Factorization Calculus and Geometric Probability

✍ Scribed by R. V. Ambartzumian

• Publisher: Cambridge University Press
• Year: 1990
• Tongue: English
• Leaves: 299
• Category: Library

✦ Synopsis

The classical subjects of geometric probability and integral geometry, and the more modern one of stochastic geometry, are developed here in a novel way to provide a framework in which they can be studied. The author focuses on factorization properties of measures and probabilities implied by the assumption of their invariance with respect to a group, in order to investigate nontrivial factors. The study of these properties is the central theme of the book. Basic facts about integral geometry and random point process theory are developed in a simple geometric way, so that the whole approach is suitable for a nonspecialist audience. Even in the later chapters, where the factorization principles are applied to geometrical processes, the only prerequisites are standard courses on probability and analysis. The main ideas presented have application to such areas as stereology and geometrical statistics and this book will be a useful reference book for university students studying probability theory and stochastic geometry, and research mathematicians interested in this area.

✦ Table of Contents

Cover
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
Title
Copyright
CONTENTS
PREFACE
1 Cavalieri principle and other prerequisites
1.1 The Cavalieri principle
1.2 Lebesgue factorization
1.3 Haar factorization
1.4 Further remarks on measures
1.5 Some topological remarks
1.6 Parametrization maps
1.7 Metrics and convexity
1.8 Versions of Crofton's theorem
2 Measures invariant with respect to translations
2.1 The space G of directed lines on R
2.2 The space G of (non-directed) lines in R
2.3 The space E of oriented planes in R
2.4 The space E of planes in R
2.5
2.6 The space G of (non-directed) lines in R
2.7 Measure-representing product models
2.8 Factorization of measures on spaces with slits
2.9 Dispensing with slits
2.10 Roses of directions and roses of hits
2.11 Density and curvature
2.12 The roses of T3-invariant measures on E
2.13 Spaces of segments and flats
2.14 Product spaces with slits
2.15 Almost sure T-invariance of random measures
2.16 Random measures on G
2.17 Random measures on E
2.18 Random measures on G
3 Measures invariant with respect to Euclidean motions
3.1 The group W2 of rotations of R2
3.2 Rotations of R3
3.3 The Haar measure on W3
3.4 Geodesic lines on a sphere
3.5 Bi-invariance of Haar measures on Euclidean groups
3.6 The invariant measure on G and G
3.7 The form of dg in two other parametrizations of lines
3.8 Other parametrizations of geodesic lines on a sphere
3.9 The invariant measure on G and G
3.10 Other parametrizations of lines in R3
3.11 The invariant measure in the spaces E and E
3.12 Other parametrizations of planes in R3
3.13 The kinematic measure
3.14 Position-size factorizations
3.15 Position-shape factorizations
3.16 Position-size-shape factorizations
3.17 On measures in shape spaces
3.18 The spherical topology of Sigma
4 Haar measures on groups of affine transformations
4.1 The group A02 and its subgroups
4.2 Affine deformations of R2
4.3 The Haar measure on A02
4.4 The Haar measure on A2
4.5 Triads of points in R2
4.6 Another representation of d(r)V
4.7 Quadruples of points in R
4.8 The modified Sylvester problem: four points in R2
4.9 The group A03 and its subgroups
4.10 The group of affine deformations of R3
4.11 Haar measures on A03 and A3
4.12 V3-invariant measure in the space of tetrahedral shapes
4.13 Quintuples of points in R3
4.14 Affine shapes of quintuples in R3
4.15 A general theorem
4.16 The elliptical plane as a space of affine shapes
5 Combinatorial integral geometry
5.1 Radon rings in G and G
5.2 Extension of Crofton's theorem
5.3 Model approach and the Gauss-Bonnet theorem
5.4 Two examples
5.5 Rings in E
5.6 Planes cutting a convex polyhedron
5.7 Reconstruction of the measure from a wedge function
5.8 The wedge function in the shift-invariant case
5.9 Flag representations of convex bodies
5.10 Flag representations and zonoids
5.11 Planes hitting a smooth convex body in R3
5.12 Other ramifications and historical remarks
6 Basic integrals
6.1 Integrating the number of intersections
6.2 The zonoid equation
6.3 Integrating the Lebesgue measure of the intersection set
6.4 Vertical windows and shift-invariance
6.5 Vertical windows and a pair of non-parallel lines
6.6 Translational analysis of realizations
6.7 Integrals over product spaces
6.8 Kinematic analysis of realizations
6.9 Pleijel identity
6.10 Chords through convex polygons
6.11 Integral functions for measures in the space of triangular shapes
7 Stochastic point processes
7.1 Point processes
7.2 k-subsets of a linear interval
7.3 Finite sets on [a, b)
7.4 Consistent families
7.5 Situation in other spaces
7.6 The example of L.Shepp
7.7 Invariant models
7.8 Random shift of a lattice
7.9 Random motions of a lattice
7.10 Lattices of random shape and position
7.11 Kallenberg-Mecke-Kingman line processes
7.12 Marked point processes: independent marks
7.13 Segment processes and random mosaics
7.14 Moment measures
7.15 Averaging in the space of realizations
8 Palm distributions of point processes in Rn
8.1 Typical mark distribution
8.2 Reduction to calculation of intensities
8.3 The space of anchored realizations
8.4 Palm distribution
8.5 A continuity assumption
8.6 Some examples
8.7 Palm formulae in one dimension
8.8 Several intervals
8.9 T1-invariant renewal processes
8.10 Palm formulae for balls in Rn
8.11 The equation Pi= O*P
8.12 Asymptotic Poisson distribution
8.13 Equations with Palm distribution
8.14 Solution by means of density functions
9 Poisson-generated geometrical processes
9.1 Relative Palm distribution
9.2 Extracting point processes on groups
9.3 Equally weighted typical polygon in a Poisson line mosaic
9.4 Solution
9.5 Derivation of the basic relation
9.6 Further weightings
9.7 Cases of infinite intensity
9.8 Thinnings yield probability distributions
9.9 Simplices in the Poisson point processes in Rn
9.10 Voronoi mosaics
9.11 Mean values for random polygons
10 Sections through planar geometrical processes
10.1 Palm distribution of line processes on R2
10.2 Palm formulae for line processes
10.3 Second order line processes
10.4 Averaging a combinatorial decomposition
10.5 Further remarks on line processes
10.6 Extension to random mosaics
10.7 Boolean models for disc processes
10.8 Exponential distribution of typical white intervals
REFERENCES
INDEX OF KEY WORDS