Exploring Periodic Orbit Expansions and Renormalisation with the Quantum Triangular Billiard

A study of the quantum triangular billiard requires consideration of a boundary value problem for the Green's function of the Laplacian on a trianglar domain. Our main result is a reformulation of this problem in terms of coupled non--singular integral equations. A non--singular formulation, via Fredholm's theory, guarantees uniqueness and provides a mathematically firm foundation for both numerical and analytic studies. We compare and contrast our reformulation, based on the exact solution for the wedge, with the standard singular integral equations using numerical discretisation techniques. We consider in detail the (integrable) equilateral triangle and the Pythagorean 3-4-5 triangle. Our non--singular formulation produces results which are well behaved mathematically. In contrast, while resolving the eigenvalues very well, the standard approach displays various behaviours demonstrating the need for some sort of ``renormalisation''. The non-singular formulation provides a mathematically firm basis for the generation and analysis of periodic orbit expansions. We discuss their convergence paying particular emphasis to the computational effort required in comparision with Einstein--Brillouin--Keller quantisation and the standard discretisation, which is analogous to the method of Bogomolny. We also discuss the generalisation of our technique to smooth, chaotic billiards.Comment: 50 pages LaTeX2e. Uses graphicx, amsmath, amsfonts, psfrag and subfigure. 17 figures. To appear Annals of Physics, southern sprin

A non-parametric and scale-independent method for cluster analysis II: the multivariate case

A general method is described for detecting and analysing galaxy systems. The multivariate geometrical structure of the sample is studied by using an extension of the method which we introduced in a previous paper. The method is based on an estimate of the probability density underlying a data sample. The density is estimated by using an iterative and adaptive kernel estimator. The used kernels have spherical symmetry, however we describe a method in order to estimate the locally optimal shape of the kernels. We use the results of the geometrical structure analysis in order to study the effects that is has on the cluster parameter estimate. This suggests a possible way to distinguish between structure and substructure within a sample. The method is tested by using simulated numerical models and applied to two galaxy samples taken from the literature. The results obtained for the Coma cluster suggest a core-halo structure formed by a large number of geometrically independent systems. A different conclusion is suggested by the results for the Cancer cluster indicating the presence of at least two independent structures both containing substructure. The dynamical consequences of the results obtained from the geometrical analysis will be described in a later paper. Further applications of the method are suggested and are currently in progress.Comment: To appear in Monthly Notices of R.A.S., 50 pages of text, latex file, aasms style, figures are available on request from the Autho

The nurse of the Mediterranean

During the First World War Malta did not take an active part in the fighting. Britain was joined in an ‘entente’ a friendship agreement with France since 1904 and later with Russia in 1907. On the other hand Germany was allied to the Austrian- Hungerian Empire, hence when the Great War started in July 1914 there were France, Britain and Russia on one side and Germany and Austria-Hungary on the other. The British fleet “ruled the waves”, hence with France and Britain as allies, to be joined later by Italy, the Mediterranean was more or less an allied lake, with Malta in the centre.peer-reviewe