Archive for May, 2010

“[it] will be written with a pencil, not with blood”

May 28, 2010

Just a quicky to highlight a really good article (in English) about the current election – it’s very consistent with everything I’ve heard from people on the ground. The false positive thing is particularly scandalous and rather makes trouser presses pale into insignificance.

(salsa) dancing in the street

May 24, 2010

So, there I was, minding my own business wondering through town for my usual Sunday carajillo in the nearby branch of Juan Valdez when I stumble across what appears to be a large free open air gig in the main square of the town. Naturally I think to myself “free live music – why ever not?” and moved in for a closer look. Next thing you know I find myself bang in the middle of this (yes, I am lost somewhere in that crowd).

Now why can’t British politicians organize free gigs???

Voting and the Harmonic series

May 21, 2010

As noted in a previous post my choice of reading material at the moment is a little restricted. As a result at the weekend I finished reading the cryptically titled Gamma – a book about the fourth most famous constant in mathematics.

Overall the book is relatively dry, but there was one bit that caught my eye – a strategy for deciding how to vote (in many ways a topical subject), the analysis of which uses the harmonic series. Now, I’m certainly not the first person to write a blog post applying a simple mathematical model to the British electoral system, but this still seems worth bring to the attention of anyone who cares to read this. We define H_n:=\sum_{r=1}^{n}\frac{1}{r} (the ‘harmonic series’) It is a common undergraduate exercise to show that H_n\rightarrow\infty as n\rightarrow\infty, the easiest way of seeing this probably being this demonstration due to Oresme. Don’t even think about trying to see this by adding up a few terms – it would take much longer than the age of the universe for this sum to exceed 100. The reason for the painfully slow divergence begin that the sum grows like the natural logarithm, indeed the constant gamma is defined by the somewhat surprising formula \gamma:=lim_{n\rightarrow\infty}(H_n-ln(n)). These concepts play an important role in many areas of mathematics, are closely related to several open problems and have many extremely interesting properties.

Anyway – how to vote! When confronted with n possible choices, how do you choose the best one? You could read every manifesto cover to cover, weigh up all the pros and cons of every possibility and make a perfectly informed decision – a process far too time consuming for anyone to actually go through (the average constituency in the last UK general election had more than 6 candidates in it, besides would you expect an employer to interview EVERY applicant for a job?) Alternatively you could simply choose randomly – an approach that is very unlikely to arrive at the best choice.

An obvious strategy most voters (at least I think) adopt is to simply ignore some of the candidates and focus on just a few of them – most people will pay no attention to, say, any candidates for the BNP, the Communist Party of Britain or the Monster Raving Looney Party, for example. But how many should we ignore? Ignore too few and we’re still confronted with a dauntingly large array of choices, too many and we’re likely to overlook the best person for the job.

Enter the harmonic series!

[On rereading, my paraphrasing isn’t very clear and certainly doesn’t do the mathematics any justice, but here goes.]

As with any model, several clearly false simiplifying assumptions have to be made and so there are numerous holes you can find in the below discussion, though for an innocent bit of fun there’s no real harm done by approximating reality in this way.

Suppose B is the best candidate and we ignore the first r candidates. If B happens to be in the (r+1)th position we are guaranteed to choose them. The probability of this is 1/n. If B happens to be in the (r+2)th position and the candidate in the (r+1)th position is the best one we have considered so far then we will not successfully choose B. We will thus vote for B if the best yet among the first r+1 choices happens to be among the first r of them – an event that occurs with probability r/(r+1). The total probability of successfully choosing B in this case is then r/n(r+1) (remember the probability of B occupying any given position is 1/n). Similarly the corresponding probabilities for the (r+3)rd position etc are

\frac{r}{n(r+2)},\mbox{ }\frac{r}{n(r+3)},\ldots,\frac{r}{n(n-1)}.

The total probability that we will successfully choose B is thus

P(n,r):=\frac{1}{n}(1+\frac{r}{r+1}+\frac{r}{r+2}+\cdots+\frac{r}{n-1}).

Which value of r maximizes this? Notice we can rewrite this as

P(n,r)=\frac{1}{n}(1+r(H_{n-1}-H_r)).

Given the definition of H_n, we can approximate H_n with ln(n)+\gamma, enabling us (after half a line of school-level algebra) to rewrite P(n,r) as

P(n,r)\approx\frac{1}{n}(1+rln(\frac{n-1}{r}))

and differentiating this gives

P'(n,r)\approx \frac{1}{n}(ln(\frac{n-1}{r})-1)

leading us to the conclusion the optimal r (that is, the value of r for which this last formula is 0) is achieved when (n-1)/r=e (Napier’s constant). Certainly if n is large we may as well take r to be n/e. In plain english, if ignore roughly the top third of the ballot paper, we’re still very likely to vote for the best candidate (I wonder which parties tend to be near the top of the ballot paper).

Well, I found that mildly amusing, even if nobody else did.

Skype interviews

May 11, 2010

Okay, so being quite junior, my job is thus very temporary and as such I am applying for several other jobs (such is the nature of the academic job market). Many of these are, of course, thousands of miles away.

In particular, I’ve been offered a job interview in London. Being at such short notice booking a flight is prohibitively expensive. Consequently they have offered an alternative – an interview via skype.

Should I accept this offer, or try to persuade them to delay by a couple of weeks, when I may well be in England anyway? Is being interviewed in this way known to disadvantage an application or not or what???

Does anyone reading this have any experience of Skype based interviews? If so, is it sensible to approach these in the same way as any others or is it worth doing anything a differently?

Any comments/suggestions/tips are all VERY VERY VERY welcome. Thanks.

BST-6

May 10, 2010

Initially it seemed that being in a timezone that was far removed from any and every thing I had ever known wouldn’t work out so well. Bizarly, th¡s seems to have recently changed to the opposite scenario – I’m in the perfect timezone for keeping up with British culture.

Firstly there is of course the world 20 20 in the windies. For most people in Britain, the games are scheduled to coincide with rush hour/sorting out dinner making keeping up with them a bit inconvenient. Here, I return to my desk after lunch just in time for the start of the marvellous TMS commentary.

Secondly, there was of course the UK’s recent descent into a banana repulic (and sadly in more ways than one). Why on Earth they still insist on doing these things on Thursdays is beyond me, but clearly very few people could watch that night’s extraordinary events unfold. In principle I could have stayed up and watched most of it, but after the first four hours or so it all got a bit too depressing (and there was a lot to be depressed about). Even so, I suspect I kept up with more of it than most.

“…you’re jewish, you’re a foreigner, and you’re too good a mathematician for these people”

May 4, 2010

(The author makes no claim regarding the relevance of the third clause of the title of this post to himself.)

So, my Spanish is not really good enough for reading books in the language just yet. Consequently, due to bad planning and all five floors of the university library being the perfect place to get lost my recent reading material has primarily consisted of popular mathematics/science. I would like to highly recommend to anyone reading this a book I finished at the weekend – The Apprenticeship of a Mathematician by the French algebraic geometer André Weil. The book is essentially an autobiography stretching from his early life to the end of world war II (ie the ‘comic opera in six acts’).

The mathematician reading this will enjoy reading about our hero’s interactions with Bourbaki, Brouer, Chevalley, Courant, Lebesgue, Noether and many many many more. The non-mathematican will enjoy reading about our hero’s interactions with the real Bourbaki, Ghandi, Trotsky, Hitler, the queen, Clementine Churchill and even WH Auden, among others.

During his early career he worked in France, Italy, Germany, England, Finland, the USA, India and even Brazil! Indeed the title of this post is a quote from a friend of Weil’s explaining to him why he had so much difficulty finding a job in the USA during the war – the very employment problems that ultimately lead to him winding up in Brazil.

There are numerous passages from this book that I would love to quote, but there are simply too many to include here (comparing the description of the Indian Postal Service to the almost non-existant Colombian analogue, certainly raised an eyebrow). In short: read it!

Mayday

May 4, 2010

Like any major city, Bogotá was filled with rioting. As previously mentioned, on a good day, the center of Bogotá is normally patrolled by numerous policemen, soldiers, guns and dogs. On Mayday, however, the security services went into overdrive.

All the real trouble took place in the morning when I happened to be in my office, barely fifteen minutes walk from the center (itself barely ten minutes walk from where I live). Indeed, I was unaware of anything having been going on until I popped into town that afternoon. They must have mobilised every policeman and soldier for miles around. Every bank had been attacked in some way, mostly with paint balls, but in some cases smashed up too. Literally, in ever direction you looked from almost any point in the center there where whole gangs of policemen guarding almost anything that wasn’t meant to be moved. Altogether they must have been an army of hundreds, if not 1000+.

I’m just glad wasn’t there in the morning. A colleague of mine had made that mistake last year…and got teargased as a consequence!