Euclidean Distance Matrices and Their Applications in Rigidity Theory - Softcover

Alfakih, Abdo Y.

Softcover

ISBN 10: 303007417X ISBN 13: 9783030074173

Publisher: Springer, 2019

Synopsis

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.

About this title

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.

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.

Edition details

• Publisher: Springer
• Publication date: 2019
• Language: English
• ISBN 10: 303007417X
• ISBN 13: 9783030074173
• Binding: Paperback
• Number of pages: 265