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An Introduction To Convex Polytopes

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An Introduction To Convex Polytopes

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Author : Arne Brondsted
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

An Introduction To Convex Polytopes written by Arne Brondsted and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

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.

An Introduction To Convex Polytopes

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Author : Arne Brøndsted
language : en
Publisher:
Release Date : 1983-01

An Introduction To Convex Polytopes written by Arne Brøndsted and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1983-01 with Convex polytopes categories.

Lectures On Polytopes

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Author : Günter M. Ziegler
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-05-03

Lectures On Polytopes written by Günter M. Ziegler and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-05-03 with Mathematics categories.

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.

Grobner Bases And Convex Polytopes

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Author : Bernd Sturmfels
language : en
Publisher: American Mathematical Soc.
Release Date : 1996

Grobner Bases And Convex Polytopes written by Bernd Sturmfels and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996 with Mathematics categories.

This book is about the interplay of computational commutative algebra and the theory of convex polytopes. It centres 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 Gröbner 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.

Convex Optimization

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Author : Stephen P. Boyd
language : en
Publisher: Cambridge University Press
Release Date : 2004-03-08

Convex Optimization written by Stephen P. Boyd and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004-03-08 with Business & Economics categories.

Convex optimization problems arise frequently in many different fields. This book provides a comprehensive introduction to the subject, and shows in detail how such problems can be solved numerically with great efficiency. The book begins with the basic elements of convex sets and functions, and then describes various classes of convex optimization problems. Duality and approximation techniques are then covered, as are statistical estimation techniques. Various geometrical problems are then presented, and there is detailed discussion of unconstrained and constrained minimization problems, and interior-point methods. The focus of the book is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. It contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance and economics.

Lectures On Polytopes

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Author : Günter M. Ziegler
language : en
Publisher: Springer
Release Date : 2012-05-03

Lectures On Polytopes written by Günter M. Ziegler and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-05-03 with Mathematics categories.

Polytopes And Symmetry

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Author : Stewart A. Robertson
language : en
Publisher: Cambridge University Press
Release Date : 1984-01-26

Polytopes And Symmetry written by Stewart A. Robertson and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984-01-26 with Mathematics categories.

This book describes a fresh approach to the classification of of convex plane polygons and of convex polyhedra according to their symmetry properties, based on ideas of topology and transformation group theory. Although there is considerable agreement with traditional treatments, a number of new concepts emerge that present classical ideas in a quite new way.

Convex Bodies And Algebraic Geometry

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Author : Tadao Oda
language : en
Publisher: Springer Verlag
Release Date : 1988

Convex Bodies And Algebraic Geometry written by Tadao Oda and has been published by Springer Verlag this book supported file pdf, txt, epub, kindle and other format this book has been release on 1988 with Mathematics categories.

Convex Polyhedra

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Author : A.D. Alexandrov
language : en
Publisher: Springer Science & Business Media
Release Date : 2005-02-10

Convex Polyhedra written by A.D. Alexandrov and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-02-10 with Mathematics categories.

This classic geometry text explores the theory of 3-dimensional convex polyhedra in a unique fashion, with exceptional detail. Vital and clearly written, the book includes the basics of convex polyhedra and collects the most general existence theorems for convex polyhedra that are proved by a new and unified method. This edition includes a comprehensive bibliography by V.A. Zalgaller, and related papers as supplements to the original text.

Introduction To Toric Varieties

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Author : William Fulton
language : en
Publisher: Princeton University Press
Release Date : 1993

Introduction To Toric Varieties written by William Fulton and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993 with Mathematics categories.

Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic geometry notions such as singularities, birational maps, cycles, homology, intersection theory, and Riemann-Roch translate into simple facts about polytopes, toric varieties provide a marvelous source of examples in algebraic geometry. In the other direction, general facts from algebraic geometry have implications for such polytopes, such as to the problem of the number of lattice points they contain. In spite of the fact that toric varieties are very special in the spectrum of all algebraic varieties, they provide a remarkably useful testing ground for general theories. The aim of this mini-course is to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications. The text concludes with Stanley's theorem characterizing the numbers of simplicies in each dimension in a convex simplicial polytope. Although some general theorems are quoted without proof, the concrete interpretations via simplicial geometry should make the text accessible to beginners in algebraic geometry.

An Introduction To Convex Polytopes

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