109 research outputs found
Non-smooth differential geometry and algebras of generalized functions
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 this field and highlights some applications in mathematical
physics.Comment: 17 pages, typos correcte
Group analysis of differential equations and generalized functions
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 Colombeau's theory of algebras of generalized functions. We show
that under some mild conditions on the differential equations, symmetries of
classical solutions remain symmetries for generalized solutions. Moreover, we
introduce a generalization of the infinitesimal methods of group analysis that
allows to compute symmetries of linear and nonlinear differential equations
containing generalized function terms. Thereby, the group generators and group
actions may be given by generalized functions themselves.Comment: 27 pages, LaTe
Microlocal properties of basic operations in Colombeau algebras
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 the algebra. Furthermore, we give explicit examples showing that
the given result is optimal, i.e. its assumptions cannot be weakened. Finally,
we discuss the interrelation of these results with the concept of pullback
under smooth maps.Comment: LaTeX, 18 page
Generalized Group Actions in a Global Setting
We study generalized group actions on differentiable manifolds in the
Colombeau framework, extending previous work on flows of generalized vector
fields and symmetry group analysis of generalized solutions. As an application,
we analyze group invariant generalized functions in this setting
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