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Diffstat (limited to 'Master/texmf-dist/source')
22 files changed, 460 insertions, 6648 deletions
diff --git a/Master/texmf-dist/source/latex/amsmath/README b/Master/texmf-dist/source/latex/amsmath/README deleted file mode 100644 index 6e449706be8..00000000000 --- a/Master/texmf-dist/source/latex/amsmath/README +++ /dev/null @@ -1,148 +0,0 @@ -README for amsmath 2.14 [2013/01/14] American Mathematical Society - -Copyright 2001-2004, 2007, 2008, 2010, 2011, 2013 American Mathematical Society. - -This work may be distributed and/or modified under the -conditions of the LaTeX Project Public License, either version 1.3c -of this license or (at your option) any later version. -The latest version of this license is in - http://www.latex-project.org/lppl.txt -and version 1.3c or later is part of all distributions of LaTeX -version 2005/12/01 or later. - -This work has the LPPL maintenance status `maintained'. - -The Current Maintainer of this work is the American Mathematical -Society. - -American Mathematical Society -Technical Support -Publications Technical Group -201 Charles Street -Providence, RI 02904 -USA -tel: (401) 455-4080 - (800) 321-4267 (USA and Canada only) -fax: (401) 331-3842 -email: tech-support@ams.org - -======================================================================== - CONTENTS -I. OVERVIEW -II. INSTALLATION AND GETTING STARTED -III. SUBMITTING BUG REPORTS -IV. REMARKS ON THIS RELEASE -V. CHANGE LOG - -======================================================================== -I. OVERVIEW - -The amsmath package is an extension package for LaTeX that provides -additional features to facilitate mathematical typesetting. It has been -developed by the American Mathematical Society and released for general -use as a service to the mathematical community. A number of smaller -auxiliary packages are also distributed with the amsmath package. - -======================================================================== -II. INSTALLATION AND GETTING STARTED - -In order to use amsmath you need to have TeX installed first. TeX is -not an AMS product. For information on getting TeX see one of the -following: - - http://www.tug.org/ - http://www.ams.org/tex/tex-resources - -If you are installing amsmath manually, see install.txt. - -The primary documentation for amsmath is in - - amsldoc.pdf - -Additional documentation files include: - - diffs-m.txt - subeqn.pdf - technote.pdf - testmath.pdf - -which are included in the collection. - -======================================================================== -III. SUBMITTING BUG REPORTS - -Technical support is available via the addresses listed above. Bug -reports should be submitted using the standard LaTeX bug reporting -system: Execute the command - - latex latexbug.tex - -and follow the resulting instructions. Select "amslatex" when asked -for a category. - -======================================================================== -IV. REMARKS ON THIS RELEASE - -Version 2.14 is a minor update to provide more fluid interoperability -with stix.sty. It should be fully compatible with version 2.13. - -======================================================================== -V. CHANGE LOG (REVERSE CHRONOLOGICAL ORDER) - -2013-01-14 amsmath.dtx 2.14 - - * amsmath.sty 2.14 - -- Minimal changes to make amsmath compatible with stix.sty. - -2000-07-18 amsmath.dtx 2.13 - -- After the numbering patches in 2.11, \notag failed in certain - circumstances: introduce some more auxiliary functions to sort - things out, and redefine \nonumber. - -2000-06-29 amstext.dtx 2.01 - -- Use \f@size instead of \tf@size because they are not - necessarily the same and the former is better for putting a few - words into a display. - -2000-06-06 amsmath.dtx 2.12 - -- Fix transposed lines in 2.11 patch. - -2000-06-02 amsmath.dtx 2.11 - - * amsmath.sty 2.11 - -- Prevent "Arithmetic overflow" error by guarding against - divide-by-zero in \x@calc@shift@lc (align environment). - - * amsdtx.dtx - -- Moved to the amscls distribution. - -2000-05-25 amsmath.dtx 2.10 - -- Clear up error message for \allowdisplaybreaks[0]. - -- Make mathdisplay re-entrant by introducing mathdisplay@stack, - to clear up numbering problems in unusual circumstances such as - \[ \] nested inside minipage inside equation. - -2000-04-21 amsmath.dtx 2.09 - -- Ensure good catcodes for " etc. - -2000-03-16 amsmath.dtx 2.08 - -- Fixed erroneous tag placement on split with fleqn/tbtags options. - -2000-03-15 amsmath.dtx 2.07 - -- Add \reset@strutbox@ to deal with the following bug: After - $...\mbox{\Huge $...$}...$, line spacing is wrong in a - following "gather" or other environment that uses \strut@. - -- Patch to fix bug with intlimits option: too much space in the - middle of \iint. - -- Overhaul math accents again to fix a couple of bugs reported by - Thimm. - -2000-03-10 amsmath.dtx 2.06 - -- Change \MathAccent to \mathaccentV so \DeclareMathAccent won't - give an error when redefining an accent. - -2000-01-06 amsmath.dtx 2.05 - -- Fixed incorrect placement of fleqn/reqno equation numbers - inside indented lists (displaywidth < columnwidth). Changed the - multline/fleqn/leqno indent to match mathmargin when possible - instead of always just using multlinetaggap. diff --git a/Master/texmf-dist/source/latex/amsmath/amsbsy.dtx b/Master/texmf-dist/source/latex/amsmath/amsbsy.dtx index a5a4124a096..0cb877b225c 100644 --- a/Master/texmf-dist/source/latex/amsmath/amsbsy.dtx +++ b/Master/texmf-dist/source/latex/amsmath/amsbsy.dtx @@ -1,21 +1,8 @@ -% \def\filename{amsbsy.dtx} -% \def\fileversion{1.2d} -% \def\filedate{1999/11/29} % % \iffalse meta-comment % -% American Mathematical Society -% Technical Support -% Publications Technical Group -% 201 Charles Street -% Providence, RI 02904 -% USA -% tel: (401) 455-4080 -% (800) 321-4267 (USA and Canada only) -% fax: (401) 331-3842 -% email: tech-support@ams.org -% % Copyright 1995, 1999 American Mathematical Society. +% Copyright 2016 LaTeX3 Project and American Mathematical Society. % % This work may be distributed and/or modified under the % conditions of the LaTeX Project Public License, either version 1.3c @@ -27,15 +14,21 @@ % % This work has the LPPL maintenance status `maintained'. % -% The Current Maintainer of this work is the American Mathematical -% Society. +% The Current Maintainer of this work is the LaTeX3 Project. % % \fi % %\iffalse %<*driver> \documentclass{amsdtx} +\def\MaintainedByLaTeXTeam#1{% +\begin{center}% +\fbox{\fbox{\begin{tabular}{@{}l@{}}% +This file is maintained by the \LaTeX{} Project team.\\% +Bug reports can be opened (category \texttt{#1}) at\\% +\url{http://latex-project.org/bugs.html}.\end{tabular}}}\end{center}} \usepackage{amsbsy} +\GetFileInfo{amsbsy.sty} \begin{document} \title{The \pkg{amsbsy} package} \author{Frank Mittelbach\and Rainer Sch\"opf\and Michael Downes} @@ -46,6 +39,7 @@ %\fi % % \maketitle +% \MaintainedByLaTeXTeam{amsmath} % % \MakeShortVerb\| % @@ -92,7 +86,7 @@ % \begin{macrocode} \NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX) [1994/12/01]% LaTeX date must December 1994 or later -\ProvidesPackage{amsbsy}[1999/11/29 v1.2d] +\ProvidesPackage{amsbsy}[1999/11/29 v1.2d Bold Symbols] % \end{macrocode} % % \section{The implementation} diff --git a/Master/texmf-dist/source/latex/amsmath/amsbsy.ins b/Master/texmf-dist/source/latex/amsmath/amsbsy.ins index 90a179b2d01..78c68fea112 100644 --- a/Master/texmf-dist/source/latex/amsmath/amsbsy.ins +++ b/Master/texmf-dist/source/latex/amsmath/amsbsy.ins @@ -1,19 +1,6 @@ -%% filename: amsbsy.ins -%% version: 1.00 -%% date: 2010/08/13 -%% -%% American Mathematical Society -%% Technical Support -%% Publications Technical Group -%% 201 Charles Street -%% Providence, RI 02904 -%% USA -%% tel: (401) 455-4080 -%% (800) 321-4267 (USA and Canada only) -%% fax: (401) 331-3842 -%% email: tech-support@ams.org %% %% Copyright 2010 American Mathematical Society. +%% Copyright 2016 LaTeX3 Project and American Mathematical Society. %% %% This work may be distributed and/or modified under the %% conditions of the LaTeX Project Public License, either version 1.3c @@ -25,8 +12,7 @@ %% %% This work has the LPPL maintenance status `maintained'. %% -%% The Current Maintainer of this work is the American Mathematical -%% Society. +%% The Current Maintainer of this work is the LaTeX3 Project. %% %% ==================================================================== @@ -39,17 +25,7 @@ \preamble Copyright 1995, 1999 American Mathematical Society. - -American Mathematical Society -Technical Support -Publications Technical Group -201 Charles Street -Providence, RI 02904 -USA -tel: (401) 455-4080 - (800) 321-4267 (USA and Canada only) -fax: (401) 331-3842 -email: tech-support@ams.org +Copyright 2016 LaTeX3 Project and American Mathematical Society. This work may be distributed and/or modified under the conditions of the LaTeX Project Public License, either version 1.3c @@ -61,8 +37,7 @@ version 2005/12/01 or later. This work has the LPPL maintenance status `maintained'. -The Current Maintainer of this work is the American Mathematical -Society. +The Current Maintainer of this work is the LaTeX3 project. \endpreamble diff --git a/Master/texmf-dist/source/latex/amsmath/amscd.dtx b/Master/texmf-dist/source/latex/amsmath/amscd.dtx index f1e97aef587..10009e4815e 100644 --- a/Master/texmf-dist/source/latex/amsmath/amscd.dtx +++ b/Master/texmf-dist/source/latex/amsmath/amscd.dtx @@ -1,21 +1,8 @@ -% \def\filename{amscd.dtx} -% \def\fileversion{2.0} -% \def\filedate{1999/11/29} % % \iffalse meta-comment % -% American Mathematical Society -% Technical Support -% Publications Technical Group -% 201 Charles Street -% Providence, RI 02904 -% USA -% tel: (401) 455-4080 -% (800) 321-4267 (USA and Canada only) -% fax: (401) 331-3842 -% email: tech-support@ams.org -% % Copyright 1995, 1999 American Mathematical Society. +% Copyright 2016 LaTeX3 Project and American Mathematical Society. % % This work may be distributed and/or modified under the % conditions of the LaTeX Project Public License, either version 1.3c @@ -27,15 +14,21 @@ % % This work has the LPPL maintenance status `maintained'. % -% The Current Maintainer of this work is the American Mathematical -% Society. +% The Current Maintainer of this work is the LaTeX3 Project. % % \fi % %\iffalse %<*driver> \documentclass{amsdtx} +\def\MaintainedByLaTeXTeam#1{% +\begin{center}% +\fbox{\fbox{\begin{tabular}{@{}l@{}}% +This file is maintained by the \LaTeX{} Project team.\\% +Bug reports can be opened (category \texttt{#1}) at\\% +\url{http://latex-project.org/bugs.html}.\end{tabular}}}\end{center}} \usepackage{amscd} +\GetFileInfo{amscd.sty} % Dummy citation to avoid error in an example. \bibcite{fre:cichon}{C} \begin{document} @@ -48,6 +41,7 @@ %\fi % % \maketitle +% \MaintainedByLaTeXTeam{amsmath} % % \MakeShortVerb\| % \newcommand{\operatorname}[1]{\mathop{\mathrm{#1}}} @@ -109,7 +103,7 @@ % \begin{macrocode} \NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX) [1994/12/01]% LaTeX date must December 1994 or later -\ProvidesPackage{amscd}[1999/11/29 v2.0] +\ProvidesPackage{amscd}[1999/11/29 v2.0 AMS Commutatitive Diagrams] % \end{macrocode} % \begin{macrocode} \RequirePackage{amsgen} diff --git a/Master/texmf-dist/source/latex/amsmath/amscd.ins b/Master/texmf-dist/source/latex/amsmath/amscd.ins index dd05ab15bde..abae05afa00 100644 --- a/Master/texmf-dist/source/latex/amsmath/amscd.ins +++ b/Master/texmf-dist/source/latex/amsmath/amscd.ins @@ -1,19 +1,6 @@ -%% filename: amscd.ins -%% version: 1.00 -%% date: 2010/08/13 -%% -%% American Mathematical Society -%% Technical Support -%% Publications Technical Group -%% 201 Charles Street -%% Providence, RI 02904 -%% USA -%% tel: (401) 455-4080 -%% (800) 321-4267 (USA and Canada only) -%% fax: (401) 331-3842 -%% email: tech-support@ams.org %% %% Copyright 2010 American Mathematical Society. +%% Copyright 2016 LaTeX3 Project and American Mathematical Society. %% %% This work may be distributed and/or modified under the %% conditions of the LaTeX Project Public License, either version 1.3c @@ -25,8 +12,7 @@ %% %% This work has the LPPL maintenance status `maintained'. %% -%% The Current Maintainer of this work is the American Mathematical -%% Society. +%% The Current Maintainer of this work is the LaTeX3 Project. %% %% ==================================================================== @@ -39,17 +25,7 @@ \preamble Copyright 1995, 1999 American Mathematical Society. - -American Mathematical Society -Technical Support -Publications Technical Group -201 Charles Street -Providence, RI 02904 -USA -tel: (401) 455-4080 - (800) 321-4267 (USA and Canada only) -fax: (401) 331-3842 -email: tech-support@ams.org +Copyright 2016 LaTeX3 Project and American Mathematical Society. This work may be distributed and/or modified under the conditions of the LaTeX Project Public License, either version 1.3c @@ -61,8 +37,7 @@ version 2005/12/01 or later. This work has the LPPL maintenance status `maintained'. -The Current Maintainer of this work is the American Mathematical -Society. +The Current Maintainer of this work is the LaTeX3 project. \endpreamble diff --git a/Master/texmf-dist/source/latex/amsmath/amsgen.dtx b/Master/texmf-dist/source/latex/amsmath/amsgen.dtx index 366b3aab041..e3c50bde63c 100644 --- a/Master/texmf-dist/source/latex/amsmath/amsgen.dtx +++ b/Master/texmf-dist/source/latex/amsmath/amsgen.dtx @@ -1,21 +1,8 @@ -% \def\filename{amsgen.dtx} -% \def\fileversion{2.0} -% \def\filedate{1999/11/30} % % \iffalse meta-comment % -% American Mathematical Society -% Technical Support -% Publications Technical Group -% 201 Charles Street -% Providence, RI 02904 -% USA -% tel: (401) 455-4080 -% (800) 321-4267 (USA and Canada only) -% fax: (401) 331-3842 -% email: tech-support@ams.org -% % Copyright 1995, 1999 American Mathematical Society. +% Copyright 2016 LaTeX3 Project and American Mathematical Society. % % This work may be distributed and/or modified under the % conditions of the LaTeX Project Public License, either version 1.3c @@ -27,14 +14,21 @@ % % This work has the LPPL maintenance status `maintained'. % -% The Current Maintainer of this work is the American Mathematical -% Society. +% The Current Maintainer of this work is the LaTeX3 Project. % % \fi % % \iffalse %<*driver> \documentclass{amsdtx} +\def\MaintainedByLaTeXTeam#1{% +\begin{center}% +\fbox{\fbox{\begin{tabular}{@{}l@{}}% +This file is maintained by the \LaTeX{} Project team.\\% +Bug reports can be opened (category \texttt{#1}) at\\% +\url{http://latex-project.org/bugs.html}.\end{tabular}}}\end{center}} +\usepackage{amsgen} +\GetFileInfo{amsgen.sty} \begin{document} \title{The \pkg{amsgen} package} \author{American Mathematical Society\\Michael Downes} @@ -45,6 +39,7 @@ % \fi % % \maketitle +% \MaintainedByLaTeXTeam{amsmath} % % \MakeShortVerb\| % @@ -64,7 +59,7 @@ % \begin{macrocode} \NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX) [1994/12/01]% LaTeX date must December 1994 or later -\ProvidesFile{amsgen.sty}[1999/11/30 v2.0] +\ProvidesFile{amsgen.sty}[1999/11/30 v2.0 generic functions] % \end{macrocode} % % \section{Implementation} diff --git a/Master/texmf-dist/source/latex/amsmath/amsgen.ins b/Master/texmf-dist/source/latex/amsmath/amsgen.ins index 8745b0a8548..67821930206 100644 --- a/Master/texmf-dist/source/latex/amsmath/amsgen.ins +++ b/Master/texmf-dist/source/latex/amsmath/amsgen.ins @@ -1,19 +1,6 @@ -%% filename: amsgen.ins -%% version: 1.00 -%% date: 2010/08/13 -%% -%% American Mathematical Society -%% Technical Support -%% Publications Technical Group -%% 201 Charles Street -%% Providence, RI 02904 -%% USA -%% tel: (401) 455-4080 -%% (800) 321-4267 (USA and Canada only) -%% fax: (401) 331-3842 -%% email: tech-support@ams.org %% %% Copyright 2010 American Mathematical Society. +%% Copyright 2016 LaTeX3 Project and American Mathematical Society. %% %% This work may be distributed and/or modified under the %% conditions of the LaTeX Project Public License, either version 1.3c @@ -25,8 +12,7 @@ %% %% This work has the LPPL maintenance status `maintained'. %% -%% The Current Maintainer of this work is the American Mathematical -%% Society. +%% The Current Maintainer of this work is the LaTeX3 Project. %% %% ==================================================================== @@ -39,17 +25,7 @@ \preamble Copyright 1995, 1999 American Mathematical Society. - -American Mathematical Society -Technical Support -Publications Technical Group -201 Charles Street -Providence, RI 02904 -USA -tel: (401) 455-4080 - (800) 321-4267 (USA and Canada only) -fax: (401) 331-3842 -email: tech-support@ams.org +Copyright 2016 LaTeX3 Project and American Mathematical Society. This work may be distributed and/or modified under the conditions of the LaTeX Project Public License, either version 1.3c @@ -61,8 +37,7 @@ version 2005/12/01 or later. This work has the LPPL maintenance status `maintained'. -The Current Maintainer of this work is the American Mathematical -Society. +The Current Maintainer of this work is the LaTeX3 project. \endpreamble diff --git a/Master/texmf-dist/source/latex/amsmath/amsldoc.tex b/Master/texmf-dist/source/latex/amsmath/amsldoc.tex deleted file mode 100644 index abb532cccb8..00000000000 --- a/Master/texmf-dist/source/latex/amsmath/amsldoc.tex +++ /dev/null @@ -1,2620 +0,0 @@ -%%% ==================================================================== -%%% @LaTeX-file{ -%%% filename = "amsldoc.tex", -%%% version = "2.09", -%%% date = "2004/04/06", -%%% time = "15:13:33 EDT", -%%% checksum = "39446 2620 11111 93831", -%%% author = "American Mathematical Society", -%%% address = "American Mathematical Society, -%%% Technical Support, -%%% Electronic Products and Services, -%%% P. O. Box 6248, -%%% Providence, RI 02940, -%%% USA", -%%% telephone = "401-455-4080 or (in the USA and Canada) -%%% 800-321-4AMS (321-4267)", -%%% FAX = "401-331-3842", -%%% email = "tech-support@ams.org (Internet)", -%%% copyright = "Copyright 1999 American Mathematical Society, -%%% all rights reserved. Copying of this file is -%%% authorized only if either: -%%% (1) you make absolutely no changes to your copy, -%%% including name; OR -%%% (2) if you do make changes, you first rename it -%%% to some other name.", -%%% supported = "yes", -%%% keywords = "latex, amsmath", -%%% abstract = "This file is the source file for the user's -%%% guide of the \LaTeX{} package amsmath .", -%%% docstring = "The checksum field above contains a CRC-16 -%%% checksum as the first value, followed by the -%%% equivalent of the standard UNIX wc (word -%%% count) utility output of lines, words, and -%%% characters. This is produced by Robert -%%% Solovay's checksum utility.", -%%% } -%%% ==================================================================== -\documentclass[leqno,titlepage,openany]{amsldoc}[1999/12/13] - -\ifx\UndEfiNed\url - \ClassError{amsldoc}{% - This version of amsldoc.tex must be processed\MessageBreak - with a newer version of amsldoc.cls (2.02 or later)}{} -\fi - -\title{User's Guide for the \pkg{amsmath} Package (Version~2.0)} -\author{American Mathematical Society} -\date{1999-12-13 (revised 2002-02-25)} - -% Use the amsmath package and amscd package in order to print -% examples. -\usepackage{amsmath} -\usepackage{amscd} - -\makeindex % generate index data -\providecommand{\see}[2]{\textit{see} #1} - -% The amsldoc class includes a number of features useful for -% documentation about TeX, including: -% -% ---Commands \tex/, \amstex/, \latex/, ... for uniform treatment -% of the various logos and easy handling of following spaces. -% -% ---Commands for printing various common elements: \cn for command -% names, \fn for file names (including font-file names), \env for -% environments, \pkg and \cls for packages and classes, etc. - -% Many of the command names used here are rather long and will -% contribute to poor linebreaking if we follow the \latex/ practice -% of not hyphenating anything set in tt font; instead we selectively -% allow some hyphenation. -\allowtthyphens % defined in amsldoc.cls - -\hyphenation{ac-cent-ed-sym-bol add-to-counter add-to-length align-at - aligned-at allow-dis-play-breaks ams-art ams-cd ams-la-tex amsl-doc - ams-symb ams-tex ams-text ams-xtra bmatrix bold-sym-bol cen-ter-tags - eqn-ar-ray idots-int int-lim-its latex med-space neg-med-space - neg-thick-space neg-thin-space no-int-lim-its no-name-lim-its - over-left-arrow over-left-right-arrow over-right-arrow pmatrix - qed-sym-bol set-length side-set small-er tbinom the-equa-tion - thick-space thin-space un-der-left-arrow un-der-left-right-arrow - un-der-right-arrow use-pack-age var-inj-lim var-proj-lim vmatrix - xalign-at xx-align-at} - -% Prepare for illustrating the \vec example -\newcommand{\vect}[1]{\mathbf{#1}} - -\newcommand{\booktitle}[1]{\textit{#1}} -\newcommand{\journalname}[1]{\textit{#1}} -\newcommand{\seriesname}[1]{\textit{#1}} - -% Command to insert and index a particular phrase. Doesn't work for -% certain kinds of special characters in the argument. -\newcommand{\ii}[1]{#1\index{#1}} - -\newcommand{\vstrut}[1]{\vrule width0pt height#1\relax} - -% An environment for presenting comprehensive address information: -\newenvironment{infoaddress}{% - \par\topsep\medskipamount - \trivlist\centering - \item[]% - \begin{minipage}{.7\columnwidth}% - \raggedright -}{% - \end{minipage}% - \endtrivlist -} - -\newenvironment{eqxample}{% - \par\addvspace\medskipamount - \noindent\begin{minipage}{.5\columnwidth}% - \def\producing{\end{minipage}\begin{minipage}{.5\columnwidth}% - \hbox\bgroup\kern-.2pt\vrule width.2pt% - \vbox\bgroup\parindent0pt\relax -% The 3pt is to cancel the -\lineskip from \displ@y - \abovedisplayskip3pt \abovedisplayshortskip\abovedisplayskip - \belowdisplayskip0pt \belowdisplayshortskip\belowdisplayskip - \noindent} -}{% - \par -% Ensure that a lonely \[\] structure doesn't take up width less than -% \hsize. - \hrule height0pt width\hsize - \egroup\vrule width.2pt\kern-.2pt\egroup - \end{minipage}% - \par\addvspace\medskipamount -} - -% The chapters are so short, perhaps we shouldn't call them by the -% name `Chapter'. We make \chaptername read an argument in order to -% remove a following \space or "{} " (both possibilities are present -% in book.cls). - -\renewcommand{\chaptername}[1]{} -\newcommand{\chapnum}[1]{\mdash #1\mdash } -\makeatletter -\def\@makechapterhead#1{% - \vspace{1.5\baselineskip}% - {\parindent \z@ \raggedright \reset@font - \ifnum \c@secnumdepth >\m@ne - \large\bfseries \chapnum{\thechapter}% - \par\nobreak - \vskip.5\baselineskip\relax - \fi - #1\par\nobreak - \vskip\baselineskip - }} -\makeatother - -% A command for ragged-right parbox in a tabular. -\newcommand{\rp}{\let\PBS\\\raggedright\let\\\PBS} - -% Non-indexed file name -\newcommand{\nfn}[1]{\texttt{#1}} - -% For the examples in the math spacing table. -%%\newcommand{\lspx}{\mbox{\rule{5pt}{.6pt}\rule{.6pt}{6pt}}} -%%\newcommand{\rspx}{\mbox{\rule[-1pt]{.6pt}{7pt}% -%% \rule[-1pt]{5pt}{.6pt}}} -\newcommand{\lspx}{\mathord{\Rightarrow\mkern-1mu}} -\newcommand{\rspx}{\mathord{\mkern-1mu\Leftarrow}} -\newcommand{\spx}[1]{$\lspx #1\rspx$} - -% For a list of characters representing document input. -\newcommand{\clist}[1]{% - \mbox{\ntt\spaceskip.2em plus.1em \xspaceskip\spaceskip#1}} - -% Fix weird \latex/ definition of rightmark. -\makeatletter -\def\rightmark{\expandafter\@rightmark\botmark{}{}} -% Also turn off section marks. -\let\sectionmark\@gobble -\renewcommand{\chaptermark}[1]{% - \uppercase{\markboth{\rhcn#1}{\rhcn#1}}} -\newcommand{\rhcn}{\thechapter. } -\makeatother - -% Include down to \section but not \subsection, in toc: -\setcounter{tocdepth}{1} - -\DeclareMathOperator{\ix}{ix} -\DeclareMathOperator{\nul}{nul} -\DeclareMathOperator{\End}{End} -\DeclareMathOperator{\xxx}{xxx} - -\begin{document} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\frontmatter - -\maketitle -\pagestyle{headings} -\tableofcontents -\cleardoublepage % for better page number placement - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\mainmatter -\chapter{Introduction} - -The \pkg{amsmath} package is a \LaTeX{} package that provides -miscellaneous enhancements for improving the information structure and -printed output of documents that contain mathematical formulas. Readers -unfamiliar with \LaTeX{} should refer to \cite{lamport}. If you have an -up-to-date version of \LaTeX{}, the \pkg{amsmath} package is normally -provided along with it. Upgrading when a newer version of the -\pkg{amsmath} package is released can be done via -\url{http://www.ams.org/tex/amsmath.html} or -\url{ftp://ftp.ams.org/pub/tex/}. - -This documentation describes the features of the \pkg{amsmath} package -and discusses how they are intended to be used. It also covers some -ancillary packages: -\begin{ctab}{ll} -\pkg{amsbsy}& \pkg{amstext}\\ -\pkg{amscd}& \pkg{amsxtra}\\ -\pkg{amsopn} -\end{ctab} -These all have something to do with the contents of math formulas. For -information on extra math symbols and math fonts, see \cite{amsfonts} -and \url{http://www.ams.org/tex/amsfonts.html}. For documentation of the -\pkg{amsthm} package or AMS document classes (\cls{amsart}, -\cls{amsbook}, etc.\@) see \cite{amsthdoc} or \cite{instr-l} and -\url{http://www.ams.org/tex/author-info.html}. - -If you are a long-time \latex/ user and have lots of mathematics in what -you write, then you may recognize solutions for some familiar problems -in this list of \pkg{amsmath} features: -\begin{itemize} - -\item A convenient way to define new `operator name' commands analogous -to \cn{sin} and \cn{lim}, including proper side spacing and automatic -selection of the correct font style and size (even when used in -sub- or superscripts). - -\item Multiple substitutes for the \env{eqnarray} environment to make -various kinds of equation arrangements easier to write. - -\item Equation numbers automatically adjust up or down to avoid -overprinting on the equation contents (unlike \env{eqnarray}). - -\item Spacing around equals signs matches the normal spacing in the -\env{equation} environment (unlike \env{eqnarray}). - -\item A way to produce multiline subscripts as are often used with -summation or product symbols. - -\item An easy way to substitute a variant equation number for a given -equation instead of the automatically supplied number. - -\item An easy way to produce subordinate equation numbers of the form -(1.3a) (1.3b) (1.3c) for selected groups of equations. - -\end{itemize} - -The \pkg{amsmath} package is distributed together with some small -auxiliary packages: -\begin{description} -\item[\pkg{amsmath}] Primary package, provides various features for - displayed equations and other mathematical constructs. - -\item[\pkg{amstext}] Provides a \cn{text} command for - typesetting a fragment of text inside a display. - -\item[\pkg{amsopn}] Provides \cn{DeclareMathOperator} for defining new - `operator names' like \cn{sin} and \cn{lim}. - -\item[\pkg{amsbsy}] For backward compatibility this package continues -to exist but use of the newer \pkg{bm} package that comes with \LaTeX{} -is recommended instead. - -\item[\pkg{amscd}] Provides a \env{CD} environment for simple - commutative diagrams (no support for diagonal arrows). - -\item[\pkg{amsxtra}] Provides certain odds and ends such as - \cn{fracwithdelims} and \cn{accentedsymbol}, for compatibility with - documents created using version 1.1. - -\end{description} - -The \pkg{amsmath} package incorporates \pkg{amstext}, \pkg{amsopn}, and -\pkg{amsbsy}. The features of \pkg{amscd} and \pkg{amsxtra}, however, -are available only by invoking those packages separately. - -\chapter{Options for the \pkg{amsmath} package}\label{options} - -The \pkg{amsmath} package has the following options: -\begin{description} - -\item[\opt{centertags}] (default) For a split equation, place equation -numbers\index{equation numbers!vertical placement} vertically centered -on the total height of the equation. - -\item[\opt{tbtags}] `Top-or-bottom tags': For a split equation, place -equation numbers\index{equation numbers!vertical placement} level with -the last (resp.\@ first) line, if numbers are on the right (resp.\@ -left). - -\item[\opt{sumlimits}] (default) Place the subscripts and -superscripts\index{subscripts and superscripts!placement}\relax -\index{limits|see{subscripts and superscripts}} of summation symbols -above and below, in displayed equations. This option also affects other -symbols of the same type\mdash $\prod$, $\coprod$, $\bigotimes$, -$\bigoplus$, and so forth\mdash but excluding integrals (see below). - -\item[\opt{nosumlimits}] Always place the subscripts and superscripts of -summation-type symbols to the side, even in displayed equations. - -\item[\opt{intlimits}] Like \opt{sumlimits}, but for -integral\index{integrals!placement of limits} symbols. - -\item[\opt{nointlimits}] (default) Opposite of \opt{intlimits}. - -\item[\opt{namelimits}] (default) Like \opt{sumlimits}, but for certain -`operator names' such as $\det$, $\inf$, $\lim$, $\max$, $\min$, that -traditionally have subscripts \index{subscripts and -superscripts!placement} placed underneath when they occur in a displayed -equation. - -\item[\opt{nonamelimits}] Opposite of \opt{namelimits}. - -\end{description} - -To use one of these package options, put the option name in the optional -argument of the \cn{usepackage} command\mdash e.g., -\verb"\usepackage[intlimits]{amsmath}". - -The \pkg{amsmath} package also recognizes the following options which -are normally selected (implicitly or explicitly) through the -\cn{documentclass} command, and thus need not be repeated in the option -list of the \cn{usepackage}|{amsmath}| statement. -\begin{description} - -\item[\opt{leqno}] Place equation numbers on the left.\index{equation -numbers!left or right placement} - -\item[\opt{reqno}] Place equation numbers on the right. - -\item[\opt{fleqn}] Position equations at a fixed indent from the left -margin rather than centered in the text column.\index{displayed -equations!centering} - -\end{description} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\chapter{Displayed equations} - -\section{Introduction} -The \pkg{amsmath} package provides a number of additional displayed -equation structures\index{displayed - equations}\index{equations|see{displayed equations}} beyond the ones -provided in basic \latex/. The augmented set includes: -\begin{verbatim} - equation equation* align align* - gather gather* flalign flalign* - multline multline* alignat alignat* - split -\end{verbatim} -(Although the standard \env{eqnarray} environment remains available, -it is better to use \env{align} or \env{equation}+\env{split} instead.) - -Except for \env{split}, each environment has both starred and unstarred -forms, where the unstarred forms have automatic numbering using -\latex/'s \env{equation} counter. You can suppress the number on any -particular line by putting \cn{notag} before the \cn{\\}; you can also -override\index{equation numbers!overriding} it with a tag of your own -using \cn{tag}|{|\<label>|}|, where \<label> means arbitrary text such -as |$*$| or |ii| used to \qq{number} the equation. There is also a -\cn{tag*} command that causes the text you supply to be typeset -literally, without adding parentheses around it. \cn{tag} and \cn{tag*} -can also be used within the unnumbered versions of all the \pkg{amsmath} -alignment structures. Some examples of the use of \cn{tag} may be found -in the sample files \fn{testmath.tex} and \fn{subeqn.tex} provided with -the \pkg{amsmath} package. - -The \env{split} environment is a special subordinate form that is used -only \emph{inside} one of the others. It cannot be used inside -\env{multline}, however. - -In the structures that do alignment (\env{split}, \env{align} and -variants), relation symbols have an \verb'&' before them but not -after\mdash unlike \env{eqnarray}. Putting the \verb'&' after the -relation symbol will interfere with the normal spacing; it has to go -before. - -\begin{table}[p] -\caption[]{Comparison of displayed equation environments (vertical lines -indicating nominal margins)}\label{displays} -\renewcommand{\theequation}{\arabic{equation}} -\begin{eqxample} -\begin{verbatim} -\begin{equation*} -a=b -\end{equation*} -\end{verbatim} -\producing -\begin{equation*} -a=b -\end{equation*} -\end{eqxample} - -\begin{eqxample} -\begin{verbatim} -\begin{equation} -a=b -\end{equation} -\end{verbatim} -\producing -\begin{equation} -a=b -\end{equation} -\end{eqxample} - -\begin{eqxample} -\begin{verbatim} -\begin{equation}\label{xx} -\begin{split} -a& =b+c-d\\ - & \quad +e-f\\ - & =g+h\\ - & =i -\end{split} -\end{equation} -\end{verbatim} -\producing -\begin{equation}\label{xx} -\begin{split} -a& =b+c-d\\ - & \quad +e-f\\ - & =g+h\\ - & =i -\end{split} -\end{equation} -\end{eqxample} - -\begin{eqxample} -\begin{verbatim} -\begin{multline} -a+b+c+d+e+f\\ -+i+j+k+l+m+n -\end{multline} -\end{verbatim} -\producing -\begin{multline} -a+b+c+d+e+f\\ -+i+j+k+l+m+n -\end{multline} -\end{eqxample} - -\begin{eqxample} -\begin{verbatim} -\begin{gather} -a_1=b_1+c_1\\ -a_2=b_2+c_2-d_2+e_2 -\end{gather} -\end{verbatim} -\producing -\begin{gather} -a_1=b_1+c_1\\ -a_2=b_2+c_2-d_2+e_2 -\end{gather} -\end{eqxample} - -\begin{eqxample} -\begin{verbatim} -\begin{align} -a_1& =b_1+c_1\\ -a_2& =b_2+c_2-d_2+e_2 -\end{align} -\end{verbatim} -\producing -\begin{align} -a_1& =b_1+c_1\\ -a_2& =b_2+c_2-d_2+e_2 -\end{align} -\end{eqxample} - -\begin{eqxample} -\begin{verbatim} -\begin{align} -a_{11}& =b_{11}& - a_{12}& =b_{12}\\ -a_{21}& =b_{21}& - a_{22}& =b_{22}+c_{22} -\end{align} -\end{verbatim} -\producing -\begin{align} -a_{11}& =b_{11}& - a_{12}& =b_{12}\\ -a_{21}& =b_{21}& - a_{22}& =b_{22}+c_{22} -\end{align} -\end{eqxample} - -\begin{eqxample} -\begin{verbatim} -\begin{flalign*} -a_{11}& =b_{11}& - a_{12}& =b_{12}\\ -a_{21}& =b_{21}& - a_{22}& =b_{22}+c_{22} -\end{flalign*} -\end{verbatim} -\producing -\begin{flalign*} -a_{11}& =b_{11}& - a_{12}& =b_{12}\\ -a_{21}& =b_{21}& - a_{22}& =b_{22}+c_{22} -\end{flalign*} -\end{eqxample} -\end{table} - -\section{Single equations} - -The \env{equation} environment is for a single equation with an -automatically generated number. The \env{equation*} environment is the -same except for omitting the number.% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\footnote{Basic \latex/ doesn't provide an \env{equation*} environment, -but rather a functionally equivalent environment named -\env{displaymath}.} - -\section{Split equations without alignment} - -The \env{multline} environment is a variation of the \env{equation} -environment used for equations that don't fit on a single line. The -first line of a \env{multline} will be at the left margin and the last -line at the right margin, except for an indention on both sides in the -amount of \cn{multlinegap}. Any additional lines in between will be -centered independently within the display width (unless the \opt{fleqn} -option is in effect). - -Like \env{equation}, \env{multline} has only a single equation number -(thus, none of the individual lines should be marked with \cn{notag}). -The equation number is placed on the last line (\opt{reqno} option) or -first line (\opt{leqno} option); vertical centering as for \env{split} -is not supported by \env{multline}. - -It's possible to force one of the middle lines to the left or right with -commands \cn{shoveleft}, \cn{shoveright}. These commands take the entire -line as an argument, up to but not including the final \cn{\\}; for -example -\begin{multline} -\framebox[.65\columnwidth]{A}\\ -\framebox[.5\columnwidth]{B}\\ -\shoveright{\framebox[.55\columnwidth]{C}}\\ -\framebox[.65\columnwidth]{D} -\end{multline} -\begin{verbatim} -\begin{multline} -\framebox[.65\columnwidth]{A}\\ -\framebox[.5\columnwidth]{B}\\ -\shoveright{\framebox[.55\columnwidth]{C}}\\ -\framebox[.65\columnwidth]{D} -\end{multline} -\end{verbatim} - -The value of \cn{multlinegap} can be changed with the usual \latex/ -commands \cn{setlength} or \cn{addtolength}. - -\section{Split equations with alignment} - -Like \env{multline}, the \env{split} environment is for \emph{single} -equations that are too long to fit on one line and hence must be split -into multiple lines. Unlike \env{multline}, however, the \env{split} -environment provides for alignment among the split lines, using |&| to -mark alignment points. Unlike the other \pkg{amsmath} equation -structures, the \env{split} environment provides no numbering, because -it is intended to be used \emph{only inside some other displayed - equation structure}, usually an \env{equation}, \env{align}, or -\env{gather} environment, which provides the numbering. For example: -\begin{equation}\label{e:barwq}\begin{split} -H_c&=\frac{1}{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 -\Bigl[(n-l )^2-\sum^p_{j=1}(n_i-l _i)^2\Bigr]. -\kern-2em % adjust equation body to the right [mjd,13-Nov-1994] -\end{split}\end{equation} - -\begin{verbatim} -\begin{equation}\label{e:barwq}\begin{split} -H_c&=\frac{1}{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 -\Bigl[(n-l )^2-\sum^p_{j=1}(n_i-l _i)^2\Bigr]. -\end{split}\end{equation} -\end{verbatim} - -The \env{split} structure should constitute the entire body of the -enclosing structure, apart from commands like \cn{label} that produce no -visible material. - -\section{Equation groups without alignment} - -The \env{gather} environment is used for a group of consecutive -equations when there is no alignment desired among them; each one is -centered separately within the text width (see Table~\ref{displays}). -Equations inside \env{gather} are separated by a \cn{\\} command. -Any equation in a \env{gather} may consist of a \verb'\begin{split}' - \dots\ \verb'\end{split}' structure\mdash for example: -\begin{verbatim} -\begin{gather} - first equation\\ - \begin{split} - second & equation\\ - & on two lines - \end{split} - \\ - third equation -\end{gather} -\end{verbatim} - -\section{Equation groups with mutual alignment} - -The \env{align} environment is used for two or more equations when -vertical alignment is desired; usually binary relations such as equal -signs are aligned (see Table~\ref{displays}). - -To have several equation columns side-by-side, use extra ampersands -to separate the columns: -\begin{align} -x&=y & X&=Y & a&=b+c\\ -x'&=y' & X'&=Y' & a'&=b\\ -x+x'&=y+y' & X+X'&=Y+Y' & a'b&=c'b -\end{align} -% -\begin{verbatim} -\begin{align} -x&=y & X&=Y & a&=b+c\\ -x'&=y' & X'&=Y' & a'&=b\\ -x+x'&=y+y' & X+X'&=Y+Y' & a'b&=c'b -\end{align} -\end{verbatim} -Line-by-line annotations on an equation can be done by judicious -application of \cn{text} inside an \env{align} environment: -\begin{align} -x& = y_1-y_2+y_3-y_5+y_8-\dots - && \text{by \eqref{eq:C}}\\ - & = y'\circ y^* && \text{by \eqref{eq:D}}\\ - & = y(0) y' && \text {by Axiom 1.} -\end{align} -% -\begin{verbatim} -\begin{align} -x& = y_1-y_2+y_3-y_5+y_8-\dots - && \text{by \eqref{eq:C}}\\ - & = y'\circ y^* && \text{by \eqref{eq:D}}\\ - & = y(0) y' && \text {by Axiom 1.} -\end{align} -\end{verbatim} -A variant environment \env{alignat} allows the horizontal space between -equations to be explicitly specified. This environment takes one argument, -the number of \qq{equation columns}: count the maximum number of \verb'&'s -in any row, add 1 and divide by 2. -\begin{alignat}{2} -x& = y_1-y_2+y_3-y_5+y_8-\dots - &\quad& \text{by \eqref{eq:C}}\\ - & = y'\circ y^* && \text{by \eqref{eq:D}}\\ - & = y(0) y' && \text {by Axiom 1.} -\end{alignat} -% -\begin{verbatim} -\begin{alignat}{2} -x& = y_1-y_2+y_3-y_5+y_8-\dots - &\quad& \text{by \eqref{eq:C}}\\ - & = y'\circ y^* && \text{by \eqref{eq:D}}\\ - & = y(0) y' && \text {by Axiom 1.} -\end{alignat} -\end{verbatim} - -\section{Alignment building blocks} - -Like \env{equation}, the multi-equation environments \env{gather}, -\env{align}, and \env{alignat} are designed to produce a structure -whose width is the full line width. This means, for example, that one -cannot readily add parentheses around the entire structure. But variants -\env{gathered}, \env{aligned}, and \env{alignedat} are provided whose -total width is the actual width of the contents; thus they can be used -as a component in a containing expression. E.g., -\begin{equation*} -\left.\begin{aligned} - B'&=-\partial\times E,\\ - E'&=\partial\times B - 4\pi j, -\end{aligned} -\right\} -\qquad \text{Maxwell's equations} -\end{equation*} -\begin{verbatim} -\begin{equation*} -\left.\begin{aligned} - B'&=-\partial\times E,\\ - E'&=\partial\times B - 4\pi j, -\end{aligned} -\right\} -\qquad \text{Maxwell's equations} -\end{equation*} -\end{verbatim} -Like the \env{array} environment, these \texttt{-ed} variants also take -an optional \verb'[t]' or \verb'[b]' argument to specify vertical -positioning. - -\qq{Cases} constructions like the following are common in -mathematics: -\begin{equation}\label{eq:C} -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} -and in the \pkg{amsmath} package there is a \env{cases} environment to -make them easy to write: -\begin{verbatim} -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{verbatim} -Notice the use of \cn{text} (cf.~\secref{text}) and the nested -math formulas. - -\section{Adjusting tag placement} - -Placing equation numbers can be a rather complex problem in multiline -displays. The environments of the \pkg{amsmath} package try hard to -avoid overprinting an equation number on the equation contents, if -necessary moving the number down or up to a separate line. Difficulties -in accurately calculating the profile of an equation can occasionally -result in number movement that doesn't look right. There is a -\cn{raisetag} command provided to adjust the vertical position of the -current equation number, if it has been shifted away from its normal -position. To move a particular number up by six points, write -|\raisetag{6pt}|. This kind of adjustment is fine tuning like line -breaks and page breaks, and should therefore be left undone until your -document is nearly finalized, or you may end up redoing the fine tuning -several times to keep up with changing document contents. - -\section{Vertical spacing and page breaks in multiline displays} - -You can use the \cn{\\}|[|\<dimension>|]| command to get extra vertical -space between lines in all the \pkg{amsmath} displayed equation -environments, as is usual in \latex/. When the \pkg{amsmath} package is -in use \ii{page breaks} between equation lines are normally disallowed; -the philosophy is that page breaks in such material should receive -individual attention from the author. To get an individual page break -inside a particular displayed equation, a \cn{displaybreak} command is -provided. \cn{displaybreak} is best placed immediately before the -\cn{\\} where it is to take effect. Like \latex/'s \cn{pagebreak}, -\cn{displaybreak} takes an optional argument between 0 and 4 denoting -the desirability of the pagebreak. |\displaybreak[0]| means \qq{it is - permissible to break here} without encouraging a break; -\cn{displaybreak} with no optional argument is the same as -|\displaybreak[4]| and forces a break. - -If you prefer a strategy of letting page breaks fall where they may, -even in the middle of a multi-line equation, then you might put -\cn{allowdisplaybreaks}\texttt{[1]} in the preamble of your document. An -optional argument 1\ndash 4 can be used for finer control: |[1]| means -allow page breaks, but avoid them as much as possible; values of 2,3,4 -mean increasing permissiveness. When display breaks are enabled with -\cn{allowdisplaybreaks}, the \cn{\\*} command can be used to prohibit a -pagebreak after a given line, as usual. - -\begin{bfseries} -Note: Certain equation environments wrap their contents in an -unbreakable box, with the consequence that neither \cn{displaybreak} nor -\cn{allowdisplaybreaks} will have any effect on them. These include -\env{split}, \env{aligned}, \env{gathered}, and \env{alignedat}. -\end{bfseries} - -\section{Interrupting a display} - -The command \cn{intertext} is used for a short interjection of one or -two lines of text\index{text fragments inside math} in the middle of a -multiple-line display structure (see also the \cn{text} command in -\secref{text}). Its salient feature is preservation of the alignment, -which would not happen if you simply ended the display and then started -it up again afterwards. \cn{intertext} may only appear right after a -\cn{\\} or \cn{\\*} command. Notice the position of the word \qq{and} in -this example. -\begin{align} -A_1&=N_0(\lambda;\Omega')-\phi(\lambda;\Omega'),\\ -A_2&=\phi(\lambda;\Omega')-\phi(\lambda;\Omega),\\ -\intertext{and} -A_3&=\mathcal{N}(\lambda;\omega). -\end{align} -\begin{verbatim} -\begin{align} -A_1&=N_0(\lambda;\Omega')-\phi(\lambda;\Omega'),\\ -A_2&=\phi(\lambda;\Omega')-\phi(\lambda;\Omega),\\ -\intertext{and} -A_3&=\mathcal{N}(\lambda;\omega). -\end{align} -\end{verbatim} - -\section{Equation numbering} - -\subsection{Numbering hierarchy} -In \latex/ if you wanted to have equations numbered within -sections\mdash that is, have -equation numbers (1.1), (1.2), \dots, (2.1), (2.2), -\dots, in sections 1, 2, and so forth\mdash you could redefine -\cn{theequation} as suggested in the \latex/ manual \cite[\S6.3, \S -C.8.4]{lamport}: -\begin{verbatim} -\renewcommand{\theequation}{\thesection.\arabic{equation}} -\end{verbatim} - -This works pretty well, except that the equation counter won't be reset -to zero at the beginning of a new section or chapter, unless you do it -yourself using \cn{setcounter}. To make this a little more convenient, -the \pkg{amsmath} package provides a command\index{equation -numbers!hierarchy} \cn{numberwithin}. To have equation numbering tied to -section numbering, with automatic reset of the equation counter, write -\begin{verbatim} -\numberwithin{equation}{section} -\end{verbatim} -As its name implies, the \cn{numberwithin} command can be applied to -any counter, not just the \texttt{equation} counter. - -\subsection{Cross references to equation numbers} - -To make cross-references to equations easier, an \cn{eqref} -command\index{equation numbers!cross-references} is provided. This -automatically supplies the parentheses around the equation number. I.e., -if \verb'\ref{abc}' produces 3.2 then \verb'\eqref{abc}' produces -(3.2). - -\subsection{Subordinate numbering sequences} - -The \pkg{amsmath} package provides also a \env{subequations} -environment\index{equation numbers!subordinate numbering} to make it -easy to number equations in a particular group with a subordinate -numbering scheme. For example -\begin{verbatim} -\begin{subequations} -... -\end{subequations} -\end{verbatim} -causes all numbered equations within that part of the document to be -numbered (4.9a) (4.9b) (4.9c) \dots, if the preceding numbered -equation was (4.8). A \cn{label} command immediately after -\verb/\begin{subequations}/ will produce a \cn{ref} of the parent -number 4.9, not 4.9a. The counters used by the subequations -environment are \verb/parentequation/ and \verb/equation/ and -\cn{addtocounter}, \cn{setcounter}, \cn{value}, etc., can be applied -as usual to those counter names. To get anything other than lowercase -letters for the subordinate numbers, use standard \latex/ methods for -changing numbering style \cite[\S6.3, \S C.8.4]{lamport}. For example, -redefining \cn{theequation} as follows will produce roman numerals. -\begin{verbatim} -\begin{subequations} -\renewcommand{\theequation}{\theparentequation \roman{equation}} -... -\end{verbatim} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\chapter{Miscellaneous mathematical features} - -\section{Matrices}\label{ss:matrix} - -The \pkg{amsmath} package provides some environments for -matrices\index{matrices} beyond the basic \env{array} environment of -\latex/. The \env{pmatrix}, \env{bmatrix}, \env{Bmatrix}, \env{vmatrix} -and \env{Vmatrix} have (respectively) $(\,)$, $[\,]$, $\lbrace\,\rbrace$, -$\lvert\,\rvert$, and -$\lVert\,\rVert$ delimiters built in. For naming consistency there is a -\env{matrix} environment sans delimiters. This is not entirely redundant -with the \env{array} environment; the matrix environments all use more -economical horizontal spacing than the rather prodigal spacing of the -\env{array} environment. Also, unlike the \env{array} environment, you -don't have to give column specifications for any of the matrix -environments; by default you can have up to 10 centered columns.% -\footnote{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -More precisely: The maximum number of columns in a matrix is determined -by the counter |MaxMatrixCols| (normal value = 10), which you can change -if necessary using \latex/'s \cn{setcounter} or \cn{addtocounter} -commands.% -}\space%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -(If you need left or right alignment in a column or other special -formats you must resort to \env{array}.) - -To produce a small matrix suitable for use in text, there is a -\env{smallmatrix} environment (e.g., -\begin{math} -\bigl( \begin{smallmatrix} - a&b\\ c&d -\end{smallmatrix} \bigr) -\end{math}) -that comes closer to fitting within a single text line than a normal -matrix. Delimiters must be provided; there are no |p|,|b|,|B|,|v|,|V| -versions of \env{smallmatrix}. The above example was produced by -\begin{verbatim} -\bigl( \begin{smallmatrix} - a&b\\ c&d -\end{smallmatrix} \bigr) -\end{verbatim} - -\cn{hdotsfor}|{|\<number>|}| produces a row of dots in a -matrix\index{matrices!ellipsis dots}\index{ellipsis dots!in -matrices}\index{dots|see{ellipsis dots}} spanning the given number of -columns. For example, -\begin{center} -\begin{minipage}{.3\columnwidth} -\noindent$\begin{matrix} a&b&c&d\\ -e&\hdotsfor{3} \end{matrix}$ -\end{minipage}% -\qquad -\begin{minipage}{.45\columnwidth} -\begin{verbatim} -\begin{matrix} a&b&c&d\\ -e&\hdotsfor{3} \end{matrix} -\end{verbatim} -\end{minipage}% -\end{center} - -The spacing of the dots can be varied through use of a square-bracket -option, for example, |\hdotsfor[1.5]{3}|. The number in square brackets -will be used as a multiplier (i.e., the normal value is 1.0). -\begin{equation}\label{eq:D} -\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} - -\section{Math spacing commands} - -The \pkg{amsmath} package slightly extends the set of math -spacing\index{horizontal space!in math mode} commands, as shown below. -Both the spelled-out and abbreviated forms of these commands are robust, -and they can also be used outside of math. -\begin{ctab}{lll|lll} -Abbrev.& Spelled out& Example & Abbrev.& Spelled out& Example\\ -\hline -\vstrut{2.5ex} -& no space& \spx{}& & no space & \spx{}\\ -\cn{\,}& \cn{thinspace}& \spx{\,}& - \cnbang& \cn{negthinspace}& \spx{\!}\\ -\cn{\:}& \cn{medspace}& \spx{\:}& - & \cn{negmedspace}& \spx{\negmedspace}\\ -\cn{\;}& \cn{thickspace}& \spx{\;}& - & \cn{negthickspace}& \spx{\negthickspace}\\ -& \cn{quad}& \spx{\quad}\\ -& \cn{qquad}& \spx{\qquad} -\end{ctab} -For the greatest possible control over math spacing, use \cn{mspace} -and `math units'. One math unit, or |mu|, is equal to 1/18 em. Thus to -get a negative \cn{quad} you could write |\mspace{-18.0mu}|. - -\section{Dots} - -For preferred placement of ellipsis dots (raised or on-line) in various -contexts there is no general consensus. It may therefore be considered a -matter of taste. By using the semantically oriented commands -\begin{itemize} -\item \cn{dotsc} for \qq{dots with commas} -\item \cn{dotsb} for \qq{dots with binary operators/relations} -\item \cn{dotsm} for \qq{multiplication dots} -\item \cn{dotsi} for \qq{dots with integrals} -\item \cn{dotso} for \qq{other dots} (none of the above) -\end{itemize} -instead of \cn{ldots} and \cn{cdots}, you make it possible for your -document to be adapted to different conventions on the fly, in case (for -example) you have to submit it to a publisher who insists on following -house tradition in this respect. The default treatment for the various -kinds follows American Mathematical Society conventions: -\begin{center} -\begin{tabular}{@{}l@{}l@{}} -\begin{minipage}[t]{.54\textwidth} -\begin{verbatim} -Then we have the series $A_1, A_2, -\dotsc$, the regional sum $A_1 -+A_2 +\dotsb $, the orthogonal -product $A_1 A_2 \dotsm $, and -the infinite integral -\[\int_{A_1}\int_{A_2}\dotsi\]. -\end{verbatim} -\end{minipage} -& -\begin{minipage}[t]{.45\textwidth} -\noindent -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{minipage} -\end{tabular} -\end{center} - -\section{Nonbreaking dashes} - -A command \cn{nobreakdash} is provided to suppress the possibility -of a linebreak after the following hyphen or dash. For example, if you -write `pages 1\ndash 9' as |pages 1\nobreakdash--9| then a linebreak will -never occur between the dash and the 9. You can also use -\cn{nobreakdash} to prevent undesirable hyphenations in combinations -like |$p$-adic|. For frequent use, it's advisable to make abbreviations, -e.g., -\begin{verbatim} -\newcommand{\p}{$p$\nobreakdash}% for "\p-adic" -\newcommand{\Ndash}{\nobreakdash--}% for "pages 1\Ndash 9" -% For "\n dimensional" ("n-dimensional"): -\newcommand{\n}[1]{$n$\nobreakdash-\hspace{0pt}} -\end{verbatim} -The last example shows how to prohibit a linebreak after the hyphen but -allow normal hyphenation in the following word. (It suffices to add a -zero-width space after the hyphen.) - -\section{Accents in math} - -In ordinary \LaTeX{} the placement of the second accent in doubled math -accents is often poor. With the \pkg{amsmath} package you -will get improved placement of the second accent: -$\hat{\hat{A}}$ (\cn{hat}|{\hat{A}}|). - -The commands \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/. - -To get a superscripted hat or tilde character, load the \pkg{amsxtra} -package and use \cn{sphat} or \cn{sptilde}. Usage is \verb'A\sphat' -(note the absence of the \verb'^' character). - -To place an arbitrary symbol in math accent position, or to get under -accents, see the \pkg{accents} package by Javier Bezos. - -\section{Roots} - -In ordinary \latex/ the placement of root indices is sometimes not so -good: $\sqrt[\beta]{k}$ (|\sqrt||[\beta]{k}|). In the -\pkg{amsmath} package \cn{leftroot} and \cn{uproot} allow you to adjust -the position of the root: -\begin{verbatim} - \sqrt[\leftroot{-2}\uproot{2}\beta]{k} -\end{verbatim} -will move the beta up and to the right: -$\sqrt[\leftroot{-2}\uproot{2}\beta]{k}$. The negative argument used -with \cn{leftroot} moves the $\beta$ to the right. The units are a small -amount that is a useful size for such adjustments. - -\section{Boxed formulas} - -The command \cn{boxed} puts a box around its -argument, like \cn{fbox} except that the contents are in math mode: -\begin{equation} -\boxed{\eta \leq C(\delta(\eta) +\Lambda_M(0,\delta))} -\end{equation} -\begin{verbatim} - \boxed{\eta \leq C(\delta(\eta) +\Lambda_M(0,\delta))} -\end{verbatim} - -\section{Over and under arrows} - -Basic \latex/ provides \cn{overrightarrow} and \cn{overleftarrow} -commands. Some additional over and under arrow commands are provided -by the \pkg{amsmath} package to extend the set: - -\begin{tabbing} -\qquad\=\ncn{overleftrightarrow}\qquad\=\kill -\> \cn{overleftarrow} \> \cn{underleftarrow} \+\\ - \cn{overrightarrow} \> \cn{underrightarrow} \\ - \cn{overleftrightarrow}\> \cn{underleftrightarrow} -\end{tabbing} - -\section{Extensible arrows} - -\cn{xleftarrow} and \cn{xrightarrow} produce -arrows\index{arrows!extensible} that extend automatically to accommodate -unusually wide subscripts or superscripts. These commands take one -optional argument (the subscript) and one mandatory argument (the -superscript, possibly empty): -\begin{equation} -A\xleftarrow{n+\mu-1}B \xrightarrow[T]{n\pm i-1}C -\end{equation} -\begin{verbatim} - \xleftarrow{n+\mu-1}\quad \xrightarrow[T]{n\pm i-1} -\end{verbatim} - -\section{Affixing symbols to other symbols} - -\latex/ provides \cn{stackrel} for placing a -superscript\index{subscripts and superscripts} above a binary relation. -In the \pkg{amsmath} package there are somewhat more general commands, -\cn{overset} and \cn{underset}, that can be used to place one symbol -above or below another symbol, whether it's a relation or something -else. The input |\overset{*}{X}| will place a superscript-size $*$ above -the $X$: $\overset{*}{X}$; \cn{underset} is the analog for adding a -symbol underneath. - -See also the description of \cn{sideset} in \secref{sideset}. - -\section{Fractions and related constructions} - -\subsection{The \cn{frac}, \cn{dfrac}, and \cn{tfrac} commands} - -The \cn{frac} command, which is in the basic command set of -\latex/,\index{fractions} takes two arguments\mdash numerator and -denominator\mdash and typesets them in normal fraction form. The -\pkg{amsmath} package provides also \cn{dfrac} and \cn{tfrac} as -convenient abbreviations for |{\displaystyle\frac| |...| |}| -and\indexcs{textstyle}\relax -\indexcs{displaystyle} |{\textstyle\frac| |...| |}|. -\begin{equation} -\frac{1}{k}\log_2 c(f)\quad\tfrac{1}{k}\log_2 c(f)\quad -\sqrt{\frac{1}{k}\log_2 c(f)}\quad\sqrt{\dfrac{1}{k}\log_2 c(f)} -\end{equation} -\begin{verbatim} -\begin{equation} -\frac{1}{k}\log_2 c(f)\;\tfrac{1}{k}\log_2 c(f)\; -\sqrt{\frac{1}{k}\log_2 c(f)}\;\sqrt{\dfrac{1}{k}\log_2 c(f)} -\end{equation} -\end{verbatim} - -\subsection{The \cn{binom}, \cn{dbinom}, and \cn{tbinom} commands} - -For binomial expressions\index{binomials} such as $\binom{n}{k}$ -\pkg{amsmath} has \cn{binom}, \cn{dbinom} and \cn{tbinom}: -\begin{equation} -2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2} -\end{equation} -\begin{verbatim} -2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2} -\end{verbatim} - -\subsection{The \cn{genfrac} command} - -The capabilities of \cn{frac}, \cn{binom}, and their variants are -subsumed by a generalized fraction command \cn{genfrac} with six -arguments. The last two correspond to \cn{frac}'s numerator and -denominator; the first two are optional delimiters (as seen in -\cn{binom}); the third is a line thickness override (\cn{binom} uses -this to set the fraction line thickness to 0\mdash i.e., invisible); and -the fourth argument is a mathstyle override: integer values 0\ndash 3 select -respectively \cn{displaystyle}, \cn{textstyle}, \cn{scriptstyle}, and -\cn{scriptscriptstyle}. If the third argument is left empty, the line -thickness defaults to `normal'. - -\begin{center}\begin{minipage}{.85\columnwidth} -\raggedright \normalfont\ttfamily \exhyphenpenalty10000 -\newcommand{\ma}[1]{% - \string{{\normalfont\itshape#1}\string}\penalty9999 \ignorespaces} -\string\genfrac \ma{left-delim} \ma{right-delim} \ma{thickness} -\ma{mathstyle} \ma{numerator} \ma{denominator} -\end{minipage}\end{center} -To illustrate, here is how \cn{frac}, \cn{tfrac}, and -\cn{binom} might be defined. -\begin{verbatim} -\newcommand{\frac}[2]{\genfrac{}{}{}{}{#1}{#2}} -\newcommand{\tfrac}[2]{\genfrac{}{}{}{1}{#1}{#2}} -\newcommand{\binom}[2]{\genfrac{(}{)}{0pt}{}{#1}{#2}} -\end{verbatim} -If you find yourself repeatedly using \cn{genfrac} throughout a document -for a particular notation, you will do yourself a favor (and your -publisher) if you define a meaningfully-named abbreviation for that -notation, along the lines of \cn{frac} and \cn{binom}. - -The primitive generalized fraction commands \cs{over}, \cs{overwithdelims}, -\cs{atop}, \cs{atopwithdelims}, \cs{above}, \cs{abovewithdelims} produce -warning messages if used with the \pkg{amsmath} package, for reasons -discussed in \fn{technote.tex}. - -\section{Continued fractions} - -The continued fraction\index{continued fractions} -\begin{equation} -\cfrac{1}{\sqrt{2}+ - \cfrac{1}{\sqrt{2}+ - \cfrac{1}{\sqrt{2}+\cdots -}}} -\end{equation} -can be obtained by typing -{\samepage -\begin{verbatim} -\cfrac{1}{\sqrt{2}+ - \cfrac{1}{\sqrt{2}+ - \cfrac{1}{\sqrt{2}+\dotsb -}}} -\end{verbatim} -}% End of \samepage -This produces better-looking results than straightforward use of -\cn{frac}. Left or right placement of any of the numerators is -accomplished by using \cn{cfrac}|[l]| or \cn{cfrac}|[r]| instead of -\cn{cfrac}. - -\section{Smash options} - -The command \cn{smash} is used to typeset a subformula with an effective height and depth of zero, which is sometimes -useful in adjusting the subformula's position with respect to adjacent -symbols. With the \pkg{amsmath} package \cn{smash} has optional -arguments |t| and |b|, because occasionally it is advantageous to be -able to \qq{smash} only the top or only the bottom of something while -retaining the natural depth or height. For example, when adjacent -radical symbols are unevenly sized or positioned because of differences -in the height and depth of their contents, \cn{smash} can be employed to -make them more consistent. Compare -$\sqrt{x}+\sqrt{y}+\sqrt{z}$ and $\sqrt{x}+\sqrt{\smash[b]{y}}+\sqrt{z}$, -where the latter was produced by -\verb"$\sqrt{x}" \verb"+" -\verb"\sqrt{"\5\verb"\smash[b]{y}}" \verb"+" \verb"\sqrt{z}$". - -\section{Delimiters} - -\subsection{Delimiter sizes}\label{bigdel} - -The automatic delimiter sizing done by \cn{left} and \cn{right} has two -limitations: First, it is applied mechanically to produce delimiters -large enough to encompass the largest contained item, and second, the -range of sizes is not even approximately continuous but has fairly large -quantum jumps. This means that a math fragment that is infinitesimally -too large for a given delimiter size will get the next larger size, a -jump of 3pt or so in normal-sized text. There are two or three -situations where the delimiter size is commonly adjusted, using a set of -commands that have `big' in their names. -\begin{ctab}{l|llllll} -Delimiter& - text& \ncn{left}& \ncn{bigl}& \ncn{Bigl}& \ncn{biggl}& \ncn{Biggl}\\ -size& - size& \ncn{right}& \ncn{bigr}& \ncn{Bigr}& \ncn{biggr}& \ncn{Biggr}\\ -\hline -Result\vstrut{5ex}& - $\displaystyle(b)(\frac{c}{d})$& - $\displaystyle\left(b\right)\left(\frac{c}{d}\right)$& - $\displaystyle\bigl(b\bigr)\bigl(\frac{c}{d}\bigr)$& - $\displaystyle\Bigl(b\Bigr)\Bigl(\frac{c}{d}\Bigr)$& - $\displaystyle\biggl(b\biggr)\biggl(\frac{c}{d}\biggr)$& - $\displaystyle\Biggl(b\Biggr)\Biggl(\frac{c}{d}\Biggr)$ -\end{ctab} -The first kind of situation is a cumulative operator with limits above -and below. With \cn{left} and \cn{right} the delimiters usually turn out -larger than necessary, and using the |Big| or |bigg| -sizes\index{big@\cn{big}, \cn{Big}, \cn{bigg}, \dots\ delimiters} -instead gives better results: -\begin{equation*} -\left[\sum_i a_i\left\lvert\sum_j x_{ij}\right\rvert^p\right]^{1/p} -\quad\text{versus}\quad -\biggl[\sum_i a_i\Bigl\lvert\sum_j x_{ij}\Bigr\rvert^p\biggr]^{1/p} -\end{equation*} -\begin{verbatim} -\biggl[\sum_i a_i\Bigl\lvert\sum_j x_{ij}\Bigr\rvert^p\biggr]^{1/p} -\end{verbatim} -The second kind of situation is clustered pairs of delimiters where -\cn{left} and \cn{right} make them all the same size (because that is -adequate to cover the encompassed material) but what you really want -is to make some of the delimiters slightly larger to make the nesting -easier to see. -\begin{equation*} -\left((a_1 b_1) - (a_2 b_2)\right) -\left((a_2 b_1) + (a_1 b_2)\right) -\quad\text{versus}\quad -\bigl((a_1 b_1) - (a_2 b_2)\bigr) -\bigl((a_2 b_1) + (a_1 b_2)\bigr) -\end{equation*} -\begin{verbatim} -\left((a_1 b_1) - (a_2 b_2)\right) -\left((a_2 b_1) + (a_1 b_2)\right) -\quad\text{versus}\quad -\bigl((a_1 b_1) - (a_2 b_2)\bigr) -\bigl((a_2 b_1) + (a_1 b_2)\bigr) -\end{verbatim} -The third kind of situation is a slightly oversize object in running -text, such as $\left\lvert\frac{b'}{d'}\right\rvert$ where the -delimiters produced by \cn{left} and \cn{right} cause too much line -spreading. In that case \ncn{bigl} and \ncn{bigr}\index{big@\cn{big}, -\cn{Big}, \cn{bigg}, \dots\ delimiters} can be used to produce -delimiters that are slightly larger than the base size but still able to -fit within the normal line spacing: -$\bigl\lvert\frac{b'}{d'}\bigr\rvert$. - -In ordinary \latex/ \ncn{big}, \ncn{bigg}, \ncn{Big}, and \ncn{Bigg} -delimiters aren't scaled properly over the full range of \latex/ font -sizes. With the \pkg{amsmath} package they are. - -\subsection{Vertical bar notations} - -The \pkg{amsmath} package provides commands \cn{lvert}, \cn{rvert}, -\cn{lVert}, \cn{rVert} (compare \cn{langle}, \cn{rangle}) to address the -problem of overloading for the vert bar character \qc{\|}. This -character is currently used in \latex/ documents to represent a wide -variety of mathematical objects: the `divides' relation in a -number-theory expression like $p\vert q$, or the absolute-value -operation $\lvert z\rvert$, or the `such that' condition in set -notation, or the `evaluated at' notation $f_\zeta(t)\bigr\rvert_{t=0}$. -The multiplicity of uses in itself is not so bad; what is bad, however, -is that fact that not all of the uses take the same typographical -treatment, and that the complex discriminatory powers of a knowledgeable -reader cannot be replicated in computer processing of mathematical -documents. It is recommended therefore that there should be a one-to-one -correspondence in any given document between the vert bar character -\qc{\|} and a selected mathematical notation, and similarly for the -double-bar command \ncn{\|}\index{"|@\verb"*+"\"|+}. This immediately -rules out the use of \qc{|} -and \ncn{\|}\index{"|@\verb"*+"\"|+} for delimiters, because left and right -delimiters are distinct usages that do not relate in the same way to -adjacent symbols; recommended practice is therefore to define suitable -commands in the document preamble for any paired-delimiter use of vert -bar symbols: -\begin{verbatim} -\providecommand{\abs}[1]{\lvert#1\rvert} -\providecommand{\norm}[1]{\lVert#1\rVert} -\end{verbatim} -whereupon the document would contain |\abs{z}| to produce $\lvert -z\rvert$ and |\norm{v}| to produce $\lVert v\rVert$. - -\chapter{Operator names} - -\section{Defining new operator names}\label{s:opname} - -Math functions\index{operator names}\relax \index{function -names|see{operator names}} such as $\log$, $\sin$, and $\lim$ are -traditionally typeset in roman type to make them visually more distinct -from one-letter math variables, which are set in math italic. The more -common ones have predefined names, \cn{log}, \cn{sin}, \cn{lim}, and so -forth, but new ones come up all the time in mathematical papers, so the -\pkg{amsmath} package provides a general mechanism for defining new -`operator names'. To define a math function \ncn{xxx} to work like -\cn{sin}, you write -\begin{verbatim} -\DeclareMathOperator{\xxx}{xxx} -\end{verbatim} -whereupon ensuing uses of \ncn{xxx} will produce {\upshape xxx} in the -proper font and automatically add proper spacing\index{horizontal -space!around operator names} on either side when necessary, so that you -get $A\xxx B$ instead of $A\mathrm{xxx}B$. In the second argument of -\cn{DeclareMathOperator} (the name text), a pseudo-text mode prevails: -the hyphen character \qc{\-} will print as a text hyphen rather than a -minus sign and an asterisk \qc{\*} will print as a raised text asterisk -instead of a centered math star. (Compare -\textit{a}-\textit{b}*\textit{c} and $a-b*c$.) But otherwise the name -text is printed in math mode, so that you can use, e.g., subscripts and -superscripts there. - -If the new operator should have subscripts and superscripts placed in -`limits' position above and below as with $\lim$, $\sup$, or $\max$, use -the \qc{\*} form of the \cn{DeclareMathOperator} command: -\begin{verbatim} -\DeclareMathOperator*{\Lim}{Lim} -\end{verbatim} -See also the discussion of subscript placement in -Section~\ref{subplace}. - -The following operator names are predefined: -\begin{ctab}{rlrlrlrl} -\cn{arccos}& $\arccos$ &\cn{deg}& $\deg$ & \cn{lg}& $\lg$ & \cn{projlim}& $\projlim$\\ -\cn{arcsin}& $\arcsin$ &\cn{det}& $\det$ & \cn{lim}& $\lim$ & \cn{sec}& $\sec$\\ -\cn{arctan}& $\arctan$ &\cn{dim}& $\dim$ & \cn{liminf}& $\liminf$ &\cn{sin}& $\sin$\\ -\cn{arg}& $\arg$ & \cn{exp}& $\exp$ & \cn{limsup}& $\limsup$ &\cn{sinh}& $\sinh$\\ -\cn{cos}& $\cos$ & \cn{gcd}& $\gcd$ & \cn{ln}& $\ln$ & \cn{sup}& $\sup$\\ -\cn{cosh}& $\cosh$ & \cn{hom}& $\hom$ & \cn{log}& $\log$ & \cn{tan}& $\tan$\\ -\cn{cot}& $\cot$ & \cn{inf}& $\inf$ & \cn{max}& $\max$ & \cn{tanh}& $\tanh$\\ -\cn{coth}& $\coth$ & \cn{injlim}& $\injlim$ &\cn{min}& $\min$\\ -\cn{csc}& $\csc$ & \cn{ker}& $\ker$ & \cn{Pr}& $\Pr$ -\end{ctab} -\begin{ctab}{rlrl} -\cn{varlimsup}& $\displaystyle\varlimsup$& - \cn{varinjlim}& $\displaystyle\varinjlim$\\ -\cn{varliminf}& $\displaystyle\varliminf$& - \cn{varprojlim}& $\displaystyle\varprojlim$ -\end{ctab} - -There is also a command \cn{operatorname} such that using -\begin{verbatim} -\operatorname{abc} -\end{verbatim} -in a math formula is equivalent to a use of \ncn{abc} defined by -\cn{DeclareMathOperator}. This may be occasionally useful for -constructing more complex notation or other purposes. (Use the variant -\cn{operatorname*} to get limits.) - -\section{\cn{mod} and its relatives} - -Commands \cn{mod}, \cn{bmod}, \cn{pmod}, \cn{pod} are provided to deal -with the special spacing conventions of \qq{mod} notation. \cn{bmod} and -\cn{pmod} are available in \latex/, but with the \pkg{amsmath} package -the spacing of \cn{pmod} will adjust to a smaller value if it's used in -a non-display-mode formula. \cn{mod} and \cn{pod} are variants of -\cn{pmod} preferred by some authors; \cn{mod} omits the parentheses, -whereas \cn{pod} omits the \qq{mod} and retains the parentheses. -\begin{equation} -\gcd(n,m\bmod n);\quad x\equiv y\pmod b; -\quad x\equiv y\mod c;\quad x\equiv y\pod d -\end{equation} -\begin{verbatim} -\gcd(n,m\bmod n);\quad x\equiv y\pmod b; -\quad x\equiv y\mod c;\quad x\equiv y\pod d -\end{verbatim} - -\chapter{The \cn{text} command}\label{text} - -The main use of the command \cn{text} is for words or -phrases\index{text fragments inside math} in a display. It is very -similar to the \latex/ command \cn{mbox} in its effects, but has a -couple of advantages. If you want a word or phrase of text in a -subscript, you can type |..._{\text{word or phrase}}|, which is slightly -easier than the \cn{mbox} equivalent: |..._{\mbox{\scriptsize| |word| -|or| |phrase}}|. The other advantage is the more descriptive name. -\begin{equation} -f_{[x_{i-1},x_i]} \text{ is monotonic,} -\quad i = 1,\dots,c+1 -\end{equation} -\begin{verbatim} -f_{[x_{i-1},x_i]} \text{ is monotonic,} -\quad i = 1,\dots,c+1 -\end{verbatim} - -\chapter{Integrals and sums} - -\section{Multiline subscripts and superscripts} - -The \cn{substack} command can be used to produce a multiline subscript -or superscript:\index{subscripts and superscripts!multi-line}\relax -\index{superscripts|see{subscripts and superscripts}} for example -\begin{ctab}{ll} -\begin{minipage}[t]{.6\columnwidth} -\begin{verbatim} -\sum_{\substack{ - 0\le i\le m\\ - 0<j<n}} - P(i,j) -\end{verbatim} -\end{minipage} -& -$\displaystyle -\sum_{\substack{0\le i\le m\\ 0<j<n}} P(i,j)$ -\end{ctab} -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{ctab}{ll} -\begin{minipage}[t]{.6\columnwidth} -\begin{verbatim} -\sum_{\begin{subarray}{l} - i\in\Lambda\\ 0<j<n - \end{subarray}} - P(i,j) -\end{verbatim} -\end{minipage} -& -$\displaystyle - \sum_{\begin{subarray}{l} - i\in \Lambda\\ 0<j<n - \end{subarray}} - P(i,j)$ -\end{ctab} - -\section{The \cn{sideset} command}\label{sideset} - -There's also a command called \cn{sideset}, for a rather special -purpose: putting symbols at the subscript and -superscript\index{subscripts and superscripts!on sums} corners of a -large operator symbol such as $\sum$ or $\prod$. \emph{Note: this -command is not designed to be applied to anything other than sum-class symbols.} The prime -example is the case when you want to put a prime on a sum symbol. If -there are no limits above or below the sum, you could just use -\cn{nolimits}: here's -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -|\sum\nolimits' E_n| in display mode: -\begin{equation} -\sum\nolimits' E_n -\end{equation} -If, however, you want not only the prime but also something below or -above the sum symbol, it's not so easy\mdash indeed, without -\cn{sideset}, it would be downright difficult. With \cn{sideset}, you -can write -\begin{ctab}{ll} -\begin{minipage}[t]{.6\columnwidth} -\begin{verbatim} -\sideset{}{'} - \sum_{n<k,\;\text{$n$ odd}} nE_n -\end{verbatim} -\end{minipage} -&$\displaystyle -\sideset{}{'}\sum_{n<k,\;\text{$n$ odd}} nE_n -$ -\end{ctab} -The extra pair of empty braces is explained by the fact that -\cn{sideset} has the capability of putting an extra symbol or symbols at -each corner of a large operator; to put an asterisk at each corner of a -product symbol, you would type -\begin{ctab}{ll} -\begin{minipage}[t]{.6\columnwidth} -\begin{verbatim} -\sideset{_*^*}{_*^*}\prod -\end{verbatim} -\end{minipage} -&$\displaystyle -\sideset{_*^*}{_*^*}\prod -$ -\end{ctab} - -\section{Placement of subscripts and limits}\label{subplace} - -The default positioning for subscripts depends on the -base symbol involved. The default for sum-class symbols is -`displaylimits' positioning: When a sum-class symbol appears -in a displayed formula, subscript and superscript are placed in `limits' -position above and below, but in an inline formula, they are placed to -the side, to avoid unsightly and wasteful spreading of the -surrounding text lines. -The default for integral-class symbols is to have sub- and -superscripts always to the side, even in displayed formulas. -(See the discussion of the \opt{intlimits} and related options in -Section~\ref{options}.) - -Operator names such as $\sin$ or $\lim$ may have either `displaylimits' -or `limits' positioning depending on how they were defined. The standard -operator names are defined according to normal mathematical usage. - -The commands \cn{limits} and \cn{nolimits} can be used to override the -normal behavior of a base symbol: -\begin{equation*} -\sum\nolimits_X,\qquad \iint\limits_{A}, -\qquad\varliminf\nolimits_{n\to \infty} -\end{equation*} -To define a command whose subscripts follow the -same `displaylimits' behavior as \cn{sum}, put -\cn{displaylimits} at the tail end of the definition. When multiple -instances of \cn{limits}, \cn{nolimits}, or \cn{displaylimits} occur -consecutively, the last one takes precedence. - -\section{Multiple integral signs} - -\cn{iint}, \cn{iiint}, and \cn{iiiint} give multiple integral -signs\index{integrals!multiple} with the spacing between them nicely -adjusted, in both text and display style. \cn{idotsint} is an extension -of the same idea that 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} - -\chapter{Commutative diagrams}\label{s:commdiag} - -Some commutative diagram commands like the ones in \amstex/ are -available as a separate package, \pkg{amscd}. For complex commutative -diagrams authors will need to turn to more comprehensive packages like -\pkg{kuvio} or \xypic/, but for simple diagrams without diagonal -arrows\index{arrows!in commutative diagrams} the \pkg{amscd} commands -may be more convenient. Here is one example. -\begin{equation*} -\begin{CD} -S^{{\mathcal{W}}_\Lambda}\otimes T @>j>> T\\ -@VVV @VV{\End P}V\\ -(S\otimes T)/I @= (Z\otimes T)/J -\end{CD} -\end{equation*} -\begin{verbatim} -\begin{CD} -S^{{\mathcal{W}}_\Lambda}\otimes T @>j>> T\\ -@VVV @VV{\End P}V\\ -(S\otimes T)/I @= (Z\otimes T)/J -\end{CD} -\end{verbatim} -In the \env{CD} environment the commands |@>>>|, -|@<<<|, |@VVV|, and |@AAA| give respectively right, left, down, and up -arrows. For the horizontal arrows, material between the first and second -|>| or |<| symbols will be typeset as a superscript, and material -between the second and third will be typeset as a subscript. Similarly, -material between the first and second or second and third |A|s or |V|s -of vertical arrows will be typeset as left or right \qq{sidescripts}. -The commands |@=| and \verb'@|' give horizontal and vertical double lines. -A \qq{null arrow} command |@.| can be used instead of a visible arrow -to fill out an array where needed. - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\chapter{Using math fonts} - -\section{Introduction} - -For more comprehensive information on font use in \latex/, see the -\latex/ font guide (\fn{fntguide.tex}) or \booktitle{The \latex/ -Companion} \cite{tlc}. The basic set of math font commands\index{math -fonts}\relax \index{math symbols|see{math fonts}} in \latex/ includes -\cn{mathbf}, \cn{mathrm}, \cn{mathcal}, \cn{mathsf}, \cn{mathtt}, -\cn{mathit}. Additional math alphabet commands such as -\cn{mathbb} for blackboard bold, \cn{mathfrak} for Fraktur, and -\cn{mathscr} for Euler script are available through the packages -\pkg{amsfonts} and \pkg{euscript} (distributed separately). - -\section{Recommended use of math font commands} - -If you find yourself employing math font commands frequently in your -document, you might wish that they had shorter names, such as \ncn{mb} -instead of \cn{mathbf}. Of course, there is nothing to keep you from -providing such abbreviations for yourself by suitable \cn{newcommand} -statements. But for \latex/ to provide shorter names would actually be a -disservice to authors, as that would obscure a much better alternative: -defining custom command names derived from the names of the underlying -mathematical objects, rather than from the names of the fonts used to -distinguish the objects. For example, if you are using bold to indicate -vectors, then you will be better served in the long run if you define a -`vector' command instead of a `math-bold' command: -\begin{verbatim} - \newcommand{\vect}[1]{\mathbf{#1}} -\end{verbatim} -you can write |\vect{a} + \vect{b}| to produce $\vect{a} + -\vect{b}$. -If you decide several months down the road that you want to use the bold -font for some other purpose, and mark vectors by a small over-arrow -instead, then you can put the change into effect merely by changing the -definition of \ncn{vect}; otherwise you would have to replace all -occurrences of \cn{mathbf} throughout your document, perhaps even -needing to inspect each one to see whether it is indeed -an instance of a vector. - -It can also be useful to assign distinct -command names for different letters of a particular font: -\begin{verbatim} -\DeclareSymbolFont{AMSb}{U}{msb}{m}{n}% or use amsfonts package -\DeclareMathSymbol{\C}{\mathalpha}{AMSb}{"43} -\DeclareMathSymbol{\R}{\mathalpha}{AMSb}{"52} -\end{verbatim} -These statements would define the commands \cn{C} and \cn{R} to produce -blackboard-bold letters from the `AMSb' math symbols font. If you refer -often to the\break complex numbers or real numbers in your document, you -might find this method more convenient than (let's say) defining a -\ncn{field} command and writing\break |\field{C}|, |\field{R}|. But for -maximum flexibility and control, define such a \ncn{field} command and -then define \ncn{C} and \ncn{R} in terms of that -command:\indexcs{mathbb} -\begin{verbatim} -\usepackage{amsfonts}% to get the \mathbb alphabet -\newcommand{\field}[1]{\mathbb{#1}} -\newcommand{\C}{\field{C}} -\newcommand{\R}{\field{R}} -\end{verbatim} - -\section{Bold math symbols} - -The \cn{mathbf} command is commonly used to obtain bold Latin letters in -math, but for most other kinds of math symbols it has no effect, or its -effects depend unreliably on the set of math fonts that are in use. For -example, writing -\begin{verbatim} -\Delta \mathbf{\Delta}\mathbf{+}\delta \mathbf{\delta} -\end{verbatim} -produces $\Delta \mathbf{\Delta}\mathbf{+}\delta -\mathbf{\delta}$; the \cn{mathbf} has no effect on the plus -sign or the small delta. - -The \pkg{amsmath} package therefore provides two additional commands, -\cn{boldsymbol} and \cn{pmb}, that can be applied to other kinds of math -symbols. \cn{boldsymbol} can be used for a math symbol that remains -unaffected by \cn{mathbf} if (and only if) your current math font set -includes a bold version of that symbol. \cn{pmb} can be used as a last -resort for any math symbols that do not have a true bold version -provided by your set of math fonts; \qq{pmb} stands for \qq{poor man's -bold} and the command works by typesetting multiple copies of the symbol -with slight offsets. The quality of the output is inferior, especially -for symbols that contain any hairline strokes. When the standard default set of -\latex/ math fonts are in use (Computer Modern), the only symbols that -are likely to require \cn{pmb} are large operator symbols like \cn{sum}, -extended delimiter symbols, or the extra math symbols provided by -the \pkg{amssymb} package \cite{amsfonts}. - -The following formula shows some of the results that are possible: -\begin{verbatim} -A_\infty + \pi A_0 -\sim \mathbf{A}_{\boldsymbol{\infty}} \boldsymbol{+} - \boldsymbol{\pi} \mathbf{A}_{\boldsymbol{0}} -\sim\pmb{A}_{\pmb{\infty}} \pmb{+}\pmb{\pi} \pmb{A}_{\pmb{0}} -\end{verbatim} -\begin{equation*} -A_\infty + \pi A_0 -\sim \mathbf{A}_{\boldsymbol{\infty}} \boldsymbol{+} - \boldsymbol{\pi} \mathbf{A}_{\boldsymbol{0}} -\sim\pmb{A}_{\pmb{\infty}} \pmb{+}\pmb{\pi} \pmb{A}_{\pmb{0}} -\end{equation*} -If you want to use only the \cn{boldsymbol} command without loading the -whole \pkg{amsmath} package, the \pkg{bm} package is recommended (this -is a standard \latex/ package, not an AMS package; you probably have it -already if you have a 1997 or newer version of \latex/). - -\section{Italic Greek letters} - -For italic versions of the capital Greek letters, the following commands -are provided: -\begin{ctab}{rlrl} -\cn{varGamma}& $\varGamma$& \cn{varSigma}& $\varSigma$\\ -\cn{varDelta}& $\varDelta$& \cn{varUpsilon}& $\varUpsilon$\\ -\cn{varTheta}& $\varTheta$& \cn{varPhi}& $\varPhi$\\ -\cn{varLambda}& $\varLambda$& \cn{varPsi}& $\varPsi$\\ -\cn{varXi}& $\varXi$& \cn{varOmega}& $\varOmega$\\ -\cn{varPi}& $\varPi$ -\end{ctab} - -\chapter{Error messages and output problems} - -\section{General remarks} - -This is a supplement to Chapter~8 of the \latex/ manual \cite{lamport} (first -edition: Chapter~6). For the reader's convenience, the set of error -messages discussed here overlaps somewhat with the set in that chapter, -but please be aware that we don't provide exhaustive coverage here. -The error messages are arranged in alphabetical order, disregarding -unimportant text such as |! LaTeX Error:| at the beginning, and -nonalphabetical characters such as \qc{\\}. Where examples are given, we -show also the help messages that appear on screen when you respond to an -error message prompt by entering |h|. - -There is also a section discussing some output errors, i.e., instances -where the printed document has something wrong but there was no \latex/ -error during typesetting. - -\section{Error messages} - -\begin{error}{\begin{split} won't work here.} -\errexa -\begin{verbatim} -! Package amsmath Error: \begin{split} won't work here. - ... - -l.8 \begin{split} - -? h -\Did you forget a preceding \begin{equation}? -If not, perhaps the `aligned' environment is what you want. -? -\end{verbatim} -\errexpl -The \env{split} environment does not construct a stand-alone displayed -equation; it needs to be used within some other environment such as -\env{equation} or \env{gather}. - -\end{error} - -\begin{error}{Extra & on this line} -\errexa -\begin{verbatim} -! Package amsmath Error: Extra & on this line. - -See the amsmath package documentation for explanation. -Type H <return> for immediate help. - ... - -l.9 \end{alignat} - -? h -\An extra & here is so disastrous that you should probably exit - and fix things up. -? -\end{verbatim} -\errexpl -In an \env{alignat} structure the number of alignment points per line -is dictated by the numeric argument given after |\begin{alignat}|. -If you use more alignment points in a line it is assumed that you -accidentally left out a newline command \cn{\\} and the above error is -issued. -\end{error} - -\begin{error}{Improper argument for math accent} -\errexa -\begin{verbatim} -! Package amsmath Error: Improper argument for math accent: -(amsmath) Extra braces must be added to -(amsmath) prevent wrong output. - -See the amsmath package documentation for explanation. -Type H <return> for immediate help. - ... - -l.415 \tilde k_{\lambda_j} = P_{\tilde \mathcal - {M}} -? -\end{verbatim} -\errexpl -Non-simple arguments for any \LaTeX{} command should be enclosed in -braces. In this example extra braces are needed as follows: -\begin{verbatim} -... P_{\tilde{\mathcal{M}}} -\end{verbatim} -\end{error} - -\begin{error}{Font OMX/cmex/m/n/7=cmex7 not loadable ...} -\errexa -\begin{verbatim} -! Font OMX/cmex/m/n/7=cmex7 not loadable: Metric (TFM) file not found. -<to be read again> - relax -l.8 $a - b+b^2$ -? h -I wasn't able to read the size data for this font, -so I will ignore the font specification. -[Wizards can fix TFM files using TFtoPL/PLtoTF.] -You might try inserting a different font spec; -e.g., type `I\font<same font id>=<substitute font name>'. -? -\end{verbatim} -\errexpl -Certain extra sizes of some Computer Modern fonts that were formerly -available mainly through the AMSFonts\index{AMSFonts collection} -distribution are considered part of standard \latex/ (as of June 1994): -\fn{cmex7}\ndash \texttt{9}, \fn{cmmib5}\ndash \texttt{9}, and -\fn{cmbsy5}\ndash \texttt{9}. If these extra sizes are missing on your -system, you should try first to get them from the source where you -obtained \latex/. If that fails, you could try getting the fonts from -CTAN (e.g., in the form of Metafont\index{Metafont source files} source -files, directory \nfn{/tex-archive/fonts/latex/mf}, or in PostScript -Type 1 format, directory -\nfn{/tex-archive/fonts/cm/ps-type1/bakoma}\index{BaKoMa fonts}\relax -\index{PostScript fonts}). - -If the font name begins with \fn{cmex}, there is a special option -\fn{cmex10} for the \pkg{amsmath} package that provides a temporary -workaround. I.e., change the \cn{usepackage} to -\begin{verbatim} -\usepackage[cmex10]{amsmath} -\end{verbatim} -This will force the use of the 10-point size of the \fn{cmex} font in -all cases. Depending on the contents of your document this may be -adequate. -\end{error} - -\begin{error}{Math formula deleted: Insufficient extension fonts} -\errexa -\begin{verbatim} -! Math formula deleted: Insufficient extension fonts. -l.8 $ab+b^2$ - -? -\end{verbatim} -\errexpl -This usually follows a previous error |Font ... not loadable|; see the -discussion of that error (above) for solutions. -\end{error} - -\begin{error}{Missing number, treated as zero} -\errexa -\begin{verbatim} -! Missing number, treated as zero. -<to be read again> - a -l.100 \end{alignat} - -? h -A number should have been here; I inserted `0'. -(If you can't figure out why I needed to see a number, -look up `weird error' in the index to The TeXbook.) - -? -\end{verbatim} -\errexpl -There are many possibilities that can lead to this error. However, one -possibility that is relevant for the \pkg{amsmath} package is that you -forgot to give the number argument of an \env{alignat} environment, as -in: -\begin{verbatim} -\begin{alignat} - a& =b& c& =d\\ -a'& =b'& c'& =d' -\end{alignat} -\end{verbatim} -where the first line should read instead -\begin{verbatim} -\begin{alignat}{2} -\end{verbatim} - -Another possibility is that you have a left bracket character |[| -following a linebreak command \cn{\\} in a multiline construction such -as \env{array}, \env{tabular}, or \env{eqnarray}. This will be -interpreted by \latex/ as the beginning of an `additional vertical -space' request \cite[\S C.1.6]{lamport}, even if it occurs on the following -line and is intended to be part of the contents. For example -\begin{verbatim} -\begin{array} -a+b\\ -[f,g]\\ -m+n -\end{array} -\end{verbatim} -To prevent the error message in such a case, you can -add braces as discussed in the \latex/ manual \cite[\S C.1.1]{lamport}: -\begin{verbatim} -\begin{array} -a+b\\ -{[f,g]}\\ -m+n -\end{array} -\end{verbatim} - -\end{error} - -\begin{error}{Missing \right. inserted} -\errexa -\begin{verbatim} -! Missing \right. inserted. -<inserted text> - \right . -l.10 \end{multline} - -? h -I've inserted something that you may have forgotten. -(See the <inserted text> above.) -With luck, this will get me unwedged. But if you -really didn't forget anything, try typing `2' now; then -my insertion and my current dilemma will both disappear. -\end{verbatim} -\errexpl -This error typically arises when you try to insert a linebreak inside a -\cn{left}-\cn{right} pair of delimiters in a \env{multline} or -\env{split} environment: -\begin{verbatim} -\begin{multline} -AAA\left(BBB\\ - CCC\right) -\end{multline} -\end{verbatim} -There are two possible solutions: (1)~instead of using \cn{left} and -\cn{right}, use `big' delimiters of fixed size (\cn{bigl} \cn{bigr} -\cn{biggl} \cn{biggr} \dots; see \secref{bigdel}); or (2)~use null -delimiters to break up the \cn{left}-\cn{right} pair into parts for each -line: -\begin{verbatim} -AAA\left(BBB\right.\\ - \left.CCC\right) -\end{verbatim} -The latter solution may result in mismatched delimiter sizes; -ensuring that they match requires using \cn{vphantom} in the line -that has the smaller delimiter (or possibly \cn{smash} in the line that -has the larger delimiter). In the argument of \cn{vphantom} put a copy -of the tallest element that occurs in the other line, e.g., -\begin{verbatim} -xxx \left(\int_t yyy\right.\\ - \left.\vphantom{\int_t} zzz ... \right) -\end{verbatim} -\end{error} - -\begin{error}{Paragraph ended before \xxx was complete} -\errexa -\begin{verbatim} -Runaway argument? - -! Paragraph ended before \multline was complete. -<to be read again> - \par -l.100 - -? h -I suspect you've forgotten a `}', causing me to apply this -control sequence to too much text. How can we recover? -My plan is to forget the whole thing and hope for the best. -? -\end{verbatim} -\errexpl -This might be produced by a misspelling in the |\end{multline}| command, -e.g., -\begin{verbatim} -\begin{multline} -... -\end{multiline} -\end{verbatim} -or by using abbreviations for certain environments, such as |\bal| and -|\eal| for |\begin{align}| and |\end{align}|: -\begin{verbatim} -\bal -... -\eal -\end{verbatim} -For technical reasons that kind of abbreviation does not work with -the more complex displayed equation environments of the \pkg{amsmath} package -(\env{gather}, \env{align}, \env{split}, etc.; cf.\@ \fn{technote.tex}). -\end{error} - -\begin{error}{Runaway argument?} -See the discussion for the error message -\texttt{Paragraph ended before \ncn{xxx} was complete}. -\end{error} - -\begin{error}{Unknown option `xxx' for package `yyy'} -\errexa -\begin{verbatim} -! LaTeX Error: Unknown option `intlim' for package `amsmath'. -... -? h -The option `intlim' was not declared in package `amsmath', perhaps you -misspelled its name. Try typing <return> to proceed. -? -\end{verbatim} -\errexpl -This means that you misspelled the option name, or the package simply -does not have an option that you expected it to have. Consult the -documentation for the given package. -\end{error} - -\begin{error}{Old form `\pmatrix' should be \begin{pmatrix}.} -\errexa -\begin{verbatim} -! Package amsmath Error: Old form `\pmatrix' should be - \begin{pmatrix}. - -See the amsmath package documentation for explanation. -Type H <return> for immediate help. - ... - -\pmatrix ->\left (\matrix@check \pmatrix - \env@matrix -l.16 \pmatrix - {a&b\cr c&d\cr} -? h -`\pmatrix{...}' is old Plain-TeX syntax whose use is -ill-advised in LaTeX. -? -\end{verbatim} -\errexpl -When the \pkg{amsmath} package is used, the old forms of \cn{pmatrix}, -\cn{matrix}, and \cn{cases} cannot be used any longer because of naming -conflicts. Their syntax did not conform with standard \LaTeX{} syntax -in any case. -\end{error} - -\begin{error}{Erroneous nesting of equation structures} -\errexa -\begin{verbatim} -! Package amsmath Error: Erroneous nesting of equation structures; -(amsmath) trying to recover with `aligned'. - -See the amsmath package documentation for explanation. -Type H <return> for immediate help. - ... - -l.260 \end{alignat*} - \end{equation*} -\end{verbatim} -\errexpl -The structures \env{align}, \env{alignat}, etc., are designed -for top-level use and for the most part cannot be nested inside some -other displayed equation structure. The chief exception is that -\env{align} and most of its variants can be used inside the -\env{gather} environment. -\end{error} - -\section{Warning messages} - -\begin{error}{Foreign command \over [or \atop or \above]} -\errexa -\begin{verbatim} -Package amsmath Warning: Foreign command \over; \frac or \genfrac -(amsmath) should be used instead. -\end{verbatim} -\errexpl The primitive generalized fraction commands of \tex/\mdash -\cs{over}, \cs{atop}, \cs{above}\mdash are deprecated when the -\pkg{amsmath} package is used because their syntax is foreign to \latex/ -and \pkg{amsmath} provides native \latex/ equivalents. See -\fn{technote.tex} for further information. -\end{error} - -\begin{error}{Cannot use `split' here} -\errexa -\begin{verbatim} -Package amsmath Warning: Cannot use `split' here; -(amsmath) trying to recover with `aligned' -\end{verbatim} -\errexpl The \env{split} environment is designed to serve as the entire -body of an equation, or an entire line of an \env{align} or \env{gather} -environment. There cannot be any printed material before or -after it within the same enclosing structure: -\begin{verbatim} -\begin{equation} -\left\{ % <-- Not allowed -\begin{split} -... -\end{split} -\right. % <-- Not allowed -\end{equation} -\end{verbatim} -\end{error} - -\section{Wrong output} - -\subsection{Section numbers 0.1, 5.1, 8.1 instead of 1, 2, 3} -\label{numinverse} - -This most likely means that you have the arguments for \cn{numberwithin} -in reverse order: -\begin{verbatim} -\numberwithin{section}{equation} -\end{verbatim} -That means `print the section number as \textit{equation -number}.\textit{section number} and reset to 1 every time an equation -occurs' when what you probably wanted was the inverse -\begin{verbatim} -\numberwithin{equation}{section} -\end{verbatim} - -\subsection{The \cn{numberwithin} command had no effect on equation -numbers} - -Are you looking at the first section in your document? Check the section -numbers elsewhere to see if the problem is the one described in -\secref{numinverse}. - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\chapter{Additional information} - -\section{Converting existing documents} - -\subsection{Converting from plain \LaTeX{}} - -A \LaTeX{} document will typically continue to work the same in most -respects if \verb'\usepackage{amsmath}' is added in the document -preamble. By default, however, the \pkg{amsmath} package suppresses page -breaks inside multiple-line displayed equation structures such as -\env{eqnarray}, \env{align}, and \env{gather}. To continue allowing page -breaks inside \env{eqnarray} after switching to \pkg{amsmath}, you will -need to add the following line in your document preamble: -\begin{verbatim} -\allowdisplaybreaks[1] -\end{verbatim} -To ensure normal spacing around relation symbols, you might also want to -change \env{eqnarray} to \env{align}, \env{multline}, or -\env{equation}\slash\env{split} as appropriate. - -Most of the other differences in \pkg{amsmath} usage can be considered -optional refinements, e.g., using -\begin{verbatim} -\DeclareMathOperator{\Hom}{Hom} -\end{verbatim} -instead of \verb'\newcommand{\Hom}{\mbox{Hom}}'. - -\subsection{Converting from \amslatex/ 1.1} -See \fn{diffs-m.txt}. - -\section{Technical notes} -The file \fn{technote.tex} contains some remarks on miscellaneous -technical questions that are less likely to be of general interest. - -\section{Getting help} - -Questions or comments regarding \pkg{amsmath} and related packages -should be sent to: -\begin{infoaddress} -American Mathematical Society\\ -Technical Support\\ -Electronic Products and Services\\ -P. O. Box 6248\\ -Providence, RI 02940\\[3pt] -Phone: 800-321-4AMS (321-4267) \quad or \quad 401-455-4080\\ -Internet: \mail{tech-support@ams.org} -\end{infoaddress} -If you are reporting a problem you should include -the following information to make proper investigation possible: -\begin{enumerate} -\item The source file where the problem occurred, preferably reduced - to minimum size by removing any material that can be removed without - affecting the observed problem. -\item A \latex/ log file showing the error message (if applicable) and - the version numbers of the document class and option files being used. -\end{enumerate} - -\section{Of possible interest}\label{a:possible-interest} -Information about obtaining AMSFonts or other \tex/-related -software from the AMS Internet archive \fn{e-math.ams.org} -can be obtained by sending a request through electronic mail to: -\mail{webmaster@ams.org}. - -Information about obtaining the \pkg{amsmath} distribution on diskette -from the AMS is available from: -\begin{infoaddress} -American Mathematical Society\\ -Customer Services\\ -P. O. Box 6248\\ -Providence, RI 02940\\[3pt] -Phone: 800-321-4AMS (321-4267) \quad or \quad 401-455-4000\\ -Internet: \mail{cust-serv@ams.org} -\end{infoaddress} - -The \tex/ Users Group\index{TeX Users@\tex/ Users Group} is a nonprofit -organization that publishes a journal -(\journalname{TUGboat}\index{TUGboat@\journalname{TUGboat}}), holds -meetings, and serves as a clearing-house of general information about -\tex/ and \tex/-related software. -\begin{infoaddress} -\tex/ Users Group\\ -PO Box 2311\\ -Portland, OR 97208-2311\\ -USA\\[3pt] -Phone: +1-503-223-9994\\ -Email: \mail{office@tug.org} -\end{infoaddress} -Membership in the \tex/ Users Group is a good way to support continued -development of free \tex/-related software. There are also many local -\tex/ user groups in other countries; information about contacting a -local user group can be gotten from the \tex/ Users Group. - -There is a Usenet newsgroup called \fn{comp.text.tex} that is a fairly -good source of information about \latex/ and \tex/ in general. If you -don't know about reading newsgroups, check with your local system -administrator to see if newsgroup service is available at your site. - -\begin{thebibliography}{9} -\addcontentsline{toc}{chapter}{Bibliography} - -\bibitem{amsfonts}\booktitle{AMSFonts version \textup{2.2}\mdash user's guide}, -Amer. Math. Soc., Providence, RI, 1994; distributed -with the AMSFonts package. - -\bibitem{instr-l}\booktitle{Instructions for preparation of -papers and monographs\mdash \amslatex/}, -Amer. Math. Soc., Providence, RI, 1996, 1999. - -\bibitem{amsthdoc}\booktitle{Using the \pkg{amsthm} Package}, -Amer. Math. Soc., Providence, RI, 1999. - -\bibitem{tlc} Michel Goossens, Frank Mittelbach, and Alexander Samarin, -\booktitle{The \latex/ companion}, Addison-Wesley, Reading, MA, 1994. - [\emph{Note: The 1994 edition is not a reliable guide for the - \pkg{amsmath} package unless you refer to the errata for Chapter - 8\mdash file \fn{compan.err}, distributed with \LaTeX{}.}] - -% Deal with a line breaking problem -\begin{raggedright} -\bibitem{mil} G. Gr\"{a}tzer, -\emph{Math into \LaTeX{}: An Introduction to \LaTeX{} and AMS-\LaTeX{}} - \url{http://www.ams.org/cgi-bin/bookstore/bookpromo?fn=91&arg1=bookvideo&itmc=MLTEX}, -Birkh\"{a}user, Boston, 1995.\par -\end{raggedright} - -\bibitem{kn} Donald E. Knuth, \booktitle{The \tex/book}, -Addison-Wesley, Reading, MA, 1984. - -\bibitem{lamport} Leslie Lamport, \booktitle{\latex/: A document preparation -system}, 2nd revised ed., Addison-Wesley, Reading, MA, 1994. - -\bibitem{msf} Frank Mittelbach and Rainer Sch\"opf, -\textit{The new font family selection\mdash user -interface to standard \latex/}, \journalname{TUGboat} \textbf{11}, -no.~2 (June 1990), pp.~297\ndash 305. - -\bibitem{jt} Michael Spivak, \booktitle{The joy of \tex/}, 2nd revised ed., -Amer. Math. Soc., Providence, RI, 1990. - -\end{thebibliography} - -\begin{theindex} - - \item {\normalfont \ttfamily \bslchar \qcbang }, 11 - \item \verb*+\,+, 11 - \item \verb*+\:+, 11 - \item \verb*+\;+, 11 - \item \verb*+\\+, 3, 5, 6, 8, 9, 24, 26 - \item \verb*+\\*+, 9 - \item \verb*+\|+, 16 - - \indexspace - - \item \verb*+\above+, 14, 29 - \item \verb*+\abovewithdelims+, 14 - \item \verb*+\accentedsymbol+, 2 - \item \texttt{accents} package, 12 - \item \verb*+\addtocounter+, 10 - \item \verb*+\addtolength+, 5 - \item \texttt{align} environment, 3, 5--7, 28--30 - \item \texttt{alignat} environment, 7, 24, 26, 29 - \item \texttt{aligned} environment, 7, 9 - \item \texttt{alignedat} environment, 7, 9 - \item \verb*+\allowdisplaybreaks+, 8, 9 - \item \texttt{amsart} class, 1 - \item \texttt{amsbook} class, 1 - \item \texttt{amsbsy} package, 1, 2 - \item \texttt{amscd} package, 1, 2, 20 - \item \texttt{amsfonts} package, 21 - \item AMSFonts collection, 25 - \item \texttt{amsmath} package, i, 1--3, 5, 8--18, 22, 23, 25, 26, - 28--32 - \item \texttt{amsopn} package, 1, 2 - \item \texttt{amssymb} package, 22 - \item \texttt{amstext} package, 1, 2 - \item \texttt{amsthm} package, 1, 32 - \item \texttt{amsxtra} package, 1, 2, 12 - \item \verb*+\arccos+, 18 - \item \verb*+\arcsin+, 18 - \item \verb*+\arctan+, 18 - \item \verb*+\arg+, 18 - \item \texttt{array} environment, 8, 10, 26 - \item arrows - \subitem extensible, 13 - \subitem in commutative diagrams, 20 - \item \verb*+\atop+, 14, 29 - \item \verb*+\atopwithdelims+, 14 - - \indexspace - - \item BaKoMa fonts, 25 - \item \cn{big}, \cn{Big}, \cn{bigg}, \dots\ delimiters, 16 - \item \verb*+\biggl+, 27 - \item \verb*+\biggr+, 27 - \item \verb*+\bigl+, 27 - \item \verb*+\bigr+, 27 - \item \verb*+\binom+, 14 - \item binomials, 14 - \item \texttt{bm} package, 2, 23 - \item \texttt{Bmatrix} environment, 10 - \item \texttt{bmatrix} environment, 10 - \item \verb*+\bmod+, 18 - \item \verb*+\boldsymbol+, 22, 23 - \item \verb*+\boxed+, 13 - - \indexspace - - \item \verb*+\C+, 22 - \item \texttt{cases} environment, 8 - \item \verb*+\cases+, 28 - \item \texttt{CD} environment, 2, 21 - \item \verb*+\cdots+, 12 - \item \texttt{centertags} option, 2 - \item \verb*+\cfrac+, 15 - \item \texttt{cmbsy5}, 25 - \item \texttt{cmex}, 25 - \item \texttt{cmex10}, 25 - \item \texttt{cmex7}, 25 - \item \texttt{cmmib5}, 25 - \item \texttt{comp.text.tex}, 31 - \item \texttt{compan.err}, 32 - \item continued fractions, 15 - \item \verb*+\cos+, 18 - \item \verb*+\cosh+, 18 - \item \verb*+\cot+, 18 - \item \verb*+\coth+, 18 - \item \verb*+\csc+, 18 - - \indexspace - - \item \verb*+\dbinom+, 14 - \item \verb*+\ddddot+, 12 - \item \verb*+\dddot+, 12 - \item \verb*+\ddot+, 12 - \item \verb*+\DeclareMathOperator+, 2, 17, 18 - \item \verb*+\deg+, 18 - \item \verb*+\det+, 18 - \item \verb*+\dfrac+, 14 - \item \texttt{diffs-m.txt}, 30 - \item \verb*+\dim+, 18 - \item \verb*+\displaybreak+, 8, 9 - \item displayed equations, 3 - \item displayed equations - \subitem centering, 3 - \item \verb*+\displaylimits+, 20 - \item \texttt{displaymath} environment, 3 - \item \verb*+\displaystyle+, 14 - \item \verb*+\documentclass+, 2 - \item \verb*+\dot+, 12 - \item dots, \see{ellipsis dots}{11} - \item \verb*+\dotsb+, 12 - \item \verb*+\dotsc+, 12 - \item \verb*+\dotsi+, 12 - \item \verb*+\dotsm+, 12 - \item \verb*+\dotso+, 12 - - \indexspace - - \item \texttt{e-math.ams.org}, 31 - \item ellipsis dots - \subitem in matrices, 11 - \item \texttt{eqnarray} environment, 1, 3, 26, 30 - \item \verb*+\eqref+, 10 - \item \texttt{equation} environment, 1, 3, 5, 7, 24, 30 - \item equation numbers - \subitem cross-references, 10 - \subitem hierarchy, 9 - \subitem left or right placement, 3 - \subitem overriding, 3 - \subitem subordinate numbering, 10 - \subitem vertical placement, 2 - \item \texttt{equation*} environment, 3 - \item equations, \see{displayed equations}{3} - \item \texttt{euscript} package, 21 - \item \verb*+\exp+, 18 - - \indexspace - - \item \verb*+\fbox+, 13 - \item \texttt{fleqn} option, 3, 5 - \item \texttt{fntguide.tex}, 21 - \item \verb*+\frac+, 14, 15 - \item fractions, 14 - \item \verb*+\fracwithdelims+, 2 - \item function names, \see{operator names}{17} - - \indexspace - - \item \texttt{gather} environment, 5--7, 24, 28--30 - \item \texttt{gathered} environment, 7, 9 - \item \verb*+\gcd+, 18 - \item \verb*+\genfrac+, 14 - - \indexspace - - \item \verb*+\hat+, 12 - \item \verb*+\hdotsfor+, 11 - \item \verb*+\hom+, 18 - \item horizontal space - \subitem around operator names, 17 - \subitem in math mode, 11 - - \indexspace - - \item \verb*+\idotsint+, 20 - \item \verb*+\iiiint+, 20 - \item \verb*+\iiint+, 20 - \item \verb*+\iint+, 20 - \item \verb*+\inf+, 18 - \item \verb*+\injlim+, 18 - \item integrals - \subitem multiple, 20 - \subitem placement of limits, 2 - \item \verb*+\intertext+, 9 - \item \texttt{intlimits} option, 2, 20 - - \indexspace - - \item \verb*+\ker+, 18 - \item \texttt{kuvio} package, 20 - - \indexspace - - \item \verb*+\label+, 6, 10 - \item \verb*+\langle+, 16 - \item \verb*+\ldots+, 12 - \item \verb*+\left+, 15, 16, 27 - \item \verb*+\leftroot+, 13 - \item \texttt{leqno} option, 3, 5 - \item \verb*+\lg+, 18 - \item \verb*+\lim+, 1, 2, 17, 18 - \item \verb*+\liminf+, 18 - \item limits, \see{subscripts and superscripts}{2} - \item \verb*+\limits+, 20 - \item \verb*+\limsup+, 18 - \item \verb*+\ln+, 18 - \item \verb*+\log+, 17, 18 - \item \verb*+\lVert+, 16 - \item \verb*+\lvert+, 16 - - \indexspace - - \item math fonts, 21 - \item math symbols, \see{math fonts}{21} - \item \verb*+\mathbb+, 21, 22 - \item \verb*+\mathbf+, 21, 22 - \item \verb*+\mathcal+, 21 - \item \verb*+\mathfrak+, 21 - \item \verb*+\mathit+, 21 - \item \verb*+\mathrm+, 21 - \item \verb*+\mathscr+, 21 - \item \verb*+\mathsf+, 21 - \item \verb*+\mathtt+, 21 - \item matrices, 10 - \subitem ellipsis dots, 11 - \item \texttt{matrix} environment, 10 - \item \verb*+\matrix+, 28 - \item \verb*+\max+, 18 - \item \verb*+\mbox+, 18 - \item \verb*+\medspace+, 11 - \item Metafont source files, 25 - \item \verb*+\min+, 18 - \item \verb*+\mod+, 18 - \item \verb*+\mspace+, 11 - \item \texttt{multline} environment, 3, 5, 27, 30 - \item \verb*+\multlinegap+, 5 - - \indexspace - - \item \texttt{namelimits} option, 2 - \item \verb*+\negmedspace+, 11 - \item \verb*+\negthickspace+, 11 - \item \verb*+\negthinspace+, 11 - \item \verb*+\newcommand+, 21 - \item \verb*+\nobreakdash+, 12 - \item \texttt{nointlimits} option, 2 - \item \verb*+\nolimits+, 19, 20 - \item \texttt{nonamelimits} option, 2 - \item \texttt{nosumlimits} option, 2 - \item \verb*+\notag+, 3, 5 - \item \verb*+\numberwithin+, 9, 29, 30 - - \indexspace - - \item operator names, 17 - \item \verb*+\operatorname+, 18 - \item \verb*+\operatorname*+, 18 - \item \verb*+\over+, 14, 29 - \item \verb*+\overleftarrow+, 13 - \item \verb*+\overleftrightarrow+, 13 - \item \verb*+\overrightarrow+, 13 - \item \verb*+\overset+, 13 - \item \verb*+\overwithdelims+, 14 - - \indexspace - - \item page breaks, 8 - \item \verb*+\pagebreak+, 8 - \item \texttt{pmatrix} environment, 10 - \item \verb*+\pmatrix+, 28 - \item \verb*+\pmb+, 22 - \item \verb*+\pmod+, 18 - \item \verb*+\pod+, 18 - \item PostScript fonts, 25 - \item \verb*+\Pr+, 18 - \item \verb*+\projlim+, 18 - - \indexspace - - \item \verb*+\qquad+, 11 - \item \verb*+\quad+, 11 - - \indexspace - - \item \verb*+\R+, 22 - \item \verb*+\raisetag+, 8 - \item \verb*+\rangle+, 16 - \item \verb*+\ref+, 10 - \item \texttt{reqno} option, 3, 5 - \item \verb*+\right+, 15, 16, 27 - \item \verb*+\rVert+, 16 - \item \verb*+\rvert+, 16 - - \indexspace - - \item \verb*+\scriptscriptstyle+, 14 - \item \verb*+\scriptstyle+, 14 - \item \verb*+\sec+, 18 - \item \verb*+\setcounter+, 9, 10 - \item \verb*+\setlength+, 5 - \item \verb*+\shoveleft+, 5 - \item \verb*+\shoveright+, 5 - \item \verb*+\sideset+, 13, 19 - \item \verb*+\sin+, 1, 2, 17, 18 - \item \verb*+\sinh+, 18 - \item \texttt{smallmatrix} environment, 11 - \item \verb*+\smash+, 15, 27 - \item \verb*+\sphat+, 12 - \item \texttt{split} environment, 3, 5, 6, 9, 24, 27--30 - \item \verb*+\sptilde+, 12 - \item \verb*+\stackrel+, 13 - \item \texttt{subarray} environment, 19 - \item \texttt{subeqn.tex}, 3 - \item \texttt{subequations} environment, 10 - \item subscripts and superscripts, 13 - \subitem multi-line, 19 - \subitem on sums, 19 - \subitem placement, 2 - \item \verb*+\substack+, 19 - \item \verb*+\sum+, 20, 22 - \item \texttt{sumlimits} option, 2 - \item \verb*+\sup+, 18 - \item superscripts, \see{subscripts and superscripts}{19} - - \indexspace - - \item \texttt{tabular} environment, 26 - \item \verb*+\tag+, 3 - \item \verb*+\tag*+, 3 - \item \verb*+\tan+, 18 - \item \verb*+\tanh+, 18 - \item \verb*+\tbinom+, 14 - \item \texttt{tbtags} option, 2 - \item \texttt{technote.tex}, 14, 28--30 - \item \texttt{testmath.tex}, 3 - \item \tex/ Users Group, 31 - \item \verb*+\text+, 2, 7--9, 18 - \item text fragments inside math, 9, 18 - \item \verb*+\textstyle+, 14 - \item \verb*+\tfrac+, 14 - \item \verb*+\theequation+, 9, 10 - \item \verb*+\thickspace+, 11 - \item \verb*+\thinspace+, 11 - \item \journalname{TUGboat}, 31 - - \indexspace - - \item \verb*+\underleftarrow+, 13 - \item \verb*+\underleftrightarrow+, 13 - \item \verb*+\underrightarrow+, 13 - \item \verb*+\underset+, 13 - \item \verb*+\uproot+, 13 - \item \verb*+\usepackage+, 2, 25 - - \indexspace - - \item \verb*+\value+, 10 - \item \verb*+\varDelta+, 23 - \item \verb*+\varGamma+, 23 - \item \verb*+\varinjlim+, 18 - \item \verb*+\varLambda+, 23 - \item \verb*+\varliminf+, 18 - \item \verb*+\varlimsup+, 18 - \item \verb*+\varOmega+, 23 - \item \verb*+\varPhi+, 23 - \item \verb*+\varPi+, 23 - \item \verb*+\varprojlim+, 18 - \item \verb*+\varPsi+, 23 - \item \verb*+\varSigma+, 23 - \item \verb*+\varTheta+, 23 - \item \verb*+\varUpsilon+, 23 - \item \verb*+\varXi+, 23 - \item \texttt{Vmatrix} environment, 10 - \item \texttt{vmatrix} environment, 10 - \item \verb*+\vphantom+, 27 - - \indexspace - - \item \verb*+\xleftarrow+, 13 - \item \verb*+\xrightarrow+, 13 - -\end{theindex} - -\end{document} diff --git a/Master/texmf-dist/source/latex/amsmath/amsmath.dtx b/Master/texmf-dist/source/latex/amsmath/amsmath.dtx index ee7fe578b1f..af2d373c627 100644 --- a/Master/texmf-dist/source/latex/amsmath/amsmath.dtx +++ b/Master/texmf-dist/source/latex/amsmath/amsmath.dtx @@ -1,21 +1,8 @@ -% \def\filename{amsmath.dtx} -% \def\fileversion{2.14} -% \def\filedate{2013/01/14} % % \iffalse meta-comment % -% American Mathematical Society -% Technical Support -% Publications Technical Group -% 201 Charles Street -% Providence, RI 02904 -% USA -% tel: (401) 455-4080 -% (800) 321-4267 (USA and Canada only) -% fax: (401) 331-3842 -% email: tech-support@ams.org -% % Copyright 1995, 2000, 2013 American Mathematical Society. +% Copyright 2016 LaTeX3 Project and American Mathematical Society. % % This work may be distributed and/or modified under the % conditions of the LaTeX Project Public License, either version 1.3c @@ -27,15 +14,24 @@ % % This work has the LPPL maintenance status `maintained'. % -% The Current Maintainer of this work is the American Mathematical -% Society. +% The Current Maintainer of this work is the LaTeX3 Project. % % \fi % %\iffalse %<*driver> \documentclass{amsdtx} +\def\MaintainedByLaTeXTeam#1{% +\begin{center}% +\fbox{\fbox{\begin{tabular}{@{}l@{}}% +This file is maintained by the \LaTeX{} Project team.\\% +Bug reports can be opened (category \texttt{#1}) at\\% +\url{http://latex-project.org/bugs.html}.\end{tabular}}}\end{center}} \raggedbottom +\let\savedarg\arg +\usepackage{amsmath} +\let\arg\savedarg +\GetFileInfo{amsmath.sty} \begin{document} \title{The \pkg{amsmath} package} \author{Frank Mittelbach\and Rainer Sch\"opf\and Michael Downes\and @@ -53,6 +49,7 @@ %\fi % % \maketitle +% \MaintainedByLaTeXTeam{amsmath} % % \MakeShortVerb\| % @@ -77,7 +74,7 @@ % \begin{macrocode} \NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX) [1994/12/01]% LaTeX date must be December 1994 or later -\ProvidesPackage{amsmath}[2013/01/14 v2.14 AMS math features] +\ProvidesPackage{amsmath}[2016/02/20 v2.15 AMS math features] % \end{macrocode} % % \section{Catcode defenses} @@ -384,6 +381,17 @@ For additional information on amsmath, use the \lq ?\rq\space option.% % %\subsection{Math spacing commands} % +% \begin{macro}{\tmspace} +% \begin{macro}{\,} +% \begin{macro}{\thinspace} +% \begin{macro}{\!} +% \begin{macro}{\negthinspace} +% \begin{macro}{\:} +% \begin{macro}{\medspace} +% \begin{macro}{\negmedspace} +% \begin{macro}{\;} +% \begin{macro}{\thickspace} +% \begin{macro}{\negthickspace} % Here we fill in some gaps in the set of spacing commands, and make them % all work equally well in or out of math. % We want all these commands to be robust but declaring them all with @@ -407,6 +415,17 @@ For additional information on amsmath, use the \lq ?\rq\space option.% \let\thickspace\; \newcommand{\negthickspace}{\tmspace-\thickmuskip{.2777em}} % \end{macrocode} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} % % \begin{macro}{\mspace} % And while we're at it, why don't we provide an equivalent of @@ -418,6 +437,10 @@ For additional information on amsmath, use the \lq ?\rq\space option.% % \end{macro} % % \subsection{Vertical bar symbols} +% \begin{macro}{\lvert} +% \begin{macro}{\rvert} +% \begin{macro}{\lVert} +% \begin{macro}{\rVert} % Add left/right specific versions of \cn{vert}, \cn{Vert}. Don't % assume the delimiter codes are the CM defaults. % \begin{macrocode} @@ -454,6 +477,10 @@ For additional information on amsmath, use the \lq ?\rq\space option.% \fi \fi % \end{macrocode} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} % % \subsection{Fractions} % Bury the generalized fraction primitives \cs{over}, \cs{atop}, @@ -468,6 +495,7 @@ For additional information on amsmath, use the \lq ?\rq\space option.% \@saveprimitive\abovewithdelims\@@abovewithdelims % \end{macrocode} % +% \begin{macro}{\primfrac} % If someone insists on using \cs{over}, give a warning the first % time and then resurrect the old definition. Laissez-faire policy. % \begin{macrocode} @@ -480,6 +508,9 @@ Foreign command \@backslashchar#1;\MessageBreak \global\@xp\let\csname#1\@xp\endcsname\csname @@#1\endcsname \csname#1\endcsname } +% \end{macrocode} +% \end{macro} +% \begin{macrocode} \renewcommand{\over}{\primfrac{over}} \renewcommand{\atop}{\primfrac{atop}} \renewcommand{\above}{\primfrac{above}} @@ -560,11 +591,18 @@ Foreign command \@backslashchar#1;\MessageBreak % make sure \cn{coprod} and the others are simple mathchars; if not, % the attempted changes will probably fail miserably. % +% \changes{v2.15}{2016/02/20}{Accept \cs{Umathchar}} % \begin{macrocode} \begingroup +% \end{macrocode} +% \begin{macrocode} \edef\@tempa{\string\mathchar"} +\edef\@tempd{\string\Umathchar"} \def\@tempb#1"#2\@nil{#1"} \edef\@tempc{\expandafter\@tempb\meaning\coprod "\@nil} +\ifx\@tempc\@tempd\let\@tempc\@tempa\fi +% \end{macrocode} +% \begin{macrocode} \ifx\@tempa\@tempc \global\let\coprod@\coprod \gdef\coprod{\DOTSB\coprod@\slimits@} @@ -596,6 +634,7 @@ Foreign command \@backslashchar#1;\MessageBreak % % \subsection{Roots and radicals} % +% \begin{macro}{\root} % This root stuff needs syntax work and implementation work. Surely % something more compact can be done?? [mjd, 1994/09/05] % \begin{macrocode} @@ -626,6 +665,7 @@ Foreign command \@backslashchar#1;\MessageBreak \mathchoice{\r@@t\displaystyle{#2}}{\r@@t\textstyle{#2}} {\r@@t\scriptstyle{#2}}{\r@@t\scriptscriptstyle{#2}}\egroup} % \end{macrocode} +% \end{macro} % % \changes{v2.0}{1999/06/17}{Normalize @@sqrt to sqrtsign} @@ -666,6 +706,7 @@ Foreign command \@backslashchar#1;\MessageBreak }{} % \end{macrocode} % +% \begin{macro}{\overline} % \amstex/ redefines \cn{overline} as shown here, for reasons that % are probably less important in \latex/: Make it read its argument % as a macro argument rather than a ``math field'' (\emph{The @@ -677,7 +718,9 @@ Foreign command \@backslashchar#1;\MessageBreak \@saveprimitive\overline\@@overline \DeclareRobustCommand{\overline}[1]{\@@overline{#1}} % \end{macrocode} +% \end{macro} % +% \begin{macro}{\boxed} % The \cs{boxed} command is specifically intended to put a box around % an equation or piece of an equation. (Not including the equation % number.) This isn't trivial for end-users to do it properly @@ -685,16 +728,24 @@ Foreign command \@backslashchar#1;\MessageBreak % \begin{macrocode} \newcommand{\boxed}[1]{\fbox{\m@th$\displaystyle#1$}} % \end{macrocode} +% \end{macro} % +% \begin{macro}{\implies} +% \begin{macro}{\impliedby} % \begin{macrocode} \newcommand{\implies}{\DOTSB\;\Longrightarrow\;} \newcommand{\impliedby}{\DOTSB\;\Longleftarrow\;} % \end{macrocode} +% \end{macro} +% \end{macro} % +% \begin{macro}{\And} % \begin{macrocode} \def\And{\DOTSB\;\mathchar"3026 \;} % \end{macrocode} +% \end{macro} % +% \begin{macro}{\nobreakdash} % The command \cn{nobreakdash} is designed only for use before a % hyphen or dash (|-|, |--|, or |---|). % Setting the hyphen in a box and then unboxing it means that the @@ -709,7 +760,9 @@ Foreign command \@backslashchar#1;\MessageBreak \FN@\next@ } % \end{macrocode} +% \end{macro} % +% \begin{macro}{\colon} % \cs{colon} is for a colon in math that resembles a text colon: % small space on the left, larger space on the right. The \qc{\:} % character by itself is treated as a \cs{mathrel} i.e. large, equal @@ -718,6 +771,7 @@ Foreign command \@backslashchar#1;\MessageBreak \renewcommand{\colon}{\nobreak\mskip2mu\mathpunct{}\nonscript \mkern-\thinmuskip{:}\mskip6muplus1mu\relax} % \end{macrocode} +% \end{macro} % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \section{Ellipsis dots} @@ -740,6 +794,36 @@ Foreign command \@backslashchar#1;\MessageBreak {\uccode`7=`c \uccode`8=`h \uccode`9=`\" \uppercase{\gdef\mathch@#1#2#3#4#5#6\mathch@{\gtest@false \ifx 7#1\ifx 8#2\ifx 9#5\gtest@true\xdef\meaning@{9#6}\fi\fi\fi}}} +% \end{macrocode} +% \changes{v2.15}{2016/02/20}{Accept \cs{Umathchar}} +% \begin{macrocode} +{\uccode`(=`U \uccode`)=`m + \uppercase{\gdef\Umathch@#1#2#3#4"#5"#6\Umathch@{\gtest@false + \ifx(#2\ifx)#3\gtest@true + \ifcase"#5 \or\or\gdef\thedots@{\dotsb@}\or\gdef\thedots@{\dotsb@}\fi + \fi\fi + }}} +% \end{macrocode} +% \changes{v2.15}{2016/02/20}{Decode the mathcode of character tokens for Unicode TeX} +% For Unicode TeXs, if the next token is a character token look up the (U)mathcode. Do not do this for classic TeX for compatibility reasons. +% \begin{macrocode} +\ifx\Umathcharnumdef\@undefined +\gdef\thecharacter@#1\thecharacter@{} +\else +{\uccode`(=`t \uccode`)=`c + \uppercase{\gdef\thecharacter@#1#2#3#4#5\thecharacter@{% + \ifx(#1\ifx)#4% + \@xp\getmathcode@\meaning@\getmathcode@ + \fi\fi +}}} +\def\getmathcode@#1 #2 #3#4\getmathcode@{% + \Umathcharnumdef\@tempa\Umathcodenum`#3\relax + \edef\meaning@{\meaning\@tempa}% + \@xp\Umathch@\meaning@\Umathch@ +} +\fi +% \end{macrocode} +% \begin{macrocode} \newcount\classnum@ \def\getmathch@#1.#2\getmathch@{\classnum@#1 \divide\classnum@4096 \ifcase\number\classnum@\or\or\gdef\thedots@{\dotsb@}\or @@ -779,6 +863,15 @@ Foreign command \@backslashchar#1;\MessageBreak \gtest@false\DN@##1\not@{}% \ifx 5#1\ifx 6#2\ifx 7#3\ifx 8#4\DN@{\FN@\nextii@}\fi\fi\fi \fi\next@}}} +% \end{macrocode} +% \changes{v2.15}{2016/02/20}{macro added to strip \cs{long} during tests} +% \begin{macrocode} +{\uccode`9=`\l % + \uppercase{\gdef\striplong@#1#2#3\relax{% + \ifx9#2 \@xp\@xp\@xp\zap@to@space\fi}}} +\def\zap@to@space#1 {} +% \end{macrocode} +% \begin{macrocode} \def\keybin@{\gtest@true \ifx\@let@token+\else\ifx\@let@token=\else \ifx\@let@token<\else\ifx\@let@token>\else @@ -790,6 +883,8 @@ Foreign command \@backslashchar#1;\MessageBreak % \begin{macrocode} \@ifundefined{@ldots}{\def\@ldots{\mathellipsis}}{} % \end{macrocode} +% \begin{macro}{\ldots} +% \begin{macro}{\dots} % Reiterate the standard definition of \cs{ldots} to keep it from % being clobbered by the redefinition of \cs{dots}. % \begin{macrocode} @@ -799,6 +894,10 @@ Foreign command \@backslashchar#1;\MessageBreak \DeclareRobustCommand{\dots}{% \ifmmode \@xp\mdots@\else \@xp\textellipsis \fi } +% \end{macrocode} +% \end{macro} +% \end{macro} +% \begin{macrocode} \def\tdots@{\leavevmode\unskip\relaxnext@ \DN@{$\m@th\@ldots\, \ifx\@let@token,\,$\else\ifx\@let@token.\,$\else @@ -807,31 +906,86 @@ Foreign command \@backslashchar#1;\MessageBreak $ \fi\fi\fi\fi\fi\fi}% \ \FN@\next@} \def\mdots@{\FN@\mdots@@} +% \end{macrocode} +% \changes{v2.15}{2016/02/20}{Indent \cs{mdots@@} for readability and add additional tests} +% \begin{macrocode} \def\mdots@@{\gdef\thedots@{\dotso@}% - \ifx\@let@token\boldsymbol \gdef\thedots@\boldsymbol{\boldsymboldots@}% - \else\ifx,\@let@token \gdef\thedots@{\dotsc}% - \else\ifx\not\@let@token \gdef\thedots@{\dotsb@}% - \else\keybin@ - \ifgtest@\gdef\thedots@{\dotsb@}% - \else\xdef\meaning@{\meaning\@let@token..........}% - \xdef\meaning@@{\meaning@}% - \@xp\math@\meaning@\math@ - \ifgtest@ - \@xp\mathch@\meaning@\mathch@ - \ifgtest@\@xp\getmathch@\meaning@\getmathch@\fi - \else\@xp\macro@\meaning@@\macro@ - \ifgtest@ - \@xp\not@\meaning@\not@\ifgtest@\gdef\thedots@{\dotsb@}% - \else\@xp\DOTS@\meaning@\DOTS@ - \ifgtest@ - \ifcase\number\DOTSCASE@\gdef\thedots@{\dotsb@}% - \or\gdef\thedots@{\dotsi}\else\fi - \else\@xp\math@\meaning@\math@ - \ifgtest@\@xp\mathbin@\meaning@\mathbin@ - \ifgtest@\gdef\thedots@{\dotsb@}% - \else\@xp\mathrel@\meaning@\mathrel@ - \ifgtest@\gdef\thedots@{\dotsb@}% - \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi + \ifx\@let@token\boldsymbol + \gdef\thedots@\boldsymbol{\boldsymboldots@}% + \else + \ifx,\@let@token \gdef\thedots@{\dotsc}% + \else + \ifx\not\@let@token + \gdef\thedots@{\dotsb@}% + \else + \keybin@ + \ifgtest@ % if \keybin@ test + \gdef\thedots@{\dotsb@}% + \else + \xdef\meaning@{\meaning\@let@token..........}% +% \end{macrocode} +% In previous versions \verb|\long| macros were not seen +% bad as this file uses \verb|\(re)newcommand| for \verb|\implies| etc. +% \begin{macrocode} + \xdef\meaning@@{\@xp\striplong@\meaning@\relax\meaning@}% +% \end{macrocode} +% \begin{macrocode} + \@xp\math@\meaning@\math@ + \ifgtest@ % if \mathxxx test + \@xp\mathch@\meaning@\mathch@ + \ifgtest@ % if \mathchar + \@xp\getmathch@\meaning@\getmathch@ + \fi % end if \mathchar + \else % \not \mathxxx +% \end{macrocode} +% Test for \verb|\Umathchar| added. +% \begin{macrocode} + \@xp\Umathch@\meaning@"0"\Umathch@ + \ifgtest@ % if \Umathchar + \else % else not \Umathcar +% \end{macrocode} +% \begin{macrocode} + \@xp\macro@\meaning@@\macro@ + \ifgtest@ % if macro test + \@xp\not@\meaning@\not@ + \ifgtest@ % if macro starts \not test + \gdef\thedots@{\dotsb@}% + \else% else not \not + \@xp\DOTS@\meaning@\DOTS@ + \ifgtest@ % \if DOTS + \ifcase\number\DOTSCASE@ %ifcase dots + \gdef\thedots@{\dotsb@}% + \or\gdef\thedots@{\dotsi}\else + \fi % endifcase dots + \else % not macro starts \DOTS + \@xp\math@\meaning@\math@ + \ifgtest@ % \if macro starts \mathxxxx + \@xp\mathbin@\meaning@\mathbin@ + \ifgtest@ % if macro starts \mathbin + \gdef\thedots@{\dotsb@}% + \else % not macro starting \mathbin + \@xp\mathrel@\meaning@\mathrel@ + \ifgtest@ % if macro starts \mathrel + \gdef\thedots@{\dotsb@}% + \fi % endif macro starts \mathrel (no else) + \fi % endif macro starts \mathbin + \fi % endif macro starts with \mathxxx (no else) + \fi % endif macro starts \DOTS else + \fi % end macro starting \not \ifgtest@ test (no else) +% \end{macrocode} +% Additional test for a catcode 12 character. +% \begin{macrocode} + \else + \@xp\thecharacter@\meaning@\thecharacter@ +% \end{macrocode} +% \begin{macrocode} + \fi % end macro \ifgtest@ test (no else) + \fi % end if \Umathchar test + \fi % end \math@ \ifgtest@ + \fi % end \keybin@ \ifgtest@ test (no else) + \fi % end if \not (no else) + \fi % end if comma (no else) + \fi % end if boldsymbol (no else) \thedots@} % \end{macrocode} % @@ -903,6 +1057,11 @@ Foreign command \@backslashchar#1;\MessageBreak \let\next@#1\fi\fi\fi\fi\next@} % \end{macrocode} % +% \begin{macro}{\cdots} +% \begin{macro}{\dotsb} +% \begin{macro}{\dotsm} +% \begin{macro}{\dotso} +% \begin{macro}{\dotsc} % The \cn{cdots} command. % \begin{macrocode} \ams@DeclareRobustCommand{\cdots}{\DN@{\extrap@\@cdots}\FN@\next@} @@ -918,6 +1077,25 @@ Foreign command \@backslashchar#1;\MessageBreak \else \extra@\@ldots \ifgtest@\,\fi \fi\fi}% \FN@\next@} +% \end{macrocode} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \begin{macro}{\longrightarrow} +% \begin{macro}{\Longrightarrow} +% \begin{macro}{\longleftarrow} +% \begin{macro}{\Longleftarrow} +% \begin{macro}{\longleftrightarrow} +% \begin{macro}{\Longleftrightarrow} +% \begin{macro}{\mapsto} +% \begin{macro}{\longmapsto} +% \begin{macro}{\hookrightarrow} +% \begin{macro}{\hookleftarrow} +% \begin{macro}{\iff} +% Various arrows. +% \begin{macrocode} \renewcommand{\longrightarrow}{% \DOTSB\protect\relbar\protect\joinrel\rightarrow} \renewcommand{\Longrightarrow}{% @@ -934,6 +1112,18 @@ Foreign command \@backslashchar#1;\MessageBreak \renewcommand{\hookleftarrow}{\DOTSB\leftarrow\joinrel\rhook} \renewcommand{\iff}{\DOTSB\;\Longleftrightarrow\;} % \end{macrocode} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \begin{macro}{\doteq} % The \cn{doteq} command formerly used \cs{buildrel}; we avoid that % because it requires `\cn{over}' as part of its syntax. Use 0pt % instead of \cs{z@} for robustitude. @@ -941,10 +1131,12 @@ Foreign command \@backslashchar#1;\MessageBreak \renewcommand{\doteq}{% \DOTSB\mathrel{\mathop{\kern0pt =}\limits^{\textstyle.}}} % \end{macrocode} +% \end{macro} % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \section{Integral signs} % +% \begin{macro}{\if@display} % The straightforward \cs{ifinner} test to see if the current math % context is non-display, fails if, for instance, we are typesetting % a multiline display within an \cs{halign}, with the pieces going @@ -959,7 +1151,14 @@ Foreign command \@backslashchar#1;\MessageBreak \newif\if@display \everydisplay\@xp{\the\everydisplay \@displaytrue} % \end{macrocode} +% \end{macro} % +% \begin{macro}{\int} +% \begin{macro}{\oint} +% \begin{macro}{\iint} +% \begin{macro}{\iiint} +% \begin{macro}{\iiiint} +% \begin{macro}{\idotsint} % Default value for integral limits is \cs{nolimits}, see the % definition of the `nointlimits' option. % \begin{macrocode} @@ -976,7 +1175,14 @@ Foreign command \@backslashchar#1;\MessageBreak \ams@newcommand{\iiiint}{\DOTSI\protect\MultiIntegral{4}} \newcommand{\idotsint}{\DOTSI\protect\MultiIntegral{0}} % \end{macrocode} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} % +% \begin{macro}{\MultiIntegral} % If the \cs{limits} option is applied, use \cs{mathop} and fudge % the left-hand space a bit to make the subscript visually centered. % @@ -993,6 +1199,7 @@ Foreign command \@backslashchar#1;\MessageBreak \futurelet\@let@token\ints@a } % \end{macrocode} +% \end{macro} % % \begin{macrocode} \def\ints@a{% @@ -1058,6 +1265,19 @@ Foreign command \@backslashchar#1;\MessageBreak % hex digit 4 indicating `open delimiter'; this allows us to make a % relatively simple function to get the correct font and character % position. +% +% \changes{v2.15}{2016/02/20}{Modify \cs{resetMathstrut@} for Unicode \TeX} +% Original code assuming |\mathcode| is kept for 8bit \TeX. Unicode \TeX{} +% uses |\Umathcharnumdef| which works for xetex and luatex, which use +% different forms for |\mathchardef|. (New luatex always reports +% definitions using |\Umathchardef| syntax even if |\mathchardef| used.) +% +% The unicode vesrion uses e-tex |\fontcharht| to avoid boxing which +% could also be done for pdftex, but not done here. +% \begin{macrocode} +\ifx\Umathcharnumdef\@undefined +% \end{macrocode} +% Original code % \begin{macrocode} \def\resetMathstrut@{% \setbox\z@\hbox{% @@ -1065,11 +1285,27 @@ Foreign command \@backslashchar#1;\MessageBreak \def\@tempb##1"##2##3{\the\textfont"##3\char"}% \expandafter\@tempb\meaning\@tempa \relax }% + \ht\Mathstrutbox@\ht\z@ \dp\Mathstrutbox@\dp\z@ +} +\else % \end{macrocode} -% These height and depth assignments are implicitly global. +% xetex/luatex version % \begin{macrocode} - \ht\Mathstrutbox@\ht\z@ \dp\Mathstrutbox@\dp\z@ +\def\resetMathstrut@{% + \begingroup + \Umathcharnumdef\@tempa\Umathcodenum`\(\relax + \def\@tempb##1"##2"##3"##4\relax{% + \endgroup + \ht\Mathstrutbox@=\fontcharht\textfont"##3 "##4\relax + \dp\Mathstrutbox@=\fontchardp\textfont"##3 "##4\relax} + \expandafter\@tempb\meaning\@tempa \relax } +% \end{macrocode} +% \begin{macrocode} +\fi +% \end{macrocode} +% These height and depth assignments are implicitly global. +% \begin{macrocode} \addto@hook\every@math@size{\resetMathstrut@} % \end{macrocode} % \end{macro} @@ -1186,6 +1422,8 @@ Foreign command \@backslashchar#1;\MessageBreak \DeclareFontEncoding{OMS}{}{\noaccents@} % \end{macrocode} % +% \begin{macro}{\dddot} +% \begin{macro}{\ddddot} % Triple and quadruple dot accents. % \begin{macrocode} \ams@newcommand{\dddot}[1]{% @@ -1195,6 +1433,8 @@ Foreign command \@backslashchar#1;\MessageBreak {\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@ \hbox{\normalfont....}\vss}}}} % \end{macrocode} +% \end{macro} +% \end{macro} % % The following code deals with support for compound accents. % By redefining \cs{set@mathaccent} we ensure that @@ -1207,6 +1447,17 @@ Foreign command \@backslashchar#1;\MessageBreak } % \end{macrocode} % +% \begin{macro}{\hat} +% \begin{macro}{\check} +% \begin{macro}{\tilde} +% \begin{macro}{\acute} +% \begin{macro}{\grave} +% \begin{macro}{\dot} +% \begin{macro}{\ddot} +% \begin{macro}{\breve} +% \begin{macro}{\bar} +% \begin{macro}{\vec} +% \begin{macro}{\mathring} % We redefine the standard math accent commands to % call \cs{mathaccentV}, using the mathgroup/encoding-number % information embedded in their previous definitions. If the @@ -1222,18 +1473,41 @@ Foreign command \@backslashchar#1;\MessageBreak % Lucida Math or other math fonts package is loaded in addition to % \pkg{amsmath}, there are greater difficulties, but those are % addressed elsewhere. +% +% +% \changes{v2.15}{2016/02/20}{Detect \cs{Umathaccent} on package load} +% Adjust the test made at package load to recognise +% |\Umathaccent|. Although currently it is just used to give a +% modified warning that the accents will not be redefined. +% +% Note that the engines behave quite differently here, luatex +% even without these definitions using the OpenType accent set up by +% unicode-math stacks |\hat{hat{f}}| correctly but xetex acts like +% classic tex and needs this adjustment. This difference is not +% addressed here at all. +% +% This test is just at package loading and has no affect on the +% definitions used in 8bit TeX. % \begin{macrocode} \def\@tempa#1{\@xp\@tempb\meaning#1\@nil#1} \def\@tempb#1>#2#3 #4\@nil#5{% \@xp\ifx\csname#3\endcsname\mathaccent \@tempc#4?"7777\@nil#5% \else + \@xp\ifx\csname#3\endcsname\Umathaccent + \@tempd#4\@nil#5% + \else \PackageWarningNoLine{amsmath}{% Unable to redefine math accent \string#5}% - \fi -} + \fi\fi} \def\@tempc#1"#2#3#4#5#6\@nil#7{% \chardef\@tempd="#3\relax\set@mathaccent\@tempd{#7}{#2}{#4#5}} +\def\@tempd#1\@nil#2{% + \PackageWarningNoLine{amsmath}{% + Unable to redefine \string\Umathaccent\space\string#2}% +} +% \end{macrocode} +% \begin{macrocode} \@tempa{\hat} \@tempa{\check} \@tempa{\tilde} @@ -1252,6 +1526,17 @@ Foreign command \@backslashchar#1;\MessageBreak %%\@tempa\widetilde %%\@tempa\widehat % \end{macrocode} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} % % Regression testing of amsmath 2.0 showed that in some documents % there occurred fragments of the form @@ -1535,6 +1820,10 @@ Foreign command \@backslashchar#1;\MessageBreak % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \section{Mods, continued fractions, etc.} % +% \begin{macro}{\bmod} +% \begin{macro}{\pmod} +% \begin{macro}{\pod} +% \begin{macro}{\mod} % The commands \cn{bmod}, \cn{pmod}, \cn{pod}, \cn{mod} aren't % currently robust. [mjd, 1994/09/05] % \begin{macrocode} @@ -1547,7 +1836,12 @@ Foreign command \@backslashchar#1;\MessageBreak \newcommand{\mod}[1]{\allowbreak\if@display\mkern18mu \else\mkern12mu\fi{\operator@font mod}\,\,#1} % \end{macrocode} +% \end{macro} +% \end{macro} +% \end{macro} +% \end{macro} % +% \begin{macro}{\cfrac} % Continued fractions. The optional arg l or r controls horizontal % placement of the numerators. The |\kern-\nulldelimiterspace| % is needed in the definition if we want the right-hand sides of the @@ -1559,7 +1853,10 @@ Foreign command \@backslashchar#1;\MessageBreak \strut\ifx r#1\hfill\fi#2\ifx l#1\hfill\fi}{#3}}% \kern-\nulldelimiterspace} % \end{macrocode} +% \end{macro} % +% \begin{macro}{\overset} +% \begin{macro}{\underset} % \cn{overset} and \cn{underset} put symbols above, respectively % below, a symbol that is not a \cs{mathop} and therefore does not % naturally accept limits. \cs{binrel@@} uses information collected @@ -1574,7 +1871,10 @@ Foreign command \@backslashchar#1;\MessageBreak \newcommand{\underset}[2]{\binrel@{#2}% \binrel@@{\mathop{\kern\z@#2}\limits_{#1}}} % \end{macrocode} +% \end{macro} +% \end{macro} % +% \begin{macro}{\sideset} % \cn{sideset} allows placing `adscript' symbols at the four % corners of a \cs{mathop}, \emph{in addition to} limits. Left-side % adscripts go into arg \arg{1}, in the form |_{...}^{...}|, and @@ -1603,6 +1903,7 @@ Foreign command \@backslashchar#1;\MessageBreak \hbox to\dimen@{}\mathop{\kern-\dimen@\box4\box6}% } % \end{macrocode} +% \end{macro} % % \begin{macro}{\smash} % We add to the \cn{smash} command an optional argument @@ -1637,10 +1938,14 @@ Foreign command \@backslashchar#1;\MessageBreak } % \end{macrocode} % +% \begin{macro}{\relbar} +% \begin{macro}{\Relbar} % \begin{macrocode} \ams@def\relbar{\mathrel{\mathpalette\mathsm@sh\std@minus}} \ams@def\Relbar{\mathrel\std@equal} % \end{macrocode} +% \end{macro} +% \end{macro} % % \begin{macrocode} \def\arrowfill@#1#2#3#4{% @@ -1722,7 +2027,7 @@ Foreign command \@backslashchar#1;\MessageBreak % structures that allow \cn{tag}, there is some cross-influence in % the internal workings of the \cn{\\} command. % -% \subsection{The \cn{substack} command} +% \subsection{The \env{subarray} environment and \cn{substack} command} % % The \cn{substack} command can be used to set subscripts % and superscripts that consist of several lines. Usage: @@ -1960,6 +2265,7 @@ ill-advised in LaTeX.% }{} % \end{macrocode} % +% \begin{environment}{subequations} % \begin{macrocode} \newenvironment{subequations}{% % \end{macrocode} @@ -1989,6 +2295,7 @@ ill-advised in LaTeX.% \ignorespacesafterend } % \end{macrocode} +% \end{environment} % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \section{Equation numbering} @@ -1999,6 +2306,7 @@ ill-advised in LaTeX.% % equation number for the current line will be processed and added to % the page. % +% \begin{macro}{\numberwithin} % Provide a convenient way to specify that equations should be % numbered within sections. % \begin{macrocode} @@ -2010,13 +2318,16 @@ ill-advised in LaTeX.% \@xp\@nx\csname the#3\endcsname .\@nx#1{#2}}}}% } % \end{macrocode} +% \end{macro} % +% \begin{macro}{\eqref} % To make references to equation numbers easier, we provide % \cn{eqref}. We almost don't need \cn{textup}, except that % \cs{tagform@} doesn't supply the italic correction. % \begin{macrocode} \newcommand{\eqref}[1]{\textup{\tagform@{\ref{#1}}}} % \end{macrocode} +% \end{macro} % % \subsection{Preliminary macros} % @@ -3151,8 +3462,8 @@ and fix things up.} % \end{macrocode} % \end{macro} % -% \begin{macro}{\math@cr@@@aligned} -% \cs{math@cr@@@aligned} checks to make sure the user hasn't put in +% \begin{macro}{\math@cr@@@alignedat} +% \cs{math@cr@@@alignedat} checks to make sure the user hasn't put in % too many \qc{\&}s in \env{alignedat}. Since \env{alignedat} % doesn't use \cs{displ@y@}, we also reset \cs{column@} here. Note % than in \env{aligned}, \cs{column@} will increase without bound, @@ -5831,4 +6142,4 @@ Cannot use `split' here;\MessageBreak trying to recover with `aligned'}% % changed in 1994 to make it user-oriented rather than % history-oriented. % -% \Finale +\endinput diff --git a/Master/texmf-dist/source/latex/amsmath/amsmath.ins b/Master/texmf-dist/source/latex/amsmath/amsmath.ins index 3ce6f639582..841088d84fc 100644 --- a/Master/texmf-dist/source/latex/amsmath/amsmath.ins +++ b/Master/texmf-dist/source/latex/amsmath/amsmath.ins @@ -1,19 +1,6 @@ -%% filename: amsmath.ins -%% version: 1.01 -%% date: 2013/01/14 -%% -%% American Mathematical Society -%% Technical Support -%% Publications Technical Group -%% 201 Charles Street -%% Providence, RI 02904 -%% USA -%% tel: (401) 455-4080 -%% (800) 321-4267 (USA and Canada only) -%% fax: (401) 331-3842 -%% email: tech-support@ams.org %% -%% Copyright 2010, 2013 American Mathematical Society. +%% Copyright 2010 American Mathematical Society. +%% Copyright 2016 LaTeX3 Project and American Mathematical Society. %% %% This work may be distributed and/or modified under the %% conditions of the LaTeX Project Public License, either version 1.3c @@ -25,8 +12,7 @@ %% %% This work has the LPPL maintenance status `maintained'. %% -%% The Current Maintainer of this work is the American Mathematical -%% Society. +%% The Current Maintainer of this work is the LaTeX3 Project. %% %% ==================================================================== @@ -38,18 +24,8 @@ \preamble -Copyright 1995, 2000, 2013 American Mathematical Society. - -American Mathematical Society -Technical Support -Publications Technical Group -201 Charles Street -Providence, RI 02904 -USA -tel: (401) 455-4080 - (800) 321-4267 (USA and Canada only) -fax: (401) 331-3842 -email: tech-support@ams.org +Copyright 1995, 1999, 2000, 2013 American Mathematical Society. +Copyright 2016 LaTeX3 Project and American Mathematical Society. This work may be distributed and/or modified under the conditions of the LaTeX Project Public License, either version 1.3c @@ -61,8 +37,7 @@ version 2005/12/01 or later. This work has the LPPL maintenance status `maintained'. -The Current Maintainer of this work is the American Mathematical -Society. +The Current Maintainer of this work is the LaTeX3 project. \endpreamble diff --git a/Master/texmf-dist/source/latex/amsmath/amsopn.dtx b/Master/texmf-dist/source/latex/amsmath/amsopn.dtx index 44ef923a39b..39c92f01bc8 100644 --- a/Master/texmf-dist/source/latex/amsmath/amsopn.dtx +++ b/Master/texmf-dist/source/latex/amsmath/amsopn.dtx @@ -1,21 +1,8 @@ -% \def\filename{amsopn.dtx} -% \def\fileversion{2.01} -% \def\filedate{1999/12/14} % % \iffalse meta-comment % -% American Mathematical Society -% Technical Support -% Publications Technical Group -% 201 Charles Street -% Providence, RI 02904 -% USA -% tel: (401) 455-4080 -% (800) 321-4267 (USA and Canada only) -% fax: (401) 331-3842 -% email: tech-support@ams.org -% % Copyright 1995, 1999 American Mathematical Society. +% Copyright 2016 LaTeX3 Project and American Mathematical Society. % % This work may be distributed and/or modified under the % conditions of the LaTeX Project Public License, either version 1.3c @@ -27,14 +14,23 @@ % % This work has the LPPL maintenance status `maintained'. % -% The Current Maintainer of this work is the American Mathematical -% Society. +% The Current Maintainer of this work is the LaTeX3 Project. % % \fi % %\iffalse %<*driver> \documentclass{amsdtx} +\def\MaintainedByLaTeXTeam#1{% +\begin{center}% +\fbox{\fbox{\begin{tabular}{@{}l@{}}% +This file is maintained by the \LaTeX{} Project team.\\% +Bug reports can be opened (category \texttt{#1}) at\\% +\url{http://latex-project.org/bugs.html}.\end{tabular}}}\end{center}} +\let\savedarg\arg +\usepackage{amsopn} +\let\arg\savedarg +\GetFileInfo{amsopn.sty} \begin{document} \title{The \pkg{amsopn} package} \author{Michael Downes} @@ -45,6 +41,7 @@ %\fi % % \maketitle +% \MaintainedByLaTeXTeam{amsmath} % % \MakeShortVerb\| % diff --git a/Master/texmf-dist/source/latex/amsmath/amsopn.ins b/Master/texmf-dist/source/latex/amsmath/amsopn.ins index e2c25bdccb4..f8b1d713fb1 100644 --- a/Master/texmf-dist/source/latex/amsmath/amsopn.ins +++ b/Master/texmf-dist/source/latex/amsmath/amsopn.ins @@ -1,19 +1,6 @@ -%% filename: amsopn.ins -%% version: 1.00 -%% date: 2010/08/13 -%% -%% American Mathematical Society -%% Technical Support -%% Publications Technical Group -%% 201 Charles Street -%% Providence, RI 02904 -%% USA -%% tel: (401) 455-4080 -%% (800) 321-4267 (USA and Canada only) -%% fax: (401) 331-3842 -%% email: tech-support@ams.org %% %% Copyright 2010 American Mathematical Society. +%% Copyright 2016 LaTeX3 Project and American Mathematical Society. %% %% This work may be distributed and/or modified under the %% conditions of the LaTeX Project Public License, either version 1.3c @@ -25,8 +12,7 @@ %% %% This work has the LPPL maintenance status `maintained'. %% -%% The Current Maintainer of this work is the American Mathematical -%% Society. +%% The Current Maintainer of this work is the LaTeX3 Project. %% %% ==================================================================== @@ -39,17 +25,7 @@ \preamble Copyright 1995, 1999 American Mathematical Society. - -American Mathematical Society -Technical Support -Publications Technical Group -201 Charles Street -Providence, RI 02904 -USA -tel: (401) 455-4080 - (800) 321-4267 (USA and Canada only) -fax: (401) 331-3842 -email: tech-support@ams.org +Copyright 2016 LaTeX3 Project and American Mathematical Society. This work may be distributed and/or modified under the conditions of the LaTeX Project Public License, either version 1.3c @@ -61,8 +37,7 @@ version 2005/12/01 or later. This work has the LPPL maintenance status `maintained'. -The Current Maintainer of this work is the American Mathematical -Society. +The Current Maintainer of this work is the LaTeX3 project. \endpreamble diff --git a/Master/texmf-dist/source/latex/amsmath/amstext.dtx b/Master/texmf-dist/source/latex/amsmath/amstext.dtx index 4abd0b84513..dae31efd464 100644 --- a/Master/texmf-dist/source/latex/amsmath/amstext.dtx +++ b/Master/texmf-dist/source/latex/amsmath/amstext.dtx @@ -1,21 +1,8 @@ -% \def\filename{amstext.dtx} -% \def\fileversion{2.01} -% \def\filedate{2000/06/29} % % \iffalse meta-comment % -% American Mathematical Society -% Technical Support -% Publications Technical Group -% 201 Charles Street -% Providence, RI 02904 -% USA -% tel: (401) 455-4080 -% (800) 321-4267 (USA and Canada only) -% fax: (401) 331-3842 -% email: tech-support@ams.org -% % Copyright 1995, 1999 American Mathematical Society. +% Copyright 2016 LaTeX3 Project and American Mathematical Society. % % This work may be distributed and/or modified under the % conditions of the LaTeX Project Public License, either version 1.3c @@ -27,15 +14,21 @@ % % This work has the LPPL maintenance status `maintained'. % -% The Current Maintainer of this work is the American Mathematical -% Society. +% The Current Maintainer of this work is the LaTeX3 Project. % % \fi % % \iffalse %<*driver> \documentclass{amsdtx} +\def\MaintainedByLaTeXTeam#1{% +\begin{center}% +\fbox{\fbox{\begin{tabular}{@{}l@{}}% +This file is maintained by the \LaTeX{} Project team.\\% +Bug reports can be opened (category \texttt{#1}) at\\% +\url{http://latex-project.org/bugs.html}.\end{tabular}}}\end{center}} \usepackage{amstext} +\GetFileInfo{amstext.sty} \begin{document} \title{The \pkg{amstext} package} \author{Frank Mittelbach \and Rainer Sch\"opf} @@ -46,6 +39,7 @@ % \fi % % \maketitle +% \MaintainedByLaTeXTeam{amsmath} % % \MakeShortVerb\| % @@ -83,7 +77,7 @@ % \begin{macrocode} \NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX) [1994/12/01]% LaTeX date must be December 1994 or later -\ProvidesPackage{amstext}[2000/06/29 v2.01] +\ProvidesPackage{amstext}[2000/06/29 v2.01 AMS text] % \end{macrocode} % % \section{The implementation} diff --git a/Master/texmf-dist/source/latex/amsmath/amstext.ins b/Master/texmf-dist/source/latex/amsmath/amstext.ins index 39334a35514..d97c166c36d 100644 --- a/Master/texmf-dist/source/latex/amsmath/amstext.ins +++ b/Master/texmf-dist/source/latex/amsmath/amstext.ins @@ -1,19 +1,6 @@ -%% filename: amstext.ins -%% version: 1.00 -%% date: 2010/08/13 -%% -%% American Mathematical Society -%% Technical Support -%% Publications Technical Group -%% 201 Charles Street -%% Providence, RI 02904 -%% USA -%% tel: (401) 455-4080 -%% (800) 321-4267 (USA and Canada only) -%% fax: (401) 331-3842 -%% email: tech-support@ams.org %% %% Copyright 2010 American Mathematical Society. +%% Copyright 2016 LaTeX3 Project and American Mathematical Society. %% %% This work may be distributed and/or modified under the %% conditions of the LaTeX Project Public License, either version 1.3c @@ -25,8 +12,7 @@ %% %% This work has the LPPL maintenance status `maintained'. %% -%% The Current Maintainer of this work is the American Mathematical -%% Society. +%% The Current Maintainer of this work is the LaTeX3 Project. %% %% ==================================================================== @@ -39,17 +25,7 @@ \preamble Copyright 1995, 1999 American Mathematical Society. - -American Mathematical Society -Technical Support -Publications Technical Group -201 Charles Street -Providence, RI 02904 -USA -tel: (401) 455-4080 - (800) 321-4267 (USA and Canada only) -fax: (401) 331-3842 -email: tech-support@ams.org +Copyright 2016 LaTeX3 Project and American Mathematical Society. This work may be distributed and/or modified under the conditions of the LaTeX Project Public License, either version 1.3c @@ -61,8 +37,7 @@ version 2005/12/01 or later. This work has the LPPL maintenance status `maintained'. -The Current Maintainer of this work is the American Mathematical -Society. +The Current Maintainer of this work is the LaTeX3 project. \endpreamble diff --git a/Master/texmf-dist/source/latex/amsmath/amsxtra.dtx b/Master/texmf-dist/source/latex/amsmath/amsxtra.dtx index 712d545dd8b..c9abb4585f3 100644 --- a/Master/texmf-dist/source/latex/amsmath/amsxtra.dtx +++ b/Master/texmf-dist/source/latex/amsmath/amsxtra.dtx @@ -1,21 +1,8 @@ -% \def\filename{amsxtra.dtx} -% \def\fileversion{1.2c} -% \def\filedate{1999/11/15} % % \iffalse meta-comment % -% American Mathematical Society -% Technical Support -% Publications Technical Group -% 201 Charles Street -% Providence, RI 02904 -% USA -% tel: (401) 455-4080 -% (800) 321-4267 (USA and Canada only) -% fax: (401) 331-3842 -% email: tech-support@ams.org -% % Copyright 1995, 1999 American Mathematical Society. +% Copyright 2016 LaTeX3 Project and American Mathematical Society. % % This work may be distributed and/or modified under the % conditions of the LaTeX Project Public License, either version 1.3c @@ -27,14 +14,21 @@ % % This work has the LPPL maintenance status `maintained'. % -% The Current Maintainer of this work is the American Mathematical -% Society. +% The Current Maintainer of this work is the LaTeX3 Project. % % \fi % %\iffalse %<*driver> \documentclass{amsdtx} +\def\MaintainedByLaTeXTeam#1{% +\begin{center}% +\fbox{\fbox{\begin{tabular}{@{}l@{}}% +This file is maintained by the \LaTeX{} Project team.\\% +Bug reports can be opened (category \texttt{#1}) at\\% +\url{http://latex-project.org/bugs.html}.\end{tabular}}}\end{center}} +\usepackage{amsxtra} +\GetFileInfo{amsxtra.sty} \begin{document} \title{The \pkg{amsxtra} package} \author{American Mathematical Society\\Michael Downes} @@ -45,6 +39,7 @@ %\fi % % \maketitle +% \MaintainedByLaTeXTeam{amsmath} % % \MakeShortVerb\| % @@ -61,7 +56,7 @@ % \StopEventually{} % % \begin{macrocode} -\ProvidesPackage{amsxtra}[1999/11/15 v1.2c] +\ProvidesPackage{amsxtra}[1999/11/15 v1.2c AMS extra commands] % \end{macrocode} % We need to ensure that either the \pkg{amstex} package or the % \pkg{amsmath} package is loaded. This means we need something a diff --git a/Master/texmf-dist/source/latex/amsmath/amsxtra.ins b/Master/texmf-dist/source/latex/amsmath/amsxtra.ins index 9edebd567dc..84e301b7b48 100644 --- a/Master/texmf-dist/source/latex/amsmath/amsxtra.ins +++ b/Master/texmf-dist/source/latex/amsmath/amsxtra.ins @@ -1,19 +1,6 @@ -%% filename: amsxtra.ins -%% version: 1.00 -%% date: 2010/08/13 -%% -%% American Mathematical Society -%% Technical Support -%% Publications Technical Group -%% 201 Charles Street -%% Providence, RI 02904 -%% USA -%% tel: (401) 455-4080 -%% (800) 321-4267 (USA and Canada only) -%% fax: (401) 331-3842 -%% email: tech-support@ams.org %% %% Copyright 2010 American Mathematical Society. +%% Copyright 2016 LaTeX3 Project and American Mathematical Society. %% %% This work may be distributed and/or modified under the %% conditions of the LaTeX Project Public License, either version 1.3c @@ -25,8 +12,7 @@ %% %% This work has the LPPL maintenance status `maintained'. %% -%% The Current Maintainer of this work is the American Mathematical -%% Society. +%% The Current Maintainer of this work is the LaTeX3 Project. %% %% ==================================================================== @@ -39,17 +25,7 @@ \preamble Copyright 1995, 1999 American Mathematical Society. - -American Mathematical Society -Technical Support -Publications Technical Group -201 Charles Street -Providence, RI 02904 -USA -tel: (401) 455-4080 - (800) 321-4267 (USA and Canada only) -fax: (401) 331-3842 -email: tech-support@ams.org +Copyright 2016 LaTeX3 Project and American Mathematical Society. This work may be distributed and/or modified under the conditions of the LaTeX Project Public License, either version 1.3c @@ -61,8 +37,7 @@ version 2005/12/01 or later. This work has the LPPL maintenance status `maintained'. -The Current Maintainer of this work is the American Mathematical -Society. +The Current Maintainer of this work is the LaTeX3 project. \endpreamble diff --git a/Master/texmf-dist/source/latex/amsmath/diffs-m.txt b/Master/texmf-dist/source/latex/amsmath/diffs-m.txt deleted file mode 100644 index 0bfa0ff69d4..00000000000 --- a/Master/texmf-dist/source/latex/amsmath/diffs-m.txt +++ /dev/null @@ -1,552 +0,0 @@ -diffs-m.txt [1999/12/15] American Mathematical Society - -This is a synopsis of the principal differences between various versions -of the amsmath package and its companion packages. - -A synopsis of changes in the "AMS-LaTeX" distribution is included at the -end of this file. See also diffs-c.txt in the amslatex/classes area. - -Those upgrading from version 1.2 to version 2.0 will probably be -interested in the question: - - After I upgrade will anything fail in my pre-existing documents? - -Certain of the bug fixes have the potential to affect the output of -existing documents in the following ways: - ----Vertical spacing for some displayed equations will change because -"abovedisplayshortskip" is now applied when possible. - ----Incorrect horizontal spacing for certain math constructs will change, -e.g., for \iint\nolimits or a subscript on a double-accented letter. - ----The default subscript placement in displayed formulas for \varinjlim, -\varprojlim, \varliminf, \varlimsup is now below rather than to the -side. To get the old behavior, add \nolimits---e.g., -\varliminf\nolimits_{...}. - ----Ellipsis dots in text (\ldots or \dots) will now behave the same as -in standard LaTeX; the surrounding space may therefore change. - -During regression testing we found some errors cropping up in documents -from incorrect usage that had not been caught with version 1.2. Although -these errors are relatively rare, documents containing such usage will -need to be corrected before they will run properly with version 2.0: - ----Use of two mutually exclusive display structures for the same -display. In this example the outer \[ \] are redundant: - - \[ - \begin{align*} - ... - \end{align*} - \] - ----Inadequate argument markup for math accents: - - \hat \mathcal{C} - -This should have been written with braces surrounding the argument of -the \hat command: \hat{\mathcal{C}}. - -For documents where error messages crop up after the upgrade, there are -various possible approaches, including: - -(a) Change instances of incorrect usage (if applicable). - -(b) Put a copy of amsmath.sty version 1.2 into the home directory of the -document. Then it should continue to work the same as before. - -(c) Use the snapshot package in combination with a shell script to -arrange that version 1.2 of amsmath is found instead of 2.0 when -processing the document. See ftp://ftp.ams.org/pub/tex/snapshot.*. This -approach requires a certain amount of technical knowledge. - -(d) Downgrade to version 1.2 again. - -======================================================================== -Changes in amsmath 2.0 [November 1999] - ----The AMS-LaTeX User's Guide (amsldoc.tex) has been turned into the - AMSmath User's Guide by moving the amsthm documentation into a - separate file (amsthdoc.tex) in the amslatex/classes area. - ----Restored normal use of "shortskip" spacing in single equations. - -[!! NOTE: This means that page breaks may change in existing documents.] - ----Restored normal use of available shrink around binary operators in - single equations. - ----Fixed poor subscript placement for double-accented letters. Math -accent handling was reworked in the process in such a way that the -capitalized forms of the math accent commands are no longer necessary. -\hat{\hat{X}} will do proper alignment of the second accent. \Hat et -cetera are retained for compabitility reasons. (Thanks to Javier Bezos -for donating some code from his accents package.) - -[!! NOTE: Improper math accent use will now produce wrong output.] - -Suppose that we have \boldH defined in the preamble as -\newcommand{\boldH}{\mathbf{H}}. Then the following examples illustrate -recommended usage. - - \hat{\mathbf{H}} OK - \hat{\boldH} OK - \hat\boldH Works OK, but deprecated by the LaTeX book - \hat\mathbf{H} Not OK! Output will be incorrect. - -Since the output will be wrong, amsmath gives an error message when the -last variant is detected. - ----The following kinds of mixed structures worked by happenstance with - amsmath 1.2, although they should have produced an error message. - With amsmath 2.0 they will produce an error or warning message. - - [!! NOTE: Potential error messages] - - % Outer \[ \] is redundant - - \[ - \begin{align*} - ... - \end{align*} - \] - - % Outer \begin{equation*} ... \end{equation*} is redundant - - \begin{equation*}\begin{alignat*}{2} - ... - \end{alignat*} - \end{equation*} - - % No printed material is allowed between \begin{equation} and - % \begin{split}: - - \begin{equation*}\left \{ - \begin{split} - ... - \end{split}\right . - \end{equation*} - ----diff12.tex is replaced by diffs-m.txt and diffs-c.txt, with the - latter moving to the amslatex/classes area. - ----Added an "?" option which gives some information about the available - documentation. - ----The amsmath definitions for matrix, pmatrix, and cases are - incompatible with the plain-TeX syntax for \pmatrix, \matrix, and - \cases, which are used occasionally by some authors in LaTeX - documents. In version 2.0 the error messages resulting from the - plain-TeX syntax have been made more intelligible. - ----If you need the boldsymbol facilities of the amsbsy package, use of - the "bm" (boldmath) package is recommended instead for new documents. - The amsbsy package is retained for compatibility with existing - documents. If you have a recent version of LaTeX (1998 or newer) you - already have the bm package; otherwise you may need to get it from - CTAN. (It is not an AMS package.) - ----\iint\nolimits: Fixed poor placement of subscript (also \iiint, -\idotsint, etc). - ----\varinjlim, \varprojlim, \varliminf, \varlimsup: Default to -\displaylimits instead of nolimits. - ----Fixed alignedat so that it doesn't take the first token of its - contents and try to interpret it as a number. - ----Leave text version of \dots and \ldots unchanged. - ----Changed \numberwithin so it no longer assumes previous definition of - \thesomething contained no prefixes. - ----\mathbf{\Gamma}: Leave cap Greek letters alone instead of converting - them from variable (fam 7) to fixed (fam 0). \mathbf{\gamma} still - fails---but that's a math font problem. - ----\mathindent is aliased to \@mathmargin so that when the fleqn option - is used, users can reset it as the LaTeX book advertises. - ----With the showkeys package: Corrected loss of marginal notes when -labels occurred inside an align environment. - ----Carried over some tweaks (\bmod, \right/leftarrowfill) from the 1996 - version of plain.tex. - ----Adjusted spacing of overarrow@ and underarrow@ to allow more room for - larger cmsy arrowheads. - ----Corrected spurious "overfull \hbox" warnings for \xrightarrow. - ----Corrected poor spacing for \xleftarrow. - ----Removed the \leavevmode in \smash which caused trouble for some - applications (e.g., the crop package) that expected the original - LaTeX definition. - ----Changed \def to \newcommand/\newenvironment where appropriate. - ----Use \sqrtsign instead of \@@sqrt (but include a backward - compatibility hack for the latter). - ----Left brace in \env{cases} environment would fail to work in - certain circumstances: Fix by using \lbrace instead of \{. - ----Use \ignorespacesafterend where applicable instead of - \global\@ignoretrue. - ----Protected commands in the definition of \theequation were - incorrectly expanded by the subequations environment when copying - the definition to \theparentequation: Fix by using \protected@edef. - ----Spurious extra space following punctuation at that end of a display: - In \math@cr, \math@cr@@, etc, use \iffalse{\fi instead of simply { to - keep an unwanted mathord atom from being added at the end of the - current math list; added \endmultline@math to work around a similar - sort of problem specific to the `multline' environment. - ----Various wrong results or no effect for \displaybreak inside "align", - "alignat", etc. Resetting \dspbrk@level in the \everycr of - \@display@init cleared most of these up. - ----Changed \displaybreak to issue an error message if enclosed in boxing - environment that will keep it from having any effect. - ----Turned off \displaybreak during the measuring phase of multline. - ----Fixed some numbering problems with multline and multline* and error - for use of \tag. - ----Use \@gtempa instead of \@tempa where assigned globally (standard - save stack precaution). - ----Save and restore \column@ to keep aligned from screwing up the column - counting of a parent align environment. Made \column@ assignments - uniformly global, except in a couple of places where the risk of side - effects seemed too great. - ----Error recovery for misspelled '\end{multline}' was not very helpful. - Improve \collect@body along lines suggested by Hellström. - ----Fixed up certain number placement problems with align - ----Add \alignsafe@testopt to work around a LaTeX bug: when a nested - alignment environment that takes an optional argument has an empty - first cell (i.e., begins with an &) LaTeX's optional argument - scanning screws up the alignment state. - ----Add `mathdisplay' environment (internal use) to correct several - problems of \[ \] and the `equation' environment arising from the - previous implementation as a degenerate case of `gather'. - ----It is possible now to use \eeq as an abbreviation for \end{equation} -(one of the consequences of the preceding change). But still not \eal -for \end{align} or certain other display structures. - ----Reorganize `split' internal calling sequence to go along with the use - of `mathdisplay' in \[ \] and `equation'. - ----Suppression of LaTeX error messages during measuring phase made - primitive TeX errors harder to understand; stop suppressing. (As a - result, users will sometimes get redundant error messages now.) - Remove obsolete Sb, Sp environments again. - ----The file amsintx.dtx was removed from the distribution pending - further development. - ----Remove the \ifinany@ switch, use \spread@equation instead to achieve - the same ends. - ----Remove unbracefonts@ function, no longer needed after changes in - math accent handling. - ----Got rid of the special trap for '^x that had been carried over from -AMS-TeX. - ----Made \overline, \doteq, \longrightarrow, etc, robust. - ----Changed \relbar to use \std@minus---otherwise \underleftrightarrow -works poorly in a \DeclareMathOperator definition. - ----Use \new@ifnextchar in matrix, pmatrix, cases so that \\<newline>[x] -doesn't trigger an error. - ----Bug fix for nested use of \label inside minipage inside equation. - ----Add minalignsep after every second column in aligned, for consistency -with align. - -======================================================================== -Changes in amsmath 1.2 [January 1995] - ----Name change; relation to the "amsfonts" package - -In version 1.1 of AMS-LaTeX, the name of this package was "amstex". The -new name is "amsmath". The amstex package automatically invoked the -amsfonts package as well; the amsmath package does not. - ----Adapted for LaTeX2e - -Old method of invocation: - - \documentstyle[amstex]{article}% LaTeX 2.09 - -New method of invocation: - - \documentclass{article} - \usepackage{amsmath} - ----Handling of options intlim, nosumlim, nonamelm, righttag, ctagsplt - -Old usage: - - \documentstyle[amstex,righttag,intlim] - -New usage: - - \usepackage[reqno,intlimits]{amsmath} - -Several option names were changed (mainly, they are no longer restricted -to eight characters): - - intlim -> intlimits - nosumlim -> nosumlimits - nonamelm -> nonamelimits - righttag -> reqno [for consistency with "leqno"] - ctagsplt -> centertags - ----Character @ is no longer used as a special command prefix - -Certain commands in the amstex pkg started with "@" instead of "\". This -special usage has been discontinued, except for the amscd package, and -then only within the CD environment. Practically speaking, this means -primarily that it is no longer necessary to use doubled @@ to get a -single printed @ character (e.g., in e-mail addresses). - ----The @, and @! micro-spacing commands are no longer provided - -As a consequence of the @ character change, "@," and "@!" are no longer -recognized spacing commands. An \mspace command is provided that can be -used instead; see the AMSmath User's Guide. - ----Extensible arrow commands @>>> and @<<< - -The commands @>>> and @<<< are no longer available [except in the CD -environment of the amscd package]; alternative commands \xleftarrow and -\xrightarrow are provided by the amsmath package. The sub and -superscripts are given through an optional resp. mandatory argument, for -example - - X\xleftarrow{b}Y % superscript "b" only - X\xrightarrow[\alpha]{\beta}Y % sub alpha, sup beta - ----The command \bold has been superseded by the command \mathbf, as the -latter is now the standard name provided by LaTeX for this purpose. - ----The commands \newsymbol, \frak, \Bbb are no longer available because -the amsfonts package is not loaded by amsmath. In any case those -commands now have new names---\frak -> \mathfrak, \Bbb -> \mathbb, etc. -See the amsfonts documentation for further details. - ----The amsmath package provides \lvert and \rvert to use for vert-bar -symbols when they act as delimiters (compare \langle, \rangle for -angle-bracket delimiters). Commands \lVert and \rVert are likewise -provided instead of \| for paired-delimiter use of the double-vert -symbol. Use of | and \| for paired delimiters can lead to incorrect -spacing in rare cases. Further discussion is found in the AMS-LaTeX -user's guide (amsldoc.tex). - ----The ~ command is now left unchanged by amsmath; the change -that makes it remove a redundant preceding or following space character -is now done only by AMS documentclasses (in version 1.1 this was done by -the amstex package). - ----The following rarely used commands are removed to the amsxtra -package: \accentedsymbol, \sphat, \spcheck, \sptilde, \spdot, \spddot, -\spdddot, \spbreve, \@))), \@(((. - ----The six generalized fraction commands \over, \overwithdelims, \atop, -\atopwithdelims, \above, \abovewithdelims are deprecated and produce -warning messages. Their functionality can be obtained through \frac, -\binom, and \genfrac instead. See amsldoc.tex and technote.tex for -further details. - ----The \fracwithdelims command is gone, as \genfrac provides -equivalent functionality (and a little more). - ----The optional argument of \frac that allowed changing the thickness of -the fraction line is gone. That functionality is now provided only -through \genfrac: the need to change the line thickness is so rare in -practice that it seems better not to burden the ubiquitous \frac command -with the somewhat time-consuming look-ahead process required for an -optional argument. I.e., this change will contribute in a small way to -making your documents run faster. - ----The commands \lcfrac and \rcfrac for left or right alignment of -continued fraction numerators are gone; they are replaced by an optional -argument of the \cfrac command. - ----A new command \DeclareMathOperator is provided for defining new -operator names: - -\DeclareMathOperator{\Tr}{Tr} -\DeclareMathOperator*{\xlim}{x-lim} - -The * form makes the operator name produce limit-style sub and -superscripts, like \max or \lim. This is now the preferred method, -instead of using the old command \operatorname (which is, however, -retained for now). The command \operatornamewithlimits has been renamed -\operatorname*. These commands now can also be obtained through a -separate package amsopn, so it's not necessary to load the entire -amsmath package just to get that feature. - ----Environments "Sb", "Sp" retired. - -The environments Sb and Sp for multiline subscripts have been replaced -by a single command \substack, used as follows: - -\sum_{\substack{first line\\second line}} - -This command can be used equally well in a subscript or superscript, and -provides better vertical positioning (as compared to Sb, Sp) if a -multiline subscript is placed on the side, as perhaps for an integral. -There is also a slightly generalized alternative, an environment -subarray that allows you to specify either centering or left alignment -for the contents: - -\sum_{\begin{subarray}{l}first line\\second line\end{subarray}} - ----New command \nobreakdash - -A command \nobreakdash is provided to suppress the possibility of a -linebreak after the following hyphen or dash. For example, if you write -"$p$\nobreakdash-adic" then a linebreak will never be taken between the -hyphen and the "adic". - ----The placement of equation numbers has been substantially improved for -align, gather, and split. Numbers will now never overlap on top of the -equation body (as far as we can ascertain), and they are much less -likely to be shifted up or down unnecessarily when there is actually -adequate space to leave the number in the normal place. In a few -remaining cases, an equation number will be shifted unnecessarily -because technical complications make accurate measurement of the -available room too difficult; for those cases a \raisetag command is -provided that allows you to manually adjust the vertical position of the -equation number. - ----The [xx]align[at] family of environments has been thoroughly revised: -now the xalignat environment is gone, as its function has been merged -into the align environment (it's no longer necessary to specify how many -side-by-side structures, as was required for xalignat, because align -automatically handles any number of them). Similarly, the xxalignat -environment has been replaced by an environment flalign that doesn't -require you to specify how many side-by-side structures will be needed. -Certain numbering problems (presence of an equation number when it -should be absent, or vice versa) have also been cleared up. - -(Thanks to David M. Jones for his substantial [volunteer] work behind -this change and the preceding one.) - ----The alignedat environment was changed to take two arguments rather -than one: a mandatory argument (as formerly) specifying the number of -align structures, and a new optional one specifying the placement of the -environment (parallel to the optional argument of aligned). However, -aligned is simpler to use, allowing any number of aligned structures -automatically, and therefore the use of alignedat is deprecated. - ----The seldom-used command \oldnos is gone; if you need it, you should -consult the LaTeX documentation on using fonts (e.g., fntguide.tex) to -find out how to construct an equivalent command. - ----A new subequations environment causes all numbered equation -environments within its scope to be numbered (4.9a) (4.9b) (4.9c) etc., -if the preceding numbered equation was 4.8. A \label command immediately -after \begin{subequations} will produce a \ref of the parent number -`4.9', not `4.9a'. The counters used by the subequations environment are -parentequation and equation and standard uses of \addtocounter, -\setcounter, etc., are possible with those counter names. - ----The amsmath package (unlike amstex) respects the setting provided by -the documentclass for putting equation numbers on the right or the left. -[Technical note: a documentclass that has leqno as the default should -explicitly pass that option to the amsmath package with -\PassOptionsToPackage in order for it to be effective.] - ----The fleqn option for left-aligned instead of centered equations is -now supported (thanks to David M. Jones). - -************************************************************************ -The remainder of this file discusses some changes that do not directly -pertain to the amsmath package but to the parent distribution known as -"AMS-LaTeX". - -======================================================================== -Changes in AMS-LaTeX 2.0 - ----The AMS-LaTeX User's Guide (amsldoc.tex) is split up into amsldoc.tex -and amsthdoc.tex, with the latter moving into the "classes" area. - ----The file amslatex.faq was split into amsmath.faq and amsclass.faq. - ----The file amslatex.bug was done away with; most of the information was -merged into diffs-m.txt, diffs-c.txt, amsmath.faq, and amsclass.faq. - -======================================================================== -Changes in AMS-LaTeX 1.2 - ----Adapted for LaTeX2e - -AMS-LaTeX 1.0 and 1.1 were designed to work with LaTeX 2.09, which -became obsolete in 1994 with the appearance of "LaTeX2e". AMS-LaTeX 1.2 -has been extensively adapted to take advantage of various features of -LaTeX2e. It is no longer a separate TeX format---i.e., it does not need -to be invoked by a separate "amslatex" command. - ----LaTeX 2.09 font commands (\bf, \it, etc) - -Pre-existing documents will for the most part continue to work fine -without requiring any changes. However, depending on the installation -options used for your copy of AMS-LaTeX 1.1 you may need to update the -options list of the \documentstyle command by adding newlfont, for -example: - - \documentstyle[newlfont]{amsart} - ----The packages amsfonts, amssymb were moved to the AMSFonts -distribution. They used to be included in the AMS-LaTeX distribution for -pragmatic reasons but their natural home is with the AMSFonts -distribution. - ----Package amstex renamed to amsmath - -The amstex package has been renamed amsmath. The old name `amstex' came -from the fact that version 1.0 was a more or less straight port into -LaTeX of selected portions from the macro package AmS-TeX written by -Michael Spivak, but the historical origins of the package are likely of -less interest to the general usership---especially looking to the -future---than a straightforward declaration in the name `amsmath' of the -sort of features that are provided. - -Another important reason for the name change was to make possible some -progress that would be out of the question (because of the impact on -existing documents) if the name `amstex' were retained. For example, -unlike amstex, amsmath doesn't automatically load the amsfonts package, -as that package was moved out to the AMSFonts distribution as described -above. If this change were not accompanied by a name change, many -existing documents would have to be updated by hand (adding -\usepackage{amsfonts}) in order to run without error. A frozen version -of amstex.sty is provided for processing pre-existing documents; it -won't have active maintainence henceforth except for bug fixes. For new -documents, authors should use the amsmath package. - -Therefore, "\usepackage{amstex}" is incorrect usage---do not rely on the -1994 edition of `The LaTeX Companion' in this respect (unless you obtain -the errata for Chapter 8; see -http://ctan.tug.org/info/compan-rev/ch8.pdf). - ----New documentclass amsproc - -A new documentclass amsproc was added for collections of articles, such as -the proceedings of a conference. diff --git a/Master/texmf-dist/source/latex/amsmath/install.txt b/Master/texmf-dist/source/latex/amsmath/install.txt deleted file mode 100644 index ef467b51c4a..00000000000 --- a/Master/texmf-dist/source/latex/amsmath/install.txt +++ /dev/null @@ -1,132 +0,0 @@ -install.txt for amsmath 2.16 [2004/08/05] - -Copyright 2004, 2013 American Mathematical Society. - -This work may be distributed and/or modified under the -conditions of the LaTeX Project Public License, either version 1.3c -of this license or (at your option) any later version. -The latest version of this license is in - http://www.latex-project.org/lppl.txt -and version 1.3c or later is part of all distributions of LaTeX -version 2005/12/01 or later. - -This work has the LPPL maintenance status `maintained'. - -The Current Maintainer of this work is the American Mathematical -Society. - -American Mathematical Society -Technical Support -Publications Technical Group -201 Charles Street -Providence, RI 02904 -USA -tel: (401) 455-4080 - (800) 321-4267 (USA and Canada only) -fax: (401) 331-3842 -email: tech-support@ams.org - -=========================================================================== - -If you have a relatively up-do-date TeX system that complies with the -TeX Directory Structure (TDS) standard, installing according to the -instructions in section I below is the easiest method. See - - http://www.tug.org/tds/tds.pdf - -for details on the TDS standard. If your TeX system does not comply -with the TDS standard, please contact the manufacturer of your TeX -system for details on how to install third-party TeX packages. - -The first step is to identify the TEXMF folder that contains files used -by LaTeX. To do this, look for the file article.cls. If you find, e.g., - - /usr/local/share/texmf/tex/latex/base/article.cls - -then we will define TEXMF to mean /usr/local/share/texmf (everything -before the tex/latex/base part). On a Windows system the file location -will probably be something like - - c:\sw\texmf\tex\latex\base\article.cls - -Then we define TEXMF to mean c:\sw\texmf. In the following instructions, -wherever you see "TEXMF" substitute the folder name determined in this -way. - -If this terminology does not make sense to you, seek assistance from a -friend or colleague, or from the company that you bought your TeX -system from, or contact us at one of the above addresses. - -======================================================================== -I. Installing the easy way. - -1. Download the following archive file: - - ftp://ftp.ams.org/pub/tex/amsmath.zip - -2. View the following files to assess the effects of the upgrade on -your system: - - README - install.txt (this file) - diffs-m.txt - -3. Backup the old version of AMS-LaTeX, preferably by moving it out of the -TEXMF tree. - -4. Extract files from the .zip archive using your favorite unzip tool, -setting the extraction target directory to the texmf root directory. - -5. To check the end result, you should find that you now have amsmath.sty -in the following location - - TEXMF/tex/latex/amsmath/amsmath.sty - -Some TeX systems also use a database to keep track of TeX input files. -If this database is not updated, your TeX system might not find the -newly installed files. Consult your TeX system's documentation to -determine how to update this database. - -6. Some files will appear in both the source area and another area. -This is because the source form and the in-use form are identical. -As distributed, the files in the source/latex/ams* areas comprise a -complete copy of the distribution from which all the other files can -be regenerated if need be. If you need to conserve disk space, you -could delete the files in the source area, since they are not needed -for normal use. - -======================================================================== -II. Installing the not-so-easy way. - -Download ftp://ftp.ams.org/pub/tex/amsmath.zip. - -Make the following directories, if not already present: - - TEXMF/source/latex/amsmath - TEXMF/tex/latex/amsmath - TEXMF/doc/latex/amsmath - -Backup your old AMS-LaTeX files if they are installed, preferably moving them -out of the TEXMF tree. - -******** revise -- pdf file amsldoc will be included in distrib - -Create a temporary directory and generate documentation files (these assume -a UNIX TeX installation): - - - make directory /tmp/amsmath - - unzip -d /tmp/amscls amsmath.zip - - cd /tmp/amsmath - - Run latex on the file ams-m1.ins to produce .cls and .sty files - - Run latex on *.tex - - [optional] Run latex on *.dtx (technical documentation) - - move *.dvi, *.pdf, *.tex to TEXMF/doc/latex/amsmath - - [optional] move *.dtx to TEXMF/doc/latex/amsmath - - [optional] convert .dvi to .ps or .pdf - - move *.sty and *.cls to TEXMF/tex/latex/amsmath - - delete /tmp/amsmath - -Some TeX systems also use a database to keep track of TeX input files. -If this database is not updated, your TeX system might not find the -newly installed files. Consult your system's documentation to determine -how to update this database. diff --git a/Master/texmf-dist/source/latex/amsmath/manifest.txt b/Master/texmf-dist/source/latex/amsmath/manifest.txt deleted file mode 100644 index e616c8e6b46..00000000000 --- a/Master/texmf-dist/source/latex/amsmath/manifest.txt +++ /dev/null @@ -1,64 +0,0 @@ -doc: -latex - -doc/latex: -amsmath - -doc/latex/amsmath: -amsbsy.pdf -amscd.pdf -amsgen.pdf -amsldoc.pdf -amsmath.pdf -amsopn.pdf -amstext.pdf -amsxtra.pdf -subeqn.pdf -technote.pdf -testmath.pdf - -source: -latex - -source/latex: -amsmath - -source/latex/amsmath: -README -amsbsy.dtx -amsbsy.ins -amscd.dtx -amscd.ins -amsgen.dtx -amsgen.ins -amsldoc.tex -amsmath.dtx -amsmath.ins -amsopn.dtx -amsopn.ins -amstext.dtx -amstext.ins -amsxtra.dtx -amsxtra.ins -diffs-m.txt -install.txt -manifest.txt -subeqn.tex -technote.tex -testmath.tex - -tex: -latex - -tex/latex: -amsmath - -tex/latex/amsmath: -amsbsy.sty -amscd.sty -amsgen.sty -amsmath.sty -amsopn.sty -amstex.sty -amstext.sty -amsxtra.sty diff --git a/Master/texmf-dist/source/latex/amsmath/subeqn.tex b/Master/texmf-dist/source/latex/amsmath/subeqn.tex deleted file mode 100644 index d0fa7d9b023..00000000000 --- a/Master/texmf-dist/source/latex/amsmath/subeqn.tex +++ /dev/null @@ -1,202 +0,0 @@ -%%% ==================================================================== -%%% @LaTeX-file{ -%%% filename = "subeqn.tex", -%%% version = "1.2c", -%%% date = "1999/11/29", -%%% time = "14:49:29 EST", -%%% checksum = "29866 202 668 6426", -%%% author = "American Mathematical Society", -%%% copyright = "Copyright 1999 American Mathematical Society, -%%% all rights reserved. Copying of this file is -%%% authorized only if either: -%%% (1) you make absolutely no changes to your copy, -%%% including name; OR -%%% (2) if you do make changes, you first rename it -%%% to some other name.", -%%% address = "American Mathematical Society, -%%% Technical Support, -%%% Electronic Products and Services, -%%% P. O. Box 6248, -%%% Providence, RI 02940, -%%% USA", -%%% telephone = "401-455-4080 or (in the USA and Canada) -%%% 800-321-4AMS (321-4267)", -%%% FAX = "401-331-3842", -%%% email = "tech-support@ams.org (Internet)", -%%% supported = "yes", -%%% keywords = "latex, amslatex, ams-latex, amstex", -%%% abstract = "This is part of the AMS-\LaTeX{} distribution. -%%% It is a sample document illustrating the use of -%%% the subequations environment and the effect of -%%% the fleqn option.", -%%% docstring = "The checksum field above contains a CRC-16 -%%% checksum as the first value, followed by the -%%% equivalent of the standard UNIX wc (word -%%% count) utility output of lines, words, and -%%% characters. This is produced by Robert -%%% Solovay's checksum utility.", -%%% } -%%% ==================================================================== -\documentclass[fleqn]{article} - -\title{Test of subequations environment} -\author{mjd} - -\usepackage{amsmath} -\numberwithin{equation}{section} - -\newcommand{\env}[1]{{\normalfont\texttt{#1}}} - -\AtEndDocument{\multipasswarning} -\newcommand{\multipasswarning}{% - \clearpage - \typeout{% - **********************************************************************} - \typeout{% - Note: This document needs to run through LaTeX three times, instead of} - \typeout{% - the usual two, to resolve indirect cross-references.} - \typeout{% - **********************************************************************} -} - -\makeatletter -% Omit the warning message if three passes have been completed (on -% first pass \ref{check} is undefined and it is set to 0; on second -% pass \ref{check} is 0 and set to 1; on third pass it is 1). -\def\checkref{\begingroup - \@ifundefined{r@check}{\def\@currentlabel{0}}{\def\@currentlabel{1}}% - \ifnum1=0\expandafter\@firstoftwo\r@check\relax - \global\let\multipasswarning\relax - \fi - \label{check}% - \endgroup} -\makeatother - -\begin{document} -\section{Prime equation numbers} - -First an equation. -\begin{equation}\label{e:previous} -A=B -\end{equation} -That was equation \eqref{e:previous}. - -Then the same, with a prime on the number.\checkref -\begin{equation} -\tag{\ref{e:previous}$'$}\label{e:prevprime} -C=D -\end{equation} -And that was equation \eqref{e:prevprime}. - -Notice, by the way, that when a \verb"\ref" occurs inside a \verb"\tag", -and that \verb"\tag" is then \verb"\label"'d, a \verb"\ref" for the the -second \verb"\label" requires \emph{three} runs of \LaTeX{} in order to -get the proper value. (If you run through the logic of \LaTeX{}'s -cross-referencing mechanisms as they apply in this case, you will see -that this is necessary.) - -\section{Subnumbered equations} - -Here is a,b,c sub-numbering. -\begin{subequations} -\begin{eqnarray} -A&=&B\\ -D&=&C \label{e:middle}\\ -E&=&F -\end{eqnarray} -\end{subequations} -That was produced with the \env{eqnarray} environment; the middle line -was labeled as \eqref{e:middle}. - -An equation following the end of the \verb"subequations" environment -should revert to normal numbering: -\begin{equation}\label{e:check} -H<K -\end{equation} -A check on the labeling: that was equation~\eqref{e:check}. - -The sub-numbered equations can be spread out through the text, like -this: -\begin{subequations} -\begin{equation} -A=B -\end{equation} -The \verb"subequations" environment can span arbitrary text between -subsidiary equations. The only restriction is that if there are any -numbered equations inside the \verb"subequations" environment that break -out of the subequation numbering sequence, they would have to be handled -specially. -\begin{equation} -D=C \label{e:newmiddle} -\end{equation} -More arbitrary text. -\begin{equation} -E=F -\end{equation} -\end{subequations} -Label check: the middle one was \eqref{e:newmiddle} - -A final equation for a numbering check. -\begin{equation}\label{e:final} -G=H -\end{equation} -That equation was labeled as \eqref{e:final}. - -\section{Tests of \env{align}, \env{gather}, and other -AMS-\protect\LaTeX{} environments} - -The \env{align} environment: -\begin{subequations} -\begin{align} -\label{align:a}A+B&=B+A\\ -\label{align:b}C&=D+E\\ -\label{align:c}E&=F -\end{align} -\end{subequations} -Label check: that was \eqref{align:a}, \eqref{align:b}, and -\eqref{align:c}. - -The \env{align} environment again: -\begin{subequations} -\begin{align} -\label{xalign:a}A+B&=B& B&=B+A\\ -\label{xalign:b}C&=D+E& C\oplus D&=E\\ -\label{xalign:c}E&=F& E'&=F' -\end{align} -\end{subequations} -Label check: that was \eqref{xalign:a}, \eqref{xalign:b}, and -\eqref{xalign:c}. - -The \env{gather} environment. For the third line we refer to one of the -numbers in the first \env{align} structure. -\begin{subequations} -\begin{gather} -\label{gather:a}A+B=B\\ -\label{gather:b}C=D+E\\ -\tag{\ref{align:c}$'$}\label{gather:c}E=F -\end{gather} -\end{subequations} -Label check: that was \eqref{gather:a}, -\eqref{gather:b}, and -\eqref{gather:c}. - -The next \env{subequations} environment encompasses two separate -equations. A \env{split} environment: -\begin{subequations} -\begin{equation} -\label{split:x} -\begin{split} -A&=B+C+F\\ -&=G -\end{split} -\end{equation} -and a \env{multline} environment: -\begin{multline}\label{multline:x} -A[B]C[D]E[F]G[[H[I]J[K]L[M]N]]=\\ -H[I]J[K]L[M]N[O]P[Q]R[S]T[U]V[W]X[Y]Z -\end{multline} -\end{subequations} -Label check: That was \eqref{split:x} and \eqref{multline:x}. - -\end{document} diff --git a/Master/texmf-dist/source/latex/amsmath/technote.tex b/Master/texmf-dist/source/latex/amsmath/technote.tex deleted file mode 100644 index aede450621e..00000000000 --- a/Master/texmf-dist/source/latex/amsmath/technote.tex +++ /dev/null @@ -1,233 +0,0 @@ -%%% ==================================================================== -%%% @LaTeX-file{ -%%% filename = "technote.tex", -%%% version = "2.0", -%%% date = "1999/11/15", -%%% time = "15:07:20 EST", -%%% checksum = "58506 233 1647 12005", -%%% author = "American Mathematical Society", -%%% copyright = "Copyright 1995, 1999 American Mathematical Society, -%%% all rights reserved. Copying of this file is -%%% authorized only if either: -%%% (1) you make absolutely no changes to your copy, -%%% including name; OR -%%% (2) if you do make changes, you first rename it -%%% to some other name.", -%%% address = "American Mathematical Society, -%%% Technical Support, -%%% Electronic Products and Services, -%%% P. O. Box 6248, -%%% Providence, RI 02940, -%%% USA", -%%% telephone = "401-455-4080 or (in the USA and Canada) -%%% 800-321-4AMS (321-4267)", -%%% FAX = "401-331-3842", -%%% email = "tech-support@ams.org (Internet)", -%%% supported = "yes", -%%% keywords = "amsmath, latex", -%%% abstract = "This is part of the AMS-\LaTeX{} distribution. It -%%% is a document discussing some technical issues in the -%%% contents and interface of AMS-\LaTeX{} packages." -%%% docstring = "The checksum field above contains a CRC-16 checksum -%%% as the first value, followed by the equivalent of -%%% the standard UNIX wc (word count) utility output of -%%% lines, words, and characters. This is produced by -%%% Robert Solovay's checksum utility.", -%%% } -%%% ==================================================================== -\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX) -[1994/12/01]% LaTeX date must December 1994 or later -\documentclass{amsdtx} - -\title{Technical notes on the \pkg{amsmath} package} -\author{American Mathematical Society\\Michael Downes} -\date{1999/10/29} - -\markboth - {Technical notes on the \pkg{amsmath} package} - {Technical notes on the \pkg{amsmath} package} -\renewcommand{\sectionmark}[1]{} - -\providecommand{\etc}[1]{etc.} - -\providecommand{\begend}[1]{% - {\ntt \symbol{92}begin\symbol{123}#1\symbol{125}} - \ldots\ {\ntt\symbol{92}end\symbol{123}#1\symbol{125}}% -} - -\MakeShortVerb{\|} - -\begin{document} -\maketitle - -\section{Introduction} - -These notes are miscellaneous remarks on some technical questions -applicable to version 2.0 of \amslatex/. - -\section{Deprecated and disallowed commands} - -Certain commands that were in the \pkg{amstex} package (the predecessor, -in \amslatex/ 1.0 and 1.1, of the \pkg{amsmath} package) were moved into -\pkg{amsxtra} because they seemed to be little-used relics: -\cn{accentedsymbol}, \qq{sup accents} (\cn{sptilde}, \cn{sphat}, etc.). - -Certain other commands\mdash e.g., \cn{over}, \cn{pmatrix}, and -\cn{cases}\mdash are changed by the \pkg{amsmath} package to produce -warning messages or error messages, depending on their history; these -are discussed in more detail in the following sections. - -\section{Why do \cn{matrix}, \cn{pmatrix}, and \cn{cases} stop working -when I add the \pkg{amsmath} package?} - -If you used the \fn{plain.tex} versions of \cn{matrix}, \cn{pmatrix}, or -\cn{cases} in a document and then later converted the document to use -the \pkg{amsmath} package (or one of the AMS documentclasses, which -automatically call the \pkg{amsmath} package internally), the instances -of those commands will produce error messages. The problem is that when -\latex/ was originally created, it adopted most of its mathematics -features straight from \fn{plain.tex}. But in the case of \cn{matrix}, -\cn{pmatrix}, \cn{cases} this was a mistake\mdash the \fn{plain.tex} -syntax for them is decidedly non-\latex/ in style, for example the fact -that they use \cs{cr} instead of \cn{\\} to mark line breaks, and they -don't use \cn{begin} and \cn{end}. In basic \latex/ this mistake will be -perpetuated at least until \latex/3 appears, in order to avoid breaking -existing documents. But no existing documents that were written with the -\pkg{amsmath} package have that syntactic problem, as \pkg{amsmath} -provides proper \latex/-syntax versions of \cn{matrix} and the others. -The possibility of optionally allowing the \fn{plain.tex} variants to -make document conversion easier seems ill-advised since those variants -are so blatantly wrong in a \latex/ context. The \env{array} environment -ought to have been used instead. - -\section{Why do \cn{over}, \cn{atop}, \cn{above} [\dots{\ntt withdelims}] - give a warning when I add the \pkg{amsmath} package?} - -Using the six generalized fraction commands \cn{over}, -\cn{overwithdelims}, \cn{atop}, \cn{atopwithdelims}, \cn{above}, -\cn{abovewithdelims} is not really a good idea in \latex/, for reasons -explained below. I construe \latex/'s provision of \cs{frac}, and the -lack of any mention in the \latex/ book of the primitive fraction -commands, as an implicit injunction against their use, although I don't -think Lamport actually spent a lot of time pondering the issue, and the -basic \latex/ version of \cn{frac} provides access only to \cn{over}, -not to \cn{atop}, \cn{above}, or the \verb'withdelims' variants. The -\pkg{amsmath} package, however, provides a \cn{genfrac} command that -gives user-level access to all six of the generalized fraction -primitives in a way that conforms to the syntactic conventions followed -by all other \latex/ commands. - -Not only is the unusual syntax of the \tex/ primitives rather out of -place in \latex/, but furthermore that syntax seems to be responsible -for one of the most significant flaws in \tex/'s mathematical -typesetting capabilities: the fact that the current mathstyle at any -given point in a math formula cannot be determined until the end of the -formula, because of the possibility that a following generalized -fraction command will change the mathstyle of the \emph{preceding} -material. To cite two of the worst side effects: \cn{mathchoice} must -actually typeset all four of its arguments, instead of being able to -immediately select only one; and, were it possible to always know the -current math style at a given point, math font selection would be -greatly simplified and the upper limit of 16 different math font -\cn{fam}s would never be a problem as \cn{text,script[script]font} -assignments for any \cn{fam} could take immediate effect and therefore -could be changed arbitrarily often within a single formula. More -concretely, math font selection difficulties are responsible for many of -the more convoluted passages in the source code of \latex/'s NFSS (that -does font loading on demand) and of the \pkg{amsmath} package, and by -extension it has historically been responsible for significant delays in -making new features available to end users and for making those features -more prone to bugs. - -There are additional bad consequences following from the syntax of those -generalized fraction commands that only become evident when you do some -writing of nontrivial macros for math use. For example, as things -currently stand you cannot measure the size of any object in math -without going through \cn{mathchoice} and \emph{leaving and reentering -math mode} via \verb'\hbox{$' (which then introduces complications -regarding \cn{everymath} and \cn{mathsurround}). And it seems that -uncertainty about the current mathstyle is the only barrier to allowing -the use of mu units with \cn{vrule}, to make vertical struts in -constructing compound symbols or notation. And so on and so forth. - -\section{The \opt{fleqn} option and \cn{mathindent}} - -Strictly speaking, the \pkg{amsmath} package doesn't use \cn{mathindent} -to control the left indent of displayed equations when the \opt{fleqn} -option is in effect: it uses an internal parameter \cs{@mathmargin} -instead. However, for compatibility with existing \latex/ documentation, -\pkg{amsmath} turns \cn{mathindent} into an alias for \cs{@mathmargin}. -There is a small risk here: In the plain \latex/ implementation, -\cn{mathindent} is a dimen register, but with \pkg{amsmath} -\cs{@mathmargin} is a skip register and, by association, so is -\cn{mathindent}. If any package or documentclass uses \cn{mathindent} in -a way that depends on it being a dimen register, when used in -conjunction with the \pkg{amsmath} package it may be vulnerable to a -well-known pitfall having to do with the primitive \tex/ lookahead for a -\texttt{plus} or \texttt{minus} key word. However if the standard -\latex/ commands \cn{setlength} and \cn{addtolength} are used to modify -\cn{mathindent} then this problem will not arise. - -\section{Why can't I use abbreviations for \protect\begend{align}?} - -Authors often like to use abbreviations such as |\beq| |\eeq| for -|\begin{equation}| |\end{equation}|. For some environments defined by -the \pkg{amsmath} package, such as \env{align}, \env{gather}, -\env{multline}, and others of the same general type, this does not work: -An attempt to define |\bal| |\eal| as shorthand for |\begin{align}| -|\end{align}| will fail with a puzzling error message. This has to do -with unfortunately nontrivial technical complications: the given -environments must read their contents as a delimited macro argument -because they do multipass processing of the contents using algorithms -inherited from Spivak's \fn{amstex.tex}. The obvious -solution\mdash substitution of different algorithms that do box shuffling -instead of token shuffling for the multipass calculations\mdash would -require rewriting these display environments from the ground up; while -that is a worthy goal, it was beyond the original scope of the -\amslatex/ project. Work is under way on an auxiliary package called -\pkg{breqn} that addresses not only this problem but a number of others; -at the time of this writing, however [September 1999] it has only -progressed as far as a beta release. - -Some workarounds: -\begin{itemize} -\item |\def\bal#1\eal{\begin{align}#1\end{align}}| -\item Define |\newcommand{\env}[2]{\begin{#1}#2\end{#1}}| and then use - |\env{align}{...}| -\end{itemize} - -\section{The \pkg{upref} package} - -The reason for splitting out the \pkg{upref} package instead of -automatically incorporating it in the \cls{amsart} and \cls{amsbook} -classes is this: It involves low-level surgery on an important \latex/ -command. This means that if ever this command changes in the future (as -it did between versions 2.09 and 2e of \latex/) we have a maintenance -problem. And the benefit that \pkg{upref} provides is something that -most end users don't care much about. - -\section{The \pkg{amsintx} package} - -After a very preliminary trial release the \pkg{amsintx} package was -withdrawn to await further development. It is still considered a worthy -project but has simply not yet made it to the head of the priority queue -in the AMS \latex/ development program, as it has had to contend with a -number of other equally worthy projects for development time. - -\section{Hyphenation in the documentation} - -Hyphenation was allowed for certain long command names in -\fn{amsldoc.tex}; this presented technical difficulties because \latex/ -normally deactivates hyphenation for tt fonts. The method chosen to -reinstate hyphenation was to turn off the encoding-specific function -\cs{OT1+cmtt} that disables the \cs{hyphenchar} for tt fonts; see the -definition of \cn{allowtthyphens} in \fn{amsdtx.dtx}. Then a list of all -tt words in the document was gathered (from the \fn{.idx} file, produced -by the \cn{cn}, \cn{fn}, \cn{pkg}, etc. commands) and \cn{showhyphens} -was applied to this list. The result was another list in the resulting -\tex/ log, containing those words in a form suitable for the argument of -\cn{hyphenation}. That list was then edited by hand to overrule -undesirable hyphenations; words with acceptable hyphenations were -dropped from the list, as they don't need to be repeated there. - -\end{document} diff --git a/Master/texmf-dist/source/latex/amsmath/testmath.tex b/Master/texmf-dist/source/latex/amsmath/testmath.tex deleted file mode 100644 index 069944ee9d0..00000000000 --- a/Master/texmf-dist/source/latex/amsmath/testmath.tex +++ /dev/null @@ -1,2342 +0,0 @@ -%%% ==================================================================== -%%% @LaTeX-file{ -%%% filename = "testmath.tex", -%%% version = "2.0", -%%% date = "1999/11/15", -%%% time = "15:09:17 EST", -%%% checksum = "07762 2342 7811 82371", -%%% author = "American Mathematical Society", -%%% copyright = "Copyright 1995, 1999 American Mathematical Society, -%%% all rights reserved. Copying of this file is -%%% authorized only if either: -%%% (1) you make absolutely no changes to your copy, -%%% including name; OR -%%% (2) if you do make changes, you first rename it -%%% to some other name.", -%%% address = "American Mathematical Society, -%%% Technical Support, -%%% Electronic Products and Services, -%%% P. O. Box 6248, -%%% Providence, RI 02940, -%%% USA", -%%% telephone = "401-455-4080 or (in the USA and Canada) -%%% 800-321-4AMS (321-4267)", -%%% FAX = "401-331-3842", -%%% email = "tech-support@ams.org (Internet)", -%%% codetable = "ISO/ASCII", -%%% keywords = "latex, amsmath, examples, documentation", -%%% supported = "yes", -%%% abstract = "This is a test file containing extensive examples of -%%% mathematical constructs supported by the amsmath -%%% package.", -%%% docstring = "The checksum field above contains a CRC-16 -%%% checksum as the first value, followed by the -%%% equivalent of the standard UNIX wc (word -%%% count) utility output of lines, words, and -%%% characters. This is produced by Robert -%%% Solovay's checksum utility.", -%%% } -%%% ==================================================================== -\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX) -[1994/12/01]% LaTeX date must December 1994 or later -\documentclass[draft]{article} -\pagestyle{headings} - -\title{Sample Paper for the \pkg{amsmath} Package\\ -File name: \fn{testmath.tex}} -\author{American Mathematical Society} -\date{Version 2.0, 1999/11/15} - -\usepackage{amsmath,amsthm} - -% Some definitions useful in producing this sort of documentation: -\chardef\bslash=`\\ % p. 424, TeXbook -% Normalized (nonbold, nonitalic) tt font, to avoid font -% substitution warning messages if tt is used inside section -% headings and other places where odd font combinations might -% result. -\newcommand{\ntt}{\normalfont\ttfamily} -% command name -\newcommand{\cn}[1]{{\protect\ntt\bslash#1}} -% LaTeX package name -\newcommand{\pkg}[1]{{\protect\ntt#1}} -% File name -\newcommand{\fn}[1]{{\protect\ntt#1}} -% environment name -\newcommand{\env}[1]{{\protect\ntt#1}} -\hfuzz1pc % Don't bother to report overfull boxes if overage is < 1pc - -% Theorem environments - -%% \theoremstyle{plain} %% This is the default -\newtheorem{thm}{Theorem}[section] -\newtheorem{cor}[thm]{Corollary} -\newtheorem{lem}[thm]{Lemma} -\newtheorem{prop}[thm]{Proposition} -\newtheorem{ax}{Axiom} - -\theoremstyle{definition} -\newtheorem{defn}{Definition}[section] - -\theoremstyle{remark} -\newtheorem{rem}{Remark}[section] -\newtheorem*{notation}{Notation} - -%\numberwithin{equation}{section} - -\newcommand{\thmref}[1]{Theorem~\ref{#1}} -\newcommand{\secref}[1]{\S\ref{#1}} -\newcommand{\lemref}[1]{Lemma~\ref{#1}} - -\newcommand{\bysame}{\mbox{\rule{3em}{.4pt}}\,} - -% Math definitions - -\newcommand{\A}{\mathcal{A}} -\newcommand{\B}{\mathcal{B}} -\newcommand{\st}{\sigma} -\newcommand{\XcY}{{(X,Y)}} -\newcommand{\SX}{{S_X}} -\newcommand{\SY}{{S_Y}} -\newcommand{\SXY}{{S_{X,Y}}} -\newcommand{\SXgYy}{{S_{X|Y}(y)}} -\newcommand{\Cw}[1]{{\hat C_#1(X|Y)}} -\newcommand{\G}{{G(X|Y)}} -\newcommand{\PY}{{P_{\mathcal{Y}}}} -\newcommand{\X}{\mathcal{X}} -\newcommand{\wt}{\widetilde} -\newcommand{\wh}{\widehat} - -\DeclareMathOperator{\per}{per} -\DeclareMathOperator{\cov}{cov} -\DeclareMathOperator{\non}{non} -\DeclareMathOperator{\cf}{cf} -\DeclareMathOperator{\add}{add} -\DeclareMathOperator{\Cham}{Cham} -\DeclareMathOperator{\IM}{Im} -\DeclareMathOperator{\esssup}{ess\,sup} -\DeclareMathOperator{\meas}{meas} -\DeclareMathOperator{\seg}{seg} - -% \interval is used to provide better spacing after a [ that -% is used as a closing delimiter. -\newcommand{\interval}[1]{\mathinner{#1}} - -% Notation for an expression evaluated at a particular condition. The -% optional argument can be used to override automatic sizing of the -% right vert bar, e.g. \eval[\biggr]{...}_{...} -\newcommand{\eval}[2][\right]{\relax - \ifx#1\right\relax \left.\fi#2#1\rvert} - -% Enclose the argument in vert-bar delimiters: -\newcommand{\envert}[1]{\left\lvert#1\right\rvert} -\let\abs=\envert - -% Enclose the argument in double-vert-bar delimiters: -\newcommand{\enVert}[1]{\left\lVert#1\right\rVert} -\let\norm=\enVert - -\begin{document} -\maketitle -\markboth{Sample paper for the {\protect\ntt\lowercase{amsmath}} package} -{Sample paper for the {\protect\ntt\lowercase{amsmath}} package} -\renewcommand{\sectionmark}[1]{} - -\section{Introduction} - -This paper contains examples of various features from \AmS-\LaTeX{}. - -\section{Enumeration of Hamiltonian paths in a graph} - -Let $\mathbf{A}=(a_{ij})$ be the adjacency matrix of graph $G$. The -corresponding Kirchhoff matrix $\mathbf{K}=(k_{ij})$ is obtained from -$\mathbf{A}$ by replacing in $-\mathbf{A}$ each diagonal entry by the -degree of its corresponding vertex; i.e., the $i$th diagonal entry is -identified with the degree of the $i$th vertex. It is well known that -\begin{equation} -\det\mathbf{K}(i|i)=\text{ the number of spanning trees of $G$}, -\quad i=1,\dots,n -\end{equation} -where $\mathbf{K}(i|i)$ is the $i$th principal submatrix of -$\mathbf{K}$. -\begin{verbatim} -\det\mathbf{K}(i|i)=\text{ the number of spanning trees of $G$}, -\end{verbatim} - -Let $C_{i(j)}$ be the set of graphs obtained from $G$ by attaching edge -$(v_iv_j)$ to each spanning tree of $G$. Denote by $C_i=\bigcup_j -C_{i(j)}$. It is obvious that the collection of Hamiltonian cycles is a -subset of $C_i$. Note that the cardinality of $C_i$ is $k_{ii}\det -\mathbf{K}(i|i)$. Let $\wh X=\{\hat x_1,\dots,\hat x_n\}$. -\begin{verbatim} -$\wh X=\{\hat x_1,\dots,\hat x_n\}$ -\end{verbatim} -Define multiplication for the elements of $\wh X$ by -\begin{equation}\label{multdef} -\hat x_i\hat x_j=\hat x_j\hat x_i,\quad \hat x^2_i=0,\quad -i,j=1,\dots,n. -\end{equation} -Let $\hat k_{ij}=k_{ij}\hat x_j$ and $\hat k_{ij}=-\sum_{j\not=i} \hat -k_{ij}$. Then the number of Hamiltonian cycles $H_c$ is given by the -relation \cite{liuchow:formalsum} -\begin{equation}\label{H-cycles} -\biggl(\prod^n_{\,j=1}\hat x_j\biggr)H_c=\frac{1}{2}\hat k_{ij}\det -\wh{\mathbf{K}}(i|i),\qquad i=1,\dots,n. -\end{equation} -The task here is to express \eqref{H-cycles} -in a form free of any $\hat x_i$, -$i=1,\dots,n$. The result also leads to the resolution of enumeration of -Hamiltonian paths in a graph. - -It is well known that the enumeration of Hamiltonian cycles and paths in -a complete graph $K_n$ and in a complete bipartite graph $K_{n_1n_2}$ -can only be found from \textit{first combinatorial principles} -\cite{hapa:graphenum}. One wonders if there exists a formula which can -be used very efficiently to produce $K_n$ and $K_{n_1n_2}$. Recently, -using Lagrangian methods, Goulden and Jackson have shown that $H_c$ can -be expressed in terms of the determinant and permanent of the adjacency -matrix \cite{gouja:lagrmeth}. However, the formula of Goulden and -Jackson determines neither $K_n$ nor $K_{n_1n_2}$ effectively. In this -paper, using an algebraic method, we parametrize the adjacency matrix. -The resulting formula also involves the determinant and permanent, but -it can easily be applied to $K_n$ and $K_{n_1n_2}$. In addition, we -eliminate the permanent from $H_c$ and show that $H_c$ can be -represented by a determinantal function of multivariables, each variable -with domain $\{0,1\}$. Furthermore, we show that $H_c$ can be written by -number of spanning trees of subgraphs. Finally, we apply the formulas to -a complete multigraph $K_{n_1\dots n_p}$. - -The conditions $a_{ij}=a_{ji}$, $i,j=1,\dots,n$, are not required in -this paper. All formulas can be extended to a digraph simply by -multiplying $H_c$ by 2. - -\section{Main Theorem} -\label{s:mt} - -\begin{notation} For $p,q\in P$ and $n\in\omega$ we write -$(q,n)\le(p,n)$ if $q\le p$ and $A_{q,n}=A_{p,n}$. -\begin{verbatim} -\begin{notation} For $p,q\in P$ and $n\in\omega$ -... -\end{notation} -\end{verbatim} -\end{notation} - -Let $\mathbf{B}=(b_{ij})$ be an $n\times n$ matrix. Let $\mathbf{n}=\{1, -\dots,n\}$. Using the properties of \eqref{multdef}, it is readily seen -that - -\begin{lem}\label{lem-per} -\begin{equation} -\prod_{i\in\mathbf{n}} -\biggl(\sum_{\,j\in\mathbf{n}}b_{ij}\hat x_i\biggr) -=\biggl(\prod_{\,i\in\mathbf{n}}\hat x_i\biggr)\per \mathbf{B} -\end{equation} -where $\per \mathbf{B}$ is the permanent of $\mathbf{B}$. -\end{lem} - -Let $\wh Y=\{\hat y_1,\dots,\hat y_n\}$. Define multiplication -for the elements of $\wh Y$ by -\begin{equation} -\hat y_i\hat y_j+\hat y_j\hat y_i=0,\quad i,j=1,\dots,n. -\end{equation} -Then, it follows that -\begin{lem}\label{lem-det} -\begin{equation}\label{detprod} -\prod_{i\in\mathbf{n}} -\biggl(\sum_{\,j\in\mathbf{n}}b_{ij}\hat y_j\biggr) -=\biggl(\prod_{\,i\in\mathbf{n}}\hat y_i\biggr)\det\mathbf{B}. -\end{equation} -\end{lem} - -Note that all basic properties of determinants are direct consequences -of Lemma ~\ref{lem-det}. Write -\begin{equation}\label{sum-bij} -\sum_{j\in\mathbf{n}}b_{ij}\hat y_j=\sum_{j\in\mathbf{n}}b^{(\lambda)} -_{ij}\hat y_j+(b_{ii}-\lambda_i)\hat y_i\hat y -\end{equation} -where -\begin{equation} -b^{(\lambda)}_{ii}=\lambda_i,\quad b^{(\lambda)}_{ij}=b_{ij}, -\quad i\not=j. -\end{equation} -Let $\mathbf{B}^{(\lambda)}=(b^{(\lambda)}_{ij})$. By \eqref{detprod} -and \eqref{sum-bij}, it is -straightforward to show the following -result: -\begin{thm}\label{thm-main} -\begin{equation}\label{detB} -\det\mathbf{B}= -\sum^n_{l =0}\sum_{I_l \subseteq n} -\prod_{i\in I_l}(b_{ii}-\lambda_i) -\det\mathbf{B}^{(\lambda)}(I_l |I_l ), -\end{equation} -where $I_l =\{i_1,\dots,i_l \}$ and $\mathbf{B}^{(\lambda)}(I_l |I_l )$ -is the principal submatrix obtained from $\mathbf{B}^{(\lambda)}$ -by deleting its $i_1,\dots,i_l $ rows and columns. -\end{thm} - -\begin{rem} -Let $\mathbf{M}$ be an $n\times n$ matrix. The convention -$\mathbf{M}(\mathbf{n}|\mathbf{n})=1$ has been used in \eqref{detB} and -hereafter. -\end{rem} - -Before proceeding with our discussion, we pause to note that -\thmref{thm-main} yields immediately a fundamental formula which can be -used to compute the coefficients of a characteristic polynomial -\cite{mami:matrixth}: -\begin{cor}\label{BI} -Write $\det(\mathbf{B}-x\mathbf{I})=\sum^n_{l =0}(-1) -^l b_l x^l $. Then -\begin{equation}\label{bl-sum} -b_l =\sum_{I_l \subseteq\mathbf{n}}\det\mathbf{B}(I_l |I_l ). -\end{equation} -\end{cor} -Let -\begin{equation} -\mathbf{K}(t,t_1,\dots,t_n) -=\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\ --a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\ -\hdotsfor[2]{4}\\ --a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix}, -\end{equation} -\begin{verbatim} -\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\ --a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\ -\hdotsfor[2]{4}\\ --a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix} -\end{verbatim} -where -\begin{equation} -D_i=\sum_{j\in\mathbf{n}}a_{ij}t_j,\quad i=1,\dots,n. -\end{equation} - -Set -\begin{equation*} -D(t_1,\dots,t_n)=\frac{\delta}{\delta t}\eval{\det\mathbf{K}(t,t_1,\dots,t_n) -}_{t=1}. -\end{equation*} -Then -\begin{equation}\label{sum-Di} -D(t_1,\dots,t_n) -=\sum_{i\in\mathbf{n}}D_i\det\mathbf{K}(t=1,t_1,\dots,t_n; i|i), -\end{equation} -where $\mathbf{K}(t=1,t_1,\dots,t_n; i|i)$ is the $i$th principal -submatrix of $\mathbf{K}(t=1,t_1,\dots,t_n)$. - -Theorem ~\ref{thm-main} leads to -\begin{equation}\label{detK1} -\det\mathbf{K}(t_1,t_1,\dots,t_n) -=\sum_{I\in\mathbf{n}}(-1)^{\envert{I}}t^{n-\envert{I}} -\prod_{i\in I}t_i\prod_{j\in I}(D_j+\lambda_jt_j)\det\mathbf{A} -^{(\lambda t)}(\overline{I}|\overline I). -\end{equation} -Note that -\begin{equation}\label{detK2} -\det\mathbf{K}(t=1,t_1,\dots,t_n)=\sum_{I\in\mathbf{n}}(-1)^{\envert{I}} -\prod_{i\in I}t_i\prod_{j\in I}(D_j+\lambda_jt_j)\det\mathbf{A} -^{(\lambda)}(\overline{I}|\overline{I})=0. -\end{equation} - -Let $t_i=\hat x_i,i=1,\dots,n$. Lemma ~\ref{lem-per} yields -\begin{multline} -\biggl(\sum_{\,i\in\mathbf{n}}a_{l _i}x_i\biggr) -\det\mathbf{K}(t=1,x_1,\dots,x_n;l |l )\\ -=\biggl(\prod_{\,i\in\mathbf{n}}\hat x_i\biggr) -\sum_{I\subseteq\mathbf{n}-\{l \}} -(-1)^{\envert{I}}\per\mathbf{A}^{(\lambda)}(I|I) -\det\mathbf{A}^{(\lambda)} -(\overline I\cup\{l \}|\overline I\cup\{l \}). -\label{sum-ali} -\end{multline} -\begin{verbatim} -\begin{multline} -\biggl(\sum_{\,i\in\mathbf{n}}a_{l _i}x_i\biggr) -\det\mathbf{K}(t=1,x_1,\dots,x_n;l |l )\\ -=\biggl(\prod_{\,i\in\mathbf{n}}\hat x_i\biggr) -\sum_{I\subseteq\mathbf{n}-\{l \}} -(-1)^{\envert{I}}\per\mathbf{A}^{(\lambda)}(I|I) -\det\mathbf{A}^{(\lambda)} -(\overline I\cup\{l \}|\overline I\cup\{l \}). -\label{sum-ali} -\end{multline} -\end{verbatim} - -By \eqref{H-cycles}, \eqref{detprod}, and \eqref{sum-bij}, we have -\begin{prop}\label{prop:eg} -\begin{equation} -H_c=\frac1{2n}\sum^n_{l =0}(-1)^{l} -D_{l}, -\end{equation} -where -\begin{equation}\label{delta-l} -D_{l}=\eval[2]{\sum_{I_{l}\subseteq \mathbf{n}} -D(t_1,\dots,t_n)}_{t_i=\left\{\begin{smallmatrix} -0,& \text{if }i\in I_{l}\quad\\% \quad added for centering -1,& \text{otherwise}\end{smallmatrix}\right.\;,\;\; i=1,\dots,n}. -\end{equation} -\end{prop} - -\section{Application} -\label{lincomp} - -We consider here the applications of Theorems~\ref{th-info-ow-ow} and -~\ref{th-weak-ske-owf} to a complete -multipartite graph $K_{n_1\dots n_p}$. It can be shown that the -number of spanning trees of $K_{n_1\dots n_p}$ -may be written -\begin{equation}\label{e:st} -T=n^{p-2}\prod^p_{i=1} -(n-n_i)^{n_i-1} -\end{equation} -where -\begin{equation} -n=n_1+\dots+n_p. -\end{equation} - -It follows from Theorems~\ref{th-info-ow-ow} and -~\ref{th-weak-ske-owf} that -\begin{equation}\label{e:barwq} -\begin{split} -H_c&=\frac1{2n} -\sum^n_{{l}=0}(-1)^{l}(n-{l})^{p-2} -\sum_{l _1+\dots+l _p=l}\prod^p_{i=1} -\binom{n_i}{l _i}\\ -&\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i}\cdot -\biggl[(n-l )^2-\sum^p_{j=1}(n_i-l _i)^2\biggr].\end{split} -\end{equation} -\begin{verbatim} -... \binom{n_i}{l _i}\\ -\end{verbatim} -and -\begin{equation}\label{joe} -\begin{split} -H_c&=\frac12\sum^{n-1}_{l =0} -(-1)^{l}(n-l )^{p-2} -\sum_{l _1+\dots+l _p=l} -\prod^p_{i=1}\binom{n_i}{l _i}\\ -&\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i} -\left(1-\frac{l _p}{n_p}\right) -[(n-l )-(n_p-l _p)]. -\end{split} -\end{equation} - -The enumeration of $H_c$ in a $K_{n_1\dotsm n_p}$ graph can also be -carried out by Theorem ~\ref{thm-H-param} or ~\ref{thm-asym} -together with the algebraic method of \eqref{multdef}. -Some elegant representations may be obtained. For example, $H_c$ in -a $K_{n_1n_2n_3}$ graph may be written -\begin{equation}\label{j:mark} -\begin{split} -H_c=& -\frac{n_1!\,n_2!\,n_3!} -{n_1+n_2+n_3}\sum_i\left[\binom{n_1}{i} -\binom{n_2}{n_3-n_1+i}\binom{n_3}{n_3-n_2+i}\right.\\ -&+\left.\binom{n_1-1}{i} -\binom{n_2-1}{n_3-n_1+i} -\binom{n_3-1}{n_3-n_2+i}\right].\end{split} -\end{equation} - -\section{Secret Key Exchanges} -\label{SKE} - -Modern cryptography is fundamentally concerned with the problem of -secure private communication. A Secret Key Exchange is a protocol -where Alice and Bob, having no secret information in common to start, -are able to agree on a common secret key, conversing over a public -channel. The notion of a Secret Key Exchange protocol was first -introduced in the seminal paper of Diffie and Hellman -\cite{dihe:newdir}. \cite{dihe:newdir} presented a concrete -implementation of a Secret Key Exchange protocol, dependent on a -specific assumption (a variant on the discrete log), specially -tailored to yield Secret Key Exchange. Secret Key Exchange is of -course trivial if trapdoor permutations exist. However, there is no -known implementation based on a weaker general assumption. - -The concept of an informationally one-way function was introduced -in \cite{imlelu:oneway}. We give only an informal definition here: - -\begin{defn} A polynomial time -computable function $f = \{f_k\}$ is informationally -one-way if there is no probabilistic polynomial time algorithm which -(with probability of the form $1 - k^{-e}$ for some $e > 0$) -returns on input $y \in \{0,1\}^{k}$ a random element of $f^{-1}(y)$. -\end{defn} -In the non-uniform setting \cite{imlelu:oneway} show that these are not -weaker than one-way functions: -\begin{thm}[\cite{imlelu:oneway} (non-uniform)] -\label{th-info-ow-ow} -The existence of informationally one-way functions -implies the existence of one-way functions. -\end{thm} -We will stick to the convention introduced above of saying -``non-uniform'' before the theorem statement when the theorem -makes use of non-uniformity. It should be understood that -if nothing is said then the result holds for both the uniform and -the non-uniform models. - -It now follows from \thmref{th-info-ow-ow} that - -\begin{thm}[non-uniform]\label{th-weak-ske-owf} Weak SKE -implies the existence of a one-way function. -\end{thm} - -More recently, the polynomial-time, interior point algorithms for linear -programming have been extended to the case of convex quadratic programs -\cite{moad:quadpro,ye:intalg}, certain linear complementarity problems -\cite{komiyo:lincomp,miyoki:lincomp}, and the nonlinear complementarity -problem \cite{komiyo:unipfunc}. The connection between these algorithms -and the classical Newton method for nonlinear equations is well -explained in \cite{komiyo:lincomp}. - -\section{Review} -\label{computation} - -We begin our discussion with the following definition: - -\begin{defn} - -A function $H\colon \Re^n \to \Re^n$ is said to be -\emph{B-differentiable} at the point $z$ if (i)~$H$ is Lipschitz -continuous in a neighborhood of $z$, and (ii)~ there exists a positive -homogeneous function $BH(z)\colon \Re^n \to \Re^n$, called the -\emph{B-derivative} of $H$ at $z$, such that -\[ \lim_{v \to 0} \frac{H(z+v) - H(z) - BH(z)v}{\enVert{v}} = 0. \] -The function $H$ is \textit{B-differentiable in set $S$} if it is -B-differentiable at every point in $S$. The B-derivative $BH(z)$ is said -to be \textit{strong} if -\[ \lim_{(v,v') \to (0,0)} \frac{H(z+v) - H(z+v') - BH(z)(v - -v')}{\enVert{v - v'}} = 0. \] -\end{defn} - - -\begin{lem}\label{limbog} There exists a smooth function $\psi_0(z)$ -defined for $\abs{z}>1-2a$ satisfying the following properties\textup{:} -\begin{enumerate} -\renewcommand{\labelenumi}{(\roman{enumi})} -\item $\psi_0(z)$ is bounded above and below by positive constants -$c_1\leq \psi_0(z)\leq c_2$. -\item If $\abs{z}>1$, then $\psi_0(z)=1$. -\item For all $z$ in the domain of $\psi_0$, $\Delta_0\ln \psi_0\geq 0$. -\item If $1-2a<\abs{z}<1-a$, then $\Delta_0\ln \psi_0\geq -c_3>0$. -\end{enumerate} -\end{lem} - -\begin{proof} -We choose $\psi_0(z)$ to be a radial function depending only on $r=\abs{z}$. -Let $h(r)\geq 0$ be a suitable smooth function satisfying $h(r)\geq c_3$ -for $1-2a<\abs{z}<1-a$, and $h(r)=0$ for $\abs{z}>1-\tfrac a2$. The radial -Laplacian -\[\Delta_0\ln\psi_0(r)=\left(\frac {d^2}{dr^2}+\frac -1r\frac d{dr}\right)\ln\psi_0(r)\] -has smooth coefficients for $r>1-2a$. Therefore, we may -apply the existence and uniqueness theory for ordinary differential -equations. Simply let $\ln \psi_0(r)$ be the solution of the differential -equation -\[\left(\frac{d^2}{dr^2}+\frac 1r\frac d{dr}\right)\ln \psi_0(r)=h(r)\] -with initial conditions given by $\ln \psi_0(1)=0$ and -$\ln\psi_0'(1)=0$. - -Next, let $D_\nu$ be a finite collection of pairwise disjoint disks, -all of which are contained in the unit disk centered at the origin in -$C$. We assume that $D_\nu=\{z\mid \abs{z-z_\nu}<\delta\}$. Suppose that -$D_\nu(a)$ denotes the smaller concentric disk $D_\nu(a)=\{z\mid -\abs{z-z_\nu}\leq (1-2a)\delta\}$. We define a smooth weight function -$\Phi_0(z)$ for $z\in C-\bigcup_\nu D_\nu(a)$ by setting $\Phi_ -0(z)=1$ when $z\notin \bigcup_\nu D_\nu$ and $\Phi_ -0(z)=\psi_0((z-z_\nu)/\delta)$ when $z$ is an element of $D_\nu$. It -follows from \lemref{limbog} that $\Phi_ 0$ satisfies the properties: -\begin{enumerate} -\renewcommand{\labelenumi}{(\roman{enumi})} -\item \label{boundab}$\Phi_ 0(z)$ is bounded above and below by -positive constants $c_1\leq \Phi_ 0(z)\leq c_2$. -\item \label{d:over}$\Delta_0\ln\Phi_ 0\geq 0$ for all -$z\in C-\bigcup_\nu D_\nu(a)$, -the domain where the function $\Phi_ 0$ is defined. -\item \label{d:ad}$\Delta_0\ln\Phi_ 0\geq c_3\delta^{-2}$ -when $(1-2a)\delta<\abs{z-z_\nu}<(1-a)\delta$. -\end{enumerate} -Let $A_\nu$ denote the annulus $A_\nu=\{(1-2a)\delta<\abs{z-z_\nu}<(1-a) -\delta \}$, and set $A=\bigcup_\nu A_\nu$. The -properties (\ref{d:over}) and (\ref{d:ad}) of $\Phi_ 0$ -may be summarized as $\Delta_0\ln \Phi_ 0\geq c_3\delta^{-2}\chi_A$, -where $\chi _A$ is the characteristic function of $A$. -\end{proof} - -Suppose that $\alpha$ is a nonnegative real constant. We apply -Proposition~\ref{prop:eg} with $\Phi(z)=\Phi_ 0(z) e^{\alpha\abs{z}^2}$. If -$u\in C^\infty_0(R^2-\bigcup_\nu D_\nu(a))$, assume that $\mathcal{D}$ -is a bounded domain containing the support of $u$ and $A\subset -\mathcal{D}\subset R^2-\bigcup_\nu D_\nu(a)$. A calculation gives -\[\int_{\mathcal{D}}\abs{\overline\partial u}^2\Phi_ 0(z) e^{\alpha\abs{z}^2} -\geq c_4\alpha\int_{\mathcal{D}}\abs{u}^2\Phi_ 0e^{\alpha\abs{z}^2} -+c_5\delta^{-2}\int_ A\abs{u}^2\Phi_ 0e^{\alpha\abs{z}^2}.\] - -The boundedness, property (\ref{boundab}) of $\Phi_ 0$, then yields -\[\int_{\mathcal{D}}\abs{\overline\partial u}^2e^{\alpha\abs{z}^2}\geq c_6\alpha -\int_{\mathcal{D}}\abs{u}^2e^{\alpha\abs{z}^2} -+c_7\delta^{-2}\int_ A\abs{u}^2e^{\alpha\abs{z}^2}.\] - -Let $B(X)$ be the set of blocks of $\Lambda_{X}$ -and let $b(X) = \abs{B(X)}$. If $\phi \in Q_{X}$ then -$\phi$ is constant on the blocks of $\Lambda_{X}$. -\begin{equation}\label{far-d} - P_{X} = \{ \phi \in M \mid \Lambda_{\phi} = \Lambda_{X} \}, -\qquad -Q_{X} = \{\phi \in M \mid \Lambda_{\phi} \geq \Lambda_{X} \}. -\end{equation} -If $\Lambda_{\phi} \geq \Lambda_{X}$ then -$\Lambda_{\phi} = \Lambda_{Y}$ for some $Y \geq X$ so that -\[ Q_{X} = \bigcup_{Y \geq X} P_{Y}. \] -Thus by M\"obius inversion -\[ \abs{P_{Y}}= \sum_{X\geq Y} \mu (Y,X)\abs{Q_{X}}.\] -Thus there is a bijection from $Q_{X}$ to $W^{B(X)}$. -In particular $\abs{Q_{X}} = w^{b(X)}$. - -Next note that $b(X)=\dim X$. We see this by choosing a -basis for $X$ consisting of vectors $v^{k}$ defined by -\[v^{k}_{i}= -\begin{cases} 1 & \text{if $i \in \Lambda_{k}$},\\ -0 &\text{otherwise.} \end{cases} -\] -\begin{verbatim} -\[v^{k}_{i}= -\begin{cases} 1 & \text{if $i \in \Lambda_{k}$},\\ -0 &\text{otherwise.} \end{cases} -\] -\end{verbatim} - -\begin{lem}\label{p0201} -Let $\A$ be an arrangement. Then -\[ \chi (\A,t) = \sum_{\B \subseteq \A} -(-1)^{\abs{\B}} t^{\dim T(\B)}. \] -\end{lem} - -In order to compute $R''$ recall the definition -of $S(X,Y)$ from \lemref{lem-per}. Since $H \in \B$, -$\A_{H} \subseteq \B$. Thus if $T(\B) = Y$ then -$\B \in S(H,Y)$. Let $L'' = L(\A'')$. Then -\begin{equation}\label{E_SXgYy} -\begin{split} -R''&= \sum_{H\in \B \subseteq \A} (-1)^{\abs{\B}} -t^{\dim T(\B)}\\ -&= \sum_{Y \in L''} \sum_{\B \in S(H,Y)} -(-1)^{\abs{\B}}t^{\dim Y} \\ -&= -\sum_{Y \in L''} \sum_{\B \in S(H,Y)} (-1)^ -{\abs{\B - \A_{H}}} t^{\dim Y} \\ -&= -\sum_{Y \in L''} \mu (H,Y)t^{\dim Y} \\ -&= -\chi (\A '',t). -\end{split} -\end{equation} - -\begin{cor}\label{tripleA} -Let $(\A,\A',\A'')$ be a triple of arrangements. Then -\[ \pi (\A,t) = \pi (\A',t) + t \pi (\A'',t). \] -\end{cor} - -\begin{defn} -Let $(\A,\A',\A'')$ be a triple with respect to -the hyperplane $H \in \A$. Call $H$ a \textit{separator} -if $T(\A) \not\in L(\A')$. -\end{defn} - -\begin{cor}\label{nsep} -Let $(\A,\A',\A'')$ be a triple with respect to $H \in \A$. -\begin{enumerate} -\renewcommand{\labelenumi}{(\roman{enumi})} -\item -If $H$ is a separator then -\[ \mu (\A) = - \mu (\A'') \] -and hence -\[ \abs{\mu (\A)} = \abs{ \mu (\A'')}. \] - -\item If $H$ is not a separator then -\[\mu (\A) = \mu (\A') - \mu (\A'') \] -and -\[ \abs{\mu (\A)} = \abs{\mu (\A')} + \abs{\mu (\A'')}. \] -\end{enumerate} -\end{cor} - -\begin{proof} -It follows from \thmref{th-info-ow-ow} that $\pi(\A,t)$ -has leading term -\[(-1)^{r(\A)}\mu (\A)t^{r(\A)}.\] -The conclusion -follows by comparing coefficients of the leading -terms on both sides of the equation in -Corollary~\ref{tripleA}. If $H$ is a separator then -$r(\A') < r(\A)$ and there is no contribution -from $\pi (\A',t)$. -\end{proof} - -The Poincar\'e polynomial of an arrangement -will appear repeatedly -in these notes. It will be shown to equal the -Poincar\'e polynomial -of the graded algebras which we are going to -associate with $\A$. It is also the Poincar\'e -polynomial of the complement $M(\A)$ for a -complex arrangement. Here we prove -that the Poincar\'e polynomial is the chamber -counting function for a real arrangement. The -complement $M(\A)$ is a disjoint union of chambers -\[M(\A) = \bigcup_{C \in \Cham(\A)} C.\] -The number -of chambers is determined by the Poincar\'e -polynomial as follows. - -\begin{thm}\label{th-realarr} -Let $\A_{\mathbf{R}}$ be a real arrangement. Then -\[ \abs{\Cham(\A_{\mathbf{R}})} = \pi (\A_{\mathbf{R}},1). \] -\end{thm} - -\begin{proof} -We check the properties required in Corollary~\ref{nsep}: -(i) follows from $\pi (\Phi_{ l},t) = 1$, and (ii) is a -consequence of Corollary~\ref{BI}. -\end{proof} - -\begin{figure} -\vspace{5cm} -\caption[]{$Q(\A_{1}) = xyz(x-z)(x+z)(y-z)(y+z)$} -\end{figure} - -\begin{figure} -\vspace{5cm} -\caption[]{$Q(\A_{2})= xyz(x+y+z)(x+y-z)(x-y+z)(x-y-z)$} -\end{figure} - - -\begin{thm} -\label{T_first_the_int} -Let $\phi$ be a protocol for a random pair $\XcY$. -If one of $\st_\phi(x',y)$ and $\st_\phi(x,y')$ is a prefix of the other -and $(x,y)\in\SXY$, then -\[ -\langle \st_j(x',y)\rangle_{j=1}^\infty -=\langle \st_j(x,y)\rangle_{j=1}^\infty -=\langle \st_j(x,y')\rangle_{j=1}^\infty . -\] -\end{thm} -\begin{proof} -We show by induction on $i$ that -\[ -\langle \st_j(x',y)\rangle_{j=1}^i -=\langle \st_j(x,y)\rangle_{j=1}^i -=\langle \st_j(x,y')\rangle_{j=1}^i. -\] -The induction hypothesis holds vacuously for $i=0$. Assume it holds for -$i-1$, in particular -$[\st_j(x',y)]_{j=1}^{i-1}=[\st_j(x,y')]_{j=1}^{i-1}$. Then one of -$[\st_j(x',y)]_{j=i}^{\infty}$ and $[\st_j(x,y')]_{j=i}^{\infty}$ is a -prefix of the other which implies that one of $\st_i(x',y)$ and -$\st_i(x,y')$ is a prefix of the other. If the $i$th message is -transmitted by $P_\X$ then, by the separate-transmissions property and -the induction hypothesis, $\st_i(x,y)=\st_i(x,y')$, hence one of -$\st_i(x,y)$ and $\st_i(x',y)$ is a prefix of the other. By the -implicit-termination property, neither $\st_i(x,y)$ nor $\st_i(x',y)$ -can be a proper prefix of the other, hence they must be the same and -$\st_i(x',y)=\st_i(x,y)=\st_i(x,y')$. If the $i$th message is -transmitted by $\PY$ then, symmetrically, $\st_i(x,y)=\st_i(x',y)$ by -the induction hypothesis and the separate-transmissions property, and, -then, $\st_i(x,y)=\st_i(x,y')$ by the implicit-termination property, -proving the induction step. -\end{proof} - -If $\phi$ is a protocol for $(X,Y)$, and $(x,y)$, $(x',y)$ are distinct -inputs in $\SXY$, then, by the correct-decision property, -$\langle\st_j(x,y)\rangle_{j=1}^\infty\ne\langle -\st_j(x',y)\rangle_{j=1}^\infty$. - -Equation~(\ref{E_SXgYy}) defined $\PY$'s ambiguity set $\SXgYy$ -to be the set of possible $X$ values when $Y=y$. -The last corollary implies that for all $y\in\SY$, -the multiset% -\footnote{A multiset allows multiplicity of elements. -Hence, $\{0,01,01\}$ is prefix free as a set, but not as a multiset.} -of codewords $\{\st_\phi(x,y):x\in\SXgYy\}$ is prefix free. - -\section{One-Way Complexity} -\label{S_Cp1} - -$\Cw1$, the one-way complexity of a random pair $\XcY$, -is the number of bits $P_\X$ must transmit in the worst case -when $\PY$ is not permitted to transmit any feedback messages. -Starting with $\SXY$, the support set of $\XcY$, we define $\G$, -the \textit{characteristic hypergraph} of $\XcY$, and show that -\[ -\Cw1=\lceil\,\log\chi(\G)\rceil\ . -\] - -Let $\XcY$ be a random pair. For each $y$ in $\SY$, the support set of -$Y$, Equation~(\ref{E_SXgYy}) defined $\SXgYy$ to be the set of possible -$x$ values when $Y=y$. The \textit{characteristic hypergraph} $\G$ of -$\XcY$ has $\SX$ as its vertex set and the hyperedge $\SXgYy$ for each -$y\in\SY$. - - -We can now prove a continuity theorem. -\begin{thm}\label{t:conl} -Let $\Omega \subset\mathbf{R}^n$ be an open set, let -$u\in BV(\Omega ;\mathbf{R}^m)$, and let -\begin{equation}\label{quts} -T^u_x=\left\{y\in\mathbf{R}^m: - y=\tilde u(x)+\left\langle \frac{Du}{\abs{Du}}(x),z -\right\rangle \text{ for some }z\in\mathbf{R}^n\right\} -\end{equation} -for every $x\in\Omega \backslash S_u$. Let $f\colon \mathbf{R}^m\to -\mathbf{R}^k$ be a Lipschitz continuous function such that $f(0)=0$, and -let $v=f(u)\colon \Omega \to \mathbf{R}^k$. Then $v\in BV(\Omega -;\mathbf{R}^k)$ and -\begin{equation} -Jv=\eval{(f(u^+)-f(u^-))\otimes \nu_u\cdot\, -\mathcal{H}_{n-1}}_{S_u}. -\end{equation} -In addition, for $\abs{\wt{D}u}$-almost every $x\in\Omega $ the -restriction of the function $f$ to $T^u_x$ is differentiable at $\tilde -u(x)$ and -\begin{equation} -\wt{D}v=\nabla (\eval{f}_{T^u_x})(\tilde u) -\frac{\wt{D}u}{\abs{\wt{D}u}}\cdot\abs{\wt{D}u}.\end{equation} -\end{thm} - -Before proving the theorem, we state without proof three elementary -remarks which will be useful in the sequel. -\begin{rem}\label{r:omb} -Let $\omega\colon \left]0,+\infty\right[\to \left]0,+\infty\right[$ -be a continuous function such that $\omega (t)\to 0$ as $t\to -0$. Then -\[\lim_{h\to 0^+}g(\omega(h))=L\Leftrightarrow\lim_{h\to -0^+}g(h)=L\] -for any function $g\colon \left]0,+\infty\right[\to \mathbf{R}$. -\end{rem} -\begin{rem}\label{r:dif} -Let $g \colon \mathbf{R}^n\to \mathbf{R}$ be a Lipschitz -continuous function and assume that -\[L(z)=\lim_{h\to 0^+}\frac{g(hz)-g(0)}h\] -exists for every $z\in\mathbf{Q}^n$ and that $L$ is a linear function of -$z$. Then $g$ is differentiable at 0. -\end{rem} -\begin{rem}\label{r:dif0} -Let $A \colon \mathbf{R}^n\to \mathbf{R}^m$ be a linear function, and -let $f \colon \mathbf{R}^m\to \mathbf{R}$ be a function. Then the -restriction of $f$ to the range of $A$ is differentiable at 0 if and -only if $f(A)\colon \mathbf{R}^n\to \mathbf{R}$ is differentiable at 0 -and -\[\nabla(\eval{f}_{\IM(A)})(0)A=\nabla (f(A))(0).\] -\end{rem} - -\begin{proof} - We begin by showing that $v\in BV(\Omega;\mathbf{R}^k)$ and -\begin{equation}\label{e:bomb} -\abs{Dv}(B)\le K\abs{Du}(B)\qquad\forall B\in\mathbf{B}(\Omega ), -\end{equation} -where $K>0$ is the Lipschitz constant of $f$. By \eqref{sum-Di} and by -the approximation result quoted in \secref{s:mt}, it is possible to find -a sequence $(u_h)\subset C^1(\Omega ;\mathbf{R}^m)$ converging to $u$ in -$L^1(\Omega ;\mathbf{R}^m)$ and such that -\[\lim_{h\to +\infty}\int_\Omega \abs{\nabla u_h}\,dx=\abs{Du}(\Omega ).\] -The functions $v_h=f(u_h)$ are locally Lipschitz continuous in $\Omega -$, and the definition of differential implies that $\abs{\nabla v_h}\le -K\abs{\nabla u_h}$ almost everywhere in $\Omega $. The lower semicontinuity -of the total variation and \eqref{sum-Di} yield -\begin{equation} -\begin{split} -\abs{Dv}(\Omega )\le\liminf_{h\to +\infty}\abs{Dv_h}(\Omega) & -=\liminf_{h\to +\infty}\int_\Omega \abs{\nabla v_h}\,dx\\ -&\le K\liminf_{h\to +\infty}\int_\Omega -\abs{\nabla u_h}\,dx=K\abs{Du}(\Omega). -\end{split}\end{equation} -Since $f(0)=0$, we have also -\[\int_\Omega \abs{v}\,dx\le K\int_\Omega \abs{u}\,dx;\] -therefore $u\in BV(\Omega ;\mathbf{R}^k)$. Repeating the same argument -for every open set $A\subset\Omega $, we get \eqref{e:bomb} for every -$B\in\mathbf{B}(\Omega)$, because $\abs{Dv}$, $\abs{Du}$ are Radon measures. To -prove \lemref{limbog}, first we observe that -\begin{equation}\label{e:SS} -S_v\subset S_u,\qquad\tilde v(x)=f(\tilde u(x))\qquad \forall x\in\Omega -\backslash S_u.\end{equation} -In fact, for every $\varepsilon >0$ we have -\[\{y\in B_\rho(x): \abs{v(y)-f(\tilde u(x))}>\varepsilon \}\subset \{y\in -B_\rho(x): \abs{u(y)-\tilde u(x)}>\varepsilon /K\},\] -hence -\[\lim_{\rho\to 0^+}\frac{\abs{\{y\in B_\rho(x): \abs{v(y)-f(\tilde u(x))}> -\varepsilon \}}}{\rho^n}=0\] -whenever $x\in\Omega \backslash S_u$. By a similar argument, if $x\in -S_u$ is a point such that there exists a triplet $(u^+,u^-,\nu_u)$ -satisfying \eqref{detK1}, \eqref{detK2}, then -\[ -(v^+(x)-v^-(x))\otimes \nu_v=(f(u^+(x))-f(u^-(x)))\otimes\nu_u\quad -\text{if }x\in S_v -\] -and $f(u^-(x))=f(u^+(x))$ if $x\in S_u\backslash S_v$. Hence, by (1.8) -we get -\begin{equation*}\begin{split} -Jv(B)=\int_{B\cap S_v}(v^+-v^-)\otimes \nu_v\,d\mathcal{H}_{n-1}&= -\int_{B\cap S_v}(f(u^+)-f(u^-))\otimes \nu_u\,d\mathcal{H}_{n-1}\\ -&=\int_{B\cap S_u}(f(u^+)-f(u^-))\otimes \nu_u\,d\mathcal{H}_{n-1} -\end{split}\end{equation*} -and \lemref{limbog} is proved. -\end{proof} - -To prove \eqref{e:SS}, it is not restrictive to assume that $k=1$. -Moreover, to simplify our notation, from now on we shall assume that -$\Omega = \mathbf{R}^n$. The proof of \eqref{e:SS} is divided into two -steps. In the first step we prove the statement in the one-dimensional -case $(n=1)$, using \thmref{th-weak-ske-owf}. In the second step we -achieve the general result using \thmref{t:conl}. - -\subsection*{Step 1} -Assume that $n=1$. Since $S_u$ is at most countable, \eqref{sum-bij} -yields that $\abs{\wt{D}v}(S_u\backslash S_v)=0$, so that -\eqref{e:st} and \eqref{e:barwq} imply that $Dv=\wt{D}v+Jv$ is -the Radon-Nikod\'ym decomposition of $Dv$ in absolutely continuous and -singular part with respect to $\abs{\wt{D} u}$. By -\thmref{th-weak-ske-owf}, we have -\begin{equation*} -\frac{\wt{D}v}{\abs{\wt{D}u}}(t)=\lim_{s\to t^+} -\frac{Dv(\interval{\left[t,s\right[})} -{\abs{\wt{D}u}(\interval{\left[t,s\right[})},\qquad -\frac{\wt{D}u}{\abs{\wt{D}u}}(t)=\lim_{s\to t^+} -\frac{Du(\interval{\left[t,s\right[})} -{\abs{\wt{D}u}(\interval{\left[t,s\right[})} -\end{equation*} -$\abs{\wt{D}u}$-almost everywhere in $\mathbf{R}$. It is well known -(see, for instance, \cite[2.5.16]{ste:sint}) that every one-dimensional -function of bounded variation $w$ has a unique left continuous -representative, i.e., a function $\hat w$ such that $\hat w=w$ almost -everywhere and $\lim_{s\to t^-}\hat w(s)=\hat w(t)$ for every $t\in -\mathbf{R}$. These conditions imply -\begin{equation} -\hat u(t)=Du(\interval{\left]-\infty,t\right[}), -\qquad \hat v(t)=Dv(\interval{\left]-\infty,t\right[})\qquad -\forall t\in\mathbf{R} -\end{equation} -and -\begin{equation}\label{alimo} -\hat v(t)=f(\hat u(t))\qquad\forall t\in\mathbf{R}.\end{equation} -Let $t\in\mathbf{R}$ be such that -$\abs{\wt{D}u}(\interval{\left[t,s\right[})>0$ for every $s>t$ and -assume that the limits in \eqref{joe} exist. By \eqref{j:mark} and -\eqref{far-d} we get -\begin{equation*}\begin{split} -\frac{\hat v(s)-\hat -v(t)}{\abs{\wt{D}u}(\interval{\left[t,s\right[})}&=\frac {f(\hat -u(s))-f(\hat u(t))}{\abs{\wt{D}u}(\interval{\left[t,s\right[})}\\ -&=\frac{f(\hat u(s))-f(\hat -u(t)+\dfrac{\wt{D}u}{\abs{\wt{D}u}}(t)\abs{\wt{D}u -}(\interval{\left[t,s\right[}))}% -{\abs{\wt{D}u}(\interval{\left[t,s\right[})}\\ -&+\frac -{f(\hat u(t)+\dfrac{\wt{D}u}{\abs{\wt{D}u}}(t)\abs{\wt{D} -u}(\interval{\left[t,s\right[}))-f(\hat -u(t))}{\abs{\wt{D}u}(\interval{\left[t,s\right[})} -\end{split}\end{equation*} -for every $s>t$. Using the Lipschitz condition on $f$ we find -{\setlength{\multlinegap}{0pt} -\begin{multline*} -\left\lvert\frac{\hat v(s)-\hat -v(t)}{\abs{\wt{D}u}(\interval{\left[t,s\right[})} -\frac{f(\hat -u(t)+\dfrac{\wt{D}u}{\abs{\wt{D}u}}(t) -\abs{\wt{D}u}(\interval{\left[t,s\right[}))-f(\hat -u(t))}{\abs{\wt{D}u}(\interval{\left[t,s\right[})}\right\rvert\\ -\le K\left\lvert -\frac{\hat u(s)-\hat u(t)} - {\abs{\wt{D}u}(\interval{\left[t,s\right[})} --\frac{\wt{D}u}{\abs{ -\wt{D}u}}(t)\right\rvert.\end{multline*} -}% end of group with \multlinegap=0pt -By \eqref{e:bomb}, the function $s\to -\abs{\wt{D}u}(\interval{\left[t,s\right[})$ is continuous and -converges to 0 as $s\downarrow t$. Therefore Remark~\ref{r:omb} and the -previous inequality imply -\[\frac{\wt{D}v}{\abs{\wt{D}u}}(t)=\lim_{h\to 0^+} -\frac{f(\hat u(t)+h\dfrac{\wt{D}u}{\abs{\wt{D}u}} -(t))-f(\hat u(t))}h\quad\abs{\wt{D}u}\text{-a.e. in }\mathbf{R}.\] -By \eqref{joe}, $\hat u(x)=\tilde u(x)$ for every -$x\in\mathbf{R}\backslash S_u$; moreover, applying the same argument to -the functions $u'(t)=u(-t)$, $v'(t)=f(u'(t))=v(-t)$, we get -\[\frac{\wt{D}v}{\abs{\wt{D}u}}(t)=\lim_{h\to 0} -\frac{f(\tilde u(t) -+h\dfrac{\wt{D}u}{\abs{\wt{D}u}}(t))-f(\tilde u(t))}{h} -\qquad\abs{\wt{D}u}\text{-a.e. in }\mathbf{R}\] -and our statement is proved. - -\subsection*{Step 2} - -Let us consider now the general case $n>1$. Let $\nu\in \mathbf{R}^n$ be -such that $\abs{\nu}=1$, and let $\pi_\nu=\{y\in\mathbf{R}^n: \langle -y,\nu\rangle =0\}$. In the following, we shall identify $\mathbf{R}^n$ -with $\pi_\nu\times\mathbf{R}$, and we shall denote by $y$ the variable -ranging in $\pi_\nu$ and by $t$ the variable ranging in $\mathbf{R}$. By -the just proven one-dimensional result, and by \thmref{thm-main}, we get -\[\lim_{h\to 0}\frac{f(\tilde u(y+t\nu)+h\dfrac{\wt{D}u_y}{\abs{ -\wt{D}u_y}}(t))-f(\tilde u(y+t\nu))}h=\frac{\wt{D}v_y}{\abs{ -\wt{D}u_y}}(t)\qquad\abs{\wt{D}u_y}\text{-a.e. in }\mathbf{R}\] -for $\mathcal{H}_{n-1}$-almost every $y\in \pi_\nu$. We claim that -\begin{equation} -\frac{\langle \wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle -}}(y+t\nu)=\frac{\wt{D}u_y} -{\abs{\wt{D}u_y}}(t)\qquad\abs{\wt{D}u_y}\text{-a.e. in }\mathbf{R} -\end{equation} -for $\mathcal{H}_{n-1}$-almost every $y\in\pi_\nu$. In fact, by -\eqref{sum-ali} and \eqref{delta-l} we get -\begin{multline*} -\int_{\pi_\nu}\frac{\wt{D}u_y}{\abs{\wt{D}u_y}}\cdot\abs{\wt{D}u_y -}\,d\mathcal{H}_{n-1}(y)=\int_{\pi_\nu}\wt{D}u_y\,d\mathcal{H}_{n-1}(y)\\ -=\langle \wt{D}u,\nu\rangle =\frac -{\langle \wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle}}\cdot -\abs{\langle \wt{D}u,\nu\rangle }=\int_{\pi_\nu}\frac{ -\langle \wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }} -(y+\cdot \nu)\cdot\abs{\wt{D}u_y}\,d\mathcal{H}_{n-1}(y) -\end{multline*} -and \eqref{far-d} follows from \eqref{sum-Di}. By the same argument it -is possible to prove that -\begin{equation} -\frac{\langle \wt{D}v,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle -}}(y+t\nu)=\frac{\wt{D}v_y}{\abs{\wt{D}u_y}}(t)\qquad\abs{ -\wt{D}u_y}\text{-a.e. in }\mathbf{R}\end{equation} -for $\mathcal{H}_{n-1}$-almost every $y\in \pi_\nu$. By \eqref{far-d} -and \eqref{E_SXgYy} we get -\[ -\lim_{h\to 0}\frac{f(\tilde u(y+t\nu)+h\dfrac{\langle \wt{D} -u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }}(y+t\nu))-f(\tilde -u(y+t\nu))}{h} -=\frac{\langle \wt{D}v,\nu\rangle }{\abs{\langle -\wt{D}u,\nu\rangle }}(y+t\nu)\] -for $\mathcal{H}_{n-1}$-almost every $y\in\pi_\nu$, and using again -\eqref{detK1}, \eqref{detK2} we get -\[ -\lim_{h\to 0}\frac{f(\tilde u(x)+h\dfrac{\langle -\wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }}(x))-f(\tilde -u(x))}{h}=\frac{\langle \wt{D}v,\nu\rangle }{\abs{\langle \wt{D}u,\nu -\rangle }}(x) -\] -$\abs{\langle \wt{D}u,\nu\rangle}$-a.e. in $\mathbf{R}^n$. - -Since the function $\abs{\langle \wt{D}u,\nu\rangle }/\abs{\wt{D}u}$ -is strictly positive $\abs{\langle \wt{D}u,\nu\rangle }$-almost everywhere, -we obtain also -\begin{multline*} -\lim_{h\to 0}\frac{f(\tilde u(x)+h\dfrac{\abs{\langle -\wt{D}u,\nu\rangle }}{\abs{\wt{D}u}}(x)\dfrac{\langle \wt{D} -u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle }}(x))-f(\tilde u(x))}{h}\\ -=\frac{\abs{\langle \wt{D}u,\nu\rangle }}{\abs{\wt{D}u}}(x)\frac -{\langle \wt{D}v,\nu\rangle }{\abs{\langle -\wt{D}u,\nu\rangle }}(x) -\end{multline*} -$\abs{\langle \wt{D}u,\nu\rangle }$-almost everywhere in $\mathbf{R}^n$. - -Finally, since -\begin{align*} -&\frac{\abs{\langle \wt{D}u,\nu\rangle }}{\abs{\wt{D}u}} -\frac{\langle \wt{D}u,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle}} -=\frac{\langle \wt{D}u,\nu\rangle }{\abs{\wt{D}u}} -=\left\langle \frac{\wt{D}u}{\abs{\wt{D}u}},\nu\right\rangle - \qquad\abs{\wt{D}u}\text{-a.e. in }\mathbf{R}^n\\ -&\frac{\abs{\langle \wt{D}u,\nu\rangle }}{\abs{\wt{D}u}} -\frac{\langle \wt{D}v,\nu\rangle }{\abs{\langle \wt{D}u,\nu\rangle}} -=\frac{\langle \wt{D}v,\nu\rangle }{\abs{\wt{D}u}} -=\left\langle \frac{\wt{D}v}{\abs{\wt{D}u}},\nu\right\rangle - \qquad\abs{\wt{D}u}\text{-a.e. in }\mathbf{R}^n -\end{align*} -and since both sides of \eqref{alimo} -are zero $\abs{\wt{D}u}$-almost everywhere -on $\abs{\langle \wt{D}u,\nu\rangle }$-negligible sets, we conclude that -\[ -\lim_{h\to 0}\frac{f\left( -\tilde u(x)+h\left\langle \dfrac{\wt{D} -u}{\abs{\wt{D}u}}(x),\nu\right\rangle \right)-f(\tilde u(x))}h -=\left\langle \frac{\wt{D}v}{\abs{\wt{D}u}}(x),\nu\right\rangle, -\] -$\abs{\wt{D}u}$-a.e. in $\mathbf{R}^n$. -Since $\nu$ is arbitrary, by Remarks \ref{r:dif} and~\ref{r:dif0} -the restriction of $f$ to -the affine space $T^u_x$ is differentiable at $\tilde u(x)$ for $\abs{\wt{D} -u}$-almost every $x\in \mathbf{R}^n$ and \eqref{quts} holds.\qed - -It follows from \eqref{sum-Di}, \eqref{detK1}, and \eqref{detK2} that -\begin{equation}\label{Dt} -D(t_1,\dots,t_n)=\sum_{I\in\mathbf{n}}(-1)^{\abs{I}-1}\abs{I} -\prod_{i\in I}t_i\prod_{j\in I}(D_j+\lambda_jt_j)\det\mathbf{A}^{(\lambda)} -(\overline I|\overline I). -\end{equation} -Let $t_i=\hat x_i$, $i=1,\dots,n$. Lemma 1 leads to -\begin{equation}\label{Dx} -D(\hat x_1,\dots,\hat x_n)=\prod_{i\in\mathbf{n}}\hat x_i -\sum_{I\in\mathbf{n}}(-1)^{\abs{I}-1}\abs{I}\per \mathbf{A} -^{(\lambda)}(I|I)\det\mathbf{A}^{(\lambda)}(\overline I|\overline I). -\end{equation} -By \eqref{H-cycles}, \eqref{sum-Di}, and \eqref{Dx}, -we have the following result: -\begin{thm}\label{thm-H-param} -\begin{equation}\label{H-param} -H_c=\frac{1}{2n}\sum^n_{l =1}l (-1)^{l -1}A_{l} -^{(\lambda)}, -\end{equation} -where -\begin{equation}\label{A-l-lambda} -A^{(\lambda)}_l =\sum_{I_l \subseteq\mathbf{n}}\per \mathbf{A} -^{(\lambda)}(I_l |I_l )\det\mathbf{A}^{(\lambda)} -(\overline I_{l}|\overline I_l ),\abs{I_{l}}=l . -\end{equation} -\end{thm} - -It is worth noting that $A_l ^{(\lambda)}$ of \eqref{A-l-lambda} is -similar to the coefficients $b_l $ of the characteristic polynomial of -\eqref{bl-sum}. It is well known in graph theory that the coefficients -$b_l $ can be expressed as a sum over certain subgraphs. It is -interesting to see whether $A_l $, $\lambda=0$, structural properties -of a graph. - -We may call \eqref{H-param} a parametric representation of $H_c$. In -computation, the parameter $\lambda_i$ plays very important roles. The -choice of the parameter usually depends on the properties of the given -graph. For a complete graph $K_n$, let $\lambda_i=1$, $i=1,\dots,n$. -It follows from \eqref{A-l-lambda} that -\begin{equation}\label{compl-gr} -A^{(1)}_l =\begin{cases} n!,&\text{if }l =1\\ -0,&\text{otherwise}.\end{cases} -\end{equation} -By \eqref{H-param} -\begin{equation} -H_c=\frac 12(n-1)!. -\end{equation} -For a complete bipartite graph $K_{n_1n_2}$, let $\lambda_i=0$, $i=1,\dots,n$. -By \eqref{A-l-lambda}, -\begin{equation} -A_l = -\begin{cases} -n_1!n_2!\delta_{n_1n_2},&\text{if }l =2\\ -0,&\text{otherwise }.\end{cases} -\label{compl-bip-gr} -\end{equation} -Theorem ~\ref{thm-H-param} -leads to -\begin{equation} -H_c=\frac1{n_1+n_2}n_1!n_2!\delta_{n_1n_2}. -\end{equation} - -Now, we consider an asymmetrical approach. Theorem \ref{thm-main} leads to -\begin{multline} -\det\mathbf{K}(t=1,t_1,\dots,t_n;l |l )\\ -=\sum_{I\subseteq\mathbf{n}-\{l \}} -(-1)^{\abs{I}}\prod_{i\in I}t_i\prod_{j\in I} -(D_j+\lambda_jt_j)\det\mathbf{A}^{(\lambda)} -(\overline I\cup\{l \}|\overline I\cup\{l \}). -\end{multline} - -By \eqref{H-cycles} and \eqref{sum-ali} we have the following asymmetrical -result: -\begin{thm}\label{thm-asym} -\begin{equation} -H_c=\frac12\sum_{I\subseteq\mathbf{n}-\{l \}} -(-1)^{\abs{I}}\per\mathbf{A}^{(\lambda)}(I|I)\det -\mathbf{A}^{(\lambda)} -(\overline I\cup\{l \}|\overline I\cup\{l \}) -\end{equation} -which reduces to Goulden--Jackson's formula when $\lambda_i=0,i=1,\dots,n$ -\cite{mami:matrixth}. -\end{thm} - -\section{Various font features of the \pkg{amsmath} package} -\label{s:font} -\subsection{Bold versions of special symbols} - -In the \pkg{amsmath} package \cn{boldsymbol} is used for getting -individual bold math symbols and bold Greek letters---everything in -math except for letters of the Latin alphabet, -where you'd use \cn{mathbf}. For example, -\begin{verbatim} -A_\infty + \pi A_0 \sim -\mathbf{A}_{\boldsymbol{\infty}} \boldsymbol{+} -\boldsymbol{\pi} \mathbf{A}_{\boldsymbol{0}} -\end{verbatim} -looks like this: -\[A_\infty + \pi A_0 \sim \mathbf{A}_{\boldsymbol{\infty}} -\boldsymbol{+} \boldsymbol{\pi} \mathbf{A}_{\boldsymbol{0}}\] - -\subsection{``Poor man's bold''} -If a bold version of a particular symbol doesn't exist in the -available fonts, -then \cn{boldsymbol} can't be used to make that symbol bold. -At the present time, this means that -\cn{boldsymbol} can't be used with symbols from -the \fn{msam} and \fn{msbm} fonts, among others. -In some cases, poor man's bold (\cn{pmb}) can be used instead -of \cn{boldsymbol}: -% Can't show example from msam or msbm because this document is -% supposed to be TeXable even if the user doesn't have -% AMSFonts. MJD 5-JUL-1990 -\[\frac{\partial x}{\partial y} -\pmb{\bigg\vert} -\frac{\partial y}{\partial z}\] -\begin{verbatim} -\[\frac{\partial x}{\partial y} -\pmb{\bigg\vert} -\frac{\partial y}{\partial z}\] -\end{verbatim} -So-called ``large operator'' symbols such as $\sum$ and $\prod$ -require an additional command, \cn{mathop}, -to produce proper spacing and limits when \cn{pmb} is used. -For further details see \textit{The \TeX book}. -\[\sum_{\substack{i<B\\\text{$i$ odd}}} -\prod_\kappa \kappa F(r_i)\qquad -\mathop{\pmb{\sum}}_{\substack{i<B\\\text{$i$ odd}}} -\mathop{\pmb{\prod}}_\kappa \kappa(r_i) -\] -\begin{verbatim} -\[\sum_{\substack{i<B\\\text{$i$ odd}}} -\prod_\kappa \kappa F(r_i)\qquad -\mathop{\pmb{\sum}}_{\substack{i<B\\\text{$i$ odd}}} -\mathop{\pmb{\prod}}_\kappa \kappa(r_i) -\] -\end{verbatim} - -\section{Compound symbols and other features} -\label{s:comp} -\subsection{Multiple integral signs} - -\cn{iint}, \cn{iiint}, and \cn{iiiint} give multiple integral signs -with the spacing between them nicely adjusted, in both text and -display style. \cn{idotsint} gives two integral signs with dots -between them. -\begin{gather} -\iint\limits_A f(x,y)\,dx\,dy\qquad\iiint\limits_A -f(x,y,z)\,dx\,dy\,dz\\ -\iiiint\limits_A -f(w,x,y,z)\,dw\,dx\,dy\,dz\qquad\idotsint\limits_A f(x_1,\dots,x_k) -\end{gather} - -\subsection{Over and under arrows} - -Some extra over and under arrow operations are provided in -the \pkg{amsmath} package. (Basic \LaTeX\ provides -\cn{overrightarrow} and \cn{overleftarrow}). -\begin{align*} -\overrightarrow{\psi_\delta(t) E_t h}& -=\underrightarrow{\psi_\delta(t) E_t h}\\ -\overleftarrow{\psi_\delta(t) E_t h}& -=\underleftarrow{\psi_\delta(t) E_t h}\\ -\overleftrightarrow{\psi_\delta(t) E_t h}& -=\underleftrightarrow{\psi_\delta(t) E_t h} -\end{align*} -\begin{verbatim} -\begin{align*} -\overrightarrow{\psi_\delta(t) E_t h}& -=\underrightarrow{\psi_\delta(t) E_t h}\\ -\overleftarrow{\psi_\delta(t) E_t h}& -=\underleftarrow{\psi_\delta(t) E_t h}\\ -\overleftrightarrow{\psi_\delta(t) E_t h}& -=\underleftrightarrow{\psi_\delta(t) E_t h} -\end{align*} -\end{verbatim} -These all scale properly in subscript sizes: -\[\int_{\overrightarrow{AB}} ax\,dx\] -\begin{verbatim} -\[\int_{\overrightarrow{AB}} ax\,dx\] -\end{verbatim} - -\subsection{Dots} - -Normally you need only type \cn{dots} for ellipsis dots in a -math formula. The main exception is when the dots -fall at the end of the formula; then you need to -specify one of \cn{dotsc} (series dots, after a comma), -\cn{dotsb} (binary dots, for binary relations or operators), -\cn{dotsm} (multiplication dots), or \cn{dotsi} (dots after -an integral). For example, the input -\begin{verbatim} -Then we have the series $A_1,A_2,\dotsc$, -the regional sum $A_1+A_2+\dotsb$, -the orthogonal product $A_1A_2\dotsm$, -and the infinite integral -\[\int_{A_1}\int_{A_2}\dotsi\]. -\end{verbatim} -produces -\begin{quotation} -Then we have the series $A_1,A_2,\dotsc$, -the regional sum $A_1+A_2+\dotsb$, -the orthogonal product $A_1A_2\dotsm$, -and the infinite integral -\[\int_{A_1}\int_{A_2}\dotsi\] -\end{quotation} - -\subsection{Accents in math} - -Double accents: -\[\Hat{\Hat{H}}\quad\Check{\Check{C}}\quad -\Tilde{\Tilde{T}}\quad\Acute{\Acute{A}}\quad -\Grave{\Grave{G}}\quad\Dot{\Dot{D}}\quad -\Ddot{\Ddot{D}}\quad\Breve{\Breve{B}}\quad -\Bar{\Bar{B}}\quad\Vec{\Vec{V}}\] -\begin{verbatim} -\[\Hat{\Hat{H}}\quad\Check{\Check{C}}\quad -\Tilde{\Tilde{T}}\quad\Acute{\Acute{A}}\quad -\Grave{\Grave{G}}\quad\Dot{\Dot{D}}\quad -\Ddot{\Ddot{D}}\quad\Breve{\Breve{B}}\quad -\Bar{\Bar{B}}\quad\Vec{\Vec{V}}\] -\end{verbatim} -This double accent operation is complicated -and tends to slow down the processing of a \LaTeX\ file. - - -\subsection{Dot accents} -\cn{dddot} and \cn{ddddot} are available to -produce triple and quadruple dot accents -in addition to the \cn{dot} and \cn{ddot} accents already available -in \LaTeX: -\[\dddot{Q}\qquad\ddddot{R}\] -\begin{verbatim} -\[\dddot{Q}\qquad\ddddot{R}\] -\end{verbatim} - -\subsection{Roots} - -In the \pkg{amsmath} package \cn{leftroot} and \cn{uproot} allow you to adjust -the position of the root index of a radical: -\begin{verbatim} -\sqrt[\leftroot{-2}\uproot{2}\beta]{k} -\end{verbatim} -gives good positioning of the $\beta$: -\[\sqrt[\leftroot{-2}\uproot{2}\beta]{k}\] - -\subsection{Boxed formulas} The command \cn{boxed} puts a box around its -argument, like \cn{fbox} except that the contents are in math mode: -\begin{verbatim} -\boxed{W_t-F\subseteq V(P_i)\subseteq W_t} -\end{verbatim} -\[\boxed{W_t-F\subseteq V(P_i)\subseteq W_t}.\] - -\subsection{Extensible arrows} -\cn{xleftarrow} and \cn{xrightarrow} produce -arrows that extend automatically to accommodate unusually wide -subscripts or superscripts. The text of the subscript or superscript -are given as an optional resp.\@ mandatory argument: -Example: -\[0 \xleftarrow[\zeta]{\alpha} F\times\triangle[n-1] - \xrightarrow{\partial_0\alpha(b)} E^{\partial_0b}\] -\begin{verbatim} -\[0 \xleftarrow[\zeta]{\alpha} F\times\triangle[n-1] - \xrightarrow{\partial_0\alpha(b)} E^{\partial_0b}\] -\end{verbatim} - -\subsection{\cn{overset}, \cn{underset}, and \cn{sideset}} -Examples: -\[\overset{*}{X}\qquad\underset{*}{X}\qquad -\overset{a}{\underset{b}{X}}\] -\begin{verbatim} -\[\overset{*}{X}\qquad\underset{*}{X}\qquad -\overset{a}{\underset{b}{X}}\] -\end{verbatim} - -The command \cn{sideset} is for a rather special -purpose: putting symbols at the subscript and superscript -corners of a large operator symbol such as $\sum$ or $\prod$, -without affecting the placement of limits. -Examples: -\[\sideset{_*^*}{_*^*}\prod_k\qquad -\sideset{}{'}\sum_{0\le i\le m} E_i\beta x -\] -\begin{verbatim} -\[\sideset{_*^*}{_*^*}\prod_k\qquad -\sideset{}{'}\sum_{0\le i\le m} E_i\beta x -\] -\end{verbatim} - -\subsection{The \cn{text} command} -The main use of the command \cn{text} is for words or phrases in a -display: -\[\mathbf{y}=\mathbf{y}'\quad\text{if and only if}\quad -y'_k=\delta_k y_{\tau(k)}\] -\begin{verbatim} -\[\mathbf{y}=\mathbf{y}'\quad\text{if and only if}\quad -y'_k=\delta_k y_{\tau(k)}\] -\end{verbatim} - -\subsection{Operator names} -The more common math functions such as $\log$, $\sin$, and $\lim$ -have predefined control sequences: \verb=\log=, \verb=\sin=, -\verb=\lim=. -The \pkg{amsmath} package provides \cn{DeclareMathOperator} and -\cn{DeclareMathOperator*} -for producing new function names that will have the -same typographical treatment. -Examples: -\[\norm{f}_\infty= -\esssup_{x\in R^n}\abs{f(x)}\] -\begin{verbatim} -\[\norm{f}_\infty= -\esssup_{x\in R^n}\abs{f(x)}\] -\end{verbatim} -\[\meas_1\{u\in R_+^1\colon f^*(u)>\alpha\} -=\meas_n\{x\in R^n\colon \abs{f(x)}\geq\alpha\} -\quad \forall\alpha>0.\] -\begin{verbatim} -\[\meas_1\{u\in R_+^1\colon f^*(u)>\alpha\} -=\meas_n\{x\in R^n\colon \abs{f(x)}\geq\alpha\} -\quad \forall\alpha>0.\] -\end{verbatim} -\cn{esssup} and \cn{meas} would be defined in the document preamble as -\begin{verbatim} -\DeclareMathOperator*{\esssup}{ess\,sup} -\DeclareMathOperator{\meas}{meas} -\end{verbatim} - -The following special operator names are predefined in the \pkg{amsmath} -package: \cn{varlimsup}, \cn{varliminf}, \cn{varinjlim}, and -\cn{varprojlim}. Here's what they look like in use: -\begin{align} -&\varlimsup_{n\rightarrow\infty} - \mathcal{Q}(u_n,u_n-u^{\#})\le0\\ -&\varliminf_{n\rightarrow\infty} - \left\lvert a_{n+1}\right\rvert/\left\lvert a_n\right\rvert=0\\ -&\varinjlim (m_i^\lambda\cdot)^*\le0\\ -&\varprojlim_{p\in S(A)}A_p\le0 -\end{align} -\begin{verbatim} -\begin{align} -&\varlimsup_{n\rightarrow\infty} - \mathcal{Q}(u_n,u_n-u^{\#})\le0\\ -&\varliminf_{n\rightarrow\infty} - \left\lvert a_{n+1}\right\rvert/\left\lvert a_n\right\rvert=0\\ -&\varinjlim (m_i^\lambda\cdot)^*\le0\\ -&\varprojlim_{p\in S(A)}A_p\le0 -\end{align} -\end{verbatim} - -\subsection{\cn{mod} and its relatives} -The commands \cn{mod} and \cn{pod} are variants of -\cn{pmod} preferred by some authors; \cn{mod} omits the parentheses, -whereas \cn{pod} omits the `mod' and retains the parentheses. -Examples: -\begin{align} -x&\equiv y+1\pmod{m^2}\\ -x&\equiv y+1\mod{m^2}\\ -x&\equiv y+1\pod{m^2} -\end{align} -\begin{verbatim} -\begin{align} -x&\equiv y+1\pmod{m^2}\\ -x&\equiv y+1\mod{m^2}\\ -x&\equiv y+1\pod{m^2} -\end{align} -\end{verbatim} - -\subsection{Fractions and related constructions} -\label{fracs} - -The usual notation for binomials is similar to the fraction concept, -so it has a similar command \cn{binom} with two arguments. Example: -\begin{equation} -\begin{split} -\sum_{\gamma\in\Gamma_C} I_\gamma& -=2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}\\ -&\quad+\dots+(-1)^l\binom{k}{l}2^{k-l} -+\dots+(-1)^k\\ -&=(2-1)^k=1 -\end{split} -\end{equation} -\begin{verbatim} -\begin{equation} -\begin{split} -[\sum_{\gamma\in\Gamma_C} I_\gamma& -=2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}\\ -&\quad+\dots+(-1)^l\binom{k}{l}2^{k-l} -+\dots+(-1)^k\\ -&=(2-1)^k=1 -\end{split} -\end{equation} -\end{verbatim} -There are also abbreviations -\begin{verbatim} -\dfrac \dbinom -\tfrac \tbinom -\end{verbatim} -for the commonly needed constructions -\begin{verbatim} -{\displaystyle\frac ... } {\displaystyle\binom ... } -{\textstyle\frac ... } {\textstyle\binom ... } -\end{verbatim} - -The generalized fraction command \cn{genfrac} provides full access to -the six \TeX{} fraction primitives: -\begin{align} -\text{\cn{over}: }&\genfrac{}{}{}{}{n+1}{2}& -\text{\cn{overwithdelims}: }& - \genfrac{\langle}{\rangle}{}{}{n+1}{2}\\ -\text{\cn{atop}: }&\genfrac{}{}{0pt}{}{n+1}{2}& -\text{\cn{atopwithdelims}: }& - \genfrac{(}{)}{0pt}{}{n+1}{2}\\ -\text{\cn{above}: }&\genfrac{}{}{1pt}{}{n+1}{2}& -\text{\cn{abovewithdelims}: }& - \genfrac{[}{]}{1pt}{}{n+1}{2} -\end{align} -\begin{verbatim} -\text{\cn{over}: }&\genfrac{}{}{}{}{n+1}{2}& -\text{\cn{overwithdelims}: }& - \genfrac{\langle}{\rangle}{}{}{n+1}{2}\\ -\text{\cn{atop}: }&\genfrac{}{}{0pt}{}{n+1}{2}& -\text{\cn{atopwithdelims}: }& - \genfrac{(}{)}{0pt}{}{n+1}{2}\\ -\text{\cn{above}: }&\genfrac{}{}{1pt}{}{n+1}{2}& -\text{\cn{abovewithdelims}: }& - \genfrac{[}{]}{1pt}{}{n+1}{2} -\end{verbatim} - -\subsection{Continued fractions} -The continued fraction -\begin{equation} -\cfrac{1}{\sqrt{2}+ - \cfrac{1}{\sqrt{2}+ - \cfrac{1}{\sqrt{2}+ - \cfrac{1}{\sqrt{2}+ - \cfrac{1}{\sqrt{2}+\dotsb -}}}}} -\end{equation} -can be obtained by typing -\begin{verbatim} -\cfrac{1}{\sqrt{2}+ - \cfrac{1}{\sqrt{2}+ - \cfrac{1}{\sqrt{2}+ - \cfrac{1}{\sqrt{2}+ - \cfrac{1}{\sqrt{2}+\dotsb -}}}}} -\end{verbatim} -Left or right placement of any of the numerators is accomplished by using -\cn{cfrac[l]} or \cn{cfrac[r]} instead of \cn{cfrac}. - -\subsection{Smash} - -In \pkg{amsmath} there are optional arguments \verb"t" and \verb"b" for -the plain \TeX\ command \cn{smash}, because sometimes it is advantageous -to be able to `smash' only the top or only the bottom of something while -retaining the natural depth or height. In the formula -$X_j=(1/\sqrt{\smash[b]{\lambda_j}})X_j'$ \cn{smash}\verb=[b]= has been -used to limit the size of the radical symbol. -\begin{verbatim} -$X_j=(1/\sqrt{\smash[b]{\lambda_j}})X_j'$ -\end{verbatim} -Without the use of \cn{smash}\verb=[b]= the formula would have appeared -thus: $X_j=(1/\sqrt{\lambda_j})X_j'$, with the radical extending to -encompass the depth of the subscript $j$. - -\subsection{The `cases' environment} -`Cases' constructions like the following can be produced using -the \env{cases} environment. -\begin{equation} -P_{r-j}= - \begin{cases} - 0& \text{if $r-j$ is odd},\\ - r!\,(-1)^{(r-j)/2}& \text{if $r-j$ is even}. - \end{cases} -\end{equation} -\begin{verbatim} -\begin{equation} P_{r-j}= - \begin{cases} - 0& \text{if $r-j$ is odd},\\ - r!\,(-1)^{(r-j)/2}& \text{if $r-j$ is even}. - \end{cases} -\end{equation} -\end{verbatim} -Notice the use of \cn{text} and the embedded math. - -\subsection{Matrix} - -Here are samples of the matrix environments, -\cn{matrix}, \cn{pmatrix}, \cn{bmatrix}, \cn{Bmatrix}, \cn{vmatrix} -and \cn{Vmatrix}: -\begin{equation} -\begin{matrix} -\vartheta& \varrho\\\varphi& \varpi -\end{matrix}\quad -\begin{pmatrix} -\vartheta& \varrho\\\varphi& \varpi -\end{pmatrix}\quad -\begin{bmatrix} -\vartheta& \varrho\\\varphi& \varpi -\end{bmatrix}\quad -\begin{Bmatrix} -\vartheta& \varrho\\\varphi& \varpi -\end{Bmatrix}\quad -\begin{vmatrix} -\vartheta& \varrho\\\varphi& \varpi -\end{vmatrix}\quad -\begin{Vmatrix} -\vartheta& \varrho\\\varphi& \varpi -\end{Vmatrix} -\end{equation} -% -\begin{verbatim} -\begin{matrix} -\vartheta& \varrho\\\varphi& \varpi -\end{matrix}\quad -\begin{pmatrix} -\vartheta& \varrho\\\varphi& \varpi -\end{pmatrix}\quad -\begin{bmatrix} -\vartheta& \varrho\\\varphi& \varpi -\end{bmatrix}\quad -\begin{Bmatrix} -\vartheta& \varrho\\\varphi& \varpi -\end{Bmatrix}\quad -\begin{vmatrix} -\vartheta& \varrho\\\varphi& \varpi -\end{vmatrix}\quad -\begin{Vmatrix} -\vartheta& \varrho\\\varphi& \varpi -\end{Vmatrix} -\end{verbatim} - -To produce a small matrix suitable for use in text, use the -\env{smallmatrix} environment. -\begin{verbatim} -\begin{math} - \bigl( \begin{smallmatrix} - a&b\\ c&d - \end{smallmatrix} \bigr) -\end{math} -\end{verbatim} -To show -the effect of the matrix on the surrounding lines of -a paragraph, we put it here: \begin{math} - \bigl( \begin{smallmatrix} - a&b\\ c&d - \end{smallmatrix} \bigr) -\end{math} -and follow it with enough text to ensure that there will -be at least one full line below the matrix. - -\cn{hdotsfor}\verb"{"\textit{number}\verb"}" produces a row of dots in a matrix -spanning the given number of columns: -\[W(\Phi)= \begin{Vmatrix} -\dfrac\varphi{(\varphi_1,\varepsilon_1)}&0&\dots&0\\ -\dfrac{\varphi k_{n2}}{(\varphi_2,\varepsilon_1)}& -\dfrac\varphi{(\varphi_2,\varepsilon_2)}&\dots&0\\ -\hdotsfor{5}\\ -\dfrac{\varphi k_{n1}}{(\varphi_n,\varepsilon_1)}& -\dfrac{\varphi k_{n2}}{(\varphi_n,\varepsilon_2)}&\dots& -\dfrac{\varphi k_{n\,n-1}}{(\varphi_n,\varepsilon_{n-1})}& -\dfrac{\varphi}{(\varphi_n,\varepsilon_n)} -\end{Vmatrix}\] -\begin{verbatim} -\[W(\Phi)= \begin{Vmatrix} -\dfrac\varphi{(\varphi_1,\varepsilon_1)}&0&\dots&0\\ -\dfrac{\varphi k_{n2}}{(\varphi_2,\varepsilon_1)}& -\dfrac\varphi{(\varphi_2,\varepsilon_2)}&\dots&0\\ -\hdotsfor{5}\\ -\dfrac{\varphi k_{n1}}{(\varphi_n,\varepsilon_1)}& -\dfrac{\varphi k_{n2}}{(\varphi_n,\varepsilon_2)}&\dots& -\dfrac{\varphi k_{n\,n-1}}{(\varphi_n,\varepsilon_{n-1})}& -\dfrac{\varphi}{(\varphi_n,\varepsilon_n)} -\end{Vmatrix}\] -\end{verbatim} -The spacing of the dots can be varied through use of a square-bracket -option, for example, \verb"\hdotsfor[1.5]{3}". The number in square brackets -will be used as a multiplier; the normal value is 1. - -\subsection{The \cn{substack} command} - -The \cn{substack} command can be used to produce a multiline -subscript or superscript: -for example -\begin{verbatim} -\sum_{\substack{0\le i\le m\\ 0<j<n}} P(i,j) -\end{verbatim} -produces a two-line subscript underneath the sum: -\begin{equation} -\sum_{\substack{0\le i\le m\\ 0<j<n}} P(i,j) -\end{equation} -A slightly more generalized form is the \env{subarray} environment which -allows you to specify that each line should be left-aligned instead of -centered, as here: -\begin{equation} -\sum_{\begin{subarray}{l} - 0\le i\le m\\ 0<j<n - \end{subarray}} - P(i,j) -\end{equation} -\begin{verbatim} -\sum_{\begin{subarray}{l} - 0\le i\le m\\ 0<j<n - \end{subarray}} - P(i,j) -\end{verbatim} - - -\subsection{Big-g-g delimiters} -Here are some big delimiters, first in \cn{normalsize}: -\[\biggl(\mathbf{E}_{y} - \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds - \biggr) -\] -\begin{verbatim} -\[\biggl(\mathbf{E}_{y} - \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds - \biggr) -\] -\end{verbatim} -and now in \cn{Large} size: -{\Large -\[\biggl(\mathbf{E}_{y} - \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds - \biggr) -\]} -\begin{verbatim} -{\Large -\[\biggl(\mathbf{E}_{y} - \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds - \biggr) -\]} -\end{verbatim} - -\newpage -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\makeatletter - -%% This turns on vertical rules at the right and left margins, to -%% better illustrate the spacing for certain multiple-line equation -%% structures. -\def\@makecol{\ifvoid\footins \setbox\@outputbox\box\@cclv - \else\setbox\@outputbox - \vbox{\boxmaxdepth \maxdepth - \unvbox\@cclv\vskip\skip\footins\footnoterule\unvbox\footins}\fi - \xdef\@freelist{\@freelist\@midlist}\gdef\@midlist{}\@combinefloats - \setbox\@outputbox\hbox{\vrule width\marginrulewidth - \vbox to\@colht{\boxmaxdepth\maxdepth - \@texttop\dimen128=\dp\@outputbox\unvbox\@outputbox - \vskip-\dimen128\@textbottom}% - \vrule width\marginrulewidth}% - \global\maxdepth\@maxdepth} -\newdimen\marginrulewidth -\setlength{\marginrulewidth}{.1pt} -\makeatother - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\appendix -\section{Examples of multiple-line equation structures} -\label{s:eq} - -\textbf{\large Note: Starting on this page, vertical rules are -added at the margins so that the positioning of various display elements -with respect to the margins can be seen more clearly.} - -\subsection{Split} -The \env{split} environment is not an independent environment -but should be used inside something else such as \env{equation} -or \env{align}. - -If there is not enough room for it, the equation number for a -\env{split} will be shifted to the previous line, when equation numbers are -on the left; the number shifts down to the next line when numbers are on -the right. -\begin{equation} -\begin{split} -f_{h,\varepsilon}(x,y) -&=\varepsilon\mathbf{E}_{x,y}\int_0^{t_\varepsilon} -L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\ -&= h\int L_{x,z}\varphi(x)\rho_x(dz)\\ -&\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\mathbf{E}_{y} - \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds - -t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\ -&\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon} - \biggl(\mathbf{E}_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)} - \varphi(x)\,ds -\mathbf{E}_{x,y}\int_0^{t_\varepsilon} - L_{x,y_\varepsilon(\varepsilon s)} - \varphi(x)\,ds\biggr)\biggr]\\ -&=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y), -\end{split} -\end{equation} -Some text after to test the below-display spacing. - -\begin{verbatim} -\begin{equation} -\begin{split} -f_{h,\varepsilon}(x,y) -&=\varepsilon\mathbf{E}_{x,y}\int_0^{t_\varepsilon} -L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\ -&= h\int L_{x,z}\varphi(x)\rho_x(dz)\\ -&\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\mathbf{E}_{y} - \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds - -t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\ -&\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon} - \biggl(\mathbf{E}_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)} - \varphi(x)\,ds -\mathbf{E}_{x,y}\int_0^{t_\varepsilon} - L_{x,y_\varepsilon(\varepsilon s)} - \varphi(x)\,ds\biggr)\biggr]\\ -&=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y), -\end{split} -\end{equation} -\end{verbatim} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\newpage -Unnumbered version: -\begin{equation*} -\begin{split} -f_{h,\varepsilon}(x,y) -&=\varepsilon\mathbf{E}_{x,y}\int_0^{t_\varepsilon} -L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\ -&= h\int L_{x,z}\varphi(x)\rho_x(dz)\\ -&\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\mathbf{E}_{y} - \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds - -t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\ -&\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon} - \biggl(\mathbf{E}_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)} - \varphi(x)\,ds -\mathbf{E}_{x,y}\int_0^{t_\varepsilon} - L_{x,y_\varepsilon(\varepsilon s)} - \varphi(x)\,ds\biggr)\biggr]\\ -&=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y), -\end{split} -\end{equation*} -Some text after to test the below-display spacing. - -\begin{verbatim} -\begin{equation*} -\begin{split} -f_{h,\varepsilon}(x,y) -&=\varepsilon\mathbf{E}_{x,y}\int_0^{t_\varepsilon} -L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\ -&= h\int L_{x,z}\varphi(x)\rho_x(dz)\\ -&\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\mathbf{E}_{y} - \int_0^{t_\varepsilon}L_{x,y^x(s)}\varphi(x)\,ds - -t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\ -&\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon} - \biggl(\mathbf{E}_{y}\int_0^{t_\varepsilon}L_{x,y^x(s)} - \varphi(x)\,ds -\mathbf{E}_{x,y}\int_0^{t_\varepsilon} - L_{x,y_\varepsilon(\varepsilon s)} - \varphi(x)\,ds\biggr)\biggr]\\ -&=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y), -\end{split} -\end{equation*} -\end{verbatim} -%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\newpage -If the option \env{centertags} is included in the options -list of the \pkg{amsmath} package, -the equation numbers for \env{split} environments will be -centered vertically on the height -of the \env{split}: -{\makeatletter\ctagsplit@true -\begin{equation} -\begin{split} - \abs{I_2}&=\left\lvert \int_{0}^T \psi(t)\left\{u(a,t)-\int_{\gamma(t)}^a - \frac{d\theta}{k(\theta,t)} - \int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\ -&\le C_6\left\lvert \left\lvert f\int_\Omega\left\lvert \wt{S}^{-1,0}_{a,-} - W_2(\Omega,\Gamma_l)\right\rvert\right\rvert - \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}} - (\Omega;\Gamma_r,T)\right\rvert\right\rvert. -\end{split} -\end{equation}}% -Some text after to test the below-display spacing. - -%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\newpage -Use of \env{split} within \env{align}: -{\delimiterfactor750 -\begin{align} -\begin{split}\abs{I_1} - &=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\ -&\le C_3\left[\int_\Omega\left(\int_{a}^x - g(\xi,t)\,d\xi\right)^2d\Omega\right]^{1/2}\\ -&\quad\times \left[\int_\Omega\left\{u^2_x+\frac{1}{k} - \left(\int_{a}^x cu_t\,d\xi\right)^2\right\} - c\Omega\right]^{1/2}\\ -&\le C_4\left\lvert \left\lvert f\left\lvert \wt{S}^{-1,0}_{a,-} - W_2(\Omega,\Gamma_l)\right\rvert\right\rvert - \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}} - (\Omega;\Gamma_r,T)\right\rvert\right\rvert. -\end{split}\label{eq:A}\\ -\begin{split}\abs{I_2}&=\left\lvert \int_{0}^T \psi(t)\left\{u(a,t) - -\int_{\gamma(t)}^a\frac{d\theta}{k(\theta,t)} - \int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\ -&\le C_6\left\lvert \left\lvert f\int_\Omega - \left\lvert \wt{S}^{-1,0}_{a,-} - W_2(\Omega,\Gamma_l)\right\rvert\right\rvert - \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}} - (\Omega;\Gamma_r,T)\right\rvert\right\rvert. -\end{split} -\end{align}}% -Some text after to test the below-display spacing. - -\begin{verbatim} -\begin{align} -\begin{split}\abs{I_1} - &=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\ -&\le C_3\left[\int_\Omega\left(\int_{a}^x - g(\xi,t)\,d\xi\right)^2d\Omega\right]^{1/2}\\ -&\quad\times \left[\int_\Omega\left\{u^2_x+\frac{1}{k} - \left(\int_{a}^x cu_t\,d\xi\right)^2\right\} - c\Omega\right]^{1/2}\\ -&\le C_4\left\lvert \left\lvert f\left\lvert \wt{S}^{-1,0}_{a,-} - W_2(\Omega,\Gamma_l)\right\rvert\right\rvert - \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}} - (\Omega;\Gamma_r,T)\right\rvert\right\rvert. -\end{split}\label{eq:A}\\ -\begin{split}\abs{I_2}&=\left\lvert \int_{0}^T \psi(t)\left\{u(a,t) - -\int_{\gamma(t)}^a\frac{d\theta}{k(\theta,t)} - \int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\ -&\le C_6\left\lvert \left\lvert f\int_\Omega - \left\lvert \wt{S}^{-1,0}_{a,-} - W_2(\Omega,\Gamma_l)\right\rvert\right\rvert - \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}} - (\Omega;\Gamma_r,T)\right\rvert\right\rvert. -\end{split} -\end{align} -\end{verbatim} - -%%%%%%%%%%%%%%%%%% - -\newpage -Unnumbered \env{align}, with a number on the second \env{split}: -\begin{align*} -\begin{split}\abs{I_1}&=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\ - &\le C_3\left[\int_\Omega\left(\int_{a}^x - g(\xi,t)\,d\xi\right)^2d\Omega\right]^{1/2}\\ -&\phantom{=}\times \left[\int_\Omega\left\{u^2_x+\frac{1}{k} - \left(\int_{a}^x cu_t\,d\xi\right)^2\right\} - c\Omega\right]^{1/2}\\ -&\le C_4\left\lvert \left\lvert f\left\lvert \wt{S}^{-1,0}_{a,-} - W_2(\Omega,\Gamma_l)\right\rvert\right\rvert - \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}} - (\Omega;\Gamma_r,T)\right\rvert\right\rvert. -\end{split}\\ -\begin{split}\abs{I_2}&=\left\lvert \int_{0}^T \psi(t)\left\{u(a,t) - -\int_{\gamma(t)}^a\frac{d\theta}{k(\theta,t)} - \int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\ -&\le C_6\left\lvert \left\lvert f\int_\Omega - \left\lvert \wt{S}^{-1,0}_{a,-} - W_2(\Omega,\Gamma_l)\right\rvert\right\rvert - \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}} - (\Omega;\Gamma_r,T)\right\rvert\right\rvert. -\end{split}\tag{\theequation$'$} -\end{align*} -Some text after to test the below-display spacing. - -\begin{verbatim} -\begin{align*} -\begin{split}\abs{I_1}&=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\ - &\le C_3\left[\int_\Omega\left(\int_{a}^x - g(\xi,t)\,d\xi\right)^2d\Omega\right]^{1/2}\\ -&\phantom{=}\times \left[\int_\Omega\left\{u^2_x+\frac{1}{k} - \left(\int_{a}^x cu_t\,d\xi\right)^2\right\} - c\Omega\right]^{1/2}\\ -&\le C_4\left\lvert \left\lvert f\left\lvert \wt{S}^{-1,0}_{a,-} - W_2(\Omega,\Gamma_l)\right\rvert\right\rvert - \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}} - (\Omega;\Gamma_r,T)\right\rvert\right\rvert. -\end{split}\\ -\begin{split}\abs{I_2}&=\left\lvert \int_{0}^T \psi(t)\left\{u(a,t) - -\int_{\gamma(t)}^a\frac{d\theta}{k(\theta,t)} - \int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\ -&\le C_6\left\lvert \left\lvert f\int_\Omega - \left\lvert \wt{S}^{-1,0}_{a,-} - W_2(\Omega,\Gamma_l)\right\rvert\right\rvert - \left\lvert \abs{u}\overset{\circ}\to W_2^{\wt{A}} - (\Omega;\Gamma_r,T)\right\rvert\right\rvert. -\end{split}\tag{\theequation$'$} -\end{align*} -\end{verbatim} -%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\newpage -\subsection{Multline} -Numbered version: -\begin{multline}\label{eq:E} -\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2] - -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\ - =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 - \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy -\end{multline} -To test the use of \verb=\label= and -\verb=\ref=, we refer to the number of this -equation here: (\ref{eq:E}). - -\begin{verbatim} -\begin{multline}\label{eq:E} -\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2] - -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\ - =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 - \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy -\end{multline} -\end{verbatim} -%%%%%%%%%%%%%%%%%%%%%%%%%%% - -Unnumbered version: -\begin{multline*} -\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2] - -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\ - =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 - \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy -\end{multline*} -Some text after to test the below-display spacing. - -\begin{verbatim} -\begin{multline*} -\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2] - -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\ - =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 - \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy -\end{multline*} -\end{verbatim} -%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\iffalse % bugfix needed, error message "Multiple \tag" - % [mjd,24-Jan-1995] -\newpage -And now an ``unnumbered'' version numbered with a literal tag: -\begin{multline*}\tag*{[a]} -\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2] - -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\ - =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 - \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy -\end{multline*} -Some text after to test the below-display spacing. - -\begin{verbatim} -\begin{multline*}\tag*{[a]} -\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2] - -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\ - =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 - \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy -\end{multline*} -\end{verbatim} - -%%%%%%%%%%%%%%%%%%%%%%%%%%% - -The same display with \verb=\multlinegap= set to zero. -Notice that the space on the left in -the first line does not change, because of the equation number, while -the second line is pushed over to the right margin. -{\setlength{\multlinegap}{0pt} -\begin{multline*}\tag*{[a]} -\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2] - -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\ - =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 - \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy -\end{multline*}}% -Some text after to test the below-display spacing. - -\begin{verbatim} -{\setlength{\multlinegap}{0pt} -\begin{multline*}\tag*{[a]} -\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2] - -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\ - =\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 - \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy -\end{multline*}} -\end{verbatim} -%%%%%%%%%%%%%%%%%%%%%%%%%%% -\fi % matches \iffalse above [mjd,24-Jan-1995] - -%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\newpage -\subsection{Gather} -Numbered version with \verb;\notag; on the second line: -\begin{gather} -D(a,r)\equiv\{z\in\mathbf{C}\colon \abs{z-a}<r\},\\ -\seg(a,r)\equiv\{z\in\mathbf{C}\colon -\Im z= \Im a,\ \abs{z-a}<r\},\notag\\ -c(e,\theta,r)\equiv\{(x,y)\in\mathbf{C} -\colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\ -C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r). -\end{gather} -\begin{verbatim} -\begin{gather} -D(a,r)\equiv\{z\in\mathbf{C}\colon \abs{z-a}<r\},\\ -\seg(a,r)\equiv\{z\in\mathbf{C}\colon -\Im z= \Im a,\ \abs{z-a}<r\},\notag\\ -c(e,\theta,r)\equiv\{(x,y)\in\mathbf{C} -\colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\ -C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r). -\end{gather} -\end{verbatim} -%%%%%%%%%%%%%%%%%%%%%%%%%%% - -Unnumbered version. -\begin{gather*} -D(a,r)\equiv\{z\in\mathbf{C}\colon \abs{z-a}<r\},\\ -\seg (a,r)\equiv\{z\in\mathbf{C}\colon -\Im z= \Im a,\ \abs{z-a}<r\},\\ -c(e,\theta,r)\equiv\{(x,y)\in\mathbf{C} - \colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\ -C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r). -\end{gather*} -Some text after to test the below-display spacing. -\begin{verbatim} -\begin{gather*} -D(a,r)\equiv\{z\in\mathbf{C}\colon \abs{z-a}<r\},\\ -\seg (a,r)\equiv\{z\in\mathbf{C}\colon -\Im z= \Im a,\ \abs{z-a}<r\},\\ -c(e,\theta,r)\equiv\{(x,y)\in\mathbf{C} - \colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\ -C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r). -\end{gather*} -\end{verbatim} -%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\newpage -\subsection{Align} -Numbered version: -\begin{align} -\gamma_x(t)&=(\cos tu+\sin tx,v),\\ -\gamma_y(t)&=(u,\cos tv+\sin ty),\\ -\gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv, - -\frac\beta\alpha\sin tu+\cos tv\right). -\end{align} -Some text after to test the below-display spacing. - -\begin{verbatim} -\begin{align} -\gamma_x(t)&=(\cos tu+\sin tx,v),\\ -\gamma_y(t)&=(u,\cos tv+\sin ty),\\ -\gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv, - -\frac\beta\alpha\sin tu+\cos tv\right). -\end{align} -\end{verbatim} -%%%%%%%%%%%%%%%%%%%%%%%%%%% - -Unnumbered version: -\begin{align*} -\gamma_x(t)&=(\cos tu+\sin tx,v),\\ -\gamma_y(t)&=(u,\cos tv+\sin ty),\\ -\gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv, - -\frac\beta\alpha\sin tu+\cos tv\right). -\end{align*} -Some text after to test the below-display spacing. - -\begin{verbatim} -\begin{align*} -\gamma_x(t)&=(\cos tu+\sin tx,v),\\ -\gamma_y(t)&=(u,\cos tv+\sin ty),\\ -\gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv, - -\frac\beta\alpha\sin tu+\cos tv\right). -\end{align*} -\end{verbatim} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -A variation: -\begin{align} -x& =y && \text {by (\ref{eq:C})}\\ -x'& = y' && \text {by (\ref{eq:D})}\\ -x+x' & = y+y' && \text {by Axiom 1.} -\end{align} -Some text after to test the below-display spacing. - -\begin{verbatim} -\begin{align} -x& =y && \text {by (\ref{eq:C})}\\ -x'& = y' && \text {by (\ref{eq:D})}\\ -x+x' & = y+y' && \text {by Axiom 1.} -\end{align} -\end{verbatim} -%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\newpage -\subsection{Align and split within gather} -When using the \env{align} environment within the \env{gather} -environment, one or the other, or both, should be unnumbered (using the -\verb"*" form); numbering both the outer and inner environment would -cause a conflict. - -Automatically numbered \env{gather} with \env{split} and \env{align*}: -\begin{gather} -\begin{split} \varphi(x,z) -&=z-\gamma_{10}x-\gamma_{mn}x^mz^n\\ -&=z-Mr^{-1}x-Mr^{-(m+n)}x^mz^n -\end{split}\\[6pt] -\begin{align*} -\zeta^0 &=(\xi^0)^2,\\ -\zeta^1 &=\xi^0\xi^1,\\ -\zeta^2 &=(\xi^1)^2, -\end{align*} -\end{gather} -Here the \env{split} environment gets a number from the outer -\env{gather} environment; numbers for individual lines of the -\env{align*} are suppressed because of the star. - -\begin{verbatim} -\begin{gather} -\begin{split} \varphi(x,z) -&=z-\gamma_{10}x-\gamma_{mn}x^mz^n\\ -&=z-Mr^{-1}x-Mr^{-(m+n)}x^mz^n -\end{split}\\[6pt] -\begin{align*} -\zeta^0 &=(\xi^0)^2,\\ -\zeta^1 &=\xi^0\xi^1,\\ -\zeta^2 &=(\xi^1)^2, -\end{align*} -\end{gather} -\end{verbatim} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -The \verb"*"-ed form of \env{gather} with the non-\verb"*"-ed form of -\env{align}. -\begin{gather*} -\begin{split} \varphi(x,z) -&=z-\gamma_{10}x-\gamma_{mn}x^mz^n\\ -&=z-Mr^{-1}x-Mr^{-(m+n)}x^mz^n -\end{split}\\[6pt] -\begin{align} \zeta^0&=(\xi^0)^2,\\ -\zeta^1 &=\xi^0\xi^1,\\ -\zeta^2 &=(\xi^1)^2, -\end{align} -\end{gather*} -Some text after to test the below-display spacing. - -\begin{verbatim} -\begin{gather*} -\begin{split} \varphi(x,z) -&=z-\gamma_{10}x-\gamma_{mn}x^mz^n\\ -&=z-Mr^{-1}x-Mr^{-(m+n)}x^mz^n -\end{split}\\[6pt] -\begin{align} \zeta^0&=(\xi^0)^2,\\ -\zeta^1 &=\xi^0\xi^1,\\ -\zeta^2 &=(\xi^1)^2, -\end{align} -\end{gather*} -\end{verbatim} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\newpage -\subsection{Alignat} -Numbered version: -\begin{alignat}{3} -V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j, - & \qquad U_i & = u_i, - \qquad \text{for $i\ne j$;}\label{eq:B}\\ -V_j & = v_j, & \qquad X_j & = x_j, - & \qquad U_j & u_j + \sum_{i\ne j} q_i u_i. -\end{alignat} -Some text after to test the below-display spacing. - -\begin{verbatim} -\begin{alignat}{3} -V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j, - & \qquad U_i & = u_i, - \qquad \text{for $i\ne j$;}\label{eq:B}\\ -V_j & = v_j, & \qquad X_j & = x_j, - & \qquad U_j & u_j + \sum_{i\ne j} q_i u_i. -\end{alignat} -\end{verbatim} -%%%%%%%%%%%%%%%%%%%%%%%%%%% - -Unnumbered version: -\begin{alignat*}3 -V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j, - & \qquad U_i & = u_i, - \qquad \text{for $i\ne j$;} \\ -V_j & = v_j, & \qquad X_j & = x_j, - & \qquad U_j & u_j + \sum_{i\ne j} q_i u_i. -\end{alignat*} -Some text after to test the below-display spacing. - -\begin{verbatim} -\begin{alignat*}3 -V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j, - & \qquad U_i & = u_i, - \qquad \text{for $i\ne j$;} \\ -V_j & = v_j, & \qquad X_j & = x_j, - & \qquad U_j & u_j + \sum_{i\ne j} q_i u_i. -\end{alignat*} -\end{verbatim} -%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\newpage -The most common use for \env{alignat} is for things like -\begin{alignat}{2} -x& =y && \qquad \text {by (\ref{eq:A})}\label{eq:C}\\ -x'& = y' && \qquad \text {by (\ref{eq:B})}\label{eq:D}\\ -x+x' & = y+y' && \qquad \text {by Axiom 1.} -\end{alignat} -Some text after to test the below-display spacing. - -\begin{verbatim} -\begin{alignat}{2} -x& =y && \qquad \text {by (\ref{eq:A})}\label{eq:C}\\ -x'& = y' && \qquad \text {by (\ref{eq:B})}\label{eq:D}\\ -x+x' & = y+y' && \qquad \text {by Axiom 1.} -\end{alignat} -\end{verbatim} -%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\newpage -\setlength{\marginrulewidth}{0pt} - -\begin{thebibliography}{10} - -\bibitem{dihe:newdir} -W.~Diffie and E.~Hellman, \emph{New directions in cryptography}, IEEE -Transactions on Information Theory \textbf{22} (1976), no.~5, 644--654. - -\bibitem{fre:cichon} -D.~H. 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