A combined Filon/asymptotic quadrature method for highly oscillatory problems

A crossing between the asymptotic expansion of an oscillatory integral and Filon-type methods is obtained by applying a Filon-type method to the error term in the asymptotic expansion, which is in itself an oscillatory integral. The efficiency of the approach is investigated through analysis and numerical experiments, revealing a method which in many cases performs better than the Filon-type method. It is shown that considerable savings in terms of the required number of potentially expensive moments can be expected. The case of multivariate oscillatory integrals is discussed briefly. AMS subject classification (2000)  65D30     

Bounded rank-one perturbations in sampling theory

Sampling theory concerns the problem of reconstruction of functions from the knowledge of their values at some discrete set of points. In this paper we derive an orthogonal sampling theory and associated Lagrange interpolation formulae from a family of bounded rank-one perturbations of a self-adjoint operator that has only discrete spectrum of multiplicity one.     

Inequalities for formal power series and entire functions

We present several integral and exponential inequalities for formal power series and for both arbitrary entire functions of exponential type and generalized Borel transforms. They are obtained through certain limit procedures which involve the multiparameter binomial inequalities, integral inequalities for continuous functions, and weighted norm inequalities for analytic functions. Some applications to the confluent hypergeometric functions, Bessel functions, Laguerre polynomials, and trigonometric functions are discussed. Also some generalizations are given.     

Inversion of the Bessel potential operator in weighted variable Lebesgue spaces

We study the inversion problem of the Bessel potential operator within the frameworks of the weighted Lebesgue spaces with variable exponent. The inverse operator is constructed by using approximative inverse operators. This generalizes some classical results to the variable exponent setting.     

Bessel functions of third order and the distribution of cyclic planar motions with three directions

The exact distribution of a cyclic planar motion with three directions is explicitly derived in terms of Bessel functions of order three (suitably combined). The absolutely continuous part of the distribution is proved to satisfy suitable boundary conditions and some of its properties are analyzed. The transformations converting the governing equations of order three is presented and its solutions (used here) derived by applying the Frobenius method.     

Quadratic nonlinear torsional hyperelastic waves in isotropic cylinders: Primary analysis of evolution

Nonlinear torsional waves propagating along an infinite isotropic cylinder are considered. The hyperelasticity of the cylinder is described by the Murnaghan potential. An approximate analytical solution of the problem is obtained. The evolution of the initial waveprofile in cylinders made of composite materials reinforced with microscale particles is simulated numerically. It is shown that the approximation used to describe the variation in the amplitude inward the cylinder is quite accurate. Three-dimensional plots of evolution are presented __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 3, pp. 75–85, March 2008.     

Hyperasymptotics and hyperterminants: Exceptional cases

A new method is introduced for the computation of hyperterminants. It is based on recurrence relations, and can also be used to compute the parameter derivatives of the hyperterminants. These parameter derivatives are needed in hyperasymptotic expansions in exceptional cases. Numerical illustrations and an application are included.     

Total curvatures of geodesic spheres associated to quadratic curvature invariants

Total scalar curvatures of geodesic spheres obtained by integrating the second-order scalar invariants of the curvature tensor are investigated. The first terms in their power-series expansions are derived and these results are used to characterize the two-point homogeneous spaces among Riemannian manifolds with adapted holonomy. Dedicated to Professor L. VanheckeMathematics Subject Classification (2000) 53C25, 53C30     

Resonance expansions in quantum mechanics

The goal of this contribution is to discuss various resonance expansions that have been proposed in the literature. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.