MSI Weekly Bulletin - Week starting Monday 14 November, 2005

Spectral factorisation is a mathematical operation which is often credited to Norbert Wiener who originally proposed it as in ingenious solution to the so-called Wiener-Hopf equation arising in linear estimation/prediction theory. The operation takes a (matrix-valued) spectral density W in L^1 (on the unit circle) and outputs an outer function P in H^2 such that W=H*H on the unit circle. Since then spectral factorisation has been discovered to play a role in other areas as well, such as in optimal and robust control theory. The important connections between spectral factorisation to the prediction theory of stationary processes, Szeg'{o}'s theory of orthogonal polynomials and to the so-called Schur algorithm have been exploited over the years by researchers to derive iterative algorithms for computing spectral factors, the output of the spectral factorisation operation. Unfortunately, in practice, spectral factorisation is often a challenging task. It is known that non-coercive (having zeros on the unit circle) spectral densities are cumbersome or difficult to factorise for many spectral factorisation algorithms, and this becomes even more so when the spectral density is also non-rational. More recently, a necessary and sufficient condition for the spectral factorisation mapping to be sequentially continuous was established. Not long before that some significant advances had also been made in analytic interpolation theory, resulting in a new theory of interpolation with a degree or complexity constraint. In this talk, we shall explain these two recent developments in some detail. We then show how they may be combined to develop a function-theoretic treatment of spectral factorisation. This leads to a new approach to spectral factorisation of a certain class of spectral densities. A new spectral factorisation algorithm is then proposed for analytic spectral densities, which may be non-coercive (has zeros on the unit circle). Results on convergence of the algorithm will be given. We show encouraging results from application of the new algorithm to compute approximate spectral factors of the Kolmogorov and von Karman spectral densities which are both non-coercive and non-rational. These two 'physically derived' spectral densities play an important role in the study of atmospheric turbulence.

Almost everybody with any influence in funding health and health research projects things that ANY randomised controlled trial gives us better evidence than ANY other evidence possible can. That's such an amazingly dumb thing to believe that I will have to spend some time showing that people do indeed believe it. I'll also, of course, explain why it's dumb, in case that's not obvious. Assuming people agree with me that it's dumb, we might then discuss how such a situation can have occurred and what we can do about it. See also: http://rsise.anu.edu.au/~jon/NotYASS

Several methods for ultra-high throughput DNA sequencing are currently under investigation. Many of these methods yield very short blocks of sequence information (reads). A key problem for these sequencing methodologies is that as the length of each individual read decreases, the probability that a read will occur more than once in the sequence increases. There has been much debate concerning the minimum length of read required to generate useful sequence information. However, despite the importance of this analysis for the utility of many proposed ultra-high throughput sequencing methods, little work has been reported on the analysis or reassembly of sequence information from very short reads. We have developed a suffix-array based methodology for the analysis of the frequency distribution of repeats in genome sized strings and have used this methodology to analyse the level of genome sequencing possible as a function of read length. Re-sequencing and de novo sequencing of the majority of a bacterial genome is possible with read lengths of 20-30 nt, and reads of 50 nt can provide reconstructed contigs (a contiguous fragment of sequence data) of 1,000 nt and greater that cover 80% of human chromosome 1. In addition we report a visualisation of the repeat structure of genomes that highlight specific structural characteristics of different genomes.

Kobayashi-hyperbolic manifolds can be viewed as a natural generalisation of bounded domains in complex space and are a standard class of manifolds considered in complex geometry. In a series of talks we will give the details of our recent results on the classification problem for such manifolds.