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% !TeX root = ../main.tex
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% 本LaTeX模板的使用示例
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\chapter{模板使用说明}
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本章给出了撰写论文时可能用到的\LaTeX 基本命令，同学们可以对照./main.tex源码和“输出示例.pdf”文件各部分的对应内容学习模板各命令的作用。
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%==============================
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||||
\section{参考文献引用}
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\BUAAThesis{} 使用BibTeX处理参考文献，方便使用者管理参考文献条目。
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参考文献的具体内容以纯文本形式保存在根目录下的ref.bib文件中，每条参考文献信息都严格按照BibTeX格式写入文件。
|
||||
大部分文献数据库均支持将参考文献导出为BibTeX格式，使用者只需将导出的文献信息顺序写入ref.bib，并在文中按索引引用即可。
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||||
参考文献引用按照参考国标7714和北航学位论文撰写规范执行，如果导出参考文献信息时缺失出版地等项目导致引用内容出现“[出版地不详]”等缺省提示，请使用本模板提供的“GBT7714-2015-NoWarning.bst”格式文件屏蔽提示信息。
|
||||
本模板提供了多种引用参考文献命令，通常在正文中使用\verb|\upcite{}|以上标形式引用文献。
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|
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%--------------------------------
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||||
\subsection{数字标注}
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\noindent
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\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
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\verb|\cite{knuth86a}| & \cite{knuth86a}\\
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\verb|\citet{knuth86a}| & \citet{knuth86a}\\
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\verb|\citet[chap.~2]{knuth86a}| & \citet[chap.~2]{knuth86a}\\[0.5ex]
|
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\verb|\citep{knuth86a}| & \citep{knuth86a}\\
|
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\verb|\citep[chap.~2]{knuth86a}| & \citep[chap.~2]{knuth86a}\\
|
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\verb|\citep[see][]{knuth86a}| & \citep[see][]{knuth86a}\\
|
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\verb|\citep[see][chap.~2]{knuth86a}| & \citep[see][chap.~2]{knuth86a}\\[0.5ex]
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\verb|\citet*{knuth86a}| & \citet*{knuth86a}\\
|
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\verb|\citep*{knuth86a}| & \citep*{knuth86a}\\
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\end{tabular}
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\par\noindent
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\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
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\verb|\citet{knuth86a,tlc2}| & \citet{knuth86a,tlc2}\\
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\verb|\citep{knuth86a,tlc2}| & \citep{knuth86a,tlc2}\\
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\verb|\cite{knuth86a,knuth84}| & \cite{knuth86a,knuth84}\\
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\verb|\upcite{knuth86a,knuth84}| & \upcite{knuth86a,knuth84}\\
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\verb|\citet{knuth86a,knuth84}| & \citet{knuth86a,knuth84}\\
|
||||
\verb|\citep{knuth86a,knuth84}| & \citep{knuth86a,knuth84}\\
|
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\verb|\cite{knuth86a,knuth84,tlc2}| & \cite{knuth86a,knuth84,tlc2}\\
|
||||
\end{tabular}
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||||
|
||||
%--------------------------------
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||||
\subsection{数字标注-上标形式}
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||||
\noindent
|
||||
\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
|
||||
\verb|\upcite{knuth86a}| & \upcite{knuth86a}\\
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||||
\verb|\upcite{knuth86a,knuth84,tlc2}| & \upcite{knuth86a,knuth84,tlc2}\\
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||||
\end{tabular}
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\par\noindent
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%--------------------------------
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||||
\subsection{著者-出版年制标}
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正文中引用参考文献的标注方法可以采用“顺序编码制”，也可以采用“著者-出版年制”。
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撰写学位论文时仅选择一种，并全文保持统一。
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||||
本模板默认的标注形式为顺序编码制，如果要切换成著者-出版年制，需采用命令\verb|\citestyle{authoryear}|切换。
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||||
著者-出版年制标注形式如下：
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||||
\citestyle{authoryear}
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\noindent
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||||
\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
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||||
\verb|\cite{db}| & \cite{db}\\
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||||
\verb|\citet{knuth86a}| & \citet{knuth86a}\\
|
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\verb|\citet[chap.~2]{knuth86a}| & \citet[chap.~2]{knuth86a}\\[0.5ex]
|
||||
\verb|\citep{knuth86a}| & \citep{knuth86a}\\
|
||||
\verb|\citep[chap.~2]{knuth86a}| & \citep[chap.~2]{knuth86a}\\
|
||||
\verb|\citep[see][]{knuth86a}| & \citep[see][]{knuth86a}\\
|
||||
\verb|\citep[see][chap.~2]{knuth86a}| & \citep[see][chap.~2]{knuth86a}\\[0.5ex]
|
||||
\verb|\citet*{knuth86a}| & \citet*{knuth86a}\\
|
||||
\verb|\citep*{knuth86a}| & \citep*{knuth86a}\\
|
||||
\end{tabular}
|
||||
\par\noindent
|
||||
\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
|
||||
\verb|\citet{knuth86a,tlc2}| & \citet{knuth86a,tlc2}\\
|
||||
\verb|\citep{knuth86a,tlc2}| & \citep{knuth86a,tlc2}\\
|
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\verb|\cite{knuth86a,knuth84}| & \cite{knuth86a,knuth84}\\
|
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\verb|\citet{knuth86a,knuth84}| & \citet{knuth86a,knuth84}\\
|
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\verb|\citep{knuth86a,knuth84}| & \citep{knuth86a,knuth84}\\
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\end{tabular}
|
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\citestyle{numbers}
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|
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%--------------------------------
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||||
\subsection{其他形式的标注}
|
||||
\noindent
|
||||
\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
|
||||
\verb|\citealt{tlc2}| & \citealt{tlc2}\\
|
||||
\verb|\citealt*{tlc2}| & \citealt*{tlc2}\\
|
||||
\verb|\citealp{tlc2}| & \citealp{tlc2}\\
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||||
\verb|\citealp*{tlc2}| & \citealp*{tlc2}\\
|
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\verb|\citealp{tlc2,knuth86a}| & \citealp{tlc2,knuth86a}\\
|
||||
\verb|\citealp[pg.~32]{tlc2}| & \citealp[pg.~32]{tlc2}\\
|
||||
\verb|\citenum{tlc2}| & \citenum{tlc2}\\
|
||||
\verb|\citetext{priv.\ comm.}| & \citetext{priv.\ comm.}\\
|
||||
\end{tabular}
|
||||
|
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\noindent
|
||||
\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
|
||||
\verb|\citeauthor{tlc2}| & \citeauthor{tlc2}\\
|
||||
\verb|\citeauthor*{tlc2}| & \citeauthor*{tlc2}\\
|
||||
\verb|\citeyear{tlc2}| & \citeyear{tlc2}\\
|
||||
\verb|\citeyearpar{tlc2}| & \citeyearpar{tlc2}\\
|
||||
\end{tabular}
|
||||
|
||||
\section{算法、表格和插图}
|
||||
|
||||
根据北航学位论文撰写规范要求，本模板重写了部分图表浮动体环境，但使用方法与官方宏包一致，使用者可查看各宏包的官方文档获取详细使用说明。
|
||||
需要注意的是图表浮动体与正文之间的距离是弹性的，撰写论文时可以根据内容进行调整。
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\subsection{算法环境}
|
||||
|
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本模板使用 \texttt{algorithm2e} 宏包实现算法环境。下面是四种算法环境示例。
|
||||
|
||||
\begin{algorithm}[htp]
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||||
%\SetAlgoLined
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||||
%\SetAlgoVlined
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\caption{A How to (plain).}
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||||
\KwData{this text}
|
||||
\KwResult{how to write algorithm with \LaTeX2e{} }
|
||||
|
||||
initialization\;
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||||
\While{not at end of this document}{
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||||
read current\;
|
||||
\eIf{understand}{
|
||||
go to next section\;
|
||||
current section becomes this one\;
|
||||
}{
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||||
go back to the beginning of current section\;
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||||
}
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||||
}
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||||
\end{algorithm}
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\RestyleAlgo{ruled}
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||||
\begin{algorithm}[htp]
|
||||
\caption{A How to (ruled).}
|
||||
\KwData{this text}
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||||
\KwResult{how to write algorithm with \LaTeX2e{} }
|
||||
|
||||
initialization\;
|
||||
\While{not at end of this document}{
|
||||
read current\;
|
||||
\eIf{understand}{
|
||||
go to next section\;
|
||||
current section becomes this one\;
|
||||
}{
|
||||
go back to the beginning of current section\;
|
||||
}
|
||||
}
|
||||
\end{algorithm}
|
||||
|
||||
\RestyleAlgo{boxed}
|
||||
\begin{algorithm}[htp]
|
||||
\caption{A How to (boxed).}
|
||||
\KwData{this text}
|
||||
\KwResult{how to write algorithm with \LaTeX2e{} }
|
||||
|
||||
initialization\;
|
||||
\While{not at end of this document}{
|
||||
read current\;
|
||||
\eIf{understand}{
|
||||
go to next section\;
|
||||
current section becomes this one\;
|
||||
}{
|
||||
go back to the beginning of current section\;
|
||||
}
|
||||
}
|
||||
\end{algorithm}
|
||||
|
||||
\RestyleAlgo{boxruled}
|
||||
\begin{algorithm}[htp]
|
||||
\caption{A How to (boxruled).}
|
||||
\KwData{this text}
|
||||
\KwResult{how to write algorithm with \LaTeX2e{} }
|
||||
|
||||
initialization\;
|
||||
\While{not at end of this document}{
|
||||
read current\;
|
||||
\eIf{understand}{
|
||||
go to next section\;
|
||||
current section becomes this one\;
|
||||
}{
|
||||
go back to the beginning of current section\;
|
||||
}
|
||||
}
|
||||
\end{algorithm}
|
||||
\vspace{5em}
|
||||
\subsection{三线表}
|
||||
学位论文中的表格推荐使用三线表形式，如表~\ref{tab:exampletable}。
|
||||
|
||||
\begin{table}[!h]
|
||||
\centering
|
||||
\caption{表的标题}
|
||||
\label{tab:exampletable}
|
||||
\begin{tabular}{>{\centering\arraybackslash}p{4cm}>{\centering\arraybackslash}p{4cm}}
|
||||
\toprule
|
||||
操作系统 & TeX 发行版 \\
|
||||
\midrule
|
||||
所有 & TeX Live \\
|
||||
macOS & MacTeX \\
|
||||
Windows & MikTeX \\
|
||||
\bottomrule
|
||||
\end{tabular}
|
||||
\end{table}
|
||||
|
||||
当表题较长时，本模板会自适应换行处理，如表~\ref{tab:example_long_table}。
|
||||
|
||||
\begin{table}[!h]
|
||||
\centering
|
||||
\caption{长表题示例\upcite{zhudaoqian}：考虑到实验中使用到的面内磁场的大小，以及得到的磁矩稳定翻转条件，在计算中使$\alpha$固定，其余参数则与实验中相同}
|
||||
\label{tab:example_long_table}
|
||||
\begin{tabular}{c c c c c c c}
|
||||
\toprule
|
||||
\multirow{2}{*}{\textbf{材料体系}} & \multicolumn{6}{c}{\textbf{参数}} \\
|
||||
& $t_F$ & $\mu_0H_{K,eff}$ & $M_s(A\cdot m^(-1))$ & $|\Theta_SH|$ & $\iota$ & $\mu_0H_x$ \\ \midrule
|
||||
W/CoFeB & 1 nm & 0.29T & $9\times 10^5$ & 0.32 & 0.31 & 24mT \\
|
||||
Ta/CoFeB & 1.2 nm & 0.25T & $1\times 10^6$ & 0.03 & 2 & 20mT \\ \bottomrule
|
||||
\end{tabular}
|
||||
\end{table}
|
||||
|
||||
\vspace{-1pt}
|
||||
\subsection{长表格}
|
||||
|
||||
超过一页的表格要使用专门的 \texttt{longtable} 环境（表~\ref{tab:longtable}）。\\
|
||||
|
||||
\begin{longtable}[h]{ccc}
|
||||
% 首页表头
|
||||
\caption[长表格演示]{长表格演示}
|
||||
\label{tab:longtable}\\
|
||||
\toprule
|
||||
名称 & 说明 & 备注\\
|
||||
\midrule
|
||||
\endfirsthead
|
||||
% 续页表头
|
||||
\caption[]{长表格演示（续）} \\
|
||||
\toprule
|
||||
名称 & 说明 & 备注 \\
|
||||
\midrule
|
||||
\endhead
|
||||
% 首页表尾
|
||||
\hline
|
||||
\multicolumn{3}{r}{\small 续下页}
|
||||
\endfoot
|
||||
% 续页表尾
|
||||
\bottomrule
|
||||
\endlastfoot
|
||||
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
AAAAAAAAAAAA & BBBBBBBBBBB & CCCCCCCCCCCCCC \\
|
||||
\end{longtable}
|
||||
|
||||
\subsection{插图}
|
||||
|
||||
本模板支持插入pdf、eps、jpg和png等格式的图片。
|
||||
|
||||
\begin{figure}[!h]
|
||||
\centering
|
||||
\includegraphics[width=.5\textwidth]{pic/logo-buaa}
|
||||
\caption{测试图片\\第二行题注}
|
||||
\label{fig:logo}
|
||||
\end{figure}
|
||||
|
||||
模板使用了subcaption子图宏包，使用者也可自行替换惯用的宏包，一个简单的子图环境如图所示。
|
||||
|
||||
\begin{figure}[!h]
|
||||
\centering
|
||||
\subcaptionbox{子图1}{\includegraphics[width = 0.45\linewidth]{pic/logo-buaa.eps}\vspace{50pt}}
|
||||
\hfill
|
||||
\subcaptionbox{子图2}{\includegraphics[width = 0.3\linewidth]{pic/buaa-mark.jpg}}
|
||||
\caption{测试图片}
|
||||
\end{figure}
|
||||
|
||||
\section{数学环境}
|
||||
|
||||
\subsection{数学符号}
|
||||
|
||||
模板定义了一些正体（upright）的数学符号：
|
||||
\begin{center}
|
||||
\begin{tabular}{>{\centering\arraybackslash}p{4cm}>{\centering \arraybackslash}p{4cm}}
|
||||
\toprule
|
||||
符号 & 命令 \\
|
||||
\midrule
|
||||
常数$\eu$ & \verb|\eu| \\
|
||||
复数单位$\iu$ & \verb|\iu| \\
|
||||
微分符号$\diff$ & \verb|\diff| \\
|
||||
$\argmax$ & \verb|\argmax| \\
|
||||
$\argmin$ & \verb|\argmin| \\
|
||||
\bottomrule
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
更多的例子：
|
||||
\begin{equation}
|
||||
\eu^{\iu\pi} + 1 = 0
|
||||
\end{equation}
|
||||
\begin{equation}
|
||||
\frac{\diff^2u}{\diff t^2} = \int f(x) \diff x
|
||||
\end{equation}
|
||||
\begin{equation}
|
||||
\argmin_x f(x)
|
||||
\end{equation}
|
||||
|
||||
\subsection{定理、引理和证明}
|
||||
模板中使用amsmath和amsthm宏包配置了定理、引理和证明等环境。举例如下。
|
||||
\begin{definition}
|
||||
如果函数$f$的积分是可测的且非负的，我们通过以下方式定义其（扩展）\textbf{Lebesgue积分}：
|
||||
\begin{equation}
|
||||
\int f = \sup_g \int g,
|
||||
\end{equation}
|
||||
其中，上确界是在所有满足 $0 \leq g \leq f$ 的可测函数 $g$ 上取得的，且 $g$ 是有界的，并且其支撑集具有有限测度。
|
||||
\end{definition}
|
||||
|
||||
\begin{example}
|
||||
$\mathbf{R}^d$ 上的可积（或不可积）函数的简单示例如下：
|
||||
\begin{equation}
|
||||
f_a(x) =
|
||||
\begin{cases}
|
||||
|x|^{-a} & \text{if } |x| \leq 1,\\
|
||||
0 & \text{if } x > 1.
|
||||
\end{cases}
|
||||
\end{equation}
|
||||
\begin{equation}
|
||||
F_a(x) = \frac{1}{1 + |x|^a}, \qquad \text{对所有~} x \in \mathbf{R}^d.
|
||||
\end{equation}
|
||||
那么 $f_a$ 在 $a < d$ 时恰好是可积的，而 $F_a$ 在 $a > d$ 时恰好是可积的。
|
||||
\end{example}
|
||||
|
||||
\begin{lemma}[Fatou]
|
||||
假设 $\{f_n\}$ 是一列满足 $f_n \geq 0$ 的可测函数。如果对几乎处处的 $x$ 都有 $\lim_{n \to \infty} f_n(x) = f(x)$，那么
|
||||
\begin{equation}
|
||||
\int f \leq \liminf_{n \to \infty} \int f_n.
|
||||
\end{equation}
|
||||
\end{lemma}
|
||||
|
||||
\begin{remark}
|
||||
我们不排除 $\int f = \infty$ 或 $\liminf_{n \to \infty} f_n = \infty$ 的情况。
|
||||
\end{remark}
|
||||
|
||||
\begin{corollary}
|
||||
假设 $f$ 是一个非负可测函数，$\{f_n\}$ 是一列非负可测函数，满足对几乎处处的 $x$ 有 $f_n(x) \leq f(x)$ 且 $f_n(x) \to f(x)$。那么
|
||||
\begin{equation}
|
||||
\lim_{n \to \infty} \int f_n = \int f.
|
||||
\end{equation}
|
||||
\end{corollary}
|
||||
|
||||
\begin{proposition}
|
||||
假设 $f$ 在 $\mathbf{R}^d$ 上可积。那么对于每一个 $\epsilon > 0$：
|
||||
\begin{enumerate}
|
||||
\renewcommand{\theenumi}{\roman{enumi}}
|
||||
\item 存在一个有限测度的集合 $B$（例如一个球），使得
|
||||
\begin{equation}
|
||||
\int_{B^c} |f| < \epsilon.
|
||||
\end{equation}
|
||||
\item 存在一个 $\delta > 0$，使得
|
||||
\begin{equation}
|
||||
\int_E |f| < \epsilon \qquad \text{每当~} m(E) < \delta.
|
||||
\end{equation}
|
||||
\end{enumerate}
|
||||
\end{proposition}
|
||||
|
||||
\begin{theorem}
|
||||
假设 $\{f_n\}$ 是一列可测函数，满足当 $n$ 趋向无穷时，$f_n(x)$ 几乎处处收敛于 $f(x)$。如果 $|f_n(x)| \leq g(x)$，其中 $g$ 是可积函数，那么
|
||||
\begin{equation}
|
||||
\int |f_n - f| \to 0 \qquad \text{当~} n \to \infty,
|
||||
\end{equation}
|
||||
因此
|
||||
\begin{equation}
|
||||
\int f_n \to \int f \qquad \text{当~} n \to \infty.
|
||||
\end{equation}
|
||||
\end{theorem}
|
||||
|
||||
\begin{proof}
|
||||
略。
|
||||
\end{proof}
|
||||
|
||||
\subsection{自定义}
|
||||
|
||||
\newtheorem*{axiomofchoice}{选择公理}
|
||||
\begin{axiomofchoice}
|
||||
假设 $E$ 是一个集合，${E_\alpha}$ 是 $E$ 的一组非空子集。那么存在一个函数 $\alpha \mapsto x_\alpha$（一个“选择函数”），使得
|
||||
\begin{equation}
|
||||
x_\alpha \in E_\alpha,\qquad \text{对所有 }\alpha.
|
||||
\end{equation}
|
||||
\end{axiomofchoice}
|
||||
|
||||
\newtheorem{observation}{定理}[chapter]
|
||||
\begin{observation}
|
||||
假设一个偏序集 $P$ 具有这样的性质，即每个链在 $P$ 中都有一个上界。那么集合 $P$ 至少包含一个最大元素。
|
||||
\end{observation}
|
||||
\begin{proof}[证明]
|
||||
略。
|
||||
\end{proof}
|
||||
|
||||
\newtheorem{observationvar2}[observation]{定理2}
|
||||
\begin{observationvar2}
|
||||
假设一个偏序集 $P$ 满足每个链在 $P$ 中都有一个上界。那么集合 $P$ 至少包含一个最大元素。
|
||||
\end{observationvar2}
|
||||
\begin{proof}[证明]
|
||||
略。
|
||||
\end{proof}
|
||||
|
||||
\clearpage
|
||||
|
||||
Reference in New Issue
Block a user