An Introduction to Convex Polytopes

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Author: Arne Brondsted

Publisher: Springer Science & Business Media

ISBN: 1461211484

Category: Mathematics

Page: 162

View: 9817

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.

Gröbner Bases and Convex Polytopes

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Author: Bernd Sturmfels

Publisher: American Mathematical Soc.

ISBN: 0821804871

Category: Mathematics

Page: 162

View: 6044

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.

Lectures on Polytopes

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Author: Günter M. Ziegler

Publisher: Springer Science & Business Media

ISBN: 9780387943657

Category: Mathematics

Page: 370

View: 7408

Based on a graduate course at the Technische Universität, Berlin, this book presents a wealth of material on the modern theory of convex polytopes. With linear algebra as a prerequisite, the text moves quickly from the basics to topics of recent research.

Lectures on Discrete Geometry

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Author: Ji?í Matoušek

Publisher: Springer Science & Business Media

ISBN: 1461300398

Category: Mathematics

Page: 486

View: 9445

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.

Combinatorial Convexity and Algebraic Geometry

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Author: Günter Ewald

Publisher: Springer Science & Business Media

ISBN: 1461240441

Category: Mathematics

Page: 374

View: 9696

The book is an introduction to the theory of convex polytopes and polyhedral sets, to algebraic geometry, and to the connections between these fields, known as the theory of toric varieties. The first part of the book covers the theory of polytopes and provides large parts of the mathematical background of linear optimization and of the geometrical aspects in computer science. The second part introduces toric varieties in an elementary way.

A Course in Convexity

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Author: Alexander Barvinok

Publisher: American Mathematical Soc.

ISBN: 0821829688

Category: Mathematics

Page: 366

View: 4326

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.

Computing the Continuous Discretely

Integer-Point Enumeration in Polyhedra

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Author: Matthias Beck,Sinai Robins

Publisher: Springer

ISBN: 1493929690

Category: Mathematics

Page: 285

View: 5540

This richly illustrated textbook explores the amazing interaction between combinatorics, geometry, number theory, and analysis which arises in the interplay between polyhedra and lattices. Highly accessible to advanced undergraduates, as well as beginning graduate students, this second edition is perfect for a capstone course, and adds two new chapters, many new exercises, and updated open problems. For scientists, this text can be utilized as a self-contained tooling device. The topics include a friendly invitation to Ehrhart’s theory of counting lattice points in polytopes, finite Fourier analysis, the Frobenius coin-exchange problem, Dedekind sums, solid angles, Euler–Maclaurin summation for polytopes, computational geometry, magic squares, zonotopes, and more. With more than 300 exercises and open research problems, the reader is an active participant, carried through diverse but tightly woven mathematical fields that are inspired by an innocently elementary question: What are the relationships between the continuous volume of a polytope and its discrete volume? Reviews of the first edition: “You owe it to yourself to pick up a copy of Computing the Continuous Discretely to read about a number of interesting problems in geometry, number theory, and combinatorics.” — MAA Reviews “The book is written as an accessible and engaging textbook, with many examples, historical notes, pithy quotes, commentary integrating the mate rial, exercises, open problems and an extensive bibliography.” — Zentralblatt MATH “This beautiful book presents, at a level suitable for advanced undergraduates, a fairly complete introduction to the problem of counting lattice points inside a convex polyhedron.” — Mathematical Reviews “Many departments recognize the need for capstone courses in which graduating students can see the tools they have acquired come together in some satisfying way. Beck and Robins have written the perfect text for such a course.” — CHOICE

Convex Polytopes

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Author: Branko Grünbaum

Publisher: Springer Science & Business Media

ISBN: 1461300193

Category: Mathematics

Page: 471

View: 6467

"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

A Course in Convexity

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Author: Alexander Barvinok

Publisher: American Mathematical Soc.

ISBN: 0821829688

Category: Mathematics

Page: 366

View: 6272

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.

Introduction to Toric Varieties. (AM-131)

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Author: William Fulton

Publisher: Princeton University Press

ISBN: 1400882524

Category: Mathematics

Page: 180

View: 1722

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.

Mathematical Developments Arising from Linear Programming

Proceedings of a Joint Summer Research Conference Held at Bowdoin College, June 25-July 1, 1988

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Author: Jeffrey C. Lagarias,Michael J. Todd

Publisher: American Mathematical Soc.

ISBN: 0821851217

Category: Mathematics

Page: 341

View: 2797

In recent years, there has been intense work in linear and nonlinear programming, much of it centered on understanding and extending the ideas underlying N. Karmarkar's interior-point linear programming algorithm, which was presented in 1984. This interdisciplinary research was the subject of an AMS Summer Research Conference on Mathematical Developments Arising from Linear Programming, held at Bowdoin College in the summer of 1988, which brought together researchers in mathematics, computer science, and operations research. This volume contains the proceedings from the conference. Among the topics covered in this book are: completely integrable dynamical systems arising in optimization problems, Riemannian geometry and interior-point linear programming methods, concepts of approximate solution of linear programs, average case analysis of the simplex method, and recent results in convex polytopes. Some of the papers extend interior-point methods to quadratic programming, the linear complementarity problem, convex programming, multi-criteria optimization, and integer programming. Other papers study the continuous trajectories underlying interior point methods. This book will be an excellent resource for those interested in the latest developments arising from Karmarkar's linear programming algorithm and in path-following methods for solving differential equations.

Geometric Invariant Theory

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Author: David Mumford,John Fogarty,Frances Kirwan

Publisher: Springer Science & Business Media

ISBN: 9783540569633

Category: Mathematics

Page: 292

View: 910

"Geometric Invariant Theory" by Mumford/Fogarty (the firstedition was published in 1965, a second, enlarged editonappeared in 1982) is the standard reference on applicationsof invariant theory to the construction of moduli spaces.This third, revised edition has been long awaited for by themathematical community. It is now appearing in a completelyupdated and enlarged version with an additional chapter onthe moment map by Prof. Frances Kirwan (Oxford) and a fullyupdated bibliography of work in this area.The book deals firstly with actions of algebraic groups onalgebraic varieties, separating orbits by invariants andconstructionquotient spaces; and secondly with applicationsof this theory to the construction of moduli spaces.It is a systematic exposition of the geometric aspects ofthe classical theory of polynomial invariants.

Graphs for Pattern Recognition

Infeasible Systems of Linear Inequalities

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Author: Damir Gainanov

Publisher: Walter de Gruyter GmbH & Co KG

ISBN: 3110481065

Category: Mathematics

Page: 158

View: 4143

Data mining and pattern recognition are areas based on the mathematical constructions discussed in this monograph. By using combinatorial and graph theoretical techniques, it is shown how to tackle infeasible systems of linear inequalities. These are, in turn, building blocks of geometric decision rules for pattern recognition.

Convex Bodies and Algebraic Geometry

An Introduction to the Theory of Toric Varieties

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Author: Tadao Oda

Publisher: Springer

ISBN: 9783642725494

Category: Mathematics

Page: 212

View: 8745

The theory of toric varieties (also called torus embeddings) describes a fascinating interplay between algebraic geometry and the geometry of convex figures in real affine spaces. This book is a unified up-to-date survey of the various results and interesting applications found since toric varieties were introduced in the early 1970's. It is an updated and corrected English edition of the author's book in Japanese published by Kinokuniya, Tokyo in 1985. Toric varieties are here treated as complex analytic spaces. Without assuming much prior knowledge of algebraic geometry, the author shows how elementary convex figures give rise to interesting complex analytic spaces. Easily visualized convex geometry is then used to describe algebraic geometry for these spaces, such as line bundles, projectivity, automorphism groups, birational transformations, differential forms and Mori's theory. Hence this book might serve as an accessible introduction to current algebraic geometry. Conversely, the algebraic geometry of toric varieties gives new insight into continued fractions as well as their higher-dimensional analogues, the isoperimetric problem and other questions on convex bodies. Relevant results on convex geometry are collected together in the appendix.

An Introduction to Measure Theory

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Author: Terence Tao

Publisher: American Mathematical Soc.

ISBN: 0821869191

Category: Mathematics

Page: 206

View: 6845

This is a graduate text introducing the fundamentals of measure theory and integration theory, which is the foundation of modern real analysis. The text focuses first on the concrete setting of Lebesgue measure and the Lebesgue integral (which in turn is motivated by the more classical concepts of Jordan measure and the Riemann integral), before moving on to abstract measure and integration theory, including the standard convergence theorems, Fubini's theorem, and the Caratheodory extension theorem. Classical differentiation theorems, such as the Lebesgue and Rademacher differentiation theorems, are also covered, as are connections with probability theory. The material is intended to cover a quarter or semester's worth of material for a first graduate course in real analysis. There is an emphasis in the text on tying together the abstract and the concrete sides of the subject, using the latter to illustrate and motivate the former. The central role of key principles (such as Littlewood's three principles) as providing guiding intuition to the subject is also emphasized. There are a large number of exercises throughout that develop key aspects of the theory, and are thus an integral component of the text. As a supplementary section, a discussion of general problem-solving strategies in analysis is also given. The last three sections discuss optional topics related to the main matter of the book.

Gröbner Bases and Convex Polytopes

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Author: Bernd Sturmfels

Publisher: American Mathematical Soc.

ISBN: 0821804871

Category: Mathematics

Page: 162

View: 2973

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.

Lectures in Geometric Combinatorics

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Author: Rekha R. Thomas

Publisher: American Mathematical Soc.

ISBN: 9780821841402

Category: Mathematics

Page: 143

View: 6402

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 state polytope 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 from commutative 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.

Convex Polytopes

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Author: Branko Grünbaum

Publisher: Springer Science & Business Media

ISBN: 1461300193

Category: Mathematics

Page: 471

View: 8638

"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

Algebra, Geometry and Software Systems

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Author: Michael Joswig,Nobuki Takayama

Publisher: Springer Science & Business Media

ISBN: 3662051486

Category: Mathematics

Page: 332

View: 4931

A collection of surveys and research papers on mathematical software and algorithms. The common thread is that the field of mathematical applications lies on the border between algebra and geometry. Topics include polyhedral geometry, elimination theory, algebraic surfaces, Gröbner bases, triangulations of point sets and the mutual relationship. This diversity is accompanied by the abundance of available software systems which often handle only special mathematical aspects. This is why the volume also focuses on solutions to the integration of mathematical software systems. This includes low-level and XML based high-level communication channels as well as general frameworks for modular systems.

Introduction to Tropical Geometry

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Author: Diane Maclagan,Bernd Sturmfels

Publisher: American Mathematical Soc.

ISBN: 0821851985

Category: Algebraic geometry -- Special varieties -- Toric varieties, Newton polyhedra

Page: 363

View: 5025

Tropical geometry is a combinatorial shadow of algebraic geometry, offering new polyhedral tools to compute invariants of algebraic varieties. It is based on tropical algebra, where the sum of two numbers is their minimum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These tropical varieties retain a surprising amount of information about their classical counterparts. Tropical geometry is a young subject that has undergone a rapid development since the beginning of the 21st century. While establishing itself as an area in its own right, deep connections have been made to many branches of pure and applied mathematics. This book offers a self-contained introduction to tropical geometry, suitable as a course text for beginning graduate students. Proofs are provided for the main results, such as the Fundamental Theorem and the Structure Theorem. Numerous examples and explicit computations illustrate the main concepts. Each of the six chapters concludes with problems that will help the readers to practice their tropical skills, and to gain access to the research literature.