Related papers: Differential algebra and mathematical physics
Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial…
This is an overview of higher structural constructions in physics. The main motivations of our current attempt are as follows: (i) to provide a brief introduction to derived algebraic geometry, (ii) to understand how derived objects…
These notes provide an introduction to the algebra and geometry of differential operators and jet bundles. Their point of view is guided by the leitmotiv that higher-spin gravity theories call for higher-order generalisations of Lie…
It is shown that properties of a discrete space-time geometry distinguish from properties of the Riemannian space-time geometry. The discrete geometry is a physical geometry, which is described completely by the world function. The discrete…
We develop a well-posedness theory for second order systems in bounded domains where boundary phenomena like glancing and surface waves play an important role. Attempts have previously been made to write a second order system consisting of…
I introduce an innovative methodology for deriving numerical models of systems of partial differential equations which exhibit the evolution of spatial patterns. The new approach directly produces a discretisation for the evolution of the…
The field equations for both generic bosonic and generic locally supersymmetric 2D dilatonic gravity theories in the absence of matter are written as free differential algebras. This constitutes a generalization of the gauge theoretic…
The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry…
A method is introduced for solving Einstein's equations using two distinct coordinate systems. The coordinate basis vectors associated with one system are used to project out components of the metric and other fields, in analogy with the…
In this second part of the paper, dedicated to theories with extra dimensions, a new physical notion about the "tensor length scale" is introduced, based on the gravitational theories with covariant and contravariant metric tensor…
Several algorithms in computer algebra involve the computation of a power series solution of a given ordinary differential equation. Over finite fields, the problem is often lifted in an approximate $p$-adic setting to be well-posed. This…
A new notion of stochastic transformation is proposed and applied to the study of both weak and strong symmetries of stochastic differential equations (SDEs). The correspondence between an algebra of weak symmetries for a given SDE and an…
We start recalling with critical eyes the mathematical methods used in gauge theory and prove that they are not coherent with continuum mechanics, in particular the analytical mechanics of rigid bodies or hydrodynamics, though using the…
Our recent arXiv preprints and published papers on the solution of the Riemann-Lanczos and Weyl-Lanczos problems have brought our attention on the importance of revisiting the algebraic structure of the Bianchi identities in Riemannian…
In Part I of this series, the author has shown how to extend the framework of Riemannian geometry so as to include infinitesimals of higher than first order. The purpose of the present contribution is to initiate an investigation into the…
This short communication develops a new numerical procedure suitable for a large class of ordinary differential equation systems found in models in physics and engineering. The main numerical procedure is analogous to those concerning the…
We exhibit differential geometric structures that arise in numerical methods, based on the construction of Cauchy sequences, that are currently used to prove explicitly the existence of weak solutions to functional equations. We describe…
In this paper we report a few examples of algebraically solvable dynamical systems characterized by 2 coupled Ordinary Differential Equations which read as follows: x_n = P(n) (x1, x2) , n = 1, 2 , with P(n) (x1, x2) specific polynomials of…
We describe a novel approach to dimensional reduction in classical field theory. Inspired by ideas from noncommutative geometry, we introduce extended algebras of differential forms over space-time, generalized exterior derivatives and…
Recently the space-time foam differential algebras of generalized functions with dense singularities were introduced, motivated by the so called space-time foam structures in General Relativity with dense singularities, and by Quantum…