Lectures On Polytopes

Author : Günter M. Ziegler
Publisher : Springer Science & Business Media
Page : 388 pages
File Size : 41,5 Mb
Release : 2012-05-03
Category : Mathematics
ISBN : 9780387943657

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Lectures on Polytopes by Günter M. Ziegler Pdf

Based on a graduate course at the Technische Universität, Berlin, these lectures present a wealth of material on the modern theory of convex polytopes. The straightforward exposition features many illustrations, and complete proofs for most theorems. With only linear algebra as a prerequisite, it takes the reader quickly from the basics to topics of recent research. The lectures introduce basic facts about polytopes, with an emphasis on methods that yield the results, discuss important examples and elegant constructions, and show the excitement of current work in the field. They will provide interesting and enjoyable reading for researchers as well as students.

Author : Günter M. Ziegler
Publisher : Springer
Page : 388 pages
File Size : 45,8 Mb
Release : 2012-05-03
Category : Mathematics
ISBN : 038794365X

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Lectures on Polytopes by Günter M. Ziegler Pdf

Based on a graduate course at the Technische Universität, Berlin, these lectures present a wealth of material on the modern theory of convex polytopes. The straightforward exposition features many illustrations, and complete proofs for most theorems. With only linear algebra as a prerequisite, it takes the reader quickly from the basics to topics of recent research. The lectures introduce basic facts about polytopes, with an emphasis on methods that yield the results, discuss important examples and elegant constructions, and show the excitement of current work in the field. They will provide interesting and enjoyable reading for researchers as well as students.

Author : Jiri Matousek
Publisher : Springer Science & Business Media
Page : 491 pages
File Size : 51,9 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9781461300397

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Lectures on Discrete Geometry by Jiri Matousek Pdf

The main topics in this introductory text to discrete geometry include basics on convex sets, convex polytopes and hyperplane arrangements, combinatorial complexity of geometric configurations, intersection patterns and transversals of convex sets, geometric Ramsey-type results, and embeddings of finite metric spaces into normed spaces. In each area, the text explains several key results and methods.

Author : Bernd Sturmfels
Publisher : American Mathematical Soc.
Page : 162 pages
File Size : 54,8 Mb
Release : 1996
Category : Mathematics
ISBN : 9780821804872

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Gröbner Bases and Convex Polytopes by Bernd Sturmfels Pdf

This book is about the interplay of computational commutative algebra and the theory of convex polytopes. It centers around a special class of ideals in a polynomial ring: the class of toric ideals. They are characterized as those prime ideals that are generated by monomial differences or as the defining ideals of toric varieties (not necessarily normal). The interdisciplinary nature of the study of Grobner bases is reflected by the specific applications appearing in this book. These applications lie in the domains of integer programming and computational statistics. The mathematical tools presented in the volume are drawn from commutative algebra, combinatorics, and polyhedral geometry.

Author : Branko Grünbaum
Publisher : Springer Science & Business Media
Page : 561 pages
File Size : 41,6 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9781461300199

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Convex Polytopes by Branko Grünbaum Pdf

"The original edition [...] inspired a whole generation of grateful workers in polytope theory. Without it, it is doubtful whether many of the subsequent advances in the subject would have been made. The many seeds it sowed have since grown into healthy trees, with vigorous branches and luxuriant foliage. It is good to see it in print once again." --Peter McMullen, University College London

Author : Rekha R. Thomas
Publisher : American Mathematical Soc.
Page : 156 pages
File Size : 44,5 Mb
Release : 2006
Category : Mathematics
ISBN : 0821841408

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Lectures in Geometric Combinatorics by Rekha R. Thomas Pdf

This book presents a course in the geometry of convex polytopes in arbitrary dimension, suitable for an advanced undergraduate or beginning graduate student. The book starts with the basics of polytope theory. Schlegel and Gale diagrams are introduced as geometric tools to visualize polytopes in high dimension and to unearth bizarre phenomena in polytopes. The heart of the book is a treatment of the secondary polytope of a point configuration and its connections to the statepolytope of the toric ideal defined by the configuration. These polytopes are relatively recent constructs with numerous connections to discrete geometry, classical algebraic geometry, symplectic geometry, and combinatorics. The connections rely on Grobner bases of toric ideals and other methods fromcommutative algebra. The book is self-contained and does not require any background beyond basic linear algebra. With numerous figures and exercises, it can be used as a textbook for courses on geometric, combinatorial, and computational aspects of the theory of polytopes.

Author : Rekha R. Thomas
Publisher : Unknown
Page : 143 pages
File Size : 52,7 Mb
Release : 2006
Category : Combinatorial analysis
ISBN : 1470421445

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Lectures in Geometric Combinatorics by Rekha R. Thomas Pdf

Author : Arne Brondsted
Publisher : Springer Science & Business Media
Page : 168 pages
File Size : 54,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461211488

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An Introduction to Convex Polytopes by Arne Brondsted Pdf

The aim of this book is to introduce the reader to the fascinating world of convex polytopes. The highlights of the book are three main theorems in the combinatorial theory of convex polytopes, known as the Dehn-Sommerville Relations, the Upper Bound Theorem and the Lower Bound Theorem. All the background information on convex sets and convex polytopes which is m~eded to under stand and appreciate these three theorems is developed in detail. This background material also forms a basis for studying other aspects of polytope theory. The Dehn-Sommerville Relations are classical, whereas the proofs of the Upper Bound Theorem and the Lower Bound Theorem are of more recent date: they were found in the early 1970's by P. McMullen and D. Barnette, respectively. A famous conjecture of P. McMullen on the charac terization off-vectors of simplicial or simple polytopes dates from the same period; the book ends with a brief discussion of this conjecture and some of its relations to the Dehn-Sommerville Relations, the Upper Bound Theorem and the Lower Bound Theorem. However, the recent proofs that McMullen's conditions are both sufficient (L. J. Billera and C. W. Lee, 1980) and necessary (R. P. Stanley, 1980) go beyond the scope of the book. Prerequisites for reading the book are modest: standard linear algebra and elementary point set topology in [R1d will suffice.

Author : Daniel Hug,Wolfgang Weil
Publisher : Springer Nature
Page : 287 pages
File Size : 50,7 Mb
Release : 2020-08-27
Category : Mathematics
ISBN : 9783030501808

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Lectures on Convex Geometry by Daniel Hug,Wolfgang Weil Pdf

This book provides a self-contained introduction to convex geometry in Euclidean space. After covering the basic concepts and results, it develops Brunn–Minkowski theory, with an exposition of mixed volumes, the Brunn–Minkowski inequality, and some of its consequences, including the isoperimetric inequality. Further central topics are then treated, such as surface area measures, projection functions, zonoids, and geometric valuations. Finally, an introduction to integral-geometric formulas in Euclidean space is provided. The numerous exercises and the supplementary material at the end of each section form an essential part of the book. Convexity is an elementary and natural concept. It plays a key role in many mathematical fields, including functional analysis, optimization, probability theory, and stochastic geometry. Paving the way to the more advanced and specialized literature, the material will be accessible to students in the third year and can be covered in one semester.

Author : Gil Kalai,Günter M. Ziegler
Publisher : Birkhäuser
Page : 228 pages
File Size : 42,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034884389

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Polytopes - Combinations and Computation by Gil Kalai,Günter M. Ziegler Pdf

Questions that arose from linear programming and combinatorial optimization have been a driving force for modern polytope theory, such as the diameter questions motivated by the desire to understand the complexity of the simplex algorithm, or the need to study facets for use in cutting plane procedures. In addition, algorithms now provide the means to computationally study polytopes, to compute their parameters such as flag vectors, graphs and volumes, and to construct examples of large complexity. The papers of this volume thus display a wide panorama of connections of polytope theory with other fields. Areas such as discrete and computational geometry, linear and combinatorial optimization, and scientific computing have contributed a combination of questions, ideas, results, algorithms and, finally, computer programs.

Author : Ezra Miller,Victor Reiner,Bernd Sturmfels
Publisher : American Mathematical Soc.
Page : 710 pages
File Size : 44,5 Mb
Release : 2024-06-16
Category : Mathematics
ISBN : 0821886959

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Geometric Combinatorics by Ezra Miller,Victor Reiner,Bernd Sturmfels Pdf

Geometric combinatorics describes a wide area of mathematics that is primarily the study of geometric objects and their combinatorial structure. This text is a compilation of expository articles at the interface between combinatorics and geometry.

Author : Alexander Barvinok
Publisher : American Mathematical Soc.
Page : 378 pages
File Size : 55,9 Mb
Release : 2002-11-19
Category : Mathematics
ISBN : 9780821829684

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A Course in Convexity by Alexander Barvinok Pdf

Convexity is a simple idea that manifests itself in a surprising variety of places. This fertile field has an immensely rich structure and numerous applications. Barvinok demonstrates that simplicity, intuitive appeal, and the universality of applications make teaching (and learning) convexity a gratifying experience. The book will benefit both teacher and student: It is easy to understand, entertaining to the reader, and includes many exercises that vary in degree of difficulty. Overall, the author demonstrates the power of a few simple unifying principles in a variety of pure and applied problems. The prerequisites are minimal amounts of linear algebra, analysis, and elementary topology, plus basic computational skills. Portions of the book could be used by advanced undergraduates. As a whole, it is designed for graduate students interested in mathematical methods, computer science, electrical engineering, and operations research. The book will also be of interest to research mathematicians, who will find some results that are recent, some that are new, and many known results that are discussed from a new perspective.

Author : Alexander Barvinok
Publisher : European Mathematical Society
Page : 204 pages
File Size : 55,8 Mb
Release : 2008
Category : Mathematics
ISBN : 3037190523

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Integer Points in Polyhedra by Alexander Barvinok Pdf

This is a self-contained exposition of several core aspects of the theory of rational polyhedra with a view towards algorithmic applications to efficient counting of integer points, a problem arising in many areas of pure and applied mathematics. The approach is based on the consistent development and application of the apparatus of generating functions and the algebra of polyhedra. Topics range from classical, such as the Euler characteristic, continued fractions, Ehrhart polynomial, Minkowski Convex Body Theorem, and the Lenstra-Lenstra-Lovasz lattice reduction algorithm, to recent advances such as the Berline-Vergne local formula. The text is intended for graduate students and researchers. Prerequisites are a modest background in linear algebra and analysis as well as some general mathematical maturity. Numerous figures, exercises of varying degree of difficulty as well as references to the literature and publicly available software make the text suitable for a graduate course.

Author : Ana Cannas da Silva
Publisher : Springer
Page : 220 pages
File Size : 53,9 Mb
Release : 2004-10-27
Category : Mathematics
ISBN : 9783540453307

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Lectures on Symplectic Geometry by Ana Cannas da Silva Pdf

The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved.

Author : Radu Laza,Matthias Schütt,Noriko Yui
Publisher : Springer
Page : 547 pages
File Size : 44,7 Mb
Release : 2015-08-27
Category : Mathematics
ISBN : 9781493928309

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Calabi-Yau Varieties: Arithmetic, Geometry and Physics by Radu Laza,Matthias Schütt,Noriko Yui Pdf

This volume presents a lively introduction to the rapidly developing and vast research areas surrounding Calabi–Yau varieties and string theory. With its coverage of the various perspectives of a wide area of topics such as Hodge theory, Gross–Siebert program, moduli problems, toric approach, and arithmetic aspects, the book gives a comprehensive overview of the current streams of mathematical research in the area. The contributions in this book are based on lectures that took place during workshops with the following thematic titles: “Modular Forms Around String Theory,” “Enumerative Geometry and Calabi–Yau Varieties,” “Physics Around Mirror Symmetry,” “Hodge Theory in String Theory.” The book is ideal for graduate students and researchers learning about Calabi–Yau varieties as well as physics students and string theorists who wish to learn the mathematics behind these varieties.