Memoirs of the American Mathematical Society 2001; 93 pp; softcover Volume: 153 ISBN10: 0821827294 ISBN13: 9780821827291 List Price: US$54 Individual Members: US$32.40 Institutional Members: US$43.20 Order Code: MEMO/153/729
 In part 1 we construct a diffeomorphism invariant (Colombeautype) differential algebra canonically containing the space of distributions in the sense of L. Schwartz. Employing differential calculus in infinite dimensional (convenient) vector spaces, previous attempts in this direction are unified and completed. Several classification results are achieved and applications to nonlinear differential equations involving singularities are given. Part 2 gives a comprehensive analysis of algebras of Colombeautype generalized functions in the range between the diffeomorphisminvariant quotient algebra \({\mathcal G}^d = {\mathcal E}_M/{\mathcal N}\) introduced in part 1 and Colombeau's original algebra \({\mathcal G}^e\). Three main results are established: First, a simple criterion describing membership in \({\mathcal N}\) (applicable to all types of Colombeau algebras) is given. Second, two counterexamples demonstrate that \({\mathcal G}^d\) is not injectively included in \({\mathcal G}^e\). Finally, it is shown that in the range "between" \({\mathcal G}^d\) and \({\mathcal G}^e\) only one more construction leads to a diffeomorphism invariant algebra. In analyzing the latter, several classification results essential for obtaining an intrinsic description of \({\mathcal G}^d\) on manifolds are derived. Readership Graduate students and research mathematicians interested in functional analysis. Table of Contents Part 1. On the Foundations of Nonlinear Generalized Functions I  Introduction
 Notation and terminology
 Scheme of construction
 Calculus
 C and Jformalism
 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
 Noninjectivity 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
