%%@texfile{% %% filename="testart.tex", %% version="1.1a", %% date="26-JUL-1991", %% filetype="AMS-LaTeX: documentation", %% copyright="Copyright (C) American Mathematical Society, all rights %% reserved. Copying of this file is authorized only if either: %% (1) you make absolutely no changes to your copy, including name; %% OR (2) if you do make changes, you first rename it to some other %% name.", %% author="American Mathematical Society", %% address="American Mathematical Society, %% Technical Support Department, %% P. O. Box 6248, %% Providence, RI 02940, %% USA", %% telephone="401-455-4080 or (in the USA) 800-321-4AMS", %% email="Internet: Tech-Support@Math.AMS.org", %% checksumtype="line count", %% checksum="2479", %% codetable="ISO/ASCII", %% keywords="latex, amslatex, ams-latex", %% abstract="This file is part of the AMS-\LaTeX{} package, version 1.1. %% It is a sample document illustrating the use of the ``amsart'' %% documentstyle and the ``amstex'' option." %%} %%% end of file header % \documentstyle[draft% ,amscd% %% ,syntonly% %,amssymb% ]{amsart} % A method of making the syntax check decision interactively. To % see how this works, remove the double %% from the following % lines and also from the `syntonly' option above, in the % documentstyle options list. \let and \csname are primitive TeX % commands documented in the TeXbook, but not in the LaTeX manual. % %% \typein[\answer]{To run syntax check, enter `S'; %% otherwise, just press return:} %% %% \let\SANSWER=\syntaxonly \let\sANSWER=\syntaxonly %% \csname\answer ANSWER\endcsname % Some definitions useful in producing this sort of % documentation: \chardef\bslash=`\\ % p. 424, TeXbook % Normalized (nonbold, nonitalic) \tt font, to avoid font substitution % warning messages if \tt is used inside section headings and other % places where odd font combinations might result. \newcommand{\ntt}{\series m\shape n\tt} % control sequence \newcommand{\cs}[1]{{\protect\ntt\bslash#1}} % LaTeX option name \newcommand{\opt}[1]{{\protect\ntt#1}} % environment name \newcommand{\env}[1]{{\protect\ntt#1}} % Indent a little on the left in the verbatim environment. \makeatletter \def\verbatim{\interlinepenalty\@M \@verbatim \leftskip\@totalleftmargin\advance\leftskip2pc \frenchspacing\@vobeyspaces \@xverbatim} \makeatother \hfuzz1pc % Don't bother to report overfull boxes if overage is < 1pc % Theorem environments %% \theoremstyle{plain} %% This is the default \newtheorem{thm}{Theorem}[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \newtheorem{ax}{Axiom} \theoremstyle{definition} \newtheorem{defn}{Definition}[section] \theoremstyle{remark} \newtheorem{rem}{Remark}[section] \newtheorem{notation}{Notation} \renewcommand{\thenotation}{} % to make the notation environment unnumbered \numberwithin{equation}{section} \newcommand{\thmref}[1]{Theorem~\ref{#1}} \newcommand{\secref}[1]{\S\ref{#1}} \newcommand{\lemref}[1]{Lemma~\ref{#1}} % Math definitions \newcommand{\A}{{\cal A}} \newcommand{\B}{{\cal B}} \newcommand{\st}{\sigma} \newcommand{\XcY}{{(X,Y)}} \newcommand{\SX}{{S_X}} \newcommand{\SY}{{S_Y}} \newcommand{\SXY}{{S_{X,Y}}} \newcommand{\SXgYy}{{S_{X|Y}(y)}} \newcommand{\Cw}[1]{{\hat C_#1(X|Y)}} \newcommand{\G}{{G(X|Y)}} \newcommand{\PY}{{P_{\cal Y}}} \newcommand{\X}{{\cal X}} \newcommand{\per}{\operatorname{per}} \newcommand{\cov}{\operatorname{cov}} \newcommand{\non}{\operatorname{non}} \newcommand{\cf}{\operatorname{cf}} \newcommand{\add}{\operatorname{add}} \newcommand{\End}{\operatorname{End}} % \interval is used to provide better spacing after a [ that % is used as a closing delimiter. \newcommand{\interval}[1]{\mathinner{#1}} \begin{document} \title[AMSTEX/AMSART Sample Paper] {Sample Paper for the `AMSTEX' Option\\ and the `AMSART' Documentstyle\\ File name: TESTART.TEX} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% This combination of author names, addresses, and e-mail %% addresses is carefully selected to illustrate most of the %% features of the commands involved. Real-life author/address %% combinations will rarely be this complicated! %% First author \author{David Belar} \address[D. Belar and L. Eifrig]{Department of Electrical Engineering\\ University of Minnesota\\ Minneapolis, Minnesota 55455} %% Note the doubled @@: \email[D.~Belar]{davidb@@egr.umn.edu} \thanks{Research of the first author was supported in part by NSF grant CCR-87-10433 and DARPA Contract N00019-89-J-1988.} %% Second author \author{Lenore Eifrig} \curraddr[L. Eifrig]{San Diego Supercomputer Center\\P.O. Box 85608\\ San Diego, California 92138} %% Note the doubled @@: \email[L.~Eifrig]{eifrig@@sds.sdcs.edu} \thanks{Research of the second author was supported in part by NSF grant CCR-86-75257 and DARPA Contract N00019-89-J-1988.} %% Third author \author{Shafi N\"a\"at\"anen} \address[S. N\"a\"at\"anen]{MIT Laboratory for Computer Science \\ 545 Technology Square\\ Cambridge, MA 02139} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \date{October 23, 1991} \subjclass{Primary 05C38, 15A15; Secondary 05A15, 15A18} \keywords{$K$-graph, random pair, negligible set, nonatomic ccc order} \dedicatory{Dedicated to the memory of Parfon Samuelson} \thanks{This paper is in final form and no version of it will be submitted for publication elsewhere.} \maketitle \begin{abstract} This paper is a sample illustrating the use of the \opt{amstex} option in \LaTeX{}, and the American Mathematical Society preprint documentstyle, \opt{amsart}. The file used to prepare this sample is {\bf testart.tex}. \end{abstract} \tableofcontents \section{Introduction} \label{intro} This paper illustrates the use of the {\tt amsart} documentstyle, as well as the use of features from the {\tt amstex} option. It consists of extracts from published papers, interspersed with sample \TeX{} coding, instructions to authors, and comments about the use of particular commands. Sections \ref{s:font} and ~\ref{s:comp} are devoted to examples of the commands described in the \AmS-\LaTeX\ users guide. Appendix~\ref{s:eq} gives comprehensive examples of the display environments \env{align}, \env{gather}, \env{split}, \env{multline}, and \env{alignat}. \subsection{Acknowledgment} It is a pleasure to thank the referee for his valuable suggestions which resulted in an improvement of the manuscript. \section{Top matter instructions} \label{s:topmatter} The term `top matter' will be used to mean the title, author, abstract and other preliminary information. All such information should be typed at the beginning of a document, between \verb=\begin{document}= and \verb=\maketitle=.% % \footnote{Actually, as stated in the \LaTeX{} manual, the top matter information can also be typed before {\tt\char`\\begin\char`\{document\char`\}}; but the placement recommended here segregrates the information nicely in its own area.} % See the examples at the beginning of this paper. Some of the top matter information will print in a footnote on the first page, or at the end of the document, but such placement is done automatically by \LaTeX{}. \subsection{Title} The \verb=\title= command is used as described in the \LaTeX{} manual, but it has an additional option like sectioning commands, to specify the text of the running head. Thus \begin{verbatim} \title[The Selberg Trace Formula] {Some Explicit Cases of the Selberg Trace\\ Formula for Vector-Valued Functions} \end{verbatim} would put the text ``The Selberg Trace Formula'' into the running head on right-hand pages, and the text in curly braces would print as the full title on the first page. \subsection{Author} Like the \verb=\title= command, the \verb=\author= command has a square bracket option used to specify the left-hand running head. If there are two or more authors, the American Mathematical Society practice is to use initials instead of full first and middle names in the running head. In addition, in the {\tt amsart} documentstyle, a separate \verb=\author= command should be used for each author. This makes it easier to group commands like \verb=\address= and \verb=\thanks= with the associated \verb=\author= command. Cf.~the example top matter section in this file. \subsection{Addresses} Provide an address for each author by using the \verb=\address= command following the \verb=\author= command. Note that no abbreviations are used in addresses. Indicate the normal line breaks that would be made on a mailing label using \verb=\\=; in the current version of the {\tt amsart} documentstyle, these \verb=\\='s will be replaced with commas, since author addresses are set in paragraph form, but other documentstyles or a future version of the {\tt amsart} documentstyle might treat the information differently. The addresses should be entered in the same order as the author names on the title page. In the {\tt amsart} documentstyle, address information prints at the end of the paper, following the references. If the current address of an author is no longer the institution where the research being described took place, the current address can be provided through the \verb=\curraddr= command. Electronic mail addresses should also be given, when applicable, using \verb=\email=. For multiple authors with assorted current addresses and e-mail addresses, square-bracket options can be used to indicate those individuals to which an address, current address, or e-mail address pertains. For example, for two authors at the same institution, with different e-mail addresses, you would enter the information as \begin{verbatim} % Address of A. Author and B. Colleague: \address{North Carolina State University\\Raleigh, North Carolina 27695} \email[A. Author]{auth@@abel.ncsu.edu} \email[B. Colleague]{bcolleague@@fourier.ncsu.edu} \end{verbatim} \subsection{First-page footnotes} Use \verb=\thanks= for acknowledgments of grant support or other research support and similar information. This information will print as a footnote on the first page. If it pertains to only one author out of two or more, then refer to `first author' or `second author,' as shown in the examples in this paper. Papers published in proceedings of conferences are often abstracts or preliminary versions, and review journals such as {\it Mathematical Reviews\/} will be more interested in reviewing the final version. In such a case, use \verb=\thanks= to produce a first-page footnote saying ``The final [detailed] version of this paper will be [has been] submitted for publication elsewhere.'' Proceedings papers that are to be considered for review by {\it Mathematical Reviews\/} should include the following statement: ``This paper is in final form and no version of it will be submitted for publication elsewhere.'' Subject classification numbers (\verb=\subjclass=) are part of the top matter and will appear as footnotes at the bottom of the first page. Subject classifications (\verb=\subjclass=) are required for submissions to the American Mathematical Society. Use the 1980 Mathematics Subject Classification (1985 Revision) that appears in annual indexes of {\it Mathematical Reviews\/} beginning in 1984. (Note: Give a complete five-digit number; the two-digit prefixes from the Contents are insufficient.) An optional key words and phrases footnote may be added using the \verb=\keywords= command. An abstract should be typed immediately after the \verb=\maketitle= command, using the standard `abstract' environment. The heading ``{\sc Abstract.}'' will be added automatically. \section{Enumeration of Hamiltonian paths in a graph} Let $\bold A=(a_{ij})$ be the adjacency matrix of graph $G$. The corresponding Kirchhoff matrix $\bold K=(k_{ij})$ is obtained from $\bold A$ by replacing in $-\bold A$ each diagonal entry by the degree of its corresponding vertex; i.e., the $i$th diagonal entry is identified with the degree of the $i$th vertex. It is well known that \begin{equation} \det\bold K(i|i)=\text{ the number of spanning trees of $G$}, \quad i=1,\dots,n \end{equation} where $\bold K(i|i)$ is the $i$th principal submatrix of $\bold K$. \begin{verbatim} \det\bold K(i|i)=\text{ the number of spanning trees of $G$}, \end{verbatim} Let $C_{i(j)}$ be the set of graphs obtained from $G$ by attaching edge $(v_iv_j)$ to each spanning tree of $G$. Denote by $C_i=\bigcup_j C_{i(j)}$. It is obvious that the collection of Hamiltonian cycles is a subset of $C_i$. Note that the cardinality of $C_i$ is $k_{ii}\det \bold K(i|i)$. Let $\widehat X=\{\hat x_1,\dots,\hat x_n\}$. \begin{verbatim} $\widehat X=\{\hat x_1,\dots,\hat x_n\}$ \end{verbatim} Define multiplication for the elements of $\widehat X$ by \begin{equation}\label{multdef} \hat x_i\hat x_j=\hat x_j\hat x_i,\quad \hat x^2_i=0,\quad i,j=1,\dots,n. \end{equation} Let $\hat k_{ij}=k_{ij}\hat x_j$ and $\hat k_{ij}=-\sum_{j\not=i} \hat k_{ij}$. Then the number of Hamiltonian cycles $H_c$ is given by the relation \cite{liuchow:formalsum} \begin{equation}\label{H-cycles} \biggl(\prod^n_{\,j=1}\hat x_j\biggr)H_c=\frac{1}{2}\hat k_{ij}\det \widehat{\bold K}(i|i),\qquad i=1,\dots,n. \end{equation} The task here is to express \eqref{H-cycles} in a form free of any $\hat x_i$, $i=1,\dots,n$. The result also leads to the resolution of enumeration of Hamiltonian paths in a graph. It is well known that the enumeration of Hamiltonian cycles and paths in a complete graph $K_n$ and in a complete bipartite graph $K_{n_1n_2}$ can only be found from {\em first combinatorial principles\/} \cite{hapa:graphenum}. One wonders if there exists a formula which can be used very efficiently to produce $K_n$ and $K_{n_1n_2}$. Recently, using Lagrangian methods, Goulden and Jackson have shown that $H_c$ can be expressed in terms of the determinant and permanent of the adjacency matrix \cite{gouja:lagrmeth}. However, the formula of Goulden and Jackson determines neither $K_n$ nor $K_{n_1n_2}$ effectively. In this paper, using an algebraic method, we parametrize the adjacency matrix. The resulting formula also involves the determinant and permanent, but it can easily be applied to $K_n$ and $K_{n_1n_2}$. In addition, we eliminate the permanent from $H_c$ and show that $H_c$ can be represented by a determinantal function of multivariables, each variable with domain $\{0,1\}$. Furthermore, we show that $H_c$ can be written by number of spanning trees of subgraphs. Finally, we apply the formulas to a complete multigraph $K_{n_1\dots n_p}$. The conditions $a_{ij}=a_{ji}$, $i,j=1,\dots,n$, are not required in this paper. All formulas can be extended to a digraph simply by multiplying $H_c$ by 2. \section{Main Theorem} \label{s:mt} \begin{notation} For $p,q\in P$ and $n\in\omega$ we write $(q,n)\le(p,n)$ if $q\le p$ and $A_{q,n}=A_{p,n}$. \begin{verbatim} \begin{notation} For $p,q\in P$ and $n\in\omega$ ... \end{notation} \end{verbatim} We will make liberal use of Cichon's Diagram \cite{fre:cichon}: \begin{equation} \begin{CD} \cov(\cal L) @>>> \non(\cal K) @>>> \cf(\cal K) @>>> \cf(\cal L)\\ @VVV @AAA @AAA @VVV\\ \add(\cal L) @>>> \add(\cal K) @>>> \cov(\cal K) @>>> \non(\cal L) \end{CD}\end{equation} % \begin{verbatim} \begin{equation}\begin{CD} \cov(\cal L) @>>> \non(\cal K) @>>> \cf(\cal K) @>>> \cf(\cal L)\\ @VVV @AAA @AAA @VVV\\ \add(\cal L) @>>> \add(\cal K) @>>> \cov(\cal K) @>>> \non(\cal L) \end{CD}\end{equation} \end{verbatim} \end{notation} Let $\bold B=(b_{ij})$ be an $n\times n$ matrix. Let $\bold n=\{1, \dots,n\}$. Using the properties of \eqref{multdef}, it is readily seen that {\samepage \begin{lem}\label{lem-per} \begin{equation} \prod_{i\in\bold n}\biggl(\sum_{\,j\in\bold n}b_{ij}\hat x_i\biggr)= \biggl(\prod_{\,i\in\bold n}\hat x_i\biggr)\per \bold B \end{equation} where $\per \bold B$ is the permanent of $\bold B$. \end{lem} } Let $\widehat Y=\{\hat y_1,\dots,\hat y_n\}$. Define multiplication for the elements of $\widehat Y$ by \begin{equation} \hat y_i\hat y_j+\hat y_j\hat y_i=0,\quad i,j=1,\dots,n. \end{equation} Then, it follows that {\samepage \begin{lem}\label{lem-det} \begin{equation}\label{detprod} \prod_{i\in\bold n}\biggl(\sum_{\,j\in\bold n}b_{ij}\hat y_j\biggr)= \biggl(\prod_{\,i\in\bold n}\hat y_i\biggr)\det\bold B. \end{equation} \end{lem} } Note that all basic properties of determinants are direct consequences of Lemma ~\ref{lem-det}. Write \begin{equation}\label{sum-bij} \sum_{j\in\bold n}b_{ij}\hat y_j=\sum_{j\in\bold n}b^{(\lambda)} _{ij}\hat y_j+(b_{ii}-\lambda_i)\hat y_i\hat y \end{equation} where \begin{equation} b^{(\lambda)}_{ii}=\lambda_i,\quad b^{(\lambda)}_{ij}=b_{ij}, \quad i\not=j. \end{equation} Let $\bold B^{(\lambda)}=(b^{(\lambda)}_{ij})$. By \eqref{detprod} and \eqref{sum-bij}, it is straightforward to show the following result: \begin{thm}\label{thm-main} \begin{equation}\label{detB} \det\bold B= \sum^n_{l =0}\sum_{I_l \subseteq n} \prod_{i\in I_l}(b_{ii}-\lambda_i) \det\bold B^{(\lambda)}(I_l |I_l ), \end{equation} where $I_l =\{i_1,\dots,i_l \}$ and $\bold B^{(\lambda)}(I_l |I_l )$ is the principal submatrix obtained from $\bold B^{(\lambda)}$ by deleting its $i_1,\dots,i_l $ rows and columns. \end{thm} \begin{rem} Let $\bold M$ be an $n\times n$ matrix. The convention $\bold M(\bold n|\bold n)=1$ has been used in \eqref{detB} and hereafter. \end{rem} Before proceeding with our discussion, we pause to note that \thmref{thm-main} yields immediately a fundamental formula which can be used to compute the coefficients of a characteristic polynomial \cite{mami:matrixth}: {\samepage \begin{cor}\label{BI} Write $\det(\bold B-x\bold I)=\sum^n_{l =0}(-1) ^l b_l x^l $. Then \begin{equation}\label{bl-sum} b_l =\sum_{I_l \subseteq\bold n}\det\bold B(I_l |I_l ). \end{equation} \end{cor} } Let \begin{equation} \bold K(t,t_1,\dots,t_n)=\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\ -a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\ \hdotsfor[2]{4}\\ -a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix}, \end{equation} \begin{verbatim} \begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\ -a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\ \hdotsfor[2]{4}\\ -a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix} \end{verbatim} where \begin{equation} D_i=\sum_{j\in\bold n}a_{ij}t_j,\quad i=1,\dots,n. \end{equation} Set \begin{equation*} D(t_1,\dots,t_n)=\frac{\delta}{\delta t}\det\bold K(t,t_1,\dots,t_n) |_{t=1}. \end{equation*} Then \begin{equation}\label{sum-Di} D(t_1,\dots,t_n)=\sum_{i\in\bold n}D_i\det\bold K(t=1,t_1,\dots,t_n; i|i), \end{equation} where $\bold K(t=1,t_1,\dots,t_n; i|i)$ is the $i$th principal submatrix of $\bold K(t=1,t_1,\dots,t_n)$. Theorem ~\ref{thm-main} leads to \begin{equation}\label{detK1} \det\bold K(t_1,t_1,\dots,t_n)=\sum_{I\in\bold n}(-1)^{|I|}t^{n-|I|} \prod_{i\in I}t_i\prod_{j\in I}(D_j+\lambda_jt_j)\det\bold A ^{(\lambda t)}(\overline{I}|\overline I). \end{equation} Note that \begin{equation}\label{detK2} \det\bold K(t=1,t_1,\dots,t_n)=\sum_{I\in\bold n}(-1)^{|I|} \prod_{i\in I}t_i\prod_{j\in I}(D_j+\lambda_jt_j)\det\bold A ^{(\lambda)}(\overline{I}|\overline{I})=0. \end{equation} Let $t_i=\hat x_i,i=1,\dots,n$. Lemma ~\ref{lem-per} yields \begin{multline} \biggl(\sum_{\,i\in\bold n}a_{l _i}x_i\biggr) \det\bold K(t=1,x_1,\dots,x_n;l |l )\\ =\biggl(\prod_{\,i\in\bold n}\hat x_i\biggr) \sum_{I\subseteq\bold n-\{l \}} (-1)^{|I|}\per\bold A^{(\lambda)}(I|I)\det\bold A^{(\lambda)} (\overline I\cup\{l \}|\overline I\cup\{l \}). \label{sum-ali} \end{multline} \begin{verbatim} \begin{multline} \biggl(\sum_{\,i\in\bold n}a_{l _i}x_i\biggr) \det\bold K(t=1,x_1,\dots,x_n;l |l )\\ =\biggl(\prod_{\,i\in\bold n}\hat x_i\biggr) \sum_{I\subseteq\bold n-\{l \}} (-1)^{|I|}\per\bold A^{(\lambda)}(I|I)\det\bold A^{(\lambda)} (\overline I\cup\{l \}|\overline I\cup\{l \}). \label{sum-ali} \end{multline} \end{verbatim} By \eqref{H-cycles}, \eqref{detprod}, and \eqref{sum-bij}, we have \begin{prop}\label{prop:eg} \begin{equation} H_c=\frac1{2n}\sum^n_{l =0}(-1)^{l} D_{l}, \end{equation} where \begin{equation}\label{delta-l} D_{l}=\left.\sum_{I_{l}\subseteq \bold n} D(t_1,\dots,t_n)\right| _{t_i=\left\{\begin{smallmatrix} 0,& \text{if }i\in I_{l}\quad\\% \quad added for centering 1,& \text{otherwise}\end{smallmatrix}\right.\;,\;\; i=1,\dots,n}. \end{equation} \end{prop} \section{Application} \label{lincomp} We consider here the applications of Theorems~\ref{th-info-ow-ow} and ~\ref{th-weak-ske-owf} to a complete multipartite graph $K_{n_1\dots n_p}$. It can be shown that the number of spanning trees of $K_{n_1\dots n_p}$ may be written \begin{equation}\label{e:st} T=n^{p-2}\prod^p_{i=1} (n-n_i)^{n_i-1} \end{equation} where \begin{equation} n=n_1+\dots+n_p. \end{equation} It follows from Theorems~\ref{th-info-ow-ow} and ~\ref{th-weak-ske-owf} that \begin{equation}\label{e:barwq} \begin{split} H_c&=\frac1{2n} \sum^n_{{l}=0}(-1)^{l}(n-{l})^{p-2} \sum_{l _1+\dots+l _p=l}\prod^p_{i=1} \binom{n_i}{l _i}\\ &\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i}\cdot \biggl[(n-l )^2-\sum^p_{j=1}(n_i-l _i)^2\biggr].\end{split} \end{equation} \begin{verbatim} ... \binom{n_i}{l _i}\\ \end{verbatim} and \begin{equation}\label{joe} \begin{split} H_c&=\frac12\sum^{n-1}_{l =0} (-1)^{l}(n-l )^{p-2} \sum_{l _1+\dots+l _p=l} \prod^p_{i=1}\binom{n_i}{l _i}\\ &\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i} \left(1-\frac{l _p}{n_p}\right) [(n-l )-(n_p-l _p)]. \end{split} \end{equation} The enumeration of $H_c$ in a $K_{n_1\dotsm n_p}$ graph can also be carried out by Theorem ~\ref{thm-H-param} or ~\ref{thm-asym} together with the algebraic method of \eqref{multdef}. Some elegant representations may be obtained. For example, $H_c$ in a $K_{n_1n_2n_3}$ graph may be written \begin{equation}\label{j:mark} \begin{split} H_c=& \frac{n_1!\,n_2!\,n_3!} {n_1+n_2+n_3}\sum_i\left[\binom{n_1}{i} \binom{n_2}{n_3-n_1+i}\binom{n_3}{n_3-n_2+i}\right.\\ &+\left.\binom{n_1-1}{i} \binom{n_2-1}{n_3-n_1+i} \binom{n_3-1}{n_3-n_2+i}\right].\end{split} \end{equation} \section{Secret Key Exchanges} \label{SKE} Modern cryptography is fundamentally concerned with the problem of secure private communication. A Secret Key Exchange is a protocol where Alice and Bob, having no secret information in common to start, are able to agree on a common secret key, conversing over a public channel. The notion of a Secret Key Exchange protocol was first introduced in the seminal paper of Diffie and Hellman \cite{dihe:newdir}. \cite{dihe:newdir} presented a concrete implementation of a Secret Key Exchange protocol, dependent on a specific assumption (a variant on the discrete log), specially tailored to yield Secret Key Exchange. Secret Key Exchange is of course trivial if trapdoor permutations exist. However, there is no known implementation based on a weaker general assumption. The concept of an informationally one-way function was introduced in \cite{imlelu:oneway}. We give only an informal definition here: \begin{defn} A polynomial time computable function $f = \{f_k\}$ is informationally one-way if there is no probabilistic polynomial time algorithm which (with probability of the form $1 - k^{-e}$ for some $e > 0$) returns on input $y \in \{0,1\}^{k}$ a random element of $f^{-1}(y)$. \end{defn} In the non-uniform setting \cite{imlelu:oneway} show that these are not weaker than one-way functions: \begin{thm}[\cite{imlelu:oneway} (non-uniform)] \label{th-info-ow-ow} The existence of informationally one-way functions implies the existence of one-way functions. \end{thm} We will stick to the convention introduced above of saying ``non-uniform'' before the theorem statement when the theorem makes use of non-uniformity. It should be understood that if nothing is said then the result holds for both the uniform and the non-uniform models. It now follows from \thmref{th-info-ow-ow} that \begin{thm}[non-uniform]\label{th-weak-ske-owf} Weak SKE implies the existence of a one-way function. \end{thm} More recently, the polynomial-time, interior point algorithms for linear programming have been extended to the case of convex quadratic programs \cite{moad:quadpro,ye:intalg}, certain linear complementarity problems \cite{komiyo:lincomp,miyoki:lincomp}, and the nonlinear complementarity problem \cite{komiyo:unipfunc}. The connection between these algorithms and the classical Newton method for nonlinear equations is well explained in \cite{komiyo:lincomp}. \section{Review} \label{computation} We begin our discussion with the following definition: \begin{defn} A function $H\colon \Re^n \to \Re^n$ is said to be {\em 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 {\em B-derivative\/} of $H$ at $z$, such that \[ \lim_{v \to 0} \frac{H(z+v) - H(z) - BH(z)v}{\| v \|} = 0. \] The function $H$ is {\em B-differentiable in set $S$} if it is B-differentiable at every point in $S$. The B-derivative $BH(z)$ is said to be {\em strong\/} if \[ \lim_{(v,v') \to (0,0)} \frac{H(z+v) - H(z+v') - BH(z)(v -v')}{\| v - v' \|} = 0. \] \end{defn} \begin{lem}\label{limbog} There exists a smooth function $\psi_0(z)$ defined for $|z|>1-2a$ satisfying the following properties\rom: \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 $|z|>1$, then $\psi_0(z)=1$. \item For all $z$ in the domain of $\psi_0$, $\Delta_0\ln \psi_0\geq 0$. \item If $1-2a<|z|<1-a$, then $\Delta_0\ln \psi_0\geq c_3>0$. \end{enumerate} \end{lem} \begin{pf} We choose $\psi_0(z)$ to be a radial function depending only on $r=|z|$. Let $h(r)\geq 0$ be a suitable smooth function satisfying $h(r)\geq c_3$ for $1-2a<|z|<1-a$, and $h(r)=0$ for $|z|>1-\tfrac a2$. 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 $D_\nu=\{z|\,|z-z_\nu|<\delta\}$. Suppose that $D_\nu(a)$ denotes the smaller concentric disk $D_\nu(a)=\{z|\, |z-z_\nu|\leq (1-2a)\delta\}$. 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 $(1-2a)\delta<|z-z_\nu|<(1-a)\delta$. \end{enumerate} Let $A_\nu$ denote the annulus $A_\nu=\{(1-2a)\delta<|z-z_\nu|<(1-a) \delta \}$, 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{pf} Suppose that $\alpha$ is a nonnegative real constant. We apply Proposition~\ref{prop:eg} with $\Phi(z)=\Phi_ 0(z) e^{\alpha|z|^2}$. If $u\in C^\infty_0(R^2-\bigcup_\nu D_\nu(a))$, assume that $\cal D$ is a bounded domain containing the support of $u$ and $A\subset \cal D\subset R^2-\bigcup_\nu D_\nu(a)$. A calculation gives \[\int_{\cal D}|\overline\partial u|^2\Phi_ 0(z) e^{\alpha|z|^2}\geq c_4\alpha \int_{\cal D}|u|^2\Phi_ 0e^{\alpha|z|^2}+c_5\delta^{-2}\int_ A|u|^2\Phi_ 0e^{ \alpha|z|^2}.\] The boundedness, property (\ref{boundab}) of $\Phi_ 0$, then yields \[\int_{\cal D}|\overline\partial u|^2e^{\alpha|z|^2}\geq c_6\alpha \int_{\cal D}|u|^2e^{\alpha|z|^2}+c_7\delta^{-2}\int_ A|u|^2e^{\alpha|z|^2}.\] Let $B(X)$ be the set of blocks of $\Lambda_{X}$ and let $b(X) = |B(X)|$. 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 | \Lambda_{\phi} = \Lambda_{X} \}, \qquad Q_{X} = \{\phi \in M | \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 \[ |P_{Y}|= \sum_{X\geq Y} \mu (Y,X)|Q_{X}|.\] Thus there is a bijection from $Q_{X}$ to $W^{B(X)}$. In particular $|Q_{X}| = w^{b(X)}$. Next note that $b(X)=\dim X$. We see this by choosing a basis for $X$ consisting of vectors $v^{k}$ defined by \[v^{k}_{i}= \begin{cases} 1 & \text{if $i \in \Lambda_{k}$},\\ 0 &\text{otherwise.} \end{cases} \] \begin{verbatim} \[v^{k}_{i}= \begin{cases} 1 & \text{if $i \in \Lambda_{k}$},\\ 0 &\text{otherwise.} \end{cases} \] \end{verbatim} \begin{lem}\label{p0201} Let $\A$ be an arrangement. Then \[ \chi (\A,t) = \sum_{\B \subseteq \A} (-1)^{|\B|} t^{\dim T(\B)}. \] \end{lem} In order to compute $R''$ recall the definition of $S(X,Y)$ from \lemref{lem-per}. Since $H \in \B$, $\A_{H} \subseteq \B$. Thus if $T(\B) = Y$ then $\B \in S(H,Y)$. Let $L'' = L(\A'')$. Then \begin{equation}\label{E_SXgYy} \begin{split} R''&= \sum_{H\in \B \subseteq \A} (-1)^{|\B|} t^{\dim T(\B)}\\ &= \sum_{Y \in L''} \sum_{\B \in S(H,Y)} (-1)^{|\B|}t^{\dim Y} \\ &= -\sum_{Y \in L''} \sum_{\B \in S(H,Y)} (-1)^ {|\B - \A_{H}|} t^{\dim Y} \\ &= -\sum_{Y \in L''} \mu (H,Y)t^{\dim Y} \\ &= -\chi (\A '',t). \end{split} \end{equation} \begin{cor}\label{tripleA} Let $(\A,\A',\A'')$ be a triple of arrangements. Then \[ \pi (\A,t) = \pi (\A',t) + t \pi (\A'',t). \] \end{cor} \begin{defn} Let $(\A,\A',\A'')$ be a triple with respect to the hyperplane $H \in \A$. Call $H$ a {\em separator} if $T(\A) \not\in L(\A')$. \end{defn} \begin{cor}\label{nsep} Let $(\A,\A',\A'')$ be a triple with respect to $H \in \A$. \begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})} \item If $H$ is a separator then \[ \mu (\A) = - \mu (\A'') \] and hence \[ |\mu (\A)| = | \mu (\A'')|. \] \item If $H$ is not a separator then \[\mu (\A) = \mu (\A') - \mu (\A'') \] and \[ |\mu (\A)| = |\mu (\A')| + |\mu (\A'')|. \] \end{enumerate} \end{cor} \begin{pf} 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{pf} 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 \[M(\A) = \bigcup_{C \in \operatorname{Cham}(\A)} C.\] The number of chambers is determined by the Poincar\'e polynomial as follows. \begin{thm}\label{th-realarr} Let $\A_{\bold R}$ be a real arrangement. Then \[ |\operatorname{Cham}(\A_{\bold R})| = \pi (\A_{\bold R},1). \] \end{thm} \begin{pf} 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{pf} \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{pf} 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{pf} If $\phi$ is a protocol for $(X,Y)$, and $(x,y)$, $(x',y)$ are distinct inputs in $\SXY$, then, by the correct-decision property, $\langle\st_j(x,y)\rangle_{j=1}^\infty\ne\langle \st_j(x',y)\rangle_{j=1}^\infty$. Equation~(\ref{E_SXgYy}) defined $\PY$'s ambiguity set $\SXgYy$ to be the set of possible $X$ values when $Y=y$. The last corollary implies that for all $y\in\SY$, the multiset% \footnote{A multiset allows multiplicity of elements. Hence, $\{0,01,01\}$ is prefix free as a set, but not as a multiset.} of codewords $\{\st_\phi(x,y):x\in\SXgYy\}$ is prefix free. \section{One-Way Complexity} \label{S_Cp1} $\Cw1$, the one-way complexity of a random pair $\XcY$, is the number of bits $P_\X$ must transmit in the worst case when $\PY$ is not permitted to transmit any feedback messages. Starting with $\SXY$, the support set of $\XcY$, we define $\G$, the {\it characteristic hypergraph\/} of $\XcY$, and show that \[ \Cw1=\lceil\,\log\chi(\G)\rceil\ . \] Let $\XcY$ be a random pair. For each $y$ in $\SY$, the support set of $Y$, Equation~(\ref{E_SXgYy}) defined $\SXgYy$ to be the set of possible $x$ values when $Y=y$. The {\it characteristic hypergraph\/} $\G$ of $\XcY$ has $\SX$ as its vertex set and the hyperedge $\SXgYy$ for each $y\in\SY$. We can now prove a continuity theorem. \begin{thm}\label{t:conl} Let $\Omega \subset\bold R^n$ be an open set, let $u\in BV(\Omega ;\bold R^m)$, and let \begin{equation}\label{quts} T^u_x=\left\{y\in\bold R^m: y=\tilde u(x)+\left\langle \frac{Du}{|Du|}(x),z \right\rangle \text{ for some }z\in\bold R^n\right\} \end{equation} for every $x\in\Omega \backslash S_u$. Let $f\colon \bold R^m\to \bold R^k$ be a Lipschitz continuous function such that $f(0)=0$, and let $v=f(u)\colon \Omega \to \bold R^k$. Then $v\in BV(\Omega ;\bold R^k)$ and \begin{equation} Jv=(f(u^+)-f(u^-))\otimes \nu_u\cdot\, \cal H_{n-1}|_{S_u}.\end{equation} In addition, for $|\widetilde{D}u|$-almost every $x\in\Omega $ the restriction of the function $f$ to $T^u_x$ is differentiable at $\tilde u(x)$ and \begin{equation} \widetilde{D}v=\nabla (f|_{T^u_x})(\tilde u)\frac{\widetilde{D}u}{| \widetilde{D}u|}\cdot|\widetilde{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 $g\colon \left]0,+\infty\right[\to \bold R$. \end{rem} \begin{rem}\label{r:dif} Let $g \colon \bold R^n\to \bold R$ be a Lipschitz continuous function and assume that \[L(z)=\lim_{h\to 0^+}\frac{g(hz)-g(0)}h\] exists for every $z\in\bold Q^n$ and that $L$ is a linear function of $z$. Then $g$ is differentiable at 0. \end{rem} \begin{rem}\label{r:dif0} Let $A \colon \bold R^n\to \bold R^m$ be a linear function, and let $f \colon \bold R^m\to \bold R$ be a function. Then the restriction of $f$ to the range of $A$ is differentiable at 0 if and only if $f(A)\colon \bold R^n\to \bold R$ is differentiable at 0 and \[\nabla(f|_{\operatorname{Im}(A)})(0)A=\nabla (f(A))(0).\] \end{rem} \begin{pf} We begin by showing that $v\in BV(\Omega;\bold R^k)$ and \begin{equation}\label{e:bomb} |Dv|(B)\le K|Du|(B)\qquad\forall B\in\bold 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 $(u_h)\subset C^1(\Omega ;\bold R^m)$ converging to $u$ in $L^1(\Omega ;\bold R^m)$ and such that \[\lim_{h\to +\infty}\int_\Omega |\nabla u_h|\,dx=|Du|(\Omega ).\] The functions $v_h=f(u_h)$ are locally Lipschitz continuous in $\Omega $, and the definition of differential implies that $|\nabla v_h|\le K|\nabla u_h|$ almost everywhere in $\Omega $. The lower semicontinuity of the total variation and \eqref{sum-Di} yield \begin{equation} \begin{split} |Dv|(\Omega )\le\liminf_{h\to +\infty}|Dv_h|(\Omega) & =\liminf_{h\to +\infty}\int_\Omega |\nabla v_h|\,dx\\ &\le K\liminf_{h\to +\infty}\int_\Omega |\nabla u_h|\,dx=K|Du|(\Omega). \end{split}\end{equation} Since $f(0)=0$, we have also \[\int_\Omega |v|\,dx\le K\int_\Omega |u|\,dx;\] therefore $u\in BV(\Omega ;\bold R^k)$. Repeating the same argument for every open set $A\subset\Omega $, we get \eqref{e:bomb} for every $B\in\bold B(\Omega)$, because $|Dv|$, $|Du|$ 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 \[\{y\in B_\rho(x): |v(y)-f(\tilde u(x))|>\varepsilon \}\subset \{y\in B_\rho(x): |u(y)-\tilde u(x)|>\varepsilon /K\},\] hence \[\lim_{\rho\to 0^+}\frac{|\{y\in B_\rho(x): |v(y)-f(\tilde u(x))|> \varepsilon \}|}{\rho^n}=0\] 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\cal H_{n-1}&= \int_{B\cap S_v}(f(u^+)-f(u^-))\otimes \nu_u\,d\cal H_{n-1}\\ &=\int_{B\cap S_u}(f(u^+)-f(u^-))\otimes \nu_u\,d\cal H_{n-1} \end{split}\end{equation*} and \lemref{limbog} is proved. \end{pf} 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 $\Omega =\bold R^n$. 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 $|\widetilde{D}v|(S_u\backslash S_v)=0$, so that \eqref{e:st} and \eqref{e:barwq} imply that $Dv=\widetilde{D}v+Jv$ is the Radon-Nikod\'ym decomposition of $Dv$ in absolutely continuous and singular part with respect to $|\widetilde{D} u|$. By \thmref{th-weak-ske-owf}, we have \begin{equation*} \frac{\widetilde{D}v}{|\widetilde{D}u|}(t)=\lim_{s\to t^+} \frac{Dv(\interval{\left[t,s\right[})} {|\widetilde{D}u|(\interval{\left[t,s\right[})},\qquad \frac{\widetilde{D}u}{|\widetilde{D}u|}(t)=\lim_{s\to t^+} \frac{Du(\interval{\left[t,s\right[})} {|\widetilde{D}u|(\interval{\left[t,s\right[})} \end{equation*} $|\widetilde{D}u|$-almost everywhere in $\bold R$. 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 $t\in\bold R$. 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\bold R \end{equation} and \begin{equation}\label{alimo} \hat v(t)=f(\hat u(t))\qquad\forall t\in\bold R.\end{equation} Let $t\in\bold R$ be such that $|\widetilde{D}u|(\interval{\left[t,s\right[})>0$ 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)}{|\widetilde{D}u|(\interval{\left[t,s\right[})}&=\frac {f(\hat u(s))-f(\hat u(t))}{|\widetilde{D}u|(\interval{\left[t,s\right[})}\\ &=\frac{f(\hat u(s))-f(\hat u(t)+\dfrac{\widetilde{D}u}{|\widetilde{D}u|}(t)|\widetilde{D}u |(\interval{\left[t,s\right[}))}% {|\widetilde{D}u|(\interval{\left[t,s\right[})}\\ &+\frac {f(\hat u(t)+\dfrac{\widetilde{D}u}{|\widetilde{D}u|}(t)|\widetilde{D} u|(\interval{\left[t,s\right[}))-f(\hat u(t))}{|\widetilde{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|\frac{\hat v(s)-\hat v(t)}{|\widetilde{D}u|(\interval{\left[t,s\right[})} -\frac{f(\hat u(t)+\dfrac{\widetilde{D}u}{|\widetilde{D}u|}(t) |\widetilde{D}u|(\interval{\left[t,s\right[}))-f(\hat u(t))}{|\widetilde{D}u|(\interval{\left[t,s\right[})}\right|\\ \le K\left| \frac{\hat u(s)-\hat u(t)}{|\widetilde{D}u|(\interval{\left[t,s\right[})} -\frac{\widetilde{D}u}{| \widetilde{D}u|}(t)\right|.\end{multline*} }% end of group with \multlinegap=0pt By \eqref{e:bomb}, the function $s\to |\widetilde{D}u|(\interval{\left[t,s\right[})$ is continuous and converges to 0 as $s\downarrow t$. Therefore Remark~\ref{r:omb} and the previous inequality imply \[\frac{\widetilde{D}v}{|\widetilde{D}u|}(t)=\lim_{h\to 0^+} \frac{f(\hat u(t)+h\dfrac{\widetilde{D}u}{|\widetilde{D}u|} (t))-f(\hat u(t))}h\quad|\widetilde{D}u|\text{-a.e. in }\bold R.\] By \eqref{joe}, $\hat u(x)=\tilde u(x)$ for every $x\in\bold R\backslash S_u$; moreover, applying the same argument to the functions $u'(t)=u(-t)$, $v'(t)=f(u'(t))=v(-t)$, we get \[\frac{\widetilde{D}v}{|\widetilde{D}u|}(t)=\lim_{h\to 0} \frac{f(\tilde u(t)+h\dfrac{\widetilde{D}u}{|\widetilde{D}u|}(t))-f(\tilde u(t))}{h}\qquad|\widetilde{D}u|\text{-a.e. in }\bold R\] and our statement is proved. \subsection*{Step 2} Let us consider now the general case $n>1$. Let $\nu\in\bold R^n$ be such that $|\nu|=1$, and let $\pi_\nu=\{y\in\bold R^n: \langle y,\nu\rangle =0\}$. In the following, we shall identify $\bold R^n$ with $\pi_\nu\times\bold R$, and we shall denote by $y$ the variable ranging in $\pi_\nu$ and by $t$ the variable ranging in $\bold R$. By the just proven one-dimensional result, and by \thmref{thm-main}, we get \[\lim_{h\to 0}\frac{f(\tilde u(y+t\nu)+h\dfrac{\widetilde{D}u_y}{| \widetilde{D}u_y|}(t))-f(\tilde u(y+t\nu))}h=\frac{\widetilde{D}v_y}{| \widetilde{D}u_y|}(t)\qquad|\widetilde{D}u_y|\text{-a.e. in }\bold R\] for $\cal H_{n-1}$-almost every $y\in \pi_\nu$. We claim that \begin{equation} \frac{\langle \widetilde{D}u,\nu\rangle }{|\langle \widetilde{D}u,\nu\rangle |}(y+t\nu)=\frac{\widetilde{D}u_y} {|\widetilde{D}u_y|}(t)\qquad|\widetilde{D}u_y|\text{-a.e. in }\bold R \end{equation} for $\cal H_{n-1}$-almost every $y\in\pi_\nu$. In fact, by \eqref{sum-ali} and \eqref{delta-l} we get \begin{multline*} \int_{\pi_\nu}\frac{\widetilde{D}u_y}{|\widetilde{D}u_y|}\cdot|\widetilde{D}u_y |\,d\cal H_{n-1}(y)=\int_{\pi_\nu}\widetilde{D}u_y\,d\cal H_{n-1}(y)\\ =\langle \widetilde{D}u,\nu\rangle =\frac {\langle \widetilde{D}u,\nu\rangle }{|\langle \widetilde{D}u,\nu\rangle|}\cdot |\langle \widetilde{D}u,\nu\rangle |=\int_{\pi_\nu}\frac{ \langle \widetilde{D}u,\nu\rangle }{|\langle \widetilde{D}u,\nu\rangle |} (y+\cdot \nu)\cdot|\widetilde{D}u_y|\,d\cal 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 \widetilde{D}v,\nu\rangle }{|\langle \widetilde{D}u,\nu\rangle |}(y+t\nu)=\frac{\widetilde{D}v_y}{|\widetilde{D}u_y|}(t)\qquad| \widetilde{D}u_y|\text{-a.e. in }\bold R\end{equation} for $\cal H_{n-1}$-almost every $y\in \pi_\nu$. By \eqref{far-d} and \eqref{E_SXgYy} we get \[ \lim_{h\to 0}\frac{f(\tilde u(y+t\nu)+h\dfrac{\langle \widetilde{D} u,\nu\rangle }{|\langle \widetilde{D}u,\nu\rangle |}(y+t\nu))-f(\tilde u(y+t\nu))}{h} =\frac{\langle \widetilde{D}v,\nu\rangle }{|\langle \widetilde{D}u,\nu\rangle |}(y+t\nu)\] for $\cal H_{n-1}$-almost every $y\in\pi_\nu$, and using again \eqref{detK1}, \eqref{detK2} we get \[ \lim_{h\to 0}\frac{f(\tilde u(x)+h\dfrac{\langle \widetilde{D}u,\nu\rangle }{|\langle \widetilde{D}u,\nu\rangle |}(x))-f(\tilde u(x))}{h}=\frac{\langle \widetilde{D}v,\nu\rangle }{|\langle \widetilde{D}u,\nu \rangle |}(x) \] $|\langle \widetilde{D}u,\nu\rangle|$-a.e. in $\bold R^n$. Since the function $|\langle \widetilde{D}u,\nu\rangle |/|\widetilde{D}u|$ is strictly positive $|\langle \widetilde{D}u,\nu\rangle |$-almost everywhere, we obtain also \[ \lim_{h\to 0}\frac{f(\tilde u(x)+h\dfrac{|\langle \widetilde{D}u,\nu\rangle |}{|\widetilde{D}u|}(x)\dfrac{\langle \widetilde{D} u,\nu\rangle }{|\langle \widetilde{D}u,\nu\rangle |}(x))-f(\tilde u(x))}{h}\\ =\frac{|\langle \widetilde{D}u,\nu\rangle |}{|\widetilde{D}u|}(x)\frac {\langle \widetilde{D}v,\nu\rangle }{|\langle \widetilde{D}u,\nu\rangle |}(x) \] $|\langle \widetilde{D}u,\nu\rangle |$-almost everywhere in $\bold R^n$. Finally, since \begin{align*} &\frac{|\langle \widetilde{D}u,\nu\rangle |}{|\widetilde{D}u|} \frac{\langle \widetilde{D}u,\nu\rangle }{|\langle \widetilde{D}u,\nu\rangle|} =\frac{\langle \widetilde{D}u,\nu\rangle }{|\widetilde{D}u|} =\left\langle \frac{\widetilde{D}u}{|\widetilde{D}u|},\nu\right\rangle \qquad|\widetilde{D}u|\text{-a.e. in }\bold R^n\\ &\frac{|\langle \widetilde{D}u,\nu\rangle |}{|\widetilde{D}u|} \frac{\langle \widetilde{D}v,\nu\rangle }{|\langle \widetilde{D}u,\nu\rangle|} =\frac{\langle \widetilde{D}v,\nu\rangle }{|\widetilde{D}u|} =\left\langle \frac{\widetilde{D}v}{|\widetilde{D}u|},\nu\right\rangle \qquad|\widetilde{D}u|\text{-a.e. in }\bold R^n \end{align*} and since both sides of \eqref{alimo} are zero $|\widetilde{D}u|$-almost everywhere on $|\langle \widetilde{D}u,\nu\rangle |$-negligible sets, we conclude that \[ \lim_{h\to 0}\frac{f\left( \tilde u(x)+h\left\langle \dfrac{\widetilde{D} u}{|\widetilde{D}u|}(x),\nu\right\rangle \right)-f(\tilde u(x))}h =\left\langle \frac{\widetilde{D}v}{|\widetilde{D}u|}(x),\nu\right\rangle, \] $|\widetilde{D}u|$-a.e. in $\bold R^n$. 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 $|\widetilde{D} u|$-almost every $x\in \bold R^n$ 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\bold n}(-1)^{|I|-1}|I| \prod_{i\in I}t_i\prod_{j\in I}(D_j+\lambda_jt_j)\det\bold 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\bold n}\hat x_i \sum_{I\in\bold n}(-1)^{|I|-1}|I|\per \bold A ^{(\lambda)}(I|I)\det\bold A^{(\lambda)}(\overline I|\overline I). \end{equation} By \eqref{H-cycles}, \eqref{sum-Di}, and \eqref{Dx}, we have the following result: {\samepage \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\bold n}\per \bold A ^{(\lambda)}(I_l |I_l )\det\bold A^{((\lambda)} (\overline I_{l}|\overline I_l ),|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\bold K(t=1,t_1,\dots,t_n;l |l )\\ =\sum_{I\subseteq\bold n-\{l \}} (-1)^{|I|}\prod_{i\in I}t_i\prod_{j\in I} (D_j+\lambda_jt_j)\det\bold 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\bold n-\{l \}} (-1)^{|I|}\per\bold A^{(\lambda)}(I|I)\det \bold 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} %% Added \normalshape here to avoid error message about %% cmtcsc10 font not found (for those who don't happen to %% have it). \section{Various font features of the {\ntt amstex} option} \label{s:font} \subsection{Bold versions of special symbols} In the \opt{amstex} option \cs{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 \cs{bold}. For example, \begin{verbatim} A_\infty + \pi A_0 \sim \bold{A}_{\boldsymbol{\infty}} \boldsymbol{+} \boldsymbol{\pi} \bold{A}_{\boldsymbol{0}} \end{verbatim} looks like this: \[A_\infty + \pi A_0 \sim \bold{A}_{\boldsymbol{\infty}} \boldsymbol{+} \boldsymbol{\pi} \bold{A}_{\boldsymbol{0}}\] \subsection{``Poor man's bold''} If a bold version of a particular symbol doesn't exist in the available fonts, then \cs{boldsymbol} can't be used to make that symbol bold. At the present time, this means that \cs{boldsymbol} can't be used with symbols from the {\sc msam} and {\sc msbm} fonts, among others. In some cases, poor man's bold (\cs{pmb}) can be used instead of \cs{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, \cs{mathop}, to produce proper spacing and limits when \cs{pmb} is used. For further details see {\it The \TeX book}. \[\sum\begin{Sb}i>>" and \verb"@<<<" produce arrows that extend automatically to accommodate unusually wide subscripts or superscripts. The text of the subscript or superscript is typed in between the \verb+>+ or \verb+<+ symbols. Example: \[F\times\triangle[n-1] @>>{\partial_0\alpha(b)}> E^{\partial_0b}\] \begin{verbatim} \[F\times\triangle[n-1] @>>{\partial_0\alpha(b)}> E^{\partial_0b}\] \end{verbatim} For users whose keyboards don't have \verb=<= and \verb=>= keys, \verb=@)))= and \verb=@(((= are available as synonyms. \subsection{\cs{overset}, \cs{underset}, and \cs{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 \cs{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 \cs{text} command} The main use of the command \cs{text} is for words or phrases in a display: \[\bold{y}=\bold{y}'\quad\text{if and only if}\quad y'_k=\delta_k y_{\tau(k)}\] \begin{verbatim} \[\bold{y}=\bold{y}'\quad\text{if and only if}\quad y'_k=\delta_k y_{\tau(k)}\] \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 \opt{amstex} option provides \cs{operatorname} and \cs{operatornamewithlimits} for producing new function names that will have the same typographical treatment. Examples: \[\|f\|_\infty= \operatornamewithlimits{ess\,sup}_{x\in R^n}|f(x)|\] \begin{verbatim} \[\|f\|_\infty= \operatornamewithlimits{ess\,sup}_{x\in R^n}|f(x)|\] \end{verbatim} \[\operatorname{meas}_1\{u\in R_+^1\:f^*(u)>\alpha\} =\operatorname{meas}_n\{x\in R^n\:|f(x)|\geq\alpha\} \quad \forall\alpha>0.\] \begin{verbatim} \[\operatorname{meas}_1\{u\in R_+^1\:f^*(u)>\alpha\} =\operatorname{meas}_n\{x\in R^n\:|f(x)|\geq\alpha\} \quad \forall\alpha>0.\] \end{verbatim} If you use a particular operator name often, you can save yourself some typing and make your \LaTeX\ file more readable by defining an abbreviation in the preamble area of your document, using \cs{newcommand}: \begin{verbatim} \newcommand{\esssup}{\operatornamewithlimits{ess\,sup}} \newcommand{\meas}{\operatorname{meas}} \end{verbatim} Some special operatornames are predefined in the \opt{amstex} option: \cs{varlimsup}, \cs{varliminf}, \cs{varinjlim}, and \cs{varprojlim}. Here's what they look like in use: \begin{align} &\varlimsup_{n\rightarrow\infty} \cal{Q}(u_n,u_n-u^{\#})\le0\\ &\varliminf_{n\rightarrow\infty} \left|a_{n+1}\right|/\left|a_n\right|=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} \cal{Q}(u_n,u_n-u^{\#})\le0\\ &\varliminf_{n\rightarrow\infty} \left|a_{n+1}\right|/\left|a_n\right|=0\\ &\varinjlim (m_i^\lambda\cdot)^*\le0\\ &\varprojlim_{p\in S(A)}A_p\le0 \end{align} \end{verbatim} \subsection{\cs{mod} and its relatives} The commands \cs{mod} and \cs{pod} are variants of \cs{pmod} preferred by some authors; \cs{mod} omits the parentheses, whereas \cs{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} In the \opt{amstex} option, \cs{frac} has a square bracket option that can be used to specify the thickness of the fraction line. For example: \[\frac[1.5pt]{H(z+v)-H(z)-BH(z)v}{\|v\|}\] \begin{verbatim} \[\frac[1.5pt]{H(z+v)-H(z)-BH(z)v}{\|v\|}\] \end{verbatim} There is also \cs{fracwithdelims}, for stacked constructions with delimiters on either side. \[\fracwithdelims[]{H(z+v)-H(z)-BH(z)v}{\|v\|}\] \begin{verbatim} \[\fracwithdelims[]{H(z+v)-H(z)-BH(z)v}{\|v\|}\] \end{verbatim} Because it is used fairly often, the construction \verb=\fracwithdelims()[0pt]= has a short form, \cs{binom}. 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} \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 {\samepage \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 \cs{lcfrac} or \cs{rcfrac} instead of \cs{cfrac}. \subsection{Smash} In \opt{amstex} there are optional arguments \verb"t" and \verb"b" for the plain \TeX\ command \cs{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 $X_j=(1/\sqrt{\smash[b]{\lambda_j}})X_j'$ \cs{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 \cs{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 \cs{text} and the embedded math. \subsection{Matrix} Here are samples of the matrix environments, \cs{matrix}, \cs{pmatrix}, \cs{bmatrix}, \cs{vmatrix} and \cs{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{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{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. \cs{hdotsfor}\verb"{"{\it 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 \env{Sb} and \env{Sp} environments} The \env{Sb} and \env{Sp} environments can be used to typeset several lines as a subscript or superscript: for example \begin{verbatim} \begin{equation} \sum\begin{Sb} 0\le i\le m\\ 0j>> T\\ @VVV @VV{\End P}V\\ (S\otimes T)/I @= (Z\otimes T)/J \end{CD}\end{equation} \begin{verbatim} \begin{equation}\begin{CD} S^{{\cal W}_\Lambda}\otimes T @>j>> T\\ @VVV @VV{\End P}V\\ (S\otimes T)/I @= (Z\otimes T)/J \end{CD}\end{equation} \end{verbatim} (assuming \cs{End} is defined as \cs{operatorname}{\tt\char`\{End\char`\}}). \subsection{Big-g-g-g delimiters} Here are some big delimiters, first in \cs{normalsize}: \[\biggl(\bold E_{y} \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds \biggr) \] \begin{verbatim} \[\biggl(\bold E_{y} \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds \biggr) \] \end{verbatim} and now in \cs{Large} size: % amsart.sty doesn't define \Large, so we have to do this: {\makeatletter\@setsize\Large{18\p@}\xivpt\@xivpt \[\biggl(\bold E_{y} \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds \biggr) \]} \begin{verbatim} {\Large \[\biggl(\bold E_{y} \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds \biggr) \]} \end{verbatim} \bigskip \noindent{\large\bf Note: Starting on the following 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.} \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.1pt \vbox to\@colht{\boxmaxdepth\maxdepth \@texttop\dimen128=\dp\@outputbox\unvbox\@outputbox \vskip-\dimen128\@textbottom}% \vrule width.1pt}% \global\maxdepth\@maxdepth} \makeatother %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \appendix \section{Examples of multiple-line equation structures} \label{s:eq} The lines indicating the margins are to make the marginal spacing stand out more clearly in some of the display examples in this appendix. \subsection{Split} The {\tt split} environment is not an independent environment but should be used inside something else such as {\tt equation} or {\tt align}. If there is not enough room for it, the equation number for a {\tt 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\bold 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(\bold 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(\bold E_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)} \varphi(x)\,ds -\bold E_{x,y}\int_0^{t_\varepsilon} L_{x,y_\varepsilon(\varepsilon s)} \varphi(x)\,ds\biggr)\biggr]\\ &=h\widehat{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\bold 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(\bold 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(\bold E_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)} \varphi(x)\,ds -\bold E_{x,y}\int_0^{t_\varepsilon} L_{x,y_\varepsilon(\varepsilon s)} \varphi(x)\,ds\biggr)\biggr]\\ &=h\widehat{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\bold 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(\bold 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(\bold E_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)} \varphi(x)\,ds -\bold E_{x,y}\int_0^{t_\varepsilon} L_{x,y_\varepsilon(\varepsilon s)} \varphi(x)\,ds\biggr)\biggr]\\ &=h\widehat{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\bold 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(\bold 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(\bold E_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)} \varphi(x)\,ds -\bold E_{x,y}\int_0^{t_\varepsilon} L_{x,y_\varepsilon(\varepsilon s)} \varphi(x)\,ds\biggr)\biggr]\\ &=h\widehat{L}_x\varphi(x)+h\theta_\varepsilon(x,y), \end{split} \end{equation*} \end{verbatim} %%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage If the option {\tt ctagsplt} is included in the documentstyle options list, the equation numbers for {\tt split} environments will be centered vertically on the height of the {\tt split}: {\makeatletter\ctagsplit@true \begin{equation} \begin{split} |I_2|&=\left|\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|\\ &\le C_6\left|\left|f\int_\Omega\left|\widetilde{S}^{-1,0}_{a,-} W_2(\Omega,\Gamma_l)\right|\right| \left||u|\overset{\circ}\to W_2^{\widetilde{A}} (\Omega;\Gamma_r,T)\right|\right|. \end{split} \end{equation}}% Some text after to test the below-display spacing. %%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage Use of {\tt split} within {\tt align}: {\delimiterfactor750 \begin{align} \begin{split}|I_1|&=\left|\int_\Omega gRu\,d\Omega\right|\\ &\le C_3\left[\int_\Omega\left(\int_{a}^x g(x\i,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|\left|f\left|\widetilde{S}^{-1,0}_{a,-} W_2(\Omega,\Gamma_l)\right|\right| \left||u|\overset{\circ}\to W_2^{\widetilde{A}} (\Omega;\Gamma_r,T)\right|\right|. \end{split}\label{eq:A}\\ \begin{split}|I_2|&=\left|\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|\\ &\le C_6\left|\left|f\int_\Omega \left|\widetilde{S}^{-1,0}_{a,-} W_2(\Omega,\Gamma_l)\right|\right| \left||u|\overset{\circ}\to W_2^{\widetilde{A}} (\Omega;\Gamma_r,T)\right|\right|. \end{split} \end{align}}% Some text after to test the below-display spacing. \begin{verbatim} \begin{align} \begin{split}|I_1|&=\left|\int_\Omega gRu\,d\Omega\right|\\ &\le C_3\left[\int_\Omega\left(\int_{a}^x g(x\i,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|\left|f\left|\widetilde{S}^{-1,0}_{a,-} W_2(\Omega,\Gamma_l)\right|\right| \left||u|\overset{\circ}\to W_2^{\widetilde{A}} (\Omega;\Gamma_r,T)\right|\right|. \end{split}\label{eq:A}\\ \begin{split}|I_2|&=\left|\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|\\ &\le C_6\left|\left|f\int_\Omega \left|\widetilde{S}^{-1,0}_{a,-} W_2(\Omega,\Gamma_l)\right|\right| \left||u|\overset{\circ}\to W_2^{\widetilde{A}} (\Omega;\Gamma_r,T)\right|\right|. \end{split} \end{align} \end{verbatim} %%%%%%%%%%%%%%%%%% \newpage Unnumbered {\tt align}, with a number on the second {\tt split}: \begin{align*} \begin{split}|I_1|&=\left|\int_\Omega gRu\,d\Omega\right|\\ &\le C_3\left[\int_\Omega\left(\int_{a}^x g(x\i,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|\left|f\left|\widetilde{S}^{-1,0}_{a,-} W_2(\Omega,\Gamma_l)\right|\right| \left||u|\overset{\circ}\to W_2^{\widetilde{A}} (\Omega;\Gamma_r,T)\right|\right|. \end{split}\\ \begin{split}|I_2|&=\left|\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|\\ &\le C_6\left|\left|f\int_\Omega \left|\widetilde{S}^{-1,0}_{a,-} W_2(\Omega,\Gamma_l)\right|\right| \left||u|\overset{\circ}\to W_2^{\widetilde{A}} (\Omega;\Gamma_r,T)\right|\right|. \end{split}\tag{\theequation$'$} \end{align*} Some text after to test the below-display spacing. \begin{verbatim} \begin{align*} \begin{split}|I_1|&=\left|\int_\Omega gRu\,d\Omega\right|\\ &\le C_3\left[\int_\Omega\left(\int_{a}^x g(x\i,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|\left|f\left|\widetilde{S}^{-1,0}_{a,-} W_2(\Omega,\Gamma_l)\right|\right| \left||u|\overset{\circ}\to W_2^{\widetilde{A}} (\Omega;\Gamma_r,T)\right|\right|. \end{split}\\ \begin{split}|I_2|&=\left|\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|\\ &\le C_6\left|\left|f\int_\Omega \left|\widetilde{S}^{-1,0}_{a,-} W_2(\Omega,\Gamma_l)\right|\right| \left||u|\overset{\circ}\to W_2^{\widetilde{A}} (\Omega;\Gamma_r,T)\right|\right|. \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} %%%%%%%%%%%%%%%%%%%%%%%%%%% \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} %%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Gather} Numbered version with \verb;\notag; on the second line: \begin{gather} D(a,r)\equiv\{z\in\bold C\colon |z-a| Question mark \? %% Commercial at \@ Left bracket \[ Backslash \\ %% Right bracket \] Circumflex \^ Underscore \_ %% Grave accent \` Left brace \{ Vertical bar \| %% Right brace \} Tilde \~} \endinput