Computational Homology

ISBN-13: 9780387408538 / Angielski / Twarda / 2004 / 482 str.

Tomasz Kaczynski; Konstantin Mischaikow; Marian Mrozek

Opis

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.

Recenzje

From the reviews:

"...This is an interesting and unusual book written with the intention of serving several purposes. One of them is to demonstrate that methods of algebraic topology, in particular homology theory, that have proved remarkably successful in several areas of pure mathematics can provide powerful, and in some cases indispensable, tools in a number of areas of applied mathematics and science. The second is to provide the necessary theory and "technology" for such applications. This means on the one hand providing all the necessary mathematical foundations of the subject, including definitions and theorems, and on the other hand efficient computational techniques capable of dealing with real life situations. Thus, the book stresses algorithmic and computational approaches; and in fact includes computer code written in a programming language specially designed for this purpose. It is addressed to a varied audience of computer scientists, experimental scientists and engineers while at the same time trying to retain the interest of mathematicians. With this in mind the authors have attempted to produce a modular book, which allows a number of different reading approaches. The basic subdivision of the book is into three parts. The last part contains all the basic pre-requisites from algebra and topology: the most essential facts about Euclidean spaces, point set topology, abelian groups, vector spaces and matrix algebras. This part also contains a description of the programming language used to describe the algorithms found in the book..." --MATHEMATICAL REVIEWS

"This is an interesting and unusual book with the intention of serving several purposes. One of them is to demonstrate that methods of algebraic topology, in particular homology theory ... . The second is to provide the necessary theory and 'technology' for such applications. ... the book admirably achieves all its stated purposes. In addition it will provide much needed ammunition for those algebraic topologists who have been feeling besieged by allegations of their subject's lack of 'useful' applications." (Andrzej Kozlowski, Mathematical Reviews, 2005g)

"This book provides the conceptual background for computational homology - a powerful tool used to study the properties of spaces and maps that are insensitive to small perturbations. The material presented here is a unique combination of current research and classical rigor, computation and application." (Corina Mohorianu, Zentralblatt Mathematik, Vol. 1039 (8), 2004)

"In addition to developing a computational homology theory which produces efficient algorithms, the authors demonstrate how these algorithms can be applied to a variety of problems ... . I certainly recommend Computational Homology to mathematicians and applied scientists who wish to learn about the potential of algebraic topological methods. ... this book is the first comprehensive effort to describe the computational aspects of homology theory ... . It is written at a level that is suitable for advanced undergraduate and early graduate courses ... ." (Thomas Wanner, SIAM Review, Vol. 48 (1). 2006)

"This is the first textbook on what is necessarily a mixture of classical mathematics, computer science, and applications. ... it is a unique feature of Computational Homology that every geometric step, however conceptually simple, is broken down into elementary operations. ... The book offers a reliable yet practical introduction to (cubical homology), with a strong emphasis on computational aspects. Hands-on experience can be gained through the many problems within the book and also by means of the software packages ... ." (Arno Berger, Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 86 (4). 2006)

 

Spis treści

Preface Part I Homology 1 Preview 1.1 Analyzing Images 1.2 Nonlinear Dynamics 1.3 Graphs 1.4 Topological and Algebraic Boundaries 1.5 Keeping Track of Directions 1.6 Mod 2 Homology of Graphs 2 Cubical Homology 2.1 Cubical Sets 2.1.1 Elementary Cubes 2.1.2 Cubical Sets 2.1.3 Elementary Cells 2.2 The Algebra of Cubical Sets 2.2.1 Cubical Chains 2.2.2 Cubical Chains in a Cubical Set 2.2.3 The Boundary Operator 2.2.4 Homology of Cubical Sets 2.3 Connected Components and H0(X) 2.4 Elementary Collapses 2.5 Acyclic Cubical Spaces 2.6 Homology of Abstract Chain Complexes 2.7 Reduced Homology 2.8 Bibliographical Remarks 3 Computing Homology Groups 3.1 Matrix Algebra over Z 3.2 Row Echelon Form 3.3 Smith Normal Form 3.4 Structure of Abelian Groups 3.5 Computing Homology Groups 3.6 Computing Homology of Cubical Sets 3.7 Preboundary of a Cycle-Algebraic Approach 3.8 Bibliographical Remarks 4 Chain Maps and Reduction Algorithms 4.1 Chain Maps 4.2 Chain Homotopy 4.3 Internal Elementary Reductions 4.3.1 Elementary Collapses Revisited 4.3.2 Generalization of Elementary Collapses 4.4 CCR Algorithm 4.5 Bibliographical Remarks 5 PreviewofMaps 5.1 Rational Functions and Interval Arithmetic 5.2 Maps on an Interval 5.3 Constructing Chain Selectors 5.4 Maps of A1 6 Homology of Maps 6.1 Representable Sets 6.2 Cubical Multivalued Maps 6.3 Chain Selectors 6.4 Homology of Continuous Maps 6.4.1 Cubical Representations 6.4.2 Rescaling 6.5 Homotopy Invariance 6.6 Bibliographical Remarks 7 Computing Homology of Maps 7.1 Producing Multivalued Representation 7.2 Chain Selector Algorithm 7.3 Computing Homology of Maps 7.4 Geometric Preboundary Algorithm (optional section) 7.5 Bibliographical Remarks Part II Extensions 8 Prospects in Digital Image Processing 8.1 Images and Cubical Sets 8.2 Patterns from Cahn-Hilliard 8.3 Complicated Time-Dependent Patterns 8.4 Size Function 8.5 Bibliographical Remarks 9 Homological Algebra 9.1 Relative Homology 9.1.1 Relative Homology Groups 9.1.2 Maps in Relative Homology 9.2 Exact Sequences 9.3 The Connecting Homomorphism 9.4 Mayer-Vietoris Sequence 9.5 Weak Boundaries 9.6 Bibliographical Remarks 10 Nonlinear Dynamics 10.1 Maps and Symbolic Dynamics 10.2 Differential Equations and Flows 10.3 Wayzewski Principle 10.4 Fixed-Point Theorems 10.4.1 Fixed Points in the Unit Ball 10.4.2 The Lefschetz Fixed-Point Theorem 10.5 Degree Theory 10.5.1 Degree on Spheres 10.5.2 Topological Degree 10.6 Complicated Dynamics 10.6.1 Index Pairs and Index Map 10.6.2 Topological Conjugacy 10.7 Computing Chaotic Dynamics 10.8 Bibliographical Remarks 11 Homology of Topological Polyhedra 11.1 Simplicial Homology 11.2 Comparison of Cubical and Simplicial Complexes 11.3 Homology Functor 11.3.1 Category of Cubical Sets 11.3.2 Connected Simple Systems 11.4 Bibliographical Remarks Part III Tools from Topology and Algebra 12 Topology 12.1 Norms and Metrics in Rd 12.2 Topology 12.3 Continuous Maps 12.4 Connectedness 12.5 Limits and Compactness 13 Algebra 13.1 Abelian Groups 13.1.1 Algebraic Operations 13.1.2 Groups 13.1.3 Cyclic Groups and Torsion Subgroup 13.1.4 Quotient Groups 13.1.5 Direct Sums 13.2 Fields and Vector Spaces 13.2.1 Fields 13.2.2 Vector Spaces 13.2.3 Linear Combinations and Bases 13.3 Homomorphisms 13.3.1 Homomorphisms of Groups 13.3.2 Linear Maps 13.3.3 Matrix Algebra 13.4 Free Abelian Groups 13.4.1 Bases in Groups 13.4.2 Subgroups of Free Groups 13.4.3 Homomorphisms of Free Groups 14 Syntax of Algorithms 14.1 Overview 14.2 Data Structures 14.2.1 Elementary Data Types 14.2.2 Lists 14.2.3 Arrays 14.2.4 Vectors and Matrices 14.2.5 Sets

Blurb

In recent years, there has been a growing interest in applying homology to problems involving geometric data sets, whether obtained from physical measurements or generated through numerical simulations. This book presents a novel approach to homology that emphasizes the development of efficient algorithms for computation.

As well as providing a highly accessible introduction to the mathematical theory, the authors describe a variety of potential applications of homology in fields such as digital image processing and nonlinear dynamics. The material is aimed at a broad audience of engineers, computer scientists, nonlinear scientists, and applied mathematicians.

Mathematical prerequisites have been kept to a minimum and there are numerous examples and exercises throughout the text. The book is complemented by a website containing software programs and projects that help to further illustrate the material described within.