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Analysis of Spherical Symmetries in Euclidean Spaces

Author: Claus Müller

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 227

ISBN-13: 1461205816

This self-contained book offers a new and direct approach to the theories of special functions with emphasis on spherical symmetry in Euclidean spaces of arbitrary dimensions. Based on many years of lecturing to mathematicians, physicists and engineers in scientific research institutions in Europe and the USA, the author uses elementary concepts to present the spherical harmonics in a theory of invariants of the orthogonal group. One of the highlights is the extension of the classical results of the spherical harmonics into the complex - particularly important for the complexification of the Funk-Hecke formula which successfully leads to new integrals for Bessel- and Hankel functions with many applications of Fourier integrals and Radon transforms. Numerous exercises stimulate mathematical ingenuity and bridge the gap between well-known elementary results and their appearance in the new formations.

Analysis of Spherical Symmetries in Euclidean Spaces

Author: Claus Muller

Publisher:

Published: 1997-11-01

Total Pages: 240

ISBN-13: 9781461205821

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Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

Author: Audrey Terras

Publisher: Springer Science & Business Media

Published: 2013-09-12

Total Pages: 430

ISBN-13: 146147972X

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This unique text is an introduction to harmonic analysis on the simplest symmetric spaces, namely Euclidean space, the sphere, and the Poincaré upper half plane. This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. Written with an informal style, the book places an emphasis on motivation, concrete examples, history, and, above all, applications in mathematics, statistics, physics, and engineering. Many corrections and updates have been incorporated in this new edition. Updates include discussions of P. Sarnak and others' work on quantum chaos, the work of T. Sunada, Marie-France Vignéras, Carolyn Gordon, and others on Mark Kac's question "Can you hear the shape of a drum?", A. Lubotzky, R. Phillips and P. Sarnak's examples of Ramanujan graphs, and, finally, the author's comparisons of continuous theory with the finite analogues. Topics featured throughout the text include inversion formulas for Fourier transforms, central limit theorems, Poisson's summation formula and applications in crystallography and number theory, applications of spherical harmonic analysis to the hydrogen atom, the Radon transform, non-Euclidean geometry on the Poincaré upper half plane H or unit disc and applications to microwave engineering, fundamental domains in H for discrete groups Γ, tessellations of H from such discrete group actions, automorphic forms, and the Selberg trace formula and its applications in spectral theory as well as number theory.

Introduction to Fourier Analysis on Euclidean Spaces

Author: Elias M. Stein

Publisher: Princeton University Press

Published: 1971-11-21

Total Pages: 318

ISBN-13: 9780691080789

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The authors present a unified treatment of basic topics that arise in Fourier analysis. Their intention is to illustrate the role played by the structure of Euclidean spaces, particularly the action of translations, dilatations, and rotations, and to motivate the study of harmonic analysis on more general spaces having an analogous structure, e.g., symmetric spaces.

Spherical Harmonics and Approximations on the Unit Sphere: An Introduction

Author: Kendall Atkinson

Publisher: Springer Science & Business Media

Published: 2012-02-17

Total Pages: 253

ISBN-13: 3642259820

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These notes provide an introduction to the theory of spherical harmonics in an arbitrary dimension as well as an overview of classical and recent results on some aspects of the approximation of functions by spherical polynomials and numerical integration over the unit sphere. The notes are intended for graduate students in the mathematical sciences and researchers who are interested in solving problems involving partial differential and integral equations on the unit sphere, especially on the unit sphere in three-dimensional Euclidean space. Some related work for approximation on the unit disk in the plane is also briefly discussed, with results being generalizable to the unit ball in more dimensions.

Robust Cluster Analysis and Variable Selection

Author: Gunter Ritter

Publisher: CRC Press

Published: 2014-09-02

Total Pages: 389

ISBN-13: 1439857970

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Clustering remains a vibrant area of research in statistics. Although there are many books on this topic, there are relatively few that are well founded in the theoretical aspects. This book presents an overview of the theory and applications of probabilistic clustering and variable selection, synthesizing the key research results of the last 50 years. It includes all the important theoretical details, and covers the probabilistic models and inference, robustness issues, optimization algorithms, validation techniques and variable selection methods. The book illustrates the different methods with simulated data and applies them to real-world data sets that can be easily downloaded from the web.

Scale Space and Variational Methods in Computer Vision

Author: Xue-Cheng Tai

Publisher: Springer Science & Business Media

Published: 2009-05-25

Total Pages: 882

ISBN-13: 3642022553

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This book constitutes the refereed proceedings of the Second International Conference on Scale Space Methods and Variational Methods in Computer Vision, SSVM 2009, emanated from the joint edition of the 5th International Workshop on Variational, Geometric and Level Set Methods in Computer Vision, VLSM 2009 and the 7th International Conference on Scale Space and PDE Methods in Computer Vision, Scale-Space 2009, held in Voss, Norway in June 2009. The 71 revised full papers presented were carefully reviewed and selected numerous submissions. The papers are organized in topical sections on segmentation and detection; image enhancement and reconstruction; motion analysis, optical flow, registration and tracking; surfaces and shapes; scale space and feature extraction.

Emerging Trends in Knowledge Discovery and Data Mining

Author: Takashi Washio

Publisher: Springer

Published: 2013-02-13

Total Pages: 168

ISBN-13: 364236778X

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This book constitutes the thoroughly refereed proceedings of the PAKDD 2012 International Workshops: Third Workshop on Data Mining for Healthcare Management (DMHM 2012), First Workshop on Geospatial Information and Documents (GeoDoc 2012), First Workshop on Multi-view data, High-dimensionality, External Knowledge: Striving for a Unified Approach to Clustering (3Clust 2012), and the Second Doctoral Symposium on Data Mining (DSDM 2012); held in conjunction with the 16th Pacific-Asia Conference on Knowledge Discovery and Data Mining (PAKDD 2012), in Kuala Lumpur, Malaysia, May/June 2012. The 12 revised papers presented were carefully reviewed and selected from numerous submissions. DMHM 2012 aimed at providing a common platform for the discussion of challenging issues and potential techniques in this emerging field of data mining for health care management; 3Clust 2012 focused on solving emerging problems such as clustering ensembles, semi-supervised clustering, subspace/projective clustering, co-clustering, and multi-view clustering; GeoDoc 2012 highlighted the formalization of geospatial concepts and relationships with a focus on the extraction of geospatial relations in free text datasets to offer to the database community a unified framework for geodata discovery; and DSDM 2012 provided the opportunity for Ph.D. students and junior researchers to discuss their work on data mining foundations, techniques and applications.

Offbeat Integral Geometry on Symmetric Spaces

Author: Valery V. Volchkov

Publisher: Springer Science & Business Media

Published: 2013-01-30

Total Pages: 596

ISBN-13: 3034805721

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The book demonstrates the development of integral geometry on domains of homogeneous spaces since 1990. It covers a wide range of topics, including analysis on multidimensional Euclidean domains and Riemannian symmetric spaces of arbitrary ranks as well as recent work on phase space and the Heisenberg group. The book includes many significant recent results, some of them hitherto unpublished, among which can be pointed out uniqueness theorems for various classes of functions, far-reaching generalizations of the two-radii problem, the modern versions of the Pompeiu problem, and explicit reconstruction formulae in problems of integral geometry. These results are intriguing and useful in various fields of contemporary mathematics. The proofs given are “minimal” in the sense that they involve only those concepts and facts which are indispensable for the essence of the subject. Each chapter provides a historical perspective on the results presented and includes many interesting open problems. Readers will find this book relevant to harmonic analysis on homogeneous spaces, invariant spaces theory, integral transforms on symmetric spaces and the Heisenberg group, integral equations, special functions, and transmutation operators theory.

Harmonic Analysis in Euclidean Spaces, Part 2

Author: Guido Weiss

Publisher: American Mathematical Soc.

Published: 1979

Total Pages: 448

ISBN-13: 0821814389

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Contains sections on Several complex variables, Pseudo differential operators and partial differential equations, Harmonic analysis in other settings: probability, martingales, local fields, and Lie groups and functional analysis.