Related papers: GRworkbench: A Computational System Based on Diffe…
The software tool GRworkbench is an ongoing project in visual, numerical General Relativity at The Australian National University. Recently, GRworkbench has been significantly extended to facilitate numerical experimentation in…
The software tool GRworkbench is an ongoing project in visual, numerical General Relativity at The Australian National University. Recently, the numerical differential geometric engine of GRworkbench has been rewritten using functional…
The software tool GRworkbench is an ongoing project in visual, numerical General Relativity at The Australian National University. This year, GRworkbench has been significantly extended to facilitate numerical experimentation. The numerical…
With the growth of machine learning algorithms with geometry primitives, a high-efficiency library with differentiable geometric operators are desired. We present an optimized Differentiable Geometry Algorithm Library (DGAL) loaded with…
Algebraic computing in relativity and gravitation dates back more than thirty years, but only relatively recently has hardware of sufficient power to tackle large scale calculations become commonplace. Whereas it is generally understood…
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…
We introduce a novel formulation for geometry on discrete points. It is based on a universal differential calculus, which gives a geometric description of a discrete set by the algebra of functions. We expand this mathematical framework so…
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…
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…
A brief characteristic of the specialized computer algebra system GRG_EC intended for symbolic computations in the field of general relativity is given.
This article provides a pedagogically oriented introduction to geometric (Clifford) calculus on pseudo-Riemannian manifolds. Unlike usual approaches to the topic, which rely on embedding the geometric algebra either within a tensor algebra…
Geometrical interpretations of deep learning models offer insightful perspectives into their underlying mathematical structures. In this work, we introduce a novel approach that leverages differential geometry, particularly concepts from…
It is often noted that many of the basic concepts of differential geometry, such as the definition of connection, are purely algebraic in nature. Here, we review and extend existing work on fully algebraic formulations of differential…
The use of geometric and symmetry techniques in quantum and classical information processing has a long tradition across the physical sciences as a means of theoretical discovery and applied problem solving. In the modern era, the emergent…
We present a generic C++ design to perform efficient and exact geometric computations using lazy evaluations. Exact geometric computations are critical for the robustness of geometric algorithms. Their efficiency is also critical for most…
The theories of strings and $D$-branes have motivated the development of non Abelian cohomology techniques in differential geometry, on the purpose to find a geometric interpretation of characteristic classes. The spaces studied here, like…
In a quantum mechanical treatment of gauge theories (including general relativity), one is led to consider a certain completion, $\agb$, of the space $\ag$ of gauge equivalent connections. This space serves as the quantum configuration…
Shape analysis and compuational anatomy both make use of sophisticated tools from infinite-dimensional differential manifolds and Riemannian geometry on spaces of functions. While comprehensive references for the mathematical foundations…
This paper establishes the basis of the quaternionic differential geometry ($\mathbbm H$DG) initiated in a previous article. The usual concepts of curves and surfaces are generalized to quaternionic constraints, as well as the curvature and…
The usual quantization of a classical space-time field does not touch the non-geometrical character of quantum mechanics. We believe that the deep problems of unification of general relativity and quantum mechanics are rooted in this poor…