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Fast integral equation methods for the modified Helmholtz equation
We present a collection of integral equation methods for the solution to the
two-dimensional, modified Helmholtz equation, u(\x) - \alpha^2 \Delta u(\x) =
0, in bounded or unbounded multiply-connected domains. We consider both
Dirichlet and Neumann problems. We derive well-conditioned Fredholm integral
equations of the second kind, which are discretized using high-order, hybrid
Gauss-trapezoid rules. Our fast multipole-based iterative solution procedure
requires only O(N) or operations, where N is the number of nodes
in the discretization of the boundary. We demonstrate the performance of the
methods on several numerical examples.Comment: Published in Computers & Mathematics with Application
Slow Convergence in Generalized Central Limit Theorems
We study the central limit theorem in the non-normal domain of attraction to
symmetric -stable laws for . We show that for i.i.d.
random variables , the convergence rate in of both the
densities and distributions of is at best
logarithmic if is a non-trivial slowly varying function. Asymptotic laws
for several physical processes have been derived using central limit theorems
with scaling and Gaussian limiting distributions. Our result
implies that such asymptotic laws are accurate only for exponentially large
.Comment: To appear in Comptes Rendus de l'Acad\'emie des Sciences,
Math\'ematique
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