On the Foundations of Nonlinear Generalized Functions I and II

Part 1. On the Foundations of Nonlinear Generalized Functions I: Introduction Notation and terminology Scheme of construction Calculus C- and J-formalism Calculus on $U_\varepsilon(\Omega)$ Construction of a diffeomorphism invariant Colombeau algebra Sheaf properties Separating the basic definition from testing Characterization results Differential equations Part 2. On the Foundations of Nonlinear Generalized Functions II: Introduction to Part 2 A simple condition equivalent to negligibility Some more calculus Non-injectivity of the canonical homomorphism from ${\mathcal G}^d(\Omega)$ into ${\mathcal G}^e(\Omega)$ Classification of smooth Colombeau algebras between ${\mathcal G}^d(\Omega)$ and ${\mathcal G}^e(\Omega)$ The algebra ${\mathcal G}^2$; classification results Concluding remarks Acknowledgments Bibliography