This volume, with an excellent introductory preface, describes the state-of-the-art of this rapidly evolving subject with special emphasis on the non-Brownian world. It collects articles written by leading experts that will appeal to the non-specialist. An extensive bibliography at the end of each article makes this an invaluable comprehensive reference text.
For the researcher and graduate student, every article contains open problems and points out directions for future research. The accessible nature of the work makes this an ideal introductory text for graduate seminars in applied probability, stochastic processes, physics, finance, and telecommunications, and a unique guide to the world of Lévy processes.
The first full text on design-based stereology opens with a review of classical and modern stereology, followed by a treatment of mathematical foundations and then on to core techniques. The final chapters discuss implementing techniques in practical sampling designs, summarize understanding of the variance of stereological estimators, and describe open problems for further research. The book also details isotropic, vertical or local sampling designs for estimating stereological parameters such as volume, surface area, particle number and spatial distribution.
This extensive text offers support to statistical consultants using examples, applications and unique Advice to Consultants sections. It contains numerous literature references, bibliographic notes and nearly 150 illustrations.
In order to reach a broader audience, the book has been written not only for specialists in stereology, integral geometry and geometric measure theory. In particular, Chapter 1 is an elementary introduction to stereology and the book contains about 75 illustrations. The theory of local steroelogy involves, however, advanced mathematical tools, which constitute an important part of the book.
Local-stereological methods are now in world-wide use in the microscopical study of biological tissue, and this invaluable book also contains a description of how the local methods are used in practice.
Contents:Introduction to StereologyThe Coarea FormulaRotation Invariant Measures on LnpThe Classical Blaschke–Petkantschin FormulaThe Generalized Blaschke–Petkantschin FormulaLocal Slice FormulaeDesign and Implementation of Local-Stereological ExperimentsThe Model-Based ApproachPerspectives and Future Trends
Readership: Researchers, teachers and graduate students in mathematical statistics and probability.
keywords:Geometric Sampling;Geometric Probability;Local Stereology;Coarea Formula;Blaschke-Petkantschin Formulae;Exercises
“This book adds weighty theoretical support to the increasingly important field of spatial sampling.”Short Book Reviews
“The whole book is very carefully written from both the mathematical and didactical points of view. This excellent style is supported by a series of suggestive figures and of exercises. It is written for researchers working in theory or in applications, for teachers and for graduate students.”Mathematical Reviews
“The book is self-contained, and well-readable with a number of explanatory figures. All chapters are followed by useful exercises and bibliographical notes. Overall, this book is going to become a standard graduate text on design-based local stereology.”Mathematics Abstracts
The present second volume contains the following lectures: "Random Walks on Finite Quantum Groups" by Uwe Franz and Rolf Gohm, "Quantum Markov Processes and Applications in Physics" by Burkhard Kümmerer, Classical and Free Infinite Divisibility and Lévy Processes" by Ole E. Barndorff-Nielsen, Steen Thorbjornsen, and "Lévy Processes on Quantum Groups and Dual Groups" by Uwe Franz.