Wiener's theorem on hypergroups

The following theorem on the circle group T is due to Norbert Wiener: If f∈L1(T) has non-negative Fourier coefficients and is square integrable on a neighbourhood of the identity, then f∈L2(T). This result has been extended to even exponents including p=∞, but shown to fail for all other p∈(1,∞]. All of this was extended further (appropriately formulated) well beyond locally compact abelian groups. In this paper we prove Wiener's theorem for even exponents for a large class of commutative hypergroups. In addition, we present examples of commutative hypergroups for which, in sharp contrast to the group case, Wiener's theorem holds for all exponents p∈[1,∞]. For these hypergroups and the Bessel-Kingman hypergroup with parameter 12 we characterise those locally integrable functions that are of positive type and square-integrable near the identity in terms of amalgam spaces

The Broadband Afterglow of GRB980329

We present radio observations of the afterglow of the bright gamma-ray burst GRB980329 made between one month and several years after the burst, a re-analysis of previously published submillimeter data, and late-time optical and near-infrared (NIR) observations of the host galaxy. From the absence of a spectral break in the optical/NIR colors of the host galaxy, we exclude the earlier suggestion that GRB980329 lies at a redshift of z >~5. We combine our data with the numerous multi-wavelength observations of the early afterglow, fit a comprehensive afterglow model to the entire broadband dataset, and derive fundamental physical parameters of the blast-wave and its host environment. Models for which the ejecta expand isotropically require both a high circumburst density and extreme radiative losses from the shock. No low density model (n << 10 cm^{-3}) fits the data. A burst with a total energy of ~ 10^{51} erg, with the ejecta narrowly collimated to an opening angle of a few degrees, driven into a surrounding medium with density ~ 20 cm^{-3}, provides a satisfactory fit to the lightcurves over a range of redshifts.Comment: 27 pages, incl. 6 figures, minor revisions (e.g. added/updated references) Accepted by Ap

Discovery of an Ultrasoft X-ray Transient Source in the 2XMM Catalog: a Tidal Disruption Event Candidate

We have discovered an ultrasoft X-ray transient source, 2XMMi J184725.1-631724, which was detected serendipitously in two XMM-Newton observations in the direction of the center of the galaxy IC 4765-f01-1504 at a redshift of 0.0353. These two observations were separated by 211 days, with the 0.2-10 keV absorbed flux increasing by a factor of about 9. Their spectra are best described by a model dominated by a thermal disk or a single-temperature blackbody component (contributing >80% of the flux) plus a weak power-law component. The thermal emission has a temperature of a few tens of eV, and the weak power-law component has a photon index of ~3.5. Similar to the black hole X-ray binaries in the thermal state, our source exhibits an accretion disk whose luminosity appears to follow the $L\propto T^4$ relation. This would indicate that the black hole mass is about 10^5-10^6 M_sun using the best-fitting inner disk radius. Both XMM-Newton observations show variability of about 21% on timescales of hours, which can be explained as due to fast variations in the mass accretion rate. The source was not detected by ROSAT in an observation in 1992, indicating a variability factor of >64 over longer timescales. The source was not detected again in X-rays in a Swift observation in 2011 February, implying a flux decrease by a factor of >12 since the last XMM-Newton observation. The transient nature, in addition to the extreme softness of the X-ray spectra and the inactivity of the galaxy implied by the lack of strong optical emission lines, makes it a candidate tidal disruption event. If this is the case, the first XMM-Newton observation would have been in the rising phase, and the second one in the decay phase.Comment: 12 pages, 6 figures. Accepted for publication in Ap

Spectral Function of 2D Fermi Liquids

We show that the spectral function for single-particle excitations in a two-dimensional Fermi liquid has Lorentzian shape in the low energy limit. Landau quasi-particles have a uniquely defined spectral weight and a decay rate which is much smaller than the quasi-particle energy. By contrast, perturbation theory and the T-matrix approximation yield spurious deviations from Fermi liquid behavior, which are particularly pronounced for a linearized dispersion relation.Comment: 6 pages, LaTeX2e, 5 EPS figure