Related papers: A Discontinuous Differential Calculus in the Frame…
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…
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 present a geometric approach to defining an algebra $\hat{\mathcal G}(M)$ (the Colombeau algebra) of generalized functions on a smooth manifold $M$ containing the space ${\mathcal D}'(M)$ of distributions on $M$. Based on differential…
This article addresses linear hyperbolic partial differential equations with non-smooth coefficients and distributional data. Solutions are studied in the framework of Colombeau algebras of generalized functions. Its aim is to prove upper…
In this paper, we establish a suitable version of the Hahn-Banach theorem within the framework of Colombeau spaces, a class of spaces used to model generalized functions. Our approach addresses the case where maps are defined…
We tackle the problem of finding a suitable categorical framework for generalized functions used in mathematical physics for linear and non-linear PDEs. We are looking for a Cartesian closed category which contains both Schwartz…
We introduce a novel formulation for geometry on discrete points. It is based on a universal differential calculus, which gives a geometric description of a discrete set by the algebra of functions. We expand this mathematical framework so…
It is often noted that many of the basic concepts of differential geometry, such as the definition of connection, are purely algebraic in nature. Here, we review and extend existing work on fully algebraic formulations of differential…
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…
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…
We present an extension of J. F. Colombeau's theory of nonlinear generalized functions to spaces of generalized sections of vector bundles. Our construction builds on classical functional analytic notions, which is the key to having a…
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…
Based on the theory of Fermat reals we introduce new topologies on spaces of Colombeau generalized points and derive some of their fundamental properties. In particular, we obtain metric topologies on the space of near-standard generalized…
Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic…
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…
We introduce the notion of generalized function taking values in a smooth manifold into the setting of full Colombeau algebras. After deriving a number of characterization results we also introduce a corresponding concept of generalized…
Co lombeau's construction of generalized functions (in its special variant) is extended to a theory of generalized sections of vector bundles. As particular cases, generalized tensor analysis and exterior algebra are studied. A point value…
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 $\hat{\mathcal G}(M)$ on the manifold $M$ given in (Grosser…
Inspired by nonstandard analysis, we define and study internal subsets and internal functions in algebras of Colombeau generalized functions. We prove a saturation principle for internal sets and provide applications to Colombeau algebras.
The concept of generalized functions taking values in a differentiable manifold is extended to a functorial theory. We establish several characterization results which allow a global intrinsic formulation both of the theory of…