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My Friend Erdös..

Posted in Biographical with tags , , , , , , on March 28, 2010 by telescoper

After one of my  lectures a few weeks ago, a student came up to me and asked whether I had an Erdős number and, if so, what it was.  I didn’t actually know what he was talking about but was yesterday reminded of it, so tried to find out.

In case you didn’t know, Paul Erdős (who died in 1996) was an eccentric Hungarian mathematician who wrote more than 1000 mathematical papers during his life but never settled in one place for any length of time. He travelled between colleagues and conference, mostly living out of a suitcase, and showed no interest at all in property or possessions. His story is a fascinating one, and his contributions to mathematics were immense and wide-ranging.  The Erdős number is a tiny part of his legacy, but one that seems to have taken hold. Some mathematicians appear to take it very seriously, but most treat it with tongue firmly in cheek, as I certainly do.

So what is the Erdős number?

It’s actually quite simple to define. First, Erdős himself is assigned an Erdős number of zero. Anyone who co-authored a paper with Erdős has an Erdős number of 1. Then anyone who wrote a paper with someone who wrote a paper with Erdős has an Erdős number of 2, and so on. The Erdős number is thus a measure of “collaborative distance”, with lower numbers representing closer connections.

I say it’s quite easy to define, but it’s rather harder to calculate. Or it would be were it not for modern bibliographic databases. In fact there’s a website run by the American Mathematical Society which allows you to calculate your Erdős number as well as a similar measure of collaborative distance with respect to any other mathematician.

A list of individuals with very low Erdős numbers (1, 2 or 3) can be found here.

Given that Erdős was basically a pure mathematician, I didn’t expect first to show up as having any Erdős number at all, since I’m not really a mathematician and I’m certainly not very pure. However, his influence is clearly felt very strongly in  physics and a surprisingly large number of physicists (and astronomers) have a surprisingly small Erdős number. According to the AMS website, mine is 5 – much lower than I would have expected. The path from me to Erdős in this case goes through G.F.R. Ellis, a renowned expert in the mathematics of general relativity (as well as a ridiculous number of other things!). I wrote a paper and a book with George Ellis some time ago.

However, looking at the list I realise that I have another route to Erdős, through the great Russian mathematician Vladimir Arnold, who has an Erdős number of 3. Arnold wrote a paper with Sergei Shandarin with whom I wrote a paper some time ago. That gives me another route to an Erdős number of 5, but I can’t find any paths  shorter than that.

I guess many researchers will have links through their PhD supervisors, so I checked mine – John D. Barrow. It turns out he also has an Erdős number of 5 so a path through him doesn’t lower my number.

I used to work in the School of Mathematical Sciences at Queen Mary, University of London, and it is there that I found some people I know well who have lower Erdős numbers than me. Reza Tavakol, for example, has an Erdős number of 3 but although I’ve known him for 20 years, we’ve never written a paper together. If we did, I could reduce my Erdős number by one. You never know….

This means that anyone I’ve ever written a paper with has an Erdős number no greater than 6. I doubt if it’s very important, but it definitely qualifies as Quite Interesting.

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Killing Vectors

Posted in The Universe and Stuff with tags , , , on February 16, 2010 by telescoper

I’ve been feeling a rant coming for some time now. Since I started teaching again three weeks ago, actually. The target of my vitriol this time is the teaching of Euclidean vectors. Not vectors themselves, of course. I like vectors. They’re great. The trouble is the way we’re forced to write them these days when we use them in introductory level physics classes.

You see, when I was a lad, I was taught to write a geometric vector in the folowing fashion:

\underline{r} =\left(\begin{array}{c} x \\ y \\ z \end{array} \right).

This is a simple column vector, where x,y,z are the components in a three-dimensional cartesian coordinate system. Other kinds of vector, such as those representing states in quantum mechanics, or anywhere else where linear algebra is used, can easily be represented in a similar fashion.

This notation is great because it’s very easy to calculate the scalar (dot) and vector (cross) products of two such objects by writing them in column form next to each other and performing a simple bit of manipulation. For example, the scalar product of the two vectors

\underline{u}=\left(\begin{array}{c} 1 \\ 1 \\ 1 \end{array} \right) and \underline{v}=\left(\begin{array}{c} 1\\ 1 \\ -2 \end{array} \right)

can easily be found by multiplying the corresponding elements of each together and totting them up:

\underline{u}\cdot \underline{v} = (1 \times 1) + (1\times 1) + (1\times -2) =0,

showing immediately that these two vectors are orthogonal. In normalised form, these two particular vectors  appear in other contexts in physics, where they have a more abstract interpretation than simple geometry, such as in the representation of the gluon in particle physics.

Moreover, writing vectors like this makes it a lot easier to transform them via the action of a matrix, by multipying rows in the usual fashion, e.g.

\left(\begin{array}{ccc} \cos \theta & \sin\theta & 0 \\ -\sin\theta & \cos \theta & 0 \\ 0 & 0 & 1\end{array} \right) \left(\begin{array}{c} x \\ y \\ z \end{array} \right) = \left(\begin{array}{c} x\cos \theta + y\sin\theta \\ -x \sin \theta + y\cos \theta \\ z \end{array} \right)

which corresponds to a rotation of the vector in the x-y plane. Transposing a column vector into a row vector is easy too.

Well, that’s how I was taught to do it.

However, somebody, sometime, decided that, in Britain at least, this concise and computationally helpful notation had to be jettisoned and students instead must be forced to write

\underline{r} = x \underline{\hat{i}} + y \underline{\hat{j}} + z \underline{\hat{k}}

Some of you may even be used to doing it that way yourself. Why is this awful? For a start, it’s incredibly clumsy. It is less intuitive, doesn’t lend itself to easy operations on the vectors like I described above, doesn’t translate easily into the more general case of a matrix, and is generally just …well… awful.

Worse still, for the purpose of teaching inexperienced students physics, it offers the possibility of horrible notational confusion. In particular, the unit vector \underline{\hat{i}} is too easily confused with i, the square root of minus one. Introduce a plane wave with a wavevector \underline{k} and it gets even worse, especially when you want to write \exp(i\underline{k}\cdot\underline{x})!

No, give me the row and column notation any day.

I would really like to know is who decided that our schools had to teach the horrible notation, rather than the nice one, and why? I think everyone who teaches physics knows that a clear and user-friendly notation is an enormous help and a bad one is an enormous hindrance.  It doesn’t surprise me that some student struggle with even simple mathematics when its presented in such a silly way. On those grounds, I refuse to play ball, and always use the better notation.

Call me old-fashioned.

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