Related papers: Numerical computation of real or complex elliptic …
The new version 2.1 of the program SecDec is described, which can be used for the factorisation of poles and subsequent numerical evaluation of multi-loop integrals, in particular massive two-loop integrals. The program is not restricted to…
Several problems which could be thought of as belonging to recreational mathematics are described. They are all such that solutions to the problem depend on finding rational points on elliptic curves. Many of the problems considered lead to…
We define a parametric variant of generalized Euler sums and construct contour integration to give some explicit evaluations of these parametric Euler sums. In particular, we establish several explicit formulas of (Hurwitz) zeta functions,…
We present several new heuristic algorithms to compute class polynomials and modular polynomials modulo a prime $p$ by revisiting the idea of working with supersingular elliptic curves. The best known algorithms to this date are based on…
A closed-form formula is derived for the generalized Clebsch-Gordan integral $ \int_{-1}^1 {[}P_{\nu}(x){]}^2P_{\nu}(-x)\D x$, with $ P_\nu$ being the Legendre function of arbitrary complex degree $ \nu\in\mathbb C$. The finite Hilbert…
We introduce a method for computing some pseudo-elliptic integrals in terms of elementary functions. The method is simple and fast in comparison to the algebraic case of the Risch-Trager-Bronstein algorithm. This method can quickly solve…
Dual numbers and their higher order version are important tools for numerical computations, and in particular for finite difference calculus. Based upon the relevant algebraic rules and matrix realizations of dual numbers, we will present a…
We give an infinite family of congruent number elliptic curves, each with rank at least two, which are related to integral solutions of $m^2=n^2+nl+l^2$.
We give explicit formulas for the number of points on reductions of elliptic curves with complex multiplication by any imaginary quadratic field. We also find models for CM $\mathbf{Q}$-curves in certain cases. This generalizes earlier…
Weakly singular Volterra integral equations of the different types are considered. The construction of accuracy-optimal numerical methods for one-dimensional and multidimensional equations is discussed. Since this question is closely…
We show how to efficiently compute Hilbert modular forms as orthogonal modular forms, generalizing and expanding upon the method of Birch.
We develop numerical methods for elliptic systems governed by partial segregation constraints, in which three nonnegative components are required to have a vanishing pointwise product throughout the domain. This constraint enforces that at…
Some methods for the numerical computation of two-loop non-infrared vertices are reviewed. A new method is also proposed and compared to the old ones. Finally, some preliminary results are presented, concerning the evaluation of the…
In the article we outline the set of Matlab functions that enable the computation of elliptic Integrals and Jacobian elliptic functions for real arguments. Correctness, robustness, efficiency and accuracy of the functions are discussed in…
We derive an analytic representation of the ten-particle, two-loop double-box integral as an elliptic integral over weight-three polylogarithms. To obtain this form, we first derive a four-fold, rational (Feynman-)parametric representation…
A mathematical model is given for the occurrence of preferred orbits and orbital velocities in a Keplerian system. The result can be extended into energies and other properties of physical systems. The values given by the model fit closely…
This paper discusses the theory and numerical method of two-scale analysis for the multiscale Landau-Lifshitz-Gilbert equation in composite ferromagnetic materials. The novelty of this work can be summarized in three aspects: Firstly, the…
The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order.…
Discrepancy between periodic orbit theory and numerical calculation of a modified Kepler problem is cleared by a quantum mechanical calculation. The diagonal approximation already gives a good fit for the numerical calculation. A better…
The complete elliptic integrals are generalized by using the generalized trigonometric functions with two parameters. It is shown that a particular relation holds for the generalized integrals. Moreover, as an application of the integrals,…