Related papers: Nested (inverse) binomial sums and new iterated in…
A sum-and-distance system is a collection of finite sets of integers such that the sums and differences formed by taking one element from each set generate a prescribed arithmetic progression. Such systems, with two component sets, arise…
In this paper, we explore identities that allow for representation of positive integers raised to positive integral powers as sums of nested sums of smaller positive integral powers. We begin by establishing the base identity involving…
In our previous paper an effective algorithm for inverting polynomial automorphisms was proposed. We extend its application to the case of formal power series over a field of arbitrary characteristic and illustrate the proposed approach…
We study properties of transfer matrices in the sl(N) spin chain models. The transfer matrices with an infinite dimensional auxiliary space are factorized into the product of N commuting Baxter Q-operators. We consider the transfer matrices…
We develop the theory of mixed finite elements in terms of special inverse systems of complexes of differential forms, defined over cellular complexes. Inclusion of cells corresponds to pullback of forms. The theory covers for instance…
Starting at 3-loop order, the massive Wilson coefficients for deep-inelastic scattering and the massive operator matrix elements describing the variable flavor number scheme receive contributions of Feynman diagrams carrying quark lines…
The computation and inversion of the binomial and negative binomial cumulative distribution functions play a key role in many applications. In this paper, we explain how methods used for the central beta distribution function (described in…
Suppose we are given a large matrix $A=(a_{i,j})$ that cannot be stored in memory but is in a disk or is presented in a data stream. However, we need to compute a matrix decomposition of the entry-wisely transformed matrix,…
An efficient way to calculate one-loop counterterms within the Feynman diagrammatic approach and dimensional regularization is to expand the propagators in the integrands of the Feynman integrals around vanishing external momentum. In this…
We consider chaining multiplicative-inverse operations in finite fields under alternating polynomial bases. When using two distinct polynomial bases to alternate the inverse operation we obtain a partition of $\mathbb F_{p^n}\setminus…
We consider multi-edge or banana graphs $b_n$ on $n$ internal edges $e_i$ with different masses $m_i$. We focus on the cut banana graphs $\Im(\Phi_R(b_n))$ from which the full result $\Phi_R(b_n)$ can be derived through dispersion. We give…
We present a new paradigm for speeding up randomized computations of several frequently used functions in machine learning. In particular, our paradigm can be applied for improving computations of kernels based on random embeddings. Above…
We extend several celebrated methods in classical analysis for summing series of complex numbers to series of complex matrices. These include the summation methods of Abel, Borel, Ces\'aro, Euler, Lambert, N\"orlund, and Mittag-Leffler,…
We present an evaluation of the two master integrals for the crossed vertex diagram with a closed loop of top quarks that allows for an easy numerical implementation. The differential equations obeyed by the master integrals are used to…
This contribution is motivated by old and recent works on matrix powers and their applications on combinatorial sequences. We give in this paper the $s$-th powers and the inverses for special upper triangular matrices and the $s$-th powers…
In the context of Higman embeddings of recursive groups into finitely presented groups we suggest an algorithm which uses Higman operations to explicitly constructs the specific recursively enumerable sets of integer sequences arising…
In this article we present a refined summation theory based on Karr's difference field approach. The resulting algorithms find sum representations with optimal nested depth. For instance, the algorithms have been applied successively to…
We propose investigating a summation analog of the paradigm for parallel integration. We make some first steps towards an indefinite summation method applicable to summands that rationally depend on the summation index and a P-recursive…
We propose a new set of Master Integrals which can be used as a basis for certain multiloop calculations in massless gauge field theories. In these theories we consider three-point Feynman diagrams with arbitrary number of loops. The…
We give explicit evaluations of the linear and non-linear Euler sums of hyperharmonic numbers $h_{n}^{\left( r\right) }$ with reciprocal binomial coefficients. These evaluations enable us to extend closed form formula of Euler sums of…