Stochastic Geometry - Jan Rataj

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Résumé :
Stochastic geometry, based on current developments in geometry, probability and measure theory, makes possible modeling of two- and three-dimensional random objects with interactions as they appear in the microstructure of materials, biological tissues, macroscopically in soil, geological sediments etc. In combination with spatial statistics it is used for the solution of practical problems such as the description of spatial arrangements and the estimation of object characteristics. A related field is stereology, which makes possible inference on the structures, based on lower-dimensional observations. Unfolding problems for particle systems and extremes of particle characteristics are studied. The reader can learn about current developments in stochastic geometry with mathematical rigor on one hand and find applications to real microstructure analysis in natural and material sciences on the other hand.

Sommaire:
Preface Acknowledgments 1: Preliminaries 1.1 Geometry and measure in the Euclidean space 1.2 Probability and statistics 2: Random Measures and Point Processes 2.1 Basic definitions 2.2 Palm distributions 2.3 Poisson process 2.4 Finite point processes 2.5 Stationary random measures on Rd 2.6 Application of point processes in epidemiology 2.7 Weighted random measures, marked point processes 2.8 Stationary processes of particles 2.9 Flat processes 3: Random Fibre And Surface Systems 3.1 Geometric models 3.2 Intensity estimators 3.3 Projection measure estimation 3.4 Best unbiased estimators of intensity 4: Vertical Sampling Schemes 4.1 Randomized sampling 4.2 Design-based approach 5: Fibre and Surface Anisotropy 5.1 Introduction 5.2 Analytical approach 5.3 Convex geometry approach 5.4 Orientation-dependent direction distribution 6: Particle Systems 6.1 Stereological unfolding 6.2 Bivariate unfolding 6.3 Trivariate unfolding 6.4 Stereology of extremes References Index

From the reviews: To write a text on stochastic geometry and its applications requires that an author successfully address a number of issues involving material and audience. ... Viktor BeneS and Jan Rataj have produced an interesting text which successfully addresses the challenges involved in writing such a book. ... The text will be a valuable addition to the library of those who have an interest in applications ... . the authors provide interesting examples. Each chapter of the text contains a number of exercises ... . (Patrick T. McDonald, Mathematical Reviews, 2005g)