What's all this, then?

Determining the number of self-avoiding paths between points on a grid is a very difficult problem; in general unsolved.

Vowels in Australian placenames In the state of Victoria, Australia, there is a town called called Euroa.

The previous blog post showed some examples of self-avoiding paths, all on orthogonal (Cartesian) grids.

Suppose we start at \((0,0)\) on the cartesian grid, and wander about randomly only in North, South, East and West directions, never visiting a vertex more than once.

A self avoiding walk is a path through a graph where no vertex is visited more than once.

The Asian Technology Conference in Mathematics is a friendly, genial conference which has been held almost every year since 1996.

Recall from the previous post that the Weierstrass-Durand-Kerner (WDK) method finds all roots of a polynomial equation in a manner similar to Newton’s method, and so converges quadratically.

This is a method for finding all the roots of a polynomial simultaneously, by applying a sort of Newton-Raphson method.

Recap, and background Two posts ago we showed how, given four points in the plane in general position, but with a few restrictions, it was possible to find two parabolas through those points.

It is (well?)

Introduction It is (or should be) well known that a parabola has the cartesian form \[ (Ax+By)^2+Cx+Dy+E = 0. \] This looks as though there are five values needed, but we can divide through in such a way as to make any of the coefficients we like equal to…

Although the method is simple to describe, the algebra becomes messy when written in full generality.

A bicentric heptagon is one for which all vertices lie on a circle, and for which all edges are tangential to another circle.

Introduction Poncelet’s porism or Poncelet’s closure theorem is one of the most remarkable results in plane geometry.

A totally different approach to dithering is error diffusion.

Image dithering, also known as half-toning, is a method for reducing the number of colours in an image, while at the same time trying to retain as much of its “look and feel” as possible.

In the previous post, we saw that a small change to the method of false position provided much faster convergence, while retaining its bracketing.

Such a long time since a last post!

Carroll originally invented his Doublets in 1877, they were published in “Vanity Fair” (the magazine, not the Thackeray novel) in 1879.

Apparently there’s a version of Doublets (see previous post) which allows you to add or delete a letter each turn.

Lewis Carroll’s game of Doublets Such a long time since my last post!

Plagiarism, text matching, and academic integrity Every modern academic teacher is in thrall to giant text-matching systems such as Ouriginal or Turnitin.

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.

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".

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.

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.

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 resu…

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).

Wordle is a pleasant game, basically Mastermind with words.

I was going to make a little post about Wordle, but I go sidetracked exploring five letter words.

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?

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…

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

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 a…

Note: This blog post is mainly computational, with a hint of proof-oriented mathematics here and there.

In all the previous discussions of voting power, we have assumed that all winning coalitions are equally likely.

As we have seen previously, it’s possible to compute power indices by means of polynomial generating functions.

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.

Recently I friend and I wrote a semi-serious paper called “The geometry of impossible objects” to be delivered at a mathematics technology conference.

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.

As we all know, American Presidential elections are done with a two-stage process: first the public votes, and then the Electoral College votes.

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.

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.

Every four years (barring death or some other catastrophe), the USA goes through the periodic madness of a presidential election.

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[\beg…

A few posts ago I showed how to do this in Python.

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.

Introduction We know that there is no general sub-exponential algorithm for computing the permanent of a square matrix.

As a first blog post for 2020, I’m dusting off one from my previous blog, which I’ve edited only slightly.

As I discussed in my last blog post, the permanent of an \(n\times n\) matrix \(M=m_{ij}\) is defined as \[ \text{per}(M)=\sum_{\sigma\in S_n}\prod_{i=1}^nm_{i,\sigma(i)} \] where the sum is taken over all permutations of the \(n\) numbers \(1,2,\ldots,n\…