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author | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
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committer | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
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diff --git a/usergrps/uktug/baskervi/5_1/bailey.tex b/usergrps/uktug/baskervi/5_1/bailey.tex new file mode 100644 index 0000000000..e736d041c3 --- /dev/null +++ b/usergrps/uktug/baskervi/5_1/bailey.tex @@ -0,0 +1,696 @@ +\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} |