scholarly journals A Convenient Notion of Compact Set for Generalized Functions

2017 ◽  
Vol 61 (1) ◽  
pp. 57-92
Author(s):  
Paolo Giordano ◽  
Michael Kunzinger
Keyword(s):  
Function Spaces ◽  
Generalized Functions ◽  
Distribution Theory ◽  
Compact Set ◽  
Compact Sets ◽  
Smooth Functions ◽  
Test Function ◽  
Compactly Supported ◽  
Nonlinear Generalized Functions

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 smooth functions on compact sets into the framework of generalized functions. Based on this concept, we introduce spaces of compactly supported generalized smooth functions that are close analogues to the test function spaces of distribution theory. We then develop the topological and functional–analytic foundations of these spaces.

scholarly journals A Global Universality of Two-Layer Neural Networks with ReLU Activations

2021 ◽  
Vol 2021 ◽  
pp. 1-3
Author(s):  
Naoya Hatano ◽  
Masahiro Ikeda ◽  
Isao Ishikawa ◽  
Yoshihiro Sawano
Keyword(s):  
Neural Networks ◽  
Global Convergence ◽  
Function Spaces ◽  
Compact Set ◽  
Compact Sets

In the present study, we investigate a universality of neural networks, which concerns a density of the set of two-layer neural networks in function spaces. There are many works that handle the convergence over compact sets. In the present paper, we consider a global convergence by introducing a norm suitably, so that our results will be uniform over any compact set.

The Mehler-Fock-Clifford transform and pseudo-differential operator on function spaces

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2457-2469
Author(s):  
Akhilesh Prasad ◽  
S.K. Verma
Keyword(s):  
Differential Operator ◽  
Function Spaces ◽  
Pseudo Differential Operator ◽  
Test Function ◽  
Basic Properties ◽  
Cone Function ◽  
Translation Operators

In this article, weintroduce a new index transform associated with the cone function Pi ??-1/2 (2?x), named as Mehler-Fock-Clifford transform and study its some basic properties. Convolution and translation operators are defined and obtained their estimates under Lp(I, x-1/2 dx) norm. The test function spaces G? and F? are introduced and discussed the continuity of the differential operator and MFC-transform on these spaces. Moreover, the pseudo-differential operator (p.d.o.) involving MFC-transform is defined and studied its continuity between G? and F?.

A generalization of the Pompeiu mean-value theorem to compact sets

2019 ◽  
Vol 35 (2) ◽  
pp. 147-152
Author(s):  
LARISA CHEREGI ◽  
VICUTA NEAGOS ◽  
◽  
Keyword(s):  
Continuous Function ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Compact Set ◽  
Compact Sets

We generalize the Pompeiu mean-value theorem by replacing the graph of a continuous function with a compact set.

On the foundations of nonlinear generalized functions I and II

2001 ◽  
Vol 153 (729) ◽  
pp. 0-0 ◽  
Author(s):  
Michael Grosser ◽  
Eva Farkas ◽  
Michael Kunzinger ◽  
Roland Steinbauer
Keyword(s):  
Generalized Functions ◽  
Nonlinear Generalized Functions

A Class of Spaces in Which Compact Sets are Finite

1981 ◽  
Vol 24 (3) ◽  
pp. 373-375 ◽  
Author(s):  
P. L. Sharma
Keyword(s):  
Hausdorff Space ◽  
Compact Set ◽  
Compact Sets

AbstractIt is shown that in a dense-in-itself Hausdorff space if every set having a dense interior is open, then every compact set is finite.

scholarly journals Spaces of Whitney Functions on Cantor-Type Sets

2002 ◽  
Vol 54 (2) ◽  
pp. 225-238 ◽  
Author(s):  
Bora Arslan ◽  
Alexander P. Goncharov ◽  
Mefharet Kocatepe
Keyword(s):  
Extension Property ◽  
Compact Set ◽  
Compact Sets ◽  
Whitney Functions ◽  
Type Sets

AbstractWe introduce the concept of logarithmic dimension of a compact set. In terms of this magnitude, the extension property and the diametral dimension of spaces Ɛ(K) can be described for Cantor-type compact sets.

GENERALISED PSEUDO-RIEMANNIAN GEOMETRY FOR GENERAL RELATIVITY

2002 ◽  
Vol 17 (20) ◽  
pp. 2776-2776
Author(s):  
R. STEINBAUER ◽  
M. KUNZINGER
Keyword(s):  
General Relativity ◽  
Riemannian Geometry ◽  
Rapid Development ◽  
Generalized Functions ◽  
Field Equations ◽  
Mathematical Methods ◽  
Colombeau Algebras ◽  
Rigorous Treatment ◽  
General Relativistic ◽  
Nonlinear Generalized Functions

The study of singular spacetimes by distributional methods faces the fundamental obstacle of the inherent nonlinearity of the field equations. Staying strictly within the distributional (in particular: linear) regime, as determined by Geroch and Traschen2 excludes a number of physically interesting examples (e.g., cosmic strings). In recent years, several authors have therefore employed nonlinear theories of generalized functions (Colombeau algebras, in particular) to tackle general relativistic problems1,5,8. Under the influence of these applications in general relativity the nonlinear theory of generalized functions itself has undergone a rapid development lately, resulting in a diffeomorphism invariant global theory of nonlinear generalized functions on manifolds3,4,6. In particular, a generalized pseudo-Riemannian geometry allowing for a rigorous treatment of generalized (distributional) spacetime metrics has been developed7. It is the purpose of this talk to present these new mathematical methods themselves as well as a number of applications in mathematical relativity.

scholarly journals The classical theory of calculus of variations for generalized functions

2017 ◽  
Vol 8 (1) ◽  
pp. 779-808 ◽  
Author(s):  
Alexander Lecke ◽  
Lorenzo Luperi Baglini ◽  
Paolo Giordano
Keyword(s):  
Calculus Of Variations ◽  
Classical Theory ◽  
Sufficient Conditions ◽  
Generalized Functions ◽  
Conjugate Points ◽  
Lagrange Equations ◽  
Smooth Functions ◽  
Nonlinear Properties ◽  
Schwartz Distributions ◽  
Necessary And Sufficient

Abstract We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions, while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and Euler–Lagrange equations, classical necessary and sufficient conditions to have a minimizer, the necessary Legendre condition, Jacobi’s theorem on conjugate points and Noether’s theorem. We close with an application to low regularity Riemannian geometry.

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