Combinatorial Algebraic Topology

Author: Dimitry Kozlov

Publisher: Springer Science & Business Media

ISBN: 9783540730514

Category: Mathematics

Page: 390

View: 3705

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.

Morse Theory

Smooth and Discrete

Author: Kevin P Knudson

Publisher: World Scientific Publishing Company

ISBN: 9814630985

Category: Mathematics

Page: 196

View: 3116

Morse Theory: Smooth and Discrete serves as an introduction to classical smooth Morse theory and to Forman's discrete Morse theory, highlighting the parallels between the two subjects. This is the first time both smooth and discrete Morse theory have been treated in a single volume. This makes the book a valuable resource for students and professionals working in topology and discrete mathematics. With a strong focus on examples, the text is suitable for advanced undergraduates or beginning graduate students.

Directed Algebraic Topology and Concurrency

Author: Lisbeth Fajstrup,Eric Goubault,Emmanuel Haucourt,Samuel Mimram,Martin Raussen

Publisher: Springer

ISBN: 3319153986

Category: Computers

Page: 167

View: 5464

This monograph presents an application of concepts and methods from algebraic topology to models of concurrent processes in computer science and their analysis. Taking well-known discrete models for concurrent processes in resource management as a point of departure, the book goes on to refine combinatorial and topological models. In the process, it develops tools and invariants for the new discipline directed algebraic topology, which is driven by fundamental research interests as well as by applications, primarily in the static analysis of concurrent programs. The state space of a concurrent program is described as a higher-dimensional space, the topology of which encodes the essential properties of the system. In order to analyse all possible executions in the state space, more than “just” the topological properties have to be considered: Execution paths need to respect a partial order given by the time flow. As a result, tools and concepts from topology have to be extended to take privileged directions into account. The target audience for this book consists of graduate students, researchers and practitioners in the field, mathematicians and computer scientists alike.

Mathematical Reviews

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: N.A

View: 9806

Distributed Computing Through Combinatorial Topology

Author: Maurice Herlihy,Dmitry Kozlov,Sergio Rajsbaum

Publisher: Newnes

ISBN: 0124047289

Category: Computers

Page: 336

View: 5656

Distributed Computing Through Combinatorial Topology describes techniques for analyzing distributed algorithms based on award winning combinatorial topology research. The authors present a solid theoretical foundation relevant to many real systems reliant on parallelism with unpredictable delays, such as multicore microprocessors, wireless networks, distributed systems, and Internet protocols. Today, a new student or researcher must assemble a collection of scattered conference publications, which are typically terse and commonly use different notations and terminologies. This book provides a self-contained explanation of the mathematics to readers with computer science backgrounds, as well as explaining computer science concepts to readers with backgrounds in applied mathematics. The first section presents mathematical notions and models, including message passing and shared-memory systems, failures, and timing models. The next section presents core concepts in two chapters each: first, proving a simple result that lends itself to examples and pictures that will build up readers' intuition; then generalizing the concept to prove a more sophisticated result. The overall result weaves together and develops the basic concepts of the field, presenting them in a gradual and intuitively appealing way. The book's final section discusses advanced topics typically found in a graduate-level course for those who wish to explore further. Named a 2013 Notable Computer Book for Computing Methodologies by Computing Reviews Gathers knowledge otherwise spread across research and conference papers using consistent notations and a standard approach to facilitate understanding Presents unique insights applicable to multiple computing fields, including multicore microprocessors, wireless networks, distributed systems, and Internet protocols Synthesizes and distills material into a simple, unified presentation with examples, illustrations, and exercises

Topology for Computing

Author: Afra J. Zomorodian

Publisher: Cambridge University Press

ISBN: 9781139442633

Category: Computers

Page: N.A

View: 4280

The emerging field of computational topology utilizes theory from topology and the power of computing to solve problems in diverse fields. Recent applications include computer graphics, computer-aided design (CAD), and structural biology, all of which involve understanding the intrinsic shape of some real or abstract space. A primary goal of this book is to present basic concepts from topology and Morse theory to enable a non-specialist to grasp and participate in current research in computational topology. The author gives a self-contained presentation of the mathematical concepts from a computer scientist's point of view, combining point set topology, algebraic topology, group theory, differential manifolds, and Morse theory. He also presents some recent advances in the area, including topological persistence and hierarchical Morse complexes. Throughout, the focus is on computational challenges and on presenting algorithms and data structures when appropriate.

Algorithms in Combinatorial Geometry

Author: Herbert Edelsbrunner

Publisher: Springer Science & Business Media

ISBN: 9783540137221

Category: Computers

Page: 423

View: 2328

Computational geometry as an area of research in its own right emerged in the early seventies of this century. Right from the beginning, it was obvious that strong connections of various kinds exist to questions studied in the considerably older field of combinatorial geometry. For example, the combinatorial structure of a geometric problem usually decides which algorithmic method solves the problem most efficiently. Furthermore, the analysis of an algorithm often requires a great deal of combinatorial knowledge. As it turns out, however, the connection between the two research areas commonly referred to as computa tional geometry and combinatorial geometry is not as lop-sided as it appears. Indeed, the interest in computational issues in geometry gives a new and con structive direction to the combinatorial study of geometry. It is the intention of this book to demonstrate that computational and com binatorial investigations in geometry are doomed to profit from each other. To reach this goal, I designed this book to consist of three parts, acorn binatorial part, a computational part, and one that presents applications of the results of the first two parts. The choice of the topics covered in this book was guided by my attempt to describe the most fundamental algorithms in computational geometry that have an interesting combinatorial structure. In this early stage geometric transforms played an important role as they reveal connections between seemingly unrelated problems and thus help to structure the field.

Combinatorial and Computational Geometry

Author: Jacob E. Goodman,Janos Pach,Emo Welzl

Publisher: Cambridge University Press

ISBN: 9780521848626

Category: Computers

Page: 616

View: 3860

This 2005 book deals with interest topics in Discrete and Algorithmic aspects of Geometry.

Combinatorial and Computational Mathematics

Author: Sungpyo Hong,Jin Ho Kwah,Ki Hang Kim,Fred W Roush

Publisher: World Scientific

ISBN: 9814490687

Category: Mathematics

Page: 288

View: 2076

This book describes and summarizes past work in important areas of combinatorics and computation, as well as gives directions for researchers working in these areas in the 21st century. It contains primarily survey papers and presents original research by Peter Fishburn, Jim Ho Kwak, Jaeun Lee, K H Kim, F W Roush and Susan Williams. The papers deal with some of the most exciting and promising developments in the areas of coding theory in relation to number theory, lattice theory and its applications, graph theory and its applications, topological techniques in combinatorics, symbolic dynamics and mathematical social science. Contents:Monte-Carlo and Quasi-Monte-Carlo Methods for Numerical Integration (H Faure)Theoretical Approaches to Judgement and Choice (P Fishburn)Combinatorial Aspects of Mathematical Social Science (K H Kim & F W Roush)Twelve Views of Matroid Theory (J P S Kung)Enumeration of Graph Coverings, Surface Branched Coverings and Related Group Theory (J H Kwak & J Lee)An Overview of the Poset of Irreducibles (G Markowsky)Number Theory and Public-Key Cryptography (D Pointcheval)Some Applications of Graph Theory (F Roberts)Duality and Its Consequences for Ordered Cohomology of Finite Type Subshifts (K H Kim et al.)Simple Maximum Likelihood Methods for the Optical Mapping Problem (V Dancík & M S Waterman) Readership: Researchers, graduate students and advanced undergraduates in combinatorics and computational mathematics. Keywords:Combinatorics;Computation;Coding Theory;Number Theory;Lattice Theory;Graph Theory;Topological Techniques;Symbolic Dynamics;Mathematical Social Science

Computational Algebraic Geometry

Author: Hal Schenck

Publisher: Cambridge University Press

ISBN: 9780521536509

Category: Mathematics

Page: 193

View: 3575

This 2003 book investigates interplay between algebra and geometry. Covers: homological algebra, algebraic combinatorics and algebraic topology, and algebraic geometry.

Effective Computational Geometry for Curves and Surfaces

Author: Jean-Daniel Boissonnat,Monique Teillaud

Publisher: Springer Science & Business Media

ISBN: 3540332596

Category: Mathematics

Page: 344

View: 6921

This book covers combinatorial data structures and algorithms, algebraic issues in geometric computing, approximation of curves and surfaces, and computational topology. Each chapter fully details and provides a tutorial introduction to important concepts and results. The focus is on methods which are both well founded mathematically and efficient in practice. Coverage includes references to open source software and discussion of potential applications of the presented techniques.

Algebra and Algebraic Topology

Author: Ched E. Stedman

Publisher: Nova Publishers

ISBN: 9781600211874

Category: Mathematics

Page: 159

View: 7701

Gathers results in pure and applied algebra, including algebraic topology from researchers around the globe.

Untersuchungen über höhere Arithmetik

Author: Carl Friedrich Gauss

Publisher: American Mathematical Soc.

ISBN: 0821842137

Category: Mathematics

Page: 695

View: 2377

In this volume are included all of Gauss's number-theoretic works: his masterpiece, Disquisitiones Arithmeticae, published when Gauss was only 25 years old; several papers published during the ensuing 31 years; and papers taken from material found in Gauss's handwriting after his death. These papers include a fourth, fifth, and sixth proof of the Quadratic Reciprocity Law, researches on biquadratic residues, quadratic forms, and other topics. This reprint of the German translation from Latin of the second edition published in 1889 includes an extensive appendix and concludes with a commentary on the papers (with references, where appropriate, to the relevant pages of the Disquisitiones).

Algorithmic and Combinatorial Algebra

Author: L.A. Bokut',G.P.. Kukin

Publisher: Springer Science & Business Media

ISBN: 9780792323136

Category: Computers

Page: 384

View: 6945

Even three decades ago, the words 'combinatorial algebra' contrasting, for in stance, the words 'combinatorial topology,' were not a common designation for some branch of mathematics. The collocation 'combinatorial group theory' seems to ap pear first as the title of the book by A. Karras, W. Magnus, and D. Solitar [182] and, later on, it served as the title of the book by R. C. Lyndon and P. Schupp [247]. Nowadays, specialists do not question the existence of 'combinatorial algebra' as a special algebraic activity. The activity is distinguished not only by its objects of research (that are effectively given to some extent) but also by its methods (ef fective to some extent). To be more exact, we could approximately define the term 'combinatorial algebra' for the purposes of this book, as follows: So we call a part of algebra dealing with groups, semi groups , associative algebras, Lie algebras, and other algebraic systems which are given by generators and defining relations {in the first and particular place, free groups, semigroups, algebras, etc. )j a part in which we study universal constructions, viz. free products, lINN-extensions, etc. j and, finally, a part where specific methods such as the Composition Method (in other words, the Diamond Lemma, see [49]) are applied. Surely, the above explanation is far from covering the full scope of the term (compare the prefaces to the books mentioned above).

Algorithms in Real Algebraic Geometry

Author: Saugata Basu,Richard Pollack,Marie-Françoise Roy

Publisher: Springer Science & Business Media

ISBN: 9783540009733

Category: Mathematics

Page: 602

View: 978

This is the first graduate textbook on the algorithmic aspects of real algebraic geometry. The main ideas and techniques presented form a coherent and rich body of knowledge. Mathematicians will find relevant information about the algorithmic aspects.

Ordered Sets

An Introduction with Connections from Combinatorics to Topology

Author: Bernd Schröder

Publisher: Birkhäuser

ISBN: 3319297880

Category: Mathematics

Page: 420

View: 8441

An introduction to the basic tools of the theory of (partially) ordered sets such as visualization via diagrams, subsets, homomorphisms, important order-theoretical constructions and classes of ordered sets. Using a thematic approach, the author presents open or recently solved problems to motivate the development of constructions and investigations for new classes of ordered sets. The text can be used as a focused follow-up or companion to a first proof (set theory and relations) or graph theory course.

Algorithmic and Quantitative Real Algebraic Geometry

DIMACS Workshop, Algorithmic and Quantitative Aspects of Real Algebraic, Geometry in Mathematics and Computer Science, March 12-16, 2001, DIMACS Center

Author: Saugata Basu,Laureano González-Vega

Publisher: American Mathematical Soc.

ISBN: 9780821871027

Category: Mathematics

Page: 219

View: 9786

Algorithmic and quantitative aspects in real algebraic geometry are becoming increasingly important areas of research because of their roles in other areas of mathematics and computer science. The papers in this volume collectively span several different areas of current research. The articles are based on talks given at the DIMACS Workshop on ''Algorithmic and Quantitative Aspects of Real Algebraic Geometry''. Topics include deciding basic algebraic properties of real semi-algebraic sets, application of quantitative results in real algebraic geometry towards investigating the computational complexity of various problems, algorithmic and quantitative questions in real enumerative geometry, new approaches towards solving decision problems in semi-algebraic geometry, as well as computing algebraic certificates, and applications of real algebraic geometry to concrete problems arising in robotics and computer graphics. The book is intended for researchers interested in computational methods in algebra.

Foundations of Computational Mathematics, Minneapolis 2002

Author: Society for the Foundation of Computational Mathematics

Publisher: Cambridge University Press

ISBN: 9780521542531

Category: Mathematics

Page: 206

View: 7057

This volume, first published in 2004, contains the plenary invited talks given at main conference in the subject.

Computational Homology

Author: Tomasz Kaczynski,Konstantin Mischaikow,Marian Mrozek

Publisher: Springer Science & Business Media

ISBN: 0387215972

Category: Mathematics

Page: 482

View: 9012

Homology is a powerful tool used by mathematicians to study the properties of spaces and maps that are insensitive to small perturbations. This book uses a computer to develop a combinatorial computational approach to the subject. The core of the book deals with homology theory and its computation. Following this is a section containing extensions to further developments in algebraic topology, applications to computational dynamics, and applications to image processing. Included are exercises and software that can be used to compute homology groups and maps. The book will appeal to researchers and graduate students in mathematics, computer science, engineering, and nonlinear dynamics.