Branching solutions to one-dimensional variational problems:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Singapore [u.a.]
World Scientific
2001
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXI, 342 S. graph. Darst. |
| ISBN: | 9810240600 |
Internformat
MARC
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| 100 | 1 | |a Ivanov, Aleksandr O. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Branching solutions to one-dimensional variational problems |c A. O. Ivanov & A. A. Tuzhilin |
| 264 | 1 | |a Singapore [u.a.] |b World Scientific |c 2001 | |
| 300 | |a XXI, 342 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 650 | 4 | |a Extremal problems (Mathematics) | |
| 650 | 4 | |a Steiner systems | |
| 650 | 0 | 7 | |a Dimension 1 |0 (DE-588)4323094-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Variationsproblem |0 (DE-588)4187419-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Steiner-Problem |0 (DE-588)4248342-6 |2 gnd |9 rswk-swf |
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| 700 | 1 | |a Tužilin, Aleksej A. |e Verfasser |4 aut | |
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Datensatz im Suchindex
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| adam_text | BRANCHING SOLUTIONS TO ONE-DIMENSIONAL VARIATIONAL PROBLEMS A. O. IVANOV
& A. A. TUZHILIN FACULTY OF MECHANICS AND MATHEMATICS, MOSCOW STATE
UNIVERSITY, RUSSIA WORLD SCIENTIFIC SINGAPORE * NEW JERSEY * LONDON *
HONG KONG CONTENTS PREFACE VII CHAPTER 1 PRELIMINARY RESULTS 1 1.1
GRAPHS 1 1.1.1 TOPOLOGICAL AND FRAMED GRAPHS, THEIR EQUIVALENCE . . . .
2 1.1.2 OPERATIONS ON GRAPHS 3 1.1.3 BOUNDARY OF GRAPH, LOCAL GRAPH 5
1.1.4 SMOOTH STRUCTURE ON TOPOLOGICAL GRAPH 6 1.2 PARAMETRIC NETWORKS 7
1.2.1 MAIN DEFINITIONS 7 1.2.2 CLASSES OF NETWORKS SMOOTHNESS 8 1.3
NETWORK-TRACES 9 1.3.1 NETWORKS-TRACES AND THEIR CANONICAL
REPRESENTATIVES . . 10 1.4 STATING OF VARIATIONAL PROBLEM 13 1.4.1
CONSTRUCTION OF EDGE FUNCTIONALS 13 1.4.1.1 CLASSICAL VARIATIONAL
FUNCTIONAL 14 1.4.1.2 CLASSICAL FUNCTIONAL OF BOLZA 16 1.4.2
CONSTRUCTION OF EDGE FUNCTIONALS FOR NETWORKS WITH FIXED TOPOLOGY 17
CHAPTER 2 NETWORKS EXTREMALITY CRITERIA 21 2.1 LOCAL STRUCTURE OF
EXTREME PARAMETRIC NETWORKS 22 2.2 LOCAL STRUCTURE OF EXTREME
NETWORKS-TRACES 27 2.2.1 SMOOTH LAGRANGIANS 27 2.2.2 QUASIREGULAR
LAGRANGIANS 29 XV XVI CONTENTS CHAPTER 3 LINEAR NETWORKS IN R N 39 3.1
MUTUALLY PARALLEL LINEAR NETWORKS WITH A GIVEN BOUNDARY . . . . 40 3.2
GEOMETRY OF PLANAR LINEAR TREES 45 3.2.1 TWISTING NUMBER OF PLANAR
LINEAR TREE 46 3.2.2 MAIN THEOREM 47 3.3 ON THE PROOF OF THEOREM 3.2 47
3.3.1 PLANAR POLYGONAL LINES I: THE CASE OF GENERAL POSITION . . 48
3.3.1.1 TWISTING AND TURNING 49 3.3.1.2 A PAIR OF POLYGONAL LINES IN
GENERAL POSITION . 51 3.3.1.3 CAPS 55 3.3.2 PLANAR POLYGONAL LINES II:
THE GENERAL CASE 56 3.3.3 TWISTING NUMBER OF A PLANAR LINEAR TREE 64
3.3.3.1 PROPER LINEAR TREES 64 3.3.3.2 QUASI-GEODESICS 65 3.3.3.3 CAPS
66 3.3.4 PROOF OF THEOREM 3.2 68 3.3.4.1 THE CASE P = Q 69 3.3.4.2 THE
CASE P Q 72 CHAPTER 4 EXTREMALS OF LENGTH TYPE FUNCTIONALS: THE CASE
OF PARAMETRIC NETWORKS 77 4.1 PARAMETRIC NETWORKS EXTREME WITH RESPECT
TO RIEMANNIAN LENGTH FUNCTIONAL 77 4.2 LOCAL STRUCTURE OF WEIGHTED
EXTREME PARAMETRIC NETWORKS ... 83 4.3 POLYHEDRON OF EXTREME WEIGHTED
NETWORKS IN SPACE, HAVING SOME GIVEN TYPE AND BOUNDARY 85 4.3.1
STRUCTURE OF THE SET OF EXTREME WEIGHTED NETWORKS ... 87 -- 4.3.2
IMMERSED EXTREME WEIGHTED STEINER NETWORKS IN THE PLANE 92 4.4 GLOBAL
STRUCTURE OF PLANAR EXTREME WEIGHTED TREES 94 4.5 GEOMETRY OF PLANAR
EMBEDDED EXTREME WEIGHTED BINARY TREES . 95 4.5.1 TWISTING NUMBER OF
EMBEDDED PLANAR WEIGHTED BINARY TREES 95 CHAPTER 5 EXTREMALS OF THE
LENGTH FUNCTIONAL: THE CASE OF NETWORKS-TRACES 99 5.1 MINIMAL NETWORKS
ON EUCLIDEAN PLANE 100 5.1.1 CORRESPONDENCE BETWEEN PLANAR BINARY TREES
AND DIAGO- NAL TRIANGULATIONS 101 CONTENTS XVII 5.1.2 STRUCTURAL
ELEMENTS OF DIAGONAL TRIANGULATIONS 103 5.1.3 TILING REALIZATION OF
BINARY TREES WHOSE TWISTING NUMBER IS AT MOST FIVE 104 5.1.4 TILINGS AND
THEIR PROPERTIES 106 5.1.4.1 DECOMPOSITIONS OF A TILING INTO A SKELETON
AND GROWTHS 106 5.1.4.2 DECOMPOSITION OF A TREE SKELETON INTO BRANCH-
ING POINTS AND LINEAR PARTS 108 5.1.4.3 AXIS 108 5.1.5 STRUCTURAL
ELEMENTS OF SKELETONS FROM W7J 110 5.1.5.1 BRANCHING POINTS OF A TREE
SKELETONS 110 5.1.5.2 LINEAR PARTS ILL 5.1.6 OPERATIONS OF REDUCTION AND
ANTIREDUCTION 112 5.1.6.1 CUTTING AND PASTING 112 5.1.6.2 REDUCTION OF
PLANAR BINARY TREE 113 5.1.6.3 REDUCTION OF TILINGS FROM W/J 114 5.1.7
PROFILES AND THEIR PROPERTIES 115 5.1.7.1 DEFINITION OF PROFILES 115
5.1.7.2 RELATION BETWEEN THE TWISTING NUMBERS OF PRO- FILES AND THE
TWISTING NUMBER OF THE TILING . . 116 5.1.7.3 TERMINOLOGICAL REMARK 116
5.1.8 CLASSIFICATION THEOREM FOR SKELETONS FROM WL% 116 5.1.8.1
DIRECTIONS OF ENDING LINEAR PARTS OF SKELETONS FROM WF$ 118 5.1.8.2
CODES OF NON-DEGENERATE 6-SKELETONS 118 5.1.9 LOCATION OF THE GROWTHS OF
TILINGS FROM WPFR ON THEIR SKELETONS 119 5.1.10 THEOREM OF REALIZATION
120 5.1.11 MINIMAL BINARY TREES WITH REGULAR BOUNDARY 121 5.1.11.1
COMPLETE CLASSIFICATION OF LOCAL MINIMAL SKELE- TONS WITH REGULAR
BOUNDARIES 121 5.1.11.2 SOME PROPERTIES OF TILINGS FROM W7J HAVING
.RM-REALIZATION 123 5.1.12 GROWTHS AND LINEAR PARTS OF MINIMAL NETWORKS
WITH CON- VEX BOUNDARIES 124 5.1.12.1 ENDING GROWTHS 125 5.1.12.2 ENDING
VERTICES 126 5.1.12.3 GEOMETRY OF ENDING LINEAR PARTS 127 CONTENTS (A)
THE LENGTH OF THE TONGUE: THE ENDING SNAKE HAS GROWTHS 128 (B) THE
LENGTH OF THE TONGUE: THE ENDING SNAKE IS NOT ATTACHED WITH GROWTHS, BUT
THE ENDING LINEAR PART HAS A BREAK. ... 131 (C) CHARACTERISTIC
HALF-PLANES 132 (D) ESTIMATIONS ON THE LENGTH OF THE TONGUE. 133 (E) THE
LENGTH OF THE TAIL: THE ENDING LINEAR PAXT HAS A BREAK, BUT THE ENDING
SNAKE IS ATTACHED WITH GROWTHS 135 (F) MUTUAL LOCATION OF ENDING LINEAR
PARTS. . 139 5.1.13 QUASIREGULAR POLYGONS WHICH CANNOT BE SPANNED BY
MIN- IMAL BINARY TREES 145 5.1.14 NON-DEGENERATE MINIMAL NETWORKS WITH
CONVEX BOUND- ARY. CYCLICAL CASE 147 5.1.14.1 PLANAR MINIMAL REALIZATION
OF NON-DEGENERATE GRAPHS AND STEINER NETWORKS 148 5.1.14.2 FUNDAMENTAL
CYCLES OF NON-DEGENERATE MINIMAL NETWORKS WITH CONVEX BOUNDARIES.
TRIVIAL NET- WORKS 149 5.1.14.3 TWISTING NUMBER OF TRIVIAL NETWORKS 149
5.1.14.4 TILING REALIZATION OF A TRIVIAL NETWORK WHOSE TWISTING NUMBER
DOES NOT EXCEED FIVE 150 5.1.14.5 DESCRIPTION OF TILINGS OF GENERAL FORM
151 (G) SKELETONS AND GROWTHS 151 (H) TILING HULLS AND KERNELS 151 (I)
BRANCHING POINTS 152 (J) LINEAR PARTS 153 (K) STRUCTURAL ELEMENTS 154
(1) MACROELEMENTS AND ENDS 155 5.1.14.6 SKELETONS FROM 7$ 156 (M)
STRUCTURAL ELEMENTS 156 (N) DIRECTIONS OF THE ENDING MACROELEMENTS OF A
SKELETON FROM 7$ 156 (O) CODES OF SKELETONS FROM 7 158 (P) POLYGROWTHS
158 CONTENTS XIX 5.1.14.7 LOCATION OF GROWTHS ON SKELETONS OF TILINGS
FROM N 159 5.1.14.8 FINAL REMARK 160 5.2 CLOSED MINIMAL NETWORKS ON
CLOSED SURFACES OF CONSTANT CURVATURE 160 5.2.1 MINIMAL NETWORKS ON
SURFACES OF CONSTANT POSITIVE CUR- VATURE . . 162 5.2.1.1 CLOSED MINIMAL
NETWORKS ON S 2 162 5.2.1.2 CLOSED MINIMAL NETWORKS ON EP 2 163 5.2.2
CLASSIFICATION OF CLOSED MINIMAL NETWORKS ON FIAT TORI . . 164 5.2.2.1
DESCRIPTION OF FLAT METRICS ON A TWO-DIMENSIONAL TORUS 165 5.2.2.2 FLAT
TORI TRANSLATIONS GROUPS, LATTICES, AND UNI- VERSAL COVERINGS 166
5.2.2.3 NET GEODESIES 167 5.2.2.4 THE TYPE OF A NETWORK 170 5.2.2.5
CHARACTERISTIC TRIANGLE 174 5.2.2.6 CLASSIFICATION THEOREMS 175 5.2.3
CLASSIFICATION OF CLOSED MINIMAL NETWORKS ON FLAT KLEIN BOTTLES 180
5.2.3.1 DESCRIPTION OF FLAT METRICS ON A KLEIN BOTTLE . 180 5.2.3.2 THE
UNIVERSAL COVERING OF A FLAT KLEIN BOTTLE . 181 5.2.3.3 THE COVERING OF
A FLAT KLEIN BOTTLE BY A FLAT TORUSL82 5.2.3.4 REGULAR NETWORKS 184
5.2.3.5 CLASSIFICATION THEOREMS 185 5.2.4 CLOSED NETWORKS ON
TWO-DIMENSIONAL SURFACES OF NEGATIVE CURVATURE 187 5.2.4.1 METRIC
RESTRICTIONS ON THE STRUCTURE OF CLOSED NETWORKS 187 5.2.4.2 EXAMPLES OF
CLOSED MINIMAL NETWORKS ON SUR- FACES OF NEGATIVE CURVATURE 188 5.2.4.3
ENUMERATION OF CLOSED LOCAL MINIMAL NETWORKS ON SURFACES OF CONSTANT
NEGATIVE CURVATURE UP TO TOPOLOGICAL EQUIVALENCE 190 5.3 CLOSED LOCAL
MINIMAL NETWORKS ON SURFACES OF POLYHEDRA . . . . 192 5.3.1 GENERAL
PROPERTIES OF LOCAL MINIMAL NETWORKS ON POLYHEDRAL92 5.3.1.1
DEVELOPMENTS 194 5.3.1.2 LOCAL GEODESIES 196 XX CONTENTS 5.3.1.3 LOCAL
STRUCTURE OF MINIMAL NETWORKS ON POLY- HEDRA 198 5.3.1.4 THE
GAUSS-BONNET THEOREM FOR POLYHEDRA . . 199 5.3.2 METRIC AND TOPOLOGICAL
RESTRICTIONS ON THE STRUCTURE OF CLOSED MINIMAL NETWORKS 199 5.3.2.1 T.
V. PAVLYUKEVICH. EXISTENCE OF CLOSED LOCAL MINIMAL NETWORKS ON CONVEX
POLYHEDRA . . . . 200 5.3.2.2 CELLS OF NETWORKS ON CONVEX POLYHEDRA . .
. . 201 5.3.2.3 THE CASE OF REGULAR POLYHEDRA 203 5.3.3 CLASSIFICATION
OF CLOSED MINIMAL NETWORKS ON REGULAR TETRA- HEDRON 205 5.3.3.1 THE
BRANCHING COVERING OF TETRAHEDRA BY THE PLANE AND BY FLAT TORI 205
5.3.3.2 REGULAR NETWORKS 208 5.3.3.3 CLASSIFICATION THEOREMS 210 5.3.4
REPRODUCTION ALGORITHM FOR CLOSED LOCAL MINIMAL NET- WORKS ON
POLYHEDRA 211 5.3.5 CLOSED GEODESIES ON THE CUBE 216 5.4 M. V. PRONIN.
MORSE INDICES OF LOCAL MINIMAL NETWORKS 218 5.4.1 INTRODUCTION 218 5.4.2
INDEX FORM 218 5.4.3 MINIMAL NETWORKS ON NON-POSITIVE CURVATURE
MANIFOLDS . 224 5.4.4 MINIMAL NETWORKS ON THE SPHERE 226 5.4.5 INDEX
THEOREM 228 5.5 G. A. KARPUNIN. MORSE THEORY FOR PLANAR LINEAR NETWORKS
. . . 232 5.5.1 INTRODUCTION . 232 5.5.2 MORSE THEORY FOR SIMPLICIAL
COMPLEXES 234 - 5.5.3 MORSE THEORY FOR SPECIAL METRIC SPACES 235 5.5.4
MORSE THEORY FOR MINIMAL NETWORKS 240 5.5.5 SOME APPLICATIONS 247
CHAPTER 6 EXTREMALS OF FUNCTIONALS GENERATED BY NORMS 253 6.1 NORMS OF
GENERAL FORM 256 6.1.1 LOCAL MINIMAL AND EXTREME NETWORKS 256 6.1.2 THE
FIRST VARIATION OF STRAIGHT SEGMENT LENGTH IN A NOR- MALIZED SPACE 257
6.1.3 STRUCTURE OF EXTREME CURVES 263 6.1.4 LOCAL STRUCTURE OF EXTREME
LINEAR PARAMETRIC NETWORKS . 264 CONTENTS XXI 6.1.5 NETWORKS-TRACES
EXTREMALITY CRITERION 271 6.2 STABILITY OF EXTREME BINARY TREES UNDER
DEFORMATIONS OF THE BOUND- ARY . , 280 6.3 PLANAR NORMS WITH STRICTLY
CONVEX SMOOTH CIRCLES 283 6.3.1 EXTREMALITY CRITERION FOR
NETWORKS-TRACES 283 6.3.2 GEOMETRY OF EXTREME NETWORKS-TRACES 288
6.3.2.1 RELATIONS WITH EUCLIDEAN NORM CASE 288 6.3.2.2 GEOMETRY OF
BOUNDARY SETS AND TWISTING NUMBER289 6.3.2.3 SET OF EXTREME NETWORKS
HAVING A FIXED TYPE . 291 6.3.2.4 SURFACES 292 6.3.2.5 NORMS WITH
ELLIPSOIDAL CIRCLES 296 6.4 MANHATTAN LOCAL MINIMAL AND EXTREME NETWORKS
297 6.4.1 GENERAL PROPERTIES 298 6.4.2 EXTREME NETWORKS AND LINEAR
NETWORKS 299 6.4.3 EXTREME NETWORKS ON THE MANHATTAN PLANE 300 APPENDIX:
SOME UNSOLVED PROBLEMS 313 BIBLIOGRAPHY 323 INDEX 331
|
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| illustrated | Illustrated |
| indexdate | 2024-07-09T18:55:23Z |
| institution | BVB |
| isbn | 9810240600 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009563874 |
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| owner_facet | DE-703 DE-11 |
| physical | XXI, 342 S. graph. Darst. |
| publishDate | 2001 |
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| publisher | World Scientific |
| record_format | marc |
| spelling | Ivanov, Aleksandr O. Verfasser aut Branching solutions to one-dimensional variational problems A. O. Ivanov & A. A. Tuzhilin Singapore [u.a.] World Scientific 2001 XXI, 342 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Extremal problems (Mathematics) Steiner systems Dimension 1 (DE-588)4323094-5 gnd rswk-swf Variationsproblem (DE-588)4187419-5 gnd rswk-swf Steiner-Problem (DE-588)4248342-6 gnd rswk-swf Variationsproblem (DE-588)4187419-5 s Dimension 1 (DE-588)4323094-5 s DE-604 Steiner-Problem (DE-588)4248342-6 s Tužilin, Aleksej A. Verfasser aut GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009563874&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ivanov, Aleksandr O. Tužilin, Aleksej A. Branching solutions to one-dimensional variational problems Extremal problems (Mathematics) Steiner systems Dimension 1 (DE-588)4323094-5 gnd Variationsproblem (DE-588)4187419-5 gnd Steiner-Problem (DE-588)4248342-6 gnd |
| subject_GND | (DE-588)4323094-5 (DE-588)4187419-5 (DE-588)4248342-6 |
| title | Branching solutions to one-dimensional variational problems |
| title_auth | Branching solutions to one-dimensional variational problems |
| title_exact_search | Branching solutions to one-dimensional variational problems |
| title_full | Branching solutions to one-dimensional variational problems A. O. Ivanov & A. A. Tuzhilin |
| title_fullStr | Branching solutions to one-dimensional variational problems A. O. Ivanov & A. A. Tuzhilin |
| title_full_unstemmed | Branching solutions to one-dimensional variational problems A. O. Ivanov & A. A. Tuzhilin |
| title_short | Branching solutions to one-dimensional variational problems |
| title_sort | branching solutions to one dimensional variational problems |
| topic | Extremal problems (Mathematics) Steiner systems Dimension 1 (DE-588)4323094-5 gnd Variationsproblem (DE-588)4187419-5 gnd Steiner-Problem (DE-588)4248342-6 gnd |
| topic_facet | Extremal problems (Mathematics) Steiner systems Dimension 1 Variationsproblem Steiner-Problem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009563874&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT ivanovaleksandro branchingsolutionstoonedimensionalvariationalproblems AT tuzilinalekseja branchingsolutionstoonedimensionalvariationalproblems |