Triangulations Structures for Algorithms and Applications için kapak resmi
Başlık:
Triangulations Structures for Algorithms and Applications
Dil:
English
ISBN:
9783642129711
Yayın Bilgileri:
Berlin, Heidelberg : Springer Berlin Heidelberg, 2010.
Fiziksel Tanımlama:
X, 539 p. 496 illus., 281 illus. in color. online resource.
Seri:
Algorithms and Computation in Mathematics, 25
İçerik:
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.
Özet:
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.

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Özet

Özet

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.


Yazar Notları

J.A. De Loera is a professor of mathematics at the University of California, Davis. His work approaches difficult computational problems in discrete mathematics and optimization using tools from algebra and convex geometry. His research has been recognized by an Alexander von Humboldt Fellowship and several national and international grants. He is an associate editor of the journal "Discrete Optimization". Jörg Rambau is the chair professor of Wirtschaftsmathematik (Business Mathematics) at the Universität of Bayreuth since 2004. Before that he was associate head of the optimization department at the Zuse Institute Berlin (ZIB). His research encompasses problems in applied optimization, algorithmic discrete mathematics and combinatorial geometry. He is the creator of the state of the art program for triangulation computations TOPCOM. He is associate editor of the "Jahresberichte der Deutschen Mathematiker-Vereinigung". Francisco Santos, a professor at the Universidad de Cantabria Spain, received the Young Researcher award from the Universidad Complutense de Madrid in 2003 and was an invited speaker in the Combinatorics Section of the International Congress of Mathematicians in 2006. He is well-known for his explicit constructions of polytopes with disconnected spaces of triangulations, some of which are featured in this book. He is an editor of Springer Verlag's journal "Discrete and Computational Geometry".


İncelemeler 1

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De Loera (Univ. of California, Davis), Rambau (Univ. of Bayreuth, Germany), and Santos (Univ. of Cantabria, Spain) refer to triangulations as decompositions of a region of Euclidean space into simplices (i.e., line segments, triangles, tetrahedra, etc.) with vertices lying in a predetermined set. Focusing on the structure of the set of all possible triangulations (subject perhaps to some hypothesis), the current study sits at the threshold of geometry and combinatorics; the familiar story of Catalan numbers and polygon triangulations offers a tiny taste of this rich story. While basic objects here possess concreteness offering terra firma to students still struggling with abstraction, the central theorem (due to Gelfand, Kapranov, and Zelevinsky) only dates to 1989, so the present elaboration carries readers to the frontiers of research. Similar to knot theory, this makes triangulations perfect for an undergraduate capstone course. It is unusual to find such a leisurely, generous exposition of a new subject, as replete with illustrations as contemporary calculus textbooks. Chapter 8 provides the computer scientist's vantage. Triangulations do have practical applications (graphics, differential equation solvers) and material here does have practical implications, but this volume concentrates on basic theory, veering occasionally into mathematical applications, e.g., real algebraic geometry and Hilbert's 16th problem. Summing Up: Recommended. Upper-division undergraduates through professionals. D. V. Feldman University of New Hampshire