What's all this, then?

• Carroll's "improved" Doublets: allowing permutations 2022-11-07
  Carroll originally invented his Doublets in 1877, they were published in "Vanity Fair" (the magazine, not the Thackeray novel) in 1879. Some years later, in an 1892 letter, Carroll added another rule: that permutations were allowed. This allows very neat chains such as: roses, noses, notes, steno, stent, scent Because the words stay the same […]
• Super Doublets: more word ladders with Julia 2022-11-05
  Apparently there's a version of Doublets (see previous post) which allows you to add or delete a letter each turn. Thus we can go from WHEAT to BREAD as WHEAT, HEAT, HEAD, READ, BREAD which is shorter than the ladder given in that previous post. However, we can easily adjust the material from that post […]
• Word ladders with Julia 2022-11-03
  Lewis Carroll's game of Doublets Such a long time since my last post! Well, that's the working life for you. Anyway, recently I was reading about Lewis Carroll - always one of my favourite people - and was reminded of his word game "Doublets" in which one word is turned into another by changing one […]
• Every academic their own text-matcher 2022-06-19
  Plagiarism, text matching, and academic integrity Every modern academic teacher is in thrall to giant text-matching systems such as Ouriginal or Turnitin. These systems are sold as "plagiarism detectors", which they are not - they are text matching systems, and they generally work by providing a report showing how much of a student's submitted work […]
• More mapping "not quite how-to" - Voronoi regions 2022-06-18
  What this post is about In the previous post we showed how to set up a simple interactive map using Python and its folium package. As the example, we used a Federal electorate situated within the city of Melbourne, Australia, and the various voting places, or polling places (also known as polling "booths") associated with […]
• A mapping "not quite how-to" 2022-06-11
  Message about the underlying software NOTE: much of the material and discussion here uses the Python package "folium", which is a front end to the Javascript package "leaflet.js". The lead developer of leaflet.js is Volodymyr Agafonkin, a Ukrainian up until recently living and working in Kyiv. Leaflet version 1.80 was released on April 18, "in […]
• Further mapping: a win and a near miss 2022-06-09
  In this post we look at two Divisions from the recent Federal election: the inner city seat of Melbourne, and the bayside seat of Macnamara. Up until the recent election, Melbourne was the only Division to have a Greens representative. Macnamara, previously known as "Melbourne Ports" has been a Labor stronghold for all of its […]
• Post-election mapping 2022-06-05
  This continues on from the previous post, trying to make some sense of the voting in my electorate of Wills and the neighbouring electorate of Cooper. Both these electorates (or more formally "Divisions"), as I mentioned in the previous post, are very similar in their geography, demography, and history. Last post I simply showed a […]
• Post-election swings 2022-05-22
  So the Australian federal election of 2022 is over as far as the public is concerned; all votes have been cast and now it's a matter of waiting while the Australian Electoral Commission tallies the numbers, sorts all the preferences, and arrives at a result. Because of the complications of the voting system, and of […]
• 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 […]