This brief post is to update further a recent post that was itself an update on the situation with EPSRC. The good news is that EPSRC postdoctoral fellowships in mathematics are now available for “intradisciplinary research” (as was already the case with the early career and established career fellowships). I am told that a certain amount of work went on behind the scenes to achieve this: we should be very grateful to the mathematicians involved, and grateful also to EPSRC for being prepared to show a degree of flexibility in this instance. I am also told, though only time will tell how true this is, that the interpretation of the word “intradisciplinary” will be generous, so unless your research is extremely narrow, you should be able to present it in a way that will qualify.
Archive for May, 2012
Since time is short, I am going to discuss a couple of Groups questions but in slightly less detail than I have been giving up to now: instead of working through the questions completely, I’ll try to zero in on the most important points. Because there wasn’t a separate Groups course until 2008, I am taking my questions from that year.
5E. For a normal subgroup of a group , explain carefully how to make the set of (left) cosets of into a group.
For a subgroup of a group , show that the following are equivalent:
(i) is a normal subgroup of ;
(ii) there exist a group and a homomorphism such that is the kernel of .
Let be a finite group that has a proper subgroup of index (in other words, ). Show that if , then cannot be simple. [Hint: Let act on the set of left cosets of by left multiplication.]
Exam day approaches, so I’ve decided to prioritize. Instead of doing question 7C, which isn’t all that interesting (in the sense that it doesn’t give me much scope to emphasize principles of more general use in exams), I’m going to skip it, and instead, if I get time, go through a groups question or two. But first I will do the final Numbers and Sets question because it involves something a lot of people dislike: the inclusion-exclusion principle. It’s worth getting comfortable with this, because it comes up year after year (either in Numbers and Sets or in Probability). You may think that applying the principle requires some ingenuity. The aim of this post is to convince you that it can be done on autopilot.
8C. Let be a finite set with elements. How many functions are there from to ? How many relations are there on ?
Show that the number of relations on such that, for each , there exists at least one with , is .
Using the inclusion-exclusion principle or otherwise, deduce that the number of such relations for which, in addition, for each , there exists at least one with , is
Now for a question on modular arithmetic. As with countability, there is a very high chance of a question on this topic. [Added after the post was written: as usual I wrote down my thoughts about this question as I had them, and I didn't spot the best approach to part (ii) of the question until after I had come up with some less good approaches. So my recommendations evolve through the post, with some of the later ones superseding some of the earlier ones.]
6C. (i) Prove Wilson’s theorem: if is prime then (mod ).
Deduce that if (mod 4) then
(ii) Suppose that is a prime of the form . Show that if (mod ) then (mod ).
(iii) Deduce that if is an odd prime, then the congruence
has exactly two solutions (modulo ) if (mod 4), and none otherwise.
Update 4th June 2012. The petition has now passed 25,000 signatures. It would still be great to push on and reach a significantly higher number by the June 19th deadline.
As you may know, there is a system in the US for setting up online petitions. Any petition that reaches 25,000 signatures in 30 days will be considered by White House staff. Recently, a petition was set up asking the Obama administration to require publications resulting from research paid for by the US taxpayer to be freely available. If such a requirement were to be put in place, it would be a huge boost to the campaign to make all academic research easily accessible.
It became possible to sign the petition last Monday, since when there have been (as I write) 14,303 signatures, well over half the number required. Even if the rate of signing goes down, the target of 25,000 by June 19th will probably be reached. However, the organizers want not just to reach the target but to go well beyond it. It is hoped that that, and reported sympathy for the idea within the Obama administration, will give it a real chance of success. So if you can spare two or three minutes (you have to give your email address so that you can receive an email and confirm your identity, and also read a Captcha), then please do the world a service and add your signature. You do not have to be a US citizen to sign. And please pass the message on.
If you want to read a bit more about the petition and the background to it, then this Guardian article is a good start.
PS During the writing of this short post, there were six new signatures, so the tally now stands at 14,309.
The obligatory question on countability/uncountability.
5C. Define what is meant by the term countable. Show directly from your definition that if is countable, then so is any subset of .
Show that is countable. Hence or otherwise, show that a countable union of countable sets is countable. Show also that for any , is countable.
A function is periodic if there exists a positive integer such that, for every , . Show that the set of periodic functions is countable.
If you read an earlier post of mine about Elsevier’s updated letter to the mathematical community then you may remember that towards the end of the post I claimed that Elsevier was lobbying heavily to have all mention of open access removed from the documents of Horizon 2020, Europe’s “Framework Programme for Research and Innovation”, a claim that was then denied by Alicia Wise, who is Elsevier’s “Director of Universal Access”.
Leaving aside who is right about this (which may depend rather sensitively on the precise words used to describe what happened, not to mention the interpretation of those words), news has broken today in the THE of potentially important developments. It seems that whatever lobbying Elsevier might have gone in for has been to no avail, because open access will be a very significant aspect of Horizon 2020.
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.
Here is the final analysis question from 2003.
12C. State carefully the formula for integration by parts for functions of a real variable.
Let be infinitely differentiable. Prove that for all and for all ,
By considering the function at , or otherwise, prove that the series
converges to .
A natural way that one might hope to bring about a genuine change to the current subscription model where libraries pay through the nose for journals is that (i) we all put our papers on the arXiv and (ii) the libraries conclude, correctly, that the benefits from their very expensive subscriptions do not justify the costs. Bundling across subjects makes this a lot more difficult of course, but it seems that some institutions in Germany do not subscribe to the Freedom Collection (see previous post for a definition), which makes it easier. And now there is an example. The Technical University of Munich mathematics department has put out an announcement that it will cancel all its Elsevier subscriptions by 2013.
Please, if you are considering submitting a paper to an Elsevier journal without putting it on the arXiv, think of the faculty members of TU Munich who will not be able to get access to your papers (or at least not conveniently), and change your mind. If you do, it will also make it easier for other departments and libraries to make similar decisions.