% Senior Project Example File
% Department of Mathematics, Bilkent University
% Aurelian Gheondea, Senior Project Coordinator
% In case you are using this file, you have to rename it!
\documentclass[12pt]{amsart}
\usepackage{amsmath,amssymb}
\usepackage{amsthm}
% The package can be used temporarily only to show the cross-reference labels
%\usepackage{showkeys} 
% This paragraph resets the sizes of the printed area of the page
\hoffset -1.5cm
\voffset -1cm
\textwidth 15.5truecm
\textheight 22.5truecm
% This paragraph defines the macros for theorems and the like
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
% This paragraph defines the macros for definitions and the like
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{remarks}[theorem]{Remarks}
% The following paragraph writes the equation numbers with two counters,
% the first is the section number and the second resets within the section.
\makeatletter
\@addtoreset{equation}{section}
\makeatother
\renewcommand\theequation{\arabic{section}.\arabic{equation}}
% The next paragraph defines macros for special roman letters to be used
\newcommand{\CC}{{\mathbb C}} % the set of complex numbers
\newcommand{\NN}{{\mathbb N}} % the set of natural numbers
\newcommand{\QQ}{{\mathbb Q}} % the set of rational numbers
\newcommand{\ZZ}{{\mathbb Z}} % the set of integer numbers
\newcommand{\DD}{{\mathbb D}} % the unit disk
\newcommand{\RR}{{\mathbb R}} % the set of real numbers
\newcommand{\TT}{{\mathbb T}} % the unit circle (the one dimensional torus)
% The next paragraph defines macros for caligraphic letters
\newcommand{\cA}{{\mathcal A}}
\newcommand{\cB}{{\mathcal B}}
\newcommand{\cC}{{\mathcal C}}
\newcommand{\cD}{{\mathcal D}}
\newcommand{\cE}{{\mathcal E}}
\newcommand{\cF}{{\mathcal F}}
\newcommand{\cG}{{\mathcal G}}
\newcommand{\cH}{{\mathcal H}}
\newcommand{\cJ}{{\mathcal J}}
\newcommand{\cK}{{\mathcal K}}
\newcommand{\cL}{{\mathcal L}}
\newcommand{\cM}{{\mathcal M}}
\newcommand{\cN}{{\mathcal N}}
\newcommand{\cP}{{\mathcal P}}
\newcommand{\cQ}{{\mathcal Q}}
\newcommand{\cR}{{\mathcal R}}
\newcommand{\cS}{{\mathcal S}}
\newcommand{\cT}{{\mathcal T}}
\newcommand{\cU}{{\mathcal U}}
\newcommand{\cV}{{\mathcal V}}
\newcommand{\cW}{{\mathcal W}}
\newcommand{\cZ}{{\mathcal Z}}
% The next paragraph defines special macros
\newcommand{\iac}{\mathrm{i}} % the imaginary i
\newcommand{\de}{\operatorname{d}} % the differential
\newcommand{\deriv}{\operatorname{D}}
% Authors macros for complicated commands can be used, but be careful!
\providecommand{\AMS}{$\mathcal{A}$\kern-.1667em%
\lower.25em\hbox{$\mathcal{M}$}\kern-.125em$\mathcal{S}$}
% Use of macros should confine the general rules for AMS-LaTeX. It is
% recommended to use \newcommand and \renewcommand instead of the TeX
% primitive \def
% Next is an example of macro I designed for numbering items that are
% different than the available ``itemize, description'' environments.
% You can use it if you want!
\newcommand{\nr}[1]{\vspace{0.1ex}\noindent\hspace*{12mm}\llap{\textup{(#1)}}}
% Topmatter produces the title, author, abstract, etc.
\begin{document}
\title[Senior Project Example]{An Example File for Senior Project in
  Mathematics} 

\author{Aurelian Gheondea}

\address{Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara,
  Turkey}
\email{aurelian@fen.bilkent.edu.tr}

\begin{abstract} An abstract of at about 150 words (10 rows) should be 
included. The abstract should summerize the results of the paper and nothing
more. The main purpose of the abstract is to enable the readers to take in
the nature and results of the papers quickly and to decide whether they are
willing to read the entire paper or not.
\end{abstract}
\maketitle
% Body of the project

\section*{Introduction} % The * is needed if you do not want this section 
% numbered; otherwise remove the * and this will be numbered by 1.

The first section is usually an introduction. It has the role of presenting
the general framework of the paper, historical notes, relevance of the
project and an outline of the main results. Repetition should be kept to
a minimum. The introduction should not have more than two
pages. Acknowledgements of support and thanks (e.g.\ to the adviser) should be
placed at the end of the introduction.

The body of the article should be usually divided in sections. Out of this
introductory section, the following section may be dedicated to preliminary
results, that is, a list of definitions and results (without proofs) that will
be used in the project. Otherwise, the following section can be the main
section, or a sequence of different sections, dedicated to the presentation of
the project from the mathematical point of view.

The results should be distinguished correspondingly as theorems, propositions,
lemmas and corollaries. Remarks, definitions, and examples can be used. 
Preferable is that all the items to be numbered  within sections: the first
figure is the section number, the latter designates the number of the item
within the same section. 

The beginning and the end of the proofs should be clearly emphasized. If
necessary, proofs can be divided in steps. Nested proofs should be avoided as
much as possible. 

If draws are necessary, the available packages for producing different
pictures in \LaTeX2e can be used. If complicated pictures are needed, there
are available packages for including graphics, e.g.\ in EPS (Encapsulated Post
Script) or PDF (Portable Document Format).

Items listed as references should be relevant to the subject of the paper. The
list of references must be in alphabetical-chronological order. Abbreviations
of names of journals and references should follow the standard form
established by the Mathematical Reviews.

\section{Preliminary Results and Notation}

The second section can be dedicated to preiliminary results, notation, and
other topics that make the background of the project.

\section{Main Section}

In this section we exemplify the use of theorems, definitions, and proofs,
with different mathematical formulas, tables, etc. By no means this file is
supposed to be exhaustive. The student should consult the textbook indicated
for \LaTeX2e by creating more complicated mathematical formulas.

In this section we give some examples of use of macros and environments. There
are two types of environments for results and the like. The first one is a
``proclaim'' type, used for theorems, corollaries, lemmas, propositions.

\begin{theorem}\label{mytheorem}
 This is an example of a proposition. It contains a statement
  in italics, equations, formulas and possibly some itemed statements like
  these:

\begin{itemize}
\item[(i)] this is the first item;
\item[(ii)] and this is the  second.
\end{itemize}
\end{theorem}
You can use an option for associating citations or names with a theorem:

\begin{theorem}[My Friends' Greatest Result, \cite{mft}]
\label{myfriendstheorem}
This is another theorem. It
  contains three items that are more or less independent.

\begin{itemize}
\item[(a)] This is the first item.
\item[(b)] This is the second item.
\item[(c)] This is the third item.
\end{itemize}
\end{theorem}

\begin{proof} This is an example of a proof where you should always use the
  environment \textit{proof} as we do.
\end{proof}

A proof may be provided with a special name.

\begin{proof}[Proof of Theorem~\ref{mytheorem}] In this proof  we emphasize a
  situation when the last row is a displayed equation. In this case, you
  should use a special command $\backslash$\texttt{qedhere}
to fix the rectangle indicated the end of the proof, like in the following:
\begin{equation}\label{nice}
B(t)=b(s).\qedhere
\end{equation}
\end{proof}

Fancy theorems, for which there is no automatic numbering and the tag is put
manually, are available.

\theoremstyle{theorem}
\newtheorem*{TA}{Theorem A}
\begin{TA} This theorem is classical.
\end{TA}

A second type of environment that we call \textit{statement} is for
definitions, remarks, and examples.

\begin{definition}\label{mydefinition} This is an example of a definition. The
  defined \emph{term} should be emphasized.
\end{definition}

Another example of this type is a remark.

\begin{remark}\label{myremark} This is an example of a remar. It will appear
  in roman letters as the defintions and examples.
\end{remark}

An example can have an emphasized text at the beginning to clearly state what
is done.

\begin{example}\label{myexample} \emph{We give an example that has an
    emphasized statement}.

Now it should follow the example which will be typed in roman letters.
\end{example}

For emphesizing text do not use boldface. Instead, the command
$\backslash$\texttt{emph} is highly recommended.

Anther important recommendation is to not use your own breaks. This class file
\texttt{amsart.cls} is designed in such a way tho produce automatically
spaces whenever
is needed, and it reflects the style used very frequently in mathematics.

You should use all-over the article labeled cross references by using
 $\backslash$\texttt{ref} command, e.g.\ citing
 Theorem~\ref{myfriendstheorem}. For equations it is recommended to use
 instead
the command $\backslash$\texttt{eqref}, like e.g.\
 \eqref{nice}, that produces the round braces as well.

It is recommended to use alignments as follows:
\begin{align}\label{bete} B(t) & = B(t+1)+B(t+2)+B(t+3) \\
                    & = A(t) \nonumber
\end{align}
Note that you can have some rows without tags.

In the following we examplify the use of matrices.
With respect to the decomposition
$\mathcal{K}=\mathcal{K}_1\oplus \mathcal{K}_2$ we have
\begin{equation*}
A+B=\left[%
\begin{array}{cc}
  0 & 0 \\
  0 & C \\
\end{array}%
\right],\;\;\;
\pi_\rho(a)=\left[%
\begin{array}{cc}
  \pi_{\rho,1}(a) & 0 \\
  0 & \pi_{\rho,2}(a) \\
\end{array}%
\right]\;\;\mbox{ for all }a\in\cA,\;\;\mbox{ and }\;\;V=\left[%
\begin{array}{c}
  V_1 \\
  V_2 \\
\end{array}%
\right],
\end{equation*}
where $C$ is an operator in $\mathcal{B(K}_2)^+$, necessarily
one-to-one, and commutes with $\pi_{\rho,2}(a)$, for all $a\in
\mathcal{A}$.

If your formulas are longer, you should organize them as follows:
\begin{align*}
 [A]B&=\left[%
\begin{array}{cc}
  0 & 0 \\
  0 & C^{1/2}FC^{1/2} \\
\end{array}%
\right]  \left[%
\begin{array}{cc}
  0 & 0 \\
  0 & C^{1/2}(1-F)C^{1/2} \\
\end{array}%
\right]\\
   &=
 \left[%
\begin{array}{cc}
  0 & 0 \\
  0 & [C^{1/2}FC^{1/2}](C^{1/2}(1-F) C^{1/2}) \\
\end{array}%
\right] \\
   &= \left[%
\begin{array}{cc}
  0 & 0 \\
  0 & C^{1/2}\;[F](1-F)\;C^{1/2} \\
\end{array}%
\right] = \deriv_\rho(\phi_{\mathrm{ac}}).
\end{align*}

For more complicated mathematical formulas, tabulars, and arrays, you should
consult the \LaTeX2e textbook indicated in the syllabus.

\section{Final Remarks}

You should be aware that complicated and longer aligned formulas can produce
rare pages, because of the page break problems.

Citations of bibliography, like \cite{mft}, should be done by labels.

When coding your \AMS-\LaTeX\ file, please bear in mind the
following general rules:

\nr{1} Use the packages \texttt{amsmath, amssymb, amsfonts},
etc.\ for quality
  mathematical typesetting.

\nr{2} Use the \LaTeXe\ and \AMS-\LaTeX\ commands and
    cross-reference capabilities
 as much as possible. Do not hardcode your own layout, for example by
 creating your own section headings, theorems and the like, displayed
 equations, etc. Do not use your own spacing: use as much as possible the
 \AMS-\LaTeX\ capabilities available in the {\tt amsmath} package.

\nr{3} Keep it simple.
Do not define macros to create a complicated layout, as
these will make the input process complicated.

\nr{4} Be consistent.
If you use a macro for a phrase, use it throughout.

\nr{5} Figures may be inserted in the usual way using an
 $\backslash${\tt includegraphics}
command at the position in the article
  where they are cited.

\nr{6} If the \LaTeX\ file is nicely written, the corrections can be easy.


\begin{thebibliography}{99}

\bibitem{mfa} \textsc{M.F. Author}, Title of the article, \textit{J. Operator
    Theory}, \textbf{79}(2009), 123-145.

\bibitem{mfc} \textsc{M. Name, Y. Name}, \textit{Title of the Book}, Theta,
Bucharest 2004.

\bibitem{mft} \textsc{M. Student}, \textit{Title of the Dissertation}, PhD
  Dissertation, My University, My Town 1450.

\end{thebibliography}

\end{document}
