Geometry of Sets and Measures in Euclidean Spaces PDF Download

Author: Pertti Mattila
Publisher: Cambridge University Press
ISBN: 9780521655958
Category : Mathematics
Languages : en
Pages : 360

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Book Description
This book studies the geometric properties of general sets and measures in euclidean space.

Author: Pertti Mattila
Publisher:
ISBN:
Category :
Languages : en
Pages :

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Author: Frank Jones
Publisher: Jones & Bartlett Learning
ISBN: 9780763717087
Category : Computers
Languages : en
Pages : 626

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Book Description
"'Lebesgue Integration on Euclidean Space' contains a concrete, intuitive, and patient derivation of Lebesgue measure and integration on Rn. It contains many exercises that are incorporated throughout the text, enabling the reader to apply immediately the new ideas that have been presented" --

Author: Jan Rataj
Publisher: Springer
ISBN: 3030181839
Category : Mathematics
Languages : en
Pages : 256

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Book Description
The book describes how curvature measures can be introduced for certain classes of sets with singularities in Euclidean spaces. Its focus lies on sets with positive reach and some extensions, which include the classical polyconvex sets and piecewise smooth submanifolds as special cases. The measures under consideration form a complete system of certain Euclidean invariants. Techniques of geometric measure theory, in particular, rectifiable currents are applied, and some important integral-geometric formulas are derived. Moreover, an approach to curvatures for a class of fractals is presented, which uses approximation by the rescaled curvature measures of small neighborhoods. The book collects results published during the last few decades in a nearly comprehensive way.

Author: K. J. Falconer
Publisher: Cambridge University Press
ISBN: 9780521337052
Category : Mathematics
Languages : en
Pages : 184

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Book Description
A mathematical study of the geometrical aspects of sets of both integral and fractional Hausdorff dimension. Considers questions of local density, the existence of tangents of such sets as well as the dimensional properties of their projections in various directions.

Author: Steven G. Krantz
Publisher: Springer Science & Business Media
ISBN: 0817646795
Category : Mathematics
Languages : en
Pages : 344

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Book Description
This textbook introduces geometric measure theory through the notion of currents. Currents, continuous linear functionals on spaces of differential forms, are a natural language in which to formulate types of extremal problems arising in geometry, and can be used to study generalized versions of the Plateau problem and related questions in geometric analysis. Motivating key ideas with examples and figures, this book is a comprehensive introduction ideal for both self-study and for use in the classroom. The exposition demands minimal background, is self-contained and accessible, and thus is ideal for both graduate students and researchers.

Author: Steven G. Krantz
Publisher: Springer Science & Business Media
ISBN: 1461215749
Category : Mathematics
Languages : en
Pages : 311

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Book Description
The analysis of Euclidean space is well-developed. The classical Lie groups that act naturally on Euclidean space-the rotations, dilations, and trans lations-have both shaped and guided this development. In particular, the Fourier transform and the theory of translation invariant operators (convolution transforms) have played a central role in this analysis. Much modern work in analysis takes place on a domain in space. In this context the tools, perforce, must be different. No longer can we expect there to be symmetries. Correspondingly, there is no longer any natural way to apply the Fourier transform. Pseudodifferential operators and Fourier integral operators can playa role in solving some of the problems, but other problems require new, more geometric, ideas. At a more basic level, the analysis of a smoothly bounded domain in space requires a great deal of preliminary spadework. Tubular neighbor hoods, the second fundamental form, the notion of "positive reach", and the implicit function theorem are just some of the tools that need to be invoked regularly to set up this analysis. The normal and tangent bundles become part of the language of classical analysis when that analysis is done on a domain. Many of the ideas in partial differential equations-such as Egorov's canonical transformation theorem-become rather natural when viewed in geometric language. Many of the questions that are natural to an analyst-such as extension theorems for various classes of functions-are most naturally formulated using ideas from geometry.

Author: Pertti Mattila
Publisher: Cambridge University Press
ISBN: 1107107350
Category : Mathematics
Languages : en
Pages : 455

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Book Description
Modern text examining the interplay between measure theory and Fourier analysis.

Author: Francesco Maggi
Publisher:
ISBN: 9781107471726
Category :
Languages : en
Pages : 454

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Book Description

Author: Francesco Maggi
Publisher: Cambridge University Press
ISBN: 1139560891
Category : Mathematics
Languages : en
Pages : 475

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Book Description
The marriage of analytic power to geometric intuition drives many of today's mathematical advances, yet books that build the connection from an elementary level remain scarce. This engaging introduction to geometric measure theory bridges analysis and geometry, taking readers from basic theory to some of the most celebrated results in modern analysis. The theory of sets of finite perimeter provides a simple and effective framework. Topics covered include existence, regularity, analysis of singularities, characterization and symmetry results for minimizers in geometric variational problems, starting from the basics about Hausdorff measures in Euclidean spaces and ending with complete proofs of the regularity of area-minimizing hypersurfaces up to singular sets of codimension 8. Explanatory pictures, detailed proofs, exercises and remarks providing heuristic motivation and summarizing difficult arguments make this graduate-level textbook suitable for self-study and also a useful reference for researchers. Readers require only undergraduate analysis and basic measure theory.