Related papers: MultivariateApart: Generalized Partial Fractions
The purpose of this short article is to announce, and briefly describe, a Maple package, PARTITIONS, that (inter alia) completely automatically discovers, and then proves, explicit expressions (as sums of quasi-polynomials) for pm(n) for…
In the analysis of large/big data sets, aggregation (replacing values of a variable over a group by a single value) is a standard way of reducing the size (complexity) of the data. Data analysis programs provide different aggregation…
In this article, we present a new R package fc that provides a streamlined, standard evaluation-based approach to function composition. Using fc, a sequence of functions can be composed together such that returned objects from composed…
Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately…
Multivariate residues appear in many different contexts in theoretical physics and algebraic geometry. In theoretical physics, they for example give the proper definition of generalized-unitarity cuts, and they play a central role in the…
A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-order fractional integral and differential equations is described. The approach is related to the ultraspherical method for…
We introduce SP$\mathbb{Q}$R, a new Mathematica package for the division and elimination of variables from polynomial systems. SP$\mathbb{Q}$R works by sampling and reconstructing results over finite fields, in an analogous manner to many…
This article provides an accessible introduction to fractional derivatives, a concept that extends classical calculus by allowing derivatives of non-integer order. It explores both the fundamental definitions and some of the most relevant…
We introduce the new sage_acsv package for the SageMath computer algebra system, allowing users to rigorously compute asymptotics for a large variety of multivariate sequences with rational generating functions. Using Sage's support for…
We perform certain alternating binomial summations with parameters that occur in the analysis of algorithms. A combination of integral and special function and special number representations is used. The results are sufficiently general to…
An algorithm is designed which decomposes a tropical univariate rational function into a composition of tropical binomials and trinomials. When a function is monotone, the composition consists just of binomials. Similar algorithms are…
A procedure to obtain differentiation matrices is extended straightforwardly to yield new differentiation matrices useful to obtain derivatives of complex rational functions. Such matrices can be used to obtain numerical solutions of some…
We present the Macaulay2 package Resultants, which provides commands for the effective computation of multivariate resultants, discriminants, and Chow forms. We provide some background for the algorithms implemented and show, with a few…
The main objective of this paper is to introduce an algorithm for solving fractional and classical differential equations based on a new generalized fractional power series. The algorithm relies on expanding the solution of an FDE or an ODE…
We prove multidimensional integration by parts formulas for generalized fractional derivatives and integrals. The new results allow us to obtain optimality conditions for multidimensional fractional variational problems with Lagrangians…
Implementing and executing numerical algorithms to solve fractional differential equations has been less straightforward than using their integer-order counterparts, posing challenges for practitioners who wish to incorporate fractional…
The Macaulay2 package SumsOfSquares decomposes polynomials as sums of squares. It is based on methods to rationalize sum-of-squares decompositions due to Parrilo and Peyrl. The package features a data type for sums-of-squares polynomials,…
We propose an algorithm for reduction of the problem of maximization of fraction of two functionals to the equivalent procedure including maximization of difference between the functionals and the solution of an equation of scalar unknown.…
We propose a simple estimator that allows to calculate the absolute value of a system's partition function from a finite sampling of its canonical ensemble. The estimator utilizes a volume correction term to compensate the effect that the…
Abramov's algorithm enables us to decide whether a univariate rational function can be written as a difference of another rational function, which has been a fundamental algorithm for rational summation. In 2014, Chen and Singer generalized…