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Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not…
The aim of this work is to present a series of concrete examples which illustrate how the computer algebra system Cadabra can be used to manipulate expressions appearing in General Relativity and other gravitational theories. We highlight…
Some aspects of Computer Algebra (notably Computation Group Theory and Computational Number Theory) have some good databases of examples, typically of the form "all the X up to size n". But most of the others, especially on the polynomial…
We study a physically motivated representation of an algebra of operators in gravitational and non gravitational theories called the covariant representation of an algebra. This is a representation where the symmetries of the operator…
Simplification of fractional powers of positive rational numbers and of sums, products and powers of such numbers is taught in beginning algebra. Such numbers can often be expressed in many ways, as this article discusses in some detail.…
Various applications of quantum algebraic techniques in nuclear structure physics and in molecular physics are briefly reviewed and a recent application of these techniques to the structure of atomic clusters is discussed in more detail.
This paper investigates centralizers and twisted centralizers in degenerate and non-degenerate Clifford (geometric) algebras. We provide an explicit form of the centralizers and twisted centralizers of the subspaces of fixed grades,…
We present the tensor computer algebra package xPert for fast construction and manipulation of the equations of metric perturbation theory, around arbitrary backgrounds. It is based on the combination of explicit combinatorial formulas for…
This thesis is concerned with formulations of the Einstein equations in axisymmetric spacetimes which are suitable for numerical evolutions. We develop two evolution systems based on the (2+1)+1 formalism. The first is a (partially)…
We discuss issues of problem formulation for algorithms in real algebraic geometry, focussing on quantifier elimination by cylindrical algebraic decomposition. We recall how the variable ordering used can have a profound effect on both…
The reader will learn how digital images are edited using linear algebra and calculus. Starting from the concept of filter towards machine learning techniques such as convolutional neural networks.
We survey large value problems, including the large value problem for Dirichlet polynomials, the restriction problem, and problems from computer science. We describe known techniques and open problems, drawing on perspectives from all three…
A tutorial of the Mathematica package CGAlgebra, for conformal geometric algebra calculations is presented. Using rule-based programming, the 5-dimensional conformal geometric algebra is implemented and defined functions simplify the…
In this paper, some real-world motivated examples are provided illustrating the power of linear algebra tools as the product of matrices, determinants, eigenvalues and eigenvectors. In this sense, some practical applications related to…
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…
Geometric Algebra and Calculus are mathematical languages encoding fundamental geometric relations that theories of physics seem to respect. We propose criteria given which statistics of expressions in geometric algebra are computable in…
I introduce yet another way to associate a C*-algebra to a graph and construct a simple nuclear C*-algebra that has irreducible representations both on a separable and a nonseparable Hilbert space.
This article presents a natural extension of the tensor algebra. In addition to "left multiplications" by vectors, we can consider "derivations" by covectors as basic operators on this extended algebra. These two types of operators satisfy…
In this survey article, we present interactions between algebraic geometry and computer vision, which have recently come under the header of algebraic vision. The subject has given new insights in multiple view geometry and its application…
In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has…