A forward second-harmonic scattering method for determining the second-order susceptibility coefficients of ZnO nanorods on the order of 100 nm in diameter is presented. The long coherence length and strong intensity of the forward second-harmonic signal allow for a simple theoretical model to match the experimental measurements; this results in getting accurate values for the second-order susc...

We present a new formulation of Maker fringes in parallel-surface films, using self-consistent boundary conditions for reflections and allowing for any degree of refractive-index dispersion. This treatment of the second-harmonic reflections and dispersion, unlike a number of previous derivations, leads correctly to the expected form for the effective second-harmonic d coefficients. Complete exp...

In line with an ongoing programme to determine accurately x-ray attenuation coefficients, we have developed a method for the quantitative determination of the effect on experimental results of monochromator harmonic components in a synchrotron beam. The technique can be adapted to suit a wide variety of experiments, and is of particular interest because it determines the effect of the harmonic ...

We show that the multipole vector decomposition, recently introduced by Copi et al., is a consequence of Sylvester’s theorem, and corresponds to the Maxwell representation. Analyzing it in terms of harmonic polynomials, we show that this decomposition results in fact from the application of the harmonic projection operator and its inverse. We derive the coefficients of the usual harmonic decomp...

This paper describes an automated model selection method for analysing the relationship between the order of the spherical harmonic basis functions used to fit high angular resolution diffusion imaging (HARDI) data and the accuracy of the fitting results. The method performs statistical inference on the spherical harmonic expansion coefficients and uses a backward elimination procedure to remov...

The harmonic polylogarithms (hpl’s) are introduced. They are a generalization of Nielsen’s polylogarithms, satisfying a product algebra (the product of two hpl’s is in turn a combination of hpl’s) and forming a set closed under the transformation of the arguments x = 1/z and x = (1−t)/(1+t). The coefficients of their expansions and their Mellin transforms are harmonic sums. AMS(1991) subject cl...

A method for the evaluation of the ε-expansion of multi-loop massless Feynman integrals is introduced. This method is based on the Gegenbauer polynomial technique and the expansion of the Gamma function in terms of harmonic sums. Algorithms for the evaluation of nested and harmonic sums are used to reduce the expressions to get analytical or numerical results for the expansion coefficients. Met...

Ordinary generating series of multiple harmonic sums admit a full singular expansion in the basis of functions {(1 − z) log(1 − z)}α∈Z,β∈N, near the singularity z = 1. A constructive proof of this result is given, and, by combinatoric aspects, an explicit evaluation of Taylor coefficients of functions in some polylogarithmic algebra is obtained. In particular, the asymptotic expansion of multip...

According to Kippenhahn’s classification, numerical ranges W (A) of unitarily irreducible 3 × 3 matrices A come in three possible shapes, an elliptical disk being one of them. The known criterion for the ellipticity of W (A) consists of several equations, involving the eigenvalues of A. It is shown herein that the set of 3× 3 matrices satisfying these conditions is nowhere dense, i.e., one of t...