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 …

Read More

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 …

Read More

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 …

Read More

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 …

Read More

Here is an interactive version of this diagram: !APF Figure 4 (click on the image to show a larger version.)

Read More

Here we show how a Tschirnhausen transformation can be used to solve a quartic equation. The steps are: Ensure the quartic is missing the cubic term, and its initial coefficient is 1. We can do this by first dividing by the initial coefficient to obtain an equation

\[ x^4+b_3x^3+b_2x^2+b_1x+b_0=0 \]

and then replace …

Read More

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 \(x^2\) …

Read More

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 …

Read More

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 …

Read More

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 …

Read More