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.
Introduction
Part
1. Medial/Skeletal Linking Structures: Multi-region configurations in
$\mathbb{R}^{n+1}$
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.
James Damon, University of North Carolina, Chapel Hill, NC.
Ellen Gasparovic, Duke University, Durham, NC.
Lisainfo e-raamatute kohta