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

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

Detection of Pulsed X-ray Emission from PSR B1706-44

We report the first detection of pulsed X-ray emission from the young, energetic radio and Gamma-ray pulsar PSR B1706-44. We find a periodic signal at a frequency of f = 9.7588088 +/- 0.0000026 Hz (at epoch 51585.34104 MJD), consistent with the radio ephemeris, using data obtained with the High Resolution Camera on-board the Chandra X-ray Observatory}. The probability that this detection is a chance occurrence is 3.5E-5 as judged by the Rayleigh test. The folded light curve has a broad, single-peaked profile with a pulsed fraction of 23% +/- 6%. This result is consistent the ROSAT PSPC upper limit of < 18% after allowing for the ability of Chandra to resolve the pulsar from a surrounding synchrotron nebula. We also fitted Chandra spectroscopic data on PSR B1706-44, which require at least two components, e.g., a blackbody of temperature T(infinity) between 1.51E6 K and 1.83E6 K and a power-law of Gamma = 2.0 +/- 0.5. The blackbody radius at the nominal 2.5 kpc distance is only R(infinity) = 3.6 +/- 0.9 km, indicating either a hot region on a cooler surface, or the need for a realistic atmosphere model that would allow a lower temperature and larger area. Because the power-law and blackbody spectra each contribute more than 23% of the observed flux, it is not possible to decide which component is responsible for the modulation in the spectrally unresolved light curve.Comment: 6 pages, 4 figures, Latex, emulateapj. Published version. Includes an updated radio ephemeris and presents the absolute radio/X-ray phase alignmen