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-%%% ====================================================================
-%%% @LaTeX-file{
-%%% filename = "testmath.tex",
-%%% version = "2.0",
-%%% date = "1999/11/15",
-%%% time = "15:09:17 EST",
-%%% checksum = "07762 2342 7811 82371",
-%%% author = "American Mathematical Society",
-%%% copyright = "Copyright 1995, 1999 American Mathematical Society,
-%%% all rights reserved. Copying of this file is
-%%% authorized only if either:
-%%% (1) you make absolutely no changes to your copy,
-%%% including name; OR
-%%% (2) if you do make changes, you first rename it
-%%% to some other name.",
-%%% address = "American Mathematical Society,
-%%% Technical Support,
-%%% Electronic Products and Services,
-%%% P. O. Box 6248,
-%%% Providence, RI 02940,
-%%% USA",
-%%% telephone = "401-455-4080 or (in the USA and Canada)
-%%% 800-321-4AMS (321-4267)",
-%%% FAX = "401-331-3842",
-%%% email = "tech-support@ams.org (Internet)",
-%%% codetable = "ISO/ASCII",
-%%% keywords = "latex, amsmath, examples, documentation",
-%%% supported = "yes",
-%%% abstract = "This is a test file containing extensive examples of
-%%% mathematical constructs supported by the amsmath
-%%% package.",
-%%% docstring = "The checksum field above contains a CRC-16
-%%% checksum as the first value, followed by the
-%%% equivalent of the standard UNIX wc (word
-%%% count) utility output of lines, words, and
-%%% characters. This is produced by Robert
-%%% Solovay's checksum utility.",
-%%% }
-%%% ====================================================================
-\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX)
-[1994/12/01]% LaTeX date must December 1994 or later
-\documentclass[draft]{article}
-\pagestyle{headings}
-
-\title{Sample Paper for the \pkg{amsmath} Package\\
-File name: \fn{testmath.tex}}
-\author{American Mathematical Society}
-\date{Version 2.0, 1999/11/15}
-
-\usepackage{amsmath,amsthm}
-
-% Some definitions useful in producing this sort of documentation:
-\chardef\bslash=`\\ % p. 424, TeXbook
-% Normalized (nonbold, nonitalic) tt font, to avoid font
-% substitution warning messages if tt is used inside section
-% headings and other places where odd font combinations might
-% result.
-\newcommand{\ntt}{\normalfont\ttfamily}
-% command name
-\newcommand{\cn}[1]{{\protect\ntt\bslash#1}}
-% LaTeX package name
-\newcommand{\pkg}[1]{{\protect\ntt#1}}
-% File name
-\newcommand{\fn}[1]{{\protect\ntt#1}}
-% environment name
-\newcommand{\env}[1]{{\protect\ntt#1}}
-\hfuzz1pc % Don't bother to report overfull boxes if overage is < 1pc
-
-% Theorem environments
-
-%% \theoremstyle{plain} %% This is the default
-\newtheorem{thm}{Theorem}[section]
-\newtheorem{cor}[thm]{Corollary}
-\newtheorem{lem}[thm]{Lemma}
-\newtheorem{prop}[thm]{Proposition}
-\newtheorem{ax}{Axiom}
-
-\theoremstyle{definition}
-\newtheorem{defn}{Definition}[section]
-
-\theoremstyle{remark}
-\newtheorem{rem}{Remark}[section]
-\newtheorem*{notation}{Notation}
-
-%\numberwithin{equation}{section}
-
-\newcommand{\thmref}[1]{Theorem~\ref{#1}}
-\newcommand{\secref}[1]{\S\ref{#1}}
-\newcommand{\lemref}[1]{Lemma~\ref{#1}}
-
-\newcommand{\bysame}{\mbox{\rule{3em}{.4pt}}\,}
-
-% Math definitions
-
-\newcommand{\A}{\mathcal{A}}
-\newcommand{\B}{\mathcal{B}}
-\newcommand{\st}{\sigma}
-\newcommand{\XcY}{{(X,Y)}}
-\newcommand{\SX}{{S_X}}
-\newcommand{\SY}{{S_Y}}
-\newcommand{\SXY}{{S_{X,Y}}}
-\newcommand{\SXgYy}{{S_{X|Y}(y)}}
-\newcommand{\Cw}[1]{{\hat C_#1(X|Y)}}
-\newcommand{\G}{{G(X|Y)}}
-\newcommand{\PY}{{P_{\mathcal{Y}}}}
-\newcommand{\X}{\mathcal{X}}
-\newcommand{\wt}{\widetilde}
-\newcommand{\wh}{\widehat}
-
-\DeclareMathOperator{\per}{per}
-\DeclareMathOperator{\cov}{cov}
-\DeclareMathOperator{\non}{non}
-\DeclareMathOperator{\cf}{cf}
-\DeclareMathOperator{\add}{add}
-\DeclareMathOperator{\Cham}{Cham}
-\DeclareMathOperator{\IM}{Im}
-\DeclareMathOperator{\esssup}{ess\,sup}
-\DeclareMathOperator{\meas}{meas}
-\DeclareMathOperator{\seg}{seg}
-
-% \interval is used to provide better spacing after a [ that
-% is used as a closing delimiter.
-\newcommand{\interval}[1]{\mathinner{#1}}
-
-% Notation for an expression evaluated at a particular condition. The
-% optional argument can be used to override automatic sizing of the
-% right vert bar, e.g. \eval[\biggr]{...}_{...}
-\newcommand{\eval}[2][\right]{\relax
- \ifx#1\right\relax \left.\fi#2#1\rvert}
-
-% Enclose the argument in vert-bar delimiters:
-\newcommand{\envert}[1]{\left\lvert#1\right\rvert}
-\let\abs=\envert
-
-% Enclose the argument in double-vert-bar delimiters:
-\newcommand{\enVert}[1]{\left\lVert#1\right\rVert}
-\let\norm=\enVert
-
-\begin{document}
-\maketitle
-\markboth{Sample paper for the {\protect\ntt\lowercase{amsmath}} package}
-{Sample paper for the {\protect\ntt\lowercase{amsmath}} package}
-\renewcommand{\sectionmark}[1]{}
-
-\section{Introduction}
-
-This paper contains examples of various features from \AmS-\LaTeX{}.
-
-\section{Enumeration of Hamiltonian paths in a graph}
-
-Let $\mathbf{A}=(a_{ij})$ be the adjacency matrix of graph $G$. The
-corresponding Kirchhoff matrix $\mathbf{K}=(k_{ij})$ is obtained from
-$\mathbf{A}$ by replacing in $-\mathbf{A}$ each diagonal entry by the
-degree of its corresponding vertex; i.e., the $i$th diagonal entry is
-identified with the degree of the $i$th vertex. It is well known that
-\begin{equation}
-\det\mathbf{K}(i|i)=\text{ the number of spanning trees of $G$},
-\quad i=1,\dots,n
-\end{equation}
-where $\mathbf{K}(i|i)$ is the $i$th principal submatrix of
-$\mathbf{K}$.
-\begin{verbatim}
-\det\mathbf{K}(i|i)=\text{ the number of spanning trees of $G$},
-\end{verbatim}
-
-Let $C_{i(j)}$ be the set of graphs obtained from $G$ by attaching edge
-$(v_iv_j)$ to each spanning tree of $G$. Denote by $C_i=\bigcup_j
-C_{i(j)}$. It is obvious that the collection of Hamiltonian cycles is a
-subset of $C_i$. Note that the cardinality of $C_i$ is $k_{ii}\det
-\mathbf{K}(i|i)$. Let $\wh X=\{\hat x_1,\dots,\hat x_n\}$.
-\begin{verbatim}
-$\wh X=\{\hat x_1,\dots,\hat x_n\}$
-\end{verbatim}
-Define multiplication for the elements of $\wh X$ by
-\begin{equation}\label{multdef}
-\hat x_i\hat x_j=\hat x_j\hat x_i,\quad \hat x^2_i=0,\quad
-i,j=1,\dots,n.
-\end{equation}
-Let $\hat k_{ij}=k_{ij}\hat x_j$ and $\hat k_{ij}=-\sum_{j\not=i} \hat
-k_{ij}$. Then the number of Hamiltonian cycles $H_c$ is given by the
-relation \cite{liuchow:formalsum}
-\begin{equation}\label{H-cycles}
-\biggl(\prod^n_{\,j=1}\hat x_j\biggr)H_c=\frac{1}{2}\hat k_{ij}\det
-\wh{\mathbf{K}}(i|i),\qquad i=1,\dots,n.
-\end{equation}
-The task here is to express \eqref{H-cycles}
-in a form free of any $\hat x_i$,
-$i=1,\dots,n$. The result also leads to the resolution of enumeration of
-Hamiltonian paths in a graph.
-
-It is well known that the enumeration of Hamiltonian cycles and paths in
-a complete graph $K_n$ and in a complete bipartite graph $K_{n_1n_2}$
-can only be found from \textit{first combinatorial principles}
-\cite{hapa:graphenum}. One wonders if there exists a formula which can
-be used very efficiently to produce $K_n$ and $K_{n_1n_2}$. Recently,
-using Lagrangian methods, Goulden and Jackson have shown that $H_c$ can
-be expressed in terms of the determinant and permanent of the adjacency
-matrix \cite{gouja:lagrmeth}. However, the formula of Goulden and
-Jackson determines neither $K_n$ nor $K_{n_1n_2}$ effectively. In this
-paper, using an algebraic method, we parametrize the adjacency matrix.
-The resulting formula also involves the determinant and permanent, but
-it can easily be applied to $K_n$ and $K_{n_1n_2}$. In addition, we
-eliminate the permanent from $H_c$ and show that $H_c$ can be
-represented by a determinantal function of multivariables, each variable
-with domain $\{0,1\}$. Furthermore, we show that $H_c$ can be written by
-number of spanning trees of subgraphs. Finally, we apply the formulas to
-a complete multigraph $K_{n_1\dots n_p}$.
-
-The conditions $a_{ij}=a_{ji}$, $i,j=1,\dots,n$, are not required in
-this paper. All formulas can be extended to a digraph simply by
-multiplying $H_c$ by 2.
-
-\section{Main Theorem}
-\label{s:mt}
-
-\begin{notation} For $p,q\in P$ and $n\in\omega$ we write
-$(q,n)\le(p,n)$ if $q\le p$ and $A_{q,n}=A_{p,n}$.
-\begin{verbatim}
-\begin{notation} For $p,q\in P$ and $n\in\omega$
-...
-\end{notation}
-\end{verbatim}
-\end{notation}
-
-Let $\mathbf{B}=(b_{ij})$ be an $n\times n$ matrix. Let $\mathbf{n}=\{1,
-\dots,n\}$. Using the properties of \eqref{multdef}, it is readily seen
-that
-
-\begin{lem}\label{lem-per}
-\begin{equation}
-\prod_{i\in\mathbf{n}}
-\biggl(\sum_{\,j\in\mathbf{n}}b_{ij}\hat x_i\biggr)
-=\biggl(\prod_{\,i\in\mathbf{n}}\hat x_i\biggr)\per \mathbf{B}
-\end{equation}
-where $\per \mathbf{B}$ is the permanent of $\mathbf{B}$.
-\end{lem}
-
-Let $\wh Y=\{\hat y_1,\dots,\hat y_n\}$. Define multiplication
-for the elements of $\wh Y$ by
-\begin{equation}
-\hat y_i\hat y_j+\hat y_j\hat y_i=0,\quad i,j=1,\dots,n.
-\end{equation}
-Then, it follows that
-\begin{lem}\label{lem-det}
-\begin{equation}\label{detprod}
-\prod_{i\in\mathbf{n}}
-\biggl(\sum_{\,j\in\mathbf{n}}b_{ij}\hat y_j\biggr)
-=\biggl(\prod_{\,i\in\mathbf{n}}\hat y_i\biggr)\det\mathbf{B}.
-\end{equation}
-\end{lem}
-
-Note that all basic properties of determinants are direct consequences
-of Lemma ~\ref{lem-det}. Write
-\begin{equation}\label{sum-bij}
-\sum_{j\in\mathbf{n}}b_{ij}\hat y_j=\sum_{j\in\mathbf{n}}b^{(\lambda)}
-_{ij}\hat y_j+(b_{ii}-\lambda_i)\hat y_i\hat y
-\end{equation}
-where
-\begin{equation}
-b^{(\lambda)}_{ii}=\lambda_i,\quad b^{(\lambda)}_{ij}=b_{ij},
-\quad i\not=j.
-\end{equation}
-Let $\mathbf{B}^{(\lambda)}=(b^{(\lambda)}_{ij})$. By \eqref{detprod}
-and \eqref{sum-bij}, it is
-straightforward to show the following
-result:
-\begin{thm}\label{thm-main}
-\begin{equation}\label{detB}
-\det\mathbf{B}=
-\sum^n_{l =0}\sum_{I_l \subseteq n}
-\prod_{i\in I_l}(b_{ii}-\lambda_i)
-\det\mathbf{B}^{(\lambda)}(I_l |I_l ),
-\end{equation}
-where $I_l =\{i_1,\dots,i_l \}$ and $\mathbf{B}^{(\lambda)}(I_l |I_l )$
-is the principal submatrix obtained from $\mathbf{B}^{(\lambda)}$
-by deleting its $i_1,\dots,i_l $ rows and columns.
-\end{thm}
-
-\begin{rem}
-Let $\mathbf{M}$ be an $n\times n$ matrix. The convention
-$\mathbf{M}(\mathbf{n}|\mathbf{n})=1$ has been used in \eqref{detB} and
-hereafter.
-\end{rem}
-
-Before proceeding with our discussion, we pause to note that
-\thmref{thm-main} yields immediately a fundamental formula which can be
-used to compute the coefficients of a characteristic polynomial
-\cite{mami:matrixth}:
-\begin{cor}\label{BI}
-Write $\det(\mathbf{B}-x\mathbf{I})=\sum^n_{l =0}(-1)
-^l b_l x^l $. Then
-\begin{equation}\label{bl-sum}
-b_l =\sum_{I_l \subseteq\mathbf{n}}\det\mathbf{B}(I_l |I_l ).
-\end{equation}
-\end{cor}
-Let
-\begin{equation}
-\mathbf{K}(t,t_1,\dots,t_n)
-=\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\
--a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\
-\hdotsfor[2]{4}\\
--a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix},
-\end{equation}
-\begin{verbatim}
-\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\
--a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\
-\hdotsfor[2]{4}\\
--a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix}
-\end{verbatim}
-where
-\begin{equation}
-D_i=\sum_{j\in\mathbf{n}}a_{ij}t_j,\quad i=1,\dots,n.
-\end{equation}
-
-Set
-\begin{equation*}
-D(t_1,\dots,t_n)=\frac{\delta}{\delta t}\eval{\det\mathbf{K}(t,t_1,\dots,t_n)
-}_{t=1}.
-\end{equation*}
-Then
-\begin{equation}\label{sum-Di}
-D(t_1,\dots,t_n)
-=\sum_{i\in\mathbf{n}}D_i\det\mathbf{K}(t=1,t_1,\dots,t_n; i|i),
-\end{equation}
-where $\mathbf{K}(t=1,t_1,\dots,t_n; i|i)$ is the $i$th principal
-submatrix of $\mathbf{K}(t=1,t_1,\dots,t_n)$.
-
-Theorem ~\ref{thm-main} leads to
-\begin{equation}\label{detK1}
-\det\mathbf{K}(t_1,t_1,\dots,t_n)
-=\sum_{I\in\mathbf{n}}(-1)^{\envert{I}}t^{n-\envert{I}}
-\prod_{i\in I}t_i\prod_{j\in I}(D_j+\lambda_jt_j)\det\mathbf{A}
-^{(\lambda t)}(\overline{I}|\overline I).
-\end{equation}
-Note that
-\begin{equation}\label{detK2}
-\det\mathbf{K}(t=1,t_1,\dots,t_n)=\sum_{I\in\mathbf{n}}(-1)^{\envert{I}}
-\prod_{i\in I}t_i\prod_{j\in I}(D_j+\lambda_jt_j)\det\mathbf{A}
-^{(\lambda)}(\overline{I}|\overline{I})=0.
-\end{equation}
-
-Let $t_i=\hat x_i,i=1,\dots,n$. Lemma ~\ref{lem-per} yields
-\begin{multline}
-\biggl(\sum_{\,i\in\mathbf{n}}a_{l _i}x_i\biggr)
-\det\mathbf{K}(t=1,x_1,\dots,x_n;l |l )\\
-=\biggl(\prod_{\,i\in\mathbf{n}}\hat x_i\biggr)
-\sum_{I\subseteq\mathbf{n}-\{l \}}
-(-1)^{\envert{I}}\per\mathbf{A}^{(\lambda)}(I|I)
-\det\mathbf{A}^{(\lambda)}
-(\overline I\cup\{l \}|\overline I\cup\{l \}).
-\label{sum-ali}
-\end{multline}
-\begin{verbatim}
-\begin{multline}
-\biggl(\sum_{\,i\in\mathbf{n}}a_{l _i}x_i\biggr)
-\det\mathbf{K}(t=1,x_1,\dots,x_n;l |l )\\
-=\biggl(\prod_{\,i\in\mathbf{n}}\hat x_i\biggr)
-\sum_{I\subseteq\mathbf{n}-\{l \}}
-(-1)^{\envert{I}}\per\mathbf{A}^{(\lambda)}(I|I)
-\det\mathbf{A}^{(\lambda)}
-(\overline I\cup\{l \}|\overline I\cup\{l \}).
-\label{sum-ali}
-\end{multline}
-\end{verbatim}
-
-By \eqref{H-cycles}, \eqref{detprod}, and \eqref{sum-bij}, we have
-\begin{prop}\label{prop:eg}
-\begin{equation}
-H_c=\frac1{2n}\sum^n_{l =0}(-1)^{l}
-D_{l},
-\end{equation}
-where
-\begin{equation}\label{delta-l}
-D_{l}=\eval[2]{\sum_{I_{l}\subseteq \mathbf{n}}
-D(t_1,\dots,t_n)}_{t_i=\left\{\begin{smallmatrix}
-0,& \text{if }i\in I_{l}\quad\\% \quad added for centering
-1,& \text{otherwise}\end{smallmatrix}\right.\;,\;\; i=1,\dots,n}.
-\end{equation}
-\end{prop}
-
-\section{Application}
-\label{lincomp}
-
-We consider here the applications of Theorems~\ref{th-info-ow-ow} and
-~\ref{th-weak-ske-owf} to a complete
-multipartite graph $K_{n_1\dots n_p}$. It can be shown that the
-number of spanning trees of $K_{n_1\dots n_p}$
-may be written
-\begin{equation}\label{e:st}
-T=n^{p-2}\prod^p_{i=1}
-(n-n_i)^{n_i-1}
-\end{equation}
-where
-\begin{equation}
-n=n_1+\dots+n_p.
-\end{equation}
-
-It follows from Theorems~\ref{th-info-ow-ow} and
-~\ref{th-weak-ske-owf} that
-\begin{equation}\label{e:barwq}
-\begin{split}
-H_c&=\frac1{2n}
-\sum^n_{{l}=0}(-1)^{l}(n-{l})^{p-2}
-\sum_{l _1+\dots+l _p=l}\prod^p_{i=1}
-\binom{n_i}{l _i}\\
-&\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i}\cdot
-\biggl[(n-l )^2-\sum^p_{j=1}(n_i-l _i)^2\biggr].\end{split}
-\end{equation}
-\begin{verbatim}
-... \binom{n_i}{l _i}\\
-\end{verbatim}
-and
-\begin{equation}\label{joe}
-\begin{split}
-H_c&=\frac12\sum^{n-1}_{l =0}
-(-1)^{l}(n-l )^{p-2}
-\sum_{l _1+\dots+l _p=l}
-\prod^p_{i=1}\binom{n_i}{l _i}\\
-&\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i}
-\left(1-\frac{l _p}{n_p}\right)
-[(n-l )-(n_p-l _p)].
-\end{split}
-\end{equation}
-
-The enumeration of $H_c$ in a $K_{n_1\dotsm n_p}$ graph can also be
-carried out by Theorem ~\ref{thm-H-param} or ~\ref{thm-asym}
-together with the algebraic method of \eqref{multdef}.
-Some elegant representations may be obtained. For example, $H_c$ in
-a $K_{n_1n_2n_3}$ graph may be written
-\begin{equation}\label{j:mark}
-\begin{split}
-H_c=&
-\frac{n_1!\,n_2!\,n_3!}
-{n_1+n_2+n_3}\sum_i\left[\binom{n_1}{i}
-\binom{n_2}{n_3-n_1+i}\binom{n_3}{n_3-n_2+i}\right.\\
-&+\left.\binom{n_1-1}{i}
-\binom{n_2-1}{n_3-n_1+i}
-\binom{n_3-1}{n_3-n_2+i}\right].\end{split}
-\end{equation}
-
-\section{Secret Key Exchanges}
-\label{SKE}
-
-Modern cryptography is fundamentally concerned with the problem of
-secure private communication. A Secret Key Exchange is a protocol
-where Alice and Bob, having no secret information in common to start,
-are able to agree on a common secret key, conversing over a public
-channel. The notion of a Secret Key Exchange protocol was first
-introduced in the seminal paper of Diffie and Hellman
-\cite{dihe:newdir}. \cite{dihe:newdir} presented a concrete
-implementation of a Secret Key Exchange protocol, dependent on a
-specific assumption (a variant on the discrete log), specially
-tailored to yield Secret Key Exchange. Secret Key Exchange is of
-course trivial if trapdoor permutations exist. However, there is no
-known implementation based on a weaker general assumption.
-
-The concept of an informationally one-way function was introduced
-in \cite{imlelu:oneway}. We give only an informal definition here:
-
-\begin{defn} A polynomial time
-computable function $f = \{f_k\}$ is informationally
-one-way if there is no probabilistic polynomial time algorithm which
-(with probability of the form $1 - k^{-e}$ for some $e > 0$)
-returns on input $y \in \{0,1\}^{k}$ a random element of $f^{-1}(y)$.
-\end{defn}
-In the non-uniform setting \cite{imlelu:oneway} show that these are not
-weaker than one-way functions:
-\begin{thm}[\cite{imlelu:oneway} (non-uniform)]
-\label{th-info-ow-ow}
-The existence of informationally one-way functions
-implies the existence of one-way functions.
-\end{thm}
-We will stick to the convention introduced above of saying
-``non-uniform'' before the theorem statement when the theorem
-makes use of non-uniformity. It should be understood that
-if nothing is said then the result holds for both the uniform and
-the non-uniform models.
-
-It now follows from \thmref{th-info-ow-ow} that
-
-\begin{thm}[non-uniform]\label{th-weak-ske-owf} Weak SKE
-implies the existence of a one-way function.
-\end{thm}
-
-More recently, the polynomial-time, interior point algorithms for linear
-programming have been extended to the case of convex quadratic programs
-\cite{moad:quadpro,ye:intalg}, certain linear complementarity problems
-\cite{komiyo:lincomp,miyoki:lincomp}, and the nonlinear complementarity
-problem \cite{komiyo:unipfunc}. The connection between these algorithms
-and the classical Newton method for nonlinear equations is well
-explained in \cite{komiyo:lincomp}.
-
-\section{Review}
-\label{computation}
-
-We begin our discussion with the following definition:
-
-\begin{defn}
-
-A function $H\colon \Re^n \to \Re^n$ is said to be
-\emph{B-differentiable} at the point $z$ if (i)~$H$ is Lipschitz
-continuous in a neighborhood of $z$, and (ii)~ there exists a positive
-homogeneous function $BH(z)\colon \Re^n \to \Re^n$, called the
-\emph{B-derivative} of $H$ at $z$, such that
-\[ \lim_{v \to 0} \frac{H(z+v) - H(z) - BH(z)v}{\enVert{v}} = 0. \]
-The function $H$ is \textit{B-differentiable in set $S$} if it is
-B-differentiable at every point in $S$. The B-derivative $BH(z)$ is said
-to be \textit{strong} if
-\[ \lim_{(v,v') \to (0,0)} \frac{H(z+v) - H(z+v') - BH(z)(v
- -v')}{\enVert{v - v'}} = 0. \]
-\end{defn}
-
-
-\begin{lem}\label{limbog} There exists a smooth function $\psi_0(z)$
-defined for $\abs{z}>1-2a$ satisfying the following properties\textup{:}
-\begin{enumerate}
-\renewcommand{\labelenumi}{(\roman{enumi})}
-\item $\psi_0(z)$ is bounded above and below by positive constants
-$c_1\leq \psi_0(z)\leq c_2$.
-\item If $\abs{z}>1$, then $\psi_0(z)=1$.
-\item For all $z$ in the domain of $\psi_0$, $\Delta_0\ln \psi_0\geq 0$.
-\item If $1-2a<\abs{z}<1-a$, then $\Delta_0\ln \psi_0\geq
-c_3>0$.
-\end{enumerate}
-\end{lem}
-
-\begin{proof}
-We choose $\psi_0(z)$ to be a radial function depending only on $r=\abs{z}$.
-Let $h(r)\geq 0$ be a suitable smooth function satisfying $h(r)\geq c_3$
-for $1-2a<\abs{z}<1-a$, and $h(r)=0$ for $\abs{z}>1-\tfrac a2$. The radial
-Laplacian
-\[\Delta_0\ln\psi_0(r)=\left(\frac {d^2}{dr^2}+\frac
-1r\frac d{dr}\right)\ln\psi_0(r)\]
-has smooth coefficients for $r>1-2a$. Therefore, we may
-apply the existence and uniqueness theory for ordinary differential
-equations. Simply let $\ln \psi_0(r)$ be the solution of the differential
-equation
-\[\left(\frac{d^2}{dr^2}+\frac 1r\frac d{dr}\right)\ln \psi_0(r)=h(r)\]
-with initial conditions given by $\ln \psi_0(1)=0$ and
-$\ln\psi_0'(1)=0$.
-
-Next, let $D_\nu$ be a finite collection of pairwise disjoint disks,
-all of which are contained in the unit disk centered at the origin in
-$C$. We assume that $D_\nu=\{z\mid \abs{z-z_\nu}<\delta\}$. Suppose that
-$D_\nu(a)$ denotes the smaller concentric disk $D_\nu(a)=\{z\mid
-\abs{z-z_\nu}\leq (1-2a)\delta\}$. We define a smooth weight function
-$\Phi_0(z)$ for $z\in C-\bigcup_\nu D_\nu(a)$ by setting $\Phi_
-0(z)=1$ when $z\notin \bigcup_\nu D_\nu$ and $\Phi_
-0(z)=\psi_0((z-z_\nu)/\delta)$ when $z$ is an element of $D_\nu$. It
-follows from \lemref{limbog} that $\Phi_ 0$ satisfies the properties:
-\begin{enumerate}
-\renewcommand{\labelenumi}{(\roman{enumi})}
-\item \label{boundab}$\Phi_ 0(z)$ is bounded above and below by
-positive constants $c_1\leq \Phi_ 0(z)\leq c_2$.
-\item \label{d:over}$\Delta_0\ln\Phi_ 0\geq 0$ for all
-$z\in C-\bigcup_\nu D_\nu(a)$,
-the domain where the function $\Phi_ 0$ is defined.
-\item \label{d:ad}$\Delta_0\ln\Phi_ 0\geq c_3\delta^{-2}$
-when $(1-2a)\delta<\abs{z-z_\nu}<(1-a)\delta$.
-\end{enumerate}
-Let $A_\nu$ denote the annulus $A_\nu=\{(1-2a)\delta<\abs{z-z_\nu}<(1-a)
-\delta \}$, and set $A=\bigcup_\nu A_\nu$. The
-properties (\ref{d:over}) and (\ref{d:ad}) of $\Phi_ 0$
-may be summarized as $\Delta_0\ln \Phi_ 0\geq c_3\delta^{-2}\chi_A$,
-where $\chi _A$ is the characteristic function of $A$.
-\end{proof}
-
-Suppose that $\alpha$ is a nonnegative real constant. We apply
-Proposition~\ref{prop:eg} with $\Phi(z)=\Phi_ 0(z) e^{\alpha\abs{z}^2}$. If
-$u\in C^\infty_0(R^2-\bigcup_\nu D_\nu(a))$, assume that $\mathcal{D}$
-is a bounded domain containing the support of $u$ and $A\subset
-\mathcal{D}\subset R^2-\bigcup_\nu D_\nu(a)$. A calculation gives
-\[\int_{\mathcal{D}}\abs{\overline\partial u}^2\Phi_ 0(z) e^{\alpha\abs{z}^2}
-\geq c_4\alpha\int_{\mathcal{D}}\abs{u}^2\Phi_ 0e^{\alpha\abs{z}^2}
-+c_5\delta^{-2}\int_ A\abs{u}^2\Phi_ 0e^{\alpha\abs{z}^2}.\]
-
-The boundedness, property (\ref{boundab}) of $\Phi_ 0$, then yields
-\[\int_{\mathcal{D}}\abs{\overline\partial u}^2e^{\alpha\abs{z}^2}\geq c_6\alpha
-\int_{\mathcal{D}}\abs{u}^2e^{\alpha\abs{z}^2}
-+c_7\delta^{-2}\int_ A\abs{u}^2e^{\alpha\abs{z}^2}.\]
-
-Let $B(X)$ be the set of blocks of $\Lambda_{X}$
-and let $b(X) = \abs{B(X)}$. If $\phi \in Q_{X}$ then
-$\phi$ is constant on the blocks of $\Lambda_{X}$.
-\begin{equation}\label{far-d}
- P_{X} = \{ \phi \in M \mid \Lambda_{\phi} = \Lambda_{X} \},
-\qquad
-Q_{X} = \{\phi \in M \mid \Lambda_{\phi} \geq \Lambda_{X} \}.
-\end{equation}
-If $\Lambda_{\phi} \geq \Lambda_{X}$ then
-$\Lambda_{\phi} = \Lambda_{Y}$ for some $Y \geq X$ so that
-\[ Q_{X} = \bigcup_{Y \geq X} P_{Y}. \]
-Thus by M\"obius inversion
-\[ \abs{P_{Y}}= \sum_{X\geq Y} \mu (Y,X)\abs{Q_{X}}.\]
-Thus there is a bijection from $Q_{X}$ to $W^{B(X)}$.
-In particular $\abs{Q_{X}} = w^{b(X)}$.
-
-Next note that $b(X)=\dim X$. We see this by choosing a
-basis for $X$ consisting of vectors $v^{k}$ defined by
-\[v^{k}_{i}=
-\begin{cases} 1 & \text{if $i \in \Lambda_{k}$},\\
-0 &\text{otherwise.} \end{cases}
-\]
-\begin{verbatim}
-\[v^{k}_{i}=
-\begin{cases} 1 & \text{if $i \in \Lambda_{k}$},\\
-0 &\text{otherwise.} \end{cases}
-\]
-\end{verbatim}
-
-\begin{lem}\label{p0201}
-Let $\A$ be an arrangement. Then
-\[ \chi (\A,t) = \sum_{\B \subseteq \A}
-(-1)^{\abs{\B}} t^{\dim T(\B)}. \]
-\end{lem}
-
-In order to compute $R''$ recall the definition
-of $S(X,Y)$ from \lemref{lem-per}. Since $H \in \B$,
-$\A_{H} \subseteq \B$. Thus if $T(\B) = Y$ then
-$\B \in S(H,Y)$. Let $L'' = L(\A'')$. Then
-\begin{equation}\label{E_SXgYy}
-\begin{split}
-R''&= \sum_{H\in \B \subseteq \A} (-1)^{\abs{\B}}
-t^{\dim T(\B)}\\
-&= \sum_{Y \in L''} \sum_{\B \in S(H,Y)}
-(-1)^{\abs{\B}}t^{\dim Y} \\
-&= -\sum_{Y \in L''} \sum_{\B \in S(H,Y)} (-1)^
-{\abs{\B - \A_{H}}} t^{\dim Y} \\
-&= -\sum_{Y \in L''} \mu (H,Y)t^{\dim Y} \\
-&= -\chi (\A '',t).
-\end{split}
-\end{equation}
-
-\begin{cor}\label{tripleA}
-Let $(\A,\A',\A'')$ be a triple of arrangements. Then
-\[ \pi (\A,t) = \pi (\A',t) + t \pi (\A'',t). \]
-\end{cor}
-
-\begin{defn}
-Let $(\A,\A',\A'')$ be a triple with respect to
-the hyperplane $H \in \A$. Call $H$ a \textit{separator}
-if $T(\A) \not\in L(\A')$.
-\end{defn}
-
-\begin{cor}\label{nsep}
-Let $(\A,\A',\A'')$ be a triple with respect to $H \in \A$.
-\begin{enumerate}
-\renewcommand{\labelenumi}{(\roman{enumi})}
-\item
-If $H$ is a separator then
-\[ \mu (\A) = - \mu (\A'') \]
-and hence
-\[ \abs{\mu (\A)} = \abs{ \mu (\A'')}. \]
-
-\item If $H$ is not a separator then
-\[\mu (\A) = \mu (\A') - \mu (\A'') \]
-and
-\[ \abs{\mu (\A)} = \abs{\mu (\A')} + \abs{\mu (\A'')}. \]
-\end{enumerate}
-\end{cor}
-
-\begin{proof}
-It follows from \thmref{th-info-ow-ow} that $\pi(\A,t)$
-has leading term
-\[(-1)^{r(\A)}\mu (\A)t^{r(\A)}.\]
-The conclusion
-follows by comparing coefficients of the leading
-terms on both sides of the equation in
-Corollary~\ref{tripleA}. If $H$ is a separator then
-$r(\A') < r(\A)$ and there is no contribution
-from $\pi (\A',t)$.
-\end{proof}
-
-The Poincar\'e polynomial of an arrangement
-will appear repeatedly
-in these notes. It will be shown to equal the
-Poincar\'e polynomial
-of the graded algebras which we are going to
-associate with $\A$. It is also the Poincar\'e
-polynomial of the complement $M(\A)$ for a
-complex arrangement. Here we prove
-that the Poincar\'e polynomial is the chamber
-counting function for a real arrangement. The
-complement $M(\A)$ is a disjoint union of chambers
-\[M(\A) = \bigcup_{C \in \Cham(\A)} C.\]
-The number
-of chambers is determined by the Poincar\'e
-polynomial as follows.
-
-\begin{thm}\label{th-realarr}
-Let $\A_{\mathbf{R}}$ be a real arrangement. Then
-\[ \abs{\Cham(\A_{\mathbf{R}})} = \pi (\A_{\mathbf{R}},1). \]
-\end{thm}
-
-\begin{proof}
-We check the properties required in Corollary~\ref{nsep}:
-(i) follows from $\pi (\Phi_{ l},t) = 1$, and (ii) is a
-consequence of Corollary~\ref{BI}.
-\end{proof}
-
-\begin{figure}
-\vspace{5cm}
-\caption[]{$Q(\A_{1}) = xyz(x-z)(x+z)(y-z)(y+z)$}
-\end{figure}
-
-\begin{figure}
-\vspace{5cm}
-\caption[]{$Q(\A_{2})= xyz(x+y+z)(x+y-z)(x-y+z)(x-y-z)$}
-\end{figure}
-
-
-\begin{thm}
-\label{T_first_the_int}
-Let $\phi$ be a protocol for a random pair $\XcY$.
-If one of $\st_\phi(x',y)$ and $\st_\phi(x,y')$ is a prefix of the other
-and $(x,y)\in\SXY$, then
-\[
-\langle \st_j(x',y)\rangle_{j=1}^\infty
-=\langle \st_j(x,y)\rangle_{j=1}^\infty
-=\langle \st_j(x,y')\rangle_{j=1}^\infty .
-\]
-\end{thm}
-\begin{proof}
-We show by induction on $i$ that
-\[
-\langle \st_j(x',y)\rangle_{j=1}^i
-=\langle \st_j(x,y)\rangle_{j=1}^i
-=\langle \st_j(x,y')\rangle_{j=1}^i.
-\]
-The induction hypothesis holds vacuously for $i=0$. Assume it holds for
-$i-1$, in particular
-$[\st_j(x',y)]_{j=1}^{i-1}=[\st_j(x,y')]_{j=1}^{i-1}$. Then one of
-$[\st_j(x',y)]_{j=i}^{\infty}$ and $[\st_j(x,y')]_{j=i}^{\infty}$ is a
-prefix of the other which implies that one of $\st_i(x',y)$ and
-$\st_i(x,y')$ is a prefix of the other. If the $i$th message is
-transmitted by $P_\X$ then, by the separate-transmissions property and
-the induction hypothesis, $\st_i(x,y)=\st_i(x,y')$, hence one of
-$\st_i(x,y)$ and $\st_i(x',y)$ is a prefix of the other. By the
-implicit-termination property, neither $\st_i(x,y)$ nor $\st_i(x',y)$
-can be a proper prefix of the other, hence they must be the same and
-$\st_i(x',y)=\st_i(x,y)=\st_i(x,y')$. If the $i$th message is
-transmitted by $\PY$ then, symmetrically, $\st_i(x,y)=\st_i(x',y)$ by
-the induction hypothesis and the separate-transmissions property, and,
-then, $\st_i(x,y)=\st_i(x,y')$ by the implicit-termination property,
-proving the induction step.
-\end{proof}
-
-If $\phi$ is a protocol for $(X,Y)$, and $(x,y)$, $(x',y)$ are distinct
-inputs in $\SXY$, then, by the correct-decision property,
-$\langle\st_j(x,y)\rangle_{j=1}^\infty\ne\langle
-\st_j(x',y)\rangle_{j=1}^\infty$.
-
-Equation~(\ref{E_SXgYy}) defined $\PY$'s ambiguity set $\SXgYy$
-to be the set of possible $X$ values when $Y=y$.
-The last corollary implies that for all $y\in\SY$,
-the multiset%
-\footnote{A multiset allows multiplicity of elements.
-Hence, $\{0,01,01\}$ is prefix free as a set, but not as a multiset.}
-of codewords $\{\st_\phi(x,y):x\in\SXgYy\}$ is prefix free.
-
-\section{One-Way Complexity}
-\label{S_Cp1}
-
-$\Cw1$, the one-way complexity of a random pair $\XcY$,
-is the number of bits $P_\X$ must transmit in the worst case
-when $\PY$ is not permitted to transmit any feedback messages.
-Starting with $\SXY$, the support set of $\XcY$, we define $\G$,
-the \textit{characteristic hypergraph} of $\XcY$, and show that
-\[
-\Cw1=\lceil\,\log\chi(\G)\rceil\ .
-\]
-
-Let $\XcY$ be a random pair. For each $y$ in $\SY$, the support set of
-$Y$, Equation~(\ref{E_SXgYy}) defined $\SXgYy$ to be the set of possible
-$x$ values when $Y=y$. The \textit{characteristic hypergraph} $\G$ of
-$\XcY$ has $\SX$ as its vertex set and the hyperedge $\SXgYy$ for each
-$y\in\SY$.
-
-
-We can now prove a continuity theorem.
-\begin{thm}\label{t:conl}
-Let $\Omega \subset\mathbf{R}^n$ be an open set, let
-$u\in BV(\Omega ;\mathbf{R}^m)$, and let
-\begin{equation}\label{quts}
-T^u_x=\left\{y\in\mathbf{R}^m:
- y=\tilde u(x)+\left\langle \frac{Du}{\abs{Du}}(x),z
-\right\rangle \text{ for some }z\in\mathbf{R}^n\right\}
-\end{equation}
-for every $x\in\Omega \backslash S_u$. Let $f\colon \mathbf{R}^m\to
-\mathbf{R}^k$ be a Lipschitz continuous function such that $f(0)=0$, and
-let $v=f(u)\colon \Omega \to \mathbf{R}^k$. Then $v\in BV(\Omega
-;\mathbf{R}^k)$ and
-\begin{equation}
-Jv=\eval{(f(u^+)-f(u^-))\otimes \nu_u\cdot\,
-\mathcal{H}_{n-1}}_{S_u}.
-\end{equation}
-In addition, for $\abs{\wt{D}u}$-almost every $x\in\Omega $ the
-restriction of the function $f$ to $T^u_x$ is differentiable at $\tilde
-u(x)$ and
-\begin{equation}
-\wt{D}v=\nabla (\eval{f}_{T^u_x})(\tilde u)
-\frac{\wt{D}u}{\abs{\wt{D}u}}\cdot\abs{\wt{D}u}.\end{equation}
-\end{thm}
-
-Before proving the theorem, we state without proof three elementary
-remarks which will be useful in the sequel.
-\begin{rem}\label{r:omb}
-Let $\omega\colon \left]0,+\infty\right[\to \left]0,+\infty\right[$
-be a continuous function such that $\omega (t)\to 0$ as $t\to
-0$. Then
-\[\lim_{h\to 0^+}g(\omega(h))=L\Leftrightarrow\lim_{h\to
-0^+}g(h)=L\]
-for any function $g\colon \left]0,+\infty\right[\to \mathbf{R}$.
-\end{rem}
-\begin{rem}\label{r:dif}
-Let $g \colon \mathbf{R}^n\to \mathbf{R}$ be a Lipschitz
-continuous function and assume that
-\[L(z)=\lim_{h\to 0^+}\frac{g(hz)-g(0)}h\]
-exists for every $z\in\mathbf{Q}^n$ and that $L$ is a linear function of
-$z$. Then $g$ is differentiable at 0.
-\end{rem}
-\begin{rem}\label{r:dif0}
-Let $A \colon \mathbf{R}^n\to \mathbf{R}^m$ be a linear function, and
-let $f \colon \mathbf{R}^m\to \mathbf{R}$ be a function. Then the
-restriction of $f$ to the range of $A$ is differentiable at 0 if and
-only if $f(A)\colon \mathbf{R}^n\to \mathbf{R}$ is differentiable at 0
-and
-\[\nabla(\eval{f}_{\IM(A)})(0)A=\nabla (f(A))(0).\]
-\end{rem}
-
-\begin{proof}
- We begin by showing that $v\in BV(\Omega;\mathbf{R}^k)$ and
-\begin{equation}\label{e:bomb}
-\abs{Dv}(B)\le K\abs{Du}(B)\qquad\forall B\in\mathbf{B}(\Omega ),
-\end{equation}
-where $K>0$ is the Lipschitz constant of $f$. By \eqref{sum-Di} and by
-the approximation result quoted in \secref{s:mt}, it is possible to find
-a sequence $(u_h)\subset C^1(\Omega ;\mathbf{R}^m)$ converging to $u$ in
-$L^1(\Omega ;\mathbf{R}^m)$ and such that
-\[\lim_{h\to +\infty}\int_\Omega \abs{\nabla u_h}\,dx=\abs{Du}(\Omega ).\]
-The functions $v_h=f(u_h)$ are locally Lipschitz continuous in $\Omega
-$, and the definition of differential implies that $\abs{\nabla v_h}\le
-K\abs{\nabla u_h}$ almost everywhere in $\Omega $. The lower semicontinuity
-of the total variation and \eqref{sum-Di} yield
-\begin{equation}
-\begin{split}
-\abs{Dv}(\Omega )\le\liminf_{h\to +\infty}\abs{Dv_h}(\Omega) &
-=\liminf_{h\to +\infty}\int_\Omega \abs{\nabla v_h}\,dx\\
-&\le K\liminf_{h\to +\infty}\int_\Omega
-\abs{\nabla u_h}\,dx=K\abs{Du}(\Omega).
-\end{split}\end{equation}
-Since $f(0)=0$, we have also
-\[\int_\Omega \abs{v}\,dx\le K\int_\Omega \abs{u}\,dx;\]
-therefore $u\in BV(\Omega ;\mathbf{R}^k)$. Repeating the same argument
-for every open set $A\subset\Omega $, we get \eqref{e:bomb} for every
-$B\in\mathbf{B}(\Omega)$, because $\abs{Dv}$, $\abs{Du}$ are Radon measures. To
-prove \lemref{limbog}, first we observe that
-\begin{equation}\label{e:SS}
-S_v\subset S_u,\qquad\tilde v(x)=f(\tilde u(x))\qquad \forall x\in\Omega
-\backslash S_u.\end{equation}
-In fact, for every $\varepsilon >0$ we have
-\[\{y\in B_\rho(x): \abs{v(y)-f(\tilde u(x))}>\varepsilon \}\subset \{y\in
-B_\rho(x): \abs{u(y)-\tilde u(x)}>\varepsilon /K\},\]
-hence
-\[\lim_{\rho\to 0^+}\frac{\abs{\{y\in B_\rho(x): \abs{v(y)-f(\tilde u(x))}>
-\varepsilon \}}}{\rho^n}=0\]
-whenever $x\in\Omega \backslash S_u$. By a similar argument, if $x\in
-S_u$ is a point such that there exists a triplet $(u^+,u^-,\nu_u)$
-satisfying \eqref{detK1}, \eqref{detK2}, then
-\[
-(v^+(x)-v^-(x))\otimes \nu_v=(f(u^+(x))-f(u^-(x)))\otimes\nu_u\quad
-\text{if }x\in S_v
-\]
-and $f(u^-(x))=f(u^+(x))$ if $x\in S_u\backslash S_v$. Hence, by (1.8)
-we get
-\begin{equation*}\begin{split}
-Jv(B)=\int_{B\cap S_v}(v^+-v^-)\otimes \nu_v\,d\mathcal{H}_{n-1}&=
-\int_{B\cap S_v}(f(u^+)-f(u^-))\otimes \nu_u\,d\mathcal{H}_{n-1}\\
-&=\int_{B\cap S_u}(f(u^+)-f(u^-))\otimes \nu_u\,d\mathcal{H}_{n-1}
-\end{split}\end{equation*}
-and \lemref{limbog} is proved.
-\end{proof}
-
-To prove \eqref{e:SS}, it is not restrictive to assume that $k=1$.
-Moreover, to simplify our notation, from now on we shall assume that
-$\Omega = \mathbf{R}^n$. The proof of \eqref{e:SS} is divided into two
-steps. In the first step we prove the statement in the one-dimensional
-case $(n=1)$, using \thmref{th-weak-ske-owf}. In the second step we
-achieve the general result using \thmref{t:conl}.
-
-\subsection*{Step 1}
-Assume that $n=1$. Since $S_u$ is at most countable, \eqref{sum-bij}
-yields that $\abs{\wt{D}v}(S_u\backslash S_v)=0$, so that
-\eqref{e:st} and \eqref{e:barwq} imply that $Dv=\wt{D}v+Jv$ is
-the Radon-Nikod\'ym decomposition of $Dv$ in absolutely continuous and
-singular part with respect to $\abs{\wt{D} u}$. By
-\thmref{th-weak-ske-owf}, we have
-\begin{equation*}
-\frac{\wt{D}v}{\abs{\wt{D}u}}(t)=\lim_{s\to t^+}
-\frac{Dv(\interval{\left[t,s\right[})}
-{\abs{\wt{D}u}(\interval{\left[t,s\right[})},\qquad
-\frac{\wt{D}u}{\abs{\wt{D}u}}(t)=\lim_{s\to t^+}
-\frac{Du(\interval{\left[t,s\right[})}
-{\abs{\wt{D}u}(\interval{\left[t,s\right[})}
-\end{equation*}
-$\abs{\wt{D}u}$-almost everywhere in $\mathbf{R}$. It is well known
-(see, for instance, \cite[2.5.16]{ste:sint}) that every one-dimensional
-function of bounded variation $w$ has a unique left continuous
-representative, i.e., a function $\hat w$ such that $\hat w=w$ almost
-everywhere and $\lim_{s\to t^-}\hat w(s)=\hat w(t)$ for every $t\in
-\mathbf{R}$. These conditions imply
-\begin{equation}
-\hat u(t)=Du(\interval{\left]-\infty,t\right[}),
-\qquad \hat v(t)=Dv(\interval{\left]-\infty,t\right[})\qquad
-\forall t\in\mathbf{R}
-\end{equation}
-and
-\begin{equation}\label{alimo}
-\hat v(t)=f(\hat u(t))\qquad\forall t\in\mathbf{R}.\end{equation}
-Let $t\in\mathbf{R}$ be such that
-$\abs{\wt{D}u}(\interval{\left[t,s\right[})>0$ for every $s>t$ and
-assume that the limits in \eqref{joe} exist. By \eqref{j:mark} and
-\eqref{far-d} we get
-\begin{equation*}\begin{split}
-\frac{\hat v(s)-\hat
-v(t)}{\abs{\wt{D}u}(\interval{\left[t,s\right[})}&=\frac {f(\hat
-u(s))-f(\hat u(t))}{\abs{\wt{D}u}(\interval{\left[t,s\right[})}\\
-&=\frac{f(\hat u(s))-f(\hat
-u(t)+\dfrac{\wt{D}u}{\abs{\wt{D}u}}(t)\abs{\wt{D}u
-}(\interval{\left[t,s\right[}))}%
-{\abs{\wt{D}u}(\interval{\left[t,s\right[})}\\
-&+\frac
-{f(\hat u(t)+\dfrac{\wt{D}u}{\abs{\wt{D}u}}(t)\abs{\wt{D}
-u}(\interval{\left[t,s\right[}))-f(\hat
-u(t))}{\abs{\wt{D}u}(\interval{\left[t,s\right[})}
-\end{split}\end{equation*}
-for every $s>t$. Using the Lipschitz condition on $f$ we find
-{\setlength{\multlinegap}{0pt}
-\begin{multline*}
-\left\lvert\frac{\hat v(s)-\hat
-v(t)}{\abs{\wt{D}u}(\interval{\left[t,s\right[})} -\frac{f(\hat
-u(t)+\dfrac{\wt{D}u}{\abs{\wt{D}u}}(t)
-\abs{\wt{D}u}(\interval{\left[t,s\right[}))-f(\hat
-u(t))}{\abs{\wt{D}u}(\interval{\left[t,s\right[})}\right\rvert\\
-\le K\left\lvert
-\frac{\hat u(s)-\hat u(t)}
- {\abs{\wt{D}u}(\interval{\left[t,s\right[})}
--\frac{\wt{D}u}{\abs{
-\wt{D}u}}(t)\right\rvert.\end{multline*}
-}% end of group with \multlinegap=0pt
-By \eqref{e:bomb}, the function $s\to
-\abs{\wt{D}u}(\interval{\left[t,s\right[})$ is continuous and
-converges to 0 as $s\downarrow t$. Therefore Remark~\ref{r:omb} and the
-previous inequality imply
-\[\frac{\wt{D}v}{\abs{\wt{D}u}}(t)=\lim_{h\to 0^+}
-\frac{f(\hat u(t)+h\dfrac{\wt{D}u}{\abs{\wt{D}u}}
-(t))-f(\hat u(t))}h\quad\abs{\wt{D}u}\text{-a.e. in }\mathbf{R}.\]
-By \eqref{joe}, $\hat u(x)=\tilde u(x)$ for every
-$x\in\mathbf{R}\backslash S_u$; moreover, applying the same argument to
-the functions $u'(t)=u(-t)$, $v'(t)=f(u'(t))=v(-t)$, we get
-\[\frac{\wt{D}v}{\abs{\wt{D}u}}(t)=\lim_{h\to 0}
-\frac{f(\tilde u(t)
-+h\dfrac{\wt{D}u}{\abs{\wt{D}u}}(t))-f(\tilde u(t))}{h}
-\qquad\abs{\wt{D}u}\text{-a.e. in }\mathbf{R}\]
-and our statement is proved.
-
-\subsection*{Step 2}
-
-Let us consider now the general case $n>1$. Let $\nu\in \mathbf{R}^n$ be
-such that $\abs{\nu}=1$, and let $\pi_\nu=\{y\in\mathbf{R}^n: \langle
-y,\nu\rangle =0\}$. In the following, we shall identify $\mathbf{R}^n$
-with $\pi_\nu\times\mathbf{R}$, and we shall denote by $y$ the variable
-ranging in $\pi_\nu$ and by $t$ the variable ranging in $\mathbf{R}$. By
-the just proven one-dimensional result, and by \thmref{thm-main}, we get
-\[\lim_{h\to 0}\frac{f(\tilde u(y+t\nu)+h\dfrac{\wt{D}u_y}{\abs{
-\wt{D}u_y}}(t))-f(\tilde u(y+t\nu))}h=\frac{\wt{D}v_y}{\abs{
-\wt{D}u_y}}(t)\qquad\abs{\wt{D}u_y}\text{-a.e. in }\mathbf{R}\]
-for $\mathcal{H}_{n-1}$-almost every $y\in \pi_\nu$. We claim that
-\begin{equation}
-\frac{\langle \wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle
-}}(y+t\nu)=\frac{\wt{D}u_y}
-{\abs{\wt{D}u_y}}(t)\qquad\abs{\wt{D}u_y}\text{-a.e. in }\mathbf{R}
-\end{equation}
-for $\mathcal{H}_{n-1}$-almost every $y\in\pi_\nu$. In fact, by
-\eqref{sum-ali} and \eqref{delta-l} we get
-\begin{multline*}
-\int_{\pi_\nu}\frac{\wt{D}u_y}{\abs{\wt{D}u_y}}\cdot\abs{\wt{D}u_y
-}\,d\mathcal{H}_{n-1}(y)=\int_{\pi_\nu}\wt{D}u_y\,d\mathcal{H}_{n-1}(y)\\
-=\langle \wt{D}u,\nu\rangle =\frac
-{\langle \wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle}}\cdot
-\abs{\langle \wt{D}u,\nu\rangle }=\int_{\pi_\nu}\frac{
-\langle \wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }}
-(y+\cdot \nu)\cdot\abs{\wt{D}u_y}\,d\mathcal{H}_{n-1}(y)
-\end{multline*}
-and \eqref{far-d} follows from \eqref{sum-Di}. By the same argument it
-is possible to prove that
-\begin{equation}
-\frac{\langle \wt{D}v,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle
-}}(y+t\nu)=\frac{\wt{D}v_y}{\abs{\wt{D}u_y}}(t)\qquad\abs{
-\wt{D}u_y}\text{-a.e. in }\mathbf{R}\end{equation}
-for $\mathcal{H}_{n-1}$-almost every $y\in \pi_\nu$. By \eqref{far-d}
-and \eqref{E_SXgYy} we get
-\[
-\lim_{h\to 0}\frac{f(\tilde u(y+t\nu)+h\dfrac{\langle \wt{D}
-u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }}(y+t\nu))-f(\tilde
-u(y+t\nu))}{h}
-=\frac{\langle \wt{D}v,\nu\rangle }{\abs{\langle
-\wt{D}u,\nu\rangle }}(y+t\nu)\]
-for $\mathcal{H}_{n-1}$-almost every $y\in\pi_\nu$, and using again
-\eqref{detK1}, \eqref{detK2} we get
-\[
-\lim_{h\to 0}\frac{f(\tilde u(x)+h\dfrac{\langle
-\wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }}(x))-f(\tilde
-u(x))}{h}=\frac{\langle \wt{D}v,\nu\rangle }{\abs{\langle \wt{D}u,\nu
-\rangle }}(x)
-\]
-$\abs{\langle \wt{D}u,\nu\rangle}$-a.e. in $\mathbf{R}^n$.
-
-Since the function $\abs{\langle \wt{D}u,\nu\rangle }/\abs{\wt{D}u}$
-is strictly positive $\abs{\langle \wt{D}u,\nu\rangle }$-almost everywhere,
-we obtain also
-\begin{multline*}
-\lim_{h\to 0}\frac{f(\tilde u(x)+h\dfrac{\abs{\langle
-\wt{D}u,\nu\rangle }}{\abs{\wt{D}u}}(x)\dfrac{\langle \wt{D}
-u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }}(x))-f(\tilde u(x))}{h}\\
-=\frac{\abs{\langle \wt{D}u,\nu\rangle }}{\abs{\wt{D}u}}(x)\frac
-{\langle \wt{D}v,\nu\rangle }{\abs{\langle
-\wt{D}u,\nu\rangle }}(x)
-\end{multline*}
-$\abs{\langle \wt{D}u,\nu\rangle }$-almost everywhere in $\mathbf{R}^n$.
-
-Finally, since
-\begin{align*}
-&\frac{\abs{\langle \wt{D}u,\nu\rangle }}{\abs{\wt{D}u}}
-\frac{\langle \wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle}}
-=\frac{\langle \wt{D}u,\nu\rangle }{\abs{\wt{D}u}}
-=\left\langle \frac{\wt{D}u}{\abs{\wt{D}u}},\nu\right\rangle
- \qquad\abs{\wt{D}u}\text{-a.e. in }\mathbf{R}^n\\
-&\frac{\abs{\langle \wt{D}u,\nu\rangle }}{\abs{\wt{D}u}}
-\frac{\langle \wt{D}v,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle}}
-=\frac{\langle \wt{D}v,\nu\rangle }{\abs{\wt{D}u}}
-=\left\langle \frac{\wt{D}v}{\abs{\wt{D}u}},\nu\right\rangle
- \qquad\abs{\wt{D}u}\text{-a.e. in }\mathbf{R}^n
-\end{align*}
-and since both sides of \eqref{alimo}
-are zero $\abs{\wt{D}u}$-almost everywhere
-on $\abs{\langle \wt{D}u,\nu\rangle }$-negligible sets, we conclude that
-\[
-\lim_{h\to 0}\frac{f\left(
-\tilde u(x)+h\left\langle \dfrac{\wt{D}
-u}{\abs{\wt{D}u}}(x),\nu\right\rangle \right)-f(\tilde u(x))}h
-=\left\langle \frac{\wt{D}v}{\abs{\wt{D}u}}(x),\nu\right\rangle,
-\]
-$\abs{\wt{D}u}$-a.e. in $\mathbf{R}^n$.
-Since $\nu$ is arbitrary, by Remarks \ref{r:dif} and~\ref{r:dif0}
-the restriction of $f$ to
-the affine space $T^u_x$ is differentiable at $\tilde u(x)$ for $\abs{\wt{D}
-u}$-almost every $x\in \mathbf{R}^n$ and \eqref{quts} holds.\qed
-
-It follows from \eqref{sum-Di}, \eqref{detK1}, and \eqref{detK2} that
-\begin{equation}\label{Dt}
-D(t_1,\dots,t_n)=\sum_{I\in\mathbf{n}}(-1)^{\abs{I}-1}\abs{I}
-\prod_{i\in I}t_i\prod_{j\in I}(D_j+\lambda_jt_j)\det\mathbf{A}^{(\lambda)}
-(\overline I|\overline I).
-\end{equation}
-Let $t_i=\hat x_i$, $i=1,\dots,n$. Lemma 1 leads to
-\begin{equation}\label{Dx}
-D(\hat x_1,\dots,\hat x_n)=\prod_{i\in\mathbf{n}}\hat x_i
-\sum_{I\in\mathbf{n}}(-1)^{\abs{I}-1}\abs{I}\per \mathbf{A}
-^{(\lambda)}(I|I)\det\mathbf{A}^{(\lambda)}(\overline I|\overline I).
-\end{equation}
-By \eqref{H-cycles}, \eqref{sum-Di}, and \eqref{Dx},
-we have the following result:
-\begin{thm}\label{thm-H-param}
-\begin{equation}\label{H-param}
-H_c=\frac{1}{2n}\sum^n_{l =1}l (-1)^{l -1}A_{l}
-^{(\lambda)},
-\end{equation}
-where
-\begin{equation}\label{A-l-lambda}
-A^{(\lambda)}_l =\sum_{I_l \subseteq\mathbf{n}}\per \mathbf{A}
-^{(\lambda)}(I_l |I_l )\det\mathbf{A}^{(\lambda)}
-(\overline I_{l}|\overline I_l ),\abs{I_{l}}=l .
-\end{equation}
-\end{thm}
-
-It is worth noting that $A_l ^{(\lambda)}$ of \eqref{A-l-lambda} is
-similar to the coefficients $b_l $ of the characteristic polynomial of
-\eqref{bl-sum}. It is well known in graph theory that the coefficients
-$b_l $ can be expressed as a sum over certain subgraphs. It is
-interesting to see whether $A_l $, $\lambda=0$, structural properties
-of a graph.
-
-We may call \eqref{H-param} a parametric representation of $H_c$. In
-computation, the parameter $\lambda_i$ plays very important roles. The
-choice of the parameter usually depends on the properties of the given
-graph. For a complete graph $K_n$, let $\lambda_i=1$, $i=1,\dots,n$.
-It follows from \eqref{A-l-lambda} that
-\begin{equation}\label{compl-gr}
-A^{(1)}_l =\begin{cases} n!,&\text{if }l =1\\
-0,&\text{otherwise}.\end{cases}
-\end{equation}
-By \eqref{H-param}
-\begin{equation}
-H_c=\frac 12(n-1)!.
-\end{equation}
-For a complete bipartite graph $K_{n_1n_2}$, let $\lambda_i=0$, $i=1,\dots,n$.
-By \eqref{A-l-lambda},
-\begin{equation}
-A_l =
-\begin{cases} -n_1!n_2!\delta_{n_1n_2},&\text{if }l =2\\
-0,&\text{otherwise }.\end{cases}
-\label{compl-bip-gr}
-\end{equation}
-Theorem ~\ref{thm-H-param}
-leads to
-\begin{equation}
-H_c=\frac1{n_1+n_2}n_1!n_2!\delta_{n_1n_2}.
-\end{equation}
-
-Now, we consider an asymmetrical approach. Theorem \ref{thm-main} leads to
-\begin{multline}
-\det\mathbf{K}(t=1,t_1,\dots,t_n;l |l )\\
-=\sum_{I\subseteq\mathbf{n}-\{l \}}
-(-1)^{\abs{I}}\prod_{i\in I}t_i\prod_{j\in I}
-(D_j+\lambda_jt_j)\det\mathbf{A}^{(\lambda)}
-(\overline I\cup\{l \}|\overline I\cup\{l \}).
-\end{multline}
-
-By \eqref{H-cycles} and \eqref{sum-ali} we have the following asymmetrical
-result:
-\begin{thm}\label{thm-asym}
-\begin{equation}
-H_c=\frac12\sum_{I\subseteq\mathbf{n}-\{l \}}
-(-1)^{\abs{I}}\per\mathbf{A}^{(\lambda)}(I|I)\det
-\mathbf{A}^{(\lambda)}
-(\overline I\cup\{l \}|\overline I\cup\{l \})
-\end{equation}
-which reduces to Goulden--Jackson's formula when $\lambda_i=0,i=1,\dots,n$
-\cite{mami:matrixth}.
-\end{thm}
-
-\section{Various font features of the \pkg{amsmath} package}
-\label{s:font}
-\subsection{Bold versions of special symbols}
-
-In the \pkg{amsmath} package \cn{boldsymbol} is used for getting
-individual bold math symbols and bold Greek letters---everything in
-math except for letters of the Latin alphabet,
-where you'd use \cn{mathbf}. For example,
-\begin{verbatim}
-A_\infty + \pi A_0 \sim
-\mathbf{A}_{\boldsymbol{\infty}} \boldsymbol{+}
-\boldsymbol{\pi} \mathbf{A}_{\boldsymbol{0}}
-\end{verbatim}
-looks like this:
-\[A_\infty + \pi A_0 \sim \mathbf{A}_{\boldsymbol{\infty}}
-\boldsymbol{+} \boldsymbol{\pi} \mathbf{A}_{\boldsymbol{0}}\]
-
-\subsection{``Poor man's bold''}
-If a bold version of a particular symbol doesn't exist in the
-available fonts,
-then \cn{boldsymbol} can't be used to make that symbol bold.
-At the present time, this means that
-\cn{boldsymbol} can't be used with symbols from
-the \fn{msam} and \fn{msbm} fonts, among others.
-In some cases, poor man's bold (\cn{pmb}) can be used instead
-of \cn{boldsymbol}:
-% Can't show example from msam or msbm because this document is
-% supposed to be TeXable even if the user doesn't have
-% AMSFonts. MJD 5-JUL-1990
-\[\frac{\partial x}{\partial y}
-\pmb{\bigg\vert}
-\frac{\partial y}{\partial z}\]
-\begin{verbatim}
-\[\frac{\partial x}{\partial y}
-\pmb{\bigg\vert}
-\frac{\partial y}{\partial z}\]
-\end{verbatim}
-So-called ``large operator'' symbols such as $\sum$ and $\prod$
-require an additional command, \cn{mathop},
-to produce proper spacing and limits when \cn{pmb} is used.
-For further details see \textit{The \TeX book}.
-\[\sum_{\substack{i<B\\\text{$i$ odd}}}
-\prod_\kappa \kappa F(r_i)\qquad
-\mathop{\pmb{\sum}}_{\substack{i<B\\\text{$i$ odd}}}
-\mathop{\pmb{\prod}}_\kappa \kappa(r_i)
-\]
-\begin{verbatim}
-\[\sum_{\substack{i<B\\\text{$i$ odd}}}
-\prod_\kappa \kappa F(r_i)\qquad
-\mathop{\pmb{\sum}}_{\substack{i<B\\\text{$i$ odd}}}
-\mathop{\pmb{\prod}}_\kappa \kappa(r_i)
-\]
-\end{verbatim}
-
-\section{Compound symbols and other features}
-\label{s:comp}
-\subsection{Multiple integral signs}
-
-\cn{iint}, \cn{iiint}, and \cn{iiiint} give multiple integral signs
-with the spacing between them nicely adjusted, in both text and
-display style. \cn{idotsint} gives two integral signs with dots
-between them.
-\begin{gather}
-\iint\limits_A f(x,y)\,dx\,dy\qquad\iiint\limits_A
-f(x,y,z)\,dx\,dy\,dz\\
-\iiiint\limits_A
-f(w,x,y,z)\,dw\,dx\,dy\,dz\qquad\idotsint\limits_A f(x_1,\dots,x_k)
-\end{gather}
-
-\subsection{Over and under arrows}
-
-Some extra over and under arrow operations are provided in
-the \pkg{amsmath} package. (Basic \LaTeX\ provides
-\cn{overrightarrow} and \cn{overleftarrow}).
-\begin{align*}
-\overrightarrow{\psi_\delta(t) E_t h}&
-=\underrightarrow{\psi_\delta(t) E_t h}\\
-\overleftarrow{\psi_\delta(t) E_t h}&
-=\underleftarrow{\psi_\delta(t) E_t h}\\
-\overleftrightarrow{\psi_\delta(t) E_t h}&
-=\underleftrightarrow{\psi_\delta(t) E_t h}
-\end{align*}
-\begin{verbatim}
-\begin{align*}
-\overrightarrow{\psi_\delta(t) E_t h}&
-=\underrightarrow{\psi_\delta(t) E_t h}\\
-\overleftarrow{\psi_\delta(t) E_t h}&
-=\underleftarrow{\psi_\delta(t) E_t h}\\
-\overleftrightarrow{\psi_\delta(t) E_t h}&
-=\underleftrightarrow{\psi_\delta(t) E_t h}
-\end{align*}
-\end{verbatim}
-These all scale properly in subscript sizes:
-\[\int_{\overrightarrow{AB}} ax\,dx\]
-\begin{verbatim}
-\[\int_{\overrightarrow{AB}} ax\,dx\]
-\end{verbatim}
-
-\subsection{Dots}
-
-Normally you need only type \cn{dots} for ellipsis dots in a
-math formula. The main exception is when the dots
-fall at the end of the formula; then you need to
-specify one of \cn{dotsc} (series dots, after a comma),
-\cn{dotsb} (binary dots, for binary relations or operators),
-\cn{dotsm} (multiplication dots), or \cn{dotsi} (dots after
-an integral). For example, the input
-\begin{verbatim}
-Then we have the series $A_1,A_2,\dotsc$,
-the regional sum $A_1+A_2+\dotsb$,
-the orthogonal product $A_1A_2\dotsm$,
-and the infinite integral
-\[\int_{A_1}\int_{A_2}\dotsi\].
-\end{verbatim}
-produces
-\begin{quotation}
-Then we have the series $A_1,A_2,\dotsc$,
-the regional sum $A_1+A_2+\dotsb$,
-the orthogonal product $A_1A_2\dotsm$,
-and the infinite integral
-\[\int_{A_1}\int_{A_2}\dotsi\]
-\end{quotation}
-
-\subsection{Accents in math}
-
-Double accents:
-\[\Hat{\Hat{H}}\quad\Check{\Check{C}}\quad
-\Tilde{\Tilde{T}}\quad\Acute{\Acute{A}}\quad
-\Grave{\Grave{G}}\quad\Dot{\Dot{D}}\quad
-\Ddot{\Ddot{D}}\quad\Breve{\Breve{B}}\quad
-\Bar{\Bar{B}}\quad\Vec{\Vec{V}}\]
-\begin{verbatim}
-\[\Hat{\Hat{H}}\quad\Check{\Check{C}}\quad
-\Tilde{\Tilde{T}}\quad\Acute{\Acute{A}}\quad
-\Grave{\Grave{G}}\quad\Dot{\Dot{D}}\quad
-\Ddot{\Ddot{D}}\quad\Breve{\Breve{B}}\quad
-\Bar{\Bar{B}}\quad\Vec{\Vec{V}}\]
-\end{verbatim}
-This double accent operation is complicated
-and tends to slow down the processing of a \LaTeX\ file.
-
-
-\subsection{Dot accents}
-\cn{dddot} and \cn{ddddot} are available to
-produce triple and quadruple dot accents
-in addition to the \cn{dot} and \cn{ddot} accents already available
-in \LaTeX:
-\[\dddot{Q}\qquad\ddddot{R}\]
-\begin{verbatim}
-\[\dddot{Q}\qquad\ddddot{R}\]
-\end{verbatim}
-
-\subsection{Roots}
-
-In the \pkg{amsmath} package \cn{leftroot} and \cn{uproot} allow you to adjust
-the position of the root index of a radical:
-\begin{verbatim}
-\sqrt[\leftroot{-2}\uproot{2}\beta]{k}
-\end{verbatim}
-gives good positioning of the $\beta$:
-\[\sqrt[\leftroot{-2}\uproot{2}\beta]{k}\]
-
-\subsection{Boxed formulas} The command \cn{boxed} puts a box around its
-argument, like \cn{fbox} except that the contents are in math mode:
-\begin{verbatim}
-\boxed{W_t-F\subseteq V(P_i)\subseteq W_t}
-\end{verbatim}
-\[\boxed{W_t-F\subseteq V(P_i)\subseteq W_t}.\]
-
-\subsection{Extensible arrows}
-\cn{xleftarrow} and \cn{xrightarrow} produce
-arrows that extend automatically to accommodate unusually wide
-subscripts or superscripts. The text of the subscript or superscript
-are given as an optional resp.\@ mandatory argument:
-Example:
-\[0 \xleftarrow[\zeta]{\alpha} F\times\triangle[n-1]
- \xrightarrow{\partial_0\alpha(b)} E^{\partial_0b}\]
-\begin{verbatim}
-\[0 \xleftarrow[\zeta]{\alpha} F\times\triangle[n-1]
- \xrightarrow{\partial_0\alpha(b)} E^{\partial_0b}\]
-\end{verbatim}
-
-\subsection{\cn{overset}, \cn{underset}, and \cn{sideset}}
-Examples:
-\[\overset{*}{X}\qquad\underset{*}{X}\qquad
-\overset{a}{\underset{b}{X}}\]
-\begin{verbatim}
-\[\overset{*}{X}\qquad\underset{*}{X}\qquad
-\overset{a}{\underset{b}{X}}\]
-\end{verbatim}
-
-The command \cn{sideset} is for a rather special
-purpose: putting symbols at the subscript and superscript
-corners of a large operator symbol such as $\sum$ or $\prod$,
-without affecting the placement of limits.
-Examples:
-\[\sideset{_*^*}{_*^*}\prod_k\qquad
-\sideset{}{'}\sum_{0\le i\le m} E_i\beta x
-\]
-\begin{verbatim}
-\[\sideset{_*^*}{_*^*}\prod_k\qquad
-\sideset{}{'}\sum_{0\le i\le m} E_i\beta x
-\]
-\end{verbatim}
-
-\subsection{The \cn{text} command}
-The main use of the command \cn{text} is for words or phrases in a
-display:
-\[\mathbf{y}=\mathbf{y}'\quad\text{if and only if}\quad
-y'_k=\delta_k y_{\tau(k)}\]
-\begin{verbatim}
-\[\mathbf{y}=\mathbf{y}'\quad\text{if and only if}\quad
-y'_k=\delta_k y_{\tau(k)}\]
-\end{verbatim}
-
-\subsection{Operator names}
-The more common math functions such as $\log$, $\sin$, and $\lim$
-have predefined control sequences: \verb=\log=, \verb=\sin=,
-\verb=\lim=.
-The \pkg{amsmath} package provides \cn{DeclareMathOperator} and
-\cn{DeclareMathOperator*}
-for producing new function names that will have the
-same typographical treatment.
-Examples:
-\[\norm{f}_\infty=
-\esssup_{x\in R^n}\abs{f(x)}\]
-\begin{verbatim}
-\[\norm{f}_\infty=
-\esssup_{x\in R^n}\abs{f(x)}\]
-\end{verbatim}
-\[\meas_1\{u\in R_+^1\colon f^*(u)>\alpha\}
-=\meas_n\{x\in R^n\colon \abs{f(x)}\geq\alpha\}
-\quad \forall\alpha>0.\]
-\begin{verbatim}
-\[\meas_1\{u\in R_+^1\colon f^*(u)>\alpha\}
-=\meas_n\{x\in R^n\colon \abs{f(x)}\geq\alpha\}
-\quad \forall\alpha>0.\]
-\end{verbatim}
-\cn{esssup} and \cn{meas} would be defined in the document preamble as
-\begin{verbatim}
-\DeclareMathOperator*{\esssup}{ess\,sup}
-\DeclareMathOperator{\meas}{meas}
-\end{verbatim}
-
-The following special operator names are predefined in the \pkg{amsmath}
-package: \cn{varlimsup}, \cn{varliminf}, \cn{varinjlim}, and
-\cn{varprojlim}. Here's what they look like in use:
-\begin{align}
-&\varlimsup_{n\rightarrow\infty}
- \mathcal{Q}(u_n,u_n-u^{\#})\le0\\
-&\varliminf_{n\rightarrow\infty}
- \left\lvert a_{n+1}\right\rvert/\left\lvert a_n\right\rvert=0\\
-&\varinjlim (m_i^\lambda\cdot)^*\le0\\
-&\varprojlim_{p\in S(A)}A_p\le0
-\end{align}
-\begin{verbatim}
-\begin{align}
-&\varlimsup_{n\rightarrow\infty}
- \mathcal{Q}(u_n,u_n-u^{\#})\le0\\
-&\varliminf_{n\rightarrow\infty}
- \left\lvert a_{n+1}\right\rvert/\left\lvert a_n\right\rvert=0\\
-&\varinjlim (m_i^\lambda\cdot)^*\le0\\
-&\varprojlim_{p\in S(A)}A_p\le0
-\end{align}
-\end{verbatim}
-
-\subsection{\cn{mod} and its relatives}
-The commands \cn{mod} and \cn{pod} are variants of
-\cn{pmod} preferred by some authors; \cn{mod} omits the parentheses,
-whereas \cn{pod} omits the `mod' and retains the parentheses.
-Examples:
-\begin{align}
-x&\equiv y+1\pmod{m^2}\\
-x&\equiv y+1\mod{m^2}\\
-x&\equiv y+1\pod{m^2}
-\end{align}
-\begin{verbatim}
-\begin{align}
-x&\equiv y+1\pmod{m^2}\\
-x&\equiv y+1\mod{m^2}\\
-x&\equiv y+1\pod{m^2}
-\end{align}
-\end{verbatim}
-
-\subsection{Fractions and related constructions}
-\label{fracs}
-
-The usual notation for binomials is similar to the fraction concept,
-so it has a similar command \cn{binom} with two arguments. Example:
-\begin{equation}
-\begin{split}
-\sum_{\gamma\in\Gamma_C} I_\gamma&
-=2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}\\
-&\quad+\dots+(-1)^l\binom{k}{l}2^{k-l}
-+\dots+(-1)^k\\
-&=(2-1)^k=1
-\end{split}
-\end{equation}
-\begin{verbatim}
-\begin{equation}
-\begin{split}
-[\sum_{\gamma\in\Gamma_C} I_\gamma&
-=2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}\\
-&\quad+\dots+(-1)^l\binom{k}{l}2^{k-l}
-+\dots+(-1)^k\\
-&=(2-1)^k=1
-\end{split}
-\end{equation}
-\end{verbatim}
-There are also abbreviations
-\begin{verbatim}
-\dfrac \dbinom
-\tfrac \tbinom
-\end{verbatim}
-for the commonly needed constructions
-\begin{verbatim}
-{\displaystyle\frac ... } {\displaystyle\binom ... }
-{\textstyle\frac ... } {\textstyle\binom ... }
-\end{verbatim}
-
-The generalized fraction command \cn{genfrac} provides full access to
-the six \TeX{} fraction primitives:
-\begin{align}
-\text{\cn{over}: }&\genfrac{}{}{}{}{n+1}{2}&
-\text{\cn{overwithdelims}: }&
- \genfrac{\langle}{\rangle}{}{}{n+1}{2}\\
-\text{\cn{atop}: }&\genfrac{}{}{0pt}{}{n+1}{2}&
-\text{\cn{atopwithdelims}: }&
- \genfrac{(}{)}{0pt}{}{n+1}{2}\\
-\text{\cn{above}: }&\genfrac{}{}{1pt}{}{n+1}{2}&
-\text{\cn{abovewithdelims}: }&
- \genfrac{[}{]}{1pt}{}{n+1}{2}
-\end{align}
-\begin{verbatim}
-\text{\cn{over}: }&\genfrac{}{}{}{}{n+1}{2}&
-\text{\cn{overwithdelims}: }&
- \genfrac{\langle}{\rangle}{}{}{n+1}{2}\\
-\text{\cn{atop}: }&\genfrac{}{}{0pt}{}{n+1}{2}&
-\text{\cn{atopwithdelims}: }&
- \genfrac{(}{)}{0pt}{}{n+1}{2}\\
-\text{\cn{above}: }&\genfrac{}{}{1pt}{}{n+1}{2}&
-\text{\cn{abovewithdelims}: }&
- \genfrac{[}{]}{1pt}{}{n+1}{2}
-\end{verbatim}
-
-\subsection{Continued fractions}
-The continued fraction
-\begin{equation}
-\cfrac{1}{\sqrt{2}+
- \cfrac{1}{\sqrt{2}+
- \cfrac{1}{\sqrt{2}+
- \cfrac{1}{\sqrt{2}+
- \cfrac{1}{\sqrt{2}+\dotsb
-}}}}}
-\end{equation}
-can be obtained by typing
-\begin{verbatim}
-\cfrac{1}{\sqrt{2}+
- \cfrac{1}{\sqrt{2}+
- \cfrac{1}{\sqrt{2}+
- \cfrac{1}{\sqrt{2}+
- \cfrac{1}{\sqrt{2}+\dotsb
-}}}}}
-\end{verbatim}
-Left or right placement of any of the numerators is accomplished by using
-\cn{cfrac[l]} or \cn{cfrac[r]} instead of \cn{cfrac}.
-
-\subsection{Smash}
-
-In \pkg{amsmath} there are optional arguments \verb"t" and \verb"b" for
-the plain \TeX\ command \cn{smash}, because sometimes it is advantageous
-to be able to `smash' only the top or only the bottom of something while
-retaining the natural depth or height. In the formula
-$X_j=(1/\sqrt{\smash[b]{\lambda_j}})X_j'$ \cn{smash}\verb=[b]= has been
-used to limit the size of the radical symbol.
-\begin{verbatim}
-$X_j=(1/\sqrt{\smash[b]{\lambda_j}})X_j'$
-\end{verbatim}
-Without the use of \cn{smash}\verb=[b]= the formula would have appeared
-thus: $X_j=(1/\sqrt{\lambda_j})X_j'$, with the radical extending to
-encompass the depth of the subscript $j$.
-
-\subsection{The `cases' environment}
-`Cases' constructions like the following can be produced using
-the \env{cases} environment.
-\begin{equation}
-P_{r-j}=
- \begin{cases}
- 0& \text{if $r-j$ is odd},\\
- r!\,(-1)^{(r-j)/2}& \text{if $r-j$ is even}.
- \end{cases}
-\end{equation}
-\begin{verbatim}
-\begin{equation} P_{r-j}=
- \begin{cases}
- 0& \text{if $r-j$ is odd},\\
- r!\,(-1)^{(r-j)/2}& \text{if $r-j$ is even}.
- \end{cases}
-\end{equation}
-\end{verbatim}
-Notice the use of \cn{text} and the embedded math.
-
-\subsection{Matrix}
-
-Here are samples of the matrix environments,
-\cn{matrix}, \cn{pmatrix}, \cn{bmatrix}, \cn{Bmatrix}, \cn{vmatrix}
-and \cn{Vmatrix}:
-\begin{equation}
-\begin{matrix}
-\vartheta& \varrho\\\varphi& \varpi
-\end{matrix}\quad
-\begin{pmatrix}
-\vartheta& \varrho\\\varphi& \varpi
-\end{pmatrix}\quad
-\begin{bmatrix}
-\vartheta& \varrho\\\varphi& \varpi
-\end{bmatrix}\quad
-\begin{Bmatrix}
-\vartheta& \varrho\\\varphi& \varpi
-\end{Bmatrix}\quad
-\begin{vmatrix}
-\vartheta& \varrho\\\varphi& \varpi
-\end{vmatrix}\quad
-\begin{Vmatrix}
-\vartheta& \varrho\\\varphi& \varpi
-\end{Vmatrix}
-\end{equation}
-%
-\begin{verbatim}
-\begin{matrix}
-\vartheta& \varrho\\\varphi& \varpi
-\end{matrix}\quad
-\begin{pmatrix}
-\vartheta& \varrho\\\varphi& \varpi
-\end{pmatrix}\quad
-\begin{bmatrix}
-\vartheta& \varrho\\\varphi& \varpi
-\end{bmatrix}\quad
-\begin{Bmatrix}
-\vartheta& \varrho\\\varphi& \varpi
-\end{Bmatrix}\quad
-\begin{vmatrix}
-\vartheta& \varrho\\\varphi& \varpi
-\end{vmatrix}\quad
-\begin{Vmatrix}
-\vartheta& \varrho\\\varphi& \varpi
-\end{Vmatrix}
-\end{verbatim}
-
-To produce a small matrix suitable for use in text, use the
-\env{smallmatrix} environment.
-\begin{verbatim}
-\begin{math}
- \bigl( \begin{smallmatrix}
- a&b\\ c&d
- \end{smallmatrix} \bigr)
-\end{math}
-\end{verbatim}
-To show
-the effect of the matrix on the surrounding lines of
-a paragraph, we put it here: \begin{math}
- \bigl( \begin{smallmatrix}
- a&b\\ c&d
- \end{smallmatrix} \bigr)
-\end{math}
-and follow it with enough text to ensure that there will
-be at least one full line below the matrix.
-
-\cn{hdotsfor}\verb"{"\textit{number}\verb"}" produces a row of dots in a matrix
-spanning the given number of columns:
-\[W(\Phi)= \begin{Vmatrix}
-\dfrac\varphi{(\varphi_1,\varepsilon_1)}&0&\dots&0\\
-\dfrac{\varphi k_{n2}}{(\varphi_2,\varepsilon_1)}&
-\dfrac\varphi{(\varphi_2,\varepsilon_2)}&\dots&0\\
-\hdotsfor{5}\\
-\dfrac{\varphi k_{n1}}{(\varphi_n,\varepsilon_1)}&
-\dfrac{\varphi k_{n2}}{(\varphi_n,\varepsilon_2)}&\dots&
-\dfrac{\varphi k_{n\,n-1}}{(\varphi_n,\varepsilon_{n-1})}&
-\dfrac{\varphi}{(\varphi_n,\varepsilon_n)}
-\end{Vmatrix}\]
-\begin{verbatim}
-\[W(\Phi)= \begin{Vmatrix}
-\dfrac\varphi{(\varphi_1,\varepsilon_1)}&0&\dots&0\\
-\dfrac{\varphi k_{n2}}{(\varphi_2,\varepsilon_1)}&
-\dfrac\varphi{(\varphi_2,\varepsilon_2)}&\dots&0\\
-\hdotsfor{5}\\
-\dfrac{\varphi k_{n1}}{(\varphi_n,\varepsilon_1)}&
-\dfrac{\varphi k_{n2}}{(\varphi_n,\varepsilon_2)}&\dots&
-\dfrac{\varphi k_{n\,n-1}}{(\varphi_n,\varepsilon_{n-1})}&
-\dfrac{\varphi}{(\varphi_n,\varepsilon_n)}
-\end{Vmatrix}\]
-\end{verbatim}
-The spacing of the dots can be varied through use of a square-bracket
-option, for example, \verb"\hdotsfor[1.5]{3}". The number in square brackets
-will be used as a multiplier; the normal value is 1.
-
-\subsection{The \cn{substack} command}
-
-The \cn{substack} command can be used to produce a multiline
-subscript or superscript:
-for example
-\begin{verbatim}
-\sum_{\substack{0\le i\le m\\ 0<j<n}} P(i,j)
-\end{verbatim}
-produces a two-line subscript underneath the sum:
-\begin{equation}
-\sum_{\substack{0\le i\le m\\ 0<j<n}} P(i,j)
-\end{equation}
-A slightly more generalized form is the \env{subarray} environment which
-allows you to specify that each line should be left-aligned instead of
-centered, as here:
-\begin{equation}
-\sum_{\begin{subarray}{l}
- 0\le i\le m\\ 0<j<n
- \end{subarray}}
- P(i,j)
-\end{equation}
-\begin{verbatim}
-\sum_{\begin{subarray}{l}
- 0\le i\le m\\ 0<j<n
- \end{subarray}}
- P(i,j)
-\end{verbatim}
-
-
-\subsection{Big-g-g delimiters}
-Here are some big delimiters, first in \cn{normalsize}:
-\[\biggl(\mathbf{E}_{y}
- \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
- \biggr)
-\]
-\begin{verbatim}
-\[\biggl(\mathbf{E}_{y}
- \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
- \biggr)
-\]
-\end{verbatim}
-and now in \cn{Large} size:
-{\Large
-\[\biggl(\mathbf{E}_{y}
- \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
- \biggr)
-\]}
-\begin{verbatim}
-{\Large
-\[\biggl(\mathbf{E}_{y}
- \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
- \biggr)
-\]}
-\end{verbatim}
-
-\newpage
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\makeatletter
-
-%% This turns on vertical rules at the right and left margins, to
-%% better illustrate the spacing for certain multiple-line equation
-%% structures.
-\def\@makecol{\ifvoid\footins \setbox\@outputbox\box\@cclv
- \else\setbox\@outputbox
- \vbox{\boxmaxdepth \maxdepth
- \unvbox\@cclv\vskip\skip\footins\footnoterule\unvbox\footins}\fi
- \xdef\@freelist{\@freelist\@midlist}\gdef\@midlist{}\@combinefloats
- \setbox\@outputbox\hbox{\vrule width\marginrulewidth
- \vbox to\@colht{\boxmaxdepth\maxdepth
- \@texttop\dimen128=\dp\@outputbox\unvbox\@outputbox
- \vskip-\dimen128\@textbottom}%
- \vrule width\marginrulewidth}%
- \global\maxdepth\@maxdepth}
-\newdimen\marginrulewidth
-\setlength{\marginrulewidth}{.1pt}
-\makeatother
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\appendix
-\section{Examples of multiple-line equation structures}
-\label{s:eq}
-
-\textbf{\large Note: Starting on this page, vertical rules are
-added at the margins so that the positioning of various display elements
-with respect to the margins can be seen more clearly.}
-
-\subsection{Split}
-The \env{split} environment is not an independent environment
-but should be used inside something else such as \env{equation}
-or \env{align}.
-
-If there is not enough room for it, the equation number for a
-\env{split} will be shifted to the previous line, when equation numbers are
-on the left; the number shifts down to the next line when numbers are on
-the right.
-\begin{equation}
-\begin{split}
-f_{h,\varepsilon}(x,y)
-&=\varepsilon\mathbf{E}_{x,y}\int_0^{t_\varepsilon}
-L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\
-&= h\int L_{x,z}\varphi(x)\rho_x(dz)\\
-&\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\mathbf{E}_{y}
- \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
- -t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\
-&\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon}
- \biggl(\mathbf{E}_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)}
- \varphi(x)\,ds -\mathbf{E}_{x,y}\int_0^{t_\varepsilon}
- L_{x,y_\varepsilon(\varepsilon s)}
- \varphi(x)\,ds\biggr)\biggr]\\
-&=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y),
-\end{split}
-\end{equation}
-Some text after to test the below-display spacing.
-
-\begin{verbatim}
-\begin{equation}
-\begin{split}
-f_{h,\varepsilon}(x,y)
-&=\varepsilon\mathbf{E}_{x,y}\int_0^{t_\varepsilon}
-L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\
-&= h\int L_{x,z}\varphi(x)\rho_x(dz)\\
-&\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\mathbf{E}_{y}
- \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
- -t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\
-&\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon}
- \biggl(\mathbf{E}_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)}
- \varphi(x)\,ds -\mathbf{E}_{x,y}\int_0^{t_\varepsilon}
- L_{x,y_\varepsilon(\varepsilon s)}
- \varphi(x)\,ds\biggr)\biggr]\\
-&=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y),
-\end{split}
-\end{equation}
-\end{verbatim}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\newpage
-Unnumbered version:
-\begin{equation*}
-\begin{split}
-f_{h,\varepsilon}(x,y)
-&=\varepsilon\mathbf{E}_{x,y}\int_0^{t_\varepsilon}
-L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\
-&= h\int L_{x,z}\varphi(x)\rho_x(dz)\\
-&\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\mathbf{E}_{y}
- \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
- -t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\
-&\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon}
- \biggl(\mathbf{E}_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)}
- \varphi(x)\,ds -\mathbf{E}_{x,y}\int_0^{t_\varepsilon}
- L_{x,y_\varepsilon(\varepsilon s)}
- \varphi(x)\,ds\biggr)\biggr]\\
-&=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y),
-\end{split}
-\end{equation*}
-Some text after to test the below-display spacing.
-
-\begin{verbatim}
-\begin{equation*}
-\begin{split}
-f_{h,\varepsilon}(x,y)
-&=\varepsilon\mathbf{E}_{x,y}\int_0^{t_\varepsilon}
-L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\
-&= h\int L_{x,z}\varphi(x)\rho_x(dz)\\
-&\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\mathbf{E}_{y}
- \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds
- -t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\
-&\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon}
- \biggl(\mathbf{E}_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)}
- \varphi(x)\,ds -\mathbf{E}_{x,y}\int_0^{t_\varepsilon}
- L_{x,y_\varepsilon(\varepsilon s)}
- \varphi(x)\,ds\biggr)\biggr]\\
-&=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y),
-\end{split}
-\end{equation*}
-\end{verbatim}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\newpage
-If the option \env{centertags} is included in the options
-list of the \pkg{amsmath} package,
-the equation numbers for \env{split} environments will be
-centered vertically on the height
-of the \env{split}:
-{\makeatletter\ctagsplit@true
-\begin{equation}
-\begin{split}
- \abs{I_2}&=\left\lvert \int_{0}^T \psi(t)\left\{u(a,t)-\int_{\gamma(t)}^a
- \frac{d\theta}{k(\theta,t)}
- \int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\
-&\le C_6\left\lvert \left\lvert f\int_\Omega\left\lvert \wt{S}^{-1,0}_{a,-}
- W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
- \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
- (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
-\end{split}
-\end{equation}}%
-Some text after to test the below-display spacing.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\newpage
-Use of \env{split} within \env{align}:
-{\delimiterfactor750
-\begin{align}
-\begin{split}\abs{I_1}
- &=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\
-&\le C_3\left[\int_\Omega\left(\int_{a}^x
- g(\xi,t)\,d\xi\right)^2d\Omega\right]^{1/2}\\
-&\quad\times \left[\int_\Omega\left\{u^2_x+\frac{1}{k}
- \left(\int_{a}^x cu_t\,d\xi\right)^2\right\}
- c\Omega\right]^{1/2}\\
-&\le C_4\left\lvert \left\lvert f\left\lvert \wt{S}^{-1,0}_{a,-}
- W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
- \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
- (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
-\end{split}\label{eq:A}\\
-\begin{split}\abs{I_2}&=\left\lvert \int_{0}^T \psi(t)\left\{u(a,t)
- -\int_{\gamma(t)}^a\frac{d\theta}{k(\theta,t)}
- \int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\
-&\le C_6\left\lvert \left\lvert f\int_\Omega
- \left\lvert \wt{S}^{-1,0}_{a,-}
- W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
- \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
- (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
-\end{split}
-\end{align}}%
-Some text after to test the below-display spacing.
-
-\begin{verbatim}
-\begin{align}
-\begin{split}\abs{I_1}
- &=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\
-&\le C_3\left[\int_\Omega\left(\int_{a}^x
- g(\xi,t)\,d\xi\right)^2d\Omega\right]^{1/2}\\
-&\quad\times \left[\int_\Omega\left\{u^2_x+\frac{1}{k}
- \left(\int_{a}^x cu_t\,d\xi\right)^2\right\}
- c\Omega\right]^{1/2}\\
-&\le C_4\left\lvert \left\lvert f\left\lvert \wt{S}^{-1,0}_{a,-}
- W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
- \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
- (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
-\end{split}\label{eq:A}\\
-\begin{split}\abs{I_2}&=\left\lvert \int_{0}^T \psi(t)\left\{u(a,t)
- -\int_{\gamma(t)}^a\frac{d\theta}{k(\theta,t)}
- \int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\
-&\le C_6\left\lvert \left\lvert f\int_\Omega
- \left\lvert \wt{S}^{-1,0}_{a,-}
- W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
- \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
- (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
-\end{split}
-\end{align}
-\end{verbatim}
-
-%%%%%%%%%%%%%%%%%%
-
-\newpage
-Unnumbered \env{align}, with a number on the second \env{split}:
-\begin{align*}
-\begin{split}\abs{I_1}&=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\
- &\le C_3\left[\int_\Omega\left(\int_{a}^x
- g(\xi,t)\,d\xi\right)^2d\Omega\right]^{1/2}\\
-&\phantom{=}\times \left[\int_\Omega\left\{u^2_x+\frac{1}{k}
- \left(\int_{a}^x cu_t\,d\xi\right)^2\right\}
- c\Omega\right]^{1/2}\\
-&\le C_4\left\lvert \left\lvert f\left\lvert \wt{S}^{-1,0}_{a,-}
- W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
- \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
- (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
-\end{split}\\
-\begin{split}\abs{I_2}&=\left\lvert \int_{0}^T \psi(t)\left\{u(a,t)
- -\int_{\gamma(t)}^a\frac{d\theta}{k(\theta,t)}
- \int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\
-&\le C_6\left\lvert \left\lvert f\int_\Omega
- \left\lvert \wt{S}^{-1,0}_{a,-}
- W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
- \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
- (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
-\end{split}\tag{\theequation$'$}
-\end{align*}
-Some text after to test the below-display spacing.
-
-\begin{verbatim}
-\begin{align*}
-\begin{split}\abs{I_1}&=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\
- &\le C_3\left[\int_\Omega\left(\int_{a}^x
- g(\xi,t)\,d\xi\right)^2d\Omega\right]^{1/2}\\
-&\phantom{=}\times \left[\int_\Omega\left\{u^2_x+\frac{1}{k}
- \left(\int_{a}^x cu_t\,d\xi\right)^2\right\}
- c\Omega\right]^{1/2}\\
-&\le C_4\left\lvert \left\lvert f\left\lvert \wt{S}^{-1,0}_{a,-}
- W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
- \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
- (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
-\end{split}\\
-\begin{split}\abs{I_2}&=\left\lvert \int_{0}^T \psi(t)\left\{u(a,t)
- -\int_{\gamma(t)}^a\frac{d\theta}{k(\theta,t)}
- \int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\
-&\le C_6\left\lvert \left\lvert f\int_\Omega
- \left\lvert \wt{S}^{-1,0}_{a,-}
- W_2(\Omega,\Gamma_l)\right\rvert\right\rvert
- \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}}
- (\Omega;\Gamma_r,T)\right\rvert\right\rvert.
-\end{split}\tag{\theequation$'$}
-\end{align*}
-\end{verbatim}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\newpage
-\subsection{Multline}
-Numbered version:
-\begin{multline}\label{eq:E}
-\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2]
- -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\
- =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2
- \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy
-\end{multline}
-To test the use of \verb=\label= and
-\verb=\ref=, we refer to the number of this
-equation here: (\ref{eq:E}).
-
-\begin{verbatim}
-\begin{multline}\label{eq:E}
-\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2]
- -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\
- =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2
- \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy
-\end{multline}
-\end{verbatim}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-Unnumbered version:
-\begin{multline*}
-\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2]
- -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\
- =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2
- \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy
-\end{multline*}
-Some text after to test the below-display spacing.
-
-\begin{verbatim}
-\begin{multline*}
-\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2]
- -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\
- =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2
- \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy
-\end{multline*}
-\end{verbatim}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\iffalse % bugfix needed, error message "Multiple \tag"
- % [mjd,24-Jan-1995]
-\newpage
-And now an ``unnumbered'' version numbered with a literal tag:
-\begin{multline*}\tag*{[a]}
-\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2]
- -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\
- =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2
- \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy
-\end{multline*}
-Some text after to test the below-display spacing.
-
-\begin{verbatim}
-\begin{multline*}\tag*{[a]}
-\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2]
- -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\
- =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2
- \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy
-\end{multline*}
-\end{verbatim}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-The same display with \verb=\multlinegap= set to zero.
-Notice that the space on the left in
-the first line does not change, because of the equation number, while
-the second line is pushed over to the right margin.
-{\setlength{\multlinegap}{0pt}
-\begin{multline*}\tag*{[a]}
-\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2]
- -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\
- =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2
- \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy
-\end{multline*}}%
-Some text after to test the below-display spacing.
-
-\begin{verbatim}
-{\setlength{\multlinegap}{0pt}
-\begin{multline*}\tag*{[a]}
-\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2]
- -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\
- =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2
- \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy
-\end{multline*}}
-\end{verbatim}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\fi % matches \iffalse above [mjd,24-Jan-1995]
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\newpage
-\subsection{Gather}
-Numbered version with \verb;\notag; on the second line:
-\begin{gather}
-D(a,r)\equiv\{z\in\mathbf{C}\colon \abs{z-a}<r\},\\
-\seg(a,r)\equiv\{z\in\mathbf{C}\colon
-\Im z= \Im a,\ \abs{z-a}<r\},\notag\\
-c(e,\theta,r)\equiv\{(x,y)\in\mathbf{C}
-\colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\
-C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r).
-\end{gather}
-\begin{verbatim}
-\begin{gather}
-D(a,r)\equiv\{z\in\mathbf{C}\colon \abs{z-a}<r\},\\
-\seg(a,r)\equiv\{z\in\mathbf{C}\colon
-\Im z= \Im a,\ \abs{z-a}<r\},\notag\\
-c(e,\theta,r)\equiv\{(x,y)\in\mathbf{C}
-\colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\
-C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r).
-\end{gather}
-\end{verbatim}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-Unnumbered version.
-\begin{gather*}
-D(a,r)\equiv\{z\in\mathbf{C}\colon \abs{z-a}<r\},\\
-\seg (a,r)\equiv\{z\in\mathbf{C}\colon
-\Im z= \Im a,\ \abs{z-a}<r\},\\
-c(e,\theta,r)\equiv\{(x,y)\in\mathbf{C}
- \colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\
-C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r).
-\end{gather*}
-Some text after to test the below-display spacing.
-\begin{verbatim}
-\begin{gather*}
-D(a,r)\equiv\{z\in\mathbf{C}\colon \abs{z-a}<r\},\\
-\seg (a,r)\equiv\{z\in\mathbf{C}\colon
-\Im z= \Im a,\ \abs{z-a}<r\},\\
-c(e,\theta,r)\equiv\{(x,y)\in\mathbf{C}
- \colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\
-C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r).
-\end{gather*}
-\end{verbatim}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\newpage
-\subsection{Align}
-Numbered version:
-\begin{align}
-\gamma_x(t)&=(\cos tu+\sin tx,v),\\
-\gamma_y(t)&=(u,\cos tv+\sin ty),\\
-\gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv,
- -\frac\beta\alpha\sin tu+\cos tv\right).
-\end{align}
-Some text after to test the below-display spacing.
-
-\begin{verbatim}
-\begin{align}
-\gamma_x(t)&=(\cos tu+\sin tx,v),\\
-\gamma_y(t)&=(u,\cos tv+\sin ty),\\
-\gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv,
- -\frac\beta\alpha\sin tu+\cos tv\right).
-\end{align}
-\end{verbatim}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-Unnumbered version:
-\begin{align*}
-\gamma_x(t)&=(\cos tu+\sin tx,v),\\
-\gamma_y(t)&=(u,\cos tv+\sin ty),\\
-\gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv,
- -\frac\beta\alpha\sin tu+\cos tv\right).
-\end{align*}
-Some text after to test the below-display spacing.
-
-\begin{verbatim}
-\begin{align*}
-\gamma_x(t)&=(\cos tu+\sin tx,v),\\
-\gamma_y(t)&=(u,\cos tv+\sin ty),\\
-\gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv,
- -\frac\beta\alpha\sin tu+\cos tv\right).
-\end{align*}
-\end{verbatim}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-A variation:
-\begin{align}
-x& =y && \text {by (\ref{eq:C})}\\
-x'& = y' && \text {by (\ref{eq:D})}\\
-x+x' & = y+y' && \text {by Axiom 1.}
-\end{align}
-Some text after to test the below-display spacing.
-
-\begin{verbatim}
-\begin{align}
-x& =y && \text {by (\ref{eq:C})}\\
-x'& = y' && \text {by (\ref{eq:D})}\\
-x+x' & = y+y' && \text {by Axiom 1.}
-\end{align}
-\end{verbatim}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\newpage
-\subsection{Align and split within gather}
-When using the \env{align} environment within the \env{gather}
-environment, one or the other, or both, should be unnumbered (using the
-\verb"*" form); numbering both the outer and inner environment would
-cause a conflict.
-
-Automatically numbered \env{gather} with \env{split} and \env{align*}:
-\begin{gather}
-\begin{split} \varphi(x,z)
-&=z-\gamma_{10}x-\gamma_{mn}x^mz^n\\
-&=z-Mr^{-1}x-Mr^{-(m+n)}x^mz^n
-\end{split}\\[6pt]
-\begin{align*}
-\zeta^0 &=(\xi^0)^2,\\
-\zeta^1 &=\xi^0\xi^1,\\
-\zeta^2 &=(\xi^1)^2,
-\end{align*}
-\end{gather}
-Here the \env{split} environment gets a number from the outer
-\env{gather} environment; numbers for individual lines of the
-\env{align*} are suppressed because of the star.
-
-\begin{verbatim}
-\begin{gather}
-\begin{split} \varphi(x,z)
-&=z-\gamma_{10}x-\gamma_{mn}x^mz^n\\
-&=z-Mr^{-1}x-Mr^{-(m+n)}x^mz^n
-\end{split}\\[6pt]
-\begin{align*}
-\zeta^0 &=(\xi^0)^2,\\
-\zeta^1 &=\xi^0\xi^1,\\
-\zeta^2 &=(\xi^1)^2,
-\end{align*}
-\end{gather}
-\end{verbatim}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-The \verb"*"-ed form of \env{gather} with the non-\verb"*"-ed form of
-\env{align}.
-\begin{gather*}
-\begin{split} \varphi(x,z)
-&=z-\gamma_{10}x-\gamma_{mn}x^mz^n\\
-&=z-Mr^{-1}x-Mr^{-(m+n)}x^mz^n
-\end{split}\\[6pt]
-\begin{align} \zeta^0&=(\xi^0)^2,\\
-\zeta^1 &=\xi^0\xi^1,\\
-\zeta^2 &=(\xi^1)^2,
-\end{align}
-\end{gather*}
-Some text after to test the below-display spacing.
-
-\begin{verbatim}
-\begin{gather*}
-\begin{split} \varphi(x,z)
-&=z-\gamma_{10}x-\gamma_{mn}x^mz^n\\
-&=z-Mr^{-1}x-Mr^{-(m+n)}x^mz^n
-\end{split}\\[6pt]
-\begin{align} \zeta^0&=(\xi^0)^2,\\
-\zeta^1 &=\xi^0\xi^1,\\
-\zeta^2 &=(\xi^1)^2,
-\end{align}
-\end{gather*}
-\end{verbatim}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\newpage
-\subsection{Alignat}
-Numbered version:
-\begin{alignat}{3}
-V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j,
- & \qquad U_i & = u_i,
- \qquad \text{for $i\ne j$;}\label{eq:B}\\
-V_j & = v_j, & \qquad X_j & = x_j,
- & \qquad U_j & u_j + \sum_{i\ne j} q_i u_i.
-\end{alignat}
-Some text after to test the below-display spacing.
-
-\begin{verbatim}
-\begin{alignat}{3}
-V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j,
- & \qquad U_i & = u_i,
- \qquad \text{for $i\ne j$;}\label{eq:B}\\
-V_j & = v_j, & \qquad X_j & = x_j,
- & \qquad U_j & u_j + \sum_{i\ne j} q_i u_i.
-\end{alignat}
-\end{verbatim}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-Unnumbered version:
-\begin{alignat*}3
-V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j,
- & \qquad U_i & = u_i,
- \qquad \text{for $i\ne j$;} \\
-V_j & = v_j, & \qquad X_j & = x_j,
- & \qquad U_j & u_j + \sum_{i\ne j} q_i u_i.
-\end{alignat*}
-Some text after to test the below-display spacing.
-
-\begin{verbatim}
-\begin{alignat*}3
-V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j,
- & \qquad U_i & = u_i,
- \qquad \text{for $i\ne j$;} \\
-V_j & = v_j, & \qquad X_j & = x_j,
- & \qquad U_j & u_j + \sum_{i\ne j} q_i u_i.
-\end{alignat*}
-\end{verbatim}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\newpage
-The most common use for \env{alignat} is for things like
-\begin{alignat}{2}
-x& =y && \qquad \text {by (\ref{eq:A})}\label{eq:C}\\
-x'& = y' && \qquad \text {by (\ref{eq:B})}\label{eq:D}\\
-x+x' & = y+y' && \qquad \text {by Axiom 1.}
-\end{alignat}
-Some text after to test the below-display spacing.
-
-\begin{verbatim}
-\begin{alignat}{2}
-x& =y && \qquad \text {by (\ref{eq:A})}\label{eq:C}\\
-x'& = y' && \qquad \text {by (\ref{eq:B})}\label{eq:D}\\
-x+x' & = y+y' && \qquad \text {by Axiom 1.}
-\end{alignat}
-\end{verbatim}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\newpage
-\setlength{\marginrulewidth}{0pt}
-
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-\end{document}
-\endinput