In this paper we present a treatment of hypersingular integral equations, which have relevant applications in many problems of wave dynamics, elasticity and fluid mechanics with mixed boundary conditions. The main goal of the present work is the development of an efficient direct numerical collocation method. The paper is completed with two examples taken from crack theory and acoustics: the st...

We present a novel path-integral Monte Carlo method for calculating ortho-positronium lifetimes in insulators and illustrate the method with calculations for positronium in hard spheres, solid argon, and a simple zeolite structure. In argon, we find that the positronium wavefunction is compressed, suggesting that the contact potential may be larger than unity. We discuss the difference in physi...

In this article we investigate the existence of extremal(maximal and minimal) solutions for the following coupled system of integrodifferential equations of fractional order with the given boundary conditions of the form  Du(t) + f1(t, v(t), I v(t)) = 0, 0 < t < 1, Dv(t) + f2(t, u(t), I u(t)) = 0, 0 < t < 1, u(0) = 0, u′(1) = 0, v(0) = 0, v′(1) = 0. where 1 < α, β ≤ 2 and f1, f2 : [0, 1] × ...

The authors have introduced recently a “microcanonical functional integral” which yields directly the density of states as a function of energy. The phase of the functional integral is Jacobi’s action, the extrema of which are classical solutions at a given energy. This approach is general but is especially well suited to gravitating systems because for them the total energy can be fixed simply...

A class of distance measures on probabilities — the integral probability metrics (IPMs) — is addressed: these include the Wasserstein distance, Dudley metric, and Maximum Mean Discrepancy. IPMs have thus far mostly been used in more abstract settings, for instance as theoretical tools in mass transportation problems, and in metrizing the weak topology on the set of all Borel probability measure...

We derive boundary conditions for the vorticity equation with solid wall boundaries. The formulation uses a Dirichlet condition for the normal component of vorticity, and Neumann type conditions for the tangential components. In a Galerkin (integral) formulation the tangential condition is natural, i.e. it is enforced by a right-hand side functional and does not impose a boundary constraint on ...

Simple cosmological models are used to show that gravitation can be quantized as an ordinary gauge system if the Hamilton-Jacobi equation for the model under consideration is separable. In this situation, a canonical transformation can be performed such that in terms of the new variables the model has a linear and homogeneous constraint, and therefore canonical gauges are admissible in the path...

In a recent paper Richards and Zheng compute the determinant of a matrix whose entries are given by beta-type integrals, thereby generalising an earlier result by Dixon and Varchenko. They then use their result to obtain a generalisation of the famous Selberg integral. In this note we point out that the Selberg-generalisation of Richards and Zheng is a special case of an integral over Jack poly...

We first summarize the path integral formulation by Veach [Veach 1998]. Using the path integral formulation, light transport simulation can be expressed as: Ij = Ω fj (¯ x)dµ(¯ x), (1) where j is an index to the pixel (or measurement), fj is the measurement contribution function, µ is a measure of paths, and Ω is the set of transport paths of all lengths. We evaluate this integral using Monte C...

In this work we consider second kind Volterra integral equations with weakly singular kernels. By introducing some appropriate function spaces we prove the existence of an asymptotic error expansion for Euler’s method. This result allows the use of certain extrapolation procedures which is illustrated by means of some numerical examples. o 1997 Elsevier Science B.V.