### Spline Collocation for Singular Integro-differential Equations over (0, 1).

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The paper is devoted to an integral equation algorithm for studying the scattering of plane waves by multilayer diffraction gratings under oblique incidence. The scattering problem is described by a system of Helmholtz equations with piecewise constant coefficients in ${\mathbb{R}}^{2}$ coupled by special transmission conditions at the interfaces between different layers. Boundary integral methods lead to a system of singular integral equations, containing at least two equations for each interface. To deal with...

The paper discusses new cubature formulas for classical integral operators of mathematical physics based on the «approximate approximation» of the density with Gaussian and related functions. We derive formulas for the cubature of harmonic, elastic and diffraction potentials approximating with high order in some range relevant for numerical computations. We prove error estimates and provide numerical results for the Newton potential.

We study the critical values of the complex standard-$L$-function attached to a holomorphic Siegel modular form and of the twists of the $L$-function by Dirichlet characters. Our main object is for a fixed rational prime number $p$ to interpolate $p$-adically the essentially algebraic critical $L$-values as the Dirichlet character varies thus providing a systematic control of denominators of critical values by generalized Kummer congruences. In order to organize this information we prove the existence of...

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