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authorNorbert Preining <norbert@preining.info>2019-09-02 13:46:59 +0900
committerNorbert Preining <norbert@preining.info>2019-09-02 13:46:59 +0900
commite0c6872cf40896c7be36b11dcc744620f10adf1d (patch)
tree60335e10d2f4354b0674ec22d7b53f0f8abee672 /info/amslatex
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+../../../macros/latex/required/amsmath/amsldoc.pdf \ No newline at end of file
diff --git a/info/amslatex/primer/README b/info/amslatex/primer/README
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+"Getting up and running with AMS-LaTeX" (contained in the file
+amshelp.tex) is a primer on using AMS-LaTeX. It's intended for people
+with at least some experience using TeX, AMS-TeX, or LaTeX.
+
+This is an attempt to get you up and running with AmS-LaTeX as quickly
+as possible. These instructions (along with the template file
+template.tex) won't be a substitute for the full documentation, but
+they may get you started quickly enough so that you'll only need to
+refer to the main documentation occasionally.
+
+In addition to descriptions of the basics of AMS-LaTeX, there are
+sections with careful descriptions of the various environments for
+displayed mathematics, how to use Xy-pic to draw commutative diagrams,
+and how to use amsrefs to create a bibliography.
+
+This material is subject to the LaTeX Project Public License. See
+http://www.latex-project.org/lppl.txt for the details of that license.
diff --git a/info/amslatex/primer/amshelp.md5 b/info/amslatex/primer/amshelp.md5
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+1edb1e82c4eccea6d8a19e54c84ab258 amshelp.tex
+62157a1ebf6be9282f5dc33a24db946d amshelp.dvi
+7691e854272e965ecd7aeec5c9c888e4 amshelp.pdf
+bf05831b44f8156360b816e08b94d521 template.tex
diff --git a/info/amslatex/primer/amshelp.pdf b/info/amslatex/primer/amshelp.pdf
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+%%% amshelp.tex
+%%% This is an AMS-LaTeX file
+
+%%% Copyright (c) 1992, 2000, 2008, 2009, 2011, 2013 Philip S. Hirschhorn
+%
+% This work may be distributed and/or modified under the
+% conditions of the LaTeX Project Public License, either version 1.3
+% 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 1.3 or later is part of all distributions of LaTeX
+% version 2003/12/01 or later.
+
+
+%%% Philip Hirschhorn
+%%% Department of Mathematics
+%%% Wellesley College
+%%% Wellesley, MA 02481
+%%% psh@math.mit.edu
+%%% psh@poincare.wellesley.edu
+
+
+%%% This is an attempt to explain how to get up and running with
+%%% AmS-LaTeX for someone having some familiarity with TeX,
+%%% AMS-TeX, or LaTeX.
+
+
+
+
+\newcommand{\filedate}{January 28, 2013}
+%\newcommand{\filedate}{\today}
+\newcommand{\fileversion}{Version 2.3}
+
+%---------------------------------------------------------------------
+%\documentclass[12pt]{amsart}
+\documentclass{amsart}
+
+\usepackage{url}
+
+%\usepackage[pdfborderstyle={/S/U/W 1},hyperfootnotes=false]{hyperref}
+%\usepackage[colorlinks=false,pdfborder={0 0 0}]{hyperref}
+%\usepackage[colorlinks,pdfborder={0 0 0}]{hyperref}
+\usepackage[colorlinks]{hyperref}
+%\usepackage{hyperref}
+
+% In case we're not using hyperref.sty:
+\providecommand{\texorpdfstring}[2]{#1}
+% The following can safely be used in \section commands
+% without causing pdf warnings:
+\newcommand{\bs}{\texorpdfstring{\char`\\}{}}
+
+% We have to load amsrefs *after* hyperref. (Most packages must
+% be loaded before hyperref; amsrefs is an exception to that.)
+
+% We have to load amsrefs *before* loading Xy-pic, or else the
+% \newcommand{\cir}{\textasciicircum} in textcmds.sty will complain.
+% We can also use the optional argument ``lite'', as in
+% \usepackage[lite]{amsrefs}, to avoid the problem by suppressing
+% the reading of textcmds.sty
+\usepackage[lite]{amsrefs}
+
+\usepackage[all,cmtip]{xy}
+\let\objectstyle=\displaystyle
+
+
+
+%---------------------------------------------------------------------
+%---------------------------------------------------------------------
+
+\numberwithin{equation}{section}
+
+
+% Theorem environments
+
+\theoremstyle{plain} %% This is the default, anyway
+\newtheorem{thm}[equation]{Theorem}
+\newtheorem{cor}[equation]{Corollary}
+\newtheorem{lem}[equation]{Lemma}
+\newtheorem{prop}[equation]{Proposition}
+
+
+\theoremstyle{definition}
+\newtheorem{defn}[equation]{Definition}
+
+\theoremstyle{remark}
+\newtheorem{rem}[equation]{Remark}
+\newtheorem{ex}[equation]{Example}
+\newtheorem{notation}[equation]{Notation}
+\newtheorem{terminology}[equation]{Terminology}
+
+
+%---------------------------------------------------------------------
+%---------------------------------------------------------------------
+%---------------------------------------------------------------------
+%---------------------------------------------------------------------
+
+\begin{document}
+
+
+\title[Running \AmS-\LaTeX]{Getting up and running\\
+ with \AmS-\LaTeX}
+
+\author{Philip S. Hirschhorn}
+
+\address{Department of Mathematics\\
+ Wellesley College\\
+ Wellesley, Massachusetts 02481}
+
+\email{psh@math.mit.edu}
+
+\date{\filedate, \fileversion}
+
+\begin{abstract}
+ Together with the template file \texttt{template.tex}, these notes
+ are an attempt to tell you enough about \LaTeX{} and \AmS-\LaTeX{}
+ so that you can get started without having to read the book.
+\end{abstract}
+
+\maketitle
+
+\tableofcontents
+%---------------------------------------------------------------------
+%---------------------------------------------------------------------
+\section{Introduction}
+
+This is an attempt to get you up and running with \AmS-\LaTeX{} as
+quickly as possible. These instructions (along with the template file
+\texttt{template.tex}) won't be a substitute for the full
+documentation, but they'll give you enough to get started quickly and
+only occasionally have to refer to the main documentation.
+
+The current version of \AmS-\LaTeX{} (version 2.2) is a collection of
+document classes and optional packages for the current version of
+standard \LaTeX. \AmS-\LaTeX{} provides the document classes
+\texttt{amsart}, \texttt{amsproc}, and \texttt{amsbook} (see
+section~\ref{sec:DocClsCom}) to replace the standard document classes
+\texttt{article}, \texttt{proc}, and \texttt{book}, and several
+optional packages (mainly \texttt{amsmath}) that can be used with the
+standard \LaTeX{} document classes. Thus, using \AmS-\LaTeX{} is
+really using a variety of \LaTeX. If you're new to \LaTeX{}, and
+these last few sentences made no sense to you at all, don't worry
+about it. You don't have to know what the standard \LaTeX{} document
+classes are in order to use the \AmS-\LaTeX{} replacements for them.
+
+I'll be assuming that you have at least some experience with either
+plain \TeX, \AmS-\TeX{} or \LaTeX, and I'll try to tell you what you
+need to know so that you can get started with \AmS-\LaTeX{}
+\emph{without} actually reading the \LaTeX{} user's
+guide~\cite{latex}, or even taking much of a look at the \AmS-\LaTeX{}
+user's guide~\cite{amslatexusersguide} or the short math guide for
+\LaTeX~\cite{mathguide}.
+
+If you've never used \emph{any} version of \TeX{} or \LaTeX, then I
+recommend ``The not so short introduction to \LaTeXe{}'' by Tobias
+Oetiker, Hubert Partl, Irene Hyna, and Elisabeth Schlegl
+\cite{NotShort}. This is intended for those with no knowledge of
+\TeX{} or \LaTeX, and concisely gives a description of what a \LaTeX{}
+document looks like and how you type text and simple mathematics in a
+\LaTeX{} document.
+
+These instructions come with a template file \verb"template.tex",
+which is an attempt to give you enough to fake your way through an
+\AmS-\LaTeX{} file \emph{almost} without even reading these
+instructions. I've included the text of that file in these
+instructions as section~\ref{sec:template}, so you might want to take
+a look at that now, and then just use the table of contents of these
+instructions to find more information on whatever in that file
+confuses you.
+
+In case you haven't guessed, these instructions were printed using the
+\texttt{amsart} document class of \AmS-\LaTeX, so you can get some
+idea what it all looks like.
+
+
+
+%---------------------------------------------------------------------
+%---------------------------------------------------------------------
+\section{Basic \LaTeX{} stuff}
+\label{sec:basicstuff}
+
+In this section, we'll describe the three commands that must appear in
+every \LaTeX{} document: \verb"\documentclass",
+\verb"\begin{document}", and \verb"\end{document}". The complete
+explanation of these can be found in the \LaTeX{} User's
+Guide~\cite{latex} or in \emph{The not so short introduction to
+ \LaTeXe}~\cite{NotShort}. We'll also explain how to begin a new
+section or subsection of the paper, and how \LaTeX{} manages to get
+the cross-references right (which is also the explanation of why you
+need to run a file through \LaTeX{} \emph{twice} to be sure that all
+the cross-references are correct).
+
+%--------------------------------------------------------------------
+\subsection{The \texttt{\bs documentclass} command}
+\label{sec:DocClsCom}
+
+Before you type anything that actually appears in the paper, you must
+include a \verb"\documentclass" command. It's easiest to just put the
+\verb"\documentclass" command at the very beginning of the file,
+possibly with a few lines of comments before it.
+
+It's the choice of document class that determines whether you're using
+\AmS-\LaTeX{} or just plain old \LaTeX. \AmS-\LaTeX{} provides the
+document classes \texttt{amsart}, \texttt{amsproc}, and
+\texttt{amsbook} as replacements for the standard \LaTeX{} document
+classes \texttt{article}, \texttt{proc}, and \texttt{book}. If for
+some reason you prefer to use the standard \LaTeX{} classes
+\texttt{article}, \texttt{proc}, or \texttt{book}, you can still get
+many of the features of \AmS-\LaTeX{} by including the command
+\verb"\usepackage{amsmath}" after your \verb"\documentclass" command.
+I'll only be discussing the \texttt{amsart} document class here. For
+the others, see the \AmS-\LaTeX{} User's
+Guide~\cite{amslatexusersguide}.
+
+The simplest version of the \texttt{\bs documentclass} command is
+\begin{center}
+ \verb"\documentclass{amsart}"
+\end{center}
+This will give you the default type size, which is 10~point type. If
+you'd like to use 12~point type, then you should include the optional
+argument \verb"[12pt]"; this makes the command
+\begin{center}
+ \verb"\documentclass[12pt]{amsart}"
+\end{center}
+
+%--------------------------------------------------------------------
+\subsection{Loading optional packages}
+\label{sec:optpack}
+
+There are at least three optional packages that are of interest. The
+first is the \texttt{amsrefs} package, which makes it much easier to
+create a bibliography (see section~\ref{sec:amsrefs}). To load the
+\texttt{amsrefs} package, you put the line
+\begin{center}
+ \verb"\usepackage[lite]{amsrefs}"
+\end{center}
+after the \verb"\documentclass" command. (For an explanation of why
+we recommend using the optional argument \texttt{lite}, see
+section~\ref{sec:refoptions}.)
+
+Another important package is for when you want to use some of the
+special symbols contained in the \AmS-Fonts package. These are
+listed, along with all of the standard \LaTeX{} symbols, in
+\texttt{symbols.pdf}, available at
+\begin{center}
+ \url{http://www.ctan.org/tex-archive/info/symbols/math/symbols.pdf}
+\end{center}
+If you want the standard names for these symbols to be defined for
+your use, then you need to use the optional package \texttt{amssymb}.
+Thus, to use the default 10~point type, use \texttt{amsrefs} to create
+a bibliography, and have the special symbols defined, use the commands
+\begin{center}
+ \begin{tabular}{l}
+ \verb"\documentclass{amsart}"\\
+ \verb"\usepackage[lite]{amsrefs}"\\
+ \verb"\usepackage{amssymb}"
+ \end{tabular}
+\end{center}
+
+Another widely used optional package is \Xy-pic, which enables you to
+draw commutative diagrams as part of your \LaTeX{} file rather than
+creating them with a graphics package and importing the graphics.
+(For commutative diagrams, see section~\ref{sec:xypic}). To use
+\Xy-pic, you should include the commands
+\begin{center}
+ \begin{tabular}{l}
+ \verb"\usepackage[all,cmtip]{xy}"\\
+ \verb"\let\objectstyle=\displaystyle"
+ \end{tabular}
+\end{center}
+That loads the \Xy-pic package and sets it so that the arrowheads used
+are the same ones used in the rest of the document and the nodes in
+the diagram are, by default, in \verb"\displaystyle". If you'd like
+the default style for the nodes to be \verb"\textstyle", you should
+omit the second of those two lines.
+
+This document uses all of those packages, and so we used the commands
+\begin{center}
+ \begin{tabular}{l}
+ \verb"\documentclass{amsart}"\\
+ \verb"\usepackage[lite]{amsrefs}"\\
+ \verb"\usepackage{amssymb}"\\
+ \verb"\usepackage[all,cmtip]{xy}"\\
+ \verb"\let\objectstyle=\displaystyle"
+ \end{tabular}
+\end{center}
+
+%---------------------------------------------------------------------
+
+\subsection{\texttt{\bs begin\{document\}} and \texttt{\bs end\{document\}}}
+
+Everything that is to appear in the document must appear in between
+the \verb"\begin{document}" and \verb"\end{document}" commands. There
+are no optional arguments for these commands, so they always look the
+same. Anything following the \verb"\end{document}" command is
+ignored. In addition to \verb"\usepackage" commands (see
+section~\ref{sec:optpack}), you are allowed to have macro definitions
+(i.e., newcommands; see section~\ref{sec:definitions}) before the
+\verb"\begin{document}", and that's actually a good place for them,
+ but that's about all.
+
+%---------------------------------------------------------------------
+\subsection{Sections and subsections}
+\label{sec:sections}
+
+To begin a new section, you give the command
+\begin{center}
+ \verb"\section{Section name}"
+\end{center}
+To begin the present section, I gave the command
+\begin{center}
+ \verb"\section{Basic \LaTeX{} stuff}"
+\end{center}
+A section number is supplied automatically. If you want to be able to
+make reference to that section, then you need to \emph{label} it.
+Since I wanted to be able to demonstrate the cross-reference commands,
+I actually began this section with the lines
+\begin{center}
+ \begin{tabular}{l}
+ \verb"\section{Basic \LaTeX{} stuff}"\\
+ \verb"\label{sec:basicstuff}"
+ \end{tabular}
+\end{center}
+This allows me to type ``\verb"section~\ref{sec:basicstuff}"'' and
+have it printed as ``section~\ref{sec:basicstuff}''.
+
+To begin a new subsection, you give the command
+\begin{center}
+ \verb"\subsection{Subsection name}"
+\end{center}
+To begin the present subsection, I gave the command
+\begin{center}
+ \verb"\subsection{Sections and subsections}"
+\end{center}
+A subsection number is supplied automatically. If you want to be able
+to make reference to that subsection, then you need to \emph{label}
+it. This subsection was begun with the lines
+\begin{center}
+ \begin{tabular}{l}
+ \verb"\subsection{Sections and subsections}"\\
+ \verb"\label{sec:sections}"
+ \end{tabular}
+\end{center}
+so if we type ``\verb"section~\ref{sec:sections}",'' it is printed as
+``section~\ref{sec:sections}''.
+
+Labels always take the number of the smallest enclosing structure.
+Thus, a \verb"\label" command that's inside a section but \emph{not}
+inside a subsection or Theorem or anything else will take the value of
+the section counter, while a \verb"\label" command that's inside the
+statement of a Theorem will take the value of that Theorem number.
+For more information on this, see section~\ref{sec:xreferences}.
+
+%--------------------------------------------------------------------
+\subsubsection{Yes, there are subsubsections too}
+
+I began this subsubsection with the command
+\begin{center}
+ \verb"\subsubsection{Yes, there are subsubsections too}"
+\end{center}
+
+%--------------------------------------------------------------------
+\subsubsection*{Sections without numbers}
+
+I began this subsubsection with the command
+\begin{center}
+ \verb"\subsubsection*{Sections without numbers}"
+\end{center}
+and got a subsubsection that wasn't numbered. If you give the command
+\begin{center}
+ \verb"\section*{A Section Title}"
+\end{center}
+then you'll begin a new section that will not have a number.
+
+
+%---------------------------------------------------------------------
+\subsection{Italics \emph{for emphasis}}
+
+If you want to use italics to emphasize a word or two, the \LaTeX{}
+convention is not to explicitly switch to italics, but rather to use
+the command \verb"\emph" (which means \emph{emphasize}). This command
+works just like a font change command, except that it switches you
+\emph{into} italics if the current font is upright and switches you
+\emph{out of} italics if the current font is italics.
+
+For example, if you type
+\begin{center}
+ \verb"The whole is \emph{more} than the sum of its parts."
+\end{center}
+you'll get
+\begin{center}
+ The whole is \emph{more} than the sum of its parts.
+\end{center}
+but if you type
+\begin{verbatim}
+\begin{thm}
+ The whole is \emph{more} than the sum of its parts.
+\end{thm}
+\end{verbatim}
+you'll get
+\begin{thm}
+ The whole is \emph{more} than the sum of its parts.
+\end{thm}
+
+\subsubsection*{Note}
+
+The \verb"\emph" command is a recent addition to \LaTeX, and it has
+the feature that it automatically inserts an italic correction where
+needed. If you don't know what an italic correction is, you can
+safely ignore this paragraph, but I will at least mention that all
+those ``\verb"\/"'' commands frequently seen in \TeX{} (and older
+\LaTeX) documents are all inserting italic corrections; the point of
+this paragraph is that, with the current version of \LaTeX, you don't
+have to do that anymore.
+
+%---------------------------------------------------------------------
+\subsection{Cross references and the table of contents}
+
+This is an explanation of how \LaTeX{} manages to fill in
+cross-references (see section~\ref{sec:xreferences}) to parts of the
+file it hasn't processed yet, and what those \verb".aux" and
+\verb".toc" files are.
+
+%--------------------------------------------------------------------
+\subsubsection*{Cross-References}
+
+Every time \LaTeX{} processes your file, it writes an \emph{auxiliary}
+file. Since the file containing these instructions is called
+\verb"amshelp.tex", the auxiliary file is called \verb"amshelp.aux".
+The auxiliary file contains the definitions of all the keys used for
+cross-references. When \LaTeX{} begins to process your file, it first
+looks for an \verb".aux" file, and reads it in if it exists. Of
+course, this is the \verb".aux" file that was produced the \emph{last}
+time that your file was processed, so the Theorem numbers, Section
+numbers, etc., are all the ones from the last time the file was
+processed.
+
+The very first time that \LaTeX{} processes your file, there is no
+\verb".aux" file, and so \LaTeX{} gives \emph{lots} of warning
+messages about undefined labels, or whatever. Ignore all of this.
+The \emph{next} time that you run \LaTeX, there \emph{will} be an
+\verb".aux" file, and all the references will be filled in. (Yes, it
+is possible, at least in theory, for some page number to change every
+time you run \LaTeX{} on your file, even without any changes in the
+source file, but this isn't very likely.)
+
+\subsubsection*{The Table of Contents}
+
+If you give the command \verb"\tableofcontents", then \LaTeX{} will
+try to write a table of contents at that point, including the page
+numbers of the sections. Obviously, \LaTeX{} can't know those page
+numbers or section titles yet, so as \LaTeX{} processes your file, it
+writes a \verb".toc" file containing the information it needs. (The
+\verb".toc" file for these instructions is \verb"amshelp.toc".) Once
+again, \LaTeX{} is always using the information from the \emph{last}
+time that it processed your file.
+
+If you \emph{do} include a table of contents in your document, and if
+the table of contents takes up at least a page or so of space, then
+you might have to run \LaTeX{} \emph{three} times in order to get all
+of the cross-references right. The reason for this is that the first
+time you run \LaTeX{} there isn't any \verb".toc" file listing the
+section titles, and so the table of contents has nothing in it. The
+second time you run \LaTeX{} you'll get a table of contents that lists
+the page numbers for the sections from the last time you ran \LaTeX,
+when the table of contents took up no space at all. Unfortunately,
+during this second run, the table of contents will be created, and
+will take up enough space to change the page numbers of the sections
+from what they were during the first run. Only during the
+\emph{third} run will the correct page numbers be written into the
+table of contents. Since this doesn't change the amount of space that
+the table of contents occupies, this version will be correct.
+
+\subsubsection*{How do I know when everything is correct?}
+
+After processing your file, \LaTeX{} checks whether all the
+cross-reference numbers that it read from the \verb".aux" file are
+correct. If any of them are incorrect, it prints a warning on the
+screen at the very end of the run advising you that labels may have
+changed and that you should run \LaTeX{} again to get the
+cross-references right. Unfortunately, \LaTeX{} doesn't seem to check
+that the table of contents entries are correct, so if you change the
+name of a section in a way that doesn't make any page references
+incorrect, you won't be warned to run \LaTeX{} again.
+
+
+%---------------------------------------------------------------------
+%---------------------------------------------------------------------
+\section{Title, Author, and the \texttt{\bs maketitle} command}
+
+This stuff should go right after the \verb"\begin{document}" command.
+I'll give a quick sketch here, which is probably all you'll ever
+need, but the full explanation is given in \emph{Instructions for
+ preparation of papers and monographs: \AmS-\LaTeX} \cite{instr-l}.
+If you are already familiar with \LaTeX, then you should be warned
+that this part is slightly different from what you do when using the
+standard \LaTeX{} \verb"article" document class.
+
+%---------------------------------------------------------------------
+\subsection{The title}
+You specify the title with the command
+\begin{center}
+ \verb"\title[Optional running title]{Actual title}"
+\end{center}
+These instructions used the title command
+\begin{verbatim}
+\title[Running \AmS-\LaTeX]{Getting up and running\\
+ with \AmS-\LaTeX}
+\end{verbatim}
+Notice that you indicate line breaks in the title with a double
+backslash. If I had decided to omit the line break and also to have
+the full title printed in the head of the odd numbered pages, I would
+have used the command
+\begin{center}
+ \verb"\title{Getting up and running with \AmS-\LaTeX}"
+\end{center}
+
+%---------------------------------------------------------------------
+\subsection{The author, and the author's address}
+
+The author is specified with an \verb"author" command:
+\begin{center}
+\verb"\author{Author's name}"
+\end{center}
+These directions used the command
+\verb"\author{Philip S. Hirschhorn}".
+The author's address is given in an address command, with double
+backslashes to indicate line breaks. These instructions used the
+command
+\begin{center}
+ \begin{tabular}{l}
+ \verb"\address{Department of Mathematics\\"\\
+ \verb"Wellesley College\\"\\
+ \verb"Wellesley, Massachusetts 02481}"
+ \end{tabular}
+\end{center}
+If the author's current address is different from the address at which
+the research was carried out, then you can specify the current address
+with the command \verb"\curraddr". For example, you might type
+\begin{center}
+ \begin{tabular}{l}
+ \verb"\curraddr{Department of Mechanics\\"\\
+ \verb"Brake and Wheel Bearing Division\\"\\
+ \verb"Serene Service Center\\"\\
+ \verb"Salem, Massachusetts 02139}"
+ \end{tabular}
+\end{center}
+You can also include an email address, with the \verb"\email"
+command. These instructions used the command
+\begin{center}
+ \verb"\email{psh@math.mit.edu}"
+\end{center}
+To acknowledge support, use the command \verb"\thanks", e.g.,
+\begin{center}
+ \verb"\thanks{Supported in part by NSF grant 3.14159}"
+\end{center}
+This will be printed as a footnote on the first page.
+
+%--------------------------------------------------------------------
+\subsubsection*{Multiple authors}
+
+If there are several authors, then each one should have a separate
+\verb"\author" command, with each individual's address, current
+address, email address, and thanks following that individual's
+\verb"\author" command, in its own \verb"\address" command (and
+\verb"\curraddr" command, and \verb"\thanks" command, and
+\verb"\email" command). If there \emph{are} several authors, and
+their combined names are too long for the running head on the even
+numbered pages, you can give an optional argument to each
+\verb"\author" command to supply a shortened form to use in the
+running head, as in
+\begin{center}
+ \verb"\author[P.S. Hirschhorn]{Philip S. Hirschhorn}".
+\end{center}
+(It's apparently a convention that the running head in a multiple
+author paper should have only initials for the first and middle names,
+but I don't think that I was invited to that convention.)
+
+
+%---------------------------------------------------------------------
+\subsection{The date}
+
+This is pretty straightforward:
+\begin{center}
+\verb"\date{Whatever date you please}"
+\end{center}
+To have the date of processing used, use the command
+\verb"\date{\today}".
+
+%---------------------------------------------------------------------
+\subsection{\texttt{\bs maketitle}}
+
+After you've given all of the commands mentioned in this section, you
+can give the command \verb"\maketitle". If you \emph{don't} give the
+command \verb"\maketitle", a title won't be made. The exact
+arrangement of all this information is determined by the document
+class. In particular, the \verb"amsart" document class puts the
+author's address at the \emph{end} of the paper.
+
+%---------------------------------------------------------------------
+%---------------------------------------------------------------------
+\section{Theorems, Propositions, Lemmas, etc.}
+
+The instructions in this section assume that you're using the
+\verb"\newtheorem" commands that I put in the file \verb"template.tex"
+(see section~\ref{sec:template}).
+
+%--------------------------------------------------------------------
+\subsection{Stating theorems, propositions, etc.}
+\label{sec:theorems}
+
+To state a theorem, you do the following:
+\begin{verbatim}
+\begin{thm}
+ The square of the hypotenuse of a right triangle is equal to the
+ sum of the squares of the two adjacent sides.
+\end{thm}
+\end{verbatim}
+If you do that, you'll get the following:
+\begin{thm}
+ \label{pythagthm}
+ The square of the hypotenuse of a right triangle is equal to the sum
+ of the squares of the two adjacent sides.
+\end{thm}
+If you thought that it was only a proposition, you'd use
+\begin{verbatim}
+\begin{prop}
+ The square of the hypotenuse of a right triangle is equal to the
+ sum of the squares of the two adjacent sides.
+\end{prop}
+\end{verbatim}
+and you'd get
+\begin{prop}
+ The square of the hypotenuse of a right triangle is equal to the sum
+ of the squares of the two adjacent sides.
+\end{prop}
+
+If you think it's a theorem again, but you'd like to make reference to
+it in some other part of the paper, you have to choose a \emph{key}
+with which you'll refer to it, and then \emph{label} the theorem. If
+you want to use the key \emph{pythagthm}, then it would look like the
+following:
+\begin{verbatim}
+\begin{thm}
+ \label{pythagthm}
+ The square of the hypotenuse of a right triangle is equal to the
+ sum of the squares of the two adjacent sides.
+\end{thm}
+\end{verbatim}
+If you later give the command \verb"\ref{pythagthm}", then that
+command will expand to the \emph{number} that was assigned to that
+theorem (in this case, \ref{pythagthm}). For more explanation of
+cross-references, see section~\ref{sec:xreferences}.
+
+If you'd like to state a theorem and give a \emph{name} to it, then
+you can add an optional argument to the \verb"\begin{thm}" command.
+If you type
+\begin{verbatim}
+\begin{thm}[Pythagoras]
+ The square of the hypotenuse of a right triangle is equal to the
+ sum of the squares of the two adjacent sides.
+\end{thm}
+\end{verbatim}
+you'll get
+\begin{thm}[Pythagoras]
+ The square of the hypotenuse of a right triangle is equal to the sum
+ of the squares of the two adjacent sides.
+\end{thm}
+
+%--------------------------------------------------------------------
+\subsubsection*{Summary of environments provided in the template}
+
+All of the following structures are numbered in the same sequence,
+in the form SectionNumber.Number. Equations (i.e., displayed
+formulas, whether they are equations or not) will be numbered in the
+same sequence.
+\begin{displaymath}
+ \begin{tabular}{c@{\hspace{4em}}l@{\hspace{4em}}c}
+ \multicolumn{3}{c}{Theorem Environments}\\*[8pt]
+ \hspace{1em}Name& Printed Form& Body font\\*[6pt]
+ \texttt{thm}& \textbf{Theorem}& Italic\\
+ \texttt{cor}& \textbf{Corollary}& Italic\\
+ \texttt{lem}& \textbf{Lemma}& Italic\\
+ \texttt{prop}& \textbf{Proposition}& Italic\\
+ \texttt{defn}& \textbf{Definition}& Normal\\
+ \texttt{rem}& \textit{Remark}& Normal\\
+ \texttt{ex}& \textit{Example}& Normal\\
+ \texttt{notation}& \textit{Notation}& Normal\\
+ \texttt{terminology}& \textit{Terminology}& Normal\\*[3pt]
+ \end{tabular}
+\end{displaymath}
+For full details, see the beginning of the template file (reproduced
+here in section~\ref{sec:template}), after the comment ``The Theorem
+Environments.''
+
+%---------------------------------------------------------------------
+\subsection{Proofs}
+
+To give a proof, you do the following:
+\begin{verbatim}
+\begin{proof}
+ As any fool can plainly see, it's true!
+\end{proof}
+\end{verbatim}
+and you'll get the following:
+\begin{proof}
+ As any fool can plainly see, it's true!
+\end{proof}
+If the theorem said that a condition was both necessary and sufficient
+for something, and you want to prove each part separately,
+you can do the following:
+\begin{verbatim}
+\begin{proof}[Proof (sufficiency)]
+ Well, it's \emph{obviously} sufficient!
+\end{proof}
+\end{verbatim}
+and you'll get
+\begin{proof}[Proof (sufficiency)]
+ Well, it's \emph{obviously} sufficient!
+\end{proof}
+That is, the \verb"proof" environment allows you to use an optional
+second argument that will appear in place of the word \verb"Proof".
+
+If the proof of Theorem~\ref{pythagthm} does not appear immediately
+after its statement, you might use the following:
+\begin{verbatim}
+\begin{proof}[Proof of Theorem~\ref{pythagthm}]
+ As any fool can plainly see, it's true!
+\end{proof}
+\end{verbatim}
+and you'd get
+\begin{proof}[Proof of Theorem~\ref{pythagthm}]
+ As any fool can plainly see, it's true!
+\end{proof}
+
+%---------------------------------------------------------------------
+%---------------------------------------------------------------------
+\section{Cross-References}
+\label{sec:xreferences}
+
+This section explains how to make reference to numbered sections,
+theorems, equations, and bibliography items, with the correct
+reference numbers filled in automatically by \LaTeX.
+
+%---------------------------------------------------------------------
+\subsection{References to sections, theorems and equations}
+\label{sec:thmrefs}
+
+For each structure in the manuscript to which you'll be making
+reference, you must assign a \emph{key} that you'll use to refer to
+that structure. For sections, theorems, numbered equations, and items
+in an enumerated list (see section~\ref{sec:enumref}), you assign the
+key using the \verb"\label" command and refer to it using either the
+\verb"\ref" command or the \verb"\eqref" command. Each of these
+commands takes one argument, which is the \emph{key} you're assigning
+to the object. The command \verb"\ref{key}" produces the number that
+was assigned to that structure and the command \verb"\eqref{key}"
+produces that number enclosed in parentheses. (The \verb"\eqref"
+command also ensures that the number and parentheses are always in an
+upright font; see section~\ref{sec:EqRef}.) The convention is to use
+\verb"\eqref" to refer to equation numbers, \verb"\cite" to refer to
+bibliography entries (see section~\ref{sec:bibreferences}), and
+\verb"\ref" to refer to everything else.
+
+Consider the following example.
+\begin{thm}
+ \label{homotopy}
+ If the maps $f\colon X \to Y$ and $g\colon X \to Y$ are homotopic,
+ then the induced homomorphisms $f_{*} \colon \mathrm{H}_{*}X \to
+ \mathrm{H}_{*}Y$ and $g_{*} \colon \mathrm{H}_{*}X \to
+ \mathrm{H}_{*}Y$ are equal.
+\end{thm}
+
+We typed that theorem as follows.
+\begin{verbatim}
+\begin{thm}
+ \label{homotopy}
+ If the maps $f\colon X \to Y$ and $g\colon X \to Y$ are homotopic,
+ then the induced homomorphisms $f_{*} \colon \mathrm{H}_{*}X \to
+ \mathrm{H}_{*}Y$ and $g_{*} \colon \mathrm{H}_{*}X \to
+ \mathrm{H}_{*}Y$ are equal.
+\end{thm}
+\end{verbatim}
+If we now type ``\verb"see Theorem~\ref{homotopy}",'' then it will be
+printed as ``see Theorem~\ref{homotopy}.''
+
+%--------------------------------------------------------------------
+\subsubsection*{So, what exactly is the label labeling?}
+
+The command \verb"\label{key}" assigns to \verb"key" the value of the
+\emph{smallest enclosing structure}. For example, we began this
+section by typing
+\begin{center}
+ \begin{tabular}{l}
+ \verb"\section{Cross-References}"\\
+ \verb"\label{sec:xreferences}"
+ \end{tabular}
+\end{center}
+and we began this subsection by typing
+\begin{center}
+ \begin{tabular}{l}
+ \verb"\subsection{References to sections, theorems and equations}"\\
+ \verb"\label{sec:thmrefs}"
+ \end{tabular}
+\end{center}
+The phrase ``\verb"See section~\ref{sec:xreferences}"'' is printed as
+``See section~\ref{sec:xreferences}'' while the phrase
+%
+``\verb"See section~\ref{sec:thmrefs}"'' is printed as ``See
+section~\ref{sec:thmrefs}'' because the key \verb"sec:xreferences" was
+defined inside of section~\ref{sec:xreferences} but outside of
+section~\ref{sec:thmrefs}, while the key \verb"sec:thmrefs" was
+defined inside of section~\ref{sec:thmrefs}.
+
+%--------------------------------------------------------------------
+\subsubsection{References to equations}
+\label{sec:EqRef}
+
+To make reference to a numbered equation, you assign the \emph{key} as
+before, but you replace \verb"\ref" with \verb"\eqref", so that
+parentheses will be printed around the equation number. For example,
+if you type
+\begin{verbatim}
+\begin{equation}
+ \label{additivity}
+ \mathrm{H}_{*} \bigvee_{\alpha\in A} X_{\alpha} \approx
+ \bigoplus_{\alpha\in A}\mathrm{H}_{*} X_{\alpha}
+\end{equation}
+\end{verbatim}
+then you'll get
+\begin{equation}
+ \label{additivity}
+ \mathrm{H}_{*} \bigvee_{\alpha\in A} X_{\alpha} \approx
+ \bigoplus_{\alpha\in A}\mathrm{H}_{*} X_{\alpha}
+\end{equation}
+If we now type
+\begin{verbatim}
+\begin{thm}
+ Equation~\eqref{additivity} is true for all sorts of functors
+ $\mathrm{H}$.
+\end{thm}
+\end{verbatim}
+then we'll get
+\begin{thm}
+ Equation~\eqref{additivity} is true for all sorts of functors
+ $\mathrm{H}$.
+\end{thm}
+Notice the parentheses around the equation number, and the fact that
+even though the theorem is set in slanted type, the equation number is
+set in an upright font? This is the difference between \verb"\eqref"
+and \verb"\ref"; the command \verb"\eqref" provides parentheses,
+arranges it so that the number and surrounding parentheses are in an
+upright font no matter what the surrounding font, and supplies an
+italic correction if it's needed.
+
+%---------------------------------------------------------------------
+\subsection{References to page numbers}
+
+If you want to make reference to the \emph{page} that contains a
+label, rather than to the structure that is labeled, use the command
+\verb"\pageref{key}". For example, if you type
+\begin{verbatim}
+See page~\pageref{homotopy} to find Theorem~\ref{homotopy}.
+\end{verbatim}
+you'll get ``See page~\pageref{homotopy} to find
+Theorem~\ref{homotopy}.''
+
+%---------------------------------------------------------------------
+\subsection{Bibliographic references}
+\label{sec:bibreferences}
+
+Each bibliography item receives a \emph{key} as part of its basic
+structure, and you refer to that item using the command
+\verb"\cite{key}".
+
+When using the \texttt{amsrefs} package, each item in the bibliography
+is begun with
+\begin{verbatim}
+\bib{key}{TypeOfItem}{
+\end{verbatim}
+For example, the bibliography of these instructions contains the entry
+\begin{verbatim}
+\bib{HA}{book}{
+ author={Quillen, Daniel G.},
+ title={Homotopical Algebra},
+ series={Lecture Notes in Mathematics},
+ volume={43},
+ publisher={Springer-Verlag},
+ address={Berlin-New York},
+ date={1967}
+}
+\end{verbatim}
+If we type ``\verb"This is the work of Quillen~\cite{HA}",'' then it
+will be printed as ``This is the work of Quillen~\cite{HA}.'' Notice
+that square brackets have been inserted around the bibliography item
+number.
+
+The \verb"\cite" command takes an optional argument, which allows you
+to annotate the reference. If we type
+``\verb"see~\cite[Chapter I]{HA}"'', then it will be printed as
+``see~\cite[Chapter I]{HA}''. If you're using \texttt{amsrefs} (which
+we strongly recommend), then there's an alternate form available: If
+we type ``\verb"\cite{HA}*{Chapter I}"'', then we also get
+``\cite{HA}*{Chapter I}'', and this second form is less likely to
+cause errors when used, e.g., in the optional argument to a
+\verb"\begin{theorem}" command (see section~\ref{sec:theorems}).
+
+%--------------------------------------------------------------------
+\subsubsection{Multiple references}
+\label{sec:mulref}
+
+If you're using \texttt{amsrefs} and when you refer to multiple
+bibliography items you want to have the item numbers automatically
+sorted and compressed (e.g., replacing ``7, 6, 5, 8'' with ``5--8''),
+you can use the \verb"\cites" command. For example, if we type
+\verb"\cites{HA,yellowmonster}" then we get \cites{HA,yellowmonster}.
+
+If you want some of the references in the list to have annotations,
+you put your \verb"\cite" commands into the argument of a
+\verb"\citelist" command. For example, if we type
+\verb"\citelist{\cite{HA}*{Chapter I} \cite{yellowmonster}}" then we
+get \citelist{\cite{HA}*{Chapter I} \cite{yellowmonster}}.
+
+%--------------------------------------------------------------------
+\subsubsection{Author-year citations}
+\label{sec:authyrcit}
+
+If you're using \texttt{amsrefs} and you use the \texttt{author-year}
+option to convert your references to the author-year format (see
+section~\ref{sec:refoptions}), then there are two variations on the
+\verb"\cite" command (\verb"\ycite" and \verb"\ocite") that are
+useful. If we were using the \texttt{author-year} option in this
+document, then
+\begin{center}
+ \begin{tabular}{l@{ would be typeset as }l}
+ ``\verb"\cite{yellowmonster}"''& ``(Bousfield and Kan, 1972)''\\
+ ``\verb"\ycite{yellowmonster}"''& ``(1972)''\\
+ ``\verb"\ocite{yellowmonster}"''& ``Bousfield and Kan (1972)''
+ \end{tabular}
+\end{center}
+Thus, you could type ``For further details, see
+\verb"\cite{yellowmonster}"'', or ``Bousfield and Kan
+\verb"\ycite{yellowmonster}" showed\ldots'', or ``This can be found in
+\verb"\ocite{yellowmonster}"''.
+
+
+%---------------------------------------------------------------------
+%---------------------------------------------------------------------
+\section{Mathematics in running text}
+
+This is pretty much exactly as it is in plain \TeX, except that you
+have an extra option (which you can ignore). The simplest thing is to
+just enclose between dollar signs any material that should be in math
+mode. Thus, if you type
+\begin{center}
+ \verb"Let $f\colon X \to Y$ be a continuous function."
+\end{center}
+you'll get
+\begin{center}
+ Let $f\colon X \to Y$ be a continuous function.
+\end{center}
+(Note that if we had typed that as ``\verb"f: X \to Y"'' then the
+spacing around the colon would be wrong, since the colon symbol is
+typeset as a binary relation, with spaces on the two sides equal.)
+The only novelty that \LaTeX{} introduces is that, instead of using a
+dollar sign to toggle math mode on and off, you can use `\verb"\("' to
+\emph{begin} math mode, and `\verb"\)"' to \emph{end} math mode.
+Thus, the example above could also be typed as
+\begin{center}
+\verb"Let \(f\colon X \to Y\) be a continuous function."
+\end{center}
+This provides a tiny bit more error checking, but can
+otherwise be safely ignored.
+
+The reference \cite{mathguide} is an excellent concise summary of the
+features of \AmS-\LaTeX{} for typesetting mathematics, both displayed
+and in running text. It also includes lists of symbols available for
+mathematics.
+
+%--------------------------------------------------------------------
+%---------------------------------------------------------------------
+\section{Displayed mathematics}
+
+For a complete discussion of the environments for displayed
+mathematics, including options for customizing equation numbers,
+see~\cite[section~3]{amslatexusersguide}.
+
+%--------------------------------------------------------------------
+\subsection{Single line displays}
+\label{sec:SngLne}
+
+To display mathematics and number the display (so that you can refer
+to it from elsewhere in the paper) you use the \verb"equation"
+environment. (\LaTeX{} calls all such numbers \emph{equation
+ numbers}, whether or not the display has anything to do with
+equations.) If you type
+\begin{verbatim}
+\begin{equation}
+ \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
+\end{equation}
+\end{verbatim}
+you'll get
+\begin{equation}
+ \label{pi1eqn}
+ \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
+\end{equation}
+
+If you'd like to be able to make reference to the equation number,
+you need to \emph{label} the equation, using a \emph{key} that you
+can use for referencing it:
+\begin{verbatim}
+\begin{equation}
+ \label{pi1eqn}
+ \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
+\end{equation}
+\end{verbatim}
+If you later type ``\verb"see formula~\eqref{pi1eqn}"'' you'll get
+``see formula~\eqref{pi1eqn}.'' (For more on cross-references to
+formulas, see section~\ref{sec:thmrefs}.)
+
+To display a single line of mathematics without an equation number,
+you use the \verb"equation*" environment. (This is a common \LaTeX
+ism: Adding an asterisk to the name of a numbered \LaTeX{} environment
+often gives the unnumbered equivalent.) If you type
+\begin{verbatim}
+\begin{equation*}
+ \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
+\end{equation*}
+\end{verbatim}
+then you'll get
+\begin{equation*}
+ \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
+\end{equation*}
+The \texttt{displaymath} environment is equivalent to this; you can
+produce the same unnumbered display by typing
+\begin{verbatim}
+\begin{displaymath}
+ \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
+\end{displaymath}
+\end{verbatim}
+You can also produce that same unnumbered display by typing
+\begin{verbatim}
+\[
+ \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
+\]
+\end{verbatim}
+but some people think that that makes your \LaTeX{} source file less
+readable. Although you can also produce the same thing by typing
+\begin{verbatim}
+$$
+ \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
+$$
+\end{verbatim}
+using double dollar signs is definitely deprecated.
+
+%--------------------------------------------------------------------
+\subsection{Text in displayed mathematics}
+\label{sec:text}
+
+To include text in a displayed mathematics environment you use the
+\verb"\text" command, as in \verb"\text{here's some text}". The
+\verb"\text" command is preferable to the standard \LaTeX{}
+\verb"\mbox" command because \verb"\text" correctly adjusts its size
+for use in subscripts and superscripts. For example, if you type
+\begin{verbatim}
+\begin{equation*}
+ \pi_{1}(X) \approx G * H
+ \quad\text{where $G$ is torsion and $H$ is torsion free}
+\end{equation*}
+\end{verbatim}
+then you'll get
+\begin{equation*}
+ \pi_{1}(X) \approx G * H
+ \quad\text{where $G$ is torsion and $H$ is torsion free}
+\end{equation*}
+and if you type
+\begin{verbatim}
+\begin{equation*}
+ X_{n} = \coprod_{\text{monomorphisms $[k] \to [n]$}} S_{k}
+\end{equation*}
+\end{verbatim}
+then you'll get
+\begin{equation*}
+ X_{n} = \coprod_{\text{monomorphisms $[k] \to [n]$}} S_{k}
+\end{equation*}
+
+
+%--------------------------------------------------------------------
+\subsection{Displaying multiple lines without alignment}
+\label{sec:gather}
+
+You can put several displayed lines together, each one centered, with
+no alignment between the different lines, using the \verb"gather"
+environment. When typing this, the lines are separated by a double
+backslash \verb"\\". For example, if you type
+\begin{verbatim}
+\begin{gather}
+ (X\otimes L) \amalg_{(X\otimes K)} (Y\otimes K)
+ \longrightarrow Y\otimes L\\
+ X^{L} \longrightarrow X^{K} \times_{Y^{K}} Y^{L}
+\end{gather}
+\end{verbatim}
+then you'll get
+\begin{gather}
+ \label{eq:push}
+ (X\otimes L) \amalg_{(X\otimes K)} (Y\otimes K)
+ \longrightarrow Y\otimes L\\
+ \label{eq:pull}
+ X^{L} \longrightarrow X^{K} \times_{Y^{K}} Y^{L}
+\end{gather}
+The \verb"gather*" environment would produce the same thing without
+the equation numbers. You can also label each line so that you can
+refer to them: If you had typed that as
+\begin{verbatim}
+\begin{gather}
+ \label{eq:push}
+ (X\otimes L) \amalg_{(X\otimes K)} (Y\otimes K)
+ \longrightarrow Y\otimes L\\
+ \label{eq:pull}
+ X^{L} \longrightarrow X^{K} \times_{Y^{K}} Y^{L}
+\end{gather}
+\end{verbatim}
+and then typed ``\verb"see \eqref{eq:push} or \eqref{eq:pull}"'',
+you'd get ``see \eqref{eq:push} or \eqref{eq:pull}''.
+
+There is also a \texttt{gathered} environment, which does not produce
+a display of its own but rather produces a block of centered lines of
+mathematics that can be used inside of another displayed mathematics
+environment (see section~\ref{sec:gathered}).
+%--------------------------------------------------------------------
+\subsection{Displays with linebreaks}
+\label{sec:multline}
+
+For a long display that must be broken across several lines, you can
+use the \texttt{multline} environment. (There is also a
+\texttt{multline*} environment, which is similar except that it omits
+the equation number.) When typing this, the lines are separated by a
+double backslash \verb"\\". The first line will be shifted left of
+center, the last will be shifted right of center, and the lines in
+between those will be centered. For example, if you type
+\begin{verbatim}
+\begin{multline}
+ \label{eq:BigComp}
+ \mathrm{F} X\otimes\Delta[n] \xrightarrow{1 \otimes D}
+ \mathrm{F} X\otimes(\Delta[n]\times\Delta[n])\\
+ \xrightarrow{\sigma}
+ \mathrm{F}\bigl(X\otimes(\Delta[n]\times\Delta[n])\bigr)
+ \approx
+ \mathrm{F}\bigl((X\otimes\Delta[n])\otimes\Delta[n]\bigr)\\
+ \xrightarrow{\mathrm{F}(\alpha\otimes 1)}
+ \mathrm{F}(Y\otimes\Delta[n]) \xrightarrow{\mathrm{F}(\beta)}
+ \mathrm{F}(Z)
+\end{multline}
+\end{verbatim}
+then you'll get
+\begin{multline}
+ \label{eq:BigComp}
+ \mathrm{F} X\otimes\Delta[n] \xrightarrow{1 \otimes D}
+ \mathrm{F} X\otimes(\Delta[n]\times\Delta[n])\\
+ \xrightarrow{\sigma}
+ \mathrm{F}\bigl(X\otimes(\Delta[n]\times\Delta[n])\bigr)
+ \approx
+ \mathrm{F}\bigl((X\otimes\Delta[n])\otimes\Delta[n]\bigr)\\
+ \xrightarrow{\mathrm{F}(\alpha\otimes 1)}
+ \mathrm{F}(Y\otimes\Delta[n]) \xrightarrow{\mathrm{F}(\beta)}
+ \mathrm{F}(Z)
+\end{multline}
+You can then type ``\verb"the composition \eqref{eq:BigComp}"" and it
+will appear as ``the composition \eqref{eq:BigComp}''.
+
+%--------------------------------------------------------------------
+\subsection{Displays with alignment}
+\label{sec:align}
+
+There are a number of ways to produce displays with multiple lines and
+horizontal alignment between the lines.
+\begin{itemize}
+\item The \texttt{align} and \texttt{align*} environments allow
+ horizontal alignment of characters chosen from each line (see
+ section~\ref{sec:SingAlign}).
+\item The \texttt{aligned} and \texttt{split} environments produce
+ alignments that can be be part of a larger display (see
+ section~\ref{sec:aligned}).
+\item The \texttt{gathered} environment produces a block of several
+ lines, each centered as in the \texttt{gather} environment (see
+ section~\ref{sec:gather}), that can be part of a larger display (see
+ section~\ref{sec:gathered}). (Of course, this doesn't involve
+ horizontal alignment, but the \texttt{gathered} environment is often
+ used in conjunction with other display environments.)
+\item The \texttt{align} and \texttt{align*} environments also allow
+ for multiple columns with alignment in each column (see
+ section~\ref{sec:MulAlign}).
+\item The \texttt{flalign} and \texttt{flalign*} environments allow
+ multiple alignment points and place the outermost aligned blocks
+ flush against the margins (see section~\ref{sec:flalign}).
+\item The \texttt{alignat} and \texttt{alignat*} environments allow
+ multiple alignment points and allow you to choose the spacing
+ between the columns (see section~\ref{sec:alignat}).
+\item The \texttt{alignedat} environment is similar to the
+ \texttt{alignat*} environment, except that it produces an alignment
+ intended to be part of a larger display (see
+ section~\ref{sec:alignedat}).
+\end{itemize}
+%--------------------------------------------------------------------
+\subsubsection{Displays with a single alignment point}
+\label{sec:SingAlign}
+
+To display several lines of mathematics with horizontal alignment, you
+use the \texttt{align} environment. (There is also an \texttt{align*}
+environment, which is similar except that it omits the equation
+numbers.) When typing this, the lines are separated by a double
+backslash \verb"\\" and each line has an ampersand \verb"&"
+immediately preceding the symbol to be aligned with the corresponding
+symbols on the other lines. For example, if you type
+\begin{verbatim}
+\begin{align}
+ \label{eq:pi1}
+ \pi_{1}(X\vee Y) &\approx \pi_{1}X * \pi_{1}Y\\
+ \label{eq:additivity}
+ \widetilde{\mathrm{H}}_{*}(X\vee Y) &\approx
+ \widetilde{\mathrm{H}}_{*}X \oplus \widetilde{\mathrm{H}}_{*}Y
+\end{align}
+\end{verbatim}
+then you'll get
+\begin{align}
+ \label{eq:pi1}
+ \pi_{1}(X\vee Y) &\approx \pi_{1}X * \pi_{1}Y\\
+ \label{eq:additivity}
+ \widetilde{\mathrm{H}}_{*}(X\vee Y) &\approx
+ \widetilde{\mathrm{H}}_{*}X \oplus \widetilde{\mathrm{H}}_{*}Y
+\end{align}
+and if you type ``\verb"see \eqref{eq:pi1} or \eqref{eq:additivity}"''
+then you'll get ``see \eqref{eq:pi1} or \eqref{eq:additivity}''. For
+another example, if you type
+\begin{verbatim}
+\begin{align*}
+ \mathcal{M}\bigl((\operatorname{colim} \boldsymbol{X})
+ \otimes K,Y\bigr) &\approx
+ \mathcal{M}(\operatorname{colim} \boldsymbol{X},Y^K)\\
+ &\approx \lim \mathcal{M}(\boldsymbol{X},Y^K)\\
+ &\approx \lim \mathcal{M}(\boldsymbol{X} \otimes K,Y)\\
+ &\approx \mathcal{M}\bigl(\operatorname{colim}
+ (\boldsymbol{X} \otimes K),Y\bigr)
+\end{align*}
+\end{verbatim}
+then you'll get
+\begin{align*}
+ \mathcal{M}\bigl((\operatorname{colim} \boldsymbol{X})
+ \otimes K,Y\bigr) &\approx
+ \mathcal{M}(\operatorname{colim} \boldsymbol{X},Y^K)\\
+ &\approx \lim \mathcal{M}(\boldsymbol{X},Y^K)\\
+ &\approx \lim \mathcal{M}(\boldsymbol{X} \otimes K,Y)\\
+ &\approx \mathcal{M}\bigl(\operatorname{colim}
+ (\boldsymbol{X} \otimes K),Y\bigr)
+\end{align*}
+Both the \texttt{align} and \texttt{align*} environments allow you to
+have multiple columns in the display; for this, see
+section~\ref{sec:MulAlign}.
+
+%--------------------------------------------------------------------
+\subsubsection{Alignments that are part of a larger display}
+\label{sec:aligned}
+
+The \texttt{aligned} environment does not create a display of its own.
+Instead, it allows you to have several lines of mathematics, with
+alignment as in the \texttt{align*} environment (see
+section~\ref{sec:SingAlign}), which can then be used inside of a
+displayed mathematics environment.
+
+The \textbf{aligned} environment doesn't produce any equation numbers
+of its own. If the enclosing display environment produces a number,
+and if the \texttt{aligned} environment doesn't use the optional
+argument of \texttt{[t]} or \texttt{[b]} (see \eqref{eq:dbltop} and
+\eqref{eq:dblbot}), then that number is centered vertically in the
+contents of the \texttt{aligned} environment. For an example of this,
+see \eqref{eq:longeq}.
+
+There is also a \texttt{split} environment, which is similar to the
+\texttt{aligned} environment except that the \texttt{aligned}
+environment has options (changing the vertical alignment (see the
+examples in this section) or having multiple alignment points (see
+section~\ref{sec:MulAlign})) that are not possible with the
+\texttt{split} environment. Thus, a \texttt{split} environment can
+always be replaced by an \texttt{aligned} environment.
+
+For example, if you type
+\begin{verbatim}
+\begin{equation}
+ \label{eq:dblaligned}
+ \begin{aligned}
+ a &= b + c\\
+ d &= e + f
+ \end{aligned}
+ \qquad\text{which, of course, leads to}\qquad
+ \begin{aligned}
+ g &= h + i\\
+ j &= k + l
+ \end{aligned}
+\end{equation}
+\end{verbatim}
+then you'll get
+\begin{equation}
+ \label{eq:dblaligned}
+ \begin{aligned}
+ a &= b + c\\
+ d &= e + f
+ \end{aligned}
+ \qquad\text{which, of course, leads to}\qquad
+ \begin{aligned}
+ g &= h + i\\
+ j &= k + l
+ \end{aligned}
+\end{equation}
+
+The above example illustrates that, by default, when an
+\texttt{aligned} (or \texttt{aligned*}) environment is used as part of
+a display involving other elements, it is vertically aligned at its
+center. The \texttt{aligned} environment can take an optional
+argument of \texttt{[t]} or \texttt{[b]} to cause the alignment to be
+vertically aligned at either the top or the bottom row. For example,
+if you type
+\begin{verbatim}
+\begin{equation}
+ \label{eq:dbltop}
+ \begin{aligned}[t]
+ a &= b + c\\
+ d &= e + f
+ \end{aligned}
+ \qquad\text{which, of course, leads to}\qquad
+ \begin{aligned}[t]
+ g &= h + i\\
+ j &= k + l
+ \end{aligned}
+\end{equation}
+\end{verbatim}
+then you'll get
+\begin{equation}
+ \label{eq:dbltop}
+ \begin{aligned}[t]
+ a &= b + c\\
+ d &= e + f
+ \end{aligned}
+ \qquad\text{which, of course, leads to}\qquad
+ \begin{aligned}[t]
+ g &= h + i\\
+ j &= k + l
+ \end{aligned}
+\end{equation}
+and if you type
+\begin{verbatim}
+\begin{equation}
+ \label{eq:dblbot}
+ \begin{aligned}[b]
+ a &= b + c\\
+ d &= e + f
+ \end{aligned}
+ \qquad\text{which, of course, leads to}\qquad
+ \begin{aligned}[b]
+ g &= h + i\\
+ j &= k + l
+ \end{aligned}
+\end{equation}
+\end{verbatim}
+then you'll get
+\begin{equation}
+ \label{eq:dblbot}
+ \begin{aligned}[b]
+ a &= b + c\\
+ d &= e + f
+ \end{aligned}
+ \qquad\text{which, of course, leads to}\qquad
+ \begin{aligned}[b]
+ g &= h + i\\
+ j &= k + l
+ \end{aligned}
+\end{equation}
+
+The display created by an \texttt{aligned} environment can be used as
+an element of a display. For example, if you type
+\begin{verbatim}
+\begin{equation*}
+ \left .
+ \begin{aligned}
+ f(x) &= \sin^{2} x\\
+ g(x) &= \cos^{2} x\\
+ h(x) &= \sin x + \cos x
+ \end{aligned}
+ \right \}
+ \qquad \text{A nice collection of functions}
+\end{equation*}
+\end{verbatim}
+then you'll get
+\begin{equation*}
+ \left .
+ \begin{aligned}
+ f(x) &= \sin^{2} x\\
+ g(x) &= \cos^{2} x\\
+ h(x) &= \sin x + \cos x
+ \end{aligned}
+ \right \}
+ \qquad \text{A nice collection of functions}
+\end{equation*}
+
+
+%--------------------------------------------------------------------
+\subsubsection{Several centered lines as part of a larger display}
+\label{sec:gathered}
+
+The \texttt{gathered} environment does not create a display of its
+own. Instead, it allows you to have several lines of mathematics,
+each one centered as in the \texttt{gather} environment (see
+section~\ref{sec:gather}), which can then be used inside of a
+displayed mathematics environment. For example, if you type
+\begin{verbatim}
+\begin{equation}
+ \begin{gathered}
+ x^{2} + y^{2} = z^{2}\\
+ a = b + c
+ \end{gathered}
+ \qquad \text{and, in addition,}\qquad
+ \begin{aligned}
+ A &= B+C\\
+ &= D+E
+ \end{aligned}
+\end{equation}
+\end{verbatim}
+then you'll get
+\begin{equation}
+ \begin{gathered}
+ x^{2} + y^{2} = z^{2}\\
+ a = b + c
+ \end{gathered}
+ \qquad \text{and, in addition,}\qquad
+ \begin{aligned}
+ A &= B+C\\
+ &= D+E
+ \end{aligned}
+\end{equation}
+
+The \texttt{gathered} environment can take the same optional argument
+of \texttt{[t]} or \texttt{[b]} that the \texttt{aligned} argument can
+take (see section~\ref{sec:aligned}) to cause the vertical alignment
+to be on either the top or the bottom line. For example, if you type
+\begin{verbatim}
+\begin{equation}
+ \begin{gathered}[t]
+ x^{2} + y^{2} = z^{2}\\
+ a = b + c
+ \end{gathered}
+ \qquad \text{and, in addition,}\qquad
+ \begin{aligned}[t]
+ A &= B+C\\
+ &= D+E
+ \end{aligned}
+\end{equation}
+\end{verbatim}
+then you'll get
+\begin{equation}
+ \begin{gathered}[t]
+ x^{2} + y^{2} = z^{2}\\
+ a = b + c
+ \end{gathered}
+ \qquad \text{and, in addition,}\qquad
+ \begin{aligned}[t]
+ A &= B+C\\
+ &= D+E
+ \end{aligned}
+\end{equation}
+
+%--------------------------------------------------------------------
+\subsubsection{Multiple alignment points}
+\label{sec:MulAlign}
+
+The \texttt{align} and \texttt{align*} environments (see
+section~\ref{sec:SingAlign}) as well as the \texttt{aligned}
+environment (see section~\ref{sec:aligned}) can be used to create
+displays in which each line is broken into several pieces, with each
+piece horizontally aligned at a chosen character. That is: they can
+be used to create a display with several columns, with alignment
+within each column.
+
+When typing this, the lines are separated by a double backslash
+\verb"\\", the different displays on each line are separated by an
+ampersand \verb"&", and the symbols to be aligned are preceded by an
+ampersand \verb"&". For example, if you type
+\begin{verbatim}
+\begin{align*}
+ K &\approx G * H& i&= j+k& B &\subset C\\
+ H &\approx A_{0}*B_{0}& i'&= j'+k'& C &= D\cap E\\
+ G &\approx \coprod_{\alpha\in A} L_{\alpha}& i''&=j''+k''&
+ A &= D \cup E
+\end{align*}
+\end{verbatim}
+then you'll get
+\begin{align*}
+ K &\approx G * H& i&= j+k& B &\subset C\\
+ H &\approx A_{0}*B_{0}& i'&= j'+k'& C &= D\cap E\\
+ G &\approx \coprod_{\alpha\in A} L_{\alpha}& i''&=j''+k''&
+ A &= D \cup E
+\end{align*}
+When typing such an alignment with $n$ columns, each line will have at
+most $2n-1$ ampersands.
+
+For another example, if you type
+\begin{verbatim}
+\begin{align*}
+ \pi_{1}(X\vee Y) &\approx \pi_{1}(X) * \pi_{1}(Y)
+ &&\text{(by the van Kampen theorem)}\\
+ &\approx G*H &&\text{(by the computation in the previous section)}
+\end{align*}
+\end{verbatim}
+then you'll get
+\begin{align*}
+ \pi_{1}(X\vee Y) &\approx \pi_{1}(X) * \pi_{1}(Y)
+ &&\text{(by the van Kampen theorem)}\\
+ &\approx G*H &&\text{(by the computation in the previous section)}
+\end{align*}
+If you want to specify the separation between the columns in the
+alignment, you should use the \texttt{alignat} environment (see
+section~\ref{sec:alignat}).
+
+
+%--------------------------------------------------------------------
+\subsubsection{Alignments flush left and flush right}
+\label{sec:flalign}
+
+To produce alignments similar to those in section~\ref{sec:MulAlign}
+except with the leftmost column flush left and the rightmost column
+flush right, you use the \texttt{flalign} environment (or, to omit the
+equation numbers, the \texttt{flalign*} environment). For example, if
+you type
+\begin{verbatim}
+\begin{flalign*}
+ K &\approx G * H& i&= j+k& B &\subset C\\
+ H &\approx A_{0}*B_{0}& i'&= j'+k'& C &= D\cap E\\
+ G &\approx \coprod_{\alpha\in A} L_{\alpha}& i''&=j''+k''&
+ A &= D \cup E
+\end{flalign*}
+\end{verbatim}
+then you'll get
+\begin{flalign*}
+ K &\approx G * H& i&= j+k& B &\subset C\\
+ H &\approx A_{0}*B_{0}& i'&= j'+k'& C &= D\cap E\\
+ G &\approx \coprod_{\alpha\in A} L_{\alpha}& i''&=j''+k''&
+ A &= D \cup E
+\end{flalign*}
+
+%--------------------------------------------------------------------
+\subsubsection{Multiple alignment points with chosen spacing}
+\label{sec:alignat}
+
+To produce alignments as in section~\ref{sec:MulAlign} except with the
+ability to choose the amount of horizontal space between the columns,
+you use the \texttt{alignat} environment (or, to omit the equation
+numbers, the \texttt{alignat*} environment). These environments don't
+insert any horizontal space between the columns, and so you can insert
+the exact amount of space you want by including it at the beginning of
+one of the columns.
+
+The format of \texttt{alignat} is slightly different from that of the
+\texttt{align} environment in that you must include an argument
+specifying the number of columns. For example, if you type
+\begin{verbatim}
+\begin{alignat}{2}
+ K &\approx G*H& \qquad&\text{(by an earlier theorem)}\\
+ A &\approx \lim_{i\in I} A_{i}& &\text{(by the definition of $A$)}
+\end{alignat}
+\end{verbatim}
+then you'll get
+\begin{alignat}{2}
+ K &\approx G*H& \qquad&\text{(by an earlier theorem)}\\
+ A &\approx \lim_{i\in I} A_{i}& &\text{(by the definition of $A$)}
+\end{alignat}
+Note that it's only necessary to insert the \verb"\qquad" space in one
+row; the alignment forces the space to appear in all rows.
+
+%--------------------------------------------------------------------
+\subsubsection{Multiple alignment points with chosen spacing as part of a larger display}
+\label{sec:alignedat}
+
+The \texttt{alignedat} environment is similar to the \texttt{aligned}
+environment (see section~\ref{sec:aligned}), in that neither of them
+create a display of their own but are instead used as a part of a
+larger display, but the \texttt{alignedat} environment allows you to
+choose the spacing between the columns. That is, the
+\texttt{alignedat} environment doesn't insert any space between the
+columns, so you can insert the exact amount of space you want by
+including it at the beginning of one of the columns. The
+\texttt{alignedat} environment also has a required argument stating
+the number of columns. For example, if you type
+\begin{verbatim}
+\begin{equation*}
+ \left \{
+ \begin{alignedat}{2}
+ a &= b + c& d &= e + f\\
+ A &= B + C \qquad& D &= E + F
+ \end{alignedat}
+ \right \}
+ \qquad \text{Twin pairs of equations}
+\end{equation*}
+\end{verbatim}
+then you'll get
+\begin{equation*}
+ \left \{
+ \begin{alignedat}{2}
+ a &= b + c& d &= e + f\\
+ A &= B + C \qquad& D &= E + F
+ \end{alignedat}
+ \right \}
+ \qquad \text{Twin pairs of equations}
+\end{equation*}
+Note that it's only necessary to insert the \verb"\qquad" space in one
+row; the alignment forces the space to appear in all rows.
+
+
+%--------------------------------------------------------------------
+\subsection{The \texttt{cases} environment}
+\label{sec:cases}
+
+There is a \texttt{cases} environment, which constructs the usual
+display of several cases, and which is used as a part of one of the
+displayed mathematics environments. For example, if you type
+\begin{verbatim}
+\begin{equation}
+ \label{eq:abs}
+ |x| =
+ \begin{cases}
+ x& \text{if $x \ge 0$}\\
+ -x& \text{if $x < 0$}
+ \end{cases}
+\end{equation}
+\end{verbatim}
+then you'll get
+\begin{equation}
+ \label{eq:abs}
+ |x| =
+ \begin{cases}
+ x& \text{if $x \ge 0$}\\
+ -x& \text{if $x < 0$}
+ \end{cases}
+\end{equation}
+
+For another example, if you type
+\begin{verbatim}
+\begin{align*}
+ d_{i} \sigma &=
+ \begin{cases}
+ \alpha_{1} \xrightarrow{\sigma_{1}} \alpha_{2}
+ \xrightarrow{\sigma_{2}} \cdots
+ \xrightarrow{\sigma_{n-1}} \alpha_{n}
+ &\text{if $i=0$}\\
+ \alpha_0 \xrightarrow{\sigma_{0}} \cdots
+ \xrightarrow{\sigma_{i-2}} \alpha_{i-1}
+ \xrightarrow{\sigma_{i}\sigma_{i-1}} \alpha_{i+1}
+ \xrightarrow{\sigma_{i+1}} \cdots
+ \xrightarrow{\sigma_{n-1}} \alpha_{n}
+ &\text{if $0<i<n$}\\
+ \alpha_{0} \xrightarrow{\sigma_{0}} \alpha_{1}
+ \xrightarrow{\sigma_1} \cdots
+ \xrightarrow{\sigma_{n-2}} \alpha_{n-1}
+ &\text{if $i=n$}
+ \end{cases}\\
+ s_{i} \sigma &= \alpha_{0} \xrightarrow{\sigma_{0}} \cdots
+ \xrightarrow{\sigma_{i-1}} \alpha_{i}
+ \xrightarrow{1_{\alpha_{i}}} \alpha_{i}
+ \xrightarrow{\sigma_{i}} \alpha_{i+1}
+ \xrightarrow{\sigma_{i+1}} \cdots
+ \xrightarrow{\sigma_{n-1}} \alpha_{n}
+\end{align*}
+\end{verbatim}
+then you'll get
+\begin{align*}
+ d_{i} \sigma &=
+ \begin{cases}
+ \alpha_{1} \xrightarrow{\sigma_{1}} \alpha_{2}
+ \xrightarrow{\sigma_{2}} \cdots
+ \xrightarrow{\sigma_{n-1}} \alpha_{n}
+ &\text{if $i=0$}\\
+ \alpha_0 \xrightarrow{\sigma_{0}} \cdots
+ \xrightarrow{\sigma_{i-2}} \alpha_{i-1}
+ \xrightarrow{\sigma_{i}\sigma_{i-1}} \alpha_{i+1}
+ \xrightarrow{\sigma_{i+1}} \cdots
+ \xrightarrow{\sigma_{n-1}} \alpha_{n}
+ &\text{if $0<i<n$}\\
+ \alpha_{0} \xrightarrow{\sigma_{0}} \alpha_{1}
+ \xrightarrow{\sigma_1} \cdots
+ \xrightarrow{\sigma_{n-2}} \alpha_{n-1}
+ &\text{if $i=n$}
+ \end{cases}\\
+ s_{i} \sigma &= \alpha_{0} \xrightarrow{\sigma_{0}} \cdots
+ \xrightarrow{\sigma_{i-1}} \alpha_{i}
+ \xrightarrow{1_{\alpha_{i}}} \alpha_{i}
+ \xrightarrow{\sigma_{i}} \alpha_{i+1}
+ \xrightarrow{\sigma_{i+1}} \cdots
+ \xrightarrow{\sigma_{n-1}} \alpha_{n}
+\end{align*}
+
+%--------------------------------------------------------------------
+\subsection{Multiple lines with equation numbers on some of them}
+\label{sec:MuEqOneNo}
+
+By default, most of the environments for displayed mathematics either
+have an equation number on every line or no equation numbers at all.
+In this section we discuss how to have multiple lines with either one
+equation number for the entire display (see
+sections~\ref{sec:MulLinOneNo} and \ref{sec:GatherCenter}) or equation
+numbers on some of the lines but not all of them (see
+section~\ref{sec:notag}).
+
+%--------------------------------------------------------------------
+\subsubsection{Multiple lines with horizontal alignment and one equation number}
+\label{sec:MulLinOneNo}
+
+To display multiple lines with horizontal alignment but with only one
+equation number for the entire display, you use the \texttt{equation}
+environment (see section~\ref{sec:SngLne}) to produce the equation
+number and you put an \texttt{aligned} environment (see
+section~\ref{sec:aligned}) inside of that to produce the alignment.
+For example, if you type
+\begin{verbatim}
+\begin{equation}
+ \label{eq:BothCoprod}
+ \begin{aligned}
+ \pi_{1}(X\vee Y) &\approx \pi_{1}X * \pi_{1}Y\\
+ \widetilde{\mathrm{H}}_{*}(X\vee Y) &\approx
+ \widetilde{\mathrm{H}}_{*}X \oplus \widetilde{\mathrm{H}}_{*}Y
+ \end{aligned}
+\end{equation}
+\end{verbatim}
+then you'll get
+\begin{equation}
+ \label{eq:BothCoprod}
+ \begin{aligned}
+ \pi_{1}(X\vee Y) &\approx \pi_{1}X * \pi_{1}Y\\
+ \widetilde{\mathrm{H}}_{*}(X\vee Y) &\approx
+ \widetilde{\mathrm{H}}_{*}X \oplus \widetilde{\mathrm{H}}_{*}Y
+ \end{aligned}
+\end{equation}
+and if you type ``\verb"the isomorphisms~\eqref{eq:BothCoprod}"'' then
+you'll get ``the isomorphisms~\eqref{eq:BothCoprod}''.
+
+For another example, if you type
+\begin{verbatim}
+\begin{equation}
+ \label{eq:longeq}
+ \begin{aligned}
+ F(a,b,c,d) &= \int_{a}^{b} \sin^{3}x \cos^{2}x \, dx
+ + \int_{c}^{d} \sin^{3}x \cos x \, dx\\
+ &\quad + \int_{a}^{b} \sin^{2} x \cos^{2} x \, dx + \int_{c}^{d}
+ \sin x \cos x \, dx
+ \end{aligned}
+\end{equation}
+\end{verbatim}
+then you'll get
+\begin{equation}
+ \label{eq:longeq}
+ \begin{aligned}
+ F(a,b,c,d) &= \int_{a}^{b} \sin^{3}x \cos^{2}x \, dx
+ + \int_{c}^{d} \sin^{3}x \cos x \, dx\\
+ &\quad + \int_{a}^{b} \sin^{2} x \cos^{2} x \, dx + \int_{c}^{d}
+ \sin x \cos x \, dx
+ \end{aligned}
+\end{equation}
+
+If you'd like to specify the exact amount of horizontal space between
+the columns, you'd use the \texttt{alignedat} environment (see
+section~\ref{sec:alignedat}) in place of the \texttt{aligned}
+environment.
+
+%--------------------------------------------------------------------
+\subsubsection{Multiple lines without horizontal alignment and with one equation number}
+\label{sec:GatherCenter}
+
+To create a display of multiple lines without horizontal alignment but
+with each line centered, and to have a single equation number for the
+entire display, you use the \texttt{equation} environment (see
+section~\ref{sec:SngLne}) to produce the equation number and you put a
+\texttt{gathered} environment (see section~\ref{sec:gathered}) inside
+of that. For example, if you type
+\begin{verbatim}
+\begin{equation}
+ \label{eq:Corner}
+ \begin{gathered}
+ (X\otimes L) \amalg_{(X\otimes K)} (Y\otimes K)
+ \longrightarrow Y\otimes L\\
+ X^{L} \longrightarrow X^{K} \times_{Y^{K}} Y^{L}
+ \end{gathered}
+\end{equation}
+\end{verbatim}
+then you'll get
+\begin{equation}
+ \label{eq:Corner}
+ \begin{gathered}
+ (X\otimes L) \amalg_{(X\otimes K)} (Y\otimes K)
+ \longrightarrow Y\otimes L\\
+ X^{L} \longrightarrow X^{K} \times_{Y^{K}} Y^{L}
+ \end{gathered}
+\end{equation}
+and if you type ``\verb"see~\eqref{eq:Corner}"'' then you'll get
+``see~\eqref{eq:Corner}''.
+
+%--------------------------------------------------------------------
+\subsubsection{Suppressing some equation numbers: {\normalfont \texttt{\bs notag}}}
+\label{sec:notag}
+
+If you want to have equation numbers on some but not all of the lines
+of a display, you can use an environment that numbers every line and
+then suppress the equation number on particular lines by typing
+\begin{center}
+ \verb"\notag"
+\end{center}
+at the end of each of those lines, either just before the double
+backslash that ends the line or at the end of the last line. For
+example, if you type
+\begin{verbatim}
+\begin{align}
+ a^{2} + b^{2} &= c^{2} \notag\\
+ \label{eq:ABC}
+ A^{2} + B^{2} &= C^{2}\\
+ p^{2} + q^{2} &- r^{2} \notag\\
+ \label{eq:PQR}
+ P^{2} + Q^{2} &= R^{2}
+\end{align}
+\end{verbatim}
+then you'll get
+\begin{align}
+ a^{2} + b^{2} &= c^{2} \notag\\
+ \label{eq:ABC}
+ A^{2} + B^{2} &= C^{2}\\
+ p^{2} + q^{2} &- r^{2} \notag\\
+ \label{eq:PQR}
+ P^{2} + Q^{2} &= R^{2}
+\end{align}
+and if you type ``\verb"equations~\eqref{eq:ABD} and \eqref{eq:PQR}"''
+then you'll get``equations~\eqref{eq:ABC} and \eqref{eq:PQR}''.
+
+%--------------------------------------------------------------------
+\subsection{Overriding equation numbers}
+\label{sec:EqNoOver}
+
+There are commands \verb"\tag" and \verb"\tag*" that allow you to
+override the normal equation numbering for any line of any of the
+displayed mathematics environments. You can also suppress the equation
+number entirely using the \verb"\notag" command (see
+section~\ref{sec:notag}).
+
+If you put the command ``\verb"\tag{stuff}"'' at the end of a line of
+a display (either just before the double backslash that ends the line
+or at the end of the last line), where ``\texttt{stuff}'' is arbitrary
+text, then that arbitrary text will be placed on the page, enclosed in
+parentheses, in place of an equation number. The command
+``\verb"\tag*{stuff}"'' is similar, but it does not add the
+parentheses. If these commands are used in a display that normally
+has unnumbered lines, then ``\texttt{stuff}'' will be inserted anyway.
+
+For example, if you type
+\begin{verbatim}
+\begin{equation}
+ \label{eq:add}
+ \widetilde{\mathrm{H}}_{*}(X\vee Y) \approx
+ \widetilde{\mathrm{H}}_{*}X \oplus \widetilde{\mathrm{H}}_{*}Y
+ \tag*{$(*)$}
+\end{equation}
+\end{verbatim}
+then you'll get
+\begin{equation}
+ \label{eq:add}
+ \widetilde{\mathrm{H}}_{*}(X\vee Y) \approx
+ \widetilde{\mathrm{H}}_{*}X \oplus \widetilde{\mathrm{H}}_{*}Y
+ \tag*{$(*)$}
+\end{equation}
+and if you type
+%
+``\verb"equation~\ref{eq:add} is additivity"'' then you'll get
+``equation~\ref{eq:add} is additivity''. Note that we used the
+command \verb"\tag*" and inserted the parentheses as part of the tag,
+rather than using the command \verb"\tag", because we wanted cross
+references to show the parentheses surrounding the asterisk.
+
+
+
+%--------------------------------------------------------------------
+%--------------------------------------------------------------------
+\section{Commutative diagrams using \texorpdfstring{\Xy-pic}{Xy-pic}}
+\label{sec:xypic}
+
+\Xy-pic is a powerful package that enables you to draw very complex
+commutative diagrams within your \LaTeX{} file, avoiding the need to
+create the graphics separately and then import them. \Xy-pic isn't a
+part of \AmS-LaTeX, but \Xy-pic and \AmS-LaTeX{} work quite well
+together. We'll describe the most important features of \Xy-pic's
+commutative diagram macros here. The full documentation for
+commutative diagrams can be found in the \emph{\Xy-pic User's Guide}
+\cite{xyguide}, and the full documentation for all of \Xy-pic, which
+can do quite a bit more than draw commutative diagrams, can be found
+in the \emph{\Xy-pic Reference Manual} \cite{xyrefer}.
+
+%--------------------------------------------------------------------
+\subsection{Basic diagrams}
+\label{sec:basicdiag}
+
+To use \Xy-pic to draw commutative diagrams, you include the lines
+\begin{center}
+ \begin{tabular}{l}
+ \verb"\usepackage[all,cmtip]{xy}"\\
+ \verb"\let\objectstyle=\displaystyle"
+ \end{tabular}
+\end{center}
+in the preamble of your document, i.e., after the
+\verb"\documentclass" command and before the \verb"\begin{document}"
+command. You can omit the second of those two lines if you want the
+nodes of your diagram to be in \verb"\textstyle" by default.
+
+You create a diagram using the \verb"\xymatrix" command, immediately
+followed by a pair of braces that enclose the diagram specification.
+The diagram is described in a manner similar to the way matrices are
+described:
+\begin{enumerate}
+\item There is a rectangular array of nodes of the diagram. A node
+ should not begin with a macro or with an asterisk, and so it's
+ safest (although not always required) to enclose each node in braces
+ \verb"{}".
+\item The nodes in a row are separated by an ampersand \verb"&",
+ and the rows are separated by a double backslash \verb"\\".
+\item An arrow is specified immediately following the node that is its
+ source (although this is not actually required; see
+ section~\ref{sec:SrcTrgt}). A straight line arrow is specified by
+ \verb"\ar[target]", where \verb"target" is
+ \begin{itemize}
+ \item \verb"r" for the node one place to the right,
+ \item \verb"l" for the node one place to the left,
+ \item \verb"d" for the node one place down,
+ \item \verb"u" for the node one place up,
+ \item \verb"rr" for the node two places to the right,
+ \item \verb"dr" for the node one place down and one place to the
+ right,
+ \item etc.
+ \end{itemize}
+\end{enumerate}
+It's also possible to have labels on the arrows (see
+section~\ref{sec:labelarrow}), arrows that are built of things other
+than a single solid line with a single arrowhead at the end (see
+section~\ref{sec:arstyle}), curved arrows (see
+section~\ref{sec:arcurve}), and arrows that pass over or under other
+arrows (see sections~\ref{sec:ArUnder} and~\ref{sec:CrossArrow}).
+
+For example,
+if you type
+\begin{verbatim}
+\begin{equation}
+ \xymatrix{
+ {A}\ar[r] \ar[d]
+ &{B} \ar[d]\\
+ {C} \ar[r]
+ &{D}
+ }
+\end{equation}
+\end{verbatim}
+then you'll get
+\begin{equation}
+ \xymatrix{
+ {A}\ar[r] \ar[d]
+ &{B} \ar[d]\\
+ {C} \ar[r]
+ &{D}
+ }
+\end{equation}
+If you'd like the equation number to be centered vertically with
+respect to the diagram, you should enclose the \verb"\xymatrix"
+command in a \verb"\vcenter", as in
+\begin{verbatim}
+\begin{equation}
+ \vcenter{
+ \xymatrix{
+ {A}\ar[r] \ar[d]
+ &{B} \ar[d]\\
+ {C} \ar[r]
+ &{D}
+ }
+ }
+\end{equation}
+\end{verbatim}
+in which case you'll get
+\begin{equation}
+ \vcenter{
+ \xymatrix{
+ {A}\ar[r] \ar[d]
+ &{B} \ar[d]\\
+ {C} \ar[r]
+ &{D}
+ }
+ }
+\end{equation}
+
+Not all nodes need appear. To omit a node in the middle of a line,
+just type the \verb"&" that moves you on to the next node. To omit a
+node at the end of the line, just type the line-ending \verb"\\" and
+go on to the next line. For example, if you type
+\begin{verbatim}
+\begin{equation*}
+ \xymatrix{
+ {A} \ar[rr] \ar[dr] \ar[d]
+ && {B} \ar[dd]\\
+ {C} \ar[d] \ar[drr]
+ & {D} \ar[l] \ar[ur]\\
+ {E} \ar[rr]
+ &&{F}
+ }
+\end{equation*}
+\end{verbatim}
+you'll get
+\begin{equation*}
+ \xymatrix{
+ {A} \ar[rr] \ar[dr] \ar[d]
+ && {B} \ar[dd]\\
+ {C} \ar[d] \ar[drr]
+ & {D} \ar[l] \ar[ur]\\
+ {E} \ar[rr]
+ &&{F}
+ }
+\end{equation*}
+Note that that diagram has three rows and three columns, but two of
+the nodes in the second column and one of the nodes in the third
+column are missing.
+
+%--------------------------------------------------------------------
+\subsection{Changing the spacing}
+\label{sec:ChgSpc}
+
+You can change the space between the rows of a diagram and the space
+between the columns of a diagram. The following commands can be
+inserted following the command \verb"\xymatrix" and before the left
+brace that begins the diagram specification:
+\begin{center}
+ \begin{tabular}{ll}
+ \texttt{@=dimen}& set row and column spacing to \texttt{dimen}\\
+ \texttt{@R=dimen}& set row spacing to \texttt{dimen}\\
+ \texttt{@C=dimen}& set column spacing to \texttt{dimen}
+ \end{tabular}
+\end{center}
+The \texttt{dimen} in all of these commands can be positive, zero, or
+negative. (Negative \texttt{dimen}'s can be useful in diagrams with
+very wide entries; see \eqref{diag:wide}.) Although you can use
+whatever units you like for \texttt{dimen} (e.g., inches
+(\texttt{in}), centimeters (\texttt{cm}), points (\texttt{pt}), etc.),
+it's a good idea to measure vertical distances in \texttt{ex} (roughly
+the height of an upper case ``X'') and horizontal distances in
+\texttt{em} (roughly the width of an upper case ``M''), so that the
+distance will be roughly correct even if you change font sizes.
+
+For example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \vcenter{
+ \xymatrix@=2ex{
+ {A} \ar[r] \ar[d]
+ & {B} \ar[d]\\
+ {C} \ar[r]
+ & {D}
+ }% \xymatrix
+ }% \vcenter
+ \qquad\text{or}\qquad
+ \vcenter{
+ \xymatrix{
+ {A} \ar[r] \ar[d]
+ & {B} \ar[d]\\
+ {C} \ar[r]
+ & {D}
+ }% \xymatrix
+ }% \vcenter
+ \qquad\text{or}\qquad
+ \vcenter{
+ \xymatrix@C=10ex{
+ {A} \ar[r] \ar[d]
+ & {B} \ar[d]\\
+ {C} \ar[r]
+ & {D}
+ }% \xymatrix
+ }% \vcenter
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \vcenter{
+ \xymatrix@=2ex{
+ {A} \ar[r] \ar[d]
+ & {B} \ar[d]\\
+ {C} \ar[r]
+ & {D}
+ }% \xymatrix
+ }% \vcenter
+ \qquad\text{or}\qquad
+ \vcenter{
+ \xymatrix{
+ {A} \ar[r] \ar[d]
+ & {B} \ar[d]\\
+ {C} \ar[r]
+ & {D}
+ }% \xymatrix
+ }% \vcenter
+ \qquad\text{or}\qquad
+ \vcenter{
+ \xymatrix@C=10ex{
+ {A} \ar[r] \ar[d]
+ & {B} \ar[d]\\
+ {C} \ar[r]
+ & {D}
+ }% \xymatrix
+ }% \vcenter
+\end{displaymath}
+For an example of the usefulness of negative column spacing: If you
+type
+\begin{verbatim}
+\newcommand{\pushout}[3]{#1\mathbin{\mathord{\amalg}_{#2}}#3}
+\begin{equation}
+ \label{diag:wide}
+ \vcenter{
+ \xymatrix@C=-12em{
+ {\pushout{(A\otimes L)}
+ {A\otimes K}
+ {(B\otimes K)}
+ } \ar[r] \ar[dr]
+ & {B\otimes L}
+ & {\pushout{(P\otimes L)}
+ {P\otimes M}
+ {(Q\otimes M)}
+ } \ar[l] \ar[dl]\\
+ & {\pushout{\bigl(\pushout{(A\otimes L)}
+ {A\otimes K}
+ {(B\otimes K)}\bigr)}
+ {\pushout{(X\otimes L)}
+ {X\otimes K}
+ {(Y\otimes K)}}
+ {\bigl(\pushout{(P\otimes L)}
+ {P\otimes M}
+ {(Q\otimes M)}\bigr)}
+ } \ar[u]
+ }
+ }
+\end{equation}
+\end{verbatim}
+then you'll get
+\newcommand{\pushout}[3]{#1\mathbin{\mathord{\amalg}_{#2}}#3}
+\begin{equation}
+ \label{diag:wide}
+ \vcenter{
+ \xymatrix@C=-12em{
+ {\pushout{(A\otimes L)}
+ {A\otimes K}
+ {(B\otimes K)}
+ } \ar[r] \ar[dr]
+ & {B\otimes L}
+ & {\pushout{(P\otimes L)}
+ {P\otimes M}
+ {(Q\otimes M)}
+ } \ar[l] \ar[dl]\\
+ & {\pushout{\bigl(\pushout{(A\otimes L)}
+ {A\otimes K}
+ {(B\otimes K)}\bigr)}
+ {\pushout{(X\otimes L)}
+ {X\otimes K}
+ {(Y\otimes K)}}
+ {\bigl(\pushout{(P\otimes L)}
+ {P\otimes M}
+ {(Q\otimes M)}\bigr)}
+ } \ar[u]
+ }
+ }
+\end{equation}
+which would never fit on the page without negative column spacing.
+
+%--------------------------------------------------------------------
+\subsection{Shifting nodes sideways}
+\label{sec:ShiftSide}
+
+You can shift a node to the right by enclosing it in braces and
+preceding the opening brace with \verb"*+[r]" and shift it to the left
+by enclosing it in braces and preceding the opening brace with
+\verb"*+[l]". This can be useful, for example, if you want a node to
+consist of both the name of something and its definition, but you want
+the arrows to the node to point to the name, rather than to the center
+of the node. For example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r] \ar[d]
+ & *+[r]{B = \{x \in W \mid f(x) > 0\}} \ar[d]\\
+ {C} \ar[r]
+ & {D}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r] \ar[d]
+ & *+[r]{B = \{x \in W \mid f(x) > 0\}} \ar[d]\\
+ {C} \ar[r]
+ & {D}
+ }
+\end{displaymath}
+Note that the large node at the upper right caused the columns to be
+far apart, even though that node was shifted to the right before being
+placed onto the page. To counteract that, you can decrease the column
+spacing: If you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix@C=-2em{
+ {A} \ar[r] \ar[d]
+ & *+[r]{B = \{x \in W \mid f(x) > 0\}} \ar[d]\\
+ {C} \ar[r]
+ & {D}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix@C=-2em{
+ {A} \ar[r] \ar[d]
+ & *+[r]{B = \{x \in W \mid f(x) > 0\}} \ar[d]\\
+ {C} \ar[r]
+ & {D}
+ }
+\end{displaymath}
+
+
+
+%--------------------------------------------------------------------
+\subsection{Arrows passing under nodes; crossing arrows}
+\label{sec:ArUnder}
+
+Arrows can be created that go to a sequence of nodes, passing under
+(i.e., leaving a small gap at) all the intermediate nodes. If an
+arrow is created that passes under an empty node while a second arrow
+crosses that node in the normal way, the effect is that the first
+arrow passes under the second arrow. (For another way to have one
+arrow pass under another, see section~\ref{sec:CrossArrow}.)
+
+To draw an arrow that passes under a sequence of nodes and then goes
+on to a final node, the \verb"\ar" is followed by \verb"'[node]" for
+each node that you pass under, followed by \verb"[finalnode]". For
+example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar '[r] '[rr] [rrr]
+ & {B}
+ & {}
+ & {D}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar '[r] '[rr] [rrr]
+ & {B}
+ & {}
+ & {D}
+ }
+\end{displaymath}
+For an example of crossing arrows, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix@=3ex{
+ {A} \ar[rr] \ar[dd] \ar'[dr][ddrr]
+ && {B} \ar[dd] \ar[ddll]\\
+ & {}\\
+ {C} \ar[rr]
+ && {D}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix@=3ex{
+ {A} \ar[rr] \ar[dd] \ar'[dr][ddrr]
+ && {B} \ar[dd] \ar[ddll]\\
+ & {}\\
+ {C} \ar[rr]
+ && {D}
+ }
+\end{displaymath}
+Note that that diagram has three rows and three columns, but nothing
+appears in either the second row or the second column. (For an
+explanation of the command \verb"@=3ex", which changes the size of the
+diagram, see section~\ref{sec:ChgSpc}.) For a more elaborate example,
+if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix@=2ex{
+ {A} \ar[rr] \ar[dr] \ar[dd]
+ && {B} \ar[dr] \ar'[d][dd]\\
+ & {A'} \ar[rr] \ar[dd]
+ && {B'} \ar[dd]\\
+ {C} \ar'[r][rr] \ar[dr]
+ && {D} \ar[dr]\\
+ & {C'} \ar[rr]
+ && {D'}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix@=2ex{
+ {A} \ar[rr] \ar[dr] \ar[dd]
+ && {B} \ar[dr] \ar'[d][dd]\\
+ & {A'} \ar[rr] \ar[dd]
+ && {B'} \ar[dd]\\
+ {C} \ar'[r][rr] \ar[dr]
+ && {D} \ar[dr]\\
+ & {C'} \ar[rr]
+ && {D'}
+ }
+\end{displaymath}
+
+
+
+%--------------------------------------------------------------------
+\subsection{Labeling the arrows}
+\label{sec:labelarrow}
+
+It's possible to label an arrow, on one or both sides of the arrow.
+(It's also possible to have the label ``break'' the arrow; for this,
+see section~\ref{sec:arbreak}.) By default, the label is located
+halfway from the center of the source to the center of the target.
+This will often be halfway along the arrow, but not if the source and
+the target are of different sizes. There are also options to place
+the label along the midpoint of the arrow (see
+section~\ref{sec:arcent}) or at an arbitrary point between the center
+of the source and the center of the target (see
+section~\ref{sec:LblArb}).
+
+To put a label above an arrow (where ``above'' means when the paper is
+oriented so that the arrow goes left to right), you type
+\verb"^{thelabel}" either before or after the target. To put a label
+below an arrow (where ``below'' means when the paper is oriented so
+that the arrow goes left to right), you type \verb"_{thelabel}" either
+before or after the target. Thus, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A}\ar[r]^{f} \ar[d]_{g}
+ &{B} \ar[d]^{h}\\
+ {C} \ar[r]_{k}^{\text{above}}
+ &{D}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A}\ar[r]^{f} \ar[d]_{g}
+ &{B} \ar[d]^{h}\\
+ {C} \ar[r]_{k}^{\text{above}}
+ &{D}
+ }
+\end{displaymath}
+
+%--------------------------------------------------------------------
+\subsubsection{Centering the labels on the arrows}
+\label{sec:arcent}
+
+If you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {X\cup_{Y}Z} \ar[r]^{f} \ar[d]_{g}
+ &{B} \ar[d]^{h}\\
+ {C} \ar[r]_{k}
+ &{D}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {X\cup_{Y}Z} \ar[r]^{f} \ar[d]_{g}
+ &{B} \ar[d]^{h}\\
+ {C} \ar[r]_{k}
+ &{D}
+ }
+\end{displaymath}
+Note that the label $f$ is halfway from the center of the arrow's
+source to the center of the arrow's target, but it is not centered
+along the arrow. To have that label centered along the arrow, we
+insert a \verb"-" immediately following the \verb"^", so that we type
+it as
+\begin{center}
+ \verb"\ar[r]^-{f}"
+\end{center}
+and we then get
+\begin{displaymath}
+ \xymatrix{
+ {X\cup_{Y}Z} \ar[r]^-{f} \ar_{g}[d]
+ &{B} \ar^{h}[d]\\
+ {C} \ar_{k}[r]
+ &{D}
+ }
+\end{displaymath}
+For another example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A \amalg_{B} C} \ar[r]^{\text{Add}}_{\sigma} \ar[d]
+ & {X} \ar[d]^{\psi}\\
+ {Y} \ar[r]^{\alpha}_{\text{Lifted}} \ar[r]
+ & {P\times_{Q}R}
+ }
+\end{displaymath}
+\end{verbatim}
+you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A \amalg_{B} C} \ar[r]^{\text{Add}}_{\sigma} \ar[d]
+ & {X} \ar[d]^{\psi}\\
+ {Y} \ar[r]^{\alpha}_{\text{Lifted}} \ar[r]
+ & {P\times_{Q}R}
+ }
+\end{displaymath}
+but if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A \amalg_{B} C} \ar[r]^-{\text{Add}}_-{\sigma} \ar[d]
+ & {X} \ar[d]\\
+ {Y} \ar[r]^-{\alpha}_-{\text{Lifted}}
+ & {P\times_{Q}R}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A \amalg_{B} C} \ar[r]^-{\text{Add}}_-{\sigma} \ar[d]
+ & {X} \ar[d]\\
+ {Y} \ar[r]^-{\alpha}_-{\text{Lifted}}
+ & {P\times_{Q}R}
+ }
+\end{displaymath}
+
+%--------------------------------------------------------------------
+\subsubsection{Arbitrary placement of labels}
+\label{sec:LblArb}
+
+In addition to the possibility of centering a label along an arrow
+(see section~\ref{sec:arcent}), it is possible to place a label at an
+arbitrary point between the center of the source and the center of the
+target. If \texttt{a} is a number between $0$ and $1$, then you can
+place a label \texttt{a} of the way from the center of the source to
+the center of the target by typing \verb"(a)" immediately following
+the \verb"^" or \verb"_". For example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix@C=8em{
+ {A} \ar[r]^(.3){f} \ar[d]
+ & {B} \ar[d]\\
+ {C} \ar[r]^{g} \ar[d]
+ & {D} \ar[d]\\
+ {E} \ar[r]_(.7){h}
+ & {F}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix@C=8em{
+ {A} \ar[r]^(.3){f} \ar[d]
+ & {B} \ar[d]\\
+ {C} \ar[r]^{g} \ar[d]
+ & {D} \ar[d]\\
+ {E} \ar[r]_(.7){h}
+ & {F}
+ }
+\end{displaymath}
+For a more elaborate example, with labels along segments of segmented
+arrows, see diagram~\ref{diag:LblCube}.
+
+
+%--------------------------------------------------------------------
+\subsubsection{Labeling each segment of a segmented arrow}
+\label{sec:LblSeg}
+
+An arrow that uses the ``arrows passing under'' feature (see
+section~\ref{sec:ArUnder}) will be in several segments, and a label
+can be attached to each of the segments. For example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar '[r]^{f} '[rr]^{g} [rrr]^{h}
+ & {B}
+ & {}
+ & {D}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar '[r]^{f} '[rr]^{g} [rrr]^{h}
+ & {B}
+ & {}
+ & {D}
+ }
+\end{displaymath}
+For a more elaborate example, if you type
+\begin{verbatim}
+\begin{equation}
+ \label{diag:LblCube}
+ \vcenter{
+ \xymatrix@=2ex{
+ {A} \ar[rr]^{f} \ar[dr] \ar[dd]_{i}
+ && {B} \ar[dr] \ar'[d][dd]^(.3){j}\\
+ & {A'} \ar[rr]^(.3){f'} \ar[dd]^(.25){i'}
+ && {B'} \ar[dd]^{j'}\\
+ {C} \ar'[r]^{g}[rr] \ar[dr]
+ && {D} \ar[dr]\\
+ & {C'} \ar[rr]_{g'}
+ && {D'}
+ }
+ }
+\end{equation}
+\end{verbatim}
+then you'll get
+\begin{equation}
+ \label{diag:LblCube}
+ \vcenter{
+ \xymatrix@=2ex{
+ {A} \ar[rr]^{f} \ar[dr] \ar[dd]_{i}
+ && {B} \ar[dr] \ar'[d][dd]^(.3){j}\\
+ & {A'} \ar[rr]^(.3){f'} \ar[dd]^(.25){i'}
+ && {B'} \ar[dd]^{j'}\\
+ {C} \ar'[r]^{g}[rr] \ar[dr]
+ && {D} \ar[dr]\\
+ & {C'} \ar[rr]_{g'}
+ && {D'}
+ }
+ }
+\end{equation}
+
+
+%--------------------------------------------------------------------
+\subsubsection{Breaking an arrow with a label}
+\label{sec:arbreak}
+
+Instead of placing a label to the side of an arrow, you can have the
+label ``break'' the arrow. For this, you use the vertical bar
+character \verb"|" in place of either \verb"^" or \verb"_".
+
+For example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r]|{f}
+ & {B} \ar[r]|{g}
+ & {C}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r]|{f}
+ & {B} \ar[r]|{g}
+ & {C}
+ }
+\end{displaymath}
+
+A label that breaks an arrow can be positioned anywhere along the
+arrow, just as for labels placed alongside the arrow (see
+sections~\ref{sec:arcent} and \ref{sec:LblArb}). For example, if you
+type
+\begin{verbatim}
+\begin{equation*}
+ \xymatrix{
+ {A \amalg_{B} C} \ar[r]|-{f} \ar[d]_{i}
+ & {X} \ar[r]|(.7){m} \ar[d]^{p}
+ & {Y} \ar[dl]|(.3)n\\
+ {D} \ar[r]|-{g}
+ & {W \times_{Z} Y}
+ }
+\end{equation*}
+\end{verbatim}
+then you'll get
+\begin{equation*}
+ \xymatrix{
+ {A \amalg_{B} C} \ar[r]|-{f} \ar[d]_{i}
+ & {X} \ar[r]|(.7){m} \ar[d]^{p}
+ & {Y} \ar[dl]|(.3)n\\
+ {D} \ar[r]|-{g}
+ & {W \times_{Z} Y}
+ }
+\end{equation*}
+%--------------------------------------------------------------------
+\subsection{More crossing arrows}
+\label{sec:CrossArrow}
+
+In section~\ref{sec:ArUnder} we presented a method of having an arrow
+pass over another arrow by having them cross at an empty node. We
+present here another method of having one arrow pass over another.
+
+To have one arrow appear to cross over another, we create a small gap
+in the second arrow at the spot at which the first arrow will cross.
+We do this by breaking that second arrow with a label, as in
+section~\ref{sec:arbreak}, using the special label \verb"\hole", which
+just leaves an empty hole in the arrow. For example, if you type
+\begin{verbatim}
+\begin{equation*}
+ \xymatrix{
+ {A} \ar[dr]|{\hole}
+ & {B} \ar[dl]\\
+ {C}
+ & {D}
+ }
+\end{equation*}
+\end{verbatim}
+then you'll get
+\begin{equation*}
+ \xymatrix{
+ {A} \ar[dr]|{\hole}
+ & {B} \ar[dl]\\
+ {C}
+ & {D}
+ }
+\end{equation*}
+
+For another example, if you type
+\begin{verbatim}
+\begin{equation*}
+ \xymatrix{
+ {A} \ar[dr] \ar[r] \ar[d]
+ & {B} \ar[r]
+ & {C}\\
+ {D} \ar[urr]|(.33){\hole} \ar[r]
+ & {E}
+ }
+\end{equation*}
+\end{verbatim}
+then you'll get
+\begin{equation*}
+ \xymatrix{
+ {A} \ar[dr] \ar[r] \ar[d]
+ & {B} \ar[r]
+ & {C}\\
+ {D} \ar[urr]|(.33){\hole} \ar[r]
+ & {E}
+ }
+\end{equation*}
+(see section~\ref{sec:LblArb}). It took some trial and error to
+figure out that \texttt{.33} was the number to use in the arrow with
+the hole, but there's a much better way specify the location of the
+hole: Instead of inserting that number between 0 and 1 that specifies
+how far from the center of the source to the center of the target the
+break should appear, you can specify the place to put the break using
+the notation \verb"!{[node1];[node2]}" to denote the point on the
+arrow at which a straight line from \verb"[node1]" to \verb"[node2]"
+would cross the arrow. (\emph{Note:} This method only works for
+straight line arrows.) That is, if you type
+\begin{verbatim}
+\begin{equation*}
+ \xymatrix{
+ {A} \ar[dr] \ar[r] \ar[d]
+ & {B} \ar[r]
+ & {C}\\
+ {D} \ar[urr]|!{[u];[r]}{\hole} \ar[r]
+ & {E}
+ }
+\end{equation*}
+\end{verbatim}
+then you'll get
+\begin{equation*}
+ \xymatrix{
+ {A} \ar[dr] \ar[r] \ar[d]
+ & {B} \ar[r]
+ & {C}\\
+ {D} \ar[urr]|!{[u];[r]}{\hole} \ar[r]
+ & {E}
+ }
+\end{equation*}
+The position of the hole is specified as \verb"!{[u];[r]}" because the
+arrow with the hole begins at $D$ and, relative to that node, the
+arrow that crosses over goes from the node \verb"[u]" to the node
+\verb"[r]".
+
+For another example, if you type
+\begin{verbatim}
+\begin{equation*}
+ \xymatrix{
+ {A} \ar[d] \ar[dr]
+ & {B} \ar[d] \ar[dr]
+ & {C} \ar[d]\\
+ {D}
+ \ar[urr]|!{[u];[r]}{\hole}
+ |!{[ur];[r]}{\hole}
+ |!{[ur];[rr]}{\hole}
+ & {E}
+ & {F}
+ }
+\end{equation*}
+\end{verbatim}
+then you'll get
+\begin{equation*}
+ \xymatrix{
+ {A} \ar[d] \ar[dr]
+ & {B} \ar[d] \ar[dr]
+ & {C} \ar[d]\\
+ {D}
+ \ar[urr]|!{[u];[r]}{\hole}
+ |!{[ur];[r]}{\hole}
+ |!{[ur];[rr]}{\hole}
+ & {E}
+ & {F}
+ }
+\end{equation*}
+
+
+%--------------------------------------------------------------------
+\subsection{Making room for large labels}
+\label{sec:labelroom}
+
+Although \Xy-pic automatically leaves enough room for large nodes, it
+ignores the size of the labels on the arrows when constructing the
+diagram. Thus, if you have a particularly large label, it may
+overwrite the nodes if you don't do something to make the diagram
+larger. You can do that either by enlarging the entire diagram (see
+section~\ref{sec:LblEnlrgeDiag}) or by creating a row (or column) all
+of whose nodes are empty and have the arrows span the empty row (or
+column) (see section~\ref{sec:emptynodes}).
+
+%--------------------------------------------------------------------
+\subsubsection{Enlarging the entire diagram}
+\label{sec:LblEnlrgeDiag}
+
+To enlarge the entire diagram, we use the spacing commands described
+in section~\ref{sec:ChgSpc}. For example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A\amalg B} \ar[d]
+ \ar[r]^-{\operatorname{amalgamate}}
+ & {X} \ar[r] \ar[d]
+ & {P} \ar[d]\\
+ {C \amalg D}
+ \ar[r]_-{\operatorname{amalgamate}}
+ & {Y} \ar[r]
+ & {Q}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A\amalg B} \ar[d]
+ \ar[r]^-{\operatorname{amalgamate}}
+ & {X} \ar[r] \ar[d]
+ & {P} \ar[d]\\
+ {C \amalg D}
+ \ar[r]_-{\operatorname{amalgamate}}
+ & {Y} \ar[r]
+ & {Q}
+ }
+\end{displaymath}
+but if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix@C=6em{
+ {A\amalg B} \ar[d] \ar[r]^-{\operatorname{amalgamate}}
+ & {X} \ar[r] \ar[d]
+ & {P} \ar[d]\\
+ {C \amalg D} \ar[r]_-{\operatorname{amalgamate}}
+ & {Y} \ar[r]
+ & {Q}
+ }
+\end{displaymath}
+\end{verbatim}
+(see section~\ref{sec:ChgSpc}) then you'll get
+\begin{displaymath}
+ \xymatrix@C=6em{
+ {A\amalg B} \ar[d] \ar[r]^-{\operatorname{amalgamate}}
+ & {X} \ar[r] \ar[d]
+ & {P} \ar[d]\\
+ {C \amalg D} \ar[r]_-{\operatorname{amalgamate}}
+ & {Y} \ar[r]
+ & {Q}
+ }
+\end{displaymath}
+If you don't want to have to make the entire diagram larger, you can
+just create a column of empty nodes (see
+section~\ref{sec:emptynodes}).
+%--------------------------------------------------------------------
+\subsubsection{Creating a column of empty nodes}
+\label{sec:emptynodes}
+
+We continue the discussion of section~\ref{sec:LblEnlrgeDiag}: Instead
+of enlarging the entire diagram, you can create an additional column
+in between the first two columns and have all the nodes in the new
+column be empty. For example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A\amalg B} \ar[d] \ar[rr]^-{\operatorname{amalgamate}}
+ && {X} \ar[r] \ar[d]
+ & {P} \ar[d]\\
+ {C \amalg D} \ar[rr]_-{\operatorname{amalgamate}}
+ && {Y} \ar[r]
+ & {Q}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A\amalg B} \ar[d] \ar[rr]^-{\operatorname{amalgamate}}
+ && {X} \ar[r] \ar[d]
+ & {P} \ar[d]\\
+ {C \amalg D} \ar[rr]_-{\operatorname{amalgamate}}
+ && {Y} \ar[r]
+ & {Q}
+ }
+\end{displaymath}
+Note that that diagram has four columns, but both of the nodes in the
+second column are empty, and the labelled arrows are created as
+\verb"\ar[rr]". If you'd like to create even more space for that
+large label, you can put an \verb"\hspace" command into one of the
+nodes in the second column. For example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A\amalg B} \ar[d] \ar[rr]^-{\operatorname{amalgamate}}
+ & {\hspace{3em}}
+ & {X} \ar[r] \ar[d]
+ & {P} \ar[d]\\
+ {C \amalg D} \ar[rr]_-{\operatorname{amalgamate}}
+ && {Y} \ar[r]
+ & {Q}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A\amalg B} \ar[d] \ar[rr]^-{\operatorname{amalgamate}}
+ & {\hspace{3em}}
+ & {X} \ar[r] \ar[d]
+ & {P} \ar[d]\\
+ {C \amalg D} \ar[rr]_-{\operatorname{amalgamate}}
+ && {Y} \ar[r]
+ & {Q}
+ }
+\end{displaymath}
+Note that we only put the \verb"\hspace" command into the first row;
+there's no need to put it into every row.
+
+%--------------------------------------------------------------------
+\subsection{Different arrow styles}
+\label{sec:arstyle}
+
+It's possible to have arrows with tails, multiple heads, dotted,
+dashed, or multiple shafts, and any combination of these (see
+Table~\ref{tab:ArSty}). You can even omit both the head and the tail,
+or omit the arrow entirely, which is useful for placing things into
+the diagram in places outside of the grid of nodes (see
+Diagram~\ref{diag:CentEq}).
+\begin{table}
+ \centering
+ \begin{tabular}{c@{\qquad\qquad}l}
+ To produce:& \multicolumn{1}{c}{Type:}\\[1ex]
+ $\xymatrix{\ar@{.>}[r]&}$& \verb"\ar@{.>}[r]"\\
+ $\xymatrix{\ar@{-->}[r]&}$& \verb"\ar@{-->}[r]"\\
+ $\xymatrix{\ar@{=>}[r]&}$& \verb"\ar@{=>}[r]"\\
+ $\xymatrix{\ar@{:>}[r]&}$& \verb"\ar@{:>}[r]"\\
+ $\xymatrix{\ar@{->>}[r]&}$& \verb"\ar@{->>}[r]"\\
+ $\xymatrix{\ar@{>->}[r]&}$& \verb"\ar@{>->}[r]"\\
+ $\xymatrix{\ar@{<->}[r]&}$& \verb"\ar@{<->}[r]"\\
+ $\xymatrix{\ar@{-->>}[r]&}$& \verb"\ar@{-->>}[r]"\\
+ $\xymatrix{\ar@{>-->}[r]&}$& \verb"\ar@{>-->}[r]"\\
+ $\xymatrix{\ar@{<-->}[r]&}$& \verb"\ar@{<-->}[r]"\\
+ $\xymatrix{\ar@{|->}[r]&}$& \verb"\ar@{|->}[r]"\\
+ $\xymatrix{\ar@{^{(}->}[r]&}$& \verb"\ar@{^{(}->}[r]"\\
+ $\xymatrix{\ar@{_{(}->}[r]&}$& \verb"\ar@{_{(}->}[r]"\\
+ $\xymatrix{\ar@{=>>}[r]&}$& \verb"\ar@{=>>}[r]"\\
+ $\xymatrix{\ar@{<=>}[r]&}$& \verb"\ar@{<=>}[r]"\\
+ $\xymatrix{\ar@{:>>}[r]&}$& \verb"\ar@{:>>}[r]"\\
+ $\xymatrix{\ar@{<:>}[r]&}$& \verb"\ar@{<:>}[r]"\\
+ $\xymatrix{\ar@{-}[r]&}$& \verb"\ar@{-}[r]"\\
+ $\xymatrix{\ar@{.}[r]&}$& \verb"\ar@{.}[r]"\\
+ $\xymatrix{\ar@{=}[r]&}$& \verb"\ar@{=}[r]"\\
+ $\xymatrix{\ar@{}[r]&}$& \verb"\ar@{}[r]"\\[1ex]
+ \end{tabular}
+ \caption{Arrow Styles}
+ \label{tab:ArSty}
+\end{table}
+
+All of these arrows can point in whatever direction you choose; we
+used \verb"[r]" in the table just for readability. For an example of
+the use of invisible arrows: If you type
+\begin{verbatim}
+\begin{equation}
+ \label{diag:CentEq}
+ \vcenter{
+ \xymatrix{
+ {A} \ar@{.>}[r] \ar@{.>}[d] \ar@{}[dr]|{=}
+ & {B} \ar[d]\\
+ {C} \ar[r]
+ & {D}
+ }
+ }
+\end{equation}
+\end{verbatim}
+then you'll get
+\begin{equation}
+ \label{diag:CentEq}
+ \vcenter{
+ \xymatrix{
+ {A} \ar@{.>}[r] \ar@{.>}[d] \ar@{}[dr]|{=}
+ & {B} \ar[d]\\
+ {C} \ar[r]
+ & {D}
+ }
+ }
+\end{equation}
+Note that the equals sign is inserted by having it break the middle of
+the invisible arrow from the upper left node to the lower right node.
+
+%--------------------------------------------------------------------
+\subsection{Curved arrows}
+\label{sec:arcurve}
+
+It's possible to have arrows curve, either by specifying the amount
+that they curve or by specifying the direction in which they leave
+their source and the direction from which they arrive at their
+target.
+
+%--------------------------------------------------------------------
+\subsubsection{Specifying the amount of the curve}
+
+To have an arrow curve in the up direction (where ``up'' means when
+the paper is oriented so that the arrow goes left to right) by a
+default amount you follow the \verb"\ar" with \verb"@/^/"; that is,
+you type \verb"\ar@/^/[target]". To have it curve in the down
+direction by a default amount you follow the \verb"\ar" with
+\verb"@/_/"; that is, you type \verb"\ar@/_/[target]" (or, if you want
+to label the arrow, \verb"\ar@/_/[target]^{label}"). Thus, if you
+type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar@/^/[drr]^{p} \ar@{.>}[dr]|{\exists!} \ar@/_/[ddr]_{q}\\
+ & {B} \ar[r] \ar[d]
+ & {C} \ar[d]\\
+ & {D} \ar[r]
+ & {E}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar@/^/[drr]^{p} \ar@{.>}[dr]|{\exists!} \ar@/_/[ddr]_{q}\\
+ & {B} \ar[r] \ar[d]
+ & {C} \ar[d]\\
+ & {D} \ar[r]
+ & {E}
+ }
+\end{displaymath}
+If you'd like to specify the amount of curve, you can specify a
+dimension following the \verb"^" or the \verb"_", as in
+\verb"\ar@/^1ex/[target]". (Although you can use whatever units you
+like for \texttt{dimen} (e.g., inches (\texttt{in}), centimeters
+(\texttt{cm}), points (\texttt{pt}), etc.), it's a good idea to
+measure vertical distances in \texttt{ex} (roughly the height of an
+upper case ``X'') and horizontal distances in \texttt{em} (roughly the
+width of an upper case ``M''), so that the distance will be roughly
+correct even if you change font sizes.) For example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar@/^3ex/[drr]^{p} \ar@{.>}[dr]|{\exists!}
+ \ar@/_3ex/[ddr]_{q}\\
+ & {B} \ar[r] \ar[d]
+ & {C} \ar[d]\\
+ & {D} \ar[r]
+ & {E}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar@/^3ex/[drr]^{p} \ar@{.>}[dr]|{\exists!}
+ \ar@/_3ex/[ddr]_{q}\\
+ & {B} \ar[r] \ar[d]
+ & {C} \ar[d]\\
+ & {D} \ar[r]
+ & {E}
+ }
+\end{displaymath}
+and if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar@{>->}[r]^{i}
+ & {B} \ar@{->>}[r]^{p}
+ & {C} \ar@{.>}@/_4ex/[l]_{s}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar@{>->}[r]^{i}
+ & {B} \ar@{->>}[r]^{p}
+ & {C} \ar@{.>}@/_4ex/[l]_{s}
+ }
+\end{displaymath}
+
+
+%--------------------------------------------------------------------
+\subsubsection{Specifying the start and end directions}
+\label{sec:CrvStEd}
+
+To specify the start and end directions of a curved arrow you type
+\verb"\ar@(start,end)[target]" where ``\verb"start"'' is the direction
+towards which the arrow begins and ``\verb"end"'' is the direction
+from which the arrow ends. For example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[dr]|{p} \ar@(r,u)[dr]^{f} \ar@(d,l)[dr]_{g}\\
+ & {B}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[dr]|{p} \ar@(r,u)[dr]^{f} \ar@(d,l)[dr]_{g}\\
+ & {B}
+ }
+\end{displaymath}
+and if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar@(ur,ul)[rr]^{1_{A}} \ar[r]_{f}
+ & {B} \ar[r]_{g} \ar@(dr,dl)[rr]_{1_{B}}
+ & {A} \ar[r]_{f}
+ & {B}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar@(ur,ul)[rr]^{1_{A}} \ar[r]_{f}
+ & {B} \ar[r]_{g} \ar@(dr,dl)[rr]_{1_{B}}
+ & {A} \ar[r]_{f}
+ & {B}
+ }
+\end{displaymath}
+Since the current node is denoted \verb"[]" (i.e., a pair of square
+brackets with nothing inside), if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar@(ur,dr)[]^{f}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar@(ur,dr)[]^{f}
+ }
+\end{displaymath}
+
+%--------------------------------------------------------------------
+\subsection{Sliding arrows sideways; multiple arrows}
+\label{sec:arslide}
+
+It's possible to slide arrows sideways, so that you can have more than
+one straight line arrow between a single pair of nodes. To do this,
+you type \verb"\ar@<dimen>" to move the arrow ``upwards'' by
+\verb"dimen" (where ``upwards'' means when the paper is held so that
+the arrow goes from left to right). (A negative \verb"dimen" will
+move the arrow ``downwards''.) Although you can use whatever units
+you like for \texttt{dimen} (e.g., inches (\texttt{in}), centimeters
+(\texttt{cm}), points (\texttt{pt}), etc.), it's a good idea to
+measure vertical distances in \texttt{ex} (roughly the height of an
+upper case ``X'') and horizontal distances in \texttt{em} (roughly the
+width of an upper case ``M''), so that the distance will be roughly
+correct even if you change font sizes. For example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar@<1ex>[r]
+ & {B} \ar[r]
+ & {C} \ar@<-1ex>[r]
+ & {D}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar@<1ex>[r]
+ & {B} \ar[r]
+ & {C} \ar@<-1ex>[r]
+ & {D}
+ }
+\end{displaymath}
+For another example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {\coprod_{i \in I} A_{i}\quad}
+ \ar@<+.7ex>[r]^-{\phi} \ar@<-.7ex>[r]_-{\psi}
+ & {\quad\coprod_{i \in I} B_{i}}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {\coprod_{i \in I} A_{i}\quad}
+ \ar@<+.7ex>[r]^-{\phi} \ar@<-.7ex>[r]_-{\psi}
+ & {\quad\coprod_{i \in I} B_{i}}
+ }
+\end{displaymath}
+It's also possible to slide curved arrows sideways. For example, if
+you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar@/^ 1ex/[r] \ar@/^ 1ex/@<1ex>[r]
+ & {B}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar@/^ 1ex/[r] \ar@/^ 1ex/@<1ex>[r]
+ & {B}
+ }
+\end{displaymath}
+
+
+%--------------------------------------------------------------------
+\subsection{Specifying both the source and the target of an arrow}
+\label{sec:SrcTrgt}
+
+Up until now, all of our examples of arrows have gone from the current
+node to a node specified in the code for the arrow. It's possible,
+however, to specify both the source and the target of an arrow, rather
+than letting the source be the current node. This allows you to save
+the description of all of the arrows of a diagram until the end.
+
+To draw an arrow who's source may not be at the current node, you type
+\begin{center}
+ \verb"\ar[source];[target]"
+\end{center}
+instead of just \verb"\ar[target]". Since the current node is denoted
+\verb"[]", typing \verb"\ar[target]" is the same as typing
+\verb"\ar[];[target]".
+
+For example, if you
+type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A}
+ & {B}\\
+ {C}
+ & {D}
+ \ar^{f}[ul];[u]
+ \ar_{g}[l];[]
+ \ar_{h}[ul];[l]
+ \ar^{k}[u];[]
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A}
+ & {B}\\
+ {C}
+ & {D}
+ \ar^{f}[ul];[u]
+ \ar_{g}[l];[]
+ \ar_{h}[ul];[l]
+ \ar^{k}[u];[]
+ }
+\end{displaymath}
+Note that when we drew the arrows the current node was $D$, and so,
+e.g., the target of the arrow labelled $g$ was specified as \verb"[]"
+(which denotes the current node).
+
+%--------------------------------------------------------------------
+\subsection{Absolute specification of the source and target of an arrow}
+\label{sec:AbsDesc}
+
+Up until now, the target and source (see section~\ref{sec:SrcTrgt}) of
+an arrow were always specified relative to the current node (see
+section~\ref{sec:basicdiag}). It is possible to give instead an
+absolute specification of the target and source of an arrow.
+
+Each node of a diagram can be specified as
+\begin{center}
+ \verb!"p,q"!
+\end{center}
+(including the double quote symbols) where \verb"p" is the row number
+(counting from the top, where the first row is row number $1$) and
+\verb"q" is the column number (counting from the left, where the first
+column is column number $1$). For example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar^{f}"1,2" \ar_{h}"2,1"
+ & {B} \ar^{k}"2,2"\\
+ {C} \ar_{g}"2,2"
+ & {D}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar^{f}"1,2" \ar_{h}"2,1"
+ & {B} \ar^{k}"2,2"\\
+ {C} \ar_{g}"2,2"
+ & {D}
+ }
+\end{displaymath}
+In that example, each arrow began at the current node. For example,
+the arrow labelled $f$ began at the current node $A$ (and so it's
+source was not specified) and its target $B$ was specified as
+\verb!"1,2"! because it is in the first row, second column.
+
+As described in section~\ref{sec:SrcTrgt}, you can also specify both
+the source and target of an arrow, and this allows you to save the
+specification of all the arrows until the end of the diagram. For
+example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A}
+ & {B}
+ & {C}\\
+ {D}
+ & {E}
+ & {F}
+ \ar_{f} "1,1";"1,2"
+ \ar_{g} "1,2";"1,3"
+ \ar@(ur,ul)^{h} "1,1";"1,3"
+ \ar"1,1";"2,1"
+ \ar"1,2";"2,2"
+ \ar"1,3";"2,3"
+ \ar^{p} "2,1";"2,2"
+ \ar^{q} "2,2";"2,3"
+ \ar@(dr,dl)_{r} "2,1";"2,3"
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A}
+ & {B}
+ & {C}\\
+ {D}
+ & {E}
+ & {F}
+ \ar_{f} "1,1";"1,2"
+ \ar_{g} "1,2";"1,3"
+ \ar@(ur,ul)^{h} "1,1";"1,3"
+ \ar"1,1";"2,1"
+ \ar"1,2";"2,2"
+ \ar"1,3";"2,3"
+ \ar^{p} "2,1";"2,2"
+ \ar^{q} "2,2";"2,3"
+ \ar@(dr,dl)_{r} "2,1";"2,3"
+ }
+\end{displaymath}
+(For a description of the curved arrows, see
+section~\ref{sec:CrvStEd}.) Note that, e.g., the arrow labelled $f$
+is specified as
+\begin{center}
+ \verb!\ar_{f} "1,1";"1,2"!
+\end{center}
+because it starts at $A$, which is in row $1$ column $1$, and goes to
+$B$, which is in row $1$ column $2$.
+
+%--------------------------------------------------------------------
+\subsection{Nodes off of the grid}
+\label{sec:NdOffGrd}
+
+It's possible to place a node at an arbitrary point in the diagram
+without affecting the positioning of the other nodes. You can also
+assign a \emph{name} to the node that you can use later to draw arrows
+to or from that node.
+
+You place a node off the grid using the \emph{excursion} commands
+\begin{center}
+ \verb"\save"\qquad and\qquad\verb"\restore"
+\end{center}
+In between those commands you can put commands to
+\begin{itemize}
+\item move from the current node using a displacement vector
+ ``\verb"[]+<Dx,Dy>"'' (where \verb"Dx" is the horizontal
+ displacement and \verb"Dy" is the vertical displacement),
+\item typeset a node there using ``\verb"*+{stuff}"'',
+\item optionally name that node using ``\verb!*+{stuff}="name"!'', and
+\item optionally draw arrows from whatever you've typeset there to
+ other nodes in the diagram. If you've named the new node, you can
+ specify the arrows later on, and the arrows can either start from
+ this node or go to this node (or both, as in the second example in
+ this section).
+\end{itemize}
+Although you can use whatever units you like for the displacements
+(e.g., inches (\texttt{in}), centimeters (\texttt{cm}), points
+(\texttt{pt}), etc.), it's a good idea to measure vertical distances
+in \texttt{ex} (roughly the height of an upper case ``X'') and
+horizontal distances in \texttt{em} (roughly the width of an upper
+case ``M''), so that the displacement will be roughly correct even if
+you change font sizes.
+
+For example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r] \ar[d]
+ & {B} \ar[d]
+ \save []+<3em, 3em>*+{H}
+ \ar@(l,ur)_{p}[l] \ar@(d,ur)^{r}[d] \ar_{q}[]
+ \restore
+ \\
+ {C} \ar[r]
+ & {D}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r] \ar[d]
+ & {B} \ar[d]
+ \save []+<3em, 3em>*+{H}
+ \ar@(l,ur)_{p}[l] \ar@(d,ur)^{r}[d] \ar_{q}[]
+ \restore
+ \\
+ {C} \ar[r]
+ & {D}
+ }
+\end{displaymath}
+We created the node $H$ off of the grid by starting at the node $B$
+(in the first row, second column) and moving \verb"3em" to the right
+and \verb"3em" up. We then drew three arrows starting at that node:
+\begin{itemize}
+\item The arrow $p$ goes to the node $A$, and so its target is
+ specified as \verb"[l]" since that is one node to the left of the
+ node $B$ (which is the current node from which we were on an
+ excursion),
+\item the arrow $q$ goes to the node $B$, and so its target is
+ specified as \verb"[]" (a pair of square brackets with nothing
+ inside) since that is the current node, and
+\item the arrow $r$ goes to the node $D$, and so its target is
+ specified as \verb"[d]" since it is one node below the current
+ node.
+\end{itemize}
+
+For another example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r] \ar[d]
+ & {B} \ar[d]\\
+ {C} \ar[r]
+ & {D}
+ \save []+<4em,1em>*+{H}="firstthing" \restore
+ \save []+<-1em,-3em>*+{K}="secondthing" \restore
+ \ar"firstthing";"1,2"_{p}
+ \ar"firstthing";"2,2"_{q}
+ \ar@(d,r)"firstthing";"secondthing"^{r}
+ \ar"secondthing";"2,2"_{s}
+ \ar"secondthing";"2,1"^{t}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r] \ar[d]
+ & {B} \ar[d]\\
+ {C} \ar[r]
+ & {D}
+ \save []+<4em,1em>*+{H}="firstthing" \restore
+ \save []+<-1em,-3em>*+{K}="secondthing" \restore
+ \ar"firstthing";"1,2"_{p}
+ \ar"firstthing";"2,2"_{q}
+ \ar@(d,r)"firstthing";"secondthing"^{r}
+ \ar"secondthing";"2,2"_{s}
+ \ar"secondthing";"2,1"^{t}
+ }
+\end{displaymath}
+In this diagram, the only nodes on the grid are $A$, $B$, $C$, and
+$D$. After creating the nodes and arrows on the grid, we created the
+nodes $H$ and $K$ and the arrows that go to and from these nodes.
+\begin{itemize}
+\item We created each of the nodes $H$ and $K$ with an excursion from
+ the node $D$, giving the name \texttt{firstthing} to the node $H$
+ and the name \texttt{secondthing} to the node \verb"K".
+\item Each of the arrows to or from the nodes $H$ and $K$ specify both
+ their source and their target (a semicolon separates the source from
+ the target; see section~\ref{sec:SrcTrgt}).
+\item Each of the arrows from either $H$ or $K$ to one of the nodes on
+ the grid describes that node on the grid absolutely (see
+ section~\ref{sec:AbsDesc}).
+\end{itemize}
+The arrow from $H$ to $K$ begins \verb"\ar@(d,r)" because it is a
+curved arrow (see section~\ref{sec:CrvStEd}). Since that arrow goes
+from the node $H$ (named \texttt{firsthing}) to the node $K$ (named
+\texttt{secondthing}), the arrow is specified as
+\begin{center}
+ \verb!\ar@(d,r)"firstthing";"secondthing"!
+\end{center}
+Finally, that arrow is labelled $r$, and so the complete specification
+is
+\begin{center}
+ \verb!\ar@(d,r)"firstthing";"secondthing"^{r}!
+\end{center}
+
+For another example, we show how to annotate ``diagram chase'' proofs
+in homological algebra. If you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r]^{\alpha} \ar[d]_{f}
+ & {B} \ar[r]^{\beta} \ar[d]_{g}
+ % Add the element b of B:
+ \save []+<2ex,3ex>*+{b}="b"\restore
+ & {C} \ar[r]^{\gamma} \ar[d]_{h}
+ % Add the element c of C:
+ \save []+<0ex,3ex>*+{c}="c"\restore
+ & {D} \ar[r]^{\delta} \ar[d]_{j}
+ % Add the element d of D:
+ \save []+<2ex,3ex>*+{d}="d"\restore
+ & {E} \ar[d]_{k}\\
+ {A'} \ar[r]^{\alpha'}
+ & {B'} \ar[r]^{\beta'}
+ % Add the element b' of B':
+ \save []+<2ex,-5ex>*+{b'}="bprime"\restore
+ & {C'} \ar[r]^{\gamma'}
+ % Add the element c' of C':
+ \save []+<0ex,-3ex>*+{c'}="cprime"\restore
+ % Add the element c'-gamma(c) of C':
+ \save []+<1.5ex,-5ex>*+{(c'-\gamma(c))}="cprimeminus"\restore
+ & {D'} \ar[r]^{\delta'}
+ % Add the element \gamma'(c') of D':
+ \save []+<2ex,-3ex>*+{\gamma'(c')}="gammacprime"\restore
+ & {E'}
+ % Add the arrows connecting the elements:
+ \ar@{|->}"b";"bprime"
+ \ar@{|->}"c";"d"
+ \ar@{|->}"d";"gammacprime"
+ \ar@{|->}"cprime";"gammacprime"
+ \ar@{|->}"bprime";"cprimeminus"
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get the following diagram, which that might be used in
+part of the proof of the ``five lemma''.
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r]^{\alpha} \ar[d]_{f}
+ & {B} \ar[r]^{\beta} \ar[d]_{g}
+ % Add the element b of B:
+ \save []+<2ex,3ex>*+{b}="b"\restore
+ & {C} \ar[r]^{\gamma} \ar[d]_{h}
+ % Add the element c of C:
+ \save []+<0ex,3ex>*+{c}="c"\restore
+ & {D} \ar[r]^{\delta} \ar[d]_{j}
+ % Add the element d of D:
+ \save []+<2ex,3ex>*+{d}="d"\restore
+ & {E} \ar[d]_{k}\\
+ {A'} \ar[r]^{\alpha'}
+ & {B'} \ar[r]^{\beta'}
+ % Add the element b' of B':
+ \save []+<2ex,-5ex>*+{b'}="bprime"\restore
+ & {C'} \ar[r]^{\gamma'}
+ % Add the element c' of C':
+ \save []+<0ex,-3ex>*+{c'}="cprime"\restore
+ % Add the element c'-gamma(c) of C':
+ \save []+<1.5ex,-5ex>*+{(c'-\gamma(c))}="cprimeminus"\restore
+ & {D'} \ar[r]^{\delta'}
+ % Add the element \gamma'(c') of D':
+ \save []+<2ex,-3ex>*+{\gamma'(c')}="gammacprime"\restore
+ & {E'}
+ % Add the arrows connecting the elements:
+ \ar@{|->}"b";"bprime"
+ \ar@{|->}"c";"d"
+ \ar@{|->}"d";"gammacprime"
+ \ar@{|->}"cprime";"gammacprime"
+ \ar@{|->}"bprime";"cprimeminus"
+ }
+\end{displaymath}
+
+%--------------------------------------------------------------------
+\subsection{Text objects}
+\label{sec:TextObj}
+
+You can insert a text object into a diagram by typing
+\begin{center}
+ \verb"*\txt{Some text}"\quad or\quad \verb"*\txt<width>{Some text}"
+\end{center}
+In the first form you create linebreaks with a double backslash
+``\verb"\\"'' and in the second form the lines are broken
+automatically. Text objects by default don't have any blank space
+around them, but you can add a bit of space by typing them as
+\verb"*+\txt{Some text}", or even more space by typing them as
+\verb"*++\txt{Some text}".
+
+For example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ *+\txt{Roses are red,\\ Violets are blue.\\
+ This is a contrived example,\\ And the rhyme is bad too.}
+ \ar@{=>}[r]
+ & *\txt<2in>{Four score and seven years ago our fathers brought
+ forth on this continent a new nation, conceived in liberty,
+ and dedicated to the proposition that all men are created
+ equal.}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ *+\txt{Roses are red,\\ Violets are blue.\\
+ This is a contrived example,\\ And the rhyme is bad too.}
+ \ar@{=>}[r]
+ & *\txt<2in>{Four score and seven years ago our fathers brought
+ forth on this continent a new nation, conceived in liberty,
+ and dedicated to the proposition that all men are created
+ equal.}
+ }
+\end{displaymath}
+We typed that first one as \verb"*+\txt" so that the arrow wouldn't
+start too close to the text.
+
+You can also shift a text object to the left (so that arrows to or
+from the object will go to its right end instead of its center) by
+typing it as
+%
+\verb"*+[l]\txt{Some text}" or to the right (so that arrows to or from
+the object will go to its left end instead of its center) by typing it
+as
+%
+\verb"*+[r]\txt{Some text}" (see section~\ref{sec:ShiftSide}). For
+example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ *+[l]\txt{Interior} \ar[r] \ar[d]
+ & *+[r]\txt{Exterior} \ar[d]\\
+ {A} \ar[r]
+ & {B}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ *+[l]\txt{Interior} \ar[r] \ar[d]
+ & *+[r]\txt{Exterior} \ar[d]\\
+ {A} \ar[r]
+ & {B}
+ }
+\end{displaymath}
+
+Text objects are mostly used for adding annotations to a diagram (see
+section~\ref{sec:Drop}).
+
+%--------------------------------------------------------------------
+\subsection{Arrows off of the grid}
+\label{sec:OffGrid}
+
+It's possible to draw arrows that begin or end at points other than
+nodes. To do this, you specify the point by choosing a node and then
+adding a displacement vector ``\verb"+<Dx,Dy>"'' from that node (where
+\verb"Dx" is the horizontal displacement and \verb"Dy" is the vertical
+displacement). (Although you can use whatever units you like for the
+displacements (e.g., inches (\texttt{in}), centimeters (\texttt{cm}),
+points (\texttt{pt}), etc.), it's a good idea to measure vertical
+distances in \texttt{ex} (roughly the height of an upper case ``X'')
+and horizontal distances in \texttt{em} (roughly the width of an upper
+case ``M''), so that the displacement will be roughly correct even if
+you change font sizes.) For example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[d]+<4em,0ex>\\
+ {B}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[d]+<4em,0ex>\\
+ {B}
+ }
+\end{displaymath}
+since the arrow is pointing to the spot 4em to the right of the $B$.
+
+For another example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A}
+ & {B}\\
+ {C}
+ & {D}
+ \ar[ul]+<-2em,0ex>;[]+<2em,0ex>
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A}
+ & {B}\\
+ {C}
+ & {D}
+ \ar[ul]+<-2em,0ex>;[]+<2em,0ex>
+ }
+\end{displaymath}
+(see section~\ref{sec:SrcTrgt}) because we specified the source as
+being \verb"[ul]+<-2em,0ex>", i.e., 2em to the left of the node that's
+one up and one to the left of the current node ($D$) and the target as
+being \verb"[]+<2em,0ex>", i.e., 2em to the right of the current node
+($D$).
+
+For an example that combines displacements of source and target along
+with curving arrows and labelled arrows: if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r]+<0em,-3ex>^{f}
+ & {B} \ar@(ur,ur)[d]+<2em,0ex>^{g}\\
+ {C}
+ & {D} \ar@/^ 2ex/[l];[]^{h}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r]+<0em,-3ex>^{f}
+ & {B} \ar@(ur,ur)[d]+<2em,0ex>^{g}\\
+ {C}
+ & {D} \ar@/^ 2ex/[l];[]^{h}
+ }
+\end{displaymath}
+
+Realistic examples of arrows off of the grid can be found in
+section~\ref{sec:Drop}, where we show how to drop objects into a
+diagram at either the source or the target of an arrow.
+
+%--------------------------------------------------------------------
+\subsection{Dropping objects into a diagram; annotations}
+\label{sec:Drop}
+
+You can drop an object at an arbitrary point of a diagram (i.e., not
+necessarily at a node) by drawing an arrow to or from that point (see
+section~\ref{sec:OffGrid}) and dropping the object at one of the ends
+of the arrow. The object can be either mathematics or a text object
+(see section~\ref{sec:TextObj}).
+
+For example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r] \ar[d]
+ & {B} \ar[d]
+ \ar[]+<5em,-5ex>*+\txt{This space\\is compact.};[]\\
+ {C} \ar[r]
+ & {D}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r] \ar[d]
+ & {B} \ar[d]
+ \ar[]+<5em,-5ex>*+\txt{This space\\is compact.};[]\\
+ {C} \ar[r]
+ & {D}
+ }
+\end{displaymath}
+Note that the text and the arrow from it was typed by first creating
+the arrow
+\begin{center}
+ \verb"\ar[]+<5em,-5ex>;[]"
+\end{center}
+whose source is displaced from the current node (see
+sections~\ref{sec:SrcTrgt} and \ref{sec:OffGrid}) and whose target is
+the current node, and then dropping the text object (see
+section~\ref{sec:TextObj}) at the source by inserting
+\begin{center}
+ \verb"*+\txt{This space\\is compact.}"
+\end{center}
+immediately after the specification of the source.
+
+Note that the arrows to or from a text object by default go to its
+center, and if a text object is dropped at a specified point by
+default it's the center of the text object that lands at that point.
+Thus, if you're typing a wide text object, you should be sure either
+to leave enough room so that it doesn't overlap other parts of the
+diagram or to shift the text object sideways (as in
+section~\ref{sec:ShiftSide}). For example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r] \ar[d]
+ & {B} \ar[d]
+ \ar[]+<5em,-5ex>*+\txt{This topological space is compact.};[]\\
+ {C} \ar[r]
+ & {D}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r] \ar[d]
+ & {B} \ar[d]
+ \ar[]+<5em,-5ex>*+\txt{This topological space is compact.};[]\\
+ {C} \ar[r]
+ & {D}
+ }
+\end{displaymath}
+but if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r] \ar[d]
+ & {B} \ar[d]
+ \ar[]+<10em,-5ex>*+\txt{This topological space is compact.};[]\\
+ {C} \ar[r]
+ & {D}
+ }
+\end{displaymath}
+\end{verbatim}
+(in which we increased the horizontal displacement to 10em) then
+you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r] \ar[d]
+ & {B} \ar[d]
+ \ar[]+<10em,-5ex>*+\txt{This topological space is compact.};[]\\
+ {C} \ar[r]
+ & {D}
+ }
+\end{displaymath}
+and if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r] \ar[d]
+ & {B} \ar[d]
+ \ar[]+<5em,-5ex>*+[r]\txt{This topological space is compact.};[]\\
+ {C} \ar[r]
+ & {D}
+ }
+\end{displaymath}
+\end{verbatim}
+(in which we typed
+\begin{center}
+ \verb"*+[r]\txt{This topological space is compact.}"
+\end{center}
+to create the text object, i.e., we added ``\verb"[r]"'' to shift the
+text object to the right; see section~\ref{sec:ShiftSide}) then you'll
+get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r] \ar[d]
+ & {B} \ar[d]
+ \ar[]+<5em,-5ex>*+[r]\txt{This topological space is compact.};[]\\
+ {C} \ar[r]
+ & {D}
+ }
+\end{displaymath}
+
+If you want to drop an object onto a spot other than at a node without
+printing an arrow to or from that spot, you can draw an invisible
+arrow going to that spot and drop the object at the end of the
+invisible arrow. For example, if you type
+\begin{verbatim}
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r] \ar[d]_{f}
+ & {B} \ar[d]^{g}
+ \ar@{}[]+<8em,-4ex>*+\txt{Both $f$ and $g$\\are fibrations.}\\
+ {C} \ar[r]
+ & {D}
+ }
+\end{displaymath}
+\end{verbatim}
+then you'll get
+\begin{displaymath}
+ \xymatrix{
+ {A} \ar[r] \ar[d]_{f}
+ & {B} \ar[d]^{g}
+ \ar@{}[]+<8em,-4ex>*+\txt{Both $f$ and $g$\\are fibrations.}\\
+ {C} \ar[r]
+ & {D}
+ }
+\end{displaymath}
+The arrow to the text object is invisible because we typed it starting
+with \verb"\ar@{}" (see section~\ref{sec:arstyle} and
+Diagram~\ref{diag:CentEq}). After that, we typed
+\begin{center}
+ \verb"[]+<8em,-4ex>*+\txt{Both $f$ and $g$\\are fibrations.}"
+\end{center}
+so that the arrow would go to the point 8em to the right and 4ex below
+the current node, and we dropped the text object at that point.
+
+
+%--------------------------------------------------------------------
+%--------------------------------------------------------------------
+\section{Including graphics}
+\label{sec:graphics}
+
+If you have graphics files that you want to include in your document,
+then you should load the \verb"graphicx" package by putting the
+command
+\begin{center}
+\verb"\usepackage{graphicx}"
+\end{center}
+after your \verb"\documentclass" command. (Note: The spelling
+\verb"graphicx" is correct; the package graphicx.sty is the updated
+version of the older package graphics.sty.) You can then include
+graphics using the \verb"\includegraphics" command.
+
+The standard practice is to include graphics inside of a \verb"figure"
+environment, so that it can float to the next page if there isn't
+enough room for it on the current page. This also allows you to give
+a caption (printed below the graphics) and a label (for use in a
+\verb"\ref" command). Thus, if you want to include the contents of
+\verb"mygraphic.eps", you might type
+\begin{verbatim}
+\begin{figure}[h]
+ \centering
+ \includegraphics{mygraphics.eps}
+ \caption{A pretty picture}
+ \label{fig:pretty}
+\end{figure}
+\end{verbatim}
+and you can then refer to it as \verb"figure~\ref{fig:pretty}". (The
+\verb"\centering" command centers the graphics in the figure; this is
+preferable to using a \verb"center" environment, which would add
+additional vertical space.) It's important to note that the
+\verb"\label" command must come \emph{after} the \verb"\caption"
+command, since it's the \verb"\caption" command that creates the
+number that the \verb"\label" command labels. You can also put the
+\verb"\label" command \emph{inside} the caption, as in
+\begin{center}
+\verb"\caption{A pretty picture\label{fig:pretty}}"
+\end{center}
+and some people find that this helps prevent errors.
+
+
+%--------------------------------------------------------------------
+\subsection{Graphic formats: Postscript or pdf}
+
+One problem to watch out for is that different versions of \LaTeX{}
+require different types of graphics files:
+\begin{itemize}
+\item Standard \verb"latex" (which produces dvi files) requires
+ encapsulationed postscript graphics files, while
+\item \verb"pdflatex" (which directly produces pdf files) requires pdf
+ graphics files.
+\end{itemize}
+This can be confusing, because some installations of \LaTeX{} give you
+a \verb"latex" command that is actually running \verb"pdflatex". In
+particular, the standard installation of \TeX Shop for Mac OS X gives
+you a \verb"latex" command that actually runs \verb"pdflatex" and
+produces a pdf file. \TeX Shop can be configured to run standard
+\verb"latex" to produce dvi files and then convert them to pdf; for
+this, see the list of frequently asked questions at
+\begin{center}
+ \url{http://www.ams.org/authors/author-faq.html}
+\end{center}
+and search that page for ``texshop''.
+
+On most systems, if you need to include encapsulated postscript
+graphics and produce a pdf file, you can use plain old \verb"latex" to
+produce a dvi file and then use \verb"dvipdf" to create the pdf file.
+Another option is to use the \verb"epstopdf" program to create a
+postscript file from your eps file.
+
+
+%---------------------------------------------------------------------
+%---------------------------------------------------------------------
+\section{Macro definitions, a.k.a.\ \texttt{\bs newcommand}}
+\label{sec:definitions}
+
+\LaTeX{} allows you to use the same \verb"\def" command that you use
+in plain \TeX, but it's considered bad style. Instead, \LaTeX{} has
+the \verb"\newcommand" and \verb"\renewcommand" commands, which do a
+little error checking for you. In plain \TeX, you might use the
+command
+\begin{center}
+ \verb"\def\tensor{\otimes}"
+\end{center}
+but in \LaTeX{} the preferred form is
+\begin{center}
+ \verb"\newcommand{\tensor}{\otimes}"
+\end{center}
+The advantage of this is that \LaTeX{} will check to see if there
+already is a command with the name \verb"\tensor" and give you an
+error message if there is. If you know that there is a previous
+definition of \verb"\tensor" but you \emph{want} to override it,
+then you use the command
+\begin{center}
+ \verb"\renewcommand{\tensor}{\otimes}"
+\end{center}
+
+If you want to use macros with replaceable parameters, both
+\verb"\newcommand" and \verb"\renewcommand" allow this. For the
+equivalent of the plain \TeX{} command
+\begin{center}
+ \verb"\def\pushout#1#2#3{#1\cup_{#2}#3}"
+\end{center}
+you use the \LaTeX{} command
+\begin{center}
+ \verb"\newcommand{\pushout}[3]{#1\cup_{#2}#3}"
+\end{center}
+i.e., the command name is enclosed in braces, and the number of
+parameters is enclosed in square brackets.
+
+
+%---------------------------------------------------------------------
+%---------------------------------------------------------------------
+\section{Lists: \texttt{itemize, enumerate, and description}}
+
+There are three list making environments:
+\begin{itemize}
+\item \texttt{itemize},
+\item \texttt{enumerate}, and
+\item \texttt{description}.
+\end{itemize}
+The \texttt{itemize} environment just lists the items with a marker in
+front of each one. If you type
+\begin{verbatim}
+\begin{itemize}
+\item This is the first item in the list, which runs on long enough
+ to spill over onto a second line.
+\item This is the second item in the list, which is a bit shorter.
+\item This is the last item.
+\end{itemize}
+\end{verbatim}
+then you'll get
+\begin{itemize}
+\item This is the first item in the list, which runs on long enough
+ to spill over onto a second line.
+\item This is the second item in the list, which is a bit shorter.
+\item This is the last item.
+\end{itemize}
+
+The \texttt{enumerate} environment looks the same, except that the
+items in the list are numbered. If you type
+\begin{verbatim}
+\begin{enumerate}
+\item This is the first item in the list, which runs on long enough
+ to spill over onto a second line.
+\item This is the second item in the list, which is a bit shorter.
+\item This is the last item.
+\end{enumerate}
+\end{verbatim}
+then you'll get
+\begin{enumerate}
+\item This is the first item in the list, which runs on long enough
+ to spill over onto a second line.
+\item This is the second item in the list, which is a bit shorter.
+\item This is the last item.
+\end{enumerate}
+
+The \texttt{description} environment requires an argument for each
+\verb"\item" command, which will be printed at the beginning of the
+item. If you type
+\begin{verbatim}
+\begin{description}
+\item[sedge] A green plant, found in both wetlands and uplands.
+ Sedges are often confused with grasses and rushes.
+\item[grass] A green plant, found in both wetlands and uplands.
+ Grasses are often confused with sedges and rushes.
+\item[rush] A green plant, found in both wetlands and uplands.
+ Rushes are often confused with sedges and grasses
+\end{description}
+\end{verbatim}
+you'll get
+\begin{description}
+\item[sedge] A green plant, found in both wetlands and uplands.
+ Sedges are often confused with grasses and rushes.
+\item[grass] A green plant, found in both wetlands and uplands.
+ Grasses are often confused with sedges and rushes.
+\item[rush] A green plant, found in both wetlands and uplands.
+ Rushes are often confused with sedges and grasses
+\end{description}
+
+These environments can be inserted within each other, and the
+\verb"enumerate" environment keeps track of what level it's at, and
+numbers its items accordingly. If you type
+\begin{verbatim}
+\begin{enumerate}
+\item I went to the dry cleaners.
+\item I went to the supermarket. I bought
+ \begin{enumerate}
+ \item bread,
+ \item cheese, and
+ \item Tabasco sauce.
+ \end{enumerate}
+\item I went to the bank.
+\end{enumerate}
+\end{verbatim}
+you'll get
+\begin{enumerate}
+\item I went to the dry cleaners.
+\item I went to the supermarket. I bought
+ \begin{enumerate}
+ \item bread,
+ \item cheese, and
+ \item Tabasco sauce.
+ \end{enumerate}
+\item I went to the bank.
+\end{enumerate}
+
+%--------------------------------------------------------------------
+
+\subsection{Referring to enumerated items by number}
+\label{sec:enumref}
+
+If you want to refer to an item in an enumerated list by item number,
+then you need to label the item using the \verb"\label" command and
+then refer to it using the \verb"\ref" command, just as you do for
+sections, subsections, theorems, etc.\ (see
+section~\ref{sec:xreferences}). For example, if you type
+\begin{verbatim}
+\begin{enumerate}
+\item \label{step:trans} Put the transmission into drive.
+\item \label{step:brake} Release the parking brake.
+\item \label{step:move} Start moving.
+\end{enumerate}
+You must complete steps \ref{step:trans} and \ref{step:brake}
+before beginning step~\ref{step:move}.
+\end{verbatim}
+then you'll get
+\begin{enumerate}
+\item \label{step:trans} Put the transmission into drive.
+\item \label{step:brake} Release the parking brake.
+\item \label{step:move} Start moving.
+\end{enumerate}
+You must complete steps \ref{step:trans} and \ref{step:brake} before
+beginning step~\ref{step:move}.
+
+%---------------------------------------------------------------------
+%---------------------------------------------------------------------
+\section{The bibliography}
+\label{sec:bibliography}
+
+The recommended way to create a bibliography is to use the
+\texttt{amsrefs} package (see section~\ref{sec:amsrefs}). This makes
+it simple to type the bibliography items and have them correctly
+formatted and automatically numbered, and makes it possible to change
+the style of the bibliography without retyping the items. At the same
+time, it allows you to preserve the structure of the information
+contained in the items, and (optionally) to create a database of
+possible references that can be reused in other documents.
+
+The \texttt{amsrefs} package is fairly new. The classical method of
+creating a bibliography from a database of possible references uses
+Bib\TeX{}, which is a separate program that must be run in between the
+runs of \LaTeX. If you already have one or more BiB\TeX{} databases,
+\texttt{amsrefs} can make use of them through a special Bib\TeX{}
+style file (see section~\ref{sec:bibtex}).
+
+If for some reason you want to avoid the use of \texttt{amsrefs} and
+of Bib\TeX, you can type and format the bibliography items any way you
+choose by creating a \texttt{thebibliography} environment (see
+section~\ref{sec:thebibliography}).
+
+
+
+%--------------------------------------------------------------------
+\subsection{Using \texttt{amsrefs}}
+\label{sec:amsrefs}
+
+To use \texttt{amsrefs}, you load it by putting the command
+\begin{center}
+ \verb"\usepackage[lite]{amsrefs}"
+\end{center}
+into the preamble of your document, i.e., after the
+\verb"\documentclass" command and before the \verb"\begin{document}"
+command. You can then
+\begin{enumerate}
+\item begin the bibliography with the commands
+\begin{verbatim}
+\begin{bibdiv}
+ \begin{biblist}
+\end{verbatim}
+\item define the bibliography items using a \verb"\bib" command for
+ each item (see section~\ref{sec:bib}), and then
+\item end the bibliography with the commands
+\begin{verbatim}
+ \end{biblist}
+\end{bibdiv}
+\end{verbatim}
+\end{enumerate}
+The reason there are two different environments (the outer environment
+\texttt{bibdiv} containing the inner environment \texttt{biblist}) is
+that the \texttt{bibdiv} environment produces the chapter or section
+heading for the bibliography (depending on what's appropriate for the
+type of your document) and the \texttt{biblist} environment produces
+the actual list of references. You can also type material in between
+the \verb"\begin{bibdiv}" and \verb"begin{biblist}" commands and it
+will be printed in between the heading for the list and the actual
+list.
+
+%--------------------------------------------------------------------
+\subsection{The \texttt{\bs bib} command}
+\label{sec:bib}
+
+The information for each bibliography item inside the \texttt{bibdiv}
+environment (see section~\ref{sec:amsrefs}) is entered using a
+\verb"\bib" command. The format of a \verb"\bib" command is
+\begin{verbatim}
+\bib{ReferenceKey}{ReferenceType}{
+ comma separated list of keyword={value} statements
+}
+\end{verbatim}
+where \texttt{ReferenceKey} is the key that will be used to refer to
+the item with a \verb"\cite" command (see
+section~\ref{sec:bibreferences}) and \texttt{ReferenceType} is either
+\begin{quote}
+ \texttt{article}, \texttt{book}, \texttt{misc}, \texttt{report}, or
+ \texttt{thesis}.
+\end{quote}
+For example, the bibliography of these notes contains the following
+\verb"\bib" commands:
+\begin{verbatim}
+\bib{yellowmonster}{book}{
+ author={Bousfield, A.K.},
+ author={Kan, D.M.},
+ title={Homotopy Limits, Completions and Localizations},
+ date={1972},
+ series={Lecture Notes in Mathematics},
+ volume={304},
+ publisher={Springer-Verlag},
+ address={Berlin-New York}
+}
+
+\bib{quil:rht}{article}{
+ author={Quillen, Daniel G.},
+ title={Rational Homotopy Theory},
+ journal={Ann. of Math. (2)},
+ volume={90},
+ date={1969},
+ pages={205--295}
+}
+\end{verbatim}
+If those \verb"\bib" commands are in your \texttt{biblist}
+environment, you can type
+\begin{center}
+ \verb"see \cite{yellowmonster} and \cite{quil:rht}"
+\end{center}
+and it will be typeset as ``see \cite{yellowmonster} and
+\cite{quil:rht}''.
+
+Some of the rules governing the \verb"\bib" command are:
+\begin{itemize}
+\item The keywords must all be typed in lower case. (This is
+ different from in Bib\TeX{} databases, in which the case of field
+ names is ignored.)
+\item The value in each ``\verb"keyword={value}"'' statement must
+ always be enclosed in braces. (This is different from in Bib\TeX{}
+ databases, in which the braces can sometimes be omitted.)
+\item Author names must be typed in the form ``von Last, First, Jr.'',
+ as in ``\verb"author={Jones, John Paul}"'', or
+ ``\verb"author={van Beethoven, Ludwig}"'', or
+ ``\verb"author={Ford, Henry, Jr.}"''. This ensures there won't be
+ errors in reversing the first and last names, or when
+ \texttt{amsrefs} replaces the first and middle names with initials
+ (when the \texttt{initials} option is used; see
+ section~\ref{sec:refoptions}), or when \texttt{amsrefs} creates a
+ label based on the last name (when the \texttt{alphabetic} option is
+ used; see section~\ref{sec:refoptions}).
+\item Multiple authors must each be listed in a separate \verb"author"
+ field.
+\item The capitalization in the title should be exactly as you want it
+ to appear in the bibliography. (This is different from in Bib\TeX{}
+ databases, in which capitalization is often changed to fit a chosen
+ style.)
+\item The date must be written in the form \verb"date={1776}", or
+ \verb"date={1776-07}", or \verb"date={1776-07-04}".
+\end{itemize}
+The full list of simple fields (i.e., fields that can appear at most
+once) is
+\begin{quote}
+ \texttt{address}, \texttt{booktitle}, \texttt{date},
+ \texttt{edition}, \texttt{eprint}, \texttt{hyphenation},
+ \texttt{journal}, \texttt{label}, \texttt{language}, \texttt{note},
+ \texttt{number}, \texttt{organization}, \texttt{pages},
+ \texttt{part}, \texttt{publisher}, \texttt{series}, \texttt{status},
+ \texttt{subtitle}, \texttt{title}, \texttt{type}, \texttt{volume},
+ and \texttt{xref}
+\end{quote}
+The full list of repeatable fields is
+\begin{quote}
+ \texttt{author}, \texttt{editor}, \texttt{translator},
+ \texttt{isbn}, \texttt{issn}, and \texttt{review}.
+\end{quote}
+The full list of compound fields (for which the value of each is a
+comma separated list of \verb"keyword={value}" statements) is
+\begin{quote}
+ \texttt{book}, \texttt{conference}, \texttt{contribution},
+ \texttt{partial}, \texttt{reprint}, and \texttt{translation}.
+\end{quote}
+A description of the less obvious ones can be found in
+\cite[section~5.2]{amsrefsguide}.
+
+%--------------------------------------------------------------------
+\subsection{Obtaining bibliographic information from MathSciNet}
+\label{sec:mathscinet}
+
+If you have access to MathSciNet (at
+\url{http://www.ams.org/mathscinet}), then you can obtain a completely
+filled out \verb"\bib" command (see section~\ref{sec:bib}) for any of
+the publications listed there. To do this:
+\begin{itemize}
+\item Use MathSciNet to find the Mathematical Reviews review of the
+ publication.
+\item Look near the upper left corner of the page for a pulldown menu
+ titled ``Select Alternative Format''.
+\item Select ``AMSRefs'' from that menu; you'll get a new page showing
+ the \verb"\bib" command for the publication.
+\item Copy that \verb"\bib" command and paste it into your \LaTeX{}
+ file.
+\end{itemize}
+
+
+%--------------------------------------------------------------------
+\subsection{Using a database of possible references}
+\label{sec:refbase}
+
+If your list of \verb"\bib" commands (see section~\ref{sec:bib}) grows
+so long that you don't want to include it in your \LaTeX{} file, or if
+you want to reuse a collection of \verb"\bib" commands from other
+papers, you can put all of your \verb"\bib" commands into a file and
+have \texttt{amsrefs} extract only the ones used in your current
+paper. To do this, you create a file whose name ends in \verb".ltb",
+for example: \verb"myrefs.ltb", and put all of your \verb"\bib"
+commands into that file. In such a file, it's important that each
+\verb"\bib" command begins on a new line, and that that line contains
+the first two arguments and the following open brace, as in
+\begin{center}
+ \verb"\bib{ReferenceKey}{ReferenceType}{"
+\end{center}
+You then delete all of the \verb"\bib" commands from your \LaTeX{}
+file and replace them with the single line
+\begin{center}
+ \verb"\bibselect{myrefs}"
+\end{center}
+and \texttt{amsrefs} will automatically read \verb"myrefs.ltb" and
+extract only those items that are cited in the current paper. Thus,
+if you have the \verb"\bib" commands for all of your possible
+references in the file \verb"myrefs.ltb", and you've loaded
+\texttt{amsrefs} with the command \verb"\usepackage[lite]{amsrefs}" in
+your preamble (see section~\ref{sec:amsrefs}), then you would create
+the bibliography with the lines
+\begin{verbatim}
+\begin{bibdiv}
+ \begin{biblist}
+ \bibselect{myrefs}
+ \end{biblist}
+\end{bibdiv}
+\end{verbatim}
+If your \LaTeX{} file contains the above lines and your file is named
+\verb"mypaper.tex", then \LaTeX{} will create the file
+\verb"mypaper.bbl" which will contain the \verb"\bib" commands for
+only those references cited in your paper, and \verb"mypaper.bbl" will
+automatically be read and used in your \LaTeX{} file.
+
+If your \verb"\bib" commands are spread over several \verb".ltb"
+files, you can either list them all, separated by commas, in the
+argument of the \verb"\bibselect" command, as in
+\begin{center}
+ \verb"\bibselect{myfirstrefs,mysecondrefs}"
+\end{center}
+or just use multiple \verb"\bibselect" commands, as in
+\begin{center}
+ \begin{tabular}{l}
+ \verb"\bibselect{myfirstrefs}"\\
+ \verb"\bibselect{mysecondrefs}"
+ \end{tabular}
+\end{center}
+
+As described above, a \verb"\bibselect" command uses only the
+references actually cited in your document. To list \emph{all} the
+references in an \verb".ltb" file, whether or not they're cited in
+your document, use the command
+\begin{center}
+ \verb"\bibselect*{myrefs}"
+\end{center}
+
+%--------------------------------------------------------------------
+\subsection{Using \texttt{amsrefs} with Bib\TeX}
+\label{sec:bibtex}
+
+If you already have a Bib\TeX{} database of possible references, you
+can use that by combining Bib\TeX{} with \texttt{amsrefs}. If your
+references are in the Bib\TeX{} file \verb"myrefs.bib", then you would
+load the \texttt{amsrefs} package by putting the command
+\verb"\usepackage[lite]{amsrefs}" in your preamble (in the same way
+that you would load it if you were using \texttt{amsrefs} without
+Bib\TeX; see section~\ref{sec:amsrefs}) and put the single line
+\begin{center}
+ \verb"\bibliography{myrefs}"
+\end{center}
+where you want the bibliography to appear. (That is, you don't create
+the \texttt{bibdiv} or \texttt{biblist} environments.) You then run
+Bib\TeX{} on your file in the normal way, and Bib\TeX{} will create
+the \verb".bbl" file. That \verb".bbl" file will contain the
+\texttt{bibdiv} and \texttt{biblist} environments, and \LaTeX{} will
+read it the next time that you run \LaTeX{} and create the
+bibliography.
+
+If you use this method, you don't use the \verb"\bibliographystyle"
+command (and any such command in your \LaTeX{} file will be ignored).
+
+%--------------------------------------------------------------------
+\subsection{Options for the \texttt{amsrefs} package}
+\label{sec:refoptions}
+
+There are a number of optional arguments that you can use when loading
+the \texttt{amsrefs} package (see section~\ref{sec:amsrefs}), most of
+which affect the formatting of the bibliography. You use them by
+listing them as optional arguments to the \verb"\usepackage" command
+(see section~\ref{sec:amsrefs}). For example, we suggested that you
+always use the optional argument \texttt{lite}, as in
+\begin{center}
+ \verb"\usepackage[lite]{amsrefs}"
+\end{center}
+We'll describe the most important options here; the full list can be
+found in \cite[section~6]{amsrefsguide}.
+\begin{description}
+\item[lite] We suggest that you always use this option in order to
+ avoid a conflict between \texttt{amsrefs} and the \Xy-pic package
+ (see section~\ref{sec:xypic}). If you don't use the \texttt{lite}
+ option, then \texttt{amsrefs} automatically loads the packages
+ \texttt{mathscinet} (which defines a number of special characters
+ and accents that are sometimes encountered when downloading data
+ from MathSciNet) and \texttt{txtcmds} (which provides shorthand
+ commands for a number of characters that are usually specified via
+ ligatures). The problem is that both \Xy-pic and the
+ \texttt{txtcmds} package define the command \verb"\cir" (they define
+ it to be entirely different things). If you load \texttt{txtcmds}
+ \emph{before} loading \Xy-pic then \Xy-pic will redefine \verb"\cir"
+ as it pleases while putting a warning into your \verb".log" file,
+ but if you load \texttt{txtcmds} \emph{after} loading \Xy-pic then
+ \LaTeX{} will stop with an error, since the \texttt{txtcmds} package
+ uses \verb"\newcommand" to define \verb"\cir". Unfortunately, if
+ you need to use the \texttt{mathscinet} package and you try to load
+ it by putting the command \verb"\usepackage{mathscinet}" after your
+ \verb"\documentclass" command, it will also load the
+ \texttt{txtcmds} package, so this doesn't avoid the conflict.
+
+ If you need to use both \Xy-pic and the definitions in the
+ \texttt{mathscinet} and \texttt{txtcmds} packages, you should load
+ \texttt{amsrefs} \emph{before} \Xy-pic, as in
+ \begin{center}
+ \begin{tabular}{l}
+ \verb"\usepackage{amsrefs}"\\
+ \verb"\usepackage[all,cmtip]{xy}"
+ \end{tabular}
+ \end{center}
+ to keep \LaTeX{} from generating an error when loading
+ \texttt{txtcmds}. (Of course, the command \verb"\cir" as defined in
+ \texttt{textcmds} will not be available.)
+\item[initials] If you use this option, then all authors, editors, and
+ translators will have their first and middle names replaced by their
+ initials.
+\item[alphabetic] If you use this option, as in
+ \begin{center}
+ \verb"\usepackage[alphabetic,lite]{amsrefs}"
+ \end{center}
+ then instead of numbers being used to label the bibliography
+ entries, you will get alphabetic labels similar to the
+ \texttt{alpha} style used by Bib\TeX, consisting of the first
+ letter(s) of each author name plus the year of publication.
+\item[shortalphabetic] This is similar to the \texttt{alphabetic}
+ option, except that you'll get a shorter alphabetic label using only
+ the first letter of each author name.
+\item[y2k] If you use the \texttt{alphabetic} option, only the last
+ two digits of the year are normally used in the label. If you also
+ use the \texttt{y2k} option, as in
+ \begin{center}
+ \verb"\usepackage[y2k,alphabetic,lite]{amsrefs}"
+ \end{center}
+ then the full year will be used.
+\item[author-year] If you use this option, then bibliography items
+ will not be labelled at all, and references to them will be in the
+ author-year format similar to that described in
+ \verb"The Chicago Manual of Style". If you do use this option, you
+ may sometimes want to use the \verb"\ycite" and \verb"\ocite"
+ commands when referring to bibliography items (see
+ section~\ref{sec:authyrcit}).
+\end{description}
+
+
+
+%--------------------------------------------------------------------
+\subsection{Avoiding both \texttt{amsrefs} and Bib\TeX}
+\label{sec:thebibliography}
+
+If for some reason you don't want to use \texttt{amsrefs} or Bib\TeX,
+you can type each bibliography item exactly as you want it to appear
+and have the items numbered automatically. To do this, you
+\begin{enumerate}
+\item begin the bibliography with the command
+ \begin{center}
+ \verb"\begin{thebibliography}{number}"
+ \end{center}
+ where \emph{number} is any number that, when printed, is as wide as
+ the widest number of any item in the bibliography,
+\item define the bibliography items using a \verb"\bibitem" command
+ for each item (see section~\ref{sec:bibitem}), and then
+\item end the bibliography with the command
+ \begin{center}
+ \verb"\end{thebibliography}"
+ \end{center}
+\end{enumerate}
+The only use made of the \verb"number" in
+\verb"\begin{thebibliography}{number}" is that \LaTeX{} assumes that
+its width when printed is at least as large as the width of any number
+of an item in the bibliography. For example, if the bibliography will
+contain between 10 and~99 items, you can use
+\verb"\begin{thebibliography}{99}".
+
+
+%--------------------------------------------------------------------
+\subsubsection{Bibliography items}
+\label{sec:bibitem}
+
+When using the commands \verb"\begin{thebibliography}" and
+\verb"\end{thebibliography}" to create the bibliography (see
+section~\ref{sec:thebibliography}), each item is begun with a
+\verb"\bibitem" command. The format is
+\begin{center}
+ \verb"\bibitem{ReferenceKey}Item entry"
+\end{center}
+For example, had we not been using \texttt{amsrefs}, the bibliography
+in these instructions might have contained the entry
+\begin{verbatim}
+\bibitem{yellowmonster}
+A. K. Bousfield and D. M. Kan, \emph{Homotopy Limits, Completions
+and Localizations,} Lecture Notes in Mathematics number 304,
+Springer-Verlag, New York, 1972.
+\end{verbatim}
+
+The above entry would allow us to type
+\begin{verbatim}
+Homotopy inverse limits are discussed
+in~\cite[Chapter 11]{yellowmonster}.
+\end{verbatim}
+and have it print as ``Homotopy inverse limits are discussed
+in~\cite[Chapter 11]{yellowmonster}.'' For more on the \verb"\cite"
+command, see section~\ref{sec:bibreferences}.
+
+
+
+
+
+
+
+
+
+
+
+%---------------------------------------------------------------------
+%---------------------------------------------------------------------
+\section{The template file}
+\label{sec:template}
+
+The following is the text of the file \verb"template.tex".
+
+\begin{verbatim}
+%%% template.tex
+%%% This is a template for making up an AMS-LaTeX file
+%%% Version of February 12, 2011
+%%%---------------------------------------------------------
+%%% The following command chooses the default 10 point type.
+%%% To choose 12 point, change it to
+%%% \documentclass[12pt]{amsart}
+\documentclass{amsart}
+
+%%% The following command loads the amsrefs package, which will be
+%%% used to create the bibliography:
+\usepackage[lite]{amsrefs}
+
+%%% The following command defines the standard names for all of the
+%%% special symbols in the AMSfonts package, listed in
+%%% http://www.ctan.org/tex-archive/info/symbols/math/symbols.pdf
+\usepackage{amssymb}
+
+%%% The following commands allow you to use \Xy-pic to draw
+%%% commutative diagrams. (You can omit the second line if you want
+%%% the default style of the nodes to be \textstyle.)
+\usepackage[all,cmtip]{xy}
+\let\objectstyle=\displaystyle
+
+%%% If you'll be importing any graphics, uncomment the following
+%%% line. (Note: The spelling is correct; the package graphicx.sty is
+%%% the updated version of the older graphics.sty.)
+% \usepackage{graphicx}
+
+
+
+%%% This part of the file (after the \documentclass command,
+%%% but before the \begin{document}) is called the ``preamble''.
+%%% This is where we put our macro definitions.
+
+%%% Comment out (or delete) any of these that you don't want to use.
+\newcommand{\tensor}{\otimes}
+\newcommand{\homotopic}{\simeq}
+\newcommand{\homeq}{\cong}
+\newcommand{\iso}{\approx}
+
+\DeclareMathOperator{\ho}{Ho}
+\DeclareMathOperator*{\colim}{colim}
+
+\newcommand{\R}{\mathbb{R}}
+\newcommand{\C}{\mathbb{C}}
+\newcommand{\Z}{\mathbb{Z}}
+
+\newcommand{\M}{\mathcal{M}}
+\newcommand{\W}{\mathcal{W}}
+
+\newcommand{\itilde}{\tilde{\imath}}
+\newcommand{\jtilde}{\tilde{\jmath}}
+\newcommand{\ihat}{\hat{\imath}}
+\newcommand{\jhat}{\hat{\jmath}}
+
+
+
+%%%-------------------------------------------------------------------
+%%%-------------------------------------------------------------------
+%%% The Theorem environments:
+%%%
+%%%
+%%% The following commands set it up so that:
+%%%
+%%% All Theorems, Corollaries, Lemmas, Propositions, Definitions,
+%%% Remarks, Examples, Notations, and Terminologies will be numbered
+%%% in a single sequence, and the numbering will be within each
+%%% section. Displayed equations will be numbered in the same
+%%% sequence.
+%%%
+%%%
+%%% Theorems, Propositions, Lemmas, and Corollaries will have the most
+%%% formal typesetting.
+%%%
+%%% Definitions will have the next level of formality.
+%%%
+%%% Remarks, Examples, Notations, and Terminologies will be the least
+%%% formal.
+%%%
+%%% Theorem:
+%%% \begin{thm}
+%%%
+%%% \end{thm}
+%%%
+%%% Corollary:
+%%% \begin{cor}
+%%%
+%%% \end{cor}
+%%%
+%%% Lemma:
+%%% \begin{lem}
+%%%
+%%% \end{lem}
+%%%
+%%% Proposition:
+%%% \begin{prop}
+%%%
+%%% \end{prop}
+%%%
+%%% Definition:
+%%% \begin{defn}
+%%%
+%%% \end{defn}
+%%%
+%%% Remark:
+%%% \begin{rem}
+%%%
+%%% \end{rem}
+%%%
+%%% Example:
+%%% \begin{ex}
+%%%
+%%% \end{ex}
+%%%
+%%% Notation:
+%%% \begin{notation}
+%%%
+%%% \end{notation}
+%%%
+%%% Terminology:
+%%% \begin{terminology}
+%%%
+%%% \end{terminology}
+%%%
+%%% Theorem environments
+
+% The following causes equations to be numbered within sections
+\numberwithin{equation}{section}
+
+% We'll use the equation counter for all our theorem environments, so
+% that everything will be numbered in the same sequence.
+
+% Theorem environments
+
+\theoremstyle{plain} %% This is the default, anyway
+\newtheorem{thm}[equation]{Theorem}
+\newtheorem{cor}[equation]{Corollary}
+\newtheorem{lem}[equation]{Lemma}
+\newtheorem{prop}[equation]{Proposition}
+
+\theoremstyle{definition}
+\newtheorem{defn}[equation]{Definition}
+
+\theoremstyle{remark}
+\newtheorem{rem}[equation]{Remark}
+\newtheorem{ex}[equation]{Example}
+\newtheorem{notation}[equation]{Notation}
+\newtheorem{terminology}[equation]{Terminology}
+
+%%%-------------------------------------------------------------------
+%%%-------------------------------------------------------------------
+%%%-------------------------------------------------------------------
+%%%-------------------------------------------------------------------
+%%%-------------------------------------------------------------------
+%%%-------------------------------------------------------------------
+%%%-------------------------------------------------------------------
+\begin{document}
+
+%%% In the title, use a double backslash "\\" to show a linebreak:
+%%% Use one of the following two forms:
+%%% \title{Text of the title}
+%%% or
+%%% \title[Short form for the running head]{Text of the title}
+\title{}
+
+
+%%% If there are multiple authors, they're described one at a time:
+%%% First author: \author{} \address{} \curraddr{} \email{} \thanks{}
+%%% Second author: \author{} \address{} \curraddr{} \email{} \thanks{}
+%%% Third author: \author{} \address{} \curraddr{} \email{} \thanks{}
+\author{}
+
+%%% In the address, show linebreaks with double backslashes:
+\address{}
+
+%%% Current address is optional.
+% \curraddr{}
+
+%%% Email address is optional.
+% \email{}
+
+
+%%% If there's a second author:
+% \author{}
+% \address{}
+% \curraddr{}
+% \email{}
+
+
+%%% To have the current date inserted, use \date{\today}:
+\date{}
+
+%%% To include an abstract, uncomment the following two lines and type
+%%% the abstract in between them:
+% \begin{abstract}
+% \end{abstract}
+
+
+\maketitle
+
+%%% To include a table of contents, uncomment the following line:
+% \tableofcontents
+
+
+%%%-------------------------------------------------------------------
+%%%-------------------------------------------------------------------
+%%% Start the body of the paper here! E.G., maybe use:
+%%% \section{Introduction}
+%%% \label{sec:intro}
+
+%%% For a numbered display, use
+%%% \begin{equation}
+%%% \label{something}
+%%% The display goes here
+%%% \end{equation}
+%%% and you can refer to it as \eqref{something}.
+
+%%% For an unnumbered display, use
+%%% \begin{equation*}
+%%% The display goes here
+%%% \end{equation*}
+
+%%% To import a graphics file, you must have said
+%%% \usepackage{graphicx}
+%%% in the preamble (i.e., before the \begin{document}).
+%%% Putting it into a figure environment enables it to float to the
+%%% next page if there isn't enough room for it on the current page.
+%%% The \label command must come after the \caption command.
+% \begin{figure}[h]
+% \includegraphics{filename}
+% \caption{Some caption}
+% \label{somelabel}
+% \end{figure}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+%%% -------------------------------------------------------------------
+%%% -------------------------------------------------------------------
+%%% This is where we create the bibliography.
+
+\begin{bibdiv}
+ \begin{biblist}
+
+%%% The format of bibliography items is as in the following examples:
+%%%
+%%% \bib{yellowmonster}{book}{
+%%% author={Bousfield, A.K.},
+%%% author={Kan, D.M.},
+%%% title={Homotopy Limits, Completions and Localizations},
+%%% date={1972},
+%%% series={Lecture Notes in Mathematics},
+%%% volume={304},
+%%% publisher={Springer-Verlag},
+%%% address={Berlin-New York}
+%%% }
+
+%%% \bib{HA}{book}{
+%%% author={Quillen, Daniel G.},
+%%% title={Homotopical Algebra},
+%%% series={Lecture Notes in Mathematics},
+%%% volume={43},
+%%% publisher={Springer-Verlag},
+%%% address={Berlin-New York},
+%%% date={1967}
+%%% }
+
+%%% \bib{serre:shfs}{article}{
+%%% author={Serre, Jean-Pierre},
+%%% title={Homologie Singuli\`ere des Espaces Fibr\'es. Applications},
+%%% journal={Ann. of Math. (2)},
+%%% date={1951},
+%%% volume={54},
+%%% pages={425--505}
+%%% }
+
+
+
+
+
+ \end{biblist}
+\end{bibdiv}
+
+\end{document}
+\end{verbatim}
+
+%---------------------------------------------------------------------
+%---------------------------------------------------------------------
+
+
+
+
+
+\begin{bibdiv}
+ \begin{biblist}
+
+\bib{amslatexusersguide}{report}{
+ author={American Mathematical Society},
+ title={User's Guide for the \texttt{amsmath} Package},
+ edition={version 2.0},
+ date={2002-02-25},
+ eprint={ftp://ftp.ams.org/pub/tex/doc/amsmath/amsldoc.pdf},
+ note={also available from CTAN
+ (the Comprehensive \TeX{} Archive Network) at
+ \url{http://www.ctan.org/tex-archive/macros/latex/required/amslatex/math/amsldoc.pdf}}
+}
+
+\bib{instr-l}{report}{
+ author={American Mathematical Society},
+ title={Instructions for Preparation of Papers and Monographs:
+ \AmS-\LaTeX},
+ date={2004-08},
+ eprint={ftp://ftp.ams.org/pub/tex/doc/amscls/instr-l.pdf}
+}
+
+\bib{testmath}{misc}{
+ author={American Mathematical Society},
+ title={Sample Paper for the \texttt{amsmath} Package},
+ date={1996-11},
+ note={available at \url{ftp://ftp.ams.org/pub/tex/amslatex/math/testmath.tex}}
+}
+
+\bib{yellowmonster}{book}{
+ author={Bousfield, A.K.},
+ author={Kan, D.M.},
+ title={Homotopy Limits, Completions and Localizations},
+ date={1972},
+ series={Lecture Notes in Mathematics},
+ volume={304},
+ publisher={Springer-Verlag},
+ address={Berlin-New York}
+}
+
+\bib{mathguide}{report}{
+ author={Downes, Michael},
+ title={Short Math Guide for \LaTeX},
+ edition={version 1.09},
+ publisher={American Mathematical Society},
+ eprint={ftp://ftp.ams.org/pub/tex/doc/amsmath/short-math-guide.pdf}
+}
+
+\bib{amsrefsguide}{report}{
+ author={Jones, David M.},
+ title={User's Guide to the \texttt{amsrefs} Package},
+ publisher={American Mathematical Society},
+ date={2007-10-16},
+ eprint={ftp://ftp.ams.org/pub/tex/amsrefs/amsrdoc.pdf}
+}
+
+\bib{latex}{book}{
+ author={Lamport, Leslie},
+ title={\LaTeX: A Document Preparation System},
+ edition={2},
+ publisher={Addison-Wesley},
+ date={1994}
+}
+
+\bib{NotShort}{report}{
+ author={Oetiker, Tobias},
+ author={Partl, Hubert},
+ author={Hyna, Irene},
+ author={Schlegl, Elisabeth},
+ title={The Not So Short Introduction to \LaTeXe},
+ edition={version 4.24},
+ eprint={http://www.ctan.org/tex-archive/info/lshort/english/lshort.pdf}
+}
+
+\bib{HA}{book}{
+ author={Quillen, Daniel G.},
+ title={Homotopical Algebra},
+ series={Lecture Notes in Mathematics},
+ volume={43},
+ publisher={Springer-Verlag},
+ address={Berlin-New York},
+ date={1967}
+}
+
+\bib{quil:rht}{article}{
+ author={Quillen, Daniel G.},
+ title={Rational Homotopy Theory},
+ journal={Ann. of Math. (2)},
+ volume={90},
+ date={1969},
+ pages={205--295}
+}
+
+\bib{xyguide}{report}{
+ author={Rose, Kristoffer H.},
+ title={\Xy-pic User's Guide},
+ edition={version 3.85},
+ eprint={http://www.ctan.org/tex-archive/macros/generic/diagrams/xypic/xy-3.8.5/doc/xyguide.pdf}
+}
+
+\bib{xyrefer}{report}{
+ author={Rose, Kristoffer H.},
+ author={Moore, Ross},
+ title={\Xy-pic Reference Manual},
+ edition={version 3.85},
+ eprint={http://www.ctan.org/tex-archive/macros/generic/diagrams/xypic/xy-3.8.5/doc/xyrefer.pdf}
+}
+
+ \end{biblist}
+\end{bibdiv}
+
+
+
+\end{document}
diff --git a/info/amslatex/primer/template.tex b/info/amslatex/primer/template.tex
new file mode 100644
index 0000000000..0e15df1f01
--- /dev/null
+++ b/info/amslatex/primer/template.tex
@@ -0,0 +1,296 @@
+%%% template.tex
+%%% This is a template for making up an AMS-LaTeX file
+%%% Version of February 12, 2011
+%%%---------------------------------------------------------
+%%% The following command chooses the default 10 point type.
+%%% To choose 12 point, change it to
+%%% \documentclass[12pt]{amsart}
+\documentclass{amsart}
+
+%%% The following command loads the amsrefs package, which will be
+%%% used to create the bibliography:
+\usepackage[lite]{amsrefs}
+
+%%% The following command defines the standard names for all of the
+%%% special symbols in the AMSfonts package, listed in
+%%% http://www.ctan.org/tex-archive/info/symbols/math/symbols.pdf
+\usepackage{amssymb}
+
+%%% The following commands allow you to use \Xy-pic to draw
+%%% commutative diagrams. (You can omit the second line if you want
+%%% the default style of the nodes to be \textstyle.)
+\usepackage[all,cmtip]{xy}
+\let\objectstyle=\displaystyle
+
+%%% If you'll be importing any graphics, uncomment the following
+%%% line. (Note: The spelling is correct; the package graphicx.sty is
+%%% the updated version of the older graphics.sty.)
+% \usepackage{graphicx}
+
+
+
+%%% This part of the file (after the \documentclass command,
+%%% but before the \begin{document}) is called the ``preamble''.
+%%% This is where we put our macro definitions.
+
+%%% Comment out (or delete) any of these that you don't want to use.
+\newcommand{\tensor}{\otimes}
+\newcommand{\homotopic}{\simeq}
+\newcommand{\homeq}{\cong}
+\newcommand{\iso}{\approx}
+
+\DeclareMathOperator{\ho}{Ho}
+\DeclareMathOperator*{\colim}{colim}
+
+\newcommand{\R}{\mathbb{R}}
+\newcommand{\C}{\mathbb{C}}
+\newcommand{\Z}{\mathbb{Z}}
+
+\newcommand{\M}{\mathcal{M}}
+\newcommand{\W}{\mathcal{W}}
+
+\newcommand{\itilde}{\tilde{\imath}}
+\newcommand{\jtilde}{\tilde{\jmath}}
+\newcommand{\ihat}{\hat{\imath}}
+\newcommand{\jhat}{\hat{\jmath}}
+
+
+
+%%%-------------------------------------------------------------------
+%%%-------------------------------------------------------------------
+%%% The Theorem environments:
+%%%
+%%%
+%%% The following commands set it up so that:
+%%%
+%%% All Theorems, Corollaries, Lemmas, Propositions, Definitions,
+%%% Remarks, Examples, Notations, and Terminologies will be numbered
+%%% in a single sequence, and the numbering will be within each
+%%% section. Displayed equations will be numbered in the same
+%%% sequence.
+%%%
+%%%
+%%% Theorems, Propositions, Lemmas, and Corollaries will have the most
+%%% formal typesetting.
+%%%
+%%% Definitions will have the next level of formality.
+%%%
+%%% Remarks, Examples, Notations, and Terminologies will be the least
+%%% formal.
+%%%
+%%% Theorem:
+%%% \begin{thm}
+%%%
+%%% \end{thm}
+%%%
+%%% Corollary:
+%%% \begin{cor}
+%%%
+%%% \end{cor}
+%%%
+%%% Lemma:
+%%% \begin{lem}
+%%%
+%%% \end{lem}
+%%%
+%%% Proposition:
+%%% \begin{prop}
+%%%
+%%% \end{prop}
+%%%
+%%% Definition:
+%%% \begin{defn}
+%%%
+%%% \end{defn}
+%%%
+%%% Remark:
+%%% \begin{rem}
+%%%
+%%% \end{rem}
+%%%
+%%% Example:
+%%% \begin{ex}
+%%%
+%%% \end{ex}
+%%%
+%%% Notation:
+%%% \begin{notation}
+%%%
+%%% \end{notation}
+%%%
+%%% Terminology:
+%%% \begin{terminology}
+%%%
+%%% \end{terminology}
+%%%
+%%% Theorem environments
+
+% The following causes equations to be numbered within sections
+\numberwithin{equation}{section}
+
+% We'll use the equation counter for all our theorem environments, so
+% that everything will be numbered in the same sequence.
+
+% Theorem environments
+
+\theoremstyle{plain} %% This is the default, anyway
+\newtheorem{thm}[equation]{Theorem}
+\newtheorem{cor}[equation]{Corollary}
+\newtheorem{lem}[equation]{Lemma}
+\newtheorem{prop}[equation]{Proposition}
+
+\theoremstyle{definition}
+\newtheorem{defn}[equation]{Definition}
+
+\theoremstyle{remark}
+\newtheorem{rem}[equation]{Remark}
+\newtheorem{ex}[equation]{Example}
+\newtheorem{notation}[equation]{Notation}
+\newtheorem{terminology}[equation]{Terminology}
+
+%%%-------------------------------------------------------------------
+%%%-------------------------------------------------------------------
+%%%-------------------------------------------------------------------
+%%%-------------------------------------------------------------------
+%%%-------------------------------------------------------------------
+%%%-------------------------------------------------------------------
+%%%-------------------------------------------------------------------
+\begin{document}
+
+%%% In the title, use a double backslash "\\" to show a linebreak:
+%%% Use one of the following two forms:
+%%% \title{Text of the title}
+%%% or
+%%% \title[Short form for the running head]{Text of the title}
+\title{}
+
+
+%%% If there are multiple authors, they're described one at a time:
+%%% First author: \author{} \address{} \curraddr{} \email{} \thanks{}
+%%% Second author: \author{} \address{} \curraddr{} \email{} \thanks{}
+%%% Third author: \author{} \address{} \curraddr{} \email{} \thanks{}
+\author{}
+
+%%% In the address, show linebreaks with double backslashes:
+\address{}
+
+%%% Current address is optional.
+% \curraddr{}
+
+%%% Email address is optional.
+% \email{}
+
+
+%%% If there's a second author:
+% \author{}
+% \address{}
+% \curraddr{}
+% \email{}
+
+
+%%% To have the current date inserted, use \date{\today}:
+\date{}
+
+%%% To include an abstract, uncomment the following two lines and type
+%%% the abstract in between them:
+% \begin{abstract}
+% \end{abstract}
+
+
+\maketitle
+
+%%% To include a table of contents, uncomment the following line:
+% \tableofcontents
+
+
+%%%-------------------------------------------------------------------
+%%%-------------------------------------------------------------------
+%%% Start the body of the paper here! E.G., maybe use:
+%%% \section{Introduction}
+%%% \label{sec:intro}
+
+%%% For a numbered display, use
+%%% \begin{equation}
+%%% \label{something}
+%%% The display goes here
+%%% \end{equation}
+%%% and you can refer to it as \eqref{something}.
+
+%%% For an unnumbered display, use
+%%% \begin{equation*}
+%%% The display goes here
+%%% \end{equation*}
+
+%%% To import a graphics file, you must have said
+%%% \usepackage{graphicx}
+%%% in the preamble (i.e., before the \begin{document}).
+%%% Putting it into a figure environment enables it to float to the
+%%% next page if there isn't enough room for it on the current page.
+%%% The \label command must come after the \caption command.
+% \begin{figure}[h]
+% \includegraphics{filename}
+% \caption{Some caption}
+% \label{somelabel}
+% \end{figure}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+%%% -------------------------------------------------------------------
+%%% -------------------------------------------------------------------
+%%% This is where we create the bibliography.
+
+\begin{bibdiv}
+ \begin{biblist}
+
+%%% The format of bibliography items is as in the following examples:
+%%%
+%%% \bib{yellowmonster}{book}{
+%%% author={Bousfield, A.K.},
+%%% author={Kan, D.M.},
+%%% title={Homotopy Limits, Completions and Localizations},
+%%% date={1972},
+%%% series={Lecture Notes in Mathematics},
+%%% volume={304},
+%%% publisher={Springer-Verlag},
+%%% address={Berlin-New York}
+%%% }
+
+%%% \bib{HA}{book}{
+%%% author={Quillen, Daniel G.},
+%%% title={Homotopical Algebra},
+%%% series={Lecture Notes in Mathematics},
+%%% volume={43},
+%%% publisher={Springer-Verlag},
+%%% address={Berlin-New York},
+%%% date={1967}
+%%% }
+
+%%% \bib{serre:shfs}{article}{
+%%% author={Serre, Jean-Pierre},
+%%% title={Homologie Singuli\`ere des Espaces Fibr\'es. Applications},
+%%% journal={Ann. of Math. (2)},
+%%% date={1951},
+%%% volume={54},
+%%% pages={425--505}
+%%% }
+
+
+
+
+
+ \end{biblist}
+\end{bibdiv}
+
+\end{document}
diff --git a/info/amslatex/vietnamese/README b/info/amslatex/vietnamese/README
new file mode 100644
index 0000000000..92bc30391b
--- /dev/null
+++ b/info/amslatex/vietnamese/README
@@ -0,0 +1,22 @@
+
+AMSLaTeX Manual v2.0, Vietnamese edition.
+
+
+Original Author: American Mathematical Society
+
+
+Translator: Ky Anh <kyanh@o2.pl>
+
+
+License of the translation: LPPL, version 1.3 or newer,
+according to http://www.latex-project.org/lppl.txt
+
+
+Files:
+ README.txt (this file)
+ TODO
+ amsldoc-vi.pdf
+ amsldoc-print-vi.pdf
+ src/Makefile
+ src/amsldoc-print-vi.tex
+ src/amsldoc-vi.tex
diff --git a/info/amslatex/vietnamese/TODO b/info/amslatex/vietnamese/TODO
new file mode 100644
index 0000000000..eeef773d40
--- /dev/null
+++ b/info/amslatex/vietnamese/TODO
@@ -0,0 +1,4 @@
+* Vietnamese version of 'testmath.tex'
+* Vietnamese version of 'subeqn.tex'
+* amsldoc: Translate 'Error Message' section into Vietnamese
+* amsldoc: make index page
diff --git a/info/amslatex/vietnamese/amsldoc-print-vi.pdf b/info/amslatex/vietnamese/amsldoc-print-vi.pdf
new file mode 100644
index 0000000000..6b333cc6d0
--- /dev/null
+++ b/info/amslatex/vietnamese/amsldoc-print-vi.pdf
Binary files differ
diff --git a/info/amslatex/vietnamese/amsldoc-vi.pdf b/info/amslatex/vietnamese/amsldoc-vi.pdf
new file mode 100644
index 0000000000..e34316c6b2
--- /dev/null
+++ b/info/amslatex/vietnamese/amsldoc-vi.pdf
Binary files differ
diff --git a/info/amslatex/vietnamese/src/Makefile b/info/amslatex/vietnamese/src/Makefile
new file mode 100644
index 0000000000..c7fd09cef0
--- /dev/null
+++ b/info/amslatex/vietnamese/src/Makefile
@@ -0,0 +1,32 @@
+SCREEN = amsldoc-vi
+PRINT = amsldoc-print-vi
+
+amsldoc: screen print
+
+screen:
+ pdflatex $(SCREEN)
+ pdflatex $(SCREEN)
+ pdflatex $(SCREEN)
+ thumbpdf $(SCREEN).pdf
+ pdflatex $(SCREEN)
+
+print:
+ pdflatex $(PRINT)
+ pdflatex $(PRINT)
+ pdflatex $(PRINT)
+
+quick:
+ pdflatex $(SCREEN)
+
+tar:
+ mkdir -p amsldoc-vi/src
+ cp -pv {$(SCREEN),$(PRINT)}.pdf README.txt TODO amsldoc-vi
+ cp -pv *.tex Makefile amsldoc-vi/src
+ tar -cjf amsldoc-vi.tar.bz2 amsldoc-vi
+
+clean:
+ rm -fv *.log *.idx *.aux *.dvi *.bbl *.blg *.toc *.idx *.tps \
+ *.ps *.ilg *.tcp *~ *.ind *.out *.tpt
+
+cleanall: clean
+ rm -fv *.pdf *~ amsldoc-vi.tar.bz2
diff --git a/info/amslatex/vietnamese/src/amsldoc-print-vi.tex b/info/amslatex/vietnamese/src/amsldoc-print-vi.tex
new file mode 100644
index 0000000000..bacc754d82
--- /dev/null
+++ b/info/amslatex/vietnamese/src/amsldoc-print-vi.tex
@@ -0,0 +1,2 @@
+\def\printversion{true}
+\input amsldoc-vi.tex
diff --git a/info/amslatex/vietnamese/src/amsldoc-vi.tex b/info/amslatex/vietnamese/src/amsldoc-vi.tex
new file mode 100644
index 0000000000..f370b88abd
--- /dev/null
+++ b/info/amslatex/vietnamese/src/amsldoc-vi.tex
@@ -0,0 +1,3330 @@
+\ifx\printversion\undefined
+\documentclass[11pt,leqno,titlepage,openany,oneside]{amsldoc}[1999/12/13]
+\else
+\documentclass[a4paper,11pt,leqno,titlepage,openany]{amsldoc}[1999/12/13]
+\usepackage[monochrome]{color}
+\advance \topmargin by -3\baselineskip
+\advance \textheight by 5\baselineskip
+\usepackage{lmodern}
+\fi
+
+% \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
+
+% ----------------------------------------------------------------------
+\pdfminorversion=3
+\usepackage[utf8]{vietnam}
+\usepackage{amsxtra}
+\usepackage{shortvrb}
+\usepackage{afterpage}
+\usepackage{amsfonts,euscript}
+
+\MakeShortVerb{|}
+
+% hack \url
+% cannot use package hyperref. Donknow why?
+% \makeatletter
+% \let\old@url=\url
+% \def\url#1{\textcolor{blue}{\old@url{#1}}}
+% \makeatother
+
+% ----------------------------------------------------------------------
+
+\DeclareMathOperator{\tg}{tg}
+
+
+% ----------------------------------------------------------------------
+
+
+\newcommand{\entrylabel}[1]{\mbox{\textsf{#1:}}\hfil}
+\newenvironment{entry}%
+ {\begin{list}{}%
+ {\renewcommand{\makelabel}{\entrylabel}%
+ \setlength{\labelwidth}{100pt}%
+ \setlength{\leftmargin}{\labelwidth}
+ \addtolength{\leftmargin}{\labelsep}%
+ }%
+ }%
+ {\end{list}}
+
+\newlength{\mylen}
+\newcommand{\lentrylabel}[1]{%
+ \settowidth{\mylen}{\textsf{#1:}}%
+ \ifthenelse{\lengthtest{\mylen >\labelwidth}}%
+ {\parbox[b]{\labelwidth}%
+ {\makebox[0pt][l]{\textsf{#1\space:}}\\}}%
+ {\hfil\textsf{#1\space:}}%
+ \relax}
+\newenvironment{lentry}%
+ {\renewcommand{\entrylabel}{\lentrylabel}%
+ \begin{entry}}%
+ {\end{entry}}
+
+% ----------------------------------------------------------------------
+% :: hack heading
+
+
+\makeatletter
+\def\ps@empty{\let\@mkboth\@gobbletwo
+ \let\@oddhead\@empty \let\@evenhead\@empty
+ \let\@oddfoot\@empty
+ \let\@evenfoot\@empty
+ \global\topskip\normaltopskip}
+\def\ps@plain{\ps@empty
+ \def\@oddfoot{\normalfont\scriptsize \hfil\thepage\hfil}%
+ \let\@evenfoot\@oddfoot}
+%\newswitch{runhead}
+\def\ps@headings{\ps@empty
+%% \def\@oddfoot{\hfil Vn\TeX\ (\url{http://vntex.org})\hfil}
+ \def\@oddfoot{}
+%% \def\@evenhead{%
+%% \normalfont\small%\scriptsize
+%% %%% \hfil
+%% \thesection.
+%% \leftmark{}{}\hfil \llap{\thepage}}%
+ \def\@evenhead{%
+ \normalfont\small\thepage\hfil
+ \leftmark{}{}}%
+ \def\@oddhead{%
+ \normalfont\small
+ \rightmark{}{}\hfil \llap{\thepage}}%
+%%% \let\@evenhead\@oddhead
+ \let\@mkboth\markboth
+ \def\partmark{\@secmark\markboth\partrunhead\partname}%
+ \def\chaptermark{%
+ \@secmark\markboth\chapterrunhead{}}%
+ \def\sectionmark{%
+ \@secmark\markright\sectionrunhead\sectionname}%
+}
+\let\sectionname\@empty
+\let\subsectionname\@empty
+\let\subsubsectionname\@empty
+\let\paragraphname\@empty
+\let\subparagraphname\@empty
+\def\leftmark{\expandafter\@firstoftwo\topmark{}{}}
+\def\rightmark{\expandafter\@secondoftwo\botmark{}{}}
+\long\def\@nilgobble#1\@nil{}
+\def\@secmark#1#2#3#4{%
+ \begingroup \let\protect\@unexpandable@protect
+ \edef\@tempa{\endgroup \toks@{\protect#2{#3}{\@secnumber}}}%
+ \@tempa
+ \toks@\@xp{\the\toks@{#4}}%
+ \afterassignment\@nilgobble\@temptokena\@themark{}\@nil
+ \edef\@tempa{\@nx\@mkboth{%
+ \ifx\markright#1\the\@temptokena\else\the\toks@\fi}{\the\toks@}}%
+ \@tempa}
+\let\@secnumber\@empty
+\def\markboth#1#2{%
+ \begingroup
+ \@temptokena{{#1}{#2}}\xdef\@themark{\the\@temptokena}%
+ \mark{\the\@temptokena}%
+ \endgroup
+ \if@nobreak\ifvmode\nobreak\fi\fi}
+\newskip\normaltopskip
+\normaltopskip=10pt \relax
+\let\sectionmark\@gobble
+\let\subsectionmark\@gobble
+\let\subsubsectionmark\@gobble
+\let\paragraphmark\@gobble
+
+\def\partrunhead#1#2#3{%
+ \@ifnotempty{#2}{\textsc{\ignorespaces#1 #2\unskip}\@ifnotempty{#3}{. }}%
+ \def\@tempa{#3}%
+ \ifx\@empty\@tempa\else
+ \begingroup \def\\{ \ignorespaces}% defend against questionable usage
+% \MakeUppercase{\@tempa}
+ \textsc{\@tempa}
+ \endgroup
+ \fi
+}
+\let\chapterrunhead\partrunhead
+\let\sectionrunhead\partrunhead
+
+\renewenvironment{thebibliography}[1]
+ {\chapter*{\bibname}%
+ \@mkboth{\textsc{\bibname}}{\textsc{\bibname}}%
+ \list{\@biblabel{\@arabic\c@enumiv}}%
+ {\settowidth\labelwidth{\@biblabel{#1}}%
+ \leftmargin\labelwidth
+ \advance\leftmargin\labelsep
+ \@openbib@code
+ \usecounter{enumiv}%
+ \let\p@enumiv\@empty
+ \renewcommand\theenumiv{\@arabic\c@enumiv}}%
+ \sloppy
+ \clubpenalty4000
+ \@clubpenalty \clubpenalty
+ \widowpenalty4000%
+ \sfcode`\.\@m}
+ {\def\@noitemerr
+ {\@latex@warning{Empty `thebibliography' environment}}%
+ \endlist}
+
+\makeatother
+% ----------------------------------------------------------------------
+
+% :: title
+%\title{User's Guide for the \pkg{amsmath} Package (Version~2.0)}
+\title{Hướng dẫn sử dụng gói \pkg{amsmath} (phiên bản 2.0)}
+%\author{American Mathematical Society}
+
+\author{Tác giả: Hội Toán học Mỹ (AMS)\\[6pt]
+ 13/12/1999 (sửa đổi 25/02/2002)\\[1cm]
+ Biên dịch: Ky Anh
+ $\langle$\href{mailto:kyanh@o2.pl}{kyanh@o2.pl}$\rangle$\\[6pt]
+ Bản dịch mới nhất 15/10/2005\\[5cm]
+ \url{http://VietTUG.org}}
+
+\date{}
+
+%%\medskip \emph{Bản dịch số {$\mathbf{\the\buildnum}$}}}
+
+% 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]{%
+%% \MakeUppercase{\markboth{\rhcn#1}{\rhcn#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}
+
+\usepackage[colorlinks,draft=false,
+pdftitle={AMSLaTeX Manual, v2.0, Vietnamese edition},
+pdfauthor={American Mathematical Society, Translator: Ky Anh <kyanh@o2.pl>},
+pdfsubject={Advanced Math Typesetting},
+pdfkeywords={math, typesetting, formulas}]{hyperref} \input pd1supp.def
+\ifx\printversion\undefined
+\RequirePackage{thumbpdf}
+\hypersetup{pdfpagemode=UseThumbs}
+\fi
+\hypersetup{plainpages=false}
+\hypersetup{pdfpagelabels=false}
+%\hypersetup{hypertexnames=true}
+\hypersetup{naturalnames=true}
+
+%\pagenumbering{roman}
+\pagestyle{headings}
+\begin{document}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\pagestyle{empty}
+\maketitle
+%% \rightline{\emph{The only way to learn mathematics is to do mathematics.}}
+%% \medskip
+%% \rightline{{\rm P. R. HALMOS}}
+\ifx\printversion\undefined
+\newpage
+\else
+\cleardoublepage
+\fi
+\tableofcontents
+\cleardoublepage % for better page number placement
+\pagestyle{headings}
+%\pagenumbering{arabic}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Thuật ngữ}
+
+Dưới đây là một số thuật ngữ dùng trong tài liệu này.
+
+\medskip
+\begin{lentry}
+\item[dimension] độ dài trong \latex/, ví dụ: |6pt|, |-2pc|, |5mm|,...
+\item[font]kiểu chữ
+\item[hyphen]tách một chữ (ở cuối dòng) với nhiều ký tự thành các phần nhỏ
+ ngăn cách bởi dấu gạch ngang (dấu |hyphen|).
+ Việc tách này giúp cho một chữ quá dài không tràn ra khỏi dòng.
+\item[number] \emph{bla bla bla\ldots}
+\item[preamble] phần nằm trước |\begin{document}| của tập tin nguồn \latex/.
+\item[tag] chỉ số phương trình
+\item[robust]\emph{bla bla bla\ldots}
+\item[text] chuỗi các mã nguồn \latex/, ví dụ: "|xem Tiên đề~\ref{ax:1}|"
+\item[typeset] sắp chữ nhờ \latex/
+% biên dịch tài liệu \latex/ nhờ chương trình |latex|, |pdflatex|,...
+\item[wrap] tự động chia một dòng quá dài thành nhiều dòng nhỏ, để chúng
+ bố trí vừa trên một chiều rộng cố định cho trước
+\item[canh cột]bố trí các phần của phương trình theo cột (chiều đứng)
+\item[dấu ngoặc](phân cách); thuật ngữ tiếng Anh là |delimiter|; là các dấu
+ (,),[,],\{,\},$\vert$,...
+\item[chỉ số phương trình]nhãn dùng để phân biệt các phương trình
+\item[phương trình] biểu thức toán học bất kỳ được biểu diễn nhờ \latex/
+\item[v-khoảng cách] khoảng cách theo chiều đứng
+\end{lentry}
+
+\chapter{Giới thiệu}
+
+% 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/}.
+Gói \pkg{amsmath} dành cho \LaTeX{} cung cấp nhiều tiện ích để |typeset|
+các tài liệu Toán học phức tạp. Gói này có trong hầu hết các bản phân phối
+mới của \LaTeX{}. Để lấy các thông tin cập nhật về gói này, bạn xem ở
+
+\medskip
+\rightline{\url{http://www.ams.org/tex/amsmath.html}}
+\rightline{\url{ftp://ftp.ams.org/pub/tex/}}
+\medskip
+
+% 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:
+Tài liệu này mô tả các tính năng và của gói \pkg{amsmath} và thảo luận
+về các hướng sử dụng chúng. Tài liệu cũng đề cập sơ lược về các gói
+\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}.
+Các gói này đều liên quan đến việc |typseset| biểu thức toán học.
+Thông tin về các ký hiệu và |font| mở rộng, xem ở \cite{amsfonts} và
+
+\medskip
+\rightline{\url{http://www.ams.org/tex/amsfonts.html}}
+\medskip
+\noindent
+Tài liệu về gói \pkg{amsthm}, các lớp\footnote{|documentclass|} AMS (\cls{amsart},
+\cls{amsbook}, etc.\@) có thể tìm thấy trong \cite{amsthdoc}, \cite{instr-l}
+và
+
+\medskip
+\rightline{\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:
+\bigskip
+Nếu bạn đã làm việc lâu dài với \latex/ và phải |typeset| rất nhiều các biểu thức
+toán học, thì với gói \pkg{amsmath}, bạn có thể tìm thấy giải pháp
+cho những vấn đề hay gặp nhất:
+
+\medskip
+\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).
+Dễ dàng định nghĩa toán tử, hàm toán học mới (tương tự như \cn{sin}, \cn{cos});
+các toán tử mới sẽ tự động canh chỉnh kích thước, kiểu |font|
+và khoảng cách tương quan với các phần tử khác trong biểu thức.
+
+\item
+% Multiple substitutes for the \env{eqnarray} environment to make
+% various kinds of equation arrangements easier to write.
+Nhiều biến thể của môi trường \env{eqnarray}
+để sắp xếp nhiều loại (hệ) phương trình khác nhau.
+
+\item
+% Equation numbers automatically adjust up or down to avoid
+% overprinting on the equation contents (unlike \env{eqnarray}).
+Các số chỉ phương trình tự động chuyển dịch lên, xuống để tránh
+tình trạng tràn trang (khắc phục nhược điểm của \env{eqnarray}).
+
+\item
+% Spacing around equals signs matches the normal spacing in the
+% \env{equation} environment (unlike \env{eqnarray}).
+Khoảng cách xung quanh dấu bằng ($=$) giống hệt khoảng cách bình thường
+trong môi trường \env{equation} (không giống như \env{eqnarray}).
+
+\item
+% A way to produce multiline subscripts as are often used with
+% summation or product symbols.
+Có thể tạo chỉ số dưới, chỉ số trên với nhiều dòng (thường gặp khi
+làm việc với các ký hiệu tổng, tích)
+
+\item
+% An easy way to substitute a variant equation number for a given
+% equation instead of the automatically supplied number.
+Dễ dàng tạo các biến thể cho việc đánh số một phương trình cho trước
+(nếu bạn không thích kiểu đánh số mặc định).
+
+\item
+% An easy way to produce subordinate equation numbers of the form
+% (1.3a) (1.3b) (1.3c) for selected groups of equations.
+Dễ dàng đánh số các phương trình con dạng (1.3a) (1.3b) (1.3c)
+từ một nhóm các phương trình. Việc đánh số này là \emph{tự động.}
+\end{itemize}
+
+% The \pkg{amsmath} package is distributed together with some small
+% auxiliary packages:
+\medskip
+Gói \pkg{amsmath} được phân phối cùng với một số gói bổ trợ
+
+\begin{description}
+\item[\pkg{amsmath}]
+% Primary package, provides various features for
+% displayed equations and other mathematical constructs.
+ Gói chính; cung cấp rất nhiều tiện ích để biễu diễn phương trình
+ và các biểu thức toán học từ đơn giản đến phức tạp.
+
+\item[\pkg{amstext}]
+% Provides a \cn{text} command for
+% typesetting a fragment of text inside a display.
+ Cung cấp lệnh \cn{text} để sắp xếp các đoạn văn bên trong
+ biểu thức toán học.
+
+\item[\pkg{amsopn}]
+% Provides \cn{DeclareMathOperator} for defining new
+% `operator names' like \cn{sin} and \cn{lim}.
+ Cung cấp lệnh \cn{DeclareMathOperator} để định nghĩa
+ các toán tử mới, như \cn{sin}, \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.
+ Gói này vẫn tồn tại để bảo đảm tính tương thích; tuy nhiên,
+ bạn nên dùng gói \pkg{bm} để thay thế cho \pkg{amsbsy}.
+
+\item[\pkg{amscd}]
+% Provides a \env{CD} environment for simple
+% commutative diagrams (no support for diagonal arrows).
+ Cung cấp môi trường \env{CD} để biểu diễn các biểu đồ giao hoán
+ đơn giản (với gói này, bạn không thể vẽ các mũi tên chéo).
+
+\item[\pkg{amsxtra}]
+% Provides certain odds and ends such as
+% \cn{fracwithdelims} and \cn{accentedsymbol}, for compatibility with
+% documents created using version 1.1.
+ Gói bổ sung, nhằm bảo đảm tương thích với tài liệu dùng phiên bản 1.1
+ của \pkg{amsmath}. Cung cấp: \cn{fracwithdelims}, \cn{accentedsymbol},...
+
+\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.
+\medskip
+Gói \pkg{amsmath} đã bao gộp các gói \pkg{amstext}, \pkg{amsopn}, and
+\pkg{amsbsy}; nghĩa là khi nạp gói \pkg{amsmath}, ba gói kia sẽ tự động
+nạp theo. Còn để dùng các gói \pkg{amscd}, \pkg{amsxtra}, bạn phải nạp riêng chúng.
+
+\chapter{Các tùy chọn của gói \pkg{amsmath}}\label{options}
+
+Để dùng tùy chọn của gói, bạn để tên của tùy chọn vào trong phần tham
+số bổ sung của lệnh nạp gói \cn{usepakage}. Các tùy chọn cách nhau bằng
+dấu phảy. Ví dụ:
+
+\medskip
+\verb"\usepackage[intlimits]{amsmath}"\\
+\indent\verb"\usepackage[sumlimits,intlimits]{amsmath}"
+
+\medskip
+% The \pkg{amsmath} package has the following options:
+Gói \pkg{amsmath} có các tùy chọn sau đây:
+\begin{description}
+
+\item[\opt{centertags}] (mặc định)
+%For a split equation, place equation
+%numbers\index{equation numbers!vertical placement} vertically centered
+%on the total height of the equation.
+Đánh số phương trình\index{equation numbers!vertical placement}
+bằng cách đặt chỉ số canh giữa theo chiều cao của phương trình.
+
+\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).
+`Top-or-bottom tags': Đặt chỉ số của phương trình ở phía bên phải,
+dòng cuối cùng; hoặc ở phía bên trái, dòng đầu tiên.
+
+\item[\opt{sumlimits}] (mặc định)
+% 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).
+Đặt các chỉ số trên và dưới của các ký hiệu tổng ($\sum$) ở trên và
+ở dưới (trong công thức riêng dòng). Tùy chọn này cũng ảnh hưởng
+đến các ký hiệu cùng loại\mdash $\prod$, $\coprod$, $\bigotimes$,
+$\bigoplus$,...\mdash (nhưng ký hiệu tích phân thì không; xem dưới đây)
+
+\item[\opt{nosumlimits}]
+% Always place the subscripts and superscripts of
+% summation-type symbols to the side, even in displayed equations.
+Luôn đặt chỉ số trên và chỉ số dưới của các ký hiệu dạng tổng ($\sum$, $\prod$,...)
+ở bên cạnh, ngay cả trong công thức riêng dòng. Ví dụ $\sum_0^1$.
+
+\item[\opt{intlimits}]
+% Like \opt{sumlimits}, but for
+% integral\index{integrals!placement of limits} symbols.
+Tương tự như \opt{sumlimits}, nhưng cho ký hiệu tích phân.
+
+\item[\opt{nointlimits}] (mặc định) Ngược với \opt{intlimits}.
+
+\item[\opt{namelimits}] (mặc định)
+% 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.
+Tương tự \opt{sumlimits}, nhưng cho một số toán tử như
+$\det$, $\inf$, $\lim$, $\max$, $\min$;
+các toán tử này theo truyền thống thường có chỉ số đặt bên dưới toán tử
+(chế độ công thức riêng dòng).
+
+\item[\opt{nonamelimits}] Ngược với \opt{namelimits}.
+
+% 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.
+
+\item[\opt{leqno}]
+% Place equation numbers on the left.\index{equation
+% numbers!left or right placement}
+Đặt chỉ số phương trình bên trái.
+
+\item[\opt{reqno}]
+% Place equation numbers on the right.
+Đặt chỉ số phương trình bên phải.
+
+\item[\opt{fleqn}]
+% Position equations at a fixed indent from the left
+% margin rather than centered in the text column.\index{displayed
+% equations!centering}
+Biểu diễn phương trình với lề trái cố định; theo mặc định, các phương
+trình được canh giữa (do đó, lề trái của chúng \emph{thay đổi}).
+
+\end{description}
+
+% 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.
+\medskip
+Đối với ba tùy chọn cuối cùng (\opt{leqno}, \opt{reqno}, \opt{fleqn}),
+bạn có thể để chúng vào phần tham số bổ sung của \cn{documentclass};
+gói \pkg{amsmath} nhận biết điều này và do đó bạn không cần lặp lại
+khi nạp gói bằng \cn{usepackage}|{amsmath}|:
+
+\medskip
+\verb"\documentclass[reqno]{report}"\par
+\indent\verb"\usepackage{amsmath}"\% có tác dụng như \verb"\usepackage[reqno]{amsmath}"
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%\chapter{Displayed equations}
+\chapter{Biểu diễn phương trình}
+
+\section{Giới thiệu}
+% 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:
+Gói \pkg{amsmath} cung cấp thêm các môi trường biểu diễn phương trình sau
+đây, bên cạnh các môi trường chuẩn của \latex/:
+\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.)
+(Mặc dù môi trường chuẩn \env{eqnarray} vẫn dùng được, nhưng tốt hơn hết
+nên dùng môi trường \env{align} hoặc tổ hợp \env{equation}+\env{split}.)
+
+\medskip
+
+% 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.
+Ngoại trừ \env{split}, mỗi môi trường đều có hai dạng:
+\emph{có sao (*)} và \emph{không sao};
+các môi trường không sao sẽ sử dụng bộ đếm \env{equation} của \latex/
+để đánh số các phương trình một cách tự động (do đó, ta gọi
+chúng là \emph{môi trường có đánh số}).
+% You can suppress the number on any
+% particular line by putting \cn{notag} before the \cn{\\};
+Bạn có thể bỏ qua việc đánh số cho bất kỳ dòng phương trình nào
+bằng cách đặt lệnh \cn{notag} trước khi dùng \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.
+cũng có thể thay đổi kiểu đánh số cho dòng phương trình cụ thể, bằng
+cách dùng \cn{tag}|{|\<label>|}|, ở đây \<label> là |text| bất kỳ,
+chẳng hạn |$*$| hoặc |ii|. Theo mặc định, \<label> của \cn{tag} sẽ được
+đặt trong cặp dấu ngoặc đơn, ví dụ (3.32);
+nếu không muốn điều này xảy ra, bạn có thể dùng \cn{tag*}.
+% There is also a
+% \cn{tag*} command that causes the text you supply to be typeset
+% literally, without adding parentheses around it.
+\emph{Để ý rằng,}
+% \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.
+\cn{tag} và \cn{tag*} có thể dùng với mọi môi trường đã liệt kê ở trên,
+chứ không phải với chỉ các môi trường có đánh số (không sao).
+Ví dụ về việc dùng \cn{tag} có thể tìm thấy trong \fn{testmath.tex}
+và \fn{subeqn.tex} được phân phối cùng với tài liệu này.
+
+% 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.
+\medskip
+Môi trường \env{split} là một dạng đặc biệt, chỉ có thể được dùng
+bên trong các môi trường khác. Tuy nhiên, nó lại không thể dùng bên
+trong \env{multline}.
+
+% 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.
+\medskip
+Với các môi trường có chức năng canh cột (\env{split}, \env{align},...),
+các ký hiệu quan hệ (dấu $=$, $>$, $\le$, ...) phải được đặt sau dấu \verb'&'.
+Đây là điểm khác biệt so với \env{eqnarray}.
+
+\begin{table}[p]
+\caption[]{
+% Comparison of displayed equation environments (vertical lines
+% indicating nominal margins)
+So sánh các môi trường biểu diễn phương trình (đường thẳng đứng
+dùng để chỉ lề trái, phải của tờ giấy tưởng tượng)
+}
+\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}
+
+\afterpage{\clearpage}
+
+%\section{Single equations}
+\section{Phương trình đơn}
+
+% 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.%
+Môi trường \env{equation} dùng biểu diễn phương trình đơn
+và tự động đánh số cho phương trình đó. Môi trường \env{equation*}
+có tác dụng tương tự, nhưng không đánh số.%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \footnote{Basic \latex/ doesn't provide an \env{equation*} environment,
+% but rather a functionally equivalent environment named
+% \env{displaymath}.}
+\footnote{\latex/ chuẩn không có môi trường \env{equation*}, mà
+chỉ có dạng tương đương là \env{displaymath}.}
+
+\medskip
+
+Hai môi trường này chỉ biểu diễn phương trình trên đúng một dòng;
+bạn không thể dùng \cn{\\} bên trong hai môi trường đó.
+Nếu biểu thức quá dài, sẽ xảy tràn trang. Hãy xem mục tiếp theo.
+
+%\section{Split equations without alignment}
+\section{Chia nhỏ phương trình nhưng không canh cột}
+
+% 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).
+Môi trường \env{multline} là biến thể của \env{equation}, cho phép
+biểu diễn các phương trình quá dài, không thể bố trí vừa khít trên một dòng.
+Trong môi trường này, bạn dùng \cn{\\} để tách các dòng.
+Dòng đầu tiên sẽ canh ở lề trái, dòng cuối cùng được canh ở lề phải;
+ngay trước dòng đầu tiên và ngay sau dòng cuối cùng là khoảng trắng |indent|
+được cung cấp bởi biến độ dài \cn{multlinegap}.
+Các dòng còn lại sẽ được canh giữa trang, trừ khi bạn dùng tùy chọn \opt{fleqn}.
+
+% 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}.
+\medskip
+Giống như \env{equation}, môi trường \env{multline} chỉ đánh số tất cả
+các dòng của nó bởi đúng một chỉ số (do đó, bạn không thể dùng \cn{notag}
+cho riêng dòng nào trong môi trường). Chỉ số được đánh sẽ được ở dòng
+cuối cùng (tùy chọn \opt{reqno}) hoặc dòng đầu tiên (tùy chọn \opt{leqno});
+kiểu đánh số canh giữ như \env{split} không được hỗ trợ.
+
+% 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
+\medskip
+Có thể làm cho các dòng giữa của môi trường dịch chuyển qua trái hoặc
+qua phải bằng cách lệnh \cn{shoveleft} hay \cn{shoveright}.
+Các lệnh này sẽ nhận cả dòng cần dịch chuyển làm tham số
+(nhưng trừ ra \cn{\\} ở cuối dòng)
+\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}.
+Giái trị của biến độ dài \cn{multlinegap} có thể thay đổi nhờ các
+lệnh của \latex/ là \cn{setlength} và \cn{addtolength}.
+
+%\section{Split equations with alignment}
+\section{Chia nhỏ phương trình và canh cột}
+
+% 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.
+Giống như \env{multline}, môi trường \env{split} dùng biểu diễn
+\emph{các phương trình đơn} quá dài, không bố trí vừa trên riêng một dòng
+và do đó phải chia chúng thành nhiều dòng.
+% Unlike \env{multline}, however, the \env{split}
+% environment provides for alignment among the split lines, using |&| to
+% mark alignment points.
+Nhưng không như \env{multline}, môi trường \env{split}
+cho phép canh cột các dòng nhờ sử dụng |&| để đánh dấu cột.
+% 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:
+Không như các môi trường phương trình khác của gói \pkg{amsmath},
+\env{split} không đánh số phương trình, bởi \emph{nó chỉ có thể dùng
+bên trong khác môi trường khác}, thường là
+\env{equation}, \env{align}, hay \env{gather}
+(các môi trường vừa nhắc đến thực hiện đánh số). Ví dụ:
+\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.
+Phần phương trình biểu diễn nhờ \env{split}
+nên được xem là một phần riêng biệt;
+lệnh \cn{label} đặt bên trong môi trường sẽ không có tác dụng.
+
+%\section{Equation groups without alignment}
+\section{Nhóm phương trình không canh cột}
+
+% 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{\bslash} command.
+% Any equation in a \env{gather} may consist of a \verb'\begin{split}'
+% \dots\ \verb'\end{split}' structure\mdash for example:
+Môi trường \env{gather} thường dùng biểu diễn nhóm các phương trình
+mà không quan tâm đến việc canh cột giữa chúng; mỗi phương trình trong nhóm
+sẽ nằm ở dòng riêng (xem Bảng~\ref{displays}). Các phương trình bên trong
+môi trường \env{gather} được phân biệt nhờ \cn{\\}. Bất kỳ phương trình
+nào bên trong \env{gather} cũng có thể là ở dạng
+\verb'\begin{split}' \dots\ \verb'\end{split}'
+\begin{verbatim}
+\begin{gather}
+ phương trình thứ nhất\\
+ \begin{split}
+ phương trình & thứ hai\\
+ & với hai dòng
+ \end{split}
+ \\
+ phương trình thứ ba
+\end{gather}
+\end{verbatim}
+
+%\section{Equation groups with mutual alignment}
+\section{Nhóm phương trình có canh cột}
+
+% 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}).
+Môi trường \env{align} biểu diễn nhóm các phương trình và cho phép
+canh cột giữa các phương trình. Xem Bảng~\ref{displays}.
+
+% To have several equation columns side-by-side, use extra ampersands
+% to separate the columns:
+\medskip
+Để có vài cột phương trình đi cùng nhau, hãy \emph{dùng thêm} các dấu |&|
+\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:
+Có thể thêm chú thích cho phương trình (nằm cùng dòng với phương trình)
+bằng cách dùng lệnh \cn{text} bên trong môi trường \env{align}:
+\begin{align}
+x& = y_1-y_2+y_3-y_5+y_8-\dots
+ && \text{theo \eqref{eq:C}}\\
+ & = y'\circ y^* && \text{theo \eqref{eq:D}}\\
+ & = y(0) y' && \text {theo Tiên đề 1.}
+\end{align}
+%
+\begin{verbatim}
+\begin{align}
+x& = y_1-y_2+y_3-y_5+y_8-\dots
+ && \text{theo \eqref{eq:C}}\\
+ & = y'\circ y^* && \text{theo \eqref{eq:D}}\\
+ & = y(0) y' && \text {theo Tiên đề 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.
+Biến thể \env{alignat} của môi trường \env{align} cho phép xác định khoảng
+cách (theo chiều ngang), nhờ đó sẽ tiết kiệm không gian hơn. Môi trường này
+nhận một tham số, là \emph{"số cột phương trình"}. Để có số này, bạn
+đếm số dấu |&| trong các phương trình, chọn ra số lớn nhất, cộng số đó
+với 1 rồi chia kết quả cho 2. So sánh ví dụ dưới đây với ví dụ ở ngay trên.
+\begin{alignat}{2}
+x& = y_1-y_2+y_3-y_5+y_8-\dots
+ &\quad& \text{theo \eqref{eq:C}}\\
+ & = y'\circ y^* && \text{theo \eqref{eq:D}}\\
+ & = y(0) y' && \text {theo Tiên đề 1.}
+\end{alignat}
+%
+\begin{verbatim}
+\begin{alignat}{2}
+x& = y_1-y_2+y_3-y_5+y_8-\dots
+ &\quad& \text{theo \eqref{eq:C}}\\
+ & = y'\circ y^* && \text{theo \eqref{eq:D}}\\
+ & = y(0) y' && \text {theo Tiên đề 1.}
+\end{alignat}
+\end{verbatim}
+
+%\section{Alignment building blocks}
+\section{Canh cột các khối phương trình}
+
+% 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.
+Giống như \env{equation}, các môi trường phương trình nhiều dòng
+\env{gather}, \env{align} và \env{alignat} bố trí phương trình \emph{trên
+cả chiều rộng của dòng.} Điều này dẫn tới, chẳng hạn, ta không thể
+thêm các dấu ngoặc để bao quanh phương trình.
+
+% 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.,
+\medskip
+Các môi trường \env{gathered}, \env{aligned} và \env{alignedat} khắc phục
+nhược điểm trên; chúng biểu diễn phương trình \emph{trên một chiều rộng đúng bằng chiều
+rộng của nội dung phương trình.} Nhờ đó, chúng có thể xem như một phần của biểu thức,
+ví dụ:
+\begin{equation*}
+\left.% dùng \left. để bỏ đi dấu ngoặc bên trái
+\begin{aligned}
+ B'&=-\partial\times E,\\
+ E'&=\partial\times B - 4\pi j,
+\end{aligned}
+\right\}
+\qquad \text{Phương trình Maxwell}
+\end{equation*}
+\begin{verbatim}
+\begin{equation*}
+\left.% dùng \left. để bỏ đi dấu ngoặc bên trái
+\begin{aligned}
+ B'&=-\partial\times E,\\
+ E'&=\partial\times B - 4\pi j,
+\end{aligned}
+\right\}
+\qquad \text{Phương trình Maxwell}
+\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.
+
+\medskip
+Giống như \env{array}, các biến thể \texttt{-ed} nhận tham số tùy chọn (bổ sung)
+\verb'[t]' hoặc \verb'[b]' để chỉ vị trí biểu diễn (`t' cho top-ở trên;
+`b' cho bottom-ở dưới)
+
+% \qq{Cases} constructions like the following are common in
+% mathematics:
+\medskip
+Biến thể \env{cases} dùng để \emph{chia các trường hợp}:
+\begin{equation}\label{eq:C}
+P_{r-j}=
+ \begin{cases}
+ 0& \text{nếu $r-j$ là số lẻ},\\
+ r!\,(-1)^{(r-j)/2}& \text{nếu $r-j$ là số chẵn}.
+ \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{nếu $r-j$ là số lẻ},\\
+ r!\,(-1)^{(r-j)/2}& \text{nếu $r-j$ là số chẵn}.
+ \end{cases}
+\end{verbatim}
+% Notice the use of \cn{text} (cf.~\secref{text}) and the nested
+% math formulas.
+Để ý đến việc dùng lệnh \cn{text} (xem Mục~\secref{text}) và biểu thức
+toán |$...$| bên trong \cn{text}.
+
+\medskip
+Cần nhớ: các biến thể \env{cases} và \env{-ed} chỉ được dùng bên trong
+các môi trường phương trình khác\mdash tương tự như \env{split}.
+
+
+%\section{Adjusting tag placement}
+\section{Thay đổi vị trí chỉ số phương trình}
+
+% 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.
+Thay đổi vị trí đặt chỉ số phương trình là vấn đề khá phức tạp trong
+biểu diễn phương trình nhiều dòng. Các môi trường của gói \pkg{amsmath}
+luôn cố gắng để không đặt chỉ số phương trình \emph{đè} lên nội dung phương
+trình, và nếu cần thiết thì đặt chỉ số ở trên hoặc dưới một chút,
+thậm chí, ở riêng một dòng.
+% Difficulties
+% in accurately calculating the profile of an equation can occasionally
+% result in number movement that doesn't look right.
+Nhưng việc tính chính xác thuộc tính của phương trình là điều khó khăn;
+do đó, đôi khi việc dịch chuyển chỉ số phương trình do gói \pkg{amsmath}
+thực hiện sẽ không mang lại kết quả đẹp mắt.
+% 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.
+Trong trường hợp đó, bạn phải \emph{làm bằng tay} nhờ lệnh \cn{raisetag}.
+%To move a particular number up by six points, write |\raisetag{6pt}|.
+Để dịch chuyển chỉ số lên phía trên một quãng $6pt$, dùng |\raisetag{6pt}|;
+còn để dịch xuống, dùng chẳng hạn |\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.
+
+\medskip
+Việc điều chỉnh ví trí chỉ số bằng \cn{raisetag} là công việc tỉ mỉ, cũng như việc
+ngắt dòng, ngắt trang bằng \cn{linebreak}, \cn{pagebreak}. Bạn \emph{chỉ nên}
+thực hiện điều chỉnh khi tài liệu của bạn gần như hoàn tất, và cần làm
+lại mỗi khi bạn thay đổi nội dung tài liệu.
+
+%\section{Vertical spacing and page breaks in multiline displays}
+\section{V-khoảng cách và ngắt trang trong biểu diễn nhiều dòng}
+
+% 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/.
+Bạn có thể dùng \cn{\\}|[|\<dimension>|]| để thêm các v-khoảng cách giữa
+các dòng trong mọi môi trường biểu diễn phương trình của gói \pkg{amsmath};
+điều này cũng tương tự trong \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.
+
+
+\medskip
+Khi dùng gói \pkg{amsmath}, việc \ii{ngắt trang} giữa các dòng của phương trình
+không (tự động) xảy ra; bởi việc ngắt trang như vậy sẽ làm cho phương trình trở nên
+khó theo dõi đối với độc giả.
+% To get an individual page break
+% inside a particular displayed equation, a \cn{displaybreak} command is
+% provided.
+Để ngắt trang bên trong phương trình, bạn phải làm bằng tay nhờ lệnh
+\cn{displaybreak}.
+% \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.
+Nơi đặt \cn{displaybreak} tốt nhất là ngay trước \cn{\\} ở dòng cần ngắt trang.
+Giống như lệnh \cn{pagebreak} của \latex/, lệnh \cn{displaybreak} nhận
+tham số bổ sung (tùy chọn) là một trong các số 0, 1, 2, 3, 4;
+tham số này cho biết mức độ ngắt trang. Dùng |\displaybreak[0]| để hàm ý
+\qq{không được ngắt trang ở đây},...; trong khi |\displaybreak|
+(không có tham số bổ sung), như |\displaybreak[4]|, hàm ý
+\qq{ngắt trang ở đây}.
+
+% 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.
+\medskip
+Việc ngắt trang dùng \cn{displaybreak} như trên chỉ có thể làm đối với
+phương trình cụ thể. Nếu muốn cung cấp cho gói \pkg{amsmath} \qq{giấy phép ngắt trang}
+một cách toàn cục (cho mọi phương trình nhiều dòng),
+bạn dùng \cn{allowdisplaybreaks}\texttt{[1]} trong phần |preamble| của tài liệu.
+Tham số bổ sung của lệnh này nhận giá trị từ 1 đến 4: |[1]| có nghĩa là
+cho phép ngắt trang, nhưng tránh việc đó nếu có thể được; các giá trị 2,3,4
+càng tăng khả năng ngắt trang. Khi dùng \cn{allowdisplaybreaks}, thì lệnh
+\cn{\\*} dùng để cấm xảy ra ngắt trang sau một dòng cụ thể.
+
+\medskip
+\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
+Các môi trường \env{split}, \env{aligned}, \env{gathered} và \env{alignedat}
+có khả năng |wrap| nội dung phương trình trong các hộp không cho phép ngắt
+(|unbreakable box|); hệ quả là trong trường hợp đó, cả hai lệnh
+\cn{displaybreak} và \cn{allowdisplaybreaks} đều không có tác dụng.
+\end{bfseries}
+
+%\section{Interrupting a display}
+\section{Xen liên từ vào giữa các phương trình}
+
+% 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.
+Lệnh \cn{intertext} dùng để xen liên từ\footnote{hay cái khác, tùy bạn!}
+vào giữa các dòng của biểu diễn nhiều dòng (xem thêm về \cn{text}
+ở Mục~\secref{text}). Nhờ đó, để bảm đảm việc canh cột các phương trình,
+bạn đỡ phải tốn công kết thúc một biểu diễn rồi sau đó bắt đầu lại.
+Lệnh \cn{intertext} chỉ có thể đặt ngay sau \cn{\\} hoặc \cn{\\*}.
+Hãy chú ý đến sự xuất hiện của liên từ \qq{và} trong ví dụ sau.
+\begin{align}
+A_1&=N_0(\lambda;\Omega')-\phi(\lambda;\Omega'),\\
+A_2&=\phi(\lambda;\Omega')-\phi(\lambda;\Omega),\\
+\intertext{và}
+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{và}
+A_3&=\mathcal{N}(\lambda;\omega).
+\end{align}
+\end{verbatim}
+
+%\section{Equation numbering}
+\section{Đánh số phương trình}
+
+%\subsection{Numbering hierarchy}
+\subsection{Hệ thống chỉ số phương trình}
+% 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}:
+Với \latex/, nếu bạn muốn đánh số phương trình theo mục\mdash
+nghĩa là chỉ số phương trình có dạng (1.1), (1.2), \dots, (2.1), (2.2), \dots,
+trong chương 1, 2, v.v...\mdash bạn có thể định nghĩa lại lệnh
+\cn{theequation} như lời khuyên trong sổ tay (manual) \latex/
+\cite[\S6.3, \S C.8.4]{lamport}:
+
+\medskip
+\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}.
+\medskip
+Cách làm như trên cho kết quả như ý, nhưng có điều bất tiện là chỉ số
+phương trình sẽ không tự động đặt về 0 khi chuyển%
+\footnote{ở mục 1, các phương trình được đánh số (1.1), (1.2), \ldots;
+còn qua mục 2, chỉ số phải bắt đầu từ (2.1) rồi đến (2.2), \ldots;
+nghĩa là ta phải đặt bộ đếm về 0 khi vừa qua mục 2.}
+% ----------------------------------------------------------------------
+từ mục này qua mục khác, và bạn phải làm điều đó bằng tay nhờ lệnh \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
+Để cuộc sống dễ dàng hơn, gói \pkg{amsmath} cung cấp lệnh \cn{numberwithin};
+nhờ lệnh này, bạn có thể đánh số phương trình theo mục, với chỉ số phương trình
+\emph{tự động đặt về 0} khi sang mục mới.
+
+\medskip
+\begin{verbatim}
+\numberwithin{equation}{section}
+\end{verbatim}
+
+\medskip
+% As its name implies, the \cn{numberwithin} command can be applied to
+% any counter, not just the \texttt{equation} counter.
+Để ý rằng, lệnh \cn{numberwithin} có tác dụng đối với mọi bộ đếm, chứ
+không riêng gì bộ đếm \texttt{equation}.
+
+%\subsection{Cross references to equation numbers}
+\subsection{Tham khảo chéo đến chỉ số phương trình}
+
+% 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).
+Việc tham khảo chéo đến phương trình được thực hiện dễ dàng hơn với lệnh
+\cn{eqref}. Lệnh này tự động đặt chỉ số phương trình vào cặp dấu ngoặc đơn.
+Ví dụ: dùng \verb'\ref{abc}' ta thu được 3.2,
+trong khi \verb'\eqref{abc}' sẽ cho (3.2).
+
+%\subsection{Subordinate numbering sequences}
+\subsection{Đánh số các phương trình con của nhóm phương trình}
+
+% 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
+Gói \pkg{amsmath} cung cấp môi trường \env{subequations} cho phép
+bạn đánh số các phương trình con của một nhóm các phương trình
+\emph{một cách tự động}. Ví dụ
+
+\medskip
+\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).
+\medskip
+\noindent
+sẽ khiến các chỉ số của các phương trình bên trong môi trường
+\env{subequations} sẽ (tự động) có dạng (4.9a) (4.9b) (4.9c) \dots,
+nếu phương trình đi trước nhóm phương trình này có chỉ số (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.
+Chú ý là, nếu bạn đặt nhãn tham chiếu (|label|) ngay sau
+\verb/\begin{subequations}/, việc tham chiếu bằng \cn{ref}
+đến nhãn đó sẽ cho ra, chẳng hạn, 4.9, chứ không phải là 4.9a.
+Vui lòng xem tài liệu \fn{subeqn.pdf} và mã nguồn \fn{subeqn.tex}
+để có các ví dụ về việc dùng môi trường này.
+
+% 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.
+\medskip
+Các bộ đếm dùng bởi môi trường \env{subequations} là \verb/parentequation/
+và \verb/equation/; các lệnh thay đổi bộ đếm
+\cn{addtocounter}, \cn{setcounter}, \cn{value}, \ldots
+đều có tác dụng bình thường với những bộ đếm đó.
+% 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.
+Chẳng hạn, các phương trình con được đánh số nhờ chữ cái |alphabet|
+(a,b,c,\ldots); để thay đổi điều này, sử dụng phương pháp chuẩn
+của \latex/ để thay đổi kiểu đánh số (xem trong \cite[\S6.3, \S C.8.4]{lamport}).
+Trong ví dụ dưới đây, các phương trình con sẽ được đánh chỉ số La Mã:
+
+\medskip
+\begin{verbatim}
+\begin{subequations}
+\renewcommand{\theequation}{\theparentequation \roman{equation}}
+...
+\end{verbatim}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%\chapter{Miscellaneous mathematical features}
+\chapter{Linh tinh, nhưng quan trọng}
+
+\section{Ma trận}\label{ss:matrix}
+
+% The \pkg{amsmath} package provides some environments for
+% matrices\index{matrices} beyond the basic \env{array} environment of
+% \latex/.
+Gói \pkg{amsmath} cung cấp một số môi trường để biểu diễn ma trận
+tiện lợi hơn\footnote{|beyond|} môi trường \env{array} của \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.
+Các môi trường \env{pmatrix}, \env{bmatrix}, \env{Bmatrix}, \env{vmatrix}
+và \env{Vmatrix} tự động có (tương tứng) cặp dấu ngoặc $(\,)$, $[\,]$,
+$\lbrace\,\rbrace$,
+$\lvert\,\rvert$, và $\lVert\,\rVert$ bao xung quanh.
+% For naming consistency there is a \env{matrix} environment sans delimiters.
+Ngoài ra, môi trường \env{matrix} cho một ma trận không có kèm dấu ngoặc nào.
+% 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.
+
+\medskip
+Các môi trường ma trận của \pkg{amsmath} sử dụng thuật toán
+điều chỉnh khoảng cách và canh cột tinh tế, tiết kiệm hơn cách làm hoang phí%
+\footnote{|prodigal|} của môi trường \env{array}.
+% 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.%
+Hơn nữa, không giống như \env{array}, bạn không cần chỉ ra \emph{số cột}
+khi dùng các môi trường \env{-matrix}; theo mặc định, số cột tối đa của
+các ma trận là 10\mdash con số này có thể thay đổi được%
+\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.%
+\emph{Cụ thể hơn:} số tối đa các cột của ma trận được xác định bởi bộ đếm |MaxMatrixCols|
+(mặc định là 10); để thay đổi số này, bạn dùng các lệnh của \latex/ là
+\cn{setcounter} hoặc \cn{addtocounter}.
+}.
+% (If you need left or right alignment in a column or other special
+% formats you must resort to \env{array}.)
+(Nếu bạn muốn canh phải, trái các cột hoặc muốn tinh chỉnh ma trận theo ý bạn,
+không còn cách nào khác hơn là bạn phải quay lại dùng môi trường \env{array}.)
+
+% To produce a small matrix suitable for use in text, there is a
+% \env{smallmatrix} environment (e.g.,
+\medskip
+Các ma trận cỡ nhỏ đặt xen vào biểu thức chung dòng được cho bởi
+môi trường \env{smallmatrix}. Ví dụ, ma trận
+\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.
+được bố trí vừa vẹn trên dòng này, tốt hẳn hơn so
+với khi bạn thay thế \env{smallmatrix} bởi \env{matrix}:
+% Delimiters must be provided; there are no |p|,|b|,|B|,|v|,|V|
+% versions of \env{smallmatrix}. The above example was produced by
+
+\medskip
+\begin{verbatim}
+\bigl( \begin{smallmatrix}
+ a&b\\ c&d
+\end{smallmatrix} \bigr)
+\end{verbatim}
+
+\medskip
+Chú ý rằng các dấu ngoặc phải được chỉ ra khi dùng \env{smallmatrix};
+không có các biến thể |p|,|b|,|B|,|v|,|V| của \env{smallmatrix}.
+
+% \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,
+\medskip
+Có một lệnh hết sức tiện lợi, là \cn{hdotsfor}|{|\<number>|}|.
+Lệnh này sẽ sinh ra dãy các dấu chấm lửng trong matrix. Tham số
+của lệnh là số cột cần bỏ qua. Quan sát ví dụ dưới đây
+
+\medskip
+\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).
+\medskip
+Khoảng cách giữa các dấu chấm cho bởi \cn{hdotsfor} có thể thay thế nhờ
+tham số bổ sung của lệnh; ví dụ: |\hdotsfor[1.5]{3}|. Con số trong ngoặc
+vuông (|[1.5]|) được dùng như một thừa số nhân; giá trị mặc định là 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}\\% tăng gấp đôi khoảng cách giữa các dấu chấm
+%
+-a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix}
+\end{verbatim}
+
+%\section{Math spacing commands}
+\section{Điều chỉnh khoảng cách}
+
+% 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.
+Gói \pkg{amsmath} cung cấp tập hợp các lệnh điều chỉnh khoảng cách
+trong chế độ toán. Phiên bản rút gọn (lệnh ngắn) hoặc đầy đủ (lệnh dài) của
+các lệnh này đều có tính |robust| và có thể dùng bên ngoài chế độ toán.
+\begin{ctab}{lll|lll}
+Lệnh ngắn& Lệnh dài& Ví dụ & Lệnh ngắn& Lệnh dài& Ví dụ\\
+\hline
+\vstrut{2.5ex}
+& {\tiny (không khoảng cách)}& \spx{}& & {\tiny (không khoảng cách)} & \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}|.
+Tổng quát hơn, để điều chỉnh khoảng cách trong chế độ toán, bạn dùng
+lệnh \cn{mspace} và \emph{đơn vị toán}. \emph{Đơn vị toán},
+ký hiệu là |mu| (|math unit|), là đơn vị độ dài của \latex/ trong chế độ toán,
+bằng 1/18 em\footnote{Thế |em| là cái chi?}. Ví dụ,
+để có khoảng \cn{quad} âm, bạn có thể dùng |\mspace{-18.0mu}|.
+
+\section{Các dấu chấm}
+
+% 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
+Vị trí của dấu ba chấm (phía trên dòng hay ở ngay chân dòng) trong các
+ngữ cảnh khác nhau thường không nhất quán, và có thể là vấn đề sở thích
+cá nhân. Bằng cách sử dụng các lệnh
+\begin{itemize}
+\item \cn{dotsc} cho \qq{dấu ba chấm đi với dấu phảy}
+\item \cn{dotsb} cho \qq{dấu ba chấm đi với toán tử nhị phân/quan hệ}
+\item \cn{dotsm} cho \qq{dấu ba chấm đi trong biểu thức nhân}
+\item \cn{dotsi} cho \qq{dấu ba chấm đi trong biểu thức tích phân}
+\item \cn{dotso} cho \qq{dấu ba chấm cho mọi trường hợp còn lại}
+\end{itemize}
+% instead of \cn{ldots} and \cn{cdots},
+thay vì dùng \cn{ldots} và \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.
+bạn có thể làm cho mã nguồn \TeX\ của tài liệu trở nên trong sáng hơn,
+dễ chỉnh sửa hơn khi cần thiết (chẳng hạn theo yêu cầu của nhà xuất bản).
+Định nghĩa của các lệnh ở trên là theo quy ước của Hội Toán học Mỹ (AMS).
+% The default treatment for the various
+% kinds follows American Mathematical Society conventions:
+
+\medskip
+\begin{center}
+\begin{tabular}{@{}l@{}l@{}}
+\begin{minipage}[t]{.54\textwidth}
+\begin{verbatim}
+Ta có chuỗi $A_1, A_2, \dotsc$,
+tổng vô hạn $A_1 +A_2 +\dotsb $,
+tích vô hạn $A_1 A_2 \dotsm $,
+và tích phân không xác định
+\[\int_{A_1}\int_{A_2}\dotsi\]
+\end{verbatim}
+\end{minipage}
+&
+\begin{minipage}[t]{.45\textwidth}
+\noindent
+Ta có chuỗi $A_1, A_2, \dotsc$,
+tổng vô hạn $A_1 +A_2 +\dotsb $,
+tích vô hạn $A_1 A_2 \dotsm $,
+và tích phân không xác định
+\[\int_{A_1}\int_{A_2}\dotsi\]
+\end{minipage}
+\end{tabular}
+\end{center}
+
+%\section{Nonbreaking dashes}
+\section{Gạch ngang không vỡ}
+
+% A command \cn{nobreakdash} is provided to suppress the possibility
+% of a linebreak after the following hyphen or dash.
+Lệnh \cn{nobreakdash} giúp ta hạn chế sự ngắt dòng sau một dấu |hyphen|
+hoặc sau dấu gạch ngang.
+% 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.,
+Ví dụ, nếu bạn viết `các trang 1\ndash 9' bằng cách dùng |các trang 1\nobreakdash--9|
+thì việc ngắt dòng sẽ không bao giờ xảy ra giữa dấu gạch ngang và số 9.
+Bạn cũng có thể dùng \cn{nobreakdash} để hạn chế việc tách chữ (|hyphenation|)
+ngoài ý muốn, ví dụ trong tổ hợp |$p$-adic|. Để tiện lợi, bạn có thể định
+nghĩa các lệnh tắt như sau:
+
+\medskip
+\begin{verbatim}
+\newcommand{\p}{$p$\nobreakdash}% cho "\p-adic"
+\newcommand{\Ndash}{\nobreakdash--}% cho "các trang 1\Ndash 9"
+% Cho không gian "\n chiều" ("n-chiều"):
+\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.)
+
+\medskip
+Ví dụ cuối cùng ở trên cho biết cách làm thế nào để ngăn cản ngắt dòng sau dấu
+|hyphen| nhưng cho phép tách chữ bình thường ở từ tiếp theo dấu |hyphen| đó.
+(Chỉ cần thêm một độ rộng 0 sau dấu |hyphen|.)
+
+%\section{Accents in math}
+\section{Dấu nhấn trong toán học}
+
+% 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:
+Thông thường, \latex/ khó có thể biểu diễn cho đẹp cùng lúc hai dấu nhấn
+trên một ký hiệu. Với gói \pkg{amsmath}, việc này khá hơn rất nhiều; ví dụ
+$\hat{\hat{A}}$ (cho bởi \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/.
+\medskip
+Các lệnh \cn{dddot} và \cn{ddddot} biểu diễn hai và ba dấu chấm
+như là dấu nhấn, ví dụ $\dddot C$. Đây là các lệnh do \pkg{amsmath} cung cấp;
+còn \cn{dot} và \cn{ddot} là các lệnh của \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).
+\medskip
+Để có được chỉ số trên là dấu mũ hoặc dấu ngã, dùng gói \pkg{amsxtra}
+và các lệnh của gói này là \cn{sphat}, \cn{sptilde}. Ví dụ,
+$A\sphat$\space\space cho bởi \verb'A\sphat' (để ý ở đây ta không dùng dấu |^|).
+
+% To place an arbitrary symbol in math accent position, or to get under
+% accents, see the \pkg{accents} package by Javier Bezos.
+\medskip
+Để đặt ký hiệu bất kỳ vào vị trí của dấu nhấn, hoặc để có các dấu nhấn
+bên dưới, vui lòng xem tài liệu hướng dẫn gói \pkg{accents}(viết bởi Javier Bezos).
+
+%\section{Roots}
+\section{Căn số}
+
+% In ordinary \latex/ the placement of root indices is sometimes not so
+% good: $\sqrt[\beta]{k}$ (|\sqrt||[\beta]{k}|).
+Cách biểu diễn căn số của \latex/ nhiều khi không thật đẹp mắt, như
+trong ví dụ $\sqrt[\beta]{k}$ (cho bởi |\sqrt||[\beta]{k}|).
+% In the \pkg{amsmath} package \cn{leftroot} and \cn{uproot} allow you to adjust
+% the position of the root:
+Gói \pkg{amsmath} cung cấp các lệnh \cn{leftroot} (dịch qua trái) và \cn{uproot}
+(dịch lên trên)
+cho phép bạn điều chỉnh vị trí của bậc căn số%
+\footnote{bậc của căn số, ví dụ $\beta$ trong $\sqrt[\leftroot{-2}\uproot{2}\beta]{k}$.}.
+% ----------------------------------------------------------------------
+Chẳng hạn
+
+\medskip
+\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.
+
+\medskip\noindent
+sẽ cho $\sqrt[\leftroot{-2}\uproot{2}\beta]{k}$. Nếu bạn dùng |dimension|
+âm làm tham số của \cn{leftroot} (như trong ví dụ trên),
+thì bậc căn số sẽ dịch chuyển qua phải.
+
+%\section{Boxed formulas}
+\section{Đóng khung biểu thức}
+
+% The command \cn{boxed} puts a box around its
+% argument, like \cn{fbox} except that the contents are in math mode:
+Biểu thức \cn{boxed} sẽ đóng khung chữ nhật quanh nội dung là tham số của nó;
+lệnh này tương tự như lệnh \cn{fbox}, nhưng nội dung của lệnh ở chế độ toán.
+\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}
+\section{Mũi tên bên trên, bên dưới}
+
+% 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:
+\latex/ chuẩn chỉ cung cấp các mũi tên bên-trên-qua-trái hoặc bên-trên-qua-phải
+nhờ các lệnh \cn{overrightarrow}, \cn{overleftarrow}. Dưới đây là các dạng
+mũi tên khác do gói \pkg{amsmath} cung cấp.
+
+\begin{tabbing}
+\qquad\=\ncn{overleftrightarrow}\qquad\=\kill
+\> \cn{overleftarrow} \> \cn{underleftarrow} \+\\
+ \cn{overrightarrow} \> \cn{underrightarrow} \\
+ \cn{overleftrightarrow}\> \cn{underleftrightarrow}
+\end{tabbing}
+
+Ví dụ, \cn{overleftarrow}|{XYZ}| sẽ cho bạn $\overleftarrow{XYZ}$.
+
+%\section{Extensible arrows}
+%\section{Mũi tên có chỉ số. Bổ sung của ccc}
+\section{Mũi tên có chỉ số.}
+
+% \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):
+Các lệnh \cn{xleftarrow} và \cn{xrightarrow} cho ta các mũi tên \emph{tự động
+điều chỉnh độ dài cho bằng với nội dung của chỉ số trên hoặc dưới}.
+Tham số bổ sung của hai lệnh này là chỉ số dưới, còn tham số bắt buộc
+là chỉ số trên. Cả chỉ số trên và chỉ số dưới của hai lệnh này đều có thể rỗng.
+\begin{equation}
+A\xleftarrow{n+\mu-1}B \xrightarrow[T]{n\pm i-1}C
+\end{equation}
+\begin{verbatim}
+A\xleftarrow{n+\mu-1}B \xrightarrow[T]{n\pm i-1}C
+\end{verbatim}
+
+%% \medskip
+%% Để có chẳng hạn,
+%% \begin{equation}
+%% A\xlongequal[ccc]{\text{theo (1)}}B,\qquad
+%% A\xLongrightarrow{\text{theo Tiên đề 2}}B,
+%% \end{equation}
+%% bạn có thể dùng gói \pkg{ccc-arrows} (của |ccc|;
+%% tải về: \url{http://vntex.org/download}).
+
+%\section{Affixing symbols to other symbols}
+\section{Gắn các ký hiệu với nhau}
+
+% \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.
+\latex/ cung cấp lệnh \cn{stackrel} để đặt chỉ số trên cho các quan hệ (toán tử)
+nhị phân. Với gói \pkg{amsmath}, các lệnh \cn{overset} và \cn{underset}
+còn làm được việc tổng quát hơn: đặt một ký hiệu bên trên hoặc bên dưới
+một ký hiệu khác, bất kể ký hiệu đó có phải là toán tử nhị phân hay không.
+Ví dụ, với |\overset{*}{X}| sẽ đặt dấu sao $*$ bên trên chữ $X$:
+$\overset{*}{X}$; lệnh \cn{underset} cũng tương tự, nhưng nó đặt chỉ số
+bên dưới ký hiệu.
+
+\medskip
+%See also the description of \cn{sideset} in \secref{sideset}.
+Xem thêm cách dùng \cn{sideset} ở Mục~\secref{sideset}.
+
+%\section{Fractions and related constructions}
+\section{Phân số và cấu trúc liên quan}
+
+\subsection{Các lệnh \cn{frac}, \cn{dfrac}, và \cn{tfrac}}
+
+% 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\index{textstyle@\cn{textstyle}}\relax
+% \index{displaystyle@\cn{displaystyle}} |{\textstyle\frac| |...| |}|.
+Lệnh \cn{frac}, là lệnh cơ bản của \latex/, nhận hai tham số\mdash tử số
+và mẫu số\mdash và biểu diễn phân số theo kiểu thông thường trong toán học.
+Để ý rằng, các phân số ở chế độ toán chung dòng thường nhỏ; để khiến chúng
+to hơn, bạn dùng |\displaystyle\frac...|. Để tiện lợi,
+gói \pkg{amsmath} cung cấp thêm các lệnh: \cn{dfrac}, \cn{tfrac} lần lượt
+là dạng viết tắt của |{\displaystyle\frac| |...| |}| và
+|{\textstyle\frac| |...| |}|. Ví dụ
+\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}
+\subsection{Ký hiệu Tổ hợp: \cn{binom}, \cn{dbinom}, \cn{tbinom}}
+
+% For binomial expressions\index{binomials} such as $\binom{n}{k}$
+% \pkg{amsmath} has \cn{binom}, \cn{dbinom} and \cn{tbinom}:
+Để biểu diễn tổ hợp dạng $\binom{n}{k}$, bạn dùng các lệnh
+\cn{binom}, \cn{dbinom} và \cn{tbinom}. Các tiếp vĩ ngữ |d| và |t|
+được hiểu tương tự như ở các lệnh về phân số.
+\begin{equation}
+2^k-\binom{k}{1}2^{k-1}+\dbinom{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}
+\subsection{Tạo phân số tổng quát với \cn{genfrac}}
+
+% The capabilities of \cn{frac}, \cn{binom}, and their variants are
+% subsumed by a generalized fraction command \cn{genfrac} with six
+% arguments.
+Khả năng của các lệnh \cn{frac}, \cn{binom} và các biến thể có thể có
+được nhờ lệnh \cn{genfrac}. Đây là lệnh tạo phân số tổng quát với
+sáu tham số.
+% The last two correspond to \cn{frac}'s numerator and
+% denominator;
+Hai tham số cuối cùng của \cn{genfrac} lần lượt là \emph{tử số} và \emph{mẫu số};
+% the first two are optional delimiters (as seen in
+% \cn{binom});
+hai tham số bổ sung đầu tiên là các dấu ngoặc trái và phải (như trong
+kết quả của \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
+tham số thứ ba là độ dày của đường kẻ ngang giữa tử số và mẫu số (lệnh
+\cn{binom} dùng độ dày 0, nghĩa là \emph{không thấy được});
+% 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'.
+tham số thứ tư xác định chế độ toán trong đó phân số được biểu diễn:
+tham số này nhận giá trị 0,1,2,3 tương ứng với các chế độ
+ \cn{displaystyle}, \cn{textstyle}, \cn{scriptstyle}, \cn{scriptscriptstyle}.
+
+\medskip
+\begin{center}\begin{minipage}{\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.
+\medskip\noindent
+Ví dụ, các lệnh \cn{frac}, \cn{tfrac} và \cn{binom} có thể được định nghĩa như sau:
+
+\medskip
+\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}.
+\medskip
+Các lệnh nguyên thủy \cn{over}, \cs{overwithdelims},
+\cs{atop}, \cs{atopwithdelims}, \cs{above}, \cs{abovewithdelims}
+khi dùng với gói \pkg{amsmath} sẽ sinh ra cảnh báo lỗi. Vui lòng
+xem thêm tài liệu \fn{technote.tex} có trong |texmf/source/latex/amsmath/|
+
+%\section{Continued fractions}
+\section{Phân số liên tục}
+
+Phân số liên tục\index{phân số liên tục}
+\begin{equation}
+\cfrac{1}{\sqrt{2}+
+ \cfrac{1}{\sqrt{2}+
+ \cfrac{1}{\sqrt{2}+\cdots
+}}}
+\end{equation}
+%can be obtained by typing
+có thể thu đựơc nhờ
+{\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}.
+\medskip\noindent
+Việc dùng \cn{cfrac} cho kết quả dễ nhìn hơn so với khi bạn tạo phân số
+liên tục bằng \cn{frac}. Bạn có thể đặt vị trí của tử số ở bên trái hoặc
+bên phải tùy ý, bằng cách dùng \cn{cfrac}|[l]| hoặc \cn{cfrac}|[r]| thay vì
+\cn{cfrac}.
+
+%\section{Smash options}
+\section{Lệnh \cn{smash} đặt độ cao/sâu về 0}
+
+% 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.
+Lệnh \cn{smash} khiến cho các biểu thức con có chiều cao (|heigth|) và chiều
+sâu (|depth|) bằng 0; việc dùng biểu thức con như vậy rất có hữu ích,
+chẳng hạn khi điều chỉnh vị trí của biểu thức con theo các ký hiệu.
+% 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.
+Gói \pkg{amsmath} cung cấp lệnh \cn{smash} với tham số tùy chọn là |t|
+và |b|, bởi đôi khi ta chỉ cần làm cho chiều cao (ứng với |t|) hoặc
+chiều sâu (ứng với |b|) trở về 0.
+% 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
+
+\medskip
+Ví dụ, chiều cao của các dấu căn bậc hai trong một biểu thức
+không bằng nhau vì chiều cao đó tùy thuộc vào nội dung bên dưới dấu căn.
+(Xem ở độ phóng đại lớn,) bạn sẽ thấy dấu căn trong $\sqrt y$ trong biểu thức
+$\sqrt{x}+\sqrt{y}+\sqrt{z}$ đi xuống phía dưới chân dòng hơn so với
+hai dấu căn còn lại. Ta sẽ dùng \cn{smash[b]} để đặt độ sâu của chữ $y$
+về 0, nhờ đó, dấu căn trong $\sqrt y$ sẽ giống hệt dấu căn trong $\sqrt x$:
+
+\medskip
+\verb"$\sqrt{x}" \verb"+"
+\verb"\sqrt{"\5\verb"\smash[b]{y}}" \verb"+" \verb"\sqrt{z}$".
+
+\medskip\noindent
+Bây giờ, bạn có $\sqrt{x}+\sqrt{\smash[b]{y}}+\sqrt{z}$. Bạn có thấy
+sự khác biệt ở biểu thức này với biểu thức trước kia không?
+
+%\section{Delimiters}
+%
+\section{Dấu ngoặc}
+
+\subsection{Kích cỡ dấu ngoặc}\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.
+Việc dùng dấu ngoặc tự động điều chỉnh kích cỡ với \cn{left} và \cn{right}
+có hai hạn chế: \emph{thứ nhất,} các lệnh này máy móc điều chỉnh cho dấu ngoặc
+có chiều cao vừa bằng với chiều cao của nội dung cao nhất,
+và \emph{thứ hai,} tập hợp các kích cỡ do chúng tạo ra không biến thiên
+đều đặn, mà có những bước nhảy quá lớn\footnote{\emph{Nghĩa là:} hai dấu ngoặc kế tiếp nhau
+trong cùng biểu thức tạo bởi \cn{left} hoặc \cn{right} có thể có chiều cao
+chênh nhau đến 3pt hoặc nhiều hơn, và như thế sẽ quá xấu!}.
+% 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.
+\medskip
+Trong thực tế, thường có hai hoặc ba trường hợp ta cần điều chỉnh kích cỡ
+của dấu ngoặc nhờ vào các lệnh `big' của gói \pkg{amsmath}.
+\begin{ctab}{l|llllll}
+Cỡ&
+Cỡ& \ncn{left}& \ncn{bigl}& \ncn{Bigl}& \ncn{biggl}& \ncn{Biggl}\\
+ngoặc&
+|text|& \ncn{right}& \ncn{bigr}& \ncn{Bigr}& \ncn{biggr}& \ncn{Biggr}\\
+\hline
+Kết quả\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:
+
+\emph{Trường hợp đầu tiên,} là khi xảy ra sự chồng chất (|cumulative|) các
+toán tử với chỉ số trên và dưới. Với \cn{left} và \cn{right}, các dấu ngoặc
+tạo ra thường lớn hơn nhiều so với ý muốn; việc dùng |Big| or |bigg| thay
+thế cho \cn{left} và \cn{right} cho kết quả khả quan hơn hẳn:
+\begin{equation*}
+\left[\sum_i a_i\left\lvert\sum_j x_{ij}\right\rvert^p\right]^{1/p}
+\quad\text{so với}\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.
+
+\medskip
+\emph{Trường hợp thứ hai,} là khi các dấu ngoặc lồng nhau liên tiếp.
+Trong trường hợp này, lệnh \cn{left} và \cn{right} cho ra các dấu ngoặc
+cùng một kích cỡ\footnote{%
+ because that is adequate to cover the encompassed material.},
+trong dấu ngoặc càng ở bên ngoài càng cần có kích thước lớn hơn.
+\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{so với}\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:
+
+\medskip
+\emph{Trường hợp thứ ba,} là khi biểu diễn các biểu thức quá khổ trong một
+dòng, chẳng hạn biểu thức $\left\lvert\frac{b'}{d'}\right\rvert$ trong dòng này.
+Các dấu ngoặc tạo bởi \cn{left} và \cn{right}, như bạn thấy, sẽ làm cho dòng
+bị dãn quá nhiều theo chiều đứng. Khi đó, việc dùng \ncn{bigl} và \ncn{bigr}
+chỉ cho ra dấu ngoặc vừa đủ cao và nhờ đó dòng không bị v-dãn nhiều quá,
+như trong $\bigl\lvert\frac{b'}{d'}\bigr\rvert$.
+
+% 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.
+\medskip
+Các lệnh \ncn{big}, \ncn{bigg}, \ncn{Big} và \ncn{Bigg} mà \latex/ cung cấp
+không thay đổi kích cỡ dấu ngoặc một cách chính xác trong các kích cỡ |font|
+khác nhau của \latex/. Gói \pkg{amsmath} khắc phục nhược điểm đó.
+
+%\subsection{Vertical bar notations}
+\subsection{Ký hiệu $\rvert$}
+
+% 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{\|}.
+Gói \pkg{amsmath} cung cấp các lệnh \cn{lvert}, \cn{rvert},
+\cn{lVert}, \cn{rVert} (so sánh với \cn{langle}, \cn{rangle}) nhằm
+giải quyết vấn đề trùng hợp ý nghĩa của ký tự \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 $f_\zeta(t)\bigr\rvert_{t=0}$.
+Ký tự này được dùng trong tài liệu \latex/ để chỉ nhiều đối tượng
+toán học khác nhau: chỉ quan hệ `ước số' trong lý thuyết số, ví dụ như
+trong biểu thứ $p\vert q$; chỉ giá trị tuyệt đối $\lvert z\rvert$; hay
+là viết tắt của `sao cho' khi biểu diễn tập hợp; hoặc hàm ý `lấy giá trị tại'
+trong ký hiệu $f_\zeta(t)\bigr\rvert_{t=0}$.
+% 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.
+Tính đa nghĩa của ký hiệu $\vert$ không phải là điều gì xấu; cái không
+hay là ở chỗ: không phải tất cả cách dùng của $\vert$ đều có cùng cách thể hiện
+xét về mặt in ấn (|typographical|), và sự phân biệt phức tạp của độc giả
+có kiến thức không thể nào thể hiện được trong |typeset|.
+% 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 \cn{\|}.
+Vì vậy, trong bất kỳ tài liệu nào, bạn cũng nên cho tương ứng ký tự \qc{\|}
+với một khái niệm toán học cụ thể; và cũng nên làm điều tương tự với ký tự
+\cn{\|}.
+% This immediately rules out the use of \qc{|}
+% and \ncn{\|}\index{"|@\cn{"\"|}} for delimiters, because left and right
+% delimiters are distinct usages that do not relate in the same way to
+% adjacent symbols;
+Điều này không bao gồm cách dùng \qc{|} và \ncn{\|}\index{"|@\cn{"\"|}}
+như các dấu ngoặc, bởi vì cách dùng dấu ngoặc trái hoặc phải khác hẳn với
+các cách dùng của $\vert$ như kể trên.
+% recommended practice is therefore to define suitable
+% commands in the document preamble for any paired-delimiter use of vert
+% bar symbols:
+
+\medskip
+Ví dụ, bạn nên định nghĩa
+\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 $\lVert v\rVert$.
+và suốt trong tài liệu của mình, bạn dùng |\abs{z}| để có
+$\lvert z\rvert$ và dùng |\norm{v}| để có $\lVert v\rVert$.
+
+%\chapter{Operator names}
+\chapter{Tên toán tử}
+
+%\section{Defining new operator names}
+\section{Định nghĩa toán tử mới}\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
+Các toán tử toán học như $\log$, $\sin$, $\lim$ theo truyền thống được
+|typeset| với kiểu chữ |roman| để có thể dễ dàng phân biệt chúng với
+các tên biến toán học (các tên biến được |typeset| với |font| nghiêng).
+Các toán tử thường gặp nhất là \cn{log}, \cn{sin}, \cn{lim},... đã được
+định nghĩa; còn các toán tử mới có thể định nghĩa dễ dàng nhờ lệnh
+\cn{DeclareMathOperator}. Lệnh này chỉ có thể đặt trong phần |preamble|
+của tài liệu. Chẳng hạn, để định nghĩa hàm số \ncn{tg} (là
+cách viết hàm số |tangement| theo kiểu Việt Nam), có thể dùng
+
+\begin{verbatim}
+\DeclareMathOperator{\tg}{tg}
+\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.
+nhờ đó, việc dùng \ncn{tg} sẽ cho ra $\tg$ với |font| chính xác
+đồng thời tự động thêm cách khoảng cách xung quanh toán tử {\upshape tg}
+khi cần thiết, nhờ đó, ta thu được $A(\tg x)B$ thay vì $A(\mathrm{tg}x)B$.
+
+% If the new operator should have subscripts and superscripts placed in
+% `limits' position above and below as with $\lim$, $\sup$, or $\max$, use
+\medskip
+Nếu toán tử mới cần các chỉ số trên và chỉ số dưới, như trong các toán tử
+$\lim$, $\sup$, or $\max$, thì bạn dụng dạng \qc{\*} của lệnh \cn{DeclareMathOperator}:
+% the \qc{\*} form of the \cn{DeclareMathOperator} command:
+
+\medskip
+\begin{verbatim}
+\DeclareMathOperator*{\Lim}{Lim}
+\end{verbatim}
+%See also the discussion of subscript placement in
+
+\medskip\noindent
+Xem thêm ở Mục~\ref{subplace} về việc đặt chỉ số dưới.
+
+\medskip
+%The following operator names are predefined:
+Các toán tử được định nghĩa trước là
+\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
+\medskip
+Ngoài ra, còn có lệnh \cn{operatorname}: ví dụ
+
+\medskip
+\begin{verbatim}
+\operatorname{abc}
+\end{verbatim}
+%in a math formula is equivalent to a use of \ncn{abc} defined by
+
+\medskip\noindent
+trong biểu thức toán học tương đương với việc khai báo nhờ
+\cn{DeclareMathOperator} rồi dùng \ncn{abc}. Lệnh \cn{operatorname}
+chỉ nên dùng nếu toán tử thi thoảng gặp trong tài liệu;
+còn nếu toán tử xuất hiện nhiều lần, bạn nên khai báo toán tử mới
+nhờ \cn{DeclareMathOperator}.
+% This may be occasionally useful for
+% constructing more complex notation or other purposes.
+
+\medskip
+%(Use the variant \cn{operatorname*} to get limits.)
+Để ý dùng \cn{operatorname*} trong trường hợp toán tử cần chỉ số.
+
+%\section{\cn{mod} and its relatives}
+\section{Ký hiệu Đồng dư}
+
+% 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.
+Các lệnh \cn{mod}, \cn{bmod}, \cn{pmod}, \cn{pod} biểu diễn các ký hiệu
+đồng dư. Lệnh \cn{bmod} và \cn{pmod} đã có trong \latex/, nhưng trong gói
+\pkg{amsmath}, chúng được tinh chỉnh để cho ra khoảng cách tốt hơn trong
+các biểu thức chung dọng. Các lệnh \cn{mod} và \cn{pod} là biến thể của
+\cn{pmod}; lệnh \cn{mod} không tạo ra cặp dấu ngoặc đơn, trong khi lệnh
+\cn{pod} lại bỏ \qq{mod} và các thứ khác vào cặp dấu ngoặc.
+\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}
+\chapter{Lệnh \cn{text} chèn \texttt{text} vào biểu thức}\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.
+Mục đích chính của lệnh \cn{text} là để chèn văn bản, |text| vào giữa
+các biểu thức toán học. Lệnh có tác dụng tương tự như lệnh \cn{mbox}
+của \latex/, nhưng dùng \cn{text} hay hơn ở nhiều điểm.
+Nếu bạn muốn văn bản thêm vào ở cỡ chữ |sub-script|, bạn có thể dùng
+|..._{\text{từ hoặc câu}}|, trong khi với \cn{mbox}, bạn phải dùng
+|..._{\mbox{\scriptsize| |từ| |hoặc| |câu}}|. Điểm hay khác, là tên của
+lệnh \cn{text} phản ánh được đầy đủ tác dụng của nó hơn lệnh \cn{mbox}.
+\begin{equation}
+f_{[x_{i-1},x_i]} \text{ là đơn điệu,}
+\quad i = 1,\dots,c+1
+\end{equation}
+\begin{verbatim}
+f_{[x_{i-1},x_i]} \text{ là đơn điệu,}
+\quad i = 1,\dots,c+1
+\end{verbatim}
+
+% \chapter{Integrals and sums}
+\chapter{Tích phân và Tổng}
+
+%\section{Multiline subscripts and superscripts}
+\section{Chỉ số trên/dưới nhiều dòng}
+
+% 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
+Nhờ lệnh \cn{substack}, bạn biểu diễn các chỉ số trên, dưới với nhiều dòng.
+Ví dụ
+\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:
+Tổng quát hơn, môi trường \env{subarray} còn cho phép bạn canh trái các
+dòng của chỉ số trên/dưới, thay vì canh giữa như mặc định.
+Hãy để ý đến tham số |{l}| của \env{subaray} trong ví dụ dưới đây.
+\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}
+\section{Lệnh \cn{sideset}}\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
+Lệnh \cn{sideset} dùng cho mục đích đặc biệt, là đặt các ký hiệu vào
+vị trí của chỉ số trên và chỉ số dưới của các toán tử lớn như $\sum$ hay $\prod$.
+\emph{Chú ý rằng, lệnh này không có mục đích áp dụng vào các ký hiệu
+không thuộc lớp ký hiệu tổng.}
+Ví dụ dễ thấy nhất là khi bạn muốn có một dấu phảy để chỉ biến thể
+của ký hiệu tổng:
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% |\sum\nolimits' E_n| in display mode:
+\begin{equation}
+\sum\nolimits' E_n
+\end{equation}
+Để được như trên, bạn có thể dùng |\sum\nolimits' E_n|; ở đây, \cn{nolimits}
+phải được dùng, bởi nếu không, dấu phảy sẽ được đặt ở bên trên chứ không phải
+là góc trên-bên-phải của ký hiệu tổng. Nếu bạn làm như vậy, thì nảy ra
+vấn đề nan giải là, bây giờ khó có thể đặt thứ gì vào bên trên và bên dưới
+của $\sum$ nữa (bởi bạn đã dùng \cn{nolimits}). Cách giải quyết vấn đề này
+nhờ lệnh \cn{sideset} được minh họa trong ví dụ sau:
+% 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
+
+\medskip
+\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
+Phải dùng cặp dấu ngoặc rỗng (|{}|) như trên, là bởi lệnh \cn{sideset}
+có khả năng đặt các ký hiệu vào bốn góc của toán tử lớn. Hãy học hỏi ví
+dụ rất thú vị sau đây:
+
+\medskip\begin{ctab}{ll}
+\begin{minipage}[t]{.6\columnwidth}
+\begin{verbatim}
+\sideset{_*^*}{_*^*}\prod_{n=1}^{\infty}
+\end{verbatim}
+\end{minipage}
+&$\displaystyle
+\sideset{_*^*}{_*^*}\prod_{n=1}^{\infty}
+$
+\end{ctab}
+
+%\section{Placement of subscripts and limits}
+\section{Vị trí của chỉ số dưới và `limit'}\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.
+Vị trí mặc định của chỉ số dưới của ký hiệu phụ thuộc vào tính chất
+ký hiệu đó. Với các ký hiệu dạng tổng ($\sum$, $\prod$), vị trí đó
+là `displaylimits': nghĩ là, khi ký hiệu dạng tổng xuất hiện trong
+biểu thức riêng dòng, chỉ số trên và chỉ số dưới được bố trí bên trên
+hoặc bên dưới ký hiệu; còn nếu ký hiệu đó xuất hiện trong chế độ chung dòng,
+thì các chỉ số trên, dưới được đặt ở bên cạnh ký hiệu, nhằm tránh việc
+dãn dòng qua mức cần thiết.
+% 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}.)
+Đối với các ký hiệu dạng tích phân, vị trí mặc định của các chỉ số luôn
+là ở bên cạnh ký hiệu, ngay cả trong các công thức riêng dòng.
+(Nhưng xem về tùy chọn \opt{intlimits} trong Mục~\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.
+\medskip
+Các toán tử như $\sin$, $\lim$ đều có vị trí chỉ số thuộc loại
+`displaylimits' hay `limits' tùy theo cách chúng được định nghĩa,
+và điều đã được chọn theo thói quen phổ biến trong giới toán học.
+
+% The commands \cn{limits} and \cn{nolimits} can be used to override the
+% normal behavior of a base symbol:
+\medskip
+Các lệnh \cn{limits} và \cn{nolimits} dùng để bật qua lại giữa hai chế
+độ đặt vị trí chỉ số:
+\begin{equation*}
+\sum\nolimits_X,\qquad \iint\limits_{A},
+\qquad\varliminf\nolimits_{n\to \infty}
+\end{equation*}
+\begin{verbatim}
+\sum\nolimits_X,\qquad \iint\limits_{A},
+\qquad\varliminf\nolimits_{n\to \infty}
+\end{verbatim}
+
+% 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.
+\medskip
+Để định nghĩa một lệnh mới mà chỉ số dưới của nó có thuộc tính
+`displaylimits', bạn đặt lệnh \cn{displaylimits} và cuối định nghĩa
+của lệnh. Trong dãy gồm các lệnh \cn{limits}, \cn{nolimits}, \cn{displaylimits}
+tiếp nhau, thì lệnh cuối cùng có tác dụng ưu tiên.
+
+%\section{Multiple integral signs}
+\section{Dấu tích phân bội}
+
+% \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.
+Các lệnh \cn{iiint}, \cn{iiint} và \cn{iiiint} cho ta các dấu tích phân bội
+với khoảng cách giữa các dấu tích phân được tinh chỉnh tốt hơn.
+Lệnh \cn{idotsint} cho ta dấu tích phân bội với số lớp tùy ý.
+\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}
+\chapter{Biểu đồ giao hoán}\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.
+Một số lệnh tạo biểu đồ giao hoán như trong \amstex/ được gộp thành
+gói riêng biệt, là \pkg{amscd}. Với các biểu đồ phức tạp, bạn nên dùng gói
+chuyên dụng hơn là \pkg{kuvio} hoặc \pkg{xypic}; nhưng với các biểu đồ
+đơn giản, không có các mũi tên chéo, thì dùng gói \pkg{amscd} sẽ nhanh
+chóng cho kết quả. Dưới đây là ví dụ.
+\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.
+Trong môi trường \env{CD}, các lệnh |@>>>|, |@<<<|, |@VVV| và |@AAA|
+lần lượt cho các mũi tên qua phải, trái, xuống, lên. Với các mũi tên ngang,
+mọi thứ đặt giữa hai ký hiệu |>| hoặc |<| đầu tiên sẽ được xem là chỉ số trên,
+các đối tượng đặt giữa ký hiệu thứ hai và thứ ba được xem là chỉ số dưới.
+Tương tự, đối tượng đặt giữa ký hiệu |A| hay |V| thứ nhất và thứ hai
+sẽ là chỉ số bên trái, còn đối tượng đặt giữa ký hiệu |A| hay |V| thứ nhì và
+thứ ba sẽ là chỉ số bên phải. Các lệnh |@=| và \verb'@|'
+sẽ cho các đường-ngang-kép và đường-đứng-kép. Một \qq{mũi tên rỗng}
+được cho bởi lệnh |@.|.
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%\chapter{Using math fonts}
+\chapter{Sử dụng `font' toán}
+
+\section{Giới thiệu}
+
+% 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).
+Mọi vấn đề về |font| có thể tìm thấy trong hướng dẫn (\fn{fntguide.tex})
+hoặc \booktitle{The \latex/ Companion} \cite{tlc}.
+
+\medskip
+Các lệnh cơ bản (của \latex/)
+điều chỉnh |font| trong chế độ toán bao gồm:
+\cn{mathbf}, \cn{mathrm}, \cn{mathcal}, \cn{mathsf}, \cn{mathtt}, \cn{mathit}.
+Ngoài ra, gói \pkg{amsfonts} cung cấp các lệnh đặc biệt:
+\cn{mathbb} cho kiểu `blackboard bold' ($\mathbb R$),
+\cn{mathfrak} cho kiểu `Fraktur' ($\mathfrak R$).
+% và \cn{mathscr} cho kiểu `Euler-script' ($\mathscr r$).
+
+%\section{Recommended use of math font commands}
+\section{Lời khuyên}
+
+% 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:
+Nếu bạn phải thường xuyên thay đổi kiểu |font| trong chế độ toán,
+bạn có thể nghĩ đến các định nghĩa vắn tắt, chẳng hạn, dùng \ncn{mb}
+thay vì \cn{mathbf}. Tất nhiên, không gì ngăn cản bạn làm điều đó,
+vì bạn có trong tay lệnh-tạo-định-nghĩa \cn{newcommand}. Nhưng với \latex/,
+việc đưa ra các dạng định nghĩa vắn tắt như vậy cần nên tránh, cần thay thế
+bởi giải pháp hay hơn: \emph{định nghĩa một lệnh dựa trên \emph{thuộc tính của
+đối tượng toán học}, hơn là dựa trên tên của |font| dùng để phân biệt
+các đối tượng.} Ví dụ, nếu bạn muốn dùng |font| đậm chỉ các véctơ, bạn
+nên định nghĩa một lệnh tạo véctơ với |font| đậm, hơn là định nghĩa một
+lệnh tạo |font| đậm để tạo véctơ; nhờ
+
+\medskip
+\begin{verbatim}
+ \newcommand{\vect}[1]{\mathbf{#1}}
+\end{verbatim}
+
+\medskip\noindent
+bạn có thể viết |\vect{a} + \vect{b}| để có $\vect{a}+\vect{b}$;
+bạn không nên dùng, chẳng hạn \cn{mb}|{a}|+\ncn{mb}|{b}| với lệnh \ncn{mb}
+là dạng viết tắt của \cn{mathbf}.
+% 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.
+Sau một vài tháng chẳng hạn, bạn thấy cần phải dùng |font| đậm cho mục đích khác,
+chứ không phải để chỉ các véctơ, và muốn các véctơ bây giờ có thêm các mũi tên,
+thì bạn có thể dễ dàng thực hiện được ý đồ của mình nếu bạn dùng định nghĩa
+\cn{vect} để tạo véctơ; còn nếu ngược lại, bạn phải thay thế toàn bộ các lệnh
+\cn{mb} trong tài liệu của bạn\mdash điều này nguy hiểm ở chỗ, việc thay thế
+có thể ảnh hưởng đến các đối tượng không là véctơ nhưng được biểu diễn bằng
+\cn{mb}.
+
+% It can also be useful to assign distinct
+% command names for different letters of a particular font:
+\medskip
+Sẽ có ích nếu bạn tạo các lệnh mới để lấy ra các ký tự đặc biệt trong
+một |font| cụ thể nào đó, ví dụ
+
+\medskip
+\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:\index{mathbb@\cn{mathbb}}
+\medskip\noindent
+Các dòng ở trên định nghĩa các lệnh \cn{C}, \cn{R} để lấy ra ký hiệu
+`blackboard' từ bộ `AMSb' là |font| các ký hiệu. Nếu bạn thường xuyên
+làm việc với các trường số thực, phức, bạn có thể có một cách tiện lợi, là
+định nghĩa lệnh \ncn{field} và sau đó dùng |\field{C}|, |\field{R}|,...
+để có các trường số mong muốn. Nhưng để \emph{tăng tối đa tính uyển chuyển
+và khả năng điều khiển}, bạn hãy định nghĩa lệnh \ncn{field} rồi sau đó
+định nghĩa các lệnh \ncn{C}, \ncn{R} dựa trên \ncn{field} như sau:
+
+\medskip
+\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}
+\section{Các ký hiệu in đậm}
+
+% 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
+Lệnh \cn{mathbf} thường được dùng để có được phiên bản đậm của các chữ cái
+trong chế độ toán, nhưng với hầu hết các ký hiệu toán khác (không phải là chữ cái),
+lệnh này không có tác dụng, hoặc có tác dụng nhưng kết quả
+của nó không có liên hệ chặt chẽ với |font| đang được dùng. Ví dụ, viết
+
+\medskip
+\begin{verbatim}
+\Delta \mathbf{\Delta}\mathbf{+}\delta \mathbf{\delta}
+\end{verbatim}
+
+\medskip\noindent
+sẽ cho bạn $\Delta \mathbf{\Delta}\mathbf{+}\delta \mathbf{\delta}$;
+%the \cn{mathbf} has no effect on the plus sign or the small delta.
+để ý rằng, trong kết quả thu được, các dấu cộng và dấu $\delta$ không
+bị ảnh hưởng bởi \cn{mathbf}.
+
+% 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}.
+\medskip
+Gói \pkg{amsmath} vì vậy cung cấp thêm hai lệnh mới, là
+\cn{boldsymbol} và \cn{pmb}; các lệnh này có tác dụng làm đậm mọi loại
+ký hiệu. Lệnh \cn{boldsymbol} có thể được dùng cho các ký hiệu toán không
+bị ảnh hưởng bởi lệnh \cn{mathbf} \emph{nếu và chỉ nếu} |font| toán
+đang dùng có phiên bản |đậm| tương ứng của ký hiệu đó. Lệnh \cn{pmb}
+được dùng, như là giải pháp cuối cùng, cho trường hợp lệnh \cn{boldsymbol}
+bất lực; \qq{pmb} là viết tắt của \qq{poor man's bold};
+nguyên lý làm việc của lệnh \cn{pmb} là |typeset| nhiều bản sao của
+ký hiệu và sắp xếp các bản sao đó sát bên nhau (giống như khi bạn
+dùng bút kẻ nhiều đường liên tiếp nhau trên giấy để có một đường đậm nét).
+Rõ ràng, chất lượng kết quả của lệnh \cn{pmb} là không tốt lắm, nhất là
+độ nét của ký hiệu. Trong khi bộ |font| chuẩn của \latex/ được dùng để
+|typeset| các biểu thức toán, là bộ |font| CM (Computer Modern), thì cần đến
+lệnh \cn{pmb} chỉ là các ký hiệu toán tử lớn, như \cn{sum},
+các dấu ngoặc quá cỡ, hay là các ký hiệu toán học bổ sung bởi gói
+\pkg{amssymb} (xem \cite{amsfonts}).
+
+% The following formula shows some of the results that are possible:
+\medskip
+Các biểu thức sau cho thấy một số kết quả khả dĩ:
+\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/).
+
+\medskip\noindent
+Nếu bạn muốn dùng lệnh \cn{boldsymbol} độc lập mà không tải gói \pkg{amsmath},
+thì hãy dùng gói \pkg{bm}; đây là gói thuộc chuẩn \latex/, không thuộc bộ
+phân phối của các gói AMS. Gói này tích hợp trong bộ \latex/ phiên
+bản năm 1997 hoặc cao hơn.
+
+%\section{Italic Greek letters}
+\section{Các chữ cái Hy Lạp in nghiêng}
+
+% 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}
+\newpage
+\chapter{Lỗi thường gặp khi dùng gói \pkg{amsmath}}
+
+\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.
+
+\newpage
+
+\begin{thebibliography}{9}
+\addcontentsline{toc}{chapter}{Tài liệu tham khảo}
+
+\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}
+
+%\end{theindex}
+
+\end{document}
+
+% Local Variables:
+% coding: utf-8
+% eval: (set-input-method "vietnamese-viqr")
+% End: