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Theory And Computation Of Tensors

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Theory and Computation of Tensors

Theory and Computation of Tensors Book
Author : Yimin Wei,Weiyang Ding
Publisher : Academic Press
Release : 2016-09-01
ISBN : 9780128039533
Language : En, Es, Fr & De

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Book Description :

"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 tensorsInvestigates theories and computations of tensors to broaden perspectives on matricesDiscusses how to extend numerical linear algebra to numerical multi-linear algebraOffers examples of how researchers and students can engage in research and the applications of tensors and multi-dimensional arrays

Theory and Computation of Tensors

Theory and Computation of Tensors Book
Author : Yimin Wei,Weiyang Ding
Publisher : Academic Press
Release : 2016-08-28
ISBN : 0128039809
Language : En, Es, Fr & De

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Book Description :

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

Theory and Computation of Complex Tensors and its Applications

Theory and Computation of Complex Tensors and its Applications Book
Author : Maolin Che,Yimin Wei
Publisher : Springer Nature
Release : 2020-04-01
ISBN : 9811520593
Language : En, Es, Fr & De

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Book Description :

The book provides an introduction of very recent results about the tensors and mainly focuses on the authors' work and perspective. A systematic description about how to extend the numerical linear algebra to the numerical multi-linear algebra is also delivered in this book. The authors design the neural network model for the computation of the rank-one approximation of real tensors, a normalization algorithm to convert some nonnegative tensors to plane stochastic tensors and a probabilistic algorithm for locating a positive diagonal in a nonnegative tensors, adaptive randomized algorithms for computing the approximate tensor decompositions, and the QR type method for computing U-eigenpairs of complex tensors. This book could be used for the Graduate course, such as Introduction to Tensor. Researchers may also find it helpful as a reference in tensor research.

Theory and Computation of Complex Tensors and its Applications

Theory and Computation of Complex Tensors and its Applications Book
Author : Maolin Che,Yimin Wei
Publisher : Springer
Release : 2020-04-29
ISBN : 9789811520587
Language : En, Es, Fr & De

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Book Description :

The book provides an introduction of very recent results about the tensors and mainly focuses on the authors' work and perspective. A systematic description about how to extend the numerical linear algebra to the numerical multi-linear algebra is also delivered in this book. The authors design the neural network model for the computation of the rank-one approximation of real tensors, a normalization algorithm to convert some nonnegative tensors to plane stochastic tensors and a probabilistic algorithm for locating a positive diagonal in a nonnegative tensors, adaptive randomized algorithms for computing the approximate tensor decompositions, and the QR type method for computing U-eigenpairs of complex tensors. This book could be used for the Graduate course, such as Introduction to Tensor. Researchers may also find it helpful as a reference in tensor research.

Tensors for Data Processing

Tensors for Data Processing Book
Author : Yipeng Liu
Publisher : Academic Press
Release : 2021-11-01
ISBN : 0323859658
Language : En, Es, Fr & De

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Book Description :

Tensors for Data Processing: Theory, Methods and Applications presents both classical and state-of-the-art methods on tensor computation for data processing, covering computation theories, processing methods, computing and engineering applications, with an emphasis on techniques for data processing. This reference is ideal for students, researchers and industry developers who want to understand and use tensor-based data processing theories and methods. As a higher-order generalization of a matrix, tensor-based processing can avoid multi-linear data structure loss that occurs in classical matrix-based data processing methods. This move from matrix to tensors is beneficial for many diverse application areas, including signal processing, computer science, acoustics, neuroscience, communication, medical engineering, seismology, psychometric, chemometrics, biometric, quantum physics and quantum chemistry. Provides a complete reference on classical and state-of-the-art tensor-based methods for data processing Includes a wide range of applications from different disciplines Gives guidance for their application

Atomic and Molecular Nonlinear Optics Theory Experiment and Computation

Atomic and Molecular Nonlinear Optics  Theory  Experiment and Computation Book
Author : G. Maroulis,T. Bancewicz,B. Champagne
Publisher : IOS Press
Release : 2011-05-27
ISBN : 1607507420
Language : En, Es, Fr & De

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Book Description :

The papers collected in this volume in honor of the late Stanisław Kielich cover an impressive range of modern subjects in molecular science. These subjects include, among others, the nonlinear optics of molecules, new approaches to the electronic structure of large molecules, the properties of carbon nanotubes, fluorescence polarization spectroscopy, computational studies of systems of fundamental interest to collision-induced spectroscopy, the simulation of fluids, NLO materials, chemical bonding in complex molecules, the NLO properties of functionalized DNA and the magnetic properties of molecular assemblies. Written by eminent specialists, the papers should offer valuable guidance to a wide community of graduate students and researchers.

Physical Components of Tensors

Physical Components of Tensors Book
Author : Wolf Altman,Antonio Marmo De Oliveira
Publisher : CRC Press
Release : 2014-11-11
ISBN : 1482263823
Language : En, Es, Fr & De

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Book Description :

Illustrating the important aspects of tensor calculus, and highlighting its most practical features, Physical Components of Tensors presents an authoritative and complete explanation of tensor calculus that is based on transformations of bases of vector spaces rather than on transformations of coordinates. Written with graduate students, professors, and researchers in the areas of elasticity and shell theories in mind, this text focuses on the physical and nonholonomic components of tensors and applies them to the theories. It establishes a theory of physical and anholonomic components of tensors and applies the theory of dimensional analysis to tensors and (anholonomic) connections. This theory shows the relationship and compatibility among several existing definitions of physical components of tensors when referred to nonorthogonal coordinates. The book assumes a basic knowledge of linear algebra and elementary calculus, but revisits these subjects and introduces the mathematical backgrounds for the theory in the first three chapters. In addition, all field equations are also given in physical components as well. Comprised of five chapters, this noteworthy text: Deals with the basic concepts of linear algebra, introducing the vector spaces and the further structures imposed on them by the notions of inner products, norms, and metrics Focuses on the main algebraic operations for vectors and tensors and also on the notions of duality, tensor products, and component representation of tensors Presents the classical tensor calculus that functions as the advanced prerequisite for the development of subsequent chapters Provides the theory of physical and anholonomic components of tensors by associating them to the spaces of linear transformations and of tensor products and advances two applications of this theory Physical Components of Tensors contains a comprehensive account of tensor calculus, and is an essential reference for graduate students or engineers concerned with solid and structural mechanics.

Tensors

Tensors Book
Author : J. M. Landsberg
Publisher : American Mathematical Soc.
Release : 2011-12-14
ISBN : 0821869078
Language : En, Es, Fr & De

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Book Description :

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.

Computational Tensor Analysis of Shell Structures

Computational Tensor Analysis of Shell Structures Book
Author : Steve Naomis,Paul C.M. Lau
Publisher : Springer Science & Business Media
Release : 2012-12-06
ISBN : 3642842437
Language : En, Es, Fr & De

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Book Description :

This book presents a method which is capable of evaluating the deformation characteristics of thin shell structures A free vibration analysis is chosen as a convenient means of studying the displacement behaviour of the shell, enabling it to deform naturally without imposing any particular loading conditions. The strain-displacement equations for thin shells of arbitrary geometry are developed. These relationships are expressed in general curvilinear coordinates and are formulated entirely in the framework of tensor calculus. The resulting theory is not restricted to shell structures characterized by any particular geometric form, loading or boundary conditions. The complete displacement and strain equations developed by Flugge are approximated by the curvilinear finite difference method and are applied to computing the natural frequencies and mode shapes of general thin shells. This approach enables both the displacement components and geometric properties of the shell to be approximated numerically and accurately. The selection of an appropriate displacement field to approximate the deformation of the shell within each finite difference mesh is discussed in detail. In addition, comparisons are made between the use of second and third-order finite difference interpolation meshes.

Polarization and Moment Tensors

Polarization and Moment Tensors Book
Author : Habib Ammari,Hyeonbae Kang
Publisher : Springer Science & Business Media
Release : 2007-06-16
ISBN : 0387715665
Language : En, Es, Fr & De

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Book Description :

This book presents important recent developments in mathematical and computational methods used in impedance imaging and the theory of composite materials. By augmenting the theory with interesting practical examples and numerical illustrations, the exposition brings simplicity to the advanced material. An introductory chapter covers the necessary basics. An extensive bibliography and open problems at the end of each chapter enhance the text.

Theory of Elasticity

Theory of Elasticity Book
Author : A.I. Lurie
Publisher : Springer Science & Business Media
Release : 2010-05-30
ISBN : 9783540264552
Language : En, Es, Fr & De

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Book Description :

The classical theory of elasticity maintains a place of honour in the science ofthe behaviour ofsolids. Its basic definitions are general for all branches of this science, whilst the methods forstating and solving these problems serve as examples of its application. The theories of plasticity, creep, viscoelas ticity, and failure of solids do not adequately encompass the significance of the methods of the theory of elasticity for substantiating approaches for the calculation of stresses in structures and machines. These approaches constitute essential contributions in the sciences of material resistance and structural mechanics. The first two chapters form Part I of this book and are devoted to the basic definitions ofcontinuum mechanics; namely stress tensors (Chapter 1) and strain tensors (Chapter 2). The necessity to distinguish between initial and actual states in the nonlinear theory does not allow one to be content with considering a single strain measure. For this reason, it is expedient to introduce more rigorous tensors to describe the stress-strain state. These are considered in Section 1.3 for which the study of Sections 2.3-2.5 should precede. The mastering of the content of these sections can be postponed until the nonlinear theory is studied in Chapters 8 and 9.

Tensor Analysis with Applications in Mechanics

Tensor Analysis with Applications in Mechanics Book
Author : L. P. Lebedev
Publisher : World Scientific
Release : 2010
ISBN : 9814313998
Language : En, Es, Fr & De

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Book Description :

The tensorial nature of a quantity permits us to formulate transformation rules for its components under a change of basis. These rules are relatively simple and easily grasped by any engineering student familiar with matrix operators in linear algebra. More complex problems arise when one considers the tensor fields that describe continuum bodies. In this case general curvilinear coordinates become necessary. The principal basis of a curvilinear system is constructed as a set of vectors tangent to the coordinate lines. Another basis, called the dual basis, is also constructed in a special manner. The existence of these two bases is responsible for the mysterious covariant and contravariant terminology encountered in tensor discussions. A tensor field is a tensor-valued function of position in space. The use of tensor fields allows us to present physical laws in a clear, compact form. A byproduct is a set of simple and clear rules for the representation of vector differential operators such as gradient, divergence, and Laplacian in curvilinear coordinate systems. This book is a clear, concise, and self-contained treatment of tensors, tensor fields, and their applications. The book contains practically all the material on tensors needed for applications. It shows how this material is applied in mechanics, covering the foundations of the linear theories of elasticity and elastic shells. The main results are all presented in the first four chapters. The remainder of the book shows how one can apply these results to differential geometry and the study of various types of objects in continuum mechanics such as elastic bodies, plates, and shells. Each chapter of this new edition is supplied with exercises and problems most with solutions, hints, or answers to help the reader progress. An extended appendix serves as a handbook-style summary of all important formulas contained in the book.

Advances in Quantum Computation

Advances in Quantum Computation Book
Author : Kazem Mahdavi,Deborah Koslover
Publisher : American Mathematical Soc.
Release : 2009-04-01
ISBN : 0821846272
Language : En, Es, Fr & De

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Book Description :

This volume represents the talks given at the Conference on Interactions between Representation Theory, Quantum Field Theory, Category Theory, Mathematical Physics, and Quantum Information Theory, held in September 2007 at the University of Texas at Tyler. The papers in this volume, written by top experts in the field, address physical aspects, mathematical aspects, and foundational issues of quantum computation. This volume will benefit researchers interested in advances in quantum computation and communication, as well as graduate students who wish to enter the field of quantum computation.

EPR of Free Radicals in Solids

EPR of Free Radicals in Solids Book
Author : Anders Lund,Masaru Shiotani
Publisher : Springer Science & Business Media
Release : 2013-06-29
ISBN : 1475751664
Language : En, Es, Fr & De

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Book Description :

EPR of Free Radicals in Solids: Trends in Methods and Applications presents methods and applications of modern EPR for the study of free radical processes in solids, which so far are only available in the journal literature. The first part of the book, covering trends in methods, contains experimentally oriented chapters on continuous wave and pulsed EPR techniques and special methods involving muon magnetic resonance and optical detection and theory for dynamic studies. New simulation schemes, including the influence of dynamics, are presented as well as advances in the calculation of hyperfine and electronic g-tensors. The second part of the book presents applications involving studies of radiation and photo-induced inorganic and organic radicals in inert matrices, including novel results of quantum effects in small radicals. High-spin molecules and complexes are also considered as well as radical processes in photosynthesis. Recent advances in EPR dosimetry are summarized.

The Metric Theory of Tensor Products

The Metric Theory of Tensor Products Book
Author : Joseph Diestel,Alexandre Grothendieck,Jan H. Fourie,Johan Swart
Publisher : American Mathematical Soc.
Release : 2008-01-01
ISBN : 9780821872697
Language : En, Es, Fr & De

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Book Description :

Famed mathematician Alexander Grothendieck, in his Resume, set forth his plan for the study of the finer structure of Banach spaces. He used tensor products as a foundation upon which he built the classes of operators most important to the study of Banach spaces and established the importance of the "local" theory in the study of these operators and the spaces they act upon. When Lintenstrauss and Pelczynski addressed his work at the rebirth of Banach space theory, they shed his Fundamental Inequality in the trappings of operator ideals by shedding the tensorial formulation. The authors of this book, however, feel that there is much of value in Grothendieck's original formulations in the Resume and here endeavor to "expose the Resume" by presenting most of Grothendieck's arguments using the mathematical tools that were available to him at the time.

Tensor Categories

Tensor Categories Book
Author : Pavel Etingof,Shlomo Gelaki,Dmitri Nikshych,Victor Ostrik
Publisher : American Mathematical Soc.
Release : 2016-08-05
ISBN : 1470434415
Language : En, Es, Fr & De

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Book Description :

Is there a vector space whose dimension is the golden ratio? Of course not—the golden ratio is not an integer! But this can happen for generalizations of vector spaces—objects of a tensor category. The theory of tensor categories is a relatively new field of mathematics that generalizes the theory of group representations. It has deep connections with many other fields, including representation theory, Hopf algebras, operator algebras, low-dimensional topology (in particular, knot theory), homotopy theory, quantum mechanics and field theory, quantum computation, theory of motives, etc. This book gives a systematic introduction to this theory and a review of its applications. While giving a detailed overview of general tensor categories, it focuses especially on the theory of finite tensor categories and fusion categories (in particular, braided and modular ones), and discusses the main results about them with proofs. In particular, it shows how the main properties of finite-dimensional Hopf algebras may be derived from the theory of tensor categories. Many important results are presented as a sequence of exercises, which makes the book valuable for students and suitable for graduate courses. Many applications, connections to other areas, additional results, and references are discussed at the end of each chapter.

Applied and Computational Matrix Analysis

Applied and Computational Matrix Analysis Book
Author : Natália Bebiano
Publisher : Springer
Release : 2018-05-04
ISBN : 9783319842912
Language : En, Es, Fr & De

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Book Description :

This volume presents recent advances in the field of matrix analysis based on contributions at the MAT-TRIAD 2015 conference. Topics covered include interval linear algebra and computational complexity, Birkhoff polynomial basis, tensors, graphs, linear pencils, K-theory and statistic inference, showing the ubiquity of matrices in different mathematical areas. With a particular focus on matrix and operator theory, statistical models and computation, the International Conference on Matrix Analysis and its Applications 2015, held in Coimbra, Portugal, was the sixth in a series of conferences. Applied and Computational Matrix Analysis will appeal to graduate students and researchers in theoretical and applied mathematics, physics and engineering who are seeking an overview of recent problems and methods in matrix analysis.

Methods in Computational Chemistry

Methods in Computational Chemistry Book
Author : Stephen Wilson
Publisher : Springer Science & Business Media
Release : 2013-12-14
ISBN : 1489916393
Language : En, Es, Fr & De

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Book Description :

Providing specialist reviews and analyses of contemporary theories, algorithms, and techniques, this series aims to facilitate the effective exploitation of available computing power. The current volume focuses on the theoretical determination of atomic and molecular properties as related to wave functions, electron densities, and total energies.

Algebraic and Computational Aspects of Real Tensor Ranks

Algebraic and Computational Aspects of Real Tensor Ranks Book
Author : Toshio Sakata,Toshio Sumi,Mitsuhiro Miyazaki
Publisher : Springer
Release : 2016-03-18
ISBN : 4431554599
Language : En, Es, Fr & De

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Book Description :

This book provides comprehensive summaries of theoretical (algebraic) and computational aspects of tensor ranks, maximal ranks, and typical ranks, over the real number field. Although tensor ranks have been often argued in the complex number field, it should be emphasized that this book treats real tensor ranks, which have direct applications in statistics. The book provides several interesting ideas, including determinant polynomials, determinantal ideals, absolutely nonsingular tensors, absolutely full column rank tensors, and their connection to bilinear maps and Hurwitz-Radon numbers. In addition to reviews of methods to determine real tensor ranks in details, global theories such as the Jacobian method are also reviewed in details. The book includes as well an accessible and comprehensive introduction of mathematical backgrounds, with basics of positive polynomials and calculations by using the Groebner basis. Furthermore, this book provides insights into numerical methods of finding tensor ranks through simultaneous singular value decompositions.

Conformal Field Theories and Tensor Categories

Conformal Field Theories and Tensor Categories Book
Author : Chengming Bai,Jürgen Fuchs,Yi-Zhi Huang,Liang Kong,Ingo Runkel,Christoph Schweigert
Publisher : Springer Science & Business Media
Release : 2013-10-30
ISBN : 3642393837
Language : En, Es, Fr & De

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Book Description :

The present volume is a collection of seven papers that are either based on the talks presented at the workshop "Conformal field theories and tensor categories" held June 13 to June 17, 2011 at the Beijing International Center for Mathematical Research, Peking University, or are extensions of the material presented in the talks at the workshop. These papers present new developments beyond rational conformal field theories and modular tensor categories and new applications in mathematics and physics. The topics covered include tensor categories from representation categories of Hopf algebras, applications of conformal field theories and tensor categories to topological phases and gapped systems, logarithmic conformal field theories and the corresponding non-semisimple tensor categories, and new developments in the representation theory of vertex operator algebras. Some of the papers contain detailed introductory material that is helpful for graduate students and researchers looking for an introduction to these research directions. The papers also discuss exciting recent developments in the area of conformal field theories, tensor categories and their applications and will be extremely useful for researchers working in these areas.