Archive for the ‘Cambridge teaching’ Category

A look at a few Tripos questions VII

May 20, 2012

The obligatory question on countability/uncountability.

5C. Define what is meant by the term countable. Show directly from your definition that if X is countable, then so is any subset of X.

Show that \mathbb{N}\times\mathbb{N} is countable. Hence or otherwise, show that a countable union of countable sets is countable. Show also that for any n\geq 1, \mathbb{N}^n is countable.

A function f:\mathbb{Z}\to\mathbb{N} is periodic if there exists a positive integer m such that, for every x\in\mathbb{Z}, f(x+m)=f(x). Show that the set of periodic functions f:\mathbb{Z}\to\mathbb{N} is countable.

A look at a few Tripos questions VI

May 11, 2012

I’m now going to turn to the Numbers and Sets questions from the same year, 2003. I’ve lost count of the number of times I’ve heard people say that the course is quite easy but the questions on the examples sheets and exams are very hard and “not very closely related to the course”. There is a grain of truth in that: the new concepts you have to grasp in Numbers and Sets are not as difficult as the new concepts you have to grasp in most of the other courses, so in order to give enough substance to Tripos questions the examiners are almost forced to put in a significant problem-solving element. However, certain styles of problem occur quite regularly, so it’s good to get a bit of practice. And perhaps a detailed discussion of the 2003 questions will be helpful as well. As I did with Analysis I, I’ll start with a post on the Section I questions and then I’ll have separate posts for each of the four Section II questions. The paper, by the way, is Paper 4.

A look at a few Tripos questions V

May 8, 2012

Here is the final analysis question from 2003.

12C. State carefully the formula for integration by parts for functions of a real variable.

Let f:(-1,1)\to\mathbb{R} be infinitely differentiable. Prove that for all n\geq 1 and for all t\in(-1,1),

\displaystyle f(t)=f(0)+f'(0)t+\frac 1{2!}f''(0)t^2+\dots
\displaystyle \dots+\frac 1{(n-1)!}f^{(n-1)}(0)t^{n-1}+\frac 1{(n-1)!}\int_0^tf^{(n)}(x)(t-x)^{n-1}dx.

By considering the function f(x)=\log(1-x) at x=1/2, or otherwise, prove that the series

\displaystyle \sum_{n=1}^\infty\frac 1{n2^n}

converges to \log 2.

A look at a few Tripos questions IV

May 2, 2012

This post belongs to a series that began here. Next up is a question about integration.

11B. Let f:[a,b]\to\mathbb{R} be continuous. Define the integral \int_a^bf(x)dx. (You are not asked to prove existence.)

Suppose that m, M are real numbers such that m\leq f(x)\leq M for all x\in [a,b]. Stating clearly any properties of the integral that you require, show that

\displaystyle m(b-a)\leq\int_a^bf(x)dx\leq M(b-a).

The function g:[a,b]\to\mathbb{R} is continuous and non-negative. Show that

\displaystyle m\int_a^bg(x)dx\leq\int_a^bf(x)g(x)dx\leq M\int_a^bg(x)dx.

Now let f be continuous on [0,1]. By suitable choice of g show that

\displaystyle \lim_{n\to\infty}\int_0^{1/\sqrt{n}}nf(x)e^{-nx}dx=f(0),

and by making an appropriate change of variable, or otherwise, show that

\displaystyle \lim_{n\to\infty}\int_0^1nf(x)e^{-nx}dx=f(0).

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.

Group actions IV: intrinsic actions

December 10, 2011

I have a confession to make. When I was an undergraduate at Cambridge (hmm, that sounds as though it might be the beginning of quite an interesting confession, so I’d better forestall any disappointment by saying right now that it isn’t), there was a third-year course in group theory, taught by John Thompson no less, on which I did not do very well. For a few weeks it seemed to cover material that we’d done in our first year, and then suddenly it got serious, with things like the Sylow theorems. And at that point I got lost, and was unable to do the questions on the examples sheets. I can’t remember much about the questions, but I think my difficulty was that there was a slightly indirect style of proof that caused me to find arguments hard to remember and even harder to come up with. And I never got round to doing anything about it: I went into a different area of maths, and even now I don’t know the proofs of the Sylow theorems. In fact, I don’t even know the statements, though I know they’re about the existence of subgroups of various cardinalities, and I know that they are proved using cleverly defined group actions. I’ve skim-read the proofs, so I have a fairly good idea of their flavour, but I don’t know the details. In particular, I don’t know which action does the job.

A short post on countability and uncountability

November 28, 2011

There is plenty I could write about countability and uncountability, but much of what I have to say I have said already in written form, and I don’t see much reason to rewrite it. So here’s a link to two articles on the Tricki, which, if you don’t know, is a wiki for mathematical techniques. The Tricki hasn’t taken off, and probably never will, but it’s still got some useful material on it that you might enjoy looking at. The articles in question are one about how to tell almost instantly whether a set is countable and another about how to find neat proofs that sets are countable when they are.

Group actions III — what’s the point of them?

November 25, 2011

Somebody told me recently that a few years ago they had a supervision with a colleague of mine (who shall remain nameless, but he or she is an applied mathematician) and asked what the point of group actions was. “I have absolutely no idea,” was the response, and the implication that one might draw from it was apparently intended.

No pure mathematician could hold such a view. I’ve stated a few times that group actions tell you a lot about groups. In this post I want to try to explain why that is, though there is far more to say than I am capable of explaining, let alone fitting into one blog post.

Several proofs that use group actions seem to depend on almost magically coming up with an action that just happens, when you analyse it the right way, to tell you what you wanted to know. I am not an algebraist and do not have a good all-purpose method for finding actions to prove given statements. I don’t rule out that such a method might exist, at least for reasonably simple statements, and would be interested to hear from anybody who thinks they can usefully add to what I have to say.