This paper reviews two simple numerical algorithms particularly useful in Computational ElectroMagnetics (CEM): the Weighted Averages (WA) algorithm and the Double Exponential (DE) quadrature. After a short historical introduction and an elementary description of the mathematical procedures underlying both techniques, they are applied to the evaluation of Sommerfeld integrals, where WA and DE c...

In these notes we review the fundamentals of three-dimensional vector calculus. We will be surveying calculus on curves, surfaces and solid bodies in three-dimensional space. The three methods of integration — line, surface and volume (triple) integrals — and the fundamental vector differential operators — gradient, curl and divergence — are intimately related. The differential operators and in...

After reduction techniques, two-loop amplitudes in N = 4 super Yang-Mills theory can be written in a basis of integrals containing scalar double-box integrals with rational coefficients, though the complete basis is unknown. Generically, at two loops, the leading singular behavior of a scalar double box integral with seven propagators is captured by a hepta-cut. However, it turns out that a cer...

The calculation of exclusive observables beyond the one-loop level requires elaborate techniques for the computation of multi-leg two-loop integrals. We discuss how the large number of different integrals appearing in actual two-loop calculations can be reduced to a small number of master integrals. An efficient method to compute these master integrals is to derive and solve differential equati...

An efficient local coupled cluster method with single and double excitation operators and perturbative treatment of triple excitations [DF-LCCSD(T)] is described. All required two-electron integrals are evaluated using density fitting approximations. These have a negligible effect on the accuracy but reduce the computational effort by 1-2 orders of magnitude, as compared to standard integral-di...

The purpose of these notes is to review the basics of vector calculus in the two dimensions. We will assume you are familiar with the basics of partial derivatives, including the equality of mixed partials (assuming they are continuous), the chain rule, implicit differentiation. In addition, some familiarity with multiple integrals is assumed, although we will review the highlights. Proofs and ...

We show that the general stable convergence results proved in Peccati and Taqqu (2007) for generalized adapted stochastic integrals can be used to obtain limit theorems for multiple stochastic integrals with respect to independently scattered random measures. Several applications are developed in a companion paper (see Peccati and Taqqu, 2008a), where we prove central limit results involving si...