Using the Borsuk-Ulam Theorem

Lectures on Topological Methods in Combinatorics and Geometry

Author: Jiri Matousek

Publisher: Springer Science & Business Media

ISBN: 3540766499

Category: Mathematics

Page: 214

View: 7944

To the uninitiated, algebraic topology might seem fiendishly complex, but its utility is beyond doubt. This brilliant exposition goes back to basics to explain how the subject has been used to further our understanding in some key areas. A number of important results in combinatorics, discrete geometry, and theoretical computer science have been proved using algebraic topology. While the results are quite famous, their proofs are not so widely understood. This book is the first textbook treatment of a significant part of these results. It focuses on so-called equivariant methods, based on the Borsuk-Ulam theorem and its generalizations. The topological tools are intentionally kept on a very elementary level. No prior knowledge of algebraic topology is assumed, only a background in undergraduate mathematics, and the required topological notions and results are gradually explained.
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Graph-Theoretic Concepts in Computer Science

37th International Workshop, WG 2011, Teplá Monastery, Czech Republic, June 21-24, 2011, Revised Papers

Author: Petr Kolman,Jan Kratochvíl

Publisher: Springer Science & Business Media

ISBN: 3642258697

Category: Computers

Page: 344

View: 2936

This book constitutes the revised selected papers of the 37th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2011, held at Teplá Monastery, Czech Republic, in June 2011. The 28 revised papers presented were carefully reviewed and selected from 52 submissions. The workshop aims at merging theory and practice by demonstrating how concepts from graph theory can be applied to various areas in computer science, and by extracting new graph theoretic problems from applications.
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Polynomial Methods in Combinatorics

Author: Larry Guth

Publisher: American Mathematical Soc.

ISBN: 1470428903

Category: Combinatorial geometry

Page: 273

View: 7680

This book explains some recent applications of the theory of polynomials and algebraic geometry to combinatorics and other areas of mathematics. One of the first results in this story is a short elegant solution of the Kakeya problem for finite fields, which was considered a deep and difficult problem in combinatorial geometry. The author also discusses in detail various problems in incidence geometry associated to Paul Erdős's famous distinct distances problem in the plane from the 1940s. The proof techniques are also connected to error-correcting codes, Fourier analysis, number theory, and differential geometry. Although the mathematics discussed in the book is deep and far-reaching, it should be accessible to first- and second-year graduate students and advanced undergraduates. The book contains approximately 100 exercises that further the reader's understanding of the main themes of the book.
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A Journey Through Discrete Mathematics

A Tribute to Jiří Matoušek

Author: Martin Loebl,Jaroslav Nešetřil,Robin Thomas

Publisher: Springer

ISBN: 3319444794

Category: Computers

Page: 810

View: 1277

This collection of high-quality articles in the field of combinatorics, geometry, algebraic topology and theoretical computer science is a tribute to Jiří Matoušek, who passed away prematurely in March 2015. It is a collaborative effort by his colleagues and friends, who have paid particular attention to clarity of exposition – something Jirka would have approved of. The original research articles, surveys and expository articles, written by leading experts in their respective fields, map Jiří Matoušek’s numerous areas of mathematical interest.
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Proofs from THE BOOK

Author: Martin Aigner,Günter M. Ziegler

Publisher: Springer

ISBN: 3662442051

Category: Mathematics

Page: 308

View: 993

This revised and enlarged fifth edition features four new chapters, which contain highly original and delightful proofs for classics such as the spectral theorem from linear algebra, some more recent jewels like the non-existence of the Borromean rings and other surprises. From the Reviews "... Inside PFTB (Proofs from The Book) is indeed a glimpse of mathematical heaven, where clever insights and beautiful ideas combine in astonishing and glorious ways. There is vast wealth within its pages, one gem after another. ... Aigner and Ziegler... write: "... all we offer is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations." I do. ... " Notices of the AMS, August 1999 "... This book is a pleasure to hold and to look at: ample margins, nice photos, instructive pictures and beautiful drawings ... It is a pleasure to read as well: the style is clear and entertaining, the level is close to elementary, the necessary background is given separately and the proofs are brilliant. ..." LMS Newsletter, January 1999 "Martin Aigner and Günter Ziegler succeeded admirably in putting together a broad collection of theorems and their proofs that would undoubtedly be in the Book of Erdös. The theorems are so fundamental, their proofs so elegant and the remaining open questio ns so intriguing that every mathematician, regardless of speciality, can benefit from reading this book. ... " SIGACT News, December 2011.
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Understanding and Using Linear Programming

Author: Jiri Matousek,Bernd Gärtner

Publisher: Springer Science & Business Media

ISBN: 3540307176

Category: Mathematics

Page: 226

View: 330

The book is an introductory textbook mainly for students of computer science and mathematics. Our guiding phrase is "what every theoretical computer scientist should know about linear programming". A major focus is on applications of linear programming, both in practice and in theory. The book is concise, but at the same time, the main results are covered with complete proofs and in sufficient detail, ready for presentation in class. The book does not require more prerequisites than basic linear algebra, which is summarized in an appendix. One of its main goals is to help the reader to see linear programming "behind the scenes".
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Numerical Optimization

Theoretical and Practical Aspects

Author: Joseph-Frédéric Bonnans,Jean Charles Gilbert,Claude Lemarechal,Claudia A. Sagastizábal

Publisher: Springer Science & Business Media

ISBN: 3662050781

Category: Mathematics

Page: 423

View: 5702

This book starts with illustrations of the ubiquitous character of optimization, and describes numerical algorithms in a tutorial way. It covers fundamental algorithms as well as more specialized and advanced topics for unconstrained and constrained problems. This new edition contains computational exercises in the form of case studies which help understanding optimization methods beyond their theoretical description when coming to actual implementation.
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Qualitative Theory of Planar Differential Systems

Author: Freddy Dumortier,Jaume Llibre,Joan C. Artés

Publisher: Springer Science & Business Media

ISBN: 9783540329022

Category: Mathematics

Page: 302

View: 5514

This book deals with systems of polynomial autonomous ordinary differential equations in two real variables. The emphasis is mainly qualitative, although attention is also given to more algebraic aspects as a thorough study of the center/focus problem and recent results on integrability. In the last two chapters the performant software tool P4 is introduced. From the start, differential systems are represented by vector fields enabling, in full strength, a dynamical systems approach. All essential notions, including invariant manifolds, normal forms, desingularization of singularities, index theory and limit cycles, are introduced and the main results are proved for smooth systems with the necessary specifications for analytic and polynomial systems.
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A Course in Topological Combinatorics

Author: Mark de Longueville

Publisher: Springer Science & Business Media

ISBN: 1441979107

Category: Mathematics

Page: 240

View: 679

A Course in Topological Combinatorics is the first undergraduate textbook on the field of topological combinatorics, a subject that has become an active and innovative research area in mathematics over the last thirty years with growing applications in math, computer science, and other applied areas. Topological combinatorics is concerned with solutions to combinatorial problems by applying topological tools. In most cases these solutions are very elegant and the connection between combinatorics and topology often arises as an unexpected surprise. The textbook covers topics such as fair division, graph coloring problems, evasiveness of graph properties, and embedding problems from discrete geometry. The text contains a large number of figures that support the understanding of concepts and proofs. In many cases several alternative proofs for the same result are given, and each chapter ends with a series of exercises. The extensive appendix makes the book completely self-contained. The textbook is well suited for advanced undergraduate or beginning graduate mathematics students. Previous knowledge in topology or graph theory is helpful but not necessary. The text may be used as a basis for a one- or two-semester course as well as a supplementary text for a topology or combinatorics class.
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Combinatorial Algebraic Topology

Author: Dimitry Kozlov

Publisher: Springer Science & Business Media

ISBN: 3540719628

Category: Mathematics

Page: 390

View: 3123

This volume is the first comprehensive treatment of combinatorial algebraic topology in book form. The first part of the book constitutes a swift walk through the main tools of algebraic topology. Readers - graduate students and working mathematicians alike - will probably find particularly useful the second part, which contains an in-depth discussion of the major research techniques of combinatorial algebraic topology. Although applications are sprinkled throughout the second part, they are principal focus of the third part, which is entirely devoted to developing the topological structure theory for graph homomorphisms.
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Extremal Graph Theory

Author: Bela Bollobas

Publisher: Courier Corporation

ISBN: 0486317587

Category: Mathematics

Page: 512

View: 394

The ever-expanding field of extremal graph theory encompasses a diverse array of problem-solving methods, including applications to economics, computer science, and optimization theory. This volume, based on a series of lectures delivered to graduate students at the University of Cambridge, presents a concise yet comprehensive treatment of extremal graph theory. Unlike most graph theory treatises, this text features complete proofs for almost all of its results. Further insights into theory are provided by the numerous exercises of varying degrees of difficulty that accompany each chapter. Although geared toward mathematicians and research students, much of Extremal Graph Theory is accessible even to undergraduate students of mathematics. Pure mathematicians will find this text a valuable resource in terms of its unusually large collection of results and proofs, and professionals in other fields with an interest in the applications of graph theory will also appreciate its precision and scope.
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Sheaves in Topology

Author: Alexandru Dimca

Publisher: Springer Science & Business Media

ISBN: 3642188680

Category: Mathematics

Page: 240

View: 8401

Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds. This introduction to the subject can be regarded as a textbook on modern algebraic topology, treating the cohomology of spaces with sheaf (as opposed to constant) coefficients. The author helps readers progress quickly from the basic theory to current research questions, thoroughly supported along the way by examples and exercises.
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SPEECH RECOGNITION: THEORY AND C++ IMPLEMENTATION (With CD )

Author: Claudio Becchetti,Klucio Prina Ricotti

Publisher: John Wiley & Sons

ISBN: 9788126517749

Category:

Page: 428

View: 5567

Special Features: · Source codes for compiling and implementing ASR algorithms in C++ are included in electronic format on an accompanying CD-ROM· Contains a practical account of the functioning of ASR· Includes implementation-oriented mathematical and technical explanations of ASR· Features a stage-by-stage explanation of how to create an ASR interface· Can be used both for teaching speech recognition techniques and testing and development of new systems on digital signal processing hardware About The Book: Automatic Speech Recognition (ASR) is becoming increasingly prevalent in such applications as private telephone exchanges and real-time on-line telephone information services. This book introduces the principles of ASR systems, including the theory and the implementation issues behind multi-speaker continuous speech ASR. The book supplies the full C++ code to further clarify the implementation details of a typical commercial/laboratory ASR system and to allow the readers to reach practical solutions for ASR-related problems.About the topic/technology Automatic Speech Recognition (ASR) is the technology behind the voice-triggered computer menus. Uses of these systems are now proliferating rapidly and include private telephone exchanges and real-time on-line telephone information services.
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Jerusalem Combinatorics '93

An International Conference in Combinatorics, May 9-17, 1993, Jerusalem, Israel

Author: Hélène Barcelo,Gil Kalai

Publisher: American Mathematical Soc.

ISBN: 0821802941

Category: Mathematics

Page: 360

View: 9930

This book contains twenty-two papers presented at the International Conference in Combinatorics, held in Jerusalem in May 1993. The papers describe some of the latest developments in algebraic combinatorics, enumeration, graph and hypergraph theory, combinatorial geometry, and geometry of polytopes and arrangements. The papers are accessible to specialists as well as nonspecialists.
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A Course on Mathematical Logic

Author: Shashi Mohan Srivastava

Publisher: Springer Science & Business Media

ISBN: 1461457467

Category: Mathematics

Page: 198

View: 8641

This is a short, modern, and motivated introduction to mathematical logic for upper undergraduate and beginning graduate students in mathematics and computer science. Any mathematician who is interested in getting acquainted with logic and would like to learn Gödel’s incompleteness theorems should find this book particularly useful. The treatment is thoroughly mathematical and prepares students to branch out in several areas of mathematics related to foundations and computability, such as logic, axiomatic set theory, model theory, recursion theory, and computability. In this new edition, many small and large changes have been made throughout the text. The main purpose of this new edition is to provide a healthy first introduction to model theory, which is a very important branch of logic. Topics in the new chapter include ultraproduct of models, elimination of quantifiers, types, applications of types to model theory, and applications to algebra, number theory and geometry. Some proofs, such as the proof of the very important completeness theorem, have been completely rewritten in a more clear and concise manner. The new edition also introduces new topics, such as the notion of elementary class of structures, elementary diagrams, partial elementary maps, homogeneous structures, definability, and many more.
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Logic and Structure

Author: Dirk van Dalen

Publisher: Springer Science & Business Media

ISBN: 3662029626

Category: Mathematics

Page: 220

View: 9573

New corrected printing of a well-established text on logic at the introductory level.
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SIAM Journal on Computing

Author: Society for Industrial and Applied Mathematics

Publisher: N.A

ISBN: N.A

Category: Electronic data processing

Page: N.A

View: 5673

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Mathematical Modeling for the Life Sciences

Author: Jacques Istas

Publisher: Springer Science & Business Media

ISBN: 354027877X

Category: Mathematics

Page: 168

View: 5506

Provides a wide range of mathematical models currently used in the life sciences Each model is thoroughly explained and illustrated by example Includes three appendices to allow for independent reading
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Algebraic Geometry

An Introduction

Author: Daniel Perrin

Publisher: Springer Science & Business Media

ISBN: 9781848000568

Category: Mathematics

Page: 263

View: 9180

Aimed primarily at graduate students and beginning researchers, this book provides an introduction to algebraic geometry that is particularly suitable for those with no previous contact with the subject; it assumes only the standard background of undergraduate algebra. The book starts with easily-formulated problems with non-trivial solutions and uses these problems to introduce the fundamental tools of modern algebraic geometry: dimension; singularities; sheaves; varieties; and cohomology. A range of exercises is provided for each topic discussed, and a selection of problems and exam papers are collected in an appendix to provide material for further study.
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