On the Foundations of Nonlinear Generalized Functions I and II
The main drawback of the Colombeau generalised functions is that the canonical embedding of the space of Schwartz distributions into the algebra (and sheaf) of generalised functions is not intrinsic. This canonical imbedding is not preserved by coordinate diffeomorphisms, so generalised functions cannot be defined on a manifold although the distributions can. How to remove this inconvenience by a slight change of definition of the Colombeau generalised functions was outlined by Colombeau and Meril in 1994 and elaborated by Jelínek in 1999.
In the first part, the Colombeau-type algebra defined by Jelínek (called diffeomorphism invariant) is carefully examined, and its description is corrected and completed. Colombeau-type algebra of generalised functions is defined as a quotient algebra G = EM / I, where EM is an algebra of so-called moderate representatives and I is an ideal of negligible representatives: some equivalent definitions of these notions are presented. In the second part, other ways of defining moderateness and negligibility are studied and only two diffeomorphism invariant algebras are found among them. The assumption that they are different is neither proved nor disproved.