├── .gitignore ├── README.md ├── budapest ├── GRT_notes.pdf ├── GRT_notes.tex ├── PRO_notes.pdf ├── PRO_notes.tex ├── RFM_notes.pdf └── RFM_notes.tex ├── co ├── 330_notes.pdf ├── 330_notes.tex ├── 463_notes.pdf ├── 463_notes.tex ├── 749_notes.pdf └── 749_notes.tex ├── pmath ├── 348_notes.pdf ├── 348_notes.tex ├── 351_notes.pdf ├── 351_notes.tex ├── 441_notes.pdf ├── 441_notes.tex ├── 445_notes.pdf ├── 445_notes.tex ├── 446_notes.pdf ├── 446_notes.tex ├── 451_notes.pdf ├── 451_notes.tex ├── 453_notes.pdf ├── 453_notes.tex ├── 465_notes.pdf ├── 465_notes.tex ├── 833_notes.pdf ├── 833_notes.tex ├── 950_notes.pdf └── 950_notes.tex └── stat ├── 902_notes.pdf └── 902_notes.tex /.gitignore: -------------------------------------------------------------------------------- 1 | */**/* 2 | !pmath/*_notes.tex 3 | !pmath/*_notes.pdf 4 | !co/*_notes.tex 5 | !co/*_notes.pdf 6 | !budapest/*_notes.tex 7 | !budapest/*_notes.pdf 8 | !stat/*_notes.tex 9 | !stat/*_notes.pdf 10 | !.gitignore 11 | !README.md 12 | .DS_Store 13 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Warning: no longer maintained 2 | I am slowly migrating the notes here to my personal webpage. 3 | You can find more updated versions of some of these notes at [https://rutar.org/writing/](https://rutar.org/writing/). 4 | The files will remain hosted here for access purposes. 5 | 6 | # UWaterloo Course Notes 7 | ## General Information 8 | The notes here are not my original work; they are derived from courses taught by the mentioned professor. 9 | The newer PDFs have embedded hyperlinks (which don't work on GitHub), so you can download the PDF or open it in its own page in your browser to use them. 10 | ## Courses 11 | - [**RFM**](budapest/RFM_notes.pdf), Real Functions and Measures: Fall 2018 - 12 | - [**GRT**](budapest/GRT_notes.pdf), Graph Theory: Fall 2018 - 13 | - [**PRO**](budapest/PRO_notes.pdf), Introductory Probability Theory: Fall 2018 - 14 | - [**PMATH 351**](pmath/351_notes.pdf), Real Analysis: Summer 2018 - Kathryn Hare 15 | - [**PMATH 441**](pmath/441_notes.pdf), Algebraic Number Theory: Winter 2019 - Matthew Satriano 16 | - [**PMATH 451**](pmath/451_notes.pdf), Measure Theory: Winter 2019 - Nico Spronk 17 | - [**PMATH 453**](pmath/453_notes.pdf), Functional Analysis: Fall 2019 - Nico Spronk 18 | - [**PMATH 445**](pmath/445_notes.pdf), Representation Theory: Fall 2019 - Blake Madill 19 | - [**PMATH 446**](pmath/446_notes.pdf), Commutative Algebra: Winter 2019 - Rahim Moosa 20 | - [**PMATH 833**](pmath/833_notes.pdf), Harmonic Analysis: Winter 2020 - Nico Spronk 21 | - [**PMATH 950**](pmath/950_notes.pdf), Fractal Geometry: Winter 2020 - Kathryn Hare 22 | - [**CO 749**](co/749_notes.pdf), Topics in Graph Theory: Fall 2019 - Carsten Thomassen 23 | - [**STAT 902**](stat/902_notes.pdf), Martingales and Stochastic Calculus: Winter 2020 - Yi 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11 | 12 | 13 | % packages --------------------------------------------------------------------- 14 | \usepackage{amsmath,amssymb,amsfonts} 15 | \usepackage{graphicx} 16 | \usepackage{etoolbox} 17 | 18 | % Set enimitem 19 | \usepackage{enumitem} 20 | \SetEnumitemKey{nl}{nolistsep} 21 | \SetEnumitemKey{r}{label=(\roman*)} 22 | 23 | % Set tikz 24 | \usepackage{tikz, pgfplots} 25 | \pgfplotsset{compat=1.15} 26 | \usetikzlibrary{intersections,positioning,cd} 27 | \usetikzlibrary{arrows,arrows.meta} 28 | \usetikzlibrary{shapes.geometric} 29 | \tikzcdset{arrow style=tikz,diagrams={>=stealth}} 30 | \tikzset{ 31 | vtx/.style={inner sep=2pt,circle,fill=black}, 32 | e/.style={thick}, 33 | de/.style={e,decoration={markings,mark=at position 0.5 with {\arrow{>}}},postaction={decorate}} 34 | } 35 | 36 | % Set hyperref 37 | \usepackage[hidelinks]{hyperref} 38 | \usepackage{xcolor} 39 | \newcommand\myshade{85} 40 | \colorlet{mylinkcolor}{violet} 41 | \colorlet{mycitecolor}{orange!50!yellow} 42 | \colorlet{myurlcolor}{green!50!blue} 43 | 44 | \hypersetup{ 45 | linkcolor = mylinkcolor!\myshade!black, 46 | citecolor = mycitecolor!\myshade!black, 47 | urlcolor = myurlcolor!\myshade!black, 48 | colorlinks = true, 49 | } 50 | 51 | 52 | % macros ----------------------------------------------------------------------- 53 | \DeclareMathOperator{\N}{{\mathbb{N}}} 54 | \DeclareMathOperator{\Q}{{\mathbb{Q}}} 55 | \DeclareMathOperator{\Z}{{\mathbb{Z}}} 56 | \DeclareMathOperator{\R}{{\mathbb{R}}} 57 | \DeclareMathOperator{\C}{{\mathbb{C}}} 58 | \DeclareMathOperator{\F}{{\mathbb{F}}} 59 | 60 | % Boldface includes math 61 | \newcommand{\mbf}[1]{{\boldmath\bfseries #1}} 62 | 63 | % proof implications 64 | \newcommand{\imp}[2]{($#1\Rightarrow#2$)\hspace{0.2cm}} 65 | \newcommand{\impe}[2]{($#1\Leftrightarrow#2$)\hspace{0.2cm}} 66 | \newcommand{\impr}{{($\Rightarrow$)\hspace{0.2cm}}} 67 | \newcommand{\impl}{{($\Leftarrow$)\hspace{0.2cm}}} 68 | 69 | % align macros 70 | \newcommand{\agspace}{\ensuremath{\phantom{--}}} 71 | \newcommand{\agvdots}{\ensuremath{\hspace{0.16cm}\vdots}} 72 | 73 | % convenient brackets 74 | \newcommand{\brac}[1]{\ensuremath{\left\langle #1 \right\rangle}} 75 | \newcommand{\norm}[1]{\ensuremath{\left\lVert#1\right\rVert}} 76 | \newcommand{\abs}[1]{\ensuremath{\left\lvert#1\right\rvert}} 77 | 78 | % arrows 79 | \newcommand{\lto}[0]{\ensuremath{\longrightarrow}} 80 | \newcommand{\fto}[1]{\ensuremath{\xrightarrow{\scriptstyle{#1}}}} 81 | \newcommand{\hto}[0]{\ensuremath{\hookrightarrow}} 82 | \newcommand{\mapsfrom}[0]{\mathrel{\reflectbox{\ensuremath{\mapsto}}}} 83 | 84 | % Divides, Not Divides 85 | \renewcommand{\div}{\bigm|} 86 | \newcommand{\ndiv}{% 87 | \mathrel{\mkern.5mu % small adjustment 88 | % superimpose \nmid to \big| 89 | \ooalign{\hidewidth$\big|$\hidewidth\cr$/$\cr}% 90 | }% 91 | } 92 | 93 | % Convenient overline 94 | \newcommand{\ol}[1]{\ensuremath{\overline{#1}}} 95 | 96 | % Big \cdot 97 | \makeatletter 98 | \newcommand*\bigcdot{\mathpalette\bigcdot@{.5}} 99 | \newcommand*\bigcdot@[2]{\mathbin{\vcenter{\hbox{\scalebox{#2}{$\m@th#1\bullet$}}}}} 100 | \makeatother 101 | 102 | % Big and small Disjoint union 103 | \makeatletter 104 | \providecommand*{\cupdot}{% 105 | \mathbin{% 106 | \mathpalette\@cupdot{}% 107 | }% 108 | } 109 | \newcommand*{\@cupdot}[2]{% 110 | \ooalign{% 111 | $\m@th#1\cup$\cr 112 | \sbox0{$#1\cup$}% 113 | \dimen@=\ht0 % 114 | \sbox0{$\m@th#1\cdot$}% 115 | \advance\dimen@ by -\ht0 % 116 | \dimen@=.5\dimen@ 117 | \hidewidth\raise\dimen@\box0\hidewidth 118 | }% 119 | } 120 | 121 | \providecommand*{\bigcupdot}{% 122 | \mathop{% 123 | \vphantom{\bigcup}% 124 | \mathpalette\@bigcupdot{}% 125 | }% 126 | } 127 | \newcommand*{\@bigcupdot}[2]{% 128 | \ooalign{% 129 | $\m@th#1\bigcup$\cr 130 | \sbox0{$#1\bigcup$}% 131 | \dimen@=\ht0 % 132 | \advance\dimen@ by -\dp0 % 133 | \sbox0{\scalebox{2}{$\m@th#1\cdot$}}% 134 | \advance\dimen@ by -\ht0 % 135 | \dimen@=.5\dimen@ 136 | \hidewidth\raise\dimen@\box0\hidewidth 137 | }% 138 | } 139 | \makeatother 140 | 141 | 142 | % macros (theorem) ------------------------------------------------------------- 143 | \usepackage[thmmarks,amsmath,hyperref]{ntheorem} 144 | \usepackage[capitalise,nameinlink]{cleveref} 145 | 146 | % Numbered Statements 147 | \theoremstyle{change} 148 | \theoremindent\parindent 149 | \theorembodyfont{\itshape} 150 | \theoremheaderfont{\bfseries\boldmath} 151 | \newtheorem{theorem}{Theorem.}[section] 152 | \newtheorem{lemma}[theorem]{Lemma.} 153 | \newtheorem{corollary}[theorem]{Corollary.} 154 | \newtheorem{proposition}[theorem]{Proposition.} 155 | 156 | % Claim environment 157 | \theoremstyle{plain} 158 | \theorempreskip{0.2cm} 159 | \theorempostskip{0.2cm} 160 | \theoremheaderfont{\scshape} 161 | \newtheorem{claim}{Claim} 162 | \renewcommand\theclaim{\Roman{claim}} 163 | \AtBeginEnvironment{theorem}{\setcounter{claim}{0}} 164 | 165 | % Un-numbered Statements 166 | \theorempreskip{0.1cm} 167 | \theorempostskip{0.1cm} 168 | \theoremindent0.0cm 169 | \theoremstyle{nonumberplain} 170 | \theorembodyfont{\upshape} 171 | \theoremheaderfont{\bfseries\itshape} 172 | \newtheorem{definition}{Definition.} 173 | \theoremheaderfont{\itshape} 174 | \newtheorem{example}{Example.} 175 | \newtheorem{remark}{Remark.} 176 | 177 | % Proof / solution environments 178 | \theoremseparator{} 179 | \theoremheaderfont{\hspace*{\parindent}\scshape} 180 | \theoremsymbol{$//$} 181 | \newtheorem{solution}{Sol'n} 182 | \theoremsymbol{$\blacksquare$} 183 | \theorempostskip{0.4cm} 184 | \newtheorem{proof}{Proof} 185 | \theoremsymbol{} 186 | \newtheorem{nmproof}{Proof} 187 | 188 | % Format references 189 | \crefformat{equation}{(#2#1#3)} 190 | \Crefformat{theorem}{#2Thm. #1#3} 191 | \Crefformat{lemma}{#2Lem. #1#3} 192 | \Crefformat{proposition}{#2Prop. #1#3} 193 | \Crefformat{corollary}{#2Cor. #1#3} 194 | \crefformat{theorem}{#2Theorem #1#3} 195 | \crefformat{lemma}{#2Lemma #1#3} 196 | \crefformat{proposition}{#2Proposition #1#3} 197 | \crefformat{corollary}{#2Corollary #1#3} 198 | 199 | 200 | % macros (algebra) ------------------------------------------------------------- 201 | \DeclareMathOperator{\Ann}{Ann} 202 | \DeclareMathOperator{\Aut}{Aut} 203 | \DeclareMathOperator{\chr}{char} 204 | \DeclareMathOperator{\coker}{coker} 205 | \DeclareMathOperator{\disc}{disc} 206 | \DeclareMathOperator{\End}{End} 207 | \DeclareMathOperator{\Fix}{Fix} 208 | \DeclareMathOperator{\Frac}{Frac} 209 | \DeclareMathOperator{\Gal}{Gal} 210 | \DeclareMathOperator{\GL}{GL} 211 | \DeclareMathOperator{\Hom}{Hom} 212 | \DeclareMathOperator{\id}{id} 213 | \DeclareMathOperator{\im}{im} 214 | \DeclareMathOperator{\lcm}{lcm} 215 | \DeclareMathOperator{\Nil}{Nil} 216 | \DeclareMathOperator{\rank}{rank} 217 | \DeclareMathOperator{\Res}{Res} 218 | \DeclareMathOperator{\Spec}{Spec} 219 | \DeclareMathOperator{\spn}{span} 220 | \DeclareMathOperator{\Stab}{Stab} 221 | \DeclareMathOperator{\Tor}{Tor} 222 | 223 | % Lagrange symbol 224 | \newcommand{\lgs}[2]{\ensuremath{\left(\frac{#1}{#2}\right)}} 225 | 226 | % Quotient (larger in display mode) 227 | \newcommand{\quot}[2]{\mathchoice{\left.\raisebox{0.14em}{$#1$}\middle/\raisebox{-0.14em}{$#2$}\right.} 228 | {\left.\raisebox{0.08em}{$#1$}\middle/\raisebox{-0.08em}{$#2$}\right.} 229 | {\left.\raisebox{0.03em}{$#1$}\middle/\raisebox{-0.03em}{$#2$}\right.} 230 | {\left.\raisebox{0em}{$#1$}\middle/\raisebox{0em}{$#2$}\right.}} 231 | 232 | 233 | % macros (analysis) ------------------------------------------------------------ 234 | \DeclareMathOperator{\M}{{\mathcal{M}}} 235 | \DeclareMathOperator{\B}{{\mathcal{B}}} 236 | \DeclareMathOperator{\ps}{{\mathcal{P}}} 237 | \DeclareMathOperator{\pr}{{\mathbb{P}}} 238 | \DeclareMathOperator{\E}{{\mathbb{E}}} 239 | \DeclareMathOperator{\supp}{supp} 240 | \DeclareMathOperator{\sgn}{sgn} 241 | 242 | \renewcommand{\Re}{\ensuremath{\operatorname{Re}}} 243 | \renewcommand{\Im}{\ensuremath{\operatorname{Im}}} 244 | \renewcommand{\d}[1]{\ensuremath{\operatorname{d}\!{#1}}} 245 | 246 | 247 | % file-specific preamble ------------------------------------------------------- 248 | \DeclareMathOperator{\Inv}{Inv} 249 | \DeclareMathOperator{\inv}{inv} 250 | \DeclareMathOperator{\OB}{OB} 251 | \newcommand{\sqbinom}[2]{\begin{bmatrix}#1\\#2\end{bmatrix}} 252 | 253 | 254 | % constants -------------------------------------------------------------------- 255 | \newcommand{\subject}{REPLACE} 256 | \newcommand{\semester}{REPLACE} 257 | 258 | 259 | % formatting ------------------------------------------------------------------- 260 | % Fonts 261 | \usepackage{kpfonts} 262 | \usepackage{dsfont} 263 | 264 | % Adjust numbering 265 | \numberwithin{equation}{section} 266 | \counterwithin{figure}{section} 267 | \counterwithout{section}{chapter} 268 | \counterwithin*{chapter}{part} 269 | 270 | % Footnote 271 | \setfootins{0.5cm}{0.5cm} % footer space above 272 | \renewcommand*{\thefootnote}{\fnsymbol{footnote}} % footnote symbol 273 | 274 | % Table of Contents 275 | \renewcommand{\thechapter}{\Roman{chapter}} 276 | \renewcommand*{\cftchaptername}{Chapter } % Place 'Chapter' before roman 277 | \setlength\cftchapternumwidth{4em} % Add space before chapter name 278 | \cftpagenumbersoff{chapter} % Turn off page numbers for chapter 279 | \maxtocdepth{section} % table of contents up to section 280 | 281 | % Section / Subsection headers 282 | \setsecnumdepth{section} % numbering up to and including "section" 283 | \newcommand*{\shortcenter}[1]{% 284 | \sethangfrom{\noindent ##1}% 285 | \Large\boldmath\scshape\bfseries 286 | \centering 287 | \parbox{5in}{\centering #1}\par} 288 | \setsecheadstyle{\shortcenter} 289 | \setsubsecheadstyle{\large\scshape\boldmath\bfseries\raggedright} 290 | 291 | % Chapter Headers 292 | \chapterstyle{verville} 293 | 294 | % Page Headers / Footers 295 | \copypagestyle{myruled}{ruled} % Draw formatting from existing 'ruled' style 296 | \makeoddhead{myruled}{}{}{\scshape\subject} 297 | \makeevenfoot{myruled}{}{\thepage}{} 298 | \makeoddfoot{myruled}{}{\thepage}{} 299 | \pagestyle{myruled} 300 | \setfootins{0.5cm}{0.5cm} 301 | \renewcommand*{\thefootnote}{\fnsymbol{footnote}} 302 | 303 | % Titlepage 304 | \title{\subject} 305 | \author{Alex Rutar\thanks{\itshape arutar@uwaterloo.ca}\\ University of Waterloo} 306 | \date{\semester\thanks{Last updated: \today}} 307 | 308 | \begin{document} 309 | \pagenumbering{gobble} 310 | \hypersetup{pageanchor=false} 311 | \maketitle 312 | \newpage 313 | \frontmatter 314 | \hypersetup{pageanchor=true} 315 | \tableofcontents* 316 | \newpage 317 | \mainmatter 318 | 319 | 320 | % main document ---------------------------------------------------------------- 321 | \chapter{REPLACE} 322 | \begin{enumerate} 323 | \item For $a,b,k\in\N$, 324 | \begin{equation}\label{e:van} 325 | \binom{a+b}{k} = \sum_{j=1}^k\binom{a}{j}\cdot\binom{b}{k-j} 326 | \end{equation} 327 | We prove this with a bijection: 328 | \begin{equation*} 329 | \mathcal{B}(a+b,k) \leftrightharpoons \bigcup_{j=0}^k\mathcal{B}(a,j)\times\mathcal{B}(b,k-j) 330 | \end{equation*} 331 | given by $S\mapsto (S\cap\{1,\ldots,a\},(S\cap\{a+1,\ldots,a+b\})^{(-a)}$ and $(P,Q)\mapsto P\cup Q^{(a)}$, where $\mathcal{B}(n,i)$ is the set of $i-$element subsets of $\{1,2,\ldots,n\}$ and for $C\subseteq\Z$ and $q\in\Z$, $C^{(q)}=\{c+q:c\in C\}$. 332 | Note that the equation in fact gives the polynomial identity 333 | \begin{equation*} 334 | \binom{x+y}{k}=\sum_{j=0}^k\binom{x}{j}\binom{y}{k-j} 335 | \end{equation*} 336 | in $\Q[x,y]$. 337 | We denote the falling factorial $(x)_i=x(x-1)(x-2)\cdots(x-i+1)$, which has degree $i$ for each $i\in\N$. 338 | In particular, $(x)_i=i!\binom{x}{i}$, so multiplying our identity by $k!$, we get 339 | \begin{equation*} 340 | (x+y)_k=\sum_{j=0}^k\binom{k}{j}(x)_j(y)_{k-j} 341 | \end{equation*} 342 | Compare this with the standard binomial theorem 343 | \begin{equation*} 344 | (x+y)^k=\sum_{j=0}^k\binom{k}{j}x^jy^{k-j} 345 | \end{equation*} 346 | These are called sequences of binomial type. 347 | \item Here's another identity. 348 | For $n\geq 0$ and $s,t\geq 1$, 349 | \begin{equation*} 350 | \binom{n+s+t-1}{s+t-1}=\sum_{k=0}^n\binom{k+s-1}{s-1}\binom{n-k+t-1}{t-1} 351 | \end{equation*} 352 | Let $\mathcal{M}(m,r)$ denote a multiset of size $m$ with elements of $r$ types, so that $|\mathcal{M}(m,r)|=\binom{m+r-1}{r-1}$. 353 | Let's define a bijection 354 | \begin{equation}\label{e:po1} 355 | \mathcal{M}(n,s+t)\rightleftharpoons\bigcup_{k=1}^n\mathcal{M}(k,s)\times\mathcal{M}(n-k,t) 356 | \end{equation} 357 | $\mu=(m_1,\ldots,m_{s+t})\mapsto ((m_1,\ldots,m_s),(m_{s+1},\ldots,m_{s+t}))$ and $(\nu,\theta)\mapsto \nu\theta$. 358 | Note that if $f,g$ are polynomials of degree $d$ and $e$ respectively, then $\sum_{k=0}^nf(k)g(n-k)$ is a polynomial in $n$ of degree $d+e-1$. 359 | 360 | Is there some way to understand \cref{e:po1}? 361 | It is unclear, with our known techniques, that this corresponds to a polynomial identity since there is a variable $n$ in the exponent. 362 | However, we can use generating functions. 363 | Define 364 | \begin{align*} 365 | \sum_{n=0}^\infty\binom{n+s+t-1}{s+t-1}z^n &= \sum_{n=0}^\infty|\mathcal{M}(n,s+t)|z^n= \sum_{(m_1,\ldots,m_{s+t})}z^{m_1+\cdots+m_{s+t}}\\ 366 | &= \left(\sum_{m=0}^\infty z^m\right)^{s+t}\\ 367 | &= \frac{1}{(1-z)^{s+t}}=\frac{1}{(1-z)^s}\frac{1}{(1-z)^t}\\ 368 | &= \sum_{k=0}^\infty\binom{k+s-1}{s-1}z^k\sum_{\ell=0}^\infty\binom{\ell+t-1}{t-1}z^\ell\\ 369 | &= \sum_{n=0}^\infty z^n\left(\sum_{k=0}^n\binom{k+s-1}{s-1}\binom{n-k+t-1}{t-1}\right) 370 | \end{align*} 371 | Similarly, \cref{e:van} is equivalent to saying $(1+z)^{a+b}=(1+z)^a(1+z)^b$. 372 | Note that $(1+z)^n=\sum_{k=0}^n\binom{n}{k}z^k=\sum_{k=0}^\infty\binom{n}{k}z^k$ for $n\in\N$. 373 | 374 | Can we substitute $\frac{1}{(1-q)^t}=(1+z)^n$ where $z=-q$ and $n=-t$? 375 | \item Consider 376 | \begin{equation*} 377 | (x_1+x_2)^n = \sum_{i=0}^n\binom{n}{i}x_1^ix_2^{n-i} 378 | \end{equation*} 379 | and 380 | \begin{equation*} 381 | (x_1+x_2)^n = \sum_{f:N_n\to\{1,2\}}\prod_{j=1}^n x_{f(j)} 382 | \end{equation*} 383 | More generally, we can consider 384 | \begin{equation*} 385 | (x_1+\cdots+x_k)^n = \sum_{f:N_n\to N_k}\prod_{j\in N_n}x_{f(j)} 386 | \end{equation*} 387 | If we set all $x_1=\cdots=x_k=1$, then $k^n$ gives the number of functions from $N_n$ to $N_k$. 388 | If we set $x_i=q^i$ for all $i\in N_k$, then we get 389 | \begin{equation*} 390 | \left(\frac{q-q^{k+1}}{1-q}\right)^n =(q+q^2+\cdots+q^k)^n = \sum_{f:N_n\to N_k}q^{f(1)+\cdots+f(k)} 391 | \end{equation*} 392 | Collect all the terms in $(x_1+\cdots+x_k)^n$ that produce the same monomial. 393 | Given a multiset $\mu$ with $m_1+\cdots+m_k=n$, write $x_1^{m_1}\cdots x_k^{m_k}=\underline{x}^\mu$. 394 | Then 395 | \begin{equation*} 396 | (x_1+\cdots+x_k)^n =\frac{n!}{m_1!\cdots m_k!}\underline{x}^\mu= \sum_{\mu\in \mathcal{M}(n,t)}\binom{n}{\mu}\underline{x}^\mu 397 | \end{equation*} 398 | 399 | \item How can we interpret 400 | \begin{equation*} 401 | P_n(q)=\prod_{i=1}^n(1+q+q^2+\cdots+q^{i-1}) 402 | \end{equation*} 403 | In general, if we set $q=1$, we see that $P_n(1)=n!$. 404 | We might hope that there is some weight function on permutations $w:\mathcal{S}_n\to\N$ such that $P_n(q)=\sum_{\sigma\in\mathcal{S}_n}q^{w(\sigma)}$. 405 | Recall the bijection $I_n:\mathcal{S}_n\to\mathcal{Q}_n$ from chapter 1. 406 | Let's find some weight function $\nu:\mathcal{Q}_n\to\N$ such that $\sum_{\rho\in\mathcal{Q}_n}x^{\nu(\rho)}=P_n(q)$, then ``pull back'' the definition of $\nu:\mathcal{Q}_n\to\N$ to get a definition for $\omega:\mathcal{S}_n\to\N$. 407 | Note that $\sum_{h\in N_r}q^{h-1}=1+q+\cdots+q^{r-1}$. 408 | Thus 409 | \begin{equation*} 410 | \sum_{\rho=(h_1,\ldots,h_n)\in\mathcal{Q}_n}q^{(h_1-1)+(h_2-1)+\cdots+(h_n-1)} = \prod_{i=1}^n(1+q+\cdots+q^{i-1})=P_n(q) 411 | \end{equation*} 412 | so we can define $\nu(\rho)=|\rho|-n$ and $\sum_{q\in\mathcal{Q}_n}q^{|\rho|-n}=P_n(q)$. 413 | We also have 414 | \begin{equation*} 415 | \sum_{\rho\in\mathcal{Q}_n}q^{(h_1-1)+\cdots+(h_n-1)} = (1+q+\cdots+q^{n-1})(1+q+\cdots+q^{n-2})\cdots(1+q)(1) 416 | \end{equation*} 417 | For notation, define $[m]_q=1+q+\cdots+q^{m-1}=\frac{1-q^m}{1-q}$. 418 | Then $[m]_q!=[m]_q[m-1]_q\cdots[1]_q$. 419 | \begin{equation*} 420 | \begin{array}{c|cccccc} 421 | &1&q&q^2&q^3&q^4\\ 422 | \hline 423 | q[3]_q & 0&1&1&1\\ 424 | \hline 425 | [2]_q[3]_q & 1&2&2&1\\ 426 | -q[2]_q[3]_q & 0&-1&-2&-2&-1\\ 427 | q^2[2]_q[3]_q & 0&0&1&2&2&1\\ 428 | \hline 429 | [6]_q & 1&1&1&1&1&1 430 | \end{array} 431 | \end{equation*} 432 | so that $[6]_q=(1-q+q^2)[2]_q[3]_q$. 433 | An \textbf{inversion} in $\sigma=a_1\ldots a_n\in\mathcal{S}_n$ is a pair $(i,j)$ of indices $1\leq ia)j$. 434 | Define $\Inv(\sigma)$ as the set of inversions of $\sigma$, and $\inv(\sigma)=|\Inv(\sigma)|$. 435 | Notice that if $\sigma=a_1\ldots a_n\mapsto\rho=(h_1,\ldots,h_n)$, then for each $1\leq i\leq n$, $h_i-1$ is the number of inversions of $\sigma$ with $i$ in the first coordinate. 436 | Recall 437 | \begin{align*} 438 | \mathcal{S}_n&\leftrightharpoons\mathcal{B}(n,k)\times\mathcal{S}_k\times\mathcal{S}_{n-k}\\ 439 | \sigma=a_1\ldots a_n&\leftrightarrow (A,\beta,\gamma)\\ 440 | \inv(\sigma) &= w(A)+\inv(\beta)+\inv(\gamma) 441 | \end{align*} 442 | Assuming such a weight fuction $w(A)$ exists, then 443 | \begin{align*} 444 | [n]!_q&=\sum_{\sigma\in\mathcal{S}_n}q^{\inv(\sigma)} =\sum_{(A,\beta,\gamma)}q^{w(A)+\inv(\beta)+\inv(\gamma)}\\ 445 | &= [k]!_q\cdot[n-k]!_q\cdot\sum_{A\in\mathcal{B}(n,k)}q^{w(A)} 446 | \end{align*} 447 | so that 448 | \begin{equation*} 449 | \sum_{A\in\mathcal{B}(n,k)}q^{w(A)}=\frac{[n]!_q}{[k]!_q\cdot[n-k]!_q}=\sqbinom{n}{k}_q 450 | \end{equation*} 451 | \end{enumerate} 452 | \begin{equation*} 453 | \sum_{S\in\mathcal{B}(n,k)} q^{\text{sum}(S)} = q^{\binom{k+1}{2}}\sqbinom{n}{k}_q 454 | \end{equation*} 455 | \begin{theorem} 456 | Let $V$ be an $n-$dimensional vector space over a finite field $\F_q$. 457 | Then for $0\leq k\leq n$, the number of $k-$dimensional subspaces of $V$ is $\sqbinom{n}{k}_q$. 458 | \end{theorem} 459 | \begin{lemma} 460 | Let $L:V\to W$ be a linear transformation that is surjective. 461 | Then $\dim V=\dim W+\dim(\ker L)$. 462 | So if this is over a finite field $\R_q$, every $w\in W$ is the image of exactly $q^{\dim(\ker(L)}$ vectors $v\in V$. 463 | \end{lemma} 464 | For every $w\in W$, is the image of exactly $q^k$ vectors in $V$. 465 | The number of ordered bases of $V$ is $q^{\binom{n}{2}}(q-1)^n[n]!_q$. 466 | \begin{theorem} 467 | Let $V$ be an $n-$dimensional vector space over a finite field $\F_q$. 468 | For $0\leq k\leq n$, the number of $k-$dimensional subspaces of $V$ is $\sqbinom{n}{k}_q$. 469 | \end{theorem} 470 | \begin{proof} 471 | Let $\OB(V)$ be the set of ordered bases of $V$, and let $G(V,k)$ be the set of $k-$dimensional subspaces of $V$. 472 | Define a function 473 | \begin{equation*} 474 | \OB(V)\to\bigcup_{U\in G(V,k)}\left(\{U\}\times\OB(U)\times\OB(V/U)\right) 475 | \end{equation*} 476 | as follows. 477 | Given $(v_1,\ldots,v_n)$ an ordered basis of $V$, let $U=\spn_{\F_q}\{v_1,\ldots,v_k\}$. 478 | Then $(v_1,\ldots,v_k)\in\OB(U)$ and $(v_{k+1}+U,\ldots,v_n+U)\in\OB(V/U)$. 479 | Consider the map $L:V\to V/U$ given by $L(v)=v+U$, so that every $v+U$ in $V/U$ is the image of $q^k$ vectors in $V$. 480 | Thus $(v_{k-1}+U,\ldots,v_n+U)$ is the image of $q^{k(n-k)}$ sequences $(z_{k+1},\ldots,z_n)$ of vectors in $V$. 481 | Thus the function $(v_1,\ldots,v_n)\mapsto(U,(v_1,\ldots,v_k),(v_{k+1}+U,\ldots,v_n+U))$ is surjective and hits everything on the RHS $q^{k(n-k)}$ times. 482 | But then counting both sides, 483 | \begin{align*} 484 | q^{\binom{n}{2}}(q-1)^n[n]!_q &= \sum_{U\in G(V,k)}1\cdot q^{\binom{k}{2}}(q-1)^k[k]!_q\cdot q^{\binom{n-k}{2}}(q-1)^{n-k}[n-k]!_q\cdot q^{k(n-k)}\\ 485 | q^{\binom{n}{2}}[n]!_q &= |G(V,k)|\cdot[k]!_q\cdot[n-k]!_q q^{\binom{k}{2}+\binom{n-k}{2}+k(n-k)}\\ 486 | [n]!_q &= |G(V,k)|\cdot[k]!_q\cdot[n-k]!_q 487 | \end{align*} 488 | giving our desired result. 489 | \end{proof} 490 | A \textbf{set partition} $\pi$ of a set $V$ is a collection of subsets $\pi=\{B_1,\ldots,B_k\}$ of $V$ such that 491 | \begin{itemize}[nl] 492 | \item Each $B_i$ is not empty 493 | \item $B_i\cap B_j=\emptyset$ if $i\neq j$ 494 | \item $B_1\cup\cdots\cup B_k=V$ 495 | \end{itemize} 496 | Let $\Pi(n,k)$ be the set of set partitions of $N_n$ with $k$ blocks, and set $S(n,k)=|\Pi(n,k)|$. 497 | Certainly $S(0,0)=1$ for the empty set partition. 498 | If $n\geq 1$, then $S(n,0)=0$, $S(n,n)=1$, and $S(n,1)=1$. 499 | We can also define a recurrence relation. 500 | Let $\Pi'(n,k)$ be those $\pi\in\Pi(n,k)$ in which $\{n\}$ is a block, and $\Pi''(n,k)$ is the set of $\pi$ in which $n$ is in a block of size at least 2. 501 | Note that $\Pi'(n,k)\leftrightharpoons\Pi(n-1,k-1)$ by removing or adding the independent element. 502 | Furthemore, the function which removes the element $n$ from a block in $\Pi''(n,k)$ is a surjective function onto $\Pi(n-1,k)$ which hits every element of $\Pi(n-1,k)$ $k$ times. 503 | Thus combing these observations, $S(n,k)=S(n-1,k-1)+k\cdot S(n-1,k)$. 504 | Thus we can compute 505 | \begin{equation*} 506 | \begin{array}{c|c|c|c|c|c|c|c} 507 | S(n,k) & 0 & 1 & 2 & 3 & 4 & 5 & 6\\ 508 | \hline 509 | 0 & 1 & X & X & X & X & X & X\\ 510 | \hline 511 | 1 & 0 & 1 & X & X & X & X & X\\ 512 | \hline 513 | 2 & 0 & 1 & 1 & X & X & X & X\\ 514 | \hline 515 | 3 & 0 & 1 & 3 & 1 & X & X & X\\ 516 | \hline 517 | 4 & 0 & 1 & 7 & 6 & 1 & X & X\\ 518 | \hline 519 | 5 & 0 & 1 & 15& 25& 10& 1 & X\\ 520 | \hline 521 | 6 & 0 & 1 & 31& & 1\\ 522 | \end{array} 523 | \end{equation*} 524 | From homework 2, we have that 525 | \begin{equation*} 526 | x^n=\sum_{k=0}^n k! S(n,k)\binom{n}{k} 527 | \end{equation*} 528 | Invert this using Binomial Inversion. 529 | \begin{theorem}[Binomial Inversion] 530 | Let $a_0,a_1,\ldots$ be a sequence. 531 | \end{theorem} 532 | \begin{proof} 533 | For $h\in\N$, let $b_h=\sum_{i=0}^h\binom{h}{i}a_i$. 534 | Let $A(t)=\sum_{i=0}^\infty a_it^i$ and $B(t)=\sum_{h=0}^\infty b_ht^h$. 535 | Then 536 | \begin{align*} 537 | B(t) &= \sum_{h=0}^\infty t^h\sum_{i=0}^h\binom{h}{i}a_i\\ 538 | &= \sum_{i=0}^\infty a_it^i\sum_{h=i}^\infty\binom{h}{h-i}t^{h-i}\\ 539 | &= \sum_{i=0}^\infty a_it^i\sum_{j=0}^\infty\binom{i+j}{j}t^j=\sum_{i=0}^\infty\frac{a_it^i}{(1-t)^{i+1})}\\ 540 | &= \frac{1}{1-t}\sum_{i=0}^\infty a_i\left(\frac{t}{1-t}\right)^i=\frac{1}{1-t}A\left(\frac{t}{1-t}\right) 541 | \end{align*} 542 | Let $z=t/(1-t)$, so that $t=z/(1+z)$. 543 | Thus 544 | \begin{equation*} 545 | B\left(\frac{z}{1+z}\right)=(1+z)A(z) 546 | \end{equation*} 547 | so that 548 | \begin{align*} 549 | \sum_{i=0}^\infty a_iz^i &= \frac{1}{1+z}B\left(\frac{z}{1+z}\right) = \sum_{h=0}^\infty b_h\frac{z^h}{(1+z)^{h+1}}\\ 550 | &= \sum_{h=0}^\infty b_h\sum_{j=0}^\infty\binom{j+h}{h}z^{h}(-z)^j\\ 551 | &= \sum_{n=0}^\infty z^n\sum_{j=0}^\infty\binom{n}{j}(-1)^{n-j}b_{j} 552 | \end{align*} 553 | Thus for all $m\in\N$, $a_m=\sum_{j=0}^m\binom{m}{j}(-1)^{m-j}b_j$. 554 | \end{proof} 555 | In particular, applying this to Stirling numbers of the second kind, for all $n\in\N$ in $\R[x]$, we have 556 | \begin{equation*} 557 | x^n=\sum_{k=0}^n S(n,k)\binom{x}{k}k! 558 | \end{equation*} 559 | Let $b_i=i^n$ for $i=0,1,2,\ldots$. 560 | If $k>n$ or $k>i$, then $S(n,k)\binom{i}{k}=0$; thus, 561 | \begin{align*} 562 | i^n &= \sum_{k=0}^nS(,nk)\binom{i}{k}k! = \sum_{k=0}^{\min(n,i)}S(n,k)\binom{i}{k}k!=\sum_{k=0}^i S(n,k)\binom{i}{k}k!\\ 563 | &= \sum_{k=0}^i\binom{i}{k}a_k 564 | \end{align*} 565 | where $a_k=k!S(n,k)$ for all $k\in\N$. 566 | But then apply binomial inversion to get 567 | \begin{equation*} 568 | a_k=\sum_{j=0}^k\binom{k}{j}(-1)^{k-j}b_j 569 | \end{equation*} 570 | Suppose $m^n=\sum_{k=0}^n S(n,k)\binom{m}{k}k!$. 571 | Then $[m]_q^n=\sum_{k=0}^\infty S[n,k]_q\sqbinom{m}{k}_q[k]!_q$, where $S[n,k]_q=\sum_{\pi\in\Pi(n,k)}q^{w(\pi)}$. 572 | Is there some function $w:\Pi(n,k)\to\N$ that makes this work? 573 | \begin{enumerate} 574 | \item[B.5] For $\mathcal{S}$ a set of BTs, let $\mathcal{R}$ be the trees in $\mathcal{S}$ with a red root and $\mathcal{B}$ be the trees in $\mathcal{S}$ with a blue root, so $\mathcal{S}=\mathcal{R}\cupdot\mathcal{B}$ disjointly. 575 | Let $r(T)$ count the number of red notes, and $b(T)$ count the number of blue nodes, and let $S(x,y)=\sum_{T\in\mathcal{S}}x^{r(T)}y^{b(T)}$. 576 | In particular, $S(t,t)=\sum_{T\in\mathcal{S}}t^{n(T)}$ where $n(T)=r(T)+b(T)$ is the total number of notes. 577 | 578 | We have bijections 579 | \begin{align*} 580 | \mathcal{R} &\leftrightharpoons \{{\color{red}\bullet}\}\times\bigcup_{k=0}^\infty\mathcal{S}^k\\ 581 | \mathcal{B} &\leftrightharpoons\{{\color{blue}\bullet}\}\times\bigl((\epsilon\cup\mathcal{R})(\mathcal{B}\mathcal{R})^*(\epsilon\cup\mathcal{B})\bigr)\\ 582 | \mathcal{S} &\leftrightharpoons\mathcal{R}\cup\mathcal{B} 583 | \end{align*} 584 | so that 585 | \begin{align*} 586 | S&=R+B\\ 587 | R &= \frac{x}{1-S}\\ 588 | B &= y(1+R)\frac{1}{1-BR}(1+B) 589 | \end{align*} 590 | Substituting $R$ and $B$ using the first two equations, we get 591 | \begin{equation*} 592 | S-\frac{x}{1-S}=\frac{y\left(1+\frac{x}{1-S}\right)\left(1+S-\frac{x}{1-S}\right)}{1-\frac{x}{1-S}\left(S-\frac{x}{1-S}\right)} 593 | \end{equation*} 594 | \begin{center} 595 | \begin{tikzpicture}[btree/.style={draw=black,regular polygon, regular polygon sides=3,rounded corners}] 596 | \node[vtx,fill=red,draw=black] (v) at (2,2) {}; 597 | \node[btree] (b1) at (0,0) {$S_1$}; 598 | \node[btree] (b2) at (2,0) {$S_2$}; 599 | \node[draw=none] (eps) at (3.5,0) {$\cdots$}; 600 | \node[btree] (b3) at (5,0) {$S_n$}; 601 | \draw (v) -- ([yshift=-0.087cm]b1.north); 602 | \draw (v) -- ([yshift=-0.087cm]b2.north); 603 | \draw (v) -- ([yshift=-0.087cm]b3.north); 604 | \end{tikzpicture} 605 | \end{center} 606 | \begin{center} 607 | \begin{tikzpicture}[btree/.style={draw=black,regular polygon, regular polygon sides=3,rounded corners}] 608 | \node[vtx,fill=blue,draw=black] (v) at (2,2) {}; 609 | \node[btree,fill=blue,fill opacity=0.1,text opacity=1] (b1) at (0,0) {$S_1$}; 610 | \node[btree,fill=red,fill opacity=0.1,text opacity=1] (b2) at (2,0) {$S_2$}; 611 | \node[draw=none] (eps) at (3.5,0) {$\cdots$}; 612 | \node[btree,fill=blue,fill opacity=0.1,text opacity=1] (b3) at (5,0) {$S_n$}; 613 | \draw (v) -- ([yshift=-0.087cm]b1.north); 614 | \draw (v) -- ([yshift=-0.087cm]b2.north); 615 | \draw (v) -- ([yshift=-0.087cm]b3.north); 616 | \end{tikzpicture} 617 | \end{center} 618 | \end{enumerate} 619 | \chapter{Power Series Identities} 620 | \begin{enumerate}[nl,r] 621 | \item $\displaystyle\frac{1}{(1-z)^h}=\sum_{k=0}^\infty\binom{k+h-1}{h-1}z^k$ 622 | \item Let $a_0,a_1,\ldots$ be a sequence, and $b_h=\sum_{i=0}^h\binom{h}{i}a_i$. 623 | Then $a_m=\sum_{i=0}^m\binom{m}{i}(-1)^{m-i}b_i$. 624 | \item General Binomial Series. 625 | For $k\in\N$, let $\binom{y}{k}=\frac{y(y-1)\cdots(y-k+1)}{k!}\in\Q[y]$. 626 | Then we define 627 | \begin{equation*} 628 | (1+x)^y=\sum_{k=0}^\infty\binom{y}{k}x^k 629 | \end{equation*} 630 | which is a power series in $x$. 631 | Each coefficient of $[x^n]$ is in $\Q[y]$. 632 | Then by Vandermone convolution, 633 | \begin{align*} 634 | (1+x)^y(1+x)^z&=\sum_{i=0}^\infty\binom{y}{i}x^i\sum_{j=0}^\infty\binom{z}{j}x^j\\ 635 | &=\sum_{n=0}^\infty x^n\left(\sum_{i=0}^n\binom{y}{i}\binom{z}{n-i}\right)\\ 636 | &= \sum_{n=0}^\infty\binom{y+z}{n}x^n=(1+x)^{y+z} 637 | \end{align*} 638 | Furthermore, if $y=-p<0$ is an integer, then 639 | \begin{align*} 640 | (1+x)^{-p}&=\sum_{k=0}^\infty\binom{-p}{k}x^k\\ 641 | &=\sum_{k=0}^\infty\binom{k+p-1}{p-1} 642 | \end{align*} 643 | For $\alpha\in\C$, $f(x)=(1+x)^\alpha$ is analytic for $|x|<1$. 644 | In particular, by Taylor's theorem, 645 | \begin{equation*} 646 | (1+x)^\alpha=\sum_{k=0}^\infty c_kx^k 647 | \end{equation*} 648 | where $c_k=\frac{1}{k!}\frac{d^k}{dx^k}(1+x)^\alpha|_{x=0}$. 649 | \end{enumerate} 650 | Consider the class $\mathcal{Q}$ of (unrooted) tres in which every vertex has odd degree. 651 | We identify $\mathcal{Q}^\bullet\equiv\chi *\xi_{\text{odd}}[\mathcal{N}]$ for some class $\mathcal{N}$ describing the components of $T\setminus \{v\}$. 652 | A structure in $\mathcal{N}$ is a rooted tree in which every vertex has an even number of children. 653 | Moreover, $\mathcal{N}\equiv\chi *\xi_{\text{even}}[\mathcal{N}]$. 654 | Note that the exponential generating function for $\xi_{\text{odd}}$ is $\sum_{j=0}^\infty\frac{e^{2j+1}}{(2j+1)!}$, and similarly for the even components. 655 | This give 656 | \begin{align*} 657 | \mathcal{Q}^\bullet &= x\cdot E_{\text{odd}}(N(x))=x\cdot\sinh(N(x))\\ 658 | N(x) &= x\cdot E_{\text{even}}(N(x))=x\cdot\cosh(N(x))\\ 659 | \end{align*} 660 | Now apply LIFT with $K=\Q$, $G(u)=\cosh(u)$, and $F(u)=\sinh(u)$, so $F'(u)=\cosh(u)$. 661 | Now for $n\geq 2$, 662 | \begin{align*} 663 | |\mathcal{Q}_n|&=\frac{1}{n}|\mathcal{Q}^\bullet_n|=\frac{1}{n}\cdot n![x^n]Q^\bullet(x)\\ 664 | &= (n-1)![x^n]x\sinh(N(x))\\ 665 | &= (n-1)![x^{n-1}]\sinh(N(x))\\ 666 | &= (n-1)!\cdot\frac{1}{n-1}[u^{n-2}]F'(u)G(u)^{n-1}\\ 667 | &= (n-2)!\cdot[u^{n-2}]\cosh(u)^n\\ 668 | &= (n-2)![u^{n-2}]\left(\frac{e^u+e^{-u}}{2}\right)^n\\ 669 | &= \frac{(n-2)!}{2^n}[u^n]\sum_{j=0}^n\binom{n}{j}(e^u)^j(e^{-u})^{n-j}\\ 670 | &= \frac{(n-2)!}{2^n}[u^{n-2}]\sum_{j=0}^n\binom{n}{j}e^{(2j-n)u}\\ 671 | &= \frac{(n-2)!}{2^n}\sum_{j=0}^n\binom{n}{j}[u^{n-2}]\sum_{i=0}^\infty\frac{((2j-n)u)^i}{i!}\\ 672 | &= \frac{(n-2)!}{2^n}\sum_{j=0}^n\binom{n}{j}\frac{(2j-n)^{n-2}}{(n-2)!}\\ 673 | &= \frac{1}{2^n}\sum_{j=0}^n\binom{n}{j}(2j-n)^{n-2} 674 | \end{align*} 675 | If $n$ is odd, then this summation is zero, as expected. 676 | \subsection{Endofunctions} 677 | An \textbf{endofunction} is any function $\phi:X\to X$. 678 | If $|X|=n$, then there are $n^n$ endofunctions $\phi:X\to X$. 679 | Call this class $\mathcal{F}$. 680 | We can define the \textbf{functional directed graph} of $\phi:X\to X$ with vertices $X$ and directed edges $v\to\phi(v)$ for $v\in X$. 681 | When we say $\phi$ is connected, we mean the underlying undirected graph is connected. 682 | Call this class $\mathfrak{C}$. 683 | Certainly $\mathcal{F}\equiv\xi[\mathfrak{C}]$. 684 | 685 | What is the expected number of components among all $n^n$ endofunctions on $\{1,2,\ldots,n\}$? 686 | Certainly $F(x)=\exp(C(x))$ for the EGFs $F(x)$ and $C(x)$ for $\mathcal{F}$ and $\mathfrak{C}$ respectively. 687 | Let $c(\phi)$ be the number of connected components of $\phi\in\mathcal{F}_X$. 688 | Then 689 | \begin{equation*} 690 | F(x,y) = \sum_{n=0}^\infty\left(\sum_{\phi\in\mathcal{F}_n}y^{c(\phi)}\right)\frac{x^n}{n!} 691 | \end{equation*} 692 | Recall $F(x)=\sum_{k=0}^\infty\frac{C(x)^k}{k!}$, where $C(x)^k$ is the generating function for a graph with $k$ connected components. 693 | Thus 694 | \begin{equation*} 695 | F(x,y)=\sum_{k=0}^\infty\frac{(C(x)y)^k}{k!}=\exp(yC(x)) 696 | \end{equation*} 697 | 698 | Let's determine the structure of a connected endofunction $\phi\in\mathfrak{C}_X$. 699 | By following arrows, the graph must contain a directed cycle; in fact, this directed cycle must be unique. 700 | The same argument allows use to decompose the graph into a set of components, one for each vertex in the directed cycle. 701 | But then each component is in fact a rooted tree. 702 | We can thus identify $\mathfrak{C}\equiv\mathcal{C}[\mathcal{R}]$ where $\mathcal{R}=\mathcal{T}^\bullet$ is the class of rooted trees and $\mathcal{C}$ is the class of cyclic permutations. 703 | Passing to EGFs, we have 704 | \begin{align*} 705 | F(x,y) &= \exp(yC(x))\\ 706 | C(x) &= \log\left(\frac{1}{1-R(x)}\right)\\ 707 | R(x) &= x\exp(R(x)) 708 | \end{align*} 709 | Thus, 710 | \begin{equation*} 711 | F(x,y)=\exp\left(\log\left(\left(\frac{1}{(1-R(x))}\right)^y\right)\right)=\left(\frac{1}{1-R(x)}\right)^y 712 | \end{equation*} 713 | Apply LIFT with $R(x)=x\exp(R(x))$, $G(u)=\exp(u)$, $F(u)=\frac{1}{(1-u)^y}$. 714 | Then $F'(u)=\prt{}{u}F(u)=\frac{y}{(1-u)^{y+1}}$ 715 | Thus the total number of components among all $n^n$ $\phi\in\mathcal{F}_n$ is 716 | \begin{align*} 717 | n![x^n]\prt{}{y}F(x,y)|_{y=1} &= \left.n!\prt{}{y}\frac{1}{n}[u^{n-1}]\frac{y}{(1-u)^{y+1}}\exp(u)^n\right|_{y=1}\\ 718 | &=(n-1)![u^{n-1}]\exp(nu)\left[\frac{(1-u)^{y+1}-y(y+1)(1-u)^y}{(1-u)^{2y+2}}\right]_{y=1}\\ 719 | &=(n-1)![u^{n-1}]\exp(nu)\left[\frac{(1-u)^{2}-2(1-u)}{(1-u)^{4}}\right] 720 | \end{align*} 721 | For each $j\geq 1$, let $M_i\subseteq\N$ be a set of \textbf{allowed multiplicites} (for parts of size $j$) 722 | \begin{equation*} 723 | \lambda\mapsto\underline{m}(\lambda)=\langle m_1,m_2,m_3,\ldots\rangle 724 | \end{equation*} 725 | We require that only finitely many $j\geq 0$ have $0\notin m_j$. 726 | Consider $\mathcal{Z}\subseteq \mathcal{Y}$ given by 727 | \begin{equation*} 728 | \mathcal{Z} = \{\lambda\in\mathcal{Y}:m_j(\lambda)\in M_j\text{ for all }j\geq 1\} 729 | \end{equation*} 730 | \begin{theorem} 731 | \begin{equation*} 732 | \Phi_{\mathcal{Z}}(x,y)=\sum_{\lambda\in\mathcal{Z}}x^{n(\lambda)}y^{k(\lambda)}=\prod_{j=1}^\infty\left(\sum_{m\in M_j}x^{jm}y^m\right) 733 | \end{equation*} 734 | \end{theorem} 735 | \begin{proof} 736 | Let $\hat{\mathcal{Z}}=\{\underline{m}(\lambda):\lambda\in\mathcal{Z}\}$. 737 | For $\ell\geq 1$, let $\mathcal{M}(\ell)=\{\rho\in\mathcal{M}:r_j=0\text{ if }j>\ell\}$. 738 | Then $\bigcup_{\ell=1}^\infty\mathcal{M}(\ell)=\mathcal{M}$. 739 | Consider partitions in $\hat{\mathcal{Z}}\cap\mathcal{M}(\ell)$ when $\ell$ is bigger than the greatest index $i$ such that $0\notin M_i$. 740 | Then $\lambda\in \hat{\mathcal{Z}}\cap\mathcal{M}(\ell)$ if and only if $\underline{m}(\lambda)=\langle m_1,m_2,\ldots\rangle$ with $m_j\in M_j$ and $m_j=0$ if $j>\ell$. 741 | Thus $\hat{\mathcal{Z}}\cap\mathcal{M}(\ell)\leftrightharpoons M_1\times\cdots\times M_\ell$. 742 | Thus 743 | \begin{align*} 744 | \sum_{\substack{\lambda\in\mathcal{Z}\\\underline{m}(\lambda)\in\mathcal{M}(\ell)}}x^{n(\lambda)}y^{k(\lambda)} &= \sum_{\rho\in M_1\times\cdots\times M_\ell}x^{r_1+2r_2+\cdots+\ell r_\ell}y^{r_1+\cdots+r_\ell}\\ 745 | &= \sum_{r\in M_1}x^ry^r\sum_{r\in M_2}x^{2r}y^r\cdots\sum_{j\in M_\ell}x^{\ell r}y^r\\ 746 | &= \prod_{j=1}^\ell\left(\sum_{r\in M_j}x^{jr}y^r\right) 747 | \end{align*} 748 | Since $\mathcal{M}(1)\subseteq\mathcal{M}(2)\subseteq\cdots\subseteq\mathcal{M}$ and $\bigcup_{\ell=1}^\infty\mathcal{M}(\ell)=\mathcal{M}$, by taking limits, 749 | \begin{align*} 750 | \Phi_{\mathcal{Z}}(x,y) &= \lim_{\ell\to\infty}\Phi_{\hat{\mathcal{Z}}\cap\mathcal{M}(\ell)}(x,y)\\ 751 | &= \lim_{\ell\to\infty}\prod_{j=1}^\infty\left(\sum_{r\in \mathcal{M}_j}x^{jr}y^r\right)\\ 752 | &= \prod_{j=1}^\infty\left(\sum_{r\in M_j}x^{jr}y^r\right) 753 | \end{align*} 754 | \end{proof} 755 | \begin{example} 756 | Partitions with distinct parts $\mathcal{D}$. 757 | Then $M_j=\{0,1\}$ for all $j\geq 1$. 758 | Then $\phi_{\mathcal{D}}(x,y)=\prod_{j=1}^\infty(1+x^jy)$ 759 | \end{example} 760 | \begin{example} 761 | Partitions with only odd parts $\mathcal{O}$. 762 | Then $M_j=\{0\}$ if $j=2i$ is even, and $M_j=\N$ if $j=2i+1$ is odd. 763 | Then 764 | \begin{equation*} 765 | \Phi_{\mathcal{O}}(x,y)=\prod_{i=0}^\infty\left(\sum_{r\in\N}x^{(2i+1)r}y^r\right)=\prod_{i=0}^\infty\frac{1}{1-x^{2i+1}y} 766 | \end{equation*} 767 | \end{example} 768 | Set $y=1$ in $\Phi_{\mathcal{D}}(x,y)$ and $\Phi_{\mathcal{O}}(x,y)$. 769 | Then 770 | \begin{align*} 771 | \Phi_{\mathcal{D}}(x,1)&=\prod_{j=1}^\infty(1+x^j)=\prod_{j=1}^\infty\frac{(1+x^j)(1-x^j)}{(1-x)^j}\\ 772 | &= \prod_{j=1}^\infty\frac{(1-x^{2j})}{(1-x^j)}\\ 773 | &= \prod_{i=0}^\infty\frac{1}{1-x^{2i+1}}=\Phi_{\mathcal{O}}(x,1) 774 | \end{align*} 775 | We thus have a bijection $\mathcal{D}\leftrightharpoons\mathcal{O}$ where if $\lambda\leftrightarrow\mu$, then $n(\lambda)=n(\mu)$ (but in general, lengths are not preserved). 776 | \begin{example} 777 | Let $\mathcal{A}$ denote the set of partitions in which each part occurs 0,1,4, or 5 times. 778 | Then 779 | \begin{align*} 780 | \Phi_{\mathcal{A}}(x,1) &= \prod_{j=1}^\infty(1+x^j+x^{4j}+x^{5j}) 781 | \end{align*} 782 | Let $\mathcal{B}$ denote the set of partitions with no parts congrudent to 2 mod 4, and parts divisible by 4 are distinct. 783 | Then 784 | \begin{equation*} 785 | M_j = 786 | \begin{cases} 787 | \N &:j\text{ odd}\\ 788 | \{0\} &: j\equiv 2\pmod{4}\\ 789 | \{0,1\} & : 4\div j 790 | \end{cases} 791 | \end{equation*} 792 | So 793 | \begin{equation*} 794 | \Phi_{\mathcal{B}}(x)=\prod_{i=1}^\infty\left(\frac{1}{1-x^{2i-1}}\right)(1+x^{4i}) 795 | \end{equation*} 796 | Now, 797 | \begin{align*} 798 | \Phi_{\mathcal{A}}(x) &= \prod_{j=1}^\infty(1+x^j+x^{4j}+x^{5j})\\ 799 | &= \prod_{j=1}^\infty(1+x^j)(1+x^{4j})\\ 800 | &= \prod_{j=1}^\infty(1+x^{4j})\frac{(1+x^j)(1-x^j)}{(1-x^j)}\\ 801 | &= \prod_{j=1}^\infty(1+x^{4j})\left(\frac{1-x^{2j}}{1-x^j}\right)\\ 802 | &= \prod_{j=1}^\infty\frac{(1+x^{4j}}{1-x^{2j-1}}=\Phi_{\mathcal{B}}(x) 803 | \end{align*} 804 | \end{example} 805 | \begin{example} 806 | Let $m,n\in\Z$. 807 | Count the number of points $(a,b,c)\in\Z^3$ such that $a^2+2b^2+c^2=m$ and $a^2+a-2b+c^2=n$. 808 | 809 | For any $r\geq 0$, the set $\{(a,b,c)\in\R^3:a^2+2b^2+c^2=r\}$ is compact and contains only finitely many points with integer coordinates. 810 | Call this number $f(m,n)$. 811 | We have 812 | \begin{align*} 813 | \sum_{m=0}^\infty\sum_{n=-\infty}^\infty f(m,n)s^mt^n &= \sum_{(a,b,c)\in\Z^3}s^{a^2+2b^2+c^2}t^{a^2+a-2b+c^2}\\ 814 | &= \left(\sum_{a\in\Z} s^{a^2}t^{a^2+a}\right)\left(\sum_{b\in\Z}s^{2b^2}t^{-2b}\right)\left(\sum_{c\in\Z}s^{c^2}t^{c^2}\right)\\ 815 | &= \vartheta(st,t)\vartheta(s^2,t^{-2})\vartheta(st,1)\\ 816 | &= \prod_{i=1}^\infty(\cdots) 817 | \end{align*} 818 | where $\vartheta(x,y)=\sum_{h=-\infty}^\infty x^{h^2}y^h$ is the Jacobi theta function, and we can get the product expression using JTPF. 819 | \end{example} 820 | Final Exam, about 1/3 of the exam on each of these topics: 821 | \begin{itemize}[nl] 822 | \item Exponential generating functions 823 | \item Integer partitions 824 | \item Everything else (LIFT, formal power series) 825 | \end{itemize} 826 | For example, let $\mathcal{H}_{a,b}$ denote the set of integer partitions $\lambda$ such that $\lambda_1\leq a$ and $k(\lambda)\leq b$. 827 | Then we have a bijection $\mathcal{H}_{a,b}\leftrightharpoons\mathcal{L}(a,b)$ taking the bounding path for $\lambda$. 828 | Note that $n(\lambda)=\operatorname{area}(P)$, so $\sum_{\lambda\in\mathcal{H}_{a,b}}q^{n(\lambda)}=\qbinom{a+b}{b}$. 829 | \end{document} 830 | 831 | 832 | -------------------------------------------------------------------------------- /co/463_notes.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/alexrutar/UWaterloo-Course-Notes/9663d05575969e4a46c1e4181f4983b10198e320/co/463_notes.pdf -------------------------------------------------------------------------------- /co/463_notes.tex: -------------------------------------------------------------------------------- 1 | % header ----------------------------------------------------------------------- 2 | % Template created by texnew (author: Alex Rutar); info can be found at 'https://github.com/alexrutar/texnew'. 3 | % version (1.13) 4 | 5 | 6 | % doctype ---------------------------------------------------------------------- 7 | \documentclass[11pt, a4paper]{memoir} 8 | \usepackage[utf8]{inputenc} 9 | \usepackage[left=3cm,right=3cm,top=3cm,bottom=4cm]{geometry} 10 | \usepackage[protrusion=true,expansion=true]{microtype} 11 | 12 | 13 | % packages --------------------------------------------------------------------- 14 | \usepackage{amsmath,amssymb,amsfonts} 15 | \usepackage{graphicx} 16 | \usepackage{etoolbox} 17 | \usepackage{braket} 18 | 19 | % Set enimitem 20 | \usepackage{enumitem} 21 | \SetEnumitemKey{nl}{nolistsep} 22 | \SetEnumitemKey{r}{label=(\roman*)} 23 | 24 | % Set tikz 25 | \usepackage{tikz, pgfplots} 26 | \pgfplotsset{compat=1.15} 27 | \usetikzlibrary{intersections,positioning,cd} 28 | \usetikzlibrary{arrows,arrows.meta} 29 | \tikzcdset{arrow style=tikz,diagrams={>=stealth}} 30 | 31 | % Set hyperref 32 | \usepackage[hidelinks]{hyperref} 33 | \usepackage{xcolor} 34 | \newcommand\myshade{85} 35 | \colorlet{mylinkcolor}{violet} 36 | \colorlet{mycitecolor}{orange!50!yellow} 37 | \colorlet{myurlcolor}{green!50!blue} 38 | 39 | \hypersetup{ 40 | linkcolor = mylinkcolor!\myshade!black, 41 | citecolor = mycitecolor!\myshade!black, 42 | urlcolor = myurlcolor!\myshade!black, 43 | colorlinks = true, 44 | } 45 | 46 | 47 | % macros ----------------------------------------------------------------------- 48 | \DeclareMathOperator{\N}{{\mathbb{N}}} 49 | \DeclareMathOperator{\Q}{{\mathbb{Q}}} 50 | \DeclareMathOperator{\Z}{{\mathbb{Z}}} 51 | \DeclareMathOperator{\R}{{\mathbb{R}}} 52 | \DeclareMathOperator{\C}{{\mathbb{C}}} 53 | \DeclareMathOperator{\F}{{\mathbb{F}}} 54 | 55 | % Boldface includes math 56 | \newcommand{\mbf}[1]{{\boldmath\bfseries #1}} 57 | 58 | % proof implications 59 | \newcommand{\imp}[2]{($#1\Rightarrow#2$)\hspace{0.2cm}} 60 | \newcommand{\impe}[2]{($#1\Leftrightarrow#2$)\hspace{0.2cm}} 61 | \newcommand{\impr}{{($\Longrightarrow$)\hspace{0.2cm}}} 62 | \newcommand{\impl}{{($\Longleftarrow$)\hspace{0.2cm}}} 63 | 64 | % align macros 65 | \newcommand{\agspace}{\ensuremath{\phantom{--}}} 66 | \newcommand{\agvdots}{\ensuremath{\hspace{0.16cm}\vdots}} 67 | 68 | % convenient brackets 69 | \newcommand{\brac}[1]{\ensuremath{\left\langle #1 \right\rangle}} 70 | \newcommand{\norm}[1]{\ensuremath{\left\lVert#1\right\rVert}} 71 | \newcommand{\abs}[1]{\ensuremath{\left\lvert#1\right\rvert}} 72 | 73 | % arrows 74 | \newcommand{\lto}[0]{\ensuremath{\longrightarrow}} 75 | \newcommand{\fto}[1]{\ensuremath{\xrightarrow{\scriptstyle{#1}}}} 76 | \newcommand{\hto}[0]{\ensuremath{\hookrightarrow}} 77 | \newcommand{\mapsfrom}[0]{\mathrel{\reflectbox{\ensuremath{\mapsto}}}} 78 | 79 | % Divides, Not Divides 80 | \renewcommand{\div}{\bigm|} 81 | \newcommand{\ndiv}{% 82 | \mathrel{\mkern.5mu % small adjustment 83 | % superimpose \nmid to \big| 84 | \ooalign{\hidewidth$\big|$\hidewidth\cr$/$\cr}% 85 | }% 86 | } 87 | 88 | % Convenient overline 89 | \newcommand{\ol}[1]{\ensuremath{\overline{#1}}} 90 | 91 | % Big \cdot 92 | \makeatletter 93 | \newcommand*\bigcdot{\mathpalette\bigcdot@{.5}} 94 | \newcommand*\bigcdot@[2]{\mathbin{\vcenter{\hbox{\scalebox{#2}{$\m@th#1\bullet$}}}}} 95 | \makeatother 96 | 97 | % Big and small Disjoint union 98 | \makeatletter 99 | \providecommand*{\cupdot}{% 100 | \mathbin{% 101 | \mathpalette\@cupdot{}% 102 | }% 103 | } 104 | \newcommand*{\@cupdot}[2]{% 105 | \ooalign{% 106 | $\m@th#1\cup$\cr 107 | \sbox0{$#1\cup$}% 108 | \dimen@=\ht0 % 109 | \sbox0{$\m@th#1\cdot$}% 110 | \advance\dimen@ by -\ht0 % 111 | \dimen@=.5\dimen@ 112 | \hidewidth\raise\dimen@\box0\hidewidth 113 | }% 114 | } 115 | 116 | \providecommand*{\bigcupdot}{% 117 | \mathop{% 118 | \vphantom{\bigcup}% 119 | \mathpalette\@bigcupdot{}% 120 | }% 121 | } 122 | \newcommand*{\@bigcupdot}[2]{% 123 | \ooalign{% 124 | $\m@th#1\bigcup$\cr 125 | \sbox0{$#1\bigcup$}% 126 | \dimen@=\ht0 % 127 | \advance\dimen@ by -\dp0 % 128 | \sbox0{\scalebox{2}{$\m@th#1\cdot$}}% 129 | \advance\dimen@ by -\ht0 % 130 | \dimen@=.5\dimen@ 131 | \hidewidth\raise\dimen@\box0\hidewidth 132 | }% 133 | } 134 | \makeatother 135 | 136 | 137 | % macros (theorem) ------------------------------------------------------------- 138 | \usepackage[thmmarks,amsmath,hyperref]{ntheorem} 139 | \usepackage[capitalise,nameinlink]{cleveref} 140 | 141 | % Numbered Statements 142 | \theoremstyle{change} 143 | \theoremindent\parindent 144 | \theorembodyfont{\itshape} 145 | \theoremheaderfont{\bfseries\boldmath} 146 | \newtheorem{theorem}{Theorem.}[section] 147 | \newtheorem{lemma}[theorem]{Lemma.} 148 | \newtheorem{corollary}[theorem]{Corollary.} 149 | \newtheorem{proposition}[theorem]{Proposition.} 150 | 151 | % Claim environment 152 | \theoremstyle{plain} 153 | \theorempreskip{0.2cm} 154 | \theorempostskip{0.2cm} 155 | \theoremheaderfont{\scshape} 156 | \newtheorem{claim}{Claim} 157 | \renewcommand\theclaim{\Roman{claim}} 158 | \AtBeginEnvironment{theorem}{\setcounter{claim}{0}} 159 | 160 | % Un-numbered Statements 161 | \theorempreskip{0.1cm} 162 | \theorempostskip{0.1cm} 163 | \theoremindent0.0cm 164 | \theoremstyle{nonumberplain} 165 | \theorembodyfont{\upshape} 166 | \theoremheaderfont{\bfseries\itshape} 167 | \newtheorem{definition}{Definition.} 168 | \theoremheaderfont{\itshape} 169 | \newtheorem{example}{Example.} 170 | \newtheorem{remark}{Remark.} 171 | 172 | % Proof / solution environments 173 | \theoremseparator{} 174 | \theoremheaderfont{\hspace*{\parindent}\scshape} 175 | \theoremsymbol{$//$} 176 | \newtheorem{solution}{Sol'n} 177 | \theoremsymbol{$\blacksquare$} 178 | \theorempostskip{0.4cm} 179 | \newtheorem{proof}{Proof} 180 | \theoremsymbol{} 181 | \newtheorem{nmproof}{Proof} 182 | 183 | % Format references 184 | \crefformat{equation}{(#2#1#3)} 185 | \Crefformat{theorem}{#2Thm. #1#3} 186 | \Crefformat{lemma}{#2Lem. #1#3} 187 | \Crefformat{proposition}{#2Prop. #1#3} 188 | \Crefformat{corollary}{#2Cor. #1#3} 189 | \crefformat{theorem}{#2Theorem #1#3} 190 | \crefformat{lemma}{#2Lemma #1#3} 191 | \crefformat{proposition}{#2Proposition #1#3} 192 | \crefformat{corollary}{#2Corollary #1#3} 193 | 194 | 195 | % macros (algebra) ------------------------------------------------------------- 196 | \DeclareMathOperator{\Ann}{Ann} 197 | \DeclareMathOperator{\Aut}{Aut} 198 | \DeclareMathOperator{\chr}{char} 199 | \DeclareMathOperator{\coker}{coker} 200 | \DeclareMathOperator{\disc}{disc} 201 | \DeclareMathOperator{\End}{End} 202 | \DeclareMathOperator{\Fix}{Fix} 203 | \DeclareMathOperator{\Frac}{Frac} 204 | \DeclareMathOperator{\Gal}{Gal} 205 | \DeclareMathOperator{\GL}{GL} 206 | \DeclareMathOperator{\Hom}{Hom} 207 | \DeclareMathOperator{\id}{id} 208 | \DeclareMathOperator{\im}{im} 209 | \DeclareMathOperator{\lcm}{lcm} 210 | \DeclareMathOperator{\Nil}{Nil} 211 | \DeclareMathOperator{\rank}{rank} 212 | \DeclareMathOperator{\Res}{Res} 213 | \DeclareMathOperator{\Spec}{Spec} 214 | \DeclareMathOperator{\spn}{span} 215 | \DeclareMathOperator{\Stab}{Stab} 216 | \DeclareMathOperator{\Tor}{Tor} 217 | 218 | % Lagrange symbol 219 | \newcommand{\lgs}[2]{\ensuremath{\left(\frac{#1}{#2}\right)}} 220 | 221 | % Quotient (larger in display mode) 222 | \newcommand{\quot}[2]{\mathchoice{\left.\raisebox{0.14em}{$#1$}\middle/\raisebox{-0.14em}{$#2$}\right.} 223 | {\left.\raisebox{0.08em}{$#1$}\middle/\raisebox{-0.08em}{$#2$}\right.} 224 | {\left.\raisebox{0.03em}{$#1$}\middle/\raisebox{-0.03em}{$#2$}\right.} 225 | {\left.\raisebox{0em}{$#1$}\middle/\raisebox{0em}{$#2$}\right.}} 226 | 227 | 228 | % macros (analysis) ------------------------------------------------------------ 229 | \DeclareMathOperator{\M}{{\mathcal{M}}} 230 | \DeclareMathOperator{\B}{{\mathcal{B}}} 231 | \DeclareMathOperator{\ps}{{\mathcal{P}}} 232 | \DeclareMathOperator{\pr}{{\mathbb{P}}} 233 | \DeclareMathOperator{\E}{{\mathbb{E}}} 234 | \DeclareMathOperator{\supp}{supp} 235 | \DeclareMathOperator{\sgn}{sgn} 236 | 237 | \renewcommand{\Re}{\ensuremath{\operatorname{Re}}} 238 | \renewcommand{\Im}{\ensuremath{\operatorname{Im}}} 239 | \renewcommand{\d}[1]{\ensuremath{\operatorname{d}\!{#1}}} 240 | 241 | 242 | % file-specific preamble ------------------------------------------------------- 243 | \newcommand{\defname}[1]{{\textit{(#1)}:}} 244 | \newcommand{\exname}[1]{{\textit{#1}:}} 245 | \newcommand{\defn}[1]{{\boldmath\bfseries #1}} 246 | % \usepackage{therefore} 247 | \newcommand{\TODO}[1]{[\textit{\textbf{TODO: #1}}]} 248 | \newcommand{\NOTE}[1]{[\textit{\textbf{NOTE: #1}}]} 249 | \DeclareMathOperator*{\esssup}{ess\,sup} 250 | \DeclareMathOperator{\ext}{ext} 251 | \DeclareMathOperator{\conv}{conv} 252 | \DeclareMathOperator{\dist}{dist} 253 | \DeclareMathOperator{\Pol}{Pol} 254 | \newcommand{\bdim}{\ensuremath{\dim_B}} 255 | \newcommand{\ubdim}{\ensuremath{\overline{\dim}_B}} 256 | \newcommand{\lbdim}{\ensuremath{\underline{\dim}_B}} 257 | \newcommand{\cwx}{\ensuremath{\overline{\operatorname{conv}}^{w^*}\,}} 258 | \newcommand{\idc}{\mathbf{1}} 259 | \newcommand{\FA}{\ensuremath{\operatorname{F}\!\operatorname{A}}} 260 | \newcommand{\cw}{\ensuremath{\overline{\operatorname{conv}}\,}} 261 | 262 | % Tons of notation: 263 | % \newcommand{\Lip}[1]{\ensuremath{\operatorname{Lip}_{\F}(#1)}} 264 | \newcommand{\Lipspace}{\ensuremath{\operatorname{Lip}_{\F}(X,d)}} 265 | 266 | 267 | \newcommand{\lp}[1]{\ensuremath{\ell^{#1}}} 268 | \newcommand{\csn}{\ensuremath{\mathbf{c}}} 269 | \newcommand{\csz}{\ensuremath{\mathbf{c}_0}} 270 | \newcommand{\lpspace}[1]{\ensuremath{\ell^{#1}_{\F}}} 271 | \newcommand{\Lp}[1]{\ensuremath{L^{#1}_{\F}}} 272 | % \newcommand{\Lpm}{\ensuremath{L^{#1}_{\F}(X,\mathcal{M},\mu)}} 273 | \DeclareMathOperator{\Lip}{Lip} 274 | \newcommand{\lbr}[1]{\ensuremath{\left[#1\right]}} 275 | \newcommand{\inr}[1]{\ensuremath{\left(#1\right)}} 276 | 277 | 278 | % constants -------------------------------------------------------------------- 279 | \newcommand{\subject}{Convex Analysis and Optimization} 280 | \newcommand{\semester}{Winter 2020} 281 | 282 | 283 | % formatting ------------------------------------------------------------------- 284 | % Fonts 285 | \usepackage{kpfonts} 286 | \usepackage{dsfont} 287 | 288 | % Adjust numbering 289 | \numberwithin{equation}{section} 290 | \counterwithin{figure}{section} 291 | \counterwithout{section}{chapter} 292 | \counterwithin*{chapter}{part} 293 | 294 | % Footnote 295 | \setfootins{0.5cm}{0.5cm} % footer space above 296 | \renewcommand*{\thefootnote}{\fnsymbol{footnote}} % footnote symbol 297 | 298 | % Table of Contents 299 | \renewcommand{\thechapter}{\Roman{chapter}} 300 | \renewcommand*{\cftchaptername}{Chapter } % Place 'Chapter' before roman 301 | \setlength\cftchapternumwidth{4em} % Add space before chapter name 302 | \cftpagenumbersoff{chapter} % Turn off page numbers for chapter 303 | \maxtocdepth{subsection} % table of contents up to section 304 | 305 | % Section / Subsection headers 306 | \setsecnumdepth{subsection} % numbering up to and including "subsection" 307 | \newcommand*{\shortcenter}[1]{% 308 | \sethangfrom{\noindent ##1}% 309 | \Large\boldmath\scshape\bfseries 310 | \centering 311 | \parbox{5in}{\centering #1}\par} 312 | \setsecheadstyle{\shortcenter} 313 | \setsubsecheadstyle{\large\scshape\boldmath\bfseries\raggedright} 314 | 315 | % Chapter Headers 316 | \chapterstyle{verville} 317 | 318 | % Page Headers / Footers 319 | \copypagestyle{myruled}{ruled} % Draw formatting from existing 'ruled' style 320 | \makeoddhead{myruled}{}{}{\scshape\subject} 321 | \makeevenfoot{myruled}{}{\thepage}{} 322 | \makeoddfoot{myruled}{}{\thepage}{} 323 | \pagestyle{myruled} 324 | \setfootins{0.5cm}{0.5cm} 325 | \renewcommand*{\thefootnote}{\fnsymbol{footnote}} 326 | 327 | % Titlepage 328 | \title{\subject} 329 | \author{Alex Rutar\thanks{\itshape arutar@uwaterloo.ca}\\ University of Waterloo} 330 | \date{\semester\thanks{Last updated: \today}} 331 | 332 | \begin{document} 333 | \pagenumbering{gobble} 334 | \hypersetup{pageanchor=false} 335 | \maketitle 336 | \newpage 337 | \frontmatter 338 | \hypersetup{pageanchor=true} 339 | \tableofcontents* 340 | \newpage 341 | \mainmatter 342 | 343 | 344 | % main document ---------------------------------------------------------------- 345 | \chapter{Convex Analysis} 346 | 347 | \end{document} 348 | -------------------------------------------------------------------------------- /co/749_notes.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/alexrutar/UWaterloo-Course-Notes/9663d05575969e4a46c1e4181f4983b10198e320/co/749_notes.pdf -------------------------------------------------------------------------------- /co/749_notes.tex: -------------------------------------------------------------------------------- 1 | % header ----------------------------------------------------------------------- 2 | % Template created by texnew (author: Alex Rutar); info can be found at 'https://github.com/alexrutar/texnew'. 3 | % version (1.13) 4 | 5 | 6 | % doctype ---------------------------------------------------------------------- 7 | \documentclass[11pt, a4paper]{memoir} 8 | \usepackage[utf8]{inputenc} 9 | \usepackage[left=3cm,right=3cm,top=3cm,bottom=4cm]{geometry} 10 | \usepackage[protrusion=true,expansion=true]{microtype} 11 | 12 | 13 | % packages --------------------------------------------------------------------- 14 | \usepackage{amsmath,amssymb,amsfonts} 15 | \usepackage{graphicx} 16 | \usepackage{etoolbox} 17 | 18 | % Set enimitem 19 | \usepackage{enumitem} 20 | \SetEnumitemKey{nl}{nolistsep} 21 | \SetEnumitemKey{r}{label=(\roman*)} 22 | 23 | % Set tikz 24 | \usepackage{tikz, pgfplots} 25 | \pgfplotsset{compat=1.15} 26 | \usetikzlibrary{intersections,positioning,cd} 27 | \usetikzlibrary{arrows,arrows.meta} 28 | \usetikzlibrary{decorations.markings} 29 | \tikzcdset{arrow style=tikz,diagrams={>=stealth}} 30 | \tikzset{ 31 | vtx/.style={inner sep=2pt,circle,fill=black}, 32 | e/.style={thick}, 33 | de/.style={e,decoration={markings,mark=at position 0.5 with {\arrow{>}}},postaction={decorate}} 34 | } 35 | 36 | % Set hyperref 37 | \usepackage[hidelinks]{hyperref} 38 | \usepackage{xcolor} 39 | \newcommand\myshade{85} 40 | \colorlet{mylinkcolor}{violet} 41 | \colorlet{mycitecolor}{orange!50!yellow} 42 | \colorlet{myurlcolor}{green!50!blue} 43 | 44 | \hypersetup{ 45 | linkcolor = mylinkcolor!\myshade!black, 46 | citecolor = mycitecolor!\myshade!black, 47 | urlcolor = myurlcolor!\myshade!black, 48 | colorlinks = true, 49 | } 50 | 51 | 52 | % macros ----------------------------------------------------------------------- 53 | \DeclareMathOperator{\N}{{\mathbb{N}}} 54 | \DeclareMathOperator{\Q}{{\mathbb{Q}}} 55 | \DeclareMathOperator{\Z}{{\mathbb{Z}}} 56 | \DeclareMathOperator{\R}{{\mathbb{R}}} 57 | \DeclareMathOperator{\C}{{\mathbb{C}}} 58 | \DeclareMathOperator{\F}{{\mathbb{F}}} 59 | 60 | % Boldface includes math 61 | \newcommand{\mbf}[1]{{\boldmath\bfseries #1}} 62 | 63 | % proof implications 64 | \newcommand{\imp}[2]{($#1\Rightarrow#2$)\hspace{0.2cm}} 65 | \newcommand{\impe}[2]{($#1\Leftrightarrow#2$)\hspace{0.2cm}} 66 | \newcommand{\impr}{{($\Rightarrow$)\hspace{0.2cm}}} 67 | \newcommand{\impl}{{($\Leftarrow$)\hspace{0.2cm}}} 68 | 69 | % align macros 70 | \newcommand{\agspace}{\ensuremath{\phantom{--}}} 71 | \newcommand{\agvdots}{\ensuremath{\hspace{0.16cm}\vdots}} 72 | 73 | % convenient brackets 74 | \newcommand{\brac}[1]{\ensuremath{\left\langle #1 \right\rangle}} 75 | \newcommand{\norm}[1]{\ensuremath{\left\lVert#1\right\rVert}} 76 | \newcommand{\abs}[1]{\ensuremath{\left\lvert#1\right\rvert}} 77 | 78 | % arrows 79 | \newcommand{\lto}[0]{\ensuremath{\longrightarrow}} 80 | \newcommand{\fto}[1]{\ensuremath{\xrightarrow{\scriptstyle{#1}}}} 81 | \newcommand{\hto}[0]{\ensuremath{\hookrightarrow}} 82 | \newcommand{\mapsfrom}[0]{\mathrel{\reflectbox{\ensuremath{\mapsto}}}} 83 | 84 | % Divides, Not Divides 85 | \renewcommand{\div}{\bigm|} 86 | \newcommand{\ndiv}{% 87 | \mathrel{\mkern.5mu % small adjustment 88 | % superimpose \nmid to \big| 89 | \ooalign{\hidewidth$\big|$\hidewidth\cr$/$\cr}% 90 | }% 91 | } 92 | 93 | % Convenient overline 94 | \newcommand{\ol}[1]{\ensuremath{\overline{#1}}} 95 | 96 | % Big \cdot 97 | \makeatletter 98 | \newcommand*\bigcdot{\mathpalette\bigcdot@{.5}} 99 | \newcommand*\bigcdot@[2]{\mathbin{\vcenter{\hbox{\scalebox{#2}{$\m@th#1\bullet$}}}}} 100 | \makeatother 101 | 102 | % Big and small Disjoint union 103 | \makeatletter 104 | \providecommand*{\cupdot}{% 105 | \mathbin{% 106 | \mathpalette\@cupdot{}% 107 | }% 108 | } 109 | \newcommand*{\@cupdot}[2]{% 110 | \ooalign{% 111 | $\m@th#1\cup$\cr 112 | \sbox0{$#1\cup$}% 113 | \dimen@=\ht0 % 114 | \sbox0{$\m@th#1\cdot$}% 115 | \advance\dimen@ by -\ht0 % 116 | \dimen@=.5\dimen@ 117 | \hidewidth\raise\dimen@\box0\hidewidth 118 | }% 119 | } 120 | 121 | \providecommand*{\bigcupdot}{% 122 | \mathop{% 123 | \vphantom{\bigcup}% 124 | \mathpalette\@bigcupdot{}% 125 | }% 126 | } 127 | \newcommand*{\@bigcupdot}[2]{% 128 | \ooalign{% 129 | $\m@th#1\bigcup$\cr 130 | \sbox0{$#1\bigcup$}% 131 | \dimen@=\ht0 % 132 | \advance\dimen@ by -\dp0 % 133 | \sbox0{\scalebox{2}{$\m@th#1\cdot$}}% 134 | \advance\dimen@ by -\ht0 % 135 | \dimen@=.5\dimen@ 136 | \hidewidth\raise\dimen@\box0\hidewidth 137 | }% 138 | } 139 | \makeatother 140 | 141 | 142 | % macros (theorem) ------------------------------------------------------------- 143 | \usepackage[thmmarks,amsmath,hyperref]{ntheorem} 144 | \usepackage[capitalise,nameinlink]{cleveref} 145 | 146 | % Numbered Statements 147 | \theoremstyle{change} 148 | \theoremindent\parindent 149 | \theorembodyfont{\itshape} 150 | \theoremheaderfont{\bfseries\boldmath} 151 | \newtheorem{theorem}{Theorem.}[section] 152 | \newtheorem{lemma}[theorem]{Lemma.} 153 | \newtheorem{conjecture}[theorem]{Conjecture.} 154 | \newtheorem{corollary}[theorem]{Corollary.} 155 | \newtheorem{proposition}[theorem]{Proposition.} 156 | 157 | % Claim environment 158 | \theoremstyle{plain} 159 | \theorempreskip{0.2cm} 160 | \theorempostskip{0.2cm} 161 | \theoremheaderfont{\scshape} 162 | \newtheorem{claim}{Claim} 163 | \renewcommand\theclaim{\Roman{claim}} 164 | \AtBeginEnvironment{theorem}{\setcounter{claim}{0}} 165 | 166 | % Un-numbered Statements 167 | \theorempreskip{0.1cm} 168 | \theorempostskip{0.1cm} 169 | \theoremindent0.0cm 170 | \theoremstyle{nonumberplain} 171 | \theorembodyfont{\upshape} 172 | \theoremheaderfont{\bfseries\itshape} 173 | \newtheorem{definition}{Definition.} 174 | \theoremheaderfont{\itshape} 175 | \newtheorem{example}{Example.} 176 | \newtheorem{remark}{Remark.} 177 | 178 | % Proof / solution environments 179 | \theoremseparator{} 180 | \theoremheaderfont{\hspace*{\parindent}\scshape} 181 | \theoremsymbol{$//$} 182 | \newtheorem{solution}{Sol'n} 183 | \theoremsymbol{$\blacksquare$} 184 | \theorempostskip{0.4cm} 185 | \newtheorem{proof}{Proof} 186 | \theoremsymbol{} 187 | \newtheorem{nmproof}{Proof} 188 | 189 | % Format references 190 | \crefformat{equation}{(#2#1#3)} 191 | \Crefformat{theorem}{#2Thm. #1#3} 192 | \Crefformat{lemma}{#2Lem. #1#3} 193 | \Crefformat{proposition}{#2Prop. #1#3} 194 | \Crefformat{corollary}{#2Cor. #1#3} 195 | \crefformat{theorem}{#2Theorem #1#3} 196 | \crefformat{lemma}{#2Lemma #1#3} 197 | \crefformat{proposition}{#2Proposition #1#3} 198 | \crefformat{corollary}{#2Corollary #1#3} 199 | 200 | 201 | % macros (algebra) ------------------------------------------------------------- 202 | \DeclareMathOperator{\Ann}{Ann} 203 | \DeclareMathOperator{\Aut}{Aut} 204 | \DeclareMathOperator{\chr}{char} 205 | \DeclareMathOperator{\coker}{coker} 206 | \DeclareMathOperator{\disc}{disc} 207 | \DeclareMathOperator{\End}{End} 208 | \DeclareMathOperator{\Fix}{Fix} 209 | \DeclareMathOperator{\Frac}{Frac} 210 | \DeclareMathOperator{\Gal}{Gal} 211 | \DeclareMathOperator{\GL}{GL} 212 | \DeclareMathOperator{\Hom}{Hom} 213 | \DeclareMathOperator{\id}{id} 214 | \DeclareMathOperator{\im}{im} 215 | \DeclareMathOperator{\lcm}{lcm} 216 | \DeclareMathOperator{\Nil}{Nil} 217 | \DeclareMathOperator{\rank}{rank} 218 | \DeclareMathOperator{\Res}{Res} 219 | \DeclareMathOperator{\Spec}{Spec} 220 | \DeclareMathOperator{\spn}{span} 221 | \DeclareMathOperator{\Stab}{Stab} 222 | \DeclareMathOperator{\Tor}{Tor} 223 | 224 | % Lagrange symbol 225 | \newcommand{\lgs}[2]{\ensuremath{\left(\frac{#1}{#2}\right)}} 226 | 227 | % Quotient (larger in display mode) 228 | \newcommand{\quot}[2]{\mathchoice{\left.\raisebox{0.14em}{$#1$}\middle/\raisebox{-0.14em}{$#2$}\right.} 229 | {\left.\raisebox{0.08em}{$#1$}\middle/\raisebox{-0.08em}{$#2$}\right.} 230 | {\left.\raisebox{0.03em}{$#1$}\middle/\raisebox{-0.03em}{$#2$}\right.} 231 | {\left.\raisebox{0em}{$#1$}\middle/\raisebox{0em}{$#2$}\right.}} 232 | 233 | 234 | % macros (analysis) ------------------------------------------------------------ 235 | \DeclareMathOperator{\M}{{\mathcal{M}}} 236 | \DeclareMathOperator{\B}{{\mathcal{B}}} 237 | \DeclareMathOperator{\ps}{{\mathcal{P}}} 238 | \DeclareMathOperator{\pr}{{\mathbb{P}}} 239 | \DeclareMathOperator{\E}{{\mathbb{E}}} 240 | \DeclareMathOperator{\supp}{supp} 241 | \DeclareMathOperator{\sgn}{sgn} 242 | 243 | \renewcommand{\Re}{\ensuremath{\operatorname{Re}}} 244 | \renewcommand{\Im}{\ensuremath{\operatorname{Im}}} 245 | \renewcommand{\d}[1]{\ensuremath{\operatorname{d}\!{#1}}} 246 | 247 | 248 | % file-specific preamble ------------------------------------------------------- 249 | 250 | % constants -------------------------------------------------------------------- 251 | \newcommand{\subject}{Topics in Graph Theory} 252 | \newcommand{\semester}{Fall 2019} 253 | \newcommand{\from}{\ensuremath{\leftarrow}} 254 | 255 | 256 | % formatting ------------------------------------------------------------------- 257 | % Fonts 258 | \usepackage{kpfonts} 259 | \usepackage{dsfont} 260 | 261 | % Adjust numbering 262 | \numberwithin{equation}{section} 263 | \counterwithin{figure}{section} 264 | \counterwithout{section}{chapter} 265 | \counterwithin*{chapter}{part} 266 | 267 | % Footnote 268 | \setfootins{0.5cm}{0.5cm} % footer space above 269 | \renewcommand*{\thefootnote}{\fnsymbol{footnote}} % footnote symbol 270 | 271 | % Table of Contents 272 | \renewcommand{\thechapter}{\Roman{chapter}} 273 | \renewcommand*{\cftchaptername}{Chapter } % Place 'Chapter' before roman 274 | \setlength\cftchapternumwidth{4em} % Add space before chapter name 275 | \cftpagenumbersoff{chapter} % Turn off page numbers for chapter 276 | \maxtocdepth{section} % table of contents up to section 277 | 278 | % Section / Subsection headers 279 | \setsecnumdepth{section} % numbering up to and including "section" 280 | \newcommand*{\shortcenter}[1]{% 281 | \sethangfrom{\noindent ##1}% 282 | \Large\boldmath\scshape\bfseries 283 | \centering 284 | \parbox{5in}{\centering #1}\par} 285 | \setsecheadstyle{\shortcenter} 286 | \setsubsecheadstyle{\large\scshape\boldmath\bfseries\raggedright} 287 | 288 | % Chapter Headers 289 | \chapterstyle{verville} 290 | 291 | % Page Headers / Footers 292 | \copypagestyle{myruled}{ruled} % Draw formatting from echisting 'ruled' style 293 | \makeoddhead{myruled}{}{}{\scshape\subject} 294 | \makeevenfoot{myruled}{}{\thepage}{} 295 | \makeoddfoot{myruled}{}{\thepage}{} 296 | \pagestyle{myruled} 297 | \setfootins{0.5cm}{0.5cm} 298 | \renewcommand*{\thefootnote}{\fnsymbol{footnote}} 299 | 300 | % Titlepage 301 | \title{\subject} 302 | \author{Alex Rutar\thanks{\itshape arutar@uwaterloo.ca}\\ University of Waterloo} 303 | \date{\semester\thanks{Last updated: \today}} 304 | 305 | \begin{document} 306 | \pagenumbering{gobble} 307 | \hypersetup{pageanchor=false} 308 | \maketitle 309 | \newpage 310 | \frontmatter 311 | \hypersetup{pageanchor=true} 312 | \tableofcontents* 313 | \newpage 314 | \mainmatter 315 | 316 | 317 | % main document ---------------------------------------------------------------- 318 | \chapter{Graph Colourings} 319 | \section{List Colourings} 320 | Recall that a colouring of a graph $G$ is an assignment to each $v\in V(G)$ an element $c(v)$ of some set $C$ called ``colors'' such that if $v$ and $v'$ are neighbours, then $c(v)\neq c(v')$. 321 | Then the \textbf{chromatic number} $\chi(G)$ is the smallest cardinality $|C|$ such that there exists a colouring of $G$ from $C$. 322 | 323 | There are some basic upper bounds on the chromatic number of a graph: 324 | \begin{enumerate}[nl] 325 | \item $\chi(G)\leq|V(G)|$, by colouring every vertex distinctly 326 | \item $\chi(G)\leq\Delta(G)+1$, by randomly colouring the graph based on colours not used on the neighbours 327 | \end{enumerate} 328 | Note that these upper bounds are in fact tight; for example, the complete graph is tight for both, and an odd cycle is tight for (2). 329 | 330 | There are some graphs for which the chromatic number is not known: consider the graph given by $V(G)=\R^2$ where vertices are adjacent if they have euclidean distance 1. 331 | This graph is not $3-$colorable, by taking for example the subgraph 332 | \begin{center} 333 | \begin{tikzpicture}[scale=3] 334 | \begin{scope}[rotate=16.78] 335 | \node[vtx] (a1) at (0,0) {}; 336 | \node[vtx] (a2) at (-.577350269,-1) {}; 337 | \node[vtx] (a3) at (.577350269,-1) {}; 338 | \node[vtx] (a4) at (0,-2) {}; 339 | \end{scope} 340 | \begin{scope}[rotate=-16.78] 341 | \node[vtx] (b1) at (0,0) {}; 342 | \node[vtx] (b2) at (-.577350269,-1) {}; 343 | \node[vtx] (b3) at (.577350269,-1) {}; 344 | \node[vtx] (b4) at (0,-2) {}; 345 | \end{scope} 346 | \draw (a1) -- (a2) -- (a4) -- (a3) -- (a1); 347 | \draw (a2) -- (a3); 348 | \draw (b1) -- (b2) -- (b4) -- (b3) -- (b1); 349 | \draw (b2) -- (b3); 350 | \draw (a4) -- (b4); 351 | \end{tikzpicture} 352 | \end{center} 353 | Recently there was a construction showing that the graph is not $4-$colourable, and there is an easy upper bound of $7$, so that $5\leq \chi(G)\leq 7$. 354 | 355 | We also define the notion of a list colouring: 356 | \begin{definition} 357 | A list assignment is an assignment of a set $L(v)$ of colors to each vertex $v$. 358 | Then a graph is $k-$list-colorable if you can always colour $V(G)$ whenever every vertex has a list of size at least $k$. 359 | \end{definition} 360 | Note that $\chi(G)\leq\chi_\ell(G)$ since asssigning an identical list of size $k$ is a valid list assignmnet and yields a standard coloring. 361 | In many cases list colorings can be hard to determine, but in some cases the exact value is known. 362 | Consider the complete bipartite graph $K_{k,q}$ where $q\geq k$. 363 | We then have the following classification: 364 | \begin{proposition} 365 | $\chi_\ell(K_{k,q})\leq k$ if and only if $q1$, $1->2$, $2->3$ 496 | 497 | For \imp{iii}{i}, colour some vertex 0. 498 | Then for any other vertex, take a path connecting the vertices and walk along the path by adding one for every forward traversal, and subtract one for each backwards traversal, modulo 3. 499 | If there are multiple paths, then the multiple paths would form a walk which is balanced modulo 3, so the lengths must be the same. 500 | \end{proof} 501 | \begin{definition} 502 | A \textbf{cut} in a graph. 503 | Partition the vertex set into two pieces. 504 | Then a cut is the set of edges between the two vertex sets. 505 | A \textbf{minimal cut} is a cut containing no other cuts. 506 | \end{definition} 507 | Note that a cut is minimal if and only if each side of the cut is connected. 508 | If $G$ is planar, then the dual graph is formed as follows: each face becomes a vertex, and the vertices are joined by an edge if the corresponding faces are adjacent. 509 | The number of edges is unchanged, and the number of vertices and faces swaps. 510 | 511 | Given an orientation on the original graph, we can pass the orientation to the dual graph by setting the orientation anticlockwise relative to the intersection. 512 | Let $E\subseteq E(G)$. 513 | Then $E$ is a minimal cut in $G$ if and only if $E^*$ is a cycle, and $E$ is a cycle in $G$ if and only if $E^*$ is a minimal cut in $G^*$. 514 | 515 | Assume $G$ is planar. 516 | If $G$ is 4-edge-connected, then each cut has at least 4 edges, $G^*$ has girth at least 4, then $\chi(G^*)\leq 3$, then the following equivalent things hold: 517 | \begin{enumerate}[nl,r] 518 | \item $G^*$ has an orientation so that all cycles are balanced modulo 3 519 | \item $G$has an orientation such that all cuts are balanced modulo 3 520 | \item $G$ has an orientation such that $d^+(v)\equiv d^-(v)$ modulo 3 521 | \end{enumerate} 522 | \begin{conjecture}[Tutte] 523 | If $G$ is 4-edge-connected, then there exists an orientation on $G$ such that all degrees are balanced modulo 3. 524 | \end{conjecture} 525 | Currently proven for 6-edge-connected. 526 | If $G$ is 4-edge connected, then there exists an orientation on $G$ and a flow 1 or 2 on each edge such that at each vertex the inflow equals the outflow. 527 | This is equivalent to the conjecture by reversing the orientation for all edges which have flow 2, or by simply placing flow 1 on every edge in the graph. 528 | 529 | In fact, one can remove the modular condition. 530 | Assume each edge has a flow 1 or 2 or 3 or 4, and assume that each inflow is equivalen to the out flow modulo 5. 531 | 532 | \begin{proposition} 533 | If $G$ is planar and 4-edge-onnected, then there exists an orientation such that $G$ is balanced modulo 3. 534 | \end{proposition} 535 | \begin{proposition} 536 | If $G$ is cubic and 3-edge-connected, there exists an orientation which is balanced modulo 3 if and only if $G$ is bipartite. 537 | \end{proposition} 538 | Does there exists an orientation on $G$ such that $G$ is balanced modulo $k$? 539 | Or such that each vertex $v$ has out degree $p(v)$ modulo $k$? 540 | 541 | If the second holds for every $p$ and $k$ is odd, then the first holds. 542 | Let $v$ be a vertex with degree $d(v)$; we want that $d^+(v)\cong d^-(v)$, in other words that $2d^+(v)\cong d(v)\pmod{k}$, s 543 | \begin{equation*} 544 | \frac{k-1}{2}\cdot 2d^+(v)\cong\frac{k-1}{2} d(v)\Rightarrow d^+(v)\cong\frac{-(k-1)}{2} d(v) 545 | \end{equation*} 546 | 547 | Suppose $k=2$. 548 | Here's a necessary condition: then $|E(G)|=\sum_{v\in V(G)}d^+(v)\cong\sum_{v\in V(G)}p(v)$, modulo 2. 549 | In fact, if $G$ is connected and $\sum_{v\in V(G)}p(v)\cong|E(G)$, then such an orientation exists. 550 | Do do this, fix any orientation. 551 | If there is a vertex which does not satisfy the requirements, by parity, there must be some other vertex which does not satisfy the requirements. 552 | Take a path connecting the vertices and flip all the edges, repeating until the graph is balanced. 553 | \begin{conjecture}[Jaeger] 554 | If $G$ is 1000-edge-connected, then there exists an orientation on $G$ balanced modulo 3. 555 | \end{conjecture} 556 | This has been proven in the affirmative for 8-edge-connected, then 6-edge-connected. 557 | It is enough to prove this for 5. 558 | \begin{conjecture}[Jaeger] 559 | If $G$ is $(2k-2)-$edge-connected, then there exists an orientation on $G$ that is balanced modulo $k$ if $k$ is odd. 560 | \end{conjecture} 561 | It has been shown that if there is a $(2k^2+2)-$edge-connected graph, then there exists an orientation on $G$ with any out degrees modulo $k$, also true for $k$ is even. 562 | If $G$ is $(3k-3)-$edge connected, then the same holds, but only for $k$ odd. 563 | 564 | Suppose $G$ is 4-edge-connected: then there exists an orientation of $G$ balanced modulo 4. 565 | This is equivalent to the 3-flow conjecture. 566 | Given an orientation balanced modulo 3, by a previous exercise, we can also balance each vertex modulo $k$ for any $k$. 567 | 568 | If $k=5$, the statement says that $G$ is 8-edge-connected implies $G$ is balanced modulo 5. 569 | 570 | Let $G$ be 2-edge-connected, then there exists an orientation on $G$ with flow values on $\{1,2,3,4\}$ such that the inflow and the outflow are equal for all $v\in V(G)$. 571 | It suffices to verify this for cubic 3-connected graphs. 572 | Note that for cubic graphs, the edge and vertex connectivity are the same. 573 | $k$ connected means there are $k$ internally vertex disjoint paths, and $k-$edge-connected means there are $k$ internally edge disjoint paths. 574 | 575 | \begin{example} 576 | Assume the 5-flow-conjecture holds for $G$ cubic 3-connected. 577 | Then prove that it holds for $G$ 2-edge-connected. 578 | There's a couple cases: if there is a vertex of degree 2 with edges going to the same vertex, simply add the same flow value going in and out. 579 | If there is a vertex of degree 2 with edges going to distinct edges, simply merge the edges, apply induction, and then apply the flow assigned to that edge to both pieces. 580 | 581 | If there is a vertex with degree large, remove two of the edges so as not to create a bridge, and apply the same argument. 582 | What happens if we have all vertex of degree 3? 583 | We need to deal with the case where $G$ is 2-edge connected. 584 | Isolate the pair of edges $e_1$ and $e_2$. 585 | First close the loops, and then multiply the flows or perhaps re-orient so that the edges agree. 586 | 587 | Now assume $G$ is cubic and 3-edge-connected. 588 | Then take the graph and replace every edge by 3 edges to get some $G'$ that is 9-edge-connected. 589 | Therefore, by the result above (Jaeger with $k=5$), it has an orientation that is balanced modulo 5. 590 | Then replace each triple of edges with the oriented net sum of the number of edges. 591 | \end{example} 592 | \begin{example} 593 | $K_8$ is 7-edge-connected nad has no orientation balanced modulo 5. 594 | \end{example} 595 | Let's consider factors modulo $k$. 596 | A $d$-factor is a spanning subgraph of $G$ such that every vertex of the subgraph has degree $d$. 597 | \begin{theorem} 598 | Let $G$ be bipartite with bipartition $V(G)=A\cup B$ with $V(G)=\{v_1,\ldots,v_n\}$. 599 | For every $v_i$, let $d_i$ be a natural number. 600 | We want a spanning subgraph of $H\subseteq G$ such that $d_H(v_i)\cong d_i\pmod{k}$ where $k$ is odd. 601 | Then $H$ exists if $G$ is $(3k-3)-$edge connected and $\sum_{v_i\in A}d_i\cong\sum_{v_i\in B}d_i\pmod{k}$. 602 | \end{theorem} 603 | \begin{proof} 604 | Apply the $(3k-3)$ result, and assign the function $p(v_i)=d_i$ for $v_i\in A$ and $p(v_j)=d(v_j)-d_j$ for $v_j\in B$. 605 | Certainly $\sum_{v_i\in V(G)}p_i\cong|E(G)|$ modulo $k$ by the modular summation condition on the $d_i$. 606 | Then we simply take all $A->B$ edges. 607 | \end{proof} 608 | Recall that if $G$ is 9-edge-connected, then there exists an orientation on $G$ balanced modulo $5$. 609 | We've shown that if $G$ is 9-edge-connected, then Tutte's 5-flow theorem follows. 610 | Jaeger conjectured that this in fact holds for 8-edge-connected graphs. 611 | 612 | \begin{example} 613 | Which $K_n$ have an orientation balanced modulo 5? 614 | If $n$ is odd, this always works, since then all the vertex degrees have even degree, and we can simply use an eulerian tour. 615 | Now $K_8$ is 7-edge connected, and does not have an orientation balanced modulo 5. 616 | 617 | We can write $7=7+0=6+1=5+2=4+3$; and if $K_8$ is balanced modulo 5, then all $d^+(v),d^-(v)\in\{1,6\}$. 618 | If such an orientation exists, we must have 4 with out degree 6, and 4 with out degree 1 by counting flows. 619 | But then on the $d^+=1$ side, the sum of the out degrees is 4, but it must be at least 6 (by counting internal vertices). 620 | 621 | We can do $K_{10}$: partition into copies of $K_5$, make each balanced modulo 5, and then add all edges from one side to the other with the same orientation. 622 | We can also generalize this, by adding two vertices. 623 | \end{example} 624 | Let $G$ be bipartite with $N(G)=A\cup B$. 625 | Set $V(G)=\{v_1,\ldots,v_n\}$, with $d_1,d_2,\ldots,d_n\in\N$. 626 | Then we want to find $G\supseteq H$ such that $d_H(v_i)\equiv d_i\pmod{k}$ where $\sum_{v_i\in A}d_i=\sum_{v_i\in B}d_i$. 627 | This is always doable if $G$ is $(3k-3)-$edge connected (for $k$ odd), else $G$ is $(2k^2+k)-$edge-connected that $k$ is even. 628 | 629 | Let $G$ be a graph and partition $G$ into sides $A$ and $B$ such that the number edges between them is maximal. 630 | Let $H$ be the graph induced by the maximum cut edges, so that $H$ is bipartite. 631 | Then the following properties hold: 632 | \begin{enumerate}[nl,r] 633 | \item $d_H(v)\geq\frac{1}{2}d_G(v)$. 634 | \item $|E(H)|\geq\frac{1}{2}|E(G)|$ 635 | \item If $G$ is $(2k-1)-$edge connected, then $H$ is $k-$edge-connected. 636 | \end{enumerate} 637 | How to see this? 638 | If $v\in B\subseteq V(G)$ is a given vertex, then the number of edges in the cut must be at least as large as the number of internal edges from $v$ on side $A$ (or we could swap $v$ to the other side and get a better cut). 639 | This shows (i) and (ii). 640 | To show (iii), suppose $H$ is not $k-$edge-connected ... (see paper). 641 | \begin{theorem} 642 | Let $k\in\N$, and $G$ a $(6k-7)-$edge connected connected graph with $k$ odd. 643 | Let $V(G)=\{v_1,\ldots,v_n\}$ and $d_1,\ldots,d_n\in\N$ given. 644 | We wish to find $H\subseteq G$ such that $d_H(v_i)\cong d_i\pmod{k}$. 645 | This can be done if for every partition $V(G)=A\cup B$, $\sum_{v_i}\in A d_i\cong\sum_{v_i\in B}d_i\pmod{k}$. 646 | \end{theorem} 647 | \begin{proof} 648 | By the previous arguments, there exists some $H'\subseteq G$ with $H'$ bipartite and $(3k-3)-$edge connectivity, and apply the previous result with $H'\supseteq H$ satisfying the result. 649 | \end{proof} 650 | \begin{conjecture} 651 | If $G$ is simple and $4-$regular, then $G$ contains a 3-regular subgraph. 652 | \end{conjecture} 653 | If $|E(G)|>2|V(G)|$, then $G\supseteq H$ all vertices degree equivalent to 0 modulo 3 where $H$ has nonempty edge set. 654 | 655 | Suppose $G$ is $4-$regular, perhaps with multiple edges. 656 | Then $G$ with an extra edge contains a 3-regular subgraph. 657 | 658 | Now consider the previous theorem where all $d_i=k$, and we work modulo $2k$. 659 | If $G$ is $[2(2(2k)^2+2k)-1]-$edge-connected with $|V(G)|$ even, then $G\supseteq H$ has all degrees congruent to $k$ modulo $2k$. 660 | 661 | \section{Group Valued Flows} 662 | Find an orientation on $G$ such that $G$ is almost balanced. 663 | There exists some small $\epsilon>0$ such that $E(A,B)\leq(1+\epsilon) E(B,A)$ ($E(A,B)$ is number of oriented edges from $A$ to $B$) and $E(B,A)\leq (1+\epsilon)E(A,B)$. 664 | Given an abelian group $\Gamma$ and $F\subseteq\Gamma$, and $G$ is a graph. 665 | We want an $F-$flow in $G$, in other words that each edge $e$ gets some $g\in G$ such that the sum of the in flow is equal to the sum of the out flow. 666 | 667 | If $G$ is 6-edge-connected, then $G$ has a $\{1,2\}-$flow. 668 | If $G$ is 2-edge connected, then $G$ has a $\{1,2,3,4,5\}-$flow. 669 | 670 | Let $f(F,\Gamma)$ denote the smallest $k$ such that every $k-$edge connected has an $F-$flow. 671 | For example, if $F=\{1,3\}\subseteq\Z$, this is not always possible. 672 | But we do have 673 | \begin{theorem} 674 | $f(F,\Gamma)$ exists if and only if the odd sum condition holds. 675 | \end{theorem} 676 | The odd sum condition is the statement that it is possible to have a sum of an even number of elements and a sum of an odd number of elements have equal value. 677 | 678 | Recall: 679 | \begin{itemize}[nl] 680 | \item 5-flow conjecture: if $G$ if 2-edge-connected, then there exists a flow with values $1,2,3,4$ 681 | \item 3-flow-conjecture: if $G$ is 4-edge-connected, then there exists a flow with values $1,2$ 682 | \item $(2+\epsilon)$-flow-conjecture: if $G$ is $\alpha(\epsilon)$-edge-connected, then there exists a flow with values $[1,1+\epsilon]$ 683 | \end{itemize} 684 | To have a flow with only 1, graph must be Eulerian. 685 | 686 | If $G$ is 3-edge-connected, the $(2+\epsilon)$ flow need not exist (for example, with a 3-strut: two components with 3 edges joining them). 687 | 688 | Let $c(x,y)$ denote the number of edge-disjoint paths from $x$ to $y$ (the \textbf{local edge connectivity}). 689 | \begin{theorem}[Mader Lifting] 690 | Let $G$ be a graph and $v$ a vertex with neighbours. 691 | Fix a neighbour of $v$ such that $d(v)\geq 4$, $w,w'$ and remove the edges $\{v,w\}$ and $\{v,w'\}$ and add the edge $\{w,w'\}$. 692 | This is called a lift. 693 | 694 | The lifting can be chosen such that all $c(x,y)$ remains the same for all $x\neq y$, $x\neq v$, $y\neq v$. 695 | \end{theorem} 696 | \begin{proof} 697 | TODO 698 | \end{proof} 699 | Let $\Gamma$ be an abelian group and $F\subseteq\Gamma$. 700 | Then a $F-$flow is an assignment of $g\in F$ to each $v\in V(G)$ such that the sum of the ingoing edges is equal to the sum of the outgoing edges. 701 | \begin{theorem} 702 | Suppose $F$ satisfies the odd-sum condition; in other words, there exists $a_i,b_i\in F$ so that $a_1+\cdots+a_{2p}=b_1+\cdots+b_{2q+1}$. 703 | Then there exists a function $f(F,\Gamma)\leq 3k-1$ where $k=2p+2q+1$ such that every $f(F,\Gamma)-$edge-connected graph has an $F-$flow. 704 | \end{theorem} 705 | \begin{proof} 706 | By induction on $|E(G)|$. 707 | First suppose $|V(G)|=2$. 708 | If there are an even number of vertices, choose half the orientations in either direction and take the same element of $F$ on all edges. 709 | If $|E(G)|$ is odd, then there are at least $3k-1$ edges, so take $k$ and use the odd-sum identity and use the even trick as before. 710 | 711 | Select a vertex $v$. 712 | Suppose $v$ has a neighbour $w$ such that there are multiple edges. 713 | Then repeatedly lift pairs of edges until there is only one pair of vertices left, which will have an even number of edges between them. 714 | Then color $G\setminus\{v\}$ inductively, and add back the pairs (if $\deg(v)$ is even). 715 | For the induction, use Mader's trick (1) if there is a vertex with even degree. 716 | 717 | If there is a vertex with odd degree, apply Mader's trick to two neighbours. 718 | But then the induction only fails if there is a vertex with degree precisely $3k$. 719 | Fix an orientation of $G$ such that all vertices have out-degree 0, modulo $k$. 720 | Then there are three cases: the out degree is $0$, $k$, $2k$, or $3k$. 721 | 722 | We can reduce this to the case where every vertex has in-degree $k$ or out-degree $k$. 723 | But then $G$ is bipartite (sorting by in or out degree). 724 | 725 | Let $G$ be bipartite and $k-$regular: then $G=M_1\cup M_2\cup\cdots\cup M_k$ where each $M_i$ is a perfect matching (by Hall's theorem repeatedly). 726 | But then write $\{1,\ldots,k\}=\{a_1,\ldots,a_{2q},-b_1,\ldots,-b_{2q+1}\}$ and apply that flow to each edge in $M_k$, and we are done. 727 | \end{proof} 728 | \begin{theorem}[Seymour] 729 | If $G$ is $2-$edge-connected, then $G$ has a $(\Gamma\setminus\{0\})$-flow for $|\Gamma|\geq 6$. 730 | \end{theorem} 731 | If $|\Gamma|\geq 3$, then take $a+a=2a$ or if $a+a=0$ or all $a$, then $a+b=c$. 732 | If $G$ is $8-$edge-connected, then $G$ has a $\Gamma-$flow using only $a,2a$. 733 | 734 | To see the $(2+\epsilon)$ conjecture, take $\Gamma=\R$ and $F=\{1,1+1/k\}$. 735 | Then 736 | \begin{equation*} 737 | \underbrace{1+1+\cdots+1}_{k+1} = \underbrace{(1+1/k)+\cdots+(1+1/k)}_k 738 | \end{equation*} 739 | If $G$ is $6k-$edge-connected, then $G$ has a $(1,1+1/k)-$flow, and the conjecture follows with $f(\epsilon)=6/\epsilon$. 740 | \subsection{Almost balanced orientation} 741 | Take a $1,1+1/k$ flow, and ignore the flows values (keep only the orientation). 742 | Take an arbitrary cut $A,B$, and let $E(A,B)$ denote the number of edges from side $A$ to side $B$. 743 | Then 744 | \begin{equation*} 745 | |E(A,B)|\leq|A\to B\text{ flow}|=|B\to A\text{ flow}|\leq\left(1+\frac{1}{k}\right)|E(B,A)| 746 | \end{equation*} 747 | and likewise in reverse; thus, it gives an almost-balanced orientation. 748 | 749 | Identify $\R^2=\C$, and consider the group $R_3=\{z:z^3=1\}$. 750 | \begin{theorem} 751 | $G$ has a $\{1,2\}-$flow if and only if $G$ has an $R_3-$flow. 752 | \end{theorem} 753 | \begin{proof} 754 | Assume that $G$ has a $\{1,2\}-$flow; we prove the claim by induction on the number of edges. 755 | Fix a vertex $v$. 756 | If $v$ has an incoming flow of $\alpha$ and an outgoing flow of $\alpha$, then we lift the two edges and use induction. 757 | The only other case is that all incoming edges have flow $2$ and all outgoing edges have flow $1$. 758 | Then there are twice as many outgoing edges as incoming edges, so we may separate the vertex into multiple vertices such that each incoming edge is 2 and the pair of outgoing edges is 1. 759 | Then the edges of flow 2 form a perfect matching, and the edges of flow 1 form a 2-factor (which is a disjoint collection of cycles), where the edges in the cycle alternate in direction. 760 | Then we give flow $1\in R_3$ to the edges with flow 2, and alternate labels $e^{\pm\frac{2\pi i}{3}}$ on the cycle. 761 | Re-identifying vertices preserves the incoming and outgoing flow, so we are done. 762 | 763 | Conversely, in the case when an incoming and outgoing flow are the same, we use the same argument as above. 764 | Arguing by sign, this forces such a vertex to either have all incoming or all outgoing edges. 765 | Then each of $1$, $e^{2\pi i/3}$, and $e^{2\pi i/3}$ must occur the same number of times. 766 | But then we can separate the graph into a cubic bipartite graph, which therefore has a $1,2-$flow. 767 | \end{proof} 768 | \begin{theorem} 769 | If $G$ is a cubic graph, then the following are equivlent: 770 | \begin{enumerate}[nl,r] 771 | \item $G$ has a $\{1,2\}-$flow 772 | \item $G$ has a $R_3-$flow 773 | \item $G$ has an $S^1-$flow 774 | \item $G$ is bipartite 775 | \end{enumerate} 776 | \end{theorem} 777 | Assume $G$ is cubic and has an $S^1-$flow. 778 | Re-orienting edges and changing sign (which preserves a $S^1-$element), we can guarantee that the flow only uses the 3 roots of unity. 779 | 780 | However, the implication (iii) implies (ii) does not work in general. 781 | If $G$ is planar, then $G$ has an $R_3-$flow if and only if $G^*$ has chromatic number at most 3 (this is Grotsch's theorem). 782 | Let's show that $G$ has an $S^1-$flow if and only if a homomorphic image of $G^*\subseteq U$ where $U$ is a unit distance graph. 783 | 784 | Assume $G$ has an $S^1-$flow, and let $e$ be an edge from $x$ to $y$ with flow $g\in S^1$. 785 | Then on the dual graph, we rotate the orientation counterclockwise and keep the same flow. 786 | Fix some vertex $x^*$ and assert $x^*$ is at $(0,0)$. 787 | Now take $v^*$, and a flow from $x^*$ to $v^*$, and place $v^*$ at $g_1+g_2-g_3+g_4$. 788 | This is well-defined since the cycles in the dual graph are balanced. 789 | The reverse construction certainly works as well. 790 | 791 | Now consider some dual graph $G^*$. 792 | If $G^*\subseteq U$, then $G$ has an $S^1-$flow. 793 | However if $\chi(G^*)\geq 4$, then $G$ has no $R_3-$flow. 794 | 795 | ... TODO: draw this dual graph explicitly (from one of the first classes), give the $S^1$ flow. 796 | Also: connection to list colouring. 797 | 798 | Recall that $R_k=\{z\in\C:z^k=1\}$. 799 | \begin{itemize}[nl] 800 | \item If there exists an orintation of $G$ balanced modulo 3, then $G$ has an $R_3-$flow. 801 | \item$G$ has an orientation balanced modulo 3 if and only if $G$ has a $\{1,2\}-$flow in $\Z$ or $\Z_3$. 802 | \item Conjecture of Kamal Jain: $f(S^1,\R^2)=4$, $f(S^2,\R^3)=2$. 803 | \item Tutte 3-flow implies conjecture of Jain 804 | \end{itemize} 805 | If there exists a balanced orientation modulo $k$, then $G$ has an $R_k-$flow. 806 | \begin{proof} 807 | Induction. 808 | The base case is straightforward (two vetices with a number of edges). 809 | Thus take a vertex $v$ with some incoming edge and outgoing edge. 810 | Then one can lift the edge, and use induction ... 811 | \end{proof} 812 | \begin{proposition} 813 | If $G$ has an $R_5-$flow, then there exists an orientation balanced modulo 5. 814 | \end{proposition} 815 | \begin{proof} 816 | Assume $G$ has an $R_5-$flow. 817 | As before, assume we have a vertex $v$ with an incoming and outgoing edge with the same flow; then we lift those edges and replace the flow and use induction. 818 | Thus we assume that all vertices $v$ have all outgoing edges with flow distinct from all incoming flow... 819 | \end{proof} 820 | 821 | Let $T$ denote the set of all vectors in $\R^3$ with one 0 and two $\pm 1$. 822 | \begin{theorem} 823 | $G$ has a $T-$flow if and only if $G=H_1\cup H_2\cup H_3$ such that every edge is covered twice and every vertex of $H_i$ has even degree. 824 | Furthermore, if $G$ is cubic, then the following are equivalent: 825 | \begin{enumerate}[nl,r] 826 | \item $G$ has a $T-$flow 827 | \item $G=M_1\cup M_2\cup M_3$ where each $M_i$ is a perfect matching. 828 | \item $G$ is 3-edge-colourable 829 | \item $G$ is class 1 (Vizing theorem, +1 case) 830 | \end{enumerate} 831 | \end{theorem} 832 | \begin{proof} 833 | \begin{center} 834 | \begin{tikzpicture}[d/.style={decoration={markings,mark=at position 0.5 with {\arrow{>}}},postaction={decorate}},scale=2] 835 | \node[vtx] (a) at (0,0) {}; 836 | \node[vtx] (b) at (1,0.5) {}; 837 | \node[vtx] (c) at (2,0) {}; 838 | \node[vtx] (d) at (1,-0.5) {}; 839 | \path[] 840 | (a) edge[d] (b) 841 | (b) edge[d] (c) 842 | (c) edge[d] (d) 843 | (d) edge[d] (a) 844 | (b) edge[d] (d); 845 | \end{tikzpicture} 846 | \end{center} 847 | Given $H_1\cup H_2\cup H_3$, to each edge, assign $(a_1,a_2,a_3)$ where 848 | \begin{equation*} 849 | a_i = 850 | \begin{cases} 851 | 1 &: e\in H_i\text{ with the same orientation}\\ 852 | -1 &:e\in H_i\text{ with the opposite orientation}\\ 853 | 0 &: e\notin H_i 854 | \end{cases} 855 | \end{equation*} 856 | clearly this has exactly two $\pm 1$ and that the edge sums work. 857 | 858 | Conversely, let 859 | \begin{align*} 860 | H_1 &= \{e:\text{first coordinate is $\pm 1$}\}\\ 861 | H_1 &= \{e:\text{second coordinate is $\pm 1$}\}\\ 862 | H_1 &= \{e:\text{third coordinate is $\pm 1$}\} 863 | \end{align*} 864 | 865 | Now suppose $G$ is cubic. 866 | If $G$ is a union of perfect matchings, take $M_1\cup M_2$, $M_1\cup M_3$, $M_2\cup M_3$. 867 | Conversely, take $M_i=E(G)\setminus E(H_i)$. 868 | \end{proof} 869 | Note that this also shows that if we have a graph covering, then we have an $S^2-$flow (scale $T-$flow by $\sqrt{2}$). 870 | \begin{theorem} 871 | Suppose $G$ is $(3k-1)-$edge-connected. 872 | Then $G$ is covered by $k$ even graphs such that every edge is covered precisely $k-1$ times. 873 | \end{theorem} 874 | \begin{proof} 875 | Let $\Z_2^k=\Z_2\times\cdots\times \Z_2$, and let $F$ be the set of all vectors with precisely one 0. 876 | Suppose $G$ has an $F-$flow; then take $H_i$ composed of the edges in $E(G)$ where the $i-$th coordinate is 1. 877 | \end{proof} 878 | \begin{theorem} 879 | Every 2-edge-connected planar graph $G$ has a $T-$flow. 880 | \end{theorem} 881 | \begin{proof} 882 | Consider $G$, so that $G^*$ is 4-colorable. 883 | Assign every color to the vertex of the tetrahedron, and place the vertices on these corners. 884 | Then given some edge $(x^*,y^*)\in E(G^*)$, give the edge the flow equal to the vector corresponding to the edge of the tetrahedron. 885 | \end{proof} 886 | 887 | Recall: 888 | \begin{conjecture} 889 | If $G$ is 4-edge-connected, it contains an $R_3$-flow. 890 | \end{conjecture} 891 | This conjecture is equivalent to Tutte's 3-flow conjecture. 892 | \begin{conjecture} 893 | If $G$ is 8-edge-connected (or maybe 9-edge-connected), it contains an $R_5$-flow. 894 | \end{conjecture} 895 | This conjecture implies Tutte's 5-flow conjecture. 896 | Is there some kind of converse? 897 | 898 | To prove the 5-flow conjecture, it is sufficient to consider cubic graphs which are 3-connected. 899 | 900 | \begin{proposition} 901 | If $G$ has a $(\Gamma\setminus\{0\})-$flow (NZ $\Gamma-$flow) and if $|\Gamma'|\geq|\Gamma|$, then $G$ has a NZ $\Gamma'-$flow. 902 | \end{proposition} 903 | Let $f(G,\Gamma)$ denote the number of NZ $\Gamma-$flows (allowing $G$ to have loops and multiple edges). 904 | Fix a non-loop edge $e$, and consider the graphs $G/e$ (edge contraction) and $G-e$. 905 | We can then generate a bijection ... TODO: type 906 | But then $f(G/e,\Gamma)=f(G,\Gamma)+f(G-e,\Gamma)$. 907 | Futhermore, for the base case, let $G$ be a graph with $1$ vertex and $q$ loops. 908 | But then $\Gamma(G,\Gamma)=(|\Gamma|-1)^q$. 909 | 910 | Now if $G$ has a $\Gamma-$flow, then $G$ has a NZ $\Z_n-$flow, so $G$ has a $\{1,2,\ldots,n-1\}-$flow, so $G$ has a NZ $Z_{n'}$-flow for any $n'\geq n$. 911 | 912 | What happens if $\Gamma'$ is infinite? 913 | If it contains an infinite copy of $\Z$, we are certainly done; otherwise, $G'$ finite groups of arbitrary large size, and we stop when we have a subgroup of size at least $|\Gamma|$. 914 | 915 | \begin{proposition} 916 | If $G$ is 4-edge-connected, then $G$ contains two edge-disjoint spanning trees. 917 | \end{proposition} 918 | If $G$ is 3-edge-connected, this is not true (except for small base cases). 919 | For example, if $G$ must have at least $2n-2$ edges, while if $G$ is cubic, then $G$ has $3n/2$ edges. 920 | 921 | \begin{proof} 922 | To do this, we use Mader's lifting theorem. 923 | \end{proof} 924 | \begin{corollary} 925 | More generally, if $G$ is $2k-$edge-connected, then $G$ contains $k$ edge-disjoint spanning trees. 926 | \end{corollary} 927 | \begin{proof} 928 | We prove this by induction on $|E(G)|$. 929 | Delete $e$ if possible such that $G-e$ is $2k-$edge-connected, and use induction. 930 | Thus assume no such edge exists. 931 | Using the proposition below, there exists some vertex $v$ with degree $2k$. 932 | Then apply Mader's lifting theorem to remove the vertex, and get $k$ loops which is $2k-$edge-connected. 933 | By induction, get $k$ edge-disjoint spanning trees. 934 | If each edge from the lift has a distinct tree, then we are done. 935 | Otherwise, suppose some pair of edges in the lift has the same tree. 936 | \end{proof} 937 | \begin{proposition} 938 | Consider the case when $G$ is minimally $k-$edge-connected. 939 | Then $\delta(G)=k$. 940 | \end{proposition} 941 | \begin{proof} 942 | For any edge $e$, there is a cut $(A,B)$ containing $e$ of size $k$. 943 | In particular, choose $(A,B)$ such that one size of the cut $B$ has minimal size. 944 | If $|B|=1$, then the result follows. 945 | Otherwise, $|B|>1$ so $B$ must have some internal edge $e'$ 946 | Now take another cut containing $e'$ of minimal size. 947 | ... see paper 948 | \end{proof} 949 | \begin{proposition} 950 | Let $A\subseteq V(T)$ such that $|A|$ is even. 951 | Then there exists some pairing of $A$ and edge disjoint paths betwen those elements. 952 | \end{proposition} 953 | \begin{proof} 954 | By induction. 955 | If there is some leaf $v$ not in $A$, remove $v$ and apply induction. 956 | Otherwise, every leaf is in $A$, so select an arbitrary one, and look at its neighbour in $T$. 957 | If its neighbour is not in $A$, remove $v$, and label its neighbour and apply induction. 958 | If its neighbour is in $A$, connect the two and apply induction to the rest of $G$. 959 | \end{proof} 960 | \begin{corollary} 961 | Let $G$ be 4 edge connected, so $G$ contains $T_1,T_2$ edge-disjoint spanning trees. 962 | Then $G-E(T_1)$ is connected, and for each odd degree vertex, add a path in $T_1$ joining odd degree vertex. 963 | Then this graph is even and connected (hence Eulerian). 964 | \end{corollary} 965 | \begin{corollary} 966 | If $G$ is 4-edge-connected, then $G=H_1\cup H_2$ where each $H_i$ is even. 967 | Then $G$ has a nowhere zero 4-flow. 968 | \end{corollary} 969 | 970 | We say $H\subseteq G$ is an $r-$factor in $G$ if $V(H)=V(G)$ and $H$ is $R-$regular. 971 | Suppose $r=p\cdot q$ and $G$ is $r-$regular. 972 | In this case, is $G$ a union of a $p$ $q-$factors? 973 | 974 | \begin{proposition} 975 | Next time, we will show that $G$ is $6-$edge-connected and 9-regular implies that $G$ is a union of 3 3-factors, and similarly for 4-edge-connected (if the 3-flow conjecture is true). 976 | \end{proposition} 977 | \begin{proposition} 978 | Let $G$ be a planar $2-$connected graph. 979 | Then the following are equivalent: 980 | \begin{enumerate}[nl,r] 981 | \item If $G$ is cubic, then $G=M_1\cup M_2\cup M_3$ where $M_i$ are 1-factors. 982 | \item $G$ is 4-colourable. 983 | \item If $G$ is 9-regular, then $G=F_1\cup F_2\cup F_3$ where $F_i$ are 3-regular. 984 | \end{enumerate} 985 | \end{proposition} 986 | \begin{proof} 987 | \imp{i}{ii} 988 | Assume the 4 color theorem. 989 | Color the faces with colours from $K_4$ (Klein 4-group), then for any edge in $G$, colour it equal to the sum of adjacent elements on the faces. 990 | But then this is a valid 3-coloring, since the sum of distinct elements of $K_4$ is 0 precisely when the elements are the same, and no two edges of the same face can be coloured the same. 991 | 992 | \imp{ii}{i} 993 | Conversely, suppose $G=M_1\cup M_2\cup M_3$. 994 | Consider $M_1\cup M_2$, which is a 2-factor and hence a collection of disjoint cycles. 995 | This $G$ gets a face coloring using colors 0 and 1, with $C_1(p)$. 996 | Repeat with $M_1\cup M_3$ and get $C_2(p)$. 997 | Then set $C(p)=(C_1(p),C_2(p))$. 998 | 999 | \imp{iii}{ii} 1000 | Let $G$ be planar cubic 2-edge-connected. 1001 | Take each vertex $v$ with edges $e_1,e_2,e_3$ and expand the vertex into three vertices each with 4 edges connecting them, and apply (iii) to this graph. 1002 | Then precisely one of $e_1,e_2,e_3$ is contained in some distinct 3-factor $F_i$. 1003 | Color that edge $i$. 1004 | \end{proof} 1005 | \begin{proposition} 1006 | Let $G$ be planar and 2-edge-connected. 1007 | Then the edges of $G$ can be colored with $\{1,2,3\}$ and $c_1(v),c_2(v),c_3(v)$ (the color count for each vertex $v$) are all equal or two are equal and the third differs by $\pm 2$. 1008 | \end{proposition} 1009 | \begin{proof} 1010 | Assume not. 1011 | Consider the following conditions: 1012 | \begin{enumerate}[nl,r] 1013 | \item The number of blocks (piece with no cut vertex) is minimum 1014 | \item The number of vertex degrees not equivalent to 0 mod 3 is minimal 1015 | \item The sum of the vertex degrees greater than 3 is minimal 1016 | \item Number of edges is minimal 1017 | \end{enumerate} 1018 | 1019 | Let $G_0$ be 2-connected and minimal with respect to the above conitions. 1020 | Let's determine some properties of $G_0$. 1021 | \begin{enumerate}[nl] 1022 | \item If there are blocks, separate and apply induction, permuting the colors in one of the components of necessary. 1023 | Thus the number of blocks is 1 1024 | \item Let $v$ be a vertex with degree not 0 modulo 3. 1025 | If $v$ has degree 7, separate the vertex into a pair $v_1,v_2$ (degrees 2 and 5) and with an additional edge between $v_1,v_2$. 1026 | Then the number of blocks is unchanged, and the number of not 0 modulo 3 degrees goes down, so we may apply induction. 1027 | If $v$ has degree 5, separate the vertex into a pair $v_1,v_2$ (degrees 1 and 4), with 2 additional edges, and repeat the same process. 1028 | Every possible degree falls into one of these degrees. 1029 | Thus, every vertex has degree equivalent to 0 modulo 3. 1030 | \item A least one vertex has degree at least 6. 1031 | \item $G$ is 3-edge connected; if not, get a bridge with 2 edges, do the standard swap. 1032 | \item For all $x$ of degree at least 3, there exists $y$ such that $\{x,y\}$ is a 2-vertex-cut. 1033 | Suppose some vertex $x$ has degree at least $n\geq 6$. 1034 | Then separate into a vertex $x'$ with degree 3 and $x''$ with degree $n-3$. 1035 | Suppose the number of blocks is now 2. 1036 | Then there exists some vertex $y$ such that every $x'$ to $x''$ path contains $y$. 1037 | \item Every vertex has at least 3 neighbours. 1038 | \end{enumerate} 1039 | \end{proof} 1040 | \begin{proposition} 1041 | We have the following increasing sequence of dependencies: 1042 | \begin{enumerate}[nl] 1043 | \item If $G$ is 4-edge-connected. 1044 | \item $G$ contains two edge-disjoint spanning trees. 1045 | \item $G\supseteq H_1\cup H_2$ where $H_i$ are eulerian 1046 | \item $G=H_1\cup H_2\cup H_3$ such $H_i$ is even and every edge is covered twice. 1047 | \item There is a cycle double cover (collection of cycles) 1048 | \item $L(G)$ is Hamiltonian 1049 | \item $G$ has an $S^2-$flow. 1050 | \end{enumerate} 1051 | \end{proposition} 1052 | \begin{proof} 1053 | We show 3 implies 4. 1054 | Put $E(H_3)$ as the symmetric difference of $E(H_1)$ and $E(H_2)$. 1055 | \end{proof} 1056 | If $G$ is 40-regular, can $G$ be factorized into 5 $8-$factors? 1057 | Yes: if $G$ is $2k-$regular, can decompose into $k$ $2-$factors (take an Euler walk and split the vertices into 2, with all the out weight or all the in-weight). 1058 | \begin{theorem} 1059 | Let $r=k\cdot q$ where $q$ is odd. 1060 | \begin{enumerate}[nl,r] 1061 | \item Suppose $k$ is odd, $G$ is $r-$regular, and $(3k-3)-$edge-connected. 1062 | Then we can factor $G$ into $k$ $q-$factors. 1063 | \item If $k$ is even, the same holds with $(2k^2+k)-$edge-connected and $|V(G)|$ is even. 1064 | \end{enumerate} 1065 | \end{theorem} 1066 | \begin{proof} 1067 | Recall if $k$ is odd and $G$ is $(3k-3)-$connected, then $G$ has an orientation with prescribed out-degree modulo $k$ (given degree constraints, which are satisfied when $|V(G)|$ is even). 1068 | Moreover, if $G$ has odd edge connectivity at least $3k-2$, then $G$ has an orientation balanced modulo $k$. 1069 | 1070 | Fix a vertex $v$ with in-degree $\alpha$ and out-degree $\beta$. 1071 | Then $\alpha\cong\beta\pmod{k}$ and $\alpha+\beta\cong 0\pmod{k}$, so $\alpha\cong 0\pmod{k}$ and $\beta\cong 0\pmod{k}$. 1072 | Thus we can split each vertex at $\alpha$ and $\beta$ ... 1073 | \end{proof} 1074 | If $G$ is a $9-$reglar graph with odd edge connectivity at least $7$, then $G=F_1\cup F_2\cup F_3$ where the $F_i$ are $3-$factors. 1075 | If Tutte's 3-flow holds, we may take $5$ instead. 1076 | \begin{proposition} 1077 | Let $n\equiv 3\pmod{6}$. 1078 | Then for any $s_1,\ldots,s_k$ with $s_i\geq 2$ and $\sum_{i=1}^k s_i=n$, $G$ can be factorized into $k$ factors $M_1,\ldots,M_k$ where each $M_i$ is $s_i-$regular. 1079 | \end{proposition} 1080 | \begin{proof} 1081 | Let $o_1,\ldots,o_\ell$ and $e_1,\ldots,e_m$ enumerate $s_i$ with $o_i$ odd and $e_i$ even. 1082 | Get $G$ as a union of $r/3$ $3-$factors, and identify ... 1083 | \end{proof} 1084 | Let $G$ have $\Delta(G)\leq r$, with $r=r_1+\cdots+r_m$ and $r_i\geq 2$ satisfying the same conditions as the previous proposition. 1085 | Then there is a decomposition $G=G_1\cup\cdots\cup G_m$ where $\Delta(G_i)\leq r_i$. 1086 | 1087 | \begin{theorem} 1088 | If $G$ is 2-edge-connected, then $G$ has a flow with values $\{1,2,\ldots,7\}$. 1089 | \end{theorem} 1090 | \begin{proof} 1091 | Since $G$ is 2-edge-connected, let's first see that $G=H_1\cup H_2\cup H_3$ where each $H_i$ is even. 1092 | By induction. 1093 | \begin{enumerate}[nl,r] 1094 | \item If a vertex has degree $2$, flatten the vertex and apply induction. 1095 | \item If $G$ has a vertex with degree at least 4, lift some pair of vertices and apply induction. 1096 | \item If $G$ is cubic and $2-$connected, fold the bridge (standard technique) and merge the pieces by induction 1097 | \item Finally, if $G$ is cubic and 3-connected, double every edge to get a 6-edge-connected, and thus contains three 3 edge disjoint spanning trees $T_1,T_2,T_3$. 1098 | Return the trees to $G$, and set $H_1=G-E(T_1)$, and for each edge $e$, there is some tree not containing $e$. 1099 | \end{enumerate} 1100 | 1101 | Then on each $H_i$, pick an orientation on the cycles and assign values $\pm 1$ on $H_1$, $\pm 2$ on $H_2$, and $\pm 4$ on $H_3$. 1102 | Flip the orientations on edges with negative values. 1103 | \end{proof} 1104 | 1105 | \begin{definition} 1106 | Let $G$ be an undirected graph and $D$ a directed graph. 1107 | \begin{itemize}[nl] 1108 | \item $G$ is $k-$connected if for all $x,y\in V(G)$, there are $k$ internally vertex disjoint paths between $x$ and $y$. 1109 | \item $G$ is $k-$edge-connected if there are $k$ edge disjoint paths between any $x$ and $y$. 1110 | \item $D$ is strongly $k-$connected if there are $k$ internally disjoint directed paths. 1111 | \item $D$ is strongly $k-$edge-connected (or $k$-arc-connected) if there are $k$ internally edge disjoint directed paths. 1112 | \end{itemize} 1113 | \end{definition} 1114 | \begin{theorem}[Robbin] 1115 | Let $G$ be a graph. 1116 | Then $G$ has an orientation that makes it strongly connected if and only if $G$ is 2-edge-connected. 1117 | \end{theorem} 1118 | \begin{proof} 1119 | Since $G$ is 2-edge-connected, begin with a cycle and orient the edges in the same direction. 1120 | If we are not done, there exists some edge adjacent to a vertex already in some oriented cycle. 1121 | This is called an ear decomposition. 1122 | \end{proof} 1123 | \begin{theorem} 1124 | Let $G$ be a mixed graph: some edges have orientations, while other edges do not. 1125 | Then $G$ has no bridge and no directed cut (a cut with all edges with the same orientation) if and only if $G$ can be extended to a strongly oriented $G$. 1126 | \end{theorem} 1127 | \begin{proof} 1128 | Let's see that every edge is in a consistent cycle. 1129 | Suppose $e$ is an edge from $x$ to $y$, and let $B$ be the set of vertices which can be reached with a directed path from $y$. 1130 | If $e$ is oriented from $x$ to $y$ with no consistent cycle, then it means that every other path connecting $x$ and $y$ must have the same orientation across some cut, a contradiction. 1131 | If $e$ is not directed, the only case we need to treat is when the $A$ to $B$ cut containing $e$ has all edges oriented from $A$ to $B$. 1132 | But then since $e$ is not a bridge, there exists some distinct edge $e'$ with orientation from $A$ to $B$, which is contained in a consistent cycle, and this cycle ust contained $e$. 1133 | \end{proof} 1134 | \begin{theorem}[Nash-Williams] 1135 | Let $G$ be a graph. 1136 | Then $G$ has an orientation that makes it $k-$arc-conected if and only if $G$ is $2k-$edge-connected. 1137 | \end{theorem} 1138 | \begin{proof} 1139 | Delete edges until $G$ is minimally $2k-$edge-connected. 1140 | Consider a vertex $x$ with degree precisely $2k$. 1141 | By mader's lifting theorem, we can lift all the pairs of edges so that $G$ remains $2k-$edge-connected, and possibly leaving some $x_0$ with multiple edges. 1142 | Apply induction on the lifted graph to get an orientation on $G'$. 1143 | Then maintain the orientations on the edges to $v$. 1144 | The only case that to verify that $x$ and $y$ are $k-$arc-connected for any $y$ that is not a neigbour of $x$. 1145 | 1146 | Let $A$ be a component containins $x$ and $B$ a component containing $y$, and suppose for contradiction $A$ and $B$ define a cut with less than $k$ directed edges in some direction. 1147 | If there is another vertex $z$ in $A$, we have a contradiction by induction on the lifted $G$. 1148 | But then if $x\in A$ is the only vertex, there are the same number of edges going out and in. 1149 | \end{proof} 1150 | \begin{conjecture} 1151 | If $G$ is $f(k)-$connected, then there exists an orientation on $G$ which is strongly $k$-connected. 1152 | \end{conjecture} 1153 | This is open for $k=3$. 1154 | 1155 | If $G$ is cubic $3-$connected, then there is no orientation which makes $G$ strongly $2-$connected. 1156 | \begin{conjecture}[Frank] 1157 | There exists an orientation on $G$ so it is strongly $k-$connected if and only if for all $S\subseteq V(G)$ with $|S|1$, for if $k=1$, this statement is immediately true. 1262 | Now let $B$ be a component with finite boundary of size $k'$. 1263 | Choose $k'$ minimal, where $k'\leq j$. 1264 | We prove this by induction on $k-k'$. 1265 | 1266 | First suppose $k-k'=0$. 1267 | Pick a $1-\infty$ path $P_0$, and separate $G$ into components. 1268 | Let $A_2$ be the boundary of the infinite connected component $G_2$ containing $P_0$, so the boundary has size $k'$. 1269 | Since $k'=k$, we may apply Menger's theorem (infinite case) to get $P_{1,1},\ldots,P_{1,k}$ edge disjoint $A_0-A_2$ paths. 1270 | Then remove $A_2$ from $G_2$, and let $A_3$ be the boundary of a connected component of $G_2$ containing $P_0$. 1271 | As before, there must be at least $k'$ edges, so we may choose $P_{2,1},P_{2,2},\ldots,P_{2,k}$. 1272 | Always do this with minimal total length. 1273 | Continue this to get an infinite family of paths rowwise edge disjoint. 1274 | Then one can construct $k$ disjoint 1-way infinite paths. 1275 | We must ensure that all paths are in the same end. 1276 | 1277 | Let $H$ be a component with a distinct end, and suppose (for example) $P_3$ has edges in $H$. 1278 | Consider $P_{i,1},\ldots,P_{i,k}$, which uses the boundary of $H$ in only finitely many ways. 1279 | Thus $P_{i,1},\ldots,P_{i,k}$ use $H$ in only finitely many ways, for if not, then they would extend arbitrarily far into $H$, contradicting minimality. 1280 | 1281 | Now suppose $k'k$, then there exists $G\supseteq H$ such that $\chi(H)>k$ and $\lambda(H)\geq k$ ($\lambda$ is edge connectivity). 1540 | \end{proposition} 1541 | \begin{proof} 1542 | Fix some minimal cut $E\subseteq G$ of size $k-1$. 1543 | We must show that one side of the cut has chromatic number greater than $k$. 1544 | Assume not, and $k-$color each side. 1545 | Create a bipartite graph $K_{k,k}$ labelled with colours, and an edge if there is an edge joining such colours. 1546 | Then if there is conflict, there are $p(k-p+1)\geq k$ edges in the complement $K_{k,k}$, so we may apply Hall's theorem. 1547 | \end{proof} 1548 | Let $G_1=G$, and $G_{i+1}=G_i-D_i$, and $G_i=\bigcap_{j\aleph_0$ and $\lambda\geq\lambda_0$. 1566 | Suppose for contradiction we can colour one of these components with colours in $\N$. 1567 | Let's show that by cut addition we can maintain the countable number. 1568 | Let $V(G)=\{v_i:i\in I\}$ where $I$ is well-ordered. 1569 | Enumerate the parts $H_i$, and focus on some $H_i$. 1570 | Fix some vertex $v$ in $B$, where $v$ has colour $c_i(v)$ and distinguished vertex $k_i$. 1571 | Then whenever we have parts $H_i,H_j$, we keep the colours in $i$ and permute the colours in $j$ if $k_i\leq k_j$. 1572 | Then define $c(v)=\lim c_i(v)$, where the limit exists since $k_i\leq k_j$ whenever $i>j$. 1573 | \end{proof} 1574 | We say a graph has colouring number $\Col(G)=\omega$ where $\omega$ is the smallest cardinality such that there exists a well-ordering on $V(G)$ so that all the edges from $v$ to some smaller vertex $w$ is strictly less than $\omega$. 1575 | Suppose we combine components $A,B$ with $k(A)k|V(H)|$. 1581 | \end{theorem} 1582 | This also holds for $k$ finite and $G$ infinite, where we require $H\subseteq G$ is finite. 1583 | \begin{theorem} 1584 | For any orientation on $G$ there exists a vertex of out-degree $\aleph_0$ implie there is some $G\supseteq H$ uncountable and of infinite edge-connectivity. 1585 | \end{theorem} 1586 | \begin{proof} 1587 | Apply finite cut deletion to get a set of disconnected components. 1588 | Certainly each has infinite edge-connectivity: let's show that some component is uncountable. 1589 | Put orientations on each component, and then when we combine $A$ with $B$ with $k(A)