Related papers: New Symbolic Tools for Differential Geometry, Grav…
Differential equations are fundamental to modeling dynamic systems in physics, engineering, biology, and economics. While analytical solutions are ideal, most real-world problems necessitate numerical approaches. This study conducts a…
We present different methods for symbolic computer algebra computations in higher dimensional (\ge9) Clifford algebras using the \Clifford\ and \Bigebra\ packages for \Maple(R). This is achieved using graded tensor decompositions,…
This article presents some aspects and experience in the use of algebraic manipulation software applied to general relativity. Some years ago certain results were reported using computer algebra platforms, but the growing popularity of…
In this paper, we demonstrate how deterministic and stochastic dynamics on manifolds, as well as differential geometric constructions can be implemented concisely and efficiently using modern computational frameworks that mix symbolic…
We introduce the C++ library Wedge, based on GiNaC, for symbolic computations in differential geometry. We show how Wedge makes it possible to use the language C++ to perform such computations, and illustrate some advantages of this…
Differential graded (DG) algebras are powerful tools from rational homotopy theory. We survey some recent applications of these in the realm of homological commutative algebra.
The differentiable programming paradigm is a cornerstone of modern scientific computing. It refers to numerical methods for computing the gradient of a numerical model's output. Many scientific models are based on differential equations,…
This paper provides a toolbox of para-differential calculus on compact Lie groups. The toolbox is based on representation theory of compact Lie groups and contains exact formulas of symbolic calculus. Para-differential operators are…
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…
The context of this work is the design of a software, called MEMSALab, dedicated to the automatic derivation of multiscale models of arrays of micro- and nanosystems. In this domain a model is a partial differential equation. Multiscale…
The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed in recent work (collaboration with H.…
Jet manifolds and vector bundles allow one to employ tools of differential geometry to study differential equations, for example those arising as equations of motions in physics. They are necessary for a geometrical formulation of…
We have developed a new tool for numerical work in General Relativity: GRworkbench. While past tools have been ad hoc, GRworkbench closely follows the framework of Differential Geometry to provide a robust and general way of computing on…
It is the aim of this work to identify and illustrate the potential and weaknesses of the computer algebra system Maple in the area of the Calculus of Variations: a classical area of mathematics that studies the methods for finding maximum…
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…
MatchingTools is a Python library for doing symbolic calculations in effective field theory. It provides the tools to construct general models by defining their field content and their interaction Lagrangian. Once a model is given, the…
Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic…
We give detailed exposition of modern differential geometry from global coordinate independent point of view as well as local coordinate description suited for actual computations. In introduction, we consider Euclidean spaces and different…
We report on our experience formalizing differential geometry with mathlib, the Lean mathematical library. Our account is geared towards geometers with no knowledge of type theory, but eager to learn more about the formalization of…
An efficient systematic procedure is provided for symbolic computation of Lie groups of equivalence transformations and generalized equivalence transformations of systems of differential equations that contain arbitrary elements (arbitrary…