mirror of
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2309 lines
79 KiB
TeX
2309 lines
79 KiB
TeX
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% This is the AMS's testmath.tex modified to check unicode-math output.
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\documentclass{article}
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\pagestyle{headings}
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\title{Sample Paper for the \texttt{XITS Math} font\\
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File name: \fn{testmath.tex}}
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\author{American Mathematical Society}
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\date{Version 2.0, 1999/11/15}
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\usepackage{amsmath,amsthm,unicode-math}
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\setmainfont[Ligatures=TeX]{XITS}
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\setsansfont[Ligatures=TeX]{TeX Gyre Heros}
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\setmonofont{Latin Modern Mono}
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\setmathfont{XITS Math}
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% Some definitions useful in producing this sort of documentation:
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\chardef\bslash=`\\ % p. 424, TeXbook
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% Normalized (nonbold, nonitalic) tt font, to avoid font
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% substitution warning messages if tt is used inside section
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% headings and other places where odd font combinations might
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% result.
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\newcommand{\ntt}{\normalfont\ttfamily}
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% command name
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\newcommand{\cn}[1]{{\protect\ntt\bslash#1}}
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% LaTeX package name
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\newcommand{\pkg}[1]{{\protect\ntt#1}}
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% File name
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\newcommand{\fn}[1]{{\protect\ntt#1}}
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% environment name
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\newcommand{\env}[1]{{\protect\ntt#1}}
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\hfuzz1pc % Don't bother to report overfull boxes if overage is < 1pc
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% Theorem environments
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%% \theoremstyle{plain} %% This is the default
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\newtheorem{thm}{Theorem}[section]
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\newtheorem{cor}[thm]{Corollary}
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\newtheorem{lem}[thm]{Lemma}
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\newtheorem{prop}[thm]{Proposition}
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\newtheorem{ax}{Axiom}
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\theoremstyle{definition}
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\newtheorem{defn}{Definition}[section]
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\theoremstyle{remark}
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\newtheorem{rem}{Remark}[section]
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\newtheorem*{notation}{Notation}
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%\numberwithin{equation}{section}
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\newcommand{\thmref}[1]{Theorem~\ref{#1}}
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\newcommand{\secref}[1]{\S\ref{#1}}
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\newcommand{\lemref}[1]{Lemma~\ref{#1}}
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\newcommand{\bysame}{\mbox{\rule{3em}{.4pt}}\,}
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% Math definitions
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\newcommand{\A}{\mathcal{A}}
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\newcommand{\cB}{\mathcal{B}}
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\newcommand{\st}{\sigma}
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\newcommand{\XcY}{{(X,Y)}}
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\newcommand{\SX}{{S_X}}
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\newcommand{\SY}{{S_Y}}
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\newcommand{\SXY}{{S_{X,Y}}}
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\newcommand{\SXgYy}{{S_{X|Y}(y)}}
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\newcommand{\Cw}[1]{{\hat C_#1(X|Y)}}
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\newcommand{\cG}{{G(X|Y)}}
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\newcommand{\PY}{{P_{\mathcal{Y}}}}
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\newcommand{\X}{\mathcal{X}}
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\newcommand{\wt}{\widetilde}
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\newcommand{\wh}{\widehat}
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\DeclareMathOperator{\per}{per}
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\DeclareMathOperator{\cov}{cov}
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\DeclareMathOperator{\non}{non}
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\DeclareMathOperator{\cf}{cf}
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\DeclareMathOperator{\add}{add}
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\DeclareMathOperator{\Cham}{Cham}
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\DeclareMathOperator{\IM}{Im}
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\DeclareMathOperator{\esssup}{ess\,sup}
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\DeclareMathOperator{\meas}{meas}
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\DeclareMathOperator{\seg}{seg}
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% \interval is used to provide better spacing after a [ that
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% is used as a closing delimiter.
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\newcommand{\interval}[1]{\mathinner{#1}}
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% Notation for an expression evaluated at a particular condition. The
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% optional argument can be used to override automatic sizing of the
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% right vert bar, e.g. \eval[\biggr]{...}_{...}
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\newcommand{\eval}[2][\right]{\relax
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\ifx#1\right\relax \left.\fi#2#1\rvert}
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% Enclose the argument in vert-bar delimiters:
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\newcommand{\envert}[1]{\left\lvert#1\right\rvert}
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\let\abs=\envert
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% Enclose the argument in double-vert-bar delimiters:
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\newcommand{\enVert}[1]{\left\lVert#1\right\rVert}
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\let\norm=\enVert
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\begin{document}
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\maketitle
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\markboth{Sample paper for the XITS Math font}
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{Sample paper for the XITS Math font}
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\renewcommand{\sectionmark}[1]{}
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\section{Introduction}
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This paper contains examples of various features from \AmS-\LaTeX{}.
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\section{Enumeration of Hamiltonian paths in a graph}
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Let $\mathbf{A}=(a_{ij})$ be the adjacency matrix of graph $G$. The
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corresponding Kirchhoff matrix $\mathbf{K}=(k_{ij})$ is obtained from
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$\mathbf{A}$ by replacing in $-\mathbf{A}$ each diagonal entry by the
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degree of its corresponding vertex; i.e., the $i$th diagonal entry is
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identified with the degree of the $i$th vertex. It is well known that
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\begin{equation}
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\det\mathbf{K}(i|i)=\text{ the number of spanning trees of $G$},
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\quad i=1,\dots,n
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\end{equation}
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where $\mathbf{K}(i|i)$ is the $i$th principal submatrix of
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$\mathbf{K}$.
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\begin{verbatim}
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\det\mathbf{K}(i|i)=\text{ the number of spanning trees of $G$},
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\end{verbatim}
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Let $C_{i(j)}$ be the set of graphs obtained from $G$ by attaching edge
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$(v_iv_j)$ to each spanning tree of $G$. Denote by $C_i=\bigcup_j
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C_{i(j)}$. It is obvious that the collection of Hamiltonian cycles is a
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subset of $C_i$. Note that the cardinality of $C_i$ is $k_{ii}\det
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\mathbf{K}(i|i)$. Let $\wh X=\{\hat x_1,\dots,\hat x_n\}$.
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\begin{verbatim}
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$\wh X=\{\hat x_1,\dots,\hat x_n\}$
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\end{verbatim}
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Define multiplication for the elements of $\wh X$ by
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\begin{equation}\label{multdef}
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\hat x_i\hat x_j=\hat x_j\hat x_i,\quad \hat x^2_i=0,\quad
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i,j=1,\dots,n.
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\end{equation}
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Let ${\hat k}_{ij}=k_{ij}\hat x_j$ and $\hat k_{ij}=-\sum_{j\neq i} \hat
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k_{ij}$. Then the number of Hamiltonian cycles $H_c$ is given by the
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relation \cite{liuchow:formalsum}
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\begin{equation}\label{H-cycles}
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\biggl(\prod^n_{\,j=1}\hat x_j\biggr)H_c=\frac{1}{2}\hat k_{ij}\det
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\wh{\mathbf{K}}(i|i),\qquad i=1,\dots,n.
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\end{equation}
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The task here is to express \eqref{H-cycles}
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in a form free of any $\hat x_i$,
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$i=1,\dots,n$. The result also leads to the resolution of enumeration of
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Hamiltonian paths in a graph.
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It is well known that the enumeration of Hamiltonian cycles and paths in
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a complete graph $K_n$ and in a complete bipartite graph $K_{n_1n_2}$
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can only be found from \textit{first combinatorial principles}
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\cite{hapa:graphenum}. One wonders if there exists a formula which can
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be used very efficiently to produce $K_n$ and $K_{n_1n_2}$. Recently,
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using Lagrangian methods, Goulden and Jackson have shown that $H_c$ can
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be expressed in terms of the determinant and permanent of the adjacency
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matrix \cite{gouja:lagrmeth}. However, the formula of Goulden and
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Jackson determines neither $K_n$ nor $K_{n_1n_2}$ effectively. In this
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paper, using an algebraic method, we parametrize the adjacency matrix.
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The resulting formula also involves the determinant and permanent, but
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it can easily be applied to $K_n$ and $K_{n_1n_2}$. In addition, we
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eliminate the permanent from $H_c$ and show that $H_c$ can be
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represented by a determinantal function of multivariables, each variable
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with domain $\{0,1\}$. Furthermore, we show that $H_c$ can be written by
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number of spanning trees of subgraphs. Finally, we apply the formulas to
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a complete multigraph $K_{n_1\dots n_p}$.
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The conditions $a_{ij}=a_{ji}$, $i,j=1,\dots,n$, are not required in
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this paper. All formulas can be extended to a digraph simply by
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multiplying $H_c$ by 2.
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\section{Main Theorem}
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\label{s:mt}
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\begin{notation} For $p,q\in P$ and $n\in\omega$ we write
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$(q,n)\le(p,n)$ if $q\le p$ and $A_{q,n}=A_{p,n}$.
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\begin{verbatim}
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\begin{notation} For $p,q\in P$ and $n\in\omega$
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...
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\end{notation}
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\end{verbatim}
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\end{notation}
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Let $\mathbf{B}=(b_{ij})$ be an $n\times n$ matrix. Let $\mathbf{n}=\{1,
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\dots,n\}$. Using the properties of \eqref{multdef}, it is readily seen
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that
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\begin{lem}\label{lem-per}
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\begin{equation}
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\prod_{i\in\mathbf{n}}
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\biggl(\sum_{\,j\in\mathbf{n}}b_{ij}\hat x_i\biggr)
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=\biggl(\prod_{\,i\in\mathbf{n}}\hat x_i\biggr)\per \mathbf{B}
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\end{equation}
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where $\per \mathbf{B}$ is the permanent of $\mathbf{B}$.
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\end{lem}
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Let $\wh Y=\{\hat y_1,\dots,\hat y_n\}$. Define multiplication
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for the elements of $\wh Y$ by
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\begin{equation}
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\hat y_i\hat y_j+\hat y_j\hat y_i=0,\quad i,j=1,\dots,n.
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\end{equation}
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Then, it follows that
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\begin{lem}\label{lem-det}
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\begin{equation}\label{detprod}
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\prod_{i\in\mathbf{n}}
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\biggl(\sum_{\,j\in\mathbf{n}}b_{ij}\hat y_j\biggr)
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=\biggl(\prod_{\,i\in\mathbf{n}}\hat y_i\biggr)\det\mathbf{B}.
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\end{equation}
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\end{lem}
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Note that all basic properties of determinants are direct consequences
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of Lemma ~\ref{lem-det}. Write
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\begin{equation}\label{sum-bij}
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\sum_{j\in\mathbf{n}}b_{ij}\hat y_j=\sum_{j\in\mathbf{n}}b^{(\lambda)}
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_{ij}\hat y_j+(b_{ii}-\lambda_i)\hat y_i\hat y
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\end{equation}
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where
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\begin{equation}
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b^{(\lambda)}_{ii}=\lambda_i,\quad b^{(\lambda)}_{ij}=b_{ij},
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\quad i\neq j.
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\end{equation}
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Let $\mathbf{B}^{(\lambda)}=(b^{(\lambda)}_{ij})$. By \eqref{detprod}
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and \eqref{sum-bij}, it is
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straightforward to show the following
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result:
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\begin{thm}\label{thm-main}
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\begin{equation}\label{detB}
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\det\mathbf{B}=
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\sum^n_{l =0}\sum_{I_l \subseteq n}
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\prod_{i\in I_l}(b_{ii}-\lambda_i)
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\det\mathbf{B}^{(\lambda)}(I_l |I_l ),
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\end{equation}
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where $I_l =\{i_1,\dots,i_l \}$ and $\mathbf{B}^{(\lambda)}(I_l |I_l )$
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is the principal submatrix obtained from $\mathbf{B}^{(\lambda)}$
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by deleting its $i_1,\dots,i_l $ rows and columns.
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\end{thm}
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\begin{rem}
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Let $\mathbf{M}$ be an $n\times n$ matrix. The convention
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$\mathbf{M}(\mathbf{n}|\mathbf{n})=1$ has been used in \eqref{detB} and
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hereafter.
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\end{rem}
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Before proceeding with our discussion, we pause to note that
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\thmref{thm-main} yields immediately a fundamental formula which can be
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used to compute the coefficients of a characteristic polynomial
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\cite{mami:matrixth}:
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\begin{cor}\label{BI}
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Write $\det(\mathbf{B}-x\mathbf{I})=\sum^n_{l =0}(-1)
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^l b_l x^l $. Then
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\begin{equation}\label{bl-sum}
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b_l =\sum_{I_l \subseteq\mathbf{n}}\det\mathbf{B}(I_l |I_l ).
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\end{equation}
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\end{cor}
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Let
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\begin{equation}
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\mathbf{K}(t,t_1,\dots,t_n)
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=\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\
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-a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\
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\hdotsfor[2]{4}\\
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-a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix},
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\end{equation}
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\begin{verbatim}
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\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\
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-a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\
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\hdotsfor[2]{4}\\
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-a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix}
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\end{verbatim}
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where
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\begin{equation}
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D_i=\sum_{j\in\mathbf{n}}a_{ij}t_j,\quad i=1,\dots,n.
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\end{equation}
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Set
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\begin{equation*}
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D(t_1,\dots,t_n)=\frac{\delta}{\delta t}\eval{\det\mathbf{K}(t,t_1,\dots,t_n)
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}_{t=1}.
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\end{equation*}
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Then
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\begin{equation}\label{sum-Di}
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D(t_1,\dots,t_n)
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=\sum_{i\in\mathbf{n}}D_i\det\mathbf{K}(t=1,t_1,\dots,t_n; i|i),
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\end{equation}
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where $\mathbf{K}(t=1,t_1,\dots,t_n; i|i)$ is the $i$th principal
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submatrix of $\mathbf{K}(t=1,t_1,\dots,t_n)$.
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Theorem ~\ref{thm-main} leads to
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\begin{equation}\label{detK1}
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\det\mathbf{K}(t_1,t_1,\dots,t_n)
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=\sum_{I\in\mathbf{n}}(-1)^{\envert{I}}t^{n-\envert{I}}
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\prod_{i\in I}t_i\prod_{j\in I}(D_j+\lambda_jt_j)\det\mathbf{A}
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^{(\lambda t)}(\overline{I}|\overline I).
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\end{equation}
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Note that
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\begin{equation}\label{detK2}
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\det\mathbf{K}(t=1,t_1,\dots,t_n)=\sum_{I\in\mathbf{n}}(-1)^{\envert{I}}
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\prod_{i\in I}t_i\prod_{j\in I}(D_j+\lambda_jt_j)\det\mathbf{A}
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^{(\lambda)}(\overline{I}|\overline{I})=0.
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\end{equation}
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Let $t_i=\hat x_i,i=1,\dots,n$. Lemma ~\ref{lem-per} yields
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\begin{multline}
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\biggl(\sum_{\,i\in\mathbf{n}}a_{l _i}x_i\biggr)
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\det\mathbf{K}(t=1,x_1,\dots,x_n;l |l )\\
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=\biggl(\prod_{\,i\in\mathbf{n}}\hat x_i\biggr)
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\sum_{I\subseteq\mathbf{n}-\{l \}}
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(-1)^{\envert{I}}\per\mathbf{A}^{(\lambda)}(I|I)
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\det\mathbf{A}^{(\lambda)}
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(\overline I\cup\{l \}|\overline I\cup\{l \}).
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\label{sum-ali}
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\end{multline}
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\begin{verbatim}
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\begin{multline}
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\biggl(\sum_{\,i\in\mathbf{n}}a_{l _i}x_i\biggr)
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\det\mathbf{K}(t=1,x_1,\dots,x_n;l |l )\\
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=\biggl(\prod_{\,i\in\mathbf{n}}\hat x_i\biggr)
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\sum_{I\subseteq\mathbf{n}-\{l \}}
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(-1)^{\envert{I}}\per\mathbf{A}^{(\lambda)}(I|I)
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\det\mathbf{A}^{(\lambda)}
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(\overline I\cup\{l \}|\overline I\cup\{l \}).
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\label{sum-ali}
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\end{multline}
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\end{verbatim}
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By \eqref{H-cycles}, \eqref{detprod}, and \eqref{sum-bij}, we have
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\begin{prop}\label{prop:eg}
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\begin{equation}
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H_c=\frac1{2n}\sum^n_{l =0}(-1)^{l}
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D_{l},
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\end{equation}
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where
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\begin{equation}\label{delta-l}
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D_{l}=\eval[2]{\sum_{I_{l}\subseteq \mathbf{n}}
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D(t_1,\dots,t_n)}_{t_i=\left\{\begin{smallmatrix}
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0,& \text{if }i\in I_{l}\quad\\% \quad added for centering
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1,& \text{otherwise}\end{smallmatrix}\right.\;,\;\; i=1,\dots,n}.
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\end{equation}
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\end{prop}
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\section{Application}
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\label{lincomp}
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We consider here the applications of Theorems~\ref{th-info-ow-ow} and
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~\ref{th-weak-ske-owf} to a complete
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multipartite graph $K_{n_1\dots n_p}$. It can be shown that the
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number of spanning trees of $K_{n_1\dots n_p}$
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may be written
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\begin{equation}\label{e:st}
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T=n^{p-2}\prod^p_{i=1}
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(n-n_i)^{n_i-1}
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\end{equation}
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where
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\begin{equation}
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n=n_1+\dots+n_p.
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\end{equation}
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It follows from Theorems~\ref{th-info-ow-ow} and
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~\ref{th-weak-ske-owf} that
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\begin{equation}\label{e:barwq}
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\begin{split}
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H_c&=\frac1{2n}
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\sum^n_{{l}=0}(-1)^{l}(n-{l})^{p-2}
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\sum_{l _1+\dots+l _p=l}\prod^p_{i=1}
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\binom{n_i}{l _i}\\
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&\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i}\cdot
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\biggl[(n-l )^2-\sum^p_{j=1}(n_i-l _i)^2\biggr].\end{split}
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\end{equation}
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\begin{verbatim}
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... \binom{n_i}{l _i}\\
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\end{verbatim}
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and
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\begin{equation}\label{joe}
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\begin{split}
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H_c&=\frac12\sum^{n-1}_{l =0}
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(-1)^{l}(n-l )^{p-2}
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\sum_{l _1+\dots+l _p=l}
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\prod^p_{i=1}\binom{n_i}{l _i}\\
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&\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i}
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\left(1-\frac{l _p}{n_p}\right)
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[(n-l )-(n_p-l _p)].
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\end{split}
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\end{equation}
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The enumeration of $H_c$ in a $K_{n_1\dotsm n_p}$ graph can also be
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carried out by Theorem ~\ref{thm-H-param} or ~\ref{thm-asym}
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together with the algebraic method of \eqref{multdef}.
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Some elegant representations may be obtained. For example, $H_c$ in
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a $K_{n_1n_2n_3}$ graph may be written
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\begin{equation}\label{j:mark}
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\begin{split}
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H_c=&
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\frac{n_1!\,n_2!\,n_3!}
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{n_1+n_2+n_3}\sum_i\left[\binom{n_1}{i}
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\binom{n_2}{n_3-n_1+i}\binom{n_3}{n_3-n_2+i}\right.\\
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&+\left.\binom{n_1-1}{i}
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\binom{n_2-1}{n_3-n_1+i}
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\binom{n_3-1}{n_3-n_2+i}\right].\end{split}
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\end{equation}
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\section{Secret Key Exchanges}
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\label{SKE}
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|
|
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
|
|
\[ \chi (\A,t) = \sum_{\cB \subseteq \A}
|
|
(-1)^{\abs{\cB}} t^{\dim T(\cB)}. \]
|
|
\end{lem}
|
|
|
|
In order to compute $R''$ recall the definition
|
|
of $S(X,Y)$ from \lemref{lem-per}. Since $H \in \cB$,
|
|
$\A_{H} \subseteq \cB$. Thus if $T(\cB) = Y$ then
|
|
$\cB \in S(H,Y)$. Let $L'' = L(\A'')$. Then
|
|
\begin{equation}\label{E_SXgYy}
|
|
\begin{split}
|
|
R''&= \sum_{H\in \cB \subseteq \A} (-1)^{\abs{\cB}}
|
|
t^{\dim T(\cB)}\\
|
|
&= \sum_{Y \in L''} \sum_{\cB \in S(H,Y)}
|
|
(-1)^{\abs{\cB}}t^{\dim Y} \\
|
|
&= -\sum_{Y \in L''} \sum_{\cB \in S(H,Y)} (-1)^
|
|
{\abs{\cB - \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 $T(\A) \notin L(\A')$.
|
|
\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 $\cG$,
|
|
the \textit{characteristic hypergraph} of $\XcY$, and show that
|
|
\[
|
|
\Cw1=\lceil\,\log\chi(\cG)\rceil\ .
|
|
\]
|
|
|
|
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} $\cG$ 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)^{\left\lvert I\right\rvert}\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
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|
Transactions on Information Theory \textbf{22} (1976), no.~5, 644--654.
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|
|
\bibitem{fre:cichon}
|
|
D.~H. Fremlin, \emph{Cichon's diagram}, 1983/1984, presented at the
|
|
S{\'e}minaire Initiation {\`a} l'Analyse, G. Choquet, M. Rogalski, J.
|
|
Saint Raymond, at the Universit{\'e} Pierre et Marie Curie, Paris, 23e
|
|
ann{\'e}e.
|
|
|
|
\bibitem{gouja:lagrmeth}
|
|
I.~P. Goulden and D.~M. Jackson, \emph{The enumeration of directed
|
|
closed {E}uler trails and directed {H}amiltonian circuits by
|
|
{L}angrangian methods}, European J. Combin. \textbf{2} (1981), 131--212.
|
|
|
|
\bibitem{hapa:graphenum}
|
|
F.~Harary and E.~M. Palmer, \emph{Graphical enumeration}, Academic
|
|
Press, 1973.
|
|
|
|
\bibitem{imlelu:oneway}
|
|
R.~Impagliazzo, L.~Levin, and M.~Luby, \emph{Pseudo-random generation
|
|
from one-way functions}, Proc. 21st STOC (1989), ACM, New York,
|
|
pp.~12--24.
|
|
|
|
\bibitem{komiyo:unipfunc}
|
|
M.~Kojima, S.~Mizuno, and A.~Yoshise, \emph{A new continuation method
|
|
for complementarity problems with uniform p-functions}, Tech. Report
|
|
B-194, Tokyo Inst. of Technology, Tokyo, 1987, Dept. of Information
|
|
Sciences.
|
|
|
|
\bibitem{komiyo:lincomp}
|
|
\bysame, \emph{A polynomial-time algorithm for a class of linear
|
|
complementarity problems}, Tech. Report B-193, Tokyo Inst. of
|
|
Technology, Tokyo, 1987, Dept. of Information Sciences.
|
|
|
|
\bibitem{liuchow:formalsum}
|
|
C.~J. Liu and Yutze Chow, \emph{On operator and formal sum methods for
|
|
graph enumeration problems}, SIAM J. Algorithms Discrete Methods
|
|
\textbf{5} (1984), 384--438.
|
|
|
|
\bibitem{mami:matrixth}
|
|
M.~Marcus and H.~Minc, \emph{A survey of matrix theory and matrix
|
|
inequalities}, Complementary Series in Math. \textbf{14} (1964), 21--48.
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|
|
|
\bibitem{miyoki:lincomp}
|
|
S.~Mizuno, A.~Yoshise, and T.~Kikuchi, \emph{Practical polynomial time
|
|
algorithms for linear complementarity problems}, Tech. Report~13, Tokyo
|
|
Inst. of Technology, Tokyo, April 1988, Dept. of Industrial Engineering
|
|
and Management.
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|
|
|
\bibitem{moad:quadpro}
|
|
R.~D. Monteiro and I.~Adler, \emph{Interior path following primal-dual
|
|
algorithms, part {II}: Quadratic programming}, August 1987, Working
|
|
paper, Dept. of Industrial Engineering and Operations Research.
|
|
|
|
\bibitem{ste:sint}
|
|
E.~M. Stein, \emph{Singular integrals and differentiability properties
|
|
of functions}, Princeton Univ. Press, Princeton, N.J., 1970.
|
|
|
|
\bibitem{ye:intalg}
|
|
Y.~Ye, \emph{Interior algorithms for linear, quadratic and linearly
|
|
constrained convex programming}, Ph.D. thesis, Stanford Univ., Palo
|
|
Alto, Calif., July 1987, Dept. of Engineering--Economic Systems,
|
|
unpublished.
|
|
|
|
\end{thebibliography}
|
|
|
|
\end{document}
|
|
\endinput
|