\documentclass{amsart} \usepackage{fullpage} %\usepackage{fullpage,amscd,amsfonts, amssymb,amsthm,mathrsfs,graphicx,stmaryrd,appendix} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{proposition}{Proposition} \newtheorem{corollary}{Corollary} \newtheorem{problem}{Problem} \theoremstyle{theorem}\newtheorem*{question}{Question} \theoremstyle{remark}\newtheorem*{definition}{Definition} \newtheorem{example}{Example} \theoremstyle{remark} \newtheorem*{solution}{Solution} \theoremstyle{remark} \newtheorem*{remark}{Remark} \newcommand{\ncr}[2]{\displaystyle{\ensuremath{\left(\begin{matrix} #1 \\ #2 \end{matrix} \right)}}} \newcommand{\zsub}[1]{\ensuremath{\mathbb{Z} / #1 \mathbb{Z}}} \newcommand{\gen}[1]{\ensuremath{\left\langle #1 \right\rangle}} \newcommand{\ceil}[1]{\ensuremath{\left \lceil #1 \right \rceil}} \newcommand{\floor}[1]{\ensuremath{\left \lfloor #1 \right \rfloor}} \newcommand{\cd}[1]{\ensuremath{\begin{CD} #1 \end{CD}}} \newcommand{\A}[0]{\ensuremath{\mathbb{A}}} \newcommand{\B}[0]{\ensuremath{\mathbb{B}}} \newcommand{\C}[0]{\ensuremath{\mathbb{C}}} \newcommand{\D}[0]{\ensuremath{\mathbb{D}}} \newcommand{\E}[0]{\ensuremath{\mathbb{E}}} \newcommand{\F}[0]{\ensuremath{\mathbb{F}}} \newcommand{\G}[0]{\ensuremath{\mathbb{G}}} \newcommand{\I}[0]{\ensuremath{\mathbb{I}}} \newcommand{\J}[0]{\ensuremath{\mathbb{J}}} \newcommand{\K}[0]{\ensuremath{\mathbb{K}}} \newcommand{\M}[0]{\ensuremath{\mathbb{M}}} \newcommand{\N}[0]{\ensuremath{\mathbb{N}}} \newcommand{\Q}[0]{\ensuremath{\mathbb{Q}}} \newcommand{\R}[0]{\ensuremath{\mathbb{R}}} \newcommand{\T}[0]{\ensuremath{\mathbb{T}}} \newcommand{\V}[0]{\ensuremath{\mathbb{V}}} \newcommand{\W}[0]{\ensuremath{\mathbb{W}}} \newcommand{\X}[0]{\ensuremath{\mathbb{X}}} \newcommand{\Y}[0]{\ensuremath{\mathbb{Y}}} \newcommand{\Z}[0]{\ensuremath{\mathbb{Z}}} \newcommand{\eps}[0]{\ensuremath{\varepsilon}} \newcommand{\inject}[0]{\ensuremath{\hookrightarrow}} \newcommand{\Aut}[0]{\operatorname{Aut}} \newcommand{\Cov}[0]{\operatorname{Cov}} \newcommand{\Deck}[0]{\operatorname{Deck}} \newcommand{\disc}[0]{\operatorname{disc}} \newcommand{\End}[0]{\operatorname{End}} \newcommand{\Frob}[0]{\operatorname{Frob}} \newcommand{\Gal}[0]{\operatorname{Gal}} \newcommand{\Harm}[0]{\operatorname{Harm}} \newcommand{\Hom}[0]{\operatorname{Hom}} \newcommand{\Id}[0]{\operatorname{Id}} \newcommand{\im}[0]{\operatorname{Im}} \newcommand{\lcm}[0]{\operatorname{lcm}} \newcommand{\Mat}[0]{\operatorname{Mat}} \newcommand{\obj}[0]{\operatorname{obj}} \newcommand{\Orb}[0]{\operatorname{Orb}} \newcommand{\ord}[0]{\operatorname{ord}} \newcommand{\re}[0]{\operatorname{Re}} \newcommand{\Res}[0]{\operatorname{Res}} \newcommand{\SL}[0]{\operatorname{SL}} \newcommand{\Stab}[0]{\operatorname{Stab}} \newcommand{\Supp}[0]{\operatorname{Supp}} \newcommand{\Top}[0]{\operatorname{top}} \newcommand{\Tr}[0]{\operatorname{Tr}} \newcommand{\Var}[0]{\operatorname{Var}} \newcommand{\legendre}[2]{\displaystyle{\ensuremath{\left(\frac{#1}{#2} \right)}}} \begin{document} \title{Introduction to \LaTeX} \author{Tex User} \maketitle \begin{abstract} This is an introduction to the use of \LaTeX\ to create mathematical documents. \end{abstract} \begin{tableofcontents} \end{tableofcontents} \section{Introduction} Good afternoon. For basic mathematical expressions, try something like $x^2 + 1$, $\sqrt{3} $, or $\frac{1}{2} $. If you want to put an expression by itself, try $$ p(x) = 6x^3 + \frac{3}{5}x^2 - 17x + \sqrt{2}, $$ or $$ e^{i \pi} + 1 = 0, $$ just not $$ e^i \pi + 1 \neq 0. $$ $(\displaystyle \frac{1}{2 + \frac{1}{x}})$ looks bad. Try, $\left(\displaystyle \frac{1}{2 + \frac{1}{x}} \right)$. Other examples include $\left[\displaystyle \frac{1}{2 + \frac{1}{x}} \right]$. For curly braces, try $\{x^2 + 1\}$. Other ways to make different size parentheses include $\Bigg(\bigg(\Big(\big((x)\big)\Big)\bigg)\Bigg)$. For other Greek letters, try $\lambda$, $\Lambda $, $\chi$, $\Phi$, $\phi$, or $\rho$. Also, try $\infty$. Is this a new paragraph? No, this is. Other methods of starting a new line include\\ this, \vspace{120pt} or this, \hspace{58pt} but not this. Different fonts include \textit{italic}, \textbf{bold}, and \texttt{terminal (good for writing GAP codes)}. Another way of writing GAP codes is \begin{verbatim} gap> f:=GF(16); GF(2^4) gap> Elements(f); [ 0*Z(2), Z(2)^0, Z(2^2), Z(2^2)^2, Z(2^4), Z(2^4)^2, Z(2^4)^3, Z(2^4)^4, Z(2^4)^6, Z(2^4)^7, Z(2^4)^8, Z(2^4)^9, Z(2^4)^11, Z(2^4)^12, Z(2^4)^13, Z(2^4)^14 ] \end{verbatim} For presentation of groups, try $$ D_n \cong \langle x,y | x^2, y^n , xy= y^{-1} x \rangle $$ \section{Equation Arrays} For long equations, write \begin{eqnarray} \left( \frac{f}{g} \right)' &=& \left(f \right)' \left( \frac{1}{g} \right) + \left( f \right) \left( \frac{1}{g} \right)',\\ &=& f' \left( \frac{1}{g} \right) + f \left( \frac{-g'}{g^2} \right),\\ &=& \frac{f'g - fg'}{g^2}, \end{eqnarray} or \begin{eqnarray*} \left( \frac{f}{g} \right)' &=& \left(f \right)' \left( \frac{1}{g} \right) + \left( f \right) \left( \frac{1}{g} \right)',\\ &=& f' \left( \frac{1}{g} \right) + f \left( \frac{-g'}{g^2} \right),\\ &=& \frac{f'g - fg'}{g^2}. \end{eqnarray*} Some properties of \texttt{eqnarray} include \begin{eqnarray*} 5x^2 + 3x &=& i + \pi - 3e,\\ 4 \geq 3 \geq 2 \geq 1 ,\\ & & 1 \leq 2 \leq 3 \leq 4,\\ g &\in& G,\\ H &\lhd& G,\\ \{1,2\} &\subset& \{1,2,3\}. \end{eqnarray*} We can make maps \begin{eqnarray*} (\; \cdot \;) \colon G \times G &\to& G,\\ (g_1, g_2) &\mapsto& g_1 g_2 \end{eqnarray*} \section{Advanced Math Commands} For analysis enthusiasts, try $$ \int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^n f(y_i) \Delta(x_i), $$ where $a = x_1 < x_2 < \ldots < x_n = b$. For linear algebra enthusiasts, try $$ \begin{matrix} 1 & x_2 & 0.5 \\ y+2 & a_1 a_2 \cdots a_n & \sum_{i=1}^n i^3 \end{matrix}. $$ For matrices with parentheses, try $$ \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}. $$ For matrices with brackets, try $$ \begin{bmatrix} a & b \\ c & d \end{bmatrix}. $$ For the determinant of a matrix, we can write $$ \det (a_{ij}) = \begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{vmatrix}. $$ \begin{proposition}[Product Formula] Let $s$ be a complex number and let $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$. Then $$ \zeta(s) = \prod_p \frac{1}{1 - \frac{1}{p^s}}. $$ \end{proposition} \begin{proof} See~\cite{apostol}. \end{proof} \section{Other Commands} Other functions include $$ \log x, \sin x, \arctan y, \exp a, \max \{a,b\}, \sup S, \ker \phi. $$ Other symbols include $$ \mathbb{C}, \mathbb{Q}, \mathbb{Z}, \mathbb{R}, \mathcal{O}, \mathfrak{F} $$ After defining the correct commands, we can now write $$ \C, \Q, \Z, \R , \Aut(\mathbb{H}), \Stab_G(x), \Orb_{S_4}((1,2)(3,4)) $$ \section{Lists and Enumeration} The mentors at this program are \begin{enumerate} \item Selin Kalaycioglu \item Ken Monks \item Belin Tsinnajinnie \item Sam Xu \end{enumerate} but if you do not want to see enumeration, then try \begin{itemize} \item Selin Kalaycioglu \item Ken Monks \item Belin Tsinnajinnie \item Sam Xu \end{itemize} \newpage \begin{thebibliography}{XXXXX} \normalsize \bibitem[Apo]{apostol}Apostol, T. {\it Introduction to Analytic Number Theory}. New York: Springer Science+Business Media, Inc., 1976. \end{thebibliography} \end{document}