Related papers: Nested Integrals and Rationalizing Transformations
Nested sums containing binomial coefficients occur in the computation of massive operator matrix elements. Their associated iterated integrals lead to alphabets including radicals, for which we determined a suitable basis. We discuss…
I consider the expansion of transcendental functions in a small parameter around rational numbers. This includes in particular the expansion around half-integer values. I present algorithms which are suitable for an implementation within a…
This paper describes algorithms to deal with nested symbolic sums over combinations of harmonic series, binomial coefficients and denominators. In addition it treats Mellin transforms and the inverse Mellin transformation for functions that…
This work deals with special nested objects arising in massive higher order perturbative calculations in renormalizable quantum field theories. On the one hand we work with nested sums such as harmonic sums and their generalizations…
Expansion of higher transcendental functions in a small parameter are needed in many areas of science. For certain classes of functions this can be achieved by algebraic means. These algebraic tools are based on nested sums and can be…
We study multiplicative nested sums, which are generalizations of harmonic sums, and provide a calculation through multiplication of index matrices. Special cases interpret the index matrices as stochastic transition matrices of random…
This paper summarizes the essential functionality of the computer algebra package HarmonicSums. On the one hand HarmonicSums can work with nested sums such as harmonic sums and their generalizations and on the other hand it can treat…
This article is written with the hope to draw attention to a method that uses integral transforms to find exact values for a large class of convergent series (and, in particular, series of rational terms). We apply the method to some series…
Harmonic sums and their generalizations are extremely useful in the evaluation of higher-order perturbative corrections in quantum field theory. Of particular interest have been the so-called nested sums,where the harmonic sums and their…
A nested coordinate system is a reassigning of independent variables to take advantage of geometric or symmetry properties of a particular application. Polar, cylindrical and spherical coordinate systems are primary examples of such a…
In this paper, we propose methods for computing the Hilbert series of multigraded right modules over the free associative algebra. In particular, we compute such series for noncommutative multigraded algebras. Using results from the theory…
In this article algebraic constructions are introduced in order to study the variety defined by a radical parametrization (a tuple of functions involving complex numbers, $n$ variables, the four field operations and radical extractions). We…
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…
In these introductory lectures we discuss classes of presently known nested sums, associated iterated integrals, and special constants which hierarchically appear in the evaluation of massless and massive Feynman diagrams at higher loops.…
Incremental computation aims to compute more efficiently on changed input by reusing previously computed results. We give a high-level overview of works on incremental computation, and highlight the essence underlying all of them, which we…
This work connects two mathematical fields - computational complexity and interval linear algebra. It introduces the basic topics of interval linear algebra - regularity and singularity, full column rank, solving a linear system, deciding…
A summation is a shift-invariant ${\rm R}$-module homomorphism from a submodule of ${\rm R}[[\sigma]]$ to ${\rm R}$ or another ring. [11] formalized a method for extending a summation to a larger domain by telescoping. In this paper, we…
A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.
An analytical-numeric calculation method of extremely complicated integrals is presented. These integrals appear often in magnet soliton theory. The appropriate analytical continuation and a corresponding integration contour allow to reduce…
This survey describes probabilistic algorithms for linear algebra computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problem instances. The paper…