Related papers: Computer-assisted proofs of "Kariya's theorem" wit…

There are quantum solutions for computational problems that make use of interference at some stage in the algorithm. These stages can be mapped into the physical setting of a single particle travelling through a many-armed interferometer.…

This article is intended to an introductory lecture in material physics, in which the modern computational group theory and the electronic structure calculation are in collaboration. The effort of mathematicians in field of the group…

We provide a visual and intuitive introduction to effectively calculating in 2-groups along with explicit examples coming from non-abelian 1- and 2-form gauge theory. In particular, we utilize string diagrams, tools similar to tensor…

The paper is a survey of some results about Weil algebras applicable in differential geometry, especially in some classification questions on bundles of generalized velocities and contact elements. Mainly, a number of claims concerning a…

We present some methods and results in the application of algebraic geometry and computer algebra to the study of algebraic vector bundles, foliations and zeta functions. A connection of the methods and results with noncommutative geometry…

We use a canonical parametrization of twisted geometries describing the classical phase space of loop quantum gravity on a fixed graph, and establish its explicit correspondence with the associated frame bases and spinorial descriptions.…

Kedlaya's algorithm (Kedlaya, J. Ramanujan Math. Soc 16, 2001) can be used to count the points of arbitrary hyperelliptic curves over finite fields of characteristic p, where p is an odd prime. The algorithm uses the cohomology of a p-adic…

Following N. Elkies ("ABC implies Mordell") we show that the abc conjecture of Masser-Oesterle implies an effective version of Siegel's theorem about integral points on algebraic curves, i.e. an upper bound for the S-integral points where…

This short expository paper outlines applications of computer algebra to the implication problem of conditional independence for Gaussian random variables. We touch on certificates for validity and invalidity of inference rules from the…

While the earliest applications of AI methodologies to pure mathematics and theoretical physics began with the study of Hodge numbers of Calabi-Yau manifolds, the topology type of such manifold also crucially depend on their intersection…

Plotting solution sets for particular equations may be complicated by the existence of turning points. Here we describe an algorithm which not only overcomes such problematic points, but does so in the most general of settings. Applications…

This paper describes the formal verification of two Turing machines using the program verifier Dafny. Both machines are deciders, so we prove total correctness. They are typical first examples of Turing machines used in any course of…

This paper considers two point boundary value problems for conservative systems defined in multiple coordinate systems, and develops a flexible a-posteriori framework for computer assisted existence proofs. Our framework is applied to the…

Field theory is an area in physics with a deceptively compact notation. Although general purpose computer algebra systems, built around generic list-based data structures, can be used to represent and manipulate field-theory expressions,…

The problem of mathematical modeling in geography is one of the most important strategies in order to establish the evolution and the prevision of geographical phenomena. Models must have a simplified structure, to reflect essential…

Geometry constitutes a core set of intuitions present in all humans, regardless of their language or schooling [1]. Could brain's built in machinery for processing geometric information take part in uncertainty representation? For decades…

We prove a centre manifold theorem for a map along a manifold-with-boundary of fixed points, and provide an application to the study of gradient descent with large step size on two-layer matrix factorisation problems.

We employ techniques of machine-learning, exemplified by support vector machines and neural classifiers, to initiate the study of whether AI can "learn" algebraic structures. Using finite groups and finite rings as a concrete playground, we…

We investigate systems of equations, involving parameters from the point of view of both control theory and computer algebra. The equations might involve linear operators such as partial (q-)differentiation, (q-)shift, (q-)difference as…

This paper presents an alternative proof of the Fundamental Theorem of Algebra that has several distinct advantages. The proof is based on simple ideas involving continuity and differentiation. Visual software demonstrations can be used to…