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\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}