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authorKarl Berry <karl@freefriends.org>2016-03-01 23:30:12 +0000
committerKarl Berry <karl@freefriends.org>2016-03-01 23:30:12 +0000
commit64fe092a8f3dc062b323f28f52dc82c7be509252 (patch)
treec047898c9e2c1fa0c7d090dd1fcab2bd85290023 /Master/texmf-dist/source/latex/amsmath
parentb37042b5f1062ab261f6837b986d39354c33bc19 (diff)
amsmath (1mar16)
git-svn-id: svn://tug.org/texlive/trunk@39904 c570f23f-e606-0410-a88d-b1316a301751
Diffstat (limited to 'Master/texmf-dist/source/latex/amsmath')
-rw-r--r--Master/texmf-dist/source/latex/amsmath/README148
-rw-r--r--Master/texmf-dist/source/latex/amsmath/amsbsy.dtx28
-rw-r--r--Master/texmf-dist/source/latex/amsmath/amsbsy.ins33
-rw-r--r--Master/texmf-dist/source/latex/amsmath/amscd.dtx28
-rw-r--r--Master/texmf-dist/source/latex/amsmath/amscd.ins33
-rw-r--r--Master/texmf-dist/source/latex/amsmath/amsgen.dtx29
-rw-r--r--Master/texmf-dist/source/latex/amsmath/amsgen.ins33
-rw-r--r--Master/texmf-dist/source/latex/amsmath/amsldoc.tex2620
-rw-r--r--Master/texmf-dist/source/latex/amsmath/amsmath.dtx409
-rw-r--r--Master/texmf-dist/source/latex/amsmath/amsmath.ins37
-rw-r--r--Master/texmf-dist/source/latex/amsmath/amsopn.dtx29
-rw-r--r--Master/texmf-dist/source/latex/amsmath/amsopn.ins33
-rw-r--r--Master/texmf-dist/source/latex/amsmath/amstext.dtx28
-rw-r--r--Master/texmf-dist/source/latex/amsmath/amstext.ins33
-rw-r--r--Master/texmf-dist/source/latex/amsmath/amsxtra.dtx29
-rw-r--r--Master/texmf-dist/source/latex/amsmath/amsxtra.ins33
-rw-r--r--Master/texmf-dist/source/latex/amsmath/diffs-m.txt552
-rw-r--r--Master/texmf-dist/source/latex/amsmath/install.txt132
-rw-r--r--Master/texmf-dist/source/latex/amsmath/manifest.txt64
-rw-r--r--Master/texmf-dist/source/latex/amsmath/subeqn.tex202
-rw-r--r--Master/texmf-dist/source/latex/amsmath/technote.tex233
-rw-r--r--Master/texmf-dist/source/latex/amsmath/testmath.tex2342
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}
-
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-% optional argument can be used to override automatic sizing of the
-% right vert bar, e.g. \eval[\biggr]{...}_{...}
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- \ifx#1\right\relax \left.\fi#2#1\rvert}
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-
-\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}
-
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-
-\end{thebibliography}
-
-\end{document}
-\endinput