| Description |
1 online resource (xxi, 342 pages) : illustrations |
| Physical Medium |
polychrome |
| Description |
text file |
| Bibliography |
Includes bibliographical references (pages 323-329) and index. |
| Summary |
This study 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) it investigates 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. |
| Contents |
Ch. 1. Preliminary results. 1.1. Graphs. 1.2. Parametric networks. 1.3. Network-traces. 1.4. Stating of variational problem -- ch. 2. Networks extremality criteria. 2.1. Local structure of extreme parametric networks. 2.2. Local structure of extreme networks-traces -- ch. 3. Linear networks in [symbol]. 3.1. Mutually parallel linear networks with a given boundary. 3.2. Geometry of planar linear trees. 3.3. On the proof of Theorem -- ch. 4. Extremals of length type functionals: the case of parametric networks. 4.1. Parametric networks extreme with respect to Riemannian length functional. 4.2. Local structure of weighted extreme parametric networks. 4.3. Polyhedron of extreme weighted networks in space, having some given type and boundary. 4.4. Global structure of planar extreme weighted trees. 4.5. Geometry of planar embedded extreme weighted binary trees -- ch. 5. Extremals of the length functional: the case of networks -- traces. 5.1. Minimal networks on Euclidean plane. 5.2. Closed minimal networks on closed surfaces of constant curvature. 5.3. Closed local minimal networks on surfaces of polyhedra. 5.4. M.V. Pronin. Morse indices of local minimal networks. 5.5. G.A. Karpunin. Morse theory for planar linear networks -- ch. 6. Extremals of functionals generated by norms. 6.1. Norms of general form. 6.2. Stability of extreme binary trees under deformations of the boundary. 6.3. Planar norms with strictly convex smooth circles. 6.4. Manhattan local minimal and extreme networks. |
| Local Note |
eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - North America |
| Subject |
Extremal problems (Mathematics)
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Extremal problems (Mathematics) |
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Steiner systems.
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Steiner systems. |
| Genre/Form |
Electronic books.
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Electronic books.
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| Added Author |
Tuzhilin, A. A.
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| Other Form: |
Print version: Ivanov, A.O. (Alexander O.). Branching solutions to one-dimensional variational problems. Singapore ; River Edge, NJ : World Scientific, ©2001 9789810240608 (DLC) 00063439 (OCoLC)44811646 |
| ISBN |
9789812810717 (electronic book) |
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9812810714 (electronic book) |
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9810240600 (alkaline paper) |
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9789810240608 (alkaline paper) |
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1281956368 |
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9781281956361 |
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