Archive for mathematics

R.I.P. John M Stewart (1943-2016)

Posted in Biographical, Education, The Universe and Stuff with tags , , , on November 23, 2016 by telescoper

john-stewartI was very sad this morning to hear of the death of distinguished mathematical physicist Dr John M. Stewart (left). Apart from a few years in Munich in the 1970s John Stewart spent most of his working life in Cambridge, having studied there as an undergraduate and postgraduate and then returning from his spell at the Max Planck Institute to the Department of Applied Mathematics and Theoretical Physics for forty years.

John’s research mostly concerned relativistic fluid dynamics. Indeed, he was one of the pioneers of numerical relativity in the United Kingdom, and he applied his knowledge to a number of problems in early Universe cosmology and structure formation. I think it is fair to say that he wasn’t the most prolific researcher in terms of publications, which is perhaps why he only got promoted to Senior Lecturer in 2000 and never made it to a Chair, retiring as Reader in Gravitational Physics in 2010. However, his work was always of a very high technical standard and presented with great clarity and he was held in a very high regard by those who knew him and worked with him.

The tributes paid to John Stewart by King’s College (of which he was a Life Fellow) here and his colleagues in the Centre for Theoretical Cosmology here give a detailed account of his research achievements, so I refer you to them for more information about that aspect of his career.

I just wanted to add a personal note not about John Stewart’s research, but about something else mentioned in the obituaries linked to above: his teaching. I was fortunate enough to have him as a lecturer when I was studying Natural Sciences at Cambridge during the early 1980s. In the second year (Part IB) I specialised in Physics and Mathematics, and John taught part of the Mathematics syllabus. He was an absolutely superb teacher. For a start he was superbly well organized and had clearly thought very deeply about how best to present some quite difficult material. But it wasn’t just that. He projected a very engaging personality, with nice touches of humour, that made him easy to listen. His lectures were also very well paced for taking notes. In fact he was one of the few lecturers I had whose material I didn’t have to transcribe into a neat form from rough notes.

I have kept all the notes from that course for over thirty years. Here are a couple of pages as an example:


Anyone who has ever seen my handwriting will know that this is about as neat as I ever get!

When I was called upon to teach similar material at Cardiff and Sussex I drew on them heavily, so anyone who has learned anything from me about complex analysis, contour integration, Green’s functions and a host of other things actually owes a huge debt to John Stewart. Anything they didn’t understand was of course my fault, not his..

I also remember that John came to Queen Mary to give a seminar when I worked there in the early 90s as a postdoc. I was still a bit in awe of him because of my experience of him in Cambridge. His talk was about a method for handling the evolution of cosmological matter perturbations based on an approach based on the Hamilton-Jacobi formalism. His visit was timely, as I’d been struggling to understand the papers that had been coming out at the time on this topic. In the bar after his talk I plucked up the courage to explain to him what it was that I was struggling to understand. He saw immediately where I was going wrong and put me right on my misconceptions straight away, plucking a simple illustrative example apparently out of thin air. I was deeply impressed, not only by his ability to identify the issue but also with his friendly and helpful demeanour.

Rest in Peace, Dr John M. Stewart (1943-2016).

Computable Numbers, 80 Years on..

Posted in History, mathematics, Uncategorized with tags , , , , on May 28, 2016 by telescoper

There’s been rather a lot of sad news conveyed via this blog recently, so I thought that today I’d mark a happier event. Eighty years ago today (i.e. on 28th May 1936), a paper by Alan Turing arrived at the London Mathematical Society. Entitled “On Computable Numbers, with an Application to the Enstscheidungsproblem“, this was not only enormously influential but also a truly beautiful piece of work. Turing was only 23 when he wrote it. It was delivered to the London Mathematical Society about 6 months after it was submitted,  i.e. in November 1936..

Here’s the first page:


The full reference is

Proc. London Math. Soc. (1937) s2-42 (1): 230-265. doi: 10.1112/plms/s2-42.1.230

You can find the full paper here. I heartily recommend reading it, it’s wonderful.


The Terror of Maths

Posted in History, The Universe and Stuff with tags , , , , on May 9, 2016 by telescoper

I’m not sure whether to be amused or appalled by the story of the Professor whose flight was delayed in order for him to be interrogated because a fellow passenger saw him doing some mathematical calculations. I know some people who find mathematics scary but that’s taking things too far! I wonder if the passenger was Simon Jenkins?

I was wondering whether the calculation was concerned with plane geometry but that seems not to be the case. The academic concerned is an Economist and he was studying a differential equation. That surprises me. I hadn’t realised economists knew about calculus. Or about anything else, for that matter.

The BBC coverage of the story used the following image:


The physicists among you will recognize this as a representation of some of Maxwell’s Equations. I very much doubt they played a part in the work of  our Economics Professor, so presumably this is just one of the  BBC’s stock of generic “scary maths” images.

Other things worth noting are that this version of Maxwell’s Equations isn’t written in SI units, the standard notation in the UK and Europe. As a matter of fact it uses cgs units, which suggests it may be an American import. Nor is it really correct anyway, because the time derivative inside the brackets should surely be partial.

All of which goes to demonstrate how Mathematics is usually viewed in the media and, by extension, the public at large: like an arcane book written in an incomprehensible  language that should be viewed with suspicion or ridicule by any sensible person.

There is nothing new about this, of course. I’m reminded that in 1870, during the Franco-Prussian Way, Norwegian mathematician Sophus Lie was arrested in France on suspicion of being a German spy because the authorities thought his mathematical notes were coded messages of some sort.

In reality, mathematics is the most open and universal language of all and, as such, is a powerful force for human good. Among many other things, quantitative reasoning and proper logic help to defend us against those who lie and distort the facts in order to gain power. Mathematics may not be the easiest language to learn, but it’s well worth the effort, even if you can only master the basics.



Geometry, by Rita Dove

Posted in Poetry with tags , , on February 23, 2016 by telescoper

I prove a theorem and the house expands:
the windows jerk free to hover near the ceiling,
the ceiling floats away with a sigh.

As the walls clear themselves of everything
but transparency, the scent of carnations
leaves with them. I am out in the open

and above the windows have hinged into butterflies,
sunlight glinting where they’ve intersected.
They are going to some point true and unproven.

by Rita Dove


17 Equations that Changed the World

Posted in History, The Universe and Stuff with tags , , on December 14, 2015 by telescoper

Yesterday I posted about a map that “changed the world”. Clearly the world changed a lot and for many different reasons because when I got home I noticed the following picture on Facebook, depicting 17 equations that also “changed the world”:


17 Equations

This is from a book by mathematician Ian Stewart.

Of course it’s actually 20 equations, because there are four Maxwell Equations. It is an interesting selection. Are there any surprising omissions?



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.


Protons for Breakfast

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 post 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.