commit c9b62cc0f548dc7368ace0d54f1caca016a5d077 Author: Yu Cong Date: Fri May 9 10:09:45 2025 +0800 first commit diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..b2b403a --- /dev/null +++ b/.gitignore @@ -0,0 +1,311 @@ +.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 \ No newline at end of file diff --git a/SourceHanSerifSC-Bold.otf b/SourceHanSerifSC-Bold.otf new file mode 100644 index 0000000..d5b31b1 Binary files /dev/null and b/SourceHanSerifSC-Bold.otf differ diff --git a/SourceHanSerifSC-Regular.otf b/SourceHanSerifSC-Regular.otf new file mode 100644 index 0000000..cb41635 Binary files /dev/null and b/SourceHanSerifSC-Regular.otf differ diff --git a/hw3.pdf b/hw3.pdf new file mode 100644 index 0000000..349146f Binary files /dev/null and b/hw3.pdf differ diff --git a/hw3.tex b/hw3.tex new file mode 100644 index 0000000..957068c --- /dev/null +++ b/hw3.tex @@ -0,0 +1,113 @@ +\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 $d'_{n-m}\leq d_{n-m+1}n/2-3\delta-1\geq -1$. 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} \ No newline at end of file diff --git a/hw4.pdf b/hw4.pdf new file mode 100644 index 0000000..c8b1ccd Binary files /dev/null and b/hw4.pdf differ diff --git a/hw4.tex b/hw4.tex new file mode 100644 index 0000000..4d3e8e6 --- /dev/null +++ b/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} \ No newline at end of file diff --git a/latexmkrc b/latexmkrc new file mode 100644 index 0000000..d78de37 --- /dev/null +++ b/latexmkrc @@ -0,0 +1 @@ +$pdflatex=q/xelatex -synctex=1 %O %S/ \ No newline at end of file diff --git a/myctex.sty b/myctex.sty new file mode 100644 index 0000000..62dfd3d --- /dev/null +++ b/myctex.sty @@ -0,0 +1,469 @@ +\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}} + +