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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."
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."
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