Related papers: CGAlgebra: a Mathematica package for conformal geo…
The area of geometry with its very strong and appealing visual contents and its also strong and appealing connection between the visual content and its formal specification, is an area where computational tools can enhance, in a significant…
The set theory relations \in, \backslash, \Delta, \cap, and \cup have corollaries in subspace relations. Geometric Algebra is introduced as the ideal framework to explore these subspace operations. The relations \in, \backslash, and \Delta…
We formulate a number of new results in Algebraic Geometry and outline their derivation from Theorem 2.12 which belongs to Algebraic Combinatorics.
We showcase applications of nonlinear algebra in the sciences and engineering. Our review is organized into eight themes: polynomial optimization, partial differential equations, algebraic statistics, integrable systems, configuration…
We review some recent applications of machine learning to algebraic geometry and physics. Since problems in algebraic geometry can typically be reformulated as mappings between tensors, this makes them particularly amenable to supervised…
Determining visibility in planar polygons and arrangements is an important subroutine for many algorithms in computational geometry. In this paper, we report on new implementations, and corresponding experimental evaluations, for two…
A method for converting the geometrical problem of rectangle packing to an algebraic problem of solving a system of polynomial equations is described.
We introduce the MathGR package, written in Mathematica. The package can manipulate tensor and GR calculations with either abstract or explicit indices, simplify tensors with permutational symmetries, decompose tensors from abstract indices…
With a large number of software tools dedicated to the visualisation and/or demonstration of properties of geometric constructions and also with the emerging of repositories of geometric constructions, there is a strong need of linking…
In this work, we are presenting an extension of the cooperative dual-task space (CDTS) in conformal geometric algebra. The CDTS was first defined using dual quaternion algebra and is a well established framework for the simplified…
We discuss the application of computer algebra to problems commonly arising in numerical relativity, such as the derivation of 3+1-splits, manipulation of evolution equations and automatic code generation. Particular emphasis is put on…
Comparison of geometric quantities usually means obtaining generally true equalities of different algebraic expressions of a given geometric figure. Today's technical possibilities already support symbolic proofs of a conjectured theorem,…
Enumerative algebraic geometry counts the solutions to certain geometric constraints. Numerical algebraic geometry determines these solutions for any given instance. This article illustrates how these two fields complement each other. Our…
Much of the success of deep learning is drawn from building architectures that properly respect underlying symmetry and structure in the data on which they operate - a set of considerations that have been united under the banner of…
Differential forms is a highly geometric formalism for physics used from field theories to General Relativity (GR) which has been a great upgrade over vector calculus with the advantages of being coordinate-free and carrying a high degree…
Numerical Algebraic Geometry uses numerical data to describe algebraic varieties. It is based on the methods of numerical polynomial homotopy continuation, an alternative to the classical symbolic approaches of computational algebraic…
This article describes an entirely algebraic construction for developing conformal geometries, which provide models for, among others, the Euclidean, spherical and hyperbolic geometries. On one hand, their relationship is usually shown…
It is nearly twenty years that there exist computer programs to reduce products of Lie algebra irreps. This is a contribution in the field that uses a modern computer language (``C'') in a highly structured and object-oriented way. This…
We describe TensoriaCalc, a tensor calculus package written to be smoothly consistent with the Wolfram Language, so as to ensure ease of usage. It allows multiple metrics to be defined in a given session; and, once a metric is computed,…
We introduce the Macaulay2 package GradedLieAlgebras for doing computations in graded Lie algebras presented by generators and relations.