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.
Author(s): Tomasz Kaczynski, Konstantin Mischaikow, Marian Mrozek (auth.)
Series: Applied Mathematical Sciences 157
Edition: 1
Publisher: Springer-Verlag New York
Year: 2004
Language: English
Pages: 482
City: New York
Tags: Applications of Mathematics; Category Theory, Homological Algebra; Dynamical Systems and Ergodic Theory; Computational Mathematics and Numerical Analysis; Algebraic Topology
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
