Archive for mathematics

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:

End of Term Balls

Posted in Biographical, Education with tags , , , , on April 18, 2015 by telescoper

I haven’t had time to post for the last couple of days because I’ve been too bust with end-of-term business (and pleasure). Yesterday (Friday) was the last day of teaching term and this week I had to get a lot of things finished because of various deadlines, as well as attending numerous meetings. It’s been quite an exhausting week, not just because of that but also because by tradition the two departments within the School of Mathematical and Physical Sciences at the University of Sussex, the Department of Mathematics and the Department of Physics & Astronomy, hold their annual Staff-Student Balls on consecutive days. When I arrived here just over two years ago I decided that I should attend both or neither, as to attend at only one would look like favouritism. In fact this is the third time I’ve attended both of them. Let no-one say I don’t take my obligations seriously.  It’s a tough job, but someone has to do it. Holding both balls so close together  poses some problems for a person of my age, but I coped and also tried to weigh them up relative to each other and see  which was  most impressive.

Actually, both were really well organized. The Mathematics Ball was held in the elegant Hilton Metropole hotel and the Physics one in the Holiday Inn, both on the seafront. As has been the case in previous years the Mathematics ball is a bit more refined and sedate, the Physics one a little more raucous. Also this year there was a very large difference in the number of people going, with over 200 at the Physics Ball and only just over half that number at the Mathematics one. In terms of all-round fun I have to declare the Physics Ball the winner last year, but both occasions were very enjoyable. I’d like to say a very public thank you to the organizers of both events, especially Sinem and Jordan for Mathematics and Francis for Physics. Very well done.

The highlight of the Physics Ball was an after-dinner speech by particle physicist Jon Butterworth, who has an excellent blog called Life and Physics on the Guardian website. I’ve actually been in contact with Jon many times through social media (especially Twitter) over a period of over six years, but we never actually met in person until last night. I think he was a bit nervous beforehand because he had never done an after-dinner speech before, in the end though his talk was funny and wise, and extremely well received. Mind you, I did make it easy for him by giving a short speech to introduce him, and after a speech by me almost anyone would look good!

Thereafter the evening continued with drinking and dancing. After a while most people present were rather tired and emotional.  I even think some might even have been drunk. I eventually got home about 2am, after declining an invitation to go to the after-party. I’m far too old for that sort of thing. Social events like this can be a little bit difficult, for a number of reasons. One is that there’s an inevitable “distance” between students and staff, not just in terms of age but also in the sense that the staff have positions of responsibility for the students. Students are not children, of course, so we’re not legally  in loco parentis, but something of that kind of relationship is definitely there. Although it doesn’t stop either side letting their hair down once in a while, I always find there’s a little bit of tension especially if the revels get a bit out of hand. To help occasions like this run smoothly I think it’s the responsibility of the staff members present to drink heavily in order to put the students at ease. United by a common bond of inebriation, the staff-student divide crumbles and a good time is had by all.

There’s another thing I find a bit strange. Chatting to students last night was the first time I had spoken to many of my students like that, i.e. outside the lecture  or tutorial. I see the same faces in my lectures day in, day out but all I do is talk to them about physics. I really don’t know much about them at all. But it is especially nice when on occasions like this students come up, as several did last night, and say that they enjoyed my lectures. Actually, it’s more than just nice. Amid all the bureaucracy and committee meetings, it’s very valuable to be reminded what the job is really all about.


P.S. Apologies for not having any pictures. I left my phone in the office on Friday when I went home to get changed. I will post some if anyone can supply appropriate images. Or, better still, inappropriate ones!



One Fine Conformal Transformation

Posted in Brighton, Cute Problems with tags , , , , , on March 25, 2015 by telescoper

It’s been a while since I posted a cute physics problem, so try this one for size. It is taken from a book of examples I was given in 1984 to illustrate a course on Physical Applications of Complex Variables I took during the a 4-week course I took in Long Vacation immediately prior to my third year as an undergraduate at Cambridge.  Students intending to specialise in Theoretical Physics in Part II of the Natural Sciences Tripos (as I was) had to do this course, which lasted about 10 days and was followed by a pretty tough test. Those who failed the test had to switch to Experimental Physics, and spend the rest of the summer programme doing laboratory work, while those who passed it carried on with further theoretical courses for the rest of the Long Vacation programme. I managed to get through, to find that what followed wasn’t anywhere near as tough as the first bit. I inferred that Physical Applications of Complex Variables was primarily there in order to separate the wheat from the chaff. It’s always been an issue with Theoretical Physics courses that they attract two sorts of student: one that likes mathematical work and really wants to do theory, and another that hates experimental physics slightly more than he/she hates everything else. This course, and especially the test after it, was intended to minimize the number of the second type getting into Part II Theoretical Physics.

Another piece of information that readers might find interesting is that the lecturer for Physical Applications of Complex Variables was a young Mark Birkinshaw, now William P. Coldrick Professor of Cosmology and Astrophysics at the University of Bristol.

As it happens, this term I have been teaching a module on Theoretical Physics to second-year undergraduates at the University of Sussex. This covers many of the topics I studied at Cambridge in the second year, including the calculus of variations, relativistic electrodynamics, Green’s functions and, of course, complex functions. In fact I’ve used some of the notes I took as an undergraduate, and have kept all these years, to prepare material for my own lectures. I am pretty adamant therefore that the academic level at which we’re teaching this material now is no lower than it was thirty years ago.

Anyway, here’s a typically eccentric problem from the workbook, from a set of problems chosen to illustrate applications of conformal transformations (which I’ve just finished teaching this term). See how you get on with it. The first correct answer submitted through the comments box gets a round of applaud.

conformal transformation



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