Computational Homology

Tomasz Kaczynski, Konstantin Mischaikow, Marian Mrozek (auth.)9780387408538, 0-387-40853-3

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.

Table of contents :
Front Matter….Pages I-XVII
Front Matter….Pages 1-1
Preview….Pages 3-37
Cubical Homology….Pages 39-92
Computing Homology Groups….Pages 93-141
Chain Maps and Reduction Algorithms….Pages 143-171
Preview of Maps….Pages 173-197
Homology of Maps….Pages 199-234
Computing Homology of Maps….Pages 235-254
Front Matter….Pages 255-255
Prospects in Digital Image Processing….Pages 257-278
Homological Algebra….Pages 279-306
Nonlinear Dynamics….Pages 307-376
Homology of Topological Polyhedra….Pages 377-393
Front Matter….Pages 395-395
Topology….Pages 397-418
Algebra….Pages 419-449
Syntax of Algorithms….Pages 451-464
Back Matter….Pages 465-482