Lectures on Polytopes

Author: Günter M. Ziegler

Publisher: Springer Science & Business Media

ISBN: 9780387943657

Category: Mathematics

Page: 370

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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.
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Lectures on Polytopes

Author: Günter M. Ziegler

Publisher: Springer Science & Business Media

ISBN: 1461384311

Category: Mathematics

Page: 370

View: 3089

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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.
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Lectures in Geometric Combinatorics

Author: Rekha R. Thomas

Publisher: American Mathematical Soc.

ISBN: 9780821841402

Category: Mathematics

Page: 143

View: 9639

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This book presents a course in the geometry of convex polytopes in arbitrary dimension. It takes readers from the basics of polytope theory to recent developments around secondary and state polytopes arising from point configurations. The most needed concepts are developed from scratch. Text illustrates the interaction among discrete geometry, computational algebra and combinatorics. This book is published in cooperation with IAS/Park City Mathematics Institute.
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Advances in Discrete Differential Geometry

Author: Alexander I. Bobenko

Publisher: Springer

ISBN: 3662504472

Category: Mathematics

Page: 439

View: 1497

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This is one of the first books on a newly emerging field of discrete differential geometry and an excellent way to access this exciting area. It surveys the fascinating connections between discrete models in differential geometry and complex analysis, integrable systems and applications in computer graphics. The authors take a closer look at discrete models in differential geometry and dynamical systems. Their curves are polygonal, surfaces are made from triangles and quadrilaterals, and time is discrete. Nevertheless, the difference between the corresponding smooth curves, surfaces and classical dynamical systems with continuous time can hardly be seen. This is the paradigm of structure-preserving discretizations. Current advances in this field are stimulated to a large extent by its relevance for computer graphics and mathematical physics. This book is written by specialists working together on a common research project. It is about differential geometry and dynamical systems, smooth and discrete theories, and on pure mathematics and its practical applications. The interaction of these facets is demonstrated by concrete examples, including discrete conformal mappings, discrete complex analysis, discrete curvatures and special surfaces, discrete integrable systems, conformal texture mappings in computer graphics, and free-form architecture. This richly illustrated book will convince readers that this new branch of mathematics is both beautiful and useful. It will appeal to graduate students and researchers in differential geometry, complex analysis, mathematical physics, numerical methods, discrete geometry, as well as computer graphics and geometry processing.
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Topics in Algebraic Geometry and Geometric Modeling

Workshop on Algebraic Geometry and Geometric Modeling, July 29-August 2, 2002, Vilnius University, Lithuania

Author: Ron Goldman,Rimvydas Krasauskas

Publisher: American Mathematical Soc.

ISBN: 0821834207

Category: Mathematics

Page: 366

View: 9832

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Algebraic geometry and geometric modeling both deal with curves and surfaces generated by polynomial equations. Algebraic geometry investigates the theoretical properties of polynomial curves and surfaces; geometric modeling uses polynomial, piecewise polynomial, and rational curves and surfaces to build computer models of mechanical components and assemblies for industrial design and manufacture. The NSF sponsored the four-day ``Vilnius Workshop on Algebraic Geometry and Geometric Modeling'', which brought together some of the top experts in the two research communities to examine a wide range of topics of interest to both fields. This volume is an outgrowth of that workshop. Included are surveys, tutorials, and research papers. In addition, the editors have included a translation of Minding's 1841 paper, ``On the determination of the degree of an equation obtained by elimination'', which foreshadows the modern application of mixed volumes in algebraic geometry. The volume is suitable for mathematicians, computer scientists, and engineers interested in applications of algebraic geometry to geometric modeling.
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Mathematics Unlimited - 2001 and Beyond

Author: Björn Engquist,Wilfried Schmid

Publisher: Springer

ISBN: 364256478X

Category: Mathematics

Page: 1236

View: 1825

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This is a book guaranteed to delight the reader. It not only depicts the state of mathematics at the end of the century, but is also full of remarkable insights into its future de- velopment as we enter a new millennium. True to its title, the book extends beyond the spectrum of mathematics to in- clude contributions from other related sciences. You will enjoy reading the many stimulating contributions and gain insights into the astounding progress of mathematics and the perspectives for its future. One of the editors, Björn Eng- quist, is a world-renowned researcher in computational sci- ence and engineering. The second editor, Wilfried Schmid, is a distinguished mathematician at Harvard University. Likewi- se the authors are all foremost mathematicians and scien- tists, and their biographies and photographs appear at the end of the book. Unique in both form and content, this is a "must-read" for every mathematician and scientist and, in particular, for graduates still choosing their specialty. Limited collector's edition - an exclusive and timeless work. This special, numbered edition will be available until June 1, 2000. Firm orders only.
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Polytopes - Combinations and Computation

Author: Gil Kalai,Günter M. Ziegler

Publisher: Birkhäuser

ISBN: 3034884389

Category: Mathematics

Page: 225

View: 8447

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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.
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Novel Approaches to Hard Discrete Optimization

Author: Panos M. Pardalos,Henry Wolkowicz

Publisher: American Mathematical Soc.

ISBN: 9780821885918

Category: Mathematics

Page: 181

View: 8608

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During the last decade, many novel approaches have been considered for dealing with computationally difficult discrete optimization problems. Such approaches include interior point methods, semidefinite programming techniques, and global optimization. More efficient computational algorithms have been developed and larger problem instances of hard discrete problems have been solved. This progress is due in part to these novel approaches, but also to new computing facilities and massive parallelism. This volume contains the papers presented at the workshop on ''Novel Approaches to Hard Discrete Optimization''. The articles cover a spectrum of issues regarding computationally hard discrete problems.
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Lectures on Discrete Geometry

Author: Ji?í Matoušek

Publisher: Springer Science & Business Media

ISBN: 1461300398

Category: Mathematics

Page: 486

View: 7298

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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.
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An Introduction to Intersection Homology Theory, Second Edition

Author: Frances Kirwan,Jonathan Woolf

Publisher: CRC Press

ISBN: 9781584881841

Category: Mathematics

Page: 248

View: 1235

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Now more that a quarter of a century old, intersection homology theory has proven to be a powerful tool in the study of the topology of singular spaces, with deep links to many other areas of mathematics, including combinatorics, differential equations, group representations, and number theory. Like its predecessor, An Introduction to Intersection Homology Theory, Second Edition introduces the power and beauty of intersection homology, explaining the main ideas and omitting, or merely sketching, the difficult proofs. It treats both the basics of the subject and a wide range of applications, providing lucid overviews of highly technical areas that make the subject accessible and prepare readers for more advanced work in the area. This second edition contains entirely new chapters introducing the theory of Witt spaces, perverse sheaves, and the combinatorial intersection cohomology of fans. Intersection homology is a large and growing subject that touches on many aspects of topology, geometry, and algebra. With its clear explanations of the main ideas, this book builds the confidence needed to tackle more specialist, technical texts and provides a framework within which to place them.
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