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+% !TeX root = ../main.tex
+
+\chapter{数学}
+
+\section{数学符号和公式}
+《写作规范》没有对数学符号和公式做具体要求,这里我们让数学符号遵循 GB/T 3102.11—1993《物理科学和技术中使用的数学符号》
+\footnote{原 GB 3102.11—1993,自 2017 年 3 月 23 日起,该标准转为推荐性标准。}。
+该标准参照采纳 ISO 31-11:1992 \footnote{目前已更新为 ISO 80000-2:2019。},
+但是与 \TeX{} 默认的美国数学学会(AMS)的符号习惯有所区别。
+具体地来说主要有以下差异:
+\begin{enumerate}
+ \item 大写希腊字母默认为斜体,如
+ \begin{equation*}
+ \Gamma \Delta \Theta \Lambda \Xi \Pi \Sigma \Upsilon \Phi \Psi \Omega.
+ \end{equation*}
+ 注意有限增量符号 $\increment$ 固定使用正体,模板提供了 \cs{increment} 命令。
+ \item 小于等于号和大于等于号使用倾斜的字形 $\le$、$\ge$。
+ \item 积分号使用正体,比如 $\int$、$\oint$。
+ \item 行间公式积分号的上下限位于积分号的上下两端,比如
+ \begin{equation*}
+ \int_a^b f(x) \dif x.
+ \end{equation*}
+ 行内公式为了版面的美观,统一居右侧,如 $\int_a^b f(x) \dif x$ 。
+ \item
+ 偏微分符号 $\partial$ 使用正体。
+ \item
+ 省略号 \cs{dots} 按照中文的习惯固定居中,比如
+ \begin{equation*}
+ 1, 2, \dots, n \quad 1 + 2 + \dots + n.
+ \end{equation*}
+ \item
+ 实部 $\Re$ 和虚部 $\Im$ 的字体使用罗马体。
+\end{enumerate}
+
+以上数学符号样式的差异可以在模板中统一设置。
+但是还有一些需要用户在写作时进行处理:
+\begin{enumerate}
+ \item 数学常数和特殊函数名用正体,如
+ \begin{equation*}
+ \uppi = 3.14\dots; \quad
+ \symup{i}^2 = -1; \quad
+ \symup{e} = \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n.
+ \end{equation*}
+ \item 微分号使用正体,比如 $\dif y / \dif x$。
+ \item 向量、矩阵和张量用粗斜体(\cs{symbf}),如 $\symbf{x}$、$\symbf{\Sigma}$、$\symbfsf{T}$。
+ \item 自然对数用 $\ln x$ 不用 $\log x$。
+\end{enumerate}
+
+模板中使用 \pkg{unicode-math} 宏包配置数学字体。
+该宏包与传统的 \pkg{amsfonts}、\pkg{amssymb}、\pkg{bm}、
+\pkg{mathrsfs}、\pkg{upgreek} 等宏包\emph{不}兼容。
+本模板作了处理,用户可以直接使用 \cs{bm}, \cs{mathscr},
+\cs{upGamma} 等命令。
+关于数学符号更多的用法,参见 \pkg{unicode-math} 宏包的使用说明和符号列表
+\pkg{unimath-symbols}。
+
+
+
+\section{量和单位}
+
+宏包 \pkg{siunitx} 提供了更好的数字和单位支持:
+\begin{itemize}
+ \item \num{12345.67890}
+ \item \num{.3e45}
+ \item \si{kg.m.s^{-1}}
+ \item \si{\micro\meter} $\si{\micro\meter}$
+ \item \si{\ohm} $\si{\ohm}$
+ \item \numlist{10;20}
+ \item \numlist{10;20;30}
+ \item \SIlist{0.13;0.67;0.80}{\milli\metre}
+ \item \numrange{10}{20}
+ \item \SIrange{10}{20}{\degreeCelsius}
+\end{itemize}
+
+
+
+\section{定理和证明}
+
+示例文件中使用 \pkg{amsthm} 宏包配置了定理、引理和证明等环境。
+用户也可以使用 \pkg{ntheorem} 宏包。\par
+
+公式引用示例:见式~\ref{eq:example}
+\begin{definition}
+ If the integral of function $f$ is measurable and non-negative, we define
+ its (extended) \textbf{Lebesgue integral} by
+ \begin{equation}
+ \int f = \sup_g \int g,
+ \end{equation}
+ where the supremum is taken over all measurable functions $g$ such that
+ $0 \le g \le f$, and where $g$ is bounded and supported on a set of
+ finite measure.
+\end{definition}
+
+\begin{assumption}
+The communication graph is strongly connected.
+\end{assumption}
+
+\begin{example}
+ Simple examples of functions on $\mathbb{R}^d$ that are integrable
+ (or non-integrable) are given by
+ \begin{equation}
+ f_a(x) =
+ \begin{cases}
+ |x|^{-a} & \text{if } |x| \le 1, \\
+ 0 & \text{if } x > 1.
+ \end{cases}
+ \end{equation}
+ \begin{equation}
+ \label{eq:example}
+ F_a(x) = \frac{1}{1 + |x|^a}, \qquad \text{all } x \in \mathbb{R}^d.
+ \end{equation}
+ Then $f_a$ is integrable exactly when $a < d$, while $F_a$ is integrable
+ exactly when $a > d$.
+\end{example}
+
+\begin{lemma}[Fatou]
+ Suppose $\{f_n\}$ is a sequence of measurable functions with $f_n \geq 0$.
+ If $\lim_{n \to \infty} f_n(x) = f(x)$ for a.e. $x$, then
+ \begin{equation}
+ \int f \le \liminf_{n \to \infty} \int f_n.
+ \end{equation}
+\end{lemma}
+
+\begin{remark}
+ We do not exclude the cases $\int f = \infty$,
+ or $\liminf_{n \to \infty} f_n = \infty$.
+\end{remark}
+
+\begin{corollary}
+ Suppose $f$ is a non-negative measurable function, and $\{f_n\}$ a sequence
+ of non-negative measurable functions with
+ $f_n(x) \le f(x)$ and $f_n(x) \to f(x)$ for almost every $x$. Then
+ \begin{equation}
+ \lim_{n \to \infty} \int f_n = \int f.
+ \end{equation}
+\end{corollary}
+
+\begin{proposition}
+ Suppose $f$ is integrable on $\mathbb{R}^d$. Then for every $\epsilon > 0$:
+ \begin{enumerate}
+ \renewcommand{\theenumi}{\roman{enumi}}
+ \item There exists a set of finite measure $B$ (a ball, for example) such
+ that
+ \begin{equation}
+ \int_{B^c} |f| < \epsilon.
+ \end{equation}
+ \item There is a $\delta > 0$ such that
+ \begin{equation}
+ \int_E |f| < \epsilon \qquad \text{whenever } m(E) < \delta.
+ \end{equation}
+ \end{enumerate}
+\end{proposition}
+
+\begin{theorem}
+ Suppose $\{f_n\}$ is a sequence of measurable functions such that
+ $f_n(x) \to f(x)$ a.e. $x$, as $n$ tends to infinity.
+ If $|f_n(x)| \le g(x)$, where $g$ is integrable, then
+ \begin{equation}
+ \int |f_n - f| \to 0 \qquad \text{as } n \to \infty,
+ \end{equation}
+ and consequently
+ \begin{equation}
+ \int f_n \to \int f \qquad \text{as } n \to \infty.
+ \end{equation}
+\end{theorem}
+
+\begin{proof}
+ Trivial.
+\end{proof}
+
+\newtheorem*{axiomofchoice}{Axiom of choice}
+\begin{axiomofchoice}
+ Suppose $E$ is a set and ${E_\alpha}$ is a collection of
+ non-empty subsets of $E$. Then there is a function $\alpha
+ \mapsto x_\alpha$ (a ``choice function'') such that
+ \begin{equation}
+ x_\alpha \in E_\alpha,\qquad \text{for all }\alpha.
+ \end{equation}
+\end{axiomofchoice}
+
+\newtheorem{observation}{Observation}
+\begin{observation}
+ Suppose a partially ordered set $P$ has the property
+ that every chain has an upper bound in $P$. Then the
+ set $P$ contains at least one maximal element.
+\end{observation}
+\begin{proof}[A concise proof]
+ Obvious.
+\end{proof}