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" New trends in intuitive geometry / "
Gergely Ambrus, Imre Bárány, Károly J. Böröczky, Gábor Fejes Tóth, János Pach, editors.
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BL
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| Record Number
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882815
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| Title & Author
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New trends in intuitive geometry /\ Gergely Ambrus, Imre Bárány, Károly J. Böröczky, Gábor Fejes Tóth, János Pach, editors.
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| Publication Statement
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Berlin, Germany :: Springer,, 2018.
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| Series Statement
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Bolyai Society mathematical studies,; volume 27
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1 online resource
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| ISBN
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3662574136
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: 9783662574133
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3662574128
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9783662574126
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| Contents
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Intro; Preface; Contents; The Tensorization Trick in Convex Geometry; 1 Introduction; 1.1 Tensor Power; 1.2 Tensor Powers and Polynomials; 1.3 Chebyshev Polynomials; 2 Packing Points in the Sphere; 2.1 Packing Points in the Sphere; 3 Approximating a Norm by a Polynomial and a Convex Body by an Algebraic Hypersurface; 3.1 Norms and Convex Bodies; 3.2 Some Geometric Corollaries; 3.3 Approximation by Convex Semi-algebraic Sets; 4 Approximation of Convex Bodies by Polytopes; 4.1 Convex Bodies and Polytopes; 4.2 Fine Approximations; 4.3 Coarse Approximations; 4.4 Intermediate Approximations
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3.3 Two Lemmas4 Proofs; 4.1 Proof of the Topological Generalized Transversal Van Kampen-Flores Theorem; 4.2 Proof of the Topological Transversal Weak Colored Tverberg Theorem; References; On the Volume of Boolean Expressions of Balls -- A Review of the Kneser-Poulsen Conjecture; 1 Introduction; 2 First Results; 3 Unions and Intersections of Balls in Spaces of Constant Curvature; 4 Jumping into Higher Dimensions -- The Leapfrog Lemma; 5 Proof of the Kneser-Poulsen Conjecture in the Euclidean Plane; 6 Monotonicity of the Volume of Weighted Flowers
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5 The Polynomial Method5.1 Constructing Neighborly Polytopes; 5.2 Bounding the Constant in the Grothendieck Inequality; 5.3 Polynomial Ham Sandwich Theorem; 5.4 Equiangular Lines; 5.5 A Counterexample to Borsuk's Conjecture; References; Contact Numbers for Sphere Packings; 1 Introduction; 2 Motivation from Materials Science; 3 Largest Contact Numbers in the Plane; 3.1 The Euclidean Plane; 3.2 Spherical and Hyperbolic Planes; 4 Largest Contact Numbers in 3-Space; 5 Empirical Approaches; 5.1 Contact Number Estimates for up to 11 Spheres; 5.2 Maximal Contact Rigid Clusters
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6 Digital and Totally Separable Sphere Packings for d=2, 37 On Largest Contact Numbers in Higher Dimensional Spaces; 7.1 Packings by Translates of a Convex Body; 7.2 Contact Graphs of Unit Sphere Packings in mathbbEd; 7.3 Digital and Totally Separable Sphere Packings in mathbbEd; 8 Contact Graphs of Non-congruent Sphere Packings; References; The Topological Transversal Tverberg Theorem Plus Constraints; 1 Introduction; 2 Statement of the Main Results; 3 A Generalized Borsuk-Ulam Type Theorem and Two Lemmas; 3.1 Fadell-Husseini Index; 3.2 A Generalized Borsuk-Ulam Type Theorem
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7 A Schläfli-Type Formula for Polytopes with Curved Faces7.1 Polytopes with Curved Faces in a Manifold and Their Variations; 7.2 Generalized Schläfli Formula in Einstein Manifolds; 8 Some Applications of the Generalized Schläfli Formula; 9 Application to a Problem of M. Kneser; 10 Alexander's Conjecture; 11 The Conjecture in More General Spaces; References; A Survey of Elekes-Rónyai-Type Problems; 1 The Elekes-Rónyai Problem; 1.1 Sums, Products, and Expanding Polynomials; 1.2 Extensions; 1.3 Applications; 1.4 About the Proof of Theorem1.1; 2 The Elekes-Szabó Problem
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| Abstract
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This volume contains 17 surveys that cover many recent developments in Discrete Geometry and related fields. Besides presenting the state-of-the-art of classical research subjects like packing and covering, it also offers an introduction to new topological, algebraic and computational methods in this very active research field. The readers will find a variety of modern topics and many fascinating open problems that may serve as starting points for research.--
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Geometry.
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Geometry.
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MATHEMATICS-- Geometry-- General.
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| Dewey Classification
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516
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| LC Classification
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QA445
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| Added Entry
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Ambrus, Gergely
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Bárány, Imre
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Böröczky, K.
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Pach, János
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Tóth, Gábor,Ph. D.
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