Archive for mathematics

A Keno Game Problem

Posted in Cute Problems with tags , , , , on July 25, 2014 by telescoper

It’s been a while since I posted anything in the Cute Problems category so, given that I’ve got an unexpected gap of half an hour today, I thought I’d return to one of my side interests, the mathematics and games and gambling.

There is a variety of gambling games called Keno games in which a player selects (or is given) a set of numbers, some or all of which the player hopes to match with numbers drawn without replacement from a larger set of numbers. A common example of this type of game is Bingo. These games mostly originate in the 19th Century when travelling carnivals and funfairs often involved booths in which customers could gamble in various ways; similar things happen today, though perhaps with more sophisticated games.

In modern Casino Keno (sometimes called Race Horse Keno) a player receives a card with the numbers from 1 to 80 marked on it. He or she then marks a selection between 1 and 15 numbers and indicates the amount of a proposed bet; if n numbers are marked then the game is called `n-spot Keno’. Obviously, in 1-spot Keno, only one number is marked. Twenty numbers are then drawn without replacement from a set comprising the integers 1 to 80, using some form of randomizing device. If an appropriate proportion of the marked numbers are in fact drawn the player gets a payoff calculated by the House. Below you can see the usual payoffs for 10-spot Keno:

tabke
If fewer than five of your numbers are drawn, you lose your £1 stake. The expected gain on a £1 bet can be calculated by working out the probability of each of the outcomes listed above multiplied by the corresponding payoff, adding these together and then subtracting the probability of losing your stake (which corresponds to a gain of -£1). If this overall expected gain is negative (which it will be for any competently run casino) then the expected loss is called the house edge. In other words, if you can expect to lose £X on a £1 bet then X is the house edge.

What is the house edge for 10-spot Keno?

Answers through the comments box please!

Time for a Factorial Moment…

Posted in Bad Statistics with tags , , on July 22, 2014 by telescoper

Another very busy and very hot day so no time for a proper blog post. I suggest we all take a short break and enjoy a Factorial Moment:

Factorial Moment

I remember many moons ago spending ages calculating the factorial moments of the Poisson-Lognormal distribution, only to find that they were well known. If only I’d had Google then…

Mathematics and Meningococcal Meningitis

Posted in Education, Science Politics with tags , , , , on June 9, 2014 by telescoper

Last week I attended a very enjoyable and informative event entitled Excellence with Impact that showcased some of the research that the University of Sussex submitted to the 2014 Research Excellence Framework. One of the case studies came from the Department of Mathematics which is part of the School of Mathematical and Physical Sciences (of which I am Head) so I thought I would showcase it here too:

The description from Youtube reads

Meningococcal meningitis is a debilitating and deadly disease, causing an estimated 10,000 deaths annually in endemic areas of sub-Saharan Africa. A novel mathematical model developed by Sussex researcher Dr Konstantin Blyuss and colleagues has helped explain the patterns of the dynamics of meningococcal meningitis in endemic areas. This model is now being used by epidemiologists and clinical scientists to design and deliver efficient public-health policies to combat this devastating disease.

You can find out more by following this link.

Lincoln – Green Shoots for Maths and Physics?

Posted in Education with tags , , , , on March 3, 2014 by telescoper

I noticed over the weekend that there’s a job being advertised at the University of Lincoln designated Founding Head of the School of Mathematics and Physics. It seems the powers that be at Lincoln University (which is in the Midlands) have decided to set up an entire new activity in Mathematics and Physics. I’m pointing this out not because of any personal connection with the position, but because it’s refreshing to see a new(ish) Higher Education Institute apparently willing to take the plunge and invest in a new venture, particularly because it includes Physics. It wasn’t at all long ago that UK Physics departments were being closed down – the University of Reading being a prominent example, in 2006. I think Reading is thinking of starting up Physics again, in fact. Perhaps these are the green shoots that presage a new spring for Physics in this country? I do hope so.

It won’t be an easy task to start up a new department from scratch in Lincoln: grant funding is tight and the competition for students among established institutions is already so intense that it will be very difficult for a brand new outfit to break through. Nevertheless, I think it’s a praiseworthy initiative and I wish it well.

From Real Time to Imaginary Time

Posted in Brighton, Education, The Universe and Stuff with tags , , , , , , , , , , , on February 24, 2014 by telescoper

Yesterday, after yet another Sunday afternoon in my office on the University of Sussex campus, I once again encountered the baffling nature of the “real time boards” at the bus-stop at Falmer Station (just over the road from the University). These boards are meant to show the expected arrival times of buses; an example can be seen on the left of the picture below, taken at Churchill Square (in the City Centre).

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The real-time board system works pretty well in central Brighton, but it’s a very different story at Falmer, especially for the Number 23 which is my preferred bus home. Yesterday provided a typical illustration of the problem: the time of the first bus on the list, a No. 23, was shown as “1 min” when I arrived at the stop. It then quickly moved to “due” (a word which I’ll comment about later). It then moved back to “2 mins” for about 5 minutes and then back to “due” again. It stayed like that for over 10 minutes at which point the bus that was second on the list (a No. 28 from Lewes) appeared. Rather than risk waiting any longer for the 23 I got on the 28 and had a slightly longer walk home from the stop at the other end. Just as well I did because the 23 vanished entirely from the screen as soon as I boarded the other bus. This apparent time-travel isn’t unusual at Falmer, although I’ve never really understood why.

By sheer coincidence when I got to the bus stop to catch a bus to campus this morning there was a chap from Brighton and Hove buses there. He was explaining what sometimes goes wrong with the real time boards to a lady, so I joined in the conversation and asked him if he knew why Falmer is so unreliable. He was happy to oblige. It turns out that the way the real-time boards work depends on each bus having a GPS system that communicates to a central computer via a radio link. If the radio link drops out for some reason – as it apparently does quite often up at Falmer (mobile phone connectivity is poor here also) – the system looks up the expected time of the bus after the one that it has lost contact with. Thus it is that a bus can apparently be “due” and then apparently go back in time. Also, if a bus has to divert from the route programmed into the GPS tracker then it is also removed from the real-time boards.

However, there is another system in operation alongside the GPS tracker. When a bus actually stops at a stop and opens its doors the onboard computer communicates this to the central system at the same time as the location signs inside the bus are updated. At this point the real-time boards are reset.

The unreliability I’ve observed at Falmer is in fact caused by two problems: (i) the patchy radio coverage as the bus wanders around the hilly environs of Falmer campus; and (ii) the No. 23 is on a new route around the back of campus which means that it vanishes from the system entirely when it wanders off the old route, as would happen if the bus were to break down.

Mystery solved then, in a sense, but it means there’s a systematic problem that isn’t going to be fixed in the short-term. Would it be better to switch off the boards than have them show inaccurate information? Perhaps, but only if it were always wrong. In fact the boards seem to work OK for the more frequent bus, the No. 25. My strategy is therefore never to rely on the information provided concerning the No. 23 and just get the first bus that comes. It’s not a problem anyway during the week because there’s a bus every few minutes, but on a Sunday evening it is quite irksome to see apparently random times on the screens.

All this talk about real-time boards reminds me of a question I was asked in a lecture last week. I was starting a new section of my Theoretical Physics module for 2nd Year students on Complex Analysis: the Cauchy-Riemann equations, Conformal Transformations, Contour Integrals and all that Jazz. To start the section I went on a bit of a ramble about the ubiquity of complex numbers in physics and whether this means that imaginary numbers are, in some sense, real. You can find an enjoyable polemic on this subject, given the answer “no” to the question here.

Anyway, I got the class to suggest examples of the use of complex numbers in physics. The things you’d expect came up such as circuit theory, wave propagation etc. Then somebody mentioned that somewhere they had heard of imaginary time. The context had probably been provided Stephen Hawking who mentioned this in his book A Brief History of Time. In fact the trick of introducing imaginary time is called a Wick Rotation and the basic idea is simple. In special relativity we deal with four-dimensional space-time intervals of the form

ds^2 = -c^2dt^2 + dx^2 + dy^2 +dz^2,

i.e. the metric describing Minkowski space. The minus sign in front of the time bit is essential to the causal structure of space-time but it causes quite a few mathematical difficulties. However if we make the substitution

\tau \rightarrow i c t

then the metric becomes

ds^2 = d\tau^2 + dx^2 + dy^2 +dz^2,

which corresponds to a four-dimensional Euclidean space which is in many situations much easier to handle mathematically.

Complex variables and complex functions provide the theoretical physicist with a host of extremely elegant techniques for solving tricky problems. But does that mean they are somehow “built in” to nature? I don’t think so. I don’t think the Brighton & Hove Bus company uses imaginary time on its display boards either, although it does sometimes seem that way.

 

POSTSCRIPT. I forgot to include my planned rant about the use of the word “due”. The boards displaying train times at railway stations usually give the destination and planned departure time of the train, e.g. “Brighton 11.15″. If things are running to schedule this information is supplemented by the phrase “On Time”. If not, which is sadly a more likely contingency in the UK, this changes to “due 11.37″ or some such. This really annoys me.: the train is due at 11.15. If it doesn’t come until after then, it’s overdue or, in other words, late.

The most beautiful equation?

Posted in The Universe and Stuff with tags , , , , on February 13, 2014 by telescoper

There’s an interesting article on the BBC website today that discusses the way mathematicians’ brains appear to perceive “beauty”. A (slightly) more technical version of the story can be found here. According to functional magnetic resonance imaging studies, it seems that beautiful equations excite the same sort of brain activity as beautiful music or art.

The question of why we think equations are beautiful is one that has come up a number of times on this blog. I suspect the answer is a slightly different one for theoretical physicists compared with pure mathematicians. Anyway, I thought it might be fun to invite people offer suggestions through the comments box as to the most beautiful equation along with a brief description of why.

I should set the ball rolling myself, and I will do so with this, the Dirac Equation:

dirac_equation

This equation is certainly the most beautiful thing I’ve ever come across in theoretical physics, though I don’t find it easy to articulate precisely why. I think it’s partly because it is such a wonderfully compact fusion of special relativity with quantum mechanics but also partly because of the great leaps of the imagination that were needed along the journey to derive it and consequent admiration for the intellectual struggle involved. I feel it is therefore as much an emotional response to the achievement of another human being – such as one feels when hearing great music or looking at great art – as it is a rational response to the mathematical structure involved.

Anyway, feel free to suggest formulae or equations through the comments box, preferably with a brief explanation of why you think they’re so beautiful.

Pseudospheres Corner..

Posted in Cute Problems with tags , , , , on November 28, 2013 by telescoper

I’m sure you have all seen a knitted pseudosphere, but this is a particularly fine collection made by the excellent Miss Lemon and briefly displayed in my office this morning.

IMG-20131128-00223

A pseudosphere is a space of negative curvature (whereas a sphere is one of constant positive curvature). There are various ways to realize a two-dimensional surface which has negative curvature everywhere; this knitted version is based on hyperbolic space. If you’re keen to have a go at making one yourself you can find some instructions here. I’m advised, though, that the better way to approach the task is to start out with a large circular ring onto which you cast about 400 stitches, gradually working your way in with fewer and fewer stitches (say 400,200,100,50 etc), which is much easier than working outwards as described in the link. The folds and crenellations are produced quite naturally as a consequence of tension in the wool.

Happy knitting!

Advice for Students on Clearing

Posted in Education, The Universe and Stuff with tags , , , , on August 15, 2013 by telescoper

1-dont-panic

We still have places in the School of Mathematical & Physical Sciences at the University of Sussex. No other Physics & Astronomy department in the UK scored more highly in the latest NSS survey than ours, so whether you’re interested in Physics, Astrophysics, Astronomy or Mathematics (or even a combination of those subjects), why not just take a look at the University’s Clearing Page and give us a ring.?

As a matter of fact, I’ll be there myself from 8am this morning to talk to interested students.

11.30 UPDATE. I finished my first shift at 11am. I’ll be going back at 5pm for the last session, until the lines close at 7pm. During the last hour a minimum of 20 overs must be bowled. Or something.  The main call centre (which has fifty phone lines) is next door to where we were sitting and is operated by admissions experts and student helpers who are processing the queries and, if necessary, routing them through to academics (i.e. people like me) to provide further information or to answer specific questions. You can take a peep behind the scenes here. Some of the calls were from very anxious prospective students, and it’s a very nice feeling being able to help them sort out their course! Now back to other things until I start again this evening.

 

19.30 UPDATE. Phew. Finally been stood down, but I’ll be back on duty tomorrow afternoon. It’s been a very interesting day which has gone very well for us. Lines stay open until 8pm tonight and re-open at 8 in the morning and we’re still in business to see if we can give just a few more students the opportunity to study in the School next academic year. Now I’m off home to chill, probably over a glass or two of wine!

(Lack of) Diversity in STEM Subjects

Posted in Science Politics with tags , , , , , , on May 10, 2013 by telescoper

Among the things I learnt over the last few days was some interesting information about the diversity (or, rather, lack of diversity) of undergraduates taking undergraduate degrees in STEM subjects in the UK universities. For those of you not up on the lingo, `STEM’ is short for Science, Technology, Engineering and Mathematics. Last year the Institute of Physics produced a report that contains a wealth of statistical information about the demographics of the undergraduate population, from which the following numbers are only a small component.

Physics

Maths

Chemistry

Engineering

Female

21%

41%

44%

12%

BME

11%

24%

20%

30%

Socio-Economic

37%

42%

43%

51%

Non-EU

5%

12%

7%

32%

For completeness I should point out that these numbers refer to first-year undergraduates in 2010-11; I have no particular reason to suppose there has been a qualitative change since then. “BME” stands for “Black and Minority Ethnic”, and “Socio-Economic” refers to students whose with parents not employed in managerial or professional positions.

Overall, the figures here at the University of Sussex are roughly in line with, but slightly better than, these national statistics; the proportion of female students in our Physics intake for 2010/11, for example, was 27%.

There are some interesting (and rather disappointing) things to remark. First is that the proportion of Physics students who are female remains low; Physics scores very badly on ethnic diversity too. Mathematics on the other hand seems a much more attractive subject for female students.  Notice also how Physics and Chemistry attract a very small proportion of overseas students compared to Engineering.

In summary, therefore, we can see that Physics is a subject largely studied by white  middle-class European males. What are we doing wrong?

Despite considerable efforts to promote Physics to a more diverse constituency,  the proportion of, e.g., female physics students seems to have been bumping along at around 20% for ages.  Interestingly, all the anecdotal evidence suggests that those women who do Physics at University do disproportionately well, in the sense that female students constitute a  much larger fraction of First-class graduates than 20%. This strongly suggests that the problem lies at school level; some additional IOP information and discussion on this can be found here.

I’m just passing these figures on for information, as I’m quite often asked about them during, e.g., admissions-related activities. I don’t have any really compelling suggestions, but I would like to invite the blogosphere to comment and/or make suggestions as to promote diversity in STEM disciplines.

Advice for Prospective Physics Students

Posted in Education with tags , , , , , , , on February 25, 2013 by telescoper

Just got time for a quickie this morning before I head up to the Big Smoke for the first meeting of the Astronomy Grants Panel of the Science and Technology Facilities Council (STFC). I had thought that the last round would be my last, but I must have misbehaved somehow and my sentence has been extended accordingly.

Anyway, I thought I’d follow up Saturday’s post with a response to a question that a few prospective Physics students asked me during our Admissions Day. Those attending these days have already applied to this University but, at this stage of the annual undergraduate admissions cycle, the applicants are still deciding which University to put as their first or firm choice. I see our job in the Admissions Days simply to make available as much information as possible to the applicants to help them make a choice. Quite a few people I spoke to on Saturday, however, said that they had already made up their minds and just wanted some advice about what to do during the summer after they have finished their A-levels to help them prepare for their undergraduate studies in Physics (and/or Astronomy).

The answer I gave was pretty simple: practice your mathematics, especially your calculus! 

The justification for exhortation is that the big difference between Physics at A-level and Physics at undergraduate University level is that the latter is taught in a much more mathematical way than the former. This is because the physical laws that underpin our understanding of the natural world are expressed in a mathematical language; the more fluent you are in this language the easier you will find it to assimilate the physical concepts. To put it another way, you will find it difficult to understand the physical meaning of what is being taught if you are struggling with the mathematical meaning of the symbols being used or the manipulations needed to obtain useful solutions to the relevant equations.

Newton’s Second Law, for example,  relates the rate of change of momentum of a body to the force exerted upon it. If you’re comfortable with calculus you don’t think twice about writing d(mv)/dt for the rate of change of momentum and then constructing a differential equation which you can (hopefully) solve. You won’t absorb the importance of laws like this unless you become so familiar with the mathematics that it ceases to occupy the part of your mind that’s needed to really think.

I think that learning to do Physics is a bit like learning to play a musical instrument. Practicing such basic mathematical procedures as integration and differentiation is analogous to the five-finger exercises you have to do when learning to play the piano. The more you practice them, the greater the extent to which they become hard-wired. Your brain can therefore concentrate on the more interesting conceptual stuff – that’s really the hard part of learning Physics. We do of course do as much as we can to help with this once you’ve got to University, but doing some preparation on your own beforehand would greatly smooth the transition.

So I’d tell any prospective physics student wondering what to do this summer to get hold of as many basic calculus exercises as they can and do them whenever they get the chance. It may not be the most exciting way to spend your post A-level holiday, but it is the single thing you can do that will best prepare you for life as a Physics student.

On the other hand, the advice I’d give to physicists rather later in their careers is to think very carefully before agreeing to be on committees or panels…

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