Euclidean Distance Matrices and Their Applications in Rigidity Theory
Authors: Alfakih, Abdo Y.
Free Preview- Offers a comprehensive and accessible exposition of Euclidean Distance Matrices (EDMs) and rigidity theory of bar-and-joint frameworks
- Highlights two parallel approaches to rigidity theory that lend themselves easily to semidefinite programming machinery
- Includes numerous examples that illustrate important theorems and concepts
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- About this book
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This book offers a comprehensive and accessible exposition of Euclidean Distance Matrices (EDMs) and rigidity theory of bar-and-joint frameworks. It is based on the one-to-one correspondence between EDMs and projected Gram matrices. Accordingly the machinery of semidefinite programming is a common thread that runs throughout the book. As a result, two parallel approaches to rigidity theory are presented. The first is traditional and more intuitive approach that is based on a vector representation of point configuration. The second is based on a Gram matrix representation of point configuration.
Euclidean Distance Matrices and Their Applications in Rigidity Theory begins by establishing the necessary background needed for the rest of the book. The focus of Chapter 1 is on pertinent results from matrix theory, graph theory and convexity theory, while Chapter 2 is devoted to positive semidefinite (PSD) matrices due to the key role these matrices play in our approach. Chapters 3 to 7 provide detailed studies of EDMs, and in particular their various characterizations, classes, eigenvalues and geometry. Chapter 8 serves as a transitional chapter between EDMs and rigidity theory. Chapters 9 and 10 cover local and universal rigidities of bar-and-joint frameworks. This book is self-contained and should be accessible to a wide audience including students and researchers in statistics, operations research, computational biochemistry, engineering, computer science and mathematics. - About the authors
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Abdo Y. Alfakih is a Professor in the Department of Mathematics and Statistics at the University of Windsor. He received his PhD in Industrial and Operations Engineering at the University of Michigan. His research interests are in the areas of combinatorial optimization, semidefinite programming. His current work focuses on new approaches to the Graph Realization Problem and its relatives (bar and tensegrity framework rigidity, global rigidity, dimensional rigidity, universal rigidity etc) using Euclidean distance matrices, projected Gram matrices, Gale transform and semidefinite programming.
- Reviews
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“This monograph is more than a standard text on matrices and rigidity theory. It is particularly important for providing the necessary information to mathematicians who are not experts in these areas. I really enjoyed the way how the topics are presented.” (Shing So, zbMATH 1422.15002, 2019)
- Table of contents (10 chapters)
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Mathematical Preliminaries
Pages 1-28
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Positive Semidefinite Matrices
Pages 29-50
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Euclidean Distance Matrices (EDMs)
Pages 51-87
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Classes of EDMs
Pages 89-113
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The Geometry of EDMs
Pages 115-120
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Table of contents (10 chapters)
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Bibliographic Information
- Bibliographic Information
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- Book Title
- Euclidean Distance Matrices and Their Applications in Rigidity Theory
- Authors
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- Abdo Y. Alfakih
- Copyright
- 2018
- Publisher
- Springer International Publishing
- Copyright Holder
- Springer Nature Switzerland AG
- eBook ISBN
- 978-3-319-97846-8
- DOI
- 10.1007/978-3-319-97846-8
- Hardcover ISBN
- 978-3-319-97845-1
- Softcover ISBN
- 978-3-030-07417-3
- Edition Number
- 1
- Number of Pages
- XIV, 251
- Number of Illustrations
- 28 b/w illustrations
- Topics
