first commit

.gitignore

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+.vscode/
+.ds_store
+
+## Core latex/pdflatex auxiliary files:
+*.aux
+*.lof
+*.log
+*.lot
+*.fls
+*.out
+*.toc
+*.fmt
+*.fot
+*.cb
+*.cb2
+.*.lb
+
+## Intermediate documents:
+*.dvi
+*.xdv
+*-converted-to.*
+# these rules might exclude image files for figures etc.
+# *.ps
+# *.eps
+# *.pdf
+
+## Generated if empty string is given at "Please type another file name for output:"
+.pdf
+
+## Bibliography auxiliary files (bibtex/biblatex/biber):
+*.bbl
+*.bbl-SAVE-ERROR
+*.bcf
+*.bcf-SAVE-ERROR
+*.blg
+*-blx.aux
+*-blx.bib
+*.run.xml
+
+## Build tool auxiliary files:
+*.fdb_latexmk
+*.synctex
+*.synctex(busy)
+*.synctex.gz
+*.synctex.gz(busy)
+*.pdfsync
+*.rubbercache
+rubber.cache
+
+## Build tool directories for auxiliary files
+# latexrun
+latex.out/
+
+## Auxiliary and intermediate files from other packages:
+# algorithms
+*.alg
+*.loa
+
+# achemso
+acs-*.bib
+
+# amsthm
+*.thm
+
+# beamer
+*.nav
+*.pre
+*.snm
+*.vrb
+
+# changes
+*.soc
+
+# comment
+*.cut
+
+# cprotect
+*.cpt
+
+# elsarticle (documentclass of Elsevier journals)
+*.spl
+
+# endnotes
+*.ent
+
+# fixme
+*.lox
+
+# feynmf/feynmp
+*.mf
+*.mp
+*.t[1-9]
+*.t[1-9][0-9]
+*.tfm
+
+#(r)(e)ledmac/(r)(e)ledpar
+*.end
+*.?end
+*.[1-9]
+*.[1-9][0-9]
+*.[1-9][0-9][0-9]
+*.[1-9]R
+*.[1-9][0-9]R
+*.[1-9][0-9][0-9]R
+*.eledsec[1-9]
+*.eledsec[1-9]R
+*.eledsec[1-9][0-9]
+*.eledsec[1-9][0-9]R
+*.eledsec[1-9][0-9][0-9]
+*.eledsec[1-9][0-9][0-9]R
+
+# glossaries
+*.acn
+*.acr
+*.glg
+*.glo
+*.gls
+*.glsdefs
+*.lzo
+*.lzs
+*.slg
+*.slo
+*.sls
+
+# uncomment this for glossaries-extra (will ignore makeindex's style files!)
+# *.ist
+
+# gnuplot
+*.gnuplot
+*.table
+
+# gnuplottex
+*-gnuplottex-*
+
+# gregoriotex
+*.gaux
+*.glog
+*.gtex
+
+# htlatex
+*.4ct
+*.4tc
+*.idv
+*.lg
+*.trc
+*.xref
+
+# hypdoc
+*.hd
+
+# hyperref
+*.brf
+
+# knitr
+*-concordance.tex
+# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files
+# *.tikz
+*-tikzDictionary
+
+# listings
+*.lol
+
+# luatexja-ruby
+*.ltjruby
+
+# makeidx
+*.idx
+*.ilg
+*.ind
+
+# minitoc
+*.maf
+*.mlf
+*.mlt
+*.mtc[0-9]*
+*.slf[0-9]*
+*.slt[0-9]*
+*.stc[0-9]*
+
+# minted
+_minted*
+*.pyg
+
+# morewrites
+*.mw
+
+# newpax
+*.newpax
+
+# nomencl
+*.nlg
+*.nlo
+*.nls
+
+# pax
+*.pax
+
+# pdfpcnotes
+*.pdfpc
+
+# sagetex
+*.sagetex.sage
+*.sagetex.py
+*.sagetex.scmd
+
+# scrwfile
+*.wrt
+
+# svg
+svg-inkscape/
+
+# sympy
+*.sout
+*.sympy
+sympy-plots-for-*.tex/
+
+# pdfcomment
+*.upa
+*.upb
+
+# pythontex
+*.pytxcode
+pythontex-files-*/
+
+# tcolorbox
+*.listing
+
+# thmtools
+*.loe
+
+# TikZ & PGF
+*.dpth
+*.md5
+*.auxlock
+
+# titletoc
+*.ptc
+
+# todonotes
+*.tdo
+
+# vhistory
+*.hst
+*.ver
+
+# easy-todo
+*.lod
+
+# xcolor
+*.xcp
+
+# xmpincl
+*.xmpi
+
+# xindy
+*.xdy
+
+# xypic precompiled matrices and outlines
+*.xyc
+*.xyd
+
+# endfloat
+*.ttt
+*.fff
+
+# Latexian
+TSWLatexianTemp*
+
+## Editors:
+# WinEdt
+*.bak
+*.sav
+
+# Texpad
+.texpadtmp
+
+# LyX
+*.lyx~
+
+# Kile
+*.backup
+
+# gummi
+.*.swp
+
+# KBibTeX
+*~[0-9]*
+
+# TeXnicCenter
+*.tps
+
+# auto folder when using emacs and auctex
+./auto/*
+*.el
+
+# expex forward references with \gathertags
+*-tags.tex
+
+# standalone packages
+*.sta
+
+# Makeindex log files
+*.lpz
+
+# xwatermark package
+*.xwm
+
+# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib
+# option is specified. Footnotes are the stored in a file with suffix Notes.bib.
+# Uncomment the next line to have this generated file ignored.
+#*Notes.bib

hw3.tex

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+\documentclass[11pt]{article}
+
+\usepackage{myctex}
+\usepackage{tikz}
+\usepackage{tikz,fullpage}
+\usetikzlibrary{arrows,%
+                petri,%
+                topaths}%
+\usepackage{tkz-berge}
+\usepackage{tkz-graph}
+\GraphInit[vstyle=Normal]
+\usepackage[position=top]{subfig}
+
+
+\title{作业 3}
+\author{11 丛宇 202411081537}
+\date{}
+
+\begin{document}
+\maketitle
+\section*{第四章}
+
+\begin{itemize}
+  \item[1.] a 四个奇点 不行; b 可以; c 两个奇点 不行; d 可以
+  \item[3.] \begin{minipage}{0.9\textwidth}
+\begin{figure}[H]
+\centering
+\subfloat[]{
+\begin{tikzpicture}
+  \grCycle[prefix=v,Math=true,RA=1]{4};
+\end{tikzpicture}
+}
+\subfloat[]{
+
+\begin{tikzpicture}
+  \SetVertexMath
+  \Vertex[x=0,y=0]{v_1}
+  \Vertex[x=2,y=0]{v_2}
+  \Vertex[x=1,y=1.5]{v_3}
+  \Vertex[x=3,y=1.5]{v_4}
+  \Vertex[x=1,y=3]{v_5}
+  \Edge(v_1)(v_2)
+  \Edge(v_1)(v_3)
+  \Edge(v_2)(v_3)
+  % \Edge(v_2)(v_4)
+  \Edge(v_3)(v_4)
+  \Edge(v_3)(v_5)
+  \Edge(v_4)(v_5)
+\end{tikzpicture}
+}
+\subfloat[]{
+
+\begin{tikzpicture}
+  \grComplete[RA=1,prefix=v,Math=true]{4}
+\end{tikzpicture}
+}
+\subfloat[]{
+
+\begin{tikzpicture}
+  \grComplete[RA=1,prefix=v,Math=true]{2}
+\end{tikzpicture}
+}
+\end{figure}
+\end{minipage}
+
+\item[4.] 归纳. $n=3$时显然成立. 假设对所有$n\leq k$成立, 现在考虑$n=k+1$. $m=\frac{k(k-1)}{2}+2$. 取 $v$ 为 $\deg(v)$ 最小的点, $G\setminus\set{v}$ 的边数 $m'\geq \frac{k(k-1)}{2}+2-(k-1)\geq \frac{(k-1)(k-2)}{2}+2$. 由归纳假设 $G\setminus\set{v}$ 包含 Hamilton cycle. 由于 $\deg(v)\geq 2$, $G$也包含 Hamilton cycle.
+\item[5.] $S=\set{i,g,k,p,b,d}$, $\omega(G\setminus S)=7>6=|S|$.
+\item[10.] 
+\begin{itemize}
+  \item $G$ 不是2-connected则存在两点 $u,v$之间只有一条 vertex disjoint path. 假设$G$有 Hamilton cycle, 则$u,v$之间存在两条vertex disjoint path. 矛盾.
+  \item 假设二分图$G$ 存在 Hamilton cycle, $(v_1,...,v_n,v_1)$, 则所有奇数编号点属于$X$, 所有偶数编号点属于$Y$且$|X|=|Y|$. 与假设$|X|\neq|Y|$矛盾.
+\end{itemize}  
+\item[12.] %假设 $G$ 无hamilton path. 对任意$(u,v)\notin E$, $G+(u,v)$ 不包含Hamilton cycle. 
+令 $d'_1\leq \dots\leq d'_n$ 为 $G+(u,v)$ 的度序列, 令$d_1\leq \dots\leq d_n$ 为 $G$ 的度序列. 由于$d'$中只有两个位置与$d$不同且都只$+1$, 不妨假设两个序列对应顶点相同. 因此 $d'_m\leq d_m+1\leq m$ 或 $d'_{n-m}\leq d_{n-m+1}<n-m$ 对 $G+(u,v)$成立. 由定理8, $G+(u,v)$ 一定包含 Hamilton cycle, 因此 $G$ 中一定有Hamilton path.
+\item[13.] 假设$G$不是Hamilton图, 那么存在$m<n/2$, $G$ 度弱于$C_{m,n}$.
+\begin{align*}
+|E(G)| &\leq |E(C_{m,n})|\\
+& \leq \frac{1}{2}[m^2+(n-2m)(n-m-1)+m(n-1)]\\
+& = \binom{n-\delta}{2}+\delta^2-\frac{1}{2}(m-\delta)(2n-3m-3\delta-1)\\
+& \leq \binom{n-\delta}{2}+\delta^2
+\end{align*}
+最后一个不等号是观察到 $m\geq \delta$, $2n-3m-3\delta -1>n/2-3\delta-1\geq -1$. 有$2n-3m-3\delta -1\geq 0$ (是整数). 
+
+因此与假设矛盾.
+\end{itemize}
+\section*{第五章}
+\begin{itemize}
+  \item[2.] 考虑$T$中任何$\deg(v)=1$的点, 如果存在完美匹配, 与叶子相连的边都在匹配中. 可以确定性的知道$T$中所有边是否在匹配中. 因此完美匹配唯一(如果存在).
+  \item[5.] 3, 5
+  \item[7.] \begin{minipage}{0.4\textwidth}
+    \begin{tikzpicture}
+      \grComplete[RA=3,prefix=,Math=true]{9}
+    \end{tikzpicture}
+  \end{minipage}
+  cycles: $\begin{aligned}
+    &\set{1, 2, 8, 3, 7, 4, 6, 5, 9},\\
+    &\set{2, 3, 1, 4, 8, 5, 7, 6, 9},\\
+    &\set{3, 4, 2, 5, 1, 6, 8, 7, 9},\\
+    &\set{4, 5, 3, 6, 2, 7, 1, 8, 9}\\
+  \end{aligned}$
+\item[13.] 找$K_{5,5}$的最小权重完美匹配. 30
+\item[14.] SDR等价于bipartite matching 等价于 Hall's Theorem, 略
+\item[18.] 等价于证明Tutte–Berge formula.
+\begin{proof}
+令$d=\frac{1}{2}\min_{S\subset V} |V|-o(G-S)+|S|$, $\nu(G)$是$G$的最大匹配的大小.
+考虑任意点集$U\subset V$, 有$\nu(G)\leq |U|+\nu(G-U)\leq |U|+1/2(|V\setminus U|-o(G-U))$ 因为$G-U$的每个odd component都至少有一个点不能被匹配到. 因此 $\nu(G)\leq d$. 下面证明$d\leq \nu(G)$.
+
+需要证明$\exists S\subset V$ 使 $|V|+|S|+o(G-S)=\nu(G)$. 满足条件的$S$是 Edmonds-Gallai decomposition. 所以$d\leq \nu(G)$.
+
+
+\end{proof}
+\end{itemize}
+\end{document}

hw4.tex

@@ -0,0 +1,53 @@
+\documentclass[11pt]{article}
+
+\usepackage{myctex}
+\usepackage{tikz}
+\usepackage{tikz,fullpage}
+\usetikzlibrary{arrows,%
+                petri,%
+                topaths}%
+\usepackage{tkz-berge}
+\usepackage{tkz-graph}
+\GraphInit[vstyle=Normal]
+\usepackage[position=top]{subfig}
+
+
+\title{作业 4}
+\author{11 丛宇 202411081537}
+\date{\today}
+
+\begin{document}
+\maketitle
+\section*{第六章}
+\begin{itemize}
+\item[3.] 由欧拉公式 $\phi = m+2-n \leq 3n-6 +2-n=2n-4$.
+\item[4.] 
+\begin{enumerate}
+\item 由于极大平面图所有face degree都是3, 有 $3\phi=2m$. 由欧拉公式 $n+2m/3-m=2$得到$m=3n-6$
+\item $\phi=2m/3=2n-4$
+\item 对 $n$ 归纳,  $n=4$时显然成立. 假设$G$为$k$个点的极大简单平面图. 若$\kappa(G)=1$, $G$ 中一定包含degree 是1 的点(简单图), 与极大简单平面图矛盾; 若$\kappa(G)=2$, 则存在两个点$u,v\in G$ s.t. $G-\{u,v\}$使 $G$ 不联通. 那么在$G$一定可以表示成两个图$G_1,G_2$ 的2-clique sum. 但是由于 $n>4$, 平面图$G_1$与$G_2$都包含$u,v$之外的点, 一定存在$x\in G_1,y\in G_2$ s.t. $G+(x,y)$ 仍然是平面图, 与极大简单平面图假设矛盾. 因此$\kappa(G)\geq 3$.
+\end{enumerate}
+\item[5.] 显然. 假设$G$并非极大可平面图, 则存在$e\notin E$ s.t. $G+e$ 仍然是平面图. 与$n$个点的平面图边数$\leq 3n-6$矛盾.
+\item[6.]
+\begin{enumerate}
+\item 容易发现$n\leq 4$时不是简单图,$n=5$时不是平面图. 所以$n\geq 6$. 假设$G$ 中只有5个顶点的度数不超过5, 有 $2m\geq 5+16+6(n-5)$, 得到$m\geq 3n-4.5$ 与 $m=3n-6$ 矛盾
+\item 若$G$中只有11个点度数为5, 有$2m\geq 11*5+6(n-11)$, 得到$m\geq 3n-11/2$ 与 $m=3n-6$ 矛盾
+\end{enumerate}
+\item[8.]
+\begin{enumerate}
+\item 如果$G$有度数为2的点$u$,由于$n\geq 4$ 一定存在一个点$v$与$u$不相邻, 显然$G+(u,v)$仍然是平面图
+\item 假设只有两个度数$\leq 5$的点, $2m\geq 6(n-2)+2$, $m\geq 3n-4$ 矛盾
+\item $2m\geq 6(n-3)+9$, $m\geq 3n-4.5$
+\end{enumerate}
+\item[25.] 1. matroid $M$ 中的cocircuit = dual matroid $M^*$ 中的circuit. planar graph 对应的 graphic matroid dual = planar dual graph 上的graphic matroid. (互为对偶,只需要证明一边. 假设$G$的cycle $C$在$G^*$中的对应$C^*$不是cut, 那么令$C$内部的某个face对应的顶点为$v^*$, 在$G^*$中存在$(v^*,u^*)\in E$ s.t. $u^*$对应的face 不在$C$中. 显然$C$不是cycle) 2. Euler 图的cut大小一定是偶数. 由1知对偶图中所有cycle长度都是偶数. 无奇环一定是二分图.
+\end{itemize}
+
+
+\section*{第七章}
+\begin{itemize}
+\item[2.] 由推论2, 简单图点数$2k+1$, 边数$m=\Delta n/2=\frac{2k+1}{2}\Delta>k\Delta$, 所以$\chi'=\Delta+1$.
+\item[3.] 如果所有cycle长度都是偶数, $G$是二分图, $\chi'=\Delta$. 如果所有cycle长度都是奇数, 由于两个cycle的对称差还是cycle, 观察到任意两个奇环不能共用边,只能共用点. 将每个奇环看成一个点, 共用顶点的奇环之间连边, 得到的图一定是 forest, 否则包含简单偶环. 由于每个奇环只需要3种颜色即可染色且奇环之间形成树(要求$G$联通), 容易看出$\chi'\leq \Delta$.
+\item[4.] 对$n$归纳, base case是$P_3$,显然成立. 假设对所有$k\leq n-1$阶满足条件的简单图都有$\chi'=k-1$. 现在考虑$k$阶满足条件的简单图$G$, 其中$\deg(v)=k-1$. $G-v$满足归纳假设, 有一个$k-2$边染色, 所以一定存在一个$k-1$边染色. 重复使用$k-1$次引理1容易推出$G$也可$k-1$边染色.
+\item[5.] 
+\end{itemize}
+\end{document}

latexmkrc

@@ -0,0 +1 @@
+$pdflatex=q/xelatex -synctex=1 %O %S/

myctex.sty

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+\ProvidesPackage{myctex}
+\usepackage[fontset=none, scheme=plain]{ctex}
+
+\setCJKmainfont{SourceHanSerifSC-Regular.otf}[BoldFont={SourceHanSerifSC-Bold.otf}, ItalicFont=SourceHanSerifSC-Regular.otf]
+
+\RequirePackage[margin=1in]{geometry}
+\RequirePackage{latexsym,amsmath}
+\RequirePackage[dvipsnames]{xcolor}
+
+\RequirePackage[colorlinks=true,urlcolor=Blue,citecolor=Green,linkcolor=BrickRed,unicode]{hyperref}
+
+\RequirePackage[hyperref]{ntheorem}   % theorem numbering
+\RequirePackage{aliascnt}
+\RequirePackage{xspace}
+\RequirePackage{minted}
+
+
+\RequirePackage{graphicx} % essentials
+% \RequirePackage{microtype}
+\RequirePackage[charter]{mathdesign}    % change fonts
+\RequirePackage{berasans, beramono}
+\RequirePackage{eucal}
+\RequirePackage[nocompress]{cite}       % other convenient stuff
+\RequirePackage{enumerate}
+%\usepackage{pgf,tikz}
+\RequirePackage{wrapfig}
+\RequirePackage[medium,compact]{titlesec}
+\RequirePackage{verbatim}
+\newtheorem{theorem}{Theorem}[section]
+
+\newaliascnt{lemma}{theorem}
+\newaliascnt{corollary}{theorem}
+\newaliascnt{conjecture}{theorem}
+\newaliascnt{proposition}{theorem}
+
+\newtheorem{lemma}{Lemma}[section]
+\newtheorem{corollary}{Corollary}[section]
+\newtheorem{conjecture}{Conjecture}[section]
+\newtheorem{proposition}{Proposition}[section]
+\newtheorem{problem}{Problem}
+
+\newcommand{\lemmaautorefname}{Lemma}
+\newcommand{\corollaryautorefname}{Corollary}
+\newcommand{\conjectureautorefname}{Conjecture}
+\newcommand{\propositionautorefname}{Proposition}
+\newcommand{\problemautorefname}{Problem}
+
+\newenvironment{definition}[1][Definition]{\begin{trivlist}
+\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
+\newenvironment{example}[1][Example]{\begin{trivlist}
+\item[\hskip \labelsep {\bfseries #1}]}{\end{trivliST}}
+\newenvironment{remark}[1][Remark]{\begin{trivlist}
+\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
+
+\widowpenalty 5000
+\clubpenalty 5000
+
+\numberwithin{figure}{section}
+\pagenumbering{arabic}
+
+\urlstyle{same}     % dammit
+
+\hyphenation{co-or-din-ate co-or-din-ates half-plane half-space stereo-iso-mers stereo-iso-mer Round-table homol-ogous homol-ogy poly-hedron poly-hedra homeo-morphic homeo-morph-ism}
+
+\pdfstringdefDisableCommands{\let\boldmath\relax} % allow \boldmath in section titles
+
+\let\subsectionautorefname\sectionautorefname
+\let\subsubsectionautorefname\sectionautorefname
+
+
+% ----------------------------------------------------------------------
+%  Common Latin abbreviations, only one of which is actually kosher
+% ----------------------------------------------------------------------
+
+\def\etal{\emph{et~al.}}            % and others
+%\def\ie{\emph{i.e.}}           % that is
+\def\ie{\textcolor{red}{that is}}
+%\def\eg{\emph{e.g.}}           % for example
+\def\eg{\textcolor{red}{for example}}
+
+% -----------------------
+% Common Names
+% -----------------------
+\def\Mobius{M\"obius\xspace}
+
+% -----------------------
+% useful shortcuts
+% -----------------------
+
+\def\cost{\cent}
+
+% ----------------------------------------------------------------------
+%  Simple math stuff
+% ----------------------------------------------------------------------
+\renewcommand{\subset}{\subseteq}
+% ---- SYMBOLS ----
+\let\e\varepsilon               % a ``real'' epsilon — better yet, just use Unicode ε.
+%
+%  I give up.  These are in the wrong font, but my kludged versions 
+%  LOOK like kludges, especially \Z, \Q, and \C.
+%
+\def\Real{\mathbb{R}}
+\def\Proj{\mathbb{P}}
+\def\Hyper{\mathbb{H}}
+\def\Integer{\mathbb{Z}}
+\def\Natural{\mathbb{N}}
+\def\Complex{\mathbb{C}}
+\def\Rational{\mathbb{Q}}
+
+\let\N\Natural
+\let\Q\Rational
+\let\R\Real
+\let\Z\Integer
+\def\Rd{\Real^d}
+\def\RP{\Real\Proj}
+\def\CP{\Complex\Proj}
+
+
+
+% ---- OPERATORS (requires amsmath) ----
+\def\aff{\operatorname{aff}}        
+\def\area{\operatorname{area}}
+\def\argmax{\operatornamewithlimits{arg\,max}}
+\def\argmin{\operatornamewithlimits{arg\,min}}
+\def\Aut{\operatorname{Aut}}        % Automorphism group
+\def\card{\operatorname{card}}  % cardinality, deprecated for \abs
+\def\conv{\operatorname{conv}}  
+\def\E{\operatorname{E}}            % Expectation: $\E[X]$ (like \Pr)
+\def\EE{\operatornamewithlimits{E}}
+\def\Hom{\operatorname{Hom}}        % Homomorphism group
+\def\id{\operatorname{id}}      % identity
+\def\im{\operatorname{im}}      % image
+\def\lcm{\operatorname{lcm}}
+\def\lfs{\operatorname{lfs}}        % local feature size
+\def\poly{\operatorname{poly}}
+\def\polylog{\operatorname{polylog}}
+\def\rank{\operatorname{rank}}
+\def\rel{\operatorname{rel\,}}  % relative (interior, boundary, etc.)
+\def\sgn{\operatorname{sgn}}
+\def\vol{\operatorname{vol}}        % volume
+
+\def\fp#1{^{\underline{#1}}}        % falling powers: $n\fp{d}$
+\def\rp#1{^{\overline{#1}}}     % rising powers:  $n\rp{d}$
+
+\def\setsymdiff{\operatorname{\triangle}}
+% --- Darts and fences ---
+% less nice replacements for stmaryrd characters
+\@ifundefined{shortrightarrow}{\let\shortrightarrow\rightarrow}{}
+\@ifundefined{shortleftarrow}{\let\shortleftarrow\leftarrow}{}
+\@ifundefined{shortuparrow}{\let\shortuparrow\uparrow}{}
+\@ifundefined{shortdownarrow}{\let\shortdownarrow\downarrow}{}
+
+\def\arcto{\mathord\shortrightarrow}
+\def\arcfrom{\mathord\shortleftarrow}
+\def\arc#1#2{#1\arcto#2}
+\def\cra#1#2{#1\mathord\shortleftarrow#2}
+\def\fence#1#2{#1\mathord\shortuparrow#2}
+\def\ecnef#1#2{#1\mathord\shortdownarrow#2}
+
+% --- Cheap displaystyle operators ---
+\def\Frac#1#2{{\displaystyle\frac{#1}{#2}}}
+\def\Sum{\sum\limits}
+\def\Prod{\prod\limits}
+\def\Union{\bigcup\limits}
+\def\Inter{\bigcap\limits}
+\def\Lor{\bigvee\limits}
+\def\Land{\bigwedge\limits}
+\def\Lim{\lim\limits}
+\def\Max{\max\limits}
+\def\Min{\min\limits}
+
+% ---- RELATORS ----
+\def\deq{\stackrel{\scriptscriptstyle\triangle}{=}} % Use := instead.
+\def\into{\DOTSB\hookrightarrow}        % = one-to-one
+\def\onto{\DOTSB\twoheadrightarrow}
+\def\inonto{\DOTSB\lhook\joinrel\twoheadrightarrow}
+\def\from{\leftarrow}
+\def\tofrom{\leftrightarrow}
+\def\mapsfrom{\mathrel{\reflectbox{$\mapsto$}}}
+\def\longmapsfrom{\mathrel{\reflectbox{$\longmapsto$}}}
+
+% ---- DELIMITER PAIRS ----
+% --- always self-scaling delmiter pairs ---
+\def\set#1{\left\{ #1 \right\}}
+\def\floor#1{\left\lfloor #1 \right\rfloor}
+\def\ceil#1{\left\lceil #1 \right\rceil}
+\def\seq#1{\left\langle #1 \right\rangle}
+\def\abs#1{\left| #1 \right|}
+\def\norm#1{\left\| #1 \right\|}
+\def\paren#1{\left( #1 \right)}     % need better macro name!
+\def\brack#1{\left[ #1 \right]}     % need better macro name!
+\def\indic#1{\left[ #1 \right]}     % indicator variable; Iverson notation
+
+%
+%  Macros to typeset sets like {foo|bar} with all three delimiters
+%  correctly scaled to fit.  What I *really* want is a \middle macro
+%  that acts just like \left and \right.  Grumble.
+%
+%  Set version is handled by package braket.
+\def\Bigbar#1{\mathrel{\left|\vphantom{#1}\right.\n@space}}
+\def\Seqbar#1#2{\Seq{#1 \Bigbar{#1 #2} #2}}
+\def\Brackbar#1#2{\Brack{#1 \Bigbar{#1 #2} #2}}
+
+%  Math mode fbox
+\def\mfbox#1{\mathchoice{{\fbox{\ensuremath{\displaystyle #1}}}}
+            {{\fbox{\ensuremath{\textstyle #1}}}}
+            {{\fbox{\ensuremath{\scriptstyle #1}}}}
+            {{\fbox{\ensuremath{\scriptscriptstyle #1}}}}}
+
+% ---- ``ACCENTS'' ----
+%   NB: Commands equivalent to \lrarrowfill and \overlrarrow are
+%       already defined in amstex!
+\def\lrarrowfill{$\m@th\mathord\leftarrow\mkern-6mu%
+  \cleaders\hbox{$\mkern-2mu\mathord-\mkern-2mu$}\hfill
+  \mkern-6mu\mathord\rightarrow$}
+\def\overlrarrow#1{\vbox{\ialign{##\crcr
+      \lrarrowfill\crcr\noalign{\kern-\p@\nointerlineskip}
+      $\hfil\displaystyle{#1}\hfil$\crcr}}}
+
+\def\Line#1{\!\overlrarrow{\vphantom{t}\smash{\,#1\,}}\!}
+\def\Ray#1{\overrightarrow{\vphantom{t}#1\,}\!}
+\def\Seg#1{\overline{\vphantom{t}#1}}
+
+% ---- TEXT STYLES ----
+\def\mathsc#1{\text{\textsc{#1}}}
+\def\mathbs#1{\text{\boldmath\ensuremath{#1}}}
+\def\EMPH#1{\textbf{\emph{\boldmath #1}}}
+
+% ----------------------------------------------------------------------
+%  \begin{bigabstract}...\end{bigabstract}
+%  For use in documents with title pages.  Use normal-sized text in
+%  the abstract!
+% ----------------------------------------------------------------------
+\newenvironment{bigabstract}%
+{\begin{center}\large\textbf{\abstractname}\end{center}\begin{quotation}}%
+{\end{quotation}}
+
+% ----------------------------------------------------------------------
+%  Make captions smaller than the text, make their titles bold.
+%  Arguments:   #1 == figure name: "Figure 5"
+%               #2 == caption: "Papers by \Erdos, 1900--2000 (projected)"
+%  Less space after floats and before caption, since captions are smaller
+% ----------------------------------------------------------------------
+\@ifundefined{abovecaptionskip}{\newlength\abovecaptionskip}
+\long\def\@makecaption#1#2{
+   \vskip \abovecaptionskip
+   \setbox\@tempboxa\hbox{{\sffamily\footnotesize \textbf{#1.} #2}}
+   \ifdim \wd\@tempboxa >\hsize                     % if more than one line,
+       {\sffamily\footnotesize \textbf{#1.} #2\par}     %     typeset normally
+     \else                                          % else
+       \hbox to\hsize{\hfil\box\@tempboxa\hfil} %     center
+   \fi}
+\dbltextfloatsep 18pt plus 2pt minus 4pt    % was 20pt plus 2pt minus 4pt
+\textfloatsep 18pt plus 2pt minus 4pt       % was 20pt plus 2pt minus 4pt
+\abovecaptionskip 6pt                   % was 10pt
+
+% ----------------------------------------------------------------------
+%  Revised theorem environment:
+%  Add a period after the theorem number, and make theorems slanted
+%  instead of italic to help distinguish text from math.  Use
+%  \boldmath in theorem label in case it includes math.
+% ----------------------------------------------------------------------
+\def\@opargbegintheorem#1#2#3{\trivlist
+    \item[\hskip\labelsep{\bfseries\boldmath #1\ #2\ (#3).}]\slshape}
+\def\@begintheorem#1#2{\trivlist
+    \item[\hskip\labelsep{\bfseries\boldmath #1\ #2.}]\slshape}
+
+% ----------------------------------------------------------------------
+%  \newproof{type}{text}(style)[post]:
+%  Define a new type of unnumbered ``theorem'' environment.  The last
+%  two arguments are optional; most environments will not use them.
+%
+%  To change label style, put new style command in label:
+%       \newproof{comment}{\textsf{Comment}}
+% ----------------------------------------------------------------------
+\def\newproof#1#2{\@ifnextchar({\@snproof{#1}{#2}}{\@snproof{#1}{#2}()}}
+\def\@snproof#1#2(#3){\@ifnextchar[{\@xnproof{#1}{#2}{#3}}
+                                   {\@xnproof{#1}{#2}{#3}[]}}
+
+\def\@xnproof#1#2#3[#4]{%\expandafter\@ifdefinable\csname #1\endcsname
+{\global\@namedef{#1}{\@prf{#2}{#3}}\global\@namedef{end#1}{\@endprf{#4}}}}
+
+\def\@prf#1#2{\@ifnextchar[{\@xprf{#1}{#2}}{\@yprf{#1}{#2}}}
+\def\@xprf#1#2[#3]{\@yprf{#1\ (#3)}{#2}}
+\def\@yprf#1#2{\begin{trivlist}\item[\hskip\labelsep{\bfseries\boldmath #1:}]#2}
+
+\def\@endprf#1{#1\end{trivlist}}
+
+%
+%  Standard proof envrionment: last line has a halmos at the right margin.
+%  Insert \qed to override default halmos placement.
+%  Incompatible with amsprf package!
+%
+\newif\ifneedqed
+\global\needqedfalse
+\@ifundefined{square}{\let\square\Box}{}    % grumble latex2e 
+\def\QED{\ensuremath{{\square}}}
+\def\markatright#1{\leavevmode\unskip\nobreak\quad\hspace*{\fill}{#1}}
+\def\qed{\ifneedqed\markatright{\QED}\global\needqedfalse\else\null\fi}
+\newproof{proof}{Proof\global\needqedtrue}[\qed]
+\newproof{sketch}{Proof Sketch\global\needqedtrue}[\qed]
+
+%
+%  'Fake' proof environment, for use with wrapfig and similar packages
+%
+\newenvironment{rawproof}%
+   {\begin{trivlist}\item[\hskip\labelsep\textbf{Proof:}\global\needqedtrue]}%
+   {\qed\end{trivlist}}
+
+%
+%  If the proof ends with a displayed equation, use \aftermath just
+%  before \end{proof} to put the halmos in the ``right'' place.  
+%  Doesn't work worth a damn. 
+%
+\def\aftermath{\par\vspace{-\belowdisplayskip}\vspace{-\parskip}\vspace{-\baselineskip}}
+
+% ----------------------------------------------------------------------
+%  One-shot theoremish environment.  For named things like ``Zorn's
+%  Lemma'' or ``The Death Leap Principle'' or ``The Zone Theorem''.
+%  To get one-shot proofish environment, put \normalfont inside.
+%  New improved version stolen from Erik Demaine.
+% ----------------------------------------------------------------------
+\newenvironment{oneshot}[1]{\@begintheorem{#1}{\unskip}}{\@endtheorem}
+
+% Meh.
+\def\Comment#1{\textsf{\textsl{$\langle\!\langle$#1\/$\rangle\!\rangle$}}}
+
+\def\beginc@de{\noindent\begin{center}\begin{minipage}{1in}\tt
+        \begin{tabbing}~~~~\=~~~~\=~~~~\=~~~~\=~~~~\=~~~~\=~~~~\=\kill}
+\def\endc@de{\end{tabbing}\end{minipage}\end{center}}
+
+\newenvironment{code}{\beginc@de}{\endc@de}
+
+%
+%  These only work with <1 line of text.  Use sparingly!!
+%
+\def\textul#1{\underline{\smash{#1}\vphantom{.}}}
+\def\strike#1{\ensuremath{\overline{\text{\smash{#1}\vphantom{.}}}}}
+
+% ----------------------------------------------------------------------
+%  Include a file verbatim.  Searches the TEXINPUTS path for the file,
+%  even though that's probably not what you really want.  Stolen from
+%  ``verbatimfiles.sty'' by Chris Rowley and others.  [Don't use their 
+%  \vertbatimlisting macro; it has some bizarre side-effects!]
+% ----------------------------------------------------------------------
+\def\verbinput#1{
+\begingroup\@verbatim\frenchspacing\@vobeyspaces\input#1\endgroup
+}
+
+% ----------------------------------------------------------------------
+%  \now  -- Current time in h:mm AM/PM format
+%  \mdyy -- Today's date in m/d/yy format.  Forget Y2K; this is for humans!
+% ----------------------------------------------------------------------
+\newcount\timehh\timehh=\time
+\divide\timehh by 60
+\newcount\timemm\timemm=\time
+\count255=\timehh
+\multiply\count255 by -60
+\advance\timemm by \count255
+\newif\iftimePM
+\ifnum\timehh>11 \timePMtrue\else\timePMfalse\fi
+\ifnum\timehh<1 \advance\timehh by 12\fi
+\ifnum\timehh>12 \advance\timehh by -12\fi
+\def\now{\number\timehh:\ifnum\timemm<10 0\fi\number\timemm
+         \iftimePM pm\else am\fi}
+\newcount\mdYY\mdYY=\year
+\count255=\year
+\divide\count255 by 100
+\multiply\count255 by 100
+\advance\mdYY by -\count255
+\def\mdyy{\number\month/\number\day/\ifnum\mdYY<10 0\fi\number\mdYY}
+
+
+% ----------------------------------------------------------------------
+%  Notes to myself.  The margin flags are broken, thanks to an
+%  incompatibility with the geometry package.
+% ----------------------------------------------------------------------
+\def\n@te#1{\textsf{\boldmath \textbf{$\langle\!\langle$#1$\rangle\!\rangle$}}\leavevmode}
+\def\n@tew@rn{\GenericWarning{}{AUTHOR WARNING: Unresolved \protect\note}}
+
+\def\n@ten@te#1{\marginpar
+    [\hfill\llap{\textcolor{red}{{#1}$\!\Longrightarrow$}}]
+    {\rlap{\textcolor{red}{$\Longleftarrow\!${#1}}}}}
+
+% Warning: The textcomp and mathdesign packages are incompatible!
+%\def\n@tedingb@t{\@ifundefined{textmusicalnote}
+%           {$\maltese$}            % if you don't have textcomp
+%           {\textmusicalnote}}     % if you do have textcomp
+
+\def\n@tedingb@t{\textcircled{$\maltese$}}
+
+\def\note#1{\textcolor{red}{\n@tew@rn\n@te{\n@ten@te{\n@tedingb@t}#1}}}
+
+\def\sidenote#1{\marginpar{\tiny\textsf{#1}}}
+
+\setlength{\marginparsep}{0.1in}
+\setlength{\marginparwidth}{\paperwidth}
+\addtolength{\marginparwidth}{-\oddsidemargin}
+\addtolength{\marginparwidth}{-\textwidth}
+\addtolength{\marginparwidth}{-1.0\marginparsep} 
+\addtolength{\marginparwidth}{-1.0in}
+
+%----------------------------------------------------------------------
+%  TODO sets enclosed text in a gray rectangle with a red border
+%  with small sans-serif text.
+%----------------------------------------------------------------------
+\definecolor{TODOcolor}{cmyk}{0.05,0,0,0}
+\definecolor{TODOtxtcolor}{cmyk}{0,1,1,0}
+
+\newenvironment{TODO}
+   {\fboxrule1.5pt\begin{lrbox}{\@tempboxa}
+    \begin{minipage}{0.9\columnwidth}\small\sffamily\color{TODOtxtcolor}}
+   {\end{minipage}
+    \end{lrbox}
+    \medskip\centerline{\fcolorbox{TODOtxtcolor}{TODOcolor}{\usebox{\@tempboxa}}}
+    \sidenote{\LARGE\color{Red}$\bigstar\bigstar\bigstar$}\medskip}
+
+
+%----------------------------------------------------------------------
+%  Bibliography aliases, so I can use mnemonic citation keys and
+%  geom.bib at the same time.  \bibalias{foo}{bar} makes \cite{foo}
+%  act (almost) exactly like \cite{bar}.  I don’t actually use this.
+%----------------------------------------------------------------------
+\def\bibalias#1#2{%
+    \global\@namedef{b@#1}{%
+        {\@ifundefined{b@#2}{\textsf{?}}{}\csname b@#2\endcsname}}}
+
+% ----------------------------------------------------------------------
+%  Draft notice.  Put \DRAFT on the title page, in place of \date.
+% ----------------------------------------------------------------------
+\def\draftnotice{Preliminary draft --- \mdyy\ --- Not for distribution}
+
+\def\DRAFT{
+\date{\fbox{\textsf{\textbf{\draftnotice}}}}
+\def\@oddfoot{\footnotesize\hss\fbox{\textbf{\textsf{\draftnotice}}}\hss}
+\let\@evenfoot\@oddfoot
+}
+
+
+%----------------------------------------------------------------------
+% 'cramped' list style, stolen from Jeff Vitter.  Doesn't always work.
+%----------------------------------------------------------------------
+\def\cramped
+  {\parskip\@outerparskip\@topsep\parskip\@topsepadd2pt\itemsep0pt
+}
+
+%----------------------------------------------------------------------
+%  More footnote symbols, please.
+%----------------------------------------------------------------------
+\def\@fnsymbol#1{\ensuremath{\ifcase#1\or *\or \dagger\or \ddagger\or
+   \mathsection\or \mathparagraph\or \|\or **\or \dagger\dagger
+   \or \ddagger\ddagger \or \mathsection\mathsection \or
+   \mathparagraph\mathparagraph \else\@ctrerr\fi}}
+
+%----------------------------------------------------------------------
+%  TeXbook style quotations -- I should really use the right font, tho.
+%----------------------------------------------------------------------
+\newenvironment{rightquote}[1]
+{\flushright\begin{minipage}{#1\textwidth}
+\parskip0pt\footnotesize\sffamily\slshape}
+{\end{minipage}\par\bigskip}
+\def\quotee#1{\par\vspace{0.5ex}\begin{flushright}\unskip\textup{--- #1}\end{flushright}\unskip}
+
+%----------------------------------------------------------------------
+%  Creative Commons license
+%----------------------------------------------------------------------
+\def\CCLicense#1{\vfill\vbox to 0pt{\centering\tiny\sffamily \textcopyright\ Copyright \number\year\ Chao Xu.\\ This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License (\url{http://creativecommons.org/licenses/by-nc-sa/4.0/}).\\  Free distribution is strongly encouraged; commercial distribution is expressly forbidden.\quad  See {#1} for the most recent revision.\vss}}
+
+