\documentclass[11pt]{article}
\usepackage[mtbold]{mathtime}
\usepackage{eqnarrays}
\input{page}

\newcommand{\defn}[1]{\textbf{\textit{#1}}}
\newcommand{\subheading}[1]{\subsubsection*{#1}}

\begin{document}

\title{Typesetting Math, Part 2}
\author{}
\date{}

\maketitle

We continue looking at issues in typesetting math.  We won't cover
everything in Lamport.  We'll cover just the trickier topics.

\subheading{Ellipsis and matrices}

In math, we often use three dots to indicate the omitted portion of a
sequence, e.g., ``for $i = 1, 2, \ldots, n$.''  The three dots are
known as an \defn{ellipsis}.  You do \emph{not} typeset them by just
typing three periods: ``for $i = 1, 2, ..., n$'' looks bad, because
the dots are too close together.  Instead, you use the \verb`\ldots`
command: \verb`for $i = 1, 2, \ldots, n$`.

A common error is to use \verb`\ldots` for dots that should be higher
above the baseline.  (The \defn{baseline} is the imaginary line on
which the text is placed.)  This error is most frequently made when
the sequence includes operators in addition to items.  For example,
\verb`$x_1 + x_2 + \ldots + x_n$` produces \mbox{$x_1 + x_2 + \ldots +
x_n$}.  Here, the dots are too low; they should be aligned with the
middle of the plus signs.  Replacing \verb`\ldots` by \verb`\cdots`
does the trick: $x_1 + x_2 + \cdots + x_n$.

What about using \verb`\cdots` within a sequence of multiplications,
when you use \verb`\cdot` ($\cdot$) to indicate multiplication?  By
analogy to the summation example, we would typeset the expression by
replacing the plus signs by \verb`\cdot`:
\verb`$x_1 \cdot x_2 \cdot \cdots \cdot x_n$`, producing \mbox{$x_1
\cdot x_2 \cdot \cdots \cdot x_n$}.  That looks weird.  Omit the
\verb`\cdot` operators immediately adjacent to the \verb`\cdots`.
That change gives \verb`$x_1 \cdot x_2 \cdots x_n$`, producing $x_1
\cdot x_2 \cdots x_n$.

There are also vertical dots $\vdots$ produced by \verb`\vdots` and
diagonal dots $\ddots$ produced by \verb`\ddots`.  These symbols are
typically used in matrices:
\[ \left( \begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{array} \right) \ . \]
Note the use of \verb`\cdots` in the third column.

\newcommand{\matx}[2]{\left[\begin{array}{*{#1}{c}}#2\end{array}\right]}

In my work, matrices come up often enough that I've made a simple
macro to help me typeset them:
{\small
\begin{verbatim}
\newcommand{\matx}[2]{\left[\begin{array}{*{#1}{c}}#2\end{array}\right]}
\end{verbatim}
}
\noindent The first argument gives the number of columns, and the
second gives the guts of the matrix.  Because my work with matrices
usually requires me to subdivide them into blocks, I like to use
square brackets rather than parentheses around the matrices; that's
why the \verb`\matx` macro uses \verb`\left[` and \verb`\right]`.
Here's the above matrix, typeset with this command:
\[ \matx{4}{
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}
} \ . \]

\subheading{Symbols}

Most of what you need to know about symbols and delimiters appears in
Sections 3.3.2 and~3.3.4 of Lamport.

The \verb`\not` command is kind of cute.  It puts a slash through the
operator that follows it.  I think the only time I ever use it is to
show that something is not a member of a set: $x \not\in S$.  Most
other ``negation'' operators are either already given to you (e.g.,
$\neq$) or are silly (e.g., why use $\not\leq$ when you can use $>$
instead?).

Generally speaking, you should avoid using the summation-like symbols
outside of math displays.  They look pretty bad in the main text.  In
other words,
\[ \sum_{i=0}^n i = \frac{n(n+1)}{2} \]
looks good, but $\sum_{i=0}^n i = n(n+1)/2$ does not look quite as
good.  You can put the limits of the in-line summation above and below
the summation sign by using the \verb`\displaystyle` command:
\mbox{$\displaystyle\sum_{i=0}^n i = \frac{n(n+1)}{2}$}.  I do not
recommend doing so normally.  Instead, a good guideline is to use a
math display for this sort of notation.

\subheading{Log-like functions}

There are two things to say about log-like functions.

First, use them.  Don't typeset $log n$.  Put in that backslash: $\log
n$.

\makeatletter
\newcommand{\rank}{\mathop{\operator@font rank}\nolimits}
\makeatother

Second, you can make your own.  For example, to make \verb`\rank`
a log-like function, define the macro
\begin{verbatim}
\newcommand{\rank}{\mathop{\operator@font rank}\nolimits}
\end{verbatim}
Now you can typeset expressions like $\rank A = 0$.

Note that some of the log-like functions take arguments.  An example
is \verb`\pmod`: $x \equiv y \pmod{n}$.  Others can take limits, for
example \verb`\lim`:
\[ \lim_{n \rightarrow \infty} x = 0 \ . \]
You get a similar behavior to summations when you use \verb`\lim` in
line: $\lim_{n \rightarrow \infty} x = 0$.

\subheading{Digressions}

There are a couple of things about the modulus functions that are
worth mentioning.  First, there are two of them, and they are for
different purposes.  \verb`\bmod` is a binary operator: $b$ is a
divisor of~$a$ if $a \bmod b = 0$.  Mathematically, this operator
takes two operands and produces a result.  In typesetting, the correct
amount of space is left on either side of the operator.  \verb`\pmod`
is \emph{not} an operator; it states a relation.  When we write $x
\equiv y \pmod{n}$, we are stating that if we have a
congruence-modulo-$n$ relation---let's call it $C$---then $(x, y) \in
C$.

\makeatletter
\def\clrspmod#1{\allowbreak\mkern12mu({\operator@font mod}\,\,#1)}
\def\imod#1{\allowbreak\mkern10mu({\operator@font mod}\,\,#1)}
\makeatother

Now, I find that the space left before the left paren when you use
\verb`\pmod` is a bit excessive for in-line use.  It's OK for
displays, but a bit too much in line.  For CLRS, I defined a related
macro, \verb`\imod`, that leaves a little less space.  Here are the
definitions of \verb`\pmod` and \verb`\imod` from the book files:
\begin{verbatim}
\def\pmod#1{\allowbreak\mkern12mu({\operator@font mod}\,\,#1)}
\def\imod#1{\allowbreak\mkern10mu({\operator@font mod}\,\,#1)}
\end{verbatim}
(These definitions use the \TeX{}-style way of defining macros.)  In
fact, even our definition of \verb`\pmod` differs from the stock
\LaTeX2e{} definition; the stock definition uses 18mu, whatever that
is.  Here's how they all differ:
\begin{center}
\begin{tabular}{ll}
Stock \LaTeX2e: & $x \equiv y \pmod{n}$ \\
CLRS \verb`\pmod`: & $x \equiv y \clrspmod{n}$ \\
CLRS \verb`\imod`: & $x \equiv y \imod{n}$
\end{tabular}
\end{center}

Now for an unrelated digression.\footnote{Does that make it a
digression from the digression?  Do two digressions cancel each other
out, like South Northfield, Vermont being just Field?}  Did you notice
how when I defined macros above, the definitions were bracketed by
\verb`\makeatletter` and \verb`\makeatother`?  The problem is that @
means something to \TeX{} some of the time.  If I were better at
\TeX{}, I could give you the particulars.  Anyway, if you're defining
a macro and it contains @, then you cannot go wrong by putting
\verb`\makeatletter` before the macro definition and
\verb`\makeatother` afterward.

\subheading{Arrays (and Tabular)}

If you look at Section~3.3.3 of Lamport, you might be under the
impression that arrays are a bit limited.  Not so.  The binding of my
Lamport is cracked at Section~C.10.  There you can see all about
arrays.  The array and tabular environments are so similar that we can
talk about them pretty much in one gulp.

Your best bet is to check out the big example on page~204.  Note in
particular the \verb`\multicolumn`, \verb`@`, \verb`p`, \verb`*`,
\verb`\hline`, and \verb`\cline` options.  A common phenomenon is,
armed with the awesome power of vertical and horizontal lines, to get
line-happy:
\begin{center}
\begin{tabular}{||l|l||}
\hline\hline
Year & Election winner \\
\hline\hline
1980 & Reagan \\
\hline
1984 & Reagan \\
\hline
1988 & G. W. H. Bush \\
\hline
1992 & Clinton \\
\hline
1996 & Clinton \\
\hline
2000 & G. W. Bush \\
\hline\hline
\end{tabular}
\end{center}
That is just too many lines.  You can make the same point with fewer:
\begin{center}
\begin{tabular}{ll}
\hline
Year & Election winner \\
\hline
1980 & Reagan \\
1984 & Reagan \\
1988 & G. W. H. Bush \\
1992 & Clinton \\
1996 & Clinton \\
2000 & G. W. Bush \\
\hline
\end{tabular}
\end{center}
And even the top and bottom lines here are only to offset the tabular
from the rest of the text; you could just as easily lose them, too.

Sometimes, you want to change the interline or intercolumn spacing in
an array or table.  For interline, add an optional extra space in the
\verb`\\` that ends each line.  For example, here's a $3 \times 3$
array typeset normally:
\[ \begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i
\end{array} \ . \]
To add more space between lines, I suggest expressing it in terms of
ex's.  An \defn{ex} is the height of a lowercase letter 'x'.  Here's
the same array with 2~ex of extra space between lines 1 and 2:
\[ \begin{array}{ccc}
a & b & c \\[2ex]
d & e & f \\
g & h & i
\end{array} \ . \]
You can use negative space to shrink the distance between lines 1 and
2:
\[ \begin{array}{ccc}
a & b & c \\[-1ex]
d & e & f \\
g & h & i
\end{array} \ . \]

To change the interline space uniformly, change \verb`\arraystretch`.
Note that \verb`\arraystretch` is actually a command, so you have to
use \verb`\renewcommand` to change it.  Here's the same array with
the interline space 1.5 times the default:
\renewcommand{\arraystretch}{1.5}
\[ \begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i
\end{array} \ . \]
\renewcommand{\arraystretch}{1}

You can use the \verb`@` specification to change the spacing between
columns.  Here's the array with extra space between the first two
columns:
\[ \begin{array}{c@{\hspace{2em}}cc}
a & b & c \\
d & e & f \\
g & h & i
\end{array} \ . \]
To change the intercolumn space uniformly, set \verb`\arraycolsep` or
\verb`\tabcolsep`.  These are lengths, so you use the \verb`\setlength`
command.  I set up macros to save and set these:
{\small
\begin{verbatim}
\newlength{\savearraycolsep}
\newcommand{\narrowarray}[1]{\setlength{\savearraycolsep}{\arraycolsep}%
    \setlength{\arraycolsep}{#1\arraycolsep}}
\newcommand{\normalarray}{\setlength{\arraycolsep}{\savearraycolsep}}
\end{verbatim}
}
\newlength{\savearraycolsep}
\newcommand{\narrowarray}[1]{\setlength{\savearraycolsep}{\arraycolsep}%
    \setlength{\arraycolsep}{#1\arraycolsep}}
\newcommand{\normalarray}{\setlength{\arraycolsep}{\savearraycolsep}}
Here's the array with all intercolumn spacing 0.5 times the default:
\narrowarray{0.5}
\[ \begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i
\end{array} \ . \]
\normalarray
If you use a factor greater than 1, you stretch the intercolumn space.
I usually am trying to reduce the intercolumn space, and so I think of
it as narrowing the array.

Another great use for arrays is for the ``cases'' notation used in
math.  There, you use a left delimiter that's a curly brace and a null
right delimiter:
\[ n! = \left\{ \begin{array}{ll}
1 & \mbox{if $n = 0$} \ , \\
n \cdot (n-1)! & \mbox{if $n > 0$} \ .
\end{array} \right. \]

There are a couple of things to note here.  First, there's a comma
after each case except the last (i.e., after ``if $n=0$'').  Second,
this comma has a space before it.  Third, since this particular
displayed equation ends a sentence, there's a period after the last
case (i.e., after ``if $n > 0$'').  Fourth, there's a space before
that period.  Do \emph{not} put the period after the null right
delimiter:
\[ n! = \left\{ \begin{array}{ll}
1 & \mbox{if $n = 0$} \ , \\
n \cdot (n-1)! & \mbox{if $n > 0$}
\end{array} \right. \ . \]
Although you might think that logically it makes sense---after all,
the displayed equation ends the sentence---it just looks bizarre.

\renewcommand{\cases}[1]{\left\{ \begin{array}{ll}#1\end{array}\right.}
\newcommand{\cif}[1]{\mbox{if $#1$}}

I have macros for doing such cases:
{\small
\begin{verbatim}
\renewcommand{\cases}[1]{\left\{ \begin{array}{ll}#1\end{array}\right.}
\newcommand{\cif}[1]{\mbox{if $#1$}}
\end{verbatim}
}
\noindent I'd use it as
\[ n! = \cases{
1 & \cif{n = 0} \ , \\
n \cdot (n-1)! & \cif{n > 0} \ . } \]

\subheading{One-line, displayed formulas}

Use either \verb`\[` \ldots \verb`\]` or the equation environment.
The former does not number equations, and the latter does.  Some folks
use \verb`$$` \ldots \verb`$$` or the displaymath environment instead
of \verb`\[` \ldots \verb`\]`; they're all equivalent, but I like
\verb`\[` \ldots \verb`\]`.

The equation environment gives you numbered equations:
\begin{equation}
n! = \left\{ \begin{array}{ll}
1 & \mbox{if $n = 0$} \ , \\
n \cdot (n-1)! & \mbox{if $n > 0$} \ .
\end{array} \right.
\label{eq:n-factorial}
\end{equation}
Now I can refer to equation~(\ref{eq:n-factorial}).

A few things to note here.  First, if you're not going to refer to an
equation, then don't number it, unless you're forced to by peer
pressure.  By that, I mean that if all the equations around it are
numbered, then it might look downright weird not to number this one.
But that usually applies to eqnarray environments, not onesy displays.

Second, \emph{use symbolic labels}.  Don't refer to equation~(1).  As
you can see, you can put all sorts of symbols in the label.  It need
not begin with ``eq:''.  That's just a nice convention that we adopted
in CLR, and I have continued using it to this day.

\newcommand{\eqnref}[1]{(\ref{#1})}

Third, when you refer to the equation, put parentheses around it.  It
is not equation~\ref{eq:n-factorial}.  In fact, I have a macro
\begin{verbatim}
\newcommand{\eqnref}[1]{(\ref{#1})}
\end{verbatim}
so now I can refer to equation~\eqnref{eq:n-factorial}.

Fourth, I like to call it equation~\eqnref{eq:n-factorial} rather than
Equation~\eqnref{eq:n-factorial}.  When referring to chapters,
sections, and parts, the name ``Chapter'', ``Section'', or ``Part''
should be capitalized.  But when referring to non-sectional units,
like equations, pages, or lines, do not capitalize.  You wouldn't
write, ``See Page~34'' or ``see Line~7''.

Fifth, remember to run the latex command twice when there are symbolic
labels anywhere (not just in numbered equations).  The first run puts
the definition of the labels in the .aux file, and the second run
resolves these definitions.

\subheading{eqnarrays and the like}

When you have multiline formulas, or you have multiple formulas that
you want to line up nicely, use the eqnarray environment.  If you use
eqnarray, every line will be numbered as an equation, unless you put
\verb`\nonumber` on the line.  Put labels on the numbered lines so
that you can refer to them.  You can suppress numbering of all lines
by using the eqnarray* environment instead.

The eqnarray environment gives you three columns (plus equation
numbers if you use them).  The left column is flush right, the center
column is centered, and the right column is flush left.  Here's an
example:
\begin{eqnarray}
T(n)
 & \leq & 2T(n/2) + \Theta(n) \label{ineq:T} \\
 & = & O(n \lg n) \ . \nonumber
\end{eqnarray}
Here, we see a few things.  First, you don't have to use anything in
particular in the center column.  Second, the first line is
equation~\eqnref{ineq:T}, but the second line is unnumbered.  Third,
I've ended the eqnarray with a period.  Fourth, no \verb`\\` goes
after the last line.

To see that the left column is flush right, here is an eqnarray* that
holds multiple equations:
\begin{eqnarray*}
\sum_{k=1}^n (c a_k + b_k) & = & c \sum_{k=1}^n a_k + \sum_{k=1}^n b_k \ , \\
\sum_{k=0}^n k^2 & = & \frac{n(n+1)(2n+1)}{6} \ .
\end{eqnarray*}
The left column looks weird that way, so I have created a package
called eqnarrays (notice that the top of this file has the line the
\verb`\usepackage{eqnarrays}`) that provides three variants on
eqnarray.  To make the left column be flush left, use eqnarrayl:
\begin{eqnarrayl*}
\sum_{k=1}^n (c a_k + b_k) & = & c \sum_{k=1}^n a_k + \sum_{k=1}^n b_k \ , \\
\sum_{k=0}^n k^2 & = & \frac{n(n+1)(2n+1)}{6} \ .
\end{eqnarrayl*}

The eqnarrayr environment makes the right column be flush right.  We
found it useful for writing systems of difference constraints:
\begin{eqnarrayr}
x_1 - x_2 & \leq & 0 \ \mbox{,} \label{ineq:d1} \\
x_1 - x_5 & \leq & -1 \ \mbox{.} \label{ineq:d2}
\end{eqnarrayr}
If you leave the \verb`\mbox` commands out, the spacing isn't quite
right.  I don't recall why, nor do I recall how we discovered that
using the \verb`\mbox` commands fixed the problem.  Notice also that
by using eqnarrayr rather than eqnarrayr*, we get numbers for
inequalities \eqnref{ineq:d1} and~\eqnref{ineq:d2}.  All the
environments in the eqnarrays package come in starred and unstarred
versions.

\makeatletter
\newcommand{\prob}[1]{\Pr\left\{ #1 \right\}}
\newcommand{\expect}[1]{{\operator@font E}\left[ #1 \right]}
\makeatother

The most useful environment in the eqnarrays package is
eqnarrayreason.  This environment adds an extra column on the right.
This column is flush left and \emph{not} in math mode.  It's great for
adding reasons why a line is valid.  For example, here's how I would
typeset a proof using indicator random variables for the hat-check
problem.  (Here, $X$ is a random variable for how many customers get
back their own hat, and $X_i$ is an indicator random variable for the
event that customer~$i$ gets back his own hat.)
\begin{eqnarrayreason*}
\expect{X}
 & = & \expect{\sum_{i=1}^n X_i} & (definition of $X$) \\
 & = & \sum_{i=1}^n \expect{X_i} & (linearity of expectation) \\
 & = & \sum_{i=1}^n \prob{\mbox{customer $i$ gets his own hat}}
		& (CLRS Lemma~5.1) \\
 & = & \sum_{i=1}^n 1/n & \\
 & = & 1 \ . &
\end{eqnarrayreason*}
Note that even when there was no reason, I put in the \verb`&`.  I
don't think it's absolutely necessary, but it seems like good form.
Note also that the period ending the equation goes after the last text
on its line, which is not necessarily the rightmost column.

Finally, a couple of things that Lamport mentions in Section~3.3.5,
but are good to point out.  First, if you continue a line and the
continuation line starts with a $+$ or $-$, it could look wrong:
\begin{eqnarray*}
X
 & = & X_1 + X_2 + X_3 + X_4 + X_5 + X_6 + X_7 \\
 &   & + X_8 + X_9 + X_{10} \ .
\end{eqnarray*} 
The $+$ before $X_8$ is assumed to be a unary operator, and so it is
closer to~$X_8$ than it would be if were a binary operator.  You can
fool \LaTeX2e{} into thinking that it's a binary operator by giving it
an empty \verb`\mbox` as a left operand:
\begin{eqnarray*}
X
 & = & X_1 + X_2 + X_3 + X_4 + X_5 + X_6 + X_7 \\
 &   & \mbox{} + X_8 + X_9 + X_{10} \ .
\end{eqnarray*} 

Sometimes you need to let the columns overlap.  This situation often
occurs when the first and some succeeding line are both long.  For
example,
\begin{eqnarray*}
O(D(n) + t(H)) + ((D(n)+1) + 2 \, m(H)) - (t(H) + 2 \, m(H)) + \Theta(n \lg n)
& = & O(D(n)) + O(t(H))-t(H) + \Theta(n \lg n) \\
& = & O(D(n))  + \Theta(n \lg n) \ ,
\end{eqnarray*}
We solve this problem by putting the left column of the first line on
its own line, surrounding it by the command \verb`\lefteqn`:
\begin{eqnarray*}
\lefteqn{O(D(n) + t(H)) + ((D(n)+1) + 2 \, m(H)) - (t(H) + 2 \, m(H))
	+ \Theta(n \lg n)} \\
& = & O(D(n)) + O(t(H))-t(H) + \Theta(n \lg n) \\
& = & O(D(n))  + \Theta(n \lg n) \ ,
\end{eqnarray*}
This still looks bad.  We correct the problem by adding some
horizontal space after the argument of \verb`\lefteqn` but before
\verb`\\`:
\begin{eqnarray*}
\lefteqn{O(D(n) + t(H)) + ((D(n)+1) + 2 \, m(H)) - (t(H) + 2 \, m(H))
	+ \Theta(n \lg n)} \hspace*{16em} \\
& = & O(D(n)) + O(t(H))-t(H) + \Theta(n \lg n) \\
& = & O(D(n))  + \Theta(n \lg n) \ ,
\end{eqnarray*}
Unfortunately, you have to use trial and error to determine the right
amount of space to add.

\end{document}