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author | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
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committer | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
commit | e0c6872cf40896c7be36b11dcc744620f10adf1d (patch) | |
tree | 60335e10d2f4354b0674ec22d7b53f0f8abee672 /info/amslatex |
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diff --git a/info/amslatex/english/amsldoc.pdf b/info/amslatex/english/amsldoc.pdf new file mode 120000 index 0000000000..324e408aa8 --- /dev/null +++ b/info/amslatex/english/amsldoc.pdf @@ -0,0 +1 @@ +../../../macros/latex/required/amsmath/amsldoc.pdf
\ No newline at end of file diff --git a/info/amslatex/primer/README b/info/amslatex/primer/README new file mode 100644 index 0000000000..0bb2eecf16 --- /dev/null +++ b/info/amslatex/primer/README @@ -0,0 +1,17 @@ +"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 new file mode 100644 index 0000000000..d738ecc54e --- /dev/null +++ b/info/amslatex/primer/amshelp.md5 @@ -0,0 +1,4 @@ +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 Binary files differnew file mode 100644 index 0000000000..57830947ee --- /dev/null +++ b/info/amslatex/primer/amshelp.pdf diff --git a/info/amslatex/primer/amshelp.tex b/info/amslatex/primer/amshelp.tex new file mode 100644 index 0000000000..af2901fed6 --- /dev/null +++ b/info/amslatex/primer/amshelp.tex @@ -0,0 +1,4919 @@ +%%% 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 Binary files differnew file mode 100644 index 0000000000..6b333cc6d0 --- /dev/null +++ b/info/amslatex/vietnamese/amsldoc-print-vi.pdf diff --git a/info/amslatex/vietnamese/amsldoc-vi.pdf b/info/amslatex/vietnamese/amsldoc-vi.pdf Binary files differnew file mode 100644 index 0000000000..e34316c6b2 --- /dev/null +++ b/info/amslatex/vietnamese/amsldoc-vi.pdf 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: |