Branching Solutions To One-dimensional Variational Problems

By: Alexandr Ivanov, Alexei Avgustinovich Tuzhilin

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Hardcover | 18 January 2001

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364 Pages

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This book deals with the new class of one-dimensional variational problems — the problems with branching solutions. Instead of extreme curves (mappings of a segment to a manifold) we investigate extreme networks, which are mappings of graphs (one-dimensional cell complexes) to a manifold. Various applications of the approach are presented, such as several generalizations of the famous Steiner problem of finding the shortest network spanning given points of the plane.

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Non-Fiction Mathematics Topology Geometry Differential & Riemannian Geometry Calculus & Mathematical Analysis Calculus of Variations Booktopia Publisher Services World Scientific Publishing

Preface
Preliminary Results	p. 1
Graphs	p. 1
Parametric networks	p. 7
Network-traces	p. 9
Stating of variational problem	p. 13
Networks Extremality Criteria	p. 21
Local structure of extreme parametric networks	p. 22
Local structure of extreme networks-traces	p. 27
Linear Networks in R[superscript N]	p. 39
Mutually parallel linear networks with a given boundary	p. 40
Geometry of planar linear trees	p. 45
On the proof of Theorem 3.2
Extremals of Length Type Functionals: The Case of Parametric Networks	p. 77
Parametric networks extreme with respect to Riemannian length functional	p. 77
Local structure of weighted extreme parametric networks	p. 83
Polyhedron of extreme weighted networks in space, having some given type and boundary	p. 85
Global structure of planar extreme weighted trees	p. 94
Geometry of planar embedded extreme weighted binary trees	p. 95
Extremals of the Length Functional: The Case of Networks-Traces	p. 99
Minimal networks on Euclidean plane	p. 100
Closed minimal networks on closed surfaces of constant curvature	p. 160
Closed local minimal networks on surfaces of polyhedra	p. 192
M. V. Pronin. Morse indices of local minimal networks	p. 218
G. A. Karpunin. Morse theory for planar linear networks	p. 232
Extremals of Functionals Generated by Norms	p. 253
Norms of general form	p. 256
Stability of extreme binary trees under deformations of the boundary	p. 280
Planar norms with strictly convex smooth circles	p. 283
Manhattan local minimal and extreme networks	p. 297
App.: Some Unsolved Problems	p. 313
Bibliography	p. 323
Index	p. 331
Table of Contents provided by Blackwell. All Rights Reserved.

Branching Solutions To One-dimensional Variational Problems

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