# The Serial, or Oxford, Comma

In the sentence

The great historical heroes of applied mathematics include Archimedes, Newton, Euler, and Gauss.

the comma before the “and” is known as a serial comma. Whether or not to include it is a matter of style.

The serial comma is also known as the Oxford comma, because Oxford University Press style rules require it to be present. The Chicago Manual of Style (CMS) requires the serial comma, as does SIAM, which follows the CMS recommendations and explicitly states, in the SIAM Style Manual, “Use the serial comma before the and or or in lists of three or more items.”

Other organizations, such as the New York Times, The Economist, and the University of Oxford, require that the serial comma is used only when necessary to avoid ambiguity. Consider the sentence

Three important techniques in the design of algorithms are bisection, divide and conquer, and recursion.

If the serial comma is omitted the final phrase becomes “are bisection, divide and conquer and recursion”, which will be confusing to anyone who does not know that “divide and conquer” is a technique.

Conversely, the serial comma is sometimes incorrect when it might appear to be optional. In the sentence

The results show that, unlike Algorithm 1, Algorithm 2 and the SVD-based algorithm exhibit forward stable behaviour in all the experiments.

a serial comma must not be put after “Algorithm 2” because the three algorithms do not form a list, so the sentence does not make sense with that extra comma.

Examples such as the last two, where the serial comma either must be used or must not be used, irrespective of style, are relatively infrequent, but they do arise from time to time.

For the last year or two I have been using the serial comma in my papers and books, partly because it is the style of the relevant publishers. In particular, I became accustomed to its use in The Princeton Companion to Applied Mathematics. But I also like the simplicity of the serial comma: I do not have to stop to think whether to use it every time I write a list. For informal writing, such as on this blog, I have not made up my mind which style to use. I think the serial comma would look fussy in the tagline at the top right corner of this page.

In the chapter “Commas the Serial Killer” in his book Making a Point: The Pernickity Story of English Punctuation, David Crystal notes that originally the use of the serial comma was standard, and it was only in the early twentieth century that it started to be avoided, “as part of the trend towards punctuation minimalism”. Interestingly, Crystal uses the serial comma in his book even though the style of his publisher (Profile Books) is to avoid it.

There is a large amount of material on the internet about the serial comma, of which the short post The Oxford, Comma has some good examples of where it is needed, and Wikipedia has a good entry. There is a song “Oxford Comma” by the American rock band Vampire Weekend (thanks to Sam Clark for pointing this out); a video is here, but beware the expletive in the first line of the song. The “comma queen” Mary Norris has produced an excellent video about the serial comma. The serial comma even has its own Twitter account, @IAmOxfordComma.

What better way to support the Oxford comma than by giving up some of your 140 characters for it in a Tweet!

# Typesetting Mathematics According to the ISO Standard

In The Princeton Companion to Applied Mathematics we used the conventions that the constants e (the base of the natural logarithm) and i (the imaginary unit), and the d in derivatives and integrals, are typeset in an upright font. These conventions are part of an ISO standard, ISO 80000-2:2009. The standard is little-known, though there is an excellent article about it in TUGboat by Claudio Beccari, and Kopka and Daly’s A Guide to $\LaTeX$ has a page on the standard (in section 7.4.10 of the fourth edition and section 5.4.10 of the third edition).

The standard goes into great detail about how all kinds of mathematical notation should be typeset. It is unclear how the typesetting choices were made or who was on the technical committees that made them. Nevertheless the recommendations are well thought-out.

The most interesting aspects of the standard concern the use of an upright versus a sloping font, which in practice usually amounts to roman versus italic.

1. Variables and generic functions are written in italic. This, of course, is standard practice.
2. Mathematical constants whose values do not change are written in roman. Thus e, i, and $\pi$ should be in roman font. However, standard $\LaTeX$ fonts do not have upright lower case Greek letters, so an italic $\pi$ is unavoidable.
3. Mathematical functions with a fixed meaning, such as exp and sin, are written in roman. Of course, $\LaTeX$ has such definitions built in for many standard functions, but it is a common error for inexperienced users to write, for example, $sin(x)$ (giving $sin(x)$) instead of $\sin(x)$ (giving $\sin(x)$). The best way to define macros for additional functions is via \DeclareMathOperator, assuming you are using the amsmath package:

\DeclareMathOperator{\diag}{diag}

4. Mathematical operators are written in roman. This includes the d in derivatives and integrals.

Although the second and fourth of these rules are not widely followed, they are appealing in that they distinguish variable quantities from fixed ones.

There are some subtleties and some dubious cases.

• A capital delta may appear in both forms: as an operator, hence roman, as in the forward difference operator $\Delta(f) = f(x+h) - f(x)$; and combined with a letter to denote a variable, hence italic, as in $A + \mathnormal{\Delta}A$ (where in $\LaTeX$ the latter delta is typed as \mathnormal{\Delta}).
• The ISO standard explicitly says that named polynomials, such as the Chebyshev polynomials, should be written in roman: $\mathrm{T}_n(x)$ instead of $T_n(x)$. This certainly follows the rules above, since such polynomials have a fixed meaning, but I have never seen the upright font being used for such polynomials in practice.

I’ve started to use rules 1–4 in my recent papers, most thoroughly in this recent EPrint on matrix functions, and intend to use them in my future writing. In doing so, I am using the following $\LaTeX$ macros, based on those suggested in Beccari’s article.

% The number e'.
\def\eu{\ensuremath{\mathrm{e}}}
% The imaginary unit.
\def\iu{\ensuremath{\mathrm{i}}}
% The differential operator.
\def\du{\ensuremath{\mathrm{d}}}


The \ensuremath is not essential, but it means that you can type \eu, etc., outside math mode—for example, in the phrase “the limit of this sequence is \eu”. You may want to rewrite the \def commands using \newcommand, so that if the \eu command has already been defined an error will be issued:

\newcommand{\eu}{\ensuremath{\mathrm{e}}}


\int_C\frac{\eu^z}{z}\,\du z = 2\pi\iu.


Note that if you are using Beamer with the recommended sans serif fonts then mathrm should be replaced by \mathsf in these definitions.



## 7. Make the Header Contain the Section and Chapter Number and Title

I like to know where I am when I am reading a book, so I expect the page headers to tell me the section number and chapter number, and preferably their titles as well. I cannot understand why some books omit this information. Without it, phrases such as “as discussed in the previous chapter” become harder to follow up, and searching for a particular section is more difficult.

# Top Five Tips on Book Writing

I’ve written four books, and am currently writing and editing a fifth (The Princeton Companion to Applied Mathematics). I am also an editor of two SIAM book series and chair the SIAM Book Committee. Based on this experience here are my top five tips about writing an (academic) book. These cover high level issues. In a subsequent post I will give some more specific tips relating to writing and typesetting a book or thesis.

Book publishers ask prospective authors to complete a proposal form, one part of which asks who is the audience for the book. This is a crucial question that should be answered before a book is written, as the answer will influence the book in many ways.

As an example, you might be contemplating writing a book about the numerical solution of a certain class of equations and intend to include computer code. Your audience might be

• readers in mathematics or a related subject who wish to learn about numerical methods for solving the equations and are most concerned with the theory or algorithms,
• readers whose primary interest is in solving the equations and who wish to have lots of sample code that they can run,
• readers in the previous class who also need to learn the language in which the examples are written.

The choice of content, and how the book is presented, will depend very much on which audience you are writing for.

## 2. Revise, Revise, Revise

Just like a paper, a book draft needs to go through multiple revisions, and you must not be afraid to make major changes at any stage. You may receive constructive criticisms from reviewers of your book proposal, but reviewers may not have time to read the complete manuscript carefully and you should not assume that they have found all errors, typos, and areas for improvement.

## 3. Take Time to Choose Your Publisher

Given the huge effort that goes into writing a book you should take the time to find the right publisher. Discuss your book with several publishers and compare what they can offer in the way of

• format (hardback, paperback, electronic) and, if more than one format, the timescale in which each is made available,
• if the publisher has branches in more than one country, how price and publication schedule will differ between the countries,
• whether you are allowed to make a PDF version of the book freely available on your website, if this interests you,
• willingness to allow you to choose the book design (page size, font, cover, etc.),
• use of colour (which increases the cost),
• royalties (including a possible advance),
• pricing,
• the publisher’s policy on translations,
• copy editing (see the next section),
• time from delivering a completed manuscript to publication,
• marketing (will the book be advertised at all, and if so how?), and
• how long your book is guaranteed to stay in print.

It is perfectly acceptable to submit a proposal to several publishers and see what they are willing to offer. However, it is only fair and proper to make clear to a publisher that you are talking to other publishers and, once you have set the wheels of a publisher’s review process in motion, to wait for an offer before making a decision to go with another publisher.

I am always surprised when I hear of authors who approach only one publisher, or who go with the first publisher to express an interest in the book. As in many contexts, it is best to make an informed choice from among the available options.

## 4. Ensure Your Book is Copy Edited

If you are an inexperienced writer, or your first language is not English, the benefits of copy editing are obvious. But even an experienced author finds it virtually impossible to think about all the little details that a copy editor will check for, such as correctness and consistency of spelling, notation, punctuation (notably the serial comma), citations, and references. For example, I sometimes mix US and UK spellings and don’t want to have to worry about finding and correcting my occasional lapses. A good copy editor will also suggest minor improvements of the text that might escape even the best writers.

Unfortunately, not all publishers copy edit all books nowadays. Notable exceptions that always do copy edit (and, as I know from experience, work to the highest standards in every respect) are Princeton University Press and SIAM.

If your publisher has a Style Manual it obviously makes sense to follow its guidelines in order to minimize changes at the copy editing stage. Here is a link to the SIAM Style Manual.

## 5. Think Twice Before Co-Authoring a Book

It might seem an attractive proposition to share authorship of a book: surely having $n$ co-authors reduces the work by a factor $1/n$? Unfortunately it often does not work out like that, despite best intentions. In fact, $n$ co-authors can easily take $n$ times as long to write a book as any one of them would. One of the biggest difficulties is timescale: one author may be willing and able to finish a book in a year but another may need twice that period to make their contribution. Indeed it is rare for the co-authors to be matched in the amount of effort they can put into the book; this is clearly problematic if initial expectations are not realized. Other potential problems are potentially differing opinions on content, notation, level, length, and almost anything else associated with a book.

Successful authorship teams often have a track record of co-authoring papers together. Although it is no guarantee that a much larger book project will run smoothly, experience with writing papers together will at least have given a good indication of where disagreements are likely to lie.

# The Spotlight Factor

In my Handbook of Writing for the Mathematical Sciences I described the spotlight factor, originally introduced by Tompa in 1989. The spotlight factor is defined for the first author of a paper in which there are $n$ authors listed alphabetically, and it is assumed that the paper is from a community where it is the custom to order authors alphabetically.

The spotlight factor is the probability that if $n-1$ coauthors are chosen independently at random they will all have surnames later in the alphabet than the first author. This definition is not precise, since it is not clear what is the sample space of all possible names, so it is better to regard the spotlight factor as being defined by the formula given by Tompa, which is implemented in the MATLAB function below.

The smallest spotlight factor I have found is the value 0.0244 for Zielinski, for the paper

Pawel Zielinski and Krystyna Zietak, The Polar Decomposition—Properties, Applications and Algorithms, Applied Mathematics, Ann. Pol. Math. Soc. 38, 23-49, 1995

This beats the best factor of 0.0251 reported by Tompa in a 1990 follow-up paper.

Can you do better?

Here is a MATLAB M-file to compute the spotlight factor, preceded by an example of its usage:

>> spotlight('zielinski',1)
ans =
2.4414e-02

function s = spotlight(x, k)
%SPOTLIGHT   Tompa's spotlight factor of authorship.
%   SPOTLIGHT(X, K) is the spotlight factor for the author whose
%   last name is specified in the string X, with K coauthors.
%   Mixed upper and lower case can be used.
%   Smaller spotlight factors correspond to rarer events.

%   Reference:
%   Martin Tompa, Figures of Merit, SIGACT News 20 (1), 62-71, 1989

if ~ischar(x), error('First argument must be a string.'), end
if nargin < 2, error('Must give two arguments.'), end

x = double(upper(x)) - double('A') + 1;
x( find(x < 0 | x > 26) ) = 0;  % Handle punctuation and spaces.

s = 0;

% Ideally use Horner's rule, but the following is clearer.

for i=1:length(x)
t = x(i);
s = s + t/27^i;
end

s = (1 - s)^k;
`