MSI Weekly Bulletin - Week starting Monday 28 August, 2006

Events such as explosions or radar pulses generate acoustic or electromagnetic signals, and measurements of the times that the associated signals arrive at known locations can be used to locate the original event. Applications range from locating artillery by sound ranging in World War I to GPS today. The equations relating N measurements xn, tn to the unknown location x, t are non-linear, but it has long been known that this system can be reduced to N-1 linear equations and one quadratic. On the face of it, however, this reduction does not appear to be much use in the over-determined case as the standard least-squares solution of the linear system has no particular relation to the desired maximum-likelihood estimate. The talk will show that nevertheless it is possible to take advantage of the existence of the equivalent, nearly linear system to simplify optimisation of the likelihood function. To do so, likelihood maximisation is recast as a constrained least squares problem, which then translates into an errors-in-variables problem for the equivalent, nearly linear system. Errors-in-variables problems are in turn solvable by total-least-squares techniques, with the present case being an apparently hitherto unstudied variant in that the same errors appear in two different columns: its solution turns out to be characterised in terms of a quadratic eigenvalue decomposition instead of the standard SVD. Finally the approach can be generalised: the talk will conclude by presenting an application in HF communications where locations and propagation paths are restricted to the surface of a sphere.

It is quite well known that some physical and biological structures can be modelled by fractals. Examples include coastlines, ferns, the human nervous system, and formations of terrain. But how do we construct mappings from one fractal to another? How may we stretch and deform space so that one coastline becomes another one? How might we match two plants of the same species to one another? This talk will show you how to construct such mappings, called fractal transformations, and illustrate some of their remarkable properties. Applications to art and computer graphics are obvious. But can they also be used to identify skin cancers?

Gene regulatory networks describe the interaction of genes and its products the proteins. Because relatively small numbers of copies of each substance are involved the dynamics of these networks are mainly driven by noise generated by the translation processes involving the genes and their products. Therefore these systems are best described by stochastic models. With these models, the stochastic master equations, one can follow the time development of the probability distribution for the states defined by copy numbers of each substance. As for each substance involved, the state space grows exponentially the challenges lie in the large discrete state spaces due to high dimensionality.

Lotka-Volterra systems are a large class of models for interaction among species. Depending on such interactions competition, cooperation or predator-prey situations can occurr, giving rise to further classifications. The dynamics depends on parameters intrinsic to the species, tipically growth rate and carrying capacity, and on the interaction coefficients, which however are often more difficult to specify. We focus here on competition among species and, differently from the classical case, we consider for them a kind of adaptive skills: the ability to compete is proportional to the average number of contacts between species in their past, however with a fade-out memory effect. For the general case of N-species a system of NxN nonlinear ordinary differential equations is obtained, investigated according to bifurcation theory, with the aim to discuss the role of interactions. The investigation takes advantage of the existence of reduced systems, where N appears as a parameter, accounting for all the equilibria and their stability. Such results are very useful in moving toward higher dimensional cases, for which not many results are available. Adaptation is shownn to be a mechanism able to establish the appearance of a variety of behaviors, different from equilibria, as distinct kinds of oscillations and chaotic patterns. In particular we provide an example of species coexistence in the form of complicated alternance between chaotic behavior and periodic one, in both cases with multiplicity of attractors.