Full and Special Colombeau Algebras

Grosser, Michael; Nigsch, Eduard A.

doi:10.1017/s001309151800010x

Cited by 8 publications

(15 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This can be done by following the ideas of the full Colombeau algebra (see e.g. [48,44,43,49]). Nevertheless, this choice decreases the simplicity of the present approach and is incompatible with properties like (3.10) and (3.11).…”

Section: Embedding Of Schwartz Distributionsmentioning

confidence: 99%

A Grothendieck topos of generalized functions I: basic theory

Giordano¹,

Kunzinger²,

Vernaeve³

2021

Preprint

View full text Add to dashboard Cite

The main aim of the present work is to arrive at a mathematical theory near to the historically original conception of generalized functions, i.e. set theoretical functions defined on, and with values in, a suitable ring of scalars and sharing a number of fundamental properties with smooth functions, in particular with respect to composition and nonlinear operations. This is as they are still used in informal calculations in Physics. We introduce a category of generalized functions as smooth set-theoretical maps on (multidimensional) points of a ring of scalars containing infinitesimals and infinities. This category extends Schwartz distributions. The calculus of these generalized functions is closely related to classical analysis, with point values, composition, non-linear operations and the generalization of several classical theorems of calculus. In this first paper, we present the basic theory; in subsequent ones the theory of ODE and PDE. Contents 1. Introduction: foundations of generalized functions as set-theoretical maps 2 2. The ring of scalars and its topologies 3 2.1. Topologies on R n 6 2.2. Open, closed and bounded sets generated by nets 8 3. Generalized functions as smooth set-theoretical maps 11 4. Differential calculus and the Fermat-Reyes theorem 25 5. Integral calculus using primitives 29 6. Some classical theorems for generalized smooth functions 33 7. Multidimensional integration and hyperlimits 39 7.1. Integration over functionally compact sets 39 7.2. Hyperfinite limits 42 7.3. Properties of multidimensional integral 45 8. Sheaf properties 48 8.1. The Lebesgue generalized number 50 8.2. The dynamic compatibility condition 51 8.3. Proof of the sheaf property 52

show abstract

Section: Embedding Of Schwartz Distributionsmentioning

confidence: 99%

A Grothendieck topos of generalized functions I: basic theory

Giordano¹,

Kunzinger²,

Vernaeve³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Our approach is based on Colombeau's space of generalized functions [Col84;Col85]. However, we incorporate in our construction several extensions which were developed during the last couple of years: a functional analytic formulation [Nig15], which led to a geometrization of the theory [Nig16a]; a representation of smoothing operators for vector distributions [Nig16b]; and a study of locality conditions, which are used to obtain the sheaf property [GN18]. Only through the combination of these results the scope of the theory of nonlinear generalized functions can be extended to tensor fields in sufficient generality while at the same time it is concrete enough such that calculations with singular can be performed.…”

Section: Generalized Sectionsmentioning

confidence: 99%

“…e key to obtaining the sheaf property of the quotient will be to employ socalled locality conditions (first introduced in [Nig15] and studied in more detail in [GN18]). Again, in this whole subsection we fix a manifold M, a vector bundle E → M and a homogeneous functor λ.…”

Section: E Sheaf Propertymentioning

confidence: 99%

Spacetimes with distributional semi-Riemannian metrics and their curvature

Nigsch

2020

Journal of Geometry and Physics

View full text Add to dashboard Cite

show abstract

“…Note that for the sake of presentation we completely omit discussion of the sheaf property; to obtain it we actually would have to restrict the basic space to a somewhat smaller one. For details, we refer to [31,34].…”

Section: The Algebra Of Generalized Tensor Fieldsmentioning

confidence: 99%

A nonlinear theory of distributional geometry

Nigsch

Vickers

2020

Proc. R. Soc. A.

Self Cite

View full text Add to dashboard Cite

This paper builds on the theory of nonlinear generalized functions begun in Nigsch & Vickers (Nigsch, Vickers 2021 Proc. R. Soc. A 20200640 ( doi:10.1098/rspa.2020.0640 )) and extends this to a diffeomorphism-invariant nonlinear theory of generalized tensor fields with the sheaf property. The generalized Lie derivative is introduced and shown to commute with the embedding of distributional tensor fields and the generalized covariant derivative commutes with the embedding at the level of association. The concept of a generalized metric is introduced and used to develop a non-smooth theory of differential geometry. It is shown that the embedding of a continuous metric results in a generalized metric with well-defined connection and curvature and that for C 2 metrics the embedding preserves the curvature at the level of association. Finally, we consider an example of a conical metric outside the Geroch–Traschen class and show that the curvature is associated to a delta function.

show abstract

Full and Special Colombeau Algebras

Cited by 8 publications

References 27 publications

A Grothendieck topos of generalized functions I: basic theory

A Grothendieck topos of generalized functions I: basic theory

Spacetimes with distributional semi-Riemannian metrics and their curvature

A nonlinear theory of distributional geometry

Contact Info

Product

Resources

About