Bibliographic Information

Computational homology

Tomasz Kaczynski, Konstantin Mischaikow, Marian Mrozek

(Applied mathematical sciences, v. 157)

Springer, c2004

  • : softcover

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Note

Includes bibliographical references (p. [465]-469) and indexes

Description and Table of Contents

Description

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.

Table of Contents

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

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Details
  • NCID
    BA66309364
  • ISBN
    • 0387408533
    • 9781441923547
  • Country Code
    us
  • Title Language Code
    eng
  • Text Language Code
    eng
  • Place of Publication
    New York ; Tokyo
  • Pages/Volumes
    xvii, 480 p.
  • Size
    24 cm
  • Classification
  • Subject Headings
  • Parent Bibliography ID
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