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＜電子ブック＞
Random Fields and Geometry / by R. J. Adler, Jonathan E. Taylor
(Springer Monographs in Mathematics. ISSN:21969922)

版	1st ed. 2007.
出版者	(New York, NY : Springer New York : Imprint: Springer)
出版年	2007
本文言語	英語
大きさ	XVIII, 454 p. 21 illus : online resource
著者標目	*Adler, R. J author Taylor, Jonathan E author SpringerLink (Online service)
件　名	LCSH:Probabilities LCSH:Statistics  LCSH:Geometry LCSH:Mathematical physics FREE:Probability Theory FREE:Statistics FREE:Geometry FREE:Mathematical Methods in Physics
一般注記	Gaussian Processes -- Gaussian Fields -- Gaussian Inequalities -- Orthogonal Expansions -- Excursion Probabilities -- Stationary Fields -- Geometry -- Integral Geometry -- Differential Geometry -- Piecewise Smooth Manifolds -- Critical Point Theory -- Volume of Tubes -- The Geometry of Random Fields -- Random Fields on Euclidean Spaces -- Random Fields on Manifolds -- Mean Intrinsic Volumes -- Excursion Probabilities for Smooth Fields -- Non-Gaussian Geometry This monograph is devoted to a completely new approach to geometric problems arising in the study of random fields. The groundbreaking material in Part III, for which the background is carefully prepared in Parts I and II, is of both theoretical and practical importance, and striking in the way in which problems arising in geometry and probability are beautifully intertwined. The three parts to the monograph are quite distinct. Part I presents a user-friendly yet comprehensive background to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness, entropy and majorizing measures, Borell and Slepian inequalities. Part II gives a quick review of geometry, both integral and Riemannian, to provide the reader with the material needed for Part III, and to give some new results and new proofs of known results along the way. Topics such as Crofton formulae, curvature measures for stratified manifolds, critical point theory, and tube formulae are covered. In fact, this is the only concise, self-contained treatment of all of the above topics, which are necessary for the study of random fields. The new approach in Part III is devoted to the geometry of excursion sets of random fields and the related Euler characteristic approach to extremal probabilities. "Random Fields and Geometry" will be useful for probabilists and statisticians, and for theoretical and applied mathematicians who wish to learn about new relationships between geometry and probability. It will be helpful for graduate students in a classroom setting, or for self-study. Finally, this text will serve as a basic reference for all those interested in the companion volume of the applications of the theory. These applications, to appear in a forthcoming volume, will cover areas as widespread as brain imaging, physical oceanography, and astrophysics HTTP:URL=https://doi.org/10.1007/978-0-387-48116-6

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