Poisson Hyperplane Tessellations

• Autore: Hug Daniel
• Pubblicazione: Cham : , : Springer, , 2024; ©2024
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource (550 pages)
• Disciplina: 519.2
• Altri autori: SchneiderRolf
• Nota di contenuto: Intro -- Preface -- Contents -- 1 Notation -- 1.1 Basic Structures -- 1.2 Measure and Integration -- 1.3 Euclidean Space and Its Structures -- 2 Hyperplane and Particle Processes -- 2.1 Point Processes -- Notes for Sect. 2.1 -- 2.2 Flat and Hyperplane Processes -- Notes for Sect. 2.2 -- 2.3 Particle Processes -- Notes for Sect. 2.3 -- 2.4 Random Tessellations -- Notes for Sect. 2.4 -- 2.5 Measurability -- 3 Distribution-Independent Density Relations -- 3.1 Functionals and Their Densities -- Notes for Sect. 3.1 -- 3.2 Density Relations for Hyperplane Tessellations -- Notes for Sect. 3.2 -- 4 Poisson Hyperplane Processes -- 4.1 Poisson Point Processes -- Notes for Sect. 4.1 -- 4.2 Poisson Hyperplane Processes -- Notes for Sect. 4.2 -- 4.3 Poisson Hyperplane Tessellations -- Notes for Sect. 4.3 -- 5 Auxiliary Functionals and Bodies -- 5.1 The Hitting Functional and the Blaschke Body -- Notes for Sect. 5.1 -- 5.2 The Matheron Zonoid -- 5.3 Intersection Densities -- Notes for Sect. 5.3 -- 5.4 Affine Intersection Densities -- 5.5 Face Intensities and Functional Densities -- 5.6 Section Processes -- Notes for Sect. 5.6 -- 6 Zero Cell and Typical Cell -- 6.1 The Zero Cell -- Notes for Sect. 6.1 -- 6.2 The Typical Cell -- Notes for Sect. 6.2 -- 6.3 K-Cells -- 7 Mixing and Ergodicity -- 7.1 Mixing Properties -- 7.2 Ergodicity and Applications -- Notes for Sect. 7.2 -- 8 Observations Inside a Window -- 8.1 The Number of k-Faces Hitting the Window -- Notes for Sect. 8.1 -- 8.2 Intersections: Expectations -- Notes for Sect. 8.2 -- 8.3 Intersections: Variances -- Notes for Sect. 8.3 -- 8.4 The Isotropic Case -- Notes for Sect. 8.4 -- 9 Central Limit Theorems -- 9.1 An Approach by Classical Methods -- Notes for Sect. 9.1 -- 9.2 Central Limit Theorems with Rates -- Notes for Sect. 9.2 -- 9.3 Stein's Method and Poisson Random Sums -- Notes for Sect. 9.3. ; 10 Palm Distributions and Related Constructions -- 10.1 Palm Distributions of Stationary Random Measures -- 10.2 Palm Distributions of Hyperplane Processes -- Notes for Sect. 10.2 -- 10.3 Reduced Second Moment Measures -- 10.4 The Isotropic Case -- 10.5 Pair-Correlation Functions -- Notes for Sect. 10.5 -- 11 Typical Faces and Weighted Faces -- 11.1 Typical and Weighted k-Faces -- Notes for Sect. 11.1 -- 11.2 Faces with Given Directions -- 11.3 More General Weights and Second Moments -- Notes for Sect. 11.3 -- 12 Large Cells and Faces -- 12.1 A General Version of Kendall's Problem -- 12.2 Limit Shapes and Special Cases -- Notes for Sect. 12.2 -- 12.3 Proof of the Main Result -- Notes for Sect. 12.3 -- 12.4 A Survey of Additional Results -- 12.5 Large Faces -- 13 Cells with a Given Number of Facets -- 13.1 The Typical Cell, Conditioned with the Facet Number -- 13.2 The Zero Cell, Conditioned with the Facet Number -- 13.3 Some Preparatory Lemmas -- 13.4 Cells with Many Facets -- Notes for Sect. 13.4 -- 14 Small Cells -- 14.1 Subspace-Free Directional Distributions -- Notes for Sect. 14.1 -- 14.2 Typical Cells with Small Circumradius -- 14.3 Typical Cells with Small Inradius -- 15 The K-Cell Under Increasing Intensities -- 15.1 The Possibility of Approximation -- 15.2 Some Preparatory Lemmas -- 15.3 The Rate of Convergence Under Increasing Intensities -- 15.4 Estimation of the Hitting Functional -- 15.5 The Mean Width Difference -- Notes for Sect. 15.5 -- 15.6 The Volume Difference -- Notes for Sect. 15.6 -- 15.7 The Facet Number -- 15.8 The Selection Expectation -- Notes for Sect. 15.8 -- 16 Isotropic Zero Cells -- 16.1 Face Numbers of the Zero Cell -- Notes for Sect. 16.1 -- 16.2 Expected Face Numbers in High Dimensions -- Notes for Sect. 16.2 -- 16.3 Lower-Dimensional Sections of the Zero Cell -- Notes for Sect. 16.3. ; 16.4 The Volume of the Zero Cell -- 16.5 Variance Inequalities for the Volume of the Zero Cell -- Notes for Sect. 16.5 -- 16.6 Asymptotic Comparison with Balls of Fixed Volume -- Notes for Sect. 16.6 -- 16.7 Second Moment of the Volume in the Stationary Case -- Notes for Sect. 16.7 -- 17 Functionals of Poisson Processes and Applications -- 17.1 Poisson U-Statistics -- 17.2 General Poisson Functionals -- Notes for Sect. 17.2 -- 17.3 Normal Approximation of Poisson Functionals -- Notes for Sect. 17.3 -- 17.4 Application to Poisson Flats in Hyperbolic Space -- Notes for Sect. 17.4 -- 18 Appendix: Some Auxiliary Results -- 18.1 Mixed Volumes and Intrinsic Volumes -- 18.2 Valuations -- 18.3 The Volume Product -- 18.4 Some Results on Zonoids -- 18.5 Proof of Lemma 17.2.5 -- 18.6 The Characteristic Functional of a Poisson Process -- References -- Author Index -- Subject Index -- Notation Index.
• Titolo autorizzato: Poisson Hyperplane Tessellations
• ISBN: 3-031-54104-9
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910864196703321
• Lo trovi qui: Univ. Federico II
• Serie: Springer Monographs in Mathematics Series