Related papers: Topics on nonlinear generalized functions
The object of this lecture is to propose a series of conjectures and problems in different fields of analysis. They have been formulated with the aim of introducing some innovative methods in the study of classical topics, as open mappings,…
We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties enjoyed by standard…
We introduce the space of grid functions, a space of generalized functions of nonstandard analysis that provides a coherent generalization both of the space of distributions and of the space of Young measures. We will show that in the space…
An approach to generalize any kind of collinear functionals in density functional theory to non-collinear functionals is proposed. This approach, for the very first time, satisfies the correct collinear limit for any kind of functionals,…
Someone knowledgeable in nonstandard analysis may get the feeling that in the nonlinear theory of generalized functions, too often one works directly on the nets and spends effort to obtain results that should be clear from general…
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…
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…
We present numerical techniques based on generalized functions adapted to nonlinear calculations. They concern main numerical engineering problems ruled by-or issued from-nonlinear equations of continuum mechanics. The aim of this text is…
We present in this paper the motivation and theory of nonlinear spectral representations, based on convex regularizing functionals. Some comparisons and analogies are drawn to the fields of signal processing, harmonic analysis and sparse…
In this paper we introduce Hausdorff locally convex algebra topologies on subalgebras of the whole algebra of nonlinear generalized functions. These topologies are strong duals of Fr\'echet-Schwartz space topologies and even strong duals of…
We expose some simple facts at the interplay between mathematics and the real world, putting in evidence mathematical objects " nonlinear generalized functions" that are needed to model the real world, which appear to have been generally…
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…
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…
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…
This article introduces a novel nonparametric methodology for Generalized Linear Models which combines the strengths of the binary regression and latent variable formulations for categorical data, while overcoming their disadvantages.…
Some concepts, such as non-compactness measure and condensing operators, defined on metric spaces are extended to uniform spaces. Such extensions allow us to locate, in the context of uniform spaces, some classical results existing in…
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…
This application of nonstandard analysis utilizes the notion of the highly-staturated enlargement. These nonstandard methods clarify many aspects of the theory of generalized functions (distributions).
This paper lays the foundations for a nonlinear theory of differential geometry that is developed in a subsequent paper which is based on Colombeau algebras of tensor distributions on manifolds. We adopt a new approach and construct a…
The present article is devoted to the generalized Salem functions, the generailed shift operator, and certain related problems. A description of further investigations of the author of this article is given.These investigations (in terms of…