Archive for April, 2012

A look at a few Tripos questions III

April 30, 2012

Here’s another one.

10F. State without proof the Integral Comparison Test for the convergence of a series \sum_{n=1}^\infty a_n of non-negative terms.

Determine for which positive real numbers \alpha the series \sum_{n=1}^\infty n^{-\alpha} converges.

In each of the following cases determine whether the series is convergent or divergent:

(i) \displaystyle \sum_{n=3}^\infty \frac 1{n\log n},

(ii) \displaystyle \sum_{n=3}^\infty \frac 1{n\log n(\log\log n)^2},

(iii) \displaystyle \sum_{n=3}^\infty \frac 1{n^{1+1/n}\log n}.

A look at a few Tripos questions II

April 28, 2012

This is the second in a series of posts that started here. In the first post I explained what I’m up to. Now let me just continue with some more questions. I’m now on to the harder Section II questions. Here’s the first one I want to look at. Even though it makes the posts shortish, I think I’m going to stick to one long question per post.

9F. Prove the Axiom of Archimedes.

Let x be a real number in [0,1] and let m,n be positive integers. Show that the limit

\displaystyle \lim_{m\to\infty}[\lim_{n\to\infty} \cos^{2n}(m!\pi x)]

exists, and that its value depends on whether x is rational or irrational.

[You may assume standard properties of the cosine function provided they are clearly stated.]

A look at a few Tripos questions I

April 28, 2012

When I was a mathematics undergraduate, I became aware of a huge cultural difference between mathematicians and engineers. That sounds like the beginning of a joke you’ve heard twenty times already, but it isn’t. The difference was that when mathematicians were set questions, they were expected to work out how to solve them, and if they couldn’t do so then it was too bad — the best they could do about it was ask their supervisors. But engineers had model answers for everything, available with the latest technology, which in those days was microfiche. In case you have no idea what I’m talking about, the answers were reduced in size by a factor of about five in each direction and printed on to some kind of transparent plastic that you could look at through a magnifying machine. There were a couple of the machines in our college library, and they were nearly always in use.

Model answers have always seemed to me to be a bad idea in mathematics, because it is hard to learn how to think for yourself when you are given the answers to all the problems you tackle. So it might seem a bit odd that in this post I’m going to attempt to help people preparing for Part IA of the Cambridge Mathematical Tripos by providing some model answers.

Polymath paper published

April 23, 2012

I’m glad to be able to report that “A new proof of the density Hales-Jewett theorem” has recently appeared in Annals of Mathematics. Unfortunately it’s behind a paywall, but you can find an almost final version on the arXiv.

I might add that my enthusiasm for this way of working is undimmed. The reason there has been no Polymathematical activity on this blog for quite a while is that I’ve been busy with more conventional projects, but in the not too distant future I’d like to do some more open research. Also, Gil Kalai and I have a plan to try soon to revive the EDP project. I won’t say any more about that now, but it seems a good moment to mention it.

A brief EPSRC update

April 13, 2012

Last summer I wrote a post about EPSRC’s plans to direct their funding towards certain areas and not others, and in particular on its effect on mathematicians, the most dramatic of which was to restrict their fellowships, which had previously been available throughout mathematics, to statistics and applied probability. The strongest argument I could see in favour of EPSRC’s position was that they were reviewing the various subareas of mathematics before deciding which should be grown, which maintained and which shrunk, and that so far only statistics and applied probability had been reviewed (with a decision that it should be grown).

It was of course a bizarre decision to remove fellowships entirely from areas that have not yet been reviewed — the obvious thing to do would surely have been to maintain the status quo in an area until the review was complete — but they promised that the reviews would be completed in November, so at least one could hope that this decision would represent no more than a brief hiatus.