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Simplicial Complexes of Graphs

Author: Jakob Jonsson

Publisher: Springer Science & Business Media

Published: 2007-11-15

Total Pages: 376

ISBN-13: 3540758585

A graph complex is a finite family of graphs closed under deletion of edges. Graph complexes show up naturally in many different areas of mathematics. Identifying each graph with its edge set, one may view a graph complex as a simplicial complex and hence interpret it as a geometric object. This volume examines topological properties of graph complexes, focusing on homotopy type and homology. Many of the proofs are based on Robin Forman's discrete version of Morse theory.

Topological Signal Processing

Author: Michael Robinson

Publisher: Springer Science & Business Media

Published: 2014-01-07

Total Pages: 245

ISBN-13: 3642361048

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Signal processing is the discipline of extracting information from collections of measurements. To be effective, the measurements must be organized and then filtered, detected, or transformed to expose the desired information. Distortions caused by uncertainty, noise, and clutter degrade the performance of practical signal processing systems. In aggressively uncertain situations, the full truth about an underlying signal cannot be known. This book develops the theory and practice of signal processing systems for these situations that extract useful, qualitative information using the mathematics of topology -- the study of spaces under continuous transformations. Since the collection of continuous transformations is large and varied, tools which are topologically-motivated are automatically insensitive to substantial distortion. The target audience comprises practitioners as well as researchers, but the book may also be beneficial for graduate students.

Algebraic L-theory and Topological Manifolds

Author: Andrew Ranicki

Publisher: Cambridge University Press

Published: 1992-12-10

Total Pages: 372

ISBN-13: 9780521420242

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Assuming no previous acquaintance with surgery theory and justifying all the algebraic concepts used by their relevance to topology, Dr Ranicki explains the applications of quadratic forms to the classification of topological manifolds, in a unified algebraic framework.

Combinatorial Algebraic Topology

Author: Dimitry Kozlov

Publisher: Springer Science & Business Media

Published: 2008-01-08

Total Pages: 416

ISBN-13: 9783540730514

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This volume is the first comprehensive treatment of combinatorial algebraic topology in book form. The first part of the book constitutes a swift walk through the main tools of algebraic topology. Readers - graduate students and working mathematicians alike - will probably find particularly useful the second part, which contains an in-depth discussion of the major research techniques of combinatorial algebraic topology. Although applications are sprinkled throughout the second part, they are principal focus of the third part, which is entirely devoted to developing the topological structure theory for graph homomorphisms.

Random Simplicial Complexes

Author: Lior Aronshtam

Publisher:

Published: 2013

Total Pages: 64

ISBN-13:

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Random graphs are vastly researched and are of great importance in modern discrete mathematics. A graph may be viewed as a one-dimensional simplicial complex.

Graphs, Surfaces and Homology

Author: Peter Giblin

Publisher: Cambridge University Press

Published: 2010-08-12

Total Pages: 273

ISBN-13: 1139491172

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Homology theory is a powerful algebraic tool that is at the centre of current research in topology and its applications. This accessible textbook will appeal to mathematics students interested in the application of algebra to geometrical problems, specifically the study of surfaces (sphere, torus, Mobius band, Klein bottle). In this introduction to simplicial homology - the most easily digested version of homology theory - the author studies interesting geometrical problems, such as the structure of two-dimensional surfaces and the embedding of graphs in surfaces, using the minimum of algebraic machinery and including a version of Lefschetz duality. Assuming very little mathematical knowledge, the book provides a complete account of the algebra needed (abelian groups and presentations), and the development of the material is always carefully explained with proofs given in full detail. Numerous examples and exercises are also included, making this an ideal text for undergraduate courses or for self-study.

Graphs, Simplicial Complexes, and Beyond

Author: Abubakr Muhammad

Publisher: VDM Publishing

Published: 2008-05

Total Pages: 0

ISBN-13: 9783836491860

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The increasing pervasiveness and accuracy of sensors, unprecedented automation of data collection, extremely cheap storage and rapid dissemination of data by communication networks have enabled researchers to think about deploying swarms of cooperating robotic agents for various applications. However, the conception of such large-scale systems is contigent on efficient methods to deal with an explosion of data. Thus the main challenge in this field has shifted from difficulties in manufacturing to the lack of theoretical foundations for provably correct design and deployment. This monograph, appearing originally as a doctoral thesis, offers a unique perspective on the solution of such problems. It introduces some novel methods for dealing with the spatial complexities in robotic networks. At the same time, it makes connections to several emerging disciplines in engineering and mathematical sciences, most notably computational algebraic topology, graph drawing, networked control, sensor networks and distributed optimization. This work received the Georgia Tech Sigma Xi Best doctoral dissertation award in 2006.

Using the Borsuk-Ulam Theorem

Author: Jiri Matousek

Publisher: Springer Science & Business Media

Published: 2008-01-12

Total Pages: 221

ISBN-13: 3540766499

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To the uninitiated, algebraic topology might seem fiendishly complex, but its utility is beyond doubt. This brilliant exposition goes back to basics to explain how the subject has been used to further our understanding in some key areas. A number of important results in combinatorics, discrete geometry, and theoretical computer science have been proved using algebraic topology. While the results are quite famous, their proofs are not so widely understood. This book is the first textbook treatment of a significant part of these results. It focuses on so-called equivariant methods, based on the Borsuk-Ulam theorem and its generalizations. The topological tools are intentionally kept on a very elementary level. No prior knowledge of algebraic topology is assumed, only a background in undergraduate mathematics, and the required topological notions and results are gradually explained.