Archive for mathematics

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:

Turing

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:

scary_maths

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!

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