From 82d8bf7ac34630d23ccfe1d09af5f1eb2892f3d2 Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Tue, 4 Nov 2025 15:25:06 +0800 Subject: [PATCH] notes --- .gitignore | 4 +- chao.sty | 533 +++++++++++++++++++++++++++++++++++++++++++++++++++++ notes.tex | 95 ++++++++++ 3 files changed, 631 insertions(+), 1 deletion(-) create mode 100644 chao.sty create mode 100644 notes.tex diff --git a/.gitignore b/.gitignore index 8efabbb..a6c9518 100644 --- a/.gitignore +++ b/.gitignore @@ -10,4 +10,6 @@ *.fls *.fdb_latexmk .vscode -__pycache__ \ No newline at end of file +__pycache__ +notes.* +!notes.tex \ No newline at end of file diff --git a/chao.sty b/chao.sty new file mode 100644 index 0000000..f939853 --- /dev/null +++ b/chao.sty @@ -0,0 +1,533 @@ +% ========================-*- LaTeX -*-========================= +% +% chao.sty -- modified from jeffe.sty +% jeffe.sty can be found at http://web.engr.illinois.edu/~jeffe/pubs/latex.html +% +% ========================================================== + + +\ProvidesPackage{chao} + + +% remarkable size +\newif\ifchao@remarkable +\DeclareOption{remarkable}{\chao@remarkabletrue} +\PassOptionsToPackage{hyphens}{url} +\PassOptionsToPackage{unicode}{hyperref} +\PassOptionsToPackage{naturalnames}{hyperref} +\ProcessOptions + + + + + +\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} + +% st mary fonts +\RequirePackage{stmaryrd} % comment out if unavailable +\SetSymbolFont{stmry}{bold}{U}{stmry}{m}{n} +\RequirePackage{wasysym} +% nagging +\RequirePackage[l2tabu, orthodox]{nag} +\RequirePackage[all,warning]{onlyamsmath} + + + +\RequirePackage{graphicx} % essentials +\RequirePackage{microtype} +\RequirePackage[mathletters]{ucs} % allow Unicode in .tex file +\RequirePackage[utf8]{inputenc} + +\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} +\usepackage{CJKutf8} + + +\newtheorem{theorem}{Theorem} + +\newaliascnt{lemma}{theorem} +\newaliascnt{corollary}{theorem} +\newaliascnt{conjecture}{theorem} +\newaliascnt{proposition}{theorem} + +\newtheorem{lemma}{Lemma} +\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 + +% line numbers (be nice to the referee) +\RequirePackage[mathlines, pagewise]{lineno} +%\linenumbers +\nolinenumbers +\renewcommand\linenumberfont{\normalfont\tiny\sffamily\color{gray}} +\setlength\linenumbersep{2em} + +\numberwithin{figure}{section} + +\ifchao@remarkable + \RequirePackage[paperheight=209.6mm, paperwidth=157.2mm, top=0.2in,bottom=0.2in,right=0.2in,left=0.2in]{geometry} + \tolerance=10000 +\else + \RequirePackage[margin=1in]{geometry} +\fi + +%\setlength{\topmargin}{-0.5in} \setlength{\textwidth}{6.5in} +%\setlength{\oddsidemargin}{0.0in} \setlength{\textheight}{9.1in} + +%\pagestyle{empty} +%\newlength{\pagewidth} +%\setlength{\pagewidth}{6.5in} %\pagestyle{empty} +\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 + + +% Chao addition +\def\email#1{\href{mailto:#1}{#1}} +% chinese +\def\zh#1{\begin{CJK}{UTF8}{gbsn}#1\end{CJK}} + + + +% ---------------------------------------------------------------------- +% 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} +\def\F{\mathbb{F}} + + + +% ---- 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}} + + diff --git a/notes.tex b/notes.tex new file mode 100644 index 0000000..29755a2 --- /dev/null +++ b/notes.tex @@ -0,0 +1,95 @@ +\documentclass[a4paper]{article} +\usepackage{chao} +\title{Cogirth of Perturbed Graphic Matroids} +\author{} +\date{} +\begin{document} +% \maketitle + +We want to use basepacking on perturbed graphic matroid. Basepacking works for deletion closed matroid classes with constant cogirth-packing gap. +PGMs are closed under deletion. It remains to show PGMs have constant gap. + +Geelen and Kapadia reduce the cogirth problem on PGMs to finding the cogirth of $(1,t)$-signed grafts. +We work on binary matroid $M$ defined on the following binary matrix: + +\[ +\begin{array}{ccc} + & \begin{array}{cc} E(G) & \tau \end{array} \\ + \begin{array}{r} V(G) \\ \omega \end{array} + & + \begin{pmatrix} + A(G) & T \\ + \sigma & \alpha + \end{pmatrix} +\end{array} +\] +where $G$ is a connect graph and $A(G)$ is the incidence matrix of $G$. +The goal is to find the cogirth of $M/\tau$. + +This matroid can be built from a graphic matroid via a series of extension, contraction and co-extension. We first show that $M([A,T])/T$ has constant gap, and then prove a constant gap of $M/\tau$. + +\begin{lemma} +Let $M$ be a binary matroid with binary representation $B\in \F_2^{n\times m}$. If $M$ has constant gap, then $M\left(\begin{bmatrix}B\\ \sigma\end{bmatrix}\right)$ has constant gap for any row vector $\sigma\in \F_2^{m}$. +\end{lemma} +\begin{proof} +Let $M'$ be $M\left(\begin{bmatrix}B\\ \sigma\end{bmatrix}\right)$ and let the ground set be $E$. +$\lambda(M')\leq \lambda(M)$ (row space). +Let $F^*\in \argmin_{F\subset E} \frac{|E-F|}{r'(E)-r'(F)}$. + +\begin{equation*} +\begin{aligned} +\sigma(M') &= \frac{|E-F^*|}{r'(E)-r'(F^*)}\\ + &\geq \frac{|E-F^*|}{r(E)+1-r(F^*)}\\ + &\geq \frac{|E-F^*|}{2(r(E)-r(F^*)}\\ + &\geq \frac{\sigma(M)}{2} +\end{aligned} +\end{equation*} +So the gap $\frac{\lambda(M')}{\sigma(M')}\leq \frac{2\lambda(M)}{\sigma(M)}$. +\end{proof} + +\begin{lemma} +Let $M$ be a binary matroid with representation $A\in \F_2^{n\times m}$. Let $A'$ be the binary matrix $[A,T]$ for any $T\in \F_2^n$. If $M$ and deletion minors of $M$ have constant gap, then $M(A')/T$ also has constant gap. +\end{lemma} + +\paragraph{Graphic case} +Before proving this lemma consider a easier case where $M$ is graphic. The new element $T$ identifies a vertex set $T$ (abusing notations). The minimum cocircuit of $M'/T$ is exactly the minimum even cut in $(G,T)$. Consider the fundamental circuits $C(B,T)$. $C(B,T)\cap E$ for any spanning tree $B$ is exactly the $T$-join in $B$. What is a base of $M'/T$? $B-e$ for all spanning tree $B$ and $e\in C(B,T)\setminus \set{T}$. This base hitting set of $M'/t$ is the even cut. +We want to prove the followings: +\[ + \lambda(M'/t)\leq \lambda_2(G)\leq 2\sigma_2(G)\leq 2(|J|+\sigma(M'/T)|_{E\setminus J}), +\] +where $\lambda_2(G)$ and $\sigma_2(G)$ are the integral and fractional 3-cut in $G$, $J$ is the minimum $T$-cut. To see the last inequality, consider for any base $B$ the size of $J\cap B$. If this is at least 2, then RHS satisfies the 2-cut constraint for $B$; otherwise, $B\setminus J$ should be a base in $M'/T$ since $T$-cut intersects every $T$-join, and we have $\sigma(M'/T)|_{B\setminus J}\geq 1$ as a constraint in the hitting set. + +Then we discuss the connections in the minimum cut of $G$, the minimum $T$-cut of $G$ and the minimum even cut of $G$. It is easy to see that the mincut is either an odd cut or an even cut. +\begin{enumerate} +\item $J$ is mincut. $\lambda(M'/t)\leq 2(\lambda(G)+\sigma(M'/t))\leq 6\sigma(M'/t)$ +\item $\lambda(M'/t)$ is the mincut. $\lambda(M'/t)=\lambda(G)\leq 2\sigma(G)\leq 2\sigma(M'/t)$ +\end{enumerate} + +Now we generalize this idea to binary matroids. Note that we need constant gap for all deletion minors of $M$ since we are going to use the gap of 2-cocycle LP of $M$. + +Instead of $T$-cut we find the minimum hitting set of $C(B,T)$ for all base $B$. An useful lemma is the following. +\begin{lemma} +Let $M$ be a binary matroid and let $M'$ be $(M+f)/f$ for a new element $f$ ($+$ is extension on binary representation). +The cogirth of $M$ is either the minimum hitting set of $\{C(B,f)|\forall B\}$ or the cogirth of $M'$. +\end{lemma} + +\begin{proof} +For binary matroid $M$, finding the cogirth of $M$ is equivalent to finding the smallest nonempty set in the cocircuit space. + +Let $A$ be the binary representation of $M$. Note that the cocircuit space of $M$ is the same as the row space of $A$. So we work on matrix instead. Any cocircuit of $M$ have a indicator vector $yA$ and thus we can use a row vector $y$ to indicate any cocircuit. $f$-extension can be also seen as adding a new column $v$ to the binary matrix. This new element gives a dichotomy on the cocircuit space: + +\begin{enumerate} +\item even cocycle. $yv=0$ (Note that cocycle here only means that the set is in the cocircuit space.) +\item odd cocycle. $yv=1$ +\end{enumerate} + +Now we consider the hitting sets. It follows from definitions that the minimum hitting set of $\{B-e |\forall B ,\forall e\in B\cap C(B,f)\}$ is exactly the minimum cocircuit of $(M+f)/f$, which is the minimum set in the cocircuit space with $yv=0$. So the minimum hitting set of $B-e$ is always an even cocycle. + +Consider a set $X$ which fails to hit all $\{C(B,f)|\forall B\}$. Then there must be a set $F\subseteq E\setminus X$ that span the new element $f$. On matrix, this is equivalent to the existence of $\chi_F$ that $A\chi_F=v$. Using Farkas' lemma on $\mathbb F_2$, this implies that there dose not exists $y$ that ${\rm supp}(yA)\in X\land yv=1$. (chatGPT says one can use Farkas lemma on finite field like this, need to verify). +So $X$ hits all $\{C(B,f)|\forall B\}$ iff there is such a $y$. Minimizing $X$ pushes ${\rm supp}(yA)$ which is an odd cocycle. Hence, the minimum hitting set for $C(B,f)$ is always an odd cocycle. + +The theorem then follows directly from the fact that any cocycle is either odd or even. + +\end{proof} + +\end{document} \ No newline at end of file