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E-raamat: Euclidean Distance Geometry: An Introduction

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Euclidean Distance Geometry: An IntroductionSuurem pilt
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This textbook, the first of its kind, presents the fundamentals of distance geometry:  theory, useful methodologies for obtaining solutions, and real world applications. Concise proofs are given and step-by-step algorithms for solving fundamental problems efficiently and precisely are presented in Mathematica®, enabling the reader to experiment with concepts and methods as they are introduced. Descriptive graphics, examples, and problems, accompany the real gems of the text, namely the applications in visualization of graphs, localization of sensor networks, protein conformation from distance data, clock synchronization protocols, robotics, and control of unmanned underwater vehicles, to name several.  Aimed at intermediate undergraduates, beginning graduate students, researchers, and practitioners, the reader with a basic knowledge of linear algebra will gain an understanding of the basic theories of distance geometry and why they work in real life.

Arvustused

The book under review is an invitation to a field with a subject as old as the ancient Greeks, with relatively new name - Euclidean Distance Geometry (EDG). The book addresses readers at undergraduate level, researchers and practioners . The textbook ends with a generous appendix covering all the prerequisites needed for reading the book which are quite modest. (Martin Lukarevski, zbMATH 1492.51002, 2022)

The authors intended audience is undergraduate students. The book is intensely mathematical. It would probably be more suitable for graduate students in mathematics than undergraduates. (Anthony J. Duben, Computing Reviews, May 14, 2019)

The authors make use of the computing system Mathematica to show step-by step proofs. Aimed at students with a solid foundation in linear algebra, this text would be appropriate for upper-level undergraduates or graduate students. (J. A. Bakal, Choice, Vol. 55 (12), August, 2018)



This textbook on distance geometry covers some relevant theory with several algorithms presented in Mathematica. The featured problems explore graph visualization, sensor networks, molecule topology and more. Beginning graduate students and researchers with a suitable foundation in graph, vector, and matrix theory as well as linear algebra will gain from the modeling explorations here. (Tom Schulte, MAA Reviews, March, 2018)

1 Motivation
1(8)
1.1 How it all started
1(2)
1.2 Setting up Mathematica
3(1)
1.3 Four examples
3(3)
1.3.1 Clock synchronization
3(1)
1.3.2 Sensor network localization
4(1)
1.3.3 Structural biology
4(1)
1.3.4 Big data
5(1)
1.3.5 What these problems have in common
5(1)
1.4 Solving the Clock Synchronization Problem
6(1)
1.5 Exercises
7(2)
2 The Distance Geometry Problem
9(10)
2.1 Computing all pairwise distances from points
9(1)
2.2 Computing points from all pairwise distances
9(2)
2.2.1 Ill-posedness
9(1)
2.2.2 No solution
10(1)
2.3 The fundamental problem of DG
11(1)
2.3.1 The input as a weighted graph
11(1)
2.3.2 Formalization of the DGP
11(1)
2.4 A quadratic system of equations
12(4)
2.4.1 The number of solutions
13(1)
2.4.2 Computational complexity of the DGP
14(2)
2.5 Direct solution methods
16(2)
2.5.1 A global optimization formulation
16(2)
2.6 Exercises
18(1)
3 Realizing complete graphs
19(12)
3.1 Cliques
19(1)
3.2 Realizing (K + 1)-cliques in RK-1
19(4)
3.2.1 The trilateration system in RK-1
20(1)
3.2.2 Solving the linear system
21(1)
3.2.3 Iterative realization of complete graphs
21(2)
3.3 Realizing (K+ l)-cliques in RK
23(5)
3.3.1 Basic and nonbasic columns
23(1)
3.3.2 Expressing basics as linear functions of nonbasics
23(1)
3.3.3 The K-lateration system in RK
24(2)
3.3.4 Differences between Rk and Rk-1
26(1)
3.3.5 The realization algorithm
27(1)
3.3.6 The assumption on the rank of A
27(1)
3.4 Exercises
28(3)
4 Discretizability
31(12)
4.1 The volume of simplices
31(1)
4.1.1 Length and area: k-volume for K ≥ 2
31(1)
4.1.2 The Cayley-Menger determinant
32(1)
4.2 Realizing quasi-cliques
32(1)
4.2.1 Flat simplices and zero volume
32(1)
4.3 Realizing K-laterative graphs in RK
33(3)
4.3.1 Trilateration orders
34(1)
4.3.2 Trilaterative DGP
35(1)
4.3.3 The number of solutions of the TDGP
35(1)
4.3.4 Sensor network localization
36(1)
4.4 Realizing (K - 1)-laterative graphs in RK
36(5)
4.4.1 The shape of protein backbones
37(1)
4.4.2 Discretizable DGP
37(1)
4.4.3 A Branch-and-Prune algorithm
38(1)
4.4.4 Some examples
39(2)
4.4.5 Finding all realizations
41(1)
4.4.6 Worst-case complexity
41(1)
4.4.7 Best-case complexity
41(1)
4.5 Exercises
41(2)
5 Molecular distance geometry problems
43(14)
5.1 Contiguous (K - 1)-lateration orders
43(2)
5.1.1 The generalized DMDGP
43(1)
5.1.2 Realizing KDMDGP graphs
44(1)
5.1.3 Feasibility of Next
44(1)
5.2 Partial reflection symmetry
45(7)
5.2.1 Isometry and congruence
46(1)
5.2.2 The discretization group
47(2)
5.2.3 The pruning group
49(2)
5.2.4 A symmetry-aware BP
51(1)
5.2.5 Number of realizations of KDMDGP graphs
52(1)
5.3 Fixed-parameter tractability
52(2)
5.3.1 BP tree width
52(2)
5.3.2 The BP seems polynomial on proteins
54(1)
5.4 Exercises
54(3)
6 Vertex orders
57(10)
6.1 Existence of trilateration orders
57(4)
6.1.1 Problem hardness
57(3)
6.1.2 A Fixed-Parameter Tractable algorithm
60(1)
6.2 Existence of contiguous trilateration orders
61(3)
6.2.1 Problem hardness
62(1)
6.2.2 A mathematical programming formulation
63(1)
6.3 Exercises
64(3)
7 Flexibility and rigidity
67(14)
7.1 Some preliminary notions
67(1)
7.2 Rigidity of frameworks
68(1)
7.3 The rigidity matrix
69(6)
7.3.1 The rank of the rigidity matrix
69(1)
7.3.2 Regular and singular realizations
70(1)
7.3.3 The nullity of the rigidity matrix: infinitesimal rigidity
71(2)
7.3.4 Asimow and Roth's theorems
73(1)
7.3.5 Generic rigidity
74(1)
7.4 Graph rigidity on the line and in the plane
75(4)
7.4.1 Graph rigidity on a line
76(1)
7.4.2 General position
76(1)
7.4.3 Abstract rigidity
76(2)
7.4.4 Laman's theorem
78(1)
7.5 Exercises
79(2)
8 Approximate realizations
81(12)
8.1 The weighted adjacency matrix
81(1)
8.2 Matrix completion
81(1)
8.3 Overall method structure
82(1)
8.4 Approximate Completion Methods
82(1)
8.4.1 Constant completion
82(1)
8.4.2 Shortest paths
83(1)
8.5 Approximate realization methods
83(4)
8.5.1 Classic Multidimensional Scaling
83(4)
8.5.2 Proximity adjustment
87(1)
8.6 Approximate projection methods
87(3)
8.6.1 Principal Components Analysis
87(1)
8.6.2 Gaussian random projections
88(1)
8.6.3 The Johnson--Lindenstrauss lemma
88(2)
8.7 Isomap
90(1)
8.8 Stochastic Proximity Embedding
90(1)
8.9 Exercises
91(2)
9 Taking DG further
93(4)
9.1 Modeling signal processing problems
94(1)
9.2 Theory of solution uniqueness
95(1)
9.3 Combinatorial methods
95(1)
9.4 Optimization-based solution methods
95(1)
9.5 Debitum Gratitudinis (DG)
96(1)
Appendix: Mathematical notions 97(20)
References 117(4)
Index 121
Leo Liberti is a research director at CNRS and a professor at Ecole Polytechnique, France. Professor Libertis mathematical and optimization-related research interests are broad and his publications are extensive. In addition to co-authorship of this present textbook, he has co-edited two volumes with Springer: Distance Geometry, © 2013, 978-1-4614-5127-3  and Global Optimization: From Theory to Implementation, © 2008,  978-0-387-28260-2.

Carlile Lavor is a Full Professor at the Department of Applied Mathematics, University of Campinas, Campinas, Brazil. His main research interests are related to theory and applications of distance geometry and geometric algebra. In addition to co-authorship of this present textbook, he is co-author of the SpringerBrief Introduction to Distance Geometry Applied to Molecular Geometry, © 2017, 978-3-319-57182-9, and co-editor of Distance Geometry, © 2013, 978-1-4614-5127-3.
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