Medial/Skeletal Linking Structures for Multi-Region Configurations.

Cover -- Title page -- Chapter 1. Introduction -- Part 1 . Medial/Skeletal Linking Structures -- Chapter 2. Multi-Region Configurations in \Rⁿ⁺¹ -- Local Models for Regions at Singular Points on Boundary -- The Space of Equivalent Configurations via Mappings of a Model -- Configurations Allowing Containment of Regions -- Chapter 3. Skeletal Linking Structures for Multi-Region Configurations in \Rⁿ⁺¹ -- Skeletal Structures for Single Regions -- Skeletal Linking Structures for Multi-Region Configurations -- Linking Between Regions and Between Skeletal Sets -- Chapter 4. Blum Medial Linking Structure for a Generic Multi-Region Configuration -- Blum Medial Axis for a Single Region with Smooth Generic Boundary -- Structure of Maxwell Set Described by \cR⁺-Versal Unfoldings -- Addendum to Generic Blum Structure for a Region with Boundaries and Corners -- Spherical Axis of a Configuration -- Blum Medial Linking Structure -- Existence of a Blum Medial Linking Structure -- Addendum: Classification of Linking Types for Blum Medial Linking Structures in \R³ -- Chapter 5. Retracting the Full Blum Medial Structure to a Skeletal Linking Structure -- Example of Evolving Skeletal Linking Structure for Simple Generic Transition -- Retracting Full Blum Linking Structure via Smoothing -- Part 2 . Positional Geometry of Linking Structures -- Chapter 6. Questions Involving Positional Geometry of a Multi-Region Configuration -- Chapter 7. Shape Operators and Radial Flow for a Skeletal Structure -- The Radial Flow -- Radial and Edge Shape Operators -- Curvature Conditions and Nonsingularity of the Radial Flow -- Evolution of the Shape Operators Under the Radial Flow -- Chapter 8. Linking Flow and Curvature Conditions -- Nonsingularity of the Linking Flow -- Special Möbius Transformations of Matrices and Operators.

Evolution of the Shape Operators Under the Linking Flow -- Chapter 9. Properties of Regions Defined Using the Linking Flow -- Medial/Skeletal Linking Structures in the Unbounded Case -- Medial/Skeletal Linking Structures for the Bounded Case -- Chapter 10. Global Geometry via Medial and Skeletal Linking Integrals -- Defining Medial and Skeletal Linking Integrals -- Computing Boundary Integrals via Medial Linking Integrals -- Computing Integrals as Skeletal Linking Integrals via the Linking Flow -- Skeletal Linking Integral Formulas for Global Invariants -- Chapter 11. Positional Geometric Properties of Multi-Region Configurations -- Neighboring Regions and Measures of Closeness -- Measuring Positional Significance of Objects Via Linking Structures -- Properties of Invariants for Closeness and Positional Significance -- Tiered Linking Graph -- Higher Order Positional Geometric Relations via Indirect Linking -- Part 3 . Generic Properties of Linking Structures via Transversality Theorems -- Chapter 12. Multi-Distance and Height-Distance Functions and Partial Multi-Jet Spaces -- Multi-Distance and Height-Distance Functions -- Partial Jet Spaces for Multi-Distance and Height-Distance Functions -- Chapter 13. Generic Blum Linking Properties via Transversality Theorems -- Transversality Theorem for Multi-Distance and Height-Distance Functions -- Submanifolds Defining Generic Properties of Blum Linking Structures -- Chapter 14. Generic Properties of Blum Linking Structures -- Properties of Transversality and Whitney Stratified Sets -- Consequences of Transversality for (Multi-) Distance Functions -- Consequences of Transversality for Height-Distance Functions -- Generic Properties of \cB_{∞} and _{∞} -- Proof of Theorem 4.17 for a Residual Set of Embeddings -- Chapter 15. Concluding Generic Properties of Blum Linking Structures.

Blum Medial Structure Near Corner Points -- Openness of the Genericity Conditions -- Relation with Infinitesimally Stable Mappings -- Openness of Generic Properties for General Configurations -- Part 4 . Proofs and Calculations for the Transversality Theorems -- Chapter 16. Reductions of the Proofs of the Transversality Theorems -- Hybrid Transversality Theorem -- Chapter 17. Families of Perturbations and their Infinitesimal Properties -- Construction of the Families of Perturbations -- Computation of Derivatives for Families of Perturbations -- Multijet Properties Implying Stratification Properties on the Boundaries -- Chapter 18. Completing the Proofs of the Transversality Theorems -- Perturbation Transversality Conditions via Fiber Jet Extension Maps -- Derivatives of Multi-Distance and Height-Distance Functions -- Verifying the Perturbation Transversality Conditions -- Concluding the Proofs -- Appendix A. List of Frequently Used Notation -- Bibliography -- Back Cover.

The authors consider a generic configuration of regions, consisting of a collection of distinct compact regions \{ \Omega_i\} in \mathbb{R}^{n+1} which may be either regions with smooth boundaries disjoint from the others or regions which meet on their piecewise smooth boundaries \mathcal{B}_i in a generic way. They introduce a skeletal linking structure for the collection of regions which simultaneously captures the regions' individual shapes and geometric properties as well as the "positional geometry" of the collection. The linking structure extends in a minimal way the individual "skeletal structures" on each of the regions. This allows the authors to significantly extend the mathematical methods introduced for single regions to the configuration of regions.

Description based on publisher supplied metadata and other sources.

Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.