Colombeau products of distributions
In this paper, some products of distributions are derived. The results are obtained in Colombeau algebra of generalized functions, which is the most relevant algebraic construction for dealing with Schwartz distributions. Colombeau algebra $${\mathcal{G}}({\mathbf{R}})$$G(R) contains the space $${\mathcal{D}'}({\mathbf{R}})$$D′(R) of Schwartz distributions as a subspace, and has a notion of 'association’ that allows us to evaluate the results in terms of distributions.
A concise introduction to Colombeau generalized functions and their applications in classical electrodynamics
The objective of this introduction to Colombeau algebras of generalized functions (in which distributions can be freely multiplied) is to explain in elementary terms the essential concepts necessary for their application to basic nonlinear problems in classical physics. Examples are given in hydrodynamics and electrodynamics. The problem of the self-energy of a point electric charge is worked out in detail: the Coulomb potential and field are defined as Colombeau generalized functions, and integrals of nonlinear expressions corresponding to products of distributions (such as the square of the Coulomb field and the square of the delta function) are calculated. Finally, the methods introduced in Gsponer (2007 Eur. J. Phys. 28 267, 2007 Eur. J. Phys. 28 1021 and 2007 Eur. J. Phys. 28 1241), to deal with point-like singularities in classical electrodynamics are confirmed.
On a class of valuation fields introduced by A. Robinson
It is shown that the nonarchimedean valuation fieldsρR introduced by A. Robinson are not only complete but are also spherically complete. Further-more, it is shown that to every normed linear space over the reals there exists a nonarchimedean normed linear spaceρE overρR in the sense of Monna which is spherically complete and extendsE.