Discrete geometry and algebraic combinatorics / Alexander Barg, Oleg R. Musin, editors.

Format:

Book

Contributor:

Barg, Alexander, 1960- editor.

Musin, O. R. (Oleg Rustamovich), editor.

Series:

Contemporary mathematics (American Mathematical Society) ; 625.

Contemporary Mathematics, 1098-3627 ; 625

Language:

English

Subjects (All):

Discrete geometry--Congresses.

Discrete geometry.

Combinatorial analysis--Congresses.

Combinatorial analysis.

Physical Description:

1 online resource (190 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2014.

Language Note:

English

Summary:

This volume contains the proceedings of the AMS Special Session on Discrete Geometry and Algebraic Combinatorics held on January 11, 2013, in San Diego, California, USA. The collection of articles in this volume is devoted to packings of metric spaces and related questions, and contains new results as well as surveys of some areas of discrete geometry. This volume consists of papers on combinatorics of transportation polytopes, including results on the diameter of graphs of such polytopes; the generalized Steiner problem and related topics of the minimal fillings theory; a survey of distance graphs and graphs of diameters, and a group of papers on applications of algebraic combinatorics to packings of metric spaces including sphere packings and topics in coding theory. In particular, this volume presents a new approach to duality in sphere packing based on the Poisson summation formula, applications of semidefinite programming to spherical codes and equiangular lines, new results in list decoding of a family of algebraic codes, and constructions of bent and semi-bent functions.

Contents:

""Cover""; ""Title page""; ""Contents""; ""Preface""; ""Plank theorems via successive inradii""; ""1. Introduction""; ""2. Extensions to Successive Inradii""; ""3. Proof of Theorem 2.1""; ""4. Proof of Theorem 2.2""; ""5. Proof of Corollary 2.3""; ""6. The equivalence of Conjectures 1.2, ???, ???, ???, ???, and ???""; ""7. Conclusion""; ""References""; ""Minimal fillings of finite metric spaces: The state of the art""; ""1. Introduction: Length-Minimizing Connections""; ""2. Combinatorial Definition of Minimal Filling""; ""3. Parametric Minimal Fillings""

""4. Realization of Minimal Filling as a Minimal Network""""5. Minimal Parametric Fillings and Linear Programming""; ""6. Generalized Fillings""; ""7. Formula for the Weight of Minimal Filling""; ""8. Uniqueness Problem""; ""9. Minimal Fillings of Additive and Pseudo-Additive Spaces""; ""10. Examples of Minimal Fillings""; ""11. Ratios""; ""12. Generalizations for Infinite Sets""; ""Acknowledgments""; ""References""; ""Combinatorics and geometry of transportation polytopes: An update""; ""1. Introduction""; ""2. Classical transportation polytopes (2-ways)""

""3. Multi-way transportation polytopes""""4. Further research directions and more open problems""; ""Acknowlegements""; ""References""; ""A Tree Sperner Lemma""; ""1. Introduction""; ""2. A Tree Sperner Lemma""; ""3. Metric Trees and Segmentations""; ""4. KKM Covers of Trees""; ""5. A Fixed Point Theorem for Finite Trees""; ""6. Infinite Settings""; ""7. A KKM Theorem for Cycles""; ""References""; ""Cliques and cycles in distance graphs and graphs of diameters""; ""1. Distance graphs: definitions and motivation""; ""2. Graphs of diameters: definitions and motivation""

""3. What is the role of cliques and cycles in geometric graphs?""""4. Counting cliques in distance graphs and graphs of diameters""; ""5. Distance graphs with exponential chromatic numbers and without cliques or cycles""; ""6. The chromatic numbers of spheres""; ""7. Counterexamples to Borsuk�s conjecture on spheres of small radii""; ""References""; ""New bounds for equiangular lines""; ""1. Introduction""; ""2. SDP bounds for equiangular lines""; ""3. Tight spherical designs of harmonic index 4 and equiangular lines""; ""References""

""Formal duality and generalizations of the Poisson summation formula""""1. Introduction""; ""2. Poisson summation formulas and duality""; ""3. Examples""; ""4. Structure theory in the cyclic case""; ""5. Non-existence of some formal duals""; ""6. Open questions""; ""Acknowledgments""; ""References""; ""On constructions of semi-bent functions from bent functions""; ""1. Introduction""; ""2. Notation and preliminaries""; ""3. Constructions of semi-bent functions from bent functions""; ""4. Conclusion""; ""References""; ""Some remarks on multiplicity codes""; ""1. Introduction""

""2. Multiplicity Codes""

Notes:

"AMS Special Session on Discrete Geometry and Algebraic Combinatorics, January 11, 2013".

Includes bibliographical references at the end of each chapters.

Description based on print version record.

ISBN:

1-4704-1949-1