math.tex

% -*- coding: utf-8 -*-
% This is part of the book TeX for the Impatient.
% Copyright (C) 2003, 2014 Paul W. Abrahams, Kathryn A. Hargreaves, Karl Berry.
% See file fdl.tex for copying conditions.

\input macros
%\chapter {Commands \linebreak for composing \linebreak math formulas}
\chapter {数学公式命令}

%\bix^^{math}
%\chapterdef{math}
\bix^^{数学}
\chapterdef{math}

%This section covers commands for constructing math formulas.
%For an explanation of the conventions used in this section,
%see \headcit{Descriptions of the commands}{cmddesc}.
这一章包括排印数学公式所需要的命令。
在\headcit{命令描述}{cmddesc}这一节中给出了这章的惯例。

\begindescriptions
%==========================================================================
%\section {Simple parts of formulas}
\section {简单公式排版}

%==========================================================================
%\subsection {Greek letters}
\subsection {希腊字母}

%\begindesc
%\bix^^{Greek letters}
%\dothreecolumns 40
%\easy\ctsdisplay alpha {}
%\ctsdisplay beta {}
%\ctsdisplay chi {}
%\ctsdisplay delta {}
%\ctsdisplay Delta {}
%\ctsdisplay epsilon {}
%\ctsdisplay varepsilon {}
%\ctsdisplay eta {}
%\ctsdisplay gamma {}
%\ctsdisplay Gamma {}
%\ctsdisplay iota {}
%\ctsdisplay kappa {}
%\ctsdisplay lambda {}
%\ctsdisplay Lambda {}
%\ctsdisplay mu {}
%\ctsdisplay nu {}
%\ctsdisplay omega {}
%\ctsdisplay Omega {}
%\ctsdisplay phi {}
%\ctsdisplay varphi {}
%\ctsdisplay Phi {}
%\ctsdisplay pi {}
%\ctsdisplay varpi {}
%\ctsdisplay Pi {}
%\ctsdisplay psi {}
%\ctsdisplay Psi {}
%\ctsdisplay rho {}
%\ctsdisplay varrho {}
%\ctsdisplay sigma {}
%\ctsdisplay varsigma {}
%\ctsdisplay Sigma {}
%\ctsdisplay tau {}
%\ctsdisplay theta {}
%\ctsdisplay vartheta {}
%\ctsdisplay Theta {}
%\ctsdisplay upsilon {}
%\ctsdisplay Upsilon {}
%\ctsdisplay xi {}
%\ctsdisplay Xi {}
%\ctsdisplay zeta {}
%\egroup
\begindesc
\bix^^{希腊字母}
\dothreecolumns 40
\easy\ctsdisplay alpha {}
\ctsdisplay beta {}
\ctsdisplay chi {}
\ctsdisplay delta {}
\ctsdisplay Delta {}
\ctsdisplay epsilon {}
\ctsdisplay varepsilon {}
\ctsdisplay eta {}
\ctsdisplay gamma {}
\ctsdisplay Gamma {}
\ctsdisplay iota {}
\ctsdisplay kappa {}
\ctsdisplay lambda {}
\ctsdisplay Lambda {}
\ctsdisplay mu {}
\ctsdisplay nu {}
\ctsdisplay omega {}
\ctsdisplay Omega {}
\ctsdisplay phi {}
\ctsdisplay varphi {}
\ctsdisplay Phi {}
\ctsdisplay pi {}
\ctsdisplay varpi {}
\ctsdisplay Pi {}
\ctsdisplay psi {}
\ctsdisplay Psi {}
\ctsdisplay rho {}
\ctsdisplay varrho {}
\ctsdisplay sigma {}
\ctsdisplay varsigma {}
\ctsdisplay Sigma {}
\ctsdisplay tau {}
\ctsdisplay theta {}
\ctsdisplay vartheta {}
\ctsdisplay Theta {}
\ctsdisplay upsilon {}
\ctsdisplay Upsilon {}
\ctsdisplay xi {}
\ctsdisplay Xi {}
\ctsdisplay zeta {}
\egroup
%\explain
%These commands produce Greek letters suitable for mathematics.
%You can only use them
%within a math formula, so if you need a Greek letter within ordinary
%text you must enclose it in dollar signs (|$|).  \TeX\ does not have
%commands for Greek letters that look like their roman
%counterparts, since you can get them by using those roman
%counterparts.  For example, you can get a lowercase
%^{omicron} in a formula by writing the letter `o', i.e.,
%`|{\rm o}|' or an uppercase ^{beta} (`B') by writing
%`|{\rm B}|'.
\explain
输入这些命令可以排印出数学公式中的相应的希腊字母符号。
你只能在数学模式中使用它们，所以如在普通的文本中使用它们时，
你必须把它们括在美元符号 (|$|) 内。
\TeX\ 并不包含这些数学中使用的希腊字母所对应的正体字符的命令，
不过你可以很方便地得到这些字符。
比如说，你可以在公式中使用 `|{\rm o}|' 来得到一个小写的 ^{omicron} `o',
又比如，你可以使用 `|{\rm B}|' 得到大写的 ^{beta} (`B').

%Don't confuse the following letters:
%\ulist \compact
%\li |\upsilon| (`$\upsilon$'), |{\rm v}| (`v'), and |\nu| (`$\nu$').
%\li |\varsigma| (`$\varsigma$') and |\zeta| (`$\zeta$').
%\endulist
注意不要混淆下面的符号：
\ulist \compact
\li |\upsilon| (`$\upsilon$'), |{\rm v}| (`v'), 和 |\nu| (`$\nu$').
\li |\varsigma| (`$\varsigma$') 和 |\zeta| (`$\zeta$').
\endulist

%You can get slanted capital Greek letters by using the math italic
%(|\mit|) \minref{font}.
使用数学的意大利\minref{字体} (|\mit|) 可以得到斜体的大写希腊字母。

%\TeX\ treats Greek letters as ordinary symbols when it's figuring how
%much space to put around them.
在计算在希腊字母周围插入多少的空白时，\TeX\ 把它们当作正常的符号。

%\example
%If $\rho$ and $\theta$ are both positive, then $f(\theta)
%-{\mit \Gamma}_{\theta} < f(\rho)-{\mit \Gamma}_{\rho}$.
%|
%\produces
%If $\rho$ and $\theta$ are both positive, then
%$f(\theta)-{\mit \Gamma}_{\theta} < f(\rho)-{\mit \Gamma}_{\rho}$.
%\endexample
%\eix^^{Greek letters}
%\enddesc
\example
如果 $\rho$ 和 $\theta$ 都是正数，那么 $f(\theta)
-{\mit \Gamma}_{\theta} < f(\rho)-{\mit \Gamma}_{\rho}$.
|
\produces
如果 $\rho$ 和 $\theta$ 都是正数，那么
$f(\theta)-{\mit \Gamma}_{\theta} < f(\rho)-{\mit \Gamma}_{\rho}$.
\endexample
\eix^^{希腊字母}
\enddesc

%==========================================================================
%\subsection {Miscellaneous ordinary math symbols}
\subsection {各种普通数学符号}

\begindesc
\xrdef{specsyms}
\dothreecolumns 34
\easy\ctsdisplay infty {}
\ctsdisplay Re {}
\ctsdisplay Im {}
\ctsdisplay angle {}
\ctsdisplay triangle {}
\ctsdisplay backslash {}
\ctsdisplay vert {}
\ctsydisplayn \| @bar {} \directidx{"\"|+@\hskip-.5em\tt\\\char`"\"|\rm}
\ctsdisplay Vert {}
\ctsdisplay emptyset {}
\ctsdisplay bot {}
\ctsdisplay top {}
\ctsdisplay exists {}
\ctsdisplay forall {}
\ctsdisplay hbar {}
\ctsdisplay ell {}
\ctsdisplay aleph {}
\ctsdisplay imath {}
\ctsdisplay jmath {}
\ctsdisplay nabla {}
\ctsdisplay neg {}
\ctsdisplay lnot {}
\actdisplay \' @prime \ (上标点)
\ctsdisplay prime {}
\ctsdisplay partial {}
\ctsdisplay surd {}
\ctsdisplay wp {}
\ctsdisplay flat {}
\ctsdisplay sharp {}
\ctsdisplay natural {}
\ctsdisplay clubsuit {}
\ctsdisplay diamondsuit {}
\ctsdisplay heartsuit {}
\ctsdisplay spadesuit {}
\egroup
\explain
^^{音符} ^^{花色}
这些命令可以排印各种符号。
为了把它们和其它的符号，比如关系符号等，区分开来，它们被称为普通数学符号。
你只能在数学模式中使用这些符号，所以如果在普通的文本中使用，你必须使用美元符号 (|$|) 把它们括起来。

当你想在 `$i$' 或 `$j$' 上加上重音符号，则需要使用 |\imath| 和 |\jmath| 命令来表示它们本身。

上标点符号 (|'|) 是一个 |\prime| 的上标的简写。
(|\prime| 本身可以排印一个很大的丑陋的撇号。)

|\!|| 和 ^|\Vert| 命令是等价的，就像 ^|\neg| 和 ^|\lnot| 命令一样。
\margin{增加了 {\tt\\vert} 的解释}
|\vert| 符号可以排印出和 `|!||' 相同的效果。
\directidx{"\"|@\tt\char`"\"|\rm}

由 |\backslash|, |\vert|, 和 |\Vert| 排印的命令叫做 \minref{分界符}.
使用 ^|\bigm| 等 (\xref \bigm) 命令可以排印大号的这些字符。

\example
The Knave of $\heartsuit$s, he stole some tarts.
|
\produces
The Knave of $\heartsuit$s, he stole some tarts.
\nextexample
如 $\hat\imath < \hat\jmath$ 则 $i' \leq j^\prime$.
|
\produces
如 $\hat\imath < \hat\jmath$ 则 $i' \leq j^\prime$.
\nextexample
$${{x-a}\over{x+a}}\biggm\backslash{{y-b}\over{y+b}}$$
|
\dproduces
$${{x-a}\over{x+a}}\biggm\backslash{{y-b}\over{y+b}}$$
\endexample
\enddesc

%==========================================================================
\subsection {二元运算符}

\begindesc
\bix^^{运算符}
\xrdef{binops}
\dothreecolumns 34
\easy\ctsdisplay vee {}
\ctsdisplay wedge {}
\ctsdisplay amalg {}
\ctsdisplay cap {}
\ctsdisplay cup {}
\ctsdisplay uplus {}
\ctsdisplay sqcap {}
\ctsdisplay sqcup {}
\ctsdisplay dagger {}
\ctsdisplay ddagger {}
\ctsdisplay land {}
\ctsdisplay lor {}
\ctsdisplay cdot {}
\ctsdisplay diamond {}
\ctsdisplay bullet {}
\ctsdisplay circ {}
\ctsdisplay bigcirc {}
\ctsdisplay odot {}
\ctsdisplay ominus {}
\ctsdisplay oplus {}
\ctsdisplay oslash {}
\ctsdisplay otimes {}
\ctsdisplay pm {}
\ctsdisplay mp {}
\ctsdisplay triangleleft {}
\ctsdisplay triangleright {}
\ctsdisplay bigtriangledown {}
\ctsdisplay bigtriangleup {}
\ctsdisplay ast {}
\ctsdisplay star {}
\ctsdisplay times {}
\ctsdisplay div {}
\ctsdisplay setminus {}
\ctsdisplay wr {}
\egroup
\explain
这些命令可以排印各种二元运算符。
二元运算符是 \TeX\ 的一种符号\minref{集}.
\TeX\ 在不同的符号集周围会插入不同的空白。
当 \TeX\ 需要在一个数学公式中间断行时，
它会考虑在二元运算符后面进行断行---不过仅在它出现在公式的最外层时，而不是在一个组中。

除了这些命令以外，\TeX\ 也把 `|+|' and `|-|' 作为二元运算符。
它把 `|/|' 当作一个普通符号，
因为虽然事实上在数学中它是一个二元运算，
但是它在周围加入的空白更少时看上去更漂亮。

\example
$$z = x \div y \quad \hbox{当且仅当} \quad
z \times y = x \;\hbox{且}\; y \neq 0$$
|
\dproduces
$$z = x \div y \quad \hbox{当且仅当} \quad
z \times y = x \;\hbox{且}\; y \neq 0$$
\endexample
\enddesc

%\begindesc
%\ctsymbol \* \ctsxrdef{@star}
%\explain
%The |\*| command indicates a discretionary multiplication symbol
%($\times$), which is a binary operation.  This multiplication symbol
%behaves like a discretionary hyphen when it appears in a formula within
%text\minrefs{text math}.  That is, \TeX\ will typeset the |\times|
%symbol \emph{only} if the formula needs to be broken at that point.
%There's no point in using |\*| in a displayed formula \minrefs{display
%math} since \TeX\ never breaks displayed formulas on its own.
\begindesc
\ctsymbol \* \ctsxrdef{@star}
\explain
命令 |\*| 表示自定断点的乘法符号 ($\times$)，它是一个二元运算符。
该符号在文内公式中功能与自定连字符类似。
这就是说，\emph{仅当}公式需要在该处断行时，\TeX\ 才会排版 |\times| 符号。
因为 \TeX\ 永远不会在陈列公式中断行，所以 |\*| 在陈列公式\minrefs{陈列公式}中是没有任何作用的。

\example
Let $c = a\*b$. In the case that $c=0$ or $c=1$, let
$\Delta$ be $(\hbox{the smallest $q$})\*(\hbox{the
largest $q$})$ in the set of approximate $\tau$-values.
|
\produces
Let $c = a\*b$. In the case that $c=0$ or $c=1$, let
$\Delta$ be $(\hbox{the smallest $q$})\*(\hbox{the
largest $q$})$ in the set of approximate $\tau$-values.

\eix^^{运算符}
\endexample
\enddesc

%==========================================================================
\subsection {关系符号}

\begindesc
\xrdef {relations}
\bix^^{关系符}
\dothreecolumns 39
\easy\ctsdisplay asymp {}
\ctsdisplay cong {}
\ctsdisplay dashv {}
\ctsdisplay vdash {}
\ctsdisplay perp {}
\ctsdisplay mid {}
\ctsdisplay parallel {}
\ctsdisplay doteq {}
\ctsdisplay equiv {}
\ctsdisplay ge {}
\ctsdisplay geq {}
\ctsdisplay le {}
\ctsdisplay leq {}
\ctsdisplay gg {}
\ctsdisplay ll {}
\ctsdisplay models {}
\ctsdisplay ne {}
\ctsdisplay neq {}
\ctsdisplay notin {}
\ctsdisplay in {}
\ctsdisplay ni {}
\ctsdisplay owns {}
\ctsdisplay prec {}
\ctsdisplay preceq {}
\ctsdisplay succ {}
\ctsdisplay succeq {}
\ctsdisplay bowtie {}
\ctsdisplay propto {}
\ctsdisplay approx {}
\ctsdisplay sim {}
\ctsdisplay simeq {}
\ctsdisplay frown {}
\ctsdisplay smile {}
\ctsdisplay subset {}
\ctsdisplay subseteq {}
\ctsdisplay supset {}
\ctsdisplay supseteq {}
\ctsdisplay sqsubseteq {}
\ctsdisplay sqsupseteq {}
\egroup
\explain
这些命令可以排印各种关系符号。
关系符号是 \TeX\ 的数学符号中的\minref{类}之一。
\TeX\ 在不同的\minref{类}之间插入不同的空白长度。
当 \TeX\ 需要在一个数学公式处断行，\minrefs{断行}
它会考虑在一个关系符后进行断行---不过仅在它出现在公式的最外层时，而不是在一个组中。

除了这里列出的命令以外，\TeX\ 也把 `|=|' \chidxref{\=} 和``箭头'' 命令 (\xref{arrows}) 作为关系运算符。

一些关系符有多种命令表达方式，你可以使用任何一个来排印它们：
\ulist \compact
\li `$\ge$' (|\ge| 和 |\geq|).
\li `$\le$' (|\le| 和 |\leq|).
\li `$\ne$' (|\ne|, |\neq|, 和 |\not=|).
\li `$\ni$' (|\ni| 和 |\owns|).
\endulist

\xrdef{\not}
在这些符号前加上 |\not|, 可以排印它们的非运算：

\nobreak
\threecolumns 21
\basicdisplay {$\not\asymp$}{\\not\\asymp}\ctsidxref{asymp}
\basicdisplay {$\not\cong$}{\\not\\cong}\ctsidxref{cong}
\basicdisplay {$\not\equiv$}{\\not\\equiv}\ctsidxref{equiv}
\basicdisplay {$\not=$}{\\not=}\chidxref{\=}
\basicdisplay {$\not\ge$}{\\not\\ge}\ctsidxref{ge}
\basicdisplay {$\not\geq$}{\\not\\geq}\ctsidxref{geq}
\basicdisplay {$\not\le$}{\\not\\le}\ctsidxref{le}
\basicdisplay {$\not\leq$}{\\not\\leq}\ctsidxref{leq}
\basicdisplay {$\not\prec$}{\\not\\prec}\ctsidxref{prec}
\basicdisplay {$\not\preceq$}{\\not\\preceq}\ctsidxref{preceq}
\basicdisplay {$\not\succ$}{\\not\\succ}\ctsidxref{succ}
\basicdisplay {$\not\succeq$}{\\not\\succeq}\ctsidxref{succeq}
\basicdisplay {$\not\approx$}{\\not\\approx}\ctsidxref{approx}
\basicdisplay {$\not\sim$}{\\not\\sim}\ctsidxref{sim}
\basicdisplay {$\not\simeq$}{\\not\\simeq}\ctsidxref{simeq}
\basicdisplay {$\not\subset$}{\\not\\subset}\ctsidxref{subset}
\basicdisplay {$\not\subseteq$}{\\not\\subseteq}\ctsidxref{subseteq}
\basicdisplay {$\not\supset$}{\\not\\supset}\ctsidxref{supset}
\basicdisplay {$\not\supseteq$}{\\not\\supseteq}\ctsidxref{supseteq}
\basicdisplay {$\not\sqsubseteq$}{\\not\\sqsubseteq}%
   \ctsidxref{sqsubseteq}
\basicdisplay {$\not\sqsupseteq$}{\\not\\sqsupseteq}%
   \ctsidxref{sqsupseteq}
\egroup

\example
我们可以得到 $AB \perp AC$,且
$\triangle ABF \not\sim \triangle ACF$.
|
\produces
我们可以得到 $AB \perp AC$,且
$\triangle ABF \not\sim \triangle ACF$.

\eix^^{关系符}
\endexample
\enddesc

%==========================================================================
%\subsection {Left and right delimiters}
\subsection {左右定界符}

%\begindesc
%\bix^^{delimiters}
%%
%\dothreecolumns 12
%\easy\ctsdisplay lbrace {}
%\ctsydisplay { @lbrace {}
%\ctsdisplay rbrace {}
%\ctsydisplay } @rbrace {}
%\ctsdisplay lbrack {}
%\ctsdisplay rbrack {}
%\ctsdisplay langle {}
%\ctsdisplay rangle {}
%\ctsdisplay lceil {}
%\ctsdisplay rceil {}
%\ctsdisplay lfloor {}
%\ctsdisplay rfloor {}
%\egroup
%\explain
%These commands produce left and right \minref{delimiter}s.
%Mathematicians use delimiters to indicate the boundaries between parts
%of a formula.  Left delimiters are also called ``^{opening}s'', and
%right delimiters are also called ``^{closing}s''.  Openings and closings
%are two of \TeX's \minref{class}es of math symbols.  \TeX\ puts
%different amounts of space around different \minref{class}es of math
%symbols. You might expect the space that \TeX\ puts around openings and
%closings to be symmetrical, but in fact it isn't.
\begindesc
\bix^^{定界符}
%
\dothreecolumns 12
\easy\ctsdisplay lbrace {}
\ctsydisplay \{ @lbrace {}
\ctsdisplay rbrace {}
\ctsydisplay \} @rbrace {}
\ctsdisplay lbrack {}
\ctsdisplay rbrack {}
\ctsdisplay langle {}
\ctsdisplay rangle {}
\ctsdisplay lceil {}
\ctsdisplay rceil {}
\ctsdisplay lfloor {}
\ctsdisplay rfloor {}
\egroup
\explain
这些命令排印各种左右\minref{定界符}。
数学家用定界符指明公式各部分的边界。
左定界符又称为``^{开符号}''，右定界符又称为``^{闭符号}''。
开符号和闭符号是 \TeX\ 数学公式中的两种字符类。
\TeX\ 在不同\minref{类}的数学符号之间留下不同大小的间隔。
你也许认为在开符号和闭符号旁边的间隔是对称的，但实际上并非如此。

%Some left and right delimiters have more than one command that you can
%use to produce them:
有些左定界符和右定界符可以用不止一个命令排印：

%\ulist\compact
%\li `$\{$' (|\lbrace| and |\{|)
%\li `$\}$' (|\rbrace| and |\}|)
%\li `$[$' (|\lbrack| and `|[|')
%\li `$]$' (|\rbrack| and `|]|')
%\endulist
%\noindent You can also use the left and right bracket characters
%(in either form) outside of math mode.
\ulist\compact
\li `$\{$' (|\lbrace| 和 |\{|)
\li `$\}$' (|\rbrace| 和 |\}|)
\li `$[$' (|\lbrack| 和 `|[|')
\li `$]$' (|\rbrack| 和 `|]|')
\endulist
\noindent 左右方括号（两种形式皆可）在数学模式之外也可以使用。

%In addition to these commands, \TeX\ treats `|(|' as a left
%delimiter and `|)|' as a right delimiter.
除这些命令之外，\TeX\ 还将 `|(|' 视为左定界符，将 `|)|' 视为右定界符。

%You can have \TeX\
%choose the size for a delimiter by using |\left| and |\right| (\xref\left).
%Alternatively,
%you can get a delimiter of a specific size by using one of the |\big|$x$
%commands (see |\big| et al., \xref{\big}).
利用 |\left| 和 |\right|（\xref\left ）命令，
你可以让 \TeX\ 选择定界符的尺寸。
或者利用某个 |\big|$x$ 命令（见 |\big| 等，\xref{\big}），
你可以选择特定尺寸的定界符。

%\example
%The set $\{\,x \mid x>0\,\}$ is empty.
%|
%\produces
%The set $\{\,x \mid x>0\,\}$ is empty.
\example
集合 $\{\,x \mid x>0\,\}$ 是空集。
|
\produces
集合 $\{\,x \mid x>0\,\}$ 是空集。

%\eix^^{delimiters}
%\endexample
%\enddesc
\eix^^{定界符}
\endexample
\enddesc

%==========================================================================
%\subsection {Arrows}
\subsection {箭头}

%\begindesc
%\bix^^{arrows}
%\xrdef{arrows}
%%
%{\symbolspace=24pt \makecolumns 34/2:
%\easy%
%\ctsdisplay leftarrow {}
%\ctsdisplay gets {}
%\ctsdisplay Leftarrow {}
%\ctsdisplay rightarrow {}
%\ctsdisplay to {}
%\ctsdisplay Rightarrow {}
%\ctsdisplay leftrightarrow {}
%\ctsdisplay Leftrightarrow {}
%\ctsdisplay longleftarrow {}
%\ctsdisplay Longleftarrow {}
%\ctsdisplay longrightarrow {}
%\ctsdisplay Longrightarrow {}
%\ctsdisplay longleftrightarrow {}
%\ctsdisplay Longleftrightarrow {}
%\basicdisplay {$\Longleftrightarrow$}{\\iff}\pix\ctsidxref{iff}\xrdef{\iff}
%\ctsdisplay hookleftarrow {}
%\ctsdisplay hookrightarrow {}
%\ctsdisplay leftharpoondown {}
%\ctsdisplay rightharpoondown {}
%\ctsdisplay leftharpoonup {}
%\ctsdisplay rightharpoonup {}
%\ctsdisplay rightleftharpoons {}
%\ctsdisplay mapsto {}
%\ctsdisplay longmapsto {}
%\ctsdisplay downarrow {}
%\ctsdisplay Downarrow {}
%\ctsdisplay uparrow {}
%\ctsdisplay Uparrow {}
%\ctsdisplay updownarrow {}
%\ctsdisplay Updownarrow {}
%\ctsdisplay nearrow {}
%\ctsdisplay searrow {}
%\ctsdisplay nwarrow {}
%\ctsdisplay swarrow {}
%}
%\explain
%These commands provide arrows of different kinds.  They
%are classified as relations (\xref{relations}).
%The vertical arrows in the list are also \minref{delimiter}s, so you can make
%them larger by using |\big| et al.\ (\xref \big).
\begindesc
\bix^^{箭头}
\xrdef{arrows}
%
{\symbolspace=24pt \makecolumns 34/2:
\easy%
\ctsdisplay leftarrow {}
\ctsdisplay gets {}
\ctsdisplay Leftarrow {}
\ctsdisplay rightarrow {}
\ctsdisplay to {}
\ctsdisplay Rightarrow {}
\ctsdisplay leftrightarrow {}
\ctsdisplay Leftrightarrow {}
\ctsdisplay longleftarrow {}
\ctsdisplay Longleftarrow {}
\ctsdisplay longrightarrow {}
\ctsdisplay Longrightarrow {}
\ctsdisplay longleftrightarrow {}
\ctsdisplay Longleftrightarrow {}
\basicdisplay {$\Longleftrightarrow$}{\\iff}\pix\ctsidxref{iff}\xrdef{\iff}
\ctsdisplay hookleftarrow {}
\ctsdisplay hookrightarrow {}
\ctsdisplay leftharpoondown {}
\ctsdisplay rightharpoondown {}
\ctsdisplay leftharpoonup {}
\ctsdisplay rightharpoonup {}
\ctsdisplay rightleftharpoons {}
\ctsdisplay mapsto {}
\ctsdisplay longmapsto {}
\ctsdisplay downarrow {}
\ctsdisplay Downarrow {}
\ctsdisplay uparrow {}
\ctsdisplay Uparrow {}
\ctsdisplay updownarrow {}
\ctsdisplay Updownarrow {}
\ctsdisplay nearrow {}
\ctsdisplay searrow {}
\ctsdisplay nwarrow {}
\ctsdisplay swarrow {}
}
\explain
这些命令提供各种箭头。它们被划分为关系符号（\xref{relations}）。
上面的竖直箭头同时也是\minref{定界符}，
因此你可以用 |\big| 等命令让它们变大（\xref \big ）。

%The command |\iff| differs from |\Longleftrightarrow| in that
%it produces extra space to the left and right of the arrow.
命令 |\iff| 和 |\Longleftrightarrow| 的差别之处在于，
它在箭头两边生成额外间隔。

%You can place symbols or other legends on top of a left or right arrow
%with |\buildrel| (\xref \buildrel).
你可以用 |\buildrel|（\xref \buildrel ）命令将符号或者其他文字放在箭头上边。

%\example
%$$f(x)\mapsto f(y) \iff x \mapsto y$$
%|
%\dproduces
%$$f(x)\mapsto f(y) \iff x \mapsto y$$
\example
$$f(x)\mapsto f(y) \iff x \mapsto y$$
|
\dproduces
$$f(x)\mapsto f(y) \iff x \mapsto y$$

%\eix^^{arrows}
%\endexample
%\enddesc
\eix^^{箭头}
\endexample
\enddesc

%==========================================================================
%\subsection {Named mathematical functions}
\subsection {已命名的数学函数}

%\begindesc
%\xrdef{namedfns}
%\bix^^{functions, names of}
%{\symbolspace = 36pt
%\threecolumns 32
%\easy\ctsdisplay cos {}
%\ctsdisplay sin {}
%\ctsdisplay tan {}
%\ctsdisplay cot {}
%\ctsdisplay csc {}
%\ctsdisplay sec {}
%\ctsdisplay arccos {}
%\ctsdisplay arcsin {}
%\ctsdisplay arctan {}
%\ctsdisplay cosh {}
%\ctsdisplay coth {}
%\ctsdisplay sinh {}
%\ctsdisplay tanh {}
%\ctsdisplay det {}
%\ctsdisplay dim {}
%\ctsdisplay exp {}
%\ctsdisplay ln {}
%\ctsdisplay log {}
%\ctsdisplay lg {}
%\ctsdisplay arg {}
%\ctsdisplay deg {}
%\ctsdisplay gcd {}
%\ctsdisplay hom {}
%\ctsdisplay ker {}
%\ctsdisplay inf {}
%\ctsdisplay sup {}
%\ctsdisplay lim {}
%\ctsdisplay liminf {}
%\ctsdisplay limsup {}
%\ctsdisplay max {}
%\ctsdisplay min {}
%\ctsdisplay Pr {}
%\egroup}
%\explain
%These commands set the names of various mathematical functions
%in roman type, as is customary.
%If you apply a superscript or subscript to one of these commands,
%\TeX\ will in most cases typeset it in the usual place.
%In display style, \TeX\ typesets superscripts and subscripts
%on |\det|, |\gcd|, |\inf|, |\lim|, |\liminf|,
%|\limsup|, |\max|, |\min|, |\Pr|, and |\sup|
%as though they were limits,
%i.e., directly above or directly below the function name.
\begindesc
\xrdef{namedfns}
\bix^^{函数名称}
{\symbolspace = 36pt
\threecolumns 32
\easy\ctsdisplay cos {}
\ctsdisplay sin {}
\ctsdisplay tan {}
\ctsdisplay cot {}
\ctsdisplay csc {}
\ctsdisplay sec {}
\ctsdisplay arccos {}
\ctsdisplay arcsin {}
\ctsdisplay arctan {}
\ctsdisplay cosh {}
\ctsdisplay coth {}
\ctsdisplay sinh {}
\ctsdisplay tanh {}
\ctsdisplay det {}
\ctsdisplay dim {}
\ctsdisplay exp {}
\ctsdisplay ln {}
\ctsdisplay log {}
\ctsdisplay lg {}
\ctsdisplay arg {}
\ctsdisplay deg {}
\ctsdisplay gcd {}
\ctsdisplay hom {}
\ctsdisplay ker {}
\ctsdisplay inf {}
\ctsdisplay sup {}
\ctsdisplay lim {}
\ctsdisplay liminf {}
\ctsdisplay limsup {}
\ctsdisplay max {}
\ctsdisplay min {}
\ctsdisplay Pr {}
\egroup}
\explain
这些命令以惯用的罗马字体排印各种数学函数的名称。
如果你给这些命令中的任何一个加上上标或下标，
\TeX\ 将在通常的位置排版它。
在陈列样式中，对于 |\det|、|\gcd|、|\inf|、|\lim|、|\liminf|、
|\limsup|、|\max|、|\min|、|\Pr| 和 |\sup|，
\TeX\ 将上标和下标当成极限那样排版，
即将它们直接放在函数名的上边或下边。

%\example
%$\cos^2 x + \sin^2 x = 1\qquad\max_{a \in A} g(a) = 1$
%|
%\produces
%$\cos^2 x + \sin^2 x = 1\qquad\max_{a \in A} g(a) = 1$
%\endexample\enddesc
\example
$\cos^2 x + \sin^2 x = 1\qquad\max_{a \in A} g(a) = 1$
|
\produces
$\cos^2 x + \sin^2 x = 1\qquad\max_{a \in A} g(a) = 1$
\endexample\enddesc

%\begindesc
%\cts bmod {}
%\explain
%This command produces a binary operation for indicating a ^{modulus}
%within a formula.
%\example
%$$x = (y+1) \bmod 2$$
%|
%\dproduces
%$$x = (y+1) \bmod 2$$
%\endexample
%\enddesc
\begindesc
\cts bmod {}
\explain
此命令排印一个标明公式内的^{模运算}的二元运算符。
\example
$$x = (y+1) \bmod 2$$
|
\dproduces
$$x = (y+1) \bmod 2$$
\endexample
\enddesc

%\begindesc
%\cts pmod {}
%\explain
%This command provides a notation for indicating a ^{modulus} in parentheses
%at the end of a formula.
%\example
%$$x \equiv y+1 \pmod 2$$
%|
%\dproduces
%$$x \equiv y+1 \pmod 2$$
\begindesc
\cts pmod {}
\explain
此命令在公式末尾排印放在圆括号中的^{模运算}。
\example
$$x \equiv y+1 \pmod 2$$
|
\dproduces
$$x \equiv y+1 \pmod 2$$

%\eix^^{functions, names of}
%\endexample
%\enddesc
\eix^^{函数名称}
\endexample
\enddesc


%==========================================================================
%\subsection {Large operators}
\subsection {巨算符}

%\begindesc
%\bix^^{operators//large}
%\threecolumns 15
%\easy\ctsdoubledisplay bigcap {}
%\ctsdoubledisplay bigcup {}
%\ctsdoubledisplay bigodot {}
%\ctsdoubledisplay bigoplus {}
%\ctsdoubledisplay bigotimes {}
%\ctsdoubledisplay bigsqcup {}
%\ctsdoubledisplay biguplus {}
%\ctsdoubledisplay bigvee {}
%\ctsdoubledisplay bigwedge {}
%\ctsdoubledisplay coprod {}
%{\symbolspace = 42pt\basicdisplay {\hskip 26pt$\smallint$}%
%   {\\smallint}\ddstrut}%
%   \xrdef{\smallint} \pix\ctsidxref{smallint}
%\ctsdoubledisplay int {}
%\ctsdoubledisplay oint {}
%\ctsdoubledisplay prod {}
%\ctsdoubledisplay sum {}
%}
%\explain
%These commands produce various large operator symbols.
%\TeX\ produces the smaller size when it's in ^{text style}
%\minrefs{math mode} and the larger size when it's in ^{display style}.
%Operators are one of \TeX's \minref{class}es of math symbols.
%\TeX\ puts different amounts of space
%around different classes of math symbols.
\begindesc
\bix^^{运算符//巨算符}
\threecolumns 15
\easy\ctsdoubledisplay bigcap {}
\ctsdoubledisplay bigcup {}
\ctsdoubledisplay bigodot {}
\ctsdoubledisplay bigoplus {}
\ctsdoubledisplay bigotimes {}
\ctsdoubledisplay bigsqcup {}
\ctsdoubledisplay biguplus {}
\ctsdoubledisplay bigvee {}
\ctsdoubledisplay bigwedge {}
\ctsdoubledisplay coprod {}
{\symbolspace = 42pt\basicdisplay {\hskip 26pt$\smallint$}%
   {\\smallint}\ddstrut}%
   \xrdef{\smallint} \pix\ctsidxref{smallint}
\ctsdoubledisplay int {}
\ctsdoubledisplay oint {}
\ctsdoubledisplay prod {}
\ctsdoubledisplay sum {}
}
\explain
这些命令排印各种巨算符。
\TeX\ 在^{文内样式}中排印小号字符，
\minrefs{math mode}而在^{陈列样式}中排印大号字符。
巨算符是 \TeX\ 数学符号的其中一\minref{类}。
\TeX\ 在不同类数学符号间留下不同大小的间隔。

%The large operator symbols with `|big|' in their names are different
%from the corresponding binary operations (see \xref{binops}) such as
%|\cap| ($\cap$) since they usually appear at the beginning
%of a formula.  \TeX\ uses different spacing for a large operator
%than it does for a binary operation.
名称中带有 `|big|' 的巨算符和对应的二元运算符%
（比如 |\cap| ($\cap$)，见\xref{binops}）不同，
因为它们通常出现公式的开头。
\TeX\ 给巨算符留下的间隔与二元运算符的不同。

%Don't confuse `$\sum$' (|\sum|) with `$\Sigma$'^^|\Sigma| (|\Sigma|)
%or confuse `$\prod$' (|\prod|) with `$\Pi$' ^^|\Pi| (|\Pi|).
%|\Sigma| and |\Pi| produce capital Greek letters, which are smaller and
%have a different appearance.
不要混淆 `$\sum$' (|\sum|) 和 `$\Sigma$'^^|\Sigma| (|\Sigma|)，
或者 `$\prod$' (|\prod|) 和 `$\Pi$' ^^|\Pi| (|\Pi|)。
|\Sigma| 和 |\Pi| 排印大写希腊字母，它们尺寸更小，外观也不同。

%A large operator can have ^{limits}.  The lower limit is specified as a
%subscript and the upper limit as a superscript.
巨算符可以带有^{极限}。下极限用下标指定，而上极限用上标指定。

%\example
%$$\bigcap_{k=1}^r (a_k \cup b_k)$$
%|
%\dproduces
%$$\bigcap_{k=1}^r (a_k \cup b_k)$$
%\endexample
%\interexampleskip
%\example
%$${\int_0^\pi \sin^2 ax\,dx} = {\pi \over 2}$$
%|
%\dproduces
%$${\int_0^\pi \sin^2 ax\,dx} = {\pi \over 2}$$
%\endexample
%\enddesc
\example
$$\bigcap_{k=1}^r (a_k \cup b_k)$$
|
\dproduces
$$\bigcap_{k=1}^r (a_k \cup b_k)$$
\endexample
\interexampleskip
\example
$${\int_0^\pi \sin^2 ax\,dx} = {\pi \over 2}$$
|
\dproduces
$${\int_0^\pi \sin^2 ax\,dx} = {\pi \over 2}$$
\endexample
\enddesc

%\begindesc
%\cts limits {}
%\explain
%In text style, \TeX\ normally places limits after a large operator.
%This command tells \TeX\ to place limits above and below a large
%operator rather than after it.
\begindesc
\cts limits {}
\explain
在文内样式中，\TeX\ 通常将极限放在巨算符后边。
此命令让 \TeX\ 将极限放在巨算符的上边和下边，而不是在后边。

%If you specify more than one of |\limits|, |\nolimits|,
%and |\display!-limits|, the last command rules.
如果你多次使用 |\limits|、|\nolimits| 或 |\display!-limits|，
仅最后一个命令生效。

%\example
%Suppose that $\bigcap\limits_{i=1}^Nq_i$ contains at least
%two elements.
%|
%\produces
%Suppose that $\bigcap\limits_{i=1}^Nq_i$ contains at least
%two elements.
%\endexample
%\enddesc
\example
Suppose that $\bigcap\limits_{i=1}^Nq_i$ contains at least
two elements.
|
\produces
Suppose that $\bigcap\limits_{i=1}^Nq_i$ contains at least
two elements.
\endexample
\enddesc

%\begindesc
%\cts nolimits {}
%\explain
%In display style, \TeX\ normally places limits above and below a large
%operator.  This command tells \TeX\ to place limits after a large
%operator rather than above and below it.
\begindesc
\cts nolimits {}
\explain
在陈列样式中，\TeX\ 通常将极限放在巨算符的上边和下边。%
此命令让 \TeX\ 将极限放在巨算符后边，而不是上边和下边。

%The integral operators |\int| and |\oint| are exceptions---\TeX\ places
%limits after them in all cases, unless overridden, as in |\int\limits|.
%(\plainTeX\ defines ^|\int| and ^|\oint| as macros that specify the
%operator symbol followed by |\nolimits|---this is what causes them to
%behave differently by default.)
%^^|\int//limits after|
积分算符 |\int| 和 |\oint| 是两个例外——除非像 |\int\limits| 这样指定，
\TeX\ 总是将极限放在算符的后边。（\plainTeX\ 将 ^|\int| 和 ^|\oint|
定义为算符后跟着 |\nolimits| 的宏——这就是其默认表现有所不同的原因。）%
^^|\int//极限放在后面|

%If you specify more than one of |\limits|, |\nolimits|,
%and |\display!-limits|, the last command rules.
如果你多次使用 |\limits|、|\nolimits| 或 |\display!-limits|，
仅最后一个命令生效。

%\example
%$$\bigcap\nolimits_{i=1}^Nq_i$$
%|
%\dproduces
%$$\bigcap\nolimits_{i=1}^Nq_i$$
%\endexample
%\enddesc
\example
$$\bigcap\nolimits_{i=1}^Nq_i$$
|
\dproduces
$$\bigcap\nolimits_{i=1}^Nq_i$$
\endexample
\enddesc

%\begindesc
%\cts displaylimits {}
%\explain
%This command tells \TeX\ to place limits above and below all operators
%(including the integrals) if in display style, and after all operators
%if in text style.
\begindesc
\cts displaylimits {}
\explain
此命令让 \TeX\ 在陈列样式中将极限放在算符（包括积分符）的上边和下边，
在文内样式中将极限放在算符的后边。

%If you specify more than one of |\limits|, |\nolimits|,
%and |\display!-limits|, the last command rules.
如果你多次使用 |\limits|、|\nolimits| 或 |\display!-limits|，
仅最后一个命令生效。

%\example
%$$a(\lambda) = {1 \over {2\pi}} \int\displaylimits
%_{-\infty}^{+\infty} f(x)e^{-i\lambda x}\,dx$$
%|
%\dproduces
%$$a(\lambda) = {1 \over {2\pi}} \int\displaylimits
%_{-\infty}^{+\infty} f(x)e^{-i\lambda x}\,dx$$
\example
$$a(\lambda) = {1 \over {2\pi}} \int\displaylimits
_{-\infty}^{+\infty} f(x)e^{-i\lambda x}\,dx$$
|
\dproduces
$$a(\lambda) = {1 \over {2\pi}} \int\displaylimits
_{-\infty}^{+\infty} f(x)e^{-i\lambda x}\,dx$$

%\eix^^{operators//large}
%\endexample
%\enddesc
\eix^^{运算符//巨算符}
\endexample
\enddesc


%==========================================================================
%\subsection {Punctuation}
\subsection {标点}

%\begindesc
%\bix^^{punctuation in math formulas}
%\cts cdotp {}
%\cts ldotp {}
%\explain
%These two commands respectively produce a centered dot and a dot
%positioned on the \minref{baseline}.  They are valid only in math
%\minref{mode}.  \TeX\ treats them as punctuation, putting no extra space in
%front of them but a little extra space after them.
%In contrast, \TeX\ puts an equal amount of space on both sides
%of a centered dot generated by the ^|\cdot| command (\xref \cdot).
%\example
%$x \cdotp y \quad x \ldotp y \quad x \cdot y$
%|
%\produces
%$x \cdotp y \quad x \ldotp y \quad x \cdot y$
%\endexample
%\enddesc
\begindesc
\bix^^{数学公式中的标点}
\cts cdotp {}
\cts ldotp {}
\explain
这两个命令分别排印居中的圆点和在\minref{基线}上的圆点。
它们仅可用于数学\minref{模式}中。
\TeX\ 将它们视为标点，在前面不留间隔而在后面留下一点间隔。
与此相反，对于用 ^|\cdot| 命令（\xref\cdot ）生成的居中圆点，
\TeX\ 在其两侧留下相同大小的间隔。
\example
$x \cdotp y \quad x \ldotp y \quad x \cdot y$
|
\produces
$x \cdotp y \quad x \ldotp y \quad x \cdot y$
\endexample
\enddesc

%\begindesc
%\cts colon {}
%\explain
%This command produces a colon punctation symbol.
%It is valid only in math mode.
%The difference between |\colon| and the colon character (|:|) is that
%`|:|' is an operator, so \TeX\ puts extra space to the left of it whereas
%it doesn't put extra space to the left of |\colon|.
%\example
%$f \colon t \quad f : t$
%|
%\produces
%$f \colon t \quad f : t$
\begindesc
\cts colon {}
\explain
此命令排印一个冒号标点，它只能用在数学模式中。
冒号标点 |\colon| 和冒号字符(|:|)的区别在于，
`|:|' 是一个运算符，因此 \TeX\ 在其左侧留下额外间隔，
然而在 |\colon| 左侧却不留额外间隔。
\example
$f \colon t \quad f : t$
|
\produces
$f \colon t \quad f : t$

%\eix^^{punctuation in math formulas}
%\endexample
%\enddesc
\eix^^{数学公式中的标点}
\endexample
\enddesc


%==========================================================================
%\secondprinting{\vfill\eject\null\vglue-30pt\vskip0pt}
%\section {Superscripts and subscripts}
\section {上标和下标}

%\begindesc
%\margin{Two groups of commands have been combined here.}
%\bix^^{superscripts}
%\bix^^{subscripts}
%\secondprinting{\vglue-12pt}
%\makecolumns 4/2:
%\easy\ctsact _ \xrdef{@underscore} {\<argument>}
%\cts sb {\<argument>}
%\ctsact ^ \xrdef{@hat} {\<argument>}
%\cts sp {\<argument>}
%\secondprinting{\vglue-4pt}
%\explain
%The commands in each column are equivalent.  The commands in the first
%column typeset \<argument> as a subscript, and those in the second
%column typeset \<argument> as a superscript.  The |\sb| and |\sp|
%commands are mainly useful if you're working on a terminal that lacks an
%underscore or caret, or if you've redefined `|_|' or `|^|' and need
%access to the original definition.  These commands are also used for
%setting lower and upper limits on summations and integrals.  ^^{lower
%limits} ^^{upper limits}
\begindesc
\margin{Two groups of commands have been combined here.}
\bix^^{上标}
\bix^^{下标}
\secondprinting{\vglue-12pt}
\makecolumns 4/2:
\easy\ctsact \_ \xrdef{@underscore} {\<argument>}
\cts sb {\<argument>}
\ctsact \^ \xrdef{@hat} {\<argument>}
\cts sp {\<argument>}
\secondprinting{\vglue-4pt}
\explain
各栏的两个命令都是等价的。第一栏的命令将 \<argument> 排版为下标，
而第二栏的命令将 \<argument> 排版为上标。
|\sb| 和 |\sp| 命令主要用于无法使用下划线和插入符的终端中，
或者用在重新定义了 `|_|' or `|^|' 但需要其原始定义的情况下。
这些命令也用于设定求和号和积分号的下极限和上极限。
^^{下极限} ^^{上极限}

%If a subscript or superscript is not a single \minref{token}, you need
%to enclose it in a \minref{group}.  \TeX\ does not prioritize subscripts
%or superscripts, so it will reject formulas such as |a_i_j|, |a^i^j|, or
%|a^i_j|.
如果下标或上标不是单个\minref{记号}，你需要将它放在\minref{编组}中。
\TeX\ 并不处理下标和上标的优先级，
因此它将拒绝类似 |a_i_j|、|a^i^j| 或 |a^i_j| 的公式。

%Subscripts and superscripts are normally typeset in ^{script style}, or
%in ^{scriptscript style} if they are second-order, e.g., a subscript on
%a subscript or a superscript on a a subscript.  You can set \emph{any}
%text in a math formula in a script or scriptscript \minref{style} with
%the ^|\scriptstyle| and ^|\scriptscriptstyle| commands (\xref
%\scriptscriptstyle).
下标和上标排版时通常用^{标号样式}，或者^{小标号样式}，
如果它们是二阶标号，比如下标中的下标或下标中的上标。
利用 ^|\scriptstyle| 和 ^|\scriptscriptstyle| 命令（\xref\scriptscriptstyle ），
你可以将数学公式的\emph{任何}文本设为标号或小标号\minref{样式}。

%You can apply a subscript or superscript to any of the commands that
%produce named mathematical functions in roman type (see
%\xref{namedfns}).  In certain cases (again, see \xref{namedfns}) the
%subscript or superscript appears directly above or under the function
%name as shown in the examples of ^|\lim| and ^|\det| below.
对任何以罗马字体排印命名数学函数（见\xref{namedfns}）的命令，
你都可以给它添加下标和上标。
在某些情形中（同样见\xref{namedfns}），
下标和上标分别出现在函数名的下边和上边，
如下面例子中的 ^|\lim| 和 ^|\det| 所示。

%\example
%$x_3 \quad t_{\max} \quad a_{i_k} \quad \sum_{i=1}^n{q_i}
%   \quad x^3\quad e^{t \cos\theta}\quad r^{x^2}\quad
%   \int_0^\infty{f(x)\,dx}$
%$$\lim_{x\leftarrow0}f(x)\qquad\det^{z\in A}\qquad\sin^2t$$
%|
%\produces
%\secondprinting{\divide\abovedisplayskip by 2}
%$x_3 \quad t_{\max} \quad a_{i_k} \quad \sum_{i=1}^n{q_i}
%   \quad x^3\quad e^{t \cos\theta}\quad r^{x^2}\quad
%   \int_0^\infty{f(x)\,dx}$
%$$\lim_{x \leftarrow 0} f(x)\qquad
%   \det^{z \in A}\qquad \sin^2 t$$
\example
$x_3 \quad t_{\max} \quad a_{i_k} \quad \sum_{i=1}^n{q_i}
   \quad x^3\quad e^{t \cos\theta}\quad r^{x^2}\quad
   \int_0^\infty{f(x)\,dx}$
$$\lim_{x\leftarrow0}f(x)\qquad\det^{z\in A}\qquad\sin^2t$$
|
\produces
%\secondprinting{\divide\abovedisplayskip by 2}
$x_3 \quad t_{\max} \quad a_{i_k} \quad \sum_{i=1}^n{q_i}
   \quad x^3\quad e^{t \cos\theta}\quad r^{x^2}\quad
   \int_0^\infty{f(x)\,dx}$
$$\lim_{x \leftarrow 0} f(x)\qquad
   \det^{z \in A}\qquad \sin^2 t$$

%\eix^^{superscripts}
%\eix^^{subscripts}
%\endexample
%\enddesc
\eix^^{上标}
\eix^^{下标}
\endexample
\enddesc

%\secondprinting{\vfill\eject}

%==========================================================================
%\subsection {Selecting and using styles}
\subsection {选用样式}

%\begindesc
%\bix^^{styles}
%\cts textstyle {}
%\cts scriptstyle {}
%\cts scriptscriptstyle {}
%\cts displaystyle {}
%\explain
%^^{text style} ^^{script style} ^^{scriptscript style} ^^{display style}
%These commands override the normal \minref{style} and hence the
%font that \TeX\ uses in setting a formula.  Like
%font-setting commands such as |\it|, they are in
%effect until the end of the group containing them.
%They are useful when \TeX's choice of style is inappropriate for the formula
%you happen to be setting.
%\example
%$t+{\scriptstyle t + {\scriptscriptstyle t}}$
%|
%\produces
%$t+{\scriptstyle t + {\scriptscriptstyle t}}$
%\endexample
%\enddesc
\begindesc
\bix^^{样式}
\cts textstyle {}
\cts scriptstyle {}
\cts scriptscriptstyle {}
\cts displaystyle {}
\explain
^^{文本样式} ^^{标号样式} ^^{小标号样式} ^^{陈列样式}
这些命令覆盖 \TeX\ 排版公式时通常使用的\minref{样式}及其字体。
如同类似 |\it| 的字体设置命令，它们在其所在编组结束前一直有效。
当 \TeX\ 给你要排版的公式选用了不合适的样式时，你可以使用这些命令。
\example
$t+{\scriptstyle t + {\scriptscriptstyle t}}$
|
\produces
$t+{\scriptstyle t + {\scriptscriptstyle t}}$
\endexample
\enddesc

%\begindesc
%\cts mathchoice {%
%   \rqbraces{\<math$_1$>}
%   \rqbraces{\<math$_2$>}
%   \rqbraces{\<math$_3$>}
%   \rqbraces{\<math$_4$>}}
%\explain
%This command tells \TeX\ to typeset one of the subformulas
%\<math$_1$>, \<math$_2$>, \<math$_3$>, or \<math$_4$>, making its choice
%according to the current \minref{style}.
%That is, if \TeX\ is in
%display style it sets the |\mathchoice| as \<math$_1$>; in text style it sets
%it as \<math$_2$>; in script style it sets it as \<math$_3$>;
%and in scriptscript style it sets it as \<math$_4$>.
%\example
%\def\mc{{\mathchoice{D}{T}{S}{SS}}}
%The strange formula $\mc_{\mc_\mc}$ illustrates a
%mathchoice.
%|
%\produces
%\def\mc{{\mathchoice{D}{T}{S}{SS}}}
%The strange formula $\mc_{\mc_\mc}$ illustrates a
%mathchoice.
%\endexample
%\enddesc
\begindesc
\cts mathchoice {%
   \rqbraces{\<math$_1$>}
   \rqbraces{\<math$_2$>}
   \rqbraces{\<math$_3$>}
   \rqbraces{\<math$_4$>}}
\explain
此命令让 \TeX\ 根据当前\minref{样式}选择并排版其中一个子公式
\<math$_1$>、\<math$_2$>、\<math$_3$> 或 \<math$_4$>。
也就是说，如果在陈列样式中，\TeX\ 将 |\mathchoice| 排版为 \<math$_1$>；
在文本样式中排版为 \<math$_2$>，在标号样式中排版为 \<math$_3$>；
而在小标号样式中排版为 \<math$_4$>。
\example
\def\mc{{\mathchoice{D}{T}{S}{SS}}}
The strange formula $\mc_{\mc_\mc}$ illustrates a
mathchoice.
|
\produces
\def\mc{{\mathchoice{D}{T}{S}{SS}}}
The strange formula $\mc_{\mc_\mc}$ illustrates a
mathchoice.
\endexample
\enddesc

%\begindesc
%\cts mathpalette {\<argument$_1$> \<argument$_2$>}
%\explain
%^^{math symbols}
%This command provides a convenient way of
%producing a math construct that works in all four \minref{style}s.
%To use it, you'll normally need to define an additional macro,
%which we'll call |\build|.
%The call on |\math!-palette| should then have the form
%|\mathpalette|\allowbreak|\build|\<argument>.
\begindesc
\cts mathpalette {\<argument$_1$> \<argument$_2$>}
\explain
^^{数学符号}
此命令提供一种生成适用于四种\minref{样式}的数学结构的简便方法。%
\footnote{译注：该宏定义为
|\def\mathpalette#1#2{\mathchoice{#1\displaystyle{#2}}|\break
|{#1\textstyle{#2}}{#1\scriptstyle{#2}}{#1\scriptscriptstyle{#2}}}|。}
要使用它，通常你需要定义一个额外的宏，假设我们称它为 |\build|。
调用 |\math!-palette| 就应该用
|\mathpalette|\allowbreak|\build|\<argument> 这种形式。


%|\build| tests what style \TeX\ is in and typesets \<argu\-ment> accordingly.
%It should be defined to have two parameters.
%When you call |\math!-palette|, it will in turn call |\build|,
%with |#1| being a
%command that selects the current style and |#2| being \<argument>.
%Thus, within the definition of |\build| you can typeset something
%in the current style by preceding it with `|#1|'.
%See \knuth{page~360} for examples of using |\mathpalette|
%and \knuth{page~151} for a further explanation of how it works.
|\build| 测试 \TeX\ 位于何种样式，并相应地排版 \<argu\-ment>。
它应该定义为有两个参数。
当你调用 |\math!-palette| 时，它以 |#1| 为选择样式的命令，
|#2| 为 \<argument> 转而调用 |\build|。
因此，在 |\build| 的定义中，
通过将某些东西放在 `|#1|' 前面，就可以用当前样式排版它。
在\knuth{第~360~页}中有如何使用 |\mathpalette| 的例子，
而在\knuth{第~151~页}中有它如何运作的进一步解释。

%\eix^^{styles}
%\enddesc
\eix^^{样式}
\enddesc

%==========================================================================
%\section {Compound symbols}
\section {复合符号}

%==========================================================================
%\subsection {Math accents}
\subsection {数学重音}

%\begindesc
%\xrdef{mathaccent}
%^^{accents}
%^^{math//accents}
%%
%\easy\ctsx acute {^{acute accent} as in $\acute x$}
%\ctsx b {^{bar-under accent} as in $\b x$}
%\ctsx bar {^{bar accent} as in $\bar x$}
%\ctsx breve {^{breve accent} as in $\breve x$}
%\ctsx check {^{check accent} as in $\check x$}
%\ctsx ddot {^{double dot accent} as in $\ddot x$}
%\ctsx dot {^{dot accent} as in $\dot x$}
%\ctsx grave {^{grave accent} as in $\grave x$}
%\ctsx hat {^{hat accent} as in $\hat x$}
%\ctsx widehat {^{wide hat accent} as in $\widehat {x+y}$}
%\ctsx tilde {^{tilde accent} as in $\tilde x$}
%\ctsx widetilde {^{wide tilde accent} as in $\widetilde {z+a}$}
%\ctsx vec {^{vector accent} as in $\vec x$}
%\explain
%These commands produce accent marks in math formulas.  You'll ordinarily
%need to leave a space after any one of them.
%A wide accent can be applied to a multicharacter subformula;
%\TeX\ will center the accent over the subformula.
%The other accents are usefully applied only to a single character.
\begindesc
\xrdef{mathaccent}
^^{重音}
^^{数学//数学重音}
%
\easy\ctsx acute {^{锐音符}，如同 $\acute x$}
\ctsx b {^{下线符}，如同 $\b x$}
\ctsx bar {^{上线符}，如同 $\bar x$}
\ctsx breve {^{短音符}，如同 $\breve x$}
\ctsx check {^{抑扬符}，如同 $\check x$}
\ctsx ddot {^{双点符}，如同 $\ddot x$}
\ctsx dot {^{上点符}，如同 $\dot x$}
\ctsx grave {^{钝音符}，如同 $\grave x$}
\ctsx hat {^{尖角符}，如同 $\hat x$}
\ctsx widehat {^{宽尖角符}，如同 $\widehat {x+y}$}
\ctsx tilde {^{波浪符}，如同 $\tilde x$}
\ctsx widetilde {^{宽波浪符}，如同 $\widetilde {z+a}$}
\ctsx vec {^{向量符}，如同 $\vec x$}
\explain
这些命令在数学公式上排印重音标记。你通常需要在它们后面留下空格。
宽重音可以应用到多字符子公式中；\TeX\ 将把重音放在子公式的中间。
其他重音仅在应用到单个字符时才有用。

%\example
%$\dot t^n \qquad \widetilde{v_1 + v_2}$
%|
%\produces
%$\dot t^n \qquad \widetilde{v_1 + v_2}$
%\endexample
\example
$\dot t^n \qquad \widetilde{v_1 + v_2}$
|
\produces
$\dot t^n \qquad \widetilde{v_1 + v_2}$
\endexample

%\begindesc
%\cts mathaccent {\<mathcode>}
%\explain
%This command tells \TeX\ to typeset a math accent
%whose family and character code are given by \<mathcode>.  (\TeX\ ignores
%the class of the \minref{mathcode}.)
%See \knuth{Appendix~G} for the details of how \TeX\ positions such an accent.
%The usual way to use |\mathaccent| is to put it in a macro definition
%that gives a name to a math accent.
%\example
%\def\acute{\mathaccent "7013}
%|
%\endexample
%\enddesc
\begindesc
\cts mathaccent {\<mathcode>}
\explain
此命令让 \TeX\ 排版字体族和字符编码由 \<mathcode> 给出的数学重音。%
（\TeX\ 忽略\minref{数学码}中的类。）
请参阅\knuth{附录~G}对 \TeX\ 如何放置该重音的详细介绍。
经常将 |\mathaccent| 放在宏定义中，以给数学重音一个名称。
\example
\def\acute{\mathaccent "7013}
|
\endexample
\enddesc

%\see ``Accents'' (\xref {accents}).
%\enddesc
\see ``Accents''（\xref {accents}）。
\enddesc

%==========================================================================
%\subsection {Fractions and other stacking operations}
\subsection {分式和其他堆叠运算}

%\begindesc
%\bix^^{fractions}
%\bix^^{stacking subformulas}
%\easy\cts over {}
%\cts atop {}
%\cts above {\<dimen>}
%\cts choose {}
%\cts brace {}
%\cts brack {}
%\explain
%{\def\fri{\<formula$_1$>}%
%\def\frii{\<formula$_2$>}%
%These commands stack one subformula on top of another one.  We will explain how
%|\over| works, and then relate the other commands to it.
\begindesc
\bix^^{分式}
\bix^^{堆叠子公式}
\easy\cts over {}
\cts atop {}
\cts above {\<dimen>}
\cts choose {}
\cts brace {}
\cts brack {}
\explain
{\def\fri{\<formula$_1$>}%
\def\frii{\<formula$_2$>}%
这些命令将一个子公式堆放在另一个子公式之上。
我们将解释 |\over| 如何作用，然后说明其他命令与它的关系。

%|\over| is the command that you'd normally use to produce a fraction.
%^^{fractions//produced by \b\tt\\over\e}
%If you write something in one of the following forms:
%\csdisplay
%$$!fri\over!frii$$
%$!fri\over!frii$
%\left!<delim>!fri\over!frii\right!<delim>
%{!fri\over!frii}
%|
%you'll get a fraction with numerator \fri\  and denominator \<for\-mu\-la$_2$>,
%i.e., \fri\ over \frii.
%In the first three of
%these forms the |\over| is not implicitly contained in a group;
%it absorbs
%everything to its left and to its right until it comes to a boundary,
%namely, the beginning or end of a group.
|\over| 命令通常用于排印分式。
^^{分式//用 \b\tt\\over\e 生成}
如果你按下面几种形式之一撰写：
\csdisplay
$$!fri\over!frii$$
$!fri\over!frii$
\left!<delim>!fri\over!frii\right!<delim>
{!fri\over!frii}
|
你将得到分子为 \fri\ 分母为 \<for\-mu\-la$_2$> 的分式，
即 \fri\ 除以 \frii 。
在前面三种形式中，|\over| 非显式地包含在一个编组中；
它吸收左边和右边的内容直到遇到边界，即编组的开头和结尾。

%You can't use |\over| or any of the other commands in this group
%more than once in a formula.
%Thus a formula such as:
%\csdisplay
%$$a \over n \choose k$$
%|
%isn't legal.
%This is not a severe restriction because
%you can always enclose one of the commands in braces.
%The reason for the restriction is that if you had two of these commands
%in a single formula, \TeX\ wouldn't know how to group them.
你不可以在一个公式中多次使用 |\over| 或这批命令的其他命令。
因此下面的公式：
\csdisplay
$$a \over n \choose k$$
|
是不合法的。这不是什么严重的限制，因为你总可以将其中一个命令放在花括号中。
作此限制的原因是，如果你把这些命令的其中两个放在同一个公式中，
\TeX\ 将不知道如何划分它们。

%The other commands are similar to |\over|, with the following exceptions:
%\ulist\compact
%\li |\atop| leaves out the fraction bar.
%\li |\above| provides a fraction bar of thickness \<dimen>.
%\li |\choose|
%leaves out the fraction bar and encloses the construct in parentheses.
%(It's called ``choose'' because $n \choose k$ is the notation for the
%number of ways of choosing $k$ things out of $n$ things.)
%\li |\brace| leaves out the fraction bar and encloses the construct in braces.
%\li |\brack|
%leaves out the fraction bar and encloses the construct in brackets.
%\endulist
%}%
%\example
%$${n+1 \over n-1}      \qquad {n+1 \atop n-1}   \qquad
%  {n+1 \above 2pt n-1} \qquad {n+1 \choose n-1} \qquad
%  {n+1 \brace n-1}     \qquad {n+1 \brack n-1}$$
%|
%\dproduces
%$${n+1 \over n-1}      \qquad {n+1 \atop n-1}   \qquad
%  {n+1 \above 2pt n-1} \qquad {n+1 \choose n-1} \qquad
%  {n+1 \brace n-1}     \qquad {n+1 \brack n-1}$$
%\endexample
%\enddesc
其他命令与 |\over| 类似，但有所不同：
\ulist\compact
\li |\atop| 去掉分式的横线。
\li |\above| 给出厚度为 \<dimen> 的分式横线。
\li |\choose| 去掉分式横线，并将结构放在圆括号中。%
（称它为``选择''，
是因为 $n \choose k$ 表示从 $n$ 个东西中任取 $k$ 个的所有选取方式的数目。）%
\li |\brace| 去掉分式横线，并将结构放在花括号中。
\li |\brack| 去掉分式横线，并将结构放在方括号中。
\endulist
}%
\example
$${n+1 \over n-1}      \qquad {n+1 \atop n-1}   \qquad
  {n+1 \above 2pt n-1} \qquad {n+1 \choose n-1} \qquad
  {n+1 \brace n-1}     \qquad {n+1 \brack n-1}$$
|
\dproduces
$${n+1 \over n-1}      \qquad {n+1 \atop n-1}   \qquad
  {n+1 \above 2pt n-1} \qquad {n+1 \choose n-1} \qquad
  {n+1 \brace n-1}     \qquad {n+1 \brack n-1}$$
\endexample
\enddesc

%\begindesc
%\cts overwithdelims {\<delim$_1$> \<delim$_2$>}
%\cts atopwithdelims {\<delim$_1$> \<delim$_2$>}
%\cts abovewithdelims {\<delim$_1$> \<delim$_2$> \<dimen>}
%\explain
%Each of these commands stacks one subformula on top of another one and
%surrounds the entire construct with \<delim$_1$> on the left and
%\<delim$_2$> on the right.  These commands follow the same rules as
%|\over|, |\atop|, and |\above|. The \<dimen> in |\abovewithdelims|
%specifies the thickness of the fraction bar.
%\example
%$${m \overwithdelims () n}\qquad
%  {m \atopwithdelims !|!| n}\qquad
%  {m \abovewithdelims \{\} 2pt n}$$
%|
%\dproduces
%$${m \overwithdelims () n}\qquad
%  {m \atopwithdelims || n}\qquad
%  {m \abovewithdelims \{\} 2pt n}$$
%\endexample
%\enddesc
\begindesc
\cts overwithdelims {\<delim$_1$> \<delim$_2$>}
\cts atopwithdelims {\<delim$_1$> \<delim$_2$>}
\cts abovewithdelims {\<delim$_1$> \<delim$_2$> \<dimen>}
\explain
这里的每个命令都将一个子公式堆放在另一个子公式之上，
并将整个结构的左边用 \<delim$_1$>，右边用 \<delim$_2$> 包围。
这些命令遵循与 |\over|、|\atop| 和 |\above| 相同的规则。
|\abovewithdelims| 后面的 \<dimen> 指定分式横线的厚度。
\example
$${m \overwithdelims () n}\qquad
  {m \atopwithdelims !|!| n}\qquad
  {m \abovewithdelims \{\} 2pt n}$$
|
\dproduces
$${m \overwithdelims () n}\qquad
  {m \atopwithdelims || n}\qquad
  {m \abovewithdelims \{\} 2pt n}$$
\endexample
\enddesc

%\begindesc
%\cts cases {}
%\explain
%^^{combinations, notation for}
%This command produces the mathematical form that denotes a choice among
%several cases.
%Each case has two parts, separated by `|&|'.
%\TeX\ treats the first part as a math formula
%and the second part as ordinary text.  Each
%case must be followed by |\cr|.
\begindesc
\cts cases {}
\explain
^^{组合数记法}
此命令排印一个表示从多个情形中选择的数学形式。
每种情形由两部分组成，两者以 `|&|' 分隔。
\TeX\ 将第一部分视为数学公式，第二部分视为普通文本。
每个情形之后必须加上 |\cr|。

%\example
%$$g(x,y) = \cases{f(x,y),&if $x<y$\cr
%                  f(y,x),&if $x>y$\cr
%                  0,&otherwise.\cr}$$
%|
%\dproduces
%$$g(x,y) = \cases{f(x,y),&if $x<y$\cr
%                  f(y,x),&if $x>y$\cr
%                  0,&otherwise.\cr}$$
%\endexample
%\enddesc
\example
$$g(x,y) = \cases{f(x,y),&if $x<y$\cr
                  f(y,x),&if $x>y$\cr
                  0,&otherwise.\cr}$$
|
\dproduces
$$g(x,y) = \cases{f(x,y),&if $x<y$\cr
                  f(y,x),&if $x>y$\cr
                  0,&otherwise.\cr}$$
\endexample
\enddesc

%\begindesc
%\cts underbrace {\<argument>}
%\cts overbrace {\<argument>}
%\cts underline {\<argument>}
%\cts overline {\<argument>}
%\cts overleftarrow {\<argument>}
%\cts overrightarrow {\<argument>}
%\explain
%These commands place extensible ^{braces}, lines, or ^{arrows}
%over or under the subformula given by \<argument>.
%\TeX\ will make these constructs as wide as they need to be for
%the context.
%When \TeX\ produces the extended braces, lines, or arrows, it considers
%only the dimensions of the \minref{box} containing \<argument>.
%If you use more than one of these commands in a single formula, the
%braces, lines, or arrows they produce
%may not line up properly with each other.
%You can use the |\mathstrut| command (\xref \mathstrut)
%to overcome this difficulty.
%\example
%$$\displaylines{
%\underbrace{x \circ y}\qquad \overbrace{x \circ y}\qquad
%\underline{x \circ y}\qquad \overline{x \circ y}\qquad
%\overleftarrow{x \circ y}\qquad
%\overrightarrow{x \circ y}\cr
%{\overline r + \overline t}\qquad
%{\overline {r \mathstrut} + \overline {t \mathstrut}}\cr
%}$$
%|
%\dproduces
%$$\displaylines{
%\underbrace{x \circ y}\qquad \overbrace{x \circ y}\qquad
%\underline{x \circ y}\qquad \overline{x \circ y}\qquad
%\overleftarrow{x \circ y}\qquad
%\overrightarrow{x \circ y}\cr
%{\overline r + \overline t}\qquad
%{\overline {r \mathstrut} + \overline {t \mathstrut}}\cr
%}$$
%\endexample
%\enddesc
\begindesc
\cts underbrace {\<argument>}
\cts overbrace {\<argument>}
\cts underline {\<argument>}
\cts overline {\<argument>}
\cts overleftarrow {\<argument>}
\cts overrightarrow {\<argument>}
\explain
这些命令将可伸长的^{花括号}、横线或^{箭头}%
放在由 \<argument> 给出的子公式的上边或下边。
\TeX\ 将让这些结构足够宽以适应内容。
当 \TeX\ 排印可伸长的花括号、横线或箭头时，
它只考虑包含 \<argument> 的 \minref{盒子}的尺寸。
如果你在一个公式中使用这些命令中的两个以上，
其中排印的花括号、横线或箭头之间可能无法恰当地对齐。
你可以使用 |\mathstrut| 命令（\xref\mathstrut ）克服此困难。
\example
$$\displaylines{
\underbrace{x \circ y}\qquad \overbrace{x \circ y}\qquad
\underline{x \circ y}\qquad \overline{x \circ y}\qquad
\overleftarrow{x \circ y}\qquad
\overrightarrow{x \circ y}\cr
{\overline r + \overline t}\qquad
{\overline {r \mathstrut} + \overline {t \mathstrut}}\cr
}$$
|
\dproduces
$$\displaylines{
\underbrace{x \circ y}\qquad \overbrace{x \circ y}\qquad
\underline{x \circ y}\qquad \overline{x \circ y}\qquad
\overleftarrow{x \circ y}\qquad
\overrightarrow{x \circ y}\cr
{\overline r + \overline t}\qquad
{\overline {r \mathstrut} + \overline {t \mathstrut}}\cr
}$$
\endexample
\enddesc

%\begindesc\secondprinting{\vglue-.5\baselineskip\vskip0pt}
%\cts buildrel {\<formula> {\bt \\over} \<relation>}
%\explain
%^^{relations//putting formulas above}
%This command produces a \minref{box} in which \<formula>
%is placed on top of \<relation>. \TeX\ treats the result as a relation
%for spacing purposes \seeconcept{class}.
%\example
%$\buildrel \rm def \over \equiv$
%|
%\produces
%$\buildrel \rm def \over \equiv$
\begindesc%\secondprinting{\vglue-.5\baselineskip\vskip0pt}
\cts buildrel {\<formula> {\bt \\over} \<relation>}
\explain
^^{关系符//将公式放在其上}
此命令将 \<formula> 所在的\minref{盒子}放在 \<relation> 上边。
\TeX\ 处理间隔时将结果视为一个关系符\seeconcept{类}。
\example
$\buildrel \rm def \over \equiv$
|
\produces
$\buildrel \rm def \over \equiv$

%\eix^^{fractions}
%\eix^^{stacking subformulas}
%\endexample
%\enddesc
\eix^^{分式}
\eix^^{堆叠子公式}
\endexample
\enddesc

%\secondprinting{\vfill\eject}


%==========================================================================
%\subsection {Dots}
\subsection {圆点}

%\begindesc
%\bix^^{dots}
%\easy\cts ldots {}
%\cts cdots {}
%\explain
%These commands produce three ^{dots} in a row.  For |\ldots|, the dots
%are on the baseline; for |\cdots|, the dots are centered with respect to
%the axis (see the explanation of |\vcenter|, \xref\vcenter).
\begindesc
\bix^^{圆点}
\easy\cts ldots {}
\cts cdots {}
\explain
这两个命令都排印三个一排的^{圆点}。对于 |\ldots|，
圆点放在基线上；对于 |\cdots|，圆点放在中轴线上%
（见 \xref\vcenter 对 |\vcenter| 的解释）。

%\example
%$t_1 + t_2 + \cdots + t_n \qquad x_1,x_2, \ldots\,, x_r$
%|
%\produces
%$t_1 + t_2 + \cdots + t_n \qquad x_1,x_2, \ldots\,, x_r$
%\endexample
%\enddesc
\example
$t_1 + t_2 + \cdots + t_n \qquad x_1,x_2, \ldots\,, x_r$
|
\produces
$t_1 + t_2 + \cdots + t_n \qquad x_1,x_2, \ldots\,, x_r$
\endexample
\enddesc

%\begindesc
%\easy\cts vdots {}
%\explain
%This command produces three vertical dots.
%\example
%$$\eqalign{f(\alpha_1)& = f(\beta_1)\cr
%   \noalign{\kern -4pt}%
%   &\phantom{a}\vdots\cr % moves the dots right a bit
%   f(\alpha_k)& = f(\beta_k)\cr}$$
%|
%\dproduces
%$$\eqalign{f(\alpha_1)& = f(\beta_1)\cr
%   \noalign{\kern -4pt}%
%   &\phantom{a}\vdots\cr
%   f(\alpha_k)& = f(\beta_k)\cr}$$
%\endexample
%\enddesc
\begindesc
\easy\cts vdots {}
\explain
此命令排印三个竖直的圆点。
\example
$$\eqalign{f(\alpha_1)& = f(\beta_1)\cr
   \noalign{\kern -4pt}%
   &\phantom{a}\vdots\cr % moves the dots right a bit
   f(\alpha_k)& = f(\beta_k)\cr}$$
|
\dproduces
$$\eqalign{f(\alpha_1)& = f(\beta_1)\cr
   \noalign{\kern -4pt}%
   &\phantom{a}\vdots\cr
   f(\alpha_k)& = f(\beta_k)\cr}$$
\endexample
\enddesc

%\begindesc
%\cts ddots {}
%\explain
%This command produces three dots on a diagonal.
%Its most common use is to indicate repetition along the diagonal of a matrix.
%\example
%$$\pmatrix{0&\ldots&0\cr
%           \vdots&\ddots&\vdots\cr
%           0&\ldots&0\cr}$$
%|
%\dproduces
%$$\pmatrix{0&\ldots&0\cr
%           \vdots&\ddots&\vdots\cr
%           0&\ldots&0\cr}$$
\begindesc
\cts ddots {}
\explain
此命令排印斜线上的三个圆点。它常用于表示沿矩阵对角线的重复。
\example
$$\pmatrix{0&\ldots&0\cr
           \vdots&\ddots&\vdots\cr
           0&\ldots&0\cr}$$
|
\dproduces
$$\pmatrix{0&\ldots&0\cr
           \vdots&\ddots&\vdots\cr
           0&\ldots&0\cr}$$

%\eix^^{dots}
%\endexample
%\enddesc
\eix^^{圆点}
\endexample
\enddesc

%\see |\dots| \ctsref\dots.
\see |\dots|\ctsref\dots 。

%==========================================================================
%\subsection {Delimiters}
\subsection {定界符}

%\begindesc
%\bix^^{delimiters}
%%
%\cts lgroup {}
%\cts rgroup {}
%\explain
%These commands produce large left and right ^{parentheses}
%that are defined as opening and closing \minref{delimiter}s.
%The smallest available size for these delimiters is |\Big|.
%If you use smaller sizes, you'll get weird characters.
%\example
%$$\lgroup\dots\rgroup\qquad\bigg\lgroup\dots\bigg\rgroup$$
%|
%\dproduces
%$$\lgroup\dots\rgroup\qquad\bigg\lgroup\dots\bigg\rgroup$$
%\endexample
%\enddesc
\begindesc
\bix^^{定界符}
%
\cts lgroup {}
\cts rgroup {}
\explain
这两个命令排印大号的左和右^{圆括号}，
它们分别作为开定界符和闭\minref{定界符}。
这两个定界符的最小可用尺寸为 |\Big|。
如果使用更小的尺寸，你将得到奇怪的字符。
\example
$$\lgroup\dots\rgroup\qquad\bigg\lgroup\dots\bigg\rgroup$$
|
\dproduces
$$\lgroup\dots\rgroup\qquad\bigg\lgroup\dots\bigg\rgroup$$
\endexample
\enddesc

%\begindesc
%\margin{{\tt\\vert} and {\tt\\Vert} were explained elsewhere.}
%\easy\cts left {}
%\cts right {}
%\explain
%These commands must be used together in the pattern:
%\display
%{{\bt \\left} \<delim$_1$> \<subformula> {\bt \\right} \<delim$_2$>}
%This construct causes \TeX\ to produce \<subformula>,
%enclosed in the \minref{delimiter}s \<delim$_1$> and \<delim$_2$>.
%The vertical size of the delimiter is adjusted to fit the
%vertical size (height plus depth) of \<subformula>.  \<delim$_1$> and
%\<delim$_2$> need not correspond.
%For instance, you could use `|]|' as a left delimiter
%and `|(|' as a right delimiter in a single use of |\left|
%and |\right|.
\begindesc
\margin{{\tt\\vert} and {\tt\\Vert} were explained elsewhere.}
\easy\cts left {}
\cts right {}
\explain
这两个命令必须按照下面模式一起使用：
\display
{{\bt \\left} \<delim$_1$> \<subformula> {\bt \\right} \<delim$_2$>}
这个构造将让 \TeX\ 排印 \<subformula>，
并用\minref{定界符} \<delim$_1$> 和 \<delim$_2$> 包围它。
\TeX\ 调整定界符的竖直尺寸以适应 \<subformula> 的竖直尺寸（高度加深度）。
\<delim$_1$> 和 \<delim$_2$> 不需要相对应。
举个例子，在使用 |\left| 和 |\right| 时，
你可以将 `|]|' 作为左定界符，而将 `|(|' 作为右定界符。

%|\left| and |\right| have the important property that they define a
%group, i.e., they act like left and right braces.  This grouping
%property is particularly useful when you put ^|\over| (\xref{\over}) or
%a related command between |\left| and |\right|, since you don't need to
%put braces around the fraction constructed by |\over|.
|\left| 和 |\right| 有个重要性质是它们定义了一个编组，
即它们能够充当左和右花括号。
当你在|\left| 和 |\right| 之间放上 ^|\over|（\xref{\over}）或其他相关命令时，
此编组性质就很有用，因为你无需在 |\over| 构造的分式两边加上花括号。

%If you want a left delimiter but not a right delimiter, you can use `|.|' in
%place of the delimiter you don't want and it will turn into empty space
%(of width ^|\nulldelimiterspace|).
%\example
%$$\left\Vert\matrix{a&b\cr c&d\cr}\right\Vert
%  \qquad \left\uparrow q_1\atop q_2\right.$$
%|
%\dproduces
%$$\left\Vert\matrix{a&b\cr c&d\cr}\right\Vert
%  \qquad \left\uparrow q_1\atop q_2\right.$$
%\endexample
%\enddesc
如果你需要左定界符但不需要右定界符，
你可以用 `|.|' 代替你不需要的定界符，
这样它就变成一个空白（宽度为 ^|\nulldelimiterspace|）。
\example
$$\left\Vert\matrix{a&b\cr c&d\cr}\right\Vert
  \qquad \left\uparrow q_1\atop q_2\right.$$
|
\dproduces
$$\left\Vert\matrix{a&b\cr c&d\cr}\right\Vert
  \qquad \left\uparrow q_1\atop q_2\right.$$
\endexample
\enddesc

%\begindesc
%\cts delimiter {\<number>}
%\explain
%This command produces a delimiter whose characteristics are given by
%\<number>.  \<number> is normally written in hexadecimal notation.
%You can use the |\delimiter| command instead of a character in any context
%where \TeX\ expects a delimiter (although the command is rarely used
%outside of a macro definition).
%Suppose that \<number> is the hexadecimal number $cs_1s_2s_3
%l_1l_2l_3$.  Then \TeX\ takes the delimiter to have
%\minref{class} $c$, small variant
%$s_1s_2s_3$, and large variant $l_1l_2l_3$.  Here $s_1s_2s_3$ indicates
%the math character found in position $s_2s_3$ of family $s_1$, and
%similarly for $l_1l_2l_3$.  This is the same convention as the one
%used for ^|\mathcode| (\xref\mathcode).
%\example
%\def\vert{\delimiter "026A30C} % As in plain TeX.
%|
%\endexample
%\enddesc
\begindesc
\cts delimiter {\<number>}
\explain
此命令排印用 \<number> 刻画其特性的定界符。\<number> 通常用十六进制表示。
在 \TeX\ 需要定界符的任何地方你都可以用 |\delimiter| 命令代替一个字符%
（尽管此命令很少在宏定义之外的地方使用）。
假设 \<number> 为十六进制数 $cs_1s_2s_3l_1l_2l_3$。
则 \TeX\ 知道该定界符属于第$c$\minref{类}，
小号变体为 $s_1s_2s_3$, 而大号变体为 $l_1l_2l_3$。
这里 $s_1s_2s_3$ 表示第 $s_1$ 族位置 $s_2s_3$ 的数学字符，
$l_1l_2l_3$ 类似。这里使用与 ^|\mathcode|（\xref\mathcode ）一样的约定。
\example
\def\vert{\delimiter "026A30C} % As in plain TeX.
|
\endexample
\enddesc

%\begindesc
%\margin{{\tt\\delcode} was explained in two places.  The
%combined explanation is now in `General operations'.}
%\cts delimiterfactor {\param{number}}
%\cts delimitershortfall {\param{number}}
%\explain
%^^{delimiters//height of}
%These parameters together tell \TeX\ how the height of a \minref{delimiter}
%should be related to the vertical size of the subformula
%with which the delimiter is associated.
%|\delimiterfactor| gives the minimum
%ratio of the delimiter size to the vertical size of the subformula, and
%|\delimitershortfall| gives the maximum by which the height of the
%delimiter will be reduced from that of the vertical size of the subformula.
\begindesc
\margin{{\tt\\delcode} was explained in two places.  The
combined explanation is now in `General operations'.}
\cts delimiterfactor {\param{number}}
\cts delimitershortfall {\param{number}}
\explain
^^{定界符//定界符高度}
这两个参数共同确定了\minref{定界符}高度与其中子公式的竖直尺寸的关系。
|\delimiterfactor| 给出定界符高度相对子公式竖直尺寸的最小比例，
而 |\delimitershortfall| 给出定界符高度相对子公式竖直尺寸的最大差距。

%Suppose that the \minref{box} containing the subformula
%has height $h$ and depth $d$, and let $y=2\,\max(h,d)$.
%Let the value of |\delimiterfactor| be $f$ and the value of
%|\delimitershortfall| be $\delta$.
%Then \TeX\ takes the minimum delimiter size to be at least $y \cdot
%f/1000$ and at least $y-\delta$.  In particular, if |\delimiterfactor|
%is exactly $1000$ then \TeX\ will try to make a delimiter at least as tall
%as the formula to which it is attached.
%See \knuth{page~152 and page~446 (Rule 19)}
%for the exact details of how \TeX\ uses these parameters.
%\PlainTeX\ sets |\delimiter!-factor| to $901$ and
%|\delimiter!-shortfall| to |5pt|.
%\enddesc
假设包含子公式的\minref{盒子}的高度为 $h$ 深度为 $d$，
且令 $y=2\,\max(h,d)$。
设 |\delimiterfactor| 的值为 $f$，|\delimitershortfall| 的值为 $\delta$。
则 \TeX\ 选取的定界符高度至少为 $y \cdot f/1000$，且至少为 $y-\delta$。
特别地，如果 |\delimiterfactor| 恰好为 $1000$，
则 \TeX\ 将试着生成一个至少和其中的子公式一样高的定界符。
见\knuth{第~152~页和第~446~页（规则19）}中 \TeX\ 如何使用这些参数的细节。
\PlainTeX\ 设定 |\delimiter!-factor| 为 $901$，
|\delimiter!-shortfall| 为 |5pt|。
\enddesc

%\see |\delcode| (\xref\delcode), |\vert|, |\Vert|,
%and |\backslash| (\xref\vert).
%\eix^^{delimiters}
\see |\delcode|（\xref\delcode )、|\vert|、|\Vert| 和 |\backslash|（\xref\vert ）。
\eix^^{定界符}

%==========================================================================
%\subsection {Matrices}
\subsection {矩阵}

%\begindesc
%\cts matrix
%   {{\bt \rqbraces{\<line> \\cr $\ldots$ \<line> \\cr}}}
%\cts pmatrix
%   {{\bt \rqbraces{\<line> \\cr $\ldots$ \<line> \\cr}}}
%\cts bordermatrix
%   {{\bt \rqbraces{\<line> \\cr $\ldots$ \<line> \\cr}}}
%\explain
%Each of these three commands produces a ^{matrix}.
%The elements of each row of the input matrix
%are separated by `|&|' and each row in turn is ended
%by |\cr|.
%(This is the same form that is used for an
%\minref{alignment}.)
%The commands differ in the following ways:
%\ulist\compact
%\li |\matrix| produces a matrix without any surrounding or inserted
%\minref{delimiter}s.
%\li |\pmatrix| produces a matrix surrounded by parentheses.
%\li |\bordermatrix| produces a matrix in which the first row and the first
%column are treated as labels.  (The first element of the first row is
%usually left blank.)  The rest of the matrix is enclosed in
%parentheses.
%\endulist
%\TeX\ can make the parentheses for |\pmatrix| and |\bordermatrix| as large as
%they need to be by inserting vertical extensions.  If you want a matrix
%to be surrounded by delimiters other than parentheses, you should use
%|\matrix| in conjunction with |\left| and |\right| (\xref \left).
\begindesc
\cts matrix
   {{\bt \rqbraces{\<line> \\cr $\ldots$ \<line> \\cr}}}
\cts pmatrix
   {{\bt \rqbraces{\<line> \\cr $\ldots$ \<line> \\cr}}}
\cts bordermatrix
   {{\bt \rqbraces{\<line> \\cr $\ldots$ \<line> \\cr}}}
\explain
这三个命令每个都排印一个^{矩阵}，
输入矩阵时各行的元素之间用 `|&|' 分隔，而各行用 |\cr| 结尾。%
（这里使用与\minref{阵列}一样的形式。）%
这些命令之间的区别如下：
\ulist\compact
\li |\matrix| 排印一个四周空白不带\minref{定界符}的矩阵。
\li |\pmatrix| 排印一个两边带圆括号的矩阵。
\li |\bordermatrix| 排印一个将第一行和第一列视为标号的矩阵。%
（第一行的第一个元素通常为空白。）%
矩阵的其他元素被圆括号包含。
\endulist
通过增加竖直延伸，\TeX\ 能够为 |\pmatrix| 和 |\bordermatrix| 制作足够大的圆括号。
如果你需要用不同于圆括号的定界符包围矩阵，你应当将
|\matrix| 与 |\left| 和 |\right|（\xref\left ）合起来使用。

%\example
%$$\displaylines{
%   \matrix{t_{11}&t_{12}&t_{13}\cr
%           t_{21}&t_{22}&t_{23}\cr
%           t_{31}&t_{32}&t_{33}\cr}\qquad
%\left\{\matrix{t_{11}&t_{12}&t_{13}\cr
%           t_{21}&t_{22}&t_{23}\cr
%           t_{31}&t_{32}&t_{33}\cr}\right\}\cr
%\pmatrix{t_{11}&t_{12}&t_{13}\cr
%           t_{21}&t_{22}&t_{23}\cr
%           t_{31}&t_{32}&t_{33}\cr}\qquad
%\bordermatrix{&c_1&c_2&c_3\cr
%           r_1&t_{11}&t_{12}&t_{13}\cr
%           r_2&t_{21}&t_{22}&t_{23}\cr
%           r_3&t_{31}&t_{32}&t_{33}\cr}\cr}$$
%|
%\dproduces
%$$\displaylines{
%   \matrix{t_{11}&t_{12}&t_{13}\cr
%   t_{21}&t_{22}&t_{23}\cr
%   t_{31}&t_{32}&t_{33}\cr}\qquad
%\left\{\matrix{t_{11}&t_{12}&t_{13}\cr
%   t_{21}&t_{22}&t_{23}\cr
%   t_{31}&t_{32}&t_{33}\cr}\right\}\cr
%\pmatrix{t_{11}&t_{12}&t_{13}\cr
%   t_{21}&t_{22}&t_{23}\cr
%   t_{31}&t_{32}&t_{33}\cr}\qquad
%\bordermatrix{&c_1&c_2&c_3\cr
%   r_1&t_{11}&t_{12}&t_{13}\cr
%   r_2&t_{21}&t_{22}&t_{23}\cr
%   r_3&t_{31}&t_{32}&t_{33}\cr}\cr}$$
%\endexample
%\enddesc
\example
$$\displaylines{
   \matrix{t_{11}&t_{12}&t_{13}\cr
           t_{21}&t_{22}&t_{23}\cr
           t_{31}&t_{32}&t_{33}\cr}\qquad
\left\{\matrix{t_{11}&t_{12}&t_{13}\cr
           t_{21}&t_{22}&t_{23}\cr
           t_{31}&t_{32}&t_{33}\cr}\right\}\cr
\pmatrix{t_{11}&t_{12}&t_{13}\cr
           t_{21}&t_{22}&t_{23}\cr
           t_{31}&t_{32}&t_{33}\cr}\qquad
\bordermatrix{&c_1&c_2&c_3\cr
           r_1&t_{11}&t_{12}&t_{13}\cr
           r_2&t_{21}&t_{22}&t_{23}\cr
           r_3&t_{31}&t_{32}&t_{33}\cr}\cr}$$
|
\dproduces
$$\displaylines{
   \matrix{t_{11}&t_{12}&t_{13}\cr
   t_{21}&t_{22}&t_{23}\cr
   t_{31}&t_{32}&t_{33}\cr}\qquad
\left\{\matrix{t_{11}&t_{12}&t_{13}\cr
   t_{21}&t_{22}&t_{23}\cr
   t_{31}&t_{32}&t_{33}\cr}\right\}\cr
\pmatrix{t_{11}&t_{12}&t_{13}\cr
   t_{21}&t_{22}&t_{23}\cr
   t_{31}&t_{32}&t_{33}\cr}\qquad
\bordermatrix{&c_1&c_2&c_3\cr
   r_1&t_{11}&t_{12}&t_{13}\cr
   r_2&t_{21}&t_{22}&t_{23}\cr
   r_3&t_{31}&t_{32}&t_{33}\cr}\cr}$$
\endexample
\enddesc

%==========================================================================
%\subsection {Roots and radicals}
\subsection {根号与根数}

%\begindesc
%\easy\cts sqrt {\<argument>}
%\explain
%This command produces the notation for the square root of \<argument>.
%\example
%$$x = {-b\pm\sqrt{b^2-4ac} \over 2a}$$
%|
%\dproduces
%$$x = {-b\pm\sqrt{b^2-4ac} \over 2a}$$
%\endexample
%\enddesc
\begindesc
\easy\cts sqrt {\<argument>}
\explain
此命令排印 \<argument> 的平方根。
\example
$$x = {-b\pm\sqrt{b^2-4ac} \over 2a}$$
|
\dproduces
$$x = {-b\pm\sqrt{b^2-4ac} \over 2a}$$
\endexample
\enddesc

%\begindesc
%\easy\cts root {\<argument$_1$> {\bt \\of} \<argument$_2$>}
%\explain
%This command produces the notation for a root of \<argument$_2$>, where the
%root is given by \<argument$_1$>.
%\example
%$\root \alpha \of {r \cos \theta}$
%|
%\produces
%$\root \alpha \of {r \cos \theta}$
%\endexample
%\enddesc
\begindesc
\easy\cts root {\<argument$_1$> {\bt \\of} \<argument$_2$>}
\explain
此命令排印 \<argument$_2$> 的 \<argument$_1$> 次根号。
\example
$\root \alpha \of {r \cos \theta}$
|
\produces
$\root \alpha \of {r \cos \theta}$
\endexample
\enddesc

%\begindesc
%\cts radical {\<number>}
%\explain
%This command produces a radical sign
%whose characteristics are given by
%\<number>.  It uses the same representation as the delimiter code
%^^{delimiter codes}
%in the ^|\delcode| command (\xref \delcode).
\begindesc
\cts radical {\<number>}
\explain
此命令排印用 \<number> 刻画其特性的根数符号。
它使用的定界码表示法与 ^|\delcode| 命令（\xref\delcode ）的相同。
^^{定界码}

%\example
%\def\sqrt{\radical "270370} % as in plain TeX
%|
%\endexample
%\enddesc
\example
\def\sqrt{\radical "270370} % as in plain TeX
|
\endexample
\enddesc


%==========================================================================
%\section {Equation numbers}
\section {方程编号}

%\begindesc
%\easy\cts eqno {}
%\cts leqno {}
%\explain
%These commands attach an equation number to a displayed formula.
%|\eqno| puts the equation number on the right and |\leqno| puts it on
%the left.
%The commands must be given at the end of the formula.
%If you have a multiline display and you want to number more than one
%of the lines, use the |\eq!-alignno| or |\leq!-alignno| command
%(\xref \eqalignno).
\begindesc
\easy\cts eqno {}
\cts leqno {}
\explain
这两个命令给陈列公式加上方程编号。
|\eqno| 将编号放在右侧，而|\leqno| 将编号放在左侧。
这两个命令必须放在公式末尾。
如果你有个多行陈列公式，而你希望给不止一行编号，
你可以用 |\eq!-alignno| 或 |\leq!-alignno| 命令（\xref\eqalignno ）。

%These commands are valid only in display math mode.
这两个命令只能在陈列数学模式中使用。

%\example
%$$e^{i\theta} = \cos \theta + i \sin \theta\eqno{(11)}$$
%|
%\produces
%$$e^{i\theta} = \cos \theta + i \sin \theta\eqno{(11)}$$
%\endexample
%\example
%$$\cos^2 \theta + \sin^2 \theta = 1\leqno{(12)}$$
%|
%\produces
%\abovedisplayskip = -\baselineskip
%$$\cos^2 \theta + \sin^2 \theta = 1\leqno{(12)}$$
%\endexample
%\enddesc
\example
$$e^{i\theta} = \cos \theta + i \sin \theta\eqno{(11)}$$
|
\produces
$$e^{i\theta} = \cos \theta + i \sin \theta\eqno{(11)}$$
\endexample
\example
$$\cos^2 \theta + \sin^2 \theta = 1\leqno{(12)}$$
|
\produces
\abovedisplayskip = -\baselineskip
$$\cos^2 \theta + \sin^2 \theta = 1\leqno{(12)}$$
\endexample
\enddesc


%==========================================================================
%\section {Multiline displays}
\section {多行陈列公式}

%\begindesc
%\bix^^{displays//multiline}
%\cts displaylines
%   {{\bt \rqbraces{\<line>\ths\\cr$\ldots$\<line>\ths\\cr}}}
%\explain
%This command produces a multiline math display in which each line is
%centered independently of the other lines.
%You can use the |\noalign| command (\xref \noalign) to change the amount
%of space between two lines of a multiline display.
\begindesc
\bix^^{陈列公式//多行陈列公式}
\cts displaylines
   {{\bt \rqbraces{\<line>\ths\\cr$\ldots$\<line>\ths\\cr}}}
\explain
此命令排印一个多行陈列公式，其中的各行独立地居中放置。
你可以使用 |\noalign| 命令（\xref\noalign ）改变多行陈列公式中两行的间隔。

%If you want to attach equation numbers to some or all of the equations
%in a multiline math display, you should use |\eqalignno| or
%|\leqalignno|.
%\example
%$$\displaylines{(x+a)^2 = x^2+2ax+a^2\cr
%                (x+a)(x-a) = x^2-a^2\cr}$$
%|
%\dproduces\centereddisplays
%$$\displaylines{
%(x+a)^2 = x^2+2ax+a^2\cr
%(x+a)(x-a) = x^2-a^2\cr
%}$$
%\endexample
%\enddesc
如果你希望给多行陈列公式的某个或某些方程添加编号，
你应当使用|\eqalignno| 或 |\leqalignno|。
\example
$$\displaylines{(x+a)^2 = x^2+2ax+a^2\cr
                (x+a)(x-a) = x^2-a^2\cr}$$
|
\dproduces\centereddisplays
$$\displaylines{
(x+a)^2 = x^2+2ax+a^2\cr
(x+a)(x-a) = x^2-a^2\cr
}$$
\endexample
\enddesc

%\begindesc
%\cts eqalign {}
%   {{\bt \rqbraces{\<line> \\cr $\ldots$ \<line> \\cr}}}
%\cts eqalignno {}
%   {{\bt \rqbraces{\<line> \\cr $\ldots$ \<line> \\cr}}}
%\cts leqalignno {}
%   {{\bt \rqbraces{\<line> \\cr $\ldots$ \<line> \\cr}}}
%\explain
%^^{equation numbers}
%These commands produce a multiline math display
%in which certain corresponding parts of the lines are lined up vertically.
%The |\eqalignno| and |\leqalignno| commands also let you
%provide equation numbers for some or all of the lines.
%|\eqalignno| puts the equation numbers on the right and
%|\leqalignno| puts them on the left.
\begindesc
\cts eqalign {}
   {{\bt \rqbraces{\<line> \\cr $\ldots$ \<line> \\cr}}}
\cts eqalignno {}
   {{\bt \rqbraces{\<line> \\cr $\ldots$ \<line> \\cr}}}
\cts leqalignno {}
   {{\bt \rqbraces{\<line> \\cr $\ldots$ \<line> \\cr}}}
\explain
^^{公式编号}
这些命令排印一个多行陈列公式，其中某些行的对应部分竖直对齐。
|\eqalignno| 和 |\leqalignno| 命令还允许你给某个或某些行添加方程编号。
|\eqalignno| 将方程编号放在右侧，
而 |\leqalignno| 将编号放在左侧。

%Each line in the display is ended by |\cr|.  Each of the parts to be aligned
%(most often an equals sign) is preceded by
%`|&|'.  An `|&|' also precedes each equation number, which comes at the
%end of a line.
%You can put more than one of these commands in a single display in order
%to produce several groups of equations.  In this case, only the rightmost
%or leftmost group can be produced by |\eqalignno| or |\leqalignno|.
陈列公式的每行用 |\cr| 结尾。
各行需要对齐的各部分（多半是等号）前面加上 `|&|'。
方程编号放在公式末尾，它的前面也要加上 `|&|'。
你可以在单个陈列公式中多次使用这些命令以排印多组方程。
在这种情形中，
只有最右边或最左边的那组方程可以用 |\eqalignno| 或 |\leqalignno| 编号。

%You can use the |\noalign| command (\xref \noalign) to change the amount
%of space between two lines of a multiline display.
%\example
%$$\left\{\eqalign{f_1(t) &= 2t\cr f_2(t) &= t^3\cr
%         f_3(t) &= t^2-1\cr}\right\}
%  \left\{\eqalign{g_1(t) &= t\cr g_2(t) &= 1}\right\}$$
%|
%\dproduces
%$$\left\{\eqalign{f_1(t) &= 2t\cr f_2(t) &= t^3\cr
%   f_3(t) &= t^2-1\cr}\right\}
%\left\{\eqalign{g_1(t) &= t\cr g_2(t) &= 1}\right\}$$
%\nextexample
%$$\eqalignno{
%\sigma^2&=E(x-\mu)^2&(12)\cr
%   &={1 \over n}\sum_{i=0}^n (x_i - \mu)^2&\cr
%   &=E(x^2)-\mu^2\cr}$$
%|
%\produces
%\abovedisplayskip = -\baselineskip
%$$\eqalignno{
%\sigma^2&=E(x-\mu)^2&(12)\cr
%   &={1 \over n}\sum_{i=0}^n (x_i - \mu)^2&\cr
%   &=E(x^2)-\mu^2\cr}$$
%\nextexample
%$$\leqalignno{
%\sigma^2&=E(x-\mu)^2&(6)\cr
%   &=E(x^2)-\mu^2&(7)\cr}$$
%|
%\produces
%\abovedisplayskip = -\baselineskip
%$$\leqalignno{
%\sigma^2&=E(x-\mu)^2&(6)\cr
%   &=E(x^2)-\mu^2&(7)\cr}$$
%\nextexample
%$$\eqalignno{
%  &(x+a)^2 = x^2+2ax+a^2&(19)\cr
%  &(x+a)(x-a) = x^2-a^2\cr}$$
%% same effect as \displaylines but with an equation number
%|
%\dproduces
%$$\eqalignno{
%&(x+a)^2 = x^2+2ax+a^2&(19)\cr
%&(x+a)(x-a) = x^2-a^2\cr
%}$$
%% same effect as \displaylines but with an equation number
你可以使用 |\noalign| 命令（\xref\noalign ）改变多行陈列公式中两行的间隔。
\example
$$\left\{\eqalign{f_1(t) &= 2t\cr f_2(t) &= t^3\cr
         f_3(t) &= t^2-1\cr}\right\}
  \left\{\eqalign{g_1(t) &= t\cr g_2(t) &= 1}\right\}$$
|
\dproduces
$$\left\{\eqalign{f_1(t) &= 2t\cr f_2(t) &= t^3\cr
   f_3(t) &= t^2-1\cr}\right\}
\left\{\eqalign{g_1(t) &= t\cr g_2(t) &= 1}\right\}$$
\nextexample
$$\eqalignno{
\sigma^2&=E(x-\mu)^2&(12)\cr
   &={1 \over n}\sum_{i=0}^n (x_i - \mu)^2&\cr
   &=E(x^2)-\mu^2\cr}$$
|
\produces
\abovedisplayskip = -\baselineskip
$$\eqalignno{
\sigma^2&=E(x-\mu)^2&(12)\cr
   &={1 \over n}\sum_{i=0}^n (x_i - \mu)^2&\cr
   &=E(x^2)-\mu^2\cr}$$
\nextexample
$$\leqalignno{
\sigma^2&=E(x-\mu)^2&(6)\cr
   &=E(x^2)-\mu^2&(7)\cr}$$
|
\produces
\abovedisplayskip = -\baselineskip
$$\leqalignno{
\sigma^2&=E(x-\mu)^2&(6)\cr
   &=E(x^2)-\mu^2&(7)\cr}$$
\nextexample
$$\eqalignno{
  &(x+a)^2 = x^2+2ax+a^2&(19)\cr
  &(x+a)(x-a) = x^2-a^2\cr}$$
% same effect as \displaylines but with an equation number
|
\dproduces
$$\eqalignno{
&(x+a)^2 = x^2+2ax+a^2&(19)\cr
&(x+a)(x-a) = x^2-a^2\cr
}$$
% same effect as \displaylines but with an equation number

%\eix^^{displays//multiline}
%\endexample
%\enddesc
\eix^^{陈列公式//多行陈列公式}
\endexample
\enddesc


%==========================================================================
%\section {Fonts in math formulas}
\section {数学公式字体}

%\begindesc
%^^{fonts}
%\xrdef{mathfonts}
%%
%\easy\ctsx cal {use calligraphic uppercase font}
%\ctsx mit {use math italic font}
%\ctsx oldstyle {use old style digit font}
%\explain
%These commands cause \TeX\ to typeset the following text in the
%specified font.  You can only use them in \minref{math mode}.
%The |\mit| command is useful for producing slanted capital ^{Greek letters}.
%You can also use the commands given in
%\headcit{Selecting fonts}{selfont} to change fonts in math mode.
%\example
%${\cal XYZ} \quad
%{\mit AaBb\Gamma \Delta \Sigma} \quad
%{\oldstyle 0123456789}$
%|
%\produces
%${\cal XYZ} \quad
%{\mit AaBb\Gamma \Delta \Sigma} \quad
%{\oldstyle 0123456789}$
%\endexample
%\enddesc
\begindesc
^^{字体}
\xrdef{mathfonts}
%
\easy\ctsx cal {use calligraphic uppercase font}
\ctsx mit {use math italic font}
\ctsx oldstyle {use old style digit font}
\explain
这些命令让 \TeX\ 用指定的字体排版之后的文本。
你只能在\minref{数学模式}中使用它们。
|\mit| 命令可用于排印斜体大写^{希腊字母}。
你还可以用\headcit{选择字体}{selfont}中的那些命令改变数学模式中的字体。
\example
${\cal XYZ} \quad
{\mit AaBb\Gamma \Delta \Sigma} \quad
{\oldstyle 0123456789}$
|
\produces
${\cal XYZ} \quad
{\mit AaBb\Gamma \Delta \Sigma} \quad
{\oldstyle 0123456789}$
\endexample
\enddesc

%^^{type styles}
%\begindesc
%\ctsx itfam {family for italic type}
%\ctsx bffam {family for boldface type}
%\ctsx slfam {family for slanted type}
%\ctsx ttfam {family for typewriter type}
%\explain
%These commands define type families \minrefs{family} for use in
%\minref{math mode}.  Their principal use is in defining the
%|\it|, |\bf|, |\sl|, and |\tt| commands so that they work in math mode.
%\enddesc
^^{字体风格}
\begindesc
\ctsx itfam {family for italic type}
\ctsx bffam {family for boldface type}
\ctsx slfam {family for slanted type}
\ctsx ttfam {family for typewriter type}
\explain
这些命令定义几种用于\minref{数学模式}的字体族\minrefs{族}。
它们主要用在 |\it|、|\bf|、|\sl| 和 |\tt| 命令的定义中，使这些命令能在数学模式中使用。
\enddesc

%\begindesc
%\cts fam {\param{number}}
%\explain
%When \TeX\ is in \minref{math mode}, it ordinarily typesets a character
%using the font family ^^{class} given in its \minref{mathcode}.
%^^{family//given by \b\tt\\fam\e}
%However, when \TeX\ is in math mode and encounters a character whose
%\minref{class} is $7$ (Variable), it typesets that character using
%the font \minref{family} given by the value of |\fam|, provided that the
%value of |\fam| is between $0$ and $15$.
%If the value of |\fam| isn't in that range, \TeX\ uses the family in
%the character's mathcode as in the ordinary case.
%\TeX\ sets |\fam| to $-1$ whenever it enters math mode.
%Outside of math mode, |\fam| has no effect.
\begindesc
\cts fam {\param{number}}
\explain
在\minref{数学模式}时，\TeX\ 通常用字符的\minref{数学码}指定的字体族排版该字符。
^^{类}^^{族//用 \b\tt\\fam\e 给出}
但是，如果 \TeX\ 在数学模式中遇到第 $7$ \minref{类}（变量）字符，
它将用由 |\fam| 的值给出的字体\minref{族}排版该字符，
只要 |\fam| 的值在 $0$ 和 $15$ 之间。
如果 |\fam| 的值不在该范围内，
\TeX\ 就像通常情形那样使用字符的数学码指定的字体族。
\TeX\ 在进入数学模式时设定 |\fam| 为 $-1$。
在数学模式之外，|\fam| 无任何效果。

%By assigning a value to
%|\fam| you can change the way that \TeX\ typesets ordinary
%characters such as variables.
%For instance, by setting |\fam| to |\ttfam|, you cause \TeX\ to typeset
%variables using a typewriter font.
%\PlainTeX\ defines |\tt| as a \minref{macro} that, among other things,
%sets |\fam| to |\ttfam|.
%\example
%\def\bf{\fam\bffam\tenbf} % As in plain TeX.
%|
%\endexample
%\enddesc
通过赋予 |\fam| 不同的值，你能让 \TeX\ 用不同方式排版普通字符，比如变量。
举个例子，设定了 |\fam| 为 |\ttfam| ，你可以让 \TeX\ 用打字机字体排版变量。
\PlainTeX\ 在定义 |\tt| \minref{宏}时，除了其他设定之外，
还设定 |\fam| 等于 |\ttfam|。
\example
\def\bf{\fam\bffam\tenbf} % As in plain TeX.
|
\endexample
\enddesc

%\begindesc
%\cts textfont {\<family>\param{fontname}}
%\cts scriptfont {\<family>\param{fontname}}
%\cts scriptscriptfont {\<family>\param{fontname}}
%\explain
%^^{text style}
%^^{script style}
%^^{scriptscript style}
%Each of these parameters specifies the font that \TeX\ is to use for
%typesetting the indicated \minref{style} in the indicated \minref{family}.
%These choices have no effect outside of \minref{math mode}.
%\example
%\scriptfont2 = \sevensy % As in plain TeX.
%|
%\endexample
%\enddesc
\begindesc
\cts textfont {\<family>\param{fontname}}
\cts scriptfont {\<family>\param{fontname}}
\cts scriptscriptfont {\<family>\param{fontname}}
\explain
^^{文本样式}
^^{标号样式}
^^{小标号样式}
这三个参数分别选择 \TeX\ 排版指定\minref{族}的指定\minref{样式}时所用的字体。
这些选择在\minref{数学模式}之外无任何效果。
\example
\scriptfont2 = \sevensy % As in plain TeX.
|
\endexample
\enddesc

%\see ``Type styles'' (\xref{seltype}).
\see ``字体风格''（\xref{seltype}）。


%==========================================================================
%\section {Constructing math symbols}
\section {构造数学符号}

%==========================================================================
%\subsection {Making delimiters bigger}
\subsection {增大定界符}

%\begindesc
%\makecolumns 16/4:
%\easy\cts big {}
%\cts bigl {}
%\cts bigm {}
%\cts bigr {}
%\cts Big {}
%\cts Bigl {}
%\cts Bigm {}
%\cts Bigr {}
%\cts bigg {}
%\cts biggl {}
%\cts biggm {}
%\cts biggr {}
%\cts Bigg {}
%\cts Biggl {}
%\cts Biggm {}
%\cts Biggr {}
%\explain
%^^{delimiters//enlarging}
%These commands make \minref{delimiter}s bigger than their normal size.
%The commands in the four columns
%produce successively larger sizes.  The difference between |\big|,
%|\bigl|, |\bigr|, and |bigm| has to do with the \minref{class} of the
%enlarged delimiter:
%\ulist\compact
%\li |\big| produces an ordinary symbol.
%\li |\bigl| produces an opening symbol.
%\li |\bigr| produces a closing symbol.
%\li |\bigm| produces a relation symbol.
%\endulist
%\noindent
%\TeX\ uses the class of a symbol in order to decide how much space to put
%around that symbol.
\begindesc
\makecolumns 16/4:
\easy\cts big {}
\cts bigl {}
\cts bigm {}
\cts bigr {}
\cts Big {}
\cts Bigl {}
\cts Bigm {}
\cts Bigr {}
\cts bigg {}
\cts biggl {}
\cts biggm {}
\cts biggr {}
\cts Bigg {}
\cts Biggl {}
\cts Biggm {}
\cts Biggr {}
\explain
^^{定界符//增大定界符}
这些命令让\minref{定界符}比它们的正常尺寸还大。
这四栏中的命令生成依次增大的尺寸。|\big|、|\bigl|、|\bigr|
和 |\bigm| 的区别在于增大的定界符所属的\minref{类}：
\ulist\compact
\li |\big| 生成一个普通符号。
\li |\bigl| 生成一个开符号。
\li |\bigr| 生成一个闭符号。
\li |\bigm| 生成一个关系符号。
\endulist
\noindent
\TeX\ 从字符所属的类确定要在该字符两边留下多大的空格。

%These commands, unlike |\left| and |\right|,
%do \emph{not} define a group.

%\example
%$$(x) \quad \bigl(x\bigr) \quad \Bigl(x\Bigr) \quad
%   \biggl(x\biggr) \quad \Biggl(x\Biggr)\qquad
%[x] \quad \bigl[x\bigr] \quad \Bigl[x\Bigr] \quad
%   \biggl[x\biggr] \quad \Biggl[x\Biggr]$$
%|
%\dproduces
%$$(x) \quad \bigl(x\bigr) \quad \Bigl(x\Bigr) \quad
%\biggl(x\biggr) \quad \Biggl(x\Biggr)\qquad
%[x] \quad \bigl[x\bigr] \quad \Bigl[x\Bigr] \quad
%\biggl[x\biggr] \quad \Biggl[x\Biggr]$$
%\endexample
%\enddesc
\example
$$(x) \quad \bigl(x\bigr) \quad \Bigl(x\Bigr) \quad
   \biggl(x\biggr) \quad \Biggl(x\Biggr)\qquad
[x] \quad \bigl[x\bigr] \quad \Bigl[x\Bigr] \quad
   \biggl[x\biggr] \quad \Biggl[x\Biggr]$$
|
\dproduces
$$(x) \quad \bigl(x\bigr) \quad \Bigl(x\Bigr) \quad
\biggl(x\biggr) \quad \Biggl(x\Biggr)\qquad
[x] \quad \bigl[x\bigr] \quad \Bigl[x\Bigr] \quad
\biggl[x\biggr] \quad \Biggl[x\Biggr]$$
\endexample
\enddesc

%==========================================================================
%\subsection {Parts of large symbols}
\subsection {大符号的一部分}

%\begindesc
%\cts downbracefill {}
%\cts upbracefill {}
%\explain
%These commands respectively produce upward-pointing
%and downward-pointing extensible ^{horizontal braces}. ^^{braces}
%\TeX\ will make the braces as wide as necessary.
%These commands
%are used in the definitions of ^|\overbrace| and ^|\underbrace|
%(\xref \overbrace).
%\example
%$$\hbox to 1in{\downbracefill} \quad
%   \hbox to 1in{\upbracefill}$$
%|
%\dproduces
%$$\hbox to 1in{\downbracefill} \quad
%   \hbox to 1in{\upbracefill}$$
%\endexample
%\enddesc
\begindesc
\cts downbracefill {}
\cts upbracefill {}
\explain
这两个命令分别排印朝上和朝下的可伸展^{水平花括号}。^^{花括号}
\TeX\ 将让花括号足够宽。
这两个命令用于定义 ^|\overbrace| 和 ^|\underbrace|（\xref\overbrace ）。
\example
$$\hbox to 1in{\downbracefill} \quad
   \hbox to 1in{\upbracefill}$$
|
\dproduces
$$\hbox to 1in{\downbracefill} \quad
   \hbox to 1in{\upbracefill}$$
\endexample
\enddesc

%\begindesc
%\cts arrowvert {}
%\cts Arrowvert {}
%\cts lmoustache {}
%\cts rmoustache {}
%\cts bracevert {}
%\explain
%These commands produce portions of certain large
%delimiters
%^^{delimiters//parts of}
%and can themselves be used as delimiters.
%They refer to characters in the ^|cmex10| math font.
%\example
%$$\cdots \Big\arrowvert \cdots \Big\Arrowvert \cdots
%  \Big\lmoustache \cdots \Big\rmoustache \cdots
%  \Big\bracevert \cdots$$
%|
%\dproduces
%$$\cdots \Big\arrowvert \cdots \Big\Arrowvert \cdots
%  \Big\lmoustache \cdots \Big\rmoustache \cdots
%  \Big\bracevert \cdots$$
%\endexample
%\enddesc
\begindesc
\cts arrowvert {}
\cts Arrowvert {}
\cts lmoustache {}
\cts rmoustache {}
\cts bracevert {}
\explain
这些命令排印某些大定界符的一部分，
^^{定界符//定界符的一部分}
把它们也用作定界符。
它们取自 ^|cmex10| 数学字体中的字符。
\example
$$\cdots \Big\arrowvert \cdots \Big\Arrowvert \cdots
  \Big\lmoustache \cdots \Big\rmoustache \cdots
  \Big\bracevert \cdots$$
|
\dproduces
$$\cdots \Big\arrowvert \cdots \Big\Arrowvert \cdots
  \Big\lmoustache \cdots \Big\rmoustache \cdots
  \Big\bracevert \cdots$$
\endexample
\enddesc


%==========================================================================
%\section {Aligning parts of a formula}
\section {对齐部分公式}

%==========================================================================
%\subsection {Aligning accents}
\subsection {对齐数学重音}

%\begindesc
%\bix^^{accents//aligning}
%\cts skew {\<number> \<argument$_1$> \<argument$_2$>}
%\explain
%This command shifts the accent \<argument$_1$> by
%\<number> \minref{mathematical unit}s to the right of its normal position
%with respect to \<argu\-ment$_2$>.
%The most common use of this command is for
%modifying the position of an accent that's over
%another accent.
%\example
%$$\skew 2\bar{\bar z}\quad\skew 3\tilde{\tilde y}\quad
%  \skew 4\tilde{\hat x}$$
%|
%\dproduces
%$$\skew 2\bar{\bar z}\quad\skew 3\tilde{\tilde y}\quad
%  \skew 4\tilde{\hat x}$$
%\endexample
%\enddesc
\begindesc
\bix^^{重音//对齐重音}
\cts skew {\<number> \<argument$_1$> \<argument$_2$>}
\explain
此命令将重音 \<argument$_1$> 相对 \<argu\-ment$_2$>
从它的正常位置往右移动 \<number> 个\minref{数学单位}。
此命令常用于调整在其他重音之上的重音的位置。
\example
$$\skew 2\bar{\bar z}\quad\skew 3\tilde{\tilde y}\quad
  \skew 4\tilde{\hat x}$$
|
\dproduces
$$\skew 2\bar{\bar z}\quad\skew 3\tilde{\tilde y}\quad
  \skew 4\tilde{\hat x}$$
\endexample
\enddesc

%\begindesc
%\cts skewchar {\<font>\param{number}}
%\explain
%The |\skewchar| of a font
%is the character in the font whose kerns,
%as defined in the font's metrics file, determine the positions
%of math accents. That is, suppose that \TeX\ is applying a math accent
%to the character `|x|'.  \TeX\ checks if the character pair
%`|x\skewchar|' has a kern; if so, it moves the accent by the amount of
%that kern. The complete algorithm that \TeX\ uses to position math
%accents (which involves many more things) is in \knuth{Appendix~G}.
\begindesc
\cts skewchar {\<font>\param{number}}
\explain
字体的 |\skewchar| 是字体中的某个字符，
它在字体度量文件中定义的紧排确定了数学重音的位置。
也就是说，假设 \TeX\ 要给字符 `|x|' 加上数学重音，
则 \TeX\ 检查字符对 `|x\skewchar|' 是否有个紧排；
如果有，它就以该紧排的值移动该重音。
\TeX\ 放置数学重音的完整算法（这涉及到很多事情）在\knuth{附录~G}中描述。

%If the value of |\skewchar| is not in the range $0$--$255$,
%\TeX\ takes the kern value to be zero.
如果 |\skewchar| 的值不在 $0$--$255$ 的范围内，\TeX\ 将紧排的值当作零。

%Note that \<font> is a control sequence
%that names a font, not a \<font\-name> that names font files.
%Beware:
%an assignment to |\skewchar| is \emph{not} undone at the end
%of a group.
%If you want to change |\skewchar| locally, you'll need to
%save and restore its original value explicitly.
%\enddesc
注意 \<font> 是一个控制序列，它是字体的名称，而不是字体文件的名称 \<font\-name>。
小心：对 |\skewchar| 的赋值在编组结束时\emph{并不会}还原。
如果你想局部改变|\skewchar|，你需要显式地保存和还原它的原始值。
\enddesc

%\begindesc
%\cts defaultskewchar {\param{number}}
%\explain
%When \TeX\ reads the metrics file
%^^{metrics file//default skew character in}
%for a font in response to a
%^|\font| command, it sets the font's ^|\skewchar| to
%|\default!-skewchar|.
%If the value of |\default!-skewchar| is
%not in the range $0$--$255$, \TeX\ does not assign any
%skew characters by default.
%\PlainTeX\ sets |\defaultskewchar| to $-1$, and it's usually best
%to leave it there.
%\margin{Misleading example deleted.}
%\eix^^{accents//aligning}
%\enddesc
\begindesc
\cts defaultskewchar {\param{number}}
\explain
在执行 ^|\font| 命令读取字体的度量文件时，
^^{度量文件//其中的默认斜字符}
\TeX\ 设定该字体的 ^|\skewchar| 等于 |\default!-skewchar|。
如果 |\default!-skewchar| 的值不在 $0$--$255$ 的范围内，
\TeX\ 默认就不设定 |\skewchar| 的值。
\PlainTeX\ 设定 |\defaultskewchar| 等于 $-1$，一般不需要改动它。
\margin{Misleading example deleted.}
\eix^^{重音//对齐重音}
\enddesc

%==========================================================================
%\subsection {Aligning material vertically}
\subsection {竖直对齐素材}

%\begindesc
%\cts vcenter {\rqbraces{\<vertical mode material>}}
%\ctsbasic {\\vcenter to \<dimen> \rqbraces{\<vertical mode material>}}{}
%\ctsbasic {\\vcenter spread \<dimen> \rqbraces{\<vertical mode material>}}{}
%\explain
%Every math formula has an invisible
%``^{axis}'' that \TeX\ treats as a kind of
%horizontal centering line for that formula.
%For instance, the axis of a formula consisting of a
%fraction is at the center of the fraction bar.
%The |\vcenter| command tells \TeX\ to place the \<vertical mode material>
%in a \minref{vbox} and to center the vbox
%with respect to the axis of the formula it is currently constructing.
\begindesc
\cts vcenter {\rqbraces{\<vertical mode material>}}
\ctsbasic {\\vcenter to \<dimen> \rqbraces{\<vertical mode material>}}{}
\ctsbasic {\\vcenter spread \<dimen> \rqbraces{\<vertical mode material>}}{}
\explain
每个数学公式都有一个不可见的``^{轴线}''，\TeX\ 将它作为该公式的水平中心线。
举个例子，由分式组成的公式的轴线就在分数线的中心。
|\vcenter| 命令让 \TeX\ 将 \<vertical mode material> 放入\minref{竖直盒子}中，
并将该竖直盒子与当前公式的轴线居中对齐。

%The first form of the command
%centers the material as given.  The second and third
%forms expand or shrink the material vertically as in the |\vbox| command
%(\xref \vbox).
此命令的第一种形式如上所述居中放置素材。
后两种形式竖直扩展或收缩素材，如同 |\vbox| 命令（\xref\vbox ）。

%\example
%$${n \choose k} \buildrel \rm def \over \equiv \>
%\vcenter{\hsize 1.5 in \noindent the number of
%combinations of $n$ things taken $k$ at a time}$$
%|
%\dproduces
%$${n \choose k} \buildrel \rm def \over \equiv \>
%\vcenter{\hsize 1.5 in \noindent the number of
%combinations of $n$ things taken $k$ at a time}$$
%\endexample
%\enddesc
\example
$${n \choose k} \buildrel \rm def \over \equiv \>
\vcenter{\hsize 1.5 in \noindent the number of
combinations of $n$ things taken $k$ at a time}$$
|
\dproduces
$${n \choose k} \buildrel \rm def \over \equiv \>
\vcenter{\hsize 1.5 in \noindent the number of
combinations of $n$ things taken $k$ at a time}$$
\endexample
\enddesc


%==========================================================================
%\section {Producing spaces}
\section {生成间隔}

%==========================================================================
%\subsection {Fixed-width math spaces}
\subsection {固定宽度数学间隔}

%\begindesc
%\bix^^{space//in math formulas}
%\ctspecial ! \ctsxrdef{@shriek}
%\ctspecial , \ctsxrdef{@comma}
%\ctspecial > \ctsxrdef{@greater}
%\ctspecial ; \ctsxrdef{@semi}
%\explain
%These commands produce various amounts of ^{extra space} in formulas.  They
%are defined in terms of \minref{mathematical unit}s, so \TeX\ adjusts
%the amount of space according to the current \minref{style}.
%\ulist
%\li |\!!| produces a negative thin space, i.e., it reduces the space
%between its neighboring subformulas by the amount of a thin space.
%\li |\,| produces a thin space.
%\li |\>| produces a medium space.
%\li |\;| produces a thick space.
%\endulist
%\example
%$$00\quad0\!!0\quad0\,0\quad0\>0\quad0\;0\quad
%{\scriptstyle 00\quad0\!!0\quad0\,0\quad0\>0\quad0\;0}$$
%|
%\dproduces
%$$00\quad0\!0\quad0\,0\quad0\>0\quad0\;0\quad
%{\scriptstyle 00\quad0\!0\quad0\,0\quad0\>0\quad0\;0}$$
%\endexample
%\enddesc
\begindesc
\bix^^{间隔//数学公式中的间隔}
\ctsymboln \! \ctsxrdef{@shriek} \pix\directidx{"\"!+@\hskip-.5em\tt\\\char`"\"!\rm}
\ctsymbol \, \ctsxrdef{@comma}
\ctsymbol \> \ctsxrdef{@greater}
\ctsymbol \; \ctsxrdef{@semi}
\explain
这些命令在公式中生成各种大小的^{额外间隔}。
它们使用\minref{数学单位}来定义，
因此 \TeX\ 会根据当前\minref{样式}调整间隔的大小。
\ulist
\li |\!!| 生成负的细小间隔，即它让相邻子公式的间隔减去该细小间隔的大小。
\li |\,| 生成细小间隔。
\li |\>| 生成中等间隔。
\li |\;| 生成较大间隔。
\endulist
\example
$$00\quad0\!!0\quad0\,0\quad0\>0\quad0\;0\quad
{\scriptstyle 00\quad0\!!0\quad0\,0\quad0\>0\quad0\;0}$$
|
\dproduces
$$00\quad0\!0\quad0\,0\quad0\>0\quad0\;0\quad
{\scriptstyle 00\quad0\!0\quad0\,0\quad0\>0\quad0\;0}$$
\endexample
\enddesc

%\begindesc
%\cts thinmuskip {\param{muglue}}
%\cts medmuskip {\param{muglue}}
%\cts thickmuskip {\param{muglue}}
%\explain
%These parameters define thin, medium, and thick spaces in
%math mode.
%\example
%$00\quad0\mskip\thinmuskip0\quad0\mskip\medmuskip0
%   \quad0\mskip\thickmuskip0$
%|
%\produces
%$00\quad0\mskip\thinmuskip0\quad0\mskip\medmuskip0
%   \quad0\mskip\thickmuskip0$
%\endexample
%\enddesc
\begindesc
\cts thinmuskip {\param{muglue}}
\cts medmuskip {\param{muglue}}
\cts thickmuskip {\param{muglue}}
\explain
这些参数定义了数学模式中细小、中等和较大间隔的大小。
\example
$00\quad0\mskip\thinmuskip0\quad0\mskip\medmuskip0
   \quad0\mskip\thickmuskip0$
|
\produces
$00\quad0\mskip\thinmuskip0\quad0\mskip\medmuskip0
   \quad0\mskip\thickmuskip0$
\endexample
\enddesc

%\begindesc
%\cts jot {\param{dimen}}
%\explain
%This parameter defines a distance that is equal to three points (unless
%you change it).
%The |\jot| is a convenient unit of measure for opening up \hbox{math displays}.
%\enddesc
\begindesc
\cts jot {\param{dimen}}
\explain
此参数定义为三个点的距离（除非你改变了它）。
在用 |\openup| 命令分开陈列公式各行时，|\jot| 是一个实用的度量单位。
\footnote{译注：下面的例子为译者所加。请参阅 |\openup| 命令（\xref\openup ）。}
\example
$$\vbox{\halign{$\hfil#\hfil$\cr x\cr y\cr}}$$
$$\openup2\jot\vbox{\halign{$\hfil#\hfil$\cr x\cr y\cr}}$$
|
\produces
$$\vbox{\halign{$\hfil#\hfil$\cr x\cr y\cr}}$$
$$\openup2\jot\vbox{\halign{$\hfil#\hfil$\cr x\cr y\cr}}$$
\endexample
\enddesc

%==========================================================================
%\subsection {Variable-width math spaces}
\subsection {可变宽度数学间隔}

%\begindesc
%\cts mkern {\<mudimen>}
%\explain
%^^{kerns//in math formulas}
%This command
%produces a \minref{kern}, i.e., blank space, of width \<mudimen>.
%The kern is measured
%in \minref{mathematical unit}s, which vary according to the style.
%Aside from its unit of measurement, this command behaves just like
%|\kern| (\xref \kern) does in horizontal mode.
\begindesc
\cts mkern {\<mudimen>}
\explain
^^{紧排//数学公式中的紧排}
此命令生成一个宽度为 \<mudimen> 的\minref{紧排}，即空白间隔。
该紧排用\minref{数学单位}表示，因此在不同样式中有不同的尺寸。
除了使用数学单位外，此命令与水平模式的|\kern|（\xref\kern ）的表现类似。

%\example
%$0\mkern13mu 0 \qquad {\scriptscriptstyle 0 \mkern13mu 0}$
%|
%\produces
%$0\mkern13mu 0 \qquad {\scriptscriptstyle 0 \mkern13mu 0}$
%\endexample
%\enddesc
\example
$0\mkern13mu 0 \qquad {\scriptscriptstyle 0 \mkern13mu 0}$
|
\produces
$0\mkern13mu 0 \qquad {\scriptscriptstyle 0 \mkern13mu 0}$
\endexample
\enddesc

%\begindesc
%\cts mskip {\<mudimen$_1$> {\bt plus} \<mudimen$_2$> {\bt minus}
%   \<mudimen$_3$>}
%\explain
%^^{glue}
%This command produces horizontal \minref{glue}
%that has natural width \<mu\-dimen$_1$>, stretch \<mudimen$_2$>,
%and shrink \<mudimen$_3$>.
%The glue is measured in \minref{mathematical unit}s, which vary according
%to the style.  Aside from its units of measurement, this command behaves
%just like |\hskip| (\xref \hskip).
\begindesc
\cts mskip {\<mudimen$_1$> {\bt plus} \<mudimen$_2$> {\bt minus}
   \<mudimen$_3$>}
\explain
^^{粘连}
此命令生成一个水平\minref{粘连}，它的自然宽度为 \<mu\-dimen$_1$>，
伸长量为 \<mudimen$_2$>，收缩量为 \<mudimen$_3$>。
该粘连用\minref{数学单位}表示，因此将随着样式的变化而变化。
除了使用数学单位外，此命令与 |\hskip|（\xref\hskip ）的表现类似。

%\example
%$0\mskip 13mu 0 \quad {\scriptscriptstyle 0 \mskip 13mu 0}$
%|
%\produces
%$0\mskip 13mu 0 \quad {\scriptscriptstyle 0 \mskip 13mu 0}$
%\endexample
%\enddesc
\example
$0\mskip 13mu 0 \quad {\scriptscriptstyle 0 \mskip 13mu 0}$
|
\produces
$0\mskip 13mu 0 \quad {\scriptscriptstyle 0 \mskip 13mu 0}$
\endexample
\enddesc

%\begindesc
%\cts nonscript {}
%\explain
%When \TeX\ is currently typesetting in script or scriptscript
%\minref{style} and encounters this command
%immediately in front of glue or a kern,
%it cancels the glue or kern.
%|\nonscript| has no effect in the other styles.
\begindesc
\cts nonscript {}
\explain
在排版标号或小标号\minref{样式}时，如果 \TeX\ 在粘连或紧排跟前遇到此命令，
它就丢弃该粘连或紧排。|\nonscript| 在其他样式中无任何效果。

%This command provides a way of ``tightening up'' the spacing in
%script and scriptscript styles, which generally are set in smaller type.
%It is of little use outside of macro definitions.
%\example
%\def\ab{a\nonscript\; b}
%$\ab^{\ab}$
%|
%\produces
%\def\ab{a\nonscript\; b}
%$\ab^{\ab}$
%\endexample
%\enddesc
此命令提供一种``收紧''标号和小标号样式中的间隔的方法；
通常用小号字体排版这两个样式。在宏定义之外的地方，此命令很少用到。
\example
\def\ab{a\nonscript\; b}
$\ab^{\ab}$
|
\produces
\def\ab{a\nonscript\; b}
$\ab^{\ab}$
\endexample
\enddesc

%\see |\kern| (\xref\kern), |\hskip| (\xref\hskip).
%\eix^^{space//in math formulas}
\see |\kern|（\xref\kern ）和 |\hskip|（\xref\hskip ）。
\eix^^{间隔//数学公式中的间隔}

%==========================================================================
%\subsection {Spacing parameters for displays}
\subsection {陈列公式的间隔参数}

%\begindesc
%\bix^^{displays//spacing parameters for}
%\cts displaywidth {\param{dimen}}
%\explain
%This parameter specifies the maximum width that
%\TeX\ allows for a math display.  If \TeX\ cannot fit the display
%into a space of this width, it sets an overfull \minref{hbox}
%and complains.
%\TeX\ sets the value of |\displaywidth| when it encounters the `|$$|'
%that starts the display.  This initial value is
%|\hsize| (\xref \hsize) unless it's overridden by changes to the
%paragraph shape.
%See \knuth{pages~188--189} for a more detailed explanation of this parameter.
%\enddesc
\begindesc
\bix^^{陈列公式//陈列公式的间隔参数}
\cts displaywidth {\param{dimen}}
\explain
此参数指定 \TeX\ 对陈列公式所允许的最大宽度。
如果 \TeX\ 无法将陈列公式放入这样宽的空间中，
它将生成一个过满的\minref{水平盒子}并给出警告。
\TeX\ 在遇到 `|$$|' 开始陈列公式时就设定 |\displaywidth| 的值。
它的初始值为 |\hsize|（\xref\hsize ），除非段落形状改变了。
见\knuth{第~188--189~页}中对此参数的更仔细的说明。
\enddesc

%\begindesc
%\cts displayindent {\param{dimen}}
%\explain
%This parameter specifies the space by which \TeX\ indents a
%math display.
%\TeX\ sets the value of |\displayindent| when it encounters the `|$$|'
%that starts the display.  Usually this initial value is zero,
%but if the paragraph shape indicates that the display should
%be shifted by an amount $s$,
%\TeX\ will set |\displayindent| to $s$.
%See \knuth{pages~188--189} for a more detailed explanation of this parameter.
%\enddesc
\begindesc
\cts displayindent {\param{dimen}}
\explain
此参数指定 \TeX\ 对陈列公式的缩进量。
\TeX\ 在遇到 `|$$|' 开始陈列公式时就设定 |\displayindent| 的值。
通常它的初始值为零，但如果段落形状表明该陈列公式需要移动距离 $s$，
\TeX\ 就设定 |\displayindent| 等于 $s$。
见\knuth{第~188--189~页}中对此参数的更仔细的介绍。
\enddesc

%\begindesc
%\cts predisplaysize {\param{dimen}}
%\explain
%\TeX\ sets this parameter to the width of the line preceding
%a math display.
%\TeX\ uses |\predisplaysize| to determine whether or not
%the display starts to
%the left of where the previous line ends, i.e., whether or not it visually
%overlaps the previous line.
%If there is overlap, it uses the |\abovedisplayskip| and
%|\belowdisplayskip| glue in setting the display;
%otherwise it uses the |\abovedisplay!-shortskip| and
%|\belowdisplay!-shortskip| glue.
%See \knuth{pages~188--189} for a more detailed explanation of this parameter.
%\enddesc
\begindesc
\cts predisplaysize {\param{dimen}}
\explain
\TeX\ 设定此参数等于陈列公式之前的文本行的宽度。
\TeX\ 利用 |\predisplaysize| 确定是否让陈列公式的起始点位于前一行结尾处的左边，
即它在外观上是否可能与前一行重叠。如果会有重叠，
\TeX\ 在排版陈列公式时使用|\abovedisplayskip| 和 |\belowdisplayskip| 粘连；
否则 \TeX\ 使用 |\abovedisplay!-shortskip| 和 |\belowdisplay!-shortskip| 粘连。
见\knuth{第~188--189~页}中对此参数的更仔细的介绍。
\enddesc

%\begindesc
%\cts abovedisplayskip {\param{glue}}
%\explain
%This parameter specifies the amount of vertical glue that
%\TeX\ inserts before a display when the display starts to
%the left of where the previous line ends, i.e., when it visually
%overlaps the previous line.
%\PlainTeX\ sets |\abovedisplayskip| to |12pt plus3pt minus9pt|.
%See \knuth{pages~188--189} for a more detailed explanation of this parameter.
%\enddesc
\begindesc
\cts abovedisplayskip {\param{glue}}
\explain
此命令指定当陈列公式的起始点位于前一行结尾处的左边时，
即它在外观上可能与前一行有重叠时，
\TeX\ 在陈列公式之前插入的竖直粘连的大小。
\PlainTeX\ 设定 |\abovedisplayskip| 等于 |12pt plus3pt minus9pt|。
见\knuth{第~188--189~页}中对此参数的更仔细的介绍。
\enddesc

%\begindesc
%\cts belowdisplayskip {\param{glue}}
%\explain
%This parameter specifies the amount of vertical glue that
%\TeX\ inserts after a display when the display starts to
%the left of where the previous line ends, i.e., when it visually
%overlaps the previous line.
%\PlainTeX\ sets |\belowdisplay!-skip| to |12pt plus3pt minus9pt|.
%See \knuth{pages~188--189} for a more detailed explanation of this parameter.
%\enddesc
\begindesc
\cts belowdisplayskip {\param{glue}}
\explain
此命令指定当陈列公式的起始点位于前一行结尾处的左边时，
即它在外观上可能与前一行有重叠时，
\TeX\ 在陈列公式之后插入的竖直粘连的大小。
\PlainTeX\ 设定 |\belowdisplay!-skip| 等于 |12pt plus3pt minus9pt|。
见\knuth{第~188--189~页}中对此参数的更仔细的介绍。
\enddesc

%\begindesc
%\cts abovedisplayshortskip {\param{glue}}
%\explain
%This parameter specifies the amount of vertical glue that
%\TeX\ inserts before a math display
%when the display starts to
%the right of where the previous line ends, i.e., when it does not visually
%overlap the previous line.
%\PlainTeX\ sets |\abovedisplay!-shortskip| to |0pt plus3pt|.
%See \knuth{pages~188--189} for a more detailed explanation of this parameter.
%\enddesc
\begindesc
\cts abovedisplayshortskip {\param{glue}}
\explain
此命令指定当陈列公式的起始点位于前一行结尾处的右边时，
即它在外观上不会与前一行有重叠时，
\TeX\ 在陈列公式之前插入的竖直粘连的大小。
\PlainTeX\ 设定 |\abovedisplay!-shortskip| 等于 |0pt plus3pt|。
见\knuth{第~188--189~页}中对此参数的更仔细的介绍。
\enddesc

%\begindesc
%\cts belowdisplayshortskip {\param{glue}}
%\explain
%This parameter specifies the amount of vertical glue that
%\TeX\ inserts after a display
%when the display starts to
%the right of where the previous line ends, i.e., when it does not visually
%overlap the previous line.
%\PlainTeX\ sets |\belowdisplay!-shortskip| to |7pt plus3pt minus4pt|.
%See \knuth{pages~188--189} for a more detailed explanation of this parameter.
\begindesc
\cts belowdisplayshortskip {\param{glue}}
\explain
此命令指定当陈列公式的起始点位于前一行结尾处的右边时，
即它在外观上不会与前一行有重叠时，
\TeX\ 在陈列公式之后插入的竖直粘连的大小。
\PlainTeX\ 设定 |\belowdisplay!-shortskip| 等于 |7pt plus3pt minus4pt|。
见\knuth{第~188--189~页}中对此参数的更仔细的介绍。

%\eix^^{displays//spacing parameters for}
%\enddesc
\eix^^{陈列公式//陈列公式的间隔参数}
\enddesc


%==========================================================================
\subsection {其他的数学间隔参数}

%\begindesc
%\cts mathsurround {\param{dimen}}
%\explain
%This parameter specifies the amount of space that \TeX\
%inserts before and after a math formula in text mode (i.e., a formula
%surrounded by single |$|'s).  See \knuth{page~162} for further details about
%its behavior.
%\PlainTeX\ leaves |\mathsurround| at |0pt|.
%\enddesc
\begindesc
\cts mathsurround {\param{dimen}}
\explain
此参数指定 \TeX\ 在文内数学公式（即放在两个|$|之间的公式）两边插入的间隔的大小。
见\knuth{第~162~页}对此行为的进一步解释。
\PlainTeX\ 设定 |\mathsurround| 为 |0pt|。
\enddesc

%\begindesc
%\cts nulldelimiterspace {\param{dimen}}
%\explain
%^^{delimiters//null, space for}
%This parameter specifies the width of the
%space produced by a null \minref{delimiter}.
%\PlainTeX\ sets |\nulldelimiterspace| to |1.2pt|.
%\enddesc
\begindesc
\cts nulldelimiterspace {\param{dimen}}
\explain
^^{定界符//空定界符的间隔}
此参数指定空\minref{定界符}生成的间隔的大小。
\PlainTeX\ 设定 |\null!-delimiterspace| 等于 |1.2pt|。
\enddesc

%\begindesc
%\cts scriptspace {\param{dimen}}
%\explain
%This parameter specifies the amount of space that \TeX\
%inserts before and after a subscript or superscript.
%The |\nonscript| command (\xref\nonscript) ^^|\nonscript|
%after a subscript or superscript cancels this space.
%\PlainTeX\ sets |\script!-space| to |0.5pt|.
%\enddesc
\begindesc
\cts scriptspace {\param{dimen}}
\explain
此参数指定 \TeX\ 在上标或下标前后插入的间隔的大小。
上标或下标之后的 |\nonscript| 命令（\xref\nonscript ）^^|\nonscript|
可以取消此间隔。
\PlainTeX\ 设定 |\script!-space| 等于 |0.5pt|。
\enddesc


%==========================================================================
%\section {Categorizing math constructs}
\section {分类数学结构}

%\begindesc
%\makecolumns 7/2:
%\cts mathord {}
%\cts mathop {}
%\cts mathbin {}
%\cts mathrel {}
%\cts mathopen {}
%\cts mathclose {}
%\cts mathpunct {}
%\explain
%These commands tell \TeX\ to treat the construct that follows as belonging
%to a particular ^{class} (see \knuth{page~154} for the definition
%of the classes).  They are listed here in the order of the class numbers,
%from $0$ to $6$.  Their primary
%effect is to adjust the spacing around the construct
%to be whatever it is for the specified class.
\begindesc
\makecolumns 7/2:
\cts mathord {}
\cts mathop {}
\cts mathbin {}
\cts mathrel {}
\cts mathopen {}
\cts mathclose {}
\cts mathpunct {}
\explain
这些命令让 \TeX\ 把随后的结构归入指定的^{类}（见\knuth{第~154~页}对类的定义）。
它们按照类编号的大小顺序排列，从 $0$ 到 $6$。
它们主要用于按照指定的类调整该结构两边的间隔大小。
%\example
%$\mathop{\rm minmax}\limits_{t \in A \cup B}\,t$
%% By treating minmax as a math operator, we can get TeX to
%% put something underneath it.
%|
%\produces
%$\mathop{\rm minmax}\limits_{t \in A \cup B}\,t$
%\endexample
%\enddesc
\example
$\mathop{\rm minmax}\limits_{t \in A \cup B}\,t$
% By treating minmax as a math operator, we can get TeX to
% put something underneath it.
|
\produces
$\mathop{\rm minmax}\limits_{t \in A \cup B}\,t$
\endexample
\enddesc

%\begindesc
%\cts mathinner {}
%\explain
%This command tells \TeX\ to treat the construct that follows
%as an ``inner formula'', e.g., a fraction, for spacing purposes.
%It resembles the class commands given just above.
%\enddesc
\begindesc
\cts mathinner {}
\explain
此命令让 \TeX\ 将随后的结构视为``内部公式''，比如分式，并据此调整间隔。
它与上面刚提到的类命令类似。
\enddesc

%==========================================================================
%\section {Special actions for math formulas}
\section {特殊处理数学公式}

%\begindesc
%\cts everymath {\param{token list}}
%\cts everydisplay {\param{token list}}
%\explain
%^^{displays//actions for every display}
%These parameters specify \minref{token} lists that \TeX\ inserts
%at the start of every text math or display math formula, respectively.
%You can
%take special actions at the start of each math formula by
%assigning those actions to |\everymath| or
%|\everydisplay|.  Don't forget that if you want both kinds of formulas to
%be affected, you need to set \emph{both} parameters.
%\example
%\everydisplay={\heartsuit\quad}
%\everymath = {\clubsuit}
%$3$ is greater than $2$ for large values of $3$.
%$$4>3$$
%|
%\produces
%\everydisplay={\heartsuit\quad}
%\everymath = {\clubsuit}
%$3$ is greater than $2$ for large values of $3$.
%$$4>3$$
%\endexample
%\enddesc
\begindesc
\cts everymath {\param{token list}}
\cts everydisplay {\param{token list}}
\explain
^^{陈列公式//作用到每个陈列公式}
这两个命令分别指定 \TeX\ 在每个文内公式或陈列公式开头插入的\minref{记号}列。
你可以利用 |\everymath| 或 |\everydisplay| 在每个数学公式开头作特殊处理。
你务必清楚，若你需要同时处理两种公式，你必须\emph{同时}设定这两个参数。
\example
\everydisplay={\heartsuit\quad}
\everymath = {\clubsuit}
$3$ is greater than $2$ for large values of $3$.
$$4>3$$
|
\produces
\everydisplay={\heartsuit\quad}
\everymath = {\clubsuit}
$3$ is greater than $2$ for large values of $3$.
$$4>3$$
\endexample
\enddesc

%\enddescriptions
%\eix^^{math}
%\endchapter
%\byebye
\enddescriptions
\eix^^{数学}

\ifcompletebook\else
\vskip4em{\sectionfonts\leftline{本章索引}}
\readindexfile{i}
\fi

\endchapter
\byebye