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 …

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 may …

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. …

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 …

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 …

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 …

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} …

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. As before we use the SIR model, …

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 are indexed …

Introduction We know that there is no general sub-exponential algorithm for computing the permanent of a square matrix. But we may very reasonably ask -- might there be a faster, possibly even polynomial-time algorithm, for some specific classes of matrices? For example, a sparse matrix will have most terms of the …