Generation, t-designs and other mathematical notation

Donald Knuth, a mathematician (his adviser was Marshall Hall, his thesis on algebraic structures related to projective planes) turned computer scientist, became dissatisfied with typesetting while producing his multi-volume The Art of Computer Programming. So he took time off to produce a computer typesetting system which would satisfy his high standards. The result was \TeX, or TeX as it is written in plain text, which he gave to the mathematical community. The name is derived from the first three letters τεχ of the Greek word “techne”, carrying the suggestion of both art and technology. (In the first chapter of The TeXbook, Knuth gives instructions on how to pronounce the name of his program.)

TeX has remained almost unchanged since the late 1970s, and still produces much higher-quality mathematical typesetting than more recent word-processors. It and its derivatives such as LaTeX are now so standard in mathematical publishing that journals specify that manuscripts should be in LaTeX, and often provide style files for the purpose. The inventor of LaTeX, Leslie Lamport, neglected to give instructions on how it should be pronounced, with the result that this is somewhat controversial.

TeX is not WYSIWYG. It gives users the possibility to “create masterpieces of the publishing art”, as its creator said; but also allows various horrors, since the computer cannot divine the operator’s intentions.

One of my pet aversions is the use of “less than” and “greater than” for angle brackets. Suppose I have a group G generated by two elements a and b. If I say, G=\langle a,b\rangle in a mathematical formula, TeX gives me G=\langle a,b\rangle, as I want. But lots of people type the shorter expression G=<a,b>, which produces G =<a> . [Sorry, the b has disappeared here, I don't know why!] If you look at this, you will see that TeX has interpreted =< as a mathematical relation, and surrounded the compound symbol by space. The formula begins with something which I suppose is G ≤ a, and the rest of the formula [even if correctly rendered] makes no sense for there is nothing related to b.

I was once asked to review a new mathematics journal for a librarians’ journal. I had to point out that, since the authors effectively typeset their own papers, they were able to produce horrors like complicated fractions in exponents, which are very hard to read and parse. (Incidentally, in traditional publishing, copy-editors were there to save us from this; but publishers have given up on this important function.)

Other problems with mathematics are not entirely the fault of the typesetting system.

A t-(v,k,λ) design (or, for short, a t-design, consists of a set of v points, with a collection B of k-element subsets called blocks, such that each set of t points is contained in a unique block. The concept was introduced by Dan Hughes in the early 1960s (though the case t = 2 was familiar to statisticians much earlier). Dan credits Donald Higman with inventing the terminology. I once asked Dan about the correct way to typeset this. His reply was, more or less, $t$-$(v,k,\lambda)$, producing t-(v,k,\lambda). But it is very common to find that, to save typing two characters, authors write $t-(v,k,\lambda)$, giving t-(v,k,\lambda): the hyphen has become a subtraction sign and is surrounded by space as a mathematical operator.

(Incidentally, Dan Hughes and Don Higman were early mentors of mine, to whom I owe a great debt. Don was my doctoral examiner in Oxford, and as I student I took and wrote up notes of his lectures on coherent configurations. Dan twice offered me a job, and also persuaded me to write my first book, with Jack van Lint. They both organised regular Oberwolfach programs at which I was a regular attendee and learned a lot.)

There is another problem with the notation, the same as for the group theorists’ usage of p-groups. A 2-design is a t-design with t = 2. But other concepts have been introduced, such as Ryser’s λ-designs; should a 2-design be a λ-design with λ = 2? It gets worse. Arnold Neumaier allowed t to be an integer plus a half; the title of his paper was t½-designs, or more precisely t\frac{1}{2}-{} designs. Oh dear.

There is another example of this. To most of the combinatorial world, a k-graph is something a bit like a graph, except that an edge contains k vertices rather than just two. (This has one of the problems I just discussed, if a number is substituted for k.) But the notion of two-graph was defined by Graham Higman, as a 2-cocycle (mod 2) on the simplex, that is, a collection of 3-element sets with the property that any 4-element set contains an even number of them. These are very important objects (another story), and I think that Higman wrote the word “two” to discourage such substitutions. In this he was not completely successful, and in any case, this is now a source of some confusion …

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More on open access

The European Mathematical Society Newsletter has often discussed new developments in academic publishing, in particular open access. In the current issue, it returns to the fray with four somewhat contrasting opinions from a two semi-retired academics, two editors of Zentralblatt MATH, and a mathematical publisher.

Garth Dales is the most uncompromising. After asking “Why is it unacceptable that the author or their institution pays for publishing?” he comments,

Although this seems quite evident, many colleagues do not seem to see clearly the serious dangers of this model,

and proceeds to list some of them. Later, he says,

Another argument comes from a comparison with novels: authors publishing at their own expense are not considered real writers. Curiously enough (but is it that curious?), commercial publishers claim that publishing is a service to authors that will help them in their careers and THUS authors should pay for this! And nobody seems to burst out laughing …

These strong opinions are backed up by facts from the zbMATH editors, Gert-Martin Greuel and Dirk Werner, who are at the sharp end. “The number of OA journals indexed in zbMATH has soared from 180 in 2005 to just short of 500 in 2012.” Zentralblatt clearly does not have the resources to review every published paper, but when there is clear evidence of lack of refereeing – for example, being spoofed by mathgen-produced papers, or publishing “A complete simple proof of the Fermat’s last conjecture” [sic] – they de-list the offending journal. But it is a difficult line: at least one subscription-based journal has also published a paper by mathgen.

Andrew Odlyzko, a man with impressive credentials in electronic publishing as well as mathematics, sees the current situation as part of a process whose end we cannot foresee. He is sure that some form of open access will prevail, and the cost of publishing will somehow be paid in the savings made by libraries. I wish I were as confident that these savings will be redirected appropriately. I have seen too many instances of universities receiving money earmarked for one purpose and spending it on something completely different.

Finally, the publisher, Klaus Peters, raises a number of difficulties with open access which have maybe not been fully discussed. To choose just two examples, self-archiving elimiates the good (as well as the harm) done by copy-editors, and journals who “have invested in the expensive editing process” may be reluctant to give long-term permission for self-archiving of these papers.

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From the archive, 8

Various diaries came to light while I was office-clearing last Christmas, most notably a diary I kept for five months, from December 1986 to May 1987. I had just started work at Queen Mary College (as it was then), and celebrated my new job by applying for a place in the London Marathon; the diary begins on the day the acceptance arrived, and ends the day after the race, in which I ran 2hrs 46mins 59secs and came 957th out of a field of over 20000.

I have just typed it up (it runs to 105 pages, so nobody wants to read it). I found the experience very evocative; apart from the training, and the new job, I was facing commuting from Oxford to London (four hours travel every workday), and doing my share of childminding (the children were 11, 9, and 7 in the time, and all sang in choirs).

The diary records day to day events, running and other injuries, reflections on my running on the past and comparisons with the present, etc.

I ran the London Marathon again the following year; the year after that, I sat on the sofa and watched it on television, and was thoroughly put in my place when I saw Gareth Jones (who was a student with me in Oxford, and at the time did no sport while I ran in the University cross-country team and earned a Full Blue) finish in a time of somewhere round 2hrs 25mins.

The exercise of typing up the diary might inspire me to start running again when I am back in St Andrews; it is in many ways a more attractive place for running than London.

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A letter to the Guardian

Don Braben is one of my heroes. He has a clear-headed but deeply-held belief, with which I agree, that the bureaucracy of modern science gets in the way of getting good science done, at a time when the world needs good clear science more than ever. He was a leader in the (mostly unsuccessful) campaign against “Impact” in the REF.

So I am very happy to be a signatory of a letter published in yesterday’s Guardian on this subject. Here is the text of the letter:

“Science is the belief in the ignorance of experts,” said Richard Feynman in the 1960s. But times change. Before about 1970, academics had access to modest funding they could use freely. Industry was similarly enlightened. Their results included the transistor, the maser-laser, the electronics and telecommunications revolutions, nuclear power, biotechnology and medical diagnostics galore that enriched the lives of virtually everyone; they also boosted 20th-century economic growth.

After 1970, politicians substantially expanded academic sectors. Peer review’s uses allowed the rise of priorities, impact etc, and is now virtually unavoidable. Applicants’ proposals must convince their peers that they serve national policies and are the best possible uses of resources. Success rates are about 25%, and strict rules govern resubmissions. Rejected proposals are usually lost. Industry too has lost its taste for the unpredictable. The 500 major discoveries, almost all initiated before about 1970, challenged mainstream science and would probably be vetoed today. Nowadays, fields where understanding is poor are usually neglected because researchers must convince experts that working in them will be beneficial.

However, small changes would keep science healthy. Some are outlined in Donald Braben’s book, Promoting the Planck Club: How Defiant Youth, Irreverent Researchers and Liberated Universities Can Foster Prosperity Indefinitely. But policies are deeply ingrained. Agencies claiming to support blue-skies research use peer review, of course, discouraging open-ended inquiries and serious challenges to prevailing orthodoxies. Mavericks once played an essential role in research. Indeed, their work defined the 20th century. We must relearn how to support them, and provide new options for an unforeseeable future, both social and economic. We need influential allies. Perhaps Guardian readers could help?

The signatories are: Donald W Braben, University College London; John F Allen, Queen Mary, University of London; William Amos, University of Cambridge; Richard Ball, University of Edinburgh; Tim Birkhead FRS, University of Sheffield; Peter Cameron, Queen Mary, University of London; Richard Cogdell FRS, University of Glasgow; David Colquhoun FRS, University College London; Rod Dowler, Industry Forum, London; Irene Engle, United States Naval Academy, Annapolis; Felipe Fernández-Armesto, University of Notre Dame; Desmond Fitzgerald, Materia Medica; Pat Heslop-Harrison, University of Leicester; Dudley Herschbach, Harvard University, Nobel Laureate; H Jeff Kimble, Caltech, US National Academy of Sciences; Sir Harry Kroto FRS, Florida State University, Tallahassee, Nobel Laureate; James Ladyman, University of Bristol; Nick Lane, University College London; Peter Lawrence FRS, University of Cambridge; Angus MacIntyre FRS, Queen Mary, University of London; John Mattick, Garvan Institute of Medical Research, Sydney; Beatrice Pelloni, University of Reading; Martyn Poliakoff FRS, University of Nottingham; Douglas Randall, University of Missouri; David Ray, Bio Astral Limited; Sir Richard J Roberts FRS, New England Biolabs, Nobel Laureate; Ken Seddon, Queen’s University of Belfast; Colin Self, University of Newcastle; Harry Swinney, University of Texas, US National Academy of Sciences; Claudio Vita-Finzi FBA Natural History Museum.

Please put your suggestions, stories, and ideas about how such people can be supported outside the funding mainstream here; I will make sure they get heard.

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Combinatorics in Scotland, group theory in Portugal

I never really wanted to retire. For various reasons which no longer matter, I decided to retire from my position at Queen Mary, University of London, on turning 65 two years ago. I hoped that I would find enough to do to keep me busy, and I have not been disappointed!

Indeed, I have substantially overdone things, and yesterday began teaching my fourth course this year …

Combinatorics in Scotland

This semester I have been teaching a course on Advanced Combinatorics at St Andrews. When they asked me for a syllabus for such a course, I provided three (roughly, one on enumeration, one on finite geometry, and one on group actions), and suggested that one of them could be approved and I would teach that. Instead, they approved all three, so I had to decide which one to teach (and I have the option of teaching the other two in successive years).

So this year it is enumeration. I felt fairly secure, having taught a much briefer course on enumeration at the London Taught Course Centre last semester. But, inevitably, this course is turning out a bit different. Something about the set-up encourages me to ramble on, and tell stories, about Paul Erdős, or Alan Sokal, or someone else. I spent much longer on basics of formal power series, since the class seemed to be well prepared in Analysis, and needed convincing that it really didn’t matter if the series converge or not. (My answer to that is in two parts: the good news is that it doesn’t matter, since the formal power series form a ring in which various other operations, such as substitution, infinite sums and products, and differentiation, all work fine (sometimes under specified conditions), and our manipulations are valid in this setting without the need for convergence; the good news, on the other hand, is that any identity between convergent power series expressed in terms of these operations is also valid in the setting of formal power series, so we can just read off from Analysis all properties of, for example, the exponential and logarithm functions that we need.)

I varied enumeration with a small dose of combinatorics of subsets early on, basically Ramsey’s theorem and Steiner systems, the second giving the opportunity to mention Peter Keevash’s result. (These are two topics where some of the formulae involve binomial coefficients, which had been discussed in the first part of the course.)

In fact, it has not quite gone as I expected. We have talked about subsets but not yet about partitions or permutations; we haven’t done any asymptotics yet, or group actions, or unimodality, and probably several of these topics are not going to get in. Yet I included things like the counting proof of the existence of finite fields, and the inclusion-exclusion formula for the chromatic polynomial of a graph; and if time permits I will do Richard Borcherds’ wonderful proof of Jacobi’s Triple Product Identity by counting states of Dirac electrons.

We are about three-quarters of the way through the course now. It is Spring break, so I have a couple of weeks to get myself ahead and produce the last few instalments of notes.

Group theory in Portugal

Meanwhile, João Araújo persuaded me to teach a course on Group Theory at the Open University in Portugal. The course started yesterday; little has happened yet apart from the students starting to introduce themselves. Already it is clear that I am not very competent with the Moodle interface (well, I did know that already – of course in St Andrews, the first thing I did was to make a course web page!), but I have two competent pairs of hands on the spot to catch me if I fall.

I like the idea of teaching open university students. If one can make invidious comparisons, they are on the whole more committed and enthusiastic than regular university students, since they are making very big sacrifices to study in their own time. I am, of course, a bit nervous that, being quite far away and not speaking their native tongue, I will give them a less good course than they deserve.

But so far, I am fairly happy with the notes I have produced. I started off knowing that there were certain things that should be put in, but after a while I came to see that putting in the things that I would like to be put in would work better. Group theory is a very technical subject; I am trying to give them an overview of the things I am interested in (of course, this means quite a big dose of group actions) without getting too bogged down. So, for example, I will cover doubly transitive groups but may not get on to saying much about primitive groups. (With doubly transitive groups, you get quite quickly to some pretty things.)

Many of these students have a strong background in computers, either a first degree or a job in IT, and several of them are skilful programmers. Maybe I will be able to set them some projects with real content, and get some good stuff done!

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Note on infinity

A common caricature of the view of the mediaeval scholastics is that they wondered whether the number of angels who can dance on the head of a pin is infinite or not. In fact, this calumny was invented much later.

But another common view is that, after Aristotle told people that it was forbidden to think about completed infinities, nobody did so until Cantor broke the barrier.

I don’t think there was ever a time when people didn’t think about infinity. So I was interested to discover the metaphysical poet John Donne, in his poem Love’s Growth, perplexed by the question whether it is possible to make an infinite set bigger by adding something to it:

Methinks I lied all winter, when I swore
My love was infinite, if spring make it more.

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Bayes again

It is always a pleasure to read David Colquhoun’s posts.

The most recent explains a simple statistical point that still escapes many health adminisitrators (and others). He describes two tests for Alzheimer’s disease. The first (which I will discuss) is actually a test for mild cognitive impairment (MCI), “a condition that may, but often isn’t, a precursor of Alzheimer’s disease”. This condition has prevalence 1% in the population; the new test has specificity 95% (so only 5% probability of a false positive) and sensitivity 80% (so 20% probability of a false negative). With the help of a tree diagram, he calculates that if the test were used for screening (as is proposed, apparently), 86% of people testing positive would not have the disease.

He is righteously (and rightly) indignant that everything from the journal’s press release to NHS Choices seems to ignore this, which as he says makes the test “worse than useless”.

This is a simple application of Bayes’ Theorem. I taught Probability to the first-year maths students for many years, and calculations like this were a standard example that I used.

How many times do you think Colquhoun mentioned Thomas Bayes (or Richard Price) in his article?

So this post is really a musing on the vagaries of fame in mathematics.

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