Archive for the mathematics Category

Halloween in LaTeX

Posted in History, mathematics on October 21, 2018 by telescoper

I forget where I found this list of spooky LaTeX commands but, with the dreaded Halloween coming up soon, I thought I’d share it here.

Anyway, it reminded of the mathematical curve known in English as The Witch of Agnesi, the witch of which is a mistranslation of the Italian versiera meaning a ‘sheet’ (ie the rope connecting to a ship’s sail) rather than a shortened version of ‘avversiera’ meaning ‘a female devil’ or ‘witch’.

Advertisements

Vector Calculus Weather

Posted in mathematics, The Universe and Stuff with tags , , on October 12, 2018 by telescoper

As it happens I did a lecture today about vector fields as part of my module on vector calculus. Whenever I did similar lectures in the past I used the day’s weather map as an illustration, so this morning I downloaded what turned out to be a particularly dramatic example. The curl of the velocity field around the weather system off the west coast of Ireland this morning was definitely non-zero…

Storm Callum turned out to be not as damaging as feared. Apparently it was rather windy in Maynooth overnight, but I slept right through it.

The Problem of the Moving Triangle

Posted in Cute Problems, mathematics with tags , , on August 16, 2018 by telescoper

I found this nice geometric puzzle a few days ago on Twitter. It’s not too hard, but I thought I’d put it in the `Cute Problems‘ folder.

In the above diagram, the small equilateral triangle moves about inside the larger one in such a way that it keeps the orientation shown. What can you say about the sum a+b+c?

Answers through the comments box please, and please show your working!

The Problem with Odd Moments

Posted in Bad Statistics, Cute Problems, mathematics with tags , , on July 9, 2018 by telescoper

Last week, realizing that it had been a while since I posted anything in the cute problems folder, I did a quick post before going to a meeting. Unfortunately, as a couple of people pointed out almost immediately, there was a problem with the question (a typo in the form of a misplaced bracket). I took the post offline until I could correct it and then promptly forgot about it. I remembered it yesterday so have now corrected it. I also added a useful integral as a hint at the end, because I’m a nice person. I suggest you start by evaluating the expectation value (i.e. the first-order moment). Answers to parts (2) and (3) through the comments box please!

Answers to (2) and (3) via the comments box please!

 

SOLUTION: I’ll leave you to draw your own sketch but, as Anton correctly points out, this is a distribution that is asymmetric about its mean but has all odd-order moments equal (including the skewness) equal to zero. it therefore provides a counter-example to common assertions, e.g. that asymmetric distributions must have non-zero skewness. The function shown in the problem was originally given by Stieltjes, but a general discussion can be be found in E. Churchill (1946) Information given by odd moments, Ann. Math. Statist. 17, 244-6. The paper is available online here.

Does Physics need Philosophy (and vice versa)?

Posted in mathematics, The Universe and Stuff with tags , , on June 1, 2018 by telescoper

There’s a new paper on the arXiv by Carlo Rovelli entitled Physics Needs Philosophy. Philosophy Needs Physics. Here is the abstract:

Contrary to claims about the irrelevance of philosophy for science, I argue that philosophy has had, and still has, far more influence on physics than is commonly assumed. I maintain that the current anti-philosophical ideology has had damaging effects on the fertility of science. I also suggest that recent important empirical results, such as the detection of the Higgs particle and gravitational waves, and the failure to detect supersymmetry where many expected to find it, question the validity of certain philosophical assumptions common among theoretical physicists, inviting us to engage in a clearer philosophical reflection on scientific method.

Read and discuss.

When Log Tables aren’t Log Tables

Posted in Education, mathematics, Maynooth with tags , , , , , on May 17, 2018 by telescoper

Every now and then – actually more frequently than that – I reveal myself in Ireland as an ignorant foreigner. The other day some students were going through a past examination paper (from 2014) and I was surprised to see that the front cover (above) mentioned  `log tables’.

Now I’m old enough to remember using tables of logarithms (and other mathematical tables  of such things as square roots and trigonometric functions, in the form of lists of numbers) extensively at school. These were provided in this book of four-figure tables (which can now buy for 1p on Amazon, plus p&p).

As a historical note I’ll point out that I was in the first year at my school that progressed to calculators rather than slide rules (in the third year) so I was never taught how to use the former. My set of four-figure tables which was so heavily used that it was falling to bits anyway, never got much use after that and I threw it out when I went to university despite the fact that I’m a notorious hoarder.

Anyway, assuming that the mention of `log tables’ was a relic of many years past, I said to the group of students going through the old examination paper that it seemed somewhat anachronistic. I was promptly corrected, and told that `log tables’ are in regular use in schools and colleges throughout Ireland, but that the term is a shorthand for a booklet containing a general collection of mathematical formulae, scientific data and other bits of stuff that might come in useful to students; for an example appropriate to the Irish Leaving Certificate, see here. One thing that they don’t contain is a table of logarithms…

Students in Physics & Astronomy at Cardiff University are also given a formula booklet for use during examinations. I don’t remember having access to such a thing as an undergraduate, but I don’t object to it. It seems to me that an examination shouldn’t be a memory test, and giving students the basic formulae as a starting point if anything allows the examiner to concentrate on testing what matters much more, i.e. the ability to formulate and solve a problem. The greatest challenge of science education at University level is, in my opinion, convincing students that their brain is much more than a memory device…

Project Work

Posted in Biographical, Education, mathematics with tags , , , , , on April 23, 2018 by telescoper

I’m progressively clearing out stuff from my office prior to the big move to Ireland. This lunchtime I opened one old box file and found my undergraduate project. This was quite an unusual thing at the time as I did Theoretical Physics in Part II (my final year) of Natural Sciences at Cambridge, which normally meant no project but an extra examination paper called Paper 5. As a member of a small minority of Theoretical Physics students who wanted to do theory projects, I was allowed to submit this in place of half of Paper 5…

The problem was to write a computer program that could solve the equations describing the action of a laser, starting with the case of a single-mode laser as shown in the diagram below that I constructed using a sophisticated computer graphics package:

The above system is described by a set of six simultaneous first-order ordinary differential equations, which are of relatively simple form to look at but not so easy to solve numerically because the equations are stiff (i.e. they involve exponential decays or growths with very different time constants). I got around this by using a technique called Gear’s method. There wasn’t an internet in those days so I had to find out about the numerical approach by trawling through books in the library.

The code I wrote – in Fortran 77 – was run on a mainframe, and the terminal had no graphics capability so I had to check the results as a list of numbers before sending the results to a printer and wait for the output to be delivered some time later. Anyway, I got the code to work and ended up with a good mark that helped me get a place to do a PhD.

The sobering thought, though, is that I reckon a decent undergraduate physics student nowadays could probably do all the work I did for my project in a few hours using Python….