Applied Mathematical Sciences

Computational Homology

Authors: Kaczynski, Tomasz, Mischaikow, Konstantin, Mrozek, Marian

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About this Textbook

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.

Reviews

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)

 


Table of contents (14 chapters)

Buy this book

eBook $69.99
price for USA (gross)
  • ISBN 978-0-387-21597-6
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $99.00
price for USA
  • ISBN 978-0-387-40853-8
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $99.00
price for USA
  • ISBN 978-1-4419-2354-7
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Computational Homology
Authors
Series Title
Applied Mathematical Sciences
Series Volume
157
Copyright
2004
Publisher
Springer-Verlag New York
Copyright Holder
Springer Science+Business Media New York
eBook ISBN
978-0-387-21597-6
DOI
10.1007/b97315
Hardcover ISBN
978-0-387-40853-8
Softcover ISBN
978-1-4419-2354-7
Series ISSN
0066-5452
Edition Number
1
Number of Pages
XVIII, 482
Topics