Computing Fresnel integrals via modified trapezium rules

In this paper we propose methods for computing Fresnel integrals based on truncated trapezium rule approximations to integrals on the real line, these trapezium rules modified to take into account poles of the integrand near the real axis. Our starting point is a method for computation of the error function of complex argument due to Matta and Reichel (J Math Phys 34:298–307, 1956) and Hunter and Regan (Math Comp 26:539–541, 1972). We construct approximations which we prove are exponentially convergent as a function of N , the number of quadrature points, obtaining explicit error bounds which show that accuracies of 10−15 uniformly on the real line are achieved with N=12 , this confirmed by computations. The approximations we obtain are attractive, additionally, in that they maintain small relative errors for small and large argument, are analytic on the real axis (echoing the analyticity of the Fresnel integrals), and are straightforward to implement.

Using Choquet integrals for kNN approximation and classification

k-nearest neighbors (kNN) is a popular method for function approximation and classification. One drawback of this method is that the nearest neighbors can be all located on one side of the point in question x. An alternative natural neighbors method is expensive for more than three variables. In this paper we propose the use of the discrete Choquet integral for combining the values of the nearest neighbors so that redundant information is canceled out. We design a fuzzy measure based on location of the nearest neighbors, which favors neighbors located all around x.

Using choquet integrals for evaluating citation indices in journal ranking

The use of citation-based indices to evaluate the quality of journals is becoming increasingly widespread. Recently, ISI Web of Knowledge has begun to include three new indices in their journal statistics, including a 5-year impact factor. Here, we continue our earlier research which modeled the behavior of journal assessors based on some of these indices and the Choquet integral. We interpret the obtained fuzzy measures of many new datasets toward understanding the importance of these newly published indices and how indicative they may be of a journal's quality. The problem is one of ordinal classification, and the values of the best-fitting fuzzy measures can be obtained using the FMtools software package.

Asymptotics of zeros of polynomials arising from rational integrals

We prove that the zeros of the polynomials P.. (a) of degree m, defined by Boros and Moll via[GRAPHICS]approach the lemmiscate {zeta epsilon C: \zeta(2) - 1\ = Hzeta < 0}, as m --> infinity. (C) 2004 Elsevier B.V. All rights reserved.

Light-cone gauge integrals: prescriptionlessness at two loops

The only calculations performed beyond one-loop level in the light-cone gauge make use of the Mandelstam-Leibbrandt (ML) prescription in order to circumvent the notorious gauge dependent poles. Recently we have shown that in the context of negative dimensional integration method (NDIM) such prescription can be altogether abandoned, at least in one-loop order calculations. We extend our approach, now studying two-loop integrals pertaining to two-point functions. While previous works on the subject present only divergent parts for the integrals, we show that our prescriptionless method gives the same results for them, besides finite parts for arbitrary exponents of propagators. (C) 2000 Elsevier B.V. B.V. All rights reserved.

One-loop n-point equivalence among negative-dimensional, Mellin-Barnes and Feynman parametrization approaches to Feynman integrals

We show that at one-loop order, negative-dimensional, Mellin-Barnes (MB) and Feynman parametrization (FP) approaches to Feynman loop integral calculations are equivalent. Starting with a generating functional, for two and then for n-point scalar integrals, we show how to reobtain MB results, using negative-dimensional and FP techniques. The n-point result is valid for different masses, arbitrary exponents of propagators and dimension.