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The problem of simplifying tensor expressions is addressed in two parts. The first part presents an algorithm designed to put tensor expressions into a canonical form, taking into account the symmetries with respect to index permutations…
In this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from…
Reduze is a computer program for reducing Feynman Integrals to master integrals employing a Laporta algorithm. The program is written in C++ and uses classes provided by the GiNaC library to perform the simplifications of the algebraic…
Cet article se veut un survol des r\'eseaux de tenseurs et s'adresse aux d\'ebutants en la mati\`ere. Nous y mettons l'accent sur les outils n\'ecessaires \`a l'impl\'ementation concr\`ete d'algorithmes. Quatre op\'erations de base…
We present Groebner.jl, a Julia package for computing Groebner bases with the F4 algorithm. Groebner.jl is an efficient, portable, and open-source software. Groebner.jl works over integers modulo a prime and over the rationals, supports…
We present in this paper an evolution of a tool from a user interface for a concrete Computer Algebra system for Algebraic Topology (the Kenzo system), to a front-end allowing the interoperability among different sources for computation and…
Modern advances in general-purpose computer algebra systems offer solutions to a variety of problems, which in the past required substantial time investments by trained mathematicians. An excellent example of such development are the…
This article is intended to an introductory lecture in material physics, in which the modern computational group theory and the electronic structure calculation are in collaboration. The effort of mathematicians in field of the group…
We present LieART 2.0 which contains substantial extensions to the Mathematica application LieART (Lie Algebras and Representation Theory) for computations frequently encountered in Lie algebras and representation theory, such as tensor…
Fulfilling the rich promise of rapid advances in time-domain astronomy is only possible through confronting our observations with physical models and extracting the parameters that best describe what we see. Here, we introduce {\sc…
A Mathematica implementation of the program LERG-I is presented that performs the reduction of tensor integrals, encountered in one-loop Feynman diagram calculations, to scalar integrals. The program was originally coded in REDUCE and in…
A fully tensorial theoretical framework for hypercomplex-valued neural networks is presented. The proposed approach enables neural network architectures to operate on data defined over arbitrary finite-dimensional algebras. The central…
A characteristic feature of differential-algebraic equations is that one needs to find derivatives of some of their equations with respect to time, as part of so called index reduction or regularisation, to prepare them for numerical…
In the present paper, we describe some experiences in using programming, commands and graphical interfaces based on computer algebra systems, as tools for learning Physics and Mathematics.
Clifford algebras have broad applications in science and engineering. The use of Clifford algebras can be further promoted in these fields by availability of computational tools that automate tedious routine calculations. We offer an…
This article documents the free computer algebra system "gTybalt". The program is build on top of other packages, among others GiNaC, TeXmacs and Root. It offers the possibility of interactive symbolic calculations within the C++…
A tensor is a multidimensional array of numbers that can be used to store data, encode a computational relation and represent quantum entanglement. In this sense a tensor can be viewed as valuable resource whose transformation can lead to…
This paper introduces a formalism that aims to describe the intricacies of quantum computation by establishing a connection with the mathematical foundations of tensor theory and multilinear maps. The focus is on providing a comprehensive…
As dynamic and control systems become more complex, relying purely on numerical computations for systems analysis and design might become extremely expensive or totally infeasible. Computer algebra can act as an enabler for analysis and…
This text is a support for different courses of the master of Mechanics of the University Paris-Saclay. The content of this text is an introduction, for graduate students, to tensor algebra and analysis. Far from being exhaustive, the text…