Archive for mathematics

Helping Blind Students with Mathematics and Physics

Posted in Education with tags , , , , on October 16, 2015 by telescoper

This short video clip features Daniel Hajas, a third-year theoretical physics student in the Department of Physics & Astronomy at the University of Sussex who has been working on technology intended to help visually impaired students to   engage with the charts, graphs and equations involved in studying mathematics and physics. Here is a news item arising from a recent poster competition for which Daniel, who is himself visually impaired, highlighted the challenges faced by blind students by exhibiting a completely blank poster, explaining that this was how a blind person would experience a complex equation. In the video he explains a little more about the work he has been doing.


The Crocodile Maths Challenge

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

I’m indebted to an anonymous informant (John Peacock) for drawing my attention to a BBC Scotland story about an allegedly challenging examination question that appeared on a “Higher Maths” paper. For those of you not up with the Scottish examination system, “Highers” are taken in the penultimate year at school so are presumably roughly equivalent to the AS levels taken in England and Wales.

Anyway, here is the question that is supposed to have been so difficult. For the record, it’s Paper 2, Question 8 of the SQA examination 2015.

crocodile_questionCall me old-fashioned, but it doesn’t seem that difficult to me  but I never took Scottish Highers and there have been many changes in Mathematics education since I did my O and A-levels; here’s the O-level Mathematics paper I took in 1979, for example.  I wonder what my readers think? Comments through the box if you please.

Feel free to give it a go. If you get stuck here’s a worked solution!

Honoris Causa: John Francis, Inventor of the QR Algorithm

Posted in The Universe and Stuff with tags , , , , , , on July 18, 2015 by telescoper

It’s been yet another busy week, trying to catch up on things I missed last week as well as preparing for Thursday’s graduation ceremony for students from the School of Mathematical and Physical Sciences. At this year’s ceremony, as well as reading out the names of graduands from the School of which I am Head, I also had the pleasant duty of presenting mathematician John G.F. Francis for an Honorary Doctorate of Science.

The story of John Francis is a remarkable one which I hope you will agree if you read the following brief account which is adapted from the oration I delivered at the ceremony. It was a special pleasure to asked to present this award because you could never wish to meet a more modest or self-effacing individual. Indeed, when I asked him at the lunch following the ceremony, what he thought of the work for which he had been awarded a degree honoris causa he shrugged it off, and said that he thought it was an obvious thing to do and anyone else could have done it had they thought of it. Maybe that’s true in hindsight, but the point is that “they” didn’t and “he” did. The fact that it has taken over fifty years for him to be recognized for something so important is regrettable to say the least, but I am glad to have been there to see him justifiably honoured. Great thanks are due to Drs Omar Lakkis and Anotida Madzvamuse of the Department of Mathematics at the University of Sussex for bringing his case to the attention of the University as eminently suitable for such an honour. So impressed were the graduating students that a number shook his hand as they passed him on the stage during their own part of the ceremony. I’ve never seen that happen before!

John Francis receiving his Honorary Doctorate from the Chancellor, Sanjeev Bhaskar.

John Francis receiving his Honorary Doctorate from the Chancellor, Sanjeev Bhaskar.

John Francis is a pioneer in the field of mathematical computation where his name is more-or-less synonymous with the so-called “QR algorithm”, an ingenious factorization procedure used to calculate the eigenvalues and eigenvectors of linear operators (represented as matrices).

Before I go on it’s probably worth explaining that the letters ‘QR’ don’t stand for any words in particular. The algorithm involves decomposing the matrix whose eigenvalues are required into the product of an orthogonal matrix (which Francis happened to call Q) and an upper-triangular matrix (which Francis happened to call R). In fact in his original manuscript, the orthogonal matrix was called O but it was subsequently changed to avoid confusion with ‘O’. At any rate, certainly has nothing to do with research funding!

The mathematics and physics graduates in the audience were probably well aware of the importance of eigenvalue problems, which crop up in a huge variety of contexts in these and other scientific disciplines, from geometry to graph theory to quantum mechanics to geology to molecular structure to statistics to engineering; the list is almost endless. Indeed here can be few people working in such fields who haven’t at one time or another turned to the QR algorithm in the course of their calculations. I know I have, in my own field of astrophysics! It has become a standard component of any theoretician’s mathematical toolkit because of its numerical stability.

The algorithm was first derived by John Francis in two papers published in 1959 and, independently a couple of years later, by the Russian mathematician Vera Kublanovskaya (who passed away in 2012). You can find both the papers online: here and here. Interestingly, the problem that John Francis was trying to solve when he devised the QR algorithm concerned the “flutter” or vibrations of aircraft wings.

But it is in the world of the World Wide Web that the QR algorithm has had perhaps its greatest impact. Many of us who were using the internet in 1998 were astonished when Google arrived on the scene because it was so much faster and more effective than all the other search engines available at the time. The secret of this success was the PageRank algorithm (named after Larry Page, one of the founders of Google) which involved applying the QR decomposition to calculate numerical factors expressing the relative “importance” of elements within a linked set (such as pages on the World Wide Web) measured by the nature of their links to other elements. The QR algorithm is not the only technique exploited by Google, but it is safe to say that it is what gave Google its edge.

The achievements of John Francis are indeed impressive, even more so when you read his biography, for he did all this pioneering work in numerical analysis without even having an undergraduate degree in Mathematics.

John Francis actually left school in 1952 and obtained a place at Christ’s College, Cambridge for entry in 1955, after two years of National Service during which he served in Germany and Korea with the Royal Artillery. On leaving the army in 1954 he worked for a time at the National Research Development Corporation which was set up in 1948 by the Attlee government in order to facilitate the transfer of new technologies developed during World War 2 into the private sector in an effort to boost British commerce and industry. Among the priority areas covered by the NRDC was computing, and it was there that John Francis cut his teeth in the field of numerical analysis. He went to University as planned but did not complete his degree, instead returning to the NRDC in 1956 after less than a year of study. It was while working there in 1958 and 1959 that he devised the QR algorithm.

He left the NRDC in 1961 to work at Ferranti Ltd after which, in 1967, he moved to Brighton and took up a position at the University of Sussex in the Laboratory of Experimental Psychology, helping to devise a new computer language for running experiments. He left the University in 1972 to work in various private sector computer service companies in Sussex. He has now retired but still lives locally, in Hove.

Having left the field of numerical analysis in the 1960s, John Francis had absolutely no idea of the impact his work on the QR algorithm had had, nor was he aware that it was widely recognized as one of the Top Ten Algorithms of the Twentieth Century, until he was traced and contacted in 2007 by the organizers of a mini-symposium that was being planned to celebrate 50 years of the QR algorithm; he was the opening speaker at that meeting in Glasgow when it took place in 2009.

More recently still, in 2011, after what he describes as “sporadic” study over many years, John Francis was awarded an undergraduate degree from the Open University, 56 years after he started one at Cambridge.  I am very glad that there was no similar delay in him proceeding to a Doctorate!

Physics is more than applied mathematics

Posted in Education, The Universe and Stuff with tags , , , on July 15, 2015 by telescoper


I thought rather hard before reblogging this, as I do not wish to cause any conflict between the different parts of my School – the Department of Mathematics and the Department of Physics and Astronomy!

I don’t think I really agree that Physics is “more” than Applied Mathematics, or at least I would put it rather differently. Physics and Mathematics intersect, but there are parts of mathematics that are not physical and parts of physics that are not mathematical.


Originally posted on Protons for Breakfast Blog:

A problem set for potential applicants in the foyer of the Cavendish Laboratory. Despite appearances - this is not physics! A problem set for potential applicants in the foyer of the Physics department of a premier UK university. It looks like physics, but it is in fact maths. The reason is that in the context of this problem, the string cannot pull a particle along at all unless it stretches slightly. Click the image for a larger diagram.

While accompanying my son on an Open Day in the Physics Department of a premier UK university, I was surprised and appalled to be told that Physics ‘was applied mathematics‘.

I would just like to state here for the record that Physics is notapplied mathematics.

So what’s the difference exactly?

I think there are two linked, but subtly distinct, differences.

1. Physics is a science and mathematics is not.

This means that physics has an experimental aspect. In physics, it is possible to disprove a hypothesis by experiment: this cannot be done in maths.

2. Physics is about…

View original 256 more words

Hannah and her Sweets: that EdExcel Examination Question…

Posted in Education with tags , , on June 5, 2015 by telescoper

You may or may not know that yesterday there was a bit of a Twitterstorm of students complaining about an “unfairly difficult” examination question on the GCSE Mathematics paper set by EdExcel.

This is the question:

There are n sweets in a bag. Six of the sweets are orange. The rest of the sweets are yellow.

Hannah takes a sweet from the bag. She eats the sweet. Hannah then takes at random another sweet from the bag. She eats the sweet.

The probability that Hannah eats two orange sweets is 1/3. Show that n²-n-90=0.

Not sure what all the fuss is about. Seems very straightforward. The question tells you that 6/n × 5/(n-1)=1/3 whence the equation follows by a trivial rearrangement. In fact I’m a little surprised the question didn’t go on to ask the students to solve the quadratic equation n²-n-90=0 to show that n=10…

I don’t really know what is on the GCSE Mathematics syllabus these days. In fact I never did GCSE Mathematics, I did O-level Mathematics which was quite a different thing. You can see the papers I took – way back in 1979 – here.


League Table Positions

Posted in Education, Football with tags , , , , , , , on May 26, 2015 by telescoper

Among the things I didn’t have time to blog about over a very busy Bank Holiday Weekend was the finish of the English Premiership season. I haven’t posted much about my own team, Newcastle United, this season because I haven’t been able to think of anything particularly positive to say. Since Alan Pardew quit in January to join Crystal Palace, Newcastle slumped to such an alarming extent that they went into their last game of the season (against West Ham) just two points above the drop zone. Had they lost their game, which did not seem unlikely on the basis of their recent form, and had Hull won against Manchester United, which did not seem unlikely on the grounds that Man Utd wwould finish in 4th place whatever happened in that game, then Newcastle would be relegated to the Championship. In the event, however, Newcastle won 2-0 which made them safe while Hull could only draw 0-0 which meant that Newcastle would have survived even if they had lost against West Ham. Moreover, Sunderland also lost their last game, which meant that the final Premier League Table looked like this:


(courtesy of the BBC Website). The important places are 15 and 16, obviously. The natural order of things has been restored….

Another League Table came out over the Bank Holiday. This was the annual Guardian University Guide. I’m deeply sceptical of the value of these league tables, but there’s no question that they’re very important to potential students so we have to take them seriously. This year was pretty good for Sussex as far as the Guardian Table is concerned: the University of Sussex rose to 19th place overall and the two departments of the School of Mathematical and Physical Sciences both improved: Physics & Astronomy is back in the top 10 (at number 9, up from 11th place last year) and Mathematics rose 22 places to take 21st place. Gratifyingly, both finished well above Sunderland.

While these results are good news in themselves, at least around my neck of the woods, as they will probably lead to increased applications to Sussex from students next year, it is important to look behind the simplistic narrative of “improvements”. Since last year there have been several substantial changes to the Guardian’s methodology. The weighting given to “spend-per-student” has been reduced from 15% to 10% of the overall score and the method of calculating “value added” has excluded specific predictions based on “non-tariff” students (i.e. those without UK entry qualifications, especially A-levels). What the Guardian consistently fails to do is explain the relative size of the effect of arbitrary methodological changes on its tables compared to actual changes in, e.g., cash spent per student.

Imagine the outrage there would be if football teams were not told until the end of a Premier League season how many points would be awarded for a win….

A Problems Class in Complex Analysis

Posted in Education, The Universe and Stuff with tags , , , , , on May 15, 2015 by telescoper

My theoretical physics examination is coming up on Monday and the students are hard at working revising for it (or at least they should be) so I thought I’d lend a hand by deploying some digital technology in the form of the following online interactive video-based learning resource on Complex Analysis:


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