What is included with this book?
Preface | p. V |
Gaussian Processes | |
Gaussian Fields | p. 7 |
Random Fields | p. 7 |
Gaussian Variables and Fields | p. 8 |
Boundedness and Continuity | p. 11 |
Examples | p. 20 |
Fields on R[superscript N] | p. 20 |
Differentiability on R[superscript N] | p. 22 |
The Brownian Family of Processes | p. 24 |
Generalized Fields | p. 30 |
Set-Indexed Processes | p. 36 |
Non-Gaussian Processes | p. 40 |
Majorizing Measures | p. 41 |
Gaussian Inequalities | p. 49 |
Borell-TIS Inequality | p. 49 |
Comparison Inequalities | p. 57 |
Orthogonal Expansions | p. 65 |
The General Theory | p. 66 |
The Karhunen-Loeve Expansion | p. 70 |
Excursion Probabilities | p. 75 |
Entropy Bounds | p. 76 |
Processes with a Unique Point of Maximal Variance | p. 86 |
Examples | p. 89 |
Extensions | p. 93 |
The Double-Sum Method | p. 95 |
Local Maxima and Excursion Probabilities | p. 96 |
Stationary Fields | p. 101 |
Basic Stationarity | p. 101 |
Stochastic Integration | p. 103 |
Moving Averages | p. 105 |
Spectral Representations on R[superscript N] | p. 109 |
Spectral Moments | p. 112 |
Constant Variance | p. 114 |
Isotropy | p. 115 |
Stationarity over Groups | p. 119 |
Geometry | |
Integral Geometry | p. 127 |
Basic Integral Geometry | p. 127 |
Excursion Sets Again | p. 134 |
Intrinsic Volumes | p. 141 |
Differential Geometry | p. 149 |
Manifolds | p. 149 |
Tensor Calculus | p. 154 |
Riemannian Manifolds | p. 160 |
Integration on Manifolds | p. 166 |
Curvature | p. 171 |
Intrinsic Volumes for Riemannian Manifolds | p. 175 |
A Euclidean Example | p. 176 |
Piecewise Smooth Manifolds | p. 183 |
Whitney Stratified Spaces | p. 184 |
Locally Convex Spaces | p. 188 |
Cone Spaces | p. 190 |
Critical Point Theory | p. 193 |
Critical Points | p. 193 |
The Normal Morse Index | p. 195 |
The Index | p. 195 |
Generalized Tangent Spaces and Tame Manifolds | p. 196 |
Regular Stratified Manifolds | p. 198 |
The Index on Intersections of Sets | p. 198 |
Morse's Theorem for Stratified Spaces | p. 206 |
Morse Functions | p. 206 |
Morse's Theorem | p. 207 |
The Euclidean Case | p. 210 |
Volume of Tubes | p. 213 |
The Volume-of-Tubes Problem | p. 215 |
Volume of Tubes and Gaussian Processes | p. 216 |
Local Geometry of Tube(M, [rho]) | p. 219 |
Basic Structure of Tubes | p. 220 |
Stratifying the Tube | p. 222 |
Computing the Volume of a Tube | p. 223 |
First Steps | p. 223 |
An Intermediate Computation | p. 224 |
Subsets of R[superscript l] | p. 225 |
Subsets of Spheres | p. 230 |
Weyl's Tube Formula | p. 231 |
Volume of Tubes and Gaussian Processes, Continued | p. 242 |
Intrinsic Volumes for Whitney Stratified Spaces | p. 244 |
Alternative Representation of the Curvature Measures | p. 249 |
Breakdown of Weyl's Tube Formula | p. 249 |
Generalized Lipschitz-Killing Curvature Measures | p. 250 |
The Generalized Curvature Measures | p. 251 |
Surface Measure on the Boundary of a Tube | p. 252 |
Series Expansions for the Gaussian Measure of Tubes | p. 254 |
The Geometry of Random Fields | |
Random Fields on Euclidean Spaces | p. 263 |
Rice's Formula | p. 263 |
An Expectation Metatheorem | p. 266 |
Suitable Regularity and Morse Functions | p. 280 |
An Alternate Proof of the Metatheorem | p. 283 |
Higher Moments | p. 284 |
Preliminary Gaussian Computations | p. 286 |
The Mean Euler Characteristic | p. 289 |
Mean Intrinsic Volumes | p. 298 |
On the Importance of Stationarity | p. 299 |
Random Fields on Manifolds | p. 301 |
The Metatheorem on Manifolds | p. 301 |
Riemannian Structure Induced by Gaussian Fields | p. 305 |
Connections and Curvatures | p. 306 |
Some Covariances | p. 308 |
Gaussian Fields on R[superscript N] | p. 310 |
Another Gaussian Computation | p. 312 |
The Mean Euler Characteristic | p. 315 |
Manifolds without Boundary | p. 315 |
Manifolds with Boundary | p. 317 |
Examples | p. 323 |
Chern-Gauss-Bonnet Theorem | p. 327 |
Mean Intrinsic Volumes | p. 331 |
Crofton's Formula | p. 332 |
Mean Intrinsic Volumes: The Isotropic Case | p. 333 |
A Gaussian Crofton Formula | p. 334 |
Mean Intrinsic Volumes: The General Case | p. 342 |
Two Gaussian Lemmas | p. 343 |
Excursion Probabilities for Smooth Fields | p. 349 |
On Global Suprema | p. 351 |
A First Representation | p. 352 |
The Problem with the First Representation | p. 354 |
A Second Representation | p. 354 |
Random Fields | p. 360 |
Suprema and Euler Characteristics | p. 362 |
Some Fine Tuning | p. 365 |
Gaussian Fields with Constant Variance | p. 368 |
Examples | p. 372 |
Stationary Processes on [0, T] | p. 372 |
Isotropic Fields with Monotone Covariance | p. 374 |
A Geometric Approach | p. 376 |
The Cosine Field | p. 382 |
Non-Gaussian Geometry | p. 387 |
A Plan of Action | p. 389 |
A Representation for Mean Intrinsic Volumes | p. 391 |
Proof of the Representation | p. 392 |
Poincare's Limit | p. 398 |
Kinematic Fundamental Formulas | p. 400 |
The KFF on R[superscript n] | p. 401 |
The KFF on S[subscript lambda] (R[superscript n]) | p. 402 |
A Model Process on the l-Sphere | p. 402 |
The Process | p. 403 |
Mean Curvatures for the Model Process | p. 404 |
The Canonical Gaussian Field on the l-Sphere | p. 410 |
Mean Curvatures for Excursion Sets | p. 411 |
Implications for More General Fields | p. 415 |
Warped Products of Riemannian Manifolds | p. 416 |
Warped Products | p. 417 |
A Second Fundamental Form | p. 419 |
Non-Gaussian Mean Intrinsic Volumes | p. 421 |
Examples | p. 425 |
The Gaussian Case | p. 426 |
The [chi superscript 2] Case | p. 427 |
The F Case | p. 430 |
References | p. 435 |
Notation Index | p. 443 |
Subject Index | p. 445 |
Table of Contents provided by Ingram. All Rights Reserved. |
The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.
The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.