%
% Lines starting with % are comments, which are ignored when
% you compile your code. This is a handy way of indicating
% the date and version of your document.
%
% Carthage Sample LaTeX File
% September, 2000
%
%
\documentclass{article}  %This tells us what type of document we're producing.
\usepackage{epsfig}     %The packages put really useful stuff at your disposal.
%                        We only use one basic packages so our code is usable
%                         on the web.
\usepackage{amsmath}
\usepackage{makeidx}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{graphics}
\usepackage{enumerate}
\usepackage{multicol}
%
% This establishes our standard structures like theorem, proof, etc.
%
\newtheorem{thm}{Theorem}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{ax}[thm]{Axiom}

\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
\newtheorem{exam}[thm]{Example}

\newtheorem{rem}[thm]{Remark}

%
% Title and author(s)
%
\title{A Sample Mathematics Document}

\author{Dee Y. Deetee\\Carthage College\\dydt@carthage.edu
        \and
        Dee X. Deetee \\
        Carthage College\\
        dxdt@carthage.edu}
%

\begin{document}
\maketitle

% Your abstract is the last thing you write.

\begin{abstract}
    \noindent Your abstract should contain a brief summary of your results.
    Aim for around 250 words.  This is probably the last thing you write.
\end{abstract}


%
% Here is the start of the paper.
%

\section{Introduction}
    The introduction introduces your reader to your project.  Technical language and notation are generally not appropriate in this section.

\section{Definitions and Development}
Here is where you get technical.  Derive your equations, develop you model, state basic definitions formally, etc.  You will
almost certainly need some definitions similar to Definition \ref{definition1}, so use this format.

\begin{defn}\label{definition1}
A \textbf{new term} is an expression that you have not used before and that your reader needs to know about.
\end{defn}

Of course, once you have a new definition, you will probably have a theorem or two telling us about what you discovered about your new term.

\begin{lem}[Little Theorem] \label{theorem1}
If $x$ is a new term, then $x$ is important to your readers.  Most of your theorems will go in the Results Section, but little lemmas could go here.
\end{lem}
\begin{proof}
If the term was not important, you would not need to define it.
\end{proof}

\begin{exam} \label{anexample}  Examples should be numbered too.
\end{exam}

Of course, we can refer to a theorem as Theorem \ref{theorem1} or as Fake Theorem.  If you need to refer to Example \ref{anexample}, do so by
number.

\section{Results}
What have you learned from what you did in the previous section?  This section could contain graphs, tables, computations, proofs, etc.
When you include equations or functions, you can put them into a sentence $f(x)=x^2 \sin x$, or you can set them apart
\[
f(x)=x^2 \sin x.
\]
Notice that you do need to punctuate.  If you refer back to the equation, label it,
\begin{equation} \label{equ}
x^2 \sin x = 0,
\end{equation}
and avoid statements like ``the equation below.''  Instead, refer to Equation (\ref{equ}).

\begin{thm}[Fake Theorem] \label{theorem1}
If $x$ is a new term, then $x$ is important to your readers.
\end{thm}
\begin{proof}
If the term was not important, you would not need to define it.
\end{proof}

The same holds for tables and graphs.  Label the graph, include a caption, and refer to it via the label.  See Figure \ref{fig}.
\begin{figure}[htb]
\centering \fbox{\epsfig{file=vennsetminus.eps, height=1 in}}
\caption{The Difference of Two Sets}\label{fig}
\end{figure}


\section{Conclusion and Directions for Further Research}
Draw some broad conclusions, and tell us what comes next.

\begin{thebibliography}{9}
     \bibitem{lamport:1}
          Lamport, L.,
         \textit{\LaTeX - A Document Preparation System},
         Addison-Wesley, Chicago, 1998.
     \bibitem{snavely:1}
         Snavely, M.,
        A Journal Example,
        \textit{Fake Mathematics Journal}, \textbf{27} (2010) 12-15.

\end{thebibliography}

\section*{Appendices}

Include appendices here if necessary.

\end{document}
