Factorization calculus and geometric probability:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
1990
|
| Edition: | 1. publ. |
| Series: | Encyclopedia of mathematics and its applications
33 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 286 S. graph. Darst. |
| ISBN: | 0521345359 |
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| 245 | 1 | 0 | |a Factorization calculus and geometric probability |c R. V. Ambartzumian |
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| adam_text | Titel: Factorization calculus and geometric probability
Autor: Ambartzumian, Ruben V
Jahr: 1990
CONTENTS Preface ix 1 Cavalieri principle and other prerequisities 1 1.1 The Cavalieri principle 1 1.2 Lebesgue factorization 3 1.3 Haar factorization 5 1.4 Further remarks on measures 9 1.5 Some topological remarks 10 1.6 Parametrization maps 14 1.7 Metrics and convexity 15 1.8 Versions of Crofton’s theorem 18 2 Measures invariant with respect to translations 20 2.1 The space G of directed lines on R 2 20 2.2 The space G of (non-directed) lines in R 2 21 2.3 The space Ë of oriented planes in R 3 22 2.4 The space E of planes in R 3 23 2.5 The space T of directed lines in R 3 23 2.6 The space T of (non-directed) lines in R 3 24 2.7 Measure-representing product models 24 2.8 Factorization of measures on spaces with slits 27 2.9 Dispensing with slits 28 2.10 Roses of directions and roses of hits 29 2.11 Density and curvature 30 2.12 The roses of T 3 -invariant measures on E 31 2.13 Spaces of segments and flats 34 2.14 Product spaces with slits 36 2.15 Almost sure T-invariance of random measures 37 2.16 Random measures on G 38 2.17 Random measures on E 40 2.18 Random measures on T 41
Contents vi 3 Measures invariant with respect to Euclidean motions 43 3.1 The group W 2 of rotations of R 2 43 3.2 Rotations of R 3 44 3.3 The Haar measure onW 3 45 3.4 Geodesic lines on a sphere 46 3.5 Bi-invariance of Haar measures on Euclidean groups 47 3.6 The invariant measure on G and G 47 3.7 The form of d g in two other parametrizations of lines 48 3.8 Other parametrizations of geodesic lines on a sphere 50 3.9 The invariant measure on T and T 51 3.10 Other parametrizations of lines in R 3 52 3.11 The invariant measure in the spaces E and E 53 3.12 Other parametrizations of planes in R 3 53 3.13 The kinematic measure - - 55 3.14 Position-size factorizations 57 3.15 Position-shape factorizations 59 3.16 Position-size-shape factorizations 61 3.17 On measures in shape spaces 67 3.18 The spherical topology of £ 70 4 Haar measures on groups of affine transformations 72 4.1 The group A® and its subgroups 72 4.2 Affine deformations of R 2 74 4.3 The Haar measure on A§ 76 4.4 The Haar measure on A 2 77 4.5 Triads of points in R 2 78 4.6 Another representation of d (r) V 8C 4.7 Quadruples of points in R 2 82 4.8 The modified Sylvester problem: four points in R 2 8 4.9 The group A® and its subgroups 8 4.10 The group of affine deformations of R 3 8 4.11 Haar measures on A® and A 3 8! 4.12 V 3 -invariant measure in the space of tetrahedral shapes 91 4.13 Quintuples of points in R 3 9 4.14 Affine shapes of quintuples in R 3 9 4.15 A general theorem 9 4.16 The elliptical plane as a space of affine shapes 9 5 Combinatorial integral geometry 10 5.1 Radon rings in G and G 10 5.2 Extension of Crofton’s theorem 10 5.3 Model approach and the Gauss-Bonnet theorem 10 5.4 Two examples 1C 5.5 Rings in E 11 5.6 Planes cutting a convex polyhedron 11 5.7 Reconstruction of the measure from a wedge function 11 5.8 The wedge function in the shift-invariant case 11
Contents vii 5.9 Flag representations of convex bodies 118 5.10 Flag representations and zonoids 119 5.11 Planes hitting a smooth convex body in S? 3 120 5.12 Other ramifications and historical remarks 123 6 Basic integrals 127 6.1 Integrating the number of intersections 127 6.2 The zonoid equation 130 6.3 Integrating the Lebesgue measure of the intersection set 131 6.4 Vertical windows and shift-invariance 133 6.5 Vertical windows and a pair of non-parallel lines 134 6.6 Translational analysis of realizations 137 6.7 Integrals over product spaces 141 6.8 Kinematic analysis of realizations 145 6.9 Pleijel identity 153 6.10 Chords through convex polygons 156 6.11 Integral functions for measures in the space of triangular shapes 158 7 Stochastic point processes 161 7.1 Point processes 161 7.2 fe-subsets of a linear interval 162 7.3 Finite sets on [a, b) 165 7.4 Consistent families 168 7.5 Situation in other spaces 171 7.6 The example of L. Shepp 173 7.7 Invariant models 174 7.8 Random shift of a lattice 175 7.9 Random motions of a lattice 177 7.10 Lattices of random shape and position 177 7.11 Kallenberg-Mecke-Kingman line processes 179 7.12 Marked point processes: independent marks 181 7.13 Segment processes and random mosaics 185 7.14 Moment measures 187 7.15 Averaging in the space of realizations 192 8 Palm distributions of point processes in IR 200 8.1 Typical mark distribution 200 8.2 Reduction to calculation of intensities 202 8.3 The space of anchored realizations 202 8.4 Palm distribution 204 8.5 A continuity assumption 205 8.6 Some examples 207 8.7 Palm formulae in one dimension 208 8.8 Several intervals 210 8.9 T t -invariant renewal processes 211 8.10 Palm formulae for balls in IR 216 8.11 The equation IT = © * P 217
Contents viii 8.12 Asymptotic Poisson distribution 218 8.13 Equations with Palm distribution 220 8.14 Solution by means of density functions 221 9 Poisson-generated geometrical processes 227 9.1 Relative Palm distribution 227 9.2 Extracting point processes on groups 229 9.3 Equally weighted typical polygon in a Poisson line mosaic 231 9.4 Solution 232 9.5 Derivation of the basic relation 234 9.6 Further weightings 235 9.7 Cases of infinite intensity 238 9.8 Thinnings yield probability distributions __ 240 9.9 Simplices in the Poisson point processes in R 243 9.10 Voronoi mosaics 244 9.11 Mean values for random polygons 247 10 Sections through planar geometrical processes 252 10.1 Palm distribution of line processes on R 2 252 10.2 Palm formulae for line processes 255 10.3 Second order line processes 256 10.4 Averaging a combinatorial decomposition 262 10.5 Further remarks on line processes 266 10.6 Extension to random mosaics 269 10.7 Boolean models for disc processes 272 10.8 Exponential distribution of typical white intervals 276 References 279 Index 283
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| discipline | Mathematik |
| edition | 1. publ. |
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| language | English |
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| spelling | Ambartzumian, R. V. 1940- Verfasser (DE-588)142103675 aut Factorization calculus and geometric probability R. V. Ambartzumian 1. publ. Cambridge [u.a.] Cambridge Univ. Press 1990 XI, 286 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Encyclopedia of mathematics and its applications 33 Stochastische Geometrie (DE-588)4133202-7 gnd rswk-swf Geometrische Wahrscheinlichkeit (DE-588)4156727-4 gnd rswk-swf Faktorisierung (DE-588)4128927-4 gnd rswk-swf Maß Mathematik (DE-588)4037856-1 gnd rswk-swf Faktorisierung (DE-588)4128927-4 s Maß Mathematik (DE-588)4037856-1 s DE-604 Geometrische Wahrscheinlichkeit (DE-588)4156727-4 s Stochastische Geometrie (DE-588)4133202-7 s Encyclopedia of mathematics and its applications 33 (DE-604)BV000903719 33 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002667228&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ambartzumian, R. V. 1940- Factorization calculus and geometric probability Encyclopedia of mathematics and its applications Stochastische Geometrie (DE-588)4133202-7 gnd Geometrische Wahrscheinlichkeit (DE-588)4156727-4 gnd Faktorisierung (DE-588)4128927-4 gnd Maß Mathematik (DE-588)4037856-1 gnd |
| subject_GND | (DE-588)4133202-7 (DE-588)4156727-4 (DE-588)4128927-4 (DE-588)4037856-1 |
| title | Factorization calculus and geometric probability |
| title_auth | Factorization calculus and geometric probability |
| title_exact_search | Factorization calculus and geometric probability |
| title_full | Factorization calculus and geometric probability R. V. Ambartzumian |
| title_fullStr | Factorization calculus and geometric probability R. V. Ambartzumian |
| title_full_unstemmed | Factorization calculus and geometric probability R. V. Ambartzumian |
| title_short | Factorization calculus and geometric probability |
| title_sort | factorization calculus and geometric probability |
| topic | Stochastische Geometrie (DE-588)4133202-7 gnd Geometrische Wahrscheinlichkeit (DE-588)4156727-4 gnd Faktorisierung (DE-588)4128927-4 gnd Maß Mathematik (DE-588)4037856-1 gnd |
| topic_facet | Stochastische Geometrie Geometrische Wahrscheinlichkeit Faktorisierung Maß Mathematik |
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