Discrete pseudo-integrals

Integration of simple functions is a corner stone of general integration theory and it covers integration over finite spaces discussed in this paper. Different kinds of decomposition and subdecomposition of simple functions into basic functions sums, as well as different kinds of pseudo-operations exploited for integration and sumation result into several types of integrals, including among others, Lebesgue, Choquet, Sugeno, pseudo-additive, Shilkret, PAN, Benvenuti and concave integrals. Some basic properties of introduced discrete pseudo-concave integrals are discussed, and several examples of new integrals are given.     

Bessel,sine and cosine functions and extrapolation methods for computing molecular multi-center integrals

In this work, we present an extremely efficient approach for a fast numerical evaluation of highly oscillatory spherical Bessel integrals occurring in the analytic expressions of the so-called molecular multi-center integrals over exponential type functions. The approach is based on the Slevinsky-Safouhi formulae for higher derivatives applied to spherical Bessel functions and on extrapolation methods combined with practical properties of sine and cosine functions. Recurrence relations are used for computing the approximations of the spherical Bessel integrals, allowing for a control of accuracy and the stability of the algorithm. The computer algebra system Maple was used in our development, mainly to prove the applicability of the extrapolation methods. Among molecular multi-center integrals, the three-center nuclear attraction and four-center two-electron Coulomb and exchange integrals are undoubtedly the most difficult ones to evaluate rapidly to a high pre-determined accuracy. These integrals are required for both density functional and ab initio calculations. Already for small molecules, many millions of them have to be computed. As the molecular system gets larger, the computation of these integrals becomes one of the most laborious and time consuming steps in molecular electronic structure calculation. Improvement of the computational methods of molecular integrals would be indispensable to a further development in computational studies of large molecular systems. Convergence properties are analyzed to show that the approach presented in this work is a valuable contribution to the existing literature on molecular integral calculations as well as on spherical Bessel integral calculations.     

Complex valued multiparameter stochastic integrals

In this paper we will consider the properties of various stochastic integrals over a complex valued, two parameter Wiener process. Integrals of this type exhibit pleasant features which do not appear within the real framework. They are stable under approximation and viewed in relation with analytic functions, they typically satisfy an ordinary chain rule. This in turn gives rise to several nice representation formulas.     

Representations of solutions of Lindblad equations by randomized Feynman formulas

Representations of solutions of Lindblad equations by randomized Feynman integrals over trajectories are obtained by averaging similar representations for solutions of stochastic Schrödinger equations (Schrödinger–Belavkin equations). An approach based on the application of Chernoff’s theorem is applied. First, (randomized) Feynman formulas approximating Feynman path integrals are obtained; these formulas contain integrals over finite Cartesian powers of the space of values of the functions over which the Feynman integrals are taken.     

A sinc quadrature rule for Hadamard finite-part integrals

Summary A Sinc quadrature rule is presented for the evaluation of Hadamard finite-part integrals of analytic functions. Integration over a general are in the complex plane is considered. Special treatment is given to integrals over the interval (–1,1). Theoretical error estimates are derived and numerical examples are included.     

The plane wave expansion, infinite integrals and identities involving spherical Bessel functions

This paper shows that the plane wave expansion can be a useful tool in obtaining analytical solutions to infinite integrals over spherical Bessel functions and the derivation of identities for these functions. The integrals are often used in nuclear scattering calculations, where an analytical result can provide an insight into the reaction mechanism. A technique is developed whereby an integral over several special functions which cannot be found in any standard integral table can be broken down into integrals that have existing analytical solutions.     

Criterium for the gradient catastrophe for the non-isentropic gas dynamics equations in the one dimensional case

A method to describe the motion of non-isentropic polytropic gas by means of integral functionals is proposed. They are as a matter of fact the integrals of the solution components over a material volume. Included among the functionals are the kinetic and potential energies, impulse, momentum of mass, and a number of specific integrals. These integral functionals are functions of time and satisfy to a system of ODE, which is closed provided we consider a linear velocity field. In 1D case the system can be reduced to a certain nonlinear second order ODE for the first derivative in space of velocity. The problem about the data yielding the gradient catastrophe (the unboundedness of the solution gradients) can be reduced to the problem of blow-up of the solutions to the latter equation. This problem can be completely solved, so, the criterium of the gradient catastrophe can be obtained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary layer approximate approximations and cubature of potentials in domains

In this article we present a new approach to the computation of volume potentials over bounded domains, which is based on the quasi‐interpolation of the density by almost locally supported basis functions for which the corresponding volume potentials are known. The quasi‐interpolant is a linear combination of the basis function with shifted and scaled arguments and with coefficients explicitly given by the point values of the density. Thus, the approach results in semi‐analytic cubature formulae for volume potentials, which prove to be high order approximations of the integrals. It is based on multi‐resolution schemes for accurate approximations up to the boundary by applying approximate refinement equations of the basis functions and iterative approximations on finer grids. We obtain asymptotic error estimates for the quasi‐interpolation and corresponding cubature formulae and provide some numerical examples. This revised version was published online in June 2006 with corrections to the Cover Date.     

Semi-analytic treatment of the three-dimensional Poisson equation via a Galerkin BIE method

A systematic treatment of the three-dimensional Poisson equation via singular and hypersingular boundary integral equation techniques is investigated in the context of a Galerkin approximation. Developed to conveniently deal with domain integrals without a volume-fitted mesh, the proposed method initially converts domain integrals featuring the Newton potential and its gradient into equivalent surface integrals. Then, the resulting boundary integrals are evaluated by means of well-established cubature methods. In this transformation, weakly-singular domain integrals, defined over simply- or multiply-connected domains with Lipschitz boundaries, are rigorously converted into weakly-singular surface integrals. Combined with the semi-analytic integration approach developed for potential problems to accurately calculate singular and hypersingular Galerkin surface integrals, this technique can be employed to effectively deal with mixed boundary-value problems without the need to partition the underlying domain into volume cells. Sample problems are included to validate the proposed approach.