28,729 research outputs found
On Sums of SL(3,Z) Kloosterman Sums
We show that sums of the SL(3,Z) long element Kloosterman sum against a
smooth weight function have cancellation due to the variation in argument of
the Kloosterman sums, when each modulus is at least the square root of the
other. Our main tool is Li's generalization of the Kuznetsov formula on
SL(3,R), which has to date been prohibitively difficult to apply. We first
obtain analytic expressions for the weight functions on the Kloosterman sum
side by converting them to Mellin-Barnes integral form. This allows us to relax
the conditions on the test function and to produce a partial inversion formula
suitable for studying sums of the long-element SL(3,Z) Kloosterman sums.Comment: 44 pages, 1 figure, Revised version accepted by the Ramanujan Journa
A fast high-order solver for problems of scattering by heterogeneous bodies
A new high-order integral algorithm for the solution of scattering problems by heterogeneous bodies is presented. Here, a scatterer is described by a (continuously or discontinuously) varying refractive index n(x) within a two-dimensional (2D) bounded region; solutions of the associated Helmholtz equation under given incident fields are then obtained by high-order inversion of the Lippmann-Schwinger integral equation. The algorithm runs in O(Nlog(N)) operations where N is the number of discretization points. A wide variety of numerical examples provided include applications to highly singular geometries, high-contrast configurations, as well as acoustically/electrically large problems for which supercomputing resources have been used recently. Our method provides highly accurate solutions for such problems on small desktop computers in CPU times of the order of seconds
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