Research

What's all this, then?

• Ramanujan's cubes 2022-04-03
  This post illustrates the working of Ramanujan's generating functions for solving Euler's diophantine equation \(a^3+b^3=c^3+d^3\) as described by Andrews and Berndt in "Ramanujan's Lost Notebook, Part IV", pp 199 - 205 (Section 8.5). The text is available from Springer. Ramanujan's result is that if \[ f_1(x) = \frac{1+53x+9x^2}{1-82x-82x^2+x^3} = a_0+a_1x+a_2x^2+a_3x^3+\cdots = \alpha_0+\frac{\alpha_1}{x}+\frac{\alpha_2}{x^2}+\frac{\alpha_3}{x^3}+\cdots\] \[ f_2(x) = […]
• Wordle 2022-01-24
  Wordle is a pleasant game, basically Mastermind with words. You choose an English word (although it can also be played in other languages), and then you're told if your letters are incorrect, correct but in the wrong place, or correct and in the right place. These are shown by the colours grey, yellow, and green. […]
• Five letter words in English 2022-01-23
  I was going to make a little post about Wordle, but I go sidetracked exploring five letter words. At the same time, I had a bit of fun with regular expressions and some simple scripting with ZSH. The start was to obtain lists of 5-letter words. One is available at the Stanford Graphbase site; the […]
• A new year (2022) 2022-01-01
  What does one do on the first day of a new year but write a blog post, and in it clearly delineate all plans for the coming year? Well, I'm doing the first part, but not the second, as I know that any plans will not be fulfilled - something always gets in the way. […]
• Exploring Tanh-Sinh quadrature 2021-04-30
  As is well known, tanh-sinh quadrature takes an integral \[ \int_{-1}^1f(x)dx \] and uses the substitution \[ x = g(t) = \tanh\left(\frac{\pi}{2}\sinh t\right) \] to transform the integral into \[ \int_{-\infty}^{\infty}f(g(t))g'(t)dt. \] The reason this works so well is that the derivative \(g'(t)\) dies away at a double exponentional rate; that is, at the rate […]
• A note on Steffensen's method for solving equations 2021-04-30
  Steffensen's method is based on Newton's iteration for solving a non-linear equation \(f(x)=0\): \[ x\leftarrow x-\frac{f(x)}{f'(x)} \] Newton's method can fail to work in a number of ways, but when it does work it displays qudratic convergence; the number of correct signifcant figures roughly doubling at each step. However, it also has the disadvntage of […]
• High precision quadrature with Clenshaw-Curtis 2021-04-21
  An article by Bailey, Jeybalan and LI, "A comparison of three high-precision quadrature schemes", and available online here, compares Gauss-Legendre quadrature, tanh-sinh quadrature, and a rule where the nodes and weights are given by the error function and its integrand respectively. However, Nick Trefethen of Oxford has shown experimentally that Clenshaw-Curtis quadrature is generally no […]
• The circumference of an ellipse 2021-04-10
  Note: This blog post is mainly computational, with a hint of proof-oriented mathematics here and there. For a more in-depth analysis, read the excellent article "Gauss, Landen, Ramanujan, the Arithmetic-Geometric Mean, Ellipses, pi, and the Ladies Diary" by Gert Akmkvist and Bruce Berndt, in The American Mathematical Monthly, vol 95 no. 7 (August-September 1988), pages […]
• Voting power (7): Quarreling voters 2021-01-24
  In all the previous discussions of voting power, we have assumed that all winning coalitions are equally likely. But in practice that is not necessarily the case. Two or more voters may be opposed on so many issues that they would never vote the same way on any issues: such a pair of voters may […]
• Voting power (6): Polynomial rings 2021-01-22
  As we have seen previously, it's possible to compute power indices by means of polynomial generating functions. We shall extend previous examples to include the Deegan-Packel index, in a way somewhat different to that of Alonso-Meijide et al (see previous post for reference). Again, suppose we consider the voting game \[ [30;28,16,5,4,3,3] \] What we'll […]
• Voting power (5): The Deegan-Packel and Holler power indices 2021-01-14
  We have explored the Banzhaf and Shapley-Shubik power indices, which both consider the ways in which any voter can be pivotal, or critical, or necessary, to a winning coalition. A more recent power index, which takes a different approach, was defined by Deegan and Packel in 1976, and considers only minimal winning coalitions. A winning […]
• Three-dimensional impossible CAD 2021-01-10
  Recently I friend and I wrote a semi-serious paper called "The geometry of impossible objects" to be delivered at a mathematics technology conference. The reviewer was not hugely complimentary, saying that there was nothing new in the paper. Well, maybe not, but we had fun pulling together some information about impossible shapes and how to […]
• Voting power (4): Speeding up the computation 2021-01-06
  Introduction and recapitulation Recall from previous posts that we have considered two power indices for computing the power of a voter in a weighted system; that is, the ability of a voter to influence the outcome of a vote. Such systems occur when the voting body is made up of a number of "blocs": these […]
• Voting power (3): The American swing states 2021-01-03
  As we all know, American Presidential elections are done with a two-stage process: first the public votes, and then the Electoral College votes. It is the Electoral College that actually votes for the President; but they vote (in their respective states) in accordance with the plurality determined by the public vote. This unusual system was […]
• Voting power (2): computation 2020-12-31
  Naive implementation of Banzhaf power indices As we saw in the previous post, computation of the power indices can become unwieldy as the number of voters increases. However, we can very simply write a program to compute the Banzhaf power indices simply by looping over all subsets of the weights: 1def banzhaf1(q,w): 2 n = […]
• Voting power 2020-12-30
  After the 2020 American Presidential election, with the usual post-election analyses and (in this case) vast numbers of lawsuits, I started looking at the Electoral College, and trying to work out how it worked in terms of power. Although power is often conflated simply with the number of votes, that's not necessarily the case. We […]
• Electing a president 2020-11-07
  Every four years (barring death or some other catastrophe), the USA goes through the periodic madness of a presidential election. Wild behaviour, inaccuracies, mud-slinging from both sides have been central since George Washington's second term. And the entire business of voting is muddied by the Electoral College, the 538 members of which do the actual […]
• Enumerating the rationals 2020-01-18
  The rational numbers are well known to be countable, and one standard method of counting them is to put the positive rationals into an infinite matrix \(M=m_{ij}\), where \(m_{ij}=i/j\) so that you end up with something that looks like this: \[ \left[\begin{array}{ccccc} \frac{1}{1}&\frac{1}{2}&\frac{1}{3}&\frac{1}{4}&\dots\\[1ex] \frac{2}{1}&\frac{2}{2}&\frac{2}{3}&\frac{2}{4}&\dots\\[1ex] \frac{3}{1}&\frac{3}{2}&\frac{3}{3}&\frac{3}{4}&\dots\\[1ex] \frac{4}{1}&\frac{4}{2}&\frac{4}{3}&\frac{4}{4}&\dots\\[1ex] \vdots&\vdots&\vdots&\vdots&\ddots \end{array}\right] \] It is clear that not only […]
• Fitting the SIR model of disease to data in Julia 2020-01-15
  A few posts ago I showed how to do this in Python. Now it's Julia's turn. The data is the same: spread of influenza in a British boarding school with a population of 762. This was reported in the British Medical Journal on March 4, 1978, and you can read the original short article here. […]
• The Butera-Pernici algorithm (2) 2020-01-06
  The purpose of this post will be to see if we can implement the algorithm in Julia, and thus leverage Julia's very fast execution time. We are working with polynomials defined on nilpotent variables, which means that the degree of any generator in a polynomial term will be 0 or 1. Assume that our generators […]