相关论文: Local Functions : Algebras, Ideals, and Reduced Po…

We give an overview of the development of algebras of generalized functions in the sense of Colombeau and recent advances concerning diffeomorphism invariant global algebras of generalized functions and tensor fields. We furthermore provide…

We present a differential algebra of generalized functions over a field of generalized scalars by means of several axioms in terms of general algebra and topology. Our differential algebra is of Colombeau type in the sense that it contains…

A differential algebra of nonlinear generalized functions is presented as a tool for a wide range of nonsmooth nonlinear problems. The power of the differential algebra is used to do mathematical calculations or proofs; then the final…

We illustrate the use of internal objects in the nonlinear theory of generalized functions by means of an application to microlocal analysis in Colombeau algebras.

Generalized Functions play a central role in the understanding of differential equations containing singularities and nonlinearities. Introducing infinitesimals and infinities to deal with these obstructions leads to controversies…

We construct a diffeomorphism invariant (Colombeau-type) differential algebra canonically containing the space of distributions in the sense of L. Schwartz. Employing differential calculus in infinite dimensional (convenient) vector spaces,…

Modelling of singularities given by discontinuous functions or distributions by means of generalized functions has proved useful in many problems posed by physical phenomena. We introduce in a systematic way generalized functions of…

We investigate density of various subalgebras of regular generalized functions in the special Colombeau algebra of generalized functions.

In this paper we review the extent to which one can use classical distribution theory in describing solutions of Einstein's equations. We show that there are a number of physically interesting cases which cannot be treated using…

Starting from a description of various generalized function algebras based on sequence spaces, we develop the general framework for considering linear problems with singular coefficients or non linear problems. Therefore, we prove…

This is a gentle introduction to Colombeau nonlinear generalized functions, a generalization of the concept of distributions such that distributions can freely be multiplied. It is intended to physicists and applied mathematicians who…

We develop the diffeomorphism invariant Colombeau-type algebra of nonlinear generalized functions in a modern and compact way. Using a unifying formalism for the local setting and on manifolds, the construction becomes simpler and more…

We extend the Colombeau algebra of generalized functions to arbitrary (infinitely differentiable, paracompact) n-dimensional manifolds M. Embedding of continuous functions and distributions is achieved with the help of a family of n-forms…

We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of {\em Colombeau type} in the sense that it contains a copy of the space of Schwartz…

Starting from the Colombeau's full generalized functions, the sharp topologies and the notion of generalized points, we introduce a new kind differential calculus (for functions between totally disconnected spaces). We study generalized…

We present an extension of the methods of classical Lie group analysis of differential equations to equations involving generalized functions (in particular: distributions). A suitable framework for such a generalization is provided by…

The Colombeau algebra of generalized functions allows to unrestrictedly carry out products of distributions. We analyze this operation from a microlocal point of view, deriving a general inclusion relation for wave front sets of products in…

Recently the space-time foam differential algebras of generalized functions with dense singularities were introduced, motivated by the so called space-time foam structures in General Relativity with dense singularities, and by Quantum…

Algebras of generalized functions offer possibilities beyond the purely distributional approach in modelling singular quantities in non-smooth differential geometry. This article presents an introductory survey of recent developments in…

In this paper, weakly homogeneous generalized functions in the special Colombeau algebras are determined up to equality in the sense of generalized distributions. This yields characterizations that are formally similar to distribution…