% LaTEX source code
% Last modified December 8, 1998.
% Steve Miller
\documentclass[12pt,letterpaper]{report}
\usepackage{times}
\usepackage[T1]{fontenc}
\usepackage{mathrsfs}
\usepackage{latexsym}
\usepackage[dvips]{graphics}
\usepackage{epsfig}
%\usepackage{hyperref, amsmath, amsthm, amsfonts, amscd, flafter,epsf}
\usepackage{amsmath,amsfonts,amsthm,amssymb,amscd}
\input amssym.def
\input amssym.tex
%=======================================================
% THIS IS WHERE YOU PUT SHORTCUT DEFINITIONS
%========================================================
% Note that we use a percent sign to comment out a line
% below are shortcut commands
% Theorem / Lemmas et cetera
\newtheorem{thm}{Theorem}[section]
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{exa}[thm]{Example}
\newtheorem{defi}[thm]{Definition}
\newtheorem{exe}[thm]{Exercise}
\newtheorem{rek}[thm]{Remark}
\newtheorem{que}[thm]{Question}
\newtheorem{prob}[thm]{Problem}
\newtheorem{cla}[thm]{Claim}
% shortcuts to environments
\newcommand\be{\begin{equation}}
\newcommand\ee{\end{equation}}
\newcommand\bea{\begin{eqnarray}}
\newcommand\eea{\end{eqnarray}}
\newcommand\bi{\begin{itemize}}
\newcommand\ei{\end{itemize}}
\newcommand\ben{\begin{enumerate}}
\newcommand\een{\end{enumerate}}
\newcommand\bc{\begin{center}}
\newcommand\ec{\end{center}}
\newcommand\ba{\begin{array}}
\newcommand\ea{\end{array}}
\newcommand{\tbf}[1]{\textbf{#1}}
\newcommand{\vars}[2]{ #1_1, \dots, #1_{#2} }
\newcommand{\ncr}[2]{{#1 \choose #2}}
\newcommand{\twocase}[5]{#1 \begin{cases} #2 & \text{#3}\\ #4
&\text{#5} \end{cases} }
\newcommand{\twocaseother}[3]{#1 \begin{cases} #2 & \text{#3}\\ 0
&\text{otherwise} \end{cases} }
% shortcut to words
\newcommand\CF{{Continued Fraction}}
\newcommand\cf{{continued fraction}}
% General Symbols
\def\notdiv{\ \mathbin{\mkern-8mu|\!\!\!\smallsetminus}}
%Blackboard Letters
\newcommand{\R}{\ensuremath{\mathbb{R}}}
\newcommand{\C}{\ensuremath{\mathbb{C}}}
\newcommand{\Z}{\ensuremath{\mathbb{Z}}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\W}{\mathbb{W}}
\newcommand{\Qoft}{\mathbb{Q}(t)} %use in linux
% Finite Fields and Groups
\newcommand{\Fp}{ \F_p }
\newcommand{\Fpf}{ \Fp^{*} }
\newcommand{\ZnZ}{ \Z / n\Z}
\newcommand{\ZnZf}{ (\Z / n\Z)^{*}}
\newcommand{\ZqZ}{ \Z / q\Z}
\newcommand{\ZqZf}{ (\Z / q\Z)^{*}}
% Fractions
\newcommand{\fof}{\frac{1}{4}} %oneforth
\newcommand{\foh}{\frac{1}{2}} %onehalf
\newcommand{\fot}{\frac{1}{3}} %onethird
\newcommand{\fop}{\frac{1}{\pi}} %1/pi
\newcommand{\ftp}{\frac{2}{\pi}} %2/pi
\newcommand{\fotp}{\frac{1}{2 \pi}} %1/2pi
\newcommand{\fotpi}{\frac{1}{2 \pi i}}
% Legendre Symbols
\newcommand{\jsi}[1]{ { \underline{#1} \choose p_i} }
\newcommand{\jso}[1]{ { \underline{#1} \choose p_1} }
\newcommand{\jst}[1]{ { \underline{#1} \choose p_2} }
\newcommand{\jsthree}[1]{ { \underline{#1} \choose 3} }
\newcommand\lag[2]{\ensuremath{\left(\frac{#1}{#2}\right)}}
\newcommand{\jsq}[1]{ { \underline{#1} \choose q} } %(*/q)
\newcommand{\jsn}[1]{ { \underline{#1} \choose n} } %(*/q)
\newcommand{\js}[1]{ { \underline{#1} \choose p} }
% Continued Fraction Shortcuts
\newcommand{\cffour}[4]{#1+\frac{1}{#2+\frac{1}{#3+\frac{1}{\ddots+\frac{1}{#4}}}}}
\newcommand{\cfthree}[3]{#1+\frac{1}{#2+\frac{1}{\ddots + \frac{1}{#3}}}}
\newcommand{\cftwo}[2]{#1+\frac{1}{\ddots + \frac{1}{#2}} }
\newcommand{\alphampq }{ \Bigg| \alpha - \frac{p}{q} \Bigg| }
\newcommand{\gi}[2]{ \Bigg[\frac{#1}{#2}\Bigg] }
\newcommand{\gismall}[2]{ \Big[\frac{#1}{#2}\Big] }
% Subscript Shortcuts for ps, qs, and as
\newcommand{\pnpo}{p_{n+1}}
\newcommand{\pmpo}{p_{m+1}}
%\newcommand{\pm}{p_m}
\newcommand{\pn}{p_n}
\newcommand{\pmmo}{p_{m-1}}
\newcommand{\pnmo}{p_{n-1}}
\newcommand{\pmmt}{p_{m-2}}
\newcommand{\pnmt}{p_{n-2}}
\newcommand{\qmpo}{q_{m+1}}
\newcommand{\qnpo}{q_{n+1}}
\newcommand{\qm}{q_m}
\newcommand{\qn}{q_n}
\newcommand{\qmmo}{q_{m-1}}
\newcommand{\qnmo}{q_{n-1}}
\newcommand{\qmmt}{q_{m-2}}
\newcommand{\qnmt}{q_{n-2}}
\newcommand{\ampo}{a_{m+1}}
\newcommand{\anpo}{a_{n+1}}
\newcommand{\am}{a_m}
\newcommand{\an}{a_n}
\newcommand{\ammo}{a_{m-1}}
\newcommand{\anmo}{a_{n-1}}
\newcommand{\ammt}{a_{m-2}}
\newcommand{\anmt}{a_{n-2}}
% matrix shortcuts
\newcommand{\mattwo}[4]
{\left(\begin{array}{cc}
#1 & #2 \\
#3 & #4
\end{array}\right) }
\newcommand{\matthree}[9]
{\left(\begin{array}{ccc}
#1 & #2 & #3 \\
#4 & #5 & #6 \\
#7 & #8 & #9
\end{array}\right) }
\newcommand{\dettwo}[4]
{\left|\begin{array}{cc}
#1 & #2 \\
#3 & #4
\end{array}\right| }
\newcommand{\detthree}[9]
{\left|\begin{array}{ccc}
#1 & #2 & #3 \\
#4 & #5 & #6 \\
#7 & #8 & #9
\end{array}\right| }
% greek letter shortcuts
\newcommand{\ga}{\alpha} %gives you a greek alpha
\begin{document}
\title{Title of the Paper}
\author{Your Name\thanks{E-mail: \texttt{whoyou@math.princeton.edu}} \\
\centerline{Mathematics Department} \\
\centerline{Princeton University} \\
\centerline{Princeton, NJ 08544}}
\date{\today}
\maketitle
\begin{abstract}
The beginning of the abstract. The middle. How do you like it so
far? Almost done. Done.
\end{abstract}
\tableofcontents
\chapter{First Steps}
\section{One Line Equations}
Garbage text for format purposes. Here is math mode: $\alpha^3 +
\beta^{12}.$ To enter math mode inside text, simply type a dollar
sign. Type another dollar sign to exit math mode.
Some people might have latex environments where it will replace a
$\alpha$ with the greek letter alpha. Note that if you type
$\alpha$ it might replace it with what a real alpha, while if I
type $\ga$, the user defined shortcut, it does not replace.
Here's how to do an equation. Once you type begin{equation} (with
the slash in front of it) you've automatically entered math mode.
Now if you type anything, say uppercase Lamda or lowercase epsilon
or even some of our user-defined shortcuts, they will be properly
formatted:
\begin{equation}
\frac{\Lambda'(s)}{\Lambda(s)} = \epsilon \int_a^b 3x^2 e^{2\pi i
x} dx.
\end{equation}
Now, if I don't want to type begin equation and all that, I can
use the shortcuts I've defined:
\be \frac{\Lambda'(s)}{\Lambda(s)} = \epsilon \int_a^b 3x^2
e^{2\pi i x} dx. \ee
\section{Labeling Equations}
What if I want to keep track of the equation number, so that I can
refer to it in the text? For example, consider
\begin{equation}\label{eqlambdaprimelambda}
\frac{\Lambda'(s)}{\Lambda(s)} = \epsilon \int_a^b 3x^2 e^{2\pi i
x} dx.
\end{equation}
Note the label. I can choose anything (as long as there are no
numbers, just letters) for the name. I choose to label all
equations with eq followed by a descriptive name; lemmas I start
lem followed by a descriptive name, and so on.
To refer to the equation, I merely have to write
\ref{eqlambdaprimelambda}; however, it's better to write Equation
\ref{eqlambdaprimelambda}.
Whenever you add equations, you have to compile Latex twice to get
the references correct.
\section{Multi-Line Equations: Eqnarray}
What if your equation is more than one line? You might want to use
eqnarray instead of equation. The slashnonumberslashslash is a
carriage return without numbering that line; personally, I like to
wait to the last line to number something. Here's an example:
\begin{eqnarray}
\frac{\Lambda'(t)}{\Lambda(t+1)} & = & f(x) g(s) + f(x-t) -
g(s)f(x) \nonumber\\ \frac{\Lambda'(t)}{\Lambda(t+1)} & = & f(x -
t)
\end{eqnarray}
Again, if I don't want to type begin eqnarray I can use the
shortcut:
\bea \frac{\Lambda'(t)}{\Lambda(t+1)} & = & f(x) g(s) + f(x-t) -
g(s)f(x) \nonumber\\ \frac{\Lambda'(t)}{\Lambda(t+1)} & = & f(x -
t) \eea
The formatting is done by the ampersand signs, $\&$. (Note: if you
have a special symbol which you want to display in Latex, you put
a slash before it. Thus, to print a percent-sign in math mode is
$\%$, or to print a pound sign is $\#$.) The eqnarray environment
has two ampersands per line, and centers the lines on what is
between the ampersands.
Usually, one does not repeat the left hand side. Thus, it is more
natural to write
\begin{eqnarray}
\frac{\Lambda'(t)}{\Lambda(t+1)} & = & f(x) g(s) + f(x-t) -
g(s)f(x) \nonumber\\ & = & f(x - t)
\end{eqnarray}
Here's a somewhat lengthier example:
\begin{eqnarray}
\frac{1}{m}\sum_p^{m^\sigma} p^{-\foh} & \leq & \frac{1}{m} (
\sum_p^{m^\sigma} \frac{1}{p})^{\foh} ( \sum_p^{m^\sigma}
1)^{\foh} \nonumber\\ & \leq & \frac{1}{m} (\log \log m^{\sigma} +
A)^{\foh} (Li(x) + O(x^{\foh} \log x))^{\foh} \nonumber\\ & \ll &
\frac{1}{m} (\log \log m)^{\foh} (\frac{2m^{\sigma}}{\log
m})^{\foh} \nonumber\\ & \ll & m^{\foh \sigma - 1} (\frac{\log
\log m}{\log m})^{\foh}.
\end{eqnarray}
\section{Lemmas, Propositions, Theorems and Corollaries}
Now let's add a lemma. Below is how one would write it. Notice all
the English text is italicized. We'll follow the lemma immediately
with a proposition.
\begin{lem} Let $\hat{\phi} (\xi) = \int_{R} \phi(x) e^{2 \pi i x} dx.$ Then
$\hat{\phi_r} (\xi) = \frac{1}{r} \hat{\phi}(\xi / r). $
\end{lem}
\begin{prop}
If $f \in \mathcal{C}^3$ and $f'(0) = 0, \ f''(0) > 0$ then $0$ is
a local minimum.
\end{prop}
Proof: this follows immediately from the well known relation
\begin{equation}
3x + 2y = 4z
\end{equation}
The following is a new lemma, and the [] give the lemma a name.
\begin{lem}[Discrete Measure Lemma]
$$\frac{1}{\log N} \sum_p \frac{\log p}{p} \phi(2 \log n) =
\frac{1}{4} \phi(0) + O(N).$$
\end{lem}
You can label lemmas just like you would equations:
\begin{thm}\label{lemimplicitequation}[The Implicit Equation] Let
$x, y, z \in \C$. Then
\begin{equation}
x^y + y^z + z^x = -1
\end{equation}
\end{thm}
By Lemma \ref{lemimplicitequation}, we see that the desired
expression equals $-1$.
\begin{cor}
$x = y = z = 1$ is not a solution
\end{cor}
\begin{cor}[Hensel's Observation]
$x = y = -1, \ z = 1$ is a solution
\end{cor}
more garbage text to just switch
\begin{equation}
\Lambda(s) = \frac{4z}{g(s)}
\end{equation}
\section{Using Subsections}
If we wanted, we could put subsections in a section.
\subsection{Pythagoras}
If a section is very long, we might want to have sub-sections in
the sections. The commands are exactly what you think.
\begin{lem}[Lengths of Sides]
The sum of the lengths of any two sides of a triangle are greater
than the third length.
\end{lem}
\subsubsection{Garbage}
Of course, you can have sub-divisions of sub-sections.
Here is some more garbage text. \\ And here is some more.
Note the double slash above forces a carriage return.
We are now going to force a new page.
\pagebreak
\subsection{Prime Number Theorem}
Below is the Prime Number Theorem. If we assume the Riemann
Hypothesis we can take $\alpha = \frac{1}{2} + \epsilon.$
\begin{equation}
\pi(x) = \frac{x}{\log x} \ + \ O(x^\alpha)
\end{equation}
We have now come to the end of the first chapter -- it will
automatically start the next chapter on a new page.
I like to put a lot of percent signs between chapters (and a few
carriage returns) to make editing easier.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Environments I}
\section{Shortcut Environments}
To do an equation, recall we need slash begin curly brackets
equation curly brackets. Thus, we write
\begin{equation}
\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6} = \frac{8}{3}
\Bigg(\int_0^1 \frac{1}{1+x^2}dx\Bigg)^2. \end{equation}
If you have a lot of equations or arrays of equations, you don't
want to keep typing begin equation and end equation.
We've created some shortcuts: slash be will be begin equation;
slash ee will end the equation; bea and eea will begin and end
arrays of equations.
Thus,
\be
\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6} = \frac{8}{3}
\Bigg(\int_0^1 \frac{1}{1+x^2}dx\Bigg)^2 \ee
does it as an equation, and
\bea \sum_{n=1}^\infty \frac{1}{n^2} &=& \frac{\pi^2}{6}
\nonumber\\ &=& \frac{8}{3} \Bigg(\int_0^1
\frac{1}{1+x^2}dx\Bigg)^2 \eea
does it as an array of equations.
Other useful commands: \textbf{this will put any text in bold}
while \emph{this will emphasize or italicize text}.
\section{General Latex and Linux Commands}
We will use the shortcuts for the enumeration environment.
\ben
\item Eduardo Duenez has been kind enough to give me his .emacs
configuration file, which configures the editor to make latexing
very easy. Save the .emacs file into your home directory.
\item in whatever directory you want to latex, save the files
template.tex, template.bbl, YL.EPS. This will give you a tex
template with a bibliography section and image. Unfortunately,
it's only configured to display the image when run in the windows
environment on my machine, so I've commented out the graphics
section. The bibliography file should have the same name as your
file, except with the extension .bbl instead of .tex.
\item at the unix prompt, move into the directory where you've
saved the templates. To edit, type xemacs template.tex \& (the
ampersand makes sure it opens in a new window). To compile is
Control-c-c. Type xdvi template.dvi \& to view your compiled file.
If you make changes to the latex source file, just clicking on the
xdvi file will (if you've compiled the latex file) automatically
update the dvi file. control-c (let go of the two keys) ` displays
error messages (it's on the same key as the tilde, don't hold down
shift).
\een
If instead of numbering I wanted bullets I would do
\bi
\item General: cd directory changes the directory, ls lists all
files and sub-directories in whatevery directory you are currently
in. Control-x-w saves without exiting; Control-x-c saves with
exiting.
\ei
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Graphics}
\section{Inserting Graphics}
Let's end by inserting a picture (image courtesy of J. Ax and S.
Kochen). The image extension should be .EPS, and in the same
directory as everything.
\begin{center}
\scalebox{.25}[.25]{\includegraphics{YL.EPS}}
\end{center}
At least on my home computer, if I try to view the .dvi file I
have trouble seeing the picture; I need to convert it to a .ps
file and then use ghostview or some such. I'm not sure exactly how
to format it in the UNIX environment to add the picture.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Environments II}
\section{Lists}
We have created the shortcut ben and een for begin and end
enumeration. We use it as follows:
\ben
\item This is the first item.
\item This is the second item.
\item This is the last item.
\een
\section{Emphasize and Bolding}
If you use \textbf{this, then whatever is inside will be in bold},
while if you use \emph{this, everything will be emphasized}, and
\underline{this will cause the text to be underlined}.
I have created a shortcut for textbf, namely \tbf{this will bold
text as well}.
\section{Centering Text}
One can also center text:
\begin{center}
Everthing typed in here is centered.
Isn't centering wonderful?
I thought so too.
\end{center}
And, of course, one can define shortcuts for centering:
\bc
This is also going to center text.
This is also wonderful. \\
The two slashes above give a extra carriage return. You can only
have one double slash at the end of a line. If you want more, use
bigskip.
\bigskip
\bigskip
\bigskip
Many of these shortcut commands are from a tex template that Alex
Barnett was kind enough to share with me.
\ec
\section{Refering to Bibliography}
The bibliography is included at the end. To refer to items, simply
type \cite{RSZ}. Note all the items in the bibliography have two
abbreviations, one in brackets, one in curly braces.
What is in brackets is what the computer will print; what is in
curly braces is how you refer to it.
Thus, you should type \cite{khinchin}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix
\chapter{Psychohistorical Dynamics of the Sayshell Republic: An Analysis of
the Rise of the Mule}
This is the first appendix, works like you would expect.
\chapter{Random Walks in High Dimensions: Choosing a Universe Interesting
for Drunks}
This is the second appendix.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}
\bibitem[CGI]{casati}
\newblock G. Casati, I. Guarneri, and F. M. Izrailev,
\emph{Statistical Properties of the Quasi-Energy Spectrum of a
Simple Integrable System}, Phys. Lett. A $124$ ($1987$),
$263-266$.
\bibitem[Ca]{cassels}
\newblock J. W. S. Cassels, \emph{An Introduction to
Diophantine Approximation}, Cambridge University Press, London
$1957$.
\bibitem[Da]{Da}
\newblock H. Davenport, \emph{Multiplicative Number Theory, $2$nd edition},
Graduate Texts in Mathematics \textbf{74}, Springer-Verlag, New York,
$1980$, revised by H. Montgomery.
\bibitem[Fe]{feller}
\newblock W. Feller, \emph{An Introduction to Probability Theory
and its Applications}, Vol. II. Second edition. John Wiley \&
Sons, Inc., New York-London-Sydney 1971.
\bibitem[GT]{GT}
\newblock A. Granville and T. Tucker, \emph{It's as easy as
$abc$}, Notices of the AMS, volume $49$, number $10$ (November
$2002$).
\bibitem[HW]{HW}
\newblock G. Hardy and E. Wright, \emph{An Introduction to the
Theory of Numbers}, fifth edition, Oxford Science Publications,
Clarendon Press, Oxford, $1995$.
\bibitem[Ki]{khinchin}
\newblock A. Y. Khinchin, \emph{Continued Fractions}, Third
Edition, The University of Chicago Press, Chicago $1964$.
\bibitem[Ru]{Ru}
\newblock W. Rudin, \emph{Principles of Mathematical Analysis},
third edition, International Series in Pure and Applied
Mathematics, McGraw-Hill Inc., New York, $1976$.
\bibitem[RSZ]{RSZ}
\newblock Ze\'ev Rudnick, P. Sarnak, and A. Zaharescu, \emph{The
Distribution of Spacings Between the Fractional Parts of
\(n^2\alpha\)}, Invent. Math. $145$ ($2001$), no. $1$, $37-57$.
\end{thebibliography}
\end{document}