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+\def\Cite#1{[\cite{#1}]}
+%%%
+%%% An article for Baskerville, intended to be the third of n parts
+%%%
+\title[Maths in \LaTeX: Part~3]{Maths in \LaTeX: Part~3, Different Sorts of Mathematical Object}
+\author[R. A. Bailey]{R.~A.~Bailey\\
+Queen Mary and Westfield College, University of London}
+\newcommand{\writer}[1]{{#1}:}
+\newcommand{\book}[1]{{\it #1},}
+\newcommand{\publish}[2]{{\rm #1, #2,}}
+\newcommand{\byear}[1]{{\rm (#1).}}
+\newtheorem{preqn}{Exercise}
+\newenvironment{qn}{\begin{preqn}\normalfont\rm}{\end{preqn}}
+%%% Dear Mr Editor, I should like the content of exercises
+%%% to come out in what all copy-editors call ROMAN, not in what
+%%% Frank Mittelbach calls roman. And this should be irrespective
+%%% of the surrounding text.
+%%%
+\newcommand{\latexword}[1]{{\normalfont\tt #1}}
+\newcommand{\lamport}{{\it \LaTeX: A Document Preparation System\/} by Leslie
+Lamport}
+\newcommand{\shortlamp}{{\it The Manual}}
+%%%
+
+\begin{Article}
+\section{Recall}
+This is the third in a sequence of tutorials on typesetting Mathematics in
+\LaTeX. The first two appeared in issues~4.4 and~4.5 of \BV. The series
+includes some things which can be found in \Cite{leslie}, but I am
+working in more things which, while straightforward and necessary for
+Mathematical work, are not in \Cite{leslie} or \Cite{newleslie}.
+In case you missed the first two
+tutorials, two warnings are now repeated.
+
+I expect you, the reader, to do some work. Every so
+often comes a group of exercises, which you are supposed to do. Use \LaTeX\ to
+typeset everything in the exercise except sentences in italics, which are
+instructions. If you are not satisfied that you can do the exercise, then tell
+me. Either write
+to me
+at
+\begin{verse}
+School of Mathematical Sciences\\
+Queen Mary and Westfield College\\
+Mile End Road\\
+London E1 4NS
+\end{verse}
+with hard copy of your input and output,
+or email me at \mbox{\tt r.a.bailey@qmw.ac.uk}
+with a copy of the
+smallest possible piece of \LaTeX\ input file that contains your
+attempt at the answer.
+In either case
+I will include a solution in the following issue of \BV: you will remain
+anonymous if you wish.
+
+A word on the controversial issue of fonts.
+Fonts in Mathematics are handled differently in \LaTeX\ 2.09,
+in NFSS, and in the new standard \LaTeX, \LaTeXe.
+Rather than compare these systems every time that I mention fonts, I
+usually limit myself to \LaTeX\ 2.09.
+When you upgrade to \LaTeXe, all these commands will still work, so long as
+you use the standard styles \latexword{article}, \latexword{report} and
+\latexword{book}. In the `Answers' section below I expand a little on the
+dangers of using the font-changing commands given in
+[\cite[Section~3.1]{newleslie}].
+
+Many of the more complicated Mathematical things in this tutorial are
+not documented in \Cite{leslie} or in \Cite{newleslie}. The
+\LaTeX\ team warns me that they feel no obligation to support commands
+that are not in \Cite{newleslie}, so there is a danger that some of
+these things may change. However, everything given here works, in
+both \LaTeX\ 2.09 and in \LaTeXe, as at January 1995.
+
+Some of the tricks described in this tutorial are at the edge of what you can
+conveniently do without using the \latexword{amstex} package. That package is
+undergoing change at the moment: I hope that by the time I reach the end of
+this sequence of tutorials the \latexword{amstex}
+ package will have stabilized enough for someone
+to write an article explaining how to use it, including giving better methods
+than I can give here.
+
+\section{Answers}
+I promised to answer all questions arising from this series of articles (as far
+as I can).
+
+\subsection{Uneven subscripts}
+In \BV~4.5 Malcolm Clark asks about uneven baselines in subscripts. He gives a
+method of ensuring that all subscripts have the same baseline. I think that
+many Mathematical writers will not require that; nonetheless, some of us are
+uncomfortable with the unevenness in a single term such as
+\[
+4z_1z_2^3
+\]
+The easy way around this is to put a dummy superscript on the $z_1$, because
+it is the superscript on the $z_2$ that is pushing the $2$ down: thus
+\begin{quote}
+\verb+4 z_1^{} z_2^3+ \qquad $4 z_1^{} z_2^3$.
+\end{quote}
+
+\subsection{Roman text in notation}
+He also muses on whether to use \verb+\textrm+ or \verb+\mathrm+ or \verb+\rm+
+in subscripts, if you are using \LaTeXe. My advice is never to use
+\verb+\textrm+ in
+Mathematical notation. In the first place, \verb+\textrm+ does {\em not\/}
+give you roman type, according to such expert references as
+\Cite{hart,chamb,chicago}, all of whom say that `roman' type is upright, as
+opposed to italic. All that \verb+\textrm+ does is give you back serifs and
+proportional spacing, if you had turned them off. Perhaps he meant
+\verb+\textup+. But, secondly, I don't think that you should use {\em any\/}
+of the commands \verb+\text...+ in Mathematical notation, because their effect
+depends on the surrounding text font but notation should be independent of the
+surrounding text. For example, try the following and compare the output:
+\begin{verbatim}
+{\rm $x_{\textup{big}} + \textup{size}_3$}
+{\bf $x_{\textup{big}} + \textup{size}_3$}
+{\bf $x_{\textrm{big}} + \textrm{size}_3$}
+\end{verbatim}
+
+%{\rm $x_{\textup{big}} + \textup{size}_3$}
+%{\bf $x_{\textup{big}} + \textup{size}_3$}
+%{\bf $x_{\textrm{big}} + \textrm{size}_3$}
+
+
+Malcolm was concerned because he wanted to obey the instruction in
+\Cite{companion} to always use commands like \verb+\textit{...}+ rather than
+switches like \verb+\it+. The trouble with that instruction is
+%In fact, I disagree quite strongly with the suggestion in \Cite{companion} that
+%we should refrain from using commands like \verb+\rm+.
+that the new commands
+\verb+\text...+ all work in a relative way. In my experience of writing
+(a lot of)
+Mathematics I have {\em never\/} needed such a relative change. I always need
+to specify my fonts absolutely, so that, say, the font chosen for long names of
+variables to be analysed does not change as the surrounding text font changes.
+Of course, it is sensible to do this with a macro such as \verb+\variablename+;
+but that macro needs to call something with a syntax similar to
+\verb+\textsl{...}+ but which makes an absolute font change. I tried to
+persuade the \LaTeX\ team to include commands like \verb+\basesl{...}+,
+\verb+\basett{...}+ for such absolute changes, but I failed. Since the team
+wants to reserve the right to remove switches like \verb+\tt+ at some future
+time, this means that most of us will have to write our own macros, with our
+own idiosyncratic names, something like the following:
+\begin{verbatim}
+\DeclareTextFontCommand{\basett}%
+ {\normalfont\ttfamily}
+\end{verbatim}
+%\newcommand{\basett}[1]%
+% {{\normalfont\ttfamily #1}}
+
+
+\subsection{Spaces in subscripts}
+Malcolm also asked how to get spaces into subscripts.
+If I need to put a verbal phrase in a subscript then I use \verb+\rm+ and put
+in the interword spaces by hand.
+\begin{quote}
+\begin{tabular}{c}
+\verb+\sum_{p{\rm\ is\ prime}} \frac{1}{p}+\\[\jot]
+$\displaystyle\sum_{p{\rm\ is\ prime}} \frac{1}{p}$
+\end{tabular}
+\end{quote}
+
+\subsection{Empty set}
+Kathleen Lyle has queried the symbol I gave last time for the empty set, with
+the command \verb+\emptyset+. She points out that \Cite{companion} shows a
+different symbol given by this command, a symbol which looks like a circle with
+a diagonal line through it and which is much closer to a
+Mathematician's idea of the empty set than is~$\emptyset$. But \Cite{companion}
+also gives the command \verb+\varnothing+, available with the package
+\latexword{amssymb}, which produces the symbol~$\emptyset$.
+It appears that Knuth made a mistake in using the name \verb+\emptyset+ for the
+glyph which most of us think of as a variant form of zero. To correct this
+mistake, the AMS has redefined the command \verb+\emptyset+ to
+produce the symbol more like the empty set and given us \verb+\varnothing+ for
+the sake of those authors who really do want a zero with a line through it.
+It is a pity that \Cite{companion} does not say that its
+\verb+\emptyset+ is the AMS one rather than the Knuth one.
+
+What to do when a software author makes a mistake like this is a controversial
+question. Personally,
+much as I prefer the AMS's empty set, I deplore such redefinition of a command,
+ because it
+destroys portability of documents. Suppose that I write a document without the
+\latexword{amssymb} package and use \verb+\emptyset+. I may send this document
+to someone (perhaps the AMS itself\/) who always uses the \latexword{amssymb}
+package when compiling documents. Even though I have made no explicit calls to
+commands defined by the package, my empty sets will come out looking different.
+A topologist may be content with the change; a computer scientist may not. In
+either case the document is printed with different symbols in the two cases,
+and this really should not happen. I think that it would have been better if
+the AMS had used a different name, such as \verb+\trueemptyset+, for their empty
+set: then authors with access to the \latexword{amssymb} package could choose
+whether or not to include
+\begin{verbatim}
+\renewcommmand{\emptyset}{\trueemptyset}
+\end{verbatim}
+at the start of their files.
+
+\addtocounter{section}{2}
+\section{A Spaced-out Interlude}
+\subsection{Quads}
+Traditionally, there are certain lengths of space (depending on the type size)
+which are always used in certain places in Mathematical typesetting. The most
+useful are the {\it quad\/} space and the two-quad space. When I was a
+copy-editor I used to just put the marks for these two types of space in the
+appropriate places in the copy; I did not have to know how big they were.
+Neither do you. In displayed Maths, use \verb+\qquad+ to obtain a two-quad
+space between a formula and a short verbal condition or justification.
+\begin{verbatim}
+y \in Y \qquad\mbox{by defintion of~$Y$}
+\end{verbatim}
+If there
+are two short formulas linked in a display by a short verbal phrase (perhaps
+only one word) use \verb+\quad+ to produce a quad space on either side of the
+phrase.
+\begin{verbatim}
+A \subseteq B \quad\mbox{and}\quad A \ne B
+\end{verbatim}
+
+\subsection{Other Spaces}
+A sequence of much smaller horizontal spaces that you can insert yourself is,
+in increasing order of magnitude,
+\begin{quote}
+\verb+\,+ \quad \verb+\:+ \quad \verb+\;+ \quad \verb*+\ +
+\end{quote}
+%\begin{quote}
+%a \thinspace b \medspace c \thickspace d
+%rubbish, the last two don't exist
+%\end{quote}
+They are called {\it thin space}, {\it medium space}, {\it thick space\/} and
+{\it interword space\/} respectively; their size also depends on the current
+type size.
+The thin space is usually needed after the \verb+!+ in factorials and often
+needed after a square root.
+\begin{quote}
+\begin{tabular}{cc}
+\verb+\sqrt{3} \, a+ & $\sqrt{3} \, a$\\
+\verb+5!\,4!+ & $5!\,4!$
+\end{tabular}
+\end{quote}
+It is also used before each $dx$~term in an integral. On the other hand, in
+multiple integrals the integral signs may be too far apart, in which case the
+{\it negative\/} thin space \verb+\!+ should be inserted between them.
+
+For consistency, these adjustments should all be made via macros. For example,
+\begin{verbatim}
+\newcommand{\sqrtsp}[1]{\sqrt{#1}\,}
+\end{verbatim}
+will make \verb+\sqrtsp+ into the command for a square root with a little extra
+space, and a macro for factorials can be made similarly. For the integral signs
+you can use
+\begin{verbatim}
+\newcommand{\intt}{\int\!}
+\end{verbatim}
+or the rather different solution provided in \latexword{amstex}. A suitable
+macro for the $dx$ is
+\begin{verbatim}
+\newcommand{\diff}[1]{\, d #1}
+\end{verbatim}
+which has the added advantage that if you believe that only variables should be
+in Maths italic then the \verb+{\, d #1}+ can be changed to
+\verb+{\, {\rm d} #1}+.
+
+%These adjustments are all rather finicky, and should usually be left until the
+%document is almost complete.
+
+\subsection{Phantoms}
+The useful command \verb+\phantom+ allows you to leave a space whose horizontal
+and vertical dimensions are those of its argument. For example, if you want to
+define the notation $[\phantom{x}]$ as the least-integer function without
+specifying a dummy variable, you can type \verb+[\phantom{x}]+.
+
+All digits are the same width, so \verb+\phantom{0}+ produces a phantom digit.
+It is very useful in tables of data when all other methods of alignment fail.
+Make yourself a macro for it.
+
+There are also horizontal and vertical phantoms \verb+\hphantom+ and
+\verb+\vphantom+ respectively. Each of these measures only one dimension of its
+argument.
+
+\subsection{Horizontal Expanders}
+In the first tutorial we saw that \verb+\widehat+ and \verb+\widetilde+ expand
+as far as necessary (up to a given upper bound) to cover their arguments. The
+following commands also expand horizontally to match the arguments:
+\begin{quote}
+\begin{tabular}{cc}
+\verb+\overline+ & \verb+\underline+\\
+\verb+\overrightarrow+ & \verb+\overleftarrow+\\
+\verb+\overbrace+ & \verb+\underbrace+
+\end{tabular}
+\end{quote}
+You can use a superscript to put a label on an overbrace, and a subscript with
+an underbrace.
+\begin{verbatim}
+n\bar{y}^2 +
+\overbrace{(y_1-\bar{y})^2 + \cdots +
+(y_n-\bar{y})^2}^{\rm sum\ of\ squares}
+\end{verbatim}
+\[
+n\bar{y}^2 +
+\overbrace{(y_1 - \bar{y})^2 + \cdots + (y_n - \bar{y})^2}^{\rm sum\
+of\ squares}
+\]
+
+\section{Exercises}
+\addtocounter{preqn}{22}
+\begin{qn}
+\[(x_1 + x_2)^3 = x_1^3 + 3x_1^2x^{}_2 + 3x_1^{}x_2^{2} + x_2^3\]
+\end{qn}
+
+\begin{qn}
+\[\sum_{n\ {\rm divides}\ 10} n = 18\]
+\end{qn}
+
+\begin{qn}
+In geometry, $\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}$.
+\end{qn}
+
+\begin{qn}
+We define $P_g$ by
+\[t(vP_g) = (t^{g^{-1}})v \qquad\mbox{for $v\in {\bf R}^{\cal T}$.}\]
+\end{qn}
+
+\begin{qn}
+\[
+2^a \times 2^b = \underbrace{2 \times \cdots \times 2}_{a\ \rm factors} \times
+ \underbrace{2 \times \cdots \times 2}_{b\ \rm factors}
+ = 2^{a+b}.
+\]
+\end{qn}
+
+\begin{qn}
+An inner product $\langle\phantom{\chi},\phantom{\chi}\rangle$ is defined
+on~$G^*$ by
+\[
+\langle\theta,\phi\rangle = \frac{1}{|G|}\sum_{g\in G} \theta(g)
+\overline{\phi}(g).
+\]
+\end{qn}
+
+
+\begin{qn}
+If~ $\overline{\phantom{\chi}}$ denotes complex conjugation, then
+\[
+\overline{\xi + \zeta} = \overline{\xi} + \overline{\zeta}\quad\mbox{and}\quad
+\overline{\xi\zeta} = \overline{\xi}\,\overline{\zeta}.
+\]
+\end{qn}
+
+\begin{qn}
+\[ \int \! \int \phi(r, \theta) \, dr \, d\theta \]
+\end{qn}
+
+\begin{qn}
+\[
+{}^6 C_2 = \frac{6!}{4!\,2!}
+\]
+\end{qn}
+
+
+\section{Operators and relations}
+\subsection{Limits}
+In the second tutorial I introduced various things that could have their
+limits, or ranges, typed in as sub- and super-scripts: standard functions with
+English names, like \verb+\log+; repeated binary operators, like \verb+\sum+;
+and the integral sign \verb+\int+. \TeX\ thinks of all of these as
+\latexword{operator}s. Some operators have the limits set above and
+below in dispayed Maths, but to the right in text; others always have their
+limits set to the right. You can override these defaults by using one of the
+commands \verb+\limits+, \verb+\nolimits+, \verb+\displaylimits+ after the
+name of the operator. The integral sign normally has its limits set to the
+right: if you want them set above and below type \verb+\int\limits+.
+\[
+\begin{tabular}{cc}
+\verb+\int_0^2 x^3 \, dx=4+ &
+$\int_0^2 x^3 \, dx = 4$\\[\jot]
+\verb+\int\limits_0^2 x^3 \, dx=4+ &
+$\int\limits_0^2 x^3 \, dx = 4$
+\end{tabular}
+\]
+If you want
+the limits to be above and below if the operator happens to be in displayed
+Maths, but to the right otherwise, use \verb+\displaylimits+ instead of
+\verb+\limits+. Finally, to ensure that the limits always come to the right,
+use \verb+\nolimits+.
+
+If you want to change the size of the operator as well as the position of its
+limits, you probably need to see the section on styles below.
+
+\subsection{Operators}
+The standard functions with English names already provided by \TeX\ cannot be
+enough for the whole of Mathematics. You make new ones by using \verb+\mathop+,
+usually inside a \verb+\newcommand+. For example,
+\begin{verbatim}
+\newcommand{\var}{\mathop{\rm Var}\nolimits}
+\var X \geq 0
+\end{verbatim}
+\newcommand{\var}{\mathop{\rm Var}\nolimits}
+\[\var X \geq 0\]
+(If you have \LaTeXe, you may feel safer using \verb+{\mathrm{Var}}+ in place of
+\verb+{\rm Var}+.)
+You may put one of \verb+\limits+, \verb+\nolimits+,
+\verb+\displaylimits+
+after the contents of the \latexword{mathop}, to specify how sub- and
+super-scripts should behave. Putting nothing is equivalent to putting
+\verb+\displaylimits+.
+
+There is a school of thought that all operators should be in the same font, so
+that the \verb+\rm+ in the definition of \verb+\var+ should be replaced by a
+command like \verb+\operatorfont+, which would, of course, be defined in the
+style file or in the preamble to the document. I do not agree with this. It is
+not at all unusal to use bold for the expectation operator while retaining
+roman for the variance.
+
+If you make a single letter into an \latexword{operator}, it will be vertically
+centred, which may not be what you intend:
+\begin{verbatim}
+\newcommand{\ee}{\mathop{\rm E}\nolimits}
+\[\ee X + \ee Y = \ee(X+Y)\]
+\end{verbatim}
+\newcommand{\ee}{\mathop{\rm E}\nolimits}
+\[\ee X + \ee Y = \ee(X+Y)\]
+To override this, put the single letter in a box:
+\begin{verbatim}
+...\mathop{\mbox{\rm E}}...
+\end{verbatim}
+
+\subsection{Novel uses of operators}
+In the first tutorial I said that you did not need to think of the
+symbol~\verb+'+ as a superscript. Usually you do not, but \TeX\ always does, so
+you occasionally get unexpected results. You might want to write
+$\mathop{\sum\nolimits'}$ for a variant of the usual summation, perhaps to
+indicate omission of all~$i$ for which $\lambda_i=0$, as in
+\[\mathop{\sum\nolimits'}_{i=1}^n \frac{1}{\lambda_i} P_i.\]
+If you use \verb+\sum'+ it will come out as
+\[\sum'\]
+in display,
+%\[\sum'_{0}^{m}\]
+and even worse things happen when you try to put limits on. Writing
+\verb+\sum\nolimits'+
+cures the problem about placing the dash, but then you no longer have an
+\latexword{operator} to put limits on. So you need to make the whole of
+$\sum\nolimits'$ into an \latexword{operator}:
+\begin{verbatim}
+\newcommand{\summ}{\mathop{\sum\nolimits'}}
+\[\summ_{i=3}^{7}\]
+\end{verbatim}
+\newcommand{\summ}{\mathop{\sum\nolimits'}}
+\[\summ_{i=3}^{7}\]
+If you look closely you will now see that the limits are centred on the whole
+of $\summ$. This is logical, but may not be exactly what you intended. I do not
+know how to do the illogical but more aesthetically pleasing version, but a
+method is provided in \latexword{amstex}.
+
+Sometimes you want to put a range of summation under (or over) the middle of a
+pair of summation signs. Do this by turning the pair of summation signs into an
+\latexword{operator}:
+\begin{verbatim}
+\newcommand{\twosum}{\mathop{\sum\sum}}
+\[\twosum_{1<i<j<n} x_i x_j\]
+\end{verbatim}
+\newcommand{\twosum}{\mathop{\sum\sum}}
+\[\twosum_{1<i<j<n} x_i x_j\]
+
+To get two ranges of summation under a summation sign, make an
+\latexword{operator} containing the summation sign and the interior range(s),
+and then put a subscript on that:
+\begin{verbatim}
+\[\mathop{\sum_{j=1}^{n}}_{j\ne i} Y_j\]
+\end{verbatim}
+\[\mathop{\sum_{j=1}^{n}}_{j\ne i} Y_j\]
+You would normally do this only in displayed Maths.
+
+%% the line below is what DEK says: it comes out just the same.
+%%\[\sum_{\scriptstyle j=1 \atop \scriptstyle j\ne i}^{n} Y_j \]
+
+\subsection{Binary operators}
+\TeX\ does not class ordinary binary operators as \latexword{operator}s.
+Use \verb+\mathbin+ to make something into an infix binary operator.
+For example,
+\verb+$n\mathbin{**}r$+ gives $n\mathbin{**}r$. (What does \verb+$n**r$+
+produce?)
+Usually this is
+done with a \verb+\newcommand+.
+Even a single symbol may need to be explicitly turned into a
+\latexword{mathbin}, if it is not one already, so that the spacing and
+linebreaks around it are correct: this is as true for single symbols that
+already exist as for those that you build up laboriously out of pieces.
+If the new binary operator is (part of) an
+English word, you will need to specify the font, just as for \verb+\mathop+.
+
+\subsection{Binary relations}
+In the same way, \verb+\mathrel+ is used to make new binary relations. The
+considerations are similar to those for \latexword{mathbin}s. Note that
+\latexword{mathbin}s and \latexword{mathrel}s are different in several subtle
+ways, such as the spacing around them, the linebreaks near them, and their
+behaviour when they do not find themselves between two ordinary symbols
+(compare \verb+$n**r$+ with \verb+$n==r$+). If you are not a Mathematician you
+will probably have to ask the author of the document whether a particular
+squiggle is an operator or a relation.
+
+If the new relation consists of two parts, one on top of the other, you can
+make the new relation in one step with \verb+\stackrel+.
+\[
+\verb+\Phi\stackrel{\rm rev}{\sim}\Psi+
+\qquad
+\Phi\stackrel{\rm rev}{\sim}\Psi
+\]
+
+\subsection{Styles}
+When \verb+\sum+ appears in displayed Maths it is larger than in text Maths,
+and has its limits in a different place. However, once it is inside a fraction
+or an array, even in displayed Maths, it reverts to its appearance in text
+Maths. To force one style or the other, precede \verb+\sum+ with either
+\verb+\displaystyle+ or \verb+\textstyle+.
+\begin{quote}
+\verb+\[\frac{\sum_i x_i}{n}\]+
+\qquad
+$\displaystyle\frac{\sum_i x_i}{n}$
+\end{quote}
+\[
+\begin{tabular}{c}
+\verb+\[\frac{\displaystyle\sum_i x_i}{n}\]+\\[2\jot]
+$\displaystyle\frac{\displaystyle\sum_i x_i}{n}$
+\end{tabular}
+\]
+These two commands can affect the appearance of many items in Maths mode,
+including \verb+\frac+.
+
+There are analogous commands \verb+\scriptstyle+, which sets the following
+items as if they were in a subscript, and \verb+\scriptscriptstyle+, which sets
+them as if they were in a second-level subscript.
+
+None of these four commands takes an argument. They are all switches, like
+\verb+\rm+ and \verb+\large+, and
+apply until the end of the current subformula
+(such as the numerator of a \verb+\frac+).
+%
+%obey the normal scoping rules. hoho, not quite, they don't obey environments
+%
+
+\section{Exercises}
+\begin{qn}
+If $f$~is a probability density function then
+\[\int\limits_{-\infty}^{\infty} f = 1.\]
+\end{qn}
+
+\begin{qn}
+\newcommand{\cov}{\mathop{\rm Cov}\nolimits}
+We assume that $Y$ is a random vector with
+$$\cov Y = \sum_{\alpha\in A} \xi_\alpha S_\alpha,$$
+where the $S_\alpha$ are known
+symmetric matrices satisfying $S_\alpha S_\beta =
+\delta_{\alpha\beta} S_\alpha$ and $\sum_{\alpha\in A} S_\alpha = I$.
+\end{qn}
+
+\begin{qn}
+\newcommand{\expect}{\mathop{\mbox{\bf E}}\nolimits}
+The definition of variance is: $\var X = \expect\left(X - \expect X\right)^2$.
+\end{qn}
+
+\begin{qn}
+\newcommand{\opt}{\mathop{\rm opt}\limits}
+The optimize function $\opt$ is defined so that
+$\opt_{i=1}^n M_i$ is equal to $\max\{\max_i M_i, 0\}$.
+\end{qn}
+
+\begin{qn}
+\[
+\mathop{\sum_{i=1}^{n} \sum_{j=1}^{n}}_{j\ne i} y_i y_j
+ = \left(\sum_{i=1}^{n} y_i \right)^2 - \sum_{i=1}^{n} y_i^2.
+\]
+\end{qn}
+
+\begin{qn}
+\[
+\mathop{\sum_{0 < i < m}}_{0 < j < n} P(i,j).
+\]
+\end{qn}
+
+\begin{qn}
+\newcommand{\fplus}{\mathbin{\framebox{+}}}
+Define the operator~$\fplus$ on finite sets of integers as follows.
+If $A$~and~$B$ are two finite sets of integers, then
+$A\fplus B$ is the multiset of integers in which the number
+of times that $n$ occurs is equal to \[\left|\{(a,b): a\in A,\ b\in B,\
+a+b=n\}\right|.\]
+\end{qn}
+
+\begin{qn}
+\newcommand{\rw}{\mathbin{\rm rw}}
+We want to write our wreath products in reverse order, so we put
+$G\rw H = H\wr G$.
+\end{qn}
+
+\begin{qn}
+\newcommand{\aunt}{\mathrel{\rho}}
+The relation~$\aunt$ is said to be symmetric if
+\[ x \aunt y \quad \mbox{implies} \quad y \aunt x.\]
+\end{qn}
+
+
+\begin{qn}
+The strong law of large numbers states that if $X_1$, $X_2$, \ldots\ are
+independent and identically distributed with finite fourth moment then
+\[
+\frac{X_1 + \cdots + X_n}{n} \stackrel{\rm a.s.}{\longrightarrow} Y,
+\]
+where $\Pr[Y=E(X_1)] = 1$.
+\label{lln}
+\end{qn}
+
+\begin{qn}
+\newcommand{\odddiv}{\mathrel{<_2}}
+Define the relation $\odddiv$
+on the natural numbers by
+$n\odddiv m$ if $n\mid m$ and $m/n$ is odd.
+This is a partial order.
+\end{qn}
+
+\begin{qn}
+\newcommand{\bincolon}{\mathbin{:}}
+\it
+Create a binary operator for the colon in $G\bincolon H$ and compare it with
+`:' and \verb+\colon+.
+%$G:H$ and $G\colon H$
+\end{qn}
+
+\begin{qn}
+({\it
+Redo Exercise~16 with a built-up fraction instead of the solidus, with the
+large operators remaining the same size.})
+\[
+\prod_{k\ge 0} \frac{1}{(1 - q^kz)} =
+\frac{\displaystyle
+\sum_{n\ge 0} z^n }{\displaystyle\prod_{1\le
+k\le n} (1 - q^k)}
+\]
+\label{dek}
+\end{qn}
+
+\begin{qn}
+\it
+Redo Question~\ref{lln} with the `a.s.'\ in normal-sized type.
+\end{qn}
+
+\begin{qn}
+More on binomial coefficients:
+\[
+\sum_{
+ \scriptstyle 1\leq n \leq m\atop
+\scriptstyle 1\leq r \leq n}
+ \frac{n!}{r!\,(n-r)!} \quad=\quad \sum_{n=1}^{m} 2^n \quad=\quad 2^{m+1} -2.
+\]
+\end{qn}
+\begin{qn}
+\newcommand{\pistar}{\mathop{\prod\nolimits^{*}}}
+\[
+\pistar_{i=0}^{m} f(\lambda_i)
+\]
+\end{qn}
+
+\begin{thebibliography}{9}
+\bibitem{hart}
+%\writer{Oxford University Press}
+\book{Hart's Rules for Compositors and Readers}
+\publish{Oxford University Press}{Oxford}
+\byear{1967}
+\bibitem{chamb}
+\book{The Chambers Dictionary}
+\publish{Chambers Harrap}{Edinburgh}
+\byear{1993}
+\bibitem{chicago}
+\book{The Chicago Manual of Style}
+\publish{The University of Chicago Press}{Chicago}
+\byear{1982}
+\bibitem{companion}
+\writer{Goossens,~M., Mittelbach,~F. \& Samarin,~A.}
+\book{The \LaTeX\ Companion}
+\publish{Addison-Wesley}{Reading, Mass.}
+\byear{1994}
+\bibitem{leslie}
+\writer{Lamport,~L.}
+\book{\LaTeX: A Document Preparation System}
+first edition,
+\publish{Addison-Wesley}{Reading, Mass.}
+\byear{1986}
+\bibitem{newleslie}
+\writer{Lamport,~L.}
+\book{\LaTeX: A Document Preparation System}
+second edition,
+\publish{Addison-Wesley}{Reading, Mass.}
+\byear{1994}
+
+\end{thebibliography}
+\end{Article}