What's all this, then?

Mathematics, software, general nonsense

Poles of inaccessibility

Just recently there was a news item about a solo explorer being the first Australian to reach the Antarctic “Pole of inaccessibility”. Such a Pole is usually defined as that place on a continent that is furthest from the sea. The South Pole is about 1300km from the nearest open sea, and can be reached by specially fitted aircraft, or by tractors and sleds along the 1600km “South Pole Highway” from McMurdo Base.

An interesting sum

I am not an analyst, so I find the sums of infinite series quite mysterious. For example, here are three. The first one is the value of \(\zeta(2)\), very well known, sometimes called the “Basel Problem” and first determined by (of course) Euler: \[ \sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6}. \] Second, subtracting one from the denominator: \[ \sum_{n=2}^\infty\frac{1}{n^2-1}=\frac{3}{4} \] This sum is easily demonstrated by partial fractions: \[ \frac{1}{n^2-1}=\frac{1}{2}\left(\frac{1}{n-1}-\frac{1}{n+1}\right) \] and so the series can be expanded as: \[ \frac{1}{2}\left(\frac{1}{1}-\frac{1}{3}+\frac{1}{2}-\frac{1}{4}+\frac{1}{3}-\frac{1}{5}\cdots\right) \] This is a telescoping series in which every term in the brackets is cancelled except for \(1+1/ 2\), which produces the sum immediately.

Runge's phenomenon in Geogebra

Runge’s phenomenon says roughly that a polynomial through equally spaced points over an interval will wobble a lot near the ends. Runge demonstrated this by fitting polynomials through equally spaced point in the interval \([-1,1]\) on the function \[ \frac{1}{1+25x^2} \] and this function is now known as “Runge’s function”. It turns out that Geogebra can illustrate this extremely well. Equally spaced vertices Either open up your local version of Geogebra, or go to http://geogebra.

Fitting the SIR model of disease to data in Python

Introduction and the problem The SIR model for spread of disease was first proposed in 1927 in a collection of three articles in the Proceedings of the Royal Society by Anderson Gray McKendrick and William Ogilvy Kermack; the resulting theory is known as Kermack–McKendrick theory; now considered a subclass of a more general theory known as compartmental models in epidemiology. The three original articles were republished in 1991, in a special issue of the Bulletin of Mathematical Biology.

Mapping voting gains between elections

So this goes back quite some time to the recent Australian Federal election on May 18. In my own electorate (known formally as a “Division”) of Cooper, the Greens, who until recently had been showing signs of winning the seat, were pretty well trounced by Labor. Some background asides First, “Labor” as in “Australian Labor Party” is spelled the American way; that is, without a “u”, even though “labour” meaning work, is so spelled in Australian English.

Tschirnhausen's solution of the cubic

A general cubic polynomial has the form \[ ax^3+bx^2+cx+d \] but a general cubic equation can have the form \[ x^3+ax^2+bx+c=0. \] We can always divide through by the coefficient of \(x^3\) (assuming it to be non-zero) to obtain a monic equation; that is, with leading coefficient of 1. We can now remove the \(x^2\) term by replacing \(x\) with \(y-a/3\): \[ \left(y-\frac{a}{3}\right)^{\negmedspace 3}+a\left(y-\frac{a}{3}\right)^{\negmedspace 2} +b\left(y-\frac{a}{3}\right)+c=0. \] Expanding and simplifying produces \[ y^3+\left(b-\frac{a^2}{3}\right)y+\frac{2}{27}a^3-\frac{1}{3}ab+c=0.

Colonial massacres, 1794 to 1928

The date January 26 is one of immense current debate in Australia. Officially it’s the date of Australia Day, which supposedly celebrates the founding of Australia. To Aboriginal peoples it is a day of deep mourning and sadness, as the date commemorates over two centuries of oppression, bloodshed, and dispossession. To them and their many supporters, January 26 is Invasion Day. The date commemorates the landing in 1788 of Arthur Phillip, in charge of the First Fleet and the first Governor of the colony of New South Wales.

Vote counting in the Australian Senate

Recently we have seen senators behaving in ways that seem stupid, or contrary to accepted public opinion. And then people will start jumping up and down and complaining that such a senator only got a tiny number of first preference votes. One commentator said that one senator, with 19 first preference votes, “couldn’t muster more than 19 members of his extended family to vote for him”. This displays an ignorance of how senate counting works.

Concert review: Lixsania and the Labyrinth

This evening I saw the Australia Brandenburg Orchestra with guest soloist Lixsania Fernandez, a virtuoso player of the viola da gamba, from Cuba. (Although she studied, and now lives, in Spain.) Lixsania is quite amazing: tall, statuesque, quite absurdly beautiful, and plays with a technique that encompasses the wildest of baroque extravagances as well as the most delicate and refined tenderness. The trouble with the viol, being a fairly soft instrument, is that it’s not well suited to a large concert hall.

Linear programming in Python (2)

Here’s an example of a transportation problem, with information given as a table: .transport table { width: 80%; table-layout: fixed;} th, td { text-align: center;} th { background: #DCDCDC;} Demands 300 360 280 340 220 750 100 150 200 140 35 Supplies  400 50 70 80 65 80 350 40 90 100 150 130 This is an example of a balanced, non-degenerate transportation problem.