Lower bounds of some singular integrals and their applications

Let and . We are interested in the lower bounds of the integral:

Quadpack computation of Feynman loop integrals

The paper addresses a numerical computation of Feynman loop integrals, which are computed by an extrapolation to the limit as a parameter in the integrand tends to zero. An important objective is to achieve an automatic computation which is effective for a wide range of instances. Singular or near singular integrand behavior is handled via an adaptive partitioning of the domain, implemented in an iterated/repeated multivariate integration method. Integrand singularities possibly introduced via infrared (IR) divergence at the boundaries of the integration domain are addressed using a version of the Dqags algorithm from the integration package Quadpack, which uses an adaptive strategy combined with extrapolation. The latter is justified for a large class of problems by the underlying asymptotic expansions of the integration error. For IR divergent problems, an extrapolation scheme is presented based on dimensional regularization.     

The use of a connection between integral transforms for the computation of integrals

A theorem is proved which establishes a connection between four well-known integral transforms: Laplace, Kantorovich-Lebedev, Mehler-Fok, and the K-transform of Meier. This theorem is used to calculate a number of integrals containing the Legendre function β_1/2+iτ(X), and also the MacDonald function K_iτ(X). Certain other integrals can be calculated by using the indicated method.     

Algorithm for the computation of Bessel function integrals

A fortran subroutine is given for the computation of integrals of the form ∫^c₀f(x)J_v(αx)dx, where v = 0, 1,…,10.     

Weak lower semicontinuity of polyconvex integrals: a borderline case

This article was processed by the author using the Springer-Verlag TEX QPMZGHB macro package 1991.     

Generation of random bivariate normal deviates and computation of related integrals

A method which transforms two random variables having rectangular distributions into a pair of bivariate normal deviates with prescribed covariance matrix is described. The same transformation is used for integrating the bivariate normal distribution over areas which are the intersection of the domain outside an equiprobability ellipse and a sector determined by two lines through the point of gravity of the normal distribution.     

Equivalence of some integrals and modulus of smoothness

Let f∈C(R). We are interested in lower and upper bounds of the integrals

     

Pseudo-abelian integrals: Unfolding generic exponential case

We consider functions of the form , with Pi, R, and Q∈R[x,y], which are (generalized Darboux) first integrals of the polynomial system MdlogH₀=0. We assume that H₀ defines a family of real cycles in a region bounded by a polycycle.To each polynomial form η one can associate the pseudo-abelian integrals I(h) of M−1η along γ(h), which is the first order term of the displacement function of the orbits of MdH₀+δη=0.We consider Darboux first integrals unfolding H₀ (and its saddle-nodes) and pseudo-abelian integrals associated to these unfoldings. Under genericity assumptions we show the existence of a uniform local bound for the number of zeros of these pseudo-abelian integrals.The result is a part of a program to extend Varchenko-Khovanskii's theorem from abelian integrals to pseudo-abelian integrals and prove the existence of a bound for the number of their zeros in function of the degree of the polynomial system only.     

Modified clenshaw-curtis method for the computation of Bessel function integrals

The numerical evaluation of Bessel function integrals may be difficult when the Bessel function is rapidly oscillating in the interval of integration. In the method presented here, the smooth factor of the integrand is replaced by a truncated Chebyshev series approximation and the resulting integral is computed exactly. The numerical aspects of this exact integration are discussed.     

Orthogonal polynomials and some q-beta integrals of Ramanujan

Two integrals of Ramanujan are used to define a q-analogue of the Euler beta integral on the real line and of the Cauchy beta-integral on the complex unit circle. Such integrals are connected to orthogonal, biorthogonal and Laurent polynomials. Explicit examples of Laurent orthogonal polynomials are given on the real line and on the circle.     

A method for efficient computation of integrals with oscillatory and singular integrand

Numerical Algorithms - A method based on modification of numerical steepest descent method to efficiently compute highly oscillatory integrals having endpoint singularities of algebraic and...     

The interpretation of fuzzy integrals and their application to fuzzy systems

Fuzzy integrals, in general, and Sugeno integrals, in particular, are well known aggregation operators. They can be used in a great variety of decision making applications. Nevertheless, their use is not easy as their interpretation is not straightforward. In this paper we study the interpretation of fuzzy integrals, focusing on Sugeno ones, and we show their application to fuzzy inference systems when the rules are not independent.     

On the evaluation of some Selberg-like integrals

Several methods of evaluation are presented for a family {In,d,p} of Selberg-like integrals that arise in the computation of the algebraic-geometric degrees of a family of spherical nilpotent orbits associated to the symmetric space of a simple real Lie group. Adapting the technique of Nishiyama, Ochiai and Zhu, we present an explicit evaluation in terms of certain iterated sums over permutation groups. The resulting formula, however, is only valid when the integrand involves an even power of the Vandermonde determinant. We then apply, to the general case, the theory of symmetric functions and obtain an evaluation of the integral In,d,p as a product of polynomial of fixed degree times a particular product of gamma factors; thereby identifying the asymptotics of the integrals with respect to their parameters. Lastly, we derive a recursive formula for evaluation of another general class of Selberg-like integrals, by applying some of the technology of generalized hypergeometric functions.