Download or read book Tensor Eigenvalues and Their Applications written by Liqun Qi and published by Springer. This book was released on 2018-03-30 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Tensor Analysis written by Liqun Qi and published by SIAM. This book was released on 2017-04-19 with total page 313 pages. Available in PDF, EPUB and Kindle. Book excerpt: Tensors, or hypermatrices, are multi-arrays with more than two indices. In the last decade or so, many concepts and results in matrix theory?some of which are nontrivial?have been extended to tensors and have a wide range of applications (for example, spectral hypergraph theory, higher order Markov chains, polynomial optimization, magnetic resonance imaging, automatic control, and quantum entanglement problems). The authors provide a comprehensive discussion of this new theory of tensors. Tensor Analysis: Spectral Theory and Special Tensors is unique in that it is the first book on these three subject areas: spectral theory of tensors; the theory of special tensors, including nonnegative tensors, positive semidefinite tensors, completely positive tensors, and copositive tensors; and the spectral hypergraph theory via tensors. ?
Download or read book Tensors Geometry and Applications written by J. M. Landsberg and published by American Mathematical Soc.. This book was released on 2011-12-14 with total page 464 pages. Available in PDF, EPUB and Kindle. Book excerpt: Tensors are ubiquitous in the sciences. The geometry of tensors is both a powerful tool for extracting information from data sets, and a beautiful subject in its own right. This book has three intended uses: a classroom textbook, a reference work for researchers in the sciences, and an account of classical and modern results in (aspects of) the theory that will be of interest to researchers in geometry. For classroom use, there is a modern introduction to multilinear algebra and to the geometry and representation theory needed to study tensors, including a large number of exercises. For researchers in the sciences, there is information on tensors in table format for easy reference and a summary of the state of the art in elementary language. This is the first book containing many classical results regarding tensors. Particular applications treated in the book include the complexity of matrix multiplication, P versus NP, signal processing, phylogenetics, and algebraic statistics. For geometers, there is material on secant varieties, G-varieties, spaces with finitely many orbits and how these objects arise in applications, discussions of numerous open questions in geometry arising in applications, and expositions of advanced topics such as the proof of the Alexander-Hirschowitz theorem and of the Weyman-Kempf method for computing syzygies.
Download or read book Tensor Analysis with Applications in Mechanics written by L. P. Lebedev and published by World Scientific. This book was released on 2010 with total page 378 pages. Available in PDF, EPUB and Kindle. Book excerpt: 1. Preliminaries. 1.1. The vector concept revisited. 1.2. A first look at tensors. 1.3. Assumed background. 1.4. More on the notion of a vector. 1.5. Problems -- 2. Transformations and vectors. 2.1. Change of basis. 2.2. Dual bases. 2.3. Transformation to the reciprocal frame. 2.4. Transformation between general frames. 2.5. Covariant and contravariant components. 2.6. The cross product in index notation. 2.7. Norms on the space of vectors. 2.8. Closing remarks. 2.9. Problems -- 3. Tensors. 3.1. Dyadic quantities and tensors. 3.2. Tensors from an operator viewpoint. 3.3. Dyadic components under transformation. 3.4. More dyadic operations. 3.5. Properties of second-order tensors. 3.6. Eigenvalues and eigenvectors of a second-order symmetric tensor. 3.7. The Cayley-Hamilton theorem. 3.8. Other properties of second-order tensors. 3.9. Extending the Dyad idea. 3.10. Tensors of the fourth and higher orders. 3.11. Functions of tensorial arguments. 3.12. Norms for tensors, and some spaces. 3.13. Differentiation of tensorial functions. 3.14. Problems -- 4. Tensor fields. 4.1. Vector fields. 4.2. Differentials and the nabla operator. 4.3. Differentiation of a vector function. 4.4. Derivatives of the frame vectors. 4.5. Christoffel coefficients and their properties. 4.6. Covariant differentiation. 4.7. Covariant derivative of a second-order tensor. 4.8. Differential operations. 4.9. Orthogonal coordinate systems. 4.10. Some formulas of integration. 4.11. Problems -- 5. Elements of differential geometry. 5.1. Elementary facts from the theory of curves. 5.2. The torsion of a curve. 5.3. Frenet-Serret equations. 5.4. Elements of the theory of surfaces. 5.5. The second fundamental form of a surface. 5.6. Derivation formulas. 5.7. Implicit representation of a curve; contact of curves. 5.8. Osculating paraboloid. 5.9. The principal curvatures of a surface. 5.10. Surfaces of revolution. 5.11. Natural equations of a curve. 5.12. A word about rigor. 5.13. Conclusion. 5.14. Problems -- 6. Linear elasticity. 6.1. Stress tensor. 6.2. Strain tensor. 6.3. Equation of motion. 6.4. Hooke's law. 6.5. Equilibrium equations in displacements. 6.6. Boundary conditions and boundary value problems. 6.7. Equilibrium equations in stresses. 6.8. Uniqueness of solution for the boundary value problems of elasticity. 6.9. Betti's reciprocity theorem. 6.10. Minimum total energy principle. 6.11. Ritz's method. 6.12. Rayleigh's variational principle. 6.13. Plane waves. 6.14. Plane problems of elasticity. 6.15. Problems -- 7. Linear elastic shells. 7.1. Some useful formulas of surface theory. 7.2. Kinematics in a neighborhood of [symbol]. 7.3. Shell equilibrium equations. 7.4. Shell deformation and strains; Kirchhoff's hypotheses. 7.5. Shell energy. 7.6. Boundary conditions. 7.7. A few remarks on the Kirchhoff-Love theory. 7.8. Plate theory. 7.9. On Non-classical theories of plates and shells
Download or read book Tensor Algebra and Tensor Analysis for Engineers written by Mikhail Itskov and published by Springer Science & Business Media. This book was released on 2012-08-13 with total page 279 pages. Available in PDF, EPUB and Kindle. Book excerpt: There is a large gap between the engineering course in tensor algebra on the one hand and the treatment of linear transformations within classical linear algebra on the other hand. The aim of this modern textbook is to bridge this gap by means of the consequent and fundamental exposition. The book primarily addresses engineering students with some initial knowledge of matrix algebra. Thereby the mathematical formalism is applied as far as it is absolutely necessary. Numerous exercises are provided in the book and are accompanied by solutions, enabling self-study. The last chapters of the book deal with modern developments in the theory of isotropic and anisotropic tensor functions and their applications to continuum mechanics and are therefore of high interest for PhD-students and scientists working in this area. This third edition is completed by a number of additional figures, examples and exercises. The text and formulae have been revised and improved where necessary.
Download or read book Theory and Computation of Tensors written by Yimin Wei and published by Academic Press. This book was released on 2016-08-28 with total page 150 pages. Available in PDF, EPUB and Kindle. Book excerpt: Theory and Computation of Tensors: Multi-Dimensional Arrays investigates theories and computations of tensors to broaden perspectives on matrices. Data in the Big Data Era is not only growing larger but also becoming much more complicated. Tensors (multi-dimensional arrays) arise naturally from many engineering or scientific disciplines because they can represent multi-relational data or nonlinear relationships. - Provides an introduction of recent results about tensors - Investigates theories and computations of tensors to broaden perspectives on matrices - Discusses how to extend numerical linear algebra to numerical multi-linear algebra - Offers examples of how researchers and students can engage in research and the applications of tensors and multi-dimensional arrays
Download or read book Generalized Continua as Models for Classical and Advanced Materials written by Holm Altenbach and published by Springer. This book was released on 2016-04-15 with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is devoted to an actual topic which is the focus world-wide of various research groups. It contains contributions describing the material behavior on different scales, new existence and uniqueness theorems, the formulation of constitutive equations for advanced materials. The main emphasis of the contributions is directed on the following items - Modelling and simulation of natural and artificial materials with significant microstructure, - Generalized continua as a result of multi-scale models, - Multi-field actions on materials resulting in generalized material models, - Theories including higher gradients, and - Comparison with discrete modelling approaches
Download or read book Visualization and Processing of Tensors and Higher Order Descriptors for Multi Valued Data written by Carl-Fredrik Westin and published by Springer. This book was released on 2014-07-17 with total page 346 pages. Available in PDF, EPUB and Kindle. Book excerpt: Arising from the fourth Dagstuhl conference entitled Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data (2011), this book offers a broad and vivid view of current work in this emerging field. Topics covered range from applications of the analysis of tensor fields to research on their mathematical and analytical properties. Part I, Tensor Data Visualization, surveys techniques for visualization of tensors and tensor fields in engineering, discusses the current state of the art and challenges, and examines tensor invariants and glyph design, including an overview of common glyphs. The second Part, Representation and Processing of Higher-order Descriptors, describes a matrix representation of local phase, outlines mathematical morphological operations techniques, extended for use in vector images, and generalizes erosion to the space of diffusion weighted MRI. Part III, Higher Order Tensors and Riemannian-Finsler Geometry, offers powerful mathematical language to model and analyze large and complex diffusion data such as High Angular Resolution Diffusion Imaging (HARDI) and Diffusion Kurtosis Imaging (DKI). A Part entitled Tensor Signal Processing presents new methods for processing tensor-valued data, including a novel perspective on performing voxel-wise morphometry of diffusion tensor data using kernel-based approach, explores the free-water diffusion model, and reviews proposed approaches for computing fabric tensors, emphasizing trabecular bone research. The last Part, Applications of Tensor Processing, discusses metric and curvature tensors, two of the most studied tensors in geometry processing. Also covered is a technique for diagnostic prediction of first-episode schizophrenia patients based on brain diffusion MRI data. The last chapter presents an interactive system integrating the visual analysis of diffusion MRI tractography with data from electroencephalography.
Download or read book Tensor Calculus With Applications written by Vladislav V Goldberg and published by World Scientific Publishing Company. This book was released on 2003-09-29 with total page 381 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook presents the foundations of tensor calculus and the elements of tensor analysis. In addition, the authors consider numerous applications of tensors to geometry, mechanics and physics.While developing tensor calculus, the authors emphasize its relationship with linear algebra. Necessary notions and theorems of linear algebra are introduced and proved in connection with the construction of the apparatus of tensor calculus; prior knowledge is not assumed. For simplicity and to enable the reader to visualize concepts more clearly, all exposition is conducted in three-dimensional space. The principal feature of the book is that the authors use mainly orthogonal tensors, since such tensors are important in applications to physics and engineering.With regard to applications, the authors construct the general theory of second-degree surfaces, study the inertia tensor as well as the stress and strain tensors, and consider some problems of crystallophysics. The last chapter introduces the elements of tensor analysis.All notions introduced in the book, and also the obtained results, are illustrated with numerous examples discussed in the text. Each section of the book presents problems (a total over 300 problems are given). Examples and problems are intended to illustrate, reinforce and deepen the presented material. There are answers to most of the problems, as well as hints and solutions to selected problems at the end of the book.
Download or read book Matrix Analysis and Applications written by Xian-Da Zhang and published by Cambridge University Press. This book was released on 2017-10-05 with total page 761 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory, methods and applications of matrix analysis are presented here in a novel theoretical framework.
Download or read book Visualization and Processing of Tensor Fields written by Joachim Weickert and published by Springer Science & Business Media. This book was released on 2007-06-25 with total page 478 pages. Available in PDF, EPUB and Kindle. Book excerpt: Matrix-valued data sets – so-called second order tensor fields – have gained significant importance in scientific visualization and image processing due to recent developments such as diffusion tensor imaging. This book is the first edited volume that presents the state of the art in the visualization and processing of tensor fields. It contains some longer chapters dedicated to surveys and tutorials of specific topics, as well as a great deal of original work by leading experts that has not been published before. It serves as an overview for the inquiring scientist, as a basic foundation for developers and practitioners, and as as a textbook for specialized classes and seminars for graduate and doctoral students.
Download or read book Medical Imaging Systems Technology Volume 5 Methods In Cardiovascular And Brain Systems written by Cornelius T Leondes and published by World Scientific. This book was released on 2005-10-25 with total page 408 pages. Available in PDF, EPUB and Kindle. Book excerpt: This scholarly set of well-harmonized volumes provides indispensable and complete coverage of the exciting and evolving subject of medical imaging systems. Leading experts on the international scene tackle the latest cutting-edge techniques and technologies in an in-depth but eminently clear and readable approach.Complementing and intersecting one another, each volume offers a comprehensive treatment of substantive importance to the subject areas. The chapters, in turn, address topics in a self-contained manner with authoritative introductions, useful summaries, and detailed reference lists. Extensively well-illustrated with figures throughout, the five volumes as a whole achieve a unique depth and breath of coverage.As a cohesive whole or independent of one another, the volumes may be acquired as a set or individually.
Download or read book MRI in Psychiatry written by Christoph Mulert and published by Springer. This book was released on 2014-06-27 with total page 430 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first comprehensive textbook on the use of MRI in psychiatry covering imaging techniques, brain systems and a review of findings in different psychiatric disorders. The book is divided into three sections, the first of which covers in detail all the major MRI-based methodological approaches available today, including fMRI, EEG-fMRI, DTI and MR spectroscopy. In addition, the role of MRI in imaging genetics and combined brain stimulation and imaging is carefully explained. The second section provides an overview of the different brain systems that are relevant for psychiatric disorders, including the systems for perception, emotion, cognition and reward. The final part of the book presents the MRI findings that are obtained in all the major psychiatric disorders using the previously discussed techniques. Numerous carefully chosen images support the informative text, making this an ideal reference work for all practitioners and trainees with an interest in this flourishing field.
Download or read book Tensor Trigonometry written by A.S. Ninul and published by FIZMATLIT. This book was released on with total page 322 pages. Available in PDF, EPUB and Kindle. Book excerpt: Resume Planimetry includes metric part and trigonometry. In geometries of metric spaces from the end of XIX age their tensor forms are widely used. However the trigonometry is remained only in its scalar form in a plane. The tensor trigonometry is development of the flat scalar trigonometry from Leonard Euler classic forms into general multi-dimensional tensor forms with vector and scalar orthoprojections and with step by step increasing complexity and opportunities. Described in the book are fundamentals of this new mathematical subject with many initial examples of its applications. In theoretic plan, the tensor trigonometry complements naturally Analytic Geometry and Linear Algebra. In practical plan, it gives the clear instrument for solutions of various geometric and physical problems in homogeneous isotropic spaces, such as Euclidean, quasi- and pseudo-Euclidean ones. In these spaces, the tensor trigonometry gives very clear general laws of motions in complete forms and with polar decompositions into principal and secondary motions, their descriptive trigonometric vector models, which are applicable also to n-dimensional non-Euclidean geometries in subspaces of constant radius embedded in enveloping metric spaces, and in the theory of relativity. In STR, these applications were considered till a trigonometric 4D pseudoanalog of the 3D classic theory by Frenet–Serret with absolute differentially-geometric, kinematic and dynamic characteristics in the current points of a world line. New methods of the tensor trigonometry can be also useful in other domains of mathematics and physics. The book is intended for researchers in the fields of multi-dimensional spaces, analytic geometry, linear algebra with theory of matrices, non-Euclidean geometries, theory of relativity and also to all those who is interested in new knowledges and applications, given by exact sciences. It may be useful for educational purposes on this new subject in the university departments of algebra, geometry and physics. This book is an updated author’s English version of the original Russian scientific monograph “Tensor Trigonometry. Theory and Applications.” – Moscow: Publisher MIR, 2004, 336p., ISBN-10: 5-03-003717-9 and ISBN-13: 978-5-03-003717-2. On the Google books there is an original Russian edition of this book (2004): https://books.google.ru/books/about?id=HGgjEAAAQBAJ
Download or read book Diffusion Weighted MR Imaging of the Brain Head and Neck and Spine written by Toshio Moritani and published by Springer Nature. This book was released on 2021-05-30 with total page 928 pages. Available in PDF, EPUB and Kindle. Book excerpt: This richly illustrated book, now in an updated and extended third edition, systematically covers the use of diffusion-weighted (DW) MR imaging in all major areas of neuroradiology, including imaging of the head and neck and the spine as well as the brain. The authors guide the reader from the basic principles of DW imaging through to the use of cutting-edge diffusion sequences such as diffusion tensor (DTI) and kurtosis (DKI), fiber tractography, high b value, intravoxel incoherent motion (IVIM), neurite orientation dispersion and density imaging (NODDI), and oscillating gradient spin echo (OGSE). Pathology, pathophysiology, and patient management and treatment are all thoroughly discussed. Since the early descriptions by LeBihan and colleagues of the ability to image and measure the micromovement of water molecules in the brain, diffusion imaging and its derivatives have contributed ever more significantly to the evaluation of multiple disease processes. In comprehensively describing the state of the art in the field, this book will be of high value not only for those who deal routinely with neuro-MR imaging but also for readers who wish to establish a sound basis for understanding diffusion images in the hope of extending these principles into more exotic areas of neuroimaging.
Download or read book Vector Analysis and Cartesian Tensors written by D. E. Bourne and published by Academic Press. This book was released on 2014-05-10 with total page 271 pages. Available in PDF, EPUB and Kindle. Book excerpt: Vector Analysis and Cartesian Tensors, Second Edition focuses on the processes, methodologies, and approaches involved in vector analysis and Cartesian tensors, including volume integrals, coordinates, curves, and vector functions. The publication first elaborates on rectangular Cartesian coordinates and rotation of axes, scalar and vector algebra, and differential geometry of curves. Discussions focus on differentiation rules, vector functions and their geometrical representation, scalar and vector products, multiplication of a vector by a scalar, and angles between lines through the origin. The text then elaborates on scalar and vector fields and line, surface, and volume integrals, including surface, volume, and repeated integrals, general orthogonal curvilinear coordinates, and vector components in orthogonal curvilinear coordinates. The manuscript ponders on representation theorems for isotropic tensor functions, Cartesian tensors, applications in potential theory, and integral theorems. Topics include geometrical and physical significance of divergence and curl, Poisson's equation in vector form, isotropic scalar functions of symmetrical second order tensors, and diagonalization of second-order symmetrical tensors. The publication is a valuable reference for mathematicians and researchers interested in vector analysis and Cartesian tensors.
Download or read book Tensor Algebra and Tensor Analysis for Engineers written by Mikhail Itskov and published by Springer Science & Business Media. This book was released on 2009-04-30 with total page 253 pages. Available in PDF, EPUB and Kindle. Book excerpt: There is a large gap between engineering courses in tensor algebra on one hand, and the treatment of linear transformations within classical linear algebra on the other. This book addresses primarily engineering students with some initial knowledge of matrix algebra. Thereby, mathematical formalism is applied as far as it is absolutely necessary. Numerous exercises provided in the book are accompanied by solutions enabling autonomous study. The last chapters deal with modern developments in the theory of isotropic and anisotropic tensor functions and their applications to continuum mechanics and might therefore be of high interest for PhD-students and scientists working in this area.
