Diffeomorphism invariant Colombeau algebras. Part II: Classification
نویسنده
چکیده
This contribution presents a comprehensive analysis of Colombeau (-type) algebras in the range between the diffeomorphism invariant algebra G = E M / N d introduced in Part I (see [Gro01]) and Colombeau’s original algebra G introduced in [Col85]. Along the way, it provides several classification results (again see [Gro01]) which are indispensable for obtaining an intrinsic description of a (full) Colombeau algebra on a manifold ([Gro99]). The latter will be the focus of Part III of this series of contributions.
منابع مشابه
Diffeomorphism invariant Colombeau algebras. Part I: Local theory
This contribution is the first in a series of three: it reports on the construction of (a fine sheaf of) diffeomorphism invariant Colombeau algebras G(Ω) = EM (Ω)/N (Ω) on open sets of Eucildean space ([Gro01]), which completes earlier approaches ([Col94, Jel99]). Part II and III will show, among others, the way to an intrinsic definition of Colombeau algebras on manifolds which, locally, repro...
متن کاملOn the foundations of nonlinear generalized functions II
This paper gives a comprehensive analysis of algebras of Colombeau-type generalized functions in the range between the diffeomorphism-invariant quotient algebra G = EM/N introduced in part I and Colombeau’s original algebra G . Three main results are established: First, a simple criterion describing membership in N (applicable to all types of Colombeau algebras) is given. Second, two counterexa...
متن کاملA geometric approach to full Colombeau algebras
We present a geometric approach to diffeomorphism invariant full Colombeau algebras which allows a particularly clear view on the construction of the intrinsically defined algebra Ĝ(M) on the manifold M given in [8]. MSC 2000: Primary: 46T30; secondary: 46F30.
متن کاملDiffeomorphism invariant Colombeau algebras. Part III: Global theory
We present the construction of an associative, commutative algebra Ĝ(X) of generalized functions on a manifold X satisfying the following optimal set of permanence properties: (i) D(X) is linearly embedded into Ĝ(X), f(p) ≡ 1 is the unity in Ĝ(X). (ii) For every smooth vector field ξ on X there exists a derivation operator L̂ξ : Ĝ(X) → Ĝ(X) which is linear and satisfies the Leibniz rule. (iii) L...
متن کاملNew ideas about multiplication of tensorial distributions
There is a huge need in general relativity for a consistent and useful mathematical theory defining the multiplication of tensor distributions in a geometric (diffeomorphism invariant) way. Significant progress has been made through the concept of Colombeau algebras, and the construction of full Colombeau algebras on differential manifolds for arbitrary tensors. Despite the fact that this goal ...
متن کامل