We show spherical completeness of the ring of Colombeau generalized real numbers endowed with the sharp norm. As an application, we establish a Hahn-Banach extension theorem for ultra-pseudo-normed modules of generalized functions in the sense of Colombeau.

Formulas for the solutions of initial value problems for ordinary differential equations with singular δ-like driving terms are derived in the framework of an algebra of generalized functions (of Colombeau type) over a field of generalized scalars. Some of the solutions might have physical meaning such as of the electrical current after lightning or under superconductivity but do not have count...

A new approach to the algebra Gτ of temperate nonlinear generalized functions is proposed, in which Gτ is based on the space OM endowed with is natural topology in contrary to previous constructions. Thus, this construction fits perfectly in the general scheme of construction of Colombeau type algebras and reveals better properties of Gτ . This is illustrated by the natural introduction of a re...

We define the algebra G̃(A) of Colombeau generalized functions on a subset A of the space of generalized points R̃d. If A is an open subset of R̃d, such generalized functions can be identified with pointwise maps from A into the ring of generalized numbers C̃. We study analyticity in G̃(A), where A is an open subset of C̃. In particular, if the domain is an open ball for the sharp norm on C̃, we chara...

Solutions of the wave equation in a space-time containing a thin cosmic string are examined in the context of non-linear generalised functions. Existence and uniqueness of solutions to the wave equation in the Colombeau algebra G is established for a conical space-time and this solution is shown to be associated to a distributional solution. A concept of generalised hyperbolicity, based on test...

We show that contrary to recent papers by S. Albeverio, A. Yu. Khrennikov and V. Shelkovich, point values do not determine elements of the so-called p-adic Colombeau-Egorov algebra G(Q p ) uniquely. We further show in a more general way that for an Egorov algebra G(M,R) of generalized functions on a locally compact ultrametric space (M, d) taking values in a nontrivial ring, a point value chara...

Inspired by nonstandard analysis, we define and study internal subsets and internal functions in algebras of Colombeau generalized functions. We prove a saturation principle for internal sets and provide applications to Colombeau algebras.

The electromagnetic scattering of a spin-0 charged particle off a fixed center is calculated in first-order quantum perturbation theory. This implies evaluating the square of a ‘Dirac delta-function,’ an operation that is not defined in Schwartz distribution theory, and which in elementary text-books is dealt with according to ‘Fermi’s golden rule.’ In this paper these conventional calculations...

‎The present paper is devoted to the existence and uniqueness result of the fractional evolution equation $D^{q}_c u(t)=g(t,u(t))=Au(t)+f(t)$‎ ‎for the real $qin (0,1)$ with the initial value $u(0)=u_{0}intilde{R}$‎, ‎where $tilde{R}$ is the set of all generalized real numbers and $A$ is an operator defined from $mathcal G$ into itself‎. Here the Caputo fractional derivative $D^{q}_c$ is used i...