notes
This commit is contained in:
4
.gitignore
vendored
4
.gitignore
vendored
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*.fls
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*.fls
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*.fdb_latexmk
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*.fdb_latexmk
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.vscode
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.vscode
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__pycache__
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__pycache__
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notes.*
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!notes.tex
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533
chao.sty
Normal file
533
chao.sty
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@@ -0,0 +1,533 @@
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% ========================-*- LaTeX -*-=========================
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%
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% chao.sty -- modified from jeffe.sty
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% jeffe.sty can be found at http://web.engr.illinois.edu/~jeffe/pubs/latex.html
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%
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% ==========================================================
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\ProvidesPackage{chao}
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% remarkable size
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\newif\ifchao@remarkable
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\DeclareOption{remarkable}{\chao@remarkabletrue}
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\PassOptionsToPackage{hyphens}{url}
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\PassOptionsToPackage{unicode}{hyperref}
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\PassOptionsToPackage{naturalnames}{hyperref}
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\ProcessOptions
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\RequirePackage{latexsym,amsmath}
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\RequirePackage[dvipsnames]{xcolor}
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\RequirePackage[colorlinks=true,urlcolor=Blue,citecolor=Green,linkcolor=BrickRed,unicode]{hyperref}
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\RequirePackage[hyperref]{ntheorem} % theorem numbering
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\RequirePackage{aliascnt}
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\RequirePackage{xspace}
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% st mary fonts
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\RequirePackage{stmaryrd} % comment out if unavailable
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\SetSymbolFont{stmry}{bold}{U}{stmry}{m}{n}
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\RequirePackage{wasysym}
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% nagging
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\RequirePackage[l2tabu, orthodox]{nag}
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\RequirePackage[all,warning]{onlyamsmath}
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\RequirePackage{graphicx} % essentials
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\RequirePackage{microtype}
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\RequirePackage[mathletters]{ucs} % allow Unicode in .tex file
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\RequirePackage[utf8]{inputenc}
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\RequirePackage[charter]{mathdesign} % change fonts
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\RequirePackage{berasans, beramono}
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\RequirePackage{eucal}
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\RequirePackage[nocompress]{cite} % other convenient stuff
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\RequirePackage{enumerate}
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%\usepackage{pgf,tikz}
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\RequirePackage{wrapfig}
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\RequirePackage[medium,compact]{titlesec}
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\RequirePackage{verbatim}
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\usepackage{CJKutf8}
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\newtheorem{theorem}{Theorem}
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\newaliascnt{lemma}{theorem}
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\newaliascnt{corollary}{theorem}
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\newaliascnt{conjecture}{theorem}
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\newaliascnt{proposition}{theorem}
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\newtheorem{lemma}{Lemma}
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\newtheorem{corollary}{Corollary}[section]
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\newtheorem{conjecture}{Conjecture}[section]
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\newtheorem{proposition}{Proposition}[section]
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\newtheorem{problem}{Problem}
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\newcommand{\lemmaautorefname}{Lemma}
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\newcommand{\corollaryautorefname}{Corollary}
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\newcommand{\conjectureautorefname}{Conjecture}
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\newcommand{\propositionautorefname}{Proposition}
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\newcommand{\problemautorefname}{Problem}
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\newenvironment{definition}[1][Definition]{\begin{trivlist}
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\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
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\newenvironment{example}[1][Example]{\begin{trivlist}
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\item[\hskip \labelsep {\bfseries #1}]}{\end{trivliST}}
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\newenvironment{remark}[1][Remark]{\begin{trivlist}
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\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
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\widowpenalty 5000
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\clubpenalty 5000
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% line numbers (be nice to the referee)
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\RequirePackage[mathlines, pagewise]{lineno}
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%\linenumbers
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\nolinenumbers
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\renewcommand\linenumberfont{\normalfont\tiny\sffamily\color{gray}}
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\setlength\linenumbersep{2em}
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\numberwithin{figure}{section}
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\ifchao@remarkable
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\RequirePackage[paperheight=209.6mm, paperwidth=157.2mm, top=0.2in,bottom=0.2in,right=0.2in,left=0.2in]{geometry}
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\tolerance=10000
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\else
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\RequirePackage[margin=1in]{geometry}
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\fi
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%\setlength{\topmargin}{-0.5in} \setlength{\textwidth}{6.5in}
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%\setlength{\oddsidemargin}{0.0in} \setlength{\textheight}{9.1in}
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%\pagestyle{empty}
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%\newlength{\pagewidth}
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%\setlength{\pagewidth}{6.5in} %\pagestyle{empty}
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\pagenumbering{arabic}
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\urlstyle{same} % dammit
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\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}
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\pdfstringdefDisableCommands{\let\boldmath\relax} % allow \boldmath in section titles
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\let\subsectionautorefname\sectionautorefname
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\let\subsubsectionautorefname\sectionautorefname
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% Chao addition
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\def\email#1{\href{mailto:#1}{#1}}
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% chinese
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\def\zh#1{\begin{CJK}{UTF8}{gbsn}#1\end{CJK}}
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% ----------------------------------------------------------------------
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% Common Latin abbreviations, only one of which is actually kosher
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% ----------------------------------------------------------------------
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\def\etal{\emph{et~al.}} % and others
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%\def\ie{\emph{i.e.}} % that is
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\def\ie{\textcolor{red}{that is}}
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%\def\eg{\emph{e.g.}} % for example
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\def\eg{\textcolor{red}{for example}}
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% -----------------------
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% Common Names
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% -----------------------
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\def\Mobius{M\"obius\xspace}
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% -----------------------
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% useful shortcuts
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% -----------------------
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\def\cost{\cent}
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% ----------------------------------------------------------------------
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% Simple math stuff
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% ----------------------------------------------------------------------
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\renewcommand{\subset}{\subseteq}
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% ---- SYMBOLS ----
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\let\e\varepsilon % a ``real'' epsilon — better yet, just use Unicode ε.
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%
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% I give up. These are in the wrong font, but my kludged versions
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% LOOK like kludges, especially \Z, \Q, and \C.
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%
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\def\Real{\mathbb{R}}
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\def\Proj{\mathbb{P}}
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\def\Hyper{\mathbb{H}}
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\def\Integer{\mathbb{Z}}
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\def\Natural{\mathbb{N}}
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\def\Complex{\mathbb{C}}
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\def\Rational{\mathbb{Q}}
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\let\N\Natural
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\let\Q\Rational
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\let\R\Real
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\let\Z\Integer
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\def\Rd{\Real^d}
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\def\RP{\Real\Proj}
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\def\CP{\Complex\Proj}
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\def\F{\mathbb{F}}
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% ---- OPERATORS (requires amsmath) ----
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\def\aff{\operatorname{aff}}
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\def\area{\operatorname{area}}
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\def\argmax{\operatornamewithlimits{arg\,max}}
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\def\argmin{\operatornamewithlimits{arg\,min}}
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\def\Aut{\operatorname{Aut}} % Automorphism group
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\def\card{\operatorname{card}} % cardinality, deprecated for \abs
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\def\conv{\operatorname{conv}}
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\def\E{\operatorname{E}} % Expectation: $\E[X]$ (like \Pr)
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\def\EE{\operatornamewithlimits{E}}
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\def\Hom{\operatorname{Hom}} % Homomorphism group
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\def\id{\operatorname{id}} % identity
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\def\im{\operatorname{im}} % image
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\def\lcm{\operatorname{lcm}}
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\def\lfs{\operatorname{lfs}} % local feature size
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\def\poly{\operatorname{poly}}
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\def\polylog{\operatorname{polylog}}
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\def\rank{\operatorname{rank}}
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\def\rel{\operatorname{rel\,}} % relative (interior, boundary, etc.)
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\def\sgn{\operatorname{sgn}}
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\def\vol{\operatorname{vol}} % volume
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\def\fp#1{^{\underline{#1}}} % falling powers: $n\fp{d}$
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\def\rp#1{^{\overline{#1}}} % rising powers: $n\rp{d}$
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\def\setsymdiff{\operatorname{\triangle}}
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% --- Darts and fences ---
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% less nice replacements for stmaryrd characters
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\@ifundefined{shortrightarrow}{\let\shortrightarrow\rightarrow}{}
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\@ifundefined{shortleftarrow}{\let\shortleftarrow\leftarrow}{}
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\@ifundefined{shortuparrow}{\let\shortuparrow\uparrow}{}
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\@ifundefined{shortdownarrow}{\let\shortdownarrow\downarrow}{}
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\def\arcto{\mathord\shortrightarrow}
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\def\arcfrom{\mathord\shortleftarrow}
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\def\arc#1#2{#1\arcto#2}
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\def\cra#1#2{#1\mathord\shortleftarrow#2}
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\def\fence#1#2{#1\mathord\shortuparrow#2}
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\def\ecnef#1#2{#1\mathord\shortdownarrow#2}
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% --- Cheap displaystyle operators ---
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\def\Frac#1#2{{\displaystyle\frac{#1}{#2}}}
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\def\Sum{\sum\limits}
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\def\Prod{\prod\limits}
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\def\Union{\bigcup\limits}
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\def\Inter{\bigcap\limits}
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\def\Lor{\bigvee\limits}
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\def\Land{\bigwedge\limits}
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\def\Lim{\lim\limits}
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\def\Max{\max\limits}
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\def\Min{\min\limits}
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% ---- RELATORS ----
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\def\deq{\stackrel{\scriptscriptstyle\triangle}{=}} % Use := instead.
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\def\into{\DOTSB\hookrightarrow} % = one-to-one
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\def\onto{\DOTSB\twoheadrightarrow}
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\def\inonto{\DOTSB\lhook\joinrel\twoheadrightarrow}
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\def\from{\leftarrow}
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\def\tofrom{\leftrightarrow}
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\def\mapsfrom{\mathrel{\reflectbox{$\mapsto$}}}
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\def\longmapsfrom{\mathrel{\reflectbox{$\longmapsto$}}}
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% ---- DELIMITER PAIRS ----
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% --- always self-scaling delmiter pairs ---
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\def\set#1{\left\{ #1 \right\}}
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\def\floor#1{\left\lfloor #1 \right\rfloor}
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\def\ceil#1{\left\lceil #1 \right\rceil}
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\def\seq#1{\left\langle #1 \right\rangle}
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\def\abs#1{\left| #1 \right|}
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\def\norm#1{\left\| #1 \right\|}
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\def\paren#1{\left( #1 \right)} % need better macro name!
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\def\brack#1{\left[ #1 \right]} % need better macro name!
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\def\indic#1{\left[ #1 \right]} % indicator variable; Iverson notation
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%
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|
% Macros to typeset sets like {foo|bar} with all three delimiters
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% correctly scaled to fit. What I *really* want is a \middle macro
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% that acts just like \left and \right. Grumble.
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%
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|
% Set version is handled by package braket.
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\def\Bigbar#1{\mathrel{\left|\vphantom{#1}\right.\n@space}}
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\def\Seqbar#1#2{\Seq{#1 \Bigbar{#1 #2} #2}}
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|
\def\Brackbar#1#2{\Brack{#1 \Bigbar{#1 #2} #2}}
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||||||
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||||||
|
% Math mode fbox
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|
\def\mfbox#1{\mathchoice{{\fbox{\ensuremath{\displaystyle #1}}}}
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{{\fbox{\ensuremath{\textstyle #1}}}}
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|
{{\fbox{\ensuremath{\scriptstyle #1}}}}
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||||||
|
{{\fbox{\ensuremath{\scriptscriptstyle #1}}}}}
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||||||
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||||||
|
% ---- ``ACCENTS'' ----
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||||||
|
% NB: Commands equivalent to \lrarrowfill and \overlrarrow are
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||||||
|
% already defined in amstex!
|
||||||
|
\def\lrarrowfill{$\m@th\mathord\leftarrow\mkern-6mu%
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||||||
|
\cleaders\hbox{$\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}}
|
||||||
|
|
||||||
|
|
||||||
95
notes.tex
Normal file
95
notes.tex
Normal file
@@ -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}
|
||||||
Reference in New Issue
Block a user