We consider in this paper asymptotic and numerical aspects of highly oscillatory integrals of the form ∫ b a f(x)g(sin[ωθ(x)])dx, where ω 1. Such integrals occur in the simulation of electronic circuits, but they are also of independent mathematical interest. The integral is expanded in asymptotic series in inverse powers of ω. This expansion clarifies the behaviour for large ω and also provide...

The paper deals with the problem of numerical integration of a function [0, 1]→ R for which only a set-membership description is known. The set of values of the integrals of all possible realizations of the uncertain function is considered as a guaranteed result of the integration. The problem is to approximate this set, with a prescribed accuracy ε, by polyhedral sets, in particular, to enclos...

An inversion procedure is used to obtain from the extended principle of corresponding - states the pair interaction potentials for argon, krypton and its mixture over a tetnperature range from absolute zero to the onset of ionization. The experimentally-reduced viscosity collision integrals obtained from the corresponding - states correlation have been inverted directly to give the reduced...

We give an overview of some p-adic algorithms for computing with elliptic and hyperelliptic curves, starting with Kedlaya’s algorithm. While the original purpose of Kedlaya’s algorithm was to compute the zeta function of a hyperelliptic curve over a finite field, it has since been used in a number of applications. In particular, we describe how to use Kedlaya’s algorithm to compute Coleman inte...

The purpose of this paper is to discuss the relationship between commutative and non-commutative integrability of Hamiltonian systems and to construct new examples of integrable geodesic flows on Riemannian manifolds. In particular, we prove that the geodesic flow of the biinvariant metric on any bi-quotient of a compact Lie group is integrable in the non-commutative sense by means of polynomia...

Integral representations are derived for the parabolic cylinder functions U(a, x), V (a, x) andW (a, x) and their derivatives. The new integrals will be used in numerical algorithms based on quadrature. They follow from contour integrals in the complex plane, by using methods from asymptotic analysis (saddle point and steepest descent methods), and are stable starting points for evaluating the ...

Abstract. A large class of Feynman integrals, like e.g., two-point parameter integrals with at most one mass and containing local operator insertions, can be transformed to multi-sums over hypergeometric expressions. In this survey article we present a difference field approach for symbolic summation that enables one to simplify such definite nested sums to indefinite nested sums. In particular...

We present new methods for the evaluation of one-loop tensor integrals which have been used in the calculation of the complete electroweak one-loop corrections to e + e − → 4 fermions. The described methods for 3-point and 4-point integrals are, in particular, applicable in the case where the conventional Passarino–Veltman reduction breaks down owing to the appearance of Gram determinants in th...

The Pfaff lattice was introduced by us in the context of a Lie algebra splitting of gl (∞) into sp (∞) and an algebra of lower-triangular matrices. The Pfaff lattice is equivalent to a set of bilinear identities for the wave functions, which yield the existence of a sequence of “τ -functions”. The latter satisfy their own set of bilinear identities, which moreover characterize them. In the semi...

In previous work the authors, using the Bethe Ansatz, found for the N -particle asymmetric simple exclusion process on the integers a formula—a sum of multiple integrals—for the probability that a system is in a particular configuration at time t given an initial configuration. The present work extends this to the case where particles are of different species, with particles of a higher species...