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Euclidean Distance Matrices and Their Applications in Rigidity Theory

  • Book
  • © 2018

Overview

Authors:
  1. Abdo Y. Alfakih

    1. Department of Mathematics and Statistics, University of Windsor, Windsor, Canada

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  • 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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Table of contents (10 chapters)

  1. Front Matter

    Pages i-xiv
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  2. Mathematical Preliminaries

    • Abdo Y. Alfakih
    Pages 1-28

  3. Positive Semidefinite Matrices

    • Abdo Y. Alfakih
    Pages 29-50

  4. Euclidean Distance Matrices (EDMs)

    • Abdo Y. Alfakih
    Pages 51-87

  5. Classes of EDMs

    • Abdo Y. Alfakih
    Pages 89-113

  6. The Geometry of EDMs

    • Abdo Y. Alfakih
    Pages 115-120

  7. The Eigenvalues of EDMs

    • Abdo Y. Alfakih
    Pages 121-143

  8. The Entries of EDMs

    • Abdo Y. Alfakih
    Pages 145-161

  9. EDM Completions and Bar Frameworks

    • Abdo Y. Alfakih
    Pages 163-184

  10. Local and Infinitesimal Rigidities

    • Abdo Y. Alfakih
    Pages 185-210

  11. Universal and Dimensional Rigidities

    • Abdo Y. Alfakih
    Pages 211-235

  12. Back Matter

    Pages 237-251
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Keywords

About this book

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 ourapproach. 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.

Reviews

“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)

Authors and Affiliations

  • Department of Mathematics and Statistics, University of Windsor, Windsor, Canada

    Abdo Y. Alfakih

About the author

​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.

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