Weighted Coorbit Spaces and Banach Frames on Homogeneous Spaces

This article is concerned with frame constructions on domains and manifolds. The starting point is a unitary group representation which is square integrable modulo a suitable subgroup and therefore gives rise to a generalized continuous wavelet transform. Then generalized coorbit spaces can be defined by collecting all functions for which this wavelet transform is contained in a weighted L_p-space. Moreover, we show that a judicious discretization of the representation leads to an atomic decomposition and to Banach frames for these coorbit spaces.     

Analytical comparison between the surface current integral equation and the second-order small-slope approximation

This paper is the third in a series discussing a new approximate bistatic model for electromagnetic scattering from perfectly conducting rough surfaces. Our previous approach supplemented the Kirchhoff model through the addition of new terms involving linear orders in slope and surface elevation differences that arise naturally from a second iteration of the surface current integral equation. This completion of the Kirchhoff was shown to provide the correct first-order small perturbation method (SPM-1) in the general bistatic context. The agreement with SPM-1 was achieved because differences of surface heights are no longer expanded in powers of surface slope. While consistent with SPM, our previous formulation fails to reconverge toward the Kirchhoff model, at some incidence and scattered angles, when the illuminated surface satisfies the high frequency roughness condition. This weakness is also shared with the first-order small slope approximation (SSA-1) which is structurally equivalent to our previous formulation where the polarization is independent of surface roughness. The second-order small slope approximation (SSA-2), which satisfies the SPM-1 and second-order small perturbation method (SPM-2) limits by construction, was shown by Voronovich to converge toward the tangent plane approximation of the Kirchhoff model under high frequency conditions. In the present paper, we show that, in addition to the linear orders in our previous model, one must now include cross-terms between slope and surface elevation to ensure convergence toward both high frequency and small perturbation limits. With the inclusion of these terms, our new formulation becomes comparable to the SSA-2 (second-order kernel) without the need to evaluate all the quadratic order slope and elevations terms. SSA-2 is more complete, however, in the sense that it guarantees convergence toward the second-order Bragg limit (SPM-2) in the fully dielectric case in addition to both SPM-1 and Kirchhoff. Our new generalization is shown to explain correctly extra depolarization in specular conditions to be caused by surface curvature and surface autocorrelation for incoherent and coherent scattering, respectively. This result will have large repercussions on the interpretation of bistatically reflected signals such as those from GPS.     

A p-version finite element method for nonlinear elliptic variational inequalities in 2D

This article introduces and analyzes a p-version FEM for variational inequalities resulting from obstacle problems for some quasi-linear elliptic partial differential operators. We approximate the solution by controlling the obstacle condition in images of the Gauss–Lobatto points. We show existence and uniqueness for the discrete solution u _p from the p-version for the obstacle problem. We prove the convergence of u _p towards the solution with respect to the energy norm, and assuming some additional regularity for the solution we derive an a priori error estimate. In numerical experiments the p-version turns out to be superior to the h-version concerning the convergence rate and the number of unknowns needed to achieve a certain exactness of the approximation.     

Primes in quadratic progressions on average

In this paper, we establish a theorem on the distribution of primes in quadratic progressions on average.     

Elliott-Morse measures and Kakutani's dichotomy theorem

Dedicated to the memory of John Oxtoby     

Isotopic production cross sections of residues in reactions induced by relativistic heavy ions with protons and deuterons

The isotopic production cross sections of heavy residues in relativistic heavy-ion collisions have been investigated in inverse kinematics. The primary reaction products were fully identified in mass and atomic number prior to beta decay using the fragment separator FRS. The huge collection of data obtained helps in the understanding of the two main reaction mechanisms involved: fragmentation and fission. These data provide basic information for future radioactive ion beam facilities and for technical applications like intense neutron sources by means of spallation targets.     

Time interval distributions of atoms in atomic beams

We report an experimental investigation of two-particle correlations between neutral atoms in a Hanbury Brown and Twiss experiment. Both an atom laser beam and a pseudo-thermal atomic beam are extracted from a Bose–Einstein condensate and the atom flux is measured with a single atom counter. We determine the conditional and the unconditional detection probabilities for the atoms in the beam and find good agreement with the theoretical predictions. PACS 03.75.Pp; 05.30.Jp; 07.77.Gx; 42.50.Pq     

Adaptive application of operators in standard representation

Recently adaptive wavelet methods have been developed which can be shown to exhibit an asymptotically optimal accuracy/work balance for a wide class of variational problems including classical elliptic boundary value problems, boundary integral equations as well as certain classes of noncoercive problems such as saddle point problems. A core ingredient of these schemes is the approximate application of the involved operators in standard wavelet representation. Optimal computational complexity could be shown under the assumption that the entries in properly compressed standard representations are known or computable in average at unit cost. In this paper we propose concrete computational strategies and show under which circumstances this assumption is justified in the context of elliptic boundary value problems. Dedicated to Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 41A25, 41A46, 65F99, 65N12, 65N55. This work has been supported in part by the Deutsche Forschungsgemeinschaft SFB 401, the first and third author are supported in part by the European Community's Human Potential Programme under contract HPRN-CT-202-00286 (BREAKING COMPLEXITY). The second author acknowledges the financial support provided through the European Union's Human Potential Programme, under contract HPRN-CT-2002-00285 (HASSIP) and through DFG grant DA 360/4–1.     

Bayesian Object Identification: Variants

We present a Bayesian theory of object identification. Here, identifying an object means selecting a particular observation from a group of observations (variants), this observation (the regular variant) being characterized by a distributional model. In this sense, object identification means assigning a given model to one of several observations. Often, it is the statistical model of the regular variant, only, that is known. We study an estimator which relies essentially on this model and not on the characteristics of the “irregular” variants. In particular, we investigate under what conditions this variant selector is optimal. It turns out that there is a close relationship with exchangeability and Markovian reversibility. We finally apply our theory to the case of irregular variants generated from the regular variant by a Gaussian linear model.