A note on quantum chaology and gamma approximations to eigenvalue spacings for infinite random matrices

Quantum counterparts of certain simple classical systems can exhibit chaotic behaviour through the statistics of their energy levels and the irregular spectra of chaotic systems are modelled by eigenvalues of infinite random matrices. We use known bounds on the distribution function for eigenvalue spacings for the Gaussian orthogonal ensemble (GOE) of infinite random real symmetric matrices and show that gamma distributions, which have an important uniqueness property, can yield an approximation to the GOE distribution. That has the advantage that then both chaotic and non chaotic cases fit in the information geometric framework of the manifold of gamma distributions, which has been the subject of recent work on neighbourhoods of randomness for general stochastic systems. Additionally, gamma distributions give approximations, to eigenvalue spacings for the Gaussian unitary ensemble (GUE) of infinite random hermitian matrices and for the Gaussian symplectic ensemble (GSE) of infinite random hermitian matrices with real quaternionic elements, except near the origin. Gamma distributions do not precisely model the various analytic systems discussed here, but some features may be useful in studies of qualitative generic properties in applications to data from real systems which manifestly seem to exhibit behaviour reminiscent of near-random processes.Comment: 9 pages, 5 figures, 2 tables, 27 references. Updates version 1 with data and references from feedback receive

An inhomogeneous stochastic rate process for evolution from states in an information geometric neighbourhood of uniform fitness

This study elaborates some examples of a simple evolutionary stochastic rate process where the population rate of change depends on the distribution of properties--so different cohorts change at different rates. We investigate the effect on the evolution arising from parametrized perturbations of uniformity for the initial inhomogeneity. The information geometric neighbourhood system yields also solutions for a wide range of other initial inhomogeneity distributions, including approximations to truncated Gaussians of arbitrarily small variance and distributions with pronounced extreme values. It is found that, under quite considerable alterations in the shape and variance of the initial distribution of inhomogeneity in unfitness, the decline of the mean does change markedly with the variation in starting conditions, but the net population evolution seems surprisingly stable.Comment: 9 pages, 11 figures, 9 reference

Some recent work in Frechet geometry

Some recent work in Frechet geometry is briefly reviewed. In particular an earlier result on the structure of second tangent bundles in the finite dimensional case was extended to infinite dimensional Banach manifolds and Frechet manifolds that could be represented as projective limits of Banach manifolds. This led to further results concerning the characterization of second tangent bundles and differential equations in the more general Frechet structure needed for applications. A summary is given of recent results on hypercyclicity of operators on Frechet spaces.Comment: 14 pages 48 reference

On the entropy flows to disorder

Gamma distributions, which contain the exponential as a special case, have a distinguished place in the representation of near-Poisson randomness for statistical processes; typically, they represent distributions of spacings between events or voids among objects. Here we look at the properties of the Shannon entropy function and calculate its corresponding flow curves. We consider univariate and bivariate gamma, as well as Weibull distributions which also include exponential distributions.Comment: Enlarged version of original. 11 pages, 6 figures, 15 reference

Two decades of pulsar timing of Vela

Pulsar timing at the Mt Pleasant observatory has focused on Vela, which can be tracked for 18 hours of the day. These nearly continuous timing records extend over 24 years allowing a greater insight into details of timing noise, micro glitches and other more exotic effects. In particular we report the glitch parameters of the 2004 event, along with the reconfirmation that the spin up for the Vela pulsar occurs instantaneously to the accuracy of the data. This places a lower limit of about 30 seconds for the acceleration of the pulsar to the new rotational frequency. We also confirm of the low braking index for Vela, and the continued fall in the DM for this pulsar.Comment: Isolated Neutron Stars conference, London, April 24-28 200