Weyl calculus and dual pairs

We consider a dual pair $(G,G')$, in the sense of Howe, with $G$ compact acting on $L^2(\mathbb R^n)$ for an appropriate $n$ via the Weil Representation. Let $\widetilde{G}$ be the preimage of $G$ in the metaplectic group. Given a genuine irreducible unitary representation $\Pi$ of $\widetilde{G}$ we compute the Weyl symbol of orthogonal projection onto $L^2(\mathbb R^n)_\Pi$, the $\Pi$-isotypic component. We apply the result to obtain an explicit formula for the character of the corresponding irreducible unitary representation $\Pi'$ of $\widetilde{G'}$ and to compute of the wave front set of $\Pi'$ by elementary means

Semisimple orbital integrals on the symplectic space for a real reductive dual pair

We prove a Weyl Harish-Chandra integration formula for the action of a reductive dual pair on the corresponding symplectic space $W$. As an intermediate step, we introduce a notion of a Cartan subspace and a notion of an almost semisimple element in the symplectic space $W$. We prove that the almost semisimple elements are dense in $W$. Finally, we provide estimates for the orbital integrals associated with the different Cartan subspaces in $W$

Nonlinear Relaxation in Population Dynamics

We analyze the nonlinear relaxation of a complex ecosystem composed of many interacting species. The ecological system is described by generalized Lotka-Volterra equations with a multiplicative noise. The transient dynamics is studied in the framework of the mean field theory and with random interaction between the species. We focus on the statistical properties of the asymptotic behaviour of the time integral of the i-th population and on the distribution of the population and of the local field.Comment: 11 pages, 4 figures, in press in Int. Journal of Fractals (2001

From topological insulators to superconductors and Confinement

Topological matter in 3D is characterized by the presence of a topological BF term in its long-distance effective action. We show that, in 3D, there is another marginal term that must be added to the action in order to fully determine the physical content of the model. The quantum phase structure is governed by three parameters that drive the condensation of topological defects: the BF coupling, the electric permittivity and the magnetic permeability of the material. For intermediate levels of electric permittivity and magnetic permeability the material is a topological insulator. We predict, however, new states of matter when these parameters cross critical values: a topological superconductor when electric permittivity is increased and magnetic permeability is lowered and a charge confinement phase in the opposite case of low electric permittivity and high magnetic permeability. Synthetic topological matter may be fabricated as 3D arrays of Josephson junctions.Comment: 5 pages, no figures, few references added, typos corrected and few comments adde

GRB Flares: UV/Optical Flaring (Paper I)

We present a new algorithm for the detection of flares in gamma-ray burst (GRB) light curves and use this algorithm to detect flares in the UV/optical. The algorithm makes use of the Bayesian Information Criterion (BIC) to analyze the residuals of the fitted light curve, removing all major features, and to determine the statistically best fit to the data by iteratively adding additional `breaks' to the light curve. These additional breaks represent the individual components of the detected flares: T_start, T_stop, and T_peak. We present the detection of 119 unique flaring periods detected by applying this algorithm to light curves taken from the Second Swift Ultraviolet/Optical Telescope (UVOT) GRB Afterglow Catalog. We analyzed 201 UVOT GRB light curves and found episodes of flaring in 68 of the light curves. For those light curves with flares, we find an average number of ~2 flares per GRB. Flaring is generally restricted to the first 1000 seconds of the afterglow, but can be observed and detected beyond 10^5 seconds. More than 80% of the flares detected are short in duration with Delta t/t of < 0.5. Flares were observed with flux ratios relative to the underlying light curve of between 0.04 to 55.42. Many of the strongest flares were also seen at greater than 1000 seconds after the burst.Comment: Submitted to ApJ. 20 pages (including 8 figures and 1 table

Integrated system to perform surrogate based aerodynamic optimisation for high-lift airfoil

This work deals with the aerodynamics optimisation of a generic two-dimensional three element high-lift configuration. Although the high-lift system is applied only during take-off and landing in the low speed phase of the flight the cost efficiency of the airplane is strongly influenced by it [1]. The ultimate goal of an aircraft high lift system design team is to define the simplest configuration which, for prescribed constraints, will meet the take-off, climb, and landing requirements usually expressed in terms of maximum L/D and/or maximum CL. The ability of the calculation method to accurately predict changes in objective function value when gaps, overlaps and element deflections are varied is therefore critical. Despite advances in computer capacity, the enormous computational cost of running complex engineering simulations makes it impractical to rely exclusively on simulation for the purpose of design optimisation. To cut down the cost, surrogate models, also known as metamodels, are constructed from and then used in place of the actual simulation models. This work outlines the development of integrated systems to perform aerodynamics multi-objective optimisation for a three-element airfoil test case in high lift configuration, making use of surrogate models available in MACROS Generic Tools, which has been integrated in our design tool. Different metamodeling techniques have been compared based on multiple performance criteria. With MACROS is possible performing either optimisation of the model built with predefined training sample (GSO) or Iterative Surrogate-Based Optimization (SBO). In this first case the model is build independent from the optimisation and then use it as a black box in the optimisation process. In the second case is needed to provide the possibility to call CFD code from the optimisation process, and there is no need to build any model, it is being built internally during the optimisation process. Both approaches have been applied. A detailed analysis of the integrated design system, the methods as well as th

Dynamical correlation functions of the mesoscopic pairing model

We study the dynamical correlation functions of the Richardson pairing model (also known as the reduced or discrete-state BCS model) in the canonical ensemble. We use the Algebraic Bethe Ansatz formalism, which gives exact expressions for the form factors of the most important observables. By summing these form factors over a relevant set of states, we obtain very precise estimates of the correlation functions, as confirmed by global sum-rules (saturation above 99% in all cases considered). Unlike the case of many other Bethe Ansatz solvable theories, simple two-particle states are sufficient to achieve such saturations, even in the thermodynamic limit. We provide explicit results at half-filling, and discuss their finite-size scaling behavior

An Estimation of the Gamma-Ray Burst Afterglow Apparent Optical Brightness Distribution Function

By using recent publicly available observational data obtained in conjunction with the NASA Swift gamma-ray burst mission and a novel data analysis technique, we have been able to make some rough estimates of the GRB afterglow apparent optical brightness distribution function. The results suggest that 71% of all burst afterglows have optical magnitudes with mR < 22.1 at 1000 seconds after the burst onset, the dimmest detected object in the data sample. There is a strong indication that the apparent optical magnitude distribution function peaks at mR ~ 19.5. Such estimates may prove useful in guiding future plans to improve GRB counterpart observation programs. The employed numerical techniques might find application in a variety of other data analysis problems in which the intrinsic distributions must be inferred from a heterogeneous sample.Comment: 15 pages including 2 tables and 7 figures, accepted for publication in Ap

The Ages of Elliptical Galaxies from Infrared Spectral Energy Distributions

The mean ages of early-type galaxies obtained from the analysis of optical spectra, give a mean age of 8 Gyr at z = 0, with 40% being younger than 6 Gyr. Independent age determinations are possible by using infrared spectra (5-21 microns), which we have obtained with the Infrared Spectrograph on the Spitzer Observatory. This age indicator is based on the collective mass loss rate of stars, where mass loss from AGB stars produces a silicate emission feature at 9-12 microns. This feature decreases more rapidly than the shorter wavelength continuum as a stellar population ages, providing an age indicator. From observations of 30 nearby early-type galaxies, 29 show a spectral energy distribution dominated by stars and one has significant emission from the ISM and is excluded. The infrared age indicators for the 29 galaxies show them all to be old, with a mean age of about 10 Gyr and a standard deviation of only a few Gyr. This is consistent with the ages inferred from the values of M/L_B, but is inconsistent with the ages derived from the optical line indices, which can be much younger. All of these age indicators are luminosity-weighted and should be correlated, even if multiple-age components are considered. The inconsistency indicates that there is a significant problem with either the infrared and the M/L_B ages, which agree, or with the ages inferred from the optical absorption lines.Comment: Accepted for publication in Ap