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%IGNORE THESE
\def\mathlexicon#1{$$\vcenter{\halign{\displayandname{##}\hfil&&\qquad
                   \displayandname{##}\hfil\cr #1}}$$}
\def\displayandname#1{\rlap{$\displaystyle\csname #1\endcsname$}%
                      \qquad \texttt{\char92 #1}}


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  \Large{\bf{Math 280 - Protecting Information}}\\
  \small{\bf{Applications of Abstract Algebra and Quantum Physics}}\\
  Sample \LaTeX\ file - Lecture 99\\
  Benjamin Lotto - {\tt <lotto@vassar.edu>}\\
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\noindent The purpose of this document is to provide some help for
when you write up the official class notes.  Most word-processors do
not provide good mathematical typesetting facilities.  Fortunately
\LaTeX\ was designed specifically for that purpose.  The best way to
use this guide is to look at the source file ({\tt sample.tex}) while
experimenting on your own.  If you'd prefer more in-depth tutorial,
several good ones exist on the web.  (One such tutorial is {\tt
http://www.maths.tcd.ie/\textasciitilde dwilkins/LaTeXPrimer/}.)

\section{Math Mode Introduction}

To typeset mathematical expressions in \LaTeX, one first has to
indicate that the expression is not regular old text. There are
several ways to do this:

$Q \wedge (P \supset Q) \vdash P$

\begin{math} %or for short, \( \)
P \Rightarrow Q
\end{math}

\begin{displaymath} %or for short, \[ ... \] 
P \supset Q
\end{displaymath}

\begin{equation} %numbers the equations
\neg P \supset \neg Q
\end{equation}

It is also easy to define theorems, conjectures, and lemmas: 

%environment defined by newtheorem command
\begin{thm}[Existence of $GF(p^{n})$]
For each prime power $p^{n}$ there exists a field $GF(p^{n})$
containing $p^{n}$ elements.
\end{thm}
\begin{lemma}
For each prime $p$, $\Z/(p)$ is a finite field.
\end{lemma}

\begin{defn}
The \emph{characteristic} of a field $F$ is defined to be the
smallest integer $n > 0$ such that
\[ n \cdot 1 = \overbrace{1+1+ \cdots + 1}^{n \mathrm{times}}=0\]
if there is such an integer, or $0$ otherwise.
\end{defn}

\LaTeX\ keeps track of all of the numbering, \emph{e.g.}:

\begin{defn}
Suppose that $F$ is a finite field. A generator of $F^{\times}$ is
called a \emph{primitive root} of $F$.
\end{defn}

\section{Arrays}

To typeset a matrix in \LaTeX, one typically uses the {\tt array}
environment.

Consider the BCH code with generating polynomial 

\[g(x) = (x^{4} + x + 1)(x^{4} + x^{3} + x^{2} + x + 1)(x^{2} + x +
1)\]
 and $d = 7$. Then

%the array environment specifies a ``table'' like environment
%the 15 l's specify the number of columns.
% & separates columns, \\ separates rows

%\left( and \right) create big ( and )'s
\[
H = \left( \begin{array}{lllllllllllllll}
\alpha^{14} & \alpha^{13} & \alpha^{12} & \alpha^{11} & \alpha^{10} & \alpha^{9} & \alpha^{8} & \alpha^{7} & \alpha^{6} & \alpha^{5} & \alpha^{4} &
\alpha^{3} & \alpha^{2} & \alpha & 1 \\
\alpha^{12} & \alpha^{9} & \alpha^{6} & \alpha^{3} & 1 & \alpha^{12} & \alpha^{9} & \alpha^{6} & \alpha^{3} & 1 & \alpha^{12} &
\alpha^{9} & \alpha^{6} & \alpha^{3} & 1 \\
\alpha^{10} & \alpha^{5} & 1 & \alpha^{10} & \alpha^{5} & 1 & \alpha^{10}
& \alpha^{5} & 1 & \alpha^{10} & \alpha^{5} & 1 & \alpha^{10} & \alpha^{5}
& 1\\
\end{array}
\right)
\]

Or to demonstrate matrix multiplication:

\[
\left[\begin{array}{ll} a & b \\ c & d\\\end{array} \right]
\left(\begin{array}{l} e \\ f \end{array} \right)
= \left(a \cdot e + b \cdot f, c \cdot e + d \cdot f\right)
\]
\section{Symbols}

\subsection{Mathematical Symbols}

This section lifted verbatim from {\tt http://www.maths.tcd.ie/\textasciitilde dwilkins/LaTeXPrimer/}

Miscellaneous Symbols:

\begin{quotation}
\mathlexicon{aleph&prime&forall\cr
hbar&emptyset&exists\cr
imath&nabla&neg\cr
jmath&surd&flat\cr
ell&top&natural\cr
wp&bot&sharp\cr
Re&|&clubsuit\cr
Im&angle&diamondsuit\cr
partial&triangle&heartsuit\cr
infty&backslash&spadesuit\cr}
\end{quotation}

``Large'' Operators:

\begin{quotation}
\mathlexicon{sum&bigcap&bigodot\cr
prod&bigcup&bigotimes\cr
coprod&bigsqcup&bigoplus\cr
int&bigvee&biguplus\cr
oint&bigwedge&\omit\hfil\cr}
\end{quotation}

Binary Operations:

\begin{quotation}
\mathlexicon{pm&cap&vee\cr
mp&cup&wedge\cr
setminus&uplus&oplus\cr
cdot&sqcap&ominus\cr
times&sqcup&otimes\cr
ast&triangleleft&oslash\cr
star&triangleright&odot\cr
diamond&wr&dagger\cr
circ&bigcirc&ddagger\cr
bullet&bigtriangleup&amalg\cr
div&bigtriangledown&\omit\hfil\cr}
\end{quotation}

Relations:

\begin{quotation}
\mathlexicon{leq&geq&equiv\cr
prec&succ&sim\cr
preceq&succeq&simeq\cr
ll&gg&asymp\cr
subset&supset&approx\cr
subseteq&supseteq&cong\cr
sqsubseteq&sqsupseteq&bowtie\cr
in&ni&propto\cr
vdash&dashv&models\cr
smile&mid&doteq\cr
frown&parallel&perp\cr}
\end{quotation}

Negated Relations:

\begin{quotation}
\def\negdisplayandname#1{\rlap{$\displaystyle\not\csname #1\endcsname$}%
\qquad \texttt{\char92 not\char92 #1}}
$$\vcenter{\halign{\negdisplayandname{#}\hfil&&\qquad
        \negdisplayandname{#}\hfil\cr
\omit\rlap{$\not<$}\qquad\texttt{\char92 not<}\hfil&\omit
\qquad\rlap{$\not>$}\qquad\texttt{\char92 not>}\hfil&\omit
\qquad\rlap{$\not=$}\qquad\texttt{\char92 not=}\hfil\cr
leq&geq&equiv\cr
prec&succ&sim\cr
preceq&succeq&simeq\cr
subset&supset&approx\cr
subseteq&supseteq&cong\cr
sqsubseteq&sqsupseteq&asymp\cr}}$$
\end{quotation}

Arrows:

\begin{quotation}
\mathlexicon{%
leftarrow&rightarrow\cr
longleftarrow&longrightarrow\cr
Leftarrow&Rightarrow\cr
Longleftarrow&Longrightarrow\cr
leftrightarrow&Leftrightarrow\cr
longleftrightarrow&Longleftrightarrow\cr
hookleftarrow&hookrightarrow\cr
leftharpoonup&rightharpoonup\cr
leftharpoondown&rightharpoondown\cr
uparrow&downarrow\cr
Uparrow&Downarrow\cr
updownarrow&Updownarrow\cr
nearrow&nwarrow\cr
searrow&swarrow\cr
mapsto&longmapsto\cr
rightleftharpoons&\omit\hfil\cr}
\end{quotation}

Openings:

\begin{quotation}
\mathlexicon{lbrack&lfloor&lceil\cr
lbrace&langle&\omit\hfil\cr}
\end{quotation}

Closings:

\begin{quotation}
\mathlexicon{rbrack&rfloor&rceil\cr
rbrace&rangle&\omit\hfil\cr}
\end{quotation}

Alternative Names:

\begin{quotation}
\def\widedisplayandname#1{\rlap{$\displaystyle\csname #1\endcsname$}%
                      \qquad\qquad \texttt{\char92 #1}}
$$\vcenter{\halign{\widedisplayandname{#}\hfil&\qquad
(same as \texttt{\char92 #})\hfil\cr
\omit\rlap{$\not=$}\qquad\qquad
      \texttt{\char92 ne} or \texttt{\char92 neq}\hfil&not=\cr
le&leq\cr
ge&geq\cr
\omit\rlap{$\{$}\qquad\qquad\texttt{\char92 \char123}\hfil&lbrace\cr
\omit\rlap{$\}$}\qquad\qquad\texttt{\char92 \char125}\hfil&lbrace\cr
to&rightarrow\cr
gets&leftarrow\cr
owns&ni\cr
land&wedge\cr
lor&vee\cr
lnot&neg\cr
vert&\omit\qquad (same as \texttt{|})\hfil\cr
Vert&\omit\qquad (same as \texttt{\char92 |})\hfil\cr
iff&\omit\qquad (same as \texttt{\char92 Longleftrightarrow}, but with\hfil\cr
\omit\hfil&\omit\qquad\ extra space at each end)\hfil\cr
colon&\omit\qquad (same as \texttt{:}, but with less space around it and\hfil\cr
\omit\hfil&\omit
\qquad\ less likelihood of a line break after it)\hfil\cr}}$$
\end{quotation}

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