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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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tree | 60335e10d2f4354b0674ec22d7b53f0f8abee672 /usergrps/uktug/baskervi/4_4/bailey2.tex |
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diff --git a/usergrps/uktug/baskervi/4_4/bailey2.tex b/usergrps/uktug/baskervi/4_4/bailey2.tex new file mode 100644 index 0000000000..eda6aee2b1 --- /dev/null +++ b/usergrps/uktug/baskervi/4_4/bailey2.tex @@ -0,0 +1,462 @@ +\hyphenation{super-script} +\title{Maths in \LaTeX: Part~1, Back to Basics} +\author[R.~A. Bailey]{R.~A. Bailey\\Goldsmiths' +College, University of London} + +\def\realrm{\rmfamily\upshape} +\newtheorem{prequestion}{Question} +\newtheorem{preanswer}{Answer} +\renewcommand{\thepreanswer}{} +\newenvironment{question}{\begin{prequestion}\realrm}{\end{prequestion}} +\newenvironment{answer}{\begin{preanswer}\realrm}{\end{preanswer}} +\newtheorem{preqn}{Exercise} +\newenvironment{qn}{\begin{preqn}\realrm}{\end{preqn}} +\newcommand{\latexword}[1]{{\ttfamily\upshape #1}} +\newcommand{\theoremheadfont}[1]{{\textbf{#1}}} + +\newcommand{\lamport}{{\it \LaTeX: A Document Preparation System\/} by Leslie +Lamport} +\newcommand{\shortlamp}{\emph{The Manual}} + +\begin{Article} +\section{Introduction} +The book \lamport\ is rather coy about Mathematics. It simply does not reveal +the full range of Mathematical expressions that can be correctly typeset +without going outside \LaTeX. The result is that some Mathematical authors, +while attracted to the generic mark-up of \LaTeX, believe that they need to use +plain \TeX\ or \AmSTeX\ +to write their documents. + +This sequence of tutorials seeks to correct that impression, by +explaining what Mathematical expressions can be typeset with \LaTeX\ +without the need for the \latexword{amstex} package. Perhaps this will +provoke someone else to write a tutorial on that package. The first +part is mostly, but not entirely, devoted to things which you can find +in \shortlamp, even though you may have overlooked some of them. +Succeeding parts (in the next and later issues of \BV) will be mostly +about Mathematical goodies provided by \TeX\ but upon which +\shortlamp\ is silent, even though they are necessary and quite easy +to use. The final part will deal with arrays, concentrating on their +use in Mathematics. + +These are tutorials, so 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 write to me with hard copy of your input and +output (no email address before we go to press, I'm afraid): I will +include a solution in the following issue of \BV. + +A word on fonts. Fonts in Mathematics are handled differently in +\LaTeX\ 2.09, in NFSS, and in \LaTeXe. Rather than compare these +systems every time that I mention fonts, I shall limit myself to +\LaTeX\ 2.09. With any luck, this will enrage some knowledgeable +person enough to write an article on handling of Maths fonts in +different flavours of \LaTeX. + +\section{What does it look like?} + +\subsection{Maths Mode} +\AllTeX\ has a special state, called {\em Maths mode}, which it must +be in to recognize Mathematical expressions and typeset them properly. +Maths mode in \LaTeX\ is everything between |\(| and~|\)|, +or, alternatively, everything between |$| and~|$|. The +parentheses are better for trapping errors, because it is obvious +whether the left or right one is missing, if any. A missing~|$| +causes \AllTeX\ to swap Maths mode and ordinary mode from then +onwards, giving strange output but no errors until it eventually meets +something like |x^2| that it cannot interpret in the wrong mode. +On the other hand, the dollar signs are easier to type, and easier to +see in your input file. + +In Maths mode most symbols are typeset as if they represent single-letter +variables. A string of three letters will be set as if those three variables +should be multiplied together. Fancy features like kerns and ligatures, which +are used in normal text to help the reader interpret letter-strings as words, +are turned off. Letters are set in the special font known as {\em Maths +italic\/} which is usually used for variables. + +Almost all spaces that you type are ignored. \AllTeX\ thinks that it +knows better than you do how Mathematics should be spaced, and it is +probably right to think so. + +Don't stay in Maths mode for too long just because you are too lazy to type a +few |$|~signs. Everything between the~|$|s should be Maths. A common +mistake by beginners is to forget that a punctuation sign, like a comma, may +have a different meaning in Maths from its meaning in text. In +\begin{quote} +the scalars $a$, $b$ and $c$ +\end{quote} +we have a textual list containing three mathematical objects, so the input file +contains +\begin{verbatim} + the scalars $a$, $b$ and $c$ +\end{verbatim} +That comma is a textual one. The lazy typist types +\begin{verbatim} + the scalars $a, b$ and $c$ +\end{verbatim} +and obtains +\begin{quote} + the scalars $a, b$ and $c$ +\end{quote} +On the other hand, in +\begin{quote} +the vector $(a,b,c)$ +\end{quote} +there is a single Mathematical object, so it is correct to type +\begin{verbatim} + the vector $(a,b,c)$ +\end{verbatim} +or, equally well, +\begin{verbatim} + the vector $(a, b, c)$ +\end{verbatim} +These commas are part of the Mathematical notation. + +\subsection{Basic symbols} +The basic symbols are the numerals {\tt 1}, {\tt 2}, \ldots, the Latin +letters {\tt a}, {\tt b}, \ldots, {\tt z}, {\tt A}, \ldots, {\tt Z}, and the +Greek letters |\alpha|, |\beta|, |\gamma|, \ldots, +|\omega|, |A|, |B|, |\Gamma|, \ldots,~|\Omega|. +If you don't know the standard English spellings of Greek letters, look on +page~43 of \shortlamp. Upper-case Greek letters which are conventionally the +same as their Latin equivalents do not have special commands. Some Greek letters +have variants: |\varepsilon|, for example. + +The obvious symbols for operators are the keyboard symbols |+| +and~{\tt -}. If you forget to go into Maths mode (a common temptation +when typing a table of data), the symbol~{\tt -} will not look like a +minus sign. Outside Maths mode the~|+| will look like a plus +sign, but the spacing will be wrong. In Maths mode \AllTeX\ knows what +is the proper spacing to put around binary operators like~\verb|+| +and~{\tt -}; it also knows the proper spacing to surround binary +relations like~{\tt =}. Try typing the following both inside Maths +mode and outside it, and compare the results. +\begin{verbatim} + 1 +2 = 3 4-1 = 3 + 1 -4 = -3 -2+7 =+5 +\end{verbatim} +Also try~{\tt >} outside Maths mode: you may be surprised. +\label{symsec} + +\subsection{Sub- and superscripts} +Subscripts are introduced with~|_|: for example, |x_n| gives~$x_n$. +If there is more than one thing in the subscript you have to use braces, as in +\verb!x_{n+1}! for~$x_{n+1}$. You can type |x_{n}| for $x_{n}$ if you +want, but it makes your input file less readable. + +Superscripts are done similarly, using~|^|: thus |y^3| for~$y^3$ and +|y^{-1}| for~$y^{-1}$. + +A sub- and superscript can be put on the same symbol in either order: +|x_n^2| and |x^2_n| both produce~$x_n^2$. Double subscripts or +superscripts are obtained by using braces in the obvious way: |x_{n_2}| +and |n^{m^2}|. + +To put a sub- or superscript {\em before\/} a symbol, precede it with +|{}|. Otherwise the sub- or superscript attaches itself to the previous +thing, which may well be something like $+$ or~$=$. + +In an expression such as ${(X+Y)^2}$, strictly speaking \TeX\ thinks it is +putting the superscript on the right parenthesis if +you type \verb!(X+Y)^2!, and +it positions the superscript in accordance with that thought. If this really +offends you, you can force \TeX\ to share your logic by typing +\verb!{(X+Y)}^2!, but you may not always prefer the result. + +\subsection{Modifying symbols} +To turn~$x$ into~$x'$ type~|x'|. You do not need to think of the prime as +a superscript. + +Some common modifiers are exemplified in +\[ +\begin{tabular}{cc@{\qquad}cc} +|\bar{x}| & $\bar{x}$ & |\tilde{x}| & $\tilde{x}$\\ +|\hat{x}| & $\hat{x}$ & |\vec{x}| & $\vec{x}$ +\end{tabular} +\] +A few more such decorations are shown on page~51 of \shortlamp. If any +of them is used over an~$i$ or a~$j$ then the dotless versions of +those letters should be used: |\imath| and |\jmath|. + +There are wide versions of |\hat| and |\tilde|: +\[ +\begin{tabular}{c@{\qquad}c} +\verb!\widehat{a+b}! & $\widehat{a+b}$\\ +|\widetilde{1-\theta}| & $\widetilde{1-\theta}$ +\end{tabular} +\] +There are also wide versions of |\bar| and |\vec| but with less +obvious names: I'll cover these in a later tutorial. + +Logically, a decoration such as |\hat| may modify the whole of a +subscripted expression such as~$x_2$; you usually mean `the estimate of~$x_2$' +rather than `the second part of~$\hat{x}$'. However, both $\hat{x_2}$ and +$\widehat{x_2}$ simply look wrong, so you have to let aesthetics triumph over +logic and type |\hat{x}_2|. + +\subsection{Dots} +To get a line of dots to show that some items have been missed out, use +|\ldots| if the missing items are normally aligned on their baselines, +such as letters, or |\cdots| if the missing items are normally aligned on +the centreline, such as binary operators. If the missing items are part of a +textual list, don't forget to come out of Maths mode and to put a comma at the +end of the dots. +\[ +\begin{tabular}{c} +|for $i=1$, $2$, \ldots, $10$|\\ +|the vector $(x_1, x_2, \ldots, x_n)$|\\ +|$a_1 + a_2 + \cdots + a_n$|\\ +|$y_1 = y_2 = \cdots = y_7$| +\end{tabular} +\] +If you think that the dollar signs round the numerals in the first example are +unnecessary, try embedding that phrase in a piece of italic text. + +\subsection{Square roots} +Type |\sqrt{2}| to obtain $\sqrt{2}$. The same technique works for more +complicated expressions than~$2$: you don't have to do anything to make the +root sign the right size. For example, +\begin{quote} +\verb!\sqrt{n^2+6}! \qquad $\sqrt{n^2+6}$ +\end{quote} +Other roots, such as cube roots, are obtained by putting in an optional +argument: +\begin{quote} +|\sqrt[3]{8} = 2| \qquad $\sqrt[3]{8} = 2$ +\end{quote} +The simple symbol for a square root is |\surd|. + +Don't abuse \TeX's wizardry by using |\sqrt| for a large expression in +text or in a complicated display. The mess obscures the message. + +\subsection{Displayed Maths} +To get a single line of displayed Maths, type the contents between |\[| +and~|\]|. You should not start a paragraph with displayed Maths, but may +end one. If the displayed Maths is in the middle of a paragraph, remember not +to leave blank lines around it in your input file. + +Displayed Maths may also be typed between |$$| and~|$$|, but the +effect is not quite the same. For example, the document option +\latexword{fleqn} aligns displayed Maths on the left if you use |\[| and +|\]|, but not if you use |$$|. + +To put a short piece of text in displayed Maths, insert it in |\mbox|, +remembering to include any necessary spaces that would be ignored in Maths +mode. +\begin{verbatim} + \[ a=b \mbox{ if } c=d \] +\end{verbatim} +Don't try to use |\mbox| in a similar way to put short text +between pieces of Maths in text: it inhibits line-breaks. + +\subsection{Words as labels} +Sometimes you want to attach natural-language words to Mathematical symbols to +label them. For example, you might have analogous quantities associated with +the rows and columns of a rectangular array, and wish to indicate this by using +the same symbol, say~$Q$, with different subscripts. It simply will not do to +type |Q_{rows}|, because this gives $Q_{rows}$, where the subscript looks +like the product of $r$~by $o$ by \ldots. And it is no good putting {\tt rows} +in an |\mbox|, because it will come out too big. Once something has been +put in a box, it doesn't change size. You have to type +|Q_{\rm rows}| to get $Q_\mathrm{rows}$. +(Did you remember the caveat about fonts?) + +If this seems too much trouble, you might decide to abbreviate to $Q_r$ +and~$Q_c$. But this will not do either, +because the subscripts look like variables +into which numbers, say, could be substituted. If you don't want to mislead +your readers, you should type |Q_{\rm r}|. + +\subsection{Fractions} +A built-up fraction is made with |\frac|: +\begin{quote} +|\frac{n}{m}| \qquad $\displaystyle\frac{n}{m}$ +\end{quote} +This comes out larger in displayed Maths than in text. It is better to use the +solidus, as in $n/m$, for most fractions in text, with the exception of a few +simple common fractions like~$\frac{1}{2}$. + +Of course, fractions can be put inside other fractions with no bother: +\[ +\begin{tabular}{c} +\verb!\frac{a(b+c)}{5 + \frac{1}{xy}}! \\[2\jot] +$\displaystyle \frac{a(b+c)}{5 + \frac{1}{xy}}$ +\end{tabular} +\] + +\subsection{Binary operators} +In the golden olden days of golf-ball typewriters, it was a luxury to +a Mathematician to have the symbol for direct sum, or for union. +\AllTeX\ not only has the symbols; it knows that they are operators, +and gives them the correct spacing for infix operators, and has +reasonably good ideas about where to break lines near them. A few of +the common ones are: +\[ +\begin{tabular}{cc@{\qquad}cc@{\qquad}cc} +\verb!+! & $+$ & \verb!-! & $-$ & \verb!\pm! & $\pm$\\ +\verb!\times! & $\times$ & \verb!\div! & $\div$ & \verb!\oplus! & $\oplus$\\ +\verb!\cup! & $\cup$ & \verb!\cap! & $\cap$ & \verb!\wedge! & $\wedge$. +\end{tabular} +\] +There are many more on page~44. + +In fact, \AllTeX\ is even cleverer than this. If a binary operator +doesn't find itself between two things it can operate on then it +becomes a simple symbol, and spaces and line-breaks adjust +accordingly. You should have noticed this if you did the exercise +suggested above.%in Section~\ref{symsec}. + +\subsection{Binary relations} +\AllTeX\ also knows about infix relations, such as +\[ +\begin{tabular}{cc@{\qquad}cc@{\qquad}cc} +\verb!=! & $=$ & \verb!\in! & $\in$ & \verb!\subset! & $\subset$\\ +\verb!<! & $<$ & \verb!\leq! & $\leq$ & \verb!\perp! & $\perp$. +\end{tabular} +\] +More are shown on page~44. Don't confuse~$\in$ with either of the epsilons. + +Compare \verb!\mid! with~\verb!|!. The former is a relation, while the latter +is just a symbol. So which should you use for `divides'? + +Relations can be negated by preceding them with |\not|: +%\begin{quote} +\[ +\verb!Z_2 \times Z_2 \not\cong Z_4! \qquad +Z_2 \times Z_2 \not\cong Z_4 +\] +%\end{quote} +This doesn't work quite right for~$\in$, so there is the special command +|\notin|. Also, |\ne| is a useful shorthand for |\not=|. + +\subsection{Fonts in Maths} +(Did you remember the caveat about fonts?) + +For something like script letters use |\cal|, as in |${\cal F}(x)$| +for $\mathcal{F}(x)$. The braces give the scope of |\cal|: for a +single Mathematical letter such as~$\mathcal H$ you can get away with +|$\cal H$|. Only upper-case Latin letters may be modified by |\cal|. + +In some branches of Mathematics, constants are shown in Roman type. So the base +of natural logarithms is |{\rm e}|. + +For bold letters, you can use |\bf| to modify +Latin letters and upper-case Greek ones: +\[ +\verb!{\bf Mv} = a{\bf w}! \qquad \mathbf{Mv} = a\mathbf{w} +\] +For lower-case Greek letters, and for non-letters, you have to use a +cumbersome construction: +\begin{quote} +\verb!\mbox{\boldmath $\lambda$}! \qquad \mbox{\boldmath $\lambda$} +\end{quote} +Because of the box, this does not change size properly in sub- and +superscripts. + +\subsection{Writing Maths} +The ability to produce beautiful Mathematical formulae is no licence to produce +poor Mathematical writing. Remember that relations are verbs. It is impossible +to parse the sentence +\begin{quote} +Therefore $n=56$ is the sample size. +\end{quote} +but +\begin{quote} +The equation $x^2+9=0$ has no real roots. +\end{quote} +is fine. + +Don't start a sentence with notation: the reader doesn't get the right visual +clue. If possible, avoid putting notation immediately after {\em any\/} +punctuation, unless it is part of a list. This saves the reader from having to +work out if the punctuation is Mathematical or textual. Similarly, avoid +abbreviations like `iid' and `e.g.'\ which might be mistaken for notation at a +first glance. + +\section{Exercises} +\begin{qn} +The zeros of the quadratic $ax^2 + bx + c$ are +\[ +\frac{-b\pm \sqrt{b^2 - 4ac}}{2a} . +\] +\end{qn} + +\begin{qn} +The upper 5\% point of the $\chi^2_6$ distribution is 12.592. +\end{qn} + +\begin{qn} +If $\nu= n_1 + n_2 -2 $ and +\[ +s^2 = \frac{(n_1 -1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 -2} +\] +then +\[ +\frac{\bar{X}_1 - \bar{X}_2}{s\surd(\frac{1}{n_1} + \frac{1}{n_2})} +\] +is distributed as $t_\nu$. +\label{tdist} +\end{qn} + +\begin{qn} +By choosing bases, it follows that the subspaces $Z_1$, \ldots, $Z_r$ span~$V$; +hence it follows that $V$ is the direct sum $V=Z_1 \oplus \cdots \oplus Z_r$, +as asserted. +\end{qn} + +\begin{qn} +If $\mathcal M$ and $\mathcal N$ are subspaces of a finite-dimensional inner product +space $\mathcal V$ then +\[ +(\mathcal{M} + \mathcal{N})^\perp = \mathcal{M}^\perp \cap \mathcal{N}^\perp +\] +and +\[ +(\mathcal{M} \cap \mathcal{N})^\perp = \mathcal{M}^\perp + \mathcal{N}^\perp. +\] +Moreover, $\mathcal{M}^\perp \cong \mathcal{V}/\mathcal{N}$. +\end{qn} + +\begin{qn} +The sum of squares for the linear model $V_\mathrm{protein} ++ V_\mathrm{fishmeal}$ +is $1559378$. +\end{qn} + +\begin{qn} +\newcommand{\bbeta}{\mbox{\boldmath $\beta$}} +\newcommand{\bep}{\mbox{\boldmath $\varepsilon$}} +The usual regression equation is +$\mathbf{Y} = \mathbf{X}\bbeta + \bep$, where +$\mathbf{Y}$ is an $n\times 1$ vector, $\mathbf{X}$ +is an $n\times p $ matrix, $\bbeta$ +is the $p\times1$ vector of unknown parameters, and $\bep$ is the $n\times1$ +vector of random errors. The least-squares estimate $\widehat{\bbeta}$ of the +parameters is given by +\[ +\widehat{\bbeta} = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{Y}. +\] +\end{qn} + +\begin{qn} +The $T$-orders are $p(x)^{e_1}$, $p(x)^{e_2}$ and $p(x)^{e_3}$, where $e_1 >e_2 +\geq e_3$. This implies that $p(x)^{e_1}\mid\eta(x)^{e_1-d}$ and hence that +$\eta(x) = \psi(x)p(x)^d$ for some polynomial $\psi(x)$. +\end{qn} + + +\begin{qn} +We have $t\in A\setminus B$ if and only if \[t \in A \mbox{ and } t\notin B.\] +\end{qn} + + +\begin{qn} +Pascal's triangle is based on the identity +\[ +{}^{n-1}\mathrm{C}_{k} + {}^{n-1}\mathrm{C}_{k-1} = {}^{n}\mathrm{C}_{k}. +\] +\end{qn} + +\end{Article} |