis;
end;
consider p being SCPartialNominativePredicate of V,A such that
A2: dom p = D & (for d being object st d in dom p holds
(P[d] implies p.d = TRUE) & (not P[d] implies p.d = FALSE))
from PARTPR_2:sch 1(A1);
take p;
thus thesis by A2;
end;
uniqueness
proof
for p,q being SCPartialNominativePredicate of V,A st
(dom p = D & (for d being object st d in dom p holds
(P[d] implies p.d = TRUE) & (not P[d] implies p.d = FALSE))) &
(dom q = D & (for d being object st d in dom q holds
(P[d] implies q.d = TRUE) & (not P[d] implies q.d = FALSE)))
holds p = q from PARTPR_2:sch 2;
hence thesis;
end;
end;
theorem Th16:
for val being size-element FinSequence holds
loc,val,size are_correct_wrt d1 &
d = LocalOverlapSeq(A,loc,val,d1,size).(size-1) & 2 <= n+1 < size &
local_overlapping(V,A,d,denaming(V,A,val.len val).d,loc/.len val) in dom p
implies
local_overlapping(V,A,LocalOverlapSeq(A,loc,val,d1,size).(size-n-1),
denaming(V,A,val.(len val-n)).
(LocalOverlapSeq(A,loc,val,d1,size).(size-n-1)),
loc/.(len val-n)) in dom(SC_Psuperpos_Seq(loc,val,p).n)
proof
let val be size-element FinSequence;
set D = denaming(V,A,val.len val);
set S = SC_Psuperpos_Seq(loc,val,p);
set L = LocalOverlapSeq(A,loc,val,d1,size);
deffunc F(Nat) = local_overlapping(V,A,L.(size-$1-1),
denaming(V,A,val.(len val-$1)).(L.(size-$1-1)),loc/.(len val-$1));
assume that
A1: loc,val,size are_correct_wrt d1 and
A2: d = LocalOverlapSeq(A,loc,val,d1,size).(size-1) and
A3: 2 <= n+1 and
A4: n+1 < size and
A5: local_overlapping(V,A,d,D.d,loc/.len val) in dom p;
A6: len val = size by CARD_1:def 7;
A7: len L = size by NOMIN_7:def 4;
A8: len S = len val by Def9;
defpred P[Nat] means 2 <= $1+1 < size implies F($1) in dom(S.$1);
A9: P[0];
A10: for k being Nat st P[k] holds P[k+1]
proof
let k be Nat such that
A11: P[k] and
2 <= k+1+1 and
A12: k+1+1 < size;
A13: 2 <= k+2 by NAT_1:11;
then
A14: 2 < size by A12,XXREAL_0:2;
per cases;
suppose
A15: k = 0;
S.1 = SC_Psuperpos(p,D,loc/.len val) by Def9;
then
A16: dom(S.1) = {d where d is TypeSCNominativeData of V,A:
local_overlapping(V,A,d,D.d,loc/.len val) in dom p & d in dom D}
by NOMIN_2:def 11;
A17: dom D = {d where d is NonatomicND of V,A: val.len val in dom d}
by NOMIN_1:def 18;
reconsider N = size-2 as Element of NAT by A13,A12,XXREAL_0:2,INT_1:5;
2-2 < size-2 by A14,XREAL_1:14;
then
A18: 0+1 <= N by NAT_1:13;
A19: N < len L by A7,XREAL_1:44; then
A20: L.(N+1) =
local_overlapping(V,A,L.N,denaming(V,A,val.(N+1)).(L.N),loc/.(N+1))
by A18,NOMIN_7:def 4;
A21: 1 <= len val by A6,A7,A18,A19,XXREAL_0:2;
A22: 1 <= N+1 by NAT_1:11;
A23: N+1 <= size by XREAL_1:44,NAT_1:13;
reconsider F1 = F(1) as NonatomicND of V,A
by A6,A7,A20,A23,NAT_1:11,NOMIN_7:def 6;
val.len val in dom(F1) by A1,A6,A21,A20,A22,A23,NOMIN_7:10;
then F(1) in dom D by A17;
hence F(k+1) in dom(S.(k+1)) by A15,A16,A5,A2,A6,A20;
end;
suppose
A24: k > 0;
then 0+1 <= k by NAT_1:13;
then
A25: 1+1 <= k+1 by XREAL_1:7;
A26: k+1 < size by A12,NAT_1:13;
set D = denaming(V,A,val.(len val-k));
A27: 0+1 <= k by A24,NAT_1:13;
k < k+2 by XREAL_1:29;
then k < size by A12,XXREAL_0:2; then
S.(k+1) = SC_Psuperpos(S.k,D,loc/.(len val-k)) by A6,A8,A27,Def9;
then
A28: dom(S.(k+1)) = {d where d is TypeSCNominativeData of V,A:
local_overlapping(V,A,d,D.d,loc/.(len val-k)) in dom(S.k) &
d in dom(D)} by NOMIN_2:def 11;
A29: dom(D) = {d where d is NonatomicND of V,A: val.(len val-k) in dom d}
by NOMIN_1:def 18;
A30: size-(k+1) < size by XREAL_1:44;
A31: k+1-k < size-k by A26,XREAL_1:9;
then
A32: 1-1 < size-k-1 by XREAL_1:9;
then
A33: 0+1 <= size-k-1 by INT_1:7;
reconsider s = size-k-1 as Element of NAT by A32,INT_1:3;
reconsider N = s-1 as Element of NAT by A33,INT_1:5;
k+1+1-k < size-k by A12,XREAL_1:9;
then 1+1-1 < size-k-1 by XREAL_1:9;
then 1-1 < N by XREAL_1:9;
then
A34: 0+1 <= N by INT_1:7;
N < s by XREAL_1:44;
then N < len L by A7,A30,XXREAL_0:2; then
A35: L.s =
local_overlapping(V,A,L.N,denaming(V,A,val.(N+1)).(L.N),loc/.(N+1))
by A34,NOMIN_7:def 4;
reconsider Fk1 = F(k+1) as NonatomicND of V,A
by A6,A7,A35,A30,A33,NOMIN_7:def 6;
reconsider M = size-k as Element of NAT by A31,INT_1:3;
M <= size by XREAL_1:43;
then val.(len val-k) in dom(Fk1) by A1,A6,A35,A30,A33,A31,NOMIN_7:10;
then F(k+1) in dom(D) by A29;
hence F(k+1) in dom(S.(k+1)) by A6,A11,A25,A28,A35,A12,NAT_1:13;
end;
end;
for k being Nat holds P[k] from NAT_1:sch 2(A9,A10);
hence thesis by A3,A4;
end;
theorem Th17:
for val being size-element FinSequence holds
loc,val,size are_correct_wrt d1 &
d = LocalOverlapSeq(A,loc,val,d1,size).(size-1) &
local_overlapping(V,A,d,denaming(V,A,val.len val).d,loc/.len val) in dom p
implies
for m,n being Nat st 1 <= m < size & 1 <= n < size holds
SC_Psuperpos_Seq(loc,val,p).m.(LocalOverlapSeq(A,loc,val,d1,size).(size-m)) =
SC_Psuperpos_Seq(loc,val,p).n.(LocalOverlapSeq(A,loc,val,d1,size).(size-n))
proof
let val be size-element FinSequence such that
A1: loc,val,size are_correct_wrt d1 and
A2: d = LocalOverlapSeq(A,loc,val,d1,size).(size-1) and
A3: local_overlapping(V,A,d,denaming(V,A,val.len val).d,loc/.len val) in dom p;
let m,n be Nat such that
A4: 1 <= m and
A5: m < size and
A6: 1 <= n and
A7: n < size;
set S = SC_Psuperpos_Seq(loc,val,p);
set L = LocalOverlapSeq(A,loc,val,d1,size);
defpred P[Nat] means
1 <= $1 < size implies S.m.(L.(size-m)) = S.$1.(L.(size-$1));
A8: P[0];
A9: for k being Nat st P[k] holds P[k+1]
proof
let k be Nat such that
A10: P[k] and
1 <= k+1 and
A11: k+1 < size;
set D = denaming(V,A,val.(len val-k));
A12: len val = size by CARD_1:def 7;
A13: len S = len val by Def9;
A14: len L = size by NOMIN_7:def 4;
A15: k < size by A11,NAT_1:13;
per cases;
suppose
A16: k = 0;
defpred R[Nat] means
1 <= $1 < size implies S.$1.(L.(size-$1)) = S.1.(L.(size-1));
A17: R[0];
A18: for x being Nat st R[x] holds R[x+1]
proof
let x be Nat such that
A19: R[x] and
1 <= x+1 and
A20: x+1 < size;
per cases;
suppose x = 0;
hence thesis;
end;
suppose x > 0;
then
A21: 0+1 <= x by NAT_1:13;
set DD = denaming(V,A,val.(len val-x));
A22: x <= x+1 by NAT_1:11;
then x < len S by A13,A12,A20,XXREAL_0:2;
then
A23: S.(x+1) = SC_Psuperpos(S.x,DD,loc/.(len val-x)) by A21,Def9;
reconsider u = size-x-1 as Element of NAT
by A20,XREAL_1:19,INT_1:5;
x+1-x < size-x by A20,XREAL_1:9;
then 1-1 < size-x-1 by XREAL_1:9;
then
A24: 0+1 <= size-(x+1) by INT_1:7;
A25: size-(x+1) < size by XREAL_1:44;
then reconsider dd = L.u as NonatomicND of V,A
by A14,A24,NOMIN_7:def 6;
A26: dom(SC_Psuperpos(S.x,DD,loc/.(len val-x))) =
{d where d is TypeSCNominativeData of V,A:
local_overlapping(V,A,d,DD.d,loc/.(len val-x)) in dom(S.x) &
d in dom DD} by NOMIN_2:def 11;
1+1 <= x+1 by A21,XREAL_1:6;
then
A27: local_overlapping(V,A,dd,DD.dd,loc/.(len val-x)) in dom(S.x)
by A1,A2,A3,A20,Th16;
A28: dom DD = {d where d is NonatomicND of V,A:val.(len val-x) in dom d}
by NOMIN_1:def 18;
A29: 1 <= u+1 by NAT_1:11;
u+1 <= size by A25,INT_1:7;
then val.(u+1) in dom(L.u) by A1,A24,A25,A29,NOMIN_7:10;
then dd in dom DD by A12,A28;
then
A30: dd in dom(SC_Psuperpos(S.x,DD,loc/.(len val-x))) by A26,A27;
L.(u+1) =
local_overlapping(V,A,L.u,denaming(V,A,val.(u+1)).(L.u),loc/.(u+1))
by A14,A24,A25,NOMIN_7:def 4;
hence thesis by A22,A23,A30,A12,A19,A21,A20,XXREAL_0:2,NOMIN_2:35;
end;
end;
for x being Nat holds R[x] from NAT_1:sch 2(A17,A18);
hence S.m.(L.(size-m)) = S.(k+1).(L.(size-(k+1))) by A4,A5,A16;
end;
suppose k > 0;
then
A31: 0+1 <= k by NAT_1:13;
then
A32: S.(k+1) = SC_Psuperpos(S.k,D,loc/.(size-k)) by A12,A13,A15,Def9;
set D1 = L.(size-(k+1));
A33: k+1-k < size-k by A11,XREAL_1:9;
then reconsider w = size-k-1 as Element of NAT by INT_1:5;
1-1 < w by A33,XREAL_1:9;
then
A34: 0+1 <= w by NAT_1:13;
A35: size-(k+1) < size by XREAL_1:44;
then
A36: L.(w+1) =
local_overlapping(V,A,L.w,denaming(V,A,val.(w+1)).(L.w),loc/.(w+1))
by A14,A34,NOMIN_7:def 4;
reconsider D1 as NonatomicND of V,A by A14,A34,A35,NOMIN_7:def 6;
A37: dom(SC_Psuperpos(S.k,D,loc/.(len val-k))) =
{d where d is TypeSCNominativeData of V,A:
local_overlapping(V,A,d,D.d,loc/.(len val-k)) in dom(S.k) & d in dom D}
by NOMIN_2:def 11;
A38: D1 = L.(size-k-1);
1+1 <= k+1 by A31,XREAL_1:6; then
A39: local_overlapping(V,A,D1,D.D1,loc/.(len val-k)) in dom(S.k)
by A1,A11,A2,A3,A38,Th16;
A40: dom D = {d where d is NonatomicND of V,A: val.(len val-k) in dom d}
by NOMIN_1:def 18;
A41: 1 <= w+1 by NAT_1:11;
w+1 <= size by A35,INT_1:7;
then val.(w+1) in dom(L.w) by A1,A34,A35,A41,NOMIN_7:10;
then D1 in dom D by A12,A40;
then D1 in dom(SC_Psuperpos(S.k,D,loc/.(len val-k))) by A37,A39;
hence thesis by A10,A31,A32,A12,A36,A11,NAT_1:13,NOMIN_2:35;
end;
end;
for k being Nat holds P[k] from NAT_1:sch 2(A8,A9);
hence thesis by A6,A7;
end;
theorem
for val being size-element FinSequence
for dx,dy being object
for NN being Nat st NN = size-2 holds
loc,val,size are_correct_wrt d1 &
dx = LocalOverlapSeq(A,loc,val,d1,size).(size-1) &
local_overlapping(V,A,dx,denaming(V,A,val.len val).dx,loc/.len val)
in dom p &
dy = local_overlapping(V,A,LocalOverlapSeq(A,loc,val,d1,size).NN,
denaming(V,A,val.(NN+1)).(LocalOverlapSeq(A,loc,val,d1,size).NN),
loc/.(NN+1)) &
local_overlapping(V,A,dy,denaming(V,A,val.len val).dy,loc/.len val) in dom p
implies
SC_Psuperpos_Seq(loc,val,p).1.(LocalOverlapSeq(A,loc,val,d1,size).(size-1)) =
p.(LocalOverlapSeq(A,loc,val,d1,size).size)
proof
let val be size-element FinSequence;
let dx,dy be object;
let NN be Nat such that
A1: NN = size-2;
set S = SC_Psuperpos_Seq(loc,val,p);
set L = LocalOverlapSeq(A,loc,val,d1,size);
set D = denaming(V,A,val.len val);
assume that
A2: loc,val,size are_correct_wrt d1 and
A3: dx = LocalOverlapSeq(A,loc,val,d1,size).(size-1) and
A4: local_overlapping(V,A,dx,D.dx,loc/.len val) in dom p and
A5: dy = local_overlapping(V,A,L.NN,denaming(V,A,val.(NN+1)).(L.NN),
loc/.(NN+1)) and
A6: local_overlapping(V,A,dy,D.dy,loc/.len val) in dom p;
A7: 0+2 <= size-2+2 by A1,XREAL_1:6;
then
A8: 1 <= size by XXREAL_0:2;
reconsider N = size-1 as Element of NAT by A7,XXREAL_0:2,INT_1:5;
A9: len L = size by NOMIN_7:def 4;
A10: len val = size by CARD_1:def 7;
A11: S.1 = SC_Psuperpos(p,D,loc/.len val) by Def9;
A12: dom(SC_Psuperpos(p,D,loc/.len val)) =
{d where d is TypeSCNominativeData of V,A:
local_overlapping(V,A,d,D.d,loc/.len val) in dom p & d in dom D}
by NOMIN_2:def 11;
A13: 2-1 <= N by A7,XREAL_1:9;
then per cases by XXREAL_0:1;
suppose
A14: N = 1;
set D1 = denaming(V,A,val.1);
A15: L.1 = local_overlapping(V,A,d1,D1.d1,loc/.1) by NOMIN_7:def 4;
A16: L.(N+1) =
local_overlapping(V,A,L.N,denaming(V,A,val.(N+1)).(L.N),loc/.(N+1))
by A9,A14,NOMIN_7:def 4;
set dd = local_overlapping(V,A,d1,D1.d1,loc/.1);
dd in dom D by A2,A9,A10,A8,A15,Th15;
then dd in dom(SC_Psuperpos(p,D,loc/.len val)) by A12,A3,A4,A14,A15;
hence S.1.(L.(size-1)) = p.(L.size) by A14,A10,A15,A16,A11,NOMIN_2:35;
end;
suppose
A17: 1 < N;
then reconsider NN = N-1 as Element of NAT by INT_1:5;
1-1 < N-1 by A17,XREAL_1:9;
then
A18: 0+1 <= NN by INT_1:7;
A19: N-1 < N by XREAL_1:44;
A20: N < len L by A9,XREAL_1:44;
then NN < len L by A19,XXREAL_0:2;
then
A21: L.(NN+1) =
local_overlapping(V,A,L.NN,denaming(V,A,val.(NN+1)).(L.NN),loc/.(NN+1))
by A18,NOMIN_7:def 4;
A22: L.(N+1) =
local_overlapping(V,A,L.N,denaming(V,A,val.(N+1)).(L.N),loc/.(N+1))
by A17,A20,NOMIN_7:def 4;
set Dn = denaming(V,A,val.(NN+1));
set dd = local_overlapping(V,A,L.NN,Dn.(L.NN),loc/.(NN+1));
dd in dom D by A2,A9,A10,A8,A21,A13,A20,Th15;
then dd in dom(SC_Psuperpos(p,D,loc/.len val)) by A1,A5,A6,A12;
hence S.1.(L.(size-1)) = p.(L.size) by A10,A22,A11,A21,NOMIN_2:35;
end;
end;
theorem
for val being size-element FinSequence
for p being SCPartialNominativePredicate of V,A holds
3 <= size &
loc,val,size are_correct_wrt d1 &
local_overlapping(V,A,LocalOverlapSeq(A,loc,val,d1,size).(size-1),
denaming(V,A,val.len val).(LocalOverlapSeq(A,loc,val,d1,size).(size-1)),
loc/.len val) in dom p
&
local_overlapping(V,A,d1,denaming(V,A,val.1).d1,loc/.1)
in dom(SC_Psuperpos_Seq(loc,val,p).(size-1))
implies
(SC_Psuperpos_Seq(loc,val,p).len SC_Psuperpos_Seq(loc,val,p)).d1 =
SC_Psuperpos(SC_Psuperpos_Seq(loc,val,p).(size-2),denaming(V,A,val.2),
loc/.2).(LocalOverlapSeq(A,loc,val,d1,size).1)
proof
let val be size-element FinSequence;
let p be SCPartialNominativePredicate of V,A;
set SE = SC_Psuperpos_Seq(loc,val,p);
set F = LocalOverlapSeq(A,loc,val,d1,size);
set dd = F.(size-1);
set D1 = denaming(V,A,val.1);
set D2 = denaming(V,A,val.2);
set P = SE.(size-2);
assume that
A1: 3 <= size and
A2: loc,val,size are_correct_wrt d1 and
A3: local_overlapping(V,A,dd,denaming(V,A,val.len val).dd,loc/.len val)
in dom p and
A4: local_overlapping(V,A,d1,D1.d1,loc/.1) in dom(SE.(size-1));
A5: len val = size by CARD_1:def 7;
A6: len SE = len val by Def9;
A7: len F = size by NOMIN_7:def 4;
A8: 2 < size by A1,XXREAL_0:2;
reconsider nn = size-2 as Element of NAT by A1,XXREAL_0:2,INT_1:5;
set N = nn+1;
A9: 3-2 <= nn by A1,XREAL_1:9;
then
A10: 1+1 <= nn+1 by XREAL_1:6;
A12: size-1 < size by XREAL_1:44;
A13: nn < size by XREAL_1:44;
A15: len val-N = 1 by A5;
A16: F.1 = local_overlapping(V,A,d1,D1.d1,loc/.1) by NOMIN_7:def 4;
A11: 1 <= N by A9,XREAL_1:29; then
A14: 1 < len F by A7,A12,XXREAL_0:2; then
A17: F.1 is NonatomicND of V,A by NOMIN_7:def 6;
A18: SE.(N+1) = SC_Psuperpos(SE.N,D1,loc/.1) by A6,A11,A12,A15,Def9;
A19: F.(1+1) =
local_overlapping(V,A,F.1,denaming(V,A,val.(1+1)).(F.1),loc/.(1+1))
by A14,NOMIN_7:def 4;
A20: dom SC_Psuperpos(P,D2,loc/.2) =
{d where d is TypeSCNominativeData of V,A:
local_overlapping(V,A,d,D2.d,loc/.2) in dom P & d in dom D2}
by NOMIN_2:def 11;
A21: local_overlapping(V,A,F.(size-nn-1),denaming(V,A,val.(len val-nn)).
(F.(size-nn-1)),loc/.(len val-nn)) in dom(P) by A2,A10,A3,XREAL_1:44,Th16;
A22: dom D2 = {d where d is NonatomicND of V,A: val.2 in dom d}
by NOMIN_1:def 18;
val.2 in dom(F.1) by A7,A2,A8,A14,NOMIN_7:10;
then F.1 in dom D2 by A17,A22;
then
A23: F.1 in dom(SC_Psuperpos(P,D2,loc/.2)) by A5,A17,A20,A21;
A24: dom SC_Psuperpos(SE.N,D1,loc/.1) =
{d where d is TypeSCNominativeData of V,A:
local_overlapping(V,A,d,D1.d,loc/.1) in dom(SE.N) & d in dom D1}
by NOMIN_2:def 11;
A25: dom D1 = {d where d is NonatomicND of V,A: val.1 in dom d}
by NOMIN_1:def 18;
A26: val is_valid_wrt d1 by A2;
1 in dom val by A5,A7,A14,FINSEQ_3:25;
then val.1 in rng val by FUNCT_1:def 3;
then d1 in dom D1 by A25,A26;
then d1 in dom(SC_Psuperpos(SE.N,D1,loc/.1)) by A24,A4;
hence (SE.len SE).d1 = SE.N.(F.(size-N)) by A5,A6,A18,A16,NOMIN_2:35
.= SE.nn.(F.(size-nn)) by A2,A3,A11,A12,A9,A13,Th17
.= SC_Psuperpos(P,D2,loc/.2).(F.1) by A17,A23,A19,NOMIN_2:35;
end;
definition
let V,A,loc,d1,pos;
let prg be FPrg(ND(V,A))-valued FinSequence such that
A1: len prg > 0;
defpred P[Nat,object,object] means
$3 = local_overlapping(V,A,$2,prg.($1+1).($2),loc/.(pos.($1+1)));
set X = local_overlapping(V,A,d1,(prg.1).d1,loc/.(pos.1));
func PrgLocalOverlapSeq(A,loc,d1,prg,pos) -> FinSequence of ND(V,A) means
:Def14:
len it = len prg &
it.1 = local_overlapping(V,A,d1,(prg.1).d1,loc/.(pos.1)) &
for n being Nat st 1 <= n < len it holds
it.(n+1) = local_overlapping(V,A,it.n,prg.(n+1).(it.n),loc/.(pos.(n+1)));
existence
proof
A2: for n being Nat st 1 <= n < len prg
for x being Element of ND(V,A)
ex y being Element of ND(V,A) st P[n,x,y]
proof
let n be Nat;
assume 1 <= n & n < len prg;
let x be Element of ND(V,A);
set y = local_overlapping(V,A,x,prg.(n+1).x,loc/.(pos.(n+1)));
y in ND(V,A);
then reconsider y as Element of ND(V,A);
take y;
thus P[n,x,y];
end;
X in ND(V,A);
then reconsider X as Element of ND(V,A);
ex p being FinSequence of ND(V,A) st len p = len prg &
(p.1 = X or len prg = 0) &
for n st 1 <= n & n < len prg holds P[n,p.n,p.(n+1)]
from RECDEF_1:sch 4(A2);
hence thesis by A1;
end;
uniqueness
proof
set size = len prg;
let F,G be FinSequence of ND(V,A) such that
A3: len F = size and
A4: F.1 = X and
A5: for n being Nat st 1 <= n < len F holds
F.(n+1) =
local_overlapping(V,A,F.n,prg.(n+1).(F.n),loc/.(pos.(n+1))) and
A6: len G = size and
A7: G.1 = X and
A8: for n being Nat st 1 <= n < len G holds
G.(n+1) =
local_overlapping(V,A,G.n,prg.(n+1).(G.n),loc/.(pos.(n+1)));
A9: for n st 1 <= n & n < size for x,y1,y2 being set st P[n,x,y1] & P[n,x,y2]
holds y1 = y2;
A10: len F = size & (F.1 = X or size = 0) & for n st 1 <= n & n < size
holds P[n,F.n,F.(n+1)] by A3,A4,A5;
A11: len G = size & (G.1 = X or size = 0) &
for n st 1 <= n & n < size holds P[n,G.n,G.(n+1)] by A6,A7,A8;
thus thesis from RECDEF_1:sch 7(A9,A10,A11);
end;
end;
registration
let V,A,loc,d1,prg,pos;
cluster PrgLocalOverlapSeq(A,loc,d1,prg,pos) -> (V,A)-NonatomicND-yielding;
coherence
proof
set F = PrgLocalOverlapSeq(A,loc,d1,prg,pos);
A1: len prg > 0;
let n such that
A2: 1 <= n <= len F;
set X = local_overlapping(V,A,d1,(prg.1).d1,loc/.(pos.1));
defpred P[Nat] means
1 <= $1 & $1 <= len F implies F.$1 is NonatomicND of V,A;
A3: P[0];
A4: for n st P[n] holds P[n+1]
proof
let n;
assume that
A5: P[n] and
1 <= n+1 and
A6: n+1 <= len F;
per cases;
suppose
A7: n = 0;
F.1 = X by A1,Def14;
hence F.(n+1) is NonatomicND of V,A by A7,NOMIN_2:9;
end;
suppose 0 < n;
then
A8: 0+1 <= n by NAT_1:13;
A9: n+0 < n+1 by XREAL_1:8;
then n < len F by A6,XXREAL_0:2;
then F.(n+1) =
local_overlapping(V,A,F.n,prg.(n+1).(F.n),loc/.(pos.(n+1)))
by A1,A8,Def14;
hence thesis by A5,A6,A8,A9,NOMIN_2:9,XXREAL_0:2;
end;
end;
for n holds P[n] from NAT_1:sch 2(A3,A4);
hence thesis by A2;
end;
end;
registration
let V,A,loc,d1,prg,pos,n;
cluster
PrgLocalOverlapSeq(A,loc,d1,prg,pos).n -> Function-like Relation-like;
coherence
proof
set F = PrgLocalOverlapSeq(A,loc,d1,prg,pos);
per cases;
suppose n in dom F;
then 1 <= n <= len F by FINSEQ_3:25;
hence thesis by NOMIN_7:def 6;
end;
suppose not n in dom F;
hence thesis by FUNCT_1:def 2;
end;
end;
end;
definition
let V,A,loc,d1,prg,pos;
pred prg_doms_of loc,d1,prg,pos means
for n being Nat st 1 <= n < len prg holds
PrgLocalOverlapSeq(A,loc,d1,prg,pos).n in dom(prg.(n+1));
end;
theorem Th20:
1 <= n <= len prg & PrgLocalOverlapSeq(A,loc,d1,prg,pos).m in dom(prg.n)
implies
prg.n.(PrgLocalOverlapSeq(A,loc,d1,prg,pos).m)
is TypeSCNominativeData of V,A
proof
set F = PrgLocalOverlapSeq(A,loc,d1,prg,pos);
set P = prg.n;
assume that
A1: 1 <= n and
A2: n <= len prg and
A3: F.m in dom(P);
n in dom prg by A1,A2,FINSEQ_3:25;
then P in rng prg by FUNCT_1:def 3;
then
A4: rng(P) c= ND(V,A) by RELAT_1:def 19;
P.(F.m) in rng P by A3,FUNCT_1:def 3;
hence thesis by A4,NOMIN_1:39;
end;
theorem Th21:
V is non empty & A is_without_nonatomicND_wrt V implies
for n being Nat st 1 <= n & n < len prg &
PrgLocalOverlapSeq(A,loc,d1,prg,pos).n in dom(prg.(n+1)) holds
dom(PrgLocalOverlapSeq(A,loc,d1,prg,pos).(n+1)) =
{ loc/.(pos.(n+1)) } \/ dom(PrgLocalOverlapSeq(A,loc,d1,prg,pos).n)
proof
set size = len prg;
set F = PrgLocalOverlapSeq(A,loc,d1,prg,pos);
assume that
A1: V is non empty and
A2: A is_without_nonatomicND_wrt V;
let n be Nat;
assume that
A3: 1 <= n and
A4: n < size and
A5: F.n in dom(prg.(n+1));
A6: len F = size by Def14;
reconsider Fn = F.n as NonatomicND of V,A by A3,A4,A6,NOMIN_7:def 6;
set v = loc/.(pos.(n+1));
set d2 = prg.(n+1).(F.n);
n+1 <= size by A4,NAT_1:13;
then d2 is TypeSCNominativeData of V,A by A5,NAT_1:11,Th20;
then dom local_overlapping(V,A,Fn,d2,v) = {v} \/ dom Fn
by A1,A2,NOMIN_4:4;
hence thesis by A3,A4,A6,Def14;
end;
theorem Th22:
V is non empty & A is_without_nonatomicND_wrt V implies
for n being Nat st 1 <= n & n < len prg &
PrgLocalOverlapSeq(A,loc,d1,prg,pos).n in dom(prg.(n+1)) holds
dom(PrgLocalOverlapSeq(A,loc,d1,prg,pos).n) c=
dom(PrgLocalOverlapSeq(A,loc,d1,prg,pos).(n+1))
proof
set F = PrgLocalOverlapSeq(A,loc,d1,prg,pos);
assume
A1: V is non empty & A is_without_nonatomicND_wrt V;
let n be Nat;
assume 1 <= n & n < len prg & F.n in dom(prg.(n+1));
then dom(F.(n+1)) = { loc/.(pos.(n+1)) } \/ dom(F.n) by A1,Th21;
hence thesis by XBOOLE_1:7;
end;
theorem
V is non empty & A is_without_nonatomicND_wrt V &
dom PrgLocalOverlapSeq(A,loc,d1,prg,pos) c= dom prg &
d1 in dom(prg.1) & prg_doms_of loc,d1,prg,pos
implies
for n being Nat st 1 <= n <= len prg holds
dom(d1) c= dom(PrgLocalOverlapSeq(A,loc,d1,prg,pos).n)
proof
set F = PrgLocalOverlapSeq(A,loc,d1,prg,pos);
assume that
A1: V is non empty and
A2: A is_without_nonatomicND_wrt V and
A3: dom F c= dom prg and
A4: d1 in dom(prg.1) and
A5: prg_doms_of loc,d1,prg,pos;
let n be Nat;
assume that
A6: 1 <= n and
A7: n <= len prg;
defpred P[Nat] means 1 <= $1 <= len prg implies dom(d1) c= dom(F.$1);
A8: P[0];
A9: for k being Nat st P[k] holds P[k+1]
proof
let k be Nat such that
A10: P[k] and
A11: 1 <= k+1 and
A12: k+1 <= len prg;
A13: len F = len prg by Def14;
per cases;
suppose
A14: k = 0;
set v = loc/.(pos.1);
set D = prg.1;
1 <= len F by A11,A12,A13,XXREAL_0:2;
then 1 in dom F by FINSEQ_3:25;
then D in rng prg by A3,FUNCT_1:def 3;
then
reconsider d2 = D.d1 as TypeSCNominativeData of V,A
by PARTFUN1:4,NOMIN_1:39,A4;
A15: F.1 = local_overlapping(V,A,d1,d2,v) by A7,Def14;
dom local_overlapping(V,A,d1,d2,v) = {v} \/ dom d1 by A1,A2,NOMIN_4:4;
hence thesis by A14,A15,XBOOLE_1:7;
end;
suppose k > 0;
then
A16: 0+1 <= k by NAT_1:13;
A17: k <= k+1 by NAT_1:12;
k+0 < k+1 by XREAL_1:8;
then
A18: k < len prg by A12,XXREAL_0:2;
then F.k in dom(prg.(k+1)) by A5,A16;
then dom(F.k) c= dom(F.(k+1)) by A1,A2,A16,A18,Th22;
hence thesis by A17,A10,A12,A16,XXREAL_0:2;
end;
end;
for k being Nat holds P[k] from NAT_1:sch 2(A8,A9);
hence thesis by A6,A7;
end;
theorem Th24:
V is non empty & A is_without_nonatomicND_wrt V &
prg_doms_of loc,d1,prg,pos
implies
for m,n being Nat st 1 <= n <= m <= len prg holds
dom(PrgLocalOverlapSeq(A,loc,d1,prg,pos).n) c=
dom(PrgLocalOverlapSeq(A,loc,d1,prg,pos).m)
proof
set F = PrgLocalOverlapSeq(A,loc,d1,prg,pos);
assume that
A1: V is non empty & A is_without_nonatomicND_wrt V and
A2: prg_doms_of loc,d1,prg,pos;
let m,n be Nat;
assume that
A3: 1 <= n and
A4: n <= m and
A5: m <= len prg;
per cases by A4,XXREAL_0:1;
suppose n = m;
hence thesis;
end;
suppose
A6: n < m;
defpred P[Nat] means n < $1 <= len prg implies dom(F.n) c= dom(F.$1);
A7: P[0];
A8: for k being Nat st P[k] holds P[k+1]
proof
let k be Nat such that
A9: P[k] and
A10: n < k+1 and
A11: k+1 <= len prg;
A12: n <= k by A10,NAT_1:13;
then
A13: 1 <= k by A3,XXREAL_0:2;
k+0 < k+1 by XREAL_1:8;
then
A14: k < len prg by A11,XXREAL_0:2;
then F.k in dom(prg.(k+1)) by A2,A13;
then dom(F.(k)) c= dom(F.(k+1)) by A1,A13,A14,Th22;
hence dom(F.n) c= dom(F.(k+1)) by A9,A12,A11,NAT_1:13,XXREAL_0:1;
end;
for k being Nat holds P[k] from NAT_1:sch 2(A7,A8);
hence thesis by A5,A6;
end;
end;
theorem
V is non empty & A is_without_nonatomicND_wrt V &
dom PrgLocalOverlapSeq(A,loc,d1,prg,pos) c= dom prg &
d1 in dom(prg.1) & prg_doms_of loc,d1,prg,pos
implies
for m,n being Nat st 1 <= n <= m <= len prg holds
loc/.(pos.n) in dom(PrgLocalOverlapSeq(A,loc,d1,prg,pos).m)
proof
set size = len prg;
set F = PrgLocalOverlapSeq(A,loc,d1,prg,pos);
assume that
A1: V is non empty and
A2: A is_without_nonatomicND_wrt V and
A3: dom F c= dom prg and
A4: d1 in dom(prg.1) and
A5: prg_doms_of loc,d1,prg,pos;
let m,n be Nat such that
A6: 1 <= n and
A7: n <= m and
A8: m <= size;
A9: 1 <= m by A6,A7,XXREAL_0:2;
A10: n <= size by A7,A8,XXREAL_0:2;
A11: len F = size by Def14;
reconsider i1 = n-1 as Element of NAT by A6,INT_1:5;
set v = loc/.(pos.n);
set D = prg.n;
A12: v in {v} by TARSKI:def 1;
n in dom F by A6,A10,A11,FINSEQ_3:25;
then
A13: D in rng prg by A3,FUNCT_1:def 3;
per cases;
suppose
A14: i1 = 0;
then reconsider d2 = D.d1 as TypeSCNominativeData of V,A
by A4,A13,PARTFUN1:4,NOMIN_1:39;
A15: F.1 = local_overlapping(V,A,d1,d2,v) by A8,A14,Def14;
A16: dom local_overlapping(V,A,d1,d2,v) = {v} \/ dom(d1) by A1,A2,NOMIN_4:4;
A17: dom(F.1) c= dom(F.m) by A1,A5,A2,A8,A9,Th24;
v in {v} \/ dom(d1) by A12,XBOOLE_0:def 3;
hence v in dom(F.m) by A15,A16,A17;
end;
suppose i1 > 0;
then
A18: 0+1 <= i1 by NAT_1:13;
n-1 < n-0 by XREAL_1:15;
then
A19: i1 < size by A10,XXREAL_0:2;
then reconsider dd = F.i1 as NonatomicND of V,A by A18,A11,NOMIN_7:def 6;
dd in dom(prg.(i1+1)) by A5,A18,A19;
then reconsider d2 = D.dd as TypeSCNominativeData of V,A
by PARTFUN1:4,NOMIN_1:39,A13;
A20: F.n = local_overlapping(V,A,dd,d2,loc/.(pos.(i1+1)))
by A11,A18,A19,Def14;
A21: dom local_overlapping(V,A,dd,d2,v) = {v} \/ dom(dd) by A1,A2,NOMIN_4:4;
A22: v in {v} \/ dom(dd) by A12,XBOOLE_0:def 3;
dom(F.n) c= dom(F.m) by A1,A5,A2,A6,A7,A8,Th24;
hence v in dom(F.m) by A22,A20,A21;
end;
end;
| {"subset_name": "curated", "file": "formal/mizar/nomin_8.miz"} |
TITLE: Inverse Fourier transform of a partial fraction decomposition?
QUESTION [2 upvotes]: For the function
$$\alpha(\omega)=\frac{R}{i\omega-\lambda}+\frac{R^*}{i\omega-\lambda^*},$$
where $R$ and $\lambda$ are both complex numbers, What is the simplest way to obtain the inverse Fourier transform (not going into Laplace transform):
$$h(t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} \alpha(\omega)\exp(i\omega t)\,\mathrm{d}\omega.$$
(The solution should be $h(t)=R\exp(\lambda t)+R^*\exp(\lambda^*t)$.)
and then go back to the frequency domain with
$$\alpha(\omega)=\int_{-\infty}^{+\infty} h(t)\exp(-i\omega t)\,\mathrm{d}t.$$
REPLY [2 votes]: $\def\i{\mathrm{i}}\def\e{\mathrm{e}}\def\d{\mathrm{d}}\def\Re{\mathop{\mathrm{Re}}}$First, for any $t > 0$, $r > |λ|$, by the residue theorem,$$
\int_{-r}^r \frac{\e^{\i zt}}{z + \i λ} \,\d z + \int\limits_{γ_r} \frac{\e^{\i zt}}{z + \i λ} \,\d z = \begin{cases}
2π\i · \e^{\i zt} \Biggr|_{z = -\i λ} = 2π\i \e^{λt}; & \Re(λ) < 0\\
0; & \Re(λ) > 0
\end{cases}
$$
where $γ_r = \{z = r\e^{\i θ} \mid 0 \leqslant θ \leqslant π\}$ with counterclockwise orientation. Note that$$
\lim_{|z| \to +\infty} \left| \frac{1}{z + \i λ} \right| = 0,
$$
making $r \to +\infty$ and by Jordan's lemma,$$
\int_{-\infty}^{+\infty} \frac{\e^{\i zt}}{z + \i λ} \,\d z = \begin{cases}
2π\i \e^{λt}; & \Re(λ) < 0\\
0; & \Re(λ) > 0
\end{cases}
$$
Analogously,$$
\int_{-\infty}^{+\infty} \frac{\e^{\i zt}}{z - \i λ} \,\d z = \begin{cases}
2π\i \e^{-λt}; & \Re(λ) > 0\\
0; & \Re(λ) < 0
\end{cases}
$$
Case 1: $\Re(λ) > 0$, then $\Re(\overline{λ}) = \Re(λ) > 0$. For $t > 0$,\begin{align*}
\int_{-\infty}^{+\infty} α(ω) \e^{\i ωt} \,\d ω &= \int_{-\infty}^{+\infty} \frac{R \e^{\i ωt}}{\i ω - λ} \,\d ω + \int_{-\infty}^{+\infty} \frac{\overline{R} \e^{\i ωt}}{\i ω - \overline{λ}} \,\d ω\\
&= -\i R \int_{-\infty}^{+\infty} \frac{\e^{\i zt}}{z + \i λ} \,\d z - \i \overline{R} \int_{-\infty}^{+\infty} \frac{\e^{\i zt}}{z + \i \overline{λ}} \,\d z\\
&= -\i R · 0 - \i \overline{R} · 0 = 0.
\end{align*}
For $t < 0$,\begin{align*}
\int_{-\infty}^{+\infty} α(ω) \e^{\i ωt} \,\d ω &= \int_{-\infty}^{+\infty} \frac{R \e^{\i ωt}}{\i ω - λ} \,\d ω + \int_{-\infty}^{+\infty} \frac{\overline{R} \e^{\i ωt}}{\i ω - \overline{λ}} \,\d ω\\
&= \int_{-\infty}^{+\infty} \frac{R \e^{\i ω'(-t)}}{-\i ω' - λ} \,\d ω' + \int_{-\infty}^{+\infty} \frac{\overline{R} \e^{\i ω'(-t)}}{-\i ω' - \overline{λ}} \,\d ω'\\
&= \i R \int_{-\infty}^{+\infty} \frac{\e^{\i z(-t)}}{z - \i λ} \,\d z + \i \overline{R} \int_{-\infty}^{+\infty} \frac{\e^{\i z(-t)}}{z - \i \overline{λ}} \,\d z\\
&= \i R · 2π\i \e^{-λ(-t)} + \i \overline{R} · 2π\i \e^{-\overline{λ}(-t)} = -2π (R \e^{λt} + \overline{R} \e^{\overline{λ}t}).
\end{align*}
Therfore,$$
h(t) = \frac{1}{2π} \int_{-\infty}^{+\infty} α(ω) \e^{\i ωt} \,\d ω = \begin{cases}
0; & t > 0\\
-(R \e^{λt} + \overline{R} \e^{\overline{λ}t}); & t < 0
\end{cases}
$$
Case 2: $\Re(λ) < 0$. Analogous calculation shows that$$
h(t) = \frac{1}{2π} \int_{-\infty}^{+\infty} α(ω) \e^{\i ωt} \,\d ω = \begin{cases}
R \e^{λt} + \overline{R} \e^{\overline{λ}t}; & t > 0\\
0; & t < 0
\end{cases}
$$
Now, if $\Re(λ) > 0$, then for any $ω \in \mathbb{R}$, $M > 0$,$$
\int_{-M}^0 \e^{λt} \e^{-\i ωt} \,\d t = \left. \frac{\e^{(λ - \i ω)t}}{λ - \i ω} \right|_{-M}^0 = -\frac{1}{\i ω - λ} (1 - \e^{-(λ - \i ω)M}).
$$
Note that $\Re(λ) > 0$ and $$
|\e^{-(λ - \i ω)M}| = \exp(\Re(-(λ - \i ω)M)) = \exp(-M \Re(λ)),
$$
making $M \to +\infty$,$$
\int_{-\infty}^0 \e^{λt} \e^{-\i ωt} \,\d t = -\frac{1}{\i ω - λ}.
$$
Analogously, if $\Re(λ) < 0$, then$$
\int_0^{+\infty} \e^{λt} \e^{-\i ωt} \,\d t = \frac{1}{\i ω - λ}.
$$
Case 1: $\Re(λ) > 0$, then $\Re(\overline{λ}) = \Re(λ) > 0$. For any $ω \in \mathbb{R}$,\begin{align*}
\int_{-\infty}^{+\infty} h(t) \e^{-\i ωt} \,\d t &= -\int_{-\infty}^0 (R \e^{λt} + \overline{R} \e^{\overline{λ}t}) \e^{-\i ωt} \,\d t\\
&= -R \int_0^{+\infty} \e^{λt} \e^{-\i ωt} \,\d t - \overline{R} \int_0^{+\infty} \e^{\overline{λ}t} \e^{-\i ωt} \,\d t\\
&= \frac{R}{\i ω - λ} + \frac{\overline{R}}{\i ω - \overline{λ}} = α(ω).
\end{align*}
Case 2: $\Re(λ) < 0$. Analogous calculation shows that$$
\int_{-\infty}^{+\infty} h(t) \e^{-\i ωt} \,\d t = α(ω).
$$ | {"set_name": "stack_exchange", "score": 2, "question_id": 2280196} |
TITLE: Determination of rank in Wilcoxon Signed Test
QUESTION [0 upvotes]: When 2 values, say -5 and +5 appears and the current allocation of rank is to be 10, we split the rank to those two as 10.5 each. And the next rank will be 12. Now if there are 3 values with magnitude 5, then what will be the ranks?
REPLY [0 votes]: The allocations will be 10 and 11 for the two values of 5. The mean of 10 and 11 will be 10.5. So each of them will have 10.5 as rank. The rank of the next higher value will be 12 as usual. Now if there are three 5's, then the ranks allocated will be 10, 11 and 12. The mean of 10, 11 and 12 is 11. So each of them will have 11 as rank. The next higher value will have 13 as rank. | {"set_name": "stack_exchange", "score": 0, "question_id": 2482665} |
TITLE: How does $\operatorname{codim}(\mathcal{N}(f))=1$ holds if $\mathcal{N}(f)=\{0\}$
QUESTION [0 upvotes]: I'm studying Functional Analysis of Kreyszig and in problem 2.8.10 I must show that for a linear functional $f\neq 0$ we have $\operatorname{codim}\mathcal{N}(f)=1$.
I am not asking for a solution to how to show this. The only question that I have is shouldn't the author of the book state that $\mathcal{N}(f)\neq \{0\}$, or am I missing something? otherwise if we would have $\mathcal{N}(f)=\{0\}$ then $\operatorname{codim}\mathcal{N}(f)=\operatorname{dim}(X/\{0\})=\operatorname{dim}(X)$ , where $X$ is the normed space we are working on. But $\operatorname{dim}(X)$ is not $1$ in general....
REPLY [2 votes]: If $\mathcal N(f) = \{0\}$, then $f : X \to \mathbb{K}$ is injective, where $\mathbb K$ is your field. Hence, $f$ is a bijection and $X$ is one-dimensional. | {"set_name": "stack_exchange", "score": 0, "question_id": 2557171} |
TITLE: Can you use indefinite integration to prove equivalence of two functions?
QUESTION [1 upvotes]: Is it always the case that if:
$$
\int f(x) dx = F_1(x) + C
$$
and
$$
\int f(x) dx = F_2(x) + C
$$
then
$$
F_1(x) = F_2(x)
$$
and why?
Is it a legitimate way to prove the equivalence of two functions, namely $F_1$ and $F_2$ ?
REPLY [0 votes]: Basically yes. We need some continuity conditions on $f$ as well. And then,
your first two lines mean $F_1'=f=F_2'$, and hence $(F_1-F_2)'=0$. Now the main theorem is that $g'=0 \Rightarrow g=$ constant $c$, that is, $F_1-F_2=c$ can be concluded.
If you also know any point $x_0$ such that $F_1(x_0)=F_2(x_0)$, or one endpoint of the integral is determined (and the other varies), and the two $C$'s mean the same, then $F_1=F_2$ follows. | {"set_name": "stack_exchange", "score": 1, "question_id": 297113} |
\begin{document}
\bibliographystyle{alpha}
\sloppy \maketitle
\tableofcontents
\begin{abstract}
Let $\clX$ a projective stack over an algebraically closed field $k$ of characteristic 0. Let $\clE$ be a generating sheaf over $\clX$ and $\clO_X(1)$ a polarization of its coarse moduli space $X$. We define a notion of pair
which is the datum of a non vanishing morphism $\Gamma\otimes\clE\to \clF$ where $\Gamma$ is a finite dimensional $k$ vector space
and $\clF$ is a coherent sheaf over $\clX$. We construct the stack and the moduli space of semistable pairs.
The notion of semistability depends on a polynomial parameter and it is dictated by the GIT
construction of the moduli space.
\end{abstract}
\section{Introduction}
Recently a lot of attention has been drawn by sheaf theoretic curve counting theories of projective threefolds.
Among them Pandharipande-Thomas invariants \cite{PandThom1-09} are computed via integration over
the virtual fundamental class of the moduli space of the so called {\em stable pairs}.
The moduli spaces used to compute PT invariants are a special case of moduli spaces of coherent systems introduced by Le Potier in \cite{LePot93}. A coherent system is the datum of a pair $(F, \Gamma)$, where $F$ is a pure $d$-dimensional
coherent sheaf and $\Gamma\subset H^0(X,F)$ is a subspace of its global sections.
The moduli spaces are constructed as projective varieties via GIT techniques. The GIT stability
condition is equivalent to a modified Gieseker stability, where the Hilbert polynomial is corrected
by a contribution proportional to $\mbox{dim}\ \Gamma$ and to a polynomial stability parameter. A coherent system can be reconstructed from the associated evaluation morphism
$ev:\Gamma\otimes\clO_X\to F$.
In this note we study a similar moduli problem over projective stacks. If we work over an algebraically closed field $k$ of
characteristic zero, a projective stack is a stack with projective coarse moduli space that can be embedded into a smooth proper Deligne-Mumford stack. Any projective stack admits a {\em generating} sheaf $\clE$, namely
a locally free sheaf whose fibers carry every representation of the automorphism group of the underlying point.
We propose a notion of {\em pair} on projective stacks which is a natural generalization of the evaluation morphism
in the setting of \cite{Nir08-Mod}. The pair is defined as a non vanishing morphism
$$
\phi:\Gamma\otimes\clE \to \clF
$$
where $\Gamma$ is a finite dimensional $k$-vector space and $\clF$ is a coherent sheaf on $\clX$.
Note that $\phi$ determines a morphism
$$
ev(\phi):\Gamma\otimes\clO_\clX \to \clF\otimes\clE^\vee
$$
which we don't require to be injective on global sections.
Such a definition is reasonable if we think of projective stacks which are banded gerbes.
In that case, there are no morphisms $\Gamma\otimes\clO_\clX\to \clF$ if $\clF$ is not
a pullback from the coarse moduli space. Then in this case by twisting coherent sheaves by the generating sheaf
we get a richer theory than the theory of the coarse moduli space.
We give a notion of semistability which depends on a Hilbert polynomial of pairs, defined as the sum of
the usual Hilbert polynomial of $\clF\otimes\clE^\vee$ plus a term depending on a polynomial $\delta$.
We follow closely two papers dealing with very similar moduli problems:
\cite{HuyLe95} and \cite{Wand10}. In this note we make the exercise of checking
that the proofs extend to our setting, by using results on sheaves over projective stacks
proven in \cite{Nir08-Mod}.
We construct the stack of semistable pairs as a global quotient stack and we obtain
its coarse moduli space with GIT techniques.\\
\subsection*{Conventions}
In this papaer we work over an algebraically closed field $k$ of characteristic zero. By an {\em algebraic stack} we mean an algebraic stack over $k$ in the sense of \cite{Art74}. By a {\em Deligne-Mumford stack} we mean an algebraic stack over $k$ in the sense of \cite{DM69}. We assume moreover all stacks and schemes
unless otherwise stated are are noetherian of finite type over $k$.
For sheaves on stacks we refer to \cite{LMBca}.
\subsection*{Notations}
When dealing with sheaves we often adopt the notation in \cite{HuyLe2}.
We denote by $\clX\xrightarrow{\pi} X\to k$ a projective stack.
We choose a polarization $\clO_X(1)$ on the coarse moduli space. Given a
sheaf $\clF$ over $\clX$ we often denote by $\clF(m)$ the sheaf $\clF\otimes\pi^*\clO_\clX(m)$.
\section{Introductory material}
\subsection{Recall on projective stacks and on generating sheaves}
\begin{defn}
A {\em projective stack} is Deligne-Mumford stack with projective coarse moduli scheme and a locally free sheaf which is a {\em generating sheaf} in the sense of \cite{OlSt03}.
\end{defn}
For the reader's convenience we recall the notion of {\em generating sheaf} following \cite{Nir08-Mod}.
\begin{defn}[{\bf Generating sheaf}]\label{gen-sheaf-defn}
A locally free sheaf $\clE$ is said to be a {\em generator} for a quasi coherent sheaf $\clF$ is the adjunction morphism (left adjoint to the identity $\pi_*\clF\otimes\clE^\vee \stackrel{\id}{\to}\pi_*\clF\otimes\clE^\vee)$
\begin{eqnarray}
\theta_\clE(\clF):\pi^*\pi_*\mathcal{H}om_{\clO_\clX}(\clE,\clF)\otimes \clE \to \clF
\end{eqnarray}
is
surjective. It is a {\em generating} sheaf of $\clX$ if it is a generator for every quasi coherent sheaf on $\clX$.
\end{defn}
A characterization of generating sheaves can be given by making use of a relative (to the base) ampleness notion for locally free sheaves on stacks introduced in \cite{OlSt03}.
\begin{defn}
A locally free sheaf on $\clX$ is {\em $\pi$-ample} if and only if for every geometric point of $\clX$ the representation on the fiber of the stabilizer group at that point is faithful.
\end{defn}
\begin{defn}
A locally free sheaf $\clE$ on $\clX $ is {\em $\pi$-very ample} if for any geometric point of $\clX$ at that point
the representation of the stabilizer group on the fiber at that point contains every irreducible representation.
\end{defn}
\begin{prop}[{\cite{Kre06}}, 5.2] Let $\clE$ be a $\pi$-ample sheaf on $\clX$, then there is a positive integer
$r$ such that the locally free sheav $\bigoplus_{i=0}^r \clE^{\otimes i}$ is $\pi$-very ample.
\end{prop}
\begin{prop}[{\bf \cite{OlSt03}, 5.2}]
A locally free sheaf on a Deligne-Mumfors stack $\clX$ is a generating sheaf if and only if it is $\pi$-very ample.
\end{prop}
We now come to the definition of projective stack.
In \cite{Kre06} it shown that for a proper Deligne-Mumford
stacl over a field the following characterizations are equivalent.
\begin{thm}[\cite{Kre06}{\bf Corollary 5.4}]
Let $\clX\to k$ be a proper Deligne-Mumford stack. Then the following are equivalent:
\begin{enumerate}
\item[1)] the stack $\clX$ has projective coarse moduli space is a quotient stack;
\item[2)] the stack $\clX$ has a projective coarse moduli scheme and there exists a generating sheaf;
\item[3)] the stack $\clX$ has a closed embedding in a smooth proper Deligne-Mumford stack over $k$
and has a projective coarse moduli scheme.
\end{enumerate}
\end{thm}
The third statement is used in \cite{Kre06} as a definition.
\begin{defn}[ {\bf \cite{Kre06} Definition 5.5 }]\label{proj-stack-defn}
A stack $\clX\to k$ is projective if it admits a closed embedding into a smooth Deligne-Mumford stacks proper over $k$
and has projective coarse moduli space.
\end{defn}
\begin{defn}[{\bf Functors $F_\clE$ and $G_\clE$}]\label{functors-defn}
Let $\clE$ be a locally free sheaf on $\clX$. Let $F_\clE:\mathfrak{QCoh}_{\clX/S}\to \mathfrak{QCoh}_{X/S}$
be the functor mapping $\clF\mapsto \pi_*\mathcal{H}om_{\clO_\clX}(\clE,\clF)$ and let $G_\clE: \mathfrak{QCoh}_{X/S}\to \mathfrak{QCoh}_{\clX/S}$ be a second functor mapping $F\mapsto \pi^*F\otimes\clE$.
\end{defn}
\begin{rmk}
The functor $F_\clE$ is exact because both $\otimes \clE^\vee$ and $\pi_*$ are exact functors.
On the other hand $G_\clE$ is not exact unless $\pi^*$ is exact, i.e. $\pi$ is flat.
Examples of stacks with flat map to the coarse moduli scheme are flat gerbes
(e.g. \cite{LMBca} {\bf Definition 3.5}) over schemes or
stacks root of line bundles (see \cite{AGV06} or \cite{Cadm03}).
\end{rmk}
\begin{rmk}
The notation $F_\clE$ is the same as in \cite{OlSt03} but $\clG_\clE$ there corresponds to $\clG_E\circ \clF_\clE$ here.
\end{rmk}
\begin{notation}\label{iota-notation}
We denote by
\begin{eqnarray}
\iota_\clE(\clF): \clF\otimes\clO_X\rightarrow \clF\otimes\clE nd(\clE)
\end{eqnarray}
the injective morphism mapping a section to its tensor product with the identity endomorphism of $\clE$.
\end{notation}
In this paper we use the following notion of slope. Given a coherent sheaf $\clF$ of dimension $d$
\begin{eqnarray}
\hat{\mu}_\clE(\clF)=\frac{\alpha_{d-1}(\clF\otimes\clE)}{\alpha_d(\clF\otimes\clE^\vee)}.
\end{eqnarray}
We often denote the multiplicity $\alpha_d(\clF\otimes\clE^\vee)$ by $r_{\clE,\clF}$.
\section{Setting up the moduli problem}
In the following $\pi:\clX\to X \to k$ will be a smooth projective stack with coarse moduli scheme $X$ over an algebraically closed field $k$.
We will fix a polarization $(\clO_X(1),\clE)$ and a rational polynomial $\delta$ such that $\delta(m)\geq 0$ for $m>>0$.
\begin{defn}\label{pairs-defn}
A {\em pair} $(\clF,\phi)$ is a non-trivial morphism
\begin{eqnarray}
\phi:\Gamma\otimes \clE\to \clF,
\end{eqnarray}
where $\Gamma$ is a finite dimensional $k$-vector space, $\clF$ is a coherent sheaf of dimension $d$, $d\in\bbN$, $d\leq \mbox{dim}\ \clX$,
and $\clE$ is the fixed generating sheaf. A morphism between two pairs $(\clF,\phi)$, $(\clF',\phi')$ is
a commutative diagram
\begin{eqnarray}\label{morphs-diag-def}
\xymatrix{
\Gamma\otimes\clE\ar[r]^{\phi}\ar[d]_{\lambda} & \clF\ar[d]^{\alpha}\\
\Gamma\otimes \clE\ar[r]_{\phi'} & \clF'
}
\end{eqnarray}
where $\lambda\in \bbC^*$ and $\alpha$ is a morphism of coherent sheaves.
\end{defn}
Note that we can relate the notion of pairs to a {\em stacky version} of coherent systems of \cite{LePot93}.
For the reader's convenience we recall the definition of coherents systems on schemes given by LePotier.
\begin{defn}{\bf \cite{LePot93} Def. 4.1}
Let $X$ be a smooth projective variety of dimension $n$. A coherent system of dimension $d$ is a pair
$(\Gamma,F)$, where $F$ is a coherent sheaf of dimension $d$ over $X$ and $\Gamma\subseteq H^0(F)$ is a vector subspace.
\end{defn}
We extend this notion to projective Deligne-Mumford stacks.
\begin{defn}
Let $\clX$ be a projective Deligne-Mumford stack over $k$. A {\em twisted coherent system} on $\clX$ is a pair
$(\Gamma,\ \clF)$ where $\clF$ is a coherent sheaf over $\clX$ and $\Gamma\subseteq H^0(\clF\otimes\clE^{\vee})$
is a vector subspace.
\end{defn}
A pair $(\clF,\phi)$ determines a subspace of $H^0(\clX,\clF\otimes\clE^\vee)$ given
by the image of $\Gamma$ along $H^0(ev(\phi))$ where
\begin{eqnarray}
\xymatrix{
ev(\phi):\Gamma\otimes\clO_\clX\ar@{^{(}->}[r]^-{\iota_\clE(\Gamma)} & (\Gamma\otimes\clE)\otimes\clE^\vee\ar[r]^-{\phi\otimes\clE^\vee} & \clF\otimes\clE^\vee
}
\end{eqnarray}
is obtained from $\phi$ by applying the functor $-\otimes\clE^\vee$ and by composing by the inclusion morphism $\iota_\clE(\Gamma)$.
Note that $H^0(ev(\phi))$ is not necessarily injective on global sections. Hence
the twisted coherent system determined as above is $(W,\clF)$ with $\mbox{dim} W < \mbox{dim} \Gamma$.
Conversely, let us consider a twisted pair $(\Gamma,\clF)$. Let
\begin{eqnarray}\label{descr-1}
ev: \Gamma\otimes\clO_\clX\to \clF\otimes\clE^\vee.
\end{eqnarray}
be the corresponding evaluation morphism.
It is possible to associate to (\ref{descr-1})
the pair $(\clF,\phi)$
\begin{eqnarray}\label{induced-ev-morphi-eqn}
\xymatrix{
\phi(ev): \Gamma\otimes\clE\ar[r]^-{ev\otimes\clE} & (\clF\otimes \clE^\vee)\otimes \clE\ar[r]^-{Tr} & \clF
}
\end{eqnarray}
obtained by applying the functor $-\otimes\clE^\vee$ and by composing with $Tr:\clE nd(\clE)\to\clO_\clX$.
It is not hard to see that $ev(\phi(ev))=ev$ and that $\phi(ev(\phi))=\phi$.
We also give the definition of family of pairs.
Let $S$ be a scheme of finite type over $k$. Let $\pi_\clX:\clX\times S\to \clX$ and
$\pi_S:\clX\times S\to S$ be the natural projections.
\begin{defn}\label{family-of-pair-defn}
A {\em pair parametrized by $S$} is a $S$-flat coherent
sheaf $\clF$ over $\clX\times S$ and a homomorphism
$$
\phi_S:\pi_\clX^* \Gamma\otimes\clE\to \clF
$$
such that for any closed point $s$ of $S$
$$
\phi_S(s):\pi_\clX^* \Gamma\otimes\clE(s)\to \clF(s)
$$
is a pair.
\end{defn}
\subsection{(Semi)stability}
We define a parameter-dependent Hilbert polynomial for a stable pair in the following way.
\begin{defn}\label{hilb-polyn-defn}
The {\em Hilbert polynomial} of a pair $(\clF,\phi)$ is
\begin{eqnarray}
P_\clE{(\clF,\phi)}:=P(F_\clE(\clF)) +\epsilon(\phi)\delta
\end{eqnarray}
where $\epsilon(\phi)=1$ if $\phi\neq 0$ and $0$ otherwise. \\
The {\em reduced Hilbert polynomial} is
\begin{eqnarray}
p_\clE(\clF,\phi):= \frac{P_\clE{(\clF,\phi)}}{r_{F_\clE(\clF)}}.
\end{eqnarray}
\end{defn}
\begin{rmk}
Note that in the above definition following \cite{Nir08-Mod} we do not use the Hilbert polynomial
$P(\clF)$ of the sheaf on the stack, rather the Hilbert polynomial $P(F_\clE(\clF))=P(\clF\otimes \clE^\vee)$.
The reason is particularly evident if we consider sheaves on gerbes. In this case the non twisted Hilbert polynomial
of any sheaf which is not a pull-back from the coarse moduli space vanishes. For more details see \cite{Nir08-Mod}.
\end{rmk}
We will define (semi)stability by using the Hilbert polynomial introduced above. We need some more preliminary remarks and notations.
\begin{defn}\label{induced-hom-defn}
Let $(\clF,\phi)$ be a stacky pair. Any subsheaf $\clF'\subset\clF$ defines a {\em induced homomorphism}
$\phi':\Gamma\otimes\clE\to \clF'$ which is equal to $\phi$ if $\mbox{Im}\ \phi\subseteq \clF'$ and zero otherwise.
The corresponding quotient $\clF''=\clF/\clF'$ also inherits an {\em induced homomorphism}
$\phi'':\Gamma\otimes \clE\to\clF''$ which is defined as the composition of $\phi$ with the quotient map. Note that it is the zero morphism if and only if $\mbox{Im}\ \phi\subseteq \clF'$.
\end{defn}
\begin{rmk}
The Hilbert polynomial of stacky pairs is additive on short exact sequences.
\end{rmk}
\begin{defn}
A stacky pair is {\em $\delta$ (semi)stable} if for any saturated submodule $\clF'\subset \clF$
\begin{eqnarray}\label{delta-ss-defn-eqn}
P_\clE(\clF',\phi') (\leq) r_{\clE,\clF'}\ p(F_\clE(\clF),\phi)
\end{eqnarray}
\end{defn}
\begin{defn}\label{ss-pair-family-defn}
A {\em $\delta$ semi stable pair} parametrized by $S$ is a stacky pair over $S$ such that for every closed point of $S$ the pair
$(\clF(s), \phi|_{\pi_\clX^*\Gamma\otimes\clE_(s)})$ is a $\delta$ semistable pair.
\end{defn}
\subsection{Properties of $\delta$ semi stable pairs.}
We note that $\delta$ semi stablity implies purity of the underlying sheaf of the pair.
\begin{prop}
Let $(\clF,\phi)$ be a $\delta$ semi stable pair. Then $\clF$ is pure.
\end{prop}
\begin{pf}
Let us assume that $\clF$ is not pure. Let $\mathcal{T}=T_{d-1}(\clF)$ be the element of the torsion filtration of maximal dimension. Then
\begin{eqnarray}
P_\clE(\mathcal{T})+\epsilon(\mathcal{T})\leq \frac{r_{F_\clE(\mathcal{T})}}{r_{F_\clE(\clF)}}(p+\delta)=0
\end{eqnarray}
where the equality on the r.h.s. holds because $\mathcal{T}$ is a sheaf at most of dimension $d-1$.
It follows that $\clT=0$.
\end{pf}
For $\delta$ semistability we have characterizations and properties analogous to usual
Gieseker semistability for sheaves on schemes. We list some of them
\begin{prop}
Let $(\clF,\phi)$ be a stacky pair. Then the following conditions are equivalent:
\begin{enumerate}
\item[i)]for all proper subsheaves $\clF'\subseteq \clF$ $$P_\clE(\clF',\phi') (\leq) r_{F_\clE(\clF')}p_\clE(\clF,\phi);$$
\item[ii)] $(\clF,\phi)$ is (semi)stable;
\item[iii)] for all proper quotient sheaves $\clF\to \clF''$ with $\alpha_d(F_\clE(\clF))>0$ $$P_\clE(\clF'',\phi'')(\geq) r_{F_\clE(\clF'')} p_\clE(\clF,\phi);$$
\item[iv)] for all proper purely $d$-dimensional quotient sheaves $\clF\to \clF''$ with $\alpha_d(F_\clE(\clF))>0$ $$P_\clE(\clF'',\phi'') (\geq) F_\clE(\clF'') p_\clE(\clF,\phi);$$
\end{enumerate}
\end{prop}
\begin{pf}
The proof is very similar to \cite{HuyLe2} {\em Prop. 1.2.6}. We use additivity of the ranks and of the modified Hilbert polynomials on short exact sequences. Note that
if inequality
(\ref{delta-ss-defn-eqn}) holds for saturated subsheaves, it also holds for arbitrary subsheaves.
Indeed, let $\clF'\subseteq \clF$ be a not necessarily saturated subsheaf. Then if $\mbox{Im}\ phi$ is contained in
$\clF'$, then $\mbox{Im}\ \phi\subseteq \clF'^s$, where $\clF'^s$ is the saturation of $\clF'$ in $\clF$. Moreover
$P_\clE(\clF')\leq P_{\clE}(\clF'^s)$.
\end{pf}
\begin{lem}\label{ss-hom-lem}
Let $(\clF,\phi)$, $(\clG,\psi)$ be two $\delta$ semistable pairs such that
$p_\clE((\clF,\phi))> p_\clE((\clG,\psi))$. Then $\mbox{Hom}(\clF,\phi),(\clG,\psi))=0$.
\end{lem}
\begin{pf}
Let us assume there is a non zero morphism $(\alpha,\lambda)$. Let $\clH=\mbox{Im}\ \alpha$.
By semi stability we get
\begin{eqnarray}\label{chain-ineq-eqn}
p_\clE(\clF,\phi)\leq p_\clE(\clH,\phi_\clH)= p_\clE(\clH,\psi_\clH)\leq p_\clE(\clG,\psi),
\end{eqnarray}
where $\phi_\clH$ and $\psi_\clH$ are the induced homomorphisms. Inequality (\ref{chain-ineq-eqn})
contradicts the assumption.
\end{pf}
\begin{lem}
Let $\alpha: (\clF,\phi)\to (\clG,\psi)$ be a homomorphism between $\delta$ stable pairs of the same reduced Hilbert polynomial.
Then $\alpha$ is 0 or an isomorphism.
\end{lem}
\begin{pf}
Analogous to {\em Lemma 1.6} in \cite{Wand10}. Cfr. also \cite{HuyLe2} {\em Proposition 1.2.7}.
\end{pf}
\begin{cor}
Let $(\clF,\delta)$ be a semi stable pair. Then $\mathcal{E}nd((\clF,\delta))$ is a finite dimensional division algebra. Since we work over an algebraically closed field $k$ $\mathcal{E}nd((\clF,\delta))\simeq k$.
\end{cor}
\begin{pf}
Same as \cite{HuyLe2} {\em Cor. 1.2.8}.
\end{pf}
\subsection{Harder-Nahrasiman and Jordan-H\"older filtration}
\begin{prop}
Let $(\clF,\phi)$ be a pair such that $\clF$ is pure. Then it admits a unique Harder-Nahrasiman filtration
\begin{eqnarray}
0\subset HN_0(\clF,\phi)\subset....\subset HN_{l-1}(\clF,\phi)\subset HN_{l}(\clF,\phi)=(\clF,\phi)
\end{eqnarray}
such that each $gr_i^{HN}(\clF,\phi)= HN_i(\clF,\phi)/HN_{i-1}(\clF,\phi)$ is $\delta$ semistable
and if $p_i:=p_\clE(gr_i^{HN}(\clF,\phi))$ then
\begin{eqnarray}
p_{max}(\clF,\phi)=p_1 > p_2 > ...>p_l=p_{min}(\clF,\phi)
\end{eqnarray}
\end{prop}
\begin{pf}
The proof proceeds as in \cite{HuyLe2} Theorem 1.3.4.
Indeed, it is possible to find a subsheaf $\clF_0\subseteq \clF$ such that
it is not contained in any subsheaf $\clF'$ of $\clF$ with
$p_\clE(\clF_0,\phi_0) < p_\clE(\clF',\phi')$, where $\phi_0$ and $\phi'$ are the induced homomorphisms.
This implies the existence part. Uniqueness is proven by using Lemma \ref{ss-hom-lem}.
\end{pf}
\begin{prop}
Let $(\clF,\phi)$ be a $\delta$ semistable pair with reduced Hilbert polynomial $p$. Then there is a Jordan-Holder filtration
\begin{eqnarray}
0= JH_0(\clF,\phi)\subset JH_1(\clF,\phi)\subset ...\subset JH_l(\clF,\phi)=(\clF,\phi)
\end{eqnarray}
such that each $gr_i^{JH}(\clF,\phi)= JH_i(\clF,\phi)/JH_{i-1}(\clF,\phi)$ is $\delta$ stable with reduced Hilbert polynomial $p$.
The graded object $gr^{JH}(\clF,\phi)=\oplus_i\ gr_i^{JH}(\clF,\phi)$ is independent of the choice of the filtration.
Note that it inherits an induced homomorphism $gr^{JH}(\phi):\Gamma\otimes\clE\to gr^{JH}(\clF,\phi)$.
\end{prop}
\begin{pf}
The proof is the same as \cite{HuyLe2} {1.5}. The same arguments hols because of additivity the modified Hilbert polynomial on short exact sequences.
\end{pf}
\begin{rmk}
It is not hard to see that $gr^{JH}(\phi)$ is non trivial if $\phi$ is not, and its image is cointained in only one summand of $gr^{JH}(\clF,\phi)$.
\end{rmk}
\begin{defn}\label{S-equiv-defn}
Two $\delta$ semistable pairs are said to be {\em $S$-equivalent} if their Jordan-H\"older graded objects are isomorphic.
\end{defn}
\begin{rmk}
From now on we will always assume that $\delta$ is strictly positive and $\phi$ is non vanishing
otherwise $\delta$ (semi)stability reduces to usual Gieseker (semi)stability for sheaves on stacks.
\end{rmk}
We introduce a symbol which is convenient to restate the (semi)stability
condition when assuming that the homomorphism of a pair is non vanishing.
\begin{defn}\label{epsilon-defn}
Let $(\clF,\phi)$ be a stacky pair. For any exact sequence
\begin{eqnarray}
0\to \clF'\to \clF\to \clF''\to 0
\end{eqnarray}
let
\begin{eqnarray}
\epsilon(\clF') := \left\{ \begin{array}{rl} 1 & \quad \mbox{if Im}\ \phi\subseteq \clF' \\
0 & \quad \mbox{otherwise}
\end{array}\right.\nonumber
\end{eqnarray}
and
\begin{eqnarray}
\epsilon(\clF''):= 1 -\epsilon(\clF')
\end{eqnarray}
\end{defn}
With the above definition we can restate the $\delta$ (semi)stability condition for stacky pairs $(\clF,\phi)$. Indeed $(\clF,\phi)$
is semi stable if and only if for every saturated subsheaf $\clF'$
\begin{eqnarray}\label{nice-notation-sst-restatement}
P_\clE(\clF') + \epsilon(\clF')\delta (\leq) \frac{r_{F_\clE(\clF')}}{r_{F_\clE(\clF)}} (P_\clE(\clF) + \delta)
\end{eqnarray}
\section{Boundedness}\label{bounded-prop}
In this section we prove boundedness of the family of $\delta$ semistable pairs.
Since we want to use a GIT construction similar to \cite{HuyLe95} and \cite{Wand10} we take $\mbox{deg}\ \delta < \mbox{dim} \clX$.
\begin{prop}
Let $P$ be a fixed polynomial of degree $d< \mbox{dim}\ \clX$. Then the family of $\delta$ semistable pairs with
Hilbert polynomial $P$ is bounded.
\end{prop}
\begin{pf}
Let $\mathfrak{F}$ be a family of coherent sheaves over $\clX$.
According to \cite{Nir08-Mod} Corollary 4.17 $\mathfrak{F}$ is bounded if and only if $F_\clE(\mathfrak{F})$ is bounded over $X$.
We use \cite{Simp-rep-I} Theorem 1.1, according to which a family $\mathfrak{F}$ of sheaves over a projective scheme $X$ with fixed Hilbert polynomial is bounded if and only there exists a constant $C$ such that for any $F\in \mathfrak{F}$ $\mu_{max}(F)\leq C$.
Let $\phi: \clE\otimes \Gamma\to \clF$ be a $\delta$ semistable pair. Let $\mbox{Supp}(\clF)=\clY$.
Let us assume first that $\mbox{Im}\phi\nsubseteq HN_{l-1}(\clF)$, where $HN_{l-1}(\clF)$ is the
maximal proper subsheaf in the Harder-Nahrasiman filtration.
Then the composition
\begin{eqnarray}
\Gamma\otimes \clE \otimes\clO_\clY\to \clF\to gr^{HN}_l(\clF)
\end{eqnarray}
is a non zero morphism between sheaves of pure dimension $d$. This implies that
\begin{eqnarray}\label{mu-min-eqn}
\hat{\mu}_{\clE,min}(\clE \otimes\clO_\clY)\leq \hat{\mu}_{\clE, min}(\clF).
\end{eqnarray}
We note first that $\hat{\mu}_{\clE,min}(\clE \otimes\clO_\clY)\geq \hat{\mu}_{min}(\pi_*\clE nd(\clE)\otimes\clO_Y)$
where $Y=\pi(\clY)$. The reason is that not all the quotient sheaves of $F_\clE(\clF)$ are obtained as images by the functor $F_\clE$
of quotient sheaves of $\clF$ (cfr.\cite{Nir08-Mod} {\bf Remark 3.15})
We want to find a lower bound for $\mu_{min}(\clO_Y)$. This is provided by a result proven in
\cite{LePot93}.
\begin{cor}[{\bf\cite{LePot93} Corollary 2.13} ]
Let $X$ be a projective scheme over $k$. Let $S$ be a subscheme of pure dimension $d$ and of degree $k$. Then $\mu_{min}(\clO_S)$ is bounded from below by a constant which only depends on
$d$, $k$ and $X$.
\end{cor}
We observe that $Y=\pi\ \mbox{Supp}\ \clF=\mbox{Supp}\ F_\clE(\clF)$ is a purely $d$-dimensional subscheme
of degree $\leq r_{\clE,\clF}^2$.
Then
\begin{eqnarray}\label{mu-min-bound-eq }
\hat{\mu}_{\clE,min}(\clF)\geq \hat{\mu}_{min}(\clO_Y)+ \hat{\mu}_{min}(\pi_*\clE nd(\clE))\geq A + \hat{\mu}_{min}(\pi_*\clE nd(\clE)):=B
\end{eqnarray}
for some constant $A$ which only depends on $X$ and on the fixed polinomial $P$.
By the barycenter formula for the slope this implies that
\begin{eqnarray}
\hat{\mu}_{\clE,max}(\clF)\leq \mbox{max}\{r_{\clE,\clF}\ \hat{\mu}_\clE(\clF) - (r_{\clE,\clF}-1)\ B, \hat{\mu}_\clE(\clF) \}.
\end{eqnarray}
Boundedness of $\mu_{max}(F_\clE(\clF))$ follows from Lemma \ref{bound-on-stack-implies-cms}.
Let us consider now the case where $\mbox{Im}\phi\subseteq HN_{l-1}(\clF)$.
Then by $\delta$ semistability
$$
p_\clE(HN_{l-1}(\clF))\leq p_\clE(\clF),
$$
which in turn implies that
$$
p_\clE(\clF)\leq p_\clE(gr_l(\clF))
$$
and by the barycenter formula that
$$
\hat{\mu}_{\clE,max}(\clF)\leq \hat{\mu}_\clE(\clF).
$$
Summing up we get
\begin{eqnarray}\label{bound-on-mu-max-final}
\hat{\mu}_{max}(F_\clE(\clF))\leq \mbox{max}\{\hat{\mu}_\clE(\clF) ,r_{\clE,\clF}\mu_\clE(\clF) - (r_{\clE,\clF}-1) B \} + \tilde{m}\ \mbox{deg} \clO_X(1),
\end{eqnarray}
where the inequality is a consequence of the above estiamtes and of {\em Lemma \ref{bound-on-stack-implies-cms}}.
\end{pf}
\begin{lem}\label{bound-on-stack-implies-cms}
Let $\clF$ be a coherent sheaf on $\clX$ of pure dimension $d$. Then
if $\mu_{max,\clE}(\clF)$ ($\mu_{min,\clE}(\clF)$) is bounded from above (below), then also $\mu_{max}(F_\clE(\clF))$ ($\mu_{max}(F_\clE(\clF))$) is bounded from above (below).
\end{lem}
\begin{pf}
The proof is similar to \cite{Nir08-Mod} {\bf Proposition 4.24}.
Let $\overline{F}\subset F_\clE(\clF)$ be the maximal destabilizing subsheaf.
Let us consider the morphism
\begin{eqnarray}
\pi^*\overline{F}\otimes \clE \longrightarrow \pi^*(\pi_*\clF\otimes\clE^\vee)\otimes \clE \xrightarrow{\theta_\clE(\clF)} \clF.
\end{eqnarray}
The right arrow is surjective by definition of generating sheaf. Let $\overline{\clF}$ be the subsheaf corresponding to the image
of the composition. By applying the functor $F_\clE$ we get the surjective morphism
\begin{eqnarray}
\overline{F}\otimes \pi_*\mathcal{E}nd(\clE)\to F_\clE(\overline{\clF})
\end{eqnarray}
Let $\tilde{m}>>1$ be an integer number such that $\pi_*\mathcal{E}nd(\clE)$ is generated by global sections.
Let $N=h^0(\pi_*\mathcal{E}nd(\clE)(\tilde{m}))$. Then there is a surjective morphism
$\overline{F}\otimes\clO_X^{\otimes N}(-\tilde{m})\to F_\clE(\overline{\clF})$. Note that since $\overline{F}$ is semistable, so is
$\overline{F}(-\tilde{m})$. Moreover, for any $k\in\bbN$, $\overline{F}(-\tilde{m})^{\oplus k}$ is also semistable.
By composition we get the surjective morphism
\begin{eqnarray}
\overline{F}(-\tilde{m})^{\oplus N} \to \overline{F}\otimes \pi_*\mathcal{E}nd(\clE)\to F_\clE(\overline{\clF}).
\end{eqnarray}
Then by semistability of $\overline{F}(-\tilde{m})^{\oplus N}$
\begin{eqnarray}
\hat{\mu}_{max}(F_\clE(\clF))\leq \hat{\mu}_\clE(\overline{\clF}) + \tilde{m}\ \mbox{deg}\ \clO_X(1)
\leq \hat{\mu}_{\clE,max}(\clF) + \tilde{m}\ \mbox{deg}\ \clO_X(1).
\end{eqnarray}
Hence if $\hat{\mu}_{\clE,max}(\clF)$ is bounded from above, $\hat{\mu}_{max}(F_\clE(\clF))$ is also bounded from above.
Boundedness from below of $\hat{\mu}_{max}(F_\clE(\clF))$ also follows.
\end{pf}
\subsection{Rephrasing semistability in terms of number of global sections}
We apply here a result due Le Potier and Simpson (cfr. e.g. \cite{HuyLe2} Corollary 3.3.1 and 3.3.8) in order
to get a bound on the number of global sections of $F_\clE(\clF)$, where $\clF$ is a coherent
sheaf on $\clX$ of pure dimension $d$.
We state it for sheaves on $X$ obtained by applying the functor $F_\clE$ to some sheaf on $\clX$.
\begin{cor}\label{glob-sections-bound-char0-coro}
Let $\clF$ be a $d$-dimensional coherent sheaf over $\clX$. Let $r=r_{\clE,\clF}$ be the
multiplicity of $F_\clE(\clF)$. Let $C:=r(r+d)/2$. Then
\begin{displaymath}
h^0(\clF\otimes\clE^\vee)\leq \frac{r-1}{r}\cdot\frac{1}{d!}[\hat{\mu}_{max}(F_\clE(\clF))+C-1+m]^d_+ +\frac{1}{r}\cdot
\frac{1}{d!}[\hat{\mu}(F_\clE(\clF))+C-1+m]^d_+,
\end{displaymath}
whre $[x]_+=\mbox{max}\ \{0,x\}$.
\end{cor}
We will need the above estimate in order to
give the following characterization of semistability.
\begin{prop}\label{num-cond-d-ss-prop}
For $m>>0$ for any pure pair $(\clF,\phi)$ the following properties are equivalent
\begin{enumerate}
\item $(\clF,\phi)$ is $\delta$ (semi)-stable,
\item $P(m)\leq h^0(\clF\otimes\clE^\vee(m))$ and for any subsheaf $\clF'\subseteq \clF$ with $0<r_{F_\clE(\clF')}<r_{F_\clE(\clF)}$
$$
h^0(\clF'\otimes\clE^\vee(m))+\epsilon(\clF')\delta(m) (\leq) < \frac{r_{F_\clE(\clF')}}{r_{F_\clE(\clF)}}(P(m)+\delta(m))
$$
\item for any quotient $\clF\to \clF''$ with $0<r_{F_\clE(\clF'')}<r_{F_\clE(\clF)}$
$$
\frac{r_{F_\clE(\clF'')}}{r_{F_\clE(\clF)}}(P(m)+\delta(m))(\leq) < h^0(\clF''\otimes\clE^\vee(m))+\epsilon(\clF'')\delta(m)
$$
\end{enumerate}
\end{prop}
\begin{pf}
We prove $(1)\Rightarrow (2)$. The family of sheaves
underlying the family of $\delta$ semistable pairs on $\clX$ is bounded.
This is equivalent to the family of sheaves on $X$ obtained by applying the functor $F_\clE$ being bounded
(cfr. \cite{Nir08-Mod} {\bf Corollary 4.17}).
Hence there exists $m$ such that
for any $\clF$ in a $\delta$ semistable pair $P_\clE(m)=h^0(F_\clE(\clF)(m))$. Let $\clF'\subseteq\clF$
be an arbitrary subsheaf of $\clF$. Let us assume that inequality (\ref{bound-on-mu-max-final})
gives $\hat{\mu}_{max}(F_\clE(\clF))\leq \hat{\mu}_{max}(F_\clE(\clF)) + \tilde{m}\mbox{deg}\ \clO_X(1)$.
We distinguish two cases:\\
\begin{enumerate}\label{cases-distinction}
\item[A)] $\hat{\mu}(F_\clE(\clF'))\geq \hat{\mu}_\clE(\clF) - (r-1)\tilde{m}\mbox{deg}\ \clO_X(1) - C\cdot r - \delta_1 r$;
\item[B)] $\hat{\mu}(F_\clE(\clF'))\leq \hat{\mu}_\clE(\clF) -(r-1)\tilde{m}\mbox{deg}\ \clO_X(1) - C\cdot r - \delta_1 r$;
\end{enumerate}
where $C=r(r+d)/2$ and $r=r_{\clE,\clF}$. If $\clF'$ is of type $A$, then $\hat{\mu}(F_\clE(\clF'))$ is bounded from below. We observe that we can assume that
$\clF'$ is saturated (which implies that also $F_\clE(\clF')$ is saturated, because the functor
$F_\clE$ maps torsion filtrations to torsion filtrations, cfr. \cite{Nir08-Mod} {\bf Corollary 3.17}). Indeed for any sheaf $\clH$ on $\clX$
$P_\clE(\clH)\leq P(\clH^{s})$,
where $\clH^{s}$ is the saturation of $\clH$. Then the family of sheaves of type $A$ is bounded by Grothendieck Lemma for stacks
(see \cite{Nir08-Mod} {\bf Lemma 4.13}). As a consequence the number of Hilbert polynomials of the family is finite and
there exists an integer number $m_0$ such that for any $m\geq m_0$ and for any subsheaf $\clF'$ of type $A$
$P(\clF'\otimes\clE^\vee)=h^0(\clF'\otimes\clE^\vee)$ and
\begin{eqnarray}
P(\clF'(m)\otimes\clE(m)) + \epsilon(\clF')\delta(m) & (\leq) & \frac{r_{F_\clE(\clF')}}{r_{F_\clE(\clF)}}[p(m)+\delta(m)]\Leftrightarrow\nonumber\\ P(\clF'\otimes\clE^\vee) + \epsilon(\clF')\delta & (\leq) & \frac{r_{F_\clE(\clF')}}{r_{F_\clE(\clF)}}[p+\delta].
\end{eqnarray}
Let us consider now sheaves of type $B$. We get
\begin{displaymath}
h^0(\clF'\otimes\clE^\vee)\leq \frac{r'-1}{r'}\cdot\frac{1}{d!}[\hat{\mu}_{max}(F_\clE(\clF'))+C'-1+m]^d_+ +\frac{1}{r'}\cdot
\frac{1}{d!}[\hat{\mu}(F_\clE(\clF'))+C'-1+m]^d_+,
\end{displaymath}
where $C'=r'(r'+d)/2$ and $r'=r_{\clE,\clF'}$. This in turn implies
\begin{eqnarray}
\frac{h^0(\clF'\otimes\clE^\vee)}{r_{\clE,\clF'}} & \leq & \frac{r-1}{r}\cdot\frac{1}{d!}[\hat{\mu}_\clE(\clF) +\tilde{m}\mbox{deg}\ \clO_X(1) +C-1+m]^d_+ \nonumber \\
& & +\frac{1}{r}\cdot
\frac{1}{d!}[\hat{\mu}(F_\clE(\clF))+ (1-r)C -(r-1)\tilde{m}\mbox{deg}\ \clO_X(1) -1 -\delta_1r + m]^d_+\nonumber\\
& \leq & \frac{m^d}{d!} + \frac{m^{d-1}}{(d-1)!}(\hat{\mu}_\clE(\clF) -1 -\delta_1)+\ .....
\end{eqnarray}
where $\delta_1$ is the degree $d-1$ coefficient of $\delta$ and
$....$ stay for lower degree polynomials.
We can conclude that
\begin{eqnarray}
\frac{1}{r_{\clE,\clF'}}(h^0(\clF'\otimes\clE^\vee(m))+\epsilon(\clF')\delta(m))
< \frac{P(m)}{r} < \frac{P(m)+\delta(m)}{r}
\end{eqnarray}
If inequality (\ref{bound-on-mu-max-final}) gives a different upper bound
it is possible to apply the same arguments used above, except that suitably modified bounds have to be chosen
to define sheaves of type $A$ and $B$ as on page
\pageref{cases-distinction}.\\
$(2)\Rightarrow (3)$
Let $\clF''$ be any quotient of $\clF$ with $0<r_{F_\clE(\clF'')}<r_{F_\clE(\clF)}$.
Let $\clF'\subseteq\clF$ denote the corresponding kernel.
Then
\begin{eqnarray}
h^0(\clF''\otimes\clE^\vee(m))+\epsilon(\clF'')\delta(m) & (\geq) & h^0(\clF\otimes\clE^\vee(m))-h^0(\clF''\otimes\clE^\vee(m))+\delta(m)-\epsilon(\clF'')\delta(m)\nonumber\\
&(\geq)& \frac{1}{r}(rP_\clE(m)-r'P_\clE(m)+r\delta(m)-r'\delta(m))\nonumber\\
& = & \frac{r''}{r'}(P_\clE(m)+\delta(m))\nonumber
\end{eqnarray}
whhere $r=r_{F_\clE(\clF)}$, $r'=r_{F_\clE(\clF')}$, $r''=r_{F_\clE(\clF'')}$.\\
$(3)\Rightarrow (1)$.
Let us show first that the underlying sheaves of $\delta$ semistabe pairs satisfying $(3)$ form a bounded family.
Let $(\clF,\phi)$ be a such a pair.
Let ${\clF}_{min}$ the minimal destabilizing quotient of $\clF$. Then by hypothesis
\begin{eqnarray}
\frac{P_\clE(m)+\delta(m)}{r_{\clE,\clF}} - \frac{\epsilon(\clF_{min})\delta(m)}{r_{\clE,\clF_{min}}}\leq
\frac{h^0(\clF_{min}\otimes \clE^\vee)}{r_{\clE,\clF_{min}}} \nonumber\\
\leq \frac{1}{d!}[\hat{\mu}_\clE(\clF_{min})+\tilde{m}\mbox{deg}\ \clO_X(1) +C -1 +m]_+^d \nonumber
\end{eqnarray}
where the second inequality can be deduced from {\em Corollary} \ref{glob-sections-bound-char0-coro}.
It follows that $\hat{\mu}_{\clE,min}(\clF)$ is bounded from below. By Lemma \ref{bound-on-stack-implies-cms}, $\hat{\mu}_{min}(F_\clE(\clF))$ is also bounded from below or equivalently $\hat{\mu}_{max}(F_\clE(\clF))$ is bounded from above,
which implies boundedness for sheaves satisfying $(3)$.
Let us consider now an arbitrary quotient
$\clF''$ of $\clF$. Then either $\hat{\mu}_{\clE}(\clF'')>\hat{\mu}_{\clE}(\clF)+\delta_1/r$ or $\hat{\mu}_{\clE}(\clF'')\leq\hat{\mu}_{\clE}(\clF)+\delta_1/r$. In the first case a strict inequality for Hilbert polynomials is satified, implying stability.
In the second case $\hat{\mu}_{\clE}(\clF'')$ is bounded from above. By Grothendieck Lemma for stacks
we get that the family of pure dimensional quotients satisfying the second inequality is bounded. Hence there exists $m$ large enough
such that for any sheaf $\clF$ satisfying $(3)$
$h^0(\clF''\otimes\clE^\vee(m))=P_\clE(\clF''(m))$. Therefore
\begin{eqnarray}
P(\clF''\otimes\clE^\vee(m))+\epsilon(\clF'')\delta(m)(\geq)\frac{r''}{r}[P_\clE(m)+\delta(m)]\nonumber\\\Leftrightarrow P(\clF''\otimes\clE^\vee)+\epsilon(\clF'')\delta (\geq) \frac{r''}{r}[P_\clE+\delta],
\end{eqnarray}
where $r=r_{\clE,\clF}$ and $r''=r_{\clE,\clF''}$.
Eventually we remark that from the proofs of $(2)$ and $(3)$ equality holds if and only if the subsheaf or
the quotient sheaf is destabilizing.
\end{pf}
We recall a technical result originally due to Le Potier which will be used in Theorem \ref{GIT-implies-dss-thm}.
The proof can be founf in \cite{Nir08-Mod} Lemma 6.10 and generalizes \cite{HuyLe2} Proposition 4.4.2.
\begin{lem}\label{deform-sheaf-lem}
Let $\clF$ be a coherent sheaf over $\clX$ that can be deformed to a sheaf of the same dimension $d$. Then there is a
pure $d$-dimensional sheaf $\clG$ on $\clX$ with a map $\clF\to \clG$ such that the kernel is
$T_{d-1}(\clF)$ and $P_\clE(\clF)=P_\clE(\clG)$.
\end{lem}
\section{The parameter space}
By {\bf Proposition \ref{bounded-prop}} we know that the family of sheaves obtained by
applying the functor $F_\clE$ to the underlying family of sheaves of $\delta$ semistable pairs with
fixed Hilbert polynomial is bounded.
Therefore there exists an integer number $m_0$ such that for any $m\geq m_0$ and for any$\delta$ semistable pair $(\clF,\phi)$,
$F_\clE(\clF)(m)$ is generated by global sections. Let $V$ be a $k$ vector space of dimension equal to $P_\clE(\clF)(m)$.
There is a surjective morphism
\begin{eqnarray}\label{q-constr-eqn}
\xymatrix{
q: V\otimes\clE(-m)\ar[r] & \pi^*\pi_*(\clF\otimes\clE^\vee)\otimes\clE \ar@{->>}[r] & \clF
}
\end{eqnarray}
obtained by applying the functor $G_\clE$ to
$$
V\isomto H^0(F_\clE(\clF(m))\twoheadrightarrow F_\clE(\clF(m)
$$
and composing with $\theta_\clE(\clF): G_\clE(F_\clE(\clF))\to \clF$.
The morphism $q$ corresponds to a closed point of $\tilde{Q}:=Quot(V\otimes\clE(-m), P_\clE(\clF))$
(existence of Quot schemes on Deligne-Mumford stacks follows from \cite{OlSt03}).
Up to changing $m$ we can assume that also $F_\clE(\clE)(m)=\pi_*\mathcal{E}nd(\clE)(m)$ is generated by global sections.
Under this assumption we get a surjection
\begin{eqnarray}
\xymatrix{
H:= H^0(\pi_*\mathcal{E}nd(\clE)(m))\otimes\clE(-m) \ar@{->>}[r] & \clE.
}
\end{eqnarray}
To any pair $(\clF,\phi)$ we can associate a commutative diagram
\begin{eqnarray}
\xymatrix{
H\times \Gamma \otimes\clE(-m)\ar[r]^-{\tilde{a}}\ar[d]_{\tilde{ev}:=\pi^*ev\otimes\clE(-m)} & V\otimes\clE(-m)\ar[d]^{q}\\
\Gamma \otimes \clE \ar[r]_{\phi} & \clF
}
\end{eqnarray}
where $\tilde{a}:=a\otimes \clE(-m)$ and $a\in \mbox{Hom}(H\times\Gamma,V)$.
In this section we will show that we can use as parameter space a suitable locus of $N\times\tilde{Q}$,
where $N$ is the projectivization of the vector space $\mbox{Hom}(H\times\Gamma,V)$.
We start by recalling some useful facts.
\subsection{Quot schemes on projective stacks}
Let $\clX\stackrel{\pi}{\to} X \stackrel{f}{\to} S$ be a projective stack.
Let $\tilde{\clQ}$ denote $\mbox{Quot}_{\clX/S}(V\otimes\clE(-m),P)$ and let $Q$ denote
$\mbox{Quot}_{X/S}(F_\clE(V\otimes\clE(-m)),P)$.
By \cite{OlSt03} Proposition 6.2 and \cite{Nir08-Mod} 4.20 there is a closed embedding
$\iota:\tilde{\clQ}\to Q$. We know that for any $l\in\bbN$ big enough there is a closed embedding into the Grassmannian
\begin{eqnarray}
j_l: \mbox{Quot}_{X/S}(F_\clE(V\otimes\clE(-m)),P)\hookrightarrow Grass(f_*F_\clE(V\otimes\clE(l-m)),P(l))
\end{eqnarray}
given by the very ample line bundles $det f_{Q*}\clU(l)$, where $\clU$ is the universal quotient sheaf over $Q$
and $f_{Q}$ is defined as in the following cartesian diagram
\begin{eqnarray}\label{notation-diag-1}
\xymatrix{
\clX_{\tilde{\clQ}}\ar[d]_{\pi_{\tilde{\clQ}}}\ar[rrrr] & & & & \clX \ar[d]^{\pi}\\
X_{\tilde{\clQ}}\ar[d]_{f_{\tilde{\clQ}}}\ar[r]^{\tilde{\iota}}& X_\clQ\ar[d]_{f_Q}\ar[r] & X_{G}\ar[r]\ar[d] & X_{\bbP}\ar[d]\ar[r] & X\ar[d]^{f}\\
\tilde{\clQ}\ar[r]_{\iota} & \clQ\ar[r]_{j_l} & G \ar[r]_{k_l} & \bbP\ar[r] & S
}
\end{eqnarray}
where $G=Grass(f_*F_\clE(V\otimes\clE(l-m)),P(l))$, $\bbP=\bbP(\wedge^{P(l)}(f_*F_\clE(V\otimes\clE(l-m)))$,
$j_l$ is the Pl\"ucker embedding and $k_l$ is the Grothendieck embedding.
A result in \cite{Nir08-Mod} identifies a (relatively) very ample line bundle
giving an equivariant embedding into the projective space $\bbP(\wedge^{P(l)}(f_*F_\clE(V\otimes\clE(l-m)))$.
\begin{prop}[\cite{Nir08-Mod} Prop 6.2]\label{very-ample-l-b-lem}
The class of invertible sheaves
\begin{eqnarray}
L_l:=det(\ f_{\tilde{Q}_*}(F_\clE(\tilde{\clU})(l)))
\end{eqnarray}
is very ample for $l$ big enough, where we refer for notation to diagram (\ref{notation-diag-1})
and $\tilde{\clU}$ is the universal quotient bundle over $\tilde{\clQ}$.
\end{prop}
As in \cite{HuyLe2} pag. 101, $\tilde{\clU}$ has a natural $GL(V)$-linearization induced by the universal
automorphism of $GL(V)$. As observed in \cite{Nir08-Mod} Lemma 6.3 there is an induced linearization of $L_l$
as its formation commutes with arbitrary base change.
\subsection{Identification of the parameter space}
Let $N:=\bbP(Hom(H\times \Gamma, V)^\vee) $ be the projective space of morphisms $H\times\Gamma\to V$,
which is polarized by $\clO_N(1)$.
\begin{lem}
Let $(\clF,\phi)$ be a $\delta$ semistable pair over $\clX$ with $P_\clE(\clF)=P$.
Then it determines a pair $(a,q)$ in $N\times \tilde{Q}$ such that $$q\circ a=\phi\circ \tilde{ev}$$
and such that $H^0(F_\clE(q(m))\circ \varphi_\clE(V\otimes\clO_X))$ is an isomorphism, where
$\varphi_\clE(V\otimes\clO_X):V\otimes\clO_\clX\hookrightarrow V\otimes\clE nd(\clE)$ is the
multiplication by the identity endomorphism.
\end{lem}
\begin{pf}
Since $\clF$ is bounded, then for $m$ large enough there is a surjection
\begin{eqnarray}
\pi^* H^0(F_\clE(\clF)(m))\otimes\clE\xrightarrow{G_\clE(ev)} \pi^* F_\clE(\clF)(m)\otimes\clE \xrightarrow{\theta_\clE(\clF)} \clF(m)
\end{eqnarray}
Moreover, $P=h^0(F_\clE(\clF)(m))$. Hence, by choosing an isomorphism $V\isomto H^0(F_\clE(\clF)(m))$
and by tensoring by $\clO_\clX(-m)$
we get a quotient
\begin{eqnarray}
V\otimes\clE(-m) \to \clF
\end{eqnarray}
namely an element of $\tilde{\clQ}$. Let us consider $\phi:\clE\otimes\Gamma\to \clF$.
By twisting by $\pi^*\clO_X(-m)$, by applying the functor $F_\clE$ and by taking global sections we get
\begin{eqnarray}
H^0( F_\clE(\clE)(m))\times \Gamma \to H^0(F_\clE(\clF)(m))
\end{eqnarray}
By choosing again an isomorphism $V\isomto H^0(F_\clE(\clF)(m))$ we get a morphism
$a:H\times\Gamma\to V$. Note that $a$ and $\lambda a$ , $\lambda\in\bbC^*$, come from isomorphic
pairs. Indeed for any $\lambda\in\bbC^*$
$(\clF,\phi)\simeq (\clF,\lambda\clF)$ by definition.
We observe moreover
$H^0(F_\clE(q(m))\circ\varphi_\clE(V\otimes\clO_X))$ is an isomorphism by construction. The same is true for the relation $q\circ a=\phi\circ \tilde{ev}$.
\end{pf}
We characterize now the locus in $N\times\tilde{Q}$ containing pairs $(a,q)$ yielding
a pair $(\clF,\phi)$.
\begin{prop}
There is a closed subscheme $\clW\subseteq N\times\tilde{Q}$ with the following property:
given a pair $(a,q)\in N\times\tilde{Q}$
the map $q\circ \tilde{a}$ factors through $\tilde{ev}$ iff $(a,q)\in\clW$.
\end{prop}
\begin{pf}
Same as \cite{Wand10} Prop. 3.4.
\end{pf}
\begin{defn}
We define $\clZ$ to be the closure in $\clW$ of the open locus of points $(a,q)$
such that $q(V\otimes\clE(-m))$ is pure.
\end{defn}
\section{GIT Construction}
We come now to the GIT construction of the moduli space of $\delta$ semistable pairs.
We observe that $\bbC^*\subset GL(V,k)$ acts trivially on both $N$ and $\tilde{Q}$.
As far as the GIT problem is concerned we can consider the action of the group $PGL(V,k)$ or
$SL(V,k)$ ($SL(V)$ from now on). Indeed $PGL(V)$ is a quotient of $SL(V)$ by a finite subgroup.
As a consequence, up to taking finite tensor powers, the line bundles linearized for the
actions of the two groups are the same.
There is a natural action of the group $SL(V)$ on $\clZ$ defined as follows
\begin{eqnarray}
g\cdot (a,q)\mapsto ( g^{-1}\cdot a, q\cdot g).
\end{eqnarray}
The line bundles $L_l$ of Lemma \ref{very-ample-l-b-lem} and $\clO_N(1)$ have a natural $SL(V)$ linearization.
We choose as linearized line bundle for the GIT construction
\begin{eqnarray}\label{very-ample-linearized-l.b.}
\clO_\clZ(n_1,n_2):=\pi_{\tilde{Q}}^*L_l^{n_1}\otimes\pi_N^*\clO_N(1)^{n_2}|_{\clZ}.
\end{eqnarray}
We choose $n_1$ and $n_2$ as
\begin{eqnarray}\label{n_1-n_2-choice-eqn}
\frac{n_1}{n_2}=\frac{P_\clE(l)\delta(m)-P_\clE(m)\delta(l)}{P_\clE(m)+\delta(m)}.
\end{eqnarray}
\begin{defn}\label{clR-defn}
Let $\clR\subseteq \clZ$ be the subset of points corresponding to $\delta$ semistable pairs and such that
$H^0(F_\clE(q(m))\circ \iota_\clE(V\otimes\clO_\clX))$ is an isomorphism.
\end{defn}
\begin{lem}
The subset $\clR$ of Definiton \ref{clR-defn} is open and $SL(V)$ invariant. Moreover there is an open subset
$\clR^s\subseteq \clR$ corresponding to $\delta$ stable pairs.
\end{lem}
\begin{pf}
The proof is almost the same as \cite{Wand10} {\bf Definition/Lemma 3.5}.
The proof relies on the fact that the set of Hilbert polynomials
of purely dimensional quotients destabilizing the pair corresponding to some $(a,q)\in\clZ$ is finite.
To prove this for projective stacks one need two ingredients. The first is
the Grothendieck Lemma for stacks. The second
is the fact that given a family of projective stacks and a coherent sheaf over it there exists
a finite stratification
such that the restriction of the given sheaf to each stratum is flat (cfr. \cite{Nir08-Mod} {\bf Proposition 1.13}).
Recall that the Hilbert polynomial over a projective stack is constant in families.
\end{pf}
\begin{defn}\label{R-bar-defn}
We define $\overline{\clR}$ as the closure of $\clR$.
\end{defn}
We define the functor of semistable pairs.
\begin{defn}\label{functor-defn}
Let the functor
\begin{eqnarray}
\underline{\clM}_{\clX,\delta}(\clE,P): (Sch/k)^o\to (Sets)
\end{eqnarray}
be defined as follows. For any $S$ in $(Sch/k)$ $\underline{\clM}_{\clX,\delta}(\clE,P)(S)$ is the set of isomorphism classes of families $(\clF,\phi)$ of $\delta$ semistable pairs parametrized by $S$ as defined in {\em Definition \ref{ss-pair-family-defn}},
such that for any closed point $s$ of $S$ $(\clF(s), \phi|_{\pi_\clX^*\clE(s)})$ has Hilbert polynomial $P$.
For any $f:S'\to S$ $$\underline{\clM}_{\clX,\delta}(\clE,P)(f):\underline{\clM}_{\clX,\delta}(\clE,P)(S)\to \underline{\clM}_{\clX,\delta}(\clE,P)(S')$$
takes a family $(\clF,\phi)$ over $S$ to $((f\times{\bf 1}_\clX)^*\clF, (f\times{\bf 1}_\clX)^*\phi)$ over $S'$.
We define
$\underline{\clM}^s_{\clX,\delta}(\clE,P)$ as the subfunctor parametrizing $\delta$ stable pairs.
\end{defn}
\begin{thm}
Let $(\clX,\clO_\clX(1),\clE)$ be a polarized smooth projective stack. Then the functor $\underline{\clM}_{\clX,\delta}(\clE,P)$ is isomorphic to $[\clR^{ss}/GL(V)]$, where $\clR^{ss}\subseteq \overline{\clR}$ is the subset of GIT semistable points.
\end{thm}
\begin{pf}
The proof uses a quite standard machinery (cfr. \cite{HuyLe2} {\bf Lemma 4.3.1}, \cite{Diac08} {\bf Proposition 3.9}, \cite{Shesh10} {\bf Theorem 4.0.7}).
We sketch it.
We will show that there is an invertible functor of categories fibered in groupoids
between $[\clR^{ss}/GL(V)]$ and $\underline{\clM}_{\clX,\delta}(\clE,P)$. Let us show that $\xi$ exists.
An object of $[\clR^{ss}/GL(V)]$ over $S$ is a diagram
\begin{eqnarray}
\xymatrix{
P\ar[r]\ar[d] & \clR^{ss}\\
S &
}
\end{eqnarray}
where the horizontal arrow is $GL(V)$ equivariant.
By pulling back the universal family over $\overline{R}\times\clX$ we get a $\delta$ semistable pair
$\phi:\clE\otimes\Gamma\to \clF$ over $P\times\clX$ with an isomorphism
$V\isomto H^0(F_\clE(\clF(m))(s))$ for any closed point $s$ of $S$. The pair comes
with a $GL(V)$ linearization.
Therefore both sheaves and the morphism between them descend to $S$.
The same is true for a morphism between $\delta$ semistable pairs over $P\times\clX$.
We show that there exists a functor $\eta$ in the opposite direction.
Let us draw a diagram to fix the notations:
\begin{eqnarray}\label{family-over-P-diag}
\xymatrix{
P\times \clX \ar[r]^{p\times{\bf 1}_\clX}\ar[d]_{\pi_P} & S\times \clX\ar[d]^{\pi_S}\\
P\ar[r]_{p} & S
}.
\end{eqnarray}
Let us consider a $\delta$ semistable pair over $\clX\times S$
$\phi:\clE\otimes\Gamma\to \clF$. Since $\clE$ and $\clF$ are flat over $S$ by definition of family,
$\pi_{S*}\clE nd(m)$ and $\pi_{S*}\clF\otimes\clE^\vee(m)$ are locally free $\clO_S$-modules and the morphisms
\begin{eqnarray}
\pi_{S}^*\mathcal{A}\otimes\clE:=\pi_{S}^*\pi_{S*}(\Gamma\otimes\clE nd(\clE)(m))\otimes\clE\to \clE(m)
\end{eqnarray}
and
\begin{eqnarray}
\pi_{S}^* V\mathcal{B}\otimes\clE:=\pi_{S}^*\pi_{S*}(\clF\otimes\clE^\vee(m))\otimes\clE\to \clF(m)
\end{eqnarray}
are surjective.
Note that $\clA\simeq H\times\Gamma\otimes \clO_S$, $H:=H^0(\clX,\clE nd(\clE)(m))$
Let $P:=\mathbb{I}som(V,\clB)$ be the frame bundle of $\clB$ (see \cite{HuyLe2} {\bf Example 4.2.3} for the definition). Then $p:P\to S$ is a $GL(V)$ principal bundle.
There is a commutative diagram over $P\times\clX$
\begin{eqnarray}
\xymatrix{
H\times\Gamma\otimes\clE\otimes \clO_{P\times \clX}\ar@{->>}[d] \ar[r] & V\otimes \clE\otimes\clO_{P\times \clX}\ar@{->>}[d]\\
(p\times{\bf 1}_\clX)^* \clE(m)\ar[r] & (p\times{\bf 1}_\clX)^*\clF(m)
}
\end{eqnarray}
where
\begin{enumerate}
\item[i)] the right vertical arrow corresponds to a morphism $P\to \tilde{Q}$ and it is obtained by composing with $(p\times{\bf 1}_\clX)^*\clF(m)$
with $\pi_P^*\mathfrak{h}^{-1}$, where $\mathfrak{h}$ is the universal isomorphism
over $P$;\\
\item[ii)] the upper horizontal arrow corresponds to a morphism $P\to N$ and it is obtained
by post-composing the natural morphism $H\times\Gamma\otimes \clO_{P\times\clX}\to (p\times{\bf 1}_\clX)^*\pi_S^*\clB$
with $\pi_P^*\mathfrak{h}^{-1}$.
\end{enumerate}
By Cohomology and base change for Deligne-Mumford stacks (see \cite{Nir08-Mod} {\bf Theorem 1.7}) there are isomorphisms
$H\times\Gamma\otimes\clO_{P} \isomto p^*\clA$
and $V\otimes \clO_{P} \isomto p^*\clB$.
Let $p$ be a closed point of $P$.
Then
\begin{eqnarray}\label{isom-cohom-prop}
H\times\Gamma\simeq p^*\clA|_p\simeq H^0(F_\clE(\clE\otimes\Gamma(m))(p))\nonumber\\
V\simeq p^*\clB|_p\simeq H^0(F_\clE(\clF(m))(p))
\end{eqnarray}
Then the family of diagram (\ref{family-over-P-diag}) with the property (\ref{isom-cohom-prop})
and its natural $GL(V)$ linearization
provides
a $GL(V)$-equivariant morphism to $\clR^{ss}$.
Such a morphism descends to a morphism $S\to [\clR^{ss}/GL(V)]$.
In a similar way we can reconstruct morphisms.
Let $f:S\to T$ be a morphism,
let $(\clF,\phi)$ and $(\clF',\phi')$ be families of pairs over $S\times\clX$ and $T\times \clX$,
$\tau:(\clF,\phi)\to (p\times{\bf 1}_\clX)^*(\clF',\phi')$ be a morphism over $S\times\clX$.
By cohomology and base change along the diagram
\begin{eqnarray}
\xymatrix{
S\times \clX \ar[r]\ar[d] & T\times\clX\ar[d]\\
S\ar[r] & T
}
\end{eqnarray}
we see that the families produce $GL(V)$ principal bundles $P$ and $P'$ such that
$P\isomto f^*P'$.
By cohomology and base change along the diagram
\begin{eqnarray}
\xymatrix{
f^*P'\times\clX\ar[r]^u\ar[d] & P'\times\clX\ar[d]\\
f^*P'\ar[r] & P'
}
\end{eqnarray}
we see that there is a canonical isomorphism between the pair over $P$ and the pullback
of the pair over $P'$. Then we get two isomorphic $GL(V)$ equivariant maps
$g:P\to \clR^{ss}$ and $g':f^*P'\to \clR^{ss}$ such that
$g'\circ u=g$.
We conclude that the functor $\eta$ is defined.
It is straightforward to check that $\xi$ and $\eta$ are the inverse of each other.
\end{pf}
\begin{lem}\label{categorical-quotient-corepr-lem}
Let $M$ be a scheme which is a categorical quotient for the $SL(V)$ action on $\clR$. Then it corepresent
the functor $\underline{\clM}_{\clX,\delta}(\clE,P)$.
\end{lem}
\begin{pf}
The functors $\underline{\clM}_{\clX,\delta}(\clE,P)$ and $[\clR/GL(V)]$
are isomorphic. Hence to corepresent either functor is the same.
\end{pf}
We state the theorem relating GIT semistability of points of the parameter space to
$\delta$ semistability of the corresponding pairs. The proof will be given by {\em Theorems \ref{GIT-implies-dss-thm} and
\ref{delta-ss-implies-GIT-ss-thm}}.
\begin{thm}\label{main-thm}
For $l$ large enough
the subset of points in the closure $\overline{\clR}$ of $\clR$ which are semistable with respect
to $\clO_{\clZ}(n_1,n_2)$ and its $SL(V)$ linearization coincides with the
subset of points corresponding to $\delta$ semistable pairs.
\end{thm}
\begin{thm}
Let $(\clX,\clO_\clX(1),\clE)$ be a polarized smooth Deligne-Mumford stack.
Then there exists a moduli space $M_{\clX,\delta}(\clE,P)$
of $\delta$ semistable pairs. Two pairs correspond to the same point in $M_{\clX,\delta}(\clE,P)$ if and only if they are S-equivalent. Moreover there is an open
subset $M_{\clX,\delta}^s(\clE,P)$ corresponding to stable pairs. It is a fine moduli space for $\delta$ semistable pairs.
\end{thm}
\begin{pf}
The prrof is an in \cite{Wand10} {\bf Theorem 3.8}.
We proved in {\em Lemma \ref{categorical-quotient-corepr-lem}}
that a categorical quotient of $\cl$ for the $SL(V)$ action
corepresent the functor $\underline{\clM}_{\clX,\delta}(\clE,P)$.
General GIT theory and {\em Theorem \ref{main-thm}}
provide a categorical quotient of $\clR$ which in particular is a good quotient.
By using arguments analogous to \cite{HuyLe95} {\bf Proposition 3.3} or \cite{HuyLe2} {\bf Theorem 4.3.3}
one proves that closure of the orbit of a $\delta$ semistable pair $(\clF,\phi)$ contains
the pair $(gr^{JH}(\clF), gr^{JH}(\phi))$. Moreover the orbit of $(\clF,\phi)$ is closed if and only if
$(\clF,\phi)$ is polystable. In proving the last fact a semicontinuity result in used, which also holds
for Deligne-Mumford stacks (cfr. \cite{Nir08-Mod}) {\bf Theorem 1.8}. The second statement follows then
from properties of good quotients.
Moreover there is a universal family over $\clX\times M^s_{\clX,\delta}(\clE,P)$, which is implied
by the existence of a line bundle over $\clR^s$ of weight $1$ for the action of $\bbG_{m}\subseteq GL(V)$
(cfr. \cite{HuyLe2} {\bf Section 4.6}). Such a line bundle is provided by $\clO_N(1)$.
\end{pf}
\subsection{GIT computations}
We give in the following proposition numerical conditions implied by GIT semistability.
We recall some useful standard results.
The action of a 1-parameter subgroup of $SL(V)$ $\lambda$ can be diagonalized. Hence $V$ splits in eigenspaces
for the eigenvectors of $\lambda$. The subspaces $V_{\leq n} =\oplus_{i=1}^n V_i\subseteq V$ give an
ascending filtration.
Let $q:V\otimes \clE(-m)\to \clF$ a closed point of $\tilde{Q}$.
We get a corresponding filtration $\clF_{\leq n}$ of $\clF$, where $\clF_{\leq n}=q(V_{\leq n}\otimes\clE(-m))$.
Let $\clF_n=\clF_{\leq n}/\clF_{\leq n-1}$ the graduate pieces.
Let
\begin{eqnarray}
\overline{q}: V\otimes\clE(-m) \to \oplus_{i=1}^n \clF_{\leq n}.
\end{eqnarray}
be a closed point of $\tilde{Q}$.
Then we have the following result generalizing \cite{HuyLe2} Lemma 4.4.3 and proven in \cite{Nir08-Mod} Lemma 6.11.
\begin{lem}
The quotient $[\overline{q}]$ is $\lim_{t\to 0}\lambda(t)q$ in the sense of the Hilbert-Mumford criterion.
\end{lem}
As in the case of scheme the action of $\lambda$ on the fiber of $L_l$ is characterized as follows.
\begin{lem}[{\bf \cite{Nir08-Mod} Lem. 6.12}]\label{quot-weight}
The action of $\bbG_{m,k}$ via the representation $\lambda$ on the fiber of $L_l$ at the point $[\overline{q}]$
is given by the weight
\begin{eqnarray}
\sum_n P_\clE(\clF_n(l)).
\end{eqnarray}
\end{lem}
We fix some notation. Let $W:=H^0(\clO_X(l-m))$. We denote by $q'$ the following morphism induced by $q$:
\begin{eqnarray}
q':V\otimes W\xrightarrow{H^0(\varphi_\clE(l))\circ H^0(\phi_\clE(l))} H^0(\clF\otimes\clE^\vee(l))
\end{eqnarray}
and by $q''$ its $r$-th antisymmetric product
\begin{eqnarray}\label{q''-eqn}
q'':\wedge^r(V\otimes W) \to det\ H^0(\clF\otimes\clE^\vee(l)).
\end{eqnarray}
\begin{prop}
Let $(a,q)$ be a point in $\overline{\clR}$. For $l$ large enough $(a,q)$ is GIT semistable with respect
to $\clO_{\clZ}(n_1,n_2)$ if and only if the following holds. For any non trivial subspace
$U\subseteq V$ we have
\begin{eqnarray}\label{GIT-num-cond-1-eqn}
dim\ U[n_1 P(l) - n_2] (\leq) P(m)[n_1 dim\ q'(U\otimes W)-\epsilon(U)n_2]
\end{eqnarray}
where $\epsilon(U)=1$ if $im\ a\subseteq U$ and 0 otherwise.
\end{prop}
\begin{pf}
We use Hilbert-Mumford criterion.
Let $\lambda:\bbC^*\to SL(V)$ be a one parameter subgroup. Let us choose a basis of $V$ $v_1,..,v_p$
such that $v_i\cdot\lambda(t)=t^{\gamma_i}v_i$, $\gamma_{i+1}\geq \gamma_i$, $i=1..,p-1$.
Recall that $\lambda$ is completely specified by a weight vector $(\gamma_1,...,\gamma_p)\in\bbZ^p$
Moreover $\sum_{i=1}^p \gamma_i=0$.
The number $\mu(q,\lambda)$ is computed by Lemma \ref{quot-weight}.
Let us compute $\mu(a,\lambda)$, $a\in N=Proj(Sym^{\bullet} Hom(H\times\Gamma,V)^\vee)$.
Sections of $\clO_N(1)$ are $Hom(H\times\Gamma,V)^\vee$.
Let $\xi:H\times\Gamma\to V$ be a homomorphism. We can write it in matricial form
as $\xi=\sum_{i,j}\alpha_{ji} w_j^\vee\otimes v_i$, $w_j\in H\times\Gamma$.
Then
\begin{eqnarray}
\mu(a,\lambda)=\mbox{max}\{\gamma_i|\alpha_{ij}\neq 0\}.
\end{eqnarray}
Equivalently $\mu(a,\lambda)=\gamma_i$ where $i=\mbox{min}\{\ i|\mbox{im}\ a\subseteq \langle v_1,.,v_i\rangle \}$.
By the Hilbert-Mumford criterion (semi) stability of $(a,q)$ requires that
\begin{eqnarray}
\mu(q'',\lambda)n_1+\mu(a,\lambda)n_2(\geq)0\quad.
\end{eqnarray}
Given a base $v_1,..,v_p$ of $V$, let consider weight vectors of the form
\begin{eqnarray}\label{sample-weight-vector}
\gamma^{(i)}=(\underbrace{i-p,..,i-p}_{i},\underbrace{i,..,i}_{p-i})
\end{eqnarray}
Any other weight vector can be expressed as a finite non negative linear combination of weight vectors in this class. In fact
\begin{displaymath}
\gamma_k=\sum_{i=1}^{p-1} \left( \frac{\gamma_{i+1}-\gamma_i}{p}\right) \gamma^{(i)}_k.
\end{displaymath}
Let us start by evaluating $\mu(q,\lambda)$. By Lemma \ref{quot-weight} we get
\begin{eqnarray}
\mu(q,\lambda)=\sum_n n P(\clF_n(l))=(i-p)\sum_{n\leq i} P(\clF_n(l)) + i \sum_{i<n\leq p} P(\clF_n(l))=\nonumber\\
i\rho - p\psi(i),
\end{eqnarray}
where $\rho=h^0(\clF\otimes\clE^\vee(l))$ and $\psi(i)= \sum_{n\leq i} P(\clF_n(l))=\mbox{dim}(q'(\langle v_1,..,v_i\rangle\otimes W))$.
Here the second equality holds because the graded pieces are also bounded.
Let us come to $\mu(a,\lambda)$.
Note that $\mu(a,\lambda)=i-p$ if $\mbox{im}\ a\subseteq \langle v_1,..,v_i\rangle$,
and that $\mu(a,\lambda)=i$ otherwise. More compactly, $\mu(a,\lambda)=i-\epsilon(i)p$,
where $\epsilon(i)=1$ if $\mbox{im}\ a\subseteq \langle v_1,..,v_i\rangle$, 0 otherwise.
Summing up the Hilbert-Mumford criterion implies
\begin{eqnarray} \label{1st-inequality}
i(\rho n_1 -n_2) (\leq ) p(\psi(i)n_1 - \epsilon(i)n_2)\quad
\end{eqnarray}
We observe that the above equation does neither depend on the base of $V$ we chosed nor
on the particular weight vector we evaluated the Hilbert-Mumford criterion on.
Let $U\subseteq V$ the vector subspace generated by $v_1,.,v_i$. Since $\rho=P_\clE(l)$
and $\mbox{dim} V= P_\clE(m)$ the inequality (\ref{1st-inequality}) can be rewritten as
\begin{eqnarray}\label{GIT-num-cond-1-eqn}
\mbox{dim}\ U(P_\clE(l)n_1 -n_2) (\leq) \mbox{dim}\ V(\mbox{dim}(q'(U\otimes W)n_1 - \epsilon(U)n_2)\quad
\end{eqnarray}
where $\epsilon(U)=1$ if $\mbox{im}\ a\subseteq U$ and 0 otherwise.
\end{pf}
\begin{notation}\label{FU-notation}
Let $U\subseteq V$ be a sub vector space. We denote by $\clF_U$ the subsheaf of $\clF$
generated by $q(G_\clE(U))$.
\end{notation}
\begin{cor}\label{injectivity-cor}
For any GIT-semistable point $(a,q)$ the induced morphism
$$q(m)\circ\iota_\clE(V\otimes\clO_\clX): V\otimes\clO_\clX\to V \otimes\mathcal{E}nd\ \clE\to \clF\otimes\clE^\vee(m)$$
is injective.
In particular $\mbox{dim}(V\cap H^0(F_\clE(\clF)(m)))\leq h^0(F_\clE(\clF)(m))$. Moreover $q'$ is injective
and for any subspace $U\subseteq V$ $\mbox{dim}\ q'(U\otimes W)\leq h^0(\clF_U(l))$
where $\clF_U=q(U(-m))$.
\end{cor}
\begin{pf}
Let $U\subseteq V$ be the kernel of $q(m)\circ\iota_\clE(V\otimes\clO_\clX)$. Then $\epsilon(U)=0$, otherwise
$\phi$ would be zero. Moreover
$\mbox{dim}\ q'(U\otimes V)=0$. Substituting in equation
(\ref{GIT-num-cond-1-eqn}) we get $\mbox{dim}\ U\leq 0$ because $n_1\rho -n_2\geq 0$
by the choice (\ref{n_1-n_2-choice-eqn}).
\end{pf}
We give another numerical characterization for GIT semi-stability.
\begin{prop}
For sufficiently large $l$ a point $(a,q)$ is GIT-semistable if for any
$U\subseteq V$ the following polynomial equation holds
\begin{eqnarray}\label{GIT-num-cond-2-eqn}
\mbox{dim}\ U(P_\clE(l)n_1 -n_2) (\leq) \mbox{dim}\ V(\mbox{dim}(P_\clE(\clF_U(l))n_1 - \epsilon(U)n_2)
\end{eqnarray}
\end{prop}
\begin{pf}
It is enough to prove that equation (\ref{GIT-num-cond-2-eqn}) implies (\ref{GIT-num-cond-1-eqn}).
Note that subsheaves of the form $\clF_U$ are bounded. Hence for $l$ large enough
$P_\clE(\clF_U(l))=h^0( \clF_U\otimes\clE^\vee(l))=\mbox{dim}\ q'(U\otimes W)$. We are left to show that $\epsilon(U)=1\Leftrightarrow \epsilon(\clF_U)=1$.
For any $U\subseteq V$ $\epsilon(U)=1\Rightarrow \epsilon(\clF_U)=1$. Let us prove the opposite implication.
It can happen that $\mbox{Im}\ a\subsetneq U$
but $\mbox{Im}\ \phi\subseteq \clF_U$. Let $U'$ be the subspace of $V$ generated by $U$
and by $\mbox{Im}\ a$. Then the inequality holds with $\mbox{dim}\ U'$ replacing $\mbox{dim}\ U$,
hence it holds a fortiori for $\mbox{dim}\ U$.
\end{pf}
\begin{prop}
For sufficiently large $l$ a point $(a,q)$ is GIT-semistable if and only if for any
$U\subseteq V$ the following polynomial equation holds
\begin{eqnarray}\label{GIT-num-cond-3-eqn}
P(\mbox{dim}\ U + \epsilon(\clF_U)\delta(m))+\delta (\mbox{dim}\ U -\epsilon(\clF_U)P(m))(\leq) P_{\clF_U}(P(m)+\delta(m))
\end{eqnarray}
\end{prop}
\begin{pf}
The proof is as in \cite{Wand10}.
Take $l$ large enough such that inequality (\ref{GIT-num-cond-2-eqn}) holds as an inequality of polynomials.
Then put
\begin{eqnarray}
\frac{n_1}{n_2}=\frac{P_\clE(l)\delta(m)-P_\clE(m)\delta(l)}{P(m)+\delta(m)}
\end{eqnarray}
\end{pf}
\begin{thm}\label{GIT-implies-dss-thm}
For sufficiently large $l$ if a point $(a,q)$ in $\overline{\clR}$ is GIT-semistable then the corresponding pair
$(\clF,\phi)$ is $\delta$ semi-stable and $H^0(q(m)\circ \iota_\clE(V\otimes\clO_\clX))$ is an isomorphism.
In particular any GIT-semistable point corresponds to a pair with torsion-free sheaf.
\end{thm}
\begin{pf}
Note that by {\em Corollary \ref{injectivity-cor}} $V\otimes\clO_\clX \to \clF\otimes\clE^\vee(m)$ is injective.
Hence for dimensional reasons $V\to H^0(\clF\otimes\clE^\vee(m))$ is an isomorphism.
Let $\clF'\subseteq \clF$ be a subsheaf. Let $U=V\cap H^0(F_\clE(\clF'))$. Then
$\mbox{dim}\ U=h^0(F_\clE(\clF'))$.
Let $\clF_U\subseteq \clF'$ as in {\em Notation \ref{FU-notation}}. We observe that
$\epsilon(\clF_U)=1$ iff $\epsilon(\clF')=1$. The ``if'' direction holds because
$\epsilon(\clF')=1$ implies $\epsilon(U)=1$ since $U=V\cap H^0(F_\clE(\clF'))$.
This in turn implies $\epsilon(\clF_U)=1$.
By taking the leading coefficients in the polinomial equation (\ref{GIT-num-cond-3-eqn})
we get
\begin{eqnarray}\label{leading-coeff-GIT-ss-eqn}
\mbox{dim} U +\epsilon(\clF')\delta(m)\leq \frac{r'}{r}(P_\clE(m)+\delta(m))
\end{eqnarray}
By {\em Lemma \ref{deform-sheaf-lem}} there exists a morphism
$\psi:\clF\to\clH$, where $\clH$ is pure, the kernel is a torsion subsheaf $\clT$ and
$P_\clE(\clF)=P_\clE(\clH)$.
There is an induced homomorphism $\phi_H:\clE\otimes\Gamma\to \clH$ which is non vanishing.
If it was, then (\ref{leading-coeff-GIT-ss-eqn}) would be violated for $\clF'=\clT$.
Let $\clH''$ be a quotient of $\clH$, and $\clH'$ the corresponding kernel.
Let $\clF''$ the image of $\clF$ in $\clH''$ and let $\clF'$ be the corresponding kernel. Then
\begin{eqnarray}\label{clH-is-ss}
h^0(\clH''\otimes\clE^\vee(m))+\epsilon(\clH'')\delta(m)\geq\nonumber\\
h^0(\clF''\otimes\clE^\vee)+\epsilon(\clF'')\delta(m)\geq\nonumber\\
\mbox{dim} V + \delta(m) - (\mbox{dim} U + \epsilon(\clF'))\nonumber\geq\\
P_\clE(m)+\delta(m) - \frac{r_{\clE,\clF'}}{r_{\clE,\clF}} (P_\clE(m)+\delta(m))\geq\nonumber\\
\frac{r_{\clE,\clF''}}{r_{\clE,\clF}} (P_\clE(m)+\delta(m))=\nonumber\\
\frac{r_{\clE,\clH''}}{r_{\clE,\clF}} (P_\clE(m)+\delta(m))\ \
\end{eqnarray}
It follows that $(\clH,\phi_H)$ is $\delta$ semistable by {\em Proposition \ref{num-cond-d-ss-prop}}, therefore belongs to a bounded family.
Consequently it is $m$-regular.
Let us consider the sheaf $\psi(\clF)\subseteq \clH$. It is a quotient of $\clF$. By
(\ref{clH-is-ss}) $h^0(\psi(\clF)\otimes\clE^\vee(m))\geq h^0(\clH\otimes\clE^\vee(m))=P_\clE(m)$,
while the opposite inequality is obvious. Hence $V\otimes\clE^\vee(-m)\to \clH$ is surjective,
which implies $\clF\to \clH$ is surjective.
By semistability of $\clH$ and $P_\clE(\clF)=P_\clE(\clH)$ it follows that $\psi$ is an isomorphism.
Eventually, it is an obvious consequence that $V\to H^0(\clH\otimes\clE^\vee(m))$ is an isomorphism, because
$V\to H^0(\clF\otimes\clE^\vee(m))$ is.
\end{pf}
We prove now the inverse implication.
\begin{thm}\label{delta-ss-implies-GIT-ss-thm}
Let $(\clF,\phi)$ be a $\delta$-semistable pair such that $H^0(F_\clE(q(m))\circ\iota_\clE(V\otimes\clO_\clX)):V\to H^0(F_\clE(\clF))$ is an isomorphism. Then the corresponding point is GIT-semistable.
\end{thm}
\begin{pf}
If $\delta$ stability holds we can prove (\ref{GIT-num-cond-3-eqn}) by only considering the leading coefficients.
In this case, given $\clF'\subseteq\clF$ we define $U=V\cap H^0(\clF'\otimes\clE^\vee(m))$.
Then $\mbox{dim} U\leq h^0(\clF'\otimes\clE^\vee(m))$.
Recall $\epsilon(\clF')=1\Leftrightarrow \epsilon(\clF_U)$ where $\clF_U$ is the sheaf generated by $U$ via $q(m)\otimes\clE^\vee\circ\varphi_\clE$.
This inequality and the $\delta$ stability condition yield
\begin{displaymath}
\mbox{dim} U + \epsilon(\clF')\delta(m) < \frac{r_{\clE,\clF'}}{r_{\clE,\clF}}(P_\clE(m)+\delta(m)),
\end{displaymath}
which implies (\ref{GIT-num-cond-3-eqn}) as a polynomial inequality.
The rest of the proof taking into account $\delta$ semistability proceeds exactly
as in \cite{Wand10} {\bf Theorem 4.7}.
In particular a pair $(\clF,\phi)$ is GIT stable only if it is $\delta$ stable.
\end{pf} | {"config": "arxiv", "file": "1105.5637.tex"} |
TITLE: Show that if $p(x)=a+bx+cx^2$ is a 2nd degree polynomial such that $p(1)=p(2)=p(3)=0$ then $p(x)=0$, using determinants.
QUESTION [1 upvotes]: Show that if $p(x)=a+bx+cx^2$ is a 2rd degree polynomial such that $p(1)=p(2)=p(3)=0$ then $p(x)=0$ (i.e. $a=b=c=0$), using the determinant of the matrix: $\left(\begin{array}{l}1&x&x^2\\1&y&y^2\\1&z&z^2\end{array}\right)$
Hey everyone. So I managed to prove that the determinant of the given matrix is $(y-x)(z-x)(z-y)$ and therefore the matrix is invertible whenever $x\neq z \lor y\neq x \lor z\neq y$
I've proved this claim using basic algebra but am confused on how to show it using the determinant. I tried showing it by plugging in the values of the polynomial by order:
$\left|\begin{array}{l}a&b&c\\a&2b&4c\\a&3b&9c\end{array}\right|$ (Column operations) $ =\left|\begin{array}{l}a&b&c+b+a\\a&2b&4c+2b+a\\a&3b&9c+3b+a\end{array}\right|$ $(p(1)=p(2)=p(3)=0)$$ = \left|\begin{array}{l}a&b&0\\a&2b&0\\a&3b&0\end{array}\right|=0$ but this does not help me. I would love to get some help on this question. Thanks in advance.
REPLY [1 votes]: Note that\begin{align}p(1)=p(2)=p(3)=0&\iff\left\{\begin{array}{l}a+b\times1+c\times1^2=0\\a+b\times2+c\times2^2=0\\a+b\times3+c\times3^2=0\end{array}\right.\\&\iff\begin{pmatrix}1&1&1\\1&2&2^2\\1&3&3^2\end{pmatrix}.\begin{pmatrix}a\\b\\c\end{pmatrix}=0.\end{align}So, if $(a,b,c)\neq(0,0,0)$, the determinant of that matrix will be $0$. But it isn't. | {"set_name": "stack_exchange", "score": 1, "question_id": 2664406} |
TITLE: Calculating the volume of a cone given the surface and $s$
QUESTION [0 upvotes]: I've been struggling with this for so long and I never got a chance to ask my teacher how to solve it.
If the surface of the cone is $360\pi$ and $s = 26 \text{cm}$, calculate the volume of that cone.I found the solution but there is no explanation, somehow you need to get to squared binomial and I'm not sure why.
Formula for the cone volume: $V = \frac13\cdot\pi\cdot r^2\cdot H$
Formula for the cone surface: $P = \pi\cdot r\cdot(r+s)$
REPLY [0 votes]: $SA = \pi rs + \pi r^2$. Thus: $360\pi = \pi 26r + \pi r^2 \to r^2 + 26r = 360 \to (r+13)^2 = 23^2 \to r = 10$. Then $h = \sqrt{s^2 - r^2} = \sqrt{26^2 - 10^2} = 24$. We can now find $V$, and $V = \dfrac{\pi r^2h}{3} = \dfrac{\pi 10^2\cdot 24}{3} = 800\pi$ | {"set_name": "stack_exchange", "score": 0, "question_id": 874579} |
TITLE: Let $(E; d)$ be a metric space. Let $A;B \subseteq E$ be nonempty disjoint subsets and suppose A is compact and B is closed.
QUESTION [0 upvotes]: Show that the sets $A$ and $B$ have a positive distance, i.e. show:
$d(A, B) := \inf \{\,d(a, b) : a \in A, b \in B\} > 0$
I've tried to show this but I couldn't succeed. Basically I was trying to derive a contradiction to $A$ and $B$ being disjoint.
REPLY [2 votes]: For each $x\in A$, we have $d(x,B)=\inf\{d(x,y):y\in B\}>0$ because $x\not\in B$ and $B$ is closed.
Moreover, the function $x\mapsto d(x,B)$ is continuous (this essentially follows from the triangle inequality), and since $A$ is compact, it must attain its minimum. Therefore there is some $x_0\in A$ such that
$$ d(A,B)=d(x_0,B)>0$$
The compactness of $A$ is essential, as the example $A=\{(x,0):x\geq0\}$ and $B=\{(x,\frac{1}{x}):x>0\}$ in $\mathbb{R}^2$ shows. | {"set_name": "stack_exchange", "score": 0, "question_id": 1955447} |
TITLE: Is there a general formula for a quadratic that is always positive?
QUESTION [1 upvotes]: I encountered a problem of quadratic. It asks for a quadratic that is in the form $f(x)=ax^2+bx+c$. It is always positive and $b$ is greater than $a$. Than it asks me to find $f(17)$ based on $f(16)=20$. Is there a fast and easy way to do this? My idea is letting $16a+b=2\sqrt{5}$. But than I don't know how to do it since $a$ nor $b$ can be irrational. Any help will be appreciated!
REPLY [1 votes]: Condition $f(x)\geq 0$ gives you that $a>0$ because, if $a<0$, then $\lim_{x\to\pm\infty}f(x) = -\infty$, so there will be some $x$ such that $f(x) < 0$.
We can write down $f(x) = a(x-\alpha)^2+\beta$.
Since $a(x-\alpha)^2 \geq 0$, it follows that $f(x)\geq \beta$ with $f(\alpha) = \beta$, so $\beta$ is the minimum value $f$ achieves. We have $f(x)\geq 0\iff \beta \geq 0$.
If we expand, we get that $f(x) = ax^2 - 2a\alpha x + a\alpha^2 + \beta$ (i.e. $b=-2a\alpha$) and condition $b>a$ gives us $-2a\alpha > a$. Since $a>0$, this means that $b>a\iff\alpha <-\frac 12$.
Finally, $f(16) = 20$ gives us two things: $\beta\leq 20$ and $a=\frac{20-\beta}{(16-\alpha)^2}$, for $\beta \neq 20$ (I will get back to case $\beta = 20$).
Thus, pick any $\alpha < -\frac 1 2$ and $0\leq \beta < 20$ and you have that $$f(x) = \frac{20-\beta}{(16-\alpha)^2}(x-\alpha)^2 + \beta$$ satisfies all the conditions. In particular, $$f(17) = \frac{20-\beta}{(16-\alpha)^2}(17-\alpha)^2 + \beta.$$
If $\beta = 20$, then we have that $a(16-\alpha)^2 = 0$, and since $a>0$, it means that $\alpha = 16$. But, this is contradiction with requirement that $\alpha <-\frac 12$. | {"set_name": "stack_exchange", "score": 1, "question_id": 2080605} |
\begin{document}
\maketitle
\begin{abstract}
An involution is usually defined as a mapping that is its own inverse. In this paper, we
study quaternion involutions that have the additional properties of distribution over
addition and multiplication. We review formal axioms for such involutions, and
we show that the quaternions
have an infinite number of involutions. We show that the conjugate of a quaternion may be expressed
using three mutually perpendicular involutions. We also show that any set of three
mutually perpendicular quaternion involutions is closed under composition. Finally, we
show that projection of a vector or quaternion can be expressed concisely using
involutions.
\end{abstract}
\section{Introduction}
Involutions are usually defined simply as self-inverse mappings. A trivial example
is conjugation of a complex number, which is obviously self-inverse.
In this paper we consider involutions of the quaternions, that
is functions of a quaternion variable that are self-inverse. Quaternion conjugation
is an obvious involution, but it is not the only quaternion involution. In fact,
the quaternions have an infinite number of involutions, as we show. The paper begins
by reviewing the classical basics of quaternions, and then presents axioms for
involutions which go beyond the simple definition of a self-inverse mapping. Section
\ref{qinvolutions} then presents the quaternion involutions, and section \ref{qinvolutionprops}
presents their properties. Section \ref{conjugate} discusses the quaternion conjugate
and shows that it may be expressed using three mutually perpendicular quaternion
involutions. Finally, section \ref{projection} shows that the projection of a vector or
quaternion may be expressed using involutions.
\section{Basics of quaternions}
A quaternion may be represented in Cartesian form $q = w + \i x + \j y + \k z$
where $\i$, $\j$ and $\k$ are mutually perpendicular unit vectors obeying the
multiplication rules below discovered by Hamilton in 1843 \cite{Hamilton:1866},
and $w$, $x$, $y$, $z$, are real.
\begin{gather}
\label{hrules} \i^2 = \j^2 = \k^2 = \i\j\k = -1\
\end{gather}
The conjugate of a quaternion is given by $\conjugate{q} = w - \i x - \j y - \k z$.
The quaternion algebra \H is a normed division algebra. The modulus of a quaternion is
the square root of its norm: $|q| = \sqrt{w^2 + x^2 + y^2 + z^2}$, and every non-zero
quaternion has a multiplicative inverse given by its conjugate divided by its norm:
$q^{-1} = \conjugate{q}/|q|^2 = (w - \i x - \j y - \k z)/(w^2 + x^2 + y^2 + z^2)$.
For a more detailed exposition of the basics of quaternions, we refer the reader
to Coxeter's 1946 paper \cite{Coxeter:1946}.
An alternative and much more powerful representation for a quaternion is as a combination
of a \emph{scalar} and a \emph{vector} part, analogous to a complex number, and this
representation will be employed in the rest of the paper: $q = a + \vmu b$, where $\vmu$
is a unit vector, and $a$ and $b$ are real. $b$ is the modulus of the vector part of the
quaternion and $\vmu$ is its direction. In terms of the Cartesian representation:
\begin{equation}
\label{svform}
a = w,\qquad b = \sqrt{x^2 + y^2 + z^2},\qquad \vmu = \frac{\i x + \j y + \k z}{b}
\end{equation}
\begin{lemma}
\label{unitvector}
The square of any unit vector is $-1$.
\end{lemma}
\begin{proof}
Let $\vmu$ be an arbitrary unit vector as defined in equation \ref{svform}.
Its square is given by:
\begin{equation*}
\begin{split}
\vmu^2 &= \frac{(\i x + \j y + \k z)^2}{x^2 + y^2 + z^2}\\
&= \frac{\i^2 x^2 + \j^2 y^2 + \k^2 z^2 +
\i\j x y + \j\i x y + \i\k x z + \k\i x z + \j\k y z + \k\j y z}{x^2 + y^2 + z^2}\\
\end{split}
\end{equation*}
Applying the rules in equation \ref{hrules} we get:
$\vmu^2 = \frac{-x^2 -y^2 -z^2}{x^2 + y^2 + z^2} = -1$
\end{proof}
A corollary of Lemma \ref{unitvector} is that there are an infinite number of
solutions to the equation $x^2 = -1$.
The conjugate of a quaternion in complex form is $\conjugate{q} = a - \vmu b$.
Geometrically, this is obviously a reversal of the direction
of the vector part. The quaternion conjugate has properties
analogous to those of the complex conjugate, with one minor
exception: the quaternion conjugate is an anti-involution whereas
the complex conjugate is an involution. We define these terms in
the next section, and we return to this point with Theorem \ref{anticonjugate}
in section \ref{conjugate}. The product of a quaternion with its conjugate
gives the norm, or square of the modulus. This follows directly
from Lemma \ref{unitvector}:
$(a + \vmu b)(a - \vmu b) = a^2 - \vmu^2 b^2 = a^2 + b^2 = w^2 + x^2 + y^2 + z^2$.
\section{Involutions}
\label{involutions}
The formal definition of an involution is not easy to find (most mathematical
reference works define it simply as a mapping which is its own inverse) but
\cite{KluwerEncylMaths:1990} gives a reasonably authoritative statement from
which we reproduce the following axioms. It is clear from what follows in the
paper that all three of these axioms are important, otherwise it is possible
to define trivial self-inverse mappings, which have uninteresting properties.
\newcommand{\involution}[1]{f(#1)}
We denote an arbitrary involution by the mapping $x \rightarrow \involution{x}$.
\begin{axiom}
\label{inverse}
$\involution{\involution{x}} = x$. An involution is its own inverse.
\end{axiom}
\begin{axiom}
\label{linearity}
An involution is linear:
$\involution{x_1 + x_2} = \involution{x_1} + \involution{x_2}$ and
$\lambda\involution{x} = \involution{\lambda x}$ where $\lambda$ is
a real constant.
\end{axiom}
\begin{axiom}
\label{product}
$\involution{x_1 x_2} = \involution{x_1}\involution{x_2}$. If the terms
on the right must be reversed (which can only be necessary if $x_1$ and $x_2$
do not commute), then $\involution{\ }$ is an \emph{anti-involution}.
\end{axiom}
\section{Quaternion involutions}
\label{qinvolutions}
Involutions over the quaternion field have been published by Chernov \cite{Chernov:1995},
and used by Bülow \cite{Bulow:1999,BulowSommer:2001}, but with an important difference
from the involutions given in this paper: we show here that there are an
infinite number of involutions over the quaternion field
whereas Chernov and Bülow wrote of
only three (plus conjugation). Chernov defined the
following involutions, which we generalize in Theorem \ref{general}:
\begin{equation}
\begin{split}
\alpha(q) &= - \i q \i = w + \i x - \j y - \k z\\
\beta(q) &= - \j q \j = w - \i x + \j y - \k z\\
\gamma(q) &= - \k q \k = w - \i x - \j y + \k z\\
\end{split}
\end{equation}
They also showed that the quaternion conjugate can be expressed in terms of these three
involutions, a result that we generalize in Theorem \ref{genconj}.
\begin{theorem}
\label{general}
The mapping $q \rightarrow -\vnu q \vnu$ where $q$ is an arbitrary quaternion is
an involution for any unit vector $\vnu$.
\end{theorem}
\begin{proof}
Axiom \ref{inverse} is easily shown to be satisfied using Lemma \ref{unitvector}:
$-\vnu (-\vnu q \vnu) \vnu = (-1)q(-1) = q$. Axiom \ref{linearity} is seen to be
satisfied from: $-\vnu (q_1 + q_2) \vnu = -\vnu q_1 \vnu - \vnu q_2 \vnu$ (multiplication
of quaternions is distributive over addition). Since reals commute with quaternions, the
second part of the axiom is trivially seen.
Axiom \ref{product} can be shown to be satisfied as follows:
\begin{equation*}
\begin{split}
\involution{q_1}\involution{q_2} & = (-\vnu q_1 \vnu)(-\vnu q_2 \vnu)\\
& = \vnu q_1 \vnu \vnu q_2 \vnu\\
& = \vnu q_1 (-1) q_2 \vnu\\
& = -\vnu q_1 q_2 \vnu = \involution{q_1 q_2}\\
\end{split}
\end{equation*}
\end{proof}
Note that a mapping $q \rightarrow \vnu_1 q \vnu_2$ ($\vnu_1 \neq \vnu_2$) is its
own inverse, but is not an involution as considered in this paper, because it does
not satisfy axiom \ref{product}.
In what follows we introduce a new notation for involutions using an overbar with a
subscript unit vector, thus $\involute{q}{\vnu} = -\vnu q \vnu$. We refer to the direction
defined by $\vnu$ as the \emph{axis of involution}. The fact that we use an overbar to
denote involutions as well as the conjugate is not coincidental and we shall see that
there is a close relationship between involutions and conjugation.
\section{Properties of quaternion involutions}
\label{qinvolutionprops}
\begin{lemma}
\label{unitvectorproduct}
The product of any two vectors with arbitrary non-zero norms is a quaternion. The scalar
part is minus the inner or scalar product of the two vectors and the vector part is the
vector product of the two vectors. Reversing the order of the product conjugates the
resulting quaternion.
\end{lemma}
\begin{proof}
Let $\vmu_1 = \i x_1 + \j y_1 + \k z_1$ and $\vmu_2 = \i x_2 + \j y_2 + \k z_2$.
Their product is given by:
\begin{equation*}
\begin{split}
\vmu_1 \vmu_2 & = (\i x_1 + \j y_1 + \k z_1)(\i x_2 + \j y_2 + \k z_2)\\
& = \i^2 x_1 x_2 + \j^2 y_1 y_2 + \k^2 z_1 z_2\\
& \quad + \i\j x_1 y_2 + \j\i y_1 x_2 + \i\k x_1 z_2 + \k\i z_1 x_2 + \j\k y_1 z_2 + \k\j z_1 y_2\\
& = - (x_1 x_2 + y_1 y_2 + z_1 z_2)\\
& \quad + \i(y_1 z_2 - z_1 y_2) + \j(z_1 x_2 - x_1 z_2) + \k(x_1 y_2 - y_1 x_2)
\end{split}
\end{equation*}
Changing the order of the product changes the order of all the products of two unit
vectors $\i$, $\j$ and $\k$. Since $\i\j = -\j\i$ and so on, this negates all the
components of the vector part of the result. The scalar part, $-(x_1 x_2 + y_1 y_2 + z_1 z_2)$,
is unchanged. Thus reversing the order of the product conjugates the result, as stated.
The scalar and vector parts can be seen to be equal to minus the scalar product and
the vector product respectively, according to standard definitions of these products.
\end{proof}
\begin{lemma}
\label{perpvec}
The product of two perpendicular vectors changes sign if the order of the product
is reversed.
\end{lemma}
\begin{proof}
Lemma \ref{unitvectorproduct} identified the scalar part of the result
$-(x_1 x_2 + y_1 y_2 + z_1 z_2)$ as minus the inner product of the two
vectors. Since the inner product is zero in the case of perpendicular vectors, the
product of perpendicular vectors is a vector and it is the vector product of the
two vectors. It follows that this vector changes sign (reverses) if the
order of the product is reversed.
\end{proof}
\begin{theorem}
\label{composition}
Composition: the composition of two perpendicular involutions is commutative. That is:
$\involute{\involute{q}{\vnu_1}}{\vnu_2} = \involute{\involute{q}{\vnu_2}}{\vnu_1}$ where
$\vnu_1\perp\vnu_2$.
\end{theorem}
\begin{proof}
\[
\involute{\involute{q}{\vnu_1}}{\vnu_2} = -\vnu_2(-\vnu_1 q \vnu_1)\vnu_2 = \vnu_2\vnu_1 q \vnu_1\vnu_2
\]
and by Lemma \ref{perpvec} we can reverse the order of the pairs of unit vectors
if we change their signs:
\[
\involute{\involute{q}{\vnu_1}}{\vnu_2} = (-\vnu_1\vnu_2) q (-\vnu_2\vnu_1) = -\vnu_1(-\vnu_2 q \vnu_2)\vnu_1 = \involute{\involute{q}{\vnu_2}}{\vnu_1}
\]
\end{proof}
\begin{theorem}
\label{perpinvolutions}
Double Composition: given a set of three mutually perpendicular unit vectors,
$\vnu_1$, $\vnu_2$, $\vnu_3$, such that $\vnu_1 \vnu_2 = \vnu_3$, then
\[
\involute{\involute{q}{\vnu_1}}{\vnu_2} = \involute{q}{\vnu_3}
\]
\end{theorem}
\begin{proof}
\begin{equation*}
\begin{split}
\involute{\involute{q}{\vnu_1}}{\vnu_2} & = -\vnu_2 (-\vnu_1 q \vnu_1) \vnu_2 = \vnu_2\vnu_1 q \vnu_1\vnu_2\\
& = -\vnu_1\vnu_2 q \vnu_1\vnu_2 \quad\text{by Lemma \ref{perpvec}}\\
& = -\vnu_3 q \vnu_3 = \involute{q}{\vnu_3}
\end{split}
\end{equation*}
\end{proof}
\begin{corollary}
Triple Composition: the composition of three mutually perpendicular involutions is an identity.
\end{corollary}
\begin{proof}
From Theorem \ref{perpinvolutions}:
$\involute{\involute{q}{\vnu_1}}{\vnu_2} = \involute{q}{\vnu_3}$.
Apply an involution about $\vnu_3$ to both sides:
$\involute{\involute{\involute{q}{\vnu_1}}{\vnu_2}}{\vnu_3} = \involute{\involute{q}{\vnu_3}}{\vnu_3} = q$
\end{proof}
Thus we see that a set of three mutually perpendicular involutions
(that is involutions about a set of three mutually perpendicular
axes) is closed under composition of the involutions
\emph{and by Theorem \ref{composition} the order of the composition is unimportant.}
We now present a geometric interpretation of quaternion involution.
\begin{theorem}
\label{geometry} Given an arbitrary quaternion $q = a + \vmu b$,
an involution $\involute{q}{\vnu}$ leaves the scalar part of $q$
(that is, $a$) invariant, and reflects the vector part of $q$
(that is, $\vmu b$) across the line defined by the
axis of involution $\vnu$. (Equivalently, the vector part of $q$
is rotated by $\pi$ radians about the axis of involution $\vnu$.)
\end{theorem}
\begin{proof}
\begin{equation*}
\begin{split}
\involute{q}{\vnu} & = -\vnu(a + \vmu b)\vnu = -\vnu a\vnu -\vnu\vmu b\vnu = -\vnu^2 a -\vnu\vmu\vnu b
\intertext{and, since $\vnu$ is a unit vector, by Lemma \ref{unitvector}:}
\involute{q}{\vnu} & = a - \vnu\vmu\vnu b
\end{split}
\end{equation*}
We recognise $\vnu\vmu\vnu$ to be a reflection of $\vmu$ in the plane $p$ normal to $\vnu$
as shown by Coxeter \cite[Theorem 3.1]{Coxeter:1946}. Therefore $-\vnu\vmu\vnu$ is a
reflection of $\vmu$ in the line defined by $\vnu$ as shown in Figure \ref{reflect}.
The result of the reflection of the vector part remains a vector, and therefore the
scalar part remains unchanged, as shown, and as stated.
\end{proof}
\begin{figure}
\centerline{\includegraphics[width=0.4\textwidth]{reflect.eps}}
\caption{\label{reflect}Reflection of a vector $\vmu$ in a line defined by a unit vector $\vnu$.
$p$ is a plane perpendicular to $\vnu$ seen edge on.}
\end{figure}
\begin{corollary}
\label{invparq} An involution applied to a quaternion with vector
part parallel to the involution axis is an identity,
\ie, $\involute{a + \vmu b}{\vnu} = a + \vmu b$, where $\vnu\parallel\vmu$.
\end{corollary}
\begin{proof}
From Theorem \ref{geometry}: $\involute{q}{\vnu} = a - \vnu\vmu\vnu b$. Since both $\vnu$
and $\vmu$ are unit vectors, and are parallel, $\vnu = \pm\vmu$ and the result on the right
reduces to $a + \vmu b = q$ in both cases.
\end{proof}
\begin{corollary}
\label{invperpq} An involution applied to a quaternion with vector
part perpendicular to the involution axis conjugates the quaternion,
\ie, $\involute{a + \vmu b}{\vnu} = a - \vmu b$, where $\vnu\perp\vmu$.
\end{corollary}
\begin{proof}
From Theorem \ref{geometry}: $\involute{q}{\vnu} = a - \vnu\vmu\vnu b$.
If $\vnu\perp\vmu$ we can use Lemma \ref{perpvec} to reverse the order of the two
unit vectors, thus obtaining $a + \vnu^2\vmu b = a - \vmu b = \conjugate{q}$ as stated.
\end{proof}
\begin{lemma}
\label{argvecprod} The product of two unit vectors is a quaternion
with argument equal to the angle between the two vectors.
\end{lemma}
The sign of the argument is significant, because it depends on the
ordering of the two vectors (the angle is measured from the first
vector to the second).
\begin{proof}
Lemma \ref{unitvectorproduct} identified the scalar and vector parts
of the product of two vectors with minus the inner product, and the vector product
respectively of the two vectors. Since these products are given for unit vectors
by $\cos\theta$ and $\vmu\sin\theta$, where $\vmu$ is perpendicular to the plane
containing the two vectors, we may write the product of two unit vectors as:
$-\cos\theta + \vmu\sin\theta = -\exp(-\vmu\theta)$.
\end{proof}
\begin{theorem}
The composition of two involutions is a rotation of the vector part of the quaternion
operated upon about an axis normal to the plane containing the two axes of involution.
The angle of rotation is twice the angle between the two involution axes.
In the case where the two involutions are perpendicular the composite result is of course
an involution, which is a rotation of the vector part of the quaternion by $\pi$.
\end{theorem}
\begin{proof}
Let $\vnu_1$ and $\vnu_2$ be two unit vectors and $q$ be an arbitrary quaternion.
Then the composition of two involutions about $\vnu_1$ and $\vnu_2$ is given by:
\[
\involute{\involute{q}{\vnu_1}}{\vnu_2} = -\vnu_2(-\vnu_1 q \vnu_1)\vnu_2 = \vnu_2\vnu_1 q \vnu_1\vnu_2
\]
From Lemma \ref{unitvectorproduct} we can write the result on the right as $p q
\conjugate{p}$, where $p = \vnu_2\vnu_1$ is a unit quaternion\footnote{$p$ is
a unit quaternion, because it is the product of two unit vectors. This follows from
the fact that the quaternion algebra is a normed algebra.}.
Separating $q = a + \vmu b$ into its scalar and vector parts we obtain:
\[
\involute{\involute{q}{\vnu_1}}{\vnu_2} = p \conjugate{p} a + p \vmu b \conjugate{p} = a + (p\vmu\conjugate{p}) b
\]
We recognise the term $p\vmu\conjugate{p}$ as a rotation as given by Coxeter \cite[Theorem
3.2]{Coxeter:1946}. The axis of rotation is given by the vector part of $p$, and the angle
of rotation is twice the argument of $p$. Therefore, from Lemma \ref{unitvectorproduct} we
know that the axis of rotation is perpendicular to the plane containing the two vectors,
and from Lemma \ref{argvecprod} we know that the angle of rotation is twice the angle
between the two vectors.
\end{proof}
\section{The quaternion conjugate}
\label{conjugate}
\begin{theorem}
\label{anticonjugate}
The quaternion conjugate is an anti-involution.
\end{theorem}
\begin{proof}
The definition of the conjugate of a quaternion $q = a + \vmu b$ is $\conjugate{q} = a -
\vmu b$. We have to show that this satisfies the three axioms given in section \ref{involutions},
and that in Axiom \ref{product} we have to reverse the terms on the
right-hand side.
It is easily seen that the quaternion conjugate satisfies Axiom \ref{inverse}.
To demonstrate that the quaternion conjugate satisfies Axiom \ref{linearity}, let
$q_1 = a + \vmu_1 b$ and $q_2 = c + \vmu_2 d$. Then $q_1 + q_2 = (a + c) + (\vmu_1 b + \vmu_2 d)$.
Since reversing two vectors also reverses their sum, we see that the quaternion conjugate
is distributive over addition, as required. The second part of the axiom is easily seen.
To show that the quaternion conjugate satisfies Axiom \ref{product}, we state
the required equality and then demonstrate that it is satisfied by expanding the left and
right hand sides until identical:
\begin{equation*}
\begin{split}
\conjugate{q_1 q_2} & = \conjugate{q_2}\,\conjugate{q_1}\\
\conjugate{(a + \vmu_1 b)(c + \vmu_2 d)} & = (c - \vmu_2 d)(a - \vmu_1 b)\\
\intertext{Expanding the left hand side, we employ Axiom \ref{linearity}:}
a c - \vmu_1\vmu_2 b d - \vmu_1 b c - \vmu_2 a d & = a c + \vmu_2\vmu_1 b d - \vmu_1 b c - \vmu_2 a d\\
\intertext{and using Lemma \ref{perpvec} we change the order of the two vectors
in the second term on the right-hand side to obtain the required result:}
a c - \vmu_1\vmu_2 b d - \vmu_1 b c - \vmu_2 a d & = a c - \vmu_1\vmu_2 b d - \vmu_1 b c - \vmu_2 a d\\
\end{split}
\end{equation*}
\end{proof}
We now show that the quaternion conjugate can be expressed using the sum of three mutually
perpendicular involutions.
\begin{lemma}
\label{treperpinv}
The sum of three mutually perpendicular involutions applied to a vector negates the
vector (reverses its direction). That is, given a set of three mutually perpendicular
unit vectors as in Theorem \ref{perpinvolutions} and an arbitary vector $\vmu$:
\begin{equation}
\label{treperinveqn}
\involute{\vmu}{\vnu_1} + \involute{\vmu}{\vnu_2} + \involute{\vmu}{\vnu_3} = -\vmu
\end{equation}
\end{lemma}
\begin{proof}
\newcommand{\veta}{\v\eta}
Let $\vmu = \veta_1 + \veta_2 + \veta_3$ where $\veta_i
\parallel \vnu_i, i \in \{1, 2, 3\}$.
In other words,
resolve $\vmu$ into three vectors\footnote{The three vectors $\veta_i$ are not, in general,
of unit modulus.} parallel to the three
mutually perpendicular unit vectors $\vnu_1$, $\vnu_2$ and
$\vnu_3$. Substitute this representation of $\vmu$ into the
left-hand side of Equation \ref{treperinveqn}:
\begin{equation*}
\involute{\veta_1 + \veta_2 + \veta_3}{\vnu_1} +
\involute{\veta_1 + \veta_2 + \veta_3}{\vnu_2} +
\involute{\veta_1 + \veta_2 + \veta_3}{\vnu_3} = -\vmu
\end{equation*}
Axiom \ref{linearity} allows us to apply the involutions separately to the three components:
\begin{equation*}
\involute{\veta_1}{\vnu_1} + \involute{\veta_2}{\vnu_1} + \involute{\veta_3}{\vnu_1} +
\involute{\veta_1}{\vnu_2} + \involute{\veta_2}{\vnu_2} + \involute{\veta_3}{\vnu_2} +
\involute{\veta_1}{\vnu_3} + \involute{\veta_2}{\vnu_3} + \involute{\veta_3}{\vnu_3} = -\vmu
\end{equation*}
We now make use of Corollaries \ref{invparq} and \ref{invperpq}. In this case we are
applying them to a vector, so the first states that an involution with axis parallel to a
vector is an identity, and the second states that an involution with axis perpendicular to
the vector reverses, or negates, the vector:
\begin{equation*}
\veta_1 - \veta_2 - \veta_3 - \veta_1 + \veta_2 - \veta_3 - \veta_1 - \veta_2 + \veta_3 = -\vmu
\end{equation*}
and cancelling out, we obtain: $- \veta_1 - \veta_2 - \veta_3 = -\vmu$, which is the assumption
we made at the start of the proof.
\end{proof}
The following theorem is a generalization of a similar result given in \cite[Definition 2.2, p.12]{Bulow:1999}.
\begin{theorem}
\label{genconj}
Given a set of three mutually perpendicular unit vectors as in Theorem \ref{perpinvolutions},
the conjugate of $q$ may be expressed as:
\begin{equation}
\label{conjinvol}
\conjugate{q}
= \frac{1}{2}\left(\involute{q}{\vnu_1} + \involute{q}{\vnu_2} + \involute{q}{\vnu_3} -q \right)
\end{equation}
\end{theorem}
\begin{proof}
Let $q = a + \vmu b$. Substituting this expression for $q$ into the right-hand side of
Equation \ref{conjinvol} we obtain:
\begin{equation*}
\conjugate{q} = \frac{1}{2}\left(\involute{a + \vmu b}{\vnu_1}
+ \involute{a + \vmu b}{\vnu_2}
+ \involute{a + \vmu b}{\vnu_3}
- \left(a + \vmu b\right)
\right)
\end{equation*}
We now apply the three involutions separately to the components of $q$ using Axiom
\ref{linearity}, and noting from Theorem \ref{geometry} that the scalar part $a$ is
invariant under involutions:
\begin{equation*}
\conjugate{q} = \frac{1}{2}\left(a + \involute{\vmu}{\vnu_1} b
+ a + \involute{\vmu}{\vnu_2} b
+ a + \involute{\vmu}{\vnu_3} b
- a - \vmu b
\right)
\end{equation*}
Gathering terms together and factoring out $b$:
\begin{equation*}
\conjugate{q} = a +\frac{1}{2}\left(\involute{\vmu}{\vnu_1}
+ \involute{\vmu}{\vnu_2}
+ \involute{\vmu}{\vnu_3}
- \vmu \right) b
\end{equation*}
and the right-hand side is equal to $a - \vmu b$ by Lemma \ref{treperpinv}.
\end{proof}
\section{Projection using involutions}
\label{projection}
Finally, we now demonstrate the utility of quaternion involutions by presenting formulae for
projection of a vector into or perpendicular to a given direction. These results have
been published in \cite{SangwineEll:2000c}, but without explicit use of involutions.
\begin{theorem}
\label{projections}
An arbitrary vector $\vmu$ may be resolved into two components parallel to,
and perpendicular to, a direction in 3-space defined by a unit vector $\vnu$:
\begin{equation*}
\vmu_{\parallel\vnu} = \frac{1}{2}(\vmu + \involute{\vmu}{\vnu})\qquad
\vmu_{\perp\vnu} = \frac{1}{2}(\vmu - \involute{\vmu}{\vnu})
\end{equation*}
where $\vmu_{\parallel\vnu}$ is parallel to $\vnu$ and $\vmu_{\perp\vnu}$ is
perpendicular to $\vnu$, and $\vmu = \vmu_{\parallel\vnu} + \vmu_{\perp\vnu}$.
\end{theorem}
\begin{proof}
From Theorem \ref{geometry}, $\involute{\vmu}{\vnu}$ is the reflection of $\vmu$ in the
line defined by $\vnu$ as shown in Figure \ref{reflect}. When $\vmu$ is added to its
reflection the components of each perpendicular to $\vnu$ cancel, and the components
parallel to $\vnu$ add to give twice the stated result. The factor of ½ gives the
result as stated. Similarly, half the difference between $\vmu$ and its reflection
gives the component of $\vmu$ perpendicular to $\vnu$.
\end{proof}
Theorem \ref{projections} may be generalised to quaternions as
well as vectors. Since the scalar part of a quaternion is
invariant under an involution, the component of the quaternion
`parallel' to $\vnu$ includes the scalar part as well as the
component of the vector part parallel to $\vnu$. In other words
the `parallel' component of the quaternion is that component which
is in the same Argand plane as the axis of involution $\vnu$. The
component of the quaternion perpendicular to $\vnu$ is a vector
(the component of the vector part perpendicular to $\vnu$, and
therefore perpendicular to the Argand plane of the `parallel'
component) since the subtraction cancels out the scalar
part. As stated earlier, the
representation $a + \vmu b$ is independent of the coordinate
system in that it expresses the quaternion in terms of the
direction in 3-space of the vector part. However, the quaternion
can be rewritten in terms of a set of orthogonal basis
vectors, $\vnu_1$, $\vnu_2$, and $\vnu_3$, without recourse to a
numerical representation. The three projections across $\vnu_1$,
$\vnu_2$, and $\vnu_3$ and the conjugate anti-involution provide
the mechanism.
\newcommand{\vb}{\v{b}}
That is, we may write a quaternion $q = a + \vmu b = a + \vb$ as
\begin{equation*}
q = a + \vnu_1\alpha + \vnu_2\beta + \vnu_3\gamma = a + \vb_1 + \vb_2 + \vb_3
\end{equation*}
where $\vb_i\parallel\vnu_i$ and $\alpha$, $\beta$ and $\gamma$ are
real:
\begin{equation*}
a = \frac{1}{2}( q + \conjugate{q});\qquad
\vb = \frac{1}{2}( q - \conjugate{q});\qquad
\vb_i = \frac{1}{2}(\vb + \involute{\vb}{\vnu_i}),\quad i\in\{1,2,3\}
\end{equation*} | {"config": "arxiv", "file": "math0506034/paper.tex"} |
\begin{document}
\title {On tradeoffs between treatment time and plan quality of volumetric-modulated arc therapy with sliding-window delivery}
\author[1,2]{Lovisa Engberg\thanks{Corresponding author: loven140@kth.se}}
\author[1] {Anders Forsgren}
\affil[1] {Optimization and Systems Theory, Department of Mathematics, \protect\\KTH Royal Institute of
Technology, Stockholm SE-100 44, Sweden}
\affil[2] {RaySearch Laboratories, Sveav\"{a}gen 44, Stockholm SE-103 65, Sweden}
\date {Manuscript\\\today}
\markboth {On tradeoffs between treatment time and plan quality of sliding-window VMAT}{On tradeoffs between treatment time and plan quality of sliding-window VMAT}
\maketitle\thispagestyle{empty}
\begin{abstract}
The purpose of this study is to give an exact formulation of optimization of volumetric-modulated arc therapy (VMAT) with sliding-window delivery, and to investigate the plan quality effects of decreasing the number of sliding-window sweeps made on the 360-degree arc for a faster VMAT treatment. In light of the exact formulation, we interpret an algorithm previously suggested in the literature as a heuristic method for solving this optimization problem. By first making a generalization, we suggest a modification of this algorithm for better handling of plans with fewer sweeps. In a numerical study involving one prostate and one lung case, plans with varying treatment times and number of sweeps are generated. It is observed that, as the treatment time restrictions become tighter, fewer sweeps may lead to better plan quality. Performance of the original and the modified version of the algorithm is evaluated in parallel. Applying the modified version results in better objective function values and less dose discrepancies between optimized and accurate dose, and the advantages are pronounced with decreasing number of sweeps.
\smallskip
\noindent{\bf Keywords: VMAT, sliding window, convex optimization, heuristics}
\end{abstract}
\section{Introduction}
A clinical advantage of volumetric-modulated arc therapy (VMAT) over static-gantry delivery of radiotherapy, is the potential to obtain a shortened treatment time without compromising plan quality. In static delivery, the treatment is limited to a few angles around the patient, and the beam is turned off while the gantry moves to the next angle. VMAT delivery, on the other hand, allows the gantry to rotate during irradiation and multileaf collimation. Treatment time savings are achieved since the beam is never turned off.
From a treatment planning perspective, a delivery technique and its clinical advantages can be implemented first when there is a way to mathematically formulate and efficiently solve (at least approximately) the associated optimization problem, so that a planning tool can eventually be developed. VMAT is clinical routine since more than a decade thanks to dedicated research and development reflected in the many approaches to VMAT treatment planning proposed in the literature. As noticed by Peng et al.~\cite{peng2012}, the literature is mainly focused on algorithms of a heuristic nature---possibly with some local-optimizing steps given a restricted optimization formulation, but seldom related to a complete VMAT optimization formulation---due to the mathematical complexity associated with the continuously rotating gantry. Suggested approaches are often divided into two categories. In two-phase algorithms, idealized fluence profiles at certain gantry angles are first generated by solving a fluence map optimization (FMO) problem. Both coarse \cite{bzdusek2009} and dense angular discretizations \cite{craft2012a} have been used. The optimized fluence profiles are then transformed into leaf trajectories in an arc-sequencing phase, where all delivery constraints are taken into account. The objective of arc-sequencing varies: algorithms and/or formulations have been suggested that either amount to find the leaf trajectories that best reproduce the optimal fluence profile (see, e.g., \cite{shepard2007,craft2012a}) or that best replicate the delivered target dose (see, e.g., \cite{wang2008}). The other approach to VMAT optimization is to directly take the deliverability of the treatment plan into account by (approximately) solving a so-called direct machine parameter optimization (DMPO) problem. DMPO formulations of VMAT delivery are in general nonconvex and considered more complex than the static-gantry counterparts. On the other hand, as reported by Shepard et al.~\cite{shepard2002} and Rao et al.~\cite{rao2010}, targeting a DMPO formulation often results in improved plan quality for both static-gantry and VMAT delivery as compared to applying two-phase algorithms, and can be motivated in that aspect. Bzdusek et al.~\cite{bzdusek2009} describe an efficient method that combines a two-phase algorithm and a nonconvex DMPO formulation, the former giving the initial solution to a gradient-based method for solving the latter to local optimality. The method has been adopted by several commercial systems for VMAT treatment planning \cite{unkelbach2015}. Peng et al.~\cite{peng2012,peng2015} suggest a column-generation and a heuristic decomposition approach to handle a fully stated DMPO formulation. Classical heuristic methods have also been applied, including simulated annealing \cite{otto2007} and tabu search \cite{ulrich2007}. A comprehensive review of approaches to VMAT optimization is given by Unkelbach et al. \cite{unkelbach2015}.
In Papp and Unkelbach~\cite{papp2014}, an algorithm is presented to handle DMPO for VMAT delivery restricted to unidirectional leaf trajectories---``sliding windows'' or sweeps. By considering as variables the times of arrival and departure of the leaves at fixed positions (bixels) along the sweeping direction, the authors demonstrate that the set of sweeps can be expressed using linear inequalities, and that the resultant radiation fluence passing through the bixels is given by a linear, hence convex, function of the arrival and departure times. This opportunity does not occur for regular (with arbitrary leaf motions) VMAT delivery, which requires nonconvex formulations to exactly model the fluence profiles as a function of leaf positions \cite{unkelbach2015}. Unfortunately, nonconvexities eventually catch also sliding-window VMAT, since the computation of dose is a nonconvex operation due to the rotation of the gantry. The algorithm suggested by Papp and Unkelbach therefore amounts to solving a sequence of simplified DMPO formulations (subproblems) with approximate linear dose computations; accurate dose is computed first as a final step. In the present study, we express the accurate dose as an explicit function of the sliding window sweeps to obtain a full DMPO formulation. The purpose is to formalize sliding-window VMAT optimization for further development of algorithms. In particular, in light of the suggested exact formulation of sliding-window VMAT optimization, we interpret the Papp and Unkelbach algorithm as a heuristic method for solving this optimization problem and suggest a generalization of the subproblem update scheme.
Except for the linear representation of leaf trajectories and resultant fluence, another notable benefit of sliding-window over regular VMAT is the opportunity to create a new deliverable plan by convex combination of other sliding-window VMAT plans~\cite{craft2014}. This property is particularly interesting for multicriteria optimization, as it enables Pareto set navigation in the domain of deliverable plans. It also motivates further development of methods to handle the associated optimization problems.
A drawback with the sliding-window approach is that the many unidirectional sweeps back and forth over the fluence field increase the treatment time as compared to regular VMAT delivery, especially for cases with large targets and thus wide fields to traverse~\cite{unkelbach2015}. In the studies by Papp and Unkelbach~\cite{papp2014} and Craft et al.~\cite{craft2012a}, delivery times of 3-6 minutes are needed to achieve the desired plan quality, whereas 1-2 minutes are expected for regular VMAT~\cite{bzdusek2009}. The criticism regarding delivery time is justified since, again, a shortened treatment was one of the main arguments for choosing VMAT over static delivery in the first place. Therefore, in our numerical study, we investigate the effect on plan quality of decreasing the number of sweeps made on the 360-degree arc for a faster treatment. Besides explicitly limiting the treatment time during optimization, controlling the number of sweeps is a potential means to trade plan quality for a more efficient VMAT delivery, analogous to limiting the number of beams in static-gantry delivery. To generate plans, we suggest and apply a version of the generalization of the Papp and Unkelbach algorithm. Our suggested version is designed to better handle a setting with fewer sweeps delivered on relatively large portions (arc segments) of the 360-degree arc.
\section{Method}\label{sec:Methods}
\subsection{Optimization formulation}
The planning objectives and constraints used in this study follow the formulation suggested in our previous works~\cite{engberg2017,engberg2018}:
\begin{equation}\label{eq:propForm}
\begin{aligned}
& \minimize{d,\,\xi}
&& \mkern-10mu \mathmakebox[0pt][l]{\big[\,\xi_1, \cdots, \xi_q, -\xi_{q+1}, \cdots, -\xi_K\,\big]^T} \\
& \subject && D^+(d;v_k,s_k) \leq \xi_k \leq u_k, && \hat{l}_k \leq \xi_k, && k = 1,\ldots,q, \\
& && D^-(d;v_k,s_k) \geq \xi_k \geq l_k, && \hat{u}_k \geq \xi_k, && k = q\!+\!1,\ldots,K, \\
& && \mathmakebox[0pt][l]{d \text{ deliverable dose distribution,}}
\end{aligned}
\end{equation}
where $D^+(\cdot\,;v,s)$ and $D^-(\cdot\,;v,s)$ denote the upper and lower mean-tail-dose functions for volume fraction $v$ in structure $s$. The mean-tail-dose functions were introduced by Romeijn et al.~\cite{romeijn2006} and are in our research used as convex approximations of the dose-at-volume function frequently used to evaluate plan quality. All dose constraints of \eqref{eq:propForm} can be expressed by a set of linear inequalities (see Appendix A of our previous work~\cite{engberg2018} for a fully expanded formulation). Depending on the deliverability constraints, the optimization problem can therefore be a linear or convex program, or a general non-convex program.
\subsection{Accurate dose computation for sliding-window VMAT}
The computation of accurate dose is added as a final step in the sliding-window VMAT optimization algorithm by Papp and Unkelbach~\cite{papp2014}, but not stated as an explicit function. In this section, we formulate the accurate dose as a function of deliverable leaf trajectories, and show how this nonsmooth function can be incorporated into a mixed integer linear program (MILP) to form an exact formulation of sliding-window VMAT optimization.
Let $b$, $b=1,\ldots,B$, enumerate the arc segments of the 360-degree arc, within each one sweep is to be delivered. Let the sweeping trajectories of the leaves be represented as in \cite{papp2014} and denoted as in \cite{engberg2018}, i.e., by the points in time $r_{b,n,j}$ and $l_{b,n,j}$ when respectively the leading and the trailing leaf of leaf pair $n$, $n=1,\ldots,N$, when regarded in the $b$th arc segment, begin traversing bixel $j$, $j=1,\ldots,J$. Furthermore, let $k$, $k=1,\ldots,K$, enumerate the control points on the 360-degree arc, equiangularly distributed at angles $k\theta$ (typical control point spacings are $\theta = 2\degree$ or $4\degree$). Each control point angle $k\theta$ has an associated dose deposition matrix $P^{k\theta}$ which, in accordance with the final accurate dose computation in \cite{papp2014}, is assumed valid in the $\theta$-neighborhood $\left[(k-\frac{1}{2})\theta,(k+\frac{1}{2})\theta\right]$.
As the gantry rotates across the arc segments, a number of control points will be traversed. Let $K_b$, $K_b \subset \{1,\ldots,K\}$, denote the set that collects the consecutive control points passed by arc segment $b$. The rotating speed of the gantry is assumed constant over each arc segment, and is determined by the time when all leaves have finished traversing the field; let $t^{k\theta}$ denote the point in time when the gantry passes angle $k\theta$ (in turn given by the speed of the gantry). To formulate the dose distribution as a function of leaf trajectories, we first note that the exposure of bixel $(b,n,j)$ at control point $k \in K_b$, illustrated in the trajectory plot of Figure~\ref{fig:leaftrajectory}, equals the quantity
\[
\max\big\{\min\big(l_{b,n,j}+\frac{\Delta}{2},\,t^{(k+\frac{1}{2})\theta}\big) - \max\big(r_{b,n,j}+\frac{\Delta}{2},\,t^{(k-\frac{1}{2})\theta}\big),\, 0\, \big\},
\]
where $\Delta$ is the constant bixel traversing time (assuming, e.g., that the leaves are always travelling with maximum speed while in motion). In this expression, the inner $\min$ and $\max$ functions localize the beginning and end of the exposure, respectively, and the outer $\max$ function transforms any negative value to zero exposure.
\begin{figure}\centering
\includegraphics[scale=.75]{leaftrajectory_new.png}
\caption{A sweep example. The leaf trajectory is represented by the points in time $r_{b,n,j}$ and $l_{b,n,j}$ when respectively the leading and the trailing leaf begin traversing bixel $(b,n,j)$, with $\Delta$ the constant traversing time (see text for further definitions). The blue ribbons mark the exposure of bixel $(b,n,j)$ at control points $k-1$, $k$, and $k+1$.}
\label{fig:leaftrajectory}
\end{figure}
The dose in voxel $i$ is given by the sum of exposure contributions from all bixels, and can thus be written
\begin{multline}\label{eq:doseComp}
d_i = \delta \sum_{b,n,j} \sum_{k \in K_b} P^{k\theta}_{i,(b,n,j)}\, \max\big\{\min\big(l_{b,n,j}+\frac{\Delta}{2},\,t^{(k+\frac{1}{2})\theta}\big)\, - \\ - \max\big(r_{b,n,j}+\frac{\Delta}{2},\,t^{(k-\frac{1}{2})\theta}\big),\, 0\, \big\},
\end{multline}
where $\delta$ is the constant dose rate.\footnote{Note that $r_{b,n,j}$ and $l_{b,n,j}$ alternate between denoting the traversing time of the right and left leaf, depending on the sweeping direction (left-to-right or right-to-left) used in arc segment $b$.} Since involving the nonsmooth $\min$ and $\max$ functions, it is not clear how to handle the accurate dose computation in \eqref{eq:doseComp} in a smooth optimization setting. It is possible, however, to transform all dose constraints of \eqref{eq:propForm} into a MILP formulation by the introduction of artificial and binary variables; its derivation is delayed to Appendix~\ref{app:MILP} as no attempt is made in this study to solve the exact MILP formulation. The MILP formulation requires the introduction of six binary variables per bixel and control point, adding up to the order of $10^5$ binary variables. While the sequential nature of the unidirectional leaf sweeps likely allows the construction of several types of valid inequalities, which has not been investigated in this study, we envisage that applying standard methods to solve the MILP formulation to proven optimality would be too time-consuming for our application. It should also be noted that the MILP formulation requires the use of the ``big $M$'' method where a large-valued parameter $M$ is included, which is known to be prone to numerical instability in combination with standard methods.
In the following, for readability, we consider a renumbering of the bixels that enables enumeration by one index instead of three. We reuse $j$ as the new index since its association with a bixel has already been established (i.e., $j=1,\ldots,BNJ$ in the following). Furthermore, for ease of notation, we will utilize the fact that $j$ determines the arc segment membership, thus the set $K_b$, given the chosen renumbering mapping.
\subsection{Heuristic methods}
We refer to the optimization formulation \eqref{eq:propForm} with the accurate dose computation given in \eqref{eq:doseComp} as the exact problem.
Given the exact problem, we are in a position to interpret the algorithm by Papp and Unkelbach~\cite{papp2014} as a heuristic method to find an approximate solution. Its heuristic nature is due to the approximate dose computation used during optimization. In the algorithm, the dose deposition \emph{column} for bixel $j$ is assumed constant over the arc segment, hence does not change as the gantry passes new control points. The fixed column, here denoted by $P_j$, is chosen among all the available columns $P^{k\theta}_j, k \in K_b$. The effect is a linear but approximate dose computation,
\[
d_i = \delta \sum_j P_j (l_j-r_j).
\]
The algorithm amounts to solving the resulting smooth subproblem repeated times with updated choices of $P_j$ depending on the previous solution. More specifically, $P^{k\theta}_j$ of the control point whose $\theta$-neighborhood contains the midpoint of the exposure interval for bixel $j$ is chosen; $P^{k\theta}_j$ of the control point closest to the midpoint of the arc segment is chosen initially. The method terminates when the new choice is sufficiently similar to the previous one.
Now, in terms of the exact problem, the Papp and Unkelbach algorithm solves a sequence of relaxations of restrictions. To see this, we introduce variables $p_j^k$ to denote the \emph{fraction} of the total exposure of bixel $j$ occurring at control point $k \in K_b$,
\begin{equation}\label{eq:exposureFraction}
p_j^k = \max\big\{\min\big(l_j+\frac{\Delta}{2},\,t^{(k+\frac{1}{2})\theta}\big) - \max\big(r_j+\frac{\Delta}{2},\,t^{(k-\frac{1}{2})\theta}\big),\, 0\, \big\}\, \frac{1}{l_j-r_j},
\end{equation}
so that the exact dose can be written
\begin{equation}\label{eq:doseComp2}
d_i = \sum_j \Big( \sum_{k \in K_b} P^{k\theta}_{ji} p_j^k \Big)\, (l_j-r_j).
\end{equation}
By construction, we have that $\sum_{k \in K_b} p_j^k = 1$ and $p_j^k \in [0,1]$ which implies that the column $\sum_{k \in K_b} P^{k\theta}_j p_j^k$ is a convex combination of the columns $P^{k\theta}_j$. Constructing a subproblem of the algorithm is equivalent to treating the $p_j^k$'s as parameters and fixing them to binary values: for each bixel, $p_j^k = 1$ for the control point for which $P_j$ was chosen and $p_j^k = 0$ for the others. The interpretation of this maneuver is that the sweeps are assumed resulting in bixel exposures concentrated to one pre-determined control point. Besides the fixation of $p_j^k$, a proper restriction of the exact problem along this assumption needs the additional bounds
\begin{equation}\label{eq:boundsForRestriction}
r_j + \frac{\Delta}{2} \geq t^{(k_j-\frac{1}{2})\theta} \quad\text{and}\quad
l_j + \frac{\Delta}{2} \leq t^{(k_j+\frac{1}{2})\theta},
\end{equation}
denoting by $k_j$ the control point $k$ for which $p_j^k = 1$. However, the bounds are not included in the subproblems, which thus may be interpreted as relaxations of this restriction. It should be noted that the restricted problem is of little interest in practice, since the underlying assumption of concentrated bixel exposure is highly conservative.
In light of the exact problem, we are also in a position to make a generalization of the Papp and Unkelbach algorithm. A generalization is obtained by allowing any fractional values when updating the fixed $p_j^k$'s. As with binary values, fractional values of $p_j^k$ give a linear approximate dose computation favorable for optimization; but also have the ability to reflect bixel exposures that span several consecutive control points within the arc segment, thus have the potential to give a better approximation of accurate dose. A benefit of using a high-quality approximate dose computation during the optimization process is less deterioration of the optimal solution after accurate dose has been computed. While only minor such dose deviations are reported in the numerical study of the original paper~\cite{papp2014}, the explanation given is the observation that the exposure of most bixels of the 18-degree arc segments lasts no more than one control point. For longer arc segments (i.e., fewer sweeps) with more control points to pass, it is likely that a similar observation can no longer be made. We therefore suggest a modification of the Papp and Unkelbach algorithm, where the $p_j^k$ are assigned the exact exposure fraction in \eqref{eq:exposureFraction} obtained for the previous solution. The sought-after benefit is better handling of long arc segments.
A note regarding convergence is needed. The Papp and Unkelbach algorithm is terminated once the updated values of the $p_j^k$'s are sufficiently close (in a given metric) to the previous values. There is thus an expectation, supported by the numerical results of \cite{papp2014}, that the algorithm reaches a state with only slight changes in updates. Unfortunately from a mathematical perspective, convergence of the algorithm in this sense cannot be related to the globally optimal solution of the exact problem; nor can it be guaranteed that the algorithm produces a monotonically improving sequence of solutions. The convergence situation does not change with our suggested modification, i.e., with $p_j^k$ set to the exact exposure fraction.
\section{Results}\label{sec:Results}
The effects of decreasing the number of sweeps in order to obtain a faster treatment is studied for one prostate and one lung case. The optimized plan quality in terms of objective function value is evaluated, as well as the performance of the suggested fractional version of the generalization of the Papp and Unkelbach algorithm compared to the original binary version. For simplicity of notation, the two versions of the algorithm are henceforth referred to as the \emph{fractional} and the \emph{binary} version, respectively.
All dose deposition matrices and patient data is exported from RayStation (RaySearch Laboratories, Stockholm, Sweden) to MATLAB. We consider a scalarized weighted-sum instance of the multicriteria formulation in \eqref{eq:propForm} constructed by accumulating the objective functions using positive weighting factors. Combined with the linear sliding-window deliverability constraints and the linear approximate dose computation, the subproblems of the two algorithm versions become linear programs. A tailored interior-point method implemented in MATLAB that exploits the structure of these linear programs is used to solve the sequence of subproblems; we refer to our previous work~\cite{engberg2018} for a description of the interior-point method. After each subproblem solve, accurate dose is computed for performance analysis purposes.
Treatment plans of different treatment time restrictions and number of sweeps are generated using both the fractional and the binary version of the algorithm; delivery of 7, 11, and 20 sweeps in a maximum time of 240, 180, and 120 seconds are considered for both patient cases (120 seconds relaxed to 150 seconds for the 20-sweep plans due to infeasibility). A 4-degree control point spacing is used. The rotating speed of the gantry is limited to between 4.8 and 0.5 degrees per second, implying a minimum of 75 seconds to rotate through the 360-degree arc.
To simulate the termination criteria used in \cite{papp2014} for the binary version of the algorithm, we use a metric that accumulates the absolute differences in index $k$ between the previous and updated control point for which $p_j^k = 1$ (thus, a quantity proportional to the angular difference between control points). For the fractional version, we use a similar metric that accumulates the absolute differences in previous and updated $p_j^k$ for all $k$. Evaluations of these two metrics are presented in Figures~\ref{fig:stopCriteria}.
\begin{figure}
\centering
\begin{subfigure}[b]{.7\textwidth}
\flushleft\hskip7pt
\hskip6pt\includegraphics[scale=.59]{legend20.png}\hskip20pt
\hskip13pt\includegraphics[scale=.59]{legend11.png}\hskip20pt
\hskip13pt\includegraphics[scale=.59]{legend7.png}
\end{subfigure}
\vskip-.25\baselineskip
\begin{subfigure}[b]{.7\textwidth}
\flushleft\hskip7pt
\includegraphics[scale=.59]{legend240.png}\hskip20pt
\includegraphics[scale=.59]{legend180.png}\hskip20pt
\raisebox{-1.5pt}{\includegraphics[scale=.59]{legend120.png}}
\end{subfigure}
\vskip\baselineskip
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[scale=.75]{VPStop_Binary_new.png}
\end{subfigure}
\hskip3pt
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[scale=.75]{VPStop_Fractional_new.png}
\end{subfigure}
\vskip\baselineskip
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[scale=.75]{VLStop_Binary_new.png}
\end{subfigure}
\hskip3pt
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[scale=.75]{VLStop_Fractional_new.png}
\end{subfigure}
\caption{Evaluation of the metrics measuring the difference between previous and updated value of the $p_j^k$'s (see text for definitions) as a function of subproblems solved. Values obtained for the prostate (top) and lung (bottom) case when solved using the binary (red) and the fractional (blue) version of the algorithm. Note that the red and blue values are given by two different metrics and cannot be compared.}
\label{fig:stopCriteria}
\end{figure}
The two algorithm versions behave similarly: the metrics stagnate after a few iterations, which is in accordance with the observations in \cite{papp2014} where only two or three iterations were required. However, while lower values in the fractional-version metric is an indication of less deterioration in dose after accurate dose computation---a consequence of choosing the exact exposure fraction as fixed $p_j^k$---the same cannot be said about low values in the binary-version metric. An observation along these lines can be made in Figure~\ref{fig:discrepancy}, where the dose discrepancy between optimized and accurate dose is illustrated. The discrepancy obtained with the binary version is almost constant, whereas it decreases during the first few iterations of the fractional version of the algorithm according to a pattern similar to the termination metric in Figure~\ref{fig:stopCriteria}. The dependence of the discrepancy on the number of sweeps appears the strongest for the binary version, with the smallest discrepancy obtained for the plans with largest number of sweeps.
\begin{figure}
\centering
\begin{subfigure}[b]{.7\textwidth}
\flushleft\hskip7pt
\hskip6pt\includegraphics[scale=.59]{legend20.png}\hskip20pt
\hskip13pt\includegraphics[scale=.59]{legend11.png}\hskip20pt
\hskip13pt\includegraphics[scale=.59]{legend7.png}
\end{subfigure}
\vskip-.25\baselineskip
\begin{subfigure}[b]{.7\textwidth}
\flushleft\hskip7pt
\includegraphics[scale=.59]{legend240.png}\hskip20pt
\includegraphics[scale=.59]{legend180.png}\hskip20pt
\raisebox{-1.5pt}{\includegraphics[scale=.59]{legend120.png}}
\end{subfigure}
\vskip\baselineskip
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[scale=.75]{VPDiscr_Binary_new.png}
\end{subfigure}
\hskip3pt
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[scale=.75]{VPDiscr_Fractional_new.png}
\end{subfigure}
\vskip\baselineskip
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[scale=.75]{VLDiscr_Binary_new.png}
\end{subfigure}
\hskip3pt
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[scale=.75]{VLDiscr_Fractional_new.png}
\end{subfigure}
\caption{The (scaled) two-norm of the dose discrepancy between optimized and accurate dose as a function of subproblems solved. Results obtained for the prostate (top) and lung (bottom) case solved using the binary (red) and the fractional (blue) version of the algorithm.}
\label{fig:discrepancy}
\end{figure}
The plan quality in terms of objective function value evaluated for the accurate dose is presented in Figures~\ref{fig:objValSweeps} and~\ref{fig:objValTimes}.
\begin{figure}
\centering
\begin{subfigure}[b]{\textwidth}
\flushleft
\hskip6pt\includegraphics[scale=.59]{legend20.png}\hskip20pt
\hskip13pt\includegraphics[scale=.59]{legend11.png}\hskip20pt
\hskip13pt\includegraphics[scale=.59]{legend7.png}
\end{subfigure}
\vskip-.25\baselineskip
\begin{subfigure}[b]{\textwidth}
\flushleft
\includegraphics[scale=.59]{legend240.png}\hskip20pt
\includegraphics[scale=.59]{legend180.png}\hskip20pt
\raisebox{-1.5pt}{\includegraphics[scale=.59]{legend120.png}}
\end{subfigure}
\vskip.5\baselineskip
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[scale=.75]{VPObjVal_20_new.png}
\end{subfigure}
\hfill
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[scale=.75]{VPObjVal_11_new.png}
\end{subfigure}
\hfill
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[scale=.75]{VPObjVal_7_new.png}
\end{subfigure}
\vskip\baselineskip
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[scale=.75]{VLObjVal_20_new.png}
20 sweeps.
\end{subfigure}
\hfill
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[scale=.75]{VLObjVal_11_new.png}
11 sweeps.
\end{subfigure}
\hfill
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[scale=.75]{VLObjVal_7_new.png}\\
7 sweeps.
\end{subfigure}
\caption{Objective function values evaluated for the accurate dose as a function of subproblems solved. Results obtained for the prostate (top) and lung (bottom) case solved using the binary (red) and the fractional (blue) version of the algorithm; results are subdivided into plots by number of sweeps.}
\label{fig:objValSweeps}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}[b]{\textwidth}
\flushleft
\hskip6pt\includegraphics[scale=.59]{legend20.png}\hskip20pt
\hskip13pt\includegraphics[scale=.59]{legend11.png}\hskip20pt
\hskip13pt\includegraphics[scale=.59]{legend7.png}
\end{subfigure}
\vskip-.25\baselineskip
\begin{subfigure}[b]{\textwidth}
\flushleft
\includegraphics[scale=.59]{legend240.png}\hskip20pt
\includegraphics[scale=.59]{legend180.png}\hskip20pt
\raisebox{-1.5pt}{\includegraphics[scale=.59]{legend120.png}}
\end{subfigure}
\vskip.5\baselineskip
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[scale=.75]{VPObjVal_240_new.png}
\end{subfigure}
\hfill
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[scale=.75]{VPObjVal_180_new.png}
\end{subfigure}
\hfill
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[scale=.75]{VPObjVal_120_new.png}
\end{subfigure}
\vskip\baselineskip
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[scale=.75]{VLObjVal_240_new.png}
240 seconds.
\end{subfigure}
\hfill
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[scale=.75]{VLObjVal_180_new.png}
180 seconds.
\end{subfigure}
\hfill
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[scale=.75]{VLObjVal_120_new.png}
120 (150) seconds.
\end{subfigure}
\caption{Objective function values from Figure~\ref{fig:objValSweeps}, here subdivided into plots by treatment time. Results obtained for the prostate (top) and lung (bottom) case when solved using the fractional version of the algorithm.}
\label{fig:objValTimes}
\end{figure}
In Figure~\ref{fig:objValSweeps}, the plans are subdivided by number of sweeps. Comparing the two algorithm versions, a slight advantage in favor of the fractional version can be observed that, as expected, becomes more pronounced with fewer number of sweeps. In Figure~\ref{fig:objValTimes}, the values obtained when using the fractional version are rearranged, subdivided by treatment time. The effect of tighter time restrictions is the clearest in the prostate case where, e.g., the objective function values of the 20-sweep plan increase to eventually make this plan a less favorable alternative than both the 11- and 7-sweep plans. For instance, in this particular case, the 7-sweep plan is the best choice in terms of objective function value if a 120-second delivery is required. The variation in treatment time has less influence in the lung case. For such situations, the question of number of sweeps becomes more discrete: a decreasing treatment time will eventually become too tight for delivery of a large number of sweeps. For instance, as already mentioned, the 120-second restriction was relaxed to 150 seconds for all 20-sweep plans in order to fulfil the dose and delivery constraints. The results for the lung case still indicate that, e.g., delivering the 11-sweep plan in 120 seconds is nearly as good an option as delivering the 20-sweep plan in 150 seconds, with respect to objective function value.
\section{Discussion}\label{sec:Discussion}
A drawback with many heuristic methods is the lack of information about the distance to the global optimum. While the observations made in the numerical study indicate less deterioration in dose and better objective function values when applying our suggested fractional version of the Papp and Unkelbach algorithm~\cite{papp2014}, the heuristic nature of the algorithm makes it difficult to evaluate the improvement in proportion to the globally optimal plan. On the other hand, choosing the fractional version over the original binary version is not associated with any costs; their computational complexity, for instance, is identical. The decision to use the fractional version should therefore be uncontroversial and, as suggested by the results, the better alternative.
In comparing plans delivered with different numbers of sweeps, we have chosen to report only objective function values. It should be mentioned that also the feasibility with respect to dose constraints has been evaluated, and that the fractional version of the algorithm resulted in plans of higher accuracy in this sense. However, a more clinical evaluation of the effects of fewer sweeps, e.g., using dose-volume histograms (DVHs), is not included in this study (though in our previous studies, we have observed good correlation with plan quality measures of the objective functions of \eqref{eq:propForm} \cite{engberg2017,engberg2018}). While DVHs give a more detailed view of the entire dose distribution, focus is easily placed on DVH features that are not controlled by any objectives or constraints.
An arc-sequencing method has been suggested by Craft et al.~\cite{craft2012a} that successively decreases the number of sweeps, thus improves the delivery efficiency, until the fluence maps are no longer reproduced with sufficient precision. A drawback with methods that are focused on reproducing fluence maps is that, for longer arc segments, even perfectly reproduced fluence maps may give large differences in dose due to the larger variations in the dose deposition matrices. Similar to the Papp and Unkelbach algorithm, such methods thus rely on the final plan having relatively short arc segments and a large number of sweeps. In the present study, we have generated plans with only 11 and 7 arc segments. Results from the two patient cases indicate that even the 7-sweep plans could show the better objective function values, in case of a treatment time restriction approaching the typical delivery time of a regular VMAT plan (note that comparison has not been made to the plan quality of regular VMAT; however, recall that, e.g., regular VMAT is not as compatible with multicriteria optimization). Development of a method to, as in \cite{craft2012a}, dynamically determine the optimal---with respect to objective function value---number of sweeps given a certain delivery time restriction was beyond the scope of this study but is a possible direction of further research.
\section{Conclusion}
We have given an exact formulation of direct machine parameter optimization of sliding-window VMAT, by expressing the accurate dose as an explicit function of the sweeping leaf trajectories while taking into account the rotation of the gantry. The exact formulation is a nonsmooth optimization problem, and while to directly solve this formulation is not considered in this study, it has enabled us to generalize an algorithm previously suggested in the literature for generation of sliding-window VMAT plans.
In the numerical study, plans have been generated with as few as 11 and 7 sliding-window sweeps, each delivered on a relatively large arc segment of the 360-degree arc. The purpose was to study the effects on the plan quality of a tight time restriction approaching the delivery time of regular (arbitrary leaf motion) VMAT. The results from the two patient cases show that, if requiring such an efficient sliding-window delivery, the few-sweep plans could give the better objective function values when compared to 20-sweep plans. While the plan quality naturally is not comparable to that obtained for 20-sweep plans with a generous time restriction, the few-sweep plans could be regarded as fast-delivery alternatives to static-gantry treatment plans for which 7-11 beam angles can be considered. The results furthermore show that our suggested version of the generalized algorithm performs better than the original algorithm in terms of better objective function value and less dose deterioration after accurate dose computation. These results are particularly pronounced for the plans with large arc segments.
\section*{Acknowledgement}
The authors thank Kjell Eriksson for valuable discussions.
\bibliography{vmat}
\bibliographystyle{myplain}
\appendix
\section{A MILP formulation of dose constraints}\label{app:MILP}
\allowdisplaybreaks
Every dose constraint of \eqref{eq:propForm} can be generalized into either an upper or lower bound on the voxel dose $d_i$. More precisely, upper bounds are obtained for the maximum dose and upper mean-tail-dose objectives/constraints, whereas the minimum dose and lower mean-tail-dose objectives/constraints imply lower bounds. To demonstrate the transformation into MILP constraints, it thus suffices to consider the two cases $d_i \leq \xi$ and $d_i \geq \xi$. For the upper-bound case, by the introduction of binary variables, we first obtain a nonlinear integer formulation:
\begin{align*}
d_i \leq \xi \\
\Leftrightarrow\enskip & \delta \sum_j \sum_{k \in K_b} P^{k\theta}_{ji}\, \max\big\{\min\big(l_j+\frac{\Delta}{2},\,t^{(k+\frac{1}{2})\theta}\big)\, - \\[-9pt]
& \qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad - \max\big(r_j+\frac{\Delta}{2},\,t^{(k-\frac{1}{2})\theta}\big),\, 0\, \big\} \leq \xi \\
\Leftrightarrow\enskip & \delta \sum_j \sum_{k \in K_b} P^{k\theta}_{ji}\, \gamma_j^k \leq \xi, \\
& \qquad\gamma_j^k \geq \min\big(l_j+\frac{\Delta}{2},\,t^{(k+\frac{1}{2})\theta}\big) - \max\big(r_j+\frac{\Delta}{2},\,t^{(k-\frac{1}{2})\theta}\big) \\
& \qquad\gamma_j^k \geq 0 \\[5pt]
\Leftrightarrow\enskip & \delta \sum_j \sum_{k \in K_b} P^{k\theta}_{ji}\, \gamma_j^k \leq \xi, \\
& \qquad \gamma_j^k \geq b_1^{jk}\,(l_j-r_j), \\
& \qquad \gamma_j^k \geq b_2^{jk}\,(l_j+\frac{\Delta}{2}-t^{(k-\frac{1}{2})\theta}), \\
& \qquad \gamma_j^k \geq b_3^{jk}\,(t^{(k+\frac{1}{2})\theta}-r_j-\frac{\Delta}{2}), \\
& \qquad \gamma_j^k \geq b_4^{jk}\,(t^{(k+\frac{1}{2})\theta}-t^{(k-\frac{1}{2})\theta}), \\
& \qquad (\gamma_j^k \geq 0 \text{ implicit}) \\[5pt]
& \qquad b_1^{jk}+b_2^{jk}+b_3^{jk}+b_4^{jk} = 1,\quad b_1^{jk},b_2^{jk},b_3^{jk},b_4^{jk} \;\text{ binary.}
\end{align*}
An equivalent MILP formulation can then be constructed by using the ``big $M$'' method, with which a nonlinear integer constraint such as
\[
\gamma_j^k \geq b_1^{jk}\,(l_j-r_j)
\]
is transformed using a sufficiently large $M$ into the linear integer constraint
\[
\gamma_j^k \geq l_j-r_j - M(1-b_1^{jk}).
\]
MILP formulations for the remaining three integer constraints are analogously constructed. For the lower-bound case, a ``big $M$'' MILP formulation is obtained directly:
\begin{align*}
d_i \geq \xi \\
\Leftrightarrow\enskip & \delta \sum_j \sum_k P^{k\theta}_{ji}\, \max\big\{\min\big(l_j+\frac{\Delta}{2},\,t^{(k+\frac{1}{2})\theta}\big)\, - \\[-9pt]
& \qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad - \max\big(r_j+\frac{\Delta}{2},\,t^{(k-\frac{1}{2})\theta}\big),\, 0\, \big\} \geq \xi \\
\Leftrightarrow\enskip & \delta \sum_j \sum_k P^{k\theta}_{ji}\, \lambda_j^k \geq \xi, \\
& \qquad\lambda_j^k \leq \max\big\{\min\big(l_j+\frac{\Delta}{2},\,t^{(k+\frac{1}{2})\theta}\big) - \max\big(r_j+\frac{\Delta}{2},\,t^{(k-\frac{1}{2})\theta}\big),\, 0\, \big\} \\[5pt]
\Leftrightarrow\enskip & \delta \sum_j \sum_k P^{k\theta}_{ji}\, \lambda_j^k \geq \xi, \\
& \qquad \lambda_j^k \leq l_j-r_j + M b_5^{jk}, \\
& \qquad \lambda_j^k \leq l_j+\frac{\Delta}{2}-t^{(k-\frac{1}{2})\theta} + M b_5^{jk}, \\
& \qquad \lambda_j^k \leq t^{(k+\frac{1}{2})\theta}-r_j-\frac{\Delta}{2} + M b_5^{jk}, \\
& \qquad \lambda_j^k \leq t^{(k+\frac{1}{2})\theta}-t^{(k-\frac{1}{2})\theta} + M b_5^{jk}, \qquad \lambda_j^k \leq M b_6^{jk}, \\[5pt]
& \qquad b_5^{jk}+b_6^{jk} = 1,\quad b_5^{jk},b_6^{jk} \;\text{ binary.}
\end{align*}
\end{document} | {"config": "arxiv", "file": "1810.08610/vmat.tex"} |
TITLE: Is this topology a subset of the Euclidean topology?
QUESTION [0 upvotes]: On $\Bbb R^2$, let $\tau$ be the collection of subsets which contain an open line segment in each direction about each if its points. We claim that $\tau$ form a topology on $\Bbb R^2$. Clearly, $\Bbb R^2$ and $\emptyset$ are in $\tau$. For any $A,B\in \tau$, suppose $x\in A\cap B$ and $m\in\Bbb R\cup \{\infty\}$. Since $A,B\in \tau$, there exist some open line segments $l_A$, $l_B$ of slope $m$ which contain $x$ in $A$ and $B$, respectively. Since $l_A\cap l_B$ is an open line segment of slope $m$ which contains $x$ in $A\cap B$, $\tau$ is closed under finite intersections. Also, $\tau$ is closed under arbitrary unions.
Clearly, The Euclidean topology is a subset of $\tau$.
Question. Is it true that $\tau$ is a subset of the Euclidean topology? Thanks.
REPLY [0 votes]: We show that set $U:=(\mathbb{R}^2 \setminus S^1) \cup \{ (1, 0) \}$ belongs to $\tau$. Let $x\in U$. Then either $x\in \Bbb R^2\setminus S^1$ or $x\in \{(1,0)\}$. Suppose $x\in \Bbb R^2\setminus S^1$. Since $R^2\setminus S^1 \in \tau_{Euclid}$, there is some $\epsilon>0$ such that the open ball $B(x,\epsilon)$ is a subset of $R^2\setminus S^1$. But $B(x,\epsilon)\in \tau$ and $B(x,\epsilon)\subseteq U$. Now, suppose $x\in\{(1,0)\}$ and $l$ be any line passes through the point $x$ in $\Bbb R^2$. If $l$ does not intersect any point in $S^1$ other than the point $(1,0)$, we are done. But if it is so, say it $p$, then we can pick $\epsilon>0$ less than the distance between $p$ and $x$. And the open line segment $B(x,\epsilon)\cap l$ is a subset of $U$. | {"set_name": "stack_exchange", "score": 0, "question_id": 2908149} |
TITLE: Confusion over Partitions in Set Theory
QUESTION [2 upvotes]: Suppose that $n\in \omega$, $m$ is a cardinal and $\kappa, \lambda$ are infinite cardinals. Suppose that any onto function $F:[\kappa]^n \to m$ has the property that there exists a set $H \subseteq \kappa$ with $|H| = \lambda$, such that $F$ is constant on $[H]^n$.
How do I show that if the above is true, then the same is true if I replace $m$ by a smaller cardinal $m'$? I attempted to prove this by taking $F:[\kappa]^n \to m'$ and simply extending the range to $m$. F then becomes a function into $m$ which takes no values in $m \setminus m'$. Then there exists a set $H$ as above and $F$ is constant on $[H]^n$. Then I restricted the range of $F$ back to $m'$.
However this does not work because obviously $F:[\kappa]^n \to m$ is not onto. Does anyone have any ideas as to how to fix this? Thanks very much in advance.
(By $[S]^n$ I mean subsets of size $n$ of $S$).
REPLY [3 votes]: The idea of the argument is very natural, but the details are slightly more annoying than one would have anticipated.
First of all, I am going to assume that $\kappa\ge m$. Otherwise, the given assumption is vacuously true, as no function $F:[\kappa]^n\to m$ is onto. However, in this case the result you want to prove is false, as we can take $m'=\kappa<m$ and let $F:[\kappa]^n\to\kappa$ be a bijection, contradicting the existence of any homogeneous set of size larger than $n$.
So, assume $\kappa\ge m$. (Then actually $\kappa>m$, or else $\kappa=m$ and a bijection as above contradicts the given assumption.)
Ok, we are ready to start the proof. Suppose $F:[\kappa]^n\to m'$ is onto.
Redefine $F(a)$ for a few values $a\in[\kappa]^n$ so the new function is onto. To be specific: For some $i<m'$ there are more than $m$ many $a\in[\kappa]^n$ with $F(a)=i$. This is because the $\kappa$-sized set $[\kappa]^n$ partitions into $m'$ many sets according to the values assigned by $F$. But $m'<m<\kappa$. Now, if
$$A_i=\{a\in[\kappa]^n\mid F(a)=i\}$$
has size $\le m$ for all $i$, then $[\kappa]^n$ would have size at most $m'\times m<\kappa$, a contradiction.
Fix some such $i$, and pick distinct sets $a_j\in A_i$ for ordinals $j\in m\setminus m'$. Note that $A_i\setminus\{a_j\mid j\in m\setminus m'\}$ is nonempty because $|A_i|>m\ge|m\setminus m'|$.
Define $F':[\kappa]^n\to m$ as follows: $F'(a)=F(a)$ unless $a=a_j\in A_i$, in which case $F'(a)=j$.
Note that $F'$ is onto $m$, because if $F(a)\ne i$ then $F'(a)=F(a)$, and there are are still values of $a$ such that $F'(a)=i$, since we only redefined $F$ at $|m\setminus m'|<|A_i|$ many places. So at least $F'$ is onto $m'$. But the $a_j$ ensure that $F'$ also takes all values in $m\setminus m'$. So it is onto $m$.
By assumption, this new function $F'$ admits a homogeneous $H$ of size $\lambda$. I claim that $H$ is also homogeneous for $F$. This will complete the proof.
The point is that if $a_j\subset H$ for some $j$ with $m'\le j<m$, then by homogeneity, $F'(a)=j$ for any $a\in[H]^n$. But the only $a\in[\kappa]^n$ with $F'(a)=j$ is $a_j$, by construction. So $|H|= n<\omega\le\lambda=|H|$, a contradiction. This means that if $a\in[H]^n$, then $F'(a)=F(a)$, so $H$ is homogeneous for $F$, as claimed. | {"set_name": "stack_exchange", "score": 2, "question_id": 120763} |
\begin{document}
\maketitle
\begin{abstract}
There is a natural connection between the class of
diffusions, and a certain class of solutions to the Skorokhod
Embedding Problem (SEP). We show that the important concept
of minimality in the SEP leads to the new and useful concept of a {\it
minimal diffusion}. Minimality is closely related to the martingale
property. A diffusion is minimal if it minimises the expected local time at
every point among all diffusions with a given distribution at an
exponential time. Our approach makes explicit the connection between
the boundary behaviour, the martingale property and the local time
characteristics of time-homogeneous diffusions.
\end{abstract}
\section{Introduction}
This article shows that there is a one-to-one correspondence between a
(generalised) diffusion in natural scale and a class of solutions to
the Skorokhod Embedding Problem (SEP) for a Brownian motion. The main
contribution of this article is to establish the concept of a {\it
minimal diffusion}, which is motivated by the correspondence.
The link between embedding and diffusion involves an additive
functional which can be used to define both a diffusion (via a time
change) and a stopping time (the first time that the additive
functional exceeds an independent exponentially distributed random
variable). The fundamental connection was first observed by Cox et.\
al.\ in \cite{CoxHobsonObloj:2011}. In that article, the authors
proved existence of a (martingale) diffusion with a given law at an
exponentially distributed time by exploiting the link between
martingale diffusions and minimal embeddings. It was then demonstrated
in \cite{MK:12} that a diffusion's speed measure can be represented in
terms of its exponential time law without recourse to embedding
theory.
The general theme of this paper is that we can exploit the
correspondence between diffusions and the SEP to introduce notions
from the theory of SEP into the diffusion setting. This leads us to
introduce the novel concept of a {\it minimal diffusion}, which we
argue is the canonical diffusion in the family of diffusions with
given law at a random time, and corresponds to the minimal embedding.
In the literature on the SEP, the concept of a minimal stopping time
is crucial. The problem of constructing stopping times so that a
stopped process has a given law degenerates unless there is a notion
of what constitutes a `good' solution. An early definition of `good'
was `finite expectation'. The concept of a `minimal solution' was
first introduced by Monroe, \cite{Monroe:72}. Loosely speaking, a
stopping time is minimal if there is no almost surely smaller stopping
time for the process which results in the same distribution. The
property of minimality connects much of the modern literature on
Skorokhod embeddings and their applications in mathematical
finance. The main contribution of this article is to show that the
concept of minimality extends naturally as a property of
diffusions. In leading up to this result, we provide a novel
characterisation of minimal stopping times in terms of the local
times, which is important in transferring the concept of minimality
from solutions of the SEP to diffusions.
From the point of view of classical diffusion theory, the article provides
insight into the additive functional that defines a diffusion and introduces
the intuitive concept of `minimality' to diffusion theory. We argue that
the property of minimality is in many ways more natural than the
martingale property. The concept of minimality dis-entangles
distributional properties of the process from its boundary behaviour. There
may not exist a martingale diffusion with a given exponential time law, but
there is always a minimal diffusion.
Our study of this problem was motivated in part by the observation
that the construction of time-homogenous diffusions via time-change
was closely related to the solution to the SEP given by Bertoin \& Le
Jan \cite{BLJ}. Indeed, one of the proofs in
\cite{CoxHobsonObloj:2011} relied on explicitly constructing a process
which enabled the use of the results of \cite{BLJ}. The Bertoin Le-Jan
(BLJ) embedding stops a process at the first time an additive
functional grows larger than the accumulated local time at the
starting point. However many of the same quantities appear in the
construction of both the BLJ embedding and the time-homogenous
diffusion. By considering the BLJ embedding, we are able to explain
why this is the case, and thereby provide new insight into the BLJ
construction, and its connection to time-homogenous diffusions.
\section{Preliminaries}
\subsection{Generalised Diffusions}
Let $m$ be a non-negative, non-zero Borel measure on an interval $I \subseteq \R$, with left endpoint $a$ and right endpoint $b$ (either or both of which may be infinite). Let $x_0 \in (a,b)$ and let $B=(B_t)_{t \geq 0}$ be a Brownian motion started at $B_0=x_0$ supported on a filtration $\F^B=({\mathcal F}_u^B)_{u\geq 0}$ with local time process $\{ L_u^x ; u \geq 0, x \in \R \}$. Define $\Gamma$ to be the continuous, increasing, additive functional
\[\Gamma_u = \int_{\R} L_u^x m(dx),\]
and define its right-continuous inverse by
\[A_t = \inf \{u \ge 0 : \Gamma_u > t \}. \] If $X_t = B_{A_t}$ then $X=(X_t)_{t \geq 0}$ is a one-dimensional generalised diffusion in natural scale with $X_0=x_0$ and speed measure $m$. Moreover, $X_t \in I$ almost surely for all $t \geq 0$. In contrast to diffusions, which are continuous by definition, generalised diffusions may be discontinuous if the speed measure places no mass on an interval. For instance, if the speed measure is purely atomic, then the process is a birth-death process in the sense of Feller \cite{feller}. See also Kotani and Watanabe \cite{kotaniwatanabe}. In the sequel, we will use diffusion to denote the class of generalised diffusions, rather than continuous diffusions.
Let $H_x=\inf\{u:X_u=x\}$. Then for $\lambda>0$ (see e.g. \cite{Salminen}),
\begin{equation} \label{eq:eigenfunction}
\E_{x}[e^{-\lambda H_y}]= \left\{\begin{array}{ll}
\frac{\varphi_\lambda(x)}{\varphi_\lambda(y)} &\; x \leq y \\
\frac{\phi_\lambda(x)}{\phi_\lambda(y)} &\; x \geq y ,
\end{array}\right.
\end{equation}
where $\varphi_\lambda$ and $\phi_\lambda$ are respectively a strictly increasing and a strictly decreasing solution to the differential equation
\begin{equation} \label{eq:differentialsc}
\frac{1}{2} \frac{d^2}{dm dx} f = \lambda f.
\end{equation}
The two solutions are linearly independent with Wronskian
$W_\lambda=\varphi_\lambda' \phi_\lambda-\phi_\lambda' \varphi_\lambda$, which is a positive constant.
The solutions to (\ref{eq:differentialsc}) are called the $\lambda$-eigenfunctions of the diffusion. We will scale the $\lambda$-eigenfunctions so that $\varphi_\lambda(x_0)=\phi_\lambda(x_0)=1$.
\subsection{The Skorokhod Embedding Problem}
We recall some important notions relating to the Skorokhod Embedding Problem (SEP). The SEP can be stated as follows: given a Brownian motion $(B_t)_{t \ge 0}$ (or, more generally, some stochastic process) and a measure $\mu$ on $\R$, a solution to the SEP is a stopping time $\tau$ such that $B_{\tau} \sim \mu$. We refer to Ob\l\'oj \cite{Obloj:04} for a comprehensive survey of the history of the Skorokhod Embedding Problem.
Many solutions to the problem are known, and it is common to require some additional assumption on the process: for example that the stopped process $(B_{t \wedge \tau})_{t \ge 0}$ is uniformly integrable. In the case where $B_0 = x_0$, this requires some additional regularity on $\mu$ --- specifically that $\mu$ is integrable, and $x_0 = \bar{x}_\mu = \int y \, \mu(dy)$. We recall a more general notion due to Monroe \cite{Monroe:72}:
\begin{definition} \label{def:minimal}
A stopping time $\tau$ is {\it minimal} if, whenever $\sigma$ is
another stopping time with $B_{\sigma} \sim B_{\tau}$ then $\sigma
\le \tau \ \Prob$-a.s.{} implies $\sigma = \tau \ \Prob$-a.s..
\end{definition}
That is, a stopping time $\tau$ is minimal if there is no strictly
smaller stopping time which embeds the same distribution. It was
additionally shown by Monroe that if the necessary condition described
above was true ($\mu$ integrable with mean $x_0$) then minimality of
the embedding $\tau$ is equivalent to uniform integrability of the
stopped process. In the case where the means do not agree, minimality
of stopping times was investigated in Cox and Hobson
\cite{CoxHobson:06} and Cox \cite{Cox:08}. Most natural solutions to
the Skorokhod Embedding problem can be shown to be minimal. In some
cases, constructions can be extended to non-minimal stopping times ---
see for example Remark~4.5 in \cite{cox_roots_2013}.
To motivate some of our later results, we give the following alternative characterisation of minimality for stopping times of Brownian motion. We note that this condition was first introduced by Bertoin and Le Jan \cite{BLJ}, as a property of the BLJ embedding. We write $L_t^a$ for the local time at the level $a$. When $a=0$, we will often simply write $L_t$.
\begin{lemma} \label{lem:minimal}
Let $\mu$ be an integrable measure, and suppose $\tau$ embeds $\mu$ in a Brownian motion $(B_{t})_{t \ge 0}$ with $B_0 = x_0$. Let $a \in \R$ be fixed. Then $\tau$ is minimal if and only if $\tau$ minimises $\E[L_{\sigma}^a]$ over all stopping times $\sigma$ embedding $\mu$.
\end{lemma}
\begin{proof}
Without loss of generality, we may assume $a=0$. Let $\tau$ be an
embedding of $\mu$. Note that $L_t-|B_t|$ is a local martingale. Let
$\tau_N$ be a localising sequence of stopping times such that $\tau_N \uparrow \tau$ and $\{L_{t\wedge \tau_N}-|B_{t\wedge \tau_N}|\}$ is a family of martingales. We have
\begin{equation*}
\E[L_{\tau_N}] = -|x_0| + \E[|B_{\tau_N}|]
\end{equation*}
and hence
\begin{equation*}
\lim_{N \to \infty}\E[L_{\tau_N}] = -|x_0| + \lim_{N \to \infty}\E[|B_{\tau_N}|].
\end{equation*}
Since $|x| = x + 2 x_-$ (where $x_- = \max\{0,-x\}$) we can write
\begin{equation*}
\lim_{N \to \infty}\E[|B_{\tau_N}|] = \lim_{N \to \infty} \E
[B_{\tau_N}] + 2\lim_{N \to \infty} \E[ ( B_{\tau_N})_-].
\end{equation*}
Since $\tau_N$ is a localising sequence, $\lim_{N \to \infty} \E
[B_{\tau_N}] = x_0$, while, by Fatou's Lemma, $\lim_{N \to \infty}
\E [( B_{\tau_N})_-] \ge \E [(B_{\tau})_-]$, and the final term depends
only on $\mu$. So $\E [ L_\tau] \ge x_0-|x_0| + 2 \E [(B_{\tau})_-]$.
This is true for any embedding $\tau$, however in the case where
$\tau$ is minimal, we observe that $\left\{(B_{\tau_N})_-\right\}_{N
\in \N}$ is a UI family, by Theorem~5 of \cite{CoxHobson:06}, and
so we have the equality: $\lim_{N \to \infty} \E [( B_{\tau_N})_-] =
\E [(B_{\tau})_-]$, and hence $\E [ L_\tau] = x_0-|x_0| + 2 \E [(B_{\tau})_-]$.
For the converse, again using Theorem~3 of \cite{CoxHobson:06}, we
observe that it is sufficient to show that $\left\{(B_{t \wedge
\tau})_-\right\}_{t \ge 0}$ is a UI family.
Note that for any integrable measure $\mu$ a minimal embedding
exists, and therefore any embedding $\tau$ which minimises
$\E[L_{\tau}]$ over the class of embeddings must have $\lim_{t \to
\infty} \E[( B_{\tau \wedge t})_-] = \E
[(B_{\tau})_-]$. However, suppose for a contradiction that
$\left\{(B_{t \wedge \tau})_-\right\}_{t \ge 0}$ is not a UI
family. Since $(B_{t \wedge \tau})_- \to (B_{\tau})_-$ in
probability as $t \to \infty$, this implies we cannot have
convergence in $\mathcal{L}^1$ (otherwise the sequence would be
UI). It follows that
\begin{equation*}
\E \left[ | (B_{t \wedge \tau})_- - (B_{\tau})_-|\right] \to \varepsilon >0,
\end{equation*}
as $t \to \infty$. But
\begin{equation*}
\E \left[| (B_{t \wedge \tau})_- - (B_{\tau})_-| \right]\le \E \left[\left|\left(
(B_{t \wedge \tau})_- - (B_{\tau})_-\right) \Indi_{\{\tau
\le t\}}\right|\right] + \E [ (B_\tau)_- \Indi_{\{\tau
> t\}}] + \E [ (B_t)_- \Indi_{\{\tau
> t\}}].
\end{equation*}
It follows that $\lim_{t \to \infty}\E[(B_{t})_- \Indi_{\{\tau >
t\}}] \ge \varepsilon$. But $\E[(B_{t \wedge \tau})_-
\Indi_{\{\tau \le t\}}] \to \E[(B_{\tau})_-]$, and hence
$\lim_{t \to \infty} \E[( B_{\tau \wedge t})_-] > \E
[(B_{\tau})_-]$.
\end{proof}
\section{Diffusions with a given law at an exponential time}
Let us begin by recalling the construction of a diffusion's speed measure in terms of its exponential time law in \cite{MK:12}.
Given an integrable probability measure $\mu$ on $I$, let
$U^\mu(x)=\int_{I} |x-y| \mu(dy)$, $C^\mu(x)=\int_{I} (y-x)^+ \mu(dy)$
and $P^\mu(x)=\int_{I}(x-y)^+ \mu(dy)$. Let $T_{\lambda}$ be an
exponentially distributed random variable, independent of $B$ with
mean $1/\lambda$. The following theorem summarises the main results in
\cite{MK:12}.
\begin{theorem} \label{t:main}
Let $X=(X_t)_{t \geq 0}$ be a diffusion in natural scale with Wronskian $W_\lambda$. Then $X_{T_{\lambda}} \sim \mu$ if and only if the speed measure of $X$ satisfies
\begin{equation} \label{eq:speeddecomp}
m(dx)= \left\{\begin{array}{ll}
\frac{1}{2\lambda} \frac{\mu(dx)}{P^\mu(x)-P^\mu(x_0)+1/W_\lambda}, &\; a < x \leq x_0 \\
\frac{1}{2\lambda} \frac{\mu(dx)}{C^\mu(x)-C^\mu(x_0)+1/W_\lambda}, &\; x_0 \leq x < b.
\end{array}\right.
\end{equation}
\end{theorem}
Since $m$ is positive, $1/W_\lambda \ge \max\{C^\mu(x_0),P^\mu(x_0)\}$. Note that $C^\mu(x_0) \geq (\leq) \ P^\mu(x_0)$ if $x_0 \leq \ (\geq) \ \bar{x}_\mu$.
\begin{comment}
\begin{example} \label{ex:WF}
Consider the neutral Wright-Fisher diffusion on $(0,1)$ started at $1/2$,
\[dX_t=\sqrt{X_t (1-X_t)} dB_t, \ \ \ X_0=1/2,\] and consider its distribution at
an exponential time of parameter $1$. The 1-eigenfunctions are given by
\[\varphi(x)=\psi(1-x)=\frac{2^{\frac{1+i \sqrt{7}}{2}} \ _2F_1\left[\frac{1-i \sqrt{7}}{2},
\frac{1+i \sqrt{7}}{2}, 2, x\right]}{_2F_1\left[ \frac{1-i \sqrt{7}}{2},
\frac{3-i \sqrt{7}}{2}, 2, -1\right]},\]
where $_2F_1$ is the generalised hypergeometric function defined \[_2F_1(a,b,c,z)=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)} \int_0^1 \frac{t^{b-1}(1-t)^{c-b-1}}{(1-tz)^a}dt.\]
We find that $\lim_{x \downarrow 0} \varphi(x) = 0$, and
$\lim_{x \downarrow 0} \varphi'(x) \approx 1.1407$. We calculate
$W_1 \approx 6.60902$. We can calculate the probability that $X$ reaches
the endpoints by time $T$, which is just the probability of observing the process at an endpoint at time $T$, $\Prob(X_T = 0)=\Prob(X_T=1)=\frac{\varphi''(0+)}{W_1} \approx \frac{2.28139}{6.60902} =0.345193$. Note that the symmetry of the process allows us to verify these calculations, since $1/2=\mu([0,1/2))=\frac{\varphi'(1/2)}{W_1}$, which gives again $W_1 \approx 6.60902$.
\end{example}
\end{comment}
The decomposition of the speed measure in Theorem \ref{t:main} is
essentially related to the $\lambda$-potential of a diffusion. Recall
that for a diffusion $X$, the $\lambda$-potential (also known as the
resolvent density) of $X$ is defined as $u_\lambda(x,y)=\E_x
\left[\int_0^\infty e^{- \lambda t} dL_{A_t}^y(t)\right]$. This has
the natural interpretation that $\lambda
u_\lambda(x_0,y)=\E_{x_0}[L_{A_{T_{\lambda}}}^y]$ is the expected
local time of $X$ at $y$ up until the exponentially distributed time
$T_{\lambda}$.
\begin{corollary} \label{c:lpotential}
The $\lambda$-potential of a diffusion $X$ with $X_{T_{\lambda}} \sim \mu$ satisfies
\[
u_\lambda(x_0,y)= \left\{\begin{array}{ll}
2(P^\mu(y)-P^\mu(x_0))+2/W_\lambda, &\; a < y \leq x_0 \\
2(C^\mu(y)-C^\mu(x_0))+2/W_\lambda, &\; x_0 \leq y < b.
\end{array}\right.
\]
Moreover, $\varphi(x) = u_{\lambda}(x_0,x)$ for $x \le x_0$ and $\phi(x) = u_{\lambda}(x_0,x)$ for $x \ge x_0$.
\end{corollary}
\begin{proof}
It follows from the calculations in \cite{MK:12} (p.~4) that
\[
\E_y[e^{-\lambda H_{x_0}}]= \left\{\begin{array}{ll}
W_\lambda(P^\mu(y)-P^\mu(x_0))+1, &\; a< y \leq x_0 \\
W_\lambda(C^\mu(y)-C^\mu(x_0))+1, &\; x_0 \leq y < b.
\end{array}\right.
\]
Since $u_\lambda(x_0,y)=\frac{2}{W_\lambda} \E_y[e^{-\lambda H_{x_0}}]$ (cf. Theorem 50.7, V.50 in Rogers and Williams \cite{rogers} for classical diffusions and It\^o and McKean \cite{mckean} for generalised diffusions), the result follows.
\end{proof}
\begin{example} \label{ex:jump} Let $\mu=\frac{1}{3} \delta_{0} +
\frac{1}{3} \delta_{1/2} + \frac{1}{3} \delta_{1}$. Then
$P^\mu(x)=\frac{1}{3} x$ for $0 \leq x \leq \frac{1}{2}$ and
$C^\mu(x)=\frac{1}{3}-\frac{1}{3}x$ for $1/2 \leq x \leq 1$. Let
$u_\lambda(1/2,x)$ be the $\lambda$-resolvent of a diffusion started
at $1/2$ such that $X_{T_{\lambda}} \sim \mu$. Then
\[
u_\lambda(1/2,x)= \left\{\begin{array}{ll}
2(x/3-1/6)+2/W_\lambda, &\; 0 \leq x \leq 1/2 \\
2(1/6-x/3)+2/W_\lambda, &\; 1/2 \leq x \leq 1.
\end{array}\right.
\]
where $W_\lambda \leq 6$. The speed measure of the consistent diffusion charges only the points $0$, $1/2$ and $1$ and is given by
\[
2 \lambda m(\{x\})= \left\{\begin{array}{ll}
\frac{3}{6/W_\lambda-1}, &\; x=0, \\
\frac{W_\lambda}{2}, &\; x=1/2, \\
\frac{3}{6/W_\lambda-1}, &\; x=1.
\end{array}\right.
\]
Consistent diffusions are birth-death processes in the sense of Feller
\cite{feller}. $\frac{d\Gamma_t}{dt} > 0$ whenever $B_t \in \{0,1,2\}$
and the process is `sticky' there. The process $\Gamma_t$ is constant
whenever $B_t \notin \{0,1,2\}$, so that $A_t$ skips over these time
intervals and $X_t=B_{A_t}$ spends no time away from these
points. Note that if $W_\lambda=6$ then $m(\{0\})=m(\{1\})=\infty$ so
that $\Gamma_t=\infty$ for $t$ greater than the first hitting time of
$0$ or $1$ implying that the endpoints are absorbing. If
$W_\lambda<6$, the endpoints are reflecting but sticky.
\end{example}
Let $H_x=\inf\{u \geq 0 :X_u = x\}$ and let $y \in (a,b)$.
\begin{lemma} \label{l:boundary}
$\int_{a+} (|x|+1) m(dx) = \infty$ $($resp. $\int^{b-} (|x|+1) m(dx) =
\infty)$ if and only if $1/W_\lambda = P^\mu(x_0)$
$($resp. $1/W_\lambda = C^\mu(x_0))$.
\end{lemma}
\begin{proof}
We prove the second statement, the first follows similarly.
Clearly, if $1/W_\lambda > C^\mu(x_0)$, then $\int^{b-} \frac{(1+|x|) \mu(dx)}{C^{\mu}(x)-C^{\mu}(x_0)+1/W_\lambda} < \infty$. Conversely, suppose that $1/W_\lambda=C^{\mu}(x_0)$. Suppose first that $b=\infty$. Then $\lim_{x \uparrow \infty} \phi_\lambda(x) = \lim_{x \uparrow \infty} u_\lambda(x_0,x) = 0$, since $C^\mu(x) \downarrow 0$ as $x \uparrow \infty$. It follows by Theorem 51.2 in \cite{rogers} (which holds in the generalised diffusion case) that $\int^{\infty} x m(dx) = \infty$. Now suppose instead that $b< \infty$. Observe that $C^\mu(x)$ is a convex function on $I$, so it has left and right derivatives, while its second derivative can be interpreted as a measure; moreover, $C^{\mu}(b) = 0 = (C^\mu)'(b+)$ and $(C^{\mu})''(dx) = \mu(dx)$.
From the convexity and other properties of $C^{\mu}$, we note that $\frac{(x-b)(C^{\mu})'(x-)}{C^{\mu}(x)} \to 1$ as $x \nearrow b$, and also $\frac{C^{\mu}(x)}{x-b} \ge (C^{\mu})'(x-)$ for $x<b$, so $\frac{1}{x-b} \ge \frac{(C^{\mu})'(x-)}{C^{\mu}(x)}$ and $\frac{(C^{\mu})'(x-)}{C^{\mu}(x)} \searrow -\infty$ as $x \nearrow b$. The claim will follow provided we can show $\lim_{v \nearrow b} \int_u^v \frac{(C^\mu)''(dx)}{C^{\mu}(x)} = \infty$ for $u<b$. Using Fubini/integration by parts, and the fact that $(C^{\mu})'(x)$ is increasing and negative, we see that:
\begin{align*}
\int_u^{v-} \frac{(C^\mu)''(dx)}{C^{\mu}(x)} & = \frac{(C^{\mu})'(v-)}{C^{\mu}(v)} - \frac{(C^{\mu})'(u-)}{C^{\mu}(u)} + \int_u^v \left(\frac{(C^{\mu})'(x-)}{C^{\mu}(x)}\right)^2 \, dx\\
& \ge \frac{(C^{\mu})'(v-)}{C^{\mu}(v)} - \frac{(C^{\mu})'(u-)}{C^{\mu}(u)} + (C^{\mu})'(v-)\int_u^v \frac{(C^{\mu})'(x-)}{C^{\mu}(x)^2} \, dx \\
& \ge \frac{(C^{\mu})'(v-)}{C^{\mu}(v)} - \frac{(C^{\mu})'(u-)}{C^{\mu}(u)} + (C^{\mu})'(v-)\left(\frac{1}{C^{\mu}(u)} - \frac{1}{C^{\mu}(v)}\right) \\
& \ge \frac{(C^{\mu})'(v-)- (C^{\mu})'(u-)}{C^{\mu}(u)}.
\end{align*}
Letting first $v \to b$, the right hand side is equal (in the limit) to $-\frac{(C^{\mu})'(u-)}{C^{\mu}(u)}$, but we observed above that this is unbounded as $u \nearrow b$, and since the whole expression is increasing in $u$, it must be infinite, as required.
\end{proof}
As Lemma \ref{l:boundary} demonstrates, the behaviour of $X$ at the
boundaries is determined by the value of $W_\lambda$. When $I$ is
unbounded, the boundary behaviour determines whether or not $X$ is a
martingale diffusion. Suppose that $a$ is finite, $b=\infty$. Then
Kotani \cite{Kotani} (see also Delbaen and Shirakawa
(\cite{delbaen2002})) show that $X_{t \wedge H_a \wedge H_b}$ is a
martingale if and only if $\int^{\infty-} x m(dx) = \infty$
(i.e. $\infty$ is not an entrance boundary). By Lemma \ref{l:boundary}
this is equivalent to the conditions $x_0 \leq \bar{x}_\mu$ and
$1/W_\lambda=C^\mu(x_0)$ being satisfied. An analogous observation
holds when $b$ is finite and $a$ is infinite.
Theorem \ref{t:main} and Lemma \ref{l:boundary} provide a natural way
of determining boundary properties by inspection of the decomposition
of the speed measure in terms of $\Prob(X_{T_{\lambda}} \in
dx)$. Furthermore, the decomposition gives us a canonical way of
constructing strict local martingales with a given law at a random
time. For instance if $b$ is infinite, we can generate strict local
martingale diffusions by choosing a measure $\mu$ and setting
$W_\lambda < 1/C^\mu(x_0)$.
\begin{example} \label{ex:Bessel}
Let $m(dx)=\frac{dx}{x^4}$ and $I = (0,\infty)$. Suppose $X_0=1$ and $\lambda=2$. Then $\phi(x)=\frac{x\sinh(\frac{1}{x})}{\sinh(1)}$ and $\varphi(x)=x e^{1-1/x}$ are respectively the strictly decreasing and strictly increasing eigenfunctions of the inverse Bessel process of dimension three, $X$, which solves the equation \[\frac{1}{2} \frac{d^2}{dm dx} f = 2 f.\]
We calculate (cf. Equations (3.2) and (3.3) in \cite{MK:12}),
\begin{equation}
\mu(dx)=\Prob(X_{T_{\lambda}} \in dx)= \left\{\begin{array}{ll}
\frac{\sinh(1)}{2 x^3} e^{-1/x}\, dx &\; 0<x \leq 1,\\
\frac{e^{-1}}{2 x^3}\sinh(1)\, dx &\; 1 \leq x.
\end{array}\right.
\end{equation}
We find $\bar{x}_\mu=1-1/e$. Further, we calculate $P^\mu(x)=\frac{1}{2}\sinh(1) x e^{-1/x}$ for $x \leq 1$ and
$C^\mu(x)=e^{-1}( x \sinh(1/x)-1)$ for $x \geq 1$. Thus
\begin{equation}
m(dx)=dx/x^4= \left\{\begin{array}{ll}
\frac{\mu(dx)}{P^\mu(x)} &\; 0<x \leq 1,\\
\frac{\mu(dx)}{C^\mu(x)-(2C^\mu(1)+2P^\mu(1))} &\; 1 \leq x.
\end{array}\right.
\end{equation}
It follows that $X$ is a strict local-martingale diffusion.
\end{example}
In the example above, the strict local martingale property of the process follows from the fact that $C^\mu$ is `shifted up' in the decomposition of the speed measure
(\ref{eq:speeddecomp}). In general, we expect reflection at the left boundary $X = a$, when we have had to `shift the function $P^\mu$ up', and we expect the process stopped at $t=H_a \wedge H_b$ to be a strict local martingale when we `shift $C^{\mu}$ up'.
\begin{example} \label{ex:Bessel2}
Let us reconsider the diffusion in Example \ref{ex:Bessel}. Let us construct a diffusion $Y$ with the same law as the inverse Bessel process of dimension three at an exponential time, such that $Y_{t \wedge H_0}$ is a martingale (and therefore, such that $Y_0 = \bar{x}_\mu$). As we are shifting the starting point, we must adjust the speed measure between $\bar{x}_\mu$ and the starting point of the diffusion in
Example \ref{ex:Bessel}. By Lemma \ref{l:boundary} we know that the speed measure of the martingale diffusion must satisfy
\begin{equation*}
m(dx)= \left\{\begin{array}{ll}
\frac{\mu(dx)}{P^\mu(x)} &\; 0<x \leq \bar{x}_\mu,\\
\frac{\mu(dx)}{C^\mu(x)} &\; \bar{x}_\mu \leq x,
\end{array}\right.
\end{equation*}
Recall that $\bar{x}_\mu=1-1/e$. We calculate
\begin{equation*}
m(dx) = \left\{\begin{array}{ll}
\frac{1}{x^4} \, dx &\; 0<x \leq \bar{x}_\mu,\\
\frac{(e-e^{-1})e^{-1/x}\, dx}{x^3((e-e^{-1}) x e^{-1/x}+2(\bar{x}_\mu-x))} &\; \bar{x}_\mu<x \leq 1,\\
\frac{\sinh(1/x)\, dx}{x^3(x\sinh(1/x)-1)} &\; 1 \leq x.
\end{array}\right.
\end{equation*}
\end{example}
\section{The BLJ embedding for Brownian motion}
The BLJ construction is remarkably general and can be used to embed distributions in general Hunt processes. Our interest, however, lies in the specific case of embedding a law $\mu$ in Brownian Motion. In this setting, the rich structure of the embedding translates into a one-to-one correspondence to the family of diffusions with a given law at an exponential time.
Suppose that $x_0 \in [a,b]$, $\mu(\{x_0\}) = 0$ and define
\begin{equation} \label{eq:Vmu}
V^{x_0}_\mu(x)=\int \E_y[L_{H_{x_0}}^x] \mu(dy).
\end{equation}
We will assume from now on that $\mu$ has a finite first moment and that
$\mu(\{x_0\})=0$. Then $\kappa_0=\sup\{V_\mu(x) ; x \in [a,b]\} < \infty$ (see, for instance, p.~547 in \cite{BLJ}). By the Corollary on p.~540 in \cite{BLJ} it follows that for each $\kappa \geq \kappa_0$ the stopping time
\begin{equation} \label{eq:BLJstop}
\tau_\kappa=\inf \left\{t>0: \kappa \int \frac{L_t^x \mu(dx)}{\kappa-V^{x_0}_\mu(x)} > L_t^{x_0} \right\}
\end{equation}
embeds $\mu$ in $B^{x_0}$, i.e. $B^{x_0}_{\tau_{\kappa}} \sim \mu$. Moreover, we have $\E_{x_0}[L^{x_0}_{\tau_{\kappa}}]=\kappa$. Finally, the stopping time $\tau_{\kappa_0}$ is optimal in the following sense; if $\sigma$ is another embedding of $\mu$ in $B_{x_0}$, then for every $x \in I$, $\E_{x_0}[L^x_\sigma] \geq \E_{x_0}[L^x_{\tau_{\kappa_0}}]$. From Lemma~\ref{lem:minimal}, it follows that the stopping time $\tau_{\kappa_0}$ is minimal in the sense of Definition~\ref{def:minimal}.
\section{Embeddings and diffusions}
We will now show that each BLJ embedding $\tau_{\kappa}$ of $\mu$ in
$B$ corresponds to a local-martingale diffusion $X$ such that
$X_{T_{\lambda}} \sim \mu$, for an independent exponentially
distributed random variable $T_{\lambda}$. The correspondence is a
consequence of the role played by the functional $\Gamma_u=\int_{\R}
L_u^x m(dx)$, where $m$ is defined as in (\ref{eq:speeddecomp}) for a
fixed starting point $x_0$ and a target law $\mu$ on $(a,b)$. We have
already seen that $B_{A_{T_\lambda}} \sim \mu$, where
$A_t=\Gamma_t^{-1}$. We will now observe that for $\kappa \geq
\kappa_0$, the BLJ embedding $\tau_\kappa$ introduced above can be
re-written as $\tau_\kappa=\inf\{t \geq 0 : \Gamma_t \lambda \kappa >
L_t^{x_0}\}$. The remarkable role played by the additive functional
$\Gamma$ in both settings, and the resulting correspondence between
embeddings and diffusions, is the subject of this section. In order to
better understand the connection, we first make the following
distinction: let $\tau_{\kappa}^* = \inf\{t \ge 0: \Gamma_t >
T_{\lambda}\}$, where $T_{\lambda}$ is an independent exponential
random variable with mean $1/\lambda$, and $m(dx)$ is given by
\eqref{eq:speeddecomp}, with $\kappa=2/W_{\lambda}$. It follows that
$B_{\tau^*_\kappa} \sim \mu$ for $\kappa \ge
2\max\{C^{\mu}(x_0),P^\mu(x_0)\}$, and (from
Corollary~\ref{c:lpotential}) we have $\E_{x_0}[L^{x_0}_{T_{\lambda}}]
= \kappa$.
The connection between the quantities that define BLJ embeddings and
those that define diffusions follows from relating the
$\lambda$-potential of diffusions, the potential $U^\mu$ of $\mu$ and
$V_\mu^{x_0}$. A number of solutions to the Skorokhod Embedding
Problem, most notably the solutions of Chacon-Walsh
\cite{ChaconWalsh:76}, Az{\'e}ma-Yor (\cite{AzemaYor:79a},
\cite{AzemaYor:79b}) and Perkins \cite{Perkins:86} can be derived
directly from quantities related to the potential $U^\mu$.
\begin{lemma} \label{l:VPo}
\begin{equation} \label{eq:VPo}
V^{x_0}_\mu(x)= \left\{\begin{array}{ll}
2(P^\mu(x_0)-P^\mu(x)) &\; x \leq x_0 \\
2(C^\mu(x_0)-C^\mu(x)) &\; x > x_0.
\end{array}\right.
\end{equation}
\end{lemma}
\begin{proof}
Observe that $\E_y[L_{H_{x_0}}^x]=|y-x_0|+|x_0-x|-|y-x|$ is simply the potential kernel for $B^{x_0}$ killed at $x_0$. Integrating we find
\[V^{x_0}_\mu(x)=\int(|y-x_0|+|x_0-x|-|y-x|)\mu(dy)=U^\mu(x_0)-U^\mu(x)+|x_0-x|.\]
The result follows after some re-arrangement.
\end{proof}
Recalling the definition of $\kappa_0$ in the BLJ embedding, we now observe that $\kappa_0=\max\{2C^\mu(x_0),2P^\mu(x_0)\}$, which is the smallest constant $c$ such that $c-V^{x_0}_\mu(x) \geq 0$ for all $x$. The situation is illustrated in Figure 1 below.
\begin{figure}[htb]
\centering
\begin{asy}[width=\textwidth]
import graph;
real xmin = -1;
real xmax = 1;
real ymax = 1;
real u0(real x) {return abs(x);}
real scale = 1.4;
real u1(real x) {return (log(2*cosh(scale*x))/scale);}
// real u0s(real x, real s) {return u0(x-s)
real skip = 0.35;
real eps = 0.15;
real shrink = 0.7;
real yfmax = ymax+eps/2;
pen q = black+0.5;
real shift = 0.2;
real x2 = skip + xmax-xmin+shift;
real T(real x) {return x-shift;}
real u0s(real x) {return u0(T(x));}
real u1s(real x) {return u1(x)+shift;}
bool3 u0smax(real x) { return (u0s(x)<=yfmax);}
bool3 u1smax(real x) { return (u1s(x)<=yfmax);}
draw(graph(u0s,xmin,xmax,u0smax),deepblue+0.5+dashed);
draw(graph(u1s,xmin,xmax,u1smax),deepgreen+0.5+dashed);
draw((0,u1(0))--(0,u1s(0)),deepred+0.5,MidArrow);
draw(graph(u0,xmin,xmax),deepblue+0.5);
label(scale(shrink)*"$|x-{\mmu}|$",(-0.5,u0(-0.5)),SW,deepblue+0.5);
draw(graph(u1,xmin,xmax),deepgreen+0.5);
label(scale(shrink)*"$U^{\mu}(x)$",(0.5,u1(0.5)),SE,deepgreen+0.5,filltype=UnFill);
label("$\mmu$",(0,0),S);
label("$x_0$",(shift,0),S);
real Pm(real x) {return u1(x) + x;}
real Cm(real x) {return u1(x) -x;}
real Pms(real x) {return Pm(x - x2);}
real Cms(real x) {return Cm(x - x2);}
real Cms2(real x) {return Cm(x - x2) + Pm(shift)-Cm(shift);}
bool3 Pmax(real x) { return (Pms(x)<=yfmax);}
bool3 Cmax(real x) { return (Cms(x)<=yfmax);}
bool3 C2max(real x) { return (Cms2(x)<=yfmax);}
draw(graph(Pms,xmin+x2,xmax+x2,Pmax),deepblue+0.5);
label(scale(shrink)*"$P^{\mu}(x)$",(x2-0.5,Pms(x2-0.5)),NW,deepblue+0.5);
draw(graph(Cms,xmin+x2,xmax+x2,Cmax),deepgreen+0.5);
label(scale(shrink)*"$C^{\mu}(x)$",(x2+0.5,Cms(x2+0.5)),NE,deepgreen+0.5);
draw(graph(Cms2,xmin+x2,xmax+x2,C2max),deepgreen+0.5+dashed);
label(scale(shrink)*"$C^{\mu}(x)+P^{\mu}(x_0)-C^{\mu}(x_0)$",(x2,Cms2(x2)),E,deepgreen+0.5,filltype=UnFill);
draw((x2+shift,0)--(x2+shift,Pm(shift)),deepred+0.5+dashed);
draw((x2,0)--(x2,Pm(0)),deepred+0.5+dashed);
draw((x2+xmin,Pm(0))--(x2,Pm(0)),gray+0.5+dotted);
label("$\mmu$",(x2,0),S);
label("$x_0$",(shift+x2,0),S);
label("$\kappa_0$",(xmin+x2,Pm(0)),W);
draw ((xmin,0)--(xmax+shift+eps,0),q,Arrow);
draw ((xmin,0)--(xmin,ymax+eps),q,Arrow);
draw ((xmin+x2,0)--(xmax+eps+x2,0),q,Arrow);
draw ((xmin+x2,0)--(xmin+x2,ymax+eps),q,Arrow);
label("$x$",(xmax+shift+eps,0),SE);
label("$x$",(xmax+eps+x2,0),SE);
\end{asy}
\caption{If $x_0=\bar{x}_\mu$, then
$\kappa_0-V_\mu^{{x_0}}(x)=U^\mu(x)-|x-\bar{x}_\mu|$. If $x_0 \neq
\bar{x}_\mu$, then the potential must be shifted upwards to lie
above $|x_0-x|$ everywhere. Thus, if $x_0 < \bar{x}_\mu$,
$\kappa_0-V_\mu^{x_0}(x)=U^\mu(x)+(2C^\mu(x_0)-U^\mu(x_0))-|x_0-x|$,
while if $x_0>\bar{x}_\mu$,
$\kappa_0-V_\mu^{x_0}(x)=U^\mu(x)+(2P^\mu(x_0)-U^\mu(x_0))-|x_0-x|$
(see left picture). Note that if $x \geq x_0>\bar{x}_{\mu}$,
$\frac{1}{2}(\kappa_0-V_\mu^{x_0}(x))=P^\mu(x_0)-C^\mu(x_0)+C^\mu(x)$
(see right figure).}
\end{figure}
The following Lemma follows from the Corollary in \cite{BLJ}, page~540.
\begin{lemma} \label{l:lambda}
\begin{equation} \label{eq:lambda}
\E_{x_0}[L^{x_0}_{\tau_{\kappa_0}}]= \left\{\begin{array}{ll}
2C^\mu(x_0) &\; x_0 \leq \bar{x}_\mu\\
2P^\mu(x_0) &\; x_0 \geq \bar{x}_\mu.
\end{array}\right.
\end{equation}
\end{lemma}
\begin{proof}
By the Corollary in \cite{BLJ}, $\kappa_0=\E_{x_0}[L^{x_0}_{\tau_{\kappa_0}}]$.
The result now follows from the fact that $\kappa_0=\max\{2C^\mu(x_0),2P^\mu(x_0)\}$.
\end{proof}
In comparison, the Corollary on p. 540 in \cite{BLJ} implies the formula $\E_{x_0}[L^x_{\tau_{\kappa_0}}]=\kappa_0-V^{x_0}_\mu(x)$. It is helpful to
visualise this relationship pictorially, see Fig. 1. The expected local time at $x$ of Brownian motion started at $x_0$ until the minimal BLJ stopping time is the distance at $x$ between the minimally shifted potential function and the potential kernel.
So far we have focused on the minimal embedding $\tau_{\kappa_0}$ for $\mu$. The picture for the suboptimal embeddings $\tau_{\kappa}$, $\kappa>\kappa_0$
is analogous. The picture simply reflects the additional expected local time accrued between the minimal and the non-minimal stopping time.
The expected local time of a non-minimal embedding is the
expression in (\ref{eq:lambda}) plus the positive constant $\kappa-\kappa_0$, i.e. $\E_{x_0}[L^x_{\tau_{\kappa}}]=\E_{x_0}[L^x_{\tau_{\kappa_0}}]+(\kappa-\kappa_0)$.
The graph of the expected local time (see Fig. 1) is given by
shifting the potential picture $\kappa_0-V^{x_0}_\mu(x)$ upwards by
$\kappa-\kappa_0$.
It follows from these observations that the local times
$\E_{x_0}[L^x_{\tau_{\kappa}}]$ and $\E_{x_0}[L^x_{\tau^*_{\kappa}}]$
are equal for all $\kappa \ge \kappa_0$ and all $x\in [a,b]$. Let us
now make the correspondence between diffusions (through the stopping
times $\tau_\kappa^*$) and the BLJ embeddings precise. Fix $x_0 \in
\supp(\mu)$ and let $\kappa \geq \max\{2C^\mu(x_0), 2P^\mu(x_0)\}$.
\begin{proposition} \label{p:correspondence} Fix $\lambda>0$ and
define a Borel measure $m$ via
$m(dx)=\frac{1}{\lambda}\frac{\mu(dx)}{\kappa-V_\mu^{x_0}(x)}$.
Define the functional $\Gamma_u=\int_\R L_u^x m(dx)$. Then:
\begin{enumerate}
\item The stopping time $\tau_\kappa=\inf\{u \geq 0: \lambda \kappa
\Gamma_u > L^{x_0}_u\}$ has $B_{\tau_\kappa} \sim \mu$ and
$\E_{x_0}[{L_{\tau_\kappa}^x}]=\kappa-V_\mu^{x_0}(x)$.
\item The stopping time $\tau_\kappa^*=\inf\{u \geq 0: \Gamma_u >
T_{\lambda}\}$ has $B_{\tau_\kappa^*} \sim \mu$ and
$\E_{x_0}[{L_{\tau_\kappa}^x}]=\kappa-V_\mu^{x_0}(x)$.
\item The diffusion with speed measure $m$, defined via
$X_t=B^{x_0}_{A_t}$, where $A$ is the right-continuous inverse of
$\Gamma_u$, has Wronskian $W_\lambda=\frac{2}{\kappa}$,
$\lambda$-potential $u_\lambda(x_0,y)=\kappa-V^{x_0}_\mu(y)$ and
$X_{T_\lambda} = B_{\tau^*} \sim \mu$.
\end{enumerate}
\end{proposition}
\begin{proof}
Since $\mu$ has a finite first moment, $\kappa_0 < \infty$ and
Hypothesis $1$ in \cite{BLJ} is satisfied. Thus by the Corollary in
\cite{BLJ}, $\tau_\kappa=\inf \left\{u>0: \kappa \int \frac{L_u^x
\mu(dx)}{\kappa-V^{x_0}_\mu(x)} > L_u^{x_0} \right\} =\inf\{u
\geq 0 : \lambda \kappa \Gamma_u > L^{x_0}_u\}$ embeds $\mu$ in
$B^{x_0}$. It follows from Lemma \ref{l:VPo} and Corollary
\ref{c:lpotential} that $u_\lambda(x_0,x)=\kappa-V_\mu^{x_0}(x)$ and
that
$\E_{x_0}[L^{x_0}_{A_{T_\lambda}}]=\frac{2}{W_\lambda}=\kappa$. By
Theorem \ref{t:main}, $X_{T_\lambda} \sim \mu$.
\end{proof}
\begin{remark}
The assumption $\mu(\{x_0\})=0$ made in Section~4 is necessary in the construction of the BLJ embedding in \cite{BLJ}, but not in the construction of diffusions with a given exponential time law in \cite{MK:12}. To make the correspondence independent of this assumption on the target law, a modified version of the BLJ embedding with external randomisation can be constructed.
\end{remark}
With the close connection indicated by
Proposition~\ref{p:correspondence}, it is natural to consider if we
can more explicitly explain the connection between $\tau_\kappa^*$ and
$\tau_\kappa$. To do this, we consider the excursion theory that is
behind the correspondence between diffusions and embeddings. We refer
to Rogers~\cite{rogers1989guided} for background on excursion
theory. Let $U:=\{f \in C(\R^+) : f^{-1}(\R \ \{0\}) = (0, \zeta),$
for some $\zeta >0\}$, be the set of excursions of $B$. Then, by a
well-known result due to It{\^o}, there exists a measure $\Xi$ on $u$
such that for any $A \subseteq U$,
\[\Xi(A):= \# \{\mbox{Excursions of} \ B_t \text{ in } A \ \mbox{between} \ L_t=l \ \mbox{and} \ L_t=r \} \sim \text{Poisson}((r-l)n(A)).\]
Consider the Brownian motion $B_t$, and $\Gamma_t=\int_\R L_u^x m(dx)$, which is increasing. For a test function $g$, define: $\phi(u):=\int n(df) g(f(\Gamma_t^{-1}(u)) \Indi_{\{\Gamma_t(\zeta(f)) \geq u\}}$, the expected value of $g(B_{\Gamma_u^{-1}})$ averaged over the excursion measure, conditioned on the event that $\Gamma_s \geq u$ on the excursion.
If ${T_\lambda}$ is an independent exponential random variable with parameter $\lambda$ then we compute, using the fact that the process $B$ is recurrent, and the memoryless property of the exponential, that
\begin{equation} \label{eq:firstexcursion}
\E[g(B_{\tau_\kappa^*})] = \E[g(B_{\Gamma_{T_\lambda}^{-1}})]=c \int \phi(u) \lambda e^{-\lambda u} du,
\end{equation}
for some constant $c$.
In the BLJ setting, consider the stopping time $\tau_\kappa=\inf\{u
\geq 0 : \Gamma_u \lambda \kappa > L_t^{x_0}\}$. Observe that
$\Gamma_{L_s^{-1}}$ is an increasing process with independent,
identically distributed increments, and hence $Y_s:=s-\lambda \kappa
\Gamma_{L_s^{-1}}$ is a L{\'e}vy process which goes below $0$ on the
excursion for which $\tau_\kappa$ occurs. In particular, we have
\begin{equation} \label{eq:BLJexcursion}
\E[g(B_{\tau_\kappa})]=\E\left[\int_0^{\tau_\kappa} \phi(L_t^{x_0} - \lambda \kappa \Gamma_t) dL_t^{x_0}\right] = \E\left[\int_0^{L^{-1}_{\tau_\kappa}} \phi(Y_s) ds \right],
\end{equation}
where $L^{-1}_{\tau_\kappa} = \inf\{t \geq 0 : Y_t \leq 0\}$ and the first
equality follows from the Maisonneuve exit system, see \cite{BLJ}, p. 544.
In addition we observe that $Y_t$ is a L{\'e}vy process and so, using
Lemma 1 in \cite{BLJ}, we can rewrite this formula as
\[\E\left[\int_0^{L^{-1}_{\tau_\kappa}} \phi(Y_s) ds \right] = \int_0^\infty \phi(u) \exp(-\psi^{-1}(0) u)du,\]
where $\psi(x)=\frac{1}{t} \log(\E^{0}[\exp(xY_t)])=x-\log(\E[\exp(-\lambda \kappa \Gamma_{L^{-1}_1})])$.
The behaviour of $\psi$ is determined by properties of $Y_t$. In
particular, it is convex and $\psi^{-1}(0)=\inf\{t : \psi(t) >0\} >0$
if and only if $\psi' (0) <0$ which is true if and only if
$\E[Y_1]<0$, i.e. $\E^{x_0}[\lambda \kappa \Gamma_t] >
\E^{x_0}[L_t^{x_0}]$. It follows that $\E[Y_1] \leq 0$, since
otherwise with positive probability, $Y_t \nearrow \infty$ and hence
$L_{\tau}^{-1} = \infty$ with positive probability, and hence the same
is true for the BLJ stopping time.
Equating the terms in (\ref{eq:BLJexcursion}) and (\ref{eq:firstexcursion}), we now have an identity involving only $\phi$ and we can choose $\kappa$ so that $\psi^{-1}(0)=\lambda$. Note that the constant $c$ appearing in
(\ref{eq:firstexcursion}) is equal to $\frac{1}{\lambda}$. Moreover, by the embedding property of $\tau_\kappa$ and the construction of $\Gamma$, both equations are equal to $\int g(x) \mu(dx)$.
\section{Minimal diffusions}
The purpose of this final section is to define a notion of minimality for diffusions which will be analogous to the notion of minimality for the BLJ Skorokhod embeddings.
Suppose we have fixed a starting point $x_0$ and a law $\mu$ and we are faced with a (Skorokhod embedding) problem of finding a stopping time $\tau$, such that $B_\tau \sim \mu$, where $B$ is a standard Brownian motion started at $x_0$. Then (and especially when we are interested in questions of optimality) it is most natural to search for stopping times $\tau$ which are minimal.
In the family of BLJ embeddings, the embedding $\tau_{\kappa_0}$ is
minimal, and hence (by Lemma~\ref{lem:minimal}) so too is
$\tau_{\kappa_0}^*$, and the notion of minimal embedding carries over
into a notion of minimality for diffusions in natural scale.
\begin{definition} \label{d:minimality} We say that a time homogeneous diffusion $X$ in natural scale and started at $x_0$ is a $\lambda$-minimal diffusion if, whenever $Y$ is another time-homogeneous diffusion in natural scale, with $X_{T_\lambda} \sim Y_{T_\lambda}$ where ${T_\lambda}$ an independent exponential random variable with mean $\frac{1}{\lambda}$, then $\E_{x_0}[L_{{T_\lambda}}^{x_0}(X)] \le \E_{x_0}[L_{{T_\lambda}}^{x_0}(Y)]$.
We say that $X$ is a minimal diffusion in natural scale if $X$ is $\lambda$-minimal for all $\lambda>0$.
\end{definition}
By Proposition \ref{p:correspondence}, the $\lambda$-minimal local-martingale diffusion corresponds to the minimal BLJ embedding $\tau_{\kappa_0}$. As an analogue of the notion of minimality for the BLJ solution to the Skorokhod Embedding Problem, minimality is a natural probabilistic property. A minimal diffusion has the lowest value of $\E_{x_0}[L^y_{T_\lambda}]$ for all $y>0$ among the diffusions with a given law $\mu$ at time ${T_\lambda}$ started at $x_0$ (this follows from Theorem~\ref{t:main}). In terms of diffusion dynamics, the minimal diffusion moves around the state space slower than all other diffusions with the same exponential time law.
Minimality is a necessary condition for $X_{t \wedge H_a \wedge H_b}$ to be a martingale. For instance, if $a$ is finite and $b$ is infinite, then $X_{t \wedge H_a \wedge H_b}$ is a martingale if and only if $X$ is minimal and $x_0 \leq \bar{x}_\mu$, i.e. $b$ is not an entrance boundary. When $I$ is a finite interval, minimality is a more natural property than the martingale property; every (stopped) diffusion on a finite interval is a martingale diffusion, but there is only one minimal diffusion for every exponential time law.
Note also that these definitions extend in an obvious way to diffusions which are natural scale: a diffusion which is not in natural scale is minimal if and only if it is minimal when it is mapped into natural scale. (See Example~\ref{ex:Jacobi}.)
We collect these observations in the following result:
\begin{theorem}
\label{thm:minimal}
Let $X$ be a time-homogenous diffusion in natural scale. Then the following are equivalent:
\begin{enumerate}
\item $X$ is $\lambda$-minimal;
\item if $X_{T_\lambda} \sim \mu$ and $W_\lambda$ is the Wronskian of $X$, then:
\begin{equation*}
1/W_\lambda=\max\{C^\mu(x_0),P^\mu(x_0)\};
\end{equation*}
\item $X$ has at most one entrance boundary;
\item $X$ is a minimal diffusion.
\end{enumerate}
\end{theorem}
\begin{proof}
The equivalence of the first two statements follows from Corollary \ref{c:lpotential}. Next, if $X$ is minimal
then $\E_a[e^{-\lambda H_{x_0}}]=0$ or $\E_b[e^{-\lambda H_{x_0}}]=0$, and hence $X$ has at most one entrance boundary.
Finally, the properties of the boundary points are independent of the choice of $\lambda$, so if $X$ has at most one entrance boundary, then $X$ is minimal.
\end{proof}
\begin{example} \label{ex:Jacobi}
A natural class of non-minimal diffusions is the following class of
Jacobi diffusions in natural scale. On the domain $[0,b]$
let \[dX_t=(\alpha-\beta X_t)dt+\sigma \sqrt{X_t(b-X_t)}, \ \ X_0
\in (0,b),\] where $\frac{2\beta}{\sigma^2}-\frac{2\alpha}{\sigma^2
b}-1 >0$ and $\frac{2\alpha}{\sigma^2 b}-1 >0$. The eigenfunctions
$\varphi_\lambda$ and $\phi_\lambda$ can be calculated explicitly as
hypergeometric functions, see Albanese and Kuznetsov \cite{AK}. It
is shown in \cite{AK} that $\lim_{x \downarrow 0} \varphi_\lambda(x)
> 0$ and $\lim_{x \uparrow b} \phi_\lambda(x) > 0$. Let $s$ be the
scale function of $X$ and consider $Y=s(X)$ with eigenfunctions
$\overline{\varphi}_\lambda$ and $\overline{\phi}_\lambda$. Then
$\lim_{x \downarrow s(0)} \overline{\varphi}_\lambda(x) = \lim_{x
\downarrow 0} \varphi_\lambda(s^{-1}(x)) > 0$. Similarly $\lim_{x
\uparrow s(b)} \overline{\phi}(x) > 0$. Hence $Y$ is non-minimal.
\end{example}
\begin{example}
Let $I=(0,1)$ and let $X=(X_t)_{t \geq 0}$ be a diffusion with $X_0=1/2$ and speed measure $m(dx)=\frac{dx}{x^2(1-x^2)}$ (sometimes known as the Kimura martingale cf. \cite{huillet2}). The increasing and decreasing eigenfunctions are $\varphi(x)=\frac{2x^2}{1-x}$ and $\phi(x)=\frac{2(1-x)^2}{x}$ respectively. Note that both boundaries are natural, so this diffusion is minimal.
Non-minimal diffusions with the same exponential time law have increasing/decreasing eigenfunctions $\varphi_\delta(x)=\varphi(x)+\delta$ and $\phi_\delta(x)=\phi(x)+\delta$, for some $\delta > 0$, and with reflection at the endpoints. Accordingly, the consistent speed measures indexed by $\delta$ are given by $m_\delta(dx)=\frac{dx}{\sigma_\delta^2(x)}$, where
\[
\sigma_\delta^2(x)=
\begin{cases}
\left(\frac{\delta}{2}(1-x)+x^2\right)(1-x)^2 & x < \frac{1}{2}\\
\left(\frac{\delta}{2}x+(1-x)^2\right)x^2 & x \ge \frac{1}{2}.
\end{cases}
\]
\end{example}
\begin{example}
Consider Brownian motion on $[0,2]$ with instantaneous reflection at the endpoints and initial value $B_0=x_0 \in [1,2)$. Let $\lambda=1/2$, then the $1/2$-eigenfunctions solve $\frac{d^2 f}{dx^2} = f$, so the increasing eigenfunction (up to a constant) is $\varphi(x)=\frac{\cosh(x)}{\cosh(x_0)}$ and
$\phi(x)=\frac{\cosh(2-x)}{\cosh(2-x_0)}$. Note that this diffusion is non-minimal. To construct the minimal
diffusion with the same marginal law at an exponential time, let $\eta=\cosh(x_0)^{-1}$ and set $\varphi_\eta(x)=\varphi(x)-\eta=\frac{\cosh(x)-1}{\cosh{x_0}}$, and similarly
$\phi_\eta(x)=\phi(x)-\eta=\frac{\cosh(2-x)}{\cosh(2-x_0)}-\frac{1}{\cosh(x_0)}$. The diffusion co-efficient of
the minimal diffusion is given by $\sigma^2(x) = \frac{\varphi_\eta(x)}{\varphi''_\eta(x)}=1-\frac{1}{\cosh(x)}$ for
$x \in [0,x_0]$ and by $\sigma^2(x)=\frac{\phi_\eta(x)}{\phi''_\eta(x)}=1-\frac{\cosh(2-x_0)}{\cosh(2-x)\cosh(x_0)}$ for $x \in [x_0,1]$.
\end{example}
\bibliography{mindiff}
\end{document} | {"config": "arxiv", "file": "1306.2873/minimaldiffusion_resub5.tex"} |
\begin{document}
\title{Interpolation of abstract Ces\`aro, Copson \newline and Tandori spaces{\rm *}}
\thanks{{\rm *}This publication has been produced during scholarship period of the first author
at the Lule{\aa} University of Technology, thanks to a Swedish Institute scholarschip (number 0095/2013).}
\author[Le\'snik]{Karol Le\'snik}
\address[Karol Le{\'s}nik]{Institute of Mathematics\\
of Electric Faculty, Pozna\'n University of Technology, ul. Piotrowo 3a, 60-965 Pozna{\'n}, Poland}
\email{\texttt{klesnik@vp.pl}}
\author[Maligranda]{Lech Maligranda}
\address[Lech Maligranda]{Department of Engineering Sciences and Mathematics\\
Lule{\aa} University of Technology, SE-971 87 Lule{\aa}, Sweden}
\email{\texttt{lech.maligranda@ltu.se}}
\begin{abstract}
We study real and complex interpolation of abstract Ces\`aro, Copson and Tandori spaces, including the description of
Calder\'on-Lozanovski{\v \i} construction for those spaces. The results may be regarded as generalizations of interpolation
for Ces\`aro spaces $Ces_p(I)$ in the case of real method, but they are new even for $Ces_p(I)$ in the case of complex
method. Some results for more general interpolation functors are also presented. The investigations show an interesting
phenomenon that there is a big difference between interpolation of Ces\`aro function spaces in the cases of finite and
infinite interval.
\end{abstract}
\footnotetext[1]{2010 \textit{Mathematics Subject Classification}: 46E30, 46B20, 46B42, 46B70.}
\footnotetext[2]{\textit{Key words and phrases}: Ces\`aro function spaces, Ces\`aro operator, Copson function spaces, Copson
operator, Tandori function spaces, Banach ideal spaces, symmetric spaces, interpolation, K-functional, K-method of interpolation,
complex method of interpolation, Calder\'on-Lozanovski{\v \i} construction.}
\maketitle
\section{\protect \medskip Definitions and basic facts}
We recall some notations and definitions which will be needed. Let $(I,\Sigma,m)$ be a $\sigma$-complete measure space. By $L^0 = L^0(I)$ we denote the set of all equivalence
classes of real-valued $m$ - measurable functions defined $I$. A {\it Banach ideal space}
$X = (X, \|\cdot\|)$ is understood to be a Banach space contained in $L^0$, which satisfies the so-called ideal
property: if $f, g \in L^0, |f| \leq |g|$ $m$-a.e. on $I$ and $g \in X$, then $ f\in X$ and $\|f\| \leq \|g\|$. Sometimes we write
$\|\cdot\|_{X}$ to be sure which norm is taken in the space. If it is not stated otherwise we understand that in a Banach ideal
space there is $f\in X$ with $f(x) > 0$ for each $x \in I$ (such an element is called the {\it weak unit} in $X$), which means that
${\rm supp}X =I$. In the paper we concentrate on three underlying measure spaces $(I,\Sigma,m)$. If we say that $X$ is a
{\it Banach function space} it means that it is a Banach ideal space where $I=[0,1]$ or $I=[0,\infty)$ and $m$ is just the Lebesgue
measure, and $X$ is a {\it Banach sequence space} when $I = \mathbb{N}$ with counting measure.
Later on by saying Banach ideal space we mean only one of those three cases.
For two Banach spaces $X$ and $Y$ the symbol $X\overset{A}{\hookrightarrow }Y$ means that the embedding $X \subset Y$
is continuous with the norm at most $A$, i.e., $\| f\|_{Y} \leq A \|f\|_{X}$ for all $f\in X$. When $X\overset{A}{\hookrightarrow }Y$
holds with some constant $A > 0$ we simply write $X\hookrightarrow Y$. Furthermore, $X = Y$ (or $X \equiv Y$) means that
the spaces are the same and the norms are equivalent (or equal).
For a Banach ideal space $X = (X, \|\cdot\|)$ the {\it K{\"o}the dual space} (or {\it associated space}) $X^{\prime}$ is the
space of all $f \in L^0$ such that the {\it associated norm}
\begin{equation} \label{dual}
\|f\|^{\prime} = \sup_{g \in X, \, \|g\|_{X} \leq 1} \int_{I} |f g | \, dm
\end{equation}
\vspace{-2mm}
is finite. The K{\"o}the dual $X^{\prime} = (X^{\prime}, \|\cdot \|^{\prime})$ is then a Banach ideal space.
Moreover, $X \overset{1}{\hookrightarrow }X^{\prime \prime}$ and we have equality $X = X^{\prime \prime}$ with
$\|f\| = \|f\|^{\prime \prime}$ if and only if the norm in $X$ has the {\it Fatou property}, that is, if the conditions
$0 \leq f_{n} \nearrow f$ a.e. on $I$ and $\sup_{n \in {\bf N}} \|f_{n}\| < \infty$ imply that $f \in X$ and $\|f_{n}\| \nearrow \|f\|$.
For a Banach ideal space $X = (X, \| \cdot\|)$ on $I$ with the K\"othe dual $X^{\prime}$ the following {\it generalized H\"older-Rogers
inequality} holds: if $f \in X$ and $g \in X^{\prime}$, then $fg$ is integrable and
\begin{equation} \label{1.2}
\int_I |f(x) g(x)| \, dx \leq \| f\|_X \| g \|_{X^{\prime}}.
\end{equation}
A function $f$ in a Banach ideal space $X$ on $I$ is said to have an {\it order continuous norm} in $X$ if, for any
decreasing sequence of $m$-measurable sets $A_{n} \subset I $ with $m(\bigcap A_n) = 0$, we have that $\|f \chi_{A_{n}} \| \rightarrow 0$
as $n \rightarrow \infty$. The set of all functions in $X$ with an order continuous norm is denoted by $X_{a}$. If $X_{a} = X$, then
the space $X$ is said to be {\it order continuous} (we write shortly $X\in (OC)$). For an order continuous Banach ideal space $X$
the K{\"o}the dual $X^{\prime}$ and the dual space $X^{*}$ coincide. Moreover, a Banach ideal space $X$ with the Fatou property
is reflexive if and only if both $X$ and its associate space $X^{\prime}$ are order continuous.
For a given {\it weight} $w$, i.e. a measurable function on $I$ with $0 < w(x) < \infty$ a.e. and for a Banach ideal space $X$
on $I$, the {\it weighted Banach ideal space} $X(w)$ is defined as $X(w)=\{f\in L^0: fw \in X\}$ with the norm
$\|f\|_{X(w)}=\| f w \|_{X}$. Of course, $X(w)$ is also a Banach ideal space and
\begin{equation} \label{dualweight}
[X(w)]^{\prime} \equiv X^{\prime}\Big(\frac{1}{w}\Big).
\end{equation}
By a {\it rearrangement invariant} or {\it symmetric space} on $I$ with the Lebesgue measure $m$, we mean a Banach
function space $X=(X,\| \cdot \|_{X})$ with additional property that for any two equimeasurable functions
$f \sim g, f, g \in L^{0}(I)$ (that is, they have the same distribution functions $d_{f}\equiv d_{g}$, where
$d_{f}(\lambda) = m(\{x \in I: |f(x)|>\lambda \}),\lambda \geq 0)$, and $f\in E$ we have that $g\in E$ and $\| f\|_{E} = \| g\|_{E}$.
In particular, $\| f\|_{X}=\| f^{\ast }\|_{X}$, where $f^{\ast }(t)=\mathrm{\inf } \{\lambda >0\colon \ d_{f}(\lambda ) < t\},\ t\geq 0$.
Similarly one can define a {\it symmetric sequence space}.
For general properties of Banach ideal spaces and symmetric spaces we refer to the books \cite{BS88}, \cite{KA77}, \cite{KPS82},
\cite{LT79} and \cite{Ma89}.
In order to define and formulate results we need the (continuous) Ces\`aro and Copson operators $C, C^*$ defined, respectively,
as
$$
Cf(x) = \frac{1}{x} \int_0^x f(t) \,dt, 0 < x \in I, ~~C^*f(x) = \int_{I \cap [x, \infty)} \frac{f(t)}{t} \,dt, x \in I,
$$
where $I=[0,1]$ or $I=[0,\infty)$.
The nonincreasing majorant $\widetilde{f}$ of a given function $f$, is defined for $x \in I$ as
$$
\widetilde{f}(x) = \esssup_{t \in I, \, t \geq x} |f(t)|.
$$
For a Banach function space $X$ on $I$ we define the {\it abstract Ces\`aro (function) space} $CX = CX(I)$ as
\begin{equation} \label{Cesaro}
CX=\{f\in L^0(I): C|f| \in X\} ~~ {\rm with ~the ~norm } ~~ \|f\|_{CX} = \| C|f| \|_{X},
\end{equation}
the {\it abstract Copson space} $C^*X = C^*X(I)$ as
\begin{equation} \label{Copson}
C^*X=\{f\in L^0(I): C^*|f| \in X\} ~~ {\rm with ~the ~norm } ~~ \|f\|_{C^*X} = \| C^*|f| \|_{X},
\end{equation}
and the {\it abstract Tandori space} $\widetilde{X} = \widetilde{X} (I)$ as
\begin{equation} \label{falka}
\widetilde{X}=\{f\in L^0(I): \widetilde{f}\in X\} ~~ {\rm with ~the ~norm } ~~ \|f\|_{\widetilde{X}}=\|\widetilde{f}\|_{X}.
\end{equation}
The {\it dilation operators} $\sigma_\tau$ ($\tau > 0$) defined on $L^0(I)$ by
$$
\sigma_\tau f(x) = f(x/\tau) \chi_{I}(x/\tau) = f(x/\tau) \chi_{[0, \, \min(1, \, \tau)]}(x), ~~ x \in I,
$$
are bounded in any symmetric function space $X$ on $I$ and $\| \sigma_\tau \|_{X \rightarrow X} \leq \max \,(1, \tau)$ (see \cite[p. 148]{BS88}
and \cite[pp. 96-98]{KPS82}). These operators are also bounded in some Banach function spaces which are not necessary symmetric.
For example, if either $X = L^p(x^{\alpha})$ or $X = C(L^p(x^{\alpha}))$, then $\| \sigma_\tau\|_{X \rightarrow X} = \tau^{1/p + \alpha}$
(see \cite{Ru80} for more examples).
In the sequence case the discrete Ces\`aro and Copson operators $C_d, C_d^*$ are defined for $n \in {\mathbb N}$ by
$$
(C_d a)_n = \frac{1}{n} \sum_{k = 1}^n a_k, ~ (C^*_d a)_n = \sum_{k = n}^{\infty} \frac{a_k}{k}.
$$
The nonincreasing majorant $\widetilde{a}$ of a given sequence $a = (a_n)$ is defined for $n \in \mathbb N$ as
$$
\widetilde{a_n} = \sup_{k \in {\mathbb N}, \, k \geq n} |a_k|.
$$
Then the corresponding {\it Ces\`aro sequence space} $C_dX$, {\it Copson sequence space} $C_d^*X$
and {\it Tandori sequence space} $\widetilde{X}_d$ are defined analogously as in (\ref{Cesaro}), (\ref{Copson})
and (\ref{falka}). Moreover, for every $m \in \mathbb N$ let $\sigma_m$ and $\sigma_{1/m}$ be the {\it dilation operators}
defined in spaces of sequences $a = (a_n)$ by (cf. \cite[p. 131]{LT79} and \cite[p. 165]{KPS82}):
$$
\sigma_m a = \left( ( \sigma_m a)_n \right)_{n=1}^{\infty} = \big (a_{[\frac{m-1+n}{m}]} \big)_{n=1}^{\infty}
= \big ( \overbrace {a_1, a_1, \ldots, a_1}^{m}, \overbrace {a_2, a_2, \ldots, a_2}^{m}, \ldots \big)
$$
\begin{eqnarray*}
\sigma_{1/m} a
&=&
\left( ( \sigma_{1/m} a)_n \right)_{n=1}^{\infty} = \Big (\frac{1}{m} \sum_{k=(n-1)m + 1}^{nm} a_k \Big)_{n=1}^{\infty} \\
&=&
\big ( \frac{1}{m} \sum_{k=1}^m a_k, \frac{1}{m} \sum_{k=m+1}^{2m} a_k, \ldots, \frac{1}{m} \sum_{k=(n-1)m + 1}^{nm} a_k, \ldots \big).
\end{eqnarray*}
These operators are discrete analogs of the dilation operators $\sigma_{\tau}$ defined in function spaces.
They are bounded in any symmetric sequence space but also in some Banach sequence spaces, for example,
$\| \sigma_{m} \|_{l^p(n^{\alpha}) \rightarrow l^p(n^{\alpha})} \leq m^{1/p} \max(1, m^{\alpha})$ and
$\| \sigma_{1/m} \|_{l^p(n^{\alpha}) \rightarrow l^p(n^{\alpha})} \leq m^{-1/p} \max \,(1, m^{-\alpha})$.
Properties of Ces\`aro sequence spaces $ces_p = Cl^p$ were investigated in many papers (see \cite{MPS07} and references
given there), while properties of Ces\`aro function spaces $Ces_p(I) = CL^p(I)$ we can find in \cite{AM09} and \cite{AM14b}.
Abstract Ces\`aro spaces $CX$ for Banach ideal spaces $X$ on $[0, \infty)$ were defined already in \cite{Ru80} and spaces
$CX, \,\widetilde{X}$ for $X$ being a symmetric space on $[0,\infty)$ have appeared, for example, in \cite{KMS07}, \cite{DS07}
and \cite{AM13}. General considerations of abstract Ces\`aro spaces began to be studied in papers \cite{LM15a, LM15b}.
Copson sequence spaces $cop_p = C^*l^p$ and Copson function spaces $Cop_p = C^*L^p$ on $[0, \infty)$ we can find in
G. Bennett's memoir \cite[pp. 25-28 and 123]{Be96}. Moreover, Copson function spaces $Cop_p = C^*L^p$ on $[0, 1]$ were
used in \cite{AM13} (see also \cite{AM14a}, \cite{AM14b}) to understand Ces\`aro spaces and their interpolation.
We put the name generalized Tandori function spaces on $\widetilde{X} = \widetilde{X}(I)$ in honour of Tandori who proved
in 1954 that the dual space to $(Ces_{\infty}[0, 1])_a$ is $\widetilde{L^1[0, 1]}$. In 1966 Luxemburg-Zaanen \cite[Theorem 4.4]{LZ66} have
found the K\"othe dual of $Ces_{\infty}[0, 1]$ as $(Ces_{\infty}[0, 1])^{\prime} \equiv \widetilde{L^1[0, 1]}$. Already in 1957,
Alexiewicz \cite[Theorem 1]{Al57} showed (even for weighted case) that $( \widetilde{l^1} )^{\prime} \equiv ces_{\infty}$.
In \cite[Theorem 7]{LM15a} we were able to give a simple proof of a generalization of the Luxemburg-Zaanen duality theorem:
$\big [ C(L^{\infty}(v)) \big]^{\prime} \equiv \widetilde{L^1(w)}$, where $v(x) = x/\int_0^x w(t)\, dt, x \in I$.
Bennett \cite[Corollary 12.17]{Be96} proved that $(ces_p)^{\prime} = \widetilde{l^{p^{\prime}}}$ for $1 < p < \infty$. Surprisingly, the dual
of Ces\`aro function space is essentaially different for $I = [0, \infty)$ and for $I = [0,1]$, as it was proved by Astashkin-Maligranda \cite{AM09} (see also \cite{LM15a} for a simpler proof).
Namely, for $1 < p < \infty$ we have $(Ces_p[0, \infty))^{\prime} = \widetilde{L^{p^{\prime}}[0, \infty)}$ (cf. \cite[Theorem 2]{AM09}) and
$(Ces_p[0, 1])^{\prime} = \widetilde{L^{p^{\prime}}(\frac{1}{1-x})[0, 1]}$ (cf. \cite[Theorem 3]{AM09}). Generalized Tandori
spaces $\widetilde{X}$ (without this name on the spaces) and their properties appeared in papers \cite[p. 935]{LM15a}
and \cite[p. 228]{LM15b}.
\vspace{3mm}
We will need the following result on duality of abstract Ces\`aro spaces proved in \cite{LM15a}.
\vspace{1mm}
{\bf Theorem A.} {\it
Let $X$ be a Banach ideal space with the Fatou property such that the Ces\`aro operator $C$ is bounded on $X$.
\begin{itemize}
\item[(i)] If $I = [0,\infty)$ and the dilation operator $\sigma_\tau$ is bounded on $X$ for some $0 < \tau < 1$, then
\begin{equation} \label{Thm2i}
(CX)^{\prime} = \widetilde{X^{\prime}}.
\end{equation}
\item[(ii)] If $X$ is a symmetric space on $[0, 1]$ such that $C, C^*: X \rightarrow X$ are bounded, then
\begin{equation} \label{Thm2ii}
(CX)^{\prime} = \widetilde{ X^{\prime}(1/v)}, ~ {\rm where} ~~ v(x) = 1-x, ~x \in [0, 1).
\end{equation}
\item[(iii)] If $X$ is a sequence space and the dilation operator $\sigma_{3}$ is bounded on $X^{\prime}$, then
\begin{equation} \label{Thm2iii}
(CX)^{\prime} = \widetilde{X^{\prime}}.
\end{equation}
\end{itemize}
}
\vspace{3mm}
The paper is organized as follows. In Section 2 we present comparisons of Ces{\`a}ro, Copson and Tandori spaces
as well as the ``iterated'' spaces $CCX$ and $C^*C^*X$.
Section 3 contains results on commutativity of the Calder\'on-Lozanovski{\v \i} construction with abstract Ces{\`a}ro spaces
and with generalized Tandori spaces. There are some differences in assumptions on Ces{\`a}ro function spaces in the
cases on $[0, \infty)$ and on $[0, 1]$, and the sequence case, as we can see in Theorem \ref{thm:cesCL}.
Important in our investigations were results proved in \cite{LM15a} on K{\"o}the duality of $CX$ (cf. Theorem A).
In the case of generalized Tandori spaces we were able to prove an analogous result in Theorem \ref{thm:tyldaspace} by using another method, that ommits the duality argument.
Results proved here are then used to described interpolation of abstract Ces{\`a}ro and Tandori spaces by the complex method.
Identifications in Theorem 5 are new even for classical Ces{\`a}ro spaces $Ces_p(I)$.
In Section 4, the commutativity of the real method of interpolation with abstract Ces{\`a}ro spaces is investigated in
Theorem \ref{thm:Cesaro=real}. We also collected here our knowledge about earlier results on interpolation of Ces{\`a}ro
spaces $Ces_p$ and their weighted versions.
Finally, in Section 5, we collected information on Calder\'on couples of Ces{\`a}ro spaces and some related spaces. Several remarks
and open problems are also formulated. From all the above discussions we can see a big difference between interpolation of abstract
Ces{\`a}ro spaces on intervals $[0, \infty)$ and $[0, 1]$.
\section{Comparison of Ces{\`a}ro, Copson and Tandori spaces}
First of all notice that $X\overset{A}{\hookrightarrow }Y$ implies $CX\overset{A}{\hookrightarrow }CY$,
$C^*X\overset{A}{\hookrightarrow }C^*Y$ and $\widetilde{X} \overset{A}{\hookrightarrow } \widetilde{Y}$.
Moreover, it can happend that spaces are different but corresponding Ces{\`a}ro, Copson and Tandori spaces
are the same, that is, there are $X \neq Y$ such that $CX = CY$, $C^*X = C^*Y$ and $ \widetilde{X} = \widetilde{Y}$.
\begin{example} \label{Ex1}
If $X = L^2[0, \frac{1}{4}] \oplus L^{\infty}[\frac{1}{4}, \frac{1}{2}] \oplus L^2[\frac{1}{2}, 1]$, then $X\hookrightarrow L^2[0, 1]$,
$CX = CL^2 = Ces_2[0,1], C^*X = C^*L^2 = Cop_2[0, 1]$ and $ \widetilde{X} = \widetilde{L^2[0, 1]}$, because
\begin{eqnarray*}
\sup_{x \in [ \frac{1}{4}, \frac{1}{2}]} \frac{1}{x} \int_0^x |f(t)| \, dt
&\leq &
4 \int_0^{1/2} |f(t)| \, dt = 4 \int_0^{1/2} |f(t)| \, dt \, (\int_{1/2}^1 x^{-2} dx)^{1/2} \\
&\leq&
4 \left[ \int_{1/2}^1 \big( \frac{1}{x} \int_0^x |f(t)| \, dt \big)^2dx \right]^{1/2},
\end{eqnarray*}
\begin{equation*}
\sup_{x \in [ \frac{1}{4}, \frac{1}{2}]} \int_x^1 \frac{|f(t)|}{t} \, dt = \int_{1/4}^1 \frac{|f(t)|}{t} \, dt \leq
2 \left[ \int_0^{1/4} \big( \int_x^1 \frac{|f(t)|}{t} \, dt \big)^2 dx\right]^{1/2},
\end{equation*}
and
\begin{equation*}
\sup_{x \in [ \frac{1}{4}, \frac{1}{2}]} \widetilde{f}(x) = \widetilde{f}(1/4) \leq 2 \, \big( \int_0^{1/4} \widetilde{f}(x)^2 \, dx \big)^{1/2}.
\end{equation*}
\end{example}
Since $\widetilde{X} \overset{1}{\hookrightarrow } X$ it follows that $C \widetilde{X} \overset{1}{\hookrightarrow }CX$ and the reverse
imbedding, under some assumptions on $X$, was proved in \cite[Theorem 1]{LM15b}. Namely, consider the maximal operator $M$
(defined for $x \in I$ by $Mf(x) = \sup_{a, b \in I, 0 \leq a \leq x \leq b} \frac{1}{b-a} \int_a^b |f(t)|\, dt$) and a Banach ideal space $X$
on $I$. In the case $I = [0, \infty)$, if $M$ is bounded on $X$, then
\begin{equation} \label{2.1}
CX \overset{B}{\hookrightarrow } C \widetilde{X} ~~ {\rm with} ~ B = 4 \, \| M \|_{X \rightarrow X}
\end{equation}
(cf. \cite[Theorem 1(i)]{LM15b}), and in the case $I = [0, 1]$ if $M, \sigma_{1/2}$ are bounded on $X$ and $L^{\infty} \hookrightarrow X$, then
\begin{equation} \label{2.2}
CX \cap L^1 \overset{B_1}{\hookrightarrow } C \widetilde{X} \overset{B_2}{\hookrightarrow } CX \cap L^1
\end{equation}
with $B_1 = 4 \, \| M \|_{X \rightarrow X} \| \sigma_{1/2} \|_{X \rightarrow X}$ and $B_2 = \max \,(1, 1/\| \chi_{[0, 1]} \|_X)$ (cf.
\cite[Theorem 1(iii)]{LM15b}). In particular, if $X$ is a symmetric space on $I$ and $C$ is bounded on $X$, then
\begin{equation} \label{2.3}
C \widetilde{X} = CX ~ {\rm for} ~ I = [0, \infty) ~ {\rm and} ~~ C \widetilde{X} = CX \cap L^1 ~ {\rm for} ~ I = [0, 1].
\end{equation}
We should mention here that the boundedness of $C$ on a symmetric space $X$ implies boundedness of the maximal operator
$M$ on $X$, which follows from the Riesz inequality $(Mf)^*(x) \leq c\, Cf^*(x)$ true for any $x \in I$ with a constant $c \geq 1$ independent
of $f$ and $x$ (cf. \cite[p. 122]{BS88}).
\vspace{0.1mm}
Now, we collect inclusions and equalities between Ces{\`a}ro spaces $CX$, Copson spaces $C^*X$, their iterations $ CCX, C^*C^*X$
and Tandori spaces $ \widetilde{X}$. Some results were proved before for $X = l^p$ by Bennett in \cite{Be96} and for $X = L^p$ by
Astashkin-Maligranda \cite{AM09}. Moreover, Curbera and Ricker in \cite{CR13} already proved point (viii) in the theorem below.
Let us recall that the {\it unilateral shift} $S$ on a sequence space is defined by $S(x_1, x_2, x_3, \ldots) = (0, x_1, x_2, x_3, \ldots)$.
\begin{theorem}\label{thm:relations} \item[(a)] Let $X$ be a Banach function space on $I = [0, \infty)$.
\begin{itemize}
\item[(i)] If $C$ is bounded on $X$, then $X \hookrightarrow CX$ and if, additionally, the dilation operator
$\sigma_{1/a}$ is bounded on $X$ for some $a > 1$, then $CX = CCX$.
\item[(ii)] If $C^*$ is bounded on $X$, then $X \hookrightarrow C^*X$ and if, additionally, the dilation operator
$\sigma_{1/a}$ is bounded on $X$ for some $0 < a < 1$, then $C^*X = C^*C^*X$.
\item[(iii)] If both operators $C$ and $C^*$ are bounded on $X$, then $CX = C^*X$.
\end{itemize}
\item[(b)] Let $X$ be a Banach function space on $I = [0, 1]$.
\vspace{-2mm}
\begin{itemize}
\item[(iv)] If $C$ is bounded on $X$, then $X \overset{A}{\hookrightarrow } CX$ and $C^*X\overset{A}{\hookrightarrow } CX$
with $A = \| C \|_{X \rightarrow X}$. The last embedding is strict even if $X = L^p[0, 1]$ with $1 < p < \infty$.
\item[(v)] If $C^*$ is bounded on $X$, then $X \hookrightarrow C^*X$. If, additionally, the dilation operator $\sigma_{1/a}$ is
bounded on $X$ for some $0 < a < 1$, then $C^*X = C^*C^*X$.
\item[(vi)] If $C^*$ is bounded on $X$ and $L^{\infty} \hookrightarrow X$, then $C^*X \hookrightarrow CX \cap L^1$. If,
additionally, $C$ is bounded on $X$ and $X \hookrightarrow L^1$, then $C^*X= CX \cap L^1$ and
$(C^*X)^{\prime} = \widetilde{X^{\prime}}$.
\item[(vii)] If both operators $C$ and $C^*$ are bounded on $X$, and $X$ is a symmetric space, then
$C^*X = CX \cap L^1 = C \widetilde{X}$.
\end{itemize}
\item[(c)] Let $X$ be a Banach sequence space.
\begin{itemize}
\item[(viii)] If $C_d$ is bounded on $X$, then $X \hookrightarrow C_dX$ and if, additionally, the dilation operator $\sigma_{1/2}$
is bounded on $X$, then $C_dX = C_dC_dX$.
\item[(ix)] If $C_d^*$ is bounded on $X$, then $X \hookrightarrow C_d^*X$ and if, additionally, the dilation operator
$\sigma_2$ is bounded on $X$, then $C_d^*X = C_d^*C_d^*X$.
\item[(x)] If operators $C_d, C_d^*$ and unilateral shift $S$ with its dual $S^*$ are bounded on $X$, then $C_dX = C_d^*X$.
\end{itemize}
\end{theorem}
\proof
(i) The first inclusion is clear from which we obtain also $CX \hookrightarrow CCX$. Then the equality $CX = CCX$
follows from Lemma 6 in \cite{LM15a}, where it was proved that $CC|f|(x) \geq \frac{\ln a}{a} \, C|f|(x/a)$ for all $x > 0$
with arbitrary $a > 1$. Thus
$$
\sigma_{1/a} CC|f|(x) = CC|f|(ax) \geq \frac{\ln a}{a} \, C|f|(x),
$$
and so
$$
\| C|f|\|_X \leq \frac{a}{\ln a} \| \sigma_{1/a} CC|f| \|_X \leq \frac{a}{\ln a} \| \sigma_{1/a}\|_{X \rightarrow X} \| CC|f| \|_X,
$$
which gives the required inclusion $CCX \hookrightarrow CX$.
(ii) Once again the first inclusion comes directly from the assumption and thus $C^*X \hookrightarrow C^*C^*X$.
To get the reverse inclusion observe that for $f \geq 0, x>0$ and $0 < a < 1$ by monotonicity of $C^*f$ we have
\begin{eqnarray*}
\sigma_{1/a}C^*C^*f(x)
&=&
C^*C^*f(ax) = \int_{ax}^{\infty} \frac{C^*f(t)}{t} \, dt \\
&\geq&
\int_{ax}^x\frac{C^*f(t)}{t} \, dt \geq C^*f(x) \ln \frac{1}{a}.
\end{eqnarray*}
Thus,
$$
\| C^* f\|_X \leq \frac{1}{\ln1/a} \| \sigma_{1/a}(C^*C^*)f \|_X \leq \frac{\| \sigma_{1/a} \|_{X \rightarrow X}}{\ln1/a} \, \| C^*C^*f \|_X,
$$
which gives the required inclusion $C^*C^*X \hookrightarrow C^*X$.
\vspace{2mm}
(iii) Since $C|f| + C^*|f| = C^*C|f|$ and $C|f| + C^*|f| = CC^*|f|$ it follows that
\begin{eqnarray*}
\| f \|_{CX}
&=&
\| C|f| \|_X \leq \| C|f| + C^*|f| \|_X = \| C^*C|f| \|_X \\
&\leq&
\| C^* \|_{X \rightarrow X} \| C|f| \|_X = \| C^* \|_{X \rightarrow X} \, \| f \|_{CX}
\end{eqnarray*}
and
\begin{eqnarray*}
\| f \|_{C^*X}
&=&
\| C^*|f| \|_X \leq \| C|f| + C^*|f| \|_X = \| CC^*|f| \|_X \\
&\leq&
\| C \|_{X \rightarrow X} \| C^*|f| \|_X = \| C \|_{X \rightarrow X} \, \| f \|_{C^*X}.
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
\| f \|_{CX}
&\leq&
\| C|f| + C^*|f| \|_X \leq \| C \|_{X \rightarrow X} \, \| f \|_{C^*X} \\
&\leq&
\| C \|_{X \rightarrow X} \, \| C|f| + C^*|f| \|_X \leq \| C \|_{X \rightarrow X} \, \| C^* \|_{X \rightarrow X} \, \| f \|_{CX},
\end{eqnarray*}
and so $C^*X\overset{A}{\hookrightarrow } CX \overset{ B}{\hookrightarrow } C^*X$ with $A = \| C \|_{X \rightarrow X}$ and
$B = \| C^* \|_{X \rightarrow X}$.
\vspace{2mm}
(iv) The first inclusion is clear. The second embedding follows from the equality $ C + C^* = C C^*$, which gives
\begin{equation} \label{imbeddingCC}
C^*X\overset{A}{\hookrightarrow } CX ~ {\rm with} ~ A = \| C \|_{X \rightarrow X}.
\end{equation}
Moreover, equality in (\ref{imbeddingCC}) does not hold in general as it was shown already in \cite[p. 48]{AM09} for $X = L^p[0, 1]$.
\vspace{2mm}
(v) The prove is the same as in $(ii)$.
\vspace{2mm}
(vi) Since for $f \geq 0$ and $x \in [0, 1]$ we have (see also \cite[p. 48]{AM13})
$$
C^*Cf(x) = Cf (x) + C^*f(x) - \int_0^1 f(t) \, dt
$$
it follows that
\begin{eqnarray*}
\| f \|_{C^*X}
&=&
\| C^*f \|_X \leq \| C^*f + Cf \|_X = \| C^* Cf + \int_0^1 f(t) \, dt \, \chi_{[0, 1]} \|_X \\
& \leq&
\| C^* \|_{X \rightarrow X} \| Cf\|_X + \| f \|_{L^1} \, \| \chi_{[0, 1]} \|_X \\
&\leq&
4 \, \max \{\| C^* \|_{X \rightarrow X}, \| \chi_{[0, 1]} \|_X \} \, \max \{ \| Cf\|_X, \| f \|_{L^1} \}\\
&=&
4 \, \max \{\| C^* \|_{X \rightarrow X}, \| \chi_{[0, 1]} \|_X \} \, \| f\|_{CX \cap L^1},
\end{eqnarray*}
that is, $CX \cap L^1\overset{D}{\hookrightarrow } C^*X$ with $D = 4 \, \max \{\| C^* \|_{X \rightarrow X}, \| \chi_{[0, 1]} \|_X \}$.
On the other hand, from (\ref{imbeddingCC}) and since $X \hookrightarrow L^1$ it follows $C^*X \hookrightarrow C^*L^1 \equiv L^1$
and we obtain $C^*X \hookrightarrow CX \cap L^1$. Therefore, $C^*X = C^*L^1 \cap L^1$.
The embedding $(C^*X)^{\prime} \hookrightarrow \widetilde{X^{\prime}}$ can be proved in the following way.
Using just mentioned identification (\ref{2.3}), equality of the K\"othe dual of the sum as the intersection of K\"othe duals
(see, for example, \cite[Lemma 3.4, p. 342]{LZ66} or \cite[Lemma 15.5]{Ma89}) and Theorem A(ii) we obtain
$$
(C^*X)^{\prime} = (CX \cap L^1)^{\prime} = (CX)^{\prime} + L^{\infty} = \widetilde{X^{\prime}(1/v)} + L^{\infty}.
$$
Then, since $v(x) = 1 - x \leq 1$ it follows that $\widetilde{X^{\prime}(1/v)} \overset{1}{\hookrightarrow } \widetilde{X'}$ and by the assumption
$X \hookrightarrow L^1$ we obtain $L^{\infty} \hookrightarrow X^{\prime}$ which gives
$L^{\infty} = \widetilde{L^{\infty}} \hookrightarrow \widetilde{X^{\prime}}$. Thus, $\widetilde{X^{\prime}(1/v)} + L^{\infty} \hookrightarrow \widetilde{X'}$.
To finish the proof we need to show the embedding $\widetilde{X'} \hookrightarrow \widetilde{X^{\prime}(1/v)} + L^{\infty}$. Let $0\leq f\in \widetilde{X'}$.
Then
$$
\|(f - \tilde f(1/2))_+\|_{\widetilde{X^{\prime}(1/v)}}=\|\frac{(\tilde f - \tilde f(1/2))_+}{v}\|_{X^{\prime}}
\leq \frac{1}{1-1/2}\|(\tilde f - \tilde f(1/2))_+\|_{X^{\prime}}\leq 2 \, \|\tilde f \|_{X^{\prime}}.
$$
Moreover, by the H\"older-Rogers inequality (\ref{1.2}), we obtain for $f \in X$ and any $0 < t < 1$,
$$
f^*(t) \leq \frac{1}{t} \int_0^t f^*(s)\, ds \leq \frac{1}{t} \| \chi_{[0, t]}\|_X \, \| f^*\|_{X^{\prime}}.
$$
Therefore, $\|\tilde f(1/2) \chi_{[0,1]}\|_{L^{\infty}} = \tilde f(1/2) \leq 2 \, \varphi_X(1/2) \|\tilde f\|_{X'} $,
and consequently
$$
\| f \|_{\widetilde{X^{\prime}(1/v)} + L^{\infty}} \leq [2+ 2\, \varphi_{X}(1/2)] \, \| f \|_{\widetilde {X'}} \,.
$$
(vii) The first equality follows from (vi) and the second from (\ref{2.3}).
(viii) Of course, $C_dX \hookrightarrow C_d C_dX$ and the reverse inclusion for $X = l^p$ was already given by
Bennett \cite[20.31]{Be96}, but it was simplified and generalized by Curbera and Ricker \cite[Proposition 2]{CR13}
who have shown that for $n \geq 2$ there holds
\begin{equation} \label{CR}
\dfrac{1}{[n/2]} \sum_{j=1}^{[n/2]} |a_j| \leq 6 \sum_{k=1}^n \dfrac{1}{k} \sum_{j=1}^k |a_j|.
\end{equation}
Thus $(C_d a)_n \leq 6\, (C_d C_d a)_{2n} \leq 12 \, (\sigma_{1/2} C_d C_d a)_n$ and
$$
\| C_d a \|_X \leq 12 \|\sigma_{1/2}\|_{X \rightarrow X} \, \| C_dC_d a\|_X,
$$
which gives the required inclusion.
(ix) Of course, $C^*_dX \hookrightarrow C^*_d C^*_dX$ and we need only to prove the reverse inclusion. Since
$ \sum_{k=n}^{2n-1} \frac{1}{k} \geq \int_n^{2n} \frac{1}{t} dt = \ln 2 \geq \frac{1}{2}$ it follows that
\begin{eqnarray*}
(C^*_d C^*_d a)_n
&=&
\sum_{k=n}^{\infty} \frac{(C^*_d a)_k}{k} \geq \sum_{k=n}^{2n-1} \frac{(C^*_d a)_k}{k} \geq
(C^*_d a)_{2n-1} \sum_{k=n}^{2n-1} \frac{1}{k} \\
&\geq&
\frac{1}{2} \, (C^*_d a)_{2n-1} \geq \frac{1}{2} \, (C^*_d a)_{2n},
\end{eqnarray*}
and
$$
(\sigma_2C^*_d C^*_d a)_n = (C^*_d C^*_d a)_{[\frac{n+1}{2}]} \geq \frac{1}{2} \, (C^*_d a)_n.
$$
Thus,
$$
\| C^*_d a \|_X \leq 2 \, \| \sigma_2 \|_{X \rightarrow X} \, \| C^*_d C^*_d a\|_X,
$$
which gives the required inclusion.
(x) This result for $X = l^p$ was proved by Bennett \cite[p. 47]{Be96} who observed that
\begin{equation*}
C_d = (C_d - S^*) C^*_d ~~{\rm and} ~~ C^*_d = (C^*_d - I) S C_d,
\end{equation*}
where $S$ is the unilateral shift and $S^*$ its dual.
Of course, his proof is also working for more general Banach sequence spaces. Namely,
$$
\| C_d a \|_X \leq \left( \| C_d \|_{X \rightarrow X} + \| S^* \|_{X \rightarrow X} \right) \| C^*_d a \|_X = A \, \| C^*_d a \|_X,
$$
which gives $C_d^*X \overset{A}\hookrightarrow C_dX$. Also
$$
\| C_d^* a \|_X \leq \left( \| C_d S\|_{X \rightarrow X} + \| S \|_{X \rightarrow X} \right) \| C^*_d a \|_X = B \, \| C^*_d a \|_X
$$
and so $C_dX \overset{B}\hookrightarrow C_d^*X$. Putting together both inclusions we obtain $C_dX = C_d^*X$.
\endproof
\begin{remark} \label{Re1}
On $[0,1]$ the space $CXX$ may be essentially bigger than $CX$. In fact, taking $f(x) = \frac{1}{(1 - x)^2}, x \in(0,1)$ we have
$f\in CCL^p$ for any $1 \leq p < \infty$ and $f \not \in CL^p$ since $Cf(x) = \frac{1}{1-x} \not \in CL^1$. Moreover, if the operator
$C$ is not bounded in $X$ on $[0, 1]$, then embedding relationships between $X, CX$ and $CCX$ may not hold.
For $X = L^1[0, 1]$ we have
\begin{equation*}
\| f \|_{CL^1} = \int_0^1 |f(x)| \, \ln (1/x) \, dx ~ {\rm and} ~ \| f \|_{CCL^1} = \int_0^1 |f(x)| \, \ln^2 (1/x) \, dx.
\end{equation*}
Spaces $L^1, CL^1$ and $CCL^1$ are not comparable. In fact, for $0 < \alpha < 1$ let $f_{\alpha}(x) = \frac{1}{x} \chi_{[\alpha, 1]}$.
Then $\| f_{\alpha} \|_{L^1} = \ln (1/\alpha), \| f_{\alpha} \|_{CL^1} = \frac{1}{2} \ln^2 (1/\alpha),
\| f_{\alpha} \|_{CCL^1} = \frac{1}{3} \ln^3 (1/\alpha)$ and
$$
\dfrac{2 \, \| f_{\alpha} \|_{CL^1}}{\| f_{\alpha} \|_{L^1}} = \dfrac{3 \, \| f_{\alpha} \|_{CCL^1}}{\| f_{\alpha} \|_{CL^1}} = \ln \frac{1}{\alpha} \rightarrow \infty ~{\rm as} ~ \alpha \rightarrow 0^+ ~( {\rm and} ~ \rightarrow 0 ~ {\rm as} ~ \alpha \rightarrow 1^-).
$$
\end{remark}
\section{Calder\'on-Lozanovski{\v \i} construction}
Let us recall the Calder\'on-Lozanovski{\v \i} construction for Banach ideal spaces. The class ${\mathcal U}$ consists of all functions
$\varphi: {\mathbb R_+} \times {\mathbb R_+} \rightarrow {\mathbb R_+}$ that are positively homogeneous (i.e.,
$\varphi(\lambda s, \lambda t) = \lambda \varphi (s, t)$ for every $s, t, \lambda \geq 0$) and concave, that is
$\varphi (\alpha s_1 + \beta s_2, \alpha t_1 + \beta t_2) \geq \alpha \varphi (s_1, t_1) + \beta \varphi(s_2, t_2)$ for all
$\alpha, \beta \in [0, 1]$ with $\alpha + \beta = 1$, and all $s_i, t_i \geq 0, i = 0, 1$. Note that any function $\varphi \in {\mathcal U}$ is
continuous on $(0, \infty) \times (0, \infty)$.
Given such $\varphi \in {\mathcal U}$ and a couple of Banach ideal spaces $(X_0, X_1)$ on the same measure space, the
{\it Calder\'{o}n-Lozanovski{\u \i} space} $\varphi (X_0, X_1)$ is defined as the set of all $f \in L^0$ such that for some $f_0 \in X_0, f_1 \in X_1$
with $\| f_0 \|_{X_0} \leq 1, \| f_1\|_{X_1} \leq 1$ and for some $\lambda > 0$ we have
\begin{equation*}
| f(x)| \leq \lambda\, \varphi(|f_0(x)|, | f_1(x)|) ~~ {\rm a.e. ~on} ~ I.
\end{equation*}
The norm $ \| f \|_{\varphi (X_0, X_1)}$ of an element $f \in \varphi(X_0, X_1)$ is defined as the
infimum of those values of $\lambda$ for which the above inequality holds and the space $(\varphi (X_0, X_1), \| \cdot \|_{\varphi})$
is then a Banach ideal space. It can be shown that
\begin{equation*}
\varphi (X_0, X_1) = \left\{ f\in L^{0}: |f| \leq \varphi (f_0, f_1) ~{\rm for ~some} ~ 0 \leq f_0 \in X_0,
~ 0 \leq f_1 \in X_1 \right \}
\end{equation*}
with the norm
\begin{equation*}
\| f \|_{\varphi \left( X_0, X_1\right) }=\inf \left\{ \max \left\{\| f_0 \|_{X_0}, \| f_1 \|_{X_1}\right\}: \text{\ }
| f| \leq \varphi (f_0, f_1)\text{,}\ 0 \leq f_0 \in X_0, 0 \leq f_1 \in X_1 \right\} \text{.}
\end{equation*}
In the case of power functions $\varphi (s, t) = s^{\theta } t^{1-\theta} $ with $0 \leq \theta \leq 1$ spaces $\varphi (X_0, X_1)$ became
the well known Calder\'on spaces $X_0^{\theta } X_1^{1-\theta }$ (see \cite{Ca64}). Another important situation, investigated
by Calder\'on and independently by Lozanovski{\u \i}, appears when $X_1\equiv L^{\infty }$. In particular,
$X^{1/p}(L^{\infty})^{1-1/p} = X^{(p)}$ for $1 < p < \infty $ is known as the {\it $p$-convexification} of $X$ (see \cite{LT79}).
The properties of $\varphi (X_0, X_1)$ were studied by Lozanovski{\v \i} in \cite{Lo73, Lo78a} and \cite{Lo78b} (see also \cite{Ma89}),
where among other facts it is proved the Lozanovski{\v \i} duality theorem: for any Banach ideal spaces $X_0, X_1$
with ${\rm supp} X_0 = {\rm supp} X_1$ and $\varphi \in {\mathcal U}$ we have
\begin{equation} \label{Lduality}
\varphi (X_0, X_1)^{\prime} = \hat{\varphi}(X_0^{\prime}, X_1^{\prime}),
\end{equation}
where the conjugate function $\hat {\varphi}$ is defined by
$$
\hat {\varphi} (s, t) := \inf \Big\{\frac {a s + b t} {\varphi (a, b)} ; \, a, b > 0\Big\}, s, t \geq 0.
$$
There hold $\hat {\varphi} \in {\mathcal U}$ and $\, \hat {{\hat \varphi}} = \varphi$ (see \cite[Lemma 2]{Lo78b}, \cite[Lemma 2]{Ma85} and
\cite[Lemma 15.8]{Ma89}). It is easy to see that the Calder\'on-Lozanovski{\v \i} construction $\varphi (\cdot)$ is homogeneous
with respect to an arbitrary weight $w$, that is, the equality
\begin{equation} \label{homogeneous}
\varphi (X_0(w), X_1(w)) = \varphi(X_0, X_1)(w),
\end{equation}
holds for arbitrary Banach ideal spaces $X_0, X_1$ and arbitrary weight $w$.
More information, especially on interpolation property, can be found in \cite{Be81, BK91, KLM13, KPS82, KMP93, Lo78b, Ma85, Ma89,
Ni85, Ov76, Ov84, Sh81}.
\vspace{3mm}
We shall now identify the Calder\'on-Lozanovski{\v \i} construction for abstract Ces\`aro spaces.
\begin{theorem}\label{thm:inclusions}
For any Banach ideal spaces $X_0, X_1$ and $\varphi \in {\mathcal U}$ the following embeddings hold
\begin{equation} \label{embeddings}
\varphi (CX_0,CX_1) \overset{1}{\hookrightarrow }C [\varphi(X_0,X_1)] ~~{\it and} ~~ \varphi (\widetilde{X_0},\widetilde{X_1})\overset{1}{\hookrightarrow } [\varphi (X_0,X_1)]^{\Large \sim }.
\end{equation}
\end{theorem}
\proof
Suppose $X_0, X_1$ are Banach function spaces on $I = [0, \infty)$ and let $f \in \varphi(C{X_0},C{X_1})$ with
$\|f \|_{\varphi(C{X_0},C{X_1})}<\lambda$.
Then $ |f|\leq \lambda \varphi (|f_0|, |f_1|)$ a.e. for some $f_i\in CX_i$ with $\| f_i\|_{C{X_i}}\leq 1$ for $i=0,1$.
Using now the Jensen inequality for concave function we obtain
\begin{eqnarray*}
C|f|(x)
&=&
\frac{1}{x}\int_0^x |f(t)| \, dt \leq \frac{\lambda}{x}\int_0^x \varphi (|f_0(t)|, |f_1(t)|) \, dt \\
&\leq&
\lambda \varphi \Big(\frac{1}{x}\int_0^x |f_0(t)| \, dt, \frac{1}{x}\int_0^x |f_1(t)| \, dt \Big) =\lambda \varphi (C|f_0|(x), C|f_1|(x)).
\end{eqnarray*}
Since $\|C|f_i| \|_{{X_i}} = \| f_i\|_{C{X_i}}\leq 1$, $i=0,1$, it follows that $f \in C [\varphi({X_0},{X_1})]$ with
$\|f\|_{C [\varphi({X_0},{X_1})]} = \|C|f| \|_{\varphi({X_0},{X_1})}\leq \lambda$ and the first embedding in (\ref{embeddings}) is proved.
To prove the second embedding in (\ref{embeddings}) let $f \in \varphi(\widetilde{X_0},\widetilde{X_1})$ with $\|f\|_{\varphi(\widetilde{X_0},\widetilde{X_1})}<\lambda$. This means that $ |f|\leq \lambda \varphi (|f_0|, |f_1|)$ a.e. for some $f_i\in \widetilde{X_i}$ with
$\|f_i\|_{\widetilde{X_i}}\leq 1$, $i=0,1$. Therefore, for all $x \in I$,
\begin{eqnarray*}
\widetilde{f}(x)
&=&
\esssup_{t \in I, \, t \geq x} |f(t)| \leq \lambda \esssup_{t \in I, \, t \geq x} \varphi (|f_0(t)|, |f_1(t)|) \\
&\leq&
\lambda \varphi \big (\esssup_{t \in I, \, t \geq x} |f_0(t)|, \esssup_{t \in I, \, t \geq x} |f_1(t)| \big) =\lambda \varphi (\widetilde{f_0}(x), \widetilde{f_1}(x)).
\end{eqnarray*}
Of course, $\widetilde{f_i} \in X_i$ and $\|\widetilde{f_i}\|_{X_i}\leq 1$ for $i = 0, 1$, which means that $\widetilde{f}\in\varphi (X_0,X_1)$
or $f \in \widetilde{\varphi (X_0,X_1)}$ with $\|f\|_{\widetilde{\varphi (X_0,X_1)}}\leq \|f\|_{\varphi(\widetilde{X_0},\widetilde{X_1})}$.
This proves the second embedding in (\ref{embeddings}) for the case of $I = [0, \infty)$.
The remaining cases of $I = [0, 1]$ or $I = \mathbb N$ require only the evident modifications and therefore will be omitted.
\endproof
Of course, the natural question is if there are equalities in (\ref{embeddings}) and, in fact, it is the case when we assume something more on spaces $X_0$ and $X_1$.
\begin{theorem}\label{thm:cesCL}
Let $X_0, X_1$ be Banach ideal spaces with the Fatou property and such that the Ces\`aro operator $C$ is bounded on $X_0$ and $X_1$. Suppose that one of the following conditions hold:
\begin{itemize}
\item[(i)] $I = [0,\infty)$ and the dilation operator $\sigma_\tau$ is bounded in $X_0$ and $X_1$ for some $0 < \tau < 1$,
\item[(ii)] $I = [0, 1]$ and $X_0, X_1$ are symmetric spaces with the Fatou property such that both operators
$C, C^*: X_i \rightarrow X_i$ are bounded for $i =0, 1$,
\item[(iii)] $I = \mathbb N$ and the dilation operator $\sigma_{3}$ is bounded on dual spaces $X_0^{\prime}$ and
$X_1^{\prime}$.
\end{itemize}
Then
\begin{equation} \label{3.4}
\varphi (CX_0, CX_1) = C [\varphi(X_0, X_1)].
\end{equation}
\end{theorem}
\proof
In view of Theorem \ref{thm:inclusions} we need to prove only the remaining inclusion. It appears however, that both inclusions
in (\ref{embeddings}) are complemented to each other by duality. Thus we have for particular cases:
(i) Let $I = [0,\infty)$. Using twice the Lozanovski{\u \i} duality theorem (\ref{Lduality}), Theorem A(i) on duality of Ces\`aro spaces and the
second imbedding from Theorem \ref{thm:inclusions} we obtain
\begin{eqnarray*}
[\varphi (CX_0, CX_1)]^{\prime}
&=&
\hat{\varphi}([CX_0]^{\prime}, [CX_1]^{\prime}) = \hat{\varphi}(\widetilde{X_0^{\prime}}, \widetilde{X_1^{\prime}})
\hookrightarrow [\hat{\varphi}(X_0^{\prime}, X_1^{\prime})]^{\thicksim} \\
&=&
[{\varphi}(X_0, X_1)^{\prime}]^{\thicksim} = [C\varphi(X_0,X_1)]^{\prime}.
\end{eqnarray*}
In the last equality, in order to use Theorem A(i), we notice that if $\sigma_{\tau}$ is bounded on $X_0$ and $X_1$ for some
$0 < \tau < 1$, then it is also bounded on $\varphi(X_0,X_1)$ (one can also use here a more general result that $\varphi(X_0,X_1)$ is an interpolation space between $X_0$ and $X_1$ for positive operators -- see \cite[Theorem 1]{Be81}, \cite[Theorem 3.1]{Sh81}, \cite[Theorem 1]{Ma85} and \cite[Theorem 15.13]{Ma89}).
Finally, by the Fatou property of both spaces, we obtain that $CX_0, CX_1, \varphi (X_0, X_1)$ and $\varphi (CX_0, CX_1)$ have
the Fatou property (cf. \cite[Theorem 1(d)]{LM15a} and \cite[Corollary 3, p. 185]{Ma89}), and so
$$
C[\varphi(X_0, X_1)] \equiv [C \big(\varphi(X_0, X_1) \big)]^{\prime \prime} \hookrightarrow
[\varphi (CX_0,CX_1)]^{\prime \prime} \equiv \varphi (CX_0,CX_1),
$$
which finishes the proof in this case.
(ii) Let $I = [0, 1]$. Similarly as before, by the Lozanovski{\u \i} duality result (\ref{Lduality}) used twice, Theorem A(ii) on duality of
Ces\`aro spaces, property (\ref{homogeneous}) and the second imbedding from Theorem \ref{thm:inclusions} we have
\begin{eqnarray*}
[\varphi(CX_0, CX_1)]^{\prime}
&=&
\hat{\varphi}([CX_0]^{\prime},[CX_1]^{\prime}) =
\hat{\varphi} \big(\widetilde{X_0^{\prime}(1/v)},\widetilde{X_1^{\prime}(1/v)} \big) \hookrightarrow
[\hat{\varphi}(X_0^{\prime}(1/v), X_1^{\prime}(1/v))]^{\thicksim} \\
&=&
[\hat{\varphi}(X_0^{\prime}, X_1^{\prime})(1/v)]^{\thicksim} = \big[{\varphi}(X_0, X_1)^{\prime}(1/v) \big]^{\thicksim} =
[C \big(\varphi(X_0,X_1) \big)]^{\prime}.
\end{eqnarray*}
where the weight $v$ is $v(x)=1-x, x \in [0, 1)$.
Observe that assumptions of the Theorem A(ii) are satisfied for ${\varphi}(X_0, X_1)$ thanks to interpolation property of the
Calder\'on-Lozanovski{\v \i} construction for positive operators. Once again, by the Fatou property of both spaces, we have
$$
C\big(\varphi(X_0, X_1)\big) \equiv [C \big(\varphi(X_0, X_1) \big)]^{\prime \prime} \hookrightarrow \varphi (CX_0,CX_1)^{\prime \prime}
\equiv \varphi (CX_0,CX_1).
$$
Proof in the sequence case is similar to the proof of case (i).
\endproof
Immediately from Theorems \ref{thm:inclusions} and \ref{thm:cesCL} by taking $\varphi(s, t) = \min(s, t)$ and
$\varphi (s, t) = \max (s, t) \approx s + t$ we obtain results for the intersection and the sum of Ces\`aro spaces. On the other hand,
taking $\varphi(s, t) = s^{1 - \theta} t^{\theta}, 0 < \theta < 1$ and $X_i = L^{p_i}$ or $X_i = l^{p_i}, i = 0, 1$ we obtain another corollary.
\begin{corollary} \label{Co1}
Under the assumptions of Theorem 3 for Banach ideal spaces $X_0, X_1$ we have
\begin{equation} \label{Ces+}
C(X_0 \cap X_1) = C(X_0) \cap C(X_1) ~~ {\rm and} ~~ C(X_0 + X_1) = C(X_0) + C(X_1).
\end{equation}
\end{corollary}
Note that one can even prove that $C(X_0 \cap X_1) \equiv C(X_0) \cap C(X_1)$ without additional assumptions on the spaces, but
the second equality in (\ref{Ces+}) is not true in general. If we consider spaces on $[0, \infty)$, then
$C(L^1) + C(L^{\infty}) = C(L^{\infty})\subsetneq C(L^1 + L^{\infty})$.
\begin{example} \label{Ex2}
Let $f(x) = x^{-1} \ln^{-3} (1/x) \, \chi_{(0, 1/e)}(x)$ for $x > 0$. Then $\| f \|_{L^1} = 1/2$ and
$f \in C(L^1 + L^{\infty}) \setminus C(L^{\infty}) $. In fact, since
$Cf(x) = \frac{1}{2 x \ln^2x} \, \chi_{(0, 1/e]}(x) + \frac{1}{2 x} \, \chi_{(1/e, \infty)}(x)$ it follows that
$$
\| f \|_{C(L^{\infty}) } = \sup_{x > 0} Cf(x) = \lim_{x \rightarrow 0^+} \frac{1}{2 x \ln^2x} = \infty,
$$
and
\begin{eqnarray*}
\| f \|_{C(L^1 + L^{\infty}) } =
\int_0^1 \big( C|f|)^*(x) \, dx = \int_0^{1/e} \frac{1}{2 x \ln^2x} \, dx + \int_{1/e}^1 \frac{1}{2 x} \, dx= 1.
\end{eqnarray*}
It seems to be of interest to investigate structure of the space $C(L^1 + L^{\infty})$ with its norm
$\| f \|_{C(L^1 + L^{\infty})} = \int_0^1 \big( C|f|)^*(x) \, dx$.
\end{example}
\begin{corollary} \label{Co2}
Let $1< p_0, p_1<\infty$ and $0<\theta<1$ be such that $\frac{1}{p}=\frac{1-\theta}{p_0}+\frac{\theta}{p_1}$. Then
\begin{equation} \label{Cestheta}
\big( Ces_{p_0} \big)^{1-\theta} \big(Ces_{p_1} \big)^{\theta} = Ces_{p} ~~ {\rm and} ~~ ( ces_{p_0})^{1-\theta} (ces_{p_1})^{\theta} = ces_{p}.
\end{equation}
\end{corollary}
Note that by the duality Theorem A and Theorem \ref{thm:cesCL} we have also equality
$\varphi(\widetilde{X_0}, \widetilde{X_1}) = [\varphi(X_0, X_1)]^{\Large \sim}$. It is however instructive to see that assumptions on $X_0$
and $X_1$ may be weakened if we do not use the duality argument in the proof.
\begin{theorem}\label{thm:tyldaspace}
Let $X_0$ and $X_1$ be two Banach ideal spaces with the Fatou property. If either $X_0, X_1$ are symmetric spaces or
$C, C^*$ are bounded on both $X_0$ and $X_1$, then
\begin{equation} \label{tilda}
\varphi(\widetilde{X_0}, \widetilde{X_1}) = [\varphi(X_0, X_1)]^{\Large \sim}.
\end{equation}
\end{theorem}
\proof
Of course, we need to prove only the inclusion ``$\hookrightarrow$" and the Fatou property is not necessary here. We restrict ourselfs to the case of function spaces on $I=[0,\infty)$, since the remaining cases are analogous. Let
$f \in \widetilde{\varphi(X_0, X_1)}$ with $\| f \|_{\widetilde{\varphi(X_0, X_1)}}<\lambda$. Then
$\widetilde{f}\in \varphi(X_0,X_1)$ and so there are $f_i\in X_i$ with $\| f_i\|_{X_i}\leq 1$ for $i=0,1$ such that
$$
\widetilde{f}(x) \leq \lambda \varphi (|f_0(x)|, |f_1(x)|) ~~{\rm a.e. ~on} ~I.
$$
Now, if we assume that both spaces $X_0$ and $X_1$ are symmetric, then the argument is as follows. Firstly, we have
$$
\widetilde{f}(x) = \big (\widetilde{f} \, \big)^*(x) \leq \lambda \varphi (f_0^*(x/2), f_1^*(x/2)) ~ {\rm a.e. ~on} ~I,
$$
where the last inequality is a consequence of the estimation
\begin{equation} \label{inequality-star}
\varphi (|f_0|, |f_1|)^*(x) \leq \varphi (f_0^*(x/2), f_1^*(x/2)) ~~{\rm for} ~x \in I.
\end{equation}
Observe that the inequality (\ref{inequality-star}) follows from the fact that the conjugation operation on ${\mathcal U}$ is
an involution. In fact, by a standard inequality for rearrangements $(f+g)^*(x) \leq f^*(x/2) + g^*(x/2)$ we obtain
$$
\varphi (|f_0(x)|, |f_1(x)|) \leq \dfrac{a |f_0(x)| + b |f_1(x)|}{\hat{\varphi}(a, b)}~~{\rm for ~all} ~~a, b > 0
$$
and
$$
\varphi (|f_0|, |f_1|)^*(x) \leq \dfrac{a f_0^*(x/2) + b f_1^*(x/2)}{\hat{\varphi}(a, b)} ~~{\rm for ~all} ~~a, b > 0.
$$
Taking the infimum over all $a, b > 0$ we get
$$
\varphi (|f_0|, |f_1|)^*(x) \leq \hat {{\hat \varphi}} (f_0^*(x/2), f_1^*(x/2)) = \varphi (f_0^*(x/2), f_1^*(x/2)).
$$
Secondly, putting $g_i(x) = f_i^*(x/2)$ for $i = 0, 1$, by symmetry of $X_i$, we obtain $g_i \in X_i$ and
$$
\| g_i\|_{X_i} \leq \|\sigma_2\|_{X_i\rightarrow X_i}\| f_i\|_{X_i} \leq \|\sigma_2\|_{X_i\rightarrow X_i}, i = 0, 1.
$$
Moreover, $\widetilde{g_i} = g_i$ and so $\| g_i\|_{\widetilde{{X_i}}} \leq \|\sigma_2\|_{X_i\rightarrow X_i}$
for $i = 0, 1$. Thus $f \in \varphi(\widetilde{X_0}, \widetilde{X_1})$ with $\| f\|_{\varphi(\widetilde{X_0}, \widetilde{X_1})}\leq A \|f\|_{\widetilde{\varphi(X_0,X_1)}}$, where
$A = \max (\| \sigma_2\|_{X_0\rightarrow X_0}, \| \sigma_2\|_{X_1\rightarrow X_1})$.
\vspace{3mm}
If we assume that both $C$ and $C^*$ are bounded on $X_0$ and $X_1$, then we use the following estimation
\begin{equation} \label{3.9}
CC^*[ \varphi(|f_0|, |f_1|)] \leq \varphi[CC^*(|f_0|), CC^*(|f_1|)] ~ {\rm a.e. ~ on } ~ I.
\end{equation}
In fact, we have for a.e. $x \in I$ and all $a, b > 0$
$$
\varphi(|f_0(x)|, |f_1(x)|) \leq \dfrac{a |f_0(x)| + b |f_1(x)|}{\hat{\varphi}(a, b)}.
$$
Then, by linearity and monotonicity of $C$ and $C^*$ we get for $x \in I$
$$
CC^*[\varphi(|f_0|, |f_1|)](x) \leq \frac{a \, CC^*|f_0(x)| + b \, CC^*|f_1(x)|}{\hat{\varphi}(a, b)}
$$
for all $a,b>0$. Using again the involution property of the conjugation we obtain
$$
CC^*[\varphi(|f_0|, |f_1|)] \leq \hat{\hat{\varphi}}(CC^*|f_0|, CC^*|f_1|) = \varphi(CC^*|f_0|, CC^*|f_1|).
$$
Consequently,
\begin{equation} \label{3.10}
\begin{split}
|f| &
\leq
\widetilde{f}\leq C\widetilde{f}\leq C\widetilde{f}+C^*\widetilde{f}= CC^*\widetilde{f} \\
&\leq
CC^*[\varphi(|f_0|, |f_1|)] \leq \varphi[CC^*(|f_0|), CC^*(|f_1|)]. \\
\end{split}
\end{equation}
However, by our assumption $CC^*(|f_i|) \in X_i$ and $\widetilde{CC^*|f_i|} = CC^*|f_i|$, which means that
$CC^*|f_i| \in \widetilde{X_i}$ for $i = 0, 1$. Therefore,
$$
f \in \varphi(\widetilde{X_0}, \widetilde{X_1}) ~ ~{\rm and} ~~ \|f \|_{\varphi(\widetilde{X_0}, \widetilde{X_1})}\leq B \| f \|_{\widetilde{\varphi(X_0, X_1)}},$$
where $B = \max (\|C\|_{X_0\rightarrow X_0}\|C^*\|_{X_0\rightarrow X_0}, \|C\|_{X_1\rightarrow X_1}\|C^*\|_{X_1\rightarrow X_1})$.
\endproof
\begin{corollary} \label{Co3}
Let $1< p_0, p_1<\infty$ and $0<\theta<1$ be such that $\frac{1}{p}=\frac{1-\theta}{p_0}+\frac{\theta}{p_1}$. Then
\begin{equation} \label{Cestheta}
\big( \widetilde{L^{p_0}} \big)^{1-\theta} \big( \widetilde{L^{p_1}} \big)^{\theta} = \widetilde{L^p} ~~ {\rm and} ~~ ( \widetilde{l^{p_0}} )^{1-\theta} (\widetilde{l^{p_1}} )^{\theta} = \widetilde{l^{p}}.
\end{equation}
\end{corollary}
Applying Theorem \ref{thm:tyldaspace} we can get some ``one-sided" result similar to (\ref{3.4}).
\begin{corollary} \label{Co4} (a) Let $I = [0, \infty)$. If $X$ is a symmetric space on $I$ with the Fatou property such that $C$ is bounded
on $X$ and on $\varphi (L^1, X)$, then
\begin{equation} \label{3.12}
\varphi (L^1, CX) = C[\varphi (L^1, X)].
\end{equation}
In particular,
\begin{equation} \label{3.13}
(L^1)^{1-\theta} (Ces_{p})^{\theta} = Ces_{q}
\end{equation}
for $1 < p \leq \infty, \frac{1}{q} = 1- \theta + \frac{\theta}{p}$ and any $0 < \theta < 1$.
(b) Let $I = [0, 1]$. Then
\begin{equation} \label{3.14}
\varphi (L^1, Ces_{\infty}) = C[\varphi (L^1, L^{\infty})].
\end{equation}
\end{corollary}
\proof
(a) Using twice the Lozanovski{\v \i} duality result, twice Theorem A and Theorem \ref{thm:tyldaspace}
we obtain
\begin{eqnarray*}
\varphi(L^1, CX)^{\prime}
&=&
\hat{\varphi}[(L^1)^{\prime}, (CX)^{\prime}] = \hat{\varphi}(L^{\infty}, \widetilde{X^{\prime}})
= \hat{\varphi}( \widetilde{L^{\infty}}, \widetilde{X^{\prime}}) \\
&=&
\widetilde{\hat{\varphi}( L^{\infty}, X^{\prime}) } = \widetilde{\hat{\varphi}( (L^1)^{\prime}, X^{\prime})} =
\widetilde{\varphi ( L^1, X)^{\prime}} = [C \varphi (L^1, X)]^{\prime}.
\end{eqnarray*}
Then, by the Fatou property of $X$, all $CX, \varphi (L^1, CX), \varphi (L^1, X)$ and $C[\varphi (L^1, X)]$ also have
the Fatou property (cf. \cite[Theorem 1(d)]{LM15a} and \cite[Corollary 3, p. 185]{Ma89}), and so
$$
\varphi(L^1, CX) \equiv \varphi(L^1, CX)^{\prime \prime} = [C \varphi (L^1, X)]^{\prime \prime} \equiv C \varphi (L^1, X).
$$
Equality (\ref{3.13}) follows from (\ref{3.12}) and the identification $(L^1)^{1-\theta} (L^p)^{\theta} = L^q$.
(b) Similarly as in (a), with Theorem A replaced by the Luxemburg-Zaanen duality result \cite{LZ66} (see also
\cite[Theorem 7]{LM15a}) $(Ces_{\infty})^{\prime} \equiv \widetilde{L^1}$, we obtain
\begin{eqnarray*}
\varphi(L^1, Ces_{\infty})^{\prime}
&=&
\hat{\varphi}[(L^1)^{\prime}, (Ces_{\infty})^{\prime}] = \hat{\varphi}(L^{\infty}, \widetilde{L^1})
= \hat{\varphi}( \widetilde{L^{\infty}}, \widetilde{L^1}) \\
&=&
\widetilde{\hat{\varphi}( L^{\infty}, L^1) } = \widetilde{\hat{\varphi}( (L^1)^{\prime}, (L^{\infty})^{\prime})} =
\widetilde{\varphi ( L^1, L^{\infty})^{\prime}} = [C \varphi (L^1, L^{\infty})]^{\prime},
\end{eqnarray*}
and by the Fatou property $\varphi(L^1, Ces_{\infty}) = C \varphi (L^1, L^{\infty})$.
\endproof
\begin{example} \label{Ex3}
In the case $I = [0,1]$ one cannot expect a general result like in Corollary \ref{Co4}(a) even for $X = L^2$. In fact,
for the weight $v(x) = 1 - x$ we have
$$
[(L^1)^{1/2}(Ces_2)^{1/2}]^{\prime} = (L^{\infty})^{1/2}(\widetilde{L^2(1/v)})^{1/2} = (\widetilde{L^2(1/v)})^{(2)}=\widetilde{L^4(1/\sqrt{v})},
$$
where $X^{(2)}$ is $2$-convexification of $X$. On the other hand,
$$
(C[(L^1)^{1/2}(L^2)^{1/2}])^{\prime} = [C(L^{4/3})]^{\prime} = (Ces_{4/3})^{\prime} = \widetilde{L^4(1/v)}.
$$
Therefore, $(L^1)^{1/2}(Ces_2)^{1/2} = [(L^1)^{1/2}(Ces_2)^{1/2}]^{\prime \prime} = [\widetilde{L^4(1/\sqrt{v})}]^{\prime}$ and
$$
C[(L^1)^{1/2}(L^2)^{1/2}] = (C[(L^1)^{1/2}(L^2)^{1/2}])^{\prime \prime} [ \widetilde{L^4(1/v)}]^{\prime} = Ces_{4/3} \subsetneq
[\widetilde{L^4(1/\sqrt{v})}]^{\prime}
$$
since, of course, $\widetilde{L^4(1/v)} \subsetneq \widetilde{L^4(1/\sqrt{v})}$.
\end{example}
Since we deal with Banach ideal spaces, the previous results one can apply to the complex method of interpolation. In order to
present it we need the following simple lemma.
\begin{lemma} \label{L1} \item[(a)] If a Banach ideal space $X$ satisfies $X \in (OC)$, then $CX \in (OC)$.
\item[(b)] If a Banach sequence space $X\subset c_0$ satisfies $X \in (OC)$, then $\widetilde{X} \in (OC)$.
\end{lemma}
\proof
(a) Let $f\in CX$, where $X$ is a Banach ideal space and let $(A_n)$ be a sequence of measurable sets such that
$A_{n+1}\subset A_n$, $n=1,2,3,...$, and $m(\bigcap A_n) = 0$. Then, by the Lebesgue domination theorem, for each $x > 0$
$$
C|f|(x) = \frac{1}{x} \int_0^x |f(t)|\chi_{A_n}(t) \, dt \rightarrow 0 \ {\rm as }\ n\rightarrow \infty.
$$
Thus, $C|f|\geq C|f|\chi_{A_n}\rightarrow 0$ a.e. and by order continuity of $X$ we obtain $\| f \chi_{A_n}\|_{CX}
= \|C|f|\chi_{A_n}\|_X\rightarrow 0$, as required. Simple modification proves the case of sequence spaces.
(b) Let $x\in \widetilde{X}$. It is enough to check whether $\|x\chi_{[n,\infty)}\|_{\widetilde{X}}\rightarrow 0$. We have
$\tilde x\geq \widetilde{x\chi_{[n,\infty)}}$. On the other hand, since $X\subset c_0$, one gets $ (\widetilde{x\chi_{[n,\infty)}})(i)\leq \widetilde{x}_n\rightarrow 0$ for each $i=1,2,3,\dots$, so that order continuity of $X$ gives the claim.
\endproof
Notice that the reverse implication in the above Lemma \ref{L1}(a) does not hold (cf. Example \ref{Ex1}): for
$X = L^2[0, 1/4] \oplus L^{\infty}[1/4, 1/2] \oplus L^2[1/2, 1]$ we have $CX[0, 1] = CL^2[0, 1] = Ces_2[0, 1] \in (OC)$ but $X\not \in (OC)$.
It is also worth to emphasize here once more conclusion of Lemma \ref{L1}(b). Namely, in contrast to the Tandori function spaces, which are never order continuous (cf. \cite{LM15a}), the construction in sequence case behaves quite well.
\vspace{2mm}
To simplify an exposition of the next theorem we assume that Banach ideal spaces contain all characteristic functions of subsets of finite measure of
underlying measure space. This assumption is equivalent to the imbedding (cf. \cite[Lemma 4.1, p. 90]{KPS82}):
\begin{equation} \label{Eq3.15}
L^1 \cap L^{\infty} \hookrightarrow X.
\end{equation}
Condition (\ref{Eq3.15}) is a little stronger than existence of a weak unity. Moreover, if $X$ is a Banach ideal space and (\ref{Eq3.15})
holds, then also $\widetilde{X}$ is a Banach ideal space with property (\ref{Eq3.15}). If additionally $C$ is bounded on $X$, then also
$CX$ satisfies property (\ref{Eq3.15}).
\vspace{2mm}
The {\it Calder\'on (lower) complex method of interpolation} is defined only for a couple of Banach spaces $(X_0, X_1)$ over
the complex field therefore we must, in fact, apply it to the couple $(X_0({\mathbb C}), X_1({\mathbb C}))$, where $X_k({\mathbb C})$
denotes the {\it complexification} of $X_k$ (namely the space of all complex-valued measurable functions $f: I \rightarrow \mathbb C$
such that $|f| \in X_k$ with the norm $\| f\|_{X_k({\mathbb C})} = \| |f| \|_{X_k}), k = 0, 1$. Let $[X_0, X_1]_{\theta}$ denote the subspace
of real-valued functions in Calder\'on's interpolation spaces $[X_0({\mathbb C}), X_1({\mathbb C})]_{\theta}$ for $0 < \theta < 1$. For
formal definition and properties of the Calder\'on (lower) method of complex interpolation we refer to original Calder\'on's paper \cite{Ca64}
and books \cite{BL76, BK91, KPS82}.
All proofs given for Banach ideal spaces of real-valued functions are true also for complexified Banach ideal spaces of measurable
functions on $I$ (cf. \cite{Ca64, CN03, Cw10}). For example, if ${\rm supp}X_0 = {\rm supp}X_1 = I$, then
${\rm supp}X_0 \cap X_1 = {\rm supp}[X_0, X_1]_{\theta} = I$.
\begin{theorem}\label{complex}
Let $0 < \theta < 1$. Assume that $X, X_0, X_1$ are Banach ideal spaces on $I$ with the Fatou property and the property (\ref{Eq3.15}),
and such that the Ces\`aro operator $C$ is bounded on all of them.
\begin{itemize}
\item[(a)] If $I = [0, \infty)$, the dilation operator $\sigma_a$ for some $0<a<1$ is bounded on $X_0$ and $X_1$ and at least one of the spaces
$X_0$ or $X_1$ is order continuous, then
$$
[CX_0, CX_1]_{\theta} = C([X_0, X_1]_{\theta}).
$$
\item[(b)] If $I = [0, 1]$, $X_0, X_1$ are symmetric spaces such that $C^*$ is bounded on both of them and at least one of the spaces
$X_0$ or $X_1$ is order continuous, then
$$
[CX_0, CX_1]_{\theta} = C([X_0, X_1]_{\theta}).
$$
\item[(c)] Let $X_0,X_1$ be Banach sequence spaces such that the dilation operator $\sigma_{3}$ is bounded on dual spaces
$X_0^{\prime}, X_1^{\prime}$ and at least one of the spaces $X_0$ or $X_1$ is order continuous, then
$$
[CX_0,CX_1]_{\theta} = C([X_0,X_1]_{\theta}).
$$
\item[(d)] If $X$ is a symmetric space on $I=[0,\infty)$, then
$$
[L^1, CX]_{\theta} = C([L^1, X]_{\theta}).
$$
\item[(e)] For $I = [0, 1]$ we have
$$
[L^1, Ces_{\infty}]_{\theta} = C([L^1, L^{\infty}]_{\theta}).
$$
\item[(f)] Let $I = [0, \infty)$ or $I = [0, 1]$ and suppose that at least one of the spaces $X_0,X_1$ is order continuous. If either $X_0$ and
$X_1$ are symmetric spaces or $C^*$ is bounded on $X_0$ and $X_1$, then
$$
[\widetilde{X_0},\widetilde{X_1}]_{\theta} = ([X_0,X_1]_{\theta})^{\sim}.
$$
\end{itemize}
\end{theorem}
\proof The main tool in the proof of all points will be Theorem \ref{thm:cesCL} and Shestakov's representation of the complex method of interpolation for Banach ideal spaces $X_0, X_1$ (cf. \cite[Theorem 1]{Sh74}; see also \cite[Theorem 9]{RT10}), i.e.,
\begin{equation} \label{Shestakov}
[X_0, X_1]_{\theta} \equiv \overline{X_0\cap X_1}^{X_0^{1-\theta}X_1^{\theta}}.
\end{equation}
In fact, the proofs of all our points relay on this theorem. It is enough to notice that in all cases from (a) to (c) we have
$$
\overline{X_0\cap X_1}^{X_0^{1-\theta}X_1^{\theta}} = {X_0^{1-\theta}X_1^{\theta}} ~~ {\rm and} ~~ \overline{CX_0\cap CX_1}^{CX_0^{1-\theta}CX_1^{\theta}} = ({CX_0)^{1-\theta} (CX_1)^{\theta}}
$$
just because $X_0^{1-\theta}X_1^{\theta}$ and $(CX_0)^{1-\theta} (CX_1)^{\theta}$ are order continuous under our assumptions. In fact,
$X_0^{1-\theta} X_1^{\theta}$ for $0 < \theta < 1$ is order continuous when at least one of $X_0$ or $X_1$ is order continuous (see, for
example, \cite[Lemma 20, p. 428]{Lo69}, \cite[Proposition 4]{Re88}, \cite[Theorem 15.10]{Ma89} and \cite{KL10}, where $(OC)$ property
of $\varphi(X_0, X_1)$ was investigated), so that simple functions are dense therein. Using all of these representations we get
\begin{equation*}
[CX_0,CX_1]_{\theta} = C([X_0,X_1]_{\theta}) = C(X_0^{1-\theta}X_1^{\theta}) = (CX_0)^{1-\theta} (CX_1)^{\theta}
\end{equation*}
with a corresponding modifications in points (d) and (e), where instead of Theorem \ref{thm:cesCL} we use Corollary \ref{Co4}.
To prove (f) we need a little more delicate argument. First of all recall that $\widetilde{X}$ are never order continuous in a function case
and even worse $\widetilde{X}_a = \{0\}$ (see \cite[Theorem 1(e)]{LM15a}). So that at the first look it seems to be hopeless to apply an argument like above here. Fortunately, order continuity means only that one can approximate any function in a norm by each majorized sequence tending to it almost everywhere, but we need to approximate a given function at least by one sequence, so that the lack of order continuous elements will not be an obstacle. Of course,
\begin{equation*}
\overline{\widetilde{X_0}\cap\widetilde{X_1}}^{\widetilde{X_0}^{1-\theta}\widetilde{X_1}^{\theta}} \hookrightarrow \widetilde{X_0}^{1-\theta}\widetilde{X_1}^{\theta}.
\end{equation*}
We will show that in the above imbedding we have equality. Firstly, consider the case when $I = [0,\infty)$. Let
$f\in \widetilde{X_0}^{1-\theta}\widetilde{X_1}^{\theta}=\widetilde{X_0^{1-\theta}X_1^{\theta}}$ be such that $\widetilde{f}(x) > 0$ for
each $x \geq 0$ (always $\widetilde{f}(x) \geq 0$ and we do proof in the worst case when $\widetilde{f}(x) > 0$). By definition $\tilde f\in X_0^{1-\theta}X_1^{\theta}$. Since $\tilde f$ is nonincreasing, for each interval $[\frac{1}{n},n]$ we can find a simple function $g_n$ with support in $[\frac{1}{n},n]$ such that, $|g_n|\leq|f|$ and $\|(f-g_n)\chi_{[\frac{1}{n},n]}\|_{\infty} \leq \widetilde{f}(n)$.
Clearly, $g_n\in \widetilde{X_0}\cap\widetilde{X_1}$. We have
$$
| f - g_n |\leq |f|\chi_{[0,\frac{1}{n}]}+\widetilde{f}(n)\chi_{[\frac{1}{n},n]}+|f|\chi_{[n,\infty)}
$$
and consequently
$$
\widetilde{|(f-g_n)|}\leq \widetilde{|f|\chi_{[0,\frac{1}{n}]}}+\widetilde{f}(n)\chi_{[0,n]}+\widetilde{|f|\chi_{[n,\infty)}}.
$$
Now, since $\widetilde{f}(n) \rightarrow 0$ with $n\rightarrow \infty$, we see that $ \widetilde{|f|\chi_{[0,\frac{1}{n}]}}\rightarrow 0$,
$ \widetilde{f}(n) \chi_{[0,n]} \rightarrow 0$ and $ \widetilde{|f|\chi_{[n,\infty)}}\rightarrow 0$ a.e. on $I$. Moreover, all these three
sequences are dominated by $\widetilde{f}$, so that order continuity of $X_0^{1-\theta}X_1^{\theta}$ guarantees that
$
\|\widetilde{f-g_n}\|_{X_0^{1-\theta}X_1^{\theta}}\rightarrow 0$ as $n \rightarrow \infty$,
which proves the claim. In case when there is $a> 0$ such that $\widetilde{f}(x)=0$ for all $x > a$, we can proceed analogously, only
replacing intervals $[\frac{1}{n},n]$ by $[\frac{1}{n},n]\cap [0,a]$, $\widetilde{f}(n)$ by $\frac{1}{n}$ and a new majorant is then
$\max\{\tilde f, \chi_{[0,a]}\}$. The same argument works as well for the case $I=[0,1]$.
\endproof
\section{Real method}
One of the most important interpolation methods is the {\it $K$-method} known also as the {\it real Lions-Peetre interpolation method}. For a Banach couple $\bar{X} = (X_0, X_1)$ the {\it Peetre K-functional} of an element $f \in X_0+X_1$ is defined for $t > 0$ by
$$
K(t, f; X_0, X_1) = \inf \{ \| f_0\|_{X_0} + t \| f_1\|_{X_1}: f = f_0 + f_1, f_0 \in X_0, f_1 \in X_1 \}.
$$
Let $G$ be a Banach ideal space on $(0, \infty)$ containing function $\min(1, t)$. Then the space of real interpolation
or the $K$-method of interpolation
$$
(X_0, X_1)_G^K = \{f \in X_0 + X_1: K(t, f; X_0, X_1) \in G \}
$$
is a Banach space with the norm $\| f \|_{(X_0, X_1)_G^K} = \| K(t, f; X_0, X_1) \|_G$. This space is an intermediate space
between $X_0$ and $X_1$, that is, $X_0 \cap X_1 \hookrightarrow (X_0, X_1)_G^K \hookrightarrow X_0 + X_1$. Moreover, $
(X_0, X_1)_G^K$ is an interpolation space between $X_0$ and $X_1$.
The most common with several applications is $K$-method, where $G$ is given by
$\| f\|_G = (\int_0^{\infty} (t^{-\theta} |f(t)|)^p \frac{dt}{t})^{1/p}$ for $0 < \theta < 1, 1 \leq p < \infty$ or
$\| f\|_G = (\sup_{t > 0} t^{-\theta} |f(t)|)$ for $0 \leq \theta \leq 1$ (when $p= \infty$) and then
$ (X_0, X_1)_{G}^K = (X_0, X_1)_{\theta, p}$.
More information about interpolation spaces, and, in particular, interpolation functors may be found in the books \cite{BS88, BL76, BK91} and \cite{KPS82}.
\begin{theorem}\label{thm:Cesaro=real}
Let $X_0,X_1$ be two Banach function spaces on $I = [0, \infty)$. If $C$ and $C^*$ are bounded on $X_i$ for $i = 0, 1$ and $F$ is an interpolation functor
with the homogenity property, that is, $F(X_0(w), X_1(w)) = F(X_0, X_1)(w)$ for any weight $w$ on $I$, then
\begin{equation} \label{Cesaro=real}
F(CX_0, CX_1) = CF(X_0, X_1).
\end{equation}
In particular,
\begin{equation} \label{4.2}
(CX_0, CX_1)_G^K = C[(X_0, X_1)_G^K].
\end{equation}
\end{theorem}
\proof
Let $X(w_0)$ be a weighted Banach ideal space on $(0, \infty)$ with the weight $w_0(t) = \frac{1}{t}$ and such that $C, C^*$ are
bounded on $X$. First of all notice that
$$
K(t, f; L^1, L^1(1/s)) = K(t, |f|; L^1, L^1(1/s)) = t \, [C|f|(t) + C^*|f|(t)]
$$
implies
\begin{equation} \label{CesaroK}
(L^1, L^1(1/s))_{X(w_0)}^K = CX.
\end{equation}
In fact, we have
$$
\| f \|_{(L^1, L^1(1/s))_{X(w_0)}^K } = \| K(t, f; L^1, L^1(1/s)) \|_{X(w_0)}^K = \| C|f| + C^*|f| \|_X
$$
and
\begin{eqnarray*}
\| f \|_{CX}
&=&
\| C|f| \|_X \leq \| C|f| + C^*|f|\|_X = \| C^*C|f| \|_X \\
&\leq&
\| C^* \|_{X \rightarrow X} \, \| C|f| \|_X = \| C^* \|_{X \rightarrow X} \, \| f \|_{CX}.
\end{eqnarray*}
Notice now that the operator $Sf(t) = \int_0^{\infty} \min(1, \frac{t}{s}) f(s) \frac{ds}{s}$ is bounded on $X(w_0)$. In fact,
\begin{eqnarray*}
\| Sf \|_{X(w_0)}
&=&
\| \frac{1}{t} \int_0^t f(s) \frac{ds}{s} + \int_t^{\infty} \frac{f(s)}{s} \frac{ds}{s} \|_X \\
&=&
\| C(f w_0) + C^*(f w_0) \|_X \leq \big( \| C \|_{X \rightarrow X} + \| C^* \|_{X \rightarrow X} \big) \| f w_0\|_X \\
&=&
\big( \| C \|_{X \rightarrow X} + \| C^* \|_{X \rightarrow X} \big) \| f \|_{X(w_0)}.
\end{eqnarray*}
Let's return to the proof of commutativity (\ref{Cesaro=real}). Since $C, C^*$ are bounded on $X_0$ and on $X_1$, then by (\ref{CesaroK})
we obtain $CX_i = (L^1, L^1(1/s))_{X_i(w_0)}^K, i = 0, 1$ and since operator $S$ is bounded on $X_0(w_0)$ and on $X_1(w_0)$,
then by Brudny{\v \i}-Dmitriev-Ovchinnikov theorem (see \cite[Theorem 1]{DO79} and \cite[Theorem 4.3.1]{BK91}): for any interpolation
functor $F$ we have
$$
F \big( (L^1, L^1(1/s))_{X_0(w_0)}^K, (L^1, L^1(1/s))_{X_1(w_0)}^K \big) = (L^1, L^1(1/s))_{F(X_0(w_0), X_1(w_0))}^K.
$$
Now, by assumption that the functor $F$ has the homogeneity property we obtain
\begin{eqnarray*}
F(CX_0, CX_1)
&=&
F \big( (L^1, L^1(1/s))_{X_0(w_0)}^K, (L^1, L^1(1/s))_{X_1(w_0)}^K \big) \\
&=&
(L^1, L^1(1/s))_{F(X_0(w_0), X_1(w_0))}^K \\
&=&
(L^1, L^1(1/s))_{F(X_0, X_1)(w_0)}^K = C[F(X_0, X_1)],
\end{eqnarray*}
where the last equality follows from (\ref{CesaroK}) and the fact that $F$ is an interpolation functor which implies boundedness
of $C$ and $C^*$ on $F(X_0, X_1)$.
To prove the second statement, note that the $K$-method of interpolation is homogeneous. It follows from the equality
$K(t, f; X_0(w), X_1(w)) = K(t, f w; X_0, X_1)$ (cf. \cite[Proposition 14]{LM15a}), since then
\begin{eqnarray*}
\| f \|_{(X_0(w), X_1(w))_G^K}
&=&
\| K(t, f; X_0(w), X_1(w)) \|_G = \| K(t, f w; X_0, X_1) \|_G \\
&=&
\| f w \|_{(X_0, X_1)_G^K} = \| f \|_{(X_0, X_1)_{G}^{K} (w)}.
\end{eqnarray*}
\endproof
\begin{remark}
Note that (\ref{4.2}) for the case of symmetric spaces $X_0, X_1$ on $[0, \infty)$ with the operator $C$ bounded on $X_i, i = 0, 1$ was already proved
in \cite[Corollary 3.2]{AM13} and (\ref{CesaroK}) for the case $X = L^p$ with $1 < p < \infty$ was proved in \cite[Theorem 2.1(ii)]{AM13}.
\end{remark}
\begin{remark}
As it was mentioned in the previous section also the Calder\'on-Lozanovski{\v \i} construction has the homogeneity property (note that it is an interpolation functor for example when all spaces have the Fatou property). Another classical functors with the homogeneity property are functors of orbit $Orb_{\bar E}(a,\bar X,L)$ and coorbit $Corb_{\bar Y}(F,\bar X)$ (cf. \cite{BK91}). Using a similar argument as in the proof of Theorem \ref{complex} one could also prove the homogeneity of the complex method.
\end{remark}
As an example we consider interpolation of weighted Ces\'aro spaces $Ces_{p, \alpha} = C(L^p(x^{\alpha}))$ on $I = [0, \infty)$. We only
need to observe that $C$ and $C^*$ are bounded on $L^p(x^{\alpha})$ if and only if $1 \leq p \leq \infty$ and $-1/p < \alpha < 1- 1/p$
(see \cite[p. 245]{HLP52} for sufficiency and \cite[pp. 38-40]{KMP07} for equivalence).
\begin{corollary} \label{Co5}
Let $1 \leq p_i \leq\infty$ and $-1/p_i <\alpha_i <1-1/p_i$ for $i = 0, 1$, then
\begin{equation} \label{Cesthetap}
\big ( Ces_{p_0, \alpha_0}, Ces_{p_1, \alpha_1} \big)_{\theta, p} = Ces_{p, \alpha},
\end{equation}
where $\frac{1}{p} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1}$ and $\alpha = (1-\theta) \alpha_0 + \theta \alpha_1$
\end{corollary}
In particular, for $\alpha_0 = \alpha_1 = 0$ and $1 < p_0 < p_1 < \infty$ we obtain from (\ref{Cesthetap}) that
$$
\big ( Ces_{p_0}, Ces_{p_1} \big)_{\theta, p} = Ces_{p} ~ {\rm for} ~~ \frac{1}{p} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1},
$$
which was already proved in \cite[Corollary 3.2]{AM13} and \cite[Theorem 2]{AM14a} using the identification from (\ref{CesaroK})
$(L^1, L^1(1/s))_{1-1/p, p} = Ces_p \, (1 < p < \infty)$ and reiteration theorem for the $K$-method $(\cdot)_{\theta, p}$ (see also
\cite[Corollary 2]{Si91}).
From Theorem \ref{thm:tyldaspace} we obtain that if $X_0, X_1$ are Banach ideal spaces with the Fatou property such that either
$X_0, X_1$ are symmetric spaces or $C, C^*$ are bounded on both $X_0$ and $X_1$, then
$\widetilde{X_0} + \widetilde{X_1} = \widetilde{X_0 + X_1}$, which gives that
$K(t, f; \widetilde{X_0}, \widetilde{X_1}) \approx K(t, \widetilde{f}; X_0, X_1)$ and we get commutativity of Tandori spaces
with the real method of interpolation
\begin{equation} \label{4.5}
(\widetilde{X_0}, \widetilde{X_1})_G^K = [(X_0, X_1)_G^K]^{\Large \sim}.
\end{equation}
\section{Additional remarks}
A fundamental problem in interpolation theory is the description of all interpolation spaces with respect to a given Banach pair.
In particular, it is not so rare that for a given Banach couple, all interpolation spaces may be generated by K-method and such
couples are referred to be {\it Calder\'on couples} or {\it Calder\'on pairs}.
A Banach couple $\bar{X} = (X_0, X_1)$ is called a {\it Calder\'on couple} if every interpolation space between $\bar{X}$ is described
by the $K$-method. Equivalently, if for each $f, g \in X_0 + X_1$ satisfying
$$
K(t, f; X_0, X_1) \leq K(t, g; X_0, X_1) ~ {\rm for ~all} ~~t > 0,
$$
there is a bounded operator $T: X_0+X_1 \rightarrow X_0+X_1$ such that $\max (\| T\|_{X_0 \rightarrow X_0},
\| T\|_{X_1 \rightarrow X_1}) < \infty$ and $Tg = f$ (cf. \cite{BK91}, Theorem 4.4.5).
There are many examples of Calder\'on couples and couples which are not Calder\'on. What can we say about this problem for
Ces\`aro, Copson and Tandori spaces?
By the Cwikel's result (cf. \cite[Theorem 1]{Cw81}) we obtain that $(Ces_p(I), Ces_q(I))$ is a Calder\'on couple for $1 < p < q < \infty$, since
$$
(L^1[0, \infty), Ces_{\infty}[0, \infty))_{1-1/p, p} = Ces_p[0, \infty)
$$
and
$$
(L^1(1-x)[0, 1], Ces_{\infty}[0, 1])_{1-1/p, p} = Ces_p[0, 1]
$$
(see Astashkin-Maligranda \cite[Proposition 3.1 and Theorem 4.1]{AM13}).
Moreover, using the Brudny{\v \i}-Dmitriev-Ovchinnikov theorem (see \cite[Theorem 1]{DO79} and \cite[Theorem 4.3.1]{BK91})
and Theorem 5 we obtain that if $X_0, X_1$ are Banach function spaces on $I = [0, \infty)$ such that both $C$ and $C^*$ are bounded on
$X_0$ and on $X_1$, then $(CX_0, CX_1)$ is a Calder\'on couple.
Let us notice also that Masty{\l}o-Sinnamon \cite{MS06} proved that $(L^1, Ces_{\infty})$ is a Calder\'on couple and Le\'snik \cite{Le15}
showed that also $(\widetilde{L^1}, L^{\infty})$ is a Calder\'on couple.
\vspace{2mm}
The following problems are natural to formulate here.
\vspace{2mm}
{\bf Problem 1}. For $I = [0, 1]$ and $1 \leq p < \infty$ identify $(L^1)^{1-\theta}(Ces_p)^{\theta}$ or $\varphi(L^1, Ces_p)$,
or even more generally, $\varphi(L^1, CX)$.
\vspace{2mm}
{\bf Problem 2}. For $I = [0, 1]$ identify $(Ces_1)^{1-\theta}(Ces_{\infty})^{\theta}$ or $\varphi (Ces_1, Ces_{\infty})$, or even more general $\varphi(Ces_1, CX)$.
Note that $(Ces_1)^{1-1/p}(Ces_{\infty})^{1/p} \neq Ces_p$ for $1 < p < \infty$. In fact, from (\ref{3.12}) we have that
$(L^1)^{1-1/p}(Ces_{\infty})^{1/p} = Ces_p$ and by the uniqueness theorem (cf. \cite[Theorem 3.5]{CN03} or \cite[Corollary 1]{BM05})
$$
Ces_p = (L^1)^{1-1/p}(Ces_{\infty})^{1/p} \neq (Ces_1)^{1-1/p}(Ces_{\infty})^{1/p},
$$
since $L^1 \neq Ces_1$. Under some mild conditions on $\varphi$, from the uniqueness theorem proved in \cite[Theorem 1]{BM05}, we
get that even $C[\varphi (L^1, L^{\infty})] \neq \varphi( Ces_1, Ces_{\infty})$.
\vspace{2mm}
{\bf Problem 3}. For $I = [0, 1]$ or $I = [0, \infty)$ identify $\widetilde{X}^{1-\theta} X^{\theta}$ or $\varphi(\widetilde{X}, X)$.
\vspace{2mm}
The last identification can be useful for factorization since there appeared for $1 < p < \infty$ the $p$-convexification of $Ces_{\infty}$,
that is, $Ces_{\infty}^{(p)}$ and we have equalities (cf. \cite{KLM14})
$$
Ces_{\infty}^{(p)} = Ces_{\infty}^{1/p} (L^{\infty})^{1-1/p}= \big[ \widetilde{L^1}^{1/p} (L^{1})^{1-1/p} \big]^{\prime}.
$$
Eventual identification in Problem 3 will suggests how to generalize factorization results presented in \cite{AM09} and \cite{KLM14}.
\vspace{2mm}
{\bf Problem 4}. What is an analogue of Theorem \ref{thm:Cesaro=real} for $I=[0,1]$? | {"config": "arxiv", "file": "1502.05732.tex"} |
\subsection{Phase Synchronization}\label{Subsection: Phase Synchronization}
For identical natural frequencies and zero phase shifts, the practical stability results in Theorem \ref{Theorem: Synchronization Condition I} and Theorem \ref{Theorem: Synchronization Condition II} imply $\subscr{\gamma}{min} \downarrow 0$, i.e., phase synchronization of the non-uniform Kuramoto oscillators\,\,\eqref{eq: Non-uniform Kuramoto model}.
\begin{theorem}[Phase synchronization]\label{Theorem: Phase Synchronization}
Consider the non-uniform Kuramoto model \eqref{eq: Non-uniform Kuramoto model}, where the graph induced by $P$ has a globally reachable node, $\subscr{\varphi}{max} = 0$, and $\omega_{i}/D_{i} = \bar \omega$ for all $i \in \until n$.
Then for every $\theta(0) \in \bar\Delta(\gamma)$ with $\gamma \in {[0,\pi[}$,
\begin{enumerate}
\item [1)] the phases $\theta_{i}(t)$ synchronize exponentially to $\theta_{\infty}(t) \in [\subscr{{\theta}}{min}(0) , \subscr{{\theta}}{max}(0)] + \bar \omega t$, and
\item [2)] if $P=P^{T}$, then the phases $\theta_{i}(t)$ synchronize exponentially to the weighted mean angle\footnote{This weighted average of angles is geometrically well defined for $\theta(0) \in \Delta(\pi)$.}
$\theta_{\infty}(t) = \sum_{i} D_{i} \theta_{i}(0)/\sum_{i} D_{i} + \bar \omega t$ at a rate no worse than
\begin{equation}
\subscr{\lambda}{ps}
=
- \lambda_{2}(L(P_{ij})) \sinc(\gamma) \cos(\angle(D\fvec 1,\fvec 1))^{2} \!/ \subscr{D}{max}
\label{eq: rate lambda_ps}
\,.
\end{equation}
\end{enumerate}
\end{theorem}
The worst-case phase synchronization rate $\subscr{\lambda}{ps}$ can be interpreted similarly as the terms in \eqref{eq: rate lambda_fe}, where $\sinc(\gamma)$ corresponds to the initial phase cohesiveness in $\bar\Delta(\gamma)$. For classic Kuramoto oscillators \eqref{eq: Kuramoto system} statements 1) and 2) can be reduced to the Kuramoto results found in \cite{ZL-BF-MM:07} and Theorem 1 in \cite{AJ-NM-MB:04}.
{\itshape Proof of Theorem \ref{Theorem: Phase Synchronization}: }
First we proof statement 1). Consider again the Lyapunov function $V(\theta(t))$ from the proof of Theorem \ref{Theorem: Synchronization Condition I}. The Dini derivative in the case $\subscr{\varphi}{max} = 0$ and $\omega_{i}/D_{i} = \bar \omega$ is simply
\begin{equation*}
D^{+} V (\theta(t))
=
- \sum\nolimits_{k=1}^{n} \Big(
\frac{P_{m k}}{D_{m}} \sin(\theta_{m}(t) - \theta_{k}(t))
+
\frac{P_{\ell k}}{D_{\ell}} \sin(\theta_{k}(t)-\theta_{\ell}(t))
\Big)
\,.
\end{equation*}
Both sinusoidal terms are positive for $\theta(t) \in \bar\Delta(\gamma)$, $\gamma \in {[0,\pi[}$. Thus, $V(\theta(t)$ is non-increasing, and $\bar\Delta(\gamma)$ is positively invariant. Therefore, the term $a_{ij}(t) = (P_{ij}/D_{i}) \sinc(\theta_{i}(t) - \theta_{j}(t))$ is strictly positive for all $t \geq 0$, and after changing to a rotating frame (via the coordinate transformation $\theta \mapsto \theta - \bar \omega\,t$) the non-uniform Kuramoto model \eqref{eq: Non-uniform Kuramoto model} can be written as the consensus time-varying consensus protocol
\begin{equation}
\dot \theta_{i}(t) = - \sum\nolimits_{j=1}^{n} a_{ij}(t) (\theta_{i}(t) - \theta_{j}(t))
\label{eq: consensus for phases}
\,,
\end{equation}
Statement 1) follows directly along the lines of the proof of statement 1) in Theorem \ref{Theorem: Frequency synchronization}. In the case of symmetric coupling $P=P^{T}$, the phase dynamics \eqref{eq: consensus for phases} can be reformulated as a {\it symmetric} time-varying consensus protocol with strictly positive weights $w_{ij}(t) = P_{ij} \sinc(\theta_{i}(t) - \theta_{j}(t))$ and multiple rates $D_{i}$\,\,as
\begin{equation}
\dt D \theta
=
- L(w_{ij}(t)) \, \theta
\label{eq: LPV concensus system for dot theta - trivial phase shifts}
\,,
\end{equation}
Statement 2) now follows directly along the lines of the proof of statement 2) in Theorem \ref{Theorem: Frequency synchronization}.
\footnote{The proof of Theorem \ref{Theorem: Synchronization Condition II} can be extended for $HD^{-1} \omega = \fvec 0$ and $X = \fvec 0$ to show statement 2) of Theorem \ref{Theorem: Phase Synchronization} with a slightly different worst-case synchronization frequency than \eqref{eq: rate lambda_ps}.}
\hspace*{\fill}~\QED\par\endtrivlist\unskip
The main result Theorem \ref{Theorem: Main Synchronization Result} can be proved now as a corollary of Theorem \ref{Theorem: Synchronization Condition I} and Theorem \ref{Theorem: singular perturbations}.
{\itshape Proof of Theorem \ref{Theorem: Main Synchronization Result}: }
The assumptions of Theorem \ref{Theorem: Main Synchronization Result} correspond exactly to the assumptions of Theorem \ref{Theorem: Synchronization Condition I} and statements 1) and 2) of Theorem \ref{Theorem: Main Synchronization Result} follow trivially from Theorem \ref{Theorem: Synchronization Condition I}.
Since the non-uniform Kuramoto model synchronizes exponentially and achieves phase cohesiveness in $\bar\Delta(\subscr{\gamma}{min}) \subsetneq \Delta(\pi/2 - \subscr{\varphi}{max})$, it follows from Lemma \ref{Lemma: Properties of grounded Kuramoto model} that the grounded non-uniform Kuramoto dynamics \eqref{eq: grounded Kuramoto model} converge exponentially to a stable fixed point $\delta_{\infty}$. Moreover, $\delta(0)=\groundedphases(\theta(0))$ is bounded and thus necessarily in a compact subset of the region of attraction of the fixed point $\delta_{\infty}$. Thus, the assumptions of Theorem \ref{Theorem: singular perturbations} are satisfied. Statements 3) and 4) of Theorem \ref{Theorem: Main Synchronization Result} follow from Theorem \ref{Theorem: singular perturbations}, where we made the following changes: the approximation errors \eqref{eq: singular perturbation error 1}-\eqref{eq: singular perturbation error 2} are expressed as the approximation\,\,errors \eqref{eq: approx-errors} in $\theta$-coordinates, we stated only the case $\epsilon < \epsilon^{*}$ and $t \geq t_{b}>0$, we reformulated $h(\bar \delta(t)) = D^{-1} Q(\bar\theta(t))$, and weakened the dependence of $\epsilon$ on $\Omega_{\delta}$ to a dependence on $\theta(0)$.
\hspace*{\fill}~\QED\par\endtrivlist\unskip | {"config": "arxiv", "file": "0910.5673/ArXiv Version 2/Sections/PhaseSynchronization.tex"} |
TITLE: What exactly does the expectation value of $x$ mean in quantum mechanics?
QUESTION [2 upvotes]: When I learn quantum mechanics （by reading Griffith's book Introduction to quantum mechanics 2ed edition (Page 15)）, I was confused by the concept of the expectation value of $x$, i.e. $\langle x\rangle=\int^{+\infty}_{-\infty}x|\Psi(x,t)|^2dx$.
He said that,
In short, the expectation value is the average of repeated measurements on an ensemble of identically prepared systems, not the average of repeated measurements on one and the same system.
I can't understand that why he said we should have a whole ensemble of identically prepared systems. I hope you could explain that to me in detail.
REPLY [4 votes]: In quantum systems it is not possible to perform a measurement without affecting the measured system. This is because, roughly speaking, the interaction with the instrument creates a correlation between the system and the instrument whose effective result is a modification of the system's state.
Therefore if you perform repeated measurements of the same observable in the same system and take the average, you would not get the same value as if you average the outcome of many measurements of the same observable in identical copies of the given initial state of such system. As a matter of fact, measuring the observable an $n$-th time in a system where a measurement has already been done, you would get exactly the same outcome as in the aforementioned previous measurement.
The expectation value is instead what you would get, in the limit of an infinite number of measurements, averaging the measured value of an observable in identical copies of the same initial configuration. Because of the above, apart from the case in which the initial state is particularly special (eigenstate of the observable), the two procedures do not coincide. | {"set_name": "stack_exchange", "score": 2, "question_id": 232063} |
\begin{document}
\title{Extrapolation-based implicit-explicit general linear methods}
\author{A. Cardone,
\thanks{Dipartimento di Matematica,
Universit\`{a} degli studi di Salerno, Fisciano (Sa),
84084 Italy,
\mbox{e-mail}: ancardone@unisa.it.
The work of this author was supported by travel fellowship from
the Department of Mathematics, University of Salerno.
} \
Z. Jackiewicz,
\thanks{Department of Mathematics, Arizona State University,
Tempe, Arizona 85287,
\mbox{e-mail}: jackiewicz@asu.edu,
and AGH University of Science and Technology, Krak\'ow, Poland.
} \ A. Sandu,
\thanks{Department of Computer Science, Virginia Polytechnic Institute \& State University,
Blacksburg, Virginia 24061,
\mbox{e-mail}: sandu@cs.vt.edu.
} \ and H. Zhang
\thanks{Department of Computer Science, Virginia Polytechnic Institute \& State University,
Blacksburg, Virginia 24061,
\mbox{e-mail}: zhang@vt.edu.}
}
\maketitle
\textbf{Abstract}
For many systems of differential equations modeling problems in science and
engineering, there are natural splittings of the right hand side into two
parts, one non-stiff or mildly stiff, and the other one stiff.
For such systems implicit-explicit (IMEX) integration combines an explicit scheme for the non-stiff part with an implicit scheme for the stiff part.
In a recent series of papers two of the authors (Sandu and Zhang) have
developed IMEX GLMs, a family of implicit-explicit schemes based on general linear methods.
It has been shown that, due to their high stage order, IMEX GLMs require no additional coupling order conditions, and are not marred by order reduction.
This work develops a new extrapolation-based approach to construct practical IMEX GLM pairs of high order.
We look for methods with large absolute stability region, assuming that the
implicit part of the method is $A$- or $L$-stable. We provide examples of
IMEX GLMs with optimal stability properties. Their application to a two dimensional test problem confirms the theoretical findings.
\vspace{3mm}
{\bf Key words.}
IMEX methods, general linear methods, error analysis, order conditions,
stability analysis
\vspace{2mm}
\newpage
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\setcounter{table}{0}
\section{Introduction} \label{sec1}
Many practical problems in science and engineering are modeled by large systems
of ordinary differential equations (ODEs) which arise from discretization in space
of partial differential equations (PDEs) by finite difference methods,
finite elements or finite volume methods, or pseudospectral methods.
For such systems there are often natural splittings of the right hand sides
of the differential systems into two parts, one of which is non-stiff or
mildly stiff, and suitable for explicit time integration, and the other part is stiff,
and suitable for implicit time integration. Such systems can be written in the form
\begin{equation} \label{eq1.1}
\left\{
\begin{array}{ll}
y'(t)=f\big(y(t)\big)+g\big(y(t)\big), & t\in[t_0,T], \\
y(t_0)=y_0,
\end{array}
\right.
\end{equation}
where $f(y)$ represents the non-stiff processes, for example advection,
and $g(y)$ represents stiff processes, for example diffusion or chemical reaction,
in semi-discretization of advection-diffusion-reaction equations \cite{hv03}.
Implicit-explicit (IMEX) integration approach discretizes the non-stiff part $f(y)$ is
with an explicit method, and the stiff part $g(y)$ with an implicit, stable method.
This strategy seeks to ensure the numerical stability of the solution of \eqref{eq1.1}
while reducing the amount of implicitness, and therefore the overall computational effort.
IMEX multistep methods were introduced by Crouzeix \cite{cro80}
and Varah \cite{var80} and further analyzed in \cite{arw95,fhv97}.
IMEX Runge-Kutta methods have been investigated in \cite{ars97,cfn01,kc03,pr00,pr05,zho96}.
In a recent series of papers the last two authors and their collaborators have
proposed the new IMEX GLM family of
implicit-explicit schemes based on general linear methods.
A general formalism for partitioned GLMs and their order conditions
was developed by Zhang and Sandu \cite{zs13}. The partitioned method formalism was then used to construct IMEX GLMs. The
starting and ending procedures, linear stability, and stiff convergence properties of the new family have been analyzed. Zhang and Sandu
examined practical methods of second
order in \cite{zs12} and of third order in \cite{zs13}.
A class of IMEX two step Runge-Kutta (TSRK)
methods was proposed by Zharovski and Sandu \cite{zhasan13}.
The results in \cite{zs12,zs13,zhasan13} prove that the general linear framework is well suited for the construction of multi-methods. Specifically, owing to the high stage orders, no coupling conditions are needed to ensure the order of accuracy of the partitioned GLM \cite{zs13}. In addition, it has been shown that IMEX GLMs are particularly attractive for solving stiff problems, where other multistage methods may suffer from order reduction \cite{zs13}.
This paper extends our previous work \cite{zs12,zs13,zhasan13} and develops a new extrapolation-based approach
for the construction of practical IMEX GLM schemes of high order and high stage order.
The organization of this paper is as follows.
General linear methods and the implicit-explicit variants are reviewed in Section \ref{sec:imex-glm}.
The new extrapolation-based IMEX GLMs are derived in Section \ref{sec:eximex}, and their order conditions are presented.
The stability analysis is performed in Section \ref{sec:stability} and specific methods are constructed in Section \ref{sec:construction}.
Numerical experiments are presented in Section \ref{sec:numerics}, and Section \ref{sec:conclusions} gives some concluding remarks and plans for future work.
\section{Implicit-explicit general linear methods}\label{sec:imex-glm}
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In this section we briefly review GLMs and the IMEX GLM family.
The GLMs for ODEs were introduced by Burrage and Butcher \cite{bb80}
and further investigated in \cite{bur95,but87,but03,cj12,cjm,hlw02,hnw93,hw96}.
We also refer the reader to the review article \cite{but06a} and the recent
monograph \cite{jac09} and references therein.
A diagonally implicit GLM for (\ref{eq1.1}) is defined by
\begin{equation} \label{eq1.2}
\left\{
\begin{array}{lcl}
Y_i^{[n+1]} & = & h\ds\sum_{j=1}^ia_{ij}\Big(f\big(Y_j^{[n+1]}\big)
+g\big(Y_j^{[n+1]}\big)\Big)+\ds\sum_{j=1}^ru_{ij}y_j^{[n]},
\quad i=1,2,\ldots,s \\ [3mm]
y_i^{[n+1]} & = & h\ds\sum_{j=1}^sb_{ij}\Big(f\big(Y_j^{[n+1]}\big)
+g\big(Y_j^{[n+1]}\big)\Big)+\ds\sum_{j=1}^rv_{ij}y_j^{[n]},
\quad i=1,2,\ldots,r, \\ [3mm]
\end{array}
\right.
\end{equation}
$n=0,1,\ldots,N-1$.
Here, $N$ is a positive integer, $h=(T-t_0)/N$, $t_n=t_0+nh$, $n=0,1,\ldots,N$,
$Y_i^{[n+1]}$ are approximations of stage order $q$ to $y(t_n+c_ih)$,
i.e.,
\begin{equation} \label{eq1.3}
Y_i^{[n+1]}=y(t_n+c_ih)+O(h^{q+1}),
\quad
i=1,2,\ldots,s,
\end{equation}
$y_i^{[n]}$ are approximations of order $p$ to the
linear combinations of the derivatives of the solution $y$ at the point $t_n$,
i.e.,
\begin{equation} \label{eq1.4}
y_i^{[n]}=\ds\sum_{k=0}^pq_{ik}h^ky^{(k)}(t_n)+O(h^{p+1}),
\quad
i=1,2,\ldots,r,
\end{equation}
and $y$ is the solution to (\ref{eq1.1}).
These methods can be characterized by the abscissa vector
$\mbf{c}=[c_1,\ldots,c_s]^T$, the coefficient matrices
$\mbf{A}=[a_{ij}]\in\R^{s\times s}$,
$\mbf{U}=[a_{ij}]\in\R^{s\times r}$,
$\mbf{B}=[a_{ij}]\in\R^{r\times s}$,
$\mbf{V}=[a_{ij}]\in\R^{r\times r}$,
the vectors $\mbf{q}_0,\ldots,\mbf{q}_s\in\R^{r}$ defined by $\mbf{q}_i = [q_{j,i}]_{1 \le j \le r}$,
and four integers: the order $p$, the stage order $q$,
the number of external approximations $r$, and the number
of stages or internal approximations $s$.
The method (\ref{eq1.2}) can be written in a compact form
\begin{equation} \label{underlying-GLM}
\left\{
\begin{array}{l}
Y^{[n+1]}=h(\mbf{A}\otimes \mbf{I})\Big(f\big(Y^{[n+1]}\big)+g\big(Y^{[n+1]}\big)\Big)
+(\mbf{U}\otimes \mbf{I})y^{[n]}, \\ [3mm]
y^{[n+1]}=h(\mbf{B}\otimes \mbf{I})\Big(f\big(Y^{[n+1]}\big)+g\big(Y^{[n+1]}\big)\Big)
+(\mbf{V}\otimes \mbf{I})y^{[n]},
\end{array}
\right.
\end{equation}
$n=0,1,\ldots,N-1$, and the relation (\ref{eq1.4}) takes the form
\begin{equation} \label{eq1.6}
y^{[n]}=\ds\sum_{k=0}^p\mbf{q}_kh^ky^{(k)}(t_n)+O(h^{p+1}).
\end{equation}
Applying \eqref{eq1.2} to the basic test equation
$y'(t)=\lambda y(t)$, $t\geq 0$, $\lambda \in \C$, leads to the recurrence
equation
$$
y^{[n+1]}=\mbf{S}(z)y^{[n]},
\quad
n=0,1,\ldots,
$$
$z=h\lambda$, with the stability matrix given by
\begin{equation} \label{eq1.7}
\mbf{S}(z)=\mbf{V}+z\mbf{B}(\mbf{I}-z\mbf{A})^{-1}\mbf{U}.
\end{equation}
We also define the stability polynomial $\eta(w,z)$ by
\begin{equation} \label{eq1.8}
\eta(w,z)=\det\big(w\mbf{I}-\mbf{S}(z)\big).
\end{equation}
The region of absolute stability of the method (\ref{eq1.2}) is
the subset of the complex plane
\begin{equation} \label{eq1.9}
\mathcal{A}=\big\{z\in\C: \ \textrm{all roots} \ w_i(z) \ \textrm{of} \ \eta(w,z) \
\textrm{are in the unit circle}\big\}.
\end{equation}
The traditional concepts of $A(\alpha)$-stability, $A$-stability, and $L$-stability
apply directly to GLMs via \eqref{eq1.9}.
In this paper we will examine only methods of high stage order, i.e., methods
where $q=p-1$ or $q=p$. It has been shown in \cite{bj93,but93,jac09} that
the GLM (\ref{eq1.2}) has order $p$ and stage order $q=p$ or $q=p-1$ if and only if
\begin{eqnarray}
\label{stage-order-cond}
\mbf{c}^k-k\,\mbf{A}\, \mbf{c}^{k-1} - k!\, \mbf{U}\, \mbf{q}_k &=& 0\,, \quad k=0,1,\dots,q\,, \quad \textnormal{and}\\
\label{order-cond}
\sum_{\ell=0}^k \frac{k!}{\ell!}\, \mbf{q}_{k-\ell} - k\,\mbf{B}\, \mbf{c}^{k-1} - k!\, \mbf{V}\, \mbf{q}_k &=& 0\,, \quad k=0,1,\dots,p\,.
\end{eqnarray}
An IMEX-GLM \cite[Definition 4]{zhasan13} has the form
\begin{equation} \label{imex-glm}
\left\{
\begin{array}{l}
Y^{[n+1]}=h (\mbf{A}^{\rm exp}\otimes \mbf{I})\,f\big(Y^{[n+1]}\big)+h (\mbf{A}^{\rm imp}\otimes \mbf{I})\,g\big(Y^{[n+1]}\big)
+(\mbf{U}\otimes \mbf{I})y^{[n]}, \\ [3mm]
y^{[n+1]}=h(\mbf{B}^{\rm exp}\otimes \mbf{I})\,f\big(Y^{[n+1]}\big)+h(\mbf{B}^{\rm imp}\otimes \mbf{I})\, g\big(Y^{[n+1]}\big)
+(\mbf{V}\otimes \mbf{I})y^{[n]},
\end{array}
\right.
\end{equation}
where $\mbf{A}^{\rm exp}$, $\mbf{B}^{\rm exp}$ correspond to the explicit part and $\mbf{A}^{\rm imp}$, $\mbf{B}^{\rm imp}$ to the implicit part. The methods share the same
abscissa $\mbf{c}^{\rm exp}=\mbf{c}^{\rm imp}$, which makes \eqref{imex-glm} internally consistent \cite[Definition 2]{zhasan13}.
The methods also share the same coefficient matrices $\mbf{U}^{\rm exp}=\mbf{U}^{\rm imp}=\mbf{U}$ and $\mbf{V}^{\rm exp}=\mbf{V}^{\rm imp}=\mbf{V}$. The coefficients $\mbf{q}^{\rm exp}_k$, $\mbf{q}^{\rm imp}_k$ in \eqref{eq1.6} can be different,
which means that the implicit and explicit components use different initialization and termination procedures.
An IMEX-GLM \eqref{imex-glm} is a special case of a partitioned GLM \cite[Definition 1]{zhasan13};
while in \eqref{imex-glm} the right hand side is split in two components, stiff and nonstiff, a partitioned GLM
allows for splitting in an arbitrary number of components.
It has been shown in \cite[Theorem 2]{zhasan13} that
an internally consistent partitioned GLM (and, in particular, the IMEX GLM \eqref{imex-glm})
has order $p$ and stage order $q \in \{p-1,p\}$
if and only if each component method $\left(\mathbf{A}^{\rm exp},\mathbf{B}^{\rm exp},\mathbf{U},\mathbf{V}\right)$
and $\left(\mathbf{A}^{\rm imp},\mathbf{B}^{\rm imp},\mathbf{U},\mathbf{V}\right)$ has order $p$ and stage order $q$.
We note that no additional ``coupling'' conditions are needed for the IMEX GLM (i.e.,
no order conditions that contain coefficients of both the implicit and the explicit schemes).
\section{Extrapolation-based IMEX GLMs} \label{sec:eximex}
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\subsection{Method formulation} \label{sec2}
In this section we derive the new extrapolation-based IMEX GLMs.
Consider the following extrapolation formula depending on
stage values $Y_k^{[n]}$ and $Y_k^{[n+1]}$ at two consecutive steps
\begin{equation} \label{eq1.11}
f_j^{[n+1]}=\ds\sum_{k=1}^s\alpha_{jk}f\big(Y_k^{[n]}\big)
+\ds\sum_{k=1}^{j-1}\beta_{jk}f\big(Y_k^{[n+1]}\big),
\quad
j=1,2,\ldots,s.
\end{equation}
Substituting $f_j^{[n+1]}$ in (\ref{eq1.11}) for $f\big(Y_j^{[n+1]}\big)$ in (\ref{eq1.2}) leads to the proposed class of
extrapolation-based IMEX GLMs. The simple example of IMEX method consisting of the explicit
Euler method combined with the $A$-stable implicit $\theta$-method
corresponding to $\theta\geq 1/2$ is presented in \cite{hv03}.
Substituting (\ref{eq1.11}) into (\ref{eq1.2}) leads to
$$
\begin{array}{lcl}
Y_i^{[n+1]} & = & h\ds\sum_{j=1}^i\ds\sum_{k=1}^sa_{ij}\alpha_{jk}f\big(Y_k^{[n]}\big)
+h\ds\sum_{j=1}^i\ds\sum_{k=1}^{j-1}a_{ij}\beta_{jk}f\big(Y_k^{[n+1]}\big) \\ [3mm]
& + &
h\ds\sum_{j=1}^ia_{ij}g\big(Y_j^{[n+1]}\big)+\ds\sum_{j=1}^ru_{ij}y_j^{[n]},
\quad i=1,2,\ldots,s, \\ [3mm]
y_i^{[n+1]} & = & h\ds\sum_{j=1}^s\ds\sum_{k=1}^sb_{ij}\alpha_{jk}f\big(Y_k^{[n]}\big)
+h\ds\sum_{j=1}^s\ds\sum_{k=1}^{j-1}b_{ij}\beta_{jk}f\big(Y_k^{[n+1]}\big) \\ [3mm]
& + &
h\ds\sum_{j=1}^ib_{ij}g\big(Y_j^{[n+1]}\big)+\ds\sum_{j=1}^rv_{ij}y_j^{[n]},
\quad i=1,2,\ldots,r, \\ [3mm]
\end{array}
$$
$n=0,1,\ldots,N-1$.
Changing the order of summation in the double sums above and then
interchanging the indices $j$ and $k$ we obtain
$$
\begin{array}{lcl}
Y_i^{[n+1]} & = & h\ds\sum_{j=1}^s\ds\sum_{k=1}^ia_{ik}\alpha_{kj}f\big(Y_j^{[n]}\big)
+h\ds\sum_{j=1}^{i-1}\ds\sum_{k=j+1}^ia_{ik}\beta_{kj}f\big(Y_j^{[n+1]}\big) \\ [3mm]
& + &
h\ds\sum_{j=1}^ia_{ij}g\big(Y_j^{[n+1]}\big)+\ds\sum_{j=1}^ru_{ij}y_j^{[n]},
\quad i=1,2,\ldots,s,\\ [3mm]
y_i^{[n+1]} & = & h\ds\sum_{j=1}^s\ds\sum_{k=1}^sb_{ik}\alpha_{kj}f\big(Y_j^{[n]}\big)
+h\ds\sum_{j=1}^{s-1}\ds\sum_{k=j+1}^sb_{ik}\beta_{kj}f\big(Y_j^{[n+1]}\big) \\ [3mm]
& + &
h\ds\sum_{j=1}^sb_{ij}g\big(Y_j^{[n+1]}\big)+\ds\sum_{j=1}^rv_{ij}y_j^{[n]},
\quad i=1,2,\ldots,r.\\
\end{array}
$$
These relations lead to IMEX GLMs of the form
\begin{equation} \label{eq2.1}
\left\{
\begin{array}{lcl}
Y_i^{[n+1]} & = & h\ds\sum_{j=1}^s\bar{a}_{ij}f\big(Y_j^{[n]}\big)
+h\ds\sum_{j=1}^{i-1}a_{ij}^{*}f\big(Y_j^{[n+1]}\big) \\ [3mm]
& + &
h\ds\sum_{j=1}^ia_{ij}g\big(Y_j^{[n+1]}\big)
+\ds\sum_{j=1}^ru_{ij}y_j^{[n]},
\quad i=1,2,\ldots,s,\\ [3mm]
y_i^{[n+1]} & = & h\ds\sum_{j=1}^s\bar{b}_{ij}f\big(Y_j^{[n]}\big)
+h\ds\sum_{j=1}^{s-1}b_{ij}^{*}f\big(Y_j^{[n+1]}\big) \\ [3mm]
& + &
h\ds\sum_{j=1}^sb_{ij}g\big(Y_j^{[n+1]}\big)
+\ds\sum_{j=1}^rv_{ij}y_j^{[n]},
\quad i=1,2,\ldots,r,
\end{array}
\right.
\end{equation}
$n=0,1,\ldots,N-1$,
where the coefficients $\bar{a}_{ij}$, $a_{ij}^{*}$,
$\bar{b}_{ij}$, and $b_{ij}^{*}$ are defined by
$$
\bar{a}_{ij}=\ds\sum_{k=1}^ia_{ik}\alpha_{kj},
\quad
a_{ij}^{*}=\ds\sum_{k=j+1}^ia_{ik}\beta_{kj},
\quad
\bar{b}_{ij}=\ds\sum_{k=1}^sb_{ik}\alpha_{kj},
\quad
b_{ij}^{*}=\ds\sum_{k=j+1}^sb_{ik}\beta_{kj}.
$$
In matrix notation
$$
\bar{\mbf{A}}=[\bar{a}_{ij}]\in\R^{s\times s},
\
\mbf{A}^{*}=[a_{ij}^{*}]\in\R^{s\times s},
\
\bar{\mbf{B}}=[\bar{b}_{ij}]\in\R^{r\times s},
\
\mbf{B}^{*}=[b_{ij}^{*}]\in\R^{r\times s}.
$$
with
$$
\bar{\mbf{A}}=\mbf{A} \bf{\alpha},
\quad
\mbf{A}^{*}=\mbf{A} \mathbf{\beta},
\quad
\bar{\mbf{B}}=\mbf{B} \mathbf{\alpha},
\quad
\mbf{B}^{*}=\mbf{B} \mathbf{\beta},
$$
where $\mathbf{\alpha}=[\alpha_{ij}]\in\R^{s\times s}$, $\mathbf{\beta}=[\beta_{ij}]\in\R^{s\times s}$.
Observe that the matrix $\mbf{A}^{*}$ is strictly lower triangular
and that the last column of the matrix $\mbf{B}^{*}$ is zero.
In matrix notation the extrapolation-based IMEX-GLM is defined by:
\begin{eqnarray}
\nonumber
Y^{[n+1]}&=&h(\bar{\mbf{A}}\otimes \mbf{I})f\big(Y^{[n]}\big)
+h(\mbf{A}^{*}\otimes \mbf{I})f\big(Y^{[n+1]}\big) \\
\label{extrap-imex}
&& +h(\mbf{A}\otimes \mbf{I}) g\big(Y^{[n+1]}\big)+(\mbf{U}\otimes \mbf{I})y^{[n]}, \\
\nonumber
y^{[n+1]}&=&h(\bar{\mbf{B}}\otimes \mbf{I})f\big(Y^{[n]}\big)
+h(\mbf{B}^{*}\otimes \mbf{I})f\big(Y^{[n+1]}\big) \\
\nonumber
&&+h(\mbf{B}\otimes \mbf{I})f\big(Y^{[n+1]}\big) +(\mbf{V}\otimes \mbf{I})y^{[n]},
\end{eqnarray}
$n=0,1,\ldots,N-1$.
The explicit part of (\ref{extrap-imex}), obtained for $g(y)=0$, can be represented as a single GLM
extended over two steps from
$t_{n-1}$ to $t_n$ and $t_n$ to $t_{n+1}$, as follows
\begin{equation} \label{explicit}
\left[
\begin{array}{c}
Y^{[n]} \\
Y^{[n+1]} \\
\hline
Y^{[n+1]} \\
y^{[n+1]}
\end{array}
\right]=\left[
\begin{array}{cc|cc}
\mbf{0} & \mbf{0} & \; \mbf{I} & \mbf{0} \\
\bar{\mbf{A}} & \; \mbf{A}^{*} & \; \mbf{0} & \mbf{U} \\
\hline
\bar{\mbf{A}} & \mbf{A}^{*} & \; \mbf{0} & \mbf{U} \\
\bar{\mbf{B}} & \mbf{B}^{*} & \; \mbf{0} & \mbf{V} \\
\end{array}
\right]\left[
\begin{array}{c}
f\big(Y^{[n]}\big) \\
f\big(Y^{[n+1]}\big) \\ \hline
Y^{[n]} \\
y^{[n]}
\end{array}
\right].
\end{equation}
The abscissa vector is
$\mbf{c}^{\rm exp}=[(\mbf{c}-\mbf{e})^T,\mbf{c}^T]^T$.
Similarly, the implicit part of the IMEX scheme (\ref{eq2.1}) corresponding
to $f(y)=0$ assumes the form
\begin{equation} \label{implicit}
\left[
\begin{array}{c}
Y^{[n]} \\
Y^{[n+1]} \\
\hline
Y^{[n+1]} \\
y^{[n+1]}
\end{array}
\right]=\left[
\begin{array}{cc|cc}
\mbf{0} & \mbf{0} & \ \mbf{I} & \mbf{0} \\
\mbf{0} & \mbf{A} & \ \mbf{0} & \mbf{U} \\
\hline
\mbf{0} & \mbf{A} & \ \mbf{0} & \mbf{U} \\
\mbf{0} & \mbf{B} & \ \mbf{0} & \mbf{V} \\
\end{array}
\right]\left[
\begin{array}{c}
g\big(Y^{[n]}\big) \\
g\big(Y^{[n+1]}\big) \\
\hline
Y^{[n]} \\
y^{[n]}
\end{array}
\right].
\end{equation}
This method has the order and stage order of the underlying GLM (\ref{eq1.2}),
since it is the same method.
The abscissa vector is
$\mbf{c}^{\rm imp}=[(\mbf{c}-\mbf{e})^T,\mbf{c}^T]^T$, and therefore the method \eqref{extrap-imex}
is internally consistent.
\subsection{Construction of the interpolant}
We define the local discretization errors
$\eta(t_n+c_jh)$ of the extrapolation formula (\ref{eq1.11})
by the relation
\begin{equation} \label{eq2.11}
\begin{array}{l}
f\big(y(t_n+c_jh)\big)=\ds\sum_{k=1}^s\alpha_{jk}f\big(y(t_{n-1}+c_kh)\big)+ \\ [3mm]
\quad\quad + \
\ds\sum_{k=1}^{j-1}\beta_{jk}f\big(y(t_n+c_kh)\big)+\eta(t_n+c_jh),
\end{array}
\end{equation}
$j=1,2,\ldots,s$.
Letting $\varphi(t)=f(y(t))$ the relation (\ref{eq2.11}) can be written
in the form
$$
\eta(t_n+c_jh)=
\varphi(t_n+c_jh)-\ds\sum_{k=1}^s\alpha_{jk}\varphi\big(t_n+(c_k-1)h\big)
-\ds\sum_{k=1}^{j-1}\beta_{jk}\varphi(t_n+c_kh),
$$
$j=1,2,\ldots,s$.
Expanding $\varphi(t_n+c_jh)$, $\varphi(t_n+(c_k-1)h)$, and $\varphi(t_n+c_kh)$
into Taylor series around $t_n$ we obtain
$$
\eta(t_n+c_jh)=\ds\sum_{l=0}^p
\bigg(\ds\frac{c_j^l}{l!}-\ds\sum_{k=1}^s\alpha_{jk}\ds\frac{(c_k-1)^l}{l!}
-\ds\sum_{k=1}^{j-1}\beta_{jk}\ds\frac{c_k^l}{l!}\bigg)h^l\varphi^{(l)}(t_n)
+O(h^{p+1}).
$$
Assuming that the extrapolation procedure given by (\ref{eq1.11}) has order $p$, i.e.,
\mbox{$\eta(t_n+c_jh)=O(h^{p})$}, leads to the following system of equations for
the interpolation coefficients:
\begin{equation} \label{eq2.12}
\ds\sum_{k=1}^s\alpha_{jk}(c_k-1)^\ell
=c_j^\ell-\ds\sum_{k=1}^{j-1}\beta_{jk}c_k^\ell,
\quad
\ell=0,1,\ldots,p-1,
\quad
j=1,2,\ldots,s.
\end{equation}
In matrix notation we have
\begin{equation}
\label{interpolation-order}
\alpha\, (\mathbf{c}-\mathbf{e})^\ell + \beta \, \mathbf{c}^\ell =\mathbf{c}^\ell\,, \quad \ell=0,1,\ldots,p-1\,.
\end{equation}
\subsection{Stage and order conditions}
\begin{lemma}
Assume that the underlying GLM \eqref{underlying-GLM} has order $p$ and stage order $q=p$ or $q=p-1$, and that the interpolation formula (\ref{eq1.11})
has order $p$ \eqref{interpolation-order}.
Then the explicit method \eqref{explicit} has order $p$ and stage order $q$.
\end{lemma}
\begin{proof}
The method \eqref{explicit} has the coefficients
\[
\mbf{A}^{\rm exp}=\begin{bmatrix}
\mbf{0} & \mbf{0} \\
\bar{\mbf{A}} & \; \mbf{A}^{*}
\end{bmatrix}\,, \quad
\mbf{B}^{\rm exp}=\begin{bmatrix}
\bar{\mbf{A}} & \mbf{A}^{*} \\
\bar{\mbf{B}} & \mbf{B}^{*}
\end{bmatrix}\,, \quad
\mbf{c}^{\rm exp}=\begin{bmatrix} \mbf{c}-\mbf{e} \\ \mbf{c} \end{bmatrix}\,.
\]
where $\mbf{e}=[1,\ldots,1]^T\in\R^s$,
$$
\mbf{U}^{\rm exp}=\left[
\begin{array}{cc}
\mbf{I} & \mbf{0} \\
\mbf{0} & \mbf{U}
\end{array}
\right],
\quad
\mbf{V}^{\rm exp}=\left[
\begin{array}{cc}
\mbf{0} & \mbf{U} \\
\mbf{0} & \mbf{V}
\end{array}
\right]\,,
$$
and the vectors
$$
{\mbf{q}}_0^{\rm exp}=\left[
\begin{array}{c}
\mbf{e} \\
\mbf{q}_0
\end{array}
\right]\,;
\quad
{\mbf{q}}_i^{\rm exp}=
\begin{bmatrix}
\frac{(\mbf{c}-\mbf{e})^i}{i!} \\
\mbf{q}_i
\end{bmatrix} \,, ~~ i=1,\dots,p\,.
$$
We verify directly that the extrapolation-based explicit method \eqref{explicit} has stage order $q$,
i.e., it satisfies equations \eqref{stage-order-cond}
\begin{eqnarray*}
&& \ds (\mbf{c}^{\rm exp})^k - k\,\mbf{A}^{\rm exp}\, (\mbf{c}^{\rm exp})^{k-1}-k!\,\mbf{U}^{\rm exp}\, \mbf{q}^{\rm exp}_k \\
&& =
\begin{bmatrix}
\mathbf{0} \\
\ds \mbf{c}^k- k\, \mbf{A} \left( \alpha\, (\mbf{c}-\mathbf{e})^{k-1} + \beta\, \mathbf{c}^{k-1} \right)-k!\, \mbf{U}\mbf{q}_k
\end{bmatrix} \\
&& =
\begin{bmatrix}
\mathbf{0} \\
\ds \mbf{c}^k-k\, \mbf{A}\mbf{c}^{k-1}-k!\,\mbf{U}\mbf{q}_k
\end{bmatrix}\quad \{ \textrm{from }\eqref{interpolation-order}\} \\
&& = \begin{bmatrix}
\mathbf{0} \\ \mathbf{0}
\end{bmatrix} \,, \quad k = 0,1,\dots,q \quad \{ \textrm{from }\eqref{stage-order-cond}\} \,.
\end{eqnarray*}
From \eqref{order-cond} we verify that the method \eqref{explicit} has order $p$
\begin{eqnarray*}
&& \sum_{\ell=0}^k \frac{k!}{\ell!}\, \mbf{q}^{\rm exp}_{k-\ell} - k\,\mbf{B}^{\rm exp}\, (\mbf{c}^{\rm exp})^{k-1} - k!\, \mbf{V}^{\rm exp}\, \mbf{q}^{\rm exp}_k \\
&& = \sum_{\ell=0}^k \frac{k!}{\ell!}\,\begin{bmatrix} \ds \frac{(\mbf{c}-\mbf{e})^{k-\ell}}{(k-\ell)!} \\ \ds \mbf{q}_{k-\ell} \end{bmatrix}
- k\, \begin{bmatrix} \mathbf{A}\, \left( \alpha\, (\mathbf{c}-\mathbf{e})^{k-1} + \beta\, \mathbf{c}^{k-1} \right) \\ \mathbf{B}\, \left( \alpha\, (\mathbf{c}-\mathbf{e})^{k-1} + \beta\, \mathbf{c}^{k-1} \right) \end{bmatrix}
- k!\,\begin{bmatrix} \mathbf{U}\, \mathbf{q}_k \\ \mathbf{V}\, \mathbf{q}_k \end{bmatrix} \\
&& = \begin{bmatrix}
\ds\sum_{\ell=0}^k \frac{k!}{\ell!} \frac{(\mbf{c}-\mbf{e})^{k-\ell}}{(k-\ell)!} - k\,\mbf{A}\mbf{c}^{k-1}- k!\, \mathbf{U}\, \mathbf{q}_k \\
\ds\sum_{\ell=0}^k \frac{k!}{\ell!} \mbf{q}_{k-\ell} - k\,\mbf{B}\mbf{c}^{k-1}- k!\, \mathbf{V}\, \mathbf{q}_k
\end{bmatrix} \quad \{ \textrm{from }\eqref{interpolation-order} \} \\
&& = \begin{bmatrix}
\mathbf{0} \\ \mathbf{0}
\end{bmatrix} \,, \quad k = 0,1,\dots,p \quad \{ \textrm{from }\eqref{stage-order-cond}, \eqref{order-cond}, \textrm{ and } \eqref{combinatorial} \}\,.
\end{eqnarray*}
For the first component we have used the fact that
\begin{equation}
\label{combinatorial}
\mbf{c}^\ell = \bigl( (\mbf{c}-\mbf{e}) + \mbf{e} \bigr)^\ell = \sum_{\ell=0}^k \frac{\ell!}{k!\,(\ell-k)! } (\mbf{c}-\mbf{e})^{k-\ell}\,.
\end{equation}
\flushright{$\square$}
\end{proof}
To analyze the order and stage order of IMEX GLMs (\ref{eq2.1}) we will
impose some conditions on the local discretization errors of the internal
and external stages of the underlying GLM (\ref{eq1.2}) and on the accuracy of
the extrapolation procedure (\ref{eq1.11}).
We have the following result.
\begin{theorem} \label{th2.3}
Assume that the underlying GLM (\ref{eq1.2}) has order $p$ and stage order $q=p$ or $q=p-1$,
and that the extrapolation procedure (\ref{eq1.11}) has order $p$. Then the IMEX GLM (\ref{eq2.1})
has order $p$ and stage order $q=p$ or $q=p-1$.
\end{theorem}
\begin{proof}
The method (\ref{extrap-imex}) is an IMEX GLM of the form \eqref{imex-glm} \cite[Definition 1]{zs13}.
The explicit (\ref{explicit}) and implicit (\ref{implicit}) components have the same order $p$ and stage order $q$, and they share
the same abscissa vector and the same coefficients
\[
\mbf{c}^{\rm exp} = \mbf{c}^{\rm imp}\,,
\quad
\mbf{U}^{\rm imp}=\mbf{U}^{\rm exp}\,,
\quad
\mbf{V}^{\rm imp}=\mbf{V}^{\rm exp}\,.
\]
The result follows directly from \cite[Theorem 2]{zs13}.
\flushright{$\square$}
\end{proof}
\subsection{Prothero-Robinson convergence of IMEX GLMs} \label{sec3}
The extrapolation IMEX-GLM schemes (\ref{eq2.1})
do not suffer from order reduction phenomenon when applied to stiff
systems of differential equations.
Following \cite{but87,zs13,zhasan13} we consider the Prothero-Robinson
(PR) \cite{pr74} test problem of the form
\begin{equation} \label{eq3.1}
\left\{
\begin{array}{ll}
y'(t)=\mu\big(y(t)-\phi(t)\big)+\phi'(t), & t\geq 0, \\
y(0)=\phi(0),
\end{array}
\right.
\end{equation}
where $\mu\in\C$ has a large and negative real part and $\phi(t)$ is a
slowly varying function.
The solution to (\ref{eq3.1}) is $y(t)=\phi(t)$.
The IMEX scheme (\ref{eq2.1}) is said to be PR-convergent if the application
of (\ref{eq2.1}) to the equation (\ref{eq3.1}) leads to the numerical
solution $y^{[n]}$ whose global error satisfies
$$
\left\|y^{[n]}-\ds\sum_{k=0}^p\mbf{q}_kh^ky^{(k)}(t_n)\right\|=\mathcal{O}(h^p)
\quad
\textrm{as}
\quad
h\rightarrow 0
\quad
\textrm{and}
\quad
h\mu\rightarrow -\infty.
$$
We have the following result.
\begin{theorem} \label{th3.1}
Assume that the implicit GLM (\ref{eq1.2}) has order
$p$ and stage order $q=p-1$ or $q=p$, and that the extrapolation
formula (\ref{eq1.11}) has order $p$.
Then the IMEX scheme (\ref{eq2.1}) is PR-convergent with
order $\min(p,q)$ as $h\rightarrow 0$, $h\mu\rightarrow -\infty$,
and $h\mu\in\mathcal{S}_I$. Here, $\mathcal{S}_I$ is the stability region of the implicit GLM \eqref{implicit}.
\end{theorem}
\begin{proof}
The result follows directly from \cite[Theorem 3]{zs13} on PR-convergence of IMEX-GLMs. \flushright{$\square$}
\end{proof}
\setcounter{equation}{0}
\setcounter{figure}{0}
\setcounter{table}{0}
\section{Stability analysis of IMEX GLMs} \label{sec:stability}
To analyze stability properties of IMEX GLMs (\ref{eq2.1}) we
use the test equation
\begin{equation} \label{eq4.1}
y'(t)=\lambda_0y(t)+\lambda_1y(t),
\quad
t\geq 0,
\end{equation}
where $\lambda_0$ and $\lambda_1$ are complex parameters.
Applying \eqref{extrap-imex} to (\ref{eq4.1}) and letting
$z_i=h\lambda_i$, $i=0,1$, we obtain
$$
\left\{
\begin{array}{c}
\big(\mbf{I}-(z_0\mbf{A}^{*}+z_1\mbf{A})\big)Y^{[n+1]}=z_0\bar{\mbf{A}}Y^{[n]}
+\mbf{U}y^{[n]}, \\ [3mm]
-(z_0\mbf{B}^{*}+z_1\mbf{B})Y^{[n+1]}+y^{[n+1]}=z_0\bar{\mbf{B}}Y^{[n]}
+\mbf{V}y^{[n]},
\end{array}
\right.
$$
$n=0,1,\ldots,N-1$. This is equivalent to the matrix recurrence relation
\begin{equation} \label{eq4.4}
\left[
\begin{array}{c}
Y^{[n+1]} \\
y^{[n+1]}
\end{array}
\right]=\mbf{M}(z_0,z_1)\left[
\begin{array}{c}
Y^{[n]} \\
y^{[n]}
\end{array}
\right],
\end{equation}
where the stability matrix $\mbf{M}(z_0,z_1)$ is defined by
$$
\mbf{M}(z_0,z_1)=\left[
\begin{array}{cc}
m_{11}(z_0,z_1) & m_{12}(z_0,z_1) \\
m_{21}(z_0,z_1) & m_{22}(z_0,z_1)
\end{array}
\right]
$$
with
$$
m_{11}(z_0,z_1)=z_0\big(\mbf{I}-(z_0\mbf{A}^{*}+z_1\mbf{A})\big)^{-1}\bar{\mbf{A}},
$$
$$
m_{12}(z_0,z_1)=\big(\mbf{I}-(z_0\mbf{A}^{*}+z_1\mbf{A})\big)^{-1}\mbf{U},
$$
$$
m_{21}(z_0,z_1)=z_0\Big(\bar{\mbf{B}}+(z_0\mbf{B}^{*}+z_1\mbf{B})
\big(\mbf{I}-(z_0\mbf{A}^{*}+z_1\mbf{A})\big)^{-1}\bar{\mbf{A}}\Big),
$$
$$
m_{22}(z_0,z_1)=\mbf{V}+(z_0\mbf{B}^{*}+z_1\mbf{B})
\big(\mbf{I}-(z_0\mbf{A}^{*}+z_1\mbf{A})\big)^{-1}\mbf{U}.
$$
We define also the stability function of the IMEX GLM (\ref{eq2.1}) as a characteristic
polynomial of the stability matrix $\mbf{M}(z_0,z_1)$, i.e.,
$$
p(w,z_0,z_1)=\det\big(w\mbf{I}-\mbf{M}(z_0,z_1)\big).
$$
For $z_0=0$ the stability matrix $\mbf{M}(0,z_1)$ and polynomial $p(w,0,z_1)=w^s\eta(w,z_1)$
are those of the underlying GLM (\ref{eq1.2}).
For $z_1=0$ we obtain $\mbf{M}(z_0,0)$,
and it can be verified that this corresponds to the stability matrix
of the explicit method (\ref{explicit}).
We say that the IMEX GLM (\ref{eq2.1}) is stable for given $z_0,z_1\in\C$ if
all the roots $w_i(z_0,z_1)$, $i=1,2,\ldots,s+r$, of the stability function
$p(w,z_0,z_1)$ are inside of the unit circle.
As observed in \cite{hv03} in the context of IMEX $\theta$-methods
of order one, a large region of absolute stability for the
explicit method (\ref{explicit}) and good stability properties
(for example $A$- or $L$-stability) for the
implicit method are not
sufficient to guarantee desirable stability
properties of the overall IMEX GLM (\ref{eq2.1}). We have to investigate the stability properties of the combined as IMEX GLM \cite{zhasan13,zs13}.
In this paper
we will be mainly interested in IMEX schemes which
are $A(\alpha)$- or $A$-stable with respect to the implicit part $z_1\in\C$.
To investigate such methods
we consider, similarly as in \cite{hv03,zs13}, the sets
\begin{equation} \label{eq4.5}
\mathcal{S}_{\alpha}=
\left\{ \begin{array}{lll}
z_0\in \C &:& \ \textrm{the IMEX GLM is stable for any } \\
&&z_1\in \C: \ \mathcal{R}(z_1)<0 ~~ \textrm{and} ~~ \big|\Im(z_1)\big|\leq \tan(\alpha)\big|\mathcal{R}(z_1)\big|
\end{array} \right\}.
\end{equation}
For fixed values of $y\in \R$ we define also the sets
\begin{equation} \label{eq4.6}
\mathcal{S}_{\alpha,y}=
\left\{
\begin{array}{lcl}
z_0\in \C\!\!\!\!&:& \textrm{the IMEX GLM is stable for fixed} \\
& & z_1=-|y|/\tan(\alpha)+iy
\end{array}
\right\}.
\end{equation}
It follows from the maximum principle that
\begin{equation} \label{eq4.7}
\mathcal{S}_{\alpha}=\bigcap_{y\in\R}\mathcal{S}_{\alpha,y}.
\end{equation}
Observe also that the region $\mathcal{S}_{\alpha,0}$ is independent of
$\alpha$, and corresponds to the region of absolute stability
of the explicit method (\ref{explicit}).
This region will be denoted by $\mathcal{S}_E$.
We have
\begin{equation} \label{eq4.8}
\mathcal{S}_{\alpha}\subset \mathcal{S}_E,
\end{equation}
and we will look for IMEX GLMs for which the stability region $\mathcal{S}_{\alpha}$ contains a
large part of the stability region $\mathcal{S}_E$ of the explicit method (\ref{explicit}),
for some $\alpha\in (0,\pi/2]$, preferably for $\alpha=\pi/2$.
The boundary $\partial \mathcal{S}_{\alpha,y}$ of the region $\mathcal{S}_{\alpha,y}$ can be determined by
the boundary locus method which computes the locus of the curve
$$
\partial \mathcal{S}_{\alpha,y}=\Big\{z_0\in\C: \ p\big(e^{i\theta},z_0,-|y|/\tan(\alpha)+iy\big)=0,
\ \theta\in[0,2k\pi]\Big\},
$$
where $k$ is a positive integer.
\begin{figure}[t!h!b!]
\begin{center}
\includegraphics[width=0.75\textwidth]{fig4_1-eps-converted-to.pdf}
\caption{Points on the intersection of the ray $y_0=mx_0$ and
$\partial\mathcal{S}_{\pi/2,y}$ for $y=-1.5,-1.0,\ldots,1.5$ (circles)
and on the intersection of $y_0=mx_0$ and $\partial\mathcal{S}_{\pi/2}$ (square).
This figure corresponds to IMEX GLM scheme with $p=q=r=s=2$ for $m=-1$, $\lambda=0.3$,
and $\beta_{21}=4.3$} \label{fig4.1}
\end{center}
\end{figure}
We have also developed an algorithm to determine the boundary $\partial \mathcal{S}_{\alpha}$
of the stability
region $\mathcal{S}_{\alpha}$. For fixed direction $m$ corresponding to the ray
$$
y_0=mx_0,
$$
and for fixed $z_1=-|y|/\tan(\alpha)+iy$ (or any $z_1\in\C$) we can compute the point
$z_0=x_0+iy_0$ of intersection of the boundary $\partial \mathcal{S}_{\alpha,y}$ of
$\mathcal{S}_{\alpha,y}$ with the ray $y_0=mx_0$ taking into account that such a point satisfies the
condition
$$
\max_{i=1,2,\ldots,s+r}\Big|w_i\big(z_0,-|y|/\tan(\alpha)+iy\big)\Big|=1.
$$
This can be done
by the bisection method which terminates when the
condition
\begin{equation} \label{eq4.9}
\Big|\max_{i=1,2,\ldots,s+r}\Big|w_i\big(z_0,-|y|/\tan(\alpha)+iy\big)\Big|-1\Big| \leq tol,
\end{equation}
with accuracy tolerance $tol$ is satisfied.
We apply this method to the interval $[\bar{x}_0,0]$ with $\bar{x}_0$ large enough so
that the condition (\ref{eq4.9}) is not satisfied for the first iteration of bisection method.
This process leads to the definition of the function
$$
x_0=f(m,\alpha,y),
$$
where $x_0$ corresponds to the points $z_0=x_0+iy_0\in \partial\mathcal{S}_{\alpha,y}$.
Then the boundary $\partial\mathcal{S}_{\alpha}$ of the region $\mathcal{S}_{\alpha}$
can be determined by minimizing the negative value of this function for
$m\in\R$ and plotting the resulting points $z_0=x_0+iy_0$.
This algorithm is illustrated in Fig.~\ref{fig4.1},
where we have plotted
points on the intersection of the ray $y_0=mx_0$ and
$\partial\mathcal{S}_{\pi/2,y}$ for $y=-1.5,-1.0,\ldots,1.5$ (circles)
and on the intersection of $y_0=mx_0$ and $\partial\mathcal{S}_{\pi/2}$ (square).
This figure corresponds to IMEX GLM scheme with $p=q=r=s=2$ for $m=-1$, $\lambda=0.3$,
and $\beta_{21}=4.3$.
This minimization can be
accomplished using the subroutine \texttt{fminsearch.m} in Matlab
applied to $f(m,\alpha,y)$ for fixed values of $m\in\R$ and $\alpha\in[0,\pi/2]$
starting with
appropriately chosen initial guesses for $y$.
This process will be next applied to specific IMEX GLMs.
\setcounter{equation}{0}
\setcounter{figure}{0}
\setcounter{table}{0}
\section{Construction of IMEX GLMs with desirable stability properties} \label{sec:construction}
In this section we describe the construction of IMEX GLMs (\ref{eq2.1}) up to the
order $p=4$ with large regions of absolute stability $\mathcal{S}$ with respect to the explicit part
assuming that the implicit part is $A$- or $L$-stable.
We will always start with implicit DIMSIM with $p=q=r=s$ and the abscissa
vector $\mbf{c}$ given in advance. After computing the coefficient
matrices $\mbf{A}$ and $\mbf{V}$ so that the resulting method has Runge-Kutta
stability with the underlying Runge-Kutta formula which is
$A$- or $L$-stable, the coefficient matrix $\mbf{B}$ is computed from the relation
\begin{equation} \label{eq5.1}
\mbf{B}=\mbf{B}_0-\mbf{A}\mbf{B}_1-\mbf{V}\mbf{B}_2+\mbf{V}\mbf{A}.
\end{equation}
Here, $\mbf{B}_0$, $\mbf{B}_1$, and $\mbf{B}_2$ are $s\times s$ matrices with
the $(i,j)$ elements
given by
$$
\ds\frac{\ds\int_0^{1+c_i}\phi_j(x)dx}{\phi_j(c_j)},
\quad
\ds\frac{\phi_j(1+c_i)}{\phi_j(c_j)},
\quad
\ds\frac{\ds\int_0^{c_i}\phi_j(x)dx}{\phi_j(c_j)},\quad
\phi_i(x)=\ds\prod_{j=1,j\neq i}^s(x-c_j),
$$
$i=1,2,\ldots,s$, compare Th.~$5.1$ in \cite{but93} or Th.~$3.2.1$ in \cite{jac09}.
\subsection{IMEX GLMs with $p=q=r=s=1$} \label{sec5.1}
Consider the implicit $\theta$-method defined by
\begin{equation} \label{eq5.2}
\left\{
\begin{array}{l}
Y^{[n+1]}=h\theta\big(f(Y^{[n+1]})+g(Y^{[n+1]})\big)+y^{[n]}, \\ [3mm]
y^{[n+1]}=h\big(f(Y^{[n+1]})+g(Y^{[n+1]})\big)+y^{[n]},
\end{array}
\right.
\end{equation}
$n=0,1,\ldots,N-1$.
This method is $A$-stable for $\theta\in[1/2,1]$ and $L$-stable for
$\theta\in(1/2,1]$. Consider also the extrapolation procedure
\begin{equation} \label{eq5.3}
f(Y^{[n+1]})=f(Y^{[n]}),
\end{equation}
$n=1,2,\ldots,N-1$.
Substituting (\ref{eq5.3}) into (\ref{eq5.2}) we obtain IMEX $\theta$-method of the form
\begin{equation} \label{eq5.4}
\left\{
\begin{array}{l}
Y^{[n+1]}=h\theta\big(f(Y^{[n]})+g(Y^{[n+1]})\big)+y^{[n]}, \\ [3mm]
y^{[n+1]}=h\big(f(Y^{[n]})+g(Y^{[n+1]})\big)+y^{[n]},
\end{array}
\right.
\end{equation}
$n=1,2,\ldots,N-1$.
We would like to point out that this variant of IMEX scheme is different
from IMEX $\theta$-method considered in \cite{hv03}.
Observe that the method (\ref{eq5.4}) requires a starting procedure
to compute $Y^{[1]}\approx y(t_0+\theta h)$ and $y^{[1]}\approx y(t_1)$.
The method (\ref{eq5.2}) can be represented by the abscissa $\mbf{c}=\theta$, the partitioned
matrix
\begin{equation*}
\left[
\begin{array}{c|c}
\mbf{A} & \mbf{U} \\
\hline
\mbf{B} & \mbf{V}
\end{array}
\right]=\left[
\begin{array}{c|c}
\theta \ & \ 1 \\
\hline
1 \ & \ 1
\end{array}
\right],
\end{equation*}
and $\mbf{q}_0=1$, $\mbf{q}_1=0$,
and the explicit formula corresponding to $g(y)=0$ in (\ref{eq5.4}) is GLM
with $\mbf{c}=[\theta-1,\theta]^T$,
\begin{equation} \label{eq5.5}
\left[
\begin{array}{c|c}
\mbf{A} & \mbf{U} \\
\hline
\mbf{B} & \mbf{V}
\end{array}
\right]=\left[
\begin{array}{cc|cc}
0 & 0 & \ 1 & 0 \\
\theta & 0 & \ 0 & 1 \\
\hline
\theta & 0 & \ 0 & 1 \\
1 & 0 & \ 0 & 1
\end{array}
\right],
\end{equation}
and $\mbf{q}_0=[1,1]^T$, $\mbf{q}_1=[\theta-1,0]^T$.
\begin{figure}[t!h!b!]
\begin{center}
\includegraphics[width=0.75\textwidth]{fig5_1-eps-converted-to.pdf}
\caption{Stability regions $\mathcal{S}_E=\mathcal{S}_E(\theta)$ of
explicit methods for $\theta=1/2$, $2/3$, $3/4$, $4/5$, and $1$} \label{fig5.1}
\end{center}
\end{figure}
\begin{figure}[t!h!b!]
\begin{center}
\includegraphics[width=0.75\textwidth]{fig5_2-eps-converted-to.pdf}
\caption{Stability regions $\mathcal{S}_{\pi/2,y}(\theta)$, $y=-1.0,-0.9,\ldots,1.0$ (thin lines),
$\mathcal{S}_{\pi/2}(\theta)$ (shaded region), and $\mathcal{S}_E(\theta)$ (thick line)
for $\theta=3/4$} \label{fig5.2}
\end{center}
\end{figure}
\begin{figure}[t!h!b!]
\begin{center}
\includegraphics[width=0.75\textwidth]{fig5_3-eps-converted-to.pdf}
\caption{Stability regions $\mathcal{S}_{\pi/2,y}(\theta)$, $y=-1.0,-0.9,\ldots,1.0$ (thin lines),
$\mathcal{S}_{\pi/2}(\theta)$ (shaded region), and $\mathcal{S}_E(\theta)$ (thick line)
for $\theta=2/3$} \label{fig5.3}
\end{center}
\end{figure}
It can be verified that the stability matrix of IMEX scheme (\ref{eq5.4}) takes the form
$$
\mbf{M}(z_0,z_1)=
\ds\frac{1}{1-\theta z_1}
\left[
\begin{array}{cc}
\theta z_0 & 1 \\
z_0 & 1+(1-\theta)z_1
\end{array}
\right].
$$
To investigate stability properties of (\ref{eq5.4}) it is
more convenient to work with the polynomial obtained by multiplying the
characteristic function $p(w,z_0,z_1)$ of $\mbf{M}(z_0,z_1)$ by a factor
$(1-\theta z_1)^2$. The resulting quadratic polynomial,
which will be denoted by the same symbol
$p(w,z_0,z_1)$, takes the form
$$
p(w,z_0,z_1)=(1-\theta z_1)^2w^2
-(1-\theta z_1)\big(1+\theta z_0+(1-\theta)z_1\big)w
-(1-\theta)z_0(1-\theta z_1).
$$
The stability polynomial of the explicit methods (\ref{eq5.5})
obtained by letting $g(y)=0$ in (\ref{eq5.4})
corresponds to $z_1=0$ and is given by
$$
p(w,z_0,0)=w^2-(1+\theta z_0)w-(1-\theta)z_0.
$$
The stability regions $\mathcal{S}_E=\mathcal{S}_E(\theta)$
corresponding to this polynomial are plotted in Fig~\ref{fig5.1} for
$\theta=1/2$, $2/3$, $3/4$, $4/5$ and $1$.
Observe that the stability region of the method (\ref{eq5.5}) corresponding
to $\theta=1$ is the unit disk
$$
\mathcal{S}_E=\mathcal{S}_E(1)=\big\{z_0\in\C: \ |z_0+1|<1\big\}.
$$
We will investigate next the regions $\mathcal{S}_{\pi/2}=\mathcal{S}_{\pi/2}(\theta)$
defined by (\ref{eq4.5}).
We analyze first the case $\theta=1$. It follows from Schur criterion applied
to the polynomial $p(w,z_0,iy)$ that $z_0\in\mathcal{S}_{\pi/2}(1)$ if
and only if
$$
y^2-2x_0-x_0^2-y_0^2>0
$$
for any $y\in\R$.
This is equivalent to
$$
(x_0+1)^2+y_0^2<1
\quad
\textrm{or}
\quad
|z_0+1|<1
$$
and it follows that in this case $\mathcal{S}_{\pi/2}(1)=\mathcal{S}_E(1)$.
For $\theta\in(1/2,1)$ $\mathcal{S}_{\pi/2}(\theta)$ is no longer equal to $\mathcal{S}_E(\theta)$,
but $\mathcal{S}_{\pi/2}(\theta)$ contains some part of $\mathcal{S}_E(\theta)$. This is
illustrated in Fig.~\ref{fig5.2} for $\theta=3/4$ and in Fig.~\ref{fig5.3}
for $\theta=2/3$.
In these figures we have plotted the regions $\mathcal{S}_{\pi/2,y}=\mathcal{S}_{\pi/2,y}(\theta)$
defined by (\ref{eq4.6}) for
$y=-1.0,-0.9,\ldots,1.0$ (thin lines), the regions $\mathcal{S}_{\pi/2}=\mathcal{S}_{\pi/2}(\theta)$
defined by (\ref{eq4.5}) (shaded regions), and the regions
$\mathcal{S}_E=\mathcal{S}_E(\theta)$ corresponding to explicit methods
(\ref{eq5.5}) for $\theta=3/4$ and $\theta=2/3$. We can see that for these values of
$\theta$ the sets $\mathcal{S}_{\pi/2}(\theta)$ contain quite large parts
of $\mathcal{S}_E(\theta)$.
These figures illustrate also
the relation (\ref{eq4.7}) and the inclusion (\ref{eq4.8}).
For $\theta=1/2$ it follows from Schur criterion that
$z_0\in\mathcal{S}_{\pi/2}(1/2)$ if and only if
$$
x_0^2+y_0^2<4
\quad
\textrm{and}
\quad
x_0y^2-4yy_0-x_0(x_0+2)^2-y_0^2(4+x_0)>0
$$
for any $y\in\R$.
This implies that
$$
x_0>0
\quad
\textrm{and}
\quad
(2+x_0)^2(x_0^2+y_0^2)<0
$$
and it follows that $\mathcal{S}_{\pi/2}(1/2)$ is empty.
\subsection{IMEX GLMs with $p=q=r=s=2$} \label{sec5.2}
Consider the implicit DIMSIM with $\mbf{c}=[0,1]^T$, the coefficient
matrices given by
$$
\left[
\begin{array}{c|c}
\mbf{A} & \mbf{U} \\
\hline
\mbf{B} & \mbf{V}
\end{array}
\right]=\left[
\begin{array}{cc|cc}
\lambda & 0 & 1 & 0 \\
\frac{2}{1+2\lambda} & \lambda & 0 & 1 \\
\hline
\frac{8\lambda^3+12\lambda^2-2\lambda+5}{4(2\lambda+1)} & \frac{1-4\lambda^2}{4} &
\frac{1}{2}+\lambda & \frac{1}{2}-\lambda \\
\frac{8\lambda^3+20\lambda^2-2\lambda+3}{4(2\lambda+1)} &
\frac{-8\lambda^3-12\lambda^2+10\lambda-1}{4(2\lambda+1)} &
\frac{1}{2}+\lambda & \frac{1}{2}-\lambda
\end{array}
\right],
$$
and the vectors $\mbf{q}_0$, $\mbf{q}_1$, and $\mbf{q}_2$ equal to
$$
\mbf{q}_0=\left[
\begin{array}{c}
1 \\
1
\end{array}
\right],
\quad
\mbf{q}_1=\left[
\begin{array}{c}
-\lambda \\
\frac{-2\lambda^2+\lambda-1}{2\lambda+1}
\end{array}
\right],
\quad
\mbf{q}_2=\left[
\begin{array}{c}
0 \\
\frac{1-2\lambda}{2}
\end{array}
\right].
$$
It was demonstrated in \cite{jac09} that this method has order $p=2$ and
stage order $q=2$. Moreover, this method is $A$-stable if $\lambda\geq 1/4$
and $L$-stable for $\lambda=(2\pm \sqrt{2})/2$.
The coefficients $\alpha_{jk}$ of the extrapolation formula (\ref{eq1.11})
of order $p=2$ computed from the system (\ref{eq2.12}) corresponding to
$p=s=2$ take the form
$$
\alpha=\left[
\begin{array}{cc}
\alpha_{11} & \alpha_{12} \\
\alpha_{21} & \alpha_{22}
\end{array}
\right]=\left[
\begin{array}{cc}
\phantom{-}0 & 1 \\
-1 & 2-\beta_{21}
\end{array}
\right],
$$
and the matrices $\bar{\mbf{A}}$, $\mbf{A}^{*}$, $\bar{\mbf{B}}$, and $\mbf{B}^{*}$
appearing in the representation of IMEX GLM (\ref{eq2.1}), and the corresponding
explicit scheme (\ref{extrap-imex}) or (\ref{explicit}), are given by
$$
\bar{\mbf{A}}=\left[
\begin{array}{cc}
0 & \lambda \\
-\lambda & \frac{2+(2-\beta_{21})\lambda+2(2-\beta_{21})\lambda^2}{1+2\lambda}
\end{array}
\right],
\quad
\mbf{A}^{*}=\left[
\begin{array}{cc}
0 & 0 \\
\beta_{21}\lambda & 0
\end{array}
\right],
$$
$$
\bar{\mbf{B}}=\left[
\begin{array}{cc}
\frac{4\lambda^2-1}{4} &
\frac{7-\beta_{21}+2(1-\beta_{21})\lambda+4(1+\beta_{21})\lambda^2-8(1-\beta_{21})\lambda^3}{4(1+2\lambda)} \\
\frac{1-10\lambda+12\lambda^2+8\lambda^3}{4(1+2\lambda)} &
\frac{1+\beta_{21}+2(9-5\beta_{21})\lambda-4(1-3\beta_{21})\lambda^2-8(1-\beta_{21})\lambda^3}{4(1+2\lambda)}
\end{array}
\right],
$$
$$
\mbf{B}^{*}=\left[
\begin{array}{cc}
\frac{\beta_{21}(1-4\lambda^2)}{4} & 0 \\
\frac{-\beta_{21}(1-10\lambda+12\lambda^2+8\lambda^3)}{4(1+2\lambda)} & 0
\end{array}
\right].
$$
\begin{figure}[t!h!b!]
\begin{center}
\includegraphics[width=0.75\textwidth]{fig5_4-eps-converted-to.pdf}
\caption{Areas of stability regions $\mathcal{S}_E=\mathcal{S}_E(\beta_{21})$ and
$\mathcal{S}_{\alpha}=\mathcal{S}_{\alpha}(\beta_{21})$ for
$\alpha=\pi/2$ and $\beta_{21}\in [0,8]$} \label{fig5.4}
\end{center}
\end{figure}
\begin{figure}[t!h!b!]
\begin{center}
\includegraphics[width=0.75\textwidth]{fig5_5-eps-converted-to.pdf}
\caption{Stability regions $\mathcal{S}_{\pi/2,y}$, $y=-2.0,-1.8,\ldots,2.0$ (thin lines),
$\mathcal{S}_{\pi/2}$ (shaded region), and $\mathcal{S}_E$ (thick line)
for $\lambda=(2-\sqrt{2})/2$ and $\beta_{21}\approx 4.64$} \label{fig5.5}
\end{center}
\end{figure}
\begin{figure}[t!h!b!]
\begin{center}
\includegraphics[width=0.75\textwidth]{fig5_6-eps-converted-to.pdf}
\caption{Contour plots of the area of stability region
$\mathcal{S}_{\pi/2}$ of IMEX GLMs for $p=q=r=s=2$} \label{fig5.6}
\end{center}
\end{figure}
\begin{figure}[t!h!b!]
\begin{center}
\includegraphics[width=0.75\textwidth]{fig5_7-eps-converted-to.pdf}
\caption{Stability regions $\mathcal{S}_{\pi/2,y}$, $y=-2.0,-1.8,\ldots,2.0$ (thin lines),
$\mathcal{S}_{\pi/2}$ (shaded region), and $\mathcal{S}_E$ (thick line)
for $\lambda \approx 0.29$ and $\beta_{21}\approx 4.64$} \label{fig5.7}
\end{center}
\end{figure}
To investigate the stability properties of the resulting IMEX scheme
we will work with the stability polynomial
$p(w,z_0,z_1)$ obtained by multiplying stability function of
the method by a factor $(1-\lambda z_1)^2$. It can be verified that
this polynomial takes the form
$$
p(w,z_0,z_1)=
w\big((1-\lambda z_1)^2w^3-p_3(z_0,z_1)w^2
+p_2(z_0,z_1)w-p_1(z_0,z_1)\big),
$$
with the coefficients $p_1(z_0,z_1)$, $p_2(z_0,z_1)$, and $p_3(z_0,z_1)$ which
depend also on $\lambda$ and $\beta_{21}$. These coefficients are given by
$$
\begin{array}{lcl}
p_1(z_0,z_1) \ = \ \ds\frac{4-\beta_{21}-2(4-\beta_{21})\lambda+4\beta_{21}\lambda^2-8\beta_{21}\lambda^3}
{4(1+2\lambda)}z_0
+\ds\frac{1-4\lambda+2\lambda^2}{2}z_0^2,
\end{array}
$$
$$
\begin{array}{lcl}
p_2(z_0,z_1) & = & \ds\frac{2(1-\lambda+(2-\beta_{21})\lambda^2-2\beta_{21}\lambda^3)}
{1+2\lambda}z_0
+\ds\frac{\beta_{21}-4\beta_{21}\lambda+2(1+\beta_{21})\lambda^2}
{2}z_0^2 \\ [5mm]
& - &
\ds\frac{2-\beta_{21}-4(2-\beta_{21})\lambda+2(2-\beta_{21})\lambda^2}{2}z_0z_1,
\end{array}
$$
$$
\begin{array}{lcl}
p_3(z_0,z_1) & = &
1+\ds\frac{8+\beta_{21}+2(4-\beta_{21})\lambda+4(4-3\beta_{21})\lambda^2-8\beta_{21}\lambda^3}
{4(1+2\lambda)}z_0 \\ [5mm]
& + &
\beta_{21}\lambda^2z_0^2-(2-\beta_{21})z_0z_1+(1-2\lambda)z_1
+\ds\frac{1-4\lambda+2\lambda^2}{2}z_1^2.
\end{array}
$$
The underlying implicit GLM is $A$- and $L$-stable for
$\lambda=(2\pm\sqrt{2})/2$, compare \cite{jac09}, and we choose $\lambda=(2-\sqrt{2})/2$
since this value leads to explicit methods and IMEX schemes with larger regions of stability
$ \mathcal{S}_E$ and $\mathcal{S}_{\pi/2}$ than those corresponding to $\lambda=(2+\sqrt{2})/2$.
We have plotted in Fig~\ref{fig5.4} the area of the stability region
$\mathcal{S}_E=\mathcal{S}_E(\beta_{21})$
of the explicit method (corresponding to $z_1=0$)
and the area of the stability region $\mathcal{S}_{\pi/2}=\mathcal{S}_{\pi/2}(\beta_{21})$
of the IMEX scheme for $\beta_{21}\in[0,8]$.
It can be verified that the explicit formula attains the maximal area of
$\mathcal{S}_E$, approximately
equal to $7.15$ for $\beta_{21}\approx 4.56$, and the IMEX scheme attains the
maximal area of $\mathcal{S}_{\pi/2}$, approximately equal to $5.75$ for
$\beta_{21}\approx 4.64$.
On Fig~\ref{fig5.5} we have plotted stability regions $\mathcal{S}_{\pi/2,y}$ for
$y=-2.0,-1.8,\ldots,2.0$ (thin lines), stability region $\mathcal{S}_{\pi/2}$ (shaded region),
and stability region $\mathcal{S}_E$ (thick line). We can see that $\mathcal{S}_{\pi/2}$ contains
a significant part of $\mathcal{S}_E$.
We have also displayed on Fig.~\ref{fig5.6} contour plots of the area of stability region
$\mathcal{S}_{\pi/2}$ of IMEX methods for $\beta_{21}\in[3,6]$ and $\lambda\in[0.25,0.35]$.
This area attains its maximum value approximately equal to $5.83$ for
$\beta_{21}\approx 4.59$ and $\lambda\approx 0.29$.
This point is marked by the symbol `$\times$' on Fig.~\ref{fig5.6}.
On Fig~\ref{fig5.7} we have plotted stability regions $\mathcal{S}_{\pi/2,y}$ for
$y=-2.0,-1.8,\ldots,2.0$ (thin lines), stability region $\mathcal{S}_{\pi/2}$ (shaded region),
and stability region $\mathcal{S}_E$ (thick line) corresponding to these
values of $\beta_{21}$ and $\lambda$. We can see again that $\mathcal{S}_{\pi/2}$ contains
a significant part of $\mathcal{S}_E$.
\subsection{IMEX GLMs with $p=q=r=s=3$} \label{sec5.3}
Let $\lambda\approx 0.43586652$ be a root of the cubic polynomial
$$
\varphi(\lambda)=\lambda^3-3\lambda^2+\ds\frac{3}{2}\lambda-\ds\frac{1}{6},
$$
and consider the implicit DIMSIM with $\mbf{c}=[0,1/2,1]^T$, the coefficient
matrix $\mbf{A}$ given by
$$
\mbf{A}=\left[
\begin{array}{ccc}
\phantom{-}0.43586652 & 0 & 0 \\
\phantom{-}0.25051488 & 0.43586652 & 0 \\
-1.2115943 & 1.0012746 & 0.43586652
\end{array}
\right],
$$
the rank one coefficient matrix $\mbf{V}=\mbf{e}\mbf{v}^T$,
where $\mbf{e}=[1,1,1]^T$ and
$$
\mbf{v}=\left[
\begin{array}{ccc}
0.55209096 & 0.73485666 & -0.28694762
\end{array}
\right]^T,
$$
and the vectors $\mbf{q}_0$, $\mbf{q}_1$, $\mbf{q}_2$, and $\mbf{q}_3$
equal to $\mbf{q}_0=\mbf{e}$,
$$
\mbf{q}_1=\left[
\begin{array}{ccc}
-0.43586652 &
-0.18638140 &
0.77445315
\end{array}
\right]^T,
$$
$$
\mbf{q}_2=\left[
\begin{array}{ccc}
0 &
-0.092933261&
-0.43650382
\end{array}
\right]^T,
$$
$$
\mbf{q}_3=\left[
\begin{array}{ccc}
0 &
-0.033649982 &
-0.17642592
\end{array}
\right]^T.
$$
Computing the coefficient matrix $\mbf{B}$ from the relation
(\ref{eq5.1}) leads to the method of order $p=3$ and stage order $q=3$.
It was demonstrated in \cite{jac09} that the resulting method is
$A$- and $L$-stable.
The coefficients $\alpha_{jk}$ of the extrapolation formula (\ref{eq1.11})
of order $p=3$ computed from the system (\ref{eq2.12}) corresponding to
$p=s=3$ take the form
$$
\alpha=\left[
\begin{array}{ccc}
\alpha_{11} & \alpha_{12} & \alpha_{13} \\
\alpha_{21} & \alpha_{22} & \alpha_{23} \\
\alpha_{31} & \alpha_{32} & \alpha_{33}
\end{array}
\right]=\left[
\begin{array}{ccc}
0 & \phantom{-}0 & 1 \\
1 & -3 & 3-\beta_{21} \\
3-\beta_{32} & 3\beta_{32}-8 & 6-\beta_{31}-3\beta_{32}
\end{array}
\right].
$$
To investigate the stability properties of the resulting IMEX scheme
we will work with the stability polynomial
$p(w,z_0,z_1)$ obtained by multiplying stability function of
the method by a factor $(1-\lambda z_1)^3$,
where $\lambda$ is the diagonal element of the matrix $\mbf{A}$.
It can be verified that
this polynomial takes the form
$$
\begin{array}{lcl}
p(w,z_0,z_1) & = &
(1-\lambda z_1)^3w^6-p_5(z_0,z_1)w^5
+p_4(z_0,z_1)w^4-p_3(z_0,z_1)w^3 \\ [2mm]
& + &
p_2(z_0,z_1)w^2-p_1(z_0,z_1)w+p_0(z_0,z_1),
\end{array}
$$
with the coefficients $p_0(z_0,z_1)$, $p_1(z_0,z_1)$, $p_2(z_0,z_1)$,
$p_3(z_0,z_1)$, $p_4(z_0,z_1)$, and $p_5(z_0,z_1)$ which
are polynomials of degree less than or equal to $3$ with respect to
$z_0$ and $z_1$. These coefficients
depend also on $\beta_{21}$, $\beta_{31}$, and $\beta_{32}$.
\begin{figure}[t!h!b!]
\begin{center}
\includegraphics[width=0.75\textwidth]{fig5_8-eps-converted-to.pdf}
\caption{Stability regions $\mathcal{S}_{\pi/2,y}$, $y=-2.0,-1.8,\ldots,2.0$ (thin lines),
$\mathcal{S}_{\pi/2}$ (shaded region), and $\mathcal{S}_E$ (thick line)
for $\beta_{21}\approx 1.13$, $\beta_{31}\approx 1.45$, and $\beta_{32}\approx -0.158$} \label{fig5.8}
\end{center}
\end{figure}
\begin{figure}[t!h!b!]
\begin{center}
\includegraphics[width=0.75\textwidth]{fig5_9-eps-converted-to.pdf}
\caption{Stability regions $\mathcal{S}_{\pi/4,y}$, $y=-2.0,-1.8,\ldots,2.0$ (thin lines),
$\mathcal{S}_{\pi/4}$ (shaded region), and $\mathcal{S}_E$ (thick line)
for $\beta_{21}\approx 1.13$, $\beta_{31}\approx 1.45$, and $\beta_{32}\approx -0.158$} \label{fig5.9}
\end{center}
\end{figure}
\begin{figure}[t!h!b!]
\begin{center}
\includegraphics[width=0.75\textwidth]{fig5_10-eps-converted-to.pdf}
\caption{Stability regions $\mathcal{S}_{\pi/2,y}$, $y=-2.0,-1.8,\ldots,2.0$ (thin lines),
$\mathcal{S}_{\pi/2}$ (shaded region), and $\mathcal{S}_E$ (thick line)
for $\beta_{21}\approx 1.39$, $\beta_{31}\approx -0.146$, and $\beta_{32}\approx 1.24$} \label{fig5.10}
\end{center}
\end{figure}
\begin{figure}[t!h!b!]
\begin{center}
\includegraphics[width=0.75\textwidth]{fig5_11-eps-converted-to.pdf}
\caption{Stability regions $\mathcal{S}_{\pi/4,y}$, $y=-2.0,-1.8,\ldots,2.0$ (thin lines),
$\mathcal{S}_{\pi/4}$ (shaded region), and $\mathcal{S}_E$ (thick line)
for $\beta_{21}\approx 1.25$, $\beta_{31}\approx 1.62$, and $\beta_{32}\approx 0.0555$} \label{fig5.11}
\end{center}
\end{figure}
We have performed a computer search in the parameter space
$\beta_{21}$, $\beta_{31}$, and $\beta_{32}$ looking first for methods
for which the stability region $\mathcal{S}_E$ of the explicit method
is maximal. This corresponds to the parameter values
$
\beta_{21}\approx 1.13
$,
$
\beta_{31}\approx 1.45
$,
$
\beta_{32}\approx -0.158
$,
for which the area of $\mathcal{S}_E$ is approximately equal to $3.54$.
The stability region $\mathcal{S}_E$ of the resulting method is plotted
on Fig.~\ref{fig5.8} by a thick line. We have also plotted stability regions
$\mathcal{S}_{\pi/2,y}$ for $y=-2.0,-1.8,\ldots,2.0$ (thin lines) and the stability
region $\mathcal{S}_{\pi/2}$ (shaded region) of the corresponding IMEX scheme.
We can see that this region $\mathcal{S}_{\pi/2}$ is substantially smaller
than the region $\mathcal{S}_E$, the area of $\mathcal{S}_{\pi/2}$ is approximately
equal to $0.39$.
However, we can obtain larger regions $\mathcal{S}_{\alpha}$ for values of
$\alpha$ smaller than $\pi/2$, i.e., if we relax the requirement that the
implicit part of IMEX scheme is $A$-stable and require instead
$A(\alpha)$-stability for $\alpha<\pi/2$. This is illustrated on
Fig.~\ref{fig5.9}, where we have plotted again the stability region
$\mathcal{S}_E$ of the explicit method (thick line), stability regions
of $\mathcal{S}_{\alpha,y}$ for $y=-2.0,-1.8,\ldots,2.0$ (thin lines),
and stability region $\mathcal{S}_{\alpha}$ (shaded region) for $\alpha=\pi/4$.
The area of this region is approximately equal to $1.91$.
We have also performed a computer search looking for methods for
which stability regions $\mathcal{S}_{\alpha}$ are maximal for some
fixed values of $\alpha$. For $\alpha=\pi/2$ this corresponds to
the parameter values $\beta_{21}\approx 1.39$, $\beta_{31}\approx -0.146$,
and $\beta_{32}\approx 1.24$ for which the area of $\mathcal{S}_{\pi/2}$
is approximately equal to $0.50$.
For $\alpha=\pi/4$ this corresponds to
the parameter values $\beta_{21}\approx 1.25$, $\beta_{31}\approx 1.62$,
and $\beta_{32}\approx 0.00555$ for which the area of $\mathcal{S}_{\pi/4}$
is approximately equal to $2.80$.
We have plotted on Fig.~\ref{fig5.10} and Fig.~\ref{fig5.11}
the stability regions $\mathcal{S}_E$ of the resulting explicit
methods (thick lines), the regions $\mathcal{S}_{\alpha,y}$ for
$y=-2.0,-1.8,\ldots,2.0$ (thin lines) and stability regions $\mathcal{S}_{\alpha}$
of IMEX schemes (shaded regions) for $\alpha=\pi/2$ and $\alpha=\pi/4$.
\subsection{IMEX GLMs with $p=q=r=s=4$} \label{sec5.4}
Let $\lambda\approx 0.57281606$ be a root of the polynomial
$$
\varphi(\lambda)=\lambda^4-4\lambda^3+3\lambda^2-\ds\frac{2}{3}\lambda+\ds\frac{1}{24},
$$
and consider the implicit DIMSIM with $\mbf{c}=[0,1/3,2/3,1]^T$, the coefficient
matrix $\mbf{A}$ given by
$$
\mbf{A}=\left[
\begin{array}{cccc}
0.57281606 & \phantom{-}0 & 0 & 0 \\
0.15022075 & \phantom{-}0.57281606 & 0 & 0 \\
0.59515808 & -0.26632807 & 0.57281606 & 0 \\
1.7717286 & -1.64234444 & 0.39147320 & 0.57281606
\end{array}
\right],
$$
the rank one coefficient matrix $\mbf{V}=\mbf{e}\mbf{v}^T$,
where $\mbf{e}=[1,1,1,1]^T$ and
$$
\mbf{v}=\left[
\begin{array}{cccc}
15.615037 & -46.967269 & 41.290082 & -8.9378502
\end{array}
\right]^T,
$$
and the vectors $\mbf{q}_0$, $\mbf{q}_1$, $\mbf{q}_2$, $\mbf{q}_3$, and $\mbf{q}_4$
equal to $\mbf{q}_0=\mbf{e}$,
$$
\mbf{q}_1=\left[
\begin{array}{cccc}
-0.57281606 &
-0.38970348 &
-0.23497940 &
-0.093673420
\end{array}
\right]^T,
$$
$$
\mbf{q}_2=\left[
\begin{array}{cccc}
0 &
-0.13538313 &
-0.070879128 &
0.21364995
\end{array}
\right]^T,
$$
$$
\mbf{q}_3=\left[
\begin{array}{cccc}
0 &
-0.025650275 &
-0.063113738 &
-0.11549405
\end{array}
\right]^T,
$$
$$
\mbf{q}_4=\left[
\begin{array}{cccc}
0 &
-0.0030214983 &
-0.018412760 &
-0.062996758
\end{array}
\right]^T.
$$
Computing the coefficient matrix $\mbf{B}$ from the relation
(\ref{eq5.1}) leads to the method of order $p=4$ and stage order $q=4$.
It was demonstrated in \cite{wri01} that the resulting method is
$A$- and $L$-stable.
\begin{figure}[t!h!b!]
\begin{center}
\includegraphics[width=0.75\textwidth]{fig5_12-eps-converted-to.pdf}
\caption{Stability regions $\mathcal{S}_{\pi/4,y}$, $y=-2.0,-1.8,\ldots,2.0$ (thin lines),
$\mathcal{S}_{\pi/4}$ (shaded region), and $\mathcal{S}_E$ (thick line)
for $\beta_{21}\approx 0.0645$, $\beta_{31}\approx -0.351$, and $\beta_{32}\approx 0.272$,
$\beta_{41}\approx -2.82$, $\beta_{42}\approx 3.47$, $\beta_{43}\approx -1.05$} \label{fig5.12}
\end{center}
\end{figure}
\begin{figure}[t!h!b!]
\begin{center}
\includegraphics[width=0.75\textwidth]{fig5_13-eps-converted-to.pdf}
\caption{Stability regions $\mathcal{S}_{\alpha}$ for $\alpha=0$, $\pi/12$, $\pi/6$,
$\pi/4$, $\pi/3$, $5\pi/12$, and $\pi/2$
for $\beta_{21}\approx 0.0645$, $\beta_{31}\approx -0.351$, and $\beta_{32}\approx 0.272$,
$\beta_{41}\approx -2.82$, $\beta_{42}\approx 3.47$, $\beta_{43}\approx -1.05$} \label{fig5.13}
\end{center}
\end{figure}
\begin{figure}[t!h!b!]
\begin{center}
\includegraphics[width=0.75\textwidth]{fig5_14-eps-converted-to.pdf}
\caption{Stability regions $\mathcal{S}_{\pi/2,y}$, $y=-2.0,-1.8,\ldots,2.0$ (thin lines),
$\mathcal{S}_{\pi/2}$ (shaded region), and $\mathcal{S}_E$ (thick line)
for $\beta_{21}\approx -0.00516$, $\beta_{31}\approx -0.939$,
$\beta_{32}\approx 1.18$, $\beta_{41}\approx -1.71$, $\beta_{42}\approx 2.07$,
and $\beta_{43}\approx 0.32$} \label{fig5.14}
\end{center}
\end{figure}
\begin{figure}[t!h!b!]
\begin{center}
\includegraphics[width=0.75\textwidth]{fig5_15-eps-converted-to.pdf}
\caption{Stability regions $\mathcal{S}_{\pi/4,y}$, $y=-2.0,-1.8,\ldots,2.0$ (thin lines),
$\mathcal{S}_{\pi/4}$ (shaded region), and $\mathcal{S}_E$ (thick line)
for $\beta_{21}\approx 0.0964$, $\beta_{31}\approx -0.278$,
$\beta_{32}\approx 0.464$, $\beta_{41}\approx -1.63$, $\beta_{42}\approx 2.73$,
and $\beta_{43}\approx -0.678$} \label{fig5.15}
\end{center}
\end{figure}
The coefficients $\alpha_{jk}$ of the extrapolation formula (\ref{eq1.11})
of order $p=4$ computed from the system (\ref{eq2.12}) corresponding to
$p=s=4$ take the form
$$
\alpha=\left[
\begin{array}{cccc}
0 & 0 & 0 & 1 \\
-1 & 4 & -6 & 4-\beta_{21} \\
\beta_{32}-4 & 15-4\beta_{32} & 2(3\beta_{32}-10) & 10-\beta_{31}-4\beta_{32} \\
\alpha_{41} & \alpha_{42} & \alpha_{43} & \alpha_{44}
\end{array}
\right],
$$
with
$$
\begin{array}{ll}
\alpha_{41}=\beta_{42}+4\beta_{43}-10, &
\alpha_{42}=36-4\beta_{42}-15\beta_{43}, \\
\alpha_{43}=6\beta_{42}+20\beta_{43}-45, &
\alpha_{44}=20-\beta_{41}-4\beta_{42}-10\beta_{43}.
\end{array}
$$
To investigate the stability properties of the resulting IMEX scheme
we will work with the stability polynomial
$p(w,z_0,z_1)$ obtained by multiplying stability function of
the method by a factor $(1-\lambda z_1)^4$,
where $\lambda$ is the diagonal element of the matrix $\mbf{A}$.
It can be verified that
this polynomial takes the form
$$
\begin{array}{l}
p(w,z_0,z_1) \ = \
(1-\lambda z_1)^4w^8-p_7(z_0,z_1)w^7
+p_6(z_0,z_1)w^6-p_5(z_0,z_1)w^5 \\ [2mm]
\quad\quad + \
p_4(z_0,z_1)w^4-p_3(z_0,z_1)w^3+p_2(z_0,z_1)w^2-p_1(z_0,z_1)w+p_0(z_0,z_1),
\end{array}
$$
with coefficients $p_0(z_0,z_1)$, $p_1(z_0,z_1)$, $p_2(z_0,z_1)$,
$p_3(z_0,z_1)$, $p_4(z_0,z_1)$, \mbox{$p_5(z_0,z_1)$}, $p_6(z_0,z_1)$, and $p_7(z_0,z_1)$ which
are polynomials of degree less than or equal to $4$ with respect to
$z_0$ and $z_1$. These coefficients
depend also on $\beta_{21}$, $\beta_{31}$, $\beta_{32}$, $\beta_{41}$,
$\beta_{42}$, and $\beta_{43}$.
We have performed a computer search in the parameter space
$\beta_{21}$, $\beta_{31}$, $\beta_{32}$, $\beta_{41}$, $\beta_{42}$,
and $\beta_{43}$ looking first for methods
for which the stability region $\mathcal{S}_E$ of the explicit method
is maximal. This corresponds to the parameter values
$
\beta_{21}\approx 0.0625
$,
$
\beta_{31}\approx -0.355
$,
$
\beta_{32}\approx 0.272
$,
$
\beta_{41}\approx -2.84
$,
$
\beta_{42}\approx 3.49
$,
$
\beta_{43}\approx -1.06
$,
for which the area of $\mathcal{S}_E$ is approximately equal to $2.82$.
The stability region $\mathcal{S}_E$ of the resulting method is plotted
on Fig.~\ref{fig5.12} by a thick line. We have also plotted stability regions
$\mathcal{S}_{\pi/4,y}$ for $y=-2.0,-1.8,\ldots,2.0$ (thin lines) and the stability
region $\mathcal{S}_{\pi/4}$ (shaded region) of the corresponding IMEX scheme.
The area of $\mathcal{S}_{\pi/4}$ is approximately equal to $0.32$.
We have also plotted on Fig.~\ref{fig5.13} stability regions
$\mathcal{S}_{\alpha}$ for $\alpha=0$, $\pi/12$, $\pi/6$,
$\pi/4$, $\pi/3$, $5\pi/12$, and $\pi/2$ corresponding to the same values
of $\beta_{ij}$.
We can see in particular that the region $\mathcal{S}_{\pi/2}$ is quite small, its area is
approximately equal to $0.0069$.
As in Section~\ref{sec5.3} we have also performed a computer search looking
directly for methods for
which stability regions $\mathcal{S}_{\alpha}$ are maximal for some
fixed values of $\alpha$. For $\alpha=\pi/2$ this corresponds to
the parameter values $\beta_{21}\approx -0.00516$, $\beta_{31}\approx -0.939$,
$\beta_{32}\approx 1.18$, $\beta_{41}\approx -1.71$, $\beta_{42}\approx 2.07$,
and $\beta_{43}\approx 0.32$, for which the area of $\mathcal{S}_{\pi/2}$
is approximately equal to $0.16$.
For $\alpha=\pi/4$ this corresponds to
the parameter values $\beta_{21}\approx 0.0964$, $\beta_{31}\approx -0.278$,
$\beta_{32}\approx 0.464$, $\beta_{41}\approx -1.63$, $\beta_{42}\approx 2.73$,
and $\beta_{43}\approx -0.678$, for which the area of $\mathcal{S}_{\pi/4}$
is approximately equal to $0.65$.
We have plotted on Fig.~\ref{fig5.14} and Fig.~\ref{fig5.15}
the stability regions $\mathcal{S}_E$ of the resulting explicit
methods (thick lines), the regions $\mathcal{S}_{\alpha,y}$ for
$y=-2.0,-1.8,\ldots,2.0$ (thin lines) and stability regions $\mathcal{S}_{\alpha}$
of IMEX schemes (shaded regions) for $\alpha=\pi/2$ and $\alpha=\pi/4$.
\section{Numerical experiments} \label{sec:numerics}
\setcounter{equation}{0}
\setcounter{figure}{0}
\setcounter{table}{0}
The extrapolation-based IMEX GLMs constructed in Section \ref{sec:construction} have been implemented in Matlab.
The required starting values $y^{[0]}$ and $Y^{[0]}$ were computed by finite difference approximations from solutions obtained with the Matlab routine \texttt{ode15s}.
The test problem is the two dimensional shallow-water equations system \cite{swe1998},
which approximates a thin layer of fluid inside a shallow basin:
\begin{eqnarray}
\frac{\partial}{\partial t} h + \frac{\partial}{\partial x} (uh) + \frac{\partial}{\partial y} (vh) &=& 0 \nonumber \\
\frac{\partial}{\partial t} (uh) + \frac{\partial}{\partial x} \left(u^2 h + \frac{1}{2} g h^2\right) + \frac{\partial}{\partial y} (u v h) &=& 0 \label{swe} \\
\frac{\partial}{\partial t} (vh) + \frac{\partial}{\partial x} (u v h) + \frac{\partial}{\partial y} \left(v^2 h + \frac{1}{2} g h^2\right) &=& 0 \;.
\nonumber
\end{eqnarray}
Here $h(t,x,y)$ is the fluid layer thickness, $u(t,x,y)$ and $v(t,x,y)$ are the components of the velocity field,
and $g$ denotes the gravitational acceleration.
The spatial domain is $\Omega = [-3,\,3]^2$ (spatial units), and the integration window is $t_0 = 0 \le t \le t_\textrm{f} = 10$ (time units).
We use reflective boundary conditions and the initial conditions at $t_0 = 0$
\begin{equation}
u(t_0,x,y)=v(t_0,x,y)=0\,,~~
h(t_0,x,y) = 1 + e^{-\|(x,y)-(c_1,c_2)\|^2_2}\,,
\end{equation}
with the Gaussian height profile $c_1 = 1/3$ and $c_2 = 2/3$.
A second order Lax-Wendroff finite difference scheme is used for space discretization, resulting in a semi-discrete ODE system of the form
\begin{equation}
\frac{\mathrm{d}}{\mathrm{d}t}U(t) = F \left(U\right) = \underbrace{ F_U \bigl(U\bigr)\cdot U(t) }_{g(U)}+ \underbrace{\left( F(U) - F_U \bigl(U\bigr)\cdot U(t) \right)}_{f(U)} \, ,
\label{semiDescreteODEswe}
\end{equation}
where $U(t)$ is a combined column vector of the discretized state variables $ (\widehat{h},\widehat{uh},\widehat{vh})$,
and $F_U=\partial F/\partial U$ be the Jacobian of right hand side.
We consider a splitting of equation \eqref{semiDescreteODEswe} into the linear part $g(U)$, considered stiff, and the nonlinear part $f(U)$,
considered non-stiff.
The linear stiff part is treated implicitly, and the non-stiff part is treated explicitly.
We compare the numerical results for the solution at the final time with a reference solution computed by the Matlab function \texttt{ode15s} with very tight tolerances $atol=rtol=10^{-14}$.
The errors are measured in $\mathcal{L}_2$ norms.
The error diagram against the time step size is presented in Fig. \ref{fig6.1}.
The observed orders for all the methods tested match the theoretical predictions.
\begin{figure}[t!h!b!]
\begin{center}
\includegraphics[width=0.70\textwidth]{swe_imex_glm-eps-converted-to.pdf}
\caption{Error vs. time step size for several extrapolation-based IMEX-GLMs applied to the shallow water equations test problem.} \label{fig6.1}
\end{center}
\end{figure}
\section{Concluding remarks} \label{sec:conclusions}
General linear methods offer an excellent framework for the construction of implicit-explicit schemes.
In this paper we develop a new extrapolation-based approach for the construction of practical IMEX GLM schemes of high order and high stage order.
The accuracy, linear stability, and Prothero-Robinson convergence are analyzed.
These schemes are particularly attractive for solving stiff problems, where other multistage methods may suffer from order reduction.
The extrapolation-based mechanism offers a systematic approach for constructing IMEX GLM schemes.
The construction starts with the selection of an implicit component method with suitable stability and order properties. The explicit component is then obtained
though an optimization procedure that maximized the combined stability region of the pair. We apply this methodology to construct IMEX pairs of orders one to four.
Future work is planned to extend the extrapolation idea to construct other types of partitioned GLMs, including parallel time integrators, and asynchronous pairs of methods
that do not share the same abscissae.
\textbf{Acknowledgements.}
The results reported in this paper were obtained during the
visit of the first author to the Arizona State University
in January--March of $2013$. This author wish to
express her gratitude to the School
of Mathematical and Statistical Sciences for hospitality during this visit.
The work of Sandu and Zhang has been supported in part by the awards NSF OCI-8670904397, NSF CCF-0916493, NSF DMS-0915047, NSF CMMI-1130667, NSF CCF-1218454, AFOSR FA9550-12-1-0293-DEF, AFOSR 12-2640-06, and by the Computational Science Laboratory at Virginia Tech. | {"config": "arxiv", "file": "1304.2276/imex_arXiv.tex"} |
\section{Mathematical solution} \label{Sec:solution}
As mentioned in the introduction, a significant obstacle associated with the current mathematical model is the fact that it is \emph{doubly connected}. This topology presents a challenge insofar as the mathematical solution is concerned since most analytic methods are only applicable to simply connected domains. To overcome this difficulty we introduce a special function known as the ``$P$-function'' which may be simply expressed as the infinite product
\begin{align}
P(\zeta) &= (1 - \zeta) \prod_{k=1}^{\infty} \left(1 - q^{2k} \zeta \right) \cdot \left( 1- q^{2k} \zeta^{-1}\right), \label{Eq:SKprod}
\end{align}
for $0<q<1$. We suppress the dependence on $q$ for notational convenience. The \mbox{$P$-function} is a special case of the Schottky--Klein prime function, which has found relevance as a fundamental tool for solving fluid problems in multiply connected domains \citep{Crowdy2010,Crowdy2020}. In particular, the significance of the \mbox{$P$-function} is that it is effectively a generalisation of the function $(1 - z)$ to doubly connected domains \citep{Baker1897}. A full discussion of this fact is well beyond the scope of the current article, but for now it is sufficient to note that the \mbox{$P$-function} is analytic in the annulus $q< |\zeta|< q^{-1}$, inside which it never vanishes except for a simple zero at $\zeta = 1$. Further details of the $P$-function are provided in section \ref{Sec:Pfun} of the supplementary material.
Remarkably, both the conformal maps \emph{and} the complex potentials for a range of flows can be expressed exclusively in terms of the $P$-function.
\input{sections/conformal-map}
\input{sections/complex-potential} | {"config": "arxiv", "file": "1912.02713/sections/solution.tex"} |
TITLE: Number of all graphs that contain at least one $k$-clique
QUESTION [1 upvotes]: There are $n$ vertices and natural number $k$.
How many are there graphs that contain at least one $k$-clique?
I tried to find number of all homomorphism between complete graph $K_{k}$ and graph with $n$-vertices. But i don't know how to do this.
A few weeks ago i asked simillar question that its answer would imply solution for this problem, but i saw that using that method some graphs are counted more than once.
REPLY [1 votes]: Exact results don't seem to be known, even for the simplest nontrivial case $k=3$ (in the case $k=2$, the answer is $2^{\binom n2} - 1$, since all $n$-vertex graphs except the empty graph contain a copy of $K_2$).
For any $k \ge 1$, almost all of the $2^{\binom n2}$ $n$-vertex graphs contain a $k$-clique (that is, the fraction of $n$-vertex graphs that don't approaches $0$ as $n \to \infty$). It is more natural to count the graphs that are $K_k$-free, and subtract the resulting number from $2^{\binom n2}$ to get the answer you want.
Let $f_k(n)$ denote the number of $K_k$-free graphs on $n$ vertices. The $(k-1)$-partite Turán graph on $n$ vertices has approximately $\binom{k-1}{2}(\frac{n}{k-1})^2 = (1 - \frac1{k-1})\frac{n^2}{2}$ edges; this is exact when $k-1 \mid n$. This graph, and every subgraph of this graph, is $K_k$-free; this gives us $2^{(1 - \frac1{k-1})\frac{n^2}{2}}$ distinct $K_k$-free graphs already.
Erdös, Frankl and Rödl (The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent) show that this is asymptotically tight in the following sense: we have $$f_k(n) = 2^{(1 - \frac1{k-1} + o(1))\frac{n^2}{2}}$$ for fixed $k$ as $n \to \infty$. Here, $o(1)$ stands for a function approaching $0$ as $n\to \infty$. More is true: the same bound holds if we consider $H$-free graphs for any fixed graph $H$ with chromatic number $k$. So the answer to your question is that asymptotically, $$2^{\binom n2} - 2^{(1 - \frac1{k-1}+o(1))\frac{n^2}{2}}$$ of the $n$-vertex graphs contain at least one $K_k$. | {"set_name": "stack_exchange", "score": 1, "question_id": 3393790} |
TITLE: Prove the following inequality using the Mean Value Theorem
QUESTION [0 upvotes]: I want to show that
$$\log(2+x) - \log(x) \lt \frac{2}{x}$$
$\forall x \in \mathbb{R^+}$
I know I need to apply the Mean Value Theorem to find an upper bound of the function to the left and show that it is smaller than $\frac{2}{x}$, but I can't find the correct upper bound. I've tried multiple variations of the inequality. My teacher also said that I only needed to check in the interval from 0 to 2, but I'm not sure why.
REPLY [1 votes]: For every real $\alpha>0$, let $f(x)=\ln(x+\alpha)$. Now with Mean Value Theorem in $[0,2]$ :
$$f(2)-f(0)=f'(c)(2-0)$$
for a $c\in(0,2)$. This gives us $\ln(2+\alpha)-\ln\alpha<\dfrac{2}{\alpha}$. | {"set_name": "stack_exchange", "score": 0, "question_id": 2847184} |