| Description |
1 online resource (355 pages). |
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text file |
| Series |
London Mathematical Society Lecture Note Series, 389 ; v. 389
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London Mathematical Society lecture note series ; 389.
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| Contents |
Cover; Title; Copyright; Contents; Dedication; Preface; 1 Introduction; 1.1 Overview; 1.2 Cosmological motivations; 1.3 Mathematical framework; 1.4 Plan of the book; 2 Background Results in Representation Theory; 2.1 Introduction; 2.2 Preliminary remarks; 2.3 Groups: basic definitions; 2.3.1 First definitions and examples; 2.3.2 Cosets and quotients; 2.3.3 Actions; 2.4 Representations of compact groups; 2.4.1 Basic definitions; 2.4.2 Group representations and Schur Lemma; 2.4.3 Direct sum and tensor product representations; 2.4.4 Orthogonality relations; 2.5 The Peter-Weyl Theorem. |
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Machine generated contents note: 1. Introduction -- 1.1. Overview -- 1.2. Cosmological motivations -- 1.3. Mathematical framework -- 1.4. Plan of the book -- 2. Background Results in Representation Theory -- 2.1. Introduction -- 2.2. Preliminary remarks -- 2.3. Groups: basic definitions -- 2.4. Representations of compact groups -- 2.5. Peter-Weyl Theorem -- 3. Representations of SO(3) and Harmonic Analysis on S2 -- 3.1. Introduction -- 3.2. Euler angles -- 3.3. Wigner's D matrices -- 3.4. Spherical harmonics and Fourier analysis on S2 -- 3.5. Clebsch-Gordan coefficients -- 4. Background Results in Probability and Graphical Methods -- 4.1. Introduction -- 4.2. Brownian motion and stochastic calculus -- 4.3. Moments, cumulants and diagram formulae -- 4.4. simplified method of moments on Wiener chaos -- 4.5. graphical method for Wigner coefficients -- 5. Spectral Representations -- 5.1. Introduction -- 5.2. Stochastic Peter-Weyl Theorem -- 5.3. Weakly stationary random fields in Rm -- 5.4. Stationarity and weak isotropy in R3 -- 6. Characterizations of Isotropy -- 6.1. Introduction -- 6.2. First example: the cyclic group -- 6.3. spherical harmonics coefficients -- 6.4. Group representations and polyspectra -- 6.5. Angular polyspectra and the structure of δl1 ... l1 -- 6.6. Reduced polyspectra of arbitrary orders -- 6.7. Some examples -- 7. Limit Theorems for Gaussian Subordinated Random Fields -- 7.1. Introduction -- 7.2. First example: the circle -- 7.3. Preliminaries on Gaussian-subordinated fields -- 7.4. High-frequency CLTs -- 7.5. Convolutions and random walks -- 7.6. Further remarks -- 7.7. Application: algebraic/exponential dualities -- 8. Asymptotics for the Sample Power Spectrum -- 8.1. Introduction -- 8.2. Angular power spectrum estimation -- 8.3. Interlude: some practical issues -- 8.4. Asymptotics in the non-Gaussian case -- 8.5. quadratic case -- 8.6. Discussion -- 9. Asymptotics for Sample Bispectra -- 9.1. Introduction -- 9.2. Sample bispectra -- 9.3. central limit theorem -- 9.4. Limit theorems under random normalizations -- 9.5. Testing for non-Gaussianity -- 10. Spherical Needlets and their Asymptotic Properties -- 10.1. Introduction -- 10.2. construction of spherical needlets -- 10.3. Properties of spherical needlets -- 10.4. Stochastic properties of needlet coefficients -- 10.5. Missing observations -- 10.6. Mexican needlets -- 11. Needlets Estimation of Power Spectrum and Bispectrum -- 11.1. Introduction -- 11.2. general convergence result -- 11.3. Estimation of the angular power spectrum -- 11.4. functional central limit theorem -- 11.5. central limit theorem for the needlets bispectrum -- 12. Spin Random Fields -- 12.1. Introduction -- 12.2. Motivations -- 12.3. Geometric background -- 12.4. Spin needlets and spin random fields -- 12.5. Spin needlets spectral estimator -- 12.6. Detection of asymmetries -- 12.7. Estimation with noise -- 13. Appendix -- 13.1. Orthogonal polynomials -- 13.2. Spherical harmonics and their analytic properties -- 13.3. proof of needlets' localization. |
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3 Representations of SO(3) and Harmonic Analysis on S23.1 Introduction; 3.2 Euler angles; 3.2.1 Euler angles for SU(2); 3.2.2 Euler angles for SO(3); 3.3 Wigner's D matrices; 3.3.1 A family of unitary representations of SU(2); 3.3.2 Expressions in terms of Euler angles and irreducibility; 3.3.3 Further properties; 3.3.4 The dual of SO(3); 3.4 Spherical harmonics and Fourier analysis on S2; 3.4.1 Spherical harmonics and Wigner's Dl matrices; 3.4.2 Some properties of spherical harmonics; 3.4.3 An alternative characterization of spherical harmonics; 3.5 The Clebsch-Gordan coefficients. |
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3.5.1 Clebsch-Gordan matrices3.5.2 Integrals of multiple spherical harmonics; 3.5.3 Wigner 3 j coefficients; 4 Background Results in Probability and Graphical Methods; 4.1 Introduction; 4.2 Brownian motion and stochastic calculus; 4.3 Moments, cumulants and diagram formulae; 4.4 The simplified method of moments on Wiener chaos; 4.4.1 Real kernels; 4.4.2 Further results on complex kernels; 4.5 The graphical method for Wigner coefficients; 4.5.1 From diagrams to graphs; 4.5.2 Further notation; 4.5.3 First example: sums of squares; 4.5.4 Cliques and Wigner 6 j coefficients. |
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4.5.5 Rule n. 1: loops are zero4.5.6 Rule n. 2: paired sums are one; 4.5.7 Rule n. 3: 2-loops can be cut, and leave a factor; 4.5.8 Rule n. 4: three-loops can be cut, and leave a clique; 5 Spectral Representations; 5.1 Introduction; 5.2 The Stochastic Peter-Weyl Theorem; 5.2.1 General statements; 5.2.2 Decompositions on the sphere; 5.3 Weakly stationary random fields in Rm; 5.4 Stationarity and weak isotropy in R3; 6 Characterizations of Isotropy; 6.1 Introduction; 6.2 First example: the cyclic group; 6.3 The spherical harmonics coefficients; 6.4 Group representations and polyspectra. |
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6.5 Angular polyspectra and the structure of?l1 ... ln6.5.1 Spectra of strongly isotropic fields; 6.5.2 The structure of?l1 ... ln; 6.6 Reduced polyspectra of arbitrary orders; 6.7 Some examples; 7 Limit Theorems for Gaussian Subordinated Random Fields; 7.1 Introduction; 7.2 First example: the circle; 7.3 Preliminaries on Gaussian-subordinated fields; 7.4 High-frequency CLTs; 7.4.1 Hermite subordination; 7.5 Convolutions and random walks; 7.5.1 Convolutions on?SO (3); 7.5.2 The cases q = 2 and q = 3; 7.5.3 The case of a general q: results and conjectures; 7.6 Further remarks. |
| Note |
7.6.1 Convolutions as mixed states. |
| Summary |
Reviews recent developments in the analysis of isotropic spherical random fields, with a view towards applications in cosmology. |
| Bibliography |
Includes bibliographical references (pages 326-337) and index. |
| Local Note |
eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - North America |
| Subject |
Compact groups.
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Compact groups. |
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Cosmology -- Statistical methods.
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Cosmology. |
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Statistics. |
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Random fields.
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Random fields. |
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Spherical harmonics.
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Spherical harmonics. |
| Genre/Form |
Electronic books.
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| Added Author |
Peccati, Giovanni, 1975-
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| Other Form: |
Print version: Marinucci, Domenico. Random Fields on the Sphere : Representation, Limit Theorems and Cosmological Applications. Cambridge : Cambridge University Press, ©2011 9780521175616 |
| ISBN |
9781139117487 |
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1139117483 |
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1283296179 |
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9781283296175 |
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9780511751677 (electronic book) |
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0511751672 (electronic book) |
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9781139128148 (electronic book) |
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1139128140 (electronic book) |
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1139115316 (electronic book) |
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9781139115315 (electronic book) |
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0521175615 |
|
9780521175616 |
|
