Euclidean Distance Matrices and Their Applications in Rigidity Theory

Abdo Y. Alfakih (Autor)

Buch | Softcover

XIV, 251 Seiten

2019 | 1. Softcover reprint of the original 1st ed. 2018
Springer International Publishing (Verlag)
978-3-030-07417-3 (ISBN)

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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 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.

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.

Chapter 1. Mathematical Preliminaries.- Chapter 2. Positive Semidefinite Matrices.- Chapter 3. Euclidean Distance Matrices (EDMs).- Chapter 4. Classes of EDMs.- Chapter 5. The Geometry of EDMs.- Chapter 6. The Eigenvalues of EDMs.- Chapter 7. The Entries of EDMs.- Chapter 8. EDM Completions and Bar Frameworks.- Chapter 9. Local and Infinitesimal Rigidities.- Chapter 10. Universal and Dimensional Rigidities.- Epilogue.

"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)

“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)

Erscheinungsdatum	14.05.2019
Zusatzinfo	XIV, 251 p. 28 illus.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Gewicht	415 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
Schlagworte	distance geometry • edms • Eigenvalues of EDMs • Euclidean Distance Matrices • Euclidean Distance Matrix • Local Rigidity • Matrix Completion Problems • Multidimensional Scaling • Semidefinite Matrices • Spherical EDMs • Universal Rigidity
ISBN-10	3-030-07417-X / 303007417X
ISBN-13	978-3-030-07417-3 / 9783030074173
Zustand	Neuware