Stochastic Geometry: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 13-18, 2004

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Springer, Oct 26, 2006 - Mathematics - 292 pages

Stochastic Geometry is the mathematical discipline which studies mathematical models for random geometric structures, as they appear frequently in almost all natural sciences or technical fields. Although its roots can be traced back to the 18th century (the Buffon needle problem), the modern theory of random sets was founded by D. Kendall and G. Matheron in the early 1970's. Its rapid development was influenced by applications in Spatial Statistics and by its close connections to Integral Geometry. The volume "Stochastic Geometry" contains the lectures given at the CIME summer school in Martina Franca in September 2004. The four main lecturers covered the areas of Spatial Statistics, Random Points, Integral Geometry and Random Sets, they are complemented by two additional contributions on Random Mosaics and Crystallization Processes. The book presents an up-to-date description of important parts of Stochastic Geometry.

 

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Contents

Spatial Point Processes and their Applications
1
Moments and Summary Statistics
26
Conditioning
42
Modelling and Statistical Inference
57
References
73
Minimal Caps and a General Result
80
Proofs of the Properties of the Mregions
87
Proof of Theorem 3 1
95
Translative Integral Geometry
151
Measures on Spaces of Flats
164
References
181
Mean Values of Additive Functionals
198
Contact Distributions
231
References
243
Voronoi and Delaunay Mosaics
253
Kendalls Conjecture
260

Expectation of fkKn
101
Lattice Polytopes
108
Segments on the Surface of K
114
From Hitting Probabilities to Kinematic Formulae
120
Localizations and Extensions
136
On the Evolution Equations of Mean Geometric Densities
267
The Evolution Equation of Mean Densities for the Stochastic
274
References
280
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