相关论文: GRworkbench: A Computational System Based on Diffe…

In this work, we introduce GRChombo: a new numerical relativity code which incorporates full adaptive mesh refinement (AMR) using block structured Berger-Rigoutsos grid generation. The code supports non-trivial "many-boxes-in-many-boxes"…

We present a new certified and complete algorithm to compute arrangements of real planar algebraic curves. Our algorithm provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic…

The paper introduces a novel representation for Generalized Planning (GP) problems, and their solutions, as C++ programs. Our C++ representation allows to formally proving the termination of generalized plans, and to specifying their…

Differential calculus on the quantum quaternionic group GL(1,H$_q$) is introduced.

We present the GRjunction package which allows boundary surfaces and thin-shells in general relativity to be studied with a computer algebra system. Implementing the Darmois-Israel thin shell formalism requires a careful selection of…

Here it is shown how to combine two generically globally rigid bar frameworks in $d$-space to get another generically globally rigid framework. The construction is to identify $d+1$ vertices from each of the frameworks and erase one of the…

This article introduces a general purpose framework and software to approximate partial differential equations (PDEs). The sparsity patterns of finite element discretized operators is identified automatically using the tools from…

Deep neural networks implement a sequence of layer-by-layer operations that are each relatively easy to understand, but the resulting overall computation is generally difficult to understand. We consider a simple hypothesis for interpreting…

Geometry problem solving, a crucial aspect of mathematical reasoning, is vital across various domains, including education, the assessment of AI's mathematical abilities, and multimodal capability evaluation. The recent surge in deep…

Computer simulations are enabling researchers to investigate systems which are extremely difficult to handle analytically. In the particular case of General Relativity, numerical models have proved extremely valuable for investigations of…

We pose a new algebraic formalism for studying differential calculus in vector bundles. This is achieved by studying various functors of differential calculus over arbitrary graded commutative algebras (DCGCA) and applying this language to…

We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed.…

We discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries. The method is then applied to the Schr\"{o}dinger equation in…

Probabilistic theory and differential equation are powerful tools for the interpretability and guidance of the design of machine learning models, especially for illuminating the mathematical motivation of learning latent variable from…

Systems consisting of spheres rolling on elastic membranes have been used to introduce a core conceptual idea of General Relativity (GR): how curvature guides the movement of matter. However, such schemes cannot accurately represent…

We develop a novel approach to gravity that we call `matrix general relativity' (MGR) or `gravitational chromodynamics' (GCD or GQCD for quantum version). Gravity is described in this approach not by one Riemannian metric (i.e. a symmetric…

Many problems in robotics are fundamentally problems of geometry, which lead to an increased research effort in geometric methods for robotics in recent years. The results were algorithms using the various frameworks of screw theory, Lie…

This note serves two purposes: 1) define actions by differential graded Lie algebras, and 2) apply such differential graded Lie symmetry in general relativity (GR) to constrain the spacetime geometry on a neighborhood of infinity.

We provide a novel new approach for aligning geometric models using a dual graph structure where local features are mapping probabilities. Alignment of non-rigid structures is one of the most challenging computer vision tasks due to the…

This is an introduction to geometric algebra, an alternative to traditional vector algebra that expands on it in two ways: 1. In addition to scalars and vectors, it defines new objects representing subspaces of any dimension. 2. It defines…