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diff --git a/macros/xetex/latex/hfutthesis/hfut-chapters/math.tex b/macros/xetex/latex/hfutthesis/hfut-chapters/math.tex new file mode 100644 index 0000000000..a421a34715 --- /dev/null +++ b/macros/xetex/latex/hfutthesis/hfut-chapters/math.tex @@ -0,0 +1,189 @@ +% !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} |