\input{template}
\input{macros}
\begin{document}
\lecture{0}{Manual for Scribes}{x}


You will need to use the latex document processing system.
This document is not a latex manual, but it does show you some basic tricks.
Several good latex references exist on the web, and your grad colleagues
should have books on latex. (My favorite is the {\em Latex Companion}.)

A file called {\tt template.tex}
 already exists in this directory that wil take care
of formatting. A file called {\tt macros.tex} contains shorthands for 
math commands as well as for frequently used formatting tricks.
%Double-check with me before editing either of these. Also, 
Send me email if you have latex questions. 

We are using the {\tt amsmath} package for typesetting, which does a few
things differently from standard latex.

Some tricks and tips appear below.
Look in the source file {\tt howto.tex} to understand how
to do these tricks.

\section{General Formatting Tips}

\begin{enumerate}

\item Theorems, lemmas, corollaries, proofs, definitions, examples, exercises,
remarks, etc. are typeset
inside special environments. (The environment names are 
{\tt Thm, Lem, Cor, proof, Def, Exa, Ex, Rem} respectively.) 
Here is how you write a theorem.

\begin{Thm} \label{einsteinthm}
If $E$ denote energy, $m$ denotes mass, and $c$ denotes the speed of light,
then
\begin{equation}
E = mc^2
\end{equation}
\end{Thm}

\item File {\tt macros.tex} also contains macros to typeset the following
(not an exhaustive list): set notation (e.g. $\set{1, 2, 3, 4}$),
cardinality of a set (e.g. $\card{\set{1,2,3}}$), Real and natural numbers
($\rea, \nat$ respectively), probabilities (e.g. $\pr[\text{coin comes up head}] =1/2$), $\var[X] = \av[X^2] - \av[X]^2$). 
If you want to add a new macro to {\tt macros.tex}, 
{\em please send me email}. Do not edit 
{\tt macros.tex}; I want all students to use the same version.

%\item 

\item There are macros for writing pseudocode. Look in the
source file to see how to generate the following piece of
pseudocode.

\begin{program}
input: $G = (V,E)$, $s$, $t$ \\
output: \textsc{yes} if it discovers that $t$ is  reachable from $s$, 
and \textsc{no} otherwise \\
\\
        \>     guess the distance $k$ between $s$ and $t$\\
        \>     $p$ := $s$\\
    	\>     \FOR \= $i$ := 1 to $k$ \DO \\
        \>     \> non-deterministically pick a neighbor $q$ of $p$ \\
        \>     \> $p$ := $q$ \\
        \>     \IF\ $p=t$ \THEN\ \ACCEPT\\ 
        \>     \> \ELSE \REJECT 
\\
\end{program}


\item You can include figures by using the {\tt ffigure} command.
You first create a figure using {\tt xfig} (on Unix)
or {\tt Adobe Illustrator},
save it as a postscript file (the subscript should be {\tt .eps}) in the same
directory as the latex files. Lets say this figure is {\tt 12sets.eps}. 
Look
in the source file to see how we can include this file and generate
Figure~\ref{figure:sets}.

\ffigure{12sets}{1in}{The bigger $S'$ is, the more likely $h(S')$ will 
hit a given point in $\B^m$.}{figure:sets}




If you get any error messages while including figures, check that the .eps file
begins with \%! and if not, edit out the first line or two of junk.

You will probably get better results if you draw the figures (in {\tt xfig} or
another program) in landscape orientation. Make it fill the entire page,
since you can resize it when using the {\tt ffigure} command.

\end{enumerate}

\section{General Math Formatting Tips}

\begin{enumerate}
\item Use {\tt align} to typeset a series of contiguous equations
such as those occuring in a long derivation.
(Do not use the old {\tt eqnarray} command; it uses nonstandard 
typographical conventions.) In the source file you will see that 
an \& tells the program which symbol to align on.
\begin{align}
E & =  mc^2\\
E+ H + G & = t 
\end{align}


Use the  {\tt equation} command for single equations.
\begin{equation}
E  =  mc^2
\end{equation}

To mix text into math formulae, use the {\tt text} command.
\begin{equation}
E  =  mc^2 \qquad \text{(Einstein)}
\end{equation}

While  presenting a sequence of calculations 
(using the {\tt align} command) we sometimes
need to say something briefly in the middle, say to explain a step.
We can do this with
the {\tt intertext} command.
\begin{align}
A+ B+ C + D + E & =  R+ S \\
\intertext{{\em intertext:} which can be upperbounded using the inductive hypothesis by}
& \leq Q + N
\end{align}

\item If no alignment is needed, we use {\tt gather} to make the group of equations look
neat.

\begin{gather}
a+b = b+a \\
(a+b)\cdot (a-b) = a^2 -b^2
\end{gather}

\item There is also {\tt alignat} for {\tt align} type structures side by side.

\begin{alignat}{2}
L_1 & = R_1 & \qquad L_2 & = R_2 \\
L_3 & =  R_3 & \qquad L_4 &= R_4
\end{alignat}

\item Equations that do not fit into a line are typeset using the {\tt split} environment,
which allows alignment between lines using \& as usual.
\begin{equation}
\begin{split}
(a+ b)^3 - (c+d)^3 - (a+ d)^3  & = a^3 + b^3 +3ab(a+b) + c^3 +d^3 +3cd(c+d) \\
& \quad - (a^3 + d^3 +3ad(a+d))
\end{split}
\end{equation}
\item To refer later to an equation, you need to label it with a {\tt label} command.
The command {\tt notag} will make the equation unnumbered. The command {\tt tag} will
replace the equation number with some other designated symbol.

\begin{align}
x^2 -y^2 &= (x-y)\cdot(x+y) \label{eq:r1} \\
x^3 -y^3 & = (x-y)(x^2 +xy + y^2). \tag{$*$} \label{mystar}\\
\intertext{Using \eqref{eq:r1} and \eqref{mystar} we obtain}
a + b &= d  \\
\intertext{Now we give an unnumbered equation; note that the numbering resumes below}
d+ e & = f \notag
\end{align}



\item You can typeset equations involving ``case'' situations with the {\tt cases}
environment.
\begin{equation}
\delta_{i,j} = \begin{cases}
	1 & \text{if $i =j$} \\
	-1 & \text{if $i<j$} \\
	0 & \text{otherwise}
	\end{cases}
\end{equation}
\item The {\tt matrix} environment produces matrices. Below, we show the 
matrices produced using {\tt matrix, pmatrix, bmatrix, vmatrix, Vmatrix} respectively.
\begin{gather*}
\begin{matrix} a & b \\ c & d \end{matrix} \quad
\begin{pmatrix} a & b \\ c & d \end{pmatrix} \quad 
\begin{bmatrix} a & b \\ c & d \end{bmatrix} \quad
\begin{vmatrix} a & b \\ c & d \end{vmatrix} \quad
\begin{Vmatrix} a & b \\ c & d \end{Vmatrix}
\end{gather*}
This environment can handle matrices with up to $10$ columns.
To produce a matrix like 
$\left( \begin{smallmatrix} a & b \\ c& d \end{smallmatrix} \right)$ 
inside a text
paragraph, use the {\tt smallmatrix} environment and enclose with the appropriate
parentheses. Try not to do this in the last line of the paragraph since that looks
untidy, as here $\left( \begin{smallmatrix} a & b \\ c& d \end{smallmatrix} \right)$.

\item To use bold-faced letters inside math equations, use {\tt mathbf} command.
To get bold numbers, greek symbols etc., use the {\tt boldsymbol}
command.
\begin{equation}
\mathbf{A}_{\boldsymbol{\infty}} = \boldsymbol{\alpha}+ d
\end{equation}

\item Function and operator symbols (in addition to standard ones like {\tt sin, cos, log})
can be defined using the {\tt operatorname} command. 

\item Typeset modular arithmetic using the {\tt pmod} command.
\begin{equation}
x \equiv y + z \pmod{n}.
\end{equation}
\item You may be familiar with the {\tt frac} command in latex. In practice one needs
to distinguish typographically between fractions inside paragraph (such as
$\tfrac{x+y}{z^2 +5}$) and  ones that occur in displayed equations, such as
\begin{equation}
\dfrac{x+ y}{z^2 + 5} = 8
\end{equation}
The former use the {\tt tfrac} command and the latter the {\tt dfrac} command.
One can also specify the parentheses around displayed fractions with 
{\tt fracp} and {\tt fracb} commands. The advantage of
these is that the size of the parentheses
will change automatically with the size of the display; compare $\fracp{x+y}{z^2 +5}$with 
$$
\fracp{\fracp{x}{u}+y}{z^2 +5} = 5 = \fracb{x/u+y}{z^2 + 5}
$$
%({\em This is my hack, but apparently there used to be  standard commands like this.})

\item Random examples of how to format some math expressions: $\binom{n}{k}$,
$\exp(-x^2)$, $x \wedge y \vee \neg z$, $f \from \nat \to \rea$.

\section{Note on labels}

{\bf IMPORTANT: When labelling equations, theorems, figures etc., use a prefix
that includes the lecture number. For example} {\tt lec5:eqn10} or {lec5:fig6}.
{\bf Similarly, call your figures something like} {\tt lec5fig1.eps} {\bf instead of just}
{\tt fig1.eps}. {\bf This will prevent naming conflicts when I try to latex the lecture notes all
together.}
\end{enumerate}
\end{document}