Inversion of Colombeau generalized functions

We introduce different notions of invertibility for generalized functions in the sense of Colombeau. Several necessary conditions for (left, right) invertibility are derived, giving rise to the concepts of compactly asymptotic injectivity and surjectivity. We analyse the extent to which these properties are also sufficient to guarantee the existence of a (left, right) inverse of a generalized function. Finally, we establish several Inverse Function Theorems in this setting and study the relation to their classical counterparts.

Algebras of generalized functions with smooth parameter dependence

We show that spaces of Colombeau generalized functions with smooth parameter dependence are isomorphic to those with continuous parametrization. Based on this result we initiate a systematic study of algebraic properties of the ring $\tilde{\mathbb{K}}_\mathrm{sm}$ of generalized numbers in this unified setting. In particular, we investigate the ring and order structure of $\tilde{\mathbb{K}}_\mathrm{sm}$ and establish some properties of its ideals.

Algebraic and Geometric Theory of the Topological Ring of Colombeau Generalized Functions

We continue the investigation of the algebraic and topological structure of the algebra of Colombeau generalized functions with the aim of building up the algebraic basis for the theory of these functions. This was started in a previous work of Aragona and Juriaans, where the algebraic and topological structure of the Colombeau generalized numbers were studied. Here, among other important things, we determine completely the minimal primes of and introduce several invariants of the ideals of (Ω). The main tools we use are the algebraic results obtained by Aragona and Juriaans and the theory of differential calculus on generalized manifolds developed by Aragona and co-workers. The main achievement of the differential calculus is that all classical objects, such as distributions, become C∞-functions. Our purpose is to build an independent and intrinsic theory for Colombeau generalized functions and place them in a wider context.