Related papers: Numerical Evaluation of Harmonic Polylogarithms
Many speech and music analysis and processing schemes rely on an estimate of the fundamental frequency $f_0$ of periodic signal components. Most established schemes apply rather unspecific signal models such as sinusoidal models to the…
Spin-weighted spheroidal harmonics are useful in a variety of physical situations, including light scattering, nuclear modeling, signal processing, electromagnetic wave propagation, black hole perturbation theory in four and higher…
An algorithm, based on numerical description of the terms of many-body perturbation theory (Goldstone diagrams), is presented. The algorithm allows the use of the same piece of computer code to evaluate any particular diagram in any…
The harmonic numbers and generalized harmonic numbers appear frequently in many diverse areas such as combinatorial problems, many expressions involving special functions in analytic number theory and analysis of algorithms. The aim of this…
We study the problem of computing the isolated regular solutions of a system \((f_1,\ldots,f_n)\) of \(n\) polynomial equations in \(n\) variables \((X_1, \dots, X_n)\) over a field of characteristic zero \(k\). We focus on systems with a…
We find a new lower bound for the maximal number of zeros to harmonic polynomials, $p(z)+\overline{q(z)}$, when ${\rm deg}\, p = n$ and ${\rm deg}\, q = n-2$.
In this paper, we describe a numerical approach to evaluate Feynman loop integrals. In this approach the key technique is a combination of a numerical integration method and a numerical extrapolation method. Since the computation is carried…
In the computation of Feynman integrals which evaluate to multiple polylogarithms one encounters quite often square roots. To express the Feynman integral in terms of multiple polylogarithms, one seeks a transformation of variables, which…
Extending ideas of A.N. Parshin and D.V. Osipov, Harmonic Analysis is developed for filtered infinite dimensional modules over a ring. We establish Pontryagin duality, the Fourier inversion formula, Plancherel formula and Poisson summation…
We study values of generalized polylogarithms at various points and relationships among them. Polylogarithms of small weight at the points 1/2 and -1 are completely investigated. We formulate a conjecture about the structure of the linear…
In this paper we focus on r-geometric polynomials, r-exponential polynomials and their harmonic versions. It is shown that harmonic versions of these polynomials and their generalizations are useful to obtain closed forms of some series…
Some years ago, the harmonic polynomial was introduced in order to understand better the harmonic topological index; for instance, it allows to obtain bounds of the harmonic index of the main products of graphs. Here, we obtain several…
Integer relation algorithms can convert numerical results for Feynman integrals to exact evaluations, when one has reason to suspect the existence of reductions to linear combinations of a basis, with rational or algebraic coefficients.…
We derive the structural relations between nested harmonic sums and the corresponding Mellin transforms of Nielsen integrals and harmonic polylogarithms at weight {\sf w = 6}. They emerge in the calculations of massless single--scale…
A simple formula for computing the generalized Hubbell radiation rectangular source integral $$H[{a,b,p,\lambda \atop \alpha,\beta,\gamma}]=\frac{\sigma a}{4\pi}\int_0^b x^\lambda(x^2+p)^{-\alpha}{}_2F_1(\alpha,\beta;\gamma;-{a^2\over…
We prove that the logarithm of the permanent of an nxn real matrix A and the logarithm of the hafnian of a 2nx2n real symmetric matrix A can be approximated within an additive error 1 > epsilon > 0 by a polynomial p in the entries of A of…
Some explicit algorithms for higher order symplectic integration of a large class of Hamilton's equations have recently been discussed by Mushtaq \emph{et. al}. Here we present a Python program for automatic numerical implementation of…
The cosmological polytope of a graph $G$ was recently introduced to give a geometric approach to the computation of wavefunctions for cosmological models with associated Feynman diagram $G$. Basic results in the theory of positive…
We present a very simple expression and a Fortran code for the fast and precise calculation of three-dimensional harmonic-oscillator transformation brackets. The complete system of symmetries for the brackets along with analytical…
A procedure for calculation of rotation-vibration states of medium sized molecules is presented. It combines the advantages of variational calculations and perturbation theory. The vibrational problem is solved by diagonalizing a…