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E-grāmata: Tensor Eigenvalues and Their Applications

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Tensor Eigenvalues and Their ApplicationsPalielināt
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This book offers an introduction to applications prompted by tensor analysis, especially by the spectral tensor theory developed in recent years. It covers applications of tensor eigenvalues in multilinear systems, exponential data fitting, tensor complementarity problems, and tensor eigenvalue complementarity problems. It also addresses higher-order diffusion tensor imaging, third-order symmetric and traceless tensors in liquid crystals, piezoelectric tensors, strong ellipticity for elasticity tensors, and higher-order tensors in quantum physics. This book is a valuable reference resource for researchers and graduate students who are interested in applications of tensor eigenvalues.

Recenzijas

This carefully written book is really useful for students, researchers, practitioners, and anybody who is interested in tensors, numerical multilinear algebra, and related subjects. This book will provide a good base for further research on tensor eigenvalue applications in these and some more areas. I definitely recommend this book! (Guyan Ni, Mathematical Reviews, December, 2018)

1 Preliminaries
1(8)
1.1 Tensors (Hypermatrices) and Tensor Products
1(3)
1.2 Eigenvalues of Tensors
4(3)
1.3 Notes
7(2)
2 Multilinear Systems
9(40)
2.1 Multilinear Systems Defined by M-Tensors
10(6)
2.2 Finding the Positive Solution of a Nonsingular M-Equation
16(9)
2.3 Tensor Methods for Solving Symmetric M-Tensor Systems
25(10)
2.4 Solution Methods for General Multilinear Systems
35(9)
2.5 Notes
44(3)
2.6 Exercise
47(2)
3 Hankel Tensor Computation and Exponential Data Fitting
49(16)
3.1 Fast Hankel Tensor--Vector Product
50(3)
3.2 Computing Eigenvalues of a Hankel Tensor
53(3)
3.3 Convergence Analysis
56(4)
3.4 Exponential Data Fitting
60(3)
3.5 Notes
63(1)
3.6 Exercises
64(1)
4 Tensor Complementarity Problems
65(70)
4.1 Preliminaries for Tensor Complementarity Problems
67(3)
4.2 An m Person Noncooperative Game
70(8)
4.3 Positive Definite Tensors for Tensor Complementarity Problems
78(5)
4.4 P and P0-Tensors
83(9)
4.5 Tensor Complementarity Problems and Semi-positive Tensors
92(6)
4.6 Tensor Complementarity Problems and Q-Tensors
98(12)
4.7 Z-Tensor Complementarity Problems
110(6)
4.8 Solution Boundedness of Tensor Complementarity Problems
116(6)
4.9 Global Uniqueness and Solvability
122(4)
4.10 Exceptional Regular Tensors and Tensor Complementarity Problems
126(7)
4.11 Notes
133(1)
4.12 Exercises
134(1)
5 Tensor Eigenvalue Complementarity Problems
135(48)
5.1 Tensor Eigenvalue Complementarity Problems
137(10)
5.2 Pareto H(Z)-Eigenvalues of Tensors
147(3)
5.3 Computational Methods for Tensor Eigenvalue Complementarity Problems
150(8)
5.4 A Unified Framework of Tensor Higher-Degree Eigenvalue Complementarity Problems
158(11)
5.5 The Semidefinite Relaxation Method
169(12)
5.6 Notes
181(1)
5.7 Exercises
182(1)
6 Higher Order Diffusion Tensor Imaging
183(24)
6.1 Diffusion Kurtosis Tensor Imaging and D-Eigenvalues
184(4)
6.2 Positive Definiteness of Diffusion Kurtosis Imaging
188(3)
6.3 Positive Semidefinite Diffusion Tensor Imaging
191(4)
6.4 Nonnegative Diffusion Orientation Distribution Function
195(4)
6.5 Nonnegative Fiber Orientation Distribution Function
199(3)
6.6 Image Authenticity Verification
202(2)
6.7 Notes
204(2)
6.8 Exercises
206(1)
7 Third Order Tensors in Physics and Mechanics
207(42)
7.1 Third Order Tensors and Hypermatrices
208(10)
7.2 C-Eigenvalues of the Piezoelectric Tensors
218(8)
7.3 Third Order Three Dimensional Symmetric Traceless Tensors and Liquid Crystals
226(5)
7.4 Algebraic Expression of the Dome Surface
231(7)
7.5 Algebraic Expression of the Separatrix Surface
238(5)
7.6 Eigendiscriminant from Algebraic Geometry
243(2)
7.7 Notes
245(2)
7.8 Exercises
247(2)
8 Fourth Order Tensors in Physics and Mechanics
249(36)
8.1 The Elasticity Tensor, Strong Ellipticity and M-Eigenvalues
250(7)
8.2 Strong Ellipticity via Z-Eigenvalues of Symmetric Tensors
257(4)
8.3 Other Sufficient Condition for Strong Ellipticity
261(8)
8.4 Computational Methods for M-Eigenvalues
269(5)
8.5 Higher Order Elasticity Tensors
274(9)
8.6 Notes
283(1)
8.7 Exercises
284(1)
9 Higher Order Tensors in Quantum Physics
285(28)
9.1 Quantum Entanglement Problems
287(1)
9.2 Geometric Measure of Entanglement of Multipartite Pure States
288(4)
9.3 Z-Eigenvalues and Entanglement of Symmetric States
292(5)
9.4 Geometric Measure and U-Eigenvalues of Tensors
297(2)
9.5 Regularly Decomposable Tensors and Classical Spin States
299(11)
9.6 Notes
310(1)
9.7 Exercises
311(2)
References 313(14)
Index 327
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