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\begin{document}
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\begin{abstract}
This paper contains an algebraic constructive and self-contained account of the invariance rule of the digital root under division for an arbitrary natural basis representation.
Both the cases of repeating and non-repeating fractionals are treated. In the preliminary section some known results such as the uniqueness in the representation of a fraction
are discussed for both the finite and infinite bases cases. Simple examples are introduced throughout the text for illustrative purposes.
\end{abstract}
\section{Introduction}
It is a known fact that behind the easily comprehensible statements in Arithmetic very commonly lies a staggeringly complex solution, when it exists. Questions such as Fermat's Last Theorem, Goldbach's conjecture, the twin prime conjecture, the odd perfect number conjecture and so many others can be easily understood by a layman. Nonetheless, for the aforementioned conjectures the last three remain open to this day, the first mentioned being the only one that has been entirely solved by Andrew Wiles \cite{WilesFermat}. The mere study of these and other profound statements has produced new mathematical tools and entire subfields have emerged in Number Theory. Although seemingly simple, Arithmetic is complex and rich enough not to escape from Gödel's Incompleteness Theorem, which haunts Mathematicians to this very day. To quote the prince of Mathematics, K. F. Gauss, " Mathematics is the queen of the sciences and Arithmetic is the queen of Mathematics".
From a very early age we are educated mathematically through the decimal numeral system. This system is so firmly rooted within us that it may be already passing through genetically to our descendants. Although very practical, since we have ten fingers to count, the decimal numeral system is no more special than any other numeral system based on a different basis. In any case, there are some curious and interesting properties associated with the digit $9$, which are frequently used by magicians to fool their audiences into guessing which number one has picked by asking one to perform certain calculations with the chosen number. An amazing example that could let people stumped is by asking them to find the digit $A$ such that $2A99561=[3(523+A)]^2$. At first it seems an impossible feature, unless the person notices that the number on the right side of the equation is divisible by 9, therefore the number on the left is also divisible by $9$ and so is the sum of digits of the corresponding number, that is $2+A+9+9+5+6+1=32+A$ and therefore, $A=4$. This and many other curiosities regarding the digit $9$ are humorously presented in Beiler's book \cite{Beiler}. Essentially, the same role played by the digit $9$ in the decimal basis representation is played by the digit $k-1$ in the representation to the basis $k$.
This paper was designed to be the more self-contained it could be in order to provide the reader with a smooth ride through the entire text, but without attempting to reinvent the wheel. With that in mind, \autoref{preliminaries}, which is divided into four subsections, intends to lay down a complete background for the comprehension of the two main theorems, \autoref{main1} and \autoref{main2}. Although most of the results in this preliminary section are well-known in the literature, we insisted on presenting them with proofs, remarks and examples, not as a pedantic act, but as a pedagogical artifice for a constructive and progressive understanding of the tools that are going to be explored in the main sections. In addition, still regarding the preliminary section, some of the results are exhibited in a unique fashion, which in and of itself makes it worth reading. Instead of describing some properties using purely calculational methods, as achieved by \cite{Izmirli} and \cite{Costa}, we have explored an algebraic approach to digital roots. The first subsection introduces some basic nomenclatures and definitions. The main purpose of the second subsection, besides presenting some common notions on modular arithmetic, is the partitioning of the ring of integers modulo $n$ by orbits obtained from the left action on it of its multiplicative subgroup, which will play a vital role in the proof of \autoref{main1}. The third subsection develops both the finite and the infinite representation theorems to a fixed base $k\in\mathbb{N}$, where the latter is going to be the starting point in the proof of \autoref{main2}. The fourth and final subsection introduces the notion of digital sum, additive persistence and digital root, which are the key elements for the attaining of our main results. \autoref{semdizima} is centered around \autoref{main1}, which essentially establishes behaviour patterns regarding the digital root function to the base $k$, and its correlation to orbits, defined in \autoref{mod-aret}, in the ring of integers modulo $k$. \autoref{comdizima} treats the case of a repeating fractionals to the basis $k$, with similar patterns emerging from the repetend. The paper is finalized with a conclusion section.
\section{Preliminaries}\label{preliminaries}
This section presents most of the tools that are going to be used throughout this paper. As mentioned in the introduction section, a good part of the results presented here is treated in classic books on Algebra and Number Theory such as \cite{Andrews}, \cite{Jacobson} and \cite{GarciaLequain}. However, we do not advise the reader to skip it.
\subsection{Basic notions}\label{basicnotions}
Throughout this entire paper we shall use the following terminologies $\mathbb{N}=\{0,1,\cdots\}$, $\{0,1,\cdots,n\}=:\langle n\rangle$, and $\mathbb{N}_p:=\mathbb{N}\setminus\{0,1,\cdots,p-1\}=\mathbb{N}\setminus \langle p-1\rangle$. Given an arbitrary set $X$, we conveniently define $X^+:=\{x\in X: x\geq0\}$. Moreover, the symbols $\lfloor\cdot\rfloor$ and $\lceil\cdot\rceil$ represent the floor and the ceiling functions, respectively, where $\lfloor\cdot\rfloor:D
\subseteq\mathbb{R}\rightarrow\mathbb{Z}$ with $\lfloor x\rfloor=\max\{m\in\mathbb{Z}: m\leq x\}$, and $\lceil\cdot\rceil:D
\subseteq\mathbb{R}\rightarrow\mathbb{Z}$ with $\lceil x\rceil=\min\{m\in\mathbb{Z}: m\geq x\}$.
\color{black}
\begin{df}
Let $a,b,c\in\mathbb{Z}$ and $c\neq 0$, then we say $a$ is congruent to $b$ modulo $c$, symbolically $a\equiv b \ (\text{mod}\ c)$, if $(a-b)/c\in\mathbb{Z}$, or equivalently, $c|(a-b)$.
\end{df}
\begin{ex}
Since $(16-4)/3=4\in\mathbb{Z}$, then $16\equiv 4 \ (\text{mod}\ 3)$. On the other hand, $(16-5)/3=11/3\notin\mathbb{Z}$, which implies $16\not\equiv 5 \ (\text{mod}\ 3)$.
\end{ex}
\begin{df}
Two numbers $a,b\in\mathbb{Z}$ are said to be \textbf{coprime} or \textbf{relatively prime} if $\text{gcd} \ (a,b)=1$.
\end{df}
\begin{lm}\label{lema}
Let $a,b,q\in\mathbb{Z}$ such that $ab\equiv 0 \ (mod \ q)$ and $a$ and $q$ are coprime, then $b\equiv 0 \ (mod \ q)$.
\end{lm}
\begin{proof}
From the first condition of the hypothesis, $\exists k\in\mathbb{Z}$ such that $ab=kq$, which implies $q \mid ab$. Since from the second hypothesis, $\text{gcd} \ (a,q)=1$, we must have $q \mid b$, that is, $b\equiv 0 \ (mod \ q)$.
\end{proof}
\begin{lm}\label{kk-1}
Let $k\in\mathbb{N}_2$, then $k$ and $k-1$ are coprime, that is, $\text{gcd} \ (k,k-1)=1$.
\end{lm}
\begin{proof}
Let $k=k_1^{l_1}\cdots k_d^{l_d}$ and $k-1=q_1^{s_1}\cdots q_e^{s_e}=:q$ be the prime factorizations of $k$ and $k-1$, respectively. Suppose $\exists m>1$ such that $\text{gcd} \ (k,k-1)=m$, then $\exists q_{j_0}$, where $1\leq j_0\leq e$ such that $q_{j_0}|m$ and consequently $q_{j_0}|k_1^{l_1}\cdots k_d^{l_d}$. However, since $k_1^{l_1}\cdots k_d^{l_d}=q_1^{s_1}\cdots q_e^{s_e}+1$
\begin{equation}
\frac{k_1^{l_1}\cdots k_d^{l_d}}{q_{j_0}}=\left(q_1^{s_1}\cdots q_{j_0}^{s_{j_0}-1} \cdots q_e^{s_e}+\frac{1}{q_{j_0}}\right)\notin\mathbb{N},
\end{equation}
which is a contradiction, therefore $\text{gcd} \ (k,k-1)=1$.
\end{proof}
\begin{rem}\label{gcdrk-1}
Note that $\text{gcd} \ (k,k-1)=1 \iff \text{gcd} \ (r,k-1)=1$ where $r$ is any divisor of $k$.
\end{rem}
\begin{df}\label{fractional}
Let $q\in\mathbb{Q}$, we say $q$ is a \textbf{non-repeating or terminating fractional to the base $k\in\mathbb{N}_2$} if $\exists \rho\in\mathbb{N}$ such that $k^\rho q\in\mathbb{Z}$ and that $q$ is a \textbf{repeating fractional to the base $k$} otherwise. Moreover, we may define $\rho_0:=\min \{\rho\in\mathbb{N}: k^\rho q\in\mathbb{\mathbb{Z}}\}$ which is called the \textbf{minimum exponent} of $q$ to the base $k$.
\end{df}
\begin{rem}
Since repeating and terminating fractionals dichotomize $\mathbb{Q}$, given $k\in\mathbb{N}_2$, we may write $\mathbb{Q}=\mathbb{Q}_{T_k}\dot\cup\mathbb{Q}_{R_k}$, where $\mathbb{Q}_{T_k}$ and $\mathbb{Q}_{R_k}$ represent the set of terminating and repeating fractionals to the base $k$, respectively. Notice also that since for every basis $k\in\mathbb{N}_2$, the minimum exponent for an integer is $\rho_0=0$, then $\forall k\in\mathbb{N}_2, \ \mathbb{Z}\subset\mathbb{Q}_{T_k}$.
\end{rem}
\begin{lm}\label{teofraciff}
Let $n,m\in\mathbb{N}_1$ be coprime numbers and $k\in\mathbb{N}_2$, then $n/m\in\mathbb{Q}_{T_k}$ if and only if $m=k_1^{l_1}k_2^{l_2}\cdots k_d^{l_d}$, where $l_i\geq 0$ for $1\leq i\leq d$ and $k_1,k_2,\cdots, k_d$ are the $d$ prime divisors of $k$.
\end{lm}
\begin{proof}
Suppose first that $m=k_1^{l_1}k_2^{l_2}\cdots k_d^{l_d}$, then
\begin{equation}
\frac{1}{m}=\frac{1}{k_1^{l_1}k_2^{l_2}\cdots k_d^{l_d}}
\end{equation}
Since $k_1,k_2,\cdots,k_d$ are the prime divisors of $k$, then by the Fundamental Theorem of Arithmetic $\exists l'_1,l'_2,\cdots, l'_d\in\mathbb{N}$ such that $k=k_1^{l'_1}k_2^{l'_2}\cdots k_d^{l'_d}$. Therefore,
\begin{equation}
\frac{k^\rho}{m}=\frac{\left(k_1^{l'_1}k_2^{l'_2}\cdots k_d^{l'_d}\right)^\rho}{k_1^{l_1}k_2^{l_2}\cdots k_d^{l_d}}=k_1^{\rho l'_1-l_1}k_2^{\rho l'_2-l_2}\cdots k_d^{\rho l'_d-l_d}.
\end{equation}
Thus, choosing
\begin{equation}
\rho\geq \underset{1 \leq i \leq d}{\max}\left\{\left\lceil\frac{l_i}{l'_i}\right\rceil\right\},
\end{equation}
we have
\begin{equation}
k^\rho\frac{n}{m}\in\mathbb{N},
\end{equation}
and consequently, by definition, $n/m$ is a terminating fractional to the base $k$. Conversely, suppose $n/m\in\mathbb{Q}_{T_k}$, then by \autoref{fractional}, $\exists \rho\in\mathbb{N}$ such that $k^\rho(\frac{n}{m})\in\mathbb{N}_1$ and since $\text{gcd} \ (n,m)=1$ and $m$ must divide $k^\rho$, $\exists l_1,l_2,\cdots, l_d\in\mathbb{N}$ such that $m=k_1^{l_1}k_2^{l_2}\cdots k_d^{l_d}$, where $l_i\geq 0$ for $1\leq i\leq d$ and $k_1,k_2,\cdots, k_d$ are the $d$ prime divisors of $k$.
\end{proof}
\subsection{Modular Arithmetic}\label{mod-aret}
Since the congruence relation is an equivalence relation on $\mathbb{Z}$, we may define the \textbf{congruence classes modulo $n$} or \textbf{residue classes modulo $n$} as
\begin{equation}
\overline{x}:=\{y\in\mathbb{Z}: \ x\equiv y \ (mod \ n)\}=\{x+jn\}_{j\in\mathbb{Z}}.
\end{equation}
From this definition we can introduce the set of all congruence classes modulo $n$, i.e. $\mathbb{Z}_n:=\{\overline{0},\overline{1},\cdots,\overline{n-1}\}$. The operations of addition and multiplication on $\mathbb{Z}$ induce the following operations on the set $\mathbb{Z}_n$:
\begin{align*}
\oplus: \mathbb{Z}_n\times\mathbb{Z}_n&\longrightarrow\mathbb{Z}_n\\
(\overline{x},\overline{y})&\longmapsto\overline{x+y}\\
\odot: \mathbb{Z}_n\times\mathbb{Z}_n&\longrightarrow\mathbb{Z}_n
\end{align*}
It is easy to check that $(\mathbb{Z}_n,\oplus,\odot)$ is a ring called the \textbf{ring of integers modulo $n$}. Moreover, let
\begin{equation}\label{gama1n}
((\mathbb{Z}_n)^*,\odot):=\{\overline{\delta}\in\mathbb{Z}_n : \text{gcd} \ (\delta,n)=1\}=:\Gamma_1^n.
\end{equation}
The set defined in \eqref{gama1n} is actually the multiplicative subgroup of $(\mathbb{Z}_n,\oplus,\odot)$. Equivalently, $\Gamma_1^n$ may be defined as the group of units of the ring of integers modulo n, that is
\begin{equation}
((\mathbb{Z}_n)^*,\odot):=\{\overline{\delta}\in\mathbb{Z}_n : \exists\gamma\in\mathbb{Z}: \delta\gamma\equiv 1 \ (mod \ n)\}
\end{equation}
In order to generalize the definition of $\Gamma_1^n$ to a set where the greatest common divisors are allowed to be different from $1$, let us introduce the following notations. Let $n\in\mathbb{N}$ and $\mathcal{D}_n:=\{\delta\in\mathbb{N}: \delta | n \}=\{\delta_1,\cdots,\delta_{|\mathcal{D}_n|}\}$ be the set of all divisors of $n$.
It is clear that $\mathcal{D}_0=\mathbb{N}_1$ with $|\mathcal{D}_0|=\aleph_0$
and $1\leq|\mathcal{D}_n|\leq n$ for $n>0$. Furthermore, let $n=n_1^{l_1}\cdots n_m^{l_m}$ be the prime decomposition of $n$, then it is also clear that $|\mathcal{D}_n|=\prod_{i=1}^{m}(l_i+1)$. \par
Let $\delta_i\in\mathcal{D}_n$, then we may define
\begin{equation}
\Gamma_{\delta_i}^n:=\{\overline{\delta}\in\mathbb{Z}_n : \text{gcd} \ (\delta,n)=\delta_i\}.
\end{equation}
For the sake of contextual harmony and parsimony we shall use the notation $\Gamma_{1}^n$ instead of $((\mathbb{Z}_n)^*,\odot)$. Notice, that since $1\notin\Gamma_{\alpha}^n$ if $\alpha\neq1$, only $\Gamma_1^n$ is a group. Also, if $\alpha$ is not a divisor of $n$, then $\Gamma_{\alpha}^n=\emptyset$. Furthermore, since the relation defined by $\overline{a}\sim \overline{b} :\Leftrightarrow \text{gcd} \ (a,n)= \text{gcd} \ (b,n)$ is an equivalence relation on $\mathbb{Z}_n$, it becomes clear that the sets $\Gamma^{n}_{\delta_i}$ partition the set $\mathbb{Z}_n$, that is
\begin{equation}\label{partition}
\coprod_{i=1}^{|\mathcal{D}_n|}\Gamma^{n}_{\delta_i}=\mathbb{Z}_n.
\end{equation}
\begin{df}
Let $(G,\odot)$ be a group and $g\in(G,\odot)$, then we denote its \textbf{cardinality} or \textbf{order} by $|G|$. We say \textbf{$g$ has order $\delta$ in $G$}, denoted by $\text{ord}(g)=\delta$, if $\delta$ is the smallest positive integer for which $g^\delta=e$, where $e$ is the identity element in $G$ and $g^\delta$ represents $g$ $\odot$-multiplied $\delta$ times. Moreover, the order of an element $g$ is equal to the order of its cyclic subgroup $\langle g\rangle:= \{g^k: \ k\in\mathbb{Z}\}$, that is, $\text{ord}(g)=|\langle g\rangle|$.
\end{df}
\begin{rem}
It has already been shown that the group $\Gamma_1^n$ is cyclic if and only if $n=1, 2, 4, p^j$ or $2p^j$ where $p$ is an odd prime number \cite{IrelandRosen}. Thus, only in these cases, $\exists g\in\Gamma_1^n$ such that $\{1,g,\cdots, g^{\varphi(n)-1}\}=\Gamma_1^n$, where $\varphi$ is the \textbf{Euler's totient function}, which gives the cardinality of $\Gamma_1^n$, that is, $\varphi(n)=|\Gamma_1^n|$.
\end{rem}
\begin{lm}\label{orderofelement}
The order of any element in $\mathbb{Z}_n$ divides $n$. In addition, if $\delta$ is a divisor of $n$, then the set of elements of order $\delta$ in $\mathbb{Z}_n$ is non-empty.
\end{lm}
\begin{proof}
Let $\bar{x}\in\mathbb{Z}_n$, then since $\langle\bar{x} \rangle := \{\bar{x}k: \ k\in\mathbb{Z}\}$ is a subgroup of $\mathbb{Z}_n$, by Lagrange's theorem, $\text{ord}(\bar{x}) | n$. Furthermore, for the second claim, let $\delta$ be a divisor of $n$ and $A_\delta$ denote the set of all elements of order $\delta$ in $\mathbb{Z}_n$, we shall prove that $\overline{\left(\frac{n}{\delta}\right)}\in A_\delta$. First, note that $\overline{\left(\frac{n}{\delta}\right)}\bar{\delta}=\bar{0}$, now suppose $\exists\delta'\in\mathbb{N}$ such that $\delta'|\delta$ and $\overline{\left(\frac{n}{\delta}\right)}\bar{\delta'}=\bar{0}$. Since $\delta'|n$, $\exists\alpha,\beta\in\mathbb{N}$ such that $n=\delta'\alpha $ and $\delta=\beta\delta'$, it follows that $\frac{n}{\delta}\delta'=\frac{n}{\beta}$ and consequently $\overline{\left(\frac{n}{\delta}\right)}\bar{\delta'}=\overline{\left(\frac{n}{\beta}\right)}$, that is $\overline{\left(\frac{n}{\beta}\right)}=\bar{0}$, which implies $\beta=1$ and $\delta=\delta'$. Thus, $\overline{\left(\frac{n}{\delta}\right)}\in A_\delta$.
\end{proof}
For future endeavors a particular group action on $\mathbb{Z}_n$ will be very useful:
\begin{pro}\label{G-action}
The map $\phi: \Gamma^n_1\times\mathbb{Z}_n\rightarrow\mathbb{Z}_n$, given by $\phi(\bar{g},\bar{x}):=\bar{g}\bar{x}$ is a left group action on $\mathbb{Z}_n$, which decomposes $\mathbb{Z}_n$ into $|\mathcal{D}_n|$-many distinct orbits given by $\Gamma_{1}^n\cdot\bar{\delta_i}$, where $\delta_i\in\mathcal{D}_n$.
\end{pro}
\begin{proof}
Let $n$ and $\delta|n$ be positive integers. Let $\psi: \Gamma^n_1\rightarrow\Gamma^\delta_1$ be the map given by $\psi(\bar{g})=\tilde{g}$, where $\bar{g}$ and $\tilde{g}$ are the congruence classes of the integer $g$ modulo $n$ and $\delta$, respectively. Then, $\psi$ is a well-defined surjective homomorphism with $|\ker\psi|=\varphi(n)/\varphi(\delta)$.
Now, according to \autoref{orderofelement}, it is always possible to pick an element $\bar{x}$ of order $\delta$ in $\mathbb{Z}_n$. Let us consider the stabilizer subgroup of $\Gamma^n_1$ with respect to $\bar{x}$, i.e. $S_{\bar{x}}:=\{\bar{g}\in\Gamma^n_1: \ \bar{g}\bar{x}=\bar{x}\}$. If $\bar{x}=\bar{0}$, then $\delta=1$ and it trivially follows that $\tilde{g}=\tilde{1}$. Let $\bar{x}\neq\bar{0}$, then from $(\bar{g}-\bar{1})\bar{x}=\bar{0}$ and $\bar{\delta}\bar{x}=\bar{0}$ it follows that $\bar{\delta}=\bar{g}-\bar{1}$, that is, $\tilde{g}=\tilde{1}$. Therefore, we may rewrite $S_{\bar{x}}$ as $S_{\bar{x}}:=\{\bar{g}\in\Gamma^n_1: \ \tilde{g}=\tilde{1}\}$, which corresponds to the kernel of the homomorphism $\psi$. Therefore, $|S_{\bar{x}}|=\varphi(n)/\varphi(\delta)$. By the orbit-stabilizer theorem, we have $|\mathbb{Z}_n|=|\Gamma_{1}^n\cdot\bar{x}||S_{\bar{x}}|$ and consequently $|\Gamma_{1}^n\cdot\bar{x}|=\varphi(\delta)$. Now, let $\bar{y}\in\Gamma_{1}^n\cdot\bar{x}$, then $\exists\bar{g}'\in\Gamma_{1}^n: \ \bar{y}=\bar{g}'\bar{x}$ which implies $\bar{\delta}\bar{y}=\bar{g}'\bar{\delta}\bar{x}=\bar{0}$. Moreover, suppose $\exists q\in\mathbb{N}: \ q|\delta$ and $\bar{q}\bar{y}=\bar{0}$, then $\bar{q}\bar{y}=\bar{g}'\bar{q}\bar{x}=0\implies \ \bar{q}\bar{x}=\bar{0}$ and since $\bar{x}$ is, by assumption, of order $\delta$, it follows that $q=\delta$ and every element of $\Gamma_{1}^n\cdot\bar{x}$ is of order $\delta$. Therefore, since the orbit $\Gamma_{1}^n\cdot\bar{x}$ has $\varphi(\delta)$ elements of order $\delta$, we may choose another representative for the orbit of $\bar{x}$, for instance let $\delta_i:=\overline{\left(\frac{n}{\delta}\right)} | n$, then $\Gamma_{1}^n\cdot\bar{x}=\Gamma_{1}^n\cdot\bar{\delta_i}$. Finally, since $\sum_{\delta_i\in\mathcal{D}_n}\varphi(\delta_i)=n$, we have
\begin{equation}\label{partition2}
\mathbb{Z}_n=\coprod_{i=1}^{|\mathcal{D}_n|}\Gamma_{1}^n\cdot\bar{\delta_i}
\end{equation}
\end{proof}
\begin{teo}\label{orbiteq}
Let $\delta_i$ be a divisor of $n$ and $\Gamma_1^n\cdot \overline{\delta_i}= \{\overline{g}\cdot\overline{\delta_i}, \ \overline{g}\in\Gamma_{1}^n \} $ be its orbit under the action of $\Gamma_1^n$. Then
\begin{equation}
\Gamma_1^n\cdot \overline{\delta_i}=\Gamma_{\delta_i}^n
\end{equation}
\end{teo}
\begin{proof}
Let us first prove that $\Gamma_1^n\cdot \overline{\delta_i}\subseteq \Gamma_{\delta_i}^n$. Let $\overline{\delta}\in\Gamma_1^n\cdot \overline{\delta_i}$, then $\exists\overline{g}\in\Gamma_1^n$ such that $\overline{\delta}=\overline{g}\overline{\delta_i}$ which implies $\delta =g\delta_i+nl$ for some $l\in\mathbb{Z}$. Therefore, $\text{gcd} \ (\delta,n)=\text{gcd} \ (g\delta_i+nl,n)=\text{gcd} \ (g\delta_i,n)=\text{gcd} \ (\delta_i,n)=\delta_i$.Thus, by definition, $\overline{\delta}\in\Gamma_{\delta_i}^n$.\\
Now let us prove the inverse inclusion, that is, $\Gamma_{\delta_i}^n\subseteq\Gamma_1^n\cdot \overline{\delta_i} $. By equations \eqref{partition} and \eqref{partition2}, we have
\begin{equation}\label{partitionequality}
\coprod_{i=1}^{|\mathcal{D}_n|}\Gamma^{n}_{\delta_i}=\coprod_{i=1}^{|\mathcal{D}_n|}\Gamma_{1}^n\cdot\bar{\delta_i}.
\end{equation}
Equation \eqref{partitionequality} together with the inclusion established in the first part of this proof imply $\Gamma_1^n\cdot \overline{\delta_i}=\Gamma_{\delta_i}^n$.
\end{proof}
\color{black}
\begin{ex}\label{Z9}
Let us consider the case of $n=9$, which will develop an important role in the proofs of our main theorems on the invariance of the digital root. In this case, we have the set $\mathbb{Z}_9=\{\overline{0},\overline{1},\cdots,\overline{8}\}$. The divisors of $9$ are $1$, $3$ and $9$, and consequently $\mathcal{D}_9=\{1,3,9\}$. Moreover, using \autoref{G-action} and \autoref{orbiteq}, we have the following $3$ distinct orbits
\begin{align*}
&\Gamma_{1}^9:=\{\overline{\delta}\in\mathbb{Z}/9\mathbb{Z} : \text{gcd} \ (\delta,9)=1\}=\{\overline{1},\overline{2},\overline{4},\overline{5},\overline{7},\overline{8}\}\\
&\Gamma_{3}^9:=\{\overline{\delta}\in\mathbb{Z}/9\mathbb{Z} : \text{gcd} \ (\delta,9)=3\}=\{\overline{3},\overline{6}\}\\
&\Gamma_{9}^9:=\{\overline{\delta}\in\mathbb{Z}/9\mathbb{Z} : \text{gcd} \ (\delta,9)=9\}=\{\overline{0}\}.
\end{align*}
\end{ex}
\begin{pro}\label{gcdr^jk-1}
Let $k\in\mathbb{N}_2$. If $r\in\mathcal{D}_k$, then $\bar{r}\in\Gamma_1^{k-1}$.
\end{pro}
\begin{proof}
Let $r\in\mathcal{D}_k$, that is, $r$ is a divisor of $k$, then as stated in \autoref{gcdrk-1}, $\text{gcd} \ (r,k-1)=1$ which implies $\bar{r}\in\Gamma_1^{k-1}$.
\end{proof}
\subsection{Finite and infinite representation theorems}\label{fin and inf rep}
\begin{teo}[Finite Basis Representation Theorem]\label{FBRT}
Let $q\in\mathbb{Q}^+$, then $q\in\mathbb{Q}^+_{T_k}$ if, and only if. there exists a unique finite representation for $q$ with respect to the base $k$ as
\begin{equation}\label{basek}
q=\sum_{j=0}^{p_i}\alpha_{p_i-j} k^{p_i-\rho_0-j}=\alpha_{p_i}k^{p_i-\rho_0}+\alpha_{p_i-1}k^{p_i-\rho_0-1}+\cdots+\alpha_{0}k^{-\rho_0},
\end{equation}
where $\alpha_{p_i}, \alpha_{p_i-1},\cdots, \alpha_{0}\in\langle k-1\rangle$, such that $0 < \alpha_{p_i}<k$ if $p_i-\rho_0> 0$ and $0 < \alpha_{0}<k$ if $\rho_0>0$.
\end{teo}
\begin{proof}
Let $q\in\mathbb{Q}^+_{T_k}$, then $k^{\rho_0}q\in\mathbb{N}$ where $\rho_0$ is the minimum exponent of $q$ with respect to the base $k$. By the Finite Basis Representation Theorem for natural numbers, $k^{\rho_0}q$ can be written as
\begin{equation}\label{equation t_0}
k^{\rho_0}q=\sum_{j=0}^{p_i}\alpha_{p_i-j} k^{p_i-j}=\alpha_{p_i}k^{p_i}+\alpha_{p_i-1}k^{p_i-1}+\cdots+\alpha_{0},
\end{equation}
where $\alpha_0,\alpha_1,\cdots \alpha_{p_i}\in\langle k-1\rangle$ and $\alpha_{p_i}\neq 0$. From equation \eqref{equation t_0} we obtain
\begin{equation}
q=\sum_{j=0}^{p_i}\alpha_{p_i-j} k^{p_i-\rho_0-j}=\alpha_{p_i}k^{p_i-\rho_0}+\alpha_{p_i-1}k^{p_i-\rho_0-1}+\cdots+\alpha_{0}k^{-\rho_0}.
\end{equation}
The uniqueness of the representation with respect to a specific basis follows directly from the Finite Basis Representation for natural numbers (see \cite{Andrews}, pages 8-10).
The reverse implication is straightforward, since if $q\in\mathbb{Q}^+$ is given by equation \eqref{basek}, then $k^{\rho_0}q\in\mathbb{N}$, which by \autoref{fractional} means $q\in\mathbb{Q}^+_{T_k}$.
\end{proof}
\color{black}
\begin{rem}
According to the preceding theorem, a natural number $n$ may be uniquely represented in terms of its digits with respect to the base $k$.
Therefore, we shall conveniently depict a natural number $n$ with respect to the base $k$ as a juxtaposition of its digits in the following fashion $[\alpha_{p_i}\alpha_{p_i-1}\cdots \alpha_{0}]_k$, and in the case of the base $10$ we write simply $[\alpha_{p_i}\alpha_{p_i-1}\cdots \alpha_{0}]_{10}:=\alpha_{p_i}\alpha_{p_i-1}\allowbreak\cdots \alpha_{0}$. For non-integer numbers with finite representation given by \eqref{basek} we may write $[\alpha_{p_i}\cdots\alpha_{\rho_0},\alpha_{\rho_0-1}\cdots \alpha_{0}]_k$.
By using this convention, the finite basis representation theorem for natural numbers may be extended for non-repeating fractionals as was done in \ref{FBRT}.
\end{rem}
\begin{ex}
Let us see some illustrative examples.
\begin{enumerate}[(i)]
\item $7205=7\cdot 10^3+2\cdot 10^2+0\cdot 10^1+5\cdot 10^0$
\item $[425]_6=4\cdot 6^2+2\cdot 6^1+5\cdot 6^0=144+12+5=161.$
\item $
[101011]_2 =1\cdot 2^5+0\cdot 2^4+1\cdot 2^3+0\cdot 2^2+1\cdot 2^1+1\cdot 2^0 $\\
$= (1\cdot 3^3+0\cdot 3^2+1\cdot 3^1+2\cdot 3^0)+(2\cdot 3^1+2\cdot 3^0)+(2\cdot 3^0)+(1\cdot 3^0)$\\
$= 1\cdot 3^3+1\cdot 3^2+2\cdot 3^1+1\cdot 3^0= [1121]_3$
\item $ 72.05=7\cdot 10^1+2\cdot 10^0+0\cdot 10^{-1}+5\cdot 10^{-2}.$
\item $[4.25]_6=4\cdot 6^0+2\cdot 6^{-1}+5\cdot 6^{-2}=\dfrac{161}{36}.$
\item Let us consider the hexadecimal basis with $A=10$, $B=11$, $C=12$, $D=13$, $E=14$ and $F=15$. Then
\begin{equation*}
[2A7E]_{16}=2\cdot 16^3+A\cdot 16^2+7\cdot 16^1+E\cdot 16^0=10878.
\end{equation*}
\end{enumerate}
\end{ex}
\begin{rem}
Notice in the last item of the previous example that the number $161/36$ is, according to \autoref{teofraciff}, a repeating fractional number to the base $10$ (i.e. a repeating decimal number), since $\text{gcd} \ (161,36)=1$ and $36=2^23^3$, where $3$ is not a divisor of $10$.
Notwithstanding, $161/36$ is a terminating fractional to the base $6$ and therefore admits a finite basis representation to this base.
\end{rem}
\begin{lm}\label{k^m-geq}
Let $k\in\mathbb{N}_2$, $a_j\in\langle k-1\rangle$ for all $j\in\mathbb{N}$, then
\begin{equation}\label{k^m-ineq}
k^m\geq \sum_{j=0}^{+\infty}a_{j} k^{m-1-j},
\end{equation}
and equality holds if and only if $a_j=k-1 \ \forall j\in\mathbb{N}$.
\end{lm}
\begin{proof}
The series in \eqref{k^m-ineq} is clearly convergent. Let $a_j=k-1, \forall j\in\mathbb{N}$, then
\begin{equation}
\sum_{j=0}^{+\infty}(k-1) k^{m-1-j}=(k^m-k^{m-1})+(k^{m-1}-k^{m-1})+\cdots=k^m.
\end{equation}
If $\exists j_0\in\mathbb{N}: \ a_{j_0}\neq k-1$, then clearly
\begin{equation}
k^m=\sum_{j=0}^{+\infty}(k-1) k^{m-1-j}>\sum_{j=0}^{+\infty}a_{j} k^{m-1-j}.
\end{equation}
\end{proof}
\begin{teo}[Infinite Basis Representation Theorem]\label{IBRT}
Let $k\in\mathbb{N}_2$. Then, $\forall x\in\mathbb{R}^+$, there exists a unique infinite representation for $x$ with respect to the base $k$ as
\begin{equation}\label{basekex}
x=\sum_{j=0}^{+\infty}a_{p_i-j} k^{p_i-j},
\end{equation}
where $p_i\in\mathbb{Z}$ and $a_{p_i-j}\in\mathbb{N}$ such that $0\leq a_{p_i-j}< k$ for all $j\geq 0$, and $\forall N>0, \ \exists j_0 > N: a_{p_i - j_0} \neq 0$. Furthermore, defining
\begin{equation}
x_n:=\sum_{j=0}^{n}a_{p_i-j} k^{p_i-j},
\end{equation}
and $\{x_n\}_{n\in\mathbb{N}_0}=:S\subset\mathbb{R}$, $x$ is a limit point of $S$.
\end{teo}
\begin{proof}
Let $x\in\mathbb{R}^+$ and $k$ be a positive integer greater than $1$, then $\exists m\in\mathbb{Z}$ such that
$k^{m}<x$. Let $m_0$ be the greatest integer with such a property, that is, $m_0:=\max\{m\in\mathbb{Z}: k^m<x\}$ and $0\leq a_{m_0}<k$ be the greatest value for which it still holds $a_{m_0}k^{m_0}<x$. Once again, let us choose $0\leq a_{m_0-1}<k$ as the greatest value for which $a_{m_0}k^{m_0}+a_{m_0-1}k^{m_0-1}<x$.
Since $\mathbb{R}$ is densely ordered, the indefinite repetition of this process is valid and we might construct a non-decreasing sequence $\{x_n\}_{n\in\mathbb{N}}$ such that its general term $x_n$ is given by
\begin{equation}
x_n:=\sum_{j=1}^{n}a_{m_0-j+1} k^{m_0-j+1}<x.
\end{equation}
Since $\{x_n\}_{n\in\mathbb{N}}$ is a non-decreasing sequence that is bounded from above by $x$, by the Monotone Convergence Theorem the sequence converges to its supremum. It still remains to show that $x=\sup_{n\in\mathbb{N}}\{x_n\}$. In this regard, let $\epsilon>0$, then, since $\mathbb{N}$ is unbounded from above, $\exists N_0\in\mathbb{N}$ such that
\begin{equation}
\epsilon>k^{m_0-N_0+1}.
\end{equation}
Thus,
\begin{equation}\label{SN0+}
x_{N_0}+\epsilon>x_{N_0}+k^{m_0-N_0+1}=a_{m_0}k^{m_0}+\cdots+(a_{m_0-N_0+1}+1)k^{m_0-N_0+1},
\end{equation}
However, by the very definition of the sequence $\{x_n\}_{n\in\mathbb{N}}$, $a_{m_0-N_0+1}$ is the greatest coefficient of $k^{m_0-N_0+1}$ for which $x_{N_0}<x$. Therefore, from equation \eqref{SN0+} we have $x_{N_0}+\epsilon > x$, that is, $x_{N_0} > x-\epsilon$.
Since $\epsilon$ is an arbitrary positive real number, it follows that $x=\sup_{n\in\mathbb{N}}\{x_n\}$.
Moreover, by construction $x_{N'} < x$ $\forall N' \in \mathbb{N}$ which means $x_{N'}\in (x-\epsilon,x+\epsilon)\setminus\{x\}$ $\forall N' \geq N_0$ which proves $x$ is a limit point of $\{x_n\}_{n\in\mathbb{N}}$.
For the uniqueness of the representation, let
\begin{equation}\label{basekex'}
x=\sum_{j=0}^{+\infty}b_{p'_i-j} k^{p'_i-j},
\end{equation}
be an infinite representation for $x$, with $p'_i\in\mathbb{Z}$ and $b_{p'_i-j}\in\mathbb{N}$ such that $0\leq b_{p'_i-j}< k$ for all $j\geq 0$, and $\forall N>0, \ \exists j_1 > N: b_{p'_i - j_1} \neq 0$. Suppose, without any loss of generality, that $p'_i>p_i$, then $\exists l\in\langle k-1 \rangle$ such that $p'_i=p_i+l$. Thus, subtracting \eqref{basekex} from \eqref{basekex'} we obtain
\begin{align}
0&=\sum_{j=0}^{+\infty}b_{p_i+l-j} k^{p_i+l-j}-\sum_{j=0}^{+\infty}a_{p_i-j} k^{p_i-j}\nonumber\\
&=\sum_{j=0}^{l-1}b_{p_i+l-j} k^{p_i+l-j}+\sum_{j=0}^{+\infty}(b_{p_i-j}-a_{p_i-j}) k^{p_i-j}\label{x-x}.
\end{align}
If we had $p'_i=p_i$, then equation \eqref{x-x} would still be valid with the omission of the first term on the right-hand side. Equation \eqref{x-x} may be rewritten as
\begin{equation}
b_{p_i+l}k^{p_i+l}+\cdots+b_{p_i+1}k^{p_i+1}=\sum_{j=0}^{+\infty}(a_{p_i-j}-b_{p_i-j}) k^{p_i-j}.
\end{equation}
From \autoref{k^m-geq}, we have
\begin{equation}
k^{p_i+l}>k^{p_i+l-1}>\cdots>k^{p_i+1}=\sum_{j=0}^{+\infty}(k-1) k^{p_i-j},
\end{equation}
Since the coefficients $b_{p_i+l-j}$ are all non-negative and $-(k-1)\leq a_{p_i-j}-b_{p_i-j}\leq k-1$,
we must have $b_{p_i+l-j}=0$ for $0\leq j\leq l-2$ and
\begin{equation}\label{bk}
b_{p_i+1}k^{p_i+1}=\sum_{j=0}^{+\infty}(a_{p_i-j}-b_{p_i-j}) k^{p_i-j},
\end{equation}
where, either $b_{p_i+1}=1$, or $b_{p_i+1}=0$. In the first case, according to \autoref{k^m-geq}, $a_{p_i-j}-b_{p_i-j}=k-1, \ \forall j\in\mathbb{N}$, which is equivalent to the assertion that $a_{p_i-j}=k-1$ and $b_{p_i-j}=0$ for all $j\in\mathbb{N}$, which in turn contradicts the hypothesis that $\forall N>0, \ \exists j_1 > N: b_{p'_i - j_1} \neq 0$. Therefore, we must have $b_{p_i+1}=0$. We want to prove that $a_{p_i-j}=b_{p_i-j}, \ \forall j\in\mathbb{N}$ and we shall do so by induction on $j$. Let us prove the initial case, that is, for $j=0$. Suppose $b_{p_i}>a_{p_i}$, then equation \eqref{bk} can be rewritten as
\begin{equation}\label{ind-j=0}
(b_{p_i}-a_{p_i})k^{p_i}=\sum_{j=1}^{+\infty}(a_{p_i-j}-b_{p_i-j}) k^{p_i-j},
\end{equation}
From \autoref{k^m-geq}, for equation \eqref{ind-j=0} to hold, either $(b_{p_i}-a_{p_i})=1$, or $(b_{p_i}-a_{p_i})=0$. In the first case, it follows that
$a_{p_i-j}=k-1$ and $b_{p_i-j}=0$ for all $j\in\mathbb{N}_1$, which again contradicts the hypothesis that $\forall N>0, \ \exists j_1 > N: b_{p'_i - j_1} \neq 0$. The case $a_{p_i}>b_{p_i}$ is completely analogous and also leads to contradiction. Therefore, $b_{p_i}=a_{p_i}$. For the final step of the induction, suppose $a_{p_i-j}=b_{p_i-j}, \ \forall j\in\langle n\rangle $, where $n\in\mathbb{N}$, then from equation \eqref{bk} it follows
\begin{equation}
\sum_{j=0}^{+\infty}(a_{p_i-j}-b_{p_i-j}) k^{p_i-j}=\sum_{j=n+1}^{+\infty}(a_{p_i-j}-b_{p_i-j}) k^{p_i-j}=0,
\end{equation}
which may be rewritten as
\begin{equation}\label{ind-j=n}
(b_{p_i-n-1}-a_{p_i-n-1})k^{p_i-n-1}=\sum_{j=0}^{+\infty}(a_{p_i-n-2-j}-b_{p_i-n-2-j}) k^{p_i-n-2-j}
\end{equation}
Again from \autoref{k^m-geq}, for equation \eqref{ind-j=n} to hold, either $(b_{p_i-n-1}-a_{p_i-n-1})=1$, or $(b_{p_i-n-1}-a_{p_i-n-1})=0$. In the first case, it follows that
$a_{p_i-j}=k-1$ and $b_{p_i-j}=0$ for all $j\in\mathbb{N}_{n+2}$, which again contradicts the hypothesis that $\forall N>0, \ \exists j_1 > N: b_{p'_i - j_1} \neq 0$. Therefore, $b_{p_i-n-1}=a_{p_i-n-1}$, hence $a_{p_i-j}=b_{p_i-j}, \ \forall j\in\langle n+1\rangle $ and the final step of the induction is done.
\end{proof}
\begin{rem}\label{repreg}
The Infinite Basis Representation Theorem allows a unique representation in a given basis $k\in\mathbb{N}_2$ for any positive real number $x$ in such a way that there are infinitely many non-zero digits, that is, $q:=[a_{p_i}\cdots a_0,a_{-1}\cdots]_k$. In the case of a positive rational number, however, there is always a finite string of digits that repeats itself indefinitely and so, a finite amount of numbers is enough to describe it. In the case $q\in\mathbb{Q}^+$, we shall use the following notation
\begin{align}
q&=[a_{p_i}\cdots a_{0},\cdots a_{-\rho_0}\overline{a_{\rho_0-1}\cdots a_{-\rho_0-T}}]_k\\
&=[a_{p_i}\cdots a_{0},\cdots a_{-\rho_0}]_k+[ 0,\cdots \overline{a_{\rho_0-1}\cdots a_{-\rho_0-T}}]_k\\
&=:Reg_k(q)+\overline{Rep_k(q)},
\end{align}
where the string of digits under the horizontal line is the repeating portion of $q$, called its \textbf{repetend} and denoted by $Rep_k(q)$, $T\in\mathbb{N}$ is the number of digits in the repeating cycle, called the \textbf{period} or \textbf{length of the repetend}, and $Reg_k(q)$ denotes the non-repeating string of digits of $q$, called its \textbf{regular part}. Clearly, there is no possibility of confusion between a finite and an infinite representation, since in the latter $T\geq 1$ and a horizontal line over a finite string of digits is always present. In the former case though, we might assume $T=0$, where $q$ reduces to $q:=[a_{p_i}\cdots a_{0},\cdots a_{-\rho_0}]_k$, and the digits are now considered with respect to its finite representation to the base $k$. Furthermore, if $q\in\mathbb{Q}_{T_k}$, then $q$ has both a finite and an infinite representation to the base $k$, which are evidently distinct from one another. As an example, $[4.25]_6=[4.24\overline{5}]_6$.
\end{rem}
\subsection{Digital sum, additive persistence and digital root}\label{dig root and dig sum}
\begin{df}
Let $n$ be a natural number to the base $k$. Then, the function $d_k:\mathbb{N}\rightarrow\mathbb{N}$ defined by
\begin{equation}
d_k(n)=d_k([a_{p_i}\cdots a_{0}]_k):=\sum_{j=0}^{p_i}a_{p_i-j}
\end{equation}
is called the \textbf{digital sum} of $n$ with respect to the base $k$.
\end{df}
\begin{ex} \
\begin{enumerate}[(i)]
\item
$d_{10}([7205]_{10})=d_{10}(7205)=7+2+0+5=[14]_{10}=14$
\item $d_6([425]_6)=4+2+5=11=[15]_6$
\item $
d_2([101011]_2)=1+0+1+0+1+1=4=[100]_2$
\item $d_{16}([2A7E]_{16})=2+A+7+E=[21]_{16}$
\end{enumerate}
\end{ex}
\begin{pro}
Let $n$ be a natural number to the base $k$. Then, $d_k(n)\leq n$ and equality holds if and only if $n$ is a single digit number to the base $k$.
\end{pro}
\begin{proof}
Using the definition of the function $d_k$ it is straightforward that
\begin{equation}
d_k(n)=\sum_{j=0}^{p_i} a_{p_i-j}\leq\sum_{j=0}^{p_i}a_{p_i-j}k^{p_i-j}=n.
\end{equation}
and clearly equality holds if and only if $p_i-j=0$ for $0\leq j\leq p_i$, which is equivalent to stating that $p_i=0$. Thus, $n=a_0=\sum_{j=0}^{0}a_{p_i-j}=d_k(n)$.
\end{proof}
\begin{rem}
Since $d_k(n)\in\mathbb{N}$ we might calculate its digital sum with respect to the base $k$, that is, $d_k([d_k(n)]_k)\leq d_k(n)\leq n$ and this process might be iterated as many times as we please by composing the digital sum function with itself multiple times. For convenience we shall define the $N$-times composition of $d_k$ with itself as
\begin{equation}
d_k^{(N)}(n):=\underset{N \text{-times}}{(d_k\circ\cdots\circ d_k})(n).
\end{equation}
Moreover, it is clear that from the definition of $d_k$ and from the fact that $\mathbb{N}$ is bounded from bellow, given a basis $k$, $\forall n\in\mathbb{N},\exists N\in\mathbb{N}:\ d_k^{(N)}(n)=b_0$, where $b_0$ is a single digit number to the base $k$, that is $0\leq b_0\leq k-1$.
\end{rem}
\begin{df}
Let $n\in\mathbb{N}$.Then, the function $\mathcal{A}_k:\mathbb{N}\rightarrow\mathbb{N}$ such that $\mathcal{A}_k(n):=\min\{N\in\mathbb{N}: 0\leq d_k^{(N)}(n)\leq k-1\}$ is called the \textbf{additive persistence} of $n$ with respect to the base $k$.
\end{df}
\begin{rem}
The additive persistence function gives the minimum number of times we need to add the digits of a particular number recursively until we obtain a single digit number.
\end{rem}
\begin{ex} \
\begin{enumerate}[(i)]
\item
$d_{10}(7205)=14$ and $ d_{10}(14)=5$ $\implies$ $ \mathcal{A}_{10}(7205)=2$;
\item $d_6([425]_6)=[15]_6$, $d_6([15]_6)=6=[10]_6$, and $d_6([10]_6)=[1]_6$ $\implies$ $\mathcal{A}_6([425]_6)=3$;
\item $
d_2([101011]_2)=[100]_2$ and $d_2([100]_2)=[1]_2$ $\implies$ $\mathcal{A}_2([101011]_2)=2$.
\item $d_{16}([2A7E]_{16})=2+A+7+E=[21]_{16}$ and $d_{16}([21]_{16})=[3]_{16}$ $\implies$ $\mathcal{A}_{16}([2A7E]_{16})=2$.
\end{enumerate}
\end{ex}
\begin{df}\label{root1}
The function $r_k: \mathbb{N}\rightarrow \langle k-1\rangle$ defined by $r_k(n):=d_k^{\mathcal{A}_k(n)}(n)$ is called the \textbf{digital root} of $n$ to the base $k$.
\end{df}
\begin{ex}From the previous example it follows that $r_{10}(7205)=5$, $r_6([425]_6)=[1]_6$ and $r_2([101011]_2)=[1]_2$.
\end{ex}
Some of the previous results may be easily extended from natural to non-repeating rational numbers.
\begin{df}\label{rootsum2}
Let $\Dt_k:\mathbb{Q}^+_{T_k}\rightarrow\mathbb{N}$ and $\bar{r}_k:\mathbb{Q}^+_{T_k}\rightarrow\langle k-1 \rangle$ be defined as $\Dt_k(q):=d_k(k^{\rho_0}q)$ and $\bar{r}_k(q):=r_k(k^{\rho_0}q)$ where $\rho_0$ is the minimum exponent of $q$ to the base $k$, then $\Dt_k$ and $\bar{r}_k$ are called the \textbf{terminating fractional digital sum} (TFDS) and the \textbf{terminating fractional digital root} (TFDR) of $q\in\mathbb{Q}^+_{T_k}$ to the base $k\in\mathbb{N}_2$.
\end{df}
\begin{ex}
\begin{enumerate}[(i)]
\item
$\Dt_{10}([72.05]_{10})=d_{10}(10^2\cdot72.05)=d_{10}(7205)=14$ and analogously $\bar{r}_{10}([72.05]_{10})=r_{10}(10^2\cdot72.05)=r_{10}(7205)=5$
\item $
\Dt_2([1.01011]_2)=d_2(2^5[1.01011]_2)=d_2([101011]_2)=[100]_2$ and analogously $\bar{r}_2([1.01011]_2)=r_2(2^5[1.01011]_2)=r_2([101011]_2)=[1]_2$.
\end{enumerate}
\end{ex}
\section{Terminating Fractionals and Digital Root}\label{semdizima}
\begin{lm}\label{lemadkrk}
Let $q\in\mathbb{Q}^+_{T_k}$, then $\Dt_k(q)\equiv \bar{r}_k(q) \ (\text{mod}\ k-1)$.
\end{lm}
\begin{proof}
If $q=0$, then evidently $\Dt_k(q)=\bar{r}_k(q)=0$,and in particular, $\Dt_k(q)\equiv \bar{r}_k(q) \ (\text{mod}\ k-1)$. Thus, let $q\in\mathbb{Q}^+_{T_k}\setminus\{0\}$ with $\Dt_k(q)\equiv \gamma \ (\text{mod}\ k-1)$ where $0\leq \gamma<k-1$, and $q=[\alpha_{p_i}\cdots\alpha_{p_i-\rho_0},\cdots\alpha_{0}]_k$ be the representation of $q$ to the base $k$, then by definition,
\begin{equation}
\Dt_k(q)=\Dt_k([\alpha_{p_i}\cdots\alpha_{p_i-\rho_0},\cdots\alpha_{0}]_k)=\sum_{j=0}^{p_i}\alpha_{p_i-j}=:q'\in\mathbb{N}.
\end{equation}
Therefore, $\Dt_k(q)\equiv \gamma \ (\text{mod}\ k-1)$ if and only if $q'\equiv \gamma \ (\text{mod}\ k-1)$. Again by the Finite Representation Theorem we may write $q'=[\alpha'_{p'_i}\cdots\alpha'_{0}]_k$ from which it follows that
\begin{equation}
q'':=\Dt_k(q')=\sum_{j=0}^{p'_i}\alpha'_{p'_i-j}\equiv\sum_{j=0}^{p'_i}\alpha'_{p'_i-j}k^{p_i-j} \ (\text{mod}\ k-1)=q'.
\end{equation}
Thus, $q''\equiv q' \ (\text{mod}\ k-1)\equiv \gamma \ (\text{mod}\ k-1)$, that is $\Dt_k(\Dt_k(q))\equiv \gamma \ (\text{mod}\ k-1)$. By induction it follows that $\forall n\in\mathbb{N}_1, \ \Dt^{(n)}_k(q)\equiv \gamma \ (\text{mod}\ k-1)$ and particularly $\bar{r}_k(q)=\Dt^{\mathcal{A}_k(q)}_k(q)\equiv \gamma \ (\text{mod}\ k-1)$.
\end{proof}
\begin{rem}
Notice that from the congruence relation $\bar{r}_k(q)\equiv \gamma \ (\text{mod}\ k-1)$ it follows that $\bar{r}_k(q)=\gamma+l(k-1)$, where $l\in\mathbb{Z}$ and since the digital root is a positive single digit number
\[ \bar{r}_k(q) =
\begin{cases}
\gamma, & \text{if} \quad 0<\gamma<k-1 \\
k-1, & \text{if} \quad \gamma=0 \ \wedge \ l=1 \\
0, & \text{if} \quad \gamma=0 \ \wedge \ l=0 \\
\end{cases}
.\]
It is also evident that $\bar{r}_k(q)=0\Leftrightarrow\Dt_k(q)=0\Leftrightarrow q=0$. From which it follows that $\bar{r}_k(q) =k-1\Leftrightarrow \Dt_k(q)$ is a multiple of $k-1$ and also if $q\in\mathbb{N}$, then it is a multiple of $k-1$.
\end{rem}
\begin{teo}\label{main1}
Let $q\in\mathbb{Q}^+_{T_k}\setminus\{0\}$ and $r\in\mathbb{N}_2$ such that $r$ is a proper divisor of $k$, then $\exists\delta_i\in\mathcal{D}_{k-1}$
\begin{equation}\label{maineq1}
\bigg\{\overline{\bar{r}_k\left(\frac{q}{r^j}\right)}\bigg\}_{j\in\mathbb{N}}\subseteq\Gamma^{k-1}_{\delta_i}.
\end{equation}
\end{teo}
\begin{proof}
Let us denote $\bar{r}_k(q/r^j)=:R_j$. Since, by \autoref{G-action}, the collection of orbits $\{\Gamma_{\delta}^{k-1}\}_{\delta\in\mathcal{D}_{k-1}}$ partitions the set $\mathbb{Z}_{k-1}$, given $j\in\mathbb{N}$, $\exists\delta\in\mathcal{D}_{k-1}$ such that $\overline{\bar{r}_k\left(\frac{q}{r^{j}}\right)}=\overline{R}_j\in\Gamma_{\delta}^{k-1}$. Let $\Gamma_{\delta_0}^{k-1}$ be the orbit containing $\overline{R}_0=\overline{\bar{r}_k(q)}$. We shall prove that $\forall j\in\mathbb{N}$, $\overline{R}_j\in\Gamma_{\delta_0}^{k-1}$. Suppose, by contradiction, $\exists j_1\neq 0: \ \overline{R}_{j_1}\notin\Gamma_{\delta_0}^{k-1}$, that is, suppose $\exists \delta_1\in\mathcal{D}_{k-1}$ with $\delta_0\neq\delta_1$ such that $\overline{R}_{j_1}\in\Gamma_{\delta_1}^{k-1}$. Denoting $\frac{q}{r^j}=:q_j$, according to \autoref{FBRT}, both $q_{0}=q$ and $q_{j_1}$ may be uniquely represented to the base $k$ as
\begin{equation}\label{q0}
q_{0}=q=\sum_{l=0}^{p_i}\alpha_{p_i-l} k^{p_i-\rho_0-l},
\end{equation}
and
\begin{equation}\label{qj1}
q_{j_1}=\frac{q}{r^{j_1}}=\sum_{l=0}^{p'_i}\alpha'_{p'_i-l} k^{p'_i-\rho'_0-l},
\end{equation}
where $\rho_0$ and $\rho'_0$ are the minimum exponents of $q_0$ and $q_{j_1}$ to the base $k$, respectively. Since $\rho_0$ is the minimum exponent of $q$, equations \eqref{q0} and \eqref{qj1} imply
\begin{equation}\label{}
k^{\rho_0}q_{0}=k^{\rho_0}q=\sum_{l=0}^{p_i}\alpha_{p_i-l} k^{p_i-l}\equiv \sum_{l=0}^{p_i}\alpha_{p_i-l} \ (\text{mod}\ k-1),
\end{equation}
that is,
\begin{equation}\label{q0'}
k^{\rho_0}q\equiv \Dt_k\left(q_0\right) \ (\text{mod}\ k-1),
\end{equation}
and
\begin{equation}\label{}
k^{\rho_0}q=r^{j_1}\sum_{l=0}^{p'_i}\alpha'_{p'_i-l} k^{p'_i+\rho_0-\rho'_0-l}\equiv r^{j_1} \sum_{l=0}^{p'_i}\alpha'_{p'_i-l} \ (\text{mod}\ k-1),
\end{equation}
that is,
\begin{equation}\label{qj1'}
k^{\rho_0}q\equiv r^{j_1}\Dt_k\left(q_{j_1}\right) \ (\text{mod}\ k-1).
\end{equation}
Moreover, by \autoref{lemadkrk}, $\Dt_k\left(q_0\right)\equiv R_0 \ (\text{mod}\ k-1)$ and $\Dt_k\left(q_{j_1}\right)\equiv R_{j_1} \ (\text{mod}\ k-1)$. Thus, from equations \eqref{q0'} and \eqref{qj1'} it follows that
\begin{equation}\label{last-eq}
r^{j_1}R_{j_1}\equiv R_0\ (\text{mod}\ k-1).
\end{equation}
Now, since $r\in\mathcal{D}_{k}$, from \autoref{gcdrk-1}, $\forall j\in\mathbb{N}, \text{gcd} \ (r^j,k-1)=1$, from which it follows, particularly, that $\forall j\in\mathbb{N},\overline{r^j}\in\Gamma_1^{k-1}$. Moreover, since the action $\phi: \Gamma_1^{k-1}\times\mathbb{Z}_{k-1}\rightarrow\mathbb{Z}_{k-1}$, given by $\phi(g,x):=gx$ defined in \autoref{G-action} leaves the orbits invariant, $\overline{r}^{j_1}\overline{R}_{j_1}\in\Gamma_{\delta_1}^{k-1}$, which is a contradiction since from equation \eqref{last-eq} $\overline{r}^{j_1}\overline{R}_{j_1}=\overline{R}_0\in\Gamma_{\delta_0}^{k-1}$ and the orbits are disjoint.
\end{proof}
\begin{cor}\label{cor1}
Let $q\in\mathbb{Q}^+_{T_k}\setminus\{0\}$ and $r\in\mathbb{N}_2$ such that $r$ is a proper divisor of $k$. If $\bar{r}_k(q)\equiv 0 \ (\text{mod}\ k-1)$, then $\bar{r}_k(q/r)\equiv 0 \ (\text{mod}\ k-1)$.
\end{cor}
\begin{proof}
From \autoref{main1} $\overline{\bar{r}_k(q)}\in\Gamma_{k-1}^{k-1}=\{\overline{0}\}$ and consequently $\overline{\bar{r}_k(q/r)}\in\Gamma_{k-1}^{k-1}=\{\overline{0}\}$, that is $\bar{r}_k(q/r)\equiv 0 \ (\text{mod}\ k-1)$.
\end{proof}
\begin{ex}
As an example let us look again at the decimal basis case. In this case, as already pointed out in \autoref{Z9}, the orbits are $\Gamma_{1}^{9}=\{\overline{1},\overline{2},\overline{4},\overline{5},\overline{7},\overline{8}\}$, $\Gamma_{3}^{9}=\{\overline{3},\overline{6}\}$, and $\Gamma_{9}^{9}=\{\overline{0}\}$. Since $2$ and $5$ are the proper divisors of $10$, the division of any non-repeating decimal number by powers of $2$ or $5$ will again result in a non-repeating decimal number. Moreover, according to \autoref{main1}, the digital root of two numbers whose ratio are integer powers of $2$ or $5$ lies in the same orbit. As a consequence of \autoref{cor1}, the orbit of $9$ is special in the sense that it contains only one element, namely, $\overline{0}$, which means that if the terminating fractional digital root of a number $q$ is in $\Gamma_{9}^9$, every number whose ratio by $q$ yields integer powers of $2$ or $5$ has digital root equal to $9$.
This example is interestingly enough, however, it might immediately lead oneself to ask if there is any similar conclusions we can draw on for repeating fractionals such as $1/7$ or $3/11$. The answer is ``yes, we can.'' We cannot apply either definitions \ref{root1} or \ref{rootsum2}, but we could turn our attention to repetends to see if any pattern emerges in this case, which may be investigated by directly looking at the repetend. The following section is devoted to this matter.
\end{ex}
\begin{ex}
For a final example, consider the non-decimal base example given by the sequence of terminating fractionals to the base $8$, $\{[25]_8,$ $[12.4]_8,$ $[5.2]_8,$ $[2.5]_8,$ $[1.24]_8,$$\cdots \}$, which is a geometric progression with respect to the base $8$ with ratio $1/2$. Since the digital root of the number $[25]_8$ is $7$, as a consequence of \autoref{cor1}, dividing $[25]_8$ by powers of $2$ will result in non-fractional numbers to the base $8$ with the same digital root $7$.
\end{ex}
\color{black}
\section{Repeating Fractionals and Digital Root}\label{comdizima}
\begin{teo}\label{main2}
Let $n\in\mathbb{N}_1$, $s\in\mathbb{N}_2$ such that $n/s$ is an irreducible fraction to the base $k$ and $s=k_1^{l_1}\cdots k_m^{l_m}p$, where $ l_1,\cdots, l_m\in\mathbb{N}$, $k_1,\dots,k_m$ are primes in the prime decomposition of $k$, and $p\in\mathbb{N}_2$ such that $\text{gcd} \ (p,k-1)=\text{gcd} \ (p,k)=1$. Then, $n/s\in\mathbb{Q}_{R_k}$ and letting $n_{s,T}$ be its repetend, $\bar{r}_k(n_{s,T})\equiv 0 \mod(k-1)$.
\end{teo}
\begin{proof}
The fact that $n/s$ is a repeating fractional is a direct consequence of \autoref{teofraciff}. Thus, by the infinite basis representation theorem, \autoref{IBRT}, we have
\begin{align}\label{q/r}
t:=\frac{n}{s}&=\sum_{j=0}^{+\infty}a_{p_i-j} k^{p_i-j},\\
&=\sum_{j=0}^{p_i+\rho_0}a_{p_i-j}k^{p_i-j}+\sum_{j=p_i+\rho_0+1}^{\infty}a_{p_i-j} k^{p_i-j}\\
&=\sum_{j=0}^{p_i+\rho_0}a_{p_i-j}k^{p_i-j}+\sum_{j=0}^{\infty}a_{-\rho_0-1-j} k^{-\rho_0-1-j}\\
&=Reg_k(t)+\overline{Rep_k(t)}\label{cyclic}
\end{align}
where $p_i\in\mathbb{Z}$, $a_{p_i-j}\in\langle k-1\rangle$ for all $j\in\mathbb{N}$, and $\rho_0$ is the minimum exponent of the regular part of $q$. According to \autoref{repreg}, the second term in the right-hand side of equation \eqref{cyclic} contains the cyclic part of $t$. In addition, $\exists T\geq 1$ such that $\forall j\in\mathbb{N} \ a_{-\rho_0-1-j}=a_{-\rho_0-1-j-T}$. Furthermore, using the Euclidean algorithm we may rewrite the summation variable $j$ in the cyclic term in \eqref{cyclic} as $j=m+lT$, where $0 \leq m \leq T-1$ and $l\geq 0$, obtaining
\begin{align}\label{t''gen}
t&=\sum_{j=0}^{p_i+\rho_0}a_{p_i-j}k^{p_i-j}+k^{-\rho_0-1}\sum_{m=0}^{T-1}\sum_{l=0}^{\infty}a_{-\rho_0-1-m-lT} k^{-(m+lT)}\\
&=\sum_{j=0}^{p_i+\rho_0}a_{p_i-j}k^{p_i-j}+k^{-\rho_0-1}\sum_{m=0}^{T-1}a_{-\rho_0-1-m}k^{-m}\sum_{l=0}^{\infty} k^{-{lT}}\\
&=\sum_{j=0}^{p_i+\rho_0}a_{p_i-j}k^{p_i-j}+\frac{k^{T-\rho_0-1}}{k^T-1}\sum_{m=0}^{T-1}a_{-\rho_0-1-m}k^{-m}\label{negpowers}
\end{align}
Since $t$ is a repeating fractional to the base $k$, $\nexists t\in\mathbb{N}$ such that $k^{r}t\in\mathbb{N}$. However, by letting $t':=k^{\rho_0}t$ we eliminate negative powers of $k$ appearing in the sums of \eqref{negpowers}, that is
\begin{equation}\label{t''''gen}
t'=k^{\rho_0}t=\sum_{j=0}^{p_i+\rho_0}a_{p_i-j}k^{p_i+\rho_0-j}+\frac{k^{T-1}}{k^T-1}\sum_{m=0}^{T-1}a_{-\rho_0-1-m}k^{-m},
\end{equation}
The term $k^T-1$ in the denominator of \eqref{t''''gen} is precisely what makes $t'$ a repeating fractional number to the base $k$. Multiplying the last equation by $k^T-1$ we obtain
\begin{equation}\label{lasteq}
(k^T-1)t'=\left[(k^T-1)\sum_{j=0}^{p_i+\rho_0}a_{p_i-j}k^{p_i+\rho_0-j}+\sum_{m=0}^{T-1}a_{-\rho_0-1-m}k^{T-1-m}\right]\in\mathbb{N}
\end{equation}
From equations \eqref{q/r}, \eqref{t''''gen} and \eqref{lasteq}, we obtain the following result
\begin{equation}\label{t''geninf}
(k^T-1)k^{\rho_0}\frac{n}{s}=(k-1)\left(\sum_{j=0}^{T-1}k^j\right)k^{\rho_0}\frac{n}{s}=:t''\in\mathbb{N}_1
\end{equation}
Since $t''\in\mathbb{N}_1$ and $n/s$ is an irreducible fraction, $s$ must divide the term $k^{\rho_0}(k-1)\sum_{j=0}^{T-1}k^j$. By definition of $k$ and $\rho_0$ we have that $k_1^{l_1}\cdots k_m^{l_m}|k^{\rho_0}$, moreover, since $s=k_1^{l_1}\cdots k_m^{l_m}p$ with $\text{gcd} \ (p,k)=\text{gcd} \ (p,k-1)=1$, it follows that $\left(\sum_{j=0}^{T-1}k^j\right)/p=:u\in\mathbb{N}_1$, that is
\begin{equation}\label{finalt''}
(k-1)u\left(\frac{k^{\rho_0}}{k_1^{l_1}\cdots k_m^{l_m}}\right)n=t'',
\end{equation}
which implies $t''$ is divisible by $k-1$. In terms of congruences, from expressions \eqref{lasteq}, \eqref{t''geninf}, and \eqref{finalt''} we have
\begin{equation}
\left[(k^T-1)\sum_{j=0}^{p_i+\rho_0}a_{p_i-j}k^{p_i+\rho_0-j}+\sum_{m=0}^{T-1}a_{-\rho_0-1-m}k^{T-1-m}\right]\equiv 0\ (mod \ k-1)
\end{equation}
and consequently
\begin{align}
\sum_{m=0}^{T-1}a_{-\rho_0-1-m}k^{T-1-m}&\equiv \sum_{m=0}^{T-1}a_{-\rho_0-1-m}\equiv 0\ (mod \ k-1)\\
&=\Dt_k(Rep_k(t))\equiv \bar{r}_k(Rep_k(t))\ (mod \ k-1)
\end{align}
\end{proof}
\begin{rem}
Since $\forall q\in\mathbb{Q}^+_{T_k}$, $k^{\rho_0}q=n\in\mathbb{N}$, where $\rho_0$ is the minimum exponent of $q$, such that $\bar{r}_k(q)=r_k(k^{\rho_0}q)=r_k(n)$, \autoref{main2} is still valid if we replace $n\in\mathbb{N}_1$, by $q\in\mathbb{Q}^+_{T_k}\setminus\{0\}$. Moreover, \autoref{main2} essentially complements \autoref{main1} when considering divisions resulting in repeating fractionals. For instance, in the $k=10$ case, for any irreducible fraction $n/s$ where $s$ is such that $\exists p\in\mathbb{N}: \ p|s \wedge \forall l_1,l_2,l_3\in\mathbb{N} \ p\neq 2^{l_1}3^{l_2}5^{l_3}$, the number $n/s$ is not only a repeating decimal, but its repetend's digital root is always null, that is, it is a multiple of $9$ irregardless of the value of the numerator's digital root $r_k(n)$ in contraposition to the terminating fractionals case treated in \autoref{main1}.
\end{rem}
\begin{ex}
For an example of a sequence of repeating decimals, consider the sequence $\{9/{p_n}\}_{n\in\mathbb{N}}$ where $\{p_n\}_{n\in\mathbb{N}}$ is the sequence of all prime numbers starting with $p_1=7$, that is, $\{9/{p_n}\}_{n\in\mathbb{N}}=\{1.\overline{285714}, 0.\overline{81}, 0.\overline{692307}, 0. \overline{5294117647058823}\allowbreak,\cdots \}$, which explicits the first 4 terms of the sequence, which have repetends whose digital root is $9$.
\end{ex}
\begin{ex}
For the final example, let us consider the following sequence to the base $8$, $\{[25]_8, [5.1\overline{7}]_8, [1.\overline{1463}]_8,\cdots \}$ which is a geometric progression of ratio $1/5$. Since $[5]_8=5$ is coprime to both $[25]_8=21$ and $[10]_8=8$, then every term, except the first, in the preceding sequence is a repeating fractional to the base $8$ whose repetends' digital root is $7$.
\end{ex}
\color{black}
\section{Conclusion}\label{aaa}
The solution of a mathematical problem is usually strongly dependent upon the problem's characterization within a given mathematical structure. By adequately characterizing a problem we may be able to solve it nicely using theorems and techniques from the structure we have been able to model the problem on. Although the Preliminary section served the purpose of laying the foundations for the resolution of the main theorems, \autoref{main1} and \autoref{main2}, it definitely has its own merit. In \autoref{mod-aret}, \autoref{G-action} establishes a partition of $\mathbb{Z}_n$ into distinct orbits under the action of the group of units contained in $\mathbb{Z}_n$. Moreover, the number of orbits is shown to be precisely the number of divisors of $n$. \autoref{orbiteq} describes the orbits in a useful and simpler form through the sets $\Gamma_{\delta_i}^n$. In \autoref{fin and inf rep}, we have presented both the finite and infinite representation theorems, \autoref{FBRT} and \autoref{IBRT}, which were useful for the main results in \autoref{main1} and \autoref{main2}, respectively. The last preliminary subsection is devoted for basic notions and definitions regarding both the digital sum and the digital root. In \autoref{semdizima} we focus on terminating fractionals by describing an invariance rule for the digital root in such cases, which is accomplished by \autoref{main1}. Essentially if $q$ and $q/r$ are two terminating fractionals to the basis $k$, then their digital roots lie in the same orbit. As an immediate consequence of this theorem, \autoref{cor1}, it follows that, assuming $r$ and $k-1$ to be coprime, if $q$'s digital root equals to $k-1$, then so does $q/r$'s, since the orbit in this case is the singleton $\Gamma_{k-1}^{k-1}=\{\bar{0}\}$. Within this last corollary lies the decimal basis case, with $k-1=9$ as the ``magic number'', which is humorously depicted in Beiler's book and basically everywhere the digital root is mentioned. Lastly, \autoref{comdizima} deals with the repeating fractionals case, exhibiting an invariance rule, which, in a sense, is stronger than the terminating fractionals case. Roughly speaking, let $n/r$ be an irreducible repeating fractional to the base $k$ with $r$ coprime to $k-1$, then \autoref{main2} asserts that the digital root of $n/r$'s repetend equals to $k-1$ irregardless of the value of $n$ within the condition pre-established in the theorem's assertion.
\section*{Acknowledgements}
The first author would like to thank José Amâncio dos Santos for his valuable comments on previous drafts of the paper.
\bibliographystyle{abbrv}
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Endometriosis is one of the most common gynecological conditions in the United States. We don't know exactly how many American women suffer from this disease, but best estimates set the number at about 5 million. Women of all ages, races and backgrounds have been found to have endometriosis. Recent information suggests that the disease is becoming more prevalent. Some 2 million women had hysterectomies for pelvic pain related to endometriosis between 1965 and 1984. Over that time period, the number of hysterectomies performed per year for this condition doubled. In addition, the proportion of all hysterectomies performed because of endometriosis rose from approximately 10% to 20%.
Endometriosis can also effect fertility, and about 30% of infertile women are found to have endometriosis. The symptoms and problems related to endometriosis lead to the hospitalization of a substantial number of women every year.
The tissue that lines the uterus and is shed during the menstrual period is called the endometrium. In some women, this same tissue can be found growing outside of the uterus, where it does not belong. When this occurs, endometriosis is said to be present..
The lining cells of the uterus normally go through cyclic changes in response to the varying levels of the female hormones estrogen and progesterone produced by the ovary throughout the month. During the menstrual cycle, as estrogen levels rise, the tissue first grows and builds up, and then, as the level of both estrogen and progesterone fall at the end of the cycle, the tissue breaks down and is shed as menstrual blood. When a woman has endometriosis, while the lining cells are present in locations where they are not intended to be, they still respond to hormonal changes in much the same way as if they were still within the uterus. infertity. Thus, the abnormal location of uterine lining cells leads to the symptoms and problems that we associate with the condition called endometriosis.
The appearance of endometriosis is variable and changes over time. Areas of endometriosis may be small, only a millimeter, or larger than a few centimeters. We think that new endometriosis appears as small, almost clear, raised areas on the surface of the uterus, tubes, ovaries or inside lining of the abdomen. Over time, these areas, called implants, continue to collect the pigment contained in the blood they secrete. As this occurs the areas become pink, then dark red, and finally a dark brown color. The darker areas have often been called "powder burns" because of their color and shape. In order to evaluate a woman for the presence of endometriosis, a careful inspection of the entire pelvis and abdomen must be performed, looking for all the possible apearances of endometriosis, some of which are fairly subtle (see fig 6.1).
If you have painful periods, chronic pelvic pain, pain during or after sex, premenstrual backache, painful bowel movements, the sudden onset of pelvic pain, or a problem with fertility, your doctor will consider endometriosis as one of the possible causes for your problem. On the other hand, many women with endometriosis have no symptoms at all, and the condition may be discovered inadvertently during surgery for another reason.
Infertility has long been felt to be associated with endometriosis, but the reason endometriosis might cause difficulty getting pregnant has not been established. In fact, it may be that the cause of the endometriosis may also independently cause infertility. We know that about 5% of women who have had children and request tubal sterilization will be noted to have areas of old endometriosis at the time of their surgery. Therefore, the presence of endometriosis does not, per se, imply that a woman can not get pregnant.
However, it does appear that the chance of getting pregnant is decreased somewhat if you have endometriosis, and the more endometriosis you have, the less likely you are to get pregnant. Endometriosis appears to start as small areas of abnormally situated endometrial lining cells. As the tissue grows and bleeds, scar tissue forms around it, increasing the amount of damaged tissue. The scar tissue may even grow around the tubes and ovaries in a way that blocks the passage of the egg down the tube.
The probability of a healthy woman getting pregnant is about 25% per month. For women with mild endometriosis, where the endometriosis is present in small amounts and has not caused any scarring, the pregnancy rate is about 7% per month. For women who have severe endometriosis, where extensive scarring, blockage of the fallopian tubes, and large cysts in the ovaries are present, it is not hard to understand why pregnancy rates are extremely low without treatment.
Endometriosis may be suspected if tender, thickened areas are felt near the uterus on a pelvic examination. If an ovarian cyst is present, sometimes a sonogram may exhibit the patterns suggestive of an endometrioma, and the diagnosis may then be suspected. Unfortunately, we do not have any test presently available that can reliably predict whether or not endometriosis is present. Neither sonography, MRI, CT scan or blood tests are accurate in this regard.
The diagnosis of endometriosis can only be confirmed by looking at the pelvic organs at the time of surgery. Areas with the characteristic appearance of endometriosis can then be seen. Usually a minor surgical procedure, called laparoscopy, is performed under general anesthesia for this purpose. A small lighted instrument is inserted through the navel, and the surgeon looks through the instrument directly or, with the aid of a camera attached to the laparoscope, the pelvis can be projected on a TV screen.
At times, the diagnosis of endometriosis is made during a laparotomy, abdominal surgery which is performed under either general or regional (such as epidural) anesthesia. The incision in the abdomen ranges from approximately two to five inches in length. This abdominal surgery may be needed when a large endometrioma has been identified by the sonogram or if a pelvic mass of uncertain cause is found on examination. In addition, endometriosis may be incidentally found during an abdominal surgery performed for another reason, such as fibroids, an ovarian cyst or even surgery for appendicitis.
Treatment is aimed at reducing the symptoms of endometriosis, usually either pain or infertility. Treatment is divided into three paths - observation, medication, or surgery.
Women who have minimal or mild endometriosis and do not have pain may not require any treatment other than careful follow-up. In practice, however, if the diagnosis of endometriosis is made during laparoscopy, most gynecologists will burn or cut away these cells. However, a few studies have demonstrated that this treatment of mild endometriosis does not enhance fertility. For women with mild endometriosis, fertility rates are good even if no treatment is performed.
It is known that estrogen causes endometriosis to grow. Endometriosis is extremely rare before a young woman begins to produce estrogen and starts to have periods and the disease usually disappears after menopause, when estrogen production stops. Therefore, one goal of treatment with medication is to lower, or stop, the production of estrogen. Reducing the levels of estrogen "starves" the endometriosis and causes it to shrink and sometimes even disappear. Two classes of drugs have been developed which lower the amount of estrogen in a woman's body - Danocrine and GnRH agonist (see details in our book). Progesterone can also be used to treat endometriosis.
Conservative surgical treatment is considered when a woman needs surgery for pain or infertility associated with endometriosis, and she desires to preserve her pelvic organs. The goal of this approach is to remove as much endometriosis and scar tissue as possible and restore the uterus, tubes, and ovaries to their normal positions. Conserative surgery can be performed using laparoscopic surgery or an abdominal incision. Newer modalities involving laparoscopic surgical techniques and use of instruments such as lasers have allowed for surgery to be performed through very small incisions with the benefit of a shorter hospital stay and quicker recovery time.
However, laparoscopic surgery requires special training, expertise, and experience on the part of the surgeon. Conservative surgery may provide a cure, but it may also provide only temporary relief of symptoms. A woman may elect to have conservative surgery in order to complete her family, and then, at a later time, she may elect to undergoing radical surgery. And, some women may require more than one conservative surgical procedure before they need to have, or are willing to consider, a more extensive operation. Yet, for some women, multiple conservative operations may provide relief of symptoms.
If a patient undergoes a conservative surgical procedure for infertility, her chance of getting pregnant is related to the amount of endometriosis found at surgery. Women who have mild endometriosis have about an 80-90% chance of becoming pregnant within 5 years whether they have the endometriosis removed surgically or not. Women who had moderate endometriosis treated surgically have about a 60% chance, and women with severe disease have about a 35% chance of getting pregnant.
Disclaimer: The ideas, procedures and suggestions contained on this web site are not intended as a substitute for consulting with your physician. All matters regarding your health require medical supervision.
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North barn, set across a private courtyard, with beautiful views of the countryside. With accommodation just under a mile away it is the perfect wedding escape. With high, wood beamed ceilings, Healey hall is one for the rustic, yet romantic wedding couple.
The atmosphere couldn’t be more romantic, as it is completely candlelit, with a fireplace at the head of the barn leaving an enchanted vibe throughout the venue. It can accommodate both large and smaller wedding numbers in both of their rooms.
Eshott Hall
A combination of historic elegant and modern comforts – it truly makes for a beautiful backdrop to any wedding day!
From planning to partying they are dedicated to making your wedding theirs! With several grand, yet inviting reception rooms, a splendid ballroom (which comfortably fits 100 guests), as well as an exquisite restaurant, numerous bedrooms, a golf course and beautiful grounds, you will be hard pressed to find anywhere as inviting as Eshott Hall. As it is only 25 miles north of Newcastle, Eshott Hall is easily accessible from a number of routes.
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Le Petit Chateau
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Alnwick Gardens
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Treehouse
Just a short distance from the gardens, the Alnwick Garden Treehouse is nested under an enchanting woodland canopy. The venue offers an intimate wedding which is both unique and quirky.
Make sure to treat your guests to an impeccable wedding menu, with first class chefs, local, fresh produce and inviting mains, your wedding will stand out from the crowd in more ways than one.
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Newton Hall
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Located in one of the most breathtaking places in the UK and in the heart of the National Trust, Newton Hall combines the best of both worlds – rolling views of the countryside and exquisite views of the natural coastline. Just a 60-minute drive, North of Newcastle and 90 minutes South of Edinburgh. We may be biased, but Newton Hall is the perfect wedding venue!
The Georgian inspired house features indoor and outdoor wedding spaces, as well as elaborate rooms and a tasteful wedding menu that you and your guests are sure to love!
If you’re interested in our wedding venue, don’t hesitate to contact our team today. They are equipped to answer any questions. Or why not come along to one of our open days? We would love to meet you and discuss your future wedding plans.
The post Places to get married in Northumberland appeared first on Newton Hall.
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TITLE: A question concerning $l_1(\mathbb{N})=c_0(\mathbb{N})'$
QUESTION [0 upvotes]: I have a question concerning the identification\begin{equation}l_1(\mathbb{N})=c_0(\mathbb{N})'
\end{equation}where $c_0(\mathbb{N}):=\{(x_j)_{j\in \mathbb{N}} : x_j \to 0, j \to \infty\}\subset l_\infty(\mathbb{N})$ and $E'$ denotes the dual space of some normed vector space $E$.
My problem is that I don't see why in this proof $c_0$ can't be replaced with $l_\infty$.
The proof which was presented to us was as follows:
First consider the map \begin{equation}I:l_1\to c_0', y=(y_j)\mapsto I(y)=\left( x=(x_j)\mapsto \sum x_j y_j\right).\end{equation} This is well defined and injective.
Now consider \begin{equation}I':c_0'\to l_1, f\mapsto (y_j),\end{equation} where $y_j:=f( (\delta_{ij})_i).$ Now for each $j$ we can choose some $\epsilon_j$ in the corresponding field ($\mathbb{R}$ or $\mathbb{C}$) with \begin{equation}\vert \epsilon_j\vert=1 \text{ and }\vert y_j\vert =\epsilon_jy_j.\end{equation} Now for all $N\in \mathbb{N}$ estimate
\begin{equation}\sum_{j=1}^N \vert y_j\vert=\sum \epsilon_j y_j=f((\epsilon_1,...,\epsilon_N,0,0,...))\leq \Vert f \Vert_{op} \Vert (\epsilon_1,...,\epsilon_N,0,0,...)\Vert_\infty= \Vert f \Vert_{op} ,
\end{equation}which proves that $I'$ is well defined. Because $I'$ is injective, we are done.
So that was the proof. I'd be glad for any hints why this proof is wrong when replacing $c_0$ by $l_\infty.$ Thanks in advance.
REPLY [2 votes]: Because $I′$ is injective, we are done.
This is the part that does not carry over to $\ell^\infty$. How do you know that $I'$ is injective? Because the linear span of basis vectors $e_j = (\delta_{ij})_i$ is dense in $c_0$, that is why. On $\ell_\infty$ this is no longer true. Consequently, one can have a linear functional that vanishes on all $e_j$ (so on all $c_0$) but is not zero on $\ell_\infty$. Such functionals are not presented explicitly, but they exist by the Hahn-Banach theorem.
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This is no April Fools deal, here. This is the real schiznit. The Big UGLY sucker in the corner no one wants to talk about., And my buddy David (goat ranchering up in Arkysaw) nails it squarely:
Funny you should mention it…
And we have to give up on planned obsolescence.
If the alleged environmentalist of our generation were the least bit honest planned obsolescence would be the first place to start an environmental movement.
It always fascinated me when the were millions available to the “movement” but you could not find an honest (useful idiot) that had contributed more then $5.
Yes, Brother Dave gets the only gold star I am going to pass out all day long. He and I both see that the reason we can’t have a happy ending to the present worldview is that no one is addressing the number one problem: You can’t have constant growth in a fixed system forever.
Something is going to break. It’s all just a matter of time. Simple as that. D’oh.
Think about this: If everything lasted three times as long, we’d all only have to work one third of the time!
Followed logically, our present trajectory nets us a planet in 100 years where everyone has a 100” TV, a car that gets 900 miles per gallon. Oh, and everyone is dead because with the LBGT movement succeeding, there are no kids being anymore and……
Hello? Can we please – just this once – answer the right question instead of going of on the BS bait and switch? This year’s car, this year’s style, this year’s phone, this year’s Office/operating system and whatever…
Please, tell me you’re not so dumb to miss it? The environmental movement is – Yet Another Distraction (YAD) – because no one starting at the get go and that because no one wants to admit planned obsolescence is what kills us all.
So bring on over-production (and let’s over-medicate, too, while we’re at it) because that way, we can all “green police” ourselves into oblivion.
Save a tree, kill a planet. Save a whale, create a government job. Ban cow farts, more regulators. Blame climate change…and here’s another tax, and more government jobs..
At some point, making the round ball rounder stops being productive. And that’s when the NewCiv begins to arise. The one that blows out of planned obsolescence by asking “Do you know anyone who has ever worn out a dinner plate or a fork?” and other logical starting points.
If PACCAR can put out a Kenworth good for a couple of million miles, how come Detroit can’t do that with cars? Money, turnover, the illusion that working hard is smart. That isn’t necessarily true, you know….
“Why is there Round-Up leftovers not only in soil, but also in air now?” wondered another friend of mine. “What’s the health linkage there?”
The best April Fools joke today is what?
A trip to the mirror with a lie detector.
Learning Your Learning Style, II
I don’t normally pick up a single topic two mornings in a row, but the whole matter of how we learn is critically important. Even more so today because the world is changing – faster and faster – so if we don’t keep up, we’re going to be trampled by the other 7 point some billion people. This is way longer than our usual posting, but the topic is terribly under-addressed in the Mainstream Media.
All Came from a Reader Email
So here we go on more notes about learning. Unlike most sections, this one will be developed just like schools develop curriculum: The start off with a list of “What’s to be learned” and then list out reading assignments, exercises, research, audio, videos, and then they test to make sure you “get it.”
My own “observation of fact” is that everyone has approximately the same mental capacity. What differs is not biologic (in most cases) but how Nature wired our brains. Some brains are impacted by Autism. They work just fine, but with a different underlying “source code” than other people. There are savants, too. See the movie “Rain Man” for details. But both cases argue that people can be exceptional in some areas, while “deficient” [in terms of judgmental society] in other aspects.
Society classifies people according to a prevailing mindset of an era. Under-appreciated is that autistic, savant, and prodigy all live right next door to one another. The Big Lie is that IQ is a simple, two-dimensional curve. It is most assuredly not.
It is a topology.
Those that are on the top of the “bump” that defines it are the self-proclaimed “normals.” Those who see this, are seize leadership are normals tending toward sociopath. At the extreme, sociopath and beyond.
But going off the other direction, people of really good heart around the top of the bump slide into empaths and believers in causes and so forth. None of which makes a person “bad” any more than having a capacitor discharge ignition system is “better” than a magneto system. All have their place.
This email from reader Dave covers a lot of ground:
George, I enjoyed your post on learning. I have been meaning to write to you for several months. Several months back you talked about “Learning to Learn” in one of your posts. In that post you remarked about playing music over and over, or watching moves multiple times as a
learning mode for you. Then more recently you mentioned learning and
had a link to “Gregorc Associates” which offer a program to individuals to find their best learning modes. Today’s post covered focus and right brain / left brain. I have had difficulty learning all my life. I attributed it to a poor memory as so many teachers would say “just memorize this” and you will be fine. With theory I could usually understand and know how to apply formula and data that was in the “Desk Reference” [that you have mentioned, and by the way is available in small pocket edition with all the same info.] and really did not have to memorize anything.
Long intro, my question. Since we are all unique, Is the material from Gregorc Associates of any value in “Learning how to learn”?.
Thanks, Keep Well and Safe,
Ole Dave, in NC.
The answer to the Gregorc Delimiter question is simple: It will give you come insight into how you learn. In other words, whether you do well with sequential or random information, or whether you gobble up abstracts or concrete blocks of information.
I also used it (no kidding here) as a style check when it came to getting married for the third time. I wanted to make sure that whoever I married was very close in abstract-concrete and random-sequential preference. As luck would have it, Elaine and I are just about perfectly matched.
Now, the reason this is important is we both enjoy about a mid-range on both axes, concrete/abstract and random/sequential, so we can both enjoy “smooth jazz” while working around the house . An oldie like Take Five by Dave Brubeck for example” but we both lose interest while Miles is off exploring one of them 8-miles high Miles, places, if you know what I mean.
Since we input similarly, we both get a kick out of the TV soaps lyrics in country music. Toby Keith, or Willy, of whatever. And as a matter of fact (if you don’t mind the commercial on the front end), Hank Junior’s “Red, White, and Pink Slip Blues” is about as poignant a commentary on what gutting America has looked like from the production line, as you’ll find. Bass line is similar to Everlast (in Turn Your Lights On), but you didn’t come for the music critiques here…so I won’t digress further (for now)…
Point is, if you pay attention to what you like to do in life, when no one else is around, when you’re not flat broke, that will tell you gobs about how you learn best.
Light-heartedly, if you read Penthouse, (or PlayGirl, ahem) then you could be visually inclined. If you read Penthouse Forum, then you would be a read-to-learn type.
If you download Earl Nightingale motivational series, like this one, and rip them into MP3s to listen to, then you may be an audio learner. Or, if you call 900 numbers, then audio learning is your mode.
If you jump on your dirt bike for jollies (or your extended forks, chopped soft-tail and know what a top rocker is, and sneer at the dirt bikers as wannabes who can’t handle more than 750 cc) then you’re a tactile learner.
Now that we know what kind of learner you are we can work on speed of learning.
If you find yourself being picked up off bar room floors after fights real often, you’re a slow tactile learner.
On the other hand, if you can shoot the wings off a fly with a Glock at 65-feet while dislodging a particularly well-entrenched bugger from your left nostril, then you are a very fast tactile learner. Provided, of course, you didn’t stop digging in your nose with your left hand to snap off the shot at the fly down-range with your right.
Alternatively, if you have flown a helicopter solo for more than 5000 hours, and have been through at least unplanned autorotation, then you’re either a fast tactile learner, of you should be buying lottery tickets because there’s a cloud of luck around you.
Next we deal with depth of learning.
If you get into bar fights and always lose, then you may not have a lot of depth to your learning.
I’ve seen enough bar fights to know the best way to win is a) keep my mouth shut so I don’t get in them, b) run faster than anyone present, or c) have 9-1-1 on speed dial and take a picture of a perp threatening to email it. This is great depth of learning.
There are lots of other measures of how deep your learning is – I mean really. If you can remember more than one memorable hangover, you may have a problem to deal with. No, make that two problems.
Toss in reading UrbanSurvival and you’ve got a counseling trifecta.
Time Compression
Seriously: It finally all comes down to this: Find how you learn best. Look at your house and your hobbies for hints.
If your wife paints your house (yours doesn’t?) what your learning style is doesn’t matter…because you’ve obviously done something right. Fast-talking audio learning works here with a heaping side of compliments on ‘artistic ability.’ You’re probably a sociopath, but leaders, lawyers, and leeches have to have something in common.
Learn the point, not the fluff.
Life is 18,000 days long by the time you’re old enough to get the humor around here. I passed the 23,800 hour mark a while back. If I were an airplane, I’d have been scrapped by now.
So believe me when I tell you learn what you can use, plus maybe 10% more than you can use immediately so you can see what’s coming.
Then move on with life. Buy someone a paint brush or a lottery ticket. Or better, figure out a way to start a business so you don’t have to work for The Man.
Education is a fine thing…but that’s something we hear from government and educators. The very people who have us locked in to the low quality, disposable crap that is screwing up the world in order to show continuing profits.
And thus, we arrive at our next email…
About That Education Programming
Of course the reason we can’t rethink the world is that planned obsolescense let’s bankers work by lending money. And the reason self-learning is so frowned on is (at least I npart) it doesn’t employ people. As subscribe Kai notices
Not sure if you ever get my emails, but I will keep writing when I find topics that hit me.
I find the “Common Core” programming of our children to be very frightening.
Lately I have seen many posts by parents and teachers of examples of this material. Take a look at this one and tell me that you don’t see the hand of the government carefully choosing the language in these tests:
Here are some very Orwellian excerpts:
“The job of a president is not easy”
“He makes sure the country’s laws are fair”
“The commands of government officials must be obeyed by all”
I really fear for the programmed children that are being brought up to be replaceable cogs in the government machine. I hope you live long enough to see a revolution that means something, but I doubt that will come before even I am grey. People can live with a lot of pain, especially when applied slowly. Take away the rights one by one, give the government more power, make the people dependent on the government, and now you have what they want.
I am a 30-something and think I might be in the last generation to grow up without this kind of influence in the U.S. education system. This is part of the reason I moved to Mexico to raise my child in private schools that do not have the influence of the U.S. government (yet).
Keep writing. And yes, we’ve gone from a country where disagreement is honored to a bunch of ADHD group thinkers on pills.
The sad hell of it is I’ll be saying the same thing tomorrow when it’s not April you-know-what Day..
A Word from the HR Department
And from a reader in one of them:
G- An interesting Trend I have been noticing the last year or so. Back when I first started in this biz I would always receive a lot of resumes over the weekend. Of course when I first started companies advertised heavily on the Sunday paper for jobs but even when things went online with Monster and like companies people by and large would apply over the weekend for the most part. But now It’s hardly any on the weekend and almost all on Monday and Tuesday. Which means in reality people are at their jobs and on company time are surfing the net for jobs and applying then. Or of course they are unemployed but don’t want to take up any valuable weekend time to look for a job. God forbid take time away from their College hoops lol But I figured you would appreciate a trend like that.
Oh, well, dude, simple. What part of mass awakening to corpgov BS and knowing there has to be a better deal than this one because it sucks, did you miss?
.
Ham Radio – the New Social
From a New Mexico reader, on the virtues of ham radio (the original social network – and still commercial-free):
Sunday I was working in my Barn and listening to the radio traffic on the Outstanding statewide 2m/70cm, “New Mexico Mega Link,” repeater network. A young man checked in looking for a contact. He was returning with his family to Colorado on I-40 inbound to ABQ and then northbound on I-25 to Santa Fe and beyond. He was first contacted by a local ABQ Ham who was mobile on his way to finish some weekend chores. The young Ham’s first question was, “are you working for your General yet”? The New Ham continued to receive contacts from other New Mexico Hams on the Mega Link network. The last is the best.
A young Ham made contact with the Colorado Ham and the conversation became very animated. They were both new Hams having both received their call signs in Feb. 2014. The New Mexico Ham asked the Colorado Ham if he had presented any demonstrations of what Ham Radio is. The Colorado Ham asked if the New Mexico Ham had done a Demonstration of Ham Radio for the Boy Scouts??? To which SHE replied, NO I AM A GIRL!!! She then continued, “The next time you are asked, what is your Social Network? Just tell them mine is, HAM RADIO”!!! She then stated that her parents are still LOL belly laughing over that one.
Age of the New Generation of Hams I was listening to—11 years old—one Male and one Female.
I am in my 7th decade and received my call sign in early October 2013. To the wimps in the crowd, the Young and the Old can pass the Technician exam. All it takes is a little perseverance. TURN OFF your TeeVee and your Pulsed Microwave Cell Phones. You will not regret the change.
73 George
P.S. I was listening on a Baofeng UV-5RA. It comforting to know that if I have an emergency on my property, I can contact the entire State of New Mexico and request Emergency assistance through the MegaLink system using my low dollar 2m/7cm Handy Talkie.
Which is why we pop our annual check in the mail to support our local ham radio club…because those reliable, robot ham radio systems need maintenance and new solar panels now and then…
Speaking of Solar Panels
No, this is not an ad. But, as you know, UrbanSurvival and Peoplenomics are both “solar powered” on the content side (the servers are a different matter). But here, we have 3.5 kW of solar panels and two grid-interactive since wave inverter/chargers in a system I designed and built myself.
So while huge numbers of people claim to be trying to change the environment, the “cut through the BS” question to me lately has been “So, how many panels do you have up?” We have 20 and I keep eyeing more.
And that’s my point. My friend John down at (where we bought our panels) has 230-watt class panels at 68-cents a watt. I’m going to be picking up a single 100-watt class panel to update the automatic tower system for our ham radio set up. But prices of solar right now are just dirt cheap and I thought I’d mention it.
They also have solar powered freezers on sale…
On the panels only, the payback period is getting down well under 10-years and while the balance of system costs will push it back to 15-years, there’s nothing like tiny power bills and a line on your bill that says “Net metering credit.” Most months ours is small ($20 bucks or so) but that 3 kw of solar reduces our home bill to about half what neighbors run.
And we took the tax break for putting it in, of course, too….
Like I said, not an ad…just a heads up.
A Bet Lost
Reader Bob thought he had me:
Bet you didn’t use alcohol-free gas, or didn’t put enough “stabilizer” in the gas. And you let the chainsaw, or the gas, sit more than 30 days. Bet the entire fuel system will need replacing soon. Two=cycle repair people love E-10 and especially E-15 for a continuing source of revenue.
Wrong! We buy 5-gallon jugs of 100 low lead (blue) that we burn in the airplane and use it in all of our farm machines. Except the diesel, of course. And an Avblend with LinKite 12 oz Bottle with each oil change on the 4-cycle stuff..
I’m proud to say we have some of the best-running equipment around, when the fuel filter is clean and the hose isn’t plum wore out. We go through 304 chains per year, even with sharpening, to give you an idea… We haven’t run meth’ed gas forever knowing the evils of crappy fuel.
OK, write when you get rich…and ask yourself “Am I the kind of person I would lend money to?”
George george@ure.net
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TITLE: How Can I Represent These Progressions in Sigma Notation?
QUESTION [0 upvotes]: I would like to represent the following finite progressions in sigma notation:
$Finding\ the \ n^{th} \ term \ of \ a \ geometric \ progression$: $a_n=a_1(r^{n-1})$, where $a_1$ is the first time and $r$ is the common ratio
$The \ sum \ of \ a \ geometric \ progression: \ S_n=a_1\frac{1-r^n}{1-r}$
$Determining \ the \ n^{th} \ term \ of \ an \ arithmetic \ progression: a_n=a_1+(n-1)d$,
$\ $ where $d$ is the common difference
And finally, the sum of an arithmetic progression: $S_n=\frac{n}{2}(2a_1+(n-1)d)$
REPLY [0 votes]: You wish to expression the sum of the first $n$ terms of an geometric progression and the sum of the first $n$ terms of an arithmetic progression in summation notation.
Sum of a geometric progression: If the initial term is $a_1$ and the common ratio is $r$, then the $k$th term of the geometric progression is $a_k = a_1r^{k - 1}$. Hence, the sum of the first $n$ terms of the geometric progression is
$$S_n = \sum_{k = 1}^{n} a_1r^{k - 1} = \begin{cases}
a_1 \dfrac{1 - r^n}{1 - r} & \text{if $r \neq 1$}\\
na_1 & \text{if $r = 1$}
\end{cases}
$$
Notice that the index of the variable must be different from the index of the upper limit. Otherwise, all $n$ terms in the sum would be equal to $a_1r^{n - 1}$.
Sum of an arithmetic progression: If the initial term is $a_1$ and the common difference is $d$, then the $k$th term of the arithmetic progression is $a_k = a_1 + (k - 1)d$. Hence, the sum of the first $n$ terms of the arithmetic progression is
$$S_n = \sum_{k = 1}^{n} [a_1 + (k - 1)d] = \frac{n}{2}[2a_1 + (n - 1)d] = \frac{n(a_1 + a_n)}{2}$$
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OK, I was looking at that quake map and noticed the “string of pearls” on the Mid Atlantic Ridge where we have 4 quakes. Got me wondering: Do quakes on the ridge mean Iceland gets some more action?
That lead to this article:
which finds just such a cycle….
Hmmm… That’s not encouraging… Maybe it’s time to invest in passenger ships out of England and Ireland to the mainland..
So we’ve got Iceland linked to the rifting strain. A long period of low activity, that’s now shifted to more active, and a historical pattern of that period spanning the (roughly) 60 year cycle… rather like the PDO / Pacific cycle. Well, looks to me like folks in the EU ought to start watching the activity on the Mid Atlantic Ridge a mite closer.
Wonder if that timing might also match the timing that was predicted by the Russians for the deepest part of the predicted cooling. About 2040, IIRC..
Oh dear…
That’s gonna leave a mark…
We seem to have a constellation of things coming together and pointing to the same mechanisms, the same timings, and the same outcomes. A cold dismal 2020-2040 with increased volcanism as well. Along with probable loss of crops and disruption of travel systems.
Probably not a good time to be buying insurance company stock… or airlines.
There are times when it’s really annoying to have clue about what’s coming.
Live Map:
North America and Mid Atlantic Ridge Quake Map
Original with clickable details
Iceland Seismicity Map
Iceland Seismicity Map
Present quake map at link:
Unfortunately, I’ve not figured out how to get a live map html from their site, so you have to click the link to see what’s up now.
Interesting, is this related to the ENSO cycle as well? There is a correlation between the Pacific rise earthquakes and El Nino
check it out.
The excellent link David supplied pre-empted my post on the latest earthquake and tsunami in Indonesia. Three Tasmanians were among the “lost” surfers but they have now reported in shaken but safe. A Tasmanian also owns the resort where the bungalows were wrecked. Interesting to note that the 7.7 quake on Monday has been followed by a 5.2 quake in the same area today.
The erupting volcano, Mt.Merapi is apparently Indonesia’s most active and has been sending out warnings for some time.
New Zealand is known locally as the “Shaky Isles” but I think the islands which make up Indonesia have become more deserving of that dubious honour.
The devastated villagers in those remote places who had little to lose in the first place, have now lost everything. The survivors are deserving of the greatest sympathy and help.
As you say, there is “a constellation of things coming together”. Makes one wonder what is coming next !
Here we go with another sixty year cycle. It’s amazing the way it keeps popping up.
And yet, there’s no physical phenomenon that we know of that causes it. Yes we have the PDO and AMO, which are roughly sixty year cycles, but these are effects which then cause other effects.
We have the sunspot cycle at 11 or 22 years, the lunar cycle of 18.6 years, the deVries sun cycle (Grand Minimums) of 210 years, but none of these seem to coincide in any meaningful way with the major ups and downs of the temperature datasets.
The Saturn orbit takes 60 years alright, but what could be the mechanism that could cause it to have an effect on Earth, when we pass between it and the Sun 60 times during that time.
I wonder is it an as yet unknown cycle in the interior of the Earth. That would go some way to explaining the increased vulcanism and earthquakes and possibly even the PDO/AMO. I think if we can find the source of that 60 year cycle, we’ll go a long way to understanding what drives the climate.
We seem to have a constellation of things coming together
And that seems to apply to your last few postings… because the same sort of thoughts were running through my mind when I was grounded in the UK by the volcanic ash earlier this year… perhaps this was a flavour of things to come… a flavour of how much society will have to adjust… just how flawed the UK governments computer models are regarding climate change, foot & mouth disease, volcanic ash, the economy…. the list just seems endless these days… and government stupidity seems to know no bounds.
The problems with the volcanic ash computer models were evident on day one as I listened to the no fly news report on the car radio while driving westward into the sunset… the problem was clear to see because the was no beautiful sunset… simply no colours in the sky as the sun sank through the clear northern air over Yorkshire… but government and the media are so far removed from reality that they literally can’t see the evidence.
Now the changes aren’t all bad… this I discovered as I travelled from the UK to Spain overland… travellers talked of their experiences and shared news and knowledge… especially in the hotel bars during the overnight stops… so instead of a few hours flying I experienced a four day adventure across Europe. The lights on but nobody home people stayed at home… there was no travel for the worker ants… while the rest of us had a good time exploring the planet.
So perhaps western society will have to adjust significantly in the next few decades… with volcanoes and a colder climate being the main drivers… unfortunately the governments responses will probably make the reality far worse… like the needless death and destruction inflicted upon the UK farm animals in the face of foot & mouth disease… which brings me to your Doctor Strangelove moment which may become increasingly relevant as the American Empire peaks and moves into terminal decline… and the possibility of riots, revolts and revolutions… which takes me back to the late 60’s while I was interned in a Military Boarding School and watched the very British film If… which depicted a savage insurrection at a public school – see…. (include the dots in the URL)
Although the years have ticked over I am still not ready to roll over…. the adolescent in me still listens to Mountain playing Rollover Beethoven and I still remember the concert where the Who blasted out Wont Get Fooled Again… which brings me right back to the present day as Iceland begins to Rock ‘n’ Roll and I keep the adrenalin flowing by ramping up the volume….
Lou Reed – Sweet Jane from Rock n Roll Animal
Sultans of Swing – Dire Straits
Perhaps this is a good time to express my thanks to E.M.Smith for all the hard work, experience and insight that goes into Musings from the Chiefio… for me E.M. is a fellow traveller who lived through the Summer of 68 and learnt from the experience… it was a time of contradictions… a time of War… and a time of Love & Peace… so little seems to have changed in that respect over the years… the contradictions still endure… but so do the travellers… we still strive and endure… we still try to walk down that Country Road while the world around us retreats to the Urban Jungle… so I bow and say thank you to my fellow travellers who continue to make my journey so enjoyable.
James Taylor – Country Road
@Malaga View: Strange you would choose “Country Road” as a touchstone. I grew up in a little farm town. About 3 miles across. (the long way) For 2 years I worked summers on a cannery “line” packing fruit (stacking boxes, running label machines). The “start time” was something God Awful like 6:10 AM, so I was “out the door” at 5:50. And EVERY SINGLE MORNING the radio station tape ran the same series. As I was dreading the day to come (12 hour work day peak season), I’d be hearing the John Denver rendition of “Take Me HOOOOOOME Country ROOOOads…” and I never wanted to go home quite so much … Yeah, not the same as James Taylor “Country Road”, but still, to this day, the phrase “Country Roads” sets off 2 years of “longing”…
FWIW, the story of “frozen on a motorcycle” was from about ’68 as I was trying to reach a “Woodstock west” event near San Francisco. Didn’t make it (too cold and too slow); but the effort makes up part of my own ‘precious moments’ store.
Glad to be a ‘fellow traveler’ and happy to swap stories and “moments”…
@Scarlet Pumpernickel & Paul Hanlon:
There is also a 60 year Length Of Day change that is in sync with the PDO cycle.. That’s the working hypothesis at least. Basically, the orbital / rotational motions have some wobble in them and wobbling a sloppy ball of magma causes some spots to leak…
Do you have a link for the ENSO / quake connection?
@David: Yeah, I saw that quake (and put a comment in the “Mexico 6.7” posting – but was too busy / lazy to make a full posting about it… though earlier in some other posting I’d said I was bailing from Indonesian investments for a while when they had 18 volcanos on warning status…) I’ve just got a “Baaaadd Feeeeeling about this” on the Indonesian volcano front. Too much history of “big ones” there during major minima events… and trying to suppress that emotional response by under-reporting it, I guess. Had thought of making it 1/2 of this posting, but resisted mixing two sides of the planet.
Good link, BTW. Nice coverage.
@Keith Hill: The price of living in a Volcanic Paradise.. Be it New Zealand, Indonesia, or a dozen other Pacific Islands. And perhaps even California and the Pacific Northwest…
I don’t think I need to point it out, but the North Island is on a Giant Volcano. Super Volcano scale. So we’re all 3 “near” the biggest scale of risks possible. (Mammoth Mountain / Long Valley for me, Taupo in New Zealand, Toba in Sumatra, Indonesia). has some nice pictures, in an over excitable kind of way.
Why it pays to watch the quake maps… so you can get on an early flight out of Dodge.
Not paranoid about it. Just keeping an eye on the “few thousand year event” Grumpy Gus to make sure he’s staying asleep for this century…
But having gotten interested in volcanoes at an early age watching our local ones (and climbing Lassen to the peak), I’ve been waiting 1/2 a Century now for one of them to erupt… After Mt. Saint Helens, I’d be happy to keep on waiting… But maybe a little eruption would be nice to watch ;-)
FWIW, went to Hawaii a few times. Volcano was dead silent. Gave up. It started erupting. Been erupting for a couple of decades now. Any time folks in Hawaii would like it to stop, just buy me a ticket for a visit. I’ve got a 100% negative correlation between my presence and volcanic eruptions… Must be my calming influence.
All together now “OOOOOOoooommmmmmmMMMMM”
;-)
Pingback: World Spinner.
If you throw into that mix the Expanding Earth theory, as articulated by Neal Adams, then I personally think we are on the right track… it is a 3 minute video that really opened my eyes and my mind to lots of possibilities that had previously seemed crazy… … like the concepts of abiogenic oil, abiogenic natural gas and especially abiogenic water because we have a lot of water in our oceans… and what I really like about this approach to science is that it is based upon the simple sort of observations and thought processes that I had as a child… before I was indoctrinated by settled science at school and university. The video makes perfect sense to me… but where it leads us to is another huge can of worms…. enjoy the challenge!
PS
The Expanding Earth theory also probably implies that the earths atmosphere is abiogenic in origin… that would mean that the atmosphere has basically been outgassed since the earth was formed… and that the composition of the atmosphere may change over time in line with the production of abiogenic gasses.
Another thought to ponder is the impact an Expanding Earth might have upon the earths speed of rotation and, thus, the length of day… so perhaps there are some simple explanations for the earths geologic history of temperatures and atmospheric CO2 concentrations… let alone an explanation for our varying cycles of warming and ice ages…
PPS
And perhaps lots of other planets have gone through a abiogenic phase of expansion that had the potential to support life forms… but when the abiogenic phase is complete then the atmosphere and oceans evaporate off into space leaving us a dead planet to look at through out telescopes.
@Malaga View: I first ran into the Expanding Earth thesis when I was about 5? and we got a Reader’s Digest World Atlas (that i still have!). Loved that book.
The problem I have with the thesis now is the same one I had with it then. Conservation of matter.
Where does all the extra “stuff” come from to make an expanding earth?
If it were meteors, we would not have the nice neat continental outlines to fit together, as 1/2 a globe of mass in-falling would make a mess of it.
If it were chemical realignments (i.e. changing one mole of dense compound into 2 moles of lighter compounds) we’d have a different chemical history in the rocks (though you can make a case for the CO2 making limestone causing “growth” as air becomes land… but the quantity is too small).
So in the end, I embraced the continental drift / subduction model.
Yes, all the edges match, as they have all broken apart from each other (and rejoined) at various times…
BTW, for that “perfect match”, it would be better if they used the continental sub-ocean edges rather than the present shoreline. Then South America has a much fatter “tail”…
The other “issue” with the expanding earth is just that we don’t see it expanding now. We see subduction. Expansion at the mid ocean ridges, subduction under my feet (an making volcanoes as it does so…). We can detect the subduction products in the land above that zone. In fact, the predictive ability pretty much tells you where to find gold and other minerals…
So in the end, as much as I think it’s an interesting alternative hypothesis to kick around (and you can learn things from thinking about it) the “issues” with it (especially when compared with how well plate tectonics works) cause me to figure it’s got it wrong.
If someone can show me where the extra “stuff” comes from to make the world expand, and explain why we see subduction today and not expansion, I’d be open to a re-visit. FWIW, at various times I’ve calculated the rate of mass gain from meteorites. It’s not much. You would need to have a MUCH higher rate of mass in-fall. And that “has issues” for things like the evolution of life.
One final note: The video has an issue or two with the fossil record too. It shows the Himalayas and the California mountains as existing for most of the history. But they both are sedimentary rocks full of marine fossils. Where’s the uplift? If there IS uplift, where does it come from in an expanding stretching world (where things would tend to sink instead… like Death Valley where the expansion is pulling it apart, so it’s sunk way below sea level… Eventually it will join the Gulf of California as that sea spreads inland through the Salton Sea…)
Not wanting to rain on your parade, but it’s that kind of stuff that caused geologists to walk away from the Expanding Earth thesis. (In the age of that Reader’s Digest atlas, it WAS the ‘settled science’ of the era. Yeah, a very old book…) I never could quite get past the ‘missing stuff’ problem and when I finally ran into the Plate Tectonics theory, it just covered so much more and had so much less ‘issues’, it was just a very pleasing “fit’ that got rid of most (all?) of the loose ends for me.
Oh, one other minor problem. We know life began in the oceans. The fossil record is pretty clear on that. So how does life begin in the ocean that does not exist for most of the history of an expanding earth? You could postulate it was in some shallow sea instead, but then you have a planet covered in almost all dirt, and life only plays in the mudpuddle for a few billion years? Odd, that.
It’s that kind of ‘every turn is an issue’ vs the plate tectonics where ‘every turn is an answer / fit’ experience that turns me from one thesis to the other…
But the edges do fit together nicely and the video is interesting to watch 8-)
Per ‘dead planets’: I think that’s part of the reason plate tectonics works. It explains the CO2 / water recycle. Cooked out of rocks in the subduction. Without it, the air and water ought to eventually end up bound in hydrates and oxides / nitrides / nitrates / nitrites and related.
So we have volcanoes venting the stuff like crazy, right over the subduction zones where the hydrated nitrogen rich sediments are being pulled down in subduction trenches, and we have evidence for water / solvent separation of minerals in the plumes over the zone (like gold and silver deposits).
If there were no subduction, what keeps the gases and water recycled… We’d end up like Mars in no time. Water and CO2 ice at the poles, everything else solids, air mostly gone or bound in rocks. We’re not massive enough to hang onto an atmosphere forever and we’re reactive enough to bind it into solids. That’s a problem… (but one I hope we don’t face for another billion years or so, depending on when the U and Th run down in the crust…)
In a very real sense, the Volcanoes keep the planet alive. We are the planet Vulcan…
And they are powered by molten rocks, powered by radioactive decay, that also powers the convective currents that give us drift and subduction. But that same power, through keeping things melted down below, tends to prevent the changes needed for an ‘expanding earth’ via recycling bound water and air in light rocks, precipitating out the heavy basalts.
I personally would not rule out subduction… especially where new ocean floor is expanding outwards and being pushed towards and underneath older continental plates… which perhaps explains the ring of fire… and the compression needed to generate mountain ranges.
I would not rule out the possibility that a smaller earth would support lakes and seas… so there is plenty of opportunity for sediments to develop on the original continent plates.
I would not rule out the possibility that the earth slowly gains mass via meteorites, asteroids and dust… although a big one might cause some moon sized damage.
I would not rule out the possibility that the forces causing the earth to expand from the inside work in three dimensions… so the older continental plates could be pushed upwards as well as apart… especially as the earth is not a perfect sphere.
I suspect most of this comes down to understanding the internal physics of stars and planets… so the expanding earth probably fits with the theories of Professor Oliver K. Manuel, Professor of Nuclear Chemistry, University of Missouri-Rolla who views the sun as a giant plasma “diffuser” that sorts ionized atoms by weight… so perhaps there is a similar process in the core of the earth.
I look for answers that fit the observations… and that need not be an either / or process… I still understand the we can have a bit of this and that.
we don’t see it expanding now
I am not so sure about that… we have Ridges in the Pacific and the Atlantic… and both the oceans seem to be getting wider at a snails pace. From what I have seen there are examples of expansion… but I haven’t come across examples of shrinkage… not that I am saying that can’t exist… it is just that there seems to more examples of expansion… its a bit like football… the score may be 5-0 in favour of expansion at the moment… but that is not to say that the other side can’t score a few goals as well…
PS
The arguments for and against put a lot of good observations and theories on the table… the challenge is to fit and adjust the theories so they fit together and support all the observations… in which case we may find that some settled science isnt so settled after all… and that would be no surprise based upon the flawed settled science of some climate scientists…
Thanks for the interesting Armageddon link Chiefio. A few years ago I stayed with a dear old friend (now deceased) who had a lovely home overlooking beautiful Lake Taupo. He blithely told me that it was not a matter of if, but when the super-volcano would blow again and that none of the locals were particularly concerned, all apparently willing to pay the “price of living in a Volcanic Paradise” .
German scientists at that time were operating in a miniature submarine researching and monitoring the activity underwater.
It was quite an experience playing golf on one of the two championship course at Lake Taupo. Walking down the fairway where little fumeroles were smoking away (free drop if your ball landed on one) was novel to say the least and one smoking fenced-off hazard was definitely a no-go area for ball retrieval, unless you wanted a real proverbial “hotfoot” !
Re Malaga View; whilst I have travelled down the “Country Road” much longer than either you or E.M, the erudite posters on this site and the wonderful links they provide have given me so much pleasure in my latter years.
I’m a mental pigmy compared to the intellects on show here, but share many of the thoughts and feelings you express and am very grateful to be still learning at 77.
The one thing that continues to astound me is the gullibility of those who believe that with all the major known and maybe unknown physical, cosmic and universal forces at play in influencing our climate, infinitesimal levels of CO2 are the major factor ! The power of propaganda !
Chiefio,
Here’s a link on cyclical phenomena I found interesting:
I have always been interested in cycles – the business cycle – the climate cycles – attitude and public opinion cycles ( I mean, what ever happened to “Love and Peace”?).
The CAGW crowd does not seem to have any feel for natural cyclical phenomena. It is very interesting to see if the recent moderate warming will be followed by a time like we had back in the ’70s when it got cold.
Maybe a little OT, but I read in one of your other blog posts you are a language fan. Are there common sayings in languages you have studied for “seven years of bad luck” and other such common observations?
@GregO. Very interesting link. I was forgetting about the eccentricites of orbit and the fact that those orbits are not on the exact same plane. There could be something there alright.
@Keith Hill. I’d join you in saying I’m a mental pygmy, but if I can still say at 77 that I’m still learning, I will have considered myself to have lived a life less ordinary.
“Maybe it’s time to invest in passenger ships out of England and Ireland to the mainland”
Nah, just buy Brit activated carbon mask making companies. Activated carbon masks do quite nicely for all kinds of volcanic smellies (SO2, HF & H2S) and dust too!
Admittedly, Laki was very very smelly when it went off.
OT, but related to matters earth scientific – just got a call from youngest daughter in St. Lucia. The hurricane warnings are out as Tomas trundles in from the east, expected to hit later this afternoon 2-3 hrs time.
It’s a Cat 1, so should be quite exciting, but relatively safe compared to the larger options…
E.M.
I was also inspired by the Readers Digest Atlas in my misspent youth, also still have it, and much read and equally treasured by the children as they grew up.
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I don't know Jake Beistel.
I've never seen in wrestle.
I'd never even heard of him until about five minutes ago.
But he's quickly become a guy that I'd love to see do well.
He's a freshman heavyweight at Southmoreland, and he's off to a pretty good start at 5-0 with a Chartiers-Houston tournament title under his belt.
But it's what he's doing to help others that makes him such a great story.
I won't recap much of it here, because Ken Wunderley did such a nice job of it in the Post-Gazette, but Beistel is raising money for his school's autistic support program that (in some cases) is dependent on how many wins he can pile up. It's a great cause and a great story BUT, it's not the first time Beistel has done something like this.
Read all the way to the end of the story. You'll be glad you did.
And, I'll bet if you see him at the Southmoreland tournament next weekend or maybe even at the Southwest Regional Tournament in Johnstown in a few months, you'll be rooting for him.
Just saw this info on Facebook (thanks to Bob Lichtenfels for sharing the story and the info):
Per
Dave Alt: For those of you wanting to help Jake achieve his goal all
you have to do is mail a check payable to the SES Activity Fund with a
note that your donation is for Autism Speaks. Mail it to Southmoreland
Elementary School, Attn: Leah Govi, 100 Scottie Way, Scottdale, PA
15683. If you would rather sponsor him per match please contact Leah
Govi at 724-887-2020 or you can email her at govil@southmoreland.net
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TITLE: Characterizing elementary embeddings of $L$ and $L_\alpha$ under 0#
QUESTION [6 upvotes]: Suppose 0# exists.
It is clear that every order preserving map from the indiscernibles to the indiscernibles gives an elementary embedding from $L$ to $L$. Furthermore, following lemmas 18.7 and 18.8 of Jech, if $\alpha$ is an infinite infinite limit ordinal, an increasing map from alpha to beta gives an elementary embedding from $L_{i_\alpha}$ to $L_{i_\beta}$, where $i_\alpha$ is the $\alpha$-th indiscernible. This is because $L_{i_\alpha}$ equals the Skolem hull in itself of the first $\alpha$ indiscernibles. However, I am not clear on the following points.
1) Is it the case that for a finite successor ordinal, n, $L_{i_n}$ is necessarily equal to the Skolem hull in $L_{i_n}$ of the first n indiscernibles? Jech only proves this result for infinite ordinals.
2) Is it possible that there could be an elementary embedding from $L$ to $L$, or from $L_{i_\alpha}$ to $L_{i_\beta}$ ($\alpha, \beta$ may be finite or infinite), that does not always map indiscernibles to indiscernibles?
This sounds weird, but I'm not convinced it's impossible. As far as I know, there's no formula in $L$ that defines "$\alpha$ is a Silver indiscernible." (In fact there is no such formula -- see Andreas Blass's comment below.)
REPLY [5 votes]: The answer to Q2 is 'No'. Suppose $j:L\rightarrow L$ is a non-trivial elementary embedding. We use the following fact:
$\bullet$ $cp(j)$ (the first ordinal moved by $j$) is always a Silver indiscernible.
Now let $I$ be the class of Silver indiscernibles, and $\delta \in I$ but $j(\delta)\notin I$ for a contradiction. Let $H$ be the Skolem hull in $L$ of $j(\delta)\cup j$''$I\backslash (\delta +1)$. $H$ is isomorphic to $L$. If $j(\delta)\notin H$ but $\pi:H \rightarrow L$ is the transitive collapse, then $\pi^{-1}:L\rightarrow L$ is non-trivial with critical point $j(\delta)$. Hence, by the Fact, $j(\delta)$ must be in $H$. Then we see that for some $\vec \xi <j(\delta)$ some $\overrightarrow{j(\zeta)} > j(\delta)$ with $\vec \zeta \in I\backslash (\delta +1)$ that
$L \models $ ''$\exists \vec \xi < j(\delta)( j(\delta) = t(\vec \xi ,\overrightarrow{j(\zeta)}))$''.
for some term $t$. But then:
$L \models $ ''$\exists \vec \xi < \delta( \delta = t(\vec \xi, \overrightarrow{\zeta}))$''
is a definition of the indiscernible $\delta$ from larger indiscernibles and smaller ordinals, which is impossible. (This works for the variant of the question, taking embeddings between sets, if $\alpha, \beta$ are limit ordinals.)
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Kevin Mccormick Named Executive Vice President, Production, and Senior Advisor, Warner Bros. Pictures
(May 15, 2017 – Burbank, CA) – Respected film and theatrical producer Kevin McCormick will join Warner Bros. Pictures as Executive Vice President, Production and Senior Advisor, it was announced today by Toby Emmerich, President and Chief Content Officer, Warner Bros. Pictures Group. McCormick will also serve as a senior consultant to Mark Kaufman, who heads Warner Bros. Theatre Ventures and reports to Emmerich. In his new role, McCormick will report to Courtenay Valenti, President, Production, Warner Bros. Pictures.
These appointments mark a homecoming for McCormick, who previously spent more than a decade at the Studio before leaving to launch his own film production company and produce live theater.
“As both a successful creative executive and producer, Kevin has developed relationships with key filmmakers across the industry, while honing his exceptional taste and sensibilities,” said Emmerich. “Most recently, he has been working with Mark Kaufman to bring Theatre Ventures’ ‘Charlie and the Chocolate Factory’ to audiences here and abroad. I’ve known Kevin for years, I trust his instincts and know he’ll be a fantastic addition to Warner Bros. Pictures.”
McCormick is a producer on Theatre Ventures’ current Broadway show, “Charlie and the Chocolate Factory,” having helped mount the show’s London production in 2013. He will also serve as a producer on the division’s upcoming “Beetlejuice.”
McCormick was President of Langley Park Productions, a studio-based production company at Warner Bros., where he served as a producer on the romantic drama “The Lucky One,” starring Zac Efron, and the 2011 comedy “Arthur,” starring Russell Brand and Helen Mirren. He also produced sixth-highest-grossing film of all time, domestically; “The Hangover,” which is, domestically, the highest grossing R-rated youth comedy of all time; and the worldwide hit “Sherlock Holmes.” Among the studio’s other films during his tenure were “Get Smart,” the top-grossing Harry Potter franchise, “The Informant!,” “Watchmen” and “Terminator Salvation.”
McCormick first joined Warner Bros. Pictures in 1999 as Executive Vice President, Production, overseeing such notable films as Tim Burton’s “Sweeney Todd: The Demon Barber of Fleet Street,” “Michael Clayton,” “Blood Diamond,” “The Last Samurai,” “Matchstick Men,” “Insomnia,” “Syriana,” “Corpse Bride” and “Charlie and the Chocolate Factory.” McCormick began his career as an executive producer on the film “Saturday Night Fever.”
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\begin{document}
\baselineskip14pt
\title*{Stationary Stochastic Viscosity Solutions of SPDEs}
\titlerunning{Stationary Stochastic Viscosity Solutions of SPDEs}
\author{Qi Zhang\inst{\ {\rm 1}, {\rm 2}, {\rm 3}}}
\authorrunning{Q. Zhang}
\institute{$^{\rm 1}$ School of Mathematical Sciences, Fudan
University, Shanghai, 200433, China. (Current address)\\$^{\rm 2}$
School of
Mathematics, Loughborough University, Loughborough, LE11 3TU, UK.\\
$^{\rm 3}$ School of Mathematics, Shandong University, Jinan,
250100, China.\\\texttt{Email: qzh@fudan.edu.cn}}
\maketitle
\newcounter{bean}
\begin{abstract} In this paper we aim to find the stationary stochastic viscosity
solutions of a parabolic type SPDEs through the infinite horizon
backward doubly stochastic differential equations (BDSDEs). For
this, we study the existence, uniqueness and regularity of solutions
of the corresponding infinite horizon BDSDEs as well as the
``perfection procedure" applied to the solutions of BDSDEs. At last
the ``perfect"
stationary stochastic viscosity solutions of SPDEs constructed by solutions of corresponding BDSDEs are obtained.
\end{abstract}
\textbf{Keywords:} stochastic partial differential equations,
backward doubly stochastic differential equations, stochastic
viscosity solutions, stationary solutions, random dynamical systems.
\vskip5pt
\noindent {AMS 2000 subject classifications}: 60H15, 60H10, 37H10.
\vskip15pt
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\section{Introduction}
\setcounter{equation}{0}
\ \ \ \ \ The pathwise stationary solution of a stochastic dynamical
system is one of the fundamental concept in the study of the long
time behaviour of the stochastic dynamical systems. It describes the
pathwise invariance of the stationary solution, over time, along the
measurable and $P$-preserving transformation
$\theta_t:\Omega\to\Omega$ and the pathwise limit of the solutions
of the random dynamical systems:
\begin{eqnarray}\label{zhao001}
u(t,Y(\omega),\omega)=Y(\theta_t\omega)\ \ \ t\geq 0,\ \rm{a.s.},
\end{eqnarray}
where $u: [0,\infty)\times U\times \Omega \to U$ is a measurable
random dynamical system on a measurable space $(U,\mathcal{B})$ over
a metric dynamical system ($\Omega$, $\cal F$, $P$,
$(\theta_t)_{t\geq0})$ and $Y:\Omega \to U$ is a $\cal F$-measurable
stationary solution. Needless to say that the ``one-force,
one-solution" setting is a natural extension of the equilibrium or
fixed point in the theory of the deterministic dynamical systems to
stochastic counterparts. Such a random fixed point consists of
infinitely many randomly moving invariant surfaces on the
configuration space due to the random external force pumped to the
system constantly. Therefore, in contrast to the deterministic
dynamical systems, the existence and stability of stationary
solutions of stochastic dynamical systems, generated e.g. by SDEs or
SPDEs, are a difficult and subtle problem.
In many works on random dynamical systems the existence of
stationary solutions is a basic assumption, e.g. in the study of
stability (Has$'$minskii \cite{ha}) and in the theory of stable and
unstable manifolds (Arnold \cite{ar}, Mohammed, Zhang and Zhao
\cite{mo-zh-zh}, Duan, Lu and Schmalfuss \cite{du-lu-sc1}). These
theories gave neither the existence of stationary solutions, nor a
way to find them. Although in \cite{mo-zh-zh}, Mohammed, Zhang and
Zhao introduced an integral equation of infinite horizon for the
stationary solutions of certain stochastic evolution equations, the
existence of the solutions of such stochastic integral equations in
general is far from clear.
Besides, from a pathwise stationary solution we can construct an
invariant measure for the skew product of the metric dynamical
system and the random dynamical system. The invariant measure
describes the invariance of a certain solution in law when time
changes, therefore it is a stationary measure of the Markov
transition probability. It is well known that an invariant measure
gives a stationary solution when it is a random Dirac measure.
Although an invariant measure of a random dynamical system on
${\mathbb{R}^{1}}$ gives a stationary solution, in general, this is
not true unless one considers an extended probability space.
However, considering the extended probability space, one essentially
regards the random dynamical system as noise as well, so the
dynamics is different. In fact, the pathwise stationary solution
gives the support of the corresponding invariant measure, so reveals
more detailed information than an invariant measure.
In spite of the importance of stationary solution, the difficulties,
arising mainly from random external force, prevent researchers form
finding a method universal to the stationary solutions of SPDEs with
great generalities. Some works on stationary solutions of certain
types of SPDEs usually under additive or linear noise include Sinai
\cite{si1}, \cite{si2} for stochastic Burgers' equations with
periodic or random forcing, Caraballo, Kloeden, Schmalfuss
\cite{kloeden} for stochastic evolution equations with small
Lipschitz constant. If one notices the solutions of infinite horizon
backward stochastic differential equations (BSDEs) give a classical
or viscosity solution of elliptic type PDEs (Poisson equations) from
the works of Peng \cite{pe} and Pardoux \cite{pa}, then it would be
natural to conjecture the stationary solutions of SPDEs can be
represented as the solutions of infinite horizon backward doubly
stochastic differential equations (BDSDEs). Inspired by this idea,
Zhang and Zhao in \cite{zh-zh1} proved that under the Lipschitz and
monotone conditions, the
$L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{1}})\otimes
L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solution of an
infinite horizon BDSDE exists and gives the stationary weak solution
of the corresponding parabolic SPDE. Zhang and Zhao
further considered this problem
under the linear growth and monotone conditions in \cite{zh-zh2}. It
is easy to see the solutions of elliptic type PDEs give the
stationary solutions of the corresponding parabolic type PDEs,
however, for SPDEs of the parabolic type, such kind of connection
does not exist, so in this sense BDSDEs (or BSDEs) can be regarded
as more general SPDEs (or PDEs).
The stochastic viscosity solution of SPDE was first put forward by
Lions and Souganidis in \cite{li-so1} through stochastic
characteristics to remove the stochastic integrals in the SPDE. Then
Buckdahn and Ma in \cite{bu-ma1}-\cite{bu-ma3} gave their definition
through the Doss-Sussmann transformation. After that a few works on
stochastic viscosity solutions of SPDEs emerge using Buckdahn and
Ma's definition and corresponding BDSDEs, such as Boufoussi, Van
Casteren and Mrhardy \cite{bo-va-mr} for the SPDEs with Neumann
boundary conditions, Boufoussi and Mrhardy \cite{bo-mr} for the
multivalued SPDEs. Then an interesting question arises: can we also
find the stationary solution of some SPDE in the sense of stochastic
viscosity solution? This paper gives this question a positive
answer. By adopting Buckdahn and Ma's definition and using its
connection with BDSDE we can find the stationary stochastic
viscosity solution of the following SPDE:
\begin{eqnarray}\label{zz20}
v(t,x)&=&v(0,x)+\int_{0}^{t}[\mathscr{L}v(s,x)+f\big(x,v(s,x),\sigma^*(x)Dv(s,x)\big)]ds+\int_{0}^{t}\langle
g\big(x,v(s,x)\big),d{B}_s\rangle.
\end{eqnarray}
Here $(B_t)_{t\geq 0}$ is a Brownian motion with values in
$\mathbb{R}^l$; $f$, $g$ satisfy the condition (A.1)-(A.3) in
Section 2; $\mathscr{L}$ is the infinitesimal generator of the
diffusion process $X_{s}^{t,x}$ generated by the SDE as follows:
\begin{eqnarray}\label{qi17}
\left\{\begin{array}{l}
dX_{s}^{t,x}=b(X_{s}^{t,x})ds+\sigma(X_{s}^{t,x})dW_s,\ \ \ s>t\\
X_{s}^{t,x}=x,\ \ \ 0\leq s\leq t,
\end{array}\right.
\end{eqnarray}
where $(W_t)_{t\geq 0}$, independent of $(B_t)_{t\geq 0}$, is a
Brownian motion with values in $\mathbb{R}^d$ and $b$, $\sigma$
satisfy the condition (A.4) in Section 2, hence $\mathscr{L}$ is
given by
\begin{eqnarray*}\label{buchong1}
\mathscr{L}={1\over2}\sum_{i,j=1}^na_{ij}(x){{\partial^2}\over{\partial
x_i\partial x_j}}+\sum_{i=1}^nb_i(x){\partial\over{\partial x_i}}
\end{eqnarray*}
with $\big(a_{ij}(x)\big)=\sigma\sigma^*(x)$ and $(b_1(x),b_2(x),\cdot\cdot\cdot,b_n(x))^*=b(x)$.
The infinite horizon BDSDEs we study as our tool can be written in
the following integration from:
\begin{eqnarray}\label{zz36}
{\rm e}^{-{K'\over2}s}Y_s^{t,x}&=&\int_{s}^{\infty}{\rm e}^{-{K'\over2}r}f(X_r^{t,x},Y_r^{t,x},Z_r^{t,x})dr+\int_{s}^{\infty}{K'\over2}{\rm e}^{-{K'\over2}r}Y_r^{t,x}dr\nonumber\\
&&-\int_{s}^{\infty}{\rm e}^{-{K'\over2}r}\langle
g(X_r^{t,x},Y_r^{t,x}),d^\dagger\hat{B}_r\rangle-\int_{s}^{\infty}{\rm
e}^{-{K'\over2}r}\langle Z_r^{t,x},dW_r\rangle,\ \ \ s\geq t.
\end{eqnarray}
Here
$\hat{B}$ is the time reverse version of ${B}$, i.e. $\hat{B}_s=
B_{T-s}-B_T$ for arbitrary $T>0$ and all $s\in\mathbb{R}^{1}$,
and the integral w.r.t. $\hat{B}$ is a backward It$\hat {\rm o}$'s
integral (see \cite{zh-zh1} for details and the relationship between
the forward and backward It$\hat {\rm o}$'s integral). Our purpose
is to prove that, for arbitrary $T>0$ and $0\leq t\leq T$,
$v(t,x)(\omega)=Y_{T-t}^{T-t,x}(\hat{\omega})$ is a stationary
stochastic
viscosity solution of SPDE (\ref{zz20}).
Five sections are organized in this paper for this purpose. In next
section we give brief introduction to the notion of stochastic
viscosity solutions of SPDEs and the connection between SPDEs and
BDSDEs in the sense of stochastic viscosity solution. In Section 3
under the assumption of the existence, uniqueness and regularity of
solution to infinite horizon BDSDE, we study its stationary
property, in which the general version ``perfection procedure" plays
an important role. The existence, uniqueness and regularity of
solution to infinite horizon BDSDE are proved in Section 4. In
Section 5 we deduce the stationary property for the stochastic
viscosity solutions of SPDEs constructed by the solutions of
infinite horizon BDSDEs.
As far as we know, the connection between the pathwise stationary
stochastic viscosity solutions of SPDEs and infinite horizon BDSDEs
in this paper is new. By the techniques as we dealt with the weak
solutions of PDEs or SPDEs in \cite{zh-zh2} and \cite{zh-zh3}, we
believe this connection can be extended to studying the stationary
stochastic viscosity solutions of more general parabolic SPDEs such
as those with linear or polynomial growth nonlinear terms, more
types of noises etc., but in this paper we only study Lipschitz
continuous nonlinear term and finite dimensional noise for
simplicity in order to initiate this method to the case of
stationary stochastic viscosity solutions of SPDEs. Finally we would
like to point out that the uniqueness of the stationary solution of
SPDE (\ref{zz20}) is still an open problem due to its complexity.
\section{Definition and Results for Stochastic Viscosity Solutions of SPDEs}\label{s25}
\setcounter{equation}{0}
\ \ \ \ The main purpose of this paper is to find the stationary
stochastic viscosity solution of SPDE (\ref{zz20}). As shown in
\cite{zh-zh1} and \cite{zh-zh2}, under appropriate conditions, for
$T\geq t\geq0$, defining $u(t,x)\triangleq v(T-t,x)$, we can obtain
the time reverse version of SPDE (\ref{zz20}) on $[0,T]$:
\begin{eqnarray}\label{zz21}
u(t,x)&=&u(T,x)+\int_{t}^{T}[\mathscr{L}u(s,x)+f\big(x,u(s,x),(\sigma^*\nabla
u)(s,x)\big)]ds-\int_{t}^{T}\langle
g\big(x,u(s,x)\big),d^\dagger\hat{B}_s\rangle.\nonumber\\
\end{eqnarray}
The BDSDE on $[t,T]$ associated with SPDE (\ref{zz21}) has the
following form:
\begin{eqnarray}\label{zz22}
Y_{s}^{t,x}=Y_{T}^{t,x}+\int_{s}^{T}f(X_{r}^{t,x},Y_{r}^{t,x},Z_{r}^{t,x})dr-\int_{s}^{T}\langle
g(X_{r}^{t,x},Y_{r}^{t,x}),d^\dagger\hat{B}_r\rangle-\int_{s}^{T}\langle
Z_{r}^{t,x},dW_r\rangle.
\end{eqnarray}
For $k,l\geq0$, we denote by $C_{b}^{k,l}$
the set of $C^{k,l}$-functions whose partial derivatives of order
for the first variable less than or equal to $k$ and for the second
variable less than or equal to $l$ are bounded. We assume
\begin{description}
\item[(A.1).] Functions $f: \mathbb{R}^d\times \mathbb{R}^1\times \mathbb{R}^{d}{\longrightarrow{\mathbb{R}^1}}$ and $g: \mathbb{R}^d\times \mathbb{R}^1{\longrightarrow{\mathbb{R}^{l}}}$ are $\mathscr{B}_{\mathbb{R}^{d}}\otimes\mathscr{B}_{\mathbb{R}^{1}}\otimes\mathscr{B}_{\mathbb{R}^{d}}$
and
$\mathscr{B}_{\mathbb{R}^{d}}\otimes\mathscr{B}_{\mathbb{R}^{1}}$
measurable respectively, and there exist constants $C_0$, $C_1$,
$C\geq0$ s.t. for any $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)\in
\mathbb{R}^d\times \mathbb{R}^1\times \mathbb{R}^{d}$,
\begin{eqnarray*}
&&|f(x_1, y_1, z_1)-f(x_2, y_2, z_2)|^2\leq C_0|x_1-x_2|^2+C_1|y_1-y_2|^2+C|z_1-z_2|^2,\nonumber\\
&&|g(x_1, y_1)-g(x_2, y_2)|^2\leq C_0|x_1-x_2|^2+C|y_1-y_2|^2;
\end{eqnarray*}
\item[(A.2).] $g(\cdot,\cdot)\in
C_{b}^{2,3}(\mathbb{R}^d\times\mathbb{R}^1;\mathbb{R}^l)$;
\item[(A.3).] There exist constants $K\in\mathbb{R}^+$, $p>d+2$, $K<K'<2K$ and $\mu>0$ with $2\mu-{p\over2}K'-{p(p+1)\over2}C>0$ s.t. for
any $y_1$, $y_2\in \mathbb{R}^1$, $x$, $z\in \mathbb{R}^{d}$,
$$(y_1-y_2)(f(x, y_1, z)-f(x, y_2, z))\leq -\mu |y_1-y_2|^2;$$
\item[(A.4).] Functions $b(\cdot):
\mathbb{R}^d\longrightarrow \mathbb{R}^d$, $\sigma(\cdot):
\mathbb{R}^d\longrightarrow \mathbb{R}^{d\times d}$ are globally
Lipschitz continuous with Lipschitz constant $L$ and for $p$, $K$ in
(A.3), $K-pL-{p(p-1)\over2}L^2>0$.
\end{description}
Denote the set of $C^0$-functions with linear growth by $C^0_{l}$.
Buckdahn and Ma proved that if $u(T,\cdot)\in
C^0_{l}(\mathbb{R}^d;\mathbb{R}^1)$ is given, the solution
$Y_t^{t,x}$ of BDSDE (\ref{zz22}),
$(t,x)\in[0,T]\times\mathbb{R}^{d}$, is a stochastic viscosity
solution of SPDE (\ref{zz21}) under Conditions (A.1), (A.2) and
(A.4), therefore it gives the stochastic viscosity solution of SPDE
(\ref{zz20}) through the time reversal argument. To benefit the
reader, we include briefly Buckdahn and Ma's definition of
stochastic viscosity solution of SPDE (\ref{zz20}) through the
Doss-Sussmann transformation in \cite{bu-ma1}-\cite{bu-ma3}.
Let ${\cal N}$ be the class of $P$ null measure sets of
${\mathscr{F}}$. For any process $(\eta_t)_{t\geq0}$,
$\mathscr{F}_{s,t}^\eta\triangleq\sigma\{\eta_r-\eta_s$; ${0\leq
s\leq r\leq t}\}\bigvee{\cal N}$,
${\mathscr{F}}_t^\eta\triangleq{\mathscr{F}}_{0,t}^\eta$,
$\mathscr{F}_{t,\infty}^{\eta}\triangleq\bigvee_{T\geq0}{\mathscr{F}_{t,T}^\eta}$. Let $\mathbb{E}$ and $\mathbb{F}$ be the generic Euclidean spaces, then we denote\\
$\bullet$ $\mathscr{M}_{0,T}^B$ to be all the $\{\mathscr{F}_{t}^B\}_{t\geq0}$ stopping times $\tau$ such that $0\leq\tau\leq T$ a.s., where $T>0$ is some fixed time horizon;\\
$\bullet$ for any sub-$\sigma$-field $\mathscr{G}\subseteq\mathscr{F}_{T}^B$ and real number $p\geq0$, $L^p(\mathscr{G};\mathbb{E})$ to be $\mathbb{E}$-valued, $\mathscr{G}$-measurable random variables $\xi$ such that $E[|\xi|^p]<\infty$;\\
$\bullet$ for any sub-$\sigma$-field $\mathscr{G}\subseteq\mathscr{F}_{T}^B$, $C^{k,l}(\mathscr{G},[0,T]\times\mathbb{E};\mathbb{F})$ to be the space of all $C^{k,l}([0,T]\times\mathbb{E};\mathbb{F})$-valued random variables that are $\mathscr{G}\otimes\mathscr{B}_{[0,T]}\otimes\mathscr{B}_{\mathbb{E}}$-measurable;\\
$\bullet$
$C^{k,l}(\{\mathscr{F}_{t}^B\}_{t\geq0},[0,T]\times\mathbb{E};\mathbb{F})$
to be the space of all random fields $\varphi\in
C^{k,l}(\mathscr{F}_{T}^B,[0,T]\times\mathbb{E};\mathbb{F})$, such
that for fixed $x\in\mathbb{E}$, the mapping
$(t,\omega)\to\varphi(t,x,\omega)$ is
$\mathscr{F}_{t}^B$-progressively measurable.
The definition of stochastic viscosity solution depends heavily on
the following stochastic flow $\lambda\in
C^{0,0}(\{\mathscr{F}_{t}^B\}_{t\geq0},[0,T]\times\mathbb{R}^d\times\mathbb{R}^1;\mathbb{R}^1)$,
defined as the unique solution of the following SDE
\begin{eqnarray*}
\lambda(t,x,y)=y+{1\over2}\int_0^t\langle
g,D_yg\rangle(x,\lambda(s,x,y))ds-\int_0^t\langle
g(x,\lambda(s,x,y)),d{B}_s\rangle.
\end{eqnarray*}
Under Condition (A.2), $\lambda(t,x,y)$ is a stochastic flow, i.e.
for fixed $x$, the random field $\lambda(t,x,y)$ is continuously
differentiable in the variable $y$, and the mapping
$y\longrightarrow\lambda(t,x,y)$ defines a diffeomorphism for all
$(t,x)$, $P$-a.s. Denote the inverse of $\lambda$ by
$\zeta(t,x,y)=(\lambda(t,x,\cdot))^{-1}(y)$.
\begin{defi}\label{zz65} \rm{({\cite{bu-ma1}})}
A random field $w\in
C^{0,0}(\{\mathscr{F}_{t}^B\}_{t\geq0},[0,T]\times\mathbb{R}^d;\mathbb{R}^1)$
is called a stochastic viscosity subsolution (resp. supersolution)
of SPDE (\ref{zz20}), if $w(0,x)\leq$ (resp.$\geq$) $v(0,x)$,
$\forall x\in\mathbb{R}^d$; and if for any
$\tau\in\mathscr{M}_{0,T}^B$, $\xi\in
L^0(\mathscr{F}_{\tau}^B;\mathbb{R}^d)$, and any random field
$\varphi\in
C^{1,2}(\mathscr{F}_{\tau}^B,[0,T]\times\mathbb{R}^d;\mathbb{R}^1)$
satisfying
$$w(t,x)-\lambda(t,x,\varphi(t,x))\leq\ {\rm(resp.\ \geq)}\ 0=w(\tau,\xi)-\lambda(\tau,\xi,\varphi(\tau,\xi)),$$
for all $(t,x)$ in a neighborhood of $(\tau,\xi)$, P-a.e. on the set
$\{0<\tau<T\}$, it holds that
$$\mathscr{L}\psi(\tau,\xi)+f\big(\xi,\psi(\tau,\xi),\sigma^*(\xi)D\psi(\tau,\xi)\big)\geq\ {\rm(resp.\ \leq)}\ D_y\lambda\big(\tau,\xi,\varphi(\tau,\xi)\big)D_t\varphi(\tau,\xi),$$
P-a.e. on $\{0<\tau<T\}$, where
$\psi(t,x)\triangleq\lambda(t,x,\varphi(t,x))$.
A random field $w\in
C^{0,0}(\{\mathscr{F}_{t}^B\}_{t\geq0},[0,T]\times\mathbb{R}^d;\mathbb{R}^1)$
is called a stochastic viscosity solution of SPDE (\ref{zz20}), if
it is both a stochastic viscosity subsolution and a supersolution.
\end{defi}
By Doss-Sussmann transformation, SPDE (\ref{zz20}) can be converted
to the following PDE
\begin{eqnarray}\label{zz43}
\tilde{v}(t,x)=\tilde{v}(0,x)+\int_{0}^{t}[\mathscr{L}\tilde{v}(s,x)+\tilde{f}\big(s,x,\tilde{v}(s,x),\sigma^*(x)D\tilde{v}(s,x)\big)]ds,
\end{eqnarray}
where
\begin{eqnarray*}
&&\tilde{f}(t,x,y,z)={1\over
D_y\lambda(t,x,y)}\big(f(x,\lambda(t,x,y),\sigma^*(x)D_x\lambda(t,x,y)+D_y\lambda(t,x,y)z)\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \
+\mathscr{L}_x\lambda(t,x,y)+\langle\sigma^*(x)D_{xy}\lambda(t,x,y),z\rangle+{1\over2}D_{yy}\lambda(t,x,y)|z|^2\big),
\end{eqnarray*}
and the stochastic viscosity solutions of (\ref{zz20}) and
(\ref{zz43}) have a kind of relationship like
$\tilde{v}(t,x)=\zeta(t,x,v(t,x))$. The Doss-Sussman transformation
plays a big role in the notion of the stochastic viscosity solution
of SPDE (\ref{zz20}). For more details, see Buckdahn and Ma
\cite{bu-ma1}-\cite{bu-ma3}.
Define
\begin{eqnarray*}
\mathscr{F}_{t,T}\triangleq{\mathscr{F}_{t,T}^{\hat{B}}}\bigvee
\mathscr{F}_t^W,\ {\rm for}\ 0\leq t\leq T;\ \ \
\mathscr{F}_t\triangleq{\mathscr{F}_{t,\infty}^{\hat{B}}}\bigvee
\mathscr{F}_t^W,\ {\rm for}\ t\geq0.
\end{eqnarray*}
For $q\geq2$, we define some useful solution spaces.
\begin{defi}\label{qi00}
Let $\mathbb{S}$ be a Banach space with norm $\|\cdot\|_\mathbb{S}$
and Borel $\sigma$-field $\mathscr{S}$. For $K\in\mathbb{R}^+$, we
denote by $M^{q,-K}([0,\infty);\mathbb{S})$ the set of
$\mathscr{B}_{\mathbb{R}^+}\otimes\mathscr{F}/\mathscr{S}$
measurable random processes $\{\phi(s)\}_{s\geq0}$ with values in
$\mathbb{S}$ satisfying
\begin{enumerate-roman}
\item $\phi(s):\Omega\rightarrow\mathbb{S}$ is $\mathscr{F}_s$ measurable for $s\geq 0$;
\item $E[\int_{0}^{\infty}{\rm e}^{-Ks}\|\phi(s)\|_\mathbb{S}^qds]<\infty$.
\end{enumerate-roman}
Also we denote by $S^{q,-K}([0,\infty);\mathbb{S})$ the set of
$\mathscr{B}_{\mathbb{R}^+}\otimes\mathscr{F}/\mathscr{S}$
measurable random processes $\{\psi(s)\}_{s\geq0}$ with values in
$\mathbb{S}$ satisfying
\begin{enumerate-roman}
\item $\psi(s):\Omega\rightarrow\mathbb{S}$ is $\mathscr{F}_s$ measurable for $s\geq0$ and $\psi(\cdot,\omega)$ is continuous $P$-a.s.;
\item $E[\sup_{s\geq0}{\rm
e}^{-Ks}\|\psi(s)\|_\mathbb{S}^q]<\infty$.
\end{enumerate-roman}
\end{defi}
Similarly, for $0\leq t\leq T<\infty$, we define
$M^{q,0}([t,T];\mathbb{S})$ and $S^{q,0}([t,T];\mathbb{S})$ on a
finite time interval.
\begin{defi}\label{zhao005}
Let $\mathbb{S}$ be a Banach space with norm $\|\cdot\|_\mathbb{S}$
and Borel $\sigma$-field $\mathscr{S}$. We denote by
$M^{q,0}([t,T];\mathbb{S})$ the set of
$\mathscr{B}_{[t,T]}\otimes\mathscr{F}/\mathscr{S}$ measurable
random processes $\{\phi(s)\}_{t\leq s\leq T}$ with values in
$\mathbb{S}$ satisfying
\begin{enumerate-roman}
\item $\phi(s):\Omega\rightarrow\mathbb{S}$ is $\mathscr{F}_{s,T}\bigvee{\mathscr{F}_{T,\infty}^{\hat{B}}}$ measurable for $t\leq s\leq T$;
\item $E[\int_{t}^{T}\|\phi(s)\|_\mathbb{S}^qds]<\infty$.
\end{enumerate-roman}
Also we denote by $S^{q,0}([t,T];\mathbb{S})$ the set of
$\mathscr{B}_{[t,T]}\otimes\mathscr{F}/\mathscr{S}$ measurable
random processes $\{\psi(s)\}_{t\leq s\leq T}$ with values in
$\mathbb{S}$ satisfying
\begin{enumerate-roman}
\item $\psi(s):\Omega\rightarrow\mathbb{S}$ is
$\mathscr{F}_{s,T}\bigvee{\mathscr{F}_{T,\infty}^{\hat{B}}}$
measurable for $t\leq s\leq T$ and $\psi(\cdot,\omega)$ is
continuous $P$-a.s.;
\item $E[\sup_{t\leq s\leq T}\|\psi(s)\|_\mathbb{S}^2]<\infty$.
\end{enumerate-roman}
\end{defi}
The following Buckdahn and Ma's result established the connection
between the solution of BDSDE (\ref{zz22}) and the stochastic
viscosity solution of SPDE (\ref{zz20}) on finite time interval
$[0,T]$.
\begin{thm}\label{theorem2.2} {\rm({\cite{bu-ma1}})}
Assume Conditions {\rm(A.1)}, {\rm(A.2)}, {\rm(A.4)} are satisfied
and the function $v(0,\cdot)\in C^0_{l}(\mathbb{R}^d)$ is given.
Then $v(t,x)=u(T-t,x)=Y_{T-t}^{T-t,x}$, where $Y^{t,x}_\cdot\in
S^{2,0}([0,T];\mathbb{R}^{1})$ is the solution of BDSDE
(\ref{zz22}), is a stochastic viscosity solution of SPDE
(\ref{zz20}) on finite time interval $[0,T]$.
\end{thm}
\begin{rmk}\label{zz24}
From the argument of Buckdahn and Ma we can see if we replace the
condition $v(0,\cdot)\in C^0_{l}(\mathbb{R}^d)$ in Theorem
\ref{theorem2.2} by that $v(0,x)$ is continuous w.r.t. $x$ and
$E[|v(0,X_T^{t,x})|^2]<\infty$, then the conclusion of Theorem
\ref{theorem2.2} remains true since
$E[|v(0,X_T^{t,x})|^2]=E[|Y_{T}^{t,x}|^2]<\infty$
guarantees the corresponding BDSDE has a square-integrable terminal
value.
\end{rmk}
\section{Stationary Property of Solutions of BDSDEs}
\setcounter{equation}{0}
\ \ \ \ The purpose of this section is to study the stationary
property of the solution to infinite horizon BDSDE (\ref{zz36}). In
order to show the main idea, we first assume that there exists a
unique solution $(Y^{t,x}_{\cdot}, Z^{t,x}_{\cdot})\in
S^{p,-K}([0,\infty); \mathbb{R}^1)\cap
M^{2,-K}([0,\infty);\mathbb{R}^1)\times M^{2,-K}([0,\infty);
\mathbb{R}^{d})$ to BDSDE (\ref{zz36}) and $(t,x)\to Y^{t,x}_{t}$ is
a.s. continuous. The study of the existence, uniqueness and
regularity of solution to BDSDE (\ref{zz36}) will be deferred to
next section.
We now construct the measurable metric dynamical system through
defining a measurable and measure-preserving shift. Let
$\hat{\theta}_t:\Omega\longrightarrow\Omega$, $t\geq0$, be a
measurable mapping on $(\Omega, {\mathscr{F}}, P)$, defined by
\begin{eqnarray*}
\hat{\theta}_{t}\circ \hat{B}_s=\hat{B}_{s+t}-\hat{B}_t,\ \ \
\hat{\theta}_{t}\circ W_s=W_{s+t}-W_t.
\end{eqnarray*}
Then for any $s,t\geq0$,
\begin{description}
\item[$(\textrm{i})$]$P\cdot\hat{\theta}_{t}^{-1}=P$;
\item[$(\textrm{i}\textrm{i})$]$\hat{\theta}_{0}=I$, where $I$ is the identity transformation on $\Omega$;
\item[$(\textrm{i}\textrm{i}\textrm{i})$]$\hat{\theta}_{s}\circ\hat{\theta}_{t}=\hat{\theta}_{s+t}$.
\end{description}
Also for an arbitrary $\mathscr{F}$ measurable random variable
$\phi$, set
\begin{eqnarray*}
\hat{\theta}\circ\phi(\omega)=\phi\big(\hat{\theta}(\omega)\big).
\end{eqnarray*}
For any $r\geq0$, $s\geq t$, $x\in\mathbb{R}^d$, applying
$\hat{\theta}_r$ to SDE (\ref{qi17}), we have
\begin{eqnarray*}
\hat{\theta}_r\circ
X_{s}^{t,x}=x+\int_{t+r}^{s+r}b(\hat{\theta}_r\circ
X_{u-r}^{t,x})du+\int_{t+r}^{s+r}\sigma(\hat{\theta}_r\circ
X_{u-r}^{t,x})dW_u.
\end{eqnarray*}
So under Condition (A.4), by the uniqueness of the solution,
we have for any $r$, $t\geq0$, $x\in\mathbb{R}^{d}$,
\begin{eqnarray}\label{qi18}
\hat{\theta}_r\circ X_{s}^{t,x}=X_{s+r}^{t+r,x},\ \ {\rm for}\ {\rm
all}\ s\geq0\ {\rm a.s.}
\end{eqnarray}
For $Y\in\mathbb{R}^{1}$, $x$, $Z\in\mathbb{R}^{d}$, let
\begin{eqnarray*}
\hat{f}(\mathcal{T},Y,Z)=f(X_{s}^{t,x},Y,Z),\ \ \
\hat{g}(\mathcal{T},Y,Z)=g(X_{s}^{t,x},Y,Z).
\end{eqnarray*}
Here we take $\mathcal{T}=(s,t)$ as a dual time variable (t is
fixed). Using (\ref{qi18}) we can verify that $\hat{f}$ and
$\hat{g}$ satisfy the stationary conditions in Proposition 2.5 in
\cite{zh-zh1} for any $\hat{\theta}_{r}$ $(r\geq0)$, $\mathcal{T}$,
$Y$ and $Z$, then using a similar argument as in Theorem 2.12 in
\cite{zh-zh1} we can deduce the following proposition by the
uniqueness of BDSDE (\ref{zz36}):
\begin{prop}\label{qi031}
Assume BDSDE (\ref{zz36}) has a unique solution $(Y^{t,x}_{\cdot},
Z^{t,x}_{\cdot})\in S^{p,-K}([0,\infty); \mathbb{R}^1)\cap
M^{2,-K}([0,\infty);\mathbb{R}^1)\times M^{2,-K}([0,\infty);
\mathbb{R}^{d})$, then under Condition (A.4), $(Y^{t,x}_s
Z^{t,x}_s)_{s\geq0}$ satisfies the following stationary property
w.r.t. $\hat{\theta}_\cdot$: for any $r$, $t\geq0$,
$x\in\mathbb{R}^{d}$,
\begin{eqnarray*}
\hat{\theta}_r\circ Y^{t,x}_s=Y^{t+r,x}_{s+r}, \ \
\hat{\theta}_r\circ Z^{t,x}_s=Z^{t+r,x}_{s+r}\ \ {\rm for}\ {\rm
all}\ s\geq0\ {\rm a.s.}
\end{eqnarray*}
In particular, for any $r$, $t\geq0$, $x\in\mathbb{R}^{d}$,
\begin{eqnarray}\label{zz18}
\hat{\theta}_r\circ Y^{t,x}_t=Y^{t+r,x}_{t+r}\ \ \ {\rm a.s.}
\end{eqnarray}
\end{prop}
If we regard $Y_t^{t,x}$ as a function of $(t,x)$, (\ref{zz18})
gives a ``very crude" stationary property of $Y$. Borrowing the idea
of perfecting crude cocycles in \cite{ar} and \cite{ar-sc}, we then
prove the following theorem which makes the ``very crude" stationary
property
of $Y$ ``perfect".
\begin{thm}\label{qi032}
Let $(\Omega,\mathscr{F},P)$ be a probability space and $\mathbb{H}$
be a separable Hausdorff topological space with $\sigma$-algebra
$\mathscr{H}$. Assume $Y(t,x,\omega)$:
$[0,\infty)\times\mathbb{R}^d\times\Omega\longrightarrow \mathbb{H}$
is
$\mathcal{B}_{\mathbb{R}^+}\otimes\mathcal{B}_{\mathbb{R}^d}\otimes{\mathscr{F}}$
measurable, a.s. continuous w.r.t. $t$, $x$ and satisfies the ``very
crude" stationary property w.r.t. $\hat{\theta}_\cdot$, i.e. for any
$t,r\geq0$, $x\in\mathbb{R}^d$
\begin{eqnarray}\label{zhang100}
\hat{\theta}_{r}\circ Y(t,x,\omega)=Y(t+r,x,\omega)\ \ \rm{a.s.}
\end{eqnarray}
Then there exists a $\hat{Y}(t,x,\omega)$ which is an
indistinguishable version of ${Y}(t,x,\omega)$ s.t.
$\hat{Y}(t,x,\omega)$ is
$\mathcal{B}_{\mathbb{R}^+}\otimes\mathcal{B}_{\mathbb{R}^d}\otimes{\mathscr{F}}$
measurable, continuous w.r.t. $t$, $x$ for all $\omega$ and
satisfies the ``perfect" stationary property w.r.t.
$\hat{\theta}_\cdot$:
\begin{eqnarray}\label{zz63}
\hat{\theta}_{r}\circ \hat{Y}(t,x,\omega)=\hat{Y}(t+r,x,\omega)\ \ \
{\rm for}\ {\rm all}\ t,r\geq0,\ x\in\mathbb{R}^d\ {\rm a.s.}
\end{eqnarray}
\end{thm}
{\em Proof}. From the continuity of $Y(t,x,\omega)$ w.r.t. $t$, $x$
and using a standard argument, we easily see that for any $r\geq0$,
\begin{eqnarray}\label{zhang331}
\hat{\theta}_{r}\circ Y(t,x,\omega)=Y(t+r,x,\omega)\ \ \ {\rm for}\
{\rm all}\ t\geq0,\ x\in\mathbb{R}^d\ {\rm a.s.}
\end{eqnarray}
Define
\begin{eqnarray*}
&&M=\{(r,\omega):\hat{\theta}_{r}\circ Y(t,x,\omega)=Y(t+r,x,\omega)\ {\rm for}\ {\rm all}\ t,x\};\\
&&\tilde{\Omega}=\{\omega:(r,\omega)\in M\ {\rm for}\ {\rm a.e.}\ r\};\\
&&{\Omega^*}=\{\omega:\hat{\theta}_r\omega\in\tilde{\Omega}\ {\rm for}\ {\rm a.e.}\ r\};\\
&&A(r,t,x,\omega)=\hat{\theta}_r\circ Y(t,x,\omega)-Y(t+r,x,\omega).
\end{eqnarray*}
Obviously, $A(r,t,x,\omega)$ is measurable w.r.t.
$\mathcal{B}_{\mathbb{R}^+}\otimes\mathcal{B}_{\mathbb{R}^+}\otimes\mathcal{B}_{\mathbb{R}^d}\otimes{\mathscr{F}}$.
If we denote by $Q$ and $\tilde{Q}$ the normalized Lebesgue measure
on $\mathbb{R}^+$ and $\mathbb{R}^d$ respectively such that
$Q(\mathbb{R}^+)=1$ and $\tilde{Q}(\mathbb{R}^d)=1$, then by
(\ref{zhang331}),
\begin{eqnarray}\label{zhang351}
Q\otimes Q\otimes\tilde{Q}\otimes
P\big(A^{-1}(0)\big)=\int_{\mathbb{R}^+}\int_{\mathbb{R}^+}\int_{\mathbb{R}^d}\int_{\Omega}I_{A^{-1}(0)}(r,t,x,\omega)dPd\tilde{Q}dQdQ=1,
\end{eqnarray}
where $I$ is the indicator function in
$(\mathbb{R}^+\times\mathbb{R}^+\times\mathbb{R}^d\times\Omega,\
\mathcal{B}_{\mathbb{R}^+}\otimes\mathcal{B}_{\mathbb{R}^+}\otimes\mathcal{B}_{\mathbb{R}^d}\otimes\mathscr{F})$.
It is easy to see that
\begin{eqnarray*}
M=\{(r,\omega):\int_{\mathbb{R}^+}\int_{\mathbb{R}^d}I_{A^{-1}(0)}(r,t,x,\omega)d\tilde{Q}dQ=1\}\in\mathcal{B}_{\mathbb{R}^+}\otimes\mathscr{F}.
\end{eqnarray*}
And by (\ref{zhang351}), we have
\begin{eqnarray*}
Q\otimes P(M)=Q\otimes
P\big(\{(r,\omega):\int_{\mathbb{R}^+}\int_{\mathbb{R}^d}I_{A^{-1}(0)}(r,t,x,\omega)d\tilde{Q}dQ=1\}\big)=1.
\end{eqnarray*}
Similarly, we can also know
\begin{eqnarray*}
\tilde{\Omega}=\{\omega:\int_{\mathbb{R}^+}I_M(r,\omega)dQ=1\}\in\mathscr{F}
\end{eqnarray*}
and
\begin{eqnarray*}
P(\tilde{\Omega})=P\big(\{\omega:\int_{\mathbb{R}^+}I_M(r,\omega)dQ=1\}\big)=1.
\end{eqnarray*}
Moreover, the measurability of $\Omega^*$ can be seen easily as
\begin{eqnarray*}
{\Omega^*}=\{\omega:\int_{\mathbb{R}^+}\int_{\mathbb{R}^+}I_M(r,\hat{\theta}_u\omega)dQdQ=1\}\in\mathscr{F}.
\end{eqnarray*}
And since $\tilde{\Omega}$ has full measure,
\begin{eqnarray*}
P({\Omega^*})&\geq&P\big(\{\omega:Y(t+r,x,\hat{\theta}_u\omega)=Y(t,x,\hat{\theta}_r\circ\hat{\theta}_u\omega)\ {\rm for}\ {\rm a.e.}\ r\ {\rm and}\ u,\ {\rm and}\ {\rm all}\ t,\ x\}\bigcap{\tilde{\Omega}}\big)\\
&=&P\big(\{\omega:Y(t+r+u,x,\omega)=Y(t,x,\hat{\theta}_{r+u}\omega)\ {\rm for}\ {\rm a.e.}\ r\ {\rm and}\ u,\ {\rm and}\ {\rm all}\ t,\ x\}\bigcap{\tilde{\Omega}}\big)\\
&=&P\big(\{\omega:Y(t+r',x,\omega)=Y(t,x,\hat{\theta}_{r'}\omega)\ {\rm for}\ {\rm a.e.}\ r',\ {\rm and}\ {\rm all}\ t,\ x\}\bigcap{\tilde{\Omega}}\big)\\
&=&P(\tilde{\Omega})\\
&=&1.
\end{eqnarray*}
One can prove $\hat{\theta}_u\Omega^*\subset\Omega^*$ for any
$u\geq0$. Indeed, for any $\omega\in\hat{\theta}_u\Omega^*$, there
exists $\hat{\omega}\in\Omega^*$ s.t.
$\omega=\hat{\theta}_u\hat{\omega}$ and
$\hat{\theta}_r\hat{\omega}\in\tilde{\Omega}$ for a.e. $r\geq0$. But
$\hat{\theta}_r\omega=\hat{\theta}_{u+r}\hat{\omega}\in\tilde{\Omega}$
for a.e. $r\geq0$, so $\omega\in\Omega^*$. That is to say
$\hat{\theta}_u\Omega^*\subset\Omega^*$. Define
\begin{eqnarray*}
\left\{\begin{array}{l}
\hat{Y}(t,x,\omega)=Y(t-r,x,\hat{\theta}_r\omega),\ \ \ {\rm where}\ r\in[0,t]\ {\rm with}\ \hat{\theta}_r\omega\in\tilde{\Omega},\ {\rm if}\ \omega\in\Omega^*,\nonumber\\
\hat{Y}(t,x,\omega)=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\rm
if}\ \omega\in{\Omega^*}^c.\nonumber
\end{array}\right.
\end{eqnarray*}
An important fact is that if $\omega\in\Omega^*$, then for an
arbitrary $r\in[0,t]$ with $\hat{\theta}_r\omega\in\tilde{\Omega}$,
$Y(t-r,x,\hat{\theta}_r\omega)$ is independent of $r$ and
\begin{eqnarray}\label{qi11}
{Y}(t-r,x,\hat{\theta}_{r}\omega)=Y(t,x,\omega).
\end{eqnarray}
To see this, as $\hat{\theta}_r\omega\in\tilde{\Omega}$, so there
exists $u\geq r$ s.t. $(u,\hat{\theta}_r\omega)\in M$ and
$(u-r,\hat{\theta}_r\omega)\in M$. If not, it means for a.e. $r$
there doesn't exist $u$ satisfying $(u,\hat{\theta}_r\omega)\in M$
and $(u-r,\hat{\theta}_r\omega)\in M$. Then one can easily get the
measure of $\{u:(u,\hat{\theta}_r\omega)\notin M\}$ is positive.
That is a contradiction. So such a $u$ certainly exists and
satisfies
\begin{eqnarray*}
\hat{\theta}_u{Y}(t-r,x,\hat{\theta}_{r}\omega)=Y(t-r+u,x,\hat{\theta}_r\omega)=Y(t,x,\hat{\theta}_{u-r}\hat{\theta}_r\omega)=Y(t,x,\hat{\theta}_u\omega).
\end{eqnarray*}
So
\begin{eqnarray*}
{Y}(t-r,x,\hat{\theta}_{r}\omega)=\hat{\theta}_u^{-1}Y(t,x,\hat{\theta}_u\omega)=Y(t,x,\omega).
\end{eqnarray*}
Therefore (\ref{qi11}) is true and $\hat{Y}(t,x,\omega)$ doesn't
depend on the choice of $r$. That is to say $\hat{Y}(t,x,\omega)$ is
well defined. Moreover (\ref{qi11}) implies that
${Y}(t,x,\omega)=\hat{Y}(t,x,\omega)$ for all $t\geq0$,
$x\in\mathbb{R}^d$ on a full measure set $\Omega^*$, thus
${Y}(t,x,\omega)$ and $\hat{Y}(t,x,\omega)$ are indistinguishable.
Define
\begin{eqnarray*}
\left\{\begin{array}{l}
B(r,t,x,\omega)=Y(t-r,x,\hat{\theta}_r\omega),\ \ \ \ \ \ {\rm if}\ r\in[0,t],\ \ \hat{\theta}_r\omega\in\tilde{\Omega},\ \ {\rm and}\ \omega\in\Omega^*,\nonumber\\
B(r,t,x,\omega)=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\rm
otherwise}.\nonumber
\end{array}\right.
\end{eqnarray*}
Then $B(r,t,x,\omega)$ is
$\mathcal{B}_{\mathbb{R}^+}\otimes\mathcal{B}_{\mathbb{R}^+}\otimes\mathcal{B}_{\mathbb{R}^d}\otimes\mathscr{F}$
measurable. By the definition of $\Omega^*$, if $\omega\in\Omega^*$,
then for a.e. $0\leq r\leq t$,
$\hat{\theta}_r\omega\in\tilde{\Omega}$. We denote $L(r)$ the
Lebesgue measure in $[0,t]$. Since the countable base of $H$
generates $\mathscr{H}$ and separates points, $(H,\mathscr{H})$ is
isomorphic as a measurable space to a subset of $[0,1]$.
Consequently, for all $t,x,\omega$,
\begin{eqnarray*}
\hat{Y}(t,x,\omega)=\int_{0}^{t}B(r,t,x,\omega)dL(r).
\end{eqnarray*}
So by Fubini's theorem, $\hat{Y}(t,x,\omega)$ is
$\mathcal{B}_{\mathbb{R}^+}\otimes\mathcal{B}_{\mathbb{R}^d}\otimes\mathscr{F}$
measurable. $\hat{Y}(t,x,\omega)$ is a.s continuous w.r.t. $t$, $x$
due to the a.s continuity of ${Y}(t-r,x,\omega)$. But there exists a
null measure set $N\in\mathscr{F}$ s.t. $\{\omega:\
\hat{Y}(t,x,\omega)\ {\rm is}\ {\rm not}\ {\rm continuous}\ {\rm
w.r.t.}\ t,x\}\subset N$. Let $\hat{Y}(t,x,\omega)$ on $N$ equal
$0$. We still denote this new version of $\hat{Y}(t,x,\omega)$ by
$\hat{Y}(t,x,\omega)$, then this version of $\hat{Y}(t,x,\omega)$ is
continuous for all $\omega$.
The remaining work is to check $\hat{Y}(t,x,\omega)$ satisfies the
``perfect" stationary property (\ref{zz63}). For $\omega\in\Omega^*$
and all $r\geq0$,
$\hat{\theta}_r\omega\in\hat{\theta}_r\Omega^*\subset\Omega^*$. Pick
a $u$ s.t. $\hat{\theta}_u\omega\in\tilde{\Omega}$,
$\hat{\theta}_{u+r}\omega\in\tilde{\Omega}$, then by (\ref{qi11}) we
have
\begin{eqnarray*}
\hat{Y}(t,x,\hat{\theta}_{r}\omega)&=&Y(t-u,x,\hat{\theta}_{u+r}\omega)=Y(t+r-u-r,x,\hat{\theta}_{u+r}\omega)\\
&=&Y(t+r,x,\omega)={Y}(t+r-u,x,\hat{\theta}_u\omega)=\hat{Y}(t+r,x,\omega).
\end{eqnarray*}
The theorem is proved.$\hfill\diamond$
From now on, we neglect the difference between two distinguishable
random processes. Then with Proposition \ref{qi031} and Theorem
\ref{qi032}, it follows immediately that
\begin{thm}\label{zz42}
If BDSDE (\ref{zz36}) has a unique solution $(Y^{t,x}_{\cdot},
Z^{t,x}_{\cdot})\in S^{p,-K}([0, \infty);\mathbb{R}^1)\cap
M^{2,-K}([0,\infty);\mathbb{R}^1)\\\times M^{2,-K}([0, \infty);
\mathbb{R}^{d})$ and $(t,x)\to Y_t^{t,x}$ is a.s. continuous, then
under Condition (A.4), $Y_t^{t,x}$ satisfies the ``perfect"
stationary property w.r.t. $\hat{\theta}_\cdot$, i.e.
\begin{eqnarray*}
\hat{\theta}_r\circ Y_t^{t,x}=Y_{t+r}^{t+r,x}\ \ {\rm for}\ {\rm
all}\ r,\ t\geq0,\ x\in\mathbb{R}^d\ {\rm a.s.}
\end{eqnarray*}
\end{thm}
\section{Infinite Horizon BDSDEs}\label{s24}
\setcounter{equation}{0}
\ \ \ \ In this section we first prove the assumption in Theorem
\ref{zz42} that BDSDE (\ref{zz36}) has a unique solution
$(Y_{\cdot}, Z_{\cdot})\in S^{p,-K}\bigcap
M^{2,-K}([0,\infty);\mathbb{R}^1)\times M^{2,-K}([0, \infty);
\mathbb{R}^{d})$ is obtainable and reasonable under Conditions
(A.1)-(A.4).
To begin with, we briefly introduce the pioneering work by Pardoux
and Peng in \cite{pa-pe3} for the following finite horizon BDSDE:
\begin{eqnarray}\label{zz27}
Y_s=Y_T+\int_{s}^{T}f(r,Y_r,Z_r)dr-\int_{s}^{T}\langle
g(r,Y_r,Z_r),d^\dagger\hat{B}_r\rangle-\int_{s}^{T}\langle
Z_r,dW_r\rangle.
\end{eqnarray}
Here we only consider $\mathbb{R}^1$-valued BDSDE for our purpose.
One can also refer to \cite{pa-pe3} for multi-dimensional BDSDE if
interested. Assume
\begin{description}
\item[(A.1)$'$.] Functions $f: \Omega\times[0,T]\times \mathbb{R}^1\times \mathbb{R}^{d}{\longrightarrow{\mathbb{R}^1}}$ and $g: \Omega\times[0,T]\times \mathbb{R}^1\times \mathbb{R}^{d}{\longrightarrow{\mathbb{R}^{l}}}$ are ${\mathscr{F}}\otimes\mathscr{B}_{[0,T]}\otimes\mathscr{B}_{\mathbb{R}^{1}}\otimes\mathscr{B}_{\mathbb{R}^{d}}$
measurable, and for any $(y, z)\in\mathbb{R}^1\times\mathbb{R}^{d}$,
$f(\cdot,y,z)\in M^{2,0}([0,T];\mathbb{R}^{1})$ and $g(\cdot,y,z)\in
M^{2,0}([0,T];\mathbb{R}^{d})$, moreover there exist constants
$C\geq0$ and $0\leq\alpha<1$ s.t. for any $r\in[0,T]$, $(y_1, z_1)$,
$(y_2, z_2)\in\mathbb{R}^1\times \mathbb{R}^{d}$,
\begin{eqnarray*}
&&|f(r, y_1, z_1)-f(r, y_2, z_2)|^2\leq C|y_1-y_2|^2+C|z_1-z_2|^2,\nonumber\\
&&|g(r, y_1, z_1)-g(r, y_2, z_2)|^2\leq
C|y_1-y_2|^2+\alpha|z_1-z_2|^2.
\end{eqnarray*}
\end{description}
\begin{thm}\label{theorem2.1}
{\rm(\cite{pa-pe3})} Under Condition {\rm(A.1)$'$}, for any given
${\cal F}_T\bigvee{\mathscr{F}_{T,\infty}^{\hat{B}}}$ measurable
$Y_T\in L^2(\Omega)$, BDSDE (\ref{zz27}) has a unique solution
\begin{center}
$(Y_{\cdot}, Z_{\cdot})\in S^{2,0}([0,T];\mathbb{R}^1)\bigotimes
M^{2,0}([0,T]; \mathbb{R}^{d})$.
\end{center}
\end{thm}
In \cite{pa-pe3}, Pardoux and Peng also discussed a type of forward
BDSDE, a special case of BDSDE (\ref{zz27}),
\begin{eqnarray}\label{zz28}
Y_s^{t,x}&=&h(X_T^{t,x})+\int_s^Tf(X_r^{t,x},Y_r^{t,x},Z_r^{t,x})dr\nonumber\\
&&-\int_s^T\langle
g(X_r^{t,x},Y_r^{t,x},Z_r^{t,x}),d^\dagger\hat{B}_r\rangle-\int_s^T\langle
Z_r^{t,x},dW_r\rangle,
\end{eqnarray}
where $(X_s^{t,x})_{t\leq s\leq T}$ is the solution of SDE
(\ref{qi17}). Assume
\begin{description}
\item[(A.2)$'$.] Functions $f: \mathbb{R}^{d}\times\mathbb{R}^1\times\mathbb{R}^{d}{\longrightarrow{\mathbb{R}^1}}$ and $g: \mathbb{R}^{d}\times\mathbb{R}^1\times\mathbb{R}^{d}{\longrightarrow{\mathbb{R}^{l}}}$ are $\mathscr{B}_{\mathbb{R}^{d}}\otimes\mathscr{B}_{\mathbb{R}^{1}}\otimes\mathscr{B}_{\mathbb{R}^{d}}$
measurable, and there exist constants $C\geq0$ and $0\leq\alpha<1$
s.t. for any $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)\in
\mathbb{R}^d\times \mathbb{R}^1\times \mathbb{R}^{d}$,
\begin{eqnarray*}
&&|f(x_1, y_1, z_1)-f(x_2, y_2, z_2)|^2\leq C|x_1-x_2|^2+C|y_1-y_2|^2+C|z_1-z_2|^2,\nonumber\\
&&|g(x_1, y_1, z_1)-g(x_2, y_2, z_2)|^2\leq
C|x_1-x_2|^2+C|y_1-y_2|^2+\alpha|z_1-z_2|^2.
\end{eqnarray*}
\end{description}
For BDSDE (\ref{zz28}), it is not difficult to deduce from Theorem
\ref{theorem2.1} that
\begin{thm}\label{zz38} Under Condition {\rm(A.2)$'$}, for each $x\in\mathbb{R}^{d}$ and any
given ${\cal F}_T\bigvee{\mathscr{F}_{T,\infty}^{\hat{B}}}$
measurable $h$ satisfying $h(X_T^{t,x})\in L^2(\Omega)$, BDSDE
(\ref{zz28}) has a unique solution
\begin{center}
$(Y^{t,x}_{\cdot}, Z^{t,x}_{\cdot})\in
S^{2,0}([t,T];\mathbb{R}^1)\bigotimes M^{2,0}([t,T];
\mathbb{R}^{d})$.
\end{center}
\end{thm}
In \cite{pa-pe3}, for the first time, Pardoux and Peng associated
the classical solution of SPDE, if any, with the solution of BDSDE
(\ref{zz28}). They proved that under some strong smoothness
conditions of $h$, $b$, $\sigma$, $f$ and $g$ (for details see
\cite{pa-pe3}), $u(t,x)=Y_t^{t,x}$, where $Y$ is the unique solution
of BDSDE (\ref{zz28}), $(t,x)\in[0,T]\times\mathbb{R}^{d}$, is
independent of ${{\cal F}_T^W}$ and is the unique classical solution
of the following backward SPDE
\begin{eqnarray*}\label{zz29}
u(t,x)&=&h(x)+\int_{t}^{T}[\mathscr{L}u(s,x)+f\big(x,u(s,x),\sigma^*(x)Du(s,x)\big)]ds\nonumber\\
&&-\int_{t}^{T}\langle
g\big(x,u(s,x),\sigma^*(x)Du(s,x)\big),d^\dagger\hat{B}_s\rangle,\ \
\ \ 0\leq t\leq T.
\end{eqnarray*}
Now let's turn to the existence and uniqueness of solution to the
following infinite horizon BDSDE:
\begin{eqnarray}\label{zz30}
{\rm e}^{-{K'\over2}t}Y_t&=&\int_{t}^{\infty}{\rm e}^{-{K'\over2}s}f(s,Y_s,Z_s)ds+\int_{t}^{\infty}{K'\over2}{\rm e}^{-{K'\over2}s}Y_sds\nonumber\\
&&-\int_{t}^{\infty}{\rm e}^{-{K'\over2}s}\langle
g(s,Y_s,Z_s),d^\dagger\hat{B}_s\rangle-\int_{t}^{\infty}{\rm
e}^{-{K'\over2}s}\langle Z_s,dW_s\rangle,
\end{eqnarray}
or equivalently, for arbitrary $T>0$ and $0\leq t\leq T$,
\begin{eqnarray}
\left\{\begin{array}{l}\label{zz31}
dY_t=-f(t,Y_t,Z_t)dt+\langle g(t,Y_t,Z_t),d^\dagger\hat{B}_t\rangle+\langle Z_t,dW_t\rangle,\nonumber\\
\lim_{T\rightarrow\infty}{\rm e}^{-{K'\over2}T}Y_T=0\ \ \
\rm{a.s.}\nonumber
\end{array}\right.
\end{eqnarray}
We assume that
\begin{description}
\item[(H.1).] Functions $f:\Omega\times[0,\infty)\times\mathbb{R}^1\times\mathbb{R}^{d}{\longrightarrow{\mathbb{R}^1}}$ and $g:\Omega\times[0,\infty)\times\mathbb{R}^1\times\mathbb{R}^{d}{\longrightarrow{\mathbb{R}^{l}}}$ are $\mathscr{F}\otimes\mathscr{B}_{[0,\infty)}\otimes\mathscr{B}_{\mathbb{R}^{1}}\otimes\mathscr{B}_{\mathbb{R}^{d}}$
measurable, and there exist constants $C_1$, $C\geq0$ and
$0\leq\alpha<{1\over2}$ s.t. for any
$(\omega,t)\in\Omega\times[0,\infty)$, $(y_1,z_1)$, $(y_2,z_2)\in
\mathbb{R}^1\times \mathbb{R}^{d}$,
\begin{eqnarray*}
&&|f(t,y_1,z_1)-f(t,y_2,z_2)|^2\leq C_1|y_1-y_2|^2+C |z_1-z_2|^2,\nonumber\\
&&|g(t,y_1,z_1)-g(t,y_2,z_2)|^2\leq C|y_1-y_2|^2+\alpha|z_1-z_2|^2;
\end{eqnarray*}
\item[(H.2).] There exist
constants $K\in\mathbb{R}^+$, $p>d+2$, $K<K'<2K$ and $\mu>0$ with
$2\mu-K'-{p(p+1)\over2}C>0$ s.t. for any
$(\omega,t)\in\Omega\times[0,\infty)$, $y_1$, $y_2\in \mathbb{R}^1$,
$z\in \mathbb{R}^{d}$,
$$(y_1-y_2)(f(t,y_1,z)-f(t,y_2,z))\leq -\mu |y_1-y_2|^2;$$
\item[(H.3).] For $p$, $K$ in (H.2), $f(\cdot, 0, 0)\in
M^{p,-K}([0,\infty); \mathbb{R}^1 )$, $g(\cdot,0,0)\in
M^{p,-K}([0,\infty);\mathbb{R}^{l})$.
\end{description}
\begin{thm}\label{theorem2.3}
Under Conditions {\rm(H.1)}--{\rm(H.3)}, BDSDE (\ref{zz30}) has a
unique solution
\begin{center}
$(Y_{\cdot}, Z_{\cdot})\in S^{p,-K}\bigcap
M^{2,-K}([0,\infty);\mathbb{R}^1)\bigotimes M^{2,-K}([0,\infty);
\mathbb{R}^{d})$,
\end{center}
where the norm in $S^{p,-K}([0,\infty);\mathbb{R}^1)\cap
M^{2,-K}([0,\infty);\mathbb{R}^1)\bigotimes M^{2,-K}([0,\infty);
\mathbb{R}^{d})$ is defined as
\begin{eqnarray*}
\big((E[\sup_{t\geq0}{\rm e}^{-Kt}|\cdot|^p])^{2\over
p}+E[\int_{0}^{\infty}{\rm
e}^{-Kr}|\cdot|^2dr]+E[\int_{0}^{\infty}{\rm
e}^{-Kr}|\cdot|^2dr]\big)^{1\over2},
\end{eqnarray*}
as in Pardoux \cite{pa}.
\end{thm}
{\em Proof}. \underline{Uniqueness}. Let $(Y_{t}^{1},Z_{t}^{1})$ and
$(Y_{t}^{2},Z_{t}^{2})$ be two solutions of BDSDE (4.1). Define
\begin{eqnarray*}
\bar{Y}_t=Y_{t}^{1}-Y_{t}^{2},\ \ \bar{Z}_t=Z_{t}^{1}-Z_{t}^{2},\ \
\ \ \ \ t\geq 0.
\end{eqnarray*}
Applying It$\hat {\rm o}$'s formula to ${\rm
e}^{-Ks}{{|\bar{Y}_s|}^2}$, we have
\begin{eqnarray}\label{zz19}
&&E[{\rm e}^{-Kt}{{|\bar{Y}_t|}^2}]+E[\int_{t}^{T}({1\over
2}-\alpha){\rm e}^{-Ks}|\bar{Z}_s|^2ds]+E[\int_{t}^{T}(2\mu-K-3C){\rm e}^{-Ks}{{|\bar{Y}_s|}^2}ds]\nonumber \\
&\leq&E[{\rm e}^{-KT}{{|\bar{Y}_T|}^2}].
\end{eqnarray}
Taking $K'$ as in Condition (H.2) and noting $2\mu-K'-3C>0$ as well,
we can see that (\ref{zz19}) remains true when $K$ replaced by $K'$.
Therefore, we have
\begin{eqnarray}\label{qi34}
E[{\rm e}^{-K't}{|\bar{Y}_t|}^2]\leq {\rm e}^{-(K'-K)T}E[{\rm
e}^{-KT}{|\bar{Y}_T|}^2].
\end{eqnarray}
Since
$\sup_{T\geq0}E[{\rm e}^{-KT}{|\bar{Y}_T|}^2]
<\infty$,
taking the limit as $T\to \infty$ in (\ref{qi34}), we have
\begin{eqnarray*}
E[{\rm e}^{-K't}{|\bar{Y}_t|}^2]=0.
\end{eqnarray*}
Then the uniqueness is proved.\\
\underline{Existence}. For each $n\in\mathbb{N}$, we define a
sequence of BDSDEs as follows
\begin{eqnarray}\label{zhao100}
Y_{t}^{n}=\int_{t}^{n}f(s, Y_{s}^{n},
Z_{s}^{n})ds-\int_{t}^{n}\langle g(s,Y_{s}^{n},
Z_{s}^{n}),d^\dagger\hat{B}_s\rangle-\int_{t}^{n}\langle
Z_{s}^{n},dW_s\rangle.
\end{eqnarray}
Let $(Y_{t}^{n}, Z_{t}^{n})_{t\geq n}=(0, 0)$, and according to
Theorem \ref{theorem2.1}, BDSDE (\ref{zhao100}) has a unique
solution $(Y_{\cdot}^{n}, Z_{\cdot}^{n})\in S^{2,-K}\bigcap
M^{2,-K}([0,\infty);\mathbb{R}^1)\bigotimes M^{2,-K}([0,\infty);
\mathbb{R}^{n})$. Also under Conditions (H.1)--(H.3), we can prove
$Y_{\cdot}^{n}\in S^{p,-K}([0,\infty);\mathbb{R}^1)$ in the
following lemma.
\begin{lem}\label{lemma2.4}
Let $(Y_{t}^{n})_{t\geq0}$ be the solution of BDSDE (\ref{zhao100}),
then under Conditions {\rm(H.1)}--{\rm(H.3)}, $Y_{\cdot}^{n}\in
S^{p,-K}([0, \infty);\mathbb{R}^1)$.
\end{lem}
{\em Proof}. Let
\begin{eqnarray*}
&&\psi_M(x)=x^2I_{\{-M\leq x<M\}}+2M(x-M)I_{\{x\geq
M\}}-2M(x+M)I_{\{x<-M\}}\\
&&\varphi_{N,p}(x)=x^{p\over2}I_{\{0\leq
x<N\}}+{p\over2}N^{{p-2}\over2}(x-N)I_{\{x\geq N\}}.
\end{eqnarray*} Applying
generalized It$\hat {\rm o}$'s formula (c.f. Elworthy, Truman and
Zhao \cite{el-tr-zh}) to ${\rm
e}^{-Kr}\varphi_{N,p}\big(\psi_M(Y_r^n)\big)$ to have the following
estimation
\begin{eqnarray}\label{zhang690}
&&{\rm e}^{-Ks}\varphi_{N,p}\big(\psi_M(Y_{s}^{n})\big)-{K}\int_{s}^{n}{\rm e}^{-Kr}\varphi_{N,p}\big(\psi_M(Y_r^n)\big)dr\nonumber\\
&&+{1\over2}\int_{s}^{n}{\rm e}^{-Kr}\varphi^{''}_{N,p}\big(\psi_M(Y_r^n)\big)|\psi_M^{'}(Y_r^n)|^2|Z_r^n|^2dr\nonumber\\
&&+\int_{s}^{n}{\rm e}^{-Kr}\varphi^{'}_{N,p}\big(\psi_M(Y_r^n)\big)I_{\{-M\leq{Y}_r^n<M\}}|{Z}_r^n|^2dr\nonumber\\
&\leq&\int_{s}^{n}{\rm e}^{-Kr}\varphi^{'}_{N,p}\big(\psi_M(Y_r^n)\big)\psi_M^{'}(Y_r^n){f}(r,Y_r^n,Z_r^n)dr\nonumber\\
&&+\int_{s}^{n}{\rm e}^{-Kr}\varphi^{'}_{N,p}\big(\psi_M(Y_r^n)\big)I_{\{-M\leq{Y}_r^n<M\}}|g(r,Y_r^n,Z_r^n)|^2dr\nonumber\\
&&+{1\over2}\int_{s}^{n}{\rm e}^{-Kr}\varphi^{''}_{N,p}\big(\psi_M(Y_r^n)\big)|\psi_M^{'}(Y_r^n)|^2|g(r,Y_r^n,Z_r^n)|^2dr\nonumber\\
&&-\int_{s}^{n}\langle{\rm
e}^{-Kr}\varphi^{'}_{N,p}\big(\psi_M(Y_r^n)\big)\psi_M^{'}(Y_r^n)g(r,Y_r^n,
Z_r^n),d^\dagger\hat{B}_r\rangle\nonumber\\
&&-\int_{s}^{n}\langle{\rm
e}^{-Kr}\varphi^{'}_{N,p}\big(\psi_M(Y_r^n)\big)\psi_M^{'}(Y_r^n){Z}_r^n,dW_r\rangle.
\end{eqnarray}
As $(Y_{\cdot}^{t,\cdot},Z_{\cdot}^{t,\cdot})\in S^{2,-K}\bigcap
M^{2,-K}([0,\infty);{\mathbb{R}^{1}})\bigotimes
M^{2,-K}([0,\infty);{\mathbb{R}^{d}})$ and
$\varphi^{'}_{N,p}\big(\psi_M(Y_r^n)\big)\psi_M^{'}(Y_r^n)$ is
bounded, taking the expectation on both sides, we know that all the
stochastic integrals have zero expectation. Using Conditions
(H.1)-(H.3) and taking first the limit as $M\to \infty$, then the
limit as $N\to \infty$, by the monotone convergence theorem, we have
\begin{eqnarray}\label{zhang2}
&&\big(p\mu-{K}-{{p(p+1)}\over2}C-(3+{{p(p-1)}\over2}C)\varepsilon\big)E[\int_{s}^{\infty}{\rm e}^{-Kr}{|{Y}_r^n|}^pr]\nonumber\\
&&+{p\over4}\big(2p-3-(2p-2)\alpha-(2p-2)\alpha\varepsilon\big)E[\int_{s}^{\infty}{\rm e}^{-Kr}{{|{Y}_r^n|}^{p-2}}|{Z}_r^n|^2dr]\nonumber\\
&\leq&C_pE[\int_{0}^{\infty}{\rm
e}^{-Kr}|f(r,0,0)|^pdr]+C_pE[\int_{0}^{\infty}{\rm
e}^{-Kr}|g(r,0,0)|^pdr]<\infty.
\end{eqnarray}
Note that here and in the following the constant $\varepsilon$ can
be chosen to be sufficiently small and $C_p$ is a generic constant.
Due to Conditions (H.1), (H.2) and the arbitrariness of
$\varepsilon$, all the terms on the left hand side of (\ref{zhang2})
are positive. Furthermore, by the B-D-G inequality, Cauchy-Schwartz
inequality and Young inequality, from (\ref{zhang690}) we have
\begin{eqnarray}\label{zhang692}
&&E[\sup_{s\geq0}{\rm e}^{-Ks}{{|{Y}_s^{n}|}^p}]\nonumber\\
&\leq&C_pE[\int_{0}^{\infty}{\rm
e}^{-Kr}{{|{Y}_r^n|}^{p-2}}|{Z}_r^n|^2dr]+C_pE[\int_{0}^{\infty}{\rm
e}^{-Kr}|Y_r^n|^pdr]\nonumber\\
&&+C_pE[\sqrt{\int_{s}^{\infty}\big({\rm
e}^{-Kr}\varphi^{'}_{N,p}\big(\psi_M(Y_r^n)\big)|\psi_M^{'}(Y_r^n)|^2\big)\big({\rm
e}^{-Kr}\varphi^{'}_{N,p}\big(\psi_M(Y_r^n)\big)|g(r,Y_r^n,
Z_r^n)|^2\big)dr}]\nonumber\\
&&+C_pE[\sqrt{\int_{s}^{\infty}\big({\rm
e}^{-Kr}\varphi^{'}_{N,p}\big(\psi_M(Y_r^n)\big)|\psi_M^{'}(Y_r^n)|^2\big)\big({\rm
e}^{-Kr}\varphi^{'}_{N,p}\big(\psi_M(Y_r^n)\big)|{Z}_r^n|^2\big)dr}]\nonumber\\
&\leq&C_pE[\int_{0}^{\infty}{\rm
e}^{-Kr}{{|{Y}_r^n|}^{p-2}}|{Z}_r^n|^2dr]+C_pE[\int_{0}^{\infty}{\rm
e}^{-Kr}|Y_r^n|^pdr]\nonumber\\
&&+\varepsilon E[\sup_{s\geq0}\big({\rm
e}^{-Ks}\varphi^{'}_{N,p}\big(\psi_M(Y_s^n)\big)|\psi_M^{'}(Y_s^n)|^2\big)]+C_pE[\int_{0}^{\infty}{\rm
e}^{-Kr}\varphi^{'}_{N,p}\big(\psi_M(Y_r^n)\big)|g(r,Y_r^n,
Z_r^n)|^2dr]\nonumber\\
&&+C_pE[\int_{0}^{\infty}{\rm
e}^{-Kr}\varphi^{'}_{N,p}\big(\psi_M(Y_r^n)\big)|Z_r^n|^2dr].
\end{eqnarray}
Taking the limits as $M$, $N\rightarrow\infty$ and applying the
monotone convergence theorem, we have
\begin{eqnarray}\label{zhang4}
E[\sup_{s\geq0}{\rm
e}^{-Kt}|Y_{s}^{n}|^p]&\leq&C_pE[\int_{0}^{\infty}{\rm
e}^{-Kr}|Y_r^n|^{p-2}|Z_r^n|^2dr]+C_pE[\int_{0}^{\infty}{\rm
e}^{-Kr}|Y_r^n|^pdr].
\end{eqnarray}
By (\ref{zhang2}), $Y_{\cdot}^{n}\in
S^{p,-K}([0,\infty);\mathbb{R}^1)$. Lemma \ref{lemma2.4} is proved.
$\hfill\diamond$
\\
\begin{rmk}\label{remark2.5}
The proof of Lemma \ref{lemma2.4} also works with $p$ replaced by
$2$. Note that if $f(\cdot,0,0)\in
M^{p,-K}([0,\infty);\mathbb{R}^1)$, then by H$\ddot{\textrm{o}}$lder
inequality, it turns out that
$f(\cdot,0,0)\in M^{2,-K}([0,\infty);\mathbb{R}^1)$ and
$g(\cdot,0,0)\in M^{2,-K}([0,\infty);\mathbb{R}^l)$. So it is easy
to see in (\ref{zhang2}) with $p$ replaced by $2$ that
\begin{center}
${(Y_{\cdot}^{n},Z_{\cdot}^{n})}\in
M^{2,-K}([0,\infty);\mathbb{R}^1)\bigotimes
M^{2,-K}([0,\infty);\mathbb{R}^{d})$.
\end{center}
For the rest of our paper, we will leave out the similar
localization argument as in the proof of Lemma \ref{lemma2.4} when
applying It$\hat {\rm o}$'s formula to save the space of this paper.
\end{rmk}
Then back to the proof of Theorem \ref{theorem2.3}. We will show
that ${(Y_{\cdot}^{n}, Z_{\cdot}^{n})}$ is a Cauchy sequence in the
space of $S^{p,-K}([0,\infty);\mathbb{R}^1)\cap
M^{2,-K}([0,\infty);\mathbb{R}^1)\bigotimes M^{2,-K}([0,\infty);
\mathbb{R}^{d})$. First we show that, for $m,n\in\mathbb{N}$ and
$m\geq n$,
\begin{eqnarray*}
\lim_{n,m\rightarrow\infty}E[\sup_{t\geq0}{\rm
e}^{-Kt}|Y_{t}^{m}-Y_{t}^{n}|^p]=0.
\end{eqnarray*}
Define $\bar{Y}_{t}^{m,n}={Y}_{t}^{m}-{Y}_{t}^{n}$,
$\bar{Z}_{t}^{m,n}={Z}_{t}^{m}-{Z}_{t}^{n}$.
($\textrm{i}$) When $n\leq t\leq m$,
\begin{eqnarray*}
\bar{Y}_{t}^{m,n}={Y}_{t}^{m}=\int_{t}^{m}f(s,Y_{s}^{m},Z_{s}^{m})ds-\int_{t}^{m}\langle
g(s,Y_{s}^{m},Z_{s}^{m}),d^\dagger\hat{B}_s\rangle-\int_{t}^{m}\langle
Z_{s}^{m},dW_s\rangle.
\end{eqnarray*}
Some similar calculations as in (\ref{zhang2}) and (\ref{zhang4})
lead to
\begin{eqnarray}\label{zhang6}
E[\sup_{n\leq t\leq m}{\rm e}^{-Kt}|Y_{t}^{m}|^p]&\leq&C_pE[\int_{n}^{m}{\rm e}^{-Kr}|Y_{r}^{m}|^{p-2}|Z_{r}^{m}|^2dr]+C_pE[\int_{n}^{m}{\rm e}^{-Kr}|Y_{r}^{m}|^pdr]\\
&&+C_pE[\int_{n}^{m}{\rm e}^{-Kr}(|f(r,0,0)|^p+|g(r,0,0)|^p)dr]
\longrightarrow0,\ {\rm as}\ n,\ m\longrightarrow\infty.\nonumber
\end{eqnarray}
$(\textrm{i}$$\textrm{i})$ When $0\leq t\leq n$,
\begin{eqnarray*}
\bar{Y}_{t}^{m,n}&=&{Y}_{n}^{m}+\int_{t}^{n}f(r,Y_{r}^{m},Z_{r}^{m})-f(r,Y_r^n,Z_r^n)\\
&&-\int_{t}^{n}\langle
g(r,Y_{r}^{m},Z_{r}^{m})-g(r,Y_r^n,Z_r^n),d^\dagger\hat{B}_r\rangle-\int_{t}^{n}\langle\bar{Z}_{r}^{m,n},dW_r\rangle.
\end{eqnarray*}
Applying It$\hat {\rm o}$'s formula to ${\rm
e}^{-Kr}|\bar{Y}_{r}^{m,n}|^p$ and following a similar calculation
as in (\ref{zhang690}) and (\ref{zhang2}), we have for $s\leq n$,
\begin{eqnarray}\label{zhang8}
E[\int_{0}^{n}{\rm
e}^{-Kr}|\bar{Y}_{r}^{m,n}|^{p-2}|\bar{Z}_{r}^{m,n}|^2dr]+E[\int_{0}^{n}{\rm
e}^{-Kr}|\bar{Y}_{r}^{m,n}|^pdr]\leq C_pE[{\rm
e}^{-Kn}|Y_{n}^{m}|^p].
\end{eqnarray}
From $(\textrm{i})$, the right hand side of the above inequality
converges to $0$ as $n$, $m\longrightarrow\infty$.
By some similar calculations as in (\ref{zhang4}), we have
\begin{eqnarray*}
E[\sup_{0\leq t\leq n}{\rm e}^{-Kt}|\bar{Y}_{t}^{m,n}|^p]\leq
C_pE[{\rm e}^{-Kn}|Y_{n}^{m}|^p]\longrightarrow0\ \ \ {\rm as} \
n,m\longrightarrow\infty.
\end{eqnarray*}
From $(\textrm{i})$ $(\textrm{i}$$\textrm{i})$, we have for
$m,n\in\mathbb{N}$,
\begin{eqnarray*}
\lim_{n,m\rightarrow\infty}E[\sup_{t\geq0}{\rm
e}^{-Kt}|Y_{t}^{m}-Y_{t}^{n}|^p]=0.
\end{eqnarray*}
It is easy to see that the above arguments also hold for $p=2$ in
(\ref{zhang6}) and (\ref{zhang8}). Noting Remark \ref{remark2.5}, we
have as $n$, $m\longrightarrow\infty$
\begin{eqnarray*}
E[\int_{0}^{\infty}{\rm
e}^{-Kr}|\bar{Y}_{r}^{m,n}|^2dr]+E[\int_{0}^{\infty}{\rm
e}^{-Kr}|\bar{Z}_{r}^{m,n}|^2dr]\longrightarrow0.
\end{eqnarray*}
Therefore, ${(Y_{\cdot}^{n}, Z_{\cdot}^{n})}$ is a Cauchy sequence in
the Banach space $S^{p,-K}([0,\infty);\mathbb{R}^1)\cap
M^{2,-K}([0,\infty);\mathbb{R}^1)\bigotimes\\
M^{2,-K}([0,\infty);\mathbb{R}^{d})$.
We take $(Y_t, Z_t)_{t\geq0}$ as the limit of
$(Y_{t}^{n},Z_{t}^{n})_{t\geq0}$ in
$S^{p,-K}([0,\infty);\mathbb{R}^1)\cap
M^{2,-K}([0,\infty);\mathbb{R}^1)\bigotimes
M^{2,-K}\\([0,\infty);\mathbb{R}^{d})$ and then show that $(Y_t,
Z_t)_{t\geq0}$ is the solution of BDSDE (\ref{zz30}). First note that
for $t\leq n$, (\ref{zhao100}) is equivalent to
\begin{eqnarray}\label{zhang9}
{\rm e}^{-{K'\over2}t}Y_{t}^{n}&=&\int_{t}^{n}{\rm e}^{-{K'\over2}s}f(s,Y_{s}^{n},Z_{s}^{n})ds+\int_{t}^{n}{K'\over2}{\rm e}^{-{K'\over2}s}Y_{s}^{n}ds\nonumber\\
&&-\int_{t}^{n}{\rm e}^{-{K'\over2}s}\langle
g(s,Y_{s}^{n},Z_{s}^{n}),d^\dagger\hat{B}_s\rangle-\int_{t}^{n}{\rm
e}^{-{K'\over2}s}\langle Z_{s}^{n},dW_s\rangle.
\end{eqnarray}
Actually BDSDE (\ref{zhang9}) converges to BDSDE (\ref{zz30}) in
$L^2(\Omega)$ as $n\longrightarrow\infty$. To see this, we verify
the convergence term by term. For the first term,
\begin{eqnarray*}
&&E[\ |{\rm e}^{-{K'\over2}t}Y_{t}^{n}-{\rm
e}^{-{K'\over2}t}Y_t|^2]\leq E[\sup_{t\geq0}{\rm
e}^{-Kt}|Y_{t}^{n}-Y_t|^2]\longrightarrow0.
\end{eqnarray*}
For the second term, by H$\ddot{\textrm{o}}$lder inequality,
\begin{eqnarray*}
&&E[\ |\int_{t}^{n}{\rm e}^{-{K'\over2}s}f(s,Y_{s}^{n},Z_{s}^{n})ds-\int_{t}^{\infty}{\rm e}^{-{K'\over2}s}f(s,Y_s,Z_s)ds|^2]\\
&\leq&2E[\int_{t}^{n}{\rm e}^{-(K'-K)s}ds\int_{t}^{n}{\rm e}^{-Ks}|f(s,Y_{s}^{n},Z_{s}^{n})-f(s,Y_s,Z_s)|^2ds]\\
&&+2E[\int_{n}^{\infty}{\rm e}^{-(K'-K)s}ds\int_{n}^{\infty}{\rm
e}^{-Ks}|f(s,Y_s,Z_s)|^2ds]\longrightarrow0.
\end{eqnarray*}
We can deal with the third term similarly as above and deal with two
stochastic integration terms by It$\hat {\rm o}$'s isometry.
Thus ${(Y_t, Z_t)}_{t\geq 0}$ is the solution of BDSDE (\ref{zz30})
and the proof of Theorem \ref{theorem2.3} is completed.
$\hfill\diamond$\\
Then we consider the existence and uniqueness of solution to the
following infinite horizon forward BDSDE:
\begin{eqnarray}\label{zz32}
{\rm e}^{-{K'\over2}s}Y_s^{t,x}&=&\int_{s}^{\infty}{\rm e}^{-{K'\over2}r}f(X_r^{t,x},Y_r^{t,x},Z_r^{t,x})dr+\int_{s}^{\infty}{K'\over2}{\rm e}^{-{K'\over2}r}Y_r^{t,x}dr\\
&&-\int_{s}^{\infty}{\rm e}^{-{K'\over2}r}\langle
g(X_r^{t,x},Y_r^{t,x},Z_r^{t,x}),d^\dagger\hat{B}_r\rangle-\int_{s}^{\infty}{\rm
e}^{-{K'\over2}r}\langle Z_r^{t,x},dW_r\rangle,\ \ s\geq0.\nonumber
\end{eqnarray}
We replace Condition (A.1) by
\begin{description}
\item[(A.1)$^*$.] Functions $f: \mathbb{R}^d\times\mathbb{R}^1\times\mathbb{R}^{d}{\longrightarrow{\mathbb{R}^1}}$ and $g: \mathbb{R}^d\times\mathbb{R}^1\times\mathbb{R}^{d}{\longrightarrow{\mathbb{R}^{l}}}$ are $\mathscr{B}_{\mathbb{R}^{d}}\otimes\mathscr{B}_{\mathbb{R}^{1}}\otimes\mathscr{B}_{\mathbb{R}^{d}}$
measurable, and there exist constants $C_0$, $C_1$, $C\geq0$ and
$0\leq\alpha<{1\over2}$ s.t. for any $(x_1, y_1, z_1)$, $(x_2, y_2,
z_2)\in \mathbb{R}^d\times \mathbb{R}^1\times \mathbb{R}^{d}$,
\begin{eqnarray*}
&&|f(x_1, y_1, z_1)-f(x_2, y_2, z_2)|^2\leq C_0|x_1-x_2|^2+C_1|y_1-y_2|^2+C|z_1-z_2|^2,\nonumber\\
&&|g(x_1, y_1, z_1)-g(x_2, y_2, z_2)|^2\leq
C_0|x_1-x_2|^2+C|y_1-y_2|^2+\alpha|z_1-z_2|^2.
\end{eqnarray*}
\end{description}
\begin{prop}\label{zz34}
Under Conditions {\rm(A.1)$^*$}, {\rm(A.3)}, {\rm(A.4)}, BDSDE
(\ref{zz32}) has a unique solution
\begin{center}
$(Y_{\cdot}^{t,x}, Z_{\cdot}^{t,x})\in
S^{p,-K}([0,\infty);\mathbb{R}^1) \cap
M^{2,-K}([0,\infty);\mathbb{R}^1)\bigotimes
M^{2,-K}([0,\infty);\mathbb{R}^{d})$.
\end{center}
\end{prop}
\begin{rmk}\label{qi315}
For $s\in[0,t]$, BDSDE (\ref{zz32}) is equivalent to the following
BDSDE
\begin{eqnarray*}\label{zhang663}
{\rm e}^{-{K'\over2}s}Y_s^{x}&=&\int_{s}^{\infty}{\rm e}^{-{K'\over2}r}f(x,Y_r^{x},Z_r^{x})dr+\int_{s}^{\infty}{K'\over2}{\rm e}^{-{K'\over2}r}Y_r^{x}dr\nonumber\\
&&-\int_{s}^{\infty}{\rm e}^{-{K'\over2}r}\langle
g(x,Y_r^{x},Z_r^{x}),d^\dagger\hat{B}_r\rangle-\int_{s}^{\infty}{\rm
e}^{-{K'\over2}r}\langle Z_r^{x},dW_r\rangle.
\end{eqnarray*}
To unify the notation, we define
$({Y}_s^{t,x},{Z}_s^{t,x})=({Y}_s^{x},{Z}_s^{x})$ when $s\in[0,t)$.
\end{rmk}
{\em Proof of Proposition \ref{zz34}}. Let
\begin{eqnarray*} \hat{f}(s,y,z)=f(X_{s}^{t,x},y,z),\ \ \
\hat{g}(s,y,z)=g(X_{s}^{t,x},y,z).
\end{eqnarray*}
We need to verify that $\hat{f}$, $\hat{g}$ satisfy Conditions
(H.1)--(H.3) in Theorem \ref{theorem2.3}. It is obvious that
$\hat{f}$, $\hat{g}$ satisfy (H.1) and (H.2), so we only need to
show that $\hat{f}$, $\hat{g}$ satisfy (H.3) as well, i.e.
\begin{eqnarray*}
E[\int_{0}^{\infty}{\rm e}^{-Ks}|\hat{f}(s,0,0)|^pds]<\infty\ {\rm
and}\ E[\int_{0}^{\infty}{\rm e}^{-Ks}|\hat{g}(s,0,0)|^pds]<\infty.
\end{eqnarray*}
Since
\begin{eqnarray*}
E[\int_{0}^{\infty}{\rm e}^{-Ks}|\hat{f}(s,0,0)|^pds]
\leq C_pE[\int_{0}^{\infty}{\rm
e}^{-Ks}C_0^p|X_{s}^{t,x}|^pds]+C_pE[\int_{0}^{\infty}{\rm
e}^{-Ks}|f(0,0,0)|^pds],
\end{eqnarray*}
we only need to prove $E[\int_{0}^{\infty}{\rm
e}^{-Ks}|X_{s}^{t,x}|^pds]<\infty$. Now applying It$\hat {\rm o}$'s
formula to ${\rm e}^{-Kr}|X_{r}^{t,x}|^p$ and noticing Condition
(A.4), we have
\begin{eqnarray*}
E[\int_{t}^{s}{\rm e}^{-Kr}|X_{r}^{t,x}|^pdr]\leq {\rm
e}^{-Kt}|x|^p+C_pE[\int_{t}^{s}{\rm
e}^{-Kr}(|b(0)|^p+\|\sigma(0)\|^p)dr]<\infty.
\end{eqnarray*}
Taking the limit of $s$ and noting that $(X_{s}^{t,x})_{s<t}=x$, we
have
$E[\int_{0}^{\infty}{\rm e}^{-Kr}|X_{r}^{t,x}|^pdr]<\infty$.
So $E[\int_{0}^{\infty}{\rm e}^{-Ks}|\hat{f}(s,0,0)|^pds]<\infty$.
Similarly, $E[\int_{0}^{\infty}{\rm
e}^{-Ks}|\hat{g}(s,0,0)|^pds]<\infty$. $\hfill\diamond$\\
Now we prove the other assumption in Theorem \ref{zz42}, i.e. the
regularity of solutions of infinite horizon BDSDEs. An simple
application of stochastic flow property proved in \cite{ku2} leads
to
\begin{lem}\label{zz35} Under Condition {\rm(A.4)}, for arbitrary $T$ and $t$, $t'\in[0,T]$,
$x$, $x'$ belonging to an arbitrary bounded set in $\mathbb{R}^d$,
the diffusion process $(X_{s}^{t,x})_{s\geq0}$ defined in SDE
(\ref{qi17}) satisfies
\begin{eqnarray*}
E[\int_{0}^{\infty}{\rm
e}^{-Kr}|X_r^{t',x'}-X_r^{t,x}|^pdr]&\leq&C_p(|x'-x|^p+|t'-t|^{p\over2})\
\ \ \rm{a.s.}
\end{eqnarray*}
\end{lem}
$\hfill\diamond$\\
We concentrate ourselves on the regularity of infinite horizon BDSDE
(\ref{zz36}), which is
a simpler form of BDSDE (\ref{zz32}). For arbitrary given terminal
time $T$, the form of BDSDE (\ref{zz36}) on $[t,T]$ is (\ref{zz22}).
\begin{prop}\label{zz39} Under Conditions {\rm(A.1)}--{\rm(A.4)}, let $(Y_{s}^{t,x})_{s\geq0}$ be the solution of BDSDE (\ref{zz36}), then for arbitrary $T$ and $t\in[0,T]$, $x\in\mathbb{R}^d$, $(t,x)\longrightarrow Y_{t}^{t,x}$ is a.s.
continuous.
\end{prop}
{\em Proof}. For $t$, $t'$, $r\geq0$, let
\begin{eqnarray*}
&&\bar{Y}_r=Y_{r}^{t',x'}-Y_{r}^{t,x},\ \ \
\bar{Z}_r=Z_{r}^{t',x'}-Z_{r}^{t,x}.
\end{eqnarray*}
Applying It$\hat {\rm o}$'s formula to ${\rm
e}^{-{{pK'}\over2}r}|\bar{Y}_r|^p$ and following a similar
calculation as in (\ref{zhang690}), we have
for $0\leq s\leq T$,
\begin{eqnarray}\label{zhang13}
&&{\rm e}^{-{{pK'}\over2}s}|\bar{Y}_s|^p+(p\mu-{{pK'}\over2}-{{p(p+1)}\over2}C-\varepsilon)\int_{s}^{T}{\rm e}^{-{{pK'}\over2}r}|\bar{Y}_{r}|^pdr\nonumber\\
&&+{{p(2p-3)}\over4}\int_{s}^{T}{\rm e}^{-{{pK'}\over2}r}|\bar{Y}_{r}|^{p-2}|\bar{Z}_r|^2dr\nonumber\\
&\leq&{\rm e}^{-{{pK'}\over2}T}|\bar{Y}_T|^p+C_p\int_{s}^{T}{\rm e}^{-{{pK'}\over2}r}|\bar{X}_{r}|^pdr-p\int_{s}^{T}{\rm e}^{-{{pK'}\over2}r}|\bar{Y}_{r}|^{p-2}\bar{Y}_{r}\langle\bar{g}_r,d^\dagger\hat{B}_r\rangle\nonumber\\
&&-p\int_{s}^{T}{\rm
e}^{-{{pK'}\over2}r}|\bar{Y}_{r}|^{p-2}\bar{Y}_{r}\langle\bar{Z}_r,dW_r\rangle.
\end{eqnarray}
Noticing Condition (A.3),
for $0\leq s\leq T$, we have
\begin{eqnarray}\label{zhang14}
&&E[{\rm e}^{-{{pK'}\over2}s}|\bar{Y}_s|^p]+E[\int_{s}^{T}{\rm e}^{-{{pK'}\over2}r}|\bar{Y}_{r}|^{p}dr]+E[\int_{s}^{T}{\rm e}^{-{{pK'}\over2}r}|\bar{Y}_{r}|^{p-2}|\bar{Z}_r|^2dr]\nonumber\\
&\leq&C_pE[{\rm
e}^{-{{pK'}\over2}T}|\bar{Y}_T|^p]+C_pE[\int_{s}^{T}{\rm
e}^{-{{pK'}\over2}r}|\bar{X}_{r}|^pdr].
\end{eqnarray}
Since
$E[{\rm e}^{-{{pK'}\over2}T}|\bar{Y}_T|^p]\leq E[\sup_{s\geq0}{\rm
e}^{-Ks}|\bar{Y}_s|^p]<\infty$,
by the Lebesgue's dominated convergence theorem, we have
\begin{eqnarray}\label{zhang15}
\lim_{T\rightarrow\infty}E[{\rm
e}^{{{pK'}\over2}T}|\bar{Y}_T|^p]=E[(\lim_{T\rightarrow\infty}{\rm
e}^{-{K'\over2}T}|\bar{Y}_T|)^p]=0.
\end{eqnarray}
So taking the limit of $T$ in (\ref{zhang14}), by Lemma \ref{zz35}
and the monotone convergence theorem, we have
\begin{eqnarray}\label{zhang16}
E[\int_{0}^{\infty}{\rm
e}^{-{{pK'}\over2}r}|\bar{Y}_{r}|^{p-2}|\bar{Z}_r|^2dr]+E[\int_{0}^{\infty}{\rm
e}^{-{{pK'}\over2}r}|\bar{Y}_{r}|^pdr]\leq
C_pE[\int_{0}^{\infty}{\rm
e}^{-Kr}|\bar{X}_{r}|^pdr].
\end{eqnarray}
From (\ref{zhang13}), by B-D-G inequality and (\ref{zhang15}), we
have
\begin{eqnarray*}
&&E[\sup_{s\geq0}{\rm e}^{-{{pK'}\over2}s}|\bar{Y}_s|^p]\\
&\leq&C_pE[\int_{0}^{\infty}{\rm
e}^{-{{pK'}\over2}r}|\bar{X}_{r}|^pdr]+C_pE[\int_{0}^{\infty}{\rm
e}^{-{{pK'}\over2}r}|\bar{Y}_{r}|^pdr]+C_pE[\int_{0}^{\infty}{\rm
e}^{-{{pK'}\over2}r}|\bar{Y}_{r}|^{p-2}|\bar{Z}_r|^2dr].
\end{eqnarray*}
By the above inequality, Lemma \ref{zz35} and (\ref{zhang16}), for
arbitrary $T>0$, $t$, $t'\in[0,T]$, $x$, $x'$ belonging to an
arbitrary bounded set in $\mathbb{R}^d$, we have
\begin{eqnarray}\label{zhang18}
E[\sup_{s\geq0}{\rm e}^{-{pK}s}|\bar{Y}_s|^p]\leq
C_pE[\int_{s}^{T}{\rm e}^{-{{pK'}\over2}r}|\bar{X}_{r}|^pdr]\leq
C_p(|x'-x|^p+|t'-t|^{p\over2}).
\end{eqnarray}
Noting $p>d+2$ in (\ref{zhang18}), by Kolmogorov Lemma (see e.g.
\cite{ku2}), we have $Y_{s}^{(\cdot,\cdot)}$ has a continuous
modification for $t\in[0,T]$ and $x$ belonging to an arbitrary
bounded set in $\mathbb{R}^d$ under the norm $\sup_{s\geq0}{\rm
e}^{-{K}s}|Y_{s}^{(\cdot,\cdot)}|$. In particular,
\begin{eqnarray*}
\lim_{t'\rightarrow t\atop x'\rightarrow x}{\rm
e}^{-{K}t'}|Y_{t'}^{t',x'}-Y_{t'}^{t,x}|=0.
\end{eqnarray*}
Thus we have a.s.
\begin{eqnarray*}
\lim_{t'\rightarrow t\atop x'\rightarrow x}|{\rm
e}^{-{K}t'}Y_{t'}^{t',x'}-{\rm
e}^{-{K}t}Y_{t}^{t,x}|\leq\lim_{t'\rightarrow t\atop x'\rightarrow
x}(|{\rm e}^{-{K}t'}Y_{t'}^{t',x'}-{\rm
e}^{-{K}t'}Y_{t'}^{t,x}|+|{\rm e}^{-{K}t'}Y_{t'}^{t,x}-{\rm
e}^{-{K}t}Y_{t}^{t,x}|)=0.
\end{eqnarray*}
The convergence of the second term follows from the continuity of
$Y_{s}^{t,x}$ in $s$. That is to say ${\rm e}^{-{K}t}Y_{t}^{t,x}$ is
a.s. continuous, therefore $Y_{t}^{t,x}$ is continuous w.r.t.
$t\in[0,T]$ and $x$ belonging to an arbitrary bounded set in
$\mathbb{R}^d$.
Denote by $\bar{B}(0,R)$ the closed ball in $\mathbb{R}^d$ of radius
$R$ centered at $0$. It is obvious that
$\bigcup_{R=1}^{\infty}\bar{B}(0,R)=\mathbb{R}^d$. $Y_{t}^{t,x}$ is
continuous w.r.t $t\in[0,T]$ and $x\in\bar{B}(0,R)$ on $\Omega^R$.
Take $\tilde{\Omega}=\bigcap_{R=1}^{\infty}{\Omega}^{R}$, then
$P(\tilde{\Omega})=1$. Now for any $t\in[0,T]$ and
$x\in\mathbb{R}^d$, there exists an $R$ s.t. $x\in\bar{B}(0,R)$. On
the other hand, for all $\omega\in\tilde{\Omega}$, it is obvious
that $\omega\in\Omega^{R}$. So $Y_{t}^{t,x}$ is continuous w.r.t.
$t\in[0,T]$ and $x\in\mathbb{R}^{d}$ on $\tilde{\Omega}$.
Proposition \ref{zz39} is proved. $\hfill\diamond$\\
\section{Stationary Property of Stochastic Viscosity Solutions of SPDEs}\label{s27}
\setcounter{equation}{0}
\ \ \ \ With the regularity of solution of BDSDE (\ref{zz36}), for
arbitrary given $T$, we can obtain a stochastic viscosity solution
of SPDE (\ref{zz20}) on the time interval $[0,T]$ through BDSDE
(\ref{zz36}).
\begin{thm}\label{zz37}
Under Conditions {\rm(A.1)}--{\rm(A.4)}, for arbitrary given $T$ and
$t\in[0,T]$, $x\in\mathbb{R}^d$, let $v(t,x)\triangleq
Y_{T-t}^{T-t,x}$, where $(Y_{s}^{t,x},Z_{s}^{t,x})$ is the solution
of BDSDE (\ref{zz36}) with $\hat{B}_s={B}_{T-s}-{B}_T$ for all
$s\geq0$. Then $v(t,x)$ is continuous w.r.t. $t$ and $x$ and is a
stochastic viscosity solution of SPDE (\ref{zz20}) on the time
interval $[0,T]$.
\end{thm}
{\em Proof}. Notice that Condition (A.1) is stronger than (A.2)$'$,
so by Theorem \ref{zz38} BDSDE (\ref{zz36}) has a unique solution
$(Y_{\cdot}^{t,x}, Z_{\cdot}^{t,x})\in
S^{p,-K}([0,\infty);\mathbb{R}^1) \cap
M^{2,-K}([0,\infty);\mathbb{R}^1)\bigotimes
M^{2,-K}([0,\infty);\mathbb{R}^{d})$. On $[t,T]$, BDSDE (\ref{zz36})
has a form of (\ref{zz22}) which can be associated with SPDE
(\ref{zz20}) on $[0,T]$ through time reversal transformation in
(\ref{zz21}). First note that by Proposition \ref{zz39}, $v(t,x)$
defined by $Y_{T-t}^{T-t,x}$ is a.s. continuous w.r.t. $t\in[0,T]$
and $x\in\mathbb{R}^d$. Moreover, since
$X_s^{T,X_T^{t,x}}=X_s^{t,x}$ for $s\geq T$, by the uniqueness of
BDSDE (\ref{zz36}) we have
$Y_T^{T,X_T^{t,x}}=Y_T^{t,x}$ a.s., where $Y_\cdot^{T,x}
$ is the solution of BDSDE (\ref{zz36}) when the diffusion process
$X$ defined in (\ref{qi17})
starts at time $T$ and point $x\in\mathbb{R}^{d}$. Therefore
$E[|v(0,X_T^{t,x})|^2]=E[|Y_T^{T,X_T^{t,x}}|^2]=E[|Y_T^{t,x}|^2]<\infty$.
By Theorem \ref{theorem2.2} and Remark \ref{zz24}, we know that
$v(t,x)$ is a stochastic viscosity solution of SPDE (\ref{zz20}) on
the time interval $[0,T]$. Theorem \ref{zz37} is proved.
$\hfill\diamond$
\\
In the following, we show that the $v(t,x)$ constructed in Theorem
\ref{zz37} is a stationary solution of SPDE (\ref{zz20}). For this,
we need first prove a claim that
$v(t,x)(\omega)=Y_{T-t}^{T-t,x}(\hat{\omega})$ is independent of the
choice of $T$. This independence can be proved by a similar argument
as in \cite{zh-zh1} (Page 186-187) since it is unrelated to which
kind of solution (weak solution or stochastic viscosity solution)
$v$ is. Therefore, for any $T'\geq T$,
$Y_{T-t}^{T-t,x}(\hat{\omega})=Y_{T'-t}^{T'-t,x}({\hat{\omega}}')$
when $0\leq t\leq T$, where $\hat{\omega}(s)={B}_{T-s}-{B}_{T}$ and
${\hat{\omega}}'(s)={B}_{T'-s}-{B}_{T'}$.
On the probability space $(\Omega,\mathscr{F},P)$, we define
${\theta}_{t}=(\hat{\theta}_{t})^{-1}$, $t\geq0$. Actually $\hat{B}$
is a two-sided Brownian motion, so
$(\hat{\theta}_{t})^{-1}=\hat{\theta}_{-t}$ is well defined (see
\cite{ar}). It is easy to see that ${\theta}_{t}$ is a shift w.r.t.
${B}$ satisfying
\begin{description}
\item[$(\textrm{i})$]$P\cdot({\theta}_{t})^{-1}=P$;
\item[$(\textrm{i}\textrm{i})$]${\theta}_{0}=I$;
\item[$(\textrm{i}\textrm{i}\textrm{i})$]${\theta}_{s}\circ{\theta}_{t}={\theta}_{s+t}$;
\item[$(\textrm{iv})$]${\theta}_{t}\circ{B}_s={B}_{s+t}-{B}_{t}$.
\end{description}
By Theorem \ref{zz42} and the relationship between $\theta$ and
$\hat{\theta}$, we have
\begin{eqnarray*}
{\theta}_rv(t,x)(\omega)=\hat{\theta}_{-r}Y_{T-t}^{T-t,x}(\hat{\omega})=\hat{\theta}_{-r}\hat{\theta}_{r}Y_{T-t-r}^{T-t-r,x}(\hat{\omega})=Y_{T-t-r}^{T-t-r,x}(\hat{\omega})=v(t+r,x)(\omega),
\end{eqnarray*}
for all $r\geq0$ and $T\geq t+r$, $x\in\mathbb{R}^{d}$ a.s. In
particular, let
$Y(x,\omega)=v(0,x)(\omega)=Y_{T}^{T,x}(\hat{\omega})$, then the
above formula implies (\ref{zhao001}):
\begin{eqnarray*}
{\theta}_tY(x,\omega)=Y(x,{\theta}_t\omega)=v(t,x)(\omega)=v(t,x,v(0,x)(\omega))(\omega)=v(t,x,Y(x,\omega))(\omega),\
\end{eqnarray*}
for all $t\geq0$, $x\in\mathbb{R}^{d}$ a.s. That is to say
$v(t,x)(\omega)=Y(x,{\theta}_t\omega)=Y_{T-t}^{T-t,x}(\hat{\omega})$
is a stationary solution of SPDE (\ref{zz20}) w.r.t. ${\theta}$.
Therefore we have the following conclusion
\begin{thm}\label{zz41} Under Conditions {\rm(A.1)}--{\rm(A.4)}, for arbitrary $T$ and $t\in[0,T]$, let $v(t,x)\triangleq Y_{T-t}^{T-t,x}$, where $(Y_{s}^{t,x},Z_{s}^{t,x})$ is the solution of BDSDE (\ref{zz36}) with $\hat{B}_s={B}_{T-s}-{B}_T$ for all $s\geq0$. Then $v(t,x)$ is
a ``perfect" stationary stochastic viscosity solution of SPDE
(\ref{zz20}).
\end{thm}
{\bf Acknowledgements}. I would like to thank Prof. H. Z. Zhao, with
whom the main ideas were formed when I was a student in
Loughborough-Shandong Universities joint Ph.D. programme in
stochastic analysis. It is also my great pleasure to thank R.
Hudson, K. Lu, J. Ma, S. Peng and S. Tang for useful conversations.
Meanwhile, I would like to acknowledge the financial support of the
National Basic Research Program of China (973 Program) with Grant
No. 2007CB814904.
| 117,884
|
\begin{document}
\begin{abstract}
We establish necessary and sufficient conditions for a quadratic
polynomial to be irreducible in the ring $\Z[[x]]$ of formal
power series with integer coefficients. For $n,m\ge 1$ and $p$ prime,
we show that $p^n+p^m\beta x+\alpha x^2$ is reducible in $\Z[[x]]$ if
and only if it is reducible in $\Z_p[x]$, the ring of polynomials over
the $p$-adic integers.
\end{abstract}
\maketitle
\newcounter{prfeqn}
\section{Introduction}
If $K$ is a field, the question of whether or not a quadratic polynomial
is reducible in the polynomial ring $K[x]$ is well understood: A
polynomial $f(x)=c+bx+ax^2$, with $a\ne 0$, can be written as a product of
two linear factors in $K[x]$ if and only if its discriminant $b^2-4ac$ is
a square in $K$. More generally, if $D$ is a unique factorization domain,
a primitive quadratic polynomial in $D[x]$ is reducible if and only
its discriminant is a square in $D$.
\par
If we consider the polynomials in $\Z[x]$ as elements of $\Z[[x]]$, the
ring of formal power series over $\Z$, the factorization theory has a
different flavor. A power series over an integral domain $D$ is a unit
in $D[[x]]$ if and only if its constant term is a unit in $D$, so
irreducible elements in $\Z[x]$, such as $1+x$, are invertible as power
series. On the other hand, any power series whose constant term is not a
unit or a prime power, is reducible in $\Z[[x]]$, hence we can produce
many examples of polynomials that are reducible as power series, yet
irreducible in $\Z[x]$.
\par
Similarly, when considering polynomials with integer coefficients as
elements of $\Z[[x]]$ and as polynomials over $\Z_p$, the ring of $p$-adic
integers, we also observe different behaviors in their arithmetic
properties. For instance, the polynomial $p^2+x+x^2$, which is irreducible
as a power series, is reducible in $\Z_p[x]$ for any prime $p$. On the
other hand, $6+2x+x^2$ is reducible in $\Z[[x]]$ and in $\Z_3[x]$,
but it is irreducible in $\Z_2[x]$ and $\Z_5[x]$.
\par
In this paper, we discuss the factorization theory of quadratic
polynomials in the ring of formal power series over $\Z$. Based on the
above examples, it is natural to ask whether the question of
reducibility of polynomials in $\Z[[x]]$ can be reduced, at least in
some cases, to the reducibility in $D[x]$ for some integral domain $D$.
The implication of such a reduction in the quadratic
case is clear: a reducibility criterion that relies on the discriminant
being a square in $D$. In this direction, we found the following
connection between $\Z[[x]]$ and $\Z_p[x]$:
\newtheorem*{t:theorem}{Main Theorem}
\begin{t:theorem}\label{t:MT}
Let $p$ be prime. Let $\alpha,\beta\in\Z$ be such that $\gcd(p,\alpha)=1$
and $\gcd(p,\beta)=1$. Let $f(x)=p^n+ p^m\beta x +\alpha x^2$ with
$n,m\ge 1$. Then $f(x)$ is reducible in $\Z[[x]]$ if and only if it is
reducible as a polynomial over $\Z_p$.
\end{t:theorem}
We present the proof of this theorem in Sections~\ref{sec:OddPrimePower}
and \ref{sec:EvenPrimePower}.
Since the conditions for being a square in $\Z_p$ are well known, our
approach provides an effective procedure for deciding whether or not a
quadratic polynomial is reducible as a power series and, in the affirmative
case, our proofs give an algorithmic method (whose foundations are based on
the Euclidean Algorithm) for finding a proper factorization.
\par
In addition to our main theorem, we give a complete picture of the
factorization theory for quadratic polynomials in $\Z[[x]]$.
Some basic cases are treated in Section~\ref{s:factorization} where we
discuss the necessary background and develop some preliminary results.
In Section~\ref{sec:FurtherCriteria} we finish with some
simple reducibility criteria that apply to power series whose quadratic
part is of the form discussed in the other sections.
\par
A standard reference for an introduction to divisibility over integral
domains is \cite{DuFo}. For an extensive treatment of the arithmetic on
the ring of formal power series over an integral domain the reader is
referred to \cite{Ei95} and \cite{Ka70}. All the necessary
material about the ring $\Z_p$ of $p$-adic numbers, can be found for
instance in \cite{BoSh,Ei95,Serre}.
\section{Factorization in the ring of power series}\label{s:factorization}
In order to place our main result in the appropriate context, and for the
reader's convenience, we review some elementary facts about the
factorization theory in $\Z[[x]]$. First, recall that $\Z[[x]]$ is a
unique factorization domain. Moreover, if $f(x)$ is a formal power series
in $\Z[[x]]$ and $f_0\in \Z$ is its constant term, then:
\begin{enumerate}[i.]
\item $f(x)$ is invertible if and only if $f_0=\pm 1$.
\item If $f_0$ is prime then $f(x)$ is irreducible.
\item If $f_0$ is not a unit or a prime power then $f(x)$ is reducible.
\item If $f(x)=f_0$ is a constant then it is irreducible if and only if
$f_0$ is prime.
\item If $f(x)=p^m+f_1 x$, with $p$ prime and $m\ge 1$,
then $f(x)$ is irreducible if and only if $\gcd(p, f_1)=1$.
\end{enumerate}
For an accessible and more detailed treatment of the divisibility theory
in $\Z[[x]]$ the reader is referred to \cite{BiGi}.
At this point, we have definitive criteria for deciding irreducibility
in $\Z[[x]]$ for constant and linear polynomials. The next natural step is
to examine quadratic polynomials, say $f(x)=f_0+f_1 x + f_2 x^2$.
Unless $f_0$ is a prime power, we know that $f(x)$ is either a unit or it
is reducible in $\Z[[x]]$. On the other hand, if $f_0=p^n$, $n> 1$, $p$
prime, and if $f(x)=a(x)b(x)$ is a proper factorization, then we must have
$a_0=p^s, b_0=p^t$ with $s,t\ge 1$, $s+t=n$. This implies
$f_1=p^s b_1+p^t a_1$, so we conclude that $f(x)$ is irreducible
unless $p\mid f_1$. Finally, if $p$ divides all coefficients, then $f(x)$
is either reducible or associate to $p$. Thus the interesting case is when
$f(x)$ is primitive.
Therefore, in the next sections we will focus on polynomials of the form
\[ f(x)=p^n+ p^m\beta x +\alpha x^2, \]
with $n,m\ge 1$, $\gcd(p,\alpha)=1$, and $\gcd(p,\beta)=1$ or $\beta=0$.
As stated in the introduction, we will analyze the factorization
of such polynomials by considering them as elements in $\Z_p[x]$ and
$\Z[[x]]$, and the main tool for establishing this link will be
the discriminant. Our strategy will be to produce explicit
factorizations in $\Z[[x]]$, when appropriate.
To this end, it will be helpful to assume that one of the factors in
$f(x)=a(x)b(x)$ has a certain simplified form.
The basis for this assumption is the following lemma.
\begin{lemma}\label{l:ChoosingLemma}
Let $a(x)\in \Z[[x]]$ such that $a_0=p$ and $\gcd(p,a_1)=1$. For every
$t\ge 2$ there exists an associate $q(x)$ to $a(x)$ such that
$q_0=a_0$, $q_1\equiv a_1\pmod p$, and $q_2=q_3=\dotsb =q_t=0$.
\end{lemma}
More precisely, we will show that there exists a polynomial
\[ u(x)= 1+u_1x+u_2x^2+\dotsb + u_tx^t, \]
invertible in $\Z[[x]]$, such that $u(x)a(x)=p+ \lambda x +
q_{t+1}x^{t+1}+q_{t+2}x^{t+2}+\dotsb$, with $\lambda\equiv a_1\pmod p$.
In order to find $u(x)$, we set up the $t \times t$ system of equations:
\begin{equation}\label{Eq:s-system}
\begin{aligned}
\lambda &= a_1+pu_1,\\
0 &= a_2+a_1u_1+pu_2,\\
0 &= a_3+a_2u_1+a_1u_2+pu_3,\\
&\;\;\vdots\\
0 &= a_t+\sum_{j=1}^{t-1}a_ju_{t-j}+pu_{t},
\end{aligned}
\end{equation}
in the unknowns $u_1,\dotsc, u_t$. Since the determinant of the
matrix associated with this system of equations is $p^t$, it is clear
that \eqref{Eq:s-system} admits a unique solution over the rationals for
any integer $\lambda$. Our goal is to prove that for any $t\ge 2$,
there exist a suitable $\lambda\in \Z$ such that \eqref{Eq:s-system}
admits a solution over the integers.
This follows from the following two propositions.
\begin{proposition}\label{P:lambda-k}
If for some $\lambda\in \Z$, the solution $(u^0_1, \dots ,u^0_t)$
of the system \eqref{Eq:s-system} is such that $u_i^0\in \Z$ for all
$1\le i \le t$, then for every $k\in\Z$ the system
\begin{equation}\label{Eq:s-system-k}\mytag
\begin{aligned}
\lambda+kp^t &=a_1+pu_1,\\
0 &= a_2+a_1u_1+pu_2,\\
0 &= a_3+a_2u_1+a_1u_2+pu_3,\\
&\;\;\vdots\\
0 &= a_t+\sum_{j=1}^{t-1}a_ju_{t-j}+pu_{t},
\end{aligned}
\end{equation}
also has a $($unique$)$ solution over the integers.
Moreover, the solution $(u^k_1,\dots ,u^k_t)$ of \eqref{Eq:s-system-k}
and the solution $(u^0_1, \dots ,u^0_t)$ of \eqref{Eq:s-system} are
related as follows:
\begin{equation*}
\begin{aligned}
u^k_i &\equiv u^0_i \pmod p \;\text{ for } 1\le i \le t-1, \\
u^k_t &\equiv u^0_t+(-1)^{t+1}k a_1^{t-1} \pmod p.
\end{aligned}
\end{equation*}
\end{proposition}
\begin{proposition}\label{P:induction-on-t} \setcounter{prfeqn}{0}
If the $t\times t$ system of equations
\begin{equation*}
\begin{aligned}
\lambda &= a_1+pu_1,\\
0 &= a_2+a_1u_1+pu_2,\\
0 &= a_3+a_2u_1+a_1u_2+pu_3,\\
&\;\;\vdots\\
0 &= a_t+\sum_{j=1}^{t-1}a_ju_{t-j}+pu_{t},
\end{aligned}
\end{equation*}
has a solution $(u_1^0, \dotsc , u_t^0)$ over the integers, then there
exists $k\in \Z$ such that the $(t+1) \times (t+1)$ system
\begin{equation*}
\begin{aligned}
\lambda+kp^t &= a_1+pu_1,\\
0 &= a_2+a_1u_1+pu_2,\\
0 &= a_3+a_2u_1+a_1u_2+pu_3,\\
&\;\;\vdots\\
0 &= a_{t+1}+\sum_{j=1}^{t}a_ju_{t+1-j}+pu_{t+1},
\end{aligned}
\end{equation*}
also has its solution $(u_1^k, \dotsc , u_{t+1}^k)$
over the integers.
\end{proposition}
\begin{proof}[\bf Proof of Lemma~\ref{l:ChoosingLemma}]
Choose $\lambda \in \Z$ such that $\lambda\equiv a_1\pmod p$. Then
the equation $\lambda=a_1+p u_1$ can be solved for $u_1\in \Z$.
Then proceed to the desired value of $t$ by applying repeatedly
Proposition~\ref{P:lambda-k} and Proposition~\ref{P:induction-on-t}.
\end{proof}
\begin{proof}[\bf Proof of Proposition~\ref{P:lambda-k}]
Let $A$ be a $(t\times t)$-matrix and let $B^k$ be a
$t$-dimensional column vector defined as
\begin{equation*}
A=\begin{pmatrix}
p & 0 & 0 &\cdots &0 &0\\
a_1 & p & 0 &\cdots & 0 &0\\
a_2 & a_1 & p &\cdots & 0 &0 \\
\vdots\\
a_{t-1} & a_{t-2} & a_{t-3} &\cdots & a_1& p
\end{pmatrix}
\quad \text{and} \quad
B^k=\begin{pmatrix}
kp^t\\
0\\
\vdots\\
0
\end{pmatrix}.
\end{equation*}
If $A_1, A_2, \dotsc , A_t$ are the columns of $A$ then,
by Cramer's rule, the unique solution (over the rationals)
of the system \eqref{Eq:s-system-k} is given by
\begin{equation*}
u^k_i= u^0_i+\frac{1}{p^t}
\det\big(A_1, \dotsc , A_{i-1}, B^k, A_{i+1}, \dotsc ,A_t\big)
=u^0_i+(-1)^{i+1}k \det A_{1i},
\end{equation*}
where $A_{1i}$ denotes the $(t-1) \times (t-1)$ matrix obtained from $A$
by deleting its first row and $i^{\text{th}}$ column.
Thus $u^k_i\in \Z$ for every $1\le i\le t$.
The entries $q^i_{rr}$, $r=1,\dots, t-1$, in the principal diagonal of
$A_{1i}$, are as given by
\begin{equation*}
q^i_{rr}=
\begin{cases}
a_1 &\text{if } \; 1\le r \le i-1, \\
p &\text{if } \; i\le r\le t-1,
\end{cases}
\end{equation*}
and the entries in the super-diagonal of $A_{1i}$ are
\begin{equation*}
q^i_{r,r+1}=
\begin{cases}
p &\text{if } \; 3\le i\le t,\; 1\le r\le i-2, \\
0 &\text{otherwise}.
\end{cases}
\end{equation*}
Of course, all the entries above the super-diagonal in $A_{1i}$ are $0$.
Thus, when expanding $\det A_{1i}$ as a sum over all permutations in
the symmetric group of $t-1$ elements, the term corresponding to the
identity is $p^{t-i}a_1^{i-1}$, $1\le i \le t$, and the term
corresponding to any other permutation is a multiple of $p$. Then,
\begin{equation*}
\begin{aligned}
u^k_i &=u^0_i+(-1)^{i+1}k \det A_{1i}\equiv u^0_i \pmod p
\;\;\text{ for } 1\le i \le t-1, \\
u^k_t &=u^0_t+(-1)^{t+1}k \det A_{1t}
\equiv u^0_t+(-1)^{t+1}k a_1^{t-1} \pmod p.
\end{aligned}
\end{equation*}
\end{proof}
\begin{proof}[\bf Proof of Proposition~\ref{P:induction-on-t}]
Since $\gcd(p,a_1)=1$, we can choose $k$ such that
\begin{equation*}
a_{t+1}+a_1u^0_t+\sum_{j=2}^{t}a_ju^0_{t+1-j}
+(-1)^{t+1}k a_1^{t}\equiv 0 \pmod p.
\end{equation*}
By Proposition~\ref{P:lambda-k}, the system
\begin{equation}
\begin{aligned}
\lambda+kp^t &= a_1+pu_1\\
0 &= a_2+a_1u_1+pu_2,\\
0 &= a_3+a_2u_1+a_1u_2+pu_3,\\
&\;\;\vdots\\
0 &= a_t+\sum_{j=1}^{t-1}a_ju_{t-j}+pu_{t},
\end{aligned}
\end{equation}
has its solution $(u_1^k, \dotsc , u_t^k)$
over the integers. Moreover
\begin{equation*}
\begin{aligned}
u^k_i &=u^0_i \pmod p, \quad 1\le i \le t-1, \\
u^k_t&=u^0_t+(-1)^{t+1}k a_1^{t-1}\pmod p.
\end{aligned}
\end{equation*}
Therefore,
\begin{align*}
a_{t+1}+\sum_{j=1}^ta_ju^k_{t+1-j} &=\\
a_{t+1}+a_1u^k_t+\sum_{j=2}^{t}a_ju^k_{t+1-j} &=\\
a_{t+1}+a_1\bigl(u^0_t+(-1)^{t+1}k a_1^{t-1}\bigr)+
\sum_{j=2}^{t}a_ju^0_{t+1-j} &=\\
a_{t+1}+a_1u^0_t+\sum_{j=2}^{t}a_ju^0_{t+1-j}+(-1)^{t+1}k a_1^{t}
&\equiv 0 \pmod p.
\end{align*}
Hence, we can solve the equation $0=a_{t+1}+\sum_{j=1}^{t-1}a_ju_{t+1-j}
+pu_{t+1}$ for $u_{t+1}$.
\end{proof}
\section{The case when the constant term is an odd prime power}
\label{sec:OddPrimePower}
Let $p$ be an odd prime, let $\alpha,\beta\in\Z$ be such that
$\gcd(p,\alpha)=1$ and $\gcd(p,\beta)=1$.
\begin{proposition}\label{p-easycase}
Let $f(x)=p^n+ p^m\beta x +\alpha x^2$ with $n,m\ge 1$.
\begin{enumerate}[$(i)$]
\item
If $2m<n$, then $f(x)$ is reducible in both $\Z_p[x]$ and $\Z[[x]]$.
\item
If $2m>n$ and $n$ is odd, then $f(x)$ is irreducible in both $\Z_p[x]$
and $\Z[[x]]$.
\end{enumerate}
\end{proposition}
\begin{proof}
$(i)$ Observe first that the discriminant of $f(x)$ is
\begin{equation*}
p^{2m}\beta^2-4\alpha p^n=p^{2m}(\beta^2-4\alpha p^{n-2m}),
\end{equation*}
a nonzero square in $\Z_p$, and so $f(x)$ is reducible in $\Z_p[x]$.
To show that $f(x)$ is reducible as a power series, we will find
sequences $\{a_k\}$ and $\{b_k\}$ such that
\begin{equation*}
f(x)=(p^m+a_1x+a_2x^2+\cdots)(p^{n-m}+b_1x+b_2x^2+\cdots).
\end{equation*}
For $k\ge 1$ let $t_k=b_k+p^{n-2m}a_k$. For the above factorization to
hold, we need
\[ p^m\beta=p^{n-m}a_1+p^m b_1, \;\text{ so we have }
t_1=\beta. \]
Since $\beta^2-4\alpha p^{n-2m}$ is a square in $\Z_p$, the
polynomial $g(x)=p^{n-2m} x^2 -\beta x+\alpha$ is reducible in
$\Z_p[x]$. In particular, $g(x)$ has a root in $\Z/p^m\Z$, hence there are
integers $a_1$ and $t_2$ such that
\[ p^{n-2m} a_1^2 -\beta a_1+\alpha = p^m t_2. \]
Suppose that we have defined $a_k$, $t_{k+1}$ for $k=1,\dots,N-1$,
$N\ge 2$, and let
\begin{equation*}
v_N=a_1t_{N}+\sum_{k=2}^{N-1}a_k(t_{N+1-k}-p^{n-2m}a_{N+1-k}).
\end{equation*}
We want to define $a_N$ and $t_{N+1}$ in such a way that
$\sum_{k=0}^{N+1}a_kb_{N+1-k}=0$ for $N\ge 2$. In other words, we need
\begin{align*}
0 &= \sum_{k=0}^{N+1} a_k b_{N+1-k} \\
&=a_0b_{N+1}+a_{N+1}b_0+a_{N}b_1+a_1b_{N}+\sum_{k=2}^{N-1}a_kb_{N+1-k}\\
&=p^m t_{N+1}+(\beta-p^{n-2m}a_1)a_{N}+a_1(t_{N}-p^{n-2m}a_{N})
+\sum_{k=2}^{N-1}a_kb_{N+1-k}\\
&=p^m t_{N+1}+(\beta-2p^{n-2m}a_1)a_{N}+v_N.
\end{align*}
At last, since $\gcd(p,\beta)=1$, this equation
can be solved for $t_{N+1}, a_{N}\in \Z$. This shows that $f(x)$ is
reducible in $\Z[[x]]$.
\medskip
$(ii)$ In this case, the discriminant of $f(x)$,
\begin{equation*}
p^{2m}\beta^2-4\alpha p^n=p^n(p^{2\mu-n}\beta^2-4\alpha),
\end{equation*}
is not a square in $\Z_p$. Thus $f(x)$ is irreducible as a polynomial
over $\Z_p$. To show that $f(x)$ is irreducible as a
power series, assume
\begin{equation*}
f(x)=(p^s+a_1x+a_2x^2+\cdots)(p^t+b_1x+b_2x^2+\cdots)
\end{equation*}
with $t>s\ge 1$, $s+t=n$. Note that $t\not=s$ because $n$ is odd.
Then we must have
\begin{align*}
p^m\beta &=p^ta_1+p^sb_1, \\
\alpha &=p^ta_2+a_1b_1+p^sb_2.
\end{align*}
Since $p$ and $\alpha$ are coprime, it follows that
$\gcd(p,a_1)=1=\gcd(p,b_1)$. Therefore, it must be $s=m$, and so
$2m=2s< s+t=n$.
\end{proof}
It remains to analyze the cases when $n$ is even, say $n=2\nu$,
and $m\ge \nu\ge 1$.
\begin{proposition}\label{p-hardcase}
Let $m\ge \nu$.
The polynomial $f(x)=p^{2\nu}+p^m\beta x +\alpha x^2$ is reducible in
$\Z[[x]]$ if and only if $\f(x)= p^2 + p^{m-\nu+1}\beta x +\alpha x^2$
is reducible in $\Z_p[x]$.
\end{proposition}
This follows from the following three lemmas.
\begin{lemma}\label{lemmaA}
If $f(x)$ is reducible in $\Z[[x]]$, then $\f(x)$ is reducible in
$\Z[[x]]$.
\end{lemma}
\begin{lemma}\label{lemmaB}
Let $\ell\ge 1$. If the polynomial $p^2 + p^{\ell}\beta x +\alpha x^2$
is reducible in $\Z[[x]]$, then it is reducible in $\Z_p[x]$.
\end{lemma}
\begin{lemma}\label{lemmaC}
If $\f(x)$ is reducible in $\Z_p[x]$, then $f(x)$ is reducible in
$\Z[[x]]$.
\end{lemma}
\begin{proof}[\bf Proof of Lemma~\ref{lemmaA}]
We first observe that if $f(x)=a(x)b(x)$ is a proper factorization
in $\Z[[x]]$, then $a_0=b_0=p^\nu$.
To see this, assume that $a_0=p^s$, $b_0=p^t$ with $s, t\ge 1$, $s+t=2\nu$.
Then we have that
\begin{equation*}
\alpha=p^s b_2+a_1 b_1 + p^t a_2.
\end{equation*}
Since $\gcd(p,\alpha)=1$, we conclude that $\gcd(p,a_1)= \gcd(p,b_1)=1$.
We also have
\begin{equation*}
p^m\beta=p^s b_1+p^t a_1.
\end{equation*}
If $s<t$, then we would have $s<\nu\le m$, implying from the above
equation that $p\mid b_1$, a contradiction.
Similarly, we can rule out the case $t<s$, hence $s=t=\nu$.
We now write $f(x)=a(x)b(x)$ with $a_0=b_0=p^{\nu}$. Since
\begin{equation*}
p^{2\nu-2}\f(x)=f(p^{\nu-1}x)=a(p^{\nu-1}x)b(p^{\nu-1}x),
\end{equation*}
it follows that
\begin{equation*}
\f(x)=\Big(\frac{a(p^{\nu-1}x)}{p^{\nu-1}}\Big)
\Big(\frac{b(p^{\nu-1}x)}{p^{\nu-1}}\Big)
\end{equation*}
is a proper factorization of $\f(x)$ in $\Z[[x]]$.
\end{proof}
\begin{proof}[\bf Proof of Lemma~\ref{lemmaB}]
To prove that $g(x)=p^2 + p^{\ell}\beta x +\alpha x^2$ is reducible in
$\Z_p[x]$, we must show that its discriminant
$p^{2\ell}\beta^2-4\alpha p^2$ is a square in $\Z_p$.
Write $p^{2\ell-2}\beta^2-4\alpha=p^t u$ with $\gcd(p,u)=1$.
Suppose that $g(x)$ is reducible in $\Z[[x]]$.
By Lemma~\ref{l:ChoosingLemma} we can assume that $g(x)$ admits a
factorization of the form
\begin{equation*}
p^2 + p^{\ell}\beta x +\alpha x^2=a(x)b(x)\quad\text{with }\;
a_0=b_0=p,\;\; a_2=a_3=\cdots =a_{t+2}=0.
\end{equation*}
With the notation $s_j=a_j+b_j$ for $j\ge 1$,
we must have
\begin{align*}
p^{\ell}\beta &=ps_1, \\
\alpha &= ps_2+a_1s_1-a_1^2.
\end{align*}
Then $s_1=p^{\ell-1}\beta$ and $a_1$ is a root of
$y^2-s_1y+\alpha\equiv 0\pmod p$. Note that $p\ndiv a_1$.
For $n=3$ we have
\begin{equation}\label{eq:base}
0=ps_3+a_1s_2.
\end{equation}
Then $p\mid s_2$ and $a_1^2-a_1s_1+\alpha\equiv 0 \pmod{p^2}$.
For $n=4$ we have
\begin{equation*}
0=ps_4+a_1s_3.
\end{equation*}
Then $p\mid s_3$, which by \eqref{eq:base} implies that
$p^2\mid s_2$, and so $a_1^2-a_1s_1+\alpha\equiv 0 \pmod{p^3}$.
Working inductively, the equation
\begin{align*}
0= ps_{t+3}+a_1s_{t+2}
\end{align*}
implies that $p^{t+1}\mid s_2$, and so $a_1^2-a_1s_1+\alpha=
p^{t+2}v$ for some $v$.
Now, since
\[ (2a_1-s_1)^2 = (p^{2\ell-2}\beta^2-4\alpha) + 4p^{t+2}v
= p^t u +4p^{t+2}v=p^t(u+4p^2v), \]
and since $\gcd(p,u)=1$, we have that $t$ is even
and that $u$ is a square mod $p$. Hence $p^{2(\ell-1)}\beta^2-4\alpha$
is a square in $\Z_p$, and so is $p^2(p^{2(\ell-1)}\beta^2-4\alpha)$.
Therefore, $g(x)$ is reducible in $\Z_p[x]$.
\end{proof}
\begin{proof}[\bf Proof of Lemma~\ref{lemmaC}]
We will consider the cases $m=\nu$ and $m>\nu$ separately.
In both cases we will prove the reducibility of $f(x)$ in $\Z[[x]]$
by providing an explicit factorization algorithm. More precisely,
we will give inductive algorithms (depending on $m$ and $\nu$) to find
sequences $\{a_k\}$ and $\{b_k\}$ in $\Z$ such that
\begin{equation}\label{fFactor}
f(x)=\Big(\sum_{k=0}^\infty a_k x^k\Big)
\Big(\sum_{k=0}^\infty b_k x^k\Big).
\end{equation}
For $k\ge 1$ we let $s_k=a_k+b_k$.
\medskip\noindent
{\sc Case 1:} Let $m>\nu$. Since $\f(x)=p^2+p^{m-\nu+1}\beta x
+\alpha x^2$ is reducible in $\Z_p[x]$, the polynomial
$\g(x)=x^2-p^{m-\nu}\beta x+\alpha$ is reducible in $\Z_p[x]$,
too. Observe that the discriminant of $\f(x)$ is $p^2$ times the
discriminant of $\g(x)$.
Let
\[ a_0=p^\nu=b_0 \quad\text{and}\quad s_1=p^{m-\nu}\beta. \]
Since $\g(x)$ is reducible in $\Z_p[x]$, it has a root in
$\Z/p^{\nu}\Z$. Let $a_1,s_2\in\Z$ be such that
\[ a_1^2 - p^{m-\nu}\beta a_1 +\alpha = p^\nu s_2. \]
Now, $m>\nu$ and $\gcd(p,\alpha)=1$ imply
$\gcd(p,a_1)=1$ and $\gcd(p^\nu,p^{m-\nu}\beta-2a_1)=1$.
We let $a_2$ and $s_3$ be integer numbers such that
\[ 0=p^\nu s_3 + (p^{m-\nu}\beta-2a_1)a_2 + a_1s_2. \]
Suppose we have defined $a_{k}$ and $s_{k+1}$ for $k=1,\dots,N-1$,
$N\ge 3$, and let
\[ v_N=a_1s_N +\sum_{k=2}^{N-1}a_k(s_{N+1-k}-a_{N+1-k}). \]
We know that $\gcd(p^\nu,p^{m-\nu}\beta-2a_1)=1$, so the equation
\[ 0=p^\nu s_{N+1}+(p^{m-\nu}\beta-2a_1)a_N + v_N \]
can be solved for $a_N, s_{N+1}\in\Z$.
For $k=1,\dots,N$ we now have $a_k$ and $b_k$, and
it can be easily checked that the sequences $\{a_k\}$
and $\{b_k\}$ give \eqref{fFactor}.
\medskip\noindent
{\sc Case 2:} If $m=\nu$,
then
\begin{equation*}
\f(x)=p^2 + p\beta x +\alpha x^2 \quad\text{and}\quad
f(x)=p^{2\nu} + p^{\nu}\beta x +\alpha x^2.
\end{equation*}
Since $\f(x)$ is reducible in $\Z_p[x]$, so is $\g(x)=x^2-\beta x+\alpha$.
Thus there are numbers $\ell\in\N_0$ and $q\in\Z$ such that
\begin{equation*}
\beta^2-4\alpha=p^{2\ell}q
\;\text{ with } \gcd(p,q)=1.
\end{equation*}
Moreover, $\g(x)$ has a root in $\Z/p^{n}\Z$ for every $n\in\N$.
In particular, for $n=3\max(\ell,\nu)$,
there are integers $a$ and $r$ such that
\begin{equation}\label{RootofP1}
a^2 -\beta a +\alpha = p^{\mu} r
\;\text{ with } \gcd(p,r)=1,
\end{equation}
for some $\mu\ge 3\max(\ell,\nu)$.
Since $(\beta-2a)^2-(\beta^2-4\alpha)=4\g(a)$, we get
\begin{equation*}
(\beta-2a)^2=4p^\mu r + p^{2\ell}q=p^{2\ell}(4p^{\mu-2\ell}r + q),
\end{equation*}
hence we can write
\begin{equation}\label{RootofP2}
\beta-2a=p^{\ell}t \;\text{ with } \gcd(p,t)=1.
\end{equation}
Again, our goal is to construct sequences $\{a_k\}$ and $\{b_k\}$ such
that \eqref{fFactor} holds. This will be done with slightly different
algorithms for $\nu>\ell$ and $\nu\le\ell$. In both cases we let
\begin{gather*}
a_0=p^\nu=b_0, \quad s_1=\beta, \\
a_1=a, \quad s_2= p^{\mu-\nu}r,
\end{gather*}
where $a$ and $r$ are the integers from \eqref{RootofP1}.
With these choices, the first three terms in the expansion of
\eqref{fFactor} coincide with $f(x)$.
Assume $\nu>\ell$. Let
\[ \tilde a_1=0, \quad u_1=s_2=p^{\mu-\nu}r, \;\;\text{ and }\;\;
u_2=-p^{\mu-2\nu}ra_1.
\]
Let $t$ be as in \eqref{RootofP2}.
For $k\ge 2$ we will define $\tilde a_k$ and
$u_{k+1}$ such that the sequences defined by
\begin{equation}\label{FactorSeq1}
a_k = p^{\nu-\ell}\tilde a_k \quad\text{and}\quad
b_k = u_{k-1}-t\tilde a_{k-1}-a_k
\end{equation}
give the factorization \eqref{fFactor}. Note that
$s_{k+1}=a_{k+1}+b_{k+1}=u_{k}-t\tilde a_{k}$.
Let $\tilde a_2= p^{\mu-2\nu}$ and
$u_3= -p^{\mu-3\nu}\big[p^{\nu-\ell}(s_2-a_2)-ta_1\big]$. Thus
\[ p^\nu u_3 + \big[p^{\nu-\ell}(s_2-a_2)-ta_1\big]\tilde a_2=0. \]
Suppose we have defined $\tilde a_k$ and $u_{k+1}$ for $k=1,\dots,N-1$,
$N\geq 3$, and let
\[ v_N= a_1 u_N + a_2 s_N + \sum_{k=3}^{N-1}a_k(s_{N+2-k}-a_{N+2-k}). \]
Since $\gcd(p,\beta)=1$, the relation \eqref{RootofP2} implies
\[ \gcd(p,a_1)=1 \;\text{ and }\; \gcd(p^{\nu-\ell},
p^{\nu-\ell}(s_2-2a_2)-ta_1)=1.
\]
Therefore, there are $\tilde a_N, u_{N+1}\in \Z$ such that
\[ p^\nu u_{N+1} + \big[p^{\nu-\ell}(s_2-2a_2)-ta_1\big]\tilde a_N
+ \tilde v_N = 0.
\]
The sequences $\{a_k\}$ and $\{b_k\}$ defined by \eqref{FactorSeq1}
give \eqref{fFactor} when $\nu>\ell$.
Assume now $\nu\le\ell$. In this case, for $k\ge 2$ we will find
$\tilde a_k$ and $\tilde s_{k+1}$ such that the sequences defined by
\begin{equation*}
a_k=p^\ell \tilde a_k \quad\text{and}\quad
b_k=p^{3\ell-\nu}\tilde s_k-a_k
\end{equation*}
give a factorization of $f(x)$. Let $r$ and $t$ be as in
\eqref{RootofP1} and \eqref{RootofP2}, respectively.
Since $\gcd(p^\nu,t)=1$, there are $y,z\in\Z$ such that
\[ p^\nu y+tz + r=0. \]
Let $\tilde a_2=p^{\mu-2\ell-\nu}z a_1$, $\tilde s_2=p^{\mu-3\ell}r$,
and $\tilde s_3= p^{\mu-3\ell}y a_1$. Note that
\[ p^\ell \tilde s_3 + t\tilde a_2 + a_1p^{\ell-\nu}\tilde s_2 =0. \]
Suppose we have defined $\tilde a_k$ and $\tilde s_{k+1}$ for
$k=1,\dots,N-1$, $N\geq 3$, and let
\[ \tilde v_N= a_1 p^{\ell-\nu}\tilde s_N + \sum_{k=2}^{N-1}
\tilde a_k(p^{2\ell-\nu}\tilde s_{N+1-k}-\tilde a_{N+1-k}).
\]
Finally, since $\gcd(p^\ell,t)=1$, the equation
\[ 0=p^\ell \tilde s_{N+1} + t\tilde a_N + \tilde v_N \]
can be solved for $\tilde a_N, \tilde s_{N+1}\in \Z$. This implies
\begin{align*}
0
&=p^{3\ell} \tilde s_{N+1} + p^{2\ell}t\tilde a_N + p^{2\ell}\tilde v_N\\
&=p^{\nu} s_{N+1} +p^{\ell}t a_N + a_1s_N +
\sum_{k=2}^{N-1} a_k(s_{N+1-k}-a_{N+1-k}) \\
&=p^{\nu} s_{N+1} +(\beta-2a_1) a_N + a_1s_N +
\sum_{k=2}^{N-1} a_k b_{N+1-k}
=\sum_{k=0}^{N+1} a_k b_{N+1-k},
\end{align*}
as desired. This completes the proof.
\end{proof}
The main result of this section is the following.
\begin{theorem}\label{t:main}
Let $p$ be an odd prime and let $n,m\ge 1$. Let
$\alpha,\beta\in\Z$ be such that $\gcd(p,\alpha)=1$ and $\gcd(p,\beta)=1$.
The polynomial $f(x)=p^n+ p^m\beta x +\alpha x^2$
is reducible in $\Z[[x]]$ if and only if it is reducible in $\Z_p[x]$.
\end{theorem}
\begin{proof}
Using the fact that $f(x)$ is reducible in $\Z_p[x]$ iff $\f(x)$ is
reducible in $\Z_p[x]$, the statement of the theorem follows from
Proposition~\ref{p-easycase} and Proposition~\ref{p-hardcase}.
\end{proof}
\begin{remark}
The previous theorem is not valid when $m=0$. In fact, if $p\ndiv \beta$,
any power series of the form $p^n+\beta x+\cdots$ is irreducible in
$\Z[[x]]$. However, any polynomial $p^n+\beta x+\alpha x^2$ with
$\gcd(p,\beta)=1$ is reducible in $\Z_p[x]$.
\end{remark}
We finish this section with the remaining case: $\beta=0$.
\begin{proposition}\label{case:pbeta=0}
Let $p$ be an odd prime and let $n\ge 1$. Let $\alpha\in\Z$ be such that
$\gcd(p,\alpha)=1$. The polynomial $f(x)=p^n+ \alpha x^2$ is reducible
in $\Z[[x]]$ if and only if it is reducible in $\Z_p[x]$.
\end{proposition}
\begin{proof}
Recall that $f(x)=p^n+ \alpha x^2$ is reducible in $\Z_p[x]$
if and only if its discriminant $-4\alpha p^n$ is a nonzero square in
$\Z_p$. This in turn is the case if and only if $n$ is even and
$-\alpha$ is a square in $\Z/p\Z$. We will show that these conditions
on $n$ and $\alpha$ are equivalent to $f(x)$ being reducible in $\Z[[x]]$.
For $f(x)$ to admit a factorization of the form
\[
p^n+\alpha x^2 = (a_0+a_1x+a_2x^2+\cdots)(b_0+b_1x+b_2x^2+\cdots)
\]
it is necessary to solve the equations
\begin{gather*}
a_0=p^t \text{ and } b_0=p^s \;\text{ with } t+s=n, \\
0=p^t b_1 + p^s a_1, \\
\alpha = p^t b_2 + a_1b_1 + p^s a_2.
\end{gather*}
Since $\gcd(p,\alpha)=1$, these three equations can be solved in $\Z$
only when $s=t$, that is, when $n$ is even. Now, if
$n=2\nu$, we must have $a_0=b_0=p^\nu$, $s_1=a_1+b_1=0$, and
$\alpha=p^\nu(a_2+b_2)-a_1^2$. Thus, if $f(x)$ is reducible
in $\Z[[x]]$, then $-\alpha$ is a square in $\Z/p\Z$. On the other
hand, if $-4\alpha p^{2\nu}$ is a nonzero square in $\Z_p$,
so is $-\alpha$, i.e., $y^2 +\alpha$ has a root in $\Z_p$. Let $a_1$
and $s_2$ be integers such that
\[ a_1^2 + \alpha = p^{\nu} s_2. \]
Note that $\gcd(p^\nu,2a_1)=1$.
Therefore, there are integers $a_2$ and $s_3$ such that
\[ 0=p^{\nu}s_3 -2a_1a_2 + a_1s_2. \]
Finally, a factorization of $f(x)$ in $\Z[[x]]$ can be obtained with
the sequences $\{a_k\}$ and $\{s_{k+1}\}$ defined inductively
for $N\ge 3$ by the equation
\[ 0= p^\nu s_{N+1} - 2a_1a_N + v_N, \]
where $v_N=a_1s_N +\sum_{k=2}^{N-1}a_k(s_{N+1-k}-a_{N+1-k})$.
\end{proof}
\section{The case when the constant term is a power of 2}
\label{sec:EvenPrimePower}
In this section we consider polynomials of the form
\begin{equation*}
f(x)=2^n+2^m\beta x+\alpha x^2 \;\text{ with $n,m\ge 1$},
\end{equation*}
where $\alpha,\beta$ are assumed to be odd integers.
Observe that $\beta^2\equiv 1 \pmod 8$.
\begin{proposition}\label{2-easycase1}
Let $f(x)=2^n+ 2^m\beta x +\alpha x^2$ with $n,m\ge 1$.
\begin{enumerate}[$(i)$]
\item
If $2m<n$, then $f(x)$ is reducible in both $\Z_2[x]$ and $\Z[[x]]$.
\item
If $2m>n$ and $n$ is odd, then $f(x)$ is irreducible in both $\Z_2[x]$
and $\Z[[x]]$.
\end{enumerate}
\end{proposition}
\begin{proof}
$(i)$ If $2m<n$, then $4\alpha 2^{n-2m}\equiv 0\pmod 8$, so
$\beta^2-4\alpha 2^{n-2m}\equiv 1\pmod 8$. Thus the discriminant
$4^m(\beta^2-4\alpha 2^{n-2m})$ of $f(x)$ is a square in $\Z_2$ and $f(x)$
is reducible in $\Z_2[x]$.
Similarly, the polynomial $g(x)=2^{n-2m}x^2-\beta x+\alpha$ is also
reducible in $\Z_2[x]$. Therefore, the factorization algorithm given in
the proof of Proposition~\ref{p-easycase}$(i)$ works here as well and we
can conclude that $f(x)$ is reducible in $\Z[[x]]$.
\medskip
$(ii)$ In this case, the discriminant of $f(x)$ can be written as
\[ \Delta=2^n(2^{2m-n}\beta^2-4\alpha). \]
Recall that $n$ is odd. If $n=2m-1$, then
$\Delta=2^{n+1}(\beta^2-2\alpha)$. Since $\alpha$ is odd and
$\beta^2\equiv 1\pmod8$, we have $\beta^2-2\alpha\not\equiv 1\pmod 8$
which implies that $\Delta$ is not a square in $\Z_2$. If $n\le 2m-3$,
then $\Delta=2^{n+2}(2^{2m-n-2}\beta^2-\alpha)$ and we get, once again,
that $\Delta$ is not a square in $\Z_2$ since $2^{2m-n-2}\beta^2-\alpha$
and $n+2$ are both odd numbers.
In conclusion, $f(x)$ is irreducible in $\Z_2[x]$.
That $f(x)$ is irreducible in $\Z[[x]]$ follows verbatim from the
arguments in the proof of Proposition~\ref{p-easycase}$(ii)$.
\end{proof}
\begin{proposition}\label{2-easycase2}
If $n=2m$, then $f(x)$ is
irreducible in both $\Z_2[x]$ and $\Z[[x]]$.
\end{proposition}
\begin{proof}
Since $\beta^2\equiv 1\pmod 8$ and $\alpha$ is odd, we have
$\beta^2-4\alpha\not\equiv 1\pmod 8$. Thus the integer
$4^m(\beta^2-4\alpha)$, the discriminant of $f(x)$, is not a square
in $\Z_2$ which implies that $f(x)$ is irreducible in $\Z_2[x]$.
Suppose now that $f(x)$ is reducible in $\Z[[x]]$. Then there are power
series $a(x)$, $b(x)\in\Z[[x]]$ such that
\[ f(x)=(2^m+a_1x+a_2x^2+\cdots)(2^m+b_1x+b_2x^2+\cdots) \]
with
\begin{align*}
\beta= a_1+b_1, \quad
\alpha= 2^m(a_2+b_2) +a_1(\beta-a_1).
\end{align*}
Since $\beta$ is odd, the number $a_1(\beta-a_1)$ is always even,
a contradiction.
\end{proof}
It remains to analyze the case when $2m>n$ and $n=2\nu$ for some
$\nu\in\N$. To this end, we will consider the cases $m>\nu+1$ and
$m=\nu+1$, separately.
\begin{proposition}\label{case:m>nu+1}
Let $m>\nu+1$. The polynomial $f(x)=4^{\nu}+2^m\beta x+\alpha x^2$
is reducible in $\Z[[x]]$ if and only if it is reducible in $\Z_2[x]$.
\end{proposition}
\begin{proof}
First of all, observe that the discriminant of $f(x)$ can be written as
\[ \Delta= 4^{\nu+1}(4^{m-(\nu+1)}\beta^2-\alpha). \]
For $\Delta$ to be a square in $\Z_2$, we need
$4^{m-(\nu+1)}\beta^2-\alpha\equiv 1\pmod 8$. If $m-(\nu+1)\ge 2$, this
holds iff $-\alpha\equiv 1\pmod 8$, and if $m-(\nu+1)=1$, the discriminant
$\Delta$ is a square in $\Z_2$ iff $4-\alpha\equiv 1\pmod 8$. In other
words,
\begin{itemize}
\item if $m-(\nu+1)\ge 2$, $f(x)$ is reducible in $\Z_2[x]$
$\Longleftrightarrow$ $\alpha\equiv 7\pmod 8$,
\item if $m-(\nu+1)= 1$, $f(x)$ is reducible in $\Z_2[x]$
$\Longleftrightarrow$ $\alpha\equiv 3\pmod 8$.
\end{itemize}
We will prove the corresponding statements in $\Z[[x]]$.
Assume first that $f(x)$ is reducible in $\Z[[x]]$ and can be factored as
\begin{equation}\label{fFactor2}\mytag
f(x)=\Big(\sum_{k=0}^\infty a_k x^k\Big)
\Big(\sum_{k=0}^\infty b_k x^k\Big).
\end{equation}
Then we must have $a_0=2^\nu=b_0$, and using the notation $s_k=a_k+b_k$,
\begin{equation}\label{first3equationsp=2}\mytag
\begin{split}
2^m\beta &=2^\nu s_1, \text{ which implies } s_1=2^{m-\nu}\beta, \\
\alpha &=2^\nu s_2 + (2^{m-\nu}\beta-a_1)a_1, \\
0 &=2^\nu s_3+(2^{m-\nu}\beta-2a_1)a_2+a_1s_2.
\end{split}
\end{equation}
The second equation implies that $a_1$ is odd, and the third one gives
that $s_2$ is even. Thus we get
\begin{equation}\label{CharEquationp=2}\mytag
a_1^2 - 2^{m-\nu}\beta a_1 +\alpha =
2^\nu s_2= 2^{\nu+1} t_2 \;\text{ for some } t_2\in\Z.
\end{equation}
Since $\nu\ge 1$ and $m-\nu\ge 2$, this equation implies
$a_1^2+\alpha\equiv 0 \pmod 4$, i.e.,
\[ \alpha\equiv 3 \pmod 8 \;\;\text{ or }\;\;
\alpha\equiv 7 \pmod 8. \]
If $\nu>1$ and $\alpha\equiv 3 \pmod 8$, then
\[ 0\equiv 2^{\nu+1} t_2=a_1^2 - 2^{m-\nu}\beta a_1
+\alpha\equiv 4-2^{m-\nu}\beta a_1 \pmod 8. \]
This implies $2^{m-\nu}\beta a_1\equiv 4\pmod 8$, and so $m-\nu=2$,
that is, $m-(\nu+1)=1$. If $\nu>1$ and $\alpha\equiv 7 \pmod 8$, then
$-2^{m-\nu}\beta a_1\equiv 0\pmod 8$, which is possible only when
$m-(\nu+1)\ge 2$.
If $\nu=1$, then $f(x)=4+2^m\beta x+\alpha x^2$ and
\eqref{CharEquationp=2} becomes
\begin{equation*}
a_1^2 - 2^{m-1}\beta a_1 +\alpha = 2 s_2.
\end{equation*}
By Lemma~\ref{l:ChoosingLemma} we can choose $a_2=a_3=0$, so the third
equation in \eqref{first3equationsp=2}, and the next one, take the form
\begin{align*}
0&=2s_3+a_1s_2, \\
0&=2s_4+a_1s_3.
\end{align*}
Thus $s_3$ is even, hence $s_2\equiv 0\pmod 4$ and
$a_1^2 - 2^{m-1}\beta a_1 +\alpha\equiv 0\pmod 8$. As above,
$\alpha\equiv 3 \pmod 8$ implies $m-1=2$, and $\alpha\equiv 7 \pmod 8$
implies $m-1\ge 3$.
We now prove the reducibility of $f(x)$ in $\Z[[x]]$ under the conditions
on $m$, $\nu$, and $\alpha$ specified above for the reducibility in
$\Z_2[x]$. To this end, consider the polynomial
$g(x)=x^2-2^{m-\nu}\beta x +\alpha$ whose discriminant is
$2^2(4^{m-(\nu+1)}\beta^2-\alpha)$.
If $m-(\nu+1)= 1$ and $\alpha\equiv 3\pmod 8$, then
$4^{m-(\nu+1)}\beta^2-\alpha=4\beta^2-\alpha\equiv 1\pmod 8$.
If $m-(\nu+1)\ge 2$ and $\alpha\equiv 7\pmod 8$, then
$4^{m-(\nu+1)}\beta^2-\alpha\equiv 1\pmod 8$. Thus, in both cases,
the discriminant of $g(x)$ is a square in $\Z_2$. Hence $g(x)$ is
reducible and thus has a root in $\Z_2$.
Let $\tilde a_1,t_2\in\Z$ be such that
\begin{equation*}
\tilde a_1^2 - 2^{m-\nu}\beta \tilde a_1 +\alpha = 2^{2\nu+1} t_2.
\end{equation*}
Let $a_0=2^\nu=b_0$, $a_1=\tilde a_1$, $s_1=2^{m-\nu}\beta$, and
$s_2=2^{\nu+1}t_2$. With these choices, $f(x)$ coincides with the first
three terms of the product in \eqref{fFactor2}.
Since $m>\nu+1$, we have that $2^{m-\nu-1}\beta-a_1$ is odd, hence there
are integers $\tilde a_2$ and $t_3$ such that
\[ 0=2^\nu t_3+ (2^{m-\nu-1}\beta-a_1)\tilde a_2 + a_1t_2. \]
If we let $a_2=2^\nu\tilde a_2$ and $s_3=2^{\nu+1}t_3$, then
\begin{align*}
0&= 2^{\nu+1}\big(2^\nu t_3+ (2^{m-\nu-1}\beta-a_1)\tilde a_2
+ a_1t_2\big)\\
&= 2^\nu(2^{\nu+1}t_3)+(2^{m-\nu}\beta-2a_1)(2^\nu \tilde a_2)
+a_1(2^{\nu+1}t_2)\\
&=2^\nu s_3+(2^{m-\nu}\beta-2a_1)a_2 +a_1s_2.
\end{align*}
Suppose we have defined $\tilde a_k$ and $t_{k+1}$ for $k=1,\dots,N-1$,
$N\ge 3$, and let
\[ w_N= a_1 t_N+2^{\nu-1}\sum_{k=2}^{N-1}\tilde a_k
(2t_{N+1-k}-\tilde a_{N+1-k}). \]
As before, there are $\tilde a_N,t_{N+1}\in \Z$ such that
\[ 0=2^\nu t_{N+1}+(2^{m-\nu-1}\beta-a_1)\tilde a_N + w_N. \]
If we let $a_k=2^\nu\tilde a_k$ and $s_k=2^{\nu+1}t_k$ for every $k\ge 3$,
then the sequences $\{a_k\}$ and $\{b_k\}$ with $b_k=s_k-a_k$ give a
factorization of $f(x)$.
\end{proof}
\begin{proposition}\label{case:m=nu+1}
The polynomial $f(x)=4^{\nu}+2^{\nu+1}\beta x+\alpha x^2$
is reducible in $\Z[[x]]$ if and only if it is reducible in $\Z_2[x]$.
\end{proposition}
\begin{proof}
Assume first that $f(x)$ is reducible in $\Z_2[x]$. Then
$4^{\nu+1}(\beta^2-\alpha)$ must be a square in $\Z_2$, which implies
\begin{equation*}
\beta^2 - \alpha = 2^{2\ell} q \;\text{ with }
\ell\in\N_0 \text{ and } q\equiv 1\!\!\pmod 8.
\end{equation*}
Thus the polynomial $g(x)=x^2-2\beta x +\alpha$ is also reducible in
$\Z_2[x]$, and so it has a root in $\Z_2$. Let $\tilde a_1, t_2\in\Z$
be such that
\[ \tilde a_1^2 - 2\beta\tilde a_1+\alpha=2^{2\ell+\nu+2}t_2. \]
Let $u$ be the odd integer such that $\beta-\tilde a_1=2^\ell u$.
As in the proof of Proposition~\ref{case:m>nu+1},
a factorization of $f(x)$ in $\Z[[x]]$ can be obtained as follows.
We let
$a_0=2^\nu=b_0$, $a_1=\tilde a_1$, $s_1=2\beta$, $s_2=2^{2\ell+2}t_2$,
and for $N\ge 2$, we define $\tilde a_N$ and $t_{N+1}$
inductively by means of the equation
\[ 0=2^\nu t_{N+1}+u\tilde a_N + w_N, \]
where
\[ w_N= a_1 t_N + \sum_{k=2}^{N-1}\tilde a_k
(2^{\ell+1}t_{N+1-k}-\tilde a_{N+1-k}). \]
For $k\ge 2$, we then let $a_k=2^{\ell+1}\tilde a_k$ and
$b_k=s_k-a_k=2^{2\ell+2}t_{k}-a_k$.
Assume now that $f(x)$ is reducible in $\Z[[x]]$ and consider
$\f(x)=4+4\beta x+\alpha x^2$. By Lemma~\ref{lemmaA}
and Lemma~\ref{lemmaB} with $p=2$, $m=\nu+1$, and $\ell=2$,
we get that $\f(x)$ is reducible in $\Z_2[x]$. Finally, since the
discriminant of $f(x)$ is $4^{\nu-1}$ times the discriminant of $\f(x)$,
we conclude that $f(x)$ is reducible in $\Z_2[x]$, too.
\end{proof}
\begin{theorem}\label{t:mainp=2}
Let $\alpha,\beta\in\Z$ be odd. The polynomial $f(x)=2^n+ 2^m\beta x
+\alpha x^2$ with $n,m\ge 1$ is reducible in $\Z[[x]]$ if and only if
it is reducible in $\Z_2[x]$.
\end{theorem}
\begin{remark}
The previous theorem is not valid when $m=0$.
In fact, $2^n+\beta x+\alpha x^2$ is always reducible in $\Z_2[x]$,
but irreducible in $\Z[[x]]$.
\end{remark}
\begin{proposition}\label{case:2beta=0}
Let $\alpha\in\Z$ be odd. The polynomial $f(x)=2^n+ \alpha x^2$ is
reducible in $\Z[[x]]$ if and only if it is reducible in $\Z_2[x]$.
\end{proposition}
\begin{proof}
As in Proposition~\ref{case:pbeta=0}, it can be easily checked that if
$n$ is odd, then $f(x)$ is irreducible in both $\Z_2[x]$ and $\Z[[x]]$.
If $n=2\nu$, then the statement follows from the arguments in the
proof of Proposition~\ref{case:m>nu+1} for the case when $m>\nu+2$.
\end{proof}
\section{Further reducibility criteria}
\label{sec:FurtherCriteria}
In this last section we briefly discuss the factorization in
$\Z[[x]]$ of power series whose quadratic part is a polynomial
like the ones studied in the previous sections. More precisely,
we consider power series of the form
\begin{equation}\label{GenPowerSeries}
f(x)=p^n + p^m\beta x + \alpha x^2 +\sum_{k=3}^\infty c_k x^k,
\end{equation}
where $\alpha$ and $\beta$ are integers such that
$\gcd(p,\alpha)=1=\gcd(p,\beta)$.
For simplicity, we only discuss the case when $p$ is an odd prime.
We will focus on the situations for which the arguments in
Section~\ref{sec:OddPrimePower} extend with little or no additional
effort. For instance, if $m\not=\frac{n}{2}$, the reducibility of $f(x)$
in $\Z[[x]]$ follows the same pattern as the reducibility of its
quadratic part. In fact, we can use the exact same arguments from
Section~\ref{sec:OddPrimePower} to prove the following two propositions.
\begin{proposition}
If $2m<n$, then \eqref{GenPowerSeries} is reducible in $\Z[[x]]$.
If $2m>n$ and $n$ is odd, then \eqref{GenPowerSeries} is irreducible.
\end{proposition}
\begin{proposition}
If $2m>n$ and $n$ is even, then \eqref{GenPowerSeries} is reducible in
$\Z[[x]]$ if and only if $-\alpha$ is a quadratic residue \textup{mod} $p$.
\end{proposition}
If $2m=n$, the situation is in general more involved and the reducibility
of $f(x)$ depends on the roots of $x^2-\beta x +\alpha$.
The following proposition is easy to prove.
\begin{proposition}
If $2m=n$ and the polynomial $x^2-\beta x +\alpha$ has a simple root
in $\Z/p^m\Z$, then \eqref{GenPowerSeries} is reducible in $\Z[[x]]$.
\end{proposition}
If $x^2-\beta x +\alpha$ has a double root in $\Z/p^m\Z$, it is not
enough to look at the quadratic part of $f(x)$ and its reducibility
depends on the coefficients $c_k$. To illustrate this fact,
consider for example the power series
\[ f(x)=p^2+p\beta x+\alpha x^2 + c_3x^3 + c_4x^4+ \cdots, \]
with $\alpha,\beta\in\Z$ such that $\beta^2-4\alpha=p^2 q$, where $q$ is
a quadratic residue mod $p$ with $\gcd(p,q)=1$.
In order to get a proper factorization
$f(x)=a(x)b(x)$ in $\Z[[x]]$, we must have $a_0=p=b_0$,
$\beta=s_1$, as well as
\begin{align*}
\alpha &= ps_2+a_1(\beta-a_1), \\
c_3 &= ps_3 + (\beta-2a_1)a_2 + a_1s_2,
\end{align*}
where $s_k=a_k+b_k$. Then $(\beta-2a_1)^2-(\beta^2-4\alpha)=4ps_2$,
which implies $p\mid s_2$. Therefore, $f(x)$ is irreducible in $\Z[[x]]$
unless $p\mid c_3$.
On the other hand, if $p^2\mid c_k$ for every $k\ge 3$, then with the
same assumptions on $\alpha$ and $\beta$ as above, we can find
$a_k,b_k\in\Z$ such that $a(x)=\sum a_kx^k$ and $b(x)=\sum b_kx^k$ give
a proper factorization $f(x)=a(x)b(x)$. Note that $\beta^2-4\alpha$
is a square in $\Z_p$, so the polynomial $g(x)=x^2-\beta x +\alpha$
is reducible in $\Z_p[x]$. In particular, $g(x)$ has a root in
$\Z/p^3\Z$, so there are $a,r\in\Z$ such that
\[ a^2 - \beta a +\alpha = p^3 r. \]
Moreover, since $(\beta-2a)^2-(\beta^2-4\alpha)=4p^3r$ and $p^2\mid
(\beta^2-4\alpha)$, we have $p\mid(\beta-2a)$. In fact, there is an
integer $t$ with $\gcd(p,t)=1$ such that
\[ \beta-2a=pt. \]
Choose $a_1=a$, $\,\tilde s_2=r$, and write $c_{k+1}= p^2\tilde c_{k+1}$.
Since $\gcd(p,t)=1$, for $k\ge 2$ we can choose $\tilde a_k$ and
$\tilde s_{k+1}$ inductively as integer solutions of the equation
\[ \tilde c_{k+1}=p\tilde s_{k+1}+t\tilde a_k+ a_1\tilde s_k +
\sum_{j=2}^{k-1} \tilde a_j(p\tilde s_{k+1-j}-\tilde a_{k+1-j}). \]
If we let $a_k=p\tilde a_k$ and $s_k=p^2\tilde s_k$, then multiplication
by $p^2$ gives
\begin{align*}
c_{k+1} &= p s_{k+1}+pt a_k+ a_1 s_k +
\sum_{j=2}^{k-1} a_j(s_{k+1-j}-a_{k+1-j}) \\
&= ps_{k+1}+(\beta-2a_1)a_k + a_1s_k + \sum_{j=2}^{k-1} a_jb_{k+1-j} \\
&= \sum_{j=0}^{k+1} a_jb_{k+1-j}.
\end{align*}
In other words, $a(x)$ and $b(x)$ provide a factorization of $f(x)$ in
$\Z[[x]]$, as claimed.
| 113,807
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Backup plan
"5 Steinways, 50 Fingers," the 5 Browns, 7 p.m., Jan. 23 at First United Methodist Church
Before you know it
Gabriel Iglesias, 8 p.m., Jan. 31 at the Pikes Peak Center
Potato pick
Trust Me series premiere, 8 p.m., Jan. 26 on TNT
Have you heard?
Download this free font that uses 20 percent less ink: ecofont.eu
| 133,291
|
Specialists in cybersecurity detected an error that allows hackers run malware code from Evernote.A[,dropcap]s a result, intruders can use specially created URI in a note that would lead to the attack. Through link they offer user to open any malware file, for instance, “../../../../malware.app“.
Such vulnerabilities united under the term “path traversal”.
While Evernote provides technical opportunity to share notes, hackers can use this vulnerability and send malware notes in .enex format to supposed victims.
Vulnerability touched Apple laptops with macOS. Interestingly, on other platforms program does not endanger user’s confidentiality.
Recently issue traced with identification number CVE-2019-10038, and patch released for Evernote versions 7.10 Beta and 7.9.1. GA on macOS.
Correction is already working. Now it looks as a notification that arises with the attempt to open suspicious link. It is important to add that similar error found in Electronic Arts Origin service.
Source:
| 311,428
|
TITLE: Why do we categorize all other (iso.) singularities as "essential"?
QUESTION [12 upvotes]: When dealing with isolated singularities, we classify each of these points as removable, pole (of order $k$), or essential. It easy to see that all isolated singularities must be of one of these three categories by construction:
We define any isolated singularity that isn't removable or pole as an essential singularity.
Why is it that we throw all other singularities into this category? Do we not care about essential singularities to classify them further? That is, are removable singularities and poles (of order $k$) the only isolated singularities we care about?
REPLY [14 votes]: An important point is that a "pole" is actually the same thing as a removable singularity, if we think of our function as a map that takes values on the Riemann sphere (which is the complex plane with a point at $\infty$ added; the complex structure near $\infty$ comes from the map $z\mapsto 1/z$).
So a function that has a removable singularity or a pole at $z_0$ doesn't have a "real" singularity there at all; rather, we can extend the function to an analytic or meromorphic function in $z_0$. If we cannot extend the function in this way, the singularity is indeed "essential"; i.e., we cannot get rid of it. Thus the terminology is not one that is merely used for convenience or pedagogical purposes; rather, it is extremely natural.
As has already been mentioned, the magic of complex numbers results in many beautiful facts about essential singularities: functions with these singularities are very far from extending continuously.
The simplest of these facts is the Casorati-Weierstraß Theorem: The image of a neighborhood of an essential singularity is dense in the complex plane.
This is just a consequence of the removable singularities theorem. (If $f$ omitted a neighborhood of $a$, we could postcompose $f$ with a Möbius transformation that takes $a$ to infinity and see that the resulting function has a removable singularity.)
The most well-known result of this type is Picard's theorem which was already mentioned.
There are various beautiful strengthenings of Picard's theorem that arise from Nevanlinna theory, and Ahlfors's theory of covering surfaces.
So all essential singularities have some things in common, but on the other hand this should not lead us to believe that they are all the same. What they have in common is complicated behaviour, but they can be complicated in very different ways! Indeed, different transcendental entire functions (those that have an essential singularity at infinity; i.e. are not polynomials) can vary very much with respect to their behavior near infinity. Just for example, for some such functions, such as $z\mapsto e^z$, there exist curves tending to infinity on which the function is bounded, while for others this is not the case.
| 122,915
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Established: 1948
Location: Bangalore, India
Website: Amrut Distilleries
Visiting: Appointment only
Facebook: Amrut Single Malt
Twitter: Unknown
Instagram: Unknown Private Limited began in 1948 under the name and style of Amrut Laboratories, with an initial investment of barely a few lakhs.
| 283,094
|
\begin{document}
\begin{frontmatter}
\title{Study of voltage cycling conditions on Pt oxidation and dissolution
in polymer electrolyte fuel cells}
\author[mymainaryaddress,mysecondaryaddress]{V.A.~Kovtunenko}
\ead{victor.kovtunenko@uni-graz.at}
\author[mythirdaddress]{L.~Karpenko-Jereb\corref{mycorrespondingauthor}}
\ead{karpenkojereb@gmail.com}
\cortext[mycorrespondingauthor]{Corresponding author}
\address[mymainaryaddress]{Institute for Mathematics and Scientific Computing,
Karl-Franzens University of Graz, NAWI Graz, Heinrichstr. 36, 8010 Graz, Austria}
\address[mysecondaryaddress]{Lavrentyev Institute of Hydrodynamics,
Siberian Division of the Russian Academy of Sciences, 630090 Novosibirsk, Russia}
\address[mythirdaddress]{Institute of Electronic Sensor Systems,
Graz University of Technology, Inffeldgasse 10/II, Graz 8010, Austria}
\begin{abstract}
This paper is devoted to study the electrochemical behavior of Pt catalyst
in a polymer electrolyte fuel cell at various operating conditions and
at different electric potential difference (also known as voltage) cycling
applied in accelerated stress tests.
The degradation of platinum is considered with respect to
the Pt ion dissolution and the Pt oxide coverage of catalyst
described by a one-dimensional model.
In the model, degradation rate increases with temperature and
decreasing particle diameter of Pt nano-particles.
The theoretical study of the underlying diffusion system with the nonlinear
reactions is presented by analytical methods and
gives explicit solutions through a first integral of the ODE system.
Numerical tests are obtained using a second order implicit-explicit scheme.
The computer simulation shows that the lifetime of the catalyst depends
on the voltage profile and the upper potential level.
By this Pt mass loss is more significant at the membrane surface than at the gas diffusion layer.
\end{abstract}
\begin{keyword}
polymer-electrolyte fuel cell\sep
platinum surface blockage \sep
platinum dissolution \sep
potential cycling \sep
reaction-diffusion \sep
Butler--Volmer reaction rate
\MSC[2010] 78A57 \sep 80A30 \sep 80A32 \sep 35K57
\end{keyword}
\end{frontmatter}
\section{Introduction}\label{sec1}
At present, the polymer electrolyte fuel cells are very extensively developed
as power sources for application in portable computers, heating systems
as well for leisure yachts, aircraft and vehicles \cite{EK/17,KJA/18,SS/18,Wee/07}.
In commercial PEM fuel cells, the most used material for catalyst is platinum.
Effective usage of this valuable metal is challenging task in elaboration
of the modern fuel cell systems with high durability and long lifetime.
Mathematical modelling helps to better understand phenomena causing
the chemical degradation on the Pt surface and allows predicting a decline
in electro-chemical surface area (ECSA) of the catalyst and effectivity
of the fuel cell \cite{KJSFT/16,KB/11,ZYLZSWZ/20}.
In the last decades, numerous publications have been devoted to develop mathematical
models prediction Pt oxidation and dissolution in the catalyst layer of PEMFCs.
In 2007, Zhang et al \cite{ZLGLG/07} suggested a simple model accounting
Pt dissolution and further precipitation within the membrane of PEMFCs.
One year late, Bi and Fuller \cite{BF/08} published a dynamic model
applied to estimate Pt mass loss during potential cycling.
In 2009, Holby et al. \cite{HSSHM/09} developed a kinetic model
taking into account effect of Pt particle size on the rate of the degradation.
It was demonstated that, due to rapid changes in the Gibbs--Thomson energy,
particle size effects dominate degradation for 2 nm particles
but play almost no role for 5 nm particles.
In 2015 Hiraoka et al. \cite{HKMM/15} proposed a model for the Pt particle
growth based on the Gibbs--Thomson equation and simulated change
in the Pt particle size distribution during electric potential cycling.
Using the developed model, the authors investigated an effect of
high potential limit and Pt particle diameter on particle size growth.
The results showed that lowing high potential in the cycling
the Pt particle size grows more slowly.
A decrease in particle size accelerates Pt dissolution.
In the same year, Li et al. \cite{LMGAW/15} published a mathematical model
taking into account effect of RH, T and catalyst layer thickness on the loss
in electrochemical active area of the Pt catalyst.
The simulation results were in good agreement with experiments.
The conducted simulation for a triangle potential cycle demonstrated that
thinning the cathode catalyst layer would induce more rapid ECSA loss.
ECSA increases with rising temperature and higher relative humidity.
In 2018, using a simplified model Baricci et al. \cite{BBYGMC/18} studied
changes in roughness through the catalyst layer thickness.
Non-uniform degradation is observed in the catalyst layer consequently
to the formation of a platinum depleted region next to the membrane,
which, according to the model, results from diffusion and precipitation
of dissolved platinum into the membrane.
Recently, Koltsiva et al. \cite{KVSFB/18} suggested a novel model,
which for the first time considers five phenomena simultaneously
proceeding on the Pt/C catalyst surface:
platinum nanoparticles electrochemical dissolution,
particle growth due to Ostwald ripening,
migration of nanoparticles along the carbon support,
coalescence of fine particles,
diffusion of platinum ions in the ionomer.
The present paper is focused on study the effect of kind of voltage cycling, where
voltage is commonly adopted as electric potential difference versus a reference of 0 (V),
on the electro-chemical behavior of Pt catalyst in a polymer electrolyte fuel cell (PEFC).
For this purpose we have developed a mathematical model based
on the physico-chemical data of the Pt dissolution and oxidation reactions
and taking account diffusion of building Pt ions into the ionomer membrane.
The two degradation phenomena of the platinum ion (Pt$^{2+}$)
dissolution and the platinum oxide (PtO) formation of Pt catalyst layer (coverage)
in polymer-electrolyte membrane fuel cells (PEMFC)
are studied theoretically at different voltage cycling conditions.
To describe these phenomena, the degradation model due to
Holby \cite{HM/12,HSSHM/09} is utilized,
which is one-dimensional (1d) across the catalyst layer (CL) thickness
and accounts for diffusion of Pt ions.
For mathematical modeling of interface reactions in multi-phase media
within complete electrokinetic Poisson--Nernst--Planck equations,
see e.g. \cite{FK/15,GGK/18,KZ/18}.
For mathematical approaches which are suitable to describe and to test
a mechanical degradation due to fracture phenomena,
we refer to \cite{HKK/07,IKR/20,KK/00}.
The paper is organized in the following way:
after the introduction we describe theoretical approach developed, where
the part ``Degradation model of Pt catalyst'' displays geometrical, physical and
chemical properties of the catalyst layer and the ionomer membrane used for further simulation;
``Theoretical and numerical methods'' introduces the mathematical and the numerical models;
``Results'' presents simulation results and their discussion;
``Discussion'' and ``Conclusion'' summarize the most important findings of the current study.
The mathematical model is given by a coupled system of nonlinear
reaction-diffusion equations with modified Butler--Volmer reaction rates
for three unknown variables:
Pt$^{2+}$ concentration, Pt particle diameter, and PtO coverage ratio.
For its numerical solution we use a second order implicit-explicit scheme
following \cite{ARS/97}.
Neglecting diffusion of Pt ions,
in \ref{A} the resulting nonlinear reaction equations are reduced
to two unknown variables with the help of a first integral of the system,
while in \ref{B} an example of analytical solution is constructed.
\section{Degradation model of Pt catalyst}\label{sec2}
\paragraph{Approximations}
The developed model of the Platinum on carbon (Pt/C) degradation
is based on the following approximations:
\begin{itemize}
\item[1.]
This is a one-dimensional and a dynamic model.
\item[2.]
The model considers two layers of a polymer electrolyte fuel cell (PEFC):
the cathode catalyst layer (CL) with length $L_{\rm CL}$ and
the polymer electrolyte membrane (PEM) with length $L_{\rm PEM}$.
The catalyst layer is filled with catalyst particles: spherical Pt nano-particles
placed on C-support bound with the membrane by perfluorinated sulfonated ionomer,
the membrane is made from the same ionomer.
\item[3.]
The degradation of Pt nano-particles are caused by Pt oxidation and Pt dissolution,
which are described by the following electro-chemical reactions:
\begin{subequations}\label{1}
\begin{equation}\label{1a}
{\rm Pt}_{\rm (s)}\longleftrightarrow {\rm Pt}^{2+}_{\rm (aq)} +2{\rm e}^-,
\end{equation}
\begin{equation}\label{1b}
{\rm Pt}_{\rm (s)} +{\rm H}_2{\rm O}_{\rm (aq)}\longleftrightarrow
{\rm PtO}_{\rm (s)} +2{\rm H}^+_{\rm (aq)} +2{\rm e}^-.
\end{equation}
\end{subequations}
\item[4.]
The rates of the degradation reactions \eqref{1} are simulated by
a modified Butler--Volmer equation (see \eqref{3} and \eqref{4})
taking into account Gibbs--Thomson's effect:
dependence of a surface potential on nano-particle size as well as
influence of the surface potential on the potential gradient in the system.
\item[5.]
The platinum ions occurring due to the Pt dissolution \eqref{1} diffuse through
the ionomer phase of the catalyst layer into the polymer electrolyte membrane.
The diffusion of Pt ions into the gas diffusion layer is impossible
because this layer does not possess ionic conductivity.
This determines boundary conditions \eqref{5e} for the Pt ions diffusion \eqref{5a}.
\item[6.]
On balance, the model takes into account an effect of potential gradient,
Pt particle size, temperature, relative humidity as well as other phenomena
(see parameter gathered in Table~\ref{tab1})
on the degradation rates of platinum catalyst and allows calculating
the platinum ion concentration (${\rm Pt}^{2+}$), the particle diameter,
and the platinum oxide (PtO) coverage ratio from governing relations \eqref{5}.
\end{itemize}
With respect to the relative humidity in point 6 we remark the following.
In our model we consider an effect of pH on the reaction rate of Pt oxidation.
The pH depends on the dissociation degree of sulfonyl groups of the ionomer.
In the simulated cases we suppose that the sulfonyl groups
are completely dissociated and $pH=0$.
Generally, pH is function of proton concentration, in PEMFCs pH depends on relative humidity.
\paragraph{Model variables}
For a semi-infinite cathode catalyst layer of thickness $L$,
we introduce a spatial variable $x\in[0,L]$ across the CL such that
one end point $x=0$ corresponds to the CL-gas diffusion layer (GDL) interface,
and the other end point $x=L$ confirms the CL-PEM interface.
By this, we allow non-steady state operating conditions for CL
with respect to time $t\ge0$ and the spatial dependence on
$x\in[0,L]$ due to diffusion phenomena.
The model of degradation is sketched in Figure~\ref{degradation}.
\begin{figure}[hbt!]
\begin{center}
\epsfig{file=degradation.pdf,width=.6\textwidth,angle=0}
\caption{The model of degradation.}
\label{degradation}
\end{center}
\end{figure}
We emphasize that Figure~\ref{degradation} shows a schematic of the model configuration,
when the Pt particles are fully surrounded by ionomer on the carbon support.
The real scenario at Pt particles has a partial coverage by ionomer (for ion transfer)
and a partial coverage by carbon network (for electron transfer)
and a partial open space to air or gas (for oxygen gas diffusion).
However, those factors are not addressed in the current model.
Let Pt particles be hemispheres posed with density $\rho_{\rm Pt}$
and loading $p_{\rm Pt}$ on a carbon support,
such that the Pt volume fraction across CL can be estimated as follows
$\varepsilon_{\rm Pt} =\frac{p_{\rm Pt}/\rho_{\rm Pt}}{L}$
with $\varepsilon_{\rm Pt}<1$.
The Pt nanoparticle is assumed to be spherical of diameter $d_{\rm Pt}$
and volume $V_{\rm Pt} =\frac{4}{3} \pi (\frac{d_{\rm Pt}}{2})^3$.
Then the Pt number concentration in CL
is $N_{\rm Pt} =\frac{\varepsilon_{\rm Pt}}{V_{\rm Pt}}$.
For the parameter values from Table~\ref{tab1},
$\varepsilon_{\rm Pt}\approx 2$\%,
$V_{\rm Pt}\approx 1.5\times10^{-20}$ $({\rm cm}^3)$,
and $N_{\rm Pt}\approx 1.32\times10^{18}$ $(1/{\rm cm}^3)$.
The unknown constituents entering \eqref{1}
are the ${\rm Pt}^{2+}$ concentration $c$ ({\rm mol/cm$^3$}),
Pt particle diameter $d$ ({\rm cm}), and PtO coverage ratio $\theta$,
that are the time-space dependent functions such that
\begin{equation}\label{2}
c(t,x)\ge0,\quad d(t,x)\ge0,\quad 0\le\theta(t,x)\le1.
\end{equation}
Based on a modified Butler--Volmer equation,
the following reaction rates in units of mol/(cm$^2\cdot$s)
are established in \cite{HM/12}:
for the Pt ion dissolution \eqref{1a}:
\begin{subequations}\label{3}
\begin{equation}\label{3a}
r_{\rm dissol}(c, d, \theta, V) =B_1(d, \theta) e^{(1-\beta_1)
B_4(d, \theta) V} -c B_2(d, \theta) e^{-\beta_1 B_4(d, \theta) V},
\end{equation}
where the quantities $B_1$ in mol/(cm$^2\cdot$s),
$B_2$ in cm/s, $B_4$ in C/J, and $\gamma_0$ in {\rm J/cm$^2$} are
\begin{equation}\label{3b}
\begin{split}
&B_1(d, \theta) =\nu_1 \mathit{\Gamma} (1 -\theta) e^{\frac{1}{R T}
(-H_{1,{\rm fit}} -n F (1 - \beta_1) (U_{\rm eq}
-\frac{4 \mathit{\Omega} \gamma_0(\theta)}{n F d}) )},\\
&B_2(d, \theta) ={\textstyle\frac{\nu_2 \mathit{\Gamma}}{c_{\rm ref}}} (1 -\theta)
e^{\frac{1}{R T} (-H_{1,{\rm fit}} +n F \beta_1 (U_{\rm eq}
-\frac{4 \mathit{\Omega} \gamma_0(\theta)}{n F d}) )},\\
&B_4(d, \theta) ={\textstyle\frac{F}{R T}} (n
-{\textstyle\frac{4\mathit{\Omega} \mathit{\Gamma} n_2 \theta}{d}}),\\
&\gamma_0(\theta) =\gamma +\mathit{\Gamma} R T \bigl( \theta
\ln( {\textstyle\frac{\nu_2^\star}{\nu_1^\star}} 10^{-2 pH})
+\theta {\textstyle\frac{2 n_2 F U_{\rm fit} +\omega \theta}{2 R T}}
+\theta \ln ({\textstyle\frac{\theta}{2}})
+(2 -\theta) \ln (1 -{\textstyle\frac{\theta}{2}}) \bigr);
\end{split}
\end{equation}
\end{subequations}
for the Pt oxide coverage \eqref{1b}:
\begin{multline}\label{4}
r_{\rm oxide}(\theta, V) =\mathit{\Gamma} e^{-\frac{1}{R T} (H_{2,{\rm fit}}
+\lambda \theta)} \bigl( \nu_1^\star (1 -{\textstyle\frac{\theta}{2}})
e^{-\frac{n_2 F (1 - \beta_2)}{R T} (U_{\rm fit}
+\frac{\omega \theta}{n_2 F} ) +(1-\beta_2) \frac{n_2 F}{R T} V}\\
-\nu_2^\star 10^{-2 pH} e^{\frac{n_2 F \beta_2}{R T} (U_{\rm fit}
+\frac{\omega \theta}{n_2 F} ) -\beta_2 \frac{n_2 F}{R T} V}
\bigr) .
\end{multline}
The terms in \eqref{3} and \eqref{4} are rearranged in such a way to express
explicitly the dependence of the governing relations on the voltage $V$.
For the parameters from Table~\ref{tab1}, the reactions rates are illustrated
in Figure~\ref{fig_rateVdT} with respect to varying $V\in[0.9,1.2]$ (V).
\begin{figure}[hbt!]
\begin{center}
\hspace*{-15mm}
\epsfig{file=fig_rateVdT.pdf,width=1.2\textwidth,angle=0}
\caption{The reaction rates $r_{\rm dissol}$ (a), (b), and
$r_{\rm oxide}$ (c) for fixed $c$ and $\theta$.}
\label{fig_rateVdT}
\end{center}
\end{figure}
This range corresponds to the operating conditions.
In Figure~\ref{fig_rateVdT}, there are fixed
$c=3\times10^{-10}$ ({\rm mol/cm$^3$}) and $\theta=0$,
the five curves $V\mapsto r_{\rm dissol}(c, d, \theta, \,\cdot\,)$
in plot (a) correspond to five Pt particle diameters
selected equidistantly in the range of $d\in[2,4]\times10^{-7}$ (cm),
while $r_{\rm oxide}$ is independent of $d$.
For fixed $d=3\times10^{-7}$ (cm),
the curves $V\mapsto r_{\rm dissol}(c, d, \theta, \,\cdot\,)$ in (b)
and $V\mapsto r_{\rm oxide}(c, \,\cdot\,)$ in (c)
are plotted when varying the temperature $T\in[323.15,363.15]$ (K).
Figure~\ref{fig_rateVdT} demonstrates rates of the forward reactions:
platinum dissolution (a, b) and formation of platinum oxide (c)
as function of applied potential gradient at different diameters of Pt particle (a)
and at different temperature (b, c).
As seen from the figures the rate of platinum dissolution increases
with decrease of the diameter of the platinum particles.
The rate of the both reactions grow with increasing temperature.
Here it is worth noting that the rates of dissolution and re-deposition reactions are not the same.
Figure~\ref{fig_rateVdT} shows the rates only forward reactions.
The model considers dissolution and re-deposition reactions,
as well as building of platinum oxide (forward reaction, platinum oxidation)
and reduction of platinum oxide to platinum (backward reaction).
Since the dependence of $r_{\rm dissol}$ on the Pt concentration $c$ in \eqref{3a}
is linear, for large values of $c$ the backward dissolution reaction rate
(having the negative sign) dominates over forward reactions (having the positive sign).
In these examples, we depict the forward dissolution reaction rates when varying $V$.
In Figure~\ref{fig_rateVdT} (a) we observe decay of the dissolution
reaction rates when the particle diameter $d$ increases.
From Figure~\ref{fig_rateVdT} (b) and (c) we conclude that increasing
the temperature $T$ follows growth of the dissolution reaction rate
and decay of the oxidation reaction rate.
\section{Theoretical and numerical methods}\label{sec3}
\paragraph{Governing relations}
For a given voltage $V$ in \eqref{3} and \eqref{4},
in \cite{LMGAW/15} the following system
of reaction-diffusion equations is formulated:
find a triple $(c, d, \theta)$ satisfying \eqref{2} such that
\begin{subequations}\label{5}
\begin{equation}\label{5a}
{\textstyle\frac{\partial c}{\partial t}} -\sqrt{\varepsilon} D_{\rm Pt}
{\textstyle\frac{\partial^2 c}{\partial x^2}} =B_3 d^2 r_{\rm dissol}(c, d, \theta)
\quad \text{for $t>0$, $x\in(0,L)$},
\end{equation}
where $B_3 ={\textstyle\frac{\pi N_{\rm Pt}}{2 \varepsilon}}$
(1/cm$^3$) is denoted for short,
\begin{equation}\label{5b}
{\textstyle\frac{\partial d}{\partial t}} =-\mathit{\Omega}
\,r_{\rm dissol}(c,d,\theta)\quad \text{for $t>0$, $x\in(0,L)$},
\end{equation}
\begin{equation}\label{5c}
{\textstyle\frac{\partial \theta}{\partial t}}
+{\textstyle\frac{2 \theta}{d}} {\textstyle\frac{\partial d}{\partial t}}
={\textstyle\frac{r_{\rm oxide}(\theta)}{\mathit{\Gamma}}}
\quad \text{for $t>0$, $x\in(0,L)$};
\end{equation}
which is endowed with the initial conditions:
\begin{equation}\label{5d}
c =0,\quad d =d_{\rm Pt},\quad \theta =0\quad \text{as $t=0$, $x\in[0,L]$};
\end{equation}
and the mixed Neumann--Dirichlet boundary conditions:
\begin{equation}\label{5e}
{\textstyle\frac{\partial c}{\partial x}} =0\quad\text{as $t>0$, $x=0$};
\quad c =0\quad \text{as $t>0$, $x=L$}.
\end{equation}
\end{subequations}
The first equality in \eqref{5e} implies no-flux condition at the CL-GDL interface,
the second condition at the CL-membrane interface assumes
that the dissolved ${\rm Pt}^{2+}$ concentration goes to zero.
Neglecting the dependence on the space variable $x$,
thus omitting the diffusion term (the second one) in the left-hand side
of \eqref{5a} and the boundary conditions \eqref{5e},
the resulting ordinary differential equations (ODE) system
is studied theoretically in \ref{A}.
In fact, the problem is reduced to the two unknowns
by finding a first integral of the system.
In \ref{B} a particular solution is constructed analytically under specific assumptions.
The exact solution is used to test numerical solvers for the problem.
Next we will investigate the initial boundary value problem \eqref{5} numerically.
\paragraph{Numerical algorithm}
In order to solve the nonlinear reaction-diffusion equation \eqref{5a},
below we develop a second order implicit-explicit (IMEX2) scheme
following \cite{ARS/97}.
Let the half-strip $[0,\infty)\times[0,L]$ be meshed with
temporal points $(t^0,\dots,t^M,\dots)$, where $t_0=0$, and with
$N+1$ spacial points $(x_0,\dots,x_N)$, where $x_0 =0$ and $x_N=L$.
For $l=1,\dots,M$, we look for a triple of discrete functions
\begin{equation}\nonumber
c^l_h =(c^l_0,\dots,c^l_N),\quad d^l_h =(d^l_0,\dots,d^l_N),\quad
\theta^l_h =(\theta^l_0,\dots,\theta^l_N),
\end{equation}
with given $c^0_h =\theta^0_h =0$ and $d^0_h =d_{\rm Pt}$
according to the initial condition \eqref{5d}.
On the space mesh, forward
$D^+ c^l_i ={\textstyle\frac{c^l_{i+1} -c^l_i}{x_{i+1} -x_i}}$
and backward $D^- c^l_i ={\textstyle\frac{c^l_i -c^l_{i-1}}{x_i -x_{i-1}}}$
differences for $i=1,\dots,N-1$
are used for the standard approximation of the second-order derivative
\begin{equation}\nonumber
[D^- D^+] c^l_i ={\textstyle\frac{1}{x_i -x_{i-1}}}
\bigl( {\textstyle\frac{c^l_{i+1} -c^l_i}{x_{i+1} -x_i}}
-{\textstyle\frac{c^l_i -c^l_{i-1}}{x_i -x_{i-1}}} \bigr).
\end{equation}
We discretize \eqref{5} and iterate for $l=1,\dots,M$ with the time size
$\tau^l =t^l -t^{l-1}$ two implicit-explicit equations as follows
\begin{subequations}\label{6}
\begin{equation}\label{6a}
\begin{cases}
c^{l-1/2}_h -w \tau^l \sqrt{\varepsilon} D_{\rm Pt} [D^- D^+] c^{l-1/2}_h
=c^{l-1}_h +w \tau^l B_3 (d^l_h)^2
r_{\rm dissol}(c^{l-1}_h, d^l_h, \theta^l_h)\\[2ex]
c^l_h =c^{l-1}_h +\tau^l \bigl(
\sqrt{\varepsilon} D_{\rm Pt} [D^- D^+] c^{l-1/2}_h
+B_3 (d^l_h)^2 r_{\rm dissol}(c^{l-1/2}_h, d^l_h, \theta^l_h) \bigr),
\end{cases}
\end{equation}
where the IMEX parameter $w=0.5$ is set in the first equation, and
\begin{equation}\label{6b}
d^l_h =d^{l-1}_h -\tau^l \mathit{\Omega}
\,r_{\rm dissol}(c^{l-1}_h, d^{l-1}_h, \theta^{l-1}_h),
\end{equation}
\begin{equation}\label{6c}
\theta^l_h =\theta^{l-1}_h +\tau^l \bigl(
{\textstyle\frac{r_{\rm oxide}(\theta^{l-1}_h)}{\mathit{\Gamma}}}
+{\textstyle\frac{2 \mathit{\Omega} \theta^{l-1}_h}{d^{l-1}_h}}
\,r_{\rm dissol}(c^{l-1}_h, d^{l-1}_h, \theta^{l-1}_h) \bigr).
\end{equation}
The diffusion-reaction equations in \eqref{6a} are endowed by
the boundary conditions according to \eqref{5e}:
\begin{equation}\label{6d}
c^{l-1/2}_1 =c^{l-1/2}_0,\quad c^l_1 =c^l_0,\quad c^{l-1/2}_N =c^l_N =0.
\end{equation}
\end{subequations}
The standard TDMA algorithm is applied for inversion of a tridiagonal matrix
in the implicit equation (the first one) in \eqref{6a}.
For solution of the nonlinear reaction equations \eqref{6b} and \eqref{6c},
we apply the standard fourth order Runge--Kutta (RK4) method.
In the numerical examples reported further we set the uniform mesh
of the time step size $\tau =[10^{-4},10^{-2}]$ (s),
and the space step size $h =\frac{L}{10}$ (cm) when $N=10$.
We note that impulse switching of voltage (see
Figure~\ref{fig_voltage_rate_total} plot (b))
requires the smaller time step $\tau = 10^{-4}$ for stable calculation,
thus increasing the computational complexity.
The time step $\tau$ was fixed during the iteration.
The $(\tau, h)$-step choice is conform to the fact, that for large time steps
the CFL-condition may be violated, thus leading to numerical instabilities.
The instability appears in such manner that the oxide coverage
as well as the particle diameter and the Pt2+ concentration becoming negative.
\section{Results}\label{sec4}
\paragraph{Simulation setup}
We start with parameter values given in Table~\ref{tab1} at
the temperature $T = 353.15$ (K) and constant voltage $V = 0.65$ (V)
that will be used for numerical simulation.
\begin{table}[hbt!]
{\small
\begin{center}
\begin{tabular}{|l|l|l|p{0.55\textwidth}|l|}\hline
Symbol & Value & Units & Description & Ref.\\\hline
\multicolumn{5}{|l|}{Catalyst layer parameters}\\\hline
$L$ & $1\times10^{-3}$ & {\rm cm} & Thickness of cathode CL & \\\hline
$d_{\rm Pt}$ & $3\times10^{-7}$ & {\rm cm} & diameter of Pt nanoparticle & \\\hline
$p_{\rm Pt}$ & $4\times10^{-4}$ & {\rm g/cm$^2$} & Pt loading & \\\hline
$\rho_{\rm Pt}$ & 21.45 & {\rm g/cm$^3$} & density of Pt nanoparticle & \\\hline
$\varepsilon$ & 0.2 & & Volume fraction of ionomer increment in cathode & \cite{LMGAW/15}\\\hline
\multicolumn{5}{|l|}{Physical constants}\\\hline
$R$ & 8.31445985 & {\rm J/mol/K} & Gas constant & \cite{Rum/19}\\\hline
$F$ & 96485.3329 & {\rm C/mol} & Faraday constant & \cite{Rum/19}\\\hline
\multicolumn{5}{|l|}{Parameters for $Pt^{2+}$ formation and diffusion}\\\hline
$\nu_1$ & $1\times10^4$ & {\rm Hz} & Dissolution attempt frequency & \cite{LMGAW/15}\\\hline
$\nu_2$ & $8\times10^5$ & {\rm Hz} & Backward dissolution rate factor & \cite{LMGAW/15}\\\hline
$\beta_1$ & 0.5 & & Butler--Volmer transfer coefficient for Pt dissolution & \cite{LMGAW/15}\\\hline
$n$ & 2 & & Electrons transferred during Pt dissolution &\\\hline
$U_{\rm eq}$ & 1.18 & {\rm V} & Pt dissolution bulk equilibrium voltage & \cite{Dob/75}\\\hline
$\mathit{\Omega}$ & 9.09 & {\rm cm$^3$/mol} & Molar volume of Pt & \cite{LMGAW/15}\\\hline
$\gamma$ & $2.4\times10^{-4}$ & {\rm J/cm$^2$} & Pt [1 1 1] surface tension & \cite{LMGAW/15}\\\hline
$c_{\rm ref}$ & $1$ & {\rm mol/cm$^3$} & reference $Pt^{2+}$ concentration & \cite{HM/12}\\\hline
$H_{1,{\rm fit}}$ & $4\times10^4$ & {\rm J/mol} & Fit Pt dissolution activation enthalpy & \cite{LMGAW/15}\\\hline
$D_{\rm Pt}$ & $1\times10^{-6}$ & {\rm cm$^2$/s} & Diffusion coefficient of Pt$^{2+}$ in the membrane & \cite{BGADKE/11}\\\hline
\multicolumn{5}{|l|}{Parameters for Pt oxide formation}\\\hline
$pH$ & 0 & & Potential of hydrogen ions (protons) & \\\hline
$\nu_1^\star$ & $1\times10^4$ & {\rm Hz} & Forward Pt oxide formation rate constant & \cite{LMGAW/15}\\\hline
$\nu_2^\star$ & $2\times10^{-2}$ & {\rm Hz} & Backward Pt oxide formation rate constant & \cite{LMGAW/15}\\\hline
$\mathit{\Gamma}$ & $2.2\times10^{-9}$ & {\rm mol/cm$^2$} & Pt surface site density & \cite{LMGAW/15}\\\hline
$\beta_2$ & 0.5 & & Butler--Volmer transfer coefficient for PtO formation & \cite{LMGAW/15}\\\hline
$n_2$ & 2 & & Electrons transferred during Pt oxide formation &\\\hline
$U_{\rm fit}$ & 0.8 & {\rm V} & Pt oxide formation bulk equilibrium voltage & \cite{Dob/75}\\\hline
$\lambda$ & $2\times10^4$ & {\rm J/mol} & Pt oxide dependent kinetic barrier constant & \cite{LMGAW/15}\\\hline
$\omega$ & $5\times10^4$ & {\rm J/mol} & Pt oxide-oxide interaction energy & \cite{LMGAW/15}\\\hline
$H_{2,{\rm fit}}$ & $1.2\times10^4$ & {\rm J/mol} & Fit partial molar oxide formation activation enthalpy & \cite{LMGAW/15}\\\hline
\end{tabular}
\vspace*{1ex}
\caption{Physical and model parameters applied in the simulation}\label{tab1}
\end{center}
}
\end{table}
\paragraph{Operation of Pt catalyst layer}
For our investigation we chose three different protocols
used by different Institutions to test durability of the catalysts
applied in low temperature fuel cells:
\begin{itemize}
\item[(a)]
Accelerated Stress Test used DOE (The U.S. Department of Energy)
--- $\Lambda$-shaped symmetric triangle wave (50 mV/sec) from 0.6 to 1.0 V
(see \cite{YBBCGBM/17});
\item[(b)]
Accelerated Durability Protocol employed by Tennessee Tech University
--- $\Pi$-shaped square wave from 0.6 to 0.9 V, 5 sec at 0.6 V and 5 sec at 0.9 V
(see \cite{UKRPHLRER/15});
\item[(c)]
Durability protocol developed by Nissan --- slow anodic wave
--- $\angle$-shaped asymmetric triangular wave from 0.6 to 0.95 V
(see \cite{SMSLMMB/18}).
\end{itemize}
These profiles are illustrated within 5 periodic cycles in Figure~\ref{fig_voltage_rate_total}
(a), (b), and (c), respectively.
\begin{figure}[hbt!]
\hspace*{-1.5cm}
\epsfig{file=fig_voltage_rate_total.pdf,width=1.2\textwidth,angle=0}
\caption{The $\Lambda$-shaped (a) and $\Pi$-shaped (b)
$\angle$-shaped (c) profiles of cyclic voltage $V(t)$.
The reaction rates $r_{\rm dissol}$ and
$r_{\rm oxide}$ for the $\Lambda$ (d), $\Pi$ (e), $\angle$ (f)
shaped $V(t)$ within 3 cycles.}
\label{fig_voltage_rate_total}
\end{figure}
Each cycle is characterized by the length $p$.
The $\Lambda$-shaped voltage profile is continuous, symmetric,
starting and finishing with the minimal voltage value $V_{\rm min}$,
attaining the maximal voltage value $V_{\rm max}$
at the half-length $\frac{p}{2}$,
thus having the slope $\alpha =\pm\frac{2(V_{\rm max} -V_{\rm min})}{p}$.
In the plot (a) in Figure~\ref{fig_voltage_rate_total}, $p =16$ (s),
$V_{\rm min}=0.6$ (V), $V_{\rm max}=1$ (V), $\alpha =\pm5\cdot10^{-2}$ (V/S).
The $\Pi$-shaped voltage profile is discontinuous, characterized by
the minimal $V_{\rm min}=0.6$ (V) and the maximal $V_{\rm max}=0.9$ (V) voltages
switching at $\frac{p}{2}$ given in Figure~\ref{fig_voltage_rate_total} (b)
for $p =10$ (s).
The $\angle$-shaped voltage profile first accelerates during $p =10$ (s)
with the slope $\alpha =3.5\cdot10^{-2}$ (V/S) from the minimal
$V_{\rm min}=0.6$ (V) to the maximal $V_{\rm max}=0.95$ (V) voltages
and then switches to $V_{\rm min}$ again, see Figure~\ref{fig_voltage_rate_total} (c).
\begin{figure}[hbt!]
\begin{center}
\epsfig{file=fig_cycle.pdf,height=.9\textheight,angle=0}
\caption{The solution $c$ in (a), $d$ in (b), $\theta$ in (c)
under the $\Lambda$-shaped voltage cycle at $T=353.15$ (K).}
\label{fig_cycle}
\end{center}
\end{figure}
\begin{figure}[hbt!]
\begin{center}
\epsfig{file=fig_impulse.pdf,height=.9\textheight,angle=0}
\caption{The solution $c$ in (a), $d$ in (b), $\theta$ in (c)
under the $\Pi$-shaped voltage cycle at $T=353.15$ (K).}
\label{fig_impulse}
\end{center}
\end{figure}
\begin{figure}[hbt!]
\begin{center}
\epsfig{file=fig_rtriag.pdf,height=.9\textheight,angle=0}
\caption{The solution $c$ in (a), $d$ in (b), $\theta$ in (c)
under the $\angle$-shaped voltage cycle at $T=353.15$ (K).}
\label{fig_angle}
\end{center}
\end{figure}
In Figure~\ref{fig_voltage_rate_total} (d), (e), (f) we plot the mean over $x\in[0,L]$ in CL of
the reaction rates $r_{\rm dissol}$ scaled by multiplying by 100,
and $r_{\rm oxide}$ for the corresponding $\Lambda$, $\Pi$, $\angle$-shaped
voltage profiles $V(t)$ within 3 periodic cycles.
We use the numerical model \eqref{6} for computer simulation
of the catalyst coverage and the CL operation under the $\Lambda$-shaped,
$\Pi$-shaped, and $\angle$-shaped voltage cycles taken from
Figure~\ref{fig_voltage_rate_total} (a), (b), and (c).
The respective solution triples $(c,d,\theta)$ are depicted in
Figure~\ref{fig_cycle}, \ref{fig_impulse}, \ref{fig_angle}
at the temperature $T = 80{}^\circ$C.
From Figures~\ref{fig_cycle}, \ref{fig_impulse}, \ref{fig_angle} (a)
we observe that at $\Lambda$-shaped voltage cycle the concentration
of platinum ions is varied from 0 to around $8\times 10^{-7}$ mol/l;
at $\Pi$-shaped voltage cycle the Pt$^{2+}$ increases maximum to
$2.3\times 10^{-7}$ mol/l, and at $\angle$-shaped voltage profile it grows
until $4\times 10^{-7}$ mol/l.
Figures~\ref{fig_cycle}, \ref{fig_impulse}, \ref{fig_angle} (b) demonstrate
the evolution in Pt size distribution through the catalyst length.
The Pt size goes down faster at $\Lambda$-shaped voltage cycle than at other two cycles.
We should mention, that for all three investigated voltage cycles
the Pt particle diameter decreases slightly faster at the membrane surface
than at the boundary with the gas diffusion layer.
The changes in the coverage of Pt surface by platinum oxide during the voltage cycling
are depicted in Figures~\ref{fig_cycle}, \ref{fig_impulse}, \ref{fig_angle} (c).
In $\Lambda$-shaped and $\angle$-shaped voltage cycles,
the part of Pt surface is permanently covered by PtO.
The ratio of the coverage depends on the voltage.
At $\Lambda$-shaped voltage cycle, the PtO coverage is varied from 42 to 82\%,
while at $\angle$-shaped voltage cycle changes from 30 to 70\%.
During the $\Pi$-shaped voltage cycle, the platinum oxide covers from 0 to 70\% of Pt surface.
In this cycle, at the high voltage (0.9 V) the formation of PtO occurs and
in the next 5 sec, at the low voltage (0.6 V) the reverse reaction proceeds:
the platinum oxide is reduced to the platinum.
As seen, at $\Pi$-shaped voltage cycle it is enough time for the reduction reaction,
while in other two cycles it does not.
\begin{figure}[hbt!]
\begin{center}
\epsfig{file=fig_mass_total09,width=.7\textwidth,angle=0}
\caption{The mean Pt mass loss ration $m_{\rm Pt}$
under various voltage cycles at $T=353.15$ (K).}
\label{fig_mass_total}
\end{center}
\end{figure}
In Figure~\ref{fig_mass_total} we plot the calculated in time steps $l$
platinum mass ratio (the mean over $x\in[0,L]$ in CL):
\begin{equation}\nonumber
m^l_{\rm Pt} = \frac{4}{3}\pi (d^l)^3/V_{\rm Pt}\in[0,1],
\end{equation}
versus the number of cycles.
For comparison, the three curves $m^l_{\rm Pt}$ are shown
corresponding to the voltage profiles of hat ($\Lambda$),
impulse ($\Pi$), and angle ($\angle$) shapes at 10 voltage cycles
during 2 min. 40 sec. in plot (a).
While in plot (b) the Pt loss is presented
at 1000 voltage cycles respectively during 4 hours 26 min. 40 sec.
There is also depicted $m^l_{\rm Pt}$ under
the constant voltage $V(t)=0.65$ (V), thus describing idle state.
The linear loss of Pt mass during voltage cycles can be clear observed
in Figure~\ref{fig_mass_total}.
The results shown here are determined by the specific choice
of profiles and by the upper potential level.
Indeed, increasing slopes $\alpha =0.03, 0.06, \infty$ (V/s) were tested
under the fixed upper potential level $V_{\rm max}=0.9$ (V)
as presented in plot (c).
This confirms that $\Pi$-shaped profile (marked by $\alpha =\infty$)
is the most damaging with respect to livetimes
(see experimental data in \cite{UK/07} and \cite{KWSSG/18}, Fig.~4).
On the other side, increasing the upper potential level
$V_{\rm max} =0.9, 0.95, 1$ (V) under the fixed slope $\alpha =0.03$ (V/s)
also shortens the lifetime as shown in plot (d)
(see the experimental confirmation in \cite{KW/19}, Fig.~5(a)).
The degradation phenomenon would be impossible
without the diffusion (when setting $D_{\rm Pt}=0$ as in \ref{A}).
In Figure~\ref{fig_mass_length} the corresponding Pt mass loss
$m^l_{\rm Pt}(x)$ versus $x\in[0,L]$ along CL is presented
in plots (a), (b), (c) during 10 voltage cycles of $\Lambda$, $\Pi$, $\angle$ profiles.
Here we can also observe the most strong degradation phenomenon
near the CL-membrane interface at $x=L$ under the Dirichlet boundary condition.
\begin{figure}[hbt!]
\hspace*{-1.2cm}
\epsfig{file=fig_mass_length,width=1.2\textwidth,angle=0}
\caption{The Pt mass loss ration $m_{\rm Pt}$ vs. CL under $\Lambda$, $\Pi$,
$\angle$-shaped voltage cycles at $T=353.15$ (K).}
\label{fig_mass_length}
\end{figure}
Finally, extrapolating the linear Pt loss based on Figure~\ref{fig_mass_total},
we calculate the prognosis of the failure when $m_{\rm Pt}$ becomes zero,
which is presented in Table~\ref{tab2}.
\begin{table}[hbt!]
{\small
\begin{center}
\begin{tabular}{|l|l|l|l|}\hline
Voltage & Pt mass loss slope & \#cycles prognosis & time prognosis \\\hline
$\Lambda$-shaped & $1.6\times10^{-4}$ & $6\times10^3$ & $27$ h\\\hline
$\Pi$-shaped & $6\times10^{-5}$ & $1.6\times10^4$ & $46$ h\\\hline
$\angle$-shaped & $2.6\times10^{-5}$ & $3.8\times10^4$ & $106$ h\\\hline
constant &$6\times10^{-8}$ & $1.7\times10^7$ & $48000$ h\\\hline
\end{tabular}
\vspace*{1ex}
\caption{The Pt mass loss ration $m_{\rm Pt}$ before failure under various
voltage cycles at $T=353.15$ (K).}\label{tab2}
\end{center}
}
\end{table}
The average voltages in the studied cycles are 0.8 V for $\Lambda$-shaped cycle;
0.75 V for $\Pi$-shaped and 0.775 V for $\angle$-shaped cycle.
\section{Discussion}\label{sec5}
At present, the degradation effects of Pt/C catalyst in PEMFCs
have been extensively studied as experimentally and theoretically.
Currently, the decrease in the electroactivity of the Pt/C catalyst
has been related with the following mechanisms:
1) Pt dissolution and diffusion into the ionomer;
2) formation of platinum oxides on the Pt particles surface;
3) Pt particle ripening;
4) coalescence of Pt particles.
The changes in the electrochemical activities of the catalyst are usually
detected using cycling voltammetry or by measuring polarization curves.
In our study we consider only two first degradation mechanisms of Pt
and analyze the Pt mass loss and coverage ratio of Pt particle surface by PtO.
The simulation presented in the paper is difficult to validate
with experimental data available in scientific literature.
However, we found some experimental facts confirming the present calculation.
For example, Takei et all. \cite{TKKTWU/16} investigated Pt degradation of
carbon supported Pt catalyst in a single fuel cell as a function of the holding
times of OCV/load (square voltage cycling) at $T = 80{}^\circ$C and RH 100\%.
Higher operating potential enhanced the Pt oxidation,
accelerated the Pt particle growth but suppressed the Pt dissolution.
The formation of Pt oxide protects the Pt particle from dissolution.
Ferreira et al. \cite{FlOSHMMKG/05} studied a platinum degradation
for a short-stack of PEMFC operated at high voltages.
Using transmission electron microscopy (TEM) they analyzed
a cross-section of MEA cathode samples and determined
relative weight percentages of platinum particles
on the carbon support as a function of cathode thickness.
The analysis showed that the weight percentage of platinum remaining
on carbon decreases with increasing distance from gas diffusion layer.
After the FC operation the weight percentages of Pt at the interface PEM/CL
was twice time lower than at the interface CL/GDL.
Yu et al. \cite{YBBCGBM/17} detected 80\% depletion of Pt at cathode/membrane
interface after accelerated stress test which was performed by
imposing a triangular wave potential cycling from 0.6 V to 1.0 V
for 30.000 cycles at 50 mV/sec scan rate.
These finding confirm our simulation results indicating that the decrease
in Pt weight at the membrane interface same higher than at the interface to GDL.
\section{Conclusion}\label{sec6}
We suggested an one-dimensional and dynamic model which describes
degradation phenomena in Pt catalyst of a polymer electrolyte fuel cell such.
The model considers Pt dissolution, oxidation as well as diffusion of platinum
ions through ionomer of the catalyst layer into the membrane.
Also it takes into account an effect of temperature, Pt particle size and
Gibbs--Thomson’s effect: dependence of a surface potential on nano-particle size
as well as influence of the surface potential on the potential gradient in the system.
The developed model is applied to study concentration profile of Pt$^{2+}$
through catalyst length, changes in Pt particle size and mass loss,
the coverage ratio of Pt surface by platinum oxide at three different voltage cycles
often used in accelerated stress tests:
$\Lambda$-shaped, $\Pi$-shaped, and $\angle$-shaped voltage profiles.
For the parameter values from Table~\ref{tab1},
we report here on some of our theoretical and numerical findings with respect
to admissible voltage operating conditions.
\begin{itemize}
\item[(i)]
In order to preserve the physical constraints \eqref{2}, the sufficient are
$V\in[0.6,1]$ (V) for the accelerated $\Lambda$ and $\angle$-shaped voltage cycles,
and $V\in[0.6,0.9]$ (V) for the impulse $\Pi$-shaped voltage cycle.
\item[(ii)]
In Figures~\ref{fig_cycle}, \ref{fig_impulse}, and \ref{fig_angle} we observe
diffusion of the Pt ion concentration $c$ and the Pt particle diameter $d$,
whereas a non-diffusive behavior of the Pt coverage ratio $\theta$.
\item[(iii)]
The reaction rates are shown in Figure~\ref{fig_mass_length} for fixed variables,
and in Figure~\ref{fig_voltage_rate_total} during the CL operation.
\item[(iv)]
The rate of the loss of Pt total mass during voltage cycles depends on the
voltage profile and the upper potential level
(see Figure~\ref{fig_mass_total} and Figure~\ref{fig_mass_length},
and its prognosis in Table~\ref{tab2}).
\end{itemize}
The study shows that the degradation rate increases with temperature
and decreasing particle diameter of Pt nano-particles.
The mass loss in platinum and decrease in Pt particle diameter are
more significant at the membrane surface than at gas diffusion layer.
\paragraph{Acknowledgments}
L. K.-J. is supported by the Austrian Research Promotion Agency (FFG) and
the Austrian Ministry for Transport, Innovation and Technology (BMVIT).\\
V. A. K. is supported by the Austrian Science Fund (FWF) project P26147-N26: PION
and the European Research Council (ERC) under European Union's Horizon 2020
Research and Innovation Programme (advanced grant No. 668998 OCLOC),
he thanks the Russian Foundation for Basic Research (RFBR)
project 18-29-10007 for partial support.
\bibliography{kjkbibfile}
\appendix
\section{Non-diffusive case}\label{A}
The non-diffusive case of \eqref{5} is described by the following
system of nonlinear reaction equations:
find a triple $c(t)\ge0$, $d(t)\ge0$, $0\le\theta(t)\le1$ such that
\begin{subequations}\label{A1}
\begin{equation}\label{A1a}
{\textstyle\frac{{\rm d} c}{{\rm d} t}} =B_3 d^2 \bigl(
B_1(d, \theta) e^{(1-\beta_1) B_4(d, \theta) V}
-c B_2(d, \theta) e^{-\beta_1 B_4(d, \theta) V} \bigr)
\quad \text{for $t>0$},
\end{equation}
\begin{equation}\label{A1b}
{\textstyle\frac{{\rm d} d}{{\rm d} t}} =-\mathit{\Omega}
\bigl( B_1(d, \theta) e^{(1-\beta_1) B_4(d, \theta) V}
-c B_2(d, \theta) e^{-\beta_1 B_4(d, \theta) V} \bigr)
\quad \text{for $t>0$},
\end{equation}
\begin{equation}\label{A1c}
{\textstyle\frac{{\rm d} \theta}{{\rm d} t}}
+{\textstyle\frac{2 \theta}{d}} {\textstyle\frac{{\rm d} d}{{\rm d} t}}
={\textstyle\frac{r_{\rm oxide}(\theta)}{\mathit{\Gamma}}}
\quad \text{for $t>0$},
\end{equation}
which is endowed with the initial conditions:
\begin{equation}\label{A1d}
c(0) =0,\quad d(0) =d_{\rm Pt},\quad \theta(0) =0.
\end{equation}
\end{subequations}
In equations \eqref{A1a} and \eqref{A1b},
the expression \eqref{3a} was inserted in $r_{\rm dissol}$.
Multiplying \eqref{A1a} with $\mathit{\Omega}$ and \eqref{A1b}
with $B_3 d^2$, after summation we obtain the homogeneous
ordinary differential equations (ODE):
\begin{equation}\nonumber
{\textstyle\frac{{\rm d}}{{\rm d} t}} \bigl(
\mathit{\Omega} c +{\textstyle\frac{B_3}{3}} d^3 \bigr) =0
\quad \text{for $t>0$},
\end{equation}
which implies the first integral of the system
\begin{equation}\label{A2}
c(t) =c(0) +{\textstyle\frac{B_3}{3\mathit{\Omega}}} d^3(0)
-{\textstyle\frac{B_3}{3\mathit{\Omega}}} d^3(t).
\end{equation}
With the help of \eqref{A2}, the system \eqref{A1a}--\eqref{A1c}
can be reduced to two equations for either $c$ and $\theta$,
or $d$ and $\theta$.
We note also the following special cases.
Dividing \eqref{A1c} with $\theta\not=0$ implies the ODE:
\begin{equation}\label{A3}
{\textstyle\frac{{\rm d}}{{\rm d} t}} \bigl( \ln ( \theta d^2 ) \bigr)
={\textstyle\frac{r_{\rm oxide}(\theta)}{\mathit{\Gamma} \theta}}
\quad \text{for $t>0$}.
\end{equation}
If the Pt oxidation reaction rate is $r_{\rm oxide}(\theta) \equiv0$
(the equilibrium state), then the equation \eqref{A3} is solved trivially as
\begin{equation}\nonumber
\theta(t) ={\textstyle\frac{\theta(0)}{d^2(t)}} d^2(0) \equiv0
\end{equation}
due to the initial conditions \eqref{A1d}.
On the other side, if $\theta(t)\equiv1$ then
$B_1(d, 1) =B_2(d, 1) =0$ in \eqref{3b}.
Henceforth, the equations \eqref{A1a} and \eqref{A1b} have zero
right-hand sides and possess the constant solutions $c(t) \equiv c(0) =0$
and $d(t) \equiv d(0) =d_{\rm Pt}$.
\section{Analytical solution}\label{B}
We assume that the Pt particle diameter $d$ and PtO coverage ratio $\theta$
are constant in time, hence the coefficients $B_1$, $B_2$, and $B_4$ are constant,
and \eqref{A1} is reduced to the following Cauchy problem
for a single linear inhomogeneous ODE:
\begin{subequations}\label{B1}
\begin{equation}\label{B1a}
{\textstyle\frac{{\rm d} c}{{\rm d} t}}(t)
=B_1 B_3 d^2 e^{(1-\beta_1) B_4 V(t)}
-c(t) B_2 B_3 d^2 e^{-\beta_1 B_4 V(t)}\quad \text{for $t>0$},
\end{equation}
\begin{equation}\label{B1b}
c(0) =0.
\end{equation}
\end{subequations}
In the following we solve \eqref{B1} analytically
for non-steady state voltage $V(t)$.
We introduce an auxiliary function $K(t)$ such that \eqref{B1a}
can be rewritten equivalently as the following system for $t>0$:
\begin{subequations}\label{B2}
\begin{equation}\label{B2a}
{\textstyle\frac{{\rm d} K}{{\rm d} t}}(t)
=B_2 B_3 d^2 K(t) e^{-\beta_1 B_4 V(t)},
\end{equation}
\begin{equation}\label{B2b}
{\textstyle\frac{{\rm d} (K c)}{{\rm d} t}}(t)
=K {\textstyle\frac{{\rm d} c}{{\rm d} t}}(t)
+c {\textstyle\frac{{\rm d} K}{{\rm d} t}}(t)
=B_1 B_3 d^2 K(t) e^{(1-\beta_1) B_4 V(t)}.
\end{equation}
\end{subequations}
The integration of the equations \eqref{B2a} and \eqref{B2b}, respectively,
\begin{subequations}\label{B3}
\begin{equation}\label{B3a}
K(t) = K(0) e^{B_2 B_3 d^2 \int_0^t e^{-\beta_1 B_4 V(s)} \,{\rm d}s},
\end{equation}
\begin{equation}\label{B3b}
(K c)(t) =(K c)(0) +B_1 B_3 d^2 \int_0^t e^{(1-\beta_1) B_4 V(\tau)}
K(\tau) \,{\rm d}\tau,
\end{equation}
\end{subequations}
and the subsequent substitution of \eqref{B3a} into \eqref{B3b}
leads to the explicit expression
\begin{multline}\label{B4}
c(t) =e^{-B_2 B_3 d^2 \int_0^t e^{-\beta_1 B_4 V(s)}
\,{\rm d}s} \bigl( c(0)\\
+B_1 B_3 d^2 \int_0^t e^{(1-\beta_1) B_4 V(\tau)
+B_2 B_3 d^2 \int_0^\tau e^{-\beta_1 B_4 V(s)}
\,{\rm d}s} \,{\rm d}\tau \bigr),
\end{multline}
which can be shortened using the homogeneous initial condition \eqref{B1b}.
\end{document}
| 191,853
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Analysis of the North American Medium Voltage companies in this market, and examines distribution structure and pricing trends. This report also studies the latest economic trends influencing the industry functions and provides revenue forecast and growth rate discussions. The study concludes with a competitive analysis of top market participants, identifying their strengths and weaknesses for 2015.
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\begin{document}
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\setlength{\parindent}{0mm} \setlength{\parskip}{2pt plus 2pt}
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{definition}{Definition}[section]
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\newcommand{\norm}[1]{\left\Vert#1\right\Vert}
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}
} \makeatother
\title[Conjugate connections on almost Norden manifolds]{Conjugate connections and statistical structures on almost Norden\\ manifolds}
\author[M. Teofilova]{Marta Teofilova}
\maketitle
{\small
\textsc{Abstract.} Relations between conjugate connections with
respect to the pair of Norden metrics and to the almost complex
structure on almost Norden manifolds are studied. Conjugate
connections of the Levi-Civita connections induced by the Norden
metrics are obtained. Statistical structures on almost Norden
manifolds are considered.
\\
Key words: Norden metric, complex structure, conjugate connection, dual connection, complex conjugate connection, statistical manifold.\\
2010 Mathematics Subject Classification: Primary 53C15, 53C50;
Secondary 32Q60.}
\section*{Introduction}
The concept of conjugate connections relative to a metric tensor
field was originally introduced by A. P. Norden in the context of
Weyl geometry \cite{AN}. Such linear connections were independently
developed by H.~Nagaoka and S.~Amari \cite{NA} under the name dual
connections and used by S.~Lauritzen in the definition of
statistical manifolds \cite{SL}. For more details on conjugate
connections and their application to information theory, statistics
and other fields see \cite{Amari}, \cite{SA}, \cite{OC}, \cite{HM},
\cite{KN}, \cite{US}.
Another kind of conjugate connections are those which are dual with
respect to an invertible (1,1)-tensor field \cite{Al}, \cite{Be}.
Conjugate connections relative to an almost complex structure are
studied by A.~M.~Blaga and M.~Crasmareanu in \cite{B}. Relations
between conjugate connections with respect to a symplectic structure
and to a complex structure on K\"ahler manifolds are investigated in
\cite{B1}. Statistical structures and relations between conjugate
connections on Hermitian manifolds are studied in \cite{F},
\cite{Noda}.
The main purpose of the present work\footnote{This work is partially
supported by project FP17--FMI--008 of the Scientific Research Fund,
University of Plovdiv Paisii Hilendarski, Bulgaria.} is to study
relations between both aforementioned types of conjugate connections
on almost complex manifolds with Norden metric (B-metric). For the
sake of brevity, such manifolds will be called almost Norden
manifolds. These manifolds were introduced by A.~P.~Norden \cite{No}
and their geometry was studied for the first time by K.~Gribachev,
D.~Mekerov and G.~Djelepov \cite{Gri} who termed them generalized
B-manifolds.
Since on such manifolds, there exists a pair of Norden metrics, we
can consider conjugate connections with respect to each of these
metrical tensors and their relations to conjugate connections
relative to the almost complex structure. Another aim of this work
is to construct and study statistical structures on almost Norden
manifolds.
The paper is organized as follows. In Section \ref{s1}, we give some
basic information about almost Norden manifolds and conjugate
connections. In Section \ref{s2}, we study the coincidence of
conjugate connections with respect to the Norden metrics and the
almost complex structure. The case of symmetric connections and
completely symmetric connections is also investigated. In Section
\ref{s3}, we study curvature properties of the conjugate connections
of the Levi-Civita connections induced by the pair of Norden
metrics. In Section \ref{s4}, we consider statistical structures on
almost Norden manifolds by constructing families of linear
connections with completely symmetric difference tensor and studying
their curvature properties.
\section{Preliminaries}\label{s1}
\subsection{Almost Norden manifolds}
The triple $(M,J,g)$ is called an \emph{almost Norden manifold}
(almost complex manifold with Norden metric) if $M$ is a
differentiable $2n$-dimensional manifold, $J$ is an almost complex
structure, and $g$ is a pseudo Riemannian metric compatible with $J$
such that
\begin{equation}\label{1}
J^2 X = -X,\qquad g(JX,JY) = - g(X,Y).
\end{equation}
Here and further $X,Y,Z,W$ will stand for arbitrary vector fields on
$M$, i.e. elements in the Lie algebra $\mathfrak{X}(M)$, or vectors
in the tangent space $T_pM$ at an arbitrary point $p\in M$.
Equalities (\ref{1}) imply $g(JX,Y) = g(X,JY)$ which means that the
tensor $\widetilde{g}$ defined by
\begin{equation}
\widetilde{g}(X,Y) = g(X,JY)
\end{equation}
is symmetric and is known as the \emph{associated (twin) metric} of
$g$ ($g$ and $\widetilde{g}$ are called a pair of twin metrics).
This tensor also satisfies the Norden metric property, i.e.
$\widetilde{g}(JX,JY) = -\widetilde{g}(X,Y)$, i.e.
$(M,J,\widetilde{g})$ is also an almost Norden manifold. Both
metrics, $g$ and $\widetilde{g}$, are necessarily of neutral
signature $(n,n)$.
Let us denote by $\nabla^0$ and $\widetilde{\nabla}^0$ the
Levi-Civita connections of $g$ and $\widetilde{g}$, respectively.
The tensor field $F$ defined by
\begin{equation}\label{F-def}
F(X,Y,Z) = (\nabla^0_X \widetilde{g})(Y,Z) = g\left((\nabla^0_X J)Y,
Z\right)
\end{equation}
plays an important role in the geometry of almost Norden manifolds.
It has the following properties
\begin{equation}\label{F-prop}
F(X,Y,Z) = F(X,Z,Y) = F(X,JY,JZ).
\end{equation}
Let $\{e_i\}$ $(i=1,2,...,2n)$ be an arbitrary basis of $T_pM$, and
$g^{ij}$ be the components of the inverse matrix of $g$ with respect
to this basis. The Lie 1-form associated with $F$ and its
corresponding vector $\Omega$ are given by
\begin{equation}\label{theta}
\theta (X) = g^{ij} F(e_i,e_j, X),\qquad \theta(X)=g(X,\Omega).
\end{equation}
A classification of the almost Norden manifolds with respect to the
properties of $F$ is obtained by G.~Ganchev and A.~Borisov in
\cite{Ga}. This classification consists of eight classes: three
basic classes $\mathcal{W}_i$ ($i=1,2,3$), their pairwise direct
sums $\mathcal{W}_i\oplus \mathcal{W}_j$, the widest class
$\mathcal{W}_1\oplus\mathcal{W}_2\oplus\mathcal{W}_3$ and the class
$\mathcal{W}_0$ of the K\"ahler Norden manifolds defined by $F=0$
(i.e. $\nabla^0 J =0$) which is contained in the intersection of
each two classes. The basic classes are distinguished by the
following characteristic conditions, respectively
\begin{equation}\label{Fcl}
\begin{array}{l}
\mathcal{W}_1: F(X,Y,Z) = \frac{1}{2n}\left\{g(X,Y)\theta(Z) +
g(X,JY)\theta(JZ) \right.\smallskip\\ \left.\phantom{\mathcal{W}_1:
F(X,Y,Z)= \frac{1}{2n}} + g(X,Z)\theta(Y) +
g(X,JZ)\theta(JY)\right\};\medskip\\
\mathcal{W}_2: F(X,Y,JZ) + F(Y,Z,JX) + F(Z,X,JY) = 0,\quad \theta =
0;\medskip\\
\mathcal{W}_3: F(X,Y,Z) + F(Y,Z,X) + F(Z,X,Y) = 0.
\end{array}
\end{equation}
The class $\mathcal{W}_1\oplus\mathcal{W}_2$ of the Norden manifolds
(complex manifolds with Norden metric) is the widest integrable
class (i.e. with a vanishing Nijenhuis tensor) and is characterized
also by the condition
\begin{equation*}
F(X,Y,JZ) + F(Y,Z,JX) + F(Z,X,JY) = 0.
\end{equation*}
Let $R^0$ be the curvature tensor of $\nabla^0$, i.e.
\begin{equation*}
R^0(X,Y)Z = \nabla^0_X \nabla^0_Y Z - \nabla^0_Y \nabla^0_X Z -
\nabla^0_{[X,Y]}Z.
\end{equation*}
Its corresponding (0,4)-tensor with respect to $g$ is defined by\\
$R^0(X,Y,Z,W) = g(R^0(X,Y)Z,W)$ and has the following properties
\begin{equation}\label{R0}
\begin{array}{l}
R^0(X,Y,Z,W) = - R^0(Y,X,Z,W) = - R^0(X,Y,W,Z), \smallskip\\
R^0(X,Y,Z,W) + R^0(Y,Z,X,W) + R^0(Z,X,Y,W) = 0.
\end{array}
\end{equation}
Any tensor of type (0,4) which satisfies all three conditions in
(\ref{R0}) is called a \emph{curvature-like tensor}. Then, the Ricci
tensor $\rho(L)$ and the scalar curvature $\tau(L)$ of $L$ are
obtained by
\begin{equation}\label{tau}
\rho(L) (X,Y) = g^{ij}L(e_i,X,Y,e_j),\qquad \tau(L) =
g^{ij}\rho(L)(e_i,e_j).
\end{equation}
A curvature tensor $L$ is called a \emph{K\"ahler tensor} if
$L(X,Y)JZ=JL(X,Y)Z$. Then, for the corresponding (0,4)-type tensor
with respect to $g$, i.e.\\ $L(X,Y,Z,W)=g(L(X,Y)Z,W)$ we have
$L(X,Y,JZ,JW)=-L(X,Y,Z,W)$.
Let $S$ be a tensor of type (0,2), and denote by $\widetilde{S}(X,Y)
= S(X,JY)$. Consider the following (0,4)-tensors:
\begin{equation}\label{psi}
\begin{array}{c}
\psi_1 (S) = g \owedge S, \qquad \psi_2 (S) = \widetilde{g} \owedge
\widetilde{S}, \smallskip\\ \pi_1 = \frac{1}{2}\psi_1(g),\quad \pi_2
= \frac{1}{2}\psi_2(g),\quad \pi_3 =
-\psi_1(\widetilde{g})=\psi_2(\widetilde{g}),
\end{array}
\end{equation}
where $\owedge$ is the Kulkarni-Nomizu product of two (0,2)-tensors,
e.g.\\ $(g\owedge S)(X,Y,Z,W)\hspace{-0.02in} = \hspace{-0.02in}
g(Y,Z)S(X,W) - g(X,Z)S(Y,W) + g(X,W)S(Y,Z) - g(Y,W)S(X,Z)$. The
tensor $\psi_1(S)$ is curvature-like iff $S$ is symmetric, and
$\psi_2(S)$ is curvature-like iff $S$ is symmetric and hybrid with
respect to $J$, i.e. $S(X,Y)=S(Y,X)=-S(JX,JY)$.
On a pseudo-Riemannian manifold $M$ ($\dim M=2n \geq 4$) the Weyl
tensor of a curvature-like tensor $L$ is given by
\begin{equation*}\label{Weyl}
W(L)=L-\frac{1}{2(n-1)}\big
\{\psi_{1}(\rho(L))-\frac{\tau(L)}{2n-1}\pi_{1}\big \}.
\end{equation*}
The square norm of $\nabla^0J$ is defined by
\begin{equation}\label{norm}
||\nabla^0
J||^{2}=g^{ij}g^{kl}g\big((\nabla^0_{e_{i}}J)e_{k},(\nabla^0_{e_{j}}J)e_{l}
\big).
\end{equation}
An almost Norden manifold is called \emph{isotropic K\"ahlerian} if
$||\nabla^0 J||^2=0$.
\subsection{Conjugate connections with respect to a metric tensor and statistical manifolds}
Let $(M,g)$ be a pseudo Riemannian manifold, and $\nabla$ be an
arbitrary linear connection on $M$. Then the linear connection
$\nabla^\ast$ defined by
\begin{equation}\label{g-conj}
Xg(Y,Z) = g(\nabla_X Y, Z) + g(Y,\nabla^{\ast}_X Z),
\end{equation}
is called the \emph{conjugate} (\emph{dual}) \emph{connection of}
$\nabla$ \emph{with respect to} $g$. From (\ref{g-conj}) it is easy
to see that ($\nabla^{\ast})^{\ast} = \nabla$. Hence, $\nabla$ and
$\nabla^\ast$ are said to be mutually conjugate. Also, from
(\ref{g-conj}) it follows that a connection $\nabla$ is
self-conjugate, i.e. $\nabla=\nabla^\ast$ if and only if it is a
\emph{metric} ($g$-\emph{compatible}) \emph{connection}, i.e.
$\nabla g= 0$.
The average connection $\overline{\nabla} = \frac{1}{2}(\nabla +
\nabla^\ast)$ of two mutually conjugate connections is a metric
connection.
Let $R$ and $R^\ast$ be the curvature tensors of $\nabla$ and
$\nabla^\ast$, respectively, and $P$ be the average curvature tensor
of $R$ and $R^\ast$, i.e.
\begin{equation}\label{K}
P(X,Y)Z = \frac{1}{2}\left\{R(X,Y)Z + R^\ast(X,Y)Z\right\}.
\end{equation}
Then, because of the relation $g(R(X,Y)Z,W) = -g(R^\ast(X,Y)W,Z)$,
the corresponding (0,4)-type tensor of $P$ is curvature-like.
Let $\nabla$ be a torsion free (symmetric) connection. Then, it is
known that its conjugate connection $\nabla^\ast$ is also torsion
free if and only if the tensor $\nabla g$ is completely symmetric,
i.e.
\begin{equation}\label{stat}
(\nabla_X g)(Y,Z) = (\nabla_Y g)(X,Z).
\end{equation}
Then the same is valid for $\nabla^\ast g$, i.e. $(\nabla, g)$ and
$(\nabla^\ast, g)$ are both Codazzi pairs. Also, in this case the
average connection of $\nabla$ and $\nabla^\ast$ is the Levi-Civita
connection of $g$.
The triple $(M,g,\nabla)$ is called a \emph{statistical manifold} if
$\nabla$ is torsion free and $\nabla g$ is completely symmetric.
Equivalently, a statistical manifold is a pseudo Riemannian manifold
$(M,g)$ equipped with a pair of symmetric conjugate connections.
Then, $(g,\nabla,\nabla^\ast)$ is called a \emph{statistical
structure} on $M$. Hence, a statistical manifolds is a
generalization of a pseudo Riemannian manifold.
An almost Norden manifold $(M,J,g)$ equipped with a statistical
structure $(g,\nabla,\nabla^\ast)$ will be called \emph{a
statistical almost Norden manifold}.
\subsection{Conjugate connections with respect to an almost complex structure}
Let $(M,g)$ be a pseudo Riemannian manifold, and $J$ be an almost
complex structure on $M$. If $\nabla$ is an arbitrary linear
connection then the connection $\nabla^\ast$ defined by
\begin{equation}\label{J-conj}
\nabla^\ast_X Y = - J\nabla_X JY = \nabla_X Y - J(\nabla_X J)Y
\end{equation}
is called the \emph{complex conjugate} \emph{connection} of $\nabla$
\cite{Al}, \cite{B}. From (\ref{J-conj}) it follows that
$(\nabla^\ast)^\ast = \nabla$, i.e. $\nabla$ and $\nabla^\ast$ are
mutually conjugate relative to $J$.
A connection $\nabla$ is self-conjugate with respect to $J$ if and
only if it is an \emph{almost complex connection}
($J$-\emph{compatible connection}), i.e. $\nabla J = 0$.
The average connection $\overline{\nabla}=\frac{1}{2}(\nabla +
\nabla^\ast)=\nabla - \frac{1}{2}J\nabla J$ of two complex conjugate
connections is $J$-compatible \cite{Al}.
By the same manner as in \cite{B}, we prove that if $g$ is a Norden
metric then $(\nabla^\ast_X g)(JY,JZ) = - (\nabla_X g)(Y,Z)$. Thus,
$\nabla^\ast g =0$ iff $\nabla g = 0$.
\section{Relations between conjugate connections\\ on almost Norden manifolds}\label{s2}
Let $(M,J,g)$ be an almost Norden manifold. In this section, we
study relations between the aforementioned types of conjugate
connections on $M$.
First, we study the coincidence of conjugate connections with
respect the pair of Norden metrics. Let us remark that if $\nabla$
and $\nabla^\ast$ are conjugate with respect to a Norden metric
tensor $g$, then by (\ref{g-conj}) it follows that $g((\nabla_X
J)Y,Z)=g((\nabla^\ast_X J)Z,Y)$. Hence, in this case $\nabla J = 0$
iff $\nabla^\ast J = 0$.
\begin{proposition}\label{p1}
Let $\nabla$ and $\nabla^\ast$ be linear connections on an almost
Norden manifold $(M,J,g)$. Then, each two of the following
conditions imply the third one:
\begin{enumerate}
\item[(i)] $\nabla$ and $\nabla^\ast$ are conjugate relative to $g$;
\item[(ii)] $\nabla$ and $\nabla^\ast$ are conjugate relative to $\widetilde{g}$;
\item[(iii)] $\nabla J = 0$ \emph{(}$\nabla^\ast J = 0$\emph{)}.
\end{enumerate}
\end{proposition}
\begin{proof}
Let us prove that conditions (i) and (ii) imply (iii). First, we
take into account that $\nabla$ and $\nabla^\ast$ are conjugate with
respect to $g$ and substitute $Y \rightarrow JY$ in (\ref{g-conj}).
Hence, by covariant differentiation and the definition of
$\widetilde{g}$, we obtain
\begin{equation}\label{p1-1}
X\widetilde{g}(Y,Z) = \widetilde{g}(\nabla_X Y, Z) +
\widetilde{g}(Y,\nabla^\ast_X Z) + g((\nabla_X J)Y,Z).
\end{equation}
Then, keeping in mind that $\nabla$ and $\nabla^\ast$ are also
conjugate with respect to $\widetilde{g}$, equality (\ref{p1-1})
implies $\nabla J = 0$.
The truthfulness of the other two statements is proved analogously.
\end{proof}
Proposition \ref{p1} yields the following
\begin{corollary}\label{c1}
Let $(M,J,g,\nabla,\nabla^\ast)$ be a statistical almost Norden
manifold. Then, $(M,J,\widetilde{g},\nabla,\nabla^\ast)$ is also a
statistical almost Norden manifold if and only if $\nabla J = 0$
($\nabla^\ast J = 0$).
\end{corollary}
Let us remark that if $(M,J,g,\nabla,\nabla^\ast)$ and
$(M,J,\widetilde{g},\nabla,\nabla^\ast)$ are simultaneously
statistical manifolds, the Levi-Civita connections $\nabla^0$ and
$\widetilde{\nabla}^0$ of $g$ and $\widetilde{g}$, respectively,
coincide with the average connection of $\nabla$ and $\nabla^\ast$
and hence $\nabla^0 J = \widetilde{\nabla}^0 J = 0$. The last
implies that $(M,J,g)$ and $(M,J,\widetilde{g})$ are both K\"ahler
Norden manifolds.
Next, we study the coincidence of conjugate connections relative to
the metric and the almost complex structure. In this regard, we
prove the following
\begin{proposition}\label{p2}
Let $(M,J,g)$ be an almost Norden manifold, and $\nabla$ be a linear
connection on $M$. Then:
\begin{itemize}
\item[(i)] the conjugate connections of $\nabla$ relative to $g$ and to $J$ coincide if and only if $\nabla \widetilde{g} = 0$;
\item[(ii)] the conjugate connections of $\nabla$ relative to $\widetilde{g}$ and to $J$ coincide if and only if $\nabla g = 0$.
\end{itemize}
\end{proposition}
\begin{proof}
Let us prove (i) (the other statement is proved analogously). The
conjugate connections of $\nabla$ relative to $g$ and to $J$
coincide if and only if the connection $\nabla^\ast$ defined by
(\ref{J-conj}) satisfies condition (\ref{g-conj}). Keeping in mind
the properties of $g$ and $\widetilde{g}$, the last condition is
equivalent to
\begin{equation}\label{p2-1}
Xg(Y,Z) = g(\nabla_X Y ,Z) - g(JY,\nabla_X JZ).
\end{equation}
Then, by substituting $Z \rightarrow JZ$ in (\ref{p2-1}), we obtain
$X\widetilde{g}(Y,Z) = \widetilde{g}(\nabla_X Y,Z) +
\widetilde{g}(Y,\nabla_X Z)$, i.e. $\nabla \widetilde{g} =0$ which
completes the proof.
\end{proof}
It is well-known that the unique linear connection which is
symmetric and metric with respect to a given metric tensor is the
Levi-Civita connection induced by this metric tensor. In light of
the last fact, Proposition \ref{p2} yields
\begin{corollary}\label{c2}
Let $(M,J,g)$ be an almost Norden manifold, and $\nabla$ be a
symmetric connection on $M$. Then:
\begin{itemize}
\item[(i)] the conjugate connections of $\nabla$ relative to $g$ and $J$ coincide if and only if $\nabla$ is the Levi-Civita connection $\widetilde{\nabla}^0$ of $\widetilde{g}$;
\item[(ii)]the conjugate connections of $\nabla$ relative to $\widetilde{g}$ and $J$ coincide if and only if $\nabla$ is the Levi-Civita connection $\nabla^0$ of $g$.
\end{itemize}
\end{corollary}
Thus, the conjugate connection of $\nabla^0$ (resp.
$\widetilde{\nabla}^0$) relative to $\widetilde{g}$ (resp., to $g$)
is its complex conjugate connection.
The case of a completely symmetric connection $\nabla$ is considered
in the following
\begin{corollary}\label{c3}
Let $(M,J,g)$ be an almost Norden manifold, and let $\nabla$ and
$\nabla^\ast$ be linear connections on $M$. Then:
\begin{itemize}
\item[(i)] If $(M,J,g,\nabla,\nabla^\ast)$ is a statistical manifold, and $\nabla^\ast$ is the conjugate connection of $\nabla$ relative to
$J$ then $(M,J,g)$ is a K\"ahler manifold;
\item[(ii)] If $(M,J,\widetilde{g},\nabla,\nabla^\ast)$ is a statistical manifold, and $\nabla^\ast$ is the conjugate connection of $\nabla$
relative to $J$ then $(M,J,g)$ is a K\"ahler manifold.
\end{itemize}
\end{corollary}
\begin{proof}
(i) Since $(M,J,g,\nabla,\nabla^\ast)$ is a statistical manifold,
the average connection of $\nabla$ and $\nabla^\ast$ is $\nabla^0$.
But because it is also the average connection of two complex
conjugate connections, $\nabla^0$ should be an almost complex
connection, i.e. $\nabla^0 J =0$. Hence, $(M,J,g)$ is a K\"ahler
manifold.
(ii) By a similar manner, we deduce that $(M,J,\widetilde{g})$ is a
K\"ahler Norden manifold, i.e. $\widetilde{\nabla}^0 J =
\widetilde{\nabla}^0\widetilde{g} =0$ which implies
$\widetilde{\nabla}^0 g = 0$. Because $\widetilde{\nabla}^0$ is
symmetric, the last equality yields $\widetilde{\nabla}^0 =
\nabla^0$ and hence $(M,J,g)$ is also K\"ahlerian.
\end{proof}
Based on the results in this section, we conclude that a pair of
linear connections $\nabla$ and $\nabla^\ast$ is conjugate with
respect to all three structural tensor $g$, $\widetilde{g}$ and $J$
simultaneously iff $\nabla g = \nabla \widetilde{g} = \nabla J = 0$
(which implies $\nabla^\ast=\nabla$). Linear connections preserving
the structural tensors of the manifold by covariant differentiation
are called \emph{natural}(\emph{adapted}). Hence, $\nabla$ is such a
connection.
\section{Conjugate Connections of the Levi-Civita
Connections induced by the pair of Norden metrics}\label{s3}
As seen in the previous section (Corollary \ref{c2}), the
Levi-Civita connections induced by the Norden metrics are the unique
symmetric linear connections on an almost Norden manifold for which
the conjugate connections relative to the associated metric tensor
and the almost complex structure coincide. In this section, we study
curvature properties of these connections.
Let us consider the conjugate connection $\nabla^\ast$ of $\nabla^0$
with respect to $\widetilde{g}$ and $J$, i.e. $\nabla^\ast_X Y =
\nabla^0_X Y - J(\nabla^0_X J) Y$. We remark that $\nabla^\ast$ is a
metric connection, i.e. $\nabla^\ast g = 0$.
If by $R^0$ and $R^\ast$ we denote the corresponding curvature
tensors, according to \cite{B}, we have $JR^\ast(X,Y)Z =
R^0(X,Y)JZ$. Hence, the average curvature tensor $P$ of $R^0$ and
$R^\ast$ defined by (\ref{K}) satisfies the property $P(X,Y)JZ =
JP(X,Y)Z$, meaning that $P$ is a K\"ahler curvature tensor. For
(0,4)-type tensors we have
\begin{equation}\label{P}
g(P(X,Y)Z,W) = \frac{1}{2}\{R^0(X,Y,Z,W)-R^0(X,Y,JZ,JW)\}.
\end{equation}
Next, we focus on the average connection of $\nabla^0$ and
$\nabla^\ast$ which we denote by $D$, i.e. $D_X Y = \nabla^0_X Y -
\frac{1}{2}J(\nabla^0_X J)Y$. Since $\nabla^\ast$ is conjugate to
$\nabla^0$ relative to $\widetilde{g}$ and $J$ simultaneously, the
average connection satisfies $D\widetilde{g}=D J =0$ and hence
$Dg=0$, i.e. $D$ is a natural connection. Moreover, it is the
well-known Lichnerowicz first canonical connection \cite{Li}. In
\cite{Teo}, we have obtained the form of the curvature tensor $K$ of
$D$ on an almost Norden manifold as follows
\begin{equation*}\label{KD}
\begin{array}{l}
g\big(K(X,Y)Z,W\big)=\frac{1}{2}\big\{R^0(X,Y,Z,W) -
R^0(X,Y,JZ,JW)\big\}\medskip\\
+\frac{1}{4}\big\{g\big((\nabla^0_{X}J)Z,(\nabla^0_{Y}J)W\big) -
g\big((\nabla^0_{X}J)W,(\nabla^0_{Y}J)Z\big)\big\}.
\end{array}
\end{equation*}
Then, the last equality and (\ref{P}) yield
\begin{proposition}
On an almost Norden manifold, the average curvature tensor $P$ of
the conjugate connections $\nabla^0$ and $\nabla^\ast$ and the
curvature tensor $K$ of their average connection $D$ are related as
follows
\begin{equation}\label{KP}
\begin{array}{l}
g(K(X,Y)Z,W) = g(P(X,Y)Z,W)\medskip\\ +
\frac{1}{4}\big\{g\big((\nabla^0_{X}J)Z,(\nabla^0_{Y}J)W\big) -
g\big((\nabla^0_{X}J)W,(\nabla^0_{Y}J)Z\big)\big\}.
\end{array}
\end{equation}
\end{proposition}
In \cite{Teo}, we have shown that
$||\nabla^0J||^2=2g^{il}g^{jk}g\big((\nabla^0_{e_{i}}J)e_{k},(\nabla^0_{e_{j}}J)e_{l}
\big)$ on a manifold in the class $\mathcal{W}_1\oplus\mathcal{W}_2$
of the Norden manifolds. Then, if by $\tau(K)$ and $\tau(P)$ we
denote the scalar curvatures of $K$ and $P$, respectively, from
(\ref{theta}) and (\ref{KP}), on a Norden manifold we have
\begin{equation}\label{tKP}
\begin{array}{l}
\tau(K) = \tau(P) + \frac{1}{8}\big(||\nabla^0 J||^2 - 2\
\theta(\Omega)\big).
\end{array}
\end{equation}
In \cite{Teo2}, we have proved that on a manifold in the class
$\mathcal{W}_1$ the relation $\theta(\Omega)= \frac{n}{2}||\nabla^0
J||^2$ is valid. Then, by (\ref{Fcl}) and (\ref{tKP}) we get
\begin{corollary}
On a Norden manifold $(M,J,g)$ belonging to the class
$\mathcal{W}_1$ ($\dim M = 2n \geq 4$) or to $\mathcal{W}_2$ is
isotropic K\"ahlerian iff $\tau(K) = \tau(P)$.
\end{corollary}
Analogous results are valid for the Levi-Civita connection
$\widetilde{\nabla}^0$ of $\widetilde{g}$ and its conjugate
connection $\widetilde{\nabla}^\ast$ relative to $g$ and $J$.
Next, using the characteristic condition (\ref{Fcl}) of the class
$\mathcal{W}_1$, the form (\ref{psi}) of the tensors $\psi_1$ and
$\psi_2$, and by straightforward calculations, we obtain
\begin{proposition}
\noindent Let $(M,J,g)$ be a $\mathcal{W}_1$-manifold. Then, the
curvature tensors $R^\ast$ and $\widetilde{R}^\ast$ of $\nabla^\ast$
and $\widetilde{\nabla}^\ast$, respectively, have the form:
\begin{equation*}
\begin{array}{l}
R^\ast = R^0 - \frac{1}{2n}[\psi_1 + \psi_2](S)
-\frac{\theta(\Omega)}{4n^2}[\pi_1 +\pi_2],\medskip\\
\widetilde{R}^\ast = \widetilde{R}^0 -
\frac{1}{2n}[\psi_1+\psi_2](\widehat{S}) -
\frac{\theta(J\Omega)}{4n^2}[\pi_1+\pi_2],
\end{array}
\end{equation*}
where $\widetilde{R}^0$ is the curvature tensor of
$\widetilde{\nabla}^0$, $S(X,Y) = (\nabla^0_X\theta)JY
+\frac{1}{2n}\theta(X)\theta(Y)$ and $\widehat{S}(X,Y) = -S(X,JY)$.
\end{proposition}
We remark that both $R^\ast$ and $\widetilde{R}^\ast$ are not
(0,4)-type curvature-like tensors.
\section{Statistical structures on almost Norden manifolds}\label{s4}
In this section, we consider statistical structures on almost Norden
manifolds by constructing and studying families of completely
symmetric linear connections.
Let $\nabla$ be a symmetric linear connection, and $Q(X,Y)$ be its
difference tensor with respect to the Levi-Civita connection
$\nabla^0$ of $g$, i.e.
\begin{equation}\label{nabla-0}
\nabla_X Y = \nabla^0_X Y + Q(X,Y).
\end{equation}
Denote $Q(X,Y,Z)=g(Q(X,Y),Z)$. Then by covariant differentiation we
obtain $(\nabla_X g)(Y,Z) = - Q(X,Y,Z) - Q(X,Z,Y)$. If
$(g,\nabla,\nabla^\ast)$ is a statistical structure, the last
equality and (\ref{stat}) imply that the tensor $Q(X,Y,Z)$ is
completely symmetric, i.e. $Q(X,Y,Z)=Q(Y,X,Z)=Q(X,Z,Y)$, and $\nabla
g = -2Q$. In this case, the connection $\nabla$ is said to be
\emph{completely symmetric}.
By (\ref{g-conj}) and (\ref{nabla-0}) we have
\begin{equation}\label{nabla-star}
\nabla^\ast_X Y = \nabla^0_X Y - Q(X,Y).
\end{equation}
Let us remark that in the theory of statistical manifolds the
(0,3)-type tensor $C(X,Y,Z)=g(\nabla^\ast_X Y - \nabla_X Y, Z) =
(\nabla_X g)(Y,Z)$, which differs from $Q$ only by a factor, is
called the \emph{cubic form} (\emph{skewness tensor}) of the
manifold.
It is known that equality (\ref{nabla-0}) and $\nabla^0 g=0$ imply
the following relation between the curvature tensors $R$ and $R^0$
of $\nabla$ and $\nabla^0$, respectively
\begin{equation}\label{R1}
\begin{array}{l}
g(R(X,Y)Z,W) = R^0(X,Y,Z,W) + (\nabla^0_X Q)(Y,Z,W) \medskip\\ -
(\nabla^0_Y Q)(X,Z,W) + Q(X,Q(Y,Z),W) - Q(Y,Q(X,Z),W).
\end{array}
\end{equation}
Analogously, (\ref{nabla-star}) yields
\begin{equation}\label{R2}
\begin{array}{l}
g(R^\ast(X,Y)Z,W) = R^0(X,Y,Z,W) - (\nabla^0_X Q)(Y,Z,W) \medskip\\
+ (\nabla^0_Y Q)(X,Z,W) + Q(X,Q(Y,Z),W) - Q(Y,Q(X,Z),W),
\end{array}
\end{equation}
where $R^\ast$ is the curvature tensor of $\nabla^\ast$. Then, by
(\ref{R1}) and (\ref{R2}) we obtain \cite{SL}
\begin{equation}\label{R3}
\begin{array}{l}
(\nabla^0_X Q)(Y,Z,W) - (\nabla^0_Y Q)(X,Z,W)\medskip\\
=\frac{1}{2}\{g(R(X,Y)Z,W) - g(R^\ast(X,Y)Z,W)\}.
\end{array}
\end{equation}
Also, since $Q$ is completely symmetric, we have
\begin{equation}\label{R4}
Q(X,Q(Y,Z),W)=g(Q(X,W),Q(Y,Z)).
\end{equation}
Let us denote
\begin{equation}\label{R5}
L(X,Y,Z,W) = g(Q(X,W),Q(Y,Z)) - g(Q(X,Z),Q(Y,W)).
\end{equation}
Since $L$ satisfies properties (\ref{R0}), $L$ is a curvature-like
tensor.
Taking into account (\ref{R3}), (\ref{R4}), (\ref{R5}) and the form
(\ref{K}) of the average curvature tensor (known as the statistical
curvature tensor \cite{FH}) $P$ of $\nabla$ and $\nabla^\ast$, from
(\ref{R1}) we verify
\begin{proposition}\label{p4}
On a statistical manifold, the statistical curvature tensor $P$ and
the curvature tensor $R^0$ are related as follows
\begin{equation}\label{PRL}
P = R^0 + L .
\end{equation}
\end{proposition}
If $\nabla$ is flat, then $\nabla^\ast$ is also flat which imply
$P=0$. Hence, for a flat statistical manifold $R^0=-L$.
If we consider $P$ as the curvature tensor jointly generated by
$\nabla$ and $\nabla^\ast$ then in the next statement we give a
necessary and sufficient condition for the Weyl tensor to be
invariant under the transformation of the Levi-Civita connection
$\nabla^0$ into the pair of symmetric conjugate connections
$(\nabla,\nabla^\ast)$.
\begin{corollary}\label{c-Weyl}
On a statistical manifold, the Weyl tensors of $P$ and $R^0$
coincide iff $W(L)=0$ where $L$ is given by (\ref{R5}).
\end{corollary}
Let $(M,J,g)$ be an almost Norden manifold, and
$(g,\nabla,\nabla^\ast)$ be a statistical structure on $M$. If we
ask for this structure to be compatible with $J$, i.e. $\nabla J =0$
(which implies $\nabla^\ast J =0$) we immediately obtain $\nabla^0 J
=0$. Hence, almost complex completely symmetric connections exist
only on K\"ahler manifolds. Thus, in order to study wider classes of
statistical almost Norden manifolds we will not aim for
$J$-compatibility.
\subsection{Completely symmetric connections constructed by the metrics and the Lie 1-forms}
According to (\ref{Fcl}), an almost Norden manifold which is not in
the class $\mathcal{W}_2\oplus\mathcal{W}_3$ has non-vanishing Lie
1-forms $\theta$ and $\widetilde{\theta}=\theta\circ J$. Thus, on
such manifolds, the pairs of Lie 1-forms and Norden metrics can be
used to construct difference tensors of completely symmetric linear
connections and thus statistical structures. One such family of
connections is introduced in the next
\begin{proposition}
On an almost Norden manifold
$(M,J,g)\not\in\mathcal{W}_2\oplus\mathcal{W}_3$, there exists a
four-parametric family of completely symmetric connections $\nabla$
defined by (\ref{nabla-0}) with difference tensor $Q$ given by
\begin{equation}\label{Q1}
\begin{array}{l}
Q(X,Y) = \lambda_1[\theta(X)Y+\theta(Y)X+g(X,Y)\Omega]\medskip\\
\phantom{Q(X,Y)}+
\lambda_2[\theta(JX)Y+\theta(JY)X+g(X,Y)J\Omega]\medskip\\
\phantom{Q(X,Y)}+\lambda_3
[\theta(X)JY+\theta(Y)JX+g(X,JY)\Omega]\medskip\\
\phantom{Q(X,Y)} +\lambda_4
[\theta(JX)JY+\theta(JY)JX+g(X,JY)J\Omega],
\end{array}
\end{equation}
$\lambda_i \in \mathbb{R}$ ($i=1,2,3,4$).
\end{proposition}
By (\ref{psi}), (\ref{R5}), (\ref{Q1}) and straightforward
calculations we obtain
\begin{proposition}
Let $(M,J,g,\nabla,\nabla^\ast)$ be the statistical almost Norden
manifold with $\nabla$ defined by (\ref{nabla-0}) and (\ref{Q1}).
Then, the statistical curvature tensor $P$ of the manifold has the
form (\ref{PRL}) where
\begin{equation}\label{L1}
\begin{array}{l}
L = \psi_1(S_1) + \psi_2(S_2) \medskip\\
\phantom{L} + [(\lambda_1^2 - \lambda_2^2)\theta(\Omega) +
2\lambda_1\lambda_2\theta(J\Omega)]\pi_1 \medskip\\
\phantom{L} + [(\lambda_3^2 -
\lambda_4^2)\theta(\Omega)+2\lambda_3\lambda_4\theta(J\Omega)]\pi_2\medskip\\
\phantom{L} - [(\lambda_1\lambda_3
-\lambda_2\lambda_4)\theta(\Omega)+(\lambda_1\lambda_4+\lambda_2\lambda_3)\theta(J\Omega)]\pi_3,
\end{array}
\end{equation}
and
\begin{equation*}
\begin{array}{l}
S_1(X,Y) = (\lambda_1^2 + \lambda_3^2 -
2\lambda_2\lambda_3)\theta(X)\theta(Y) + (\lambda_2^2 + \lambda_4^2
+ 2\lambda_1\lambda_4)\theta(JX)\theta(JY)\medskip\\
\phantom{P(X,Y)}+(\lambda_1(\lambda_2+\lambda_3)+\lambda_4(\lambda_3-\lambda_2))[\theta(X)\theta(Y)+\theta(JX)\theta(JY)],\bigskip\\
S_2(X,Y) = (\lambda_3^2-\lambda_4^2)[\theta(X)\theta(Y) -
\theta(JX)\theta(JY)]\medskip\\
\phantom{S(X,Y)}-2\lambda_3\lambda_4[\theta(X)\theta(Y)+\theta(JX)\theta(JY)].
\end{array}
\end{equation*}
\end{proposition}
Since for the Weyl of $\psi_1(S)$ it is valid $W(\psi_1(S))=0$, by
Corollary \ref{c-Weyl}, equalities (\ref{psi}), (\ref{Weyl}) and
(\ref{L1}) we get the following
\begin{proposition}
\noindent Let $\nabla$ be the family of linear connections defined
by (\ref{nabla-0}) and (\ref{Q1}) with the condition
$\lambda_3=\lambda_4=0$. Then, the Weyl tensors of $P$ and $R^0$
coincide.
\end{proposition}
\subsection{Completely symmetric connections constructed by the Lie 1-forms}
A family of completely symmetric linear connections with difference
tensor depending only on the Lie 1-forms $\theta$ and
$\widetilde{\theta}=\theta\circ J$ is presented in the following
\begin{proposition}
On an almost Norden manifold
$(M,J,g)\not\in\mathcal{W}_2\oplus\mathcal{W}_3$, there exists a
four-parametric family of completely symmetric connections $\nabla$
defined by (\ref{nabla-0}) with difference tensor $Q$ given by
\begin{equation}\label{Q2}
\begin{array}{l}
Q(X,Y) = \lambda_1\theta(X)\theta(Y)\Omega + \lambda_2
\theta(JX)\theta(JY)J\Omega \medskip\\
\phantom{Q(X,Y)}+\lambda_3[\theta(X)\theta(Y)J\Omega
+\theta(X)\theta(JY)\Omega + \theta(JX)\theta(Y)\Omega] \medskip\\
\phantom{Q(X,Y)} + \lambda_4[\theta(JX)\theta(Y)J\Omega +
\theta(JX)\theta(JY)\Omega+\theta(X)\theta(JY)J\Omega],
\end{array}
\end{equation}
$\lambda_i \in \mathbb{R}$ ($i=1,2,3,4$).
\end{proposition}
By (\ref{R5}), (\ref{Q2}) and straightforward calculations we obtain
\begin{proposition}
Let $(M,J,g,\nabla,\nabla^\ast)$ be the statistical almost Norden
manifold with $\nabla$ defined by (\ref{nabla-0}) and (\ref{Q2}).
Then, the statistical curvature tensor $P$ of the manifold has the
form (\ref{PRL}) where
\begin{equation*}
L(X,Y,Z,W) = \alpha
[\theta(X)\theta(JY)-\theta(JX)\theta(Y)][\theta(Z)\theta(JW)-\theta(JZ)\theta(W)],
\end{equation*}
where $\alpha =
[\lambda_3^2-\lambda_4^2-\lambda_1\lambda_4+\lambda_2\lambda_3]\theta(\Omega)-
(\lambda_1\lambda_2+\lambda_3\lambda_4)\theta(J\Omega)$.
\end{proposition}
A direct consequence of the last statement and (\ref{PRL}) is that
on manifolds with isotropic Lie vector field $\Omega$ with respect
to both $g$ and $\widetilde{g}$, i.e. satisfying
$\theta(\Omega)=\theta(J\Omega)=0$, we obtain $L=0$, and thus the
statistical curvature tensor $P$ of the statistical structure
defined by (\ref{nabla-0}) and (\ref{Q2}) coincides with the
curvature tensor $R^0$ of $\nabla^0$.
\textbf{Acknowledgement.} The author would like to express her
gratitude to Professor Dr. C.~Udri\c{s}te for his suggestion on the
topic of this paper.
| 95,213
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\section{Open-closed TQFTs and the Cardy Condition}
\label{sec:cardy}
We conclude with a discussion of open-closed topological field theories in the B-model and we prove that a condition holds for
Hochschild homology which is equivalent to the Cardy Condition in the
Calabi-Yau case. Appropriate references for open-closed 2d
topological field theories include Moore-Segal~\cite{MooSeg},
Costello~\cite{Cos} and Lauda-Pfeiffer~\cite{LaudaPfeifer}.
\subsection{Open-closed 2d TQFTs.}
Consider the open and closed 2-cobordism category $\TwoCob$ whose
objects are oriented, compact one-manifolds --- in other words,
disjoint unions of circles and intervals --- and whose morphisms are
(diffeomorphism classes of) cobordisms-with-corners between the source
and target one-manifolds. A morphism can be drawn as a vertical
cobordism, from the source at the bottom to the target at the top. As
well as parts of the boundary being at the top and the bottom, there
will be parts of the boundary in between, corresponding to the fact
that this is a cobordism with corners. An example is shown below.
\[\pic{cobordism}\]
Disjoint union makes $\TwoCob$ into a symmetric monoidal category and
an open-closed two-dimensional topological quantum field theory (2d
TQFT) is defined to be a symmetric monoidal functor from $\TwoCob$ to
some appropriate symmetric monoidal target category, which we will
take to be the category of vector spaces or the category of graded
vector spaces. The category $\TwoCob$ has a simple description in
terms of generators and relations which means that there is a
reasonably straight forward classification of open-closed 2d TQFTs up
to equivalence. This is what we will now describe. The following
morphisms generate $\TwoCob$ as a symmetric monoidal category.
\[\pic{fig1}\]
We will come back to the relations below.
To specify an open-closed 2d TQFT up to equivalence on objects it
suffices to specify the image $\closed$ of the circle and the image
$\open$ of the interval. The former is called the space of closed
string states and the latter is called the space of open-string
states. Using the four \emph{planar} generating morphisms pictured
above, together with the relations between them, it transpires that
$\open$, the space of open-string states is precisely a symmetric, but
not-necessarily commutative Frobenius algebra. This means that it is a
unital algebra with a non-degenerate, symmetric, invariant
inner-product. It is useful to note here that the inner product is
symmetric because the two surfaces pictured below are diffeomorphic,
however these surfaces are \emph{not} ambient isotopic --- so one
cannot be deformed to the other in three-space while the bottom
boundary is fixed.
\[\pic{syminnerprod1}\stackrel{\text{diffeo}}{=}\pic{syminnerprod2}\]
On the other hand, the first four generating morphisms, along with
their relations, mean that $\closed$, the space of closed string
states, is a \emph{commutative} Frobenius algebra. The last two
morphisms mean that there are maps $i_*\colon \closed\to\open$ and
$i^*\colon \open \to \closed$, and by the relations these are adjoint
with respect to the pairings on these spaces. Moreover, $i_*$ is an
algebra map, such that its image lies in the centre of $\open$. The
final relation that these must satisfy is the \emph{Cardy Condition}.
In terms of the generators pictured above this is the following
relation:
\[\pic{cardy1}=\pic{cardy2}.\]
Note again that these surfaces are diffeomorphic but not isotopic in
three-space. In terms of maps, writing $\mu\colon \open\otimes
\open\to \open$ and $\delta\colon\open\to\open\otimes\open$ for the
product and coproduct of the open string state space and writing
$\tau\colon\open\otimes\open\to\open\otimes\open$ for the symmetry in
the target category, the Cardy Condition is the equality of maps from
$\open$ to $\open$:
\[\mu\circ \tau\circ \delta=i_*\circ i^*.\]
We will have reason to use an equivalent condition below.
To summarize, having an open-closed 2d TQFT is equivalent to having
the data of a commutative Frobenius algebra $\closed$, a symmetric
Frobenius algebra $\open$, and an algebra map $i_*\colon \closed \to
\open$ with central image, such that the Cardy Condition is satisfied.
\subsection{Open-closed 2d TQFTs with D-branes.}
A more interesting model of string theory is obtained when we specify
a set of `boundary conditions' or `D-branes' for the open strings. For
a mathematician this just means a set of labels for the boundary
points of objects. So fix a set $\Lambda$ of labels, and consider the
category $\TwoCobL$ of open-closed cobordisms such that the objects
are compact, oriented one-manifolds with the boundary points labelled
with elements of the set $\Lambda$, and morphisms having their
internal boundaries labelled compatibly with their boundaries. Here is
an example of a morphism from the union of the circle and the interval
labelled $(B,A)$, to the interval labelled $(B,A)$.
\[\pic{labelledcobordism}\]
Now a $\Lambda$-labelled open-closed TQFT is a symmetric monoidal
functor to some appropriate target category which we will again take
to be the category of vector spaces or the category of graded vector
spaces. Moreover, the category $\TwoCobL$ is similarly generated by
morphisms as listed above, but now they must all be labelled, and the
relations are just labelled versions of the previous relations. Thus
we can similarly classify $\Lambda$-labelled open-closed TQFTs. Once
again the image of the circle is a commutative Frobenius algebra,
$\closed$. However, rather than getting a single vector space $\open$
associated to an interval, we get a vector space $\open_{BA}$
associated to each ordered pair $(B,A)$ of elements of $\Lambda$; so
we do not get a single Frobenius algebra, but rather something which
could be called a `Frobenius algebra with many objects' or a
`Frobenius algebroid', but, for the reason explained below, such a
thing is commonly known as a Calabi-Yau category. It is a category in
the following sense. We take the category whose objects are
parametrized by $\Lambda$ and, for $A,B\in \Lambda$, the morphism set
$\Hom(A,B)$ is taken to be $\open_{BA}$ (this is consistent with us
reading diagrams from right to left). The composition
$\mu_{CBA}\colon\open_{CB}\otimes\open_{BA}\to\open_{CA}$ is given by
the image of the appropriately labelled version of the morphism
pictured.
\[\pic{labelledproduct}\]
The Frobenius or Calabi-Yau part of the structure is a --- possibly
graded --- perfect pairing $\open_{AB}\otimes\open_{BA}\to \k$: the
grading degree of this map is called the dimension of the Calabi-Yau
category.
So to specify a labelled open-closed 2d TQFT it suffices to specify a
commutative Frobenius algebra $\closed$, a Calabi-Yau category $\open$
and an algebra map $i_A\colon \closed\to \open_{AA}$ with central
image, for each object $A$, such that the labelled version of the
Cardy Condition holds.
\subsection{The open-closed 2d TQFT from a Calabi-Yau
manifold.}
Associated to a Calabi-Yau manifold $X$ there are two standard 2d TQFTs
coming from string theory, imaginatively named the
A-model and the B-model: it is the B-model we will be
interested in here. In the B-model the boundary conditions are
supposed to be ``generated'' by complex submanifolds of $X$ so the boundary conditions are
taken to be complexes of coherent sheaves on $X$; the open string
category is then supposed to be the derived category of coherent sheaves on
$X$. This is indeed a Calabi-Yau category, which is why such
categories are so named: for each $\cE$ and $\cF$, the requisite pairing
$\Hom^\blob_{\D(X)}(\cE,\cF)\otimes \Hom^\blob_{\D(X)}(\cF,\cE)\to
\k[-\dim X]$ comes from the Serre pairing as a Calabi-Yau
manifold is precisely a manifold with a trivial canonical bundle.
According to the physics, the closed string state space $\closed$ should be $\Hom^\blob_{\D(X\times X)}(\cO_\Delta,\cO_\Delta)$, in other words, the
Hochschild cohomology algebra
$\HH^\blob(X)$.
As $X$ is Calabi-Yau, a trivialization of the canonical bundle induces
an isomorphism between Hochschild cohomology and Hochschild homology,
up to a shift. This means that the closed string space $\closed$
has both the cohomological product and the Mukai pairing, and these make $\closed$ into a
Frobenius algebra.
We need to specify the algebra maps $i_\cE\colon \closed \to
\open_{\cE\cE}$. These are maps
\[ i_\cE\colon \Hom^\blob(\cO_\Delta,\cO_\Delta)\to
\Hom^\blob(\cE,\cE) \]
which can be given by interpreting $\cE$ as a kernel $\pt\to X$ and taking $i_\cE$ to be convolution with the identity on $\cE$. This is given
diagrammatically on an element
$\phi\in\Hom^\blob(\cO_\Delta,\cO_\Delta)$ as follows.
\[\pic{phiinHH}\quad\mapsto\quad \pic{iEofphiinHH}.\]
At this point it should be noted that $\closed$ is to be thought of as the centre of the category $\open$. The notion of centre is generalized from algebras to categories by taking the centre of a category to be the natural transformations of the identity functor; however, in a 2-category an appropriate notion of the centre of an object is the set of 2-endomorphisms of the identity morphism on that object. This means that $\closed$ is the centre, in this sense, of the category $\open$ in the 2-category $\gVar$.
The map going the other way, $i^\cE\colon \Hom^\blob(\cE,\cE)\to
\Hom^\blob(\cO_\Delta,\cO_\Delta)$ is given by taking the trace, namely for
$e\in \Hom_{\D(X)}(\cE,\cE)$ the map is given by
\[\pic{ereallyinHomEE}\quad\mapsto \quad\pic{iEofeinHomEE}.\]
This definition relies on the fact that $X$ is Calabi-Yau, so that the
Serre kernel is, up to a shift, just the identity 1-morphism $\Id_X$.
An argument similar to Proposition~\ref{prop:iotasadjoint} shows that
$i_\cE$ and $i^\cE$ are adjoint. In order to argue that we indeed
have an open-closed TQFT it remains to show that the Cardy Condition
holds. In fact, we will prove a more general statement, the Baggy
Cardy Condition.
\subsection{The Baggy Cardy Condition.}
In the case of a manifold $X$ that is not necessarily Calabi-Yau we
don't have the same coincidence of structure as above: we no longer
have a Frobenius algebra $\HH^\blob(X)$; rather we have an algebra
$\HH^\blob(X)$ and an inner product space $\HH_\blob(X)$. This means
that we can not formulate the Cardy Condition as it stands. We now state a
condition which makes sense for an arbitrary, non-Calabi-Yau manifold and
which is equivalent to the Cardy Condition in the Calabi-Yau case.
\begin{Theorem}
Suppose that $\open$ is a Calabi-Yau category and $\closed$ is an
inner product space, such that for each $A\in \open$ there are
adjoint maps $i^A\colon \open_{AA}\to \closed$ and $i_A\colon
\closed \to \open_{AA}$. Then the Cardy Condition
\[\mu_{BAB}\circ\tau\circ\delta_{ABA}=i_B\circ i^A\]
is equivalent to the following equality holding for all $a\in
\open_{AA}$ and $b\in \open_{BB}$, where the map ${}_{a}m_{b}\colon
\open_{AB}\to \open_{AB}$ is the map obtained by pre-composing with
$a$ and post-composing with $b$:
\[\left\langle i^B{-},i^A{-}\right\rangle_\closed=
\Tr{}_{-}m_{-}.\]
\end{Theorem}
\begin{Proof}
The first thing to do is examine the left-hand side of the Cardy
Condition. As $\open$ is a Calabi-Yau category there is the
following equality of morphisms $\open_{AA}\to \open_{BB}$.
\[\pic{cardyA}=\pic{cardyB}\]
Note that this does not require any reference to $\closed$, but it
does fundamentally require the symmetry of the inner product. This
is reflected in the fact that the surfaces underlying the above
pictures are diffeomorphic but not ambient isotopic.
This means that the Cardy Condition is equivalent to the following
equality.
\[\pic{cardyC}=\pic{cardyD}\]
By the non-degeneracy of the inner product on $\open_{BB}$ this is
equivalent to the equality of two maps $\open_{BB}\otimes
\open_{AA}\to\k$ which are drawn as follows.
\[\pic{cardyE}=\pic{cardyF}\]
The right-hand side is instantly identifiable as $\left\langle
i^B{-},i^A{-}\right\rangle_\closed$. The left-hand side is
identifiable as the trace of the triple composition map
$\open_{BB}\otimes\open_{BA}\otimes\open_{AA}\to \open_{BA}$ which
gives the required result.
\end{Proof}
We can now show that the alternative condition given in the above
theorem holds for the derived category and Hochschild homology of
\emph{any} space: in particular, the Cardy Condition holds for
Calabi-Yau spaces.
\begin{Theorem}[The Baggy Cardy Condition]
\label{thm:cardy}
Let $X$ be a space, let $\cE$ and $\cF$ be objects in $\D(X)$ and
consider morphisms
\[ e\in \Hom_{\D(X)}(\cE, \cE) \quad\text{and}\quad f\in
\Hom_{\D(X)}(\cF, \cF). \]
Define the operator
\[ {}_f m_e \colon\Hom^\blob_{\D(X)}(\cE, \cF) \ra
\Hom^\blob_{\D(X)}(\cE, \cF) \]
to be post-composition by $f$ and pre-composition by $e$. Then we
have
\[ \Tr {}_f m_e=\MP{\iota^\cE(e)}{\iota^\cF(f)}, \]
where $\iota^\cE$, $\iota^\cF$ are the maps defined in
Section~\ref{subsec:iotae}, and $\Tr$ denotes the (super) trace.
\end{Theorem}
\begin{Proof}
The proof is very similar to the proof of the
Semi-Hirzebruch-Rie\-mann-Roch Theorem (Theorem~\ref{thm:hrr}). The first thing
to observe is that $\Hom(\cE,\cF)\cong \cE^\chk\circ \cF$ and that
${}_f m_e$ is just $e^\chk\circ f$. However, we have
$\tau_R(e)=\SerK_\pt\circ e^\chk$ and $\SerK_\pt$ is trivial so
$e^\chk=\tau_R(e)$. Putting this together with the invariance of
the Serre trace under the partial trace we get the following
sequence, and hence the required result.
\begin{align*}
\Tr {}_f m_e&=\Tr\left(\pic{baggy1}\right)
=\Tr\left( \pic{baggy2}\right)\\
&=\Tr\left( \pic{baggy3}\right)
=\Tr\left( \pic{baggy4}\right)\\
&=\MPbig{\pic{baggy5a}}{\pic{baggy5b}}
=\MP{\iota^\cE\!(e)}{\iota^\cF\!(f)}.\qedhere
\end{align*}
\end{Proof}
\medskip
\noindent
Observe that the Semi-Hirzebruch-Riemann-Roch Theorem is a direct
consequence of the Baggy Cardy Condition, with $e=\id_\cE$,
$f=\id_\cF$.
| 102,094
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. different causes why people moving from one city to another. Whatever the cause is for the moving, you have to make sure that you are ready and prepared for the moving day.
Moving and Packing Tips
| 195,777
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\begin{document}
\begin{abstract}
Given a Hopf algebra in a symmetric monoidal category with duals, the
category of modules inherits the structure of a monoidal category with
duals. If the notion of algebra is replaced with that of monad on a
monoidal category with duals then Brugui\`eres and Virelizier showed
when the category of modules inherits this structure of being monoidal
with duals, and this gave
rise to what they called a Hopf monad. In this paper it is shown that
there are good diagrammatic descriptions of dinatural transformations
which allows the three-dimensional, object-free nature of their
constructions to become apparent.
\end{abstract}
\maketitle
\section*{Introduction}
\subsection*{Overview}
\thispagestyle{empty}
An algebra, i.e., a monoid in the category of vector spaces,
has an associated category of modules (or representations, if you
prefer). A Hopf algebra is an algebra equipped with extra structure
which ensures that its category of modules inherits the monoidal
structure and
duals from the category of vector spaces. These notions work similarly in braided monoidal categories
other than that of vector spaces. A monad on a monoidal category can
be thought of as a generalization of an algebra (or monoid) in that category and has an associated
category of modules (also known, confusingly, as its category of
algebras).
Brugi\`eres and Virelizier~\cite{BruguieresVirelizier:Hopf} defined, following Moerdijk~\cite{Moerdijk:MonadsTensorCategories},
a \emph{Hopf monad} structure which
ensures that a monad's category of modules is monoidal with duals.
The goal of this paper is to put some of the work of Brugui\`eres and
Virelizier into a diagrammatic context, which also means to put it
into appropriate framework of monoidal two-categories. One of the
purposes of this was to make their constructions essentially
object-free. To do this it was necessary to do various things
including using an object-free formulation of categories with duals,
which here means describing evaluation and coevaluation as dinatural
transformations and then extending string diagrammatics to include
dinatural transformations, something which appears to work rather
well. Such dinatural transformations exist in the landscape of the
monoidal two-category of categories, and so are three-dimensional in
nature, thus are better manipulated, I would argue, using the
three-dimensional algebra presented here.
In terms of results on Hopf monads, many of the results are just
slight simplifications of those of Brugui\`eres and Virelizier. The
example of a strong monoidal functor with a left adjoint is given a
more explicit treatment, this example being of
primary importance to me. My
motivation lies in the specific case of a Hopf monad on the derived
category of coherent sheaves on a complex manifold; the monad coming
from a strong monoidal functor with a left adjoint.
\subsection*{Three-dimensional string diagrams}
The utility of string diagram notation in describing adjunctions and
monads is well-known (see for example
\cite{Lauda:FrobeniusAmbidextrous} though it undoubtably has its roots in the
Australian school). Adjunctions and monads live in the two-category
of categories and their counterparts in the monoidal category (or
one-object two-category) of vector spaces are duals and algebras, and
quantum topologists know that these are well notated using string
diagrams, with the distinction between notation and application
becoming blurred in the case of knot invariants arising from ribbon
categories. Low dimensional topology and low dimensional category
theory seem closely linked.
When one is considering monads on monoidal categories, one is led to
considering three-dimensional notation and this works similarly well.
There is some precedent in the use of surface
diagrams by the Australian school but this is not well
represented in the literature: see, for
example,~\cite{Street:FunctorialCalculus}. There is also the mythical,
unavailable~\cite{McIntyreTrimble}, but I have not seen a copy.
\subsection*{Hopf monads}
A monad \(T\colon \CC\to\CC\) on a category is an endofunctor together
with a \emph{multiplication} natural transformation \(T^2\nattrans T\)
and a \emph{unit} natural transformation \(\Id_\C\nattrans T\)
satisfying some appropriate associativity and unital conditions. One
then has the category of \(T\)-modules, consisting of pairs \((m,r)\)
where \(m\) is an object in \(\CC\) and \(r\colon T(m)\to m\) is an action map in a
suitable sense. If \(\CC\) is a monoidal category then it is natural
to ask if the tensor product of any two \(T\)-modules can be given a
natural \(T\)-module structure. For instance, if \(T\) is of the form
\(A\otimes {-}\) for some algebra object \(A\) of \(\CC\) then this
occurs when \(A\) has the structure of a bialgebra. The question was
considered by Moerdijk \cite{Moerdijk:MonadsTensorCategories} and he
showed that lifting the monoidal structure on \(\CC\) to a monoidal structure
on the module category corresponds precisely to giving \(T\) some
extra structure to form what he called, following operad terminology,
a Hopf monad. However in this arena this is not the best nomenclature
and a better term would be either \emph{bimonad}, by analogy with
bialgebras, or, more accurately, but less succinctly,
\emph{opmonoidal monad}, as the extra structure is the
same as making \(T\) into a monad in the category of opmonoidal
functors, as was observed by McCrudden
\cite{McCrudden:OpmonoidalMonads}.
A natural progression from this is to ask if duals from \(\CC\) lift
to duals on the category of modules, so if \(\CC\) is a monoidal
category with duals and \(M\) is object of \(\CC\) with an action of
the bimonad \(T\) on it, does the dual \(M^\vee\) also naturally have an
action of \(T\) on it? In the case where \(T=A\otimes{-}\) is a monad
on, say, the category of vector spaces for a bialgebra \(A\), then
\(A\) being a \emph{Hopf} algebra, i.e., having an antipode, suffices
to make the module category have a lift of the duality on vector
spaces. Brugui\`eres and Virelizier defined the notion of antipode
for a bimonad on a category with duals, such that it corresponds to
the duals lifting to the module category.
\subsection*{An example}
As explained in Section~4, a strong monoidal functor with a left
adjoint gives rise to a Hopf monad by composing it with its left
adjoint. Suppose $G$ is a finite group, then a good example of such a
functor is the functor $\Delta^*$ from the category of representations
of $G\times G$ to the category of representations of $G$ defined by
restricting to the diagonal copy of $G$. This is a strong monoidal
functor and has a left adjoint $\Delta_*$ which is induction from $G$
to $G\times G$ along the diagonal embedding. Then $\Delta^*\Delta_*$
is a Hopf monad on the category of representations of $G$, in fact it
is basically the tensor-with-the-group-algebra functor, the group
algebra (with the conjugation action) is a Hopf algebra in this
category. Similar examples arise in other areas such as when a
complex manifold takes the place of the finite group and the derived
category of coherent sheaves takes the place of the representation
category. These examples naturally arise in topological quantum field
theory. Details will appear elsewhere.
\subsection*{Synopsis} In the first section we recall the notion of
string diagrams and how they are useful for denoting adjoints. We
then enhance the notation in a third dimension to denote products of
categories. We then see how this is used with monoidal and opmonoidal
functors and natural transformations. Finally opposite categories are
introduced to the notation.
In Section~2 we recall the string diagrammatic approach to monads and
see how Moerdijk's notion of bimonad fits in. In Section~3 we recall
the notion of dinatural transformations and how they are used in the
definition of a monoidal category with duals.
Section~4 is where we get to the definition of Hopf monad and
Brugui\`eres and Virilizier's theorem about their categories of
modules being monoidal with duals. Finally, following~\cite{BruguieresVirelizier:Hopf}, we see how a strong
monoidal functor with a left adjoint gives rise to a Hopf monad.
\subsection*{Terminology}
There is a split in terminology between category theorists and, say,
quantum algebraists, in that an object in a monoidal category with a
unital, associative multiplication is called a monoid by the former
and an algebra by the latter, primarily because the typical categories
for the two groups of mathematicians are respectively the category of sets and the
category of vector spaces. We will stick to the latter terminology as
we are close to areas in which Lie algebra objects and universal
enveloping algebra objects are considered. Similarly it makes sense
from this perspective to talk of \emph{modules} over a monad, rather
than \emph{algebras} over a monad as the category theorists prefer.
My apologies go to any reader to whom this seems ridiculous, but there
will be a good proportion of the audience to whom it will seem very sensible.
\subsection*{Acknowledgements}
Thanks to Alain Brugui\`eres, Eugenia Cheng, Aaron Lauda and Alexis
Virelizier for useful comments.
\section{The diagrammatics of natural transformations}
In this section we introduce the basic string diagram notation for
natural transformations. In particular we will be interested in
monoidal categories and opmonoidal functors, so will need to extend
the notation in an extra dimension. We will also need to consider
contravariant functors so we end this section with a way of denoting
opposite categories.
\subsection{String diagrams}
Firstly, here is a very quick reminder on the use of string diagrams to
represent natural transformations. These diagrammatics have a history
going back to Feynmann and Penrose, but were formalized in the context
of monoidal categories by Joyal and Street \cite{JoyalStreet}, and this formalism extends
to arbitrary two-categories; though here we will
just be interested in the two-category of categories, functors and
natural transformations. The
idea of string diagrams for natural transformations
is that the usual globular pictures of functors and natural transformations are
replaced by their Poincar\'e duals, so that categories are represented by
two-cells, functors by one-cells, and natural transformations by
zero-cells. The basic example is as follows:
\[\pstex{H1_1_0}\qquad\text{becomes}\qquad\pstex{H1_1_1}.\]
A more complicated example is the following:
\[\pstex{H1_1_0b}\qquad\text{becomes}\qquad\pstex{H1_1_1b}.\]
Note that the identity functor is omitted from the string diagram
notation.
The standard example of the utility of string diagrams is with regard
to adjoint functors and this will turn out to be useful to us later.
So suppose that \(F\colon \CC\to\DD\) and \(U\colon \DD \to \CC\) form
an adjunction \(F\dashv U\), then we have the unit and counit natural
transformations, \(\epsilon \colon F\circ U\nattrans \Id_\DD\) and
\(\eta \colon \Id_\CC\nattrans U\circ F\), which are drawn as follows:
\[\epsilon\equiv\pstex{H1_1_2};\qquad\eta\equiv{\pstex{H1_1_3}}.\]
Then the required conditions on the unit and counit are drawn as
\[\pstex{H1_1_4}=\pstex{H1_1_5}\quad\text{and}\quad
\pstex{H1_1_6}=\pstex{H1_1_7},\]
where the vertical line marked with \(F\) means the identity natural
transformation on the functor \(F\).
\subsection{Monoidal categories and monoidal functors}
The notation above can be enhanced to also encode cartesian products
of categories, so can denote, for instance, functors of the form
\(\otimes\colon \CC\times\CC\to\CC\). In this context we can think
of the two-category \(\CAT\) with its cartesian product as being a
one-object three-category, and so
should expect to have to use three-dimensions in our notation.
We will use the direction out of the page for the `product direction'.
So, for example, given functors \(G,H\colon \CC\times \CC'\to \CC''\)
and \(K\colon \CC''\to \CC''\), we denote \(G\) and \(K\circ H\) as
follows.
\[\pstex{H1_2_2a}\quad\pstex{H1_2_2b}\]
A natural transformation \(\theta\colon G \nattrans K\circ H\) will
then be denoted
\[\pstex{H1_2_2c}\quad\text{or}\quad\pstex{H1_2_2d},\]
the latter being used if the labels are clear from the
context.
Note that diagrams are read front to back, right to left and bottom to top.
\subsubsection{Monoidal categories}
A monoidal category \((\CC,\otimes,\one)\) consists, as is well known,
of a
category \(\CC\), a functor \(\otimes\colon \CC\times \CC\to
\CC\), and a unit object \(\one\) --- here considered as a functor
\(\one\colon \pnt\to \CC\) from the one object, one morphism category
\(\pnt\) --- together with an associativity natural transformation
\(\alpha\colon \otimes \circ (\Id\times \otimes)\nattrans \otimes
\circ (\otimes \times \Id)\),
and unit natural isomorphisms \(\nu_l\colon {\tensor}\circ
(\one\times \Id_\CC)\nattrans \Id_\CC\) and \(\nu_r\colon {\tensor}\circ
(\Id_\CC\times \one)\nattrans \Id_\CC\). The natural transformations have to satisfy the pentagon and
triangle relations.
The associativity will be drawn as
\[\alpha\equiv\pstex{H1_2_3},\]
but coherence means that notationally we can draw three-fold tensor
products with the understanding that to make sense they must be
resolved into the composition of two-fold tensor products, but that, up
to canonical identification, it is independent of the choice of
resolution, so we could instead draw the following triple tensor
product \({-}\otimes{-}\otimes{-}\colon \CC\times \CC\times \CC\to \CC\).
\[\pstex{H1_2_4a}\]
The
category \(\pnt\) is the unit object for the product \(\times\) on
\(\CAT\); we will make the canonical identifications \(\CC\times
\pnt\cong \CC\cong \pnt\times \CC\) and thus allow ourselves to denote
\(\pnt\) by the empty surface. So for instance the unit
\(\one\colon \pnt\to \CC\) will be denoted by the picture on the left,
from which the picture on the right will be understood.
\[\pstex{H1_2_5}\equiv\pstex{H1_2_5a}.\]
The unit natural isomorphisms \(\nu_l\colon {\tensor}\circ
(\one\times \Id_\CC)\nattrans \Id_\CC\) and \(\nu_r\colon {{}\tensor{}}\circ
(\Id_\CC\times \one)\nattrans\Id_\CC\) are drawn as
\[\nu_l\equiv\pstex{H1_2_6}\qquad\text{and}\qquad\nu_r\equiv\pstex{H1_2_7}.\]
The inverses of these are drawn the same but the other way up.
\subsubsection{Monoidal and opmonoidal functors}
\label{Subsection:MonoidalFunctors}
If \((\CC,\otimes,\one)\) and \((\DD,\otimes,\one)\) are monoidal
categories then a monoidal functor \(M\colon \DD\to \CC\) ---
sometimes called a weak- or lax-monoidal functor --- means a functor
equipped with a natural family of morphisms \(M(d_1)\otimes M(d_2)\to
M(d_1\otimes d_2)\), parameterized by pairs of objects, and a morphism
\(\one\to M(\one)\) satisfying coherence conditions. In other words, there are natural transformations
\(\sigma_M^2\colon{\tensor}\circ(M\times M)\nattrans M\circ \otimes\) and
\(\sigma_M^0\colon\one\nattrans M\circ\one\), satisfying some
constraints. These natural transformations will be drawn as follows:
\[\sigma_M^2\equiv\pstex{H1_2_8}\qquad\text{and}\qquad\sigma_M^0\equiv\pstex{H1_2_9}.\]
The conditions that they are required to satisfy are the
diagrammatically appealing:
\[\pstex{H1_2_10}=\pstex{H1_2_11}\mytag{Mondl1}\label{Monoidal1}\]
\[\pstex{H1_2_12}=\pstex{H1_2_13}\mytag{Mondl2}\label{Monoidal2}
\]\[
\pstex{H1_2_14}=\pstex{H1_2_15}.\mytag{Mondl3}\label{Monoidal3}\]
The first condition together with associativity of the tensor product
means that again we can unambiguously draw triple
tensor products, with the understanding that such a triple tensor
product should be resolved into one of the two compositions of
ordinary tensor products. Thus we think of the two natural
transformations pictured as a single natural transformation
\(({-}\otimes{-}\otimes{-})\circ(M\times M \times M)\nattrans
M\circ ({-}\otimes{-}\otimes{-})\), drawn as
\[\pstex{H1_2_4}.\]
A functor \(M\) as above is said to be \emph{strong} monoidal if both the
natural transformations drawn above are natural \emph{isomorphisms}.
Similarly an opmonoidal functor \(Q\colon \CC\to \DD\) --- sometimes
called a comonoidal functor --- is a functor equipped with natural
transformations \(\sigma^Q_2\colon Q\circ \otimes\nattrans \tensor\circ(Q\times Q)\)
and \( \sigma^Q_0\colon Q\circ\one\nattrans \one\), drawn as
\[\sigma^Q_2\equiv\pstex{H1_2_16}\qquad\text{and}\qquad\sigma^Q_0\equiv\pstex{H1_2_17},\]
which satisfy the above relations inverted, which will be called (\textsf{Opmondl1--3}).
Note the important situation in which \(U\) and \(F\) form an adjoint pair
\(F\dashv U\) of
functors between monoidal categories. Then there is a
bijection between pairs of natural transformations making \(U\)
monoidal and pairs of natural transformations making \(F\) opmonoidal;
or more informally, \(U\) is monoidal if and only if \(F\) is
opmonoidal. To see this, suppose that \(U\) is monoidal, then an
opmonoidal structure is defined on \(F\) using the following two
natural transformations:
\[\pstex{H1_2_18}\qquad\text{and}\qquad\pstex{H1_2_19}.\]
The proof that they satisfy the requisite
relations is easily derived diagrammatically.
\subsubsection{Opmonoidal natural transformations}\label{Section:OpMonNT}
We will need the notion of an opmonoidal natural transformation. So
suppose that $P,Q\colon \CC\to \DD$ are two opmonoidal functors
between monoidal categories, then an opmonoidal natural transformation
$\theta\colon P\nattrans Q$ is a natural transformation which commutes
with the opmonoidal structure transformations, in other words, the
following relations hold:
\begin{align*}
\pstex{H1_2_20}&=\pstex{H1_2_21}\mytag{OpmonNT1}\label{OpmonNT1};\\
\pstex{H1_2_23}&=\pstex{H1_2_22}\mytag{OpmonNT2}\label{OpmonNT2}.
\end{align*}
\subsection{Opposite categories and contravariance}
Later on we will be interested in looking at duals in categories.
There is a slight problem from the point of view of diagrams in that a duality
\({}^\vee\) on a category \(\CC\) is actually a contravariant functor,
so is not a functor \(\CC\to \CC\) but rather can be considered a
covariant functor \(\CC^\op\to \CC\). This means it will be extremely
convenient to be able to denote opposites in the diagrammatic
language. First it is imperative to think about the operation of
taking opposites inside the two-category \(\CAT\). Given a category
\(\CC\) the category \(\CC^\op\) is the category whose objects are in
canonical bijection with the objects of \(\CC\) but whose arrows are
reversed. This means given a functor \(G\colon\CC\to \DD\) one
obtains a functor \(G^\op\colon\CC^\op \to \DD^\op\). However, given
a natural transformation \(\theta\colon G_1\nattrans G_3\circ G_2\)
one obtains a natural transformation \(\theta^\op\colon G_3^\op\circ
G_2^\op\nattrans G_1^\op\). This is an easy and informative exercise
for the reader. Put another way, this gives a two-functor
\({}^\op\colon \CAT\to \CAT^\text{co}\) where \(\CAT^\text{co}\) denotes the
two-category of categories with the two-morphisms reversed.
In traditional notation we get a correspondence as follows:
\[\pstex{H1_3_1}\qquad\text{gives rise to}\qquad\pstex{H1_3_2}.\]
To get this into the string pictures we will adopt the useful
convention that a shaded region means that it is the opposite
category, and the functors and natural transformations will be the
opposites of their labels. Thus
\[\pstex{H1_3_3}\quad\text{gives rise to}\quad\pstex{H1_3_4}
\quad\text{denoted}\quad\pstex{H1_3_5}.\]
Note the essential difference that ``shaded'' natural transformations are ``turned
upside-down''. When we get to dinatural transformations, the shading
can be given the interpretation of the `other-side' of the surface.
This notational convention means that a contravariant functor
\({}^\vee\) on a category \(\CC\) can be denoted in the following way:
\[\pstex{H1_3_6}.\]
Note that the conditions needed to be satisfied by a duality on a
monoidal category will be stated in terms of dinatural
transformations, so we will not do that properly until later.
\section{Monads and bimonads}\label{Section:MonadsAndBimonads}
In this section we will look at monads and bimonads using string
diagrams. In particular we consider the category of modules over a
monad from this perspective; this can be seen as a `formal' or
two-categorical point of view in that we discuss the category of
modules over a monad on a category without talking about the internal
structure of the category. We go on to look at bimonads and how the
cateogry of modules in this case is monoidal, analogous to the
category of modules for a bialgebra. Finally we will see how a
bimonad arises from a pair of adjoint functors.
\subsection{Monads}
This will be a quick diagrammatic recap on monads. A monad on a
category \(\CC\) is an endofunctor \(T\colon \CC\to \CC\) together
with natural transformations \(\mu\colon T^2\nattrans T\) and
\(\iota\colon \Id_\CC\nattrans T\), known as the multiplication and unit and
drawn as
\[\mu\equiv\pstex{H2_1_1}\qquad\text{and}\qquad\iota\equiv\pstex{H2_1_2}.\]
These have to satisfy the associativity and unit laws, namely
\begin{gather}
\pstex{H2_1_3}=\pstex{H2_1_4};\mytag{Monad1}\label{Monad1}\\
\pstex{H2_1_5}=\pstex{H2_1_6}=\pstex{H2_1_7}.\mytag{Monad2}\label{Monad2}
\end{gather}
There is an associated category \(T\CC\) of \emph{\(T\)-modules}, this
is sometimes written
\(\CC^T\). The objects of this category
are pairs \(\bigl(m,\ \left(r\colon T(m)\to m\right)\bigr)\) where \(m\) is an object of
\(\CC\), such that the diagrams
\[\xymatrix{
T\circ T(m)\ar[r]^{Tr}
\ar[d]_{\mu_m}
&T(m)
\ar[d]^r
\\
T(m)
\ar[r]^r
&m
}
\quad\text{and}\quad \xymatrix{ m\ar[r]^{\iota_m} \ar[dr]_{\id_m}
&T(m) \ar[d]^r
\\
&m}
\]
both commute,
as one would expect from anything befitting the name `module'. The
morphisms in the category of \(T\)-modules are morphisms between
the underlying objects of \(\CC\) that commute with the
\(T\)-action. The example that I have in mind here is where
\(\CC\) is a monoidal category, \(A\) is a unital algebra object in
\(\CC\) and \(T\) is the monad \(A\otimes{-}\); in this case the
category of \(T\)-modules is precisely the category of
\(A\)-modules in the usual sense.
We fit the category of modules into the graphical calculus by
considering associated functors and natural transformations. An
object of the
category of modules consists of an object of \(\CC\) and an action
morphisms, an alternative view of such a pair is that we have a
forgetful functor \(U_T\colon T\CC\to \CC\) given
by \(U_T(m,r):=m\), which just forgets the action, and we have a
natural transformation \(\rho\colon T\circ U_T\nattrans U_T\)
defined by \(\rho_{(m,r)}:=r\) which encodes the action. We will
denote the forgetful functor $U_T$ by a dashed-dotted line and draw
the natural transformation \(\rho\) as follows.
\[\rho\equiv\pstex{H2_1_21}\]
The module conditions above become the following:
\begin{gather}
\pstex{H2_1_22}=\pstex{H2_1_23};\mytag{Module1}\label{module1}\\
\pstex{H2_1_24}=\pstex{H2_1_25}.\mytag{Module2}\label{module2}
\end{gather}
We also have a
free module functor \(F_T\colon \CC\to T\CC\). This is given on objects by
\[ F_T(x):= \left(T(x),\ \left(\mu_{T(x)}\colon T(T(x))\to T(x)\right)\right).\]
Note that \(T\) is precisely the composite \(U_T\circ F_T\),
so we have identity natural transformations $\Id\colon U_T\circ
F_T\stackrel\sim\nattrans T$ and $\Id \colon T\stackrel\sim\nattrans
U_T\circ F_T$ which we draw as
\[\pstex{H2_1_8}\qquad\text{and}\qquad\pstex{H2_1_9}.\]
Here it should be observed that the graphical language fails to
distinguish between identity natural transformations and natural
isomorphisms.
Note that the multiplication $\mu$ on $T$ is recovered from these
identifications together with the action natural transformation $\rho$
in the following way:
\[\pstex{H2_1_1}=\pstex{H2_1_11}.\]
\subsection{Monads from adjoint functors}
A standard way of obtaining monads is via pairs of adjoint functors.
Suppose that $F$ and $U$ form such a pair, \(F\dashv U\), then
\(U\circ F\colon \CC\to\CC\) forms a monad. The multiplication and
unit of the monad are obtained from the unit and counit of the
adjunction in the following easily drawn fashion:
\[\mu\equiv\pstex{H2_2_1};\qquad \iota\equiv\pstex{H2_2_2}.\]
If the reader has not seen this before then they should immediately
verify diagrammatically that the axioms of a monad are satisfied.
It should be noted that every monad \(T\) actually arises in this way,
as the composite of a left and a right adjoint; for example,
there is an adjunction \(F_T\dashv U_T\) between the free and
forgetful functors described above. In general there will be
several different adjoint decompositions of a monad.
\subsection{Bimonads}\label{section:bimonads}
We now bring monads and monoidal categories together. Suppose that
\(T\colon \CC\to\CC\) is a monad on a monoidal category
\((\CC,\otimes,\one)\). We can then ask the question ``Under what
circumstances does the monoidal structure on \(\CC\) lift to a monoidal structure on the category of \(T\)-modules \(T\CC\)?'' Or, we could ask the weaker and less precise question ``Given two \(T\)-modules \(\left(m,\left(r\colon T(m)\to m\right)\right)\) and
\(\left(m',\left(r'\colon T(m')\to m'\right)\right)\) how do we obtain a natural \(T\)-module structure
on \(m\otimes m'\)?''
The answer to the first question (and hence the second) was found by Moerdijk, but before stating the answer we should introduce the following piece of terminology.
\begin{defn} A monad \(T\colon\CC\to \CC\) on a monoidal category has
a \emph{bimonad} (or \emph{opmonoidal monad}) structure if the
functor \(T\) has an opmonoidal structure with respect to which both
the multiplication
\(\mu\) and the unit \(\iota\) are opmonoidal natural
transformations (in the sense of Section~\ref{Section:OpMonNT}).
\end{defn}
Bimonads are so named because of the analogy with bialgebras given by
Theorem~\ref{Thm:Moerdijk} below, though they were called Hopf monads
in his original paper. Before stating the Theorem it is worth
unpacking this rather concise definition a
little. A monad has a bimonad structure if firstly there is an
opmonoidal structure for $T$, that is there are specified
natural transformation
\[\sigma_2^T\colon \otimes\circ (T\times
T)\nattrans T\circ \otimes
\quad\text{and} \quad
\sigma_0^T\colon T\circ \one \nattrans \one,\]
drawn as follows, in which hopefully it is clear where the hidden
lines go.
\[\sigma_2^T\equiv\pstex{H2_3_21}, \qquad
\sigma_0^T\equiv\pstex{H2_3_22}\] These natural transformations must
satisfy the axioms (\textsf{Opmondl1--3}) and the multiplication and
unit must be opmonoidal with respect to this which means that
the following must hold:
\[\pstex{H2_3_13}=\pstex{H2_3_14};
\qquad
\pstex{H2_3_15}=\pstex{H2_3_16}\mytag{BM1--2}\label{BM1};
\]
\[\pstex{H2_3_24}=\pstex{H2_3_23}; \qquad \pstex{H2_3_25}=\pstex{H2_3_26}.
\mytag{BM3--4}\label{BM2}\]
The result of Moerdijk can now be stated.
\begin{thm}[Moerdijk \cite{Moerdijk:MonadsTensorCategories}]
\label{Thm:Moerdijk}
Let \(T\) be a monad on a monoidal category \(\CC\) and let \(T\CC\)
be its category of modules. Then specifying a
lift of the monoidal structure on \(\CC\) to a monoidal structure on
\(T\CC\) is precisely the same as
specifying a bimonad structure on \(T\).
\end{thm}
We will spend the rest of this section seeing why this is true. To do
this, the following is an extremely useful observation.
\begin{prop}\label{Lemma:LiftingToModules}
For \(T\colon \CC\to \CC\) a monad, \(T\CC\) its category of modules
and \(\DD\) any category, specifying a functor \(H\colon \DD\to
T\CC\) is precisely the same as specifying a functor
\(\pre{U}H\colon \DD\to \CC\) and a natural transformation
\(\rho_H\colon T\circ \pre{U}H\nattrans \pre{U}H\), drawn as
\[\pstex{H2_3_1},\]
such that the following two relations are satisfied:
\[
\pstex{H2_3_2}=\pstex{H2_3_3}
\quad\text{and}\quad
\pstex{H2_3_4}=\pstex{H2_3_5}.
\mytag{R1--2}\label{R12}
\]
\end{prop}
\begin{proof}[Proof (sketch)]
This is just the fact that \(T\CC\) consists of pairs \((m,r)\),
where \(m\in \CC\) and \(r\colon T(m)\to m\) is an action. So given
such a functor \(H\) define \(\pre{U}H\) to be \(U_T\circ H\) and
\(\rho^H\colon T\circ U_T\circ H\nattrans U_T\circ H\) to be \(\rho\circ
\id_H\).
Conversely, given such a pair, define \(H(d):=\left(\pre{U}H(d),\ \left(
\rho^H_d\colon T(\pre{U}H(d))\to \pre{U}H(d)\right)\right)\).
\end{proof}
Now we will see how to prove Moerdijk's Theorem. To lift the monoidal
structure of \(\CC\) to a monoidal structure on \(T\CC\) we need to
specify the tensor product and unit on \(T\CC\), together with the
associativity and unital natural transformations. We will concentrate
on the tensor product.
For a functor \({\otimes^T}\colon \CC\times T\CC\to T\CC\) to be a
lift of a functor \({\otimes}\colon \CC\times \CC\to \CC\) (with
respect to the forgetful functor \(U_T\colon T\CC \to \CC\)) the
following diagram must commute.
\[\xymatrix{
T\CC\times T\CC
\ar[d]_{U_T\times U_T}
\ar[r]^{\otimes^T}
&
T\CC
\ar[d]^{U_T}
\\
\CC\times\CC
\ar[r]^{\otimes}
&\CC
}
\]
Algebraically this means that
\(U_T\circ \otimes^T = {\otimes}\circ (U_T\times U_T)\), so in the
notation of Proposition~\ref{Lemma:LiftingToModules} we have \(\pre{U}\otimes^T{}={\otimes}\circ (U_T\times
U_T)\). By the proposition then, lifting the functor \({\otimes}\) is precisely the same as specifying
a natural transformation \[\rho_{\otimes}\colon T\circ {\otimes}\circ (U_T\times
U_T)\nattrans {\otimes}\circ (U_T \times U_T)\]
which satisfies the two conditions of the lemma.
(Intuitively this says that to specify a lifted tensor product of two
modules we need to specify an action of $T$ on the tensor product of
the two objects underlying the modules.)
We draw the natural transformation as
\[\rho_\otimes\equiv\pstex{H2_3_6},\]
In general for an adjunction \(F\dashv U\), and with $G$ and $H$
functors with suitable source and target, there is a bijection between
sets of natural transformations:
\[
\Nat(G,H\circ F)
\cong \Nat(G\circ U, H).
\]
Hence, because there is the adjunction \(F_T\dashv
U_T\) and the monad factorizes as \(T=U_T\circ F_T\), for any functors $G,H\colon \CC\to\DD$ there are the following
identifications of sets of natural transformations
\[
\Nat(G,H\circ T)
=\Nat(G, H\circ U_T\circ F_T)
\cong \Nat(G\circ U_T, H\circ U_T).
\]
In the diagrammatic notation, the isomorphism between the two outside sets is given by
\[\pstex{H2_3_7}\mapsto \pstex{H2_3_8}\qquad\text{and}\qquad
\pstex{H2_3_9}\mapsto \pstex{H2_3_10}.\]
Similarly there is an identification
\[
\Nat(T\circ{\otimes},{\otimes}\circ(T\times T))
\cong
\Nat(T\circ{\otimes}\circ(U_T\times U_T),{\otimes}\circ(U_T\times U_T))
\]
so the natural transformation \(\rho_{\otimes}\) above is equivalent to a natural transformation
\[\pstex{H2_3_21}\equiv\pstex{H2_3_12}\]
satisfying the two conditions \bref{R12} of the proposition, but these
are clearly axioms \bref{BM1}.
(Thus we have a natural transformation \(T\circ \otimes \nattrans
\otimes \circ(T\times T)\) which is used to write down an action on
the tensor product, as if we have $T$-modules \((m,r)\) and
\((m',r')\) then we will define the action on \(m\otimes m'\) by
\(T(m\otimes m')\to Tm\otimes Tm'\xleftarrow{r\otimes r'} m\otimes
m'.\))
Having lifted the tensor product we also need to lift the unit
$\one\colon \pnt\to \CC$; this gives the morphism $\sigma_0^T$
satisfying \handtag{BM3--4}. Now for these two functors to combine to
give a tensor product structure they must give an opmonoidal structure
for $T$. This gives Theorem~\ref{Thm:Moerdijk}. Full details are
found in \cite{Moerdijk:MonadsTensorCategories}.
\subsection{Bimonads from adjoint functors}
\label{Subsection:BimonadsFromAdjoint}
Suppose now that \(U\colon \DD\to \CC \) is a strong monoidal functor
between monoidal categories, so we have the following pairs of inverse natural transformations:
\[\pstex{H2_4_1}\qquad\text{and}\qquad\pstex{H2_4_2};\]
\[\pstex{H2_4_11}\qquad\text{and}\qquad\pstex{H2_4_12}.\]
And suppose that \(U\) has a left adjoint \(F\colon\CC\to\DD\) then, as
mentioned in Section~\ref{Subsection:MonoidalFunctors}, \(F\) is opmonoidal; and as \(U\) is also opmonoidal,
so the composite \(U\circ F\) is opmonoidal, with the opmonoidal
structure given explicitly as follows:
\[\sigma^{U\circ F}_2\equiv\pstex{H2_4_3}; \qquad \sigma^{U\circ F}_0\equiv\pstex{H2_4_13}.\]
Of course, \(U\circ F\) is also a monad, and it is very easy to see in
this pictorial language that it is an opmonoidal monad, ie.\ that the
product and unit of the monad are opmonoidal transformations.
For instance, the following proves that \handtag{BM1} holds:
\begin{align*}
\pstex{H2_4_4}
&=\pstex{H2_4_5}=\pstex{H2_4_6}\\
&=\pstex{H2_4_7}.
\end{align*}
Thus in the case that \(U\) is strongly monoidal, \(U\circ F\) is a
bimonad.
Note that if \(T\) is a bimonad, then the category \(T\CC\) of
\(T\)-modules is monoidal and in the decomposition \(T=U_T\circ F_T\)
the forgetful functor \(U_T\) is strongly monoidal.
\section{The diagrammatics of dinatural transformations}
In this section we introduce dinatural transformations so that we can
give a diagrammatic description of duality on a monoidal category, the
point being that evaluation and coevaluation are dinatural rather than
natural transformations.
\subsection{Motivating example: a monoidal category with duals}
The first question to address is ``What is an appropriate notion of a
monoidal category with duals?'' To simplify the situation, we
will just consider \emph{left} duals: right duals can be handled
similarly. So suppose that \((\CC,\otimes,\one)\) is a monoidal
category, a left duality on \(\CC\) will be a functor \({}^\vee\colon
\CC^\op\to \CC\) together with evaluation and coevaluation maps for
every object \(a\) in the category,
\[\ev_a\colon \prevee a\otimes a\to \one
\quad\text{and}\quad
\coev_a\colon \one \to a\otimes \prevee a,\]
such that for every morphism \(f\colon a\to a'\) in the category
the following naturality conditions hold:
\[
\ev_a\circ (\prevee f\otimes \id)
=\ev_{a'}\circ ( \id\otimes f)
\quad\text{and}\quad
(f\otimes \id)\circ \coev_{a}
=( \id\otimes \prevee f)\circ\coev_{a'},
\]
and such that the following ``snake'' relations hold,
\begin{align*}(\id_a\otimes \ev_a)\circ (\coev_a\otimes \id_a)&=\id_a\\
(\ev_a\otimes \id_{\prevee a}) \circ (\id_{\prevee a}\otimes \coev_a)
& =\id_{\prevee a.}
\end{align*}
One would like to interpret \(\ev\) and \(\coev\) as some sort of natural
transformations, so that, for instance, \(\ev\) would be a natural
transformation from the ``functor'' \(\CC\to\CC\) given by
\(a\mapsto \prevee a\otimes
a\) to the functor \(\CC\to\CC\) given by \(a\mapsto \one\): however the former is \emph{not} a functor. So we
consider the functor \(\CC^\op\times \CC\to \CC;\ (b,a)\mapsto
\prevee b \otimes a\) and the functor \(\one\colon \pnt\to \CC\).
Evaluation is then
a family of morphisms \(\ev_a\colon \prevee a\otimes a\to \one\), indexed by the
objects of \(\CC\), such that the following diagrams commutes:
\[\xymatrix{
&\prevee a\otimes a\ar[dr]^{\ev_a}\\
\prevee a'\otimes a\ar[ur]^{\prevee f\otimes \id_a}
\ar[dr]_{\id_{a'}\otimes f }
&&\quad\one\quad.\\
&\prevee a'\otimes a'\ar[ur]_{\ev_{a'}}
}\]
Now, because, amongst other reasons, we also want to deal with
coevaluation \(\coev_a\colon \one \to a\otimes \prevee a\), and the
constraint \((\id_a\otimes \ev_a)\circ (\coev_a\otimes \id_a)=\id_a\),
we introduce the more general notion of dinatural transformation.
\subsection{Dinatural transformations}
Motivated by the above example, we are led to Eilenberg and Kelly's
notion of dinatural transformation.
Suppose
\[P\colon \CC\times \CC^\op\times \AA\to \BB
\quad\text{and}\quad
Q\colon \AA\times \DD^\op\times \DD\to \BB\]
are two functors, then a \emph{dinatural transformation} \(\beta\colon
P\dinat Q\) is a family of morphisms
\(\beta_{c,a,d}\colon P(c,c,a)\to Q(a,d,d)\) which satisfies
the following naturality condition. If $f\colon a\to a'$, $g\colon c\to c'$ and $h\colon d \to d'$ are morphisms in \(\AA\), \(\CC\) and \(\DD\) respectively then the diagram below commutes.
\[\xymatrix{
&P(c,c,a')\ar[r]^{\beta_{c,a',d}}&
Q(a',d,d)\ar[dr]^{Q(\id_{a'},\id_d,h)}\\
P(c,c',a)\ar[ur]^{P(\id_c,g,f)}\ar[dr]_{P(g,\id_c,\id_a)}&&&
Q(a',d,d')\\
&P(c',c',a)\ar[r]^{\beta_{c',a,d'}}&
Q(a,d',d')\ar[ur]_{Q(f,h,\id_{d'})}
}\]
Note that each of the categories \(\AA\), \(\CC\), \(\DD\) and \(\DD\) can be products of
other categories or can indeed be the terminal category \(\pnt\) and can
all be permuted in the definition. Thus the
case of evaluation described above occurs when \(\CC=\BB\) and
\(\AA=\DD=\pnt\), and a usual natural transformation is the case where
\(\CC=\DD=\pnt\).
We can then denote such a dinatural transformation as follows. For functors
\[P\colon \CC\times \CC^\op\times \AA\to \BB
\quad\text{and}\quad
Q\colon \AA\times \DD^\op\times \DD\to \BB\]
a dinatural transformation \(\beta\colon
P\dinat Q\) is denoted
\[\pstex{H3_2_1},\]
where as usual the diagram is read upwards.
The first thing to note is that in the case of a natural
transformation, when \(\CC=\DD=\pnt\), we recover the usual string
diagram notation.
The next thing to note is the right-hand profile of the surface. This
is the so-called Eilenberg-Kelly graph of the dinatural
transformation, consisting of arcs with the end-points of an arc
labelled by the same category. These graphs are important in the
composition of dinatural transformations as we will see below.
It can also be pointed out that a dinatural transformation can
actually be written as a natural transformation. For example a
dinatural transformation \(\beta\colon P\dinat Q\) as above is
equivalent to a natural transformation
\[ \Hom_\CC(-,-) \times\Hom_\AA(-,-)\times
\Hom_\DD(-,-) \nattrans \Hom_\BB\left(P({-},{-},{-}),Q({-},{-},{-})\right)\]
between functors from \(\CC^\op\times \CC\times\AA^\op\times \AA\times
\DD^\op\times \DD\) to \(\Set\), so this also allows a diagrammatic
description, but it is rather messier and the composition is not as
straightforward.
\subsection{Vertical composition}
In order to make sense of the snake condition on \(\ev\) and \(\coev\)
we will need to define the vertical composition of dinatural
transformations.
If \(\beta'\colon P\dinat Q\) and \(\beta\colon Q\dinat
R\) are two dinatural transformations then the composite
\(\beta\circ\beta'\) can not always be defined; however the composite
can be defined,
resulting in a dinatural transformation, in the case that the composite
of the Eilenberg-Kelly graphs contains no loops, as I will now
explain.
This is best illustrated by our motivating example.
If we have a monoidal category \(\CC\) with a functor \(\prevee\colon
\CC^\op\to \CC\) and dinatural transformations
\(\ev\colon\prevee\otimes \Id_\CC\dinat \one\) and
\(\coev\colon\one\dinat \Id_\CC\otimes\prevee\), where, for instance \(\prevee\otimes \Id_\CC\) means \({\otimes}\circ(\Id_\CC\times \prevee)\). Then we can draw these dinatural transformations as
\[\ev\equiv \pstex{H4_1_12b}\qquad \coev\equiv \pstex{H4_1_14b}.\]
We then have a dinatural transformation
\(\id\otimes\ev\colon\Id_\CC\otimes (\prevee\otimes \Id_\CC)\dinat
\Id_\CC\) whose components are given by \( (\id\otimes
\ev)_{a'',a}:=\id_{a''}\otimes\ev_a\colon a''\otimes(\prevee a \otimes
a)\to a''\). This should be drawn with binary tensor products,
but, by identifying \({\otimes}\circ(\otimes\times \Id_\CC)\) with a
triple tensor product \({-}\otimes{-}\otimes{-}\), it will be drawn as
\[\id\otimes\ev\equiv\pstex{H3_2_2}.\]
Similarly we have \(\coev\otimes\id\colon \Id_\CC\dinat (\Id_\CC\otimes \prevee)\otimes \Id_\CC\) with its components given by \([\coev\otimes\id]_{a',a}:=\coev_{a'}\otimes \id_a\colon a\to (a'\otimes \prevee (a'))\otimes a\) which is drawn as
\[\coev\otimes\id\equiv\pstex{H3_2_3}.\]
The two dinatural transformations pictured above can be vertically composed to give a dinatural transformation, which is actually a natural transformation, \(\id\otimes \ev)\circ(\coev\otimes \id)\colon \Id_\CC\nattrans\Id_\CC\)
given by \[[(\id\otimes\ev)\circ(\coev\otimes \id)]_a:=
(\id\otimes\ev)_{a,a}\circ(\coev\otimes \id)]_{a,a}\colon
a\to a\]
and drawn as
\[\pstex{H3_2_4}.\]
One of the conditions for \(({}^\vee, \ev, \coev)\) to form a duality on the monoidal category $\CC$ is that the above should be the identity natural transformation. The two conditions are drawn in the next subsection, below.
More generally, we can define the composite of two dinatural
transformations provided the composite Eilenberg-Kelly graph has no
loops. For example we can form a dinatural transformation from two
dinatural transformations of the following form. Suppose we have
functors
\[P\colon\pt\to \BB,\quad
R\colon \CC\times \CC^\op\to \BB,\quad
Q\colon \CC\times\CC^\op\times\CC\times\CC^\op\to \BB
\]
together with dinatural transformations \(\beta' \colon P\dinat Q\) and
\(\beta\colon Q\dinat R\) which pair up the categories as pictured
below, then there is a composite \(\beta\circ\beta'\colon P\dinat R\).
\[\begin{matrix}\pstex{H3_3_7}\\\pstex{H3_3_8}\end{matrix}
\mapsto \pstex{H3_3_9}.\]
However, we can not form a dinatural transformation from the composite
\(\ev\circ \coev\) as we would get a loop in the Eilenberg-Kelly graph
as can be seen here:
\[\pstex{H3_3_10}.\]
See
\cite{EilenbergKelly} for more details.
\subsection{Definition of a monoidal category with left duals}
We can now state the definition of a monoidal category with left duals in this language. Suppose that $\CC$ is a monoidal category, ${}^\vee\colon \CC^{\op}\to \CC$ is a functor and $\ev\colon {}^\vee\otimes \Id_\CC\nattrans \Id_\CC$ and $\coev\colon \Id_\CC \nattrans \Id_\CC\otimes{}^\vee$ are dinatural transformations drawn as
\[\ev\equiv \pstex{H4_1_12b}\qquad \coev\equiv \pstex{H4_1_14b}.\]
Then $({}^\vee,\ev,\coev)$ forms a \emph{left duality} of $\CC$ if the following snake relations hold:
\begin{align*}\pstex{H3_2_4}&=\pstex{H3_2_5};\mytag{Duality1}\label{Duality1}\\
\pstex{H3_3_5}&=\pstex{H3_3_6}.\mytag{Duality2}\label{Duality2}\end{align*}
\subsection{Composition with natural transformations}
\renewcommand{\tilde}{\widetilde}
Suppose that \(F\colon \tilde \AA\to \AA\), \(G\colon \tilde \CC\to
\CC\), \(H\colon \tilde \DD\to \DD\), and \(K\colon \BB\to
\tilde\BB\) are functors and that \(\beta \colon P\dinat Q\) is a
dinatural transformation of the above form then we get a dinatural
transformation
\[\tilde \beta\colon K\circ P\circ (G\times G^\op\times F)\dinat
K\circ Q\circ (F\times H^\op\times H),\]
given by
\(\tilde\beta_{\tilde a, \tilde c, \tilde d}:=K(\beta_{F(\tilde a),
G(\tilde c), H(\tilde d)})\).
This is denoted graphically as
\[\pstex{H3_4_1}.\]
Note that in the case that \(\beta\) is an ordinary natural
transformation this recovers the ordinary horizontal composition of
natural transformations.
Suppose now that we have
natural transformations
\[\pstex{H3_4_2},\quad\pstex{H3_4_3},\quad\pstex{H3_4_4}\quad
\text{and}\quad\pstex{H3_4_5},\]
together with
a dinatural transformation
\(\beta \colon P\dinat Q\)
of the above form
then the following dinatural
transformations are equal:
\[\pstex{H3_4_7}\quad=\quad\pstex{H3_4_8}.\]
This just follows from the definitions. In traditional notation it is
quite a mess to write down, thus this does show how nicely the
diagrammatics capture the essence of composition of dinatural
transformations.
\section{Hopf monads}
In this section we use the diagrammatic language to first give
Brugui\`eres and Virilizier's definition of a Hopf monad and a minor
simplification of their proof that such thing is equivalent to a lift
of duals to the module category. We then use the diagrammatics to be
more explicit than them in their example of a monad coming from a
strongly monoidal functor with a left adjoint.
\subsection{Hopf monads}
The difference between a bialgebra and a Hopf algebra is that the
latter has an antipode. In the current context, the principal
consequence of this is that the vector space dual of a module over a
Hopf algebra carries a canonical action. More precisely,
the duality on the base category of vector spaces lifts to a duality
on the category of modules. It is this property that we wish to
examine for monads, and we can do this by asking the question ``What
structure is required of a bimonad on a monoidal category with duals
so that the category of modules has a lift of the duals?'' This was
answered by Brugui\`eres and Virelizier
\cite{BruguieresVirelizier:Hopf} and here we will describe their
solution in the diagrammatic language, something that they themselves
would have liked to have done.
Brugui\`eres and Virelizier \cite{BruguieresVirelizier:Hopf} gave the
definition of `left antipode'. We will restate this definition using
the diagrammatic notation developed above.
\begin{defn} If \(T\) is a bimonad on a monoidal category with left
duals then a \emph{left antipode} for \(T\) is a natural
transformation \(\SSS\colon T\circ {}^\vee\circ T\nattrans {}^\vee\),
denoted as follows,
\[\SSS\equiv \pstex{H4_1_2},\]
which satisfies the following two relations:
\begin{align*}
\pstex{H4_1_12}&=\pstex{H4_1_13};\mytag{HM1}\label{HM1}\\
\pstex{H4_1_14}&=\pstex{H4_1_15}.\mytag{HM2}\label{HM2}
\end{align*}
Here the dinatural transformation parts of the diagrams are the
evaluation and coevaluation of the duality on the category.
A bimonad equipped with a left antipode is called a (left) \emph{Hopf monad}.
\end{defn}
The question asked above is then fully answered by the following
theorem which tells us that, in a certain specific sense, Hopf monads
are analogous to Hopf algebras.
\begin{thm}[Brugui\`eres and Virelizier
\cite{BruguieresVirelizier:Hopf}]\label{thm:antipodeduality}
Suppose that \(T\) is a bimonad on a monoidal category \(\CC\) with
a left duality and that \(T\CC\) is the monoidal category of
\(T\)-modules. Then specifying a lift of the left duality on \(\CC\)
to a left duality on \(T\CC\) is the same as specifying a left
antipode for \(T\).
\end{thm}
The rest of this section will consist of a diagrammatic proof of the
above theorem. We essentially translate Brugui\`eres and Virelizier's
proof into our diagrammatic language, with some minor simplification
making the proof more transparent.
We begin with a lemma similar to results in Section~\ref{section:bimonads} on bimonads.
\begin{lemma}
If \(T\) is a monad on a category $\CC$ and ${}^\vee\colon\CC^\op\to
\CC$ is a functor then lifts of this to a functor ${}^\wedge\colon
T\CC^\op\to T\CC$ on the category of modules correspond to natural
transformations \( T\circ {}^\vee\circ T\nattrans {}^\vee\), drawn
as
\[\pstex{H4_1_2},\]
which satisfy
\[
\pstex{H4_1_6}=\pstex{H4_1_7};
\quad\text{and}\quad
\pstex{H4_1_8}=\pstex{H4_1_9}.
\mytag{HM0}\label{HM0}
\]
\end{lemma}
\begin{proof}
The functor ${}^\wedge$ being a lift of the functor ${}^\vee$ means
that the following diagram commutes.
\[
{\xymatrix
{
T\CC^\op
\ar[r]^{{}^\wedge}
\ar[d]_{U_T^\op}
&T\CC
\ar[d]^{U_T}\\
\CC^\op
\ar[r]^{{}^\vee}
& \CC
}}\]
By
Lemma~\ref{Lemma:LiftingToModules}, specifying a functor
${}^\wedge\colon
T\CC^\op\to T\CC$
is the same as specifying a functor $U_T\circ{}^\wedge\colon
T\CC^\op\to T\CC$ and a natural transformation \(T\circ U_T\circ
{}^\wedge\nattrans U_T\circ {}^\wedge\) satisfying the two module
conditions. But as the above diagram commutes, we know that \(U_T\circ
{}^\wedge= {}^\vee\circ U_T^\op\), so we just need to specify a
natural transformation
\(T\circ {}^\vee\circ U_T^\op\nattrans
{}^\vee\circ U_T^\op\), drawn as
\[\pstex{H4_1_1},\]
and which satisfies the two module conditions. Analogously to the bimonad
case we can use the identity \(T=U_T\circ F_T\) and the reversed
adjunction \(U_T^\op\dashv F_T^\op\) to obtain a bijection
\[
\Nat(G\circ U_T^\op, H\circ U_T^\op)\cong
\Nat(G\circ T^\op,H).
\]
So the above natural transformation corresponds to a natural
transformation \(\SSS\colon T\circ {}^\vee\circ T^\op\nattrans {}^\vee\):
\[
\SSS \equiv\pstex{H4_1_2}:=\pstex{H4_1_3}.
\]
The original natural transformation is recovered in the following way:
\[
\pstex{H4_1_1}=\pstex{H4_1_5}.
\]
Concretely, we can recover the lift \({}^\wedge\) from \(\SSS \) via
\[
\pre{\wedge}(m,r)=\left(\prevee m,\ \SSS _{m}\circ T(\prevee
r)\right).
\]
The two module conditions translate to \bref{HM0} as required.
\end{proof}
In order to show that such a lift of a left duality is itself a left duality we need to show that \(\ev\) and
\(\coev\) define \(T\)-module maps. This is precisely where
\handtag{HM1} and \handtag{HM2} come into play.
\begin{thm}\label{thm:HM1evalHM2coeval}
Suppose that \(T\) is a bimonad on a monoidal category \(\CC\) with
duals, and that \(\SSS \colon T\circ {}^\vee\circ T^\op\nattrans
{}^\vee\) is a natural transformation satisfying the conditions
\bref{HM0}, so it gives rise to a functor \({}^\wedge\colon
T\CC^\op\to T\CC\).
\begin{enumerate}
\item The evaluation dinatural transformation on \(\CC\) lifts to a
dinatural transformation on the module category \(T\CC\) if and only
if \bref{HM1} is satisfied.
\item The coevaluation dinatural transformation on \(\CC\)
lifts to a dinatural transformation on the module category \(T\CC\)
if and only if \bref{HM2} is satisfied.
\end{enumerate}
\end{thm}
\begin{proof}
Consider the evaluation case. For \(\ev\) to lift to an evaluation
on \(T\CC\) its components must be maps of \(T\)-modules, that is
they must commute with the \(T\)-action, so for each \(T\)-module
\((m,r)\in T\CC\) the following diagram must commute.
\[
\xymatrix{
T(\prevee m\otimes m) \ar[d]\ar[r]^{T\ev_m}
&T(\one)\ar[d]
\\
\prevee m \otimes m \ar[r]^{\ev_m}
&\one
}
\]
Here, of course, the action on \(\prevee m \otimes m\) is using
\(\SSS\) and the bimonad structure.
In terms of the diagrammatic calculus this means that the following
must hold:
\[\pstex{H4_1_18}=\pstex{H4_1_19}.\]
Now we can use the same machinery as before to remove \(U_T\) from
the statement: namely using we have a bijection
\[\text{Dinat}\left(G\circ(U_T^\op \times U_T), H\right)
\cong
\text{Dinat}\left(G\circ(T^\op \times \Id), H\right).
\]
Via this correspondence, the above equality becomes
\[\pstex{H4_1_20}=\pstex{H4_1_21},\]
Moving things `over the top' this condition is seen to be equivalent to
\[\pstex{H4_1_22}=\pstex{H4_1_23},\]
which is, by the properties of \(U_T\) and \(F_T\) from
Section~\ref{Section:MonadsAndBimonads}, is just
\[\pstex{H4_1_13}=\pstex{H4_1_12}\]
and this \bref{HM1} as required.
The coevaluation case is similar.
\end{proof}
We have now seen that if \(T\) is a bimonad on a monoidal category with
duals then specifying a lift of the duality on \(\CC\) to a duality
on \(T\CC\) the category of \(T\)-modules is equivalent to
specifying a natural transformation \( T\circ {}^\vee\circ
T^\op\nattrans {}^\vee\) such that conditions \bref{HM0}, \bref{HM1} and \bref{HM2}
are satisfied. To prove Theorem~\ref{thm:antipodeduality} we just need to see that \bref{HM0} is actually a redundant condition.
\begin{thm}
\label{Theorem:HM1HM2impliesHM0}
Suppose that \(T\) is a bimonad on a monoidal category with duals,
then any natural transformation \( T\circ {}^\vee\circ
T^\op\nattrans {}^\vee\) which satisfies \bref{HM1} and \bref{HM2} also
satisfies \bref{HM0}.
\end{thm}
\begin{proof}
We first prove that the following equation holds:
\[\pstex{H4_1_25}=\pstex{H4_1_26}.\eqno{(\dagger)}\]
This is true for the following reason:
\begin{align*}
\text{LHS}&:=\quad\pstex{H4_1_27}\quad
\stackrel{\handtag{BM1}}=\quad\pstex{H4_1_28}\\
&\stackrel{\bref{HM2}}=\quad\pstex{H4_1_29}\quad
\stackrel{\handtag{BM3}}=\quad\pstex{H4_1_30}\\
&\stackrel{\bref{HM2}}=\quad\pstex{H4_1_31}\quad
\stackrel{\bref{Monad2}}=\quad\pstex{H4_1_32}\\
&\stackrel{\bref{HM2}}=\quad\pstex{H4_1_33}\quad
\stackrel{\bref{Monad1}}=\quad\pstex{H4_1_34}
=\quad\text{RHS}.
\end{align*}
Also, by \handtag{HM1} we have
\[\pstex{H4_1_35}=\pstex{H4_1_36}.\eqno{(\dagger\dagger)}\]
Thus
\begin{align*}
\pstex{H4_1_37}
&\stackrel{\bref{Duality2}}=\pstex{H4_1_38}
\stackrel{\handtag{$\dagger\dagger$}}\quad=\quad\pstex{H4_1_39}\\
&\stackrel{\handtag{$\dagger$}}=\quad\pstex{H4_1_40}\quad
\stackrel{\handtag{$\dagger\dagger$}}=\quad\pstex{H4_1_41}\\
&\stackrel{\bref{Duality2}}=\quad\pstex{H4_1_42}\quad.
\end{align*}
\end{proof}
Now Theorem~\ref{thm:HM1evalHM2coeval} and Theorem~\ref{Theorem:HM1HM2impliesHM0} immediately imply Theorem~\ref{thm:antipodeduality} as required.
\subsection{Hopf monads from adjoint functors}
Suppose that there is an adjunction \(F\dashv U\) where
\(U\colon \DD\to \CC\) is a strong monoidal functor
between monoidal categories with (left) duals. We know from Section~\ref{Subsection:BimonadsFromAdjoint} that \(U\circ F\) is
a bimonad; we will see that it is also naturally a Hopf monad, i.e.,
that it naturally comes equipped with an antipode. This is due to
Brugi\`eres and Virelizier but we make the structure more explicit
than in their paper.
\subsubsection{Overview}\label{section:HMAFoverview}
The key point for the definition of the antipode is that if \(U\) is strong monoidal then it commutes
with taking duals. More precisely, we will see below that there is a natural isomorphism
\({}^\vee\circ U^\op \cong U\circ \prevee\) and we will draw the
mutually inverse
transformations as
\[\pstex{H4_2_1}
\qquad\text{and}\qquad
\pstex{H4_2_2}.\]
We will also see below how to define these from the monoidal structure
of \(U\), but first we can use these together with the unit and counit
of the adjunction to
define \(\SSS\colon U\circ F\circ{}^\vee\circ U^\op\circ
F^\op\nattrans{}^\vee\), the antipode for the bimonad \(U\circ F\), in the following way:
\[\SSS :=\pstex{H4_2_3}.\]
This will be shown to indeed be an antipode in Section~\ref{Subsubsection:SIsAntipode}, but the reader is invited to check diagrammatically that this satisfies \bref{HM0}.
\subsubsection{Strong monoidal functors commute with taking duals} We
will now show that \(U\) commutes with taking duals, i.e., that there exists a natural isomorphism \({}^\vee\circ U^\op \cong U\circ \prevee\). On the level of objects
the idea is that for \(d\in \DD\) the objects \(\prevee U^\op(d)\) and
\(U(\prevee d)\) are both left duals of \(U(d)\), and so they are
canonically isomorphic via the standard yoga: namely, being somewhat
fastidious, we have
\begin{align*}
\prevee U^\op(d)
&\to
\prevee U^\op(d)\otimes \one
\to
\prevee U^\op(d)\otimes U(\one)
\to
\prevee U^\op(d)\otimes U(d\otimes \prevee d)
\\
&\to
\prevee U^\op(d)\otimes \left(U(d)\otimes U(\prevee d)\right)
\to
\left( \prevee U^\op(d)\otimes U(d)\right)\otimes U(\prevee d)\\
&\to
\one \otimes U(\prevee d)
\to
U(\prevee d).
\end{align*}
This gives rise to a natural isomorphism \({}^\vee\circ U^\op \nattrans U\circ
\prevee\) constructed from the dinatural transformations \(\ev\) and
\(\coev\) as follows:
\[\pstex{H4_2_2}:=\pstex{H4_2_4}.\]
The inverse transformation is defined similarly.
Using these, we can now show that \(U\) also commutes with evaluation
and coevaluation.
\begin{thm}\label{Thm:UcommutesEv}
If \(U\) is a strong monoidal functor as above then \(U\) commutes
with evaluation and coevaluation in the following sense:
\[\pstex{H4_4_5}=\pstex{H4_4_6};\qquad\pstex{H4_4_7}=\pstex{H4_4_8}.\]
In more traditional notation this is expressing the commutativity of
the following diagrams:
{\footnotesize\[\raisebox{25pt}{\xymatrix@C10pt{
\prevee U^\op(d)\otimes U(d)
\ar[rr]
\ar[d]
&
&\one\\
U(\prevee d)\otimes U(d)\ar[r]
& U(\prevee d\otimes d)\ar[r]
&U(\one)\ar[u]
}};\quad\raisebox{25pt}{
\xymatrix@C10pt{
\one
\ar[rr]
\ar[d]
&
& U^\op(d)\otimes\prevee U(d)\\
U(\one)\ar[r]
& U( d\otimes \prevee d)\ar[r]
&U( d)\otimes U(\prevee d)\ar[u]
}}
\]}
\end{thm}
\begin{proof}
We will consider the evaluation case as the coevaluation case is similar. The left hand diagram is seen to commute as soon as its left hand arrow is unpacked as in the following diagram.\bigskip
{\footnotesize
\[\xymatrix@C10pt{
\prevee U^\op(d)\otimes U(d)
\ar@/^2em/[rr]
\ar[d]
\ar[rd]
\ar@/^4em/[rrdd]
&&\one
\\
\prevee U^\op(d)\otimes U(\one)\otimes U(d)
\ar[d]\ar[r]
&
\prevee U^\op(d)\otimes U(\one\otimes d)
\ar[d]\ar[rd]
\\
\prevee U^\op(d)\otimes U(d\otimes\prevee d)\otimes U(d)
\ar[d]\ar[r]
&\prevee U^\op(d)\otimes U(d \otimes\prevee d \otimes d)
\ar[d]\ar[r]
&\prevee U^\op(d)\otimes U(d)\ar[uu]
\\
\prevee U^\op(d)\otimes U(d)\otimes U(\prevee d)\otimes U(d)
\ar[d]
\ar[r]
& \prevee U^\op(d)\otimes U(d)\otimes U(\prevee d\otimes d)
\ar[d]
\\
\one\otimes U(\prevee d)\otimes U(d)
\ar[d]
&\prevee U^\op(d)\otimes U(d)\otimes U(\one)
\ar@/_2em/[uur]
\\
U(\prevee d)\otimes U(d)\ar[r]
&
U(\prevee d\otimes d)\ar[r]
&U(\one)\ar@/_5em/[uuuuu]
}
\]}
\end{proof}
\subsubsection{The antipode and Hopf monad}
\label{Subsubsection:SIsAntipode}
We can now prove that the natural transformation \(\SSS\) defined above
does give a left antipode for the bimonad \(U\circ F\). This is
essentially the proof of Theorem~3.14 in~\cite{BruguieresVirelizier:Hopf}.
\begin{thm}
If \(F\dashv U\) is an adjunction where \(U\) is a strong monoidal
functor between monoidal categories with left duals, then the
natural transformation \(\SSS \colon T\circ{}^\vee\circ
T^\op\nattrans {}^\vee\) (defined in
Section~\ref{section:HMAFoverview}) is a left antipode for the
bimonad \(U\circ F\).
\end{thm}
\begin{proof} By Theorem~\ref{Theorem:HM1HM2impliesHM0}, it
suffices to show that \bref{HM1} and \bref{HM2} are satisfied. I will just
give the proof of \bref{HM1}; the proof of \bref{HM2} is analogous.
\begin{align*}
\pstex{H4_4_9}&=\pstex{H4_4_10}
\stackrel{\text{Thm~\ref{Thm:UcommutesEv}}}=\pstex{H4_4_11}\\
&=\pstex{H4_4_12}=\pstex{H4_4_13}\\
&\stackrel{\text{Thm~\ref{Thm:UcommutesEv}}}=\pstex{H4_4_14}=\pstex{H4_4_15}.
\end{align*}
\end{proof}
\noindent Thus we get that in this case \(U\circ F\) is indeed a Hopf
monad.
| 134,155
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Jackets are $99 (originally $365), dresses are $99 (originally $350), and shirts are $65 (originally $225) at JNBY’s sample sale. 75 Greene St., nr Spring St.; T–Th (10–8).
Take 20 percent off No. 6 clogs at the store’s end of summer sale. New designer clothes are 50 to 70 percent off, vintage items are 20 percent off, and the No. 6 collection is 50 percent off. 6 Centre Market Place , nr Grand St. (212-226-5759); M–S (noon–7), Su (noon–6).
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Variables Influencing Medical Usage Rates Injury Patterns and Levels of Care for Mass Gatherings
Milsten, A.M., Seaman, K.g., Liu, P., et al. (2003). Variables Influencing Medical Usage Rates, Injury Patterns, and Levels of Care for Mass Gatherings. (Abstract only.) Prehospital and Disaster Medicine. 18(4):334-46.
The authors examined medical utilization rates during three types of mass gatherings over a three-year period. They found that event type and temperature best predicted specific injuries and medical utilization rates.
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\begin{document}
\begin{frontmatter}{}
\title{Approximation order of Kolmogorov diameters via $L^{q}$-spectra
and applications to polyharmonic operators}
\author{Marc~Kesseböhmer\corref{cor1}}
\ead{mhk@uni-bremen.de}
\author{Aljoscha~Niemann\corref{cor1}}
\ead{niemann1@uni-bremen.de}
\address{Fachbereich 3 -- Mathematik und Informatik, University of Bremen,
Bibliothekstr. 1, 28359 Bremen, Germany}
\cortext[cor1]{Corresponding authors}
\begin{abstract}
We establish a connection between the $L^{q}$-spectrum of a Borel measure
$\nu $ on the $m$-dimensional unit cube and the approximation order of
Kolmogorov diameters of the unit sphere with respect to Sobolev norms in
$L_{\nu }^{p}$. This leads to improvements of classical results of Borzov
and Birman/Solomjak for a broad class of singular measures. As an application,
we consider spectral asymptotics of polyharmonic operators and obtain improved
upper bounds of the decay rate of their eigenvalues. For measures with non-trivial absolutely continuous parts as well as for self-similar measures the exact approximation orders are stated.\end{abstract}
\begin{keyword}
Kolmogorov diameters/widths, polyharmonic operator; spectral asymptotics;
$L^{q}$-spectrum; piecewise polynomial approximations; Kre\u{\i}n--Feller
operator, Sobolev spaces, adaptive approximation algorithms. \MSC[2020]
46A32; 35P20; 42B35; 31B30; 28A80
\end{keyword}
\end{frontmatter}{}
\tableofcontents{}
\section{Introduction and statement of main results}
\label{sec1}
Let us start with some basic notations. Let $\mathbb{R}^{m}$ denote the
$m$-dimensional euclidean space, $m\in \mathbb N$. For a multi-index
$k := \left (k_{1},\dots ,k_{m}\right )\in \mathbb N^{m}$ we define
$|k| := \sum _{i=1}^{m}k_{i}$ and
$x^{k} := \prod _{i=1}^{m}x_{i}^{k_{i}}$ and for a bounded open subset
$\Omega \subset \mathbb{R}^{m}$ and $p\geq 1$, we let
$L^{p}(\Omega )$ denote the set of real-valued $p$-integrable functions
on $\Omega $ with respect to the Lebesgue measure $\Lambda $ restricted
to $\Omega $. The Sobolev space $W_{p}^{\ell}(\Omega )$ (see e.g.
\citep{MR1125990} and \citep{1961RuMa}) is defined to be the set of all
functions $f\in L^{p}(\Omega )$ for which the weak derivatives up to order
$\ell \in \mathbb N$ lie in $L^{p}(\Omega )$ and
\begin{equation*}
\left \Vert f\right \Vert _{W_{p}^{\ell}(\Omega )} := \left
\Vert f\right \Vert _{L^{p}(\Omega )}+\left \Vert f\right \Vert _{L^{
\ell ,p}(\Omega )}<\infty ,
\end{equation*}
where we set
$\left \Vert f\right \Vert _{L^{\ell ,p}(\Omega )} := \left (
\int _{\Omega}\left |\nabla _{\ell}f\right |^{p}\;\mathrm d\Lambda
\right )^{1/p}$ with
$\left |\nabla _{\ell}f\right | := \left (\sum _{|k|=\ell}
\left |D^{k}f\right |^{2}\right )^{1/2}$ and
$D^{k}f := \partial ^{|k|}/\left (\partial _{x_{1}}^{k_{1}}
\cdot \cdot \cdot \partial _{x_{\ell}}^{k_{\ell}}\right )f$. We let
$W_{0,p}^{\ell}(\text{$\Omega$)}$ denote the completion of
$\mathcal{C}_{c}^{\infty}(\Omega )$ with respect to
$\left \Vert \,\cdot \,\right \Vert _{W_{p}^{\ell}(\Omega )}$, where
$\mathcal{C}_{c}^{\infty}(\Omega )$ denotes the set of all infinitely differentiable
functions with compact support in $\Omega $. For any half-open cube
$Q\subset \mathbb{R}^{m}$, which---throughout the paper---are assumed to
have edges parallel to the coordinate axes, we have by definition of the
weak derivatives that $W_{p}^{\ell}(Q)=W_{p}^{\ell}(\mathring{Q})$, where
$\mathring{Q}$ denotes the interior of $Q$. Note that for the space
$W_{0,p}^{\ell}(Q)$ an equivalent norm is given by
$\left \Vert \,\cdot \,\right \Vert _{L^{\ell ,p}(Q)}$. If
$\ell p/m>1$, then $W_{p}^{\ell}\left (Q\right )$ is compactly embedded
into
$\left (\mathcal{C}(\overline{Q}),\left \Vert \,\cdot \,\right \Vert _{
\mathcal{C}(\overline{Q})}\right )$, with
$\left \Vert \,\cdot \,\right \Vert _{\mathcal{C}(\overline{Q})}$ denoting
the uniform norm, and therefore we will always pick a continuous representative
of $W_{p}^{\ell}\left (Q\right )$. For the set of continuous function from
$A\subset \mathbb{R}^{m}$ to $\mathbb{R}$ we write
$\mathcal{C}\left (A\right )$.
For a normed vector space
$\left (V,\left \Vert \,\cdot \,\right \Vert _{V}\right )$ and a subset
$K\subset V$, the \textit{Kolmogorov $n$-diameter} (or $n$\emph{-widths})\emph{
of $K$ in $V$}, $n\in \mathbb N$, is given by
\begin{align*}
d_{n}\left (K,V\right ) & := \inf \left \{ \sup _{x\in K}\inf _{y
\in V_{n}}\left \Vert x-y\right \Vert _{V}:V_{n}
\text{ is $n$-dimensional subspace of }V\right \} .
\end{align*}
If $K$ is pre-compact, then the $n$-diameters converge to zero and one
could say that the $n$-diameter $d_{n}\left (K,V\right )$ measures the
extend to which $K$ can be approximated by $n$-dimensional subspaces of
$V$. We call the value
\begin{equation*}
\overline{\mathbf{ord}}\left (K,V\right ) := \limsup _{n\to
\infty}\frac{\log \left (d_{n}\left (K,V\right )\right )}{\log n}
\end{equation*}
the \emph{upper} \emph{approximation order of $K$ in $V$}. If the upper approximation
order coincides with the \emph{lower approximation order}
$\mathscr{\underline{\mathbf{ord}}}\left (K,V\right )$ defined by replacing
the limit superior with the limit inferior in the above definition, we
call the common value $\mathscr{\mathbf{ord}}\left (K,V\right )$ the
\emph{approximation order}. See \citep{MR774404} for further details on
this topic. For the \emph{unit sphere} in $V$ we write
$\mathscr{S}V := \left \{ f\in V:\left \Vert f\right \Vert _{V}=1
\right \} $. In the following we will concentrate on the particular choice
$V=L_{\nu}^{q}\left (\textbf{Q}\right )$ for a Borel measure $\nu $ on the
half-open unit cube $\textbf{Q} := \left (0,1\right ]^{m}$ and
$K\in \left \{ \mathscr{S}W_{p}^{\ell}\left (\textbf{Q}\right ),
\mathscr{S}W_{0,p}^{\ell}\left (\textbf{Q}\right )\right \} $,
$q\geq p>1$. Throughout, we will assume that
\begin{equation}
\varrho := q\left (\ell -m/p\right )>0\;\:\text{and }\:q\geq p>1.
\label{eq:StandingAssumption}
\end{equation}
Under this condition, using the Landau symbols, it has been shown in
\citep{MR0217487} that
\begin{equation}
d_{n}\left (\mathscr{S}W_{p}^{\ell},L_{\nu}^{q}\right )=O\left (n^{-
\left (\ell /m-1/p+1/q\right )}\right )
\label{eq:BirmanSolomanjakOld}
\end{equation}
and in the case that $\nu $ is a singular measure with respect to the Lebesgue
measure, we know from \citep{Borzov1971} that even
\begin{equation*}
d_{n}\left (\mathscr{S}W_{p}^{\ell},L_{\nu}^{q}\right )=o\left (n^{-
\left (\ell /m-1/p+1/q\right )}\right ).
\end{equation*}
In this paper we want to address the question to what extent these estimates
can be effectively improved for arbitrary Borel measures on
$\textbf{Q}$. We will see how our main result can be obtained from auxiliary
measure-geometric quantities involving the $L^{q}$-spectrum of
$\nu $ combined with some ideas from \citep{MR0217487} dealing with piecewise
polynomial approximation in $L_{\nu}^{q}$ of elements in
$W_{p}^{\ell}(\textbf{Q})$ (see Section~\ref{subsec:Theorems-on-approximation}).
For $n\in \mathbb N$, we set
\begin{equation*}
\mathcal{D}_{n} := \left \{ Q=\prod _{k=1}^{m}\left (l_{k}2^{-n},(l_{k}+1)2^{-n}
\right ]:(l_{k})_{k=1,\dots ,m}\in \mathbb{Z}^{m},\nu \left (Q\right )>0
\right \} ,\mathcal{D} := \bigcup _{n\in \mathbb N}\mathcal{D}_{n}
\end{equation*}
and the \emph{$L^{q}$-spectrum} of $\nu $ is given, for
$s\in \mathbb{R}$, by
\begin{equation*}
\beta _{\nu}(s) := \limsup _{n\rightarrow \infty}\beta _{n}^{
\nu}(s)\,\text{\,with\,\, }\beta _{n}^{\nu}\left (s\right ) :=
\log \left (\sum _{C\in \mathcal{D}_{n}}\nu (C)^{s}\right )/\log
\left (2^{n}\right ).
\end{equation*}
The $L^{q}$-spectrum has gained some high attention from various authors
in recent years, e.g. \citep{MR3897401,MR3919361,KN2022}. Note that
$\beta _{\nu}$ is---as a limit superior of convex functions---itself a
convex function and that $\beta _{\nu}(0)$ is equal to the
\emph{upper Minkowski dimension} of $\supp \nu $ denoted by
$\overline{\dim}_{M}\left (\nu \right )$. Before stating our main result,
we introduce the following key quantity
\begin{equation*}
s_{b} := \inf \left \{ s>0:\beta _{\nu}(s)-bs\leq 0\right \} \;
\text{ for }b>0.
\end{equation*}
\begin{thm}
\label{thm:Estimation-n-Diameter}
Assuming \textup{\reftext{(\ref{eq:StandingAssumption})}}, we have
\begin{align*}
\overline{\mathbf{ord}}\left (\mathscr{S}W_{p}^{\ell},L_{\nu}^{q}
\right ) & \leq -\frac{1}{q\cdot s_{\varrho }}.
\end{align*}
\end{thm}
\begin{rem}
\label{rem1.2}
Note that
\begin{equation*}
-\frac{1}{q\cdot s_{\varrho }}\leq -
\frac{\varrho }{q\overline{\dim}_{M}\left (\nu \right )}-\frac{1}{q}
\leq \frac{1}{p}-\frac{1}{q}-\frac{\ell}{m}<0,
\end{equation*}
and that $-1/\left (q\cdot s_{\varrho }\right )=-\ell /m+1/p-1/q$ if and
only if $\beta _{\nu}(\text{s})=m\left (1-s\right )$ for some and hence
for all $s\in \left (0,1\right )$. These claims follow readily from the
convexity of $\beta _{\nu}$ by observing that for all
$s\in \left [0,1\right ]$ we have
$\beta _{\nu}(s)\leq \beta _{\nu}(0)\left (1-s\right )\leq m(1-s)$.
If $\nu $ has a non-trivial absolutely continuous part with respect to
Lebesgue, then an application of Jensen inequality guarantees
$\beta _{\nu}(s)=m\left (1-s\right )$, for all
$s\in \left (0,1\right )$. Hence, we gain a new perspective on the estimate
in \textup{\reftext{(\ref{eq:BirmanSolomanjakOld})}} in terms of the $L^{q}$-spectrum.
Namely, the intersection with the line through the origin with slope
$\varrho $ and $s\mapsto m(1-s)$ is given by $m/(m+\varrho )$, which leads
to the general upper bound $-\ell /m+1/p-1/q$ as obtained in
\citep{MR0217487} (for an illustration of this observation see \reftext{Fig.~\ref{fig:Moment-generating-function}} on page
\pageref{fig:Moment-generating-function}). Consequently, whenever
$\beta _{\nu}(s)<m\left (1-s\right )$, for some
$s\in \left (0,1\right )$, \reftext{Theorem~\ref{thm:Estimation-n-Diameter}} improves
the classical result of \citep{MR0217487} and
\citep[Theorem 5.1]{Borzov1971}. Indeed, a strict inequality occurs for
many singular measures, for example if
$\overline{\dim}_{M}\left (\nu \right )<m$. Roughly speaking, the more
nonuniform the mass of $\nu $ is distributed compared to the Lebesgue measure,
the faster
$\left (d_{n}\left (\mathscr{S}W_{p}^{\ell},L_{\nu}^{q}\right )
\right )_{n}$ decreases.
\begin{figure}
\center{\begin{tikzpicture}[scale=1, every node/.style={transform shape},line cap=round,line join=round,>=triangle 45,x=1cm,y=1cm] \begin{axis}[ x=3.7cm,y=2.3cm, axis lines=middle, axis line style={very thick},ymajorgrids=false, xmajorgrids=false, grid style={thick,densely dotted,black!20}, xlabel= {$s$}, ylabel= {\;$\beta_\nu (s)$}, xmin=-0.4 , xmax=1.5 , ymin=-0.3, ymax=3.2,x tick style={color=black}, xtick={0, .425,0.6,1},xticklabels = {0,$s_\varrho$,$\frac{m}{m+\varrho}$,1}, ytick={0,1, 2,3},yticklabels = {0,1, $\overline{\dim}_M(\nu)$ ,3}] \clip(-0.5,-0.3) rectangle (4,4);
\draw[line width=1pt,smooth,samples=180,domain=-0.3:3.4] plot(\x,{log10(0.001^((\x))+0.28^((\x))+0.06^((\x))+0.659^((\x)))/log10(2)});
\draw [line width=01pt,dotted, domain=-0.05 :1.3] plot(\x,{3*(1-\x)});
\draw [line width=01pt,dashed, domain=-0.15 :1.3] plot(\x,{2*\x});
\node[circle,draw] (c) at (2.48 ,0 ){\,};
\draw [line width=.7pt,dotted, gray] (0.425 ,0.)--(0.425,1);
\draw [line width=.7pt,dotted, gray] (0.6 ,0 )-- (0.6,1.5);
\draw (1 ,1.91) node[anchor=north west] {{$s\mapsto\rho \cdot s$}};
\end{axis}
\end{tikzpicture}}
\caption{For $m=3$ the solid line illustrates the $L^{q}$-spectrum $\beta
_{\nu}$ for the self-similar measure $\nu $ supported on the\emph{
Sierpi\'{n}ski tetraeder} with all four contraction ratios equal $1/2$ and with
probability vector $\left (0.659,0.28,0.001,0.06\right )$; $\beta _{\nu}\left
(0\right )=\overline{\dim}_{M}\left (\nu \right )=2$. For $\varrho =2$ (slope of
the dashed line) the intersection of the spectrum and the dashed line determines
$s_{\varrho }$. The dotted line $s\protect\mapsto 3\left (1-s\right )$, which
coincides with the graph of $\beta _{\Lambda |_{\textbf{Q}}}$, intersects the
dashed line in $m/\left (m+\varrho \right )$ giving the upper
bound for $s_{\varrho }$.}
\label{fig:Moment-generating-function}
\end{figure}
\end{rem}
As a direct application of our result we consider polyharmonic operators,
i.e. we restrict to the Hilbert spaces setting $p=q=2$,
$H^{\ell} := W_{2}^{\ell}(\textbf{Q})$ and
$H_{0}^{\ell} := W_{0,2}^{\ell}(\textbf{Q})$: By \reftext{Theorem~\ref{thm:MainPolyharm_n-diameter}},
$d_{n-1}\left (\mathscr{S}H_{0}^{\ell},L_{\nu}^{2}\right )$ can be identified
with the square root of the $n$-th eigenvalue of the associated polyharmonic
operator with respect to $\nu $. This gives rise to improved upper bounds
of the decay rate of their eigenvalues (\reftext{Theorem~\ref{thm:Main_Polyharmonic}} in Section~\ref{subsec:General-setup-and_EVasymp}).
Further, using this connection for $\nu $ with non-trivial absolutely continuous
part with respect to Lebesgue, we deduce from
\citep[Theorem 5.1]{MR0278126} that $2s_{\varrho }=m/\ell $ and moreover,
for some explicit constant $c>0$,
\begin{equation*}
d_{n}\left (\mathscr{S}H_{0}^{\ell},L_{\nu}^{2}\right )\sim cn^{-
\ell /m}.
\end{equation*}
In Section~\ref{subsec:Application-to-self-similar} we consider self-similar
measures $\nu $ under the open set condition (cf. \reftext{Example~\ref{exa:IFS}}
for definitions); it follows from \citep{Nazarov} for $m=1$ and from \reftext{Theorem~\ref{thm:Nazarov_m>1}} for $m>1$ that
\begin{equation*}
\mathbf{ord}\left (\mathscr{S}H_{0}^{\ell},L_{\nu}^{2}\right )=-
\frac{1}{2s_{\varrho }}.
\end{equation*}
In Section~\ref{sec:Krein-Feller-operators-in} we finally consider polyharmonic
operators for the special case $\ell =m=1$. We will show that the associated
spectral problem is equivalent to the spectral problem of the classical
Kre\u{\i}n--Feller operator (see e.g.
\citep{KN21,MR2563669,MR2828537,MR3809018}). Using the superadditivity
established in \reftext{Theorem~\ref{thm:MainPolyharm_n-diameter}} (see also
\citep[Theorem 1.1]{KN2022}), we have equality in \reftext{Theorem~\ref{thm:Estimation-n-Diameter}} for \emph{all} finite Borel measure on
$\left (0,1\right )$, i.e.
\begin{equation*}
\overline{\mathbf{ord}}\left (\mathscr{S}H^{1},L_{\nu}^{2}\right )=
\overline{\mathbf{ord}}\left (\mathscr{S}H_{0}^{1},L_{\nu}^{2}\right )=-
\frac{1}{2s_{\varrho }}.
\end{equation*}
\section{Optimal partitions}
\label{sec:OptimalPartitions}
Fix $a>0$ and a Borel probability measure $\nu $ on $\textbf{Q}$. Let
$\Upsilon _{n}$ denote the set of all finite partitions consisting of at
most $n\in \mathbb N$ half-open $m$-dimensional subcubes of
$\textbf{Q}$. As in \citep{MR0217487,MR2864649}, we introduce an auxiliary
target quantity for the underlying optimisation problem given by the Kolmogorov
$n$-diameter: For $n\in \mathbb N,a>0$, and with
$\mathfrak{J}_{a}(Q) := \Lambda (Q)^{a}\nu (Q)$, $Q$ half-open subcube,
we let
\begin{align}
\gamma _{a,n} & := \inf _{\Xi \in \Upsilon _{n}}\max _{Q\in
\Xi}\mathfrak{J}_{a}(Q),
\label{eq:optimierung}
\end{align}
and define the exponential growth rate of its reciprocal
\begin{equation*}
\alpha _{a} := \liminf _{n\rightarrow \infty}
\frac{\log \left (1/\gamma _{a,n}\right )}{\log (n)}.
\end{equation*}
\begin{rem}
\label{rem2.1}
The quantity $\gamma _{a,n}$ naturally arises in the study of approximation
order in $L_{\nu}^{2}$ of functions in $W_{p}^{\ell}(\textbf{Q})$ by piecewise
polynomial approximations (see for instance \citep{MR0278126} and \reftext{Proposition~\ref{prop:_PiecewiseApprox}})
as well as in the study of the spectral behaviour of polyharmonic operators
as defined in Section~\ref{subsec:Application-to-polyharmonic}, see also
\citep{MR0278126,MR0482138}. This common ground reveals a deep connection
between these two aspects. It is also worth pointing out that the so-called
quantization problem, that is the speed of approximation of a compactly
supported Borel probability measure by finitely supported measures (see
\citep{MR1764176} for an introduction), has also close links to the growth
rate of $n\gamma _{a,n}$. This will be subject of the forthcoming paper
\citep{KN22b}.
We make use of the fact that the asymptotic optimisation problem in
\textup{\reftext{(\ref{eq:optimierung})}} can, as a result of \reftext{Proposition~\ref{prop:_Elementary_Lem}},
be transformed into the following counting problem. Motivated by
\citep{KN2022,MR0217487,MR2083820}, we introduce follow quantities. Let
$\Pi $ denote the sets of all partitions of $\textbf{Q}$ by half-open
$m$-dimensional cubes. Then the exponential growth rate of
\begin{equation*}
\mathcal{N}_{a}\left (t\right ) := \inf \left \{ \card \left (P
\right ):P\in \Pi :\max _{Q\in P}\mathfrak{J}_{a}(Q)<1/t\right \} ,t>0,
\end{equation*}
given by
\begin{equation*}
h_{a} := \limsup _{t\to \infty}
\frac{\log \mathcal{N}_{a}\left (t\right )}{\log t},
\end{equation*}
will be called the\emph{ (upper) $\nu $-partition entropy with parameter
$a$. }Let us begin with the preparatory observation that for $a>0$, the
sequence $\left (\gamma _{a,2^{mn}}\right )$ is either strictly decreasing
or eventually constant zero.
\end{rem}
\begin{lem}
\label{lem:induction_betaN}
For all $n\in \mathbb N$ and $a>0$, we have
$\gamma _{a,2^{m(n+1)}}\leq \frac{1}{2^{ma}}\gamma _{a,2^{mn}}$.
\end{lem}
\begin{proof}
For $\Xi \in \Upsilon _{2^{mn}}$ we can divide each $Q\in \Xi $ into
$2^{m}$ disjoint, equally sized, half-open cubes. The new resulting partition
denoted by $\Xi '$ satisfies
$\card (\Xi ')\leq 2^{m\left (n+1\right )}$ and
$\Lambda \left (Q\right )^{a}/2^{ma}=\Lambda \left (Q_{i}\right )^{a}$,
for all $Q'\subset Q\in \Xi $ with $Q'\in \Xi '$. This implies
\begin{equation*}
\max _{Q\in \Xi '}\mathfrak{J}_{a}(Q)\leq \frac{1}{2^{ma}}\max _{Q
\in \Xi}\mathfrak{J}_{a}(Q).\qedhere
\end{equation*}
\end{proof}
\begin{prop}
\label{prop:_Elementary_Lem}
For $a>0$ we have $h_{a}=1/\alpha _{a}$.
\end{prop}
\begin{proof}
The proof follows along the same lines as the proof of the elementary Lemma
\citep[Lemma 2.2]{MR2083820}. First note that for $0<\varepsilon <1$ we
have
$\mathcal{N}_{a}\left (1/\varepsilon \right )=\inf \left \{ n\in
\mathbb N\mid \gamma _{a,n}<\varepsilon \right \} $. By \reftext{Lemma~\ref{lem:induction_betaN}} we have that
$\left (\gamma _{a,2^{mn}}\right )_{n}$ is a strictly decreasing null sequence
or eventually constant zero. The latter case is immediate. For the first
case the strict monotonicity gives as in \citep[Lemma 2.2]{MR2083820} for
$B\left (\varepsilon \right ) := \inf \left \{ n\in \mathbb N
\mid \gamma _{a,2^{mn}}<\varepsilon \right \} $
\begin{equation*}
\frac{1}{\alpha _{a}}=\limsup _{k\to \infty}
\frac{-\log k}{\log \gamma _{a,k}}=\limsup _{n\to \infty}
\frac{mn\log 2}{-\log \gamma _{a,2^{mn}}}=\limsup _{\varepsilon
\searrow 0}
\frac{B\left (\varepsilon \right )m\log 2}{-\log \varepsilon }=h_{a},
\end{equation*}
where the second equality follows by squeezing
$2^{m(n-1)}<k\leq 2^{mn}$, and the last equality by noting that
$2^{m\left (B\left (\varepsilon \right )-1\right )}\leq \mathcal{N}_{a}
\left (1/\varepsilon \right )\leq 2^{mB\left (\varepsilon \right )}$.
\end{proof}
\subsection{The $L^{q}$-spectrum and optimal partitions}
\label{sec2.1}
For $b>0$ and $n\in \mathbb N$, by the monotonicity and continuity of
$\beta _{n}^{\nu}$ there exists a unique number
$s_{n,b}\in \left [0,1\right ]$ such that
\begin{equation*}
\beta _{n}^{\nu}\left (s_{n,b}\right )=b\cdot s_{n,b}.
\end{equation*}
\begin{lem}
\label{lem2.4}
For all $b>0$,
\begin{equation*}
s_{b}=\limsup _{n\rightarrow \infty}s_{n,b}
\end{equation*}
and if $s_{b}>0$, then
$\beta _{\nu}\left (s_{b}\right )=b\cdot s_{b}$.
\end{lem}
\begin{rem}
\label{rem2.5}
The assumption $s_{b}>0$ cannot be dropped to guarantee the equality
$\beta _{\nu}\left (s_{b}\right )=b\cdot s_{b}$. In fact, for the finite
measure $\eta := \sum p_{k}\delta _{x_{k}}$ with
$p_{k} := \mathrm{e}^{-k}$ and $x_{k} := 1/k$ where
$\delta _{x}$ denotes the Dirac measure on $x$, we have
$\overline{\dim}_{M}(\eta )=1/2$,
\begin{equation*}
\beta _{\nu}(s)=
\begin{cases}
1/2 & ,s=0,
\\
0 & ,s>0
\end{cases}
\end{equation*}
and therefore $\beta _{\nu}\left (0\right )=1/2\neq b\cdot s_{b}=0$.
\end{rem}
\begin{proof}
Recall
$s_{b}=\inf \left \{ s>0:\beta _{\nu}\left (s\right )-b\cdot s\leq 0
\right \} $ and define
$s_{*} := \limsup _{n\rightarrow \infty}s_{n,b}$. Then for every
$\varepsilon >0$ and $n$ large enough we have
$s_{n,b}\leq s_{*}+\varepsilon $ and consequently
$\beta _{n}^{\nu}\left (s_{*}+\varepsilon \right )\leq \beta _{n}^{
\nu}\left (s_{n,b}\right )$. This implies
\begin{equation*}
\beta _{\nu}\left (s_{*}+\varepsilon \right )\leq b\cdot s_{*}.
\end{equation*}
If $s_{*}=0$, then $\beta _{\nu}\left (s\right )=0$ for all $s>0$, which
shows $s_{b}=0$. Assuming $s_{*}>0$, the continuity of
$\beta _{\nu}$ in $(0,1)$ gives
$\beta _{\nu}\left (s_{*}\right )\leq b\cdot s_{*}$. Let
$\left (n_{k}\right )_{k\in \mathbb N}$ be such that
$\lim _{k}s_{n_{k},b}=s_{*}$. Then for all
$\eta \in \left (0,s_{*}/2\right )$ and for $k$ large we have
$s_{n_{k},b}\geq s_{*}-\eta $, which gives
$\beta _{n_{k}}^{\nu}\left (s_{n_{k},b}\right )\leq \beta _{n_{k}}^{
\nu}\left (s_{*}-\eta \right )$. This implies
\begin{equation*}
bs_{*}\leq \limsup _{k}\beta _{n_{k}}^{\nu}\left (s_{*}-\eta \right )
\leq \beta _{\nu}\left (s_{*}-\eta \right ).
\end{equation*}
The continuity of $\beta _{\nu}$ in (0,1) gives
$bs_{*}=\beta _{\nu}\left (s_{*}\right )$ and therefore
$s_{b}=s_{*}$.
\end{proof}
The following lemma is the key to estimate $h_{a}$ in terms of the
$L^{q}$-spectrum $\beta _{\nu}$. To this end, for a given threshold
$t\in \left (0,\mathfrak{J}_{a}(\textbf{Q})\right )$, we will construct
partitions by dyadic cubes as a function of $t$ via an
\emph{adaptive approximation algorithm} in the sense of \citep{MR939183} (see
also \citep{MR1781213}) as follows. We say $Q\in \mathcal{D}$ is
\emph{bad, }if $\mathfrak{J}_{a}\left (Q\right )\geq t$, otherwise\emph{
}we call $Q$ \emph{good. }The goal is to construct a partition of
$\textbf{Q}$ with minimal cardinality, denoted by $P_{a,t}$, consisting
of elements of half-open dyadic cubes that are good. In the first step,
we divide $\textbf{Q}$ into $2^{d}$ half open cubes of equal size and move
good cubes among them to $P_{a,t}$. Now, repeat this procedure with respect
to each of the remaining bad cubes until no bad cubes are left. Since for
each $Q\in P_{a,t}$, we have
$\mathfrak{J}_{a}(Q)<t\leq \mathfrak{J}_{a}(Q')$, where $Q'$ denotes the
unique dyadic cube such that $Q'$ is the predecessor of $Q$. This ensures
that the procedure terminates after finitely many steps. The resulting
finite partition $P_{a,t}$ is optimal (in the sense of minimizing the cardinality)
among all partitions $P$ by half-open dyadic cubes fulfilling
$\max _{Q\in P}\mathfrak{J}_{a}(Q)<t$. This indicates that
$\card P_{a,t}$ provides a good approximation of
$\mathcal{N}_{a}\left (1/t\right )$. Now, the remaining task is to connect
the asymptotic behaviour of $\card (P_{a,t})$ with the $L^{q}$-spectrum
$\beta _{\nu}$. Motivated by ideas from large derivation theory and the
thermodynamic formalism \citep{MR2129258} we are able to bound
$h_{a}$ from above by $s_{am}$, namely, by comparing the cardinality of
$P_{a,t}$ and
$Q_{a,t} := \left \{ Q\in \mathcal{D}:\mathfrak{J}_{a}(Q)\geq t
\right \} $. This will be the key idea in the proof of \reftext{Lemma~\ref{lem:_EstimateGamma_n}}.
\begin{lem}
\label{lem:_EstimateGamma_n}
For all $a>0$, we have
\begin{equation*}
h_{a}\leq s_{am}.
\end{equation*}
\end{lem}
\begin{proof}
Without loss of generality, we assume that $\nu $ is a probability measure.
For $t\in \left (0,1\right )$,
\begin{equation*}
P_{a,t}=\left \{ C\in \mathcal{D}:\mathfrak{J}_{a}\left (C\right )<t
\,\&\,\exists C'\in \mathcal{D}_{\left |\log _{2}\left (\Lambda (C)
\right )\right |/m-1}:C'\supset C\,\&\,\mathfrak{J}_{a}\left (C'
\right )\geq t\right \}
\end{equation*}
is a partition of $\textbf{Q}$ by dyadic cubes. With $Q_{a,t}$ as defined
above, we note that for $C\in P_{a,t}$ there is exactly one
$C'\in Q_{a,t}\cap \mathcal{D}_{\left |\log _{2}\left (\Lambda (C)
\right )\right |/m-1}$ with $C\subset C'$ and for each
$C'\in Q_{a,t}\cap \mathcal{D}_{\left |\log _{2}\left (\Lambda \left (C
\right )\right )\right |/m-1}$ there are at most $2^{m}$ elements of
$P_{a,t}\cap \mathcal{D}_{\left |\log _{2}\left (\Lambda (C)\right )
\right |/m}$ which are subsets of $C'$. Hence,
\begin{equation*}
\card P_{a,t}\leq 2^{m}\card Q_{a,t}.
\end{equation*}
By the definition of $s_{n,am}$, we have
\begin{equation*}
\sum _{C\in \mathcal{D}_{n}}\nu (C)^{s_{n,am}}=2^{am\cdot s_{n,am}}.
\end{equation*}
For $s>s_{am}$ and $\varepsilon := \left (s-s_{am}\right )/2$, there
exists $K\in \mathbb N$ such that for all $k\geq K$ we have
$s_{am}+\varepsilon >s_{k,am}$. This gives
$s-\varepsilon =s_{am}+(s-s_{am})/2>s_{k,am}$ and we obtain for all
$0<t<1$,
\begin{align*}
t^{s}\card P_{a,t} & =\sum _{k=1}^{\infty}\sum _{C\in P_{a,t}\cap
\mathcal{D}_{k}}t^{s}\leq 2^{m}\sum _{k=1}^{K}\sum _{C\in Q_{a,t}
\cap \mathcal{D}_{k-1}}1
\\
&\quad {}+\sum _{k=K+1}^{\infty}2^{m}t^{s}\sum _{C
\in Q_{a,t}\cap \mathcal{D}_{k-1}}
\frac{\left (\mathfrak{J}_{a}\left (C\right )\right )^{s_{k-1,am}+\varepsilon}}{t^{s_{k-1,am}+\varepsilon}}
\\
& \leq 2^{m}\sum _{k=1}^{K}\sum _{C\in Q_{a,t}\cap \mathcal{D}_{k-1}}1
\\
&\quad {}+
\sum _{k=K+1}^{\infty}2^{m}t^{s}2^{-a(k-1)m\varepsilon}
\frac{2^{-am(k-1)s_{k-1,am}}2^{(k-1)am\cdot s_{k-1,am}}}{t^{s_{k-1,am}+\varepsilon}}
\\
& =2^{m}\sum _{k=1}^{K}\sum _{C\in Q_{a,t}\cap \mathcal{D}_{k-1}}1+
\sum _{k=K+1}^{\infty}2^{m(1-a(k-1)\varepsilon )}t^{s-\varepsilon -s_{k-1,am}}
\\
& \leq 2^{m}\sum _{k=0}^{K-1}\sum _{C\in \mathcal{D}_{k}}1+2^{m(1+a
\varepsilon )}\sum _{k=K+1}^{\infty}2^{-akm\varepsilon}<\infty .
\end{align*}
This implies
\begin{equation*}
\limsup _{t\searrow 0}
\frac{\log \left (\card P_{a,t}\right )}{-\log (t)}\leq s
\end{equation*}
and since $s>s_{am}$ was arbitrary,
\begin{equation*}
h_{a}=\limsup _{t\searrow 0}
\frac{\log \left (\mathcal{N}_{a}\left (1/t\right )\right )}{-\log (t)}
\leq \limsup _{t\searrow 0}
\frac{\log \left (\card P_{a,t}\right )}{-\log (t)}\leq s_{am}.\qedhere
\end{equation*}
\end{proof}
Now, we are in the position state one of our core results needed in the
proof of \reftext{Theorem~\ref{thm:Estimation-n-Diameter}}.
\begin{prop}
\label{prop:_CoreResult}
For all $a>0$,
\begin{equation*}
\limsup _{n\rightarrow \infty}
\frac{\log \left (\gamma _{a,n}\right )}{\log (n)}=-\alpha _{a}=-
\frac{1}{h_{a}}\leq -\frac{1}{s_{am}}.
\end{equation*}
\end{prop}
\begin{proof}
This follows immediately from \reftext{Proposition~\ref{prop:_Elementary_Lem}} and
\reftext{Lemma~\ref{lem:_EstimateGamma_n}}.
\end{proof}
\begin{rem}
\label{rem2.8}
In the case $m=a=1$ we have shown in \citep{KN2022} that even equality
holds, i.e.
\begin{equation*}
\limsup _{n\rightarrow \infty}
\frac{\log \left (\gamma _{1,n}\right )}{\log (n)}=-\frac{1}{s_{1}}.
\end{equation*}
\end{rem}
\section{Approximation order}
\label{sec3}
\subsection{Piecewise polynomial approximations of functions of the Sobolev space
in the metric of $L_{\nu}^{q}$}
\label{subsec:Theorems-on-approximation}
In this section we recall some results of
\citep[\S 3]{MR0217487}
and \citep{MR0482138} which will be important for our applications to
$n$-diameters and polyharmonic operators. Let $Q\subset \textbf{Q}$ denote
a cube. As pointed out in the introduction our standing assumption
$\ell p/m>1$ ensures that $W_{p}^{\ell}\left (Q\right )$ is compactly embedded
in
$\left (\mathcal{C}(\overline{Q}),\left \Vert \,\cdot \,\right \Vert _{
\mathcal{C}(\overline{Q})}\right )$. In the case $\ell p/m\leq 1$ the situation
becomes more involved; in general, we have no compact embedding from
$W_{p}^{\ell}\left (Q\right )$ into $L_{\nu}^{2}(Q)$ (e.g.
\citep{MR2261337,MR1338787,MR817985}). Further, without loss of generality
we assume that $\nu $ is Borel probability measure on $\textbf{Q}$. For
every $u\in W_{p}^{\ell}\left (Q\right )$, we associate a polynomial
$r\in \mathbb{R}\left [x_{1},\ldots ,x_{m}\right ]$ of degree at most
$\ell -1$ satisfying the conditions
\begin{equation}
\int _{Q}x^{k}r(x)\;\mathrm d\Lambda (x)=\int _{Q}x^{k}u(x)\;
\mathrm d\Lambda (x)\,\,\text{for all}\,\,|k|\leq \ell -1.
\label{eq:Polynomial}
\end{equation}
By an application of\emph{ Hilbert's Projection Theorem} with respect to
$L^{2}(Q)$, we have that $r$ is uniquely determined by
\textup{\reftext{(\ref{eq:Polynomial})}} and set $P_{Q}u := r$. Note that
$P_{Q}$ defines a linear projection operator which maps from
$W_{p}^{\ell}\left (Q\right )$ to the finite-dimensional space of polynomials
in $m$ variables of degree not exceeding $\ell -1$ and we denote the dimension
of this finite dimensional space of polynomials by $\kappa $.
We finish this section with two crucial observations which follow from
\citep{MR0217487}.
\begin{lem}[{\citep[Lemma 3.1]{MR0217487}}]
\label{lem:_approxSupNorm}
Let $Q\subset \textbf{Q}$ be a cube. Then there exists $C_{1}>0$ independent
of $Q$ such that for all $u\in W_{p}^{\ell}\left (Q\right )$
\begin{equation*}
\left \Vert u-P_{Q}u\right \Vert _{\mathcal{C}(\overline{Q})}\leq C_{1}
\Lambda (Q)^{\ell /m-1/p}\left \Vert u\right \Vert _{L^{\ell ,p}(Q)}.
\end{equation*}
\end{lem}
\begin{defn}
\label{defn3.2}
Let $\Xi $ be a partition of $\textbf{Q}$ into half open cubes and we define
$\mathcal{P}\left (\Xi ,\ell -1\right )$ to be the space of piecewise-polynomial
functions which restrict on each cube $Q\in \Xi $ to a polynomial of degree
$\ell -1$. We define
\begin{align*}
P_{\Xi}:W_{p}^{\ell}\left (Q\right ) & \rightarrow \mathcal{P}\left (
\Xi ,\ell -1\right )
\\
u & \mapsto \sum _{Q\in \Xi}\mathbbm{1}_{Q}P_{Q}u,
\end{align*}
where $\mathbbm{1}_{Q}$ denotes the characteristic function on the cube
$Q$.
\end{defn}
\begin{prop}
\label{prop:_PiecewiseApprox}
For a finite Borel measure $\nu $ on $\textbf{Q}$ and $1\le p\leq q$, there
exists $C_{2}>0$ such for all partitions $\Xi $ of $\textbf{Q}$ of half
open cubes and every $u\in W_{p}^{\ell}\left (\textbf{Q}\right )$, we have
\begin{equation*}
\left \Vert u-P_{\Xi}u\right \Vert _{L_{\nu}^{q}(\textbf{Q})}\leq C_{2}
\left \Vert u\right \Vert _{L^{\ell ,p}(\textbf{Q})}\left (\max _{Q
\in \Xi}\mathfrak{J}_{\varrho /m}(Q)\right )^{1/q}.
\end{equation*}
\end{prop}
\begin{proof}
This follows from the proof of \citep[Theorem 3.3]{MR0217487} using \reftext{Lemma~\ref{lem:_approxSupNorm}}.
\end{proof}
\subsection{Approximation order of Kolmogorov $n$-diameters}
\label{subsec:Approximation-order-of}
In this section we will prove our main result.
\begin{lem}
\label{lem:EstimateD_n}
Under the assumption \textup{\reftext{(\ref{eq:StandingAssumption})}}, there exists
a constant $C_{3}>0$ depending only on $p,q,m,\ell $ such for all
$n\in \mathbb N$ we have
\begin{equation*}
d_{\kappa n}\left (\mathscr{S}W_{p}^{\ell},L_{\nu}^{q}\right )\leq C_{3}
\left (\gamma _{\varrho /m,\kappa n}\right )^{1/q}.
\end{equation*}
\end{lem}
\begin{proof}
For $n\in \mathbb N$ and $\Xi \in \Upsilon _{n}$ we have
$\text{dim}\mathcal{P}\left (\Xi ,\alpha \right )=\card (\Xi )\kappa
\leq n\kappa $ and therefore
\begin{equation*}
d_{\kappa n}\left (\mathscr{S}W_{p}^{\ell},L_{\nu}^{q}\right )\leq d_{
\kappa \card (\Xi )}\left (\mathscr{S}W_{p}^{\ell},L_{\nu}^{q}\right ).
\end{equation*}
Hence, we obtain by \reftext{Proposition~\ref{prop:_PiecewiseApprox}}
\begin{align*}
d_{\kappa \card (\Xi )}\left (\mathscr{S}W_{p}^{\ell},L_{\nu}^{q}
\right ) & \leq \sup _{u\in \mathscr{S}W_{p}^{\ell}}\inf _{y\in
\mathcal{P}\left (\Xi ,\ell -1\right )}\left \Vert u-y\right \Vert _{L_{
\nu}^{q}}\leq \sup _{u\in \mathscr{S}W_{p}^{\ell}}\left \Vert u-P_{
\Xi}u\right \Vert _{L_{\nu}^{q}}
\\
& \leq C_{2}\sup _{u\in \mathscr{S}W_{p}^{\ell}}\left \Vert u\right
\Vert _{L^{\ell ,p}(Q)}\left (\max _{Q\in \Xi}\mathfrak{J}_{q\ell /m-q/p}(Q)
\right )^{1/q}
\\
& \leq C_{2}\left (\max _{Q\in \Xi}\mathfrak{J}_{q\ell /m-q/p}(Q)
\right )^{1/q}.
\end{align*}
Taking the infimum over all partitions with cardinality less than or equal
to $n$ proves the lemma.
\end{proof}
We are now in the position to prove our main theorem.
\begin{proof}
[Proof of \reftext{Theorem~\ref{thm:Estimation-n-Diameter}}]
For $N\in \mathbb N$ and
$n(N) := \left \lfloor N/\kappa \right \rfloor $, \reftext{Lemma~\ref{lem:EstimateD_n}} gives
\begin{equation*}
d_{N}\left (\mathscr{S}W_{p}^{\ell},L_{\nu}^{q}\right )\leq d_{
\kappa n(N)}\left (\mathscr{S}W_{p}^{\ell},L_{\nu}^{q}\right )\leq C_{3}
\left (\gamma _{\varrho /m,\kappa n(N)}\right )^{1/q}.
\end{equation*}
Using $\kappa n(N)\leq N$, we obtain
\begin{equation*}
\frac{\log \left (d_{N}\left (\mathscr{S}W_{p}^{\ell},L_{\nu}^{q}\right )\right )}{\log (N)}
\leq
\frac{q \log (C_{3})+\log \left (\gamma _{\varrho /m,\kappa n(N)}\right )}{q\log (\kappa n(N))}
\end{equation*}
and with \reftext{Proposition~\ref{prop:_CoreResult}},
\begin{equation*}
\limsup _{N\rightarrow \infty}
\frac{\log \left (d_{N}\left (\mathscr{S}W_{p}^{\ell},L_{\nu}^{q}\right )\right )}{\log (N)}
\leq \limsup _{N\rightarrow \infty}
\frac{\log \left (\gamma _{\varrho /m,\kappa n(N)}\right )}{q\log (\kappa n(N))}
\leq -\frac{1}{q\cdot s_{\varrho }}.\qedhere
\end{equation*}
\end{proof}
In the following example we consider self-similar measure under the open
set condition. In this case the $L^{q}$-spectrum is well-known, allowing
us to give a formula of $s_{\varrho }$ only in terms of the probability
weights and contraction ratios.
\begin{example}
\label{exa:IFS}
For fixed $n\in \mathbb N$ let $\left (T_{1},\ldots ,T_{n}\right )$ be
a set of contracting \emph{similarities} of $\mathbb{R}^{m}$ with ratios
$r_{1},\ldots ,r_{n}\in (0,1)$ that is for all
$x,y\in \mathbb{R}^{d}$ and $i=1,\ldots ,n$
\begin{equation*}
\left |T_{i}(x)-T_{i}(y)\right |=r_{i}\left |x-y\right |.
\end{equation*}
Furthermore, we assume the \emph{open set condition} (OSC) is fulfilled,
i.e. there exists an open set $O\subset \mathbb{R}^{m}$ such that
\begin{equation*}
T_{i}(O)\subset O\text{\:and\:}T_{i}\left (O\right )\cap T_{j}\left (O
\right )=\varnothing ,\:i\neq j.
\end{equation*}
Moreover, there exists a unique compact set $K$ such that
\begin{equation*}
K=\bigcup _{i=1}^{n}T_{i}(K).
\end{equation*}
Without loss of generality, we assume $K\subset (0,1)^{m}$. For
$(p_{1},\ldots ,p_{n})\in (0,1)^{n}$ let $\nu $ be the unique Borel measure
with
\begin{equation*}
\nu =\sum _{i=1}^{n}p_{i}\nu \circ T_{i}^{-1}.
\end{equation*}
The measure $\nu $ is called \emph{self-similar} measure with respect to
the weights $(p_{1},\ldots ,p_{n})$ and ratios
$(r_{1},\ldots ,r_{n})$ and we have $\supp \nu =K$. By
\citep[Theorem 16]{MR1312056} the $L^{q}$-spectrum $\beta _{\nu}$ on
$\mathbb{R}_{\geq 0}$ is given by the unique solution $s$ of
\begin{equation*}
\sum _{i=1}^{n}p_{i}^{s}r_{i}^{\beta _{\nu}(s)}=1.
\end{equation*}
Hence, under our standing assumptions ($p\leq q$ and $p\ell /m>1$) and
applying \reftext{Theorem~\ref{thm:Estimation-n-Diameter}}, we have
\begin{equation}
\overline{\mathbf{ord}}\left (\mathscr{S}W_{p}^{\ell},L_{\nu}^{q}
\right )\leq -\frac{1}{q\cdot s_{\varrho }},
\label{eq:LqUpperbound}
\end{equation}
where $s_{\varrho }$ is the unique solution $s$ of the equation
$\sum _{i=1}^{n}p_{i}^{s}r_{i}^{\varrho s}=1$. In particular, for the
\emph{`geometric'} choice of the weights
$p_{i} := r_{i}^{\delta}$, $i=1,\ldots ,n$, where
$\delta \in \left [0,m\right ]$ is Hausdorff dimension
$\dim _{H}\left (K\right )$ of $K$ determined as the unique solution of
$\sum _{i=1}^{n}r_{i}^{\delta}=1$, we obtain
\begin{equation*}
s_{\varrho }=\frac{\delta}{\varrho +\delta}.
\end{equation*}
Consequently, inserting $\varrho =\ell q-mq/p$ we get
\begin{equation*}
\overline{\mathbf{ord}}\left (\mathscr{S}W_{p}^{\ell},L_{\nu}^{q}
\right )\leq \frac{\ell}{\delta}\left (\frac{m}{\ell p}-1\right )-
\frac{1}{q}\leq \frac{\ell}{m}\left (\frac{m}{\ell p}-1\right )-
\frac{1}{q}=-\frac{\ell}{m}+\frac{1}{p}-\frac{1}{q}.
\end{equation*}
\end{example}
\section{Application to polyharmonic
operators}
\label{subsec:Application-to-polyharmonic}
\subsection{General setup and eigenvalue
asymptotics}
\label{subsec:General-setup-and_EVasymp}
In this section let $\nu $ be a finite Borel measure on
$\mathring{\textbf{Q}}$ and we restrict to the Hilbert space setting
$H_{0}^{\ell}$, respectively $H^{\ell}$. First, let us define the polyharmonic
operator as in \citep{MR0278126,Borzov1971,MR1298682,MR1328700}. We define the following quadratic forms
\begin{equation*}
J_{\nu}(u) := \int \left |u\right |^{2}\;\mathrm d\nu ,\, u\in L^2_\nu,\:\:\:I_{
\ell}(u) := \int _{\textbf{Q}}\sum _{|\alpha |=\ell}\left |D^{
\alpha}u\right |^{2}\;\mathrm d\Lambda ,\:u\in H_{0}^{\ell}
\end{equation*}
and let $J_{\nu}(u,v)$ and $I_{\ell}(u,v)$ denote the corresponding bilinear
forms. Observe that $I_{\ell}^{1/2}$ is an equivalent norm in
$H_{0}^{\ell}$ (see \citep[6.30 Theorem]{2003167}) and in virtue of our
standing assumption $2\ell /m>1$, we obtain that $H_{0}^{\ell}$ is compactly
embedded into
$\left (\mathcal{C}\left (\overline{\textbf{Q}}\right ),\left \Vert
\,\cdot \,\right \Vert _{\mathcal{C}\left (\overline{\textbf{Q}}
\right )}\right )$ (see e.g.
\citep[Theorem 6.3, Part II]{2003167}). In particular, there is a constant
$C>0$ such that for all $u\in H_{0}^{\ell}$
\begin{equation*}
J_{\nu}(u)\leq CI_{\ell}(u)
\end{equation*}
and by an application of the Cauchy-Schwarz inequality, for fixed
$u\in H_{0}^{\ell}$, the map $w\mapsto J_{\nu}(u,w)$ defines a bounded
linear functional. By the Riesz Representation Theorem, we can define a
bounded linear non-negative self-adjoint operator $T_{\nu}$ mapping from
$\left (H_{0}^{\ell},I_{\ell}\left (\cdot ,\cdot \right )\right )$ to itself
such that, for all $u,w\in H_{0}^{\ell}$,
\begin{equation*}
J_{\nu}(u,w)=I_{\ell}\left (T_{\nu}(u),w\right )
\end{equation*}
and
\begin{equation*}
\sqrt{I_{\ell}\left (T_{\nu}(u),T_{\nu}(u)\right )}=\left \Vert J_{
\nu}\left (u,\,\cdot \,\right )\right \Vert := \sup _{y\in H_{0}^{
\ell}\setminus \left \{ 0\right \} }
\frac{\left |J_{\nu}(u,y)\right |}{I_{\ell}(y,y)^{1/2}}\leq C
\sqrt{I_{\ell}(u,u)}.
\end{equation*}
To finally show that $T_{\nu}$ is compact, let
$\left (u_{n}\right )_{n\in \mathbb N}$ be a bounded sequence in
$\left (H_{0}^{\ell},I_{\ell}\right )$. Then, by the compact embedding
of $H_{0}^{\ell}$ into
$\left (\mathcal{C}\left (\overline{\textbf{Q}}\right ),\left \Vert
\,\cdot \,\right \Vert _{\mathcal{C}\left (\overline{\textbf{Q}}
\right )}\right )$, there exists a subsequence
$\left (u_{n_{k}}\right )_{k\in \mathbb N}$, which is a Cauchy in
$\left (\mathcal{C}\left (\overline{\textbf{Q}}\right ),\left \Vert
\,\cdot \,\right \Vert _{\mathcal{C}\left (\overline{\textbf{Q}}
\right )}\right )$. Hence, for all $k,m\in \mathbb N$, we have
\begin{align*}
&I_{\ell}\left (T_{\nu}(u_{n_{k}})-T_{\nu}(u_{n_{m}})\right )
\\
&\quad =\int _{
\textbf{Q}}\left (u_{n_{k}}-u_{n_{m}}\right )T_{\nu}\left (u_{n_{k}}-u_{n_{m}}
\right )\;\mathrm d\nu
\\
&\quad \leq \left \Vert u_{k}-u_{n_{m}}\right \Vert _{\mathcal{C}\left (
\overline{\textbf{Q}}\right )}\sqrt{\nu (\textbf{Q})}
\frac{\left (\int _{Q}T_{\nu}\left (u_{n_{k}}-u_{n_{m}}\right )^{2}\;\mathrm d\nu \right )^{1/2}}{\sqrt{I_{\ell}\left (u_{n_{k}}-u_{n_{m}}\right )}}
\sqrt{I_{\ell}\left (u_{n_{k}}-u_{n_{m}}\right )}
\\
&\quad \leq \left \Vert u_{k}-u_{n_{m}}\right \Vert _{\mathcal{C}\left (
\overline{\textbf{Q}}\right )}\sqrt{C\nu (\textbf{Q})}
\frac{\sqrt{I_{\ell}\left (T_{\nu}\left (u_{n_{k}}-u_{n_{m}}\right )\right )}}{\sqrt{I_{\ell}\left (u_{n_{k}}-u_{n_{m}}\right )}}
\sqrt{I_{\ell}\left (u_{n_{k}}-u_{n_{m}}\right )}
\\
&\quad \leq \left \Vert u_{k}-u_{n_{m}}\right \Vert _{\mathcal{C}\left (
\overline{\textbf{Q}}\right )}
\sqrt{C^{3}\nu (\textbf{Q})I_{\ell}\left (u_{n_{k}}-u_{n_{m}}\right )},
\end{align*}
taking into account that the sequence
$\left (u_{n}\right )_{n\in \mathbb N}$ is bounded with respect to
$I_{\ell}$, we deduce that
$\left (T_{\nu}(u_{n_{k}})\right )_{k\text{$\in \mathbb N$}}$ is a Cauchy
sequence in the Hilbert space $\left (H_{0}^{\ell},I_{\ell}\right )$.
\begin{defn}
\label{defn4.1}
An element $f\in H_{0}^{\ell}\setminus \left \{ 0\right \} $ is called
\emph{eigenfunction }of $T_{\nu}$ with \emph{eigenvalue} $\lambda $, if for
all $g\in H_{0}^{\ell}$, we have
\begin{equation*}
J_{\nu}(f,g)=\lambda I_{\ell}(f,g).
\end{equation*}
\end{defn}
From the spectral theorem for self-adjoint compact operator we deduce that
there is a decreasing sequence of non-negative eigenvalues
$\left (\lambda _{n}^{\nu}\right )_{n\in \mathbb N}$ tending to $0$. We
are interested in the decay rate of the sequence
$\left (\lambda _{n}^{\nu}\right )_{n\in \mathbb N}$. Note that
$\ker \left (T_{\nu}\right )$ can be quite large; for example in the case
that $\nu $ equals the Dirac measure $\delta _{1/2}$ on $(0,1)$, we have
$\ker \left (T_{\nu}\right )=\left \{ f\in H_{0}^{1}(0,1)\mid f(1/2)=0
\right \} $ and there is exactly one eigenvalue not equal to zero, namely
$\lambda =1/4$ with (normalized) eigenfunction
$f_{1/4}(x) := \mathbbm{1}_{(0,1/2)}2x+\mathbbm{1}_{[1/2,1)}
\left (1-2x\right )$.
As we will see, the growth rate of the eigenvalues is encoded by the $L^{q}$-spectrum.
Using the variational principle it can be shown that the eigenvalues can
be computed in terms of $n$-diameter (e.g.
\citep[Theorem 4.5]{MR2517942}).
\begin{thm}
\label{thm:MainPolyharm_n-diameter}
Let $\lambda _{n}^{\nu}\searrow 0$ be the decreasing sequence of eigenvalues
of the polyharmonic operator $T_{\nu}$ with respect to the Borel measure
$\nu $. Then we have
\begin{equation*}
\sqrt{\lambda _{n+1}^{\nu}}=d_{n}\left (\mathscr{S}H_{0}^{\ell},L_{
\nu}^{2}\right ).
\end{equation*}
\end{thm}
Combining \reftext{Theorem~\ref{thm:MainPolyharm_n-diameter}}, \reftext{Theorem~\ref{thm:Estimation-n-Diameter}} and the fact
$d_{n}\left (\mathscr{S}H_{0}^{\ell},L_{\nu}^{2}\right )\leq d_{n}
\left (\mathscr{S}H^{\ell},L_{\nu}^{2}\right )$, we obtain the following
upper bound.
\begin{thm}
\label{thm:Main_Polyharmonic}
We have
\begin{equation*}
\limsup _{n\rightarrow \infty}
\frac{\log \left (\lambda _{n}^{\nu}\right )}{\log (n)}\leq -
\frac{1}{s_{2\ell -m}}\leq -\left (
\frac{2\ell -m}{\overline{\dim}_{M}\left (\nu \right )}+1\right )
\leq -\frac{2\ell}{m}.
\end{equation*}
\end{thm}
\begin{rem}
\label{rem4.4}
If $\nu $ is a singular measure with respect to the Lebesgue measure, the
result of \citep{Borzov1971} gives us
\begin{equation*}
\lambda _{n}^{\nu}=o\left (n^{^{-2\ell /m}}\right ).
\end{equation*}
In the case $\beta _{\nu}\left (s\right )<m(1-s)$, for some
$s\in (0,1)$, we can improve this estimate, in fact we have for every
$\varepsilon >0$
\begin{equation*}
\lambda _{n}^{\nu}=O\left (n^{-1/s_{2\ell -m}+\varepsilon}\right ).
\end{equation*}
In general one cannot expect that $\lambda _{n}^{\nu}\asymp n^{-s}$ for
some $s>0$ (see for example \citep{Arzt_diss} or \citep{KN2022}).
\end{rem}
\subsection{Application to self-similar measures}
\label{subsec:Application-to-self-similar}
To treat self-similar measures $\nu $ in more detail, we need the following
characterisation of the eigenvalues of the linear self-adjoint compact
operator $T_{\nu}$ by the well-known max-min principle (see for example
\citep[Section 4]{MR831201}, \citep[Theorem 2.1, page 64]{MR774404}).
\begin{prop}
\label{prop:max-min}
For all $i\in \mathbb N$, we have
\begin{equation*}
\lambda _{i}^{\nu}=\sup \left \{ \inf _{\psi \in G\setminus \left \{ 0
\right \} }\left \{ J_{\nu}\left (\psi ,\psi \right ):\,I_{\ell}(
\psi ,\psi )=1\right \} \colon G\subset H_{0}^{1},\:i
\text{-dimensional\,subspace}\right \} .
\end{equation*}
\end{prop}
\begin{proof}
Let $(e_{j})_{j\in \mathbb N}$ be an orthonormal basis of eigenfunctions
of $T_{\nu}$ corresponding to the eigenvalues
$\left (\lambda _{j}^{\nu}\right ){}_{j\in \mathbb N}$. Let $G_{i}$ be
an $i$-dimensional subspace of $H_{0}^{1}$ and define
$E_{i} := \overline{\spann \left (e_{j}:j\geq i\right )}$. Observe
that $\dim \left (H_{0}^{1}/E_{i}\right )=i-1$. Using the first and second
isomorphism theorem,
\begin{equation*}
i-\dim (G_{i}\cap E_{i})=\dim (G_{i}/G_{i}\cap E_{i})=\dim \left (
\left (G_{i}+E_{i}\right )/G_{i}\right )\leq \dim \left (H_{0}^{1}/G_{i}
\right )=i-1.
\end{equation*}
Therefore, there exists
$u\in E_{i}\cap G_{i}\neq \left \{ 0\right \} $ with
$I_{\ell}(u,u)=1$ and we may write $u=\sum _{k\geq i}c_{k}e_{k}$ with
$\sum _{k\geq i}c_{k}^{2}=1$. Consequently,
\begin{align*}
\inf _{\psi \in G_{i}\setminus \left \{ 0\right \} }\left \{ J_{\nu}
\left (\psi ,\psi \right ):\,I_{\ell}(\psi ,\psi )=1\right \} & \leq J_{
\nu}\left (u,u\right )=I_{\ell}\left (T_{\nu}(u),u\right )
\\
& =I_{\ell}\left (\sum _{k\geq i}\lambda _{k}^{\nu}c_{k}e_{k},\sum _{k
\geq i}c_{k}e_{k}\right )\leq \lambda _{i}^{\nu}.
\end{align*}
Conversely, for
$G_{i} := \spann \left \{ e_{1},\ldots ,e_{i}\right \} $, we have
$\inf _{\psi \in G_{i}\setminus \left \{ 0\right \} }\left \{ J_{\nu}
\left (\psi ,\psi \right ):\,I_{\ell}(\psi ,\psi )=1\right \} =
\lambda _{i}^{\nu}$.
\end{proof}
The following theorem has been established in
\citep[Theorem 3.1]{Nazarov} for dimension $m=1$. We prove the corresponding
result for arbitrary dimension $m\in \mathbb N$ of the ambient space.
\begin{thm}
\label{thm:Nazarov_m>1}
Let $\nu $ be a self-similar measure under OSC with feasible open set
$O\subset (0,1)^{m}$ as defined in \reftext{Example~\ref{exa:IFS}},
$\ell \in \mathbb N$, and assume $\nu \left (\partial O\right )=0$. Then,
\begin{equation*}
\mathbf{ord}\left (\mathscr{S}H_{0}^{\ell},L_{\nu}^{2}\right )=
\mathbf{ord}\left (\mathscr{S}H^{\ell},L_{\nu}^{2}\right )=\lim _{l
\to \infty}\frac{\log \left (\lambda _{l}^{\nu}\right )}{\log (l)}=-
\frac{1}{2s_{\varrho }}.
\end{equation*}
\end{thm}
\begin{proof}
Here, we follow partially \citep{MR1338787}. For $t$ with
$0<t<\min p_{i}^{s_{\varrho }}r_{i}^{(2\ell -m)s_{\varrho }}$ let
\begin{equation*}
E_{t} := \left \{ \omega \in I^{*}\mid p_{\omega}r_{\omega}^{2
\ell -m}<t\leq p_{\omega ^{-}}r_{\omega ^{-}}^{2\ell -m}\right \} ,
\end{equation*}
where
$I^{*}=\bigcup _{l\in \mathbb N}\left \{ 1,\ldots ,n\right \} ^{l}$ and
$p_{\omega}r_{\omega}^{2\ell -m} := \prod _{i=1}^{k}p_{\omega _{i}}r_{
\omega _{i}}^{2\ell -m}$,
$\omega =\omega _{1}\cdot \ldots \cdot \omega _{k}\in I^{*}$,
$k\in \mathbb N$. Then, by our assumption
$\nu \left (\partial O\right )=0$, for all $\omega \in E_{t}$, it follows
that $\nu \left (\partial T_{\omega}(O)\right )=0$ and
\begin{equation*}
\nu \left (\bigcup _{\omega \in E_{t}}T_{\omega}(O)\right )=1
\text{\:and\:}T_{\omega}(O)\cap T_{\omega '}(O)=\varnothing
\text{\:for all\:}\omega ,\omega '\in E_{t}\:\text{with}\:\omega \neq
\omega '.
\end{equation*}
Hence,
\begin{equation*}
\sum _{\omega \in E_{t}}p_{\omega}^{s_{\varrho }}r_{\omega}^{(2\ell -m)s_{
\varrho }}=1\leq t^{s_{\varrho }}\card \left (E_{t}\right ).
\end{equation*}
Fix $a\in K$ and choose $u_{0}\in \mathcal{C}_{c}^{\infty}(O)$ such that
$u_{0}(a)>0$. For $\omega \in E_{t}$, we set
\begin{equation*}
u_{\omega}(x) :=
\begin{cases}
u_{0}\left (T_{\omega}^{-1}(x)\right ) & ,x\in T_{\omega}(O)
\\
0 & ,x\in [0,1]^{d}\setminus T_{\omega}(O)
\end{cases}
.
\end{equation*}
Then
$u_{\omega}\in \mathcal{C}_{c}^{\infty}\left (T_{\omega}(O)\right )$. Since
the supports of $(u_{\omega})_{\omega \in E_{t}}$ are disjoint, it follows
that the $(u_{\omega})_{\omega \in E_{t}}$ are mutually orthogonal both
in $L_{\nu}^{2}$ and in $H_{0}^{\ell}$, and
$\spann (u_{\omega}:\omega \in E_{t})$ is therefore a
$\card (E_{t})$-dimensional subspace of $H_{0}^{\ell}$. Moreover, we have
\begin{equation*}
J_{\nu}\left (u_{\omega}\right )=p_{\omega}\int u_{0}^{2}\;\mathrm d
\nu \:\text{and}\:I_{\ell}\left (u_{\omega}\right )=r_{\omega}^{d-2
\ell}I_{\ell}\left (u_{0}\right ).
\end{equation*}
We obtain
\begin{equation*}
\frac{J_{\nu}\left (u_{\omega}\right )}{I_{\ell}\left (u_{\omega}\right )}=r_{
\omega}^{2\ell -m}p_{\omega}
\underbrace{\frac{J_{\nu}\left (u_{0}\right )}{I_{\ell}\left (u_{0}\right )}}_{
=: R}.
\end{equation*}
Now, for
$u=\sum _{\omega \in E_{t}}c_{\omega}u_{\omega}\in H_{0}^{\ell}
\setminus \left \{ 0\right \} $ with $c_{\omega}\in \mathbb{R}$. we have
\begin{equation*}
\frac{J_{\nu}\left (u_{\omega}\right )}{I_{\ell}\left (u_{\text{$\omega$}}\right )}=
\frac{\sum _{\omega \in E_{t}}c_{\omega}^{2}J_{\nu}\left (u_{\omega}\right )}{\sum _{\omega \in E_{t}}c_{\omega}^{2}I_{\ell}\left (u_{\omega}\right )}=R
\frac{\sum _{\omega \in E_{t}}c_{\omega}^{2}p_{\omega}r_{\omega}^{2\ell -m}I_{\ell}\left (u_{\omega}\right )}{\sum _{\omega \in E_{t}}c_{\omega}^{2}I_{\ell}\left (u_{\omega}\right )}
\geq tR\min p_{i}r_{i}^{2\ell -m}.
\end{equation*}
The min-max principle stated in \reftext{Proposition~\ref{prop:max-min}} gives
\begin{equation*}
tR\min p_{i}r_{i}^{2\ell -m}\leq \lambda _{\card (E_{t})}^{\nu}\leq
\lambda _{\left \lfloor t^{-s_{\varrho }}\right \rfloor }^{\nu}.
\end{equation*}
In particular, for $t=l^{-1/s_{\varrho }}$ and $l\in \mathbb N$ large,
\begin{equation*}
l^{-1/s_{\varrho }}R\min p_{i}r_{i}^{2\ell -m}\leq \lambda _{l}^{\nu},
\end{equation*}
Combining this with \textup{\reftext{(\ref{eq:LqUpperbound})}}, gives
\begin{equation*}
-\frac{1}{s_{\varrho }}\leq \liminf _{l\rightarrow \infty}
\frac{\log \left (\lambda _{l}^{\nu}\right )}{\log (l)}\leq \limsup _{m
\rightarrow \infty}
\frac{\log \left (\lambda _{l}^{\nu}\right )}{\log (l)}\leq -
\frac{1}{s_{\varrho }}.\qedhere
\end{equation*}
\end{proof}
\subsection{One-dimensional Kre\u{\i}n--Feller operators}
\label{sec:Krein-Feller-operators-in}
In this final section, we show that the spectral problem of the Kre\u{\i}n--Feller
operator in dimension one is actually equivalent to the spectral problem
of polyharmonic operator $T_{\nu}$ (excluding the eigenvalue zero) for
the case $\ell =m=1$ and $p=q=2$. Let us recall the general setting for
the one-dimensional Kre\u{\i}n--Feller Operator with respect to the finite
Borel measure $\nu $ on $\left (0,1\right )$. We set
\begin{equation*}
\mathcal{C}_{\nu}([0,1]) := \left \{ f\in \mathcal{C}([0,1])
\mid f\:\text{\text{is}\:affine\:linear\:on\:the\:components\:of\:}[0,1]
\setminus \supp (\nu )\right \}
\end{equation*}
and
$\dom (\mathcal{E}_{\nu}) := H_{0}^{1}\cap \mathcal{C}_{\nu}([0,1])$
with Dirichlet form
$\mathcal{E}_{\nu}\left (f,g\right ) := \int _{(0,1)}\nabla f
\nabla u\;\mathrm d\Lambda $. Now, we consider the spectral problem of
the classical Kre\u{\i}n--Feller operator considered in
\citep{KN21,KN2022,MR2828537}. We call
$u\in \dom (\mathcal{E}_{\nu})\setminus \left \{ 0\right \} $ an\emph{ eigenfunction
}with \emph{eigenvalue} $\lambda $ if
\begin{equation}
\int _{(0,1)}\nabla f\nabla u\;\mathrm d\Lambda =\lambda \int fu\;
\mathrm d\nu ,
\label{eq:Eigenfunction}
\end{equation}
for all $f\in \dom (\mathcal{E}_{\nu})$. We need a decomposition
\begin{equation*}
[0,1]\setminus \supp (\nu ) =: A_{1}\cup A_{2}\cup \bigcup _{i
\in I}(a_{i},b_{i}),
\end{equation*}
where $I\subset \mathbb N$, $A_{1} := [0,d_{1})$ if
$0\notin \supp (\nu )$ otherwise $A_{1}=\varnothing $,
$A_{2} := (d_{2},1]$ if $1\notin \supp (\nu )$ otherwise
$A_{2}=\varnothing $, and the intervals $[0,c_{1})$, $(c_{2},1]$,
$(a_{i},b_{i})$, $i\in I$, are mutually disjoint.
The following Lemma will provide a map from $H_{0}^{1}$ to
$\dom \left (\mathcal{E}_{\nu}\right )$.
\begin{lem}[{\citep[Lemma 2.1]{KN2022}}]
\label{lem:RepraesentantL2NU}
The map
$\tau _{\nu}:H_{0}^{1}\rightarrow \dom \left (\mathcal{E}_{\nu}
\right )$
\begin{equation*}
\tau _{\nu}(f)(x) :=
\begin{cases}
f(a_{i})+\frac{f(b_{i})-f(a_{i})}{b_{i}-a_{i}}\left (x-a_{i}\right ), &
x\in (a_{i},b_{i}),\:i\in N,
\\
f(x), & x\in \supp (\nu ),
\\
\frac{f(c_{1})}{c_{1}}(x), & x\in [0,c_{1}),\:0\notin \supp (\nu ),
\\
\frac{f(c_{2})}{1-c_{2}}\left (1-x\right ), & x\in (c_{2},1],\:1
\notin \supp (\nu ),
\end{cases}
\end{equation*}
is surjective, $\tau _{\nu}(f)=f$ as elements of $L_{\nu}^{2}$, and we
have
\begin{equation*}
\nabla \tau _{\nu}(f)(x) :=
\begin{cases}
\frac{f(b_{i})-f(a_{i})}{b_{i}-a_{i}}, & x\in (a_{i},b_{i}),\ i\in N,
\\
\nabla f(x), & x\in \supp (\nu ),
\\
\frac{f(c_{1})}{c_{1}}, & x\in [0,c_{1}),\ 0\notin \supp (\nu ),
\\
-\frac{f(c_{2})}{1-c_{2}}, & x\in (c_{2},1],\ 1\notin \supp (\nu ).
\end{cases}
\end{equation*}
\end{lem}
\begin{lem}
\label{lem:AequivalenzSolo}
We have $\varphi \in \dom (\mathcal{E}_{\nu})$ is an eigenfunction
of \textup{\reftext{(\ref{eq:Eigenfunction})}} with eigenvalue $\lambda $, if and
only if, for all $g\in H_{0}^{1}$, we have
\begin{equation*}
\int _{[0,1]}\nabla \varphi \nabla g\;\mathrm d\Lambda =\lambda
\cdot \int \varphi g\;\mathrm d\nu .
\end{equation*}
\end{lem}
\begin{proof}
Recall,
$[0,1]\setminus \supp (\nu )=A_{1}\cup A_{2}\cup \bigcup _{i\in I}(a_{i},b_{i})$
and define
$c_{i}=\nabla \varphi \left (\frac{a_{i}+b_{i}}{2}\right )$. For simplicity
we assume $0,1\in \supp (\nu )$ and let $g\in H_{0}^{1}$ be, then we have
\begin{align*}
\int _{[0,1]}\nabla \varphi \nabla g\;\mathrm d\text{$\Lambda$} & =
\int _{\supp (\nu )}\nabla \varphi \nabla g\;\mathrm d
\text{$\Lambda$}+\sum _{i\in I}\int _{(a_{i},b_{i})}c_{i}\nabla g\;
\mathrm d\Lambda
\\
& =\int _{\supp (\nu )}\nabla \varphi \nabla g\;\mathrm d
\text{$\Lambda$}+\sum _{i\in I}c_{i}(g(b_{i})-g(a_{i}))
\\
& =\int _{\supp (\nu )}\nabla \varphi \nabla g\;\mathrm d
\text{$\Lambda$}+\sum _{i\in I}c_{i}(b_{i}-a_{i})\left (
\frac{g(b_{i})-g(a_{i})}{b_{i}-a_{i}}\right )
\\
& =\int _{\supp (\nu )}\nabla \varphi \nabla \tau (g)\;\mathrm d
\text{$\Lambda$}+\sum _{i\in I}\int _{(a_{i},b_{i})}\nabla \varphi
\nabla \tau (g)\;\mathrm d\Lambda
\\
& =\int _{(0,1)}\nabla \varphi \nabla \tau (g)\;\mathrm d
\text{$\Lambda$}=\lambda \int fg\;\mathrm d\text{$\nu$.}\qedhere
\end{align*}
\end{proof}
\begin{prop}
\label{Prop:SolomanyakOperator}
Let $\varphi $ be an eigenfunction of $T_{\nu}$ with eigenvalue
$\lambda >0$. Then $\varphi $ is affine linear on the connected components
of $[0,1]\setminus \supp (\nu )$.
\end{prop}
\begin{proof}
Here we closely follow \citep[Proposition 3.2]{MR4241300}. Let
$(a,b)$ be a component $[0,1]\setminus \supp (\nu )$. Then we have for
all $f\in H_{0}^{1}$
\begin{equation*}
\lambda \int _{[0,1]}\nabla f\nabla \varphi \;\mathrm d\Lambda =\int f
\varphi \;\mathrm d\nu .
\end{equation*}
For $x_{1}x_{2}\in (a,b)$, $x_{1}<x_{2}$ and for all $\delta >0$ sufficiently
small such that
\begin{equation*}
a<x_{1}-\delta <x_{1}<x_{1}+\delta <x_{2}-\delta <x_{2}<x_{2}+\delta <b,
\end{equation*}
we have that
\begin{equation*}
g:x\mapsto \frac{x-\left (x_{1}-\delta \right )}{2\delta}\mathbbm{1}_{(x_{1}-
\delta ,x_{1}+\delta )}(x)+\mathbbm{1}_{\left [x_{1}+\delta ,x_{2}-
\delta \right ]}(x)+\frac{x_{2}+\delta -x}{2\delta}\mathbbm{1}_{(x_{2}-
\delta ,x_{2}+\delta )}(x).
\end{equation*}
defines an element in $H_{0}^{1}$. Hence, we obtain
\begin{align*}
0 & =\int _{[0,1]}\nabla g\nabla \varphi \;\mathrm d\Lambda =
\frac{1}{2\delta}\int _{(x_{1}-\delta ,x_{1}+\delta )}\nabla \varphi
\;\mathrm d\Lambda -\frac{1}{2\delta}\int _{(x_{2}-\delta ,x_{2}+
\delta )}\nabla \varphi \;\mathrm d\Lambda .
\end{align*}
The Lebesgue differentiation theorem forces $\nabla \varphi (x)=c$ almost
everywhere. For all $x\in (a,b)$, we obtain
\begin{equation*}
\varphi (x)=\varphi (a)+\int _{[a,x]}\nabla \varphi \;\mathrm d
\Lambda =\varphi (a)+c(x-a).
\end{equation*}
The following proposition shows that the equivalence of the spectral problems.
\end{proof}
\begin{prop}
\label{prop:_ProblemeAequivlaent}
We have that $\lambda >0$ is an eigenvalue of $T_{\nu}$ if and only if
$1/\lambda $ is an eigenvalue of \textup{\reftext{(\ref{eq:Eigenfunction})}}.
\end{prop}
\begin{proof}
Let $\varphi $ be an eigenfunction of $T_{\nu}$ with eigenvalue
$\lambda >0$. Using \ref{Prop:SolomanyakOperator} it follows
$\varphi \in \dom \left (\mathcal{E}_{\nu}\right )$ and by definition we
have for all
$f\in \dom \left (\mathcal{E}_{\nu}\right )\subset H_{0}^{1}$
\begin{equation*}
\lambda \int _{[0,1]}\nabla f\nabla \varphi \;\mathrm d\Lambda =\int f
\varphi \;\mathrm d\nu .
\end{equation*}
Hence, $\varphi $ is an eigenfunction of
\textup{\reftext{(\ref{eq:Eigenfunction})}} with eigenvalue $1/\lambda $.
Reversely, let $\varphi \in \dom (\mathcal{E}_{\nu})$ be an eigenfunction
of \textup{\reftext{(\ref{eq:Eigenfunction})}} with eigenvalue $\lambda $. Then it
follows $\lambda >0$ and by \reftext{Lemma~\ref{lem:AequivalenzSolo}} we have for
all $f\in H_{0}^{1}$
\begin{equation*}
\int _{[0,1]}\nabla f\nabla \varphi \;\mathrm d\Lambda =\lambda \int f
\varphi \;\mathrm d\nu ,
\end{equation*}
which shows that $\varphi $ is an eigenfunction of $T_{\nu}$ with eigenvalue
$1/\lambda $.
\end{proof}
We end this section by using the above observation to give a short proof
of the sub-/superadditivity of the eigenvalue counting function announced
in the introduction.
Now, for $d_{0}=0<d_{1}<\dots <d_{n}<d_{n+1}=1$ with
$\nu \left (\left \{ d_{k}\right \} \right )=0$, we define the following
closed subspace of $H_{0}^{1}$ given by
\begin{equation*}
F := \left \{ u\in H_{0}^{1}:u(d_{i})=0,i\in \left \{ 1,\dots ,n
\right \} \right \} .
\end{equation*}
Note that $F$ can be identified with
\begin{align*}
F & \simeq H_{0}^{1}\left (\left (d_{0},d_{1}\right )\right )\times H_{0}^{1}
\left (\left (d_{2},d_{3}\right )\right )\times \dots \times H_{0}^{1}
\left (\left (d_{n},d_{n+1}\right ).\right )
\end{align*}
Furthermore, let $T_{k,\nu}$ denote the operator on
$H_{0}^{1}\left (\left (d_{k},d_{k+1}\right )\right )$ with respect to
the form
$\left (f,g\right )\mapsto \int _{\left (d_{k},d_{k+1}\right )}fg\;
\mathrm d\nu $ and let $T_{F,\nu}$ be the operator on $F$ with respect
to the form $\left (f,g\right )\mapsto \int fg\;\mathrm d\nu $. We define
the eigenvalue counting function of
$S\in \left \{ T_{\text{$\nu$}},T_{k,\nu},T_{F,\nu}\right \} $ by
\begin{equation*}
N\left (x,S\right ) := \card \left \{ n\in \mathbb N:\lambda _{n}^{S}
\geq x\right \} ,\:x>0.
\end{equation*}
Then the following sub-/superadditivity holds true.
\begin{prop}
\label{prop:Addiitivity}
For all $x\geq 0$, we have
\begin{equation*}
\sum _{k=0}^{n}N\left (x,T_{k,\nu}\right )=N\left (x,T_{F,\nu}\right )
\leq N\left (x,T_{\nu}\right )\leq \sum _{k=0}^{n}N\left (x,T_{k,\nu}
\right )+n.
\end{equation*}
\end{prop}
\begin{proof}
From max-min principle we deduce
\begin{equation*}
N\left (x,T_{F,\nu}\right )\leq N\left (x,T_{\nu}\right ).
\end{equation*}
Moreover, we have $\dim \left (H_{0}^{1}/F\right )=n$. Hence, it follows
from \citep[Proposition 1]{MR1328700}
\begin{equation*}
N\left (x,T_{\nu}\right )\leq N\left (x,T_{F,\nu}\right )+n.
\end{equation*}
It remains to show
$N\left (x,T_{F,\nu}\right )=\sum _{k=0}^{n}N\left (x,T_{k,\nu}
\right )$. Let $f$ be an eigenfunction with eigenvalue $\lambda >0$ of
$T_{k,\nu}$ with $\nu \left (\left (d_{k},d_{k+1}\right )\right )>0$. Then
define
\begin{equation*}
g(x)=\mathbbm{1}_{(d_{k},d_{k+1})}f,
\end{equation*}
it follows $g\in F$ and we have for all $h\in F$
\begin{equation*}
\lambda \int _{(0,1)}\nabla h\nabla g\;\mathrm d\Lambda =\lambda
\int _{(d_{k},d_{k+1})}\nabla h\nabla g\;\mathrm d\Lambda =\int _{(d_{k},d_{k+1})}hg
\;\mathrm d\nu =\int _{(0,1)}hg\;\mathrm d\nu .
\end{equation*}
This implies
$N\left (x,T_{F,\nu}\right )\geq \sum _{k=0}^{n}N\left (x,T_{k,\nu}
\right )$. On the other hand, if $f$ is an eigenfunction with eigenvalue
$\lambda >0$ of $T_{F,\nu}$, then $g=\mathbbm{1}_{(d_{k},d_{k+1})}f$ is
an eigenfunction with eigenvalue $\lambda $ of $T_{k,\nu}$ provided
$g\neq 0$ and $\nu \left (\left (d_{k},d_{k+1}\right )\right )>0$. Hence,
we obtain
$N\left (x,T_{F,\nu}\right )=\sum _{k=0}^{n}N\left (x,T_{k,\nu}
\right )$.
\end{proof}
\section*{Acknowledgment}
This research was supported by the {DFG} grant {Ke 1440/3-1}. We would like
to thank the anonymous referee for her/his valuable comments, which have
contributed to a significant improvement of the presentation.
\phantomsection\addcontentsline{toc}{section}{\refname}
| 92,624
|
TITLE: A common misunderstanding regarding which path information in a double slit and Mach-Zehnder interferometer?
QUESTION [0 upvotes]: I just can't get my head around this. What does he mean when he says that a hit at one of the detectors doesn't imply that the particle went though a certain path?
He calls this the "separation fallacy" which he claims is common in quantum experiments like this.
I couldn't find any literature on this fallacy except this paper and a few lines on the wikipedia page on the delayed choice quantum eraser.
Original paper (https://www.google.com/url?sa=t&source=web&rct=j&url=http://philsci-archive.pitt.edu/10216/1/SeparationFallacy-rev.pdf&ved=2ahUKEwji77vQs5LkAhVEt3EKHf--DBsQFjAIegQICBAB&usg=AOvVaw3rWfDwXSOzoBjDIhSdWDaD&cshid=1566337034114)
REPLY [4 votes]: The separation fallacy, as presented by this author, is simply the idea that a superposition of states cannot be considered as giving those states any reality by themselves. They are not separated enough, precisely because they are superposed, to allow logical articulations such as $OR$ and $AND$ to be used to talk meaningfully about the way they interfere. And when we do so, we assume a classical viewpoint that leads to nonsensical conclusions.
Let's see this with the double-slit experiment. When no which-path detector is present, we do not know what the particle does between its emission and its detection in the pattern recorder. When a which-path detector is present, we know that the particle has been seen at one of the slits, and then we say "it went this way".
The separation fallacy consists in saying, in the absence of which-path information, that the particle went in $BOTH$ ways at the same time, which supposes that trajectories are meaningful. But a trajectory is a classical concept that does not apply in this case: there is no way to describe the behavior of a particle going through both slits in term of trajectories, and saying $BOTH$ ways means following one trajectory $AND$ also the other one - that's the fallacy. Even though the state is formally a superposition of classical descriptions (the two trajectories), you cannot use these descriptions to make sense of the state.
But this is only part of the point. Even when the particle has been detected at one slit, one still cannot say that it went though that slit! Because, fundamentally, the number of slits does not matter: in the path integral picture, any outcome has a probability computed by taking into account all the possible ways this outcome can come to be. So even a basic straight trajectory from point $A$ to point $B$ is, in the quantum view, the interference of all possible paths taken by the particle.
So in a sense, from a QM perspective it is not more mysterious for a particle to go through both slits than to go through one, and even to just go along a straigh line in the absence of any obstacle. The only difference is that in the latter cases, what we have is compatible with a classical description. So for the first case, we use those classical descriptions, the trajectories, in a semi-classical picture, the superposition, to describe the quantum state. But doing so forces us to keep in mind the notion of trajectory, and this is the fallacy. There are no trajectories - and even when there is a trajectory, well in fact in the QM view there is still no trajectory.
| 93,178
|
TITLE: Find a given logarithmic definite integral
QUESTION [1 upvotes]: Find the following integral, where $a$ is a real number bigger than $1$:
$$\int_1^{a^2} \frac{\ln x}{\sqrt x(x + a)}\,\mathrm dx.$$
By using the substitution $t = \sqrt x$, I got this new integral which seems to be easier to solve, but I haven't found any way to do it yet:
$$4\int_1^a \frac{\ln t}{t^2 + a}\,\mathrm dt.$$
Thank you in advance!
REPLY [2 votes]: Let $t=\sqrt{x}$
\begin{equation}
I = \int\limits_{1}^{a^{2}} \frac{\ln x}{\sqrt{x}(x+a)} dx
= 4 \int\limits_{1}^{a} \frac{\ln t}{t^{2}+a} dt
\end{equation}
Integrating by parts, we have
\begin{align}
I_{1} &= \int\limits_{1}^{a} \frac{\ln t}{t^{2}+a} dt \\
&= \frac{\ln t}{\sqrt{a}} \tan^{-1}\left( \frac{t}{\sqrt{a}} \right) \Big|_{1}^{a}
\, - \frac{1}{\sqrt{a}} \int\limits_{1}^{a} \frac{1}{t} \tan^{-1}\left( \frac{t}{\sqrt{a}} \right) dt \\
&= \frac{\ln a}{\sqrt{a}} \tan^{-1}(\sqrt{a})
\, - \frac{1}{\sqrt{a}} \int\limits_{1}^{a} \frac{1}{t} \tan^{-1}\left( \frac{t}{\sqrt{a}} \right) dt
\end{align}
Let $y=t/ \sqrt{a}$
\begin{align}
I_{2} &= \int\limits_{1}^{a} \frac{1}{t} \tan^{-1}\left( \frac{t}{\sqrt{a}} \right) dt
= \int\limits_{1/\sqrt{a}}^{\sqrt{a}} \frac{1}{y} \tan^{-1}(y) dy \\
\tag{a}
&= \frac{i}{2} \left[ \int\limits_{1/\sqrt{a}}^{\sqrt{a}} \frac{\ln (1-iy)}{y} dy
\, - \int\limits_{1/\sqrt{a}}^{\sqrt{a}} \frac{\ln (1+iy)}{y} dy \right] \\
\tag{b}
&= \frac{i}{2} \left[ \mathrm{Li}_{2}(-iy) - \mathrm{Li}_{2}(iy) \right] \Big|_{1/\sqrt{a}}^{\sqrt{a}} \\
&= \frac{i}{2} \left( \left[ \mathrm{Li}_{2}(-i\sqrt{a}) + \mathrm{Li}_{2}\left(\frac{i}{\sqrt{a}}\right) \right]
- \left[ \mathrm{Li}_{2}(i\sqrt{a}) + \mathrm{Li}_{2}\left(\frac{-i}{\sqrt{a}}\right) \right] \right) \\
\tag{c}
&= \frac{i}{2} \left( \left[ -\frac{\pi ^{2}}{6} - \frac{1}{2} \ln ^{2}(i\sqrt{a}) \right]
- \left[ -\frac{\pi ^{2}}{6} - \frac{1}{2} \ln ^{2}(-i\sqrt{a}) \right] \right) \\
\tag{d}
&= \frac{\pi}{4} \ln a
\end{align}
a. $\tan^{-1}(y) = \frac{i}{2} [\ln (1-iy) - \ln (1+iy)]$
b. Dilogarithm function
\begin{equation}
\mathrm{Li}_{2}(z) = -\int_{0}^{z} \frac{\ln (1-x)}{x} dx
\end{equation}
c. Use the identity
\begin{equation}
\mathrm{Li}_{2}(z) + \mathrm{Li}_{2}(1/z) = -\frac{\pi ^{2}}{6} - \frac{1}{2} \ln ^{2}(-z)
\end{equation}
d. $\ln (\pm iz) = \ln z \pm i\pi /2$
Now we have
\begin{equation}
I = 4I_{1} = \frac{4}{\sqrt{a}} (\ln a) \tan^{-1}(\sqrt{a}) \, - \frac{\pi}{\sqrt{a}} \ln a
\end{equation}
| 36,995
|
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| 385,907
|
TITLE: Distinct Sylow $p$-subgroups intersect only at the identity, which somehow follows from Lagrange's Theorem. Why?
QUESTION [30 upvotes]: It seems that often in using counting arguments to show that a group of a given order cannot be simple, it is shown that the group must have at least $n_p(p^n-1)$ elements, where $n_p$ is the number of Sylow $p$-subgroups.
It is explained that the reason this is the case is because distinct Sylow $p$-subgroups intersect only at the identity, which somehow follows from Lagrange's Theorem.
I cannot see why this is true.
Can anyone quicker than I tell me why? I know it's probably very obvious.
Note: This isn't a homework question, so if the answer is obvious I'd really just appreciate knowing why.
Thanks!
REPLY [5 votes]: In some situations, to prove that groups of order $n$ cannot be simple, you can use the counting argument if all Sylow subgroups have trivial intersection, and a different argument otherwise.
For example let $G$ be a simple group of order $n=144 = 16 \times 9$. The number $n_3$ of Sylow 3-subgroups is 1, 4 or 16. If $n_3 = 1$ then there is normal Sylow subgroup and if $n_3= 4$ then $G$ maps nontrivially to $S_4$, so we must have $n_3 = 16$.
If all pairs of Sylow 3-subgroups have trivial intersection, then they contain in total $16 \times 8$ non-identity elements, so the remaining 16 elements must form a unique and hence normal Sylow 2-subgroup of $G$.
Otherwise two Sylow 3-subgroups intersect in a subgroup $T$ of order 3. Then the normalizer $N_G(T)$ of $T$ in $G$ contains both of these Sylow 3-subgroups, so by Sylow's theorem it has at least 4 Sylow 3-subgroups, and hence has order at least 36, so $|G:N_G(T)| \le 4$ and $G$ cannot be simple.
| 101,136
|
Hola. Merhaba. Kamusta.
There are many different ways to greet this year's Pearson Scholars, who come to the University of Toronto from 24 different countries. They are – in U of T President Meric Gertler’s words – a welcome addition to “U of T's unique culture of double-diversity.”
International students make up more than one-in-five U of T students, he pointed out at a welcome reception at Hart House Monday evening – in a city where half the population was born outside of Canada. “The University of Toronto, and the city-region around our campuses, is an ideal place for brilliant students to develop as globally minded leaders in every field of endeavour,” he said.
“So, over the next four years, make the most of the countless learning opportunities we offer, both inside and outside the classroom. Finally, please remember that we are here to support you.”
The 2018 class of Lester B. Pearson scholars. Sitting in the front row are, from left: Sandy Welsh, vice-provost, students; Cheryl Regehr, vice-president and provost; U of T President Meric Gertler; and Joseph Wong, associate vice-president and vice-provost, international student experience (photo by Nick Iwanyshyn)
The current cohort of scholars were joined by last year's class, the first group.
The award recognizes students who demonstrate exceptional academic achievement, creativity and leadership, as well as a commitment to making an impact in their community. The scholarship is highly competitive since only one student can be nominated by a high school.
The 40 new scholars come with lofty aspirations, from discovering new treatments for Parkinson's to using actuarial science to help the poor in rural Kenya and advancing sustainable development.
U of T News spoke to four scholars about their first impression of Toronto and their goals.
Adriana DÍaz Lozano Patiño
From Mexico
Faculty of Applied Science & Engineering, engineering science
Patiño was a precocious kid. As early as age 12, she knew she wanted to be an engineer. “I wanted to learn how to tackle problems in an analytical way, something to allow me to create efficient solutions for the world,” she says.
The student from Mexico City chose U of T for its engineering science program, which she describes as a perfect mix of theoretical and practical courses that will prepare her for graduate school.
One of the most striking things about her first days on campus has been the diversity of the student body. “You get people from Asia, Africa, Europe and the U.S. They bring all these perspectives into your life that you didn't have before,” she observes.
Patiño doesn't want to limit her U of T experience to the classroom. Far from it. After dabbling in theatre back home – having played Timon in The Lion King and Belle in Beauty and the Beast – she wants to continue acting while also exploring engineering clubs and practising debating.
Hannah Godrey-Clarke
From the United Kingdom
In the Faculty of Music, music performance
Godfrey-Clarke comes to U of T from Halifax, north of Manchester, England, where she was the first female principal bassist in the National Youth Orchestra of Great Britain. “There shouldn't be ‘first female’ anythings in 2018, but I was very lucky to get that,” she says.
Her first instrument was the cello, but she picked up the double bass at 11 years old through a local music service and she never looked back.
If you wonder about the mini harmonica she wears around her neck, it's a nod to an inside joke with her family, who kid that she should play the harmonica so she doesn't have to lug around an instrument that can weigh over 50 pounds in its case.
She practises four or five hours a day, and hopes to join as many U of T ensembles as possible. At the same time, she wants to become involved with music outreach since the music program that inspired her was later cut through lack of funding. “Music has opened so many doors to me. I wouldn't be here without the opportunities that I've had,” she says.
Devansh Khare
From India
Faculty of Applied Science & Engineering, mechanical engineering
In his first month in Toronto after arriving from Mumbai, Khare was confronted with a problem.
He took part in the engineering student tradition of painting himself completely purple during orientation, only to remember he had a photoshoot with his fellow Pearson Scholars the next day.
“I completely forgot and had to spend two hours scrubbing my face,” he recalls with a smile.
Having represented India at international environmental events, such as the Sunburst Environment Program in Singapore, he's looking forward to a career in the clean energy sector. U of T – with its strong engineering program and diverse student population – is the ideal launching pad, he says.
“I've realized the importance of understanding different cultures to tackle the various issues faced by our society today,” he writes in his scholarship bio, “and with the Lester B. Pearson Scholarship, I hope to gain a broader perspective for improving education, environment and lifestyle in the future.”
Ami Alexis
From Trinidad and Tobago
U of T Scarborough, co-op management
Alexis's first weeks on campus have been a bit of a cultural shock particularly when it comes to cuisine.
“The food is different because at home we use a lot of seasoning,” she says, adding she may bring some back after visiting home at Christmas.
The 18-year-old from the twin island Republic of Trinidad and Tobago has a passion for water sports, especially swimming and water polo. She chose to study business at U of T Scarborough to keep her options open after graduation.
“I'm excited for what's ahead,” she says – though maybe not for her first Canadian winter.
“Monday was really cold. It was 14 degrees. I thought it was freezing. Imagine minus 40.”
| 134,117
|
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| 242,317
|
The goal of this section is to describe explicitly trimodules $\TDDD$
and $\TDDA$ which allow one to do cornered-type gluings of bordered
modules. To explain this precisely, we fix some notation:
\begin{definition}\label{def:connect-sum-pmc}
Fix matched intervals $\PMC_1$ and $\PMC_2$. As described in
Section~\ref{sec:glue-surf}, we can glue $\PMC_1$ and $\PMC_2$ to
get a pointed matched circle $\PMC=\PMC_1\cup\PMC_2$.
Writing $\PMC_i=(Z_i,\CircPts_i,M_i,z_i)$, $i=1,2$, there are
projection maps $p_*^{i}\co H_1(Z_1\cup Z_2\setminus z,
\CircPts_1\cup\CircPts_2)\to H_1(Z_i,\CircPts_i)$.
Given a pointed matched circle $\PMC=(Z,\CircPts,M,z)$ (respectively matched interval $\PMC=(Z,\CircPts,M)$), let $-Z$
denote $Z$ with its orientation reversed and let
$-\PMC=(-Z,\CircPts,M,z)$ (respectively $-\PMC=(-Z,\CircPts,M)$). Let $r\co \PMC\to -\PMC$ denote the
identity map (which is orientation-reversing).
\end{definition}
For the rest of this section, fix matched intervals $\PMC_1$, $\PMC_2$, and
$\PMC_3$. Write $\PMC_i=(Z_i,\CircPts_i,M_i)$, where $\CircPts_i$ is
a subset of $Z_i$ of cardinality $4k_i$, and $M_i\co \CircPts_i\to [2k]$
is a $2$-to-$1$ map. Let:
\begin{equation}\label{eq:PMC-ij}
\begin{aligned}
\PMC_{12}&= \PMC_1\cup(-\PMC_2) & \PMC_{23}&= \PMC_2\cup(-\PMC_3)
&
\PMC_{31}&= \PMC_3\cup(-\PMC_1)\\
\PMC_{21}&= \PMC_2\cup(-\PMC_1)=-\PMC_{12} & \PMC_{32}&=
\PMC_3\cup(-\PMC_2)=\PMC_{-23} & \PMC_{13}&=
\PMC_1\cup(-\PMC_3)=-\PMC_{31}.
\end{aligned}
\end{equation}
There are associated surfaces $\PunctF(\PMC_{ij})$ with boundary
$S^1$, and closed surfaces $F(\PMC_{ij})=\PunctF(\PMC_{ij})\cup_\bdy
\bD^2$ (see Section~\ref{sec:am} or, e.g.,~\cite[Construction 3.2]{LOT2}). These surfaces satisfy $F(-\PMC_{ij})=-F(\PMC_{ij}).$
\begin{definition}
There is a $3$-dimensional cobordism $\CT(\PMC_1,\PMC_2,\PMC_3)$ from
$F(\PMC_{12})\amalg F(\PMC_{23})$ to $F(\PMC_{13})$ defined as
follows. Let $\Delta$ denote a 2-simplex, with boundary edges $e_1$, $e_2$, and
$e_3$ (in clockwise order). Consider the surfaces $\PunctF(\PMC_i)$, each of which has boundary $S^1$. Then $\CT(\PMC_1,\PMC_2,\PMC_3)$ is obtained from
\[
\bigl([0,1]\times \PunctF(\PMC_1))\amalg
\bigl([0,1]\times \PunctF(\PMC_2))\amalg
\bigl([0,1]\times \PunctF(\PMC_3))\amalg
\bigl( \Delta\times S^1\bigr)
\]
by gluing $[0,1]\times \bdy \PunctF(\PMC_i)$ to $e_i\times S^1$.
\end{definition}
The manifold $\CT(\PMC_1,\PMC_2,\PMC_3)$ has boundary
\[
\bdy \CT(\PMC_1,\PMC_2,\PMC_3)=F(-\PMC_{12})\amalg F(-\PMC_{23})\amalg F(-\PMC_{31}).
\]
In particular, given bordered $3$-manifolds $Y_{12}$ and $Y_{23}$ with
boundaries $F(\PMC_{12})$ and $F(\PMC_{23})$, respectively, we can
glue $Y_{12}$ and $Y_{23}$ to $\CT(\PMC_1,\PMC_2,\PMC_3)$ to give a
$3$-manifold $(Y_{12}\amalg Y_{23})\cup \CT(\PMC_1,\PMC_2,\PMC_3)$
with boundary $F(\PMC_{13})$.
\begin{lemma}\label{lem:Co-glue}
Let $Y'_{12}$ (respectively $Y'_{23}$) be a cornered $3$-manifold
with vertical boundary $F(\PMC_{1})$ (respectively $F(\PMC_3)$) and
horizontal boundary $F(\PMC_2)$ (respectively $-F(\PMC_2)$). Let
$Y_{ij}$ be the smoothing of $Y'_{ij}$. Then
\[
Y'_{12}\cup_{F(\PMC_2)}Y'_{23}\cong (Y_{12}\amalg Y_{23})\cup \CT(\PMC_1,\PMC_2,\PMC_3)
\]
as bordered $3$-manifolds.
\end{lemma}
\noindent This is immediate from the definitions.
\begin{corollary}\label{cor:corn-pair-via-bord}
Let $Y'_{12}$ (respectively $Y'_{23}$) be a cornered $3$-manifold
with vertical boundary $F(\PMC_{1})$ (respectively $F(\PMC_3)$) and
horizontal boundary $F(\PMC_2)$ (respectively $-F(\PMC_2)$). Let
$Y_{ij}$ be the smoothing of $Y'_{ij}$. Then
\begin{align*}
\CFAa(Y'_{12}\cup_{F(\PMC_2)}Y'_{23})&\simeq \CFAa(Y_{12})\DTP_{\Alg(\PMC_{12})} \bigl(\CFAa(Y_{23})\DTP_{\Alg(\PMC_{23})} \CFDDAa(\CT(\PMC_1,\PMC_2,\PMC_3))\bigr)\\
\CFDa(Y'_{12}\cup_{F(\PMC_2)}Y'_{23})&\simeq \CFAa(Y_{12})\DTP_{\Alg(\PMC_{12})} \bigl(\CFAa(Y_{23})\DTP_{\Alg(\PMC_{23})} \CFDDDa(\CT(\PMC_1,\PMC_2,\PMC_3))\bigr).
\end{align*}
\end{corollary}
\noindent This is immediate from Lemma~\ref{lem:Co-glue} and the pairing
theorem for bordered Floer homology. (The notations $\CFDDDa$ and $\CFDDAa$ denote type \DDD\ and \DDA\ trimodules, respectively. See~\cite{LOT2} for the definitions of type \DD, \DA, and \AAm\ bimodules in bordered Floer homology. The extension from bimodules to trimodules is obvious; compare~\cite[Remark 5.7]{LOT2}.)
\begin{definition} \label{def:THD}
There is a Heegaard diagram $\THD{\PMC_1}{\PMC_2}{\PMC_3}$ as in
Figure~\ref{fig:THD}, with boundary
\begin{equation}\label{eq:bdy-THD}
\bdy\THD{\PMC_1}{\PMC_2}{\PMC_3}=\PMC_{12}\amalg \PMC_{23}\amalg\PMC_{31}.
\end{equation}
This Heegaard diagram is constructed as follows. Start with the
canonical arced, bordered Heegaard diagrams $\HD_i$ for the identity
map of $\PMC_i$~\cite[Definition 5.35]{LOT2}. Let $\arcz_i$ denote
the arc in $\HD_i$ connecting the two boundary components. Then
$\THD{\PMC_1}{\PMC_2}{\PMC_3}$ is obtained from $\Delta\amalg \Delta\amalg \coprod_i
(\HD_i\setminus \nbd(\arcz_i))$ by gluing the arcs in
$\bdy\Delta\amalg \bdy\Delta\amalg \coprod_i
(\bdy\nbd\arcz_i)$ together, in such a way that the boundary of
$\THD{\PMC_1}{\PMC_2}{\PMC_3}$ is given by
Formula~\eqref{eq:bdy-THD}. (We place the basepoint in one of the two triangles $\Delta$.)
\end{definition}
\begin{figure}
\centering
\includegraphics[scale=1.2]{CorneredTrimodules}
\caption{\textbf{Diagrams for cornering and cornered-type gluing.}
Left: The cornered Heegaard diagram for
turning bordered invariants into cornered invariants.
Right: The bordered Heegaard diagram
$\THD{\PMC_1}{\PMC_2}{\PMC_3}$ for cornered-type gluing. The black circles indicate handles attached, according to the letter pairs. The red lines are $\alpha$-arcs and the blue circles are $\beta$-circles.}
\label{fig:THD}
\end{figure}
\begin{lemma}
The bordered Heegaard diagram $\THD{\PMC_1}{\PMC_2}{\PMC_3}$
represents the bordered $3$-manifold $\CT(\PMC_1,\PMC_2,\PMC_3)$.
\end{lemma}
\noindent This is immediate from the definitions.
The goal of the rest of the section is to compute explicitly the
invariants
\[\CFDDDa(\CT(\PMC_1,\PMC_2,\PMC_3))=\CFDDDa(\THD{\PMC_1}{\PMC_2}{\PMC_3})
\]
and
\[
\CFDDAa(\CT(\PMC_1,\PMC_2,\PMC_3))=\CFDDAa(\THD{\PMC_1}{\PMC_2}{\PMC_3}).
\]
The trimodule $\CFDDDa(\CT(\PMC_1,\PMC_2,\PMC_3))$ is similar to the
bimodule $\CFDDa(\Id)$, and the computation of
$\CFDDDa(\CT(\PMC_1,\PMC_2,\PMC_3))$ is essentially the same as the
computation of $\CFDDa(\Id)$ given in~\cite[Section 3]{LOT4}. The
trimodule $\CFDDAa(\CT(\PMC_1,\PMC_2,\PMC_3))$ shares features with
both $\CFDDa(\Id)$ and $\CFDAa(\Id)\simeq \Alg(F)$.
The
combinatorial answers for $\CFDDDa(\CT(\PMC_1,\PMC_2,\PMC_3))$ and
$\CFDDAa(\CT(\PMC_1,\PMC_2,\PMC_3))$
are given in
Section~\ref{sec:describe-trimodules}, under the names
$\TDDD(\PMC_1,\PMC_2,\PMC_3)$ and $\TDDA(\PMC_1,\PMC_2,\PMC_3)$, respectively. We
prove that the answers are correct in Section~\ref{sec:compute-TDDD}
(for $\CFDDDa$) and Section~\ref{sec:compute-TDDA} (for $\CFDDAa$).
\subsection{Combinatorial descriptions of the
trimodules}\label{sec:describe-trimodules}
\subsubsection{Description of \texorpdfstring{$\TDDD$}{TDDD}}
\begin{definition}\label{def:complementary-idems}
Choose subsets $\SetS_i\subset [2k_i]$ for $i=1,2,3$. The
sets $\SetS_i$ specify idempotents
\begin{align*}
I_{12}&=I(\SetS_1\cup ([2k_2]\setminus \SetS_2))\in \Alg(\PMC_{12})\\
I_{23}&=I(\SetS_2\cup ([2k_3]\setminus \SetS_3))\in \Alg(\PMC_{23})\\
I_{31}&=I(\SetS_3\cup ([2k_1]\setminus \SetS_1))\in \Alg(\PMC_{31}).
\end{align*}
We call the idempotents $I_{12}, I_{23}, I_{31}$ a \emph{complementary
idempotent triple} for $\PMC_1, \PMC_2, \PMC_3$. We will also
sometimes write $I_{12}\otimes I_{23}\otimes I_{31}$ for the
complementary idempotent triple.
Let $\Idem_{\DDD}$ denote the set of complementary idempotent triples.
\end{definition}
\begin{figure}
\centering
\begin{overpic}[tics=5, scale=.5]{DDDChordTriples}
\put(20, -2){$-\PMC_{12}$}
\put(34, -2){$\PMC_{31}$}
\put(-10, 35){$-\PMC_{23}$}
\put(7,7){$\xi_{12}$}
\put(43,32){$\xi_{31}$}
\put(7,59){$\xi_{23}$}
\end{overpic}
\caption{\textbf{Chords contributing to the differential on $\TDDD$.} Left: a \DDD\ chord
triple term. Right: a chord contributing the term $a(\xi_1)\otimes \bOne_{23}\otimes
a(-\xi_1)$ to the differential.}
\label{fig:DDD-chord-triple}
\end{figure}
\begin{definition}\label{def:DDD-chord-triple}
Given a pointed matched circle $\PMC$, let $\Chord(\PMC)$ denote the
set of chords in $\PMC$.
A \emph{\DDD\ chord triple} for $\PMC_1, \PMC_2, \PMC_3$ consists of
chords $\xi_{12}\in\Chord(\PMC_{12})$, $\xi_{23}\in\Chord(\PMC_{23})$ and
$\xi_{31}\in\Chord(\PMC_{31})$ such that
\begin{align*}
p_*^2([\xi_{23}])&=r_*p_*^2([\xi_{12}]) \qquad
p_*^1([\xi_{12}])=r_*p_*^1([\xi_{31}]) \qquad
p_*^3([\xi_{31}])=r_*p_*^3([\xi_{23}]) \\
\shortintertext{and}
n_{z'_{23}}(\xi_{23})&=n_{z'_{12}}(\xi_{12})=n_{z'_{13}}(\xi_{13}),
\end{align*}
where $z'_{ij}$ is the point where $\PMC_i$ and $-\PMC_j$ are glued
together, $n_{z'}$ denotes the local multiplicity at $z'$, and
$[\xi]$ denotes the relative homology class represented by $\xi$.
(See Definition~\ref{def:connect-sum-pmc} for the maps $p_*$ and
$r_*$.)
Let $\Chord_{\DDD}$ denote the set of \DDD\ chord triples.
\end{definition}
A \DDD\ chord triple is illustrated in
Figure~\ref{fig:DDD-chord-triple}.
Abusing terminology, we will not distinguish between a chord triple
$(\xi_{12},\xi_{23},\xi_{31})$ and the associated algebra element $a(\xi_{12})\otimes a(\xi_{23})\otimes a(\xi_{31})$ of
$\Alg(\PMC_{12})\otimes\Alg(\PMC_{23})\otimes\Alg(\PMC_{31})$.
\begin{definition} \label{def:TDDD}
The left-left-left trimodule
$\TDDD(\PMC_1,\PMC_2,\PMC_3)$ is projectively generated by the set of complementary idempotent triples
$I_{12}\otimes I_{23}\otimes I_{31}$ for $\PMC_1, \PMC_2,
\PMC_3$, i.e.,
\[
\TDDD(\PMC_1,\PMC_2,\PMC_3)=\bigoplus_{I_{12}\otimes I_{23}\otimes I_{31}}\Alg(\PMC_{12})I_{12}\otimes \Alg(\PMC_{23})I_{23}\otimes \Alg(\PMC_{31})I_{31}.
\]
Define an element $A\in \Alg(\PMC_{12})\otimes
\Alg(\PMC_{23})\otimes\Alg(\PMC_{31})$ by
\begin{multline}\label{eq:A}
A=\sum_{\xi_1\in \Chord(\PMC_1)}a(\xi_1)\otimes \bOne_{23}\otimes
a(r(\xi_1))
+\sum_{\xi_2\in \Chord(\PMC_2)}a(r(\xi_2))\otimes a(\xi_2)\otimes
\bOne_{31}\\
+\sum_{\xi_3\in \Chord(\PMC_3)}\bOne_{12}\otimes a(r(\xi_3))\otimes
a(\xi_3)
+\sum_{(\xi_{12},\xi_{23},\xi_{31})\in\Chord_{\DDD}}
a(\xi_{12})\otimes a(\xi_{23})\otimes a(\xi_{31}).
\end{multline}
(Here, $\bOne_{ij}$ denotes the unit in $\Alg(\PMC_{ij})$.)
The differential on $\TDDD(\PMC_1,\PMC_2,\PMC_3)$ is
defined by
\[
\bdy(I_{12}\otimes I_{23}\otimes I_{31}) = \sum_{ (J_{12},
J_{23}, J_{31})\in\Idem_{\DDD}} \bigl((I_{12}\otimes I_{23}\otimes
I_{31})A (J_{12}\otimes J_{23}\otimes J_{31})\bigr)
\]
and the Leibniz rule.
\end{definition}
\noindent Two terms in the differential on $\TDDD(\PMC_1,\PMC_2,\PMC_3)$ are
illustrated in Figure~\ref{fig:DDD-chord-triple}.
Section~\ref{sec:compute-TDDD} will be devoted to proving:
\begin{theorem}\label{thm:compute-TDDD} There is an isomorphism
$\CFDDDa(\THD{\PMC_1}{\PMC_2}{\PMC_3})\cong
\TDDD(\PMC_1,\PMC_2,\PMC_3)$ of $\Field$-vector spaces intertwining the trimodule structures and the operators $\bdy$.
\end{theorem}
\noindent The reason for the convoluted phrasing in Theorem~\ref{thm:compute-TDDD} is that we will not verify directly that $\bdy^2=0$ on $\TDDD$; however, this follows from Theorem~\ref{thm:compute-TDDD}:
\begin{corollary}\label{cor:TDDD-is-trimodule}
Definition~\ref{def:TDDD} defines a differential trimodule.
\end{corollary}
\begin{remark}
It is not hard to prove Corollary~\ref{cor:TDDD-is-trimodule}
directly; compare~\cite[Proposition 3.4]{LOT4}.
\end{remark}
\subsubsection{Description of \texorpdfstring{$\TDDA$}{TDDA}}
\begin{lemma}\label{lem:DDA-factor}
Any basic generator $a=a(\rhos)\in\Alg(\PMC_1\cup(-\PMC_3))$ can be factored
uniquely as a product of basic generators
$a=b\cdot a(\xi_1)\cdots a(\xi_k) \cdot c$ where $b\in
\Alg(-\PMC_3)\subset\Alg(\PMC_1\cup(-\PMC_3))$,
$c\in\Alg(\PMC_1)\subset\Alg(\PMC_1\cup(-\PMC_3))$, and each
$\xi_i$ is a chord in $\PMC_1\cup(-\PMC_3)$ with initial point in
$\PMC_1$ and terminal point in $-\PMC_3$.
\end{lemma}
\begin{proof}
Suppose that $\rhos=\{\rho_1,\dots,\rho_n\}$ is a collection of
chords in $\PMC$. Write $\rhos=\rhos'\cup \rhos''$ where
$\rhos'=\{\rho_1,\dots,\rho_m\}$ and
$\rhos''=\{\rho_{m+1},\dots,\rho_n\}$. Then $a(\rhos)=a(\rhos')\cdot
a(\rhos'')$ if the following condition is met:
\begin{itemize}
\item For each pair $(i,j)$ with $1\leq i \leq m < j\leq n$, the
terminal point of $\rho_i$ is not the initial point of $\rho_j$
and is not matched to the initial point of $\rho_j$.
\end{itemize}
Now, consider a basic generator $a(\rhos)\in
\Alg(\PMC_1\cup(-\PMC_3))$. Write $\rhos=\rhos_1\cup
\rhos_2\cup\rhos_3$ where:
\begin{itemize}
\item Each of the chords in $\rhos_1$ is completely contained in $\PMC_3$.
\item Each of the chords in $\rhos_2$ has its initial point in
$\PMC_1$ and its terminal point in $-\PMC_3$.
\item Each of the chords in $\rhos_3$ is completely contained in $-\PMC_3$.
\end{itemize}
Then, by the observation above,
\[
a(\rhos_1\cup\rhos_2\cup\rhos_3)=a(\rhos_2\cup\rhos_3)\cdot
a(\rhos_1)=a(\rhos_3)\cdot a(\rhos_2)\cdot a(\rhos_1).
\]
This proves existence of the factorization. The uniqueness statement
is clear.
\end{proof}
The following are the analogues of
Definitions~\ref{def:complementary-idems}
and~\ref{def:DDD-chord-triple} for the \DDA\ case:
\begin{definition}
Choose subsets $\SetS_i\subset [2k_i]$ for $i=1,2,3$. The
sets $\SetS_i$ specify idempotents
\begin{align*}
I_{12}&=I(\SetS_1\cup ([2k_2]\setminus \SetS_2))\in \Alg(\PMC_{12})\\
I_{23}&=I(\SetS_2\cup ([2k_3]\setminus \SetS_3))\in \Alg(\PMC_{23})\\
I_{31}&=I(\SetS_1\cup ([2k_3]\setminus \SetS_3))\in \Alg(-\PMC_{31}).
\end{align*}
We call the idempotents $I_{12}, I_{23}, I_{31}$ a \emph{\DDA\
idempotent triple} for $\PMC_1, \PMC_2, \PMC_3$. We will also
sometimes write $I_{12}\otimes I_{23}\otimes I_{31}$ for the \DDA\
idempotent triple.
Let $\Idem_{\DDA}$ denote the $\Field$-vector space spanned by the
set of \DDA\ idempotent triples. The vector space $\Idem_{\DDA}$ has
obvious left actions of the sub-rings of idempotents
$\Idem(\PMC_{12})\subset\Alg(\PMC_{12})$ and
$\Idem(\PMC_{23})\subset\Alg(\PMC_{23})$ and an obvious right
action by $\Idem(-\PMC_{31})\subset\Alg(-\PMC_{31})$.
\end{definition}
\begin{definition}
A \emph{\DDA\ chord triple} for $\PMC_1, \PMC_2, \PMC_3$ consists of
chords $\xi_{12}$ for $\PMC_{12}$, $\xi_{23}$ for
$\PMC_{23}$ and $\xi_{31}$ for $-\PMC_{31}$ such that
the following holds:
\begin{align*}
p_*^2([\xi_{23}])&=r_*p_*^2([\xi_{12}])\qquad
p_*^1([\xi_{12}])=p_*^1([\xi_{31}])\qquad
p_*^3([\xi_{31}])=p_*^3([\xi_{23}])\\
\shortintertext{and}
n_{z'_{23}}(\xi_{23})&=n_{z'_{12}}(\xi_{12})=n_{z'_{13}}(\xi_{13}).
\end{align*}
Let $\Chord_{\DDA}$ denote the set of \DDA\ chord triples.
\end{definition}
Graphically, \DDA\ chord triples look like \DDD\ chord triples
(Definition~\ref{def:DDD-chord-triple}); the difference is merely in
how we are interpreting one of the boundary components.
\begin{figure}
\centering
\begin{overpic}[tics=5]{SplitXi}
\put(4,2){$\PMC_1$}
\put(4,20){$\PMC_2$}
\put(4,38){$\PMC_3$}
\put(16,3){$-\PMC_{31}$}
\put(30,3){$\PMC_{12}$}
\put(39, 17){$\PMC_{23}$}
\put(37,27){$\otimes$}
\put(46,27){$+$}
\put(55,27){$\otimes$}
\put(64,27){$+$}
\put(73,27){$\otimes$}
\put(82,27){$+$}
\put(91,27){$\otimes$}
\put(26,46){$\xi$}
\put(36,11){$\xi_{12}$}
\put(38,46){$\xi_{23}$}
\put(25,24){\tiny split}
\end{overpic}
\caption{\textbf{The splitting operation.} We have suppressed the
sum over the idempotents; the output should in fact include a sum
over all sensible ways of adding horizontal lines.}
\label{fig:split-chord}
\end{figure}
\begin{definition}
Let $\xi$ be a chord in $\PMC_1\cup(-\PMC_3)$. Define an element
$\chordsplit(\xi)\in \Alg(\PMC_{12})\otimes \Alg(\PMC_{23})$ by:
\begin{align*}
\chordsplit(\xi)=
\sum_{\substack{(\xi_{12},\xi_{23},\xi)\in\Chord_{\DDA}\\
(I_{12},I_{23},I_{31})\in\Idem_{\DDA}\\ (J_{12},J_{23},J_{31})\in\Idem_{\DDA}\\
I_{31}a(\xi)J_{31}\neq 0}} (I_{12}a(\xi_{12})J_{12})\otimes(I_{23}a(\xi_{23})J_{23}).
\end{align*}
(See Figure~\ref{fig:split-chord}.)
\end{definition}
\begin{figure}
\centering
\includegraphics{DDA_ops}
\caption{\textbf{Action and differential on $\TDDA$.} Top: the
action by an element of the algebra $\Alg(\PMC_{31})$. Bottom: the
differential of a generator of $\TDDA$.}
\label{fig:DDA-action}
\end{figure}
Our next goal is to define the trimodule
$\TDDA(\PMC_1,\PMC_2,\PMC_3)$, which we can view as a left-right
$(\Alg(\PMC_{12})\otimes\Alg(\PMC_{23}),\
\Alg(-\PMC_{31}))$-bimodule.
\begin{definition}\label{def:module-TDDA}
As a left module,
$\TDDA(\PMC_1,\PMC_2,\PMC_3)$
is just
\[ \bigl(\Alg(\PMC_{12})\otimes_{\Field}\Alg(\PMC_{23})\bigr)\otimes_{\Idem(\PMC_{12})\otimes\Idem(\PMC_{23})}\Idem_{\DDA}
=\Alg(\PMC_{12})\otimes_{\Idem(\PMC_2)}\Alg(\PMC_{23}).
\]
The idempotents of $\Alg(-\PMC_{31})$ act on the right via their
obvious action on $\Idem_{\DDA}$. It remains to define the
differential and the right action of non-idempotent elements.
The differential on $\TDDA(\PMC_1,\PMC_2,\PMC_3)$ is defined by
\[
\bdy(I_{12},I_{23},I_{31})=\sum_{\xi_2\in\Chord(\PMC_{2})}\sum_{(J_{12},J_{23},J_{31}) \in\Idem_{\DDA}}
\bigl(I_{12}\cdot a(r(\xi_2))\cdot J_{12} \otimes I_{23}\cdot
a(\xi_2)\cdot J_{23}\bigr)\otimes (J_{12},J_{23},J_{31}).
\]
and the Leibniz rule.
By Lemma~\ref{lem:DDA-factor}, to define the right module structure
on $\TDDA$ it suffices to define the actions of $\Alg(\PMC_1)$,
$\Alg(-\PMC_3)$ and elements $a(\xi)$ where $\xi$ is a chord
starting in $\PMC_1$ and terminating in $-\PMC_3$.
Given an element $x\otimes y\in
\TDDA=\Alg(\PMC_{12})\otimes\Alg(\PMC_{23})$
and elements $a\in \Alg(\PMC_1)\subset\Alg(\PMC_{12})$, $b\in\Alg(-\PMC_3)\subset\Alg(\PMC_{23})$, define
\begin{align*}
(x\otimes y)\cdot a&:=(xa)\otimes y\\
(x\otimes y)\cdot b&:=x\otimes (yb).
\end{align*}
Given a chord $\xi$ starting in $\PMC_1$ and ending in
$-\PMC_3$, define
\[
(x\otimes y)\cdot a(\xi)=(x\otimes y)\chordsplit(\xi).
\]
\end{definition}
We will verify in Proposition~\ref{prop:TDDA-is-module} that
Definition~\ref{def:module-TDDA} defines a
trimodule. This, in turn, will be the main work in proving:
\begin{theorem}\label{thm:compute-TDDA} There is a quasi-isomorphism of $\Ainf$-trimodules
$\CFDDAa(\THD{\PMC_1}{\PMC_2}{\PMC_3})\cong
\TDDA(\PMC_1,\PMC_2,\PMC_3)$.
\end{theorem}
\noindent Note here that though $\TDDA(\PMC_1,\PMC_2,\PMC_3)$ is an honest differential trimodule, $\CFDDAa(\THD{\PMC_1}{\PMC_2}{\PMC_3})$ may be only an $\Ainf$-trimodule.
\subsection{Computation of \texorpdfstring{$\TDDD$}{TDDD}}\label{sec:compute-TDDD}
This section is devoted to proving Theorem~\ref{thm:compute-TDDD}. The
proof is an adaptation of techniques
from~\cite{LOT4}, and in this section we assume familiarity with that paper. As there, the proof has two components. First, one
uses the grading to restrict what terms can occur in the
differential. Second, one uses the fact that $\bdy^2=0$ on $\CFDDDa(\CT(\PMC_1,\PMC_2,\PMC_3)$ and a few
simple computations to show that all terms in the correct grading do,
in fact, appear.
For the first half of the argument, the paper~\cite{LOT4} has two
different approaches. When computing $\CFDDa$ of the identity
cobordism, it uses a factorization argument; for $\CFDDa$ of an
arc-slide it uses the notion of a coefficient algebra (see
Definition~\ref{def:coeff-alg}, below). (In fact, both arguments can
be made to work for both computations, but the factorization argument
for arc-slides involves a massive case analysis.) Here, we will use
the coefficient algebra approach; so, even though the diagrams look
closer to the diagram for the identity cobordism, the proof is more in
the spirit of the arc-slide argument from~\cite{LOT4}.
The following is analogous to~\cite[Definitions 3.1 and 4.3]{LOT4}:
\begin{definition}
The \emph{diagonal subalgebra $\DiagAlg$} of
$\Alg(\PMC_{12})\otimes_\Field \Alg(\PMC_{23})\otimes_\Field
\Alg(\PMC_{31})$ is the subalgebra of
$\Alg(\PMC_{12})\otimes_\Field \Alg(\PMC_{23})\otimes_\Field
\Alg(\PMC_{31})$ with basis the triples $a_{12}\otimes
a_{23}\otimes a_{31}$ of strand diagrams such that
\begin{align*}
p_*^2([a_{23}])&=r_*p_*^2([a_{12}]) \qquad\qquad
p_*^1([a_{12}])=r_*p_*^1([a_{31}]) \qquad\qquad
p_*^3([a_{31}])=r_*p_*^3([a_{23}])\\
\shortintertext{and}
n_{z'_{23}}(a_{23})&=n_{z'_{12}}(a_{12})=n_{z'_{13}}(a_{13}).
\end{align*}
(Here, $[a]$ denotes the relative homology class represented by $a$.)
\end{definition}
The relevance of the diagonal subalgebra comes from the following:
\begin{lemma}\label{lem:diff-in-diag-alg}
For each complementary idempotent triple $I_{12}\otimes
I_{23}\otimes I_{31}$ there is a unique generator
$\x=\x_{I_{12}\otimes I_{23}\otimes I_{31}}$ of
$\CFDDDa(\THD{\PMC_1}{\PMC_2}{\PMC_3})$ so that $(I_{12}\otimes
I_{23}\otimes I_{31})\cdot \x_{I_{12}\otimes I_{23}\otimes
I_{31}}=\x_{I_{12}\otimes I_{23}\otimes I_{31}}$, and
every generator of
$\CFDDDa(\THD{\PMC_1}{\PMC_2}{\PMC_3})$ arises this way.
Moreover, for
any generator $\x=\x_{I_{12}\otimes I_{23}\otimes I_{31}}$ of
$\CFDDDa(\THD{\PMC_1}{\PMC_2}{\PMC_3})$, the differential $\bdy \x$
has the form
\[
\bdy\x = \sum_\y a^{\x,\y}\otimes \y
\]
where each $a^{\x,\y}$ is an element of $\DiagAlg$.
\end{lemma}
\begin{proof}
The statement about generators is clear. For the statement about the differential,
let $B$ be a domain in $\THD{\PMC_1}{\PMC_2}{\PMC_3}$ and let
$\bdy_{ij}B$ denote the part of $\bdy B$ lying in $\PMC_{ij}$. Then
\begin{align*}
p_*^2(\bdy_{23}B)&=r_*p_*^2(\bdy_{12}B) \qquad\qquad
p_*^1(\bdy_{12}B)=r_*p_*^1(\bdy_{31}B) \qquad\qquad
p_*^3(\bdy_{31}B)=r_*p_*^3(\bdy_{23}B) \\
\shortintertext{and}
n_{z'_{23}}(B)&=n_{z'_{12}}(B)=n_{z'_{13}}(B).
\end{align*}
The result follows.
\end{proof}
To prove Theorem~\ref{thm:compute-TDDD}, we need some further
properties of $\DiagAlg$. First we explain gradings. There will be a detailed discussion of gradings in Section~\ref{sec:gradings}; for now, we need a $\ZZ$-grading $\gr_{\DiagAlg}$ on $\DiagAlg$ defined as follows. Recall that the algebra $\Alg(\PMC_{ij})$ is graded by a
group $\bigGroup(\PMC_{ij})$, which is a $\ZZ$ central extension of
$H_1(Z_{ij}\setminus\{z_{ij}\},\CircPts_{ij})$; see \cite[Section 3.3]{LOT1} or the summary in Section~\ref{sec:GradingBordered} below. We can write elements
of this group as pairs $(m;x)$ where $m\in\OneHalf\ZZ$ and $x\in
H_1(Z_{ij}\setminus\{z_{ij}\},\CircPts_{ij})$. Suppose $a_{12}\otimes
a_{23}\otimes a_{31}\in\DiagAlg$, with
$\gr(a_{ij})=(m_{ij};x_{ij})$. Then define
\begin{equation}\label{eq:gr-diag-1}
\gr_{\DiagAlg}(a_{12}\otimes a_{23}\otimes
a_{31})=m_{12}+m_{23}+m_{31}+\frac{1}{2}n_{z'_{23}}(a_{23}).
\end{equation}
(Since $n_{z'_{23}}(a_{23})=n_{z'_{12}}(a_{12})=n_{z'_{13}}(a_{13})$,
this expression is symmetric in the $ij$'s.)
The grading $\gr_{\DiagAlg}$ has a more invariant description as
follows. First, we recall~\cite[Definition 2.15]{LOT4}:
\begin{definition}\label{def:coeff-alg}
Let $M$ be a type $D$ module over a differential algebra $\Alg$, graded by a $G$-set $S$ (where $G$ is a group with distinguished central element $\lambda$).
The \emph{coefficient algebra} of $M$ is generated
over $\Field$ by triples $(\x,a,\y)$ with $\x,\y$ generators of $M$
and ``$a$" a generator of $\Alg$ satisfying:
\begin{enumerate}
\item If $a=I\cdot a\cdot J$ for basic idempotents $I$ and $J$ then
$I\cdot \x=\x$ and $J\cdot\y =\y$; and
\item\label{item:coeff-alg-2} There is a $k\in\ZZ$ so that $\lambda^k\gr(\x)=\gr(a)\gr(\y)$.
\end{enumerate}
The differential is given by $\bdy(\x,a,\y)=(\x,\bdy(a),\y)$ and the
product is given by
\[
(\x_1,a_1,\y_1)\cdot(\x_2,a_2,\y_2)=
\begin{cases}
(\x_1,a_1\cdot a_2,\y_2) & \y_1=\x_2\\
0 &\text{otherwise}.
\end{cases}
\]
The grading on the coefficient algebra is given by $\gr(\x,a,\y)=k$
where $\lambda^k\gr(\x)=\gr(a)\gr(\y)$. This is well-defined if
$\lambda$ acts freely on $S$~\cite[Lemma 2.16]{LOT4}.
This extends to type left-left \DD\ bimodules $M$ over $\Alg$ and
$\Blg$ (respectively left-left-left type \DDD\ trimodules $M$ over
$\Alg$, $\Blg$, and $\Clg$) by viewing $M$ as a module over
$\Alg\otimes \Blg$ (respectively $\Alg\otimes\Blg\otimes\Clg$).
\end{definition}
The following lemma is analogous to~\cite[Lemma 4.12]{LOT4}.
\begin{lemma}
With respect to a consistent choice of grading refinement data, the
coefficient algebra of $\CFDDDa(\THD{\PMC_1}{\PMC_2}{\PMC_3})$ is
exactly the diagonal subalgebra $\DiagAlg$.
\end{lemma}
\begin{proof}
Recall that $\Alg(\PMC)$ has a canonical grading by a group
$G'(\PMC)$ consisting of pairs $(m;a)$ with $m\in\frac{1}{2}\ZZ$ and
$a\in H_1(Z\setminus \{z\},\CircPts)$; we refer to $m$ as the
\emph{Maslov component} of the grading and $a$ as the
\emph{$\SpinC$-component} of the grading. The central element
$\lambda$ is $(1;0)$.
The trimodule
$\CFDDDa(\THD{\PMC_1}{\PMC_2}{\PMC_3})$ is graded by the
$\bigl(G'(\PMC_{12})\times_\ZZ G'(\PMC_{23})\times_\ZZ
G'(\PMC_{31})\bigr)$-set
\[
\bigl(G'(\PMC_{12})\times_\ZZ G'(\PMC_{23})\times_\ZZ
G'(\PMC_{31})\bigr)/\langle g(B)\mid B\in\pi_2(\x_0,\x_0)\rangle.
\]
Here, $g(B)=(m;\bdy^\bdy(B))$ where the $\SpinC$-component $\bdy^\bdy(B)$ is given by the multiplicities of
$B$ at $\bdy\THD{\PMC_1}{\PMC_2}{\PMC_3}$. Hence, the
condition~(\ref{item:coeff-alg-2}) in the definition of the
coefficient algebra is equivalent to $\gr(\x)$ and $\gr(a)\gr(\y)$
having the same $\SpinC$-component, up to adding the boundaries of
periodic domains.
Given generators $\x,\y\in \CFDDDa(\THD{\PMC_1}{\PMC_2}{\PMC_3})$,
the grading satisfies
\[
\gr(\y)=g(B)\gr(\x)
\]
for any $B\in\pi_2(\x,\y)$. In particular, given an algebra element
$a$, $\gr(\x)$ and $\gr(a)\gr(\y)$ have the same $\SpinC$-component
if and only if the support of $a$ is the boundary of some domain
connecting $\x$ and $\y$. Inspecting the diagram, this occurs if and
only if $a$ lies in the diagonal subalgebra.
\end{proof}
Turning to the grading on the coefficient algebra, the
following lemma and corollary are analogous to~\cite[Proposition 4.15]{LOT4}:
\begin{lemma}\label{lem:g-of-domains} Let $\x$ and $\y$ be generators
of $\CFDDDa(\THD{\PMC_1}{\PMC_2}{\PMC_3})$, and let $B\in
\pi_2(\x,\y)$ be a domain. Let $e(B)$ denote the Euler measure of
$B$ and let $n_\x(B)$ denote the point measure of $B$ with respect
to $\x$. Then
\begin{equation}\label{eq:TDDD-index}
e(B)+n_\x(B)+n_\y(B)=-\frac{1}{2}n_{z'_{23}}(B).
\end{equation}
\end{lemma}
\begin{proof}
The diagram $\THD{\PMC_1}{\PMC_2}{\PMC_3}$ has two kinds of regions:
$8$-sided regions $R_i$ running between two boundary components of
the diagram and a single $12$-sided region $T$ in the middle
touching all three boundary components. (See
Figure~\ref{fig:THD-region-labels}.) For any generators $\x$ and
$\y$, each $R_i$ has $n_\x(R_i)=n_\y(R_i)=1/2$, while
$e(R_i)=-1$. Thus, $R_i$ does not contribute to the left side of
Formula~(\ref{eq:TDDD-index}). Similarly, for any generators $\x$
and $\y$, the region $T$ has $n_\x(T)=n_\y(T)=3/4$ and
$e(T)=-2$. Thus, $T$ contributes $-1/2$ to the left side of
Formula~(\ref{eq:TDDD-index}). Of course, $n_{z'_{23}}(R_i)=0$ while
$n_{z'_{23}}(T)=1$. This proves the result.
\end{proof}
\begin{figure}
\centering
\includegraphics{THD_region_labels}
\caption{\textbf{Labeling of regions in
$\THD{\PMC_1}{\PMC_2}{\PMC_3}$.} Each $R_i$ has $8$ sides, and
$T$ has $12$ sides. The ordering of the $R_i$ is not important.}
\label{fig:THD-region-labels}
\end{figure}
\begin{corollary}\label{cor:gr-on-diag}
The $\ZZ$-grading on the coefficient algebra
$\Coeff(\CFDDDa(\THD{\PMC_1}{\PMC_2}{\PMC_3}))$ is given by
$\gr_\DiagAlg$.
\end{corollary}
\begin{proof}
Associated to each generator $a_{12}\otimes a_{23}\otimes a_{31}$ of
the diagonal algebra is a domain $B(a_{12}\otimes a_{23}\otimes
a_{31})$ in $\THD{\PMC_1}{\PMC_2}{\PMC_3}$ so that the boundary of
$B(a_{12}\otimes a_{23}\otimes a_{31})$ is the same as the support
of $a_{12}\otimes a_{23}\otimes a_{31}$. The grading of
$a_{12}\otimes a_{23}\otimes a_{31}$, viewed as an element of the
diagonal subalgebra, is the Maslov component of
\begin{equation}\label{eq:gr-diag-proof}
(\gr(a_{12})\times \gr(a_{23})\times \gr(a_{31}))\cdot g(B)^{-1}\in
G'(\PMC_{12})\times_\ZZ G'(\PMC_{23})\times_\ZZ G'(\PMC_{31}).
\end{equation}
(The $\SpinC$-component of this product is zero.) We have
\[
g(B)=(-e(B)-n_\x(B)-n_\y(B);\bdy^\bdy(B))
\]
(where $\bdy^\bdy(B)$ denotes the part of
$\bdy B$ lying in the boundary of the Heegaard diagram).
Write $B$ as a linear combination $aT + \sum_i b_iR_i$, where
$T$ is the middle, $12$-sided region in the diagram
$\THD{\PMC_1}{\PMC_2}{\PMC_3}$ and the $R_i$ are the other,
$8$-sided regions. (See Figure~\ref{fig:THD-region-labels}.)
Each of $e(B)$, $n_\x(B)$ and $n_\y(B)$ is linear. Observe that
\begin{align*}
e(R_i)+n_\x(R_i)+n_\y(R_i)&=-1+1/2+1/2=0\\
e(T)+n_\x(T)+n_\y(T)&=-2+3/4+3/4=-1/2.
\end{align*}
Thus, $g(B)=(-a/2,\bdy^\bdy(B))$. So, by Formula~\ref{eq:gr-diag-proof}, we have
\[
\gr_\DiagAlg(a_{12}\otimes a_{23}\otimes a_{31})=m_{12}+m_{23}+m_{31}+a/2,
\]
in agreement with Formula~\ref{eq:gr-diag-1}.
\end{proof}
The following lemma is analogous to~\cite[Lemmas 4.20 and 4.36]{LOT4}.
\begin{lemma}\label{lem:DDD-gr}
If $a_{12}\otimes a_{23}\otimes a_{31}$ is a basic, non-idempotent element of
$\DiagAlg$, then $\gr_\DiagAlg(a_{12}\otimes a_{23}\otimes
a_{31})=-1$ if and only if either
\begin{itemize}
\item $a_{12}\otimes a_{23}\otimes a_{31}$ is a chord triple, or
\item $a_{12}\otimes a_{23}\otimes a_{31}$ has the form
$I_{12}a(\xi_1)\otimes I_{23} \otimes I_{31}a(r(\xi_1))$, $I_{12}
a(\xi_2)\otimes I_{23}a(\xi_2)\otimes I_{31}$, or $I_{12}\otimes I_{23}
a(r(\xi_3))\otimes I_{31}a(\xi_3)$.
\end{itemize}
\end{lemma}
\begin{proof}
If $a_{12}\otimes a_{23}\otimes a_{31}$ is a chord triple then the
Maslov components of the gradings are $m_{ij}=-1/2$, and we have
\[
\gr_{\DiagAlg}(a_{12}\otimes a_{23}\otimes a_{31})=-1/2 -1/2 -1/2 +
1/2=-1.
\]
In the second case, given a chord $\xi_1$, say, we have
\[
\gr_{\DiagAlg}(I_{12}a(\xi_1)\otimes I_{23} \otimes
I_{31}a(r(\xi_1))) = -1/2 +0 -1/2 + 0=-1.
\]
Conversely, if $a_{ij}$ has $n_{ij}$ moving strands then,
by~\cite[Lemma 3.6]{LOT2}, the Maslov component of the grading of
$a_{12}\otimes a_{23}\otimes a_{31}$ is at most
$-(n_{12}+n_{23}+n_{31})/2$; and
$n_{z'_{23}}(a_{23})\leq\min\{n_{12}, n_{23}, n_{31}\}.$
Thus,
\[
\gr_{\DiagAlg}(a_{12}\otimes a_{23}\otimes a_{31})\leq \frac{1}{2}\bigl(-n_{12}-n_{23}-n_{31}+\min\{n_{12}, n_{23}, n_{31}\}\bigr).
\]
At least two of the $n_{ij}$'s are $1$ or larger. So, the only two
cases in which $\gr_{\DiagAlg}(a_{12}\otimes a_{23}\otimes
a_{31})\geq -1$ are when $\{n_{12},n_{23},n_{31}\}=\{1,1,0\}$ or
when $\{n_{12},n_{23},n_{31}\}=\{1,1,1\}$ (and
$n_{z'_{23}}(a_{23})=1$). This proves the result.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:compute-TDDD}]
The isomorphism of modules
\[
\CFDDDa(\THD{\PMC_1}{\PMC_2}{\PMC_3})\cong
\TDDD(\PMC_1,\PMC_2,\PMC_3);
\]
is clear (compare
Lemma~\ref{lem:diff-in-diag-alg}).
It remains to show that this
isomorphism entwines the differentials on the two sides. Again by
Lemma~\ref{lem:diff-in-diag-alg}, the coefficients occurring in the
differential on $\CFDDDa(\THD{\PMC_1}{\PMC_2}{\PMC_3})$ lie in the
diagonal algebra. Write $\bdy \x = \sum_\y a^{\x,\y}\otimes \y$. By
Lemma~\ref{lem:DDD-gr} (and the definition of
the coefficient algebra~\cite[Definition 2.15]{LOT4}), the basic
elements of $\DiagAlg$ occurring in $a^{\x,\y}$ are a subset of the terms in
the element $A$ (Formula~\eqref{eq:A}).
It remains to show that every term in $I_\x A I_\y$ occurs in
$a^{\x,\y}$. To keep terminology simple, we will say that a term $a$
in $A$ occurs in $a^{\x,\y}$ if either $I_\x a I_\y=0$ or $a$
occurs in $a^{\x,\y}$.
Then, since $\THD{\PMC_1}{\PMC_2}{\PMC_3}$ contains the
identity Heegaard diagram for $\PMC_i$ as a sub-diagram, it follows
from~\cite[Theorem 1]{LOT4} that all of the terms of the form
$a(\xi_1)\otimes \bOne_{23}\otimes a(r(\xi_1))$, $a(r(\xi_2))\otimes
a(\xi_2)\otimes \bOne_{31}$, and $\bOne_{12}\otimes
a(r(\xi_3))\otimes a(\xi_3)$, where $\xi_i$ is a chord in $\PMC_i$,
occur in $a^{\x,\y}$.
Any chord $\xi$ in a pointed matched circle $\PMC$ has a length
$|\xi|\in\NN$. To prove that the remaining chords occur in the
differential we proceed by induction on
$|\xi_{12}|+|\xi_{23}|+|\xi_{31}|$. The base case is the unique
chord triple $(\xi_{12}^1,\xi_{23}^1,\xi_{31}^1)$ for which each
$\xi_{ij}^1$ has length $1$. The corresponding domain in
$\THD{\PMC_1}{\PMC_2}{\PMC_3}$ is a polygon (with 12
sides). Consequently, for any compatible generators (generators whose idempotents $I_{12}\otimes I_{23}\otimes I_{31}$ and $J_{12}\otimes J_{23}\otimes J_{31}$ satisfy $(I_{12}a(\xi_{12})J_{12})\otimes (I_{23}a(\xi_{23})J_{23})\otimes (I_{31}a(\xi_{31})J_{31})\neq 0$)
this domain has a unique holomorphic representative. Thus,
$a^{\x,\y}$ contains $a(\xi_{12}^1)\otimes a(\xi_{23}^1)\otimes
a(\xi_{31}^1)$.
\begin{figure}
\centering
\includegraphics{DDD_compute}
\caption{\textbf{Inductive argument used to prove
Theorem~\ref{thm:compute-TDDD}.} The four thick arrows exist by
induction (or, in one case, the definition of the
algebra). Combined with $\bdy^2=0$, this forces the thin arrow to
exist as well.}
\label{fig:prove-thm-1}
\end{figure}
The rest of the argument is outlined in
Figure~\ref{fig:prove-thm-1}. Suppose that
$(\xi_{12},\xi_{23},\xi_{31})$ is a chord triple with
$|\xi_{12}|+|\xi_{23}|+|\xi_{31}|>3$. Without loss of generality,
assume that $\xi_{12}$ has length greater than $1$. Then there is a
point $p$ in either $\PMC_1$ or $\PMC_2$ so that:
\begin{itemize}
\item $p$ is in the interior of $\xi_{12}$ and
\item $p$ is not matched to an endpoint of $\xi_{12}$.
\end{itemize}
For definiteness, suppose $p\in \PMC_1$. Let $p'$ be the point
matched to $p$.
Suppose that $\x$ (respectively $\y$) corresponds to the
complementary idempotent triple $I=I_{12}\otimes I_{23}\otimes
I_{31}$ (respectively $J=J_{12}\otimes J_{23}\otimes J_{31}$) and
that $I\cdot (a(\xi_{12})\otimes a(\xi_{23})\otimes
a(\xi_{31}))\cdot J\neq 0$. We must show that $a^{\x,\y}$ contains
the term $I\cdot (a(\xi_{12})\otimes a(\xi_{23})\otimes
a(\xi_{31}))\cdot J$.
Either $I_{12}$ contains the matched pair $\{p,p'\}$ or $I_{31}$
contains $\{p,p'\}$. For definiteness, suppose that $I_{12}$
contains $\{p,p'\}$; the other case is similar.
The element $I_{12}a(\xi_{12})J_{12}$ has a horizontal
strand at $p$, which crosses the strand $\xi_{12}$. Let
$\{\eta,\eta'\}$ denote the two chords obtained by smoothing this
crossing, with $\eta\subset \PMC_1$ ending at $p$, and $\eta'$
running from $p$ into $\PMC_2$. Write $\xi_{31}=\zeta\cup\zeta'$
where $\zeta\cap\zeta' = p$ and $\zeta\subset\PMC_1$. Consider the algebra element
\[
I\cdot (a(\{\eta,\eta'\})\otimes a(\xi_{23})\otimes
a(\xi_{31}))\cdot J.
\]
There are exactly two ways this element can occur in $\bdy^2\x$:
\begin{align}
\x &\stackrel{\bdy}{\longrightarrow} I\cdot (a(\eta')\otimes
a(\xi_{23}) \otimes a(\zeta')) \otimes \z
\stackrel{\bdy}{\longrightarrow} I\cdot (a(\eta')\otimes
a(\xi_{23}) \otimes a(\zeta'))\cdot (a(\eta)\otimes
\bOne_{23}\otimes a(\zeta))\otimes \y\label{eq:DDD-diff-exists}\\
\x & \stackrel{\bdy}{\longrightarrow} I\cdot (a(\xi_{12})\otimes
a(\xi_{23})\otimes a(\xi_{31}))\otimes \y \stackrel{\bdy}{\longrightarrow} I\cdot (\bdy(a(\xi_{12}))\otimes
a(\xi_{23})\otimes a(\xi_{31}))\otimes \y.\label{eq:DDD-diff-want}
\end{align}
(Here, the generator $\z$ is determined uniquely.) We want to show
that the first arrow in Formula~\eqref{eq:DDD-diff-want} actually exists.
In Formula~\eqref{eq:DDD-diff-exists}, both differentials use pairs of chords
that we already proved contribute to the differential (the first by
induction and the second using~\cite[Theorem 1]{LOT4}). Thus,
$a(\xi_{12})\otimes a(\xi_{23})\otimes a(\xi_{31})$
occurs in the differential as well. This completes the proof.
\end{proof}
\begin{remark}
It is immediate from the gradings that the isomorphism of
Theorem~\ref{thm:compute-TDDD} is the only graded isomorphism
between $\CFDDDa(\THD{\PMC_1}{\PMC_2}{\PMC_3})$ and
$\TDDD(\PMC_1,\PMC_2,\PMC_3)$.
\end{remark}
\subsection{Computation of \texorpdfstring{$\TDDA$}{TDDA}}\label{sec:compute-TDDA}
\begin{proposition}\label{prop:TDDA-is-module}
$\TDDA(\PMC_1,\PMC_2,\PMC_3)$ is a differential trimodule.
\end{proposition}
\begin{proof}
This is immediate from the facts that
\[
\TDDA(\PMC_1,\PMC_2,\PMC_3)\cong \rvertme{\cornAA(\PMC_1,-\PMC_2)}{\cornAD(-\PMC_2,-\PMC_3)}{\RAlg(-\PMC_2)}{0pt}{0pt}
\]
(Proposition~\ref{prop:corn-gives-trimod}) and that $\cornAA$ and
$\cornDA$ are well-defined module-$2$-modules
(Proposition~\ref{prop:cornering-mods-defined}).
\end{proof}
\noindent (Even though
Proposition~\ref{prop:corn-gives-trimod} appears later in the text,
its proof does not depend on Proposition~\ref{prop:TDDA-is-module}.)
\begin{proposition}\label{prop:TDDA-DT-Id}
$\TDDA(\PMC_1,\PMC_2,\PMC_3)\otimes_{\Alg(\PMC_{31})} \CFDDa(\Id_{\PMC_{31}})\cong
\TDDD(\PMC_1,\PMC_2,\PMC_3).$
\end{proposition}
\begin{proof}
Recall that $\CFDDa(\Id_{\PMC_{31}})$ has one generator
$\x=\x_{I,I'}$ for each pair of complementary idempotents $I\otimes
I'$ in $\Alg(-\PMC_{31})\otimes \Alg(\PMC_{31})$, and the
differential on $\CFDDa(\Id_{\PMC_{31}})$ is given by
\[
\bdy(\x_{I,I'})=\sum_{\substack{J\otimes J'\\\text{complementary}\\\text{idempotents}}}\sum_{\xi\in\Chord(\PMC_{31})} (I\otimes I')\cdot
(a(r(\xi))\otimes a(\xi))\otimes \x_{J,J'}.
\]
So, the generators of $\TDDA(\PMC_1,\PMC_2,\PMC_3)\otimes_{\Alg(\PMC_{31})}
\CFDDa(\Id_{\PMC_{31}})$ are in bijection with complementary
idempotent triples, via the correspondence
\begin{equation}\label{eq:idem-corresp}
(I_{12}\otimes I_{23}\otimes I_{31})\otimes\x_{I_{31},I'_{31}} \longleftrightarrow
(I_{12}\otimes I_{23}\otimes I'_{31})
\end{equation}
where $I'_{31}$ is the unique idempotent so that $I_{31}\otimes
I'_{31}$ is a pair of complementary idempotents.
The differential on $\TDDA(\PMC_1,\PMC_2,\PMC_3)\otimes_{\Alg(\PMC_{31})}
\CFDDa(\Id_{\PMC_{31}})$ has four kinds of terms:
\begin{itemize}
\item Terms coming from the differential on
$\TDDA(\PMC_1,\PMC_2,\PMC_3)$. These correspond exactly to the
second sum in Formula~\eqref{eq:A}.
\item Terms in the differential on $\CFDDa(\Id_{\PMC_{31}})$ in
which the chord $\xi$ is entirely contained in $\PMC_1$. These
correspond exactly to the first sum in Formula~\eqref{eq:A}.
\item Terms in the differential on $\CFDDa(\Id_{\PMC_{31}})$ in
which the chord $\xi$ is entirely contained in $\PMC_3$. These
correspond exactly to the third sum in Formula~\eqref{eq:A}.
\item Terms in the differential on $\CFDDa(\Id_{\PMC_{31}})$ in
which the chord $\xi$ runs between $-\PMC_1$ and $\PMC_3$. With
respect to the correspondence~\eqref{eq:idem-corresp}, such terms
contribute terms $(I_{12}\otimes I_{23}\otimes I'_{31})\cdot(
\chordsplit(\xi)\otimes a(r(\xi)))\otimes (J_{12}\otimes J_{23}\otimes
J'_{31})$. But
\[
\sum_{\xi_{12}\otimes\xi_{23}\in\chordsplit(\xi)}\hspace{-1.25em}
a(\xi_{12})\otimes a(\xi_{23})\otimes a(r(\xi)) =
\sum_{\substack{\text{chord triples}\\ (\xi_{12},\xi_{23},r(\xi))}}
a(\xi_{12})\otimes a(\xi_{23})\otimes a(r(\xi)),
\]
so these terms correspond exactly to the fourth sum in
Formula~\eqref{eq:A}.
\end{itemize}
Thus, the correspondence~\eqref{eq:idem-corresp} intertwines the
differentials. This proves the result.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:compute-TDDA}]
This follows from Theorem~\ref{thm:compute-TDDD},
Proposition~\ref{prop:TDDA-is-module},
Proposition~\ref{prop:TDDA-DT-Id} and the fact that tensoring with
$\CFDDa(\Id)$ gives an equivalence of derived categories of
$\Ainf$-trimodules (cf.~\cite[Section 9]{LOT2}). (Recall that any zig-zag of $\Ainf$ quasi-isomorphisms can be replaced by a single $\Ainf$ quasi-isomorphism; see, e.g.,~\cite[Section 2.4.1]{LOT2}.)
\end{proof}
| 143,108
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\begin{document}
\title[NLS approximation for the Euler-Poisson equation]
{Justification of the NLS approximation\\ for the Euler-Poisson equation}
\author{Huimin Liu and Xueke Pu}
\address{Huimin Liu \newline
Faculty of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, P.R.China}
\email{hmliucqu@163.com}
\address{Xueke Pu \newline
School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, P.R.China} \email{puxueke@gmail.com}
\thanks{This work is supported by NSFC (11871172).}
\subjclass[2000]{35M20; 35Q35} \keywords{Modulation approximation; Nonlinear Schr\"odinger equation; Euler-Poisson equation}
\begin{abstract}
The nonlinear Schr\"{o}dinger (NLS) equation can be derived as a formal approximation equation describing the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the ion Euler-Poisson equation. In this paper, we rigorously justify such approximation by giving error estimates in Sobolev norms between exact solutions of the ion Euler-Poisson system and the formal approximation obtained via the NLS equation. The justification consists of several difficulties such as the resonances and loss of regularity, due to the quasilinearity of the problem. These difficulties are overcome by introducing normal form transformation and cutoff functions and carefully constructed energy functional of the equation.
\end{abstract}
\maketitle \numberwithin{equation}{section}
\newtheorem{proposition}{Proposition}[section]
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{hypothesis}[theorem]{Hypothesis}
\newtheorem{definition}{Definition}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{assumption}{Assumption}[section]
\section{\textbf{Introduction}}
\setcounter{section}{1}\setcounter{equation}{0}
The nonlinear Schr\"{o}dinger (NLS) equation plays important roles in describing approximately slow modulations in time and space of an underlying spatially and temporarily oscillating wave packet in more complicated systems, such as the equations describing surface water waves \cite{Z} or the Euler-Poisson system describing the motion of plasma composed of ions and isothermal electrons \cite{SI}. Indeed, even at the linearized level, there are electron waves, ion acoustic waves in the Euler-Poisson system. In the current paper, we consider the NLS approximation for the amplitude of the ion oscillation in the Euler-Poisson system
\begin{subequations}\label{equation1}
\begin{numcases}{}
\partial_{t}n+\partial_{x}(nv)=0,\\
\partial_{t}v+v\partial_{x} v=-\partial_{x}{P(n)}-\partial_{x}\phi,\\
\partial_{x}^{2}\phi=e^{\phi}-n,
\end{numcases}
\end{subequations}
where $(x, t)\in\mathbb{R}\times\Bbb R^+$, $n$ is the ion density, $v$ is the ion velocity, the self-consistent field $-\partial_{x}\phi$ satisfies the Poisson equation, {the ion pressure $P$ satisfies $P(n)=\ln n$.} Taking $\rho=n-1$ and
\begin{equation*}
\begin{split}
\begin{pmatrix} \rho \\ v \end{pmatrix}=\epsilon\Psi_{NLS}+\mathcal{O}(\epsilon^{2}),
\end{split}
\end{equation*}
with
\begin{equation}
\begin{split}\label{LS}
\epsilon\Psi_{NLS}=\epsilon A\big(\epsilon(x-c_{g}t),\epsilon^{2}t\big)e^{i(k_{0}x-\omega_{0}t)}\varphi(k_{0})+c.c.,
\end{split}
\end{equation}
the nonlinear Schr\"odinger equation (NLS) can be derived for the complex amplitude $A$,
\begin{equation}
\begin{split}\label{A}
\partial_{T}A=i\nu_{1}\partial_{X}^{2}A+i\nu_{2}A|A|^{2},
\end{split}
\end{equation}
where $T=\epsilon^{2}t\in\mathbb{R}$ is the slow time scale and $X=\epsilon(x-c_{g}t)\in\mathbb{R}$ is the slow spatial scale and coefficients $\nu_{j}=\nu_{j}(k_{0})\in \mathbb{R}$ with $j\in\{1,2\}$. In the above modulation approximation, $0<\epsilon\ll1$ is a small perturbation parameter, $\omega_{0}>0$ is the basic temporal wave number associated to the basic spatial wave number $k_{0}>0$ of the underlying temporally and spatially oscillating wave train $e^{i(k_{0}x-\omega_{0}t)}$, $c_{g}$ is the group velocity and `c.c.' denotes the complex conjugate. The NLS equation is derived in order to describe the slow modulations in time and in space of the wave train $e^{i(k_{0}x-\omega_{0}t)}$ and the time and space scales of the modulations are $\mathcal{O}(1/\epsilon^{2})$ and $\mathcal{O}(1/\epsilon)$, respectively. For the Euler-Poisson equation \eqref{equation1}, the basic spatial wave number $k=k_{0}$ and the basic temporal wave number $\omega=\omega_{0}$ are related via the following linear dispersion relation
\begin{equation}
\begin{split}\label{equation3}
\omega(k)=k\sqrt{\frac{2+k^{2}}{1+k^{2}}}=k\widehat{q}(k), \ \widehat{q}(k)=\sqrt{\frac{2+k^{2}}{1+k^{2}}}.
\end{split}
\end{equation}
From the dispersion relation, the group velocity $c_{g}=\frac{\partial w}{\partial k}(k_{0})$ of the wave packet can be found, and $\varphi(k_{0})=\big(1,-\widehat{q}(k_{0})\big)^{T}$ is an eigenvector for the linearized equation. Our ansatz leads to waves moving to the right. To obtain waves moving to the left, $-\omega_{0}$ and $c_{g}$ have to be replaced with $\omega_{0}$ and $-c_{g}$, respectively. The NLS equation is a completely integrable Hamiltonian system, which can be solved explicitly with the help of inverse scattering method (see \cite{A} for example). We also note that in the year of 1968, V.E. Zakharov \cite{Z} derived the nonlinear Schr\"{o}dinger equation from the equations of hydrodynamics for an ideal fluid with a free surface of a deep fluid.
In this paper, we will consider such a NLS approximation for the ion Euler-Poisson equation. But before we state the main result concerning such an approximation in this paper, we first make some literature review on some important aspects of the Euler-Poisson equation and the modulational approximation.
In recent years, many efforts have been made to study the global existence and approximation of solutions to the Euler-Poisson equation for ions as well as the Euler-Poisson equation for electrons. We did not mention the Euler-Poisson for electrons above, but it is worth mentioning some recent results on this equation. The Euler-Poisson equations for ions as well as electrons are both important PDE models arising in plasma physics, share some basic difficulties as other important fluid models, and are far from being well understood. Guo firstly obtained global irrotational solutions with small velocity for the 3D electron fluid, based on the Klein-Gordon effect \cite{Guo98}. For the two dimensional electron fluid in Euler-Poisson system, Ionescu and Pausader obtained the global stability of the constant equilibrium solution \cite{IP13}. Jang considered the global solution with spherical symmetry initial data \cite{Jang12}. Furthermore, Jang, Li and Zhang obtained the smooth global solutions \cite{JLZ14}. Finally, Li and Wu solved the Cauchy problem for the two dimensional electron Euler-Poisson system \cite{LW14}. Recently, Guo, Han and Zhang \cite{GHZ15} finally completely settled this problem of global existence and proved that no shocks form for the 1D Euler-Poisson system for electrons. For the Euler-Poisson equation for ions, Guo and Pausader \cite{GP11} constructed global smooth irrotational solutions with small amplitude for ion dynamics in $\Bbb R^3$. For the long wave approximation, Guo and Pu \cite{GP14} established rigorously the KdV limit for the ion Euler-Poisson system in 1D for both cold and hot plasma cases, where the electron density satisfies the classical Maxwell-Boltzmann law. This result was generalized to the higher dimensional cases, and the 2D Kadomtsev-Petviashvili-II (KP-II) equation and the 3D Zakharov-Kuznetsov (ZK) equation are derived under different scalings \cite{P}. Almost at the same time, \cite{LLS} also established the Zakharov-Kuznetsov equation in 3D from the Euler-Poisson system. Recently, the authors in the present paper \cite{LP,LP17} obtained rigorously the quantum KdV limit in 1D and the KP-I and KP-II equations in 2D for the Euler-Poisson system for cold as well as hot plasma taking quantum effect into account, where the electron equilibrium is given by a Fermi-Dirac distribution. Han-Kwan \cite{HK} also introduced a long wave scaling for the Vlasov-Poisson equation and derived the KdV equation in 1D, the KP-II equation in 2D and the Zakharov-Kuznetsov equation in 3D using the modulated energy method. These long wavelength limit results are interesting, since the oscillatory solution of the KdV equation is nothing but a solution of the NLS equation in the small wave number region with frequency induced from the dispersive relation. In this sense, derivation of the NLS equation in the current paper is a more interesting problem in the context of Euler-Poisson system \eqref{equation1}.
Showing error estimates for the NLS approximation of dispersive wave systems with quadratic terms is not a trivial task, while in the absence of quadratic terms a simple application of Gronwall's inequality yields the desired result \cite{KS}. If semilinear quadratic terms are present in the original system they can be removed by the so called normal form transformation, a near identity change of variables, if the eigenvalues of the linearized problem satisfy a non-resonance condition \cite{K}. Note that the normal form transformation was first introduced to eliminate semilinear quadratic terms in quadratic nonlinear Klein-Gordon systems to study global solutions in \cite{S85}. The normal form method has been widely used and extended in decay estimate and to extend the existence time for small solutions in some quasilinear systems in recent years. The interested readers may refer to \cite{W} for almost global existence for the 2D infinite depth full water wave equation , Germain and Masmoudi for the global existence of the 3D Euler-Maxwell system \cite{GM}, Alazard, Delort and Szeftel for the 2D gravity water waves and for the nonlinear Klein-Gordon equation \cite{AD,De,D09}, Hunter, Ifrim and Tataru for the water waves and the Burgers-Hilbert equation \cite{H,H16,IT}, Ionescu and Pusateri for the water wave system in 2D \cite{IP15,IP16,IP18}. Besides, the normal form transformation was also developed to study the long wavelength approximation and the modulational approximation of nonlinear systems. For the long wavelength limit, one can refer to D\"ull \cite{D12} and Schneider and Wayner \cite{SW,SW02} for the water wave problem. For the modulational approximation, one can refer to D\"ull \cite{D1} for the quasilinear Klein-Gordon eqution and Schneider \cite{S1998, S98} for the hyperbolic systems and the KdV equation.
The quasilinearity causes two basic problems in common in justifying the NLS approximation. One is loss of derivatives, which finally makes it not easy to close energy estimates. The other one is resonances (violating the non-resonance condition \eqref{equation1,1}), and due to the continuous spectrum of the linearized problem of many physics systems such as the ion-acoustic wave problem studied in this paper and the water wave problems, the eigenvalues are continuous functions and it is hard to avoid resonances. Thus some attempts were made to overcome these two main difficulties as well as many other difficulties induced by structures of a particular quasilinear system, to justify the NLS approximation. As far as the authors know, only a few special examples are known to work up to date. One is the quasilinear dispersive wave system in which the right-hand side only loses half a derivative, in which case the elimination of the quadratic terms is still possible with the help of a normal form transformation. In this case, the transformed system loses only one derivative in total and can be handled with the Cauchy-Kowalevskaya theorem \cite{D,S,SW}. It was realized that it is not easy to close the energy estimates once the order of total loss of derivatives is greater than one. Recently, justification of the NLS approximation for a quasilinear Klein-Gordon equation has been obtained in \cite{D1} by using the normal form transformation to define a new energy to simplify the error estimates, in which the nonlinearities lose one derivative causing that the transformed system to lose two derivatives in total. But luckily there is no resonance in such a quasilinear Klein-Gordon model \cite{D1}, i.e. the non-resonance condition is satisfied. Besides, the NLS equation has been justified for the 2D water wave problem in the special case of zero surface tension and infinite depth \cite{T} by finding a transformation adapted to this problem which allows to eliminate all quadratic terms. See also the two dimensional hyperbolic NLS approximation for the 3D water wave problem \cite{T15}. Another example is in the context of the Korteweg-de Vries equation where the result can be obtained by applying a Miura transformation \cite{S1} to eliminate the dangerous quadratic terms.
Due to the quasilinearity of the system and the dispersive relation \eqref{equation3}, we face all the following principal difficulties in justifying the NLS approximation, namely,
\begin{enumerate}
\item the quasilinearity,
\item a quadratic nonlinearity,
\item trivial resonance at the wave number $k=0$ as well as nontrivial resonance at $k=k_{0}$,
\item loss of one derivative in the quadratic nonlinear term that finally causes the transformed system to lose two derivatives.
\end{enumerate}
It is worth highlighting that the water waves problem treated by Totz-Wu \cite{T} present similar difficulties, where the authors found some special transformation adapted to their problem whereas we used the normal form transformation and modified energy functional in this paper. Precisely, to handle the quadratic nonlinearity, we may need to introduce the normal form transformation, but then the quasilinearity causes the quadratic nonlinearity to lose one derivative and eventually causes the loss of two derivatives in total in the transformed system. Therefore it is not easy to close the energy estimates. On the other hand, both trivial and non-trivial resonances occur, i.e. the non-resonance condition is not satisfied. {Hence, the validity of the NLS approximation to the ion Euler-Poisson equation is far from a trivial problem.} In this paper, we will consider such an interesting problem and justify such a modulation approximation to the Euler-Poisson equation for ions \eqref{equation1}.
The main result of this paper is the following
\begin{theorem}\label{Thm1}
Fix $s_{A}\geq6$. Then for all $k_{0}\neq0$ and for all $C_{1}, \ T_{0}>0$, there exist $C_{2}>0, \ \epsilon_{0}>0$ such that for all solutions $A\in C([0,T_{0}],H^{s_{A}}(\mathbb{R},\mathbb{C}))$ of the NLS equation \eqref{A} with
\begin{equation*}
\begin{split}
\sup_{T\in[0,T_{0}]}\big\|A(\cdot,T)\big\|_{H^{s_{A}}}(\mathbb{R},\mathbb{C})\leq C_{1},
\end{split}
\end{equation*}
the following holds. For all $\epsilon\in(0,\epsilon_{0})$, there are solutions
\begin{equation*}
\begin{split}
\begin{pmatrix} n-1 \\ v \end{pmatrix}\in \Big(C\big([0,T_{0}/\epsilon^{2}],H^{s_{A}}(\mathbb{R},\mathbb{R})\big)\Big)^{2},
\end{split}
\end{equation*}
of the ion Euler-Poisson equation \eqref{equation1} that satisfy
\begin{equation*}
\begin{split}
\sup_{t\in[0,T_{0}/\epsilon^{2}]}\Big\|\begin{pmatrix} n-1 \\ v \end{pmatrix}-\epsilon\Psi_{NLS}(\cdot,t)\Big\|_{H^{s_{A}}(\mathbb{R},\mathbb{R})^{2}}
\leq C_{2}\epsilon^{3/2},
\end{split}
\end{equation*}
\end{theorem}
\noindent where $\varphi(k_{0})=\big(1,-\widehat{q}(k_{0})\big)^{T}$ is given in the definition of $\varepsilon\Psi_{NLS}$, $\widehat{q}$ comes from the equation \eqref{equation3}.
\begin{remark}
First, compared with the solution $(n-1,v)$ and the approximation $\epsilon\Psi_{NLS}$, which are both of order $\mathcal{O}(\epsilon)$ in $L^{\infty}$, the error of order $\mathcal{O}(\epsilon^{3/2})$ is small enough such that the dynamics of the NLS equation can be found in the ion Euler-Poisson system \eqref{equation1}. Secondly, we note that the Fourier transform of $\epsilon\Psi_{NLS}$ is sufficiently strongly concentrated around the wave numbers $\pm k_{0}$, hence by using a modified approximation that has compact support in Fourier space but differs only slightly from $\epsilon\Psi_{NLS}$, the smoothness of the error bound can be made equal to the assumed smoothness of the amplitude. Finally, in the following proof of Theorem \ref{Thm1}, we always assume that $s_A$ is an integer to simplify the proof, although it can be generalized to all real numbers $s_A\geq6$.
\end{remark}
\begin{remark}
In the momentum equation for ion-Euler-Poisson equation \eqref{equation1}, we choose the ion pressure $P=\ln n$. Indeed, the result in this paper can be generalized to the general $\gamma$-law of the ion pressure $P$, i.e., when $P(n)=n^{\gamma}$ for $\gamma\geq1$.
\end{remark}
We would like to end the introduction by giving the structure of this paper here. {In Section 2 we derive the NLS equation formally, and then we estimate the terms that remain after inserting the approximation into \eqref{equation6}. In Section 3 we outline the basic ideas for the diagonalized system \eqref{equation7} of the Euler-Poisson equation for ions when justifying the NLS approximation.} In Section 4 we perform the normal form transformation that is invertible and does not lose any derivatives for $|k|\leq \delta$, and then we present the transformed error equations for $(\mathcal{R}^{0},R^{1})$. In Section 5 we construct our energy and perform the error estimates to prove Theorem \ref{Thm1}.
\textbf{Notation}. We denote the Fourier transform of a function $u\in L^{2}(\mathbb{R},\mathbb{K})$, with $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$ by
\begin{equation*}
\begin{split}
\widehat{u}(k)=\frac{1}{2\pi}\int_{\mathbb{R}}u(x)e^{-ikx}dx.
\end{split}
\end{equation*}
Let $H^{s}(\mathbb{R},\mathbb{K})$ be the space of functions mapping from $\mathbb{R}$ into $\mathbb{K}$ for which the norm
\begin{equation*}
\begin{split}
\|u\|_{H^{s}(\mathbb{R},\mathbb{K})}=\Big(\int_{\mathbb{R}}|\widehat{u}(k)|^{2}(1+|k|^{2})^{s}dk\Big)^{1/2}
\end{split}
\end{equation*}
is finite. Usually we write $L^{2}$ and $H^{s}$ instead of $L^{2}(\mathbb{R},\mathbb{R})$ and $H^{s}(\mathbb{R},\mathbb{R})$. We use the space $L^{p}(m)(\mathbb{R},\mathbb{K})$ defined by $u\in L^{p}(m)(\mathbb{R},\mathbb{K})$ such that $\sigma^{m}u\in L^{p}(\mathbb{R},\mathbb{K})$, where $\sigma(x)=(1+x^{2})^{1/2}$. Finally, we write $A\lesssim B$, if $A\leq CB$ for a constant $C>0$, and $A=\mathcal{O}(B)$, if $|A|\lesssim B$.
\section{\textbf{Derivation of the NLS approximation and estimates for the residual}}
NLS type equation has been given formally for the one dimensional motion of plasma composed of cold ions and isothermal electrons in \cite{SI}. In the following we will obatin NLS equation for the Euler-Poisson equation \eqref{equation1}.
In order to isolate \eqref{equation1} into linear, quadratic and higher order terms, we can rewrite the normalized Euler-Poisson system \eqref{equation1} as follows,
\begin{subequations}\label{equation4}
\begin{numcases}{}
\partial_{t}\rho+\partial_{x}v+\partial_{x}(\rho v)=0,\\
\partial_{t}v+\partial_{x}\rho+\partial_{x}\phi+v\partial_{x} v-\partial_{x}\frac{\rho^{2}}{2}=-\partial_{x}\big[\ln(1+\rho)-\rho+\frac{\rho^{2}}{2}\big],\\
\rho=(1-\partial_{x}^{2})\phi+\frac{\phi^{2}}{2}+\big[e^{\phi}-1-\phi-\frac{\phi^{2}}{2}\big].
\end{numcases}
\end{subequations}
For small $\rho$, the last line defines an inverse operator $\rho\mapsto\phi(\rho)$. We further expand this inverse operator up to third order as
\begin{equation}
\begin{split}\label{equation5}
\phi(\rho)=(1-\partial_{x}^{2})^{-1}\rho-\frac{1}{2}(1-\partial_{x}^{2})^{-1}\big[(1-\partial_{x}^{2})^{-1}\rho\big]^{2}+\mathcal{M}(\rho),
\end{split}
\end{equation}
where $\mathcal{M}$ satisfies some good properties. We note that $v\partial_{x}v=\partial_{x}({v^{2}}/{2})$ and we can rewrite the above system \eqref{equation4} as
\begin{equation}
\begin{split}\label{equation6}
\partial_{t}&\begin{pmatrix}\rho\\ v\end{pmatrix}+\begin{pmatrix}0&\partial_{x}\\ \partial_{x}(1-\partial_{x}^{2})^{-1}+\partial_{x}&0\end{pmatrix}\begin{pmatrix}\rho\\ v\end{pmatrix}\\
=&\begin{pmatrix}-\partial_{x}(\rho v)\\-\partial_{x}\frac{v^{2}}{2}+\partial_{x}\frac{\rho^{2}}{2}
+\frac{1}{2}\partial_{x}(1-\partial_{x}^{2})^{-1}\big[(1-\partial_{x}^{2})^{-1}\rho\big]^{2}-\partial_{x}\mathcal{M}(\rho)
-\partial_{x}\big(\ln(1+\rho)-\rho+\frac{\rho^{2}}{2}\big)\end{pmatrix}.
\end{split}
\end{equation}
Let $S=\begin{pmatrix}1& 1\\-q(|\partial_{x}|)& q(|\partial_{x}|)\end{pmatrix}$ and $\begin{pmatrix}\rho\\ v\end{pmatrix}=S\begin{pmatrix}U_{1}\\ U_{-1}\end{pmatrix}$.
{We define the operator $\Omega$ by $\widehat{\Omega u}(k)=i\omega(k)\widehat{u}(k)$.} We can diagonalize the linear part of the equation \eqref{equation6} as
\begin{equation}
\begin{split}\label{equation7}
\partial_{t}U_{j}=j\Omega U_{j}+Q_{j}(U,U)+N_{j},
\end{split}
\end{equation}
where $j\in\{1,-1\}$ and the quadratic term $Q_{j}$ and the high order term $N_{j}$ take the form
\begin{equation}
\begin{split}\label{equation8}
Q_{j}&=-\frac{\partial_{x}(\rho v)}{2}-j\frac{\partial_{x}}{4q(|\partial_{x}|)}\big[-|v|^{2}+\rho^{2}+(1-\partial_{x}^{2})^{-1}[(1-\partial_{x}^{2})^{-1}\rho]^{2}\big],\\
N_{j}&=j\frac{\partial_{x}}{2q(|\partial_{x}|)}\big[\ln(1+\rho)-\rho+\frac{\rho^{2}}{2}+\mathcal{M}(\rho)\big].
\end{split}
\end{equation}
Plugging $\rho=U_{1}+U_{-1}$ and $v=-q(|\partial_{x}|)(U_{1}-U_{-1})$ into $Q_{j}$ and $N_{j}$, we now compute the Fourier transform of $U_{j}$ as
\begin{equation}
\begin{split}\label{equation9}
\partial_{t}\widehat{U}_{j}
=&ij\omega(k)\widehat{U}_{j}
+\frac{ik}{2}\int_{\mathbb{R}}\widehat{q}(m)\big(\widehat{U}_{1}(k-m)
+\widehat{U}_{-1}(k-m)\big)\big(\widehat{U}_{1}(m)-\widehat{U}_{-1}(m)\big)dm\\
&+\frac{ijk}{4\widehat{q}(k)}\int\widehat{q}(k-m)\widehat{q}(m)\big(\widehat{U}_{1}(k-m)
-\widehat{U}_{-1}(k-m)\big)\big(\widehat{U}_{1}(m)-\widehat{U}_{-1}(m)\big)dm\\
&-\frac{ijk}{4\widehat{q}(k)}\int\big(1+\frac{1}{\langle k\rangle^{2}}\frac{1}{\langle k-m\rangle^{2}}\frac{1}{\langle m\rangle^{2}}\big)\big(\widehat{U}_{1}(k-m)+\widehat{U}_{-1}(k-m)\big)\big(\widehat{U}_{1}(m)+\widehat{U}_{-1}(m)\big)dm\\
&+\frac{ijk}{2\widehat{q}(k)}\sum_{n\geq3}(-1)^{n+1}\frac{1}{n}(\widehat{U}_{1}+\widehat{U}_{-1})^{\ast n}
+\frac{ijk}{2\widehat{q}(k)}\mathcal{M}(\widehat{U}_{1}+\widehat{U}_{-1})(k),
\end{split}
\end{equation}
with $j\in\{1,-1\}$ and $\langle k\rangle=\sqrt{(1+k^{2})}$. In order to derive the NLS equation as an approximation equation for system \eqref{equation9}, we make the ansatz
\begin{equation}
\begin{split}\label{equation10}
\begin{pmatrix} U_{1} \\ U_{-1} \end{pmatrix}=\epsilon\widetilde{\Psi}
=\epsilon\widetilde{\Psi}_{1}+\epsilon\widetilde{\Psi}_{-1}+\epsilon^{2}\widetilde{\Psi}_{0}
+\epsilon^{2}\widetilde{\Psi}_{2}+\epsilon^{2}\widetilde{\Psi}_{-2},
\end{split}
\end{equation}
with
\begin{equation*}
\begin{split}
&\epsilon\widetilde{\Psi}_{\pm1}=\epsilon\widetilde{A}_{\pm1}\big(\epsilon(x-c_{g}t),\epsilon^{2}t\big)E^{\pm1}\begin{pmatrix} 1 \\ 0 \end{pmatrix},\\
&\epsilon^{2}\widetilde{\Psi}_{0}=\begin{pmatrix}\epsilon^{2}\widetilde{A}_{01}\big(\epsilon(x-c_{g}t),\epsilon^{2}t\big) \\ \epsilon^{2}\widetilde{A}_{02}\big(\epsilon(x-c_{g}t),\epsilon^{2}t\big)\end{pmatrix},\\
&\epsilon^{2}\widetilde{\Psi}_{\pm2}=\begin{pmatrix}\epsilon^{2}\widetilde{A}_{(\pm2)1}\big(\epsilon(x-c_{g}t),\epsilon^{2}t\big)E^{\pm2} \\ \epsilon^{2}\widetilde{A}_{(\pm2)2}\big(\epsilon(x-c_{g}t),\epsilon^{2}t\big)E^{\pm2}\end{pmatrix},
\end{split}
\end{equation*}
where $0<\epsilon\ll1$, $E^{\pm j}=e^{\pm ij(k_{0}x-\omega_{0}t)}$, $\omega_{0}=\omega(k_{0})$, $\widetilde{A}_{-j}=\overline{\widetilde{A}}_{j}$ and $\widetilde{A}_{-j\ell}=\overline{\widetilde{A}}_{j\ell}$. The ansatz leads to waves that move to the right. If one replaces in the above ansatz the vector $(1,0)^{T}$ by $ (0,1)^{T}$, $-\omega_{0}$ by $\omega_{0}$ and $c_{g}$ by $-c_{g}$, it leads to waves that move to the left.
We insert our ansatz \eqref{equation10} into the system \eqref{equation9} and then replace the dispersion relation $\omega=\omega(k)$ in all terms of the form $\omega\widetilde{A}_{j}E^{j}$ or $\omega\widetilde{A}_{j\ell}E^{j}$ by their Taylor expansions around $k=jk_{0}$. By equating the coefficients of the $\epsilon^{m}E^{j}$ to zero, we find that the coefficients of $\epsilon , \ \epsilon^{2} E^{0}$ and $\epsilon^{2} E^{1}$ vanish identically due to the definition of $\omega$ and $c_{g}$.
For $\epsilon^{2} E^{2}$ we obtain
\begin{equation*}
\begin{split}
\big(-2\omega_{0}+\omega(2k_{0})\big)\widetilde{A}_{21}=\gamma_{21}\big(\widetilde{A}_{1}\big)^{2},\\
\big(-2\omega_{0}-\omega(2k_{0})\big)\widetilde{A}_{22}=\gamma_{22}\big(\widetilde{A}_{1}\big)^{2},
\end{split}
\end{equation*}
where the coefficients $\gamma_{2\ell}\in\mathbb{R}$ for $\ell=1,2$. From the explicit form of $\omega(k)$, we see $-2\omega_{0}-\omega(2k_{0})\neq0$, and hence $\widetilde{A}_{2\ell}$ are well-defined in terms of $(\widetilde{A}_{1})^{2}$.
For $\epsilon^{3}E^{0}$ we obtain
\begin{equation*}
\begin{split}
\big(c_{g}-(\partial_{k}\omega)(0)\big)\partial_{X}\widetilde{A}_{01}=\gamma_{31}\partial_{X}(\widetilde{A}_{1}\widetilde{A}_{-1}),\\
\big(c_{g}+(\partial_{k}\omega)(0)\big)\partial_{X}\widetilde{A}_{02}=\gamma_{32}\partial_{X}(\widetilde{A}_{1}\widetilde{A}_{-1}),
\end{split}
\end{equation*}
where the coefficients $\gamma_{3\ell}\in\mathbb{R}$. Recall $c_{g}=(\partial_{k}\omega)(k_{0})$, then $c_{g}\pm(\partial_{k}\omega)(0)\neq0$, and the $\widetilde{A}_{0\ell}$ are well-defined in terms of $\widetilde{A}_{1}\widetilde{A}_{-1}$.
For $\epsilon^{3}E^{1}$ we obtain
\begin{equation*}
\begin{split}
\partial_{T}\widetilde{A}_{1}=\frac{i}{2}\partial_{k}^{2}\omega(k_{0})\partial_{X}^{2}\widetilde{A}_{1}+g_{1},
\end{split}
\end{equation*}
where $g_{1}$ is a sum of multiples of $\widetilde{A}_{1}\widetilde{A}_{0\ell}$ and $\widetilde{A}_{-1}\widetilde{A}_{2\ell}$. From the above steps we obtain algebraic relations such that $\widetilde{A}_{0\ell}$ and $\widetilde{A}_{2\ell}$ can be expressed in terms of $\widetilde{A}_{1}\widetilde{A}_{-1}$ and $(\widetilde{A}_{1})^{2}$, respectively. Eliminating $\widetilde{A}_{0\ell}$ and $\widetilde{A}_{2\ell}$, then gives the NLS equation
\begin{equation}
\begin{split}\label{NLS}
\partial_{T}\widetilde{A}_{1}=i\frac{\partial_{k}^{2}\omega(k_{0})}{2}\partial_{X}^{2}\widetilde{A}_{1}
+i\nu_{2}(k_{0})\widetilde{A}_{1}\big|\widetilde{A}_{1}\big|^{2},
\end{split}
\end{equation}
with some $\nu_{2}(k_{0})\in\mathbb{R}$.
Considering the residual
\begin{equation}
\begin{split}\label{equat11}
Res_{U}(\epsilon\widetilde{\Psi})=\begin{pmatrix} Res_{U_{1}}(\epsilon\widetilde{\Psi}) \\ Res_{U_{-1}}(\epsilon\widetilde{\Psi}) \end{pmatrix},
\end{split}
\end{equation}
which contains all terms that do not cancel after inserting ansatz \eqref{equation10} into system \eqref{equation9}. To prove the approximation property of the NLS equation \eqref{NLS}, the resdual term $Res_{U}(\epsilon\widetilde{\Psi})$ needs to be small enough such that the error term $R$ is of order $\mathcal{O}(1)$ for the time scaling $\mathcal{O}(1/\epsilon^{2})$. We take the following standard way to modify $\epsilon\widetilde{\Psi}$ as $\epsilon\Psi$ such that the resdual term $Res_{U}(\epsilon\Psi)$ small enough. Firstly, extend the above approximation $\epsilon\widetilde{\Psi}$ to its higher order terms. Secondly, restrict the modified approximation in Fourier space to small neighborhoods of a finite number of integer multiples of the basic wave number $k_0>0$, by some cutoff function. By such a modification, the approximation will not change too much, but will lead to a simpler control of the error and make the approximation an analytic function.
Since $\pm\omega(mk_{0})\neq\pm m\omega(k_{0})$ for all integers $m\geq2$, we can proceed analogously as in \cite{D} to replace $\epsilon\widetilde{\Psi}$
by a new approximation $\epsilon\Psi$ of the form
\begin{equation}
\begin{split}\label{equation122}
\epsilon\Psi=\epsilon\Psi_{1}+\epsilon\Psi_{-1}+\epsilon^{2}\Psi_{p},
\end{split}
\end{equation}
where
\begin{align*}
\epsilon\Psi_{\pm1}=&\epsilon\psi_{\pm1}\begin{pmatrix} 1 \\ 0 \end{pmatrix} =\epsilon A_{\pm1}\big(\epsilon(x-c_{g}t),\epsilon^{2}t\big)E^{\pm1}\begin{pmatrix} 1 \\ 0 \end{pmatrix},\\
\epsilon^{2}\Psi_{p}=&\begin{pmatrix} \epsilon^{2}\psi_{p_{1}} \\ \epsilon^{2}\psi_{p_{-1}} \end{pmatrix}
=\epsilon^{2}\Psi_{0}+\epsilon^{2}\Psi_{2}+\epsilon^{2}\Psi_{-2}+\epsilon^{2}\Psi_{h},\\
\epsilon^{2}\Psi_{0}=&\begin{pmatrix} \epsilon^{2}\psi_{01} \\ \epsilon^{2}\psi_{02} \end{pmatrix}
=\begin{pmatrix} \epsilon^{2}A_{01}(\epsilon(x-c_{g}t),\epsilon^{2}t) \\ \epsilon^{2}A_{02}\big(\epsilon(x-c_{g}t),\epsilon^{2}t\big) \end{pmatrix} ,\\
\epsilon^{2}\Psi_{\pm2}=&\begin{pmatrix} \epsilon^{2}\psi_{(\pm2)1} \\ \epsilon^{2}\psi_{(\pm2)2} \end{pmatrix}
=\begin{pmatrix} \epsilon^{2}A_{(\pm2)1}\big(\epsilon(x-c_{g}t),\epsilon^{2}t\big)E^{\pm2}
\\ \epsilon^{2}A_{(\pm2)2}\big(\epsilon(x-c_{g}t),\epsilon^{2}t\big) E^{\pm2} \end{pmatrix} ,\\
\epsilon^{2}\Psi_{h}=&\sum_{j=\pm1, n=1,2,3}\begin{pmatrix} \epsilon^{1+n}A_{j1}^{n}\big(\epsilon(x-c_{g}t),\epsilon^{2}t\big)E^{j}
\\ \epsilon^{1+n}A_{j2}^{n}\big(\epsilon(x-c_{g}t),\epsilon^{2}t\big) E^{j} \end{pmatrix} \\
&+\sum_{j=\pm2, n=1,2}\begin{pmatrix} \epsilon^{2+n}A_{j1}^{n}\big(\epsilon(x-c_{g}t),\epsilon^{2}t\big)E^{j}
\\ \epsilon^{2+n}A_{j2}^{n}\big(\epsilon(x-c_{g}t),\epsilon^{2}t\big) E^{j} \end{pmatrix} \\
&+\sum_{n=1,2}\begin{pmatrix} \epsilon^{2+n}A_{01}^{n}\big(\epsilon(x-c_{g}t),\epsilon^{2}t\big)
\\ \epsilon^{2+n}A_{02}^{n}\big(\epsilon(x-c_{g}t),\epsilon^{2}t\big) \end{pmatrix} \\
&+\sum_{j=\pm3, n=0,1}\begin{pmatrix} \epsilon^{3+n}A_{j1}^{n}\big(\epsilon(x-c_{g}t),\epsilon^{2}t\big)E^{j}
\\ \epsilon^{3+n}A_{j2}^{n}\big(\epsilon(x-c_{g}t),\epsilon^{2}t\big) E^{j} \end{pmatrix} \\
&+\sum_{j=\pm4}\begin{pmatrix} \epsilon^{4}A_{j1}\big(\epsilon(x-c_{g}t),\epsilon^{2}t\big)E^{j}
\\ \epsilon^{4}A_{j2}\big(\epsilon(x-c_{g}t),\epsilon^{2}t\big) E^{j} \end{pmatrix} ,
\end{align*}
where $A_{-j}=\overline{A}_{j}$ and $A_{-j\ell}=\overline{A}_{j\ell}$ have compact support in Fourier space for all $0<\epsilon\ll1$. Then, exactly as in Section 2 of \cite{D}, the following estimates hold for the modified residual.
\begin{lemma}\label{L4}
Let $s_{A}\geq6$ and $\widetilde{A}_{1}\in C\big([0,T_{0}],H^{s_{A}}(\mathbb{R},\mathbb{C})\big)$ be a solution of the NLS equation \eqref{NLS} with
\begin{equation*}
\begin{split}
\sup_{T\in[0,T_{0}]}\big\|\widetilde{A}_{1}\big\|_{H^{s_{A}}}\leq C_{A},
\end{split}
\end{equation*}
then for all $s\geq0$, there exist $C_{Res},C_{\Psi},\varepsilon_{0}>0$ depending on $C_{A}$ such that the following holds for all $\varepsilon\in(0,\varepsilon_{0})$. The approximation $\epsilon\Psi$ defined in \eqref{equation122} exists for all $t\in[0,T_{0}/\epsilon^{2}]$ and satisfies
\begin{subequations}\label{equation12}
\begin{numcases}{}
\sup_{t\in[0,T_{0}/\epsilon^{2}]}\big\|Res_{U}(\epsilon\Psi)\big\|_{H^{s}}\leq C_{Res}\epsilon^{9/2},\label{equation12-1}\\
\sup_{t\in[0,T_{0}/\epsilon^{2}]}\big\|S(\epsilon\Psi)-\epsilon\Psi_{NLS}\big\|_{H^{s_{A}}}\leq C_{\Psi}\varepsilon^{3/2},\label{equation12-2}\\
\sup_{t\in[0,T_{0}/\epsilon^{2}]}\Big(\big\|\widehat{\Psi}_{\pm1}\big\|_{L^{1}(s+1)(\mathbb{R},\mathbb{C})}
+\big\|\widehat{\Psi}_{p}\big\|_{L^{1}(s+1)(\mathbb{R},\mathbb{C})}\Big)\leq C_{\Psi}.\label{equation12-3}
\end{numcases}
\end{subequations}
\end{lemma}
The proof of Lemma \ref{L4} is analogous to that of Lemma 2.6 in \cite{D} (see also \cite{D1}). In fact, the first and the third estimates are valid for appropriate constants $C_{Res}$ and $C_{\Psi}$ for all $s>0$, this is a consequence of the fact that our approximation $\epsilon\Psi$ has compact support in Fourier space. Besides, the approximation $\epsilon\Psi$ differs so slightly from the actual NLS approximation $\epsilon(\widetilde{\Psi}_{1}+\widetilde{\Psi}_{-1})$ and higher order asymptotic expansions of the exact solution, which are needed to make the residual $Res_{U}(\epsilon\Psi)$ sufficiently small, that the bounds in \eqref{equation12-1} and \eqref{equation12-2} hold if $s_{A}\geq6$. This is shown by using the estimate
\begin{equation*}
\begin{split}
\big\|(\chi_{[-\delta,\delta]-1})\epsilon^{-1}\widehat{f}(\epsilon^{-1}\cdot)\big\|_{L^{2}(m)}\leq C\epsilon^{m+M-1/2}\|f\|_{H^{m+M}},
\end{split}
\end{equation*}
for all $m,M\geq0$, where $\chi_{[-\delta,\delta]}$ is the characteristic function on $[-\delta,\delta]$.
The bound \eqref{equation12-3} will be used to estimate
\begin{equation*}
\begin{split}
\|\psi_{j}f\|_{H^{s}}\leq C\|\psi_{j}\|_{C_{b}^{s}}\|f\|_{H^{s}}\leq C\big\|\widehat{\psi}_{j}\big\|_{L^{1}(s)(\mathbb{R},\mathbb{C})}\|f\|_{H^{s}},
\end{split}
\end{equation*}
without loss of powers in $\epsilon$ as it would be the case with $\|\psi_{j}\|_{L^{2}(s)(\mathbb{R},\mathbb{C})}$. Moreover, by an analogous argument as in the proof of Lemma 3.3 in \cite{D}, we have
\begin{lemma}\label{L5}
For all $s\geq0$ there exists a constant $C_{\psi}>0$ such that
\begin{equation}
\begin{split}\label{equation13}
\big\|\partial_{t}\widehat{\psi}_{\pm1}+i\omega\widehat{\psi}_{\pm1}\big\|_{L^{1}(s)}\leq C_{\psi}\epsilon^{2}.
\end{split}
\end{equation}
\end{lemma}
\section{\textbf{The basic ideas}}
In order to explain our method to prove Theorem \ref{Thm1}, we use the diagonalized system \eqref{equation7} of the {normalized Euler-Poisson equation \eqref{equation1},}
\begin{equation}
\begin{split}\label{abstract}
\partial_{t}U=\Lambda U+Q(U,U)+N(U),
\end{split}
\end{equation}
where $Q(U,U)$ and $N(U)$ are given by \eqref{equation8}. Recalling that $U=U(x,t)\in\mathbb{R}^{2}$, $x\in\Bbb R$, $t\in\mathbb{R}^+$, $\Lambda$ being a operator whose symbol is a diagonal matrix of the form
\begin{equation*}
\begin{split}
\widehat{\Lambda}(k)=diag\{i\omega(k),-i\omega(k)\}.
\end{split}
\end{equation*}
We write the $j_{1}$-th component of $Q(U,U)$ as
\begin{equation*}
\begin{split}
\widehat{Q}_{j_{1}}(U,U)
=\sum_{j_{2},j_{3}=\pm1}\int \widehat{\eta}_{j_{2},j_{3}}^{j_{1}}(k,k-m,m)\widehat{U}_{j_{2}}(k-m)\widehat{U}_{j_{3}}(m)dm,
\end{split}
\end{equation*}
where $j_{1}\in\{\pm1\}$ and $\widehat{\eta}_{j_{2},j_{3}}^{j_{1}}(k,k-m,m)$ is the kernel function of $\widehat{Q}_{j_{1}}(U,U)$. Let $\widehat{\widetilde{\eta}}_{j_{2},j_{3}}^{j_{1}}(k,k-m,m)$ be the kernel function of $\widehat{N}_{j_{1}}(U)$, according to the equation \eqref{equation9} in Fourier form, we have
\begin{equation}
\begin{split}\label{quu}
\big|\widehat{\eta}_{j_{2},j_{3}}^{j_{1}}(k,k-m,m)\big|, \ \big|\widehat{\widetilde{\eta}}_{j_{2},j_{3}}^{j_{1}}(k,k-m,m)\big|\leq C |k|,
\end{split}
\end{equation}
for any $k,m\in\mathbb{R}$.
We know that $U$ is formally approximated by $\epsilon\widetilde{\Psi}$ according to the Section 2, i.e. the residual
\begin{equation}
\begin{split}\label{resU}
Res(U)=-\partial_{t}U+\Lambda U+Q(U,U)+N(U),
\end{split}
\end{equation}
is small for $U=\epsilon\widetilde{\Psi}$. And the residual can be made arbitrarily small by modifying the formal approximation $\epsilon\widetilde{\Psi}$ (refer to the equation \eqref{equation122} and the Lemma \ref{L4}), i.e. for all $\gamma>0$ there exists a formal approximation $\epsilon\Psi$ that has compact support set comparing to the approximation $\epsilon\widetilde{\Psi}$,
such that
\begin{equation*}
\begin{split}
Res_{U}(\epsilon\Psi)=\mathcal{O}(\epsilon^{\gamma}).
\end{split}
\end{equation*}
{As in Section 2, the modified approximation $\epsilon\Psi$ satisfies}
\begin{equation}
\begin{split}\label{Phic}
&\epsilon\Psi=\epsilon\Psi_{c}+\epsilon^{2}\Psi_{p},\\
&\Psi_{c}:=\Psi_{1}+\Psi_{-1}=(\psi_{1}+\psi_{-1},0)^{\top}=:(\phi_{c},0)^{\top},
\Psi_{p}=(\psi_{p_{1}},\psi_{p_{-1}})^{\top},\\
&\text{\text{supp}}\widehat{\psi}_{\pm1}=\big\{k\mid|k\pm k_{0}|\leq\delta\big\},\\
&\text{supp}\widehat{\psi}_{p_{\pm1}}=\big\{k\mid|k\pm jk_{0}|\leq\delta, \ j=0,\pm2,\pm3,\pm4\big\},
\end{split}
\end{equation}
and
\begin{equation*}
\begin{split}
\epsilon\Psi-\epsilon\widetilde{\Psi}=\mathcal{O}(\epsilon^{2}),
\end{split}
\end{equation*}
where $\delta>0$ sufficiently small, but independent of $0<\epsilon\ll1$.
In order to prove Theorem \ref{Thm1} we have to estimate the error
\begin{equation}
\begin{split}\label{before}
\epsilon^{\beta}R=U-\epsilon\Psi,
\end{split}
\end{equation}
to be of order $\mathcal{O}(\epsilon^{\beta})$ for some $\beta>1$ on a time scale $t\in[0,T_{0}/\epsilon^{2}]$, i.e. we have to prove that $R$ is of order $\mathcal{O}(1)$ for all $t\in[0,T_{0}/\epsilon^{2}]$. Inserting \eqref{before} into \eqref{abstract} we find that the error $R$ satisfies
\begin{equation}
\begin{split}\label{error}
\partial_{t}R=&\Lambda R+2\epsilon Q(\Psi_{c},R)+2\epsilon^{2} Q(\Psi_{p},R)+\epsilon^{\beta} Q(R,R)+\epsilon^{2}N(R)+\epsilon^{-\beta}Res_{U}(\epsilon\Psi).
\end{split}
\end{equation}
For our equation, the linear operator $\Lambda$ generates a uniformly bounded semigroup. If $\beta>2$, which we assume henceforth, the terms $\epsilon^{2} Q(\Psi_{p},R)$, $\epsilon^{\beta} Q(R,R)$ and $\epsilon^{2}N(R)$ can be controlled over the relevant time interval. Also, the residual term $\epsilon^{-\beta}Res_{U}(\epsilon\Psi)$ can be made of $\mathcal{O}(\epsilon^{2})$ by choosing the approximation $\epsilon\Psi$ appropriately. However, the remaining linear term $\epsilon Q(\Psi_{c},R)$ can perturb the linear evolution such that the solutions begin to grow on time scales $\mathcal{O}(\epsilon^{-1})$ and hence we would lose all control over the size of $R$ on the desired time scale $\mathcal{O}(\epsilon^{-2})$.
To show that the error remains small over the desired time intervals $\mathcal{O}(\epsilon^{-2})$, we need to eliminate the quadratic term $2\epsilon Q(\Psi_{c},R)$ from \eqref{error} via a normal form transformation. That is to say, we make a change of dependent variable of the form
\begin{equation}
\begin{split}\label{equation45}
\widetilde{R}_{j_{1}}:=R_{j_{1}}+\epsilon \sum_{j_{2}\in\{\pm1\}}B_{j_{1},j_{2}}(\Psi_{c},R_{j_{2}}), \ j_{1}\in\{\pm1\},
\end{split}
\end{equation}
where
\begin{equation}
\begin{split}\label{equation46}
\widehat{B}_{j_{1}j_{2}}(\Psi_{c},R_{j_{2}})=\int \widehat{b}_{j_{1},j_{2}}(k,k-\ell,\ell)\widehat{\phi}_{c}(k-\ell)\widehat{R}_{j_{2}}(\ell)d\ell,
\end{split}
\end{equation}
where we have used the equation \eqref{Phic} and $\widehat{R}_{j_{2}}$ refers to the $j_{2}$ component of $\widehat{R}$. Careful calculations show that the kernel function $\widehat{b}_{j_{1},j_{2}}$ of the normal form transformation can be written as a quotient whose denominator is
\begin{equation}
\begin{split}\label{15'}
-j_{1}\omega(k)-\omega(k-\ell)+j_{2}\omega(\ell).
\end{split}
\end{equation}
As long as the denominator remains away from zero, such a normal form transformation is well defined. That is to say, a non-resonance condition has to be satisfied:
\begin{equation}
\begin{split}\label{equation1,1}
|j_{1}\omega(k)+\omega(k_{0})-j_{2}\omega(k-k_{0})|>0,
\end{split}
\end{equation}
for $j_{1},j_{2}\in\{\pm1\}$ and for all $k\in \mathbb{R}$ uniformly. It is easy to see that $\omega(k)=k\widehat{q}(k)$ in \eqref{equation3} for the ion Euler-Poisson equation does not satisfy \eqref{equation1,1}. In particular, there is a resonance at the wave number $k=0$ (whenever $j_2=-1$), which is trivial and eventually causes no problems for the definition of the normal form transformation because the nonlinear term also vanishes linearly at $k=0$ and hence $\widehat{B}_{j_{1}j_{2}}$ of \eqref{equation46} can be well defined for all $|k|\leq\delta$. However, there is always another resonance for the wave number $k=k_{0}$ (for $j_1=-1$), which turns out to be nontrivial. Therefore, we can not take the normal form method of \cite{Z} directly. For this, we introduce a suitable rescaling of the error function $R$ dependent on the wave number, and then to use a number of special normal form transformations to treat such a nontrivial resonance, as did in \cite{S,S98}.
More precisely, we scale the variable $R$ to reflect the fact that the nonlinearity vanishes at $k=0$. For some $\delta>0$ above sufficiently small, but independent of $0<\epsilon\ll1$, define a weight function $\vartheta$ by its Fourier transform
\begin{equation}
\begin{split}\label{v}
\widehat{\vartheta}(k)=\Big\{\begin{matrix} 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{for} \ \ |k|>\delta, \\ \epsilon+(1-\epsilon)| k|/\delta \ \ \ \ \ \ \ \ \ \ \text{for} \ \ |k|\leq\delta.\end{matrix}
\end{split}
\end{equation}
This makes $\widehat{\vartheta}(k)\widehat{R}(k)$ small at the wave numbers close to zero to reflect the fact that the nonlinearity vanishes at $k=0$.
Rewrite the solution $U$ of \eqref{before} as a sum of the approximation and error, i.e.,
\begin{equation}
\begin{split}\label{equation48}
U=\epsilon\Psi+\epsilon^{\beta}\vartheta R,
\end{split}
\end{equation}
with a $\beta>2$. Here and hereafter, we define $\vartheta R$ by $\widehat{\vartheta R}=\widehat{\vartheta} \widehat{R}$ to avoid writing the convolution $\vartheta\ast R$.
Inserting \eqref{equation48} into \eqref{abstract}, we find that the error $R$ satisfies
\begin{equation}
\begin{split}\label{equation49}
\partial_{t}R=& \Lambda R+2\epsilon \vartheta^{-1}Q(\Psi_{c},\vartheta R)+2\epsilon^{2}\vartheta^{-1}Q(\Psi_{p},\vartheta R)+\epsilon^{\beta} \vartheta^{-1}Q(\vartheta R, \vartheta R)\\
&+\epsilon^{2}\vartheta^{-1}N(\vartheta R)+\epsilon^{-\beta}\vartheta^{-1}Res_{U}(\epsilon\Psi).
\end{split}
\end{equation}
Due to the inequality \eqref{quu} and $\vartheta^{-1}$ is at most of order $\mathcal{O}(1/\epsilon)$, we find that the terms $\epsilon^{2}\vartheta^{-1}Q(\Psi_{p},\vartheta R)$ and $\epsilon^{2}\vartheta^{-1}N(\vartheta R)$ are at least of order $\mathcal{O}(\epsilon^{2})$. Therefore, all terms on the RHS of \eqref{equation49} are at least of $\mathcal{O}(\epsilon^{2})$ except for the linear term $2\epsilon \vartheta^{-1}Q(\Psi_{c},\vartheta R)$. We also note that the term $\Lambda R$ can be explicitly computed and causes no growth in $R$.
Besides, note that whether $k$ is close to or far away from zero not will directly influences the size of the nonlinear term in \eqref{equation49} in Fourier space due to the derivative acting on it. Hence to separate the behavior in these two regions, we define projection operators $P^{0}$ and $P^{1}$ by the Fourier multiplier
\begin{equation}
\begin{split}\label{equation58}
\widehat{P}^{0}(k)=\chi_{\mid k\mid\leq\delta}(k)\ \ \ \text{and} \ \ \ \widehat{P}^{1}(k)=\mathbf{1}-\widehat{P}^{0}(k),
\end{split}
\end{equation}
for a $\delta>0$ sufficiently small (the same constant $\delta$ in the definition of $\vartheta$), but independent of $0<\epsilon\ll1$. When necessary we will write $R=R^{0}+R^{1}$ with $R^{j}=P^{j}R$, for $j=0,1$.
Acting the projection operators $P^{0}$ and $P^{1}$ on the equation \eqref{equation49}, we will obtain the following evolutionary equations for $R^{0}$ and $R^{1}$,
\begin{equation}
\begin{split}\label{R0}
\partial_{t}R^{0}=\Lambda R^{0}+2\epsilon P^{0}\vartheta^{-1} Q(\Psi_{c},\vartheta R^{0})+2\epsilon P^{0}\vartheta^{-1} Q(\Psi_{c},\vartheta R^{1})+\mathcal{O}(\epsilon^{2}),
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}\label{R1,1}
\partial_{t}R^{1}=\Lambda R^{1}+2\epsilon P^{1}\vartheta^{-1} Q(\Psi_{c},\vartheta_{0} R^{0})+2\epsilon P^{1}\vartheta^{-1} Q(\Psi_{c},\vartheta R^{1})+\mathcal{O}(\epsilon^{2}),
\end{split}
\end{equation}
where $\vartheta_{0}=\vartheta-\epsilon$. Noting that since $\widehat{\Psi}_{c}(k-m)=0$ unless $|(k-m)\pm k_{0}|<\delta$ and $\widehat{R}^{0}(m)=0$ for $|m|>\delta$, we see that $P^{0}\vartheta^{-1} Q(\Psi_{c},\vartheta R^{0})=0$. We want to estimate the error $(R^{0},R^{1})$ on a time scale $\mathcal{O}(1/\epsilon^{2})$, thus we need to eliminate the $\mathcal{O}(\epsilon)$ terms from \eqref{R0} and \eqref{R1,1} by using the normal form transformation
\begin{equation}
\begin{split}\label{B01}
\widetilde{R}_{j_{1}}^{0}:=R_{j_{1}}^{0}+\epsilon \sum_{j_{2}\in\{\pm1\}}B^{0,1}_{j_{1},j_{2}}(\Psi_{c},R_{j_{2}}^{1}),
\end{split}
\end{equation}
\begin{equation}
\begin{split}\label{B1a}
\widetilde{R}_{j_{1}}^{1}:=R_{j_{1}}^{1}+\epsilon \sum_{j_{2}\in\{\pm1\}}\big(B_{j_{1},j_{2}}^{1,0}(\Psi_{c},R_{j_{2}}^{0})+ B_{j_{1},j_{2}}^{1,1}(\Psi_{c},R_{j_{2}}^{1})\big),
\end{split}
\end{equation}
where
\begin{equation*}
\begin{split}
B^{0,1}_{j_{1},j_{2}}(\Psi_{c},R_{j_{2}}^{1})=\int_{\mathbb{R}}\widehat{b}^{0,1}_{j_{1},j_{2}}(k,k-m,m)\widehat{\phi}_{c}(k-m)\widehat{R}_{j_{2}}^{1}(m)dm.
\end{split}
\end{equation*}
Here $j_{1},j_{2}\in\{\pm1\}$ and we have used the equation \eqref{Phic}. Similarly for $B^{1,0}(\Psi_{c},R^{0})$ and $B^{1,1}(\Psi_{c},R^{1})$.
Inserting \eqref{B01} and \eqref{B1a} into \eqref{R0} and \eqref{R1,1} respectively, and then letting the $\mathcal{O}(\epsilon)$ terms equal to zero formally, we will obtain
\begin{equation}
\begin{split}\label{equation111}
&\partial_{t}\widetilde{R}^{0}=\Lambda\widetilde{R}^{0}+\epsilon^{2}f(\Psi,\widetilde{R})+\epsilon^{-\beta}Res_{U^{0}}(\epsilon\Psi),\\
&\partial_{t}\widetilde{R}^{1}=\Lambda\widetilde{R}^{1}+\epsilon^{2}g(\Psi,\widetilde{R})+\epsilon^{-\beta}Res_{U^{1}}(\epsilon\Psi),
\end{split}
\end{equation}
provided these normal form transformations are all invertible and well-defined.
Fortunately, by careful analysis we find that these normal form transformations are all invertible and well-defined in the Euler-Poisson system considered in the present paper. Summarizing, by introducing the cutoff function $\vartheta$ and in particular $\vartheta_{0}$, both the trivial resonance $k=0$ and the non-trivial resonances $k=\pm k_{0}$ do not cause problems for the definition of the normal form transformations.
Besides, the normal form transformation $B^{1,0}(\Psi_{c},\widetilde{R}^{0})$ satisfies
\begin{equation*}
\begin{split}
\|\epsilon B^{1,0}_{j_{1},j_{2}}(\Psi_{c},\widetilde{R}_{j_{2}}^{0})\|_{H^{s'}}\lesssim\epsilon\|\widetilde{R}^{0}\|_{H^{s}},
\end{split}
\end{equation*}
for any $s,s'\geq6$.
However, there still exist some difficulties in order to obtain uniform estimates for the remainder $\widetilde{R}$ in \eqref{equation111}. First, though the trivial resonance $k=0$ associated to $B^{0,1}(\Psi_{c},R^{1})$ does not cause difficulties, $B^{0,1}(\Psi_{c},R^{1})$ will lose one $\epsilon$ because $\vartheta^{-1}=\mathcal{O}(1/\epsilon)$ for $|k|<\delta$, i.e. for arbitrary $s,s'\geq6$, we have
\begin{equation*}
\begin{split}
\|\vartheta B^{0,1}_{j_{1},j_{2}}(\Psi_{c},\widetilde{R}_{j_{2}}^{1})\|_{H^{s'}}\lesssim\|\widetilde{R}^{1}\|_{H^{s}}.
\end{split}
\end{equation*}
This means that $\epsilon^{2}f(\Psi,\widetilde{R})$ is indeed of order $\mathcal{O}(\epsilon)$, although it looks like $\mathcal{O}(\epsilon^2)$. Thus we still need to eliminate such an $\mathcal{O}(\epsilon)$ term in the first equation of \eqref{equation111} by a second normal form transformation but applied only to $\widetilde{R}^0$ to obtain a new error function $\mathcal{R}^{0}$, which together with $\widetilde{R}^{1}$ satisfies
\begin{subequations}\label{R01}
\begin{numcases}{}
\partial_{t}\mathcal{R}^{0}=\Lambda\mathcal{R}^{0}+\epsilon^{2}\widetilde{f}(\Psi,\mathcal{R}^{0},\widetilde{R}^{1})+\epsilon^{-\beta}Res_{U^{0}}(\epsilon\Psi),\label{R01-1}\\
\partial_{t}\widetilde{R}^{1}=\Lambda\widetilde{R}^{1}+\epsilon^{2}\widetilde{g}(\Psi,\mathcal{R}^{0},\widetilde{R}^{1})+\epsilon^{-\beta}Res_{U^{1}}(\epsilon\Psi),
\label{R01-2}
\end{numcases}
\end{subequations}
where $\widetilde{f}(\Psi,\mathcal{R}^{0},\widetilde{R}^{1})$ is of order $\mathcal{O}(1)$ and does not lose derivative. Traditionally, one would then consider the energy estimates starting from \eqref{R01}, since all the terms on the right are of order $\mathcal{O}(\epsilon^2)$. However this will cause the following second problem.
Secondly, we can not use the variation of constants formula and Gronwall's inequality to bound $(\mathcal{R}^{0},\widetilde{R}^{1})$ for the equation \eqref{R01}, not because of missing powers of $\epsilon$ but due to regularity problems. Note that the quadratic term $\epsilon\vartheta^{-1}P^{1}Q(\Psi_{c},\vartheta R^{1})$ of \eqref{R1,1} is quasilinear and loses one derivative, i.e., $R^{1}\mapsto\epsilon\vartheta^{-1}P^{1}Q(\Psi_{c},\vartheta R^{1})$ maps $H^{m+1}(\mathbb{R},\mathbb{C})$ into $H^{m}(\mathbb{R},\mathbb{C})$ or $C^{m+1}(\mathbb{R},\mathbb{C})$ into $C^{m}(\mathbb{R},\mathbb{C})$. This causes the term $B^{1,1}(\Psi_{c},R^{1})$ to lose one derivative, which implies that the term $\widetilde{g}(\Psi,\mathcal{R}^{0},\widetilde{R}^{1})$ loses even two derivatives in total.
In order to overcome the above two difficulties at the same time, we still make the normal form transformation twice on the error function $R^{0}$ but not on $R^{1}$. By a similar procedure, we obtain the evolutionary equations for $(\mathcal{R}^{0}, R^{1})$,
\begin{subequations}\label{R01,10}
\begin{numcases}{}
\partial_{t}\mathcal{R}^{0}=\Lambda\mathcal{R}^{0}+\epsilon^{2}\overline{f}(\Psi,\mathcal{R}^{0},R^{1})
+\epsilon^{-\beta}Res_{U^{0}}(\epsilon\Psi),\label{R01,10-1}\\
\partial_{t}R^{1}=\Lambda R^{1}+\epsilon\overline{g}(\Psi,\mathcal{R}^{0},R^{1})+\epsilon^{-\beta}Res_{U^{1}}(\epsilon\Psi),\label{R01,10-2}
\label{R01-2}
\end{numcases}
\end{subequations}
where $\overline{f}$ and $\overline{g}$ do not lose $\epsilon$ but $\overline{g}$ may only lose one derivative. Then we use the transformed remainder $\mathcal{R}^{0}$ of \eqref{R01,10-1} for low frequency $|k|\leq\delta$, and the non-transformed remainder $R^{1}$ of \eqref{R01,10-2} for high frequency $|k|\geq\delta$ combined with $\epsilon B^{1,0}(\Psi,\mathcal{R}^{0})$ and $\epsilon B^{1,1}(\Psi,R^{1})$ to define the energy
\begin{equation}
\begin{split}\label{Es}
\mathcal{E}_{s}=&\sum_{\ell=0}^{s}\Big[\frac{1}{2}\big(\int_{\mathbb{R}}(\partial_{x}^{\ell}\mathcal{R}^{0})^{2}dx+
\int_{\mathbb{R}}(\partial_{x}^{\ell}R^{1})^{2}dx\big)\\
&+\epsilon\big(
\int_{\mathbb{R}}\partial_{x}^{\ell}R^{1}\partial_{x}^{\ell}B^{1,0}(\Psi_{c},\mathcal{R}^{0})dx
+\int_{\mathbb{R}}\partial_{x}^{\ell}R^{1}\partial_{x}^{\ell}B^{1,1}(\Psi_{c},R^{1})dx\big)\Big],
\end{split}
\end{equation}
where $s=s_{A}\geq6$. Although there is an $\mathcal O(\epsilon)$ term $\epsilon\bar g(\cdot,\cdot,\cdot)$ in \eqref{R01,10-2}, we can eliminate the $\mathcal O(\epsilon)$ terms and keep only the $\mathcal O(\epsilon^2)$ {terms in the evolutionary equation of $\mathcal E_s$} by the carefully constructed energy functional $\mathcal E_s$ in \eqref{Es}. {The strategy of definition of the energy using the normal form transformation was already used in previous papers \cite{D1,D12,H,H16,IT,SW,SW02}.} But here we would like to remark that there are some basic differences in this paper.
Let us explain why we use $(\mathcal R^0,R^1)$ to construct the energy functional $\mathcal E_s$. On one hand, for the high frequency component $R^1$, if we start form \eqref{R01}, although both the evolution equation of $\mathcal{E}_{s}$ and $\|(\mathcal{R}^{0},\widetilde{R}^{1})\|_{H^{s}}^{2}$ in \eqref{R01} share the property that their right-hand side terms are all of order $\mathcal{O}(\epsilon^{2})$, $\tilde g(\cdot,\cdot,\cdot)$ loses two derivatives in total, which will leads to the difficulty for closing energy estimate of $\widetilde{R}^1$. On the other hand, the method is not unique to deal with the low frequency component $R^0$. In this paper, we do twice normal-form translations for $R^0$ to obtain the equation for $\mathcal R^0$. However, since the normal-form leading from $R^0$ to $\mathcal R^0$ only involves bounded frequencies, then we can also put $R^0$ with the related normal-form transforms in the energy argument $\mathcal E_s$ as done for $R^1$.
We can show $\|B^{1,0}(\Psi_{c},\mathcal{R}^{0})\|_{H^{s}}\lesssim\|\mathcal{R}^{0}\|_{H^{s}}$ in Lemma \ref{L10} as well as the equivalence between $\|(\mathcal{R}^{0},R^{1})\|_{H^{s}}^{2}$ and $\|(R^{0},R^{1})\|_{H^{s}}^{2}$ by the form of the normal form transformations for $|k|\leq\delta$. See \eqref{equation92} for details. Besides, $B^{1,1}(\Psi_{c},R^{1})$ can be split into a term of the form $diag\big\{h_{1}(\Psi),h_{2}(\Psi)\big\}\partial_{x}R^{1}$ and terms that do not lose regularity in Lemma \ref{L8}, which is very important to obtain closed energy estimates. Then by integration by parts and using the inequality $\|B^{1,0}(\Psi_{c},\mathcal{R}^{0})\|_{H^{s}}\lesssim\|\mathcal{R}^{0}\|_{H^{s}}$, we can obtain the equivalence between ${\mathcal{E}_{s}}$ and $\|(\mathcal{R}^{0},R^{1})\|_{H^{s}}^{2}$ for sufficiently small $\epsilon$, which finally yields the equivalence between the energy $\mathcal E_s$ and the original remainder $\|(R^{0},R^{1})\|_{H^{s}}^{2}$ in \eqref{before}.
Finally, the structure of $\Lambda$ and the properties of $\omega$ allow us to construct a modified energy
\begin{equation}
\begin{split}\label{Ess}
\widetilde{\mathcal{E}}_{s}=\mathcal{E}_{s}+\epsilon^{2}h,
\end{split}
\end{equation}
where $h=\mathcal{O}\big(\|(\mathcal{R}^{0},{R}^{1})\|_{H^{s}}^{2}\big)$ as long as $\|(\mathcal{R}^{0},{R}^{1})\|_{H^{s}}$ is $\mathcal{O}(1)$. Consequently, we obtain
\begin{equation*}
\begin{split}
\partial_{t}\widetilde{\mathcal{E}}_{s}\leq C\epsilon^{2}(\widetilde{\mathcal{E}}_{s}+1),
\end{split}
\end{equation*}
as long as $\|(\mathcal{R}^{0},{R}^{1})\|_{H^{s}}$ is $\mathcal{O}(1)$ such that Gronwall's inequality yields the $\mathcal{O}(1)$ boundedness of $\widetilde{\mathcal{E}}_{s}$ and hence of $R$ for all $t\in[0,T_{0}/\epsilon^{2}]$.
\section{\textbf{The normal form transformation}}
As mentioned in Section 3, we need to eliminate the $\mathcal{O}(\epsilon)$ terms in the error equation for $R^{0}$ (i.e. low frequency $|k|<\delta$) by normal form transformations. We define a weight function $\vartheta$ to reflect the fact that the nonlinearity vanishes at $k=0$,
\begin{equation}
\begin{split}\label{va}
\widehat{\vartheta}(k)=\Big\{\begin{matrix} 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{for} \ \ |k|>\delta, \\ \epsilon+(1-\epsilon)| k|/\delta \ \ \ \ \ \ \ \ \ \ \text{for} \ \ |k|\leq\delta,\end{matrix}
\end{split}
\end{equation}
for some $\delta>0$ sufficiently small, but independent of $0<\epsilon\ll1$,
write the solution $U$ of \eqref{equation9} as a sum of the approximation and the error, i.e.,
\begin{equation}
\begin{split}\label{equat48}
U=\epsilon\Psi+\epsilon^{5/2}\vartheta R,
\end{split}
\end{equation}
where to avoid writing the convolution $\vartheta\ast R$, $\vartheta R$ is defined by $\widehat{\vartheta R}=\widehat{\vartheta} \widehat{R}$ in a slight abuse of notation. Note that $\widehat{\vartheta}(k) \widehat{R}(k)$ is small at the wave numbers close to zero, since the nonlinearity vanishes at $k=0$.
Then we define two projection operators $P^{0}$ and $P^{1}$ by the Fourier multiplier
\begin{equation}
\begin{split}\label{equation58}
\widehat{P}^{0}(k)=\chi_{\mid k\mid\leq\delta}(k)\ \ \ and \ \ \ \widehat{P}^{1}(k)=\mathbf{1}-\widehat{P}^{0}(k),
\end{split}
\end{equation}
for $\delta>0$ sufficiently small (the same constant $\delta$ in the definition of $\vartheta$), but independent of $0<\epsilon\ll1$. When necessary we will write $R=R^{0}+R^{1}$ with $R^{j}=P^{j}R$, for $j=0,1$. In the following, the superscripts $0,1$ always denote the spectrum projections, and should not confused with the subscripts that denote the component of $R$.
Recall the form of $\Psi$ in the equation \eqref{equation122}. For simplicity, let
\begin{equation*}
\begin{split}
\phi_{c}:=\psi_{1}+\psi_{-1}, \ \phi_{p_{1}}:=\psi_{p1}+\psi_{p-1}, \ \phi_{p_{2}}:=\psi_{p1}-\psi_{p-1}.
\end{split}
\end{equation*}
Then we have $\text{supp}\widehat{\phi}_{c}=\big\{k\mid|k\pm k_{0}|<\delta\big\}$ and $\text{supp}\widehat{\phi}_{p_{1,2}}=\big\{k\mid|k\pm jk_{0}|<\delta,j=0,\pm2,\pm3,\pm4\big\}$. Besides,
noting that since $\widehat{\phi}_{c}(k-m)=0$ unless $|(k-m)\pm k_{0}|<\delta$ and since $\widehat{R}_{j}^{0}(m)=0$ for $|m|>\delta$, we see that $P^{0}(\phi_{c}R_{j}^{0})=0$ for $j=\pm1$.
Inserting \eqref{equat48} into \eqref{equation9} and by the projection operators $P^{0}$ and $P^{1}$, we obtain evolutionary equations for $R^{0}$ and $R^{1}$ in Fourier transformation with $j_{1}\in\{\pm1\}$:
\begin{align}\label{equ1}
\partial_{t}\widehat{R}_{j_{1}}^{0}=&ij_{1}\omega(k)\widehat{R}_{j_{1}}^{0}
+\epsilon\frac{\widehat{P}^{0}(k)ik}{2\widehat{\vartheta}(k)}\widehat{\phi}_{c}\ast \big(\widehat{q}\widehat{\vartheta}(\widehat{R}_{1}^{1}-\widehat{R}_{-1}^{1})\big)
+\epsilon\frac{\widehat{P}^{0}(k)ik}{2\widehat{\vartheta}(k)}(\widehat{q}\widehat{\phi}_{c})\ast \big(\widehat{\vartheta}(\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1})\big)\nonumber\\
&+j_{1}\epsilon\frac{\widehat{P}^{0}(k)ik}{2\widehat{\vartheta}(k)\widehat{q}(k)}(\widehat{q}\widehat{\phi}_{c})\ast \big(\widehat{q}\widehat{\vartheta}(\widehat{R}_{1}^{1}-\widehat{R}_{-1}^{1})\big)
-j_{1}\epsilon\frac{\widehat{P}^{0}(k)ik}{2\widehat{\vartheta}(k)\widehat{q}(k)}(\widehat{\phi}_{c})
\ast\big(\widehat{\vartheta}(\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1})\big)\nonumber\\
&-j_{1}\epsilon\frac{\widehat{P}^{0}(k)ik}{2\widehat{\vartheta}(k)\langle k\rangle^{2}\widehat{q}(k)}\big(\langle\widehat{\partial_{x}}\rangle^{-2}\widehat{\phi}_{c}\big)\ast \big(\langle\widehat{\partial_{x}}\rangle^{-2}\widehat{\vartheta}(\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1})\big)\nonumber\\
&+\epsilon^{2}\frac{\widehat{P}^{0}(k)ik}{2\widehat{\vartheta}(k)}\widehat{\phi}_{p_{1}}\ast\big(\widehat{q}\widehat{\vartheta}
(\widehat{R}_{1}^{0}-\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}-\widehat{R}_{-1}^{1})\big)\nonumber\\
&+j_{1}\epsilon^{2}\frac{\widehat{P}^{0}(k)ik}{2\widehat{\vartheta}(k)\widehat{q}(k)}\widehat{q}\widehat{\phi}_{p_{2}}\ast\big(\widehat{q}\widehat{\vartheta}
(\widehat{R}_{1}^{0}-\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}-\widehat{R}_{-1}^{1})\big)\nonumber\\
&+\epsilon^{2}\frac{\widehat{P}^{0}(k)ik}{2\widehat{\vartheta}(k)}\widehat{q}\widehat{\phi}_{p_{2}}\ast\big(\widehat{\vartheta}
(\widehat{R}_{1}^{0}+\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1})\big)\nonumber\\
&-j_{1}\epsilon^{2}\frac{\widehat{P}^{0}(k)ik}{2\widehat{\vartheta}(k)\widehat{q}(k)}(\widehat{\phi}_{p_{1}})\ast\big(\widehat{\vartheta}
(\widehat{R}_{1}^{0}+\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1})\big)\nonumber\\
&-j_{1}\epsilon^{2}\frac{\widehat{P}^{0}(k)ik}{2\widehat{\vartheta}(k)\langle k\rangle^{2}\widehat{q}(k)}\big(\langle\widehat{\partial_{x}}\rangle^{-2}\widehat{\phi}_{p_{1}}\big)\ast \big(\langle\widehat{\partial_{x}}\rangle^{-2}\widehat{\vartheta}(\widehat{R}_{1}^{0}+\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1})\big)\nonumber\\
&+\epsilon^{5/2}\frac{\widehat{P}^{0}(k)ik}{2\widehat{\vartheta}(k)}
\big(\widehat{\vartheta}(\widehat{R}_{1}^{0}+\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1})\big)\ast
\widehat{q}\widehat{\vartheta}\big(\widehat{R}_{1}^{0}-\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}-\widehat{R}_{-1}^{1})\big)\nonumber\\
&-j_{1}\epsilon^{5/2}\frac{\widehat{P}^{0}(k)ik}{4\widehat{\vartheta}(k)\widehat{q}(k)}\big(\widehat{\vartheta}
(\widehat{R}_{1}^{0}+\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1})\big)^{\ast^{2}}\nonumber\\
&-j_{1}\epsilon^{5/2}\frac{\widehat{P}^{0}(k)ik}{4\widehat{\vartheta}(k)\langle k\rangle^{2}\widehat{q}(k)}\big(\langle\widehat{\partial_{x}}\rangle^{-2}\widehat{\vartheta}
(\widehat{R}_{1}^{0}+\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1})\big)^{\ast^{2}}\nonumber\\
&+j_{1}\epsilon^{5/2}\frac{\widehat{P}^{0}(k)ik}{4\widehat{\vartheta}(k)\widehat{q}(k)}\big(\widehat{q}\widehat{\vartheta}
(\widehat{R}_{1}^{0}+\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1})\big)^{\ast^{2}}\nonumber\\
&+j_{1}\epsilon^{2}\frac{\widehat{P}^{0}(k)ik}{2\widehat{\vartheta}(k)\widehat{q}(k)}\widehat{\mathcal{G}}
\widehat{\vartheta}\big(\widehat{R}_{1}^{0}+\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1}\big)
+j_{1}\epsilon^{2}\frac{\widehat{P}^{0}(k)ik}{2\widehat{\vartheta}(k)\widehat{q}(k)}\widehat{\mathcal{M}}
\widehat{\vartheta}\big(\widehat{R}_{1}^{0}+\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1}\big)\nonumber\\
&+\epsilon^{-5/2}\widehat{Res}_{U_{j_{1}}^{0}(\epsilon\Psi)}\nonumber\\
=&:ij_{1}\omega(k)\widehat{R}_{j_{1}}^{0}+\epsilon\frac{\widehat{P}^{0}(k)ik}{2\widehat{\vartheta}(k)}
\sum_{j_{2}\in\{\pm1\}}\sum_{n=1}^{5}\widehat{\alpha}_{j_{1},j_{2}}^{n}(k,k-m,m)\widehat{\phi}_{c}\ast \widehat{\vartheta}\widehat{R}_{j_{2}}^{1}
+\epsilon^{2}\mathcal{F}^{1},
\end{align}
and
\begin{align}\label{equ1,-1}
\partial_{t}\widehat{R}_{j_{1}}^{1}=&ij_{1}\omega(k)\widehat{R}_{j_{1}}^{1}
+\epsilon\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)}\widehat{\phi}_{c}\ast \big(\widehat{q}\widehat{\vartheta}(\widehat{R}_{1}^{0}-\widehat{R}_{-1}^{0})\big)
+\epsilon\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)}(\widehat{q}\widehat{\phi}_{c})\ast \big(\widehat{\vartheta}(\widehat{R}_{1}^{0}+\widehat{R}_{-1}^{0})\big)\nonumber\\
&+j_{1}\epsilon\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)\widehat{q}(k)}(\widehat{q}\widehat{\phi}_{c})\ast \big(\widehat{q}\widehat{\vartheta}(\widehat{R}_{1}^{0}-\widehat{R}_{-1}^{0})\big)
-j_{1}\epsilon\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)\widehat{q}(k)}(\widehat{\phi}_{c})\ast
\big(\widehat{\vartheta}(\widehat{R}_{1}^{0}+\widehat{R}_{-1}^{0})\big)\nonumber\\
&-j_{1}\epsilon\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)\langle k\rangle^{2}\widehat{q}(k)}\big(\langle\widehat{\partial_{x}}\rangle^{-2}\widehat{\phi}_{c}\big)\ast \big(\langle\widehat{\partial_{x}}\rangle^{-2}\widehat{\vartheta}(\widehat{R}_{1}^{0}+\widehat{R}_{-1}^{0})\big)\nonumber\\
&+\epsilon\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)}\widehat{\phi}_{c}\ast \big(\widehat{q}\widehat{\vartheta}(\widehat{R}_{1}^{1}-\widehat{R}_{-1}^{1})\big)
+\epsilon\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)}(\widehat{q}\widehat{\phi}_{c})\ast \big(\widehat{\vartheta}(\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1})\big)\nonumber\\
&+j_{1}\epsilon\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)\widehat{q}(k)}(\widehat{q}\widehat{\phi}_{c})\ast \big(\widehat{q}\widehat{\vartheta}(\widehat{R}_{1}^{1}-\widehat{R}_{-1}^{1})\big)
-j_{1}\epsilon\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)\widehat{q}(k)}
(\widehat{\phi}_{c})\ast\big(\widehat{\vartheta}(\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1})\big)\nonumber\\
&-j_{1}\epsilon\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)\langle k\rangle^{2}\widehat{q}(k)}\big(\langle\widehat{\partial_{x}}\rangle^{-2}\widehat{\phi}_{c}\big)\ast \big(\langle\widehat{\partial_{x}}\rangle^{-2}\widehat{\vartheta}(\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1})\big)\nonumber\\
&+\epsilon^{2}\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)}\widehat{\phi}_{p_{1}}\ast\big(\widehat{q}\widehat{\vartheta}
(\widehat{R}_{1}^{0}-\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}-\widehat{R}_{-1}^{1})\big)\nonumber\\
&+\epsilon^{2}\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)}\widehat{q}\widehat{\phi}_{p_{2}}\ast\big(\widehat{\vartheta}
(\widehat{R}_{1}^{0}+\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1})\big)\nonumber\\
&+j_{1}\epsilon^{2}\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)\widehat{q}(k)}\widehat{q}\widehat{\phi}_{p_{2}}\ast\big(\widehat{q}\widehat{\vartheta}
(\widehat{R}_{1}^{0}-\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}-\widehat{R}_{-1}^{1})\big)\nonumber\\
&-j_{1}\epsilon^{2}\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)\widehat{q}(k)}(\widehat{\phi}_{p_{1}})\ast\big(\widehat{\vartheta}
(\widehat{R}_{1}^{0}+\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1})\big)\nonumber\\
&-j_{1}\epsilon^{2}\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)\langle k\rangle^{2}\widehat{q}(k)}\big(\langle\widehat{\partial_{x}}\rangle^{-2}\widehat{\phi}_{p_{1}}\big)\ast \big(\langle\widehat{\partial_{x}}\rangle^{-2}\widehat{\vartheta}
(\widehat{R}_{1}^{0}+\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1})\big)\nonumber\\
&+\epsilon^{5/2}\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)}
\big(\widehat{\vartheta}(\widehat{R}_{1}^{0}+\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1})\big)\ast
\widehat{q}\widehat{\vartheta}\big(\widehat{R}_{1}^{0}-\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}-\widehat{R}_{-1}^{1})\big)\nonumber\\
&+j_{1}\epsilon^{5/2}\frac{\widehat{P}^{1}(k)ik}{4\widehat{\vartheta}(k)\widehat{q}(k)}\big(\widehat{q}\widehat{\vartheta}
(\widehat{R}_{1}^{0}-\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}-\widehat{R}_{-1}^{1})\big)^{\ast^{2}}\nonumber\\
&-j_{1}\epsilon^{5/2}\frac{\widehat{P}^{1}(k)ik}{4\widehat{\vartheta}(k)\widehat{q}(k)}
\big(\widehat{\vartheta}(\widehat{R}_{1}^{0}+\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1})\big)^{\ast^{2}}\nonumber\\
&-j_{1}\epsilon^{5/2}\frac{\widehat{P}^{1}(k)ik}{4\widehat{\vartheta}(k)\langle k\rangle^{2}\widehat{q}(k)}\big(\langle\widehat{\partial_{x}}\rangle^{-2}\widehat{\vartheta}
(\widehat{R}_{1}^{0}+\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1})\big)^{\ast^{2}}\nonumber\\
&+j_{1}\epsilon^{2}\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)\widehat{q}(k)}\widehat{\mathcal{G}}
\widehat{\vartheta}\big(\widehat{R}_{1}^{0}+\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1}\big)
+j_{1}\epsilon^{2}\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)\widehat{q}(k)}\widehat{\mathcal{M}}
\widehat{\vartheta}\big(\widehat{R}_{1}^{0}+\widehat{R}_{-1}^{0}+\widehat{R}_{1}^{1}+\widehat{R}_{-1}^{1}\big)\nonumber\\
&+\epsilon^{-5/2}\widehat{Res}_{U_{j_{1}}^{1}}(\epsilon\Psi)\nonumber\\
=:&ij_{1}\omega(k)\widehat{R}_{j_{1}}^{1}+\epsilon\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)}
\sum_{j_{2}\in\{\pm1\}}\sum_{n=1}^{5}\widehat{\alpha}_{j_{1},j_{2}}^{n}(k,k-m,m)(\widehat{\phi}_{c}\ast \widehat{\vartheta}\widehat{R}_{j_{2}}^{0})\nonumber\\
& \ \ +\epsilon\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)}\sum_{j_{2}\in\{\pm1\}}
\sum_{n=1}^{5}\widehat{\alpha}_{j_{1},j_{2}}^{n}(k,k-m,m)(\widehat{\phi}_{c}\ast \widehat{\vartheta}\widehat{R}_{j_{2}}^{1})
+\epsilon^{2}\mathcal{F}^{2},
\end{align}
where the notation $\mathcal{G}$ may depend on the error $(R^{0},R^{1})$, but do not lose regularity. The notation $\epsilon^{2}\mathcal{F}^{1}$ means the $H^{s'}$ norm of this term can be bounded by $C\epsilon^{2}$ if $R$ is in some bounded neighborhood of the origin in $H^{s}$ for any $s,s'>0$, since the coefficients of $\epsilon^{2}\mathcal{F}^{1}$ are equal to $C\big|\epsilon^{2}\frac{k}{\vartheta}\big|\leq C\epsilon^{2}$ for $|k|<\delta$. Similarly, $\epsilon^{2}\mathcal{F}^{2}$ means the $H^{s-1}$ norm of this term can be bounded by $C\epsilon^{2}$ if $R$ is in some bounded neighborhood of the origin in $H^{s}$.
Note that $|\widehat{\alpha}_{j_{1},j_{2}}^{n}(k,k-m,m)|\leq C$ for all $k,m\in\mathbb{R}$.
In the following we will attempt to construct normal form transformations to eliminate the $\mathcal{O}(\epsilon)$ terms from \eqref{equ1} and examine their effect on the full equation \eqref{equ1}. For this purpose, we consider the first normal form transformation of the form
\begin{equation}
\begin{split}\label{equ3}
\widetilde{R}_{j_{1}}^{0}=R_{j_{1}}^{0}+\epsilon \sum_{j_{2}\in\{\pm1\}}\sum_{n=1}^{5}B_{j_{1},j_{2}}^{0,1,n}(\phi_{c},R_{j_{2}}^{1}),
\end{split}
\end{equation}
where
\begin{equation}
\begin{split}\label{equ4}
\widehat{B}_{j_{1},j_{2}}^{0,1,n}(\phi_{c},R_{j_{2}}^{1})=\int \widehat{b}_{j_{1},j_{2}}^{0,1,n}(k,k-m,m)\widehat{\phi}_{c}(k-m)\widehat{R}_{j_{2}}(m)dm.
\end{split}
\end{equation}
\emph{Construction of $B_{j_{1},j_{2}}^{0,1,n}$}. Differentiating the $\widetilde{R}_{j_{1}}^{0}$ in \eqref{equ3} w.r.t. $t$, we obtain
\begin{equation}
\begin{split}\label{equ5}
\partial_{t}\widetilde{R}_{j_{1}}^{0}=\partial_{t}R_{j_{1}}^{0}+\epsilon\sum_{j_{2}\in\{\pm1\}}\sum_{n=1}^{5}B_{j_{1},j_{2}}^{0,1,n}(\partial_{t}\phi_{c},R_{j_{2}}^{1})
+\epsilon\sum_{j_{2}\in\{\pm1\}}\sum_{n=1}^{5} B_{j_{1},j_{2}}^{0,1,n}(\phi_{c},\partial_{t}R_{j_{2}}^{1}).
\end{split}
\end{equation}
Recall that $\widehat{\Omega u}(k)=i\omega(k)\widehat{u}(k)$ and $\|\partial_{t}\widehat{\phi}_{c}+i\omega\widehat{\phi}_{c}\|_{L^{1}(s)}\leq C\epsilon^{2}$ in Lemma \ref{L5}. Then provided the transformation $B_{j_{1},j_{2}}^{0,1,n}$ is well-defined and bounded, we have
\begin{equation}
\begin{split}\label{equ6}
\partial_{t}\widetilde{R}_{j_{1}}^{0}=&j_{1}\Omega\widetilde{R}_{j_{1}}^{0}
+\epsilon \sum_{j_{2}\in\{\pm1\}}\sum_{n=1}^{5}\Big[-j_{1}\Omega B_{j_{1},j_{2}}^{0,1,n}(\phi_{c},R_{j_{2}}^{1})
+\frac{P^{0}\partial_{x}}{2\vartheta}\alpha_{j_{1},j_{2}}^{n}(\phi_{c} \vartheta R_{j_{2}}^{1})\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -B_{j_{1},j_{2}}^{0,1,n}(\Omega\phi_{c},R_{j_{2}}^{1})
+ j_{2}B_{j_{1},j_{2}}^{0,1,n}(\phi_{c},\Omega R_{j_{2}}^{1})\Big]\\
&+\epsilon^{2}\sum_{j_{2},j_{3}\in\{\pm1\}}\sum_{n,\tilde{n}=1}^{5}B_{j_{1},j_{2}}^{0,1,n}
\big(\phi_{c},\frac{P^{1}\partial_{x}}{2\vartheta}\alpha_{j_{2},j_{3}}^{\tilde{n}}(\phi_{c}\vartheta R_{j_{3}}^{1})\big)
+\epsilon^{2}\mathcal{F}^{3},
\end{split}
\end{equation}
where $\epsilon^{2}\mathcal{F}^{3}$ have the same property with $\epsilon^{2}\mathcal{F}^{1}$.
To eliminate all terms which are formally $\mathcal{O}(\epsilon)$ of \eqref{equ6}, we choose $B^{0,1,n}$ so that
\begin{equation*}
\begin{split}
-j_{1}\Omega B_{j_{1},j_{2}}^{0,1,n}(\phi_{c},R_{j_{2}}^{1})
+\frac{P^{0}\partial_{x}}{2\vartheta}\alpha_{j_{1},j_{2}}^{n}(\phi_{c} \vartheta R_{j_{2}}^{1})
-B_{j_{1},j_{2}}^{0,1,n}(\Omega\phi_{c},R_{j_{2}}^{1})
+ j_{2}B_{j_{1},j_{2}}^{0,1,n}(\phi_{c},\Omega R_{j_{2}}^{1})=0.
\end{split}
\end{equation*}
It is equivalent to require that the kernel of $B^{0,1,n}$ be of the form
\begin{equation}
\begin{split}\label{equ7}
\widehat{b}_{j_{1},j_{2}}^{0,1,n}(k,k-m,m)=\frac{ -k\widehat{P}^{0}(k)\widehat{\alpha}_{j_{1},j_{2}}^{n}(k,k-m,m)}
{-j_{1}\omega(k)-\omega(k-m)+j_{2}\omega(m)}\frac{\widehat{\vartheta}(m)}{2\widehat{\vartheta}(k)}.
\end{split}
\end{equation}
Note that the kernel $\widehat{b}_{j_{1},j_{2}}^{0,1,n}$ has to be analyzed only for $|(k-m)\pm k_{0}|<\delta$, \ $|k|\leq\delta$ and $|m|>\delta$, since $\widehat{P}^{0}$ and $\widehat{\phi}_{c}$ are localized near $k=0$ and $k-m=\pm k_{0}$ respectively. Thus the resonance at $k=0$ will play a role for $B_{j_{1},j_{2}}^{0,1,n}$. Note also that if we consider the denominator of this kernel $\widehat{b}_{j_{1},j_{2}}^{0,1,n}$ near $k=0$, then
\begin{equation*}
\begin{split}
-j_{1}\omega(k)-\omega(k-m)+j_{2}\omega(m)
=-j_{1}\omega'(0)k-\big(\omega(-m)+\omega'(-m)k\big)+j_{2}\omega(m)+\mathcal{O}(k^{2}).
\end{split}
\end{equation*}
If $j_{2}=1$, this equality is bounded by some $\mathcal{O}(1)$ constant for all $|k|<\delta$. On the other hand, if $j_{2}=-1$, there exists a positive constant $C$ such that
\begin{equation}
\begin{split}\label{equ8}
\big|-j_{1}\omega(k)-\omega(k-m)+j_{2}\omega(m)\big|\geq C|k|.
\end{split}
\end{equation}
Thus there exists some $C\geq 0$ such that
\begin{equation}
\begin{split}\label{equ9}
\big|\widehat{\vartheta}(k)\widehat{b}_{j_{1},j_{2}}^{0,1,n}(k,k-m,m)\big|\leq C,
\end{split}
\end{equation}
for all $|k|\leq \delta$ and $n=1,2,3,4,5$. Since the factor of $\widehat{P}^{0}(k)$ makes $\widehat{b}_{j_{1},j_{2}}^{0,1,n}(k,k-m,m)=0$ if $|k|>\delta$, for any $s'>0$, there exists $C_{s'}$ such that
\begin{equation}
\begin{split}\label{equ1,1}
\big\|\varepsilon B_{j_{1},j_{2}}^{0,1,n}(\phi_{c},R_{j_{2}}^{1})\big\|_{H^{s'}}\leq C_{s'}\|R_{j_{2}}^{1}\|_{H^{s}},
\end{split}
\end{equation}
given $R_{j_{2}}^{1}\in H^{s}$ for some $s\geq6$. In particular, it holds when $s'=s$. However, we cannot assume that $C_{s'}\sim\mathcal{O}(\varepsilon)$ since $\widehat{\vartheta}^{-1}(k)\sim\varepsilon^{-1}$ for $k\approx0$, in spite of the factor of $\varepsilon$ in front of $B_{j_{1},j_{2}}^{0,1,n}$.
It is worth noting that although the terms $\epsilon^{2}B_{j_{1},j_{2}}^{0,1,n}\big(\phi_{c},\frac{P^{1}\partial_{x}}{2\vartheta}\alpha_{j_{2},j_{3}}^{\tilde{n}}(\phi_{c}\vartheta R_{j_{3}}^{1})\big)$ appearing in \eqref{equ6} are formally to be $\mathcal{O}(\epsilon^{2})$, the kernel of the transformation $B^{0,1,n}_{j_{1},j_{2}}$ is indeed $\mathcal{O}(\epsilon^{-1})$ for certain wave numbers so that this term is in fact only $\mathcal{O}(\epsilon)$ for those waves numbers and must therefore be retained. Therefore a second normal form transformation is needed.
Before constructing the second normal form transformation, we prove the following Lemma which will simplify the discussion in the sequel and will allow us to extract the real dangerous terms from $\epsilon^{2}B_{j_{1},j_{2}}^{0,1,n}
(\phi_{c},\frac{P^{1}\partial_{x}}{2\vartheta}\alpha_{j_{2},j_{3}}^{\tilde{n}}(\phi_{c}\vartheta R_{j_{3}}^{1}))$. This lemma take advantage of the strong localization of $\phi_{c}$ near the wave numbers $\pm k_{0}$ in Fourier space \cite{S}.
\begin{lemma}\label{L6}
Fix $p\in\mathbb{R}$ and assume that $\kappa=\kappa(k,k-m,m)\in C(\mathbb{R},\mathbb{C})$. Assume further that $\psi$ has a finitely supported Fourier transform and that $R\in H^{s}$. Then, \\
(i)\ if $\kappa$ is Lipschitz with respect to its second argument in some neighborhood of $p\in\mathbb{R}$, there exists $C_{\psi,\kappa,p}>0$ such that
\begin{equation}
\begin{split}\label{equation56}
\Big\|\int&\kappa(\cdot,\cdot-m,m)\widehat{\psi}\big(\frac{\cdot-m-p}{\epsilon}\big)\widehat{R}(m)dm
-\int\kappa(\cdot,p,m)\widehat{\psi}\big(\frac{\cdot-m-p}{\epsilon}\big)\widehat{R}(m)dm\Big\|_{H^{s}}\\
&\leq C_{\psi,\kappa,p}\epsilon\|R\|_{H^{s}},
\end{split}
\end{equation}
(ii) if $\kappa$ is globally Lipschitz with respect to its third argument, there exists $D_{\psi,\kappa}>0$ such that
\begin{equation}
\begin{split}\label{equation57}
\Big\|\int&\kappa(\cdot,\cdot-m,m)\widehat{\psi}\big(\frac{\cdot-m-p}{\epsilon}\big)\widehat{R}(m)dm
-\int\kappa(\cdot,\cdot-m,\cdot-p)\widehat{\psi}\big(\frac{\cdot-m-p}{\epsilon}\big)\widehat{R}(m)dm\Big\|_{H^{s}}\\
&\leq D_{\psi,\kappa}\epsilon\|R\|_{H^{s}}.
\end{split}
\end{equation}
\end{lemma}
\begin{proof}
For (i), we have
\begin{equation*}
\begin{split}
\Big\|\int&\kappa(\cdot,\cdot-m,m)\widehat{\psi}\big(\frac{\cdot-m-p}{\epsilon}\big)\widehat{R}(m)dm
-\int\kappa(\cdot,\cdot-m,\cdot-p)\widehat{\psi}\big(\frac{\cdot-m-p}{\epsilon}\big)\widehat{R}(m)dm\Big\|^{2}_{H^{s}}\\
&=\int\Big(\int(\kappa(k,k-m,m)-\kappa(k,p,m))\widehat{\psi}(\frac{k-m-p}{\epsilon})\widehat{R}(m)dm\Big)^{2}(1+k^{2})^{s}dk\\
&\leq\int\Big(C_{\kappa}\int|k-m-p|\widehat{\psi}(\frac{k-m-p}{\epsilon})\widehat{R}(m)dm\Big)^{2}(1+k^{2})^{s}dk\\
&\leq C_{\kappa}^{2}\Big(\int(1+\ell)^{s/2}|\ell||\widehat{\psi}(\frac{\ell}{\epsilon})|d\ell\Big)^{2}\|R\|^{2}_{H^{s}}\\
&\leq C_{\psi,\kappa,p}\epsilon^{2}\|R\|^{2}_{H^{s}},
\end{split}
\end{equation*}
thanks to the Young's inequality and the fact that $\widehat{\psi}$ has compact support. The second one (ii) is similar.
\end{proof}
We now construct the second normal form transformation to remove the remaining term $\epsilon^{2}B_{j_{1},j_{2}}^{0,1,n}\big(\phi_{c},\frac{P^{1}\partial_{x}}{2\vartheta}\alpha_{j_{2},j_{3}}^{\tilde{n}}(\phi_{c}\vartheta R_{j_{3}}^{1})\big)$ with $j_{1},j_{2},j_{3}\in\{\pm1\}$ from \eqref{equ6}. First recall that $\psi_{\pm1}$ is supported in a neighborhood of size $\delta$ of $\pm k_{0}$ in Fourier space and $\phi_{c}=\psi_{1}+\psi_{-1}$. Thus
\begin{equation}
\begin{split}\label{equ12}
&\epsilon^{2} B_{j_{1},j_{2}}^{0,1,n}\big(\phi_{c}, \frac{P^{1}\partial_{x}}{2\vartheta}\alpha_{j_{2},j_{3}}^{\tilde{n}}(\phi_{c}\vartheta R_{j_{3}}^{1})\big)\\
=&\epsilon^{2} B_{j_{1},j_{2}}^{0,1,n}\big(\psi_{1},\frac{P^{1}\partial_{x}}{2\vartheta}\alpha_{j_{2},j_{3}}^{\tilde{n}}(\psi_{1}\vartheta R_{j_{3}}^{1})\big)
+\epsilon^{2} B_{j_{1},j_{2}}^{0,1,n}\big(\psi_{-1},\frac{P^{1}\partial_{x}}{2\vartheta}\alpha_{j_{2},j_{3}}^{\tilde{n}}(\psi_{-1}\vartheta R_{j_{3}}^{1})\big)\\
&+\epsilon^{2} B_{j_{1},j_{2}}^{0,1,n}\big(\psi_{1},\frac{P^{1}\partial_{x}}{2\vartheta}\alpha_{j_{2},j_{3}}^{\tilde{n}}(\psi_{-1}\vartheta R_{j_{3}}^{1})\big)
+\epsilon^{2} B_{j_{1},j_{2}}^{0,1,n}\big(\psi_{-1},\frac{P^{1}\partial_{x}}{2\vartheta}\alpha_{j_{2},j_{3}}^{\tilde{n}}(\psi_{1}\vartheta R_{j_{3}}^{1})\big).
\end{split}
\end{equation}
Recalling from \eqref{equ7} that
\begin{equation*}
\begin{split}
\widehat{b}_{j_{1},j_{2}}^{0,1,n}(k,k-\ell,\ell)
=\frac{ -k\widehat{P}^{0}(k)\widehat{\alpha}_{j_{1},j_{2}}^{n}(k,k-\ell,\ell)}{-j_{1}\omega(k)
-\omega(k-\ell)+j_{2}\omega(\ell)}\frac{\widehat{\vartheta}(\ell)}{2\widehat{\vartheta}(k)},
\end{split}
\end{equation*}
each of the four terms on the RHS of \eqref{equ12} can be rewritten as
\begin{equation}
\begin{split}\label{equ13}
\epsilon^{2} B_{j_{1},j_{2}}^{0,1,n}&(\psi_{l},\frac{P^{1}\partial_{x}}{2\vartheta}\alpha_{j_{2},j_{3}}^{\tilde{n}}(\psi_{\nu}\vartheta R_{j_{3}}^{1}))\\
=&\frac{\epsilon^{2}}{2}\sum_{j_{1},j_{2}\in\{1,-1\}}\int\widehat{b}_{j_{1},j_{2}}^{0,1,n}(k,k-\ell,\ell)\widehat{\psi}_{l}(k-\ell)\\
&\times\widehat{\vartheta}^{-1}(\ell)\widehat{P}^{1}(\ell)i\ell\Big(
\int\widehat{\alpha}_{j_{2},j_{3}}^{\tilde{n}}(\ell,\ell-m,m)\widehat{\psi}_{\nu}(\ell-m)
\widehat{\vartheta}(m)\widehat{R}_{j_{3}}^{1}(m)dm\Big)d\ell,
\end{split}
\end{equation}
where $l,\nu\in\{+,-\}$. Applying Lemma \ref{L6} to obtain
\begin{equation}
\begin{split}\label{equation85}
\frac{\epsilon^{2}}{2} \widehat{B}_{j_{1},j_{2}}^{0,1,n}&\Big(\psi_{l}, \ P^{1}(\cdot-lk_{0})i(\cdot-lk_{0})\vartheta^{-1}(\cdot-lk_{0})\alpha^{\tilde{n}}_{j_{2},j_{3}}(\cdot,\cdot-lk_{0},\cdot-nk_{0})
\psi_{\nu}\vartheta(\cdot-\nu k_{0})R_{j_{3}}^{1}\Big)(k)\\
=&\frac{\epsilon^{2}}{2}\int\widehat{b}_{j_{1},j_{2}}^{0,1,n,l,\nu}(k)\widehat{\psi}_{l}(k-\ell)
\vartheta^{-1}(k-lk_{0})\widehat{P}^{1}(k-lk_{0})i(k-lk_{0})\\
&\times\Big(
\int\widehat{\alpha}_{j_{2},j_{3}}^{\tilde{n}}\big(k-lk_{0},lk_{0},k-(l+\nu)k_{0}\big)\widehat{\psi}_{\nu}(\ell-m)
\widehat{\vartheta}\big(k-(l+\nu)k_{0}\big)\widehat{R}_{j_{3}}^{1}(m)dm\Big)d\ell\\
&+\epsilon^{2}\mathcal{F}^{4},
\end{split}
\end{equation}
where $\epsilon^{2}\mathcal{F}^{4}$ has the same property with $\epsilon^{2}\mathcal{F}^{1}$, and $l+\nu$ is interpreted as if $l$ and $\nu$ were $+1$ and $-1$. We also use the abbreviation
\begin{equation*}
\begin{split}
\widehat{b}_{j_{1},j_{2}}^{0,1,n,l,\nu}(k)
=\frac{ -k\widehat{P}^{0}(k)\widehat{\alpha}_{j_{1},j_{2}}^{n}(k,lk_{0},k-lk_{0})}{-j_{1}\omega(k)
-\omega(lk_{0})+j_{2}\omega(k-lk_{0})}\frac{\widehat{\vartheta}\big(k-(l+\nu)k_{0}\big)}{2\widehat{\vartheta}(k)}.
\end{split}
\end{equation*}
\begin{lemma}\label{L7}
There exists $C>0$ such that
\begin{equation*}
\begin{split}
&\Big\|\epsilon^{2} B_{j_{1},j_{2}}^{0,1,n}\big(\psi_{1}, P^{1}\partial_{x}\vartheta^{-1}\alpha_{j_{2},j_{3}}^{\tilde{n}}(\psi_{-1}\vartheta R_{j_{3}}^{1})\big)\Big\|_{H^{s}}
\leq C\epsilon^{2}\|R_{j_{3}}^{1}\|_{H^{s}},\\
&\Big\|\epsilon^{2} B_{j_{1},j_{2}}^{0,1,n}(\psi_{-1}, P^{1}\partial_{x}\vartheta^{-1}\alpha_{j_{2},j_{3}}^{\tilde{n}}(\psi_{1}\vartheta R_{j_{3}}^{1})\big)\Big\|_{H^{s}}
\leq C\epsilon^{2}\|R_{j_{3}}^{1}\|_{H^{s}},
\end{split}
\end{equation*}
with $j_{1},j_{2},j_{3}\in\{\pm1\}$.
\end{lemma}
\begin{proof}
Since $B_{j_{1},j_{2}}^{0,1,n}$ contains the factor $\widehat{P}^{0}(k)$, the integral over $k$ occurring in the $H^{s}$ norm runs only over $|k|<\delta$. Thus we can bound the $H^{s}$ norm by bounding the maximum of the kernel. The first term in Lemma \ref{L7} has the modified kernel
\begin{equation}
\begin{split}\label{equation86}
\epsilon^{2}\widehat{b}_{j_{1},j_{2}}^{0,1,1,+,-}(k)
\widehat{\vartheta}^{-1}(k-k_{0})\widehat{P}^{1}(k-k_{0})i(k-k_{0})\widehat{\alpha}_{j_{2},j_{3}}^{\tilde{n}}(k-k_{0},k_{0},k)\widehat{\vartheta}(k).
\end{split}
\end{equation}
Since $\widehat{\vartheta}(k)\widehat{b}_{j_{1},j_{2}}^{0,1,n,+,-}(k)$ is $\mathcal{O}(1)$ bounded, and all other terms in \eqref{equation86} are $\mathcal{O}(1)$ bounded for $|k|<\delta$, we have an $\mathcal{O}(\epsilon^{2})$ bound for the kernel \eqref{equation86}. The second term in Lemma \ref{L7} can be estimated similarly.
\end{proof}
Thanks to Lemma \ref{L7}, we do not need to eliminate the third and fourth terms in \eqref{equ12} by the normal form transformation. Therefore we turn to the first term in \eqref{equ12}, whose modified kernel has the form
\begin{equation}
\begin{split}\label{equation87}
\frac{\epsilon^{2}}{2}\widehat{b}_{j_{1},j_{2}}^{0,1,n,+,+}(k)
\widehat{\vartheta}^{-1}(k-k_{0})\widehat{P}^{1}(k-k_{0})i(k-k_{0})\widehat{\alpha}_{j_{2},j_{3}}^{\tilde{n}}(k-k_{0},k_{0},k-2k_{0})\widehat{\vartheta}(k-2k_{0}),
\end{split}
\end{equation}
plus some $\mathcal{O}(\epsilon^{2})$ error terms. A similar expression for the kernel of the second term in \eqref{equ12} can be obtained. In contrast to the terms considered in Lemma \ref{L7}, this expression does not contain a factor of $\widehat{\vartheta}(k)$ to offset the $\widehat{\vartheta}(k)$ in the denominator of $\widehat{b}_{j_{1},j_{2}}^{0,1,n,+,+}(k)$ and hence they must be eliminated by a second normal form transformation. We look for a transformation of the form
\begin{equation}
\begin{split}\label{equation88}
\mathcal{R}_{j_{1}}^{0}:=\widetilde{R}_{j_{1}}^{0}+\epsilon\sum_{j_{2},j_{3}\in\{\pm1\}}\sum_{n,\tilde{n}=1}^{5} \big(D_{j_{1},j_{2},j_{3}}^{n,\tilde{n},+}(\psi_{1},\psi_{1},R_{j_{3}}^{1})+ D_{j_{1},j_{2},j_{3}}^{n,\tilde{n},-}(\psi_{-1},\psi_{-1},R_{j_{3}}^{1})\big).
\end{split}
\end{equation}
Differentiating the expression for $\mathcal{R}_{j_{1}}^{0}$, we find that the terms of $\mathcal{\mathcal{O}(\epsilon)}$ in \eqref{equ6} will be eliminated if $D_{j_{1},j_{2},j_{3}}^{n,\tilde{n},+}(\psi_{1},\psi_{1},R_{j_{3}}^{1})$ satisfies
\begin{equation}
\begin{split}\label{equation89}
-j_{1}\Omega& D_{j_{1},j_{2},j_{3}}^{n,\tilde{n},+}(\psi_{1},\psi_{1},R_{j_{3}}^{1})-D_{j_{1},j_{2},j_{3}}^{n,\tilde{n},+}(\Omega\psi_{1},\psi_{1},R_{j_{3}}^{1})
-D_{j_{1},j_{2},j_{3}}^{n,\tilde{n},+}(\psi_{1},\Omega\psi_{1},R_{j_{3}}^{1})\\
&+D_{j_{1},j_{2},j_{3}}^{n,\tilde{n},+}(\psi_{1},\psi_{1},j_{3}\Omega R_{j_{3}}^{1})
=-\frac{\epsilon}{2}B_{j_{1},j_{2}}^{0,1,n}(\psi_{1},\vartheta^{-1}P^{1}\partial_{x}\alpha_{j_{2},j_{3}}^{\tilde{n}}\big(\psi_{1}\vartheta R_{j_{3}}^{1})\big),
\end{split}
\end{equation}
with similar expression for $D_{j_{1},j_{2},j_{3}}^{n,\tilde{n},-}$. We find that we have to set
\begin{equation}
\begin{split}\label{equation90}
\epsilon &\widehat{D}_{j_{1},j_{2},j_{3}}^{n,\tilde{n},+}(\psi_{1},\psi_{1},R_{j_{3}}^{1})\\
=&\frac{\epsilon^{2}}{2}\int_{\mathbb{R}}\widehat{b}_{j_{1},j_{2}}^{0,1,n,+,+}(k)\widehat{\psi}_{1}(k-\ell)
\vartheta^{-1}(k-k_{0})\widehat{P}^{1}(k-k_{0})(k-k_{0})\\
&\times\Big(
\int_{\mathbb{R}}\frac{-\widehat{\alpha}_{j_{2},j_{3}}^{\tilde{n}}(k-k_{0},k_{0},k-2k_{0})\widehat{\psi}_{1}(\ell-m)
\widehat{\vartheta}(k-2k_{0})\widehat{R}_{j_{3}}^{1}(m)}{-j_{1}\omega(k)-\omega(k_{0})
-\omega(k_{0})+j_{3}\omega(k-2k_{0})}dm\Big)d\ell,
\end{split}
\end{equation}
where we have used in the kernel that $k-\ell\approx\ell-m\approx k_{0}$ due to the localization of $\psi_{1}$ so we have $m\approx-2k_{0}$, which is made rigorous with Lemma \ref{L6}. We have to estimate the kernel w.r.t. the sup norm, thanks to the Young's inequality. Since the numerator in this expression is already $\mathcal{O}(\varepsilon)$, the denominator satisfies
\begin{equation*}
\begin{split}
-j_{1}\omega(k)-\omega(k_{0})-\omega(k_{0})+j_{3}\omega(k-2k_{0})\approx-2\omega(k_{0})-j_{3}\omega(2k_{0})\neq0.
\end{split}
\end{equation*}
Therefore the mapping $\epsilon D_{j_{1},j_{2},j_{3}}^{n,\tilde{n},+}$ is well-defined and $\mathcal{O}(\epsilon)$-bounded. The expression for $\epsilon D_{j_{1},j_{2},j_{3}}^{n,\tilde{n},-}$ can be constructed and estimated in a very similar fashion and hence, the normal form is well defined and invertible. We also have for any $s\geq6$, there exists $C>0$ such that
\begin{equation}
\begin{split}\label{equ1,19}
\epsilon\big\|D_{j_{1},j_{2},j_{3}}^{n,\tilde{n},\pm}(\psi_{\pm1},\psi_{\pm1},R_{j_{3}}^{1})\big\|_{H^{s}}
\leq C\epsilon\big\|R_{j_{3}}^{1}\big\|_{H^{s}}.
\end{split}
\end{equation}
Finally, we consider the composition of the two normal form transformations \eqref{equ3} and \eqref{equation88}, i.e.,
\begin{equation}
\begin{split}\label{equation92}
\mathcal{R}_{j_{1}}^{0}&=\widetilde{R}_{j_{1}}^{0}+\epsilon \sum_{j_{2},j_{3}\in\{\pm1\}}\sum_{n,\tilde{n}=1}^{5}\big(D_{j_{1},j_{2},j_{3}}^{n,\tilde{n},+}(\psi_{1},\psi_{1},R_{j_{3}}^{1})+\epsilon D_{j_{1},j_{2},j_{3}}^{n,\tilde{n},-}(\psi_{-1},\psi_{-1},R_{j_{3}}^{1})\big)\\
&=R_{j_{1}}^{0}+\epsilon \sum_{j_{2}\in\{\pm1\}}\sum_{n=1}^{5}B_{j_{1},j_{2}}^{0,1,n}(\phi_{c},R_{j_{2}}^{1})\\
& \ \ \ \ \ \ \ \ \ \ +\epsilon \sum_{j_{2},j_{3}\in\{\pm1\}}\sum_{n,\tilde{n}=1}^{5}\big(D_{j_{1},j_{2},j_{3}}^{n,\tilde{n},+}(\psi_{1},\psi_{1},R_{j_{3}}^{1})+\epsilon D_{j_{1},j_{2},j_{3}}^{n,\tilde{n},-}(\psi_{-1},\psi_{-1},R_{j_{3}}^{1})\big)\\
&= R_{j_{1}}^{0}+\epsilon F_{j_{1}}(R^{1}),
\end{split}
\end{equation}
where $\|\epsilon \vartheta F(R^{1})\|_{H^{s'}}\lesssim\|\epsilon R^{1}\|_{H^{s}}$ for any $s', s\geq6$ due to \eqref{equ9}, \eqref{equ1,1} and \eqref{equ1,19}.
Then we have
\begin{equation}
\begin{split}\label{equation91}
\partial_{t}\mathcal{R}^{0}=\Lambda\mathcal{R}^{0}+\epsilon^{2}\mathcal{F}^{5},
\end{split}
\end{equation}
where $\epsilon^{2}\mathcal{F}^{5}$ has the same property with $\epsilon^{2}\mathcal{F}^{1}$.
Let
\begin{equation}
\begin{split}\label{phi3}
\phi_{3}:=&\phi_{p_{1}}+\epsilon^{1/2}\big(\vartheta (R_{1}^{0}+R_{-1}^{0})+\frac{1}{2}(R_{1}^{1}+R_{-1}^{1})\big),\\
\phi_{4}:=&\phi_{p_{2}}+\epsilon^{1/2}\big(\vartheta (R_{1}^{0}-R_{-1}^{0})+\frac{1}{2}(R_{1}^{1}-R_{-1}^{1})\big).\\
\end{split}
\end{equation}
Recall $\vartheta_{0}=\vartheta-\epsilon$ and rewrite the equation \eqref{equ1,-1} as
\begin{align}\label{equ11-1}
\partial_{t}\widehat{R}_{j_{1}}^{1}
=&ij_{1}\omega(k)\widehat{R}_{j_{1}}^{1}+\epsilon\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)}
\sum_{j_{2}\in\{\pm1\}}\sum_{n=1}^{5}\widehat{\alpha}_{j_{1},j_{2}}^{n}(k,k-m,m)(\widehat{\phi}_{c}\ast \widehat{\vartheta}_{0}\widehat{\mathcal{R}}_{j_{2}}^{0})\nonumber\\
& \ \ +\epsilon\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)}\sum_{j_{2}\in\{\pm1\}}
\sum_{n=1}^{5}\widehat{\alpha}_{j_{1},j_{2}}^{n}(k,k-m,m)\big(\widehat{\phi}_{c}\ast \widehat{\vartheta}\widehat{R}_{j_{2}}^{1}\big)\nonumber\\
&+\epsilon^{2}j_{1}\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)}\widehat{\phi}_{3}\ast\big(\widehat{q}\widehat{\vartheta}
(\widehat{R}_{j_{1}}^{1}-\widehat{R}_{-j_{1}}^{1})\big)
+\epsilon^{2}\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)}\widehat{q}\widehat{\phi}_{4}\ast\big(\widehat{\vartheta}
(\widehat{R}_{j_{1}}^{1}+\widehat{R}_{-j_{1}}^{1})\big)\nonumber\\
&+\epsilon^{2}\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)\widehat{q}(k)}\widehat{q}\widehat{\phi}_{4}\ast\big(\widehat{q}\widehat{\vartheta}
(\widehat{R}_{j_{1}}^{1}-\widehat{R}_{-j_{1}}^{1})\big)
-\epsilon^{2}j_{1}\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)\widehat{q}(k)}\widehat{\phi}_{3}\ast\big(\widehat{\vartheta}
(\widehat{R}_{j_{1}}^{1}+\widehat{R}_{-j_{1}}^{1})\big)\nonumber\\
&-\epsilon^{2}\frac{\widehat{P}^{1}(k)ij_{1}k}{2\widehat{\vartheta}(k)\langle k\rangle^{2}\widehat{q}(k)}\big(\langle\widehat{\partial_{x}}\rangle^{-2}\widehat{\phi}_{3}\big)\ast \big(\langle\widehat{\partial_{x}}\rangle^{-2}\widehat{\vartheta}
(\widehat{R}_{j_{1}}^{1}+\widehat{R}_{-j_{1}}^{1})\big)\nonumber\\
&+\epsilon^{2}\frac{\widehat{P}^{1}(k)ij_{1}k}{2\widehat{\vartheta}(k)\widehat{q}(k)}\widehat{\mathcal{G}}
\widehat{\vartheta}\big(\widehat{R}_{j_{1}}^{1}+\widehat{R}_{-j_{1}}^{1}\big)
+\epsilon^{2}\frac{\widehat{P}^{1}(k)ij_{1}k}{2\widehat{\vartheta}(k)\widehat{q}(k)}\widehat{\mathcal{M}}
\widehat{\vartheta}\big(\widehat{R}_{j_{1}}^{1}+\widehat{R}_{-j_{1}}^{1}\big)
+\epsilon^{2}\mathcal{F}_{j_{1}}^{5}\nonumber\\
&+\epsilon^{-5/2}\widehat{Res}_{U_{j_{1}}^{1}}(\epsilon\Psi)\nonumber\\
=:&ij_{1}\omega(k)\widehat{R}_{j_{1}}^{1}+\epsilon\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)}
\sum_{j_{2}\in\{\pm1\}}\sum_{n=1}^{5}\widehat{\alpha}_{j_{1},j_{2}}^{n}(k,k-m,m)\big(\widehat{\phi}_{c}\ast \widehat{\vartheta}_{0}\widehat{\mathcal{R}}_{j_{2}}^{0}\big)\nonumber\\
& \ \ +\epsilon\frac{\widehat{P}^{1}(k)ik}{2\widehat{\vartheta}(k)}\sum_{j_{2}\in\{\pm1\}}
\sum_{n=1}^{5}\widehat{\alpha}_{j_{1},j_{2}}^{n}(k,k-m,m)\big(\widehat{\phi}_{c}\ast \widehat{\vartheta}\widehat{R}_{j_{2}}^{1}\big)\nonumber\\
&+(\sum_{m=1}^{7}\widehat{G}_{m})_{j_{1}}+\epsilon^{2}\mathcal{F}_{j_{1}}^{5}+\epsilon^{-5/2}\widehat{Res}_{U_{j_{1}}^{1}}(\epsilon\Psi),
\end{align}
where $\|\mathcal{F}^{5}\|_{H^{s'}}\leq C\|\mathcal{R}^{0},R^{1}\|_{H^{s}}$ for any $s, \ s'>0$.
Recall that $\widehat{\vartheta}(k)=1$ for $|k|>\delta$ and \eqref{phi3}, let $\phi_{1}:=\phi_{c}+\epsilon\phi_{3}$ and $\phi_{2}:=\phi_{c}+\epsilon\phi_{4}$ and insert \eqref{equation92} into \eqref{equ1,-1}, we have
\begin{equation*}
\begin{split}
\phi_{1}=&\phi_{c}+\epsilon\phi_{p_{1}}+\epsilon^{3/2}\Big[\vartheta\big(\mathcal{R}_{1}^{0}+\mathcal{R}_{-1}^{0}
-\epsilon F_{1}(R^{1})-\epsilon F_{-1}(R^{1})
+\frac{1}{2}(R_{1}^{1}+R_{-1}^{1})\Big],\\
\phi_{2}
=&\phi_{c}+\epsilon\phi_{p_{2}}+\epsilon^{3/2}\Big[\vartheta\big(\mathcal{R}_{1}^{0}-\mathcal{R}_{-1}^{0}
-\epsilon F_{1}(R^{1})+\epsilon F_{-1}(R^{1})
+\frac{1}{2}(R_{1}^{1}-R_{-1}^{1})\Big],
\end{split}
\end{equation*}
then we have
\begin{align}\label{R1}
\partial_{t}R_{j_{1}}^{1}=&j_{1}\Omega R_{j_{1}}^{1}
+\epsilon j_{1}\frac{P^{1}\partial_{x}}{2}\big(\phi_{1} q(R_{j_{1}}^{1}-R_{-j_{1}}^{1})\big)
+\epsilon\frac{P^{1}\partial_{x}}{2}\big(q\phi_{2} (R_{j_{1}}^{1}+R_{-j_{1}}^{1})\big)\nonumber\\
&+\epsilon \frac{P^{1}\partial_{x}}{2q}\big(q\phi_{2} q(R_{j_{1}}^{1}-R_{-j_{1}}^{1})\big)
-\epsilon j_{1}\frac{P^{1}\partial_{x}}{2q}\big(\phi_{1} (R_{j_{1}}^{1}+R_{-j_{1}}^{1})\big)\nonumber\\
&-\epsilon j_{1}\frac{P^{1}\partial_{x}\langle\partial_{x}\rangle^{-2}}{2q}\big(\langle\partial_{x}\rangle^{-2}\phi_{1} (\langle\partial_{x}\rangle^{-2}(R_{j_{1}}^{1}+R_{-j_{1}}^{1})\big)\nonumber\\
&+\epsilon^{2}j_{1}\frac{P^{1}\partial_{x}}{2q}\mathcal{G}(R_{j_{1}}^{1}+R_{-j_{1}}^{1})
+\epsilon^{2}j_{1}\frac{P^{1}\partial_{x}}{2q}\mathcal{M}(R_{j_{1}}^{1}+R_{-j_{1}}^{1})\nonumber\\
&+\epsilon^{2}\mathcal{F}_{j_{1}}^{6}
+\epsilon^{-5/2}Res_{U_{j_{1}}^{1}}(\epsilon\Psi)\nonumber\\
=:&j_{1}\Omega R_{j_{1}}^{1}+\epsilon(\sum_{m=1}^{7}D_{m})_{j_{1}}
+\epsilon^{2}\mathcal{F}_{j_{1}}^{6}
+\epsilon^{-5/2}Res_{U_{j_{1}}^{1}}(\epsilon\Psi),
\end{align}
where for arbitrary $s', \ s\geq6$, we have
\begin{equation*}
\begin{split}
\|\mathcal{G}\|_{H^{s}}&\lesssim\|\mathcal{R}^{0},R^{1}\|^{2}_{H^{s}},\\
\|\mathcal{M}\|_{H^{s+2}}&\lesssim\|\mathcal{R}^{0},R^{1}\|^{2}_{H^{s}},\\
\|\mathcal{F}^{6}\|_{H^{s'}}&\lesssim\|\mathcal{R}^{0},R^{1}\|_{H^{s}}.
\end{split}
\end{equation*}
\section{The error estimates}
From the above procedure, we have obtained the equation \eqref{equation91} and \eqref{equ11-1} (or \eqref{R1}) for $(\mathcal{R}^{0},R^{1})$. In order to control the error we define the energy
\begin{equation*}
\begin{split}
\mathcal{E}_{s}=\sum_{\ell=0}^{s}E_{\ell},
\end{split}
\end{equation*}
with
\begin{equation}
\begin{split}\label{equ14}
E_{\ell}=&\sum_{j_{1}\in\{\pm1\}}\bigg[\frac{1}{2}\Big(\int_{\mathbb{R}}(\partial_{x}^{\ell}\mathcal{R}_{j_{1}}^{0})^{2}dx+
\int_{\mathbb{R}}(\partial_{x}^{\ell}R_{j_{1}}^{1})^{2}dx\Big)\\
&+\epsilon\sum_{j_{2}\in\{\pm1\}}\sum_{n=1}^{5}\Big(
\int_{\mathbb{R}}\partial_{x}^{\ell}R_{j_{1}}^{1}\partial_{x}^{\ell}B_{j_{1},j_{2}}^{1,0,n}(\phi_{c},\mathcal{R}_{j_{2}}^{0})dx
+\int_{\mathbb{R}}\partial_{x}^{\ell}R_{j_{1}}^{1}\partial_{x}^{\ell}B_{j_{1},j_{2}}^{1,1,n}(\phi_{c},R_{j_{2}}^{1})dx\Big)\bigg],
\end{split}
\end{equation}
where
\begin{equation}
\begin{split}\label{B10}
\widehat{B}_{j_{1},j_{2}}^{1,0,n}(\phi_{c},\mathcal{R}_{j_{2}}^{0})
=\int_{\mathbb{R}}\widehat{b}_{j_{1},j_{2}}^{1,0,n}(k,k-m,m)\widehat{\phi}_{c}(k-m)\widehat{\mathcal{R}}_{j_{2}}^{0}(m)dm,
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}\label{B1,1}
\widehat{B}_{j_{1},j_{2}}^{1,1,n}(\phi_{c},R_{j_{2}}^{1})
=\int_{\mathbb{R}}\widehat{b}_{j_{1},j_{2}}^{1,1,n}(k,k-m,m)\widehat{\phi}_{c}(k-m)\widehat{R}_{j_{2}}^{1}(m)dm,
\end{split}
\end{equation}
with
\begin{equation}
\begin{split}\label{b10}
b_{j_{1},j_{2}}^{1,0,n}=&\frac{-\frac{1}{2}k\widehat{P}^{1}(k)\widehat{\alpha}_{j_{1},j_{2}}^{n}(k,k-m,m)}{-j_{1}\omega(k)-\omega(k-m)+j_{2}\omega(m)}
\frac{\widehat{\vartheta}_{0}(m)}{\widehat{\vartheta}(k)}\\
=&\frac{-\frac{1}{2}k\widehat{\alpha}_{j_{1},j_{2}}^{n}(k,k-m,m)}{-j_{1}\omega(k)-\omega(k-m)+j_{2}\omega(m)}\widehat{\vartheta}_{0}(m),
\end{split}
\end{equation}
for $|k|>\delta$, $|m|<\delta$ and $|k-m\pm k_{0}|<\delta$,
and
\begin{equation}
\begin{split}\label{b1,1}
b_{j_{1},j_{2}}^{1,1,n}=&\frac{-\frac{1}{2}k\widehat{P}^{1}(k)\widehat{\alpha}_{j_{1},j_{2}}^{n}(k,k-m,m)}{-j_{1}\omega(k)-\omega(k-m)+j_{2}\omega(m)}
\frac{\widehat{\vartheta}(m)}{\widehat{\vartheta}(k)}\\
=&\frac{-\frac{1}{2}k\widehat{\alpha}_{j_{1},j_{2}}^{n}(k,k-m,m)}{-j_{1}\omega(k)-\omega(k-m)+j_{2}\omega(m)},
\end{split}
\end{equation}
for $|k|>\delta$, $|m|>\delta$ and $|k-m\pm k_{0}|<\delta$.
For the normal form transformation $B^{1,0,n}$, we have the following good properties.
\begin{lemma}\label{L10}
For the operators $B_{j_{1},j_{2}}^{1,0,n}(\phi_{c},\mathcal{R}_{j_{2}}^{0})$ for $1\leq n\leq5$, there exists a constant $C$ such that for any $s, s'\geq6$, we have\\
(a)
\begin{equation}
\begin{split}\label{154}
\epsilon\|B_{j_{1},j_{2}}^{1,0,n}(\phi_{c},\mathcal{R}_{j_{2}}^{0})\|_{H^{s'}}\lesssim\epsilon\|\mathcal{R}_{j_{2}}^{0}\|_{H^{s}},
\end{split}
\end{equation}\\
(b) and for all $f\in H^{1}(\mathbb{R},\mathbb{R})$, we have
\begin{equation}
\begin{split}\label{16'}
-j_{1}\Omega B_{j_{1},j_{2}}^{1,0,n}(\phi_{c},f)-B_{j_{1},j_{2}}^{1,0,n}(\Omega\phi_{c},f)
+j_{2} B_{j_{1},j_{2}}^{1,0,n}(\phi_{c},\Omega f)
=-\frac{P^{1}\partial_{x}}{2\vartheta}\alpha_{j_{1},j_{2}}^{n}(\phi_{c}f),
\end{split}
\end{equation}
where the operator $\Omega$ satisfies $\widehat{\Omega u}(k)=i\omega(k)\widehat{u}(k)$, $\omega$ is defined by the symbols \eqref{equation3}.
\end{lemma}
\begin{proof}
By Lemma \ref{L6} and recalling $\phi_{c}=\psi_{1}+\psi_{-1}$, we can rewrite
\begin{equation*}
\begin{split}
\widehat{B}_{j_{1},j_{2}}^{1,0,n}(\phi_{c},\mathcal{R}_{j_{2}}^{0})
=&\widehat{B}_{j_{1},j_{2}}^{1,0,n}(\psi_{1},\mathcal{R}_{j_{2}}^{0})+\widehat{B}_{j_{1},j_{2}}^{1,0,n}(\psi_{-1},\mathcal{R}_{j_{2}}^{0})\\
=&\int_{\mathbb{R}}\widehat{b}_{j_{1},j_{2}}^{1,0,n,+}(k)\widehat{\psi}_{1}(k-m)\widehat{\mathcal{R}}_{j_{2}}^{0}(m)dm\\
&+\int_{\mathbb{R}}\widehat{b}_{j_{1},j_{2}}^{1,0,n,-}(k)\widehat{\psi}_{-1}(k-m)\widehat{\mathcal{R}}_{j_{2}}^{0}(m)dm +\epsilon^{2}\mathcal{F}^{8},
\end{split}
\end{equation*}
where
\begin{equation}
\begin{split}\label{155}
\widehat{b}_{j_{1},j_{2}}^{1,0,n,+}(k)
=\frac{-\frac{1}{2}k\widehat{P}^{1}(k)\widehat{\alpha}_{j_{1},j_{2}}^{n}(k,k_{0},k-k_{0})}{-j_{1}\omega(k)-\omega(k_{0})+j_{2}\omega(k-k_{0})}
\frac{\widehat{\vartheta}_{0}(k-k_{0})}{\widehat{\vartheta}(k)},
\end{split}
\end{equation}
with a similar expression for $\widehat{b}_{j_{1},j_{2}}^{1,0,n,-}$ and the notation $\epsilon^{2}\mathcal{F}^{8}$ has the same property with $\epsilon^{2}\mathcal{F}^{1}$.
Due to the support properties of $\widehat{\phi}_{c}$ and the definitions of $\widehat{P}^{1}$ and $\widehat{\vartheta}(k)=1$, the expression \eqref{155} only has to be analysed for $|k-k_{0}|<\delta$ and $|k|>\delta$. So there is only a resonance at $k=k_{0}$ with $j_{1}=-1$ in the denominator of \eqref{155}, since $\widehat{P}^{1}(k)=0$ for $|k|<\delta$, while the resonance at $k=0$ does not play an essential role in the analysis of $B_{j_{1},j_{2}}^{1,0,n}$. However, since the derivative of $\omega$ at $k_{0}$ is $\mathcal{O}(1)$, we have a bound on the denominator of the form
\begin{equation}
\begin{split}\label{156}
|-j_{1}\omega(k)-\omega(k_{0})+j_{2}\omega(k-k_{0})|\geq C|k-k_{0}|.
\end{split}
\end{equation}
This singularity is offset thanks to $|\widehat{\vartheta}_{0}(k-k_{0})|\leq C|k-k_{0}|$ and hence the kernel $\widehat{b}_{j_{1},j_{2}}^{1,0,n,+}$ can be extended continuously at $k=k_{0}$ with an $\mathcal{O}(1)$ bound on its size. Thus the kernel can be bounded by an $\mathcal{O}(1)$ bound for all values of $k$ and $m$.
Applying Young's inequality, due to the compact support of $\widehat{\mathcal{R}}^{0}$, the loss of regularity is not present in the estimate for $B_{j_{1},j_{2}}^{1,0,n}$ and we have the estimate \eqref{154}. (b) is a direct consequence of the construction of $B_{j_{1},j_{2}}^{1,0,n}$.
\end{proof}
Particularly, according to the form of $\widehat{\alpha}_{j_{1},j_{2}}^{5}(k,k-m,m)$ in the equation \eqref{equ1,-1}, we have
\begin{equation}
\begin{split}\label{b1,15}
b_{j_{1},j_{2}}^{1,1,5}=&\frac{j_{1}k\widehat{P}^{1}(k)}{-j_{1}\omega(k)-\omega(k-m)+j_{2}\omega(m)}\frac{1}{\langle k\rangle^{2}\langle k-m\rangle^{2}\langle m\rangle^{2}}
\frac{\widehat{\vartheta}(m)}{2\widehat{\vartheta}(k)}\\
=&\frac{j_{1}k}{-j_{1}\omega(k)-\omega(k-m)+j_{2}\omega(m)}\frac{1}{\langle k\rangle^{2}\langle k-m\rangle^{2}\langle m\rangle^{2}},
\end{split}
\end{equation}
for $|k|>\delta$, $|m|>\delta$ and $|k-m\pm k_{0}|<\delta$.
Thus we have the following estimates
\begin{equation}
\begin{split}\label{B1,15}
\epsilon\|B_{j_{1},j_{2}}^{1,1,5}(\phi_{c},R_{j_{2}}^{1})\|_{H^{s+3}}\lesssim\epsilon\|R_{j_{2}}^{1}\|_{H^{s}},
\end{split}
\end{equation}
provided with $R_{j_{2}}^{1}\in H^{s}$ for $s\geq6$ and $j_{1}, \ j_{2}\in\{\pm1\}$.
However, there is a loss of one derivative due to the growth of $k\widehat{\alpha}_{j_{1},j_{2}}^{n}\sim k$ as $|k|\rightarrow\infty$ for $1\leq n\leq4$ and hence, we can only bound $B_{j_{1},j_{2}}^{1,1,n}$ in $H^{s-1}$ rather than $H^{s}$ if $R_{j_{2}}^{1}\in H^{s}$ for $1\leq n\leq4$. In the following we analyse carefully the construction of the normal form transformation $B_{j_{1},j_{2}}^{1,1,n}$ for $1\leq n\leq4$.
\begin{lemma}\label{L8}
The operators $B_{j_{1},j_{2}}^{1,1,n}$ for $1\leq n\leq4$ have the following properties:\\
(a) Fix $h\in L^{2}(\mathbb{R},\mathbb{R})$, then the mapping $f\rightarrow B_{j,j}^{1,1,n}(h,f)$ defines a continuous linear map from $H^{1}(\mathbb{R},\mathbb{R})$ into $L^{2}(\mathbb{R},\mathbb{R})$ and $f\rightarrow B_{j,-j}^{1,1,n}(h,f)$ defines a continuous linear map from $L^{2}(\mathbb{R},\mathbb{R})$ into $L^{2}(\mathbb{R},\mathbb{R})$. In particular, for all $f\in H^{1}(\mathbb{R},\mathbb{R})$ we have
\begin{equation}
\begin{split}\label{15}
&B_{j,j}^{1,1,1}(h,f)=-j\partial_{x}(G_{j,j}h \ qf)+Q_{j,j}^{1}(h,f),\\
&B_{j,-j}^{1,1,1}(h,f)=-G_{j,-j}h \ qf+Q_{j,-j}^{1}(h,f),
\end{split}
\end{equation}
\begin{equation}
\begin{split}\label{151}
&B_{j,j}^{1,1,2}(h,f)=-\partial_{x}(G_{j,j}qh \ f)+Q_{j,j}^{2}(h,f),\\
&B_{j,-j}^{1,1,2}(h,f)=jG_{j,-j}qh \ f+Q_{j,-j}^{2}(h,f),
\end{split}
\end{equation}
\begin{equation}
\begin{split}\label{152}
&B_{j,j}^{1,1,3}(h,f)=-\partial_{x}(G_{j,j}qh \ qf)+Q_{j,j}^{3}(h,f),\\
&B_{j,-j}^{1,1,3}(h,f)=-jG_{j,-j}qh \ qf+Q_{j,-j}^{3}(h,f),
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}\label{153}
&B_{j,j}^{1,1,4}(h,f)=j\partial_{x}(G_{j,j}h \ f)+Q_{j,j}^{4}(h,f),\\
&B_{j,-j}^{1,1,4}(h,f)=-G_{j,-j}h \ f+Q_{j,-j}^{4}(h,f),
\end{split}
\end{equation}
where
\begin{equation*}
\begin{split}
&\widehat{G_{j,j}h}(k)=\frac{\chi(k)}{-2i(jk+\omega(k))}\widehat{h}(k),\\
&\widehat{G_{j,-j}h}(k)=\frac{1}{2}\chi(k)\widehat{h}(k),\\
&\|Q_{j\pm j}^{n}(h,f)\|_{H^{1}}=\mathcal{O}(\|h\|_{L^{2}},\|f\|_{L^{2}}), \ n=1,2,3,4.
\end{split}
\end{equation*}
(b) For all $f\in H^{1}(\mathbb{R},\mathbb{R})$ we have
\begin{equation}
\begin{split}\label{16}
-j_{1}\Omega B_{j_{1},j_{2}}^{1,1,n}(\phi_{c},f)-B_{j_{1},j_{2}}^{1,1,n}(\Omega\phi_{c},f)
+j_{2} B_{j_{1},j_{2}}^{1,1,n}(\phi_{c},\Omega f)
=-\frac{P^{1}\partial_{x}}{2\vartheta}\alpha_{j_{1},j_{2}}^{n}(\phi_{c}f),
\end{split}
\end{equation}
where $n=1,2,3,4,5$, $\Omega$ is defined by $\widehat{\Omega u}(k)=i\omega(k)\widehat{u}(k)$ and the operator $\omega$ is defined by the symbols \eqref{equation3}.\\
(c) For all $f,g,h\in H^{1}(\mathbb{R},\mathbb{R})$, we have
\begin{equation}
\begin{split}\label{17}
\int_{\mathbb{R}}fB_{j_{1},j_{2}}^{1,1,\{1,4\}}(h,g)=-\frac{j_{1}}{j_{2}}\int_{\mathbb{R}}B_{j_{2},j_{1}}^{1,1,\{1,4\}}(h,f)gdx
+\int_{\mathbb{R}}S_{j_{2},j_{1}}^{\{1,4\}}(\partial_{x}h,f)gdx,
\end{split}
\end{equation}
\begin{equation}
\begin{split}\label{18}
\int_{\mathbb{R}}fB_{j_{1},j_{2}}^{1,1,\{2,3\}}(h,g)=-\int_{\mathbb{R}}B_{j_{2},j_{1}}^{1,1,\{2,3\}}(h,f)gdx
+\int_{\mathbb{R}}S_{j_{2},j_{1}}^{\{2,3\}}(\partial_{x}h,f)gdx,
\end{split}
\end{equation}
where
\begin{equation*}
\begin{split}
\widehat{S}_{j_{2},j_{1}}^{n}(\partial_{x}h,f)g(k)=\int_{\mathbb{R}}\widehat{s}_{j_{2},j_{1}}^{n}(k,k-m,m)\widehat{\partial_{x}h}(k-m)\widehat{f}(m)dm,
\end{split}
\end{equation*}
with
\begin{equation*}
\begin{split}
&\widehat{s}_{j_{2},j_{1}}^{1}(k,k-m,m)=\frac{\mp j_{1}(kq(m)-mq(k))}{2(k-m)i(-j_{2}\omega(k)-\omega(k-m)+j_{1}\omega(m))},\\
&\widehat{s}_{j_{2},j_{1}}^{2}(k,k-m,m)=\frac{-q(k-m)}{2i(-j_{2}\omega(k)-\omega(k-m)+j_{1}\omega(m))},\\
&\widehat{s}_{j_{2},j_{1}}^{3}(k,k-m,m)=\frac{\mp (kq(m)-mq(k))q(k-m)}{2(k-m)i(-j_{2}\omega(k)-\omega(k-m)+j_{1}\omega(m))},\\
&\widehat{s}_{j_{2},j_{1}}^{4}(k,k-m,m)=\frac{j_{1}}{2i(-j_{2}\omega(k)-\omega(k-m)+j_{1}\omega(m))},
\end{split}
\end{equation*}
where $\mp$ means that we take $-$ sign when $j_{1}$ and $j_{2}$ with the same sign and $+$ sign when $j_{1}$ and $j_{2}$ with the opposite sign.
In particular, we have
\begin{equation}
\begin{split}\label{19}
\widehat{S}_{j,j}^{1}(\partial_{x}h,f)=-jG_{j,j}\partial_{x}h \ qf+\widetilde{Q}_{j,j}^{1}(\partial_{x}h,f),
\end{split}
\end{equation}
\begin{equation}
\begin{split}\label{191}
\widehat{S}_{j,j}^{2}(\partial_{x}h,f)=-G_{j,j}\partial_{x}qh \ f+\widetilde{Q}_{j,j}^{2}(\partial_{x}h,f),
\end{split}
\end{equation}
\begin{equation}
\begin{split}\label{192}
\widehat{S}_{j,j}^{3}(\partial_{x}h,f)=-G_{j,j}\partial_{x}qh \ qf+\widetilde{Q}_{j,j}^{3}(\partial_{x}h,f),
\end{split}
\end{equation}
\begin{equation}
\begin{split}\label{193}
\widehat{S}_{j,j}^{4}(\partial_{x}h,f)=jG_{j,j}\partial_{x}h \ f+\widetilde{Q}_{j,j}^{4}(\partial_{x}h,f),
\end{split}
\end{equation}
with
\begin{equation*}
\begin{split}
\big\|\widetilde{Q}_{j,j}^{n}(\partial_{x}h,f)\big\|_{H^{2}}=\mathcal{O}\big(\|h\|_{L^{2}},\|f\|_{L^{2}}\big).
\end{split}
\end{equation*}
\end{lemma}
\begin{proof}
(a). Because of the support properties of $\widehat{\phi}_{c}$ and $\widehat{P}^{1}$, the kernel $b^{1,1,n}_{j_{1},j_{2}}$ \eqref{b1,1} of the normal form transformation $B^{1,1,n}_{j_{1},j_{2}}$ only has to be analysed for $|k-m\pm k_{0}|<\delta$, $|k|>\delta$ and $|m|>\delta$. We have
\begin{equation*}
\begin{split}
|-j_{1}\omega(k)-\omega(k-m)+j_{2}\omega(m)|\geq C>0,
\end{split}
\end{equation*}
with a constant $C$, which implies $|\widehat{b}_{j_{1},j_{2}}^{1,1,n}(k,k-m,m)|<\infty$ for $1\leq n\leq5$.
Next, we analyze the asymptotic behavior of the $|\widehat{b}_{j_{1},j_{2}}^{1,1,n}(k,k-m,m)|<\infty$ for $|k|\rightarrow\infty$. We have
\begin{equation}
\begin{split}\label{omega}
\omega(k)=kq(k)=k+\mathcal{O}(|k|^{-1}), \ \text{for} \ |k|\rightarrow\infty,
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}\label{omega'}
\omega'(k)=1+\mathcal{O}(|k|^{-2}), \ \text{for} \ |k|\rightarrow\infty.
\end{split}
\end{equation}
By the mean value theorem we get
\begin{equation*}
\begin{split}
\widehat{b}_{j,j}^{1,1,1}(k,k-m,m)=&\frac{jkq(m)\chi(k-m)}{2(j(\omega(k)-\omega(m))+\omega(k-m))}\\
=&\frac{jkq(m)\chi(k-m)}{2(j(k-m)\omega'(k-\theta(k-m))+\omega(k-m))},
\end{split}
\end{equation*}
for some $\theta\in[0,1]$. Using again the fact that \text{supp}$\chi$ is compact, we conclude with the help of the expressions \eqref{omega'} and $\widehat{\alpha}^{1}_{j_{1},j_{2}}(k,k-m,m)$ that
\begin{equation*}
\begin{split}
\widehat{b}_{j,j}^{1,1,1}(k,k-m,m)
=&\frac{jkq(m)\chi(k-m)}{2\big(j(k-m)(1+\mathcal{O}(|k|^{-2}))+\omega(k-m)\big)}\\
=&\frac{jkq(m)\chi(k-m)}{2\big(j(k-m)+\omega(k-m)\big)}+\mathcal{O}(|k|^{-1}), \ \ \text{for} \ |k|\rightarrow\infty.
\end{split}
\end{equation*}
Exploiting once more the compactness of \text{supp}$\chi$ as well as \eqref{omega}, \eqref{omega'} and $\widehat{\alpha}^{1}_{j_{1},j_{2}}(k,k-m,m)$ yields
\begin{equation*}
\begin{split}
\widehat{b}_{j,-j}^{1,1,1}(k,k-m,m)=&\frac{-jkq(m)\chi(k-m)}{2\big(j(\omega(k)+\omega(m))+\omega(k-m)\big)}\\
=&\frac{-jkq(m)\chi(k-m)}{2j\omega(k)\big(1+\mathcal{O}(|k|^{-1})\big)}\\
=&\frac{-q(m)}{2}+\mathcal{O}(|k|^{-1}),
\end{split}
\end{equation*}
for $|k|\rightarrow\infty$. Similarly, when $|k|\to\infty$ we have
\begin{equation*}
\begin{split}
&\widehat{b}_{j,j}^{1,1,2}(k,k-m,m)
=\frac{kq(k-m)\chi(k-m)}{2\big(j(k-m)+\omega(k-m)\big)}+\mathcal{O}(|k|^{-1}),\\
&\widehat{b}_{j,-j}^{1,1,2}(k,k-m,m)
=j\frac{q(k-m)}{2}+\mathcal{O}(|k|^{-1}),\\
&\widehat{b}_{j,j}^{1,1,3}(k,k-m,m)
=\frac{kq(k-m)q(m)\chi(k-m)}{2\big(j(k-m)+\omega(k-m)\big)}+\mathcal{O}(|k|^{-1}),\\
&\widehat{b}_{j,-j}^{1,1,3}(k,k-m,m)
=\frac{-jq(k-m)q(m)}{2}+\mathcal{O}(|k|^{-1}),\\
&\widehat{b}_{j,j}^{1,1,4}(k,k-m,m)
=\frac{-jk\chi(k-m)}{2\big(j(k-m)+\omega(k-m)\big)}+\mathcal{O}(|k|^{-1}),\\
&\widehat{b}_{j,-j}^{1,1,4}(k,k-m,m)
=-\frac{1}{2}+\mathcal{O}(|k|^{-1}).
\end{split}
\end{equation*}
These asymptotic expansions of the $\widehat{b}_{j_{1},j_{2}}^{1,1,n}(k,k-m,m)$ imply \eqref{15}-\eqref{153}.
Finally, since
\begin{equation*}
\begin{split}
\widehat{b}_{j_{1},j_{2}}^{1,1,n}(-k,-(k-m),-m)=\widehat{b}_{j_{1},j_{2}}^{1,1,n}(k,k-m,m)\in\mathbb{R},
\end{split}
\end{equation*}
and $\phi_{c}$ is real-valued, all assertions of (a) follow.
(b) is a direct consequence of the construction of the operators $B_{j_{1},j_{2}}^{1,1,n}$.
In order to prove (c), we compute for all $f, g, h\in H^{1}(\mathbb{R},\mathbb{R})$ that
\begin{equation*}
\begin{split}
(f,B_{j_{1},j_{2}}^{1,1,1}(h,g)) =&\int_{\mathbb{R}}\overline{\widehat{f}(k)}\widehat{B}_{j_{1},j_{2}}^{1,1,1}(h,g)(k)dk\\
=&\int_{\mathbb{R}}\int_{\mathbb{R}}\overline{\widehat{f}(k)}\frac{\mp\frac{j_{1}}{2}k\widehat{q}(m)}
{-j_{1}\omega(k)-\omega(k-m)+j_{2}\omega(m)}\widehat{h}(k-m)\widehat{g}(m)dmdk\\
=&\int_{\mathbb{R}}\int_{\mathbb{R}}\overline{\widehat{g}(-m)}\frac{\mp\frac{j_{1}}{2}k\widehat{q}(m)}
{-j_{1}\omega(k)-\omega(k-m)+j_{2}\omega(m)}\widehat{h}(k-m)\widehat{f}(-k)dkdm\\
=&\int_{\mathbb{R}}\int_{\mathbb{R}}\overline{\widehat{g}(k)}\frac{\pm\frac{j_{1}}{2}m\widehat{q}(k)}
{-j_{2}\omega(k)-\omega(k-m)+j_{1}\omega(m)}\widehat{h}(k-m)\widehat{f}(m)dmdk\\
=&\int_{\mathbb{R}}\int_{\mathbb{R}}\overline{\widehat{g}(k)}\frac{\pm\frac{j_{1}}{2}k\widehat{q}(m)}
{-j_{2}\omega(k)-\omega(k-m)+j_{1}\omega(m)}\widehat{h}(k-m)\widehat{f}(m)dmdk\\
&+\int_{\mathbb{R}}\int_{\mathbb{R}}\overline{\widehat{g}(k)}\frac{\mp\frac{j_{1}}{2}(k\widehat{q}(m)
-m\widehat{q}(k))}
{-j_{2}\omega(k)-\omega(k-m)+j_{1}\omega(m)}\widehat{h}(k-m)\widehat{f}(m)dmdk\\
=&-\frac{j_{1}}{j_{2}}\int_{\mathbb{R}}\overline{\widehat{g}(k)}\widehat{B}_{j_{2},j_{1}}^{1,1,1}(h,f)(k)dk
+\int_{\mathbb{R}}\overline{\widehat{g}(k)}\widehat{S}_{j_{2},j_{1}}^{1}(\partial_{x}h,f)(k)dk\\
=&-\frac{j_{1}}{j_{2}}\big(g,B_{j_{2},j_{1}}^{1,1,1}(h,f)\big)+\big(g,S_{j_{2},j_{1}}^{1}(h,f)\big).
\end{split}
\end{equation*}
Similarly, we have
\begin{equation*}
\begin{split}
\big(f,B_{j_{1},j_{2}}^{1,1,2}(h,g)\big)
=-\big(g,B_{j_{2},j_{1}}^{1,1,2}(h,f)\big)+\big(g,S_{j_{2},j_{1}}^{2}(h,f)\big),
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\big(f,B_{j_{1},j_{2}}^{1,1,3}(h,g)\big)
=-\big(g,B_{j_{2},j_{1}}^{1,1,3}(h,f)\big)+\big(g,S_{j_{2},j_{1}}^{3}(h,f)\big),
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
\big(f,B_{j_{1},j_{2}}^{1,1,4}(h,g)\big)
=-\frac{j_{1}}{j_{2}}\big(g,B_{j_{2},j_{1}}^{1,1,4}(h,f)\big)+\big(g,S_{j_{2},j_{1}}^{4}(h,f)\big),
\end{split}
\end{equation*}
yielding \eqref{17} and \eqref{18}. Thanks to \eqref{15}-\eqref{153}, we obtain \eqref{19}-\eqref{193}.
\end{proof}
The transformation \eqref{equation92}, assertions of Lemma \ref{L10}, Lemma \ref{L8} (a), (c) and the Cauchy-Schwarz inequality imply
\begin{corollary}\label{C1}
$\sqrt{\mathcal{E}_{s}}$ is equivalent to $\|R^{0}\|_{H^{s}}+\|R^{1}\|_{H^{s}}$ for sufficiently small $\epsilon>0$.
\end{corollary}
Since the right-hand side of the error equation \eqref{R1} for $j_{1}\in\{\pm1\}$ loses one derivative, we will need the following identities to control the time evolution of $\mathcal{E}_{s}$. See also \cite{D1}.
\begin{lemma}\label{L9}
Let $j\in\{\pm1\}$, $a_{j}\in H^{2}(\mathbb{R},\mathbb{R})$, and $f_{j}\in H^{1}(\mathbb{R},\mathbb{R})$. Then we have
\begin{equation}
\begin{split}\label{part1}
&\int_{\mathbb{R}}a_{j}f_{j}\partial_{x}f_{j}dx=-\frac{1}{2}\int_{\mathbb{R}}\partial_{x}a_{j}f_{j}^{2}dx,
\end{split}
\end{equation}
\begin{equation}
\begin{split}\label{part2}
\sum _{j\in\{\pm1\}}&\int_{\mathbb{R}}a_{j}f_{j}\partial_{x}f_{-j}dx =\frac{1}{2}\int_{\mathbb{R}}(a_{-1}-a_{1})(f_{1}+f_{-1})\partial_{x}(f_{1}-f_{-1})dx\\
&+\mathcal{O}\Big(\big(\|a_{1}\|_{H^{2}(\mathbb{R},\mathbb{R})}+\|a_{-1}\|_{H^{2}(\mathbb{R},\mathbb{R})}\big)
\big(\|f_{1}\|_{L^{2}(\mathbb{R},\mathbb{R})}+\|f_{-1}\|_{L^{2}(\mathbb{R},\mathbb{R})})\Big),
\end{split}
\end{equation}
\begin{equation}
\begin{split}\label{part3}
\sum _{j\in\{\pm1\}}j&\int_{\mathbb{R}}a_{j}f_{j}\partial_{x}f_{-j}dx =\frac{1}{2}\int_{\mathbb{R}}(a_{1}+a_{-1})(f_{1}+f_{-1})\partial_{x}(f_{-1}-f_{1})dx\\
&+\mathcal{O}\Big(\big(\|a_{1}\|_{H^{2}(\mathbb{R},\mathbb{R})}+\|a_{-1}\|_{H^{2}(\mathbb{R},\mathbb{R})}\big)
\big(\|f_{1}\|_{L^{2}(\mathbb{R},\mathbb{R})}+\|f_{-1}\|_{L^{2}(\mathbb{R},\mathbb{R})}\big)\Big),
\end{split}
\end{equation}
\begin{equation}
\begin{split}\label{part4}
(\widehat{G}_{-1,-1}+\widehat{G}_{1,1})(k)=(\frac{1}{-ik}+ik)q(k),
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}\label{part5}
(\widehat{G}_{-1,-1}-\widehat{G}_{1,1})(k)=(\frac{1}{-ik}+ik).
\end{split}
\end{equation}
\end{lemma}
\begin{proof}
By integration by parts, \eqref{part1} follows directly. Using again integration by parts, Cauchy-Schwarz inequality, and \eqref{part1}, we obtain
\begin{equation*}
\begin{split}
\sum_{j\in\{\pm1\}}&\int_{\mathbb{R}}a_{j}f_{j}\partial_{x}f_{-j}dx\\
=&\frac{1}{2}\sum_{j\in\{\pm1\}}\Big[\int_{\mathbb{R}}a_{j}f_{j}\partial_{x}f_{-j}dx-\int_{\mathbb{R}}a_{j}\partial_{x}f_{j}f_{-j}dx
-\int_{\mathbb{R}}\partial_{x}a_{j}f_{j}f_{-j}dx\Big]\\
=&\frac{1}{2}\Big[\int_{\mathbb{R}}(a_{-1}-a_{1})f_{-1}\partial_{x}f_{1}dx-\int_{\mathbb{R}}(a_{-1}-a_{1})f_{1}\partial_{x}f_{-1}dx\Big]\\
&+\mathcal{O}\big((\|a_{1}\|_{H^{2}}+\|a_{-1}\|_{H^{2}})(\|f_{1}\|_{H^{2}}+\|f_{-1}\|_{H^{2}})\big)\\
=&\frac{1}{2}\int_{\mathbb{R}}(a_{-1}-a_{1})(f_{1}+f_{-1})\partial_{x}(f_{1}-f_{-1})dx\\
&+\mathcal{O}\big((\|a_{1}\|_{H^{2}}+\|a_{-1}\|_{H^{2}})(\|f_{1}\|_{H^{2}}+\|f_{-1}\|_{H^{2}})\big),
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
\sum_{j\in\{\pm1\}}&j\int_{\mathbb{R}}a_{j}f_{j}\partial_{x}f_{-j}dx\\
=&\frac{1}{2}\sum_{j\in\{\pm1\}}\Big[j\int_{\mathbb{R}}a_{j}f_{j}\partial_{x}f_{-j}dx-j\int_{\mathbb{R}}a_{j}\partial_{x}f_{j}f_{-j}dx
-j\int_{\mathbb{R}}\partial_{x}a_{j}f_{j}f_{-j}dx\Big]\\
=&\frac{1}{2}\Big[\int_{\mathbb{R}}(a_{-1}+a_{1})f_{1}\partial_{x}f_{-1}dx-\int_{\mathbb{R}}(a_{-1}+a_{1})f_{-1}\partial_{x}f_{1}dx\Big]\\
&+\mathcal{O}\big((\|a_{1}\|_{H^{2}}+\|a_{-1}\|_{H^{2}})(\|f_{1}\|_{H^{2}}+\|f_{-1}\|_{H^{2}})\big)\\
=&-\frac{1}{2}\int_{\mathbb{R}}(a_{1}+a_{-1})(f_{1}+f_{-1})\partial_{x}(f_{1}-f_{-1})dx\\
&+\mathcal{O}\big((\|a_{1}\|_{H^{2}}+\|a_{-1}\|_{H^{2}})(\|f_{1}\|_{H^{2}}+\|f_{-1}\|_{H^{2}})\big).
\end{split}
\end{equation*}
Recalling $\omega(k)=k\widehat{q}(k)$ and $\widehat{q}(k)=\sqrt{\frac{2+k^{2}}{1+k^{2}}}$, we have
\begin{equation*}
\begin{split}
&\widehat{G}_{1,1}+\widehat{G}_{-1,-1}=\frac{1}{-2i}\Big(\frac{1}{k+\omega(k)}+\frac{1}{-k+\omega(k)}\Big)=\big(\frac{1}{-ik}+ik\big)\widehat{q}(k),\\
&\widehat{G}_{-1,-1}-\widehat{G}_{1,1}=\frac{1}{-2i}\Big(\frac{1}{-k+\omega(k)}-\frac{1}{k+\omega(k)}\Big)=\big(\frac{1}{-ik}+ik\big).
\end{split}
\end{equation*}
The proof is complete.
\end{proof}
Now, we are prepared to analyze $\partial_{t}E_{\ell}$. We compute
\begin{equation*}
\begin{split}
\partial_{t}E_{\ell}=&\sum_{j_{1}\in\{\pm1\}}\Big[\int_{\mathbb{R}}\partial_{x}^{\ell}\mathcal{R}_{j_{1}}^{0}
\partial_{t}\partial_{x}^{\ell}\mathcal{R}_{j_{1}}^{0}dx
+\int_{\mathbb{R}}\partial_{x}^{\ell}R_{j_{1}}^{1}
\partial_{t}\partial_{x}^{\ell}R_{j_{1}}^{1}dx\\
&+\epsilon\sum_{j_{2}\in\{\pm1\}}\sum_{n=1}^{5}\big(\int_{\mathbb{R}}\partial_{t}\partial_{x}^{\ell}R_{j_{1}}^{1}
\partial_{x}^{\ell}B_{j_{1},j_{2}}^{1,0,n}(\phi_{c},\mathcal{R}_{j_{2}}^{0})dx
+\int_{\mathbb{R}}\partial_{x}^{\ell}R_{j_{1}}^{1}
\partial_{x}^{\ell}B_{j_{1},j_{2}}^{1,0,n}(\partial_{t}\phi_{c},\mathcal{R}_{j_{2}}^{0})dx\\
&+\int_{\mathbb{R}}\partial_{x}^{\ell}R_{j_{1}}^{1}
\partial_{x}^{\ell}B_{j_{1},j_{2}}^{1,0,n}(\phi_{c},\partial_{t}\mathcal{R}_{j_{2}}^{0})dx
+\int_{\mathbb{R}}\partial_{t}\partial_{x}^{\ell}R_{j_{1}}^{1}
\partial_{x}^{\ell}B_{j_{1},j_{2}}^{1,1,n}(\phi_{c},R_{j_{2}}^{1})dx\\
&+\int_{\mathbb{R}}\partial_{x}^{\ell}R_{j_{1}}^{1}
\partial_{x}^{\ell}B_{j_{1},j_{2}}^{1,1,n}(\partial_{t}\phi_{c},R_{j_{2}}^{1})dx
+\int_{\mathbb{R}}\partial_{x}^{\ell}R_{j_{1}}^{1}
\partial_{x}^{\ell}B_{j_{1},j_{2}}^{1,1,n}(\phi_{c},\partial_{t}R_{j_{2}}^{1})dx
\big)\Big].
\end{split}
\end{equation*}
Using the error equations \eqref{equation91}, \eqref{equ11-1} and \eqref{R1}, we get
\begin{equation*}
\begin{split}
&\partial_{t}E_{\ell}=\sum_{j_{1}\in\{\pm1\}}\Big[j_{1}\int_{\mathbb{R}}\partial_{x}^{\ell}\mathcal{R}_{j_{1}}^{0}
\Omega\partial_{x}^{\ell}\mathcal{R}_{j_{1}}^{0}dx
+\epsilon^{2}\int_{\mathbb{R}}\partial_{x}^{\ell}\mathcal{R}_{j_{1}}^{0}
\mathcal{F}_{j_{1}}^{5}dx
+j_{1}\int_{\mathbb{R}}\partial_{x}^{\ell}R_{j_{1}}^{1}
\Omega\partial_{x}^{\ell}R_{j_{1}}^{1}dx\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\epsilon^{2}\int_{\mathbb{R}}\partial_{x}^{\ell}R_{j_{1}}^{1}\mathcal{F}_{j_{1}}^{6}dx
+\int_{\mathbb{R}}\partial_{x}^{\ell}R_{j_{1}}^{1}(\epsilon^{-5/2}Res_{U^{1}_{j_{1}}}(\epsilon\Psi))\Big]\\
+\epsilon&\sum_{j_{1},j_{2}\in\{\pm1\}}\sum_{n=1}^{5}\Big[\frac{1}{2}\int_{\mathbb{R}} \partial_{x}^{\ell}R_{j_{1}}^{1}
\partial_{x}^{\ell+1}P^{1}\vartheta^{-1}\alpha_{j_{1},j_{2}}^{n}(\phi_{c}\vartheta_{0}\mathcal{R}_{j_{2}}^{0})dx
+j_{1}\int_{\mathbb{R}}\Omega\partial_{x}^{\ell}R_{j_{1}}^{1}
\partial_{x}^{\ell}B_{j_{1},j_{2}}^{1,0,n}(\phi_{c},\mathcal{R}_{j_{2}}^{0})dx\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\int_{\mathbb{R}} \partial_{x}^{\ell}R_{j_{1}}^{1}
\partial_{x}^{\ell}B_{j_{1},j_{2}}^{1,0,n}(\Omega\phi_{c},\mathcal{R}_{j_{2}}^{0})dx
+j_{2}\int_{\mathbb{R}} \partial_{x}^{\ell}R_{j_{1}}^{1}
\partial_{x}^{\ell}B_{j_{1},j_{2}}^{1,0,n}(\phi_{c},\Omega\mathcal{R}_{j_{2}}^{0})dx\Big]\\
+\epsilon&\sum_{j_{1},j_{2}\in\{\pm1\}}\sum_{n=1}^{5}\Big[\frac{1}{2}\int_{\mathbb{R}}\partial_{x}^{\ell}R_{j_{1}}^{1}
\partial_{x}^{\ell+1}P^{1}\vartheta^{-1}\alpha_{j_{1},j_{2}}^{n}(\phi_{c}R_{j_{2}}^{1})dx
+j_{1}\int_{\mathbb{R}} \Omega\partial_{x}^{\ell}R_{j_{1}}^{1}
\partial_{x}^{\ell}B_{j_{1},j_{2}}^{1,1,n}(\phi_{c},R_{j_{2}}^{1})dx\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\int_{\mathbb{R}} \partial_{x}^{\ell}R_{j_{1}}^{1}
\partial_{x}^{\ell}B_{j_{1},j_{2}}^{1,1,n}(\Omega\phi_{c},R_{j_{2}}^{1})dx
+j_{2}\int_{\mathbb{R}} \partial_{x}^{\ell}R_{j_{1}}^{1}
\partial_{x}^{\ell}B_{j_{1},j_{2}}^{1,1,n}(\phi_{c},\Omega R_{j_{2}}^{1})dx\Big]\\
+\epsilon&\sum_{j_{1},j_{2}\in\{\pm1\}}\sum_{n=1}^{5}\Big[\int_{\mathbb{R}} \partial_{x}^{\ell}R_{j_{1}}^{1}
\partial_{x}^{\ell}B_{j_{1},j_{2}}^{1,0,n}(\partial_{t}\phi_{c}+\Omega\phi_{c},\mathcal{R}_{j_{2}}^{0})dx
+\int_{\mathbb{R}} \partial_{x}^{\ell}R_{j_{1}}^{1}
\partial_{x}^{\ell}B_{j_{1},j_{2}}^{1,1,n}(\partial_{t}\phi_{c}+\Omega\phi_{c},R_{j_{2}}^{1})dx\Big]\\
+\epsilon^{2}&\sum _{j_{1},j_{2}\in\{\pm1\}}\sum_{n=1}^{5}\Big[
\int_{\mathbb{R}}\partial_{x}^{\ell}((\sum_{m=1}^{7}D_{m})_{j_{1}}+\epsilon\mathcal{F}_{j_{1}}^{6}+\epsilon^{-7/2}Res_{U_{j_{1}}^{1}}) \partial_{x}^{\ell}B_{j_{1},j_{2}}^{1,0,n}(\phi_{c}, \mathcal{R}_{j_{2}}^{0})dx
+\int_{\mathbb{R}}\partial_{x}^{\ell}R_{j_{1}}^{1}\partial_{x}^{\ell}\mathcal{F}_{j_{2}}^{4}\Big]\\
+\epsilon&\sum _{j_{1},j_{2}\in\{\pm1\}}\sum_{n=1}^{5}\Big[
\int_{\mathbb{R}}\partial_{x}^{\ell}(\epsilon^{2}\mathcal{F}_{j_{1}}^{6}+\epsilon^{-5/2}Res_{U_{j_{1}}^{1}}) \partial_{x}^{\ell}B_{j_{1},j_{2}}^{1,1,n}(\phi_{c}, R_{j_{2}}^{1})dx\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\int_{\mathbb{R}}\partial_{x}^{\ell}R_{j_{1}}^{1} \partial_{x}^{\ell}B_{j_{1},j_{2}}^{1,1,n}(\phi_{c}, \epsilon^{2}\mathcal{F}_{j_{2}}^{6}+\epsilon^{-5/2}Res_{U_{j_{2}}^{1}})dx\Big]\\
+\epsilon^{2}&\sum _{j_{1},j_{2}\in\{\pm1\}}\sum_{n=1}^{5}\Big[
\int_{\mathbb{R}}\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} \partial_{x}^{\ell}B_{j_{1},j_{2}}^{1,1,n}(\phi_{c}, R_{j_{2}}^{1})dx
+\int_{\mathbb{R}}\partial_{x}^{\ell}R_{j_{1}}^{1} \partial_{x}^{\ell}B_{j_{1},j_{2}}^{1,1,n}(\phi_{c}, (\sum_{m=1}^{7}D_{m})_{j_{2}})dx\Big]\\
+\epsilon^{3}&\sum _{j_{1}\in\{\pm1\}}
\int_{\mathbb{R}}\partial_{x}^{\ell}R_{j_{1}}^{1}\partial_{x}^{\ell}(\sum_{m=1}^{7}G_{m})_{j_{1}}dx.\\
\end{split}
\end{equation*}
Due to the skew symmetry of $\Omega$, the first and third integrals equal to zero. Since the operators $B_{j_{1},j_{2}}^{1,0,n}$ and $B_{j_{1},j_{2}}^{1,1,n}$ satisfy \eqref{16'} and \eqref{16}, the sixth integral cancels with the sum of the seventh, the eighth and the ninth integral. Similarly, the tenth integral cancels with the sum of the eleventh, the twelfth and the thirteenth integral. Moreover, because of the estimates \eqref{equation12-1} and \eqref{equation12-3} for the residual, the form of the terms $\mathcal{F}^{4}$ and $\mathcal{F}^{5}$ in equations \eqref{equation91} and \eqref{equ11-1}, the Lemma \ref{L5} for the bound $\partial_{t}\widehat{\psi}_{\pm1}+i\omega\widehat{\psi}_{\pm1}$, the regularity properties of the operators $B_{j_{1},j_{2}}^{1,0,n}$ and $B_{j_{1},j_{2}}^{1,1,n}$ from Lemma \ref{L10}(a) and Lemma \ref{L8}(a), identity \eqref{part1} and the Corollary \ref{C1}, the second, the fourth, the fifth, the fourteenth to the nineteenth integrals can be bounded by $C\epsilon^{2}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}+\epsilon^{3}\mathcal{E}_{s}^{2}+1)$ for some constant $C$. Hence, we have
\begin{align*}
\partial_{t}E_{\ell}=&\epsilon^{2}\sum _{j_{1}\in\{\pm1\}}\sum_{n=1}^{5}\Big[
\int_{\mathbb{R}}\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} \partial_{x}^{\ell}B_{j_{1},j_{1}}^{1,1,n}(\phi_{c}, R_{j_{1}}^{1})dx\\
&+\int_{\mathbb{R}}\partial_{x}^{\ell}R_{j_{1}}^{1} \partial_{x}^{\ell}B_{j_{1},j_{1}}^{1,1,n}(\phi_{c}, (\sum_{m=1}^{7}D_{m})_{j_{1}})dx\Big]\\
&+\epsilon^{2}\sum _{j_{1}\in\{\pm1\}}\sum_{n=1}^{5}\Big[\int_{\mathbb{R}}\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} \partial_{x}^{\ell}B_{j_{1},-j_{1}}^{1,1,n}(\phi_{c}, R_{-j_{1}}^{1})dx\\
&+\int_{\mathbb{R}}\partial_{x}^{\ell}R_{j_{1}}^{1} \partial_{x}^{\ell}B_{j_{1},-j_{1}}^{1,1,n}(\phi_{c}, (\sum_{m=1}^{7}D_{m})_{-j_{1}})dx\Big]\\
&+\epsilon^{3}\sum _{j_{1}\in\{\pm1\}}
\int_{\mathbb{R}}\partial_{x}^{\ell}R_{j_{1}}^{1}\partial_{x}^{\ell}(\sum_{m=1}^{7}G_{m})dx\\
&+\epsilon^{2}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}+\epsilon^{3}\mathcal{E}_{s}^{2}+1)\\
=&:\sum_{n=1}^{5}I_{1}^{n}+\sum_{n=1}^{5}I_{2}^{n}+I_{3}
+\epsilon^{2}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}+\epsilon^{3}\mathcal{E}_{s}^{2}+1).
\end{align*}
Firstly, we analyse $I_{1}^{n}$. To extract all terms with more than $\ell$ spatial derivatives falling on $R_{1}^{1}$ or $R_{-1}^{1}$ we apply Leibniz's rule.
For $I_{1}^{1}$, recalling the equation \eqref{R1}, \ \eqref{17} and the asymptotic expansion \eqref{15} and \eqref{19}, we have
\begin{align*}
I_{1}^{1}=&\epsilon^{2}\sum _{j_{1}\in\{\pm1\}}\Big[
\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} B_{j_{1},j_{1}}^{1,1,1}(\phi_{c}, \partial_{x}^{\ell}R_{j_{1}}^{1})dx\\
&+\ell\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} B_{j_{1},j_{1}}^{1,1,1}(\partial_{x}\phi_{c}, \partial_{x}^{\ell-1}R_{j_{1}}^{1})dx
+\int\partial_{x}^{\ell}R_{j_{1}}^{1} B_{j_{1},j_{1}}^{1,1,1}(\phi_{c}, \partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}})dx\\
&+\ell\int\partial_{x}^{\ell}R_{j_{1}}^{1} B_{j_{1},j_{1}}^{1,1,1}(\partial_{x}\phi_{c}, \partial_{x}^{\ell-1}(\sum_{m=1}^{7}D_{m})_{j_{1}})dx\Big]
+\epsilon^{2}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}+\epsilon^{3}\mathcal{E}_{s}^{2}+1)\\
=&\epsilon^{2}\sum _{j_{1}\in\{\pm1\}}\Big[
\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} S_{j_{1},j_{1}}^{1}(\partial_{x}\phi_{c}, \partial_{x}^{\ell}R_{j_{1}}^{1})dx\\
& \ \ \ \ \ \ \ \ \ +2\ell\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} B_{j_{1},j_{1}}^{1,1,1}(\partial_{x}\phi_{c}, \partial_{x}^{\ell-1}R_{j_{1}}^{1})dx\Big]+\epsilon^{2}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}+\epsilon^{3}\mathcal{E}_{s}^{2}+1)\\
=&-\epsilon^{2}(2\ell+1)\sum _{j_{1}\in\{\pm1\}}j_{1}\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}}(G_{j_{1},j_{1}}\partial_{x}\phi_{c}) (q\partial_{x}^{\ell}R_{j_{1}}^{1})dx\\
& \ \ \ \ \ \ \ \ +\epsilon^{2}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}+\epsilon^{3}\mathcal{E}_{s}^{2}+1)\\
=:&\sum_{i=1}^{7}I_{1i}^{1}+\epsilon^{2}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}+\epsilon^{3}\mathcal{E}_{s}^{2}+1).
\end{align*}
For $I_{11}^{1}$, because of \eqref{part1} and \eqref{part3}, we have
\begin{equation*}
\begin{split}
I_{11}^{1}=&-\epsilon^{2}(\ell+1/2)\sum _{j_{1}\in\{\pm1\}}\int\partial_{x}^{\ell+1}\big(\phi_{1}q(R_{j_{1}}^{1}-R_{-j_{1}}^{1})\big)G_{j_{1},j_{1}}\partial_{x}\phi_{c} q\partial_{x}^{\ell}R_{j_{1}}^{1}dx\\
=&\epsilon^{2}(\ell+1/2)\sum _{j_{1}\in\{\pm1\}}\int (G_{j_{1},j_{1}}\partial_{x}\phi_{c})\phi_{1} (q\partial_{x}^{\ell}R_{j_{1}}^{1})\partial_{x}^{\ell+1}(qR_{-j_{1}}^{1}) dx
+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2})\\
=&\frac{\epsilon^{2}(\ell+1/2)}{2}\int (G_{-1,-1}-G_{1,1})\partial_{x}\phi_{c}\phi_{1} \partial_{x}^{\ell}q(R_{1}^{1}+R_{-1}^{1})\partial_{x}^{\ell+1}q(R_{1}^{1}-R_{-1}^{1}) dx\\
&+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}).
\end{split}
\end{equation*}
Recalling \eqref{equation3} and \eqref{part4}, we have
\begin{equation*}
(\widehat{G}_{1,1}-\widehat{G}_{-1,-1})(k)=(\frac{1}{-ik}+ik)\chi(k),
\end{equation*}
and
\begin{equation*}
\widehat{q}(k)-1=\mathcal{O}(k^{-2}).
\end{equation*}
Then we have
\begin{equation*}
\begin{split}
I_{11}^{1}=&\frac{\epsilon^{2}(\ell+1/2)}{2}\int (-\phi_{c}+\partial_{x}^{2}\phi_{c})\phi_{1} \partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1})\partial_{x}^{\ell+1}(R_{1}^{1}-R_{-1}^{1}) dx\\
&+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}).
\end{split}
\end{equation*}
By the equation \eqref{equ1,-1}, we have
\begin{equation}
\begin{split}\label{equ}
\partial_{t}(R_{1}^{1}+R_{-1}^{1})=&\partial_{x}q(R_{1}^{1}-R_{-1}^{1})+\frac{\epsilon}{2} \partial_{x}\big(\phi_{1}q(R_{1}^{1}-R_{-1}^{1})\big)
+\frac{\epsilon}{2} \partial_{x}\big(q\phi_{2}(R_{1}^{1}+R_{-1}^{1})\big)\\
&+\epsilon^{-5/2}\big(Res_{U_{1}^{1}}(\epsilon\Psi)+Res_{U_{-1}^{1}}(\epsilon\Psi)\big).
\end{split}
\end{equation}
Taking $\partial_{x}^{\ell}$ on the equation \eqref{equ}, we have
\begin{equation}
\begin{split}\label{equu}
\partial_{x}^{\ell+1}&(R_{1}^{1}-R_{-1}^{1})\\
=&\frac{1}{1+\frac{\epsilon\phi_{1}}{2}}\Big[
\partial_{x}^{\ell}\partial_{t}(R_{1}^{1}+R_{-1}^{1})-\frac{\epsilon q\phi_{2}}{2}\partial_{x}^{\ell+1}(R_{1}^{1}+R_{-1}^{1})
+(1+\frac{\epsilon\phi_{1}}{2})(1-q)\partial_{x}^{\ell+1}(R_{1}^{1}-R_{-1}^{1})\\
&-\frac{\epsilon}{2}\sum_{i=1}^{\ell+1}C_{\ell+1}^{i}\partial_{x}^{i}\phi_{1}\partial_{x}^{\ell-i+1}q(R_{1}^{1}-R_{-1}^{1})
-\frac{\epsilon}{2}\sum_{i=1}^{\ell+1}C_{\ell+1}^{i}\partial_{x}^{i}q\phi_{2}\partial_{x}^{\ell-i+1}q(R_{1}^{1}+R_{-1}^{1})\\
&-\epsilon^{-5/2}\partial_{x}^{\ell}(Res_{U_{1}^{1}}(\epsilon\Psi)+Res_{U_{-1}^{1}}(\epsilon\Psi))\Big].
\end{split}
\end{equation}
Then by \eqref{equu} and integration by parts, we have
\begin{align*}
I_{11}^{1}=&\frac{\epsilon^{2}(\ell+1/2)}{2}\int \frac{(-\phi_{c}+\partial_{x}^{2}\phi_{c})\phi_{1}}{1+\frac{\epsilon\phi_{1}}{2}} \partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1})\partial_{x}^{\ell}\partial_{t}(R_{1}^{1}+R_{-1}^{1}) dx\\
&-\frac{\epsilon^{3}(\ell+1/2)}{4}\int \frac{(-\phi_{c}+\partial_{x}^{2}\phi_{c})\phi_{1}q\phi_{2}}{1+\frac{\epsilon\phi_{1}}{2}} \partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1})\partial_{x}^{\ell+1}(R_{1}^{1}+R_{-1}^{1}) dx\\
&+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2})\\
=&\frac{\epsilon^{2}(\ell+1/2)}{2}\frac{d}{dt}\int \frac{(-\phi_{c}+\partial_{x}^{2}\phi_{c})\phi_{1}}{2+\epsilon\phi_{1}}
\big(\partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1})\big)^{2}dx +\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}).
\end{align*}
Similarly, we have
\begin{equation*}
\begin{split}
I_{12}^{1}=&-\epsilon^{2}(\ell+1/2)\sum _{j_{1}\in\{\pm1\}}j_{1}\int\partial_{x}^{\ell+1}\big(q\phi_{2}(R_{j_{1}}^{1}+R_{-j_{1}}^{1})\big)G_{j_{1},j_{1}}\partial_{x}\phi_{c} (q\partial_{x}^{\ell}R_{j_{1}}^{1})dx\\
=&-\epsilon^{2}(\ell+1/2)\sum _{j_{1}\in\{\pm1\}}j_{1}\int G_{j_{1},j_{1}}\partial_{x}\phi_{c}q\phi_{2} \partial_{x}^{\ell}R_{j_{1}}^{1}\partial_{x}^{\ell+1}R_{-j_{1}}^{1} dx
+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2})\\
=&\frac{\epsilon^{2}(\ell+1/2)}{2}\int (G_{1,1}+G_{-1,-1})\partial_{x}\phi_{c}q\phi_{2} \partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1})\partial_{x}^{\ell+1}(R_{1}^{1}-R_{-1}^{1}) dx\\
&+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2})\\
=&\frac{\epsilon^{2}(\ell+1/2)}{2}\frac{d}{dt}\int \frac{q(-\phi_{c}+\partial_{x}^{2}\phi_{c})q\phi_{2}}{2+\epsilon\phi_{1}}
\big(\partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1})\big)^{2}dx +\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}),
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
I_{13}^{1}=-\frac{\epsilon^{2}(\ell+1/2)}{2}\frac{d}{dt}\int \frac{q(-\phi_{c}+\partial_{x}^{2}\phi_{c})q\phi_{2}}{2+\epsilon\phi_{1}}
\big(\partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1})\big)^{2}dx
+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}),
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
I_{14}^{1}
=\frac{\epsilon^{2}(\ell+1/2)}{2}\frac{d}{dt}\int \frac{(-\phi_{c}+\partial_{x}^{2}\phi_{c})\phi_{1}}{2+\epsilon\phi_{1}}
\big(\partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1})\big)^{2}dx
+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}),
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
I_{16}^{1}
=-\frac{\epsilon^{3}(\ell+1/2)}{2}\frac{d}{dt}\int \frac{\mathcal{G}(-\phi_{c}+\partial_{x}^{2}\phi_{c})}{2+\epsilon\phi_{1}}
\big(\partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1})\big)^{2}dx
+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}).
\end{split}
\end{equation*}
Because of the well properties for $I_{15}^{1}$ and $I_{17}^{1}$, we obtain $I_{15}^{1}+I_{17}^{1}
=\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2})$.
Then we have
\begin{equation*}
\begin{split}
I_{1}^{1}=&\frac{\epsilon^{2}(\ell+1/2)}{2}\frac{d}{dt}\int \frac{(-\phi_{c}+\partial_{x}^{2}\phi_{c})(2\phi_{1}-\epsilon\mathcal{G})}{2+\epsilon\phi_{1}}
\big(\partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1})\big)^{2}dx\\
&+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}).
\end{split}
\end{equation*}
Similarly, by the equations \eqref{151}, \eqref{18}, \eqref{191}, \eqref{part1}, \eqref{part2}, \eqref{part5} and \eqref{equu}, we have
\begin{equation*}
\begin{split}
I_{1}^{2}=&\epsilon^{2}\sum _{j_{1}\in\{\pm1\}}\Big[
\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} B_{j_{1},j_{1}}^{1,1,2}(\phi_{c}, \partial_{x}^{\ell}R_{j_{1}}^{1})dx
+\ell\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} B_{j_{1},j_{1}}^{1,1,2}(\partial_{x}\phi_{c}, \partial_{x}^{\ell-1}R_{j_{1}}^{1})dx\\
&+\int\partial_{x}^{\ell}R_{j_{1}}^{1} B_{j_{1},j_{1}}^{1,1,2}\big(\phi_{c}, \partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}}\big)dx
+\ell\int\partial_{x}^{\ell}R_{j_{1}}^{1} B_{j_{1},j_{1}}^{1,1,2}\big(\partial_{x}\phi_{c}, \partial_{x}^{\ell-1}(\sum_{m=1}^{7}D_{m})_{j_{1}}\big)dx\Big]\\
&+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2})\\
=&\epsilon^{2}\sum _{j_{1}\in\{\pm1\}}\Big[
\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} S_{j_{1},j_{1}}^{2}(\partial_{x}\phi_{c}, \partial_{x}^{\ell}R_{j_{1}}^{1})dx
+2\ell\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} B_{j_{1},j_{1}}^{1,1,2}(\partial_{x}\phi_{c}, \partial_{x}^{\ell-1}R_{j_{1}}^{1})dx\Big]\\
&+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2})\\
=&-\epsilon^{2}(2\ell+1)\sum _{j_{1}\in\{\pm1\}}\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}}G_{j_{1},j_{1}}\partial_{x}q\phi_{c} \partial_{x}^{\ell}R_{j_{1}}^{1}dx
+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2})\\
=&\frac{\epsilon^{2}(\ell+1/2)}{2}\frac{d}{dt}\int \frac{-q^{2}(-\phi_{c}+\partial_{x}^{2}\phi_{c})(2\phi_{1}-\epsilon\mathcal{G})}{2+\epsilon\phi_{1}}
(\partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1}))^{2}dx\\
&+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}).
\end{split}
\end{equation*}
By the equations \eqref{152}, \eqref{18}, \eqref{192}, \eqref{part1}, \eqref{part2}, \eqref{part3}, \eqref{part5} and \eqref{equu}, we have
\begin{equation*}
\begin{split}
I_{1}^{3}=&\epsilon^{2}\sum _{j_{1}\in\{\pm1\}}\Big[
\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} B_{j_{1},j_{1}}^{1,1,3}(\phi_{c}, \partial_{x}^{\ell}R_{j_{1}}^{1})dx
+\ell\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} B_{j_{1},j_{1}}^{1,1,3}(\partial_{x}\phi_{c}, \partial_{x}^{\ell-1}R_{j_{1}}^{1})dx\\
&+\int\partial_{x}^{\ell}R_{j_{1}}^{1} B_{j_{1},j_{1}}^{1,1,3}\big(\phi_{c}, \partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}}\big)dx
+\ell\int\partial_{x}^{\ell}R_{j_{1}}^{1} B_{j_{1},j_{1}}^{1,1,3}\big(\partial_{x}\phi_{c}, \partial_{x}^{\ell-1}(\sum_{m=1}^{7}D_{m})_{j_{1}}\big)dx\Big]\\
&+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2})\\
=&\epsilon^{2}\sum _{j_{1}\in\{\pm1\}}\Big[
\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} S_{j_{1},j_{1}}^{3}(\partial_{x}\phi_{c}, \partial_{x}^{\ell}R_{j_{1}}^{1})dx
+2\ell\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} B_{j_{1},j_{1}}^{1,1,3}(\partial_{x}\phi_{c}, \partial_{x}^{\ell-1}R_{j_{1}}^{1})dx\Big]\\
&+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2})\\
=&-\epsilon^{2}(2\ell+1)\sum _{j_{1}\in\{\pm1\}}\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}}G_{j_{1},j_{1}}\partial_{x}q\phi_{c} q\partial_{x}^{\ell}R_{j_{1}}^{1}dx
+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2})\\
=&\frac{\epsilon^{2}(\ell+1/2)}{2}\frac{d}{dt}\int \frac{-q^{2}(-\phi_{c}+\partial_{x}^{2}\phi_{c})(2\phi_{1}-\epsilon\mathcal{G})}{2+\epsilon\phi_{1}}
(\partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1}))^{2}dx\\
&+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}).
\end{split}
\end{equation*}
By the equations \eqref{153}, \eqref{17}, \eqref{193}, \eqref{part1}, \eqref{part2}, \eqref{part3}, \eqref{part5} and \eqref{equu}, we have
\begin{equation*}
\begin{split}
I_{1}^{4}=&\epsilon^{2}\sum _{j_{1}\in\{\pm1\}}\Big[
\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} B_{j_{1},j_{1}}^{1,1,4}(\phi_{c}, \partial_{x}^{\ell}R_{j_{1}}^{1})dx
+\ell\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} B_{j_{1},j_{1}}^{1,1,4}(\partial_{x}\phi_{c}, \partial_{x}^{\ell-1}R_{j_{1}}^{1})dx\\
&+\int\partial_{x}^{\ell}R_{j_{1}}^{1} B_{j_{1},j_{1}}^{1,1,4}\big(\phi_{c}, \partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}}\big)dx
+\ell\int\partial_{x}^{\ell}R_{j_{1}}^{1} B_{j_{1},j_{1}}^{1,1,4}\big(\partial_{x}\phi_{c}, \partial_{x}^{\ell-1}(\sum_{m=1}^{7}D_{m})_{j_{1}}\big)dx\Big]
+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s})\\
=&\epsilon^{2}\sum _{j_{1}\in\{\pm1\}}\Big[
\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} S_{j_{1},j_{1}}^{4}(\partial_{x}\phi_{c}, \partial_{x}^{\ell}R_{j_{1}}^{1})dx
+2\ell\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} B_{j_{1},j_{1}}^{1,1,4}(\partial_{x}\phi_{c}, \partial_{x}^{\ell-1}R_{j_{1}}^{1})dx\Big]\\
=&\epsilon^{2}(2\ell+1)\sum _{j_{1}\in\{\pm1\}}j_{1}\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}}G_{j_{1},j_{1}}\partial_{x}\phi_{c} \partial_{x}^{\ell}R_{j_{1}}^{1}dx
+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s})\\
=&\frac{\epsilon^{2}(\ell+1/2)}{2}\frac{d}{dt}\int \frac{-(-\phi_{c}+\partial_{x}^{2}\phi_{c})(2\phi_{1}-\epsilon\mathcal{G})}{2+\epsilon\phi_{1}}
\big(\partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1})\big)^{2}dx\\
&+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}).
\end{split}
\end{equation*}
For $I_{1}^{5}$, recall
\begin{equation*}
\begin{split}
b_{j_{1},j_{2}}^{1,1,5}=&\frac{-\frac{1}{2}k\langle k\rangle^{-2}\langle k-m\rangle^{-2}\langle m\rangle^{-2}}{-j_{1}\omega(k)-\omega(k-m)+j_{2}\omega(m)}
\ \ for\ |k|>\delta, \ |m|>\delta.
\end{split}
\end{equation*}
By integration by parts, we have
\begin{equation*}
\begin{split}
I_{1}^{5}=&\epsilon^{2}\sum _{j_{1}\in\{\pm1\}}\Big[
\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} \partial_{x}^{\ell}B_{j_{1},j_{1}}^{1,1,5}(\phi_{c}, R_{j_{1}}^{1})dx
+\int\partial_{x}^{\ell}R_{j_{1}}^{1} \partial_{x}^{\ell}B_{j_{1},j_{1}}^{1,1,5}\big(\phi_{c}, (\sum_{m=1}^{7}D_{m})_{j_{1}}\big)dx\Big]\\
=&\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}).
\end{split}
\end{equation*}
Then we have the estimate
\begin{equation*}
\begin{split}
\sum_{n=1}^{5}I_{1}^{n}=&-\frac{\epsilon^{2}(\ell+1/2)}{2}\frac{d}{dt}\int \frac{2q^{2}(-\phi_{c}+\partial_{x}^{2}\phi_{c})(2\phi_{1}-\epsilon\mathcal{G})}{2+\epsilon\phi_{1}}
\big(\partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1})\big)^{2}dx\\
&+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}).
\end{split}
\end{equation*}
Next, we estimate $I_{2}^{n}$.
For $I_{2}^{1}$, by \eqref{15}, \eqref{17} and Lemma \ref{L9}, we have
\begin{equation*}
\begin{split}
I_{2}^{1}=&\epsilon^{2}\sum _{j_{1}\in\{\pm1\}}\Big[
\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} B_{j_{1},-j_{1}}^{1,1,1}(\phi_{c}, \partial_{x}^{\ell}R_{-j_{1}}^{1})dx
+\int\partial_{x}^{\ell}R_{j_{1}}^{1} B_{j_{1},-j_{1}}^{1,1,1}\big(\phi_{c}, \partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{-j_{1}}\big)dx\Big]\\
&+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2})\\
=&\epsilon^{2}\sum _{j_{1}\in\{\pm1\}}\Big[
\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{j_{1}} B_{j_{1},-j_{1}}^{1,1,1}(\phi_{c}, \partial_{x}^{\ell}R_{-j_{1}}^{1})dx
+\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{-j_{1}} B_{-j_{1},j_{1}}^{1,1,1}(\phi_{c}, \partial_{x}^{\ell}R_{j_{1}}^{1})dx\Big]\\
&+\int\partial_{x}^{\ell}(\sum_{m=1}^{7}D_{m})_{-j_{1}} S_{-j_{1},j_{1}}^{1}(\phi_{c}, \partial_{x}^{\ell}R_{j_{1}}^{1})dx
+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2})\\
=:&\sum_{i=1}^{7}I_{2i}^{1}+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}).
\end{split}
\end{equation*}
For $I_{21}^{1}$, we have
\begin{equation*}
\begin{split}
I_{21}^{1}=&\frac{\epsilon^{2}}{2}\sum _{j_{1}\in\{\pm1\}}j_{1}\int\partial_{x}^{\ell+1}\big(\phi_{1}q(R_{j_{1}}^{1}-R_{-j_{1}}^{1})\big)B^{1,1,1}_{j_{1},-j_{1}}(\phi_{c} , \partial_{x}^{\ell}R_{-j_{1}}^{1})dx\\
&-j_{1}\int\partial_{x}^{\ell+1}\big(\phi_{1}q(R_{j_{1}}^{1}-R_{-j_{1}}^{1})\big)B^{1,1,1}_{-j_{1},j_{1}}(\phi_{c} , \partial_{x}^{\ell}R_{j_{1}}^{1})dx\\
=&0.
\end{split}
\end{equation*}
For $I_{22}^{1}$, we have
\begin{equation*}
\begin{split}
I_{22}^{1}=&\frac{\epsilon^{2}}{2}\sum _{j_{1}\in\{\pm1\}}\int\partial_{x}^{\ell+1}\big(q\phi_{2}(R_{j_{1}}^{1}+R_{-j_{1}}^{1})\big)B^{1,1,1}_{j_{1},-j_{1}}(\phi_{c} , \partial_{x}^{\ell}R_{-j_{1}}^{1})dx\\
&+\int\partial_{x}^{\ell+1}\big(q\phi_{2}\big(R_{j_{1}}^{1}+R_{-j_{1}}^{1})\big)B^{1,1,1}_{-j_{1},j_{1}}(\phi_{c} , \partial_{x}^{\ell}R_{j_{1}}^{1})dx\\
=&\epsilon^{2}\sum _{j_{1}\in\{\pm1\}}\int\partial_{x}^{\ell+1}\big(q\phi_{2}(R_{j_{1}}^{1}+R_{-j_{1}}^{1})\big)B^{1,1,1}_{-j_{1},j_{1}}(\phi_{c} , \partial_{x}^{\ell}R_{j_{1}}^{1})dx\\
=&-\frac{\epsilon^{2}}{2}\sum _{j_{1}\in\{\pm1\}}\int\phi_{c}q\phi_{2}\partial_{x}^{\ell}R_{j_{1}}^{1}\partial_{x}^{\ell+1}R_{-j_{1}}^{1}dx
+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2})\\
=&\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}).
\end{split}
\end{equation*}
For $I_{23}^{1}$, we have
\begin{equation*}
\begin{split}
I_{23}^{1}=&\frac{\epsilon^{2}}{2}\sum _{j_{1}\in\{\pm1\}}\int\frac{1}{q}\partial_{x}^{\ell+1}(q\phi_{2}q\big(R_{j_{1}}^{1}-R_{-j_{1}}^{1})\big)B^{1,1,1}_{j_{1},-j_{1}}(\phi_{c} , \partial_{x}^{\ell}R_{-j_{1}}^{1})dx\\
&+\int\frac{1}{q}\partial_{x}^{\ell+1}\big(q\phi_{2}q(R_{j_{1}}^{1}-R_{-j_{1}}^{1})\big)B^{1,1,1}_{-j_{1},j_{1}}(\phi_{c} , \partial_{x}^{\ell}R_{j_{1}}^{1})dx\\
=&0.
\end{split}
\end{equation*}
For $I_{24}^{1}$, we have
\begin{equation*}
\begin{split}
I_{24}^{1}=&\frac{\epsilon^{2}}{2}\sum _{j_{1}\in\{\pm1\}}\Big[-j_{1}\int\frac{1}{q}\partial_{x}^{\ell+1}\big(\phi_{1}(R_{j_{1}}^{1}+R_{-j_{1}}^{1})\big)B^{1,1,1}_{j_{1},-j_{1}}(\phi_{c} , \partial_{x}^{\ell}R_{-j_{1}}^{1})dx\\
&+j_{1}\int\frac{1}{q}\partial_{x}^{\ell+1}\big(\phi_{1}(R_{j_{1}}^{1}+R_{-j_{1}}^{1})\big)B^{1,1,1}_{-j_{1},j_{1}}(\phi_{c} , \partial_{x}^{\ell}R_{j_{1}}^{1})dx\Big]\\
=&\epsilon^{2}\sum _{j_{1}\in\{\pm1\}}j_{1}\int\frac{1}{q}\partial_{x}^{\ell+1}\big(\phi_{1}(R_{j_{1}}^{1}+R_{-j_{1}}^{1})\big)B^{1,1,1}_{-j_{1},j_{1}}(\phi_{c} , \partial_{x}^{\ell}R_{j_{1}}^{1})dx\\
=&-\frac{\epsilon^{2}}{2}\sum _{j_{1}\in\{\pm1\}}j_{1}\int\phi_{c}\phi_{1}\partial_{x}^{\ell}R_{j_{1}}^{1}\partial_{x}^{\ell+1}R_{-j_{1}}^{1}dx
+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2})\\
=&-\frac{\epsilon^{2}}{2}\int\phi_{c}\phi_{1}\partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1})\partial_{x}^{\ell+1}(R_{1}^{1}-R_{-1}^{1})dx
+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2})\\
=&-\frac{\epsilon^{2}}{2}\frac{d}{dt}\int\frac{\phi_{c}\phi_{1}}{2+\epsilon\phi_{1}}(\partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1}))^{2}
+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}).
\end{split}
\end{equation*}
For $I_{25}^{1}$, we have
\begin{equation*}
\begin{split}
I_{25}^{1}=&\frac{\epsilon^{2}}{2}\sum _{j_{1}\in\{\pm1\}}-j_{1}\int\frac{1}{q}\partial_{x}^{\ell+1}\langle|\partial_{x}|\rangle^{-2}
\big(\langle|\partial_{x}|\rangle^{-2}\phi_{1}\langle|\partial_{x}|\rangle^{-2}(R_{j_{1}}^{1}+R_{-j_{1}}^{1})\big)B^{1,1,1}_{j_{1},-j_{1}}(\phi_{c} , \partial_{x}^{\ell}R_{-j_{1}}^{1})dx\\
&+j_{1}\int\frac{1}{q}\partial_{x}^{\ell+1}\langle|\partial_{x}|\rangle^{-2}
\big(\langle|\partial_{x}|\rangle^{-2}\phi_{1}\langle|\partial_{x}|\rangle^{-2}(R_{j_{1}}^{1}+R_{-j_{1}}^{1})\big)B^{1,1,1}_{-j_{1},j_{1}}(\phi_{c} , \partial_{x}^{\ell}R_{j_{1}}^{1})dx\\
=&\epsilon^{2}j_{1}\int\frac{1}{q}\partial_{x}^{\ell+1}\langle|\partial_{x}|\rangle^{-2}
\big(\langle|\partial_{x}|\rangle^{-2}\phi_{1}\langle|\partial_{x}|\rangle^{-2}(R_{j_{1}}^{1}+R_{-j_{1}}^{1})\big)B^{1,1,1}_{-j_{1},j_{1}}(\phi_{c} , \partial_{x}^{\ell}R_{j_{1}}^{1})dx\\
=&\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}).
\end{split}
\end{equation*}
For $I_{26}^{1}$, we have
\begin{equation*}
\begin{split}
I_{26}^{1}=&\frac{\epsilon^{3}}{2}\sum _{j_{1}\in\{\pm1\}}j_{1}\int\frac{1}{q}\partial_{x}^{\ell+1}\big(\mathcal{G}(R_{j_{1}}^{1}+R_{-j_{1}}^{1})\big)B^{1,1,1}_{j_{1},-j_{1}}(\phi_{c} , \partial_{x}^{\ell}R_{-j_{1}}^{1})dx\\
&-j_{1}\int\frac{1}{q}\partial_{x}^{\ell+1}\big(\mathcal{G}(R_{j_{1}}^{1}+R_{-j_{1}}^{1})\big)B^{1,1,1}_{-j_{1},j_{1}}(\phi_{c} , \partial_{x}^{\ell}R_{j_{1}}^{1})dx\\
=&-\epsilon^{3}\sum _{j_{1}\in\{\pm1\}}j_{1}\int\frac{1}{q}\partial_{x}^{\ell+1}\big(\mathcal{G}(R_{j_{1}}^{1}+R_{-j_{1}}^{1})\big)B^{1,1,1}_{-j_{1},j_{1}}(\phi_{c} , \partial_{x}^{\ell}R_{j_{1}}^{1})dx\\
=&-\frac{\epsilon^{3}}{2}\int\phi_{c}\mathcal{G}\partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1})\partial_{x}^{\ell+1}(R_{1}^{1}-R_{-1}^{1})dx
+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2})\\
=&-\frac{\epsilon^{3}}{2}\frac{d}{dt}\int\frac{\phi_{c}\mathcal{G}}{2+\epsilon\phi_{1}}(\partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1}))^{2}dx
+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}).
\end{split}
\end{equation*}
Similar to $I_{25}^{1}$, we have $I_{27}^{1}=\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2})$.
Then, we have
\begin{equation*}
\begin{split}
I_{2}^{1}
=-\frac{\epsilon^{2}}{2}\frac{d}{dt}\int\frac{\phi_{c}(\phi_{1}+\epsilon\mathcal{G})}{2+\epsilon\phi_{1}}\big(\partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1})\big)^{2}dx
+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}).
\end{split}
\end{equation*}
Similarly, by \eqref{151}, \eqref{18} and Lemma \ref{L9}, we have
\begin{equation*}
\begin{split}
I_{2}^{2}
=\frac{\epsilon^{2}}{2}\frac{d}{dt}\int\frac{\phi_{c}q\phi_{2}}{2+\epsilon\phi_{1}}\big(\partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1})\big)^{2}dx
+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}).
\end{split}
\end{equation*}
By \eqref{152}, \eqref{18} and Lemma \ref{L9}, we have
\begin{equation*}
\begin{split}
I_{2}^{3}
=\frac{\epsilon^{2}}{2}\frac{d}{dt}\int\frac{\phi_{c}q\phi_{2}}{2+\epsilon\phi_{1}}\big(\partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1})\big)^{2}dx
+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}).
\end{split}
\end{equation*}
By \eqref{153}, \eqref{17} and Lemma \ref{L9}, we have
\begin{equation*}
\begin{split}
I_{2}^{4}&=-\frac{\epsilon^{2}}{2}\frac{d}{dt}\int\frac{\phi_{c}(\phi_{2} -\epsilon\mathcal{G})}{2+\epsilon\phi_{1}}\big(\partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1})\big)^{2}dx+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}),\\
I_{2}^{5}&=\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}).
\end{split}
\end{equation*}
Then we have
\begin{equation*}
\begin{split}
\sum_{n=1}^{5}I_{2}^{n}
=-\frac{\epsilon^{2}}{2}\frac{d}{dt}\int\frac{\phi_{c}(\phi_{1}+\phi_{2} +2q\phi_{2})}{2+\epsilon\phi_{1}}\big(\partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1})\big)^{2}dx
+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}).
\end{split}
\end{equation*}
Recall \eqref{equ11-1}, similarly, by the Lemma \ref{L9}, we have
\begin{equation*}
\begin{split}
I_{3}
=\frac{\epsilon^{2}}{2}\frac{d}{dt}\int\frac{\mathcal{G}}{2+\epsilon\phi_{1}}\big(\partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1})\big)^{2}dx
+\epsilon^{2}\mathcal{O}(\mathcal{E}_{s}+\epsilon^{3/2}\mathcal{E}_{s}^{3/2}).
\end{split}
\end{equation*}
Hence, we define the modified energy
\begin{equation*}
\begin{split}
\widetilde{\mathcal{E}_{s}}=\mathcal{E}_{s}+\frac{\epsilon^{2}}{4}\sum_{\ell=1}^{s}h_{\ell},
\end{split}
\end{equation*}
with
\begin{equation*}
\begin{split}
h_{\ell}=\int_{\mathbb{R}}&\big((2\ell+1)q^{2}(-\phi_{c}+\partial_{x}^{2}\phi_{c})(2\phi_{1}-\epsilon\mathcal{G})
+\phi_{c}(\phi_{1}+\phi_{2}-2q\phi_{2})-\mathcal{G}\big)\big(\partial_{x}^{\ell}(R_{1}^{1}+R_{-1}^{1})\big)^{2}dx,
\end{split}
\end{equation*}
to obtain
\begin{equation*}
\begin{split}
\partial_{t}\widetilde{\mathcal{E}_{s}}\lesssim\epsilon^{2}(\widetilde{\mathcal{E}_{s}}
+\epsilon^{1/2}\widetilde{\mathcal{E}_{s}}^{3/2}+\epsilon\widetilde{\mathcal{E}_{s}}^{2}+1).
\end{split}
\end{equation*}
Consequently, Gronwall's inequality yields the $\mathcal{O}(1)$ boundedness of $\widetilde{\mathcal{E}_{s}}$ for all $t\in[0,T_{0}/\epsilon^{2}]$, for sufficiently small $\epsilon>0$. Theorem \ref{Thm1} then follows thanks to the fact that $\|\mathcal{R}^{0}+R^{1}\|_{H^{s}}\lesssim\sqrt{\widetilde{\mathcal{E}_{s}}}$ for sufficiently small $\epsilon>0$ and estimate \eqref{equation12-2} .
\bigskip
\paragraph
{\bf Aknowledgement.} The second author thanks Professor Yan Guo for his valuable discussions and suggestions on this paper.
| 191,109
|
\begin{document}
\title{A Note on the Rate Region of Exact-Repair Regenerating Codes}
\author{Chao Tian}
\maketitle
\begin{abstract}
The rate region of the $(5,4,4)$ exact-repair regenerating codes is provided. The outer bound is obtained through extension of the computational approach developed in an earlier work, and this region is indeed achievable using the canonical layered codes. This result is part of the online collection of ``Solutions of Computed Information Theoretic Limits (SCITL)''.
\end{abstract}
\section{Introduction}
The precise storage-repair-bandwidth tradeoff, or the rate region, of exact-repair regenerating codes turns out to be more difficult to characterize than that of the functional-repair version, the latter of which has been known for a few years \cite{Dimakis:10}. In a recent work \cite{Tian:JSAC13}, the author provided a characterization of the $(n,k,d)=(4,3,3)$ exact-repair regenerating codes, {\em i.e.,} when there are $n=4$ storage nodes, any $k=3$ of the nodes can completely recover the stored data, and any failed node can be repaired by the remaining $d=3$ nodes. This result conclusively answered in the affirmative the question whether there is a material separation between the tradeoff of the exact-repair version and that of the functional-repair version. The converse proof in this characterization was obtained using a less conventional computational approach, based on a strategic application of Yeung's linear programming formulation for information inequalities \cite{Yeung:97}.
Several analytically derived outer bounds for general exact-repair regenerating codes, or for the restricted setting of {\em linear} exact-repair regenerating codes, were later discovered in recent works \cite{Sasidharan:14, Duursma:14,Prakash:15, Mohajer:15}. In the restricted linear code setting, the rate region for the $(n,k=n-1,d=n-1)$ codes was given in \cite{Prakash:15}. For general coding functions, a partial characterization was given in \cite{Mohajer:15} for the $(5,4,4)$ codes. The complete characterization of the fundamental tradeoff for the $(5,4,4)$ case is however not yet available. In this short note, we provide the characterization of the $(5,4,4)$ exact-repair regenerating codes, the converse of which is obtained using the computational approach developed in \cite{Tian:JSAC13}. The proof is given as tabulation without a translation into the conventional chains of inequalities.
Though this particular piece of result was obtained by the author much earlier, the motivation to put it in writing was a recent conversation with Dr. Tie Liu, who pointed out that its availability in public domain may be helpful for future researchers, though from the perspective of developing the computational approach it is an extension of \cite{Tian:JSAC13} with more variables and constraints. This characterization and outer bounds for other cases of exact-repair regenerating codes ({\em e.g.,} the $(5,3,4)$ case), as well as bounds for several other storage and communication problems, are (or will be) included in the online collection of ``Solutions of Computed Information Theoretic Limits (SCITL)'' hosted at \cite{TianWebpage}.
\section{The Rate Region of $(5,4,4)$ Regenerating Codes}
\label{sec:main}
The main result of this note is the following theorem, where $\bar{\alpha}$ and $\bar{\beta}$ are the per-node storage capacity and per-helper repair bandwidth, respectively, which are normalized by the total amount of data $B$.
\begin{theorem}
The rate region of the $(5,4,4)$ exact-repair codes is the collection of $(\bar\alpha,\bar\beta)$ pairs satisfying the following conditions,
\begin{align}
4\bar{\alpha}\geq 1,\quad 3\bar\alpha+\bar\beta\geq 1, \quad 15\bar\alpha+10\bar\beta\geq 6,\quad 5\bar\alpha+10\bar\beta\geq 3,\quad 10\bar{\beta}\geq 1. \label{eqn:bounds}
\end{align}
\end{theorem}
\begin{figure}
\centering
\includegraphics[width=10cm]{result}
\caption{Rate region of $(5,4,4)$ exact-repair regenerating codes. \label{fig:region}}
\end{figure}
This region is illustrated in Fig. \ref{fig:region}. It can be verified straightforwardly that this region is achievable using the canonical layered codes proposed in \cite{Tian:14layered}. The first, the second and the last inequalities in (\ref{eqn:bounds}) are previously known by viewing the functional-repair tradeoff as an outer bound to the exact-repair case. The remaining two are proved in the next section.
This region is the same as identified in \cite{Prakash:15} for the restricted setting of linear regenerating codes, and thus there is no loss of optimality by considering only linear codes in this case.
\section{A Converse Proof by Tabulation}
We use ${\alpha}$ and ${\beta}$ to denote the per-node storage capacity and per-helper repair bandwidth, respectively, before normalization by the data size $B$. The (random) helper information sent from node $i$ to node $j$ is denoted as $S_{i\rightarrow j}$ and the (random) information stored on node $i$ is denoted as $W_i$. Due to the symmetry of the problem \cite{Tian:JSAC13}, we can restrict the proof to symmetric codes without loss of optimality.
The two inequalities are presented as two propositions, in the form of tabulation. Each inequality is given as two tables, the first of which lists the joint entropy terms in the proof, and the second of which gives the known sub-modular inequalities (Shannon-type inequalities) and the coefficients of these inequalities, whose last row, as the summation of all the other rows, is exactly the sought-after inequality.
Note that each row in the second table is a sub-modular inequality (or an integer multiple of such an inequality), possibly after permutation of the indices for each entropy term.
These tables are given in the form obtained almost directly from the computation without much human simplification. Thus in a sense they are the ``raw data'', and certain steps can be taken to make it more concise and human-friendly, the automation of which is part of our ongoing work. This is also the motivation to establish SCITL online collection, where the matrix version of the solutions can be accessed such that further data analysis can be done more conveniently, and future researchers can interpret the data in manners more meaningful to them. In addition, for more complex problems the tabulation we use here may become too cumbersome to present, and a data file is a more appropriate media.
\begin{prop}
\label{prop:bound1}
Any $(5,4,4)$ exact-repair codes must satisfy $15\alpha+10\beta\geq 6B$.
\end{prop}
\begin{proof}
See Table \ref{tab:correspondence} and Table \ref{table:cancellation}.
\end{proof}
\begin{prop}
\label{prop:bound2}
Any $(5,4,4)$ exact-repair codes must satisfy $5\alpha+10\beta\geq 3B$.
\end{prop}
\begin{proof}
See Table \ref{tab:correspondence2} and Table \ref{table:cancellation2}.
\end{proof}
\begin{table}[ct]
\begin{center}
\caption{The entropy terms used in the proof of Proposition \ref{prop:bound1}.}
\label{tab:correspondence}
\begin{tabular}{|c|c|}
\hline
$T_{ 1}$ & $\beta$ \\
$T_{ 2}$ & $\alpha$ \\
$T_{ 3}$ & $H(S_{5\rightarrow4},W_{1})$ \\
$T_{ 4}$ & $H(S_{5\rightarrow2},S_{4\rightarrow5},W_{2})$ \\
$T_{ 5}$ & $H(S_{5\rightarrow3},S_{4\rightarrow3},W_{1})$ \\
$T_{ 6}$ & $H(S_{5\rightarrow2},S_{4\rightarrow3},S_{3\rightarrow5},W_{2})$ \\
$T_{ 7}$ & $H(S_{5\rightarrow2},S_{4\rightarrow5},S_{4\rightarrow2},S_{3\rightarrow5},W_{1})$ \\
$T_{ 8}$ & $H(W_{2},W_{1})$ \\
$T_{ 9}$ & $H(S_{5\rightarrow4},W_{2},W_{1})$ \\
$T_{10}$ & $H(S_{5\rightarrow2},W_{2},W_{1})$ \\
$T_{11}$ & $H(S_{5\rightarrow2},S_{4\rightarrow5},W_{2},W_{1})$ \\
$T_{12}$ & $H(S_{5\rightarrow2},S_{4\rightarrow3},S_{3\rightarrow5},W_{2},W_{1})$ \\
$T_{13}$ & $H(S_{4\rightarrow5},W_{5},W_{2},W_{1})$ \\
$T_{14}$ & $H(S_{5\rightarrow3},S_{4\rightarrow3},W_{3},W_{2},W_{1})$ \\
$T_{15}$ & $B$ \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table*}[tcb]
\setlength{\tabcolsep}{4pt}
\begin{center}
\caption{Proof of Proposition \ref{prop:bound1} with terms defined in Table \ref{tab:correspondence}.}
\label{table:cancellation}
\begin{tabular}{|ccccc ccccc ccccc|}
\hline
$\beta$ &$\alpha$ &$T_{ 3}$ &$T_{ 4}$ &$T_{ 5}$ &$T_{ 6}$ &$T_{ 7}$ &$T_{ 8}$ &$T_{ 9}$ &$T_{10}$ &$T_{11}$ &$T_{12}$ &$T_{13}$ &$T_{14}$ &$B$ \\
\hline
$ 5$ &$ 5$ &$ -5$ & & & & & & & & & & & & \\
&$ 1$ &$ 1$ & & & & & &$ -1$ & & & & & & \\
&$ 4$ & & &$ 4$ & & & & & & & & &$ -4$ & \\
$ 5$ & &$ 5$ & &$ -5$ & & & & & & & & & & \\
&$ 2$ & & &$ 2$ & & & & & & & &$ -2$ & & \\
& & & & & & & & & &$ -2$ & &$ 2$ &$ 2$ &$ -2$ \\
&$ 2$ & & & & & &$ -1$ & & & & & & & \\
& & &$ -1$ & &$ 1$ & & & & &$ 1$ &$ -1$ & & & \\
& & & & &$ -1$ & & & & &$ 1$ & & &$ 1$ &$ -1$ \\
& & & &$ -1$ & &$ 1$ & &$ 1$ & & & & & &$ -1$ \\
&$ 1$ & & & & & & & &$ 1$ & & & & &$ -1$ \\
& &$ -1$ &$ 1$ & & & &$ 1$ & &$ -1$ & & & & & \\
& & & & & &$ -1$ & & & & &$ 1$ & &$ 1$ &$ -1$ \\
\hline
\hline
$10$ &$15$ & & & & & & & & & & & & & -6 \\\hline
\end{tabular}
\end{center}
\end{table*}
\begin{table}[ct]
\begin{center}
\caption{The entropy terms used in the proof of Proposition \ref{prop:bound2}.}
\label{tab:correspondence2}
\begin{tabular}{|c|c|}
\hline
$T_{ 1}$ & $\beta$ \\
$T_{ 2}$ & $H(S_{5\rightarrow3},S_{4\rightarrow3})$ \\
$T_{ 3}$ & $H(S_{5\rightarrow4},S_{5\rightarrow3},S_{4\rightarrow3})$ \\
$T_{ 4}$ & $H(S_{5\rightarrow4},S_{4\rightarrow5},S_{3\rightarrow5})$ \\
$T_{ 5}$ & $H(S_{5\rightarrow4},S_{5\rightarrow2},S_{4\rightarrow2},S_{3\rightarrow2})$ \\
$T_{ 6}$ & $H(S_{5\rightarrow4},S_{5\rightarrow1},S_{4\rightarrow1},S_{3\rightarrow1},S_{2\rightarrow1})$ \\
$T_{ 7}$ & $\alpha$ \\
$T_{ 8}$ & $H(S_{5\rightarrow4},W_{1})$ \\
$T_{ 9}$ & $H(S_{5\rightarrow2},W_{2})$ \\
$T_{10}$ & $H(S_{5\rightarrow2},S_{4\rightarrow2},W_{2})$ \\
$T_{11}$ & $H(S_{5\rightarrow4},S_{4\rightarrow5},W_{1})$ \\
$T_{12}$ & $H(S_{5\rightarrow3},S_{4\rightarrow3},W_{1})$ \\
$T_{13}$ & $H(S_{5\rightarrow2},S_{4\rightarrow5},W_{2})$ \\
$T_{14}$ & $H(S_{5\rightarrow2},S_{4\rightarrow2},S_{3\rightarrow5},W_{2})$ \\
$T_{15}$ & $H(S_{5\rightarrow4},S_{5\rightarrow2},S_{4\rightarrow2},W_{2})$ \\
$T_{16}$ & $H(S_{5\rightarrow4},S_{5\rightarrow3},S_{4\rightarrow3},W_{1})$ \\
$T_{17}$ & $H(S_{5\rightarrow3},S_{4\rightarrow5},S_{4\rightarrow3},S_{3\rightarrow5},W_{1})$ \\
$T_{18}$ & $H(W_{2},W_{1})$ \\
$T_{19}$ & $H(S_{5\rightarrow2},W_{2},W_{1})$ \\
$T_{20}$ & $H(S_{5\rightarrow2},S_{4\rightarrow2},W_{2},W_{1})$ \\
$T_{21}$ & $H(S_{5\rightarrow4},S_{4\rightarrow5},W_{2},W_{1})$ \\
$T_{22}$ & $H(S_{5\rightarrow2},S_{4\rightarrow5},W_{2},W_{1})$ \\
$T_{23}$ & $H(S_{5\rightarrow1},S_{4\rightarrow2},W_{2},W_{1})$ \\
$T_{24}$ & $H(S_{5\rightarrow2},S_{4\rightarrow2},S_{3\rightarrow2},W_{1})$ \\
$T_{25}$ & $H(S_{5\rightarrow2},S_{4\rightarrow2},S_{3\rightarrow5},W_{2},W_{1})$ \\
$T_{26}$ & $H(S_{5\rightarrow2},S_{5\rightarrow1},S_{4\rightarrow2},S_{3\rightarrow2},W_{1})$ \\
$T_{27}$ & $H(S_{5\rightarrow4},S_{5\rightarrow2},S_{4\rightarrow2},S_{3\rightarrow2},W_{1})$ \\
$T_{28}$ & $H(S_{5\rightarrow2},S_{5\rightarrow1},S_{4\rightarrow5},S_{4\rightarrow2},S_{3\rightarrow2},W_{1})$ \\
$T_{29}$ & $H(S_{4\rightarrow5},W_{5},W_{2},W_{1})$ \\
$T_{30}$ & $H(S_{5\rightarrow3},S_{4\rightarrow3},W_{2},W_{1})$ \\
$T_{31}$ & $H(S_{5\rightarrow3},S_{5\rightarrow2},S_{4\rightarrow3},W_{2},W_{1})$ \\
$T_{32}$ & $H(S_{5\rightarrow3},S_{5\rightarrow2},S_{4\rightarrow3},S_{4\rightarrow2},W_{2},W_{1})$ \\
$T_{33}$ & $B$ \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{landscape}
\begin{table*}[tcb]
\setlength{\tabcolsep}{2.7pt}
\begin{center}
\caption{Proof of Proposition \ref{prop:bound2} with terms defined in Table \ref{tab:correspondence2}.}
\label{table:cancellation2}
\begin{tabular}{|ccccc ccccc ccccc ccccc ccccc ccccc ccc|}
\hline
$\beta$ &$T_{ 2}$ &$T_{ 3}$ &$T_{ 4}$ &$T_{ 5}$ &$T_{ 6}$ &$\alpha$ &$T_{ 8}$ &$T_{ 9}$ &$T_{10}$ &$T_{11}$ &$T_{12}$ &$T_{13}$ &$T_{14}$ &$T_{15}$ &$T_{16}$ &$T_{17}$ &$T_{18}$ &$T_{19}$ &$T_{20}$ &$T_{21}$ &$T_{22}$ &$T_{23}$ &$T_{24}$ &$T_{25}$ &$T_{26}$ &$T_{27}$ &$T_{28}$ &$T_{29}$ &$T_{30}$ &$T_{31}$ &$T_{32}$ &$B$ \\
\hline
$ 1$ & & & & & & & & & & &$ 1$ & & & & & & & &$ -1$ & & & & & & & & & & & & & \\
&$ 9$ & & & & &$ 9$ & & & & &$ -9$ & & & & & & & & & & & & & & & & & & & & & \\
$ 22$ &$-11$ & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\
& & & & & & &$ -1$ & & & &$ 2$ & & & & & & & & & & & &$ -1$ & & & & & & & & & \\
& & & & & & &$ -3$ & & & &$ 6$ & & & & & & & &$ -3$ & & & & & & & & & & & & & \\
&$ 3$ & & & & & & & & & & & & & & & &$ 3$ & & & & & & & & & & & &$ -3$ & & & \\
& & & & & & & & & & & & &$ -2$ & & & & & &$ 2$ & & &$ 2$ & & & & & &$ -2$ & & & & \\
$ -1$ & &$ 1$ & & & & &$ 1$ & & & & & & & &$ -1$ & & & & & & & & & & & & & & & & & \\
& & & & & & & & &$ -2$ & & & &$ 2$ & & & & & &$ 2$ & & & & &$ -2$ & & & & & & & & \\
& & & & & & & & & & & & & & & & & &$ -2$ & & &$ 2$ & & & & & & &$ 2$ & & & &$ -2$ \\
& & & & & & & & & & & & & & & & & & & & &$ -2$ & & &$ 2$ & & & & &$ 2$ &$ -2$ & & \\
& & & & & & & & & & & & & & & & &$ -2$ &$ 4$ & & & &$ -2$ & & & & & & & & & & \\
$ -1$ & & & & & &$ 1$ & &$ 1$ & & & & & & & & &$ -1$ & & & & & & & & & & & & & & & \\
& &$ -1$ & & & & &$ 1$ & & & & & & &$ 1$ & & & &$ -1$ & & & & & & & & & & & & & & \\
& & &$ -1$ & & & & & &$ 1$ &$ 1$ & & & & & & & &$ -1$ & & & & & & & & & & & & & & \\
& & & & &$ -1$ & & & & & & & & & & & & & & & & & &$ 1$ & &$ 1$ & & & &$ -1$ & & & \\
&$ -1$ & &$ 1$ & & & &$ 1$ & & & & &$ -1$ & & & & & & & & & & & & & & & & & & & & \\
$ -1$ & & & &$ 1$ & & &$ 1$ & & & & & & & & & & & & & & & & & & &$ -1$ & & & & & & \\
& & & &$ -1$ &$ 1$ & & & & & & & & & &$ 1$ & & & & & & & & & & &$ -1$ & & & & & & \\
& & & & & & & & & &$ -1$ & & & & & &$ 1$ & & & &$ 1$ & & & & & & & & & & & &$ -1$ \\
& & & & & & & & & & & & & & & & & & & & & & & & &$ -1$ & &$ 1$ & & &$ 1$ & &$ -1$ \\
& & & & & & & & & & & & & & & & & & & & & & & & & & &$ -1$ & & &$ 1$ &$ 1$ &$ -1$ \\
& & & & & & & & & & & & & & & &$ -1$ & & & & & & & & & &$ 2$ & & & & &$ -1$ & \\
& & & & & & & & & & & & & & & & & & & &$ -1$ & & & & & & & & &$ 2$ & & &$ -1$ \\
& & & & & & & &$ -1$ &$ 1$ & & &$ 1$ & &$ -1$ & & & & & & & & & & & & & & & & & & \\
\hline
\hline
$20$ & & & & & & $10$ & & & & & & & & & & & & & & & & & & & & & & & & & & -6 \\
\hline
\end{tabular}
\end{center}
\end{table*}
\end{landscape}
\section{Conclusion}
The rate region of the $(5,4,4)$ exact-repair regenerating codes is characterized in this note. This is part of the online collection of ``Solutions of Computed Information Theoretic Limits (SCITL)'' hosted at \cite{TianWebpage}, which hopefully in the future can serve as a data depot for information theoretic limits obtained through computational approaches. Several results in this collection requires non-trivial generalization or variation of the approach outlined in \cite{Tian:JSAC13}, the details of which will be presented elsewhere. We welcome contributions from researchers in the field who have developed information-theoretic bounds using computational approaches, either through approaches similarly to or completely different from that in \cite{Tian:JSAC13}.
\bibliographystyle{IEEEbib}
| 31,036
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New Delhi [India], Oct 10 (ANI) Three members of a family were found dead in New Delhi's Vasant Kunj area on Wednesday morning.
Mithlesh, his wife Siya and their daughter Neha were found with stab wounds, while their teenaged.
According to Mithilesh's sister, Suraj was kidnapped a few years ago.
"Suraj is not in a condition to speak anything right now. He was even kidnapped three-four years ago. He was 12-13 years old at that time," she said.
Further investigation in the matter is underway. (ANI)
Couple, daughter stabbed to death in Delhi's Vasant Kunj
ANI | Updated: Oct 10, 2018 12:39 IST
New Delhi [India], Oct 10 (ANI) Three members of a family were found dead in New Delhi's Vasant Kunj area on Wednesday morning.
| 213,863
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\begin{document}
\maketitle
\begin{abstract}
This paper presents knowledge-aided space-time adaptive processing
(KA-STAP) algorithms that exploit the low-rank dominant clutter and
the array geometry properties (LRGP) for airborne radar
applications. The core idea is to exploit the fact that the clutter
subspace is only determined by the space-time steering vectors,
{red}{where the Gram-Schmidt orthogonalization approach is employed
to compute the clutter subspace. Specifically, for a side-looking uniformly
spaced linear array, the}
algorithm firstly selects a group of linearly independent space-time
steering vectors using LRGP that can represent the clutter subspace.
By performing the Gram-Schmidt orthogonalization procedure, the
orthogonal bases of the clutter subspace are obtained, followed by
two approaches to compute the STAP filter weights. To overcome the
performance degradation caused by the non-ideal effects, a KA-STAP
algorithm that combines the covariance matrix taper (CMT) is
proposed. For practical applications, a reduced-dimension version of
the proposed KA-STAP algorithm is also developed. The simulation
results illustrate the effectiveness of our proposed algorithms, and
show that the proposed algorithms converge rapidly and provide a
SINR improvement over existing methods when using a very small
number of snapshots.
\end{abstract}
\begin{keywords}
Knowledge-aided space-time adaptive processing, low-rank techniques,
array geometry, reduced-dimension methods, covariance matrix tapers.
\end{keywords}
\section{Introduction}
Space-time adaptive processing (STAP) is considered to be an
efficient tool for detection of slow targets by airborne radar
systems in strong clutter environments
\cite{JWard1994,Klemm2006,Guerci2003,Melvin2004}. However, due to
the very high degrees of freedom (DoFs), the full-rank STAP has a
slow convergence and requires about twice the DoFs of the
independent and identically distributed (IID) training snapshots to
yield an average performance loss of roughly $3$dB \cite{JWard1994}.
In real scenarios, it is hard to obtain so many IID training
snapshots, especially in heterogeneous environments. Therefore, it
is desirable to develop STAP techniques that can provide high
performance in small training support situations.
Reduced-dimension and reduced-rank methods have been considered to
counteract the slow convergence of the full-rank STAP
\cite{JWard1994,Klemm2006,Guerci2003,Melvin2004,
Haimovich1997,Guerci2002,Wang1994,JScott1997,JScott1999,delamare_esp,RuiSTAP2010,fa2011}.
These methods can reduce the number of training snapshots to twice
the reduced dimension, or twice the clutter rank {blue}{if we
assume that the degrees of freedom of the reduced dimension
correspond to the rank of the clutter}. The parametric adaptive
matched filter (PAMF) based on a multichannel autoregressive model
\cite{Roman2000} provides another alternative solution to the slow
convergence of the full-rank STAP. Furthermore, the sparsity of the
received data and filter weights has been exploited to improve the
convergence of a generalized sidelobe canceler architecture in
\cite{ZcYangTSP2011}. However, it still fundamental for radar
systems to improve the convergence performance of STAP algorithms or
reduce their required sample support in heterogeneous environments
because the number of required snapshots is large relative to those
needed in IID scenarios.
Recently developed knowledge-aided (KA) STAP algorithms have
received a growing interest and become a key concept for the next
generation of adaptive radar systems
\cite{R.Guerci2006,Rangaswamy2006,Gerlach2003,Bergin2006,Stoica2008,Xumin2011,
Ruifa2010, Melvin2006,
Xie2011,IvanW2010,Chen2008,Bidon2011,Tang2011,Wu2011}. The core idea
of KA-STAP is to incorporate prior knowledge, provided by digital
elevation maps, land cover databases, road maps, Global Positioning
System (GPS), previous scanning data and other known features, to
compute estimates of the clutter covariance matrix (CCM) with high
accuracy \cite{R.Guerci2006,Rangaswamy2006}. Among the previously
developed KA-STAP algorithms, there is a class of approaches that
exploit the prior knowledge of the clutter ridge to form the STAP
filter weights in \cite{Melvin2006,Xie2011, IvanW2010} and
\cite{Chen2008}. The authors in \cite{Melvin2006} introduced a
knowledge-aided parametric covariance estimation (KAPE) scheme by
blending both prior knowledge and data observations within a
parameterized model to capture instantaneous characteristics of the
cell under test (CUT). A modified sample matrix inversion (SMI)
procedure to estimate the CCM using a least-squares (LS) approach
has been described in \cite{Xie2011} to overcome the range-dependent
clutter non-stationarity in conformal array configurations. However,
both approaches require the pseudo-inverse calculation to estimate
the CCM and this often requires a computationally costly singular
value decomposition (SVD) \cite{Melvin2006}.
{red}{Although two weighting approaches with lower
computations are discussed in \cite{Melvin2006}, they are suboptimal
approaches to the LSE by the SVD and the performance of these
approaches relative to the LSE by the SVD depends on
{blue}{the radar system parameters, especially the array
characteristics \cite{Melvin2006}}.} Moreover, the latter approach
has not considered the situation when the prior knowledge has
uncertainties. Under the assumption of the known clutter ridge in
the angle-Doppler plane, the authors in \cite{IvanW2010} imposed the
sparse regularization to estimate the clutter covariance excluding
the clutter ridge. Although this kind of method can obtain a
high-resolution even using only one snapshot, it requires the
designer to know the exact positions of the clutter ridge resulting
in being sensitive to the prior knowledge. Furthermore, the
computational complexity caused by sparse recovery is expensive. A
data independent STAP method based on prolate spheroidal wave
functions (PSWF) has been considered in MIMO radar by incorporating
the clutter ridge \cite{Chen2008}, where the computational
complexity is significantly reduced compared with the approaches in
\cite{Melvin2006} and \cite{Xie2011}. However, it is highly
dependent on the ideal clutter subspace and is not robust against
clutter subspace mismatches.
In this paper, we propose KA-STAP algorithms using prior knowledge
of the clutter ridge that avoid the pseudo-inverse calculation,
require a low computational complexity, and mitigate the impact of
uncertainties of the prior knowledge. {red}{Specifically, for a
side-looking uniformly spaced linear array (ULA),} the proposed
method selects a group of linearly independent space-time steering
vectors that can represent the ideal clutter subspace using prior
knowledge of the dominant low-rank clutter and the array geometry
properties (LRGP). The orthogonal bases of the clutter ideal
subspace are computed by a Gram-Schmidt orthogonalization procedure
on the selected space-time steering vectors. Two robust approaches
to compute the STAP filter weights are then presented based on the
estimated clutter subspace. To overcome the performance degradation
caused by the {red}{internal} clutter motion (ICM), we employ a
covariance matrix taper (CMT) to the estimated CCM. The array
calibration methods discussed in \cite{Melvin2006} can be applied to
our proposed algorithm to mitigate the
{blue}{impact} of non-ideal factors, such as channel
mismatching.
Moreover, a reduced-dimension
version of the proposed KA-STAP algorithm is devised for practical
applications. Finally, simulation results demonstrate the
effectiveness of our proposed algorithms.
The main contributions of our paper are:
(i) A KA-STAP algorithm using {blue}{prior knowledge} of the LRGP
is proposed for airborne radar applications.
(ii) A KA-STAP combining CMT is introduced to counteract the
performance degradation caused by ICM {red}{and prior knowledge
uncertainty,}
and a reduced-dimension
version is also presented for practical applications.
{red}{Furthermore, the proposed algorithm provides evidence for the
KAPE approach to directly use the received data and the calibrated
space-time steering vectors (only the spatial taper without the
temporal taper) to compute the assumed clutter amplitude.}
(iii) A detailed comparison is presented to show the computational
complexity of the proposed and existing algorithms.
(iv) A study and comparative analysis of our proposed algorithms
including the impact of inaccurate prior knowledge and non-ideal
effects on the SINR performance, the convergence speed and the
detection performance with other STAP algorithms is carried out.
The work is organized as follows. Section II introduces the signal
model in airborne radar applications. Section III details the
approach of the proposed KA-STAP algorithms and also discusses the
computational complexity. The simulated airborne radar data are used
to evaluate the performance of the proposed algorithms in Section
IV. Section V provides the summary and conclusions.
\section{Signal Model}
The system under consideration is a side-looking pulsed Doppler
radar with a {red}{ULA} consisting of $M$ elements on the airborne
radar platform, as shown in Fig.\ref{radar_geometry_ula}. The
platform is at altitude $h_p$ and moving with constant velocity
$v_p$. The radar transmits a coherent burst of pulses at a constant
pulse repetition frequency (PRF) $f_r=1/T_r$ , where $T_r$ is the
pulse repetition interval (PRI). The transmitter carrier frequency
is $f_c=c/\lambda_c$, where $c$ is the propagation velocity and
$\lambda_c$ is the wavelength. The number of pulses in a coherent
processing interval (CPI) is $N$. The received signal from the
iso-range of interest is represented by a space-time $NM\times1$
data vector ${\bf x}$.
{red}{The received space-time clutter plus noise return
from a single range bin can be represented by} \cite{Melvin2004}
\begin{eqnarray}\label{model1}
{\bf x} = \sum^{N_a}_{m=1}\sum^{N_c}_{n=1}\sigma_{m,n}
{red}{{\bf v}(f_{s,m,n},f_{d,m,n}) \odot {\boldsymbol \alpha}(m,n)} + {\bf n},
\end{eqnarray}
where ${\bf n}$ is the Gaussian white thermal noise vector with the
noise power $\sigma^2_n$ on each channel and pulse; $N_a$ is the
number of range ambiguities; $N_c$ is the number of independent
clutter patches over the iso-range of interest; $f_{s,m,n}$ and
$f_{d,m,n}$ are the {red}{spatial and Doppler frequencies} of the
$mn$th clutter patch, respectively;
{red}{$\sigma_{m,n}$ is the complex amplitude for the
$mn$th clutter patch; ${\boldsymbol \alpha}(m,n)={\boldsymbol
\alpha}_d(m,n) \otimes {\boldsymbol \alpha}_s(m,n)$ is the
space-time random taper vector characterizing the voltage
fluctuation caused by {blue}{non-ideal factors}, such as ICM and
channel mismatch (where ${\boldsymbol \alpha}_d(m,n)$ and
${\boldsymbol \alpha}_s(m,n)$ are the temporal and spatial random
tapers);} and ${\bf v}(f_{s,m,n},f_{d,m,n})$ is the $NM\times 1$
space-time steering vector for a clutter patch with
{red}{$f_{s,m,n}$ and $f_{d,m,n}$}.
\begin{figure}[!htb]
\centering
\includegraphics[width=80mm]{fig1.eps}
\caption{
Airborne radar geometry with a ULA antenna.} \label{radar_geometry_ula}
\end{figure}
The space-time steering vector is given as the Kronecker product of
the temporal and spatial steering vectors, ${\bf
v}(f_{s,m,n},f_{d,m,n})={\bf v}_t(f_{d,m,n}) \otimes {\bf
v}_s(f_{s,m,n})$, which are given by \cite{JWard1994}
\begin{eqnarray}\label{model2}
{\bf v}_t(f_{d,m,n})= [1,\cdots, \exp(j 2\pi (N-1)f_{d,m,n})]^T,
\end{eqnarray}
\begin{eqnarray}\label{model3}
{\bf v}_s(f_{s,m,n})=[1, \cdots, \exp( j 2\pi(M-1)f_{s,m,n})]^T,
\end{eqnarray}
where $()^T$ denotes the transposition operation,
$f_{s,m,n}=\frac{d_a}{\lambda_c}\cos \theta_{m,n} \sin \phi_{m,n}$,
$f_{d,m,n}=\frac{2 v_p T_r}{\lambda_c} \cos \theta_{m,n}
\sin\phi_{m,n}$, and $d_a$ is the inter-sensor spacing of the ULA. If we stack all clutter patches' amplitudes into a vector
\begin{eqnarray}\label{model4}
{\boldsymbol \sigma}=[\sigma_{1,1},\cdots,\sigma_{1,N_c},\cdots,\sigma_{N_a,1},\cdots,\sigma_{N_a,N_c}]^T,
\end{eqnarray}
{red}{and assume there are no non-ideal factors,} then the
clutter plus noise received data denoted by (\ref{model1}) can be
{red}{described} as
\begin{eqnarray}\label{model5}
{\bf x} = {\bf x}_{c} + {\bf n} = {\bf V}{\boldsymbol \sigma} + {\bf n},
\end{eqnarray}
where ${\bf V}$ denotes the clutter space-time steering matrix,
given by
\begin{eqnarray}\label{model6}
\begin{split}
{\bf V}= & [{\bf v}(\phi_{1,1},\theta_{1,1},f_{1,1}),\cdots, \\
& {\bf v}(\phi_{1,N_c},\theta_{1,N_c},f_{1,N_c}), \cdots, \\
& {\bf v}(\phi_{N_a,1},\theta_{N_a,1},f_{N_a,1}), \cdots,\\
& {\bf v}(\phi_{N_a,N_c},\theta_{N_a,N_c},f_{N_a,N_c})].
\end{split}
\end{eqnarray}
Thus, the CCM based on (\ref{model5}) can be expressed as
\begin{eqnarray}\label{model7}
{\bf R}_{c} = {\bf V}{\boldsymbol \Sigma}{\bf V}^H,
\end{eqnarray}
where ${\boldsymbol \Sigma}=E[{\boldsymbol \sigma}{\boldsymbol
\sigma}^H]$. Under the condition that the clutter patches are
independent from each other, ${\boldsymbol \Sigma}={\rm diag}({\bf
a})$, ${\bf a}=[a_{1,1},a_{1,2},\cdots,a_{N_a, N_c}]^T$ and $a_{m,n}
= E[|{\sigma}_{m,n}|^2]$ ($m = 1,\cdots,N_a$, $n=1,\cdots,N_c$) for
the statistics of the clutter patches. Here, $E[\cdot]$ denotes the
expectation operator, ${\rm diag}({\bf a})$ stands for a diagonal
matrix with the main diagonal taken from the elements of the vector
${\bf a}$ and $()^H$ represents the conjugate transpose of a matrix.
The optimal filter weight vector on maximizing the output SINR for
the Gaussian distribution clutter which is given by the full-rank
STAP processor can be written as \cite{Melvin2004}
\begin{equation}\label{model8}
{\bf w}_{\rm opt}=\mu {\bf R}^{-1} {\bf s},
\end{equation}
where $\mu$ is a constant which does not affect the SINR
performance, ${\bf s}$ is the $NM \times1$ space-time steering
vector in the target direction, and ${\bf R}=E[{\bf x}{\bf
x}^H]={\bf R}_{c}+\sigma^2_n{\bf I}$ is the clutter plus noise
covariance matrix (${\bf I}$ is the identity matrix).
\section{KA-STAP Algorithms Using LRGP}
In this section, we firstly review the method that estimates the CCM
using a LS technique in \cite{Melvin2006,Xie2011} and point out the
existing problems of this method. Then, we detail the design and the
computational complexity of the proposed KA-STAP algorithms using
LRGP.
\subsection{CCM estimated by LS}
In practice, prior knowledge of certain characteristics of the radar
system and the aerospace platform, such as platform heading, speed
and altitude, array normal direction, and antenna phase steering,
etc., can be obtained from the Inertial Navigation Unit (INU) and
the GPS data \cite{Melvin2006,Bidon2011}. In other words, we can
obtain the values of the number of range ambiguities $N_a$, the
platform velocity $v_p$, and the elevation angle $\theta$. Thus, we
can develop KA-STAP algorithms based on {red}{these} prior
knowledge, e.g., the methods described in \cite{Melvin2006,Xie2011,
IvanW2010} and \cite{Chen2008}. {red}{In reality, the clutter
consists of returns over a continuous geographical region, which we
divide into a discrete set of clutter patches for analytical and
computational convenience. The rest of the discussion is on the
issues associated with choosing the number of clutter patches $N_c$.
A} possible approach is to assume a value of $N_c$ and discretize
the whole azimuth angle evenly into $N_c$ patches for each range bin
\cite{Melvin2006,Xie2011}. {red}{In addition, it usually ignores
range ambiguities, i.e., $N_a=1$, where the justification can be
seen in \cite{Melvin2006}.} Then, the parameter ${\boldsymbol
\sigma}$ {red}{in (\ref{model5})} can be estimated using the
observation data by solving the LS problem as follows
\cite{Melvin2006,Xie2011},
\begin{eqnarray}\label{ls5}
\hat{{\boldsymbol \sigma}} = {\rm arg}\min_{{\boldsymbol \sigma}}\|{\bf x}-{\bf V}{\boldsymbol \sigma}\|^2,
\end{eqnarray}
where {red}{$\hat{{\boldsymbol
\sigma}}=[\hat{{\sigma}}_{1},\hat{{\sigma}}_{2}, \cdots,
\hat{{\sigma}}_{N_c}]^T$.} Herein, the solution for the above
problem based on an LS technique is given by
\begin{eqnarray}\label{ls6}
\hat{{\boldsymbol \sigma}} = \big[{\bf V}^H{\bf V}\big]^{-1}{\bf V}^H{\bf x}.
\end{eqnarray}
Because ${\boldsymbol \sigma}$ depends only on the clutter
distribution, it does not vary significantly with the range under
homogeneous clutter environments. Furthermore, to avoid the effect
of the target signal at CUT, the near range bins of the CUT are used
to estimate ${\boldsymbol \sigma}$ \cite{Xie2011}, which is given by
\begin{eqnarray}\label{ls7}
\hat{{\sigma}}^2_{m,n} = \frac{1}{L}\sum^{L}_{l=1}|\hat{{\sigma}}_{m,n;l}|^2,
\end{eqnarray}
where $2L$ is the total number of the secondary data. Then, the
estimated CCM by the LS method (we call it least-squares estimator
(LSE) in the following) is
\begin{eqnarray}\label{ls8}
\hat{\bf R}_{c} = {\bf V}\hat{\boldsymbol \Sigma}{\bf V}^H.
\end{eqnarray}
{red}{Then the clutter plus noise covariance matrix is
estimated as
\begin{eqnarray}\label{ls9}
\hat{\bf R} = \hat{\bf R}_{c} + \hat{\sigma}^2_n{\bf I},
\end{eqnarray}
where $\hat{\sigma}^2_n$ is the estimated noise power level which
can be collected by the radar receiver when the radar transmitter
operates in a passive mode \cite{Klemm2006}.} Finally, the STAP
filter weights can be computed according to (\ref{model8}) using
{red}{$\hat{\bf R}$ instead of ${\bf R}$}.
However, there are several aspects that should be noted. First, the
above approach requires the designer to {red}{choose the suitable
azimuth angle $\phi$ and the suitable number $N_c$} of the clutter
patches, which are difficult to obtain in practice. The
{red}{selection of $N_c$ and $\phi$ will affect the
space-time steering vectors of the clutter patches, which affects
the estimation accuracy of the estimated CCM.} Specifically, if the
assumed number of clutter patches $N_c>NM$, then $\big[{\bf V}^H{\bf
V}\big]^{-1}$ does not exist. {red}{Second}, the computational
complexity of the terms $\big[{\bf V}^H{\bf V}\big]^{-1}$ is very
high, i.e., $O((N_c)^3)+O(N_c(NM)^2)$, which should be avoided in
practice. {red}{Two weighting approaches with lower computations
are discussed in \cite{Melvin2006}. {blue}{However, the solutions
obtained by the weighting approaches are suboptimal approximations
to the LSE obtained by the SVD. The performance of these approaches
relative to the LSE computed by the SVD depends on the radar system
parameters, especially the array characteristics}
\cite{Melvin2006}.}
{blue}{In the presence of non-ideal factors in the clutter
component and despite the inclusion of the estimated
angle-independent channel mismatch in the space-time steering
vectors ${\bf V}$ and the use of the modified ${\bf V}$ to solve the
problem (\ref{ls5}), the techniques do not consider the impact of
the temporal random taper ${\boldsymbol \alpha}_d$. Nevertheless,
the received data vector ${\bf x}$ is formed by all non-ideal
factors. Thus, whether it is suitable to compute the parameter
${\boldsymbol \sigma}$ only considering the spatial random taper is
worth being investigated, as will be discussed in Section III.C.}
\subsection{Proposed KA-STAP Algorithm}
To overcome the rank-deficiency and the inverse of the matrix ${\bf
V}^H{\bf V}$, in the following, we will detail the proposed KA-STAP
algorithm to estimate the CCM using prior knowledge of LRGP.
{red}{In this subsection, we only consider the ideal case of
the received data, i.e., the signal model in (\ref{model5}).}
From (\ref{model5}), we know that the clutter return is a linear combination of returns from all clutter patches. Thus, we have
\begin{eqnarray}\label{ebka1}
{\rm span}({\bf R}_{c}) = {\rm span}({\bf V}) = {\rm span}({\bf V}{\bf V}^H).
\end{eqnarray}
\textit{Proof:} The first equation can be obtained from
(\ref{model7}). With regard to the second equation, let us denote
the SVD of the matrix ${\bf V}$ by {red}{${\bf V}={\bf U}{\bf
C}{\bf D}^H$}. Then, we have
\begin{eqnarray}\label{ebka1.1}
\begin{split}
& {\bf U}^H{\bf V}{\bf D}\big({\bf U}^H{\bf V}{\bf D}\big)^H = {\bf C}{\bf C}^H = \tilde{\bf C} \\
\Rightarrow & {\bf U}^H\big({\bf V}{\bf V}^H\big){\bf U}=\tilde{\bf C},
\end{split}
\end{eqnarray}
where $\tilde{\bf C}={\bf C}{\bf C}^H$ is a real-valued diagonal
matrix. Thus, ${\bf U}$ is the orthogonal basis of the matrix ${\bf
V}{\bf V}^H$, i.e., ${\rm span}({\bf V}) = {\rm span}({\bf V}{\bf
V}^H)$.
Note that the orthogonal basis of the clutter subspace ${\bf U}$ can
be calculated by ${\bf V}$, or ${\bf V}{\bf V}^H$, herein we will
not need to compute that via the CCM. From (\ref{ebka1}), it also
results that the clutter subspace is independent from the power of
the clutter patches and is only determined by the clutter space-time
steering vectors. Moreover, from the above subsection, it is seen
that the clutter space-time steering vectors can be obtained using
the prior knowledge from the INU and GPS data. Therefore, it is
easier to compute the orthogonal bases of the clutter subspace ${\bf
U}$ by ${\bf V}$, or ${\bf V}{\bf V}^H$ than that by the CCM due to
the unknown power of the clutter patches. The other problem to
calculate the clutter subspace arising is that one should know the
clutter rank first. Fortunately, some rules for estimating the
clutter rank was discussed in previous literature, such as
\cite{JWard1994,Klemm2006,Zhang1997} and \cite{Nathan2007}.
Specially, for a side-looking ULA, the estimated clutter rank is
{red}{approximated} by Brennan's rule as
\begin{eqnarray}\label{ebka2}
{\rm rank}({\bf R}_{c}) \approx N_r= {red}{\lceil M + \beta(N-1) \rceil},
\end{eqnarray}
where $\beta = 2v_pT_r/d_a$ and the brackets $\lceil \rceil$
indicate rounding to the {red}{nearest largest integer}. In
\cite{Nathan2007}, this rule has been extended to the case with
arbitrary arrays. Usually, $N_r \ll NM$ and the STAP algorithms can
be performed in a low dimensional space so that the computational
complexity and the convergence can be significantly improved
\cite{Chen2008}. After the clutter rank is determined, there are
several approaches to compute the orthogonal bases of the clutter
subspace.
First, we can use the Lanczos algorithm \cite{Xu1994} applied to
${\bf V}{\bf V}^H$ to compute the clutter subspace eigenvectors. The
computational complexity of that using the Lanczos algorithm is on
the order of $O((NM)^2N_r + N^2_rNM) \ll O(N^3_c + N^2_cNM)+O(N_c(NM)^2)$. Moreover,
the computational complexity can be significantly reduced for the
case of ULA and constant PRF by exploiting the
Toepliz-block-Toeplitz structure of ${\bf V}{\bf V}^H$
\cite{Xu1994}.
Second, an alternative low-complexity approach is to perform the
Gram-Schmidt orthogonalization procedure on the space-time steering
vectors ${\bf V}$, {red}{where the implementation steps of the
Gram-Schmidt orthogonalization are listed in Table \ref{tabel.ebka}
and interested readers are referred to \cite{Horn1985} for further
details. Note that this procedure is at the computational cost of
$O(\frac{(N_c+1)N_cNM}{2}+N^2_rNM) \ll O(N^3_c +
N^2_cNM)+O(N_c(NM)^2)$. It should be also noted that the approach of
the Gram-Schmidt orthogonalization can be applied to arbitrary
arrays if we can obtain the prior knowledge of the array geometry,
some radar system parameters and some information of the platform. }
{red}{In particular, for the case of side-looking ULA, we
can further reduce the computational complexity to compute the
clutter subspace eigenvectors. Since the} dimension of the columns
of ${\bf V}$ should satisfy $N_c \gg N_r$, if we carry out the
Gram-Schmidt orthogonalization procedure on the columns of ${\bf V}$
one by one, this will result in unnecessary computations due to the
linear correlation among the columns. Thus, it is desirable to
directly find a group of vectors that are linear independent or
nearly linear independent (i.e., most of the vectors are linearly
independent and only very few vectors are linearly correlated).
{red}{Fortunately,} for a ULA we have the following
proposition.
\textit{Proposition 1:} For the case of side-looking ULA and
constant PRF, the clutter subspace belongs to the subspace computed
by a group of space-time steering vectors $\{\bar{\bf
v}_p\}^{N_r}_{p=0}$, which are given by
\begin{eqnarray}\label{ebka3}
\bar{\bf v}_p(n,m) = \exp( j 2\pi f_s (\beta n + m)),
\end{eqnarray}
where
\begin{eqnarray}\label{ebka4}
f_s = \frac{p}{N_r}, p=0,1,\cdots,N_r-1.
\end{eqnarray}
\textit{Proof:} Let us stack the above space-time steering vectors into a $N_r \times NM$ matrix $\tilde{\bf V}$, which is shown as
\begin{eqnarray}\label{ebka5}
\tilde{\bf V} = \left[\begin{array}{rccl} 1 & 1 & \cdots & 1 \\
z_{0,0} & z_{0,1} & \cdots & z_{N,M} \\
\vdots & \vdots & \vdots & \vdots \\
z^{N_r-1}_{0,0} & z^{N_r-1}_{0,1} & \cdots & z^{N_r-1}_{N,M}
\end{array}\right],
\end{eqnarray}
where
\begin{eqnarray}\label{ebka6}
z_{n,m} = \exp( j 2\pi \frac{\beta n + m}{N_r}).
\end{eqnarray}
Note that $\tilde{\bf V}$ is a Vandermonde matrix of dimension $N_r
\times NM$. {red}{For $z_{n,m}$, $n=0,\cdots,N-1$ and
$m=0,\cdots,M-1$, the number of linearly independent columns of
$\tilde{\bf V}$ is determined by the number of distinct values of
$\beta n + m$. If $\beta$ is an integer, the number of distinct
values of $\beta n + m$ is $N_r = \beta (N-1) + M$. If $\beta$ is a
rational value (not an integer), the number of distinct values of
$\beta n + m$ is larger than $N_r = \left\lceil M + \beta(N-1)
\right\rceil$.} Therefore, $\tilde{\bf V}$ has full rank, which is
equal to \cite{JWard1994}
\begin{eqnarray}\label{ebka7}
{\rm rank}(\tilde{\bf V}) = \min (N_r, NM) = N_r.
\end{eqnarray}
The dimension of the clutter subspace is also $N_r$. Herein, the
clutter subspace shares the same subspace with $\tilde{\bf V}$. We
can then compute the clutter subspace by taking the Gram-Schmidt
orthogonalization procedure on the rows of $\tilde{\bf V}$.
Moreover, it should be noted that the computational complexity of
the second approach is on the order of
{red}{$O(\frac{(N_r+1)N_rNM}{2} + N^2_rNM)\ll
O(\frac{(N_c+1)N_cNM}{2}+N^2_rNM) \ll O(N^3_c +
N^2_cNM)+O(N_c(NM)^2$,} which exhibits a much lower complexity
compared with the LSE resulting in a very useful tool for practical
applications. It also avoids the procedure to determine the values
of the number of clutter patches $N_c$ and the azimuth angle $\phi$.
After computing the orthogonal basis of the clutter subspace, we try
to design the STAP filter weights by two different kinds of methods.
One is to use the minimum norm eigen-canceler (MNE) derived in
\cite{Haimovich1997} to form the filter weights. Specifically, the
MNE method is a linearly constrained beamformer with a minimum norm
weight vector appearing orthogonal to the clutter subspace, which is
described by \cite{Haimovich1997}
\begin{eqnarray}\label{ebka5}
\begin{split}
&\quad\min_{\bf w}\quad{\bf w}^H{\bf w},\\
{\rm subject \,\, to} \quad &{\bf U}^H{\bf w}=0 \quad {\rm and} \quad {\bf w}^H{\bf s}=1,
\end{split}
\end{eqnarray}
The solution to the above optimization problem in (\ref{ebka5}) is
provided by \cite{Haimovich1997}
\begin{eqnarray}\label{ebka6}
\hat{\bf w}=\frac{\big({\bf I}-{\bf U}{\bf U}^H\big){\bf s}}{{\bf s}^H\big({\bf I}-{\bf U}{\bf U}^H\big){\bf s}}.
\end{eqnarray}
The other method tries to design the filter weights using both the
orthogonal bases of the computed clutter subspace and the
observation data. Let us first calculate the root-eigenvalues by
projecting the data on the clutter subspace ${\bf U}$, formulated as
\begin{eqnarray}\label{ebka7}
\hat{\boldsymbol \gamma}={\bf U}^H{\bf x},
\end{eqnarray}
Then, the clutter plus noise covariance matrix $\hat{\bf R}$ can be estimated by
\begin{eqnarray}\label{ebka8}
\hat{\bf R}={\bf U}\hat{\boldsymbol \Gamma}{\bf U}^H + \hat{\sigma}^2_n{\bf I},
\end{eqnarray}
where $\hat{\boldsymbol \Gamma}={\rm diag}(\hat{\boldsymbol \gamma}
\odot \hat{\boldsymbol \gamma}^\ast)$ and $\odot$ denotes the
Hadamard product. Finally, the STAP filter weights can be computed
by
\begin{eqnarray}\label{ebka9}
\hat{\bf w} = \mu {\bf U} \big(\hat{\boldsymbol \Gamma}^{-1} + \frac{1}{\hat{\sigma}^2_n}{\bf I}\big){\bf U}^H{\bf s},
\end{eqnarray}
where we use the fact that $\hat{\bf R}^{-1}={\bf U}
\big(\hat{\boldsymbol \Gamma}^{-1} + \frac{1}{\hat{\sigma}^2_n}{\bf
I}\big){\bf U}^H$. The whole procedure of the proposed KA-STAP
algorithm is summarized in Table \ref{tabel.ebka}.
\begin{table}[!ht]
\centering
\caption{The Proposed KA-STAP Algorithm}\label{tabel.ebka}
\small
\begin{tabular}{|l|}
\hline
\textbf{Initialization:}\\
$\beta = 2v_pT_r/d_a$, $N$, $M$, {red}{$\hat{\sigma^2_n}$}.\\
\hline
\textbf{Select a group of space-time steering vectors}\\
\quad $\{\bar{\bf v}_p\}^{N_r}_{p=0}$,\\
\quad \textbf{where}\quad $\bar{\bf v}_p(n,m) = \exp( j 2\pi f_s (\beta n + m))$,\\
\quad $N_r= M + \beta(N-1)$, \textbf{and} $f_s = \frac{p}{N_r}, p=0,1,\cdots,N_r-1$,\\
\hline
{red}{
\textbf{Compute calibrated space-time steering vectors}}\\
{red}{\quad Estimate $\hat{\boldsymbol \alpha}_s$ using the methods in \cite{Melvin2006},}\\
{red}{\quad where columns of $\hat{\boldsymbol \Xi}_s$ are all equivalent to
${\bf 1}_N \otimes \hat{\boldsymbol \alpha}_s$,}\\
{red}{\quad ${\bf V}_s = {\bf V} \odot \hat{\boldsymbol \Xi}_s$,}\\
{red}{\quad (In RD version, $\bar{\bf V}_s = {\bf S}^H_D{\bf V}_s$),}\\
\hline
{red}{\textbf{Compute ${\bf U}_s$}}\\
{red}{\quad ${\bf u}_{s,0}=\bar{\bf v}_{s,0}/\|\bar{\bf v}_{s,0}\|$,}\\
{red}{\quad $\tilde{\bf u}_{s,p}=\bar{\bf v}_{s,p} - \sum^{p-1}_{i=0} ({\bf u}^H_{s,i}\bar{\bf v}_{s,p}){\bf u}_{s,i}$,}\\
{red}{\quad ${\bf u}_{s,p} = \tilde{\bf u}_{s,p}/\|\tilde{\bf u}_{s,p}\|, p=1,\cdots,N_r-1$,}\\
{red}{\quad ${\bf U}_s = [{\bf u}_{s,0},\cdots,{\bf u}_{s,N_r-1}]$.}\\
{red}{\quad (In RD version, $\bar{\bf U}_s$ instead of ${\bf U}_s$),}\\
\hline
\textbf{For each snapshot $l=1,\cdots,L$}\\
{red}{\quad $\hat{\boldsymbol \gamma}_{l,s} ={\bf U}^H_s{\bf x}_l$,}\\
\hline
{red}{\textbf{Compute $\hat{\bf R}_c$}}\\
{red}{\quad $\hat{\boldsymbol \Gamma}={\rm diag}(\frac{1}{L}\sum^L_{l=1}\hat{\boldsymbol \gamma}_{l,s} \odot \hat{\boldsymbol \gamma}^\ast_{l,s})$,}\\
{red}{\quad $\hat{\bf R}_s = \hat{\bf U}_s\hat{\boldsymbol \Gamma}_s\hat{\bf U}^H_s$,}\\
{red}{\quad Estimate $\hat{\bf T}_d$,}\\
{red}{\quad $\hat{\bf R}_c = \hat{\bf R}_s \odot \hat{\bf T}_d$,}\\
{red}{\quad (In RD version, $\bar{\bf U}_s$ instead of ${\bf U}_s$, $\hat{\bar{\bf T}}_d$ instead of $\hat{\bf T}_d$)}\\
\hline
{red}{\textbf{Compute ${\bf U}$}}\\
{red}{\quad Adopt the Lanczos algorithm to $\hat{\bf R}_c$ to compute ${\bf U}$,}\\
{red}{\quad (In RD version, $\hat{\bar{\bf R}}_c$ instead of $\hat{\bf R}_c$),}\\
\hline
\textbf{Filter weights computation}\\
\quad $\hat{\bf w}=\frac{\big({\bf I}-{\bf U}{\bf U}^H\big){\bf s}}{{\bf s}^H\big({\bf I}-{\bf U}{\bf U}^H\big){\bf s}}$,\\
\textbf{Or:}\\
\quad $\hat{\bf w} = \mu {\bf U} \big(\hat{\boldsymbol \Gamma}^{-1} + \frac{1}{\hat{\sigma}^2_n}{\bf I}\big){\bf U}^H{\bf s}$.\\
{red}{\quad (In RD version, $\bar{\bf U}$ instead of ${\bf U}$)}\\
\hline
\end{tabular}
\end{table}
\subsection{Proposed KA-STAP Employing CMT}
In practice, there are many non-ideal effects, such as the internal
clutter motion (ICM) and {blue}{the channel mismatch}
\cite{Guerci2003}, which result in mismatch between the actual
clutter subspace and that computed by our proposed algorithm. In
this case, the performance of our proposed algorithm will
significantly degrade. {red}{In the following, we will detail the
proposed KA-STAP employing CMT.}
{red}{For the angle-dependent channel mismatch under normal circumstances,
the transmit and receive antenna
patterns {blue}{point} in the same direction and have a significant
maximum in the look direction. {blue}{The energy from the sidelobes
is generally several orders of magnitude lower than that from the
mainbeam}. This will lead to clutter subspace leakage mainly coming
from the main beam \cite{Guerci2003}. Thus, the angle-dependent
channel mismatch can be approximated by spatial random tapers only
related to the main beam. Since the main beam is usually fixed in a
CPI, then this random tapers can be seen as angle-independent. For
the angle-independent channel mismatch, we assume the spatial taper
${\boldsymbol \alpha}_s$ is a random vector but stable over a CPI
due to the narrowband case considered in the paper. Herein, when in
presence of channel mismatch, the clutter plus noise
{blue}{received data vector} is given by \cite{Guerci2003}
\begin{eqnarray}\label{cmt_1}
{\bf x} = ({\bf V} \odot {\boldsymbol \Xi}_s){\boldsymbol \sigma} + {\bf n},
\end{eqnarray}
where {blue}{the} columns of ${\boldsymbol \Xi}_s$ are all
equivalent to ${\bf 1}_N \otimes {\boldsymbol \alpha}_s$ and ${\bf
1}_N$ denotes the all $1$ vector with dimension $N$. When
considering ICM, the received data can be represented as
\cite{Guerci2003}
\begin{eqnarray}\label{cmt_2}
{\bf x} = ({\bf V}_s{\boldsymbol \sigma}) \odot
({\boldsymbol \alpha}_d \otimes {\bf 1}_M) + {\bf n},
\end{eqnarray}
where ${\bf V}_s = {\bf V} \odot {\boldsymbol \Xi}_s$ and
${\boldsymbol \alpha}_d$ is the temporal taper accounting for the ICM. Then,
the clutter plus noise covariance matrix is
\begin{eqnarray}\label{cmt_3}
{\bf R} = {\bf R}_s \odot {\bf T}_d
+ \sigma^2_n{\bf I},
\end{eqnarray}
where
\begin{eqnarray}\label{cmt_4}
{\bf R}_s = {\bf V}_s{\boldsymbol \Sigma}{\bf V}^H_s,
\end{eqnarray}
\begin{eqnarray}\label{cmt_5}
{\bf T}_d = E[{\boldsymbol \alpha}_d{\boldsymbol \alpha}^H_d] \otimes {\bf 1}_{M,M},
\end{eqnarray}
where ${\bf T}_d$ denotes the space-time CMT accounting for the ICM and
${\bf 1}_{M,M}$ is the $M \times M$ all $1$ matrix. In order to obtain
the clutter plus noise covariance matrix, we should estimate ${\bf R}_s$
and ${\bf T}_d$ in (\ref{cmt_3}).}
{red}{Regarding the estimation of ${\bf R}_s$,} we can
firstly use the array calibration methods discussed in
\cite{Melvin2006} to {red}{estimate the spatial taper (denoted as
$\hat{\boldsymbol \alpha}_s$)}, which will not be discussed here due
to space limitations. The reader is
{blue}{referred} to the literature \cite{Melvin2006} for
further details. {red}{Then, substituting $\hat{\boldsymbol
\alpha}_s$ into ${\bf V}_s$, we obtain the
{blue}{estimate} $\hat{\bf V}_s$. On the other hand, since
the elements of ${\boldsymbol \alpha}_d$ do not equate
{blue}{to} zeros, we assume $\bar{\boldsymbol
\alpha}_d=[\frac{1}{\alpha_{d,1}},\cdots,\frac{1}{\alpha_{d,N}}]^T$.
If we multiply both side of (\ref{cmt_2}) by $\bar{\boldsymbol
\alpha}_d \otimes {\bf 1}_M$ and use the estimate $\hat{\bf V}_s$
instead of ${\bf V}_s$, then it becomes
\begin{eqnarray}\label{cmt_6}
{\bf x}_s = {\bf x} \odot ({\boldsymbol \alpha}_d \otimes {\bf 1}_M) \approx
\hat{\bf V}_s{\boldsymbol \sigma} + {\bf n}_s,
\end{eqnarray}
where ${\bf n}_s={\bf n} \odot ({\boldsymbol \alpha}_d \otimes {\bf
1}_M)$. In this situation, similarly as the analysis in Section
III.B, we can {blue}{employ} the Gram-Schmidt orthogonalization
procedure to compute {blue}{a matrix with} eigenvectors of
$\hat{\bf V}_s$, which is denoted as $\hat{\bf U}_s$. Then the
root-eigenvalues ${\boldsymbol \gamma}_s$ can be calculated by
\begin{eqnarray}\label{cmt_7}
\hat{\boldsymbol \gamma}_s = \hat{\bf U}_s{\bf x}_s.
\end{eqnarray}
{blue}{We can then} estimate ${\bf R}_s$ as
\begin{eqnarray}\label{cmt_8}
\hat{\bf R}_s = \hat{\bf U}_s\hat{\boldsymbol \Gamma}_s\hat{\bf U}^H_s,
\end{eqnarray}
where
\begin{eqnarray}\label{cmt_9}
\begin{split}
\hat{\boldsymbol \Gamma}_s &= {\rm diag}(\hat{\boldsymbol \gamma}_s \odot \hat{\boldsymbol \gamma}^\ast_s) \\
& = {\rm diag}((\hat{\bf U}_s{\bf x}_s) \odot (\hat{\bf U}_s{\bf x}_s)^\ast) \\
& = {\rm diag}((\hat{\bf U}_s \odot \hat{\bf U}^\ast_s) ({\bf x}_s \odot {\bf x}^\ast_s)) \\
& = {\rm diag}((\hat{\bf U}_s \odot \hat{\bf U}^\ast_s) ({\bf x} \odot {\bf x}^\ast)) \\
& = {\rm diag}((\hat{\bf U}_s {\bf x}) \odot (\hat{\bf U}_s{\bf x})^\ast)
\end{split}
\end{eqnarray}
Here, it uses the fact that the amplitude of the temporal taper
caused by the ICM is one. This fact can be seen the ICM models
{blue}{reported in} \cite{JWard1994,Guerci2003}, which
will be also detailed afterwards. From (\ref{cmt_9}), we observe
that $\hat{\bf R}_s$ can be estimated using the received data ${\bf
x}$ directly without ${\boldsymbol \alpha}_d$. It also provides
evidence for the KAPE approach to directly use the received data and
the calibrated space-time steering vectors (only the spatial taper
without the temporal taper) to compute the parameter ${\boldsymbol
\sigma}$.}
{red}{Regarding the estimation of ${\bf T}_d$, it can be obtained}
via a rough knowledge of the interference environment (e.g., forest versus desert, bandwidth,
etc.)\cite{Guerci2002}. One common model, referred to as the
Billingsley model, is suitable for a land scenario. The only
parameters required to specify the clutter Doppler power spectrum
are essentially the operating wavelength and wind speed. The
operating wavelength is usually known, while the wind speed should
be estimated. Another common model, presented by J. Ward in
\cite{JWard1994}, is suitable for a water scenario. The temporal
autocorrelation of the fluctuations is Gaussian in shape with the
form:
\begin{eqnarray}\label{icm}
\zeta(m)=\exp \big \{ - \frac{8\pi^2\sigma^2_vT^2_r}{\lambda^2_c}m^2\big\},
\end{eqnarray}
where $\sigma_v$ is the variance of the clutter spectral spread in
$m^2/s^2$. In the following simulations, we consider the CMT model
of the latter one.
{red}{After computing the estimates $\hat{\bf R}_s$
and $\hat{\bf T}_d$, we can compute the CCM as $\hat{\bf R}_c = \hat{\bf R}_s \odot \hat{\bf T}_d$.
Since $\hat{\bf R}_c$ is still of low rank, we adopt
the Lanczos algorithm to compute the clutter subspace ${\bf U}$, where
the computational complexity is on the order of $O((NM)^2N'_r)$
($N'_r$ is the clutter rank of $\hat{\bf R}_c$).
Finally, the STAP filter weights are computed according to
(\ref{ebka6}) or (\ref{ebka9}). The whole procedure
can be seen in Table \ref{tabel.ebka}.}
{red}{\textit{Prior knowledge uncertainty impact.} In the proposed algorithms,
the prior knowledge uncertainty, such as velocity misalignment and yaw angle misalignment, will have a great
impact on the performance. However, the scheme that employs the CMT will mitigate this impact.
To illustrate this, we take a typical airborne radar system for example. The parameters
of the radar system are listed at the beginning of Section IV. Consider a far field
scenario, the elevation angle will be close to zero resulting in $\cos \theta \approx 1$.
Let $v_{pu}$ and $\phi_u$ denote the velocity deviation and the yaw angle deviation, respectively.
Then, for a discretized azimuth angle $\phi$, the spatial frequency $f_s$ and Doppler frequency $f_d$
can be represented as
\begin{eqnarray}\label{f_s}
f_s=\frac{d_a}{\lambda_c}\sin \phi,
\end{eqnarray}
\begin{eqnarray}\label{f_d}
f_d = \frac{2(v_p+v_{pu})T_r}{\lambda_c}\sin(\phi + \phi_u).
\end{eqnarray}
From (\ref{f_s}) and (\ref{f_d}), we see that the prior knowledge
uncertainty will affect the position and shape of the clutter ridge,
which leads to the mismatch between the exact and the assumed
space-time steering vectors. Fig.\ref{impactsprior} provides a more
direct way to illustrate the impact of prior knowledge uncertainty
to the clutter ridge in the spatio-temporal plane. By employing a
CMT, the clutter spectra will become wider along the clutter ridge
in the figure including the exact clutter ridge. From this point of
view, the impact of prior knowledge uncertainty is mitigated.
Because the methods in \cite{Xie2011,IvanW2010} and \cite{Chen2008}
do not consider any strategies to mitigate the impact of prior
knowledge uncertainty, the performance will depend highly on the
accuracy of the prior knowledge. The KAPE approach in
\cite{Melvin2006} also adopts the CMT and can mitigate the impact of
prior knowledge uncertainty in a sense. But the differences between
the proposed algorithm and the KAPE approach lie in three aspects.
First, the KAPE approach estimates the CCM using the LS or some
approximate approaches. While the proposed algorithm estimates the
CCM using the Gram-Schmidt orthogonalization procedure (that is not
an approximate approach) by exploiting that the clutter subspace is
only determined by the space-time steering vectors. Furthermore, for
a side-looking ULA radar, the proposed algorithm directly selects a
group of linearly independent space-time steering vectors using the
LRGP and then takes the Gram-Schmidt orthogonalization procedure to
compute the clutter subspace. Second, the proposed algorithm shows
evidence {blue}{that is feasible to directly use the received data
vector} and the calibrated space-time steering vectors (only the
spatial taper without the temporal taper) to compute the parameter
${\boldsymbol \sigma}$. Third, the proposed algorithm with an RD
version in the following section is presented to further reduce the
complexity. }
\begin{figure*}[ht]
\centering
\subfigure[]{\label{prior.sub1}
\includegraphics[width=7.8cm]{fig15.eps}}
\hspace{0.5in}
\subfigure[]{\label{prior.sub2}
\includegraphics[width=7.8cm]{fig16.eps}}
\caption{Impact of prior knowledge uncertainty to the clutter ridge in the spatio-temporal plane with (a) velocity deviation $v_{pu}=2$m/s and (b) $\phi_{u} = 1^\circ$.}\label{impactsprior}
\end{figure*}
\subsection{Proposed Reduced-Dimension (RD) KA-STAP Algorithms}
From the above discussions, one aspect to be noted is that it is
impractical to use all the DoFs available at the ULA for reasons of
computational complexity when $NM$ is too large. In such situations,
a common approach is to break the full DoFs problem into a number of
smaller problems via the application of an $MN \times D$ (with
$D\ll MN$) transformation matrix ${\bf S}_D$ to the data
\cite{JWard1994}. Our proposed KA-STAP algorithms can be easily
extended to this kind of approach. By applying the reduced-dimension
transformation matrix ${\bf S}_D$ to the data and the space-time
steering vectors, we obtain
\begin{eqnarray}\label{rdka1}
\bar{\bf x}={\bf S}^H_D{\bf x}, \quad \quad \bar{\bf V}={\bf S}^H_D{\bf V},
\end{eqnarray}
where $\bar{}$ denotes the results after the transformation. Then,
the reduced-dimension CCM $\bar{\bf R}_{c}$ becomes
\begin{eqnarray}\label{rdka2}
\bar{\bf R}_{c}={\bf S}^H_D{\bf R}_{c}{\bf S}_D= \bar{\bf V} {\boldsymbol \Sigma}\bar{\bf V} = \bar{\bf U} \bar{\boldsymbol \Gamma} \bar{\bf U}^H.
\end{eqnarray}
In a manner similar to that of the proposed full-DoF KA-STAP
algorithm described in Section III.B, we compute the
orthogonal bases of the clutter subspace ${\bar{\bf U}}$, estimate
the CCM $\hat{\bar{\bf R}}_{c}={\bar{\bf U}} \hat{\bar{\boldsymbol
\Gamma}} {\bar{\bf U}}^H$, and then calculate the STAP filter
weights according to (\ref{ebka6}) or (\ref{ebka9}). When employing
a CMT to the ideal clutter covariance matrix, the final RD clutter
covariance matrix can be estimated as
\begin{eqnarray}\label{rdka3}
\hat{\bar{\bf R}}_{c}= {red}{\hat{\bar{\bf R}}_s \odot \hat{\bar{\bf T}}_d
= (\hat{\bar{\bf U}}_s \hat{\bar{\boldsymbol \Gamma}}_s \hat{\bar{\bf U}}^H_s) \odot \hat{\bar{\bf T}}_d,}
\end{eqnarray}
{red}{where $\hat{\bar{\bf U}}_s$ is computed by taking
the Gram-Schmidt orthogonalization procedure to $\hat{\bar{\bf V}}_s={\bf S}^H_D\hat{\bf V}_s$,
$\hat{\bar{\boldsymbol \Gamma}}_s$ is calculated via (\ref{cmt_9}) using
$\bar{\bf x}$ and $\hat{\bar{\bf U}}_s$ instead of ${\bf x}$ and $\hat{{\bf U}}_s$,
and $\hat{\bar{\bf T}}_d$ denotes the estimated RD CMT. Again, the STAP
filter weights can be computed according to (\ref{ebka6}) or (\ref{ebka9}).}
By inspecting (\ref{rdka2}) and (\ref{rdka3}), we find that the computational
complexity of our proposed RD-KA-STAP algorithm is related to $D$
instead of $NM$ ($D \ll NM$), which leads to great computation
savings.
In this paper, we focus on the reduced-dimension technique known as
extended factored (EFA) algorithm or multibin element-space
post-Doppler STAP algorithm\cite{JWard1994}. The simulations with
this technique will show the performance of our proposed RD-KA-STAP
algorithm.
\subsection{Complexity Analysis}
Here we illustrate the computational complexity of the proposed
algorithms (shortened as LRGP KA-STAP and LRGP RD-KA-STAP) and other
existing algorithms, namely, the sample matrix inversion algorithm
(SMI), the EFA algorithm in \cite{JWard1994}, the
joint-domain-localized (JDL) algorithm in \cite{Wang1994}, the
CSMIECC algorithm in \cite{Xie2011}, and the KAPE algorithm in
\cite{Melvin2006}. In Table \ref{table.complexity}, $D$ denotes the
size of the reduced dimension. We can see that the computational
complexity of our proposed algorithms is significantly lower than
the CSMIECC and the KAPE algorithms {red}{($N_r \ll N_c, NM$)},
which require the pseudo-inverse of the matrix ${\bf V}^H{\bf V}$.
With regard to the SMI algorithm, our proposed algorithms also show
a lower computational complexity because the number of snapshots
used for training the filter weights of the SMI is in the order of
$2NM$.
\begin{table*}[ht]
\centering
\caption{Computational Complexity of Algorithms}\label{table.complexity}
\small
\begin{tabular}{|l|c|c|}
\hline
Algorithm & Estimate the CCM & Compute filter weights\\
\hline
SMI & $O\left(L(NM)^2\right)$ & $O\left((NM)^3 \right)$\\
\hline
EFA & $O\left(L(D)^2\right) + O\left(L\frac{N}{2}\log_2(N)\right)$ & $O\left(D^3 \right)$\\
\hline
JDL & $O\left(L(D)^2\right) + O\left(L\frac{NM}{2}\log_2(NM) \right)$ & $O\left(D^3 \right)$ \\
\hline
CSMIECC & {red}{$O\left(L(NM)^2\right) + O\left(N_c(NM)^2\right)+ O\left(N^3_c+ N^2_cNM \right)$} & $O\left((NM)^3 \right)$ \\
\hline
KAPE & {red}{$O\left(N_c(NM)^2\right)+ O\left(N^3_c + N^2_cNM\right)$} & $O\left((NM)^3 \right)$\\
\hline
LRGP KA-STAP & {red}{$O\left(\frac{(N_r+1)N_rNM}{2} + N^2_rNM\right) + O\left(N_r(NM)^2\right)$} & {red}{$O\left(N_r(NM)^2 \right)$}\\
\hline
LRGP RD-KA-STAP & {red}{$O\left(\frac{(N_r+1)N_rD}{2} + N^2_rD\right) + O\left(N_rD^2\right)$} & $O\left(D^3 \right)$\\
\hline
\end{tabular}
\end{table*}
Although the computational complexity of the EFA and JDL algorithms
is lower than our proposed LRGP KA-STAP algorithm, two aspects
should be noted. One is that the number of snapshots used for
training filter weights is much larger than our proposed algorithms.
The other is that the computational complexity of EFA and JDL is
proportional to the number of Doppler frequencies of interest (we
only list the computation complexity for one Doppler frequency).
While our proposed algorithms only have to compute the CCM once for
different Doppler frequencies of interest. Besides, the
computational complexity of our proposed LRGP RD-KA-STAP is lower
than the EFA since $L$ in EFA is in the order of $2D$, where $D$ is
usually larger than $N_r$.
\section{Performance Assessment}
In this section, we assess the proposed KA-STAP algorithms by
computing the output SINR performance and probability of detection
performance using simulated radar data. The output SINR is defined
by
\begin{eqnarray}\label{eqsinr}
{\rm SINR} = \frac{\left|\hat{\bf w}^H{\bf s}\right|^2}{\left|\hat{\bf w}^H{\bf R}\hat{\bf w}\right|}.
\end{eqnarray}
{red}{Throughout the simulations, unless otherwise stated}, the simulated scenarios
use the following parameters: side-looking ULA, uniform
transmit pattern, $M=8$, $N=8$, $f_c=450$MHz, $f_r = 300$Hz, $v_p=50$m/s,
{red}{$d_a=\lambda_c/2$,
$\beta = 1$, $N_r = \lceil M + \beta(N-1) \rceil = 15$,}
$h_p=9000$m, signal-to-noise ratio (SNR) of $0$dB, the
target located at $0^\circ$ azimuth with Doppler frequency $100$Hz,
clutter-to-noise ratio (CNR) of $50$dB, and unitary thermal noise
power. All presented results are averaged over $100$ independent
Monte Carlo runs.
\begin{figure*}[ht]
\centering
\subfigure[]{\label{icm.sub1}
\includegraphics[width=7.8cm]{fig2.eps}}
\hspace{0.5in}
\subfigure[]{\label{icm.sub2}
\includegraphics[width=7.8cm]{fig3.eps}}
\subfigure[]{\label{icm.sub3}
\includegraphics[width=7.8cm]{fig4.eps}}
\hspace{0.5in}
\subfigure[]{\label{icm.sub3}
\includegraphics[width=7.8cm]{fig5.eps}}
\caption{Impacts of ICM on SINR performance against Doppler frequency with $4$ snapshots and the target Doppler frequency space from $-150$ to $150$Hz. (a): $\sigma_v=0$; (b): $\sigma_v=0.05$; (c): $\sigma_v=0.1$; (d): $\sigma_v=0.5$.}\label{impactsICM}
\end{figure*}
\subsection{Impact of ICM on the SINR Performance}
In this subsection, we evaluate the impact on the SINR performance
with different ICM for our proposed algorithms. In the examples, we
consider four different ICM cases with $\sigma_v=0$,
$\sigma_v=0.05$, $\sigma_v=0.1$ and $\sigma_v=0.5$. The number of
snapshots for training is $4$. In Fig.~\ref{impactsICM}(a), (b), (c)
and (d), we show the SINR performance against the target Doppler
frequency of our proposed LRGP KA-STAP algorithm both with and
without a CMT. From the figures, we observe the following
conclusions. (i) When there is non-ICM, the proposed LRGP KA-STAP
algorithm without a CMT can obtain the optimum performance since the
computed clutter subspace is exact. However, it degrades the SINR
performance with the increase of $\sigma_v$ resulting in extra
sensitivity to the ICM. That is because the computed clutter
subspace can not represent the true clutter subspace. (ii) Our
proposed LRGP KA-STAP algorithm with a CMT illustrates a robust
characteristic to the ICM. When the estimated parameter $\sigma_v$
of CMT is correct, we can achieve the optimum SINR performance.
Furthermore, it is demonstrated the range of values of CMT mismatch
in which the estimated spreading exhibit acceptable SINR
performance, which can be useful in applications. This can be
interpreted as that the computed clutter subspace via the
application of the CMT to the ideal clutter subspace, spans a
similar space to the true clutter subspace.
\subsection{Impact of Inaccurate Prior Knowledge on the SINR Performance}
In this subsection, we focus on the impact of inaccurate prior
knowledge on the SINR performance of our proposed algorithms. In the
first example, we consider the impact of the velocity misalignment
by showing the SINR performance against the target Doppler
frequency, as shown in Fig.\ref{impactsVelocity}. Consider three
different cases: the velocity misalignments of prior knowledge are
(a) $0.5$m/s; (b) $1$m/s; (c) $2$m/s, compared with true platform
velocity. The potential Doppler frequency space from $-150$ to
$150$Hz is examined and $4$ snapshots are used to train the filter
weights. The plots show that the proposed LRGP KA-STAP algorithm
without a CMT is sensitive to the velocity misalignment, while the
LRGP KA-STAP algorithm with a CMT is robust to that. The reason for
this is that the velocity misalignment of prior knowledge will lead
to the mismatch between the computed clutter subspace and the true
clutter subspace. Although the computed clutter subspace via the CMT
can not avoid this situation, it can mitigate this impact. Because
the velocity misalignment between the clutter patches and the
platform can be seen as the Doppler spreading of the clutter
patches. Moreover, the results also show that a slightly larger
value of the estimated parameter $\sigma_v$ will result in an
improved SINR performance for the velocity misalignment case.
The evaluation of the impact caused by the yaw angle misalignment is
shown in Fig.\ref{impactsAngle}, where we also consider three
different cases: the yaw angle misalignments of prior knowledge are
(a) $0.2^\circ$; (b) $0.5^\circ$; (c) $1^\circ$. The curves also
indicate that: (i) the proposed LRGP KA-STAP algorithm without a CMT
is sensitive to the yaw angle misalignment, while the LRGP KA-STAP
algorithm with a CMT is robust to that; (ii) a slightly larger value
of the estimated parameter $\sigma_v$ will result in an improved
SINR performance. The misalignment of the yaw angle will lead to a
Doppler frequency mismatch between the radar platform and the
clutter patches. While the CMT mainly aims at mitigating the
performance degradation caused by the clutter Doppler spreading, the
CMT will lead to an improved estimated clutter subspace and will
exhibit robustness against the yaw angle misalignment.
\begin{figure*}[ht]
\centering
\subfigure[]{\label{v.sub1}
\includegraphics[width=5.5cm]{fig6.eps}}
\subfigure[]{\label{v.sub2}
\includegraphics[width=5.5cm]{fig7.eps}}
\subfigure[]{\label{v.sub3}
\includegraphics[width=5.5cm]{fig8.eps}}
\caption{Impacts of velocity misalignment of the prior knowledge on SINR performance against Doppler frequency with $4$ snapshots and the target Doppler frequency space from $-150$ to $150$Hz. (a): velocity misalignment $0.5$m/s; (b): velocity misalignment $1$m/s; (c): velocity misalignment $2$m/s.}\label{impactsVelocity}
\end{figure*}
\begin{figure*}[ht]
\centering
\subfigure[]{\label{angle.sub1}
\includegraphics[width=5.5cm]{fig9.eps}}
\subfigure[]{\label{angle.sub2}
\includegraphics[width=5.5cm]{fig10.eps}}
\subfigure[]{\label{angle.sub3}
\includegraphics[width=5.5cm]{fig11.eps}}
\caption{Impacts of yaw angle misalignment of the prior knowledge on SINR performance against Doppler frequency with $4$ snapshots and the target Doppler frequency space from $-150$ to $150$Hz. (a): yaw angle misalignment $0.2^\circ$; (b): yaw angle misalignment $0.5^\circ$; (c): yaw angle misalignment $1^\circ$.}\label{impactsAngle}
\end{figure*}
\subsection{Comparison With Conventional STAP Algorithms}
To provide further investigation about the performance of our
proposed algorithms, we compare the SINR performance versus the
snapshots of our proposed LRGP KA-STAP and LRGP RD-KA-STAP
algorithms with the Loaded SMI (LSMI), the EFA algorithm ($3$
Doppler bins), the $3 \times 3$ JDL algorithm,
{red}{Stoica's scheme in \cite{Stoica2008} (the prior
knowledge covariance matrix is computed in the same way as the
CSMIECC {blue}{algorithm}),} and the CSMIECC algorithm (the
combination parameter is set to $0.6$) in \cite{Xie2011}, where the
simulation results are shown in Fig. \ref{sinr_snapshots}. Here, we
consider a scenario of ICM with $\sigma_v=0.5$, and assume the
diagonal loading factors for all algorithms are set to the level of
the thermal noise power. The parameter $\sigma_v$ for our proposed
algorithms is {red}{assumed to} $1$. The curves in the figure
illustrate that our proposed algorithms have a very fast SINR
convergence speed which only needs three snapshots for training, and
offer significant better SINR steady-state performance compared with
the LSMI, EFA, JDL, {red}{Stoica's scheme} and CSMIECC algorithms.
This is because the proposed algorithms provide a much better
estimation of the CCM by using prior knowledge of the data, the low
clutter rank property, the geometry of the array and the
interference environment. It should be noted that the SINR
performance of the LRGP RD-KA-STAP algorithm is worse than that of
LRGP KA-STAP with full-DOFs. This is due to the fact that the
reduced DOFs will lead to lower computational complexity at the cost
of performance degradation.
\begin{figure}[!htb]
\centering
\includegraphics[width=80mm]{fig12.eps}
\caption{
SINR performance against the number of snapshots considering ICM, where $\sigma_v=0.5$.} \label{sinr_snapshots}
\end{figure}
The results in Fig.\ref{sinr_doppler} illustrate the SINR
performance versus the target Doppler frequency. The number of
snapshots used for training in the LSMI, EFA, JDL,
{red}{Stoica's scheme} and CSMIECC
algorithms is set to $48$, while $4$ in our proposed algorithms. It
is found that our proposed LRGP KA-STAP algorithm provides the best
SINR performance among all algorithms, and forms the narrowest
clutter null resulting in improved performance for the detection of
slow targets. It is also shown that the performance of the proposed
LRGP RD-KA-STAP algorithm is worse than that of LRGP KA-STAP with
full-DOFs, but better than other algorithms in most Doppler bins.
Note that although the LRGP RD-KA-STAP algorithm performs slightly
worse than other algorithms in Doppler range of $-60$ to $60$Hz, it
requires much smaller snapshots for training filter weights.
\begin{figure}[!htb]
\centering
\includegraphics[width=80mm]{fig13.eps}
\caption{ SINR performance versus the target Doppler frequency. The
number of snapshots used for training in the LSMI, EFA, JDL and
CSMIECC algorithms is set to $48$, while we only use $4$ snapshots
for our proposed algorithms.} \label{sinr_doppler}
\end{figure}
In the next example, as shown in Fig.\ref{pd_snr}, we present the
probability of detection performance versus the target SNR for all
algorithms. The false alarm rate is set to $10^{-3}$ and for
simulation purposes the threshold and probability of detection
estimates are based on $10,000$ samples. We suppose the target is
injected in the the boresight with Doppler frequency $100$Hz. We
note that the proposed algorithms provide suboptimal detection
performance using very short snapshots, but remarkably, obtain much
higher detection rate than other algorithms at an SNR level from
$-8$dB to $0$dB.
\begin{figure}[!htb]
\centering
\includegraphics[width=80mm]{fig14.eps}
\caption{ Probability of detection performance against the target
SNR. Suppose the target is injected in the the boresight with
Doppler frequency $100$Hz, and other parameters setting for all
algorithms are the same as that in the second example.}
\label{pd_snr}
\end{figure}
\section{Conclusions}
In this paper, novel KA-STAP algorithms have been proposed by using
prior knowledge of LRGP to obtain an accurate estimation of the CCM
with a very small number of snapshots. By exploiting the fact that
the clutter subspace is only determined by the space-time steering
vectors, we {red}{have developed a Gram-Schmidt orthogonalization
approach to compute the clutter subspace. In particular, for a
side-looking ULA,} we have proposed a scheme to directly select a
group of linearly independent space-time steering vectors to compute
the orthogonal bases of the clutter subspace. Compared with the LSE
algorithm, it has not only exhibited a low complexity, but also
shown a simple way to compute the CCM. To overcome the performance
degradation caused by the non-ideal effects
{red}{and the prior knowledge uncertainty}, the proposed
KA-STAP algorithm that combines the CMT has been presented and a
reduced-dimension version has been devised for practical
applications. {blue}{This has also provided evidence that is
feasible to directly use the received data vector and the calibrated
space-time steering vectors (only the spatial taper without the
temporal taper) to compute the assumed clutter amplitude.} The
simulation results have shown that our proposed algorithms
outperform other existing algorithms in terms of SINR steady-state
performance, SINR convergence speed and detection performance for a
very small number of snapshots, and also exhibit robustness against
errors in prior knowledge.
| 95,177
|
shoulder strap when you are ready to go to your next photo adventure.
| 24,614
|
\begin{document}
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\title{Liouville-Arnold integrability of the pentagram map on closed polygons}
\author{Valentin Ovsienko
\and
Richard Evan Schwartz
\and
Serge Tabachnikov}
\date{}
\maketitle
\begin{abstract}
The pentagram map is a discrete dynamical system defined on the moduli space
of polygons in the projective plane.
This map has recently attracted a considerable interest, mostly because
its connection to a number of different domains, such as:
classical projective geometry, algebraic combinatorics,
moduli spaces, cluster algebras and integrable systems.
Integrability of the pentagram map was conjectured in \cite{Sch2}
and proved in \cite{OST} for a larger space of twisted polygons.
In this paper, we prove the initial conjecture
that the pentagram map is completely integrable
on the moduli space of closed polygons.
In the case of convex polygons in the real projective plane, this result
implies the existence of a toric foliation on the moduli space.
The leaves of the foliation carry affine structure and the dynamics of the pentagram map
is quasi-periodic.
Our proof is based on an invariant Poisson structure on the space of twisted polygons.
We prove that the Hamiltonian vector fields corresponding to the monodoromy invariants
preserve the space of closed polygons and define an invariant affine structure on the level surfaces of the monodromy invariants.
\end{abstract}
\thispagestyle{empty}
\tableofcontents
\section{Introduction}
The {\it pentagram map} is a geometric
construction which carries one polygon to another.
Given an $n$-gon $P$, the vertices of
the image $T(P)$ under the pentagram map
are the intersection points of consecutive shortest diagonals of~$P$.
The left side of Figure \ref{5and6} shows the basic construction.
The right hand side shows the second iterate of the
pentagram map. The second iterate has the virtue that it acts
in a canonical way on a labeled polygon, as indicated.
The first iterate also acts on labeled polygons, but one must
make a choice of labeling scheme; see Section \ref{LEFT}.
The simplest example of the pentagram map for pentagons was considered in \cite{Mot}.
In the case of arbitrary $n$, the map was introduced in \cite{Sch1} and
further studied in \cite{Sch2,Sch3}.
The pentagram map is defined on any polygon whose points are
in general position, and also on some polygons whose points are not in general
position. One sufficient condition for the pentagram map to be
well defined is that every consecutive triple of points is not collinear.
However, this last condition is not invariant under the pentagram map.
The pentagram map commutes with projective transformations
and thereby induces a (generically defined) map
\begin{equation}
\label{TheMap}
T:\cC_n\to\cC_n
\end{equation}
where $\cC_n$ is the moduli space of projective
equivalence classes of $n$-gons in the projective plane.
Mainly we are interested in the subspace $\cC_n^0$ of
projective classes convex $n$-gons. The
pentagram map is entirely defined on
$\cC_n^0$ and preserves this subspace.
\begin{figure}[hbtp]
\centering
\includegraphics[height=150pt]{penta2.pdf}
\newline
\caption{The pentagram map and its second iterate defined on a convex $7$-gon}
\label{5and6}
\end{figure}
Note that the pentagram map can be defined over an arbitrary field.
Usually, we restrict our considerations to the geometrically natural real case
of convex $n$-gons in $\RP^2$.
However, the complex case represents a special interest since
the moduli space of $n$-gons in $\CP^2$ is a higher analog of the moduli space
$\mathcal{M}_{0,n}$.
Unless specified,
we will be using the general notation $\pP^2$ for the projective plane
and $\PGL_3$ for the group of projective transformations.
\subsection{Integrability problem and known results}
Assuming that the labeling schemes have been chosen
carefully, the map $T:\cC_5\to\cC_5$ is the identity map and
the map $T:\cC_6\to\cC_6$ is an involution.
See \cite{Sch1}.
The conjecture that the map (\ref{TheMap}) is completely integrable
was formulated roughly in \cite{Sch1} and then
more precisely in \cite{Sch2}. This conjecture was inspired
by computer experiments in the case $n=7$.
Figure \ref{Torus} presents (a two-dimensional projection of) an orbit of a convex heptagon in $\RP^2$.
\begin{figure}[hbtp]
\centering
\includegraphics[height=150pt]{orbit2.pdf}
\newline
\caption{An orbit of the pentagram map on a heptagon}
\label{Torus}
\end{figure}
The first results regarding the integrability of the pentagram map were proved for
the pentagram map defined on a larger space,~$\cP_n$, of \textit{twisted} $n$-gons.
A series of $T$-invariant functions (or first integrals)
called the \textit{monodromy invariants},
was constructed in \cite{Sch3}.
In \cite{OST} (see also \cite{OST1} for a short version),
the complete integrability of $T$ on $\cP_n$ was proved
with the help of a $T$-invariant Poisson structure,
such that the monodromy invariants Poisson-commute.
In~\cite{Sol}, F. Soloviev found a Lax representation of the pentagram map
and proved its algebraic-geometric integrability.
The space of polygons (either $\cP_n$ or $\cC_n$)
is parametrized in terms of a spectral curve with marked points and a divisor.
The spectral curve is determined by the monodromy invariants,
and the divisor corresponds to a point on a torus -- the Jacobi variety of the spectral curve. These results allow one to construct explicit solutions formulas using Riemann theta functions (i.e., the variables that determine the polygon as explicit functions of time). Soloviev also deduces the invariant Poisson bracket of~\cite{OST}
from the Krichever-Phong universal formula.
Our result below has the same dynamical implications as that of Soloviev,
in the case of real convex polygons. Soloviev's approach is by way of
algebraic integrability, and it has the advantage that it identifies the invariant tori explicitly as certain Jacobi varieties. Our proof is in the framework of Liuoville-Arnold integrability, and it is more direct and self-contained.
\subsection{The main theorem}
The main result of the present paper is to give a purely geometric proof of the following
result.
\begin{theorem}
\label{Main}
Almost every point of $\cC_n$ lies on a $T$-invariant algebraic submanifold
of dimension
\begin{equation}
\label{Dim}
d=\left\{
\begin{array}{l}n-4, \;n \; \hbox{is odd}\\
n-5, \; n \; \hbox{is even}.
\end{array}\right.
\end{equation}
that has
a $T$-invariant affine structure.
\end{theorem}
Recall that an affine structure on a $d$-dimensional manifold
is defined by a locally free action of the $d$-dimensional Abelian Lie algebra,
that is, by $d$ commuting vector fields linearly independent at every point.
In the case of convex $n$-gons in the real projective plane,
thanks to the compactness of the space established in \cite{Sch2}, our result reads:
\begin{corollary}
\label{ac}
Almost every orbit in $\cC^0_n$ lies on a finite union
of smooth $d$-dimensional tori, where $d$ is
as in equation (\ref{Dim}). The union of
these tori has
a $T$-invariant affine structure.
\end{corollary}
\noindent
Hence, the orbit of almost every convex
$n$-gon undergoes quasi-periodic motion under the pentagram map.
The above statement is precisely the integrability theorem in the
Liouville--Arnold sense~\cite{Arn}.
Let us also mention that the dimension of the invariant sets given by (\ref{Dim})
is precisely a half of the dimension of $\cC_n$, provided $n$ is odd,
which is a usual, generic, situation for an integrable system.
If $n$ is even, then $d=\frac{1}{2}\dim\cC_n-1$ so that one can talk of
``hyper-integrability''.
Our approach is based on the results of \cite{Sch3} and \cite{OST}.
We prove that the level sets of the monodromy invariants on the subspace $\cC_n\subset\cP_n$
are algebraic subvarieties of $\cC_n$ of dimension~(\ref{Dim}).
We then prove that the Hamiltonian vector fields corresponding to the
invariant functions are tangent to $\cC_n$
(and therefore to the level sets).
Finally, we prove that the Hamiltonian vector fields define an
affine structure on a generic level set. The main calculation, which
establishes the needed independence of the monodromy invariants
and their Hamiltonian vector fields, uses a
trick that is similar in spirit to tropical algebra.
One point that is worth emphasizing is that our proof does not
actually produce a symplectic (or Poisson) structure on the
space ${\cal C\/}_n$. Rather, we use the Poisson structure
on the ambient space $\cP^n$, together with the invariants, to produce
enough commuting flows on ${\cal C\/}_n$ in order to fill
out the level sets.
\subsection{Related topics}\label{Relation}
The pentagram map is a particular example of a discrete integrable system.
The main motivation for studying this map is its relations to
different subjects, such as:
a) projective differential geometry;
b) classical integrable systems and symplectic geometry;
c) cluster algebras;
d) algebraic combinatorics of Coxeter frieze patterns.
All these relations may be beneficial not only for the study of the pentagram map,
but also for the above mentioned subjects.
Let us mention here some recent developments involving the pentagram map.
\begin{itemize}
\item The relation of~$T$ to the classical Boussinesq equation was essential for \cite{OST}.
In particular, the Poisson bracket was obtained as a discretization of the
(first) Adler-Gelfand-Dickey bracket related to the Boussinesq equation.
We refer to \cite{TN,LN} and references therein for more information about
different versions of the discrete Boussinesq equation.
\item In~\cite{ST1}, surprising results of elementary projective geometry are obtained
in terms of the pentagram map, its iterations and generalizations.
\item In \cite{ST2}, special relations amongst the monodromy
invariants are established for polygons that are inscribed into
a conic.
\item In \cite{FM}, the pentagram map is related to Lie-Poisson loop groups.
\item The paper \cite{MB} concerns discretizations of Adler-Gelfand-Dickey flows as multi-dimensional generalizations of the pentagram map.
\item A particularly interesting feature of the pentagram map is its relation to
the theory of cluster algebras developed by Fomin and Zelevinsky, see \cite{FZ1}.
This relation was noticed in \cite{OST} and developed
in \cite{Gli}, where the pentagram map on the space of twisted $n$-gons is interpreted as a
sequence of cluster algebra mutations, and an explicit formula for the iterations of $T$
is calculated\footnote{This can be understood as a version of integrability
or ``complete solvability''.}.
\item The structure of cluster manifold on the space $\cC_n$ and the related notion of $2$-frieze pattern are
investigated in \cite{SVS}.
\end{itemize}
\section{Integrability on the space of twisted $n$-gons}\label{TwiS}
In this section, we explain the proof of the main result in
our paper \cite{OST},
the Liouville-Arnold integrability of the pentagram map on the space of twisted $n$-gons.
While we omit some technical details, we take the opportunity to fill a gap
in \cite{OST}: there we claimed that the monodromy invariants
Poisson commute, but our proof there had a flaw. Here we present a correct
proof of this fact.
\subsection{The space $\cP_n$}
We recall the definition of the space of twisted $n$-gons.
A \textit{twisted $n$-gon} is a
map $\phi: \Z \to\pP^2$ such that
\begin{equation}
\label{PoT}
v_{i+n}=M \circ v_i,
\end{equation}
for all $i \in \Z$ and some fixed element $M\in\PGL_3$ called the \textit{monodromy}.
We denote by $\cP_n$
the space of twisted $n$-gons modulo
projective equivalence.
The pentagram map extends to a generically
defined map $T:\cP_n\to\cP_n$. The same geometric
definition given for ordinary polygons works here (generically)
and commutes with projective transformations.
In the next section we will describe coordinates on $\cP_n$.
These coordinates identify $\cP_n$ as an open dense
subset of $\R^{2n}$. Sometimes we will simply
identify $\cP^n$ with $\R^{2n}$.
The space $\cC_n$ is much more complicated; it is an
open dense subset of a codimension $8$ subvariety of
$\R^{2n}$.
\begin{remark}
{\rm
If $n\not=3m$, then it seems useful to impose the simple
condition that $v_i,v_{i+1},v_{i+2}$ are in general
position for all $i$. With this condition,
$\cP_n$ is isomorphic to the space of
difference equations of the form
\begin{equation}
\label{recur}
V_{i}=a_i \,V_{i-1}-b_i\,V_{i-2} + V_{i-3},
\end{equation}
where $a_i,b_i\in\C$ or $\R$ are $n$-periodic:
$a_{i+n}=a_i$ and $b_{i+n}=b_i$, for all $i$.
Therefore, $\cP_n$ is just a $2n$-dimensional vector space, provided $n\not=3m$.
Let us also mention that the spectral theory of difference operators of type
(\ref{recur}) is a classical domain (see \cite{Kri} and references therein).
}
\end{remark}
\subsection{The corner coordinates}\label{coord}
\label{CORNERDEF}
\label{LEFT}
Following \cite{Sch3}, we define local coordinates $(x_1,\ldots,x_{2n})$ on the space $\cP_n$
and give the explicit formula for the pentagram map.
Recall that the (inverse) \textit{cross ratio} of $4$ collinear points
in $\pP^2$ is given by
\begin{equation}
\label{ICR}
[t_1,t_2,t_3,t_4]=
\frac{(t_1-t_2)\,(t_3-t_4)}{(t_1-t_3)\,(t_2-t_4)},
\end{equation}
where $t$ is (an arbitrary) affine parameter.
\begin{figure}[hbtp]
\centering
\includegraphics[height=1.8in]{invariants.pdf}
\newline
\caption{Definition of the corned invariants}
\label{Fig2}
\end{figure}
We define
\begin{equation}
\label{CoD}
\begin{array}{rcl}
x_{2i-1}&=&
\displaystyle
\left[
v_{i-2},\,v_{i-1},\,
\left(
(v_{i-2},v_{i-1})\cap(v_i,v_{i+1})
\right),\,
\left(
(v_{i-2},v_{i-1})\cap(v_{i+1},v_{i+2})
\right)
\right]\\[10pt]
x_{2i+0}&=&
\displaystyle
\left[
v_{i+2},\,v_{i+1},\,
\left(
(v_{i+2},v_{i+1})\cap(v_i,v_{i-1})
\right),\,
\left(
(v_{i+2},v_{i+1})\cap(v_{i-1},v_{i-2})
\right)
\right]\\[10pt]
\end{array}
\end{equation}
where $(v,w)$ stands for the line through $v,w\in\pP^2$,
see Figure \ref{Fig2}.
The functions $(x_1,\ldots,x_{2n})$ are cyclically ordered:
$x_{i+2n}=x_i$. They provide a system of local coordinates
on the space $\cP_n$ called the \textit{corner invariants}, cf.~\cite{Sch3}.
\begin{remark}
{\rm
a) The index $2i+0$ just means $2i$. The zero is present to
align the equations.
b) The right hand side of the second equation is obtained from
the right hand side of the first equation just by swapping the
roles played by $(+)$ and $(-)$. In light of this fact, it
might seem more natural to label the variables so that the
second equation defines $x_{2i+1}$ rather than $x_{2i+0}$.
The corner invariants would then be indexed by odd integers.
In Section \ref{TAN} we will present an alternate labelling
scheme which makes the indices work out better.
c) Continuing in the same vein, we remark that
there are two useful ways to label the corner
invariants. In \cite{Sch3} one uses the variables
$x_1,x_2,x_3,x_4,...$ whereas in \cite{OST,ST2} one uses
the variables $x_1,y_1,x_2,y_2,...$. The explicit
correspondence between the two labeling schemes is
$x_{2i-1}\to{}x_i,\,x_{2i}\to{}y_i$. We call the former
convention the {\it flag convention\/} whereas we call the
latter convention the {\it vertex convention\/}. The reason
for the names is that the variables $x_1,x_2,x_3,x_4$
naturally correspond to the flags of a polygon, as
we will see in Section \ref{TAN}. The variables $x_i,y_i$
correspond to the two flags incident to the $i$th vertex.
}
\end{remark}
Let us give an explicit formula for the pentagram map in the corner coordinates.
Following~\cite{OST}, we will choose the \textit{right} labelling\footnote{
To avoid this choice between the left or right labelling one can consider the square
$T^2$ of the pentagram map.
.}
of the vertices
of $T(P)$, see Figure \ref{choices}.
One then has (see \cite{Sch3}):
\begin{figure}[hbtp]
\centering
\includegraphics[height=130pt]{map.pdf}
\newline
\caption{Left and right labelling}
\label{choices}
\end{figure}
\begin{equation}
\label{ExpXEq}
T^*x_{2i-1}=x_{2i-1}\,\frac{1-x_{2i-3}\,x_{2i-2}}{1-x_{2i+1}\,x_{2i+2}},
\qquad
T^*x_{2i}=x_{2i+2}\,\frac{1-x_{2i+3}\,x_{2i+4}}{1-x_{2i-1}\,x_{2i}},
\end{equation}
where $T^*x_i$ stands for the pull-back of the coordinate functions.
\subsection{Rescaling and the spectral parameter}\label{SpeC}
Equation (\ref{ExpXEq}) has an immediate consequence:
a \textit{scaling symmetry} of the pentagram map.
Consider a one-parameter group $\R^*$ (or~$\C^*$ in the complex case)
acting on the space $\cP_n$ multiplying the coordinates
by $s$ or $s^{-1}$ according to parity:
\begin{equation}
\label{rescal}
\textstyle
R_t:\,(x_1,x_2,x_3\ldots,x_{2n})\to
(s\,x_1,\,s^{-1}\,x_2,\,s\,x_3,\,\ldots,\,s^{-1}x_{2n}).
\end{equation}
It follows from (\ref{ExpXEq}),
that the pentagram map commutes with the rescaling operation.
We will call the parameter $s$ of the rescaling symmetry the \textit{spectral parameter} since it
defines a one-parameter deformation of the monodromy, $M_s$.
Note that the notion of spectral parameter is extremely useful in the theory
of integrable systems.
\subsection{The Poisson bracket}\label{PoS}
Recall that a \textit{Poisson bracket} on a manifold is a Lie bracket $\{.,.\}$
on the space of functions satisfying the Leibniz rule:
$$
\{F,GH\}=\{F,G\}H+G\{F,H\},
$$
for all functions $F,G$ and $H$.
The Poisson bracket is an essential ingredient of the Liouville-Arnold integrability \cite{Arn}.
Define the following Poisson structure
on~$\cP_n$.
For the coordinate functions we set
\begin{equation}
\label{PoBr}
\{x_i,\,x_{i+2}\}=(-1)^i\,x_i\,x_{i+2},
\end{equation}
and all other brackets vanish.
In other words, the Poisson bracket $\{x_i,x_j\}$ of two coordinate functions
is different from zero if and only if $|i-j|=2$.
The Leibniz rule then allows one to extend the Poisson bracket to all polynomial
(and rational) functions.
Note that the Jacobi identity obviously holds.
Indeed, the bracket (\ref{PoBr}) has constant coefficients
when considered in the logarithmic coordinates $\log{}x_i$.
\begin{proposition}
\label{PreProp}
The pentagram map preserves the Poisson bracket (\ref{PoBr}).
\end{proposition}
\proof
This is an easy consequence of formula (\ref{ExpXEq}),
see \cite{OST} (Lemma 2.9), for the details.
\proofend
Recall that a Poisson structure is a way to associate a vector field to a function.
Given a function $f$ on $\cP_n$, the corresponding vector field $X_f$ is called
the \textit{Hamiltonian vector field} defined by $X_f(g)=\{f,g\}$
for every function $g$.
In the case of the bracket (\ref{PoBr}), the explicit formula is as follows:
\begin{equation}
\label{Ham}
X_f=\sum_{i-j=2}
(-1)^{\frac{i+j}{2}}\,
\,x_i\,x_j\left(
\frac{\partial{}f}{\partial{}x_i}\,\frac{\partial}{\partial{}x_j}
-\frac{\partial{}f}{\partial{}x_j}\,\frac{\partial}{\partial{}x_i}
\right).
\end{equation}
Note that the definitions of the Poisson structure in terms of
the bracket of coordinate functions (\ref{PoBr}) and in terms of the Hamiltonian vector fields (\ref{Ham}) are equivalent.
Geometrically speaking, Hamiltonian vector fields are defined as the image of the map
\begin{equation}
\label{HaMap}
X: T^*_x\,\cP_n\to{}T_x\,\cP_n
\end{equation}
at arbitrary point $x\in\cP_n$.
The kernel of $X$ at a generic point is spanned by the differentials of the Casimir functions, that is, the functions that Poisson commute with all functions.
\begin{remark}
{\rm
The cluster algebra approach of \cite{Gli} also provides a Poisson bracket,
invariant with respect to the pentagram map (see the book \cite{GSV}).
It can be checked that this cluster Poisson bracket is induced by the bracket (\ref{PoBr}).
}
\end{remark}
\subsection{The rank of the Poisson bracket and the Casimir functions}\label{PoC1}
The corank of a Poisson structure is the dimension of the kernel of the map $X$ in (\ref{HaMap}), that is, the dimension of the space generated by the differentials of the Casimir functions.
\begin{proposition}
\label{CasProp}
The Poisson bracket (\ref{PoBr}) has corank $2$ if $n$ is odd and corank $4$ is $n$ is even;
the functions
\begin{equation}
\label{casimir0}
O_n=x_1x_3\cdots{}x_{2n-1},
\qquad
E_n=x_2x_4\cdots{}x_{2n}
\end{equation}
for arbitrary $n$ and the functions
\begin{equation}
\label{casimir1}
O_{\frac{n}{2}}=
\prod_{1\leq{}i \leq\frac{n}{2}}x_{4i-1}+
\prod_{1\leq{}i \leq\frac{n}{2}} x_{4i+1},
\qquad
E_{\frac{n}{2}}=
\prod_{1\leq{}i \leq\frac{n}{2}}x_{4i} +
\prod_{1\leq{}i \leq\frac{n}{2}} x_{4i+2},
\end{equation}
for even $n$,
are the Casimirs of the Poisson bracket (\ref{PoBr}).
\end{proposition}
\proof
First, one checks that the functions (\ref{casimir0}) and (\ref{casimir1}) are indeed Casimir functions
(for arbitrary $n$ and for even $n$, respectively).
To this end, it suffices to consider the brackets of (\ref{casimir0}) and (\ref{casimir1}), if $n$ is even, with the coordinate functions $x_i$.
Second, one checks that the corank of the Poisson bracket is
equal to $2$, for odd $n$ and $4$, for even $n$.
The corank is easily calculated in the coordinates $\log{}x_i$,
see \cite{OST}, Section 2.6 for the details.
\proofend
It follows that the Casimir functions are of the form $F \left(O_n,E_n \right)$, if $n$ is odd,
and of the form $F(O_{n/2},E_{n/2},O_n,E_n)$, if $n$ is even. In both cases
the generic symplectic leaves of the Poisson structure
have dimension $4[(n-1)/2]$.
\begin{remark}
{\rm
If $n$ is even, then the Casimir functions can be written in a more simple manner:
$$
\Big\{
\prod_{1\leq{}i \leq\frac{n}{2}}x_{4i-1},
\quad
\prod_{1\leq{}i \leq\frac{n}{2}} x_{4i+1},
\quad
\prod_{1\leq{}i \leq\frac{n}{2}}x_{4i},
\quad
\prod_{1\leq{}i \leq\frac{n}{2}} x_{4i+2}
\Big\}
$$
instead of (\ref{casimir0}) and (\ref{casimir1}).
}
\end{remark}
\subsection{Two constructions of the monodromy invariants}\label{MoIn}
The second main ingredient of the Liouville-Arnold theory is a set of
Poisson-commuting invariant functions.
In this section, we recall the construction \cite{Sch3} of a set of first integrals
of the pentagram map
$$
O_1,\ldots,O_{\left[\frac{n}{2}\right]},O_n,\;
E_1,\ldots,E_{\left[\frac{n}{2}\right]},E_n
$$
called the \textit{monodromy invariants}.
In other words, we will define $n+1$ invariant function on~$\cP_n$, if $n$ is odd,
and $n+2$ invariant function on $\cP_n$, if $n$ is even.
The monodromy invariants are polynomial in the coordinates (\ref{CoD}).
Algebraic independence of these polynomials was proved in \cite{Sch3}.
Note that $O_n$ and $E_n$ are the Casimir functions (\ref{casimir0}) and, for even $n$,
the functions $O_{\frac{n}{2}}$ and $E_{\frac{n}{2}}$ are as in (\ref{casimir1}).
The indexing of the function $O_i,E_j$ corresponds to their \textit{weight}.
More precisely, we define the weight of the coordinate functions by
\begin{equation}
\label{Weight}
\left|x_{2i+1}\right|=1,
\qquad
\left|x_{2i}\right|=-1.
\end{equation}
Then, $|O_k|=k$ and $|E_k|=-k$.
We give two definitions of the monodromy invariants.
In \cite{Sch3} it is proved that the two definitions
are equivalent.
\bigskip
{\bf A}.
{\it The geometric definition}.
Given a twisted $n$-gon (\ref{PoT}), the corresponding monodromy has a unique lift
to $\SL_3$. By slightly abusing notation, we again denote this
matrix by $M$.
The two traces, $\mathrm{tr}(M)$ and $\mathrm{tr}(M^{-1})$,
are preserved by the pentagram map (this is a consequence of the projective invariance of $T$).
These traces are rational functions in the corner invariants.
Consider the following two functions:
$$
\widetilde\Omega_1=\mathrm{tr}(M)\,O_n^{\frac{2}{3}}\,E_n^{\frac{1}{3}},
\qquad
\widetilde\Omega_2=\mathrm{tr}(M^{-1})\,O_n^{\frac{1}{3}}\,E_n^{\frac{2}{3}}.
$$
It turns out that
$\widetilde \Omega_1$ and $\widetilde \Omega_2$ are
polynomials in the corner invariants (see \cite{Sch3}).
Since the pentagram map preserves the monodromy, and
$O_n$ and $E_n$ are invariants, the two functions
$\widetilde \Omega_1$ and $\widetilde \Omega_2$
are also invariants.
We then have:
\begin{equation}
\label{OandE}
\widetilde \Omega_1=\sum_{k=0}^{[n/2]} O_k,
\qquad
\widetilde \Omega_2=\sum_{k=0}^{[n/2]} E_k,
\end{equation}
where $O_k$ has weight $k$ and $E_k$ has weight $-k$
and where we set
$$
O_0=E_0=1,
$$
for the sake of convenience.
The pentagram map preserves each homogeneous component individually
because it commutes with the rescaling (\ref{rescal}).
Notice also that, if $n$ is even, then $O_\frac{n}{2}$ and $E_\frac{n}{2}$
are precisely the Casimir functions (\ref{casimir1}).
However, the invariants $O_n$ and $E_n$ do not enter the formula (\ref{OandE}).
\bigskip
{\bf B}.
{\it The combinatorial definition}.
Together with the coordinate functions $x_i$, we consider the following ``elementary monomials''
\begin{equation}
\label{uniT}
X_i:=x_{i-1}\,x_{i}\,x_{i+1},
\qquad i=1,\ldots,2n.
\end{equation}
Let $O(X,x)$ be a monomial of the form
$$
O=X_{i_1}\cdots{}X_{i_s}\,x_{j_1}\cdots{}x_{j_t},
$$
where $i_1,\ldots,i_s$ are even and $j_1,\ldots,j_t$ are odd.
Such a monomial is called \textit{admissible} if the Poisson brackets
$\{X_{i_r},X_{i_u}\}$ and $\{X_{i_r},x_{j_u}\}$ and $\{x_{j_r},x_{j_u}\}$
of all the elementary monomials entering~$O$ vanish.
The weight of the above monomial is
$$
|O|=s+t,
$$
see (\ref{Weight}).
For every admissible monomial, we also define the \textit{sign} of $O$ via
$$
\mathrm{sign}(O):=(-1)^t.
$$
The invariant $O_k$ is defined as the alternated sum of all the admissible
monomials of weight $k$:
\begin{equation}
\label{InvOk}
O_k=\sum_{|O|=k}\mathrm{sign}(O)\,O,
\qquad
k\in\left\{1,2,\ldots,\left[\frac{n}{2}\right]\right\}.
\end{equation}
It is proved in \cite{Sch3} that this definition of $O_k$ coincides with (\ref{OandE}).
\begin{example}
{\rm
The first two invariants are:
$$
O_1=\sum_{i=1}^n
\left(
X_{2i}-x_{2i+1}
\right),
\qquad
O_2=\sum_{|i-j|\geq2}
\left(
x_{2i+1}\,x_{2j+1}-
X_{2i}\,x_{2j+1}+X_{2i}\,X_{2j+2}
\right),
$$
for $n\leq5$ the above formulas simplify, see \cite{OST}.
}
\end{example}
The definition of the functions $E_k$ is exactly the same, except that
the roles of {\it even\/} and {\it odd\/} are swapped.
\begin{remark}
{\rm There is an elegant way to
define the monodromy invariants in terms of determinants.
See \cite{ST2}.}
\end{remark}
\subsection{The monodromy invariants Poisson commute}\label{PoC2}
In this section we give a complete proof of the following
result, which was claimed in \cite{OST}.
\begin{theorem}
\label{PoCThm}
The monodromy invariants Poisson commute with each other, i.e.,
$$
\{O_i,O_j\}=\{O_i,E_j\}=\{E_i,E_j\}=0,
$$
for all $i,j$ indexing the monodromy invariants.
Hence, the Hamiltonian vector fields corresponding to the monodromy
invariants $X_{O_i},\,X_{E_i}$ commute with each other.
\end{theorem}
\proof
The second statement is a consequence of the
first statement. So, we will just prove the first statement.
We begin with a prelimianry discussion of
how the Poisson bracket interacts with the
elementary monomials defined above.
The Poisson brackets of elementary monomials
\begin{equation}
\label{Element1}
\left\{X_i,X_{i+2}\right\}=(-1)^{i+1}\,X_iX_{i+2},
\qquad
\left\{X_i,X_{i+4} \right\}=(-1)^{i+1}\,X_iX_{i+4},
\end{equation}
together with
\begin{equation}
\label{Element2}
\left\{x_i,X_j \right\}=
\left\{
\begin{array}{rl}
(-1)^i\,x_iX_j, & j=i+1,\,i+2,\,i+3,\\[8pt]
(-1)^{i+1}\,x_iX_j, & j=i-3,\,i-2,\,i-1,
\end{array}
\right.
\end{equation}
immediately follow from the definition (\ref{PoBr}).
All other brackets $\{X_i,X_j\}$, as well as~$\{X_i,x_j\}$, vanish.
\newline
Now we are ready for the main argument.
Consider first the Poisson bracket $\{O_k,O_m\}$.
This is a sum of the monomials of the form
$$
m=X_{i_1}\cdots{}X_{i_s}x_{j_1}\cdots{}x_{j_t},
$$
where $i_1,\ldots,i_s$ are even and $j_1,\ldots,j_t$ are odd.
Indeed, by definition of the Poisson structure~(\ref{PoBr}), the bracket
of two monomials is proportional to their product, so that
the above bracket contains only the monomials entering $O_k$ and $O_m$.
The monomial $m$ is not necessarily admissible.
There can be squares (some $i$'s or $j$'s may coincide), but no cubes
or higher degrees.
We want to prove that the numeric coefficient of every such monomial in $\{O_k,O_m\}$ is zero.
We define an \textit{oriented graph} with the set of vertices
$\{X_{i_1},\ldots,X_{i_s},\,x_{j_1},\ldots,x_{j_t}\}$ corresponding to the elementary monomials
in $m$; the oriented arrows joining the
vertices whenever their Poisson bracket is different from zero, the orientation being given
by the sign of the bracket.
Recall that all the non-zero brackets of elementary monomials
are listed in (\ref{Element1}) and (\ref{Element2}).
\begin{lemma}
\label{Single}
If two indices coincide, $i_r=i_u$ or $j_r=j_u$, then the
corresponding connected component of the graph consists of one element.
\end{lemma}
\proof
If $i_r=i_u$, then $X_{i_r}=X_{i_u}$ belongs both to $O_i$ and $O_j$.
By the admissibility condition, this implies all the Poisson brackets of $X_{i_r}$
with the other elementary monomials from $m$ vanish.
\proofend
The above lemma allows one to assume that all the indices in $m$ are different:
$i_r\not=i_u$ and $j_r\not=j_u$.
\begin{lemma}
\label{GraphL}
The above defined graph has
(i)
no $3$-cycles;
(ii)
no vertices with more than one outgoing or ingoing arrows;
in other words, the graph does not have the following vertices:
$$
\begin{CD}
\;@<<< a @> >> \;
\qquad
\;@>>> a @<<< \;
\end{CD}
$$
where $a=X_{i_r}$ or $x_{j_u}$.
\end{lemma}
\proof
(i) Assume there is a $3$-cycle.
Then at least two of the corresponding elementary monomials
belong to the decomposition of either $O_i$ or $O_j$.
The monomials are joined by an arrow, thus their Poisson bracket does not vanish.
This leads to a contradiction since all the
monomials in $O_i$ are admissible (see Section \ref{MoIn}, definition B).
(ii) To show that no vertex of the graph
can have more than one outgoing or ingoing arrows, one has to analyze
formulas (\ref{Element1}) and (\ref{Element2}).
Since $i_r$ are even and $j_u$ are odd, a vertex~$X_{i_r}$ can be joined by an outgoing arrow
to the following vertices (provided they belong to the graph):
$X_{i_r-2},\,X_{i_r-4},\,x_{i_r+1},x_{i_r+3}$.
In all of these cases, we obtain a 3-cycle, which is a contradiction to part (i) of the lemma.
\proofend
The above lemma implies the following statement.
\begin{corollary}
\label{GraphC}
The graph has no branching (i.e., vertices with three or more adjacent arrows).
\end{corollary}
\noindent
Indeed, a branching point has more than one out- or ingoing arrows:
$$
\begin{CD}
@.@AAA \\
@> >>a@> >>
\end{CD}
\qquad
\begin{CD}
@.@AAA \\
@> >>a@<<<
\end{CD}
$$
\begin{remark}
{\rm
One can also show that the constructed graph has no $k$-cycles for arbitrary $k$, that is,
every connected component of the graph
is of type $A_k$ oriented in the standard way:
$$
\begin{CD}
a_1@> >>a_2@> >> \cdots @> >> a_k
\end{CD}
$$
but we will not use this in the proof.
}
\end{remark}
Let us finally deduce $\{O_i,O_j\}=0$ from Lemma \ref{GraphL} and Corollary \ref{GraphC}.
Every element $X_{i_r}$ and~$x_{i_u}$ in the monomial $m$ belongs either to $O_i$, or to $O_j$.
If the constructed graph contains at least three elements, then is has a fragment:
$$
\begin{CD}
a_1@> >>a_2@> >> a_3
\end{CD}
$$
where either $a_1,a_3\in O_i$ and $a_2\in O_j$ or the other way around.
It follows from the Leibniz identity that the element $a_2$ contributes twice
in $\{O_i,O_j\}$, namely in
$\{a_1,a_2\}$ and in $\{a_3,a_2\}$, with the opposite signs.
We proved $\{O_i,O_j\}=0$, except the case where the graph is
of type $A_2$, i.e., contains only two elements:
$$
\begin{CD}
a_1@> >>a_2
\end{CD}
$$
with, say, $a_1\in O_i, a_2\in O_j$.
But in this last case, the Poisson brackets of the elementary monomials $a_1$ and $a_2$
with all the other elementary monomials in $m$ vanish.
By construction of the invariants, $O_i$ and $O_j$ are symmetric
with respect to the monomials $a_1$ and $a_2$.
It follows that the monomial $m$ appears twice in $\{O_i,O_j\}$,
with the opposite signs.
This completes the proof that $\{O_i,O_j\}=0$.
The proof of $\{E_i,E_j\}=0$ is identically the same (with odd and even indices exchanged).
It remains to consider the bracket $\{O_i,E_j\}$.
We will apply the same idea and construct a graph for every monomial in $\{O_i,E_j\}$.
Recall that $E_j$ contains the admissible monomials
$E=X_{i_1}\cdots{}X_{i_s}\,x_{j_1}\cdots{}x_{j_t},$
where the indices $i_1,\ldots,i_s$ are odd and $j_1,\ldots,j_t$ are even.
Analyzing the brackets (\ref{Element1}) and (\ref{Element2}),
we see that the graph corresponding to any monomial in $\{O_i,E_j\}$
is of the form
$$
\begin{CD}
\cdots@> >>x_i@> >>X_{i+2}@> >> x_{i+4}@> >>X_{i+6}@> >>\cdots
\end{CD}
$$
and the $X$'s and $x$'s belong to the different functions.
We observe that
$$
\{x_i,\,X_{i+2}\}=-\{X_{i+1},\,x_{i+3}\},
$$
and if $x_i\in{}O_i$ and $X_{i+2}\in{}E_j$ then
$O_i$ and $E_j$ are symmetric with respect to the exchange of
$x_i$ with $X_{i+1}$ and of $X_{i+2}$ with $x_{i+3}$,
respectively.
The monomial $m$ appears twice with the opposite signs.
This completes the proof of Theorem \ref{PoCThm}.
\proofend
In \cite{Sch3} it is proved that the monodromy invariants are algebraically
independent. The argument is rather complicated, but it is very similar
in spirit to the related independence proof we give in Section \ref{INDEP}.
The algebraic independence result combines with
Theorem \ref{PoCThm} to establish the
integrability of the pentagram map on the space $\cP_n$.
Indeed, the Poisson bracket (\ref{PoBr}) defines a symplectic foliation on $\cP_n$, the
symplectic leaves being locally described as levels of the Casimir functions, see Proposition \ref{CasProp}.
The number of the remaining invariants is exactly half of the dimension of the symplectic leaves.
The classical Liouville-Arnold theorem~\cite{Arn} is then applied.
\section{Integrability on $\cC_n$ modulo a calculation}
The general plan of the proof of Theorem \ref{Main} is as follows.
\begin{enumerate}
\item
We show that the Hamiltonian vector fields on $\cP_n$ corresponding
to the monodromy invariants are \textit{tangent} to the subspace $\cC_n$,
\item
We restrict the monodromy invariants to $\cC_n$
and show that the dimension of a generic level set is $n-4$ if $n$ is odd
and $n-5$ if $n$ is even.
\item
We show that there are exactly the same number
of independent Hamiltonian vector fields.
\end{enumerate}
In this section, we prove the first statement and also show that
the dimension of the level sets is \textit{at most} $n-4$ if $n$ is odd
and $n-5$ if $n$ is even, and similarly for the number
of independent Hamiltonian vector fields.
The final step of the proof that this upper bound is
actually the lower one will be done in the next two sections.
This final step is a nontrivial calculation that
comprises the bulk of the paper.
\subsection{The Hamiltonian vector fields are tangent to $\cC_n$}\label{TanS}
The space $\cC_n$ is a subvariety of $\cP_n$ having codimension $8$.
It turns out that one can give explicit equations for this
variety. See Lemma \ref{variety}. (These equations do not
play a role in our proof, but they are useful to have.)
The following statement is essentially a consequence of Theorem \ref{PoCThm}.
This is an important step of the proof of Theorem \ref{Main}.
\begin{proposition}
\label{TanProp}
The Hamiltonian vector field on $\cP_n$ corresponding to a monodromy invariant
is tangent to $\cC_n$.
\end{proposition}
\proof
The space $\cP_n$ is foliated by isomonodromic submanifolds
that are generically of codimension $2$ and are defined by the condition that the monodromy has fixed eigenvalues. Hence the isomonodromic submanifolds can be defined as the level surfaces of two functions,
$\mathrm{tr}(M)$ and $\mathrm{tr}(M^{-1})$.
This foliation is singular, and $\cC_n$ is a singular leaf of codimension $8$.
We note that the versal deformation of $\cC_n$
is locally isomorphic to $\SL(3)$ partitioned into the conjugacy equivalence classes.
Consider a monodromy invariant, $F\,(=O_i$ or $E_i$), and its Hamiltonian vector field, $X_F$.
We know that the Poisson bracket $\{F,\,\mathrm{tr}(M)\}=0$,
since all monodromy invariants Poisson commute
and $\mathrm{tr}(M)$ is a sum of monodromy invariants.
Hence $X_F$ is tangent to the generic leaves of the isomonodromic foliation on $\cP_n$.
Let us show that $X_F$ is tangent to $\cC_n$ as well.
In a nutshell, this follows from the observation that the tangent space to $\cC_n$ at a smooth point $x_0$ is the intersection of the limiting positions of the tangent spaces to the isomonodromic leaves at points $x$ as $x$ tends to $x_0$. Assume then that $X_F$ is transverse to $\cC_n$ at point $x_0\in \cC_n$. Then $X_F$ will be also transverse to an isomonodromic leaf at some point $x$ close to $x_0$, yielding a contradiction.
More precisely, we can apply a projective transformation so that the vertices $V_1, V_2,V_3,V_4$ of a twisted $n$-gon $V_1,V_2,\dots$ become the vertices of a standard square. This gives a local identification of $\cP_n$ with the set of tuples $(V_5,\dots,V_n; M)$ where $M$ is the monodromy, the projective transformation that takes the quadruple
$(V_1,V_2,V_3,V_4)$ to $(V_{n+1}, V_{n+2}, V_{n+3}, V_{n+4}).$
The space of closed $n$-gons is characterized by the condition that $M$ is the identity. Thus we have locally identified $\cP_n$ with $\cC_n \times \SL(3)$. In particular, we have a projection $\cP_n \to \SL(3)$, and the preimage of the identity is $\cC_n$. The isomonodromic leaves project to the conjugacy equivalent classes in $\SL(3)$.
Thus our proof reduces to the following fact about the group $\SL(3)$ (which holds for $\SL(n)$ as well).
\begin{lemma}
\label{limpos}
Consider the singular foliation of $\SL(3)$ by the conjugacy equivalence classes, and let $T_X$ be the tangent space to this foliation at $X\in \SL(3)$. Then the intersection, over all $X$, of the limiting positions of the spaces $T_X$, as $X \to 1\!\!1$, is trivial (here $1\!\!1\in \SL(3)$ is the identity).
\end{lemma}
\proof Let $B\in\SL(3)$, and let $B+\varepsilon C$ be an
infinitesimal deformation within the conjugacy equivalence class. Then
$$
\mathrm{tr}\left(B+\varepsilon C\right)=
\mathrm{tr}(B), \qquad \mathrm{tr} \left((B+\varepsilon C)^2 \right)=
\mathrm{tr} \left(B^2 \right),
$$
hence $\mathrm{tr}(C)=0$ and $\mathrm{tr}(BC)=0$,
and also $\mathrm{tr}(B^{-1} C)=0$ since $\det (B+\varepsilon C)=1$.
Thus the tangent space to a conjugacy
equivalent class of $B$ is given by
$$
\mathrm{tr}(C)=\mathrm{tr}(BC)=\mathrm{tr}(B^{-1} C)=0.
$$
Now let $B=1\!\!1+\varepsilon A$, a point in an infinitesimal neighborhood of the identity $1\!\!1$; we have $\mathrm{tr}(A)=0$.
Then our conditions on~$C$ implies $\mathrm{tr}(C)=\mathrm{tr}(AC)=0$.
Since $\mathrm{tr}(AC)$ is a non-degenerate quadratic form,
an element $C\in \mathrm{sl}(3)$ satisfying $\mathrm{tr}(AC)=0$ for all $A\in \mathrm{sl}(3)$
has to be zero.
\proofend
In view of what we said above, this implies the proposition.
\proofend
\subsection{Identities between the monodromy invariants}\label{IdenS}
In this section, we consider the restriction of the monodromy invariants from the space
of all twisted $n$-gons to the space $\cC_n$ of closed $n$-gons.
We show that these restrictions satisfy 5 non-trivial relations,
whereas their differentials, considered as covectors in $\cP_n$ whose foot-points belong to $\cC_n$,
satisfy 3 non-trivial relations.
These relations are also mentioned in \cite{OST} and \cite{Sol}.
In Sections \ref{ThEnd} and \ref{TAN}, we will prove that there are no other relations
between the monodromy invariants on $\cC_n$ and their differentials along $\cC_n$.
We remark that, strictly speaking, the identities established in this section
are not needed for the proof of our main result. For the main result, all we need to know is that
there are enough commuting flows to fill out what could be ({\it a priori\/}, with out the
results in this section) a union of level sets of the monodromy invariants. Thus,
the reader interested only in the main result can skip this section.
\begin{theorem}
\label{5and3xy}
(i) The restrictions of the monodromy integrals to $\cC_n$ satisfy the following five identities:
\begin{equation}
\label{Rel1}
\begin{array}{rcl}
\displaystyle
\sum_{j=0}^{[n/2]} O_j = 3\,E_n^{\frac{1}{3}}\,O_n^{\frac{2}{3}},
&&\displaystyle
\sum_{j=0}^{[n/2]} E_j = 3\,E_n^{\frac{2}{3}}\,O_n^{\frac{1}{3}},\\[16pt]
\displaystyle
\sum_{j=1}^{[n/2]} j\, O_j = n\, E_n^{\frac{1}{3}}\,O_n^{\frac{2}{3}},
&&\displaystyle
\sum_{j=1}^{[n/2]} j\, E_j = n\, E_n^{\frac{2}{3}}\,O_n^{\frac{1}{3}},\\[16pt]
\displaystyle
E_n^{\frac{1}{3}}\, \sum_{j=1}^{[n/2]} j^2 \,O_j&=&
\displaystyle
O_n^{\frac{1}{3}}\, \sum_{j=1}^{[n/2]} j^2\, E_j.
\end{array}
\end{equation}
(ii) The differentials of the monodromy integrals along $\cC_n$ satisfy the three identities:
\begin{equation}
\label{Rel2}
\begin{array}{rcl}
\displaystyle
\sum_{j=1}^{[n/2]} dO_j&=&
\displaystyle
2\, E_n^{\frac{1}{3}}\,O_n^{-\frac{1}{3}}\, dO_n +
E_n^{-\frac{2}{3}}\,O_n^{\frac{2}{3}} dE_n,\\[16pt]
\displaystyle
\sum_{j=1}^{[n/2]} dE_j&=&
\displaystyle
2\, E_n^{-\frac{1}{3}}\,O_n^{\frac{1}{3}} dE_n +
E_n^{\frac{2}{3}}\,O_n^{-\frac{2}{3}}\, dO_n, \\[16pt]
\displaystyle
O_n^{\frac{1}{3}}\, \Big(
\sum_{j=1}^{[n/2]} j\ dE_j
\Big) +
E_n^{\frac{1}{3}} \,\Big(
\sum_{j=1}^{[n/2]} j\ dO_j
\Big)&=&
\displaystyle
n\, E_n^{\frac{2}{3}}O_n^{\frac{2}{3}} \left(
E_n^{-1} dE_n + O_n^{-1} dO_n
\right).
\end{array}
\end{equation}
\end{theorem}
\proof
Recall that the monodromy invariants $O_j$ are the homogeneous components of the polynomial
$O_n^{2/3}E_n^{1/3}\,\mathrm{tr}(M)$ with respect to the rescaling (\ref{rescal}),
where $s=e^t$ for convenience.
Likewise, the monodromy invariants~$E_j$ are homogeneous components of
$O_n^{1/3}E_n^{2/3}\,\mathrm{tr}(M^{-1})$.
Recall also that $O_0=E_0=1$.
Denote for simplicity $O_n^{1/3}E_n^{2/3}=U,\, O_n^{2/3}E_n^{1/3}=V$.
Notice that the monodromy matrix $M$ has the unit determinant. Let
$e^{\lambda_1},\ e^{\lambda_2},\ e^{\lambda_2}$
be the eigenvalues of $M$.
One has
\begin{equation}
\label{zero}
\lambda_1+\lambda_2+\lambda_3\equiv 0.
\end{equation}
We consider a one-parameter family of $n$-gons
depending on the rescaling parameter $t$, such that for $t=0$, the $n$-gon belongs to $\cC_n$.
The monodromy $M=M_t$ also depends on $t$ so that we think of $\lambda_i$ as
functions of the corner coordinates
$(x_1,\ldots,x_{2n})$ and of $t$.
For $t=0$, one has: $\lambda_i=0,\, i=1,2,3$ since $M_0=\mathrm{Id}$.
The eigenvalues of $M^{-1}$ are
$e^{-\lambda_1},\ e^{-\lambda_2},\ e^{-\lambda_2}. $
Since the weights of $O_i$ and $E_j$ are $j$ and $-j$ respectively, the definition of the integrals writes as follows:
$$
e^{\frac{nt}{3}}\, V \left(e^{\lambda_1}+ e^{\lambda_2}+ e^{\lambda_2}\right)
=\sum_{j=0}^{[n/2]} e^{tj}\, O_j,
\qquad
e^{-\frac{nt}{3}}\, U\left(e^{-\lambda_1}+ e^{-\lambda_2}+ e^{-\lambda_2}\right)
=\sum_{j=0}^{[n/2]} e^{-tj}\, E_j.
$$
which we rewrite as
\begin{equation}
\label{def}
V \left(e^{\lambda_1}+ e^{\lambda_2}+ e^{\lambda_2}\right)=
\sum_{j=0}^{[n/2]} e^{t(j-\frac{n}{3})} \,O_j,\qquad
U\left(e^{-\lambda_1}+ e^{-\lambda_2}+ e^{-\lambda_2}\right)=
\sum_{j=0}^{[n/2]} e^{-t(j-\frac{n}{3})}\, E_j.
\end{equation}
Setting $t=0$ in these formulas yields the first two identities in (\ref{Rel1}).
Next, differentiate these equations in $t$ :
$$
V\, \sum_{i=1}^3 \lambda_i' \,e^{\lambda_i}
=\sum_{j=0}^{[n/2]} \left(j-\frac{n}{3}\right) e^{tj}\, O_j,
$$
where $\lambda_i' =d\lambda_i/dt$,
and similarly for $E_j$.
Set $t=0$, then
the left-hand-side vanishes because $\sum \lambda_i'=0$ due to (\ref{zero}).
Hence
$$
\sum_{j=0}^{[n/2]} j \, O_j = \frac{n}{3}\, \sum_{j=0}^{[n/2]} O_j= n\,V
$$
due to the first identity in (\ref{Rel1}) and similarly for $E_j$.
One thus obtains the third and the fourth identity in (\ref{Rel1}).
To obtain the fifth equation in (\ref{Rel1}), differentiate the equations (\ref{def})
with respect to $t$ twice to get
$$
\begin{array}{rcl}
\displaystyle
V\,\Big(\sum_{i=1}^3 (\lambda_i''+\lambda_i'^2)\, e^{\lambda_i}\Big)&=&
\displaystyle
\sum_{j=0}^{[n/2]} \left(j-\frac{n}{3}\right)^2 e^{tj}\, O_j,\\[16pt]
\displaystyle
U\,\Big(\sum_{i=1}^3 \left(-\lambda_i''+\lambda_i'^2\right)\, e^{\lambda_i}\Big)&=&
\displaystyle
\sum_{j=0}^{[n/2]} \left(j-\frac{n}{3}\right)^2 e^{-tj}\, E_j.
\end{array}
$$
Divide the first equality by $V$, the second by $U$, subtract one from another, and set $t=0$:
$$
2 \sum_{i=1}^3 \lambda_i''=V^{-1} \sum_{j=0}^{[n/2]}
\left(j-\frac{n}{3}\right)^2 O_j - U^{-1} \sum_{j=0}^{[n/2]} \left(j-\frac{n}{3}\right)^2 E_j.
$$
The left hand side vanishes, due to (\ref{zero}), so
\begin{equation}
\label{useful}
V^{-1}\, \sum_{j=0}^{[n/2]} \left(j-\frac{n}{3}\right)^2 O_j =
U^{-1} \,\sum_{j=0}^{[n/2]} \left(j-\frac{n}{3}\right)^2 E_j.
\end{equation}
Therefore
$$
\begin{array}{l}
\displaystyle
V^{-1} \sum_{j=0}^{[n/2]} j^2\, O_j - \frac{2n}{3}\, V^{-1} \,
\sum_{j=0}^{[n/2]} j\, O_j + V^{-1}\, \frac{n^2}{9}\, \sum_{j=0}^{[n/2]} O_j
=\\[10pt]
\displaystyle
\hskip 2cm
U^{-1} \sum_{j=0}^{[n/2]} j^2\, E_j - \frac{2n}{3}\, U^{-1} \,
\sum_{j=0}^{[n/2]} j\, E_j + U^{-1}\, \frac{n^2}{9}\, \sum_{j=0}^{[n/2]} E_j.
\end{array}
$$
The second and the third terms on the left and the right hand sides are pairwise equal,
due to the first four identities in (\ref{Rel1}). This implies the fifth identity (\ref{Rel1}).
To prove (\ref{Rel2}), take differentials of (\ref{def}):
$$
\label{diff1}
\begin{array}{l}
\displaystyle
V \sum_{i=1}^3 e^{\lambda_i}\, d\lambda_i +
\Big(\sum_{i=1}^3 e^{\lambda_i}\Big)\, dV =\\ [10pt]
\displaystyle
\Big(\sum_{j=0}^{[n/2]} \left(j-\frac{n}{3}\right) e^{t(j-\frac{n}{3})}\, O_j\Big) dt +
\sum_{j=0}^{[n/2]} e^{t(j-\frac{n}{3})}\, dO_j,
\end{array}
$$
and
$$
\begin{array}{l}
\displaystyle
-U \sum_{i=1}^3 e^{-\lambda_i}\, d\lambda_i +
\Big(\sum_{i=1}^3 e^{-\lambda_i}\Big)\, dU = \\[10pt]
\displaystyle
-\Big(\sum_{j=0}^{[n/2]} \Big(j-\frac{n}{3}\Big) e^{-t(j-\frac{n}{3})} E_j\Big)\, dt +
\sum_{j=0}^{[n/2]} e^{-t(j-\frac{n}{3})}\, dE_j.
\end{array}
$$
Set $t=0$: the first terms on the right hand sides vanish due to (\ref{zero}),
and the first parentheses on the right hand sides vanish due to (\ref{Rel1}).
We get
$$
\sum_{j=0}^{[n/2]} dO_j=3\,dV,
\qquad
\sum_{j=0}^{[n/2]} dE_j=3\,dU,
$$
the first two identities in (\ref{Rel2}).
Finally, differentiate the above equations with respect to $t$ and set $t=0$ to obtain:
$$
\begin{array}{rcl}
\displaystyle
V\, \sum_{i=1}^3 \lambda_i'\, d\lambda_i +
V\, \sum_{i=1}^3 d(\lambda_i') + \Big(\sum_{i=1}^3 \lambda_i' \Big)\, dV&=&
\displaystyle
\Big(\sum_{j=0}^{[n/2]} \left(j-\frac{n}{3}\right)^2 O_j\Big)\, dt +
\sum_{j=0}^{[n/2]} \left(j-\frac{n}{3}\right) d O_j,\\[14pt]
\displaystyle
U\, \sum_{i=1}^3 \lambda_i'\, d\lambda_i -
U\, \sum_{i=1}^3\, d(\lambda_i') +
\Big(\sum_{i=1}^3 \lambda_i' \Big)\, dU &=&
\displaystyle
\Big(\sum_{j=0}^{[n/2]} \left(j-\frac{n}{3} \right)^2 E_j\Big)\, dt -
\sum_{j=0}^{[n/2]} \left(j-\frac{n}{3}\right) d E_j.
\end{array}
$$
Once again, the second and the third sums on the left hand sides vanish, due to (\ref{zero}). Divide the first equation by $V$, the second by $U$, and subtract one from another, using (\ref{useful}):
$$
V^{-1}\, \sum_{j=0}^{[n/2]} \left(j-\frac{n}{3}\right) d O_j +
U^{-1}\, \sum_{j=0}^{[n/2]} \left(j-\frac{n}{3}\right) d E_j=0.
$$
Hence
$$
V^{-1}\, \sum_{j=0}^{[n/2]} j\, d O_j + U^{-1}\, \sum_{j=0}^{[n/2]} j\, d E_j =
\frac{n}{3}\, \Big(V^{-1} \sum_{j=0}^{[n/2]} d O_j + U^{-1} \sum_{j=0}^{[n/2]} d E_j \Big).
$$
Due to the first two identities in (\ref{Rel2}), the right-hand-side equals
$n\, (O_n^{-1}\, dO_n + E_n^{-1}\, dE_n)$.
This yields the third identity in (\ref{Rel2}).
Theorem \ref{5and3xy} is proved.
\proofend
\begin{remark}
\label{rmk}
{\rm
a)
Let ${\cal E}$ be the Euler vector field that generates the scaling. Then
$$
{\cal E} (O_j)=j\,O_j,
\qquad
\ {\cal E} (E_j)=-j\,E_j.
$$
If one evaluates the differentials in the identities (\ref{Rel2}) on ${\cal E}$,
one obtains the last three identities in (\ref{Rel1}).
This is a check that (\ref{Rel1}) and (\ref{Rel2}) are consistent with each other.
b)
Equivalently, (\ref{Rel2}) can be rewritten as
$$
\begin{array}{rcl}
\displaystyle
3\, d O_n &=& 2\, E_n^{-\frac{1}{3}}\,O_n^{\frac{1}{3}} \,
\Big( \sum_{j=1}^{[n/2]} dO_j \Big)-
E_n^{-\frac{2}{3}}\,O_n^{\frac{2}{3}}\, \Big( \sum_{j=1}^{[n/2]} dE_j \Big),\\[16pt]
\displaystyle
3\, d E_n &=& 2\, E_n^{\frac{1}{3}}\,O_n^{-\frac{1}{3}} \,
\Big( \sum_{j=1}^{[n/2]} dE_j \Big)-
E_n^{\frac{2}{3}}\,O_n^{-\frac{2}{3}}\, \Big( \sum_{j=1}^{[n/2]} dO_j \Big),\\[12pt]
0&=&
\displaystyle
O_n^{\frac{1}{3}} \,
\Big(3 \sum_{j=1}^{[n/2]} j\ dE_j -n \sum_{j=1}^{[n/2]} dE_j \Big) +
E_n^{\frac{1}{3}} \,\Big(3 \sum_{j=1}^{[n/2]} j\, dO_j -n \sum_{j=1}^{[n/2]} dO_j \Big).
\end{array}
$$
c)
The identities (\ref{Rel1}) and (\ref{Rel2}) are satisfied in a larger subspace than $\cC_n$,
consisting of twisted polygons whose monodromy has \textit{equal eigenvalues}.
This subspace has codimension 2 in~$\cP_n$.
d) In both cases, $n$ odd and $n$ even, the kernel of the Poisson map $X$ (\ref{HaMap}) (spanned by the differentials of the Casimir functions) has zero intersection with the subspace of $T^*\cP_n$ spanned by the relations \ref{Rel2}.
}
\end{remark}
\subsection{Reducing the proof to a one-point computation}\label{ReS}
For ease of exposition, we will give our proof only in
the odd case, and we set $n \geq 7$ odd. Modulo
changing some of the indices, the even case is similar.
We will explain everything in terms of the odd case and,
at the end of this section, briefly explain what happens
in the even case.
Let ${\cal M\/}$ denote the algebra generated by
the monodromy invariants. In the Section \ref{ThEnd} we make
the following calculations.
\begin{enumerate}
\item There exist elements $F_1,...,F_{n-2} \in \cal M$
and a point $p \in \cC_n$ such that the differentials
$dF,...,dF_{n-2}$ are linearly independent at $p$.
Therefore,
$dF,...,dF_{n-2}$ are linearly independent at almost all
$q \in \cC_n$.
\item There exists elements $G_1,...,G_{n-4} \in \cal M$
and a point $p \in \cC_n$ such that the differentials
$dG_1|_{T_p \cC_n},...,dG_{n-4}|_{T_p \cC_n}$ are linearly independent.
Therefore, $dG_1|_{T_q \cC_n},...,dG_{n-4}|_{T_q \cC_n}$ are linearly independent
at almost all $q \in \cC_n$.
\end{enumerate}
In Calculation 1,
we are computing the differentials
on the ambient space ${\cal P\/}_n$ but evaluating them
at a point of $\cC_n$. In Calculation 2, we are
computing the differentials on the ambient space,
evaluating them at a point of $\cC_n$, {\it and\/}
restricting the resulting linear functionals to the
tangent space of $\cC_n$. In both calculations,
we are actually evaluating at points in
$\C_n^0$. In each case, what allows us to make
a conclusion about generic points is that the
monodromy invariants are algebraic.
Calculation 2 combines with Theorem~\ref{5and3xy} to show that
there are exactly $n-4$ algebraically independent monodromy
invariants, when restricted to $\cC_n$.
Hence, the generic common level set of the monodromy
invariants $O_i,E_i$, restricted to $\cC_n$,
has dimension $n-4$.
Next, we wish to prove that these level sets
have locally free action of the abelian group~$\R^{d}$ (or~$\C^{d}$ in the complex case).
For $F \in \cal M$, the Hamiltonian vector field $X_F$ is tangent to $\cC_n$,
by Proposition \ref{TanProp}, and also tangent to the common level set
of functions in $\cal M$. Finally, by Theorem~\ref{PoCThm},
the Hamiltonian vector fields all commute with each other
(i.e., define an action of the Abelian Lie algebra).
The following lemma finishes our proof.
\begin{lemma}
The Hamiltonian vector fields of the monodromy invariants
generically span the monodromy level sets on $\cC_n$.
\end{lemma}
\startproof
Let $\wedge^1\cP_n$ denote the space of
$1$-forms on $\cP^n$. Let $\cal X$ denote the space
of vector fields on $\cC_n$.
Let $d{\cal M\/} \subset \wedge^1\cP_n$
denote the image of $\cal M$ under the $d$-operator.
Calculation 1 shows that the vector space $d{\cal M\/}$
generically has dimension $n-2$ when evaluated at
points of $\cC^n$. At the same time, we have the Poisson
map $X: d{\cal M\/} \to {\cal X\/}$, given by
$$
X(dF)=X_F,
$$
see (\ref{HaMap}).
In the odd
case, the map $X$ has $2$ dimensional kernel, see Remark \ref{rmk} d). Hence,
$X$ has $n-4$ dimensional image, as desired.
\proofend
Now we explain explicitly how the results above
give us the quasi-periodic motion in the case of
closed convex polygons. We know from the work in
\cite{Sch1} that the monodromy level sets on
$\cC_n^0$ are compact. By Sard's Theorem, and by the
calculations above, almost every level set is
a smooth compact manifold of dimension $m=n-4$. By
Sard's Theorem again, and by the dimension count above,
almost every level set $L$ possesses a framing by
Hamiltonian vector fields. That is, there are
$m$ Hamiltonian vector fields on $L$ which are
linearly independent at each point and which define
commuting flows. These vector fields define
local coordinate charts from $L$ into $\R^m$,
such that the overlap functions are translations.
Therefore $L$ is a finite union of affine
$m$-dimensional tori. The whole structure
is invariant under the pentagram map, and so
the pentagram map is a translation of $L$
relative to the affine structure on $L$. This is the
quasi-periodic motion. Even more explicitly,
some finite power of the pentagram map preserves
each connected component of $L$ and is a constant
shift on each connected component.
\newline
\newline
\noindent
{\bf The Even Case:\/}
In the even case, we have the following calculations:
\begin{enumerate}
\item There exist elements $F_1,...,F_{n-1} \in \cal M$
and a point $p \in \cC_n$ such that the differentials
$dF,...,dF_{n-1}$ are linearly independent at $p$.
Therefore,
$dF,...,dF_{n-1}$ are linearly independent at almost all
$q \in \cC_n$.
\item There exists elements $G_1,...,G_{n-3} \in \cal M$
and a point $p \in \cC_n$ such that the differentials
$dG_1|_{T_p \cC_n},...,dG_{n-3}|_{T_p \cC_n}$ are linearly independent.
Therefore, $dG_1|_{T_p \cC_n},...,dG_{n-3}|_{T_p \cC_n}$ are linearly independent
at almost all $q \in \cC_n$.
\end{enumerate}
In this case, the common level sets generically
have dimension $n-5$ and, again, the Hamiltonian
vector fields generically span these level sets. The situation is summarized in the following table.
\begin{displaymath}
\begin{array}{c|c|c|c|}
&\hfill{\rm Invariants}& \hfill{\rm Casimirs} &
\hfill{\rm Level\ sets}\,/\, \hfill{\rm Hamiltonian}\, \hfill{\rm fields}\\
\cline{1-1} \cline{2-2}\cline{3-3}\cline{4-4}
n\, \hfill{\rm odd}&n+1& 2 & d=n-4 \\
\cline{1-1}\cline{2-2} \cline{3-3}\cline{4-4}
n\, \hfill{\rm even}& n+2 & 4 & d=n-5 \\
\cline{1-1}\cline{2-2} \cline{3-3}\cline{4-4}
\end{array}
\end{displaymath}
\section{The linear independence calculation}\label{ThEnd}
\label{INDEP}
\subsection{Overview}
For any given (smallish) value of $n$, one can make the calculations
directly, at a random point, and see that it works. The
difficulty is that we need to make one calculation for each $n$.
One might say that the idea behind our calculations is
tropicalization. The monodromy invariants and their
gradients are polynomials with an enormous number
of terms. We only need to make our calculation at
one point, but we will consider a $1$-parameter family
of points, depending on a parameter $u$. As $u \to 0$,
the different variables tend to $0$ at different rates. This
sets up a kind of hierarchy (or filtration) on the the monomials
comprising the polynomials of interest to us, and
only the ``heftiest'' monomials
in this hierarchy matter. This reduces the whole
problem to a combinatorial exercise.
We take $n \geq 7$ odd.
Let $m=(n-1)/2$. Recall that
$\cal M$ is spanned by
$$O_1,...,O_m,O_n,E_1,...,E_m,E_n.$$ We define
\begin{equation}
A_{k,\pm} = O_k \pm E_k.
\end{equation}
For the first calculation, we use the monodromy invariants
\begin{equation}
\label{inv1}
A_{3,+},...,A_{m,+},A_{n,+},A_{2,-},...,A_{m,-},A_{n,-}.
\end{equation}
For the second calculation, we use the monodromy invariants
\begin{equation}
\label{inv2}
A_{3,-},...,A_{m,-},A_{3,+},...,A_{m,+},A_{n,+}.
\end{equation}
The point we use is of the form $p=P^u$, where
$P^u$ is an $n$-gon having
corner invariants
\begin{equation}
a,b,c,d,u^1,u^2,u^3,u^4,...,u^4,u^3,u^2,u^1,d,c,b,a,
\end{equation}
Here \begin{itemize}
\item $a = O(u^{(n-4)(n-3)/2})$.
\item $b = 1 +O(u)$
\item $c = 1 +O(u)$.
\item $d = 1+ O(u)$.
\end{itemize}
We will show that the results hold when $u$ is
sufficiently small.
Here we are using the big O notation, so that
$O(u)$ represents an expression that is at most $Cu$
in size, for a constant $C$ that does not depend on $u$.
We will construct $P^u$ in the next section.
Our first calculation requires only the information
presented above. The second calculation, which
is almost exactly the same as the first calculation,
requires some auxilliary justification. In order
to justify the calculation we make, we need to
make some estimates on the tangent space $T_{P^u}$
to $\cC_n$ at $P^u$.
We will also do this in the next section.
In Section \ref{calc1} and Section \ref{calc2} we will explain our
two calculations in general terms.
In Section \ref{heft} we will define the concept of
the {\it heft\/} of a monomial, and we will use
this concept to put a kind of ordering on the
monomials that appear in the monodromy invariants
of interest to us. Following the analysis of
the heft, we complete the details of our calculations.
\subsection{The first calculation in broad terms}
\label{calc1}
Let $\nabla$ denote the gradient on $\R^{2n}$.
Let $\widetilde \nabla$ denote the {\it normalized gradient\/}:
\begin{equation}
\widetilde \nabla F= \lambda^{-1}\nabla F; \hskip 30 pt
\lambda=\|\nabla F\|_{\infty}.
\end{equation}
In practice, we never end up dividing by zero.
So, the largest entry in $\widetilde \nabla F$ is
$\pm 1$.
If $F$ is a monodromy invariant, the coordinates
of $\widetilde \nabla F(P^u)$ have a power series in $u$.
We define $\Psi F$ to be the result of setting all
terms except the constant term to $0$. We call
$\Psi F$ the {\it asymptotic gradient\/}. Thus,
if $$\widetilde \nabla F(P^u)=(1-u^3\cdots,-1+u\cdots,u^2\cdots,...)$$ then
$\Psi F = (1,-1,0,...)$.
\begin{lemma}
\label{degen}
Suppose that
$\Psi F_1,...,\Psi F_k$ are
linearly independent.
Then likewise
$\nabla F_1,...,\nabla F_k$ are
linearly independent at $P^u$ for
$u$ sufficiently small.
Equivalently, the same goes for
$dF_1,...,dF_k$.
\end{lemma}
\startproof
Since $\Psi F_1,...,\Psi F_k$ are independent there is
some $\epsilon>0$ such that a sum of the form
$$\bigg|\sum b_j \Psi F_j\bigg|<\epsilon; \hskip 30 pt
\max |b_j|=1$$
is impossible.
Suppose for the sake of contradiction that
the gradients are linearly dependent at $P^u$
for all sufficiently small $u$. Then the
normalized gradients are also linearly dependent
at $P^u$ for all sufficiently small $u$. We may write
\begin{equation}
\sum b_j \widetilde \nabla F_j \cdot e_i=0;
\hskip 30 pt \max |b_j|=1.
\end{equation}
for the standard basis vectors $e_1,...,e_{2n}$.
The coefficients $b_j$ possibly depend on $u$,
but this doesn't bother us.
We have the bound
\begin{equation}
\bigg|b_j \widetilde \nabla F_j - b_j \Psi F_j\bigg|= O(u).
\end{equation}
Hence
\begin{equation}
\sum_j b_j \Psi F_j \cdot e_i=O(u)
\end{equation}
for all basis vectors $e_i$. Therefore,
we can take $u$ small enough so that
$$\bigg|\sum b_j \Psi F_j\bigg|<\epsilon; \hskip 30 pt
\max |b_j|=1,$$
in contradiction to what we said at the beginning of
the proof.
\proofend
\begin{remark}
{\rm The idea of the proof of the previous lemma is simple: given a matrix, algebraically dependent on a parameter $u$, the rank of the matrix is greatest in a Zariski open subset of the parameter space and can only drop for special values of the parameter (zero, in our case).}
\end{remark}
We form a matrix $M_+$ whose rows
are $\Psi F$, where $F$ is each of
the $A_+$ invariants.
We similarly form the matrix $M_-$.
\begin{lemma}
\label{ortho}
Each row of $M_+$ is orthogonal to each row of $M_-$.
\end{lemma}
\startproof
Consider the map
$T: \R^{2n} \to \R^{2n}$ which simply reverses
the coordinates. We have $E_k \circ T=O_k$ for all $k$
and moreover $T(P^u)=P^u$.
Letting $dT$ be the differential of $T$, we have
\begin{equation}
dT(\nabla A_{k,\pm})= \pm \nabla A_{k,\pm}.
\end{equation}
Our lemma follows immediately from this equation,
and from the fact that $T$ is an isometric involution.
\proofend
In view of Lemmas \ref{degen} and Lemma \ref{ortho}, our
first calculation follows from the statements that
$M_+$ and $M_-$ have full rank.
For the matrix $M_+$, we consider the minor
$m_+$ consisting of columns
$$1,6,7,10,11,14,15,18,19,...$$
until we have a square matrix.
We will prove below that $m_+$ has the following form
(shown in the case $n=13$.)
\begin{equation}
\label{special}
\left[\begin{matrix}
0&\pm 1&\pm 1&\pm 1&\pm 1\cr
0&0&\pm 1&\pm 1&\pm 1\cr
0&0&0&\pm 1&\pm 1\cr
0&0&0&0&\pm 1\cr
\pm 1&0&0&0&0 \end{matrix}\right]
\end{equation}
This matrix always has full rank.
Hence $M_+$ has full rank.
For the matrix $M_-$ we consider the minor
$m_-$ consisting of columns
$$1,{\bf 3\/},6,7,10,11,14,15,18,19,...$$
The only difference here is that column $3$ is
inserted. The resulting matrix has
exactly the same structure as just
described.
Hence $M_-$ has full rank.
\subsection{The second calculation in broad terms}
\label{calc2}
Let $T=T_{P^u}(\cC_n)$ denote the tangent space
to $\cC_n$ at $P^u$. Let
$\{e_k\}$ denote the standard basis for
$\R^{2n}$. Let $\pi: \R^{2n} \to \R^{2n-8}$ denote the map
which strips off the first and last $4$ coordinates.
Define
\begin{equation}
\nabla_8=\pi \circ \nabla.
\end{equation}
We define the normalized version
$\widetilde \nabla_8$ exactly as we defined
$\widetilde \nabla$. Likewise we define
$\Psi_8 G$ for any monodromy function $G$.
For a collection of vectors $v_5,\dots,v_{2n-4}$ to be specified in the next lemma, we form the vector
\begin{equation}
\Upsilon_8 G=(D_{v_5}G,...,D_{v_{2n-4}}G)
\end{equation}
made from the directional derivatives of
$G$ along these vectors. Note, by way of
analogy, that
\begin{equation}
\nabla_8 G=(D_{e_5}G,...,D_{e_{2n-4}}G).
\end{equation}
We define
the normalized version $\widetilde \Upsilon_8$ exactly
as we defined $\widetilde \nabla_8$.
In the next section, we will
establish the following result.
\begin{lemma}[Justification]
\label{just}
There is a basic
$v_5,...,v_{2n-4}$ for $T_{P^u}(\cC_n)$
such that
$\pi(v_k)=e_k$ for all $k$ and
$$\widetilde \Upsilon_8 G - \widetilde \nabla_8 G = O(u).
$$
\end{lemma}
\begin{corollary}
Suppose that
$\Psi_8 G_1,...,\Psi_8 G_k$ are
linearly independent. Then
the restrictions of
$dG_1,...,dG_k$ to $T_{P^u}(\cC_n)$ are
linearly independent for $u$ sufficiently small.
\end{corollary}
\startproof
Given our basis, $\Psi_8$ represents the constant
term approximation of both
$\widetilde \Upsilon_8$
and $\widetilde \nabla_8$. So, the same proof
as in Lemma \ref{degen} shows that
the vectors
$\widetilde \Upsilon_8 G_j$ are linearly independent.
This is equivalent to the conclusion of
our corollary.
\proofend
Using the invariants listed in (\ref{inv2}),
we form the matrices $M_+$ and $M_-$ just as
above, using $\Psi_8$ in place of $\Psi$.
Lemma \ref{ortho} again shows that
each row of $M_+$ is orthogonal to
each row of $M_-$.
Hence, we can finish the second calculation by
showing that both $M_+$ and $M_-$ have full rank.
For $M_-$ we create a square minor $m_-$ using the
columns
$$2,3,6,7,10,11,14,15,...$$
Again, we continue until we have a square.
It turns out that $m_-$ has the form
\begin{equation}
\label{special2}
\left[\begin{matrix}
\pm 1&\pm 1&\pm 1&\pm 1\cr
0&\pm 1&\pm 1&\pm 1\cr
0&0&\pm 1&\pm 1\cr
0&0&0&\pm 1\cr \end{matrix}\right]
\end{equation}
Hence $M_-$ has full rank.
For $M_+$ we create a square minor $m_+$ using the
same columns, but extending out one further
(on account of the larger matrix size.)
It turns out that $m_+$ has the same form as $m_-$.
Hence $M_+$ has full rank.
\subsection{The heft}
\label{heft}
Any monomial in
the variables $x_1,...,x_{2n}$, when
evaluated at $P^u$, has a power
series expansion in $u$. We define the
{\it heft\/} of the monomial to be the
smallest exponent that appears in this series.
For instance, the heft of $u^2+u^3$ is $2$.
We define the heft of a polynomial to be the
minimum heft of the monomials that comprise it.
Given a polynomial $F$, we define
heft of $\nabla F$ to be the minimum
heft, taken over all partial
derivatives $\partial F/\partial x_j$.
We call a monomial term of
$\partial F/\partial x_k$ {\it hefty\/} if its heft realizes
the heft of $\nabla F$.
We define $H_kF$ to be the sum of the
hefty monomials in $\partial F/\partial x_k$.
Each monomial occurs with sign $\pm 1$. We
define $|H_kF| \in \Z$ to be the sum of the
coefficients of the hefty terms in $H_kF$.
We say that $F$ is {\it good\/} if
$|H_kF| \not = 0$ for at least one index $k$.
If $F$ is good then
\begin{equation}
\label{good}
\Psi F= C (|H_1F|,...,|H_{2n}F|),
\end{equation}
for some nonzero constant $C$ that depends on $F$.
It turns out that $C=\pm 1$ in all cases.
We say that $F$ is {\it great\/} if $F$ is
{\it good\/} and $|H_k F| \not = 0$ for
at least one index $k$ which is not amongst
the first or last $4$ indices. When $F$
is great, not only does equation (\ref{good})
hold, but we also have
\begin{equation}
\label{great}
\Psi_8 F= C (|H_5F|,...,|H_{2n-4}F|),
\end{equation}
\begin{lemma}
Let $k=2,3$. Then
$A_{k,\pm}$ is great and
$\nabla A_{k,\pm}$ has heft $0$.
\end{lemma}
\startproof
Let $F=A_{k,\pm}$.
Consider the case $k=2$.
The argument turns out to be the same in the
$(+)$ and $(-)$ cases. We say that an {\it outer variable\/} is one of the
first or last $4$ variables in $\R^{2n}$, and we call
the remaining variables {\it inner\/}.
Since $x_2x_6$ and $x_6x_{2n-2}$ are both terms of
$F$, we see that $$H_6F=x_2+x_{2n-2}+...$$
In particular, $\nabla F$ has heft $0$. Any term
in $H_6F$ involves only the outer $8$ variables,
and a short case-by-case analysis shows that there
are no other possibilities besides the two terms
listed above. Hence $|H_6F|=2$. This shows that
$F$ is great.
Now consider the case $k=3$.
The argument turns out to be the same in the
$(+)$ and $(-)$ cases.
Since $x_2x_6x_{2n-2}$ is a term of $F$
we see that $$H_6F=x_2x_{2n-2}+...$$
The rest of the proof is as in the previous
case, with the only difference being
that $|H_6F|=1$ in this case.
\proofend
From now on, we fix some $F=A_{k,\pm}$ with $3<k \leq m$.
Let $\alpha_1,\alpha_2,...$ be the terms of
the following sequence
\begin{equation}
0,0,0,2,3,6,7,10,11,14,15,...
\end{equation}
\begin{lemma}
\label{array}
$\nabla F$ has heft
at most $\alpha_1+...+\alpha_k$.
\end{lemma}
\startproof
We describe a specific term in $\nabla F$ having heft
$\alpha_1+...+\alpha_k$. We make a monomial
using the indices
\begin{equation}
\label{packing}
2,2n-2,6,2n-6,10,2n-10,...
\end{equation}
stopping when we have used $k-1$ numbers.
The monomial corresponding to these
indices has heft
$$0+0+0+2+3+6+7+10+11...=\alpha_1+...+\alpha_k.$$
Thinking of our indices cyclically, we see that
our integers lie in an interval of length
$4k-7$. So, between the largest index in
(\ref{packing}) that is less than
$n$ and the smallest index greater than $n$ there
is an unoccupied stretch of at least $9$ integers.
The point here is that
$$9 + (4k-7) \leq 9 + 4m-7 = 9 + 2(n-1)-7=2n.$$
Given that the unoccupied stretch has at least
$9$ consecutive integers, there is at least
$1$ (and in fact at least $2$) even indices
$j$ such that the monomial
$$m=\pm x_j x_2 x_{2n-2} x_6 x_{2n-6}x_{10}...$$
is a term of $F$.
But then $\partial m/\partial x_j$ has heft
$\alpha_1+...+\alpha_k$.
\proofend
We mention that (\ref{packing}) is one
of two obvious ways to make a term of
heft $\alpha_1+...+\alpha_k$. The other
way is to take the {\it mirror image\/}, namely
\begin{equation}
\label{packing2}
2n-1,3,2n-5,7,2n-9,11,...
\end{equation}
\begin{lemma}
\label{inner}
If $\partial F/\partial x_j$ has a hefty term, then
$j$ is an inner variable.
\end{lemma}
\startproof
For ease of exposition, we will consider the
case when $j$ is one of the first $4$ variables.
Let $(i_1,...,i_d)$ be the sequence of indices which
appear in a term $m'$ of $\partial F/\partial x_j$.
The corresponding term $m$ in $F$ has
index sequence $(j,i_1,...,i_d)$, where these
numbers are not necessarily written in order.
We know that at least one of the indices, say $a$,
is an inner variable. By construction
$\partial m/\partial x_a$ has smaller heft than
$m'$. Hence $\partial F/\partial x_j$ has
no hefty terms. Hence $j$ is an inner variable.
\proofend
\begin{lemma}
Suppose
the monomial $\pm x_{i_1}...x_{i_a}$ is a
hefty term of $\partial F/\partial x_j$.
Then $a=k-1$ and
$i_1,...,i_{k-1}$ are either
as in (\ref{packing}) or as in
equation (\ref{packing2}).
\end{lemma}
\startproof
We have to play the following game: We have
a grid of $2n$ dots. The first and last dot
are labelled $(n-3)(n-4)/2$. The remaining
$6$ outer dots are labelled $0$. The inner
dots are labelled $1,2,3,...,3,2,1$.
Say that a {\it block\/} is a collection
of $d$ dots in a row for $d=1,2,3$. We
must pick out either $k$ or $k-1$ blocks
in such a way that the total sum of the
corresponding dots is as small as possible,
and the (cyclically reckoned)
spacing between consecutive blocks
is at least $4$. That is, at least $3$
``unoccupied dots'' must appear between
every two blocks.
It is easy to see that one should use $k-1$
blocks, all having size $1$. Moreover,
half (or half minus one) of the blocks should
crowd as much as possible to the left and
half minus one (or half) of the blocks
should crowd as much as possible to the
right. A short case by case analysis of
the placement of the first and last blocks
shows that one must have precisely the
choices made in (\ref{packing})
and (\ref{packing2}).
\proofend
\begin{corollary}
\label{GREAT}
Let $F=A_{k,\pm}$, with $k \geq 2$. Then $F$ is good.
If $k \leq m$ then $F$ is great, and the heft of $\nabla F$
is $\alpha_1+...+\alpha_k$.
\end{corollary}
\startproof
In light of the results above, the only nontrivial
result is that $F$ is great when $3<k \leq m$. The construction
in connection with (\ref{packing}) produces
a hefty term of $\partial F/\partial x_j$ for
some inner index $j$. The key observation is that,
for parity considerations, the mirror term corresponding
to (\ref{packing2}) is not a term of
$\partial F/\partial x_j$. In one case $j$ must
be odd and in the other case $j$ must be even.
Hence, there is only $1$ hefty term in
$\partial F/\partial x_j$.
\proofend
As regards the heft,
we have done everything but analyze the Casimirs.
Recall that
\begin{equation}
O_n=x_1x_3...x_{2n-1}; \hskip 30 pt
E_n=x_2x_4...x_{2n}.
\end{equation}
\begin{lemma}
\label{cas1}
$A_{n,\pm}$ is good and $\nabla A_{n,+}$ has heft
$$\frac{(n-3)(n-4)}{2}.$$
Moreover,
$$\Psi A_{n,\pm} = (1,0,...,0,\pm 1).$$
\end{lemma}
\startproof
Let $F$ be either of these functions.
Clearly the hefty terms of
$\nabla F$ are the ones which omit
the first and last variables.
From here, this lemma is an exercise in arithmetic.
\proofend
A similar argument proves
\begin{lemma}
\label{cas2}
$A_{n,\pm}$ is good and
$\nabla_8 A_{n,+}$
has heft $(n-4)^2$. Moreover,
$$\Psi A_{n,\pm} = (0,...,0,\pm 1,1,0,...,0),$$
with the $2$ middle indices being nonzero.
\end{lemma}
\subsection{Completion of the first calculation}
To complete the first calculation, we need to analyze the
matrix made from the asymptotic gradients
$\Psi F_1,\Psi F_2,...$. We deal with the first
two in a calculational way.
\begin{lemma}
\label{triv1}
$\Psi A_{2,\pm }=(0,0,\pm 1,0,0,1,\pm 1,...,1,\pm 1,0,0,1,0,0)$.
\end{lemma}
\startproof
Let $F=A_{2,\pm}$. We know that $F$ has heft $0$,
so the hefty terms in $\nabla F$ are monomials
which only involve the outer indices. Hence,
when $8 \leq j \leq 2n-8$ the result only depends
on the parity of $j$ and neither the value of $j$
nor the value of $n$. For the remaining indices,
the result is also independent of $n$. Thus, a
calculation in the case (say) $n=13$ is general
enough to rigorously establish the whole pattern.
This is what we did.
\proofend
\begin{lemma}
\label{triv2}
$\Psi A_{3,\pm }=(0,0,0,0,0,1,\pm 1,...,1,\pm 1,0,0,1,0,0)$.
\end{lemma}
\startproof
Same method as the previous result.
\proofend
Now we are ready to analyze the minors $m_+$ and $m_-$
described in connection with the first calculation.
When we say that a certain part of one of these
matrices has the form given by (\ref{special}),
we understand that (\ref{special}) gives a
smallish member of an infinite family of matrices,
all having the same general type. So, we mean to
take the corresponding member of this family which
has the correct size.
We say that a given
row or column of one of our matrices {\it checks\/} if
it matches the form given by (\ref{special}).
We will give the argument for $m_+$. The case for $m_-$ is
essentially the same.
\begin{lemma}
The first column of $m_+$ checks.
\end{lemma}
\startproof
By Lemma \ref{cas1},
the first coordinate of $\Psi A_{n,+}$ is $\pm 1$.
By Lemmas \ref{inner}, \ref{triv1},
and \ref{triv2}., we have
$\Psi A_{k,+}$ is zero for $k<n$. This is
equivalent to the lemma.
\proofend
\begin{lemma}
The first row of $m_+$ checks and the last row of
$m_+$ checks.
\end{lemma}
\startproof
The first statement follows immediately from Lemma \ref{triv2}.
The second statement follows immediately from
Lemma \ref{cas1}.
\proofend
Now we finish the proof.
Consider the $i$th row of $m_+$.
Let $k=i+2$. In light of the trivial
cases taken care of above, we can assume that
$3<k \leq m$.
Let $F=A_{k,+}$. As we discussed
in the proof of Corollary \ref{GREAT}, each
polynomial $\partial A/\partial F_j$ has
either $0$ or $1$ hefty terms.
Assume that $j$ is even. Let $J \subset \{1,...,2n\}$
be the unoccupied stretch from Lemma \ref{array}.
Let $J' \subset J$ denote the smaller set obtained
by removing the first and last $3$ members
from $J$. It follows from the construction
in Lemma \ref{array} that $\partial F/\partial j$
has a hefty term if and only if $j \in J'$.
Thus the $j$th entry of the $k$th row is
$\pm 1$ if and only if $j \in J'$. Similar
considerations hold when $j$ is odd.
It is an exercise to show that the conditions
we have given translate precisely into
the form given in (\ref{special}).
Hence $m_+$ checks.
\begin{remark}
{\rm One can approach the proof differently.
When we move from row $k$ to row $k+2$ the
corresponding interval $J'=(a,b)$ changes to the
new interval $J'=(a+4,b-4)$. From this fact, and
from our choice of minors, it
follows easily that row $k$ checks if and only
if row $k+2$ checks. At the same time, when $n$
is replaced by $n+2$, the interval $J'=(a,b)$ changes
to $J'=(a,b+4)$. This translates into the statement
that row $k$ checks for $n$ if and only if
row $k$ checks for $n+2$. All this reduces the
whole problem to a computer calculation of the
first few cases. We did the calculation up to the
case $n=13$ and this suffices.}
\end{remark}
\subsection{Completion of the second calculation}
We make all the same definitions and conventions for
the second calculation, using the matrix (family) in
(\ref{special2}) in place of the
matrix (family) in (\ref{special}).
The argument for the second calculation is really
just the same as the argument for the first
calculation. Essentially, we just ignore the
outer $8$ coordinates and see what we get.
What makes this work is that all the functions
except $A_{n,\pm}$ are great -- the inner
indices determine the heft. To handle
the last row of $m_+$, which involves the
Casimir $A_{n,+}$,
we use Lemma \ref{cas2} in place of
Lemma \ref{cas1}.
It remains to establish the Justification Lemma \ref{just}.
It is convenient to define
\begin{equation}
\delta=\frac{(n-4)(n-5)}{2}.
\end{equation}
We also mention several other pieces of notation
and terminology.
When we line up the indices
$5,...,2n-4$, there are $2$ {\it middle indices\/}.
When $n=7$ the middle indices of $5,6,7,8,9,10$ are
$7$ and $8$.
Let $\pi^{\perp}$ denote the projection
from $\R^{2n}$ onto $\R^8$ obtained by stringing
out the first and last $4$ coordinates.
\begin{lemma}[Tangent Estimate]
\label{est}
The following properties of $\pi^{\perp}(v_j)$ hold:
\begin{itemize}
\item All coordinates are $O(1)$.
\item Coordinates $3$ and $6$ are $O(u)$.
\item Except when $j$ is one of the middle two indices,
coordinates $1$ and $8$ are $O(u^{\delta+1})$.
\item When $j$ is the first middle index,
coordinate $1$ is $u^{\delta}+O(u^{\delta+1})$ and
coordinate $8$ is $O(u^{\delta+1})$.
\item When $j$ is the second middle index,
coordinate $8$ is $u^{\delta}+O(u^{\delta+1})$ and
coordinate $1$ is $O(u^{\delta+1})$.
\end{itemize}
\end{lemma}
\startproof
We prove this in the next section.
\proofend
\begin{lemma}
The Justification Lemma holds for $F=A_{n,+}$.
\end{lemma}
\startproof
A direct calculation shows that, up to
$O(u^{\delta+1})$,
\begin{equation}
\widetilde \nabla F =
(1,0,...,0,u^{\delta},u^{\delta},0,...,0,1)
\end{equation}
Hence
\begin{equation}
\widetilde \nabla_8 F=(0,...,0,1,1,0,....0)+ O(u).
\end{equation}
Let $Z$ be the first coordinate of $\nabla F$.
If $j$ is not a middle index, we have
\begin{equation}
D_{v_j}F=\nabla F \cdot v_j=Z \times O(u^{\delta+1}).
\end{equation}
This estimate comes from the Tangent Estimate Lemma \ref{est}.
If $j$ is the first middle index, then
\begin{equation}
D_{v_j}F=\nabla F \cdot v_j=Z \times 2 O(\delta).
\end{equation}
The first contribution comes from coordinate 1,
and is justified by the Tangent Estimate Lemma,
and the second contribution comes from coordinate $j$.
The above calculations show that
\begin{equation}
\widetilde \Upsilon_8 F= (0,...,0,1,1,0,....0)+ O(u).
\end{equation}
Hence $\widetilde \nabla_8 F = \widetilde \Upsilon_8 F+O(u)$.
\proofend
Now suppose that $F$ is one of the relevant monodromy
invariants, but not the Casimir.
Our analysis establishes
\begin{lemma}
\label{grad}
Both $\pi^{\perp}(\widetilde \nabla F)$ and
$\pi^{\perp}(\nabla F)$ have the following properties.
\begin{enumerate}
\item All coordinates
are at most $1+O(u)$ in size.
\item All coordinates
except coordinates $3$ and $6$ are $O(u)$.
\end{enumerate}
\end{lemma}
\startproof
This is immediate from our analysis of the heft
of $\nabla F$.
\proofend
\begin{lemma} One has
$$\widetilde \nabla_8 F \cdot e_j =
\widetilde \nabla F \cdot v_j + O(u).
$$
\end{lemma}
\startproof
Combining the Tangent Estimate Lemma with Lemma \ref{grad},
we see that
$$
\pi^{\perp}(\widetilde \nabla F) \cdot \pi^{\perp}(v_j)=O(u).
$$
Hence
\begin{equation}
\label{zoop1}
\widetilde \nabla F \cdot v_j = \pi \circ \widetilde \nabla F
\cdot e_j + O(u).
\end{equation}
From Property 1 above, we see that
$$\|\nabla_8F\|_{\infty}=
\|\nabla F\|_{\infty}+O(u).$$
Therefore
\begin{equation}
\label{zoop2}
\widetilde \nabla_8 F=\pi \circ \widetilde \nabla F+ O(u).
\end{equation}
Combining equations (\ref{zoop1}) and (\ref{zoop2}), we get
the result of the lemma.
\proofend
\begin{lemma}
Setting
$\lambda=\|\nabla F\|_{\infty}$, we have
$$
\lambda^{-1}(\Upsilon_8 F)_j = (\widetilde \Upsilon_8 F)_j + O(u).
$$
Here $(X)_j$ is the $j$th coordinate of $X$.
\end{lemma}
\startproof
Combining the Tangent Estimate Lemma \ref{est} with Lemma \ref{grad}, we have
$$\pi^{\perp} \circ \nabla F \cdot \pi^{\perp}(v_j) = O(u).$$
Therefore
$$
\|\Upsilon_8 F\|_{\infty} = \|\nabla_8 F\|_{\infty}+O(u).
$$
Combining this with equation (\ref{zoop2}), we have
$$
\|\Upsilon_8 F\|_{\infty} =\|\nabla F\|_{\infty} + O(u).
$$
Our lemma follows immediately.
\proofend
By definition, we have
\begin{equation}
\widetilde \nabla F \cdot v_j =
\lambda^{-1} \nabla F \cdot v_j=\lambda^{-1}(\Upsilon_8 F)_j; \hskip 30 pt
\lambda=\|\nabla F\|_{\infty}.
\end{equation}
Combining this last equation with our two lemmas, we have
\begin{equation}
(\widetilde \nabla_8 F)_j=\widetilde \nabla_8 F \cdot e_j =
(\widetilde \Upsilon_8 F)_j + O(u).
\end{equation}
This holds for all $j$. This completes the proof of
the Justification Lemma.
\section{The polygon and its tangent space}
\label{TAN}
The goal of this section is to construct the polygon
$P^u$ and prove the Tangent Lemma, which estimates
the tangent space $T_{P^u}(C)$. We will begin by
repackaging some of the material worked out
in \cite{Sch3}. The results here are self-contained,
though our main formula relies on the work done
in \cite{Sch3}. In order to remain consistent with
the formulas in \cite{Sch3}, we will use a slightly
different labelling convention for polygons.
\subsection{Polygonal rays}
We say that a {\it polygonal ray\/} is an infinite
list of points $P_{-7}, P_{-3}, P_1,P_5,...$ in the projective plane.
We normalize so that (in homogeneous coordinates)
\begin{equation}
P_{-7}=(0,0,1), \hskip 30 pt
P_{-3}=(1,0,1), \hskip 30 pt
P_1 = (1,1,1), \hskip 30 pt
P_5=(0,1,1).
\end{equation}
The first $4$ points are normalized to be the vertices of
the positive unit square, starting at the origin, and
going counterclockwise. Here we are interpreting these
points in the usual affine patch $z=1$.
This polygonal ray defines lines:
\begin{equation}
L_{-5+k}=P_{-7+k}P_{-3+k}; \hskip 30 pt k=0,4,8,...
\end{equation}
We denote by $LL'$ the intersection $L \cap L'$.
Similarly, $PP'$ is the line containing $P$ and $P'$.
The pairs of points and lines determine flags, as follows:
\begin{equation}
F_{-6+k}=(P_{-7+k},L_{-5_k}), \hskip 30 pt
F_{-4+k}=(P_{-3+k},L_{-5+k}), \hskip 30 pt
k=0,4,8,12...
\end{equation}
The corner invariants were defined in Section \ref{CORNERDEF}.
In this section we relate the definition there to
our labelling convention here.
We define
\begin{equation}
\label{flag2}
\chi(F_{0+k})=[P_{-7+k},P_{-3+k},L_{-5+k}L_{3+k},L_{-5+k}L_{7+k}],
\hskip 30 pt k=0,4,8,...
\end{equation}
\begin{equation}
\chi(F_{2+k})=[P_{9+k},P_{5+k},L_{7+k}L_{-1+k},L_{7+k}L_{-5+k}],
\hskip 30 pt k=0,4,8,...
\end{equation}
Here we are using the inverse cross ratio, as in equation
\ref{ICR}.
Referring to the corner invariants, we have
\begin{equation}
x_k=\chi(F_{2k}); \hskip 30 pt
x_{k+1}=\chi(F_{2k+2}); \hskip 40 pt k=0,2,4,...
\end{equation}
\begin{remark}
{\rm Notice that it is impossible to define $\chi(F_{-2})$ because we
would need to know about a point $P_{-11}$, which we have
not suppled. Likewise, it is impossible to define
$\chi(F_{-4})$ because we would need to know about $L_{-9}$,
which we have not supplied. Thus, the invariants
$x_0,x_1,x_2,...$ are well defined for our polygonal ray.}
\end{remark}
{\bf Cross product in vector form:\/}
Since we are going to be computing a lot of these
cross ratios, we mention a formula that works
quite well. We represent both points and lines
in homogeneous coordinates, so that
$(a,b,c)$ represents the line corresponding
to the equation $ax+bx+cz=0$. We define
$V*W$ to be the coordinate-wise product of
$V$ and $W$. Of course, $V*W$ is also a
vector. Let $(\times)$ denote the cross
product. We have
\begin{equation}
\label{cross}
(\chi,\chi,\chi)=\frac{(A \times B)*(C \times D)}{(A \times C)*(B \times D)}.
\end{equation}
Here $\chi$ is the inverse cross ratio of the points or
lines represented by these vectors.
It may happen that some coordinates in the denominator vanish.
In this case, one needs to interpret this equation as a kind
of limit of nearby perturbations.
This formula works whenever $A,B,C,D$ represent either
collinear points or concurrent lines in the projective plane.
\subsection{The reconstruction formulas}
\label{RECON}
Referring to the definition of the monodromy invariants, we
define $O_a^b$ to be the sum over all odd admissible monomials
in the variables $x_0,x_1,x_2,...$ which do not involve
any variables with indices $i \leq a$ or $i \geq b$. For instance
$$O_1^1=1, \hskip 20 pt O_1^3=1, \hskip 20 pt
O_1^5=1-x_3, \hskip 20 pt O_1^7 = 1 -x_3 +x_3x_4x_5.$$
We also note that, when $a<0$, the polynomial
$O_a^b$ is independent of the value of $a$.
For this reason,
when $a<0$ we simply write $O^b$ in place of $O_a^b$.
The corresponding set $S^b$ consists of admissible
sequences, all of terms are less than $b$.
Given a list $(x_0,x_1,x_2,...)$ we seek a polygonal ray which
has this list as its corner invariants. Here is the formula.
\begin{equation}
\label{reconstruct}
P_{9+2k}=(O^{3+k} - O_1^{3+k} + x_0x_1 O_3^{3+k},O^{3+k},O^{3+k} + x_0x_1 O_3^{3+k}),
\hskip 15 pt k=0,2,4,...
\end{equation}
We would also like a formula for reconstructing the lines
of a polygonal ray.
We start with the obvious:
\begin{equation}
L_{-5}=(0,1,0); \hskip 30 pt
L_{-1}=(-1,0,1); \hskip 30 pt
L_3=(0,-1,1).
\end{equation}
For the remaining points, we define
polynomials
$E_a^b$ exactly as we defined $O_a^b$ except we
interchange the uses of {\it even\/} and {\it odd\/}.
Thus, for instance $E_2^6=1-x^4$. Here is the formula.
\begin{equation}
\label{reconstruct3}
L_{7+2k} = (E^{2+k}-E_0^{2+k},E_0^{2+k}-x_0 E_2^{2+k},-E^{2+k}),
\hskip 30 pt k=0,2,4...
\end{equation}
\begin{remark}
{\rm These formulas are equivalent to equations 19
and 20 in \cite{Sch3}, but the normalization
of the first $4$ points is different, and
the roles of points and lines have been switched.
We got the above formulas by applying a suitable
projective duality to the polygonal ray in
\cite{Sch3}. }
\end{remark}
We mention one important connection between our
various reconstruction formulas. The
following is an immediate consequence of
Lemma 3.2 in \cite{Sch3}:
\begin{equation}
\label{reconstruct4}
P_{5+k} \times P_{9+k} = -(x_1x_3x_5,...,x_{k/2+1}) L_{7+k},
\hskip 20 pt k=0,4,8...
\end{equation}
We close this section with a characterization of the
moduli space of closed polygons within $X$. We
do not need this result for our proofs, but it
is nice to know.\footnote{One could give an alternative proof of Proposition \ref{TanProp} computing the Poisson bracket of the polynomials of Lemma \ref{variety} with the monodromy invariants.}
\begin{lemma}
\label{variety}
The invariant $x_1,...,x_{2n}$ define a closed polygon
if and only if $O^{2n-5}$ and all its cyclic shifts
vanish.
\end{lemma}
\startproof
We can think of a closed polygon as an $n$-periodic
infinite ray. The periodicity implies that
$P_{4n-7}=P_{-7}=(0,0,1)$. Since
$4n-7 = 2k+9$ for $k=2n-8$,
equation (\ref{reconstruct}) tells us that
$O^{2n-5}=0$. Considering equation
\ref{reconstruct3}, we see that
$E^{2n-6}=E_0^{2n-6}=0$. But $E_0^{2n-6}$ is a cyclic
shift of $O^{2n-5}$. Hence, if $P$ is closed then
$O^{2n-5}$ and all its cyclic shifts vanish.
Conversely, if $O^{2n-5}$ and all its shifts vanish then
$P_{4n-7} \in L_{-5}$ and $P_{-3} \in L_{4n-5}$. Likewise
$P_{4n-3} \in L_{-1}$ and $P_{1} \in L_{4n-1}$, and so on.
This situation forces $P_{4n-3}=P_{-3}$. Shifting
the indices, we see that $P_{4n+1}=P_1$, and so on.
\proofend
\begin{remark}
{\rm Observe that $O^{2n-5}$ involves exactly $2n-7$
consecutive corner invariants. If the
first $2n-8$ are specified, then the next
variable can be found by solving $O^{2n-5}=0$.
Thus, Lemma \ref{variety} gives an algorithmic
way to find a closed $n$-gon whose first
$2n-8$ corner invariants are specified.}
\end{remark}
\subsection{The polygon}
We start with an infinite periodic list of variables
which starts out
\begin{equation}
\label{corner}
(u,u^2,...,u^{n-4},u^{n-4},...,u^2,u^1,...)
\end{equation}
and has period $2n-8$.
We let $X_u$ denote the polygonal ray associated to this
infinite list. Once $u$ is sufficiently small, the
first $n$ points of $X_u$ are well defined. We define $P^u$ to be the $n$-gon
made from the first $n$-points of $X_u$, and we take
$u$ small enough so that this definition makes sense.
The first $2n-8$ corner invariants of $P^u$,
which we now identify with $x_0,...,x_{2n-9}$, are
the ones listed in equation (\ref{corner}).
However, when it comes time to compute
$x_{2n-8},...,x_{2n-1}$, we do not use the
relevant points of $X_u$ but rather substitute
in the corresponding point of $P^u$.
Thus, the remaining $8$ corner invariants change.
We write the corner invariants of $P^u$ as
\begin{equation}
a,b,c,d,u,u^2,u^3,...,u^3,u^2,u,d',c',b',a'.
\end{equation}
It follows
from symmetry that $e=e'$ for each $e \in \{a,b,c,d\}$.
This symmetry here is that the
first $2n-8$ invariants determine $P$, and their
palindromic nature forces $P$ to be
self-dual: the projective duality carries $P$
to the dual polygon made from the lines
extending the sides of $P$.
\begin{lemma}
\label{ccc}
$e=1+O(u)$ for each $e \in \{b,c,d\}$.
\end{lemma}
\startproof
We set $P_{-11}=(X,Y,Z)$ and $L_{-13}=(U,V,W)$. We have
\begin{equation}
L_{-9}=(1,0,0) \times (X,Y,Z)=(-Y,X,Z).
\end{equation}
Equations \ref{reconstruct} and \ref{reconstruct3} tell us
\begin{equation}
\label{orders}
(X,Y,Z)=(1,0,0)+O(u); \hskip 30 pt (U,V,W)=(0,1,-1)+O(u).
\end{equation}
We compute
\begin{equation}
\label{go3}
b=\chi(F_{-6})=\chi(P_{1},P_{-3},L_{-1}L_{-9},L_{-1}L_{-13})=
\frac{UX+WX+VY}{(U+W)(X-Y)}.
\end{equation}
\begin{equation}
\label{go2}
c=\chi(F_{-4})=\chi(P_{-11},P_{-7},L_{-9}L_{-1},L_{-9}L_3)=
\frac{X-Y}{X-Z}
\end{equation}
\begin{equation}
\label{go1}
d=\chi(F_{-2})=\chi(P_{5},P_{1},L_{3}L_{-5},L_{3}L_{-9})=
d=\frac{X}{X+Y+Z}.
\end{equation}
Our result is immediate from these formulas
and from equation (\ref{orders}).
\proofend
\begin{lemma}
\label{prefinal}
$a=u^s + O(u^{s+1})$, where $s=(n-4)(n-3)/2$.
\end{lemma}
\startproof
We have
\begin{equation}
\label{go4}
a=\chi(F_{-8})=\chi(P_{-15},P_{-11},L_{-13}L_{-5},L_{-13}L_{-1})=
\chi(A,B,C,D).
\end{equation}
We will estimate $a$ by considering the middle coordinate of
equation (\ref{cross}).
Calculations similar to the ones above give
$$
A = (0,1,1) + O(u), \hskip 10 pt
B = (0,1,1) + O(u), $$
\begin{equation}
C = (1,0,0) + O(u), \hskip 10 pt
D = (1,1,1) + O(u).
\end{equation}
Hence
\begin{equation}
\label{prefinal1}
(A \times C)_2 = + 1+O(u); \hskip 10 pt
(B \times D)_2 = + 1+O(u); \hskip 10 pt
(C \times D)_2 = - 1+O(u).
\end{equation}
Recall that
\begin{equation}
P_{-15}=P_{-15+4n}; \hskip 30 pt
P_{-11}=P_{-11+4n}.
\end{equation}
According to equation (\ref{reconstruct4}), we have
$$A \times B = -(x_1x_3,...x_{2n-9}) L_{-13+2n} = $$
\begin{equation}
\label{prefinal2}
-u_2 u_4 ... u_3 u_1 L_{-13+2n} =
-u^s L_{-13+4n}.
\end{equation}
But
\begin{equation}
L_{-13+4n} = (0,1,-1) + O(u).
\end{equation}
Therefore
$$(A \times B)_2=-u^s + O(u^{s+1}).$$
Looking at the signs in equation (\ref{prefinal1}), we see that
$a=u^s + O(u^{s+1})$.
\proofend
\subsection{The tangent space}
Recall that $\pi: \R^{2n} \to \R^{2n-8}$
is the projection which strips off the outer $4$ coordinates.
Let $\pi^{\perp}$ be as in the Tangent Estimate Lemma \ref{est}.
Recall that $\{v_k\}$ is the special basis of $T_{P}(C)$
such that $\pi(v_k)=e_k$ for $k=5,...,2n-4$.
\begin{lemma}
The following holds concerning the coordinates
of $\pi^{\perp}(v_j)$:
\begin{itemize}
\item Coordinates $2,4,5,7$ of $\pi^{\perp}(v_j)$ have
size $O(1)$
\item Coordinates $3,6$ have size
$O(u)$.
\end{itemize}
\end{lemma}
\startproof
As above, we will just consider coordinates $2,3,4$.
The other cases follow from symmetry.
We refer to the quantities used in the proof of Lemma \ref{ccc}.
Each of these quantities is a polynomial in the coordinates,
depending only on $n$. Hence $dX/dt$, etc., are all of size
at most $O(1)$. Moreover, the denominators on the right
hand sides of equations (\ref{go3}), (\ref{go2}), and (\ref{go1})
are all $O(1)$ in size. Our first claim now follows from
the product and quotient rules of differentiation.
For our second claim, we differentiate equation (\ref{go2}):
$$
\frac{dc}{dt}=\frac{X'(Y-Z) - X(Y'-Z') + ZY' - YZ'}{(X+Z)^2}=^*
$$
\begin{equation}
\label{last}
Y' - Z' + O(u)=
\frac{d}{dt} (-x_0x_1) O_3^{3+k}.
\end{equation}
The starred equality comes from the fact that
$(X,Y,Z)=(0,1,1)+O(u)$.
The claim now follows from the fact
that $x_0(0)=u$ and $x_1(0)=u^2$ and
$(O_3^{3+k})'(0)=O(1)$.
\proofend
\begin{lemma}
The following holds concerning the coordinates
of $\pi^{\perp}(v_j)$:
\begin{itemize}
\item When $j$ is not a middle index,
coordinates 1 and 8 of are of
size $O(u^{\delta+1})$.
\item When $j$ is the first middle index,
coordinate 1 equals $u^{\delta}(1+O(u))$ and
coordinate 8 is of size $O(u^{\delta+1})$.
\item When $j$ is the second middle index,
coordinate 8 equals $u^{\delta}(1+O(u))$ and
coordinate 1 is of size $O(u^{\delta+1})$.
\end{itemize}
\end{lemma}
\startproof
We will just deal with coordinate 1.
The statements about coordinate 8 follow from symmetry.
Let us revisit the proof of Lemma \ref{prefinal}.
Let $f=-(A \times B)_2$.
We have $a=fg$, where
\begin{equation}
g=-\frac{(C \times D)_2}{(A \times C)_2(B \times D)_2}.
\end{equation}
We imagine that we have taken some variation, and all these
quantities depend on $t$.
Each of the factors in the equation for $g$ has
derivative of size $O(1)$. Moreover, the denominator
in $g$ has size $O(1)$. From this, we conclude that
\begin{equation}
g(0)=1+O(u); \hskip 30 pt g'(0)=O(1).
\end{equation}
It now follows from the product
rule that
\begin{equation}
\frac{da}{dt}=\frac{df}{dt}(1+O(u)).
\end{equation}
Equations \ref{prefinal1} and \ref{prefinal2} tell us that
\begin{equation}
f(t)=(x_1x_3,...,x_{2n+9})\lambda(t); \hskip 30 pt
\lambda(t)=(L_{-13+2n})_2.
\end{equation}
By equation (\ref{reconstruct3}), we have
\begin{equation}
\lambda(0)=1+O(u); \hskip 30 pt
\lambda'(0)=O(1).
\end{equation}
Hence, by the product rule,
\begin{equation}
\frac{da}{dt} = \frac{d}{dt}(x_1x_3...x_{2n+9})(1+O(u)).
\end{equation}
Using the variables
\begin{equation}
x_1=u,...,x_j=u^j+t,x_{j+1}=u^{j+1},...
\end{equation}
we get the result of this lemma as a simple exercise in
calculus.
\proofend
The results above combine to prove the Tangent Space Lemma.
\bigskip
\noindent \textbf{Acknowledgments}.
Some of this research was carried out in
May, 2011, when all three authors were
together at Brown University. We would
like to thank Brown for its hospitality during this period.
ST was partially supported by a Simons Foundation grant.
RES was partially supported by N.S.F. Grant DMS-0072607,
and by the Brown University
Chancellor's Professorship.
\vskip 1cm
| 207,845
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1780010210
| 89,550
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TITLE: a multiple choice question on non-negative definite matrices
QUESTION [2 upvotes]: A symmetric matrix in $\mathbb{M}_n(\mathbb{R})$ is said to be non-negative definite if
$x^TAx≥0$ for all (column) vectors $x \in \mathbb{R}^n$. Which of the following statements
are true?
a. If a real symmetric $n×n$ matrix is non-negative definite, then all of its
eigenvalues are non-negative.
b. If a real symmetric $n×n$ matrix has all its eigenvalues non-negative, then
it is non-negative definite.
c. If $A\in\mathbb{M}_n(\mathbb{R})$, then $AA^T$ is non-negative definite.
I know that all of the above options are correct but I did not have proof of any one of them.can anybody tell me about the proof of them.
REPLY [2 votes]: Hints: The basic fact you need to know in order to prove part (b) is that $A$ is real symmetric if and only if it is orthogonally diagonalizable, i.e. $A$ is real symmetric if and only if $A=QDQ^T$ for some real diagonal matrix $D$ and real orthogonal matrix $Q$ (such that $QQ^T=Q^TQ=I$, i.e. $Q^T=Q^{-1}$).
(b) The eigenvalues of $A$ are the diagonal entries of $D$ (why?). So, if all eigenvalues of $A$ are nonnegative, $D$ has nonnegative diagonal entries. Now, for any vector $x$, let $y=Q^Tx$. Then $x^TAx=y^TDy=\sum_i d_iy_i^2$ and so ...
(a) You may use orthogonal diagonalization to finish this part too. If $x^TAx\ge0$ for every vector $x\in\mathbb{R}^n$, then in particular the inequality holds for $x=Q^Te_i$, where $e_i$ is the $i$-th vector in the standard basis of $\mathbb{R}^n$. Now, for such a vector $x$, what is $x^TAx$?
(a) But part (a) can be proved without orthogoanl diagonalization. If $\lambda$ is a real eigenvalue of $A$ and $v$ is an associated eigenvector, by considering $v^TAv$, you can show that $\lambda\ge0$.
(c) Note that $x^TAA^Tx=(A^Tx)^T(A^Tx)$. In general, for a vector $y$, what is $y^Ty$?
| 12,189
|
\begin{document}
\maketitle
\vspace{1cm}
\begin{abstract}
We prove the universality of the $\beta$-ensembles with convex
analytic potentials and for any $\beta>0$, i.e. we show
that the spacing distributions of log-gases at any inverse
temperature $\beta$ coincide with those of
the Gaussian $\beta$-ensembles.
\end{abstract}
\vspace{1.5cm}
{\bf AMS Subject Classification (2010):} 15B52, 82B44
\medskip
\medskip
{\it Keywords:} $\beta$-ensembles, universality, log-gas.
\medskip
\newpage
\section{Introduction}
The central concept of the random matrix theory as envisioned by E. Wigner
is the general hypothesis that
the distributions of eigenvalue spacings
of large complicated quantum systems are universal
in the sense that they depend only on the symmetry classes
of the physical systems but not on other detailed structures.
The simplest case for this hypothesis is for ensembles of large but finite dimensional
matrices. The general hypothesis in this setting thus asserts that
the eigenvalue spacing distributions of random matrices should be independent of
the probability distribution of the ensemble, up to scaling.
This is generally referred to as the universality of
random matrices.
In this paper we will focus only on the bulk behavior
i.e., on eigenvalue distribution in the interior of the spectrum,
although similar questions
regarding the edge distribution are also important.
Over the past two decades, spectacular progress (see, e.g.,
\cite{BI, DKMVZ1, DKMVZ2, PS:97, PS, DG, Lub} and \cite{AGZ, De1, DG} for a review)
on bulk universality
was made for classical invariant ensembles, i.e., matrix models with probability measure given by $e^{- N \beta { \rm Tr} V(H)/2 }/Z$
where $N$ is the size of the matrix $H$, $V$ is a real valued potential and $Z$
is the normalization.
It is well-known that the probability distribution of the ordered eigenvalues of $H$
on
the simplex determined by $\lambda_1\leq \dots\leq \lambda_N$
is given by
\begin{equation}\label{01}
\mu^{(N)}\sim e^{- \beta N \cH},
\quad
\cH = \sum_{k=1}^N \frac{1}{2}V(\lambda_k)-
\frac{1}{N} \sum_{1\leq i<j\leq N}\log (\lambda_j-\lambda_i) ,
\end{equation}
where the parameter $\beta=1,2,4$ is determined by the symmetry type of the matrix,
corresponding respectively to the classical orthogonal, unitary or symplectic ensemble.
With $\beta$ taking these special values, the correlation functions can be explicitly expressed
in terms of polynomials orthogonal to the measure $e^{- \beta V(x)/2}$.
Thus the analysis of the correlation functions
relies heavily on the asymptotic properties of the corresponding orthogonal polynomials. In the pioneering work
of Gaudin, Mehta and Dyson (see \cite{M} for a review), the potential
$V$ is the quadratic polynomial $V(x) = x^2$ and the orthogonal polynomials are the Hermite polynomials
for which asymptotic properties are well-known.
The major input of the recent work
is the asymptotic analysis of the orthogonal polynomials w.r.t. the measure $e^{- \beta V(x)/2}$ for general
classes of potentials. The formulas for orthogonal and symplectic cases, i.e., $\beta=1, 4$, are much more difficult to use than
the one for the unitary case. While universality for $\beta=2$ was proved for very general potential,
the best results for $\beta = 1, 4$ \cite{DG, KS, Sch} are still restricted to analytic $V$ with additional conditions.
For non-classical values of $\beta$, i.e., $\beta \not \in \{1, 2, 4\}$,
one can still consider the measure (\ref{01}), but
there is no simple expression of the correlation functions in
terms of orthogonal polynomials. Furthermore, the
measure \eqref{01} does not arise from mean-field type matrix models
like Wigner matrices with independent entries.
Nevertheless, $\mu$ is a Gibbs measure of particles
in $\bR$ with a logarithmic interaction, where the parameter $\beta$
is interpreted as the inverse temperature and a priori can be an arbitrary positive number.
These measures are called general $\beta$-ensembles. We will often
refer to the variables $\lambda_j$ as particles or points
and the system is called log-gas.
It was proved \cite{DumEde} that in the Gaussian case, i.e.,
when $V$ is quadratic, the measure \eqref{01} describes eigenvalues of
tri-diagonal matrices. This observation allowed one to establish
detailed properties, including the local
spacing distributions of the Gaussian $\beta$-ensembles \cite{VV}.
Gibbs measures in the continuum with long range or singular interactions are notoriously hard to analyze since
they are very far from the perturbative regime. For non-classical values of $\beta$, and
if we are not in the Gaussian case $V(\lambda)=\lambda^2$,
no simple explicit formula
is known to express the correlation functions in terms of orthogonal polynomials,
and one cannot rely on any explicit known matrix model.
In this paper we undertake
the direct analysis of the Gibbs measure and we
prove the universality for invariant models for any $\beta > 0$.
In other words, we will prove
that the local spacing distributions of \eqref{01} are independent of the potential $V$ for certain class of $V$.
There are two major ingredients in our new approach.
\medskip
\noindent
{\it Step 1. Uniqueness of local Gibbs measures with logarithmic interactions.}
The main result in this step asserts that if the particles are not too far
from their classical locations then
the spacing distributions are given by the corresponding Gaussian ones (We will take the uniqueness of the spacing
distributions as our definition of the uniqueness of Gibbs state).
More precisely, denote by $\rho$ the limiting density of the particles under the measure $\mu^{(N)}$ \eqref{01} as $N\to\infty$.
Let $\gamma_j =\gamma_{j,N}$ denote the location of the $j$-th point under $\rho$, i.e.,
$\gamma_j$ is defined by
\be\label{def:gamma}
N \int_{-\infty}^{\gamma_j} \rho (x) \rd x = j, \qquad 1\leq j\le N. \quad
\ee
We will call $\gamma_j$ the {\it classical location} of the $j$-th particle. The basic assumption is the following:
\medskip
\noindent
{\bf Assumption A.} For some $ {\frak b} < \frac 1 {38} $ and any $\alpha>0$, there exists $\epsilon_0>0$ such that
\be\label{assA}
\P_{\mu^{(N)}} ( |\lambda_k-\gamma_k|\le N^{-1 + {\frak b}} ) \ge 1 - \exp(-N^{\e_0})
\ee
for large enough $N$ and any $k \in [\alpha N, (1-\alpha) N]$.
\medskip
\noindent
Under this assumption
(under some minor and easily verifiable assumptions near the edges of the
limiting measure),
we will prove that the spacing distributions of $\mu$ are given
by the corresponding Gaussian model
with $V(x) = x^2$. We will use the Gaussian case as our reference
ensemble only for the convenience of definiteness. In fact, no detailed
properties of the Gaussian measures are used in the proof and any
other reference ensemble would have worked as well.
Furthermore, in this step
we make no assumption on the convexity of $V$, which is needed in the next step.
\medskip
\noindent
{\it Step 2. Particle location estimate.}
The second step is to verify Assumption A. For non-classical $\beta$, Assumption A
is only proved for $\frak b$ near one \cite{Joh, PS, KS}
for analytic potential $V$ under certain constraint.
This is far from sufficient to complete Step 1. We will prove Assumption A
for all $\beta > 0$ under the assumption that $V$ is convex and analytic. Our method uses the following three ideas: (1) The analysis of
the loop equation
in \cite{Joh, KS, Sch} to control the density.
(2) The logarithmic Sobolev
inequality guaranteed by the convexity of $V$.
(3) A multiscale analysis of the probability measures
of invariant ensembles.
We note that the assumption of analyticity on $V$ is needed only for using
the loop equation in (1).
The basic idea of our proof is to use the following tool from \cite{ESY4}: For two probability
measures $\mu$ and $\om$ define the Dirichlet form by
\[
D(\mu\mid \om) : =
\frac{1}{2 N} \int \Big|\nabla \sqrt{ \frac{\rd\mu}{\rd\om}}\Big|^2\rd \om.
\]
Then the difference of the local spacing distributions
of the two measures is negligible provided that
the Dirichlet form per particle is sufficiently small in the large $N$ limit \cite{ESY4}.
Notice that if we used the relative entropy of the two measures, then the uniqueness
of the Gibbs measures would require the total entropy, which is an extensive quantity, to be small.
To apply this Dirichlet form inequality,
we first localize the measure by fixing $\lambda_j$
for $j$ outside, say, the interval $[L+1, L + K]$ for $L$ in the bulk and $K = N^k$ for
some $k > 0$.
We will call these data of $\lambda_j$ outside the interval
$[L+1, L + K]$ the boundary condition.
We then compare this measure to a local Gaussian $\beta$-ensemble
with a fixed boundary condition by showing that
the Dirichlet form per particle of these two measures is small
for typical boundary conditions
w.r.t. $\mu$.
Our approach shares some philosophy from the recent
method on the universality of Wigner matrices \cite{ESY4, ESYY}.
In this approach, the key condition to establish is
\bigskip
{\bf Assumption III.} There exists an ${\frak a} > 0$ such that we have
\be
\E_{\mu_W} \frac{1}{N}\sum_{j=1}^N(x_j-\gamma_j)^2
\le CN^{-1-2{\frak a}}
\label{assum3}
\ee
with a constant $C$ uniformly in $N$. Here $\mu_W$ is the law given by the Wigner ensemble.
Under this assumption, a strong estimate on the local ergodicity of
Dyson Brownian motion (DBM) was established in \cite{ESY4, ESYY}.
DBM \cite{Dy} establishes a dynamical interpolation between Wigner matrices and
the invariant equilibrium measure $\mu$.
This estimate then implies the universality of Wigner matrices. Thus the main task
in proving the universality of Wigner matrices is reduced to verifying Assumption III.
There are several similarities between the method used for the universality of
Wigner matrices \cite{ESY4, EYYrigi} and the current proof for $\beta$-ensembles:
(i) Both rely on crude estimates such as \eqref{assA} and
\eqref{assum3} on the location of the
eigenvalues to establish the local spacing distributions are the same
as in the Gaussian cases.
(ii) Both use estimates on the Dirichlet form to identify the
local spacing distributions. (iii) The main model dependent argument
is to prove these crude bounds on the eigenvalues.
The precision of these a-priori estimates
on the eigenvalues
is weaker than the local spacing, but better than previously known results on eigenvalue locations:
we have to develop
new methods to prove \eqref{assA} and
\eqref{assum3}.
There are, however, substantial differences between the proofs of universalities for Wigner and $\beta$-ensembles.
First, since the $\beta$-ensembles are already in equilibrium,
there is no dynamical relaxation mechanism to exploit and the
local statistics need to be identified directly without dynamical argument.
Second, we obtain the crude estimate \eqref{assA}
by a method completely different from the Wigner matrices, as there is no underlying
matrix ensemble with independent entries to analyze. The accuracy result we obtain by this new method is actually optimal,
i.e. \eqref{assA} will be shown to hold for any ${\frak b}>0$.
\section{Statement of the main result}
Consider a probability measure
\begin{equation}\label{eqn:measure}
\mu^{(N)}_{\beta, V}=\mu^{(N)}(\rd\lambda)=\frac{1}{Z_N}
\prod_{1\leq i<j\leq N}|\lambda_i-\lambda_j|^\beta\prod_{k=1}^N e^{-N\frac{\beta}{2}V(\lambda_k)}
\rd\lambda_1\dots\rd\lambda_N,
\end{equation}
where $\lambda=(\lambda_1,\dots,\lambda_N)$, $\lambda_1\leq \dots\leq \lambda_N$.
Here the inverse temperature satisfies $\beta>0$ and the external potential $V$ is any
convex real analytic function in $\RR$, and such that
\begin{equation}\label{eqn:LSImu}
\varpi =\frac{\beta}{2} \inf_{x\in\RR}V''(x) > 0.
\end{equation}
For such a convex potential, as noted in the next section the equilibrium measure, denoted by $\rho(s)\rd s$, is supported on a single interval $[A,B]$.
In the following, we omit the superscript $N$ and we will write $\mu$ for $\mu^{(N)}$.
We will use $\P_\mu$ and $\E_\mu$ to denote the probability and the
expectation with respect to $\mu$.
The Gaussian case corresponds to
$V(\lambda)=\lambda^2$; the expectation with respect to this
Gaussian measure will be denoted by $\E_{\rm{Gauss}}$, and the equilibrium measure is
known to be
$$
\rho_{sc}(E):=\frac{1}{2\pi}\sqrt{(4-E^2)_+},
$$
the semicircle density. The Gaussian case includes
the classical GUE, GOE and GSE ensembles for the special choice of $\beta=1,2,4$, but
our result holds for all $\beta > 0$.
Now we state our main theorem which will be proven
at the end of Section~\ref{sec:loceq}:
\begin{theorem}\label{thm:Main} Assume
$V$ is any
real analytic function with
$\inf_{x\in\RR}V''(x) > 0$.
Let $\beta> 0$.
Consider the $\beta$-ensemble $\mu=\mu_{\beta, V}$.
Let $G:\RR\to \RR$ be a smooth, compactly supported function.
Let $E\in (A,B)$ lie in the interior of the support of $\rho$, and
similarly let $E'\in (-2,2)$ be inside the support of $\rho_{sc}$. Define $L$ and $L'$ by
$$
\frac{L}{N} = \int_{A}^{E} \rho(x) \rd x , \qquad
\frac{L'}{N} = \int_{-2}^{E'} \rho_{sc}(x) \rd x.
$$
Fix a parameter $K=N^k$ where $0<k\le \frac{1}{2}$ is an arbitrary constant. Let
$I$ and $I'$ be two intervals of natural numbers,
$I=[L+1, L+K]$, $I'=[L'+1, L'+K]$
with length $K=|I|$.
Then
\be
\lim_{N\to\infty}\Bigg| \E_\mu \frac{1}{K\rho(E)}\sum_{i\in I} G\Big(
\frac{N(\la_i-\la_{i+1})}{\rho(E)}\Big)
- \E_{\rm Gauss}
\frac{1}{K\rho_{sc}(E')}
\sum_{i\in I'} G\Big( \frac{N(\la_i-\la_{i+1})}{\rho_{sc}(E')}\Big)\Bigg|=0,
\label{eq:Main}
\ee
i.e. the appropriately normalized particles gap distribution of the
measure $\mu_{\beta, V}$ at the level $E$ in the bulk of the limiting density asymptotically coincides with that
for the Gaussian case and it is independent of the value of $E$ in the bulk. In particular the gap distribution is universal.
\end{theorem}
{\it Remark.} The same result (with the same proof) holds for higher order correlation functions
of particles gaps. More precisely,
fix $n\ge 1$ and an array of positive integers, $\bm = (m_1, m_2, \ldots, m_n)\in
\NN^n_+$.
Let $G:\bR^n\to\bR$ be a bounded smooth function with compact
support and we
define
\be
\cG_{i,\bm}(\bla) :=
\frac{1}{\rho(E)^n}G\Big( \frac{N(\la_i-\la_{i+m_1})}{\rho(E)} \, ,
\frac{N(\la_{i+m_1}-\la_{i+m_2})}{\rho(E)} \, ,
\ldots, \frac{N(\la_{i+m_{n-1}}-\la_{i+m_n})}{\rho(E)}\Big).
\label{cG}
\ee
Then, under the conditions of Theorem~\ref{thm:Main} and
using its notations,
we have
\be
\lim_{N\to\infty}\Bigg| \E_\mu \frac{1}{K}\sum_{i\in I} \cG_{i,\bm}(\bla)
- \E_{\rm Gauss}
\frac{1}{K}
\sum_{i\in I'} \cG_{i,\bm}'(\bla)\Bigg|=0,
\label{eq:Mainm}
\ee
where $\cG_{i, \bm}'$ is defined exactly as $\cG_{i,\bm}$ but
$\rho(E)$ is replaced with $\rho_{sc}(E')$.
The limit \eqref{eq:Mainm} can be reformulated as
the convergence of the correlation functions.
Let $\rho^{(N)}_n$ denote the $n$-point correlation
function of the measure $\mu =\mu_{\beta, V}^{(N)}$ defined by
\begin{equation}\label{eqn:corrFunct}
\rho^{(N)}_n(x_1,\ldots,x_n)=
\int_{\RR^{N-n}}\tilde\mu(x)\rd x_{n+1}\dots\rd x_{N},
\end{equation}
where $\tilde \mu$ is the symmetrized version of $\mu$ given in \eqref{eqn:measure}
but defined on $\RR^N$ instead of the simplex:
$$
\tilde\mu^{(N)}(\rd\lambda)=\frac{1}{N!}\mu(\rd\lambda^{(\sigma)}),
$$
where
$\lambda^{(\sigma)}=(\lambda_{\sigma(1)},\dots,\lambda_{\sigma(N)})$, with
$\lambda_{\sigma(1)}<\dots<\lambda_{\sigma(N)}$.
{F}rom \eqref{eq:Mainm}
we have the convergence of the correlation functions, stated as the following corollary.
Since the proof is a standard argument and it
is essentially identical to the one given in Section 7 of \cite{ESYY},
we omit it.
\begin{corollary}
Under the assumption of Theorem \ref{thm:Main} and with the same notations,
for any smooth test functions $O$ with compact support and for any $0<k\le \frac{1}{2}$,
we have, with $s:=N^{-1+k}$, that
\begin{align*}
\lim_{N \to \infty} \int & \rd \alpha_1 \cdots \rd \alpha_n\, O(\alpha_1,
\dots, \alpha_n) \Bigg [
\int_{E - s}^{E + s} \frac{\rd x}{2 s} \frac{1}{ \varrho (E)^n } \rho_n^{(N)} \Big ( x +
\frac{\alpha_1}{N\varrho(E)}, \dots, x + \frac{\alpha_n}{N\varrho(E)} \Big ) \\
&
- \int_{E' - s}^{E' + s} \frac{\rd x}{2 s} \frac{1}{\varrho_{sc}(E)^n}
\rho_{{\rm Gauss}, n}^{(N)}
\Big ( x +
\frac{\alpha_1}{N\varrho_{sc}(E)}, \dots, x + \frac{\alpha_n}{N\varrho_{sc}(E)} \Big ) \Bigg ]
\;=\; 0\,.
\end{align*}
\end{corollary}
The local statistics of the $\lambda_i's$ in the Gaussian case have been explicitly computed by Gaudin, Mehta and Dyson
(see, e.g., \cite{M})
for the classical value $\beta \in \{1, 2, 4\}$. For general $\beta> 0$, there is
an explicit description in terms of some stochastic differential equations, the {\it Brownian carousel} \cite{VV}.
Theorem \ref{thm:Main} will be proved in two steps as explained in the introduction.
For logical reasons, we will first present Step 2 on particle location estimates in
Section 3 and then
Step 1 on the uniqueness of Gibbs measure in a finite interval in Sections 4 and 5.
\section{Optimal accuracy for particle locations}
Along this section, we assume that $V$ satisfies the same conditions as in Theorem \ref{thm:Main}.
Let the typical position $\gamma_k$ be
defined by
$$
\int_{-\infty}^{\gamma_k}\rho(s)\rd s=\frac{k}{N}.
$$
Moreover, all constants in this section depend on the potential $V$, which is fixed.
In the following, we will denote $\llbracket x,y\rrbracket=\NN\cap[x,y]$,
The purpose of this section is to prove that accuracy holds
for the measure $\mu$ at the optimal scale $1/N$, in the following sense.
\begin{theorem}\label{thm:accuracy}
Take any $\alpha>0$ and $\epsilon>0$. There are constants
$\delta,c_1,c_2>0$ such that for any $N\geq 1$ and $k\in\llbracket \alpha N,(1-\alpha) N\rrbracket$,
$$
\P_\mu\left(|\lambda_k-\gamma_k|> N^{-1+\epsilon}\right)\leq c_1e^{-c_2N^\delta}.
$$
\end{theorem}
After some initial estimates relying on large deviations results, the proof consists in comparing
$\mu$ to some {\it locally constrained measures} for which better concentration estimates can be proved for the differences between particles. This measure is related to the pseudo-equilibrium measure in \cite{ESY4}, but has distinctly different properties.
Iterations of these comparisons will give optimal accuracy.
\subsection{Initial estimates}
The purpose of this paragraph it to prove the following crude estimate. It will be the initial step in the induction of Subsection \ref{subsec:induction}.
\begin{proposition}\label{prop:InitialEstimate}
For any $\alpha, \epsilon>0$
there are constants $c_1,c_2,\delta>0$ such that for any $N$ and $k\in\llbracket \alpha N,(1-\alpha)N\rrbracket$
\begin{equation}\label{eqn:InitialEstimate}
\P_\mu\left(|\lambda_k-\gamma_k|>N^{-\frac{1}{2}+\epsilon}\right)\leq c_1e^{- c_2 N^\delta}.
\end{equation}
\end{proposition}
This result is a direct consequence of the following equation
(\ref{eqn:initialconcentration}) and Corollary \ref{cor:InitialAccuracy}, whose proofs are the purpose of this section. We
first state well-known facts about the equilibrium measure.
For convex analytic potential $V$ satisfying the asymptotic growth condition (\ref{eqn:LSImu}) (or even with weaker hypotheses on $V$, see e.g. \cite{BPS, APS}), the equilibrium measure $\rho(s)\rd s$ associated with $(\mu^{(N)})_{N\geq 0}$
can be defined as the unique minimizer (in the set of probability measures on $\RR$ endowed with the weak topology) of the
functional
$$
I(\nu)=
\int V(t)\rd\nu(t)-
\iint\log|t-s|\rd\nu(s)\rd\nu(t)
$$
if $\int V(t)\rd\nu(t)<\infty$, and $I(\nu)=\infty$ otherwise.
Moreover, $\rho$
has the following properties:
\begin{enumerate}[(a)]
\item The support of $\rho$ is a single interval $[A,B]$.
\item This equilibrium measure satisfies
\be
\frac{1}{2}V'(t) = \int \frac{\rho(s)\rd s}{t-s}.
\label{equilibrium}
\ee
for any $t\in(A,B)$.
\item For any $t\in[A,B]$,
\begin{equation}\label{eqn:rho}
\rho(t)\ind t=\frac{1}{\pi}r(t)\sqrt{(t-A)(B-t)}\mathds{1}_{[A,B]}\ind t,
\end{equation}
where $r$ can be extended into an analytic function in $\CC$ satisfying
\begin{equation}\label{eqn:r}
r(z)=\frac{1}{2\pi}\int_A^B\frac{V'(z)-V'(t)}{z-t}\frac{\rd t}{\sqrt{(t-A)(B-t)}}.
\end{equation}
In particular, for convex $V$, $r$ has no zero in $\RR$.
\end{enumerate}
It is known that the particles locations cannot be far from its classical location \cite{BenGui, Sch}: for any
$\epsilon>0$ there are positive constants $C$, $c$, such that, for all $N\geq 1$,
\begin{align}
\P_\mu\left( \exists k\in\llbracket1,N\rrbracket\mid
| \lambda_k- \gamma_k| \ge \epsilon \right)\leq C e^{-c N^c }. \label{eqn:largDev1}
\end{align}
In order to have density strictly in a compact support,
for given $R>0$,
define the following variant of $\mu^{(N)}$ conditioned to have all particles in $[-R,R]$:
\begin{equation}\label{eqn:truncMeasure}
\mu^{(N,R)}(\rd\lambda)=\frac{1}{Z_{N,R}}
\prod_{1\leq i<j\leq N}|\lambda_i-\lambda_j|^\beta\prod_{k=1}^N e^{-N\frac{\beta}{2}V(\lambda_k)}\mathds{1}_{|\lambda_k|<R}
\rd\lambda_1\dots\rd\lambda_N.
\end{equation}
Let $\rho_k^{(N,R)}$ denote the marginals of the measure $\mu^{(N,R)}$, i.e. the
same definition as (\ref{eqn:corrFunct}), but with $\mu^{(N)}$ replaced by $\mu^{(N,R)}$.
Then Lemma 1 in \cite{BPS} states that
under condition (\ref{eqn:LSImu}) there exist some $R>0$ and $c>0$, depending only on $V$, such that for any $|x_1|,\dots,|x_k|\leq R$
\begin{equation}\label{eqn:BPS1}
\left|\rho^{(N,R)}_k(x_1,\dots,x_k)-\rho^{(N)}_k(x_1,\dots,x_k)\right|\leq \rho^{(N,R)}_k(x_1,\dots,x_k)e^{-c N},
\end{equation}
and for $|x_1|,\dots,|x_j|\geq R$, $|x_{j+1}|,\dots,|x_k|\leq R$,
\begin{equation}\label{eqn:BPS2}
\rho^{(N)}_k(x_1,\dots,x_k)\leq e^{-c N\sum_{i=1}^k\log |x_i|}.
\end{equation}
The last type of estimates we need are concentration and accuracy of the particles location at scale $N^{-1/2}$, in the bulk.
Concentration is a simple consequence of the Bakry-\'Emery convexity criterion for the logarithmic
Sobolev inequality (\cite{BakEme}, see also \cite{AGZ}):
define $\mathcal{H}$ by $\mu(\rd\lambda)=\frac{1}{Z_N}e^{- N \mathcal{H}(\lambda)}\rd \lambda$, and assume
\begin{equation}\label{eqn:BakEme}
\nabla^2 \mathcal{H}\geq \sigma \,\Id_N
\end{equation}
in the sense of partial order for positive
definite operators. Then $\mu$ satisfies a logarithmic Sobolev inequality with constant $2/(\sigma N)$:
for any probability density $f$ we have
\be\label{eqn:lsi}
\E_\mu f \log f \le \frac 2 { \sigma N} \E_\mu |\nabla \sqrt f |^2 .
\ee
It is well-known that the logarithmic Sobolev inequality implies the spectral gap and,
together with Herbst's lemma, it also implies that
for any $k\in\llbracket 1,N\rrbracket$ and $x>0$
$$
\P_\mu\left(|\lambda_k-\E_\mu(\lambda_k)|>x\right)\leq 2e^{-\sigma N x^2/2 }.
$$
In our case where $\mu$ is defined by (\ref{eqn:measure}), for any $v\in\RR^N$
\be
v^*(\nabla^2 H)v=\frac { \beta} N \sum_{i<j}\frac{(v_i-v_j)^2}{(\lambda_i-\lambda_j)^2}+\frac{\beta}{2}\sum_iV''(\lambda_i)v_i^2\geq \varpi
|v|^2
\label{eqn:Hconv},
\ee
where $\varpi$ is defined in (\ref{eqn:LSImu}).
Thus there is a
constant $\tilde c>0$ such that for any $k$
\begin{equation}\label{eqn:initialconcentration}
\P_\mu\left(|\lambda_k-\E_\mu(\lambda_k)|>x\right)\leq 2e^{-\tilde c Nx^2},
\end{equation}
i.e. concentration at scale $1/\sqrt{N}$ holds. We now prove that accuracy at the same scale
holds inside the bulk.
The proof of the following lemma is based on an argument in \cite{Joh} for the polynomial case.
In the form presented here, it follows very closely the proof in \cite{Sch} for
the analytic case except that we use the logarithmic Sobolev inequality to have a more precise estimate.
We now introduce some notations needed in the proof.
\begin{itemize}
\item $m_N$ is the Stieljes transform of $\rho^{(N)}_1(s)(\rd s)$, evaluated at some $z$ with $\Im(z)>0$, and $m$ its
limit:
$$
m_N(z)=\E_\mu\left(\frac{1}{N}\sum_{k=1}^N\frac{1}{z-\lambda_i}\right)=\int_\RR\frac{1}{z-t}\rho_1^{(N)}(t)\rd t,\
m(z)=\int_\RR\frac{1}{z-t}\rho(t)\rd t.
$$
It is well-known that uniformly in any $\{\Im(z)>\epsilon\}$, $\epsilon>0$,
$|m_N-m|\to 0$ (see e.g. \cite{AGZ}).
Along the proof of the next Lemma \ref{lem:Johansson}
we will see that this convergence holds at speed $1/N$.
\item $s(z)=-2r(z)\sqrt{(A-z)(B-z)}$, where the square root is defined such that
$$
f(z)=\sqrt{(A-z)(B-z)}\sim z \quad \text{ as } \quad z\to\infty;
$$
\item $b_N(z)$ is an analytic function defined by
$$
b_N(z)=\int_{\RR}\frac{V'(z)-V'(t)}{z-t}(\rho_1^{(N)}-\rho)(t)\ind t;
$$
\item finally, $c_N(z)=\frac{1}{N^2}k_N(z)+\frac{1}{N}\left(\frac{2}{\beta}-1\right)m_N'(z)$, where
$$
k_N(z)=\var_\mu\left(\sum_{k=1}^N\frac{1}{z-\lambda_i}\right).
$$
Here the $\var$ of a complex random variable
denotes $\var(X)=\E(X^2)-\E(X)^2$, i.e. without absolute value
unlike for the usual variance.
Note
that $|\var(X)|\leq \E(|X-\E(X)|^2)$.
\end{itemize}
The equation used by Johansson (which can be obtained by a change of variables
in (\ref{eqn:measure}) \cite{Joh} or by integration by parts \cite{Sch}), is a variation of the loop equation (see, e.g., \cite{Ey})
used in physics literatures and it takes the form
\begin{equation}\label{eqn:firstLoop}
(m_N-m)^2+s(m_N-m)+b_N=c_N.
\end{equation}
Equation \eqref{eqn:firstLoop} expresses the difference $m_N - m$ in terms of $(m_N-m)^2$, $b_N$ and $c_N$.
In the regime where $|m_N - m|$ is small,
we can neglect the quadratic term. The term $b_N$ is the same order as
$|m_N-m|$ and is difficult to treat. As observed in \cite{APS,Sch}, for analytic $V$, this term vanishes
when we perform a contour integration. So we have roughly the relation
\be\label{55}
(m_N-m) \sim \frac 1 { N^2} \var_\mu\left(\sum_{k=1}^N\frac{1}{z-\lambda_k}\right),
\ee
where we dropped the less important error involving $m_N'(z)/N $ due to the extra $1/N$ factor.
In the convex setting,
the variance can be estimated by the logarithmic Sobolev
inequality and we immediately obtain an estimate on $m_N - m$.
We then follow the method in \cite {ErdRamSchYau}
to use the Helffer-Sj\"ostrand functional calculus to have an estimate on the particle locations.
Although it is tempting
to use this new accuracy information on the particle locations to estimate the variance again in \eqref{55},
this naive bootstrap
is difficult to implement. The main reason is, roughly speaking, that
the particle location estimate obtained from knowing only the size of $m_N-m$
is not strong enough in the bootstrap.
The key idea in this section is the observation that {\it accuracy information on particle locations can be used to improve the local convexity
of the measure $\mu$ in the direction involving the differences of particle locations}, see Lemma \ref{lem:localConvexity}.
Now we are able to complete the bootstrap argument and obtain a more accurate estimate on $m_N-m$. Since this argument can be repeated, we can estimate the locations of particles up to the optimal scale in the bulk.
\begin{lemma}\label{lem:Johansson}
Let $\delta>0$.
For $z=E+\ii \eta$ with $A+\delta<E<B-\delta$
assume that
\begin{equation}\label{eqn:kNTo0}
\frac{1}{N^2}k_N(z)\to 0
\end{equation}
as $N\to\infty$ uniformly in $\eta\geq N^{-1+a}$ for some $0<a<1$. Then there
are constants $c,\e>0$ such that for any
$N^{-1+a}\le\eta\le\e$,
$A+\delta<E<B-\delta$,
\begin{equation}\label{eqn:lemJohansson}
|m_N(z)-m(z)|\leq c\left(\frac{1}{N\eta}+\frac{1}{N^2}k_N(z)\right).
\end{equation}
\end{lemma}
\begin{proof}
First, for technical contour integration reasons, it will be easier to consider the measure (\ref{eqn:truncMeasure})
instead of $\mu^{(N)}$ here. More precisely, define
\begin{align*}
m^{(R)}_N(z)&=
\E_{\mu^{(N,R)}}\left(\frac{1}{N}\sum_{k=1}^N\frac{1}{z-\lambda_i}\right)=
\int_\RR\frac{1}{z-t}\rho_1^{(N,R)}(t)\rd t,\\
k_{N}^{(R)}(z)&=\var_{\mu^{(N,R)}}\left(\sum_{k=1}^N\frac{1}{z-\lambda_i}\right),\\
c_N^{(R)}(z)&=\frac{1}{N^2}k_N^{(R)}(z)+\frac{1}{N}\left(\frac{2}{\beta}-1\right){m_N^{(R)}}'(z).
\end{align*}
Then it is a direct consequence of (\ref{eqn:BPS1}) and (\ref{eqn:BPS2}) that there are constants $c>0$ and
$R>0$ such that
uniformly on
$\eta\geq N^{-10}$ (or any power of $N$),
\begin{equation}\label{eqn:expDiff}
|m^{(R)}_N-m_N|=\OO\left(e^{-c N}\right),\ \ \ |k_{N}^{(R)}-k_{N}|=\OO(e^{-c N}).
\end{equation}
Consider the rectangle with vertices
$
2R+\ii N^{-10},- 2R+\ii N^{-10}, -2R-\ii N^{-10}, 2R-\ii N^{-10}
$,
call $\mathcal{L}$ the corresponding clockwise closed contour and $\mathcal{L}'$
the one consisting only in the horizontal pieces, with the same orientation.
{F}rom $(\ref{eqn:firstLoop})$, we obviously have, for $z\not\in\mathcal{L}'$,
$$
\frac{1}{ 2\pi\ii}\int_{\mathcal{L}'}\frac{(m_N(\xi)-m(\xi))^2+s(\xi)(m_N(\xi)-m(\xi))
+b_N(\xi)-c_N(\xi)}
{r(\xi)(z-\xi)}\rd\xi=0.
$$
Note that the above expression makes sense for large enough $N$, because then $r$ has no zero on $\mathcal{L}$.
Using (\ref{eqn:expDiff}), this implies, for $\eta\geq N^{-1}$,
$$
\frac{1}{2\pi\ii}\int_{\mathcal{L}'}\frac{(m^{(R)}_N(\xi)-m(\xi))^2
+s(\xi)(m^{(R)}_N(\xi)-m(\xi))
+b_N(\xi)-c^{(R)}_N(\xi)}{r(\xi)(z-\xi)}\rd\xi=
\OO(e^{-c N}).
$$
Now, as $\rho_1^{(N,R)}$ and $\rho$ are supported on $[-R,R]$, $m_N^{(R)}-m$
and $c_N^{(R)}$ are uniformly $\OO(1)$ in the vertical segments of $\mathcal{L}$. Consequently, from the above equation
$$
\frac{1}{2\pi\ii}\int_{\mathcal{L}}\frac{(m^{(R)}_N(\xi)-m(\xi))^2+s(\xi)(m^{(R)}_N(\xi)-m(\xi))+b_N(\xi)-c^{(R)}_N(\xi)}{r(\xi)(z-\xi)}\rd\xi=
\OO(N^{-10}).
$$
As $b_N$ and $r$ are analytic inside $\mathcal{L}$, for $z$ outside
$\mathcal{L}$ we get
$$
\frac{1}{2\pi\ii}\int_{\mathcal{L}}\frac{(m^{(R)}_N(\xi)-m(\xi))^2+s(\xi)(m^{(R)}_N(\xi)-m(\xi))-c^{(R)}_N(\xi)}{r(\xi)(z-\xi)}\rd\xi=
\OO(N^{-10}).
$$
Remember we define $f(z)=\sqrt{(A-z)(B-z)}$ uniquely by $f(z)\sim z$ as $z\to\infty$. Moreover, $|m_N^{(R)}-m|(z)=\OO(z^{-2})$ as $|z|\to\infty$ because
$\rho$ and $\rho_1^{(N,R)}$ are compactly supported:
\begin{multline*}
|m_N^{(R)}(z)-m(z)|=\left|\int_{-R}^R\frac{\rho(t)-\rho^{(N,R)}(t)}{z-t}\rd t\right|\\=\left|\int_{_R}^R(\rho(t)-\rho^{(N,R)}(t))\left(\frac{1}{z}+\OO\left(\frac{1}{z^2}\right)\right)\rd t\right|=\OO\left(z^{-2}\right).
\end{multline*}
Consequently, the function $s(m^{(R)}_N-m)/r=-2f(m^{(R)}_N-m)$ is $\OO(z^{-1})$ as $|z|\to\infty$. Moreover, it is analytic outside $\mathcal{L}$, so
the Cauchy integral formula yields
$$
\frac{1}{2\pi\ii}\int_{\mathcal{L}}\frac{s(\xi)(m^{(R)}_N(\xi)-m(\xi))}{r(\xi)(z-\xi)}\rd\xi=-2f(z)(m_N^{(R)}-m)(z),
$$
proving
\begin{equation}\label{eqn:withoutB}
-2f(z)(m_N^{(R)}(z)-m(z))=
-\frac{1}{2\pi\ii}\int_{\mathcal{L}}
\frac{(m^{(R)}_N(\xi)-m(\xi))^2-c^{(R)}_N(\xi)}{r(\xi)(z-\xi)}\rd\xi+
\OO(N^{-10}).
\end{equation}
Consider now the following rectangular contours, defined by their vertices:
\begin{align}
{\mathcal{L}}_1:\ &\notag
R+\epsilon+\ii \epsilon,- R-\epsilon+\ii \epsilon, -R-\epsilon-\ii \epsilon, R+\epsilon-\ii \epsilon,\\
{\mathcal{L}}_2:\ &\label{eqn:contour}
R+2\epsilon+2\ii \epsilon,- R-2\epsilon+2\ii \epsilon, -R-2\epsilon-2\ii \epsilon, R+2\epsilon-2\ii \epsilon,
\end{align}
where $\epsilon>0$ is fixed, small enough such that all zeros of $r$ are strictly outside $\mathcal{L}_2$.
For $z$ inside $\mathcal{L}_2$ and $\Im(z)\geq N^{-1}$, by the Cauchy formula, equation (\ref{eqn:withoutB}) implies that
\begin{multline}\label{eqn:onL1}
-2s(z)(m_N^{(R)}(z)-m(z))\\=-(m^{(R)}_N(z)-m(z))^2+c^{(R)}_N(z)
-\frac{1}{2\pi\ii}\int_{\mathcal{L}_2}\frac{(m^{(R)}_N(\xi)-m(\xi))^2-c^{(R)}_N(\xi)}
{r(\xi)(z-\xi)}\rd\xi+
\OO(N^{-10}).
\end{multline}
In the above expression, if now $z$ is on $\mathcal{L}_1$,
$|z-\xi|>\epsilon$, and on $\mathcal{L}_2$
$|r|$
is separated away
from zero by a positive universal constant. Moreover,
$c_N^{(R)}(\xi)$ can be bounded in the following way: for any constants $\alpha_1,\dots,\alpha_N\in [-R-\epsilon,R+\epsilon]$,
\begin{multline*}
\frac{1}{N^2}\left|\var_{\mu^{(N,R)}}\left(\sum_{k=1}^N\frac{1}{\xi-\lambda_k}\right)\right|
\leq
\frac{1}{N^2}
\E_{\mu^{(N,R)}}\left(\left|\sum_{k=1}^N
\frac{1}{\xi-\lambda_k}-\frac{1}{\xi-\alpha_k}\right|^2\right)\\\leq \frac{1}{\epsilon^4N}
\sum_{k=1}^N\E_{\mu^{(N,R)}}\left(|\lambda_k-\alpha_k|^2\right),
\end{multline*}
because for any $k$, we have $|\xi-\lambda_k|>\epsilon$, $|\xi-\alpha_k|>\epsilon$.
Now, choose $\alpha_k=\E_{\mu}(\lambda_k)$. By (\ref{eqn:BPS2}),
for large enough $N$ any $\alpha_k$, $1\leq k\leq N$,
is in $[-R-\epsilon,R+\epsilon]$ indeed. Moreover, by (\ref{eqn:BPS1}),
$$
\left|\E_{\mu^{(N,R)}}\left(|\lambda_k-\alpha_k|^2\right)
-\E_{\mu}\left(|\lambda_k-\alpha_k|^2\right)
\right|
\leq e^{-cN}\E_{\mu^{(N,R)}}\left(|\lambda_k-\alpha_k|^2\right),
$$
and, by the spectral gap inequality for $\mu$,
$\E_{\mu}\left(|\lambda_k-\alpha_k|^2\right)=\OO(N^{-1})$.
This together proves that $k^{(R)}_N(\xi)$ is $\OO(N^{-1})$, uniformly on the contour $\mathcal{L}_2$.
Moreover,
$\frac{1}{N}{m^{(R)}_N}'=\OO(N^{-1})$, so finally $c_N^{(R)}(\xi)$ is
uniformly $\OO(N^{-1})$ on $\mathcal{L}_2$ and (\ref{eqn:onL1}) implies
$$
-2s(z)(m_N^{(R)}(z)-m(z))=-(m^{(R)}_N(z)-m(z))^2(z)+\OO\left(\sup_{\mathcal{R}_2}|m_N^{(R)}-m|^2\right)+
\OO(N^{-1}).
$$
Moreover, from the maximum principle for analytic functions,
$\sup_{\mathcal{L}_2}|m_N^{(R)}-m|
\leq
\sup_{\mathcal{L}_1}|m_N^{(R)}-m|$, so the previous equation implies
$$
\sup_{\mathcal{L}_1}|m_N^{(R)}-m|=\OO\left(\sup_{\mathcal{L}_1}|m_N^{(R)}-m|^2+\frac{1}{N}\right).
$$
We know that $\rho_1^{(N)}(s)\rd s$ converges weakly to $\rho (s)\rd s$ (see \cite{AGZ}),
so by (\ref{eqn:BPS1}) and (\ref{eqn:BPS2}) $\rho_1^{(N,R)}(s)\rd s$ converges weakly to $\rho (s)\rd s$.
On $\mathcal{L}_1$, $z$ is at distance at least $\epsilon$ from the support of both $\rho_1^{(N,R)}(s)\rd s$
and $\rho(s)\rd s$ so, on $\mathcal{L}_1$, $m_N^{(R)}-m$ converges
uniformly to $0$. Together with the above equation, this implies that
$$
\sup_{\mathcal{L}_1}|m_N^{(R)}-m|=\OO\left(\frac{1}{N}\right).
$$
By the maximum principle the same estimate holds outside $\mathcal{L}_1$,
in particular on $\mathcal{L}_2$, so
equation (\ref{eqn:onL1}) implies that for $z$ inside $\mathcal{L}_1$
\begin{equation}\label{eqn:ReFirstLoop}
-2s(z)(m_N^{(R)}(z)-m(z))=-(m^{(R)}_N(z)-m(z))^2+c^{(R)}_N(z)+\OO\left(\frac{1}{N}\right).
\end{equation}
Moreover,
\begin{multline}\label{eqn:mNPrime}
\frac{1}{N} | {m_N^{(R)}}'(z)| = \frac{1}{N^2 } \left |
\E_{\mu^{(N,R)}} \sum_j \frac {1} {(z-\lambda_j)^2} \right |\\
\le \frac{1}{N \eta } \Im \, m^{(R)}_N(z)\leq \frac{1}{N \eta } |m^{(R)}_N(z)-m(z)|+\frac{1}{N\eta}|\Im\ m(z)|\leq
\frac{1}{N \eta } |m^{(R)}_N(z)-m(z)|+\frac{c}{N\eta}
\end{multline}
for some constant $c$.
We used the well-known fact that $\Im\ m$ is uniformly bounded on the upper half plane\footnote{This follows for example from properties of the Cauchy operator, see p 183 in \cite{De1}.}.
On the set $A+\delta <E<B-\delta$ and $|\eta|<\epsilon$, we have $\inf |s|>0$. Therefore
(\ref{eqn:ReFirstLoop}) takes the form
\begin{equation}\label{eqn:upBoundSt}
\left(1+\OO\left(\frac{1}{N\eta}\right)\right)(m^{(R)}_N(z)-m(z))=
\OO\left(|m^{(R)}_N(z)-m(z)|^2+\frac{1}{N^2}k^{(R)}_N(z)+\frac{1}{N\eta}\right).
\end{equation}
{F}rom the hypothesis (\ref{eqn:kNTo0}), if $N^{-1+a}\leq \eta\leq\epsilon$
and $A+\delta <E<B-\delta$, then
\begin{equation}\label{eqn:quadratic}
|m^{(R)}_N-m|\leq c|m^{(R)}_N-m|^2+\epsilon_N,
\end{equation}
for some $c>0$ and $\epsilon_N\to 0$ as $N\to\infty$.
For large $N$, (\ref{eqn:quadratic}) implies that
$|m^{(R)}_N-m|\leq 2\epsilon_N$ or $|m^{(R)}_N-m|\geq 1/c-2\epsilon_N$.
Together with $|m^{(R)}_N-m|(E+\epsilon \ii)\to 0$
and the continuity of
$|m^{(R)}_N-m|$ in the upper half plane, this implies that
$|m^{(R)}_N-m|\leq 2\epsilon_N$ and therefore
$|m^{(R)}_N-m|\to 0$ uniformly on $N^{-1+a}\leq \eta\leq\epsilon$, $A+\delta <E<B-\delta$.
Consequently, using (\ref{eqn:upBoundSt}), this proves that
there is a constant $c>0$ such that for any
$\eta\geq N^{-1+a}$,
$A+\delta<E<B-\delta$,
$$
|m^{(R)}_N(z)-m(z)|\leq c\left(\frac{1}{N\eta}+\frac{1}{N^2}k^{(R)}_N(z)\right).
$$
The same conclusion remains when substituting $m_N^{(R)}$ (resp. $k_N^{(R)}$)
by $m_N$ (resp. $k_N$) thanks to (\ref{eqn:BPS1}) and (\ref{eqn:BPS2}).
\end{proof}
\vspace{0.3cm}
To prove accuracy results for $\mu$, the above Lemma \ref{lem:Johansson} will be combined with the following one.
\begin{lemma}\label{lem:HS} Let $\delta<(B-A)/2$ and $E\in[A+\delta,B-\delta]$ and $0<\eta<\delta/2$.
Define a function $f= f_{E,\eta}$: $\R\to \R$
such that $f(x) = 1$ for $x\in (-\infty, E-\eta]$, $f(x)$ vanishes
for $x\in [E+\eta, \infty)$, moreover
$|f'(x)|\leq c\eta^{-1}$ and $|f''(x)|\leq c\eta^{-2}$, for some constant $c$.
Let $\wt\rho$ be an arbitrary signed measure
and let $S(z)= \int (z-x)^{-1}\wt\rho(x)\rd x$ be its Stieltjes transform.
Assume that, for any $x\in[A+\delta/2,B-\delta/2]$,
\begin{equation}\label{eqn:cond1}
\left| S(x+iy)\right|\leq \frac{ U}{Ny}\;\; \mbox{for}\;\; \eta <y<1 ,\;\;\mbox{and}\;\;
|\Im\, S(x+iy)|\leq \frac{ U}{Ny}\;\; \mbox{for}\;\; 0<y<\eta.
\end{equation}
Assume moreover that $\int_\RR\wt\rho(\lambda)\rd\lambda=0$ and that there is a real
constant $\mathcal{T}$ such that
\begin{equation}\label{eqn:cond2}
\int_{[-\mathcal{T},\mathcal{T}]^c} |\lambda\wt\rho(\lambda)|\rd\lambda \le
\frac{U}{N}.
\end{equation}
Then for some constant $C>0$, independent of $N$ and $E\in [A+\delta,B-\delta]$, we have
$$
\left|\int f_E(\lambda)\wt\rho(\lambda)\rd\lambda \right| \le
\frac{C U|\log\eta| }{N}.
$$
\end{lemma}
\begin{proof}
Our starting point, relying on the
Helffer-Sj\"ostrand functional calculus, is formula (B.13) in \cite{ErdRamSchYau}:
\begin{align}\label{eqn:HSbound1}
\left|\int_{-\infty}^\infty f_E(\lambda)\wt\rho(\lambda)\rd\lambda\right|
\leq&
C\left|\iint y f_E''(x)\chi(y)\Im S(x+\ii y)\rd x\rd y\right|\\
&+\label{eqn:HSbound2}
C\iint\left(|f_E(x)|+|y||f_E'(x)|\right)|\chi'(y)|\left|S(x+\ii y)\right|\rd x\rd y,
\end{align}
for some universal $C>0$, and
where $\chi$ is a smooth cutoff function with support in $[-1, 1]$, with $\chi(y)=1$ for $|y| \leq 1/2$ and with bounded derivatives.
Using (\ref{eqn:cond1}) and (\ref{eqn:cond2}), the support of $\chi'$ being
included in $1/2\leq|y|\leq 1$, and the fact that $f_E'$ is $\OO(\eta^{-1})$
on an interval of size $\OO(\eta)$, the term (\ref{eqn:HSbound2})
is easily bounded by $\OO\left(\frac{U}{N}\right)$.
Concerning the right hand side of (\ref{eqn:HSbound1}), following \cite{ErdRamSchYau} we split it depending on $0<y<\eta$ and $\eta<y<1$. Note that
by symmetry we only need to consider positive $y$.
The integral on the first integration regime is easily bounded by
$$
\left|\iint_{0<y<\eta}y f_E''(x)\chi(y)\Im S(x+\ii y)\rd x\rd y\right|=\OO\left(\iint_{|x-E|<\eta, 0<y<\eta}y\eta^{-2}\frac{U}{Ny}\rd x\rd y\right)=\OO\left(\frac{U}{N}\right).
$$
For the second integral, as $f_E''$ and $\chi$ are real, we can substitute $\Im m$ by $m$ and use the analyticity of $m$ when integrating by parts (first in $x$, then in $y$):
\begin{align*}
\left|\iint_{\eta<y}y f_E''(x)\chi(y)\Im S(x+\ii y)\rd x\rd y\right|
\leq&\left|\iint_{\eta<y}y f_E''(x)\chi(y)S(x+\ii y)\rd x\rd y\right|\\
=&\left|\iint_{\eta<y}y f_E'(x)\chi(y)S'(x+\ii y)\rd x\rd y\right|\\
\leq&
\left|\iint_{\eta<y}\partial_y(y\chi(y)) f_E'(x)S(x+\ii y)\rd x\rd y\right|\\
&+
\left|\int\eta f_E'(x)\chi(\eta)S(x+\ii \eta)\rd x\right|.
\end{align*}
This last integral is easily bounded by $\OO(U/N)$ using $(\ref{eqn:cond1})$. Concerning the previous one, as $f_E'=\OO(\eta^{-1})$, $|x-E|<\eta$ for non vanishing $f_E'$, $\partial_y(y\chi(y))=\OO(1)$ and $S(x+\ii y)=\OO(U/(Ny))$,
this is bounded by
$$\OO\left(\frac{U}{N}\int_{\eta}^1\frac{\rd y}{y}\right)=\OO\left(\frac{U|\log\eta|}{N}\right),$$
concluding the proof.
\end{proof}
As a corollary of Lemmas \ref{lem:Johansson} and \ref{lem:HS}, we get the accuracy at scale $1/\sqrt{N}$ for
the $\lambda_k$'s in the bulk.
\begin{corollary}\label{cor:InitialAccuracy}
For any $\alpha>0$ and $\epsilon>0$ we have
$$
|\gamma^{(N)}_k-\gamma_k|=\OO\left(N^{-1/2+\epsilon}\right)
$$
uniformly in $k\in\llbracket \alpha N,(1-\alpha)N\rrbracket$ where
$\gamma_k^{(N)}$ and $\gamma_k$ are defined by
$$
\int_{-\infty}^{\gamma^{(N)}_k}
\rho_1^{(N)}(s)\rd s=\frac{k}{N},\;\; \mbox{and} \;\; \int_{-\infty}^{\gamma_k}
\rho(s)\rd s=\frac{k}{N}.
$$
\end{corollary}
\begin{proof}
We will apply Lemma \ref{lem:HS} to $\wt\rho=\rho-\rho_1^{(N)}$ with $\eta=N^{-1/2+\epsilon}$,
and check the conditions on $S=m-m_N$.
We denote $z=x+\ii y$.
By the spectral gap inequality for the measure $\mu$, we get
\begin{equation}\label{eqn:vg}
\frac{1}{N^2}\left|\var_\mu\left(\sum_{k=1}^N\frac{1}{z-\lambda_k}\right)\right|
\leq
\frac{c}{N^3}\E_\mu\left(\left|\nabla\left(\sum_{k=1}^N
\frac{1}{z-\lambda_k}\right)\right|^2\right)\leq \frac{c}{N^2 y^4}.
\end{equation}
Together with Lemma \ref{lem:Johansson}, this implies that uniformly in
$N^{-1/2+\epsilon}\leq y\leq 1$ and $x\in[A+\delta/2,B-\delta/2]$, we have
$$
|m_N(z)-m(z)|=\OO\left(\frac{1}{N}+\frac{1}{N^2 y^4}\right)=\OO\left(\frac{\sqrt{N}}{N y}\right).
$$
For $0< y<N^{-1/2+\epsilon}$, $m$ is uniformly bounded and
$$
y\mapsto y\,\Im m_N(x+\ii y)=\int\frac{y^2}{(x-t)^2+y^2}\rho^{(N)}(t)\rd t
$$
is an increasing function, so denoting $y_0=N^{-1/2+\epsilon}$ we have
$$
y\,|\Im(m_N(x+\ii y)-m(x+\ii y))|
\leq
y_0\,\Im m_N(x+\ii y_0)+\OO(y)
\leq y_0|\Im(m_N(x+\ii y_0)-m(x+\ii y_0))|+\OO(y_0)
.$$
Therefore, for any $0<y<N^{-1/2+\epsilon}$,
$$
|\Im(m_N(x+\ii y)-m(x+\ii y))|=\OO\left(\frac{N^{1/2+\epsilon}}{Ny}\right).
$$
Finally, the condition (\ref{eqn:cond2}) with $U=\OO(N^{1/2+\epsilon})$
and with the choice of any $\mathcal{T} \ge \max (|A|, |B|)+\delta$ follows from the large deviation
estimate \eqref{eqn:largDev1}.
Using the conclusion of Lemma \ref{lem:HS}
for functions $f_E$ and $f_{E+\eta}$ defined in the same lemma,
and subtracting both results,
we get that uniformly in $E\in[A+\delta,B-\delta]$,
\begin{equation}\label{eqn:convRho}
\left|\int_{-\infty}^E(\rho^{(N)}(t)-\rho(t))\rd t\right|=\OO(N^{-1/2+\epsilon}),
\end{equation}
so if $\gamma_k^{(N)}\in[A+\delta,B-\delta]$, then
$|\gamma^{(N)}_k-\gamma_k|=\OO\left(N^{-1/2+\epsilon}\right)$. This estimate holds
uniformly in $k\in\llbracket \alpha N,(1-\alpha)N\rrbracket$:
as a consequence of (\ref{eqn:convRho}) and the smooth form (\ref{eqn:rho}) of $\rho$, for any $k\in\llbracket \alpha N,(1-\alpha)N\rrbracket$ and sufficiently large $N$ we have $\gamma_k^{(N)}\in[A+\delta,B-\delta]$, for $\delta>0$ small enough, concluding the proof.
\end{proof}
\begin{lemma}\label{edge}
For any $\epsilon>0$
there exists $c_1, c_2, \eps'$ positive
constants such that for any
$ N^{3/5+ \e}\le j \le N - N^{3/5+ \e} $, we have
$$
\P_\mu \left ( |\lambda_j-\gamma_j|\ge N^{-4/15+\epsilon}\right) \le c_1 e^{ - c_2 N^{\e'}} \; .
$$
\end{lemma}
\begin{proof}
We
will assume that $j< N/2$ in the following, i.e. we will estimate the accuracy near the edge $A$,
the proof close to the other edge $B$ being analogous.
We follow the notations used in Corollary \ref{cor:InitialAccuracy} and Lemma \ref{lem:HS}.
For $E \in [A-\delta, A+\delta] \cup [B-\delta, B + \delta]$, we have $\inf |f|(z) \ge \sqrt \eta $.
Therefore we can divide $-2 f(z)$ on both side of
(\ref{eqn:ReFirstLoop}) to have
\begin{equation}
\left(1+\OO\left(\frac{1}{N\eta ^{3/2}}\right)\right)(m^{(R)}_N(z)-m(z))=
\OO\left( \frac {|m^{(R)}_N(z)-m(z)|^2} {\sqrt \eta}+
\frac{1}{N^2 \sqrt \eta}k^{(R)}_N(z)+\frac{1}{N\eta^{3/2}}\right).
\end{equation}
By (\ref{eqn:vg}), (\ref{eqn:BPS1}) and (\ref{eqn:BPS2}) we can bound the variance term by
\be
\frac{1}{N^2 }k^{(R)}_N(z) \le \frac { c} { N^2 \eta^4}.
\ee
for $\eta\geq N^{-10}$ for example.
Following the same continuity argument in the proof of Lemma \ref{lem:Johansson}, we obtain that for any $\epsilon>0$
$$
|\Im(m_N-m)(x+\ii \eta)|=\OO\left(\frac{N^{\epsilon}}{N^2 \eta^{9/2} }\right),
$$
provided that
\be
\frac {1} {\sqrt \eta}\left [ \frac{1}{N^2 \sqrt \eta}k^{(R)}_N(z)+\frac{1}{N\eta^{3/2}} \right ] \le
\frac { c} { N^2 \eta^5 } \ll 1.
\ee
We can now follow the argument in the proof of Corollary \ref{cor:InitialAccuracy} so that
\eqref{eqn:cond1} holds with $U = N^{3/5}$. Since the condition \eqref{eqn:cond2} is easy to verify,
we thus have
$$
\left|\int f_E(t ) \big [ \rho^{(N)}(t)-\rho(t) \big ] \rd t \right| \le
\frac{C |\log\eta| }{N^{2/5}}, \quad \eta = N^{-2/5},
$$\
where $f_E$ is defined in Lemma \ref{lem:HS}. This proves that
$$
\int^{E - \eta}_{-\infty} \rho^{(N)}(t ) \rd t \le \int^{E + \eta}_{-\infty} \rho(t) \rd t +
\frac{C |\log\eta| }{N^{2/5}}.
$$
In particular,
$$
\frac j N -
\frac{C |\log\eta| }{N^{2/5}}
= \int^{\gamma_j^{(N)} }_{-\infty} \rho^{(N)}(t ) \rd t -
\frac{C |\log\eta| }{N^{2/5}} \le \int^{\gamma_j^{(N)} + 2 \eta}_{-\infty} \rho(t) \rd t,
$$
and we have, by definition of $\gamma_i$, that
$$
\gamma_{j - N^{3/5 + \e}} \le \gamma_j^{(N)} + 2 \eta.
$$
Similarly, the reverse inequality holds and we have
$$
\gamma_{j - N^{3/5 + \e}}- 2 \eta \le \gamma_j^{(N)} \le \gamma_{j + N^{3/5 + \e}} + 2 \eta.
$$
Since $\int_A^E \rho(t) \rd t \sim (E-A)^{3/2}$,
for $j \ge N^{3/5 + \e}$ we have
$$
|\gamma_{j - N^{3/5 + \e}} - \gamma_j |\le C \left ( \frac j N \right )^{-1/3}
N^{-2/5 + \e/2} \le N^{-4/15 + \e}.
$$
Moreover, by (\ref{eqn:initialconcentration}),
$\lambda_j$ is concentrated around $\E_\mu(\lambda_j)$
at scale $N^{-1/2}$, so
$|\E_\mu(\lambda_j)-\gamma^{(N)}_j|=\OO(N^{-1/2})$,
concluding the proof of the lemma.
\end{proof}
\subsection{The locally constrained measures}
In this section some arbitrary $\epsilon,\alpha>0$ are fixed.
Let $\theta$ be a continuous nonnegative function with $\theta=0$ on $[-1,1]$ and $\theta''\geq 1$ for $|x|>1$.
We can take for example $\theta(x)=(x-1)^2 \mathds{1}_{x>1}+(x+1)^2 \mathds{1}_{x<-1}$ in the following.
\begin{definition} \label{def:locallyConstrained}
For a given $k\in\llbracket\alpha N,(1-\alpha) N\rrbracket$ and any integer $1\leq M\leq \alpha N$, we denote $I^{(k,M)}=\llbracket k-M,k+M\rrbracket$ and $i_M=|I^{(k,M)}|=2M+1$.
Moreover, let
$$
\phi^{(k,M)}=
\beta\sum_{i<j,i,j\in I^{(k,M)}}\theta\left(\frac{N^{1-\epsilon}(\lambda_i-\lambda_j)}
{i_M}\right).
$$
We define the probability measure
\begin{equation}\label{eqn:omega}
\rd\omega^{(k,M)}:=\frac{1}{Z}e^{-\phi^{(k,M)}}\rd\mu,
\end{equation}
where $Z=Z_{\omega^{(k,M)}}$.
The measure
$\omega^{(k,M)}$ will be referred to as locally constrained
transform of $\mu$, around $k$, with width $M$.
The dependence of
the measure on $\epsilon$ will be suppressed in the notation.
\end{definition}
We will also frequently use the following notation
for block averages in any sequence $x_1, x_2, \ldots $
\begin{equation}\label{eqn:average}
x_k^{[M]}:=\frac{1}{i_M}\sum_{i\in I^{(k,M)}}x_i.
\end{equation}
The reason for introducing these locally constrained measures is that they improve the convexity in $I^{(k,M)}$
up to a common shift, as explained in the following lemma.
\begin{lemma}\label{lem:localConvexity}
Consider the previously defined
probability measure
$$\omega^{(k,M)}=\frac{1}{Z}e^{-\phi^{(k,M)}}\rd\mu=\frac{1}{\tilde Z}
e^{-N(\mathcal{H}_1+\mathcal{H}_2)}\rd\lambda,$$
where we denote
\begin{align*}
\mathcal{H}_1&:=\frac{1}{N}\phi^{(k,M)}-
\frac{\beta}{N}\sum_{i< j, i,j\in I^{(k,M)}}\log|\lambda_i-\lambda_j|
,\\
\mathcal{H}_2&:=-\frac{\beta}{N}\sum_{ (i, j) \in J^{(k,M)}}\log|\lambda_i-\lambda_j|+\frac{\beta}{2}\sum_{i=1}^NV(\lambda_i)
\end{align*}
where $J^{(k,M)}$ is the set of pairs of points $i<j$ in $\llbracket 1,N \rrbracket$ such that $i$ or $j$ is not in $I^{(k,M)}$,
and $\mathcal{H}_1=\mathcal{H}_1(\lambda_{k-M},\dots,\lambda_{k+M})$.
Then $\nabla^2 \mathcal{H}_2\geq 0$
and denoting $v=(v_i)_{i\in I^{(k,M)}}$, we also have
\begin{equation}\label{eqn:convexity}
v^*(\nabla^2 \mathcal{H}_1)v\geq
\frac{\beta}{2}\frac{N^{1-2\epsilon}}{i_M} \sum_{i,j\in I^{(k,M)}} (v_i-v_j)^2.
\end{equation}
\end{lemma}
\begin{proof}
Since $V$ is convex,
to prove the convexity of $\mathcal{H}_2$, it suffices to prove it for the Coulomb interaction terms;
this relies on the calculation, for any $u\in\RR^N$,
$$
u^*(\nabla^2\mathcal{H}_2(\lambda))u=\frac{\beta}{N}\sum_{J^{(k,M)}}\frac{(u_i-u_j)^2}{(\lambda_i-\lambda_j)^2}\geq 0.
$$
Moreover, for any $v\in\RR^{i_M}$, a similar calculation yields
\begin{equation}\label{eqn:calculationConvexity}
v^*(\nabla^2 \mathcal{H}_1)v\geq\frac{\beta}{N} \sum_{i< j, i,j\in I^{(k,M)}}\frac{(v_i-v_j)^2}{(\lambda_i-\lambda_j)^2}
+
\beta\frac{N^{1-2\epsilon}}{i_M^2}\sum_{i< j, i,j\in I^{(k,M)}}(v_i-v_j)^2
\theta''\left(\frac{N^{1-\epsilon}(\lambda_i-\lambda_j)}{i_M}\right).
\end{equation}
{F}rom our definition of $\theta$,
$$
\frac{1}{(\lambda_i-\lambda_j)^2}+
\frac{N^{2-2\epsilon}}{i_M^2}\theta''\left(\frac{N^{1-\epsilon}(\lambda_i-\lambda_j)}{i_M}\right)
\geq
\frac{N^{2-2\epsilon}}{i_M^2},
$$
which implies
\begin{equation}\label{iInfj}
v^*(\nabla^2 \mathcal{H}_1)v\geq \beta \frac{N^{1-2\epsilon}}{i_M^2}\sum_{i< j, i,j\in I^{(k,M)}}(v_i-v_j)^2.
\end{equation}
which completes the proof of (\ref{eqn:convexity}),
noting that the above factor $1/2$ comes from the strict ordering of $i$ and $j$ indexes in (\ref{iInfj}).
\end{proof}
The above convexity bound, associated with the following local criterion for the logarithmic Sobolev inequality,
will yield a strong concentration for $\sum_{i\in I^{(k,M)}} v_i\lambda_i$ under $\omega^{(k,M)}$, if $\sum_i v_i=0$.
This lemma is a consequence of the Brascamp-Lieb
inequality \cite{BraLie}. Notice that the original inequality applied only to measures on $\RR^N$,
but a mollifying argument in Lemma 4.4 of \cite{EKYY2}
has extended it to the measures on the simplex $\{ \lambda_1 < \lambda_2 < \ldots < \lambda_N\}$ considered in this paper.
\begin{lemma}\label{lem:localLSI} Decompose the coordinates $\lambda=(\lambda_1, \ldots ,\lambda_N)$
of a point in $\RR^N = \RR^m \times \RR^{N-m}$ as $\lambda=(x,y)$, where $x\in \RR^m$, $y\in \RR^{N-m}$.
Let $\omega=\frac{1}{Z}e^{-N \mathcal{H}}$ be a probability measure on $\RR^N = \RR^m \times \RR^{N-m}$ such that
$\mathcal{H}=\mathcal{H}_1+\mathcal{H}_2$, with $\mathcal{H}_1=\mathcal{H}_1(x)$ depending only on
the $x$ variables and $\mathcal{H}_2 =\mathcal{H}_2(x,y)$
depending on all coordinates. Assume that, for any $\lambda\in\RR^N$,
$\nabla^2\mathcal{H}_2(\lambda)\geq 0$.
Assume moreover that $\mathcal{H}_1(x)$ is independent of
$x_1 + \ldots + x_m$, i.e. $\sum_{i=1}^m \partial_i \mathcal{H}_1(x) =0$
and that for any $x, v\in\RR^m$,
\be
v^* (\nabla^2\mathcal{H}_1(x))v\geq \frac{\xi}{m} \sum_{i, j =1}^m |v_i-v_j|^2 \;
\ee
with some positive $\xi>0$.
Then for any function of the form $f(\lambda)=F( \sum_{i=1}^mv_i x_i)$, where
$\sum_iv_i=0$ and $F:\RR\to\RR$ is any smooth function,
we have
\be\label{lsii}
\int f^2\log f^2\rd\omega-\left(\int f^2\rd\omega\right)\log\left(\int f^2\rd\omega\right)
\leq \frac{1}{\xi N}\int|\nabla f|^2\rd\omega.
\ee
\end{lemma}
\begin{proof} In the space $\RR^m$ we
introduce new coordinates
$ (z, w) = M^* (x_1, \ldots, x_m)$ with
$z= (z_1, \ldots, z_{m-1}) \in \RR^{m-1}$ ,
\be
w:= m^{-1/2} \sum_i x_i,
\ee
and $M$ is an orthogonal matrix.
Since $\mathcal{H}_1(x)$ is independent of $x_1 + \ldots + x_m$,
we can define $\wh {\mathcal H}_1 (z) : = {\mathcal H}_1(x)$.
Similarly, the function $f(\lambda)=F( \sum_{i=1}^mv_i x_i)$ with $\sum_iv_i=0$
depends only on the $z$ coordinates, i.e. it can be written as
$g (z)= f(\lambda)$.
Hence we can rewrite
$$
\int_{\RR^N} f^2\log f^2\rd\omega = \int_{\RR^{m-1}} g^2\log g^2\rd \nu,\qquad
\int_{\RR^{N}} f^2\rd\omega= \int_{\RR^{m-1}} g^2\rd\nu,
$$
where $\rd \nu = \nu ( z ) \rd z $ with
$$
\nu ( z ) :=
\frac{1}{\tilde Z}e^{-N\tilde { \mathcal{H}}(z )}=\frac{ 1 } {Z} \int_{\RR\times \RR^{N-m}}
e^{-N \mathcal{H}(x,y)}\rd w \rd y.
$$
Introduce the variable $q = (w,y)\in \RR\times \RR^{N-m}$ and denote by
$ \mathcal{H}_{qq}, \mathcal{H}_{zq}, \mathcal{H}_{zz}$ the matrices of second partial derivatives.
As $\mathcal{H}_2$ is convex, the Brascamp-Lieb inequality yields
$$
\tilde{ \mathcal{H}}_{zz} \geq \frac{ 1 } {Z} \int_{\RR\times \RR^{N-m}}
e^{-N \mathcal{H}(x,y)} \Big [ \mathcal{H}_{zz} - \mathcal{H}_{zq} [ \mathcal{H}_{qq}]^{-1} \mathcal{H}_{zq} \Big ] \rd w \rd y .
$$
Since $\mathcal{H}_1$ is independent of $q$, we have, by assumption of the positivity
of the Hessian of ${\mathcal H}_2$, that for any $q$ fixed,
\be
( \mathcal{H}_2)_{zz} - \mathcal{H}_{zq} [ \mathcal{H}_{qq}]^{-1} \mathcal{H}_{zq} =
( \mathcal{H}_2)_{zz} - (\mathcal{H}_2)_{zq} [ (\mathcal{H}_2)_{qq}]^{-1} (\mathcal{H}_2)_{zq} \ge 0.
\ee
Thus we have, for any $u \in \RR^{m-1}$, that
\be
u^* \tilde{ \mathcal{H}}_{zz} u \geq u^* (\hat {\mathcal {H}}_1)_{zz} u
= u^* \wt M^* (\mathcal{H}_1)_{xx} \wt M u \ge \frac \xi m \sum_{i, j} [(\wt Mu)_i - (\wt M u )_j]^2 ,
\ee
where $\wt M$ denotes the first $m-1$ columns of $M$. Since the last column of $M$ is parallel with
$(1,1,\ldots, 1)\in \RR^m$ and $M$ is an orthogonal matrix, we have
$ \sum_i (\wt Mu)_i = 0$ and
\be
\frac \xi m \sum_{i, j=1}^m [(\wt Mu)_i - (\wt M u )_j]^2 = 2\xi \sum_{i=1}^m [(\wt Mu)_i ]^2
= 2 \xi \sum_{i=1}^{m-1} u_i ^2.
\ee
Hence the measure $\nu\sim \exp(-N {\mathcal {\wt H}})$
is log-concave with a lower bound $2N\xi$ on the Hessian of $N {\mathcal {\wt H}}$,
and we can apply the
Bakry-Emery argument to prove the logarithmic Sobolev inequality for $\nu$.
Without loss of generality we can assume
that $\int f^2\rd\omega=\int g^2\rd\nu= 1$.
Therefore, we have
\be
\int_{\RR^N} f^2\log f^2\rd\omega = \int_{\RR^{m-1}} g^2\log g^2\rd \nu
\le \frac{1}{N \xi} \int_{\RR^{m-1}}
| \nabla_z g |^2 \rd \nu = \frac{1}{N \xi} \int_{\RR^N} | \nabla_x f |^2 \rd \om,
\ee
where we have used the orthogonality of $M$ to show that $| \nabla_z g |^2=| \nabla_x f |^2$.
This proves the estimate \eqref{lsii}.
\end{proof}
\bigskip
It is now immediate, from Lemma \ref{lem:localConvexity}, Lemma \ref{lem:localLSI}
and Herbst's lemma, that the following concentration holds.
\begin{corollary}\label{cor:ConcentrationDifferences}
For any function $f(\{\lambda_i,i\in I^{(k,M)}\})=\sum_{I^{(k,M)}}v_i\lambda_i$ with $\sum_i v_i=0$ we have
$$
\P_{\omega^{(k,M)}}(|f-\E_{\omega^{(k,M)}}(f)|>x)
\leq 2\exp\left(-\frac{\beta}{4}\frac{ N^{2-2\epsilon}}{i_M |v|^2}x^2\right).
$$
\end{corollary}
Choosing $v_j = -v_{j+1} = 1$ and all other $v_i$'s being zero, this corollary shows that
the particle differences $\lambda_j-\lambda_{j+1}$ concentrate around their mean with respect to the
$\omega^{(k,M)}$ measure. By choosing $\epsilon$ small and $M$ almost
order one, we obtain concentration
almost up to the optimal scale $1/N$.
If we can justify that the measures $\omega^{(k,M)}$ and $\mu$ are very
close (in a sense to be defined), we will have concentration of differences at the optimal scale for
$\mu$. We will then separately show, by using the loop equation, that accuracy will hold
at the same scale as well. This is the purpose of the next subsection, through an inductive
argument.
\subsection{The induction}\label{subsec:induction}
The purpose of this paragraph is to prove the following proposition: if accuracy holds at scale $N^{-1+a}$, it holds also at scale $N^{-1+\frac{3}{4}a}$.
\begin{proposition}\label{prop:induction}
Assume that for some $a\in(0,1)$ the following property holds:
for any $\alpha,\epsilon>0$, there are constants
$\delta,c_1,c_2>0$ such that for any $N\geq 1$ and $k\in\llbracket \alpha N,(1-\alpha) N\rrbracket$,
\begin{equation}\label{eqn:IndHyp}
\P_\mu\left(|\lambda_k-\gamma_k|> N^{-1+a+\epsilon}\right)\leq c_1e^{-c_2N^\delta}.
\end{equation}
Then the same property holds also replacing $a$ by $3a/4$:
for any $\alpha,\epsilon>0$, there are constants
$\delta,c_1,c_2>0$ such that for any $N\geq 1$ and $k\in\llbracket \alpha N,(1-\alpha) N\rrbracket$,
we have
$$
\P_\mu\left(|\lambda_k-\gamma_k|> N^{-1+\frac{3}{4}a+\epsilon}\right)\leq c_1e^{-c_2N^\delta}.
$$
\end{proposition}
\noindent{\bf Proof of Theorem \ref{thm:accuracy}.}
This is an immediate consequence of the initial estimate, Proposition \ref{prop:InitialEstimate}, and iterations
of Proposition \ref{prop:induction}.\hfill\qed
Two steps are required in the proof of the above Proposition \ref{prop:induction}.
First we will prove that concentration holds at the smaller scale $N^{-1+\frac{a}{2}}$.
\begin{proposition}\label{prop:concentration}
Assume that (\ref{eqn:IndHyp}) holds. Then for any $\alpha>0$ and $\epsilon>0$,
there are constants $c_1,c_2,\delta>0$ such that for any $N\geq 1$ and $k\in\llbracket \alpha N,(1-\alpha)N\rrbracket$,
$$
\P_\mu\left(|\lambda_k-\E_\mu(\lambda_k)|>\frac{N^{\frac{a}{2}+\epsilon}}{N}\right)\leq
c_1e^{-c_2N^\delta}.
$$
\end{proposition}
The above step builds on the locally constrained measures of the previous subsection.
Then, knowing this better concentration, the accuracy can be improved to the scale $N^{-1+\frac{3a}{4}}$.
\begin{proposition}\label{prop:accuracy}
Assume that (\ref{eqn:IndHyp}) holds.
Then for any $\alpha>0$ and $\epsilon>0$,
there is a constant $c>0$ such that for any $N\geq 1$ and
$k\in\llbracket \alpha N,(1-\alpha)N\rrbracket$,
$$
\left|\gamma_k^{(N)}-\gamma_k\right|\leq c\frac{N^{\frac{3a}{4}+\epsilon}}{N}.
$$
\end{proposition}
Proposition \ref{prop:induction} is an immediate consequence of the last two propositions. The proofs of these two propositions are postponed to the end of this section, after
the following necessary series of lemmas.
\begin{lemma}\label{lem:concGapsOmega} Take any $\epsilon>0$ and $\alpha>0$. There are constants $c_1,c_2>0$ such that for any
$N\geq 1$, any integers $1\leq M_1\leq M\leq \alpha N$, any $k\in\llbracket\alpha N,(1-\alpha)N\rrbracket$,
and $\omega^{(k,M)}$ from Definition \ref{def:locallyConstrained} associated with $k,M,\epsilon$,
$$
\P_{\omega^{(k,M)}}\left(\left|\lambda_k^{[M_1]}-\lambda_k^{[M]}
-\E_{\omega^{(k,M)}}\left(\lambda_k^{[M_1]}- \lambda_k^{[M]}\right)\right|>\frac{x}{N^{1-\epsilon}}\sqrt{\frac{M}{M_1}}\right)\leq c_1e^{-c_2 x^2}.
$$
\end{lemma}
\begin{proof}
Note that
$\lambda_k^{[M_1]}-\lambda_k^{[M]}$ is of type $\sum_{I^{(k,M)}}v_i\lambda_i$ with $\sum v_i=0$ and
$$|v|^2=\sum_{1}^{M_1}\left(\frac{1}{M_1}-\frac{1}{M}\right)^2+\sum_{M_1+1}^M\frac{1}{M^2}
\leq \sum_{1}^{M_1}\left(\frac{2}{M_1^2}+\frac{2}{M^2}\right)+\sum_{M_1+1}^M\frac {2}{M^2}
\leq \frac{4}{M_1}.$$
Together with Corollary \ref{cor:ConcentrationDifferences}, this concludes the proof.
\end{proof}
\begin{lemma}\label{lem:diffExpectations}
Assume that for $\mu$ accuracy holds at scale $N^{-1+a}$, i.e. (\ref{eqn:IndHyp}). Take arbitrary $\alpha,\epsilon>0$.
There exist constants $c_1,c_2,\delta>0$ such that for any
$N\geq 1$, for any integer $M$ satisfying $N^a\leq M\leq \alpha N/2$, for any $k\in\llbracket\alpha N,(1-\alpha)N\rrbracket$
and for any $j\in\llbracket 1,N\rrbracket$, we have
$$
|\E_\mu(\lambda_j)-\E_{\omega^{(k,M)}}(\lambda_j)|\leq c_1 e^{-c_2 N^{\delta}},
$$
where the measure $\omega^{(k,M)}$ is defined in (\ref{eqn:omega}) from Definition \ref{def:locallyConstrained} with parameters $k,M,\epsilon$.
\end{lemma}
\begin{proof}
First, the total variation norm is bounded by the square root of the entropy, and by (\ref{eqn:BPS2}) the particles are bounded with very high probability, so we have
$$
|\E_\mu(\lambda_j)-\E_{\omega^{(k,M)}}(\lambda_j)|\leq \tilde C \sqrt { S(\mu\mid \omega^{(k,M)})} +\OO(e^{-\tilde c N})
$$
for some $\tilde c,\tilde C>0$ independent of $k, j$.
For the measures we are interested in, using the logarithmic Sobolev inequality for $\mu$, we have for some $c,C>0$
$$
S(\mu\mid \omega^{(k,M)})
\leq
C N^c\,\E_\mu\left(\theta'\left(\frac{(\lambda_{k+M}-\lambda_{k-M})N^{1-\epsilon}}{i_M}\right)^2\right).
$$
Now, as $\theta''(x)=0$ if $|x|<1$ and $\theta'(x)^2<4 x^2$, for some new and universal constants $c,C>0$
\begin{align}\label{eqn:entropyBound}
S(\mu\mid \omega^{(k,M)})
&\leq
C N^c\, \E_\mu\left((\lambda_{k+M}-\lambda_{k-M})^2
{\bf 1}_{|\lambda_{k+M}-\lambda_{k-M}|>\frac{i_M}{N^{1-\epsilon}}}\right)\nonumber\\
&\leq C N^c\,
\Big[ \E_\mu\left((\lambda_{k+M}-\lambda_{k-M})^4\right)\Big]^{1/2}
\Big[\P_\mu\left(
|\lambda_{k+M}-\lambda_{k-M}|>\frac{i_M}{N^{1-\epsilon}}\right)\Big]^{1/2}.
\end{align}
This moment of order 4 is polynomially bounded, for example just by concentration of order $N^{-1/2}$ for all $\lambda_j$'s under $\mu$.
Concerning the above probability, as $|\gamma_{k+M}-\gamma_{k-M}|=\OO(M/N)$, for sufficiently large $N$
if
$|\lambda_{k+M}-\lambda_{k-M}|>\frac{i_M}{N^{1-\epsilon}}$ then either
$|\lambda_{k+M}-\gamma_{k+M}|>M/N^{1-\epsilon}$ or $|\lambda_{k-M}-\gamma_{k-M}|>M/N^{1-\epsilon}$.
But accuracy holds at scale $N^{-1+a}<M/N$, so both previous events have exponentially small probabilities, uniformly in $k$.
Indeed, one has $k-M,k+M\in\llbracket \alpha N/2,(1-\alpha/2)N\rrbracket$ and by (\ref{eqn:IndHyp})there are constants
$\delta,c_1,c_2>0$ such that for any $N\geq 1$ and $k\in\llbracket \alpha N/2,(1-\alpha/2) N\rrbracket$,
$$
\P_\mu\left(|\lambda_k-\gamma_k|> M/N^{1-\epsilon}\right)\leq c_1e^{-c_2N^\delta}.
$$
This concludes the proof.
\end{proof}
\begin{lemma}\label{lem:toMu}
Assume that for $\mu$ accuracy and concentration hold at scale $N^{-1+a}$. Take arbitrary $\alpha,\tilde\epsilon>0$.
There are constants $c_1,c_2,\delta>0$ such that for any
$N\geq 1$, any integers $N^a\leq M\leq \alpha N$, $1\leq M_1\leq M$, and $k\in\llbracket 2\alpha N,(1-2\alpha)N\rrbracket$,
$$
\P_{\mu}\left(\left|\lambda_k^{[M_1]}-\lambda_k^{[M]}
-\E_{\mu}\left(\lambda_k^{[M_1]}-
\lambda_k^{[M]}\right)\right|>\frac{N^{\tilde\epsilon}}{N}\sqrt{\frac{M}{M_1}}\right)\leq c_1e^{-c_2 N^\delta}.
$$
\end{lemma}
\begin{proof}
Consider the measure $\omega^{(k,M)}$ associated with the choice $\epsilon=\tilde\epsilon/2$.
First note that, by Lemma \ref{lem:diffExpectations},
$$
\left|\E_{\mu}\left(\lambda_k^{[M_1]}-
\lambda_k^{[M]}\right)-\E_{\omega^{(k,M)}}\left(\lambda_k^{[M_1]}-
\lambda_k^{[M]}\right)\right|<ce^{-CN^{\delta_1}},
$$
for some coefficients $c,C,\delta_1$, uniformly in $N,k,M,M_1$.
As a consequence of this exponentially small difference of expectations, the probability bound to prove is equivalent to
the existence of $c_1,c_2,\delta>0$ such that
$$
\P_{\mu}\left(A\right)\leq c_1e^{-c_2 N^\delta},\ A=\left\{
\left|\lambda_k^{[M_1]}-\lambda_k^{[M]}-\E_{\omega^{(k,M)}}\left(\lambda_k^{[M_1]}-
\lambda_k^{[M]}\right)\right|>\frac{N^{\tilde\epsilon}}{N}\sqrt{\frac{M}{M_1}}
\right\},
$$
with the same uniformity requirements. By Lemma \ref{lem:concGapsOmega}, we know that there are such constants with
$$
\P_{\omega^{(k,M)}}\left(A\right)\leq c_1e^{-c_2 N^\delta},
$$
so the proof will be complete if we can prove that $|\P_{\omega^{(k,M)}}\left(A\right)-\P_{\mu}\left(A\right)|$
is uniformly exponentially small.
By the total variation\slash entropy inequality we have:
\be
|\P_{\omega^{(k,M)}}\left(A\right)-\P_{\mu}\left(A\right)|
\leq
\int|\rd\omega^{(k,M)}-\rd\mu|\leq
\sqrt{2S (\mu\mid \omega^{(k,M)})}.
\label{entropyineq}
\ee
This entropy was shown to be exponentially small in the proof of Lemma \ref{lem:diffExpectations}, see equation (\ref{eqn:entropyBound}).
\end{proof}
\begin{lemma}\label{lem:concA2}
Assume that for $\mu$ accuracy and concentration hold at scale $N^{-1+a}$.
For any $\tilde\epsilon>0$ and $\alpha>0$, there are constants $c_1,c_2,\delta>0$ such that for any $N\geq 1$
and $k\in\llbracket 2\alpha N,(1-2\alpha)N\rrbracket$,
$$
\P_\mu\left(\left|\lambda_k-\lambda_k^{[\alpha N]}
-\E_\mu(\lambda_k-\lambda_k^{[\alpha N]}) \right|
> \frac{N^{\frac{a}{2}+\tilde \epsilon}}{N} \right)
\leq
c_1e^{-c_2N^\delta}.
$$
\end{lemma}
Note that in this lemma and its proof, for non-integer $M$ we still
write $\lambda_k^{[M]}$ for
$\lambda_k^{[\lfloor M\rfloor]}$, where $\lfloor M \rfloor$ means
the lower integer part of $M$.
\begin{proof}
Note first that
\begin{multline*}
\left|\lambda_k-\lambda_k^{[\alpha N]}-\E_\mu(\lambda_k-\lambda_k^{[\alpha N]})
\right|\leq
\left|\lambda_k-\lambda_k^{[N^a]}-\E_\mu(\lambda_k-\lambda_k^{[N^a]})
\right|\\+
\left|\lambda_k^{[N^a]}-\lambda_k^{[\alpha N]}-\E_\mu(\lambda_k^{[N^a]}
-\lambda_k^{[\alpha N]})
\right|.
\end{multline*}
By the choice $M_1=1$, $M=N^a$ in Lemma \ref{lem:toMu}, the probability that the
first term is greater than $\frac{N^{\frac{a}{2}+\tilde \epsilon}}{N}$ is exponentially
small, uniformly in $k$, as desired.
Concerning the second term, given some $r>0$ and $q\in\NN$ defined by $1-r\leq a+q r<1$, it is bounded by
\begin{multline*}
\sum_{\ell=0}^{q-1}
\left|\lambda_k^{[N^{a+(\ell+1)r}]}-\lambda_k^{[N^{a+\ell r}]}
-\E_\mu\left(\lambda_k^{[N^{a+(\ell+1)r}]}-\lambda_k^{[N^{a+\ell r}]}\right)
\right|\\
+
\left|\lambda_k^{[a+qr]}-\lambda_k^{[\alpha N]}
-\E_\mu\left(\lambda_k^{[N^{a+qr}]}-\lambda_k^{[\alpha N]}\right)
\right|.
\end{multline*}
By Lemma \ref{lem:toMu}, for any $\epsilon>0$, each one of these $q+1$ terms has an exponentially small probability of being greater than
$
\frac{N^{\epsilon+\frac{r}{2}}}{N}
$. Consequently, choosing any $\epsilon$ and $r$ (and therefore $q$) such that $\epsilon+\frac{r}{2}<a/2$ concludes the proof.
\end{proof}
\bigskip
\noindent{\bf Proof of Proposition \ref{prop:concentration}.}
We just need to write
$$
|\lambda_k-\E_\mu(\lambda_k)|\leq
\left|\lambda_k-\lambda_k^{[\alpha N]}-
\E_\mu(\lambda_k-\lambda_k^{[\alpha N]})\right|
+
\left|\lambda_k^{[\alpha N]}-\E_\mu(\lambda_k^{[\alpha N]})\right|.
$$
By Lemma \ref{lem:concA2}, the first term has exponentially small probability to be greater than $\frac{N^{\frac{a}{2}+\epsilon}}{N}$. By the logarithmic Sobolev inequality for $\mu$ with constant $\OO(1)$ (see (\ref{eqn:LSImu})), the second term has an even better concentration, at scale $1/N$:
$$
\P\left(|\lambda_k^{[\alpha N]}-\E_\mu(\lambda_k^{[\alpha N]})|>\frac{x}{N}\right)\leq C\, e^{-c x^2}.
$$
This concludes the proof of the proposition.
\hfill\qed
\vspace{0.3cm}
\noindent{\bf Proof of Proposition \ref{prop:accuracy}.}
Thanks to Lemmas \ref{lem:Johansson}, \ref{lem:HS}, and reproducing the proof of
Corollary \ref{cor:InitialAccuracy}, we know it is sufficient to prove that for any $\delta>0$
$$
\frac{1}{N^2}\left|\var\left(\sum_k\frac{1}{z-\lambda_k}\right)\right|
$$
goes uniformly to 0 where $z=E+\ii\eta$, $E\in[A+\delta,B-\delta]$ and
$\eta\geq N^{-1+\frac{3a}{4}+\epsilon}$.
Let $i_0$ be the index in $[0,N]$ such that the typical position $\gamma^N_{i_0}$ is the closest to $E$.
Define the indexes of particles close to $E$, far from $E$ and in the edge as
\begin{align*}
\Int&=\{i:|i-i_0|<N^{a+\epsilon}\},\\
\Ext&=\{i:|i-i_0|\geq N^{a+\epsilon}, i\in\llbracket \alpha N,(1-\alpha)N\rrbracket\},\\
\Edg&=\{i: i\not\in\llbracket \alpha N,(1-\alpha)N\rrbracket\},
\end{align*}
where $\alpha$ is small enough such that
\begin{equation}\label{eqn:alphaGamma}
\gamma_{\alpha N}<A+\frac{\delta}{2}<A+\delta<E<B-\delta<B-\frac{\delta}{2}<\gamma_{(1-\alpha)N}.
\end{equation}
We choose $\alpha_k=\E_\mu(\lambda_k)$ in the following equations.
Then
\begin{multline}\label{eqn:EdgExtInt}
\frac{1}{N^2}\left|\var_\mu\left(\sum_{k}\frac{1}{z-\lambda_k}\right)\right|
\leq
\frac{C}{N^2}\E_\mu\left(\left|\sum_{k\in \Edg}\frac{1}{z-\lambda_k}-\frac{1}{z-\alpha_k}\right|^2\right)\\
+
\frac{C}{N^2}\E_\mu\left(\left|\sum_{k\in \Ext}\frac{1}{z-\lambda_k}-\frac{1}{z-\alpha_k}\right|^2\right)
+
\frac{C}{N^2}\E_\mu\left(\left|\sum_{k\in \Int}\frac{1}{z-\lambda_k}-\frac{1}{z-\alpha_k}\right|^2\right).
\end{multline}
The edge term is bounded by
\begin{equation}\label{e:edge}
\frac{C}{N}\sum_{\Edg}\E_\mu\left(\left|\frac{1}{z-\lambda_k}-\frac{1}{z-\alpha_k}\right|^2\right)
\leq
\frac{C'}{N\eta^2}\sum_{\Edg}\P\left(|E-\lambda_k|<\frac{\delta}{3}\right)
+
\frac{C'}{N\delta^2}\sum_{\Edg}\E_\mu\left(|\lambda_k-\alpha_k|^2\right).
\end{equation}
{F}rom the condition (\ref{eqn:alphaGamma}) and the large deviation estimate
(\ref{eqn:largDev1}), the above probability is exponentially small. Moreover,
the above $L^2$ moments are $\OO(1/N)$
by the spectral gap inequality for $\mu$, see e.g. equation (\ref{eqn:initialconcentration}).
Hence the edge term goes to 0 uniformly.
Using the accuracy at scale $N^{-1+a}$ and
the concentration at scale $N^{-1+a/2}$ (Proposition \ref{prop:concentration}),
the second term in (\ref{eqn:EdgExtInt})
is bounded,
up to constants, for some $c_1,c_2,\delta>0$ by
\begin{multline*}
\frac{1}{N^2}\E\left(\left|\sum_{k\geq N^{a+\epsilon}}\frac{\lambda_{i_0+k}-\alpha_{i_0+k}}
{(\frac{k}{N})^2}\right|^2\right)+c_1 e^{-c_2 N^\delta}\\
\leq
N^2\E\left(\sum_{k\geq N^{a+\epsilon}}
\frac{|\lambda_{i_0+k}-\alpha_{i_0+k}|^2}{k^2}\right)\sum_{k\geq N^{a+\epsilon}}
\frac{1}{k^2}+c_1 e^{-c_2 N^\delta}
\leq
N^2N^{-2+a}(N^{-a})^2=N^{-a}.
\end{multline*}
In particular, it converges uniformly to 0.
For the third term, for some $c>0$ it is less than
\begin{multline*}
\frac{1}{N^2\eta^4}\E\left(\left(\sum_{Int}|\lambda_k-\alpha_k|\right)^2\right)
\leq
\frac{c}{N^2\eta^4}N^{a+\epsilon}\E\left(\sum_{Int}|\lambda_k-\alpha_k|^2\right)\\
\leq c\frac{(N^{a+\epsilon})^2}{N^2\eta^4}N^{-2+a}=c\frac{N^{3a+2\epsilon}}{N^4\eta^4}.
\end{multline*}
This goes to 0 if $\eta\gg N^{-1+\frac{3a}{4}+\frac{\epsilon}{2}}$, concluding the proof.
\hfill\qed
\section{Local equilibrium measure}\label{sec:loceq}
\subsection{Construction of the local measure}
Let $0<\kappa<1/2$. Choose $q\in [\kappa, 1-\kappa]$ and set $L=[Nq]$ (integer part).
Fix an integer $K$ with $K\le (N-L)/2$, in fact we will always assume
that $K$ depends on $N$ as $K=N^k$ with $k<1$.
We will study the local spacing statistics
of $K$ consecutive particles
$$
\{ \lambda_j\; : \; j\in I\}, \qquad I=I_L:=
\llbracket L+1, L+K \rrbracket.
$$
These particles are typically located near
$E_q$ determined by the relation
$$
\int_{-\infty}^{E_q} \rho(t) \rd t = q.
$$
Note that $|\gamma_L- E_q|\le C/N$.
We will distinguish the inside and outside particles
by renaming them as
\be\label{35}
(\lambda_1, \lambda_2, \ldots,
\lambda_N):=(y_{1}, \ldots y_{L}, x_{L+1}, \ldots x_{L+K}, y_{L+K+1},
\ldots y_{N}) \in \Xi^{(N)},
\ee
but note that they keep their original indices.
The notation $\Xi^{(N)}$ refers to the simplex
$\{\bz \; :\; z_1<z_2< \ldots < z_N\}$ in $\RR^N$.
In short we will write
$$
\bx=( x_{L+1}, \ldots x_{L+K} ), \qquad \mbox{and}\qquad
\by=
(y_{1}, \ldots y_{L}, y_{L+K+1},
\ldots y_{N}),
$$
all in increasing order, i.e. $\bx\in \Xi^{(K)}$ and
$\by \in \Xi^{(N-K)}$.
We will refer to the $y$'s as the external
points and to the $x$'s as internal points.
We will fix the external points (often called
as boundary conditions) and study
the conditional measures on the internal points.
We consider the parameters $L$ and $K$ fixed and we
will not indicate them in the notation.
We first define the
{\it local equilibrium measure} on $\bx$ with boundary condition $\by$ by
\begin{equation}\label{eq:muyde}
\quad
\mu_{\by} (\rd\bx) = u_\by(\bx) \rd \bx, \qquad
u_\by(\bx):= u (\by, \bx) \left [ \int u (\by, \bx) \rd \bx \right ]^{-1},
\end{equation}
where $u$ is the density of $\mu_V$.
Note that for any fixed $\by\in \Xi^{(N-K)}$, $x_j$
lies in the interval $[y_{L}, y_{L+K+1}]$.
Given the classical locations, $\gamma=\{\gamma_1,\gamma_2, \ldots , \gamma_N\}$
with respect to the $\mu$-measure, we define the {\it relaxation measure}
$\mu_N^{\tau, \gamma} =\mu^\tau$ by
\be\label{232}
\rd\mu^\tau :=\frac{Z}{ Z_{\mu^\tau}}e^{-N Q^\tau }\rd\mu,
\quad Q^\tau (\bx) = \sum_{j\in I}
Q_j^\tau(x_j) , \qquad Q_j^\tau (x) = \frac{1}{2 \tau } ( x - \gamma_j)^2,
\ee
where $Z_\mu$ is chosen such that $\mu$ is a probability measure.
Here $0< \tau < 1$ is a parameter which may even depend
on $\by$, i.e., $\tau=\tau(\by)$ is allowed. Note that an artificial
quadratic confinement has been added to the equilibrium measure.
We define {\it the local relaxation measure}
$\mu_{\by}^\tau$ to be conditional measure of $\mu^\tau$.
Define the Dyson Brownian motion
reversible with respect to $\mu_{\by}^\tau$,
by the Dirichlet form
\be
D_{\mu_{\by}^\tau}(f) =
\sum_{i\in I} \frac{1}{2N}
\int (\partial_i f)^2 \rd \mu_{\by}^\tau ,
\label{def:dirmu}
\ee
where $\partial_i=\partial_{x_i}$.
The Hamiltonian $\cH_\by^\tau$ of the measure $\mu_\by^\tau (\rd \bx) \sim \exp(-N\cH_\by^\tau)$
is given by
\be\label{24}
\cH_{\by}^\tau (\bx) =
\sum_{i\in I} \frac{\beta}{2}V_\by (x_i)
- \frac{ \beta }{N} \sum_{i,j\in I\atop i< j}
\log |x_{j} - x_{i}| + \sum_{i \in I} Q_i^\tau (x_i ),
\ee
\be
V_\by (x) = V(x) - \frac{ 1}{ N} \sum_{j \not \in I}
\log |x - y_{j}|.
\ee
We now define the set of {\it good boundary configurations}
with a parameter $\e_0>0$ and a parameter $\delta=\delta(N)>0$ that in the applications
may depend on $N$:
\begin{align}\label{goodset}
\cG_{\delta,\e_0}=\cG:=
\Big\{ & \by \in \Xi^{(N-K)}\; :\; |y_k-\gamma_k|\le \delta, \; \forall \, k\in
\llbracket N\kappa/2, L\rrbracket \cup \llbracket L+K+1,
N(1-\kappa/2)\rrbracket, \\
& \mbox{and}\;\; |y_k-\gamma_k|\le 1, \; \forall \, k\in \llbracket 1, N\rrbracket, \nonumber \\
& \mbox{and}\;\; \E_{\mu_\by} (x_j-\gamma_j)^2\le \delta^2 \;\;\mbox{for all}
\;\; j\in \llbracket L+1, L+K \rrbracket \nonumber\\
& \mbox{and}\;\; y_L-y_{L-1}\ge \exp(-N^{\e_0}), \;\;\; y_{L+K+2}-y_{L+K+1}\ge \exp(-N^{\e_0})
\Big\} .\nonumber
\end{align}
First we show that the good configurations have overwhelmingly large probability
\begin{lemma}
For any $\e_0>0$ and for any choice $\delta= N^{-d}$
with $d\in (1-k,1)$, there is an $\e'>0$ depending on $d$ such that
\be
\P_\mu (\cG^c)\le Ce^{-cN^{\e'}} + Ce^{-cN^{\e_0}} .
\label{goodsetprob}
\ee
\end{lemma}
\begin{proof}
We have proved in Theorem~\ref{thm:accuracy} that for any choice $\delta= N^{-d}$
with $d\in (0,1)$ the probability that the first condition
in \eqref{goodset} is violated is bounded by $C\exp(-cN^{\e'})$
with some $\e'>0$ depending on $d$.
Similarly, the second condition is violated
with an analogous very small probability by \eqref{eqn:largDev1}.
To check the probability to violate the third requirement in the definition
of $\cG$, we use that
\begin{align}
\P_\mu \Big\{ \E_{\mu_\by} (x_j-\gamma_j)^2 \ge\delta^2\Big\}
& \le \P_\mu \Big\{ \P_{\mu_\by} \{ |x_j-\gamma_j| \ge\delta/2\}\ge 3\delta^2/4
\Big\}+C\exp(-cN^{\e'})\nonumber\\
& \le C\delta^{-2} \E_\mu \P_{\mu_\by} \big\{ |x_j-\gamma_j|\ge \delta/2\big\}+C\exp(-cN^{\e'})
\nonumber\\
& \le C\delta^{-2} \P_\mu\big\{ |x_j-\gamma_j|\ge \delta/2\big\}
\le c_1 e^{-c_2 N^{\e'}},
\end{align}
since for $\by$ satisfying the first two conditions of \eqref{goodset} we have
$$
\E_{\mu_\by} (x_j-\gamma_j)^2 \le \delta^2/4 +
\P_{\mu_\by} \{ |x_j-\gamma_j| \ge\delta/2\}
$$
as $x_j-\gamma_j\le y_{L+K+1} - \gamma_j \le \delta +\gamma_{L+K+1}-\gamma_1\le 1$
and also a similar lower bound holds.
Finally, we show that
$$
\P_\mu\big( y_L-y_{L-1}\le \exp(-N^{\e_0})\big)\le Ce^{-cN^{\e_0}},
$$
and a similar bound holds for the other condition in the fourth
line of \eqref{goodset}. For simplicity of the presentation
and to avoid introducing new notations, we will actually prove
$$
\P_\mu\big( y_{L+1}-y_L\le \exp(-N^{\e_0})\big)\le Ce^{-cN^{\e_0}}
$$
from which the previous inequality follows just by shifting the
indices. With the events
$$
A:= \big\{ y_{L+1}-y_L\le \exp(-N^{\e_0})\big\}, \qquad
\Omega:=\big\{ y_{L+K+1}-y_L\le 2a\big\},
$$
we write
\be
\P_\mu (A) = \E_\mu \big[{\bf 1}(\Omega)\P_{\mu_\by}(A)\big] + \P_\mu(\Omega^c).
\label{decc}
\ee
Choosing $a=C_0 K/N$ with a sufficiently large fixed constant $C_0$
Theorem~\ref{thm:accuracy} and $\delta \ll K/N$
guarantee that $\P_\mu(\Omega^c)$ is subexponentially small.
We will prove that
\be\label{tailprob1}
\P_{\mu_\by} ( x_{L+1} - y_L \le N^{-2} r ) \le C_Vr
\ee
for any $r\in (0,1)$. The constant depends on $V$, more precisely
\be
C_V= C + C\sup\big\{ |V'(x)|\; :\; x\in [y_L, y_{L+K+1}]\big\}.
\label{CVdef}
\ee
{F}rom \eqref{tailprob1} the necessary subexponential
estimate on the first term in \eqref{decc} follows by choosing $r=N^{-2}\exp(-N^{\e_0})$.
To prove \eqref{tailprob1},
on the set $\Omega$ we can shift the measure such that $y_L=-y_{L+K+1}$
and denote $a:=-y_L$.
Then we have
\begin{align}
\int\ldots\int_{-a+ a \varphi }^{a- a \varphi} & \rd \bx
\prod_{i,j\in I\atop i < j} (x_i-x_j)^\beta
e^{- N\frac{\beta}{2} \sum_j V_\by (x_j) } \nonumber \\
& = (1-\varphi)^{ K+\beta K(K-1)/2} \int\ldots\int_{-a }^{a} \rd \bw
\prod_{i < j} (w_i-w_j)^\beta e^{- N \frac{\beta}{2}\sum_j V_\by ((1-\varphi) w_j)},
\nonumber
\end{align}
where we set $w_j:=(1-\varphi)^{-1}x_{L+j}$, $\rd \bx = \rd x_{L+1} \ldots \rd x_{L+K}$
and $\rd\bw = \rd w_1 \ldots \rd w_K$.
By definition,
\begin{align}
e^{- N \frac{\beta}{2} V_\by ((1-\varphi) w_j) }
& = e^{- N \frac{\beta}{2} V ((1-\varphi) w_j)}
\prod_{i \le L} ( (1-\varphi) w_j - y_i)^\beta
\prod_{i \ge L+K+1} ( y_i-(1-\varphi) w_j)^\beta
\nonumber \\
& \ge e^{- N\frac{\beta}{2} V ( w_j)
- C_V\varphi N }
(1 - \varphi)^N \prod_{i \le L} ( w_j - y_i)^\beta \prod_{i \ge L+K+1} ( y_i- w_j)^\beta .
\nonumber
\end{align}
Note that we only used that $V$ is a $C^1$-function with bounded derivative
in performing a Taylor expansion and using that $w_j\le a$ is finite.
Hence
$$
\frac{1}{Z}\int\ldots \int_{-a+ a \varphi }^{a- a \varphi}
\rd \bx \prod_{i,j\in I\atop i < j} (x_i-x_j)^\beta e^{- N\frac{\beta}{2}\sum_j V_\by (x_j)}
\ge (1-\varphi)^{N K+CK^2} e^{- C_V NK \varphi }
$$
with
$$
Z := \int_{-a }^{a} \rd \bw \prod_{i,j\in I\atop i < j} (w_i-w_j)^\beta e^{- N
\frac{\beta}{2}\sum_j V_\by ( w_j)}.
$$
Therefore the $\mu_\by$-probability of $y_{L+1} -y_L = x_{L+1}-y_L\ge a(1-\varphi)$
can be estimated by
$$
\P_{\mu_\by} ( x_{L+1} \ge -a+ \varphi a )\ge (1-\varphi)^{N K+CK^2} e^{- C_V NK \varphi }
\ge 1- (C_V +C)NK\varphi
$$
by using $K\le N$. Choosing $\varphi=N^{-2}r/a$ and recalling
that $a\sim K/N$, we arrive at \eqref{tailprob1}.
\end{proof}
\begin{proposition}\label{prop:mumu} Let $\varphi>0$ be fixed.
For any smooth, compactly supported
function $G:\RR\to \RR$ we have
\be
\lim_{N\to\infty}\Bigg|\E_\mu \big [\E_{ \mu_{\by}} -\E_{ \mu_{\by}^\tau} \big ]
\frac 1 K \sum_{i \in I} G\Big( N(x_i-x_{i+1}) \Big)\Bigg| =0,
\ee
provided
\be
\frac{1}{2}\hat\tau\le \tau(\by)\le 2\hat \tau
\qquad \mbox{for any $\by\in \cG$}
\label{taubound}
\ee
holds for the function $\tau=\tau(\by)$
with some constant $\hat\tau=\hat \tau_N$ such that
\be\label{cond1}
\frac{N \delta^2}{ \hat\tau } \le N^{-\varphi}.
\ee
\end{proposition}
We remark that, with a slight abuse of notation, the last term, $i=L+K$ in
the sum involving the non-existing $x_{i+1}=x_{L+K+1}$ is defined to be zero.
We also point out that
the notation $\E_\mu \E_{ \mu_{\by}}$ means that
the law of $\by$ is given by $\mu$ in
the first expectation and we are using the measure $\mu_\by$ in the second
one. Of course, we have $\E_\mu=\E_\mu \E_{ \mu_{\by}}$.
\medskip
\begin{proof}
For any configuration $\by$, any $\tau$ (may depend on $\by)$ and
for any smooth function $G$ with compact support, we have
\be\label{relax}
\Bigg| \big [\E_{ \mu_{\by}} -\E_{ \mu_{\by}^\tau} \big ]
\frac 1 K \sum_{i \in I} G\Big( N(x_i-x_{i+1}) \Big)\Bigg|
\le C \Big( \frac {\tau N^{\varphi/2}}{ K}
D \big (\mu_{\by} | \mu_{\by}^\tau \big ) \, \Big)^{1/2}
+ Ce^{-cN^{\e/2}} \sqrt{S (\mu_{\by} | \mu_{\by}^\tau \big )},
\ee
Here we also introduced the notations
\be\label{def:Dirform}
D(\mu\mid \om) : = \frac{1}{2N}\int
\Big|\nabla \log \Big(\frac{\rd\mu}{\rd\om}\Big)\Big|^2 \rd \mu =
\frac{1}{2N} \int \Big|\nabla \sqrt{ \frac{\rd\mu}{\rd\om}}\Big|^2\rd \om
\ee
and
$$
S(\mu\mid \om): = \int \log \Big(\frac{\rd \mu}{\rd\om}\Big) \rd \mu
$$
for any probability measures $\mu, \om$. The estimate \eqref{relax} follows
from our the local relaxation to equilibrium argument
that in this form first appeared in Theorem 4.3 of \cite{ESYY}.
We will neglect the exponentially small entropy term since
it can be estimated by the Dirichlet form, i.e. by the first term
as long as $\tau\ge N^{-C}$.
We thus obtain
\be
\Bigg|\E_\mu \big [\E_{ \mu_{\by}} -\E_{ \mu_{\by}^\tau} \big ]
\frac 1 K \sum_{i \in I} G\Big( N(x_i-x_{i+1}) \Big)\Bigg|
\le
C \Big( \frac {N^{\varphi/2}}{ K} \E_\mu \big[ {\bf 1}_\cG\;
\tau(\by) D \big (\mu_{\by} | \mu_{\by}^\tau \big )\big] \, \Big)^{1/2}
+ Ce^{-cN^{\e'}}.
\label{EE}
\ee
To obtain the estimate \eqref{EE} we separated good and bad configurations;
we used \eqref{relax} for $\by\in \cG$. On the complement $\cG^c$
we just used the trivial estimate on $G$, and this yields
the subexponentially small second term.
Assuming \eqref{taubound},
we have
\be \frac {1}{ K} \E_\mu \big[ {\bf 1}_\cG\;
\tau(\by) D \big (\mu_{\by} | \mu_{\by}^\tau \big )\big] \,
\le \frac{N}{K}\E^\mu\Big[ {\bf 1}_\cG\;
\frac{1}{ \tau(\by) } \sum_{j \in I } ( x_j - \gamma_j)^2\Big]
\le \frac{N\delta^2}{\hat \tau}
\le N^{-\varphi}
\ee
by \eqref{cond1}.
Inserting this estimate into \eqref{EE}
we completed the proof of the proposition.
\end{proof}
\subsection{Matching the boundary conditions}\label{sec:match}
Suppose we have measures $\g $ and $ \mu$ with potentials $W$ and $V$
given by \eqref{eqn:measure} with
densities $\rho=\rho_V$ and $\rho_W$, respectively.
For our purpose $W(x)=x^2$, i.e, $\g$ is the Gaussian $\beta$-ensemble
and $\rho_W(t) =\frac{1}{2\pi}\sqrt{[4-t^2]_+}$ is the Wigner semicircle law.
Let the sequence $\gamma_j$ be the classical location for $\mu$ and
the sequence $\theta_j$ be the classical locations for $\g$.
We will match the boundary conditions for the local
measure on $J_\by:=[y_{L}, y_{L+K+1}]$ around $E_q$
with those of the $\g$ measure.
For definiteness we choose the interval $J'=[\theta_{L'}, \theta_{L'+K+1}]$
with $L' = \frac{1}{2}(N - K -1)$ as our reference interval. Note
that $J'$ is symmetric to the origin.
The local density $\rho_V(E_q)$ at the point $E_q$ we
look at may be different from the density $\rho_W(0)$ at
the origin. Thus the typical length of $J_\by$, which is
$\gamma_{L+K+1} - \gamma_L \sim [\rho_V(E_q)]^{-1} N^{-1}$, may
not be close to the length of $J'$ which is very close to
$[\rho_W(0)]^{-1} N^{-1} = \pi N^{-1}$, so we will have
to rescale the $\g$ measure by a factor
$$
s_q \approx \frac{\rho_V(E_q)}{\rho_W(0)}.
$$
In fact, we need to match not only the interval of classical locations
$\gamma$ with $J'$, but the exact interval $I_\by$. This requires
a $\by$-dependendent scaling factor $s=s(\by)$.
{F}rom now on we assume that $\by$ is a good boundary condition
with a parameter $\delta$ that satisfies
\be\label{dNK}
\frac {\delta N } K \to 0.
\ee
We can shift the coordinates
so that
\be\label{t}
- y_L = y_{L+K+1}.
\ee
Since our observable is translationally invariant, we will not track
the translation and we assume that \eqref{t} holds.
We define
\be
s(\by):= \frac{\th_{L'}}{y_L}= \frac{\th_{L'+K+1}}{y_{L+K+1}}, \qquad
s_q: = \frac{\th_{L'}}{\gamma_L}.
\label{sdef}
\ee
We have
\be\label{s-1}
|s(\by)- s_q | =\Big| \frac { \th_{L'}} { y_L}-\frac { \th_{L'}} { \gamma_L}\Big|
\le C \frac { \delta N } K \to 0
\ee
since
\be
\theta_{L'}\approx -[\varrho_W(0)]^{-1} \frac{K}{2N}, \qquad
\gamma_L \approx -[\varrho_V(E_q)]^{-1} \frac{K}{2N}, \qquad
y_L \approx -[\varrho_V(E_q)]^{-1} \frac{K}{2N}, \qquad
\label{appr}
\ee
by using $\by\in \cG$ and \eqref{dNK}. Similar formulas hold
for $\th_{L'+K+1}$, $\gamma_{L+K+1}$ and $y_{L+K+1}$ at
the upper edge of the interval. Here the $A\approx B$
is understood in the sense that the approximation error
at most of order $(K/N)^2$, recalling that $K=o(N)$.
For simplicity of the presentation, we can first shift the original $\mu$-ensemble
such that $E_q=0$. Second, we can perform
an initial rescaling of the Gaussian $\beta$-ensemble so that $s_q=1$.
\begin{lemma} Assuming $E_q=0$, $s_q=1$, we have
\be\label{38}
|\gamma_{L+j}-\th_{L'+j}|\le C\frac{j^2}{N^2} + C\delta, \qquad |j|\le \frac{1}{100} N\kappa.
\ee
\end{lemma}
\begin{proof}
The classical locations $\gamma_{L+j}$ and $\th_{L'+j}$ are given by the equation
\be\label{jN}
\int_{\gamma_L}^{\gamma_{L+j}} \rho_V(x) = \frac { j} N,
\qquad \int_{\th_{L'}}^{\th_{L'+j}} \rho_W(x) = \frac { j} N.
\ee
We will use the approximations
\be\label{ap}
\rho_V(x) = \rho_V(0) + O( x), \qquad \rho_W(x) = \rho_W(0) + O( x)
\ee
for small $x$ (to stay away from the spectral edge).
Since $ |y_j - \gamma_j |\le \delta$, we have
\be
\frac { K+1} N = \int_{\gamma_L}^{\gamma_{L+K+1}} \rho_V(x) \rd x
= \int_{y_L}^{y_{L+K+1}} \rho_V(x) \rd x + O(\delta)
= \rho_V(0) (y_{L+K+1}-y_L) + O \left ( \frac {K^2} { N^2} \right ) + O(\delta).
\ee
Similarly,
\be
\frac { K+1} N = \int_{\th_{ L'}}^{\th_{ L'+K+1}} \rho_W (x) \rd x
= \rho_W(0) (\th_{ L'+K+1} - \th_{L'} ) + O \left ( \frac {K^2} { N^2} \right ).
\ee
Since $- y_L = y_{L+K+1}=- \th_{ L'}/s = \th_{ L'+ K + 1}/s$
which is comparable with $K/N$ by \eqref{appr},
and since $|s- 1| \le C \frac { \delta N } K$ from \eqref{s-1}, we have
\be\label{VW0}
| \rho_W(0) - \rho_V(0) | \le \frac { C K } N + \frac {C\delta N } K.
\ee
{F}rom \eqref{jN}, \eqref{ap} and \eqref{appr} we get
$$
\varrho_V(0)(\gamma_{L+j}-\gamma_L) + O\Big(\frac{j^2}{N^2}\Big)
= \frac{j}{N} = \varrho_W(0)(\th_{L'+j}-\th_{L'}) + O\Big(\frac{j^2}{N^2}\Big),
$$
which combining with \eqref{VW0} and $\rho_W(0)\ge c$ gives
$$
\gamma_{L+j}-\gamma_L
= \th_{L'+j}-\th_{L'} + O\Big(\frac{j^2}{N^2}\Big) + O(\delta).
$$
Since $\gamma_L=\th_{L'}$, this completes the proof of the lemma.
\end{proof}
\newcommand{\bt}{\mbox{\boldmath $\theta$}}
\newcommand{\htau}{{\hat\tau}}
\subsection{Rescaling of the reference problem}
Throughout this section we fix a good boundary configuration.
$\by\in \cG$ and a number $\tau(\by)$ depending on
this configuration and satisfying \eqref{taubound}.
We will approximate the local relaxation measure $\mu_\by^\tau$
on $[y_L, y_{L+K+1}]$ by a fixed reference measure.
Given the collection of classical locations $\th_j$ corresponding
to the Gaussian potential $W(x)=x^2$ we define a {\it reference
local relaxation measure} $\g_\th^\htau$ via the Hamiltonian
\be\label{241}
\cH_{\th}^\htau (\bx) =
\sum_{i \in I'} \Big [ \frac{\beta}{2} x_i^2 - \frac{ \beta }{N} \sum_{j \not \in I'}
\log |x_i - \th_{j} | \Big ]
- \frac{ \beta }{N} \sum_{ i,j \in I'\atop i<j }
\log |x_{j} - x_{i}| + \frac{1}{2 \htau } \sum_{ i \in I'} ( x_i - \theta_i )^2,
\ee
on the set $[\th_{L'}, \th_{L'+K+1}]$
where $I':= \llbracket L'+1, L'+K\rrbracket$.
Note that if $\g$ is the equilibrium measure given by \eqref{eqn:measure} corresponding to $W$
and $\g^\htau$ denotes the corresponding relaxation measure
\be\label{232sigma}
\rd\g^\htau :=\frac{Z}{ Z_{\g^\htau}}e^{-N Q^\htau }\rd\g,
\quad Q^\htau (\bx) = \sum_{j\in I'}
Q_j^\htau(x_j) , \qquad Q_j^\htau (x) = \frac{1}{2 \htau } ( x - \th_j)^2,
\ee
defined analogously to \eqref{232}, then $\g_\th^\htau$ is the conditional
measure of $\g^\htau$ under the condition that the outside points
are exactly at their classical locations, i.e. $\lambda_j=\th_j$, $j\not\in I'$.
We make three simplifications in the presentation. First, as already in
Section~\ref{sec:match}, we assume that both the configuration
space $[y_L, y_{L+K+1}]$ for the original measure $\mu_\by^\htau$
and the configuration space $[\theta_{L'}, \theta_{L'+K+1}]$ of the reference
measure $\g_\th^\htau$ are symmetric around the origin; this can be achieved
by an irrelevant shift. Second, we assumed $s_q=1$, which can be achieved
by an irrelevant rescaling of $W$.
Finally, we will set $L'=L$.
This last assumption expresses an irrelevant shift in the labelling
of one of the ensembles. Strictly speaking, shifting would mean that
the original set of particles indices $\llbracket 1, N\rrbracket$ gets shifted as well.
However, in our argument this shift does not play any active role;
the only information we use about the set of indices is that $L$ is
macroscopically separated from its boundary and that its cardinality is $N$.
We now rescale the measure $\g_\theta^\htau$ from $[\theta_{L}, \theta_{L+K+1}]
=[\theta_{L}, -\theta_{L}] $ to $[y_L, y_{L+K+1}]= [y_{L}, -y_{L}]$
by the factor $s=s(\by)$ defined in \eqref{sdef} (note that $y_L, \th_L<0$).
With the rescaled boundary conditions
$\theta_j\to\th_j':= \theta_j/s$,
we define the {\it reference local relaxation measure}, or {\it reference measure}
in short, to be
\be\label{def:ref}
\g_{\th}^{\htau, s}: =\frac { 1 } { Z^{\htau, \th, s } }
e^{-N\cH_{\th}^{\htau, s} (\bx)}\rd \bx,
\ee
a measure on the set
$[y_L, y_{L+K+1}]$
with Hamiltonian
\be\label{242}
\cH_{\th}^{\htau, s} (\bx) =
\sum_{i \in I} \Big [ \frac { \beta s^2 x_i^2 } 2 - \frac{ \beta}{N} \sum_{j \not \in I}
\log | x_i - \th_{j}/s | \Big ]
- \frac{ \beta }{N} \sum_{ i,j \in I\atop i<j }
\log | x_{j} - x_{i}| + \frac{s^2}{2 \htau } \sum_{ i \in I} ( x_i - \theta_i/s )^2.
\ee
The rescaled potential associated with this Hamiltonian is $ W_s(x)= s^2 x^2$.
\newcommand{\non}{\nonumber}
For any smooth function $G$ with compact support, we have
\be\label{diff}
\E_{ \g_{\th}^{\htau, s} } \frac 1 K \sum_{i \in I} G\Big( N(x_i-x_{i+1}) \Big)
= \frac { 1 } { Z^{\htau, \th, s } }
\int_{\th_L/s}^{- \th_L/s} \rd \bx \; e^{- N \cH_{\th}^{\htau,s}
(\bx)} \; \frac 1 K \sum_{i \in I} G\Big( N(x_i-x_{i+1}) \Big),
\ee
where $\int_a^b \rd \bx$ stands for the $K$-dimensional integral
$\int_{[a,b]^K}\rd x_{L+1}\ldots \rd x_{L+K}$ and $ Z^{\htau, \th, s }$ is the normalization
factor.
Let $x_j = w_j/s$, then the right side becomes
\begin{align}
\frac { 1 } { Z^{\htau, \th} } \int_{\th_L }^{- \th_L}
\rd\bw \; e^{- N \cH_{\th}^{\htau} (\bw)}
\frac 1 K \sum_{i \in I} G\Big( \frac { N(w_i-w_{i+1})} s \Big)
& = \E_{ \sigma_{\th}^\htau } \frac 1 K \sum_{i \in I} G\Big( \frac{ N (x_i-x_{i+1})}{s}
\Big )\label{44} \\
\non
& = \E_{ \sigma_{\th}^\htau} \frac 1 K \sum_{i \in I} G\Big( N (x_i-x_{i+1}) \Big) + o(1),
\end{align}
where we renamed the $w$-variables to $x$-variables in the first step and
in the second step we have used that
\[
\Big| G\Big( N (x_i-x_{i+1})/s \Big) - G\Big( N (x_i-x_{i+1}) \Big) \Big|\le
C|1-s| \| G'\|_\infty
\]
by Taylor expansion and from the fact that $G$ is compactly supported.
Clearly, the difference vanishes as long as $s \to 1$.
Thus we are free to scale the measure with factor converging to 1.
The condition $s\to 1$ will be guaranteed by \eqref{s-1}.
Our main result is the following theorem.
\begin{theorem}\label{thm:mi} Let $0<\varphi\le \frac{1}{38}$.
Fix $K=N^k$, $\delta = N^{-d}$, $\hat \tau = N^{-t}$
with $d=1-\varphi$, $t=2d-1-\varphi=1-3\varphi$ and $k=\frac{39}{2}\varphi$,
in particular such that \eqref{cond1},
\eqref{dNK} are satisfied.
Then
\be\label{381}
\Bigg| \E_\mu \E_{ \mu_{\by}^{\htau} } \frac 1 K \sum_{i \in I} G\Big( N(x_i-x_{i+1}) \Big)
-\E_{ \g_{\th}^{\htau}}
\frac 1 K \sum_{i \in I} G\Big( N(x_i-x_{i+1}) \Big)\Bigg|
\to 0
\ee
as $N \to \infty$
for any smooth and compactly supported test function $G$.
Here the law of $\by$ is given by $\mu$ in the expectation.
\end{theorem}
\begin{proof}
{F}rom the rescaling estimates, \eqref{diff}-\eqref{44},
it suffices to prove that
\be\label{372}
\E_\mu \big [\E_{ \mu_{\by}^{\htau} } -\E_{ \g_{\th}^{\htau, s(\by)}} \big ]
\frac 1 K \sum_{i \in I} G\Big( N(x_i-x_{i+1}) \Big)
\to 0
\ee
as $N \to \infty$. Notice that after the rescaling
both measures $\mu_{\by}^{\htau}$ and $ \g_{\th}^{\htau, s(\by)}$
live on the same interval $[y_L, y_{L+K+1}]$.
In Proposition~\ref{prop:mumu} we already showed that
\be\label{371}
\E_\mu \big [\E_{ \mu_{\by}^{\htau} } -\E_{ \mu_{\by}^{\htau/s(\by)^2} }
\big ]
\frac 1 K \sum_{i \in I} G\Big( N(x_i-x_{i+1}) \Big)
\to 0,
\ee
since \eqref{s-1} with $s_q=1$ and
$\delta N/K\to 0$
guarantee that $\tau(\by):= \htau/s(\by)^2$ satisfies \eqref{taubound}.
Thus the limit \eqref{372} will follow
from the following Proposition that we will prove
in Sections~\ref{sec:conv}:
\begin{proposition}\label{prop:musig}
Under the assumptions of Theorem~\ref{thm:mi}, we have
\be\label{37}
\E_\mu \big [\E_{ \mu_{\by}^{\htau/s(\by)^2} }
-\E_{ \g_{\th}^{\htau, s(\by)}} \big ]
\frac 1 K \sum_{i \in I} G\Big( N(x_i-x_{i+1}) \Big)
\to 0.
\ee
\end{proposition}
This completes the proof of Theorem~\ref{thm:mi}.
\end{proof}
\bigskip
{\bf Proof of Theorem~\ref{thm:Main}.}
Finally, combining Theorem~\ref{thm:mi} with
Proposition~\ref{prop:mumu} and noticing that \eqref{cond1}
is satisfied since $t=2d-1-\varphi$, we have
\be\label{3811}
\Bigg| \E_\mu \frac 1 K \sum_{i \in I} G\Big( N(x_i-x_{i+1}) \Big)
-\E_{ \g_{\th}^{\htau}}
\frac 1 K \sum_{i \in I} G\Big( N(x_i-x_{i+1}) \Big)\Bigg|
\to 0
\ee
as $N \to \infty$. This holds for $K=N^k$ with any $0<k\le\frac{1}{2}$
by selecting a suitable $\varphi$ in Theorem~\ref{thm:mi}.
However, the measure $\sigma_\theta^{\htau}$ is independent of $V$,
the only information we used was that the local density matches.
So we obtain that any two measures $\mu_{\beta, V}$ and $\mu_{\beta, W}$
have the same local gap statistics assuming that the local
densities of the two ensembles coincide.
\qed
\section{Comparison with the reference problem}\label{sec:conv}
In this section we prove Proposition~\ref{prop:musig}.
On the set $\by\in \cG^c$ with subexponentially small
probabality \eqref{goodsetprob} a trivial estimate on $G$
suffices. For the sequel we therefore assume that $\by\in \cG$ and
we set $\tau(\by):= \htau/s(\by)^2$ which clearly satisfies \eqref{taubound}.
In the first step we will soften the boundary condition $\by$
for the measure local relaxation measure $\mu_\by^\tau$.
\subsection{Regularizing the boundary conditions}
We know that the boundary condition
$\by\in \cG$ is regularly spaced on the scale
$\delta= N^{-d}$, but it does not exclude that $N\delta = N^{1-d}\gg 1$
points of the colletion $\by$ pile up near the edges
of the interval
$[y_L, y_{L+K+1}]$. This would substantially influence
the local relaxation measure $\mu_\by^\tau$ near the corresponding edge
inside $[y_L, y_{L+K+1}]$. We therefore first replace the
boundary conditions near the edges by the regularly spaced
ones given by $\theta'=\theta/s(\by)$. This change will be controlled only
in the entropy sense. The local relaxation measure with regularized
boundary conditions will then be compared with
the reference measure in the stronger Dirichlet form sense.
Set a parameter
\be\label{def:B}
B = N^b \quad \mbox{with} \quad 1+\varphi -d \le b < k,
\ee
in particular $\delta N\ll B\ll K$.
Given a boundary condition $\by\in\cG$,
we define a new boundary condition $\by^B = \{ y_i^B\; : \; i\not\in I\}$ as
\be
y^B_i: = \left\{
\begin{array}{lll}
\max \{ \th_i', y_{L-4B}\} & \mbox{for} & L - 4B \le i \le L \cr
y_i & \mbox{for} & i < L -4B, \quad \mbox{or}\quad i> L+K+4B\cr
\min\{ \th_i', y_{L+K+4B}\} & \mbox{for} & L +K+1 \le i \le L+K+4B,
\end{array}\right.
\ee
i.e., we replace at most $4B$ boundary
conditions $y_i$ with the rescaled classical ones $\th_i'=\th_i/s(\by)$
near the edges of the interval $[y_L,y_{L+K+1}]=[\theta_L', \theta_{L+K+1}']$.
Note that the configuration space is unchanged.
We have
$$
y_{L-4B}\le\gamma_{L-4B}+ \delta \le \theta'_{L-4B} + CB^2N^{-2}+C\delta \le \theta'_{L-2B},
$$
where we used that $\by\in \cG$ in the first step and \eqref{38} in the second.
In the last inequality we used that $\theta'_{L-2B} - \theta'_{L-4B}\ge cBN^{-1}$
(by regular spacing) and the definition of $B$ from \eqref{def:B}.
Thus we obtain
\be\label{ybth}
y^B_i = \th'_{ i}, \quad L - 2B \le i \le L,
\ee
and similarly at the upper edge.
In other words, we do replace at least $2B$ boundary condition points near the edges with
the classical ones. Although it may happen that a few $y_i^B$ pile up,
but this occurs away from the edges.
The key property of the family $y_i^B$ is the following bound
\be
\#\{ i\; : \; y_i^B\in J \} \le CN|J|
\label{ybregular}
\ee
for any interval $J$ such that $|J|\ge cN^{-1}$
and $c|J|\le \mbox{dist} (J, [y_L, y_{L+K+1}])\le |J|/c$
with some small constant $c$.
\medskip
Consider the {\it regularized local relaxation measure}, which is defined as
the probability measure
\be\label{def:regloc}
\mu^{B,\tau}_\by(\rd \bx) = Z^{-1}e^{- N \cH_\by^{B,\tau}}\rd\bx
\ee
of $K$ ordered points $\bx=(x_{L+1}, \ldots , x_{L+K})$
in $[y_L, y_{L+K+1}]$, with Hamiltonian
\be
\cH_{\by}^{B,\tau} (\bx) :=
\sum_{i\in I} \frac{\beta}{2}V^i_{\by^B} (x_i)
- \frac{ \beta }{N} \sum_{i,j\in I\atop i< j}
\log |x_{j} - x_{i}| + \sum_{i \in I} Q_i^\tau (x_i ),
\ee
with a quadratic confinement $Q_i^\tau(x) = (2\tau(\by))^{-1} (x-\theta'_i)^2$
as in \eqref{242} and $\tau(\by)= \hat \tau/s(\by)^2$.
The potential $V^i$ is given
by
$$
V^i_{\by^B} (x) = V(x) - \frac{ 2 }{ N} \sum_{ j \le L}
\log |x - y_{j}^B| - \frac{ 2 }{ N} \sum_{ j \ge L+K+1}
\log |x - y_{j}|
\qquad \mbox{for}\quad L +1\le i \le L + 4B
$$
$$
V^i_{\by^B} (x) = V(x) - \frac{ 2 }{ N} \sum_{ j \le L}
\log |x - y_{j}| - \frac{ 2 }{ N} \sum_{ j \ge L+K+1}
\log |x - y_{j}|
\qquad \mbox{for}\quad L +4B+1\le i \le L +K -4B
$$
and
$$
V^i_{\by^B} (x) = V(x) - \frac{ 2 }{ N} \sum_{ j \le L}
\log |x - y_{j}| - \frac{ 2 }{ N} \sum_{ j \ge L+K+1}
\log |x - y_{j}^B|
\qquad \mbox{for}\quad L+K-4B+1 < i \le L +K .
$$
In other words, we replace the boundary condition $\by $ with
$\by^B$ for the points $x_i$ with $L+1\le i \le L + 4B$
at the lower edge and similarly for the other edge.
The boundary conditions for
the middle points $x_i$ with $L+4B+1\le i\le L+K-4B$
remain unchanged. Recalling \eqref{24}, we have in particular
\be
\cH_{\by}^{B,\tau} (\bx)-\cH_{\by}^{\tau} (\bx)
= \frac{2}{N} \sum_{L - 4B \le j < L}
\sum_{L < i \le L+4B }
\left [ - \log |x_i- y^B_{j} | + \log |x_i- y_{j} |\right ]
+\big( \mbox{Upper edge}\big),
\ee
where {\it (Upper edge)} refers to an analogous term
collecting interactions
near the upper edge.
\medskip
\begin{lemma}\label{lm:SS} Let $\by\in \cG$.
The relative entropies of the measures
$\mu_\by^\tau$ and $\mu^{ B,\tau}_\by$ satisfy
\be \label{77}
S( \mu_\by^\tau | \mu^{ B,\tau}_\by )+S( \mu_\by^{B,\tau} | \mu^{\tau}_\by )
\le C B^2 \log N.
\ee
\end{lemma}
\begin{proof} We start with the following lemma that estimates the relative entropy of
any two measures:
\begin{lemma}\label{lm:HH}
Suppose $\mu_i(\rd x)=Z_i^{-1}e^{-H_i}\rd x$, $i= 1, 2$ are probability measures
with Hamiltonians $H_i$ on a common measure space.
Then
\be\label{852}
S(\mu_1 | \mu_2) \le \E_{\mu_1} [ H_2 - H_1 ] + \E_{\mu_2} [ H_1 - H_2 ].
\ee
We also have the inequality
\be
\E_{\mu_2} [ H_2 - H_1 ] \le \log Z_1- \log Z_2
\le \E_{\mu_1} [ H_2 - H_1 ].
\ee
\end{lemma}
\begin{proof}
By Jensen inequality, we have
\begin{align*}
0\le S(\mu_1 | \mu_2) & = \int \rd \mu_1 \log \left ( \frac {\rd \mu_1} {\rd \mu_2} \right )
= \int \rd \mu_1 [ H_2 - H_1 ] + \log \left ( \frac {Z_2} {Z_1} \right ) \\
&
\le \E_{\mu_1} [ H_2 - H_1 ] - \log \left [ \int e^{-H_1} \frac { \rd x }
{\int e^{-H_2} \rd x } \right ] \\
& \le \E_{\mu_1} [ H_2 - H_1 ] + \E_{\mu_2} [ H_1 - H_2 ].
\end{align*}
This completes the proof of Lemma~\ref{lm:HH}.
\end{proof}
\medskip
Hence we have $S( \mu_\by^\tau | \mu^{B,\tau}_\by ) \le \beta\Omega_1 $, where
\begin{align}\label{e4}
\Omega_1 & : =
\Big ( \E_{\mu^\tau_\by} - \E_{ \mu^{ B,\tau}_\by} \Big ) \sum_{L - 4B \le j < L}
\sum_{L < i \le L+4B }
\left [ - \log ( x_i- y^B_{j} ) + \log ( x_i- y_{j} )\right ]
+\big(\mbox{Upper edge}\big). \nonumber \\
\end{align}
Using that $x_i-y_j\ll 1$,
we clearly have
\begin{align}
\Omega_1 & \le - \sum_{L - 4B \le j < L}\sum_{L < i \le L+4B }
\left [ \E_{\mu^\tau_\by} \log ( x_i- y^B_{j} ) + \E^{ \mu^{B,\tau}_\by}
\log ( x_i- y_{j} ) \right ]+\big(\mbox{Upper edge}\big)\nonumber \\
& \le C B^2 \log N - B^2
\E_{ \mu^{B,\tau}_\by}
\log ( x_{L+1}- y_L )+\big(\mbox{Upper edge}\big) . \label{Om1}
\end{align}
In the first term we used the trivial estimate
$x_i-y_j^B\ge \th_L' - \th_{L-1}' \ge cN^{-1}$
for any $j<L$.
The second term will be estimated by Lemma~\ref{lm:rep} below
and this completes the estimate for
$S( \mu_\by^\tau | \mu^{ B,\tau}_\by )$.
The other relative entropy, $S( \mu_\by^{B,\tau} | \mu^{\tau}_\by )$
can be treated similarly and this
proves Lemma~\ref{lm:SS}.
\end{proof}
\begin{lemma}\label{lm:rep}
Suppose $\tau \ge N^{-1}$, then for any $p\ge 1$ we have
\be\label{trivgap}
\E_{\mu_\by^\tau} |\log( x_{L+1} - y_L)|^p\le C_p\log N
\ee
and the same estimate holds w.r.t the measure $\mu_\by^{ B,\tau}$.
\end{lemma}
\begin{proof}
We will need that
\be\label{tailprob}
\P_{\mu_\by^\tau} ( x_{L+1} - y_L \le N^{-3} r ) \le Cr
\ee
for any $r\in (0,1)$. Then \eqref{trivgap} follows from integrating in $r$ from 0 to $1$
and treating the regime $ x_{L+1}-y_L\ge N^{-3}$ trivially
by using $ x_{L+1}-y_L\le y_{L+K+1}-y_L\le CK/N\le 1$.
The estimate \eqref{tailprob} can be proven essentially
in the same way as \eqref{tailprob1}, just the potential $\frac{\beta}{2}V(x_j)$
of the $j$-th point
in that proof is replaced with $\frac{\beta}{2}V(x_j)+ Q^\tau_j(x_j)$.
The final estimate is somewhat weaker since now the bound on the constant $C_V$
defined in \eqref{CVdef} deteriorates to
$C_V\le C\tau^{-1}\le CN$. This accounts for the change from $N^{-2}$ to $N^{-3}$
in \eqref{tailprob}.
The argument for the measure $\mu_\by^{ B,\tau}$ is analogous and this
proves Lemma~\ref{lm:rep}.
\end{proof}
\subsection{Regularization does not change spacing statistics}
Given that the local relaxation measure $\mu_\by^\tau$ and its regularized
version $\mu^{ B,\tau}_\by$
are close in relative entropy sense, the next proposition shows that their
local spacing statistics coincide.
\begin{proposition}\label{prop:spacingB}
Let $\by\in \cG$, $\tau=\tau(\by)=\htau/s(\by)^2$ and assume
that for the parameters $B=N^{b}$, $K=N^k$ and $\htau = N^{-t}$
it holds
that
\be\label{cond2}
1+ 2b -t-k< 0.
\ee
Then
\be\label{38B}
\Bigg| \big[ \E_{ \mu_{\by}^{\tau} } -\E_{ \mu_{\by}^{B, \tau}} \big]
\frac 1 K \sum_{i \in I} G\Big( N(x_i-x_{i+1}) \Big)
\Bigg|
\to 0
\ee
as $N \to \infty$
for any smooth and compactly supported test function $G$.
\end{proposition}
\begin{proof}
Since Lemma~\ref{lm:SS} and \eqref{cond2} guarantee that
\be\label{22}
\frac {N S( \mu_{\by}^{B, \tau} \mid \mu_{\by}^{\tau})
\tau } K \le \frac {C N B^2 \tau } K \log N \le N^{-\e'}
\ee
with some $\e'>0$,
Proposition~\ref{prop:spacingB} is a direct consequence of
the following comparison
lemma which was first stated in a remark after
Lemma 3.4 in \cite{ESY4}, see also Lemma 4.4 in \cite{ESYY}.
\end{proof}
\begin{lemma}\label{lm:Scomparison}
Let $G:\bR\to\bR$
be a bounded smooth function with compact support and let
a sequence $E_i$ be fixed. Let $I$ be an interval of indices with
$|I|=K$. Consider a measure $\om$ with relaxation time
$\tau$ and let $q\rd \om$ be another probability measure.
Then for any $\e_1>0$ and for any smooth compactly supported function we have
\be\label{Diff}
\Big|\frac 1 K \sum_{i \in I} \int G\big( N(x_i-E_i )\big) [q-1]\rd \omega\Big|
\le C\sqrt { \frac{ N^{1+\e_1} S_\om( q) \tau}{K}}+ Ce^{-cN^{\e_1}}\sqrt {S_\om(q)}
\ee
and
\be\label{Diff1}
\Big|\frac 1 K \sum_{i \in I} \int G\big( N(x_i-x_{i+1} )\big) [q-1]\rd \omega
\Big|
\le C\sqrt { \frac{ N^{1+\e_1} S_\om( q) \tau}{K}}+ Ce^{-cN^{\e_1}}\sqrt {S_\om(q)} ,
\ee
where $S_\om(q):=S(q\om\mid \om)$.
\end{lemma}
\begin{proof} Let $q$ evolve by the dynamics $\pt_t q_t = \cL q_t$, where
$\cL$ is the generator defined by
\be\label{def:dir}
\int -f \cL f \rd \om = D_\om(f)=\frac{1}{2N}\int |\nabla f|^2 \rd \om.
\ee
Let $\tau_1 = N^{\e_1} \tau$.
Since $q_{\tau_1}$ is already subexponentially close to $\om$ in entropy sense,
$S_\om(q_{\tau_1})\le C\exp(-cN^{\e_1})S_\om(q)$,
and the total variation norm can be estimated by the relative entropy,
we only have to compare $q$ with $q_{\tau_1}$.
By differentiation, we have (the summation over $i$ always runs $i\in I$)
\begin{align}
\int \frac 1 K \sum_{i} & G\Big( N(x_i-E_i ) \Big) q_{\tau_1} \rd \omega -
\int \frac 1 K \sum_{i} G\Big( N(x_i-E_i) \Big) q \rd \omega \\
&= \int_0^{\tau_1} \rd s \int \frac 1 K \sum_{i}
\pt_i G\Big( N(x_i-E_i))\Big)
\pt_{i} q_s \rd \omega.
\end{align}
Here we used the definition of $ \cL$ from \eqref{def:dir} and note that
the $1/N$ factor present in \eqref{def:dir} cancels the factor $N$
from the argument of $G$.
{F}rom the Schwarz inequality and $\pt q = 2 \sqrt{q}\pt\sqrt{q}$,
the last term is bounded by
\begin{align}\label{4.1}
\Big[ \frac {N} { K^2} \int_0^{\tau_1} \rd s \int &
\sum_{i} \Big[\pt_i G \big(N(x_i -E_i ) \big)\Big] ^2
\, q_s \rd \omega
\Big]^{1/2} \left [ \int_0^{\tau_1} \rd s \int \frac 1 {N } \sum_{i}
(\pt_{i}\sqrt {q_s})^2 \rd \omega \right ]^{1/2} \nonumber \\
\le & \; C \sqrt { \frac{ N S_\omega(q) \tau_1}{K}}
\end{align}
by integrating $\pt_s S_\om(q_s) =-4 D_\om(\sqrt{q_s})$.
This proves \eqref{Diff} and the proof of \eqref{Diff1} is analogous.
\end{proof}
\subsection{Accuracy of block averages}
In the next Section~\ref{sec:dir} we will compare the regularized local relaxation measure
$\mu_\by^{B,\tau}$ with the reference measure $\g_\th^{\htau,s(\by)}$
in Dirichlet form sense. As a preparation for this step, we give an estimate
on the location of the block averages $x_j^{[B]}$. Recall their definition
$$
x_j^{[B]}:= \frac{1}{2B+1} \sum_{|k-j|\le B} x_k
$$
for any $j\in \llbracket L+B+1, L+K-B\rrbracket$.
The following lemma shows concentration on a scale $\zeta$ for $x_j^{[B]}$
w.r.t. $\mu_\by^\tau$ and $\mu_\by^{B, \tau}$. The scale $\zeta$ is
larger than $\delta$ but will be smaller than $K/N$, the length of
configuration space interval. Thus that the accuracy of the position
of $x_j$ decreases from $\delta$ to $\zeta$, but
the accuracy of $y_k$ is still $\delta$.
\begin{lemma} \label{lm:accuracy}
Set $\zeta = N^{-z}$, $t=2d-1-\varphi$ and fix $\by\in \cG$.
For any $j\in \llbracket L+B+1, L+K-B\rrbracket$ we have
\be
\label{cont1}
\P_{\mu_\by^\tau}\Big( \big| x_j^{[B]}
- \E_{\mu_\by^\tau} x_j^{[B]}\big|\ge \zeta\Big)\le c_1 e^{-c_2 N^{\e'}}
\ee
and
\be
\label{cont2}
\P_{\mu_\by^{B,\tau}}\Big( \big| x_j^{[B]}
- \E_{\mu_\by^{B,\tau}} x_j^{[B]}\big|\ge \zeta\Big)\le c_1 e^{-c_2 N^{\e'}}
\ee
provided
\be\label{66}
z \le -\varphi+ \min \Big( d- \frac{b}{2}-\frac{\varphi}{2},
\; d-\frac{k}{2}+ \frac{b}{2}\Big)
\ee
for some $\e'=\e'(d,\varphi)>0$ depending only on $d$ and $\varphi$.
Furthermore, we have
\be\label{E-E}
\Big | \E_{\mu_\by^{B,\tau}} x_j^{[B]} - \gamma_j^{[B]}
\Big | \le 5\zeta, \qquad
\Big | \E_{\mu_\by^\tau} x_j^{[B]} - \gamma_j^{[B]}\Big|\le 5\zeta, \qquad
\Big | \E_{\mu_\by} x_j^{[B]} - \gamma_j^{[B]}\Big|\le 5\zeta.
\ee
\end{lemma}
\begin{proof}
We will need two standard inequalities from probability theory.
The first one is
\be
\P_\mu (A) \cdot \log \frac{1}{\P_\nu(A)} \le \log 2 + S(\mu| \nu)
\label{PPS}
\ee
for any set $A$ and probability measures $\mu, \nu$.
This can be obtained from the entropy inequality
$$
\int f \rd \mu \le S(\mu| \nu) + \log \Big[ \int e^f \rd \nu\Big]
$$
by choosing $f(x) = b \cdot {\bf 1}_A(x)$ with $b= - \log \P^\nu(A)$.
Using Lemma~\ref{lm:SS} we thus obtain
\be\label{Acomp}
\P_{\mu_\by^{B,\tau}} (A)\le \frac {\log 2
+ CB^2\log N} { -\log \P_{\mu_\by^\tau} (A)}.
\ee
The second inequality is a concentration
estimate. Suppose that the
probability measure $\om$
satisfies the logarithmic Sobolev inequality (LSI), i.e.
\be
S_\om(f) \le C_{\text{s}} \int |\nabla \sqrt{f}|^2 \rd \om
\label{lsi}
\ee
holds for any $f\ge0$ with $\int f\rd \om=1$. Then for any
random variable $X$ with $\E_\om X=0$ and any number $T>0$ we have
\be
\E_\om e^{TX}\le \E_\om
\exp \left( \frac{C_{\text{s}} T^2}{2} \, |\nabla X|^2 \right).
\label{conc}
\ee
Since the Hamiltonian $\cH_\by^\tau$ is convex with
$\nabla^2 \cH_\by^\tau\ge \tau^{-1}$,
by the Bakry-Em\'ery criterion the measure $\mu_\by^\tau\sim
\exp(-N \cH_\by^\tau)$
satisfies \eqref{lsi} with Sobolev constant $C_s= 2\tau/N$,
i.e.
\be\label{SleqD}
S(\nu\mid \mu_\by^\tau)\le 4\tau D(\nu\mid \mu_\by^\tau)
\ee
for any probability measure $\nu$ (recall that the
definition of the Dirichlet form \eqref{def:Dirform} contains
a $1/2N$ prefactor).
The same statements hold for the regularized measure
$\mu_\by^{B,\tau}$.
For $L+B+1\le j \le L+K-B$ define the event
\be
A= A_j = \big\{ \big|x_j^{[B]} - \E_{\mu_\by^\tau} x_j^{[B]} \big|
\ge \zeta \big\},
\qquad \mbox{with}\quad \zeta = N^{-z},
\ee
with a parameter $z\in (0,1)$ chosen later.
Using \eqref{conc} for $X=\pm(x_j^B - \E_{\mu_\by^\tau} x_j^B)$
and noticing that $|\nabla X|^2 =(2B+1)^{-1}$, we obtain
\be\label{Atau}
\P_{\mu_\by^\tau} (A) \le 2e^{ - \frac{1}{2}N B \zeta^2 \tau^{-1} }.
\ee
Using now \eqref{Acomp}, we get
\be\label{BA}
\P_{\mu_\by^{B,\tau}} (A)\le \frac{ CB\tau}{N\zeta^2} \to 0
\ee
assuming
\be
b -t + 2 z -1 < 0.
\ee
Using $t=2d-1-\varphi$,
we need
\be\label{662}
z < d-\frac{b}{2}-\frac{\varphi}{2}.
\ee
Under this condition we have from \eqref{Atau} that
\be
\P_{\mu_\by^\tau} \Big( \big|x_j^{[B]} - \E_{\mu_\by^\tau} x_j^{[B]} \big|
\ge \zeta \Big) \le 2e^{ - B^2 }.
\ee
Since the measure $\mu_\by^{B,\tau}$ is also concentrated by the LSI, we have
$$
\P_{\mu_\by^{B,\tau}}\Big( \big| x_j^{[B]}- \E_{\mu_\by^{B,\tau}} x_j^{[B]}
\big|\ge \zeta
\Big)\le 2e^{ - B^2 }\to0
$$
and together with \eqref{BA} we have
\be\label{E-E1}
\Big | \E_{\mu_\by^{B,\tau}} x_j^{[B]} - \E_{\mu_\by^\tau} x_j^{[B]}
\Big | \le 2\zeta.
\ee
Therefore $x_j^{[B]}$ is concentrated on a scale $\zeta$ around the same
point w.r.t both measures $\mu_\by^\tau$ and $\mu_\by^{B,\tau}$.
Using \eqref{SleqD} and that $\by\in \cG$ we get
\be\label{332}
S( \mu_\by | \mu_\by^\tau ) \le
4\tau D( \mu_\by | \mu_\by^\tau)
\le \frac {4N}{ \tau} \E_{ \mu_\by} \sum_{j \in I} (x_j - \gamma_j)^2
\le \frac { 4N \delta^2 K} \tau.
\ee
Hence by \eqref{PPS} and \eqref{Atau} we obtain
\be\label{PPPA}
\P_{\mu_\by} (A)\le \frac {\log 2 +
\frac { 4N \delta^2 K} \tau } {- \log \P_{\mu_\by^\tau} (A) }
\le \frac { C\delta^2 K }{ B\zeta^2}
\to 0
\ee
provided that
\be
z < d-\frac{k}{2}+\frac{b}{2}.
\ee
Now by the definition of $\by\in \cG$ in \eqref{goodset} we have
\begin{align}
\P_{\mu_\by}\Big( \big|x_j^{[B]}&
- \E_{\mu_\by} x_j^{[B]}\big|\ge \zeta\Big) \nonumber
\le \zeta^{-2}
\E_{\mu_\by}\big|x_j^{[B]} - \E_{\mu_\by} x_j^{[B]}\big|^2 \\
& \le\frac{1}{(2B+1)\zeta^2} \sum_{|k-j|\le B}
\E_{\mu_\by}\big|x_k - \E_{\mu_\by} x_k\big|^2 \nonumber \\
& \le\frac{1}{(2B+1)\zeta^2} \sum_{|k-j|\le B}
\E_{\mu_\by}\big|x_k - \gamma_k\big|^2
\le\frac{\delta^2}{\zeta^2}\to 0 \nonumber
\end{align}
using \eqref{662}. Combining it with \eqref{PPPA} we
obtain
\be\label{E-E2}
\Big | \E_{\mu_\by} x_j^{[B]} - \E_{\mu_\by^\tau} x_j^{[B]}
\Big | \le 2\zeta.
\ee
Finally, since $\by\in\cG$, we have
$$
\Big( \E_{\mu_\by} x_j^{[B]}- \gamma_j^{[B]}\Big)^2 \le
\E_{\mu_\by} \Big( x_j^{[B]}- \gamma_j^{[B]}\Big)^2 \le \delta^2\le\zeta^2,
$$
which, combined with \eqref{E-E1} and \eqref{E-E2}, yields \eqref{E-E}.
This completes the proof of Lemma~\ref{lm:accuracy}.
\end{proof}
\subsection{Proof of Proposition~\ref{prop:musig}}\label{sec:musig}
Now we will compare the regularized local relaxation measure
$\mu_\by^{B,\tau}$
with the reference measure $\g_\th^{\htau,s}$
in Dirichlet form sense. Recall their definitions from
\eqref{def:regloc} and \eqref{def:ref}, respectively,
and recall that $\tau=\tau(\by):= \htau/s(\by)^2$.
Here $s=s(\by)$ is a function that is approximately 1 for
good external configurations $\by\in \cG$ (see \eqref{s-1}).
The result is the following comparison of local gap statistics.
Combining this result with
Proposition~\ref{prop:spacingB} and checking that the condition
\eqref{cond2} is satisfied with the choice of parameters given below,
we arrive at the proof of Proposition~\ref{prop:musig}. \qed
\begin{proposition}\label{prop:spacingD} Fix $\varphi\le \frac{1}{38}$.
Let $\by\in \cG$, $\tau=\tau(\by)=\htau/s(\by)^2$ and assume
that for the parameters $\delta=N^{-d}$, $B=N^{b}$, $K=N^k$
with $d=1-\varphi$, $b=8\varphi$, $k=\frac{39}{2}\varphi$. Then with $t=2d-1-\varphi=1-3\varphi$
let $\htau = N^{-t}$ with $t:=2d-1-\varphi=1-3\varphi$.
Then
\be\label{38D}
\Bigg| \big[ \E_{ \mu_{\by}^{B, \tau} } -\E_{ \g_{\by}^{\htau,s}} \big]
\frac 1 K \sum_{i \in I} G\Big( N(x_i-x_{i+1}) \Big)
\Bigg|
\to 0
\ee
as $N \to \infty$
for any smooth and compactly supported test function $G$.
\end{proposition}
\begin{proof}
The key technical estimate is the following lemma whose proof will
take up most of this section.
\begin{lemma}\label{eb} Let $\varphi>0$. Suppose $B=N^b$, $K=N^k$ with $0<b<k<1$, and $\delta= N^{-d}$
with $d\in (0,1)$. Suppose that these parameters satisfy
\be
1-b < -\varphi+ \min \Big( d- \frac{b}{2}-\frac{\varphi}{2},
\; d-\frac{k}{2}+ \frac{b}{2}\Big),
\label{p1}
\ee
i.e. one can choose a number $z>1-b$ and satisfying \eqref{66}.
Let $\by\in\cG=\cG_{\delta, \e_0}$
be a good configuration. Assume that $\e_0\le \e'/10$, where $\e'=\e'(d,\varphi)$
is obtained in Lemma~\ref{lm:accuracy}.
Assume that the equilibrium measure $\rho_V$ is $C^1$ away from the edges.
Let $\htau = N^{-t}$
with $t=2d-1-\varphi$.
Then the Dirichlet form of $ \mu_{\by}^{B,\tau}$ with respect to the reference measure
is bounded by
\be\label{largebound}
\frac{\tau}{K} D \big(\mu_\by^{B,\tau}\mid \g_\th^{\htau,s})
\le C\htau (\log N) \Big [
\frac {K^2 } {N} +
\frac {N \delta^2} { \htau^2}+ \frac {K^4} {N^3\htau^2}
+ \frac { \delta^2 N^3} { BK} + N^{3/5 + \varphi} \Big ]
+ c_1 e^{-c_2N^{\e'/3}}.
\ee
\end{lemma}
The prefactor $\tau/K$ is for convenience; the local
gap statistics of two measures are approximately the same
if $\tau D/K\to 0$. More precisely, we have the following
general theorem which is a slight modification of
Lemma 3.4 \cite{ESY4} (see also Theorem~4.3 in \cite{ESYY}).
This result was originally proven for $\beta\ge 1$,
but by a regularization argument
it extends to any $\beta>0$, see Lemma A.2 of \cite{EKYY2}
for details.
\begin{lemma}\label{lm:Dcomparison}
Let $G:\bR\to\bR$
be a bounded smooth function with compact support.
Consider a measure on $\Sigma_K:=\{ \bx\; :\; x_1< \ldots < x_K\} \subset \RR^K$ defined by
\begin{equation}\label{01}
\rd \om\sim e^{- \beta N \wh \cH}\rd\bx,
\quad
\wh \cH(\bx) = \cH_0(\bx)
-\frac{1}{N} \sum_{1\leq i<j\leq K}\log (x_j-x_i) ,
\end{equation}
with the property that $\nabla^2 \cH_0 \ge \tau^{-1}$ holds for some positive constant $\tau$.
Let $q\rd \om$ be another probability measure.
Let $I\subset\{1, 2, \ldots, K-1\}$ be an interval of indices.
Then for any $\e_1>0$ and for any smooth compactly supported function we have
\be\label{Diff1new}
\Big|\frac{1}{|I|} \sum_{i \in I} \int G\big( N(x_i-x_{i+1} )\big) [q-1]\rd \omega
\Big|
\le C\sqrt { \frac{ N^{\e_1} D_\om( \sqrt q) \tau}{|I|}}+ Ce^{-cN^{\e_1}}\sqrt {S_\om(q)} ,
\ee
where $D_\om(\sqrt q):=D(q\om\mid \om)$.
\end{lemma}
We will apply Lemma~\ref{lm:Dcomparison}
for the measure $\om =\g_\th^{\htau,s}$.
It has the form \eqref{01} except that
$\g_\th^{\htau,s}$ is restricted to the interval $[y_L, y_{L+K+1}]$, i.e.
the relations $y_L< x_{L+1}$ and $x_{L+K}< y_{L+K+1}$ also hold in
addition to the ordering relation $ x_{L+1} < x_{L+2} < \ldots
< x_{L+K}$. Notice that the Hamiltonian
$\cH_{\th}^{\htau, s}$ of the measure $\g_\th^{\htau,s}$
contains a term $\frac{1}{N}
\big[ \log (x_{L+1} - y_L ) + \log (y_{L+K+1}-x_{L+K})\big] $.
This term
confines the particles in the interval $[y_L, y_{L+K+1}]$
exactly as the term $\log (x_{i+1}-x_{i})$ guarantees
the ordering constraint $x_i< x_{i+1}$. Hence the regularization argument
in Lemma A.2 of \cite{EKYY2} can be used to treat the additional constraints, $y_L < x_{L+1}$
and $x_{L+K}< y_{L+K+1}$.
The proof of Proposition~\ref{prop:spacingD} now follows
from Lemma~\ref{eb} and Lemma~\ref{lm:Dcomparison}
with $\om =\g_\th^{\htau,s}$ and $q\rd\om = \mu_\by^{B,\tau}$.
The parameters $b,k,d\in (0,1)$ have to satisfy the following relations
from \eqref{p1} and from the requirement that
the right side of \eqref{largebound} converges to zero:
\begin{align}
b &< k \non \\
1-b+\varphi&< d-\frac{b}{2}-\frac{\varphi}{2} \non \\
1-b+\varphi&< d-\frac{k}{2}+\frac{b}{2} \non\\
1-2d+\varphi +2k -1&< 0\non\\
1-2d + (2d-1-\varphi)&< 0\non\\
4k-3+(2d-1-\varphi)&< 0\non\\
1-2d+\varphi -2d+3-b-k&< 0\non \\
1-2 d + 2 \varphi + \frac 3 5 &< 0. \non
\end{align}
It is easy to check that all these conditions are satisfied
if, e.g.
$$
d=1-\varphi, \qquad b=8\varphi, \qquad k=\frac{39}{2}\varphi, \qquad 0<\varphi \le \frac{1}{38}.
$$
This choice is not optimal for the above system of inequalities,
but we took into account that the parameters will also have
to satisfy \eqref{cond2} so that we could combine
Proposition~\ref{prop:spacingD} and Proposition~\ref{prop:spacingB}
to arrive at Proposition~\ref{prop:musig}.
Finally, the entropy term $S \big(\mu_\by^{B,\tau}\mid \g_\th^{\htau,s})$
in \eqref{Diff1new} can be estimated by the Dirichlet form via
the logarithmic Sobolev inequality.
This completes the proof of Proposition~\ref{prop:spacingD}.
\end{proof}
\bigskip
\subsection{Dirichlet form estimate: proof of Lemma~\ref{eb}}\label{sec:dir}
By definition,
\[ \frac{\tau}{K} D \big(\mu_\by^{B,\tau}\mid \g_\th^{\htau,s})
= \frac{\tau }{2N K }\int \Big|\nabla \log \Big( \frac{\mu_\by^{B,\tau}}{\g_\th^{\htau,s}}
\Big)\Big|^2 \rd \mu_{\by}^{B,\tau} \le
\frac{\tau N}{ K} \int \sum_{ L+1\le j \le L+ K } Z_j^2 \rd \mu_{\by}^{B,\tau},
\]
where $Z_j$ is defined as follows:
For $L+1 < j \le L + 4B$, we set
\begin{align}
Z_j := &
\frac{\beta}{2}V'(x_j) -
\frac \beta N \sum_{ k < L-2B\atop k >L+K }
\frac 1 {x_j- y_{k}^B}
- \frac{\beta}{2} W_s '( x_j) +
\frac \beta N \sum_{ k < L-2B\atop k>L+K} \frac 1 {x_j- \th_{k}'}
+ \frac { \gamma_{j} - \theta_{j}' } { \tau } \nonumber
\end{align}
(recall that $\th'_j=\th_j/s$ and we set $W_s(x)=s^2x^2$).
Note that the summation at the lower edge is only for $k< L-2B$ instead of $k\le L$
because the interaction terms near the
boundary cancel by \eqref{ybth}. Moreover, notice that the linear terms,
coming from the derivative of the quadratic confinements (see \eqref{232}
and \eqref{242}), cancel each other
$$
\frac{s(\by)^2}{\htau}(x_j- \th_j')- \frac{1}{\tau} (x_j- \gamma_j)
= \frac { \gamma_{j} - \theta_{j}' } { \tau }
$$
by the choice of $\tau(\by) = \htau/s(\by)^2$.
Similarly, for $L+K-4B < j \le L + K$, we set
\begin{align}
Z_j := &
\frac{\beta}{2}V'(x_j) -
\frac \beta N \sum_{ k > L+K+2B \atop k<L }
\frac 1 {x_j- y_{k}^B}
- \frac{\beta}{2} W_s '( x_j) + \frac \beta N \sum_{ k > L+K+2B\atop k<L}
\frac 1 {x_j- \th_{k}'}
+ \frac { \gamma_{j} - \theta_{j}' } { \tau }. \nonumber
\end{align}
Finally,
for $L +4B < j \le L +K- 4B$, we define
\[
Z_j: =
\frac{\beta}{2} V'(x_j) - \frac \beta N \sum_{ k < L \atop k>L+K+1} \frac 1 {x_j- y_{k}}
- \frac{\beta}{2} W_s '( x_j) + \frac \beta N \sum_{ k < L\atop k>L+K+1}
\frac 1 {x_j- \th_{k}'}
+ \frac { \gamma_{j} - \theta_{j}' } { \tau }.
\]
Notice that here $y_k$ is not replaced with $y_k^B$ since
only interactions for $x_j$'s near the edges have been regularized.
Moreover, the interactions with the boundary terms $y_k$, with $k=L$ and
$k=L+K+1$ cancel out since $y_L=\th_L'$ and $y_{L+K+1}=\th'_{L+K+1}$ by
the matching construction.
\bigskip
Now we estimate the size of $Z_j$ in each case.
\bigskip
\underline{\it Case 1: $L +4B < j \le L +K- 4B$}.
The first step is to decompose $Z_j$ as
\be
Z_j
= \beta \sum_{a=1}^5 \Omega_j^a,
\ee
where
\begin{align}
\Omega_j^1 &
: = \left [ \frac{1}{2} V'(x_j) -\int \rd y \frac {\rho_V(y) } {x_j- y } \right ]
- \left [ \frac{1}{2} W_s '( x_j) - \int \rd y \frac {\rho_{W_s}( y) } { x_j- y } \right ]
\nonumber\\
\Omega^2_j & = \Omega^{2,low}_j+\Omega^{2,up}_j \non\\
& : = - \Bigg( \frac 1 N \sum_{ k< L }
\frac 1 {x_j- y_{k}} - \int_{ -\infty}^{y_{L}} \frac {\rho_V(y) } {x_j- y } \rd y\Bigg)
- \Bigg( \frac 1 N \sum_{ k > L+K+1 }
\frac 1 {x_j- y_{k}} - \int_{y_{L+K+1}}^{\infty} \frac {\rho_V(y) } {x_j- y } \rd y\Bigg)
\nonumber\\
\Omega^3_j & = \Omega^{3,low}_j+\Omega^{3,up}_j \non \\
& := \Bigg( \frac 1 N \sum_{ k< L } \frac 1 {x_j- \th_{k}'} -
\int_{ -\infty}^{ \theta_{ L}'} \frac {\rho_{W_s}(y) } { x_j- y } \rd y\Bigg)
+ \Bigg( \frac 1 N \sum_{ k> L+K+1 } \frac 1 {x_j- \th_{k}'} -
\int_{ \th_{L+K+1}'}^{ \infty} \frac {\rho_{W_s}(y) } { x_j- y } \rd y\Bigg)
\nonumber\\
\Omega^4_j & : = \int_{y_{L}}^{ y_{L+K+1} } \frac {\rho_V(y) - \rho_{W_s}(y) } {x_j- y } \rd y
\nonumber\\
\Omega^5_j & : = \frac { \gamma_{j} - \theta_{j}' } { \beta \tau }.
\label{fiveom}
\end{align}
Here we also used that $[y_L, y_{L+K+1}]=[\th_L', \th_{L+K+1}']$ when
establishing the limits of integrations.
By the equilibrium relation \eqref{equilibrium}
between $V$ and $\rho_V$, we have
\be \Omega^1_j=0.
\label{om1}
\ee
{F}rom \eqref{38}, we have
\be\label{Om2}
[\Omega^5_j]^2 = C\frac {( \gamma_{j}- \theta_{j}')^2} {\tau^2}
\le \frac C {\tau^2} \left [ \delta^2+ \frac { K^4 } { N^4 } \right ].
\ee
Since $\rho_V\in C^1$ away from the edge, and so is the semicircle
density $\rho_{W_s}$, we have by Taylor expansion
\begin{align}
|\Omega^4_j| & = \Big|\int_{y_{L}}^{ y_{L+K+1} }
\frac {\rho_V(y) - \rho_{W_s}(y) } {x_j- y } \rd y\Big|
\label{Om4}\\
& \le
\Bigg| \int_{y_{L}}^{ y_{L+K+1} }
\frac {\rho_V(x_j) - \rho_{W_s}(x_j) + O( x_j-y ) } {x_j- y } \rd y
\Bigg| \nonumber\\
& \le C \big[ |\log (x_j-y_L)|+|\log (y_{L+K+1}-x_j)|\big] \left [
\frac K N + \frac { \delta N } K \right ]. \nonumber
\end{align}
Here we used \eqref{ap} and \eqref{VW0} and the fact that
$\rho_{W_s}(x)-\rho_W(x) = O(|s-1|)$ away from the edge
together with \eqref{s-1} to estimate
$$
|\rho_V(x) - \rho_{W_s}(x)|\le C\left[ \frac K N + \frac { \delta N } K \right]
$$
for any $x\in [y_L, y_{L+K+1}]$.
The logarithmic terms after taking square and expectation w.r.t.
will give rise to an irrelevant $\log N$ factor by
using Lemma~\ref{lm:rep}
$$
\E_{\mu_\by^{B,\tau}} \big[ |\log (x_j-y_L)|+|\log (y_{L+K+1}-x_j)|\big]^2\le C\log N.
$$
\medskip
We now estimate the main error $\Omega^2_j $ and we will deal with
the first term only, coming from the lower edge, the second one can be treated similarly.
We write it as
$$
\Omega_j^{2,low}=-\Bigg( \frac 1 N \sum_{ k< L }
\frac 1 {x_j- y_{k}} - \int_{ -\infty}^{y_{L}} \frac {\rho_V(y) } {x_j- y } \rd y\Bigg)
=\Omega_j^{2,1}+\Omega_j^{2,2} + \Omega_j^{2,3}
$$
with
\begin{align}
\Omega_j^{2,1} & := -\Bigg( \frac 1 N \sum_{ k< L }
\frac 1 {x_j- \gamma_{k}} - \int_{ -\infty}^{\gamma_{L}} \frac {\rho_V(y) } {x_j- y } \rd y\Bigg)
\non\\
\Omega_j^{2,2} & := \int_{\gamma_{L}}^{y_L} \frac {\rho_V(y) } {x_j- y } \rd y
\non\\
\Omega_j^{2,3} & := \frac 1 N \sum_{ k< L }
\Big[ \frac 1 {x_j- \gamma_{k}}- \frac 1 {x_j- y_{k}} \Big].
\label{Om3decomp}
\end{align}
With $\zeta= N^{-z}$ with $z$ is given in Lemma~\ref{eb},
define the event
$$
\Lambda=\Big\{ |x_i^{[B]} -\gamma_i^{[B]}|\le 6\zeta, \quad
\forall i \in\llbracket L+B+1, L+K-B\rrbracket\Big\},
$$
then its complement has very small probability,
$$
\P_{\mu_\by^{B,\tau}}\big( \Lambda^c\big)\le c_1 e^{-c_2N^{\e'}}
$$
from \eqref{cont1} and \eqref{E-E}.
On the event $\Lambda^c$ we simply estimate
$$
\Big| \frac 1 N \sum_{ k< L }
\frac 1 {x_j- y_{k}} - \int_{ -\infty}^{y_{L}} \frac {\rho_V(y) } {x_j- y } \rd y\Big|
\le \frac{1}{y_L-y_{L-1}}+ C\big|\log (x_j-y_L)\big| ,
$$
therefore
\begin{align}
\E_{\mu_\by^{B,\tau}} {\bf 1}(\Lambda^c) & \Big| \frac 1 N \sum_{ k< L }
\frac 1 {x_j- y_{k}} - \int_{ -\infty}^{y_{L}} \frac {\rho_V(y) } {x_j- y } \rd y\Big|^2
\nonumber\\
& \le C\Big(\frac{1}{(y_L-y_{L-1})^2}+ \E_{\mu_\by^{B,\tau}}\big|\log (x_j-y_L)\big|^4\Big)^{1/2}
\big( \P_{\mu_\by^{B,\tau}}\big( \Lambda^c\big)\big)^{1/2}\le c_1 e^{-c_2N^{\e'/3}}
\label{subb}
\end{align}
by using Lemma~\ref{lm:rep} and $|y_L-y_{L-1}|\ge \exp(- N^{\e_0})$ from
$\by\in \cG$. Here we used that $\e_0\le \e'/10$.
\medskip
Now we continue the estimate on the set $\Lambda$
and we consider the three terms in \eqref{Om3decomp} separately.
For the first term we write
$$
\Omega_j^{2,1} = \frac 1 N \sum_{ k< L } \Big[
\frac 1 {x_j- \gamma_{k}} -
\int_{ \gamma_{k}}^{\gamma_{k+1}} \frac {N\rho_V(y) } {x_j- y } \rd y\Big]
= \frac{1}{N} \sum_{ k< L }
\int_{\gamma_k}^{\gamma_{k+1}} \Gamma_{j}^k N\rho_V(y) \rd y
$$
where we have used that $\int_{\gamma_k}^{\gamma_{k+1}} N\rho_V=1$ and
$$
\Gamma_{j}^k = \frac {\gamma_k-y } {(x_j- y)(x_j-\gamma_k) }.
$$
Recall that $L\ge \kappa N\gg \delta$ and
$x_j\in [y_L, y_{L+K+1}] = [\gamma_L,\gamma_{L+K+1}] + O(\delta)$.
For $k\le \frac{1}{2}\kappa N$ we know that $|\gamma_k-x_j|\ge c$ with some positive constant. Hence we have
$$
\frac{1}{N}\sum_{k\le \kappa N/2} \Gamma_{j}^k \le
\frac C N \sum_{ k\le \kappa N/2 }
\int_{\gamma_k}^{\gamma_{k+1}} |\gamma_k-y| N\rho_V(y) \rd y
\le \frac C N \sum_{ k\le \kappa N/2 } |\gamma_{k+1}-\gamma_k|\le CN^{-1},
$$
since $\gamma_{k+1}-\gamma_k \le CN^{-2/3}k^{-1/3}$ near a square root
singularity of $\rho_V$ at the edge. For the regime $k\ge \frac{1}{2}\kappa N$ we can use
$|\gamma_{k+1}-\gamma_k|\le CN^{-1}$ to get
\begin{align}
\frac{1}{N}\sum_{ \kappa N/2\le k< L} \Gamma_{j}^k & \le
\frac{1}{N}\sum_{ \kappa N/2\le k< L} \frac{C}{N} \frac{1}{(x_j-\gamma_k)^2} \nonumber \\
& \le \frac{1}{N}\sum_{ \kappa N/2\le k< L} \frac{C}{N} \frac{1}{(x_{j-B}^{[B]}-\gamma_k)^2}
\nonumber \\
& \le \frac{C}{N} \frac{1}{(x_{j-B}^{[B]}-\gamma_L)}\le \frac{C}{B}. \nonumber
\end{align}
Here in the second inequality we used that
on the set $\Lambda$ we have
\be\label{xy}
x_j\ge x_{j-B}^{[B]}> \gamma_{j-B}^{[B]}-6\zeta
\ge \gamma_{j-2B}-6\zeta\ge \gamma_L + cBN^{-1}> \gamma_k+cBN^{-1}
\ee
for $k<L$
using $j\ge L+4B$ and thus $\gamma_{j-2B}-\gamma_L\ge cBN^{-1}\gg 6\zeta$,
since $z>1-b$. Therefore $x_j-\gamma_k\ge x_{j-B}^{[B]}-\gamma_k>0$.
In the third inequality we performed the summation and used that $\gamma_k$
is regularly spaced. In the last inequality we again used \eqref{xy}.
In summary, we have shown that
\be\label{om21}
|\Om_j^{2,1}|\le \frac{C}{B}+\frac{C}{N}
\ee
on the set $\Lambda$ and we have seen that the contribution
from $\Lambda^c$ is subexponentially small \eqref{subb}.
\medskip
Now we consider $\Om_j^{2,2}$ on $\Lambda$.
We have
\be\label{om22}
|\Om_j^{2,2}|\le C \int_{y_L}^{\gamma_L} \frac{\rd y}{x_j-y}\le
\frac{C\delta}{\gamma_{j-2B}-\gamma_L},
\ee
by using $x_j-\gamma_L \ge \gamma_{j-2B}-\gamma_L -6\zeta
\ge c( \gamma_{j-2B}-\gamma_L) $ from \eqref{xy}
and from $\gamma_{j-2B}-\gamma_{L}\ge cBN^{-1}\gg 6\zeta$, moreover
$x_j-y_L\ge x_j-\gamma_L - \delta \ge c( \gamma_{j-2B}-\gamma_L)$
by
$|\gamma_L-y_L|\le \delta$ (from $\by\in \cG$)
and $\delta\ll BN^{-1}$ (from \eqref{p1}).
Thus
$$
\sum_{L+4B\le j\le L+K-4B} |\Om_j^{2,2}|^2
\le \frac{C\delta^2N^2}{B}.
$$
\medskip
For the third term $\Om_j^{2,3}$ we have
\begin{align}
\E_{\mu_\by^{B,\tau} } {\bf 1}(\Lambda)
& \sum_{L+4B < j \le L+ K-4B} [\Omega^{2,3}_j ]^2 \label{om333} \\
& \le \E_{\mu_\by^{B,\tau} } {\bf 1}(\Lambda)
\sum_{L+4B < j \le L + K-4B}
\left [ \frac 1 N \sum_{k < L } \Big( \frac 1 {x_j- y_k} - \frac 1 {x_j- \gamma_k}
\Big)\right ] ^2
\nonumber \\
& \le \E_{\mu_\by^{B,\tau} } {\bf 1}(\Lambda) \sum_{L+4B < j \le L + K-4B}
\left [ \frac 1 N \sum_{ k < L }
\frac {(y_k-\gamma_k) } { (x_j- y_k ) (x_j- \gamma_k )} \right ] ^2.
\non
\end{align}
We split the summation over $k$ into two terms: $\kappa N/2\le k <L$ and $k<\kappa N/2$
and separate by a Schwarz inequality.
First we consider the case $\kappa N/2 \le k < L$.
Expanding the square, we need to bound
\begin{align}\label{expsq}
\E_{\mu_\by^{B,\tau} }& {\bf 1}(\Lambda) \frac {1} {N^2} \sum_{ \kappa N/2\le k < L }
\sum_{\kappa N/2\le a< L}
\sum_{L +4B < j \le L+K }
\frac { |y_k-\gamma_k| |y_a-\gamma_a| } { (x_j- y_k) (x_j-\gamma_k) (x_j- y_a)
(x_j- \gamma_a)} \\
& \le 2 \E_{\mu_\by^{B,\tau} } {\bf 1}(\Lambda)
\frac 1 {N^2} \sum_{ \kappa N/2\le k < L } |y_k-\gamma_{k}|^2
\sum_{L+4B \le j \le L + K } \frac { 1 } { (x_j- y_{k})^2 }
\sum_{\kappa N/2\le a < L} \frac { 1 } { (x_j- \gamma_a)^2} ,\non
\end{align}
where we used another Schwarz inequality and
the factor 2 accounts for a similar term with the role of $k$ and $a$
interchanged.
In the case $\kappa N/2 \le k < L$ we have $|\gamma_k-y_k|\le \delta$.
Then \eqref{expsq} is bounded by
\begin{align}\label{frr}
\frac {2\delta^2} {N^2} \sum_{L+4B \le j \le L + K } &
\E_{\mu_\by^{B,\tau} } {\bf 1}(\Lambda) \sum_{ k < L } \frac { 1 } { (x_j- y_{k})^2 }
\sum_ {a <L} \frac { 1 } { (x_j- \gamma_{a})^2}
\\
& \le C \delta^2 \sum_{L+4B \le j \le L + K }
\E_{\mu_\by^{B,\tau} } {\bf 1}(\Lambda) \frac { 1 } { (x_j- y_{L})^2 }
\le C\delta^2 \frac { N^2} { B}. \non
\end{align}
Here we used
$$
\frac{1}{N} \sum_ {a <L} \frac { 1 } { (x_j- \gamma_{a})^2} \le \frac{C}{x_j-\gamma_L}
\le \frac{C}{x_j-y_L}
$$
relying on the regularity of $\gamma_a$ and using, from \eqref{xy},
that $x_j-\gamma_a\ge x_j-\gamma_L
\ge cBN^{-1}$ which is much larger than the spacing of order $N^{-1}$ of the $\gamma$-sequence.
In the last estimate $x_j-\gamma_L\gg |\gamma_L-y_L|$ was used (since $BN^{-1}\gg \delta$).
Similarly we could perform the $k$ summation
$$
\sum_{ k < L } \frac { 1 } { (x_j- y_{k})^2 } \le \frac{C}{x_j-y_L}
$$
since $x_j-y_k\ge x_j-\gamma_k -\delta \ge c(x_j-\gamma_k)$.
To perform the $j$ summation in \eqref{frr}, we use
$$
\frac { 1 } { (x_j- y_{L})^2 }
\le \frac { 1 } { (x_{j-B}^{[B]}- y_{L})^2 },
$$
and then we recall that apart from a set of subexponentially small
probability, we have
$$
|x_{j-B}^{[B]}-\gamma_{j-B}^{[B]}|\le 6\zeta
$$
from Lemma~\ref{lm:accuracy}. Since $\zeta\ll BN^{-1}$ and
$x_{j-B}^{[B]}- y_{L}\ge cBN^{-1}$ from \eqref{xy},
we see that
$$
\sum_{L+4B \le j \le L + K } \frac { 1 } { (x_j- y_{L})^2 }
\le\sum_{L+4B \le j }\frac { C } { (\gamma_{j-B}^{[B]}- y_{L})^2 }
\le \frac{CN}{\gamma_{L+3B}^{[B]}- y_{L}}\le \frac{CN^2}{B}.
$$
On the exceptional set one can just use the trivial bound
$(x_j-y_L)^{-2}\le C(x_j-\gamma_L)^{-2}\le CN^2B^{-2}$ from \eqref{xy}.
\bigskip
Consider now the case $ k \le \kappa N/2$ in \eqref{om333}. We have
\begin{align} \label{6141}
\E_{\mu_\by^{B,\tau }} {\bf 1}(\Lambda) \frac 1 {N^2} &
\sum_{ k \le \kappa N/2 } \sum_{a \le \kappa N/2 }
\sum_{L +4B \le j \le L+K }
\frac { |y_k-\gamma_{k}| |y_a-\gamma_{a}| } { (x_j- y_{k}) (x_j-\gamma_{k}) (x_j- y_{a})
(x_j- \gamma_{a})} \\
& \le 2\E_{\mu_\by^{B,\tau} } {\bf 1}(\Lambda) \frac 1 {N^2}
\sum_{ k \le \kappa N/2 } |y_k-\gamma_{k}|^2
\sum_{L+4B \le j \le L + K } \frac { 1 } { (x_j- y_{k})^2 }
\sum_ {a \le \kappa N/2 } \frac { 1 } { (x_j- \gamma_{a})^2}\non \\
& \le \frac {CK} {N} \E_{\mu_\by^{B,\tau} } {\bf 1}(\Lambda)
\sum_{ k \le \kappa N/2 } |y_k-\gamma_{k}|^2 \le C KN^{-2/5+ \varphi}, \non
\end{align}
where we used that all denominators are separated away from zero
and Lemma \ref{edge}.
Furthermore, in the last inequality, we have used Lemma \ref{edge}
for $k \ge N^{3/5 + \varphi}$ and
we used $ |y_k-\gamma_{k}| \le O(1)$ for $k \le N^{3/5 + \varphi}$ from
$\by\in \cG$ and \eqref{goodset}.
Similar comment applies to all edge terms in this proof and we will not repeat it.
Summarizing, we have shown that
\be\label{Om23}
\E_{\mu_\by^{B,\tau} } {\bf 1}(\Lambda)
\sum_{L+4B < j \le L+ K-4B} [\Omega^{2,3}_j ]^2\le \frac{C\delta^2N^2}{B}+ C KN^{-2/5+ \varphi}.
\ee
Finally, we need to estimate $\Om_j^3$ in \eqref{fiveom}. It can
be treated exactly as $\Om_j^{2,1}$ and the result is
\be
\label{om3}
|\Om_j^{3}|\le \frac{C}{B}+\frac{C}{N}
\ee
on the set $\Lambda$ and the contribution
from $\Lambda^c$ is subexponentially small as in \eqref{subb}.
\bigskip
\underline{\it Case 2: $L < j \le L +4B$.}
(There is a third case $L+K-4B\le j\le L+K$ which
is identical to Case 2 and will not be treated separately).
We decompose $Z_j$ as before and the only modifications are
\begin{align}
\Omega^{2,low}_j& := -\Bigg( \frac 1 N \sum_{ k\le L-2B }
\frac 1 {x_j- y_{k}^B} - \int_{ -\infty}^{y_{L-2B}} \frac {\rho_V(y) } {x_j- y } \rd y\Bigg)
\non\\
\Omega^{3,low}_j& := \frac 1 N \sum_{ k\le L-2B } \frac 1 {x_j- \th_{k}'} -
\int_{ -\infty}^{ \theta_{ L-2B}'} \frac {\rho_{W_s}(y) } { x_j- y } \rd y
\non\\
\Omega^4_j & := \int_{y_{L-2B}}^{ y_{L+K+1} } \frac {\rho_V(y) - \rho_{W_s}(y) } {x_j- y } \rd y
+\int_{y_{L-2B}}^{ \th'_{l-2B} } \frac {\rho_{W_s}(y) } {x_j- y } \rd y.
\non
\end{align}
We now estimate the main error term $\Omega^{2,low}_j $, and we write it, as before
$$
\Omega_j^{2,low}=\Omega_j^{2,1}+\Omega_j^{2,2} + \Omega_j^{2,3}
$$
with
\begin{align}
\Omega_j^{2,1} & := -\Bigg( \frac 1 N \sum_{ k< L-2B }
\frac 1 {x_j- \gamma_{k}} - \int_{ -\infty}^{\gamma_{L-2B}}
\frac {\rho_V(y) } {x_j- y } \rd y\Bigg)
\non\\
\Omega_j^{2,2} & := \int_{\gamma_{L-2B}}^{y_{L-2B}} \frac {\rho_V(y) } {x_j- y } \rd y
\non\\
\Omega_j^{2,3} & := \frac 1 N \sum_{ k< L-2B }
\Big[ \frac 1 {x_j- \gamma_{k}}- \frac 1 {x_j- y_{k}} \Big].
\label{Om3newdecomp}
\end{align}
We have
\begin{align}
|\Omega_j^{2,1}| & =\Bigg| \frac 1 N \sum_{ k< L-2B }
\int_{ \gamma_k}^{\gamma_{k+1}}
\frac {y-\gamma_k} {(x_j- y)(x_j-\gamma_k) } N\rho_V(y) \rd y\Bigg| \non\\
& \le \frac C N \sum_{ k< L-2B }
\frac {\gamma_{k+1}-\gamma_k } {(x_j-\gamma_k)^2 } \non\\
&\le \frac C B +\frac{C}{N}\sum_{k\le \kappa N/2} |\gamma_{k+1}-\gamma_k|\le \frac{C}{B} +\frac{C}{N}
\non
\end{align}
using that $x_j\ge y_L\ge \gamma_L-\delta \ge \gamma_{L-2B} +cBN^{-1}$.
The estimate of $\Omega_j^{2,2}$ is trivial
$$
|\Omega_j^{2,2}|\le \frac{|\gamma_{L-2B}-y_{L-2B}|}{cBN^{-1}}\le \frac{CN\delta}{B}.
$$
Finally
\begin{align}\label{613new}
\E_{\mu_\by^{B,\tau} } \sum_{L \le j \le L + 4B} [\Omega^{2,3}_j ]^2
& \le \E_{\mu_\by^{B,\tau} } \sum_{L \le j \le L + 2B} \left [ \frac 1 N \sum_{k < L-2B }
\Big(\frac 1 {x_j- y_k} - \frac 1 {x_j- \gamma_k}\Big)\right ] ^2
\non\\
& \le \frac {C\delta^2} {N^2} \sum_{L \le j \le L + 4B }
\E_{\mu_\by^{B,\tau} } \sum_{\kappa N/2\le k < L-2B } \frac { 1 } { (x_j- y_k)^2 }
\sum_ {\kappa N/2\le a < L-2B} \frac { 1 } { (x_j- \gamma_{a})^2}
\non\\
& + \frac{C}{N^2}\sum_{k\le \kappa N/2}|\gamma_k-y_k|^2
\non\\
& \le C\delta^2 \frac { N^2} { B} + C KN^{-2/5+ \varphi},
\non
\end{align}
where we again split the summation over $k$ into $\kappa N/2\le k\le L-2B$ and
$k\le \kappa N/2$, yielding the two terms, similarly to \eqref{frr} and \eqref{6141}.
The estimate $\Omega_j^{3,low}$ is analogous to that of $\Omega_j^{2,1}$.
The first term of $\Omega_j^4$ is estimated as before in \eqref{Om4}.
The additional second term in $\Omega_j^4$ is trivial
by recalling $|\gamma_{L-2B}-\th'_{L-2B}|\le CB^2N^{-2}+C\delta$ from \eqref{38}:
$$
\Bigg| \int_{y_{L-2B}}^{ \th'_{L-2B} } \frac {\rho_{W_s}(y) } {x_j- y } \rd y\Bigg|
\le \frac{|y_{L-2B}-\th'_{L-2B}|}{cBN^{-1}}\le \frac{C\delta + CB^2N^{-2}}{cBN^{-1}}
\le\frac{C\delta N}{B} + \frac{CB}{N},
$$
since the denominator can be estimated by using
$x_j-y_{L-2B}\ge y_L-y_{L-2B}\ge \gamma_L-\gamma_{L-2B}- 2\delta\ge cBN^{-1}$
and
$$
x_j-\th'_{L-2B}\ge y_L-\th'_{L-2B} \ge \gamma_L-\gamma_{L-2B} - 2\delta +(\gamma_{L-2B}-
\th'_{L-2B})\ge cBN^{-1}
$$
where we used $\delta \ll BN^{-1}$ and $B\ll N$.
Collecting all the error terms into \eqref{largebound}
and removing some redundant terms, we
have thus proved Lemma \ref{eb}. \qed
\bigskip
\noindent{\bf Acknowledgement.} We would like to thank M. Ledoux
for pointing out an error in the statement of Lemma \ref{lem:localLSI}
in an early version of this paper.
| 85,374
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Go to the first, previous, next, last section, table of contents.
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Recall that the dual of a simplicial descent category is by definition a cosimplicial descent category. In this case, a \textit{path}
functor in $\mc{D}$ is induced by $path (f,g)=\mbf{s}Path (f,g)$. More concretely, if $X\in\simp\mc{D}$ and $K$ is a pointed simplicial finite set, then $X^{K}$ is in degree $n$ equal to $\prod_{K_n}X^n$. If $C\stackrel{g}{\rightarrow}A\stackrel{f}{\leftarrow}B$ are maps in $\mc{D}$, then $Path(f,g)$ is the pullback of $A^{\Dl[1]}\stackrel{d^0\times d^1}{\rightarrow}A\times A$ and $B\times C\stackrel{f\times g}{\rightarrow} A\times A$.
The dual to the suspension is then the \textit{loop} functor
$\Omega:\mc{D}\rightarrow\mc{D}$, defined as
$\Omega(X)=path(0\rightarrow X \leftarrow 0)$, and this time the
fiber sequences considered are
$$ \Omega X \rightarrow path(f) \rightarrow X\stackrel{f}{\rightarrow} Y$$
All results concerning cofiber sequences can be dualized for fiber sequences, inducing a `left' triangulated structure on $Ho\mc{D}$.
\begin{ejs}\mbox{}\\
\indent \textbf{Differential graded algebras:} Let $\mbf{Dga}(R)$ be the category of differential graded $R$-algebras (not necessarily commutative, and positively graded) over a commutative ring $R$. The (normalized) Alexander-Whitney simple $\mbf{s}_{AW}:\simp\mbf{Dga}(R)\rightarrow \mbf{Dga}(R)$ comes from the normalized simple of cochain complexes of $R$-modules and endows $\mbf{Dga}(R)$ with a structure of cosimplicial descent category, where $\mrm{E}$=quasi-isomorphisms \cite{R1}.\\
If $f:B\rightarrow A$ and $g:C\rightarrow A$ are morphisms of differential graded algebras, then $path_{dga}(f,g)$ has as underlying cochain complex the object $path(f,g)$ in $Ch (R-mod)$, while the product in $path_{dga}(f,g)$ is induced by those of $A$, $B$ and $C$ (see \cite{G} for details). The deduced fiber sequences are studied in loc. cit. as well, where an analogous result to \ref{TR2} is proven.\\
\indent \textbf{Commutative differential graded algebras:} Denote by $\mbf{Cdga}(k)$ the category of commutative dif{f}erential
graded algebras over a field $k$ of characteristic 0.
Navarro's Thom-Whitney simple
\cite{N} $\mbf{s}_{TW}:\simp\mbf{Cdga}(k)\rightarrow\mbf{Cdga}(k)$ gives rise to a cosimplicial descent
structure on $\mbf{Cdga}(k)$, where $\mrm{E}=$quasi-isomorphisms \cite{R1}.
We deduce in this case a functor $path_{cdga}$, such that the underlying cochain complex of $path_{cdga}$ is quasi-isomorphic to the usual path functor in cochain complexes. The resulting distinguished triangles verify then theorem \ref{struct triangulada}, and the forgetful functor from $\mbf{Cdga}$ to cochain complexes of $k$-vector spaces sends a cofiber sequence to a distinguished triangle of $D(k-mod)$.
\end{ejs}
If the category $\mc{D}$ is an additive simplicial descent category, we deduce a \textit{suspended}
(or right triangulated) category structure \cite{KV} on $Ho\mc{D}$. We will prove later that $Ho\mc{D}$ is always additive for a (not necessarily additive) stable simplicial descent category $\mc{D}$. Stable means that the suspension is an equivalence of categories. But the additive and non stable case is still interesting. It covers, for instance, the case $\mrm{CF}^c\mc{A}$ of uniformly bounded-below (regularly) filtered cochain complexes, described later. The induced suspended structure gives rise to a Verdier's triangulated structure on the homotopy category of bounded-below filtered complexes. It is not obtained directly as the homotopy category of a simplicial descent category. The reason is that the simple $\mbf{s}:\Dl\mrm{CF}^c\mc{A}\rightarrow \mrm{CF}^c\mc{A}$ does not preserve regular filtrations in the (non-uniformly) bounded-below case.
\begin{defi}\cite{R1}
An \textit{additive} simplicial descent category is by definition
an additive category $\mc{D}$ endowed with a simplicial descent structure $(\mc{D},\mrm{E},\mbf{s},\mu,\lambda)$ such that $\mbf{s}$ is an additive functor and $\mu$ is an
isomorphism in $Fun_{ad}(\simp\simp\mc{D},\mc{D})[\mrm{E}^{-1}]$.
Here $Fun_{ad}(\simp\simp\mc{D},\mc{D})$ is the category of
additive functors from $\simp\simp\mc{D}$ to $\mc{D}$.
\end{defi}
\begin{cor} If $\mc{D}$ is an additive simplicial descent category, then $Ho\mc{D}$ is a suspended category \cite{KV}. In addition, a descent functor $\mc{D}\rightarrow\mc{D}'$ induces a functor of suspended categories $F:Ho\mc{D}\rightarrow Ho\mc{D}'$, that is, it preserves cofiber sequences.
\end{cor}
\begin{proof} If $\mc{D}$ is additive, one easily checks that $Ho\mc{D}$ is additive as well using \ref{MorfHoD}. The suspension $\Sigma$ is additive since it is composition of additive functors. To finish it remains to see that the `abstract' minus sign $m_B:\Sigma B\rightarrow \Sigma B$ is equal to $-Id_{\Sigma B}$ in $Ho\mc{D}$. This can be done directly, finding a homotopy between $m$ and $-Id$, or it can be deduced from proposition \ref{A1cogrupo}. Indeed, we have two group structures on $Hom_{Ho\mc{D}}(\Sigma B,\Sigma B)$. The cogroup object structure on $\Sigma B$ given by the map $d_B:\Sigma B\rightarrow \Sigma B\sqcup \Sigma B=\Sigma B\oplus\Sigma B$ of \ref{A1cogrupo} induces the first one. The group object structure given by the codiagonal $\Sigma B\oplus\Sigma B=\Sigma B\times \Sigma B \rightarrow \Sigma B$ induces the second one, that is the one coming from the sum in $\mc{D}$. By standard arguments these group structures agree, so in particular $m_B=-Id_{\Sigma B}$.
\end{proof}
\begin{ejs} We exhibit the suspended structures coming from additive (and non-stable) simplicial descent categories.
\textbf{Simplicial objects in additive$/$abelian categories:} If $\mc{A}$ is an additive or abelian category, consider $\mc{D}=\simp\mc{A}$. The simple functor $\mrm{D}:\simp \simp\mc{A}\rightarrow\simp\mc{A}$ is the diagonal functor of a bisimplicial object. Hence, our cofiber sequences are just the \textit{cofibration sequences} of \cite[5.3]{Vo}. They correspond to the usual `exact triangles' in $Ch_+\mc{A}$, the category of positive chain complexes of $\mc{A}$, through the functor $K:\simp\mc{A}\rightarrow Ch_+\mc{A}$ that takes as boundary map the alternate sum of the face maps in a simplicial object. The same holds for the normalized version of $K$, which is an equivalence of categories \cite[$\S 22$]{May}.
\textbf{Filtered cochain complexes:} Given an abelian category $\mc{A}$ and an integer $c$, let $\mrm{CF}^c\mc{A}$ be the category of filtered cochain complexes $(A,\mrm{F})$, where
$A$ is a cochain complex over $\mc{A}$ that is equal to 0 in degrees lower than $c$, and $\mrm{F}$ is a decreasing biregular filtration of $A$.\\
The normalized simple $(\mbf{s}_N,\mbf{s}_N)$ comes from the one of cochain complexes, and $\mrm{E}$=filtered quasi-isomorphisms are part of an additive simplicial descent structure on $\mrm{CF}^c\mc{A}$ (see \cite{R1} for more detail). The deduced path object of the filtered maps $(C,\mrm{H})\stackrel{g}{\rightarrow}(A,\mrm{F})\stackrel{f}{\leftarrow}(B,\mrm{G})$ is the filtered cochain complex $(path(f,g),\mrm{M})$, where $path(f,g)^n=B^n\oplus A^{n-1}\oplus C^n$ and
$$\mrm{M}^kpath(f,g)^n=\mrm{G}^kB^n\oplus \mrm{F}^kA^{n-1}\oplus \mrm{H}^kC^n\ .$$
As a corollary we get the classical left triangulated (or cosuspended) structure on the filtered derived category of uniformly bounded-below complexes. Note that the category of (arbitrarily) bounded-below complexes $\mrm{CF}^b\mc{A}$ is the union of the $\mrm{CF}^c\mc{A}$ as the integer $c$ varies. The suspended structures on the filtered derived categories $\mrm{CF}^c\mc{A}$ are compatible. Therefore they induce a Verdier's triangulated structure on the usual filtered derived category $\mrm{CF}^b\mc{A}[\mrm{E}^{-1}]$.\\
We can also consider another additive simplicial descent structure on $\mrm{CF}^c\mc{A}$ \cite{R1}. The equivalences considered now are the $E_2$-isomorphisms, that is, those maps inducing isomorphism on the second term of the associated spectral sequences. In this case the simple functor is $(\mbf{s}_N,\delta)$, where $\delta$ denotes the diagonal filtration. The path object of $f$ and $g$ is $(path(f,g),\mrm{N})$, where $path(f,g)^n=B^n\oplus A^{n-1}\oplus C^n$ as before, but
$$\mrm{N}^kpath(f,g)^n=\mrm{G}^kB^n\oplus \mrm{F}^{k-1}A^{n-1}\oplus \mrm{H}^kC^n\ .$$
We get a cosuspended structure on the localized category of $\mrm{CF}^c\mc{A}$ with respect to the $E_2$-isomorphisms. Therefore, the localized category of $\mrm{CF}^c\mc{A}[E_2^{-1}]$ is a triangulated category. In addition, we deduce that Deligne's decalage functor $\mrm{Dec}:\mrm{CF}^c\mc{A}[\mrm{E}^{-1}]\rightarrow\mrm{CF}^c\mc{A}[E_2^{-1}]$ preserves this structures, since it is a functor of additive simplicial descent categories.
As before, we can induce a Verdier's triangulated structure on $\mrm{CF}^b[E_2^{-1}]$. Then, $\mrm{Dec}:\mrm{CF}^b\mc{A}[\mrm{E}^{-1}]\rightarrow\mrm{CF}^b\mc{A}[E_2^{-1}]$ is a functor of triangulated categories.
\end{ejs}
\section{Cogroup structures in simplicial descent categories.}
In this section we deal with a simplicial descent category $\mc{D}$ such that $Ho\mc{D}$ is pointed, that is,
the map $0\rightarrow \ast \in\mrm{E}$. In this case the following proposition holds, similarly to the case of pointed topological spaces, or more generally, of pointed model categories.
\begin{defi} Denote by $\simp {}_f Set$ the category of simplicial finite sets, and by $\simp{}_fSet_\ast$ the one of pointed simplicial finite sets. The action $\boxtimes:\simp {}_f Set\times\simp\mc{D}\rightarrow\simp\mc{D}$ induces an action $\otimes:\simp{}_f Set_\ast\times\simp\mc{D}\rightarrow\simp\mc{D}$ \cite{Vo}. Given $K\in\simp {}_f Set_\ast$ and $X\in\simp\mc{D}$, then $K\otimes X$ is the pushout of the maps $\Dl[0]\boxtimes X\rightarrow \ast$ and $\Dl[0]\boxtimes X\rightarrow K\boxtimes X$, where $\Dl[0]\rightarrow K$ is the distinguished point of $K$.\\
Recall that the coproduct in $\simp{}_fSet_\ast$ of $K$ and $L$ is $K\vee L$, the quotient of $K\sqcup L$ by the set of the two base points of $K$ and $L$.
Then the natural map
$$(K\otimes X)\sqcup(L\otimes X)\rightarrow (K\vee L)\otimes X\ ,$$
is a degree-wise equivalence (since $\ast\sqcup \ast\rightarrow \ast$ is in $\mrm{E}$).\\
If we choose $d^1(\Dl[0])$ as the base point of $\Dl[1]$, then $X\otimes\Dl[1]$ is just $CX$, the cone of $X$. Also, if $A$ is in $\mc{D}$ then $\Sigma A$ is canonically isomorphic to $\mbf{s}(A\otimes S^1)$ in $Ho\mc{D}$, where $S^1= \Dl[1]/\mrm{sk}^0 \Dl[1]$ is the simplicial circle.
Indeed, $\ast\sqcup \ast\rightarrow\ast$ induces $\nu: \Lambda A= C(A\rightarrow\ast)\rightarrow A\otimes S^1$, so $\mbf{s}\nu :\Sigma A\rightarrow \mbf{s}(A\otimes S^1)$ is in $\mrm{E}$.
If $K\in\simp {}_f Set_\ast$, define $K\wedge \Dl[1]$ as the quotient of $K\times\Dl[1]$ by $\Dl[0]\times \Dl[1]$, where $\Dl[0]\subseteq K$ is the base-point of $K$. Then, $K\wedge\Dl\in\simp {}_f Set_\ast$ with the obvious base-point, and we have maps $d^0,d^1:K\rightarrow K\wedge\Dl[1]$ in $\simp {}_f Set_\ast$ induced by $d^0,d^1,\Dl[0]\rightarrow\Dl[1]$. Two maps $f,g:K\rightarrow L$ are \textit{simplicially homotopic} if there exists $H:K\wedge\Dl[1]\rightarrow L$ with $Hd^0=f$ and $Hd^1=g$.
\end{defi}
\begin{lema} If $f,g:K\rightarrow L$ are simplicially homotopic in $\simp {}_f Set_\ast$ and $A\in\mc{D}$, then $\mbf{s}(f\otimes A)=\mbf{s}(g\otimes A)$ in $Ho\mc{D}$.
\end{lema}
\begin{proof} Since $Ho\mc{D}$ is pointed, the natural map $Cyl(A\otimes K)\rightarrow A\otimes (K\wedge \Dl[1])$ is a degree-wise equivalence, and the statement follows from lemma \ref{relHomotopia}.
\end{proof}
\begin{prop}\label{A1cogrupo}
The correspondence $A\rightarrow Hom_{Ho\mc{D}}(\Sigma A,-)$ is a functor from $Ho\mc{D}$ to $Groups$. That is, $\Sigma A$ is a cogroup object in $Ho\mc{D}$. In addition, $\Sigma^k A$ is an abelian cogroup object if $k\geq 2$.
If $f:A\rightarrow B$ is any map in $\mc{D}$, then $\Sigma A$ coacts on $c(f)$.
\end{prop}
\begin{proof}
Define $\Omega\in\simp{}_f Set$ as the pushout
$$ \xymatrix@M=4pt@H=4pt@R=14pt@C=17pt{
\Dl[0]\sqcup\Dl[0] \ar[r]^-{d^0\sqcup d^1}\ar[d]_{d^1\sqcup d^0} & \Dl[1]\ar[d]^p \\
\Dl[1]\ar[r]^-q & \Omega}$$
Let $\Dl[0]\stackrel{d^1}{\rightarrow} \Dl[1]\stackrel{q}{\rightarrow} \Omega$ be the base-point of $\Omega$. Consider the map $\alpha:\Omega\rightarrow S^1$ such that $\alpha p$ is the projection $P:\Dl[1]\rightarrow S^1=\Dl[1]/\mrm{sk}^0\Dl[1]$, and $\alpha q$ is the trivial map factoring through $\Dl[0]$. If $A\in\mc{D}$, then $\Omega\otimes A$ agrees with $Cyl(CA\leftarrow A\rightarrow \ast)$ after identifying $\ast\sqcup \ast$ with $\ast$. The map $\alpha\otimes A$ sends $CA$ to $\ast$, so $\mbf{s}(\alpha\otimes A):\mbf{s}(\Omega\otimes A)\rightarrow\mbf{s}(S^1\otimes A)\equiv\Sigma A$ is an equivalence.\\
On the other hand, let $\pi:\Omega\rightarrow S^1\vee S^1$ be the map with $\pi q =i_1 P$ and $\pi p= i_2 P $, where $i_1,i_2:S^1\rightarrow S^1\vee S^1$ are the canonical inclusions. If $A\in\mc{D}$, define the map $d_A:\Sigma A\rightarrow \Sigma A\sqcup \Sigma A$ of $Ho\mc{D}$ through the composition
$$ \xymatrix@C=30pt{
\mbf{s}(S^1\otimes A)\ar[r]^-{\mbf{s}(\alpha\otimes A)^{-1}} & \mbf{s}\Omega\ar[r]^-{\mbf{s}(\pi\otimes A)} & \mbf{s}((S^1\vee S^1)\otimes A)}$$
Recall that $d_A$ endows $Hom_{Ho\mc{D}}(\Sigma A,-)$ with a group structure if and only if $d_A$ is associative, it has unit and inverse element.
Let us prove that $d_A$ has unit element, that is, that $\pi_1 d_A,\pi_2 d_A:\Sigma A\rightarrow \Sigma A$ are the identity in $Ho\mc{D}$.\\
Clearly $\pi_2 \pi =\alpha :\Omega\rightarrow S^1$, so $\pi_2 d_A=Id$. We state that $\pi_1\pi$ is simplicially homotopic to $\alpha$ in $\simp{}_f Set_\ast$. Indeed, consider the maps $r,\widetilde{r}:\Dl[1]\otimes\Dl[1]\rightarrow\Dl[1]$ in $\simp{}_f Set$ given by $r(f,g)(i)=\mrm{max}\{f(i),g(i)\}$ and $\widetilde{r}(f,g)(i)=\mrm{min}\{f(i),g(i)\}$.
Then $r(d^0\times Id)$, $r(Id\times d^0)$, $\widetilde{r}(d^1\times Id)$ and $\widetilde{r}(Id\times d^1)$ are equal to $Id_{\Dl[1]}$. Also, $r(d^1\times Id)=r(Id\times d^1)$ factors through $d^1:\Dl[0]\rightarrow\Dl[1]$, while $\widetilde{r}(d^0\times Id)=\widetilde{r}(Id\times d^0)$ factors through $d^0$.\\
Let $H:\Omega\times\Dl[1]\rightarrow S^1$ be the map such that $H(q\times Id)=P r$ and $H(p\times Id)=P\widetilde{r}$. If $\ast\subset \Omega$ is the distinguished point of $\Omega$, then $H(\ast\times \Dl[1])$ is the base-point of $S^1$. Therefore, $H$ defines $\overline{H}:\Omega\wedge \Dl[1]\rightarrow S^1$, that is a homotopy between $\pi_1\pi$ and $\alpha$. It follows that $\pi_1 d_A=Id$ in $Ho\mc{D}$.\\
We will see next that the map $m_A:\Sigma A \rightarrow \Sigma A$ given in definition \ref{signoMenos} is indeed an inverse element for $d_A$. That is, the following composition factors through $\ast$ in $Ho\mc{D}$
\begin{equation}\label{mNeutro} \xymatrix@M=4pt@H=4pt@R=14pt@C=18pt{
\Sigma A\ar[r]^-{d_A}& \Sigma A\sqcup \Sigma A\ar[r]^-{Id\sqcup m_A}& \Sigma A\sqcup \Sigma A \ar[r]^-{\delta_{\Sigma A}} & \Sigma A}\end{equation}
where $\delta_{\Sigma A}$ is the codiagonal. Set $\Omega A=Cyl(CA\leftarrow A \rightarrow \ast)\simeq \Omega\otimes A$, $\pi_A:\mbf{s}\Omega A\rightarrow \Sigma A\sqcup \Sigma A$ the map induced by $\mbf{s}(\pi\otimes A)$. Denote $\delta_{\Sigma A}(Id\sqcup m_A)$ by $(Id,m_A)$. By definition, the composition (\ref{mNeutro}) factors through $\ast$ if and only if $(Id,m_A)\pi_A:\mbf{s}\Omega A\rightarrow \Sigma A$ does. We will see that $(Id,m_A)\pi_A$ is isomorphic to the composition of consecutive maps in a cofiber sequence, so it is the trivial map.\\
Consider the maps $\delta'=(d^0, d^1):A\sqcup A\rightarrow Cyl(A)$ and $s^0:Cyl(A)\rightarrow A$. By lemma \ref{CilindroIterado}, $C(\delta')$ is isomorphic to $Cyl(CA\leftarrow A\rightarrow CA)$. Therefore, sending the $CA$ on the right to $\ast$ produces $f:C(\delta')\rightarrow \Omega A$ with $\mbf{s}f\in\mrm{E}$.
Consider the following diagram
$$ \xymatrix@M=4pt@H=4pt@R=12pt@C=23pt{
\mbf{s}C(\delta') \ar[d]^{\mbf{s}f} \ar[r]^-p & (\Sigma A, \Sigma A) \ar[r]^-{\Sigma d^0\sqcup \Sigma d^1} \ar[d]_{Id\sqcup m_A} &\Sigma cyl(A)\ar[d]^{\Sigma s^0} \\
\mbf{s}\Omega A\ar[r]^-{\pi_A} & \Sigma A\sqcup \Sigma A \ar[r]^-{(Id,m_A)} & \Sigma A}$$
The right square commutes since $m_A^2=Id_{\Sigma A}$ and $(\Sigma d^0, \Sigma d^1)\Sigma s^0$ is the codiagonal $\delta_{\Sigma A}$. The left square is commutative as well. Indeed, $m_A=p_1^{-1}p_2=p_2^{-1}p_1$, where $p_1,p_2:\Sigma^1_2 A\rightarrow \Sigma A$ are the maps defined in \ref{signoMenos}.
There is a map $h:\mbf{s}C(\delta')\rightarrow \Sigma A \sqcup \Sigma^1_2 A \simeq\mbf{s}Cyl(\Lambda A\leftarrow A\rightarrow CA)$ induced by $CA\rightarrow \Lambda A$. It follows from the definitions that $p_2h=p$ and $p_1h=\pi_Af$. Therefore $(Id\sqcup m_A)p=\pi_A \mbf{s}f$, and $d_A$ has inverse element.\\
It remains to check that $d_A$ is associative, that is, that $(d_A\sqcup Id)d_A=(Id_\sqcup d_A)d_A$ in $Ho\mc{D}$. It can be proved that $(Id\sqcup Id\sqcup m)(d_A\sqcup Id)d_A$ and $(Id\sqcup Id\sqcup m)(Id\sqcup d_A)d_A$ are both equal to the composition $\Sigma A\stackrel{\beta^{-1}}{\rightarrow} \mbf{s}C(\delta')\stackrel{\sigma}{\rightarrow} \Sigma A\sqcup \Sigma A\sqcup \Sigma A$, where $\beta$ sends both $CA$ in $C(\delta')$ to $\ast$, and $\sigma$ sends them to $\Lambda A$. The details are left to the reader.\\
It follows formally from the properties of $d_A$ that $\Sigma^2 A$ is an abelian cogroup object. Denote by $\tau_B:B\sqcup B\rightarrow B\sqcup B$ the isomorphism that interchanges the factors. Note that $d_{\Sigma A}\equiv\Sigma d_A:\Sigma^2A\rightarrow \Sigma^2 A\sqcup \Sigma^2 A$, since $Cyl(\Lambda f,\Lambda g)\equiv\Lambda Cyl(f,g)$ by lemma \ref{CilindroIterado}. Then, $\tau _{\Sigma^2 A} d_{\Sigma A}\equiv \Sigma (\tau_{\Sigma A}d_A)$. On the other hand,
$(\Sigma \pi_1\sqcup \Sigma \pi_2) d_{\Sigma A\sqcup \Sigma A}\equiv(\pi_1\sqcup \pi_2)(d_{\Sigma A}\sqcup d_{\Sigma A})=Id$. Then
$$\Sigma (\tau_{\Sigma A}d_A)=(\Sigma \pi_1\sqcup \Sigma \pi_2) d_{\Sigma A\sqcup \Sigma A} \Sigma (\tau_{\Sigma A}d_A)= (\Sigma \pi_1\sqcup \Sigma \pi_2)(\Sigma (\tau_{\Sigma A}d_A)\sqcup \Sigma (\tau_{\Sigma A}d_A))d_{\Sigma A}$$
The last equality holds since $d_{-}$ is a natural transformation between the functors $\Sigma$, $\Sigma \sqcup \Sigma:Ho\mc{D}\rightarrow Ho\mc{D}$. But $(\Sigma \pi_1\sqcup \Sigma \pi_2)(\Sigma (\tau_{\Sigma A}d_A)\sqcup \Sigma (\tau_{\Sigma A}d_A))=\Sigma (\pi_2 d_A)\sqcup \Sigma (\pi_1 d_A)=Id$. Therefore $d_{\Sigma A}=\tau_{\Sigma^2 A}d_{\Sigma A}$, and $\Sigma^2 A$ is abelian.\\
To finish, there is a natural coaction $w_A:cA=\mbf{s}(\Dl[1]\otimes A)\rightarrow \Sigma A \sqcup cA \simeq \mbf{s}(( S^1 \vee \Dl[1])\otimes A)$ described as follows. Consider $\overline{\Omega}\in\simp{}_f Set_\ast$ given by the pushout
$$ \xymatrix@M=4pt@H=4pt@R=14pt@C=17pt{
\Dl[0] \ar[r]^-{ d^1}\ar[d]_{ d^0} & \Dl[1]\ar[d]^{\overline{p}} \\
\Dl[1]\ar[r]^-{\overline{q}} & {\overline{\Omega}} {} }$$
and with base-point $\Dl[0]\stackrel{d^1}{\rightarrow} \Dl[1]\stackrel{\overline{q}}{\rightarrow} \overline{\Omega}$. We have maps $\overline{\alpha}:\overline{\Omega}\rightarrow \Dl[1]$ and $\overline{\pi}:\Omega\rightarrow S^1\vee\Dl[1]$ with $\overline{\alpha}\,\overline{p}=Id$, $\overline{\alpha}\,\overline{q}=d^1s^0$, $\overline{\pi}\,\overline{p}=i_2$, $\overline{\pi}\,\overline{q}=i_1 P$. As before, $\mbf{s}(\overline{\alpha}\otimes A)\in\mrm{E}$. Indeed, $\mbf{s}(\overline{\Omega}\otimes A)\simeq\mbf{s}Cyl(CA\leftarrow A\rightarrow A)$ and $\mbf{s}(\overline{\alpha}\otimes A)$ comes from $CA\rightarrow \ast$. Then, $w_A$ is defined as $\mbf{s}(\overline{\pi}\otimes A)\mbf{s}(\overline{\alpha}\otimes A)^{-1}$. By definition $\pi_2 w_A=Id_{cA}$, while the proof of the equality $(Id\sqcup w_A)w_A=(d_A\sqcup Id)w_A$ is analogous to the one of the associativity of $d_A$.
If $f:A\rightarrow B$, then $\omega_A$ extends to a coaction $w_f: c(f)\rightarrow \Sigma A \sqcup c(f)$, since $C(f)$ is the pushout in $\simp\mc{D}$ of $A\rightarrow CA$ along $f:A\rightarrow B$.
\end{proof}
\begin{obs} In \cite[p. 75]{GZ}, the author proves that $S^1$
is a cogroup object in the category $\simp Set_\ast$ modulo homotopy, localized by anodyne extensions. The corresponding map $\varphi:S^1\rightarrow S^1\vee S^1$ is described by means of $\Lambda^1[2]\hookrightarrow\Dl[2]$. We give here an alternative description of $\varphi$, better suited for our class of equivalences.
\end{obs}
\begin{obs} We can redefine cofiber sequences (\ref{distingTriangl}), by forgetting $p:c(f)\rightarrow \Sigma A$ and considering instead the previous coaction of $\Sigma A$ over $c(f)$. It can be proved that the reformulations of TR1 to TR4 by means of coactions still hold for this new cofiber sequences, but we do not go into this task here.
\end{obs}
\section{Stable simplicial descent categories.}
\begin{defi} A simplicial descent category $(\mc{D},\mrm{E},\mbf{s},\mu,\lambda)$ is called \textit{stable} if the induced suspension functor $\Sigma:Ho\mc{D}\rightarrow Ho\mc{D}$ is an equivalence of categories.
\end{defi}
\begin{prop} If $\mc{D}$ is a stable simplicial descent category then $Ho\mc{D}$ is an additive category.
\end{prop}
\begin{proof} Let us see first that stability implies that $Ho\mc{D}$ is pointed. Note that the initial and final objects of $\mc{D}$ are so in $Ho\mc{D}$ by proposition \ref{MorfHoD}. As $\Sigma$ is an equivalence of categories, then $\Sigma 0$ should be isomorphic to $0$. But $\Sigma 0$ is $\ast\sqcup \ast$ by definition. Composing with an inclusion $\ast\rightarrow \ast \sqcup\ast$ we get a map $\ast\rightarrow 0$ in $Ho\mc{D}$. As compositions $0\rightarrow \ast\rightarrow 0$ and $\ast\rightarrow 0\rightarrow\ast$ are identities then $0\equiv\ast$ in $Ho\mc{D}$.\\
Since $\Sigma$ is an equivalence, it follows from proposition \ref{A1cogrupo} that each object in $Ho\mc{D}$ is an (abelian) cogroup object in a natural way. But this formally implies that $Ho\mc{D}$ is additive. Indeed, the sum of two morphisms $\Sigma f,\Sigma g:\Sigma A\rightarrow \Sigma B$ is
$$\Sigma A \stackrel{d_A}{\rightarrow} \Sigma A\sqcup\Sigma A\stackrel{\Sigma f\sqcup \Sigma g}{\longrightarrow} \Sigma B\sqcup\Sigma B\stackrel{\varsigma}{\rightarrow} \Sigma B$$
where $\varsigma$ denotes the codiagonal of $\Sigma B$.
\end{proof}
The previous proposition and theorem \ref{struct triangulada} give rise to the following result.
\begin{thm}\label{StablTriang} If $\mc{D}$ is a stable simplicial descent category then $Ho\mc{D}$ is a triangulated category. In addition, a descent functor $F:\mc{D}\rightarrow \mc{D}'$ induces a functor of triangulated categories $F:Ho\mc{D}\rightarrow Ho\mc{D}'$.
\end{thm}
\begin{cor}\label{cORStablTriang} Let $\mc{D}$ be a stable simplicial descent category such that $\Sigma:\mc{D}\rightarrow \mc{D}$ is an equivalence of categories. Then, if $I$ is a small category, the category of diagrams $(I,\mc{D})$ is a stable simplicial descent category. In particular, $Ho(I,\mc{D})$ is a Verdier's triangulated category, and any functor $f:I\rightarrow J$ induces a triangulated functor $f^\ast: Ho(J,\mc{D})\rightarrow Ho(I,\mc{D})$.
\end{cor}
\begin{proof} If $\mc{D}$ is a simplicial descent category, then $(I,\mc{D})$ inherits a simplicial descent category defined object-wise. In addition, a functor $f$ as above induces a descent functor $f^\ast$. By assumption, $\Sigma^{-1}:\mc{D}\rightarrow\mc{D}$ provides ${\Sigma}^{-1}:(I,\mc{D})\rightarrow (I,\mc{D})$ preserving object-wise equivalences, for all $I$. Therefore, the induced functor $Ho(I,\mc{D})\rightarrow Ho(I,\mc{D})$ is an inverse of $\Sigma:Ho(I,\mc{D})\rightarrow Ho(I,\mc{D})$. Hence $(I,\mc{D})$ is stable of all $I$.
\end{proof}
The above corollary remains valid for a simplicial descent category $\mc{D}$ such that $\Sigma_{I}:Ho(I,\mc{D})\rightarrow Ho(I,\mc{D})$ is an equivalence of categories for all $I$. This holds, for instance, in the following case. Assume the quasi-inverse of $\Sigma$, $\Sigma^{-1}:Ho\mc{D}\rightarrow Ho\mc{D}$ comes from a functor $\underline{\Sigma}:\mc{D}\rightarrow\mc{D}$. Note that $\underline{\Sigma}$ may not be a quasi-inverse of $\Sigma:\mc{D}\rightarrow\mc{D}$. Moreover, assume that the isomorphisms $\alpha:\Sigma\Sigma^{-1}\simeq Id_{Ho\mc{D}}$ and $\beta:\Sigma\Sigma^{-1}\simeq Id_{Ho\mc{D}}$ are isomorphisms of $Fun(\mc{D},\mc{D})[\mrm{E}^{-1}]$. In this case, $\Sigma_I$ is an equivalence of categories as well.
\begin{ejs}\mbox{}\\
\indent \textbf{DG-modules over a DG-category} The cosimplicial descent structure on the category of DG-modules over a fixed DG-category is based on the one of cochain complexes. We then recover the well-known triangulated structure of the derived category of DG-modules over a DG-category \cite{K}.
\textbf{Mixed Hodge complexes} Let $\mc{H}dg$ be the category of mixed Hodge complexes defined in \cite[4.8 and 4.11]{R1}. Consider mixed Hodge complexes
$K=((K_\mathbb{Q},\mrm{W}),(K_\mathbb{C},\mrm{W},\mrm{F}),\alpha)$, $S=((S_\mathbb{Q},\mrm{U}),(S_\mathbb{C},\mrm{U},\mrm{G}),\beta)$ and $T=((T_\mathbb{Q},\mrm{V}),(T_\mathbb{C},\mrm{V},\mrm{H}),\gamma)$. Given morphisms $T\stackrel{g}{\rightarrow}K\stackrel{f}{\leftarrow}S$, then $P=path(f,g)$ is the mixed Hodge complex $((P_\mathbb{Q},\mrm{N}),(P_\mathbb{C},\mrm{N},\mrm{M}),\delta)$ where
$$P^n_\ast=S^n_\ast\oplus K^{n-1}_\ast\oplus T^n_\ast \ \ \ \ \mrm{N}_kP^n_\ast=\mrm{U}_k S^n_\ast\oplus \mrm{W}_{k+1}K_\ast^{n-1}\oplus \mrm{V}_kT^n_\ast\mbox{ for }\ast=\mathbb{Q},\mathbb{C}$$
$$\mrm{M}^kP^n_\mathbb{C}=\mrm{G}^k S^n_\mathbb{C}\oplus \mrm{F}^{k}K_\mathbb{C}^{n-1}\oplus \mrm{H}^kT^n_\mathbb{C}$$
and $\delta$ is the direct sum of $\beta$, $\alpha$ and $\gamma$.\\
The cofiber sequences defined through the $path$ functor induce a left triangulated structure on the derived category $\mc{H}dg[\mrm{E}^{-1}]$, $\mrm{E}$=quasi-isomorphisms. As in the case of filtered complexes, it becomes a Verdier's triangulated category if we consider bounded-below cochain complexes. This triangulated structure is related to the one given in \cite{Be} and \cite{H} (recall that a mixed Hodge complex in our sense becomes a mixed Hodge complex in the sense of loc. cit. after applying decalage functor $Dec$ to the weight filtration $\mrm{W}$).
\textbf{Fibrant spectra} The category $Sp$ of fibrant spectra has a structure of cosimplicial descent category where the simple functor is the homotopy limit. In the proof of \cite[proposition 5.14]{R1} it is checked that the resulting fiber sequences are the usual `homotopy fiber sequences' coming from the Quillen model structure on $Sp$. Therefore, we deduce the classical triangulated structure of the stable homotopy category of fibrant spectra.
\end{ejs}
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Wednesday, December 2, 200912 noon - 1 p.m.
Ethics RoundsResearch on the DyingDaniel P. Sulmasy, OFM, MD, PhDKilbride-Clinton Professor of Medicine and Ethics,Department of Medicine and the Divinity SchoolAssociate Director, MacLean Center for Clinical Medical EthicsUniversity of Chicago
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Contemporary Clinical Medicine: Great TeachersTreatment of Parkinson’s DiseaseJoseph Jankovic, MDProfessor of Neurology and Distinguished Chair in Movement DisordersDirector, Parkinson’s Disease Center and Movement Disorders Clinic, Department of NeurologyBaylor College of Medicine
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Ninth Annual John Doppman Memorial Lecture for Imaging SciencesCTA and 3-D Visualization: Its Evolving Role in Oncologic ImagingElliot Fishman, MDProfessor of Radiology and Oncology, Johns Hopkins University School of MedicineDirector, Division of Diagnostic ImagingDirector, Division of Abdominal Imaging and Computed Body TomographyThe Johns Hopkins Hospital
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\begin{document}
\title[CAL and RG operators for classifiers]{Categorical Representation Learning and RG flow operators for algorithmic classifiers}
\maketitle
\begin{center}
\author{}{Artan Sheshmani${}^{1,2,3, 5,6}$ and Yi-Zhuang You${}^{4,5}$ and Wenbo Fu${}^{5}$ and Ahmadreza Azizi${}^{5}$}
\end{center}
\address{${}^1$ Center for Mathematical Sciences and\\ Applications, Harvard University, Department of Mathematics, and Harvard University Physics department, Jefferson Laboratory, 17 Oxford St, Cambridge, MA 02138}
\address{${}^2$ IMSA, University of Miami}
\address{${}^3$ National Research University Higher School of Economics, Russian Federation, Laboratory of Mirror Symmetry, NRU HSE, 6 Usacheva str.,Moscow, Russia, 119048}
\address{${}^4$ University of California San Diego, Department of Physics, Condensed matter group, UC San Diego 9500 Gilman Dr. La Jolla, CA 92093}
\address{${}^5$ QGNai INC. (Quantum Geometric networks for artificial intelligence), 83 Cambridge Parkway, Unit W806. Cambridge, MA, 02142}
\address{${}^6$ NSF AI Institute for Artificial Intelligence and Fundamental Interactions}
\date{\today}
\begin{abstract}Following the earlier formalism of the categorical representation learning \cite{categorifier} by the first two authors, we discuss the construction of the ``RG-flow based categorifier". Borrowing ideas from theory of renormalization group flows (RG) in quantum field theory, holographic duality, and hyperbolic geometry, and mixing them with neural ODE's, we construct a new algorithmic natural language processing (NLP) architecture, called the RG-flow categorifier or for short the RG categorifier, which is capable of data classification and generation in all layers. We apply our algorithmic platform to biomedical data sets and show its performance in the field of sequence-to-function mapping. In particular we apply the RG categorifier to particular genomic sequences of flu viruses and show how our technology is capable of extracting the information from given genomic sequences, find their hidden symmetries and dominant features, classify them and use the trained data to make stochastic prediction of new plausible generated sequences associated with new set of viruses which could avoid the human immune system. The content of the current article is part of the recent US patent application submitted by first two authors (U.S. Patent Application No.: 63/313.504).
\smallskip
\noindent{\bf MSC codes:} 03B70, 03-04, 03D10, 11Y16
\noindent{\bf Keywords:} Renormalization Group Flow, Neural ODE, Hyperbolic geometry, Holographic duality, Category theory, Categorical representation learning, Natural language processing (NLP), Genomic sequence classification and generation.
\end{abstract}
\tableofcontents
\section{Introduction}
The renormalization group (RG) \cite{Polchinski1984Renormalization} is a powerful and useful set of methods developed in statistical physics and quantum field theory to deal with many-body problems. It helps the physicists to establish the connection between the microscopic laws of physics and the macroscopic collective behaviors of the system. It starts with a many-body system at the microscopic scale, and then performs the coarse graining iteratively to group the fundamental building blocks together into larger and larger clusters. Meanwhile, it constructs the effective descriptions of the clusters at each different scale and extracts the effective interaction among them. At the end, the many-body system can be reduced to a few-body system at the highest scales, which enables the understanding of complex systems and their collective behaviors at large scale.
This idea can be particularly useful for representation learning and classification tasks in machine learning. There are many examples of many-body systems in machine learning tasks. For instance, an image can be viewed as a system of many pixels, and a sequence can be viewed as a system of many tokens. It is desired to see whether the idea of renormalization group can also be applied to extract the overall representations of images and sequences from their microscopic representations.
In terms of mathematics, the existence of profound connections between quantum field theory and geometry/ topology has been a source of many exciting research activities. One of them, as an example, is the interesting connection between theory of RG flows, for a particular set of quantum field theories in physics, and the geometric theory of Ricci flows in mathematics. The theory of Ricci flows was developed by Richard Hamilton in the 80's \cite{Hamilton1, Hamilton2, Hamilton3, Hamilton4, Hamilton5}. Given a smooth manifold, $M$, a Riemannian metric, $g$, on $M$ defines a bilinear positive-definite product on tangent space, $T_{p}M$, for each point $p\in M$. This bilinear form is a 2-tensor which locally in an open neighborhood $U\subset M$ of $p$, will have a matrix representation. One can then investigate whether infinitesimal deformations of the metric on $M$ would provide interesting information about its geometry or topology. For instance given a 1-parameter family, $g_{t}, t\in (a,b)$ of metrics on $M$, one can study the variation of $g$ with respect to the parameter $t$. The derivative $\displaystyle{\frac{\partial g_{t}}{\partial t}}$ will then provide for every fixed choice of $t$ and fixed point $p$ a bilinear inner product form (i.e. a 2-tensor) on $T_{p}M$. It turns out that variation of metric in a 1-parameter family provides one with a differential equation $$\displaystyle{\frac{\partial g_{t}}{\partial t}}=-2\text{Ric}^{g_{t}},$$ where the term on the right hand side is the Ricci curvature tensor, $\text{Ric}^{g_{t}}$, named after Gregorio Ricci-Curbastro, measuring that, how for each fixed choice of $g_{t}$, the geometry of space is curved as one moves along the geodesics on the manifold $M$. \\
The connection between RG flows for nonlinear sigma models in physics and the Ricci flow for Riemannian manifolds in mathematics is quite known for a while, since the earlier work of Daniel Freidan \cite{Freidan}, Zamalodchikov \cite{Zomolodchikov}, Tseytlin \cite{Tseytlin}, as well as ground breaking work of Gregory Perelman \cite{Perelman} in proof of Poincare conjecture, and more recently Carfora \cite{Carfora}. In he next section we briefly provide an expository account of RG flows in the context of Ricci geometry following the work of Carfora \cite{Carfora}. It must be noted that our focus in the current article is to implement RG flows for developing algorithmic architectures in mathematical artificial intelligence, therefore later we quickly diverge from its connection to Ricci geometry, and focus solely on RG networks. We encourage the interested readers to study the resources provided above to gain a deeper understanding of the connections between the two frameworks in physics and mathematics.\\
When it comes to implementing RG flow theory in machine learning, the key challenge lies in the difficulty in constructing the coarse graining transformation at each RG step. In physics, the RG rules are usually specified by human, such as the majority vote in real-space RG or the momentum-shell integration in field theoretic RG. These intuitions may not be immediately applicable to realistic dataset of images and sequences, as the underlying coarse graining rules may be much more complicated compared to physics systems. This calls for machine learning methods to enable algorithm to design and optimize the RG transformation in adaptation to the given dataset. One important idea is borrowed from the holographic duality in physics, which states that the RG transformation can be viewed as a holographic mapping of a field configuration from a flat (boundary) space to a hyperbolic (bulk) space with one-higher dimension, such that the long-range correlation in the original field configuration can be equivalently represented as short-range correlation in the bulk space. So the optimal RG transformation can be defined as a bijective holographic mapping that disentangles the features at different hierarchies as much as possible. This allows us to embed the bijective holographic map in the flow-based generative model, and use the unsupervised machine learning technique to train the optimal RG transformation. This idea is first proposed in Ref.\,\cite{Li2018Neural} and further developed in later works \cite{Hu2020Machine,Hu2020RG-Flow}. The current article further develops the machine-learning RG method by combining the RG-flow model with neural ODE techniques \cite{Chen2018Neural}, and explores its application to representation learning of sequential data.
\section*{Acknowledgements}
The first author would like to aknowledge support by National Science Foundation SBRI grant No: 2109928, as well as support by the National Science Foundation under Cooperative Agreement PHY-2019786 (the NSF AI Institute for Artificial Intelligence and Fundamental Interactions, http://iaifi.org/). The second author was supported by a startup fund provided by UCSD and the UC Hellman fellowship. We acknowledge the discussion with Hong-Ye Hu.
\section{Analytic construction of RG flow operator on moduli space of smooth maps}
\subsection{A special example of our construction, using Laplacians and curvature form}
A large part of current section is based on work of Carfora \cite{Carfora} in relating RG flows for a specific set of QFT's and the Ricci flow construction for Riemannian manifolds. Moreover, the content in this section owes its existence to another highly recommended source, specially for a working mathematician, that is the outstanding work of Kevin Costello \cite{Costello} in his mathematical formulation of perturbative quantum field theory.
For the time being, we use the introduction to a geometric construction of the RG flow, outlined below, as it is suitably intuitive, pleasantly elegant, and mainly since it will provide us later with the shortest pathways to generalize our constructions in several ways, for instance: by deviating from the classical setup, via altering (and generalizing) our action integrals made with Laplacians and curvature forms to more general actions, or by altering the base geometrical spaces from smooth manifolds to non-smooth algebraic varieties, or discrete lattices. \\
Let $C, X$ denote respectively a compact oriented Riemann surface and a compact oriented smooth manifold of dimension at least 2, both equipped with a Riemannian metric, and defined over a base number field $\mathbb{K}$. Let $\text{Map}(C, X):=\{f: C\to X\}$ be the associated space of all continuous maps from domain $C$ to $X$. The construction of RG flow is based on considering a family of Lagrangians $\mathcal{L}(f, \phi_{i}, i=1,\cdots, n)$ associated to this space, defined as a morphism $$\mathcal{L}(f, \phi_{i}, i=1,\cdots, n):= \text{Map}(C, X)\times H^{*}(X, \mathbb{K})^{\otimes n} \to C^{\infty}(C,X)$$ taking a tuple of fields $(f, \phi_{1},\cdots, \phi_{n})$ on $X$ to the space of smooth integrable functions on $X$. Note that here the notatation $H^{*}(X, \mathbb{K})^{\otimes n}$ means that the fields $\phi_{i}, i=1, \cdots, n$ are realized as sections of a sheaf of differentially graded algebras over $X$, sitting in appropriate cohomological degrees on $X$. Moreover, we require the Lagrangians to be invariant under the action of diffeomorphism groups, $\mathcal{D}\textit{iff}(C), \mathcal{D}\textit{iff}(X)$ on $C$ and $X$ respectively.\\
Integrating the Lagrangian over the associated domain Riemann surface induces the Lagrangian action integral $$\int_{C}\mathcal{L}(f, \phi_{i}, i=1,\cdots, n).$$Let the metric tensors on $C$ and $X$ be respectively denoted by $\mu_{mn}, m,n=1,2$ and $g_{ij}, i,j=1,\cdots n$. Suppose that the local coordinates on $C$ are given by $x$ (that is $x:=(x_{1}, x_{2})$). Then a typical form of such Lagrangian action integral as defined above is given as
\begin{align}\label{eq:Laplacian}
&\int_{C}\mathcal{L}(f, \phi_{i}, i=1,\cdots, n)=\int_{C} \lambda^{-1} \left[\mu_{mn}(x)\partial_{m} f_{i}(x)\partial_{n} f_{j}(x)g_{ij}(f(x))+ \lambda \rho(f)\mathcal{K}\right]d\nu_{C}\notag\\
& m,n=1,2\,\,\,\, \text{and}\,\,\,\, i,j=1,\cdots, n.\notag\\
\end{align}
where $\lambda$ is a coupling parameter, $\nu_{C}$ is a measure on $C$, $\rho: X\to \mathbb{K}\in C^{\infty}(X)$ is a smooth function on $X$, and $\mathcal{K}$ is the Gaussian curvature on $C$ with respect to the metric $\mu$. Here the fields associated to the Lagrangian action integral are given as $$\phi=\lambda^{-1}(g, \lambda \rho).$$
\begin{rmk}
By this notation we mean that the coupling constant $\lambda$ has dependence on parameters $g, \rho$.
\end{rmk}
\begin{rmk}
By writing the action in terms of the Laplacian + curvature form in \eqnref{eq:Laplacian}, we have assumed to study the RG flow of this particular conformal field theory (CFT). However, RG flow can be more generally defined for any field theory with any action to start with, not necessarily near a conformal fixed point. See Sec.\,\ref{sec:fixed point} for more discussions of RG flow around general fixed points.
\end{rmk}
\subsection{Deformation family of Lagrangian action integrals} Let us denote$$\mathcal{S}(f, \phi_{0}):=\int_{C}\mathcal{L}(f, \phi_{i}, i=1,\cdots, n)=\int_{C} \lambda^{-1} \mu_{mn}(x)\partial_{m} f_{i}(x)\partial_{n} f_{j}(x)g^{ij}_{0}(f(x)),$$where $\phi_{0}:=\lambda^{-1}(g_{0},0)$ is a field associated to the fixed choice of $g_{0}$. One interesting case of study is to identify the moduli space (the geometric space representing the family) of smooth maps $f: C\to X$ which minimize the action integral $\mathcal{S}(f, \phi_{0})$ for fixed choice of metric $g_{0}$ over $X$. These are often identified with vacuum states of the underlying governing physical theory for our system of particles. A rather more interesting question is whether the vacuum states of the underlying theory are stable with respect to infinitesimal deformations of the geometry of $C$ and $X$ respectively, specially in quantum physics where fields and geometry of space undergo algebraic or analytic fluctuations. This question could be rigorously studied via inducing deformations of the fields involved in our physical theory, that is $$\phi_{0}\to \phi_{0}+\partial \phi_{0}=\lambda^{-1}(g_{0}+h, 0+\lambda \rho),$$where the function $h\in C^{\infty}(X, T^{\vee}X^{\otimes 2})$ is a seymmetric bilinear smooth differential form on $X$ and $\rho\in C^{\infty}(X, \mathbb{K})$ is a smooth function on $X$. Introducing these deformation parameters, one can study the set of extremizing maps $f: C\to X$ of the action integral $\mathcal{S}(f, \phi)$, that is smooth harmonic maps minimizing $\mathcal{S}(f, \phi)$, where $\mathcal{S}(f, \phi)$ is obtained as a local deformation around $\mathcal{S}(f, \phi_{0})$ induced by deforming the geometry of $C, X$. Let us consider a generalized deformed Lagrangian action
\begin{align}
&\mathcal{S}(f, \phi)=\mathcal{S}(f, \phi_{0})+\int_{C} \lambda^{-1} \mu_{mn}(x)\partial_{m} f_{i}(x)\partial_{n} f_{j}(x)g_{ij}(f(x))d\nu_{C}\notag\\
&+\lambda^{-1}\int_{C} \lambda \rho(f)\mathcal{K}d\nu_{C}+\lambda^{-1}\int_{C}\Gamma(f)d\nu_{C}+\lambda^{-1}\int_{C}f^{*}\omega d\nu_{C}
\end{align}
where as before $h\in C^{\infty}(X, T^{\vee}X^{\otimes 2})$, $\Gamma\in C^{\infty}(X, \mathbb{K})$, and $\omega\in C^{\infty}(X, \wedge^{2}T^{\vee}X)$ an antisymmetric bilinear form are all regarded as infinitesimal induced deformation parameters. Note that here the deformation parameters $\phi_{1}:=\lambda^{-1}h$, $\phi_{2}=\lambda^{-1} (\lambda \rho)$, $\phi_{3}:=\lambda^{-1}U$ and $\phi_{4}:= \lambda^{-1}\omega$ may, roughly speaking, be regarded as local coordinates in the space of deformations of $\mathcal{S}(f, \phi_{0})$. Hence we can rewrite one such deformation in terms of the other as an extension
\begin{equation}\label{local-coord}
\mathcal{S}(f, \phi)= \mathcal{S}(f, \phi_{0})+\sum_{i\geq 1}\int_{C}O_{i}(f, \phi_{i}).
\end{equation}
Moreover, it must be noted that depending on the underlying physical theory, one may consider situations where $\mathcal{S}(f, \phi_{0})$ is required to be invariant under conformal transformations $(C, \mu_{mn})\to (C, e^{-\psi}\mu_{mn})$, in which case, shall one be interested to preserve the conformal invariance of the deformed Lagrangian action $\mathcal{S}(f, \phi)$, one requires that the deformation fields $\rho$ and $U$ vanish, as they break the conformal symmetry, however the deformations $h, \omega$ can be non-vanishing, as their associated integrals are preserved under conformal group action on $C$.
\subsection{Moduli functors associated to deforming fields and maps simultaneously} We mimic the approach of algebraic geometers for constructing our moduli spaces. Consider the following situation. Let $\mathcal{T}\to \text{Spec}{\mathbb{K}}$ be a finite type parametrizing scheme. The notation means that $\mathcal{T}$ is a space (known as parametrizing scheme in algebraic geometry terms) constructed over field of numbers $\mathbb{K}$ that is topologically compact. Let $\mathfrak{M}\text{ap}(C, X): \mathcal{S}\text{ch}/\mathbb{K}\to \mathcal{A}\text{b}/ \mathbb{K}$ be defined as a two-category (i.e. a category which contains objects, their morphisms, and their morphisms of morphisms , also known as 2-morphisms), such that the category is fibered over a base category of finite type (parametrizing) schemes over $\mathbb{K}$. The objective of such functor is to produce families of maps from $C$ to $X$ parametrized by schemes such as $\mathcal{T}$. To state the latter functionality of $\mathfrak{M}\text{ap}(C, X)$ in more mathematical formal terms, we say that the groupoid sections of $\mathfrak{M}\text{ap}(C, X)$ over any $\mathcal{T}$ are given by the sheaf of Abelian groups of $\mathcal{T}$-families of smooth maps from $C$ to $X$, that is the groupoid sections of our moduli functor are given by families of maps
\begin{equation}\label{T-family}
\mathfrak{M}\text{ap}(C, X)(\mathcal{T})\cong \{\tilde{f}: C_{\mathcal{T}}:=\mathcal{T} \times_{\mathbb{K}} C\to X\}
\end{equation} such that for any $t\in \mathcal{T}$ the $t$-fibers of the family $\tilde{f}\mid_{t}\cong \{ f_{t}: C\to X\}$ are given by smooth continuous maps from domain Riemann surface $C$ to $X$. Roughly speaking, the functor $\mathfrak{M}\text{ap}(C, X)$ provides us with a platform to parametrize the smooth maps from $C$ to $X$ in a systematic way over any chosen parametrizing scheme. For instance, given any $\mathcal{T}:=\text{Spec}(\mathbb{K})$, geometric reduced point, the groupoid sections of $\mathfrak{M}\text{ap}(C, X)(\mathcal{T})$ are given by single maps $f:C\to X$. Similarly, the fibers of $\mathfrak{M}\text{ap}(C, X)$ over a line, $L$ (which as a geometric scheme belongs to our category, $\mathcal{S}\text{ch}/\mathbb{K}$, of schemes of $\mathbb{K}$) provides a one dimensional family of maps $f_{L}: C_{L}\to X$, and the fibers of $\mathfrak{M}\text{ap}(C, X)$ over a surface provides a two dimensional family of maps, etc.\\
Now as the geometric structure of $C,X$, and hence $f$ undergo deformations in our theory, similar to Feynman path integration formalism, we compute the vaccum states of the theory, by taking a stochastic average over all admissible weighted morphisms $f: C\to X$ which satisfy smoothness property. In doing so, we further allow certain induced correlation fields, defined in our theory, induced by evaluating the map $f$ at a finite number of smooth distinct marked points $p_{1}, \cdots, p_{l} \in C$. Moreover, we use the Lagrangian action integral constructed in previous section as a weight function associated to each single map $f:C\to X$. Doing so, we obtain an integral over the space parametrizing tuples $(f: C\to X, p_{1}, \cdots, p_{l})$, where $p_{i}, i=1, \cdots ,l$ are distinct smooth marked points on $C$
\begin{equation}\label{correlation}
Z[C, X, p_{1}, \cdots, p_{n}, \phi]:= \frac{1}{Z_{0}} \int_{\mathfrak{M}\text{ap}(C, p_{1}, \cdots, p_{l},X)} D_{\phi}[f](f(p_{1}, \cdots, f(p_{l}))) e^{-\mathcal{S}(f, \phi)}.
\end{equation}
Here $D_{\phi}(f)$ is a measure over $\mathfrak{M}\text{ap}(C, p_{1}, \cdots, p_{l},X)$. Note that by construction $\mathcal{S}(f, \phi)$ is regarded as a deformation of $\mathcal{S}(f, \phi_{0})$, hence following the construction in \eqref{local-coord}, one may rewrite correlation function \eqref{correlation} in terms of $\mathcal{S}(f, \phi_{0})$ as follows
\begin{align}\label{correlator}
&Z[C, X, p_{1}, \cdots, p_{n}, \phi]=\notag\\
&\frac{1}{Z_{0}} \int_{\mathfrak{M}\text{ap}(C, p_{1}, \cdots, p_{l},X)} D_{\phi}[f](f(p_{1}, \cdots, f(p_{l}))) e^{-\mathcal{S}(f, \phi_{0})} \prod_{i\geq 1}\int_{C} O_{i}(f, \phi_{i})
\end{align}
\section{Renormalization semi-group flow}
The construction of the renormalization semi-group flow is based on the fact that, in order to make the above integrals well-defined, one may merely consider certain controllable deformation regimes for the fields $\phi_{i}$, that is; one would like to consider a family of the fields $\phi_{i}(\mathcal{T})$, where the scheme $\mathcal{T}$ is the parametrizing scheme, used in \eqref{T-family} governing the geometric deformations of maps $f_{\mathcal{T}}: C_{\mathcal{T}} \to X$ induced by perturbation of geometric structures of $C$ and $X$. The idea is to consider an infinitesimal deformation flow, called renormalization semi-group flow (as it turns out that our construction in this example only provides a semi-group rather than a group), over the moduli space of maps and field deformations, that is, to consider a morphism
\begin{align}\label{TT-family}
&\mathcal{RG}_{\mathcal{T}}:=\mathfrak{M}\text{ap}(C, X)(\mathcal{T})\times H^{*}(X, \mathbb{K})^{\otimes n} \to \mathfrak{M}\text{ap}(C, X)(\mathcal{T})\times H^{*}(X, \mathbb{K})^{\otimes n}\notag\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (f, \phi_{1}, \cdots, \phi_{n}) \to (f_{\mathcal{T}}, \phi_{1,\mathcal{T}},\cdots, \phi_{n,\mathcal{T}})
\end{align}
which has a lift to a morphism on moduli space of action integrals
\begin{align}
&\overline{\mathcal{RG}}_{\mathcal{T}}:=\mathcal{A}\text{ct}(C,X)\cong \left(\mathfrak{M}\text{ap}(C, X)(\mathcal{T})\times H^{*}(X, \mathbb{K})^{\otimes n}\right)^{\vee}\notag\\
&\to \mathcal{A}\text{ct}(C,X)\cong \left(\mathfrak{M}\text{ap}(C, X)(\mathcal{T})\times H^{*}(X, \mathbb{K})^{\otimes n}\right)^{\vee}\notag\\
\end{align}
taking $\mathcal{S}(f, \phi_{0})$ to $\mathcal{S}(f_{\mathcal{T}}, \phi_{\mathcal{T}})$, which satisfies the semi-group property.
\begin{rmk}
We remark again that we are considering, generally speaking, our fields $\phi_{i}, i=1,\cdots, n$ as living in our field algebra, that is the vector space $H^{*}(X, \mathbb{K})^{\otimes n}$ generated by differentially graded forms on $X$. Moreover, the action integrals are regarded as morphisms from $\mathfrak{M}\text{ap}(C, X)(\mathcal{T})\times H^{*}(X, \mathbb{K})$ to the underlying ground field $\mathbb{K}$, and hence, realized as the dual space $\left(\mathfrak{M}\text{ap}(C, X)(\mathcal{T})\times H^{*}(X, \mathbb{K})^{\otimes n}\right)^{\vee}$.
\end{rmk}
We now elaborate further on renormalization flow. In order to define it we need to formulate a deformation process, applied to geometry of $C, X$, then compute the induced deformations of associated fields $\phi_{i}$ and $f$ with support on deformed $X$ as shown in Equation \eqref{TT-family}. Note that the functorial construction of the moduli space of maps allows us to perform this task in a rigorous algebraic manner. Take a scheme $\mathcal{T}$ (naively speaking schemes have as their skeleton, the geometrical spaces however, they come further equipped with extra topological or algebraic properties). As we noted above, the fibers of the moduli functor $\mathfrak{M}\text{ap}(C, X)$ over $\mathcal{T}$ (i.e. $\mathfrak{M}\text{ap}(C, X)(\mathcal{T})$) provide us with a $\mathcal{T}$-family of maps from $C\to X$ as in \eqref{T-family}. Now choose an algebraic deformation (a perturbation) of $\mathcal{T}$ and denote it by $\mathcal{T}'$. Then the fibers $\mathfrak{M}\text{ap}(C, X)(\mathcal{T}')$ provide a $\mathcal{T}'$-family $$\{\tilde{f}': C_{\mathcal{T}'}:=\mathcal{T}' \times_{\mathbb{K}} C\to X\},$$realized as a deformation of the former $\mathcal{T}$-family maps from $C$ to $X$.
One way of constructing such algrbraic deformation is to construct $\mathcal{T}'$ as a \textit{nilpotent thickening} of $\mathcal{T}$. We elaborate on this notion, using the language of ideals over the ring of polynomial functions.
Take the polynomial ring $\mathbb{C}[x_{1}, \cdots, x_{n}]$. In classical algebraic geometry, the set of prime ideals generated by different expressions involving the variables $x_{1}, \cdots, x_{n}$ makes a space, isomorphic to the ``\textit{affine}" space $\mathbb{C}^{n}$. Now in order to obtain more interesting spaces, one may consider an ideal, say as an example $\mathcal{I}=(x_{1}x_{2}-x_{3}^2)$, and consider the quotient ring $\mathbb{C}[x_{1}, \cdots, x_{n}]/ \mathcal{I}$. This expression means that all polynomials generated by the expression $x_{1}x_{2}-x_{3}^2$ vanish on this quotient ring. Now the set of prime ideals $p\subset \mathbb{C}[x_{1}, \cdots, x_{n}]/ \mathcal{I} $ provides us with set of geometric points of the algebraic space (algebraic variety) given as the solution set to the polynomial equation $x_{1}x_{2}-x_{3}^2=0$. Let us denote this algebraic variety as $\mathcal{T}$. In order to obtain a nilpotent thickening of $\mathcal{T}$ one can simply construct the quotient ring $\mathbb{C}[x_{1}, \cdots, x_{n}]/ \mathcal{I}^{l}$ for some $l$. The set of prime ideals in the latter provides one with the set of geometric points of the variety obtained as the solution set to $(x_{1}x_{2}-x_{3}^2)^{l}=0$, call the latter space as $\mathcal{T}'$. Due to the natural inclusion of ideals $\mathcal{I}^{l}\subset \mathcal{I}$, one can immediately obtain a natural inclusion of $\mathcal{T}\hookrightarrow \mathcal{T}'$. This deformation is called a nilpotent extension of $\mathcal{T}$ of order $l$. Given a deformation as such nilpotent extension, $\iota_{TT'}:T\hookrightarrow T'$, as we elaborated earlier, the renormalization flow must satisfy the property that$$\iota_{TT'}^{*}\overline{\mathcal{RG}}_{\mathcal{T}}\left(\mathcal{S}(f, \phi_{0})\right)=\mathcal{S}(\iota_{TT'}^{*}\mathcal{RG}_{\mathcal{T'}}(f, \phi_{0})).$$Since the action of the RG flow is realized as a pullback in our construction, one is able to define its induced action on the correlation function defined in \eqref{correlator} as follows
\begin{align}
&\overline{\mathcal{RG}}_{\mathcal{T}}(Z[C, X, p_{1}, \cdots, p_{n}, \phi])= \frac{1}{Z_{0}} \int_{\mathcal{RG}_{\mathcal{T}}(\mathfrak{M}\text{ap}(C, p_{1}, \cdots, p_{l},X))} D_{\phi}[f](f(p_{1}, \cdots, f(p_{l}))) e^{-\mathcal{S}(f, \phi)}\notag\\
&=\frac{1}{Z_{0}} \int_{\mathfrak{M}\text{ap}(C, p_{1}, \cdots, p_{l},X)} \mathcal{RG}_{\mathcal{T}}^{*}(D_{\phi}[f](f(p_{1}, \cdots, f(p_{l})))) e^{-\overline{\mathcal{RG}}_{\mathcal{T}}^{*}(\mathcal{S}(f, \phi))} \notag\\
\end{align}
Let us work out a concrete example.
\begin{exam}
For simplicity, let us assume that $\mathbb{K}$ is given as a field of characteristic zero, such as $\mathbb{C}$, the field of complex numbers. Consider the case where $\mathcal{T}:= \text{Spec}(\mathbb{K}[x_{1}, x_{2}, \cdots, x_{n}]/ (x_{2}, \cdots, x_{n}))\cong \mathbb{A}^{1}$ is given by taking the Zariski spectrum of the affine line in the direction $x_{1}$, given by ideal $\mathcal{I}=(x_{2}, \cdots, x_{n})$ over $\mathbb{K}$. Locally, after choosing a coordinate chart $(x_{1}, \cdots, x_{n})$, the set geometric points in $\mathcal{T}$ is the set of points on the $x_{1}$ axis in $\mathbb{C}^{n}$. Now we introduce an infinitesimal deformation of $\mathcal{T}\hookrightarrow \mathcal{T}'$, induced by a nilpotent extension of order 2, by taking $\mathcal{T}':=\text{Spec}(\mathbb{K}[x_{1}, \cdots, x_{n}]/\mathcal{I}^{2})$. There exists a canonical short exact sequence $$0\to \mathcal{I}/\mathcal{I}^{2}\to \mathbb{K}[x_{1}, \cdots, x_{n}]/\mathcal{I}^{2} \to \mathbb{K}[x_{1}, \cdots, x_{n}]/\mathcal{I}\to 0$$whose kernel is governed by the conormal sheaf (which in here is identified by sheaf of differential one forms on $\mathcal{T}$, that is $\Omega_{\mathcal{T}}$). This roughly speaking realizes the second order nilpotent thickening of $\mathcal{T}$, as the cotangent bundle, $\Omega_{\mathcal{T}}$, of $\mathcal{T}$. We would like to deform the correlation function \eqref{correlator} in the direction of fibers of $\Omega_{\mathcal{T}}$. This amounts to setting $\mathcal{RG}_{\mathcal{T}}$ as the differential operator which deforms the fields in direction of fibers of cotangent bundle of $\mathcal{T}$, that is, RG flow acts on the fields as a map $\phi\to \phi+d\phi$ and hence its induced action on the action integral is given by
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\begin{align}
&\overline{\mathcal{RG}}_{\mathcal{T}}(Z[C, X, p_{1}, \cdots, p_{n}, \phi])= \frac{1}{Z_{0}} \int_{\mathcal{RG}_{\mathcal{T}}(\mathfrak{M}\text{ap}(C, p_{1}, \cdots, p_{l},X))} D_{\phi}[f](f(p_{1}, \cdots, f(p_{l}))) e^{-\mathcal{S}(f, \phi)}\notag\\
&=\frac{1}{Z_{0}} \int_{\mathfrak{M}\text{ap}(C, p_{1}, \cdots, p_{l},X)} \mathcal{RG}_{\mathcal{T}}^{*}(D_{\phi}[f](f(p_{1}, \cdots, f(p_{l})))) e^{-\overline{\mathcal{RG}}_{\mathcal{T}}^{*}(\mathcal{S}(f, \phi))} \notag\\
&\frac{1}{Z_{0}} \int_{\mathfrak{M}\text{ap}(C, p_{1}, \cdots, p_{l},X)} (D_{\phi+d\phi}[f](f(p_{1}, \cdots, f(p_{l})))) e^{-(\mathcal{S}(f, \phi+d\phi))}\notag\\
&=-\frac{1}{Z_{0}} \int_{\mathfrak{M}\text{ap}(C, p_{1}, \cdots, p_{l},X)} \frac{\partial \phi(x_{n})}{\partial x_{n}} \frac{\partial}{\partial \phi} \left( D_{\phi}[f](f(p_{1}, \cdots, f(p_{l}))) e^{-\mathcal{S}(f, \phi)}\right)\notag\\
\end{align}
Therefore, viewing the RG flow as a differential operator acting on the action integral, $Z$, and rewriting the variation of $Z$, induced by nilpotent deformation of $\mathcal{T}$, in terms of $Z$ itself, we obtain a differential equation governing the change in $Z$, that is
\begin{equation}
\left[ \overline{\mathcal{RG}}_{\mathcal{T}}+\frac{\partial \phi(x_{n})}{\partial x_{n}} \frac{\partial}{\partial \phi}\right]Z[C, X, p_{1}, \cdots, p_{n}, \phi]=0
\end{equation}
We will come back to the discrete version of the differential equation above, when discussing the construction of RG flow in AI.
\end{exam}
\section{From Conventional RG to Machine-Learning RG}
The above section formulates the mathematical foundation for conventional RG. However, there are several aspects that should be upgraded before the idea of RG can find useful applications in machine learning. The main differences between the conventional RG and the machine-learning RG are summarized in \tabref{tab:RG_compare}, and discussed as follows.
\begin{table}[htp]
\caption{Differences between conventional RG and machine-learning RG proposed in Ref.\,\cite{Li2018Neural,Hu2020Machine}}
\begin{center}
\begin{tabular}{ccc}
& Conventional RG & Machine-Learning RG\\
\hline
base space & smooth manifold & discrete lattice\\
RG flow & continuous & discrete \\
RG equation & differential equation & recurrence equation \\
RG fixed point & conformal & general \\
RG scheme & human-specified (fixed) & machine-designed (learnable) \\
data driven & no & yes \\
algebraic structure & semigroup & group \\
invertibility & non-invertible & invertible \\
holographic bulk & not available & available \\
hyperbolic geometry & not defined & emergent
\end{tabular}
\end{center}
\label{tab:RG_compare}
\end{table}
\subsection{Continuous v.s. Discrete}
The conventional RG in the quantum field theory typically assumes that the field is defined on a smooth base manifold. However, this assumption is typically not the case for machine learning applications. For example, images are defined on discrete pixels, and texts are defined on discrete words. The discrete nature of most datasets in machine learning requires us to generalize the base manifold from continuous space to discrete lattice.
The discretization of the base manifold also forces the RG flow to be discrete, because it is no longer possible to perform infinitesimal dilation on a discrete lattice. Therefore, instead of writing down a differential equation to describe the continuous RG flow, the discrete RG flow should be described by a \emph{recurrent} equation. However, in the continuum limit (when the lattice spacing approaches to zero), the recurrent equation should converge to the differential equation, which will be shown in Sec.\,\ref{sec:conventional RG}.
\subsection{Semigroup v.s. Group} The conventional RG keeps decimating information in each step of coarse-graining. As a result, the conventional RG is not invertible and only forms a \emph{semigroup} instead of a \emph{group}, despite of its inaccurate name of renormalization ``group''. Recent development in physics\cite{Qi2013Exact} reveals that the RG flow can actually be viewed as a holographic mapping, which is invertible. This not only makes a profound connection from RG to quantum gravity, but also promotes RG to a group.
The conventional RG studies how perturbations of the action (or deformations of the field configuration) gets renormalized at larger and larger scales. The invertible RG has a completely different mindset: it aims to answer how the correlated field configurations on the holographic boundary can be disentangled to uncorrelated noises in the holographic bulk, or how the strongly-coupled quantum field theory on the holographic boundary can be reformulated as the weakly-coupled dual gravitational theory in the holographic bulk. By establishing the holographic mapping, any deformation of the field configuration on the holographic boundary can be translated into an excitation in the holographic bulk and analyzed more conveniently. Therefore invertible RG is a more powerful paradigm of RG. Nevertheless, it can always fall back to the conventional RG by a forgetful map that forgets about the holographic bulk degrees of freedom.
\subsection{Human v.s. Machine} The conventional RG scheme is designed by human. Due to the limitation of human intelligence, the conventional RG always assumes that the action must take a fixed form with specific types of terms, and the RG flow only change the coefficients of these terms, such that the action can only flow within a predefined moduli space. Although the moduli space allows us to parameterize the action conveniently, it also restricts our imagination. A more general RG flow can go beyond the moduli space, as new terms can be generated under RG and even the field content can change under RG (microscopic and macroscopic descriptions of a system can be fundamentally different as advocated by the emergence principle). However, such a general RG scheme is not analytically tractable by human. It is not even clear how to design the RG scheme if the form of the action and the field content are all unknown. Thus it becomes desirable to introduce artificial intelligence to learn the optimal RG scheme automatically from the big data of field configurations generated by a field theory. By learning to generate similar field configurations from independent random noise in the holographic bulk, the machine will create the optimal holographic mapping, which also specifies the optimal (invertible) RG scheme.
\subsection{Conformal v.s. General Fixed Point}\label{sec:fixed point} Conventional RG typically assume a conformal fixed point to start with. Given the conformal symmetry at the fixed point, the RG transformation is always taken to be the dilation operator in the conformal group, which corresponds to the rescaling of spacetime and fields together. Given the RG transformation, one can study how a perturbation (or deformation) of the field evolves under dilation. If the perturbation grows stronger/weaker at larger scales, then the perturbation is said to be \emph{relevant}/\emph{irrelevant} (with respect to the conformal fixed point). More quantitatively, the conformal dimensions can be defined as the eigenvalues of the dilation generator, such that relevant/irrelevant fields are simply distinguished by their positive/negative conformal dimensions. Intuitively, relevant fields are low-energy/slow-varying modes to be kept under coarse-graining, and irrelevant fields are high-energy/fast-varying modes to be decimated (or integrated out).
However, the more general machine-learning RG do not assume a conformal fixed point, because the real-world data (like images or texts) may not be scale-invariant and hence not respecting the conformal symmetry. Therefore, the dilation operator is not well-defined, and one can not prescribe an explicit RG scheme from the beginning. The RG scheme has to be learned from data using a data-driven approach. In fact, the real-world data is more likely to be closer to Gaussian fixed points. So even if one learns the RG scheme, it is not immediately clear whether the RG transformation can be used to infer the conformal dimension, as the data could be far from any conformal fixed point.
\subsection{Relevant vs. Irrelevant}
Therefore, the traditional idea of calculating scaling dimension as eigenvalues of the dilation generator no longer works in more general RG approaches. We need a different way to define what is relevant and what is irrelevant. Ref.\,\cite{Hu2020Machine} proposes an elegant and universal definition of irrelevant degrees of freedom using holographic duality and information theory. The key idea is that irrelevant fields are those degrees of freedoms that should be decimated under coarse-graining, so they should appear to us as random noise (i.e.~independent/uncorrelated random variables). Since the irrelevant fields are actually the holographic bulk field under the holographic duality, the above idea can also be rephrased to a statement that holographic bulk fields are almost uncorrelated. The goal of machine-learning RG is to learn the RG transformation that automatically identify and separate such irrelevant degrees of freedom in a field theory. We will explain this approach in more details in Sec.\,\ref{sec:irrelevance}, after introducing the concrete construction of the machine-learning RG algorithm.
For now, we would like to comment that the information theoretic definition of the irrelevant field is consistent with the conformal dimension definition in the conformal limit. Because a \emph{negative} conformal dimension in the conformal field theory (CFT) indicates that the field correlation will \emph{decay} exponentially in the dual anti-de Sitter (AdS) holographic bulk, which is equivalent to the statement that the holographic bulk field are \emph{short-range correlated}, which look like independent random noises beyond a finite correlation length, and are therefore \emph{irrelevant} in the information theoretic sense.
\section{Machine-Learning RG via Flow-Based Generative Models}
\subsection{Sequential Data and Quantum Field on One-Dimensional Lattice}\label{1dlattice}
The idea of renormalization group can be used to construct novel generative models for unsupervised learning. The discussion will mainly focus on sequential data, although generalizations to images and graphs are possible. A sequence is an ordered set of objects $a=(a_1,a_2,\cdots)$, where each object $a_i\in A$ is taken from an object set $A$ (also known as the vocabulary). In machine learning, each object $a_i$ is usually embedded as a vector $\phi_i$ in a finite-dimensional vector space $\dsR^n$ (assuming the dimension to be $n$). Denote the embedding map as $E:A\to\dsR^n$, the sequence can be represented as a ordered set of vectors $\phi=(\phi_1,\phi_2,\cdots)$, where $\phi_i = E(a_i)\in\dsR^n$.
One can also view $\phi_i$ as a quantum field on one-dimensional discrete lattice, as described by the mapping $\phi:I\to\dsR^n$, where $I\subset\dsN$ denotes the index set (equipped with an ordering). Each index $i\in I$ labels an object (or its vector embeding) in the sequence and the set $I$ describes the one-dimensional lattice. The size (cardinality) $|I|$ of the index set corresponds to the length of the sequence. Let $\mathsf{Map}(I, \dsR^n):=\{\phi: I\to \dsR^n\}$ be the associated space of all maps from the index set $I$ to the vector space $\dsR^n$. The objective of unsupervised machine learning is to model the probability measure $p(\phi)\scD\phi$ given the dataset of sequences.
\subsection{Conventional Renormalization froms a Semigroup}
The conventional notion of renormalization group transformation $\scR:\mathsf{Map}(I,\dsR^n)\to\mathsf{Map}(I',\dsR^n)$ corresponds to a coarse-graining map that extract the relevant (coarse-grained) field $\phi'=\scR(\phi)$ from the original (fine-grained) field $\phi$ and discard the remaining (irrelevant) field degrees of freedom. The renormalization transformation always reduces the degrees of freedom, therefore the index set will become smaller $|I'|\leq |I|$ under the renormalization transformation. Because of the information loss, it is no-longer possible to recover the original field configuration $\phi$ from the coarse-grained configuration $\phi'$. Therefore the renormalization transformation $\scR$ is not invertible, and only forms a \emph{semigroup}.
\subsection{Invertible Renormalization forms a Group}
The key idea to make the renormalization transformation invertible is to keep the irrelevant field $\zeta'$ together with the relevant field $\phi'$ as the joint output of the renormalization transformation. Intuitively, the relevant/irrelevant fields are the low-/high-energy modes in the field configuration. What the renormalization transformation does is to separate the irrelevant field $\zeta'$ and the relevant $\phi'$ field given the original field $\phi$ as input. The criterion to separate irrelevant fields will be elaborated in Sec.\,\ref{sec:irrelevance}.
Invertible renormalization was first proposed under the name of exact holographic mapping (EHM) \cite{Qi2013Exact}, which further leads to applications in flow-base generative models for unsupervised machine learning \cite{Li2018Neural,Hu2020Machine,Hu2020RG-Flow}. An invertible renormalization transformation is a bijective map $\hat{\scR}:\mathsf{Map}(I,\dsR^n)\to\mathsf{Map}(I', \dsR^n)\otimes\mathsf{Map}(J', \dsR^n)$, under which the original field $\phi$ splits to the relevant field $\phi'$ and the irrelevant field $\zeta'$
\begin{equation}
(\phi',\zeta') = \hat{\scR}(\phi),
\end{equation}
where $\phi'=(\cdots,\phi'_{i'},\cdots)_{i'\in I'}\in \mathsf{Map}(I', \dsR^n)$ and $\zeta'=(\cdots,\zeta'_{j'},\cdots)_{j'\in J'}\in \mathsf{Map}(J', \dsR^n)$. The bijectivity requires $|I'|+|J'|=|I|$, i.e. the numbers of relevant and irrelevant features must add up to the total number of features in the original field. The inverse renormalization transformation $\hat{\scR}^{-1}$ will also be called the generation transformation $\hat{\scG}$, denoted as
\begin{equation}
\phi = \hat{\scG}(\phi',\zeta'):=\hat{\scR}^{-1}(\phi',\zeta').
\end{equation}
As the transformation is invertible, the renormalization group (RG) is promoted from a semigroup to a \emph{group}.
\subsection{Renormalization Group Flow} The invertible renormalization transformation enables us to define invertible renormalization group (RG) flow on both the field configuration level and the probability measure (or the action) level.
\subsubsection{RG Flow on the Field Level}
Repeating the invertible renormalization transformation, an RG flow can be defined (on the field configuration level) via the following iteration
\begin{equation}\label{eq:Rk}
(\phi^{(k)},\zeta^{(k)})=\hat{\scR}^{(k)}(\phi^{(k-1)}),
\end{equation}
where $\phi^{(k)}\in\mathsf{Map}(I^{(k)},\dsR^n)$, $\zeta^{(k)}\in\mathsf{Map}(J^{(k)},\dsR^n)$ are the relevant and irrelevant fields, and $\hat{\scR}^{(k)}:\mathsf{Map}(I^{(k-1)},\dsR^n)\to\mathsf{Map}(I^{(k)}, \dsR^n)\otimes\mathsf{Map}(J^{(k)}, \dsR^n)$ is the (bijective) renormalization transformation at the $k$-th step. The condition $|I^{(k)}|+|J^{(k)}|=|I^{(k-1)}|$ is always satisfied as a necessary condition for the bijectivity. The iteration defines a flow of quantum fields, called the \emph{renormalization flow} ($\scR$-flow):
\begin{equation}\label{eq:R flow}
\begin{tikzcd}
{\phi\equiv\phi^{(0)}} & {\phi^{(1)}} & {\phi^{(2)}} & \cdots \\
& {\zeta^{(1)}} & {\zeta^{(2)}} & \cdots
\arrow["\hat{\scR}^{(1)}"', from=1-1, to=1-2]
\arrow[from=1-1, to=2-2]
\arrow["\hat{\scR}^{(2)}"', from=1-2, to=1-3]
\arrow[from=1-2, to=2-3]
\arrow["\hat{\scR}^{(3)}"', from=1-3, to=1-4]
\arrow[from=1-3, to=2-4]
\end{tikzcd}.
\end{equation}
Along the $\scR$-flow, the field configuration will be coarse-grained progressively $\phi^{(0)}\to\phi^{(1)}\to\phi^{(2)}\to\cdots$, and the relevant degrees of freedom will be reduced (as $|I^{(0)}|\geq|I^{(1)}|\geq|I^{(2)}|\geq\cdots$). Through this process, a sequence of irrelevant fields $\zeta^{(1)},\zeta^{(2)},\cdots$ is also produced, which was discarded in the conventional renormalization approach, but kept in the invertible renormalization approach. Suppose all the relevant degrees of freedom are eliminated after $K$ steps of the renormalization transformation (i.e. $|I^{(K)}|=0$), the entire $\scR$-flow corresponds to a map that encodes the original field $\phi\equiv \phi^{(0)}$ to the collection of irrelevant fields $\zeta\equiv \{\zeta^{(k)}\}_{k=1:K}$, denoted as $\zeta=\hat{\scR}(\phi)$.
Retaining these irrelevant fields allows the RG flow to be inverted. The inverse flow is also called the \emph{generation flow} ($\scG$-flow) that reconstructs the original field, as defined by the following inverse iteration
\begin{equation}
\phi^{(k-1)}=\hat{\scG}^{(k)}(\phi^{(k)},\zeta^{(k)}):=(\hat{\scR}^{(k)})^{-1}(\phi^{(k)},\zeta^{(k)}),
\end{equation}
or given by the dual diagram of \eqnref{eq:R flow}
\begin{equation}\label{eq:G flow}
\begin{tikzcd}
\cdots & {\phi^{(2)}} & {\phi^{(1)}} & {\phi^{(0)}\equiv\phi} \\
\cdots & {\zeta^{(2)}} & {\zeta^{(1)}}
\arrow["{\hat{\scG}^{(3)}}"', from=1-1, to=1-2]
\arrow["{\hat{\scG}^{(2)}}"', from=1-2, to=1-3]
\arrow["{\hat{\scG}^{(1)}}"', from=1-3, to=1-4]
\arrow[from=2-3, to=1-4]
\arrow[from=2-2, to=1-3]
\arrow[from=2-1, to=1-2]
\end{tikzcd}.
\end{equation}
The entire $\scG$-flow corresponds to a map that decodes the irrelevant fields $\zeta$ to the original field $\phi$, denoted as $\phi=\hat{\scG}(\zeta)$.
\subsubsection{RG Flow on the Probability Measure (Action) Level} The RG flow of field $\phi\to\zeta$ induces a flow of the associated probability distribution over $\mathsf{Map}(I,\dsR^n)$. Under the bijective map between the original field $\phi$ and the irrelevant field $\zeta$, the probability measure must remain invariant
\begin{equation}\label{eq:measure invariance}
p_{\Phi}(\phi)\scD\phi=p_{\Zeta}(\zeta)\scD\zeta.
\end{equation}
Given $\zeta=\hat{\scR}(\phi)$ and $\phi=\hat{\scG}(\zeta)$, \eqnref{eq:measure invariance} implies that the probability distributions are related by
\begin{equation}
p_{\Zeta}(\zeta)=p_{\Phi}(\phi)\Big\vert\det\frac{\partial\hat{\scR}(\phi)}{\partial\phi}\Big\vert^{-1}, \quad p_{\Phi}(\phi)=p_{\Zeta}(\zeta)\Big\vert\det\frac{\partial\hat{\scG}(\zeta)}{\partial\zeta}\Big\vert^{-1},
\end{equation}
where $|\det\partial_\phi\hat{\scR}(\phi)|$ denotes the absolute value of the Jacobian determinant of the transformation $\hat{\scR}$, and similarly for $|\det\partial_\zeta\hat{\scG}(\zeta)|$. More specifically, in each step of the transformation, the probability measure is deformed by (along the $\scG$-flow)
\begin{equation}
p_{\Phi}^{(k-1)}(\phi^{(k-1)})=p_{\Phi}^{(k)}(\phi^{(k)}) p_{\Zeta}^{(k)}(\zeta^{(k)})\Big\vert\det\frac{\partial\hat{\scG}^{(k)}(\phi^{(k)},\zeta^{(k)})}{\partial(\phi^{(k)},\zeta^{(k)})}\Big\vert^{-1}.
\end{equation}
In quantum field theory, the field action is defined as the negative log-likelihood of the field configuration, i.e. $S_{\Phi}^{(k)}=-\log p_{\Phi}^{(k)}$ and $S_{\Zeta}^{(k)}=-\log p_{\Zeta}^{(k)}$. In terms of the field action, the transformation relates
\begin{equation}\label{eq:recursive action}
S_{\Phi}^{(k-1)}(\phi^{(k-1)})=S_{\Phi}^{(k)}(\phi^{(k)})+ S_{\Zeta}^{(k)}(\zeta^{(k)})+S^{(k)}_{\Phi\Zeta}(\phi^{(k)},\zeta^{(k)}),
\end{equation}
where the coupling action $S^{(k)}_{\Phi\Zeta}(\phi^{(k)},\zeta^{(k)})$ is defined to be the log Jacobian determinant of the $\hat{\scG}^{(k)}$ transformation,
\begin{equation}\label{eq:S_PhiZeta}
S^{(k)}_{\Phi\Zeta}(\phi^{(k)},\zeta^{(k)}):=\log\Big\vert\det\frac{\partial\hat{\scG}^{(k)}(\phi^{(k)},\zeta^{(k)})}{\partial(\phi^{(k)},\zeta^{(k)})}\Big\vert.
\end{equation}
Therefore the renormalization transformation $\hat{\scR}$ of the relevant field $\phi^{(k)}$ induces the deformation $\bar{\scG}$ of the relevant field action $S_{\Phi}^{(k)}(\phi^{(k)})$ along the generative direction
\begin{equation}
\begin{tikzcd}
{\phi=\phi^{(0)}} & \cdots & {\phi^{(k-1)}} & {\phi^{(k)}} & \cdots & {\phi^{(K)}=\emptyset}\\
{S_{\Phi}=S_{\Phi}^{(0)}} & \cdots & {S_{\Phi}^{(k-1)}} & {S_{\Phi}^{(k)}} & \cdots & {S_{\Phi}^{(K)}=0}
\arrow[from=1-1, to=1-2]
\arrow[from=1-2, to=1-3]
\arrow["{\hat{\scR}^{(k)}}", from=1-3, to=1-4]
\arrow[from=1-4, to=1-5]
\arrow[from=1-5, to=1-6]
\arrow[from=2-6, to=2-5]
\arrow[from=2-5, to=2-4]
\arrow["{\bar{\scG}^{(k)}}", from=2-4, to=2-3]
\arrow[from=2-3, to=2-2]
\arrow[from=2-2, to=2-1]
\arrow[from=1-1, to=2-1]
\arrow[from=1-3, to=2-3]
\arrow[from=1-4, to=2-4]
\arrow[from=1-6, to=2-6]
\end{tikzcd}.
\end{equation}
In this way, the renormalization flow of the action $\bar{\scR}:=\bar{\scG}^{-1}$ is defined as the pullback of the renormalization flow $\hat{\scR}$ of the field. The RG transformation is invertible on both the field and the action level, making renormalization group literally a group.
\subsection{Criterion to Separate Irrelevant Fields}\label{sec:irrelevance} What has not been explained so far is the criterion to separate relevant fields from irrelevant fields. Ref.\,\cite{Hu2020Machine} argues that the irrelevant field should look like independent random variables (or random maps), because the irrelevant fields are supposed to be discarded under the conventional RG flow, meaning that (in the ideal limit) they do not contain information and should appear like random noise. Guided by this intuition, Ref.\,\cite{Hu2020Machine} further proposes the minimal bulk mutual information (minBMI) principle as the designing principle of renormalization flow, that the optimal renormalization transformations $\{\hat{\scR}^{(k)}\}_{k=1:K}$ should be defined as the maps that minimize the mutual information among all irrelevant fields
\begin{equation}
\sum_{k,k',j,j'}I(\zeta^{(k)}_{j}:\zeta^{(k')}_{j'})=\sum_{k,k',j,j'}\int\scD\zeta\;p_{\Zeta}(\zeta)\log\frac{p_{\Zeta}(\zeta^{(k)}_{j},\zeta^{(k')}_{j'})}{p_{\Zeta}(\zeta^{(k)}_{j})p_{\Zeta}(\zeta^{(k')}_{j'})}.
\end{equation}
The minimum is achieved when the irrelevant fields are statistically independent, i.e.
\begin{equation}\label{eq:independent prior}
p_{\Zeta}(\zeta)=\prod_{k}\prod_{j\in J^{(k)}}p_{\Zeta}(\zeta^{(k)}_{j}),
\end{equation}
such that all mutual information vanishes.
The optimal solution of $\hat{\scR}$ that converges to this limit can be found using machine learning approaches, by constructing a trainable bijective map $\hat{\scG}:=\hat{\scR}^{-1}$ (as the composition of smaller bijective maps $\hat{\scG}^{(k)}$ at each RG step) to reproduce the data distribution $p_{\Phi}(\phi)$ starting from the independent prior distribution $p_{\Zeta}(\zeta)$ in \eqnref{eq:independent prior}. The related methods were developed in Refs.\,\cite{Li2018Neural,Hu2020Machine,Hu2020RG-Flow} under the name of neural-RG. A conventional choice is to take each $p_{\Zeta}(\zeta^{(k)}_{j})=(2\pi)^{-n/2}\exp(-\frac{1}{2}\Vert\zeta^{(k)}_{j}\Vert^2)$ to be the standard normal distribution (Gaussian with zero mean and unit variance), such that
\begin{equation}\label{eq:SZ}
S_{\Zeta}^{(k)}(\zeta^{(k)})=\frac{1}{2}\sum_{j\in J^{(k)}}\Vert\zeta^{(k)}_{j}\Vert^2+\text{const.}.
\end{equation}
This action describes that the irrelevant field fluctuation is massive in the holographic bulk, which is compatible with the idea of holographic duality.
\subsection{Hierarchical Structure and Hyperbolic Space}
\label{Hierarchical Structure}
As the renormalization transformation reduces the relevant degrees of freedom, the size of the relevant index set gradually reduces $|I^{(k)}|\leq |I^{(k-1)}|$. To be more concrete, we can restrict our discussion to the case where the degrees of freedom is reduced by half under each renormalization transformation, i.e. $|I^{(k)}|=|I^{(k-1)}|/2$, such that
\begin{equation}
|I^{(k)}|=2^{-k}|I^{(0)}|.
\end{equation}
Then the condition $|I^{(k)}|+|J^{(k)}|=|I^{(k-1)}|$ implies $|J^{(k)}|=2^{-k}|I^{(0)}|$. The RG flow will stop when $|I^{(K)}|<1$, which sets the total number $K$ of RG steps to be
\begin{equation}
K=\log_2|I^{(0)}|.
\end{equation}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.75\columnwidth]{fig_hyperbolic}
\caption{Hierarchical structure of the RG flow. The renormalization/generation flows can be viewed as the encoding/decoding maps between the original field in the flat space (holographic boundary) and the irrelevant field in the hyperbolic space (holographic bulk).}
\label{fig:hyperbolic}
\end{center}
\end{figure}
As illustrated in \figref{fig:hyperbolic}, the hierarchical structure of the RG flow generates a ordered collection of index sets $\{J^{(k)}\}_{k=1:K}$, which can be union into a hyperbolic lattice (a discrete hyperbolic space), described by
\begin{equation}
J=\bigcup_{k=1:K} J^{(k)}.
\end{equation}
Instead of thinking the irrelevant fields as separate mappings $\zeta^{(k)}\in\mathsf{Map}(J^{(k)},\dsR^n)$, we can treat them jointly as a field $\zeta\in\mathsf{Map}(J,\dsR^n)$ defined on the hyperbolic lattice $J$. Therefore, the $\scR$-flow $\zeta=\hat{\scR}(\phi)$ and the $\scG$-flow $\phi=\hat{\scG}(\zeta)$ respectively define the encoding and decoding maps that connect the field $\phi$ in one-dimensional flat space to the field $\zeta$ in two-dimensional hyperbolic space, which explicitly realize the holographic duality in quantum gravity.
\subsection{Realization of Bijective Transformation}
To optimize the renormalization transformation $\hat{\scR}$, one relies on the construction of a trainable bijective map to model $\hat{\scR}$. Machine learning community has provided several realizations of trainable bijective maps, including real NVP\cite{Dinh2016Density} and neural ODE\cite{Chen2018Neural}. In the following, we will focus on the neural ODE realization, as it can capture multi-modular features better than real NVP, which is more suitable for processing sequences of discrete objects.
\subsubsection{Neural ODE}
\label{Neural ODE}
Each single-step renormalization transform $(\phi',\zeta')=\hat{\scR}(\phi)$ can be realized by an ordinary differential equation (ODE). Starting from $\phi(0)=\phi$, first evolve $\phi(t)$ from $t=0$ to $t=1$ following
\begin{equation}\label{eq:ODE}
\frac{\dd\phi(t)}{\dd t}=f_\theta(\phi(t),t),
\end{equation}
where $f_\theta$ is a trainable function (realized as a neural network) parameterized by neural network parameters $\theta$. Then split the result as $\phi(1)=(\phi',\zeta')$ to obtain $\phi'$ and $\zeta'$. $t$ is considered as an auxiliary time. The inverse transformation is simply given by the time-reversal evolution, therefore the mapping is indeed bijective as desired.
Apart from the transformation, the log Jacobian determinant of $\hat{\scR}$ can also be evaluated. Based on the ODE in \eqnref{eq:ODE}, one have
\begin{equation}
\log\Big\vert\det\frac{\partial\phi(t+\dd t)}{\partial\phi(t)}\Big\vert = \Tr\Big(\frac{\partial f_\theta(\phi(t),t)}{\partial \phi(t)}\Big)\dd t,
\end{equation}
which can be integrated to
\begin{equation}
\log\Big\vert\det\frac{\partial\hat{\scR}(\phi)}{\partial\phi}\Big\vert = \int_{0}^{1}\Tr\Big(\frac{\partial f_\theta(\phi(t),t)}{\partial \phi(t)}\Big)\dd t.
\end{equation}
Given that $\hat{\scG}:=\hat{\scR}^{-1}$, its log Jacobian determinant is simply given by a negation,
\begin{equation}
\log\Big\vert\det\frac{\partial\hat{\scG}(\phi',\zeta')}{\partial(\phi',\zeta')}\Big\vert = -\int_{0}^{1}\Tr\Big(\frac{\partial f_\theta(\phi(t),t)}{\partial \phi(t)}\Big)\dd t,
\end{equation}
which will be useful for the evaluation of the coupling action in \eqnref{eq:S_PhiZeta}.
\subsubsection{Locality and Translational Symmetry}
It is possible to design the ODE function $f_\theta$ to respect locality and translational symmetry. The idea is to realize $f_\theta$ using layers of convolutional neural networks (CNN) with finite kernel followed by element-wise activations.
\subsection{Objective Function}
The objective is to train the generative model, such that the model distribution $p_{\Phi}(\phi)$ matches the data distribution $p_\text{dat}(\phi)$ as much as possible. The objective can be achieved by minimizing the Kullback-Leibler (KL) divergence
\begin{equation}
\begin{split}
\scL&=D_\text{KL}(p_\text{dat}\Vert p_{\Phi})=\int\scD\phi\;p_\text{dat}(\phi)\log\Big(\frac{p_\text{dat}(\phi)}{p_{\Phi}(\phi)}\Big)\\
&=\mathop{\dsE}_{\phi\sim p_\text{dat}}S_{\Phi}(\phi)-H(p_\text{dat}),
\end{split}
\end{equation}
where $S_{\Phi}(\phi)=-\log p_{\Phi}(\phi)$ is the model action (as the negative log-likelihood), and $H(p_\text{dat})=-\int\scD\phi\; p_\text{dat}(\phi)\log p_\text{dat}(\phi)$ is the data entropy. As the data entropy $H(p_\text{dat})$ is independent of the model parameter, it can be dropped from the loss function $\scL$. Therefore the loss function is essentially the ensemble average of the model action on the dataset. By minimizing the average action, the ODE function $f_\theta$ in each RG transformation will get trained. Upon convergence, the algorithm will find the optimal invertible RG flow that maps the (presumably) strongly coupled original field $\phi$ on the holographic boundary to the weakly coupled irrelevant field $\zeta$ in the holographic bulk.
\subsection{Summary of the Algorithm} \label{Algorithm}
Given a set of sequences from the data, the learning algorithm goes as follows.
\begin{enumerate}
\item For each given sequence $a=(a_1,a_2,\cdots)$, represent each object $a_i$ in the sequence as a vector $\phi_i=E(a_i)\in\dsR^n$. Denote the sequence of vectors as a vector field $\phi=(\phi_1,\phi_2,\cdots)\in\mathsf{Map}(I,\dsR^n)$.
\item Starting with $\phi^{(0)}=\phi$, apply the renormalization transformation iteratively,
\begin{equation*}
(\phi^{(k)},\zeta^{(k)})=\hat{\scR}^{(k)}(\phi^{(k-1)}),
\end{equation*}
for $K=\log_2|I|$ steps (until all relevant fields are eliminated).
\begin{enumerate}
\item Each step of the transformation is implemented by solving an ODE
\begin{equation*}
\frac{\dd\phi^{(k-1)}(t)}{\dd t}=f^{(k)}_\theta(\phi^{(k-1)}(t),t),
\end{equation*}
starting from the initial condition $\phi^{(k-1)}(0)=\phi^{(k-1)}$, integrating from $t=0$ to $t=1$, and then splitting the final result into $\phi^{(k-1)}(1)=(\phi^{(k)},\zeta^{(k)})$.
\item While solving the ODE, simultaneously integrate along the time evolution to obtain the coupling action
\begin{equation*}
S^{(k)}_{\Phi\Zeta}(\phi^{(k)},\zeta^{(k)}) = -\int_{0}^{1}\Tr\Big(\frac{\partial f^{(k)}_\theta(\phi^{(k-1)}(t),t)}{\partial \phi^{(k-1)}(t)}\Big)\dd t.
\end{equation*}
\end{enumerate}
\item Starting from the initial condition $S_{\Phi}^{(K)}=0$, collect the action in the reverse order (along the generation flow)
\begin{equation*}
S_{\Phi}^{(k-1)}(\phi^{(k-1)})=S_{\Phi}^{(k)}(\phi^{(k)})+ S_{\Zeta}^{(k)}(\zeta^{(k)})+S^{(k)}_{\Phi\Zeta}(\phi^{(k)},\zeta^{(k)}),
\end{equation*}
where $S_{\Zeta}^{(k)}(\zeta^{(k)})$ is given by
\begin{equation*}
S_{\Zeta}^{(k)}(\zeta^{(k)})=\frac{1}{2}\sum_{j\in J^{(k)}}\Vert\zeta^{(k)}_{j}\Vert^2.
\end{equation*}
The resulting total action will be denoted as $S_{\Phi}(\phi):=S_{\Phi}^{(0)}(\phi^{(0)})$.
\item Train the model to minimize the loss function
\begin{equation*}
\scL=\mathop{\dsE}_{\phi\sim p_\text{dat}}S_{\Phi}(\phi).
\end{equation*}
\end{enumerate}
\subsection{Potential Applications and Advantages} After training, the model could potentially be used for the following tasks.
\begin{itemize}
\item Inference of hierarchical latent representation. Using $\zeta=\hat{\scR}(\phi)$, one can infer the hierarchical latent representation $\zeta$ of any sequence encoding $\phi$. The high-level representations ($\zeta^{(k)}$ with a large $k$) can be viewed as the encoding of the entire sequence, which can be used in downstream tasks like classification and translation.
\item Likelihood estimation. Using $S_{\Phi}(\phi)$, one can estimate the probability density $p_{\Phi}(\phi)\propto \exp(-S_{\Phi}(\phi))$ for any field configuration $\phi$. This will be useful for anomaly detection.
\item Sample generation. As a generative model, new samples can be generated by first sampling $\zeta$ in the hyperbolic space and then transforming to $\phi=\hat{\scG}(\zeta)$ using the generation flow, which may find applications in completing missing objects in a sequence.
\end{itemize}
The propose algorithm is advantageous in the following aspects.
\begin{itemize}
\item Disentangled features in scales. The optimal RG flow distills features at different scales, allowing the model to capture the long-range and multi-scale correlation in the sequential data. The features are automatically arranged in a hyperbolic spaces, making it easy to access/control.
\item Efficient inference/generation. The hierarchical and iterative approach enables the model to infer latent fields or generate original fields in $\Theta(N)$ complexity (given the sequence length $N$), which is superior compared to the $\Theta(N^2)$ complexity of transformer-based approaches, especially when the sequence is long.
\item Ability to process hierarchical structure. The renormalization transformation can progressively extract coarse-grained features from fine-grained features, making it capable of capturing global features (such as the parity of bit strings). In comparison, as shown in Ref.\,\cite{Hahn2019Theoretical}, self-attention-based models can not efficiently model hierarchical structures, unless the number of layers/heads increases with sequence length.
\end{itemize}
\subsection{Recovering Conventional RG by Integrating out Irrelevant Fields.}\label{sec:conventional RG} Finally, we would like to comment that the invertible renormalization can fall back to the conventional renormalization by integrating out irrelevant fields. Recall \eqnref{eq:Rk} that in each step of the invertible renormalization transformation, the original (fine-grained) field $\phi^{(k-1)}$ is separated into the relevant $\phi^{(k)}$ and irrelevant $\zeta^{(k)}$ fields by $(\phi^{(k)},\zeta^{(k)})=\hat{\scR}^{(k)}(\phi^{(k-1)})$. The invertible renormalization $\hat{\scR}^{(k)}$ can be downgraded to a non-invertible renormalization $\scR^{(k)}$ by a forgetful map which forgets about the irrelevant field $\zeta^{(k)}$, such that $\phi^{(k)}=\scR^{(k)}(\phi^{(k-1)})$ only transforms the the fine-grained field $\phi^{(k-1)}$ to the coarse-grained field $\phi^{(k)}$.
According to \eqnref{eq:recursive action}, the actions are related by
\begin{equation}
S_{\Phi}^{(k-1)}(\phi^{(k-1)})=S_{\Phi}^{(k)}(\phi^{(k)})+ S_{\Zeta}^{(k)}(\zeta^{(k)})+S^{(k)}_{\Phi\Zeta}(\phi^{(k)},\zeta^{(k)}),
\end{equation}
where the irrelevant field $\zeta^{(k)}$ is massive, and is described by the Gaussian action
$S_{\Zeta}^{(k)}(\zeta^{(k)})=\frac{1}{2}\Vert\zeta^{(k)}\Vert^2$ as in \eqnref{eq:SZ}. Because $\zeta^{(k)}$ represents the high-energy modes that should be integrated out under renormalization, one can argue that the fluctuation of $\zeta^{(k)}$ can be treated perturbatively due to its large mass, which justifies the expansion of the action around $\zeta^{(k)}\to 0$,
\begin{equation}
\begin{split}
S_{\Phi}^{(k-1)}(\phi^{(k-1)})&\simeq S_{\Phi}^{(k)}(\phi^{(k)})+S^{(k)}_{\Phi\Zeta}(\phi^{(k)},0)+\zeta^{(k)}\cdot\partial_{\zeta^{(k)}}S^{(k)}_{\Phi\Zeta}(\phi^{(k)},0)\\
&+\frac{1}{2}\Vert\zeta^{(k)}\Vert^2+\frac{1}{2}\zeta^{(k)}\cdot\partial_{\zeta^{(k)}}\partial_{\zeta^{(k)}}S^{(k)}_{\Phi\Zeta}(\phi^{(k)},0)\cdot \zeta^{(k)}+\cdots.
\end{split}
\end{equation}
As the approximate action is quadratic in $\zeta^{(k)}$, one can perform a Gaussian integration for $\zeta^{(k)}$, under which the action becomes
\begin{equation}\label{eq:effective action}
S_{\Phi}^{(k-1)}(\phi^{(k-1)})= S_{\Phi}^{(k)}(\phi^{(k)})+S^{(k)}_{\Phi\Zeta}(\phi^{(k)},0)+\frac{1}{2}\big(\partial_{\zeta^{(k)}}^2 S^{(k)}_{\Phi\Zeta}(\phi^{(k)},0)-\big(\partial_{\zeta^{(k)}}S^{(k)}_{\Phi\Zeta}(\phi^{(k)},0)\big)^2\big).
\end{equation}
Therefore one can define the renormalization transformation $\bar{\scR}^{(k)}$ on the action via $S_{\Phi}^{(k)}=\bar{\scR}^{(k)}(S_{\Phi}^{(k-1)})$, in correspondence to the field renormalization $\phi^{(k)}=\scR^{(k)}(\phi^{(k-1)})$. Based on \eqnref{eq:effective action}, the explicit form of the renormalization operator $\bar{\scR}$ can be given
\begin{equation}
\bar{\scR}(S_{\Phi})=S_{\Phi}-S_{\Phi\Zeta}-\frac{1}{2}(\partial_{\zeta}^2 S_{\Phi\Zeta}-(\partial_{\zeta}S_{\Phi\Zeta})^2),
\end{equation}
such that
\begin{equation}
\bar{\scR}(S_{\Phi})(\phi) = S_{\Phi}(\scR(\phi)),
\end{equation}
which reproduces the pullback construction of the action renormalization. If one further define the infinitesimal generator of $\bar{\scR}$ as $\bar{\mathfrak{r}}=\log\bar{\scR}$, the renormalization flow can be expressed as a differential equation\cite{Polchinski1984Renormalization,Ma2020Constraints}
$\partial_{\ell}S_{\Phi}=\bar{\mathfrak{r}}S_{\Phi}=-S_{\Phi\Zeta}-\frac{1}{2}(\partial_{\zeta}^2 S_{\Phi\Zeta}-(\partial_{\zeta}S_{\Phi\Zeta})^2).$
\section{Experiments on genomic sequences}
\subsection{Problem Overview}
Extracting the hidden information of genomic sequences has been a critical subject in biological research, with relevance to epidemiology, immuniology, protein design and many other subfields. With its great similarity to the natural language processing problems, there are numerous studies on applying machine learning techniques to extract information from genomic sequences. Various machine learning architectures include word2vec \cite{du2019gene2vec}\cite{wu2021replication}, bidirectional long short-term memory\cite{hie2021escape}, transformer\cite{ji2021transformer} etc. However while the existing algorithms can provide a single-gene level embedding, they do not provide a canonical sequence embedding and the hierarchical information is not clear from the natural language models. Thus we apply the renormalization group idea from the previous sections to the genomic sequence representation problem, where the hierachial structure provides the biological information at different energy levels, i.e. the deeper layer can capture the longer correlation in the sequence, and thus provide a canonical embedding of the sequence with the deepest layer. We take the Influenza HA amino acid sequences as an example\footnote{can be downlaoded from the “Protein Sequence Search'” section of \url{https://www.fludb.org}}, where the sequences are regarded as one-dimensional lattices as described in Sec.~\ref{1dlattice}. Below shows samples of the sequence data, there is clear global features embedded as one can see the similarities between different sequences.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.85\columnwidth]{influenza_seq.jpg}
\caption{Sample Influenza HA amino acid sequences.}
\label{fig:Influenza_seq}
\end{center}
\end{figure}
\subsection{Single Amino Acid Distribution Learning}
Before proceeding with the sequence into the RG-scheme, we need to verify that local features are efficiently learned. Thus we first look at the single amino acid distribution learning. As shown in Fig.~\ref{fig:Influenza_seq}, at a fixed location $i$ among the sequences, there is a discrete distribution labeled by amino acid, we pick up that amino acid from each sequence. Then each sample is labeled by $a=(a_i)$, where $a_i\in A$ represents the single amino acid. We apply the pre-trained single-amino-acid level embedding $E: A\rightarrow \mathbb{R}^n$ from Ref.~\cite{hie2021escape} where $n=20$. After embedding, the boundary field is $\phi = (\phi_i)$, where $\phi_i=E(a_i)\in \mathbb{R}^n$. To remove the difficulty in transforming the boundary discrete distribution to the bulk uncorrelated continuous Gaussian distribution, we add a small randomness on the boundary field, i.e. $\overline{\phi_i} = \phi_i + \epsilon$, where $p_Z(\epsilon)$ takes the standard normal distribution with small variance. For simplicity, in the following we still use $\phi_i$ to denote the fields with small randomness.
With this setup, we train a neural ODE model to realize the bijective transformation between the data distribution to a Gaussian distribution as described in Sec.~\ref{Neural ODE}. To further speed up the training process, we have added the Jacobian and Kinetic regularization to find an optimal bijective map as in Ref.~\cite{finlay2020train}. The input data is the 4th single amino acid from 1000 Influenza HA sequences.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.25\columnwidth]{NODE_structure.jpg}
\caption{Neural ODE structure. The input is the single amino acid vector representation $\phi\in \mathcal{R}^n$ with $n=20$. The hidden layers are concatsquash(CS) layers. The latent varible has dimension $64$ and the output has dimension $n=20$.}
\label{fig:NODE_structure}
\end{center}
\end{figure}
The ODE transformation $f_\theta(x,t)$ is constructed by a feed forward network with 4 sequential hidden layers as shown in Fig.~\ref{fig:NODE_structure}. The hidden layers are the concatsquash(CS) layers defined in Ref.~\cite{grathwohl2018ffjord}:
\begin{equation}
f_{CS}(x,t) = W_0 x \cdot \sigma(W_1 t) + W_2 t
\end{equation}
where $W_0$, $W_1$, $W_2$ are parameter matrices with shape $(d_{o},d_{i})$, $(d_{o},1)$, $(d_{o},1)$ respectively, $d_i$ and $d_o$ are input and output dimensions. We can consider $x$ as a $d_i$-dimenional vector, and concate the time variable, then $f_{CS}: \mathbb{R}^{d_i+1}\rightarrow \mathbb{R}^{d_o}$. The hyperbolic tangent activation functions are applied after the first three CS layers. Then the model is trained such that $f_{\theta}(x,t)$ describes the flow from the data to a Gaussian variable.
As shown in Fig.~\ref{fig:single_amino_acid}, a 2-dimensional feature space can be obtained by applying the t-distributed stochastic neighbor embedding (t-SNE) algorithm\cite{JMLR:v9:vandermaaten08a} to the original data and the flow generated data embedding vectors. The flow generated data are obtained by taking the inverse transformation $\hat{\mathcal{R}}^{-1}$ from vectors with the Gaussian distribution. The original data distribution with the multi-modular feature can be perfectly captured after training.
\begin{figure}[htbp]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{single_amino_acid_initial.pdf}
\caption{Initial distribution}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{single_amino_acid_final.pdf}
\caption{Distribution after training}
\end{subfigure}
\caption{Training result on a single amino acid with t-SNE representation. (A): The original data distribution(red) has a multi-modular struture, while the initial flow model's distribution(blue) is standard normal. (B): After training with neural ODE, two distributions coincide.}
\label{fig:single_amino_acid}
\end{figure}
\subsection{Amino Acid Sequence Distribution Learning}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.5\columnwidth]{mera.pdf}
\caption{An illustration of the mera structure with kernel $l=2$. Green blocks are disentangler blocks, yellow blocks are decimator blocks. After each decimator layer, half of the fields are redefined as bulk fields $\zeta$. }
\label{fig:MERA}
\end{center}
\end{figure}
To train on sequences, we adapt the hierachical RG scheme described in Sec.~\ref{Hierarchical Structure}. As discussed in the previous section, the input sequence is represented as labels $a = (a_1,a_2,\cdots, a_I)$ with the cardinality $I$ denotes the length of the sequence. With the pretrained embedding $\phi_i=E(a_i)\in \mathbb{R}^n$, the boundary fields are represented as $\phi = (\phi_1,\phi_2,\cdots, \phi_I)$. Thus we have the initial boundary fields $\phi^{(0)}=\phi$, we can run the renormalization flow using Eq.~\ref{eq:R flow}. On the other hand, the generation flow Eq.~\ref{eq:G flow} reconstruct the original field. Following the notation of MERA networks, each renormalization transformation layer consists of a disentangler layer and a decimator layer:
\begin{equation}
\hat{\scR}^{(k)} = \hat{\scR}^{\text{dec},(k)}\circ \hat{\scR}^{\text{dis},(k)}
\end{equation}
where a disentangler layer disentangles the local correlations and a decimator layer separate the decimated fields out as the bulk fields. In Fig.~\ref{fig:MERA}, we show an illustration of the model structure with green blocks as disentanglers and yellow blocks as decimators. Each block is a bijective transformation with the neural ODE structure as in Fig~\ref{fig:NODE_structure}. We can further explicitly write down the transformation equations: the covering length of a disentangler or a decimator is defined as the kernel length $l$. Then there are $\frac{I}{2^k l}$ blocks in the $k$-th layer. For the $m$-th block in the $k$-th layer, where $m\in \lbrace 0,\cdots,\frac{I}{2^k l} -1\rbrace$, the transformation is given by
\begin{equation}
\begin{split}
&(\psi^{(k)}_{2^k(ml+a)})_{a\in\lbrace 1,\cdots,l\rbrace} = \hat{\mathcal{R}}^{\text{dis},(k)}_m((\phi^{(k)}_{2^k(ml+a)})_{a\in \lbrace 1,\cdots,l\rbrace})\\
&(\phi^{(k+1)}_{2^k(ml+l/2+a)})_{a\in\lbrace 1,\cdots,l\rbrace} = \hat{\mathcal{R}}^{\text{dec},(k)}_m((\psi^{(k)}_{2^k(ml+l/2+a)})_{a\in \lbrace 1,\cdots,l\rbrace})
\end{split}
\end{equation}
Half of the resulting fields are redefined as the bulk fields: $\zeta^{(k+1)}_{2^{k+1}(ml+a)}\coloneqq\phi^{(k+1)}_{2^{k+1}(ml+a)}$. Here we have chosen a scheme that after each layer, every other existing fields are redefined as new bulk fields. Since there are position-dependent features among the sequences, to respect the local features of the sequence, we take independent block transformations as they are labeled by both layer index $k$ and block index $m$. With this setup, we train on the objective $\scL=\mathop{\dsE}_{\phi\sim p_\text{dat}}S_{\Phi}(\phi)$ as described in Sec.~\ref{Algorithm}.
In Fig.~\ref{fig:4_amino_acid} and Fig.~\ref{fig:16_amino_acid_4500}, we show the result when $I=4$, $l=2$ and when $I=16$, $l=4$ with the same set of data in the previous section. To compare the joint distribution, we concatenate the vector embeddings of the 4 and 16 amino acids for each sequence and train the t-SNE algorhim with these concated vectors. We also computed the normalized logarithmic probability defined as $\log_n p = \log p /(nI)$ with $n$ the embedding dimension and $I$ the sequence length. The numbers in the parentheses are normalized logarithmic probability before training. Both results shows that the original data joint distribution can be captured using the RG-scheme with local neural ODE blocks.
\begin{table}[htbp]
\centering
\begin{tabular}{ccc}
\toprule
& \multicolumn{2}{c}{normalized log prob}\\
\cmidrule{2-3}
& seq length $ =4$ & seq length $ =16$\\
\midrule
original data & 1.52(0.45) & 1.52(0.25)\\
generated data & 1.50(0.19) & 1.50(0.19)\\
\bottomrule
\end{tabular}
\caption{\label{tab:log_prob}Average normalized logarithmic probability from original data and generated data with sequence length 4 and 16. }
\end{table}
\begin{figure}[htbp]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{4_amino_acid_initial.pdf}
\caption{Initial distribution}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{4_amino_acid_final.pdf}
\caption{Distribution after training}
\end{subfigure}
\caption{Training result on a length 4 amino acid sequence dataset with t-SNE representation. (A): The original data distribution(red) has a multi-modular struture, while the initial flow model's distribution is standard normal(blue). (B): After training RG scheme with neural ODE blocks, two distributions concide.}
\label{fig:4_amino_acid}
\end{figure}
\begin{figure}[htbp]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{16_amino_acid_initial.pdf}
\caption{Initial distribution}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{16_amino_acid_final.pdf}
\caption{Distribution after training}
\end{subfigure}
\caption{Training result on a length 16 amino acid sequence dataset with t-SNE representation. (A): The original data distribution(red) has a multi-modular struture, while the initial flow model's distribution is standard normal(blue). (B): After training RG scheme with neural ODE blocks, two distributions concide.}
\label{fig:16_amino_acid_4500}
\end{figure}
With training on the full sequence, one can have hierarchical information from each layer. This may give a natural way for escaping virus search. The shallow layers mainly capture the local information, while the deeper layers hold the global information. Escaping virus should have a good local fitness while mean a different content compared with the existing dataset. Then one can use this separation of information levels to design rules for escaping virus or train on a downstream classification task.
\subsection{Learning Viral Escape Mutation}
We conclude our investigation on learning protein sequences distribution by studying the predictive performance of viral escapes with our model. Viral escapes are those mutations in viral protein sequences that make them unrecognizable for human immune system. In other words, although they are still effective on human body and cause infection, the immune system does not flag the mutated sequence as a threat to body. Such mutations can be single or multiple, that is, only one or few amino acids can be instantly mutated, hence, identifying underlying patterns in viral escape mutations will be essential for viral vaccine development. As described in Ref.~\cite{hie2021escape}, in terms of language models, a viral sequence can be regarded as a textual data and a viral escape mutation is seen as a word change in a sentence that does changes the semantic of the sentence however the sentence is still grammatically meaningful. With this analogy, a viral escape is the one capable of making the immune system falsely flag the mutant as a harmless sequence (change in sentiment), while the mutant preserves the virus evolutionary structure (grammatically correct). Therefore, among all possible mutations in a viral sequence, we search for viral escapes which result in a high semantic change and high gramaticality in our model. Figure~\ref{fig:viral_esc} depicts an example of all possible mutations in the test sequence.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.65\columnwidth]{viral_seq.png}
\caption{Sample of possible mutations in the test sequence. First column "word" represents the mutant amino acid, "position" indicates the mutation position in the sequence.}
\label{fig:viral_esc}
\end{center}
\end{figure}
Following this idea (constrained semantic change search (CSCS)~\cite{hie2021escape}) we first train our model on a corpus of viral sequences in an unsupervised fashion, then take a given viral protein sequence with its known viral escape mutations and rank the mutations based on their gramaticality and semantic change. In our construction, the semantic change, caused by a mutation, is regarded as the change in the internal representation of the deepest layer in our construction before and after mutation happens. In other words, given the test sequence $a =(a_1, a_2, ,a_i,.., a_I)$ and its mutant counterpart $\bar{a} =(a_1, a_2, ,\bar{a}_i,.., a_I)$, a semantic change will be noted as $\Delta \zeta= |\zeta^{(K)}(a) -\zeta^{(K)}(\bar{a})|$ where $K$ indicates the deepest layer in the hierarchical structure of our model.
According to the CSCS objective, gramaticality can be defined as how probable is a mutation in $a$, i.e, the probability value that model evaluates for a mutation. With this definition, one natural definition of gramaticality is the conditional probability $p(\bar{a_{i}}|a)$ on the mutation $\bar{a_{i}}$ in the test protein~\cite{hie2021escape}. In our model however, the joint probability $p_{\Phi}(a)$ is the optimization objective which is used to evaluate the input gramaticality. Therefore the final score for each mutation is defined as:
\begin{equation}
Score \coloneqq \Delta \zeta + p(\bar{a}_i|a)
\label{eq::score}
\end{equation}
Note that throughout evaluating our model on viral escape mutations, we only consider single mutation in test data. We also keep the size of our samples to 32 amino acids, and with only 25 different amino acids as the building blocks of sequences, there will be 768 (24x32) possible mutations. Among those, a small subset will be viral escape mutations that is already given to us. For this experiment we used escape mutations dataset in \cite{doud2018single} that indicates 65 out of those 768 mutations are viral escapes. After calculating both sentiment change and grammatically of mutations, we ranked each mutation based on their ranking score in Eq.\ref{eq::score}. Mutations with highest ranking value will be considered as predicted viral escapes and consequently, lower rankings indicate less probable for a mutation to be a viral escape. Fig~\ref{fig::deltap_z_table} and Fig~\ref{fig::deltap_z} illustrate gramaticality and semantic change of all mutations including the viral escapes (red points) which clearly shows that viral escapes tend to show a high gramaticality.
\begin{figure}[htbp]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{z_deltap_table.png}
\caption{}
\label{fig::deltap_z_table}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{z_deltap.png}
\caption{}
\label{fig::deltap_z}
\end{subfigure}
\caption{(A) Table of possible mutations with their Gramaticality and Semantic change. In the list of columns, "pos" means the position in sequence to be mutated, "sub" refers the substitution word, "mut" is the mutant amino acid, and "is-escape" shows which substitution is a viral escape.(B) Gramaticality vs Semantic change of all mutations. Red points indicates the viral escape mutations. Note that the graph is not scaled.}
\end{figure}
We also calculated the area under curve (AUC) of the ranking scores of mutations in Fig.~\ref{fig::auc}. Our results clearly indicate that both grammaticality and semantic change quantities have similar impact on the AUC value.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\textwidth]{auc.png}
\caption{AUC graph of mutations in the test viral protein sequence with 32 amino acids. Gramaticality ($p$) and semantic change ($\Delta \zeta $) have equivalent contribution to the final AUC value.}
\label{fig::auc}
\end{figure}
\newpage
\bibliographystyle{plain}
\bibliography{ref}
\noindent{\small{\tt{asheshmani@fas.harvard.edu, yzyou@physics.ucsd.edu, \\\\sd052fenber@gmail.com}, razizi@qgnai.com}
\end{document}
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TITLE: Integral as a limit of an infinite sum of infinitely narrow rectangles
QUESTION [2 upvotes]: I tried to approach integrating by filling the space with infinitely many infinitely narrow rectangles.
$a$ is the left bound
$b$ is the right bound
$n$ means the number of rectangles, approaches $\infty$
$k$ means the k-th rectangle
My integral is described by the following formula:
$$
\lim_{n \to \infty} \sum_{k=1}^{n} \frac{b-a}{n} \cdot f(a + \frac{k(b-a)}{n})
$$
The rough idea in the picture below:
Let's solve for $f(x) = \frac{1}{x}$, a = 1, b = 3
$$
\lim_{n \to \infty} \sum_{k=1}^{n} \frac{2}{n} \cdot \frac{1}{1 + \frac{2k}{n}} = \lim_{n \to \infty} \left(\frac{2}{n+2}+\frac{2}{n+4}+\frac{2}{n+6}+ \dots\right)
$$
The sum clearly diverges, but I don't know why.. any ideas? Is this way of integrating possible? If not, why? Are my formulas correct?
REPLY [1 votes]: Your sum should only go up to $n$. You have otherwise properly described the right hand Riemann sum. The sum has a finite number of terms, so it does not diverge. You take the limit after you do the sum and should get $\log 3$.
| 152,749
|
TITLE: Prove equality of two vectors if they have equal divergence and equal curls
QUESTION [5 upvotes]: I have following question:
Fields with equal divergence and equal curls $F_1$ and $F_2$ are two vectors fields, you may write them as $F_1 = M_1i+N_1j+P_1k$, $F_2 = M_2i+N_2j+P_2k$. Suppose that $\nabla \cdot F_1 = \nabla \cdot F_2$ and $\nabla \times F_1 = \nabla \times F_2$ over a region D enclosed by the oriented surface S with outward unit normal n and that $F_1 \cdot n = F_2 \cdot n$ on S. Prove that $F_1=F_2$ throughout D.
I came across this question when studying "Stokes's Theorem and Divergence Theorem" in Thomas Calculus, so I suppose we should use either of them to solve this, except I don't know how. All I could figure out is
$$F_1 \cdot n = F_2 \cdot n\Rightarrow\iint\limits_s F_1 \cdot n \, d\sigma = \iint\limits_s F_2 \cdot n \, d\sigma\Rightarrow\iiint\limits_D \nabla \times F_1\, dv = \iiint\limits_D \nabla \times F_2 \, dv,$$ the last step using divergence theorem. Or
$$\nabla \times F_1 = \nabla \times F_2 \Rightarrow\iint\limits_s \nabla \times F_1\cdot n \, d\sigma= \iint\limits_s \nabla \times F_2 \cdot n \, d\sigma \Rightarrow \oint\limits_c F_1 \cdot dr =\oint\limits_c F_2 \cdot dr,$$ the last step using Stokes's theorem.
This is all I get, then what? Am I on a wrong path? Can anyone solve this problem? Thanks.
REPLY [7 votes]: Setting $F=F_1-F_2$, we're supposed to prove that: if $F \cdot n=0$ on $S$ and $\nabla\cdot F=0$, $\nabla\times F=0$ in $D$ then $F=0$ in $D$. We actually need one more assumption: $D$ is simply connected (the statement is not true e.g. for the solid torus). $\nabla\times F=0$ implies that there is a function $f$ such that $F=\nabla f$ (here we use that $D$ is simply-connected). The condition $\nabla\cdot F=0$ becomes $\Delta f=0$.
Now the standard trick is $\nabla\cdot(f\nabla f)=\nabla f\cdot\nabla f+f\Delta f=\nabla f\cdot\nabla f$, hence
$$\iiint_D \nabla f\cdot\nabla f\, dV =\iiint_D\nabla \cdot (f\nabla f)\, dV=\iint_S f\nabla f \cdot n\,dS=0$$
as $\nabla f \cdot n=F \cdot n=0$. Since $\nabla f\cdot\nabla f\geq0$ and the integral is $0$, we get $\nabla f=0$, i.e. $F=0$.
| 208,069
|
The summer is looming, and we all know what that means. Bikinis? Don't remind us, Sam. Yes bikinis are looming. So when Virgin Active invited us down to try out their new exercise class. Come on. Let's go. We had to move it, move it. Moo-ve it. Terrible, terrible. Okay guys, this is called a frog squat. Feet are a little bit wider than shoulder width. Poke your arms inside your legs, and we just go down to a nice, deep squat so you have to find your own range and then you stand up again. And you'll work out very quickly which muscles it's working, which is obviously your quads. #NAME? Come on. We're nearly finished. Alright guys, this next move is called the Bear Crawl, so we'll start on the ground. We just crawl forward, just move forward. Then we'll go in reverse. Just focus on the push through your arms.
The next move is called a Gorilla, so we just go down to a deep squat. We lean forward onto our hands. Your objective is to get your feet to the outside of your hands. Do you get more flexible? So the more you do that, the more mobility you'll get through your hip. This last move is called an Iguana. Left hand goes forward. Right leg just up and to the side. Just do a half push up. I'll show you the power of ZUU, okay? So what moves...
Can you tell me what moves we've just been through? #NAME? #NAME? And an Iguana. Just listen to the commands, okay? Gorillas. Let's go. Frogs where you are. You know how to do these. Keep your arms tucked in. Reverse bears. Reverse bears and push push. Up to a frog, nice and deep. Let's rip them out team. Rip them out. Into Iguanas, set it up. Left hand forward, right leg to the side. Keep moving forward. There's about 26, 27 moves in the first level. I don't think you realise quite how much you're working your muscles. Do you know what I mean? Because I'm just trying to focus on doing an Iguana, and I'm actually toning up my arms, my core.
And the more you do it, the harder it becomes. What's your hardest move? The hardest move. One of the hardest moves is probably the crocodile. Ooh, could you do that? Crocodile sounds good. All the way down, and keep your arms tucked in nice and tight. Move forward so it's. I can't even move! I can't even do it. And then you do can do Fleas going back. Oh my God. That is ridiculous. You burn a lot of calories, if you put your heart into it. I've burnt 1000 calories in a class. So bikini bodies. I have a kangaroo pouch. How do I get rid of my kangaroo pouch? The first one I would do for you would be... I'd probably give you three. Yup, I'll do Frog Squats, and then I'd do Gorillas. Yup, most definitely.
And the last one I'd probably give you is called a Kick Sit. All I'm going to do is sweep my leg through and sit down. Up and down. Up and down. We'll just do four 30 second rounds. Good luck. That's round one. I'm so tired... Oooh! This is just getting more embarrassing as time goes by. Film Sam. We even stopped early and that was tough. Deceptively. Deceptively. Way deceptively tough. You just think sitting on your ass, shoving your leg out. How difficult is that? And then you do it, and I can't even last longer than 40 seconds. I'm a convert. The sign for ZUU is top hand and left hand on top of the right.
| 368,845
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TITLE: Why is the concept of potential used so loosely?
QUESTION [1 upvotes]: What does potential of a conductor actually mean? As I have understood it potential is the amount of work done by an external agent to bring a unit positive charge to a point of consideration. It is also something like a scalar version of electric field. However when I see the concept of a potential being used for things like an infinitely charged metallic sheet or items in an electrical circuit, I get confused. Can anybody connect the real definition of potential to these things? Is potential not a rigorous concept and meant to be used so loosely?
REPLY [1 votes]: The other answers are great, however I will add some of my analogy. We can think of potential as some peaks of mountain at different points in a space permeated by some $\vec E$. The geography (lack of better term) of all such mountains depends on $\vec E $.
If some particle descends to smaller peak (potential), from higher peak, (potential) then particle gains some energy. If particle ascends from smalller peak to higher peak, then particle loses some energy.
Now, coming to 'loose definition', actually it is not so. It is rigorously defined as, work done to bring one coloumb/kilogram charge/mass from infinity to some point in space. Mathematically: $$\Delta V=-\int \vec E \cdot \vec {dr}$$
Or: $$\vec E=\frac{dV}{d\vec{r}}$$
In atomic physics, the term 'eV', electron volt, is quite popular unit for measurement of energy. You would find even masses of particles written in this unit of measurement.
It is defined as energy required to accelerate one electron across 1 volt potential.
Or energy released/gained by electron descending/ascending a peak. where potential differnece between peaks is 1 volt.
$$1 \text{eV}=1.6 \times 10^{-19} \text{J}$$
So potential is a rigorous term and is quite useful in physics.
| 181,189
|
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Head Over Heels
Head Over Heels by Westside Navy navy sandals from Head Over Heels are designed with cross-over straps and a white trim on the outer sole. Pair them with all your favourite summer looks.
- Sole Material - PU
- Heel Type - Platform
- Color Family - Navy
- Upper Material - Manmade Patent Leather
- Product Listing Id :MP00000000025
| 67,092
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Blog Archives
Safety tips for keeping your pet safe this holiday season
Thanksgiving is fast approaching, and delicious food is everywhere to be found. Human foods smell and taste so good to our pets, particularly at holiday time; however, in many cases, human food can be toxic to animals. ASPCA.org, pets.webmd.com, and several other websites list foods that pet owners should avoid feeding their animals because of […]
ANIMALS IN SERVICE
In addition to honoring our veterans on this Veterans Day, we also honor all animals who provide support for their owners. Service Dogs Service dogs are dogs that have been individually trained to perform a specific task for individuals who have disabilities. The disabilities can vary greatly, and so do the tasks that the service […]
Adopt a Senior Pet
Since beginning this blog, we’ve come to recognize that there are pet holidays or celebrations in every month of the year. November is Adopt a Senior Pet Month, and the perfect time to consider bringing a special older animal companion into your life. After all, what could be more in the spirit of the season […]
Halloween is not just for kids!
Will you be dressing your pet in costume this Halloween? According to a survey by Pet360 Inc., a company that runs “pet parenting” websites, 42 percent of 1,000 pet owners the company surveyed nationwide dress their pets up for Halloween, and 71 percent of pet owners favor funny costumers. Dogs are the animals most commonly […] […]
FALL PET FESTIVITIES
Is your pet ready for all the area festivities this fall? • Saturday, October 18: Check out the 2014 Top Dog Show at Sarasota Memorial Auditorium from 5:30-8:30 p.m. The area’s premier dog lovers event features a Westminster-like dog show where owners can enter their four-legged friends in playful categories including: Best Dressed, Best Hair, […]
Gizmoe’s Funeral and Headstone
The first pet burial at Venice Memorial Gardens Pet Cemetery was in 2013 for Gizmoe, beloved pet of one of our funeral directors. Her burial was a celebration of life with a circus theme, complete with clowns, balloons, cotton candy, popcorn, and snow cones. It was held during the pet cemetery’s grand opening. The memorial […]
Pet Blessings
As autumn arrives, you may glimpse a procession of animals being led to a church for a special ceremony. This event is called the Blessing of Pets, and it is held in commemoration of St. Francis of Assisi, who had a great love of all creatures. The Feast of St. Francis is October 4, and […]
Pet Friendly Hotels & Restaurants
I WANT TO VISIT VENICE, BUT WHAT WILL I DO WITH FIDO? If you’re looking for a pet-friendly hotel*, you’ll find several places to stay with your pet in Venice or nearby, including: Bayside Villas 303 Colonia Lane West, Nokomis, FL 34275 941-726-9885 Holiday Inn Express Jacaranda, Venice 380 Commercial Court, Venice, FL 34292 (I-75 […]
Did you know that Venice is a pet friendly city?
Most people don’t know that Venice is pet-friendly. Take a stroll down Venice Avenue, and you’ll find dog bowls and dog treats to quench Fido’s thirst and keep your pet happy while you shop. If your pet needs some exercise after your visit to downtown Venice, a tip to Woodmere Dog Park may be on […]
| 355,036
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Improving markets for recycled plastics: trends, prospects and policy response
On behalf of the Organisation for Economic Co-operation and Development (OECD), Resource Futures conducted a detailed review of global recycled plastics markets. This included evaluating trends, challenges, key issues and policy interventions.
Read the report: Improving Markets for Recycled Plastics: Trends, Prospects and Policy Responses
Objectives
- Assess trends and factors affecting global trade in recycled plastics
- Identify barriers to effectively functioning recycled plastics markets
- Identify and evaluate potential policies for improving recycled plastic markets
Approach
Resource Futures conducted an extensive, desk-based review of recycled plastic markets. The study comprised a review of data available from a wide range of available publicly and proprietary sources, and information provided by OECD member governments. A ‘policy mapping’ approach was used to identify and evaluate potential policies for improving recycled plastics markets.
Outcomes
The study provided a detailed, evidence-based assessment of recycled plastics markets and the barriers that prevent them functioning properly. The study also identified appropriate policies for improving markets and serves as a key reference for OECD member governments on this issue. The full report can be found on the OECD website.
Project Information
Services involved
International development
Team involved
David Lerpiniere
Director
| 390,420
|
Recently we saw the new gameplay trailer for Metal Gear Solid: Ground Zeroes and now we have some screenshots to check. It really does show the amazing detail in the foreground elements … as well as all those lens flares. Nice shot of the 3D in-game GUI as well.
Tag Archives: Kojima Productions.
| 227,859
|
TITLE: Show that the curve $y^2 = x^3 + 2x^2$ has a double point, and find all rational points
QUESTION [5 upvotes]: Show that the curve $y^2 = x^3 + 2x^2$ has a double point. Find all rational points on this curve.
By implicit differentiation of $x$, $-3x^2 - 4x$ vanishes iff $x = -4/3$ and $0$.
By implicit differentiation of $y$, $2y$ vanishes iff $y = 0$.
Taking the second derivative, I got $-6x-4$ and then using the point on the curve $(0,0)$ I got $-4$. Is this my double point?
Thanks for any help!
REPLY [2 votes]: You may find the following fact useful: suppose that a plane curve is described by a polynomial equation $F(x,y)=0$. Write
$$
F(x,y)=\sum_{d\geq0}F_d(x,y)\qquad\qquad(*)
$$
where in $F_d(x,y)$ you collect all the monomials in $F$ of degree exactly $d$. Then:
$(0,0)$ belongs to the curve if and only if $F_0\equiv0$ (by which, I mean that is $0$ identically).
$(0,0)$ is a singular point of the curve if and only if $F_0\equiv0$ and $F_1\equiv0$.
Moreover if $d_0\geq2$ is the smallest degree appearing in $(*)$, then $F_d=0$ is the tangent cone at $(0,0)$. Note that this answers immediately and trivially to your first question.
Finally, the method applies to just any point $(x_0,y_0)$ after a simple change of variables $X=x-x_0$, $Y=y-y_0$.
| 68,992
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Last Updated: Dec 30,2009
ok, not really, I just had a few extra minutes, so I created one
haha Nice pic
thats cool
Beautiful!
lol I thought by the title that it was a new phone coming out. lol
Nice pic.
TS
That's some nice work there
=P Not bad padawon.
Separate names with a comma.
| 209,929
|
\begin{document}
\begin{abstract}
We obtain stochastic stability of $C^2$ non-uniformly
expanding one-dimensional endomorphisms, requiring only
that the first hyperbolic time map be $L^{p}$-integrable
for $p>3$. We show that, under this condition (which
depends only on the unperturbed dynamics), we can
construct a random perturbation that preserves the
original hyperbolic times of the unperturbed map and,
therefore, to obtain non-uniform expansion for random
orbits. This ensures that the first hyperbolic time map
is uniformly integrable for all small enough noise levels,
which is known to imply stochastic stability. The method
enables us to obtain stochastic stability for a class of
maps with infinitely many critical points. For higher
dimensional endomorphisms, a similar result is obtained,
but under stronger assumptions.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
\label{sec:introduction}
The main goal of Dynamical systems theory is the description
of the typical behaviour of orbits as time goes to infinity,
and to understand how this behaviour changes under small
perturbations of the system.
Given
a map $f$ from a manifold $M$ into itself, a central
concept is that of \emph{physical measure}, a
$f$-invariant probability measure $\mu$ whose
\emph{ergodic basin}
\begin{align}\label{eq:ergbasin}
B(\mu)=\big\{x\in M: n^{-1}\sum_{j=0}^{n-1}\phi(f^j(x))
\xrightarrow[n\to+\infty]{} \int\phi\,d\mu\mbox{ for all
continuous } \phi: M\to\R\big\}
\end{align}
has positive \emph{volume} or \emph{Lebesgue measure}, which
we write $\lambda$ and take as the measure associated with any
non-vanishing volume form on $M$.
The stability of physical measures under small variations of
the map
allows for small errors along orbits not to disturb too much
the long term behavior, as measured by
asymptotic time averages of continuous functions along
orbits. When considering practical systems we cannot avoid
external noise, so every realistic mathematical model should
exhibit these stability features to be able to deal with
uncertainty about parameter values, observed initial states
and even the specific mathematical formulation of the model
itself.
We investigate, under the probabilistic point of view, which
asymptotic properties of a dynamical system are preserved
under random perturbation.
Random perturbations and their features were first studied
in 1945 by Ulam and von Neumann, in \cite{UvN}. The focus
of this work are non-uniformly expanding transformations
which were introduced by Alves-Bonatti-Viana in
\cite{ABV00}, and whose ergodic properties are now well
established; see for instance
\cite{Ze03,AA03,AlV13,AlLuPi03}. Here we show that the
asymptotic behavior of these transformations is preserved
when randomly perturbed in an adapted way to their first
times of expansion, under a condition: that the first time
of expansion is $L^p$-integrable with respect to Lebesgue
measure; see next sections for precise definitions and
statements.
The interest in this kind of stochastic stability condition
lies in the fact that \emph{known conditions of stochastic
stability for non-uniformly expanding maps are expressed
as conditions on the random perturbations of the map} and
not solely on the original unperturbed dynamics. We mention
the joint works with Alves \cite{AA03} and Vasquez
\cite{alarva}, and also the recent work by Alves and
Vilarinho \cite{AlV13}.
The uniformly hyperbolic case, studied by Kifer in
\cite{Ki88,Ki86} (among others), is much simpler: uniformly
hyperbolic systems are stochastically stable under a broad
range of random perturbations without further
conditions. Other cases with the same features, which we may
say are ``almost uniformly hyperbolic systems'', where
studied in joint works with Tahzibi, in \cite{ArTah,ArTah2}.
Here, we present a sufficient condition for stochastic
stability of non-uniformly expanding transformations that
relies only on the dynamics of the unperturbed map, for a
simple type of random perturbation that is adapted to the
dynamics. This allows us to treat some exceptional cases.
Recently Shen \cite{Shn13} obtained stochastic stability for
unimodal transformations under very weak assumptions, but
does not cover the case of transformations with infinitely
many critical points; and Shen together with van Strien in
\cite{ShvS13} obtained strong stochastic stability for the
Manneville-Pomeaux family of intermittent maps, answering
questions raised in \cite{ArTah}.
Our method allows us to obtain stochastic stability for
non-uniformly expanding endomorphisms having slow recurrence
to the critical set, encompassing the family of
infinite-modal applications presented in \cite{PRV98}. We
also obtain stochastic stability (in the weak$^*$ sense, see
precise statements in the next sections) for intermittent
maps but in a restricted interval of parameter values; see
Section~\ref{sec:intermitent-maps}.
\subsection{Setting and statement of results}
\label{sec:statement-results}
We consider $M$ to be a $n$-torus, $\mathbb{T}^n=
(\mathbb{S}^{1})^n$, for some $n\ge1$ and $\lambda$ a
normalized volume form in $\mathbb{T}^n$, which we call
Lebesgue measure. This can be identified with the
restriction of Lebesgue measure on $\R^n$ to the unit cube.
We write $d$ for the standard distance function on $\T^n$ in
what follows and $\|\cdot\|$ for the standard Euclidean norm
on $\R^n$ which can be identified with the tangent space at
any point of $\T^n$.
We let $f:\T^n\to\T^n$ be a local $C^2$
diffeomorphism outside a \emph{non-degenerate critical set}
$\SC$, that is, $\SC=\{x\in M:\det Df(x)=0\}$ and $f$
behaves as the power distance to $\SC$: there are constants
$B>1$ and $\beta>0$ satisfying
\begin{itemize}
\item[(S1)]
$\frac{1}{B}\cdot d(x,\SC)^{\beta}\leq
\frac{\|Df(x)\cdot v\|}{\|v\|}
\leq
B \cdot d(x,\SC)^{-\beta}, \vec0\neq v\in T_x M$;
\item[(S2)]$\big|\log\| Df(x)^{-1}\|-\log\|Df(y)^{-1}\|\big|
\leq
B\frac{ d(x,y)}{d(x,\SC)^{\beta}}$;
\item[(S3)]
$\big|\log|\det Df(x)^{-1}|-
\log|\det Df(y)^{-1}|\:\big|\leq
B\frac{d(x,y)}{d(x,\SC)^{\be}}$;
\end{itemize}
for all $x,y\in M\setminus\SC$ with
$\d(x,y)<\frac12\d(x,\SC)$.
We say that $f$ is \emph{non-uniformly expanding} if there
is a constant $c>0$ such that:
\begin{align}\label{nue}
\limsup_{n\to+\infty}\frac1n\sum_{j=0}^{n-1}
\log\| Df(f^{j}(x))^{-1}\|
\leq-c<0 \quad\text{for}\quad \lambda-\text{a.e. }\,x\in M.
\end{align}
We need to control the recurrence to the critical set in
order to obtain nice ergodic properties. We say that $f$ has a
\emph{slow recurrence to critical set} if, for any given $\gamma>0$
there exists $\delta>0$ such that
\begin{align}\label{eq:slowrec}
\limsup_{n\to+\infty}\frac1n
\sum_{j=0}^{n-1}-\log d_{\delta}(f^{j}(x),\SC)
\leq \gamma, \quad\text{for}\quad \lambda-\text{a.e. }\,x\in M,
\end{align}
where $d_\delta$ is the \emph{$\delta$-truncated distance to
$\SC$}, defined as $d_{\delta}(x,\SC)=d(x,\SC)$ if
$d(x,\SC)<\delta$ and $d_{\delta}(x,\SC)=1$ otherwise.
We recall the concept of physical measure.
For any $f$-invariant probability measure $\mu$ we write
$B(\mu)$ for the \emph{basin} of $\mu$ as in~(\ref{eq:ergbasin}).
We say that a $f$-invariant measure $\mu$ is \emph{physical}
if its basin $B(\mu)$ has positive Lebesgue measure:
$\lambda(B(\mu))>0$.
Roughly speaking, physical measures are those that can be
``seen'' by calculating the time average of the values of a
continuous observable along the orbits the points on a
subset with positive Lebesgue measure. Clearly Birkhoff's
Ergodic Theorem ensures that $\mu(B(\mu))=1$ whenever $\mu$
is $f$-ergodic. We note that every $f$-invariant ergodic
probability measure $\mu$ which is also absolutely
continuous with respect to Lebesgue measure,
i.e. $\mu\ll\lambda$, is a physical measure.
The previous conditions on $f$ ensure that Lebesgue almost
all points behave according to some physical measure.
\begin{theorem}[Theorem C, \cite{ABV00}]
\label{thm:abv}
Let $f$ be $C^2$ diffeomorphism away from a non-degenerate
critical set, which is also a non-uniformly expanding map
whose orbits have slow recurrence to the critical
set. Then there is a finite number of $f$-invariant
absolutely continuous ergodic (physical) measures
$\mu_{1}, \ldots,\mu_{p}$ whose basins cover a set of full
measure, that is
$\lambda\big(M\setminus(B(\mu_1)\cup\dots\cup
B(\mu_p)\big)=0$. Moreover, each $f$-invariant absolutely
continuous probability measure $\mu$ can be written as a
convex linear combination the physical measures: there are
$\alpha_1=\alpha_1(\mu),\dots,\alpha_p=\alpha_p(\mu)\ge0$
such that $\sum\alpha_i=1$ and $\mu=\sum\alpha_i\mu_i$.
\end{theorem}
\begin{remark}\label{rmk:pinheiro}
Pinheiro~\cite{Pinheiro05} showed that the same
conclusions of Theorem \ref{thm:abv} can be obtained by
replacing the of non-uniform expansion condition
(\ref{nue}) by the weaker condition
\begin{eqnarray}\label{liminf}
\liminf_{n\to+\infty}\frac1n\sum_{j=0}^{n-1}
\log\| Df(f^{j}(x))^{-1}\|\leq
-c<0, \quad\text{for}\quad \lambda-\text{a.e. }\,x\in M.
\end{eqnarray}
The proof of this fact involves showing that
(\ref{liminf}) implies (\ref{nue}). Therefore, all the
arguments used in this paper remain valid in the more
general setting of condition \eqref{liminf} replacing
condition~(\ref{nue}).
\end{remark}
\subsection{Random perturbations and stochastic stability}
\label{sec:random-perturb-stoch}
We let $B=B(0,1)$ denote the unitary ball centered at the
origin $0$ in $\R^n$, set $X=\overline{B}$ and $\mathcal{F}=\{
f_{t}:M\to M; t\in X\}$ a parametrized family of maps. We
write $f_t(x)=f(t,x),(t,x)\in X\times M$ and assume in what
follows that $f_{0}=f$ is a map in the setting of
Theorem~\ref{thm:abv}.
We consider also the family of probability measures
$(\theta_{\epsilon})_{\epsilon>0}$ in $X$ given by the
normalized restriction of Lebesgue measure to the
$\epsilon$-ball $B(0,\epsilon)$ centered at $0$ in $\R^n$,
as follows
\begin{align}\label{eq:theta_ep}
\theta_{\epsilon}=\frac{\lambda\mid_{B(0,\epsilon)}}{\lambda(B(0,\epsilon))}.
\end{align}
This family is such that $\supp(\theta_{\epsilon})_{\epsilon>0}$ is a nested
family of compact and convex sets satisfying
$\supp(\theta_{\epsilon})\xrightarrow[\epsilon\rightarrow0]{}0$.
Setting $\Omega=X^\N$ the space of sequences in $X$,
the random iteration of $\SF$ is defined by
\begin{align*}
f^{n}_{\underline{t}}(x)=\left(f_{t_{n}}\circ
f_{t_{n-1}}\circ\ldots\circ f_{t_{1}}\right)(x), \quad
\underline{t}=(t_{1},t_{2},\ldots)\in\Omega, x\in M.
\end{align*}
To define the notions of stationary and ergodic measure we
consider the skew-product
\begin{align}\label{continuidade dupla}
\begin{array}{cccc}F:&\Omega\times M&\rightarrow& \Omega\times M\\
&(\underline{t},x)&\mapsto&(\sigma(\underline{t}),f_{t_{1}}(x))
\end{array}
\end{align}
where $\sigma:\Omega\rightarrow \Omega$ is a standard left
shift, and the infinite product measure $\theta_\epsilon^\N$
on $\Omega$, which is a probability measure on the Borel
subsets of $\Omega$ in the standard product topology.
From now on, for each $\epsilon>0$, we refer to
$(f_{\underline{t}},\theta_{\epsilon}^{\mathbb{N}})$ as a
\emph{random dynamical system} with noise of level $\epsilon$.
\begin{definition} [Stationary measure]
\label{eq:defstationary}
A measure $\mu^\epsilon$ is a {\it stationary measure} for the
random system
$(f_{\underline{t}},\theta_{\epsilon}^{\mathbb{N}})$ if
\begin{align*}
\int\phi\,d\mu^\epsilon =
\int\int\phi(f_{t}(x))\,d\mu^\epsilon(x)\,d\theta_{\epsilon}(t),
\quad\text{for all}\quad \phi\in C^0(M,\R).
\end{align*}
\end{definition}
\begin{remark}
\label{re:accinvariant}
If $(\mu^\epsilon)_{\epsilon>0}$ is a family of stationary
measures having $\mu^0$ as a weak$^*$ accumulation point
when $\ep\searrow0$, then from~(\ref{eq:defstationary})
and the convergence of $\supp(\theta_\epsilon)$ to $\{0\}$
it follows that $\mu^0$ must be invariant by $f=f_{0}$;
see e.g. \cite{AA03}.
\end{remark}
We say that $\mu$ is a \emph{stationary measure} if the
measure $\theta_{\epsilon}^{\mathbb{N}}\times\mu$ is
$F$-invariant. Moreover, we say that \emph{a stationary
measure $\mu$ is ergodic} if
$\theta_{\epsilon}^{\mathbb{N}}\times\mu$ is $F$-ergodic.
\begin{definition}
We say that $f$, in the setting of
Theorem~\ref{thm:abv}, is {\em stochastically stable under
the random perturbation given by} $(f_{\un
t},\theta_\epsilon^\N)_{\epsilon>0}$ if, for all
weak$^*$ accumulation points $\mu^0$ of families
$(\mu^\epsilon)_{\epsilon>0}$ of stationary measures for
the random dynamical system $(f_{\un
t},\theta_\epsilon^\N)$ when $\epsilon\searrow 0$, we
have that $\mu^0$ belongs to the closed convex hull of
$\{\mu_{1},\ldots,\mu_p\}$. That is, for all such weak$^*$
accumulation points $\mu^0$ there are
$\alpha_1=\alpha_1(\mu^0),\dots,\alpha_p=\alpha_p(\mu^0)\ge0$
such that $\sum\alpha_i=1$ and $\mu^0=\sum\alpha_i\mu_i$.
\end{definition}
In this work we consider additive perturbations given by
families of maps with the following form
\begin{eqnarray}\label{pertaditiva}
f_{t}(x)=f(x)+t\zeta(x)
\end{eqnarray}
where $\zeta:M\rightarrow\mathbb{R}^{+}$ is Borel measurable
and locally constant at $\lambda$-almost every point.
\begin{remark}\label{rmk:Dft_Df}
For such additive perturbations we have $Df_{t}(x)=Df(x)$
for all $t\in\Omega$ and $\lambda$-a.e. $x\in M$.
\end{remark}
\subsection{First hyperbolic time map and adapted random
perturbations}
\label{sec:hyperb-times-random}
The following is the fundamental concept in this work.
\begin{definition}[Hyperbolic time]
\label{def:hyptimes}
Given $\sigma<1$ and $\delta>0$, we say that $n$ is a
$(\sigma,\delta)$-hyperbolic time for $x\in
M$ if
\begin{align*}
\prod_{j=n-k}^{n-1}\| Df(f^{j}(x))^{-1}\| \leq \sigma^{k}
\quad\mbox{and}\quad d_{\delta}(f^{n-k}(x),\SC)\geq
\sigma^{bk}\quad\text{for all}\quad 1\leq k\leq n,
\end{align*}
where $b=\min\{1/2, 1/2\beta\}$ and $\beta$ is the constant
given in the non-degenerate conditions (S1)-(S2).
\end{definition}
The notion of hyperbolic times was defined in \cite{ABV00}.
To explain our Main Theorem we cite the following technical
result.
\begin{lemma}[Lemma 5.4 in \cite{ABV00}]\label{le:infhyptimes}
Let $f$ be a $C^2$ local diffeomorphism away from a
non-degenerate critical set, which satisfies the
non-uniform expansion condition $(\ref{nue})$ with
$c=3\log\sigma$ for some $0<\sigma<1$ and also the slow
recurrence condition~(\ref{eq:slowrec}).
Then there exist $\theta_{0},\delta>0$ depending on
$\sigma$ and $f$, such that for $\lambda$-a.e. $x$ and
each big enough $N\geq 1$, there are
$(\sigma,\delta)$-hyperbolic times $1\leq n_{1}<
\dots<n_{l}\leq N$ for $x$ with
$l\geq\theta_{0}N$. Moreover, the hyperbolic times $n_{i}$
satisfy
\begin{align}\label{eq:hip-times-prop}
\sum_{j=n_{i}-k}^{n_{i}-1}\log d_{\delta}(f^{j}(x),\SC)
\geq bk\log\sigma,
\quad\text{for all}\quad 0\leq k\leq n_{i}, 1\leq i\leq l.
\end{align}
\end{lemma}
\begin{remark}\label{rmk:infhyptimes}
Let $\mathcal{G}$ the set of points $x\in M$ that have no
hyperbolic time. Then $\lambda(\mathcal{G})=0$ after Lemma
$\ref{le:infhyptimes}$. Thus, if $x$ has only finitely
many hyperbolic times, then some iterate of $x$ belongs to
$\mathcal{G}$. Hence, the subset of points with finitely
many hyperbolic times is contained in
$\cup_{j\ge0}f^{-j}(\mathcal G)$. Moreover,
$\lambda(f^{-j}(\mathcal G))=0$ because $f$ is a local
diffeomorphism away from a critical/singular set with zero
$\lambda$-measure. Therefore, $\lambda$-a.e. $x\in M$ has
infinitely many hyperbolic times.
\end{remark}
Hence, in our setting we have that Lebesgue almost every
point has infinitely many $(\sigma,\delta)$-hyperbolic
times. Thus we may define the map $h:M\to\Z^{+}$ such that
for $\lambda$-a.e. point $x$ the positive integer $h(x)$ is
the first hyperbolic time of $x$. We say $h$ is the {\it
first~hyperbolic~time~map}.
In our main theorem, we will see that is possible to
randomly perturb a non-uniformly expanding map so that
almost all randomly perturbed orbits have infinitely many
hyperbolic times but also the same hyperbolic times as
the non-perturbed map. We start with a one-dimensional version.
\begin{maintheorem}\label{principal}
Let $f:\T^1\to\T^1$ be a $C^2$ diffeomorphism away from a
non-degenerate critical set, which is also a non-uniformly
expanding map whose orbits have slow recurrence to the
critical set.
Let us assume that $f$ has a dense orbit and that the
first hyperbolic time map is $L^p$-integrable for some
$p>3$, that is, $\int h(x)^p\,d\lambda(x)<\infty$.
Then $f$ is stochastically stable for a family of adapted
random perturbations.
More precisely, there exists $\zeta:\T^1\rightarrow
\mathbb{R}^{+}$ mensurable and locally constant such that
the family $(\ref{pertaditiva})$ generates a family of
random perturbations $(f_{\un
t},\theta_\epsilon^\N)_{\epsilon>0}$ for which $f$ is
stochastically stable.
\end{maintheorem}
The same proof gives the following result for endomorphisms
of compact manifolds in higher dimension, with a technical
assumption on the rate of decay of the measure of sets of
points with first hyperbolic time.
\begin{maintheorem}\label{mthm:higher}
Let $f:\T^n\to\T^n$ be a $C^2$ diffeomorphism away from a
non-degenerate critical set, which is also a non-uniformly
expanding map whose orbits have slow recurrence to the
critical set, where $n>1$.
If the first hyperbolic time map satisfies
\begin{align}\label{eq:summa}
\sum_{n\geq
1}\sum_{j=0}^{n-1}\lambda(f^{j}(h^{-1}(n)))<\infty,
\end{align}
then $f$ is stochastically stable for a family of adapted
random perturbations given by (\ref{pertaditiva}).
\end{maintheorem}
\subsection{Comments and organization of the text}
\label{sec:organization-text}
The method of proof relies on showing that the random
adapted perturbation preserve hyperbolic times in such a way
that the first hyperbolic time map of the random system is
the same as the first hyperbolic time map of the original
system. In this way, we can use the main result of
\cite{AA03} to prove (weak$*$) stochastic stability.
This construction of the adapted random perturbation depends
on an assumption of integrability of the first hyperbolic
time map for one-dimensional non-uniformly expanding
maps. For higher dimensional maps,
condition~(\ref{eq:summa}) is needed and apparently much
difficult to check.
\begin{conjecture}
\label{sec:higherdim}
A non-uniformly expanding map having a sufficiently
fast rate of decay of correlations satisfies the
summability condition~(\ref{eq:summa}).
\end{conjecture}
We presented the results using a uniform measure for
$\theta_\epsilon$ but many simple generalizations are
possible assuming only that $\theta_\epsilon\ll\lambda$ and
$\supp(\theta_\epsilon)\xrightarrow[\epsilon\searrow0]{}\{0\}$.
We also avoided technical complexities by considering only
maps on tori, on which it is clear how to make additive
perturbations in the form~(\ref{pertaditiva}). However, it
is possible (although technically more involved) to make
similar perturbations in any compact manifold, arguing along
the lines of \cite[Example 2]{vdaraujo2000}. We focus on
additive perturbations on parellelizable manifolds to
present the ideas in a simple form.
\subsection*{Acknowledgments}
This is M.P. PhD thesis prepared at the Federal University
of Rio de Janeiro, Brazil. All authors are indebted to the
research facilities provided by the Mathematics Institute at
this University.
\section{Examples of Application}
\label{sec:examples-application}
Theorem~\ref{principal} ensures stochastic stability for
any non-uniformly expanding map that has slow recurrence to
the critical set with the first hyperbolic function in
$L^{p}$ for $p>3$. We present natural conditions on
the speed of expansion that imply this integrability
condition and use this to obtain examples where our
results apply.
We note that, from slow recurrence to the critical set
and non-uniform expansion, Lemma~\ref{le:infhyptimes}
ensures that for $c=-\log\sigma>0$ and small
$\gamma,\delta>0$ the following values are well
defined $\lambda$-a.e.
\begin{align*}
\SD(x)
&=
\min\left\{k\ge1: \frac1n\sum_{j=0}^{n-1} -\log
d_{\delta}(f^{j}(x),\SC)\leq\gamma\quad\text{for
all}\quad n\ge k\right\};\quad\text{and}
\\
\SE(x)
&=\min\left\{k\ge1: \frac1n\sum_{j=0}^{n-1}\log
\|Df(f^j(x))^{-1}\|\ge \frac{c}3\quad\text{for
all}\quad n\ge k\right\}.
\end{align*}
We combine these two estimates in the set
\begin{align*}
\Gamma_n=\{x\in M:\SD(x)>n\quad\text{and}\quad\SE(x)>n\}.
\end{align*}
We now observe that, trivially from the definitions,
every point in $\Gamma_n$ has a first
$(\sigma,\delta)$-hyperbolic time of at most $n$, thus
\begin{align*}
h^{-1}(\{n\})\subset h^{-1}(\{1,2,\dots,n\})\subset\Gamma_n.
\end{align*}
\begin{remark}
\label{rmk:Gamma_tail}
If for some constant $\kappa>0$ and $q>4$ we have
$\lambda(\Gamma_n)\le\kappa n^{-q}$ for all
sufficiently large $n$, then $h\in L^p(\lambda)$ for
some $p>3$, since for all small enough $\epsilon>0$
we have $\sum_{n>m}
n^{q-1-\epsilon}\lambda(h^{-1}(\{n\}))
\le
\kappa\sum_{n>m}n^{-1-\epsilon}<\infty$ for some $m>1$.
\end{remark}
\subsection{Non-uniformly expanding maps with infinitely
many critical points}
\label{sec:non-uniformly-expand}
We now present the main motivating example of application of
Theorem~\ref{principal}: maps with infinite critical points.
We consider the family $f_t:\sS^1\to\sS^1$ from
the work of Pacifico-Rovella-Viana \cite{PRV98}. This
family is obtained from the map
$\hat{f}:[-\epsilon_1,\epsilon_1]\to[-1,1]$ given by
\begin{equation}
\label{e2.3}
\hat{f}(z)=\left\{
\begin{array}{ll}
az^{\alpha}\sin(\beta\log(1/z)) & \mbox{ if }z>0\\
-a|z|^{\alpha}\sin(\beta\log(1/|z|)) & \mbox{ if }z<0,
\end{array}
\right.
\end{equation}
where $a>0$, $0<\alpha<1, \, \beta>0$ and $\ep_1>0$, see
Figure~\ref{Fig4}.
\begin{figure}[h!] \centering
\includegraphics[height=5.5cm]{Fig4}
\caption{\label{Fig4} Graph of the circle map $f$.}
\end{figure}
Maps $\hat{f}$ as above have infinitely many critical points,
of the form
\begin{equation}
\label{e2.4}
x_{k}=\hat{x}\exp(-k\pi/\beta)
\mbox{ and }x_{-k}=-x_{k}
\mbox{ for each large }k>0
\end{equation}
where
$\hat{x}=\exp\big(-\frac1{\beta}\tan^{-1}\frac{\beta}{\alpha}\big)>0$
is independent of $k$. Let $k_0\ge 1$ be the smallest
integer such that $x_k$ is defined for all $|k|\ge k_0$, and
$x_{k_0}$ is a local minimum.
We extend this expression to the whole circle $\sS^1=I/\{-1\sim
1\}$, where $I=[-1,1]$, in the following way. Let $\tilde{f}$ be an
orientation-preserving expanding map of $\sS^1$ such that
$\tilde{f}(0)=0$ and $\tilde{f}'>\tilde\sigma$ for some constant
$\tilde\sigma>>1$. We define
$\epsilon=2\cdot x_{k_0}/(1+e^{-\pi/\beta})$,
so that $x_{k_0}$ is the middle point of the interval
$(e^{-\pi/\beta}\epsilon,\epsilon)$ and
fix two points $x_{k_0}<\hat{y}<\tilde{y}<\epsilon$, with
\begin{equation}\label{eq:condhaty}
|\hat{f}'(\hat{y})|>>1\quad\mbox{and also}\quad
2 \frac{1-\epsilon^\tau}{1+e^{-\pi/\beta}} x_{k_0}
> \hat y > x_{k_0},
\end{equation}
where $\tau$ is a small positive constant and we take
$k_0=k_0(\tau)$ sufficiently big (and $\epsilon$ small
enough) in order that \eqref{eq:condhaty} holds. Then
we take $f$ to be any smooth map on $S^1$ coinciding
with $\hat{f}$ on $[-\hat{y},\hat{y}]$, with
$\tilde{f}$ on $S^1\setminus[-\tilde{y},\tilde{y}]$,
and monotone on each interval $\pm[\hat{y},\tilde{y}]$.
Finally let $f_t$ be the following one-parameter family of circle
maps unfolding the dynamics of $f=f_0$
\begin{equation}
\label{e2.4,8}
f_{t}(z)=\left\{
\begin{array}{ll}
f(z)+t & \mbox{ for } z\in (0,\epsilon]\\
f(z)-t & \mbox{ for } z\in [-\epsilon,0)
\end{array}
\right.
\end{equation}
for $t\in(-\epsilon,\epsilon)$. For
$z\in\sS^1\setminus[-\epsilon,\epsilon]$ we assume
only that $\big|\frac{\partial}{\partial z}
f_t(z)\big|\ge2$.
From the works \cite{PRV98} together with \cite{ArPa04},
it is known that for a positive Lebesgue measure subset $P$
of parameters $t$ the map $f_t$ has a dense orbit, is
non-uniformly expanding with slow recurrence to the
critical set $\SC=\{0\}\cup\{x_k: |k|\ge k_0\}$,
admits a unique absolutely continuous invariant
probability measure $\mu_t$ and the corresponding tail set
$\Gamma_n^t$ satisfies $\lambda(\Gamma_n^t)\le C
e^{-\xi n}$ for some constants $C,\xi>0$; see
\cite[Theorem A]{PRV98} and \cite[Theorems A, B and C]{ArPa04}.
Hence, from Remark~\ref{rmk:Gamma_tail} we can apply
Theorem~\ref{principal} to each of these maps $f_t$.
\begin{corollary}
\label{cor:infinite-modal}
Given $t_0\in P$, the map $f=f_{t_0}$ is
stochastically stable for the adapted family of
random perturbations $(f_t,\theta_\epsilon^\N)$
obtained according to Theorem~\ref{principal}.
\end{corollary}
This is the first result on stochastic stability of
one-dimensional maps with infinitely many critical points.
\subsection{Non-uniformly expanding quadratic maps}
\label{sec:non-uniformly-expand-1}
The quadratic family $f_{a}:[-1,1]\rightarrow[-1,1]$
given by $f_{a}=1-ax^{2}$ for $0<a\leq 2$ provides a
class of maps satisfying the hypothesis of
Theorem~\ref{principal}. Indeed, Jakobson~\cite{Ja81} and
Benedicks-Carleson~\cite{BC85} prove the existence of a
physical measure for a positive Lebesgue measure subset
of parameters $a\in (0,2]$ for which $f_a$ is
non-uniform expanding with slow recurrence to the
critical point; Young~\cite{Yo92} and,
more recently, Freitas~\cite{freitas} obtain exponential
decay of the tail sets $\Gamma_n$. From
Remark~\ref{rmk:Gamma_tail} we can apply
Theorem~\ref{principal} for all the maps in the
positive Lebesgue measure subset of parameters found by
Jacobson and Benedicks-Carleson, obtain stochastic stability
for this class of maps. We note that strong stochastic
stability was obtained for the same class in the work of
Baladi-Viana~\cite{BaV96}.
\subsection{Intermitent Maps}
\label{sec:intermitent-maps}
Our results enables us also to deduce stochastic
stability for a class of intermittent applications
\cite{manneville1980}, where this property was obtained
for maps $C^{1+\alpha}$ but with the condition that
$\alpha\geq 1$; see \cite{ArTah}. Recently Shen, together with van
Strien in \cite{ShvS13}, obtained strong stochastic stability
for the Manneville-Pomeaux family of intermittent maps,
answering the questions raised in \cite{ArTah}.
Consider $\alpha>0$ and the map $T_\alpha: [0,1]\rightarrow [0,1]$ given by:
$$
T_\alpha(x)=\left\{
\begin{array}{ccc}
x+2^{\alpha}x^{1+\alpha}, & \mbox{~if~} & x\in[0,\frac{1}{2})\\
x-2^{\alpha}(1-x)^{1+\alpha}, & \mbox{~if~} & x\in[\frac{1}{2},1].
\end{array}
\right.
$$
This map is a $C^{1+\alpha}$ local diffeomorphism of
$\sS^1:=[0,1]/\{0\sim 1\}$, so there are no critical
points. The unique fixed point is $0$ with
$DT_\alpha(0)=1$. If $\alpha\geq 1$, then the Dirac mass in
zero $\delta_{0}$ is the unique physic probability measure
and so the Lyapunov exponent in Lebesgue almost every point
is zero; see \cite{thaler1983}. But, for $0<\alpha<1$, there
exists a unique absolutely continuous invariant probability
$\mu$ which is physical and whose basin has full Lebesgue
measure. To deduce stochastic stability for $\alpha$ in a
subinterval of $(0,1)$, we need some definitions and
results.
Given a $T_\alpha$-invariant and ergodic probability measure
$\mu$ and $\epsilon>0$ we define the \emph{large deviation
in time $n$ of the time average of the observable
$\varphi$ from its spatial average} as
\begin{align*}
\mathrm{LD}_{\mu}(\varphi,\epsilon,n)
=
\mu\left\{x : \left|\frac{1}{n}\sum_{i=0}^{n-1}\varphi(
f^{i}(x))-\int\varphi d\mu\,\right|>\epsilon\right\}
\end{align*}
We note that Birkhoff's Ergodic Theorem ensures
$\mathrm{LD}_{\mu}(\varphi,\epsilon,n)\xrightarrow[n\rightarrow\infty]{}0$
and the rate of this convergence is a relevant quantity.
Since $T_\alpha$ is a local diffeomorphism we have
$\Gamma_{n}=\{x\in\sS^1: \SE(x)> n \}$ and this is naturally
a deviation set for the time averages of
$\varphi=\log|DT_\alpha|$: if $\mu_\alpha$ is the unique
absolutely continuous $T_\alpha$-invariant probability, then
the Lyapunov exponent $\lambda=\int\varphi\,d\mu>c$, where
$c>0$ is the constant in the definition of non-uniform
expansion (\ref{nue}), and so for all large enough $n>1$ and
small enough $\epsilon>0$
\begin{align}\label{eq:LD-Gamma}
\mathrm{LD}_{\mu}(\log|DT_\alpha(x)|,\epsilon,n)\ge\mu(\Gamma_{n}).
\end{align}
To estimate $\mu(\Gamma_n)$ we now relate
$\mathrm{LD}_{\mu}$ with the rate of decay of correlations.
Let \( \mathcal B_{1}, \mathcal B_{2} \) denote Banach
spaces of real valued measurable functions defined on
\( M \). We denote the \emph{correlation} of non-zero
functions $\varphi\in \mathcal B_{1}$ and \( \psi\in
\mathcal B_{2} \) with respect to a measure $\mu$ as
\begin{align*}
\mathrm{Cor}_\mu (\varphi,\psi)
=
\frac{1}{\|\varphi\|_{\mathcal
B_{1}}\|\psi\|_{\mathcal B_{2}}}\left|\int
\varphi\, \psi\, d\mu-\int \varphi\, d\mu\int
\psi\, d\mu\right|.
\end{align*}
We say that we have \emph{decay of correlations}, with
respect to the measure $\mu$, for observables in
$\mathcal B_1$ \emph{against} observables in $\mathcal
B_2$ if, for every $\varphi\in\mathcal B_1$ and every
$\psi\in\mathcal B_2$ we have
\begin{align*}
\mathrm{Cor}_\mu(\varphi,\psi\circ f^n)\xrightarrow[n\rightarrow\infty]{}0.
\end{align*}
The following result from~\cite{Melb09} allows us to relate
decay of correlations with large deviations; see also
\cite{AFLV11}. We say that a measure $\mu$ is
\emph{$f$-non-singular} if for all measurable sets $A$ such
that $\mu(A)=0$, then $\mu(f^{-1}(A))=0$.
\begin{theorem}[\cite{Melb09,AFLV11}]\label{exemplo 2}
Let \( f: M\to M \) preserve an ergodic probability
measure \( \mu \) with respect to which $f$ is
non-singular. Let \( \SB\subset L^{\infty}(\mu) \) be a
Banach space with norm \( \|\cdot \|_{\SB} \) and
$\varphi\in \SB$.
Let $\beta>0$ and suppose that there exists $\kappa>0$
such that for all $\psi\in
L^\infty(\mu)$ we have
\(
\mathrm{Cor}_\mu(\varphi,\psi\circ f^n) \le \kappa\cdot n^{-\beta}.
\)
Then, for every
\( \epsilon>0 \), there exists \( C=C(\varphi, \epsilon)>0 \)
such that \(
\mathrm{LD}_{\mu}(\varphi,\epsilon,n) \leq C n^{-\beta}.
\)
\end{theorem}
We now observe that the absolutely continuous
$T_\alpha$-invariant probability measure $\mu_\alpha$
$f$-non-singular and that the following estimate for the
rate of decay of correlations is known.
\begin{theorem}[Theorem 4.1 in \cite{LSV99}]
For all $\psi\in L^{\infty}$ and $\varphi\in C^{1}([0,1])$
such that $\int \varphi d\mu=0$ we have: $\left|\int
(\psi\circ T_\alpha^{n})\cdot\varphi\, d\mu\right| \le
A(\|\varphi\|_{C^1})\cdot \|\psi\|_{\infty}\cdot
n^{1-1/\alpha}(\log n)^{1/\alpha}$, where
$A:\mathbb{R}\rightarrow\mathbb{R}$ is an affine map.
\end{theorem}
Hence, since $\log|DT_\alpha(x)|$ is a bounded continuous
function on $[0,1]$, there is a constant $C>0$ such that
$$
\mathrm{Cor}_{\mu}(\log|DT_\alpha(x)|,\psi\circ
T_\alpha^{n}) < Cn^{1-1/\alpha}.\log n^{1/\alpha}.$$ From
Theorem~\ref{exemplo 2} and relation~(\ref{eq:LD-Gamma}) we
deduce that, for every $\delta>0$, we have a constant
$C_1>0$ such that
\begin{eqnarray}\label{eq:mu-gamma}
\mu(\Gamma_{n})<C_{1}\cdot n^{(1-1/(\alpha+\delta))}.
\end{eqnarray}
Since $\mu\ll\lambda$, we have $d\mu=h\,d\lambda$ with a
density function $h$ which, from \cite[Theorem A]{Hu04}, is
bounded, strictly positive and, for a neighborhood $I_0$ of
$0$ there are constants $R>0$ and $\sigma_{0}=
\lim_{x\rightarrow 0}\sum_{x_{1}\in
T_\alpha^{-1}(x)\setminus
I_{0}}\frac{h(x_{1})}{DT_\alpha(x_{1})}$ such that
$|x^{\alpha}\cdot h(x)-\sigma_0|\leq R\cdot x^{\alpha}$.
This enables us to find $\kappa>0$ such that
$\lambda(\Gamma_n)\le \kappa \mu(\Gamma_n)$ which, together
with~(\ref{eq:mu-gamma}) provides a constant $C>0$ such that
for all small $\delta>0$ and large $n$
$$\lambda(\Gamma_{n})<C \cdot n^{(1-1/(\alpha+\delta))}.$$
We therefore have for $p>3$, since $\delta>0$ may be take arbitrarily small
\begin{align*}
\sum_{n=1}^{\infty}n^{p}\cdot\lambda(\Gamma_{n})
<
C\sum_{n=1}^{\infty}n^{(p+1-1/(\alpha+\delta))}<\infty
\quad\text{for all}\quad 0<\alpha\le\frac{1}{p+2}.
\end{align*}
Thus, for any $p>3$, we get for $0<\alpha<\frac{1}{5}$ the
$L^p$ integrability of the first hyperbolic time map with respect
to $\lambda$ and, from Theorem~\ref{principal} we obtain
\begin{corollary}\label{cor:intermittentstochstab}
All intermitent maps $T_\alpha$ with parameters
$0<\alpha<\frac{1}{5}$ are stochastically stable under
adapted random perturbations.
\end{corollary}
\section{Adapted random perturbations}
\label{sec:adapted-random-pertu}
Here we construct adapted random perturbations. These
perturbations are constructed by an adequate choice of
hyperbolic times along almost all orbits. Then we show that
these specially chosen hyperbolic times are preserved under
the adapted random perturbations in such a way that the
random map is non-uniformly expanding and has
slow-recurrence for random orbits. In addition, the
hyperbolic times for a point $(\underline t,x)\in\Omega$
under the adapted random perturbations are the same as the
hyperbolic times of $x$ for the unperturbed dynamics.
The only assumption is that the original unperturbed map
admits a pair $(\sigma,\delta)$, with $0<\delta,\sigma<1$,
satisfying: the first $(\sigma,\delta)$-hyperbolic time map
$h$ is defined $\lambda$-almost everywhere and $h$ is
$L^p$-integrable for some $p>3$, i.e.,
$\sum_{n\ge1}n^p\lambda(h^{-1}(n))<\infty$.
In what follows we fix $(\sigma,\delta)$ as above and write
hyperbolic time to mean $(\sigma,\delta)$-hyperbolic time.
\begin{definition}\label{th adapt} The adapted hyperbolic time of
$x\in M\setminus\mathcal{C}$ is the number
$$H(x):=\sup\{h(z)-l;
x=f^l(z), z\in M ~\mbox{and}~l\geq 0\}$$ where
$h:M\rightarrow\mathbb{Z}^{+}$ is the first hyperbolic time
function.
\end{definition}
Note that $H(x)$ is a hyperbolic time for $x$. In fact, if
$x=f^l(z)$ for $l\ge1$ and some point $z$, and $h(z)$ is the
first hyperbolic time of $z\in M$, then $h(z)-l$ is a
hyperbolic time for $f^{l}(z)=x$. Moreover, it is clear that
$H(x)\ge h(x)$ if $h(x)$ is finite.
To check that $H$ is finite almost everywhere, we note that
\begin{eqnarray}\label{controle do tempo adaptado}
H(x)\leq\sup\left\{n\in\mathbb{Z}^{+}:
x\in\bigcup_{i=0}^{n-1} f^{i}(h^{-1}(n))\right\}.
\end{eqnarray}
Since for a one-dimensional map $f$ we have $|\det
Df|=\|Df\|=|Df|$, then the assumption $h\in L^{p}(\lambda)$
with $p>3$ implies (\ref{eq:summa}) in the one-dimensional
setting.
\begin{lemma}
\label{le:summa1d}
Let $f$ be a non-uniformly expanding one-dimensional map
having slow recurrence to the non-degenerate critical set.
Let us assume that the first hyperbolic time map satisfies
$h\in L^p(\lambda)$ for some $p>3$. Then $ \sum_{n\geq
1}\sum_{j=0}^{n-1}\lambda(f^{j}(h^{-1}(n)))<\infty.$
\end{lemma}
\begin{proof}
We follow~\cite[Section 3]{AlCasPin}. We note that if $n\ge1$
is a $(\sigma,\delta)$-hyperbolic time, then $|\det
Df^n(x)|\ge a_n=\sigma^{-n}$. Let $q(x)=\min\{k\ge1:|\det
Df^k(x)|\ge a_k\}$. Then $q(x)\le h(x)$ and so $q\in
L^p(\lambda)$ if $h\in L^p(\lambda)$.
Let $W_n=\{x\in M:q(x)>n\}$. Then
$W_n\subset\cup_{m>n}h^{-1}(m)$ and so we can find
constants $\kappa,C>0$ such that
\begin{align*}
\lambda(W_n) \le \sum_{m>n}\lambda(h^{-1}(m)) \le
\sum_{m>n}\frac{\kappa}{m^{p}} \le \frac{C}{n^{p-1}}.
\end{align*}
Hence there exists $\beta>0$ and $N\in\N$ such that
$b_n=n^\beta$ satisfies
$b_n\le\min\{a_n,\lambda(W_n)^{-\epsilon}\}$ for all $n\ge
N$ and some $0<\epsilon<\frac{p-3}{p-1}$. In addition, we
clearly have $b_nb_k\ge b_{k+n}$ for all big enough
$k,n\in\N$.
In this setting, $U_n=\{x\in M:|\det Df^n(x)|\ge b_n\}$
is such that
\begin{itemize}
\item $\cup_{n\ge1} U_n$ has full Lebesgue measure, since
$T_n=\{x\in M: n$ is a $(\sigma,\delta)$-hyperbolic
time$\}$ satisfies $h^{-1}(n)\subset T_n\subset U_n$; and
\item if
$x\in U_n$ and $f^n(x)\in U_m$, then $x\in U_{n+m}$
\end{itemize}
(i.e., $(U_n)_{n\ge1}$ is a \emph{concatenated collection} as
defined in \cite{AlCasPin}). In addition, letting
$\hat q(x)=\min\{n\ge1:x\in U_n\}$, we have again $\hat
q(x)\le h(x)$ in general. However, if $f$ is one-dimensional,
then we obtain equality $\hat q(x)=h(x)$.
The choices of $U_n$ and the sequence $b_n$ ensure that
$\sum_{n\ge n_0}\sum_{j=0}^{n-1}\lambda\big(f^j(\hat
q^{-1}(n))\big)<\infty$; see \cite[Section 3]{AlCasPin}.
Moreover, in the one-dimensional setting, this series
coincides with the one in the statement of the lemma.
\end{proof}
Under this summability condition we obtain the following.
\begin{lemma}[Lemma 2.1 in \cite{AlCasPin}]
\label{tempo adaptado finito}
If (\ref{eq:summa}) is true, then $H(x)<\infty$ to
$\lambda$-almost every $x\in M$.
\end{lemma}
\begin{proof}
For $\lambda$-almost every $x\in M$ we consider the set
$\mathcal{K}(x)=\{f^{j}(x)\}_{j=0}^{h(x)-1}$, which we
call a \emph{chain}. Suppose that for some $z\in M$ we
have that $z$ belongs to infinitely many chains
$\mathcal{K}_{j}(x_{j})=\{x_{j},f(x_{j}),\ldots,f^{s_{j}-1}(x_{j})\}$
for $j\geq 1$ where $s_{j}=h(x_{j})$ is the first
hyperbolic time for $x_{j}$ and $s_{j}\rightarrow \infty$.
Now for each $j\geq 1$ we take $1\leq r_{j}<s_{j}$
such that $z=f^{r_{j}}(x_{j})$ and claim that $\lim
r_{j}=\infty$. Indeed, otherwise, taking a subsequence of
$r_{j}$, we can assume that there is $N>0$ such that
$r_{j_{k}}<N$, $\forall k\geq 1$. But this implies that
$x_{j}\in\cup_{i=1}^{N}f^{-i}(z)$, $\forall j\geq 1$ and
so the number of elements of $\cup_{i=1}^{N}f^{-i}(z)$ is finite:
$\#(\cup_{i=1}^{N}f^{-i}(z))<\infty$. However we are
assuming that the number of chains is infinite. This
contradiction proves the claim.
Hence $r_{j}\rightarrow\infty$ and $z=f^{r_{j}}(x_{j})\subset
f^{r_{j}}(h^{-1}(s_{j}))$ and so we get
$$z\in\displaystyle\cup_{n\geq k} \cup_{j=0}^{n-1}f^{j}(h^{-1}(s_{j})),~~
\forall k\geq 0.$$ Since $\sum_{n\geq
1}\sum_{j=0}^{n-1}\lambda(f^{j}(h^{-1}(n)))<\infty$, we
obtain $\lambda(\cup_{n\geq
k}\cup_{j=0}^{n-1}f^{j}(h^{-1}(n)))\xrightarrow[k\rightarrow\infty]{}
0$. Then the set of points belonging to infinitely many
chains has null Lebesgue measure. Finally, from relation
$(\ref{controle do tempo adaptado})$ the proof of the lemma
is complete.
\end{proof}
Note that it is not possible ensure that $H(f(x))=H(x)-1$ in
general, because $x$ and $f(x)$ can be in orbits of
different points, namely $z\neq w$ whose first hyperbolic
times do not satisfy the relation $h(w)=h(z)-1$. Then the
adapted hyperbolic time for $f(x)$ can be bigger than
$H(x)-1$. However, note that $H(f(x))$ can not be smaller
than $H(x)-1$ because $x$ already has $H(x)$ as hyperbolic
time. In any case we have the following important
\emph{monotonicity property}
of our choice of adapted hyperbolic time
\begin{align}\label{eq:adapted-hyp-time}
H(f(x))\geq H(x)-1.
\end{align}
Similarly we obtain $H(f^{j}(x))\geq H(x)-j$ for $0\le j <
H(x)$ as long as $H(x)$ is finite.
\begin{lemma}[Lemm 5.2 in \cite{ABV00}]\label{lema 5.2}
Given $\sigma<1$ and $\delta>0$, there is $\delta_{1}>0$
such that if $n$ is a $(\sigma,\delta)$-hyperbolic time for
$x\in M\setminus\mathcal{C}$ then there exits a neighborhood
$V_{n}(x)$ of $x$ such that:
\begin{enumerate}
\item[1.] $f^{n}$ maps $V_{n}$ diffeomorphically into the
ball of radius $\delta_{1}$ centered at $f^{n}(x)$.
\item[2.] For all $1\leq k<n$ and $y,z\in V_{n}(x)$
$$\mbox{dist}(f^{n-k}(y),f^{n-k}(z))\leq\sigma^{k/2}.\mbox{dist}(f^{n}(y),f^{n}(z)).$$
\end{enumerate}
\end{lemma}
By the definition of hyperbolic time, if $n$ is a
$\sigma$-hyperbolic time for a point $x\in M$, then there are
neighborhoods $V_{n-j}\subset B_{\delta_{1}\sigma^{j}}(f^{j}(x))$ of
$f^{j}(x)$ which are sent in time $j$ diffeomorphically into the ball
$B_{\delta_{1}}(f^{n}(x))$ for all $0\leq j\leq n$.
\begin{lemma}\label{le:Hconst}
In our setting, for $\lambda$-almost every $x$, there
exists an open neighborhood $V_H(x)$ of $x$ such that
$H\mid V_H(x)$ is constant.
\end{lemma}
\begin{proof}
The subset $Y$ of $M$ of points having some hyperbolic
time is such that $\lambda(Y)=1$. Hence $f^{-1}(Y)$ also
has full $\lambda$-measure since $f$ is a local
diffeomorphism away from a critical/singular set with zero
$\lambda$-measure. Therefore $\lambda(\cap_{n\ge1}(Y\cap
f^{-n}(Y)))=1$ and we conclude that every point in the
pre-orbit $\cup_{n\ge1}f^{-n}(\{x\})$ of Lebesgue almost
every point $x$ has some hyperbolic time.
Let $X$ be the subset of $M$ such that $H(x)<\infty$ for
all $x\in X$. We know that $\lambda(X)=1$.
Let us now fix $x\in Y\cap X$. Hence we have $h(y)<\infty$
for every point $y$ in the pre-orbit of $x$ and,
moreover, if $x=f^{k}(y)$ then $h(y)-k\le H(x)$ by
definition of $H(x)$.
It follows that the neighborhood $V_{h(y)}(y)$ of $y$
associated to the hyperbolic time $h(y)$ is such that
$f^k(V_{h(y)}(y))\supset V_{H(x)}(x)$, since $h(y)-k\le
H(x)$.
Therefore, for $x'\in V_{H(x)}(x)\subset f^k(V_{h(y)}(y))$
the inverse map $\vfi$ of $f^k\mid V_{h(y)}(y)$ is such
that $\vfi(x')=y'\in V_{h(y)}(y)$. Thus $h(y')\le h(y)$
(recall that $h(y')$ is the first hyperbolic time of $y'$
and $h(y)$ is already a hyperbolic time for $y'$). It
follows that $h(y')-k\le h(y)-k\le H(x)$.
This argument is true of any element $y$ of the pre-orbit of
$x$, whose neighborhood $V_{h(y)}(y)$ is sent by $f^k$ to
a set covering $V_{H(x)}(x)$. Hence all pre-images of
points $x'\in V_{H(x)}(x)$ respect the same inequality,
that is, $H(x')\le H(x)$. But the reverse inequality is
also true by definition of $H$, since $x'\in V_{H(x)}(x)$
has $H(x)$ as an hyperbolic time. This completes the proof.
\end{proof}
\begin{remark}
\label{rmk:thadapt}
We make the convention that $H(x)=1$ wherever the supremum
in Definition~\ref{th adapt} is not finite.
\end{remark}
\begin{remark}\label{rmk:HnearC}
Besides the obvious relation $H(x)\ge h(x)$ almost
everywhere, we can say more in certain regions. Let us
assume that $V$ is the largest open neighborhood of the
critical set $\SC$ such that $|Df\mid (M\setminus
V)|>\sigma^{-1}$ and $V\cap f^{-1}(V)=\emptyset$. Then
$H=h$ in $V$, since $h(x)\ge2$ for almost all points $x\in
V$ and all pre-orbits of $x$ have $1$ as a first
$\sigma$-hyperbolic time, which is smaller that $h(x)-1$.
The above conditions on a neighborhood of the critical set
are easily checked for non-uniformly expanding quadratic
maps and, by \cite[Section 4]{PRV98}, this is also true
for the infinite-modal family $f_\mu$ at every parameter
of the positive Lebesgue measure subset $P$; see
Section~\ref{sec:examples-application}.
\end{remark}
\subsection{Preservation of hyperbolic times}
\label{sec:preserv-hyperb-times}
Now we show that hyperbolic times are preserved if we define
a random perturbation adapted to the structure of hyperbolic
times using $H$, as in (\ref{pertaditiva}) with
$\zeta(x)=\xi e^{-\eta H(x)^2}$ for suitably chosen
constants $\xi,\eta>0$. We first define the notions of
hyperbolic times and slow recurrence in our random setting.
\subsubsection{Random non-uniformly expanding maps and random
slow recurrence}
\label{sec:random-non-uniformly}
We now define the analogous notions of non-uniform
expansion and slow recurrence for random dynamical systems
in our setting.
\begin{definition}\label{nue para o caso aleatorio}
We say that a map is non-uniformly expanding map for
random orbits if there exists a constant $c>0$ such that
for $\epsilon>0$ sufficiently small and
$\theta^{\mathbb{N}}_{\epsilon}\times
\lambda$-a.e. $(\underline{t},x)$ we
have
$\limsup_{n\to+\infty}\frac1n
\sum_{j=0}^{n-1}\log\| Df(f^{j}_{\un t}(x))^{-1}\|\leq -c<0.$
\end{definition}
\begin{definition}\label{rec sist alea}
We say that a random dynamical system
$(f_{\underline{t}},\theta_{\epsilon})$ has slow
recurrence to the critical set for random orbits if, for
all small enough $\gamma>0$, there exists
$\delta>0$ such that $\theta^{\mathbb{N}}_{\epsilon}\times
\lambda$-a.e. $(\underline{t},x)$ we have
$\limsup_{n\to+\infty}\frac{1}{n}\sum_{j=0}^{n-1}-\log
d_{\delta}(f_{\un t}^{j}(x),\SC)\leq \gamma.$
\end{definition}
\subsubsection{Random hyperbolic times}
\label{sec:random-hyperb-times}
An definition of hyperbolic analogous to \ref{def:hyptimes}
can be made for the random system
$(f_{t},\theta_{\epsilon})$.
\begin{definition}[Random Hyperbolic Time]
\label{def:hyptimesrand}
Given $\sigma\in(0,1)$ and $\delta>0$, we say that n is a
$(\sigma,\delta)$-hyperbolic time for a point
$(\underline{t},x)\in \Omega\times M$ if:
\begin{align*}
\prod_{j=n-k}^{n-1}\|
Df_{t_{j+1}}(f_{\underline{t}}^{j}(x))^{-1}\|
\leq\sigma^{k}
\quad\mbox{and}\quad
d_{\delta}(f_{\underline{t}}^{n-k}(x),\SC)\geq\sigma^{bk},
\quad\text{for all}\quad 1\leq k\leq n.
\end{align*}
\end{definition}
\begin{theorem}\label{nuero}
If $f$ is non-uniformly expanding with slow recurrence to
the critical set in the interval or the circle, then for
each $\delta>0$ there is
$\zeta:M\rightarrow\mathbb{R}^{+}$ mensurable and locally
constant such that the adapted random perturbation
$(\ref{pertaditiva})$ satisfies: there exists
$0<\sigma<\hat\sigma <1$ such that for
$\lambda$-almost every point $x$ and all
$\underline t\in[-1/2,1/2]^\nat$ has $H(x)$ as
$(\hat\sigma,\delta)$-hyperbolic time.
\end{theorem}
We assume that $f$ has a non-degenerate critical set
$\mathcal C$. We also assume without loss of generality in
what follows that $B\delta^{1-\beta}\le\log\sigma^{-1/2}$
and $\delta_1=\frac12\delta\le\frac12$, where $B,\beta>0$
are given in the non-degeneracy conditions of $\mathcal
C$.
\begin{remark}\label{rmk:localdiffeo}
The same arguments and constructions presented in
this section enable us to trivially obtain a version
of Theorem~\ref{nuero} for the local diffeomorphism
case, that is, the case where there are no critical
(or singular) points: $\mathcal C=\emptyset$.
\end{remark}
\begin{remark}
\label{rmk:hbdd}
Since by construction $h(x)\le H(x)$, whenever $h(x)$ is
finite, then we have for $\lambda$-a.e. $x$ that
$V_H(x)\subset V_n(x)$ for all hyperbolic times $n$ of $x$
such that $h(x)\le n\le H(x)$.
Moreover, we have that the random orbit of $(\un t,x)$ has
the same hyperbolic times $n$ of the unperturbed orbit of
$x$ as long as $h(x)\le n\le H(x)$. In particular, the
first hyperbolic time of $(\un t,x)$ is given by $h(x)$.
\end{remark}
\begin{lemma}
\label{le:bdd-der-above}
There exists $\omega>\sigma^{-1/2}$ such that, if $n$ is a
$(\sigma,\delta)$-hyperbolic time for $x$, then
$\|Df^n(x)\|\le\omega^n$.
\end{lemma}
\begin{proof}
Using the non-degenerate condition (S1) we get
$\log\|Df(x)\|\le\log B-\beta\log d(x,\SC)$. Hence,
since $n$ is a hyperbolic time, we have from their
construction that they satisfy
(\ref{eq:hip-times-prop}) which implies
\begin{align*}
\log\|Df^n(x)\|
&\le
\sum_{j=0}^{n-1}\log\|Df(f^j(x))\|
\le
n\log B-\beta\sum_{j=0}^{n-1} \log d(f^j(x),\SC)
\\
&\le
\log B^n +\beta\sum_{j=0}^{n-1} -\log
d_\delta(f^j(x),\SC)
+\beta\sum_{d(f^j(x),\SC)\ge\delta} -\log d(x,\SC)
\\
&\le
\log B^n +\beta\epsilon n -\beta n \log\delta
=
n(\log B +\beta(\epsilon-\log\delta))
\end{align*}
and so $\|Df^n(x)\|\le\omega^n$, where $\omega=\max\{\log B
+\beta(\epsilon-\log\delta), \sigma^{-1/2}\}$.
\end{proof}
\begin{lemma}\label{le:innerball}
If $n$ is a $(\sigma,\delta)$-hyperbolic time for $x$, then
$B_{\delta_{1}\omega^{-(n-j)}}(f^{j}(x))\subset
V_{n-j}(f^j(x))\subset
B_{\delta_1\sigma^{(n-j)/2}}(f^{j}(x))$ for each $0\leq
j\leq n$.
\end{lemma}
This result is essential to show that to keep the hyperbolic
time under perturbation all that we need is to maintain the
random orbits within a certain distance to the unperturbed
orbit during the iterated of the adapted hyperbolic time.
\begin{proof}[Proof of Lemma~\ref{le:innerball}]
We have
$d_{\delta}(f^{j}(x),\mathcal{C})\geq\sigma^{b(n-j)}$
for all $0\leq j\leq n$ and so either
$d(f^{j}(x),\mathcal{C})\geq\sigma^{b(n-j)}$ with
$f^j(x)\in B_\delta(\mathcal C)$, or
$d(f^{j}(x),\mathcal{C})\geq\delta$.
Hence for $y\in B_{\delta_{1}\sigma^{(n-j)/2}}(f^{j}(x))$
we have either
$\frac{d(y,f^{j}(x))}{d(f^{j}(x),\mathcal{C})}\le
\delta_1\sigma^{(1/2-b)(n-j)}\le\frac12$ or
$\frac{d(y,f^{j}(x))}{d(f^{j}(x),\mathcal{C})}\le
\frac{\delta_1}{\delta}\sigma^{(n-j)/2}\le\frac12$ for
$0\leq j\leq n$ (recall that $0<b\le1/2$ from the
definition of non-degenerate critical set). This enables
us to use non-degeneracy conditions (S1) and (S2).
For $y\in B_{\delta_{1}\sigma^{(n-j)/2}}(f^{j}(x))$
since $b\beta\le1/2$, the value of
$B\frac{d(y,f^{j}(x))}{d(f^{j}(x),\mathcal{C})^{\beta}}$ is
bounded above by either
$B\delta_1\sigma^{(1/2-b\beta)(n-j)/2}$ or
$B\delta_1\delta^{-\beta}\sigma^{(n-j)/2}=\frac{B}2\delta^{1-\beta}\sigma^{(n-j)/2}$,
and both are smaller than $\log\sigma^{-1/2}$. Thus from
(S2) for all $y\in B_{\delta_{1}\sigma^{(n-j)/2}}(f^{j}(x))$
\begin{align}\label{eq:df-y}
\sigma^{1/2}\|Df(f^{j}(x))^{-1}\| \le\| Df(y)^{-1}\|\le\sigma^{-1/2}\|
Df(f^{j}(x))^{-1}\|.
\end{align}
For $j= n-1$ above, we get for every $y\in
B_{\delta_{1}\sigma^{1/2}}(f^{n-1}(x))$
$$\sigma^{1/2}=\sigma^{-1/2}\|Df(f^{n-1}(x))^{-1}\|
\ge\|Df(y)^{-1}\|\ge\sigma^{1/2}\|Df(f^{n-1}(x))^{-1}\|\ge\sigma^{3/2}.$$
Hence, a smooth curve $\gamma$ from $f^n(x)$ to the boundary
of $B_{\delta_1}(f^n(x))$ and inside this ball must be such
that the unique curve $\tilde\gamma$ contained in
$V_{1}(f^{n-1}(x))$ such that $f^{n-1}(x)\in\tilde\gamma$
and $f(\tilde\gamma)=\gamma$ satisfies
$\sigma^{3/2}\delta_1=\sigma^{3/2}\ell(\gamma)\le\ell(\tilde\gamma)\le
\sigma^{1/2}\ell(\gamma)=\delta_1\sigma^{1/2}$, where
$\ell(\cdot)$ denotes the length of any smooth curve on $M$
and, recall, $f^{n-j}\mid
V_{n-j}(f^{j}(x)):V_{n-j}(f^{j}(x))\to B_{\delta_1}(f^n(x))$
is a diffeomorphism for all $j=0,\dots,n-1$. Thus
$B_{\delta_1\sigma^{1/2}}(f^{n-1}(x))\supset
V_{1}(f^{n-1}(x))\supset
B_{\delta_1\sigma^{3/2}}(f^{n-1}(x))$. In particular this
shows that the statement of the Lemma is true for
$n=1$, since $\omega>\sigma^{-1/2}$.
Now we argue by induction assuming the Lemma to be true for
all hyperbolic times up to some $n\ge1$ and consider $x$
having $n+1$ as a hyperbolic time. Then for each $1\leq j <n$
\begin{align*}
B_{\delta_{1}\omega^{-{n-j}}}(f^{j}(x))\subset
V_{n-j}(f^j(x))\subset
B_{\delta_1\sigma^{(n-j)/2}}(f^{j}(x))
\end{align*}
since $f(x)$ has $n$ as a hyperbolic time. For all $y\in
V_{n+1}(x)\cap B_{\delta_1\sigma^{(n+1)/2}}(x)$ we have $f(y)\in
V_1(f(x))$ and so by the induction assumption together with (\ref{eq:df-y})
\begin{align*}
\|Df^{n+1}(y)^{-1}\|
\le\prod_{i=0}^{n}\|Df(f^i(y))^{-1}\|
\le\prod_{i=0}^{n}(\sigma^{-1/2}\|Df(f^i(x))^{-1}\|)
\le \sigma^{(n+1)/2}
\end{align*}
Therefore, for any smooth curve $\gamma$ from $f^{n+1}(x)$
to the boundary of $B_{\delta_1}(f^{n+1}(x))$ and inside
this ball we have that the unique curve $\tilde\gamma$ contained in
$V_{n}(x)\cap B_{\delta_1\sigma^{(n+1)/2}}(x)$ such that $x\in\tilde\gamma$
and $f^{n+1}(\tilde\gamma)=\gamma$ satisfies
$\ell(\tilde\gamma)\le
\sigma^{(n+1)/2}\ell(\gamma)=\delta_1\sigma^{(n+1)/2}$. Hence
$V_{n+1}(x)\subset B_{\delta_1\sigma^{(n+1)/2}}(x)$.
Finally, from Lemma~\ref{le:bdd-der-above}, we obtain
for the same curves $\gamma,\tilde\gamma$ as above
$\ell(\gamma)=\ell(f^{n+1}\circ\tilde\gamma)\le\omega^{n+1}\ell(\tilde\gamma)$,
or
$\ell(\tilde\gamma)\ge\omega^{-(n+1)}\ell(\gamma)$. Since
this holds for any smooth curve $\gamma$ from
$f^{n+1}(x)$ to the boundary of
$B_{\delta_1}(f^{n+1}(x))$ and inside this ball, we
conclude that $V_{n+1}(x)$ contains
$B(x,\delta_1\omega^{-(n+1)})$. This completes the
inductive step and concludes the proof.
\end{proof}
\begin{remark}
\label{rmk:bdderivative}
From condition (S1) we obtain using the estimate (\ref{eq:df-y})
\begin{align*}
|Df(y)^{-1}|\ge\sigma^{1/2}|Df(f^j(x))^{-1}|\ge\frac{\sigma^{1/2}}B
d(f^j(x),\mathcal C)^{\beta}
\ge \frac{\sigma^{1/2}}B \sigma^{\beta b (n-j)}
\ge \frac{\sigma^{1/2}}B \sigma^{(n-j)/2}
\end{align*}
because $b\beta\le1/2$. Then we arrive at
\begin{align*}
|Df(y)|\le C\sigma^{-(n-j)/2}, \quad y\in V_{n-j}(f^j(x))
\end{align*}
where $C=B\sigma^{-1/2}$, whenever $x$ has $n\ge1$ as an
hyperbolic time and $0\le j < n$.
\end{remark}
\begin{proposition}\label{pr:rand-hyp-inside}
Let $f$ is a $C^2$ non-uniformly expanding
endomorphism having slow recurrence to the critical
set. There exist constants $\xi,\eta>0$ such that for
$\zeta(x)=\xi \omega^{-\eta H(x)^2}$ and the family
$f_t(x)=f(x)+t\cdot\zeta(x)$, if $x$ is such that
$H(x)$ is a hyperbolic time, then we have
$f_{\underline{t}}^{j}(x)\in V_{H(x)-j}(x)$ for all
$0\leq j\leq H(x)$ and each
$\underline{t}\in\Omega\subset[-1/2,1/2]^{\mathbb
N}$.
In particular, $H(x)$ is a
$(\hat\sigma,\delta)$-hyperbolic time for
$(\underline t,x)\in\Omega\times M$ whenever
$H(x)<\infty$, for a constant
$0<\sigma<\hat\sigma<1$.
Moreover, if
$\Omega\subset[-\epsilon_0,\epsilon_0]^\nat$ for some
$0<\epsilon_0<1/2$ and $H(x)<\infty$, then
$f_{\underline{t}}^{j}(x)\in
B_{\epsilon_0\delta_1\omega^{-\eta(H(x)-j)}}(f^j(x))$
for each $0\leq j\leq H(x)$ and for all $\underline
t\in\Omega$.
\end{proposition}
\begin{proof}
Let $\eta>3/2$ be big enough such that
$\max\{C\sigma^{2\eta-1/2},\sigma^{2\eta}\}<1/2$, choose
$\xi=\min\{\delta_1/2,1/2\}$ and fix $\underline
t=(t_1,t_2,\dots)\in\Omega$. Then
\begin{align*}
|f_{t_{1}}(x)-f(x)|\leq|t_{1}\zeta(x)|<\xi
\omega^{-\eta H(x)^2}<\delta_1\omega^{-(H(x)-1)}
\end{align*}
and so $f_{t_1}(x)\in
B_{\delta_1\omega^{-(H(x)-1)}}(f^{H(x)-1}(x))\subset
V_{H(x)-1}(f(x))$. Observe that there is $z\in
V_{H(x)}(x)$ such that $f_{t_1}(x)=f(z)$ and so
$H(f_{t_1}(x))=H(f(z))\ge H(z)-1=H(x)-1$.
Now we argue by induction on $k$ and assume that for $1\le
j\le k<n-1$ we have
\begin{enumerate}
\item $f^j_{\underline t}(x)\in
B_{\xi\omega^{-\eta(H(x)-j)^2}}(f^j(x))\subset
V_{H(x)-j}(f^j(x))$, and
\item $H(f^j_{\underline t}(x))\ge H(x)-j$.
\end{enumerate}
It is easy to see that this is true for $k=1$. For
$j=k+1$ we get, for some $w\in
B_{\delta_1\omega^{-\eta(H(x)-k)}}(f^k(x))$ in a segment
between $f^k_{\underline t}(x)$ and $f^k(x)$, according to
Remark~\ref{rmk:bdderivative}
\begin{align*}
|f^{k+1}_{\underline t}(x)-f^{k+1}(x)|
&\le
|f_{t_{k+1}}(f^k_{\underline t}(x))-f(f^k_{\underline
t}(x))|+|f(f^k_{\underline t}(x)) -f(f^k(x))|
\\
&\le
|t_{k+1}\zeta(f^k_{\underline t}(x))|+
|Df(w)|\cdot|f^k_{\underline t}(x)-f^k(x)|
\\
&<
\xi\omega^{-\eta H(f^k_{\underline t}(x))^2}
+ C\sigma^{-(H(x)-k)/2}\cdot\xi\omega^{-\eta(H(x)-k)^2}
\\
&\le
\xi\omega^{-\eta(H(x)-k)^2}(1+C\sigma^{-(H(x)-k)/2})
\\
&=
\xi\omega^{-\eta(H(x)-k-1)^2}\omega^{-\eta(2(H(x)-k)+1)}
(1+C\sigma^{-(H(x)-k)/2})
\\
&\le
\xi\omega^{-\eta (H(x)-k-1)^2}
(\omega^{-\eta(2(H(x)-k)+1)}+C\sigma^{(2\eta-1/2)(H(x)-k)})
\\
&\le\xi\omega^{-\eta (H(x)-k-1)^2}.
\end{align*}
The last inequality comes from the choice of $\eta$
and because $H(x)-k\ge1$ and
$\omega>\sigma^{-1/2}>\sigma^{-1}$. This proves that
part (1) of the inductive step. Then there exists
$z\in V_{H(x)}(x)$ such that
$f^{k+1}(z)=f^{k+1}_{\underline t}(x)$ and so
$H(f^{k+1}_{\underline
t}(x))=H(f^{k+1}(z))=H(z)-(k+1)=H(x)-(k+1)$,
completing the proof of the inductive step.
Now we check that $H(x)$ is still a hyperbolic time for
$(\underline t,x)$. This follows easily from the statement
of Proposition~\ref{pr:rand-hyp-inside} together with the
estimate~(\ref{eq:df-y}) and
Remark~\ref{rmk:Dft_Df}. However we have to relax the
constants: for $1\le k <H(x)$
\begin{align}\label{eq:pert-prod}
\prod_{j=n-k}^{H(x)-1} |Df_{t_{j+1}}(f^j_{\underline
t}(x))^{-1}|
=
\prod_{j=n-k}^{H(x)-1}|Df(f^j_{\underline t}(x))^{-1}|
\le
\prod_{j=n-k}^{H(x)-1}(\sigma^{-1/2}|Df(f^j(x))^{-1}|)
\le
\sigma^{k/2}
\end{align}
and
\begin{align}\label{eq:dist-crit-ratio}
d(f^{H(x)-j}_{\underline t}(x),\mathcal C)
&\ge
d(f^{H(x)-j}(x),\mathcal C)-d(f^{H(x)-j}_{\underline
t}(x),f^{H(x)-j}(x))\ge\sigma^{bj}-\delta_1\sigma^{j/2}
\\
&=\sigma^{bj}(1-\delta_1\sigma^{(b-1/2)j})
\ge
(1-\delta_1)\sigma^{bj}\nonumber
\end{align}
whenever $d(f^{H(x)-j}_{\underline t}(x),\mathcal C)<\delta$.
Hence $H(x)$ is a $(\hat\sigma,\delta)$-hyperbolic
time, for some $\sigma<\hat\sigma<1$ for all $x$ such
that $H(x)$ is finite.
Up until now, the proof of was done with a fixed
maximum size $1/2$ for the perturbation. If we consider
$\Omega\subset[-\epsilon_0,\epsilon_0]^\nat$ with
$0<\epsilon_0<1/2$, then the size of $t\cdot\zeta(x)$ is
reduced proportionally in all the previous estimates,
so that we obtain the last part of the statement.
\end{proof}
This concludes the proof of Theorem~\ref{nuero}.
\subsection{Asymptotic rates of expansion and recurrence on
random orbits}
\label{sec:asympt-rates-expans}
As a consequence of preservation of hyperbolic times,
we have the following uniform estimates for the
asymptotic rate of non-uniform expansion and slow
recurrence for random orbits, i.e., the estimates we
obtain do not depend on the perturbation as long as the
perturbation is small enough.
\begin{proposition} \label{pr:RSlow} If $f$ is a
non-uniformly expanding map with slow recurrence to
the critical set having a first hyperbolic time map
$L^p$-integrable for some $p>3$ then there is
$\epsilon_{0}\in(0,1/2)$ such that, for all
$0<r<\epsilon_{0}$, for $\lambda$-almost every point
$x$ and for all $\underline{t}\in[-r,r]^\nat$, we
have the bound $\liminf_{n\to+\infty} \frac{1}{n}
\sum_{j=0}^{n-1} -\log
d_{\delta}(f_{\underline{t}}^{j}(x),\mathcal{C})<2\epsilon$
and also $\liminf_{n\to+\infty}\frac1n
\sum_{j=0}^{n-1} \log \|Df(f^j_{\underline
t}(x))^{-1}\| \le\frac12\log\sigma$.
\end{proposition}
\begin{proof}
The last limit inferior is clear: since we have
infinitely many hyperbolic times $H(x)$ for
$\lambda$-almost every $x$, we also have infinitely
many hyperbolic times $H(x)$ for $\lambda$-almost
every $x$ and every $\underline
t\in[r,r]^\nat$. Hence
from~(\ref{eq:pert-prod}) we obtain infinitely many
hyperbolic times $n_1=H(x),
n_2=n_1+H(f^{n_1}_{\underline t}(x)),
n_3=n_2+H(f^{n_2}_{\underline t}(x)),\dots$ along the
random orbit of $x$ with the average rate
$\frac12\log\sigma$, which implies the stated bound
for the limit inferior.
For the limit inferior of slow approximation, we use
(\ref{eq:dist-crit-ratio}) to write for all $0\le j<H(x)$
\begin{align}\label{distcritica}
\frac{d(f_{\underline{t}}^{j}(x),\SC)}{d(f^{j}(x),\SC)}
&\geq
1-\frac{d(f_{\underline{t}}^{j}(x),f^{j}(x))}{d(f^{j}(x),
\SC)}
\ge 1-r\sigma^{(1/2-b)(H(x)-j)}.
\end{align}
From the definition of $d_\delta$ we can write, since
$0<r<1/2$ and $H(x)$ is a hyperbolic time
\begin{align*}
\sum_{j=0}^{H(x)-1}
-\log d_\delta(f_{\underline{t}}^{j}(x),\SC)
&\le
\sum_{j=0}^{H(x)-1}
-\log (1-r\sigma^{(1/2-b)(H(x)-j)})
+
\sum_{j=0}^{H(x)-1}
-\log d_\delta(f^j(x),\SC)
\\
&\le
\sum_{j=0}^{H(x)-1} 2r\sigma^{(1/2-b)(H(x)-j)}
+
\epsilon n
=
\frac{2r\sigma^{1/2-b}}{1-\sigma^{1/2-b}}+\epsilon n
\le 2\epsilon n
\end{align*}
if we take $0<r<\epsilon_0<1/2$ small enough.
The bound on the limit inferior follows again from the
existence of infinitely many hyperbolic times along the
orbit of $(\underline t,x)$ for $\lambda$-almost every
$x$ and all $\underline t\in[-r,r]^\nat$.
\end{proof}
\section{Uniqueness of absolutely continuous
stationary measure}
\label{sec:uniquen-absolut-cont}
As a consequence of the choice of adapted perturbations from
Theorem~\ref{nuero} and the family
$(\theta_\epsilon)_{\epsilon>0}$ of probability measures in
(\ref{eq:theta_ep}), we obtain the following.
\begin{theorem}\label{thm:uniquestationary}
For each sufficiently small $\epsilon>0$ in the choice of
$\zeta$ in the construction of an adapted random perturbation
from (\ref{pertaditiva}) as in Theorem~\ref{nuero}, there
exists a unique absolutely continuous and ergodic
stationary measure for the random dynamical system
$(f_{\underline{t}},\theta_\epsilon^{\mathbb{N}})$.
\end{theorem}
Consider the measure
$(f_{x})_{*}\theta_{\epsilon}^{\mathbb{N}}$ which is the
{\it push-foward} of the measure
$\theta_{\epsilon}^{\mathbb{N}}$ by $f_{t}:M\rightarrow M$
for a fixed $t\in\mbox{supp} \theta_{\epsilon}$, where we
write $f_{x}(\underline{t})$ for $f_{\underline{t}}(x)$. We
first mention a simple way to ensure the existence of a
stationary measure for
$(f_{\underline{t}},\theta_\epsilon^{\mathbb{N}})$.
\begin{lemma}\label{le:existencestationary}
For each sufficiently small $\epsilon>0$ in the choice of
$\zeta$ in the construction of an adapted random
perturbation from (\ref{pertaditiva}) as in
Theorem~\ref{nuero} and for $x\in M$ fixed, each weak$^*$
accumulation point of the sequence
$\mu_{n}^{\epsilon}(x)
=\frac{1}{n}\sum_{j=1}^{n}(f_{x}^{j})_{*}\theta_{\epsilon}^{\mathbb{N}}$
is a stationary measure.
\end{lemma}
\begin{proof}
Let $\mu^{\epsilon}$ be a weak$^*$ accumulation point of
the sequence $(\mu_{n}^{\epsilon}(x))_{n}$. For each
continuous $\phi: M\rightarrow\mathbb{R}$, we have by the
Dominated Convergence Theorem
\begin{eqnarray}\label{eq:weak-conv}
\int\int\phi(f_{t}(y))d\mu^{\epsilon}(y)\theta_{\epsilon}(t)
&=&
\lim_{k\rightarrow
+\infty}\int\int\phi(f_{t}(y))d\left(\frac{1}{n_{k}}\sum_{j=1}^{n_{k}}(f_{y}^{j})_{*}\theta_{\epsilon}^{\mathbb{N}}\right)d\theta_{\epsilon}(t)\nonumber
\\
&=&
\lim_{k\rightarrow+\infty}\frac{1}{n_{k}}\sum_{j=1}^{n_{k}}\int\int\phi(f_{t}(f_{\underline{t}}^{j}(x)))d\theta_{\epsilon}^{\mathbb{N}}(\underline{t})d\theta_{\epsilon}(t).
\end{eqnarray}
By definition of the perturbed iteration and of the infinite
product $\theta_\epsilon^{\mathbb{N}}$, and because
$\mu_{n_{k}}^{\epsilon}(x)\xrightarrow[n_{k}\to
+\infty]{}\mu^{\epsilon}$ in the weak$^{*}$ topology, the
limit in (\ref{eq:weak-conv}) equals
\begin{align*}
\lim_{k\rightarrow
+\infty}\frac{1}{n_{k}}\sum_{j=1}^{n_{k}}
\int\phi(f_{\underline{t}}^{j+1}(x))\,
d\theta_{\epsilon}^{\mathbb{N}}(\underline{t})
=\int\phi \,d\mu^{\epsilon}.
\end{align*}
Hence
$\int\int
\phi(f_{t}(y))\,d\mu^{\epsilon}(y)d\theta_{\epsilon}(t)
=\int\phi \, d\mu^{\epsilon}$ and $\mu^{\epsilon}$ is a
stationary measure.
\end{proof}
\subsection{Absolutely continuity and support with nonempty
interior}
\label{sec:absolut-contin-suppo}
We now show that each stationary measure is absolutely
continuous with respect to Lebesgue measure $\lambda$ (a
volume form) in $M$.
\begin{proposition}\label{pr:abscontpullback} We have
$(f_{x})_{*}\theta_{\epsilon}^{\mathbb{N}}<<\lambda$ for
all $x\in M$.
\end{proposition}
We recall that from~\ref{rmk:thadapt} we have that $H$ is
never zero on $M$, and so $\zeta(x)\neq0$ for all $x\in M$.
\begin{proof}In fact, consider $A\subset M$ some ball in $M$
which (we assume is a parallelizable manifold, e.g. an
interval, the circle or a $n$-torus). We have
$$
\begin{array}{cll}
(f_{x})_{*}\theta_{\epsilon}^{\mathbb{N}}(A)
&=&
\theta_{\epsilon}^{\mathbb{N}}\{ \underline{t}:
f_{\underline{t}}(x)\in A\}
\\
&=&
\theta_{\epsilon}^{\mathbb{N}}\{\underline{t}; f(x)+t_1
\cdot\zeta(x)\in A\}
\\
&=&
\theta_{\epsilon}\Big\{ t_1; t_1\in
\frac{A-f(x)}{\zeta(x)}\Big\}
\\
&=&
\frac1{\lambda(B_{\epsilon}(0))}\cdot
\lambda\left(\frac{A-f(x)}{\zeta(x)}\cap B_{\epsilon}(0)\right)
\\
&=&
\frac{1}{\zeta(x)}
\cdot
\frac1{\lambda(B_{\epsilon}(0))}
\cdot
\lambda\left((A-f(x))\cap B_{\epsilon}(0)\right)
\end{array}
$$
which shows that, if $\lambda(A)=0$, then
$(f_x)_*(\theta_\epsilon^{\mathbb N})(A)=0$.
\end{proof}
We observe that $B_\epsilon(0)\ni t\mapsto f_{t}(x)\in M$ is
continuous for each fixed $x\in M$. We also note that, since
the space $C^{0}(M,\mathbb{R})$ of continuous functions is
dense in the space $L^1(\mu^\epsilon)$ of Borel integrable
functions with respect to $\mu^\epsilon$, with the
$L^1$-norm, then the stationary condition in
Definition~\ref{eq:defstationary} holds also for any
$\mu$-integrable $\phi: M\rightarrow\mathbb{R}$.
\begin{lemma} Every stationary probability measure
$\mu^{\epsilon}$ is absolutely continuous with respect to
$\lambda$.
\end{lemma}
\begin{proof}
From the above observation that the relation in Definition
\ref{eq:defstationary} is true for all integrable
functions, we have that for any Borel measurable subset
$B\subset M$
\begin{align*}
\mu^{\epsilon}(B)
=
\int\chi_{B}\,d\mu^{\epsilon}
=
\int\int \chi_{B}\circ f_{t}(y)\,d\mu^{\epsilon}(y)d\theta_{\epsilon}(t)
=
\int(f_{y})_{*}\theta^{\mathbb{N}}_{\epsilon}(B)\,d\mu^{\epsilon}(y)
\end{align*}
and if $\lambda(B)=0$, then we obtain $\mu^\epsilon(B)=0$
from Proposition~\ref{pr:abscontpullback}.
\end{proof}
From this we are able to show that the support of any
stationary measure has non-empty interior. Let
$\mu^{\epsilon}$ be a stationary measure and let us write
$S=\mathrm{supp}(\mu^{\epsilon})$. Using again that the
relation in Definition~\ref{eq:defstationary} holds for
$\mu^\epsilon$-integrable functions
\begin{align*}
1
=
\int \chi_{S}(y)\,d\mu^{\epsilon}(y)
&=
\int\int \chi_{S}(f_{t}(y))\,
d\mu^{\epsilon}(y) d\theta_{\epsilon}(t)
\\
&=
\int\int\chi_{S}(f_{t}(y))\,
d\theta_{\epsilon}(t)d\mu^{\epsilon}(y)
\end{align*}
we conclude (since $0\le\chi_{S}\le1$) that
$\int\chi_{S}(f_{t}(y))\, d\theta_{\epsilon}(t)=1$ for
$\mu^\epsilon$-a.e. $y$. Therefore we get
$\chi_{S}(f_{t}(y))=1$, that is, $f_{t}(y)\in S$ for
$\theta_\epsilon$-a.e. $t$ and $\mu^\epsilon$-a.e. $y$.
In particular, $f_{t}(y)\in S$ for $t$ is a dense subset $D$
of $B_\epsilon(0)=\mathrm{supp}(\theta_\epsilon)$ by
definition of $\theta_\epsilon$. In addition, since
$B_\epsilon(0)\ni t\mapsto f_{t}(y)\in M$ is continuous, we
also have $f_y(D)$ is dense in $f_y(B_\epsilon(0))$ and so
the closed set $S$ contains $B_{\zeta(y)}(f(y))$, the
closure of $f_y(D)$. We obtain that \emph{$f_t(y)\in S$ for
all $t\in B_\epsilon(0)$ and $\mu^\epsilon$-a.e. $y$.}
From the definition of $f_t(y)$ in (\ref{pertaditiva}), we
see that the image of $f_y(B_\epsilon(0))$ is the ball
around $f(y)$ with radius $\zeta(y)\neq0$. Hence $S$ has
non-empty interior, as claimed.
\subsection{Every stationary measure is ergodic with full
support}
\label{sec:every-station-measur}
Now we use that the unperturbed transformation $f$ has a
dense orbit. Let $\mu^\epsilon$ be a stationary probability
measure. We have already shown that the support $S$ of
$\mu^\epsilon$ has non-empty interior and that $S$ is
\emph{almost invariant}.
\begin{lemma}
\label{le:fullinv}
Let $(f_{\underline{t}},\theta_\epsilon^{\mathbb N})$ be a
random dynamical system such that the unperturbed map
$f=f_0$ is a local diffeomorphism outside a
$\lambda$-measure zero set, has a dense positive orbit
and the parameter $0$ belongs to the support of
$\theta_\epsilon$.
Then $\mu^\epsilon$ has full support:
$S=\mathrm{supp}(\mu^\epsilon)=M$.
\end{lemma}
\begin{proof}
Let $S_0\subset S$ be such that $\mu^\epsilon(S\setminus
S_0)=0$ and $f_t(S_0)\subset S$ for all $t\in
B_\epsilon(0)$ -- this was proved in the previous
subsection. Hence we also have $\lambda(S\setminus S_0)=0$
and so $\overline{S_0}=S$.
We have that $f$ is locally a diffeomorphism outside a critical
set $\mathcal{C}$ with $\lambda$-measure zero. Then
$\lambda(f(S\setminus S_0))=0$ and, because
$f(S)\setminus f(S_0)\subset f(S\setminus S_0)$, we get
$\lambda(f(S)\setminus f(S_0))=0$.
Thus
$f(S)=f(\overline{S_0})\subseteq\overline{f(S_0)}\subseteq\overline
S=S$, and \emph{$S$ is a positively $f$-invariant subset}.
We also know that the interior of $S$ is non-empty. Let
$w\in M$ have a positive dense $f$-orbit. Then there exists
$n>1$ such that $f^n(w)$ interior to $S$ and so
$M=\omega_f(x)\subset \overline{S}=S\subset M$.
\end{proof}
To show ergodicity of any stationary measure, we need some
known auxiliary results already obtained for maps with
hyperbolic times for random orbits, as stated below.
The first result gives properties of random hyperbolic times
similar to those of Lemma~\ref{lema 5.2}.
\begin{proposition}[Proposition 2.6 and Corollary 2.7 in
\cite{AA03}]\label{pr:randhyptimes}
There exist $\delta_{1},C_1>0$ such that, if $n$ is a
$(\sigma,\delta)$-hyperbolic time for
$(\underline{t},x)\in \Omega\times M$, then there exists a
neighborhood $V_{n}(\underline{t},x)$ of $x$ in $M$ such
that:
\begin{enumerate}
\item $f_{\underline{t}}^{n}$ maps $V_{n}(\underline{t},x)$
diffeomorphically onto the ball of radius $\delta_{1}$
centered at $f_{\underline{t}}^{n}(x)$;
\item
$d(f_{\underline{t}}^{n-k}(y),f_{\underline{t}}^{n-k}(z))
\leq
\sigma^{k/2}\cdot d(f_{\underline{t}}^{n}(y),f_{\underline{t}}^{n}(z))$
for all $1\leq k\leq n$ and $y,z\in
V_{k}(\underline{t},x)$;
\item $C_1^{-1}\leq\frac{|\det
Df_{\underline{t}}^{n}(y)|}{|\det
Df_{\underline{t}}^{n}(z)|}\leq C_{1}$ for all $y,z\in
V_{n}(\underline{t},x)$.
\end{enumerate}
\end{proposition}
The next result says that every non-trivial positively
invariant subset for random non-uniformly expanding
dynamical system must contain a ball of a definite size.
\begin{definition}[Random positively invariant set]
We say that a subset $A\subset M$ is random positively
invariant if, for $\mu^{\epsilon}$-almost every $x\in A$,
we have that $f_{t}(x)\in A$ for
$\theta_{\epsilon}$-almost every t.
\end{definition}
We note that if $A$ is random positively invariant and
$\lambda(A)>0$, then the closure of its Lebesgue density
points $A^+$ is also random positively invariant, since $A$
is dense in $A^+$.
\begin{proposition}[Proposition 2.13 in \cite{AlV13}]\label{pr:nucleus}
For $\delta_{1}$ given by previous proposition, given any
random positively invariant set $A\subset M$ with
$\mu^{\epsilon}(A)>0$, there is a ball of radius
$\delta_{1}/4$ such that $\lambda(B\setminus A^+)=0$.
\end{proposition}
The following is well-known from the theory of Markov chains.
\begin{lemma}[Lemma 8.2 in
\cite{vdaraujo2000}] \label{le:restrict} The normalized
restriction of a stationary measure to a random positively
invariant set is a stationary measure.
\end{lemma}
Now we can prove that each stationary probability measure
$\mu^\epsilon$ for our random dynamical systems is ergodic.
Arguing by contradiction, let us assume that
$\mu^{\epsilon}$ is not ergodic.
Hence, there are random (positively) invariant sets $S_{1}$
and $S_{2}= M\setminus S_{1}$ such that both have
$\mu^{\epsilon}$-positive measure. From
Proposition~\ref{pr:nucleus} both sets contain a
$\delta_1/4$-ball. Thus there exist $n_1,n_2>1$ such that
$f^{n_1}(w)\in S_1$ and $f^{n_2}(w)\in S_2$, where $w$ is a
point with dense positive $f$-orbit. Therefore,
$\overline{S_1}=M=\overline{S_2}$ which is a contradiction.
\section{Stochastic stability}
Now we combine the results of the previous sections to prove
our main Theorem~\ref{principal}. We use the same strategy
as \cite{AA03} taking advantage of the uniformity of the
first hyperbolic time with respect to the adapted random
perturbations. Indeed, from the previous constructions and
from Remark~\ref{rmk:hbdd}, we
have that there exist $0<\sigma,\delta<1$ such that
\begin{align*}
\hat h:\Omega\times M\to M, \quad (\underline t,x)\mapsto
\inf\{k\ge1: k \text{ is a }
(\sigma,\delta)-\text{hyperbolic time for } (\underline t,x)\}
\end{align*}
satisfies $\hat h(\underline t,x)=\hat h(\un 0,x)=h(x)\le
H(x)$ for all $\underline
t\in\supp\theta_\epsilon^{\mathbb{N}}$ for
$\lambda$-a.e. $x\in M$, where $\underline 0$ is the
constant sequence equal to zero and $h(x)$ denotes the first
hyperbolic time map associated to the unperturbed dynamics
of $f$, as defined in
Section~\ref{sec:adapted-random-pertu}.
Hence, if we assume that $h\in L^p(\lambda)$ for some $p>3$,
then we have also that the series
\begin{equation}\label{c.unif}
\|\hat h\|_1
=
\int \hat h\,d(\theta_\epsilon^{\mathbb{N}}\times \lambda)
=
\sum_{k=0}^{\infty}k\cdot
(\theta_\epsilon^{\mathbb{N}}\times \lambda)
\big(\{(\underline t,x)\colon \hat h(\underline t,x)=k\}\,\big)
\end{equation}
has {\em uniform $L^1$-tail}, that is, the series in the
right hand side of \eqref{c.unif} converges uniformly to
$\|\hat h\|_1$ (as a series of functions of the variable
$\epsilon$).
\begin{remark}\label{rmk:L1enough}
For this argument it is enough that we assume $h\in
L^1(\lambda)$, as long as $\hat h(\cdot,x)=h(x)$ for
$\lambda$-a.e. $x\in M$ is established.
\end{remark}
Now we can follow the same arguments as in \cite[Section
5]{AA03}. We sketch them here for the convenience of the
reader. Since there exists a unique ergodic absolutely
continuous stationary measure $\mu^\epsilon$ for all small
enough $\epsilon>0$, we have that
\begin{align*}
\mu_n^\epsilon =
\frac1n\sum_{j=0}^{n-1}\int (f^j_{\underline
t})_{*} \lambda\,
d\theta_\epsilon^{\mathbb{N}}(\underline t).
\end{align*}
converges in the weak$^*$ topology to $\mu^\epsilon$ as
$n\to+\infty$. We define for each $\underline t\in
\Omega^{\mathbb{N}}$ and $n\geq 1$
\begin{align*}
H_n(\un t)
&=
\{ x\in B(\mu^\epsilon)\colon \mbox{ $n$ is a
$(\sigma,\delta)$-hyperbolic time for $(\un t,x)$ }\},
\quad\text{and}
\\
H^*_n(\un t)
&=
\{ x\in B(\mu^\epsilon)\colon \mbox{ $n$ is the first
$(\sigma,\delta)$-hyperbolic time for $(\un t,x)$ }\}.
\end{align*}
Here $H^*_n(\un t)$ is the set of points $x$ for which $\hat
h(\un t,x)=n$. For $n,k\geq 1$ we define $R_{n,k}(\un t)$
as the set of points $x$ for which $n$ is a
$(\sigma,\delta)$-hyperbolic time and $n+k$ is the first
$(\sigma,\delta)$-hyperbolic time after $n$, that is
\begin{align*}
R_{n,k}(\un t)= \left\{x\in H_n(\un t)\colon \:f^n_{\un
t}(x)\in H^*_k(\sigma^n\un t)\: \right\},
\end{align*}
where $\sigma:\Omega\circlearrowleft$ is the left shift map.
Now using the measures
\begin{align*}
\nu^\epsilon_n
&=
\int ( f_{\un t}^n)_*\big(\lambda\mid H_n(\un
t)\big)\,d\theta_\epsilon^\N(\un t)
\quad\text{and}\quad
\eta_n^\epsilon
=
\sum_{k=2}^\infty\sum_{j=1}^{k-1}\int (f_{\un
t}^{n+j})_*\big(\lambda\mid R_{n,k}(\un t)\big)
\,d\theta_\epsilon^\N(\un t),
\end{align*}
we obtain the bound
$ \mu_n^\epsilon\leq
\frac{1}{n}\sum_{j=0}^{n-1}(\nu_j^\epsilon+\eta_j^\epsilon).$
The bounded distortion property of hyperbolic times provides
the following.
\begin{proposition}\cite[Proposition 5.2]{AA03}\label{pr:dens1}
There is a constant $C_2>0$ such that for every $n\geq 0$
and $\un t\in\Omega$ we have
$ \frac{d}{d\lambda}(f_{\un t}^n)_*\big(\lambda\mid H_n(\un
t)\big)\leq C_2.$
\end{proposition}
Hence we have $\frac{d\nu_n^\epsilon}{d\lambda}\leq C_2$ for
every $n\geq 0$ and small $\epsilon>0$. We now control the
density of the measures $\eta_n^\epsilon$ so that we ensure
the absolute continuity of the weak$^*$ accumulation point
of $\mu^\epsilon$ when $\epsilon\searrow0$.
\begin{proposition}\cite[Proposition 5.3]{AA03}\label{pr:dens2}
Given $\zeta>0$, there is $C_3(\zeta)>0$ such that for
every $n\geq 0$ and $\epsilon>0$ we may bound
$\eta_n^{\epsilon}$ by the sum of two measures
$\eta_n^{\epsilon} \leq
\omega^{\:\epsilon}+\rho^{\:\epsilon}$ satisfying
$\frac{d\omega^{\:\epsilon}}{d\lambda}\leq C_3(\zeta)$ and
$\rho^{\:\epsilon}(M)<\zeta.$
\end{proposition}
It follows from Propositions \ref{pr:dens1} and
\ref{pr:dens2} that the weak$^*$ accumulation points $\mu^0$
of $\mu^\ep$ when $\ep\searrow0$ cannot have singular part,
and so are absolutely continuous with respect to
$\lambda$. Moreover, from Remark~\ref{re:accinvariant} we
have that the weak$^*$ accumulation points $\mu^0$ of a
family of stationary measures are always $f$-invariant
measures.
From the properties of non-uniformly expanding maps stated
in Theorem~\ref{thm:abv}, we conclude that $\mu^0$ is
a convex linear combination of finitely many physical
measures of $f$. This proves stochastic stability under
adapted random perturbations.
In our setting, where $f$ is transitive, we
have a unique physical measure $\mu$ for $f$, thus
$\mu^0=\mu$.
\def\cprime{$'$}
| 195,689
|
\begin{document}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\newcommand{\sect}[1]{\setcounter{equation}{0}\section{#1}}
\title{Small amplitude solitary waves in the Dirac-Maxwell system}
\author{
{\sc Andrew Comech}
\\
{\small\it
Texas A\&M University, College Station, TX 77843, USA
and
IITP, Moscow 101447, Russia}
\\~\\
{\sc David Stuart}
\\
{\small\it University of Cambridge, Cambridge CB3 0WA, UK}
}
\date{\version}
\maketitle
\begin{abstract}
We study nonlinear bound states, or solitary waves,
in the Dirac-Maxwell system proving the
existence of solutions in which the Dirac wave function is of the form
$\phi(x,\omega)e^{-i\omega t}$,
$\omega\in(-m,\omega\sb \ast)$, with some $\omega\sb \ast>-m$,
such that
$\phi(\,\cdot\,,\omega)\in H\sp 1(\R^3,\C^4)$,
$\norm{\phi(\,\cdot\,,\omega)}^2\sb{L\sp 2}=O(m-\abs{\omega})$,
and
$\norm{\phi(\,\cdot\,,\omega)}\sb{L\sp\infty}=O(m-\abs{\omega})$. The method of
proof is an implicit function theorem argument based on
an identification of the nonrelativistic limit as the ground state of the
Choquard equation.
\end{abstract}
\section{Introduction and results}
\label{sec-dm}
The Dirac equation, which appeared in \cite{dirac-1928}
just two years after the Schr\"odinger equation,
is the correct Lorentz invariant equation to
describe particles with nonzero spin
when relativistic effects cannot be ignored.
The Dirac equation predicts accurately the
energy levels of an electron in the Hydrogen atom, yielding
relativistic corrections to the spectrum of the Schr\"odinger equation.
Further higher order corrections arise on account of interactions with
the electromagnetic field, described mathematically by the
Dirac-Maxwell Lagrangian,
which aims to provide
a self-consistent description of the dynamics
of an electron interacting with its own electromagnetic field.
The perturbative treatment of the Dirac-Maxwell
system in the framework of second quantization
allows computation of quantities such as the
energy levels and scattering cross-sections, which have been
compared successfully with experiment; of course this quantum formalism
does not provide the type of tangible description of
particles and dynamical processes familiar
from classical physics.
Mathematically, the quantum theory (QED) has not been
constructed, and indeed may not exist in the generally understood
analytical sense.
In particular it is a curious fact that although the electron
is the most stable elementary particle known to physicists today,
there is no mathematically precise formulation and proof of its
existence and stability.
This has resulted in an enduring interest in the classical Dirac-Maxwell
system, both in the physics and mathematics literature. Regarding
the former, the relevance of the {\em classical} equations of motion
for QED has been widely debated. The prevalent view
seems to be that the Dirac fermionic field does not have a direct
meaning or limit in classical physics, and hence that the classical system is
not really directly relevant to the world of observation.
Nevertheless,
there have been numerous attempts,
both by Dirac himself
and by many others - see
\cite{MR0139402,wakano-1966,MR1364144} and references therein -
to construct localized solutions of the classical system or some modification
thereof, with the aim of obtaining a more cogent mathematical description
of the electron (or other fundamental particles).
We consider the system of Dirac-Maxwell equations,
where the electron,
described by the standard ``linear'' Dirac equation,
interacts with its own electromagnetic field
which is in turn required to obey the Maxwell equations:
\begin{equation}\label{dirac-maxwell-classical}
\begin{cases}
\gamma\sp\mu(i\p\sb\mu-e A\sb\mu)\psi-m\psi=0,
\\
\p\sp\mu\p\sb\mu A\sp\nu=J\sp\nu,
\quad \p\sb\mu A\sp\mu=0,
\end{cases}
\end{equation}
with the charge-current density
$J\sp\mu=(\rho,\mathbf{J})$ generated by the spinor field:
\begin{equation}\label{current-density}
J\sp\mu=e\bar\psi\gamma\sp\mu\psi.
\end{equation}
Above, $\rho$ and $\mathbf{J}$
are the charge and current respectively.
We denote $\bar\psi=(\gamma\sp{0}\psi)\sp\ast=\psi\sp\ast\gamma\sp{0}$,
with
$\psi\sp\ast$
the hermitian conjugate of $\psi$.
The charge is denoted by $e$
(so that for the electron $e<0$);
the fine structure constant is the dimensionless coupling constant
$\alpha\equiv\frac{e^2}{\hbar c}\approx 1/137$.
We choose the units so that $\hbar=c=1$.
We have written the Maxwell equations using the Lorentz gauge condition
$\p\sb\mu A\sp\mu=0$.
The Dirac $\gamma$-matrices satisfy the
anticommutation relations
\[
\{\gamma\sp\mu,\gamma\sp\nu\}=2g\sp{\mu\nu},
\]
with $g\sp{\mu\nu}=\mathop{\rm diag}[1,-1,-1,-1]$.
The four-vector potential $A\sp\mu$ has components
$({\scp},\mathbf{A})$,
with
$\mathbf{A}=(A\sp 1,\,A\sp 2,\,A\sp 3)$,
so that the lower index version
$A\sb\mu=g\sb{\mu\nu}A\sp\nu$ has components
$({\scp},-\mathbf{A})$ so $A\sb 0={\scp}$.
Following
\cite{MR0187641} and \cite{MR0441102},
we define the Dirac $\gamma$-matrices by
\begin{equation}
\gamma\sp{j}
=\left(
\begin{matrix} 0&\sigma\sb{j} \\ -\sigma\sb{j}&0\end{matrix}
\right),
\qquad
\gamma\sp{0}
=\left(
\begin{matrix} I\sb{2}&0 \\ 0&-I\sb{2}\end{matrix}
\right),
\end{equation}
where $I\sb{2}$ is the $2\times 2$ unit matrix
and
$\sigma\sb{j}$ are the Pauli matrices:
$\sigma\sb{1}=\left(\begin{matrix} 0&1 \\ 1&0\end{matrix}\right)$,
$\sigma\sb{2}=\left(\begin{matrix} 0&-i \\ i&0\end{matrix}\right)$,
$\sigma\sb{3}=\left(\begin{matrix} 1&0 \\ 0&-1\end{matrix}\right)$.
After introduction of a space time splitting,
the system \eqref{dirac-maxwell-classical}
takes the form
\begin{equation}\label{stationary-eqns}
i\p\sb t\psi
=\bm\alpha\cdot(-i\bm\nabla-e \mathbf{A})\psi
+m\beta\psi
+e A\sp 0\psi,
\qquad
(\p\sb t^2-\Delta)A\sp 0=e\psi\sp\ast\psi,
\qquad
(\p\sb t^2-\Delta)\mathbf{A}=e\psi\sp\ast\bm\alpha\psi.
\end{equation}
Here
$\bm\alpha=(\alpha\sp 1,\alpha\sp 2,\alpha\sp 3)$,
and
$\alpha\sp j$ and $\beta$ are the $4\times 4$
Dirac matrices:
\begin{equation}
\alpha\sp j
=\left(
\begin{matrix} 0&\sigma\sb{j} \\ \sigma\sb{j}&0\end{matrix}
\right),
\qquad
\beta
=\left(
\begin{matrix} I\sb{2}&0 \\ 0&-I\sb{2}\end{matrix}
\right)\,,
\end{equation}
with $\{\sigma\sb j\}_{j=1}^3$ the Pauli matrices.
We will not distinguish lower and upper indices $j$
of $\alpha$ and $\sigma$, so that
$\alpha\sb{j}=\alpha\sp{j}$, $\sigma\sb{j}=\sigma\sp{j}$.
The $\alpha$-matrices and $\gamma$-matrices are related by
\[
\gamma\sp j=\beta\alpha\sp j,
\quad
1\le j\le 3;
\qquad
\gamma\sp{0}=\beta.
\]
Numerical justification for the existence of
solitary wave solutions
to the Dirac-Maxwell system \eqref{dirac-maxwell-classical}
was obtained in \cite{MR1364144},
where it was suggested that
such solutions are formed
by the Coulomb repulsion
from the negative part of the essential spectrum
(the Klein paradox).
The numerical results of \cite{MR1364144}
showed that
the Dirac-Maxwell system
has infinitely many families of
{\em solitary wave} solutions
$\phi\sb{N}(x,\omega)e^{-i\omega t}$,
$\omega\gtrsim -m$.
Here the nonnegative integer $N$
denotes the number of nodes of the positronic component
of the solution
(number of zeros of the corresponding
spherically symmetric
solution to the Choquard equation; see
\S\ref{sect-nr}).
A variational proof of existence of solitary waves
for $\omega\in(-m,0)$
and with $N=0$
first appeared in
\cite{MR1386737},
and the generalization to handle $\omega\in(-m,m)$
is in \cite{MR1618672}.
In the present paper, we give a proof of existence
of solitary wave solutions to the Dirac-Maxwell system
based on the perturbation from the nonrelativistic limit
and also
obtain the precise asymptotics for the solution in this limit.
The physical significance of these types
of solitary wave solutions requires not only their existence but
also stability, and it is to be hoped that the type of
detailed information about the solutions
which is a consequence of the existence proof in this article, but
does not seem to be so easily accessible from the original variational
constructions, will be helpful in future stability analysis
(see Remark~\ref{remark-stability} below).
The second motivation for presenting this proof is to realize mathematically
the physical intuition explained in \cite{MR1364144} which explains
the existence of these bound state solutions in terms of the Klein paradox
(\cite[\S3.3]{MR0187641}).
Moreover, once one knows
that the excited eigenstates of the Choquard equation
are nondegenerate
(currently this nondegeneracy
is established only for the ground state, $N=0$ \cite{MR2561169}),
our argument will yield the existence
of excited
solitary wave solutions in Dirac-Maxwell system,
extending the results of \cite{MR1386737}
to $N\ge 1$.
We will construct solitary wave solutions by deforming
the solutions to the nonrelativistic limit
(represented by the Choquard equation)
via the implicit
function theorem.
Such a method was employed
in \cite{MR1750047,2008arXiv0812.2273G}
for the nonlinear Dirac equation
and in \cite{MR2593110,MR2647868,MR2671162}
for Einstein-Dirac and Einstein-Dirac-Maxwell systems.
The solitary wave
$(\phi e^{-i\omega t},A\sp\mu(x))$
satisfies the stationary system
\begin{equation}\label{omega-phi-is}
\omega\phi
=\bm\alpha\cdot(-i\bm\nabla-e \mathbf{A})\phi+m\beta\phi+e A\sp 0\phi,
\qquad
-\Delta A\sp\mu=e\bar\phi\gamma\sp\mu\phi.
\end{equation}
\begin{theorem}\label{theorem-sw-dm}
There exists ${\omega\sb\ast}>-m$ such that
for $\omega\in(-m,{\omega\sb\ast})$
there is a solution to
\eqref{omega-phi-is}
of the form
\[
\phi(x,\omega)=\begin{bmatrix}
\epsilon^3\varPhi\sb 1(\epsilon x,\epsilon)
\\
\epsilon^2\varPhi\sb 2(\epsilon x,\epsilon)
\end{bmatrix},
\qquad
\epsilon=\sqrt{m^2-\omega^2},
\]
with
\[
\varPhi
=\begin{bmatrix}\varPhi\sb 1\\\varPhi\sb 2\end{bmatrix}
\in C^\infty\bigl(
(0,{\epsilon\sb\ast})\,;\,
\big(H^1(\R^3;\C^2)\oplus H^2(\R^3;\C^2)\big)
\bigr),
\qquad
{\epsilon\sb\ast}=\sqrt{m^2-{\omega\sb\ast}^2},
\]
and with
\[
A\sp\mu\in C^\infty\bigl(
(0,{\epsilon\sb\ast})\,;\, \dot H\sp 1(\R^3,\R)\cap L^\infty(\R^3,\R)\bigr),
\qquad
0\le \mu\le 3.
\]
Above,
$\dot H^1=\dot H^1(\R^3,\R)$ is the homogeneous Dirichlet
space of $L^6$ functions with $\|f\|_{\dot H^1}^2=\int\,|\nabla f|^2\,
dx<\infty$.
For small $\epsilon>0$, one has
\[
\norm{{\varPhi\sb{1}}-\hat\varPhi\sb{1}}\sb{H\sp 2}
=O(\epsilon^2),
\qquad
\norm{{\varPhi\sb{2}}-\hat\varPhi\sb{2}}\sb{H\sp 1}
=O(\epsilon^2),
\]
where
\[
\hat\varPhi\sb 1(y)=\lim\sb{\epsilon\to 0}
\varPhi\sb 1(y,\epsilon),
\qquad
\hat\varPhi\sb 2(y)=\lim\sb{\epsilon\to 0}\varPhi\sb 2(y,\epsilon)
\]
are of Schwartz class.
The solutions could be chosen so that
in the nonrelativistic limit $\epsilon=0$
one has
\begin{equation}
\hat\varPhi\sb 2(y)=\varphi\sb 0(y)\bm{n},
\qquad
\hat\varPhi\sb 1(y)=\frac{i}{2m}
{\bm\sigma\cdot}\bm\nabla\sb{y}\hat\varPhi\sb 2(y),
\end{equation}
where $\bm{n}\in\C^2$, $\abs{\bm{n}}=1$,
and $\varphi\sb 0$
is a strictly positive spherically symmetric solution
of Schwartz class
to the Choquard equation
\begin{equation}\label{phi0}
-\frac{1}{2m}\varphi\sb 0=-\frac{1}{2m}\Delta\varphi\sb 0
-\Big(\frac{1}{4\pi\abs{x}}\ast\varphi_0^2\Big)\varphi_0,
\qquad
\varphi_0(x)\in\R,
\quad
x\in\R^3.
\end{equation}
\end{theorem}
\begin{remark}
The existence of a positive spherically-symmetric solution
$\varphi_0\in\mathscr{S}(\R^3)$
to \eqref{phi0}
was proved in
\cite{MR0471785}.
\end{remark}
Here is the plan of the paper.
We give the heuristics in \S\ref{sect-heur}.
The Choquard equation,
which is the nonrelativistic limit of the Dirac-Maxwell system,
is considered in \S\ref{sect-nr}.
In \S\ref{sect-exist}, we complete
the proof of existence of solitary waves
via the implicit function theorem.
\section{Heuristics on the nonrelativistic limit}
\label{sect-heur}
The small amplitudes waves constructed in
Theorem~\ref{theorem-sw-dm}
are best understood physically in terms of the non-relativistic limit. Since we
have set the speed of light and other physical constants equal to one, the
relevant small parameter is the excitation energy (or frequency) as compared
to the mass $m$.
To develop some preliminary intuition regarding the non-relativistic limit,
following \cite{MR1364144}, we neglect the magnetic field
described by the vector-potential $A\sp j$,
getting
\[
i\p\sb t\psi=-i\bm\alpha{\cdot\bm\nabla}\psi
+m\beta\psi
+e A\sp 0\psi,
\qquad
(\p\sb t^2-\Delta)A\sp 0=e\psi\sp\ast\psi\,.
\]
Let us consider a solitary wave solution
$\psi(x,t)=\phi(x)e^{-i\omega t}$,
with
$
\phi(x)
=
\begin{bmatrix}
\phi\sb 1(x)\\\phi\sb 2(x)
\end{bmatrix}
$,
where
$\phi\sb 1,\,\phi\sb 2\in\C^2$ and $A\sp 0=A\sp 0(x)$ only.
Then $\phi\sb 1$, $\phi\sb 2$, and $A\sp 0$ satisfy
\begin{equation}
\label{cs}
(\omega-m)\phi\sb 1
=-i\bm\sigma{\cdot\bm\nabla}\phi\sb 2
+e A\sp 0\phi\sb 1,
\qquad
(m+\omega)\phi\sb 2
=-i\bm\sigma{\cdot\bm\nabla}\phi\sb 1
+e A\sp 0\phi\sb 2,
\qquad
-\Delta A\sp 0=e
(\phi\sb 1\sp\ast\phi\sb 1+\phi\sb 2\sp\ast\phi\sb 2)\,,
\end{equation}
where $\bm\sigma=(\sigma\sb 1,\sigma\sb 2,\sigma\sb 3)$,
the vector formed from the Pauli matrices.
Consider small amplitude solitary waves with $\omega\approx -m$.
Then $A\sp 0$ is small and
$-2m\phi\sb 1\approx -i\bm\sigma{\cdot\bm\nabla}\phi\sb 2$,
\[
(m+\omega)\phi\sb 2
\approx
\frac{1}{2m}\Delta\phi\sb 2
+e A\sp 0\phi\sb 2,
\qquad
-\Delta A\sp 0
\approx
e
(\phi\sb 1\sp\ast\phi\sb 1+\phi\sb 2\sp\ast\phi\sb 2).
\]
Denoting
$\epsilon^2=m^2-\omega^2$, $0<\epsilon\ll m,\,$
the above suggests the following scaling:
\begin{equation}\label{scaling}
y=\epsilon x,
\qquad
\p\sb x=\epsilon\p\sb y,
\qquad
A\sp 0(x)=\epsilon^2 \eurA\sp 0(\epsilon x),
\qquad
\phi\sb 1(x)=\epsilon^{3}\varPhi\sb 1(\epsilon x),
\qquad
\phi\sb 2(x)=\epsilon^{2}\varPhi\sb 2(\epsilon x).
\end{equation}
Note that
since $\phi\sb j$ and $A\sp 0$ depend on $\omega$ and $x$, the scaled
functions
$\eurA\sp 0$ and $\varPhi_j$ are functions of $y$ and of $\epsilon$.
In the limit $\epsilon\to 0$,
denoting
\[
\hat\varPhi=\lim\sb{\epsilon\to 0}\varPhi,
\qquad
\hat\eurA\sp 0=\lim\sb{\epsilon\to 0}\eurA\sp 0,
\]
we arrive at
\begin{equation}\label{phi1phi2}
-2m\hat\varPhi\sb 1
=-i\bm\sigma{\cdot\bm\nabla}\sb{y}\hat\varPhi\sb 2,
\qquad
-\frac{1}{2m}\hat\varPhi\sb 2
=-i\bm\sigma{\cdot\bm\nabla}\sb{y}\hat\varPhi\sb 1
+e \hat\eurA\sp 0\hat\varPhi\sb 2,
\qquad
-\Delta\sb y \hat\eurA\sp 0=e\hat\varPhi\sb 2\sp\ast\hat\varPhi\sb 2\,,
\end{equation}
which
can be rewritten as the following equation for $\varPhi_2$ only:
\begin{equation}\label{phi2}
-\frac{1}{2m}\hat\varPhi\sb 2
=-\frac{1}{2m}\Delta\sb y\hat\varPhi\sb 2
-e \hat\eurA\sp 0\varPhi\sb 2,
\qquad
-\Delta\sb y \hat\eurA\sp 0=e\hat\varPhi\sb 2\sp\ast\hat\varPhi\sb 2\,,
\end{equation}
with the understanding that $\varPhi_1$
is then obtained from the first equation
of \eqref{phi1phi2}.
\begin{remark}
Regarding self-consistency of this approximation: one can check that,
when using the scaling \eqref{scaling},
the magnetic field vanishes to higher order
in the limit $\epsilon\to 0$,
in agreement with \cite{MR1364144}.
Indeed,
$
\mathbf{A}
=-\Delta^{-1}\mathbf{J},
$
where
$\mathbf{J}=e\psi\sp\ast\bm\alpha\psi=O(\epsilon^5)$,
hence $\mathbf{A}=-\Delta^{-1}\mathbf{J}=O(\epsilon^3)$.
The second equation from \eqref{cs}
would then take the form
\[
(m+\omega)\phi\sb 2
=-i\bm\sigma{\cdot\bm\nabla}\phi\sb 1-e {\mathbf{A}\cdot}\bm\sigma\phi\sb 1
+e A\sp 0\phi\sb 2,
\]
where
${\mathbf{A}\cdot}\bm\sigma\phi\sb 1=O(\epsilon^6)$
while other terms are $O(\epsilon^4)$. Thus the approximation is
al least self-consistent, and the analysis in \S\ref{sect-exist}
justifies this rigorously.
\end{remark}
\begin{remark}
Regarding symmetry: while
it is clear that radial symmetry
of both $\phi_1$ and $\phi_2$ is inconsistent with \eqref{phi1phi2},
solutions of the form
\begin{equation}\label{ansatz}
\begin{bmatrix}
i f(r)
\begin{pmatrix}\cos\theta\\ e^{i\phi}\sin\theta\end{pmatrix}
\\
g(r)\begin{pmatrix}
1\\0
\end{pmatrix}
\end{bmatrix},
\qquad
\begin{bmatrix}
i f(r)
\begin{pmatrix}\cos\theta\\ -e^{-i\phi}\sin\theta\end{pmatrix}
\\
g(r)\begin{pmatrix}0\\1\end{pmatrix}
\end{bmatrix}\,,
\end{equation}
are permitted in principle,
suggesting that in the non-relativistic limit $\hat\varPhi_2$
could be radial, or to be more precise
of the form
$
\hat \varPhi\sb 2(y)=\begin{bmatrix}
\Phi(y)\\0
\end{bmatrix}\in\C^2,
$
where
the spherically symmetric function
$\Phi(y)\in\C$ is to satisfy
\begin{equation}\label{phiphi}
-\frac{1}{2m}\Phi
=-\frac{1}{2m}\Delta\sb y\Phi
-e \hat\eurA\sp 0\Phi,
\qquad
-\Delta\sb y \hat\eurA\sp 0=e|\Phi|^2\,.
\end{equation}
The starting point for our perturbative construction of solitary
wave solutions to \eqref{stationary-eqns} is indeed
a radial solution of \eqref{phiphi}, although the exact form of these
solitary waves
has to be modified from \eqref{ansatz} when the
effect of the magnetic field
$\mathbf{B}=\nabla\times\mathbf{A}$
is included, see \cite[\S5]{MR1364144}. The method of proof
we employ does not require any particular symmetry class of the
solitary wave.
\end{remark}
The above discussion suggests that the system \eqref{phiphi}
will determine the non-relativistic limit to highest order.
The system \eqref{phiphi} describes a Schr\"odinger wave function with an
attractive self-interaction determined by the Poisson equation.
Because the sign of the interaction is attractive
it is often referred to as the stationary Newton-Schr\"odinger system.
It is equivalent to a nonlocal equation for $\Phi$ known as the Choquard
equation, which is the subject of the next section.
\section{The nonrelativistic limit:
the Choquard equation}
\label{sect-nr}
The system \eqref{phiphi}
can also be obtained by looking for solitary wave
solutions in the system
\begin{equation}\label{nrl}
i\p\sb t\psi=-\frac{1}{2m}\Delta\psi
-e V\psi,
\qquad
-\Delta V=e\psi\sp\ast\psi,
\qquad
\psi(x,t)\in\C,
\quad
V(x,t)\in\R,
\quad
x\in\R^3.
\end{equation}
This is the time-dependent Newton-Schr\"odinger system.
If $\big(\phi e^{-i\omega t},V(x)\big)$
is a solitary wave solution, then
$\phi$ and $V$ satisfy the stationary system
\begin{equation}\label{nss}
\omega\phi=-\frac{1}{2m}\Delta\phi-e V\phi,
\qquad
-\Delta V=e\abs{\phi}^2.
\end{equation}
We rewrite the system \eqref{nrl}
in the non-local form, called the {\em Choquard equation}:
\begin{equation}\label{choquard}
i\p\sb t\psi=-\frac{1}{2m}\Delta\psi+e^2\Delta^{-1}(\abs{\psi}^2)\psi,
\qquad
\psi(x,t)\in\C,
\qquad
x\in\R^3,
\end{equation}
where
$\Delta^{-1}$ is the operator of convolution with $-\frac{1}{4\pi \abs{x}}$.
The solitary waves are solutions are of the form
$
\psi(x,t)=\phi\sb\omega(x)e^{-i\omega t},
$
with $\phi\sb\omega$ satisfying the non-local scalar equation
\begin{equation}\label{lambda-phi}
\omega\phi
=-\frac{1}{2m}\Delta \phi+e^2\Delta^{-1}(\abs{\phi}^2)\phi\,.
\end{equation}
This suggests the following variational formulation for the problem:
find critical points of
\begin{equation}
{E}\sb{\mathrm{Choquard}}(\phi)
=\frac{1}{2m}\int |\nabla\phi|^2 dx-\frac{e^2}{8\pi}\iint\,
\frac{|\phi(x)|^2|\phi(y)|^2}{|x-y|}\,dx\,dy\,,
\end{equation}
subject to the constraint $\int\,|\phi(x)|^2\,dx\,=\const$. This formulation
is the basis of the existence and uniqueness proofs in the references
which are summarized in the following theorem.
\begin{lemma}[\cite{MR0471785,MR591299,MR2592284}]
For all $\omega<0$
and $N\in\Z$, $N\ge 0$,
the equation \eqref{choquard}
admits solitary wave solutions
\[
\psi(x,t)=\varphi\sb N(x,\omega)e^{-i\omega t},
\qquad
\lim\sb{\abs{x}\to\infty}\varphi\sb N(x,\omega)
=0,
\]
with $\varphi\sb N(x,\omega)$
a spherically symmetric solution of
\eqref{lambda-phi}.
These solutions
differ by the number $N$ of zeros (or nodes),
of the profile functions
$\varphi\sb N(x,\omega)$,
considered as a function of $r=\abs{x}$.
The profile function $\varphi_0$
with no zeros minimizes the value of the energy functional
${E}\sb{\mbox{\footnotesize\it Choquard}}(\phi)$
amongst functions
with fixed $L^2$ norm,
and is the unique (up to translation) positive $H^1$ solution of
\eqref{lambda-phi}; the corresponding solitary wave is known as the
ground state.
\end{lemma}
\begin{remark}Together with the heuristics in the previous section,
this result suggests that
for $\omega$ sufficiently close to $-m$
there might exist
infinitely many
families
of solitary waves
to the Dirac-Maxwell system,
which differ by the number of nodes.
As mentioned in \cite{MR1386737},
the variational methods used in that paper are hard to
generalize to prove the existence of multiple solitary
waves for each $\omega$
(such a multiplicity result is obtained
in \cite{MR1386737} for the Dirac -- Klein-Gordon system).
\end{remark}
\begin{remark}\label{remark-scaling}
The $\phi(x)$ and $V(x)$ for different
values of $\omega<0$ can be scaled to produce a standard form
as follows. Let $\zeta>0$ satisfy
$\zeta^2=-\omega$ and write
$y=\zeta x\,,\phi(x)=\zeta^2 u(\zeta x)\,,$
and $V(x)=\zeta^2 v(\zeta x)$. Then \eqref{nss}
is equivalent to the following
system for $u(y)$, $v(y)$:
\begin{equation}\label{Phi-W}
-u=-\frac{1}{2m}\Delta\sb y u-e v u,
\qquad
-\Delta\sb y v=e\abs{u}^2\,.
\end{equation}
\end{remark}
In the remainder of this
section we summarize the properties of the linearized
Choquard equation which follow from \cite{MR2561169} and
are needed in \S\ref{sect-exist}.
Consider a solution to the
Choquard equation of the form
\[
\psi(x,t)=(\varphi_0(x)+R(x,t)+i S(x,t))e^{-i{\omega\sb 0} t},
\]
with $R$, $S$ real-valued.
The linearized equation for $R$, $S$
is:
\begin{equation}\label{lin}
\p\sb t
\begin{bmatrix}R\\S\end{bmatrix}
=
\begin{bmatrix}0&L\sb 0\\-L\sb 1& 0\end{bmatrix}
\begin{bmatrix}R\\S\end{bmatrix},
\qquad
L\sb 0=-\frac{1}{2m}\Delta-{\omega\sb 0}
+e^2\Delta^{-1}(\varphi_0^2),
\qquad
L\sb 1=L\sb 0+2e^2\Delta^{-1}(\varphi_0\,\cdot\,)\varphi_0.
\end{equation}
Both $L\sb 0$ and $L\sb 1$ are unbounded operators
$L^2\to L^2$ which are self-adjoint with domain $H^2\subset L^2$.
Clearly $L\sb 0 \varphi_0=0$,
with $0\in\sigma\sb d(L\sb 0)$
an eigenvalue
corresponding to a positive eigenfunction
$\varphi_0$;
it follows that $0$ is a simple eigenvalue of $L\sb 0$,
with the rest of the spectrum separated from zero. The range of
$L\sb 0$ is $\{\varphi_0\}^{\perp}$, the $L^2$ orthogonal complement of
the linear span of $\varphi_0$.
Notice that $L\sb 1=\frac{1}{2}\bigl(E''(\varphi_0)-{\omega\sb 0} Q''(\varphi_0)
\bigr)$.
\begin{lemma}
The self-adjoint operator $L\sb 1:H^2\to L^2$
has exactly one negative eigenvalue,
which we denote $-\Lambda\sb 0$, and
has a three dimensional kernel $\ker L\sb 1$
spanned by $\{\partial_j \varphi_0\}_{j=1}^3$.
The range of
$L\sb 1$ is $(\ker L\sb 1)^{\perp}$, the $L^2$ orthogonal complement of
the linear span of the $\{\partial_j \varphi_0\}_{j=1}^3$.
\end{lemma}
\begin{proof}
We proceed similarly to \cite[Lemma 5.4.3]{kikuchi-thesis}.
The $n=0$ ground state solution $\varphi_0$ to \eqref{lambda-phi}
is characterized in \cite{MR0471785} as the solution,
unique up to translation and
phase rotation,
to the following minimization problem:
\begin{equation}\label{e-is-i}
E(\varphi_0)=I\sb\mu:=
\inf\{E({\phi})\sothat {\phi}\in H\sp 1(\R^3),
\ \norm{{\phi}}\sb{L\sp 2}^2=\mu\},
\end{equation}
for certain $\mu>0$.
We claim that this implies that $L\sb 1\geq 0$ on
$\{\varphi_0\}^{\perp}$. Indeed,
let $\norm{v}\sb{L\sp 2}=\norm{\varphi_0}\sb{L\sp 2}$,
$\langle v,\varphi_0\rangle=0$.
For $s\in(-1,1)$,
define ${\phi}\sb s=(1-s^2)^{1/2}\varphi_0+s v$,
so that $Q({\phi}\sb s)=Q(\varphi_0)$.
Calculating that ${\phi}\sb s\at{s=0}=\varphi_0$,
$\p\sb s\at{s=0}{\phi}\sb s=v$,
$\p\sb s^2\at{s=0}{\phi}\sb s=-\varphi_0$ we deduce from
\eqref{e-is-i},
\[
0\le\p\sb s^2\at{s=0}E({\phi}\sb s)
=\langle E'(\varphi_0),-\varphi_0\rangle
+\langle E''(\varphi_0)v,v\rangle
=
-{\omega\sb 0}\langle Q'(\varphi_0),\varphi_0\rangle
+\langle E''(\varphi_0)v,v\rangle
=
\langle v,(E''-{\omega\sb 0} Q'')v\rangle,
\]
establishing the claim.
We took into account that
$\varphi_0$ satisfies
the stationary equation
$E'(\varphi_0)={\omega\sb 0} Q'(\varphi_0)$
and also that
$
\langle Q'(\varphi_0),\varphi_0\rangle
=2\norm{\varphi_0}\sb{L\sp 2}^2
=2\norm{v}\sb{L\sp 2}^2
=\langle Q'(v),v\rangle
=\langle Q''v,v\rangle
$.
So $L\sb 1$ is non-negative on a codimension one subspace.
On the other hand,
since the integral kernel of $\Delta^{-1}$
is strictly negative, while $\varphi_0$
is strictly positive and $L\sb 0 \varphi_0=0$, it follows that
$\langle \varphi_0 L\sb 1\,\varphi_0\rangle<0$ so that there
certainly exists one negative eigenvalue characterized as
\[
-\Lambda\sb 0
:=\inf\{\langle {v},L\sb 1 {v}\rangle\sothat\norm{{v}}\sb{L\sp 2}=1\}<0.
\]
Let ${\eta\sb 0}$ be the corresponding eigenfunction,
$L\sb 1 {\eta\sb 0}=-\Lambda\sb 0 {\eta\sb 0}$. To prove that
$(-\Lambda\sb 0,0)\subset\rho(L\sb 1)$, the resolvent set,
consider the minimization problem
\begin{equation}\label{mu-positive}
\inf\{
\langle v,L\sb 1 v\rangle\sothat
\norm{v}^2=1,\ \langle \eta\sb 0, v\rangle=0
\}\, .
\end{equation}
Now the relation
$L\sb 0 \varphi_0=0$, together with translation invariance,
implies that
$L\sb 1\p\sb {j}\varphi_0=0$.
Moreover,
it is proved in \cite{MR2561169} that
$\varphi_0$ is nondegenerate,
in the sense that
the kernel of $L\sb 1$ is spanned by the $\p\sb {j}\varphi_0$,
$1\le j\le 3$.
Hence,
by consideration of linear combinations of the eigenfunctions $\eta_0$
and $\p\sb {j}\varphi_0$, that
the number defined by \eqref{mu-positive} is $\leq 0$. In fact it must
equal zero since if it were negative a simple compactness argument
(based on the negativity of ${\omega\sb 0}$) would imply the existence
of a negative eigenvalue in the interval $(-\Lambda_0,0)$ with
corresponding eigenfunction ${\eta\sb 1}$ orthogonal to ${\eta\sb 0}$.
But since
${\eta\sb 0},{\eta\sb 1}$ would then be an orthogonal pair of
eigenfunctions of $L\sb 1$ with negative eigenvalues, and
both having non-zero inner product with
$\varphi_0$, this would immediately contradict the fact that $L\sb 1$
is non-negative on $\{\varphi_0\}^{\perp}$.
\end{proof}
We conclude with a few remarks on the stability
of solitary waves to the Choquard equation.
By Remark~\ref{remark-scaling} we know the
$\omega$-dependence of
a localized solution
$\phi\sb\omega(x)e^{-i\omega t}$
to \eqref{choquard}:
one has
$\phi\sb\omega(x)=\zeta^2 u(\zeta\abs{x})$,
where $\zeta=\sqrt{-\omega}$. From this
we can obtain the frequency dependence of the charge:
\[
Q(\omega)
=\int\sb{\R^3}\phi^2(x)\,dx
=\zeta^4\int\sb{\R^3}u^2(\zeta x)\,dx
=\zeta\int\sb{\R^3}u^2(y)\,dy
=(-\omega)^{1/2}\int\sb{\R^3}u^2(y)\,dy.
\]
It follows that for all negative frequencies
$
\frac{dQ}{d\omega}<0\,.
$
By the Vakhitov-Kolokolov stability criterion (\cite{VaKo}),
this leads us to expect the linear stability
of no-node solitary waves (the ground states)
in the Choquard equation.
\begin{proposition}\label{prop-stability-ns}
The ground state solitary wave
$\varphi_0(x)e^{-i{\omega\sb 0} t}$
of the Choquard equation \eqref{choquard}
is linearly stable.
\end{proposition}
To determine the point spectrum of
$
\begin{bmatrix}0&L\sb 0\\-L\sb 1&0\end{bmatrix}$
observe that if $\begin{bmatrix}R\\S\end{bmatrix}$ is an eigenfunction
corresponding to the eigenvalue $\lambda\in\C$,
then $-\lambda^2 R=L\sb 0 L\sb 1 R$.
If $\lambda\ne 0$,
then one concludes that
$R$ is orthogonal to $\ker L\sb 0=\{\varphi_0\}$,
hence we can apply $L\sb 0^{-1}$;
taking then the inner product with $R$,
we deduce that:
\[
-\lambda^2
\langle R,L\sb 0^{-1}R\rangle
=
\langle R,L\sb 1 R\rangle\,,
\]
which implies that $\lambda^2\in\R$.
Moreover, by \eqref{mu-positive},
$\lambda^2\le 0$,
leading to the conclusion that the point spectrum
$\spec_d(JL)\subset i\R$, and hence the absence of
growing modes at the linearized level.
The (nonlinear) orbital stability of the ground state solitary
wave
was proved in \cite{MR677997}.
\begin{remark}\label{remark-stability}
In view of \cite{dirac-nd-arxiv,dirac-spectrum},
one expects that the linear stability or instability
of small amplitude solitary waves
is directly related to the linear stability or instability
of the corresponding nonrelativistic limit,
which for Dirac-Maxwell is given by the Choquard equation.
We hope that this may provide a route to understanding
stability of small solitary waves solutions for
the Dirac-Maxwell system.
\end{remark}
\section{Proof of existence of solitary waves in Dirac-Maxwell system}
\label{sect-exist}
In this section, we complete the proof of Theorem~\ref{theorem-sw-dm}.
It is
obtained as a consequence of Proposition~\ref{prop-sw-dm} after
the application of a rescaling motivated by the discussion in \S\ref{sect-nr}.
We write
$
\phi(x)
=
\begin{bmatrix}
\phi\sb 1(x)\\\phi\sb 2(x)
\end{bmatrix}
$,
where for $j=1,2$ the $\phi_j\in\C^2$
are essentially the components of $\phi$
in the range of the projection operators
$\varPi\sb{1}=\frac 1 2(1+\beta)$,
and $\varPi\sb{2}=\frac 1 2(1-\beta)$ (under obvious
isomorphisms of these subspaces with $\C^2$).
Applying $\varPi\sb{1}$ and $\varPi\sb{2}$
to \eqref{omega-phi-is},
we have:
\begin{equation}\label{mds1}
\omega{\phi\sb{1}}
={\bm\sigma\cdot}(-i\bm\nabla-e \mathbf{A}){\phi\sb{2}}+m{\phi\sb{1}}+e A\sp 0{\phi\sb{1}},
\end{equation}
\begin{equation}\label{mds2}
\omega\phi\sb 2
={\bm\sigma\cdot}(-i\bm\nabla-e \mathbf{A}){\phi\sb{1}}-m{\phi\sb{2}}+e A\sp 0{\phi\sb{2}},
\end{equation}
\begin{equation}\label{mds3}
-\Delta A\sp 0=e(\phi_1\sp\ast\phi_1+\phi_2\sp\ast\phi_2),
\qquad
-\Delta\mathbf{A}
=e\phi\sp\ast\bm\alpha\phi
=e\big(
\phi\sb{1}\sp\ast\bm\sigma{\phi\sb{2}}
+\phi\sb{2}\sp\ast\bm\sigma{\phi\sb{1}}\big).
\end{equation}
We write \eqref{mds3} as
\begin{equation}
A\sp 0=e\N*\,(\phi_1\sp\ast\phi_1+\phi_2\sp\ast\phi_2)\,,
\qquad
\mathbf{A}
=e\N*\,(\phi\sb{1}\sp\ast\bm\sigma{\phi\sb{2}}
+\phi\sb{2}\sp\ast\bm\sigma{\phi\sb{1}})\,,
\label{nl2}
\end{equation}
and regard the potentials $A\sp 0$ and $\mathbf{A}=(A\sp j)$
as non-local functionals of $\phi=\begin{bmatrix}\phi_1\\\phi_2\end{bmatrix}$.
Above,
$\N(x)=(4\pi|x|)^{-1}$ is the Newtonian potential.
In abstract terms, the equations are of the form
$\omega\mathcal{Q}'=\mathcal{E}'$ where
the charge functional is
\begin{equation}\label{def-q}
\mathcal{Q}(\phi)=\int\,
\phi\sp\ast(x)\phi(x)\,dx,
\end{equation}
and, regarding $A\sp 0,\,\mathbf{A}$
as fixed non-local functionals \eqref{nl2}
of $\phi
$,
the Hamiltonian
$\mathcal{E}(\phi)$ is given by
\begin{equation}\label{def-e}
\mathcal{E}(\phi)=\int\,
\Bigl(
-i\phi\sp\ast{\bm\alpha\cdot}\bm\nabla\phi
+m\phi\sp\ast\beta\phi
+\frac{
A\sp 0\phi\sp\ast\phi-\mathbf{A}\cdot(\phi\sp\ast\bm\alpha\phi)
}{2}
\,\Bigr)\,dx.
\end{equation}
For future reference we recall the following trick from \cite{MR1708440}:
\begin{lemma}\label{lemma-trick}
Let ${\xi}^\alpha$ be a finite
collection of vector fields on the phase space which are infinitesimal
symmetries, in the sense that
$\langle\mathcal{Q}'\,,\,{\xi}^\alpha\rangle=0
=\langle\mathcal{E}'\,,\,{\xi}^\alpha\rangle.$
Then any solution of the
equation $\omega\mathcal{Q}'-\mathcal{E}'-\sum a_\alpha {\xi}^\alpha=0\,,$
for some set $a_\alpha\in\R$,
is also
a solution of $\omega\mathcal{Q}'-\mathcal{E}'=0$,
as long as
the matrix $\langle {\xi}^\alpha,{\xi}^\beta\rangle$ is
well defined and nondegenerate.
\end{lemma}
\begin{proof}
For sufficiently regular ${\xi}^\beta$ it is possible to take the inner
product, yielding $\sum\,a_\alpha\langle {\xi}^\alpha,{\xi}^\beta\rangle=0$
which gives the result. (The precise meaning of \emph{sufficiently regular}
is just that this computation is valid; it would be sufficient for
$\xi^\alpha$ to lie in a subspace $F$ of $L^2$ with the property that
the equation $\omega\mathcal{Q}'-\mathcal{E}'-\sum a_\alpha {\xi}^\alpha=0$
holds in the dual of $F$.)
\end{proof}
\begin{example}
For $\psi:\R\to\C$ and $\mathcal{Q}=\frac{1}{2}\int\,|\psi|^2$ and
$\mathcal{E}=\int\,\frac{1}{2}|\nabla\psi|^2-\frac{1}{p+1}|\psi|^{p+1}$
the symmetry of phase rotation corresponds to the infinitesimal
symmetry ${\xi}(\psi)=i\psi$, and it is easy to check that given
an $H^1$ distributional
solution of $\omega\mathcal{Q}'-\mathcal{E}'-a{\xi}=0$, i.e. a
weak solution of $-\Delta\psi-|\psi|^p\psi=\omega\psi-i a\psi$,
for any $a\in\R$,
one necessarily has $a=0$. The same holds in higher dimensions
as long as $p$ is such that the equation holds as an equality in $H^{-1}$.
\end{example}
\begin{remark}
\label{is}
The advantage of solving the more general
equation with the unknown ``multipliers'' $a_\alpha$
is that in an implicit function
theorem setting the multipliers can be varied to fill out the part of
the cokernel corresponding to the symmetries. It is then shown after the
fact that the multipliers are in fact zero.
The choice of ${\xi}^\alpha$
is determined by the symmetry group;
in the case of Dirac-Maxwell the relevant group is the seven dimensional
group generated by translations, rotations and phase rotation.
The infinitesimal versions of these actions give the following vector fields
(\cite{MR0187641}):
\begin{eqnarray}\label{def-xi-eta-zeta}
\bm\xi=\bm\nabla\phi\,,
\qquad
\bm\eta
=\frac{i}{2}
\begin{bmatrix} \bm\sigma&0 \\ 0&\bm\sigma\end{bmatrix}
\phi\,,
\qquad
\zeta
=i\phi.
\end{eqnarray}
\end{remark}
In accordance with the heuristics in
\S\ref{sect-heur} we introduce functions
${\varPhi\sb{1}}(y,\epsilon),\ {\varPhi\sb{2}}(y,\epsilon)\in\C^2$
and
$\eurA\sp\mu(y,\epsilon)$
by the following scaling relations:
\begin{equation}\label{ansatz-2}
{\phi\sb{1}}(x,\omega)=\epsilon^3{\varPhi\sb{1}}(\epsilon x,\epsilon),
\quad
{\phi\sb{2}}(x,\omega)=\epsilon^2{\varPhi\sb{2}}(\epsilon x,\epsilon),
\quad
e A\sp 0(x,\omega)=\epsilon^2\eurA\sp 0(\epsilon x,\epsilon),
\quad
e A\sp j(x,\omega)=\epsilon^3\eurA\sp j(\epsilon x,\epsilon),
\end{equation}
where
$\epsilon$ and $\omega$ are related by
$\omega=-\sqrt{m^2-\epsilon^2}$.
Then, writing $\bm\nabla\sb{y}$ for the
gradient with respect to $y^j=\epsilon x^j$,
$1\le j\le 3$,
we have:
\begin{equation}\label{sys-phi1}
-2m{\varPhi\sb{1}}
+i\bm\sigma{\cdot\bm\nabla}\sb{y}{\varPhi\sb{2}}-\epsilon^2\eurA\sp 0{\varPhi\sb{1}}
=
-(m+\omega){\varPhi\sb{1}}
-\epsilon^2{\eubA\cdot}\bm\sigma{\varPhi\sb{2}},
\end{equation}
\begin{equation}\label{sys-phi2}
\frac{1}{2m}{\varPhi\sb{2}}
+i\bm\sigma{\cdot\bm\nabla}\sb{y}{\varPhi\sb{1}}-\eurA\sp 0{\varPhi\sb{2}}
=
\Big(\frac{1}{2m}-\frac{1}{m-\omega}\Big){\varPhi\sb{2}}
-\epsilon^2{\eubA\cdot}\bm\sigma{\varPhi\sb{1}},
\end{equation}
\begin{equation}\label{sys-a}
\eurA\sp 0
=e^2\N\ast\Bigl(\varPhi\sb{2}\sp\ast{\varPhi\sb{2}}
+\epsilon^2\varPhi\sb{1}\sp\ast{\varPhi\sb{1}}\Bigr)\,
\qquad
\eubA
=e^2\N\ast\Bigl(\varPhi\sb{1}\sp\ast\bm\sigma{\varPhi\sb{2}}
+\varPhi\sb{2}\sp\ast\bm\sigma{\varPhi\sb{1}}\Bigr)\,.
\end{equation}
Let $\varphi_0\in\mathscr{S}(\R^3)$
be the ground state solution to
the Choquard equation with $\omega_0=-\frac{1}{2m}$:
\begin{equation}\label{NR1}
\frac{1}{2m}\varphi_0
-\frac{1}{2m}\Delta \varphi_0-
\big(\N\ast\varphi_0^2\big)\varphi_0=0.
\end{equation}
That is,
$\varphi_0$ is a strictly positive, spherically symmetric,
smooth, and exponential decaying function.
As discussed in the previous section,
such a solution exists by \cite{MR0471785};
the value $\omega_0=-(2m)^{-1}$
is chosen for our convenience.
Using $\varphi_0$, we can
produce a solution
to \eqref{sys-phi1}-\eqref{sys-a}
in the nonrelativistic limit $\epsilon=0$:
\begin{equation}\label{NR2}
\hat{\varPhi}=\begin{bmatrix}\hat\varPhi\sb 1\\
\hat\varPhi\sb 2\end{bmatrix}\in\C^4,
\qquad
\mbox{with}
\quad
\hat\varPhi\sb 2=\varphi_0\,\begin{bmatrix}1\\0\end{bmatrix}\,,\qquad
\hat\varPhi\sb 1=\frac{i}{2m}\bm\sigma{\cdot\bm\nabla}\sb{y}\hat\varPhi\sb 2\,;
\end{equation}
\begin{equation}\label{NR3}
\hat\eurA\sp 0=e^2\N*\varphi_0^2\,,
\qquad
\hat\eurA\sp 1=-\frac{e^2}{m}\N*\varphi_0\partial_2\varphi_0\,,\qquad
\hat\eurA\sp 2=+\frac{e^2}{m}\N*\varphi_0\partial_1\varphi_0\,,\qquad
\hat\eurA\sp 3=0.
\end{equation}
The symmetry of this configuration
is axial, with the magnetic field along the $z$ axis of symmetry.
\begin{lemma}\label{lemma-lum}
Let $\varPhi=\begin{bmatrix}\varPhi_1\\\varPhi_2\end{bmatrix}
\in H\sp 1(\R^3,\C^{4})$.
Then $\eurA\sp\mu$ defined by \eqref{sys-a}
satisfy
\[
\eurA\sp\mu
\in
L\sp\infty(\R^3),
\qquad
0\le \mu\le 3.
\]
\end{lemma}
\begin{proof}
The functions $\eurA\sp\mu$
defined by \eqref{sys-a}
are of the form $\N* h$ with
$h:=f g$,
where $f,\,g\in H^1(\R^3)$.
Due to the Sobolev embedding
$H\sp 1(\R^3)\subset L\sp 6(\R^3)$,
we have
$h\in L\sp p(\R^3)\,,1\le p\le 3$.
By the H\"older inequality,
one has
$\abs{(\abs{x}^{-1}\chi\sb{B\sb 1}(x))\ast h}
\le\norm{\abs{x}^{-1}}\sb{L\sp{3/2}(B\sb 1)}
\norm{h}\sb{L^3}<\infty,
$
$\abs{(1-\chi\sb{B\sb 1}(x))\ast h}
\le
\norm{h}\sb{L^1}<\infty,
$
where
$B\sb 1$
is the unit ball in $\R^3$
and $\chi\sb{B\sb 1}$
is its characteristic function,
hence
$\abs{x}^{-1}\ast h\in L\sp\infty(\R^3)$.
Furthermore the structure of
\eqref{sys-a}
makes it clear that
the mappings
$(\varPhi,\epsilon)\mapsto\eurA\sp\mu$,
$0\le \mu\le 3$,
are smooth from $H\sp 1(\R^3,\C^{4})\times\R$ to $L^\infty(\R^3)$.
\end{proof}
Define
\begin{eqnarray}
&&
X=H^1(\R^3;\C^2)\oplus H^2(\R^3;\C^2)\subset L^2(\R^3;\C^2)
\oplus L^2(\R^3;\C^2)\,,
\phantom{\int\sb a}
\label{def-x}
\\
&&
Y=H^1(\R^3;\C^2)\oplus L^2(\R^3;\C^2)
\subset L^2(\R^3;\C^2)\oplus L^2(\R^3;\C^2).
\label{def-y}
\end{eqnarray}
Introducing ${\mom}=-i\bm\sigma{\cdot\bm\nabla}\sb{y}$
and substituting $\omega=-\sqrt{m^2-\epsilon^2}$,
we rewrite
\eqref{sys-phi1}, \eqref{sys-phi2}
as the equation $\mathcal{F}=0$,
where
\[
\mathcal{F}:\;
X\times(-m,+m)\longrightarrow Y,
\]
\begin{equation}\label{df}
\mathcal{F}:\;
\Big(
\begin{bmatrix}\varPhi_1\\\varPhi_2\end{bmatrix},
\,\epsilon
\Big)
\mapsto
\begin{bmatrix}
\displaystyle
2m{\varPhi\sb{1}}
+{\mom}
{\varPhi\sb{2}}+\epsilon^2\eurA\sp 0{\varPhi\sb{1}}
-\big(m-\sqrt{m^2-\epsilon^2}\big){\varPhi\sb{1}}
-\epsilon^2{\eubA\cdot}\bm\sigma{\varPhi\sb{2}}
\\
\displaystyle
-\frac{1}{2m}{\varPhi\sb{2}}
+{\mom}{\varPhi\sb{1}}+\eurA\sp 0{\varPhi\sb{2}}
+
\Big(
\frac{1}{2m}-\frac{1}{m+\sqrt{m^2-\epsilon^2}}
\Big)
{\varPhi\sb{2}}
-\epsilon^2{\eubA\cdot}\bm\sigma{\varPhi\sb{1}}\,
\end{bmatrix}.
\end{equation}
As above, we regard the
$\eurA\sp \mu=(\eurA\sp 0,\eubA)$,
$\eubA=(\eurA\sp j)$,
as non-local functionals
$\eurA\sp \mu=\eurA\sp \mu(\varPhi,\epsilon)$
determined by \eqref{sys-a}.
With this understood,
the entire system is encapsulated
in the equation $\mathcal{F}(\varPhi,\epsilon)=0$ for
$\varPhi=\begin{bmatrix}\varPhi_1\\\varPhi_2\end{bmatrix}$ only.
In terms of the original variables:
\begin{equation}
\mathcal{F}(\varPhi,\epsilon)\,=\,
\begin{bmatrix}
\epsilon^{-3}& 0\\
0&\epsilon^{-4}
\end{bmatrix}
(\mathcal{E}'-\omega\mathcal{Q}')(
\epsilon^3{\varPhi\sb{1}},
\epsilon^2{\varPhi\sb{2}}),
\end{equation}
where the functionals
$\mathcal{Q}$,
$\mathcal{E}$
are defined by \eqref{def-q}, \eqref{def-e}.
The nonrelativistic limit satisfies
$\mathcal{F}(\hat\varPhi,0)=0$
(cf.~\eqref{NR2},~\eqref{NR3}),
so that
to obtain solutions for small $\epsilon$ it is necessary to
compute the derivative of $\mathcal{F}$ at the point
$(\hat\varPhi,0)$. This is
determined by the set of directional derivatives.
Let
$\e_1=\begin{bmatrix}1\\0\end{bmatrix}$ and
$\e_2=\begin{bmatrix}0\\1\end{bmatrix}$, and
let
$g\in H^1(\R^3,\C^2)$.
To compute the directional derivatives first note that
$\eurA\sp j$ drops out on putting $\epsilon=0$, and then
note further that by \eqref{sys-a} only the derivative
of $\eurA\sp 0$ at
$(\hat\varPhi\sb 1,0)$ with respect to
$\varPhi\sb 2$ is nonzero, with derivative given by
$$
\frac{d}{dt}\eurA\sp 0
\Big(\begin{bmatrix}\hat\varPhi_1\\\hat\varPhi_2+t g\end{bmatrix},\epsilon
\Big)
\big|_{t=0,\epsilon=0}
=2e^2\N\ast\bigl(\varphi_0\Re\langle \e_1,g\rangle\sb{\C^2} \bigr).
$$
We deduce that
for $\C^2$-valued functions $U$ and $V$,
\[
\frac{d}{dt}\mathcal{F}
\Big(
\begin{bmatrix}
\hat\varPhi_1+t U\\
\hat\varPhi_2+t V\end{bmatrix}
,\epsilon
\Big)
\big|_{t=0,\epsilon=0}
=\;\eurM\begin{bmatrix}U\\V\end{bmatrix}
\,,
\]
where
\begin{equation}\label{def-m}
\eurM=
\begin{bmatrix}
2m
&{\mom}
\\
{\mom}&-\frac{1}{2m}+\hat\eurA\sp 0
+2
\varphi_0 \e_1
\N\ast(\varphi_0\Re\langle \e_1,\,\cdot\,\rangle\sb{\C^2})
\end{bmatrix},
\qquad
{\mom}=-i\bm\sigma{\cdot\bm\nabla}\sb{y},
\qquad
\N=\frac{1}{4\pi\abs{y}}.
\end{equation}
Thus, the derivative of $\mathcal{F}$ at
the nonrelativistic limit point $(\hat\varPhi,0)$
is the linear map
$D\mathcal{F}(\hat\varPhi,0)$
given by the matrix
$
\eurM
$. This is a differential operator, which we consider as an
unbounded operator on $L^2(\R^3;\C^2)
\oplus L^2(\R^3;\C^2)$.
\begin{lemma}
\label{lemma-fpc}
\begin{enumerate}
\item
The map
$\eurM:\,\begin{bmatrix}U\\V\end{bmatrix}
\mapsto\begin{bmatrix}F\\G\end{bmatrix}$
is a Hermitian operator
with domain
$X$.
\item
$\eurM$ maps
$X$ continuously into $Y$
(cf.~\eqref{def-x},~\eqref{def-y}).
\item
The kernel
of $\eurM$
is given by
\[
\ker\eurM
=
\left\{
\Big(-\frac{{\mom\,} V}{2m},\,V\Big):\ V
=
\mathbf{a}{\cdot\bm\nabla}\sb{y}
\varphi_0\,\e_1+ib\varphi_0\,\e_1+c\varphi_0\,\e_2,
\ \
\mathbf{a}\in\R^3,
\ \ b\in\R\,,\ \ c\in\C
\right\}.
\]
\item
The range of $\eurM:\,
\begin{bmatrix}U\\V\end{bmatrix}
\mapsto
\begin{bmatrix}F\\G\end{bmatrix}$
is closed in the topology of $Y$
and is given by
\begin{eqnarray}
\range \eurM
=(\ker\eurM)\sp\perp
&=&
\left\{\begin{bmatrix}F\\G\end{bmatrix}\in Y:
\,\Re \Big(\frac{{\mom\,} F}{2m}-G\Big)_1\in(\ker L_1)^\perp,
\right.
\nonumber
\\
&&
\left.
\quad
\,\Im\Big(\frac{{\mom\,} F}{2m}-G\Big)_1
\in(\ker L_0)^\perp,
\,\Big(\frac{{\mom\,} F}{2m}-G\Big)_2\in(\ker L_0)^\perp
\right\},
\nonumber
\end{eqnarray}
where $^\perp$ means the orthogonal complement with
respect to the inner product in $L^2\oplus L^2$.
\item
The inverse of
$\eurM:\,\begin{bmatrix}U\\V\end{bmatrix}
\mapsto \begin{bmatrix}F\\G\end{bmatrix}$
is given by
\[
U=\frac{1}{2m}\bigl({F}-{{\mom}\, V}\bigr)\,,
\]
\[
V=\e_1 V_1+\e_2 V_2
=
\left(
L_1^{-1}\,\Re\,\Big(\frac{{\mom\,} F}{2m}-G\Big)_1
+i L_0^{-1}\,\Im\,\Big(\frac{{\mom\,} F}{2m}-G\Big)_1
\right)
\e_1
+L_0^{-1}\,\Big(\frac{{\mom\,} F}{2m}-G\Big)_2\e_2\,,
\]
\end{enumerate}
where
the definitions
and properties of the operators $L_0,\,L_1$ are given in \S\ref{sect-nr}.
\end{lemma}
\begin{proof}
The proof depends on some properties of the linearized Choquard equation
from \cite{MR2561169} which are stated
in \S\ref{sect-nr}.
The fact in (1) that $\eurM$ is Hermitian follows from the fact that $\mom$
is Hermitian. From Lemma~\ref{lemma-lum} the assertion (2) is immediate from
the properties of
$\N$ and the fact that $\varphi_0$ and its
partial derivatives are smooth and exponentially decreasing.
To prove (3),(4) and (5) we consider how
to solve
$\eurM\begin{bmatrix}U\\V\end{bmatrix}=
\begin{bmatrix}F\\G\end{bmatrix}$, i.e. the system
\[
\eurM
\begin{bmatrix}
U\\V
\end{bmatrix}
=
\begin{bmatrix}
2m U+{\mom\,} V
\\
{\mom\,} U
-\frac{V}{2m}
+
\hat\eurA\sp 0 V
+2 \varphi_0\e_1\N\ast(\varphi_0\Re V_1)
\end{bmatrix}
=\begin{bmatrix}F\\G\end{bmatrix}.
\]
We first express $U$ in terms of $V$
by
$
U=\frac{1}{2m}(F-{{\mom\,} V})\,,
$
and, writing $V=V_1\e_1+V_2\e_2$,
\[
\frac{{\mom\,} F}{2m}
+
\frac{\Delta V}{2m}-\frac{V}{2m}
+\hat\eurA\sp 0 V
+2u_0\e_1\N\ast(u_0\Re V_1)
=G.
\]
Referring to the definitions in
\S\ref{sect-nr} of $L_0$ and $L_1$, with
${\omega\sb 0}$ set equal to $-(2m)^{-1}$,
we arrive at the following equations:
\begin{equation}
L_1 V_1=\Big(\frac{{\mom\,} F}{2m}-G\Big)_1\,,
\qquad
L_0 V_2=\Big(\frac{{\mom\,} F}{2m}-G\Big)_2\,.
\end{equation}
It is useful here that the components
with respect to $\e_1$ and $\e_2$ are decoupled.
Noting also from the form of $L_0,L_1$ that these operators
take real/imaginary valued
functions to real/imaginary valued
functions, and further that
$L_1=L_0$ on pure imaginary
functions, we obtain the given formula for $V$, and hence for $U$,
immediately from \S\ref{sect-nr}. The identification of the kernel in (3)
is then a specialization of this, given the information on
$\ker L\sb 0$ and $\ker L\sb 1$ in \S\ref{sect-nr}, and also (4)
is a consequence of the identification of the ranges of $L\sb 0$ and
$L\sb 1$ given in \S\ref{sect-nr}.
\end{proof}
The statement of Theorem~\ref{theorem-sw-dm}
will follow from the following result.
\begin{proposition}\label{prop-sw-dm}
There is ${\epsilon\sb\ast}>0$ such that
for $\epsilon\in(-\epsilon\sb\ast,{\epsilon\sb\ast})$
there is a solution to \eqref{mds1}-\eqref{mds3},
with $\omega=-\sqrt{m^2-\epsilon^2}$,
given by the ansatz \eqref{ansatz-2} with $(\varPhi\sb 1,\varPhi\sb 2)$
obtained as the image of a $C^\infty$-function
\[
\varPhi\in C^\infty\bigl(
(-{\epsilon\sb\ast},{\epsilon\sb\ast})\,;\,
\big(H^1(\R^3;\C^2)\oplus H^2(\R^3;\C^2)\big)
\cap \ker\eurM
\bigr)
\]
satisfying $\varPhi(0)=\hat{\varPhi}$,
and with
$\eurA\sp 0\in C^\infty\bigl(
(-{\epsilon\sb\ast},{\epsilon\sb\ast})\,;\, \dot H\sp 1\cap L^\infty\bigr)$,
$\eurA\sp j\in C^\infty\bigl(
(-{\epsilon\sb\ast},{\epsilon\sb\ast})\,;\,
\dot H\sp 1\cap L^\infty\bigr)
$
given by \eqref{sys-a}:
\begin{eqnarray*}
&&
\eurA\sp 0=e^2\N*\,(\varPhi_1\sp\ast\varPhi_1+
\epsilon^2\varPhi_2\sp\ast\varPhi_2)
\in C^\infty\bigl(
(-{\epsilon\sb\ast},{\epsilon\sb\ast})\,;\, \dot H\sp 1\cap L^\infty\bigr)
\,,
\\
&&
\eubA=e^2\N*\,(\varPhi\sb{1}\sp\ast\bm\sigma{\varPhi\sb{2}}
+\varPhi\sb{2}\sp\ast\bm\sigma{\varPhi\sb{1}})
\in C^\infty\bigl(
(-{\epsilon\sb\ast},{\epsilon\sb\ast})\,;\,
\dot H\sp 1\cap L^\infty\bigr)
\,.
\end{eqnarray*}
Above,
$\dot H^1=\dot H^1(\R^3,\R)$ is the homogeneous Dirichlet
space of $L^6$ functions with $\|f\|_{\dot H^1}^2=\int\,|\nabla f|^2\,
dx<\infty$.
For small $\epsilon$, one has
\begin{equation}\label{phi1-phi2-epsilon-2}
\norm{{\varPhi\sb{1}}-\hat\varPhi\sb{1}}\sb{H\sp 2}
=O(\epsilon^2),
\qquad
\norm{{\varPhi\sb{2}}-\hat\varPhi\sb{2}}\sb{H\sp 1}
=O(\epsilon^2)\,.
\end{equation}
The $\varPhi_j$ are even in $\epsilon$.
\end{proposition}
\begin{proof}
Solutions of \eqref{mds1}-\eqref{mds3} for small $\epsilon$ can be produced
by solving $\mathcal{F}=0.$
The proof of existence of solutions to this equation
is by the implicit function theorem and Lemma~\ref{lemma-trick},
perturbing from the nonrelativistic limit point
$\mathcal{F}(\hat\varPhi,0)=0$.
To start we claim that $\mathcal{F}$, as defined in
\eqref{df}, is a $C^\infty$ function $X\times (-m,+m)\to Y$. To
prove this notice that the expression for $\mathcal{F}$
is manifestly smooth in $\epsilon$ for $\epsilon^2<m^2,$ and
its dependence on $\varPhi_j$ is built up from compositions of
certain multilinear maps and linear operators;
the structure of the expressions obtained after
successive differentiation is the same. Referring to the specific formulae,
the fact that these expressions are all $C^\infty$
is an immediate consequence of the fact that multiplication
gives continuous bilinear ($\implies$ smooth) maps $H^1\times H^2\to H^1$ and
$H^2\times H^2\to H^2$ (Moser inequalities) and Lemma~\ref{lemma-lum}.
We are looking for $\varPhi(\epsilon)$ in the form
\begin{equation}\label{varphi-varpsi}
\varPhi(\epsilon)=\hat\varPhi+\varPsi(\epsilon),
\qquad
\varPsi(0)=0.
\end{equation}
We use the same component notation as above:
$
\hat\varPhi=\begin{bmatrix}\hat\varPhi\sb 1\\\hat\varPhi\sb 2\end{bmatrix}\,,
$
$
\varPsi=\begin{bmatrix}\varPsi\sb 1\\\varPsi\sb 2\end{bmatrix}\,.
$
We apply the implicit function theorem to the function
\[
\mathcal{G}:\,X\times
\R^3\times\R^3\times (-m,+m)\longrightarrow Y,
\]
\begin{equation}\label{def-g}
\mathcal{G}(\varPsi,\mathbf{a},\mathbf{b},\epsilon)
=
\mathcal{F}(\hat\varPhi+\varPsi,\epsilon)\,+
\mathbf{a}{\cdot\bm\nabla}\sb{y}
\begin{bmatrix}\epsilon(\hat\varPhi\sb 1+\varPsi\sb 1)\\
\hat\varPhi\sb 2+\varPsi\sb 2\end{bmatrix}
\,+\,
\frac{i}{2}
\mathbf{b}\cdot\begin{bmatrix}
\epsilon\bm\sigma(\hat\varPhi\sb 1+\varPsi\sb 1)\\
\bm\sigma(\hat\varPhi\sb 2+\varPsi\sb 2)\end{bmatrix}.
\end{equation}
\begin{remark}
Referring to \eqref{is} we have introduced a linear combination
of the six infinitesimal symmetries corresponding to translation and
rotation. The action of phase rotation is not independent of rotation
in the nonrelativistic limit, which is why the seventh parameter does not
appear. In terms of the original variables
(cf. \eqref{def-xi-eta-zeta}):
\begin{equation*}
\mathcal{G}(\varPsi,\mathbf{a},\mathbf{b},\epsilon)\,=\,
\begin{bmatrix}
\epsilon^{-3}& 0\\
0&\epsilon^{-4}
\end{bmatrix}
\Bigl(\mathcal{E}'-\omega\mathcal{Q}'
+\epsilon \mathbf{a}\cdot\bm\xi+\epsilon^2 \mathbf{b}\cdot\bm\eta\Bigr)
\quad\hbox{evaluated at}\;
\phi=\begin{bmatrix}\epsilon^3(\hat\varPhi\sb 1+\varPsi\sb 1)
\\
\epsilon^2 (\hat\varPhi\sb 2+\varPsi\sb 2)
\end{bmatrix}
\,.
\end{equation*}
\end{remark}
Computing the derivatives of \eqref{def-g} at $\epsilon=0$,
we see that the linear span
$\langle
\{
\p\sb{\mathbf{a}\sb j}\mathcal{G},
\p\sb{\mathbf{b}\sb j}\mathcal{G}
\sothat
1\le j\le 3
\}\rangle$
is equal to $\ker\eurM$.
Referring to Lemma~\ref{lemma-fpc},
this establishes that the derivative of $\mathcal{G}$ at
$\epsilon=0,\varPsi=0$,
$\mathbf{a}=0$, $\mathbf{b}=0$
with respect to
$(\Psi,\mathbf{a},\mathbf{b})$ is a linear homeomorphism from
$\big((\ker\eurM)^\perp\cap X\big)\times \R^3\times\R^3$ onto $Y$.
It follows that there is $\epsilon\sb\ast>0$
such that
there exist $C^\infty$ functions
$\epsilon\mapsto(\varPsi(\epsilon),\mathbf{a}(\epsilon),\mathbf{b}(\epsilon))
\in X\times \R^3\times\R^3$,
defined for $\epsilon\in(-\epsilon\sb\ast,\epsilon\sb\ast)$,
such that
\begin{equation}\label{varphi-perp-ker-m}
\mathcal{G}
\big(
\varPsi(\epsilon),\mathbf{a}(\epsilon),\mathbf{b}(\epsilon),\epsilon
\big)=0,
\qquad
\varPsi(\epsilon)
\perp
\ker\eurM,
\qquad
\epsilon\in(-\epsilon\sb\ast,\epsilon\sb\ast).
\end{equation}
This latter condition serves to divide out
by the action of the symmetry group, giving a local slice.
Referring to Lemma~\ref{lemma-trick}, to deduce that these in
fact generate solutions of $\mathcal{F}=0$,
for sufficiently small $\epsilon>0$,
it is sufficient to verify that
$\mathbf{a}(\epsilon)=0$,
$\mathbf{b}(\epsilon)=0$,
which
is in turn a consequence of the nondegeneracy of the matrix of inner products
of the infinitesimal vector fields, scaled as above. This amounts
to the need to verify nondegeneracy of the $6\times 6$ matrix
\begin{equation}\label{xi-xi}
\begin{bmatrix}
\langle\p\sb{y\sp j}{}\phi,\p\sb{y\sp k}{}\phi\rangle
\phantom{\int\sb\int}
&
\langle\p\sb{y\sp j}{}\phi,\frac i 2\Sigma\sb{k'}\phi\rangle
\\
\langle\frac i 2\Sigma\sb{j'}\phi,\p\sb{y\sp k}\phi\rangle
&
\langle
\p\sb{j'}{}\phi
\frac i 2\Sigma\sb{j'}{}\phi
,\frac i 2\Sigma\sb{k'}{}\phi\rangle
\end{bmatrix}
\end{equation}
for small $\epsilon$.
(In the matrix \eqref{xi-xi} the indices $j$, $j'$, $k$, $k'$ run
between $1$ and $3$.)
\begin{lemma}\label{lemma-xi-xi}
The matrix
given by \eqref{xi-xi},
evaluated at
$\phi(x)=\begin{pmatrix}\epsilon^3(\hat\varPhi\sb 1+\varPsi\sb 1)
\\
\epsilon^2 (\hat\varPhi\sb 2+\varPsi\sb 2)
\end{pmatrix}\biggr |_{y=\epsilon x}
$,
is nondegenerate for small $\epsilon$.
\end{lemma}
\begin{proof}
Clearly
the dominant terms arise from the second (``large'') component
giving rise to diagonal matrix elements which, referring to
the block form in \eqref{xi-xi}, are
$O(\epsilon^4)$.
Since $\varPsi_j=O(\epsilon)$,
the result will follow from
nondegeneracy of the matrix with $\varPsi\sb j$
set equal to zero.
Using
$\epsilon^{-2}\phi
=
\begin{bmatrix}-\frac{\epsilon}{2m}{\mom\,}
\hat\varPhi\sb 2
\\
\hat\varPhi\sb 2
\end{bmatrix}
$
and
$
\hat\varPhi\sb 2=\begin{bmatrix}\varphi_0\\0\end{bmatrix}$,
we calculate the first diagonal term:
\[
\epsilon^{-4}\langle\p\sb{y\sp j}{}\phi,\p\sb{y\sp k}{}\phi\rangle
\,=\,
\epsilon^2\Big\langle
\p\sb{y\sp j} \varphi_0,
(-\frac{\Delta_y}{4m^2})\p\sb{y\sp k} \varphi_0
\Big\rangle
+
\Big\langle
\p\sb{y\sp j} \varphi_0,
\p\sb{y\sp k} \varphi_0
\Big\rangle\,+\,O(\epsilon)
\,=\,
\frac{\delta\sb{j k}}{3}
\Big\langle
\varphi_0,
(-\Delta_y)\varphi_0
\Big\rangle\, +\,O(\epsilon),
\]
where we took into account the spherical symmetry
of $\varphi_0$,
which leads to
$\langle \p\sb{y\sp 1} \varphi_0,\p\sb{y\sp 1}\varphi_0\rangle
=\frac 1 3\langle \varphi_0,(-\Delta_y)\varphi_0\rangle
$. Next for the off-diagonal terms we compute, again using
the same expression for $\epsilon^{-2}\phi$:
\[
\epsilon^{-4}
\langle\p\sb{y\sp j}{}\phi,\frac i 2\Sigma\sb{k'}{}\phi\rangle
=
-
\frac{\epsilon^2}{4m^2}
\Big\langle
\p\sb{y\sp j}
{\mom\,}
\begin{bmatrix}\varphi_0\\0\end{bmatrix},
\frac{i}{2}
\sigma\sb{k'}
{\mom\,}
\begin{bmatrix}\varphi_0\\0\end{bmatrix}
\Big\rangle
+
\Big\langle
\p\sb{y\sp j}\begin{bmatrix}\varphi_0\\0\end{bmatrix},
\frac{i}{2}
\sigma\sb{k'}\begin{bmatrix}\varphi_0\\0\end{bmatrix}
\Big\rangle\,+O(\epsilon)\,=\,O(\epsilon)\,.
\]
The first two terms are identically zero
since $\varphi_0$ is spherically symmetric
(so that by parity considerations
it is $L^2$ orthogonal to all of its first partial derivatives, which
are in turn orthogonal to all of the second partial derivatives).
Finally, for the second diagonal term:
\[
\epsilon^{-4}\Big\langle
\frac i 2\Sigma\sb{j'}\phi
,\frac i 2\Sigma\sb{k'}\phi\Big\rangle
=
\frac{\delta\sb{j' k'}}{4}
\langle
\varphi_0,\varphi_0\rangle+O(\epsilon).
\]
The non-degeneracy
of the matrix \eqref{xi-xi} for small $\epsilon$
follows.
\end{proof}
Since the implicit function theorem proves that
$\varPhi\sb 1(y,\epsilon)$, $\varPhi\sb 2(y,\epsilon)$
are $C^\infty$-functions of $\epsilon\in(-\epsilon\sb\ast,\epsilon\sb\ast)$,
one has
\begin{equation}\label{phi1-phi2-epsilon-1}
\norm{{\varPhi\sb{1}}-\hat\varPhi\sb{1}}\sb{H\sp 2}
=O(\epsilon),
\qquad
\norm{{\varPhi\sb{2}}-\hat\varPhi\sb{2}}\sb{H\sp 1}
=O(\epsilon)\,.
\end{equation}
To prove a stronger estimate
\eqref{phi1-phi2-epsilon-2},
we take the derivative of
\eqref{df}
with respect to $\epsilon$ at $\epsilon=0$;
this yields
\[
\eurM
\p\sb\epsilon\varPhi\at{\epsilon=0}
=0,
\]
with $\eurM$ given by \eqref{def-m}.
Due to \eqref{varphi-varpsi},
one has
$\p\sb\epsilon\varPhi\at{\epsilon=0}
=\p\sb\epsilon\varPsi\at{\epsilon=0}$;
and the requirement
\eqref{varphi-perp-ker-m}
leads to
$\p\sb\epsilon\varPhi\at{\epsilon=0}
=\p\sb\epsilon\varPsi\at{\epsilon=0}=0$,
and hence
\[
\norm{\varPhi\sb 1-\hat\varPhi\sb 1}\sb{H^2}=O(\epsilon^2),
\qquad
\norm{\varPhi\sb 2-\hat\varPhi\sb 2}\sb{H^1}=O(\epsilon^2).
\]
Finally, notice that since the explicit dependence of $\mathcal{F}$
is on $\epsilon^2$, we have
\begin{equation}\label{pm}
\mathcal{F}
\big(\varPhi_1(-\epsilon),\varPhi_2(-\epsilon),-\epsilon\big)=
\mathcal{F}\big(\varPhi_1(-\epsilon),\varPhi_2(-\epsilon),+\epsilon\big)
=0
\end{equation}
and hence,
$\varPhi_j(\epsilon)=\varPhi_j(-\epsilon)$ since otherwise it would
be possible to contradict
the local uniqueness part of the conclusion of the implicit function theorem
(applied to $\mathcal{G}$ with $\mathbf{a}=0$, $\mathbf{b}=0$).
This completes the proof
of Proposition~\ref{prop-sw-dm}
and thus of Theorem~\ref{theorem-sw-dm}.
\end{proof}
\begin{remark}
The solutions of $\mathcal{F}=0$ are obtained for
both positive and negative epsilon close to zero, but
the $\epsilon$ negative branch apparently gives rise to
solutions of the Maxwell-Dirac system via \eqref{ansatz-2}
which are related to the positive branch as follows.
By \eqref{ansatz-2},
the branch which corresponds
to negative $\epsilon$
has the form
\[
\tilde\phi(x,\omega)
=\begin{bmatrix}
\tilde\phi\sb 1(x,\omega)
\\
\tilde\phi\sb 2(x,\omega)
\end{bmatrix}
=\begin{bmatrix}
(-\epsilon)^3
\varPhi\sb 1(-\epsilon x,-\epsilon)
\\
(-\epsilon)^2
\varPhi\sb 2(-\epsilon x,-\epsilon)
\end{bmatrix}
=\begin{bmatrix}
-\epsilon^3
\varPhi\sb 1(-\epsilon x,\epsilon)
\\
\epsilon^2
\varPhi\sb 2(-\epsilon x,\epsilon)
\end{bmatrix},
\qquad
\omega=-\sqrt{m^2-\epsilon^2},
\qquad
\epsilon\ge 0,
\]
where we took into account that
$\varPhi_1(y,\epsilon)$,
$\varPhi_2(y,\epsilon)$
obtained from Proposition~\ref{prop-sw-dm} are even in
$\epsilon$.
Comparing to
\eqref{ansatz-2},
we conclude that
this branch is related to the
$\epsilon$-positive
branch $\phi(x,\omega)$
by
\[
\tilde\phi\sb 1(x,\omega)=-\phi\sb 1(-x,\omega),
\qquad
\tilde\phi\sb 2(x,\omega)=\phi\sb 2(-x,\omega),
\]
so that
$\tilde A\sp 0(x)=A\sp 0(-x)$,
$\tilde{\mathbf{A}}(x)=-\mathbf{A}(-x)$.
That is, these two branches
have
the same magnetic field but opposite
electric field (see \cite[\S2.3 and \S5.4]{MR0187641}).
\end{remark}
\begin{remark}
We briefly consider the symmetry properties of the solitary wave
solutions:
in \cite[Section 5]{MR1364144} Lisi gives an
ansatz for the solitary waves, using cylindrical
coordinates $(\rho,z,\theta)$, from which
symmetry properties can be deduced. For our situation the
relevant ansatz for the Dirac wave function is
\begin{equation}\label{cyl}
\phi=\left(\begin{array}{l}
\psi_1(\rho,z)\\
\psi_2(\rho,z)e^{i\theta}\\
\psi_3(\rho,z)\\
\psi_4(\rho,z)e^{i\theta}\\
\end{array}
\right)\,.
\end{equation}
It seems likely that the solutions constructed via
Proposition~\ref{prop-sw-dm} have this symmetry and
that this fact could be proved via an application of the
implicit function theorem within the symmetry class
of \eqref{cyl}.
\end{remark}
| 90,814
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Difference between revisions of "Release notes/0.47"
Revision as of 06:31, 5 October Tool switching by input device
- 11.6 Layers
- 11.7 Editing bitmaps in an external editor
- 11.8 Command for relinking clones
- 11.9 Automatic relinking of clones on Duplicate
- 11.10 Pattern editing
- 11.11 Transform dialog: spacing out option
- 11.12 Converting text to path produces a group
- 11.13 Combine works on groups
- 11.14 Exclusion works on multiple paths
- 11.15 No more Whiteboard
- 11.16 Save As directory
- 12 User interface
- 13 Notable bug fixes
- 14 Known issues
- 15 Previous releases
Inkscape 0.47
(not released yet - AnnouncePlanning047)
Inkscape 0.47 brings a host of important improvements all across the program:
-
Refactoring effort
The focus of the 0.47 release was to clean up legacy code and push forward the migration to clean object-oriented C++. The goal of this effort was to increase reliability and maintainability of Inkscape. In the long run, it will mean fewer bugs and more new features, because it will be easier to develop and find bugs in Inkscape.
Migration to lib2geom
Many parts of the code have been changed to use the 2geom library for geometrical calculations instead of the old libnr and livarot libraries.
Preferences
Instead of global functions directly manipulating an XML document, the preferences API is now exposed through the
Inkscape::Preferences singleton. It abstracts away the way the preferences are stored in memory. In the future it may allow for different user settings storage backends (like GConf or the upcoming dconf on GNOME desktops or .plist files on OS X). Previously, Inkscape directly manipulated an internal XML document.
Tools
Node tool
- In previous versions of Inkscape, no visual feedback was given back to the user when hovering over paths when using the Node tool. In this update, hovering over a path with the Node tool now results in a highlighted path outline being displayed. Note: the duration and color of the new path outline feature can be configured in the Tools > Node section of the Inkscape Preferences dialog.
- pressed, it now snaps not only to the 15 degree increments starting from 0 and to the original handle direction, but also to the direction of the opposite handle (if it exists) or of the opposite line segment (if it is a straight line).
- The behavior of the buttons/shortcuts that make a node smooth or cusp has been improved:
- If a node is already a: this is a new node, the node loses its auto state and becomes simply smooth; for this reason, it is recommended to edit smooth nodes with the node handles hidden via a toggle button on the Node tool.
- Push mode moves those selected objects that are under the brush in the direction in which you move the brush. This is similar to the Push path mode, except that the Move mode affects entire objects and not parts of the paths under the brush.
- Attract/Repel Objects mode moves those selected objects that are under the brush towards the cursor (default) or away from cursor (with Shift pressed). This is similar to the Attract/repel path mode, except that the Move in/out mode affects entire objects and not parts of the paths under cursor.
-. Like with the regular Duplicate command, duplicating with Tweak tool places the copies right over the originals, and you may need to use the attract/repel objects mode Shift+z, Shift+2 paint mode Shift+j color jitter mode Shift+b blur mode
In Paint mode, painting with Shift inverts the color you're applying (e.g. when painting with yellow, Shift will switch the applied color to blue).
Calligraphy tool
- The tool's settings can now be set from a preset (see the drop-down list in the controls bar). Several presets are provided that imitate various drawing implements and styles.
- When drawing with Alt pressed, Inkscape subtracts the new path you are creating from the selected path. With Shift, it unions the new path with the selected path. This allows you to quickly patch or erase defects in a stroke you have drawn, without leaving the tool.
- The behavior of the tool when tracking a guide (drawing with Ctrl) has been improved:
- The initial "jerk" when you start drawing is suppressed.
- The undesired flipping of the stroke to the other side of the guide path, when drawing along closed paths, is fixed.
- If you lose connection with your guide path, the tool tries to continue moving in the same direction as if by inertia, so as to minimize the tearoff jerk.
Paint Bucket tool
- Paint Bucket is now more tightly integrated with potrace. As a result, memory and CPU usage on each fill operation have been reduced significantly.
Eraser Tool
A new tool, Eraser, has been added to the main toolbox. Its shortcut is Shift+E. It has two main modes, selectable by toggle buttons on its controls bar:
- Delete Objects mode: any shape touched by dragging is deleted completely. This is analogous to "touch selection" in Selector followed by Del.
- Cut mode: dragging subtracts, using a boolean Subtract operation, parts of paths it touches. On the controls bar, you can adjust the Width of the trace left by the tool. If nothing is selected, it acts on all objects in the current layer, whether selected or not; if there's a selection, its action is limited to selection. This mode is similar to Alt+drag in Calligraphic tool.
Pen and Pencil
New modes
Apart from the regular Bezier mode, these drawing tools now provide several new modes:
- Spiro mode: This mode automatically applies the new Spiro Splines path effect (see the section on new effects) to any newly drawn path. As mentioned below, it is not yet possible to preview a spiro before it is finished. This mode is available in both Pen and Pencil tools.
- Polyline mode (Pen only): This mode makes it easy to draw many straight line segments by disallowing any curves, even when you accidentally drag with the mouse instead of clicking.
- Paraxial mode (Pen only): create straight line segments that are parallel to one of the coordinate axes. This works similar to the Polyline mode with Ctrl. Normally, each next line segment is drawn perpendicular to the previous one, but the direction of the line segment being drawn can be toggled by pressing Shift. If you click on the start anchor, the path gets closed with an L-shaped segment (its direction of which can also be flipped with Shift).
Stroke shapes
As a first step towards this blueprint, it is now possible to automatically apply predefined vector shapes to path strokes in Pen and Pencil tools. The choice of shapes in the drop-down list on the controls bar includes:
- Triangle in and Triangle out: tapering out in both directions
- Ellipse: smooth elliptic shape stretching along path
- From clipboard: any path you had previously selected and copied to clipboard
To adjust the width of the stroke, open the Path Effect Editor, choose "Pattern along path" effect, and edit its Width parameter. It is measured in units of the original size of the applied shape; the triangles and ellipse are all 10px in size, and the clipboard size can be any size. The default value of Width is 1.00, which means the triangle/elliptic strokes will be 10px wide and the from-clipboard stroke will be as wide as the copied object was tall.
Pencil smoothing
In Pencil tool, the controls bar now provides the Smoothing parameter, changeable in the range from 1 to 100, which controls how much smoothing is applied to the freehand lines you draw. Small Smoothing values produce rough lines with many nodes; large values give smooth lines with few nodes. Previously, this control was only available in Inkscape Preferences.
Pencil sketch mode
The sketch mode is still experimental. In essence, it enables the artist to draw many strokes, which Inkscape then averages into a single path. It tries to mimic sketching with a real pencil and paper, taking the 'visual average' of many strokes and condensing it into one stroke.
Pick the pencil tool, press Alt, and sketch away; release Alt to finalize the result. After each stroke (a stroke starts when the mouse button is pressed down, and ends when it is released), the resulting path will be an average of the old result and the newly drawn stroke. In Inkscape's Pencil tool preferences, one can choose to either average between all drawn strokes (so that all stroke have the same weight), or just the new stroke and the old result (so that later strokes have greater weight).
Currently, going back and forth between A and B in one stroke does not give the expected result; it will result in a long path going back and forth, instead of the visually expected path going from A to B just once. We are working on this (difficult!) issue. For best results, draw strokes only from A to B (and not from B to A).
Text tool
- When editing multiline or flowed text, the PgUp and PgDn keys now work to move the cursor by one screen (i.e. by as many lines as fit into the screen at current zoom).
- The usability of the font family drop-down list in the Text tool controls bar has been improved: it no longer steals focus, all keyboard shortcuts work as designed (Alt+X to access the family control, Alt+down to open the drop-down list with font previews, arrows to move in the list, Enter to set chosen font) and the completion feature works (start typing a family name and a pop-up list with possible completions appears).
- A remaining problem that may be fixed in a future version is that the first opening of the drop-down list of family names may be slow (several seconds) if you have many fonts installed (the delay is Inkscape generating the previews for all fonts). Subsequent openings of the list are much faster.
Connector tool
- Connectors are now drawn to the edges of shapes, rather than to the bounding box of shapes.
- The routing buffer around shapes is now altered in the correct direction when the user changes this via the spacing control fon the connector toolbar.
- A bug has been fixed where the spinboxes on the connector toolbar did not properly respond to single clicks of their up and down arrows.
Path effects
Path Effects stacking
More than one Path Effect can be assigned to an object. A new UI was created to control the stack in the Path Effects Editor (Shift+Ctrl+7): the Effect list shows the stack of effects for the selected object; below, buttons allow you to move a selected effect in the stack up or down or remove it from stack.
The stack works from top to bottom, i.e. the topmost listed effect is applied first, the second one works on the result of the first, and so on until the bottom effect which produces the final visible result. A new effect that you choose in the "Apply new effect" list and created by the Add button gets added to the end of the current stack.
Path Effects for groups
Path effects can now be assigned to a group. In most cases, the effect is applied recursively to the member paths, but for Bend Path and Envelope Deformation the result that the distortion applies to the group as a whole.
- You can, as usual, enter the group by double-clicking on it, and edit the paths there watching the transformed result update live.
- Path effects can be assigned to groups of groups, applying recursively to all grouped paths.
- The Convert Object to Path command (Ctrl+Shift+C), when applied to a group with effects applied, removes these effects from group and converts all its member paths to effect-less paths looking exactly as before.
Misc new features
- The Paste Path Effect command in Path menu can now assign the path effect of the clipboard's path to any number of paths, going recursively into groups if necessary.
- A new command, Remove Path Effect in Path menu, removes any path effects from all selected objects, going recursively into groups if necessary.
- Path effects can now be assigned to the sides of a 3D box (use Ctrl+click to select individual sides) without breaking its 3D functionality.
- The Pen and Pencil tools now correctly work with paths with effects: you can continue such a path or add a new subpath to it by drawing with Shift, while preserving the effects applied to it.
- Path parameters of effects can now link to existing shapes or texts. For example, now it is possible to use a text as the pattern for the Pattern Along Path effect, or shape a path with the Envelope Deformation where one of the envelope sides is text! Since the effect links to the text, not copies it, the result will update live if you edit the text.
- Lib2geom now has an implementation for EllipticalArc. For Inkscape, this means that it is now possible to directly copy-paste ellipse shapes to path parameters (e.g. 'pattern' in Pattern along Path), without the need to convert the ellipse to path first.
New effects
- Sketch: Simulates hand-drawn lines. A set of parameters lets you tune the effect. They are all summarized in this picture.
- Hatches: Fills the shape with rough, randomized hatches, simulating a quick hand drawing. It is configurable through numeric parameters as well as on-screen handles visible in Node tool.
- von Koch: This effect creates fractal pictures. A collection of transformations (rotations, rescalings, etc...) is recursively applied to the input path. The transforms are derived from a "reference" path (a line segment) and a "generating" path (basically a collection of segments): the transforms are those needed to move the reference segment onto each segment of the generating path (components in the generating path having more than one segment might be used to define shearing/mirroring transforms). A famous example is the von Koch's snowflake.
- Warning: the complexity of the output path grows exponentially fast with the number of generations. As a guardrail, an editable complexity bound is provided, above which the effect is disabled.
- Knot:: Should not be applied to groups yet, unless you want to edit the crossing signs.
-: Draws a grid using the first three nodes of a path. The center node defines the origin. The other two nodes define the direction and length of the two adjacent sides of the first cell. If a path has more than three nodes, the other nodes are ignored. One can select the number of cells in the two orthogonal directions.
- Envelope Deformation: Allows you to deform an object (or a group of object) by deforming its sides. Modifications are done by deforming the four path parameters: Top, Bottom, Left and Right; for each, you can edit it with Node tool, copy, paste, or link to an existing path in the document.
- Ruler: Draws ruler marks along the path; you can set distance between the marks, their length for major and minor marks, the major/minor ratio, and other parameters.
- Interpolate Subpaths: Creates a given number of interpolated paths between the (first) two subpaths of a path (the subpaths after the second subpath are ignored). The interpolations are spread along an editable trajectory path.
Notable effect bugfixes and changes
We try to refrain from changing the behavior of existing path effects, because it will change appearance in old files when opened in a new version of Inkscape (but not in any other SVG viewer or editor). However, when an effect is really broken, we have to fix it:
- Pattern Along Path:
- The pattern used to be stretched across discontinuities (separate subpaths). This has been fixed; now it treats a discontinuous path as a group of continuous paths and applies the effect separately to each.
- Successive copies of the pattern can now be fused into continuous paths (using the new "fuse nearby ends" feature) so that "filling" the result works as expected.
- Bend Path:
- Closed input paths would sometimes result in unclosed output paths. This has been fixed.
Import/Export
PostScript and EPS import
Inkscape's PS and EPS import now uses Ghostscript instead of pstoedit. If you need to open files of these types, install Ghostscript and make sure the directory with the
ps2pdf utility from Ghostscript installation is in your PATH. On importing a file, you will see a preferences dialog, similar to PDF import; for multipage PS files, this dialog allows you to select which page to open.
PDF import
- A new checkbox on the PDF import dialog, Replace PDF fonts with closest-named installed fonts (on by default) attempts to replace all font names in the imported PDF with the most similar names of those fonts installed on your system. For example, if the PDF uses the font "TimesNewRomanPSMT" and you have "Times New Roman" installed, that font will be used, which will likely give you a more correct appearance than the unknown font "TimesNewRomanPSMT" that will be displayed as the default sans serif font. This is a temporary fix necessary because Inkscape cannot yet extract the fonts from the PDF files it imports nor can it embed them into SVG; when it gets these capabilities, such font name conversion will become unnecessary because all fonts will be preserved.
- Importing PDF files now works from the command line. For example,
inkscape file.pdf --export-plain-svg=file.svg
- will take the first page of the PDF and use the default import options, and save the result to SVG. If you try to import PDF without an
--exportcommand, it will show the import options dialog as before and open the file in the UI.
PDF, PostScript, and EPS export
The new Cairo-based PS and EPS exporter provides much better support for various vector features, including clipping paths, patterns, and non-ASCII characters. Those features that are not supported by the targeted format become embedded bitmaps that preserve the appearance. In particular:
- transparency is always rasterized in PS or EPS but not PDF, as PDF supports vector transparency;
- filters, such as blur, are by default rasterized in all three formats (PS, EPS, PDF). This can be turned off by unchecking the "Rasterize filter effects" option in the UI or adding the --export-ignore-filters option via the command line. In this case, filtered objects are rendered as vectors without filters and without rasterization.
- The resolution for rasterizing the filters can be set in the UI in the "Resolution for rasterization (dpi)" parameter or on the command line by --export-dpi parameter (same as used for exporting SVG documents to bitmaps). The default is 90 dpi. The rendering quality of filters for rasterization, as well as for bitmap export, is always the best possible regardless of what you have set in the Filters tab of Preferences (which only affects on-screen rendering). For example,
inkscape --export-pdf=out.pdf --export-dpi=300 file.svg
- will export file.svg to out.pdf, rasterizing any filtered objects in it at 300 dpi. (If there are no filtered objects, the --export-dpi has no effect.)
All of PS, EPS, and PDF export formats uniformly support the export area options (page). The BoundingBox (page size) of the exported PS/EPS/PDF file will correspond to the bounding box of that object. You can override this with "Export area is page" (GUI) or --export-area-page (command line) option which forces the output to have the size of the SVG document's page (this may not be possible with EPS, see below).
- The "Export area is page" (GUI) or --export-area-page (command line) option forces the output to have the size of the SVG document's page. This is the default for PS and PDF but not for EPS.
- Note, the specification of the EPS format does not allow a bounding box to extend beyond the content. This is enforced by the Cairo graphics library which means that when
--export-area-page
is used with EPS export, the page bounding box will be trimmed inwards (but never expanded has been removed; use --export-area-page instead.
- The "Export area is drawing" (GUI) or --export-area-drawing (command line) option forces the output to have the size of the exported objects' bounding box, regardless of page size. If no --export-id is specified, this means the bounding box of the entire drawing; with --export-id, this means the bounding box of the exported object only. This is the default for EPS. Note that checking "Export area is page" or using --export-area-page overrides this setting for PS and PDF output.
- The --export-embed-fonts option is removed. Inkscape now always embeds and subsets all fonts used in the document when exporting PS, EPS, or PDF.
UniConvertor-based import and export
Inkscape can now use UniConvertor to import files of the following types:
- Corel DRAW versions 7 to X4 document files (CDR)
- Corel DRAW versions 7 to X4 Template files (CDT)
- Corel DRAW Presentation Exchange files (CMX)
- Corel DRAW Compressed Exchange files (CCX)
- sK1 files (SK1)
- Computer Graphics Metafiles (CGM)
- Windows Metafiles (WMF)
- HPGL (AutoCAD) Plot files (PLT) (requires UniConvertor 1.1.4)
Inkscape can now use UniConvertor to export files of the following types:
- Windows Metafiles (WMF)
- sK1 files (SK1)
- HPGL (AutoCAD) Plot files (PLT) (requires UniConvertor 1.1.4)
Text objects are not supported as of UniConvertor 1.1.4.
On Windows, UniConvertor is included with Inkscape distribution and does not require separate installation.
HPGL export
In addition to the HPGL export via UniConvertor listed above, Inkscape can now export to HPGL (Hewlett-Packard Graphics Language) via an internal routine that is geared towards various cutters/plotters.
JavaFX export
Inkscape can export drawings to JavaFX format (.fx file extension).
DXF import and export
- DXF export for desktop cutting plotters is much faster than in previous versions. A new option was added to provide support for RoboMaster desktop cutting software. Also, polylines and polysplines are now supported.
- DXF import is new. It supports a number of the simpler DXF shapes: line, Bezier spline, ellipse, circular arc, text.
PNG export
- PNG export has been updated to include metadata if present in the source SVG. This includes the Author, Copyright, Creation Time, Description, and Title fields. PNG metadata can be viewed using the ImageMagick
identifycommand.
- Export filenames that are relative (e.g.
../file.pngor simply
file.png) are now resolved relative to the document's location. This applies to the filenames you type in the Export dialog as well as those stored in export hints in the document itself (and used by the "Batch export" checkbutton as well as in command line export with
--export-use-hints). However, export filenames specified directly on the command line are not resolved, which in effect means they work, as before, from the current working directory from which you run the export command.
OCAL (Open Clip Art Library) Export
- Export to Openclipart.org has been disabled for 0.47 as it had become non-functional and needs to be re-written.
Autosave
The new autosave feature allow for automatic timed backups as work goes on. Saved versions are put in a designated directory and do not overwrite the original SVG file nor each other.
In Inkscape Preferences (Ctrl+Shift+P), Save tab, you can enable this feature and specify various options:
- the time interval between backups, in minutes;
- the directory where you want the backups to be stored;
- and the maximum number of saved backups (if this number is exceeded, old backups will start to be deleted).
Extensions
- The former Effects menu is renamed to Extensions. This is less confusing and better reflects the content of the menu: a collection of extensions, written mostly in Python, which perform various tasks with or without selection.
New and improved extensions
- The new Arrange > Restack extension restacks (changes the z-order of) selected objects, with options including: left to right, top to bottom (or vice versa), radial outward or inward, or at an arbitrary angle. You can also specify what point of an object is used to calculate its position for restacking.
- The new Generate from Path > Extrude extension is similar to the old Extrude effect, which has been renamed Motion. The new effect requires two paths and draws connection lines or polygons between their nodes. If you want more dense extrusion, add more nodes to the paths. All the extrusion polygons are separate (grouped) objects, so they can be easily painted by the Tweak tool to get nicely shaded ribbons, 3D letters, and the like.
- The new Generate from Path > Scatter extension spreads copies of pattern along arbitrary "skeleton" paths. The pattern must be the topmost object in the selection. Groups of paths, shapes, clones are allowed.
- The improved Modify Path > Add Nodes extension now allows segments to be divided into a given number of subsegments.
- The new Modify Path > Convert to Dashes extension takes the dash pattern of the stroke and explicitly cuts the curve to duplicate this pattern. This can be used to allow desktop cutting plotters, which don't understand dashed stroke style, to cut dashed paths. You can also achieve interesting effects with smoothly varying dash length if you edit the resulting path with Node Sculpting technique (Alt+drag with Node tool).
- The new Render > Add printing marks extension adds printing marks and color bars required by print bureaus. You can either manually define margins by which cut marks are created.
- The new Render > 3D Polyhedron extension draws 2D projections of 3D polyhedrons and other 3D shapes. You can choose one of a number of predefined shapes (cube, octahedron, truncated dodecahedron and others) or load a shape definition from an OBJ file. The shape can be rotated around any of the three axes by arbitrary amount; you can also define various style options such as color of the faces and stroke width, and enable shading with adjustable light source position.
- The new Render > Cartesian Grid extension plots Cartesian (square) grids that do not fill the page, but offer three levels of division, logarithmic scales (with clutter-reduction and arbitrary base) and customizable line width. All like elements (e.g. x-axis subminor divisions) are put into subgroups together. A proper border is also drawn, with an independent line width.
- The new Render > Polar Grid extension plots a polar coordinate grid, with options for arbitrary-base logarithmic subdivisions, clutter-reduction around the origin, circumferential labels and custom line widths.
- The new Render > Draw from Triangle extension takes a triangle drawn as a path (only the first three nodes of a path are counted) and allows to draw many triangle-related geometrical objects such as circumcircles, excentral triangles, etc. It also lets you specify custom trilinear coordinates and triangle centre functions, as well as compute basic triangle properties such as area and semiperimeter.
- The new Render > Guides Creator extension quickly creates horizontal and vertical guides for subdividing the canvas. You can choose the divisions from None, 1/2, 1/3 ... to 1/10.
- The new Render > Calendar extension draws a calendar for a given year with localizable month/weekday names, colors, and many other options.
- The new Render > Foldable Box extension creates foldouts for paper boxes.
- The new Modify Path > Interpolate Attribute in a Group extension takes a group of objects and assigns to its members interpolated values of an attribute of your choice, such as width, height, opacity, etc.
- The new Web > JavaScript extensions allow you to set various interactive JavaScript attributes, such as onclick or onfocus, on SVG elements. Inkscape does not support them on rendering but you might need them for other SVG viewers such as Firefox.
Extension API changes
- While the "Live preview" checkbox is useful for most effects, for some it just does not make sense. Now, you can add the attribute
needs-live-preview="false"in the
effectelement in the .inx file of the effect to suppress this checkbox for your effect.
- Parameters passed to extensions (via the <param> element) now can have a boolean attribute,
gui-hidden, to indicate that the parameter should not be represented in the GUI. If all parameters are marked as hidden, no GUI is presented for such extension.
- All .inx files are now properly formatted XML files with their own namespace of: a Relax NG schema to define it. More information can be found on the Extensions page.
Filters
The Filter Editor (former Filter Effects) and Remove Filters commands are moved from the Object menu to the new Filters top-level menu, which also contains a collection of preset filters.
Preset filters
The Filter Editor is powerful, but can be quite cumbersome. You can now apply complex preset filters to selected objects with a single command by choosing it from the new Filters top-level menu. Submenus categorize the filters by function or appearance. To view a sampler of all preset filters, open filters.svg document from Inkscape's examples (
share/examplesin the Inkscape tree).
Most filters apply immediately after selecting the command; some present a dialog where you can adjust some of the parameters before applying the filter (such filters have "..." at the end of the command in the menu).
By default, if the selected object already has some filter applied, the chosen filter will be merged with the existing filter for combined effect. However you can also overlay several filters to an object while keeping them separated: simply press Ctrl+G after applying any filter and then apply another one; the filters will then display separately in the Filters Editor.
You can easily add your own filters to these menus. Simply place any SVG file with the filters to the
filterssubdirectory of your config directory (
~/.config/Inkscape/on Linux) and the filters will be picked up from it when you start Inkscape. By default, they will be placed in the Personal submenu under Effects > Filters. If you want to control this, add the following attributes to the
filterelement:
inkscape:label is the command label inkscape:menu is the submenu to place the command into inkscape:menu-tooltip is the tooltip (displayed in the statusbar as you select the command)
No Filters rendering mode
In order to facilitate editing documents that use lots of SVG filter effects, filter effects can now be disabled for a particular document window by selecting View > Display mode > No Filters from the menu. This provides an intermediate step between Normal and Outline view modes.
The Toggle View command in the Display Mode submenu (Ctrl+keypad 5) toggles between all 3 modes in a loop: Normal, No filters, Outline.
Filter quality setting
In addition to the Blur Quality setting, Inkscape now has a general Filter Effects Quality setting on the Filters tab of Inkscape Preferences. It affects all filters and gives you an opportunity to seek optimum balance between speed and accuracy when rendering filters.
SVG support
pathelements is reduced by about 10%. Inkscape generates the shortest possible path strings by avoiding repeated operators and using relative coordinates (when it helps).
This is controlled by the options on the SVG output page of Inkscape Preferences dialog. Also, you can change".
Horizontal and vertical path segments
If an SVG contains paths with shorthands for horizontal and vertical path segments ('H' or 'V'), then Inkscape will try to maintain those shorthands if possible, so the saved file will also contain them.
<script> tag preserved. You can design fonts within Inkscape, but using them to render text on the canvas is not yet supported. We are waiting for libpango to implement proper support of the user-fonts feature.
Currently, the main benefit of this feature is to improve the font design workflow when working with FontForge: You can save SVG files with fonts embedded and import them into Fontforge, and you only need one file per font instead of one file per glyph.
An SVG font is a mapping of chunks of SVG drawing to characters. When a certain character is used in a string, its respective glyph is rendered. If no glyph is declared for a certain character, then there is a default "missing glyph" that is rendered. You can set the drawing that defines this missing glyph; this is done by clicking on the Missing Glyph: From Selection... button at the top of the dialog.
Here's an example of a font design workflow:
- Open the SVG Font dialog by _Text > SVG Fonts_.
- Click New under the font list. Select the new font in list; you can rename it by clicking on its name and typing a new name. You will see a set of black squares in the text preview area. This is the preview text being rendered. It only uses the default missing glyph (which is initially defined as a black square) because no specific glyphs were defined yet.
- Draw something that you want to use for the missing glyph
- Click Missing Glyph: From selection...
- Draw a glyph for the "a" character (character matching is case sensitive)
- On the Glyphs tab, click Add glyph
- Type "a" in the Matching String column (at the moment, handling of the glyph-name attribute is not implemented)
- With the row selected, click Get curves from selection...
- Now, you will see the "a" glyph in the preview rendering if the preview text contains it. You can edit the preview text as needed to view different characters.
Repeat steps 5 through 9 for every glyph you wish to add to your font, then save the SVG file and open it in FontForge for further editing.
SVG Test Suite Compliance page or this page for up-to-date rendering results. Also see TestingInkscape for information on running and creating rendering tests.
Editing Aids
Grids
- The dotted rectangular grid now shows small crosses at the intersection points of emphasis lines.. freedesktop.org recent document lists (used by Gnome, KDE and Xfce). The list can be cleared from the Preferences dialog (the Interface tab).
Shell mode
If you run inkscape with
--shell, it will enter a..
Note that this may cause the clone to move if the new original and the old original objects are in different positions.
Automatic relinking of clones on Duplicate
If you turn on the When duplicating original+clones: Relink duplicated clones option on of the Transform dialog (Ctrl+Shift+M). px. The order of selecting the objects or their z-order do. open the Save As dialog with the current directory or, if the document has not been saved, with the previously used directory (the last place the Open or Save As dialogs had visited)..
Previous releases
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\begin{document}
\maketitle
\begin{abstract}
We consider in this paper the mathematical and numerical modelling of
reflective boundary conditions (BC) associated to Boltzmann - Poisson systems, including diffusive reflection in addition to specularity, in the context of electron transport in semiconductor device modelling at nano scales, and their implementation in Discontinuous Galerkin (DG) schemes.
We study these BC on the physical boundaries of the device and develop a numerical approximation to model an insulating boundary condition, or equivalently,
a pointwise zero flux mathematical condition for the electron transport equation.
Such condition balances
the incident and reflective momentum flux at the microscopic level,
pointwise at the boundary, in the case of a more general mixed reflection with momentum dependant specularity probability $p(\vec{k})$.
We compare the computational prediction of physical observables
given by the numerical implementation of these different reflection conditions in our DG scheme for BP models, and
observe that the diffusive condition influences the kinetic moments over the whole domain in position space. \\
\textbf{Keywords:} Galerkin; Boltzmann-Poisson; boundary; reflection; diffusive; specular.
\end{abstract}
\section{Introduction}
The dynamics of electronic transport in modern semiconductor devices can be described by the semiclassical Boltzmann-Poisson (BP) model
\begin{equation}
\frac{\partial f_i}{\partial t} + \frac{1}{\hbar} \nabla_{\vec{k}} \,
\varepsilon_{i} \cdot \nabla_{\vec{x}} f_i -
\frac{q_i}{\hbar} \vec{E} \cdot \nabla_{\vec{k}} f_i = \sum_{j} Q_{i,j},
\label{BE}
\end{equation}
\begin{equation}
\nabla_{\vec{x}} \cdot \left( \epsilon \, \nabla_{\vec{x}} V \right)
= \sum_{i} q_i \rho_{i} - N(\vec{x}), \quad
\vec{E} = - \nabla_{\vec{x}} V,
\label{pois}
\end{equation}
where
$f_{i}(\vec{x},\vec{k},t)$ is the probability density function (pdf) over phase space $(\vec{x},\vec{k})$ of a carrier in the $i$-th energy band in position $\vec{x}$,
with crystal momentum $\hbar \vec{k}$ at time $t$. The collision operators $Q_{i,j}(f_i,f_j)$ model
$i$-th and $j$-th carrier recombinations, collisions with phonons or generation effects. $\vec{E}(\vec{x},t)$ is the electric field,
$V(\vec{x},t)$ is the electric potential,
$\varepsilon_{i}(\vec{k})$ is the $i$-th energy band surface, the $i$-th charge density $\rho_{i}(t,\vec{x})$ is the k-average of $f_i$, $-q_i$ is the electric charge of the $i$-th carrier,
$N(\vec{x})$ is the doping profile, and $\epsilon$ is the electric permittivity of the material.\\
The BP model for electron transport on a single conduction energy band for electrons has the form
\begin{equation}
\frac{\partial f}{\partial t} + \frac{1}{\hbar} \nabla_{\vec{k}} \,
\varepsilon(\vec{k}) \cdot \nabla_{\vec{x}} f -
\frac{q}{\hbar} \vec{E}(\vec{x},t) \cdot \nabla_{\vec{k}} f = Q(f),
\label{BE1band}
\end{equation}
\begin{equation}
\nabla_{\vec{x}} \cdot \left( \epsilon \, \nabla_{\vec{x}} V \right)
= q \left[ \rho(\vec{x},t) - N(\vec{x}) \right] ,
\quad \vec{E} = - \nabla_{\vec{x}} V,
\label{pois1band}
\end{equation}
with the quantum mechanical electron group velocity
$\frac{1}{\hbar} \nabla_{\vec{k}} \, \varepsilon (\vec{k}) $, and the
electron density $ \rho(\vec{x},t) = \int_{\Omega_{\vec{k}}} f(\vec{x},\vec{k}, t) \, d\vec{k} $.
The collision integral operator $Q(f)$ describes the
scattering over the electrons,
where several mechanisms of quantum nature can be taken into account.
In the low density regime, the collisional
integral operator can be approximated as linear in $f$, having the form
\begin{equation}
Q(f) = \int_{\Omega_{\vec{k}}} \left[ S(\vec{k}', \vec{k}) f(t, \vec{x}, \vec{k}') - S(\vec{k}, \vec{k}') f(t, \vec{x}, \vec{k}) \right] d \vec{k}' \, ,
\label{ope_coll}
\end{equation}
where $S(\vec{k},\vec{k}')$ is the scattering kernel,
representing non-local interactions of electrons with a background
density distribution.
For example, in the case of silicon, one of the most important collision mechanisms are electron-phonon scatterings due to lattice vibrations of the crystal, which are modeled by acoustic (assumed elastic) and optical (non-elastic) non-polar modes, the latter with a single frequency $\omega_{p}$, given by
\begin{eqnarray}
S(\vec{k}, \vec{k}') & = & (n_{q} + 1) \, K \, \delta(\varepsilon(\vec{k}') - \varepsilon(\vec{k}) + \hw) \nonumber
\\
&&
\mbox{} + n_{q} \, K \, \delta(\varepsilon(\vec{k}') - \varepsilon(\vec{k}) - \hw) + K_{0} \, \delta(\varepsilon(\vec{k}') - \varepsilon(\vec{k})) \, ,
\label{Skkarrow}
\end{eqnarray}
with $K$, $K_{0}$ constants for silicon.
The symbol $\delta$ indicates the usual Dirac delta distribution
corresponding to the well known Fermi's Golden Rule \cite{book:Lundstrom}.
The constant $n_q$ is related to the phonon occupation factor
$$
n_q = \left[ \exp \left( \frac{\hw}{k_B T_L} \right) - 1 \right] ^{-1},
$$
where $k_B$ is the Boltzmann constant and $T_L = 300 K$ is the lattice temperature. \\
\begin{comment}
Deterministic solvers for the BP system using Discontinuous Galerkin (DG) FEM have been proposed in \cite{CGMS-CMAME2008, CGMS-IWCE13} to model
electron transport along the conduction band for 1D diodes and 2D double gate MOSFET devices. In \cite{CGMS-CMAME2008} , the energy band $\varepsilon(\vec{k})$ model used
was the nonparabolic Kane band model. These solvers are shown to be competitive with Direct Simulation Monte Carlo methods \cite{CGMS-CMAME2008} . The energy band models
used in \cite{CGMS-IWCE13} were the Kane and Brunetti, $\varepsilon(|\vec{k}|)$ analytical models, but implemented numerically for benchmark tests.\\
\end{comment}
{
The semi-classical Boltzmann description of electron transport in semiconductors is, for a truly 3-D
device, an equation in six dimensions plus time when the device is not in steady state.
The heavy computational cost is the main reason why the BP system had been
traditionally solved numerically by means of Direct Simulation Monte Carlo (DSMC) methods
\cite{jaco89}. However, after the pioneer work \cite{Fatemi_1993_JCP_Boltz}, in recent years,
deterministic solvers to the BP system were proposed in
\cite{MP, carr02, cgms03, carr03, carr021, cgms06, gm07}.
These methods provide accurate results which, in general, agree well with those obtained from Monte
Carlo (DSMC) simulations, often at a fractional computational time. Moreover, these type of solvers can resolve
transient details for the electron probability density function $f$, which are
difficult to compute with DSMC simulators. \\
The initial methods proposed in \cite{cgms03, carr03, carr021, cgms06} using weighted essentially non-oscillatory (WENO) finite
difference schemes to solve the Boltzmann-Poisson system, had the advantage that the scheme is relatively simple to code and very stable even on coarse meshes for solutions containing sharp
gradient regions. However, a disadvantage of the WENO methods is that it requires smooth meshes to achieve high order accuracy, hence it is not very flexible for adaptive meshes. \\
Motivated by the easy {\it hp-}adaptivity and the simple communication pattern of
the discontinuous Galerkin (DG) methods for macroscopic (fluid level) models \cite{Chen_95_JCP_Q,
Chen_95_VLSI, LS1,LS2}, it was proposed in \cite{chgms-sispad07, Cheng_08_JCE_BP} to implement a DG
solver to the full Boltzmann equation, that is capable of capturing transients of the probability
density function. \\
In the previous work \cite{chgms-sispad07,Cheng_08_JCE_BP}, the first DG solver for
(\ref{BE})-(\ref{pois}) was proposed, and some numerical calculations were shown for one and
two-dimensional devices. In \cite{CGMS-CMAME2008}, the DG-LDG scheme for the Boltzmann-Poisson
system was carefully formulated, and extensive numerical studies were performed to validate the
calculations. Such scheme
models electron transport along the conduction band for 1D diodes and 2D double gate MOSFET devices
with an analytic Kane energy band model.\\
A DG method for full conduction bands BP models was proposed
in \cite{CGMS-IWCE13}, following the lines of the schemes in \cite{chgms-sispad07, Cheng_08_JCE_BP, CGMS-CMAME2008},
generalizing the solver that uses the Kane non-parabolic band and adapting it to treat the full energy band case. A preliminary benchmark of numerical results shows that the direct evaluation of the Dirac delta function can be avoided, and so an accurate high-order simulation with comparable computational cost to the analytic band
cases is possible. It would be more difficult or even unpractical to produce the full band computation with other
transport scheme. It is worth to notice that a high-order positivity-preserving DG
scheme for linear Vlasov-Boltzmann transport equations, under the action of quadratically confined
electrostatic potentials, independent of the electron distribution, has been developed in
\cite{CGP}. The authors there show that these DG schemes conserve mass and preserve the positivity of the solution without
sacrificing accuracy. In addition, the standard semi-discrete schemes
were studied showing stability and error estimates. \\
The type of DG method discussed in this paper, as was
done in \cite{CGMS-CMAME2008}, belongs to a class of
finite element methods originally devised to solve hyperbolic conservation laws containing only
first order spatial derivatives, e.g. \cite{cs2,cs3,cs4,cs1,cs5}. Using a piecewise
polynomial space for both the test and trial functions in the spatial variables, and coupled with
{explicit and nonlinearly stable high order Runge-Kutta time discretization}, the DG method is a
conservative scheme that has the advantage of flexibility for arbitrarily unstructured meshes, with
a compact stencil, and with the ability to easily accommodate arbitrary {\it hp-}adaptivity. For
more details about DG scheme for convection dominated problems, we refer to the review paper
\cite{dgsurvey}, later generalized to the Local DG (LDG) method to solve the
convection diffusion equations \cite{cs6} and elliptic equations \cite{abcm}.
}\\
Regarding Boundary Conditions (BC),
there are several kinds of BC for BP semiconductor models.
They vary according to the considered device and physical situation. We list below several examples of BC that could arise
in the case of electron transport along a single conduction band.
\ \\
{\bf Charge neutrality} boundary conditions, given by \cite{CGL}
\begin{equation} \label{NeutralChargeBCmath}
\left. f_{out}(t,\vec{x},\vec{k}) \right|_{\Gamma} =
\left. N_D(\vec{x})
\frac{f_{in}(t,\vec{x},\vec{k})}{\rho_{in}(t,\vec{x})}
\right|_{\Gamma} , \quad \Gamma \quad \mbox{subset of} \, \partial \Omega_{\vec{x}} \, .
\end{equation}
This BC is imposed in source and drain boundaries, where electric currents enter or exit the device, to achieve neutral charges there,
as
$\rho_{out}(\vec{x},t) - N_D (\vec{x}) = 0$.
\ \\
{\bf Reflective} BC happen in insulating boundaries, usually defined by a Neumann boundary $\Gamma_N$, of 2D and 3D devices. In general, reflective BC can be formulated as the values of the pdf at the inflow boundary being dependent on the outflow boundary values
\begin{equation}
f(\vec{x},\vec{k}, t) |_{\Gamma_{N^-}} =
F_R \left( f
|_{\Gamma_{N^+}} \right) ,
\end{equation}
where the Neumann Inflow Boundary is defined as
\begin{equation}
\Gamma_N^- = \{(\vec{x},\vec{k}) \, | \, \vec{x} \in \Gamma_N, \, \vec{k} \in \Omega_{k}, \, \vec{v}(\vec{k}) \cdot \eta(\vec{x}) < 0 \},
\end{equation}
\begin{equation}
\quad \vec{v}(\vec{k}) = \frac{1}{\hbar} \nabla_{\vec{k}} \, \varepsilon(\vec{k}) \, ,
\end{equation}
$ \eta(\vec{x})$ outward unit normal.
The Neumann Outflow Boundary is defined as
\begin{equation}
\Gamma_N^+ = \{(\vec{x},\vec{k}) \, | \, \vec{x} \in \Gamma_N, \, \vec{k} \in \Omega_{k}, \, \vec{v}(\vec{k}) \cdot \eta(\vec{x}) > 0 \} \, .
\end{equation}
{\it Specular Reflection} BC over the Neumann Inflow Boundary is given by
\begin{equation} \label{eq:defSpecReflex}
f |_- (\vec{x},\vec{k},t) = F_S (f|_+) = f |_+ (\vec{x},\vec{k}',t) \quad \mbox{for} \quad (\vec{x},\vec{k}) \in \Gamma_N^-, \quad t>0 ,
\end{equation}
\begin{equation}
(\vec{x},\vec{k}') \in \Gamma_N^+ , \quad
\vec{k'} \quad \mbox{s.t.}
\quad \vec{v}(\vec{k}') = \vec{v}(\vec{k}) - 2 \, \eta(\vec{x}) \cdot \vec{v}(\vec{k}) \, \eta(\vec{x}) \, .
\end{equation}
{\it Diffusive reflection} is a known condition from kinetic theory, in which the distribution function at the Inflow boundary is proportional
to a Maxwellian \cite{Sone}, \cite{Jungel} with $T = T_W = T_W(\vec{x})$ the temperature at the wall
\begin{equation}
f |_- (\vec{x},\vec{k},t) = F_D (f|_+) = C \,
\sigma \left\{ f|_+ \right\}(\vec{x},t) \,
e^{-\varepsilon(\vec{k})/K_B T} \, ,
\quad (\vec{x},\vec{k}) \in \Gamma_N^- \, ,
\end{equation}
\begin{equation}
\sigma \left\{ f|_+ \right\}(\vec{x},t) =
\int_{\vec{v}(\vec{k}) \cdot \eta > 0 } \vec{v}(\vec{k}) \cdot \eta(\vec{x}) f |_+ (\vec{x},\vec{k},t) dk \, .
\end{equation}
{\it Mixed reflection} BC models the
effect of a physical surface on electron transport
in metals and semiconductors,
giving the reflected pdf representing the electrons as a linear convex combination of specular and diffuse components, as in the formula
\begin{eqnarray}
f|_- (\vec{x},\vec{k},t) &=& F_M ( f |_+ ) =
p \, F_S(f|_+) + (1-p) \, F_D(f|_+) \\
& = & p \, f |_+ (\vec{x},\vec{k}',t) \, + \, (1-p) \, C' \, \sigma' \left\{ f|_+ \right\}(\vec{x},t) \, e^{-\frac{\varepsilon(\vec{k})}{K_B T}} \, ,
\quad (\vec{x},\vec{k}) \in \Gamma_N^- \, .
\nonumber
\end{eqnarray}
$p$ is sometimes called specularity parameter. It can either be constant or a function, dependant of the momentum.
For example,
the work by Soffer \cite{Soffer}
studies a statistical model for the
reflection from a rough surface in
electrical conduction. It
derives a specularity parameter $p(\vec{k})$ which depends on the momentum, given by
\begin{equation}\label{SofferPformula}
p(\vec{k}) = e^{-4 l_r^2 |k|^2 \cos^2 \Theta} \, ,
\end{equation}
where $l_r$ is the rms height of the rough interface, and
$\Theta_B $ is th angle between the incident electron and the interface surface normal. \\
Reflection BC is a widely studied topic
in the context of the kinetic theory of gases
modelled by Boltzmann Equations.
However, in the context of kinetic models for electron transport in semiconductors,
there is less extensive previous work related
to the study of the effect of reflection boundary conditions such
as diffusive, specular, or mixed reflection.
An example of the list of references where reflection BC are studied for Boltzmann equations in the context of { kinetic theory of gases}
would include the works of
Cercignani \cite{CercigBE} and Sone \cite{Sone},
where the specular, diffusive, and mixed reflection BC
are formulated for the Boltzmann Eq. for gases.
V. D. Borman, S. Yu. Krylov, A. V. Chayanov \cite{ref:BKC}
study the nonequilibrium phenomena at a gas-solid interface.
The recent paper of Brull, Charrier, Mieussens \cite{ref:Mieussens}
studies the gas-surface interaction at a nano-scale and the boundary conditions for the associated Boltzmann equation. The recent work of
Struchtrup \cite{ref:Struchtrup} studies as well the
Maxwell boundary condition and velocity dependent accommodation
coefficients in the context of gases mentioned.
It considers the convex combination of specular reflection, isotropic scattering, and diffusive reflection, incorporating
velocity dependent coefficients into a Maxwell-type reflection kernel. It develops a modification of Maxwell's BC, extending the Maxwell model by allowing it to incorporate velocity dependent accomodation coefficients into the microscopic description
and satisfying conditions of reciprocity and unitary probability normalization. \\
Regarding reflectivity in the context of Boltzmann models of electron transport,
Fuchs \cite{FuchsBC} proposed a boundary condition for the probability density function of free electrons incident in the material surface, which is a convex combination of specular \& diffuse reflection with a constant specularity parameter $p$. Greene (\cite{Greene1}, \cite{Greene2}) studied conditions for the Fuchs BC in which the specularity parameter $p(\vec{k})$ is dependant on the angle of the momentum $\vec{k}$, deriving a boundary condition for electron distributions at crystal surfaces valid for metal, semimetal, \& semiconductor surfaces, and showing that Fuchs’ reflectivity parameter differs from the kinetic specularity parameter in physical significance and in magnitude. It considers
the unperturbed electron states of a crystal with an ideal perfectly specular surface as standing wave states, and
the diffusive reflection killing partially the incoming wave function.
Soffer \cite{Soffer} studies a statistical model for the electrical conduction, and derives under certain assumptions, such as
a rough surface random model with a Gaussian probability
of height above or below a horizontal plane,
analytical formulas for a momentum dependant specularity parameter $p(\vec{k}) = \exp(-4 l_r^2 |k|^2 \cos^2 \Theta) $ associated to this physical phenomena, abovementioned in (\ref{SofferPformula}).
As mentioned before, $l_r$ is the rms height of rough interface, and
$\Theta_B $ is th angle between the incident electron and the interface surface normal. \\
The reference book of
Markowich, Ringhofer, \& Schmeiser \cite{ref:MarkowichRS}
for semiconductor equations
discusses the mathematical definition of boundaries according
to the physical phenomena, and defines accordingly the kind
of BC to be imposed at those boundaries: Dirichlet, Neumann, Inflow and Outflow boundaries.
A work of particular importance for us is the one by
Cercignani, Gamba, and Levermore \cite{CGL}.
They study high field approximations to a Boltzmann-Poisson system and boundary conditions in a semiconductor. The BP system for electrons in a semiconductor in the case of high fields and small devices is considered. Boundary conditions are proposed at the kinetic level that yield charge neutrality at ohmic contacts, which are Dirichlet boundaries, and at insulating Neumann boundaries.
Both BC, either the one yielding charge neutrality at Dirichlet boundaries, or the one rendering zero flux of electrons at the boundary, assume that the pdf is proportional to a ground state
associated to an asymptotic expansion of a dimensionless Boltzmann-Poisson system. Then they study closures of moment equations and BC for both the pdf and for the moment closures.
J\"{u}ngel mentions in his semiconductors book \cite{Jungel}
the different kinds of reflection BC common on the
kinetic theory of gases, specular, diffusive, and mixed reflection
but no further study of diffusive and mixed reflection BC in the context of semiconductors is pursued. \\
We intend to present in this work
a mathematical, numerical, and computational study
of the effect of diffusive, specular, and mixed reflection BC
in Boltzmann-Poisson models of electron transport in semiconductors,
solved by means of Discontinuous Galerkin FEM solvers.
We study the mathematical formulation of these reflection BC
in the context of BP models for semiconductors, and derive equivalent numerical formulations of the diffusive and mixed reflection BC with non-constant $p(\vec{k})$,
such that an equivalent numerical zero flux condition
is satisfied pointwise at the insulating Neumann boundaries
at the numerical level. We present numerical simulations
for a 2D silicon diode and a 2D double gated MOSFET, comparing the effects of specular, diffusive, and mixed reflection boundary conditions in the physical observable quantities obtained from the simulations.
\section{BP system with $\vec{k}$ coordinate transformation assuming a Kane Energy Band}
The Kane Energy Band Model is a dispersion relation between the conduction energy band $\varepsilon$ (measured from a local minimum)
and the norm of the electron wave vector $|k|$, given by the analytical function ($\alpha$ is a constant parameter,
$m^*$ is the electron reduced mass for Si, and $\hbar$ is Planck's constant)
\begin{equation}
\varepsilon (1 + \alpha \varepsilon) = \frac{\hbar^2 |k|^2}{2m^*} \quad .
\end{equation}
For our preliminary numerical studies we will use a Boltzmann-Poisson model as in \cite{CGMS-CMAME2008} ,
in which the conduction energy band is assumed to be given by a Kane model.
We use the following dimensionalized variables, with the related characteristic parameters
$$
\dm
t = {\ti}/{t_*} ,
(x,y) = {\vec{x}}/{\ell_*},
\ell_* = 10^{-6} m, t_* = 10^{-12} s, V_* = 1 \mbox{V} \, .
$$
A transformed Boltzmann transport equation is used as in \cite{CGMS-CMAME2008} as well,
where the coordinates used to describe $\vec{k}$ are: $\mu$, the cosine of the polar angle, the azimuthal angle $\varphi$,
and the dimensionless Kane Energy $ w = {\varepsilon}/{K_B T} $,
which is assumed as the conduction energy band.
$K_B$ is Boltzmann's constant, $T$ is the wall temperature,
which we will assume to be equal to the lattice temperature
$T_L$, so $T_W = T = T_L$, and $ \ak = \alpha {K_B T} $. So $\vec{k}(w,\mu,\varphi)$, where
\begin{equation}
\vec{k}=
\frac{\sqrt{2 m^* \kbt }}{\hbar} \sqrt{w(1+\ak w)}
\left(
\mu,\sqrt{1-\mu^2} \cos \ph, \sqrt{1-\mu^2} \sin \ph
\right).
\end{equation}
A new unknown function $\Phi$ is used in the transformed Boltzmann Eq. \cite{CGMS-CMAME2008} , which is proportional to the Jacobian of the transformation
and to the density of states (up to a constant factor)
$$
\Phi(t, x, y, w, \mu, \ph) = s(w) f(\ti, \vec{x}, \vec{k}) \, ,
$$
where
\begin{equation}
s(w) = \sqrt{w(1+\ak w)}(1+2\ak w) \, .
\label{sw}
\end{equation}
The transformed Boltzmann transport equation for $\Phi$ used in \cite{CGMS-CMAME2008} is
\begin{equation}
\frac{\partial\Phi}{\partial t} + \frac{\partial}{\partial x} (g_1
\Phi) + \frac{\partial}{\partial y} (g_2 \Phi) +
\frac{\partial}{\partial w} (g_3 \Phi) + \frac{\partial}{\partial
\mu} (g_4 \Phi) + \frac{\partial}{\partial \ph} (g_5 \Phi) = C(\Phi).
\, \label{eqPhi}
\end{equation}
The vector
$(g_1,g_2)$ represent the 2D cartesian components of the electron velocity $\frac{1}{\hbar} \nabla_{\vec{k}} \varepsilon (\vec{k}) $,
in the coordinate system ($w$, $\mu$, $\ph$).
The triplet
$(g_3,g_4,g_5)$ represent the transport in the phase space of the new momentum coordinates ($w$, $\mu$, $\ph$) due to the self consistent electric field
$$\, \vec{E}(t,x,y) = \left(E_x(t,x,y), E_y(t,x,y), 0\right),$$
with
\begin{eqnarray*}
g_1 \argf & = & c_x \frac{ \sqrt{w(1+\ak w)}}{1+2 \ak w} \mu \, ,
\\
g_2 \argf & = & c_x \frac{\sqrt{w(1+\ak w)}}{1+2 \ak w} \sqrt{1-\mu^2} \cos\ph \, ,
\\
g_3 \argf & = & \mbox{} - c_k \frac{2 \sqrt{w(1+\ak w)}}{1+2 \ak w}
\left[ \mu \, E_x(t,x,y) + \sqrt{1-\mu^2} \cos\ph \, E_y(t,x,y)
\right] ,
\\
& = & \mbox{} - c_k \frac{2 \sqrt{w(1+\ak w)}}{1+2 \ak w}
\, \hat{e}_w \cdot \vec{E}(t,x,y) \, ,
\\
g_4 \argf & = & \mbox{} - c_k \frac{\sqrt{1-\mu^2}}{\sqrt{w(1+\ak
w)}}
\left[ \sqrt{1-\mu^2} \, E_x(t,x,y) - \mu \cos\ph \, E_y(t,x,y) \right] \, ,
\\
& = & \mbox{} - c_k \frac{\sqrt{1-\mu^2}}{\sqrt{w(1+\ak
w)}} \, \hat{e}_{\mu} \cdot \vec{E}(t,x,y) \, ,
\\
g_5 \argf & = & - c_k \frac{ - \sin\ph}{\sqrt{w(1+\ak w)} \sqrt{1-\mu^2}}
\, E_y(t,x,y)
\\
& = & -c_k \frac{1}{\sqrt{w(1+\ak w)} \sqrt{1-\mu^2}} \,
\hat{e}_{\ph} \cdot \, \vec{E}(t,x,y) \, ,
\end{eqnarray*}
$$
\dm
c_x = \frac{t_*}{\ell_*} \sqrt{\frac{2 \, \kbt}{\mass}}
\mbox{ and} \quad
c_k = \frac{t_* q E_*}{\sqrt{2 \mass \kbt}} \, ,
$$
and $\hat{e}_{w}, \, \hat{e}_{\mu}, \, \hat{e}_{\varphi} $
the orthonormal vector basis in our momentum coordinate space.
The right hand side of (\ref{eqPhi}) is the collision
operator (having applied the Dirac Delta's due to electron-phonon scattering,
which depend on the energy differences between transitions)
\begin{eqnarray*}
&& C(\Phi)(t,x,y,w,\mu,\ph) = s(w) \left\{ c_{0} \Ipm
\Phi(t,x,y,w,\mu ',\ph')
\right. \\
&& \left. + \Ipm [ c_{+} \Phi(t,x,y,w + \qe,\mu ',\ph')
+ c_{-} \Phi(t,x,y,w - \qe,\mu ',\ph') ] \right\}
\\[5pt]
&& \mbox{} - \Phi(t,x,y,w,\mu,\ph) \, 2 \pi \, [c_0 s(w) + c_+ s(w - \qe) + c_- s(w + \qe)] \, ,
\end{eqnarray*}
with the dimensionless parameters
$$
\dm
(c_0, c_+, c_-) = \frac{2 \mass \, t_*}{\hbar^3} \sqrt{2 \, \mass \, \kbt}
\left(K_0 , (n_{q} + 1) K , n_{q} K \right) ,
\quad
\qe = \frac{\hw}{\kbt} \, .
$$
The electron density is
$$
\dm n(t_* t, \ell_* x, \ell_* y)
= \Itre f(t_* t, \ell_* x, \ell_* y, \bk) \: d \bk
= \left( \frac{\sqrt{2 \,\mass \kbt }}{\hbar} \right)^{\! \! 3}
\rho(t,x,y) \, ,
$$
where
\begin{equation}
\rho(t,x,y) = \Iwmp \Phi (t,x,y,w,\mu,\ph) \, .
\label{dens}
\end{equation}
Hence, the dimensionless Poisson equation is
\begin{equation}
\label{pois}
\frac{\partial}{\partial x} \left( \epsilon_{r} \frac{\partial \nV}{\partial x}
\right)
+
\frac{\partial}{\partial y} \left( \epsilon_{r} \frac{\partial \nV}{\partial y}
\right)
= c_{p} \left[ \rho(t,x,y) - \mathcal{N}_{D}(x,y) \right] \, ,
\end{equation}
with
$$
\mathcal{N}_{D}(x,y) =
\left( \frac{\sqrt{2 \,\mass \kbt }}{\hbar} \right)^{\! \! -3}
N_{D}(\ell_* x, \ell_* y) \, \mbox{ and }
c_p = \left( \frac{\sqrt{2 \,\mass \kbt }}{\hbar} \right)^{\! \! 3}
\frac{\ell_*^{2} q}{\epsilon_{0}} \, .
$$
\section{Discontinuous Galerkin Method for Transformed Boltzmann - Poisson System and Implementation of Boundary Conditions}
The domain of the devices to be considered
can be represented by means of a
{rectangular grid}
in both position and momentum space. This rectangular grid, bidimensional in position space and tridimensional in momentum space, is defined as
$$
\Omega_{ijkmn} =
{X_{ij}}\times
{K_{kmn}} ,
$$
$$
X_{ij} =
\left[ x_{i - \ot} , \, x_{i + \ot} \right] \times
\left[ y_{j - \ot} , \, y_{j + \ot} \right] ,
$$
$$
K_{kmn} =
\left[ w_{k - \ot} , \, w_{k + \ot} \right] \times
\left[ \mu_{m - \ot} , \, \mu_{m + \ot} \right]
\times
\left[ \ph_{n - \ot} , \, \ph_{n + \ot} \right],
$$
where $i=1, \ldots N_x$, $j=1, \ldots N_y$, $k=1, \ldots N_w$, $m=1,
\ldots N_\mu$, $n=1, \ldots N_\ph$,
$$
x_{i \pm \ot} = x_{i} \pm \frac{\Delta x_{i}}{2} \, , \quad
y_{j \pm \ot} = y_{j} \pm \frac{\Delta y_{j}}{2}\, ,
$$
$$
w_{k \pm \ot} = w_{k} \pm \frac{\Delta w_{k}}{2}\, , \quad
\mu_{m \pm \ot} = \mu_{m} \pm \frac{\Delta \mu_{m}}{2}\, , \quad
\ph_{n \pm \ot} = \ph_{n} \pm \frac{\Delta \ph_{n}}{2}.
$$
The finite dimensional space used to approximate the functions is
the space of piecewise continuous polynomials which are piecewise linear in $(x,y)$ and piecewise constant in $(w,\mu,\varphi)$ ,
\begin{equation}
V_h=\{ v :
v|_{\Omega_{ijkmn}} \in Q^{1,0}(\Omega_{ijkmn})
= P^{1}(X_{ij}) \otimes P^{0}(K_{kmn})
\},
\end{equation}
with the set
$Q^{1,0}(\Omega_{ijkmn})$ of tensor product polynomials, linear over the element \\ $X_{ij} = \left[ x_{i - \ot} , \, x_{i + \ot} \right] \times \left[ y_{j - \ot} , \, y_{j + \ot} \right] $, and constant over the element\\ $K_{kmn} = \left[ w_{k - \ot} , \, w_{k + \ot} \right] \times\left[ \mu_{m - \ot} , \, \mu_{m + \ot} \right] \times\left[ \ph_{n - \ot} , \, \ph_{n + \ot} \right]$. \\
The function
$\Phi_h$ will denote the piecewise polynomial approximation of $\Phi$ over elements ${\Omega}_{I}$,
\begin{eqnarray*}
\Phi_h
& = &
\sum_{I}
\chi_{I}
\left[
T_{I}(t) +
X_{I}(t) \, \frac{(x - x_{i})}{\Delta x_{i}/2} +
Y_{I}(t) \, \frac{(y - y_{j})}{\Delta y_{j}/2}
\right], \quad I = (i,j,k,m,n).
\end{eqnarray*}
The density $\rho_h(t,x,y) $ on the cell
$[ x_{i - \ot} , \, x_{i + \ot} ] \times
[ y_{j - \ot} , \, y_{j + \ot} ] $
is, under this approximation,
\begin{eqnarray}
\quad \rho_h
&=&
\sum_{k=1}^{N_{w}}
\sum_{m=1}^{N_{\mu}} \sum_{n=1}^{N_{\varphi}}
\left[ T_{ijkmn} +
X_{ijkmn} \frac{(x - x_{i})}{\Delta x_{i}/2} +
Y_{ijkmn} \frac{(y - y_{j})}{\Delta y_{j}/2}
\right] \Delta w_{k} \Delta \mu_{m} \Delta \varphi_{n}
\nonumber\\
& = &
\quad
\sum_{k=1}^{N_{w}}
\sum_{m=1}^{N_{\mu}} \sum_{n=1}^{N_{\varphi}}
T_{ijkmn}
\Delta w_{k} \Delta \mu_{m} \Delta \varphi_{n}
\nonumber\\
& + &
\left(
\sum_{k=1}^{N_{w}}
\sum_{m=1}^{N_{\mu}} \sum_{n=1}^{N_{\varphi}}
X_{ijkmn}
\Delta w_{k} \Delta \mu_{m} \Delta \varphi_{n}
\right)
\frac{(x - x_{i})}{\Delta x_{i}/2}
\nonumber\\
& + &
\left(
\sum_{k=1}^{N_{w}}
\sum_{m=1}^{N_{\mu}} \sum_{n=1}^{N_{\varphi}}
Y_{ijkmn}
\Delta w_{k} \Delta \mu_{m} \Delta \varphi_{n}
\right)
\frac{(y - y_{j})}{\Delta y_{j}/2} \, .
\nonumber
\end{eqnarray}
\subsection{DG Formulation for Transformed Boltzmann Eq.}
The Discontinuous Galerkin formulation for the Boltzmann equation (\ref{eqPhi}) is as follows.
Find $\Phi_h \in V_h$, s.t.
\begin{eqnarray}
\label{dgb}
&&\int_{\Omega_{ijkmn}} (\Phi_h)_t \,v_h \,d \Omega
- \int_{\Omega_{ijkmn}} g_1 \Phi_h \,(v_h)_x \,d \Omega
- \int_{\Omega_{ijkmn}} g_2 \Phi_h \,(v_h)_y \,d \Omega
\nonumber
\\
&+& \mbox{} F_x^+ - F_x^- +F_y^+-F_y^- +F_w^+-F_w^-
+F_\mu^+-F_\mu^- +F_\ph^+-F_\ph^- \nonumber\\
&= & \int_{\Omega_{ijkmn}} C(\Phi_h)
\,v_h \,d \Omega .
\end{eqnarray}
for any test function $v_h \in V_h$. In (\ref{dgb}), the boundary integrals are given by
$$
F_x^\pm=\iy \iw \imu \iphi \, \widehat{ g_1 \Phi} \, v_h^\mp(x_{i\pm
\ot}, y, w, \mu, \ph)dy \, dw \, d\mu \, d\ph ,
$$
$$
F_y^\pm=\ix \iw \imu \iphi \, \widehat{ g_2 \Phi} \, v_h^\mp (x, y_{j\pm
\ot}, w, \mu, \ph)dx \, dw \, d\mu \, d\ph ,
$$
$$
F_w^\pm=\ix \iy \imu \iphi \widehat{ g_3 \, \Phi} \, v_h^\mp (x, y,
w_{k\pm \ot}, \mu, \ph)dx \, dy \, d\mu \, d\ph ,
$$
$$
F_\mu^\pm=\ix \iy \iw \iphi \widehat{g_4 \, \Phi} \, v_h^\mp (x,
y, w, \mu_{m\pm \ot}, \ph)dx \, dy \, dw \, d\ph ,
$$
$$
F_\ph^\pm=\ix \iy \iw \imu \, \widehat{ g_5 \Phi} \, v_h^\mp (x, y, w,
\mu, \ph_{n\pm \ot})dx \, dy \, dw \, d\mu ,
$$
where the upwind numerical fluxes $\widehat{g_s \Phi},\,s=1,...,5 $ are defined as
\begin{eqnarray}
\widehat{g_1 \Phi}|_{x_{i\pm 1/2}} & = &
\left( \frac{g_1 + |g_1|}{2} \right) \Phi_h |_{x_{i\pm 1/2}}^{-}
+
\left( \frac{g_1 - |g_1|}{2} \right) \Phi_h |_{x_{i\pm 1/2}}^{+} \, ,
\nonumber\\
\widehat{g_2 \Phi}|_{y_{j\pm 1/2}} & = &
\left( \frac{g_2 + |g_2|}{2} \right) \Phi_h |_{y_{j\pm 1/2}}^{-}
+
\left( \frac{g_2 - |g_2|}{2} \right) \Phi_h |_{y_{j\pm 1/2}}^{+} \, ,
\nonumber\\
\widehat{g_3 \Phi}|_{w_{k\pm 1/2}} & = &
\left( \frac{g_3 + |g_3|}{2} \right) \Phi_h |_{w_{k\pm 1/2}}^{-}
+
\left( \frac{g_3 - |g_3|}{2} \right) \Phi_h |_{w_{k\pm 1/2}}^{+} \, ,
\nonumber\\
\widehat{g_4 \Phi}|_{\mu_{m\pm 1/2}} & = &
\left( \frac{g_4 + |g_4|}{2} \right) \Phi_h |_{\mu_{m\pm 1/2}}^{-}
+
\left( \frac{g_4 - |g_4|}{2} \right) \Phi_h |_{\mu_{m\pm 1/2}}^{+} \, ,
\nonumber\\
\widehat{g_5 \Phi}|_{\varphi_{n\pm 1/2}} & = &
\left( \frac{g_5 + |g_5|}{2} \right) \Phi_h |_{\varphi_{n\pm 1/2}}^{-}
+
\left(\frac{g_5 - |g_5|}{2}\right)\Phi_h |_{\varphi_{n\pm 1/2}}^{+} \, .
\end{eqnarray}
\begin{comment}
\begin{itemize}
\item The sign of $g_1$ only depends on $\mu$,
if $\mu_m >0$, then $ \check{\Phi} =\Phi^-$; otherwise, $
\check{\Phi} =\Phi^+.$
\item The sign of $g_2$ only depends on $\cos \ph $,
if $\cos \ph_n >0$, then $ \bar{\Phi} =\Phi^- $; otherwise, $
\bar{\Phi} =\Phi^+ .$ Note that in our simulation, $N_\ph$ is always
even.
\item For $\widehat{ g_3 \, \Phi}$, we let $$\widehat{ g_3 \, \Phi}
=- 2 c_k \frac{\sqrt{w(1+\ak w)}}{1+2 \ak w} \left[ \mu \,
E_x(t,x,y) \hat{\Phi} +
\sqrt{1-\mu^2} \cos\ph \, E_y(t,x,y) \tilde{\Phi} \right] ,
$$
If $\mu_m E_x(t, x_i, y_j)<0$, then $\hat{\Phi}=\Phi^-$; otherwise,
$ \hat{\Phi} =\Phi^+.$
If $(\cos \ph_n ) E_y(t, x_i, y_j)<0$, then $\tilde{\Phi}=\Phi^-$;
otherwise, $ \tilde{\Phi} =\Phi^+.$
\item For $\widetilde{ g_4 \, \Phi}$, we let $$\widetilde{ g_4 \, \Phi}
=- c_k \frac{\sqrt{1-\mu^2}}{\sqrt{w(1+\ak w)}} \left[
\sqrt{1-\mu^2} \, E_x(t,x,y) \hat{\Phi} -
\mu \cos\ph \, E_y(t,x,y) \tilde{\Phi} \right] ,
$$
If $ E_x(t, x_i, y_j)<0$, then $\hat{\Phi}=\Phi^-$; otherwise, $
\hat{\Phi} =\Phi^+$.
If $\mu_m \cos(\ph_n) E_y(t, x_i, y_j)>0$, then
$\tilde{\Phi}=\Phi^-$; otherwise, $ \tilde{\Phi} =\Phi^+$.
\item The sign of $g_5$ only depends on $E_y(t, x, y)$,
if $E_y(t, x_i, y_j)>0$, then $ \breve{\Phi} =\Phi^- $; otherwise, $
\breve{\Phi} =\Phi^+ .$
\end{itemize}
\end{comment}
\subsection{Poisson Equation - Local Discontinuous Galerkin (LDG) Method}
The Poisson equation (\ref{pois}) is solved by the LDG method as in
\cite{CGMS-CMAME2008} .
By means of this scheme we find a solution
$\nV_h, q_h, s_h \in W_h^1$, where
$(q,s) = (\partial_{x} \nV , \, \partial_{y} \nV ) $
and
$W_h^1=\{ v : v|_{X_{ij}} \in P^1(X_{ij})\}$,
$P^1(X_{ij})$ the set of linear polynomials on $X_{ij}$.
It involves rewriting the equation into the form
\begin{equation}
\label{pois2} \left\{\begin{array} {l}
\displaystyle q= \frac{\partial \nV}{\partial x} , \qquad s=\frac{\partial \nV}{\partial y} , \\
\displaystyle \frac{\partial}{\partial x} \left( \epsilon_{r} q \right)
+ \frac{\partial}{\partial y} \left( \epsilon_{r} s \right)
= R(t,x,y) \, ,
\end{array}
\right.
\end{equation}
where $ R(t,x,y)=c_{p} \left[ \rho(t,x,y) - \mathcal{N}_{D}(x,y)
\right]$ is a known function that can be computed at each time step
once $\Phi$ is solved from (\ref{dgb}), and the coefficient
$\epsilon_r$ depends on $x, y$. The Poisson system is only on the
$(x,y)$ domain. Hence, we use the grid $I_{ij}=\left[ x_{i - \ot} ,
\, x_{i + \ot} \right] \times
\left[ y_{j - \ot} , \, y_{j + \ot} \right] $, with $i=1,\ldots, N_x$, $j=1, \ldots,
N_y+M_y$, where $j=N_y+1, \ldots, N_y+M_y$ denotes
the oxide-silicon region, and the grid in $j=1, \ldots, N_y$ is consistent with the
five-dimensional rectangular grid for the Boltzmann equation in the
silicon region. The approximation space is defined as
\begin{equation}
W_h^\kpol=\{ v : v|_{I_{ij}} \in P^\kpol(I_{ij})\}.
\end{equation}
Here $P^\kpol(I_{ij})$ denotes the set of all polynomials of degree
at most $\kpol$ on $I_{ij}$. The LDG scheme for (\ref{pois2}) is:
to find $q_h, s_h, \nV_h \in V_h^\kpol$, such that
\begin{eqnarray}
\label{ldgpois} 0&=& \mbox{}
\int_{I_{ij}} \left[q_h v_h
+
\nV_h (v_h)_x \right]dxdy
+\int_{y_{j - \ot}}^{y_{j + \ot}}
\left[
\left. \hat{\nV}_h v_h^+ \right| ( x_{i - \ot}, y)
-
\left. \hat{\nV}_h v_h^- \right| ( x_{i + \ot}, y)
\right]
dy , \nonumber \\
0&=& \mbox{}
\int_{I_{ij}} \!
\left[
s_h w_h
+
\nV_h (w_h)_y
\right]
dxdy
+\int_{x_{i - \ot}}^{x_{i + \ot}} \left[ \left. \tilde{\nV}_h w_h^+ \right| (x, y_{j - \ot})
-
\left.
\tilde{\nV}_h w_h^- \right| (x, y_{j + \ot}) \right] dx
, \nonumber \\
&-& \mbox{} \int_{I_{i,j}} \epsilon_{r} q_h (p_h)_x dxdy +\int_{y_{j
- \ot}}^{y_{j + \ot}} \widehat{ \epsilon_{r} q}_h p_h^-( x_{i +
\ot}, y) dy -\int_{y_{j - \ot}}^{y_{j + \ot}} \widehat{\epsilon_{r}
q}_h p_h^+( x_{i - \ot}, y) dy
\nonumber \\
&-& \mbox{} \int_{I_{i,j}} \epsilon_{r} s_h (p_h)_y dxdy +
\int_{x_{i - \ot}}^{x_{i + \ot}} \widetilde{ \epsilon_{r} s}_h
p_h^-(x, y_{j + \ot}) dx -\int_{x_{i - \ot}}^{x_{i + \ot}}
\widetilde{ \epsilon_{r} s}_h p_h^+(x, y_{j - \ot}) dx
\nonumber \\
& =& \mbox{} \int_{I_{i,j}} R(t,x,y) p_h dxdy \, ,
\end{eqnarray}
hold true for any $v_h, w_h, p_h \in W_h^\kpol$. In the above
formulation, we choose the flux as follows,
in the $x$-direction, we use $\hat{\nV}_h=\nV^-_h$, $\widehat{ \epsilon_{r} q}_h=\epsilon_{r} q_h^+ -[\nV_h]$.
In the $y$-direction, we use $\tilde{\nV}_h=\nV^-_h$, $\widetilde{ \epsilon_{r} s}_h = \epsilon_{r} s_h^+ -[\nV_h]$.
On some part of the domain boundary, the above flux needs to be
changed to accommodate various boundary conditions.
For example, in the case of a double gate MOSFET device,
for the boundary condition of the Poisson equation, $\nV=0.52354$ at
source, $\nV=1.5235$ at drain and $\nV=1.06$ at gate. For the rest
of the boundary regions, we have homogeneous Neumann boundary conditions,
i.e., $\frac{\partial \nV}{\partial n}=0$. The relative dielectric
constant in the oxide-silicon region is $\epsilon_r=3.9$, in the
silicon region is $\epsilon_r=11.7$.
Near the drain then, we are given Dirichlet boundary condition, so we need to flip
the flux in $x-$direction: let $\hat{\nV}_h ( x_{i + \ot},
y)=\nV^+_h ( x_{i + \ot}, y)$ and $\widehat{ \epsilon_{r} q}_h (
x_{i + \ot}, y)=\epsilon_{r} q_h^-( x_{i + \ot}, y) -[\nV_h]( x_{i +
\ot}, y),$ if the point $( x_{i + \ot}, y)$ is at the drain. For the
gate, we need to flip the flux in $y-$direction: let
$\tilde{\nV}_h(x,y_{j + \ot})=\nV^+_h(x,y_{j + \ot})$ and
$\widetilde{ \epsilon_{r} s}_h(x,y_{j + \ot}) = \epsilon_{r}
s_h^-(x,y_{j + \ot}) -[\nV_h](x,y_{j + \ot})$, if the point
$(x,y_{j + \ot})$ is at the gate. For the bottom, we need to use
the Neumann condition, and flip the flux in y-direction, i.e.,
$\tilde{\nV}_h=\nV^+_h$, $\widetilde{ \epsilon_{r} s}_h =
\epsilon_{r} s_h^-$. This scheme described above will enforce the
continuity of $\nV$ and $\epsilon_r \frac{\partial \nV}{\partial n}$
across the interface of silicon and oxide-silicon interface. The
solution of (\ref{ldgpois}) gives us approximations to both the
potential $\nV_h$ and the electric field $(E_x)_h=-c_v q_h$,
$(E_y)_h=-c_v s_h$.
\subsection{RK-DG Algorithm for BP, from $t^{n}$ to $t^{n+1}$}
The following RK-DG algorithm for BP is a
dynamic extension of the Gummel iteration map.
We write below the steps to evolve from time $t^n$ to time $t^{n+1}$.
\begin{enumerate}
\item Compute the electron density {$\rho_h(x,y,t)$}.
\item Solve Poisson Eq. for the given $\rho_h(x,y,t)$ by Local DG,
obtaining the potential $\nV_h$ and the electric field {$\mathbf{E}_h = -(q_h,s_h)$}. Compute then the respective transport terms {$g_s, \, s=1,...,5$}.
\item Solve by DG the advection and collision part of the Boltzmann Equation. A Method of Lines (an ODE system) for the time dependent coefficients of {$\Phi_h$} (degrees of freedom) is obtained.
\item Evolve ODE system by Runge-Kutta from {$t^{n}$} to { $t^{n+1}$}. (If partial time step necessary, repeat Step 1 to 3 as
needed).
\end{enumerate}
\section{Boundary Conditions Implementation for 2D-$\vec{x}$, 3D-$\vec{k}$ devices at $x,w,\mu,\varphi$ Boundaries}
We will consider in this work 2D devices in position space,
which need a 3D momentum description for kinetic equations modeling semiconductors. For example, a common device of interest is a 2D double gate MOSFET. A schematic plot of it is given in Figure \ref{mosfet}. The shadowed region denotes the oxide-silicon region, whereas the rest is the silicon region. Potential bias are applied at the source, drain, and gates. The problem is symmetric about the x-axis.\\
Another possible 2D problem is the case of a bi-dimensional bulk silicon diode, for which the doping is constant all over the physical domain, and which would have just an applied potential (bias) between the source $x=0$ and the drain $x=L_x$ (no gates), with insulating reflecting boundaries at $y=0$ and $y=L_y$.
\begin{figure}[htb]
\centering
\includegraphics[width=0.95\linewidth]{dgmos.jpg}
\caption{Schematic representation of a 2D double gate MOSFET device. From
Y. Cheng, I. M. Gamba, A. Majorana and C.-W. Shu, 'A discontinuous Galerkin solver for Boltzmann Poisson systems in nano devices', Computer Methods in Applied Mechanics and Engineering, v198 (2009), p. 3143.
}
\label{mosfet}
\end{figure}
We consider in the following sections the different kinds of boundary conditions for 2D devices and their numerical implementation, either at $\vec{x}$-boundaries or at $\vec{w}$-boundaries.
\subsection{Poisson Eq. Boundary Condition}
The BC for Poisson Eq. are imposed over the $(x,y)$-domain.
For example, for the case of a 2D Double gated MOSFET,
Dirichlet BC would be imposed to the potential $\nV$, as we
have three different applied potentials biases,
$\nV = 0.5235 $ Volts at the source $x=0$,
$\nV = 1.5235 $ Volts at the drain $x=L_x$,
$\nV = 1.06 $ Volts at the gates.
Homogeneous Neumann BC would be imposed for the rest of the boundaries, that is, $ \partial_{\hat{n}} \nV = 0$.
For the case of a 2D bulk silicon diode,
we impose Dirichlet BC for the difference of potential $\nV$
between source and drain,
$\nV = 0.5235 $ Volts at the source $x=0$,
$\nV = 1.5235 $ Volts at the drain $x=L_x$.
For the boundaries $y=0, \, L_y$ we impose Homogeneous Neumann BC too, that is, $ \partial_{y} \nV |_{y_0} = 0, \, y_0 = 0, \, L_y$. \\
\subsection{Charge Neutrality BC}
As in \cite{CGMS-CMAME2008}, at the source and drain contacts, we implement the charge neutrality boundary condition (\ref{NeutralChargeBCmath}).
Ghost cells for $i=0$ and $i=N_x+1$ at the respective boundaries
are used, implementing numerically the boundary conditions
$$\Phi(i=0)=\Phi(i=1)\frac{N_D(i=1)}{\rho(i=1)},$$
$$\Phi(i=N_x+1)=\Phi(i=N_x)\frac{N_D(i=N_x)}{\rho(i=N_x)}.$$
\begin{comment}
\subsubsection{Specular Reflection BC}
At the top and bottom of the computational domain (the silicon
region), we impose the classical elastic specular reflection
BC (symmetry w.r.t. $y$).
\begin{equation}
\Phi(x,y,w,\mu,\varphi,t) = \Phi(x,-y,w,\mu,\pi - \varphi,t) ,
\quad y=0, \quad y = L_y.
\end{equation}
So, for example, if $(x,y,w,\mu,\varphi) \in \Omega_{i0kmn}$ then $(x,-y,w,\mu,\pi - \varphi) \in \Omega_{i1kmn'}$,
with $n' = N_{\varphi} -n + 1 $. \\
\end{comment}
\subsection{Cut - Off BC}
In the $(w,\mu,\ph)$-space, we only need to apply a cut-off Boundary Condition. At $w=w_{\tiny \mbox{max}}$, $\Phi_h$ is made machine zero,
\begin{equation}
\Phi_h(x,y,w,\mu,\ph,t)|_{w = w_{\tiny \mbox{max}}} = 0.
\end{equation}
No other boundary condition is necessary for $\vec{w}$-boundaries, since analytically we have that
\begin{itemize}
\item at $w=0$, $g_3=0$,
\item at $\mu=\pm 1$, $g_4=0$,
\item at $\ph=0, \pi$, $g_5=0$,
\end{itemize}
so, at such regions, the numerical flux always vanishes.
\begin{comment}
\begin{figure}[ht]
\centering
\includegraphics[width=3in,angle=0]{density.eps}
\includegraphics[width=3in,angle=0]{energy.eps}\\
\includegraphics[width=3in,angle=0]{v_x.eps}
\includegraphics[width=3in,angle=0]{v_y.eps}
\caption{Macroscopic quantities of double gate MOSFET device at
$t=0.5$ps. Top left: density in ${cm}^{-3}$; top right: energy in
$eV$; bottom left: x-component of velocity in $cm/s$; bottom right:
y-component of velocity in $cm/s$. Solution reached steady state.}
\label{transhyd}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=3in,angle=0]{e_x.eps}
\includegraphics[width=3in,angle=0]{e_y.eps}\\
\includegraphics[width=3in,angle=0]{potential.eps}
\caption{Macroscopic quantities of double gate MOSFET device at
$t=0.5$ps. Top left: x-component of electric field in $kV/cm$; top
right: y-component of electric field in $kV/cm$; bottom: electric
potential in $V$. Solution has reached steady state.}
\label{transelec}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=2.8in,angle=0]{412.eps}
\includegraphics[width=2.8in,angle=0]{1212.eps}\\
\includegraphics[width=2.8in,angle=0]{2012.eps}
\includegraphics[width=2.8in,angle=0]{226.eps}\\
\includegraphics[width=2.8in,angle=0]{1510.eps}
\includegraphics[width=2.8in,angle=0]{151.eps}
\caption{PDF of double gate MOSFET device at $t=0.5$ps. Top left: at
$(0.025 \mu m, 0.012 \mu m)$; top right: at $(0.075 \mu m, 0.012
\mu m)$; middle left: at $(0.125 \mu m, 0.012 \mu m)$ ; middle
right: at $(0.1375 \mu m, 0.006 \mu m)$; bottom left: at $(0.09375
\mu m, 0.010 \mu m)$ ; bottom right: at $(0.09375 \mu m, 0 \mu m.)$
. Solution reached steady state.} \label{transpdf}
\end{figure}
\end{comment}
\begin{comment}
\begin{figure}[htb]
\centering
\includegraphics[width=4in,angle=0]{cart2012.eps}
\caption{PDF for 2D double gate MOSFET at $t=0.5$ps, $(x,y)=(0.125
\mu m, 0.012 \mu m)$.} \label{2dcart}
\end{figure}
\end{comment}
\begin{comment}
\begin{figure}[htb]
\centering
\includegraphics[width=4in,angle=0]{cart1510.eps}
\caption{PDF for 2D double gate MOSFET at $t=0.5$ps, $(x,y)=(0.9375
\mu m, 0.10 \mu m)$.} \label{2dcart}
\end{figure}
\end{comment}
\section{Reflection BC on BP}
Reflection Boundary Conditions can be expressed in the form
\begin{equation}
f(\vec{x}, \vec{k}, t) |_{\Gamma_{N^-}} = F_R(f|_{\Gamma_{N^+}}),
\end{equation}
such that the following pointwise zero flux condition
is satisfied at reflecting boundaries, so
\begin{eqnarray}
0 &=& \eta(\vec{x}) \cdot J(\vec{x}, t) = \eta(\vec{x}) \cdot \int_{\Omega_{\vec{k}}} \vec{v}(\vec{k}) \, f(\vec{x}, \vec{k}, t) \, d \vec{k} \, , \\
0 &=& \int_{\eta \cdot \vec{v} > 0 } \eta(\vec{x}) \cdot \vec{v}(\vec{k}) \, f(\vec{x}, \vec{k}, t)|_{\Gamma_{N^+}} \, d \vec{k} \, + \,
\int_{\eta \cdot \vec{v} < 0 } \eta(\vec{x}) \cdot \vec{v}(\vec{k}) \, f(\vec{x}, \vec{k}, t)|_{\Gamma_{N^-}} \, d \vec{k} \, , \nonumber\\
0 &=& \int_{ \vec{v} \cdot \eta > 0 } \vec{v} \cdot \eta \, f|_{\Gamma_{N^+}} \, d \vec{k} \, + \,
\int_{ \vec{v} \cdot \eta < 0 } \vec{v} \cdot \eta \, F_R(f|_{\Gamma_{N^+}}) \, d \vec{k}
\, ,
\nonumber
\end{eqnarray}
as in Cercignani, Gamba, Levermore\cite{CGL},
where the given BC at Neumann boundary regions at the kinetic level is such that the particle flow vanishes.
For simplicity we write $ \vec{v} = \vec{v}(\vec{k}) = { \nabla_{\vec{k}} \varepsilon(\vec{k}) }/{\hbar} $.
We will study three kinds of reflective boundary conditions: specular, diffusive, and mixed reflection.
The last one is a convex combination of the previous two,
but the convexity parameter can be either constant or momentum dependant, $p(\vec{k})$. We go over the mathematics and numerics related to these conditions below.
\subsection{Specular Reflection}
It is clear that, at the analytical level, the specular reflection BC (\ref{eq:defSpecReflex}) satisfies the zero flux condition pointwise at reflecting boundaries, since
\begin{equation}
\int_{_{\eta \cdot \vec{v} \, > 0 }} \!\!\! | \eta(\vec{x}) \cdot \vec{v}(\vec{k}) | \left. f(\vec{x}, \vec{k}, t)\right|_{_{\Gamma_{N^+}}} \, d \vec{k} -
\int_{_{-{\eta \cdot \vec{v} \, < 0 }}} \!\!\! | \eta(\vec{x}) \cdot \vec{v}(\vec{k}) | \left. f(\vec{x}, \vec{k}', t) \right|_{_{\Gamma_{_{N^+}}}} d \vec{k} = 0 . \nonumber
\end{equation}
Specular reflection BC in our transformed Boltzmann Eq. for the new coordinate system is mathematically formulated in our problem as
\begin{equation}
\Phi|_- (x,y,w,\mu,\varphi,t) = \Phi|_+ (x,y,w,\mu,\pi - \varphi,t),\quad (x,y,w,\mu,\varphi) \in \Gamma_N^-.
\end{equation}
To impose numerically specular reflection BC at $y=0, \, L_y$
in the DG method, we follow the procedure of \cite{CGMS-CMAME2008}.
We relate the inflow values of the pdf, associated to the outer ghost cells, to the outflow values of the pdf, which are associated to the interior cells adjacent to the boundary, as given below by
\begin{eqnarray}
&&
\Phi_h |_- (x,y_{{1}/{2}},w,\mu,\varphi,t) = \Phi_h |_+ (x,y_{{1}/{2}},w,\mu,\pi - \varphi,t) ,
\quad y_{{1}/{2}} = 0,
\\
&&
\Phi_h |_- (x,y_{N_y + \frac{1}{2}},w,\mu,\varphi,t) = \Phi_h |_+ (x,y_{N_y + \frac{1}{2}},w,\mu,\pi - \varphi,t) ,
\quad y_{N_y + \frac{1}{2}} = L_y .
\nonumber
\end{eqnarray}
In the case of the boundary $y_{1/2} = 0 $,
assuming $\Delta y_{0} = \Delta y_{1}$,
$\Delta \varphi_{n'} = \Delta \varphi_{n}$,
with $n' = N_{\varphi} -n + 1 $,
if $(x,y_{1/2}-y,w,\mu,\varphi) \in \Omega_{i0kmn}$ then $(x,y_{1/2}+y,w,\mu,\pi - \varphi) \in \Omega_{i1kmn'}$.
The values of $\Phi_h |^{\pm}_{y_{1/2}}$ at the
related inner and outer boundary
cells $\Omega_{i0kmn}$ ($j=0$) and $\Omega_{i1kmn'}$ ($j=1$) must be equal
at the boundary $y_{1/2} = 0$. Indeed
\begin{eqnarray}
&&
\Phi_h |^-_{\Omega_{i0kmn}} (x,y_{{1}/{2}},w,\mu,\varphi,t) =
\Phi_h |^+_{\Omega_{i1kmn'}} (x,y_{{1}/{2}},w,\mu,\pi - \varphi,t) \,
\implies
\nonumber\\
&&
T_{i0kmn} + X_{i0kmn} \frac{(x-x_i)}{\Delta x_i/2} + Y_{i0kmn} \frac{(y_{1/2}-y_0)}{\Delta y_0 /2} =
\nonumber\\
&&
T_{i1kmn'} + X_{i1kmn'} \frac{(x-x_i)}{\Delta x_i/2} + Y_{i1kmn'} \frac{(y_{1/2}-y_1)}{\Delta y_1 /2} \, .
\nonumber
\end{eqnarray}
Therefore, from the equality above we find the relation
between the coefficients of $\Phi_h$ at inner and outer adjacent boundary cells, given by
\begin{equation}
T_{i0kmn} = T_{i1kmn'}, \, X_{i0kmn} = X_{i1kmn'}, \, Y_{i0kmn} = -Y_{i1kmn'} \, .
\end{equation}
Following an analogous procedure for the boundary $y_{N_y + 1/2} $, we have
\begin{eqnarray}
&&
\Phi_h |^-_{\Omega_{i,N_y + 1,kmn}} (x,y_{N_y+ \frac{1}{2}},w,\mu,\varphi,t) =
\Phi_h |^+_{\Omega_{i,N_y,kmn'}} (x,y_{N_y + \frac{1}{2}},w,\mu,\pi - \varphi,t) \, .
\nonumber
\end{eqnarray}
Then
\begin{eqnarray}
&&
T_{i,N_y + 1,kmn} + X_{i,N_y + 1,kmn} \frac{(x-x_i)}{\Delta x_i/2} + Y_{i,N_y + 1,kmn} \frac{(y_{N_y + \frac{1}{2}}-y_{N_y + 1})}{\Delta y_{N_y + 1} /2} =
\nonumber\\
&&
T_{i,N_y,kmn'} + X_{i,N_y,kmn'} \frac{(x-x_i)}{\Delta x_i/2} + Y_{i,N_y,kmn'}\frac{(y_{N_y+\frac{1}{2}}-y_{N_y})}{\Delta y_{N_y}/2}
\, ,
\end{eqnarray}
and hence
$$
T_{i,N_y+1,kmn} = T_{i,N_y,kmn'}, \, X_{i,N_y+1,kmn} = X_{i,N_y,kmn'}, \, Y_{i,N_y+1,kmn} = -Y_{i,N_y,kmn'} \, .
$$
\subsection{Diffusive Reflection}
The diffusive reflection BC can be formulated as
\begin{equation}
f(\vec{x},\vec{k},t) |_- =
F_D (f|_+) =
C \,\sigma \left\lbrace f|_+ \right\rbrace (\vec{x},t) \, e^{-\varepsilon(\vec{k})/K_B T_L} \, , \quad
(\vec{x},\vec{k}) \in \Gamma_{N}^- \, ,
\end{equation}
where $\sigma \left\lbrace f|_+ \right\rbrace (\vec{x},t) = \sigma(\vec{x},t)$ and $C = C\{\eta(\vec{x})\}$ are the function and parameter such that the zero flux condition is satisfied at each of the points of the Neumann Boundary, so
\begin{eqnarray}
0 &=& \int_{ \vec{v} \cdot \eta > 0 } \vec{v} \cdot \eta \, f|_{\Gamma_{N^+}} \, d \vec{k} \, + \,
\int_{ \vec{v} \cdot \eta < 0 } \vec{v} \cdot \eta \,
\left[ C \sigma(\vec{x},t) e^{-\varepsilon(\vec{k})/K_B T_L} \right] \, d \vec{k} \, , \nonumber\\
0 &=& \int_{ \vec{v} \cdot \eta > 0 } \vec{v} \cdot \eta \,
f|_{\Gamma_{N^+}} \, d \vec{k} \, - \, \sigma(\vec{x},t) \cdot
C \int_{ \vec{v} \cdot \eta < 0 } |\vec{v} \cdot \eta| \, e^{-\varepsilon(\vec{k})/K_B T_L} \, d \vec{k} \, .
\nonumber
\end{eqnarray}
It follows then that
\begin{equation}
\sigma \left\lbrace f|_+ \right\rbrace (\vec{x},t) =
\int_{ \vec{v} (\vec{k}) \cdot \eta > 0 } \vec{v} \cdot \eta \,
f|_{\Gamma_{N^+}} (\vec{x},\vec{k},t) \, d \vec{k} \, ,
\end{equation}
\begin{equation}
C \left\lbrace \eta(\vec{x}) \right\rbrace = \left( \int_{ \vec{v} \cdot \eta < 0 } |\vec{v} \cdot \eta| \, e^{-\varepsilon(\vec{k})/K_B T_L} \, d \vec{k} \right)^{-1} \, ,
\end{equation}
\begin{equation}
f(\vec{x},\vec{k},t) |_- =
\frac{ e^{-\varepsilon(\vec{k})/K_B T_L} \int_{ \vec{v} (\vec{k}) \cdot \eta > 0 } \vec{v} \cdot \eta \, f|_{\Gamma_{N^+}} (\vec{x},\vec{k},t) \, d \vec{k}}{\int_{ \vec{v} \cdot \eta < 0 } |\vec{v} \cdot \eta| \, e^{-\varepsilon(\vec{k})/K_B T_L} \, d \vec{k} }
\, .
\end{equation}
The diffusive reflection BC, formulated in terms of the unknown function $\Phi$ of the transformed Boltzmann Equation \ref{eqPhi}, is expressed as
\begin{equation}\label{PhiDiffReflexBC}
\Phi |_- (x,y,w,\mu,\varphi,t) =
F_D ( \Phi |_+ ) =
C \, \sigma \left\lbrace \Phi |_+ \right\rbrace (x,y,t) \, e^{-w} s(w) \, ,
\end{equation}
\begin{equation}
\sigma(x,y,t) =
\int_{(g_1,g_2) \cdot \eta > 0 } \eta \cdot (g_1,g_2)(w,\mu,\varphi) \, \Phi |_+ \, dw d\mu d\varphi \, ,
\end{equation}
\begin{equation}
C(\eta) =
\left(
\int_{(g_1,g_2) \cdot \eta < 0 } | (g_1,g_2) \cdot \eta | \, e^{-w} s(w) \, dw d\mu d\varphi
\right)^{-1} \, .
\end{equation}
We have, over the portion of the boundary considered, that
$\eta = (0,-1,0)$ for $y=0$ and $\eta = (0,1,0)$ for $y = L_y$. Therefore
\begin{equation}\label{PhiDiffReflexBCnormaliz}
\Phi |_- (x,y_b,w,t) =
\frac{ e^{-w} s(w) \, \int_{- g_2 > 0 } |g_2| \, \Phi |_+ \, dw d\mu d\varphi }{ \int_{- g_2 < 0 } |g_2| \, e^{-w} s(w) \, dw d\mu d\varphi } \, ,
\quad y_b = 0 \, ,
\end{equation}
\begin{equation}\label{PhiDiffReflexBCnormalizLy}
\Phi |_- (x,y_b,w,t) =
\frac{ e^{-w} s(w) \, \int_{+ g_2 > 0 } |g_2| \, \Phi |_+ \, dw d\mu d\varphi }{ \int_{+ g_2 < 0 } |g_2| \, e^{-w} s(w) \, dw d\mu d\varphi } \, ,
\quad y_b = L_{y} \, .
\end{equation}
\subsubsection{Numerical Formulation of Diffusive BC for DG}
For the DG numerical method,
we have to project the boundary conditions to be imposed in the space $V_h$.
Our goal is to have at the numerical level an equivalent pointwise zero flux condition at the reflection boundary regions.
We formulate then the diffusive BC for the DG method as
\begin{eqnarray}\label{PhiDiffReflexBC}
\Phi_h |_- (x,y_b,w,\mu,\varphi,t) & = &
\Pi_h \left\lbrace F_D(\Phi_h |_+) \right\rbrace
\nonumber\\
& = &
\Pi_h \left\lbrace
C \, \sigma_h \left\lbrace \Phi_h |_+ \right\rbrace (x,y_b,t) \, e^{-w} s(w) \right\rbrace ,
\quad
y_b = 0, L_y.
\nonumber
\end{eqnarray}
where $\sigma_h \in V_h$ is a function in our piecewise polynomial space for $(x,y)$ and $C$ is a parameter such that the zero flux condition is satisfied numerically, so
\begin{eqnarray}
0 & = & \int_{ \vec{g} \cdot \eta > 0 } \vec{g} \cdot \eta \, \Phi_h |_+ d\vec{w} +
\int_{ \vec{g} \cdot \eta < 0 } \vec{g} \cdot \eta \, \Phi_h |_- d\vec{w}
\nonumber\\
& = & \int_{ \vec{g} \cdot \eta > 0 } \vec{g} \cdot \eta \, \Phi_h |_+ d\vec{w} +
\int_{ \vec{g} \cdot \eta < 0 } \vec{g} \cdot \eta \, \Pi_h \left\lbrace F_D( \Phi_h |_+ ) \right\rbrace d\vec{w}
\\
& = & \int_{ \vec{g} \cdot \eta > 0 } \vec{g} \cdot \eta \, \Phi_h |_+ d\vec{w} +
\int_{ \vec{g} \cdot \eta < 0 } \vec{g} \cdot \eta \,
\Pi_h \left\lbrace
C \, \sigma_h \left\lbrace \Phi_h |_+ \right\rbrace (x,y_b,t) \, e^{-w} s(w) \right\rbrace
d\vec{w} \, .
\nonumber
\end{eqnarray}
In the space $V_h$ of piecewise continuous polynomials
which are tensor products of polynomials of degree $p$ in $\vec{x}$
and of degree $q$ in $\vec{w}$, it holds that
\begin{eqnarray}
&&
\Pi_h \left\lbrace f_1( \vec{x} ) f_2(\vec{w}) \right\rbrace =
\Pi_h \left\lbrace f_1(\vec{x}) \right\rbrace \,
\Pi_h \left\lbrace f_2(\vec{w}) \right\rbrace \, ,
\\
&&
V_h = \{ v :
v|_{\Omega_{ijkmn}} \in Q^{p,q}(\Omega_{ijkmn})
= P^{p}(X_{ij}) \otimes P^{q}(K_{kmn})
\}.
\nonumber
\end{eqnarray}
Therefore, for our particular case we have
\begin{equation}
\Pi_h \left\lbrace
C \, \sigma_h (x,y_b,t) \,
e^{-w} s(w) \right\rbrace
=
C \, \sigma_h (x,y_b,t)
\,
\Pi_h \left\lbrace
e^{-w} s(w) \right\rbrace \, ,
\end{equation}
so for the numerical zero flux condition pointwise we have that
\begin{eqnarray}
0 & = &
\int_{ \vec{g} \cdot \eta > 0 } \vec{g} \cdot \eta \, \Phi_h |_+ d\vec{w} +
\int_{ \vec{g} \cdot \eta < 0 } \vec{g} \cdot \eta \,
C \, \sigma_h \left\lbrace \Phi_h |_+ \right\rbrace (x,y_b,t) \,
\Pi_h \left\lbrace
e^{-w} s(w) \right\rbrace
d\vec{w}
\nonumber\\
0 & = &
\int_{ \vec{g} \cdot \eta > 0 } \vec{g} \cdot \eta \, \Phi_h |_+ d\vec{w} -
\, \sigma_h \left\lbrace \Phi_h |_+ \right\rbrace (x,y_b,t) \, C \,
\int_{ \vec{g} \cdot \eta < 0 } |\vec{g} \cdot \eta| \,
\Pi_h \left\lbrace
e^{-w} s(w) \right\rbrace
d\vec{w} \, .
\nonumber
\end{eqnarray}
We observe then that we can obtain a numerical equivalent of the pointwise zero flux condition if we define
\begin{eqnarray}
&&
C \left\lbrace \eta \right\rbrace
=
C \left\lbrace \pm \hat{y} \right\rbrace
=
\left(
\int_{ \pm g_2 = \vec{g} \cdot \eta < 0 } |\vec{g} \cdot \eta| \,
\Pi_h \left\lbrace
e^{-w} s(w) \right\rbrace
d\vec{w}
\right)^{-1}
, \quad \eta = \pm \hat{y} \, .
\nonumber\\
&&
\sigma_h \left\lbrace \Phi_h |_+ \right\rbrace (x,y_b,t) =
\int_{ \pm \hat{y} \cdot \vec{g} > 0 } \vec{g} \cdot \eta \, \Phi_h |_+ d\vec{w}
=
\sigma \left\lbrace \Phi_h |_+ \right\rbrace (x,y_b,t)
\, , \, y_b = 0, \, L_y \, .
\nonumber
\end{eqnarray}
In our particular case, in which we have chosen our function space
as piecewise linear in $(x,y)$ and piecewise constant in $(w,\mu,\varphi)$, the projection of the Maxwellian is a piecewise constant approximation representing its average value over each momentum cell , that is
\begin{equation}
\Pi_h \left\lbrace
e^{-w} s(w) \right\rbrace
=
\sum_{k,m,n} \chi_{kmn}
\frac{
\int_{kmn} e^{-w} s(w) dw d\mu d\varphi
}{\Delta w_k \Delta \mu_m \Delta \varphi_n }
=
\sum_{k,m,n} \chi_{kmn}
\frac{
\int_{w_{k-}}^{w_{k+}} e^{-w} s(w) dw
}{\Delta w_k } .
\nonumber
\end{equation}
Therefore, for the particular space we have chosen, we have that
\begin{eqnarray}
&&
\sigma_h \left\lbrace \Phi_h |_+ \right\rbrace (x,y_b,t)
=
\int_{ \pm g_2 > 0 } \pm g_2 \, \Phi_h |_+ d\vec{w}
=
\sigma \left\lbrace \Phi_h |_+ \right\rbrace (x,y_b,t)
\, ,
\label{SIGMAhPHIh+}\\
&&
\quad
y_b = 0 = y_{1/2} \quad (\eta = - \hat{y}) \, ,
\quad \mbox{or} \quad
y_b = L_y = y_{N_y + 1/2} \quad (\eta = + \hat{y}) \, ,
\nonumber\\
&&
C^{-1}
=
\sum_{k,m,n}^{ \pm g_2 < 0 }
\frac{
1}{\Delta w_k}
{
\int_{w_{k-1/2}}^{w_{k+1/2}} e^{-w} s(w) \, dw
}
\int_{k,m,n} |g_2| \,
dw \, d\mu \, d\varphi
, \quad \eta = \pm \hat{y} \, ,
\nonumber\\
&&
\Phi_h |_- (x,y_b,w,\mu,\varphi,t) =
C \, \sigma_h \left\lbrace \Phi_h |_+ \right\rbrace (x,y_b,t) \,
\Pi_h \left\lbrace
e^{-w} s(w) \right\rbrace ,
\quad
y_b = 0, \, L_y,
\nonumber\\
&&
\Phi_h |_- (x,y_b,w,\mu,\varphi,t) =
\frac{ \int_{ \pm g_2 > 0 } | g_2 | \, \Phi_h |_+ d\vec{w}\,
\,
\sum_{k,m,n}^{\pm g_2 < 0} \chi_{kmn} \frac{ \int_{k} e^{-w} s(w) \, dw }{ \Delta w_k}
}{ \sum_{k,m,n}^{ \pm g_2 < 0 }
{
}
\int_{kmn} |g_2| \,
dw \, d\mu \, d\varphi \,
\frac{
\int_k e^{-w} s(w) \, dw
}{\Delta w_k}
} \, .
\nonumber
\end{eqnarray}
We notice that the polynomial approximation $ \sigma_h$
is equal to the analytical function $\sigma$ operating on the polynomial approximation $ \Phi_h|_+$. However,
the constant $C$ needed in order to achieve the zero flux condition numerically is not equal to the value of this parameter in the analytical solution. In this case $C$ is an approximation of the
analytical value using a piecewise constant approximation of the Maxwellian (its average over cells).
The approximate operator
$\sigma_h \left\lbrace \Phi_h |_+ \right\rbrace (x,y,t)$ gives a piecewise linear polynomial dependant on $(x,y)$ with time dependent coefficients. We have that
\begin{equation*}
\Phi_h |_+ \in V_h \implies
\sigma_h \left\lbrace \Phi_h |_+ \right\rbrace (x,y,t) =
\int_{ \pm \cos \varphi > 0 } |g_2| \, \Phi_h |_+ \, dw d\mu d\varphi \, \in V_h \, ,
\end{equation*}
where
$\Phi_h|_+$ is such that, at the boundary $y=y_b $ of the cell $\Omega_{ijkmn}$, it is given by
\begin{eqnarray*}
&& \Phi_h |_+ (t,x,y,w,\mu,\ph) =
T_{ijkmn}(t) +
X_{ijkmn}(t) \, \frac{2(x - x_{i})}{\Delta x_{i}} +
Y_{ijkmn}(t) \, \frac{2(y - y_{j})}{\Delta y_{j}} \, .
\end{eqnarray*}
We define $ I = ijkmn$, so in $\Omega_I = X_{ij} \times K_{kmn} $. Then,
\begin{equation}
\sigma_h(x,y,t) = \sigma_I^0(t) + \sigma_I^x(t) \frac{(x - x_{i})}{\Delta x_{i}/2} + \sigma_I^y(t) \frac{(y - y_{j})}{\Delta y_{j}/2} \, .
\end{equation}
We summarize the main results of these calculations for $\sigma_h$ and $\Phi_h |_- $, by showing just the ones related to $y=L_y$ (the case $y=0$ is analogous).
{At the boundary $y=L_y$, the inner cells associated to
outflow have $j=N_y$, adjacent to the boundary, whereas the ghost cells related to inflow have the index
$j= N_y + 1$}. We compute the integral $\sigma_h$ as
\begin{eqnarray}
\sigma_h \left\lbrace \Phi_h |_+ \right\rbrace (x,y,t)
&=&
\int_{\cos\varphi \geq 0} \frac{\sqrt{w(1+\ak w)}}{1+2\ak w} \sqrt{1-\mu^2} \cos\varphi \, \, \Phi_h |_+ \, dw d\mu d\varphi
\nonumber\\
& = &
\sum_{k,m,n}^{n\leq \frac{ N_p}{2}} \int_{K_{kmn}}
\frac{\sqrt{w(1+\ak w)}}{1+2\ak w} \sqrt{1-\mu^2} \cos\varphi\, \Phi_h |_+ dw d\mu d\varphi .
\nonumber
\end{eqnarray}
Therefore, we have, with $I = (i,j,k,m,n)$, $\, j = N_y $ below, that
\begin{equation*}
\sigma_I^0 = \sum_{k,m,n}^{n\leq\frac{ N_{\varphi}}{2}}
T_{i N_y kmn}
\int_{w_{k-1/2}}^{w_{k+1/2}} \frac{\sqrt{w(1+\ak w)}}{1+2\ak w} dw \int_{\mu_{m-1/2}}^{\mu_{m+1/2}} \sqrt{1-\mu^2} d\mu
\int_{\varphi_{n-1/2}}^{\varphi_{n+1/2}} \cos\varphi d\varphi ,
\end{equation*}
\begin{comment}
\begin{equation*}
+ \left( \int \frac{\sqrt{w(1+\ak w)}}{1+2\ak w} \frac{2(w - w_{k})}{\Delta w_{k}} dw \int \sqrt{1-\mu^2} d\mu \int \cos\varphi d\varphi \right) W_{ijkmn}(t)
\end{equation*}
\begin{equation*}
+ \left( \int \frac{\sqrt{w(1+\ak w)}}{1+2\ak w} dw \int \sqrt{1-\mu^2} \frac{2(\mu - \mu_{m})}{\Delta \mu_{m}} d\mu \int \cos\varphi d\varphi \right) M_{ijkmn}(t)
\end{equation*}
\begin{equation*}
+ \left( \int \frac{\sqrt{w(1+\ak w)}}{1+2\ak w} dw \int \sqrt{1-\mu^2} d\mu \int \cos\varphi \frac{2(\ph - \ph_{n})}{\Delta \ph_{n}} d\varphi \right) P_{ijkmn}(t)
\end{equation*}
\end{comment}
\begin{equation*}
\sigma_I^x = \sum_{k,m,n}^{n\leq \frac{ N_{\varphi}}{2}}
X_{i N_y kmn}
\int_{w_{k-1/2}}^{w_{k+1/2}} \frac{\sqrt{w(1+\ak w)}}{1+2\ak w} dw
\int_{\mu_{m-1/2}}^{\mu_{m+1/2}} \sqrt{1-\mu^2} d\mu
\int_{\varphi_{n-1/2}}^{\varphi_{n+1/2}} \cos\varphi d\varphi ,
\end{equation*}
\begin{equation}
\sigma_I^y = \sum_{k,m,n}^{n\leq \frac{ N_{\varphi}}{2}}
Y_{i N_y kmn}
\int_{w_{k-1/2}}^{w_{k+1/2}} \frac{\sqrt{w(1+\ak w)}}{1+2\ak w} dw \int_{\mu_{m-1/2}}^{\mu_{m+1/2}} \sqrt{1-\mu^2} d\mu \int_{\varphi_{n-1/2}}^{\varphi_{n+1/2}} \cos\varphi d\varphi. \nonumber
\end{equation}
Once the coefficients of $\sigma_h$ have been computed,
we use them to obtain the polynomial approximation $\Phi_h|_-$,
with $ j = N_y + 1$, from (\ref{SIGMAhPHIh+})
\begin{equation}
\Phi_h |^-_{y= L_y}
=
\, \sum_{i}
\sum_{k,m,n}^{n\geq \frac{ N_{\varphi} }{2}} \chi_{i N_y kmn}
C
\left[
\sigma_I^0 + \sigma_I^x \frac{(x-x_i)}{\Delta x_i/2}
+ \sigma_I^y \cdot 1
\right]
\frac{
\int_{k} e^{-w} s(w) dw }{ \Delta w_k} .
\nonumber
\end{equation}
We have at the same time, by definition, that
\begin{equation}
\Phi_h |^-_{y= L_y}
=
\sum_{ikmn}^{n\geq \frac{N_{\varphi}}{2}} \chi_{i,N_y + 1,kmn}
\left[
T_{i,N_y+1,k,m,n} +
X_{i,N_y+1,k,m,n} \frac{(x-x_i)}{\Delta x_i/2} -1\cdot
Y_{i,N_y+1,k,m,n}
\right] .
\nonumber
\end{equation}
Therefore, the coefficients for
$\Phi_h |^-_{y= L_y} $ are
\begin{equation}
T_{i,N_y+1,kmn}(t) = C \sigma_{iN_ykmn}^0(t) \frac{\int_k e^{-w} s(w) dw }{\Delta w_k} \, ,
\end{equation}
\begin{equation}
X_{i,N_y+1,kmn}(t) = C \sigma_{iN_ykmn}^x(t) \frac{\int_k e^{-w} s(w) dw }{\Delta w_k} \, ,
\end{equation}
\begin{equation}
Y_{i,N_y+1,kmn}(t) = -1\cdot C \sigma_{iN_ykmn}^y(t) \frac{\int_k e^{-w} s(w) dw }{\Delta w_k} \, ,
\end{equation}
keeping in mind that our parameter $C$ is given by the formula
\begin{equation}
C^{-1} =
\sum_{kmn}^{n\geq\frac{ N_p}{2}}
\frac{\int_k e^{-w} s(w) dw }{\Delta w_k}
\int_{k} \frac{\sqrt{w(1+\ak w)}}{1+2\ak w} dw
\int_{m} \sqrt{1-\mu^2} d\mu
\int_{n} \cos\varphi d\varphi .
\nonumber
\end{equation}
\subsection{Mixed Reflection}
The mixed reflection condition is a convex combination of the specular and diffusive reflections:
$$
f(\vec{x},\vec{k},t) |_- = p f|_+(\vec{x},\vec{k}',t) \, + \,
(1-p) C \sigma \left\lbrace f|_{+} \right\rbrace (\vec{x},t) e^{-\varepsilon(\vec{k})/K_B T} \, ,
\quad (\vec{x},\vec{k}) \in \Gamma_N^- \, ,
$$
$p$ is the Specularity Parameter, $\, 0 \leq p \leq 1$.
$p$ can be either constant or $p=p(\vec{k})$, a function of the wave vector momentum.
For $p$ constant, it can be shown easily that the previous formulas obtained for the specular and diffusive BC, in particular the previous formulas for $\sigma$ $C(x)$, works also in this case to obtain a zero flux condition at the Neumann boundaries:
\begin{eqnarray}
\eta \cdot J &=& \int_{ \vec{v} \cdot \eta > 0 } \vec{v} \cdot \eta f|_{+} d \vec{k} +
\int_{ \vec{v} \cdot \eta < 0 } \vec{v} \cdot \eta \left[ p f(\vec{x}, \vec{k}', t)|_{+} + (1-p) C e^{ \frac{-\varepsilon(\vec{k})}{K_B T_L}} \sigma(\vec{x},t) \right] d \vec{k} \nonumber\\
&=& \int_{ \vec{v} \cdot \eta > 0 } \vec{v} \cdot \eta f|_{+} d \vec{k} +
p \int_{ \vec{v} \cdot \eta < 0 } \vec{v} \cdot \eta f'|_{+} d \vec{k} +
\left(1-p\right) \sigma C \int_{ \vec{v} \cdot \eta < 0 } \vec{v} \cdot \eta e^{\frac{-\varepsilon(\vec{k})}{K_B T_L}} d \vec{k} \nonumber\\
&=& \sigma(\vec{x},t) \, - \,
p \sigma(\vec{x},t)
+ \left(1-p\right) \sigma(\vec{x},t) \left( -1 \right) = 0 \, . \nonumber
\end{eqnarray}
However, for $p(\vec{k})$ a function of the crystal momentum the same choice of $\sigma(\vec{x},t)$ and $C(x)$ as in the diffusive case does not necessarily guarantee that the zero flux condition will be satisfied at Neumann boundaries. Therefore, a new condition for $C$ in order to satisfy this condition must be derived.
We derive it below.
The general mixed reflection BC can be formulated as
\begin{equation}
f(\vec{x},\vec{k},t) |_- = p(\vec{k}) f |_+ (\vec{x},\vec{k}',t) \, + \, (1-p(\vec{k})) \, C' \sigma' \left\lbrace f |_+ \right\rbrace (\vec{x},t) \, e^{-\varepsilon(\vec{k})/K_B T}
\, ,
\quad (\vec{x},\vec{k}) \in \Gamma_N^-
\nonumber
\end{equation}
where $\sigma'\left\lbrace f|_+ \right\rbrace (\vec{x},t)$ and $C'$ are the function and parameter such that the pointwise zero flux condition is satisfied at the Neumann boundaries
\begin{eqnarray}
0 &=& \eta(\vec{x}) \cdot J(\vec{x}, t ) \nonumber\\
&=& \int_{ \vec{v} \cdot \eta > 0 } \vec{v} \cdot \eta \, f|_{+} \, d \vec{k} \, + \,
\int_{ \vec{v} \cdot \eta < 0 } \vec{v} \cdot \eta \, \left[ p(\vec{k}) f(\vec{x}, \vec{k}', t)|_{+} + (1-p(\vec{k}) ) C' e^{\frac{-\varepsilon(\vec{k})}{K_B T_L}} \sigma'(\vec{x},t) \right] d \vec{k} .
\nonumber
\end{eqnarray}
Since
\begin{equation}
0 = \int_{ \vec{v} \cdot \eta > 0 } \vec{v} \cdot \eta f|_{+} \, d \vec{k} +
\int_{ \vec{v} \cdot \eta < 0 } \vec{v} \cdot \eta \, p(\vec{k}) f'|_{+} \, d \vec{k} \, - \,
\sigma' (\vec{x},t) \, C' \int_{ \vec{v} \cdot \eta < 0 } (1-p(\vec{k}))
|\vec{v} \cdot \eta | \, e^{\frac{-\varepsilon}{K_B T_L}} \, d \vec{k} \, ,
\nonumber
\end{equation}
we conclude then that
\begin{equation}
\sigma' \left\lbrace f |_+ \right\rbrace (\vec{x},t) =
\int_{ \vec{v} \cdot \eta > 0 } \vec{v} \cdot \eta \, f|_{+} \, d \vec{k} \, - \,
\int_{ \vec{v} \cdot \eta < 0 } | \vec{v} \cdot \eta | \, p(\vec{k}) \, f(\vec{x},\vec{k}',t)|_{+} \, d \vec{k} \, ,
\end{equation}
\begin{equation}
C' \left\lbrace \eta(\vec{x}) \right\rbrace = \left( \int_{ \vec{v} \cdot \eta < 0 } (1-p(\vec{k}))
|\vec{v} \cdot \eta | \, e^{\frac{-\varepsilon}{K_B T_L}} \, d \vec{k} \right)^{-1} \, .
\end{equation}
The general mixed reflection BC then has the specific form
\begin{eqnarray}
f(\vec{x},\vec{k},t) |_- & = & p(\vec{k}) \, f |_+ (\vec{x},\vec{k}',t) \nonumber\\
& + & \,
(1-p(\vec{k}))
e^{-\frac{\varepsilon(\vec{k})}{K_B T}}
\frac{
\left(
\int_{ \vec{v} \cdot \eta > 0 } \vec{v} \cdot \eta f|_{+} d \vec{k} \, - \,
\int_{ \vec{v} \cdot \eta < 0 } | \vec{v} \cdot \eta | p(\vec{k}) f(\vec{x},\vec{k}',t)|_{+} d \vec{k}
\right)
}{\int_{ \vec{v} \cdot \eta < 0 } (1-p(\vec{k}))
|\vec{v} \cdot \eta | \, e^{\frac{-\varepsilon(\vec{k})}{K_B T_L}} \, d \vec{k} }
\, , \nonumber
\end{eqnarray}
with $\quad (\vec{x},\vec{k}) \in \Gamma_N^- \, , \quad (\vec{x},\vec{k}') \in \Gamma_N^+ \, $ s.t. $\vec{v}(\vec{k}') = \vec{v}(\vec{k}) - 2(\vec{v}(\vec{k})\cdot\eta)\eta \, .$ \\
Notice that the product $C' \sigma'(\vec{x},t)$ has the form
\begin{equation}
C' \sigma ' (\vec{x},t)
=
\frac{\left(
\int_{ \vec{v} \cdot \eta > 0 } \vec{v} \cdot \eta \, f|_{+} \, d \vec{k} \, - \,
\int_{ \vec{v} \cdot \eta < 0 } | \vec{v} \cdot \eta | \, p(\vec{k}) \, f(\vec{x},\vec{k}',t)|_{+} \, d \vec{k}
\right)
}{\int_{ \vec{v} \cdot \eta < 0 } (1-p(\vec{k}))
|\vec{v} \cdot \eta | \, e^{\frac{-\varepsilon(\vec{k})}{K_B T_L}} \, d \vec{k} }
\end{equation}
which for the case of $p$ constant, it reduces to the original function $\sigma(\vec{x},t)$ and parameter $C\left\lbrace \eta (\vec{x}) \right\rbrace $.
\begin{eqnarray}
\mbox{If} \quad p & = & \mbox{ct,} \nonumber\\
C' \sigma ' (\vec{x},t)
& = &
\frac{\left(
\int_{ \vec{v} \cdot \eta > 0 } \vec{v} \cdot \eta \, f|_{+} \, d \vec{k} \, - \, p \,
\int_{ \vec{v} \cdot \eta < 0 } | \vec{v} \cdot \eta | \, f(\vec{x},\vec{k}',t)|_{+} \, d \vec{k}
\right)
}{\int_{ \vec{v} \cdot \eta < 0 } (1-p)
|\vec{v} \cdot \eta | \, e^{\frac{-\varepsilon(\vec{k})}{K_B T_L}} \, d \vec{k} } \nonumber\\
& = &
\frac{\left( 1 - p \right) \,
\int_{ \vec{v} \cdot \eta > 0 } \vec{v} \cdot \eta \, f|_{+} \, d \vec{k} \,
}{ \left(1 - p \right) \, \int_{ \vec{v} \cdot \eta < 0 }
|\vec{v} \cdot \eta | \, e^{\frac{-\varepsilon(\vec{k})}{K_B T_L}} \, d \vec{k} } \nonumber\\
& = &
\frac{
\int_{ \vec{v} \cdot \eta > 0 } \vec{v} \cdot \eta \, f|_{+} \, d \vec{k} \,
}{ \int_{ \vec{v} \cdot \eta < 0 }
|\vec{v} \cdot \eta | \, e^{\frac{-\varepsilon(\vec{k})}{K_B T_L}} \, d \vec{k} } \nonumber\\
& = &
C \, \sigma\left(\vec{x},t\right) .
\nonumber
\end{eqnarray}
However, for the non-constant case $p(\vec{k})$ the new function and parameter $\sigma'(\vec{x},t)$, $C'(\eta)$ need to be used instead, as the previous $\sigma(\vec{x},t)$, $C(\eta)$ will not satisfy the zero flux condition in general for $p(\vec{k}) $, since
\begin{eqnarray}
0 &=& \int_{ \vec{v} \cdot \eta > 0 } \vec{v} \cdot \eta f|_{+} d \vec{k} +
\int_{ \vec{v} \cdot \eta < 0 } \vec{v} \cdot \eta p(\vec{k}) f'|_{+} d \vec{k} -
\sigma' C' \int_{ \vec{v} \cdot \eta < 0 } (1-p(\vec{k}))
|\vec{v} \cdot \eta | e^{\frac{-\varepsilon}{K_B T_L}} d \vec{k}
\nonumber\\
C' \sigma' &=&
\frac{
\int_{ \vec{v} \cdot \eta > 0 } \vec{v} \cdot \eta \, f|_{+} \, d \vec{k} \, + \,
\int_{ \vec{v} \cdot \eta < 0 } \vec{v} \cdot \eta \, p(\vec{k}) \, f(\vec{x},\vec{k}',t)|_{+} \, d \vec{k} \,
}{
\int_{ \vec{v} \cdot \eta < 0 } (1-p(\vec{k}))
|\vec{v} \cdot \eta | \, e^{\frac{-\varepsilon}{K_B T_L}} \, d \vec{k}
}
\nonumber\\
& \neq &
\frac{
\int_{ \vec{v} \cdot \eta > 0 } \vec{v} \cdot \eta \, f|_{+} \, d \vec{k} \,
}{ \int_{ \vec{v} \cdot \eta < 0 }
|\vec{v} \cdot \eta | \, e^{\frac{-\varepsilon(\vec{k})}{K_B T_L}} \, d \vec{k} } = C \sigma \nonumber(\vec{x},t) \quad \mbox{in } \, \, \mbox{general} \, \, \mbox{for} \, \, p(\vec{k}) .
\end{eqnarray}
A more general possible case of mixed reflection BC would have a specularity parameter $p(\vec{x},\vec{k},t)$
dependent on position, momentum, and time.
The related reflective BC would then be
\begin{eqnarray}
f |_- (\vec{x},\vec{k},t) &=& p(\vec{x},\vec{k},t) f|_+(\vec{x},\vec{k}',t) + \left(1 - p(\vec{x},\vec{k},t) \right) C^*(\vec{x},t) \sigma^*(\vec{x},t) M(\vec{x},\vec{k})
\nonumber\\
(\vec{x},\vec{k}) \in \Gamma_{N^-},
&\mbox{and}&
(\vec{x},\vec{k}') \in \Gamma_{N^+} \, ,
\end{eqnarray}
where $M(\vec{x},\vec{k})$ is the equilibrium probability distribution
(not necessarily a Maxwellian) according to which the electrons diffusively reflect on the physical boundary. $\sigma^*(\vec{x},t)$ and $C^*(\vec{x},t)$ are the functions such that the zero flux condition is satisfied pointwise at insulating boundaries
\begin{eqnarray}
0 &=& \eta(\vec{x}) \cdot \int_{\Omega_{\vec{k}}} \vec{v}(\vec{k}) f d \vec{k} =
\int_{\vec{v}\cdot\eta > 0 } \eta(\vec{x}) \cdot \vec{v}(\vec{k}) f|_+ d \vec{k} \, + \, \int_{ \vec{v}\cdot\eta < 0 } \eta(\vec{x}) \cdot \vec{v}(\vec{k}) f|_- d \vec{k}
\nonumber\\
&=&
\int_{\vec{v}\cdot\eta > 0 } \eta \cdot \vec{v} f|_+ d \vec{k} + \int_{ \vec{v}\cdot\eta < 0 } \eta \cdot \vec{v}
\left[
p(\vec{x},\vec{k},t) f'|_+ + \left(1 - p \right) C^*(\vec{x},t) \sigma^*(\vec{x},t) M(\vec{x},\vec{k})
\right]
d \vec{k}
\nonumber\\
&=&
\int_{\vec{v}\cdot\eta > 0 } \eta \cdot \vec{v} f|_+ d \vec{k} \, + \, \int_{ \vec{v}\cdot\eta < 0 } \eta \cdot \vec{v} \,
p(\vec{x},\vec{k},t) f|_+(\vec{x},\vec{k}',t) d \vec{k} \nonumber\\
&-& \sigma^*(\vec{x},t) \, C^*(\vec{x},t)
\, \int_{ \vec{v}\cdot\eta < 0 } | \eta \cdot \vec{v} |
\left(1 - p(\vec{x},\vec{k},t) \right) M(\vec{x},\vec{k})
d \vec{k} \, .
\nonumber
\end{eqnarray}
Therefore we conclude for this reflection case that
\begin{equation}
\sigma^*\left\lbrace f|_+ \right\rbrace (\vec{x},t) =
\int_{\vec{v}\cdot\eta > 0 } | \eta \cdot \vec{v}| f|_+ d \vec{k} \, - \, \int_{ \vec{v}\cdot\eta < 0 } |\eta \cdot \vec{v}| \,
p(\vec{x},\vec{k},t) f|_+(\vec{x},\vec{k}',t) d \vec{k} \, ,
\end{equation}
\begin{equation}
C^* (\vec{x},t) =
\left(
\int_{ \vec{v}\cdot\eta < 0 } | \eta \cdot \vec{v} |
\left(1 - p(\vec{x},\vec{k},t) \right) M(\vec{x},\vec{k})
d \vec{k}
\right)^{-1} \, ,
\end{equation}
and then the full BC formula for the $p(\vec{x},\vec{k},t)$ reflection case is
\begin{eqnarray}
&&
f |_- (\vec{x},\vec{k},t) = p(\vec{x},\vec{k},t) f|_+(\vec{x},\vec{k}',t) \quad +
\nonumber\\
&&
\left(1 - p(\vec{x},\vec{k},t) \right)
M(\vec{x},\vec{k})
\frac{
\left[
\int_{\vec{v}\cdot\eta > 0 } | \eta \cdot \vec{v}| f|_+ d \vec{k} \, - \, \int_{ \vec{v}\cdot\eta < 0 } |\eta \cdot \vec{v}| \,
p(\vec{x},\vec{k},t) f|_+(\vec{x},\vec{k}',t) d \vec{k}
\right]
}{
\int_{ \vec{v}\cdot\eta < 0 } | \eta \cdot \vec{v} |
\left(1 - p(\vec{x},\vec{k},t) \right) M(\vec{x},\vec{k})
d \vec{k} \, .
}
\nonumber
\end{eqnarray}
Remark: $p(\vec{x},\vec{k},t)$ can be any iid random variable
in $(\vec{x},\vec{k},t)$.
\subsubsection{Numerical Implementation}
The numerical implementation of the general mixed reflection with specularity parameter $p(\vec{k})$ is done in such a way that a numerical equivalent of the pointwise zero flux condition is achieved.
The general mixed reflection boundary condition in our DG numerical scheme is
\begin{eqnarray}
\left.\Phi_h\right|_{-} & = &
\Pi_h \left\lbrace F_M \left( \left.\Phi_h\right|_{+} \right) \right\rbrace \\
& = &
\Pi_h \left\lbrace
p(\vec{w})\Phi_h|_{+}(\vec{x},\vec{w}',t) +
(1 - p(\vec{w}) ) C'
\sigma'_h \left\lbrace \Phi_h|_{+} \right\rbrace (\vec{x}, t) \,
e^{-w} s(w)
\right\rbrace .
\nonumber
\end{eqnarray}
We will be using the notation
\begin{equation}
\vec{w} = (w, \mu, \varphi), \quad
d \vec{w} = dw \, d\mu \, d\varphi \, , \quad
\vec{w}' = (w, \mu, \pi - \varphi).
\end{equation}
The specific form of $C'$ and $\sigma'$ will be deduced from the numerical analogous of the mixed reflection boundary condition. We want to satisfy numerically the zero flux condition
\begin{eqnarray}
0 & = & \eta(\vec{x}) \cdot \int_{\Omega_{\vec{w}}} \vec{v}(\vec{w}) \, \Phi_h d\vec{w} \\
& = &
\int_{\vec{v}\cdot\eta>0}\vec{v} (\vec{w}) \cdot\eta\,\Phi_h|_+ d\vec{w}
\, +
\int_{\vec{v} \cdot \eta < 0 } \vec{v}(\vec{w})\cdot\eta\,\Phi_h|_- d\vec{w}
\nonumber\\
& = &
\int_{\vec{v}\cdot\eta>0}\vec{v}\cdot\eta\,\Phi_h|_+ d\vec{w}
\, +
\int_{\vec{v} \cdot \eta < 0 } \vec{v}\cdot\eta \,
\Pi_h \left\lbrace
p(\vec{w})\Phi_h '|_{+}
+
(1 - p(\vec{w}) ) C' \sigma'_h (\vec{x},t) e^{-w} s(w)
\right\rbrace
d\vec{w}
\nonumber\\
& = &
\int_{\vec{v}\cdot\eta>0}\vec{v}\cdot\eta\,\Phi_h|_+ d\vec{w}
\, -
\int_{\vec{v} \cdot \eta < 0 } |\vec{v} \cdot\eta | \,
\Pi_h \left\lbrace
p(\vec{w})\Phi_h|_{+}(\vec{x},\vec{w}',t)
\right\rbrace d \vec{w}
\nonumber\\
& + &
\int_{\vec{v} \cdot \eta < 0 } \vec{v}\cdot\eta \,
\Pi_h \left\lbrace
(1 - p(\vec{w}) ) C' \sigma'_h(\vec{x},t) e^{-w} s(w)
\right\rbrace
d\vec{w} .
\end{eqnarray}
In the space $V_h$ of piecewise continuous polynomials
which are tensor products of polynomials of degree $p$ in $\vec{x}$
and of degree $q$ in $\vec{w}$, it holds that
\begin{eqnarray}
&&
\Pi_h \left\lbrace f_1(\vec{x}) f_2(\vec{w}) \right\rbrace =
\Pi_h \left\lbrace f_1(\vec{x}) \right\rbrace \,
\Pi_h \left\lbrace f_2(\vec{w}) \right\rbrace \, ,
\\
&&
V_h = \{ v :
v|_{\Omega_{ijkmn}} \in Q^{p,q}(\Omega_{ijkmn})
= P^{p}(X_{ij}) \otimes P^{q}(K_{kmn})
\}.
\nonumber
\end{eqnarray}
Therefore, we have for our particular case that
\begin{equation*}
\Pi_h \left\lbrace
(1 - p(\vec{w}) ) C' \sigma'_h(\vec{x},t) e^{-w} s(w)
\right\rbrace
=
C' \sigma'_h(\vec{x},t)
\left[
\sum_{k,m,n} \chi_{kmn}
\frac{
\int_{K_{kmn}}
(1-p(\vec{w})) e^{-w} s(w) d\vec{w}
}{\int_{K_{kmn}} d\vec{w} }
\right]
\end{equation*}
Using this, our numerical pointwise zero flux condition is
\begin{eqnarray}
0 & = &
\int_{\vec{v}\cdot\eta>0}\vec{v}\cdot\eta\,\Phi_h|_+ d\vec{w}
\, -
\int_{\vec{v} \cdot \eta < 0 } |\vec{v} \cdot\eta | \,
\Pi_h \left\lbrace
p(\vec{w})\Phi_h|_{+}(\vec{x},\vec{w}',t)
\right\rbrace d \vec{w}
\nonumber\\
& + &
\int_{\vec{v} \cdot \eta < 0 } \vec{v}\cdot\eta \,
C' \sigma'_h(\vec{x},t)
\left[
\sum_{k,m,n} \chi_{kmn}
\frac{
\int_{K_{kmn}}
(1-p(\vec{w})) e^{-w} s(w) d\vec{w}
}{\int_{K_{kmn}} d\vec{w} }
\right]
d\vec{w}
\nonumber\\
& = &
\int_{\vec{v}\cdot\eta>0}\vec{v}\cdot\eta\,\Phi_h|_+ d\vec{w}
\, -
\int_{\vec{v} \cdot \eta < 0 } |\vec{v} \cdot\eta | \,
\Pi_h \left\lbrace
p(\vec{w})\Phi_h|_{+}(\vec{x},\vec{w}',t)
\right\rbrace d \vec{w}
\nonumber\\
& + & C' \sigma'_h(\vec{x},t)
\int_{\vec{v} \cdot \eta < 0 } \vec{v}\cdot\eta \,
\left[
\sum_{k,m,n} \chi_{kmn}
\frac{
\int_{K_{kmn}}
(1-p(\vec{w})) e^{-w} s(w) d\vec{w}
}{\int_{K_{kmn}} d\vec{w} }
\right]
d\vec{w}
\nonumber\\
& = &
\int_{\vec{v}\cdot\eta>0}\vec{v}\cdot\eta\,\Phi_h|_+ d\vec{w}
\, -
\int_{\vec{v} \cdot \eta < 0 } |\vec{v} \cdot\eta | \,
\Pi_h \left\lbrace
p(\vec{w})\Phi_h|_{+}(\vec{x},\vec{w}',t)
\right\rbrace d \vec{w}
\nonumber\\
& - & C' \sigma'_h(\vec{x},t)
\sum_{k,m,n} \chi_{kmn}
\int_{\vec{v} \cdot \eta < 0 } | \vec{v}\cdot\eta | \, d\vec{w} \,
\frac{
\int_{K_{kmn}} (1-p(\vec{w})) e^{-w} s(w) d\vec{w}
}{\int_{K_{kmn}} d\vec{w}}
\nonumber \\
& = &
\int_{\vec{v}\cdot\eta>0}\vec{v}\cdot\eta\,\Phi_h|_+ d\vec{w}
\, -
\int_{\vec{v} \cdot \eta < 0 } |\vec{v} \cdot\eta | \,
\Pi_h \left\lbrace
p(\vec{w})\Phi_h|_{+}(\vec{x},\vec{w}',t)
\right\rbrace d \vec{w}
\nonumber\\
& - & \sigma'_h(\vec{x},t) \, C'
\sum_{k,m,n, \, \vec{v} \cdot \eta < 0 }
\int_{K_{kmn}} | \vec{v}\cdot\eta | \, d\vec{w} \,
\frac{
\int_{K_{kmn}} (1-p(\vec{w})) e^{-w} s(w) d\vec{w}
}{\int_{K_{kmn}} d\vec{w}} \, .
\nonumber
\end{eqnarray}
We conclude then that we can achieve a numerical equivalent of the pointwise zero flux condition by defining
\begin{equation}
\sigma'_h \left\lbrace \Phi_h|_+ \right\rbrace (\vec{x},t) =
\int_{\vec{v}\cdot\eta>0}\vec{v}\cdot\eta\,\Phi_h|_+ d\vec{w}
\, -
\int_{\vec{v} \cdot \eta < 0 } |\vec{v} \cdot\eta | \,
\Pi_h \left\lbrace
p(\vec{w})\Phi_h|_{+}(\vec{x},\vec{w}',t)
\right\rbrace d \vec{w} ,
\nonumber
\end{equation}
\begin{equation}
\left( C' \left\lbrace \eta \right\rbrace \right)^{-1} =
\sum_{k,m,n, \, \vec{v} \cdot \eta < 0 }
\int_{K_{kmn}} | \vec{v}\cdot\eta | \, d\vec{w} \,
\frac{
\int_{K_{kmn}} (1-p(\vec{w})) e^{-w} s(w) d\vec{w}
}{\Delta w_k \Delta \mu_m \Delta \varphi_n} \, .
\end{equation}
Therefore, the inflow BC in our DG numerical method
is given by the expression
\begin{eqnarray}
\left.\Phi_h\right|_{-} & = &
\Pi_h \left\lbrace
p(\vec{w})\Phi_h|_{+}(\vec{x},\vec{w}',t)
\right\rbrace
\nonumber\\
& + &
\Pi_h \left\lbrace
(1 - p(\vec{w}) ) C'
\left(
\int_{\vec{v}\cdot\eta>0}\vec{v}\cdot\eta\,\Phi_h|_+ d\vec{w}
\right. \right.
\nonumber\\
&-& \,
\left. \left.
\int_{\vec{v} \cdot \eta < 0 } |\vec{v} \cdot\eta | \,
\Pi_h \left\lbrace
p(\vec{w})\Phi_h|_{+}(\vec{x},\vec{w}',t)
\right\rbrace d \vec{w}
\right)
e^{-w} s(w)
\right\rbrace .
\nonumber
\end{eqnarray}
The particular form of the coefficients defining the piecewise polynomial approximation $\Phi_h |_-$ for the general mixed reflection BC is presented below for the boundary $y = L_y$,
since the calculations for the case of the boundary $y=0$ are analogous.
For the boundary $y_{N_y + 1/2}=L_y \, , \quad \eta \cdot \vec{v} \propto + \hat{y} \cdot \vec{g} = g_2 \propto \cos\varphi \, $,
which defines the sign of $g_2$.
Outflow cells have the index $ \, j=N_y \,$. They are cells inside the domain adjacent to the boundary.
Inflow cells have the index $ j = N_y + 1 $. They are
ghost cells adjacent to the boundary. We have in our case that
\begin{eqnarray}
\sigma_h'
&=&
\int_{\cos\varphi>0} g_2\, {\Phi}_h|_+ d\vec{w}
\, -
\int_{\cos\varphi < 0 } |g_2 | \,
\Pi_h \left\lbrace
p(\vec{w}) {\Phi}_h|_{+}(\vec{x},\vec{w}',t)
\right\rbrace d \vec{w}
\nonumber\\
&=&
\sum_{k,m,n}^{n\leq \frac{N_{\varphi}}{2}}
\int_{K_{kmn}} g_2\, {\Phi}_h|_+ d\vec{w}
\, -
\sum_{k,m,n}^{n> \frac{ N_{\varphi}}{2}}
\int_{K_{kmn}} |g_2 | \,
\Pi_h \left\lbrace
p(\vec{w}) {\Phi}_h|_{+}(\vec{x},\vec{w}',t)
\right\rbrace d \vec{w} \, .
\nonumber
\end{eqnarray}
If $ I = (i, N_y+1, k, m , n)$ (inflow),
$\,I' = (i, N_y, k, m , n') , \, n' = N_{\varphi}' - n + 1 $
(outflow), the projection integrand is given by
\begin{equation}
\Pi_h \left\lbrace
p(\vec{w}) {\Phi}_h|_{+}(\vec{x},\vec{w}',t)
\right\rbrace
=
\sum_{I}^{n > N_{\varphi}/2}
\chi_{I} \,
\frac{\int_{{kmn}} p(\vec{w}) d\vec{w} }{\int_{{kmn}} d\vec{w}}
\left[
T_{I'} + X_{I'} \frac{(x-x_i)}{\Delta x_i/2}
+ Y_{I'}(+1)
\right] .
\nonumber
\end{equation}
The coefficients of
$\sigma_h' $ are given below. We have now that
$
I = (i, N_y, k, m , n) , \,
$
$ \,
I' = (i, N_y, k, m , n') , \quad n' = N_{\varphi}' - n + 1 \,
$, so from the previous two formulas then
\begin{eqnarray}
{\sigma'}_{i,N_y}^0 &=& \sum_{k,m,n}^{n\leq N_p/2}
\, T_{I}(t)
\int_k \frac{\sqrt{w(1+\ak w)}}{1+2\ak w} dw \int_m \sqrt{1-\mu^2} d\mu \int_n \cos\varphi d\varphi
\\
& - & \sum_{k,m,n}^{n> N_p/2}
\, T_{I'}(t)
\int_k \frac{\sqrt{w(1+\ak w)}}{1+2\ak w} dw \int_m \sqrt{1-\mu^2} d\mu \int_n |\cos\varphi | d\varphi \,
\frac{\int_{kmn} p(\vec{w}) d\vec{w} }{\int_{kmn} d\vec{w} }, \nonumber\\
{\sigma'}_{i,N_y}^x &=& \sum_{k,m,n}^{n\leq N_p/2}
X_{I}(t)
\int_k \frac{\sqrt{w(1+\ak w)}}{1+2\ak w} dw \int_m \sqrt{1-\mu^2} d\mu \int_n \cos\varphi d\varphi
\nonumber\\
& -& \sum_{k,m,n}^{n> N_p/2}
\, X_{I'}(t)
\int_k \frac{\sqrt{w(1+\ak w)}}{1+2\ak w} dw \int_m \sqrt{1-\mu^2} d\mu \int_n |\cos\varphi | d\varphi \,
\frac{\int_{kmn} p(\vec{w}) d\vec{w} }{\int_{kmn} d\vec{w} }, \nonumber\\
{\sigma'}_{i,N_y}^y &=& \sum_{k,m,n}^{n\leq N_p/2}
Y_{I}(t)
\int_k \frac{\sqrt{w(1+\ak w)}}{1+2\ak w} dw
\int_m \sqrt{1-\mu^2} d\mu
\int_n \cos\varphi d\varphi
\nonumber\\
&-& \sum_{k,m,n}^{n> N_p/2}
\, Y_{I'}(t)
\int_k \frac{\sqrt{w(1+\ak w)}}{1+2\ak w} dw \int_m \sqrt{1-\mu^2} d\mu \int_n |\cos\varphi | d\varphi \,
\frac{\int_{kmn} p(\vec{w}) d\vec{w} }{\int_{kmn} d\vec{w} } \, . \nonumber
\end{eqnarray}
Since on one hand we have
\begin{eqnarray*}
\left. {\Phi}_h\right|^{-}_{L_y} & = &
\Pi_h \left\lbrace
p(\vec{w}) {\Phi}_h|_{+}(\vec{x},\vec{w}',t)
\right\rbrace
+
\Pi_h \left\lbrace
(1 - p(\vec{w}) ) C'
\sigma'_h \left\lbrace {\Phi}_h|_{+} \right\rbrace (\vec{x}, t) \,
e^{-w} s(w)
\right\rbrace
\nonumber\\
&=&
\sum_{ikmn}^{n > \frac{N_{\varphi}}{2}}
\chi_{i,N_y + 1, kmn}
\frac{\int_{{kmn}} p\, d\vec{w} }{\int_{{kmn}} d\vec{w}}
\left[
T_{i,N_y,k,m,n'} + X_{i,N_y,k,m,n'} \frac{(x-x_i)}{\Delta x_i/2}
+ Y_{i,N_y,k,m,n'}
\right] \\
&+&
\sum_{i,k,m,n}^{n > N_{\varphi}/2}
\chi_{i,N_y+1,k,m,n} \,
\frac{\int_{{kmn}} (1- p(\vec{w})) e^{-w} s(w) d\vec{w} }{\int_{{kmn}} d\vec{w}}
\times
\nonumber\\
&\times&
C'
\left[
{\sigma'}_{i,N_y}^0 + {\sigma'}_{i,N_y}^x \frac{(x-x_i)}{\Delta x_i/2}
+ {\sigma'}_{i,N_y}^y (+1)
\right] ,
\end{eqnarray*}
and on the other hand
\begin{equation}
\left. {\Phi}_h\right|^{-}_{y_{N_y+1/2}} =
\sum_{i,k,m,n}^{n > \frac{N_{\varphi}}{2}}
\chi_{i,N_y+1,k,m,n}
\left[
T_{i,N_y+1,k,m,n} +
X_{i,N_y+1,k,m,n} \frac{(x - x_{i})}{\Delta x_{i}/2} -
Y_{i,N_y+1,k,m,n}
\right] ,
\nonumber
\end{equation}
we conclude that the coefficients for $\Phi_h|_-$ are
\begin{eqnarray}
T_{i,N_y+1,k,m,n} &=&
T_{I'} \frac{\int_{kmn} p(\vec{w}) d\vec{w} }{\int_{kmn} d\vec{w}}
+
C' {\sigma'}_{i,N_y}^0 \frac{\int_{kmn} (1 - p(\vec{w})) e^{-w} s(w) d\vec{w} }{\int_{kmn} d\vec{w}},
\nonumber\\
X_{i,N_y+1,k,m,n} &=&
X_{I'} \frac{\int_{kmn} p(\vec{w}) d\vec{w} }{\int_{kmn} d\vec{w}}
+
C' {\sigma'}_{i,N_y}^x \frac{\int_{kmn} (1 - p(\vec{w})) e^{-w} s(w) d\vec{w} }{\int_{kmn} d\vec{w}},
\nonumber\\
Y_{i,N_y+1,k,m,n} &=&
- \left(
Y_{I'} \frac{\int_{kmn} p(\vec{w}) d\vec{w} }{\int_{kmn} d\vec{w}}
+
C' {\sigma'}_{i,N_y}^y \frac{\int_{kmn} (1 - p(\vec{w})) e^{-w} s(w) d\vec{w} }{\int_{kmn} d\vec{w}}
\right),
\nonumber\\
I' &=& (i,N_y,k,m,n'), \quad
I = (i,N_y,k,m,n) \, .
\end{eqnarray}
\begin{comment}
{\color{blue}
Soffer's specularity parameter, when expressed in terms of our transformed coordinates for $\vec{k}$, takes the form:
$$p(\vec{k}) = e^{-4 l_r^2 |k|^2 \cos^2 \Theta} = \exp(-4 l_r^2 w(1+\ak w) \sin^2\varphi) = p(w,\varphi)$$
}
\end{comment}
\section{Numerical Results }
\subsection{2D bulk silicon}
We present results of numerical
simulations for the case of n
2D bulk silicon diode with an applied bias
between the boundaries $ x=0, \, L_x $,
and reflection BC at the boundaries
$y=0, \, L_y$ (Figs. \ref{fig_2Dbulksilicon}).
The required dimensionality in momentum space is a
3D $\vec{k}(w,\mu,\varphi)$.
The specifics of our simulations are:
\ \\
Initial Condition: {$ \left. \Phi(w) \right|_{t=0} = \Pi_h \left\lbrace N e^{-w}s(w) \right\rbrace $}. Final Time: 1.0ps\\[7pt]
{ Boundary Conditions (BC):}\\[5pt]
{ $\vec{k}$}-space: { Cut-off - at $w=w_{max}$,
{ $\Phi$} is machine zero.} \\[2pt]
Only needed BC in $(w,\mu,\varphi)$: transport normal to the boundary analitically zero at 'singular points' boundaries: \\%[3pt]
At { $w=0$, $g_3 = 0$}.
At { $\mu = \pm 1$, $g_4$ = 0.}
At { $\varphi = 0, \pi$, $g_5$ = 0.}\\[5pt]
{ $\vec{x}$}-space: { Charge Neutrality at boundaries
$x=0,\, x=0.15 \mu m$}.\\[2pt]
Bias - Potential: $\left. V \right|_{x=0} = 0.5235$ V, $\, \left. V \right|_{x=0.15 \mu m} = 1.5235$ V.
\\[2pt]
Neumann BC for Potential at $y=0, \, L_y =12 nm$: $\partial_y V |_{y=0, \, L_y} = 0$.\\
{ Reflection BC at $y=0, y=12 nm$}: Specular, Diffusive, Mixed Reflection with constant specularity $p=0.5$,
and Mixed Reflection using a momentum dependent specularity $p(\vec{k}) = \exp(-4 \eta^2 |k|^2 \sin^2 \varphi) $, the nondimensional roughness rms height coefficient being $\eta = 0.5 $.
We observe an influence of the Diffusive and Mixed Reflection in macroscopic observables. It is particularly noticeable in the kinetic moments. For example, the charge density slightly increases with diffusivity close to the reflecting boundaries, and, due to mass conservation, alters the density profile over the domain. Momentum \& mean velocity increase with diffusive reflection over the domain, while the energy is decreased by diffusive reflection over the domain. There is a negligible difference in the electric field $x$ component below its orders of magnitude for the different reflection cases.
\begin{figure}[htb]
\centering
\includegraphics[angle=0,width=.495\linewidth]
{densitySDMcGbulk2d.pdf}
\includegraphics[angle=0,width=.495\linewidth]{energySDMcGbulk2D.pdf}
\includegraphics[angle=0,width=.495\linewidth]{UxSDMcGbulk2D.pdf}
\includegraphics[angle=0,width=.495\linewidth]{UySDMcGbulk2D.pdf}
\includegraphics[angle=0,width=.495\linewidth]{ExSDMcGbulk2D.pdf}
\includegraphics[angle=0,width=.495\linewidth]{EySDMcGbulk2D.pdf}
\includegraphics[angle=0,width=.495\linewidth]
{potentialSDMcGbulk2D.pdf}
\caption{ Density $\rho$ ($m^{-3}$), $\quad$
Mean energy $e (eV)$,
Momentum $U_x, U_y \, (10^{28} \frac{cm^{-2}}{s})$,
Electric Field Components $E_x$ and $E_y$,
and Potential $V (Volts)$
vs Position $(x,y)$ in $(\mu m)$ plot for Specular, Diffusive, Mixed $p=0.5$ \& Mixed $p(\vec{k}) = \exp(-4 \eta^2 |k|^2 \sin^2 \varphi), \, \eta = 0.5$ Reflection
for 2D bulk silicon.}
\label{fig_2Dbulksilicon}
\end{figure}
\subsection{2D double gated MOSFET}
We present as well the results of numerical
simulations for the case of a
2D double gated MOSFET device (Figs. \ref{fig_DoubleGateMOSFET}).
On one hand, the BC for the Poisson Eq. for this device would be
the Dirichlet BC
$\nV = 0.5235 $ Volts at the source $x=0$,
$\nV = 1.5235 $ Volts at the drain $x=L_x$, and
$\nV = 1.06 $ Volts at the gates.
On the other hand, Homogeneous Neumann BC $ \partial_{\hat{n}} \nV = 0$ are imposed at the rest of the boundaries.
Specular reflection
is applied at the boundary $y=0$
because the solution is symmetric with respect
to $y=0$ for our 2D double gate MOSFET (Fig. \ref{mosfet}).
At the boundary $y= L_y$ we apply
specular, diffusive, and mixed reflection BC, both
with constant $p=0.5$, and with a momentum dependent $p(\vec{k}) = \exp(-4 \eta^2 |k|^2 \sin^2 \varphi) $ with roughness coefficient $\eta = 0.5$.
We use again the initial condition: {$ \left. \Phi(w) \right|_{t=0} = \Pi_h \left\lbrace N_D(x,y) e^{-w}s(w) \right\rbrace $},
running the simulations up to the physical time of 1.0ps.
We use again as well a cut-off BC in the boundary of the momentum domain, so $\Phi$ is machine zero at $w=w_{max}$,
and we apply charge neutrality BC at
$x=0,\, x=0.15 \mu m$.
We observe a quantitative difference in the kinetic moments
and other observables between the different cases of reflective BC,
with the physical quantities being of the same order of magnitude.
The electron density increases close to the gates with diffusive reflection, and close to the center of the device, given by the boundary $y=0$, the density profile is greater for specular reflection. The energy moment clearly decreases with diffusive reflection over the physical domain. The momentum $x$-component
for specular reflection is less than for the other reflective cases.
There is a difference in the profile of the electric field $x$-component between the specular reflection and the other cases that include diffusivity, increasing it with diffusive reflection close to the drain. The electric field $y$-component increases with diffusive reflection close to the boundary $y=0$ representing the center of the device. The electric potential is greater for the cases including diffusive reflection than for the perfectly specular case.
\begin{figure}[htb]
\centering
\includegraphics[angle=0,width=.495\linewidth]{density.pdf}
\includegraphics[angle=0,width=.495\linewidth]{energy.pdf}
\includegraphics[angle=0,width=.495\linewidth]{UX.pdf}
\includegraphics[angle=0,width=.495\linewidth]{UY.pdf}
\includegraphics[angle=0,width=.495\linewidth]{EX.pdf}
\includegraphics[angle=0,width=.495\linewidth]{EY.pdf}
\includegraphics[angle=0,width=.495\linewidth]{potential.pdf}
\caption{ Density $\rho$ ($m^{-3}$), $\quad$
Mean energy $e (eV)$,
Momentum $U_x, U_y \, (10^{28} \frac{cm^{-2}}{s})$,
Electric Field Components $E_x$ and $E_y$,
and Potential $V (Volts)$
vs Position $(x,y)$ in $(\mu m)$ plot for Specular, Diffusive, Mixed $p=0.5$ \& Mixed $p(\vec{k}) = \exp(-4 \eta^2 |k|^2 \sin^2 \varphi) , \, \eta = 0.5$ Reflection
for a 2D double gated MOSFET.}
\label{fig_DoubleGateMOSFET}
\end{figure}
\subsection{Electrons reentering the 2D domain with reflective BC in $y$ and periodic BC in $x$: comparison of bulk silicon with collisionless plasma}
We consider in this case almost the same physical situation and parameters for the previous section on the 2D bulk silicon,
except that instead of using the charge neutrality conditions we apply periodic boundary conditions in the $x$-boundaries,
simulating then that the electrons reenter the material on the opposite $x$-boundary after the outflow exits the domain.
We compare these simulations with ones in which no collisions are considered, corresponding the latter to the case of a collisionless
plasma with reflective BC in $y$ and periodic BC in $x$.
For both cases, bulk silicon with electron-phonon collisions and the collisionless electron gas,
we still apply an external potential such that $V=0$ at $x=0$ and $V=1$Volt at $x=L_x$. This can be understood in the framework
of periodic BC in $x$ as a periodic sawtooth wave with period equal to the length of the $x$-domain.
We do this comparison in order to study the effect of the reflective boundary conditions in $y$, with and without the influence of the collisions over electrons, and we let the electrons re-enter the domain under periodic boundary conditions in $x$, eliminating then the charge neutrality conditions in $x$ and any possible effect due to the latter. Since due to the periodic BC in $x$ the electrons re-enter the domain after they exit it in outflow, the effect of boundary conditions is exclusively related to the reflection in the transport domain in the $y$-boundaries.
For example, in Figs. \ref{fig_MassConserv} we present the plots of Relative Mass vs Time (ps) for Specular, Diffusive, Mixed with constant and momentum dependent specularity for different sets of simulations.
The top figure is related to simulations for bulk silicon with charge neutrality conditions on the non-reflecting boundaries,
the middle figure is associated to simulations for bulk silicon with periodic boundary conditions on the non-reflecting boundaries, and the bottom figure is related to the simulations for collisionless electron transport with periodic boundary conditions on the non-reflecting boundaries.
The last two sets of simulations mentioned conserve the mass during all the time, and these sets isolate the effect of reflection boundary conditions by using periodic boundary conditions instead of charge neutrality conditions. The first set associated to charge neutrality conditions in adition to reflection boundary conditions, however, have a slight increase in the relative mass
of less than 0.5\%. This slight increase then is associated only to the inclusion of charge neutrality conditions and a possible accumulation of numerical error due solely to it.\\
We notice in our comparison then the following effects of the collision operator in comparison with the collisionless plasma case.
As expected, the main effect of collisions is to decrease the magnitude of the average energy, average velocity and momentum (therefore the current) of electrons over the domain (Fig. \ref{fig_2DbulksiliconPeriodBC}). The effect of collisions on the distribution of the electron density profile over the domain is negligible.
Regarding the isolated effects of the reflection boundary conditions in the kinetic moments and other physical observables of interest
by considering the collisionless plasma with periodic BC in $x$, we notice, as earlier in the section for bulk silicon, the slight increase of the density profile close to the reflecting boundaries when adding diffusivity in the boundary conditions, and by conservation of mass, a decrease of the density profile over the center of the domain. The mean energy decreases over the position domain with the inclusion of diffusive reflection BC, as well as the $x$ components (which are the dominant) of the momentum and velocity (Fig. \ref{fig_2DcollisionlessPeriodBC}). It is important to notice this expected effect of the isolated reflection BC in the collisionless plasma case, since for the case that includes electron-phonon collisions combined with adding diffusive reflection BC gives actually an increase in the $x$ components of the momentum and velocity compared to the purely specular reflection case (Fig. \ref{fig_2DbulksiliconPeriodBC}). The collisionless plasma with periodic BC in $x$ and reflection BC in $y$ isolates the effect of the latter then and shows the expected behaviour of a decrease in the mean energy, velocity and momentum $x$-compoments when adding diffusivity in the reflection boundary conditions.
\begin{figure}[htb]
\centering
\includegraphics[angle=0,width=.495\linewidth]
{PCdensitySDMcGbulk2d.pdf}
\includegraphics[angle=0,width=.495\linewidth]{PCenergySDMcGbulk2D.pdf}
\includegraphics[angle=0,width=.495\linewidth]{PCUxSDMcGbulk2D.pdf}
\includegraphics[angle=0,width=.495\linewidth]{PCUySDMcGbulk2D.pdf}
\includegraphics[angle=0,width=.495\linewidth]{PCVxSDMcGbulk2D.pdf}
\includegraphics[angle=0,width=.495\linewidth]
{PCpotentialSDMcGbulk2D.pdf}
\caption{ Density $\rho$ ($m^{-3}$), $\quad$
Mean energy $e (eV)$,
Momentum $U_x, U_y \, (10^{28} \frac{cm^{-2}}{s})$,
Average Velocity Component $V_x$,
and Potential $V (Volts)$
vs Position $(x,y)$ in $(\mu m)$ plot for Specular, Diffusive, Mixed $p=0.5$ \& Mixed $p(\vec{k}) = \exp(-4 \eta^2 |k|^2 \sin^2 \varphi), \, \eta = 0.5$ Reflection
for electrons in 2D bulk silicon with reflective BC in $y$ and periodic BC in $x$.}
\label{fig_2DbulksiliconPeriodBC}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[angle=0,width=.495\linewidth]
{pncdensitySDMcGbulk2d.pdf}
\includegraphics[angle=0,width=.495\linewidth]{pncenergySDMcGbulk2D.pdf}
\includegraphics[angle=0,width=.495\linewidth]{pncUxSDMcGbulk2D.pdf}
\includegraphics[angle=0,width=.495\linewidth]{pncUySDMcGbulk2D.pdf}
\includegraphics[angle=0,width=.495\linewidth]{pncVxSDMcGbulk2D.pdf}
\includegraphics[angle=0,width=.495\linewidth]
{pncpotentialSDMcGbulk2D.pdf}
\caption{ Density $\rho$ ($m^{-3}$), $\quad$
Mean energy $e (eV)$,
Momentum $U_x, U_y \, (10^{28} \frac{cm^{-2}}{s})$,
Average Velocity Component $V_x$,
and Potential $V (Volts)$
vs Position $(x,y)$ in $(\mu m)$ plot for Specular, Diffusive, Mixed $p=0.5$ \& Mixed $p(\vec{k}) = \exp(-4 \eta^2 |k|^2 \sin^2 \varphi), \, \eta = 0.5$ Reflection
for 2D collisionless electrons with reflective BC in $y$ and periodic BC in $x$.}
\label{fig_2DcollisionlessPeriodBC}
\end{figure}
\begin{figure}[htb]
\includegraphics[angle=0,width=.64\linewidth]{./2DbsNCreflexFIGSconservedMassSDMcG-MassVStDGBPbulk2D.pdf}
\includegraphics[angle=0,width=.64\linewidth]{./PlasmasCollFIGSconservedMassSDMcG-MassVStDGBPbulk2D.pdf}
\includegraphics[angle=0,width=.64\linewidth]{./plasmasNoColl-FIGSconservedMassSDMcG-MassVStDGBPbulk2D.pdf}
\caption{
Relative Mass vs Time (ps) plot for Specular, Diffusive, Mixed $p=0.5$ \& Mixed $p(\vec{k}) = \exp(-4 \eta^2 |k|^2 \sin^2 \varphi), \, \eta = 0.5$ Reflection. The figure on top is related to the simulations for bulk silicon with charge neutrality conditions on the non-reflecting boundaries.
The figure in the middle is associated to the simulations for bulk silicon with periodic boundary conditions on the non-reflecting boundaries.
The bottom figure is related to the simulations for collisionless electron transport with periodic boundary conditions on the non-reflecting boundaries.
The figures show the conservation of mass when isolating the effect of reflection boundary conditions in the simulations, as the slight increase in the relative mass
of less than 0.5\% is associated only to simulations that also include charge neutrality conditions and possibly an accumulation of numerical error due solely to it.
}
\label{fig_MassConserv}
\end{figure}
\section{Conclusions}
We have considered the mathematical and numerical modeling of Reflective Boundary Conditions in 2D devices and their implementation in DG-BP schemes. We have studied the specular, diffusive and mixed reflection BC on the boundaries of the position domain of the device.
We developed a numerical equivalent of the zero flux condition at the position domain boundaries for the case of a more general mixed reflection with a momentum dependant specularity parameter $p(\vec{k})$.
We compared the influence of these different reflection cases in the computational prediction of moments
after implementing numerical BC equivalent to the respective
reflective BC, each one satisfying a mathematical zero flux condition at insulating boundaries.
There are effects due to the inclusion of diffusive reflection boundary conditions over the moments and physical observables of the probability density function, whose influence is not only restricted to the boundaries but actually to the whole domain. Particularly noticeable effects of the inclusion of diffusivity in kinetic moments are the increase of the density close to the reflecting boundary, the decrease of the mean energy over the domain {and, in the case when electron-phonon collisions for silicon are included, the increase of the $x$-components of the mean velocity and momentum over the domain, whereas for the collisionless case, for which only the effects of the reflection boundary conditions are considered (such as when electrons are allowed to reenter the material via periodic boundary conditions in $x$), a decrease in those $x$-components of mean velocity and momentum is observed, as expected when adding diffusivity to the reflection boundary conditions.}\\
Future research will consider, for example, the inclusion of surface roughness scattering mechanisms in the collision operator for our diffusive reflection problem in silicon devices. It will be related as well to the inclusion of diffusive reflection BC with a DG-BP-EPM full energy band.
More importantly, another line of work of our interest for future research
will be the more general case of a $p(\vec{x},\vec{k})$
specular probability dependant on momentum and position as well,
considering in addition to its mathematical aspects the related numerical issues
and the respective computational modelling,
intending to use experimental values of
$p(\vec{x},\vec{k})$ as input for the simulations.
\section*{Acknowledgment}
The authors' research was partially supported by NSF grants NSF CHE-0934450, NSF-RNMS DMS-1107465 and DMS 143064, and the ICES Moncrief Grand Challenge Award.
The computational work was partially performed by means of TACC resources under project A-ti4. Support from the
Institute of Computational Engineering and Sciences and the University of Texas Austin is gratefully acknowledged.
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TITLE: Left multiplications for each linear map
QUESTION [3 upvotes]: Problem
During studying linear algebra, I've stumbled upon linear maps. So obviously, for any $(m \times m)$-matrix $A$, the map
$$
\lambda_A (B)=AB
$$
is a linear map from $\mathfrak{M}_{m,n}$ (the set of all $(m \times n)$-matrices) to itself. Now we are looking for the converse:
Let $L:\mathfrak M_{m,n}\to\mathfrak M_{m,n}$ be a linear map. Is $L=\lambda_A$ for some $A\in\mathfrak M_{m,m}$?
My Attempt #1
First, I tried using the natural projections $\pi_j:V_1\times\cdots\times V_n\to V_j$ and embeddings $\iota_j:V_j\to V_1\times\cdots\times V_n$ defined as
$$
\pi_j(v_1,\cdots,v_n)=v_j,\qquad\iota_j(v_j)=(0,\cdots,v_j,\cdots,0)
$$
(only the $j$-th coordinate of $\iota_j(v_j)$ equals $v_j$, everything else is zero). So first we could identify $\mathfrak M_{m,n}$ as $\mathbb R^m\times\cdots\times\mathbb R^m$ ($n$ times). Then I came up with this small example:
Let $L:\mathfrak M_{2,2}\to\mathfrak M_{2,2}$ be defined as
$$
L\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}=\begin{pmatrix}
3a + 2c & 3b + 2d \\
a & b\end{pmatrix}.
$$
Then
$$
(\pi_1 \circ L\circ\iota_1)\mathbf e_1=\begin{pmatrix}3 \\ 1\end{pmatrix},
\qquad
(\pi_2 \circ L\circ\iota_2)\mathbf e_2=\begin{pmatrix}2 \\ 0\end{pmatrix}
$$
(where $\{\mathbf e_1,\mathbf e_2\}$ is the standard basis for $\mathbb R^2$) and thus $L=\lambda_A$ where
$$
A=\begin{pmatrix}3 & 2 \\ 1 & 0\end{pmatrix}.
$$
Now seeing this example, I guessed that for a given linear map $L$, the matrix $A\in\mathfrak M_{m,m}$ defined as
$$
A=
\begin{pmatrix}
(\pi_1 \circ L \circ \iota_1)\mathbf e_1 & \cdots & (\pi_m \circ L \circ \iota_m)\mathbf e_m
\end{pmatrix}
$$
would satisfy the equation $L=\lambda_A$, but I don't really get how to prove this.
My Attempt #2
In the example listed above, every row of the resulting matrix is a linear combination of the rows of the original matrix. So I thought about finding the coefficients of the linear combination; for example,
$$
\pi_i(L(B^\mathbf t))=\sum_{j=1}^m a_{ij}\pi_j(B^\mathbf t) \qquad(i=1,\cdots,m).
$$
Then defining $A=(a_{ji})$ would do. But does the equation above always have a solution?
Overall
This post has became quite long for me just writing up everything I could think of. To sum up, I would like to know how to carry on my attempts, or come up with a completely new one. Also in my attempts, I didn't really use the fact that $L$ is linear, so I wonder how to utilize it in my answer.
REPLY [2 votes]: For a concrete example, in $\mathfrak M_{2,2}$ the map of matrix transposition (easily seen to be linear) is not given by left multiplication by any fixed matrix$~A$. If it were, applying the requirement to the identity matrix gives the necessary condition $AI=I$ or $A=I$, but this only remaining candidate evidently does not work for non-symmetric matrices.
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\begin{document}
\author[A.~Gusakova]{Anna Gusakova}
\address{Anna Gusakova, Institute of Mathematical Stochastics, M\"unster University, Germany}
\email{gusakova@uni-muenster.de}
\author[E.~Spodarev]{Evgeny Spodarev}
\address{Evgeny Spodarev, Institute of Stochastics, Ulm University, Germany}
\email{evgeny.spodarev@uni-ulm.de}
\author[D.~Zaporozhets]{Dmitry Zaporozhets}
\address{Dmitry Zaporozhets, St.~Petersburg Department of Steklov Institute of Mathematics, Russia}
\email{zap1979@gmail.com}
\title[Intrinsic volumes of ellipsoids]{Intrinsic volumes of ellipsoids}
\keywords{Convex body, intrinsic volume, Querma{\ss}integral, Minkowski functional, support function, mixed discriminant, polar body}
\subjclass[2010]{52A20, 52A38, 52A39}
\thanks{The work of ES was supported by EIMI (Leonard Euler International Mathematical Institute, Saint Petersburg), grant 075-15-2019-1620 as of 08/11/2019. AG was supported by the DFG under Germany's Excellence Strategy EXC 2044 -- 390685587, \textit{Mathematics M\"unster: Dynamics - Geometry - Structure}.}.
\begin{abstract}
We deduce explicit formulae for the intrinsic volumes of an ellipsoid in $\mathbb R^d$, $d\ge 2$, in terms of elliptic integrals. Namely, for an ellipsoid ${\mathcal E}\subset \mathbb R^d$ with semiaxes $a_1,\ldots, a_d$ we show that
\begin{align*}
V_k({\mathcal E})=\kappa_k\sum_{i=1}^da_i^2s_{k-1}(a_1^2,\dots,a_{i-1}^2,a_{i+1}^2,\dots,a_d^2)\int_0^{\infty}{t^{k-1}\over(a_i^2t^2+1)\prod_{j=1}^d\sqrt{a_j^2t^2+1}}\,\rm{d}t
\end{align*}
for all $k=1,\ldots,d$,
where $s_{k-1}$ is the $(k-1)$-th elementary symmetric polynomial and $\kappa_k$ is the volume of the $k$-dimensional unit ball.
Some examples of the intrinsic volumes $V_k$ with low and high $k$ are given where our formulae look particularly simple. {As an application we derive new formulae for the expected $k$-dimensional volume of random $k$-simplex in an ellipsoid and random Gaussian $k$-simplex.}
\end{abstract}
\maketitle
\section{Introduction and main result}
For a non-empty convex compact set $K\subset \R^d$, consider its parallel set of radius $r>0$ defined as
$K+ r\B$, where $\B$ is the $d$-dimensional unit ball and the operation of {\it Minkowski addition} means the pointwise sum of two sets.
The well-known {\it Steiner formula} writes the volume $|\cdot|_d$ (i.e., the Lebesgue measure) of $K+r\B$ as a polynomial of degree $d$ in $r$:
\begin{align}\label{1800}
|K+ r\B|_d=\sum\limits_{k=0}^d \kappa_{d-k}V_{k}(K) r^{d-k}, \quad r\ge 0,
\end{align}
where $\kappa_k:=\pi^{k/2}/\Gamma\left(\frac k2+1\right)$ is the volume of the $k$-dimensional unit ball. The coefficients $V_k(K)$, $k=0,\ldots, d,$ above are called {\it intrinsic volumes} of $K$. They are normalized in a way that if $K$ is $k$--dimensional, then $V_k(K)$ coincides with the $k$-dimensional volume of $K$.
The intrinsic volumes as well as related Minkowski functionals
(or querma{\ss}integrals) and tensors of (poly)convex
bodies and sets with positive reach play an important role in convex geometry and in the applied fields, such as fractal and topological data analysis, compare e.g. \cite{MeckeStoyan00,MeckeStoyan02,Torq02,Planck2016,collaboration2015planck,SpoStrWin15}.
Although intrinsic volumes are basic and fundamental quantities of convex bodies, their computation is not a simple task even for those classical shapes like ellipsoids.
So far, only indirect results containing the computation of the surface area \cite{Tee05} as an Abelian integral and the expression of intrinsic volumes in terms of Gaussian determinants \cite{KZ12} are available. Thus, Theorem 1.1 of \cite{KZ12} states that for an arbitrary ellipsoid ${\mathcal E}\subset\R^d$ with semiaxes $a_1,\ldots, a_d$ we have
$$
V_k({\mathcal E})=\frac{(2\pi)^{k/2}}{k!} \E \sqrt{ \det(\langle A \xi_i, A\xi_j \rangle)_{i,j=1}^k},
$$
where $A=\mathrm{diag}(a_1,\ldots, a_d)$ and $\xi_1,\dots, \xi_k$ are i.i.d. standard Gaussian vectors in $\R^d$. Decomposing $\xi_i$ into independent spherical and radial parts as
$
\xi_i=\eta_i\cdot\|\xi_i\|
$
and noting that $\eta_1,\dots, \eta_d$ are i.i.d. random vectors in $\R^d$ uniformly distributed on the unit sphere $\s^{d-1}$ equipped with the Hausdorff measure $\sigma(\cdot)$ normalized by $\sigma(\s^{d-1})$ yields
\begin{align}\label{1113b}
V_k({\mathcal E})&=\frac{(2\pi)^{k/2}}{k!} \E\|\xi_1\|\dots \E\|\xi_k\|\cdot\E \sqrt{ \det(\langle A\eta_i, A\eta_j \rangle)_{i,j=1}^k}
\\\notag&
=\frac{1}{k!\kappa_{d-1}^k} \int_{(\s^{d-1})^k} \sqrt{\det(\langle A \bu_i, A\bu_j \rangle)_{i,j=1}^k}\, \sigma(\dd\bu_1)\ldots \sigma(\dd\bu_k),
\end{align}
where in the second equation we used that
\begin{align*}
\E \|\xi_i\|=\frac{\sqrt{2}\,\Gamma(\frac{d+1}{2})}{\Gamma(\frac{d}{2})},\quad \sigma(\s^{d-1})=\frac{2\pi^{d/2}}{\Gamma(\frac{d}{2})}.
\end{align*}
The later $k$-fold integration in~\eqref{1113b} makes an explicit computation of $V_k({\mathcal E})$ particularly complex.
The problem of deriving explicit formulae for $V_k({\mathcal E})$ is also deeply connected to the hypothesis that ellipsoids are uniquely determined (up to a rigid motion) by their intrinsic volumes. It is solved positively so far only in $d=2,3$ \cite{PetTar20} as well as for the dual volumes \cite{myroshnychenko2020unique}.
The main result of our paper gives the formula for $V_k(\mathcal E)$ in terms of one-dimensional elliptic integrals. Before formulating it, for a tuple $(b_1,\dots,b_n)$ denote by $s_m(b_1,\dots,b_n)$ (where $m\leq n$) the $m$-th elementary symmetric polynomial of $b_1,\dots,b_n$ defined as
\begin{align*}
s_{m}(b_1,\dots,b_n):=\sum_{1\leq i_1<\ldots<i_m\leq n}\prod_{j=1}^mb_{i_j}.
\end{align*}
Now we are ready to formulate the main theorem.
\begin{theorem}\label{0948}
For all $k\in\{1,\dots,d\}$, it holds
\begin{align*}
V_k({\mathcal E})=\kappa_k\sum_{i=1}^da_i^2s_{k-1}(a_1^2,\dots,a_{i-1}^2,a_{i+1}^2,\dots,a_d^2)\int_0^{\infty}{t^{k-1}\over(a_i^2t^2+1)\prod_{j=1}^d\sqrt{a_j^2t^2+1}}\dd t.
\end{align*}
\end{theorem}
To obtain this result, we derived an auxiliary formula expressing $V_k(\mathcal E)$ in terms of the integrals over the unit sphere which might be of independent interest.
\begin{theorem}\label{1406}
For all $k\in\{1,\ldots,d\}$, one gets
\begin{equation}\label{eq:IntrVolE_main}
V_k({\mathcal E})={1\over k\kappa_{d-k}}\sum_{i=1}^da_i^2s_{k-1}(a_1^2,\dots,a_{i-1}^2,a_{i+1}^2,\dots,a_d^2)\int\limits_{\s^{d-1}}{u_i^2\over h^{k}_{\mathcal E}(\bu)}\,\sigma(\dd\bu),
\end{equation}
where $\bu=(u_1,\dots,u_d)$ and $h_{\mathcal E}(\bu)=\sqrt{a^2_1u^2_1+\dots+a^2_du^2_d}$ is the support function of $\mathcal E$ (see Section~\ref{sect:Basic}).
\end{theorem}
{
Notice that the formulae from Theorem \ref{0948} and Theorem \ref{1406} are valid for $k=d$ as well. In this case we have $V_d({\mathcal E})=\kappa_da_1\cdot\ldots\cdot a_d$, which means that some of our formulae can presumably be further simplified} {for some $k$, see Section~\ref{sect:Examples}.}
Theorem~\ref{0948} readily follows from Theorem~\ref{1406} and the following proposition applied with $\alpha=2,\beta=k$. The idea of its proof is taken from~\cite[Lemma~2]{myroshnychenko2020unique}.
\begin{proposition}\label{0831}
For $i\in\{1,\dots, d\}$ and $\alpha, \beta\in\R^1$ such that $\beta>0$, $\alpha>d-\beta$ we have
\begin{align}\label{1112}
\int\limits_{\s^{d-1}}{u_i^\alpha \over h^{\beta}_{\mathcal E}(\bu)}\,\sigma(\dd\bu)
=
{4\pi^{(d-1)/2} \Gamma(\frac{\alpha+1}{2})\over \Gamma({d+\alpha-\beta\over 2})\Gamma({\beta\over 2})}\int_0^{\infty}{t^{\beta-1}\over(a_i^2t^2+1)^{\alpha/2}\prod_{j=1}^d\sqrt{a_j^2t^2+1}}\dd t.
\end{align}
\end{proposition}
The paper is organized as follows:
In Section \ref{sect:Examples}, examples of the intrinsic volumes $V_k({\mathcal E})$ of low ($k=1,2$) and high ($k=d-1,d-2$) orders are given. {In Section \ref{sect:ExpVol}, our Theorem \ref{0948} is applied to get a more explicit form of the expected $k$--dimensional volume of convex hulls of $k+1$ {idependent} random points uniformly distributed in an ellipsoid or having {an arbitrary centered} Gaussian distribution law. } Section~\ref{sect:Basic}
contains some preliminaries from convex and differential geometry which are used in the proofs of our results located in
Section \ref{1122}.
\section{Examples}\label{sect:Examples}
It is well known that $V_0({\mathcal E})=1$, $V_d({\mathcal E})=\kappa_d a_1\cdot\ldots\cdot a_d.$ From~\eqref{eq:IntrVolE_main} for $k=1$ we get
\begin{align}\label{1058}
V_1(\mathcal E)={1\over \kappa_{d-1}}\sum_{i=1}^da_i^2\int\limits_{\s^{d-1}}{u_i^2\over h_{\mathcal E}(\bu)}\,\sigma(\dd\bu)={1\over \kappa_{d-1}}\int\limits_{\s^{d-1}}\sqrt{\sum_{i=1}^d {a_i^2}{u_i^2}}\,\sigma(\dd\bu),
\end{align}
which agrees with~\eqref{1113b} and with the Kubota formula for the intrinsic volumes: in case $k=1$
it states that
\begin{align*}
V_1(K)={1\over \kappa_{d-1}}\int\limits_{\s^{d-1}}{ h_{K}(\bu)}\,\sigma(\dd\bu)
\end{align*}
for any convex body $K$.
On the other hand, from Theorem~\ref{0948} we obtain
\begin{align*}
V_1({\mathcal E})=2\sum_{i=1}^d\int_0^{\infty}{a_i^2\over(a_i^2t^2+1)\prod_{j=1}^d\sqrt{a_j^2t^2+1}}\dd t.
\end{align*}
Taking $k=2$ in~\eqref{eq:IntrVolE_main} gives
\begin{align*}
V_2({\mathcal E})&={1\over 2\kappa_{d-2}}\sum_{i=1}^d\bigg[a_i^2\bigg(\sum_{j=1}^da_j^2-a_i^2\bigg)\int\limits_{\s^{d-1}}{u_i^2\over h^{2}_{\mathcal E}(\bu)}\,\sigma(\dd\bu)\bigg]
\\
&=\pi\sum\limits_{i=1}^d a_i^2-\frac{\pi}{\sigma(\s^{d-1}) }
\int\limits_{\s^{d-1}}\frac{\sum\limits_{i=1}^d a_i^4u_i^2}{\sum\limits_{i=1}^d a_i^2u_i^2} \sigma(\dd\bu),
\end{align*}
where we used that $\sigma(\s^{d-1})=2\pi\kappa_{d-2}$. Applying Proposition~\ref{0831} with $\alpha=\beta=2$ leads to
\begin{align*}
V_2({\mathcal E})=\pi\sum\limits_{i=1}^d a_i^2-\pi\sum_{i=1}^d\int_0^{\infty}{a_i^4t\over(a_i^2t^2+1)\prod_{j=1}^d\sqrt{a_j^2t^2+1}}\dd t.
\end{align*}
Now let us use the following duality relation for ellipsoids which can be found in \cite[Prop. 4.8]{KZ16}:
\begin{equation}\label{eq:Dual}
V_k({\mathcal E})=\frac{\kappa_k}{\kappa_d \kappa_{d-k} }
V_d({\mathcal E})V_{d-k}({\mathcal E}^o),
\quad k=0,\ldots,d,
\end{equation}
where ${\mathcal E}^o$ is the ellipsoid dual to ${\mathcal E}$:
\begin{align*}
{\mathcal E}^o=\{ x\in\R^d:\langle x,y\rangle \le 1, \quad y\in {\mathcal E}\}.
\end{align*}
Using this relation and the fact that $\mathcal E^o$ has semiaxes $a_1^{-1},\dots,a_d^{-1}$ we can easily derive the formulae {for $V_{d-1}({\mathcal E})$ and $V_{d-2}({\mathcal E})$ from the formulae for $V_{1}({\mathcal E}^o)$ and $V_{2}({\mathcal E}^o)$, respectively}:
\begin{align*}
V_{d-1}(\mathcal E)&={a_1\dots a_d\over 2}\int\limits_{\s^{d-1}}\sqrt{\sum_{i=1}^d {a_i^{-2}}{u_i^2}}\,\sigma(\dd\bu)
\\
&=\kappa_{d-1}a_1^2\dots a_d^2\sum_{i=1}^d\int_0^{\infty}{\dd t \over(t^2+a_i^2)\prod_{j=1}^d\sqrt{t^2+a_j^2}}
\end{align*}
and
\begin{align*}
V_{d-2}({\mathcal E})&=\kappa_{d-2}a_1\dots a_d\sum\limits_{i=1}^d a_i^{-2}-\frac{a_1\dots a_d}{2\pi}
\int\limits_{\s^{d-1}}\frac{\sum\limits_{i=1}^d a_i^{-4}u_i^{2}}{\sum\limits_{i=1}^d a_i^{-2}u_i^2} \sigma(\dd\bu)
\\
&=\kappa_{d-2}a_1\dots a_d\sum\limits_{i=1}^d a_i^{-2}-
\kappa_{d-2}a_1^2\dots a_d^2
\sum_{i=1}^d\int_0^{\infty}{a_i^{-2}t\over(t^2+a_i^2)\prod_{j=1}^d
\sqrt{t^2+a_j^2}}\dd t.
\end{align*}
\section{Expected volumes of random simplices}\label{sect:ExpVol}
The intrinsic volumes of ellipsoids have a remarkable connection to the average volume of random $k$-simplex, which is formed as a convex hull of $k+1$ isotropic random points. More precisely, let $X_0,\ldots, X_k$, $1\leq k\leq d$, be random points in $\R^d$, whose joint distribution is invariant with respect to rotations.
We consider their convex hull $\conv(X_0,\ldots, X_k)$, which is a random (possibly degenerate) simplex, and its $k$-dimensional volume $|\conv(X_0,\ldots, X_k)|_k$, which is a well-defined random variable. In \cite[Corollary 1.1]{GGZ}, it was shown that for any non-degenerate matrix $A\in\R^{d\times d}$ we have
\begin{equation}\label{eq_simpl_ell}
\E |\conv(AX_0,\ldots, AX_k)|_k={\kappa_{d-k}\over {d\choose k}\kappa_d}V_k(\mathcal{E}_A)\cdot\E|\conv(X_0,\ldots, X_k)|_k,
\end{equation}
where $\mathcal{E}_A:=\{\bx\in\R^d\colon \bx^{\top}(A^\top A)^{-1}\bx\le 1\}$ is an ellipsoid.
There are two particularly interesting models which formula \eqref{eq_simpl_ell} can be applied to. For the first model consider a bit more general setup. Let $K\subset \R^d$ be a $d$-dimensional convex body, and let $Y_0,\ldots, Y_k$, $1\leq k\leq d$, be independent random points uniformly distributed over $K$. The classical problem to evaluate $M_k(K):=\E |\conv(Y_0,\ldots, Y_k)|_k$ for given $K$ goes back to Klee \cite{Klee69}. By now, not so many exact formulae for $M_k(K)$ have been obtained, and those mostly for $d=2$ and $d=3$. In dimension $2$, the exact formulae for $M_2(K)$ are available for triangles \cite{Reed74}, regular planar $n$-gones \cite{Buch84} and parallelograms \cite{Reed74}. In dimension $3$, the formulae for $M_3(K)$ have been derived in case of tetrahedron \cite{Mann94} and cube \cite{Zin03}. In arbitrary dimension $d$, $M_d(K)$ is known only for the ball \cite{Kin96} and, hence, due to affine invariance for any ellipsoid. The situation is more complicated if $k<d$ since $M_k(K)$ is not affine invariant anymore. For any $d$ and any $1\leq k\leq d$, the exact formula for $M_k(K)$ is known only in case of a ball \cite{Miles71} (see also \cite[Theorem 8.2.3]{SW08}). For the functional $M_1(K)$ describing the average distance between two points chosen uniformly in $K$, some formulae are known in planar case \cite{bG51, ZP, Baesel21, tS85}. A formula for the cube \cite{BBC07} (via the so-called {\it box integral}) is also available in dimension $d=3$. However, the box integral does not have a closed form expression for $d\ge 4$. Heinrich \cite{Hein14} has also obtained a representation of $M_1(\mathcal{E})$ for a $d$-dimensional ellipsoid $\mathcal{E}$ with semi-axes $0<a_1\leq \ldots \leq a_d$ in terms of the elliptic integral \eqref{1058}. Applying Theorem \ref{0948} to relation \eqref{eq_simpl_ell}, we are able to obtain more {explicit} formulae for $M_k(\mathcal{E})$ and all $1\leq k\leq d$:
\begin{theorem}
Let $Y_0,\ldots, Y_d$ be independent random points uniformly distributed in $\mathcal{E}$. Then for any $k\in \{1,\ldots, d\}$, the expected volume $M_k(\mathcal{E}):=\E|\conv(Y_0,\ldots, Y_k)|_k$ equals
\begin{align*}
M_k(\mathcal{E})&={1\over{ 2^k \Gamma({k\over 2}+1)}}{\Gamma\big({(d+1)(k+1)\over 2}+1\big)\over \Gamma \big({(d+1)(k+1)\over 2}+{1\over 2}\big)}\Big({\Gamma({d\over 2}+1)\over \Gamma({d+1\over 2}+1)}\Big)^{k+1}\sum_{i=1}^da_i^2s_{k-1}(a_1^2,\dots,a_{i-1}^2,a_{i+1}^2,\dots,a_d^2)\\
&\qquad\qquad\times\int_0^{\infty}{t^{k-1}\over(a_i^2t^2+1)\prod_{j=1}^d\sqrt{a_j^2t^2+1}}\dd t.
\end{align*}
\end{theorem}
\begin{proof}
The result follows directly from \eqref{eq_simpl_ell} applied to $X_0,\ldots, X_k$ distributed uniformly inside the unit ball $\mathbb{B}^d$ and an affine transformation mapping $\mathbb{B}^d$ to $\mathcal{E}$. Substituting the exact values for $\E|\conv(X_0,\ldots, X_k)|_k$ from \cite[Theorem 8.2.3]{SW08} (see also \cite[Corollary 1.5]{GGZ}) and the formula for $V_k(\mathcal{E})$ from Theorem \ref{0948} finishes the proof. It should be noted that in order to simplify the constant we have used the Legendre duplication formula $\Gamma(z)\Gamma(z+1/2)=2^{1-2z}\sqrt{\pi}\Gamma(2z)$.
\end{proof}
The second model is the so-called Gaussian random simplex. Let $X_0,\ldots, X_k$, $1\leq k\leq d,$ be independent standard Gaussian random vectors in $\R^d$. Their convex hull $\conv(X_0,\ldots, X_k)$ is almost surely a $k$-simplex. Its expected $k$-dimensional volume was calculated by Miles \cite[Equation (70)]{Miles71}. Using \eqref{eq_simpl_ell} the later result can be generalized to the convex hull of non-standard Gaussian random vectors.
\begin{theorem}
Let $Y_0,\ldots, Y_d$ be independent Gaussian centered random vectors in $\R^d$ with non--degenerate covariance matrix $\Sigma$. Let $\lambda_1,\ldots, \lambda_d>0$ be the eigenvalues of $\Sigma$. For any $k\in \{1,\ldots, d\}$ we have
\begin{align*}
\E|\conv(Y_0,\ldots, Y_k)|_k&={\sqrt{k+1}\over\Gamma({k\over 2}+1)2^{k/2}}\sum_{i=1}^d\lambda_is_{k-1}(\lambda_1,\dots,\lambda_{i-1},\lambda_{i+1},\dots,\lambda_d)\\
&\qquad\qquad\times\int_0^{\infty}{t^{k-1}\over(\lambda_it^2+1)\prod_{j=1}^d\sqrt{\lambda_jt^2+1}}\dd t.
\end{align*}
\end{theorem}
\begin{proof}
Let $X_0,\ldots, X_k$ be independent standard Gaussian random vectors. Consider an affine transformation with matrix $\Sigma^{1/2}$ (which is well defined since $\Sigma$ is symmetric and positive definite). It is clear that {$\Sigma^{1/2}X_i$}, $0\leq i\leq k$, are independent centered Gaussian random vectors with covariance matrix $\Sigma$. Thus,
{
first applying~\eqref{eq_simpl_ell} and then~\cite[Equation (70)]{Miles71} we have
\begin{align*}
\E&|\conv(Y_0,\ldots, Y_k)|_k=\E|\conv(\Sigma^{1/2}X_0,\ldots, \Sigma^{1/2}X_k)|_k
\\
&={\kappa_{d-k}\over {d\choose k}\kappa_d}V_k(\mathcal{E}_\Sigma)\cdot\E|\conv(X_0,\ldots, X_k)|_k
={\kappa_{d-k}\over {d\choose k}\kappa_d}V_k(\mathcal{E}_\Sigma)\cdot{2^{k/2}\sqrt{k+1}\over k!}{\Gamma({d+1\over 2})\over \Gamma({d-k+1\over 2})},
\end{align*}
}where $\mathcal{E}_\Sigma:=\{\bx\in\R^d\colon \bx^{\top}\Sigma^{-1}\bx\leq 1\}$ {is an ellipsoid}
with semi-axes $a_i=\sqrt{\lambda_i}>0$, $1\leq i\leq d$. Thus, by Theorem \ref{0948} we conclude that
\begin{align*}
\E|\conv(Y_0,\ldots, Y_k)&|_k={(d-k)!\over d!}{2^{k/2}\sqrt{k+1}\over \Gamma({k\over 2}+1)}{\Gamma({d\over 2}+1)\Gamma({d+1\over 2})\over \Gamma({d-k\over 2}+1)\Gamma({d-k+1\over 2})}\\
&\times\sum_{i=1}^d\lambda_is_{k-1}(\lambda_1,\dots,\lambda_{i-1},\lambda_{i+1},\dots,\lambda_d)\int_0^{\infty}{t^{k-1}\over(\lambda_it^2+1)\prod_{j=1}^d\sqrt{\lambda_jt^2+1}}\dd t.
\end{align*}
Applying the Legendre duplication formula $\Gamma(z)\Gamma(z+1/2)=2^{1-2z}\sqrt{\pi}\Gamma(2z)$ twice finishes the proof.
\end{proof}
\section{Basic facts from convex geometry}\label{sect:Basic}
The {\it mixed volumes} $V(K_1,\ldots, K_d)$ of convex bodies $K_1,\ldots, K_d \subset \R^d$
are introduced via the \emph{Minkowski theorem} as the coefficients of the polynomial expansion
\begin{align}\label{1801}
|r_1 K_1+\ldots+ r_d K_d|_d=\sum\limits_{k_1,\ldots,k_d=1}^d r_{k_1} \ldots r_{k_d} V(K_{k_1},\ldots, K_{k_d}), \quad r_1,\ldots, r_d\ge 0,
\end{align}
which is the generalisation of the Steiner formula, see~\eqref{1800}. They are non-negative and symmetric with respect to all permutations of the indices of $K_1,\ldots, K_d$, cf. e.g. \cite[Theorem 5.1.6]{rS14}. From~\eqref{1800} and~\eqref{1801} we immediately have
\begin{align}\label{1805}
V_k(K)=\frac 1{\kappa_{d-k}}{d\choose k} W_{d-k}(K), \quad k=0,\ldots,d,
\end{align}
where
\begin{align}\label{2026}
W_k(K)= V\Big(\underbrace{K, \ldots, K}_{k}, \underbrace{\B,\ldots,\B}_{d-k} \Big)
\end{align}
are called the \emph{querma{ss}integrals} of $K$, see more on them e.g. in \cite[Chapter 4]{rS14}.
Let $h_{K}(\bx)=\sup_{\by\in K} \langle \bx,\by \rangle$, $\bx\in\R^d$ be the support function of a convex body $K$.
For an ellipsoid
\begin{align}\label{1138}
{\mathcal E}=\bigg\{ \bx=(x_1,\ldots, x_d)\in\R^d: \sum_{i=1}^d \frac{x_i^2}{a_i^2}\le 1 \bigg\}\quad \text{we have}\quad
h_{{\mathcal E}}(\bx)=\sqrt{\sum_{i=1}^d a_i^2 x_i^2}.
\end{align}
The main ingredient of the proof of Theorem~\ref{1406} is the formula expressing the mixed volume of convex bodies in terms of their support functions. To formulate it, for arbitrary matrices $Q_1,\dots,Q_{d-1}\in\R^{(d-1)\times (d-1)}$ denote by $D(Q_1,\dots,Q_{d-1})$ their \emph{mixed discriminant} defined as (see, e.g.,~\cite{Bapat89})
\begin{align}\label{0950}
D_{d-1}(Q_1,\dots, Q_{d-1})=\frac{1}{(d-1)!}\sum_{\tau\in S_{d-1}}\det Q(\tau),
\end{align}
where $S_{d-1}$ is the symmetric group on $d-1$ elements, $\tau=(\tau(1),\dots,\tau(d-1))$, and $Q(\tau)$ is a matrix whose $i$-th column coincides with the $i$-th column of $Q_{\tau(i)}$.
Essentially, we compose a matrix using the columns of $Q_1,\dots, Q_{d-1}$ with pairwise different indices according to a random permutation (uniformly chosen from $S_{d-1}$), and then take its expected determinant. Given an arbitrary matrix $Q\in\R^{d\times d}$ denote by $Q^{i}$ its principal minor produced by the deleting the $i$-th row and the $i$-th column from $Q$.
Now suppose that $K_1,\dots,K_d$ are some convex bodies having $C^2$--smooth boundaries with positive Gaussian curvatures at each point. For a convex body $K\subset\R^d$ denote by $H_{K}(\bx)\in\R^{d\times d}$ the Hessian matrix of its support function $h_K$ at point $\bx$.
Then it is known (see e.g. \cite[p. 1061]{leicht93}) that
\begin{align*}
V\left(K_1, \ldots, K_d\right)=\frac{1}{d}
\sum_{i=1}^d\int\limits_{\s^{d-1}} h_{K_1}(\bu) D_{d-1}\left(H_{K_2}^{i}(\bu),\dots,H_{K_{d}}^{i}(\bu)\right)
\,\sigma(\dd\bu).
\end{align*}
Combining this with~\eqref{1805} and~\eqref{2026} for $k\geq 1$ leads to
\begin{align}\label{2038}
V_k(K)&=\frac{{d\choose k}}{d\kappa_{d-k}}
\sum_{i=1}^d\int\limits_{\s^{d-1}}h_{K}(\bu) D_{d-1}\Big(\underbrace{H_{K}^{i}(\bu),\dots,H_{K}^{i}(\bu)}_{k-1}, \underbrace{H_{{\B}}^{i}(\bu),\dots,H_{{\B}}^{i}(\bu)}_{d-k} \Big)\,\sigma(\dd\bu).
\end{align}
\section{Proofs}\label{1122}
\subsection{Proof of Theorem~\ref{1406}}
We are going to apply~\eqref{2038} with $K=\mathcal E$. To this end, let us first calculate the mixed discriminant under the integral in the right-hand side of~\eqref{2038}.
It follows by straightforward double differentiation of the right-hand side of~\eqref{1138} that $$H_{\mathcal E}(\bu)= h_{{\mathcal E}}^{-3}(\bu) A(\bu),$$ where
\begin{align*}
A(\bu)&=
h_{\mathcal E}^2(\bu) \diag(a_1^2,\dots,a_d^2)-(a_1^2u_1,\ldots,a_{d}^2u_{d})^{\top}(a_1^2u_1,\ldots,a_{d}^2u_{d})
\\[10pt]
&=\left(\begin{array}{cccc}
a_1^2 (h_{\mathcal E}^2(\bu) - a_1^2 u_1^2) & -a_1^2 a_2^2 u_1 u_2 & \ldots & -a_1^2 a_d^2 u_1 u_d \\
- a_2^2 a_1^2u_2 u_1 & a_2^2 (h_{\mathcal E}^2(\bu) - a_2^2 u_2^2) & \ldots & -a_2^2 a_d^2 u_2 u_d \\
\ldots & \ldots & \ldots & \ldots \\
- a_d^2 a_1^2 u_d u_1 & - a_d^2a_{2}^2 u_d u_{2}& \ldots & a_d^2 (h_{\mathcal E}^2(\bu) - a_d^2 u_d^2 ) \end{array}
\right).
\end{align*}
In particular, recalling that $\bu\in\s^{d-1}$ we have
\begin{align*}
H_{{\B}}(\bu)&=\|\bu\|^{-3}(I_d-\bu\bu^\top)=I_d-\bu\bu^\top
\\[10pt]
&=\left(\begin{array}{cccc}
1- u_1^2 & - u_1 u_2 & \ldots & - u_1 u_{d} \\
- u_1 u_2 & 1-u_2^2 & \ldots & - u_2 u_{d} \\
\ldots & \ldots & \ldots & \ldots \\
- u_1 u_{d} & - u_{2} u_{d} & \ldots & 1- u_{d}^2
\end{array}
\right).
\end{align*}
Using this and the linearity of the mixed discriminants we arrive at
\begin{align}\label{0456}
D_{d-1}\Big(&\underbrace{H_{\mathcal E}^{i}(\bu),\dots,H_{\mathcal E}^{i}(\bu)}_{k-1}, \underbrace{H_{{\B}}^{i}(\bu),\dots,H_{{\B}}^{i}(\bu)}_{d-k} \Big)
\\\notag
&=h_{\mathcal E}^{3-3k}(\bu) D_{d-1}\Big(\underbrace{A^i(\bu),\dots,A^i(\bu)}_{k-1}, \underbrace{H^i_{{\B}}(\bu),\dots,H^i_{{\B}}(\bu)}_{d-k} \Big).
\end{align}
By~\eqref{0950} we have
\begin{align}\label{1605}
D_{d-1}\Big(\underbrace{A^i(\bu),\dots,A^i(\bu)}_{k-1}, \underbrace{H^i_{{\B}}(\bu),\dots,H^i_{{\B}}(\bu)}_{d-k} \Big)=\frac{1}{(d-1)!}\sum_{\tau\in S_{d-1}} q^{(i)}_\tau(\bu),
\end{align}
where $q^{(i)}_\tau(\bu)$ is the determinant of the matrix composed of $k-1$ columns of $A^i(\bu)$ and $d-k$ columns of $H^i_{{\B}}(\bu)$ chosen according to the permutation $\tau$. Since for $i=1,\dots,d$ and $\tau\in S_{d-1}$ all $q^{(i)}_\tau(\bu)$ look similar, it is enough to compute only one of them: the rest will be derived by changing the variables. Consider for simplicity the one corresponding to $i=d$ and to the identical permutation $q^{(d)}_{\mathbf{id}}(\bu)$. We have
\begin{align*}
q^{(d)}_{\mathbf{id}}(\bu)=\det\left(\begin{array}{ccc|ccc}
a_1^2 (h_{\mathcal E}^2(\bu) - a_1^2 u_1^2) & \ldots & -a_1^2 a_{k-1}^2 u_1 u_{k-1} & -u_1u_k&\dots &-u_1u_{d-1}\\
\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\
- a_{k-1}^2a_1^2 u_{k-1}u_1 & \ldots& a_{k-1}^2 (h_{\mathcal E}^2(\bu) - a_{k-1}^2 u_{k-1}^2) &-u_{k-1}u_k& \ldots & - u_{k-1} u_{d-1} \\[6pt]
\hline \rule{0pt}{1\normalbaselineskip}
- a_k^2a_1^2 u_ku_1 &\ldots&-a_k^2 a_{k-1}^2 u_ku_{k-1}&1-u_k^2&\ldots&
- u_ku_{d-1}
\\
\ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\
-a_{d-1}^2a_1^2 u_{d-1} u_1 & \ldots&- a_{d-1}^2 a_{k-1}^2 u_{d-1}u_{k-1} &-u_{d-1}u_k& \ldots & 1-u_{d-1}^2
\end{array}
\right).
\end{align*}
Taking the factor $a_j^2$ out of the $j$-th column, $j=1,\dots,k-1,$ leads to
\begin{equation} \label{1945}
q^{(d)}_{\mathbf{id}}(\bu)=\prod_{j=1}^{k-1}a^2_j\cdot \det\left( \begin{array}{c|c}
C_1 & C_2 \\\hline
C_3 & C_4
\end{array}\right),
\end{equation}
where
\begin{equation}\label{0623}
\begin{aligned}
C_1&=h_{\mathcal E}^2(\bu) I_{k-1}-(a_1^2u_1,\ldots,a_{k-1}^2u_{k-1})^{\top}(u_1,\ldots,u_{k-1})
= h_{\mathcal E}^2(\bu)I_{k-1}-\bv_1^\top \bu_1,
\\%\notag
C_2&=-(u_1,\ldots,u_{k-1})^{\top}(u_k,\ldots,u_{d-1})
=-\bu_1^\top \bu_2,
\\%\notag
C_3&=-(a_k^2u_k,\ldots,a_{d-1}^2u_{d-1})^{\top}(u_1,\ldots,u_{k-1})
=-\bv_2^\top \bu_1,
\\%\notag
C_4&=I_{d-k}-(u_k,\ldots,u_{d-1})^{\top}(u_k,\ldots,u_{d-1})
=I_{d-k}-\bu_2^\top \bu_2.
\end{aligned}
\end{equation}
Here, we used the notation
\begin{align*}
\bu_1=(u_1,\ldots,u_{k-1}),\quad &\bu_2=(u_k,\ldots,u_{d-1}),\\
\bv_1=(a_1^2u_1,\ldots,a_{k-1}^2u_{k-1}),\quad &\bv_2=(a_k^2u_k,\ldots,a_{d-1}^2u_{d-1}).
\end{align*}
Matrix $C_4$ is invertible with
\begin{equation}\label{eq:invC4}
C_4^{-1}=I_{d-k}+{\bu_2^\top \bu_2\over 1-\|\bu_2\|^2}.
\end{equation}
Using the well-known formula for the determinant of a block matrix (see
e.g.~\cite[Lemma~5]{LM17}), we obtain
\begin{equation}\label{eq:detBlock}
q^{(d)}_{\mathbf{id}}(\bu)=\prod_{j=1}^{k-1}a^2_j\cdot\det(C_1-C_2 C_4^{-1} C_3) \det C_4.
\end{equation}
First of all, using the Weinstein-Aronszajn identity
\begin{equation}\label{eq:WAidentity}
\det(I_m+M_1M_2)=\det(I_n+M_2M_1),\quad M_1\in\R^{m\times n}, M_2\in\R^{n\times m},
\end{equation}
with $M_1=(1-\|\bu_2\|^2)^{-1}\bu_2^\top , M_2=\bu_2$
we obtain from~\eqref{eq:invC4} that
\begin{equation}\label{eq:detC4}
\det C_4=1-\|\bu_2\|^2.
\end{equation}
Further using~\eqref{0623} and \eqref{eq:invC4}, we calculate
\begin{align*}
C_2 C_4^{-1} C_3&=\bu_1^\top \bu_2\left(I_{d-k}+{\bu_2^\top \bu_2\over 1-\|\bu_2\|^2}\right)\bv_2^\top \bu_1
\\&=\left(1+{\|\bu_2\|^2\over 1-\|\bu_2\|^2}\right)\bu_1^\top(\bu_2\bv_2^\top) \bu_1
=\lambda\,\bu_1^\top \bu_1,
\end{align*}
where
\begin{align*}
\lambda:={\bu_2\bv_2^\top\over 1-\|\bu_2\|^2}\in\R.
\end{align*}
Therefore,
\begin{align*}
\det(C_1-C_2 C_4^{-1} C_3)&= \det\left(
h_{\mathcal E}^2(\bu)I_{k-1}-\bv_1^\top \bu_1-\lambda\,\bu_1^\top \bu_1\right)
\\
&=h_{\mathcal E}^{2k-2}(\bu)\det\left(I_{k-1}-h_{\mathcal E}^{-2}(\bu)(\bv_1^\top+\lambda\bu_1^\top)\bu_1\right).
\end{align*}
Again applying the Weinstein-Aronszajn identity with $M_1=\bv_1^\top+\lambda\bu_1^\top$ and $M_2=\bu_1$ and noting that $h_{\mathcal E}^{2}(\bu)=\bu_1\bv_1^\top+\bu_2\bv_2^\top+a_d^2u_d^2$ leads to
\begin{align*}
\det(C_1-C_2 C_4^{-1} C_3) &=h_{\mathcal E}^{2dk-2}(\bu)\left(1-h_{\mathcal E}^{-2}(\bu)\bu_1(\bv_1^\top+\lambda\bu_1^\top)\right)
\\&=h_{\mathcal E}^{2k-4}(\bu)\left(h_{\mathcal E}^{2}(\bu)-\bu_1\bv_1^\top-\lambda\bu_1\bu_1^\top\right)
\\&=h_{\mathcal E}^{2k-4}(\bu)\left(\bu_2\bv_2^\top+a_d^2u_d^2-{\|\bu_1\|^2\over 1-\|\bu_2\|^2}(\bu_2\bv_2^\top)\right)
\\&=h_{\mathcal E}^{2k-4}(\bu)\frac{u_d^2\bu_2\bv_2^\top+a_d^2u_d^2(u_d^2+\|\bu_1\|^2)}{1-\|\bu_2\|^2}.
\end{align*}
Finally, combining this with~\eqref{eq:detBlock} and~\eqref{eq:detC4} gives
\begin{align}\label{0821}
q^{(d)}_{\mathbf{id}}(\bu)&=\prod_{j=1}^{k-1}a^2_jh_{\mathcal E}^{2k-4}(\bu)\left(u_d^2\bu_2\bv_2^\top+a_d^2u_d^2(u_d^2+\|\bu_1\|^2)\right)
\\\notag
&=u_d^2h_{\mathcal E}^{2k-4}(\bu)\prod_{j=1}^{k-1}a^2_j\left(\sum_{j=k}^{d}a_j^2u_j^2+a_d^2\sum_{j=1}^{k-1}u_j^2\right)
\\\notag
&=u_d^2h_{\mathcal E}^{2k-4}(\bu)\prod_{j=1}^{k-1}a^2_j\left(h_{\mathcal E}(\bu)-\sum_{j=1}^{k-1}a_j^2u_j^2+a_d^2\sum_{j=1}^{k-1}u_j^2\right).
\end{align}
Now consider $q^{(d)}_\tau$ for an arbitrary permutation $\tau\in S_{d-1}$. By the simultaneous rearrangement of the rows and columns (thus non-changing the determinant) it is possible to bring the matrix corresponding to $q^{(d)}_\tau$ to the same block form as in~\eqref{1945}, and repeating the above reasoning we get
\begin{align*}
q^{(d)}_\tau(\bu)=u_d^2h_{\mathcal E}^{2k-4}(\bu)\prod_{j=1}^{k-1}a^2_{\tau(j)}\left(h_{\mathcal E}(\bu)-\sum_{j=1}^{k-1}a^2_{\tau(j)}u_{\tau(j)}^2+a_d^2\sum_{j=1}^{k-1}u_{\tau(j)}^2\right).
\end{align*}
Summing this up over $\tau\in S_{d-1}$ leads to
\begin{align*}
\sum_{\tau\in S_{d-1}}q^{(d)}_\tau(\bu)&=(k-1)!(d-k)!u_d^2h_{\mathcal E}^{2k-4}(\bu) \sum_{\substack{I\subset J_d \\ |I| =k-1}}
\left[\prod_{j\in I}a^2_j\left(a_d^2u_d^2+\sum_{j\in J_d\setminus I}a^2_ju_j^2+a_d^2\sum_{j\in I}u_j^2\right)\right],
\end{align*}
where we used the notation $J_i:=\{1,\dots,d\}\setminus \{i\}$ for $i=1,\dots,d$.
Similarly, for arbitrary $i$ we have
\begin{align}\label{1023}
\sum_{\tau\in S_{d-1}} \!\!\! q_\tau^{(i)}(\bu)&=(k-1)!(d-k)!h_{\mathcal E}^{2k-4}(\bu)
u_i^2 \!\!\!\! \sum_{\substack{I\subset J_i \\ |I| =k-1}} \!\!\!
\left[\prod_{j\in I}a^2_j\left(a_i^2u_i^2+\sum_{j\in J_i\setminus I}a^2_ju_j^2+a_i^2\sum_{j\in I}u_j^2\right) \!\! \right] \!\! .
\end{align}
Let us sum this up over $i$ dealing with each summand in the inner brackets separately. To simplify the notation, we will write
\begin{align*}
\ba^i:=(a_1^2,\dots,a_{i-1}^2,a_{i+1}^2,\dots,a_d^2), &\quad 1\leq i\leq d,
\\
\ba^{i,j}:=(a_1^2,\dots,a_{i-1}^2,a_{i+1}^2,\dots,a_{j-1}^2,a_{j+1}^2,\dots,a_d^2),&\quad 1\leq i<j\leq d,
\end{align*}
and $\ba^{i,j}=\ba^{j,i}$ for $i>j$.
Firstly,
\begin{align}\label{1022}
\sum_{i=1}^d u_i^2
\sum_{\substack{I\subset J_i \\ |I| =k-1}}
\left[\prod_{j\in I}a^2_j\cdot (a^2_iu_i^2)\right]
=
\sum_{i=1}^da^2_iu_i^4\cdot s_{k-1}(\ba^i).
\end{align}
Secondly, by exchanging the summation order we get
\begin{align*}
\sum_{i=1}^d u_i^2
\sum_{\substack{I\subset J_i \\ |I| =k-1}}
\left[\prod_{j\in I}a^2_j\sum_{j\in J_i\setminus I}
a^2_ju_j^2\right]&=\sum_{i=1}^d u_i^2
\sum_{j\in J_i}a^2_ju_j^2\sum_{\substack{I\subset J_{i,j} \\ |I| =k-1}}\prod_{j\in I}a^2_j\\
&=
\sum_{\substack{i,j=1 \\ i\ne j}}^da^2_iu_i^2u_j^2\cdot s_{k-1}(\ba^{i,j}),
\end{align*}
where {$J_{i,j}:=\{1,\ldots,d\}\setminus\{i,j\}$ for any $i\neq j$}.
Finally, again by exchanging the summation order
\begin{align*}
\sum_{i=1}^d u_i^2
\sum_{\substack{I\subset J_i \\ |I| =k-1}}
\left[\prod_{j\in I}a^2_j\left(a_i^2\sum_{j\in I}u_j^2\right)\right]
&=
\sum_{i=1}^d a_i^2u_i^2\sum_{j\in J_i}a_j^2u_j^2
\sum_{\substack{I\subset J_{i,j} \\ |I| =k-2}}
\prod_{j\in I}a^2_j
\\&=
\sum_{\substack{i,j=1 \\ i\ne j}}^da^2_iu_i^2a_j^2u_j^2\cdot s_{k-2}(\ba^{i,j}).
\end{align*}
Further we note that
\begin{align*}
s_{k-1}(\ba^i)=s_{k-1}(\ba^{i,j})+a_j^2s_{k-2}(\ba^{i,j})
\end{align*}
and thus
\begin{align*}
\sum_{i=1}^d u_i^2
\sum_{\substack{I\subset J_i \\ |I| =k-1}}
\left[\prod_{j\in I}a^2_j\sum_{j\in J_i\setminus I}
a^2_ju_j^2\right]
+
\sum_{i=1}^d u_i^2
\sum_{\substack{I\subset J_i \\ |I| =k-1}}
\left[\prod_{j\in I}a^2_j\left(a_i^2\sum_{j\in I}u_j^2\right)\right]
\\
=\sum_{\substack{i,j=1 \\ i\ne j}}^da^2_iu_i^2u_j^2\cdot (s_{k-1}(\ba^{i,j})+a_j^2s_{k-2}(\ba^{i,j}))
=
\sum_{\substack{i,j=1 \\ i\ne j}}^da^2_iu_i^2u_j^2\cdot s_{k-1}(\ba^i).
\end{align*}
Combining this with~\eqref{1023} and~\eqref{1022} we arrive at
\begin{align*}
\sum_{i=1}^d\sum_{\tau\in S_{d-1}}q_\tau^{(i)}(\bu)
&=
(k-1)!(d-k)!h_{\mathcal E}^{2k-4}(\bu)
\left(\sum_{i=1}^da^2_iu_i^4\cdot s_{k-1}(\ba^i)+\sum_{\substack{i,j=1 \\ i\ne j}}^da^2_iu_i^2u_j^2\cdot s_{k-1}(\ba^i)\right)
\\&=
(k-1)!(d-k)!h_{\mathcal E}^{2k-4}(\bu)\sum_{i=1}^da^2_iu_i^2s_{k-1}(\ba^i),
\end{align*}
where in the last step we used that $\sum_{i=1}^d u_i^2=1$. Recalling~\eqref{2038}--\eqref{1605} concludes the proof.
\subsection{Proof of Proposition~\ref{0831}}
The key ingredient of the proof is the following simple observation: for $c,\gamma>0$ we have
\begin{align}\label{1113}
\int_{0}^{\infty}t^{\gamma-1} e^{-ct^2}\dd t
=
\frac{c^{-\gamma/2}}2\int_{0}^{\infty}(ct^2)^{\gamma/2-1} e^{-ct^2}\dd (ct^2)
=
\frac{c^{-\gamma/2}}2\Gamma\bigg(\frac{\gamma}2\bigg).
\end{align}
Passing to the spherical coordinates and having in mind that $h_{\mathcal E}(\cdot)$ is homogeneous of degree 1 we can write
\begin{align}\label{1137}
\int\limits_{\R^d}|x_i|^\alpha h_{\mathcal E}^{-\beta}(\bx)e^{-\|\bx\|^2} \dd\bx&=\int\limits_{\s^{d-1}}u_i^\alpha h_{\mathcal E}^{-\beta}(\bu)\int_0^\infty r^{d+\alpha-\beta-1} e^{-r^2}\dd r\sigma(\dd\bu)
\\\notag
&=\frac12\Gamma\bigg({d+\alpha-\beta\over 2}\bigg)\int\limits_{\s^{d-1}}u_i^\alpha h_{\mathcal E}^{-\beta}(\bu)\sigma(\dd\bu),
\end{align}
where in the last step we used~\eqref{1113} with $c=1, \gamma=d+\alpha-\beta$. Now using~\eqref{1137} and applying~\eqref{1113} with $c=h_{\mathcal E}^{2}(\bx), \gamma=\beta$ leads to
\begin{align}\label{1507}
\int\limits_{\s^{d-1}}u_i^\alpha &h^{-\beta}_{\mathcal E}(\bu)\,\sigma(\dd\bu)={2\over \Gamma({d+\alpha-\beta\over 2})}\int\limits_{\R^d}|x_i|^\alpha h_{\mathcal E}^{-\beta}(\bx)e^{-\|\bx\|^2} \dd\bx
\\\notag
&={2\over \Gamma({d+\alpha-\beta\over 2})}
\int\limits_{\R^d}
\bigg[{2\over \Gamma({\beta\over 2})}\int_{0}^{\infty}t^{\beta-1}e^{-h_{\mathcal E}^2(\bx)t^2}\dd t\bigg]|x_i|^\alpha e^{-\|\bx\|^2} \dd\bx
\\\notag
&={4\over \Gamma({d+\alpha-\beta\over 2})\Gamma({\beta\over 2})}\int_{0}^{\infty}t^{\beta-1}\bigg[\int_{-\infty}^{\infty}|x_i|^\alpha e^{-(a_i^2t^2+1)x_i^2}\dd x_i\prod_{j\neq i}\int_{-\infty}^{\infty}e^{-(a_j^2t^2+1)x_j^2}\dd x_j\bigg]\dd t,
\end{align}
where in the last step we used that $h_{\mathcal E}^2(\bx)=a^2_1x^2_1+\dots+a^2_dx^2_d$ and the Fubini theorem. Applying~\eqref{1113} to the inner integrals gives
\begin{align*}
\int_{-\infty}^{\infty}|x_i|^\alpha e^{-(a_i^2t^2+1)x_i^2}\dd x_i
=
2\int_{0}^{\infty}x_i^\alpha e^{-(a_i^2t^2+1)x_i^2}\dd x_i
=
{\Gamma(\frac{\alpha+1}{2})\over (a_i^2t^2+1)^{(\alpha+1)/2}}
\end{align*}
and
\begin{align*}
\int_{-\infty}^{\infty}e^{-(a_j^2t^2+1)x_j^2}\dd x_j
=
2\int_{0}^{\infty} e^{-(a_j^2t^2+1)x_j^2}\dd x_j
=
{\sqrt\pi \over (a_j^2t^2+1)^{1/2}},
\end{align*}
which together with~\eqref{1507} concludes the proof.
\subsection{Proof of Theorem~\ref{0948}}
Theorem~\ref{0948} follows from Theorem~\ref{1406} and Proposition~\ref{0831} applied with $\alpha=2,\beta=k$, and the observation that for such $\alpha, \beta$ we have
\begin{align*}
{1\over k\kappa_{d-k}}\cdot
{4\pi^{(d-1)/2} \Gamma(\frac{3}{2})\over \Gamma({d+2-k\over 2})\Gamma({k\over 2})}=\kappa_k.
\end{align*}
\bibliographystyle{plain}
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In these challenging economic times, it is either encouraging or unsettling to observe the way that English and European clubs can still continue to spend vast amounts of money on new players, and even explore new sources of potential revenue to help them push on to that elusive next success.
However, such actions have come at a price. Arguably—particularly in the Premiership—the league has never been less competitive at the top, and many clubs exist in real financial peril.
TV money keeps many clubs afloat—but for how long will BSkyB be able to offer such outlandish sums for coverage that is becoming less and less engaging?
Looking ahead, then, it seems highly probable that something about the infrastructure of football will have to change. Perhaps when the time comes it can learn something from the sports across the Atlantic—those that seem to operate on a wholly more stable financial, ethical, and entertaining footing.
Below are just five potential US-influenced changes that could improve the sport:
1) The Salary Cap
In many ways, European football—and the Premier League in particular—has already learned a lot from the NFL about the art of promotion and expansion. While the much-discussed “39th Game” is yet to happen (unlike gridiron’s International Series), the reality is that it will surely materialise sooner rather than later.
In the meantime, perhaps football would do well to take another leaf out of America’s game, and adopt a salary cap.
The reasoning—especially in such financially insecure times—is sound. A salary cap would level the playing field amongst European clubs, and hopefully end the ever-increasing monopoly on trophies that just a handful of clubs possess.
Even better, limiting the salary cap to a percentage of the club’s income would ensure that all teams run on sound economic principles, and would prevent the unfair advantage created by billionaire foreign owners and their “soft” loans.
With all this in mind, arguably baseball’s version of the salary cap would fit best. Applying a luxury tax on clubs that break the salary cap would allow the teams with greater incomes (e.g. Manchester United) to spend some of their money gained, but the fine they would then have to pay to the league (which would then be re-distributed amongst the cap-abiding clubs) would ensure that the playing field remained as level as possible, allowing the smaller clubs to still attract top players.
How the transfer fee mechanism would fit into this system is more difficult to assess, but if clubs are limited by their wage bills then it is unlikely they will be able to complete the same number of multi-million pound transfer deals without the requisite multi-million pound player wages.
A redistribution of footballing wealth would be inevitable.
2) The Distinction Between “Personal” and “Technical” Fouls
A doff of the cap to Paul Gardner of World Soccer Magazine (a man I have not always agreed with) on this idea. In recent times, the rise in dissent and cynical fouls in football has gone unnoticed by very few people. But for the referees, doing something to counter it has proved more difficult.
Following basketball’s lead by creating a distinction between types of fouls (technical and personal) could provide the answer. At the moment, a player who takes his shirt off while celebrating a goal (technical) is likely to receive equal punishment to a player who goes in late and high on an opposition striker (personal).
This is quite clearly wrong. So why not create a distinction?
For example, it could be possible to allow players four technical fouls (dissent etc.) before dismissal, but just two (depending on severity) for personal fouls. That way, perhaps players will concentrate less on shackling their opponents, and more on getting their own creative juices flowing.
3) The Video Challenge
It is commonplace in tennis. It arguably first found favour in cricket. It is now an integral part of American football.
The video replay has permeated almost all the mainstream sports that could benefit from it, and FIFA’s reluctance to embrace the modern technology is a stance that should change.
Following the system as it works in the NFL, allowing club managers to challenge a referee’s decision (or lack thereof) at the next available dead-ball situation would ensure that questionable judgments (Stuart Atwell’s “ghost goal”, anyone?) could be overruled by a TV-assisted fourth official.
In the modern game, where managers’ careers can hinge on the decision of the man in black, they at least deserve the right of appeal.
Limiting the number of challenges to three a game, and removing a substitution for every failed challenge, would ensure the privilege was not abused. And maybe, just maybe, it would finally silence those managers so happy to blame the referees for their side’s shortcomings.
4) The Commissioner
The Football Association is hardly the most popular of organisations with fans and clubs alike—which is no mean feat. Years of poor decision making and woeful demonstrations of weakness has eroded its power to the point where, in all honesty, the bigger clubs seem to be able to run riot.
So bad is the situation that Sir Alex Ferguson was last month named the most influential person in British sport by a prominent daily newspaper. What does that say about the integrity and impartiality of the Premiership?
In this respect, restructuring the organisation to create a Commissioner could be beneficial.
In American sports, the Commissioner is the powerholder in the sport he heads. What he says, goes. Cross him at your peril.
He hands out fines, and oversees the sports administration. Given enough power, he would quickly be able to ensure that the likes of Ferguson and Rafa Benitez refrain from lambasting the FA’s fixture scheduling to the press, amongst other frequent sideswipes that demean the game’s organisation.
More importantly, he would be able to move the Premiership in the best direction for all concerned—not just for the benefit of the “Big Four” that currently seem to instigate change.
5) The Hall of Fame
Believe it or not, Britain already has a football Hall of Fame, at the country’s National Football Museum. Unfortunately, it is a completely different animal to the likes seen in North America.
For a start, David Beckham is already in the British version of the HOF, despite still being an active player in both domestic and international football. His legacy is yet to be cemented, and yet he has already been immortalised. Interesting.
Furthermore, the HOF was only introduced in 2002, and yet already has over enshrined over 40 members.
Devaluing the award—or making up for lost time? Perhaps the jury is still out.
Whatever the case, an end of season HOF celebration—like the kind seen in the NFL and other top American sports—would be a great chance for the sport to recognise those that have brought the sport to where it is today. Inducting four or five retired players in front of a TV audience every year would give both the players and the fans another chance to recognise the achievements of yesteryear—and perhaps force those young fans who idolise the latest superstar winger to realise that, hey, the old boys could play a bit too.
Yes, it may be harder to implement than in America, but it is somewhat ironic that a country that is so often criticised for its lack of history is so much better at remembering and lauding its sporting past than a country that prides itself on such tradition.
And Finally, One From Rugby Union…
6) Reward Goalscoring
As the number of goals scored in tournaments and leagues continues to decrease as coaches place an ever greater premium on defence, inevitably the quality and excitement of football has also begun to decline, especially—and most disappointingly—in the biggest and most significant matches of all.
A system of bonus points, as the Guinness Premiership so gainfully employs for its egg-chasers, could prove the perfect antidote. If a point was awarded for every goal scored—meaning a 3-2 defeat would be more beneficial than a 0-0 draw—then maybe, just maybe, we might see teams start with more attacking intent.
And wouldn’t that be great?
So there you have it, just a few potential changes that perhaps the EPL, and football in general, might do well to adopt. As it is, the bigwigs probably only have eyes for two—the controversial game abroad, and the removal of relegation (preserving the money-making ability of a privileged 20). But would both changes really be in the best interests of the game?
Any other American sporting elements that football could adopt? Let me know in the comments below!
| 276,168
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\begin{document}
\title{Continuous-time performance limitations for overshoot and related tracking measures }
\author{R. B. Wenczel and R. D. Hill\thanks{
Department of Mathematical and Geospatial Sciences, RMIT University, GPO Box 2476V,
Melbourne, Victoria, 3001, Australia. e-mail: robert.wenczel@rmit.edu.au, r.hill@rmit.edu.au }}
\maketitle
\begin{abstract}
A dual formulation for the problem of determining
absolute performance limitations on overshoot, undershoot, maximum amplitude
and fluctuation minimisation for continuous-time feedback systems is
constructed. Determining, for example, the minimum possible overshoot attainable by all
possible stabilising controllers is an optimisation task that cannot be
expressed as a minimum-norm problem. It is this fact, coupled with the
continuous-time rather than discrete-time formulation, that makes these problems
challenging.
We extend previous results to include more general reference functions, and derive new results (in continuous time)
on the influence of pole/zero locations on achievable time-domain performance.
\end{abstract}
\section{Introduction}\label{sec:cont:intro}
In this paper we study the problem of finding limits on the performance of the error-response performance, for a specific input, for lumped continuous-time {\scriptsize SISO} systems. We are trying to find the best possible tracking performance achievable by a lumped (or rational) {\scriptsize BIBO}--stabilising feedback controller. The theory to be presented is applicable to a large class of performance measures of practical significance, including overshoot for example. The simultaneous imposition of hard bounds on the output, in conjunction with overshoot minimisation, can also be handled, allowing consideration of the inevitable trade-off between rise-time and overshoot performance. These issues have been examined in a discrete-time setting in \cite{HEHW} etc. The investigation of time-domain performance limitations is mathematically more challenging in continuous time than in discrete time. In the continuous-time case it is harder to derive conditions under which the performance limit for rational controllers is the same as for non-rational controllers ({\em rational approximability\/}).
The usual approach to these questions, is to express this performance--limit in the form of an optimization problem (to be computed via duality methods), over an {\em extended\/} ambient signal--space, usually a standard {\em dual\/} Banach space, to ensure
{\em existence of optimal elements\/}
and adequate duality properties. In general,
these optimal elements will be non-rational, and hence suffer from implementability issues. It is then of importance to find if optimal performance may be approached using rational controllers.
Since these nonrational elements may be of doubtful practical utility, one may well abandon the search for optima, and consider only the class of rational controllers
in the optimization. This will permit more flexibility in the choice of the ambient Banach space since it no longer need be a dual space.
In earlier works this underlying space was chosen to be $L^\infty$ (dual of $L^1$), but this choice of space does not, of itself, enforce the zero--steady--state--error condition.
We shall incorporate this condition by using
the space $C_0$
(of continuous functions of time that tend to zero at infinity) as our choice---it does the trick, but it is not a dual, so that existence of optimal elements is not assured. This choice of space is dictated by the continuity properties of the basic performance measures we study here.
This is the approach followed in this paper. Moreover, to side-step the rational--approximation issue, we consider
directly the set of error signals due to the {\em rational\/} stabilizing controllers, and
the optimization over its closure. Under the conditions we assume in this paper, passage to this closure has no effect on the limit-of-performance.
This clarifies the relation between the limit-of-performance and rational approximation, but requires an explicit characterization of the closure of the feasible set of
error--signals. However, it is found that the closure itself has pleasant form, well-suited to application of the duality theory.
That is, by exploiting the structure of this set we have that
rational approximability holds by definition.
Further, we consider a general class of performance functionals and set up a general
duality framework for the analysis of such problems (in the style of \cite{HEHW} for the case of discrete time). These functionals include as special cases the well-known classical criteria of overshoot, undershoot and others. This treatment also allows a general fixed input (not just a step). The classical criteria just mentioned, are continuous with respect to the supremum--norm
on the signal space---this fact motivating our choice of $C_0$ as the ambient space.
In the literature to date the optimization is performed over error signals in the bigger space $L_\infty$, with attendant consideration of rational approximability.
In \cite{wang-sznaier-94, wang-sznaier-96, wang-sznaier-97} the set of candidate controllers is expanded, moreover, to include those that may not be {\scriptsize BIBO}--stabilising, and the limiting error is
unconstrained. Asymptotic stability and zero steady state error are then enforced through selection of suitable output weighting.
Use of the space $L_\infty$ to formulate the primal has the advantage that the optimal solution for the primal problem is guaranteed to exist for the $L_\infty$ norm minimisation problem.
For the more general cost functions considered in this paper, however, it is clearly not to be expected that an optimal solution will in general exist, neither in $L_\infty$, nor in $C_0$.
We provide a
derivation of a dual formulation covering a wide range of problems, extending known results. The continuous time $L_\infty$ norm minimisation problem for a fixed input was first considered in \cite{DP2},
but rational approximability was not addressed. Miller \cite{Miller} gave a rational approximation result for response to a step reference input. The construction of rational suboptimal controllers for the continuous time $L_\infty$ norm minimisation problem has been considered in \cite{wang-sznaier-96}, \cite{wang-sznaier-94} and \cite{Halp}. Yoon \cite{MGY} and \cite{YK} extended the class of optimisation criteria, by considering convex combinations of overshoot and undershoot (with a step-input) and in these latter papers is found the first application of conjugate--duality techniques to continuous-time control.
The constraints for the dual formulation presented here can be interpreted as arising from an open--loop dynamic system. These structural insights are exploited to derive new results on the influence of plant pole/zero locations, or rise--time constraints on achievable performance, giving results of identical form to those obtained for discrete time in \cite{HEHW}.
The results presented here are applicable for general reference inputs (\cite{MGY, YK} assumes step-input) and to systems with more general exogenous inputs, for example a fixed disturbance entering at the plant output. For definiteness and simplicity our set-up is the tracking problem shown in Figure~\ref{fig:2}.
An important design consideration is the extent to which performance objectives are in conflict. It may not be possible, for example, to find a single controller that reduces both the $L_{\infty}$--norm of the error, and the negative--error (overshoot), in response to a step, to close to their fundamental limits. Furthermore, the extent to which there is conflict depends on the location of the unstable poles and non-minimum-phase zeros of the plant. The dual provides such information. Another trade-off important in practice is that between rise--time and transient performance measures such as overshoot and undershoot. The answer to this question also depends on the location of the poles and zeros of the plant, and again the interpretation of the dual as an open-loop dynamic system provides insight and new results. Also in this paper we extend known results on the minimization of the ``maximum-peak to minimum-peak" value of the error response, that is the difference between the maximum and minimum values of the error response; we term this quantity {\em fluctuation\/}. For some applications fluctuation minimization is of more concern than $L_{\infty}$--norm minimization. For some plants there then arises an unavoidable trade-off; either a small $L_{\infty}$--norm of the error response, or a low fluctuation, must be sacrificed. To what extent they are in conflict depends on the location of the poles and zeros of the plant, and again it is the dual that provides answers.
\section{Mathematical Preliminaries}\label{sec:cont:mathprelim}
\subsection{basic notation}
We shall write $\R_+$ for the real interval $[0,+\infty)$. $\R[s]$ and $\R(s)$ denote respectively, the
spaces of real polynomials, and real--rational functions, in the complex variable $s$. The set of all proper stable
members of $\R(s)$ (i.e.\ those with no pole in the closed complex right-half-plane nor at infinity) is denoted by ${\mathcal S}$.
The Laplace transform of a function $f$, will be written as $\hat{f}$. We let $L^p(\R_+)$ ($1\le p \le \infty$) stand for
the space of Lebesgue $p$-integrable functions on $\R_+$. The space of continuous functions $\varphi:\R_+\to \R$ for which
$\lim_{t\to\infty}\varphi(t)=0$, endowed with the supremum norm $\| \varphi \|_\infty:= \sup_{t\ge 0}|\varphi(t)|$, shall be denoted by
$C_0(\R_+)$. For any $\alpha\in\R$, write $C_{0,\alpha}(\R_+)$ for the subset consisting of those elements $\varphi\in C_0(\R_+)$ for
which $\varphi(0)=\alpha$. For a subset $A$ of a space $X$, the indicator function $\iota_A$ of $A$ is defined on $X$ by
$$ \iota_A(x):= \left\{ \begin{array}{cl}
0 & \text{ if } x\in A \\
+\infty & \text{ if } x\notin A
\end{array}\right. \,. $$
If $f:X\to \R\cup \{+\infty\}$, then $\func{dom}f$ denotes the set $\{x\in X\mid f(x)<+\infty\}$.
\subsection{Fenchel duality theorem}
\begin{defn} Let $X$ be a topological linear space.
For any $f:X\to \R\cup\{ +\infty\}$, the (Young--Fenchel)
conjugate $f^*:X^*\to \R\cup\{ \pm\infty\}$ is defined by:
$$ \!\!\!\!\!\text{for all }x^*\in X^*,\qquad
f^*(x^*):=\sup_{x\in X}\,(\langle x,x^*\rangle-f(x))
$$
\end{defn}
It follows that $f^*$ is convex and weak${}^*$
lower--semicontinuous.
We shall require a Fenchel duality theorem in the following form (see
\cite[Theorem 18(a) and Example $11'$]{Rock} for a more general formulation)
\begin{prop}\label{prop:BL}
Let $f:X\to \R\cup\{ +\infty\} $ and $g:\R^n \to\R\cup\{ +\infty\}$ be
convex, with $X$ a locally-convex topological vector space.
Let $A:X\to\R^n$ be bounded linear. Assume also that $g$ is
finite-valued at some point in $A(\func{int dom}f)$. Then,
$$
\inf\{f(x)+g(Ax)\mid x\in X\}=\max\{-f^*(A^T\xi)-g^*(-\xi)\mid
\xi\in\R^n\}
$$
\end{prop}
In our applications, we will take $g=\iota_{\{b\}}$, the indicator function for a singleton $\{b\}$,
and $f$ will be of the form $f_0+\iota_T$ for a finite-valued $f_0$ and a convex set $T$.
\subsubsection{duality for a space of continuous functions}
The application of the Fenchel theorem shall require a characterisation of the dual of the Banach space
of signals under consideration. In this paper, the ambient space of (error) signals will be $C_0(\R_+)$,
which has a well-known dual space.
Indeed, the dual of $C_0(\R_+)$
is isometrically isomorphic with the space $ \mathbf{M}(\R_+)$ of all
regular finite signed Borel measures
$\mu$ on $\R_+$,
with the variation norm $\|\mu\|:=|\mu|(\R_+)$ (For formal
definitions of these properties, and a statement of the duality
for $C_0(\R_+)$ see, for example, \cite{Cohn}). The action of
$\mu\in\mathbf{M}(\R_+)$ as a dual functional on $C_0(\R_+)$, is indicated by the pairing
$$ \la \mu,e\ra = \int_{\scriptsize \R_+}\!\! e\,d\mu $$
for $e\in C_0(\R_+)$. (Note also that this expression is well--defined
whenever $e$ is bounded and (Borel-)measurable, since $\mu$ is a finite measure.)
\section{Problem Formulation}\label{sec:cont:prelim}
\subsection{primal feasible set}\label{sect:primfeasset:cont}
Consider the set ${\mathcal F}$ of all error sequences achievable with a rational stabilising controller (for the plant $P$)
in the standard one-degree-of-freedom feedback configuration, see Fig~\ref{fig:2}. Here the plant $P$ is a scalar, linear, proper,
finite-dimensional system. Mathematically, this condition on $P$ can be expressed by the requirement that $P\in\R(s)$ has a zero
at $\infty$ of order $\theta_p\ge 0$ (in the sense that the degree of the denominator is the sum of
$\theta_p$ with the degree of the numerator). The reference input $w$ is assumed to be an ordinary function
$w :\mathbf{\R_+}\to\mathbf{\R }$ (no delta-function terms) with rational Laplace transform.
A typical linear stabilising controller is denoted $C$. The plant output is $y$, and the error signal is $e =w-y$
\begin{figure}[h]
\begin{center}
\begin{tabular}{c}
\epsfxsize= 9 cm
\epsffile{sisoC.ps}
\end{tabular}
\end{center}
\caption{\label{fig:2} A closed--loop control system
}
\end{figure}
Let $\hat{n}$,
$\hat{d}$ in $\mathbf{{\mathcal S}}$ be a coprime factorization
for $P$. Then $\hat{d}(\infty):=\lim_{|s|\to\infty} \hat{d}(s)$ is
nonzero and finite, and $\hat{n}$ has a zero of order $\theta_p$
at $\infty$, so that
$$ \lim_{|s|\to\infty}s^{\theta_p-1}\hat{n}(s)=0\text{ and }
\lim_{|s|\to\infty}s^{\theta_p}\hat{n}(s)\text{ is nonzero and
finite}\,.$$ Such a factorization for $P$ is readily found; place
$P=q/r$ for coprime polynomials $q$, $r$, with $\func{deg}
r=\func{deg}q+\theta_p$. Then
$\hat{n}:=q/(\cdot+1)^{\func{{\scriptsize deg}} r}$ and
$\hat{d}:=r /(\cdot+1)^{\func{{\scriptsize deg}} r}$ both are in
$\mathbf{{\mathcal S}}$, and by \cite[Chapter 2, fact 20]{Vidy1}, are coprime
(in $\mathbf{{\mathcal S}}$).
\smallskip
Given the reference-input $w$, then $e$ is a closed--loop error--signal (for some
stabilizing controller $C$ for $P$) if
and only if
$\hat{e}=\hat{w}\hat{d}(\hat{v}-\hat{q}\hat{n})=\hat{w}-\hat{w}\hat{n}(\hat{x}+\hat{q}\hat{d})$
for some $\hat{q}\in \mathbf{{\mathcal S}}$, where $\hat{x}$,
$\hat{v}$ in $\mathbf{{\mathcal S}}$ arise from the coprimeness of
$\hat{n}$ and $\hat{d}$, and satisfy
$$ \hat{x}\hat{n}+\hat{v}\hat{d}=1 \text{ in } \mathbf{{\mathcal S}}\,.$$
From the assumption on $w$, its Laplace transform then satisfies
$$\lim_{\func{{\scriptsize Re}}s\to +\infty}\hat{w}(s)=0\,,$$ and
$\hat{w}(s)$ has a zero of order $\theta_w\ge 1$ at infinity, implying
that $\lim_{|s|\to\infty}s^{\theta_w-1}\hat{w}(s)=0$ and that
$\lim_{|s|\to\infty}s^{\theta_w}\hat{w}(s)$ is nonzero and finite.
Hence, for $e$ as above, $\hat{e}(s)$ has a zero of order at
least $\theta_w$ at $\infty$, and from
$\hat{w}-\hat{e}=\hat{w}\hat{n}(\hat{x}+\hat{q}\hat{d})$ follows
that $\hat{e}-\hat{w}$ has a zero of order at least $\theta_p
+\theta_w$ at $\infty$.
Define our feasible set of possible error signals by the affine
set (where the star $*$ denotes convolution of functions)
$$ {\mathcal F}:= \{ e\in C_0(\R_+)\mid e=w*d*(v-q*n) \text{ for some }
q \text{ such that } \hat{q}\in \mathbf{{\mathcal S}}\}\,.$$
(We are constraining $e$ to be in $C_0(\R_+)$, as a criterion for zero steady--
state tracking error). Then, for $\underline{\theta}:=(\theta_w,
\theta_p)$,
\begin{align*}
{\mathcal F}\subseteq X_{\underline{\theta}} :=\{ e\in C_0(\R_+)\mid
\hat{e} &\text{ rational, and has a zero }\\
&\text{of order at least $\theta_w$ at infinity, and } \\
&\text{$\hat{e}-\hat{w}$ has zero at $\infty$ of order at least
}\theta_p+\theta_w \}.
\end{align*}
Note that $X_{\underline{\theta}}$ is an affine subspace of
$C_0(\R_+)$. By considering the partial--fraction expansions of
$\hat{e}$, it follows that any $e\in L^1(\R_+) \cup C_0(\R_+)$ with
rational Laplace transform may be expressed in the form (for all
$t\ge 0$)
$$ e(t)=\func{Re} \sum_i c_i t^{k_i} e^{\lambda_i t} $$
where the above sum consists of a finite number of terms, and $c_i\in\C$,
$\func{Re}\lambda_i<0$, $k_i\ge 0$.
\smallskip
Introduce the notation for right half--planes
\begin{align*}
\C_+ &:= \{s\in\C\mid \func{Re}s>0\}\\
\overline{\C_+}&:= \{s\in\C\mid \func{Re}s\ge 0\}\\
\overline{\C_+}_e&:=\overline{\C_+}\cup \{\infty\}\,.
\end{align*}
We shall now follow an analogue of the developments of
\cite{HEHW}.
Let $P$
have poles $p_1,\ldots, p_m$ and zeros $z_1,\ldots, z_n$ in the
right--hand plane $\overline{\C_+}$. Also, let the
reference--input $\hat{w}$ have zeros $v_1,\ldots, v_l$ in
$\overline{\C_+}$. It is assumed that none of these lie on the
imaginary axis $\{s\mid\func{Re}s=0\}$. Further, all these
poles/zeros are assumed to be mutually distinct, and simple.
Place
\begin{equation}
M_{\underline{\theta}}=\left\{e\in
X_{\underline{\theta}}\,\Bigg{\vert}
\begin{array}{ll}
\hat{e}(z_i) =\hat{w}(z_i) & i=1,2,\ldots ,n \\
\hat{e}(p_i) =\,0 & i=1,2,\ldots ,m\\
\hat{e}(v_i) =\,0 & i=1,2,\ldots ,l\\
\end{array}
\right\} .
\end{equation}
\begin{lem} \label{lem:F2:cont}
With the assumptions as above, ${\mathcal
F}=M_{\underline{\theta}}$.
\end{lem}
\begin{Proof}
If $e\in{\mathcal F}$, then $e\in X_{\underline{\theta}}$ as
argued earlier. The constraints on $\hat{e}$ at the $z_i$, $p_i$,
$v_i$ follow as in the discrete--time case. Conversely, if $e\in
M_{\underline{\theta}}$, form $\hat{q}:=
\hat{v}/\hat{n}-\hat{e}/(\hat{n}\hat{w}\hat{d})=\frac{1}{\hat{d}}\left(
\frac{\hat{w}-\hat{e}}{\hat{w}\hat{n}}-\hat{x}\right)$. The only
possible $\overline{\C_+}_e$--poles of $\hat{q}$ are at the
$\overline{\C_+}$--poles/zeros of $P$, the
$\overline{\C_+}$--zeros of $\hat{w}$, and at infinity. The
constraints at each of the $\overline{\C_+}$--points ensure that
$\hat{e}$ has no poles there. It remains to check the behavior at
$\infty$. Now, $\hat{d}(\infty)\ne 0$, and the prescribed behavior
of $\hat{e}$ and $\hat{e}-\hat{w}$ means that
$\frac{\hat{w}-\hat{e}}{\hat{w}\hat{n}}=O(\frac{s^{-\theta_p-\theta_w}}{s^{-\theta_w}s^{-\theta_p}})=O(1)$
for $|s|\to\infty$. Thus $\hat{q}(\infty)$ is finite, and since
$\hat{q}$ has no poles in $\overline{\C_+}$, it is in
$\mathbf{{\mathcal S}}$. It now follows that $e\in{\mathcal F}$.
\end{Proof}
As in the discrete--time case, whenever $z_i$ and $z_j$ form a
conjugate pair, we retain only one of these in the list of
constraints characterizing $M_{\theta}$. We make a similar
reduction for the $p_i$ and $v_i$ also. This entails no loss of
information from $M_{\theta}$ (since $\hat{e}(\bar{z})=
\overline{\hat{e}(z)}$ for any $z$, and any real--valued function
$e$.)
\smallskip
For each interpolation point $z_j=x_j+iy_j$ (recall $x_j>0$), define
$$ \mathbf{a}_j(t):= e^{-x_j t}\cos y_jt\,,\,\, \mathbf{a}_{j+1}(t):=
e^{-x_j t}\sin y_jt \,.$$
If $ \mathbf{b}_j$ and $ \mathbf{c}_j$ are also defined similarly with
respect to the
$p_j$, $v_j$ respectively,
$M_{\theta}$ takes the form
\begin{equation}
M_{\underline{\theta}}=\left\{e\in X_{\underline{\theta}}\,
\Bigg{|}
\begin{tabular}{l}
$\langle e,\mathbf{a}_i\rangle =\langle w,\mathbf{a}_i\rangle
\quad i=1,2,\ldots ,n$ \\
$ \langle e,\mathbf{b}_i\rangle =\,\,0\qquad\quad \! i=1,2,\ldots ,m$ \\
$ \langle e,\mathbf{c}_i\rangle =\,\,0\qquad\quad i=1,2,\ldots ,l$
\end{tabular}
\right\} , \label{setMtheta}
\end{equation}
where for functions $u\in L^{\infty}$ and $v\in L^1$, $\la u, v \ra:=\int_0^\infty
v(t)u(t)dt$.
Let $A:C_0(\R_+)\to \R^{m+n+l}$ be defined by
\begin{equation}\label{eq:A}
Ae:= (\langle e,\mathbf{a}_1\rangle,\ldots,\langle e,\mathbf{a}_n\rangle,
\langle e,\mathbf{b}_1\rangle,\ldots,\langle
e,\mathbf{b}_m\rangle, \langle
e,\mathbf{c}_1\rangle,\ldots,\langle e,\mathbf{c}_l\rangle)^T\in
\R^{m+n+l}\,,
\end{equation}
and let
\begin{equation}\label{eq:b}
b:=(\langle w,\mathbf{a}_1\rangle,
\ldots,\langle w, \mathbf{a}_n\rangle,
0,0,\ldots)^T\in\R^{m+n+l}\,.
\end{equation}
Then, $M_{\underline{\theta}}$ has the form $\{e\in X_{\underline{\theta}}\mid Ae=b\}$.
Given a performance measure
$f:X_{\underline{\theta}}\to \R\cup\{+\infty \}$, our question is to evaluate
$$\text{{\bf (P)}: }\quad\inf_{e\in M_{\underline{\theta}}} f(e)\,,$$
which represents a
theoretical limit of performance for "physically realizable"
controllers (in the sense of having rational Laplace transform).
This is the central object of study in this paper.
\smallskip
\subsection{primal time-domain performance objectives}\label{sectsometdperf
:cont}
We shall principally consider functionals $f_0$ on $C_0(\R_+)$ of the form:
\begin{eqnarray*}
f_{ma}(e) \!\!\!&:&\!\!\!=\sup_t |e(t) |=\| e\| _\infty \quad\quad\text{(maximum amplitude)}\\
f_{pos}(e) \!\!\!&:&\!\!\!=\sup_t\left[ (e(t))_{+}\right] \qquad\qquad\text{(positive error)}\\
f_{os}(e) \!\!\!&:&\!\!\!=\sup_t\left[ (-e(t))_{+}\right] \quad\qquad\,\,\text{(overshoot)}\\
f_{fl}(e) \!\!\!&:&\!\!\!=\frac12\left[ \sup_t(e(t))-\inf_t(e(t))\right] \quad\text{(fluctuation)}\\
f_{us}(e)\!\!\! &:&\!\!\!=\sup_t\left[ (e(t)-w(t))_{+}\right] \qquad\quad\,\,\text{(undershoot)}.
\end{eqnarray*}
where for real $\lambda$, we define $\lambda_+:=\max(\lambda,0)$ and
$\lambda_-:=\min(\lambda,0)$. It is straightforward to verify that these functionals are all convex, and also
continuous (in fact Lipschitz, with constant 1) with respect to the $\|\cdot\|_\infty$--norm on $C_0(\R_+)$.
As in discrete--time, we have $f_{os}+f_{pos}=2f_{fl}$ and
that
\begin{equation}
\left.
\begin{array}{c}
f_{pos} \\
f_{os}
\end{array}
\right\} \leq f_{ma}\leq 2f_{fl}\leq 2f_{ma}.
\label{ineq1:cont}
\end{equation}
Also, for $e\in C_0(\R_+)$,
\begin{equation}
f_{fl}(e)=\min_{\xi \in {\scriptsize \R} }\sup_t |e(t)-\xi |\, .
\label{argmin:cont}
\end{equation}
whose proof follows by trivial modification of the proof of its
discrete counterpart in \cite{HEHW}. Later, we will consider some simple time-domain
constraints, so require treatment of functionals of form $f_0+\iota_T$ for
appropriate choices of sets $T$ representing these additional conditions.
\begin{remark}
Recall that a {\em rational\/} error--signal $e(\cdot)$ satisfies (the zero--steady--state condition) $\lim_{t\to\infty}e(t)=0$ iff $e\in C_0(\R_+)$
iff $e\in L^p(\R_+)$ ($p\ne\infty$). Thus, we {\em could\/} have chosen $L^p$ ($p\ne\infty$) as our ambient space---
however, the functionals $f_0$ above are not continuous relative to these $L^p$--norms, negating the usefulness of this choice.
\end{remark}
\subsection{initial statement of duality for our primal problem}\label{sec:dual:statement}
The analysis of problem {\bf (P)} will proceed by recasting it in dual form via Proposition~\ref{prop:BL}, as is given
below in Proposition~\ref{thmfench:cont}. As preparation, we require the following characterisation of the
closure $\overline{X_{\underline{\theta}}}$ (proved in the Appendix).
\begin{prop}\label{prop:summary}\,\,\,\quad $\overline{X_{(\theta_w,\theta_p)}}=
\left\{
\begin{array}{ll}C_{0,\alpha}(\R_+)\,\,(\text{with }\alpha=w(0+)) & \,\mbox{ if }\,\,\theta_p>0,\,\theta_w=1\\
C_{0}(\R_+) & \,\mbox{ if }\,\,\theta_p=0,\,\theta_w=1\\
C_{0,0}(\R_+) & \,\mbox{ if }\,\,\theta_w>1
\end{array}
\right.$
(where $C_{0,\alpha}(\R_+)\subseteq C_0(\R_+)$ denotes the set of functions $\varphi$ for which $\varphi(0)=\alpha$).
\end{prop}
The formulation of the dual problem will also require the following spaces:
$$U:=\func{span}[ \mathbf{a}_1,\ldots , \mathbf{a}_n],\
V:=\func{span}[ \mathbf{b}_1,\ldots , \mathbf{b}_m],\
W:=\func{span}[ \mathbf{c}_1,\ldots , \mathbf{c}_l],\,$$
From the blanket assumptions on poles and zeros, we have
$U,\ V,\ W$ contained in $C_0(\R_+)$. Also, from the resulting integrability of the $\mathbf{a}_n$, $\mathbf{b}_n$, $\mathbf{c}_n$ , each
$e^*\in U\oplus
V\oplus W$ defines a measure $\mu\in \mathbf{M}(\R_+)$ by
\begin{equation}\label{eq:meas}
\mu (E)=\int_E e^* \qquad\text{ for Borel sets $E$}\,.
\end{equation}
Thus $U\oplus V\oplus W $ may also be considered as a subspace of
$X^*={\mathbf M}(\R_+)$.
The next basic duality result is the foundation for all subsequent analysis.
\begin{prop}
\label{thmfench:cont}
Let $T\subseteq C_0(\R_+)$ be convex, $f_0:C_0(\R_+)\to \R$ convex and continuous, with
$b\in A(X_{\underline{\theta}}\cap\func{int} T)$, where $A$ and $b$ are given in (\ref{eq:A}) and
(\ref{eq:b}).
If $\theta_w + \theta_p=1$ (i.e.\ $\overline{X_{\underline{\theta}}}=C_0(\R_+)$ via Prop~\ref{prop:summary}) then
\begin{equation}\label{eq:duality:1}
\inf_{ T\cap X_{\underline{\theta}}\cap A^{-1}b }\ f_0 =\max_{
{\scriptsize
\begin{array}{c}
\mu\in U\oplus V\oplus W \\
\end{array}
} } \left[ \langle \func{Proj}_U(\mu),w\rangle -(f_0 +\iota_T)^{*}(\mu)\right]
,
\end{equation}
and if instead, $\theta_w + \theta_p>1$ (so that by Prop~\ref{prop:summary}, $\overline{X_{\underline{\theta}}}=C_{0,\alpha}(\R_+)$ with $\alpha=w(0+)$) then
\begin{equation}\label{eq:duality:2}
\inf_{ T\cap X_{\underline{\theta}}\cap A^{-1}b }\ f_0 =\max_{
{\scriptsize
\begin{array}{c}
\mu\in U\oplus V\oplus W \\
\end{array}
} } \left[ \langle \func{Proj}_U(\mu),w\rangle +(f_0 +\iota_T)^{\#}(\mu)\right]
,
\end{equation}
where: $\func{Proj}_U (\cdot)$ denotes the natural projector from
$U\oplus V\oplus W$ onto $U$; and, for any $\mu$, and any $f$,
\begin{equation}\label{eq:duality:3}
f^{\#}(\mu):= \max_ {\lambda\in{\scriptsize
\R}}\,[\alpha\lambda-f^*(\mu+\lambda\delta)]
\end{equation}
with $\delta$ denoting the Dirac measure at $0\in\R_+$.
\end{prop}
\begin{Proof}
Now, $b\in A(X_{\underline{\theta}}\cap\func{int} T)$ implies, by convexity, that
$\overline{T\cap X_{\underline{\theta}}\cap A^{-1}b}\supseteq T\cap \overline{ X_{\underline{\theta}}\cap A^{-1}b}$.
But $\overline{ X_{\underline{\theta}}\cap A^{-1}b}=\overline{ X_{\underline{\theta}}}\cap A^{-1}b $ by a simple argument using
the finite-codimensionality of $A^{-1}b$. Thus $\overline{T\cap X_{\underline{\theta}}\cap A^{-1}b}\supseteq
T\cap \overline{ X_{\underline{\theta}}}\cap A^{-1}b$ and hence by continuity of $f_0$,
$$
\inf_{ T\cap X_{\underline{\theta}}\cap A^{-1}b }\ f_0 =\inf_{ T\cap \overline{X_{\underline{\theta}}}\cap A^{-1}b }\ f_0 \,.
$$
We now apply Proposition~\ref{prop:BL} to the latter minimization in $\overline{X_{\underline{\theta}}}$. Place $f:=f_0+\iota_T$
and $g:=\iota_{\{b\}}$.
If $\overline{X_{\underline{\theta}}}=C_0$, and taking $X=C_0$, then $b\in A(\func{int dom}f)$ and after some simple manipulations of
the resulting dual, we obtain the form
(\ref{eq:duality:1}).
If instead, $\overline{X_{\underline{\theta}}}=C_{0,\alpha}$, note that
$ A^{-1}b \cap\overline{ X_{\underline{\theta}}}=\widetilde{A}^{-1}\widetilde{b}$, where
$\widetilde{A}e:=(Ae,e(0))^T$ (for all $e\in C_0(\R_+)$) and $\widetilde{b}:=(b,\alpha)^T$.
We also have $\widetilde{b}\in \widetilde{A}(\func{int}T)$, so we may apply the duality to
$\inf_{ T\cap C_{0,\alpha}\cap A^{-1}b }\ f_0 =\inf_{ T\cap \widetilde{A}^{-1}\widetilde{b} }\ f_0 $,
with $X=C_0$ again, with the new $\widetilde{A}$, $\widetilde{b}$, which eventually yields the dual in
(\ref{eq:duality:2}).
\end{Proof}
\section{Dual Formulation}
\label{dualformulation:cont}
The dual characterisation will be completed by evaluation of the conjugate functionals
appearing on the right-hand-side of equations (\ref{eq:duality:1}) and (\ref{eq:duality:2}) above.
We now study the forms of the conjugates for a general class of
cost--functions which includes those of interest in this paper.
We assume the objective functionals $f:C_0(\R_+)\to \R\cup\{+\infty\}$ take the form
\begin{equation}\label{eq:fF}
f(e)=\sup_{t\ge 0}F_t(e(t))
\end{equation}
where $F$ satisfies the following assumptions.
\begin{enumerate}
\item\label{ass:F:1}
For each $t\geq 0,\, F_t:\R\to [0,+\infty]$
\item\label{ass:F:2}
For all $t\ge 0$, all $L\ge 0$, the sublevel--set $[F_t\le L]$ is a nonempty closed (possibly unbounded) interval in $\R$, with endpoints
$e_t^{-}(L)\in \R\cup\{ -\infty\}$, $e_t^{+}(L)\in \R\cup\{ +\infty\}$ respectively, with, further,
\[
\inf_{t\ge 0}e_t^{+}(L)>-\infty\,,\quad \sup_{t\ge 0}e_t^{-}(L)<+\infty\,.
\]
\item\label{ass:F:3}
For each $L\ge 0$, the functions $t\mapsto e_t^\pm (L)$ are piecewise continuous
(in appropriate sense for extended-real-valued functions)
with at worst a countable set of jump-discontinuities, where at all such jumps $\bar{t}$, have
\[
\min\{ e_{\bar{t}+}^+(L), e_{\bar{t}-}^+(L) \} \ge \max \{e_{\bar{t}+}^-(L), e_{\bar{t}-}^-(L) \}\,.
\]
\item \label{ass:F:4} For each $L\ge 0$, there is $t_0(L)\ge 0$ such that for all $t\ge t_0$, have that
$e_t^-(L)\le 0\le e_t^+(L)$.
\end{enumerate}
Before we proceed further, a review of some relevant measure
theory is appropriate. (Our reference shall be \cite{Cohn}.)
Given $\mu\in\mathbf{M}(\R_+)$, we can find a
{\it Hahn decomposition} of $\R_+$ into disjoint Borel sets $P$,
$N$ such that $\mu(E\cap P)\ge 0$ and $\mu(E\cap N)\le 0$ for each
Borel set $E$. From this are obtained regular finite Borel
measures $\mu_{\pm}$ given by $\mu_+(E):=\mu(E\cap P)$ and
$\mu_-(E):=\mu(E\cap N)$, with $\mu_+\ge 0$ and $\mu_-\le 0$. We
then have $\mu=\mu_++\mu_-$ and the {\it variation} of $\mu$
(denoted $|\mu|$) is defined by $|\mu|:=\mu_+-\mu_-$. Note that
our sign convention has been chosen to conform with that used in
the discrete-time analysis of \cite{HEHW}, but differs from the
usual choice in measure theory, where $\mu_+$ and $\mu_-$ are both
non--negative, whereas here we have $\mu_-\le 0$. Recall that we
assume $\mathbf{M}(\R_+)$ to be normed by $\|\mu\|:=|\mu|(\R_+)$.
Note that if measure $\mu\in U\oplus V\oplus W$ and function $e^*$
are related via \eqref{eq:meas}, then by standard arguments follows that
$$
\mu_\pm (E)=\int_E e_\pm ^* \,\quad\text{ for Borel sets $E$}\,,
$$
and that $\mu\ge 0$ if and only if $e^*(t)\ge 0$ for all $t$, with a similar
relation for the reverse inequality $\mu\le 0$.
We also recall the following standard measure-theoretic convention: Since we need to integrate
real functions taking $+\infty$ as a possible value, the definition of the integral
(with respect to a measure) incorporates the convention
$0\cdot\infty:=0$ whenever one of the factors is the value of a (unsigned) measure.
\subsection{The Conjugate for the Assumed Form of the Primal Objective}
The following result provides a continuous--time analogue
of part of \cite[Theorem 9]{HEHW}. (The proof is deferred to the Appendix.)
\begin{thm}\label{thmfstar1:cont}
For $f:C_0(\R_+)\to \R\cup\{+\infty\}$ defined as in
\eqref{eq:fF}, where $F$ satisfies the assumptions 1---4 above. Then,
\begin{equation}\label{eq:conj:cont}
f^*(\mu)=\sup_{ L\ge 0 } \left(\int_0^{+\infty}\!\! \!e_t^+(L)d\mu_+(t)+
\int_0^{+\infty }\!\!\!e_t^-(L)d\mu_-(t)-L\right)
\end{equation}
for all $\mu\in \mathbf{M}(\R_+)$ such that $|\mu|(\{\bar{t}\})=0$ for each $\bar{t}$ for which there is some $L$
for which at least one of $e^+(L)$ or $e^+(L)$ is discontinuous at $\bar{t}$.
\end{thm}
\begin{remark}\label{rem:thmfstar1:cont}
If $F_t\equiv F$ (independent of $t$), assumption 3.\ is inactive, and assumption 4.\ can only be valid for $L$ satisfying
$L\ge F(0)$ (which is equivalent to $[f\le L]\neq\emptyset$) and from the proof (see Appendix) it follows that the supremum in
(\ref{eq:conj:cont}) is then to be taken over $L\ge F(0)$.
\end{remark}
\subsection{Duals for some Time-Domain Minimization Problems}
We may now derive the duals of the continuous--time versions of
MA, OS, POS, FL in the manner of \cite{HEHW}.
Because of Proposition~\ref{prop:summary}, this separates into
cases where $\overline{X_\theta}=C_0(\R_+)$ and
$\overline{X_\theta}=C_{0,\alpha}(\R_+)$. We begin with the former case.
\subsubsection{Maximum Amplitude}
Clearly
$e^{+}(L)=L$ and $e^{-}(L)=-L.$ By Theorem~\ref{thmfstar1:cont} and Remark~\ref{rem:thmfstar1:cont},
$$\func{dom}
f_{ma}^{*}=\left\{ \mu\in \mathbf{M}(\R_+)\mid\| \mu\|\leq 1\right\}
\text{ and }f_{ma}^{*}(\mu)=0\,.$$
Thus by Proposition~\ref{thmfench:cont}, the dual of the problem of
maximum--amplitude
minimization, denoted MA DUAL, can be written as
\begin{equation}
\max_{{\scriptsize \begin{array}{c} \mu\in U\oplus V\oplus W \\
\|\mu \|\leq 1
\end{array}}} \langle \func{Proj}_U(\mu),w\rangle .
\label{madual:cont}
\end{equation}
\subsubsection{Positive Error}
For this case we have $e^{-}(L)=-\infty$ and $e^{+}(L)=L$. By Theorem~\ref
{thmfstar1:cont} and Remark~\ref{rem:thmfstar1:cont}, $\func{dom}f_{pos}^{*}=\left\{ \mu\in
\mathbf{M}(\R_+)\mid \mu\geq 0
\text{ and }\mu(\R_+)\leq 1\right\} $ and $
f_{pos}^{*}(\mu)=0$. The dual of POS is
\begin{equation}
\max_{{\scriptsize \begin{array}{c} \mu\in U\oplus V\oplus W
\\ \mu\geq 0\text{, }\|
\mu\|\leq 1 \end{array}}} \langle \func{Proj}_U(\mu),w\rangle ,
\label{posdual:cont}
\end{equation}
\subsubsection{
Overshoot} For overshoot minimization $e^{+}(L)=+\infty$ and
$e^{-}(L)=-L$. By Theorem~\ref{thmfstar1:cont} (and the Remark),
$\func{dom}f_{os}^{*}=\left\{ \mu\in \mathbf{M}(\R_+)\mid \mu\leq
0\text{ and }\mu(\R_+)\geq -1\right\} $
and $
f_{os}^{*}(\mu)=0.$ The dual of OS is
\begin{equation}
\max_{{\scriptsize \begin{array}{c} \mu\in U\oplus V\oplus W
\\ \mu\leq 0\text{, }\|
\mu\|\leq 1 \end{array}}} \langle \func{Proj}_U(\mu),w\rangle
\text{.} \label{osdual:cont}
\end{equation}
\subsubsection{Fluctuation}
Also, as in the discrete--time case, we may use
\eqref{argmin:cont} to deduce that $f_{fl}^*$ is the indicator
function of
$$ \{\mu\in\mathbf{M}(\R_+)\mid \mu_+(\R_+)\le \frac12,\,
\mu_-(\R_+)\ge -\frac12\}\,,$$
(see proof of \cite[Theorem 11]{HEHW}). Hence, by
Proposition~\ref{thmfench:cont}, the dual of FL is
$$ \max_{\scriptsize \begin{array}{c} \mu\in U\oplus V\oplus W\\
\mu_+(\R_+)\le\frac12\\
\mu_-(\R_+)\ge -\frac12 \end{array} }
\la \func{Proj}_U (\mu),w\rangle \,. $$
We observe in passing that for the fluctuation--minimization problem,
that the minimum is never achieved (even over $\overline{M_{\underline{\theta}}}$),
except in the trivial case where $P$ has no
poles in $\overline{\C_+}$, so $0\in M:=\overline{M_{\underline{\theta}}}$ is optimal.
Indeed, suppose an optimal $e\in M$ is attained, and let $\mu$ denote the optimal dual
element. These must satisfy the
alignment condition $ f_{fl}(e)+f_{fl}^*(\mu)= \la e,\mu\ra $,
which yields $$ \la e,\mu\ra= f_{fl}(e)=\frac12 (\sup e-\inf e)$$
since $f_{fl}^*(\mu)=0$. If $\text{FL}_{opt}=0$ this immediately
yields the contradiction $e=0$ (since $0\notin M$). If
$\text{FL}_{opt}$ is positive, note that the optimal $\mu$ is then
nonzero.
Now as $\int d\mu_+\le\frac12$ and
$\int d\mu_-\ge -\frac12$, and $\sup e\ge 0\ge\inf e$, we obtain
\begin{align*}
\int e\,d\mu_+ + \int e\,d\mu_-=\la e,\mu\ra &= \frac12 (\sup e-\inf e)\\
& \ge \sup e \int d\mu_+ + \inf e \int d\mu_- \,,
\end{align*}
so that
$$ 0\le \int (e-\sup e)\,d\mu_+ +\int (e-\inf e)\,d\mu_- \le 0\,.$$
Since both terms in the above are nonpositive, we conclude that
$$ \int (e-\sup e)\,d\mu_+ = \int (e-\inf e)\,d\mu_-=0\,.$$
Writing this in terms of the function $e^*$ associated to $\mu$ via
\eqref{eq:meas}, we have for almost all $t$ (w.r.t.\ Lebesgue measure),
and hence by continuity, for all $t$, that
$$ (e(t)-\sup e)e^*(t)=0=(e(t)-\inf e)e^*(t)\,. $$
Thus
$$ e(t)=\left\{ \begin{array}{cl}
\sup e & \text{if }e^*(t)>0\\
\inf e &\text{if }e^*(t)<0
\end{array} \right. .$$
By analyticity of $e^*\ne 0$ (being a finite sum of sinusoids), each of
its zeros is isolated, so by continuity of $e$ follows that $\inf
e=\sup e$, implying again the contradiction $e=0$.
\smallskip
\subsubsection{Undershoot}
We assume $w(t)\ge 0$ for all $t$. (Note: To derive an expression for
$f^*_{us}$,
we do not yet need to assume a rational
Laplace transform for $w$). Place
$F_t(\xi):=(\xi-w(t))_+$. Then $F_t(0)=0$ for all $t\ge 0$, and
$0\in\{F_t\le L\}=(-\infty, w(t)+L]$ for each $L\ge 0$. Theorem~\ref{thmfstar1:cont}
gives, for all $\mu\in {\bf M}(\R_+)$, that
$$
f^*_{us}(\mu)=\int w\,d\mu_+ + (-\infty)\cdot\mu_-(\R_+) +\sup_{L\ge 0} L(\mu_+(\R_+)-1)
$$
which yields the following
\begin{cor}\label{prop:undershoot:conj}
Let $w\ge 0$. Then,
$$
\func{dom} f^*_{us}\subseteq \{ \mu\in {\bf M}(\R_+)\mid \mu\ge
0\}
$$
and for each $\mu\ge 0$,
$$
f^*_{us}(\mu)=\left\{ \begin{array}{cl}
\int w \,d\mu & \text{if }\mu\ge 0\text{ and } \|\mu\|\le 1\\
+\infty &\text{otherwise }
\end{array} \right.\,.
$$
Note that $\int w \,d\mu $ may take an infinite value.
\end{cor}
Consequently, the dual for US has the form
\begin{equation}
\max_{{\scriptsize \begin{array}{c} \mu\in U\oplus V\oplus W
\\ \mu\leq 0\text{, }\|
\mu\|\leq 1 \end{array}}} \langle \func{Proj}_{V\oplus
W}(\mu),w\rangle \text{.} \label{osdual:cont}
\end{equation}
\smallskip
\bigskip
\smallskip
Recall that each $\mu\in U\oplus V\oplus W$ corresponds to a
function $e^*$ by Equation~\eqref{eq:meas}. This has the
consequences: $\|\mu\|=|\mu|(\R_+)=\int |e^*|=\|e^*\|_1$; $\mu\ge
0$ (as a measure) if and only if $e^*(t)\ge 0$ for all $t$; and
$\mu\le 0$ if and only if $e^*(t)\le 0$ for all $t$. We may
therefore restate these duals in a form identical to those in
discrete time:
\begin{align*}
\text{(MA DUAL)}\qquad &
\max_{{\scriptsize \begin{array}{c} e^*\in U\oplus V\oplus W \\
\|e^* \|_1\leq 1
\end{array}}} \langle \func{Proj}_U(e^*),w\rangle \\
\text{(POS DUAL)}\qquad &
\max_{{\scriptsize \begin{array}{c} e^*\in U\oplus V\oplus W
\\ e^*\geq 0\text{, }\|
e^*\|_1\leq 1 \end{array}}} \langle \func{Proj}_U(e^*),w\rangle \\
\text{(OS DUAL)}\qquad &
\max_{{\scriptsize \begin{array}{c} e^*\in U\oplus V\oplus W
\\ e^*\leq 0\text{, }\|
e^*\|_1\leq 1 \end{array}}} \langle \func{Proj}_U(e^*),w\rangle \text{} \\
\text{(US DUAL)}\qquad & \max_{{\scriptsize \begin{array}{c}
e^*\in U\oplus V\oplus W
\\ e^*\leq 0\text{, }\|
e^*\|_1\leq 1 \end{array}}} \langle \func{Proj}_{V\oplus W}(e^*),w\rangle \text{} \\
\text{(FL DUAL)}\qquad &
\max_{\scriptsize \begin{array}{c} e^*\in U\oplus V\oplus W\\
\int_{\tiny \R_+}e_+^*\le\frac12\\
\int_{\tiny \R_+}e_-^*\ge -\frac12
\end{array} }
\la \func{Proj}_U (e^*),w\rangle \,.
\end{align*}
\subsubsection{case of $\overline{X_\theta}=C_{0,\alpha}$ }
When
$\overline{X_\theta}=C_{0,\alpha}$, we apply the duality formula
(\ref{eq:duality:2}), which requires calculation of $f^{\#}$,
given by (\ref{eq:duality:3}). The forms of all the duals, except
for FL, will be unchanged, because of the next result, proved in
the Appendix.
\begin{lem}\label{lem:fhash}
If $f$ stands for any of the functionals treated (except for
$f_{fl}$), then for $\mu\in U\oplus V\oplus W$, have
$$f^{\#}(\mu)=-f^*(\mu) \,, $$
whereas for FL, we have
$$
f_{fl}^{\#}(\mu)=-f_{fl}^*(\mu)+\alpha(
1/2-\mu_+(\R_+))\,.\qquad\qquad \square
$$
\end{lem}
This, for FL, yields the dual
$$
\text{(FL DUAL)}\qquad
\max_{\scriptsize \begin{array}{c} e^*\in U\oplus V\oplus W\\
\int_{\tiny \R_+}e_+^*\le\frac12\\
\int_{\tiny \R_+}e_-^*\ge -\frac12
\end{array} }
\la \func{Proj}_U (e^*),w\rangle +\alpha(1/2-\int_{\tiny \R_+} e_+^*) \,.
$$
\subsection{Overshoot/Undershoot Minimization}\label{sectosusmin:cont}
We follow the procedure of \cite{HEHW}. The first lemma is
essentially the continuous--time counterpart of \cite[Proposition
15]{HEHW}.
\begin{lem}\label{prop:sinus:cont}
Let $y_1,\ldots, y_N\ne 0$ with $y_i\ne\pm y_j$ whenever $i\ne j$. If
$\alpha_1,\ldots,\alpha_N$ and $\beta_1,\ldots,\beta_N$ are real scalars
such that for some $C,\ \rho>0$ we have
$$ \sum_{i=1}^N (\alpha_i \cos y_it +\beta_i \sin y_it)\ge -Ce^{-\rho t}
\text{ for all $t$ large} $$
then $\alpha_i=0=\beta_i$ for $i=1,\ldots,N$.
\end{lem}
\begin{Proof}
Place $a(t):= \sum_{i=1}^N (\alpha_i \cos y_it +\beta_i \sin y_it)$.
For $r>0$, set $$S(r):=\int_0^\infty a(t)e^{-rt}dt=
\sum_{i=1}^N \frac{\alpha_ir+\beta_i y_i}{r^2+y_i^2}\,.$$
Since $y_i\ne 0$ for all $i$, there exists $\lambda>0$ such that
$|S(r)|\le \lambda$ for all $r>0$.
Thus for some $t_0$ and any $N\in\R$ and $r>0$,
\begin{align*}
0\le \int_0^N a_+(t)e^{-rt}dt &\le \int_0^\infty a_+(t)e^{-rt}dt \\
& = S(r)-\int_0^\infty a_-(t)e^{-rt}dt\\
& \le \lambda + \int_0^{t_0}|a_-(t)|dt+\int_{t_0}^\infty |a_-(t)|dt \\
&\le \lambda +\int_0^{t_0}|a_-(t)|dt+ C\int_0^\infty e^{-\rho t}dt\\
&\le \lambda +\int_0^{t_0}|a_-(t)|dt+ C/\rho :=\lambda '
\end{align*}
For each fixed $N$, we may let $r\to 0$ in the above, to obtain
by the Dominated Convergence Theorem that
$0\le\int_0^N a_+(t)dt \le \lambda '$. Since $|a_-(t)|\le Ce^{-\rho t}$ for all
large $t$, it follows that $\int_0^N|a(t)|dt$ is bounded above for all $N$,
so that $a(\cdot)$ is in $L^1$. This implies that the Laplace transform
$\hat{a}(s)$ is defined for all $s$ such that $\func{Re}s\ge 0$, and
is bounded in this region. However, for all $s$ with positive real part,
$$ \hat{a}(s)= \sum_{i=1}^N\frac{\alpha_is+\beta_i y_i}{s^2+y_i^2} $$
and so if $(\alpha_k, \beta_k)\ne (0,0)$, then from the assumed distinctness
of the $y_i$, follows that $\hat{a}$ must have a pole at $s=\pm iy_k$,
contradicting the boundedness of the transform at points of the imaginary
axis. Thus necessarily $(\alpha_k, \beta_k)=(0,0)$.
\end{Proof}
\smallskip
Lemma~\ref{prop:sinus:cont} can now yield an analogue of
\cite[Proposition 17]{HEHW} in continuous time. Given the complex
numbers $z_1,.., p_1,..,$ and $v_1,..$, let $\gamma$ denote the
smallest value among those poles/zeros that lie on the positive
real axis; if there are none, set $\gamma$ to $\infty$.
\begin{thm}\label{cor1:cont}
Consider the `equivalence' class $\mathcal{A}$, of all plants
which: \begin{enumerate}
\item have the same $\gamma$ value; and
\item have the same poles and zeros in the region $\{s\mid \func{Re}s\ge
\gamma\}$.
\end{enumerate}
Then, for any fixed minimum--phase reference input
(i.e.\ having no zeros in the closed right half--plane), all plants
in $\mathcal{A}
$ have the same value for $\funcc{OS}_{\scriptsize{\funcc{opt}}}$. If the
reference--input is also non-negative, then all plants
in $\mathcal{A}
$ have the same value for $\funcc{US}_{\scriptsize{\funcc{opt}}}$.
\end{thm}
\begin{Proof}
We show that the poles/zeros in $\{s\mid\func{Re}s<\gamma\}$ do
not affect the value of the dual.
For simplicity, we assume that all the
poles/zeros in $\{s\mid\func{Re}s<\gamma\}$ have equal
real part. The general case is handled by iteration of the argument given below.
Let $e^*\in U\oplus V\oplus W$ with $e^*\ge 0$ (or $e^*\le 0$). Regrouping
the modes of $e^*$ according to the size of the real parts of the associated
poles and zeros gives
$$ e^*(t)=e^{-\rho_1 t}
\sum_{i=1}^{N_1}(\alpha _i^{(1)}\sin (y_i^{(1)}t)+
\beta _i^{(1)}\cos (y_i^{(1)}t)) +
e^{-\rho_2 t}\sum_{i=1}^{N_2}(\alpha _i^{(2)}\sin (y_i^{(2)}t)+
\beta _i^{(2)}\cos (y_i^{(2)}t)) + \ldots
$$
where $0<\rho_1<\rho_2<...$, and in the first sum, which corresponds to the
poles/zeros in $\{s\mid\func{Re}s<\gamma\}$, we have
$y_i^{(1)}\ne 0$, and $y_i^{(1)}\ne\pm y_j^{(1)}$ whenever $i\ne j$.
Hence
\begin{align*}
\sum_{i=1}^{N_1}(\alpha _i^{(1)}\sin (y_i^{(1)}t)+
\beta _i^{(1)}\cos (y_i^{(1)}t)) &= e^*(t)/e^{-\rho_1 t} +
e^{(\rho_1-\rho_2)t}\sum_{i=1}^{N_2}\ldots \\
& \ge 0 + O(e^{(\rho_1-\rho_2)t})\,.
\end{align*}
By Lemma~\ref{prop:sinus:cont}, $\alpha _i^{(1)}\!=0=\beta
_i^{(1)}$ for $i=1,..,N_1$. Thus the modes associated with poles
or zeros with real part smaller than $\gamma$ do not contribute to
the dual, as claimed.
\end{Proof}
\bigskip
\bigskip
\subsection{Analytical Results for a First--Order Plant}
We first derive analytical expressions
for performance limitations in terms of pole/zero locations for first-order plants.
Thanks to Theorem~\ref{cor1:cont},
in the case of overshoot and undershoot limitations on performance, these analytical results
can be extended to include cases where the plant has an arbitrary number of oscillatory poles
or zeros (i.e.\ those off the real axis), and the reference input has an arbitrary number of oscillatory zeros.
\begin{prop}
\label{propcontex}Suppose the plant, $P=\frac{s-z_{1}}{s-p_{1}},$
has one real positive zero, $z_{1,}$ and one real positive pole,
$p_{1},$ where $z_{1}>p_{1}.$ Let $\widehat{w}(s)=1/s$. Define
$h:=p_{1}/(z_{1}-p_{1}).$ Then
\begin{align*}
(i)\,\,\qquad \funcc{OS}_{\scriptsize{\funcc{opt}}} & \, =\,h\\
(ii) \qquad \funcc{MA}_{\scriptsize{\funcc{opt}}} & \,=\,\frac1{1-2^{-1/h}}\\
(iii)\,\qquad \funcc{FL}_{\scriptsize{\funcc{opt}}} & \, =\,\frac{(h+1)^{(h+1)}}{2h^{h}}\\
\!(iv) \,\,\quad \funcc{POS}_{\scriptsize{\funcc{opt}}} & \, =\, 1.
\end{align*}
\end{prop}
\begin{remark}
It can be verified using elementary calculus that, for $h\in(0,\infty),$
\[
\max(1,h)\leq\frac1{1-2^{-1/h}}\leq\frac{(h+1)^{(h+1)}}{h^{h}}\leq
\frac2{1-2^{-1/h}}
\]
From (ii) and (iii) it follows that
$2\funcc{MA}_{\scriptsize{\funcc{opt}}}=\funcc{FL}_{\scriptsize{\funcc{opt}}}$ if and only if $h=1$, that is
$z_{1}=2p_{1}.$ If $h\neq 1$, then $\funcc{FL}_{\scriptsize{\funcc{opt}}}$ is strictly
less than $2\funcc{MA}_{\scriptsize{\funcc{opt}}}$; in this example, when $h\neq 1$, a
minimal fluctuation response is obtainable only by allowing the
infinity--norm to be larger than optimal.
\end{remark}
\begin{Proof}
We have $A=\operatorname*{span}\left[ e^{-z_{1}t}\right] ,\
B=\operatorname*{span}\left[ e^{-p_{1}t}\right] ,\ C$ is empty
and $w=\underline{1}$ (the unit step function).
(i) The dual optimal vector for (OS DUAL),
$e^{*}=\zeta(t)-\eta(t)$ where $\zeta\in A$ and $\eta\in B,$ will
satisfy $\zeta(0)=\eta\left( 0\right) ,$ so $\zeta(t)=\alpha
e^{-z_{1}t}$ for some real number $\alpha$ and
$\eta(t)=\alpha e^{-p_{1}t}.$ Hence $\alpha\int_{0}^{\infty}(e^{-p_{1}
t}-e^{-z_{1}t})dt\leq1,$ from which $\alpha\leq
p_{1}z_{1}/(z_{1}-p_{1}).$ Then $\funcc{OS}_{\scriptsize{\funcc{opt}}}=\max_{\zeta\in
A}\langle\zeta,r\rangle=\max_{\alpha}\int
_{0}^{\infty}\zeta(t)\underline{1}dt=\max_{\alpha}\left[
\alpha/z_{1}\right] =h.$
(ii) (MA DUAL) is
\begin{align*}
& \max_{\alpha,\beta}\left[ \alpha/z_{1}\right] \\
\text{subject to } & \int_0^\infty |\alpha e^{-z_{1}t}-\beta e^{-p_{1}t}|\,dt
\leq1.
\end{align*}
It is clear that, at optimality, $\alpha>\beta>0,$ and there will
exist a positive number $t_{0}$ such that
$\zeta(t_{0})=\eta(t_{0}).$ It is obvious also that at optimality
the constraint inequality will be satisfied as an equality. The
dual becomes
\begin{align*}
& \max_{\alpha,\beta,t_{0}}\left[ \alpha/z_{1}\right] \\
\text{s.t. }\alpha e^{-z_{1}t} & =\beta e^{-p_{1}t}\text{ and
}\int _{0}^{t_{0}}\left( \alpha e^{-z_{1}t}-\beta
e^{-p_{1}t}\right) dt+\int_{t_{0}}^{\infty}\left( \beta
e^{-p_{1}t}-\alpha e^{-z_{1}t}\right) dt=1.
\end{align*}
This is a simple finite-dimensional constrained optimization
problem, which can be solved using the method of Lagrange
multipliers. After some elementary algebra one obtains
$t_{0}=\frac{\log(\alpha-\beta)}{z_{1}-p_{1}}=-\frac
{\log1/2}{p_{1}},$ $\alpha/\beta=2^{h}$ and $\alpha=z_{1}/[1-2(1/2)^{z_{1}
/p_{1}}]$, from which the result follows.
(iii) At optimality the inequalities in the two constraint
equations for (FL DUAL) will be satisfied as equalities. Then (FL
DUAL) becomes
\begin{align*}
& \max_{\alpha,\beta}\left[ \alpha/z_{1}\right] \\
\text{s.t. } & \int_0^\infty |\alpha e^{-z_{1}t}-\beta
e^{-p_{1}t}|\,dt =1\text{ and }\int_{0}^{\infty}\left(
\alpha e^{-z_{1}t}-\beta e^{-p_{1}t}\right) dt=0.
\end{align*}
The second constraint gives immediately that $\alpha p_{1}=\beta
z_{1}.$ Let
$t_{0} $ be the positive number with the property that $\zeta(t_{0}
)=\eta(t_{0}).$ After performing the integration in the first
constraint, and writing it as an equality, the dual becomes
\begin{align*}
& \max_{\alpha,t_{0}}\left[ \alpha/z_{1}\right] \\
\text{s.t. }1 & =\frac{2\alpha}{z_{1}}\left( e^{-p_{1}t_{0}}-e^{-z_{1}
t_{0}}\right) .
\end{align*}
After some algebra the optimizing $t_{0}$ and $\alpha$ are found
to be
$t_{0}=\frac{\log(z_{1}-p_{1})}{z_{1}-p_{1}}$ and $\alpha=z_{1}(1+1/h)^{h}
(1+h)/2.$
(iv) In (POS DUAL), if $r\geq0,$ an optimizing $\eta$ will be
identically zero. Then the optimizing $\zeta$ will be positive and
satisfy $\parallel \zeta\parallel_{1}=1.$ For $w=$\underline{$1$}
the optimal dual cost is
$\int_{0}^{\infty}\zeta$\underline{$1$}$dt=1.$\quad
\end{Proof}
\bigskip
\bigskip
\subsection{A time-domain constraint effect}
We consider some relationships between overshoot (and undershoot)
with some simple finite-time-horizon constraints. (We have in mind the effect
of rise-time constraints on optimal overshoot performance.)
Time-domain signal bounds will be represented by the set
\begin{equation}
T:=\{e\in C_0(\R_+)\mid \phi_-(t)\le e(t)\le \phi_+(t) \,\,\forall t\ge 0\}
\end{equation}
for suitable bounding functions $\phi_{\pm}:\R_+\to \R\cup \{ -\infty, +\infty \}$.
For simplicity, we restrict to the finite-horizon case, where for some positive
$\bar{t}>0$, have $\phi_{\pm}(t)\equiv \pm\infty$ for all $t\ge \bar{t}$.
Also, assume that $\phi_{\pm}$ are finite-valued and continuous on $[0,\bar{t}]$,
with $\phi_{-}(t) < \phi_{+}(t)$ for all $t$, and $\phi_{+}(t)\ge 0$. In this case, $T$ has the form
\begin{equation}
T=\{e\in C_0(\R_+)\mid \phi_-(t)\le e(t)\le \phi_+(t) \,\,\forall t\le \bar{t} \}\,.
\end{equation}
By selecting a specific $\bar{e}\in X_{\underline{\theta}}$ such that $A\bar{e}=b$, and
selecting $\phi_{\pm}(t)$ such that $\phi_{-}(t)<\bar{e}(t)<\phi_{+}(t)$ for all $t\le \bar{t}$,
we may ensure that
\begin{equation}\label{eq:CQ:1}
b\in A( X_{\underline{\theta}}\cap \func{int}T )
\end{equation}
which suffices for duality to hold, if the cost-function is of the form $f_0+\iota_T$
(where $f_0$ is any finite-valued functional, such as one of those listed in
Section~\ref{sectsometdperf :cont}).
Suppose also that the reference signal $w$ is minimum-phase, and that all the unstable poles and zeros
of the plant are oscillatory (that is, they all have non-vanishing imaginary part). As observed in Theorem~\ref{cor1:cont},
when $T$ is absent, have OS${}_{ \text{\scriptsize{\funcc{opt}}} }=0$. Now introduce the further constraint represented by the set $T$,
and consider
$$
\text{OS}^T_{ \text{opt} } := \inf_{e\in T\cap X_{\underline{\theta}} \cap A^{-1}b}f_{os}(e)
= \inf_{ \overline{X_{\underline{\theta}} } \cap A^{-1}b}(f_{os}+\iota_T )\,.
$$
(where the latter equality follows from (\ref{eq:CQ:1})).
With reference to Proposition~\ref{prop:summary}, make the additional assumption
in the case when
$\overline{X_{\underline{\theta}} }=C_{0,\alpha}$, that $\alpha\ge 0$, in which case we then have
$$
\phi_+(0)>\alpha\ge\max\{\phi_-(0) , 0 \}\,.
$$
(remember that in this case $\alpha=w(0+)=\bar{e}(0)$)
\begin{prop}\label{prop:tradeoff}
With the above assumptions, $\funcc{OS}^T_{ \scriptsize{\funcc{opt}} }=0$.
\end{prop}
Thus, in particular, for step-input, the imposition of rise-time constraints does not degrade the
optimal overshoot performance.
\begin{Proof}
Now, $f(e)$ is of the form $\sup_{t\ge 0}F_t(e(t))$, where
$F_t(\xi) =F^{\text{os}}(\xi)+ \iota_{\overline{(\phi_-(t),\phi_+(t))} }(\xi)$ from which follows
(since $\phi_+(t)\ge 0$) that $e_t^+(L) =\phi_+(t)$ and $e_t^-(L) =\max\{ \phi_-(t), -L\}$.
and that Assumptions 1---4 for Theorem~\ref{thmfstar1:cont} are satisfied (note, for assumption 2, needed $\phi_+(t)\ge 0$
to get $[F_t\le L]\neq\emptyset$), and hence for any $\mu\in U\oplus V\oplus W$, have
\begin{align*}
f^*(\mu) & = \int_0^{\bar{t}}\phi_+\,d\mu_+ +(+\infty)\cdot\mu_+((\bar{t},+\infty)) +\\
& \qquad +\sup_{L\ge 0}\left[ \int_0^{\bar{t}}\max\{\phi_-, -L \}d\mu_- - L( \mu_-((\bar{t},+\infty))+1) \right]\\
& \ge \int_0^{\bar{t}}\phi_+\,d\mu_+ + \int_0^{\bar{t}}\max\{\phi_-, 0 \}d\mu_- + (+\infty)\cdot\mu_+((\bar{t},+\infty))
\end{align*}
where the first two terms of the latter are clearly finite. Thus, whenever $\mu\in\func{dom}f^*$, it must follow that $\mu_+((\bar{t},+\infty))=0$,
and so $\func{dom}f^*\subseteq \{\mu\mid \mu|_{(\bar{t},+\infty)} \le 0\}$.
The dual then has the form (when $\overline{X_{\underline{\theta}} }=C_{0 }$)
$$
\max_{
{\scriptsize
\begin{array}{c}
e^*\in U\oplus V\oplus W \\
e^*\in\func{dom}f^{*} \\
e^*|_{(\bar{t},+\infty)} \le 0\\
\end{array}
} } \left[ \langle \func{Proj}_U(e^*),w\rangle -f^{*}(e^*)\right]
,
$$
but each dual $e^*$ now satisfies $e^*=0$ by Lemma~\ref{prop:sinus:cont}, by the same argument as used in Theorem~\ref{cor1:cont}.
Hence, since $f^*(0)=\sup_{L\ge 0}-L =0$, the dual has value 0.
If, instead, $\overline{X_{\underline{\theta}} }=C_{0,\alpha}$, the dual takes the form
$$ (D)\,=
\max_{
{\scriptsize
\begin{array}{c}
e^*\in U\oplus V\oplus W \\
e^*\in\func{dom}(-f^{\#}) \\
\end{array}
} } \left[ \langle \func{Proj}_U(e^*),w\rangle +f^{\#}(e^*)\right]\,,
$$
where
$$
f^{\#}(\mu)=\max_{\lambda\in\R}(\alpha\lambda-f^*(\mu+\lambda\delta))\,.
$$
We want to show that $\func{dom}(-f^{\#})\subseteq\{ \mu \mid \mu|_{(\bar{t},+\infty) }\le 0 \}$.
Now, we may apply Theorem~\ref{thmfstar1:cont} for $f^*(\mu)$ for any $\mu\in U\oplus V\oplus W\oplus\R\delta$
(since for any $L\ge 0$, $t\mapsto e_t^{\pm}(L)$ has no discontinuities at $t=0$---only at $t=\bar{t}>0$, where
$|\mu|(\{ \bar{t}\})=0$). Thus,
\begin{align*}
-f^{\#}(\mu) & = \min_{\lambda\in\R}(-\alpha\lambda+f^*(\mu+\lambda\delta))\\
& \ge \min_{\lambda\in\R} \bigg(-\alpha\lambda +\int_0^{\bar{t}} \phi_+ d(\mu+\lambda\delta)_+ \, +\,
\int_0^{\bar{t}} \max\{\phi_-, 0\}d(\mu+\lambda\delta)_- \,\,\,\, + \\
& \qquad\qquad (+\infty)\cdot(\mu+\lambda\delta)_+((\bar{t},+\infty)) \bigg) \\
& = (+\infty)\cdot\mu _+((\bar{t},+\infty)) + \int_0^{\bar{t}} \phi_+d\mu_+ \, + \,
\int_0^{\bar{t}} \max\{\phi_-, 0\}d\mu_- \,\,\,\,+ \\
& \qquad\qquad +\, \min_{\lambda\in\R}\big(-\alpha\lambda+ \lambda_+\phi_+(0) + \lambda_- \max\{\phi_-(0),0 \}\big)\qquad\text{ (by Lemma~\ref{lem:muFL})}\\
& = (+\infty)\cdot\mu _+((\bar{t},+\infty)) + \int_0^{\bar{t}} \phi_+d\mu_+ \, + \,
\int_0^{\bar{t}} \max\{\phi_-, 0\}d\mu_-
\end{align*}
where we used the relation $\phi_+(0)>\alpha\ge\max\{\phi_-(0) , 0 \}$ to ensure the latter minimization has value zero.
Therefore, if $-f^{\#}(\mu)<+\infty$, then $\mu _+((\bar{t},+\infty)) =0$ so that $\mu|_{ (\bar{t},+\infty)}\le 0$, which implies,
since $\mu\in U\oplus V\oplus W$, that $\mu=0$, whence (D)$\,=f^{\#}(0)=\max_{\lambda\in\R}(\alpha\lambda-f^*( \lambda\delta))$. But, $f^*( \lambda\delta))=
\sup_{L\ge 0}(\lambda_+e_0^+(L)-L) =\lambda_+\phi_+(0)$ since $e_0^+(L)=\phi_+(0)$ for all $L$, yielding
(D)=$\max_{\lambda\in\R}(\alpha\lambda-\lambda_+\phi_+(0))=0$ as $\alpha<\phi_+(0)$ (and $\alpha\ge 0$).
Thus, $\text{OS}^T_{ \text{opt} }=0$ in this case also.
\end{Proof}
\section{Conclusions}
Using a dual formulation, new results on
fundamental time-domain performance limitations for
continuous-time systems have been presented. For the problem of
designing a feedback system to optimally track a specific input,
or reject a specified disturbance, there are many time-domain
performance measures of the output signal that can be used. In
addition to overshoot, undershoot and the infinity norm of the
error signal, a performance measure of practical significance,
termed fluctuation, has been investigated for the first time in a
continuous-time setting.
\bigskip
\section{Appendix A: Proof of Theorem~\ref{thmfstar1:cont}}
Firstly, we consider the case where, for each $L$, the $e_{t}^\pm(L)$ are bounded in $t\ge 0$.
Now,
$
f^*(\mu) = \sup_{L\ge 0}\sup_{e\in C_0,\,
f(e)\le L} (\la\mu,e\ra-L)\,,
$
noting that if $F$ is independent of $t$, the supremum is restricted to $L$ satisfying $L\ge F(0)$, as indicated in Remark~\ref{rem:thmfstar1:cont}.
We fix a value of $L$ and evaluate the inner supremum, and will show that
$$ \sup_{\scriptsize \begin{array}{c}
e\in C_0\\
f(e)\le L \end{array} } \!\!\!\la\mu,e\ra=
\int_0^{+\infty}\!\! \!e_t^+(L)d\mu_+(t)+
\int_0^{+\infty }\!\!\!e_t^-(L)d\mu_-(t)
$$
from which \eqref{eq:conj:cont} then follows.
For $\mu\in \mathbf{M}(\R_+)$, let $(P,N)$ be a Hahn decomposition
\cite{Cohn} of $\R_+$ relative to the measure $\mu$. Now, among all
$e(\cdot):\R_+\to\R$ satisfying $e(t)\in [e_t^-(L),e_t^+(L)]$ for all $t$,
$\la\mu,e\ra$ is maximized at $\bar{e}$, where
$$ \bar{e}(t):= \left\{ \begin{array}{cl}
e_t^+(L) & t\in P \\
e_t^-(L) & t\in N
\end{array}\right. \,. $$
This maximal value
$\la\mu,\bar{e}\ra$ at majorizes the required quantity.
Note that $\bar{e}$ is bounded, so $\int|\bar{e}|\,\,d|\mu|<\infty$, but generally not continuous,
nor does it decay as $t\to+\infty$. It remains to
show that this upper bound is approached for
$e\in C_0$.
Let $\e>0$. By regularity \cite{Cohn} of the measure $|e^+-e^-|d\mu$, there are
$K_P\subseteq P\subseteq U_P$ and
$K_N\subseteq N\subseteq U_N$ ($U$ open, $K$ compact) for which
$$
\int_{(U_P\backslash K_P) \cup (U_N\backslash K_N) }
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!|e^+-e^-|\,\,d|\mu| \le \epsilon\,.
$$
Suitable shrinkage of $U_P$, $U_N$ can ensure that $U_P\cap K_N=\emptyset=U_N\cap K_P$
(since we can take $P$ and $N$ to be disjoint).
Let $\{ \varphi_P, \varphi_N \}$ be a partition-of-unity (see \cite{Dug}) relative to the open covering
$\{U_P, U_N\}$ for $\R_+$. We could now propose that $\widetilde{e}(t):=e_t^+(L)\varphi_P(t)+
e_t^-(L)\varphi_N(t)$, so that $\widetilde{e}(t)\in [e_t^-, e_t^+]$ for all $t$. However,
$\widetilde{e}$ again may not be continuous. To remedy this, we seek suitable continuous
approximations for $e^+$ and $e^-$.
Let $\bar{t}>0$ be any point of (jump-)discontinuity for $e^\pm$ ($e^+$ or $e^-$ or both).
For ease of presentation, we assume strict inequality in assumption 3.
For definiteness, assume $e_{\bar{t}-}^+> e_{\bar{t}+}^+ $. Form a line-segment, of
large negative slope $-\lambda$, from $ (\bar{t}, e_{\bar{t}+}^+)\in\R^2 $ to the point of first
contact with the graph of $e^+|_{[0,\bar{t}) }$. Using this, we define a function $\rho^+$,
whose {\em epigraph}\footnote{the epigraph of a function $f:X\to [-\infty, +\infty]$ is defined to be the set $\{(x,\lambda)\in X\times\R \mid f(x)\le\lambda \}$
of points `above' the graph of $f$. }
is the union of the epigraph of $e^+$ with the set of points above the
indicated line-segment. This yields a function continuous at $\bar{t}$, satisfying $\rho^+\le e^+$
on $\R_+$, with $\rho^+\equiv e^+$ except on a small interval
$[ \bar{t}-\alpha(\lambda),\bar{t}]$ where $\alpha(\lambda)\to 0$ as $\lambda\to +\infty$.
Further, since $e_{\bar{t}-}^+> e_{\bar{t}+}^+ $, on taking $\lambda$ large enough, we ensure, by
the continuity of $e^-_\cdot$ near (but not at) $\bar{t}$, that also $\rho^+(t)\ge e_t^-$, for all $t$.
An analogous construction applies (using line-segments of large {\em positive\/} slope) if instead
$e_{\bar{t}-}^+< e_{\bar{t}+}^+ $.
Similarly, we obtain $\rho^-$ continuous at $\bar{t}$, for which $e^-\le \rho^- \le e^+$ on $\R_+$
and $\rho^- \equiv e^-$ off a small interval at $\bar{t}$. It now follows that
\[
\lim_{\lambda\to +\infty }\int |\rho^\pm - e^\pm|\,d|\mu| =
\text{constant}\cdot[\text{jump at $\bar{t}$ for }e^\pm]\cdot |\mu|(\{0\}) =0\,.
\]
Repeating this process at each such $\bar{t}$ (these are countable in number) yields continuous functions on $\R_+$,
again denoted by $\rho^\pm$, such that $e^-\le \rho^\pm \le e^+$, with
$\int |\rho^\pm - e^\pm|\,d|\mu|\le \e/2$.
[Remark: if instead, have equality in Assumption 3.\ then the above line-segment construction will
not do---in this case one may
easily adapt this, by using nonlinear segments that are local graphs of continuous functions.]
Now, we may define continuous functions $\widetilde{e}$, by
$ \widetilde{e}:=\rho^+ (t)\varphi_P(t)+\rho^- (t)\varphi_N(t)$. Since the $0\le \varphi_{P, N}\le 1$ and
$\varphi_P + \varphi_N \equiv 1$ we get that $e^- \le \widetilde{e} \le e^+$. Also,
\begin{align*}
|\la\mu,\bar{e}\ra-\la\mu,\widetilde{e}\ra|&\le \int |\bar{e}-
\widetilde{e}|\,d|\mu| \\
& \le \int_{(U_P\backslash K_P) \cup (U_N\backslash K_N) }\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!\!|e^+-e^-|\,\,d|\mu|+
\int_{ K_P }\!\!\!|\rho^+-e^+|\,\,d|\mu| +
\int_{ K_N }\!\!\!|\rho^--e^-|\,\,d|\mu| \\
&\le 2\e
\end{align*}
At this stage, the proof is not quite complete, since
$\widetilde{e}$ may not tend to 0 as $t\to\infty$. For the final
step we form a sequence of truncations $\widetilde{e}_n$ such that:
$\widetilde{e}_n(t)=0$ for all $t\ge n+1$; they are continuous; and satisfy
$\la\mu,\widetilde{e}_n\ra \to \la\mu,\widetilde{e}\ra$ as $n\to\infty$.
We will ensure continuity for these truncations by again using steep
interpolating line-segments.
If $\widetilde{e}(n+1)=0$, we merely take $\widetilde{e}_n(t)=0$ for $t\ge n+1$. If instead,
$\widetilde{e}(n+1)$ is nonzero, we join the point $(n+1,0)\in\R^2$ to the graph of
$\widetilde{e}|_{[0,n+1)}$ by a steep-but-nonvertical line segment, to yield a function
$\widetilde{e}_n$ that vanishes on $[n+1,+\infty)$ and agrees with $\widetilde{e}$ on $[0,n]$,
with $| \widetilde{e}_n |\le |\widetilde{e} |$ on $\R_+$. Also, since $0\in [F_t\le L]$ for all
$t\ge t_0$, then on taking $n\ge t_0$ and noting that $\widetilde{e}(t)\in [F_t\le L]$ for all $t$,
we have $\widetilde{e}(t)\in \func{conv}\{0, \widetilde{e}(t) \} \subseteq[F_t\le L]$ for all $t$.
Thus, $ \widetilde{e}_n \in C_0(\R_+)$, $f(\widetilde{e}_n)\le L$, and
\begin{align*}
\int|\widetilde{e}_n-\widetilde{e}|\,\,d|\mu| & = \int_{n }^\infty\!\!\!|\widetilde{e}|\,\,d|\mu|+
\int_{n}^{n+1} \!\!\!|\widetilde{e}_n-\widetilde{e}|\,\,d|\mu| \\
&\le 3 \int_{n}^{\infty}\!\!\! |\widetilde{e}|\,\,d|\mu|\qquad \text{ since }\,|\widetilde{e}_n |\le | \widetilde{e}|\\
& \,\,\to 0 \quad\text{ as }n\to\infty\,,
\end{align*}
and so it follows that
$$
\la\mu,\bar{e}\ra \le \la\mu,\widetilde{e}\ra+2\e \le \la\mu,\widetilde{e}_n\ra +3\e \le
\sup_{\scriptsize \begin{array}{c}
e\in C_0\\
f(e)\le L \end{array} }\la\mu,e\ra
+3\e\,.
$$
As $\e>0$ is arbitrary, this establishes the required equality, and hence the formula for $f^*(\mu)$, for the case
of bounded $e^\pm$.
For the unbounded case, for each $R>0$, let $F_t^R:= F_t +\iota_{[-R,+R ]}$, where the latter denotes the indicator--function for
an interval. Then the sublevel--sets of $F_t^R$ have the form $[F_t^R\le L]=[ e_t^{R-}(L) , e_t^{R+}(L) ]$ where
\[
e_t^{R+}(L) :=\min\{R, e_t^{ +}(L)\}\,, \quad\text{ and } \quad e_t^{R-}(L) :=\max\{-R, e_t^{ -}(L)\}\,.
\]
Define $f_R(e):= \sup_{t\ge 0}F_t^R(e(t))$. Then, for any allowed $\mu$ (since the $e^{R\pm}$ are bounded and $f_R$ satisfies
Assumptions 1---4) we may apply the previous argument to $f_R$, to yield
\[
f_R^*(\mu)=\sup_{ L\ge 0 } \left(\int_0^{+\infty}\!\! \!e_t^{R+}(L)d\mu_+(t)+
\int_0^{+\infty }\!\!\!e_t^{R-}(L)d\mu_-(t)-L\right)
\]
for any allowed $\mu$.
By an easy check, for any $e\in C_0$ and any $R>\sup|e|$,
$
f(e)=f_R(e)=\inf_{R>0} f_R (e)=\lim_{R\to\infty}f_R(e)\quad(\text{since }f_R\downarrow \text{ as } R\uparrow)
$
so that
\begin{align*}
f^*(\mu) & =\sup_{R\ge 0} f_R^*(\mu) =\sup_{L\ge 0} \,\sup_{R\ge 0}
\left(\int_0^{+\infty}\!\! \!e ^{R+}(L)d\mu_+ +
\int_0^{+\infty }\!\!\!e ^{R-}(L)d\mu_- -L\right)\\
& =
\sup_{L\ge 0}
\left(\int_0^{+\infty}\!\! \!e ^{+}(L)d\mu_+ +
\int_0^{+\infty }\!\!\!e ^{-}(L)d\mu_- -L\right)\\
\end{align*}
the latter equality (in $\R \cup\{+\infty\}$) following from the monotone convergence $e^{R+}\uparrow e^+$ and $e^{R-}\downarrow e^-$.
\section{Appendix B: Proof of Proposition~\ref{prop:summary}}
The proof will
require a couple of preparatory lemmas. We remind the reader of
the following {\bf Notation:}
$$\R_+:= [0,\infty)$$
$$ C_0 (\R_+):=\{\varphi:\R_+\to \R\mid \varphi \text{ continuous,
} \varphi(\infty)=0 \}$$
$$ C_{0,\alpha}(\R_+):=\{\varphi\in C_0(\R_+)\mid\varphi(0)=\alpha \}$$
$$
\underline{\theta}:= (\theta_p, \theta_w),\quad\text{where}\quad \theta_p\geq
0,\,\,\theta_w\geq 1; \quad\text{ and }\quad
\theta:=\theta_p+\theta_w
$$
It will
be seen that the closure of $X_{\underline{\theta}}$ is always one of
$C_0$ or $C_{0,\alpha}$ (for suitable $\alpha$), depending on the
value of $\theta_w,\theta_p$.
\bigskip
We start with a simple result for Laplace integrals.
\begin{lem}\label{lem:A1} Let $\varphi:\R_+ \to \R$ such that
$\varphi^{(k)}(t)=O(e^{at})$ (for some $a>0$)
and $\varphi^{(i)}(0+)$ exists and finite, for $i=0,1,\ldots k$.
Then for all $s\in \C$ for which $\func{Re}s>a$,
$$
\widehat{\varphi^{(k)}}(s)=\widehat{\varphi}(s)s^k-\varphi(0+)s^{k-1}
-\varphi^{'}(0+)s^{k-2} - \ldots - \varphi^{(k-2)}(0+)s - \varphi^{(k-1)}(0+) \,.
$$ \end{lem}
\begin{Proof} Integrating by
parts yields
$\widehat{\varphi^{(k)}}(s)=s\widehat{\varphi^{(k-1)}}(s)-
\varphi^{(k-1)} (0+)$ for such $s$. Repeat for $\varphi^{(k-1)}$
etc.
\end{Proof}
\smallskip
We assume that the function $w$ has rational Laplace
transform. This ensures that $\widehat{w^{(i)}}(s)\to 0 $ as
$\text{Re }s\to +\infty$, and $w^{(i)}(0+)=w^{(i)}(0)$, for all
$i=0,1,2 \ldots $. Moreover, Lemma~\ref{lem:A1} applies, for every
$k$.
Now, define
\begin{align*}
X_{\underline{\theta}} :=\{ e\in C_0(\R_+)\mid
\hat{e} & \text{ rational, } \hat{e}(\infty)=0 \text{ (order at least }\theta_w) \\
& \widehat{(e-w)}(\infty)=0 \text{ (order at least
}\theta=\theta_w +\theta_p) \}.
\end{align*}
Also, form the subspace
$$
\begin{array}
[c]{cc}
Y_{\underline{\theta}}:= & \left\{ e: t\mapsto\text{Re}\sum_{i\in I}c_i
t^{k_i}e^{-\lambda_i t}\,\bigg{|}\,
\begin{array}
[l]{l}
c_i\in \C,\,k_i\geq \theta_w-1,\, \text{Re
}\lambda_i>0,\,
I \text{ finite,} \\
e^{(j)}(0)=w^{(j)}(0+), \,j=0,1,\ldots ,\theta-2
\end{array}
\right\}
\end{array}
$$
\begin{lem}\label{lem:A2}\quad $Y_{\underline{\theta}}\subseteq
X_{\underline{\theta}}$.
\end{lem}
\begin{Proof} If $e\in Y_{\underline{\theta}}$, then clearly
$\hat{e}$ has a zero of order $\theta_w$ at infinity. Now,
$[(e-w)^{(\theta-1)}]\,\widehat{}\, (s)\to 0$ when Re
$s\to\infty$. By Lemma~\ref{lem:A1} applied to $\varphi:=e-w$, we
obtain, when $s\to\infty$,
\begin{align*}
0\leftarrow [(e-w)^{(\theta-1)}]\,\widehat{}\, (s)
&=s^{\theta-1}(\hat{e}(s)-\hat{w}(s))-s^{\theta-2}(e-w)(0+)-\ldots
-(e-w)^{(\theta-2)}(0+)\\ & =s^{\theta-1}(\hat{e}(s)-\hat{w}(s))
\end{align*}
on
using the constraints at $t=0+$. Hence $\hat{e}-\hat{w}$ has a
$\theta$--order zero at $\infty$.
\end{Proof}
\begin{lem}\label{lem:A3} If
$e\in X_{\underline{\theta}}$, then:
$e^{(j)}(0)=0$ for $j=0,\ldots ,\theta_w\!-2$; and also
$e^{(j)}(0)=w^{(j)}(0+)$ for $j=0,\ldots ,\theta-2$.
\end{lem}
\begin{Proof}
Let $k:=\theta\!-\!2$. Now $[(e-w)^{(k+1)}]\,\widehat{}\, (s)\to
0$ at infinity. From Lemma~\ref{lem:A1},
$s^{k+1}(\hat{e}(s)-\hat{w}(s))-(e(0)-w(0+))s^k-\ldots -
(e^{(k)}(0)-w^{(k)}(0+))=[(e-w)^{(k+1)}]\,\widehat{}\, (s)\to 0$.
By assumption on $e$, have $s^{k+1}(\hat{e}-\hat{w})(s)\to 0$ at
infinity, and this forces the equalities $e(0)=w(0+),\ldots
,e^{(k)}(0)=w^{(k)}(0+)$. For $j=0,\ldots, \theta_w \! -\!2$,
argue as above using $e$ instead of $e-w$, to get $e^{(j)}(0)=0$.
\end{Proof}
\begin{remark}\label{rem:1} If $\theta_w\ge 2$, then Lemma~\ref{lem:A3}
requires $w^{(j)}(0+)=0$ for $j=0,\ldots,\theta_w-2$, otherwise
the statement of the lemma entails a contradiction, and
$X_{\underline{\theta}}$ will be empty, hence of no interest. If
$\theta_w=1$, the conditions $e^{(j)}(0)=0$ ($j\le \theta_w-2$)
are absent, so $w(0+)$ is unrestricted in this case.
\end{remark}
For any $\underline{\theta}$, form the set
$W_{\underline{\theta}}\supseteq Y_{\underline{\theta}}$ by
removing the constraints at $t=0$ in the latter, that is,
$$
W_{\underline{\theta}}:=\left\{e(t)=\text{Re}\sum_{i\in I}c_i
t^{k_i}e^{-\lambda_i t}\, \bigg{|}\, c_i\in \C,\,k_i\geq
\theta_w-1,\, \text{Re }\lambda_i>0,\,
I \text{ finite}
\right\}.
$$
Observe that $e(t)=O(t^m)$ at $t=0$ (where $m:=\theta_w-1$), it
follows that $e^{(j)}(0)=0$ for $j=0,1,\ldots,m-1$ when
$\theta_w>1$.
\bigskip
We turn now to the proof of the Proposition. Because of the
preceding lemmas, it suffices to prove density for the subsets
$Y_{\underline{\theta}}$ of $X_{\underline{\theta}}$.
\subsubsection{Case of $\theta_w> 1$}
If $\theta_w>1$, note that $\varphi(0)=0=\varphi(+\infty)$
whenever $\varphi\in W_{\underline{\theta}}$. Also,
$W_{\underline{\theta}} $ forms a algebra of continuous functions
on the {\em one-point compactification\/} (see \cite{Dug})
$[0,+\infty]$ of $\R_+$, since
$\varphi(+\infty)=\lim_{t\to\infty}\varphi(t)$ exists for each
such $\varphi$. We seek to apply the Stone--Weierstrass Theorem
(see, for instance, \cite{Dug}, or any functional analysis text)
to establish the density of $W_{\underline{\theta}}$ in a suitable
subspace of $C([0,+\infty])$. However, there is a difficulty, in
that $W_{\underline{\theta}}$ does not `separate all points' of
$[0,+\infty]$ in the sense of the Theorem---indeed, by definition
of $W_{\underline{\theta}}$, the points $0$ and $+\infty$ cannot
be so separated (if $\varphi\in W_{\underline{\theta}}$, it
attains equal values at $0$ and at $+\infty$). This is the only
pair of points that cannot be separated in this sense. We resolve
this issue by insisting that $0$ and $+\infty$ are `the same
point'.
To do this, let $X$ denote the the quotient space, formed by
identifying the points $0$ and $+\infty$ in $[0,+\infty]$, and
endowing this with the quotient topology \cite{Dug}. Then $X$ is
also compact, being the image of $[0,+\infty]$ under the quotient
map: $[0,+\infty]\to X$, which is continuous by definition of the
quotient topology. It is easily verified that $X$ is also
Hausdorff. Each member of $W_{\underline{\theta}} $ may now be
viewed as a continuous function on $X$, since it has a
well-defined value (zero) at $[0]=[+\infty]\in X$.
Clearly, $W_{\underline{\theta}}\oplus\R :=\{\varphi + c\!\mid \!\varphi\in W_{\underline{\theta}},\,\,
c\in\R\} $ forms a subalgebra of $C(X)$ that contains the constant
functions, and now separates the points of $X$ (since the
offending pair in $[0,+\infty]$ have been merged). We may now
apply Stone--Weierstrass, to conclude that
$W_{\underline{\theta}}\oplus\R$ is dense in $C(X)$ under the
supremum--norm.
Returning to $[0,+\infty]$, this means that any $\varphi\in
C([0,+\infty])$ for which $\varphi(0)= \varphi(+\infty)$ may be
uniformly approximated by functions from
$W_{\underline{\theta}}\oplus\R$, whence follows that
$\overline{W_{\underline{\theta}} }=C_{0,0}(\R_+)$.
\smallskip
\smallskip
To finally prove the required property, let $\varphi\in
C_{0,0}(\R_+)$, and let $\e
>0$. By the density of $W_{\underline{\theta}}$, there is $\widetilde{\varphi}\in
W_{\underline{\theta}}$ such that
$\|\widetilde{\varphi}-\varphi\|_\infty \le \e $. If $\theta_p=0$,
we are done, since in this case,
$W_{\underline{\theta}}=Y_{\underline{\theta}}$. If $\theta_p>0$,
then $\widetilde{\varphi}$ may not satisfy the conditions at
$t=0$ required for membership of the subset
$Y_{\underline{\theta}}$, so we need to perturb
$\widetilde{\varphi}$ to achieve this. Place
$$
\xi := \left( w^{ }(0+)-\widetilde{\varphi}^{ }(0),\ldots,
w^{(\theta_w+\theta_p-2)
}(0+)-\widetilde{\varphi}^{(\theta_w+\theta_p-2 ) }(0)\right)
=(0,\ldots,0,\xi_{\theta_w-1}, \ldots,\xi_{\theta_w+\theta_p-2 })
$$
the latter obtaining since $\widetilde{\varphi}^{(j ) }(0)=0$ for
$j\le \theta_w-2$ and also if $X_\theta\neq\emptyset$ we must have
$w^{(j) }(0+)=0$ for such $j$.
We now seek $p(\cdot)\in W_{\underline{\theta}}$ such that
$\widetilde{\varphi}+p\in Y_{\underline{\theta}}$ and
$\|p\|_\infty$ is suitably small. We slightly modify a procedure
used
in \cite{Miller}. Try $p(t)=t^m p_0(t)$ where $m=\theta_w-1$, and
$p_0(t)=\sum_{l=0}^q c_l e^{-(l+1)\sigma t}$ with $q:=\theta_p-1$,
and $\sigma >0$ so large that $\sup_{t\ge 0}t^m e^{-\sigma t}\le
\e/(1+|\xi|)$. Then, for $k$ such that $m\le k\le
m+q=\theta_w+\theta_p-2$, have $p^{(k)}(0)=
[k!/(k-m)!]p_0^{(k-m)}(0)$ (as well as $p^{(k)}(0)=0$ for $k< m$).
For $\widetilde{\varphi}+p$ to satisfy the required constraint at
$t=0$, we need
$$p_0^{(k-m)}(0)=[(k-m)!/k!]\xi_k\qquad k=m,m+1,\ldots,m+q$$
or, equivalently,
$$
\left[\begin{array}{cccc}
1 & 1 & \dots & 1\\
1 & 2 & \dots & q+1\\
1 & 2^2 & \dots & (q+1)^2\\
\vdots & \vdots & \ddots & \vdots\\
1 & 2^q & \dots & (q+1)^q
\end{array}\right]
\left[\begin{array}{c}
c_0 \\
c_1 \\
\vdots \\
c_q
\end{array}\right]
= \left[\begin{array}{c}
0!\xi_m/m! \\
1! \xi_{m+1}/(- (m+1)!\sigma)\\
\vdots\\
q!\xi_{m+q}/((-1)^q (m+q)!\sigma^q)
\end{array}\right]
$$
Let $\gamma$ denote a suitable norm of the inverse of the above
square matrix. Place $K(q)=(\gamma(q)+1)(q+1)$---note that it
depends {\em only\/} on $q$. Solving the above for the
coefficients $c_l$ yields $p_0$ satisfying $ |p_0(t)|\le
K(q)|\xi|e^{-\sigma t} $ for all $t\ge 0$, so that
$$
|p(t)|\le K(q)t^m e^{-\sigma t}|\xi|\le K(q)\e \quad\text{ for all
}t\ge 0\,.
$$
Thus it follows for this choice of $p$ that
$\widetilde{\varphi}+p\in Y_{\underline{\theta}}$, and
$$
\|\widetilde{\varphi}+p-\varphi\|_\infty \le \e + \|p\|_\infty \le
(1+K(q))\e\,.
$$
Since $K$ does not depend on the choice of $\varphi$ or
$\widetilde{\varphi}$ etc, it follows from the arbitrariness of
$\e>0$ that $Y_{\underline{\theta}}$, and hence
$X_{\underline{\theta}}$, is dense in $C_{0,0}(\R_+)$, as claimed.
\subsubsection{Case of $\theta_w= 1$}
The same reasoning applies, but without recourse to quotient
spaces of $[0,+\infty]$.
When $p>0$, we obtain that $Y_{\underline{\theta}}$ is dense in
$C_{0,\alpha}(\R_+)$, with $\alpha=w(0+)$.
If $p=0$, the constraints at $t=0$ in the definitions of
$Y_{\underline{\theta}}$ and $W_{\underline{\theta}}$ are absent,
and arguing as above shows that $Y_{\underline{\theta}}$ is dense
in the entire space $C_0(\R_+)$.
\bigskip
This completes the proof of Proposition~\ref{prop:summary}.
\section {Appendix C: Proof of Lemma~\ref{lem:fhash}}
We start with
a lemma.
\begin{lem}\label{lem:mu}
Let $\mu\in {\bf M}(\R_+)$. Then, $$ \|\mu
+\lambda\delta\|=\|\mu\|+|\mu(\{0\})+\lambda|-|\mu(\{0\})|\,.$$ In
particular, if $\mu(\{0\})=0$ (for example, when $\mu\in U\oplus
V\oplus W$) we have $ \|\mu +\lambda\delta\|=\|\mu\|+ |\lambda| $.
\end{lem}
\begin{Proof} By definition,
\begin{align*}
\|\mu +\lambda\delta\|&=|\mu +\lambda\delta|(\R_+)=\sup
\left\{\sum_{I\in\Pi}|\mu +\lambda\delta(I)| : \Pi \text{
partitions $\R_+$ into intervals} \right\}\\
&=\lim_{\|\Pi\|\to 0}\left( \sum_{I:0\notin I}|\mu(I)| +
|\mu(I_0)+\lambda|\right)
\end{align*}
where $I_0$ denotes the unique member of $\Pi$ that contains $0$.
Since $I_0\downarrow \{0\}$ for a subsequence of partitions, we
have $\mu(I_0)\to \mu(\{0\})$, so that
\begin{align*}
\|\mu +\lambda\delta\|&= \lim_{\|\Pi\|\to 0}\left( \sum_{I \in
\Pi}|\mu(I)| + |\mu(I_0)+\lambda|-|\mu(I_0)|\right)\\
&=\|\mu\|+|\mu(\{0\})+\lambda|-|\mu(\{0\})|
\end{align*}
\end{Proof}
Now, consider $f=f_{os}$. Then,
\begin{align*}
f_{os}^{\#}(\mu) &= \max_ {\lambda\in{\scriptsize
\R}}\,[\alpha\lambda-f_{os}^*(\mu+\lambda\delta)]\\
&=\max\{\alpha\lambda\mid \mu+\lambda\delta\le
0,\,\,\|\mu+\lambda\delta\|\le 1\}\\
&=\alpha\max\{\lambda \mid \mu+\lambda\delta\le 0,\,\,\|\mu\|+
|\lambda|\le 1\} \text{ by Lemma~\ref{lem:mu}}
\end{align*}
Let $\mu\in U\oplus V\oplus W$. If $\|\mu\|>1$, then $\|\mu
+\lambda\delta\|=\|\mu\|+ |\lambda|>1$ for each $\lambda$, so
$f_{os}^*(\mu+\lambda\delta)=+\infty$ for all $\lambda$, and
$f_{os}^{\#}(\mu)=-\infty=-f_{os}^*(\mu)$. If $\mu$ is not a
negative measure, then $\mu(E)>0$ for a Borel set $E$ in $\R$
(with $0\notin E$), implying for all $\lambda\in\R$, that
$(\mu+\lambda\delta)(E)=\mu(E)>0$, so $\mu+\lambda\delta$ also not
negative, and again, $f_{os}^*(\mu+\lambda\delta)=+\infty$ for
such $\lambda$, yielding
$f_{os}^{\#}(\mu)=-\infty=-f_{os}^*(\mu)$.
If $\mu\le 0$ and $\|\mu\|\le 1$ (so $\mu\in\func{dom}f_{os}^*$),
then, since $\mu+\lambda\delta\le 0$ implies $\lambda\le 0$ (take
$E=\{0\}$), it follows that $f_{os}^{\#}(\mu)=0$, and hence equals
$-f_{os}^{*}(\mu)$.
This proves that $f_{os}^{\#} =-f_{os}^*$ on $ U\oplus V\oplus W$.
The proof that $f_{us}^{\#} =-f_{us}^*$ is similar.
\smallskip
To treat $f_{fl}$, we note an elementary lemma.
\begin{lem}\label{lem:muFL}
Let $\mu\in {\bf M}(\R_+)$ with $\mu(\{0\})=0$. Then,
$(\mu+\lambda\delta)_\pm= \mu_\pm + \lambda_\pm\delta$ for any
$\lambda\in\R$.
\end{lem}
Then, by similar arguments to those above,
$f^{\#}_{\text{fl}}(\mu)=-\infty$ if $\mu\notin\func{dom}f^{*}_{\text{fl}}=
\{\mu\mid \mu_+(\R_+)\le 1/2 \, \, \& \,\,\mu_-(\R_+)\ge -1/2\}$,
and for $\mu\in\func{dom}f^{*}_{\text{fl}}$,
$f^{\#}_{\text{fl}}(\mu)=\alpha(1/2-\mu_+(\R_+))$. Thus the required
relation for $f_{\text{fl}}$ is also established.
\smallskip
Similar reasoning applies for the remaining functionals.
\bigskip
| 110,687
|
By Charles F. Bass, Concord Monitor.
As a former member of the New Hampshire House and Senate, I have watched with dismay efforts undertaken by Republican legislators to roll back the laws that were passed to encourage energy diversity and native energy production in our state. I am equally saddened by efforts to withdraw from the Regional Greenhouse Gas Initiative. These efforts are neither good policy nor consistent with traditional Republican values.
I would remind my fellow Republicans that it was under our leadership that the White Mountain National Forest was established. John H. Sununu was governor when the Land Conservation Investment Program was established. Governor Judd Gregg doubled the size of the LCIP program, and authored the Rivers Protection and Shoreline Protection acts. And at the federal level over the last 100 years, almost all major environmental initiatives, including the creation of the EPA, occurred under Republican presidents.
Indeed the very words “conservation” and “conservative” share the same derivation. Good environmental and energy policy should be a core conservative value.
| 11,201
|
\begin{document}
\def\proofend{\hbox to 1em{\hss}\hfill $\blacksquare
$\bigskip }
\def\powser#1{\lbrack \lbrack #1 \rbrack \rbrack }
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{remarks}[theorem]{Remarks}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\newtheorem{assumption}[theorem]{Assumption}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{question}[theorem]{Question}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{rigiditytheorem}[theorem]{Rigidity Theorem}
\newtheorem{observation}[theorem]{Observation}
\def\Z{{\mathbb Z}}
\def\R{{\mathbb R}}
\def\Q{{\mathbb Q}}
\def\C{{\mathbb C}}
\def\N{{\mathbb N}}
\def\H{{\mathbb H}}
\def\Zp #1{{\mathbb Z }/#1{\mathbb Z}}
\def\cpt{compact}
\def\wt{wt}
\def\cowt{co\wt}
\def\achtel{\frac {\dim M} 8}
\def\torus{$T$}
\def\mathtorus{T}
\def\codim{{\rm{codim}\ }}
\def\b{bun\-dle}
\def\pb{principal \b }
\def\vb{vector \b }
\def\mfd{manifold}
\def\LFF{Lefschetz fixed point formula}
\def\isorank{symrank}
\def\ell{\varphi _{ell}}
\def\eell{\widetilde \ell }
\def\ddelta {{\widetilde \delta }}
\def\eepsilon{{\widetilde \epsilon }}
\def\CC{C_0}
\def\oorder{o}
\def\oha{\cal H}
\def\paperref#1#2#3#4#5#6{\text{#1:} #2, {\em #3} {\bf#4} (#5)#6}
\def\bookref#1#2#3#4#5#6{\text{#1:} {\em #2}, #3 #4 #5#6}
\def\preprintref#1#2#3#4{\text{#1:} #2 #3 (#4)}
\hyphenation{man-i-fold equiv-a-riant in-te-ger mod-u-lo tor-sion
re-pre-sen-ta-tion di-men-sion-nal}
\title{Cyclic actions and elliptic genera}
\author{Anand Dessai}
\date{}
\maketitle
\begin{abstract}
\noindent
Let $M$ be a $Spin$-manifold with $S^1$-action and let $\sigma \in S^1$ be of finite order. We show
that the indices of certain twisted Dirac operators vanish if the action of $\sigma $ has
sufficiently large fixed point codimension. These indices occur in the Fourier expansion of the
elliptic genus of $M$ in one of its cusps. As a by-product we obtain a new proof of a theorem of
Hirzebruch and Slodowy on involutions.
\end{abstract}
\section{Introduction}\label{section intro}
Let $M$ be a smooth closed connected $Spin$-manifold with smooth $S^1$-action and let $\sigma
\in S^1$ be the element of order two. Hirzebruch and Slodowy \cite{HiSl} showed that the elliptic
genus of $M$ can be computed in terms of the transversal self-intersection of the fixed point
manifold $M^\sigma $ and used this property to deduce a vanishing theorem for certain characteristic numbers which occur in the Fourier expansion of the elliptic genus of $M$ in
one of its cusps.
In this note we extend this vanishing theorem from involutions to cyclic
actions of arbitrary order. Our main result (see Theorem \ref{theorem cyclic}) is used in \cite{De}
to exhibit obstructions against the existence of positively curved metrics with symmetry on $Spin$-manifolds. The proof of Theorem \ref{theorem cyclic} relies on the rigidity theorem for the elliptic genus which we shall
recall first. As a general reference for the theory of elliptic genera we recommend \cite{HiBeJu,
La}.
The elliptic genus $\Phi $, in the normalization considered in \cite{HiSl, Wi}, is a ring
homomorphism from the oriented bordism ring to the ring of modular functions (with $\Zp 2$-character)
for $\Gamma
_0(2):=\{A\in SL_2(\Z )\; \mid \; A\equiv (\begin{smallmatrix} * & *\\0 &
*\end{smallmatrix})\bmod 2\}$. In one of the cusps of $\Gamma _0(2)$ (the
signature cusp) the Fourier expansion of $\Phi (M)$ has an interpretation as a series of twisted
signatures
$$sign (M,\bigotimes _{n=1}^\infty
S_{q^n}TM \otimes
\bigotimes
_{n=1}^\infty \Lambda_{q^n}TM)=sign(M)+2\cdot sign(M,TM)\cdot q +\ldots .$$
Here $sign(M,E)$ denotes the index of the signature
operator twisted with the complexified \vb \ $E_\C $, $TM$ denotes the tangent bundle and $\Lambda
_t=\sum \Lambda
^i\cdot t^i$ (resp. $S_t=\sum S^i\cdot t^i$) denotes the exterior
(resp. symmetric) power operation.
Following Witten \cite{Wi} the series above is best thought of as the ``signature'' of the free loop
space ${\cal L}M$ of $M$ formally localized at the manifold $M$ of constant loops. We denote the
series of twisted signatures by $sign(q,{\cal L}M)$.
The main feature of the elliptic genus is its rigidity under $S^1$-actions. This phenomenon was first
explained by Witten \cite{Wi} using standard conjectures from quantum field theory and then shown
rigorously by Taubes and Bott-Taubes in \cite{BoTa, Ta} (cf. also \cite{Hi2, Li}).
If $S^1$ acts by isometries\footnote{This is the case after averaging a given Riemannian metric over
the $S^1$-action.} on $M$ and if $E$ is a vector bundle associated to $TM$ then the signature
operator twisted with the complexified \vb \ $E_\C $ refines to an $S^1$-equivariant operator. Its
index is a virtual $S^1$-representation which we denote by $sign_{S^1}(M,E)
\in R(S^1)$. In particular, the expansion of the elliptic genus in the signature cusp refines to a
series of equivariant twisted signatures $sign_{S^1}(q,{\cal L}M)\in R(S^1)\lbrack \lbrack q\rbrack
\rbrack $.
\begin{theorem}[Rigidity theorem \cite{BoTa, Ta}]\label{rigidity theorem} Let $M$ be a closed manifold with $S^1$-action. If $M$ is $Spin$ then each equivariant twisted
signature occurring as coefficient in the series $sign_{S^1}(q,{\cal L}M)$ is constant as a character
of $S^1$.\proofend
\end{theorem}
\noindent We use the rigidity theorem
to study the action of cyclic subgroups of $S^1$. Our investigation is inspired by work of Hirzebruch
and Slodowy \cite{HiSl} on elliptic genera and involutions. As a motivation we shall briefly recall relevant aspects of their work.
Let $M$ be a $Spin$-manifold with $S^1$-action and let $\sigma
\in S^1$ be of order two. By the rigidity theorem the expansion of the
elliptic genus in the signature cusp is equal to the $S^1$-equivariant expansion evaluated at $\sigma
\in S^1$, i.e. $sign(q,{\cal L}M)=sign_{S^1}(q,{\cal L}M)(\sigma )$. The latter can be computed via the
\LFF \ \cite{AtSi} as a sum of local contributions $a_F$ at the connected components $F$ of the fixed point manifold
$M^{\sigma }$. Hirzebruch and Slodowy showed that $a_F$ is equal to the expansion of the
elliptic genus (in the signature cusp) of the transversal self-intersection $F
\circ F $ (cf. \cite{HiSl} for details):
\begin{equation}\label{formula elliptic}
sign (q,{\cal L}M)=sign _{S^1}(q,{\cal L}M)(\sigma ) \end{equation} $$=\sum
_{F\subset M^\sigma }sign(q,{\cal L}(F
\circ F ))=sign(q,{\cal L}(M^\sigma \circ
M^\sigma ))$$
Note that, by taking constant terms, one obtains the
classical formula $sign (M)=sign(M^\sigma
\circ M^\sigma )$ for the ordinary signature which holds for the
larger class of oriented manifolds (cf. \cite{Hi1,JaOs}).
Formula (\ref{formula elliptic}) has two immediate consequences. If the codimension of $M^\sigma $,
$\codim M^\sigma
:=\min _{F\subset M^\sigma } \codim F$, is greater than half of the dimension
of $M$ then the series $sign (q,{\cal L}M)$ vanishes identically. If the codimension of $M^\sigma $
is equal to half of the dimension of $M$ then all the twisted
signatures occurring as coefficients of $q^n$, $n>0$, in the series $sign (q,{\cal L}M)$ vanish,
i.e. $sign (q,{\cal L}M)=sign(M)$.
If the codimension of $M^\sigma $ is less than half of the dimension of $M$ then formula
(\ref{formula elliptic}) still gives some information on the action of the involution $\sigma $.
Namely it implies that certain twisted Dirac operators have vanishing index provided that the
codimension of $M^\sigma $ is sufficiently large. These indices are related to the elliptic genus in
the following way. Recall that the $q$-series $sign (q,{\cal L}M)$ is the expansion of the elliptic
genus $\Phi (M)$ in one of the cusps of $\Gamma
_0(2)$. In a different cusp (the $\hat A$-cusp) the expansion
of $\Phi (M)$ may be described (using a suitable change of cusps) by
$$\Phi _0(M):=q^{-\dim M/8}\cdot \hat A(M,\bigotimes _{n=2m+1>0}\Lambda _{-q^n}TM \otimes
\bigotimes
_{n=2m>0}S_{q^n}TM)$$
$$=q^{-\dim M/8}\cdot (\hat A(M) -\hat A(M,TM)\cdot q +\hat A(M,\Lambda ^2{TM}+TM)\cdot
q^2+\ldots ).$$ Here $\hat A(M,E)$ is a characteristic number of the pair $(M,E)$ which, in the
presence of a $Spin$-structure, is equal to the index of the Dirac operator twisted with the
complexified
\vb \ $E_\C $. We call the series above the expansion of $\Phi (M)$ in the $\hat
A$-cusp.
Note that $\Phi
_0(M)$ and $sign(q,{\cal L}(M))$ are different expansions of the same
modular function $\Phi (M)$ and determine each other. By formula (\ref
{formula elliptic}) $\Phi _0(M)=\Phi _0(M^\sigma \circ M^\sigma )$ which implies the following generalization of the Atiyah-Hirzebruch vanishing
theorem for the $\hat A$-genus \cite{AtHi}.
\begin{theorem}[\cite{HiSl}]\label{theorem involution} Let $M$ be a $Spin$-manifold with
$S^1$-action and let $\sigma \in S^1$ be of order two. If $\codim M^\sigma >4r$ then the expansion of
the elliptic genus of $M$ in the $\hat A$-cusp has a pole of order less than $\achtel -r$.\proofend
\end{theorem}
\noindent
The reasoning indicated above also leads to obstructions against the existence of $S^1$-actions on highly
connected manifolds which might be of independent interest.
\begin{theorem}\label{observation} Let $M$ be a $k$-connected $Spin$-manifold. Assume $k\geq 4r$. If $M$ admits a non-trivial $S^1$-action then the
expansion of the
elliptic genus of $M$ in the
$\hat A$-cusp has a pole of order less than $\achtel -r$.
\end{theorem}
\bigskip
\noindent
Note that for $r>0$ the $Spin$-condition follows from the connectivity assumption. We remark that the
conclusion of Theorem \ref{observation} also holds if $M$ is a connected $Spin$-manifold with
non-trivial $S^1$-action and $H^{4*}(M;\Q )=0$ for $0<*\leq r $ (see Section \ref{highly connected}
for a proof).
The next result extends Theorem \ref{theorem involution} to finite cyclic actions of arbitrary order .
\begin{theorem}\label{specialcase}
Let $M$ be a $Spin$-manifold with
$S^1$-action and let $\sigma \in S^1$ be of order $\oorder \geq 2 $. If $\codim M^\sigma
> 2\oorder \cdot r$ then the expansion of the elliptic genus of $M$ in the
$\hat A$-cusp has a pole of order less than $\achtel -r$.
\end{theorem}
\noindent
The theorem follows from a more general result (see Theorem \ref{theorem cyclic} and the proof in Section \ref{section proof}). As indicated above the proof of Theorem \ref{theorem involution} given in \cite{HiSl} is specific to
actions of order two. To deal with the general situation we consider the expansion of the equivariant
elliptic genus in the $\hat A$-cusp and study the local contributions of the $S^1$-fixed point
components using the rigidity theorem. We close this section with some consequences of Theorem
\ref{specialcase}.
\begin{corollary} Let $M$ be a $Spin$-manifold with
$S^1$-action.
\begin{enumerate}
\item Let $\sigma \in S^1$ be of order $3$. If $\codim M^\sigma
> 0$ then $\hat A(M)$ vanishes. If $\codim M^\sigma
> 6$ then $\hat A(M)$ and $\hat A(M,TM)$ vanish. If $\sigma $ acts with isolated fixed points then $\Phi (M)$ vanishes identically.
\item Let $\sigma \in S^1$ be of order $4$. If $\codim M^\sigma
> 0$ then $\hat A(M)$ vanishes. If $\codim M^\sigma
> 8$ then $\hat A(M)$ and $\hat A(M,TM)$ vanish. If $\sigma $ acts with isolated fixed points then $\Phi (M)$ is equal to the signature of $M$.
\item Let $\sigma \in S^1$ be of order $\oorder <\frac {dim M}2$. If $\sigma $ acts with isolated fixed points then $\hat A(M)$ and $\hat A(M,TM)$ vanish.\proofend
\end{enumerate}
\end{corollary}
\section{Cyclic actions}\label{section cyclic actions}
In this section we state the main result of this note. Let $M$ be a connected $S^1$-manifold and let
$\oorder \geq 2$ be a natural number. At a connected component $Y$ of the fixed point manifold
$M^{S^1}$ the tangent bundle $TM$ splits equivariantly as the direct sum of $TY$ and the normal
bundle $\nu $. The latter splits (non-canonically) as a direct sum $\nu =\bigoplus_{k\neq 0} \nu _k$
corresponding to the irreducible real $2$-dimensional $S^1$-representations $e^{i\cdot
\theta }\mapsto \left (\begin{smallmatrix}
\cos k\theta & -\sin k\theta \\
\sin k\theta &\cos k\theta
\end{smallmatrix}\right )$, $k\neq 0$. We fix such a decomposition of $\nu $.
For each $k\neq 0$ choose $\alpha
_k\in \{\pm 1\}$ such that $\alpha _k k\equiv \tilde k \bmod \oorder $,
$\tilde k\in
\{0,\ldots , \lbrack
\frac o 2\rbrack \}$.
On each vector bundle
$\nu _k$ introduce a complex structure such that $\lambda \in S^1$ acts on $\nu _k$ by scalar
multiplication with $\lambda ^{\alpha _k k}$. The $\alpha _k k's$ (taken with multiplicities) are
called the rotation numbers of the $S^1$-action at
$Y$. Finally define
$$m_\oorder (Y):=(\sum _k d_k
\cdot \tilde k )/\oorder \quad \text{ and }\quad m_\oorder :=\min _{Y} m_\oorder (Y),$$
where $d_k$ denotes the complex dimension of $\nu_k$ and $Y$ runs over the connected components of
$M^{S^1}$ (to keep notation light we have suppressed the dependence of $\nu $, $\nu _k$, $d_k$ on
$Y$). We are now in the position to state
\begin{theorem}\label{theorem cyclic} Let $M$ be a $Spin$-manifold with
$S^1$-action. If $m_\oorder > r$ then the expansion of the elliptic genus of $M$ in the $\hat A$-cusp
has a pole of order less than $\achtel -r$.
\end{theorem}
\bigskip
\noindent
If $\sigma \in S^1$ has order $\oorder
=2$ then $\tilde k\in
\{0,1
\}$ and $4\cdot m_2 (Y)$ is the codimension of the connected component
of $M^\sigma $ which contains $Y$. Thus $\codim M^\sigma
\leq 4\cdot m_2$ and one recovers Theorem \ref{theorem
involution}. In general if $\sigma \in S^1$ has order $\oorder
\geq 2$ then $\codim M^\sigma \leq 2\oorder \cdot m_\oorder $
and one obtains Theorem \ref{specialcase}. Note that without the $Spin$ condition the conclusion of
the theorem fails in general, e.g. for complex projective spaces of even complex dimension (see
however Remark \ref{rigidity remark}).
\section{Proof of Theorem \ref{theorem cyclic}}\label{section proof}
We may assume that the dimension of $M$ is divisible by $4$ and that the fixed point manifold
$M^{S^1}$ is not empty since otherwise $M$ is rationally zero bordant by the \LFF \ \cite{AtSi} and $\Phi (M)$ vanishes. We may also assume that the $S^1$-action lifts to the
$Spin$-structure (otherwise the action is odd which forces the elliptic genus to vanish, see for
example \cite{HiSl}). We fix an $S^1$-equivariant Riemannian metric on $M$. The proof is divided into
three steps.
Step 1: We describe the equivariant elliptic genus at $M^{S^1}$. Consider the expansion of $\Phi (M)$
in the $\hat A$-cusp. Recall that the coefficients are indices of twisted Dirac operators associated
to the $Spin$-structure. Since the $S^1$-action lifts to the $Spin$-structure each index refines to a
virtual $S^1$-representation and the series refines to an element of $R(S^1)\lbrack q^{-\frac 1
2}\rbrack \lbrack
\lbrack q\rbrack
\rbrack $ which we denote by $\Phi _{0,S^1}(M)$. Note that $sign_{S^1}(q,{\cal L}M)$
and $\Phi _{0,S^1}(M)$ are different expansions of the same function. Hence the rigidity of
$sign_{S^1}(q,{\cal L}M)$ (see Theorem \ref{rigidity theorem}) is equivalent to the rigidity of $\Phi
_{0,S^1}(M)$, i.e. each coefficient of the series
$\Phi _{0,S^1}(M)$ is constant as a character of $S^1$.
Let $\lambda _0\in S^1$ be a fixed topological generator. By the
\LFF \ \cite{AtSi} the series $\Phi _{0,S^1}(M)(\lambda _0)\in \C \lbrack q^{-\frac 1 2}\rbrack \lbrack
\lbrack q\rbrack \rbrack $
is equal to a sum of local
data
$$\Phi_{0,S^1}(M)(\lambda _0)=\sum _Y \mu _Y(q,\lambda _0),$$
where $Y$ runs over the connected components of $M^{S^1}$.
Recall from Section \ref{section cyclic actions} that we have decomposed the normal bundle $\nu $ of
$Y$ as a direct sum $\bigoplus_{k\neq 0} \nu _k$ of complex vector bundles. Fix the orientation for
$Y$ which is compatible with the orientation of $M$ and the complex structure of $\nu $. Let $\{\pm
x_i\}$ denote the set of roots of $Y$ and let $\{x_{k,j}\}_{j=1,\ldots ,d_k}$ denote the set of roots
of the complex vector bundle $\nu
_k$. The local datum $\mu _Y(q,\lambda
_0)$ may be described in cohomological terms as (cf.
\cite{AtSi}, Section 3):
\begin{equation}\label{local data}\mu _Y(q,\lambda _0)=\left \langle
\prod _i \frac {x_i}
{f (q, x_i)}\cdot \prod _{k\neq 0\atop j=1,\ldots , d_k }\frac 1 {f (q, x_{k,j}+\alpha _k k\cdot
z_0)}, \lbrack Y\rbrack \right \rangle \end{equation} Here $f (q,x)\in \C \lbrack
\lbrack q^\frac 1 4\rbrack \rbrack \lbrack
\lbrack x\rbrack \rbrack $ is equal to
$$(e^{x/2}-e^{-x/2})\cdot q^{1/4}\cdot \frac {\prod _{n=2m>0}(1-q^n\cdot e^x)
\cdot (1-q^n\cdot e^{-x})}
{\prod _{n=2m+1>0}(1-q^n\cdot e^x)\cdot (1-q^n\cdot e^{-x})},$$ $\lambda _0=e^{z_0}$, $\lbrack
Y\rbrack $ denotes the fundamental cycle of $Y$ and $\langle \quad ,\quad \rangle $ is the Kronecker
pairing. In general each local datum $\mu
_Y(q,\lambda _0)$ depends on $\lambda _0$. However, the sum
$\sum _Y \mu _Y(q,\lambda _0)$ is equal to $\Phi_{0,S^1}(M)(\lambda _0)$ and therefore independent of
$\lambda
_0$ by the rigidity theorem.
Step 2: Each local datum is the expansion of a meromorphic function on $\oha \times \C $ where $\oha
$ denotes the upper half plane. As in the proof of the rigidity theorem given in \cite{BoTa} (cf.
also \cite{DeJu, Hi2,Li}) modularity properties of these functions will be central for
the argument. In this step we examine some of their properties.
We begin to recall relevant properties of the series $f$ (see for example \cite{DeJu, HiBeJu}). For $0<\vert q\vert <1 $ and $z\in
\C $ satisfying $\vert q\vert <\vert e^{z}\vert < \vert q\vert ^{-1}$
the series $f(q,z)$ converges normally to a holomorphic function. This function extends to a
meromorphic function $\widetilde f(\tau ,z)$ on $\oha
\times \C $ after the change of variables $q=e^{2\pi i\cdot
\tau}$ where $\tau $ is in $\oha $. The function
$\widetilde f(\tau ,z)$ is
elliptic in $z$ for the lattice $L:=4\pi i\cdot \Z
\langle 1,\tau \rangle $ and satisfies
$$\widetilde f(\tau
,z+2\pi i)=- \widetilde f(\tau ,z), \widetilde f(\tau ,z+2\pi i\cdot
\tau )=\widetilde f(\tau ,z)^{-1}, \widetilde f(\tau +2,z)=-\widetilde f(\tau ,z).$$ The zeros of $\widetilde f(\tau ,z)$ are simple and located at $L$ and $L+2\pi i$.
Let $q=e^{2\pi i\cdot \tau}$ and let $\lambda _0=e^{z_0}$ be a topological generator of $S^1$. In
view of formula (\ref{local data}) and the properties of $f$ the local datum $\mu _Y(q,\lambda _0)$
converges to a meromorphic function $\widetilde \mu
_Y$ on $\oha
\times \C $ evaluated at $(\tau , z_0)$. We proceed to explain how this function is related to $\widetilde f$. For a function $F$ in the variables $x_i, x_{k,j}$ which is smooth in the origin let ${\cal
T}(F)$ denote the Taylor expansion of $F$ with respect to $x_i, x_{k,j}=0$. It follows from formula
(\ref{local data}) that $\widetilde \mu
_Y$ is related to $\widetilde f$ by (see for example \cite{DeJu}):
$$\widetilde \mu _Y(\tau , z_0)=\left \langle {\cal T}
\left (\prod _i \frac {x_i} {\widetilde f(\tau ,
x_i)}\cdot \prod _{k\neq 0\atop j=1,\ldots ,d_k} \frac 1 {\widetilde f(\tau , x_{k,j}+\alpha _k
k\cdot z_0)}\right ), \lbrack Y\rbrack \right \rangle $$ The properties of $\widetilde f$ stated
above imply corresponding properties for $\widetilde \mu
_Y$. In particular, $\widetilde \mu _Y$ is elliptic for
the lattice $L$ and satisfies
$$\widetilde
\mu _Y(\tau +1,z)=(-1)^{\dim M/4}\cdot \widetilde
\mu _Y(\tau ,z),\quad \widetilde
\mu _Y(\tau ,z+2\pi i)=\pm \widetilde \mu _Y(\tau ,z).$$
For fixed $\tau \in \oha $ the poles of $\widetilde \mu _Y$ are contained in $\frac 1 n\cdot L$ for
some $n\in
\N $ depending on the rotation numbers of the $S^1$-action at $Y$ (see for example \cite{DeJu, HiBeJu}).
In general $\widetilde \mu _Y(\tau ,z)$ depends on $z$. If $\lambda
=e^z$ is a topological generator of $S^1$, i.e. if $z/(2\pi i )$ is irrational, then $\Phi
_{0,S^1}(M)(\lambda )$ converges to the sum $\sum
_Y \widetilde \mu _Y(\tau ,z)$ by the \LFF \ and the latter is independent of $z$ by the rigidity theorem. Note that the original data may be recovered from $\widetilde \mu
_Y(\tau ,z)$ by taking the
expansion of $\widetilde \mu _Y(\tau ,z)$ with respect to $\tau
\mapsto
\tau +2$.
Step 3: In the final step we study the series $\sum _Y \mu _Y$ in terms of the sum $\sum
_Y \widetilde \mu _Y(\tau ,s(\tau ))$ where $s:\oha \to \C $ approximates $\tau \mapsto \frac 2 \oorder
\cdot 2\pi i\cdot \tau $. We choose
$s(\tau )$ in such a way that $\widetilde
\mu _Y(\tau ,s(\tau ))$ is periodic with respect to $\tau \mapsto
\tau +N$ for some $N\in \N $ (see below).
Note that in general the series $\mu
_Y(q,\lambda )$ does not converge if $\lambda $ is close to $e^{\frac
2 \oorder \cdot 2\pi i\cdot \tau }$ and the $q^{\frac 1 N}$-expansion of $\widetilde
\mu _Y(\tau ,s(\tau ))$, denoted by $a_Y$, is different from the corresponding contribution $\mu _Y(q,\lambda _0)$ in the \LFF \ for $\Phi_{0,S^1}(M)(\lambda _0)$.
In particular, we cannot compare $\mu
_Y(q,e^{s(\tau )})$ and $\widetilde \mu _Y(\tau ,s(\tau ))$ directly.
However, since the sum $\sum _Y \widetilde \mu _Y(\tau ,z)$ is independent of $z$ the sum $\sum _Y
a_Y$ is equal to the elliptic genus in the $\hat A$-cusp (see last step). Using the properties of
$\widetilde
\mu _Y$ described above and the assumption on $m_\oorder $ we will show
that $\sum _Y a_Y$ has a pole of order less than $\achtel -r$. This
will complete the proof.
Here are the details. The discussion in the last step implies that the poles of $\widetilde
\mu
_Y$, $Y\subset M^{S^1}$, are contained in $\frac 1 n \cdot L$ for some
$n\in \N $. Choose $s(\tau ):=(1-\beta )\cdot \frac 2 {\oorder }\cdot 2\pi i \cdot \tau $, where
$\beta $ is a fixed rational positive number $\ll \frac 1 n$. Hence, $s(\tau )$ is close to $\frac 2
\oorder
\cdot 2\pi i\cdot \tau $ and $\tau \mapsto \widetilde \mu _Y(\tau ,s(\tau ))$ is
holomorphic on $\oha $ for every $Y$. Using $\alpha
_k k\equiv \tilde k \bmod \oorder $, $\tilde k\in
\{0,\ldots , \lbrack
\frac o 2\rbrack \}$, and the transformation property
$\widetilde f(\tau ,z+4\pi i\cdot\tau )=\widetilde f(\tau ,z)$ one computes that $\widetilde \mu
_Y(\tau, s(\tau ))$ is (up to sign) equal to $\langle {\cal T}(A_Y), \lbrack Y\rbrack \rangle $,
where
$$A_Y:=\prod _i \frac {x_i} {\widetilde f(\tau ,
x_i)}\cdot \prod _{k\neq 0\atop j=1,\ldots ,d_k} \frac 1 {\widetilde f(\tau , x_{k,j}+2\cdot (\frac
{\widetilde k}{\oorder}\cdot (1-\beta )-\beta _k)
\cdot (2 \pi i \cdot \tau ) )}$$
and $\beta _k:=\beta \cdot \frac {\alpha
_kk - \tilde k }\oorder $.
Note that for some $N\in \N $ (depending on $\beta $ and the rotation numbers) every summand
$\widetilde \mu _Y(\tau, s(\tau ))$ is periodic with respect to $\tau \mapsto \tau +N$. We claim that
its expansion $a_Y\in \C
\lbrack q^{-\frac 1 N}\rbrack \lbrack \lbrack q^{\frac 1 N}\rbrack
\rbrack $ has a pole of order less than $\achtel
-r$.
Since the expansion of ${\cal T}\left (x_i/\widetilde f(\tau , x_i)\right )$ (with respect to $\tau
\mapsto \tau +4$) is equal to $x_i/f(q, x_i)$ the expansion of
\begin{equation}\tag{$\ast$}{\cal T}\left (\frac 1 {\widetilde f(\tau , x_{k,j}+2\cdot
(\frac {\widetilde k}{\oorder}\cdot (1-\beta )-\beta _k)\cdot
(2 \pi i \cdot \tau ) )}\right )\end{equation}
can be easily computed in terms of $f$. The computation shows that the expansion of ($\ast$) has a
pole of order $\leq \frac 1 4 - \frac {\widetilde k}{\oorder}\cdot (1-\beta )+\beta _k$. Since
$m_\oorder (Y)\geq m_\oorder >r$ and $\beta $, $\beta _k$ are arbitrarily small it follows
that $a_Y\in \C \lbrack q^{-\frac 1 N}\rbrack \lbrack \lbrack
q^{\frac 1 N}\rbrack \rbrack $ has a pole of order less than $\achtel- r$. As explained in the
beginning of this step the sum $\sum _Y a_Y$ is equal to the expansion of the elliptic genus in the
$\hat A$-cusp. Hence, $\Phi_{0}(M)\in
\C
\lbrack q^{-\frac 1 2}\rbrack\lbrack \lbrack q\rbrack \rbrack $ has a pole of order less than
$\achtel- r$. This completes the proof.\proofend
\begin{remark}\label{rigidity remark} Essentially the same reasoning applies to
orientable $S^1$-manifolds (not necessarily $Spin$) for which the equivariant elliptic genus is rigid. The rigidity theorem is known
to hold for oriented manifolds
with finite second
homotopy group \cite{Her} and for $Spin^c$-manifolds with first Chern class a torsion
class \cite{Despinc}. Theorem \ref{theorem cyclic} is also true for these manifolds.
\end{remark}
\section{Highly connected ${\mathbf {S^1}}$-manifolds}\label{highly connected}
In this section we adapt the arguments of \cite{HiSl} to study the elliptic genus of certain
$S^1$-manifolds including highly connected manifolds. To begin with we recall the \LFF \ for twisted
signatures. Let $M$ be an oriented closed $S^1$-manifold, $E$ an $S^1$-equivariant \vb \ over $M$ and
$\sigma \in S^1$ the element of order $2$ . In the following we shall always assume that the fixed
point manifold $M^\sigma $ is orientable (this is the case if $M$ is $Spin$ \cite{BoTa}). By the \LFF
\ the equivariant twisted signature $sign _{S^1}(M,E)\in R(S^1)$ evaluated at $\sigma$ is equal to a sum of
local data $a_{F,E}$ at the connected components $F$ of the fixed point manifold $M^\sigma $
$$sign_{S^1}(M,E)(\sigma )=\sum _F a_{F,E}.$$
The local contributions are given by (cf. \cite{HiSl})
$$a_{F,E}=\left \langle A_{F,E}, \lbrack F\rbrack \right \rangle $$
where
$$A_{F,E}=\prod _i\left (x_i\cdot \frac {1+e^{-x_i}}{1-e^{-x_i}}\right )\cdot \prod _j\left (y_j\cdot\frac {1+e^{-y_j}}{1-e^{-y_j}}\right )^{-1}
\cdot ch (E_{\vert F})(\sigma )\cdot e(\nu _F).$$
Here $\pm x_i$ (resp. $\pm y_j$) denote the formal roots of $F$ (resp. the normal bundle $\nu _F$ of
$F$) for compatible orientations of $F$ and $\nu _F$, $e(\nu _F)$ is the Euler class of $\nu _F$ and
$ch(E_{\vert F})$ denotes the equivariant Chern character of $E_{\vert F}$. The local datum $a_{F,E}$ is obtained by
evaluating the cohomology class $A_{F,E}$ on the fundamental cycle $\lbrack F\rbrack $ via the
Kronecker pairing $\langle
\quad ,\quad \rangle $. Note that $a_{F,E}$ vanishes if $e(\nu _F)$ is a torsion class. Hence, the
following lemma is immediate.
\begin{lemma}\label{first lemma} Let $M$ and $E$ be as above and let $F\subset M^\sigma$ be
of codimension $k$. If $H^{k}(F ;\Q )=0$ then the local datum $a_{F,E}$ vanishes.\proofend
\end{lemma}
\noindent
For the proof of the next lemma recall that the Euler class of the normal bundle of
$i:F\hookrightarrow M$ is equal to $i^*(i_!(1))$, where $i_!:H^*(F;\Z )\to H^{*+k}(M;\Z )$ denotes
the push forward in cohomology for the oriented normal bundle $\nu _F$.
\begin{lemma}\label{second lemma} Let $M$ and $E$ be as above. If $H^{k}(M ;\Q )=0$ then $a_{F,E}$
vanishes for any connected component $F\subset M^\sigma$ of codimension $k$.\proofend
\end{lemma}
\noindent
We shall now apply these observations to the elliptic genus.
\begin{theorem} Let $M$ be a $Spin$-manifold. Assume that $H^{4*}(M;\Q )=0$ for $0<*\leq r $.
If $M$ admits a non-trivial $S^1$-action then the
expansion of $\Phi(M)$ in the $\hat A$-cusp has a pole of order less than $\achtel -r$.
\end{theorem}
\bigskip
\noindent
{\bf Proof:} Let $\sigma \in S^1$ denote the element of order two. Arguing as in the proof of Theorem
\ref{theorem cyclic} we may assume that the dimension of $M$ and the dimension of each connected
component $F\subset M^\sigma $ is divisible by $4$. Consider the expansion $sign _{S^1}(q,{\cal L}M)$
of the $S^1$-equivariant elliptic genus in the signature cusp. By the rigidity theorem $sign
_{S^1}(q,{\cal L}M)(\sigma )$ is equal to the non-equivariant expansion $sign (q,{\cal L}M)$. By the
\LFF \ $sign _{S^1}(q,{\cal L}M)(\sigma )$ is a
sum of local contributions $a_F$ at the connected components $F$ of $M^\sigma $:
$$sign (q,{\cal L}M)=sign _{S^1}(q,{\cal L}M)(\sigma )=\sum _F a_F.$$
Note that each coefficient of the $q$-power series $a_F$ is the local contribution in the
\LFF \ of an equivariant twisted signature evaluated at $\sigma \in S^1$. Since $H^{4*}(M;\Q )=0$ for
$0<*\leq r $ the contribution $a_F$ vanishes if $\codim F\leq 4r$ (see Lemma \ref{second lemma}). If
$\codim F> 4r$ then $a_F$ is equal to $sign (q,{\cal L}(F\circ F))$ (see formula (\ref{formula
elliptic})). Hence,
$$sign (q,{\cal L}M)=\sum _{\codim F> 4r} a_F=\sum _{\codim F\circ F>8r}sign (q,{\cal L}(F\circ F)).$$
This implies that the expansion of $\Phi(M)$ in the $\hat A$-cusp has a pole of order less than
$\achtel -r$.\proofend
\bigskip
\noindent
Finally note that Theorem \ref {observation} is a direct consequence of the theorem above.
| 91,407
|
\begin{document}
\title{Weak Hopf Algebras unify the Hennings--Kauffman--Radford\\
and the Reshetikhin--Turaev invariant}
\author{Hendryk Pfeiffer\thanks{E-mail: \texttt{pfeiffer@math.ubc.ca}}}
\date{\small{Department of Mathematics, The University of British Columbia,\\
1984 Mathematics Road, Vancouver, BC, V2T 1Z2, Canada}\\[1ex]
February 24, 2012}
\maketitle
\begin{abstract}
We present an invariant of connected and oriented closed $3$-manifolds based
on a coribbon Weak Hopf Algebra $H$ with a suitable left-integral. Our
invariant can be understood as the generalization to Weak Hopf Algebras of
the Hennings--Kauffman--Radford evaluation of an unoriented framed link
using a dual quantum-trace. This quantum trace satisfies conditions that
render the link evaluation invariant under Kirby moves. If $H$ is a suitable
finite-dimensional Hopf algebra (not weak), our invariant reduces to the
Kauffman--Radford invariant for the dual of $H$. If $H$ is the Weak Hopf
Algebra Tannaka--Kre\v\i n reconstructed from a modular category $\sym{C}$,
our invariant agrees with the Reshetikhin--Turaev invariant. In particular,
the proof of invariance of the Reshetikhin--Turaev invariant becomes as
simple as that of the Kauffman--Radford invariant. Modularity of $\sym{C}$
is only used once in order to show that the invariant is non-zero; apart
from this, a fusion category with ribbon structure would be sufficient. Our
generalization of the Kauffman--Radford invariant for a Weak Hopf Algebra
$H$ and the Reshetikhin--Turaev invariant for its category of
finite-dimensional comodules $\sym{C}\simeq\sym{M}^H$ always agree by
construction. There is no need to consider a quotient of the representation
category modulo 'negligible morphisms' at any stage, and our construction
contains the Reshetikhin--Turaev invariant for an arbitrary modular category
$\sym{C}$, whether its relationship with some quantum group is known or not.
\end{abstract}
\msc{
57M27,
16T05,
18D10
}
\keywords{$3$-manifold, quantum invariant, modular category, Weak Hopf Algebra}
\section{Introduction}
In some special cases, the following two quantum invariants of connected and
oriented closed (smooth) $3$-manifolds are related: the invariant of
Reshetikhin--Turaev~\cite{ReTu91} and the invariant of
Hennings~\cite{He96}. Since the latter has been studied mainly for
finite-dimensional unimodular ribbon Hopf algebras, we focus on the
reformulation of the Hennings invariant according to Kauffman and
Radford~\cite{KaRa95}. This reformulation exploits the fact that these special
Hopf algebras have a unique cointegral with suitable properties, and this
results in a substantial simplification of Hennings' original construction.
It has already been demonstrated by Hennings~\cite{He96} that the
Reshetikhin--Turaev invariant for the modular categories associated with
$U_q(\ssl_2)$ at suitable roots of unity $q$ appears as a special case
of his construction. In order to show this, one needs to understand the
quotient of the category of tilting modules over $U_q(\ssl_2)$ modulo
so-called \emph{negligible} morphisms. This quotient is finitely semisimple
and has the structure of a modular category.
This example of Hennings raises the question of whether the agreement of the
Hennings invariant with the Reshetikhin--Turaev invariant is a coincidence or
whether there is an explanation in conceptual terms. One answer to this
question was given by Lyubashenko~\cite{Ly95}. Starting from the fusion
category with ribbon structure $\sym{C}$ used in the Reshetikhin--Turaev
invariant, he uses Majid's~\cite{Ma93} universal coend
$F=\coend(\sym{C},1_\sym{C})$ over the identity functor
$1_\sym{C}\colon\sym{C}\to\sym{C}$ which forms a Hopf algebra object
$F\in|\sym{C}|$ (\emph{braided group} in Majid's terminology). If
$\sym{C}\simeq{}_H\sym{M}$ is the category of modules over a
finite-dimensional Hopf algebra $H$, Lyubashenko's invariant for
$F\in|\sym{C}|$ coincides with the Kauffman--Radford invariant for $H$. Let us
summarize the idea of Lyubashenko's unification of the invariants as follows:
He forces the Hopf algebra $H$ of the Kauffman--Radford invariant into the
language of the Reshetikhin invariant by transmuting~\cite[Section 4]{Ma93} it
into a Hopf algebra object $F\in|\sym{C}|$.
The purpose of the present article is to reverse this approach and to present
a second way of unifying the Reshetikhin--Turaev with the Kauffman--Radford
invariant, this time by forcing the fusion category with ribbon structure
$\sym{C}$ into the language of the Kauffman--Radford invariant. Thanks to the
recent generalization of Tannaka--Kre\v\i n reconstruction to fusion
categories~\cite{Pf09a,Pf11}, we know that each fusion category $\sym{C}$, in
particular each modular category, is equivalent to the category of
finite-dimensional comodules $\sym{C}\simeq\sym{M}^H$ over a Weak Hopf Algebra
(WHA) $H=\coend(\sym{C},\omega)$, the universal coend over the long canonical
functor $\omega\colon\sym{C}\to\Vect_k$. We are therefore able to recover
$\sym{C}$ from $H$ and can therefore express the Reshetikhin--Turaev invariant
entirely in terms of $H$, \ie\ in the language of the Kauffman--Radford
invariant, something that is not possible in Lyubashenko's approach. One of
the advantages of this point of view is that it renders the proof of
invariance of the Reshetikhin--Turaev invariant as easy as that of the
Kauffman--Radford invariant whereas Lyubashenko's approach renders the proof
of invariance of the Kauffman--Radford invariance as difficult as that of the
Reshetikhin--Turaev invariant.
Let us now sketch how one can find a (co)algebra with extra structure $H$ such
that the Reshetikhin--Turaev invariant for $\sym{C}$ agrees with the
Kauffman--Redford invariant for $H$. It appears that all modular categories
that yield interesting $3$-manifold invariants, \ie\ invariants that are
stronger than invariants of homotopy type, have objects of non-integer
Frobenius--Perron dimension\footnote{This is an observation about the existing
literature, and we are not aware of any counter-example.}. These categories
therfore do not form the categories of modules over any Hopf algebra, see, for
example~\cite[Theorem 8.33]{EtNi05}. The naive conjecture that the Hennings
invariant for a Hopf algebra $H$ might agree with the Reshetikhin--Turaev
invariant for the category ${}_H\sym{M}$ of modules over $H$, is therefore not
even well phrased.
Any conjecture on a coincidence of the Hennings with the Reshetikhin--Turaev
invariant has a chance of being true only if it is not ${}_H\sym{M}$ itself,
but rather a quotient of ${}_H\sym{M}$ modulo suitable negligible morphisms,
that forms the modular category. This is how Hennings' original example
works. But since there exist modular categories for which no relationship with
one of the standard quantum groups is known~\cite{HoRo08}, any conceptual
approach to relating the Hennings with the Reshetikhin--Turaev invariant needs
to avoid taking a quotient of the representation category.
Although the topologically interesting modular categories are not the
categories of modules over any Hopf algebra, every modular category is the
category of (co)modules over a Weak Hopf Algebra (WHA)~\cite{Pf09a} --- no
quotient required: for each modular category $\sym{C}$, one can
Tannaka--Kre\v\i n reconstruct a WHA $H$ whose category of finite-dimensional
comodules $\sym{M}^H$ is equivalent as a $k$-linear additive ribbon category
to the original modular category $\sym{C}$.
In the present article, we define an invariant of $3$-manifolds for a suitable
class of WHAs (see Theorem~\ref{thm_invariant} below). Our invariant can be
understood as the generalization of the Kauffman--Radford invariant from Hopf
algebras to WHAs. In the special case in which the WHA $H$ is a Hopf algebra,
our invariant reduces to the Kauffman--Radford invariant (for the
dual\footnote{For technical reasons, we work with comodules rather than
modules, and so our WHA $H$ corresponds to the dual of the Hopf algebra
featuring in the Kauffman--Radford invariant, and it is the category of
finite-dimensional comodules $\sym{M}^H$ of $H$ that forms the modular
category.} of $H$). In the special case in which $H$ is the WHA
reconstructed from a modular category $\sym{C}$, our invariant agrees with the
Reshetikhin--Turaev invariant for $\sym{C}$.
In fact, by construction, our generalization of the Kauffman--Radford
invariant for a WHA $H$ always agrees with the Reshetikhin--Turaev invariant
for the ribbon category $\sym{M}^H$. Modularity of $\sym{M}^H$ is sufficient
for the invariant to be non-zero. As a by-product, we obtain a new proof of
invariance of the Reshetikhin--Turaev invariant for an arbitrary modular
category $\sym{C}\simeq\sym{M}^H$ using computations in $H$ rather than
computations in $\sym{C}$. This new proof is substantially shorter than the
original proof presented in~\cite[Section II.3]{Tu10}.
In the near future, there will be a companion article relating the
Turaev--Viro invariant~\cite{TuVi92} with the Kuperberg invariant~\cite{Ku91}
along the same lines. Apart from tidying up some twenty-year-old results,
these identities between quantum invariants can be expected to prove useful if
one tries to categorify these invariants. Whereas categorifying a modular
category to some $2$-category with extra structure is still far beyond reach,
categorifying the reconstructed WHAs appears to be much more promising. In
particular, the WHAs reconstructed from the modular categories associated with
$U_q(\ssl_2)$ are substantially easier to understand than
$U_q(\ssl_2)$ at roots of unity itself together with the relevant
quotient of its category of modules.
The present article is structured as follows. In Section~\ref{sect_prelim}, we
briefly summarize some background material on WHAs and on Tannaka--Kre\v\i n
reconstruction. Section~\ref{sect_reconstructed} contains results on integrals
and cointegrals in the WHA reconstructed from a fusion category with ribbon
structure. We then define various ways of evaluating unoriented framed links
in $S^3$ in Section~\ref{sect_ribbon}. Our new invariant is presented in
Section~\ref{sect_invariant} in which we also show that in encompasses both
the Reshetikhin--Turaev invariant and the Kauffman--Radford
invariant. Appendix~\ref{app_wha} contains more details on WHAs with extra
structure, and Appendix~\ref{app_reconstruction} on their Tannaka--Kre\v\i n
reconstruction from fusion categories with extra structure.
\section{Preliminaries}
\label{sect_prelim}
In this section, we fix our notation and sketch some background
material on Weak Hopf Algebras, their categories of finite-dimensional
comodules, and on the canonical Weak Hopf Algebra associated to each
modular category via generalized Tannaka--Kre\v\i n reconstruction.
We use the following notation. If $\sym{C}$ is a category, we write
$X\in|\sym{C}|$ for the objects $X$ of $\sym{C}$, $\Hom(X,Y)$ for the
collection of all morphisms $f\colon X\to Y$ and $\End(X)=\Hom(X,X)$. We
denote the identity morphism of $X$ by $\id_X\colon X\to X$ and the
composition of morphisms $f\colon X\to Y$ and $g\colon Y\to Z$ by $g\circ
f\colon X\to Z$. If two objects $X,Y\in|\sym{C}|$ are isomorphic, we write
$X\cong Y$. If two categories $\sym{C}$ and $\sym{D}$ are equivalent, we write
$\sym{C}\simeq\sym{D}$. The identity functor on $\sym{C}$ is denoted by
$1_{\sym{C}}$. The category of vector spaces over a field $k$ is denoted by
$\Vect_k$ and its full subcategory of finite-dimensional vector spaces by
$\fdVect_k$. Both are $k$-linear, abelian and symmetric monoidal, and
$\fdVect_k$ is autonomous. The $n$-fold tensor power of some object
$X\in|\sym{C}|$ of a monoidal category
$(\sym{C},\otimes,\one,\alpha,\lambda,\rho)$ is denoted by $X^{\otimes n}$,
$n\in\N_0$. We set $X^{\otimes 0}:=\one$. We use the notation $\N$ and $\N_0$
for the positive integers and the non-negative integers, respectively.
\subsection{Weak Hopf Algebras and their corepresentations}
For the basics of Weak Bialgebras (WBAs) and Weak Hopf Algebras (WHAs), we
refer to~\cite{BoNi99,BoSz00} and to Appendix~\ref{app_wha}. In a WHA $H$ over
some field $k$, we denote by $\mu\colon H\otimes H\to H$, $\eta\colon k\to H$,
$\Delta\colon H\to H\otimes H$, $\epsilon\colon H\to k$ and $S\colon H\to H$
the multiplication, unit, comultiplication, counit and antipode,
respectively. The source and target counital maps are denoted by $\epsilon_s$
and $\epsilon_t$, and the source and target base algebras by $H_s$ and $H_t$,
respectively. The opposite comultiplication is given by
$\Delta^\op=\tau_{H,H}\circ\Delta$ where $\tau_{H,H}(x\otimes y)=y\otimes x$
for all $x,y\in H$.
The category of right $H$-comodules that are finite-dimensional over $k$, is
denoted by $\sym{M}^H$. It is a $k$-linear abelian and left-autonomous
monoidal category equipped with a $k$-linear, faithful and exact forgetful
functor $U^H\colon\sym{M}^H\to\fdVect_k$. This functor is in general not
strong monoidal and thereby not a fibre functor in the technical sense, but it
is equipped with a separable Frobenius structure.
A copivotal form $w\colon H\to k$ for $H$ is a dual group-like linear form
such that $S^2(x)=w(x^\prime)x^\pprime\bar w(x^\ppprime)$ for all $x\in H$,
\ie\ one that implements the square of the antipode by dual conjugation. Here
$\bar w\colon H\to k$ denotes the convolution inverse of $w$. We call such a
linear form $w$ \emph{copivotal} because it is this structure that renders the
category $\sym{M}^H$ a pivotal category.
The universal $r$-form of a coquasi-triangular WHA and its weak convolution
inverse are denoted by $r\colon H\otimes H\to k$ and $\bar r\colon H\otimes
H\to k$, respectively. Similarly, $\nu\colon H\to k$ and $\bar\nu\colon H\to
k$ denote the universal ribbon form and its convolution inverse in a coribbon
WHA. Recall that each coribbon WHA is copivotal with
$w(x)=v(x^\prime)\nu(x^\pprime)$ for all $x\in H$, involving the second dual
Drinfel'd element $v\colon H\to k, x\mapsto r(S(x^\prime)\otimes x^\pprime)$
and the universal ribbon form $\nu$.
For the convenience of the reader, we have collected in Appendix~\ref{app_wha}
the basic definitions and more detailed references to the literature as well
as the basic facts about the relevant additional structure on $\sym{M}^H$. For
example, if $H$ is coribbon, then $\sym{M}^H$ is a ribbon category. If $H$ is
finite-dimensional, split cosemisimple and pure, then $\sym{M}^H$ is fusion,
and if $H$ is in addition coribbon and weakly cofactorizable, then $\sym{M}^H$
forms a modular category.
\subsection{Tannaka--Kre\v\i n reconstruction}
For every multi-fusion category $\sym{C}$ that is $k$-linear over some
field $k$, there is a canonical functor
\begin{equation}
\label{eq_longfunctor}
\omega\colon\sym{C}\to\Vect_k,\quad X\mapsto \Hom_k(\hat V,\hat V\otimes X).
\end{equation}
Here we use the small progenerator
\begin{equation}
\label{eq_progenerator}
\hat V=\bigoplus_{j\in I}V_j,
\end{equation}
where the biproduct is over a set $I$ of one representative $V_j$, $j\in I$,
for each isomorphism class of simple objects of $\sym{C}$. The functor
$\omega$ is known as the \emph{long canonical functor}~\cite{Ha99b,Sz05} and
can be used in order to Tannaka--Kre\v\i n reconstruct a finite-dimensional
split cosemisimple coassociative counital coalgebra $H$ over $k$. It is given
by the universal coend $H=\coend(\sym{C},\omega)$ of $\omega$, and
$\sym{M}^H\simeq\sym{C}$ are equivalent as $k$-linear additive
categories. Since $\omega$ has a separable Frobenius
structure~\cite{Pf09a,Pf11}, $H$ forms a WHA, and $\sym{M}^H\simeq\sym{C}$ are
equivalent as monoidal categories as well.
If $\sym{C}$ carries a pivotal, spherical or ribbon structure, then $H$ is
copivotal, cospherical or coribbon, respectively. If $\sym{C}$ is fusion, then
$H$ is copure, and if $\sym{C}$ is modular, then $H$ is weakly
cofactorizable~\cite{Pf09a,Pf09b}. In all of these cases, the equivalence
$\sym{M}^H\simeq\sym{C}$ is compatible with the extra structure. In
Appendix~\ref{app_reconstruction}, we have compiled more details on how the
extra structure of $H$ is related to that of $\sym{C}$ and on how to perform
computations in $H$.
Note in particular that $\omega X=\Hom_k(\hat V,\hat V\otimes X)$ and
$\Hom_k(\hat V\otimes X,\hat V)$ are dually paired for all $X\in|\sym{C}|$,
see~\eqref{eq_gx}, and that most computations can be conveniently phrased in
terms of pairs of dual bases ${(e_m^{(X)})}_m$ and ${(e^m_{(X)})}_m$ of
$\omega X$ and ${(\omega X)}^\ast$, respectively. The vector space underlying
the reconstructed WHA $H$ is given in~\eqref{eq_coendvect}. Also note the
reconstruction of the copivotal form $w$ of~\eqref{eq_copivotal} and the
isomorphism $D_{\hat V}\in\End(\hat V)$ of~\eqref{eq_dtransformation}
involved.
\section{The reconstructed Weak Hopf Algebra}
\label{sect_reconstructed}
In order to see how our invariant encompasses the Reshetikhin--Turaev
invariant~\cite{ReTu91}, we need to develop some of the integral
theory of the WHA reconstructed from a modular category
$\sym{C}$. This is done in the present section. For background
material on the integral theory of WHAs, we refer to~\cite{BoNi99}.
Let $H$ be a coribbon WHA. A \emph{dual trace} is an element $\chi\in H$ such
that $\Delta^\op(\chi)=\Delta(\chi)$. It is called $S$-invariant if
$S(\chi)=\chi$. A \emph{dual quantum trace} is an element $t\in H$ such that
$\Delta^\op(t)=(\id_H\otimes S^2)\circ\Delta(t)$. A dual quantum trace $t\in
H$ is called $S$-\emph{compatible} if
\begin{equation}
\label{eq_scompatible}
\bar w(t^\prime)S(t^\pprime)=\bar w(t^\prime)t^\pprime.
\end{equation}
Observe that $t=w(\chi^\prime)\chi^\pprime$ is a dual quantum trace if and
only if $\chi$ is a dual trace. In this situation, $t$ is $S$-compatible if
and only if $\chi$ is $S$-invariant.
Note that for each $V\in|\sym{M}^H|$, its dual character $\chi_V\in H$
(see~\eqref{eq_dualchar}) forms a dual trace, and its dual quantum character
$T_V\in H$ (see~\eqref{eq_dualqchar}) is a dual quantum trace. Observe that
$S(\chi_V)=\chi_{V^\ast}$ and if $V^\ast\cong V$ in $\sym{M}^H$, then we have
in addition that $\chi_{V^\ast}=\chi_V$, \ie\ the dual character $\chi_V$ is
$S$-invariant.
An element $t\in H$ of a finite-dimensional WHA is called
\emph{non-degenerate} if $H^\ast\to k,\phi\mapsto\phi(t)$ is non-degenerate as
a functional on the dual WHA $H^\ast$, \ie\ its kernel does not contain any
non-zero left-ideal of $H^\ast$. This holds if and only if the bilinear form
$H^\ast\otimes H^\ast\to
k,\phi\otimes\psi\mapsto\phi(t^\prime)\psi(t^\pprime)$ is non-degenerate. A
\emph{left-integral} $\ell\in H$ is an element that satisfies
$x\ell=\epsilon_t(x)\ell$ for all $x\in H$. A \emph{right-integral} $r\in H$
is an element that satisfies $rx=r\epsilon_s(x)$ for all $x\in H$. A
\emph{two-sided integral} is both a left- and a right-integral. A
[left-,right-]cointegral of $H$ is a [left-,right-]integral of $H^\ast$.
A linear form $\zeta\colon H\to k$ is called \emph{dual central} if
$\zeta(x^\prime)x^\pprime=x^\prime\zeta(x^\pprime)$ for all $x\in H$. If $H$
is the WHA reconstructed from a multi-fusion category $\sym{C}$, then it is
split cosemisimple, and so its dual central linear forms can be computed as
follows.
\begin{proposition}
\label{prop_dualcentral}
Let $\sym{C}$ be a multi-fusion category over the field $k$ and
$H=\coend(\sym{C},\omega)$ be the finite-dimensional and split cosemisimple
WHA reconstructed from $\sym{C}$ using the long canonical
functor~\eqref{eq_longfunctor}. Then a basis for the vector space of all dual
central linear forms is given by ${\{\zeta_j\}}_{j\in I}$ where
\begin{equation}
\zeta_j({[\theta|v]}_{X}) = \left\{\begin{array}{ll}
\epsilon({[\theta|v]}_X),\quad&\mbox{if}\quad X\cong V_j,\\
0,\quad&\mbox{else},
\end{array}\right.
\end{equation}
for all simple $X\in|\sym{C}|$. A dual central linear form $\zeta=\sum_{j\in
I}c_j\zeta_j$ with coefficients $c_j\in k$ is convolution invertible if and
only if $c_j\neq 0$ for all $j\in I$.
\end{proposition}
The following theorem demonstrates that the canonical WHA reconstructed from a
spherical multi-fusion category $\sym{C}$ contains a very special
left-integral $\ell\in H$. This integral will feature in the construction of
the invariant below.
\begin{theorem}
\label{thm_integral}
Let $\sym{C}$ be a spherical multi-fusion category over $k$ and
$H=\coend(\sym{C},\omega)$ be the finite-dimensional and split cosemisimple
cospherical WHA reconstructed from $\sym{C}$ using the long canonical
functor~\eqref{eq_longfunctor}.
\begin{myenumerate}
\item
The following element $\ell\in H$ is a left-integral:
\begin{equation}
\label{eq_leftintegral}
\ell=\sum_{j\in I}\dim V_j\,\sum_m{[D^{-1}_{\hat V}\circ e^m_{(V_j)}\circ(D_{\hat V}\otimes\id_{V_j})|e^{(V_j)}_m]}_{V_j}.
\end{equation}
\item
The following linear form $c\colon H\to k$ is a two-sided cointegral:
\begin{equation}
\label{eq_cointegral}
c({[\theta|v]}_X) = \left\{
\begin{array}{ll}
\epsilon({[\theta|\rho^{-1}_{\hat V}]}_\one)\epsilon({[\rho_{\hat V}|v]}_\one) &
\quad\mbox{if}\quad X\cong\one,\\
0 &\quad\mbox{else},
\end{array}
\right.
\end{equation}
for all simple $X\in|\sym{C}|$.
\item
Both $\ell$ and $c$ are non-degenerate.
\item
The cointegral $c$ is $S$-invariant, and $\ell$ is a dual quantum
trace.
\item
The integral can be expressed as $\ell=\zeta(t_{\hat
V}^\prime)t_{\hat V}^\pprime=w(\chi_{\hat
V}^\prime)\zeta(\chi_{\hat V}^\pprime)\chi_{\hat V}^\ppprime$
where $\chi_{\hat V}\in H$ is the dual character~\eqref{eq_dualchar}
associated with the small progenerator $\hat V$ of~\eqref{eq_progenerator};
$t_{\hat V}=w(\chi_{\hat V}^\prime)\chi_{\hat V}^\pprime\in H$ is its dual
quantum character~\eqref{eq_dualqchar}; and $\zeta\colon H\to k$ is the
convolution invertible and dual central linear form given by
\begin{equation}
\label{eq_dualcentral}
\zeta({[\theta|v]}_X) = (\dim X)\,\epsilon({[\theta|v]}_X)
\end{equation}
for all simple $X\in|\sym{C}|$.
\item
The integral $\ell$ is $S$-compatible.
\end{myenumerate}
\end{theorem}
\begin{proof}
\begin{myenumerate}
\item
Note that the canonical left-integral $\ell_{\mathrm{can}}\in H$
that exists in every WHA, turns out to be a multiple of our $\ell$:
\begin{equation}
\ell_{\mathrm{can}} = \sum_jb_j^\prime\beta_j(S^2(b_j^\pprime))=|I|\,\ell.
\end{equation}
Here we have written $\sum_jb_j\otimes\beta_j$ for the canonical
element in $H\otimes H^\ast$. Therefore, $\ell_{\mathrm{can}}=0$
whenever the characteristic of $k$ divides the number $|I|$ of
isomorphism classes of the simple objects of $\sym{C}$. Our integral
$\ell\in H$ avoids this problem and never vanishes as we show in Part~(3)
below. The proof that it indeed forms a left-integral is by a direct
calculation and is most transparent if one uses the isomorphism
$H\cong\End({\hat V}^\ast\otimes \hat V)$ of~\cite[Section~4.1]{Pf09b}.
\item
By the result dual to~\cite[Lemma 3.3]{BoNi99}, the set of
right-cointegrals of $H$ is isomorphic as a left-$H$-comodule to
$\Hom_{\sym{M}^H}(H,H_s)$ where both $H$ and $H_s\cong\one$ are viewed as
right-$H$-comodules. Since $H$ is split cosemisimple, the set
$\Hom_{\sym{M}^H}(H,H_s)$ and thereby the set of right-cointegrals is
known explicitly. A direct computation shows that such a right-cointegral is
two-sided if and only if it is a scalar multiple of $c$ of~\eqref{eq_cointegral}.
\item
A direct computation shows that
$c\rightharpoonup\ell=1$. By~\cite[Theorem~3.18]{BoNi99}, both
$\ell$ and $c$ are therefore non-degenerate. In particular, $\ell\neq
0$. The theorem also implies that $\ell\rightharpoonup c=1^\ast$, where
$1^\ast\in H^\ast$ is the unit of the dual WHA, a result that is needed in
Part~(4) below.
\item
Since there exists the two-sided non-degenerate cointegral $c$ of
$H$, by the result dual to~\cite[Lemma~3.21]{BoNi99}, all two-sided
cointegrals are $S$-invariant. In particular, $c$ is. Since $c$ is a
non-degenerate left-cointegral and $\ell\rightharpoonup c=1^\ast$,
$\ell$ is its dual left-integral, and we can apply the result dual
to~\cite[Theorem~3.20]{BoNi99}. Since $c$ is $S$-invariant, this
theorem implies that $\ell$ is a dual quantum trace.
\item
Since the coalgebra underlying $H$ is finite-dimensional and split
cosemisimple, we know the coefficients of the right-$H$ comodule $\hat
V\in|\sym{M}^H|$ and can compute its dual character~\eqref{eq_dualchar}:
\begin{equation}
\chi_{\hat V}
= \sum_{j\in I}\chi_{V_j}
= \sum_{j\in I}\sum_m{[e^m_{(V_j)}|e_m^{(V_j)}]}_{V_j}\in H.
\end{equation}
A direct computation using the copivotal form~\eqref{eq_copivotal} and
equation~\eqref{eq_dualcentral}, proves the claim.
\item
Since ${\hat V}^\ast\cong\hat V$, its dual character is
$S$-invariant, \ie\ $S(\chi_{\hat V})=\chi_{\hat V}$. In order to prove the
claim, we use that $\zeta\circ S=\zeta$.
\end{myenumerate}
\end{proof}
\section{Evaluation of (unoriented) framed links}
\label{sect_ribbon}
\subsection{Ribbon diagrams}
\label{sect_evaluation}
Let $\sym{C}$ be a ribbon category (see
Appendix~\ref{app_coribbon}). Every morphism of $\sym{C}$ can be
represented by a composition of tensor products of the following
string diagrams,
\begin{align}
\id_X &= \tikzsymb{\identity{X}}\qquad\qquad&
\id_{X^\ast} &= \tikzsymb{\iddual{X}}\nn\\
\ev_X &= \tikzsymb{\leftev{X}}\qquad\qquad&
\coev_X &= \tikzsymb{\leftcoev{X}}\nn\\
\bar\ev_X &= \tikzsymb{\rightev{X}}\qquad\qquad&
\bar\coev_X &= \tikzsymb{\rightcoev{X}}\nn
\end{align}
\begin{align}
\nu_X &= \tikzsymb{\twistpos{X}}\qquad\qquad&
\nu^{-1}_X &= \tikzsymb{\twistneg{X}}\nn\\
\nu_{X^\ast} &= \tikzsymb{\twistposdual{X}}\qquad\qquad&
\nu^{-1}_{X^\ast} &= \tikzsymb{\twistnegdual{X}}\nn\\
\sigma_{X,Y} &= \tikzsymb{\braiddd{X}{Y}}\qquad\qquad&
\sigma^{-1}_{X,Y} &= \tikzsymb{\braidinvdd{Y}{X}},\nn
\end{align}
in which the components of the ribbon are labeled by objects
$X,Y,\ldots\in|\sym{C}|$. The monoidal unit $\one\in|\sym{C}|$ is invisible in
these diagrams. If an object label $X\in|\sym{C}|$ is replaced by its dual
$X^\ast$, the arrow is reversed. Note that we read composition from top to
bottom and the tensor product from left to right. This agrees with about half
of the literature, but notably differs from Turaev~\cite{Tu10} who reads
composition from the bottom up and who calls the right-handed rather than the
left-handed twist $\nu_X$. Our choice of diagrams turns out to be convenient
in the present context as we study corepresentations rather than
representations of (Weak) Hopf Algebras.
By a result of Reshetikhin--Turaev~\cite{ReTu90} which can be viewed as a
coherence theorem for ribbon categories, every plane projection of an oriented
framed tangle in $S^3$ can be arranged to agree with such a diagram, but
without labels. If one now labels the components of the tangle with objects of
$\sym{C}$, the resulting morphism of $\sym{C}$ can be shown to be independent
of the chosen projection~\cite{ReTu90}. Therefore, a given ribbon category
$\sym{C}$ associates morphisms of $\sym{C}$ with labeled oriented framed
tangles and, more specially, endomorphisms of the monoidal unit
$\one\in|\sym{C}|$ with labeled oriented framed links.
\subsection{Reshetikhin--Turaev evaluation for ribbon categories}
Let $\sym{C}$ be a ribbon category, $V\in|\sym{C}|$ be an object of $\sym{C}$
such that $V^\ast\cong V$, and $\zeta\colon 1_{\sym{C}}\Rightarrow
1_{\sym{C}}$ be a natural isomorphism of the identity functor such that
$\zeta^\ast=\zeta$. Given a plane projection of an (unoriented) framed link
$L$ in $S^3$, we label all components by $V$ and insert $\zeta_V$ somewhere
(anywhere) into each component of the link. We call the resulting morphism
\begin{equation}
\label{eq_rtevaluation}
{\left<L\right>}^{(\sym{C})}_{V,\zeta}\colon\one\to\one,
\end{equation}
the \emph{Reshetikhin--Turaev evaluation} of the link $L$. Notice
that~\eqref{eq_rtevaluation} is independent of the orientation of each
component of $L$ and therefore well defined. The link
evaluation~\eqref{eq_rtevaluation} for $V=\hat V$ of~\eqref{eq_progenerator}
and $\zeta$ the natural transformation associated with the linear form $\zeta$
of~\eqref{eq_dualcentral} is the one that features in the Reshetikhin--Turaev
invariant~\cite{ReTu91}.
\subsection{Evaluation for coribbon Weak Hopf Algebras}
In this section, we evaluate an (unoriented) framed link in the ribbon
category $\sym{M}^H$ for a suitable coribbon WHA $H$.
\begin{remark}[Summary of results from~\cite{Pf09a}]
\label{rem_coribbon}
Let $H$ be a coribbon WHA over some field $k$. Then the category $\sym{M}^H$
of finite-dimensional right $H$-comodules is a ribbon category with
\begin{alignat}{3}
\ev_V &\colon V^\ast\otimes V\to H_s, &&\quad \theta\otimes v\to\theta(v_0)\epsilon_s(v_1),\\
\coev_V &\colon H_s\to V\otimes V^\ast, &&\quad x\mapsto ({(e_j)}_0\otimes e^j)\epsilon(x{(e_j)}_1),\\
\nu_V &\colon V\to V, &&\quad v\mapsto v_0\nu(v_1),\\
\nu^{-1}_V &\colon V\to V, &&\quad v\mapsto v_0\bar\nu(v_1),\\
\sigma_{V,W} &\colon V\otimes W\to W\otimes V,&&\quad v\otimes w\mapsto (w_0\otimes v_0)r(w_1\otimes v_1),\\
\sigma^{-1}_{V,W}&\colon W\otimes V\to V\otimes W,&&\quad w\otimes v\mapsto (v_0\otimes w_0)\bar r(w_1\otimes v_1),\\
\end{alignat}
where we have used Sweedler notation for comodules (see
Appendix~\ref{app_wba}), $e_j\otimes e^j\in V\otimes V^\ast$ denotes the
canonical element, and $H_s$ plays the role of the monoidal unit of
$\sym{M}^H$.
The right evaluation and coevaluation $\bar\ev_V$ and $\bar\coev_V$
can be computed as in~\eqref{eq_barevribbon}
and~\eqref{eq_barcoevribbon}, respectively. Replacing one object
$V\in|\sym{M}^H|$ by its dual is is done as in~\eqref{eq_dualaction}.
\end{remark}
In the remainder of this section, we show how the Reshetikhin--Turaev
evaluation ${\left<L\right>}^{(\sym{M}^H)}_{V,\zeta}$ can be computed using
only the WHA $H$. In this computation, the dual quantum character $T_V$
appears once for each component of $L$, in conjunction with the linear form
associated with $\zeta$. The element
\begin{equation}
\ell=\zeta(T_V^\prime)T_V^\pprime\in H
\end{equation}
turns out to be an $S$-invariant dual quantum trace.
\begin{proposition}
\label{prop_universalnat}
Let $H$ be a WBA and $U^H\colon\sym{M}^H\to\Vect_k$ be the usual forgetful
functor. Every natural transformation $f\colon 1_{\sym{M}^H}\Rightarrow
1_{\sym{M}^H}$ is of the form
\begin{equation}
\label{eq_universalnat}
f_V(v) = v_0\alpha^{(f)}(v_1)
\end{equation}
for all $V\in|\sym{M}^H|$ and $v\in V$. Here $\alpha^{(f)}\colon H\to k$ is a
uniquely determined dual central linear form. In addition, $f\colon
1_{\sym{M}^H}\Rightarrow 1_{\sym{M}^H}$ is a natural equivalence if and only
if $\alpha^{(f)}$ is convolution invertible.
\end{proposition}
\begin{proof}
By the universal property of the universal coend
$H\cong\coend(\sym{M}^H,U^H)$, condition~\eqref{eq_universalnat} defines a unique linear
form $\alpha^{(f)}$ for the natural transformation $\alpha$. Since each
$f_V\colon V\to V$ is a morphism, $\alpha^{(f)}$ is dual central. The
convolution inverse $\overline{\alpha^{(f)}}$ is given by $f^{-1}_V(v) =
v_0\overline{\alpha^{(f)}}(v_1)$ if it exists, again using the universal
property of the coend.
\end{proof}
\begin{proposition}
Let $H$ be a WBA and $U^H\colon\sym{M}^H\to\Vect_k$ be the usual forgetful
functor. Every natural transformation $f\colon-\otimes-\Rightarrow-\otimes-$
of the functor
$-\otimes-\colon\sym{M}^H\times\sym{M}^H\to\sym{M}^H\times\sym{M}^H$ is of the
form
\begin{equation}
\label{eq_universalnat2}
f_{V,W}(v\otimes w) = v_0\otimes w_0\,\alpha^{(f)}(v_1\otimes w_1)
\end{equation}
for all $V,W\in|\sym{M}^H|$ and $v\in V$, $w\in W$. Here $\alpha^{(f)}\colon
H\otimes H\to k$ is a uniquely determined linear form that satisfies
\begin{gather}
\label{eq_universalnata}
\epsilon(x^\prime y^\prime)\alpha(x^\pprime\otimes y^\pprime)
= \alpha(x\otimes y)
= \alpha(x^\prime\otimes y^\prime)\epsilon(x^\pprime y^\pprime),\\
\label{eq_universalnatb}
x^\prime y^\prime\,\alpha(x^\pprime\otimes y^\pprime)
= \alpha(x^\prime\otimes y^\prime)\,x^\pprime y^\pprime,
\end{gather}
for all $x,y\in H$.
\end{proposition}
\begin{proof}
By the universal property of the universal coend $H\otimes
H\cong\coend(\sym{M}^H\times\sym{M}^H,U^H\otimes U^H)$,
\eqref{eq_universalnat2} defines a unique linear form $\alpha^{(f)}$ for the
natural transformation $\alpha$, and~\eqref{eq_universalnata} is
satisfied. Since each $f_{V,W}\colon V\otimes W\to V\otimes W$ is a morphism,
condition~\eqref{eq_universalnatb} holds.
\end{proof}
In the following, we use the same symbol for the natural transformation and
for the associated linear form, for example, $\alpha\colon
1_{\sym{M}^H}\Rightarrow 1_{\sym{M}^H}$ and $\alpha\colon H\to k$ or
$\beta\colon-\otimes-\Rightarrow-\otimes-$ and $\beta\colon H\otimes H\to k$.
\begin{lemma}
\label{la_partialtrace}
Let $H$ be a coribbon WHA and $f\colon X\otimes V\to X\otimes V$ be a morphism
in $\sym{M}^H$. Then
\begin{equation}
\tikzsymb{\partialtrace{X}{V}{f}}(x)
= \sum_{i,j} g_i(x)\otimes e^j({(h_i(e_j)}_0)w({(h_i(e_j))}_1)
\end{equation}
for all $x\in X$ where we have written $f=\sum_i g_i\otimes h_i$ with
$g_i\in\End(X)$ and $h_i\in\End(V)$, $\coev_V=\sum_j e_j\otimes e^j$, and
where $w\colon H\to k$ denotes the copivotal form. The diagram is in the
ribbon category $\sym{M}^H$ and is drawn in blackboard framing.
\end{lemma}
\begin{proof}
Use the definitions in Remark~\ref{rem_coribbon} as well as
\begin{equation}
(x_0\otimes v_0)\epsilon(\epsilon_s(x_1)v_1) = x\otimes v
\end{equation}
and
\begin{equation}
(x_0\otimes \theta_0)\epsilon(x_1\epsilon_s(\theta_1)) = x\otimes\theta
\end{equation}
for all $x\in X$, $v\in V$, and $\theta\in V^\ast$.
\end{proof}
\begin{proposition}
\label{prop_tracelemma}
Let $H$ be a coribbon WHA and $\alpha\colon-\otimes-\Rightarrow-\otimes-$ be a
natural transformation. Then for all $X,V\in|\sym{M}^H|$,
\begin{equation}
\label{eq_tracelemma}
\tikzsymb{\partialtrace{X}{V}{\alpha_{X,V}}}(x)
= x_0\,\alpha(x_1\otimes T_V)
\end{equation}
for all $x\in X$. The diagram is in $\sym{M}^H$ and drawn in blackboard
framing, and $T_V\in H$ denotes the dual quantum character
of~\eqref{eq_dualqchar}.
\end{proposition}
\begin{proof}
Apply Lemma~\ref{la_partialtrace} to $f_{X,V}=\alpha_{X,V}$.
\end{proof}
\begin{lemma}
\label{la_tracelemma}
Let $H$ be a coribbon WHA and $\alpha\colon 1_{\sym{M}^H}\Rightarrow
1_{\sym{M}^H}$ be a natural transformation. Then for $V\in|\sym{M}^H|$,
\begin{equation}
\label{eq_trace}
\tikzsymb{\trace{V}{\alpha_V}}(h) = \alpha(T_V^\prime)\epsilon_s(h\epsilon_t(T_V^\pprime))
\end{equation}
for all $h\in H_s$.
\end{lemma}
\begin{proof}
Recall that the trace~\eqref{eq_trace} is a linear map $H_s\to H_s$ where
$H_s\cong\one\in|\sym{M}^H|$ is the monoidal unit. Using the definitions in
Remark~\ref{rem_coribbon}, we obtain
\begin{equation}
\eqref{eq_trace} = \alpha(T_V^\prime)\epsilon(hT_V^\pprime)\epsilon_s(S(T_V^\ppprime))
\end{equation}
which can be shown to agree with the right hand side of~\eqref{eq_trace}.
\end{proof}
\begin{theorem}
\label{thm_evaluationform}
Let $H$ be a coribbon WHA, $L$ be an (unoriented) framed link in $S^3$ with
$m$ components, $V\in|\sym{M}^H|$ and $\zeta\colon 1_{\sym{M}^H}\Rightarrow
1_{\sym{M}^H}$ be a natural equivalence such that $\zeta^\ast=\zeta$. Then the
Reshetikhin--Turaev evaluation is of the form
\begin{equation}
\label{eq_computeeval}
{\left<L\right>}^{(\sym{M}^H)}_{V,\zeta}(h) = \phi^{(L)}(h\otimes\underbrace{\ell\otimes\cdots\otimes\ell}_{m})
\end{equation}
for all $h\in H_s$ with a linear map $\phi^{(L)}\colon H_s\otimes H^{\otimes m}\to H_s$.
Here, the element
\begin{equation}
\ell = \zeta(T_V^\prime) T_V^\pprime\in H
\end{equation}
is an $S$-compatible dual quantum trace.
\end{theorem}
\begin{proof}
We prove a slightly stronger claim in which we insert into each component
$L_j$, $1\leq j\leq m$, of $L$ a natural transformation $\gamma^{(j)}_V\colon
V\to V$ that is of the form $\gamma_V^{(j)}=\xi^{(j)}_V\circ\zeta_V$ with
arbitrary $\xi_V^{(j)}\colon V\to V$. The proof proceeds by induction on the
number of components $m$.
If $m=1$, the link evaluation is of the form
\begin{equation}
{\left<L\right>}^{(\sym{M}^H)}_{V,\zeta} = \tikzsymb{\inductionstart{V}{\gamma_V^{(1)}}{\nu_V^z}}
\end{equation}
where $\nu_V^z$, $z\in\Z$, is the appropriate power of the twist. Putting
$\alpha_V=\nu_V^z\circ\gamma^{(1)}_V=\nu_V^z\circ\xi^{(1)}_V\circ\zeta_V$
in Lemma~\ref{la_tracelemma} shows that
\begin{equation}
{\left<L\right>}^{(\sym{M}^H)}_{V,\zeta}(h)
= \zeta(T_V^\prime)\psi^{(1)}(T_V^\pprime)\epsilon_s(h\epsilon_t(T_V^\ppprime))
= \psi^{(1)}(\ell^\prime)\epsilon_s(h\epsilon_t(\ell^\pprime))
\end{equation}
for all $h\in H_s$ where the linear form $\psi^{(1)}\colon H\to k$ implements
the natural transformation $\psi^{(1)}_V=\nu_V^z\circ\xi^{(1)}_V$. This proves
our stronger proposition. For $\xi^{(1)}_V=\id_V$, we obtain the claim of the
theorem for $m=1$ as a special case.
We now assume that our stronger assumption holds for some $m\in\N$ and consider
a link $L$ with $m+1$ components. If we select one of the components, labeled
$V$ and, without loss of generality, numbered $m+1$, the diagram of $L$ can be
arranged in such a way that the selected component $V$ appears in the
following fashion:
\begin{equation}
f_{V^{\otimes m}} = \tikzsymb{\inductionstep{V^{\otimes m}}{V}{\gamma^{(m+1)}_V}{\nu_V^z}{\beta_{V^{\otimes m},V}}},
\end{equation}
with some natural transformation $\beta\colon-\otimes-\Rightarrow-\otimes-$
and some $z\in\Z$. We now apply Proposition~\ref{prop_tracelemma} with
$\alpha_{V^{\otimes m},V}=(\id_{V^{\otimes
m}}\otimes(\nu_V^z\circ\gamma^{(m+1)}_V))\circ\beta_{V^{\otimes m},V}$ and
$X=V^{\otimes m}$. This proposition shows that
\begin{equation}
f_{V^{\otimes m}}(x) = x_0\,\omega(x_1),
\end{equation}
for all $x\in V^{\otimes m}$ where
\begin{equation}
\label{eq_omega}
\omega(y)=\zeta(T_V^\prime)\psi^{(m+1)}(T_V^\pprime)\alpha(y\otimes T_V^\ppprime)
=\psi^{(m+1)}(\ell^\prime)\alpha(y\otimes\ell^\pprime)
\end{equation}
for all $y\in H$. Here, $\psi^{(m+1)}\colon H\to k$ is the linear form that
implements the natural transformation
$\psi^{(m+1)}_V=\nu_V^z\circ\xi^{(m+1)}$. In particular, the map
$f_{V^{\otimes m}}$ is natural in $V^{\otimes m}$. Even stronger, we can split
$X=V^{\otimes m}=Y\otimes V$ with $Y=V^{\otimes(m-1)}$ and see that
$f_{Y\otimes V}(y\otimes v)=y_0\otimes v_0\,\omega(y_1v_1)$ for all $y\in Y$
and $v\in V$ which shows that $f_{Y\otimes V}$ is natural both in $Y$ and in
$V$. By induction, we see that $f_{V^{\otimes m}}$ is natural in each component
$V$ of the tensor power $V^{\otimes m}$.
The evaluation ${\left<L\right>}^{(\sym{M}^H)}_{V,\zeta}$ for the
$(m+1)$-component link is therefore equal to the evaluation for an
$m$-component link in which we have inserted different natural transformations
into every component: the composition of $f_{V^{\otimes m}}$ which is natural
in each tensor factor $V$ with the $\zeta_V$ from the original claim. By the
assumption of our induction, ${\left<L\right>}^{(\sym{M}^H)}_{V,\zeta}$ is of
the form of our stronger claim. Note that for each component of $L$,
\eqref{eq_omega} is applied once and yields one tensor factor $\ell$.
Depending on the order in which we consider the components of $L$, we may
obtain different linear maps $\phi^{(L)}$. The right hand side
of~\eqref{eq_computeeval}, however, always agrees with the Reshetikhin--Turaev
evaluation.
\end{proof}
\subsection{Generalized dual Hennings--Kauffman--Radford evaluation}
The idea of Hennings~\cite{He96} in the case of Hopf algebras (not weak) is
that the $S$-compatible dual quantum trace $\ell\in H$ that features in
Theorem~\ref{thm_evaluationform} can be replaced by an arbitrary
$S$-compatible dual quantum trace and still yields a well-defined link
evaluation, \ie\ an evaluation that gives the same value irrespective of the
diagram that is used in order to represent the link. The same holds for
WHAs. We summarize this result in the following.
\begin{definition}
Let $H$ be a coribbon WHA, $\ell\in H$ be an $S$-compatible dual quantum
trace, and $L$ be an $m$-component (unoriented) framed link in $S^3$. The
\emph{generalized Hennings--Kauffman--Radford evaluation}
${\left<L\right>}^{(H)}_\ell$ is defined as follows. Consider a diagram of
$L$. Label each component of $L$ with a formal symbol $\sym{X}$ which stands
for a finite-dimensional right $H$-comodule with coaction
$\beta_\sym{X}\colon\sym{X}\to\sym{X}\otimes H$. We impose only the relations
that $\sym{X}$ be a rigid object of $\Vect_k$ and that $\beta_\sym{X}$ be a
right-$H$ comodule, \ie\
\begin{eqnarray}
\alpha_{\sym{X},H,H}\circ(\beta_\sym{X}\otimes\id_H)\circ\beta_\sym{X}
&=& (\id_\sym{X}\otimes\Delta)\circ\beta_\sym{X},\\
\rho_\sym{X}\circ(\id_\sym{X}\otimes\epsilon)\circ\beta_\sym{X}
&=& \id_\sym{X}.
\end{eqnarray}
Theorem~\ref{thm_evaluationform} applies, and so
\begin{equation}
{\left<L\right>}^{(\sym{M}^H)}_{\sym{X},\id}(h)
=\phi^{(L)}(h\otimes\underbrace{\tilde\ell\otimes\cdots\otimes\tilde\ell}_{m})
\end{equation}
for all $h\in H_s$ with a linear map $\phi^{(L)}\colon H_s\otimes H^{\otimes
m}\to H_s$. The theorem computes the $S$-compatible dual quantum trace
as $\tilde\ell=T_\sym{X}\in H$, the dual quantum character of the formal comodule
$\sym{X}$. The Hennings--Kauffman--Radford evaluation is then defined by
replacing this element $\tilde\ell$ with the given $S$-compatible dual
quantum trace $\ell\in H$:
\begin{equation}
\label{eq_karaevaluation}
{\left<L\right>}^{(H)}_\ell(h) = \phi^{(L)}(h\otimes\underbrace{\ell\otimes\cdots\otimes\ell}_{m}).
\end{equation}
for all $h\in H_s$.
\end{definition}
\begin{theorem}
\label{thm_kara}
Let $H$ be a coribbon WHA, $\ell\in H$ be an $S$-compatible dual quantum
trace, and $L$ be an $m$-component (unoriented) framed link in $S^3$. The
Hennings--Kauffman--Radford evaluation ${\left<L\right>}^{(H)}_\ell$ is well
defined, \ie\ it is independent of the diagram used to represent $L$.
\end{theorem}
\begin{proof}
The proof is dual of the proof of Hennings~\cite{He96} or the proof of
Kauffman--Radford~\cite{KaRa95}. The idea is that
in~\eqref{eq_karaevaluation}, the linear map $\phi^{(L)}\colon H_s\otimes H^{\otimes
m}\to H_s$ may initially depend on the diagram that represents $L$ and on the
order of the components of $L$ used in the proof of
Theorem~\ref{thm_evaluationform}, but the linear map
\begin{equation}
\phi^{(L)}(-\otimes\underbrace{\ell\otimes\cdots\otimes\ell}_{m})\colon
H_s\to H_s
\end{equation}
is independent of the diagram that represents the link
$L$. Kauffman--Radford~\cite{KaRa95} show that the condition that $\ell\in H$
be an $S$-compatible dual quantum trace is sufficient in order to establish
independence. Since they work with left modules whereas we use
right-comodules, we can simply rotate all their diagrams by $180^\circ$ in
order to prove our claim. The coherence theorem for the monoidal category
$\sym{M}^H$ ensures that the monoidal unit $H_s\cong\one$ can be inserted at
an arbitrary position in all tensor products.
Finally, note that since $\ell$ is $S$-compatible, the dual trace $\chi=\bar
w(\ell^\prime)\ell^\pprime$ is $S$-invariant, and so the evaluation is
independent of the orientation of each individual component of $L$.
\end{proof}
The following corollary to Theorem~\ref{thm_evaluationform} establishes the
relation between the two link evaluations.
\begin{corollary}
Let $\sym{C}$ be a multi-fusion category over $k$ which has a ribbon
structure, and let $H=\coend(\sym{C},\omega)$ be the finite-dimensional and
split cosemisimple coribbon WHA reconstructed from $\sym{C}$ using the long
canonical functor~\eqref{eq_longfunctor}. Then the Reshetikhin--Turaev and the
Hennings--Kauffman--Radford evaluations agree for every (unoriented) framed
link $L$ in $S^3$:
\begin{equation}
{\left<L\right>}^{(\sym{C})}_{\hat V,\zeta}={\left<L\right>}^{(H)}_\ell
\end{equation}
with $\hat V$ of~\eqref{eq_progenerator}, $\zeta$ of~\eqref{eq_dualcentral}
and $\ell$ of~\eqref{eq_leftintegral}.
\end{corollary}
\begin{proof}
Theorem~\ref{thm_evaluationform} and Theorem~\ref{thm_integral}(5).
\end{proof}
\section{Invariants of $3$-manifolds}
\label{sect_invariant}
\subsection{The invariant for a Weak Hopf Algebra}
We proceed in analogy to the work of Hennings~\cite{He96} and
Kauffman--Radford~\cite{KaRa95} and show that if the $S$-compatible dual
quantum trace $\ell$ is a left-integral, then the link evaluation can be made
invariant under Kirby moves.
\begin{theorem}
\label{thm_invariant}
Let $H$ be a coribbon WHA over some field $k$ and $\ell\in H$ be a
left-integral which is an $S$-compatible dual quantum trace. Let $L$
be an (unoriented) framed link in $S^3$ with components
$L_1,\ldots,L_m$, $m\in\N$. If there exist $\beta,\gamma\in
k\backslash\{0\}$ such that the following two conditions,
\begin{eqnarray}
\label{eq_kirbyplus}
\nu(x^\prime)\bar\nu(\epsilon_t(x^\pprime)\ell) &=& \frac{\gamma^2}{\beta}\,\nu(x),\\
\label{eq_kirbyminus}
\bar\nu(x^\prime)\nu(\epsilon_t(x^\pprime)\ell) &=& \beta\,\bar\nu(x),
\end{eqnarray}
hold for all $x\in H$, then
\begin{equation}
\label{eq_invariant}
I(M_L) = \beta^\sigma\,\gamma^{-\sigma-m-1}\,{\left<L\right>}_\ell^{(H)}\in\End(\one)
\end{equation}
forms an invariant of connected and oriented closed (smooth)
$3$-manifolds. Here, $\one\cong H_s$ denotes the monoidal unit of $\sym{M}^H$.
In~\eqref{eq_invariant}, $M_L$ is the $3$-manifold obtained from $S^3$ by
surgery along $L$; ${\left<L\right>}_\ell^{(H)}$ denotes the generalized
Hennings--Kauffman--Radford evaluation of the link $L$ using the dual quantum
trace $\ell$; and $\sigma$ is the signature of the linking matrix of $L$ with
framing numbers on its diagonal.
\end{theorem}
\begin{proof}
By the theorems of Wallace and Lickorish~\cite{Wa60,Li62}, of
Kirby~\cite{Ki78} and of Fenn--Rourke~\cite{FeRo79}, $M_L$ is diffeomorphic as
an oriented manifold to $M_{\tilde L}$ if and only if the (unoriented) framed
link $L$ can be transformed into $\tilde L$ using a finite sequence of
Kirby-$(+1)$- and Kirby-$(-1)$-moves.
Note that $I(M_L)$ is independent of the numbering of the components of the
link, and so $I(M_L)$ is well-defined for each given Kirby diagram $L$ of some
connected and oriented closed $3$-manifold.
We first show that~\eqref{eq_kirbyplus} and~\eqref{eq_kirbyminus}
imply that ${\left<L\right>}_\ell^{(H)}$ is invariant up to the specified scalar
factors $\gamma^2/\beta$ and $\beta$ under Kirby-$(+1)$-moves and under
Kirby-$(-1)$-moves, respectively. For the Kirby-$(+1)$-move, we show that
\begin{equation}
\biggl<\tikzsymb{\kirbyplus{\sym{X}^{\otimes n}}{\sym{X}}}\biggr>_\ell^{(H)}
= \frac{\gamma^2}{\beta}\biggl<\tikzsymb{\lefttwist{\sym{X}^{\otimes n}}}\biggr>_\ell^{(H)},
\end{equation}
for all $n\in\N_0$. The left-hand side evaluates to
$(\id_\sym{X}\otimes f)\circ\beta_\sym{X}\colon\sym{X}\to\sym{X}$ with
\begin{eqnarray}
\label{eq_kirbypluscalc}
f(x)&=&\bar\nu(\ell^\prime)\bar q(x\otimes\ell^\pprime)\nn\\
&=&\nu(x^\prime)\bar\nu(x^\pprime)\bar\nu(\ell^\prime)\bar q(x^\ppprime\otimes\ell^\pprime)\nn\\
&=&\nu(x^\prime)\bar\nu(x^\pprime\ell)\nn\\
&=&\nu(x^\prime)\bar\nu(\epsilon_t(x^\pprime)\ell)\nn\\
&=&\frac{\gamma^2}{\beta}\,\nu(x),
\end{eqnarray}
for all $x\in H$ with $\bar q(-\otimes-)$ of~\eqref{eq_defbarq}. We have used
convolution invertibility of $\nu$, a consequence of
equation~\eqref{eq_ribbontensor}, that $\ell$ is a left-integral, and
equation~\eqref{eq_kirbyplus}. The result agrees with the evaluation of the
right-hand side. Recall that with our definition of a coribbon WHA $H$, the
universal ribbon form $\nu\colon H\to k$ gives rise to the isomorphisms
$\nu_X\colon X\to X$ in $\sym{M}^H$ which represent the left-handed (!)
twist. For the Kirby-$(-1)$-move, we show that
\begin{equation}
\label{eq_kirbyminuspic}
\biggl<\tikzsymb{\kirbyminus{\sym{X}^{\otimes n}}{\sym{X}}}\biggr>_\ell^{(H)}
= \beta\,\biggl<\tikzsymb{\righttwist{\sym{X}^{\otimes n}}}\biggr>_\ell^{(H)},
\end{equation}
for all $n\in\N_0$. The left-hand side evaluates to
$(\id_\sym{X}\otimes g)\circ\beta_\sym{X}\colon\sym{X}\to\sym{X}$ with
\begin{eqnarray}
g(x)&=&\nu(\ell^\prime)q(x\otimes\ell^\pprime)\nn\\
&=&\bar\nu(x^\prime)\nu(x^\pprime)\nu(\ell^\prime) q(x^\ppprime\otimes\ell^\pprime)\nn\\
&=&\bar\nu(x^\prime)\nu(x^\pprime\ell)\nn\\
&=&\bar\nu(x^\prime)\nu(\epsilon_t(x^\pprime)\ell)\nn\\
&=&\beta\,\bar\nu(x),
\end{eqnarray}
for all $x\in H$ with $q(x\otimes y)=$ of~\eqref{eq_defq}. We have used
convolution invertibility of $\nu$, equation~\eqref{eq_ribbontensor}, that
$\ell$ is a left-integral, and equation~\eqref{eq_kirbyminus}. The result
agrees with the evaluation of the right-hand side.
Since the Kirby-$(+1)$-move decreases both $m$ and $\sigma$ by one
whereas the Kirby-$(-1)$-move decreases $m$ by one and increases
$\sigma$ by one, the expression $I(M_L)$ in~\eqref{eq_invariant} is
invariant under both moves.
\end{proof}
The above argument does not simplify much if we restrict the proof to special
Kirby-$(-1)$-moves. The special Kirby-$(-1)$-move is obtained for $n=0$
in~\eqref{eq_kirbyminuspic}, \ie\ by inserting the monoidal unit for
$\sym{X}^{\otimes n}$ which is an invisible line. Note that the coribbon WHA
we use is not required to be copure, and so the invariant takes its values in
$\End(\one)$ which need not be isomorphic to $k$.
\subsection{The Hennings--Kauffman--Radford invariant}
In this section, we show that if $H$ is a finite-dimensional unimodular ribbon
Hopf algebra, then its dual $H^\ast$ satisfies the assumptions of
Theorem~\ref{thm_invariant}. In this case, our invariant~\eqref{eq_invariant}
for $H^\ast$ agrees up to a factor with the Kauffman--Radford
formulation~\cite{KaRa95} of the Hennings invariant~\cite{He96} for $H$.
First, if we work with a Hopf algebra (not weak), then
Theorem~\ref{thm_invariant} reduces to the following
\begin{corollary}
Let $H$ be a coribbon Hopf algebra over some field $k$ and $\ell\in H$
be a left-integral which is an $S$-compatible dual quantum trace. Let
$L$ be an (unoriented) framed link in $S^3$ with components
$L_1,\ldots,L_m$, $m\in\N$. If there exist $\beta,\gamma\in
k\backslash\{0\}$ such that $\bar\nu(\ell)=\gamma^2/\beta$ and
$\nu(\ell)=\beta$, then
\begin{equation}
\label{eq_invarianthopf}
I(M_L) = \beta^\sigma\,\gamma^{-\sigma-m-1}\,{\left<L\right>}^{(H)}_\ell\in k
\end{equation}
forms an invariant of connected and oriented closed (smooth) $3$-manifolds.
\end{corollary}
\begin{proof}
In a Hopf algebra, we have $\epsilon_t=\eta\circ\epsilon$. This
simplifies~\eqref{eq_kirbyplus} and~\eqref{eq_kirbyminus}. Our assumptions
$\bar\nu(\ell)=\gamma^2/\beta$ and $\nu(\ell)=\beta$ then imply these two
conditions. Finally, $\one\cong H_s=k$, and so $\End(\one)=k$.
\end{proof}
The following proposition shows that this corollary applies to the Hopf
algebra dual to the one featuring in Kauffman--Radford~\cite{KaRa95}:
\begin{proposition}
Let $H$ be a finite-dimensional unimodular ribbon Hopf algebra over some field
$k$ and $H^\ast$ be its dual Hopf algebra. Then $H^\ast$ is coribbon and has a
unique (up to a scalar) non-zero left-integral $\ell\in H^\ast$ which forms an
$S$-compatible dual quantum trace. Furthermore, the invariant $I(M_L)$
of~\eqref{eq_invarianthopf} is $\gamma$ times the invariant $\mathrm{INV}(K)$
of~\cite[page 147]{KaRa95}.
\end{proposition}
\begin{proof}
The ribbon Hopf algebra $H$ is pivotal with some pivotal element $\mu\in H$, \ie\
$\mu$ is group-like, the ribbon element is given by $r=\mu^{-1}u$ where
$u=\sum S(b_i)a_i$ is the first Drinfel'd element and $R=\sum_i a_i\otimes b_i$
denotes the universal $R$-matrix, and we have $S^2(x)=\mu x\mu^{-1}$ for all
$x\in H$. Note that in~\cite{KaRa95}, our $\mu$ and $r$ are called $G$ and $\nu$,
respectively.
By~\cite{Ra94,KaRa95}, there exists a right-cointegral $\rho\colon H\to k$,
unique up to a scalar, such that
\begin{eqnarray}
\label{eq_kara1}
\rho(xy)&=&\rho(S^2(y)x),\\
\label{eq_kara2}
\rho(\mu^2 x)&=&\rho(S(x)),
\end{eqnarray}
for all $x,y\in H$. Note that in~\cite{KaRa95}, our $\rho$ is called
$\lambda.$
We pair $H$ with its dual $H^\ast$ using the evaluation map $H^\ast\otimes
H\to k$, $\phi\otimes x\mapsto \left<\phi,x\right>=\phi(x)$. The ribbon
structure of $H$ then gives rise to a coribbon structure on $H^\ast$. In order
to match the ribbon structure of $H$ in the terminology of~\cite{KaRa95} with
our definition of a coribbon (weak) Hopf algebra, we extend the canonical
pairing to tensor products such that $\left<\phi\otimes\psi,x\otimes
y\right>=\left<\phi,y\right>\left<\psi,x\right>$ for all $\phi,\psi\in
H^\ast$ and $x,y\in H$. Note that when writing down formulas, this is the
uncommon choice, but when drawing string diagrams in $\fdVect_k$, these can
now be drawn without crossings. Also, with this choice, a left-integral of $H$
is related to a right-cointegral of $H^\ast$.
The right-cointegral $\rho\colon H\to k$ then gives rise to a left-integral
$\ell\in H^\ast$, defined by $\ell=\sum_i\rho(b_i)\beta_i$ where we have
used the canonical element $\sum_i b_i\otimes\beta_i\in H\otimes H^\ast$. The
ribbon element $r\in H$ gives rise to a universal ribbon form $\nu\colon
H^\ast\to k$ such that $\nu(\phi)=\phi(r^{-1})$ for all $\phi\in H^\ast$, and
the pivotal element $\mu\in H$ is related to the copivotal form $w\colon
H^\ast\to k$ by $w(\phi)=\phi(\mu^{-1})$ for all $\phi\in H^\ast$.
The condition~\eqref{eq_kara1} implies that $\ell$ is a dual quantum-trace,
and condition~\eqref{eq_kara2} ensures that $\ell^\prime\bar
w(\ell^\pprime)\bar w(\ell^\ppprime)=S(\ell)$ which, given that $\ell$ is a
dual quantum trace, can be shown to be equivalent to $\ell$ being
$S$-compatible.
Whereas~\cite{KaRa95} uses left-modules of $H$, we work with right-comodules
of $H^\ast$. As in the proof of Theorem~\ref{thm_kara}, our diagrams are
obtained by rotating the diagrams of~\cite{KaRa95} by $180^\circ$. Also note
that in~\cite[page 147]{KaRa95}, $\lambda(\nu)=\gamma^2/\beta$ and
$\lambda(\nu^{-1})=\beta$ which shows precisely how our invariant is related
with that of Kauffman--Radford.
\end{proof}
\subsection{The Reshetikhin--Turaev invariant}
Let $\sym{C}$ be a modular category. We now specialize our
invariant~\eqref{eq_invariant} to the case in which $H$ is the canonical WHA
obtained from $\sym{C}$ by Tannaka--Kre\v\i n reconstruction. First, we show
that the conditions~\eqref{eq_kirbyplus} and~\eqref{eq_kirbyminus} are almost
satisfied as soon as $\sym{C}$ is a multi-fusion category with a ribbon
structure.
\begin{proposition}
\label{prop_rtinvariance}
Let $\sym{C}$ be a multi-fusion category over $k$ which has a ribbon
structure, and let $H=\coend(\sym{C},\omega)$ be the finite-dimensional and
split cosemisimple coribbon WHA reconstructed from $\sym{C}$ using the long
canonical functor~\eqref{eq_longfunctor}.
Let $V\in|\sym{M}^H|$ be such that $V^\ast\cong V$ and $\zeta\colon H\to k$ be
a convolution invertible and dual central linear form that satisfies
$\zeta\circ S=\zeta$. Let finally
$\ell=w(\chi_V^\prime)\zeta(\chi_V^\pprime)\chi_V^\ppprime$ where $\chi_V$ is
the dual character of $V$. Then
\begin{eqnarray}
\label{eq_kirbypluspre}
\nu(x^\prime)\bar\nu(\epsilon_t(x^\pprime)\ell) &=& \alpha\,\nu(x),\\
\label{eq_kirbyminuspre}
\bar\nu(x^\prime)\nu(\epsilon_t(x^\pprime)\ell) &=& \beta\,\bar\nu(x),
\end{eqnarray}
for all $x\in H$ where
\begin{eqnarray}
\alpha &=& \sum_{j\in I}c_jm_j\nu_j^{-1}\dim V_j,\\
\beta &=& \sum_{k\in I}c_jm_j\nu_j\dim V_j.
\end{eqnarray}
Here the $c_j$ are the coefficients of $\zeta$ as in
Proposition~\ref{prop_dualcentral}, the $m_j\in\N_0$ are the multiplicities of
the simple objects in $V$, \ie\ $V\cong\bigoplus_{j\in I}m_jV_j$, and the
$\nu_j$ are the eigenvalues of the ribbon twist $\nu\colon V_j\to V_j$ (the
left-handed one) on the simple objects, respectively.
\end{proposition}
\begin{proof}
First, from
\begin{equation}
\chi_V=\sum_{j\in I}m_j\sum_\ell{[e^\ell_{(V_j)}|e^{(V_j)}_\ell]}_{V_j},
\end{equation}
we compute
\begin{equation}
\ell=\sum_{j\in I}c_jm_j\sum_\ell
{[D_{\hat V}^{-1}\circ e^\ell_{(V_j)}\circ(D_{\hat V}\otimes\id_{V_j})|e^{(V_j)}_\ell]}_{V_j}.
\end{equation}
Then we evaluate the left-hand-side of~\eqref{eq_kirbypluspre} for arbitrary
$x={[\theta|v]}_X\in H$, $X\in|\sym{C}|$, $\theta\in{(\omega X)}^\ast$,
$v\in\omega X$:
\begin{gather}
(\bar\nu\circ\mu\circ(\nu\otimes\epsilon_t\otimes\id_H)\circ(\Delta\otimes\id_H))({[\theta|v]}_X\otimes\ell)\\
= \sum_{j\in I}c_jm_j\sum_{\ell,p}\nu({[\theta|e^{(X)}_p]}_X)\,\,\bar\nu
\bigl(\mu(\epsilon_t({[e^p_{(X)}|v]}_X)\otimes{[D_{\hat V}^{-1}\circ e^\ell_{(V_j)}\circ(D_{\hat V}\otimes\id_{V_j})|e^{(V_j)}_\ell]}_{V_j})\bigr).\notag
\end{gather}
We now use the expressions for $\nu$ and $\bar\nu$ of~\eqref{eq_nu}
and~\eqref{eq_nubar} and that $g_Y(\xi\otimes w)=\tr_{\hat V}(D_{\hat
V}\circ\xi\circ w)$ for all $Y\in|\sym{C}|$, $\xi\in{(\omega Y)}^\ast$,
$w\in\omega Y$, because $\sym{C}$ is spherical (\cf~\eqref{eq_bilfspherical}),
as well as $\epsilon_t$ from~\eqref{eq_epsilontnew} and obtain
\begin{gather}
\mathrm{(5.17)} = \sum_{j\in I}c_jm_j\sum_{\ell,p}
\tr_{\hat V}\biggl(D_{\hat V}\circ\theta\circ\bigl((\id_{\hat V}\otimes\nu_X)\circ e^{(X)}_p\bigr)\biggr)\\
\cdot\tr_{\hat V}\biggl(e^\ell_{(V_j)}\circ(D_{\hat V}\otimes\id_{V_j})
\circ\bigl((\Phi_X(v)\circ\Psi_X(e^p_{(X)}))\otimes\id_{V_j}\bigr)
\circ(\id_{\hat V}\otimes\nu^{-1}_{V_j})\circ e^{(V_j)}_\ell\biggr).\notag
\end{gather}
Using~\eqref{eq_dualbasispair} and the fact that the trace is cyclic and
multiplicative for tensor products of morphisms, we see that
\begin{equation}
\mathrm{(5.17)} = \sum_{j\in I}c_jm_j\sum_m
g_X\biggl(\theta\otimes\bigl((\id_{\hat V}\otimes\nu_X)\circ e^{(X)}_m\bigr)\biggl)\,
g_X(e^m_{(X)}\otimes v)\,\tr_{V_j}(\nu^{-1}_{V_j}).
\end{equation}
With the dual basis lemma and exploiting that the $V_j$ are simple, we arrive
at the right hand side of~\eqref{eq_kirbypluspre}:
\begin{equation}
\mathrm{(5.17)} = \nu({[\theta|v]}_X)\,\sum_{j\in I}c_j m_j\nu_j^{-1}\,\dim V_j.
\end{equation}
The proof of~\eqref{eq_kirbyminuspre} is identical except that $\nu_X$ and
$\nu^{-1}_X$ as well as $\nu$ and $\bar\nu$ are interchanged.
\end{proof}
\begin{corollary}
\label{cor_rtinvariance}
Let $\sym{C}$ be a multi-fusion category over $k$ which has a ribbon
structure, and let $H=\coend(\sym{C},\omega)$ be the finite-dimensional and
split cosemisimple coribbon WHA reconstructed from $\sym{C}$ using the long
canonical functor~\eqref{eq_longfunctor}. If $V=\hat V$ and $c_j=\dim V_j$,
the element $\ell$ in Proposition~\ref{prop_rtinvariance} agrees with the
left-integral~\eqref{eq_leftintegral} which is known to be an $S$-compatible
dual quantum trace. In this case, in~\eqref{eq_invariant},
\begin{equation}
{\left<-\right>}_\ell^{(H)}={\left<-\right>}^{(\sym{C})}_{\hat V,\zeta}
\end{equation}
is the Reshetikhin--Turaev evaluation. Furthermore, we compute that
\begin{equation}
\alpha=\sum_{j\in I}\nu_j^{-1}{(\dim V_j)}^2,\qquad
\beta=\sum_{j\in I}\nu_j{(\dim V_j)}^2.
\end{equation}
\end{corollary}
Finally, the following proposition shows that if $\sym{C}$ is modular, then
$\alpha\neq0$ and $\beta\neq0$.
\begin{proposition}
Let $\sym{C}$ be a modular category, linear over the field $k$, and let
$H=\coend(\sym{C},\omega)$ be the finite-dimensional and split cosemisimple
coribbon WHA reconstructed from $\sym{C}$ using the long canonical
functor~\eqref{eq_longfunctor}. Assume that in
Proposition~\ref{prop_rtinvariance}, we have $V=\hat V$ and $c_j=\dim
V_j$. Then $\alpha\neq 0$ and $\beta\neq 0$ in that proposition.
\end{proposition}
\begin{proof}
Let $i,j\in I$, and denote by $S_{ij}$ the coefficients of the
$S$-matrix. Then
\begin{eqnarray}
\alpha\,S_{ij}
&=&\alpha\,{\biggl<\tikzsymb{\hopflink{i}{j}}\biggr>}_\ell^{(H)}
=\sum_{p\in I}\dim V_p\,{\biggl<\tikzsymb{\triplelink{i}{p}{j}}\biggr>}_\ell^{(H)}\notag\\
&=&\sum_{p\in I}\nu_i^{-1}\nu_p^{-1}\nu_j^{-1} S_{ip}S_{pj},
\end{eqnarray}
where we have used that the $V_i$, $i\in I$ are simple; a Kirby-(+1)-move; and
again that the $V_i$ are simple. This implies that
\begin{equation}
\alpha S_{ij}\nu_i\nu_j
= \sum_{p\in I}\nu_p^{-1} S_{ip} S_{pj}.
\end{equation}
If $\alpha=0$, then the right hand side would vanish for all $i,j\in I$,
\ie\ the matrix product $ST=0$ where $T_{pj}=\nu^{-1}_p S_{pj}$. This
contradicts the invertibility of $S$. Interchanging $\nu$ and $\nu^{-1}$ as
well as $\alpha$ and $\beta$ in the above argument establishes that
$\beta=0$.
\end{proof}
\begin{corollary}
Let $\sym{C}$ be a modular category, linear over the field $k$, such that its
global dimension has a square root $\sym{D}\in k$, \ie\
\begin{equation}
\sym{D}^2 = \sum_{j\in I}{(\dim V_j)}^2.
\end{equation}
Let $H=\coend(\sym{C},\omega)$ be the finite-dimensional and split
cosemisimple coribbon WHA reconstructed from $\sym{C}$ using the long
canonical functor~\eqref{eq_longfunctor}. In
Proposition~\ref{prop_rtinvariance}, assume that $V=\hat V$ and $c_j=\dim
V_j$. Then~\eqref{eq_kirbyplus} and~\eqref{eq_kirbyminus} hold for
$\gamma=\sym{D}$, and the invariant~\eqref{eq_invariant} agrees with the
Reshetikhin--Turaev invariant.
\end{corollary}
\begin{proof}
Recall from Corollary~\ref{cor_rtinvariance} that with these choices of $V$
and $c_j$, ${\left<-\right>}_\ell^{(H)}={\left<-\right>}^{(\sym{C})}_{\hat V,\zeta}$.
In~\cite[Section II.3]{Tu10}, our $\beta$ is called $\Delta$, and our $\alpha$
corresponds to $d_0^{-1}$ (recall that our $\nu$ is the left-handed ribbon
twist, but the $\nu$ in~\cite{Tu10} is the right-handed one). Furthermore,
in~\cite[Section II.3]{Tu10}, it is shown that
$\alpha\beta=\Delta/d_0=\sym{D}^2$, \ie\ with our choice of $\gamma=\sym{D}$,
we arrive at $\alpha=\gamma^2/\beta$ as required in~\eqref{eq_kirbyplus}.
\end{proof}
The preceding corollary shows in particular that our proofs of
Theorem~\ref{thm_invariant} and Proposition~\ref{prop_rtinvariance} can be
combined to a new proof of invariance of the Reshetikhin--Turaev invariant,
\cf~\cite[Section II.3]{Tu10}.
\subsection{Contrast with the Lyubashenko invariant}
Let us finally point out what is the difference between our invariant $I(M_L)$
(Theorem~\ref{thm_invariant}) for the WHA $H=\coend(\sym{C},\omega)$
reconstructed from a modular category $\sym{C}$ and Lyubashenko's
invariant~\cite{Ly95} that uses Majid's coend
$F=\coend(\sym{C},1_\sym{C})$. Although these two invariants take the same
numerical value whenever each of them agrees with the Reshetikhin--Turaev
invariant, their computation is rather different.
Whereas our $H$ is a vector space equiped with linear structure maps that make
it a WHA, Lyubashenko's $F$ is a Hopf algebra object $H\in|\sym{C}|$, \ie\ an
object of $\sym{C}$ whose structure maps are morphisms in $\sym{C}$.
Let, for example, $\sym{C}_3$ be a modular category associated with
$U_q(\mathrm{sl}_2)$ with $3$ simple objects up to isomorphism. We denote
representatives of the classes of simple objects by $X_0\cong\one$, $X_1$ and
$X_2$. The fusion rules are given by $X_1\otimes X_1\cong X_0\oplus X_2$,
$X_1\otimes X_2\cong X_1$ and $X_2\otimes X_2\cong X_0$. We know that
$\sym{C}_3\simeq\sym{M}^H$ for our reconstructed WHA.
Lyubashenko's coend is the $H$-comodule
\begin{equation*}
F\cong (X_0\otimes X_0^\ast) \oplus (X_1\otimes X_1^\ast) \oplus (X_2\otimes X_2^\ast)
\cong 3X_0 \oplus X_2,
\end{equation*}
where we write $nX=X\oplus\cdots\oplus X$ (direct sum of $n$ terms). Note that
its decomposition does not contain $X_1$ and that the $X_1$-term is included
as $X_1\otimes X_1^\ast\hookrightarrow X_0\oplus X_2$ (as $H$-comodules). My
coend, however, is $H$ itself with the regular coaction, \ie\ the following
$H$-comodule:
\begin{equation*}
H\cong 3X_0 \oplus 4X_1 \oplus 3X_2.
\end{equation*}
Therefore, obviously, the two coends differ as objects of $\sym{M}^H$ and also
have different $k$-dimensions. Note that $\dim_k X_0=3$, $\dim_k X_1=4$ and
$\dim_k X_2=3$ in $\sym{C}_3\simeq\sym{M}^H$.
\appendix
\section{Weak Hopf Algebras and their corepresentations}
\label{app_wha}
In this appendix, we collect the relevant definitions on WHAs with
additional structure.
\subsection{Weak Bialgebras}
\label{app_wba}
Given a WHA $H$ over some field $k$, the source and target
\emph{counital maps} are given by
\begin{eqnarray}
\epsilon_s &:=&
(\id_H\otimes\epsilon)\circ(\id_H\otimes\mu)\circ(\tau_{H,H}\otimes\id_H)
\circ(\id_H\otimes\Delta)\circ(\id_H\otimes\eta)\colon H\to H,\\
\epsilon_t &:=&
(\epsilon\otimes\id_H)\circ(\mu\otimes\id_H)\circ(\id_H\otimes\tau_{H,H})
\circ(\Delta\otimes\id_H)\circ(\eta\otimes\id_H)\colon H\to H.
\end{eqnarray}
Here $\tau_{V,W}(v\otimes w)=w\otimes v$, $v\in V$, $w\in W$, denotes
the symmetric braiding in $\Vect_k$. The mutually commuting
subalgebras $H_s:=\epsilon_s(H)$ and $H_t:=\epsilon_t(H)$ are called
the source and target \emph{base algebra} of $H$, respectively.
A WBA [WHA] is a bialgebra [Hopf algebra] if and only if
$\epsilon_s=\eta\circ\epsilon$, if and only if
$\epsilon_t=\eta\circ\epsilon$, if and only if $H_s\cong k$, and if
and only if $H_t\cong k$.
If $H$ is a finite-dimensional WBA [WHA], then so is its dual vector
space $H^\ast$. Every finite-dimensional WHA $H$ has an invertible
antipode.
We abbreviate $\eta(1)=1$ and use Sweedler's notation
$\Delta(x)=x^\prime\otimes x^\pprime$ for the comultiplication of
$x\in H$ and $\beta(v)=v_0\otimes v_1$ for the coaction $\beta\colon
V\to V\otimes H$ of a right $H$-comodule $V$. If $H$ is
finite-dimensional, we also use the Sweedler arrows
$x\rightharpoonup\phi=\phi^\prime(x)\phi^\pprime$ and
$\phi\rightharpoonup x=x^\prime\phi(x^\pprime)$ for $x\in H$, $\phi\in
H^\ast$.
The category $\sym{M}^H$ of finite-dimensional right $H$-comodules of a WBA
forms a $k$-linear abelian monoidal category
$(\sym{M}^H,\otimes,\one,\alpha,\lambda,\rho)$ as follows~\cite{Pf09a}. The
source base algebra $H_s$ is a right $H$-comodule with
\begin{equation}
\beta_{H_s}\colon H_s\to H_s\otimes H,\qquad
x\mapsto x^\prime\otimes x^\pprime.
\end{equation}
It forms the monoidal unit object, $\one=H_s$ of $\sym{M}^H$.
Given $V,W\in|\sym{M}^H|$, their tensor product is the vector space
\begin{equation}
V\otimes W := \{\,v\otimes w\in V\otimes W\mid\quad v\otimes w= (v_0\otimes w_0)\epsilon(v_1w_1)\,\}
\end{equation}
with the coaction
\begin{equation}
\beta_{V\otimes W}\colon V\otimes W\to (V\otimes W)\otimes H,\quad
v\otimes w \mapsto (v_0\otimes w_0)\otimes (v_1w_1).
\end{equation}
The associator $\alpha_{U,V,W}\colon(U\otimes V)\otimes W\to
U\otimes(V\otimes W)$ is induced from that of $\Vect_k$. The left
and right unit constraints are given by
\begin{alignat}{3}
\lambda_V &\colon H_s\otimes V\to V,&&\quad h\otimes v\mapsto v_0\epsilon(hv_1),\\
\rho_V &\colon V\otimes H_s\to V,&&\quad v\otimes h\mapsto v_0\epsilon(v_1h).
\end{alignat}
For convenience, we list their inverses, too:
\begin{alignat}{3}
\lambda^{-1}_V &\colon V\to H_s\otimes V,&&\quad v\mapsto (1^\prime\otimes v_0)\epsilon(1^\pprime v_1),\\
\rho^{-1}_V &\colon V\to V\otimes H_s,&&\quad v\mapsto v_0\otimes\epsilon_s(v_1).
\end{alignat}
\subsection{Weak Hopf Algebras}
A \emph{left-autonomous}~\cite{FrYe92} category $\sym{C}$ is a
monoidal category in which every object $X\in|\sym{C}|$ is equipped
with a \emph{left-dual} $(X^\ast,\ev_X,\coev_X)$, \ie\ an object
$X^\ast\in|\sym{C}|$ and morphisms $\ev_X\colon X^\ast\otimes
X\to\one$ (\emph{left evaluation}) and $\coev_X\colon\one\to X\otimes
X^\ast$ (\emph{left coevaluation}) that satisfy the triangle
identities
\begin{eqnarray}
\rho_X\circ(\id_X\otimes\ev_X)\circ\alpha_{X,X^\ast,X}
\circ(\coev_X\otimes\id_X)\circ\lambda_X^{-1} &=& \id_X,\\
\lambda_{X^\ast}\circ(\ev_X\otimes\id_{X^\ast})\circ\alpha^{-1}_{X^\ast,X,X^\ast}
\circ(\id_{X^\ast}\otimes\coev_X)\circ\rho_{X^\ast}^{-1} &=& \id_{X^\ast}.
\end{eqnarray}
In a left-autonomous category $\sym{C}$, the \emph{left-dual} of a
morphism $f\colon X\to Y$ is defined as
\begin{equation}
f^\ast := \lambda_{X^\ast}\circ(\ev_Y\otimes\id_{X^\ast})\circ\alpha^{-1}_{Y^\ast,Y,X^\ast}
\circ(\id_{Y^\ast}\otimes(f\otimes\id_{X^\ast}))\circ(\id_{Y^\ast}\otimes\coev_X)\circ\rho^{-1}_{Y^\ast}.
\end{equation}
Let $\sym{C}$ be a left-autonomous category. The functor
$\ast\colon\sym{C}\to\sym{C}^\op$ that sends each object and each
morphism to their left-dual, is strong monoidal.
Let $H$ be a WHA. The category $\sym{M}^H$ is \emph{left-autonomous}
as follows. For each $V\in|\sym{M}^H|$, the dual vector space $V^\ast$
forms a right $H$-comodule with
\begin{equation}
\label{eq_dualaction}
\beta_{V^\ast}\colon V^\ast\to V^\ast\otimes H,\qquad
\theta\mapsto (v\mapsto \theta(v_0)\otimes S(v_1)).
\end{equation}
The left-dual of $V$ is given by $(V^\ast,\ev_V,\coev_V)$ where
\begin{alignat}{3}
\label{eq_ev}
\ev_V &\colon V^\ast\otimes V\to H_s,&&\quad \theta\otimes v\to\theta(v_0)\epsilon_s(v_1),\\
\label{eq_coev}
\coev_V &\colon H_s\to V\otimes V^\ast,&&\quad x\mapsto\sum_j ({(e_j)}_0\otimes e^j)\epsilon(x{(e_j)}_1).
\end{alignat}
Here we have used the evaluation and coevaluation maps that turn
$V^\ast$ into a left-dual of $V$ in $\fdVect_k$:
\begin{alignat}{3}
\ev_V^{(\fdVect_k)} &\colon V^\ast\otimes V\to k,&&\quad \theta\otimes v\mapsto \theta(v),\\
\coev_V^{(\fdVect_k)} &\colon k\to V\otimes V^\ast,&&\quad 1\mapsto \sum_j e_j\otimes e^j.
\end{alignat}
\subsection{Copivotal Weak Hopf Algebras}
Let $H$ be a WBA. A linear form $f\colon H\to k$ is said to be
\emph{convolution invertible} if there exists some linear form $\bar
f\colon H\to k$ such that $f(x^\prime)\bar
f(x^\pprime)=\epsilon(x)=\bar f(x^\prime)f(x^\pprime)$ for all $x\in
H$. The linear form $f$ is called \emph{dual central} if
$f(x^\prime)x^\pprime = x^\prime f(x^\pprime)$ for all $x\in H$. It is
called \emph{dual group-like} if $\epsilon(x^\prime
y^\prime)f(x^\pprime)f(y^\pprime) = f(xy) =
f(x^\prime)f(y^\prime)\epsilon(x^\pprime y^\pprime)$ for all $x,y\in
H$ and $f(\epsilon_t(x))=\epsilon(x)=f(\epsilon_s(x))$ for all $x\in
H$. Note that in a WHA, every dual group-like linear form is
convolution invertible with $\bar f(x)=f(S(x))$.
A WHA $H$ is called \emph{copivotal}~\cite{Pf09b} if there exists a
dual group-like linear form $w\colon H\to k$ such that $S^2(x) =
w(x^\prime)x^\pprime\bar w(x^\ppprime)$ for all $x\in H$.
A \emph{pivotal} category~\cite{FrYe92} $\sym{C}$ is a left-autonomous
category with a monoidal natural equivalence $\tau\colon
1_\sym{C}\Rightarrow \ast\circ\ast$ such that ${(\tau_X)}^\ast =
\tau_{X^\ast}^{-1}$ for all $X\in|\sym{C}|$.
Given a copivotal WHA $H$, the category $\sym{M}^H$ is pivotal with
$\tau_V\colon V\to {V^\ast}^\ast$ given by
\begin{equation}
\tau_V(v) = \tau_V^{(\fdVect_k)}(v_0)w(v_1)
\end{equation}
for all $V\in|\sym{M}^H|$ and $v\in V$. Here we denote by
$\tau_V^{(\fdVect_k)}\colon V\to {V^\ast}^\ast$ the pivotal structure of
$\fdVect_k$ which is just the usual canonical identification $V\cong
{V^\ast}^\ast$.
A \emph{right-autonomous} category $\sym{C}$ is a monoidal category in
which every object $X\in|\sym{C}|$ is equipped with a \emph{right-dual}
$(\bar X,\bar\ev_X,\bar\coev_X)$, \ie\ an object $\bar X\in|\sym{C}|$
with morphisms $\bar\ev_X\colon X\otimes\bar X\to\one$ (\emph{right
evaluation}) and $\bar\coev_X\colon\one\to\bar X\otimes X$
(\emph{right coevaluation}) that satisfy the triangle identities
\begin{eqnarray}
\lambda_X\circ(\bar\ev_X\otimes\id_X)\circ\alpha^{-1}_{X,\bar X,X}
\circ(\id_X\otimes\bar\coev_X)\circ\rho_X^{-1} &=& \id_X,\\
\rho_{\bar X}\circ(\id_{\bar X}\otimes\bar\ev_X)\circ\alpha_{\bar X,X,\bar X}
\circ(\bar\coev_X\otimes\id_{\bar X})\circ\lambda^{-1}_{\bar X} &=& \id_{\bar X}.
\end{eqnarray}
Note that every pivotal category $\sym{C}$ is not only left-, but also
right-autonomous with $\bar X=X^\ast$ and
\begin{eqnarray}
\label{eq_barev}
\bar\ev_X &=& \ev_{X^\ast}\circ(\tau_X\otimes\id_{X^\ast}),\\
\label{eq_barcoev}
\bar\coev_X &=& (\id_{X^\ast}\otimes\tau_X^{-1})\circ\coev_{X^\ast}
\end{eqnarray}
for all $X\in|\sym{C}|$. We can therefore define the \emph{right-dual}
of a morphism $f\colon X\to Y$ as
\begin{equation}
\bar f:=\rho_{\bar X}\circ(\id_{\bar X}\otimes\bar\ev_Y)\circ\alpha_{\bar X,Y,\bar Y}
\circ((\id_{\bar X}\otimes f)\otimes\id_{\bar Y})\circ(\bar\coev_X\otimes\id_{\bar Y})\circ\lambda^{-1}_{\bar Y}.
\end{equation}
It can be shown to agree with the left-dual, \ie\ $\bar f=f^\ast$.
Using both left- and right-duals, we can define two traces of a morphism $f\colon X\to Y$, the
\emph{left-trace}
\begin{equation}
\tr^{(L)}_X(f) = \ev_X\circ(\id_{X^\ast}\otimes f)\circ\bar\coev_X\colon\one\to\one
\end{equation}
and the \emph{right-trace}
\begin{equation}
\tr^{(R)}_X(f) = \bar\ev_X\circ(f\otimes\id_{X^\ast})\circ\coev_X\colon\one\to\one.
\end{equation}
Both left- and right-traces are cyclic, \ie\ $\tr^{(L)}_X(g\circ
f)=\tr^{(L)}_Y(f\circ g)$ for all $f\colon X\to Y$ and $g\colon Y\to
X$ and similarly for the right-trace. In general, however, left- and
right-traces need not agree.
\subsection{Cospherical Weak Hopf Algebras}
A \emph{spherical category}~\cite{BaWe99} is a pivotal category in
which $\tr^{(L)}_X(f) = \tr^{(R)}_X(f)$ for all morphisms $f\colon
X\to X$ in $\sym{C}$. In this case, the above expression is just
called the \emph{trace} of $f$ and denoted by $\tr_X(f)$, and $\dim(X)
= \tr_X(\id_X)$ is called the \emph{dimension} of $X$. Note that in a
spherical category, $\tr_X(f)=\tr_{X^\ast}(f^\ast)$ for every morphism
$f\colon X\to X$ and thus $\dim(X)=\dim(X^\ast)$. Finally,
$\tr_{X_1\otimes X_2}(h_1\otimes h_2)=\tr_{X_1}(h_1)\tr_{X_2}(h_2)$
for all $h_j\colon X_j\to X_j$, $j\in\{1,2\}$.
A \emph{cospherical} WHA~\cite{Pf09b} $H$ is a copivotal WHA for which
$\tr^{(L)}_V(f) = \tr^{(R)}_V(f)$ for all morphisms $f\colon V\to V$
of $\sym{M}^H$. If $H$ is a cospherical WHA, then $\sym{M}^H$ is
therefore spherical.
\subsection{Coquasitriangular Weak Hopf Algebras}
A \emph{coquasitriangular} WHA~\cite{Pf09a} is a WHA with a linear
form $r\colon H\otimes H\to k$, the \emph{universal $r$-form}, that
satisfies the following conditions:
\begin{myenumerate}
\item
For all $x,y\in H$,
\begin{equation}
\label{eq_coquasidef}
\epsilon(x^\prime y^\prime)r(x^\pprime\otimes y^\pprime)=r(x\otimes y)
=r(x^\prime\otimes y^\prime)\epsilon(y^\pprime x^\pprime).
\end{equation}
\item
There exists some linear $\bar r\colon H\otimes H\to k$ that is a
weak convolution inverse of $r$, \ie\
\begin{eqnarray}
\label{eq_coquasiinv1}
\bar r(x^\prime\otimes y^\prime)r(x^\pprime\otimes y^\pprime)&=&\epsilon(yx),\\
\label{eq_coquasiinv2}
r(x^\prime\otimes y^\prime)\bar r(x^\pprime\otimes y^\pprime)&=&\epsilon(xy).
\end{eqnarray}
\item
For all $x,y,z\in H$,
\begin{eqnarray}
x^\prime y^\prime r(x^\pprime\otimes y^\pprime)
&=&r(x^\prime\otimes y^\prime)y^\pprime x^\pprime,\\
r((xy)\otimes z)&=&r(y\otimes z^\prime) r(x\otimes z^\pprime),\\
r(x\otimes (yz))&=&r(x^\prime\otimes y) r(x^\pprime\otimes z).
\end{eqnarray}
\end{myenumerate}
Note that $\bar r$ in~(2) is uniquely determined by $r$ if one
imposes~\eqref{eq_coquasidef}, \eqref{eq_coquasiinv1}
and~\eqref{eq_coquasiinv2}.
In a coquasitriangular WHA $H$, we define the linear form $q\colon H\otimes H\to k$ by
\begin{equation}
\label{eq_defq}
q(x\otimes y) = r(x^\prime\otimes y^\prime)r(y^\pprime\otimes x^\pprime),
\end{equation}
for all $x,y\in H$. Its weak convolution inverse $\bar q\colon
H\otimes H\to k$ is then given by
\begin{equation}
\label{eq_defbarq}
\bar q(x\otimes y) = \bar r(y^\prime\otimes x^\prime)\bar r(x^\pprime\otimes y^\pprime),
\end{equation}
for all $x,y\in H$. The \emph{dual Drinfel'd elements} are the linear
forms $u\colon H\to k$ and $v\colon H\to k$ given by
$u(x)=r(S(x^\pprime)\otimes x^\prime)$ and $v(x)=r(S(x^\prime)\otimes
x^\pprime)$ for all $x\in H$.
A \emph{braided monoidal category} $\sym{C}$ is a monoidal category
with natural isomorphisms $\sigma_{X,Y}\colon X\otimes Y\to Y\otimes
X$ for all $X,Y\in|\sym{C}|$ that satisfy the two hexagon axioms
\begin{eqnarray}
\sigma_{X\otimes Y,Z} &=& \alpha_{Z,X,Y}\circ(\sigma_{X,Z}\otimes\id_Y)
\circ\alpha^{-1}_{X,Z,Y}\circ(\id_X\otimes\sigma_{Y,Z})\circ\alpha_{X,Y,Z},\\
\sigma_{X,Y\otimes Z} &=& \alpha^{-1}_{Y,Z,X}\circ(\id_Y\otimes\sigma_{X,Z})
\circ\alpha_{Y,X,Z}\circ(\sigma_{X,Y}\otimes\id_Z)\circ\alpha^{-1}_{X,Y,Z}
\end{eqnarray}
for all $X,Y,Z\in|\sym{C}|$.
If $H$ is a coquasitriangular WHA, then the category $\sym{M}^H$ is
braided monoidal with braiding
$\sigma_{V,W}\colon V\otimes W\to W\otimes V$ given by
\begin{equation}
\label{eq_braiding}
\sigma_{V,W}(v\otimes w)=(w_0\otimes v_0)r(w_1\otimes v_1)
\end{equation}
for all $V,W\in|\sym{M}^H|$ and $v\in V$, $w\in W$. Note that
\begin{equation}
\label{eq_defQ}
Q_{V,W}=\sigma_{W,V}\circ\sigma_{V,W}
\end{equation}
can be computed as $Q_{V,W}(v\otimes w)=(v_0\otimes w_0)q(v_1\otimes w_1)$ for
all $v\in V$, $w\in W$, and similarly $Q_{V,W}^{-1}(v\otimes w)=(v_0\otimes
w_0)\bar q(v_1\otimes w_1)$.
\subsection{Coribbon Weak Hopf Algebras}
\label{app_coribbon}
A \emph{coribbon} WHA~\cite{Pf09a} is a coquasitriangular WHA with a
convolution invertible and dual central linear form $\nu\colon H\to
k$, the \emph{universal ribbon twist}, such that
\begin{eqnarray}
\label{eq_ribbontensor}
\nu(xy)&=&\nu(x^\prime)\nu(y^\prime)q(x^\pprime\otimes y^\pprime),\\
\nu(S(x))&=&\nu(x)
\end{eqnarray}
all $x,y\in H$.
A \emph{ribbon category} is a braided monoidal category that is
left-autonomous with natural isomorphisms $\nu_X\colon X\to X$, the
\emph{ribbon twist}, such that
\begin{equation}
\nu_{X\otimes Y} = \sigma_{Y,X}\circ\sigma_{X,Y}\circ(\nu_X\otimes\nu_Y)
\end{equation}
and
\begin{equation}
(\nu_X\otimes\id_{X^\ast})\circ\coev_X = (\id_X\otimes\nu_{X^\ast})\circ\coev_X
\end{equation}
for all $X,Y\in|\sym{C}|$.
Note that every ribbon category is pivotal with $\tau_X\colon X\to
{X^\ast}^\ast$ given by
\begin{equation}
\tau_X = \lambda_{{X^\ast}^\ast}\circ(\ev_X\otimes\id_{{X^\ast}^\ast})
\circ(\sigma_{X,X^\ast}\otimes\id_{{X^\ast}^\ast})
\circ(\nu_X\otimes\coev_{X^\ast})\circ\rho_X^{-1},
\end{equation}
and furthermore spherical. For convenience, we give the right
evaluation and coevaluation of~\eqref{eq_barev}
and~\eqref{eq_barcoev}:
\begin{eqnarray}
\label{eq_barevribbon}
\bar\ev_X &=& \ev_X\circ\sigma_{X,X^\ast}\circ(\nu_X\otimes\id_{X^\ast}),\\
\label{eq_barcoevribbon}
\bar\coev_X &=& (\id_{X^\ast}\otimes\nu_X)\circ\sigma_{X,X^\ast}\circ\coev_X.
\end{eqnarray}
If $H$ is a coribbon WHA, then $\sym{M}^H$ is a ribbon category with
ribbon twist
\begin{equation}
\label{eq_twist}
\nu_V\colon V\to V,\qquad v\mapsto v_0\nu(v_1),
\end{equation}
for all $V\in|\sym{M}^H|$ and $v\in V$. Every coribbon WHA is
cospherical with the copivotal form $w(x)=v(x^\prime)\nu(x^\pprime)$
for all $x\in H$, involving the second Drinfel'd element and the
universal ribbon form.
Using the definition of a ribbon category as in this Appendix, we can
draw the corresponding string diagrams. If we draw composition from
top to bottom and the tensor product from left to right, we arrive at
the diagrams shown in Section~\ref{sect_evaluation}.
\subsection{Cosemisimple Weak Hopf Algebras and fusion categories}
\label{sect_fusion}
A monoidal category $\sym{C}$ is called $k$-linear over some field $k$
if the underlying category is $k$-linear, \ie\ enriched in the
category $\Vect_k$, and the tensor product of morphisms is
$k$-bilinear. A $k$-linear category is called \emph{additive} if it
has a terminal object and all binary products. Such a category
automatically has all finite biproducts. A $k$-linear category is
\emph{abelian} if it is additive, has all finite limits, and if every
monomorphism is a kernel and every epimorphism a cokernel. Note that
in $k$-linear pivotal categories, the traces
$\tr_X^{(L)},\tr_X^{(R)}\colon\End(X)\to\End(\one)$ are $k$-linear.
In a $k$-linear additive category $\sym{C}$, we call an object
$X\in|\sym{C}|$ \emph{simple} if $\End(X)\cong k$. A $k$-linear
additive monoidal category is called \emph{pure} if the monoidal unit
$\one$ is simple. A $k$-linear additive category is called \emph{split
semisimple} if every object $X\in|\sym{C}|$ is isomorphic to a
finite biproduct of simple objects. It is called \emph{finitely split
semisimple} if it is split semisimple and there exist only a finite
number of simple objects up to isomorphism. If $\sym{C}$ is a
$k$-linear additive category that is split semisimple, we denote by
${\{V_j\}}_{j\in I}$ a family of representatives of the isomorphism
classes of the simple objects $V_j$, $j\in I$, of $\sym{C}$, indexed
by the set $I$.
A \emph{multi-fusion category} $\sym{C}$ over $k$, see, for
example~\cite{EtNi05}, is a finitely split semisimple $k$-linear
additive autonomous monoidal category such that $\Hom(X,Y)$ is
finite-dimensional over $k$ for all $X,Y\in|\sym{C}|$. A \emph{fusion
category} over $k$ is a pure multi-fusion category. Note that every
multi-fusion category is abelian and essentially small, and that in
every multi-fusion category, if $X\in|\sym{C}|$ is simple, then so is
$X^\ast$.
If $C$ is a coalgebra and $V$ a finite-dimensional right $C$-comodule
with coaction $\beta\colon V\to V\otimes C$ and basis ${(v_j)}_j$,
then there are elements $c^{(V)}_{\ell j}\in C$ uniquely determined by
the condition that $\beta_V(v_j)=\sum_\ell v_\ell\otimes c^{(V)}_{\ell
j}$. They are called the \emph{coefficients of} $V$ with respect to
that basis. They span the \emph{coefficient coalgebra}
$C(V)=\Span_k\{c^{(V)}_{\ell j}\}$, a subcoalgebra of $C$.
Let $W$ be a finite-dimensional vector space over $k$ with dual space
$W^\ast$ and a pair of dual bases ${(e_j)}_j$ and ${(e^j)}_j$ of $W$
and $W^\ast$, respectively. We abbreviate $c^{(W)}_{jk}=e^j\otimes
e_k\in W^\ast\otimes W$. The coalgebra $(W^\ast\otimes
W,\Delta,\epsilon)$ with $\Delta(c^{(W)}_{jk})=\sum_\ell
c^{(W)}_{j\ell}\otimes c^{(W)}_{\ell k}$ and
$\epsilon(c^{(W)}_{jk})=\delta_{jk}$ is called the \emph{matrix
coalgebra} associated with $W$. In this case, $W$ is a right
$W^\ast\otimes W$-comodule, and $W^\ast\otimes W$ is its coefficient
coalgebra.
A coalgebra $C$ is called \emph{cosimple} if $C$ has no subcoalgebras
other than $C$ and $\{0\}$. The coalgebra $C$ is called
\emph{cosemisimple} if it is a coproduct in $\Vect_k$ of cosimple
coalgebras. The coalgebra $C$ is called \emph{split cosemisimple} if
it is cosemisimple and every cosimple subcoalgebra is a matrix
coalgebra. A right $C$-comodule $V$ of some coalgebra $C$ is called
\emph{irreducible} if $V\neq\{0\}$ and $V$ has no subcomodules other
than $V$ and $\{0\}$.
If $H$ is a WHA over some field $k$, then $\sym{M}^H$ is a $k$-linear
abelian autonomous monoidal category such that $\Hom(X,Y)$ is
finite-dimensional over $k$ for all $X,Y\in|\sym{C}|$. If $H$ is in
addition [finite-dimensional and] split cosemisimple, then $\sym{M}^H$
is [finitely] split semisimple. A WHA is called \emph{copure} if its
base algebras intersect trivially, \ie\ if $H_s\cap H_t\cong k$, see,
for example~\cite{Pf09a}. In this case, $\sym{M}^H$ is pure.
A WHA over $k$ is said to be \emph{multi-fusion} if it is
finite-dimensional and split cosemisimple. It is called \emph{fusion}
if it is in addition copure. In these cases, $\sym{M}^H$ is multi-fusion
or fusion over $k$, respectively.
\subsection{Comodular Weak Hopf Algebras}
Let $\sym{C}$ be a ribbon category and $V,W\in|\sym{C}|$. We call the
evaluation of the Hopf link with components labeled by $V$ and $W$,
\begin{equation}
S_{V,W} := \tr_{V\otimes W}(Q_{V,W})\in\End(\one).
\end{equation}
If $\sym{C}$ is in addition multi-fusion with a family ${\{V_j\}}_{j\in I}$ of
representatives of the isomorphism classes of objects, we write
$S_{j\ell}:=S_{V_jV_\ell}$, $j,\ell\in I$. If $\sym{C}$ is fusion,
$\End(\one)\cong k$, and so $S_{j\ell}\in k$. A \emph{modular category} is a
fusion category that has the structure of a ribbon category and for which the
$|I|\times|I|$-matrix with coefficients $S_{j\ell}$, $j,\ell\in I$, is
non-degenerate.
Let $H$ be a copure coribbon WHA with copivotal form $w\colon H\to k$ and
$V\in|\sym{M}^H|$, $n:=\dim_kV$. The \emph{dual character} of $V$ is the
element
\begin{equation}
\label{eq_dualchar}
\chi_V=\sum_{j=1}^n c^{(V)}_{jj}\in H,
\end{equation}
and the \emph{dual quantum character} the element
\begin{equation}
\label{eq_dualqchar}
T_V=\sum_{j,\ell=1}^n c^{(V)}_{j\ell}w(c^{(V)}_{\ell j})\in H.
\end{equation}
We denote the space of dual quantum characters of $H$ by
\begin{equation}
T(H) = \Span_k\{\,T_V\colon\quad V\in|\sym{M}^H|\,\}.
\end{equation}
If $H$ is a copure coribbon WHA and the morphism $f^{(\gamma)}_{V,W}\colon
V\otimes W\to V\otimes W$ is of the form
\begin{equation}
f^{(\gamma)}_{V,W}
=(\id_{V\otimes W}\otimes\gamma)\circ(\id_V\otimes\tau_{H,W}\otimes\id_H)
\circ(\beta_V\otimes\beta_W)
\end{equation}
with a linear form $\gamma\colon H\otimes H\to k$ that satisfies
\begin{eqnarray}
x^\prime y^\prime\gamma(x^\pprime\otimes y^\pprime)
&=& \gamma(x^\prime\otimes y^\prime)x^\pprime y^\pprime,\\
\epsilon(x^\prime y^\prime)\gamma(x^\pprime\otimes y^\pprime)
&=& \gamma(x\otimes y)
\end{eqnarray}
for all $x,y\in H$, then
\begin{equation}
\tr_{V\otimes W}(f^{(\gamma)}_{V,W}) = c^{(\gamma)}_{V,W}\,\id_{H_s},
\end{equation}
where the element $c^{(\gamma)}_{V,W}\in k$ is determined by
\begin{equation}
\gamma(T_V^\prime\otimes T_W^\prime)\epsilon_s(S(T_V^\pprime T_W^\pprime))
= c^{(\gamma)}_{V,W}\eta(1).
\end{equation}
Given a copure coribbon WHA, we can therefore define a linear form $\tilde
q\colon T(H)\otimes T(H)\to k$, $T_V\otimes T_W\to \tilde q_{V,W}$ where the
$\tilde q_{V,W}\in k$ are determined by
\begin{equation}
q(T_V^\prime\otimes T_W^\prime)\epsilon_s(S(T_V^\pprime T_W^\pprime))
= \tilde q_{V,W}\,\eta(1).
\end{equation}
$H$ is called \emph{weakly cofactorizable} if every linear form $\phi\colon
T(H)\to k$ can be written as $\phi(-)=\tilde q(-\otimes x)$ for some $x\in
T(H)$.
A \emph{comodular} WHA~\cite{Pf09a} is a coribbon WHA that is fusion and
weakly cofactorizable. If $H$ is a comodular WHA, then $\sym{M}^H$ is a
modular category~\cite{Pf09a}.
\section{Tannaka--Kre\v\i n reconstruction for a fusion categories}
\label{app_reconstruction}
\subsection{Fusion categories}
Let $\sym{C}$ be a multi-fusion category (see Appendix~\ref{sect_fusion}) over
some field $k$. By ${\{V_j\}}_{j\in I}$ we denote a (finite) set of
representatives of the isomorphism classes of simple objects of $\sym{C}$. We
use the small progenerator $\hat V=\bigoplus_{j\in I}V_j$. The \emph{long
canonical functor}
\begin{eqnarray}
\omega\colon\sym{C}\to\Vect_k,\quad X &\mapsto&\Hom(\hat V,\hat V\otimes X),\\
f &\mapsto& (\id_{\hat V}\otimes f)\circ-,\nn
\end{eqnarray}
is $k$-linear, faithful and exact~\cite{Ha99b}, takes values in
$\fdVect_k$, and has a separable Frobenius
structure~\cite{Sz05,Pf09a,Pf11}.
The algebra $R:=\End(\hat V)\cong\omega\one\cong k^{|I|}$ has a basis
${(\lambda_j)}_{j\in I}$ of orthogonal idempotents given by
$\lambda_j=\id_{V_j}\in R$, $j\in I$. It forms a Frobenius algebra
$(R,\circ,\id_R,\Delta_R,\epsilon_R)$ with comultiplication
$\Delta_R\colon R\to R\otimes R$ and counit $\epsilon_R\colon R\to k$
given by $\Delta(\lambda_j)=\lambda_j\otimes\lambda_j$ and
$\epsilon(\lambda_j)=1$ for all $j\in I$. The element $\Delta(\id_R)$
is a separability idempotent. Such a Frobenius algebra is called
\emph{index one} or \emph{Frobenius separable}~\cite{KaSz03}.
The category $\sym{C}$ is equipped with a family of non-degenerate
bilinear forms
\begin{equation}
\label{eq_gx}
g_X\colon\Hom(\hat V\otimes X,\hat V)\otimes\Hom(\hat V,\hat V\otimes X)\to k,\qquad
\theta\otimes v\mapsto\epsilon_R(\theta\circ v),
\end{equation}
which is associative, \ie\ compatible with composition,
$g_X((\theta\circ\omega f)\otimes v)=g_Y(\theta\otimes(\omega f\circ
v))$ for all $v\colon\hat V\to\hat V\otimes X$, $\theta\colon\hat
V\otimes Y\to\hat V$ and $f\colon X\to Y$. We use $g_X$ in order to
identify ${(\omega X)}^\ast\cong\Hom_k(\hat V\otimes X,\hat V)$.
By ${(e_m^{(X)})}_m$ and ${(e^m_{(X)})}_m$, we denote a pair of dual
bases of $\omega X=\Hom(\hat V,\hat V\otimes X)$ and $\Hom(\hat
V\otimes X,\hat V)$ with respect to $g_X$. In particular, we can
choose $e_j^{(\one)}=\rho_{\hat V}^{-1}\circ\lambda_j$ and
$e^j_{(\one)}=\lambda_j\circ\rho_{\hat V}$. Many computations are
particularly convenient if performed in these bases.
In addition to the dual basis lemma, \ie\ the triangle identities for
evaluation with the bilinear form $g_X$, the pair of dual bases satisfies
\begin{equation}
\label{eq_dualbasispair}
\sum_je^{(X)}_j\circ e^j_{(X)} = \id_{\hat V}\otimes\id_X
\end{equation}
for all simple $X\in|\sym{C}|$.
By a generalization of Tannaka--Kre\v\i n reconstruction from strong
monoidal functors to functors with separable Frobenius structure, we
obtain a finite-dimensional, split cosemisimple WHA
\begin{equation}
\label{eq_coendvect}
H = \coend(\sym{C},\omega) = \bigoplus_{j\in I}{(\omega V_j)}^\ast\otimes\omega V_j,
\end{equation}
such that $\sym{C}\simeq\sym{M}^H$ are equivalent as $k$-linear
additive monoidal categories. If $\sym{C}$ is fusion, \ie\ pure, then
in addition $H$ is copure, \ie\ $H_s\cap H_t\cong k$. The operations
of $H$ are given as follows~\cite{Pf09a},
\begin{eqnarray}
\mu({[\theta|v]}_X\otimes {[\zeta|w]}_Y)
&=& {[\zeta\circ(\theta\otimes\id_Y)\circ\alpha_{\hat V,X,Y}^{-1}|
\alpha_{\hat V,X,Y}\circ(v\otimes\id_Y)\circ w]}_{X\otimes Y},\\
\eta(1) &=& {[\rho_{\hat V}|\rho_{\hat V}^{-1}]}_\one,\\
\Delta({[\theta|v]}_X)
&=& \sum_j{[\theta|e_j^{(X)}]}_X\otimes {[e^j_{(X)}|v]}_X,\\
\epsilon ({[\theta|v]}_X) &=& \epsilon_R(\theta\circ v),\\
S({[e^j_{(X)}|e_\ell^{(X)}]}_X)
&=& {[\tilde e^\ell_{(X^\ast)}|\tilde e_j^{(X^\ast)}]}_{X^\ast},
\end{eqnarray}
where we write ${[\theta|v]}_X\in{(\omega X)}^\ast\otimes\omega X$
with $v\in\omega X$, $\theta\in{(\omega X)}^\ast$ and simple
$X\in|\sym{C}|$ for the homogeneous elements of $H$. The precise form
of the universal coend as a colimit also allows us to use the same
expression for arbitrary objects of $\sym{C}$, but subject to the
relations that $[{\zeta|(\omega f)(v)]}_Y ={[{(\omega
f)}^\ast(\zeta)|v]}_X$ for all $v\in\omega X$, $\zeta\in{(\omega
Y)}^\ast$ and for all morphisms $f\colon X\to Y$ of
$\sym{C}$. Recall that $(\omega f)(v)=(\id_{\hat V}\otimes f)\circ v$
and ${(\omega f)}^\ast(\zeta)=\zeta\circ(\id_{\hat V}\otimes
f)$. Furthermore, by ${(\tilde e_j^{(X^\ast)})}_j$ we denote the basis
of $\omega(X^\ast)$ defined by
\begin{equation}
\label{eq_antibasis}
\tilde e_j^{(X^\ast)} = \Psi_X(e^j_{(X)}),
\end{equation}
where
\begin{equation}
\label{eq_psi}
\Psi_X(\theta) = (\theta\otimes\id_{X^\ast})
\circ\alpha^{-1}_{\hat V,X,X^\ast}\circ(\id_{\hat V}\otimes\coev_X)
\circ\rho_{\hat V}^{-1}.
\end{equation}
By ${(\tilde e^j_{(X^\ast)})}_j$,we denote its dual basis with respect to the
bilinear form $g_{X^\ast}$, \cf~\eqref{eq_gx}.
The source and target counital maps are given by
\begin{eqnarray}
\epsilon_s({[\theta|v]}_X) &=&
{[\rho_{\hat V}|\rho^{-1}_{\hat V}\circ\theta\circ v]}_\one,\\
\label{eq_epsilont}
\epsilon_t({[\theta|v]}_X) &=&
\sum_\ell{[e^\ell_{(\one)}|\rho^{-1}_{\hat V}]}_\one\,
g_X(\theta\otimes(((\rho_{\hat V}\circ e^{(\one)}_\ell)\otimes\id_X)\circ v)),
\end{eqnarray}
for all simple $X\in|\sym{C}|$ and $\theta\in{(\omega X)}^\ast$, $v\in\omega X$.
Note that if $X\in|\sym{C}|$ is an arbitrary object, then $\omega X$
forms a right-$H$ comodule with the coaction
\begin{equation}
\beta_{\omega X}\colon\omega X\to\omega X\otimes H,\quad
v\mapsto\sum_je^{(X)}_j\otimes{[e^j_{(X)}|v]}_X.
\end{equation}
Its coefficient coalgebra is given by $C(X)={({(\omega
X)}^\ast\otimes\omega X)}/N_X\subseteq H$ where the subspace
$N_X\subseteq{(\omega X)}^\ast\otimes\omega X$ is generated by the
elements
\begin{equation}
{[\theta|(\omega f)(v)]}_X-{[{(\omega f)}^\ast(\theta)|v]}_X
\end{equation}
for all $v\in\omega X$, $\theta\in{(\omega X)}^\ast$ and
$f\in\End(X)$.
\subsection{Additional structure}
If $\sym{C}$ is pivotal with the monoidal natural isomorphism
$\tau_X\colon X\to{X^\ast}^\ast$, then $H=\coend(\sym{C},\omega)$ is
copivotal~\cite{Pf09b} with copivotal form and its convolution inverse
given by
\begin{eqnarray}
\label{eq_copivotal}
w({[\theta|v]}_X)
&=& g_X((D_{\hat V}^{-1}\circ\theta\circ(D_{\hat V}\otimes\id_X))\otimes v),\\
\bar w({[\theta|v]}_X)
&=& g_X(\theta\otimes((D_{\hat V}\otimes\id_X)\circ v\circ D_{\hat V}^{-1})),
\end{eqnarray}
for all $v\in\omega X$, $\theta\in{(\omega X)}^\ast$ and
$X\in|\sym{C}|$. Here
\begin{equation}
\label{eq_dtransformation}
D_{\hat V}=\sum_{j\in I}{(\dim V_j)}^{-1}\,\id_{V_j}\colon \hat V\to\hat V.
\end{equation}
More generally, there is a natural equivalence $D\colon 1_\sym{C}\Rightarrow
1_\sym{C}$ given by isomorphisms $D_X\colon X\to X$ for all $X\in|\sym{C}|$ as
follows. If
\begin{equation}
X\cong\bigoplus_{j\in I}m_jV_j
\end{equation}
with multiplicities $m_j\in\N_0$, then $D_X(v)=(\dim V_j)\,\id_{V_j}$ for all
homogeneous $v\in m_jV_j$.
If $\sym{C}$ is spherical, then $H$ is cospherical~\cite{Pf09b}. In
this case, the bilinear forms $g_X$ are related to the traces in
$\sym{C}$ as follows,
\begin{equation}
\label{eq_bilfspherical}
g_X(\theta\otimes v) = \epsilon_R(\theta\circ v) = \tr_{\hat V}(D_{\hat V}\circ\theta\circ v),
\end{equation}
for all $v\in\omega X$, $\theta\in{(\omega X)}^\ast$, $X\in|\sym{C}|$.
Furthermore, the basis dual to the $\tilde e^{(X^\ast)}_j$
of~\eqref{eq_antibasis} can be computed as
\begin{equation}
\tilde e^j_{(X^\ast)} = \Phi_X(e^{(X)}_j),
\end{equation}
where
\begin{equation}
\Phi_X(v) =
D^{-1}_{\hat V}\circ\rho_{\hat V}\circ(\id_{\hat V}\otimes\bar\ev_X)
\circ\alpha_{\hat V,X,X^\ast}\circ(v\otimes\id_{X^\ast})
\circ(D_{\hat V}\otimes\id_{X^\ast}),
\end{equation}
and~\eqref{eq_epsilont} simplifies to
\begin{equation}
\label{eq_epsilontnew}
\epsilon_t({[\theta|v]}_X) =
{[\Phi_X(v)\circ\Psi_X(\theta)\circ\rho_{\hat V}|\rho^{-1}_{\hat V}]}_\one
\end{equation}
with $\Psi_X$ as in~\eqref{eq_psi}.
If $\sym{C}$ is braided with braiding $\sigma_{X,Y}\colon X\otimes
Y\to Y\otimes X$, then $H$ is coquasi-triangular~\cite{Pf09a} with
universal $r$-form and its weak convolution inverse
\begin{eqnarray}
&&r({[\theta|v]}_X\otimes{[\zeta|w]}_Y)\nn\\
&=& g_{X\otimes Y}((\zeta\circ(\theta\otimes\id_Y)\circ\alpha^{-1}_{\hat V,X,Y})\otimes
((\id_{\hat V}\otimes\sigma_{Y,X})\circ\alpha_{\hat V,Y,X}\circ(w\otimes\id_X)\circ v),\\
&&\bar r({[\theta|v]}_X\otimes{[\zeta|w]}_Y)\nn\\
&=& g_{X\otimes Y}((\theta\circ(\zeta\otimes\id_X)\circ\alpha^{-1}_{\hat V,Y,Z}\circ(\id_{\hat V}\otimes\sigma^{-1}_{Y,X}))\otimes
(\alpha_{\hat V,X,Y}\circ(v\otimes\id_Y)\circ w)),
\end{eqnarray}
for all $v\in\omega X$, $w\in\omega Y$, $\theta\in{(\omega X)}^\ast$,
$\zeta\in{(\omega Y)}^\ast$ and $X,Y\in|\sym{C}|$.
If $\sym{C}$ is ribbon with twist $\nu_X\colon X\to X$, then $H$ is
coribbon~\cite{Pf09a} with universal ribbon form and its convolution
inverse given by
\begin{eqnarray}
\label{eq_nu}
\nu({[\theta|v]}_X)
&=& g_X(\theta\circ((\id_{\hat V}\otimes\nu_X)\circ v)),\\
\label{eq_nubar}
\bar\nu({[\theta|x]}_X)
&=& g_X(\theta\circ((\id_{\hat V}\otimes\nu^{-1}_X)\circ v)),
\end{eqnarray}
for all $v\in\omega X$, $\theta\in{(\omega X)}^\ast$, $X\in|\sym{C}|$.
Every ribbon category is pivotal, and in this case, the copivotal form
is given by $w(x)=v(x^\prime)\nu(x^\pprime)$ where $v\colon H\to k,
x\mapsto r(S(x^\prime)\otimes x^\pprime)$ denotes the second dual
Drinfel'd element. If $\sym{C}$ is modular, then $H$ is fusion,
coribbon and weakly cofactorizable~\cite{Pf09a}.
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\begin{document}
\title{Practical bounds for a Dehn parental test}
\author{Robert C. Haraway, III}
\date{\today}
\maketitle
\begin{abstract}
In their article ``The shape of hyperbolic Dehn surgery space,''
Hodgson and Kerckhoff proved a powerful theorem, half of which
they used to make Thurston's Dehn surgery theorem effective.
The calculations derived here use both halves of Hodgson and
Kerckhoff's theorem to give bounds leading towards a practical
algorithm to tell, given two orientable
hyperbolic 3-manifolds $M,N$ of finite volume,
whether or not $M$ is a Dehn filling of $N.$
\end{abstract}
\section{Introduction}
Thurston's Dehn surgery theorem is one of the
brightest gems in the crown of modern geometric
3-manifold topology. It runs as follows:
\begin{Thm}[\cite{Th82}, 2.6]
Let $L \subset M^3$ be a link such that
$M \smallsetminus L$ has a hyperbolic structure.
There is a finite set $S$ of filling slopes
of components of $L$ such that all Dehn fillings
of $M\smallsetminus L$ without a slope from $S$
are hyperbolic.
\end{Thm}
Following the natural order of mathematics,
one wishes to quantify this existence result into
something more palpable. One way to make this
result more concrete is to provide
some measure of length for Dehn filling slopes,
then to say that all slopes of large
length (for a suitable definition of ``large'')
yield hyperbolic Dehn fillings. Such theorems
include the asymptotics of \cite{NZ85},
the Thurston-Gromov $2\pi$-theorem proved in \cite{BH96}, and
the 6-theorem of \cite{Ag00} and \cite{La00}.
Now, the $2\pi$- and 6-theorems just use
the na\"{i}ve notion of length of a slope,
viz. length of a geodesic representative in
an embedded horospherical torus. In \cite{HK08} however,
Hodgson and Kerckhoff
defined \emph{normalized length}\footnote{This
definition was anticipated in \cite{NZ85}.} $\hat{L}$, which
for a single slope $c$ on a horospherical cusp
torus $T$ is just
\[
\hat{L}(c) = \frac{length_T(c)}{\sqrt{area(T)}}.
\]
For a more general Dehn filling $c$, letting $c_T$ be
the slope along which $c$ fills $T$, Hodgson
and Kerckhoff define $\hat{L}(c)$ by requiring $\hat{L} \geq 0$ and
\[
\frac{1}{\hat{L}(c)^2} = \sum_{T \subset \partial X} \frac{1}{\hat{L}(c_T)^2},
\]
Using this definition of length, they proved
the following theorem:
\begin{Thm}[\cite{HK08}, Thm. 5.11, Cor. 5.13]\label{thm:mot}
Let $X$ be a compact orientable 3-manifold whose
interior admits a complete hyperbolic metric of
finite volume. Let $c$ be a Dehn filling of $X$
such that $\hat{L}(c) > 7.5832$. Then
\begin{itemize}
\item $X(c)$ (the filled manifold) itself admits
a complete hyperbolic metric on its interior;
\item $X \approx X(c) \smallsetminus \gamma$, where
$\gamma$ is a simple closed geodesic of $X(c)$
of length at most $0.156012$ and admitting an
embedded tube of radius at least $\artanh(1/\sqrt{3})$; and
\item $volume(X) - volume(X(c)) < 0.198$.
\end{itemize}
\end{Thm}
After suitably rephrasing this, it seems
to give a practical method for determining
Dehn filling heritage:
\begin{Cor}
Let $M$, $N$ be orientable 3-manifolds admitting
complete hyperbolic metrics of finite volume on their
interior. $N$ is a Dehn filling of $M$ if
and only if\footnote{The only-if part is the
content of Theorem \ref{thm:mot}.} either
\begin{itemize}
\item $N$ is a Dehn filling of $M$ along a slope
of normalized length less than or equal to 7.5832, or
\item $M$ is isometric to $N\setminus\gamma$ for
a simple closed geodesic of length less than 0.156012.
\end{itemize}
\end{Cor}
The collection of slopes of $\partial M$
with normalized length less than 7.5832 is computable,
and likewise the length spectrum of $N$ is computable,
and SnapPy can drill out curves and determine isometries,
so that is that. Right?
Unfortunately not. The problem is in drilling out curves.
SnapPy can only drill out simple closed curves in the dual
1-complex of an ideal triangulation. As explained in
\cite{HW94} on page 264, these may or may not be isotopic
(or even homotopic) to a given geodesic which one wishes
to drill out.
Fortunately, Theorem \ref{thm:mot} is a corollary of a much more
powerful theorem, Theorem \ref{thm:main},
about how much Dehn filling decreases volume.
Theorem \ref{thm:mot} follows from
the upper bounds in Theorem \ref{thm:main},
but Theorem \ref{thm:main} contains lower bounds as well.
The calculations below lead to the following
theorem:
\begin{Thm}\label{thm:rewrite}
Let $M,N$ be orientable 3-manifolds
admitting complete hyperbolic metrics
of finite volume on their interiors.
Let $\Delta V = Vol(M) - Vol(N)$.
$N$ is a Dehn filling of $M$ if and
only if either
\begin{itemize}
\item $N$ is a Dehn filling of $M$
along a slope of normalized length
less than or equal to $7.5832$, or
\begin{itemize}
\item $N$ has a closed simple geodesic
of length less than $2.879 \cdot \Delta V$, and
\item $N$ is a Dehn filling of $M$ along
a slope of normalized length $\hat{L}$
such that
\begin{equation}\label{eqn:finalbds}
\frac{4.563}{\Delta V} \leq \hat{L}^2 \leq \frac{20.633}{\Delta V}.
\end{equation}
\end{itemize}
\end{itemize}
\end{Thm}
This puts Dehn filling heritage for hyperbolic manifolds
in terms of procedures either already made rigorous
or with a reasonable hope of being made rigorous soon,
viz. estimates on volume, length spectra, cusp area,
slope length, and (to a lesser
extent) isometry testing.
\section{Rewriting the Hodgson-Kerckhoff Bounds}
The stronger theorem alluded to above is
\begin{Thm}[\cite{HK08}, 5.12]\label{thm:main}
Let $X, \hat{L},$ and $c$ be as in Theorem \ref{thm:mot}.
Let $\Delta V = vol(X) - vol(X(c))$. Let $\ell$
be the length of the geodesic core of the filling. Then
\begin{equation}\label{eqn:lobd}
\frac{1}{4} \cdot \int_{\tilde{z}}^1
\frac{H'(z)}{H(z) \cdot (H(z) - \tilde{G}(z))}\,dz
\leq \Delta V,
\end{equation}
\begin{equation}\label{eqn:upbd}
\Delta V \leq
\frac{1}{4} \cdot \int_{\hat{z}}^1
\frac{H'(z)}{H(z) \cdot (H(z) + G(z))}\,dz,
\end{equation}
and
\begin{equation}\label{eqn:lenbds}
1/H(\tilde{z}) \leq 2 \pi \cdot \ell
\leq 1/H(\hat{z}),
\end{equation}
where $H,G,\tilde{G},\tilde{z}$, and $\hat{z}$
have the following definitions.
\end{Thm}
\begin{Def}
\begin{align*}
K = 3.3957,&\qquad h(z) = \frac{1+z^2}{z\cdot (1 - z^2)},\\
\tilde{g}(z) = \frac{(1+z^2)^2}
{2\cdot z^3 \cdot (3 - z^2)},&\qquad
g(z) = \frac{1+z^2}{2\cdot z^3},\\
H = h/K,\qquad G &= g/K, \qquad\tilde{G} = \tilde{g}/K,\\
F(z) = \frac{H'(z)}{H(z) + G(z)} - \frac{1}{1-z}
&= \frac{h'(z)}{h(z) + g(z)} - \frac{1}{1-z},\\
\tilde{F}(z) = \frac{H'(z)}
{H(z) - \tilde{G}(z)} - \frac{1}{1-z}
&= \frac{h'(z)}{h(z) - \tilde{g}(z)} - \frac{1}{1-z},\\
f(z) = K \cdot (1-z) \cdot e^{-\Phi(z)},&\quad
\Phi(z) = \int_1^z F(w)\,dw,\\
\tilde{f}(z)=K \cdot (1-z) \cdot e^{-\tilde{\Phi}(z)},&\quad
\tilde{\Phi}(z) = \int_1^z \tilde{F}(w)\,dw,\\
f(\hat{z}) = \frac{(2\pi)^2}{\hat{L}(c)^2}, &\qquad
\tilde{f}(\tilde{z}) = \frac{(2\pi)^2}{\hat{L}(c)^2}
\end{align*}
\end{Def}
These definitions are from
pp. 1079, 1080, and 1088 of \cite{HK08}. The reader
should note that the above theorem has
$2 \pi \cdot \ell$ in place of $\mathcal{A}$.
This is valid---see, e.g., Corollary 5.13 of \cite{HK08}.
This gives complicated bounds on
$\Delta V$ and $\ell$ in terms
of $\hat{z}$ and $\tilde{z}$. We require simple but
not necessarily tight upper and
lower bounds on $\ell$ and $\hat{L}(c)$ in terms
of $\Delta V$. In a Dehn parental test,
the bounds on $\ell$ will be used
most often; the upper bounds on $\hat{L}(c)$ will be
used when the volumes of the putative parent and
child $P$ and $C$ are close, \emph{and} $C$ has
a very short geodesic. For instance, $P$ might be
the Whitehead link complement, and $C$ might be
a high-order Dehn filling on the $(-2,3,8)$ pretzel link complement.
To get these bounds, we will approximate solutions to
inequalities (\ref{eqn:lobd}) and (\ref{eqn:upbd})
in $\tilde{z}$ and $\hat{z}$, respectively,
for given $\Delta V$.
\subsection{Monotonicities}
Let
\begin{equation}\label{eqn:lb}
LB(z) = \frac{1}{4} \cdot \int_{z}^1
\frac{H'(w)}{H(w) \cdot (H(w) - \tilde{G}(w))}\,dw
\end{equation}
and
\begin{equation}
UB(z) = \frac{1}{4} \cdot \int_{z}^1
\frac{H'(w)}{H(w) \cdot (H(w) + G(w))}\,dw.
\end{equation}
We intend to solve the inequalities by
inverting $LB$ and $UB$. This will work if
we know the monotonicity of $LB$ and $UB$.
We will require the monotonicity of several
other functions as well, and the (very
calculational) proofs are in proof-hint notation.
It behooves us then to introduce ``$\sim$.''
\begin{Def}
For all real $x$ and $y$, $x \sim y$ when
$sgn(x) = sgn(y)$, where $sgn(x)$ is the
signum function $sgn(0) = 0$, else $sgn(x) = |x|/x$.
\end{Def}
\begin{Lem}\label{lem:lb}
$LB$ is decreasing on $\left(\sqrt{\sqrt{5}-2},1\right)$.
\end{Lem}
\begin{Lem}\label{lem:ub}
$UB$ is decreasing on
$\left(\sqrt{\sqrt{5}-2},\infty\right)$.
\end{Lem}
\begin{proof}[Proof of Lemma \ref{lem:lb}.]
\begin{calculation}[\sim]
LB'(z)
\step[=]{by definition of $LB$}
-\frac{1}{4} \cdot
\frac{H'(z)}{H(z)\cdot(H(z)-\tilde{G}(z))}
\step[=]{algebra}
- K/4 \cdot \frac{h'(z)}{h(z)\cdot(h(z)-\tilde{g}(z))}
\step{$K > 0$}
- \frac{h'(z)}{h(z)\cdot(h(z)-\tilde{g}(z))}
\step[=]{algebra}
-\frac{z^2\cdot(z^2-3)\cdot(z^4+4\cdot z^2 -1)}
{(z^2+1)^2\cdot(z^2-2\cdot z-1)\cdot(z^2+2\cdot z-1)}
\step{ $z > \sqrt{\sqrt{5}-2} \Rightarrow z^2 - 3 < 0$}
\frac{z^2\cdot(z^4+4\cdot z^2 -1)}
{(z^2+1)^2\cdot(z^2-2\cdot z-1)\cdot(z^2+2\cdot z-1)}
\step{$z > \sqrt{\sqrt{5}-2} \Rightarrow z^2/(z^2+1)^2 > 0$}
\frac{z^4+4\cdot z^2 -1}
{(z^2-2\cdot z-1)\cdot(z^2+2\cdot z-1)}
\step{$z > \sqrt{\sqrt{5}-2}\Rightarrow z^4+4\cdot z^2-1>0$}
\frac{1}{(z^2-2\cdot z-1)\cdot(z^2+2\cdot z-1)}
\step{$z > \sqrt{\sqrt{5}-2}
\Rightarrow z > \sqrt{2} - 1$,\\
$z>\sqrt{2}-1
\Rightarrow z^2+2\cdot z - 1 > 0$}
\frac{1}{z^2-2\cdot z-1}
\step{$z > \sqrt{\sqrt{5}-2} \Rightarrow z > 1-\sqrt{2}$,\\
$z < 1 \Rightarrow z < 1 + \sqrt{2}$,\\
$1-\sqrt{2}<z<1+\sqrt{2}
\Rightarrow z^2-2\cdot z - 1 < 0$}
-1.
\end{calculation}
By calculus, therefore, $LB$ is decreasing on
$\left(\sqrt{\sqrt{5}-2},1\right)$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:ub}.]
\begin{align*}
&\ UB'(z)\\
=&\qquad\left\{\mbox{by definition of }UB\right\}\\
&\ -\frac{1}{4}\cdot\frac{H'(z)}{H(z)\cdot(H(z)+G(z))}\\
=&\qquad\left\{\mbox{algebra}\right\}\\
&\ -\frac{K}{4}\cdot\frac{h'(z)}{h(z)\cdot(h(z)+g(z))}\\
=&\qquad\left\{\mbox{more algebra}\right\}\\
&\ -\frac{K}{2}\cdot\frac{z^2\cdot(z^4+4\cdot z^2 - 1)}
{(z^2+1)^3}\\
\sim&\qquad\left\{K > 0;\quad z \neq 0\right\}\\
&\ -(z^4 + 4\cdot z^2 - 1)\\
\sim&\qquad\left\{z > \sqrt{\sqrt{5}-2}
\Rightarrow z^4 + 4\cdot z^2 - 1 > 0\right\}\\
&\ -1.
\end{align*}
Again, by calculus, $UB$ is decreasing,
on $\left(\sqrt{\sqrt{5}-2},\infty\right)$.
\end{proof}
Therefore, the first two inequalities of
Theorem \ref{thm:main} are equivalent,
respectively, to $\tilde{z} \geq LB^{-1}(\Delta V)$
and $UB^{-1}(\Delta V) \geq \hat{z}$.
Next, we should do the same
to the inequalities (\ref{eqn:lenbds}),
getting bounds for $\tilde{z}$ and $\hat{z}$ in terms
of $\ell$. To do that we need $H$'s monotonicity.
We can then play the various inequalities
off one another to get our desired result. Also, we
should determine the monotonicities of $f$ and $\tilde{f}$;
they will prove useful later.
\begin{Lem}\label{lem:H}
$H$ is increasing on
$\left(\sqrt{\sqrt{5}-2},\infty\right)$.
\end{Lem}
\begin{Lem}\label{lem:f}
$f$ is decreasing on
$\left(\sqrt{\sqrt{5}-2},\infty\right)$.
\end{Lem}
\begin{Lem}\label{lem:ftilde}
$\tilde{f}$ is decreasing on
$\left(\sqrt{\sqrt{5}-2},\sqrt{3}\right)$.
\end{Lem}
\begin{proof}[Proof of Lemma \ref{lem:H}.]
\begin{align*}
&\ H'(z)\\
=&\qquad\left\{\mbox{by Def}\right\}\\
&\ h'(z)/K\\
\sim&\qquad\left\{K > 0\right\}\\
&\ h'(z)\\
=&\qquad\left\{\mbox{calculus}\right\}\\
&\ \frac{z^4 + 4\cdot z - 1}
{(z-1)^2\cdot z^2 \cdot (z+1)^2}\\
\sim&\\
&\ z^4 + 4 \cdot z - 1\\
\sim&\qquad\left\{z > \sqrt{\sqrt{5}-2}
\Rightarrow z^4 + 4 \cdot z - 1 > 0\right\}\\
&\ 1.
\end{align*}
By calculus, $H$ is increasing if
$z > \sqrt{\sqrt{5}-2}$---in
particular, if
$z \in \left(\sqrt{\sqrt{5}-2},1\right)$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:f}.]
\begin{align*}
&\ f'(z)\\
=&\qquad\left\{\mbox{calculus, algebra}\right\}\\
&\ K\cdot e^{-\Phi(z)}\cdot
\left(-1 +(1-z)\cdot(-\Phi'(z))\right)\\
\sim&\qquad\left\{K > 0;\quad e^{-\Phi(z)} > 0\right\}\\
&\ (z-1)\cdot\Phi'(z) - 1\\
=&\qquad\left\{\mbox{fund. thm. of calculus}\right\}\\
&\ (z-1)\cdot F(z) - 1\\
=&\qquad\left\{\mbox{algebra}\right\}\\
&\ -\frac{2\cdot z\cdot(z^4+4\cdot z^2 - 1)}
{(z+1)\cdot(z^2+1)^2}\\
\sim&\\
&\ -(z^4 + 4\cdot z^2 - 1)\\
\sim&\qquad\left\{z > \sqrt{\sqrt{5}-2}
\Rightarrow z^4 + 4 \cdot z - 1 > 0\right\}\\
&\ -1.
\end{align*}
By calculus, $f$ is decreasing if
$z > \sqrt{\sqrt{5}-2}$---in
particular, if $z \in \left(\sqrt{\sqrt{5}-2},1\right)$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:ftilde}.]
\begin{align*}
&\ \tilde{f}'(z)\\
=&\qquad\left\{\mbox{calculus, algebra}\right\}\\
&\ K\cdot e^{-\tilde{\Phi}(z)} \cdot
\left(-1 + (1-z)\cdot(-\tilde{F}(z))\right)\\
\sim&\qquad\left\{K > 0;\quad
e^{-\tilde{\Phi}(z)} > 0\right\}\\
&\ (z-1)\cdot\tilde{F}(z) - 1\\
=&\qquad\left\{\mbox{algebra}\right\}\\
&\ \frac{-2\cdot z\cdot(z^2-3)\cdot(z^4+4\cdot z-1)}
{(z+1)\cdot(z^2+1)\cdot(z^2-2\cdot z-1)
\cdot(z^2+2\cdot z-1)}\\
\sim&\qquad\left\{z \in
\left(\sqrt{\sqrt{5}-2},\sqrt{3}\right)
\Rightarrow z^2 - 3 < 0\right\}\\
&\ \frac{2\cdot z \cdot (z^4+4 \cdot z - 1)}
{(z+1)\cdot(z^2+1)\cdot(z^2-2\cdot z-1)
\cdot(z^2+2\cdot z-1)}\\
\sim&\qquad\{z > \sqrt{\sqrt{5}-2} \Rightarrow z > 0\}\\
&\ \frac{z^4 + 4 \cdot z-1}
{(z^2-2\cdot z-1)\cdot(z^2+2\cdot z-1)}\\
\sim&\qquad\left\{\mbox{latter half of Lemma \ref{lem:lb}
}\right\}\\
&\ -1.
\end{align*}
\end{proof}
\subsection{Complicated upper bound on $\ell$}
Plainly we already have an upper bound on $\ell$, viz.
$\ell \leq 1/(2\pi \cdot H(\hat{z}))$. We just need
to put the right-hand side in terms of $\Delta V$.
In fact, since $H$ is increasing, $1/(2\pi \cdot H)$
is decreasing. Therefore we just need a lower bound
on $\hat{z}$; applying $1/(2\pi \cdot H)$ to
this lower bound will give us a bound on $\ell$.
At this point, one could use the standing assumption
in \cite{HK08} after p. 1079 that all variables named
$z$ represent $\tanh \rho$ for some
$\rho > \artanh(1/\sqrt{3})$.
Therefore, $\hat{z} > \sqrt{1/3}$. As
a matter of fact, this is where the bounds
in Theorem \ref{thm:mot} come from.
But we would like a better bound for small $\Delta V$.
Now, $UB(\hat{z}) \geq \Delta V.$
Unfortunately $UB$ is decreasing,
so this doesn't give a lower bound on $\hat{z}$.
Also, $\hat{z}$ is defined by
$f(\hat{z}) = (2\pi)^2/(\hat{L}(c)^2)$,
but all we know about $\hat{L}(c)$ is $\hat{L}(c) > 7.5832$.
In fact, this bound is taken from the standing assumption
on $z$.
However, we also know $f(\hat{z}) = \tilde{f}(\tilde{z})$
$f$ and $\tilde{f}$ both are decreasing. Therefore,
if we can get a lower bound on $\tilde{z}$, we get
a lower bound on $\hat{z}$, via upper bounds
on $f(\hat{z}) = \tilde{f}(\tilde{z})$.
Finally, (\ref{eqn:lobd}) from Theorem \ref{thm:main}
says $LB(\tilde{z}) \leq \Delta V$, and $LB$ is
decreasing on $\left(\sqrt{\sqrt{5}-2},1\right)$.
$\sqrt{1/3} > \sqrt{\sqrt{5}-2}$,
so this yields a lower bound on $\tilde{z}$,
and hence an upper bound on $\ell$, in terms
of $\Delta V$; to wit,
\begin{equation}
\ell \leq \frac{1}
{2\pi\cdot (H \circ s \circ \tilde{f} \circ BL)
(\Delta V)},\label{eqn:lbd}
\end{equation}
where $s(f(\hat{z})) = \hat{z}$
and $BL(LB(\tilde{z})) = \tilde{z}$ for
$\tilde{z}, \hat{z} \in \left(\sqrt{1/3},1\right)$,
and $s:(0,f(\sqrt{1/3})) \to (\sqrt{1/3},1)$,
$BL:(0,LB(\sqrt{1/3}))\to(\sqrt{1/3},1)$.
This bound is valid only when $\Delta V$ is
in the domain of $BL$. If this is not the case,
then the right-hand side should be replaced
by Hodgson and Kerckhoff's original bound
0.156012.
\subsection{Complicated bounds on $\hat{L}(c)$}
We know
$\frac{(2\pi)^2}{\hat{L}(c)^2} = f(\hat{z})
= \tilde{f}(\tilde{z})$.
We just got upper bounds on this, yielding
a lower bound for $\hat{L}(c)$. More explicitly,
\begin{equation}
\hat{L}(c)^2 \geq \frac{(2\pi)^2}
{\tilde{f}(BL(\Delta V))}.\label{eqn:lblc}
\end{equation}
To get an upper bound on $\hat{L}(c)$, we
can get a lower bound on $f(\hat{z})$, which
would result from an upper bound on $\hat{z}$
(since $f$ is decreasing), which would
result from a lower bound on $UB(\hat{z})$
(since $UB$ is decreasing). But
$\Delta V \leq UB(\hat{z})$ by assumption. So
\begin{equation}
\hat{L}(c)^2 \leq \frac{(2\pi)^2}
{f(BU(\Delta V))},\label{eqn:ublc}
\end{equation}
where $BU: (0,UB(\sqrt{1/3})) \to (\sqrt{1/3},1)$
satisfies $BU(UB(\hat{z})) = \hat{z}$ for
$\hat{z} \in (\sqrt{1/3},1)$.
\subsection{Nice bounds}
Since these bounds depend upon inverting
functions defined by integrals, one cannot
expect a computer to calculate the bounds
very quickly. But if we approximate the
functions and relax the bounds, we can
get decent running times.
The conditions which the approximations
should satisfy (in order to accord with
(\ref{eqn:lbd}), (\ref{eqn:lblc}), and
(\ref{eqn:ublc})) are not difficult to
derive. For instance, an approximation
$\eta$ to $1/(2\pi \cdot H)$ should
be decreasing, since $1/(2\pi\cdot H)$
is itself decreasing and we want a
reasonable approximation; and $\eta$
should be greater than $1/(2\pi\cdot H)$
so that we can deduce
\[
\ell \leq
(\eta \circ s \circ \tilde{f} \circ BL)(\Delta V)
\]
from (\ref{eqn:lbd}). In fact,
$\eta(z) = K\cdot(1-z)/(2\pi)$
suffices. Useful approximations for
all the necessary functions are as follows:
\begin{Lem}\label{lem:bounds}
\begin{align}
1/h(z) &\leq 1-z,\label{eqn:rech}\\
f(z) &\geq A \cdot (1-z),\label{eqn:fbd}\\
\tilde{f}(z) &\leq B\cdot(1-z),\label{eqn:ftbd}\\
LB(z) &\geq C\cdot(1-z),\label{eqn:blbd}\\
UB(z) &\leq D\cdot(1-z).\label{eqn:bubd}
\end{align}
where
\begin{align*}
A &= K\cdot e^{-\Phi(\sqrt{1/3})};\\
\tilde{F}(\beta) &= 0,\ \beta \in (\sqrt{1/3},1);\\
B &= K\cdot e^{-\tilde{\Phi}(\beta)};\\
t &= \frac{h'}{h\cdot(h-\tilde{g})};\\
C &= K \cdot t(\sqrt{1/3})/4;\\
D &= K/4;
\end{align*}
\end{Lem}
\begin{proof}[Proof of (\ref{eqn:rech}).]
\[
1-z-\frac{1}{h(z)} = \frac{(1-z)^2}{1+z^2} \geq 0.
\]
\end{proof}
\begin{proof}[Proof of (\ref{eqn:fbd}).]
Assume $z \in (\sqrt{1/3},1)$. Now,
by definition,
\[
F(z) = -\frac{z^4+6\cdot z^2 + 4\cdot z +1}
{(z+1)\cdot(z^2+1)^2}.
\]
But
\begin{calculation}[\Rightarrow]
F(z) = -\frac{z^4+6\cdot z^2 + 4\cdot z +1}
{(z+1)\cdot(z^2+1)^2}
\step{algebra}
F < 0 \mbox{ on } (\sqrt{1/3},1)
\step{calculus; $z \in (\sqrt{1/3},1)$}
\int_z^1 F(w)\,dw \geq \int_{\sqrt{1/3}}^1 F(w)\,dw
\step[\equiv]{calculus, algebra}
\int_1^z F(w)\,dw \leq \int_1^{\sqrt{1/3}} F(w)\,dw
\step[\equiv]{definition of $\Phi$}
\Phi(z) \leq \Phi(\sqrt{1/3})
\step[\equiv]{$x \mapsto e^{-x}$ is decreasing}
e^{-\Phi(z)} \geq e^{-\Phi(\sqrt{1/3})}
\step[\equiv]{$z \in (\sqrt{1/3},1) \Rightarrow 1-z > 0$; $K > 0$}
K\cdot e^{-\Phi(z)}\cdot (1-z)
\geq K \cdot e^{-\Phi(\sqrt{1/3})} \cdot (1-z)
\step[\equiv]{definition of $f$}
f(z) \geq K \cdot e^{-\Phi(\sqrt{1/3})} \cdot (1-z)
\step[\equiv]{definition of $A$}
f(z) \geq A \cdot (1-z)
\end{calculation}
\end{proof}
\begin{proof}[Proof of (\ref{eqn:ftbd}).]
$\tilde{F}(1) = 1$, $\tilde{F}(\sqrt{1/3}) < 0$,
and $\tilde{F}$ has exactly one root $\beta$
in $(\sqrt{1/3},1)$. Thus if $z \in (\sqrt{1/3},1)$, then
\begin{calculation}[\equiv]
\int_z^1 \tilde{F}(w)\,dw \leq \int_\beta^1 \tilde{F}(w)\,dw
\step{calculus}
\int_1^z \tilde{F}(w)\,dw \geq \int_1^\beta \tilde{F}(w)\,dw
\step{definition of $\tilde{\Phi}$}
\tilde{\Phi}(z) \geq \tilde{\Phi}(\beta)
\step{algebra}
-\tilde{\Phi}(z) \leq -\tilde{\Phi}(\beta)
\step{$x \mapsto e^x$ is increasing}
e^{-\tilde{\Phi}(z)} \leq e^{-\tilde{\Phi}(\beta)}
\step{algebra}
K\cdot (1-z) \cdot e^{-\tilde{\Phi}(z)}
\leq K\cdot (1-z) \cdot e^{-\tilde{\Phi}(\beta)}
\step{definition of $\tilde{f}$}
\tilde{f}(z) \leq K\cdot (1-z)
\cdot e^{-\tilde{\Phi}(\beta)}
\step{definition of $B$}
\tilde{f}(z) \leq B\cdot(1-z).
\end{calculation}
But the initial statement is just equation (\ref{eqn:lobd}).
\end{proof}
\begin{proof}[Proof of (\ref{eqn:blbd}).]
For variety, we do this proof backwards.
We seek a $C$ such that for all $z \in (\sqrt{1/3},1),$
$LB(z) \geq C \cdot (1 - z)$:
\begin{calculation}[\Leftarrow]
\langle \forall z:
LB(z) \geq C \cdot (1 - z) \rangle
\step[\equiv]{let $lb(z)=\int_z^1 h'/(h\cdot(h-\tilde{g}))$}
\langle \forall z:
K \cdot lb(z) / 4 \geq C \cdot (1-z) \rangle
\step[\equiv]{algebra}
\langle \forall z:
lb(z) \geq 4 \cdot C \cdot (1-z) / K \rangle
\step{calculus}
h'/(h\cdot(h-\tilde{g})) \geq 4 \cdot C / K
\mbox{ on } (\sqrt{1/3},1).
\end{calculation}
In other words, we just need a lower bound
on $t = h'/(h\cdot(h-\tilde{g}))$ over
$(\sqrt{1/3},1)$. Now,
\[
t'(z) = \frac{4 \cdot(1-z) \cdot (z+1) \cdot p(z)}
{(z^2+1)^3\cdot(z^2-2\cdot z-1)^2\cdot
(z^2+2\cdot z-1)^2},
\]
where
\[
p(z) = 5 \cdot z^8 - 6 \cdot z^6 + 88 \cdot z^4
- 26\cdot z^2 + 3.
\]
It is clear that on $(\sqrt{1/3},1)$, $t' \sim p$.
Now,
\begin{calculation}[=]
p(z)
\step{}
5 \cdot z^8 - 6 \cdot z^6 + 2 \cdot z^4
+ 86 \cdot z^4 - 26 \cdot z^2 + 3
\step{}
z^4\cdot(5 \cdot (z^2)^2 - 6 \cdot (z^2) + 2)
\step[+]{}
86 \cdot (z^2)^2 - 26 \cdot z^2 + 3.
\end{calculation}
$(-6)^2 - 4\cdot 5 \cdot 2 < 0$ and $(-26)^2 -4 \cdot 86 \cdot 3 < 0$.
Therefore, $5\cdot z^2 - 6\cdot z +2$ has constant sign,
and $86 \cdot z^2 - 26 \cdot z + 3$ does too. By evaluation
at 0, this sign is positive on both. Therefore $p$ is positive.
That is, $t' > 0$ on $(\sqrt{1/3},1)$. Consequently, $t$
achieves its smallest value at $\sqrt{1/3}$. That
is, $t \geq t(\sqrt{1/3})$. So we have, finally,
\begin{calculation}[\Leftarrow]
\langle \forall z:
LB(z) \geq C\cdot(1-z)
\rangle
\step{see above}
C = K \cdot t(\sqrt{1/3}) / 4.
\end{calculation}
\end{proof}
\begin{proof}[Proof of (\ref{eqn:bubd}).]
Likewise, we do this proof backwards.
We seek a $D$ such that for all $z \in (\sqrt{1/3},1),$
$UB(z) \leq D \cdot (1-z)$:
\begin{calculation}[\Leftarrow]
\langle \forall z: UB(z) \leq D \cdot (1-z) \rangle
\step[\equiv]{let $ub(z) = \int_z^1 h'/(h\cdot(h+g))$}
\langle \forall z: K \cdot ub(z)/4 \leq D\cdot(1-z) \rangle
\step[\equiv]{algebra}
\langle \forall z: ub(z) \leq 4\cdot D \cdot (1-z) / K \rangle
\step{calculus}
h'/(h\cdot(h+g)) \leq 4\cdot D / K \mbox{ on } (\sqrt{1/3},1).
\end{calculation}
In other words, we just need an upper bound on
$T = h'/(h\cdot(h+g))$ over $(\sqrt{1/3},1)$. Now,
\[
T'(z) = -\frac{4\cdot z \cdot (z^4 - 10 \cdot z^2 + 1)}{(z^2+1)^4}.
\]
Plainly, $T'(z) \sim - z^4 + 10 \cdot z^2 - 1$ on $(\sqrt{1/3},1)$.
This has four real roots, $\pm \sqrt{5 \pm 2 \cdot \sqrt{6}}$,
none of which lies in $(\sqrt{1/3},1)$. $-1 + 10 -1 > 0$, so
$T'$ is positive on $(\sqrt{1/3},1)$. That is, $T$ is
increasing on $(\sqrt{1/3},1)$. So it takes its maximum at 1,
where its value is just $1$! In conclusion, then,
\begin{calculation}[\Leftarrow]
\langle \forall z: UB(z) \leq D\cdot(1-z) \rangle
\step{see above}
D = K/4.
\end{calculation}
\end{proof}
\begin{Lem}\label{lem:nicebds}
\begin{equation}\label{eqn:lnice}
\frac{1}{2 \pi \cdot (H \circ s \circ \tilde{f} \circ BL)(\Delta V)}
\leq \alpha \cdot \Delta V,
\end{equation}
\begin{equation}\label{eqn:lclnice}
\frac{(2 \pi)^2}{\tilde{f}(BL(\Delta V))}
\geq \delta \cdot \frac{1}{\Delta V},
\end{equation}
and
\begin{equation}\label{eqn:lcunice}
\frac{(2 \pi)^2}{f(BU(\Delta V))}
\leq \gamma \cdot \frac{1}{\Delta V},
\end{equation}
where
\[
\alpha = \frac{2 \cdot e^{\Phi(\sqrt{1/3}) -\tilde{\Phi}(\beta)}}
{\pi \cdot t(\sqrt{1/3})},
\]
\[
\delta = \frac{(2 \pi)^2 \cdot e^{\tilde{\Phi}(\beta)} \cdot t(\sqrt{1/3})}{4},
\]
\[
\gamma = \frac{(2 \pi)^2 \cdot e^{\Phi(\sqrt{1/3})}}{4},
\]
and as above,
\[
\tilde{F}(\beta) = 0, \beta \in (\sqrt{1/3}, 1).
\]
\end{Lem}
\begin{proof}[Proof of (\ref{eqn:lnice}).]
\begin{calculation}[\leq]
1/(h \circ s \circ \tilde{f} \circ BL)(\Delta V)
\step[=]{algebra}
\left( \frac{1}{h} \circ s \circ \tilde{f} \circ BL \right)
(\Delta V)
\step{(\ref{eqn:rech})}
1 - ( s \circ \tilde{f} \circ BL )(\Delta V)
\step{$s,f$ inverse; (\ref{eqn:fbd})}
1 - ( 1 - \tilde{f}(BL(\Delta V))/A)
\step[=]{algebra}
\tilde{f}(BL(\Delta V))/A
\step{(\ref{eqn:ftbd})}
\frac{B}{A} \cdot (1 - BL(\Delta V))
\step{$BL,LB$ inverse; (\ref{eqn:blbd})}
\frac{B}{A} \cdot (1 - (1 - \Delta V/C))
\step[=]{algebra}
\frac{B}{A \cdot C}\cdot \Delta V.
\end{calculation}
Consequently,
\begin{calculation}[\leq]
1/(2 \pi \cdot (H \circ s \circ \tilde{f} \circ BL)(\Delta V))
\step[=]{definition of $H$}
K/(2 \pi \cdot (h \circ s \circ \tilde{f} \circ BL)(\Delta V))
\step[=]{algebra}
\frac{K}{2 \pi} \cdot 1/(h \circ s \circ \tilde{f} \circ BL)(\Delta V)
\step{see above}
\frac{K}{2 \pi} \cdot \frac{B}{A \cdot C} \cdot \Delta V.
\end{calculation}
Finally,
\begin{calculation}[=]
\frac{K}{2 \pi} \cdot \frac{B}{A \cdot C}
\step{definitions of $A,B,C$}
\frac{K \cdot K \cdot e^{-\tilde{\Phi}(\beta)}\cdot 4}
{2 \pi \cdot K \cdot e^{-\Phi(\sqrt{1/3})} \cdot K \cdot t(\sqrt{1/3})}
\step{algebra}
2 \cdot e^{\Phi(\sqrt{1/3})-\tilde{\Phi}(\beta)}/(\pi \cdot t(\sqrt{1/3}))
\step{definition of $\alpha$}
\alpha.
\end{calculation}
\end{proof}
\begin{proof}[Proof of (\ref{eqn:lclnice}).]
\begin{calculation}[\geq]
1/(\tilde{f}(BL(\Delta V))
\step{(\ref{eqn:ftbd}); algebra}
1/(B\cdot(1 - BL(\Delta V)))
\step{(\ref{eqn:blbd}); algebra}
1/(B\cdot(1 - (1-\Delta V/C)))
\step[=]{algebra}
(C/B) \cdot (1/\Delta V)
\end{calculation}
Consequently,
\[
\frac{(2\pi)^2}{\tilde{f}(BL(\Delta V)}
\leq \frac{(2\pi)^2 \cdot C}{B} \cdot \frac{1}{\Delta V}.
\]
Finally,
\begin{calculation}[=]
(2\pi)^2 \cdot C / B
\step{definitions of $B,C$}
\frac{(2\pi)^2 \cdot K \cdot t(\sqrt{1/3})}
{K \cdot e^{-\tilde{\Phi}(\beta)} \cdot 4}
\step{algebra}
(2\pi)^2 \cdot t(\sqrt{1/3}) \cdot e^{\tilde{\Phi}(\beta)} / 4.
\step{definition of $\delta$}
\delta.
\end{calculation}
\end{proof}
\begin{proof}[Proof of (\ref{eqn:lcunice}).]
\begin{calculation}[\leq]
1/(f(BU(\Delta V))
\step{(\ref{eqn:fbd}); algebra}
1/(A \cdot (1 - BU(\Delta V)))
\step{(\ref{eqn:bubd})}
1/(A \cdot (1 - (1 - \Delta V/D)))
\step[=]{algebra}
(D/A) \cdot (1/\Delta V).
\end{calculation}
Consequently,
\[
\frac{(2\pi)^2}{f(BU(\Delta V))}
\leq
\frac{(2\pi)^2 \cdot D}{A} \cdot \frac{1}{\Delta V}.
\]
Finally,
\begin{calculation}[=]
(2\pi)^2 \cdot D/A
\step{definitions of $A,D$}
\frac{(2\pi)^2 \cdot K}
{K \cdot e^{-\Phi(\sqrt{1/3})} \cdot 4}
\step{algebra}
(2\pi)^2 \cdot e^{\Phi(\sqrt{1/3})} / 4
\step{definition of $\gamma$}
\gamma.
\end{calculation}
\end{proof}
\subsection{Numerical approximations}
To make the bounds from Lemma
\ref{lem:nicebds} implementable in
software, we just need some simple
estimates on $\alpha,\delta,\gamma$.
Using a computer algebra system
one may show
\begin{Lem}\label{lem:appx}
$\alpha \leq 2.879$,
$\delta \geq 4.563,$ and
$\gamma \leq 20.633$.
\end{Lem}
For instance, the following code
in \texttt{Maxima} suffices:
\begin{verbatim}
K : 3.3957;
h : (1+z^2)/(z*(1-z^2)); H : h/K;
g : (1+z^2)/(2*z^3); G:g/K;
gt : (1+z^2)^2/(2*z^3*(3-z^2)); Gt : gt/K;
hh : factor(ratsimp(derivative(h,z)));
F : partfrac(ratsimp(hh/(h+g) -1/(1-z)),z);
Ft : partfrac(ratsimp(hh/(h-gt)-1/(1-z)),z);
assume(z>sqrt(1/3.0)); assume(z<1.0);
Phi : integrate(ev(F ,z=w),w,1,z);
Phit : integrate(ev(Ft,z=w),w,1,z);
f : K*(1-z)*exp(-Phi);
ft : K*(1-z)*exp(-Phit);
lbintegrand : partfrac(ratsimp(hh/(h*(h-gt))));
t : lbintegrand;
beta : rhs(realroots(Ft)[4]);
alpha : bfloat( 2 * exp( ev(Phi,z=sqrt(1/3.0))
-ev(Phit,z=beta) )
/ (
delta : bfloat( (2 *
* ev(t,z=sqrt(1/3.0)) / 4);
gamma : bfloat( (2 *
* exp(ev(Phi,z=sqrt(1/3.0))) / 4);
\end{verbatim}
The reader running this code is reminded
that \texttt{Maxima} displays big-floats in
scientific notation with, e.g,
\texttt{1.0b1} denoting 10, instead of
\texttt{1.0e1}. The \texttt{realroots} command
is based on Sturm sequences. Sturm's theorem
applies because the numerator of $\tilde{F}$
is a univariate polynomial over $\mathbf{Z}$.
\begin{proof}[Proof of Theorem \ref{thm:rewrite}.]
The if-direction is plain. For the only-if direction,
suppose $N$ is a Dehn filling of $M$. Then
either\footnote{This can be made constructive
and feasible to code by testing
$\hat{L}(c) < 7.5832 + \epsilon$
for some reasonably small $\epsilon > 0$.}
$N$ is a Dehn filling of $M$ along a
slope $c$ with $\hat{L}(c) \leq 7.5832$
or $N$ is Dehn filling of $M$ along a slope $c$
with $\hat{L}(c) > 7.5832$. The former case
is the first disjunct in Theorem \ref{thm:rewrite}.
In the latter case, Theorem \ref{thm:main} applies.
So $\Delta V$ is in the domain of $BL$ and $BU$,
and Lemma \ref{lem:nicebds} applies. Therefore,
by equations \ref{eqn:lbd} and \ref{eqn:lnice}, the core geodesic
of the filling has length $\ell$ satisfying
$\ell < 2.879 \cdot \Delta V$. So $N$ has a geodesic
satisfying the first conjunct of the second
disjunct of Theorem \ref{thm:rewrite}.
Furthermore, by equations \ref{eqn:lblc} and \ref{eqn:lclnice},
$4.563/\Delta V \leq \hat{L}(c)^2$, and by equations
\ref{eqn:ublc} and \ref{eqn:lcunice}, $\hat{L}(c)^2 \leq 20.633/\Delta V$.
This is the second and last conjunct of the second and last
disjunct of Theorem \ref{thm:rewrite}.
\end{proof}
\section{Prospects}
The above bounds are all ready to be implemented
in code, and finish the theoretical work necessary
for a Dehn parental test, modulo the estimates
mentioned before---to wit, estimates on volume
and normalized length. M. Trnkova, N. Hoffman,
the author, and M. G\"orner are working on implementing such estimates in code.
Once the Dehn parental test is finished,
one will be able to calculate the complexity
of 3-manifolds for certain notions of complexity,
among which is Gabai, Meyerhoff, and Milley's
Mom-number $m$ (see \cite{GMM11}). $m$ has
the following nice properties:
\begin{itemize}
\item if $M$ is a Dehn filling of $N$, then $m(M) \leq m(N)$;
\item for $0 < B < \infty$, there is a finite set
$S_m(B)$ such that if $m(N) < B$, then $N$ is a Dehn
filling of some element of $S_m$; and
\item $S_m: [0,\infty) \to \mathcal{T}$ has an
implementation in Regina.
\end{itemize}
The reader will note that the volume function $v$
(by Theorems 3.4 and 3.5 of \cite{Th82}) is known to satisfy
these properties as well, except for the last property.
Therefore we propose the following natural challenge:
\begin{Chal}
To implement, in one's 3-manifold software suite
of choice, a function $S_v$ which, given as input the number
4, runs for at most one week on a 2Ghz processor, and
such that for all $B$, the Dehn fillings of $S_v(B)$
include all orientable hyperbolic 3-manifolds of
volume at most $B$.
\end{Chal}
A solution to this challenge would be a significant step
towards a proper formulation and eventual solution of the hyperbolic
complexity conjecture (see \cite{GMM11}).
\section*{Acknowledgments}
Martin Bridgeman suggested that
Hodgson and Kerckhoff's work could be
used to develop a Dehn parental test.
Craig Hodgson pointed out that
the notion of normalized length used
in \cite{HK08} has
a precursor in the asymptotics of \cite{NZ85}.
I thank them both. Also I would like to thank
Andrew Yarmola for a helpful conversation.
| 161,514
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Can Adrian Peterson Break 2,500?
Adrian Peterson has set some lofty goals for himself. Last year, he was 9 yards away from breaking the record for most rushing yards in a single season. In the upcoming season, Peterson doesn’t want to just break the record, he wants to shatter it. Peterson wants to acquire 2,500 rushing yards, 395 yards more than the current record of 2,105.
Now, some may find this ridiculous and unattainable. However, this is Adrian Peterson we’re talking about. This guy is in a class by himself. After last season, anyone who questions his status as the best running back in the NFL is naive. The way he runs through defenses who are specifically geared to stop him is incredible, and to get 2,096 the season after tearing your ACL is unheard of, so when this man sets a goal that is unheard of, you better believe that he will obtain it.
With Greg Jennings now in the mix and Christian Ponder hopefully having grown as a quarterback, some of the attention may turn to the Vikings improved aerial attack. Peterson may get some help from this, but make no mistake. Defenses know what’s coming when they play the Minnesota Vikings. They’ll be facing a guy who is determined to make history, and when this man is determined, good luck trying to contain him.
(7)
| 374,815
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When you work for a London escorts agency, you are faced with many challenges. Personally, I have always found it very hard to stay on top of my very busy schedule. I realised that working for London escorts meant keeping a diary but I need not appreciate that I would have to fit so much. Or rather should I say, I did not know how busy I was going to end up being. That is a problem a lot of London escorts have and often find it hard to deal with.
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Are London escorts so busy that they need to organise their schedules? I guess not all London escorts are super busy. If you have just started out working for London escorts you may be less busy, but believe me, if you play your cards right, you will soon be really busy. I have found that my trusted Filofax does everything that I love it to do. It even has a section for business cards and putting in essential personal details such as your bra size. That is important when you want to look something up quickly. I love my Filofax and I would not want to be without it.
| 227,054
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80 Langton Street
Richard Alpert, Finger (2nd August 1975)
Geolocation
Description
This piece was a static installation. The room was in general darkness except for lights aimed at several areas in the space. An unidentifiable sound could be heard from the rear of the room. The first illuminated area approached upon entering the space showed the sentence “Chris – I went to the hospital – I think I cut my hand bad” alongside a knife and some bread. In the second spot was found a photograph placed on the floor. At the rear of the room was a spotlighted door out of which protruded my two fingers as my only visible presence in the room. As this area was approached, the sound of the space could be heard coming from an adjacent room through an open doorway and could be identified as that of a ball bouncing in a confined space. The lighting was set so that it was difficult to see into the room without entering it. Upon entering, the source of the sound, a tape recording playing, could be seen in the almost totally dark room. (Alpert 1976)
| 80,768
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Non-bailable warrant issued against Zakir Naik
A special court has issued a non-bailable warrant against controversial Islamic preacher Zakir Naik. The court said that the warrant would be an open dated one. On Tuesday the Enforcement Directorate had moved the court seeking issuance of a non-bailable warrant against controversial against Naik. The application was moved in the Special Prevention of Money Laundering Court in Mumbai.
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We have in our midst many grassroots political practitioners, former councillors, retired civil servants and professionals of diverse background. Prior to now, our members have joined the APC in their various wards and have been attending meetings. This declaration today is therefore to formalize our entry and to declare to the whole world that we in the APC.
Your Excellency Sir, there is no doubt that Ogun West keenly contested the last Gubernatorial election with you but because you carry a divine mission which no contestation can overcome, you were overwhelmingly elected by the mass of Ogun State electorates. Not only did you win the popular vote on the field, you also trounced your challenger at every level of our judicial system. On behalf of the teeming members of WOPMOV across the 5 local government of Ogun West, I want to congratulate you for the well-deserved victory.
WOPMOV have been closely following your government since you were sworn-in in May last year and we make bold to say that your unique approach to governance anchored on the principles of justice, fairness, equity, inclusiveness and respect for rule of law is not only resonating very loudly with us, it has become irresistible hence the decision of WOPMOV to come on board and be part of the APC. We are doing this so that we can optimally support your government and contribute our quota to your resolve to wipe the tears of Ogun West Senatorial District through massive infrastructural improvement, human capital development and empowerment of our people. I therefore have the honour and privilege on behalf of the leaders of MOPMOV to pledge the absolute loyalty of the over 5000 members of the organization who are here represented to the All Progressive Congress and to the government of Prince Dapo Abiodun mfr
It said that as morning shows the day so does childhood shows the man. Yoruba says “arise larika, arika baba iregun. We sincerely appreciate your Excellency for the good work you are doing in Ogun State and in Ogun West. I recall that sometimes in early May, 2019, I had the privilege of being part of delegation on behalf Yewa Traditional Council led by Emeritus Professor Anthony Asiwaju that visited your Excellency at your Iperu home. I had the honour of presenting the Ogun West Developmental Blueprint to you on behalf of the delegation and I recall the studiousness with which you took note of our demands at the interaction and how you responded diligently to every points raised.
Today, WOPMOV bear witness of your good policies and practical actions to make life more meaningful for our people. We thank your Excellency for the attention you gave to the State Hospital, Ilaro which is now a secondary public health facility we can be proud off, the reconstruction of the Ota-AIT road, the Ota-Command road, the rehabilitation of many secondary schools and primary health centres are worthy of commendation. We salute your Excellency for the employment and wealth creation initiatives through such innovations like the Anchor borrowers programme, Fadama guys, the Ogun job portal, Ogun Techhub and Okowo Dapo. Our prayer is that God Almighty will continue to strengthen you, inspire your thoughts and guide your actions to take Ogun State to that greater future you are envisaging.
Your Excellency Sir, Ogun West is like child who was starved for many years while his peers when being reasonably fed. Such a child will not only be malnourished, he will be stunted and small for age. Ogun West is currently suffering from infrastructural kwashiorkor, empowerment marasmus and growth stunting. We appeal to your Excellency, to please target Ogun West preferentially in order to change the narrative of the district positively. May Almighty God assist you to do it sir.
In conclusion, I want to on behalf of West Ogun Progressive Movement thank the soft spoken but action packed Caretaker Chairman of the All Progressive Congress in Ogun State, Chief Yemi Sanusi and the members of his team at the party level. Without the peaceful and cooperative atmosphere that you have created in the party, all the achievements we are crediting our Governor for may have been very difficult if not impossible to achieve. WOPMOV is very proud of you, the party Secretary, Hon. Ayo Olubori, the PRO, Ambassador Tunde Oladunjoye and all the party hierarchy working with Mr. Governor, Prince Dapo Abiodun to make the APC in Ogun State attractive and enticing.
Thank you all for your kind attention
Dr. Iziaq Kunle SALAKO
For West Ogun Progressive Movement (WOPMOV)
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Eclipse Plug-ins: An Interview With the Authors
Eclipse Plug-ins: An Interview With the Authors
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[img_assist|nid=7268|title=|desc=|link=none|align=right|width=199|height=101]The book, Eclipse Plug-ins, just released its third edition. It’s guaranteed to be a big seller among Eclipse developers and those looking to get into plug-in development for Eclipse. I met with authors Eric Clayberg and Dan Rubel to ask them a few questions about the book and plug-in development in general. You can look forward to my review of the book here on EclipseZone tomorrow..
James Sugrue: Can you both please introduce yourselves to our readers?
Eric Clayberg: Officially, I’m a co-founder and the Senior Vice President for Product Development for Instantiations, Inc., which means when I’m not writing or updating the Eclipse Plug-ins book, I’m using my seventeen years of commercial software development experience to continue to drive innovation in Instantiations’ WindowBuilder GUI-building products, our CodePro AnalytiX automated testing and code quality tool, as well as our other products.
Dan Rubel: As a co-founder and Chief Technology Officer for Instantiations, Inc. I play a key design and leadership role in our products, including our RCP Developer tool, WindowTester Pro, the automated GUI testing product, and CodePro AnalytiX, our code quality product. I have over 15 years of experience in object oriented technologies, as well as years on Java and Eclipse to help make our commercial software development products.
Sugrue: How long have you been working with Eclipse?
Clayberg & Rubel: We were briefed on Eclipse in late 1999 and have been using it full time since early 2000 (before it was even named Eclipse).
Sugrue: I've really been looking forward to this book. Was it difficult to write a new edition?
Clayberg & Rubel: Yes. The new 3rd Edition required over 500 hours of effort. This is a substantially bigger change than the 2nd edition was from the 1st.
Sugrue: Why the change in name from the second edition, which was “Eclipse: Building Commercial Quality Plug-ins”?
Clayberg & Rubel: We’ve simplified it to “Eclipse Plug-ins” to make it more succinct and make it more visible on bookstore shelves where only the spine is showing. It is considered the definitive book of Eclipse plug-in development so why not give it the most direct and succinct title?
Sugrue: What is the number one tip you would give to plug-in developers?
Clayberg & Rubel: Read our book ;-). Why? developers.
Our goal was to accomplish several objectives with this book:
• Provide a quick introduction to using Eclipse for new users
• Provide a reference for experienced Eclipse users wishing to expand their knowledge and improve the quality of their Eclipse-based products
• Provide a detailed tutorial on creating sophisticated Eclipse plug-ins suitable for new and experienced users
Sugrue: What is the most impressive Eclipse plug-in you have seen?
Clayberg & Rubel: Our WindowBuilder Pro product ;-). Okay, I admit that I am a little biased but it’s my favorite tool because it is such a unique product with so much depth. It’s not only me that believes this it’s the thousands of users and the fact that it has won awards. The bottom-line is that it does a brilliant job solving the needs of Java GUI developers. Our recent white paper on GUI Building enumerates many reasons for why WindowBuilder has hit the sweet spot in the Eclipse developer community.
And while we’re tooting our own horn, our CodePro AnlaytiX product adds hundreds of enhancements to Eclipse and any Eclipse-based IDE. Developing CodePro over the last several years has provided us with an opportunity to learn the details of Eclipse development at a level matched by very few others. It has also served as a test bed for many of the ideas and techniques presented in this book, providing us with a unique perspective from which to write.
Sugrue: Do you see Eclipse plug-in development getting more popular this year?
Clayberg & Rubel: Yes, definitely. Eclipse is entering its 8th year and has a substantial following. Eclipse, and open source in general seems to be growing within large enterprises, as well as companies of all sizes. There is so much potential with OSGi and Equinox, and we see some excellent opportunity. And with Eclipse continually growing and expanding, this is a great time to focus on plug-in development.
Sugrue: What are your thoughts on e4?
Clayberg & Rubel: It looks very promising and will re-invigorate the Eclipse community. It offers an excellent opportunity to fix long-standing issues and jettison old, deprecated APIs.
We will, of course, be updating our book for e4 when it comes out.
Sugrue: What advice would you give to developers who are just starting out on developing on top of Eclipse?
Clayberg & Rubel:Read our book and look at the Eclipse source for excellent examples of how to do just about anything. There is also a lot of good information on great developer portals like this one.
Sugrue: I read through the book cover to cover. But to developers who have read the last edition, and are stuck for time, what are the key chapters to read?
Clayberg & Rubel: Chapters 6, 7, 8, 19 and 20 had the most changes (due to Commands, p2, etc.) and chapter 20 (GEF) is all new.
Here’s more detail on the major changes in this third edition:
Eclipse Command Framework
The Eclipse command framework replaces the older Action framework. Throughout the book, use of the older Action framework has been replaced with new content describing how to accomplish the same thing with the new command framework. This is most obvious in Chapter 6 where the first half of the chapter is entirely devoted to the command framework. Chapter 7 and Chapter 8 also have lots of new material describing use of commands with views and editors.
Eclipse 3.4 and Java 5
All of the screen shots, text and code examples throughout the book have been updated to use the latest Eclipse 3.4 API and Java 5 syntax. Chapter 1 has been overhauled to include descriptions of new capabilities in Eclipse 3.4 including a new overview of using Mylyn and discussion of new preferences. Both Chapter 1 and Chapter 2 cover cool PDE and SWT tools available in Eclipse 3.4.
New GEF Chapter
GEF, the Graphical Editing Framework from Eclipse.org, provides a toolkit for building dynamic interactive graphical user interface elements. Chapter 20 takes you step by step through the process of building a GEF-based view for graphically presenting the relationships between the favorites items and their underlying resources. We then go further, building a GEF-based editor with the ability to add, move, resize, and delete the graphical elements representing those favorites items.
Put PDE Build through its paces
Over the past several years, the PDE build process has been steadily maturing. The proprietary Ant scripts in Chapter 19 of earlier versions of our book have been completely replaced with step-by-step instructions for getting an Eclipse PDE Build process up and running to automate assembly of your product for distribution..
Opinions expressed by DZone contributors are their own.
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- Teams»
- Kolkata Knight Riders»
- News
Kolkata Knight Riders batsman reflects on his new approach
20 April 2014
Kallis, Gambhir join Amit Mishra atop duck-scorers list
20 April 2014
Duminy snatches thriller for Daredevils
19 April 2014
Daredevils take on confident Knight Riders in Dubai
18 April 2014
Knight Riders bowling coach and mentor happy with the bowlers in their squad
18 April 2014
KKR’s new half-centurion pleased to script perfect reunion with old RCB mate
17 April 2014
| 338,208
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TITLE: What is the expansion $(x+a)^b$?
QUESTION [1 upvotes]: I don't have strong math background. Whats is the expansion of the following equation: $(x+a)^b$.
REPLY [3 votes]: This seems a difficult problem for someone without a strong mathematical background. One way to approach this is to arrange $b$ copies of $(x+a)$, as follows:
$$
\underbrace{(x+a)(x+a) \cdots (x+a)}_{\text{$b$ copies}}
$$
The product of all of these sums $(x+a)$ will be a bunch of terms of the form $x^ka^{b-k}$, where $k$ is some integer between $0$ and $b$, inclusive. Each of the copies of $(x+a)$ can contribute either a factor of $x$, or a factor of $a$. We then count up the number of ways to choose $k$ factors of $x$ and $b-k$ factors of $a$, and that will give us the coefficient of $x^ka^{b-k}$. This works, ultimately, because multiplication distributes over addition: $u(v+w) = uv+uw$.
So, for instance, there is only one way to choose the $x$ from each of $b$ copies of $x+a$; those $b$ factors of $x$ multiply to $x^b$, so the coefficient of $x^b$ is just $1$. On the other hand, since there are $b$ copies, there are $b$ ways to choose the $a$ from one of those $b$ copies (and consequently choose the $x$ from the remaining $b-1$ copies), so the coefficient of $x^{b-1}a$ is $b$. Therefore, the expansion of $(x+a)^b$ must begin
$$
x^b+bx^{b-1}a+\cdots
$$
We can continue along the rest of the terms in the same way. It turns out that there is a fairly simple formula for the number of ways to choose $k$ factors of $x$ and $b-k$ factors of $a$ from a total of $b$ copies of $(x+a)$; that formula is
$$
\binom{b}{k} = \frac{b!}{k!(b-k)!}
$$
which is in fact read "$b$ choose $k$", and where $b!$ is read "$b$ factorial" and represents the product $b \times (b-1) \times (b-2) \times \cdots \times 2 \times 1$. Thus, the overall formula for $(x+a)^b$ is
$$
\binom{b}{b}x^ba^0 + \binom{b}{b-1}x^{b-1}a^1 + \binom{b}{b-2}x^{b-2}a^2 +
\cdots + \binom{b}{1}x^1a^{b-1} + \binom{b}{0}x^0a^b
$$
which we write in shorthand as simply
$$
(x+a)^b = \sum_{k=0}^b \binom{b}{k} x^ka^{b-k}
$$
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[Federal Register Volume 78, Number 166 (Tuesday, August 27, 2013)] [Rules and Regulations] [Pages 52852-52854] From the Federal Register Online via the Government Publishing Office [] [FR Doc No: 2013-20538] ======================================================================= ----------------------------------------------------------------------- DEPARTMENT OF HEALTH AND HUMAN SERVICES Food and Drug Administration 21 CFR Parts 510, 520, 524, 556, and 558 [Docket No. FDA-2013-N-0002] New Animal Drugs; Carprofen; Enrofloxacin; Florfenicol; Tildipirosin; Zilpaterol AGENCY: Food and Drug Administration, HHS. ACTION: Final rule; technical amendment. ----------------------------------------------------------------------- SUMMARY: The Food and Drug Administration (FDA) is amending the animal drug regulations to reflect approval actions for new animal drug applications (NADAs) and abbreviated new animal drug applications (ANADAs) during June 2013. FDA is also informing the public of the availability of summaries of the basis of approval and of environmental review documents, where applicable. DATES: This rule is effective Augustune 2013, as listed in table 1.:. In addition, the animal drug regulations are being amended at 21 CFR 510.600 to correct a sponsor's name and at 21 CFR 556.733 to correct the acceptable daily intake of total residues of tildipirosin. This is being doneune 2013 ---------------------------------------------------------------------------------------------------------------- New animal drug 21 CFR NADA/ANADA Sponsor product name Action section FOIA summary NEPA review ---------------------------------------------------------------------------------------------------------------- 200-524.......... Putney, Inc., Mupirocin Original 524.1465 yes.......... CE.\1\ 400 Congress Ointment 2%. approval as a St., suite 200, generic copy Portland, ME of NADA 140- 04101. 839. 200-517.......... Novartis Animal ZOBUXA Original 520.812 yes.......... CE.\1\ Health US, (enrofloxacin) approval as a Inc., 3200 Flavored generic copy Northline Ave., Antibacterial of NADA 140- suite 300, Tablets. 441. Greensboro, NC 27408. 200-519.......... Novartis Animal FLORVIO Original 520.995 yes.......... CE.\1\ Health US, (florfenicol) approval as a Inc., 3200 2.3% generic copy Northline Ave., Concentrate of NADA 141- suite 300, Solution. 206. Greensboro, NC 27408. 200-547.......... Huvepharma AD, ZILMAX Original 558.665 yes.......... CE.\1\ 5th Floor, 3A (zilpaterol approval as a Nikolay Haytov hydrochloride) generic copy Str., 1113 plus RUMENSIN of NADA 141- Sophia, (monensin USP) 276. Bulgaria. plus TYLOVET 100 (tylosin phosphate) Type A medicated articles. [[Page 52853]] 200-555.......... Piedmont Animal LIBREVIA Original 520.309 yes.......... CE.\1\ Health, 204 (carprofen) approval as a Muirs Chapel Soft Chewable generic copy Rd., suite 200, Tablets. of NADA 141- Greensboro, NC 111. 27410. ---------------------------------------------------------------------------------------------------------------- and 524 Animal drugs. 21 CFR Part 556 Animal drugs, Foods. 21 CFR Part 558 Animal drugs, Animal feeds. Therefore, under the Federal Food, Drug, and Cosmetic Act and under authority delegated to the Commissioner of Food and Drugs and redelegated to the Center for Veterinary Medicine, 21 CFR parts 510, 520, 524, 556, entry for ``Purina Nutrition LLC'', and alphabetically add entries for ``Piedmont Animal Health'' and ``Purina Animal Nutrition LLC''; and in the table in paragraph (c)(2), in the entry for ``017800'', remove ``Purina Nutrition'' and in its place add ``Purina Animal Nutrition'', and numerically add an entry for ``058147'' to read as follows: Sec. 510.600 Names, addresses, and drug labeler codes of sponsors of approved applications. * * * * * (c) * * * (1) * * * ------------------------------------------------------------------------ Drug labeler Firm name and address code ------------------------------------------------------------------------ * * * * * * * ------------------------------------------------------------------------ Piedmont Animal Health, 204 Muirs Chapel Rd., suite 058147 200, Greensboro, NC 27410............................ ------------------------------------------------------------------------ * * * * * * * ------------------------------------------------------------------------ Purina Animal Nutrition LLC, 1080 County Road F West, 017800 Shoreview, MN 55126-2910............................. ------------------------------------------------------------------------ * * * * * * * ------------------------------------------------------------------------ (2) * * * ------------------------------------------------------------------------ Drug labeler code Firm name and address ------------------------------------------------------------------------ * * * * * * * ------------------------------------------------------------------------ 058147....................... Piedmont Animal Health, 204 Muirs Chapel Rd., suite 200, Greensboro, NC 27410 ------------------------------------------------------------------------ * * * * * * * ------------------------------------------------------------------------ PART 520--ORAL DOSAGE FORM NEW ANIMAL DRUGS 0 3. The authority citation for 21 CFR part 520 continues to read as follows: Authority: 21 U.S.C. 360b. Sec. 520.309 [Amended] 0 4. In paragraph (b)(2) of Sec. 520.309, remove ``Nos. 000115, 055529, and 062250'' and in its place add ``Nos. 000115, 055529, 058147, and 062250''. 0 5. In Sec. 520.812, revise paragraphs (a) and (b) to read as follows: Sec. 520.812 Enrofloxacin. (a) Specifications. Each tablet contains: (1) 22.7, 68.0, or 136.0 milligrams (mg) enrofloxacin; or (2) 22.7, 68.0, 136.0, or 272 mg enrofloxacin. (b) Sponsors. See sponsor numbers in Sec. 510.600(c) of this chapter for use as in paragraph (c) of this section. (1) Nos. 000859 and 026637 for use of product described in paragraph (a)(1) of this section. [[Page 52854]] (2) No. 058198 for use of product described in paragraph (a)(2) of this section. * * * * * Sec. 520.955 [Amended] 0 6. In paragraph (b) of Sec. 520.955, remove ``No. 000061'' and in its place add ``Nos. 000061 and 058198''. PART 524--OPHTHALMIC AND TOPICAL DOSAGE FORM NEW ANIMAL DRUGS 0 7. The authority citation for 21 CFR part 524 continues to read as follows: Authority: 21 U.S.C. 360b. Sec. 524.1465 [Amended] 0 8. In paragraph (b) of Sec. 524.1465, add ``026637,'' after ``025463,''. PART 556--TOLERANCES FOR RESIDUES OF NEW ANIMAL DRUGS IN FOOD 0 9. The authority citation for 21 CFR part 556 continues to read as follows: Authority: 21 U.S.C. 342, 360b, 371. Sec. 556.733 [Amended] 0 10. In paragraph (a) of Sec. 556.733, remove ``10 micrograms'' and in its place add ``50 micrograms''. PART 558--NEW ANIMAL DRUGS FOR USE IN ANIMAL FEEDS 0 11. The authority citation for 21 CFR part 558 continues to read as follows: Authority: 21 U.S.C. 360b, 371. 0 12. In Sec. 558.665, in the table, revise paragraph (e)(5) to read as follows: Sec. 558.665 Zilpaterol. * * * * * (e) * * * ---------------------------------------------------------------------------------------------------------------- Combination in Zilpaterol in grams/ton grams/ton Indications for use Limitations Sponsor ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- * * * * * * * (5) 6.8 to provide 60 to 90 mg/ Monensin 10 to Cattle fed in As in paragraph 000061 016592 head/day. 40, plus tylosin confinement for (e)(1) of this 8 to 10. slaughter: As in section; see Sec. paragraph (e)(1) of Sec. 558.355(d) this section; for and 558.625(c) of prevention and this chapter. control of Monensin as provided coccidiosis due to by No. 000986; Eimeria bovis and E. tylosin as provided zuernii; and for by Nos. 000986 or reduction of 016592 in Sec. incidence of liver 510.600(c) of this abscesses caused by chapter. Fusobacterium necrophorum and Arcanobacterum (Actinomyces) pyogenes. ---------------------------------------------------------------------------------------------------------------- * * * * * * * ---------------------------------------------------------------------------------------------------------------- Dated: August 19, 2013. Bernadette Dunham, Director, Center for Veterinary Medicine. [FR Doc. 2013-20538 Filed 8-26-13; 8:45 am] BILLING CODE 4160-01-P
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Accommodation
The hotel offers 50 comfortable and well-equipped guestrooms. Room facilities include climate control, clock radio, coffee/tea maker, free local calls, in-room safe and voice mail. Iron/ironing boards are supplied in the rooms. Entertainment: In-room entertainment options at Quality Inn & Suites include cable television service and premium television channel(s). Internet connection options: Business guests will appreciate Internet access (complimentary) and wireless Internet access. Housekeeping services are also available.
Facilities
Dining facilities at Quality Inn & Suites include a cafeteria. The hotel boasts a 24-hour front desk service. Leisure amenities: There is an indoor swimming pool at the hotel. Other services: laundry facilities and safe-deposit box - front desk. Business/Internet: The following Internet options are available at the hotel: wireless access; high-speed wireless access is complimentary.
Transportation/Parking facilities
For guest convenience, parking is offered at no extra cost. Valet parking is also available.
Hotel Services and Facilities
Hotel facilities
- Business Center
- 24 hour front desk
How to bookMaking your reservation in Quality Inn &
- Springfield, MO, United States (SGF-Springfield-Branson National) - 7.54 mi
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- Branson, MO, United States (BKG) - 43.00 mi
- Harrison, AR, United States (HRO-Boone County Regional) - 61.55 mi
- Discovery Ctr Sprngfeld Scence - 4.82 mi
- History Museum For Springfield-Greene Co - 5.25 mi
- Prize Ctter Chrity Chmpionship - 5.67 mi
| 200,222
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Location : Kampung Nyegol, Padawan, Kuching, Sarawak, Malaysia.
It’s been more than 2 years we have not been back here to this tranquil village. Noticed houses are more crowded now. It has grown from 12 to 19 and headman Mr.Simo expects it to reach 30+ soon.
Perhaps, those who abandoned their homes to move to Bengoh Dam Resettlement Scheme Site (BRS) must have realised their folly. The mountain is where their hearts should be. Simo disclosed that 2 homes at BRS had their electricity cut off due to non payment. Life must be tough. Our album here also covers the old and abandoned villages of both Sait and Rejoi.
Trekking to Kampung Nyegol
Medical service at Headman house
Abandoned Kampung Sait
Abandoned Kampung Rejoi
To see more photos, please click our Facebook Page as below
Related Links:
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TITLE: Ordered Bell numbers
QUESTION [12 upvotes]: The ordered Bell numbers (also known as Fubini numbers, sequence A000670 in OEIS) count the number of ordered partitions of an n-element set. Experimentally I have found the following expression for the n-th ordered Bell number $a_n$:
$$a_n = \sum_{\sigma \in S_n}\prod_{i=1}^n \binom{i}{\sigma(i)-1}$$ where the sum ranges over all permutations of $\{1,2,\ldots,n\}$. Even though there are $n!$ terms in the sum, only $2^{n-1}$ are non-zero.
More generally, letting $S_n^m$ denote the set of permutations of $\{1,2,\ldots,n\}$ with exactly $m$ fixed points, I believe the following is also true: the number of ordered partitions of an n-element set having exactly $m$ blocks of cardinality one is given by
$$\sum_{\sigma \in S_n^m}\prod_{i=1}^n \binom{i}{\sigma(i)-1}$$. For example, for $m=0$ the formula appears to yield OEIS sequence A032032.
Is this known? Any ideas how to prove it or references to an existing proof?
REPLY [4 votes]: I would accept Sam's and lambda's comments as the answer. For the record, I'll just flesh it out a bit for the first formula.
In terms of compositions of $n$, the following is all but self-evident
$$a_n = \sum_{n_1+n_2+\ldots+n_k=n} \binom{n}{n_k}\binom{n-n_k}{n_{k-1}}\binom{n-n_k-n_{k-1}}{n_{k-2}}\ldots \binom{n-n_k-n_{k-1}-n_{k-2}-\ldots-n_2}{n_1}$$
where the sum is over all compositions of $n$. Now define a mapping from the compositions of $n$ to the permutations $\sigma$ with $\sigma(i) \le i+1$ for $1 \le i \le n$ as follows: Map composition $n_1+n_2+\ldots+n_k = n$ to
\begin{align}
\sigma(n_1) &= 1\\
\sigma(n_1+n_2) &= n_1+1\\
\sigma(n_1+n_2+n_3) &= n_1+n_2+1\\
&\ldots\\
\sigma(n_1+n_2+\ldots+n_k) &= n_1+n_2+\ldots+n_{k-1}+1\\
\sigma(i) &= i+1\ \text{for}\ i \not\in \{n_1,n_1+n_2,\ldots,n_1+n_2+\ldots+n_k\}
\end{align}
After checking this mapping is indeed a 1-1 correspondence between the compositions of $n$ and the permutations $\sigma$ with $\sigma(i) \le i+1$, we can now rewrite
\begin{align}
&\binom{n}{n_k}\binom{n-n_k}{n_{k-1}}\binom{n-n_k-n_{k-1}}{n_{k-2}}\ldots \binom{n-n_k-n_{k-1}-n_{k-2}-\ldots-n_2}{n_1} = \\
&\binom{n_1+n_2+\ldots+n_k}{n_k}\binom{n_1+n_2+\ldots+n_{k-1}}{n_{k-1}}
\binom{n_1+n_2+\ldots+n_{k-2}}{n_{k-2}}\ldots \binom{n_1}{n_1} =\\
&\binom{n_1+n_2+\ldots+n_k}{n_1+n_2+\ldots+n_{k-1}}
\binom{n_1+n_2+\ldots+n_{k-1}}{n_1+n_2+\ldots+n_{k-2}}
\binom{n_1+n_2+\ldots+n_{k-2}}{n_1+n_2+\ldots+n_{k-3}}\\
&\ldots\binom{n_1}{0}=\\
&\binom{n_1+n_2+\ldots+n_k}{\sigma(n_1+n_2+\ldots+n_k)-1}
\binom{n_1+n_2+\ldots+n_{k-1}}{\sigma(n_1+n_2+\ldots+n_{k-1})-1}\\
&\binom{n_1+n_2+\ldots+n_{k-2}}{\sigma(n_1+n_2+\ldots+n_{k-2})-1}
\ldots \binom{n_1}{\sigma(n_1)-1} = \prod_{i=1}^n \binom{i}{\sigma(i)-1}
\end{align}
The more general formulas follow by noticing that the mapping from compositions to permutations described above is a bijection between compositions with exactly m ones and permutations with exactly m fixed points.
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Last Thursday January 6, the Morgan High Wrestling team traveled to Ogden to take on the Ben Lomond Scots of Ben Lomond. Morgan dominated the dual winning 58-12.
The next day the team traveled to Cache Valley for the Bobcat Brawls. Morgan’s Varsity team finished the two-day tournament in 4th place with three undefeated wrestlers. They were Connor Jensen, Colton Walker, and Jackson Lake. For the JV team, Nate Losee was the only undefeated wrestler.
Morgan hosted its first home wrestling match this week when the Grantsville Cowboys came to town. Their Trojan home stay was short lived, as the wrestling team is back on the road again. They will participate in the Richardson Memorial Tournament at Box Elder High School. Results for both events will be in next weeks paper.
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Natalie Simons
Priest
At age 29 Natalie is the youngest female priest in the Church of the Province of Southern Africa, which is spread over six Southern African countries. She serves in the St Andrews Anglican church in Newlands, Cape Town.
Natalie has a master’s degree in theology from Stellenbosch, where she specialised in pastoral care and counselling. Her particular passion is for inter-faith dialogue and, as well as participating in an inter-faith group with Jews and Muslims, she coordinates a programme responsible for sending 12 young South Africans on a two-week interfaith camp in New York once a year.
In Cape Town she helps run meetings of up to 50 senior school children at a time, the purpose of which is to help thern “learn to live together with difference”. She also helps to run Hope Africa, the developmental arm of the Cape Town diocese that exists to alleviate poverty, to educate and train and to help create selfsustaining communities.
Lunch Spot: Chaiyo,Cape Town
| 334,914
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Monthly Archives: December 2012
Pumpkin Pasta with Hazelnuts (gluten-free & vegan)
What a wonderful Christmas! I still can’t believe it’s already passed us by! My hubby, stepson and I spent our holiday relaxing with family and eating TONS of food
Honestly, I don’t think it gets much better than that.
Winter Granola
Merry Christmas Eve!
Yesterday, my hubby, stepson, and I drove to Houston to spend Christmas with my hubs’ family. Even though Houston is only 4 hours away, it felt so good to get out of Dallas for a change.
Anytime I go on a trip I always bring tons of food along with me. My husband’s family is so sweet and accommodating, always trying to prepare food that I can enjoy, too. But, like most families, all of their meals are centered around some kind of meat, dairy, and/or eggs, leaving me picking at what’s left on the side. As a result, I’ve found that it’s just easier for me–and for everyone else–if I prepare some food ahead of time to take with me. I look like a crazy lady arriving with my smoothies, pre-cooked quinoa, and bags of fresh fruits, veggies, and nuts. Thankfully, my hubs’ family is used to my special brand of crazy by now.
One of my favorite things to prepare is granola. It’s the easiest food to carry around in my purse for when I get hungry (which is all the time) and to munch on at night when my sweet-tooth strikes.
White Chocolate Cookies with Cranberries and Macadamia Nuts
The name is a mouthful, but these White Chocolate Cookies with Cranberries and Macadamia Nuts will be devoured so quickly that you won’t have time to say the name anyway.
When I first cut out gluten, dairy, and refined sugar from my diet, I was determined not to cross cookies off of my list of foods that I could still safely eat. For a while, I was on a mission to find the perfect cookie recipe that still tasted like authentic cookies–like the kind my Ma-Ma makes: decadent, melt-in-your-mouth, and sinfully delicious. I think I must have tried hundreds of different cookie recipes (or at least it seemed like hundreds). Some were a complete disaster and many were pretty darn good, but none of them tasted like the “real” thing. That is, not until these amazing cookies came around and completely changed my life. OK, that may be a slight exaggeration, but in all honestly, these are the BEST cookies I’ve had in a very long time–perhaps the best ever!!
Cranberry Persimmon Sauce and My Holiday-Filled WIAW
The past few days have been packed with holiday flavor! It finally occurred to me–and just in time, thank goodness–that Christmas is almost here! I’ve been celebrating by eating (a lot), bundling up and watching Christmas movies, and spending hours working on gigantic Christmas puzzles. Yes, I’m 29 going on 80.
My favorite part of all has got to be the food. Hands down, holiday food it the best. Yesterday, I tried to cram as much holiday flavor into my day as possible. In addition to everything else I ate, I kept sneaking spoonfuls of my Cranberry Persimmon Sauce (recipe below) throughout the day.
| 45,078
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