MSI Weekly Bulletin - Week starting Monday 22 October, 2007

The motivation behind this work is the solution of the Hasegawa-Wakatani equations that are used to model plasma flow. To design or predict the behaviour of turbulent flows prolonged time series is required. The challenge is to find a numerical method that can allow long time steps without polluting the results with numerical instabilities. The primary focus of the talk is the solution of systems of non-linear ordinary differential using linear multistep methods. We will present the theory behind the solution of non-linear equations and use it to predict the maximum possible time steps sizes for a number of different linear multistep methods. The results will be applied to Hasegawa-Wakatani equations. BIO: http://wwwmaths.anu.edu.au/~stals/

The famous Gelfand-Mazur theorem states that a complex normed division algebra is isomorphic to the complex numbers. The real case is a little more complicated with the reals, complexes or quaternions possible. I will discuss an application of this theorem, first proven by Ingelstam, that a unital Hilbert algebra is in fact a division algebra and hence isomorphic to the complexes if the algebra is complex or the reals, complexes or quaternions if real. Given time I may also touch on some generalisations and a C*-algebra analogue.