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Theory and Computation of Tensors

Theory and Computation of Tensors
Multi-Dimensional Arrays

by Yimin Wei,Weiyang Ding

  • Publisher : Academic Press
  • Release : 2016-08-28
  • Pages : 148
  • ISBN : 0128039809
  • Language : En, Es, Fr & De
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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
A Book

by Maolin Che,Yimin Wei

  • Publisher : Springer Nature
  • Release : 2020-04-01
  • Pages : 250
  • ISBN : 9811520593
  • Language : En, Es, Fr & De
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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.

An Introduction to Tensors and Group Theory for Physicists

An Introduction to Tensors and Group Theory for Physicists
A Book

by Nadir Jeevanjee

  • Publisher : Springer Science & Business Media
  • Release : 2011-08-26
  • Pages : 242
  • ISBN : 0817647147
  • Language : En, Es, Fr & De
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An Introduction to Tensors and Group Theory for Physicists provides both an intuitive and rigorous approach to tensors and groups and their role in theoretical physics and applied mathematics. A particular aim is to demystify tensors and provide a unified framework for understanding them in the context of classical and quantum physics. Connecting the component formalism prevalent in physics calculations with the abstract but more conceptual formulation found in many mathematical texts, the work will be a welcome addition to the literature on tensors and group theory. Advanced undergraduate and graduate students in physics and applied mathematics will find clarity and insight into the subject in this textbook.

Tensor Numerical Methods in Scientific Computing

Tensor Numerical Methods in Scientific Computing
A Book

by Boris N. Khoromskij

  • Publisher : Walter de Gruyter GmbH & Co KG
  • Release : 2018-06-11
  • Pages : 379
  • ISBN : 311036591X
  • Language : En, Es, Fr & De
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The most difficult computational problems nowadays are those of higher dimensions. This research monograph offers an introduction to tensor numerical methods designed for the solution of the multidimensional problems in scientific computing. These methods are based on the rank-structured approximation of multivariate functions and operators by using the appropriate tensor formats. The old and new rank-structured tensor formats are investigated. We discuss in detail the novel quantized tensor approximation method (QTT) which provides function-operator calculus in higher dimensions in logarithmic complexity rendering super-fast convolution, FFT and wavelet transforms. This book suggests the constructive recipes and computational schemes for a number of real life problems described by the multidimensional partial differential equations. We present the theory and algorithms for the sinc-based separable approximation of the analytic radial basis functions including Green’s and Helmholtz kernels. The efficient tensor-based techniques for computational problems in electronic structure calculations and for the grid-based evaluation of long-range interaction potentials in multi-particle systems are considered. We also discuss the QTT numerical approach in many-particle dynamics, tensor techniques for stochastic/parametric PDEs as well as for the solution and homogenization of the elliptic equations with highly-oscillating coefficients. Contents Theory on separable approximation of multivariate functions Multilinear algebra and nonlinear tensor approximation Superfast computations via quantized tensor approximation Tensor approach to multidimensional integrodifferential equations

University of Wisconsin Center for Plasma Theory and Computation Report

University of Wisconsin Center for Plasma Theory and Computation Report
A Book

by Anonim

  • Publisher : Unknown Publisher
  • Release : 1989
  • Pages : 329
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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Theory of Holors

Theory of Holors
A Generalization of Tensors

by Parry Hiram Moon,Domina Eberle Spencer

  • Publisher : Cambridge University Press
  • Release : 2005-09-08
  • Pages : 416
  • ISBN : 9780521019002
  • Language : En, Es, Fr & De
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Establishes a method by which students and teachers can learn vector and tensor analysis by a uniformed treatment.

Tensors

Tensors
The Mathematics of Relativity Theory and Continuum Mechanics

by Anadi Jiban Das

  • Publisher : Springer Science & Business Media
  • Release : 2007-10-05
  • Pages : 290
  • ISBN : 0387694692
  • Language : En, Es, Fr & De
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Here is a modern introduction to the theory of tensor algebra and tensor analysis. It discusses tensor algebra and introduces differential manifold. Coverage also details tensor analysis, differential forms, connection forms, and curvature tensor. In addition, the book investigates Riemannian and pseudo-Riemannian manifolds in great detail. Throughout, examples and problems are furnished from the theory of relativity and continuum mechanics.

Tensor Analysis with Applications in Mechanics

Tensor Analysis with Applications in Mechanics
A Book

by L. P. Lebedev

  • Publisher : World Scientific
  • Release : 2010
  • Pages : 380
  • ISBN : 9814313998
  • Language : En, Es, Fr & De
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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.

A Primer in Tensor Analysis and Relativity

A Primer in Tensor Analysis and Relativity
A Book

by Ilya L. Shapiro

  • Publisher : Springer Nature
  • Release : 2019-08-30
  • Pages : 324
  • ISBN : 3030268950
  • Language : En, Es, Fr & De
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This undergraduate textbook provides a simple, concise introduction to tensor algebra and analysis, as well as special and general relativity. With a plethora of examples, explanations, and exercises, it forms a well-rounded didactic text that will be useful for any related course. The book is divided into three main parts, all based on lecture notes that have been refined for classroom teaching over the past two decades. Part I provides students with a comprehensive overview of tensors. Part II links the very introductory first part and the relatively advanced third part, demonstrating the important intermediate-level applications of tensor analysis. Part III contains an extended discussion of general relativity, and includes material useful for students interested primarily in quantum field theory and quantum gravity. Tailored to the undergraduate, this textbook offers explanations of technical material not easily found or detailed elsewhere, including an understandable description of Riemann normal coordinates and conformal transformations. Future theoretical and experimental physicists, as well as mathematicians, will thus find it a wonderful first read on the subject.

Algebraic and Computational Aspects of Real Tensor Ranks

Algebraic and Computational Aspects of Real Tensor Ranks
A Book

by Toshio Sakata,Toshio Sumi,Mitsuhiro Miyazaki

  • Publisher : Springer
  • Release : 2016-03-18
  • Pages : 108
  • ISBN : 4431554599
  • Language : En, Es, Fr & De
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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.

Tensor Categories

Tensor Categories
A Book

by Pavel Etingof,Shlomo Gelaki,Dmitri Nikshych,Victor Ostrik

  • Publisher : American Mathematical Soc.
  • Release : 2016-08-05
  • Pages : 344
  • ISBN : 1470434415
  • Language : En, Es, Fr & De
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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.

Physical Components of Tensors

Physical Components of Tensors
A Book

by Wolf Altman,Antonio Marmo De Oliveira

  • Publisher : CRC Press
  • Release : 2014-11-11
  • Pages : 200
  • ISBN : 1482263823
  • Language : En, Es, Fr & De
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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.

Damage Mechanics: Theory, Computation and Practice

Damage Mechanics: Theory, Computation and Practice
A Book

by Khemais Saanouni

  • Publisher : Trans Tech Publications Ltd
  • Release : 2015-08-18
  • Pages : 534
  • ISBN : 3035700214
  • Language : En, Es, Fr & De
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Collection of selected, peer reviewed papers from the 2nd International Conference on Damage Mechanics (ICDM2), July 8-11, 2015, Troyes, France. The 63 papers are grouped as follows: Chapter 1: Theoretical Modeling in Damage Mechanics; Chapter 2: Numerical Simulations in Damage Mechanics; Chapter 3: Engineering Application

Polarization and Moment Tensors

Polarization and Moment Tensors
With Applications to Inverse Problems and Effective Medium Theory

by Habib Ammari,Hyeonbae Kang

  • Publisher : Springer Science & Business Media
  • Release : 2007-06-16
  • Pages : 314
  • ISBN : 0387715665
  • Language : En, Es, Fr & De
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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.

Perspectives in Mathematical and Computational Music Theory

Perspectives in Mathematical and Computational Music Theory
A Book

by Guerino Mazzola,Thomas Noll,Emilio Lluis

  • Publisher : Unknown Publisher
  • Release : 2004
  • Pages : 454
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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Tensor Algebra and Tensor Analysis for Engineers

Tensor Algebra and Tensor Analysis for Engineers
With Applications to Continuum Mechanics

by Mikhail Itskov

  • Publisher : Springer Science & Business Media
  • Release : 2009-04-30
  • Pages : 247
  • ISBN : 3540939075
  • Language : En, Es, Fr & De
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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.

Cosmology in Scalar-Tensor Gravity

Cosmology in Scalar-Tensor Gravity
A Book

by Valerio Faraoni

  • Publisher : Springer Science & Business Media
  • Release : 2004-03-31
  • Pages : 274
  • ISBN : 9781402019883
  • Language : En, Es, Fr & De
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Cosmology in Scalar-Tensor Gravity covers all aspects of cosmology in scalar-tensor theories of gravity. Considerable progress has been made in this exciting area of physics and this book is the first to provide a critical overview of the research. Among the topics treated are: -Scalar-tensor gravity and its limit to general relativity, -Effective energy-momentum tensors and conformal frames, -Gravitational waves in scalar-tensor cosmology, -Specific scalar-tensor theories, -Exact cosmological solutions and cosmological perturbations, -Scalar-tensor scenarios of the early universe and inflation, -Scalar-tensor models of quintessence in the present universe and their far-reaching consequences for the ultimate fate of the cosmos.

I. Tensor-force Effects in the Nuclear Shell Theory

I. Tensor-force Effects in the Nuclear Shell Theory
II. Nuclear Spectroscopic Studies of Isomeric States in Odd-odd Yttrium Nuclei

by Yeong E. Kim

  • Publisher : Unknown Publisher
  • Release : 1963
  • Pages : 180
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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The Scalar-Tensor Theory of Gravitation

The Scalar-Tensor Theory of Gravitation
A Book

by Yasunori Fujii,Kei-ichi Maeda

  • Publisher : Cambridge University Press
  • Release : 2003-01-02
  • Pages : 240
  • ISBN : 9781139436021
  • Language : En, Es, Fr & De
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The scalar-tensor theory of gravitation is one of the most popular alternatives to Einstein's theory of gravitation. This book provides a clear and concise introduction to the theoretical ideas and developments, exploring scalar fields and placing them in context with a discussion of Brans-Dicke theory. Topics covered include the cosmological constant problem, time variability of coupling constants, higher dimensional space-time, branes and conformal transformations. The authors emphasize the physical applications of the scalar-tensor theory and thus provide a pedagogical overview of the subject, keeping more mathematically detailed sections for the appendices. This book is suitable for graduate courses in cosmology, gravitation and relativity. It will also provide a valuable reference for researchers.

Manifolds, Tensor Analysis, and Applications

Manifolds, Tensor Analysis, and Applications
A Book

by Ralph Abraham,Jerrold E. Marsden,Tudor Ratiu

  • Publisher : Springer Science & Business Media
  • Release : 2012-12-06
  • Pages : 656
  • ISBN : 1461210291
  • Language : En, Es, Fr & De
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The purpose of this book is to provide core material in nonlinear analysis for mathematicians, physicists, engineers, and mathematical biologists. The main goal is to provide a working knowledge of manifolds, dynamical systems, tensors, and differential forms. Some applications to Hamiltonian mechanics, fluid me chanics, electromagnetism, plasma dynamics and control thcory arc given in Chapter 8, using both invariant and index notation. The current edition of the book does not deal with Riemannian geometry in much detail, and it does not treat Lie groups, principal bundles, or Morse theory. Some of this is planned for a subsequent edition. Meanwhile, the authors will make available to interested readers supplementary chapters on Lie Groups and Differential Topology and invite comments on the book's contents and development. Throughout the text supplementary topics are given, marked with the symbols ~ and {l:;J. This device enables the reader to skip various topics without disturbing the main flow of the text. Some of these provide additional background material intended for completeness, to minimize the necessity of consulting too many outside references. We treat finite and infinite-dimensional manifolds simultaneously. This is partly for efficiency of exposition. Without advanced applications, using manifolds of mappings, the study of infinite-dimensional manifolds can be hard to motivate.