Medial/Skeletal Linking Structures for Multi-Region Configurations

ISBN-13: 9781470426804 / Angielski

James Damon;Ellen Gasparovic

Opis

The authors consider a generic configuration of regions, consisting of a collection of distinct compact regions ${ Omega_i}$ in $mathbb^{n+1}$ which may be either regions with smooth boundaries disjoint from the others or regions which meet on their piecewise smooth boundaries $mathcal_i$ 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.

Spis treści

• Introduction
• Part 1. Medial/Skeletal Linking Structures: Multi-region configurations in $\mathbb^{n+1}$
• Skeletal linking structures for multi-region configurations in ${\mathbb R}^{n+1}$
• Blum medial linking structure for a generic multi-region configuration
• Retracting the full Blum medial structure to a skeletal linking structure
• Part 2. Positional Geometry of Linking Structures: Questions involving positional geometry of a multi-region configuration
• Shape operators and radial flow for a skeletal structure
• Linking flow and curvature conditions
• Properties of regions defined using the linking flow
• Global geometry via medial and skeletal linking integrals
• Positional geometric properties of multi-region configurations
• Part 3. Generic Properties of Linking Structures via Transversality Theorems: Multi-distance and height-distance functions and partial multi-jet spaces
• Generic Blum linking properties via transversality theorems
• Generic properties of Blum linking structures
• Concluding generic properties of Blum linking structures
• Part 4. Proofs and Calculations for the Transversality Theorems: Reductions of the proofs of the transversality theorems
• Families of perturbations and their infinitesimal properties
• Completing the proofs of the transversality theorems
• Appendix A. List of frequently used notation
• Bibliography.