scholarly journals Nonlinear generalized functions on manifolds

2020 ◽  
Vol 476 (2244) ◽  
pp. 20200640
Author(s):  
E. A. Nigsch ◽  
J. A. Vickers
Keyword(s):  
Covariant Derivative ◽  
Scalar Fields ◽  
Generalized Functions ◽  
Lie Derivative ◽  
Subsequent Paper ◽  
Schwartz Distributions ◽  
Colombeau Algebras ◽  
Algebras Of Generalized Functions ◽  
Tensor Distributions ◽  
Distributional Geometry

In this work, we adopt a new approach to the construction of a global theory of algebras of generalized functions on manifolds based on the concept of smoothing operators. This produces a generalization of previous theories in a form which is suitable for applications to differential geometry. The generalized Lie derivative is introduced and shown to extend the Lie derivative of Schwartz distributions. A new feature of this theory is the ability to define a covariant derivative of generalized scalar fields which extends the covariant derivative of distributions at the level of association. We end by sketching some applications of the theory. This work also lays the foundations for a nonlinear theory of distributional geometry that is developed in a subsequent paper that is based on Colombeau algebras of tensor distributions on manifolds.

scholarly journals A nonlinear theory of distributional geometry

2020 ◽  
Vol 476 (2244) ◽  
pp. 20200642 ◽  
Author(s):  
E. A. Nigsch ◽  
J. A. Vickers
Keyword(s):  
Differential Geometry ◽  
Nonlinear Theory ◽  
Delta Function ◽  
Generalized Functions ◽  
Lie Derivative ◽  
Tensor Fields ◽  
Generalized Metric ◽  
Nonlinear Generalized Functions ◽  
Distributional Geometry ◽  
Conical Metric

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.

scholarly journals Regular generalized solutions to semilinear wave equations

2020 ◽  
Author(s):  
Hideo Deguchi ◽  
Michael Oberguggenberger
Keyword(s):  
Wave Equations ◽  
Uniqueness Result ◽  
Generalized Functions ◽  
Generalized Solutions ◽  
Classical Solutions ◽  
Colombeau Algebra ◽  
Semilinear Wave Equations ◽  
Global Classical Solutions ◽  
Colombeau Algebras ◽  
Algebras Of Generalized Functions

Abstract The paper is devoted to proving an existence and uniqueness result for generalized solutions to semilinear wave equations with a small nonlinearity in space dimensions 1, 2, 3. The setting is the one of Colombeau algebras of generalized functions. It is shown that for a nonlinearity of arbitrary growth and sign, but multiplied with a small parameter, the initial value problem for the semilinear wave equation has a unique solution in the Colombeau algebra of generalized functions of bounded type. The proof relies on a fixed point theorem in the ultra-metric topology on the algebras involved. In classical terms, the result says that the semilinear wave equations under consideration have global classical solutions up to a rapidly vanishing error.

Vanishing Theorems in Colombeau Algebras of Generalized Functions

2008 ◽  
Vol 51 (4) ◽  
pp. 618-626 ◽  
Author(s):  
V. Valmorin
Keyword(s):  
Integral Operators ◽  
Generalized Functions ◽  
Vanishing Theorems ◽  
Smooth Functions ◽  
Linear Maps ◽  
Colombeau Algebras ◽  
Algebras Of Generalized Functions ◽  
Linear Embedding ◽  
Linear Integral ◽  
Linear Integral Operators

AbstractUsing a canonical linear embedding of the algebra of Colombeau generalized functions in the space of -valued ℂ-linear maps on the space of smooth functions with compact support, we give vanishing conditions for functions and linear integral operators of class . These results are then applied to the zeros of holomorphic generalized functions in dimension greater than one.

scholarly journals An embedding of Schwartz distributions in the algebra of asymptotic functions

1998 ◽  
Vol 21 (3) ◽  
pp. 417-428 ◽  
Author(s):  
Michael Oberguggenberger ◽  
Todor Todorov
Keyword(s):  
Differential Algebra ◽  
Generalized Functions ◽  
Schwartz Distributions ◽  
Asymptotic Functions ◽  
Space Of Distributions ◽  
Asymptotic Function

We present a solution of the problem of multiplication of Schwartz distributions by embedding the space of distributions into a differential algebra of generalized functions, called in the paper “asymptotic function,” similar to but different from J. F Colombeau's algebras of new generalized functions.

scholarly journals Algebras of generalized functions with smooth parameter dependence

2012 ◽  
Vol 55 (1) ◽  
pp. 105-124 ◽  
Author(s):  
Annegret Burtscher ◽  
Michael Kunzinger
Keyword(s):  
Systematic Study ◽  
Generalized Functions ◽  
Parameter Dependence ◽  
Order Structure ◽  
Colombeau Generalized Functions ◽  
Algebras Of Generalized Functions ◽  
Algebraic Properties

AbstractWe show that spaces of Colombeau generalized functions with smooth parameter dependence are isomorphic to those with continuous parametrization. Based on this result we initiate a systematic study of algebraic properties of the ring $\tilde{\mathbb{K}}_\mathrm{sm}$ of generalized numbers in this unified setting. In particular, we investigate the ring and order structure of $\tilde{\mathbb{K}}_\mathrm{sm}$ and establish some properties of its ideals.

scholarly journals On Products of Distributions in Colombeau Algebra

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Marija Miteva ◽  
Biljana Jolevska-Tuneska ◽  
Tatjana Atanasova-Pacemska
Keyword(s):  
Differential Algebra ◽  
Generalized Functions ◽  
Colombeau Algebra ◽  
Schwartz Distributions ◽  
Products Of Distributions ◽  
Renormalization Theory

Results on products of distributionsx+-kandδ(p)(x)are derived. They are obtained in Colombeau differential algebra𝒢(R)of generalized functions that contains the space𝒟'(R)of Schwartz distributions as a subspace. Products of this form are useful in quantum renormalization theory in Physics.

Sign in / Sign up

Export Citation Format

Share Document