Euclidean Distance Matrices and Their Applications in Rigidity Theory
暫譯: 歐幾里得距離矩陣及其在剛性理論中的應用
Abdo Y. Alfakih
- 出版商: Springer
- 出版日期: 2018-10-22
- 售價: $4,890
- 貴賓價: 9.5 折 $4,646
- 語言: 英文
- 頁數: 251
- 裝訂: Hardcover
- ISBN: 3319978454
- ISBN-13: 9783319978451
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商品描述
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.
商品描述(中文翻譯)
本書提供了對歐幾里得距離矩陣（Euclidean Distance Matrices, EDMs）及桿接框架的剛性理論的全面且易於理解的闡述。它基於EDMs與投影Gram矩陣之間的一對一對應關係。因此，半正定規劃的工具貫穿整本書。結果，提出了兩種平行的剛性理論方法。第一種是基於點配置的向量表示的傳統且更直觀的方法。第二種則是基於點配置的Gram矩陣表示。
《歐幾里得距離矩陣及其在剛性理論中的應用》首先建立了本書其餘部分所需的背景知識。第一章的重點是矩陣理論、圖論和凸性理論中的相關結果，而第二章則專注於正半定（Positive Semidefinite, PSD）矩陣，因為這些矩陣在我們的方法中扮演著關鍵角色。第三至第七章提供了對EDMs的詳細研究，特別是它們的各種特徵、類別、特徵值和幾何。第八章作為EDMs與剛性理論之間的過渡章節。第九章和第十章涵蓋了桿接框架的局部和全局剛性。本書是自成體系的，應該能夠讓包括統計學、運籌學、計算生物化學、工程學、計算機科學和數學的學生和研究人員在內的廣泛讀者群體輕鬆理解。
