%Next year: more real examples of originality: e.g. games playing
%examples, noticed mistake in protocol.
%Notation joke: epsilon, a, alpha, t_3.
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\begin{document}
%Title stuff, if required
\title{Writing Mathematics for MSc/MSci Projects}
\author{Mark Wildon}
\date{4 March 2014 \\ \indent This document is available online at \url{http://www.ma.rhul.ac.uk/~uvah099/Maths/ProjectAdvice.pdf}.}
\maketitle
%New section: what is a good project
\thispagestyle{empty}
\section{Project rules / guidelines}
See Section 6 of the postgraduate handbook, available at
\url{http://www.rhul.ac.uk/mathematics/documents/pdf/pgtstudenthandbook.pdf}.
Some main points to note:
\medskip
\begin{itemize}
\item MSc projects should be between 8000 and 16000 words. This usually corresponds
to between 25 and 50 pages of normally typeset text.
\smallskip
\item In \S 6.6 of the handbook there are some things that the dissertation \textbf{must} include.
On the title page please include your: \textbf{name, candidate number, project title and supervisor's name}.
\smallskip
\item The marking template is in \S 6.8. You should
note that 30 out of 100 marks are allocated to the bibliography,
and style, spelling, punctuation and grammar.
\smallskip
\item You can demonstrate originality by creating your own examples,
by giving your own version of existing proofs, by giving original analysis
of existing results and methods, and by making new connections
between published papers. Originality is encouraged and rewarded,
but you are not expected to solve an open
research problem.
\smallskip
\item Your supervisor will comment on drafts. Be sure to
give him or her plenty of time to do this.
\smallskip
\item \textbf{Deadlines:} you \textbf{must} submit your project by
by 2pm on the first Thursday of September (September 4th 2014).
%(Please check the handbook
%for the deadline.)
%2pm on 5 September (the first Thursday of September).
You should already have submitted an outline of your project. Working from an outline will make
the eventual writing up much easier. It also acts as a check that your plans
are realistic. %From the handbook:
%\smallskip
%\begin{quote}
%This should give an introduction to the general subject area and of the
%more specific problems and objectives to be studied. It should mention which literature has been studied so far and how
%the research will continue. This will usually be prepared in consultation with the project supervisor.
%\end{quote}
\end{itemize}
\medskip
\section{Advice on reading and writing mathematics}
\medskip
\subsection*{Reading papers}
Reading mathematical papers or advanced textbooks is hard work and will take a long time.
Try to work through examples in parallel: you can use these later
in your project. Force yourself to think about the material. For example, imagine you
have to give a talk on some part: what are the main points you
would want to hear yourself saying?
\medskip
\subsubsection*{Finding papers}
MathSciNet is the main database of mathematics papers. It also has reviews;
these can be very useful when finding your way into a new field: see
\url{www.ams.org/mathscinet}. (You will need to be on the college network or use VPN.)
%\enlargethispage{24pt}
\thispagestyle{empty}
\medskip
\subsection*{General writing advice}
In summary: \textbf{respect the reader}. Ideally it should be a pleasure
to read your project.
\medskip
\begin{itemize}
\item \textbf{Maybe most important:} Try to create your own examples,
and give your own approach to the material.
Do not just regurgitate the papers you have read. You should feel free to skip
add more details to some arguments, and maybe to skip others. Do not leave
the reader feeling that he or she might just as well have struggled through the
original sources.
\smallskip
\item Think of your reader as an
interested and intelligent friend who knows a reasonable
amount of general
mathematics, but nothing about your particular subject.\smallskip
\item Your project should read like a well-written piece of
English. So write in complete sentences, not in the style
of abbreviated lecture notes. A formal style is appropriate.
Contractions such
as `don't, can't'
and other forms common in speech will grate on the reader.
\smallskip
\item Think about the logical structure. Either write the
introduction last, or be prepared to rewrite it as you proceed.\smallskip
\item Try to provide a convincing narrative. This can be informed
by the problems you are considering, or by the logical structure
of a proof, or by the history of the subject. There should be coherent reason,
explained in the introduction,
for the scope of your project.
\smallskip
\item For each sentence you write, make sure that you understand it, that it is true, and
that it makes sense. Reading aloud is a good way to expose poorly worded sentences.
\smallskip
\end{itemize}
\subsubsection*{Further sources of advice}
There is much good advice available on writing
available free online. In no particular order I recommend:
\begin{itemize}
\item \url{http://web.mit.edu/jrickert/www/mathadvice.html}.\medskip
\item Notes from a course run by Donald Knuth:
\url{http://tex.loria.fr/typographie/mathwriting.pdf}.\medskip
\item Steven G. Krantz, \emph{A Primer of Mathematical Writing},
American Mathematical Society (1997).\medskip
\item T.~W.~K{\"o}rner, \emph{How to Write a Part III Essay},
\url{http://www.dpmms.cam.ac.uk/~twk/Essay.pdf}. \medskip
\item George Orwell, \emph{Politics and the English Language},
\url{http://www.mtholyoke.edu/acad/intrel/orwell46.htm}.
\end{itemize}
\medskip
\noindent For many other links and thoughts see:
\url{http://mathoverflow.net/questions/1243/how-to-write-math-well}.
\medskip
\subsubsection*{It will take longer than you expect\protect\footnote{This is Hofstadter's Law
\cite[page 152]{Hofstadter}.
A stronger version of Hofstadter's Law states that Hofstadter's Law still holds
if one applies it.}}
Even if you have a very good idea what you are going
to say, \textbf{do not expect to
average more than one page of typeset text
per day}. If you have never written a project before, this may seem absurdly low; note however
that at this rate, you will still have your project
written in one month. If you have done something similar before,
then you will be able to make a better estimate.
\bigskip
\subsection*{Specific writing advice}
See also Knuth's notes and the other links above.
%I have probably
%omitted others that are more important.
\medskip
\subsubsection*{Definitions}
Clear and consistent definitions help to unify your project. Some technical
terms are part of the common language of mathematics and do not need definitions
(for example, graphs, prime numbers). If in doubt consult your supervisor.
\medskip
\subsubsection*{Signposts} The outward structure of your
project should be clear to the reader. You are writing a mathematics paper, not
a detective story! Outline what you are going to do
in your introduction. Make forward and backward references to connect
related ideas.
\medskip
\subsubsection*{Notation}
Think carefully about the mathematical notation you use.
You may need to change
it from your sources in order to be consistent throughout
your project.
Mathematical notation is very dense and needs words to pad
it out. Do not start sentences with symbols. So
rather than
\begin{quote}`$C$ has length
$8$ because \ldots'\end{quote}
write
\begin{quote}
`The code $C$ has length $8$ because \ldots'.
\end{quote}
Do not
use punctuation to separate two mathematical clauses. For example
\begin{quote}`Hence $v_i= 0$, $i \in \{1,\ldots, 8\}$ \ldots'
\end{quote} should
be replaced with \begin{quote}
`Hence $v_i = 0$ for $i \in \{1,\ldots, 8\}$ \ldots'.
\end{quote}
Write out numbers when they are used as adjectives in sentences. This helps to avoid
some potential ambiguities. For example,
`there are two symbols equal to $1$ in the codeword 00112'. Do not use
$=$, $\le$, $\ge$ or similar symbols as the verb in a long sentence.
For example,
\begin{quote}
`Hence $|C| \le 4$.'
\end{quote}
is fine. But
\begin{quote}
`Therefore the number of codewords $=5$'
\end{quote}
is terrible compared to
\begin{quote}
`Therefore the number of codewords is $5$.'
\end{quote}
\medskip
\subsubsection*{We versus I}
Say `We' if you are doing something with the reader, for example~`We now
prove that the minimum distance \ldots'. Use `I' more rarely, and only
if you are expressing your personal opinion.
\medskip
\subsubsection*{Suspension}
A small amount of `suspension', as in `we now prove that $d_n/n! \rightarrow 1/\mathrm{e}$
as $n\rightarrow \infty$, where $d_n$ is the number of derangements of $\{1,2,\ldots,n\}$'
can help the text flow, but used unnecessarily
it can be annoying. If there is a lengthy `where \ldots'
clause following a statement, it would probably be better rephrased.
\medskip
\subsubsection*{Displayed equations} Use displayed equations (centred
in the text) for long equations, or any equation that needs to be emphasised.
Each displayed equation should be presented as part of a sentence.
\medskip
\subsubsection*{Punctuation} The semicolon is very tempting, but becomes
ridiculous if overused. A displayed equation at the end of a sentence should
end with a full-stop; a comma may be appropriate if a displayed equation
is in the middle of a sentence.
\medskip
\subsubsection*{Proper nouns}
Write `Chapter 2', `Theorem 3' as these are proper names.
It is still correct to write `in the next chapter', `the
following theorem'.
\medskip
\subsubsection*{Proof reading}
Get a friend to proof read your project for its style and
to catch typographic errors.
\medskip
\subsection*{Avoid accidentally committing plagiarism}
Do not
paraphrase someone else's account without acknowledging that
you are following them closely.
For some
technical definitions, mathematical algorithms or cryptographic protocols, which
have to be specified very precisely, it may be best to quote verbatim, making it clear
you are doing this.
If you use someone else's examples
or ideas, you must say so, and indicate this by a precise reference. Conversely, make it clear when you
have created your own examples, or are adding details to someone else's
account.
\medskip
\begin{quote}
\textbf{Even if you cite the original paper, if you %have
present
as your own words a block of text that follows another author's paragraph and sentence
structure, making only minor word changes or re-orderings, then
you are committing plagiarism.
}
\end{quote}
\medskip
\noindent
If you leave things to the last minute, then it is probably
inevitable you will write something very similar to the original source. So instead leave plenty of time:
read
the material one day, think about it some more on the next day, write your own
account of it later. Doing this will make
your own work seem more interesting to you, and it will lead to a much better project.
\subsubsection*{Citation}
Accurate citation is vital (and worth 10 marks):
refer to specific pages or theorems
in a paper, for example [4, Theorem 3.3], or [7, page~2].
(Imagine you are the reader: you do not want to read an
entire textbook to find the one result the author requires.)
It is often appropriate to mention the author's name in the text,
for example `as Erd{\H{o}}s proved in [1]', but titles and other
information are best left to the bibliography.
Definitions also need citations, unless they are basic and can
be expected to be known to all readers.
Make sure the style of your bibliography is consistent.
Unless the journal specifies otherwise, I use
\smallskip
\begin{quote}
\begin{itemize}
\item[{[1]}] P.~Erd{\H{o}}s,
\emph{On an elementary proof of some asymptotic formulas in the theory of
partitions},
\newblock {Ann.~of Math.}, \textbf{43} (1942) 437--450.
\end{itemize}
\end{quote}
\smallskip
\noindent Here \textbf{43} is the volume number.
If use \LaTeX\ then you can use the bibliography environment
to help you organize references and citations. (See example at end.)
%\enlargethispage{12pt}
\medskip
\section{Typesetting}
Almost every mathematician uses \LaTeX\ to typeset their
papers and lecture notes. I strongly advise you to do the same.
If you use Microsoft Word, or a similar word-processing package, then things may
seem easy at the start, but you risk ending up with a structureless
and unattractive document that gets increasingly hard to edit as
it grows in size. By the end you will wish you had used \LaTeX.
\subsection*{Getting started with \LaTeX}
See \url{http://www.ma.rhul.ac.uk/latex-help}.
\subsubsection*{Installation}
\begin{itemize}
\item
On Mac OS X, I recommend TeXShop: \url{http://pages.uoregon.edu/koch/texshop/obtaining.html}.
\item
Almost any Linux distribution will have it installed already.
\item
On Windows try \url{http://miktex.org/2.9/setup}.
\end{itemize}
\medskip
\subsubsection*{First steps}
\begin{itemize}
\item
Read \emph{The (Not So) Short Introduction to \LaTeX}.
\url{http://www.ctan.org/tex-archive/info/lshort/english/lshort.pdf}.
\item
Rather than start from scratch, you should modify someone else's file.
The \LaTeX\ source for this document
is available from
\url{http://www.ma.rhul.ac.uk/~uvah099/Maths/ProjectAdvice.tex}.
If you look at the end you'll find some small examples of how to
typeset equations. The \LaTeX\ source for the MT5461 notes
is available from \url{http://www.ma.rhul.ac.uk/~uvah099/Codes/MScNotes.tex}.
\end{itemize}
\medskip
\subsection*{Diagrams}
Diagrams and other figures can be very useful to the reader. Drawing
your own figures to illustrate your arguments or examples is time-consuming,
but it is a good way to show your originality. Any drawing program that
can produce scalable graphics in pdf or postscript format is suitable.
There are \LaTeX\ packages, for example, xypic,
that make it easier to draw many sorts of diagrams commonly used in mathematics.
\bigskip
\subsection*{Acknowledgements}
My thanks to many members of the Mathematics Department and ISG
for helpful suggestions.
\medskip
\begin{thebibliography}{99}
\bibitem{Erdos}
P. Erd{\H{o}}s,
\emph{On an elementary proof of some asymptotic formulas in the theory of partitions},
\newblock {Ann.~of Math.}, \textbf{43} (1942) 437--450.
\bibitem{Hofstadter}
D. Hofstadter,
\emph{G{\"o}del, Escher, Bach: An eternal golden braid}, Basic Books 1999,
new edition.
\end{thebibliography}
\end{document}
\newpage
%Move \end{document} to the end of the file to get the examples below.
%For displayed equations (i.e.~centred between text) use \[ and \].
%for aligned displays on equals use \begin{align*} and \end{align*}
Here is a displayed equation:
\[ |C| \le \frac{2d}{n-2d} \quad \text{provided $2d > n$}. \]
And here is an aligned display
\begin{align*}
\sum_{\ell = \lceil n/2\rceil}^{n-k-1} \frac{1}{\ell^2(n-\ell)^2}
&\le \int_{n/2}^{n-k} \frac{\mathrm{d}y}{y^2(n-y)^2} \\
&=
\frac{1}{n^3} \int_{1/2}^{1-k/n} \frac{ \mathrm{d}x}{x^2(1-x)^2} \\
&= \frac{1}{n^3} \int_{1/2}^{1-k/n} \Bigl( \frac{1}{x^2}
+ \frac{1}{(1-x)^2}
+ \frac{2}{x} + \frac{2}{1-x} \Bigr) \mathrm{d}x \\
&= \frac{1}{n^3}\Bigl(- \frac{1}{x} +
\frac{1}{1-x} %- \frac{1}{x}
+ 2\log x - 2\log (1-x)
\Bigr) \Bigl|_{1/2}^{1-k/n} \\
&= \frac{1}{n^3} \Bigl( \frac{n}{k} - \frac{1}{1-k/n} + 2\log (1-k/n)
- 2\log (k/n)\Bigr) \\
&\le \frac{1}{n^2k} + \frac{2 \log (n/k)}{n^3}
\end{align*}
\end{document}