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2018 | Buch

Preliminaries Kapitel lesen Erstes Kapitel lesen

Tensor Eigenvalues and Their Applications

verfasst von: Liqun Qi, Haibin Chen, Yannan Chen

Verlag: Springer Singapore

Buchreihe : Advances in Mechanics and Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Preliminaries
Abstract
In this chapter, we review some basic knowledge about tensors.
Liqun Qi, Haibin Chen, Yannan Chen
Chapter 2. Multilinear Systems
Abstract
A central problem in both pure and applied mathematics is solving various kinds of equations. Every progress in this discipline makes a big step in mathematics, especially in applied mathematics, such as Gaussian elimination method for linear equations, the simplex method for linear inequalities, and Gröbner bases for polynomial equations.
Liqun Qi, Haibin Chen, Yannan Chen
Chapter 3. Hankel Tensor Computation and Exponential Data Fitting
Abstract
Hankel structures are widely used in real-world problems arising from signal processing, automatic control, and geophysics. For example, a Hankel matrix was formulated to analyze the time-domain signals in nuclear magnetic resonance spectroscopy, which is crucial for brain tumour detection.
Liqun Qi, Haibin Chen, Yannan Chen
Chapter 4. Tensor Complementarity Problems
Abstract
Complementarity problems encompass several important classes of mathematical optimization problems, e.g., linear programming, quadratic programming, linear conic optimization problems, etc. Actually, we always solve an optimization problem via its optimality condition, which usually turns out to be a complementarity problem, e.g., KKT system.
Liqun Qi, Haibin Chen, Yannan Chen
Chapter 5. Tensor Eigenvalue Complementarity Problems
Abstract
This chapter is a companion chapter of Chap. 4. In this chapter, we mainly discuss tensor eigenvalue complementarity problems (TEiCP). It is a generalization of the matrix eigenvalue complementarity problem (EiCP), which possess a broad range of interesting applications.
Liqun Qi, Haibin Chen, Yannan Chen
Chapter 6. Higher Order Diffusion Tensor Imaging
Abstract
Diffusion tensor imaging (DTI) is one of the most promising medical imaging models, and the most applicable technique in modern clinical medicine. While, there are limitations to DTI, which becomes useless in non-isotropic materials. As a resolution, diffusion kurtosis imaging (DKI) is proposed as a new model in medical engineering, which can characterize the non-Gaussian diffusion behavior in tissues, and in which a diffusion kurtosis (DK) tensor is involved. A DK tensor is a fourth order three dimensional symmetric tensor. In this chapter, we will apply the spectral theory of tensors to this particular type of tensors arising from medical imaging and derive some applications.
Liqun Qi, Haibin Chen, Yannan Chen
Chapter 7. Third Order Tensors in Physics and Mechanics
Abstract
Third order tensors have wide applications in physics and mechanics. Examples include piezoelectric tensors in crystal study, third order symmetric traceless tensors in liquid crystal study and third order susceptibility tensors in nonlinear optics study. On the other hand, the Levi-Civita tensor is famous in tensor calculus.
Liqun Qi, Haibin Chen, Yannan Chen
Chapter 8. Fourth Order Tensors in Physics and Mechanics
Abstract
Fourth order tensors have also wide applications in physics and mechanics. Examples include the piezo-optical tensor, the elasto-optical tensor and the flexoelectric tensor. The most well-known fourth order tensor is the elasticity tensor (Huang et al., Tensor Analysis (in Chinese). Tsinghua University Press, Beijing, 2003, [134], Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices. Clarendon Press, Oxford, 1985, [212], Zou et al., J. Mech. Phys. Solids. 58:346–372, 2010, [318]). It is closely related to the strong ellipticity condition in nonlinear mechanics.
Liqun Qi, Haibin Chen, Yannan Chen
Chapter 9. Higher Order Tensors in Quantum Physics
Abstract
In this chapter, we will apply tensor analysis to the quantum entanglement problem and the classicality problem of spin states in quantum physics.
Liqun Qi, Haibin Chen, Yannan Chen
Backmatter
Metadaten
Titel
Tensor Eigenvalues and Their Applications
verfasst von
Liqun Qi
Haibin Chen
Yannan Chen
Copyright-Jahr
2018
Verlag
Springer Singapore
Electronic ISBN
978-981-10-8058-6
Print ISBN
978-981-10-8057-9
DOI
https://doi.org/10.1007/978-981-10-8058-6