Archive for mathematics

My Mathematical Valentines Message

Posted in Cute Problems with tags , on February 14, 2015 by telescoper

Here’s a little mathematical exercise with a Valentines theme:

Sketch the curve in the x-y plane described by the equation

\left(x^2 +y^2  + 2ay \right)^2 = 4a^2 \left( x^2 + y^2 \right)

for

x<3.

Geddit?

Answer: the equation is that of a cardioid:

cardioid-2

Mistaken Identity

Posted in Uncategorized with tags , on January 29, 2015 by telescoper

 

wpid-wp-1422539203608.jpeg

Achilles and the Tortoise(s)

Posted in Uncategorized with tags , , , on January 16, 2015 by telescoper

The inestimable Dorothy Lamb has yet again been doing her bit for our outreach effort here in the School of Mathematical and Physical Sciences at the University of Sussex. Charged with the task of coming up with some props to explain Zeno’s Paradox to schoolchildren. One famous version of this paradox features in the form of a story about Achilles and the Tortoise:

Achilles, the fleet-footed hero of the Trojan War, is engaged in a race with a lowly tortoise, which has been granted a head start. Achilles’ task initially seems easy, but he has a problem. Before he can overtake the tortoise, he must first catch up with it. While Achilles is covering the gap between himself and the tortoise that existed at the start of the race, however, the tortoise creates a new gap. The new gap is smaller than the first, but it is still a finite distance that Achilles must cover to catch up with the animal. Achilles then races across the new gap. To Achilles’ frustration, while he was scampering across the second gap, the tortoise was establishing a third. The upshot is that Achilles can never overtake the tortoise. No matter how quickly Achilles closes each gap, the slow-but-steady tortoise will always open new, smaller ones and remain just ahead of the Greek hero.

Anyway, we now have a splendid knitted Achilles along with not one but three lovely tortoises…DSC_0035

Achilles is a bit short in the leg, although I suppose that doesn’t necessarily mean that he can’t be fleet of foot, and we had to prop him up against the wall lest he should heel over, but nevertheless I think these are great. Could this be the next big thing in toys for mathematically inclined students?

Puzzle Time: Playing With Matches

Posted in Cute Problems with tags , , on January 13, 2015 by telescoper

matchesPuzzle time! Move three (and only three) matches and position them to create just four, i.e four and only four,  (identical) triangles. No cutting the matches, either!

Click on to see the answer…

Continue reading

That Was The REF That Was..

Posted in Finance, Science Politics with tags , , , , , , on December 18, 2014 by telescoper

I feel obliged to comment on the results of the 2014 Research Excellence Framework (REF) that were announced today. Actually, I knew about them yesterday but the news was under embargo until one minute past midnight by which time I was tucked up in bed.

The results for the two Units of Assessment relevant to the School of Mathematical and Physical Sciences are available online here for Mathematical Sciences and here for Physics and Astronomy.

To give some background: the overall REF score for a Department is obtained by adding three different components: outputs (quality of research papers); impact (referrring to the impact beyond academia); and environment (which measures such things as grant income, numbers of PhD students and general infrastructure). These are weighted at 65%, 20% and 15% respectively.

Scores are assigned to these categories, e.g. for submitted outputs (usually four per staff member) on a scale of 4* (world-leading), 3* (internationally excellent), 2* (internationally recognised), 1* (nationally recognised) and unclassified and impact on a scale 4* (outstanding), 3* (very considerable), 2* (considerable), 1* (recognised but modest) and unclassified. Impact cases had to be submitted based on the number of staff submitted: two up to 15 staff, three between 15 and 25 and increasing in a like manner with increasing numbers.

The REF will control the allocation of funding in a manner yet to be decided in detail, but it is generally thought that anything scoring 2* or less will attract no funding (so the phrase “internationally recognised” really means “worthless” in the REF, as does “considerable” when applied to impact). It is also thought likely that funding will be heavily weighted towards 4* , perhaps with a ratio of 9:1 between 4* and 3*.

We knew that this REF would be difficult for the School and our fears were born out for both the Department of Mathematics or the Department of Physics and Astronomy because both departments grew considerably (by about 50%) during the course of 2013, largely in response to increased student numbers. New staff can bring outputs from elsewhere, but not impact. The research underpinning the impact has to have been done by staff working in the institution in question. And therein lies the rub for Sussex…

To take the Department of Physics and Astronomy, as an example, last year we increased staff numbers from about 23 to about 38. But the 15 new staff members could not bring any impact with them. Lacking sufficient impact cases to submit more, we were obliged to restrict our submission to fewer than 25. To make matters worse our impact cases were not graded very highly, with only 13.3% of the submission graded 4* and 13.4% graded 3*.

The outputs from Physics & Astronomy at Sussex were very good, with 93% graded 3* or 4*. That’s a higher fraction than Oxford, Cambridge, Imperial College and UCL in fact, and with a Grade Point Average of 3.10. Most other departments also submitted very good outputs – not surprisingly because the UK is actually pretty good at Physics – so the output scores are very highly bunched and a small difference in GPA means a large number of places in the rankings. The impact scores, however, have a much wider dispersion, with the result that despite the relatively small percentage contribution they have a large effect on overall rankings. As a consequence, overall, Sussex Physics & Astronomy slipped down from 14th in the RAE to 34th place in the REF (based on a Grade Point Average). Disappointing to say the least, but we’re not the only fallers. In the 2008 RAE the top-rated physics department was Lancaster; this time round they are 27th.

I now find myself in a situation eerily reminiscent of that I found myself facing in Cardiff after the 2008 Research Assessment Exercise, the forerunner of the REF. Having been through that experience I’m a hardened to disappointments and at least can take heart from Cardiff’s performance this time round. Spirits were very low there after the RAE, but a thorough post-mortem, astute investment in new research areas, and determined preparations for this REF have paid dividends: they have climbed to 6th place this time round. That gives me the chance not only to congratulate my former colleagues there for their excellent result but also to use them as an example for what we at Sussex have to do for next time. An even more remarkable success story is Strathclyde, 34th in the last RAE and now top of the REF table. Congratulations to them too!

Fortunately our strategy is already in hand. The new staff have already started working towards the next REF (widely thought to be likely to happen in 2020) and we are about to start a brand new research activity in experimental physics next year. We will be in a much better position to generate research impact as we diversify our portfolio so that it is not as strongly dominated by “blue skies” research, such as particle physics and astronomy, for which it is much harder to demonstrate economic impact.

I was fully aware of the challenges facing Physics & Astronomy at Sussex when I moved here in February 2013, but with the REF submission made later the same year there was little I could do to alter the situation. Fortunately the University of Sussex management realises that we have to play a long game in Physics and has been very supportive of our continued strategic growth. The result of the 2014 REF result is a setback but it does demonstrate that the stategy we have already embarked upon is the right one.

Roll on 2020!

Mathematical and Physical Sciences Open Day at Sussex

Posted in Biographical, Education with tags , , , , on October 4, 2014 by telescoper

It’s another open day at the University of Sussex so I’m on campus again to help out as best I can, although I have to admit that all the hard work is being done by others! It’s been extremely busy so far; in fact, I’m told that about 6000 visitors are on campus today. This a good sign for the forthcoming admissions round, probably buoyed by the improved position of the University of Sussex in the latest set of league tables and in excellent employment prospects for graduates.

Anyway the good folks of  the Department of Physics & Astronomy  and Department of Mathematics were here bright and early to get things ready:

IMG-20141004-00413

IMG-20141004-00414

All morning we’ve had a steady stream of visitors to the School of Mathematical and Physical Sciences (which comprises both Departments mentioned above). While I’m at it let me just give a special mention to Darren Baskill’s Outreach Team (seen in the team photograph below).
outreachThey have had absolutely amazing year, running a huge range of events and activities that have reached a staggering 14,000 people of all ages (including 12,000 of school age).

Anyway, I think I’ll toddle off and see if I can sit in on one of today’s lectures. It’s about time I learned something.

 

UPDATE: Here is Mark Hindmarsh about to get started on his lecture.

IMG-20141004-00415

You could have knocked me down with a feather when I saw that he had included a quote from this blog in his talk:

I’ve worked in some good physics departments in my time, but the Department of Sussex is completely unique both for the level of support it offers students and the fact that so many of the undergraduates are so highly motivated.

And, yes, I did mean every word of that.

Bayes, Laplace and Bayes’ Theorem

Posted in Bad Statistics with tags , , , , , , , , on October 1, 2014 by telescoper

A  couple of interesting pieces have appeared which discuss Bayesian reasoning in the popular media. One is by Jon Butterworth in his Grauniad science blog and the other is a feature article in the New York Times. I’m in early today because I have an all-day Teaching and Learning Strategy Meeting so before I disappear for that I thought I’d post a quick bit of background.

One way to get to Bayes’ Theorem is by starting with

P(A|C)P(B|AC)=P(B|C)P(A|BC)=P(AB|C)

where I refer to three logical propositions A, B and C and the vertical bar “|” denotes conditioning, i.e. P(A|B) means the probability of A being true given the assumed truth of B; “AB” means “A and B”, etc. This basically follows from the fact that “A and B” must always be equivalent to “B and A”.  Bayes’ theorem  then follows straightforwardly as

P(B|AC) = K^{-1}P(B|C)P(A|BC) = K^{-1} P(AB|C)

where

K=P(A|C).

Many versions of this, including the one in Jon Butterworth’s blog, exclude the third proposition and refer to A and B only. I prefer to keep an extra one in there to remind us that every statement about probability depends on information either known or assumed to be known; any proper statement of probability requires this information to be stated clearly and used appropriately but sadly this requirement is frequently ignored.

Although this is called Bayes’ theorem, the general form of it as stated here was actually first written down not by Bayes, but by Laplace. What Bayes did was derive the special case of this formula for “inverting” the binomial distribution. This distribution gives the probability of x successes in n independent “trials” each having the same probability of success, p; each “trial” has only two possible outcomes (“success” or “failure”). Trials like this are usually called Bernoulli trials, after Daniel Bernoulli. If we ask the question “what is the probability of exactly x successes from the possible n?”, the answer is given by the binomial distribution:

P_n(x|n,p)= C(n,x) p^x (1-p)^{n-x}

where

C(n,x)= \frac{n!}{x!(n-x)!}

is the number of distinct combinations of x objects that can be drawn from a pool of n.

You can probably see immediately how this arises. The probability of x consecutive successes is p multiplied by itself x times, or px. The probability of (n-x) successive failures is similarly (1-p)n-x. The last two terms basically therefore tell us the probability that we have exactly x successes (since there must be n-x failures). The combinatorial factor in front takes account of the fact that the ordering of successes and failures doesn’t matter.

The binomial distribution applies, for example, to repeated tosses of a coin, in which case p is taken to be 0.5 for a fair coin. A biased coin might have a different value of p, but as long as the tosses are independent the formula still applies. The binomial distribution also applies to problems involving drawing balls from urns: it works exactly if the balls are replaced in the urn after each draw, but it also applies approximately without replacement, as long as the number of draws is much smaller than the number of balls in the urn. I leave it as an exercise to calculate the expectation value of the binomial distribution, but the result is not surprising: E(X)=np. If you toss a fair coin ten times the expectation value for the number of heads is 10 times 0.5, which is five. No surprise there. After another bit of maths, the variance of the distribution can also be found. It is np(1-p).

So this gives us the probability of x given a fixed value of p. Bayes was interested in the inverse of this result, the probability of p given x. In other words, Bayes was interested in the answer to the question “If I perform n independent trials and get x successes, what is the probability distribution of p?”. This is a classic example of inverse reasoning, in that it involved turning something like P(A|BC) into something like P(B|AC), which is what is achieved by the theorem stated at the start of this post.

Bayes got the correct answer for his problem, eventually, but by very convoluted reasoning. In my opinion it is quite difficult to justify the name Bayes’ theorem based on what he actually did, although Laplace did specifically acknowledge this contribution when he derived the general result later, which is no doubt why the theorem is always named in Bayes’ honour.

 

This is not the only example in science where the wrong person’s name is attached to a result or discovery. Stigler’s Law of Eponymy strikes again!

So who was the mysterious mathematician behind this result? Thomas Bayes was born in 1702, son of Joshua Bayes, who was a Fellow of the Royal Society (FRS) and one of the very first nonconformist ministers to be ordained in England. Thomas was himself ordained and for a while worked with his father in the Presbyterian Meeting House in Leather Lane, near Holborn in London. In 1720 he was a minister in Tunbridge Wells, in Kent. He retired from the church in 1752 and died in 1761. Thomas Bayes didn’t publish a single paper on mathematics in his own name during his lifetime but was elected a Fellow of the Royal Society (FRS) in 1742.

The paper containing the theorem that now bears his name was published posthumously in the Philosophical Transactions of the Royal Society of London in 1763. In his great Philosophical Essay on Probabilities Laplace wrote:

Bayes, in the Transactions Philosophiques of the Year 1763, sought directly the probability that the possibilities indicated by past experiences are comprised within given limits; and he has arrived at this in a refined and very ingenious manner, although a little perplexing.

The reasoning in the 1763 paper is indeed perplexing, and I remain convinced that the general form we now we refer to as Bayes’ Theorem should really be called Laplace’s Theorem. Nevertheless, Bayes did establish an extremely important principle that is reflected in the title of the New York Times piece I referred to at the start of this piece. In a nutshell this is that probabilities of future events can be updated on the basis of past measurements or, as I prefer to put it, “one person’s posterior is another’s prior”.

 

 

 

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