\sectiontitle{Rules for Ordinary Text (without mathematics)}
\subsectiontitle{Special Characters}
All characters on the keyboard have their standard meaning
in ordinary text with the exception of the
special characters
\begin{quote}
\begin{verbatim}
# $ % & ~ _ ^ \ { } '
\end{verbatim}
\end{quote}
which have special functions within \TeX.  On the rare
occasions when these special characters are required in
the final document they must be  produced by an appropriate
control sequence.  Thus you should type \verb?\#?, \verb?\$?,
\verb?\%?, \verb?\&?, \verb?\_?, \verb?\{? and \verb?\}? to
obtain $\#$, $\$$, $\%$, $\&$, $\_$, $\{$ and $\}$
respectively.

\subsectiontitle{Paragraphs}
Successive paragraphs in the input file should be separated
by a completely blank line.  All paragraphs will be
automatically indented by \TeX\ with the exception of the
first paragraph of a new section.  (One can override the
conventions of \TeX\ by placing the control sequence
\verb?\noindent? of the control sequence \verb?\indent?
at the beginning of the paragraph.)

\subsectiontitle{Quotation marks}
To produce single quotation marks use the characters
\verb?`? (left quote) and \verb?'? (right quote).  For
double quotation marks use \verb?``? (two left quotes)
and \verb?''? (two right quotes).  {\it Do not use\/}
\verb?"? (undirected double quote).  Thus to obtain
\begin{quotation}
\small
``This is easy'' he said.
\end{quotation}
you should type
\begin{quote}
\begin{verbatim}
``This is easy'' he said.
\end{verbatim}
\end{quote}

The control sequence \verb?\,? can be used to
separate single quotes from double quotes where
necessary.

\subsectiontitle{Dashes}
Dashes of various lengths are obtained using \verb?-?,
\verb?--? and \verb?---?.  You should use \verb?-? for
hyphenation, \verb?--? when specifying ranges of numbers,
and \verb?---? to obtain a punctuation dash.  Thus we obtain
\begin{quotation}
\small
The Cayley-Hamilton Theorem.
\end{quotation}
\begin{quotation}
\small
See pages 95--104.
\end{quotation}
\begin{quotation}
\small
Use three dashes to obtain a punctuation dash---like this.
\end{quotation}
by typing
\begin{quote}
\begin{verbatim}
The Cayley-Hamilton Theorem.
\end{verbatim}
\end{quote}
\begin{quote}
\begin{verbatim}
See pages 95--104.
\end{verbatim}
\end{quote}
\begin{quote}
\begin{verbatim}
Use three dashes to obtain a punctuation dash---like this.
\end{verbatim}
\end{quote}

\subsectiontitle{Changing Fonts}
The control sequences \verb?\rm?, \verb?\sl?, \verb?\it?, \
\verb?\tt? and \verb?\bf? change to {\rm roman}, {\sl slanted},
{\it italic}, {\tt teletype} and {\bf boldface} fonts respectively.
Any change of font made within a group enclosed within curly
brackets \verb?{? and \verb?}? will only apply to text within
that group.  On leaving the group, the current font is restored
to what it was before entering the group.  Thus we can obtain
\begin{quotation}
\small
This sentence contains a word set in {\bf boldface} type
\end{quotation}
by typing
\begin{quote}
\begin{verbatim}
This sentence contains a word set in {\bf boldface} type
\end{verbatim}
\end{quote}

   The control sequence \verb?\/? produces the so-called
`italic correction'.  It is sometimes desirable when changing
from a slanted font (such as {\it italic\/} or {\it slanted\/})
back to a non-slanted font such as {\rm roman} or
{\bf boldface}, in order to produce a small amount of extra
space to compensate for the slantedness of the font, and thus
improve the appearance of the final document.  However the
italic correction should not be applied before a period
(full stop) or a comma.  To obtain
\begin{quotation}
\small
Here is some {\it italicized\/} text.
\end{quotation}
one should type
\begin{quote}
\begin{verbatim}
Here is some {\it italicized\/} text.
\end{verbatim}
\end{quote}
However it usually does not matter all that much if you forget
about this italic correction.

   In \LaTeX\ there is also a control sequence \verb?\em?
for {\em emphasizing text}.  This control sequence changes to the
{\em italic\/} font, {\it unless we are already in the italic
font, in which case it converts to the {\em roman} font.}

\subsectiontitle{Accents in Text}
These are produced by control sequences such as \verb?\'?,
\verb?\`? and \verb?\"?.  Thus one types \verb?Se\'{a}n?
and \verb?H\"{o}lder? to obtain `Se\'{a}n' and `H\"{o}lder'
respectively.  For a full list of such accents, see
Appendix~\ref{la-txtcs}.  Note however that accents within
mathematics are produced in a different fashion.

\subsectiontitle{Producing Blank Space in \LaTeX}
To produce (horizontal) blank space within a paragraph, use
\verb?\hspace? and \verb?\hspace*?, followed by the length
of the blank space enclosed within curly brackets.  The length
of the skip should be expressed in a unit recognized by \TeX.
These recognized units are given in the following table:
\begin{quote}
\begin{tabular}{lll}
\verb?pt?  & point         & (1 in = 72.27 pt) \\
\verb?pc?  & pica          & (1 pc = 12 pt) \\
\verb?in?  & inch          & (1 in = 25.4 mm) \\
\verb?bp?  & big point     & (1 in = 72 bp) \\
\verb?cm?  & centimetre    & (1 cm = 10 mm) \\
\verb?mm?  & millimetre    & \\
\verb?dd?  & didot point   & (1157 dd = 1238 pt) \\
\verb?cc?  & cicero        & (1 cc = 12 dd) \\
\verb?sp?  & scaled point  & (65536 sp = 1 pt)
\end{tabular}
\end{quote}
Thus to produce a horizontal blank space of 20 mm in the middle
of a paragraph one would type \verb?\hspace{20 mm}? (or
\verb?\hspace*{20 mm}?.

   The difference between \verb?\hspace? and \verb?\hspace*?
is that if \TeX\ decides to break between lines at the point
where an \verb?\hspace? is specified, then the \verb?\hspace?
is ignored.  Using \verb?\hspace*? forces \TeX\ to produce a
horizontal space, whether of not \TeX\ breaks between lines.

To produce (vertical) blank space between paragraphs, use
\verb?\vspace? and \verb?\vspace*?, followed by the length
of the blank space enclosed within curly brackets.  A
\verb?\vspace? will be ignored if it comes at a
break between pages, whereas blank space will always be
produced by \verb?\vspace*?, whether or not there is a
page break.

\subsectiontitle{Forcing Blank Spaces and Preventing Line Breaks}
To force \TeX\ to produce a blank space where it might not
otherwise put one, one should precede the blank space with a
\verb?\? (backslash).  It is often advisable to precede
with a backslash blank spaces after certain abbreviations
such as `Dr.', `etc.', and `Math.\ Soc.' (so that one
should type \verb?Dr.\ Smith? etc.).

If you wish to ensure that \TeX\ does not start a new line
at a particular blank space, then you can use \verb?~? in
place of the blank space.  Thus if you type
\verb?I.~Newton? or \verb?Example~4? then you prevent
a line break at these places.

\sectiontitle{Rules for obtaining Mathematical Formulae}
\subsectiontitle{Mathematics embedded in Text}
Any mathematical expressions embedded in text should be
preceded and followed by the character \verb?$?.  Thus
to obtain
\begin{quotation}
\small
Let $f$ be the function defined by $f(x) = x + 7$.
\end{quotation}
one should type
\begin{quote}
\begin{verbatim}
Let $f$ be the function defined by $f(x) = x + 7$.
\end{verbatim}
\end{quote}

\subsectiontitle{Displayed Mathematical Formulae}
Any displayed mathematical formula should be preceded by
\verb?\[?
and followed by \verb?\]?.  Thus to obtain
\begin{quotation}
\small
Let $g$ be the function defined by
\[ g(x,y) = xy + x + y + 2. \]
The function $g$ is positive when both $x$ and $y$ are positive.
\end{quotation}
one should type
\begin{quote}
\begin{verbatim}
Let $g$ be the function defined by
\[ g(x,y) = xy + x + y + 2. \]
The function $g$ is positive when both $x$ and $y$
are positive.
\end{verbatim}
\end{quote}

\subsectiontitle{Special Characters in Mathematics Mode}
All characters on the keyboard have their standard meaning
in mathematical expressions with the exception of the
special characters
\begin{quote}
\begin{verbatim}
# $ % & ~ _ ^ \ { } '
\end{verbatim}
\end{quote}
which have special functions within \TeX.  On the rare
occasions when these special characters are required in
the final document they must be  produced by an appropriate
control sequence.  Thus you should type \verb?\#?, \verb?\$?,
\verb?\%?, \verb?\&?, \verb?\_?, \verb?\{? and \verb?\}? to
obtain $\#$, $\$$, $\%$, $\&$, $\_$, $\{$ and $\}$
respectively.  To obtain $\backslash$ in mathematics mode,
type \verb?\backslash?.

   The character \verb?'? is used to put a superscript
prime after a character.  Thus if we type \verb?$f'$? and
\verb?$g''$? we obtain $f'$ and $g''$ respectively.

\subsectiontitle{Superscripts and Subscripts}
Superscripts and subscripts are produced using the characters
\verb?^? and \verb?_? respectively.  Thus we obtain
$t^2 + x_1 - x^3_1$ by typeint \verb?$t^2 + x_1 - x^3_1$?.
If a superscript or subscript consists of more than one
character then the superscripts and subscripts should be
enclosed in curly brackets.  Thus one obtains $a_{i,j}$ by
typing \verb?$a_{i,j}$?.  One can obtain double subscripts:
we obtain $s_{n_j}$ by typing \verb?$s_{n_j}$?.

\subsectiontitle{Greek Letters}
Greek letters are obtained by preceding the name of the
letter by a backslash.  Thus we obtain
$\alpha, \beta, \gamma$ by typing
\verb?$\alpha, \beta, \gamma$?.  See Appendix~\ref{la-mthcs}
for a list of Greek letters.  Some Greek letters have variant
forms---see Appendix~\ref{la-mthcs}.

\subsectiontitle{Mathematical Symbols}
Mathematical symbols such as $\div$, $\equiv$, $\otimes$,
$\sum$, $\in$, $\cup$, $\cap$ and $\to$ are obtained using
the appropriate control sequences---see Appendix~\ref{la-mthcs}.

\subsectiontitle{Accents in Mathematics}
These are produced using the appropriate control
sequence---see Appendix~\ref{la-mthcs}.

\subsectiontitle{Standard Functions}
Certain standard functions such as $\sin$ and $\log$ are
obtained by preceding the name with a backslash---see
Appendix~\ref{la-mthcs} for a full list of these.  To obtain
a function or similar expression not on this list you should
convert to the roman font (e.g., to obtain ${\rm Aut}(G)$ one
should type \verb?${\rm Aut}(G)$?).

\subsectiontitle{Fractions}
Fractions are obtained in \LaTeX\ using the control sequence
\verb?\frac?.  We type
\begin{quote}
\verb?\frac{? {\it numerator\/} \verb?}{? {\it denominator\/} \verb?}?
\end{quote}
to obtain the required fraction.  Thus to obtain
\[ f(x) = \frac{2 x}{(1 + x^2)^2} \]
we type
\begin{quote}
\begin{verbatim}
\[ f(x) = \frac{2 x}{(1 + x^2)^2} \]
\end{verbatim}
\end{quote}

\subsectiontitle{Roots}
Square roots are obtained using the control sequence
\verb?\sqrt?.  Thus to obtain $\sqrt{3x + 7}$ we
type \verb?$\sqrt{3x + 7}$?.  To obtain an $n$th
root in \LaTeX\ we use the construction
\begin{quote}
\verb?\sqrt[?$n$\verb?]{? {\it expression} \verb?}?
\end{quote}
Thus $\sqrt[3]{3x + 7}$ is obtained by typing
\verb?$\sqrt[3]{3x + 7}$?.

\subsectiontitle{Ellipsis}
Ellipsis (three dots) is obtained in mathematical formulae
using the control sequences \verb?\cdots? (centred ellipsis)
and \verb?\ldots? (lowered ellipsis).  Thus to obtain
$x_1 + x_2 + \cdots + x_n$ and $x_1, x_2, \ldots, x_n$
we type \verb?$x_1 + x_2 + \cdots + x_n$? and
\verb?$x_1, x_2, \ldots, x_n$? respectively.

\subsectiontitle{Delimiters}
To surround a subformula with delimiters large enough to enclose
the subformula we use the construction
\begin{quote}
\verb?\left(? $\ldots$ {\it subformula} $\ldots$ \verb?\right)?
\end{quote}
(where the parentheses \verb?(? $\ldots$ \verb?)? may be replaced
by any other pair of delimiters such as
\verb?[? $\ldots$ \verb?]? or \verb?\{? $\ldots$ \verb?\}?).
Thus to obtain the equation
\[ f(x) = \left( 1 + \frac{2x}{x^2 + 1} \right) - \sin(x) \]
we type
\begin{quote}
\begin{verbatim}
\[ f(x) = \left( 1 + \frac{2x}{x^2 + 1} \right) - \sin(x) \]
\end{verbatim}
\end{quote}

\subsectiontitle{Embedding Text in Mathematics}
Text can be embedded in mathematics using the control sequence
\verb?\mbox?.  Thus if we type
\begin{quote}
\begin{verbatim}
\[ V' = \{ f \in X' : f(v) = 0 \mbox{ for all } v \in V \} \]
\end{verbatim}
\end{quote}
we obtain
\[ V' = \{ f \in X' : f(v) = 0 \mbox{ for all } v \in V \} \]

\subsectiontitle{Inserting and Removing Blank Space in Formulae}
The control sequence \verb?\quad? produces a `quad'
of blank space (a `quad' is approximately the width of the
letter `m').  The control sequence \verb?\,? inserts a thin
blank space and the control sequence \verb?\!? removes a
thin space.  One uses \verb?\,? and \verb?\!? to improve the
appearance of mathematical formulae. For example, if we
type
\begin{quote}
\begin{verbatim}
\[ \int_0^\pi \sin x dx = 2, \]
\end{verbatim}
\end{quote}
we obtain
\[ \int_0^\pi \sin x dx = 2, \]
whereas if we type
\begin{quote}
\begin{verbatim}
\[ \int_0^\pi \sin x \,dx = 2, \]
\end{verbatim}
\end{quote}
we obtain
\[ \int_0^\pi \sin x \,dx = 2, \]
and this equation has a more satisfactory appearance.

\subsectiontitle{Further Features of \LaTeX}
There are plenty of control sequences and `environments'
in \LaTeX\ for accomplishing various tasks.  Among the most
widely used environments are
\begin{quote}
\begin{flushleft}
\begin{verbatim}
\begin{equation}  ...  \end{equation}
\end{verbatim}
\end{flushleft}
\begin{flushright}
(produces a numbered displayed formula)
\end{flushright}
\begin{flushleft}
\begin{verbatim}
\begin{eqnarray}  ...  \end{eqnarray}
\end{verbatim}
\end{flushleft}
\begin{flushright}
(produces a numbered displayed multiline formula)
\end{flushright}
\begin{flushleft}
\begin{verbatim}
begin{eqnarray*}  ...  \end{eqnarray*}
\end{verbatim}
\end{flushleft}
\begin{flushright}
(produces an unnumbered displayed multiline formula)
\end{flushright}
\begin{flushleft}
\begin{verbatim}
\begin{array}  ...  \end{array}
\end{verbatim}
\end{flushleft}
\begin{flushright}
(produces an array or matrix)
\end{flushright}
\end{quote}