Related papers: A Discontinuous Differential Calculus in the Frame…
Let $\mathcal{A}$ denote a real, $n$-dimensional, unital, associative algebra.This paper provides an introductory exposition of calculus over $\mathcal{A}$. An $\mathcal{A}$-differentiable function is one for which the differential is…
The main purpose of this article is to introduce a comprehensive, unified theory of the geometry of all connections. We show that one can study a connection via a certain, closely associated second-order differential equation. One of the…
Formulas for the solutions of initial value problems for ordinary differential equations with singular $\delta^{(n)}$-like driving terms are derived in the framework of an algebra of generalized functions (of Colombeau type) over a field of…
We remark a variant of the existence part of the fundamental theorem of calculus, which, together with the Lebesgue differentiation theorem, constitute a new proof that every Riemann-integrable function on a compact interval having limit…
Through the subsequent discussion we consider a certain particular sort of (topological) algebras, which may substitute the `` structure sheaf algebras'' in many--in point of fact, in all--the situations of a geometrical character that…
In this paper, we give a geometrization and a generalization of a lemma of differential Galois theory. This geometrization, in addition of giving a nice insight on this result, offers us the occasion to investigate several points of…
We define two isomorphic algebras of differential operators: the first algebra consists of ordinary differential operators and contains the hypergeometric differential operator, while the second one consists of partial differential…
We review the multivariate holomorphic functional calculus for tuples in a commutative Banach algebra and establish a simple "na\"ive" extension to commuting tuples in a general Banach algebra. The approach is na\"ive in the sense that the…
This is the first out of two papers on independent resolutions for totally disconnected dynamical systems. In the present paper, we discuss independent resolutions from an algebraic point of view. We also present applications to group…
In [1], an operator was introduced which acts parallel to the Riemann-Liouville differintegral on a transformation of the space of real analytic functions and commutes with itself. This paper aims to extend the technique - and its defining…
We study symplectic linear algebra over the ring $\Rt$ of Colombeau generalized numbers. Due to the algebraic properties of $\Rt$ it is possible to preserve a number of central results of classical symplectic linear algebra. In particular,…
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…
A new method is presented for assigning distributional curvature, in an invariant manner, to a space-time of low differentiability, using the techniques of Colombeau's `new generalised functions'. The method is applied to show that…
In this paper we connect classical differential geometry with the concepts from geometric calculus. Moreover, we introduce and analyze a more general Laplacian for multivector-valued functions on manifolds. This allows us to formulate a…
In [Discrete differential calculus on simplicial complexes and constrained homology, Chin. Ann. Math. Ser. B 44(4), 615-640, 2023], the constrained (co)homology for simplicial complexes and independence hypergraphs is constructed via…
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…
In this paper, we continue to develop the theory of free holomorphic functions on noncommutative regular polydomains. We find analogues of several classical results from complex analysis such as Abel theorem, Hadamard formula, Cauchy…
We show how one can construct a differential calculus over an algebra where position variables x and momentum variables p have be defined. As the simplest example we consider the one-dimensional q-deformed Heisenberg algebra. This algebra…
There is a deformation of the ordinary differential calculus which leads from the continuum to a lattice (and induces a corresponding deformation of physical theories). We recall some of its features and relate it to a general framework of…
We have established a coherent framework for applying variational methods to partial differential equations on hypergraphs, which includes the propositions of calculus and function spaces on hypergraphs. Several results related to the…