\documentclass[11pt, reqno]{amsart}
%reqno means equation numbers go on the right

%Comments go like this.


\numberwithin{equation}{section} % For numbering equations.

\usepackage{graphicx, amsthm, amssymb, amsmath, color, cite}


\newcommand{\R}{\mathbb{R}}
\renewcommand{\S}{\mathbb{S}}
\newcommand{\qtq}[1]{\quad\text{#1}\quad}




%% Theorem environments:

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{problem}{Problem}[section]

%%% Adding [section] makes it list as 2.1, 2.2, 3.1, rather than Theorem 1, 2, 3, etc.


\begin{document}

\title[Template]{\LaTeX\quad  Template for students who are interested in learning LaTeX}
\author[J. Murphy]{Jason Murphy}
\author[E. Archenko]{Elena Archenko}

\email{jamu@uoregon.edu}
\address{Department of Mathematics, University of Oregon}

\maketitle

\begin{abstract} This is the abstract.
\end{abstract}

\tableofcontents

%%%%%%%%%%%sadfsdfasfaaf
\section{Introduction}\label{S:Introduction}

This is an introduction.

In Section~\ref{S:Introduction} we have an introduction.



%\vspace{2in} my suggestion is not to use vspace...

Since $\displaystyle\lim_{k\to\infty} x_k = \ell...$

Since $\displaystyle\int_0^1 f(x)\,dx = 0$... 

\[
\smallint_0^1 f(x)\,dx
\]

\subsection{A subsection} This is a subsection.

\subsubsection{A subsubsection} This is a subsubsection.


``This is a bad example''

To leave spaces, you can use smallskip, medskip, bigskip.

\bigskip

\bigskip

\bigskip



That was a bigskip.  You can make a newpage by typing $\backslash$newpage:

\newpage

%%%%%%%%%%% that's right 
% I have a bunch of comments that
% I want to make and so I am writing them here

\section{Math symbols}

Let A be a matrix. Let $A$ be a matrix. %$If you write a long sentence in math mode it looks terrible.$

\noindent Let $x\in X$, and take $y\in Y$. 

Since 
\[
\int_0^1 x\,dx = \tfrac12 x^2|_{x=0}^1 = \tfrac12...
\]

Put math symbols between \$ signs. Variable names like, $x,y,z$ should be in math mode.  Longer or important formulas should go between the following symbols:
 \begin{verbatim} \[ \] \end{verbatim}  

$$
\text{This also works.} This doesn't.
$$

\[ 
x +4 = y
\]

\[
\{ \quad \Biggl\{ \Biggr.
\] 
Parenthesis size: $($\quad $\big($ \quad $\Big($ \quad $\Bigg($

\[
\frac34\quad \tfrac34\quad x^{\frac{3}{4}}
\]

\[
\left(\frac{3x+4}{2x+5}\right.
\]

\[
\Biggl(x+3\Biggr)*4 = 12
\]

$\Biggl(3x+2\Biggr)$ 

$\text{Text}$

$\{...\}$

For example,
\[
\int_0^{10} x^2 \,dx = \tfrac13 x^3\bigg|_{x=0}^{10} = \tfrac{1000}{3}. 
\]

\subsection{Theorems, etc}

To state theorems, propositions, lemmas, etc., use:

\begin{verbatim}
\begin{theorem}\label{T:main} Blah blah blah. \end{theorem}
\end{verbatim} 

\begin{theorem}\label{T:main} Blah blah blah. \end{theorem}

\begin{problem} This is the first problem.
\end{problem}

To prove Theorem~\ref{T:main}, we don't need to do anything because it doesn't say anything.\label{test-label}

\begin{proof} uhh..
\end{proof}


\subsection{Equations}
To make a numbered equation, use:


\begin{verbatim}
\begin{equation}\label{E:example}
x=3.
\end{equation}
\end{verbatim}

\begin{equation}\label{the-star-one}\tag{$*$}
x=3
\end{equation}

By equation \eqref{the-star-one} 

\begin{equation}\label{E:example}
x=3.
\end{equation}

By equation \eqref{E:example} 

To remove the number, put ``nonumber'' instead of `label...'

Go back and look at page \pageref{test-label}. 
\bigskip

\subsection{Align}
If you need to have a string of identities or inequalities, use align or align${}^*$. Within the align environment, use \& to mark a certain point and then refer to it later. Use $\backslash\backslash$ to make a new line. 

If you use align, each line will have a number (unless you include $\backslash$nonumber.  You can label each line separately.  If you don't want numbers, use align${}^*$. 

Example:  To produce the following

\begin{align}
\int_0^1 x^2\,dx & = \tfrac13 x^3\bigg|_{x=0}^1 \nonumber \\
& = \tfrac13, \label{E:final}
\end{align}

\begin{align*}
x & = 3 \\ 
& = 2 + 1 \\ 
& = 1 + 1 + 1.
\end{align*}

Here is an ugly example: 
\begin{align}
x = 3 \nonumber \\
= 2 + 1 \\
= 1 + 1 + 1 
\end{align}

use the code

\begin{verbatim}
\begin{align}
\int_0^1 x^2\,dx & = \tfrac13 x^3\bigg|_{x=0}^1 \nonumber \\
& = \tfrac13, \label{E:final}
\end{align}
\end{verbatim}

You can also use the ``aligned'' environment inside of an equation environment. For example, the code for 
\begin{equation}\label{E:another_example}
\begin{aligned}
\int_0^1 x^2\,dx & = \tfrac13 x^3\bigg|_{x=0}^1  \\
& = \tfrac13 \\
& = \dots \\
& = \dots
\end{aligned}
\end{equation}
is

\begin{verbatim}
\begin{equation}\label{E:another_example}
\begin{aligned}
\int_0^1 x^2\,dx & = \tfrac13 x^3\bigg|_{x=0}^1  \\
& = \tfrac13
\end{aligned}
\end{equation}
\end{verbatim}

\[
\frac{1}{n}\to 0 \qtq{as}n\to\infty. 
\]

\[
e^{\frac{1}{x^2}} \quad e^{\tfrac{1}{x^2}} \quad \exp\{\tfrac{1}{x^2}\}
\]
\subsection{References}  To reference the Theorem that you labeled T:main, use 

\begin{verbatim}
Theorem~\ref{T:main}
\end{verbatim}

This produces Theorem~\ref{T:main}.

This works for Sections, Theorems, Lemmas, equations, and so on.  But for equations, it's better to use ``eqref'', e.g. equation \eqref{E:another_example} is produced by 

\begin{verbatim}
\eqref{E:another_example}
\end{verbatim}



%%%%%%%%%%%%%%%
\section{Some more formatting items}




To make an itemized list, use the following
\begin{verbatim}
\begin{itemize}
\item First item
\item[2.] Second item, with a 2. instead of a bullet.
\end{itemize}
\end{verbatim}


\begin{itemize}
\item[1.] First
\item[b.] Second
\item[(iii)] Third...
\end{itemize}

\bigskip

another example

\begin{itemize}
\item First item
\begin{itemize}
\item Nested itemize
\end{itemize}
\item[2.] Second item, with a 2. instead of a bullet.
\end{itemize}

To include a graphic, you can use the following.

\begin{verbatim}
\begin{center}
\includegraphics{uo.jpg}
\end{center}
\end{verbatim} 

This produces:

%\begin{center}
%\includegraphics[scale=.4]{uo.jpg}
%\end{center}


\begin{center}
\includegraphics[width=.9\linewidth]{uo.jpg}
%[scale=.5]{mst.png}
\end{center}

\begin{figure}[h!]
\includegraphics[width=.5\linewidth]{uo.jpg}
%\begin{center}
\caption{I don't know}
%\end{center}
\end{figure}


To change text color, size and style...

\bigskip

\textbf{Bold text,} \emph{italicized text,} {\color{blue}blue text}, {\large large text}, {\Large Large text }, {\small small text}, {\sc Smallcaps...}, {\footnotesize this is small}

\[
{\color{red} x +3 = 5}
\]

The code is:
\begin{verbatim}
\textbf{Bold text,} \emph{italicized text,} {\color{blue}blue text}, 

{\large large text},  {\Large Large text}, {\small small text}, 

{\sc Smallcaps...} (Look up Latex font sizes for more info...)
\end{verbatim}

To make a matrix, you can do something like the following:
\[
\left[\begin{array}{ccc} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \\ & & 5\end{array}\right].
\]
\newpage

The code for this is:
\begin{verbatim}
\[
\left[\begin{array}{ccc} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \\ 4 \end{array}\right].
\]
\end{verbatim}
The `ccc' means the numbers will be centered.  You can also put in `l' or `r' there.  You can also put $\vert$ in between to produce vertical lines in the matrix, e.g.
\[
\left[\begin{array}{cc|c} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5\end{array}\right].
\]
The code is
\begin{verbatim}
\[
\left[\begin{array}{cc|c} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5\end{array}\right].
\]
\end{verbatim}

\[
\binom{n}{k}
\]


\[
f(x) = \begin{cases} x & x>0 \\ x^2 & x\leq 0.\end{cases}
\]
%%%%%%%%%%%%
\section{Beamer code}

Here's some of the basic code needed for a beamer presentation... in the preamble:

\begin{verbatim}
\documentclass[10pt]{beamer}
\usetheme{Warsaw}
\mode<presentation>
\end{verbatim}

To make a slide:
\begin{verbatim}
\begin{frame}\frametitle{Title of the frame}
Content of the slide
\end{frame}
\end{verbatim} 

See the code below for the bibliography:

\begin{verbatim}
\begin{thebibliography}{100}
\bibitem{Bibitem} J. Murphy, \emph{A LaTeX Template}. 
\end{thebibliography}
\end{verbatim}

For more information, see \cite{Bibitem, Bibitem2}.

Pages 23-26 versus Pages 23--26.  Murphy--Smith 

\bigskip

23--36

$23--36$
The `right way' to do bibliographies is to use a .bibtex file, but I don't know how to do it so I can't tell you about it. Sorry. See \cite{Bibitem2, Bibitem}. 

\cite{Bibitem}


Please refer to \cite{Bibitem2}. 

%%%%%%%%%%%%%%
\begin{thebibliography}{1}
\bibitem[Murphy2021]{Bibitem} J. Murphy, \emph{A LaTeX Template}. 
\bibitem{Bibitem2} J. Murphy, \emph{Another LaTeX Template}. 
\end{thebibliography}



\end{document}