Please sir, can have some more notes?

June 2, 2010

So it seems that over here, following the marking of exams the students are actually allowed to dispute the ‘notes’ (if you take the spanish word they use for marks/results/grades/scores/whatever you want to call them and translate it into English this comes out as notes and you can imagine the confusion this has caused at various times). The result is a constant stream of students coming to my office wanting to see their exam script and have a quick argument. Even if they find nothing wrong with the marking (or should that be noting?) they start disputing the calculation of their grades. Remember they really go OTT with continuous assesment here and consequently actually calculating the final grade entails considerable amounts of tearing my hair out over a spreadsheet since there are so many different numbers with very different weights that have to be plugged into an overcomplicated formula to compute the final grade (three exams each accounting 15%, three other exams whose MEAN accounts for another 15%, a mark out of five given to them by the postgraduates overseeing the examples classes – which I have the power to tweek – accounting for another 15% and the final exam accounting for a mere 25%).

Thankfully this is all over now, and if half the horror stories my colleagues have told me are true I’ve actually got off pretty lightly. I’ve heard stories of students stealing scripts, changing their answers and then claiming the script was marked wrong. I’ve heard of students, realising that they need to find not one or two extra marks, but TEN extra marks (out of 25) to push their grade up proceeding to actually waste huge amounts of time scrutinising every square inch in an effort to pull off a miracle. There have been students who turn on the water works in an effort to gain the sympathy vote.

At least I don’t have to go through this again until December.

“[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!

Militry Check Point

April 26, 2010

So, this weekend a friend and I took a coach to Villa de Leyva – a completely unnotable place that happens to be full of small bars, overpriced restaurants, gift/tat shops, tourists, crap museums and some breathtaking mountain views. A basically unremarkable trip overall.

What was notable was what happened on the journey there. Along the motorways en route there were several places where a soldier was stood at the side of the road. As the coach passed the soldier would give a thumbs up and an inane grin. At the time we were completely mystified about what this was about. Then we encountered one point where the soldier did not give us the thumbs up. We pulled up to the side of the road where there was rather more than just one soldier on his own. In terms of buildings the army appeared to have little more than what could be described as a small gazebo erected next to one of these road side “junk food shacks” (selling not particularly nice sausages).

A soldier came onto the bus and (en español) asked the men to get off the bus. I repeat – just the men – if you want to smuggle drugs in Colombia, make sure you get a woman to do it. We got off the bus and one at a time were given an illusory ‘patting down’ (I can think of at least half a dozen places about my person that I could have hidden almost anything and they would not have found it) followed by (what I assume) was a request to open my bag. I opened it, he peered in and moved me along. Note that nobody gave two tosses about any of the side pockets. To be fair they did have dogs, so trace quantities of anything dodgy would probably have been picked up on.

Whilst this all sounds a bit scarey, it was less scary than you would think. Soldiers, policemen, guns and dogs on basically every street corner, usually in multiples of two, often more, are quite commonplace even central Bogotá.

We went back on the bus and the otherwise uneventful journey continued. The Colombians seemed to take all this in their stride – it was a seemingly everyday occurence to them.

Election Fever

April 20, 2010

So, as a mathematician, I could at this point drivel on about game theory or why Arrow’s theorem tells us that democracy doesn’t work. Instead I’ll point out that because of elections, as previously noted it is quite an odd time for a British person to be here. You see, not only there the general “OOOO PICK ME SIR, ME SIR, PICK MEEEE!!!! game, there´s also the ongoing Colombian presidential elections.

Under the Colombian constitution the current president cannot stand for a third term, so there’s everything to play for. Amusingly, one candidate that has actually been doing pretty well in the polls recently is an eccentric mathematician. To put you in the picture, while this guy was lord mayor of Bogotá he

– hired 20 mimes to make fun of traffic violators (traffic fatalities dropped by over 50%);
– took a shower in a TV ad campaign to promote reducing water consumption (which has remained 40% lower ever since – well how would you respond to seeing your mayor naked on TV?) and
– put in place a “Women’s Night”, on which the city’s men were asked to stay home for an evening to look after the house and the children.

Somehow seems it makes you wonder why British politicians cannot be more creative and less like bafoons.