\documentclass{amsart}[11pt]

\usepackage{AMSfonts}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}

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\begin{document}

\thispagestyle{empty}

\pagestyle{empty}

\hfill \textsc{2007 Summer REU}

\hfill \textsc{Groups, Graphs, and Geometry}

\bigskip

\centerline{\large The Fine Art of Writing in \LaTeX, Part II}

\bigskip

\noindent Unlike non-technical typesetting environments like MS
Word, \LaTeX\ is enormously powerful when it comes to writing in
``math,'' enabling us to render nice-lookin' mathematical material,
ranging from the fairly elementary: \[ f(x)=x^2 \Rightarrow F(x) =
\int x^2 dx = \frac{x^3}{3}+C, \] \noindent to the relatively deep:
\[ \displaystyle\lim_{p \rightarrow \infty, p \ {\rm prime}}
\kappa_a \Bigl( \Z_p,\bigl\{ [x,x \pm 1 \ {\rm mod} \ p],[x,x^{-1} \
{\rm mod} \ p] \ | \ x=0,1,...,p-1 \bigr\} \Bigr) = 3. \]

I'd like to ease you into using \LaTeX, but with so much math out
there, where do we start?  In this handout I've included a number of
the sorts of things you'll need to know how to do to begin writing;
for some of you, these pages might nearly suffice.  I'm giving you
two files, one consisting of the output .pdf document, all nice `n'
purty, and the other consisting of the raw \LaTeX\ commands, so you
can see the code that made the file look the way it does.  By
reading them in parallel, you should be able to get a sense as to
how \LaTeX\ works much of its mathematical magic.

Let's get things rolling with some...

\medskip

\begin{enumerate}\addtolength{\itemsep}{0.5\baselineskip}

\item \textbf{Special letters.} First note that there is a
difference between letters in ``text mode'' and the same letters in
``math mode,'' the environment lying between ``\$'' pairs or
``$\backslash [,\backslash ]$'' in your \LaTeX\ code: it's a matter
of a,b,c,..., versus $a,b,c,...$.  If you're writing a math formula,
you're definitely going to want all of your formula to appear in
math mode: ``$f(x)=x^2$ is a function'' is correct, whereas
``f(x)$=x^2$ is a function'' is not.  (Side note: you might take a
look at how quotation marks are rendered!  What happens if you just
type quotes as you'd ordinarily do it?)

Of course, there are Greek letters, too, constructed as you would
probably expect them to be: $\alpha, \beta, \gamma$,..., and
corresponding capitals: $\Gamma,\Delta,\Theta...$.  (Some capital
letters are identical to their Roman counterparts, and therefore
have no special symbol.)

There are also ``calligraphic'' characters: ${\mathcal A,B,C,...}$.
Certain flavors of \LaTeX\ even have built-in Gothic fonts, and
other crazy stuff you probably won't need right now.  In normal text
mode, you may want to use \textit{italic} font (useful for
\textit{definitions}) and \textbf{boldface} font, so should note how
these fonts are obtained.

\item \textbf{Special characters.} You've got some pretty standard
stuff: \[ \infty,
+,-,\pm,\times,\div,=,\neq,\leq,\geq,<,>,\sim,\equiv,\cong, \]
\noindent and so forth.  If you want to ``negate'' one of the
relations listed above, the command $\backslash$\texttt{not} comes
in handy: \[ \not\leq, \not\geq, \not<, \not>, \not\sim, \not\equiv,
\not\cong. \]

There are also more heavy-duty operators: \[ \circ, \cdot, \star,
\bullet, \oplus, \ominus, \otimes, \odot, \heartsuit, \diamondsuit,
\clubsuit, \spadesuit. \]

You've got all kinds of arrows, useful for indicating where
functions go, and for illustrating implication and equivalence: \[
\rightarrow, \longrightarrow, \leftarrow, \longrightarrow,
\Rightarrow, \Leftarrow, \Leftrightarrow, \uparrow, \downarrow,
\nearrow, \searrow, \nwarrow, \swarrow. \]

Oh yeah, here are some special sets we all know and love: \[ \N, \Z,
\Q, \R, \ {\rm and} \ \C.\] \noindent Please note that in some
flavors of \LaTeX\ you have to load these special ``blackboard
bold'' characters in, much as I've done in the \textit{preamble} of
this document.  You've also got your other set theory basics: \[
\cup, \cap, \bigcup, \bigcap, \vee, \wedge, \subset, \subseteq,
\supset, \supseteq, \in, \emptyset, \setminus, \] \noindent where
this last is set difference: \[ A \setminus B = \{x \in A \ | \ x
\not\in B \}. \]

There are hundreds of other interesting characters, but this last
example leads us naturally to the next topic...

\item \textbf{Grouping symbols.} With \LaTeX\ you've got a veritable
plethora (is there ever any other kind of plethora?) of grouping
characters at your fingertips: \[ \Biggl( \Bigl( \bigl( ( \ ) \bigr)
\Bigr) \Biggr), \Biggl[ \Bigl[ \bigl[ [ \ ] \bigr] \Bigr] \Biggr],
\Biggl\{ \Bigl\{ \bigl\{ \{ \ \} \bigr\} \Bigr\} \Biggr\}. \]

Note that you can control the size of the brackets and parentheses
you want, but oftentimes \LaTeX\ is smart enough to do that for you,
with commands like ``$\backslash$\texttt{left}('' and
``$\backslash$\texttt{right})'': \[ f(x) =
\left(\displaystyle\sum_{i=0}^{\infty} \frac{x^i}{i!} \right)^2. \]

Grouping characters (generally called \textit{delimiters}) also
appear in constructing matrices: \[ \left[ \begin{array}{ccc} 1 & x
& \alpha \\ -1 & \frac{1}{\beta} & 0 \\ \gamma & \alpha^2 & 2 \\
\end{array} \right] \left[ \begin{array}{c} a \\ b \\ c \\
\end{array} \right] = \cdots . \]

The ``$\backslash$\texttt{begin\{array\}}'' and
``$\backslash$\texttt{end\{array\}}'' commands are ways of defining
an \textit{environment} in \LaTeX; here we see the ``array''
environment at work.  There are also ``figure'' environments, and
``equation'' environments.  (``Math mode'' is really just an
environment, too.)  I'll have more to say about environments at the
end of this handout.  By the way, you could use the array
environment to construct a nice-looking table of values without the
delimiters, too: \[ \begin{array}{ccc} 1 & x & \alpha \\ -1 &
\frac{1}{\beta} & 0 \\ \gamma & \alpha^2 & 2 \\ \end{array}. \]

\item \textbf{Subscripts and superscripts.} As you've probably
noticed above, superscripts (like those in expressing powers) are
implemented using ``\texttt{\^}'': \[ x^2, x^3, x^4, ..., 15^3,
3^{15}, \heartsuit^{\spadesuit}. \] \noindent Notice the reason we
need to include curly braces around the ``15'' in the penultimate
example above: if we just type in ``\$3\texttt{\^}15\$'', what do we
get?

If you need to use double superscripts (sometimes it happens, no?),
you'll need to group things the way you want them: $2^{3^4}$ is all
right, but typing ``\$2\texttt{\^}3\texttt{\^}4\$'' will probably
give you an error.

The same is true for subscripts; here you use the character
``\texttt{\_}'' instead of ``\texttt{\^}'': \[ x_1, x_2, x_3, ...,
f_x, \clubsuit_{\diamondsuit}, y_{x^7}, {y_x}^7. \] \noindent Note
the difference between the last two; what do you get if you don't
group anything in this last example?

\item \textbf{Fractions.} One more mathematical point, and then I'll
mention a thing or two about some useful environments.  It won't be
long before you need to be able to type in quotients of mathematical
quantities.  One of the most useful \LaTeX\ commands is
$\backslash$\texttt{frac}: \[ \frac{1}{2}, \frac{2}{3},
\frac{2}{(4-x)^2}, \frac{2}{\sum_{n=0}^{\infty} \frac{1}{n^2}}. \]
\noindent Note in this last example which command lives inside of
which.  It's easy to get carried away in nesting fractions: \[ \pi =
\frac{4}{1+\frac{1}{3+\frac{4}{5+\frac{9}{7+\frac{16}{9+\cdots
}}}}}. \] Ouch!  That gets hard to read!  Fortunately there's the
command ``$\backslash$\texttt{cfrac}'', designed especially for use
in such \textit{continued fractions}: \[ \pi =
\cfrac{4}{1+\cfrac{1}{3+\cfrac{4}{5+\cfrac{9}{7+\cfrac{16}{9+\cdots
}}}}}. \]  \textit{Much} nicer!

\item \textbf{Autonumbering.} All right, let's wrap up with one
advanced topic.

As you know, math papers and texts are \textit{full} of all kinds of
lemmas, theorems, propositions, corollaries, \textit{etc.}, all of
which are generally nicely numbered and cross-referenced. Obtaining
this orderly state can be a nightmare, potentially: what happens if
you've written the bulk of a long, long paper and suddenly your
colleague points out that you really need to include a short lemma
to prove a minor fact that you use in the proof of your main
theorem?  Gadzooks!  The new lemma's \textit{got} to go smack dab in
the \textit{middle} of your paper's text...this means you've got to
renumber \textit{everything} that comes after it!  Alas!  Alack!

Here, \LaTeX's automatic numbering feature comes in handy.  If
you're using a standard \LaTeX\ document class, like ``article'' or
``amsart'' (short for ``AMS article''), you've got predefined
environments for all of the above objects.  As I've done for this
document, though, you might have to do a little tweaking in the
preamble should you want the numbering to work out okay.  (If you
don't want to bother with all of this, you can simply cut `n' paste
the preamble I've written for this document and use it for your
own.)

\begin{lemma} \label{theLemma} All humans are mortal. \end{lemma}

\begin{proof} Duh, this is obvious. \end{proof}

\begin{proposition} \label{theProposition} Socrates is a human.
\end{proposition}

Our proof will begin in a moment.

\begin{proof}[Proof of Proposition~\ref{theProposition}.] Just
kidding: this is obvious, too. \end{proof}

\begin{theorem} \label{theTheorem} Socrates is mortal. \end{theorem}

\begin{proof} Combine Lemma~\ref{theLemma} and
Proposition~\ref{theProposition}. \end{proof}

\begin{corollary} \label{theCorollary} Socrates will die at 5:14
p.m. \end{corollary}

\begin{proof} This follows immediately from
Theorem~\ref{theTheorem}, after applying a result of Karp.
\end{proof}

Notice that with astute use of the commands
``$\backslash$\texttt{label}'' and ``$\backslash$\texttt{ref}'' we
are able to refer to our numbered results by keywords that have
inherent meaning for us, rather than the numbers that will later
appear in the finished text.  Thus, if we've got to update things by
including a new lemma, the numbers are automatically updated!

The numbering is typically done by ``section'' inside of an article.
Since I haven't formally declared any sections yet, \LaTeX\ is
assuming we're in Section~0.  Let's change that and see what
happens:

\section{This is a new section}

We begin the new section with a big, important theorem:

\begin{theorem} \label{bigImportantTheorem} The moon is mostly made
of moth-eaten Madras cotton. \end{theorem}

And here's an immediate corollary:

\begin{corollary} \label{stupidCorollary} It is impossible to eat
walk on the moon with ease. \end{corollary}

See how easy?  You should definitely get in the habit of labelling
and cross-referencing your results as you type them up.  Good
writing style, that's what that is.

\end{enumerate}

Well, that's about all I'd like to say for now, although I'll soon
put together a page or two on writing up bibliographies (another
feature \LaTeX\ makes simpler by including means to implement
efficient cross-referencing). There's much more to learn about
\LaTeX, but you'll pick up much of it as you need it while writing
up your results.  Do play with it a bit, though, and as practice see
if you can \LaTeX\ up a page from one of my class note sets,
available online, or some other suitable mathematical material.

\medskip

\centerline{\large Have fun!}

\end{document}