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% % Lines starting with % are comments, which are ignored.
% % This is a handy way of indicating the date and version of
% % your document, to wit:
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% % LaTeX sample file
% % Modified March, 2002
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\title{The \LaTeX\ template file for an article in the
$\Pi$ME Journal}
\author{Primus Scriber\thanks{
College of the Enlightenment}
\and
Theeco Author\thanks{
The Virtual University
}\thanks{This work was
supported by NSB grant number G983578765401.}
}
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\documentclass{article}
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% %
% % The next command allows your in import encapsulated
% % postscript files, .epsf or .eps files, which
% % contain vector graphic image data.
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\usepackage{graphicx}
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% % We use newtheorem to define theorem-like structures
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% % Here are some common ones. . .
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\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{scolium}{Scolium} %% And a not so common one.
\newtheorem{definition}{Definition}
\newenvironment{proof}{{\sc Proof:}}{~\hfill QED}
\newenvironment{AMS}{}{}
\newenvironment{keywords}{}{}
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% % The first thanks indicates your affiliation
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% % Just the name here.
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% % Your mailing address goes at the end.
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% % \thanks is also how you indicate grant support
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\begin{document}
\newpage
\maketitle
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% abstract, keywords and Subject classification are optional.
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\begin{abstract}
This is a sample file.
You can use it as a guide for your submission.
\end{abstract}
% Most people don't use these, so they are "commented out"
% by starting the lines with a "%"
%\begin{keywords}
% \LaTeX, typesetting
%\end{keywords}
%\begin{AMS}
% 50C60, 18C25
%\end{AMS}
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% % Here is the start of the Text
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\section{\LaTeX}
Use sectioning commands for headings. Often longer articles are divided into a few sections.
\LaTeX knows that a new paragraph has started if you skip a line in the input file.
It will automatically indent the proper amount.
You must use labelling commands, e.g. \verb2\label2, \verb2\ref2,
\verb2\bibitem2, and \verb2\cite2, to refer to sections of your document, such as
see Section~\ref{AbelSection}, see Figure~\ref{SelfdualFig}, or bibliography entries,
such as see~\cite{MyFavorite}. Otherwise the look of the numbers, and sometimes the numbers
themselves, will be wrong in the final version at the printer.
\section{Abel's Theorem}\label{AbelSection}
In this section we give some background. For instance
the following definition.
\begin{definition}
A {\em semi-quaver} is defined to be half a quaver.
\end{definition}
If that definition is not enough, here is another:
\begin{definition}
The {\em order} of a note $n$ in a quaver $Q$,
$O(n,Q)$, is defined by the equation
$$ O(n,Q) = \int_{0}^{\infty} \sin(n^{2} t)/(1-Qn) dt. $$
\end{definition}
Some equations one writes inline, such as
{\em Pythagoras' Theorem,}
$c^{2} = a^{2} + b^{2}$,
while others are better off as displayed
equations, like the quadratic formula,
$$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$
which solves the quadratic $ax^{2} + bx + c = 0$. If the quadratic
formula is written inline,
$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$,
it is readable but not very nice.
This form,
$x = (-b \pm \sqrt{b^{2} - 4ac})/2a$,
is harder to read as a fraction, but better because of the larger type.
Every inline equation must be part of a sentence: Since $x < 1/2$ we have
$x + y < x + 1/2$. Inline fractions, such as $x < \frac{1}{2}$, are discouraged but
not prohibited.
You can use formulae in theorems, as in the following.
\begin{theorem}\label{ContThrm}
If $f(x)$ is defined by the equation
\begin{equation}\label{SplitFunc}
f(x) = \left\{ \begin{array}{rl}
x^{2}, & \mbox{~for $x \geq 0$.} \\
-x^{2}, & \mbox{~for $x < 0$}
\end{array} \right.
\end{equation}
Then $f(x)$ is continuous at $x=0$.
\end{theorem}
\begin{proof}
Since
$\lim_{x \rightarrow 0^{-}} f(x) =
\lim_{x \rightarrow 0^{-}} -x^{2} = 0$
and
$\lim_{x \rightarrow 0^{+}} f(x) =
\lim_{x \rightarrow 0^{+}} x^{2} = 0$
it follows that
$\lim_{x \rightarrow 0} f(x) = 0 = f(x)$,
as required.
\end{proof}
Did you notice the grammatical error in
Theorem~\ref{ContThrm}? The sentence leading
into equation~\ref{SplitFunc} is never completed.
%%%% Notice references are done with \ref
%%%% you can label any structure that produces a number
The following
theorem is worded correctly.
\begin{theorem}
If $f(x)$ is defined by the equation
\begin{equation}
f(x) = \left\{ \begin{array}{rl}
x^{2}, & \mbox{~for $x \geq 0$} \\
-x^{2}, & \mbox{~for $x < 0$}
\end{array}\right.,
\end{equation}
then $f(x)$ is continuous at $x=0$.
\end{theorem}
One rule of thumb of mathematical composition is to
use mathematical notation inside sentences only for
nouns. For example, one writes that ``$R$ is the the
radius of a circle'', but not that ``the radius of the
circle ='s the side length of the square''. According to
this rule it is correct to write that
``$(x>0) \Rightarrow (x^{3} > 0)$'' since the double arrow is part of
an equation, but not to write
``$x$ is positive $\Rightarrow$ $x^{3}$ is positive'', since the
double arrow is acting as a verb.
Here is another kind of common structure:
\begin{theorem}
The following are equivalent.
\begin{enumerate}
\item $a \leq b$
\item $b \geq a$
\item $a = b$ or $a