I shall soon be delivering the final copies of my thesis to the University of Edinburgh, who will in time make it available in both hard copy and electronically. But for convenience, you can already obtain the pdf version here.

We generalise the study of cyclotomic matrices – those with all eigenvalues in the interval [-2,2] – from symmetric rational integer matrices to Hermitian matrices with entries from rings of integers of imaginary quadratic fields. As in the rational integer case, a corresponding graph-like structure is defined.

We introduce the notion of `4-cyclotomic’ matrices and graphs, prove that they are necessarily maximal cyclotomic, and classify all such objects up to equivalence. The six rings of integers for Q(√(d)) for d=-1, -2, -3, -7, -11, -15 give rise to examples not found in the rational-integer case; in four (d=-1, -2, -3, -7) we recover infinite families as well as sporadic cases.

For d=-15, -11, -7, -2, we demonstrate that a maximal cyclotomic graph is necessarily 4-cyclotomic and thus the presented classification determines all cyclotomic matrices/graphs for those fields. For the same values of d we then identify the minimal noncyclotomic graphs and determine their Mahler measures; no such graph has Mahler measure less than 1.35 unless it admits a rational-integer representative.

At The Diverse Faces of Arithmetic there were a pair of (early morning!) overview lectures for postgraduates. I’ve finally got around to typesetting my notes from the first, Tom Ward’s session on recurrence sequences, available as pdf via the above link. The topics included are divisibilty sequences and primitive divisors; linear recurrences; elliptic divisibility sequences; integrability/ Laurent phenomena; growth rates and Lehmer’s problem.

There is also video from the panel discussion and some of the accessible talks in the various themed sessions. All of which should help convince you to sign up for next year’s Young Researchers In Mathematics conference, running 25-27 March again at Cambridge.

You can find my slides here, and an audio recording may become available in the future- the conference was held in Toronto and Vancouver via videoconference (which worked well) so hopefully all the talks will be archived online.

Update 24/x/09: Audio of all talks at Fields (hence, including mine) can now be found on their website.

It’s easy to cook up a dodgy mathematical formula in support of a cause, and that particular flavour of bad science seems fairly popular with the media, so I wanted to set things on a vaguely valid theoretical basis for a change. Plus I knew I’d recently seen a similar question – what was the probability of the 44th President of the United States being a white male? – and its solution at a lecture during Beyond Part III; I just couldn’t remember the result or its originator.

Much googling of half-remembered formulae and likely candidate long-dead French mathematicians later, I’d recovered the answer. The desired theorem is the rule of succession, due to Laplace, and it can be described as follows-

If a trial can only succeed or fail, but nothing is known about the probability of either outcome except that there have been s successful trials out of n in total, then the probability of the next trial being a success is (s+1)/(n+2).

As an immediate corollary, if you know nothing about an event except that so far it has happened n times in a row, then the probability it will happen next time is (n+1)/(n+2). (This more specific version is also sometimes refered to as the rule of succession.) Laplace was trying to solve the sunrise problem: as the sun has risen every day, what is the probability of it rising tomorrow? Armed with the rule, we still require an estimate of how many successful sunrises there have been; Laplace, working in the 18th century, took a literal reading of the bible for this, a practice which still appeals to young earth creationists. But although a more modern figure gives a probablity much closer to 1, it still admits a 1/(n+2) chance of the sun not rising tomorrow.

This has often been used as a criticism of the rule of succession, but as often occurs the problem is more one of inappropriate application of a model than a flaw in the model itself: Laplace himself immediately cautioned that “…[the probability of the sun rising tomorrow] is far greater for him who, seeing in the totality of phenomena the principle regulating the days and seasons, realizes that nothing at present moment can arrest the course of it.”

In other words, our astronomical knowledge means that we have more to go on than just observed sunrises in estimating the chance of another, and we should defer to that. The rule of succession is to be used when you have little or no knowledge of the underlying processes or probability of an event. It’s particularly useful when there have only been a few trials, or no successes have been observed at all – the rule of succession provides a non-zero estimate in that case, which is desirable by Cromwell’s rule.

With the small sample space of a pair of nodermeets, and noder romance being infinitely more mysterious than celestial mechanics, I was thus happy to apply the rule of succession and declare the probability of a third marriage to be 3/4.

Proof of the Rule of Succession
This proof is lifted from here, which is easier to read anyway…

Laplace’s assumptions were

• The event has some chance of happening, between 0 and 1.
• All possible values of this chance, from 0 to 1, are equally
  probable a priori.
• His sixth principle of probability: for E an event, C_1…C_n possible causes of E,
  
  P(C_i|E) = P(E|C_i)*P(C_i) / (Σ_{k=1..n}P(E|C_k)P(C_k)) (this is just Bayes’ Theorem.)
• His seventh principle of probability: for E an event, F a possible future event and C_1…C_n possible causes,
  
  P(F|E)=Σ_k=1..n P(F|C_1)P(C_1|E)

We may then derive the special case of the rule of succession. Let E indicate that the event has occurred n times in a row; F that the event will occur next time; and C_x that the chance of the event occurring is x. The C_x are then considered as the possible causes of the event- so P(E|C_x)=x^n and P(F|C_x) is just x. Since there are infinitely many x in [0,1], we pass from summations to integrals in the sixth and seventh principles to obtain infinite versions and thus find

and so

as claimed.