Related papers: Regularity of nonlinear generalized functions: a c…
We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz…
We present new types of regularity for nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of Colombeau's simplified model. This generalizes the notion of G^{\infty }-regularity…
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…
We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions. We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from…
We consider distributions on a closed compact manifold $M$ as maps on smoothing operators. Thus spaces of certain maps between $\Psi^{-\infty}(M)\to \mathcal{C}^{\infty}(M)$ are considered as generalized functions. For any collection of…
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…
In these lecture notes we present an introduction to non-standard analysis especially written for the community of mathematicians, physicists and engineers who do research on J. F. Colombeau' theory of new generalized functions and its…
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…
This paper gives a comprehensive analysis of algebras of Colombeau-type generalized functions in the range between the diffeomorphism-invariant quotient algebra $\mathcal{G}^d = \mathcal{E}_M/\mathcal{N}$ introduced in part I and…
We investigate density of various subalgebras of regular generalized functions in the special Colombeau algebra of generalized functions.
We define the algebra of Colombeau generalized functions on the space of generalized points of {\mathbb R}^d which naturally contains the tempered generalized functions. The subalgebra of \mathscr{S}-regular generalized functions of this…
In this review article we present regularity properties of generalized functions which are useful in the analysis of non-linear problems. It is shown that Schwartz distributions embedded into our new spaces of generalized functions, with…
Colombeau algebras constitute a convenient framework for performing nonlinear operations like multiplication on Schwartz distributions. Many variants and modifications of these algebras exist for various applications. We present a…
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…
We introduce an intrinsic notion of Hoelder-Zygmund regularity for Colombeau generalized functions. In case of embedded distributions belonging to some Zygmund-Hoelder space this is shown to be consistent. The definition is motivated by the…
As the title ``Generalized regularity and solution concepts for differential equations'' suggests, the main topic of my thesis is the investigation of generalized solution concepts for differential equations, in particular first order…
The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new…
This paper is a tutorial that demonstrates various methods from the Colombeau theory of generalized functions in the context of semilinear wave equations. The Colombeau generalized functions constitute differential algebras that contain the…
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…
Colombeau generalized functions invariant under smooth (additive) one-parameter groups are characterized. This characterization is applied to generalized functions invariant under orthogonal groups of arbitrary signature, such as groups of…