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Case Western Reserve University DEPARTMENT OF MATHEMATICS |
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DEPARTMENT OF MATHEMATICS SEMINARS and COLLOQUIUMANALYSIS |
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Professor Matthias SCHULTE ABSTRACT Every square integrable random variable depending on a Poisson point process has a representation as an (infinite) sum of multiple Wiener-Ito integrals, namely the so called Wiener-Ito chaos expansion. Combining this with the closely related Malliavin calculus and Stein's method yields central limit theorems for Poisson functionals. The aim of this talk is to give a friendly introduction to this field and to show how these general results can be applied to problems from stochastic geometry.
Professor Catalin TURC ABSTRACT The mathematical modeling of time-harmonic scalar scattering problems leads to Helmholtz equations. The partial differential operators associated with Helmholtz equation are not coercive, and this fact complicates the numerical analysis of discretization methods for the Helmholtz equation. Interestingly, there is mounting evidence that the boundary integral operators associated with integral formulations of scattering problems with Dirichlet boundary conditions are coercive. However, the important cases of Neumann and impedance boundary conditions received little attention in the literature. We investigate the coercivity properties and the norm dependence on the wavenumber of certain boundary integral operators that we recently introduced for the solution of two and three-dimensional acoustic scattering problems with Neumann boundary conditions. We show that in the case of circular and spherical boundaries, these boundary integral operators are coercive, and that their norms are bounded. We provide numerical evidence that these boundary integral operators are coercive for two-dimensional starlike boundaries.
Case Western Reserve University |
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