1650101732  

Triangulations : Structures for Algorithms and Applications / by Jesús A. Loera; Jörg Rambau; Francisco Santos

De Loera, Jesús A. *1966-* ; Rambau, Jörg *1966-* ; Santos, Francisco

Englisch

Berlin, Heidelberg : Springer-Verlag Berlin Heidelberg, 2010

Online-Ressource (X, 539p. 496 illus., 281 illus. in color, digital)

Includes bibliographical references and index

Buchausg. u.d.T.: De Loera, Jesús A. *1966-*: Triangulations. - Berlin, Heidelberg : Springer, 2010. - XIII, 535 S.

978-3-642-12971-1

1-280-39177-4 ebk

978-1-280-39177-4 MyiLibrary

Weitere Ausgaben: 978-3-642-12970-4 (Druckausgabe)

DOI: 10.1007/978-3-642-12971-1

""Triangulations ""; ""Preface""; ""Contents ""; ""1 Triangulations in Mathematics""; ""1.1 Combinatorics and triangulations""; ""1.2 Optimization and triangulations""; ""1.3 Algebra and triangulations""; ""1.4 The rest of this book""; "" Exercises""; ""2 Configurations, Triangulations, Subdivisions, and Flips""; ""2.1 The official languages in the land of triangulations""; ""2.1.1 Polyhedra and cones""; ""2.1.2 Point configurations""; ""2.1.3 Geometry of point configurations""; ""2.2 A closer look at the definition of triangulation""; ""2.2.1 There is always a triangulation""

""2.2.2 A famous example: the Delaunay triangulation""""2.2.3 Regular subdivisions and their structure""; ""2.3 A bullet-proof definition of polyhedral subdivisions""; ""2.3.1 Polyhedral subdivisions""; ""2.3.2 Regular subdivisions, again""; ""2.4 Flips and the graph of triangulations""; ""2.4.1 Corank-one configurations and circuits""; ""2.4.2 Almost-triangulations and flips""; ""2.5 Vector configurations and their triangulations""; ""2.5.1 Vector configurations""; ""2.5.2 Polyhedral subdivisions of vector configurations""; ""2.5.3 Regular subdivisions of vector configurations""

""2.6 Triangulations as simplicial complexes""""2.6.1 Simplicial complexes""; ""2.6.2 The f-vector of a simplicial complexes""; ""2.6.3 Linear constraints on the f-vector""; "" Exercises""; ""3 Life in Two Dimensions""; ""3.1 Some basic properties""; ""3.2 A few examples of triangulations in the plane""; ""3.2.1 Placing and pulling triangulations""; ""3.2.2 Delaunay triangulations""; ""3.2.3 Greedy and minimum weight triangulations""; ""3.3 The set of all triangulations of a point set""; ""3.3.1 The exact number of triangulations""; ""3.3.2 The maximum possible number of triangulations""

""3.3.3 The minimum possible number of triangulations""""3.3.4 The poset of subdivisions""; ""3.4 Flips in triangulations""; ""3.4.1 All triangulations of a point set in the plane are connected by flips""; ""3.4.2 Effective enumeration of triangulations""; ""3.4.3 Further properties of the graph of flips""; ""3.5 Pseudo-triangulations""; ""3.6 Life in three dimensions""; ""3.6.1 The number of tetrahedra""; ""3.6.2 Monotone flipping does not (always) work""; ""3.6.3 The number of flips""; ""3.7 Notes and References""; "" Exercises""; ""4 A Tool Box""; ""4.1 Combinatorics of configurations""

""4.1.1 Dependences, circuits, and the intersection property""""4.1.2 Evaluations, cocircuits, and the union property""; ""4.1.3 Gale transforms and the duality between circuits and cocircuits""; ""4.2 Manipulating vector configurations""; ""4.2.1 Pyramids and joins""; ""4.2.2 Prisms and products""; ""4.2.3 Deletion""; ""4.2.4 Contraction""; ""4.2.5 One-point suspension""; ""4.3 Generating polyhedral subdivisions""; ""4.3.1 The placing (or pushing) triangulation""; ""4.3.2 The pulling triangulation""; ""4.3.3 Lexicographic triangulations""; ""4.3.4 Pushing and pulling refinements""

""4.4 Two equivalent characterizations of flips""

*Triangulation

*Triangulation

*Computer science ; Algorithms ; Combinatorics ; Discrete groups ; Mathematics ; Computer science -- Mathematics ; Discrete mathematics. ; Computers. ; Convex geometry . ; Discrete geometry. ; Algebraic geometry. ; Computer mathematics. ; Mathematics ; Computer science ; Computer science -- Mathematics ; Algorithms ; Combinatorics ; Discrete groups

Klassifikation der Library of Congress: QA639.5-640.7 ; QA640.7-640.77 ; TA583

Dewey Dezimal-Klassifikation: 516.35; ; 516.1;

Regensburger Verbund-Klassifikation: SK 890: SA-SP Mathematik / SK Monografien / Ganzzahlige und kombinatorische Optimierung, Graphentheorie

Book Industry Communication: PBD

bisacsh: MAT012020

bisacsh: MAT008000

Book Industry Communication: PBMW

bisacsh: MAT012010

Mathematics Subject Classification: *52-02

Mathematics Subject Classification: 52B05

Mathematics Subject Classification: 52B11

Mathematics Subject Classification: 52B45

Mathematics Subject Classification: 52B55

Mathematics Subject Classification: 68R05

Triangulations in Mathematics -- Configurations, Triangulations, Subdivisions, and Flips -- Life in Two Dimensions -- A Tool Box -- Regular Triangulations and Secondary Polytopes -- Some Interesting Configurations -- Some Interesting Triangulations -- Algorithmic Issues -- Further Topics.

Triangulations appear everywhere, from volume computations and meshing to algebra and topology. This book studies the subdivisions and triangulations of polyhedral regions and point sets and presents the first comprehensive treatment of the theory of secondary polytopes and related topics. A central theme of the book is the use of the rich structure of the space of triangulations to solve computational problems (e.g., counting the number of triangulations or finding optimal triangulations with respect to various criteria), and to establish connections to applications in algebra, computer science, combinatorics, and optimization. With many examples and exercises, and with nearly five hundred illustrations, the book gently guides readers through the properties of the spaces of triangulations of "structured" (e.g., cubes, cyclic polytopes, lattice polytopes) and "pathological" (e.g., disconnected spaces of triangulations) situations using only elementary principles.

https://zbmath.org/?q=an:1207.52002 [Zentralblatt MATH]

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https://doi.org/10.1007/978-3-642-12971-1 [E-books (Springer)]