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diff --git a/Master/texmf-dist/source/latex/amsmath/testmath.tex b/Master/texmf-dist/source/latex/amsmath/testmath.tex deleted file mode 100644 index 069944ee9d0..00000000000 --- a/Master/texmf-dist/source/latex/amsmath/testmath.tex +++ /dev/null @@ -1,2342 +0,0 @@ -%%% ==================================================================== -%%% @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} - -\begin{thebibliography}{10} - -\bibitem{dihe:newdir} -W.~Diffie and E.~Hellman, \emph{New directions in cryptography}, IEEE -Transactions on Information Theory \textbf{22} (1976), no.~5, 644--654. - -\bibitem{fre:cichon} -D.~H. 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