Parametric Lie Group Actions on Global Generalised Solutions of Nonlinear PDEs (Paperback)

Elemer E. Rosinger

ISBN 10: 9048150930 / ISBN 13: 9789048150939

Editore: Springer, Dordrecht, 2010

Paperback. This book presents global actions of arbitrary Lie groups on large classes of generalised functions by using a novel parametric approach. This new method extends and completes earlier results of the author and collaborators, in which global Lie group actions on generalised functions were only defined in the case of projectable or fibre-preserving Lie group actions. The parametric method opens the possibility of dealing with vastly larger classes of Lie semigroup actions which still transform solutions into solutions. These Lie semigroups can contain arbitrary noninvertible smooth mappings. Thus, they cannot be subsemigroups of Lie groups. Audience: This volume is addressed to graduate students and researchers involved in solving linear and nonlinear partial differential equations, and in particular, in dealing with the Lie group symmetries of their classical or generalised solutions. This book presents a solution of the harder part of the problem of defining globally arbitrary Lie group actions on such nonsmooth entities as generalised functions. Earlier, in part 3 of Oberguggenberger & Rosinger, Lie group actions were defined globally - in the projectable case - on the nowhere dense differential algebras of generalised functions An, as well as on the Colombeau algebras of generalised functions, and also on the spaces obtained through the order completion of smooth functions, spaces which contain the solutions of arbitrary continuous nonlinear PDEs. Further details can be found in Rosinger & Rudolph, and Rosinger & Walus [1,2]. To the extent that arbitrary Lie group actions are now defined on such nonsmooth entities as generalised functions, this result can be seen as giving an ans wer to Hilbert's fifth problem, when this problem is interpreted in its original full generA ality, see for details chapter 11 Shipping may be from multiple locations in the US or from the UK, depending on stock availability. Codice articolo 9789048150939

Riassunto:

This book presents global actions of arbitrary Lie groups on large classes of generalised functions by using a novel parametric approach. This new method extends and completes earlier results of the author and collaborators, in which global Lie group actions on generalised functions were only defined in the case of projectable or fibre-preserving Lie group actions.
The parametric method opens the possibility of dealing with vastly larger classes of Lie semigroup actions which still transform solutions into solutions. These Lie semigroups can contain arbitrary noninvertible smooth mappings. Thus, they cannot be subsemigroups of Lie groups.
Audience: This volume is addressed to graduate students and researchers involved in solving linear and nonlinear partial differential equations, and in particular, in dealing with the Lie group symmetries of their classical or generalised solutions.

Recensione:

E.E. Rosinger

Parametric Lie Group Actions on Global Generalised Solutions of Nonlinear PDEs

Including a Solution to Hilbert’s Fifth Problem

"This book presents a novel approach to Lie group actions on ordinary and generalized functions, based on parametric representation. This allows a global definition of arbitrary nonlinear Lie group actions on functions, including generalized functions. The parametric approach also makes possible a global definition for Lie semigroup actions. It is shown that the usual Lie group symmetries of classical solutions of smooth nonlinear PDEs will remain Lie group symmetries of generalized solutions of such equations."

―MATHEMATICAL REVIEWS

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Titolo: Parametric Lie Group Actions on Global ...
Casa editrice: Springer, Dordrecht
Data di pubblicazione: 2010
Legatura: Paperback
Condizione: new
Edizione: prima edizione