Over the past few years a certain shift of focus within the theory of algebras of generalized functions (in the sense of J. F. Colombeau) has taken place. Originating in infinite dimensional analysis and initially applied mainly to problems in nonlinear partial differential equations involving singularities, the theory has undergone a change both in in ternal structure and scope of applicability, due to a growing number of applications to questions of a more geometric nature. The present book is intended to provide an in-depth presentation of these develop ments comprising its structural aspects within the theory of generalized functions as well as a (selective but, as we hope, representative) set of applications. This main purpose of the book is accompanied by a number of sub ordinate goals which we were aiming at when arranging the material included here. First, despite the fact that by now several excellent mono graphs on Colombeau algebras are available, we have decided to give a self-contained introduction to the field in Chapter 1. Our motivation for this decision derives from two main features of our approach. On the one hand, in contrast to other treatments of the subject we base our intro duction to the field on the so-called special variant of the algebras, which makes many of the fundamental ideas of the field particularly transpar ent and at the same time facilitates and motivates the introduction of the more involved concepts treated later in the chapter.
This main purpose of the book is accompanied by a number of sub ordinate goals which we were aiming at when arranging the material included here.
Provides a more cohesive and sharply focused treatment of fundamental concepts and theoretical background material, with particular attention given to better delineating connections to varying applications Exposition driven by additional ...
With a focus on applications, this work provides a compilation of approaches to the problem of singularities in nonlinear models.
... Proceedings of the Conference in Honor of te 90th Birthday of Freeman Dyson, World Scientific, 2014, pp. 148–172. M. Kontsevich, D. Zagier, Periods, in:Mathematics - 20101 and beyond, B. Engquist, W. Schmid, eds., Springer, Berlin et al ...
The work is profusely illustrated with diagrams aiding in the understandingof proofs and explanations.
5 Conclusion and Perspectives In this work we considered fractional diffusion equation using Caputo–Fabrizio operator. ... Ge, F., Chen, Y.Q., Kou, C.: Regional boundary controllability of time fractional diffusion processes.
This volume offers an integrated understanding of how the theory of general relativity gained momentum after Einstein had formulated it in 1915.
This book is also suitable as an introduction to pseudo-Riemannian geometry with emphasis on geometrical concepts. A significant part of the text is devoted to the discussion of causality and singularity theorems.
... Applications. Kluwer, Dordrecht (1988) 4. Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R.: Geometric Theory of Generalized Functions with Applications to General Relativity. Kluwer, Dordrecht (2001) 5. Hrusheuski, U ...
This is involved at a most elementary level in the very definition of the notion of hyperbolicity for an Euler-Lagrange system of partial differential equations.