diff --git a/fai/bibliography.bib b/fai/bibliography.bib new file mode 100644 index 0000000..f6a2e6d --- /dev/null +++ b/fai/bibliography.bib @@ -0,0 +1,16 @@ +@book{taylor, + author = "Taylor, Joseph L.", + title = "Complex Variables", + publisher = {AMS}, + year = {2011}, + isbn = {978-0-8218-6901-7}, + keywords = {complex,variables,analysis} +} + +@unpublished{sekhon, + author = "Sekhon, Senan", + title = "Metric and Topological Spaces", + year = {2019}, + note = "Unpublished", + keywords = {metric spaces,topological spaces} +} \ No newline at end of file diff --git a/fai/main.tex b/fai/main.tex new file mode 100644 index 0000000..b12ea47 --- /dev/null +++ b/fai/main.tex @@ -0,0 +1,428 @@ +\documentclass[10pt]{extarticle} + +\usepackage[english]{babel} +\usepackage{graphicx} +\usepackage{framed} +\usepackage[normalem]{ulem} +\usepackage{indentfirst} +\usepackage{amsmath,amsthm,amssymb,amsfonts} +\usepackage[italicdiff]{physics} +\usepackage[T1]{fontenc} +%\usepackage{pifont} %For unusual symbols +%\usepackage{mathdots} %For unusual combinations of dots +\usepackage{wrapfig} +\usepackage{lmodern,mathrsfs} +\usepackage[inline,shortlabels]{enumitem} +\setlist{topsep=2pt,itemsep=2pt,parsep=0pt,partopsep=0pt} +\usepackage[dvipsnames]{xcolor} +\usepackage[utf8]{inputenc} +\usepackage[a4paper, top=0.5in,bottom=0.2in, left=0.5in, right=0.5in, footskip=0.3in, includefoot]{geometry} +\usepackage[most]{tcolorbox} +\usepackage{tikz,tikz-3dplot,tikz-cd,tkz-tab,tkz-euclide,pgf,pgfplots} +\pgfplotsset{compat=newest} +\usepackage{multicol} +\usepackage[bottom,multiple]{footmisc} %ensures footnotes are at the bottom of the page, and separates footnotes by a comma if they are adjacent +\usepackage[backend=bibtex,style=numeric]{biblatex} +\renewcommand*{\finalnamedelim}{\addcomma\addspace} %forces authors' names to be separated by comma, instead of "and" +\addbibresource{bibliography} +\usepackage{hyperref} +\usepackage[nameinlink]{cleveref} %nameinlink ensures that the entire element is clickable in the pdf, not just the number + +\newcommand{\remind}[1]{\textcolor{red}{\textbf{#1}}} %To remind me of unfinished work to fix later +\newcommand{\hide}[1]{} %To hide large blocks of code without using % symbols + +\newcommand{\ep}{\varepsilon} +\newcommand{\vp}{\varphi} +\newcommand{\lam}{\lambda} +\newcommand{\Lam}{\Lambda} +%\newcommand{\abs}[1]{\ensuremath{\left\lvert#1\right\rvert}} % This clashes with the physics package +%\newcommand{\norm}[1]{\ensuremath{\left\lVert#1\right\rVert}} % This clashes with the physics package +\renewcommand{\ip}[1]{\ensuremath{\left\langle#1\right\rangle}} +\newcommand{\floor}[1]{\ensuremath{\left\lfloor#1\right\rfloor}} +\newcommand{\ceil}[1]{\ensuremath{\left\lceil#1\right\rceil}} +\newcommand{\A}{\mathbb{A}} +\newcommand{\B}{\mathbb{B}} +\newcommand{\C}{\mathbb{C}} +\newcommand{\D}{\mathbb{D}} +\newcommand{\E}{\mathbb{E}} +\newcommand{\F}{\mathbb{F}} +\newcommand{\K}{\mathbb{K}} +\newcommand{\N}{\mathbb{N}} 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new QED symbol +\renewcommand{\qedsymbol}{\coolqed{0.32}} %Implements the new QED symbol +\theoremstyle{mystyle}{\newtheorem{definition}{Definition}[section]} +\theoremstyle{mystyle}{\newtheorem{proposition}[definition]{Proposition}} +\theoremstyle{mystyle}{\newtheorem{theorem}[definition]{Theorem}} +\theoremstyle{mystyle}{\newtheorem{lemma}[definition]{Lemma}} +\theoremstyle{mystyle}{\newtheorem{corollary}[definition]{Corollary}} +\theoremstyle{mystyle}{\newtheorem*{remark}{Remark}} +\theoremstyle{mystyle}{\newtheorem*{remarks}{Remarks}} +\theoremstyle{mystyle}{\newtheorem*{example}{Example}} +\theoremstyle{mystyle}{\newtheorem*{examples}{Examples}} +\theoremstyle{definition}{\newtheorem*{exercise}{Exercise}} +\theoremstyle{cstyle}{\newtheorem*{cthm}{}} + +%Warning environment +\newtheoremstyle{warn}{}{}{}{}{\normalfont}{}{ }{} +\theoremstyle{warn} +\newtheorem*{warning}{\warningsign{0.2}\relax} + +%Symbol for the warning environment, designed to be easily scalable 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+\tcolorboxenvironment{examples}{boxrule=0pt,boxsep=0pt,blanker,borderline west={2pt}{0pt}{Black},left=8pt,right=8pt,sharp corners,before skip=10pt,after skip=10pt,breakable} +\tcolorboxenvironment{cthm}{boxrule=0pt,boxsep=0pt,colback={gray!10},left=8pt,right=8pt,enhanced jigsaw, borderline west={2pt}{0pt}{gray},sharp corners,before skip=10pt,after skip=10pt,breakable} + +%align and align* environments with inline size +\newenvironment{talign}{\let\displaystyle\textstyle\align}{\endalign} +\newenvironment{talign*}{\let\displaystyle\textstyle\csname align*\endcsname}{\endalign} + +\usepackage[explicit]{titlesec} +\titleformat{\section}{\fontsize{24}{30}\sffamily\bfseries}{\thesection}{20pt}{#1} +\titleformat{\subsection}{\fontsize{16}{18}\sffamily\bfseries}{\thesubsection}{12pt}{#1} +\titleformat{\subsubsection}{\fontsize{10}{12}\sffamily\large\bfseries}{\thesubsubsection}{8pt}{#1} + +\titlespacing*{\section}{0pt}{5pt}{5pt} +\titlespacing*{\subsection}{0pt}{5pt}{5pt} +\titlespacing*{\subsubsection}{0pt}{5pt}{5pt} + +%\newcommand{\sectionbreak}{\clearpage} %Start every section on a new page + +\newcommand{\Disp}{\displaystyle} +\newcommand{\qe}{\hfill\(\bigtriangledown\)} +\DeclareMathAlphabet\mathbfcal{OMS}{cmsy}{b}{n} +\setlength{\parindent}{0.2in} +\setlength{\parskip}{0pt} +\setlength{\columnseprule}{0pt} + +\title{\huge\sffamily\bfseries A Fun Template} +\author{\Large\sffamily Senan Sekhon} +\date{\sffamily October 4, 2020} + +\begin{document} + +\setlength{\abovedisplayskip}{3pt} +\setlength{\belowdisplayskip}{3pt} +\setlength{\abovedisplayshortskip}{0pt} +\setlength{\belowdisplayshortskip}{0pt} +\maketitle + +%Custom colors for different environments +\definecolor{contcol1}{HTML}{72E094} +\definecolor{contcol2}{HTML}{24E2D6} +\definecolor{convcol1}{HTML}{C0392B} +\definecolor{convcol2}{HTML}{8E44AD} + +\begin{tcolorbox}[title=Contents, fonttitle=\huge\sffamily\bfseries\selectfont,interior style={left color=contcol1!40!white,right color=contcol2!40!white},frame style={left color=contcol1!80!white,right color=contcol2!80!white},coltitle=black,top=2mm,bottom=2mm,left=2mm,right=2mm,drop fuzzy shadow,enhanced,breakable] +\makeatletter +\@starttoc{toc} +\makeatother +\end{tcolorbox} + +\vspace*{10mm} + +\begin{tcolorbox}[title=Conventions, fonttitle=\large\sffamily\bfseries\selectfont,interior style={left color=convcol1!40!white,right color=convcol2!40!white},frame style={left color=convcol1!80!white,right color=convcol2!80!white},coltitle=black,top=2mm,bottom=2mm,left=2mm,right=2mm,drop fuzzy shadow,enhanced,breakable] +$\F$ denotes either $\R$ or $\C$.\\ +$\N$ denotes the set $\{1,2,3,...\}$ of natural numbers (excluding $0$). +\end{tcolorbox} + +\newpage + +\section{Sample Chapter} +Let's dive right in! + +\subsection{Some Definitions} +\begin{definition} +The \textbf{derivative} of a function $f:I\to\R$ at $a\in I$ is given by: +\begin{equation*} + f'(x)=\limx\frac{f(x)-f(a)}{x-a} +\end{equation*} +\end{definition} + +\begin{center} +You know those awesome commutative diagrams? + +\begin{tikzcd} +A \arrow[r,"p"] \arrow[d,red,"q"'] & B \arrow[d,"r" red] \\ +C \arrow[r,red,"s"' blue] & D +\end{tikzcd} + +The derivative has \emph{nothing} to do with them! +\end{center} + +\begin{proposition}\label{diffcont} +If $f$ is differentiable at $a$, then $f$ is continuous at $a$. +\end{proposition} +\begin{proof} +Exercise (but only because this is a template). +\end{proof} + +The converse of \Cref{diffcont} is not true in general. + +\begin{examples}\leavevmode % This is needed to start the list in the next line so it won't be misaligned +\begin{enumerate} + \item $f(x)=\abs{x}$ + \item $f(x)=\begin{cases} \sin(x) & x\ge 0 \\ 0 & x<0 \end{cases}$ +\end{enumerate} +\end{examples} + +\begin{theorem} +The following statements are true: +\begin{enumerate} + \item First statement + \item Second statement +\end{enumerate} +\end{theorem} +\begin{proof}% For some reason, the proof environment does not need \leavevmode +\begin{enumerate} + \item Trivial. + \item Trivial.\qedhere % qedhere is to place the qed symbol here instead of in the next line +\end{enumerate} +\end{proof} + +\begin{corollary} +We are both very lucky to have each other as a collaborator. +\end{corollary} +\begin{proof} +We simply note that: +\begin{equation*} + \frac{1}{1}+\frac{1}{1}\gg\frac{1}{1} \qedhere +\end{equation*} +\end{proof} +\begin{remark} +This corollary is also obvious from empirical evidence. +\end{remark} + +\begin{lemma} +$(a+b)^2=a^2+2ab+b^2$ +\end{lemma} +\begin{proof} +Expand the left side. +\end{proof} +\begin{remarks}\leavevmode +\begin{enumerate} + \item It's also kind of obvious. + \item No extra points for guessing what $(a-b)^2$ is. +\end{enumerate} +\end{remarks} + +\begin{example} +$(2+4)^2=2^2+2\cdot 2\cdot 4+4^2=36$ +\end{example} + +\begin{theorem}[Pythagoras' Theorem]\label{pythagoras} +If $c$ is the hypotenuse of a right triangle and $a$ and $b$ are the other two sides, then $a^2+b^2=c^2$. +\end{theorem} +\begin{proof} +Draw a picture and convince yourself. +\end{proof} + +\hyperref[pythagoras]{Pythagoras' theorem} helps motivate the study of metric spaces, which you can learn about in \cite{sekhon}.\\ + +A lot of nice integrals can be computed using the residue theorem, see \cite[Section 5.2]{taylor}. + + +\newpage +\appendix +\section{Bonus Material} +The \verb!talign! and \verb!talign*! environments work like the \verb!align! and \verb!align*! environments, except they render equations in inline size. For example, \verb!\begin{align*}...\end{align*}! yields: +\begin{align*} + \sumn\frac{1}{n^2}=\frac{\pi^2}{6} +\end{align*} +While \verb!\begin{talign*}...\end{talign*}! yields: +\begin{talign*} + \sumn\frac{1}{n^2}=\frac{\pi^2}{6} +\end{talign*} +As usual, the purpose of \verb!*! is to prevent numbering of the equation.\\ + +Some commands, like \verb!\sumn!, can be used with or without a starting value (the default starting value is 1). For example, \verb!$\sumn\frac{1}{n^2}$! yields $\sumn\frac{1}{n^2}$, while \verb!$\sumn[69]\frac{1}{n^2}$! yields $\sumn[69]\frac{1}{n^2}$. This can be used in inline mode as well as display mode. + +\newpage + +\section{Combinatorial Optimization} +previous chapters: classical state-space search +- find action sequence (path) from initial to goal state +- difficulty: large number of states (“state explosion”) +next chapters: combinatorial optimization +$\rightsquigarrow$ similar scenario, but: +- no actions or transitions +- don’t search for path, but for configuration (“state”) +with low cost/high quality +German: Zustandsraumexplosion, kombinatorische Optimierung, +Konfiguration +\subsection{Intro and Hill-Climbing} + +\begin{definition}\label{cop} +A \textbf{combinatorial optimization problem \emph{COP}} is given by a tuple +$\langle C, S, opt, v\rangle$ +consisting of: +\begin{itemize} + \item a finite set of (solution) \textbf{candidates} $C$ + \item a finite set of \textbf{solutions} $S\subseteq C$ + \item an \textbf{objective sense} $opt\in\{min,max\}$ + \item an \textbf{objective function} $v:S\rightarrow \R$ +\end{itemize} +\textbf{Remarks:} "problem" here means "instance". Interesting COPs usually have to many candidates to enumerate explicitly. +\end{definition} +\begin{definition}\label{opt} +Let $\Os = \langle C,S,opt,v\rangle$ be a COP. The \textbf{optimal solution quality} $v^*$ of $\Os$ is defined as $v^*= \mathsf{min/max}_{c\in S}v(c) | \ opt=\mathsf{min/max}$ (undefined if $S=\emptyset$). A solution $s$ of $\Os$ is called \textbf{optimal} if $v(s)=v^*$. +\end{definition} +\textbf{Algorithmic Problem we want to solve:} Find a \emph{solution} to a COP $\Os$ which is as close to $v^*$ as possible. \\ +\textbf{Special cases:} \emph{pure search:} all sols same quality; finding any sol is hard to begin with; formal: $v$ is a constant function and $opt$ can be chosen arbitrarily. \emph{pure optimization:} all candidates are sols; difficulty is finding sol of high quality; formal: $S = C$. \\ + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%% D1 CSP +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Constraint Satisfaction Problems} +\subsection{Intro} +\textbf{Summary:} {\color{red} Constrain Satisfaction} is the problem of finding an {\color{red}assignment} for a set of \emph{variables} from a given \emph{domain}, which satisfies a given set of \emph{constraints}. +\begin{definition} + +\end{definition} + + +\printbibliography + +\end{document} \ No newline at end of file diff --git a/fai/sunglasses_emoji.png b/fai/sunglasses_emoji.png new file mode 100644 index 0000000..be85582 Binary files /dev/null and b/fai/sunglasses_emoji.png differ diff --git a/patterns/main_patterns.tex b/patterns/main_patterns.tex index b496197..620258b 100644 --- a/patterns/main_patterns.tex +++ b/patterns/main_patterns.tex @@ -1,4 +1,4 @@ -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\ref{}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % writeLaTeX Example: A quick guide to LaTeX % % Source: Dave Richeson (divisbyzero.com), Dickinson College