In distribution theory the pullback of a general distribution by a C∞function is well-defined whenever the normal bundle of the C∞-function does not intersect the wavefront set of the distribution. However, the Colombeau theory of generalized functions allows for a pullback by an arbitrary c-bounded generalized function. It has been shown in previous work that in the case of multiplication of C...

Based on the theory of Fermat reals we introduce new topologies on spaces of Colombeau generalized points and derive some of their fundamental properties. In particular, we obtain metric topologies on the space of near-standard generalized points that induce the standard Euclidean topology on the reals. We also give a new description of the sharp topology in terms of the natural extension of th...

The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new notions of ellipticity and hypoellipticity, study their interrelation, and give a number of new examples and counterexamples. Using the concept of G∞-regularity of...

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 non-linear problems in classical physics. Examples are given in hydrodynamics and electrodynamics. The problem of the selfenergy of a point electric charge is worked o...

In this paper, the structure of the ideals in the ring of Colombeau generalized numbers is investigated. Connections with the theories of exchange rings, Gelfand rings and lattice-ordered rings are given. Characterizations for prime, projective, pure and topologically closed ideals are given, answering in particular the questions about prime ideals in [1]. Also z-ideals in the sense of [21] are...

We give characterizations for generalized functions invariant under certain group actions, such as groups of rotations and the Lorentz group. In the case of rotations, it answers an open question by M. Oberguggenberger in [3]. Also general one-parameter groups are considered.