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The Kinematic Formula in Riemannian Homogeneous Spaces

1993 Mathematics
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Release : 1993
Genre : Mathematics
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Book Rating : 690/5 ( reviews)

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Book Synopsis The Kinematic Formula in Riemannian Homogeneous Spaces by : Ralph Howard

Download or read book The Kinematic Formula in Riemannian Homogeneous Spaces written by Ralph Howard. This book was released on 1993. Available in PDF, EPUB and Kindle. Book excerpt: This memoir investigates a method that generalizes the Chern-Federer kinematic formula to arbitrary homogeneous spaces with an invariant Riemannian metric, and leads to new formulas even in the case of submanifolds of Euclidean space.

Kinematic Formula in Riemannian Homogeneous Spaces

2014-08-31 MATHEMATICS
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Release : 2014-08-31
Genre : MATHEMATICS
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Book Rating : 866/5 ( reviews)

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Book Synopsis Kinematic Formula in Riemannian Homogeneous Spaces by : D H Howard Howard

Download or read book Kinematic Formula in Riemannian Homogeneous Spaces written by D H Howard Howard. This book was released on 2014-08-31. Available in PDF, EPUB and Kindle. Book excerpt: This book shows that much of classical integral geometry can be derived from the coarea formula by some elementary techniques. Howard generalizes much of classical integral geometry from spaces of constant sectional curvature to arbitrary Riemannian homogeneous spaces. To do so, he provides a general definition of an integral invariant'' of a submanifold of the space that is sufficiently general enough to cover most cases that arise in integral geometry. Working in this generality makes it clear that the type of integral geometric formulas that hold in a space does not depend on the full group of isometries, but only on the isotropy subgroup. As a special case, integral geometric formulas that hold in Euclidean space also hold in all the simply connected spaces of constant curvature. Detailed proofs of the results and many examples are included. Requiring background of a one-term course in Riemannian geometry, this book may be used as a textbook in graduate courses on differential and integral geometry.

Littlewood-Paley Theory on Spaces of Homogeneous Type and the Classical Function Spaces

1994 Mathematics
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Release : 1994
Genre : Mathematics
Kind : eBook
Book Rating : 925/5 ( reviews)

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Book Synopsis Littlewood-Paley Theory on Spaces of Homogeneous Type and the Classical Function Spaces by : Yongsheng Han

Download or read book Littlewood-Paley Theory on Spaces of Homogeneous Type and the Classical Function Spaces written by Yongsheng Han. This book was released on 1994. Available in PDF, EPUB and Kindle. Book excerpt: In this work, Han and Sawyer extend Littlewood-Paley theory, Besov spaces, and Triebel-Lizorkin spaces to the general setting of a space of homogeneous type. For this purpose, they establish a suitable analogue of the Calder 'on reproducing formula and use it to extend classical results on atomic decomposition, interpolation, and T1 and Tb theorems. Some new results in the classical setting are also obtained: atomic decompositions with vanishing b-moment, and Littlewood-Paley characterizations of Besov and Triebel-Lizorkin spaces with only half the usual smoothness and cancellation conditions on the approximate identity.

Shortest Paths for Sub-Riemannian Metrics on Rank-Two Distributions

1995 Mathematics
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Release : 1995
Genre : Mathematics
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Book Rating : 049/5 ( reviews)

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Book Synopsis Shortest Paths for Sub-Riemannian Metrics on Rank-Two Distributions by : Wensheng Liu

Download or read book Shortest Paths for Sub-Riemannian Metrics on Rank-Two Distributions written by Wensheng Liu. This book was released on 1995. Available in PDF, EPUB and Kindle. Book excerpt: A sub-Riemannian manifold ([italic capitals]M, E, G) consists of a finite-dimensional manifold [italic capital]M, a rank-two bracket generating distribution [italic capital]E on [italic capital]M, and a Riemannian metric [italic capital]G on [italic capital]E. All length-minimizing arcs on ([italic capitals]M, E, G) are either normal extremals or abnormal extremals. Normal extremals are locally optimal, i.e., every sufficiently short piece of such an extremal is a minimizer. The question whether every length-minimizer is a normal extremal was recently settled by R. G. Montgomery, who exhibited a counterexample. The present work proves that regular abnormal extremals are locally optimal, and, in the case that [italic capital]E satisfies a mild additional restriction, the abnormal minimizers are ubiquitous rather than exceptional. All the topics of this research report (historical notes, examples, abnormal extremals, Hamiltonians, nonholonomic distributions, sub-Riemannian distance, the relations between minimality and extremality, regular abnormal extremals, local optimality of regular abnormal extremals, etc.) are presented in a very clear and effective way.

Stochastic Models, Information Theory, and Lie Groups, Volume 2

2011-11-15 Mathematics
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Release : 2011-11-15
Genre : Mathematics
Kind : eBook
Book Rating : 433/5 ( reviews)

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Book Synopsis Stochastic Models, Information Theory, and Lie Groups, Volume 2 by : Gregory S. Chirikjian

Download or read book Stochastic Models, Information Theory, and Lie Groups, Volume 2 written by Gregory S. Chirikjian. This book was released on 2011-11-15. Available in PDF, EPUB and Kindle. Book excerpt: This unique two-volume set presents the subjects of stochastic processes, information theory, and Lie groups in a unified setting, thereby building bridges between fields that are rarely studied by the same people. Unlike the many excellent formal treatments available for each of these subjects individually, the emphasis in both of these volumes is on the use of stochastic, geometric, and group-theoretic concepts in the modeling of physical phenomena. Stochastic Models, Information Theory, and Lie Groups will be of interest to advanced undergraduate and graduate students, researchers, and practitioners working in applied mathematics, the physical sciences, and engineering. Extensive exercises, motivating examples, and real-world applications make the work suitable as a textbook for use in courses that emphasize applied stochastic processes or differential geometry.

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