Minimal networks: the Steiner problem and its generalizations
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton u.a.
CRC Press
1994
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 414 S. graph. Darst. |
| ISBN: | 084938642X 9780849386428 |
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MARC
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| 245 | 1 | 0 | |a Minimal networks |b the Steiner problem and its generalizations |c Alexandr O. Ivanov ; Alexei A. Tuzhilin |
| 264 | 1 | |a Boca Raton u.a. |b CRC Press |c 1994 | |
| 300 | |a XVIII, 414 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 700 | 1 | |a Tužilin, Aleksej A. |e Verfasser |4 aut | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Introduction xi
Chapter 1 Some Necessary Results from
Graph Theory and Geometry 1
1 Metric and Topological Spaces / 1
1.1 Metric Spaces
1.2 Topological Spaces
1.3 Mappings
1.4 Connectedness, Separability, Compactness
1.5 Homotopy
1.6 Cell Complexes
1.7 Triangulation
1.8 Fundamental Group
1.9 Locally Euclidean Spaces and Manifolds
2 Graphs: Topological Approach / 24
2.1 Topological Graphs and Equivalence
2.2 Routes, Paths, Cycles
2.3 Subgraphs, Frames
2.4 Weighted Graphs, Minimal Weight Frames
3 Smooth Manifolds / 28
3.1 Smooth Structure, Smooth Mappings
3.2 Tangent Vectors and Vector Fields
3.3 Tensors
3.4 Differential Forms
3.5 Integral of Differential Forms
3.6 Riemannian Metric on Smooth Manifolds
3.7 Covariant Differentiation
3.8 Geodesies
3.9 Curvature of the Riemannian Manifold
3.10 The First and Second Variations of Curve Length
3.11 Geodesic Deformations and Jacobi Fields
4 Networks on Manifolds / 56
4.1 Parametric Networks, Reduced Parametric Networks,
Degeneracy Components
v
vi Contents
4.2 Smooth, Piecewise Smooth, Embedded and Immersed
Parametric Networks
4.3 The Boundary of a Network: Closed Networks
4.4 Network Equivalence
4.5 The Length of a Network in a Riemannian Manifold
4.6 Topological Space of Parametric Networks
4.7 Deformations of Parametric Networks
4.8 The First Variation Formula for Geodesic Networks
4.9 The Second Local Geodesic Variation
of Immersed Parametric Networks
4.10 Two dimensional Surface: Euler s Formula
4.11 Planar Graphs: Pontryagin Kuratowski Theorem
5 Networks and Traces / 67
5.1 Space of Traces
5.2 Boundary of a Trace: Closed Traces
5.3 Length of a Trace
5.4 Canonical Representative
5.5 Trace Deformations
5.6 Local Structure of Traces
6 Convex Function Properties / 79
6.1 Definition of Convex Functions
6.2 Support Plane and Subgradient
6.3 Convexity of the Length Function on Euclidean Space
6.4 Minimax Theorem for Families of Concave Functions
Chapter 2 The Steiner Problem
and Its Modifications 87
1 Families of Networks / 89
1.1 Types of Minimality
1.2 Traditional Families of Networks
1.3 Important Nonclassical Families of Networks
2 Existence Theorems / 97
2.1 Parametric Networks with a Fixed Boundary
2.2 Closed Parametric Networks
2.3 Traces with Fixed Boundary
2.4 Closed Traces
3 Questions of Uniqueness / 102
3.1 Closed Networks
3.2 Networks with Boundaries
Chapter 3 Local Structure of Minimal Networks 105
1 Local Structure of Minimal Parametric Networks / 105
1.1 Local Structure of Immersed Minimal Networks
1.2 Weighted Parametric Networks
1.3 General Case
1.4 Proof of Uniqueness Theorem
2 Local Structure of Minimal Traces / 119
Contents vii
Chapter 4 Global Structure of Minimal Networks 123
1 The Minimal Realization Problem for Networks
of Given Topology / 124
1.1 Trees
1.2 Nondegenerate Networks with Cycles
1.3 Steiner Networks of General Type
2 Convex Minimal Realization of Networks of Given
Topologies / 129
2.1 Twisting Number of 2 Trees
2.2 Fundamental Cycles of Nondegenerate Minimal Networks
with Convex Boundaries: Trivial Networks
2.3 Twisting Number of Trivial Networks
3 ^ Convex Minimal Realizations of Networks / 138
4 The Minimal Realization Problem for Special
Boundary Sets / 139
5 Bifurcations Problem / 140
Chapter 5 Global Minimal Networks on the Plane 147
1 Minimal Steiner Trees / 157
1.1 Steiner Hull
1.2 Nondegenerate Steiner Trees
1.3 Hexagonal Coordinate System
1.4 Minimal Steiner Trees with Boundaries of Special Type
2 Minimal Spanning Trees / 160
2.1 Delaunay Triangulation and the Voronoi diagram
2.2 Euclidean Traveling Salesman Problem
3 Steiner Ratio / 166
3.1 Three Points Case
3.2 Gilbert Pollak Conjecture and Minimax Problems
3.3 Four Points Case
3.4 Minimal Hexagonal Trees
3.5 Inner Spanning Trees
Chapter 6 Planar Local Minimal Networks
with Convex Boundaries 179
1 Complete Classification of Minimal 2 Trees
with Convex Boundaries / 179
1.1 Tiling Realization of 2 trees whose Twisting Number
Does Not Exceed 5
1.2. Tilings and Their Properties
1.3. Structural Elements of Skeletons from ~ VP5
1.4. Reduction and Antireduction
1.5. Profiles and their Properties
1.6. Classification Theorem for Skeletons from VP5
viii Contents
1.7. Location of the Growths of Tilings from ~WP5
on Their Skeletons
1.8. Theorem of Realization
2 Nondegenerate Minimal Networks with Convex Boundaries:
Cyclical Case / 240
2.1. Dual Complexes
2.2 The Twisting Number of the Contour Edges
of the Dual Complex of a Trivial Network
2.3. Kernels of a Dual Complex
2.4. Tiling Realization of a Trivial Network
whose Twisting Number Does Not Exceed 5
2.5. Description of a Tiling of General Form
2.6. Skeletons from T5
2.7. Location of Growths on Skeletons of Tilings from P5
2.8. Degenerate Networks
Chapter 7 Planar Local Minimal Networks
with Regular Boundaries 273
1 Rains / 273
2 Construction of a Minimal Realization of a Snake
on an Arbitrary Set / 275
2.1. The Characteristic Arc
2.2. The Characteristic Arc in the Case of Three Points
2.3. The Characteristic Arc in the General Case
2.4. The Totally Characteristic Arc
3 An Existence Theorem for a Snake Spanning
a Regular n gon / 285
3.1. Characteristic Arcs and Characteristic
Triangles
3.2. Proof of the Realization Theorem of a Snake
4 Skeletons and Regular n gons / 292
4.1. Location of Ends
4.2. The Structure of Ends
4.3. The Structure of Profiles
5 Criterion for the Existence of an ^/ Realization
of a Nonlinear Skeleton / 298
6 RM Realization of Skeletons with Three Ends / 303
7 Appendix / 309
Chapter 8 Closed Minimal Networks on
Closed Surfaces of Constant Curvature 313
1 Minimal Networks on Surfaces of Constant Positive
Curvature / 315
Contents jx
1.1. Closed Minimal Networks on S2
1.2. Closed Minimal Networks on RP2
2 Classification of Closed Minimal Networks on Flat Tori / 321
2.1. Description of Flat Metrics on a Two Dimensional Torus
2.2. Flat Tori Translation Groups, Lattices,
and Universal Coverings
2.3. Net Geodesies
2.4. Partitions of the Plane into Hexagons
2.5. The Type of a Network
2.6. The Characteristic Triangle
2.7. Primary Results
3 Classification of Closed Minimal Networks
on Flat Klein Bottles / 353
3.1. Description of Flat Metrics on a Klein Bottle
3.2. The Universal Covering of a Flat Klein Bottle
3.3. The Covering of a Flat Klein Bottle by a Rat Torus
3.4. Networks of the First Discrete Type
3.5. Networks of the Second Discrete Type
3.6. Regular Networks
3.7. Classification Theorems
4 Closed Networks on Two Dimensional Surfaces
of Negative Curvature / 368
4.1. Metric Restrictions on the Structure of Closed Networks:
The Gauss Bonnet Theorem
4.2. Examples of Closed Minimal Networks
on Surfaces of Negative Curvature
Chapter 9 Minimal Networks in Other Spaces 373
1 The Case of Polyhedra / 374
1.1. Developments
1.2. Local Geodesies
1.3. Local Structure of Minimal Networks on Polyhedra
1.4. The Gauss Bonnet Theorem for Polyhedra
1.5. Metric and Topological Restrictions on the Structure
of Closed Minimal Networks
1.6. The Case of Convex Polyhedra
1.7. The Case of Regular Polyhedra
2 Classification of Closed Minimal Networks
on Regular Tetrahedra / 391
2.1. The Branching Covering of Tetrahedra
by the Plane and by the Flat Torus
2.2. Regular Networks
2.3. Classification Theorems
3 Networks on the Lobachevskian Plane / 398
References 401
Index 405
|
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| language | English |
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| spelling | Ivanov, Aleksandr O. Verfasser aut Minimal networks the Steiner problem and its generalizations Alexandr O. Ivanov ; Alexei A. Tuzhilin Boca Raton u.a. CRC Press 1994 XVIII, 414 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Steiner systems Steiner-Baum (DE-588)4228498-3 gnd rswk-swf Steiner-Problem (DE-588)4248342-6 gnd rswk-swf Opazität (DE-588)4494369-6 gnd rswk-swf Steiner-Baum (DE-588)4228498-3 s DE-604 Opazität (DE-588)4494369-6 s Steiner-Problem (DE-588)4248342-6 s DE-188 Tužilin, Aleksej A. Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006430906&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ivanov, Aleksandr O. Tužilin, Aleksej A. Minimal networks the Steiner problem and its generalizations Steiner systems Steiner-Baum (DE-588)4228498-3 gnd Steiner-Problem (DE-588)4248342-6 gnd Opazität (DE-588)4494369-6 gnd |
| subject_GND | (DE-588)4228498-3 (DE-588)4248342-6 (DE-588)4494369-6 |
| title | Minimal networks the Steiner problem and its generalizations |
| title_auth | Minimal networks the Steiner problem and its generalizations |
| title_exact_search | Minimal networks the Steiner problem and its generalizations |
| title_full | Minimal networks the Steiner problem and its generalizations Alexandr O. Ivanov ; Alexei A. Tuzhilin |
| title_fullStr | Minimal networks the Steiner problem and its generalizations Alexandr O. Ivanov ; Alexei A. Tuzhilin |
| title_full_unstemmed | Minimal networks the Steiner problem and its generalizations Alexandr O. Ivanov ; Alexei A. Tuzhilin |
| title_short | Minimal networks |
| title_sort | minimal networks the steiner problem and its generalizations |
| title_sub | the Steiner problem and its generalizations |
| topic | Steiner systems Steiner-Baum (DE-588)4228498-3 gnd Steiner-Problem (DE-588)4248342-6 gnd Opazität (DE-588)4494369-6 gnd |
| topic_facet | Steiner systems Steiner-Baum Steiner-Problem Opazität |
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