# Theory And Computation Of Tensors

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## Theory and Computation of Tensors

Author | : Yimin Wei,Weiyang Ding |

Publsiher | : Academic Press |

Total Pages | : 148 |

Release | : 2016-08-28 |

ISBN 10 | : 0128039809 |

ISBN 13 | : 9780128039809 |

Language | : EN, FR, DE, ES & NL |

**Theory and Computation of Tensors Book Review:**

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

Author | : Maolin Che,Yimin Wei |

Publsiher | : Springer Nature |

Total Pages | : 250 |

Release | : 2020-04-01 |

ISBN 10 | : 9811520593 |

ISBN 13 | : 9789811520594 |

Language | : EN, FR, DE, ES & NL |

**Theory and Computation of Complex Tensors and its Applications Book Review:**

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

Author | : Nadir Jeevanjee |

Publsiher | : Springer Science & Business Media |

Total Pages | : 242 |

Release | : 2011-08-26 |

ISBN 10 | : 0817647147 |

ISBN 13 | : 9780817647148 |

Language | : EN, FR, DE, ES & NL |

**An Introduction to Tensors and Group Theory for Physicists Book Review:**

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 Analysis

Author | : Liqun Qi,Ziyan Luo |

Publsiher | : SIAM |

Total Pages | : 318 |

Release | : 2017-04-19 |

ISBN 10 | : 1611974755 |

ISBN 13 | : 9781611974751 |

Language | : EN, FR, DE, ES & NL |

**Tensor Analysis Book Review:**

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.

## Theory of Holors

Author | : Parry Hiram Moon,Domina Eberle Spencer |

Publsiher | : Cambridge University Press |

Total Pages | : 416 |

Release | : 2005-09-08 |

ISBN 10 | : 9780521019002 |

ISBN 13 | : 0521019001 |

Language | : EN, FR, DE, ES & NL |

**Theory of Holors Book Review:**

Establishes a method by which students and teachers can learn vector and tensor analysis by a uniformed treatment.

## Tensors

Author | : Anadi Jiban Das |

Publsiher | : Springer Science & Business Media |

Total Pages | : 290 |

Release | : 2007-10-05 |

ISBN 10 | : 0387694692 |

ISBN 13 | : 9780387694696 |

Language | : EN, FR, DE, ES & NL |

**Tensors Book Review:**

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 Numerical Methods in Quantum Chemistry

Author | : Venera Khoromskaia,Boris N. Khoromskij |

Publsiher | : Walter de Gruyter GmbH & Co KG |

Total Pages | : 297 |

Release | : 2018-06-11 |

ISBN 10 | : 3110365839 |

ISBN 13 | : 9783110365832 |

Language | : EN, FR, DE, ES & NL |

**Tensor Numerical Methods in Quantum Chemistry Book Review:**

The conventional numerical methods when applied to multidimensional problems suffer from the so-called "curse of dimensionality", that cannot be eliminated by using parallel architectures and high performance computing. The novel tensor numerical methods are based on a "smart" rank-structured tensor representation of the multivariate functions and operators discretized on Cartesian grids thus reducing solution of the multidimensional integral-differential equations to 1D calculations. We explain basic tensor formats and algorithms and show how the orthogonal Tucker tensor decomposition originating from chemometrics made a revolution in numerical analysis, relying on rigorous results from approximation theory. Benefits of tensor approach are demonstrated in ab-initio electronic structure calculations. Computation of the 3D convolution integrals for functions with multiple singularities is replaced by a sequence of 1D operations, thus enabling accurate MATLAB calculations on a laptop using 3D uniform tensor grids of the size up to 1015. Fast tensor-based Hartree-Fock solver, incorporating the grid-based low-rank factorization of the two-electron integrals, serves as a prerequisite for economical calculation of the excitation energies of molecules. Tensor approach suggests efficient grid-based numerical treatment of the long-range electrostatic potentials on large 3D finite lattices with defects.The novel range-separated tensor format applies to interaction potentials of multi-particle systems of general type opening the new prospects for tensor methods in scientific computing. This research monograph presenting the modern tensor techniques applied to problems in quantum chemistry may be interesting for a wide audience of students and scientists working in computational chemistry, material science and scientific computing.

## Tensor Categories

Author | : Pavel Etingof,Shlomo Gelaki,Dmitri Nikshych,Victor Ostrik |

Publsiher | : American Mathematical Soc. |

Total Pages | : 344 |

Release | : 2016-08-05 |

ISBN 10 | : 1470434415 |

ISBN 13 | : 9781470434410 |

Language | : EN, FR, DE, ES & NL |

**Tensor Categories Book Review:**

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.

## University of Wisconsin Center for Plasma Theory and Computation Report

Author | : Anonim |

Publsiher | : Unknown |

Total Pages | : 329 |

Release | : 1989 |

ISBN 10 | : |

ISBN 13 | : WISC:89095020574 |

Language | : EN, FR, DE, ES & NL |

**University of Wisconsin Center for Plasma Theory and Computation Report Book Review:**

## Polarization and Moment Tensors

Author | : Habib Ammari,Hyeonbae Kang |

Publsiher | : Springer Science & Business Media |

Total Pages | : 314 |

Release | : 2007-06-16 |

ISBN 10 | : 0387715665 |

ISBN 13 | : 9780387715667 |

Language | : EN, FR, DE, ES & NL |

**Polarization and Moment Tensors Book Review:**

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.

## Physical Components of Tensors

Author | : Wolf Altman,Antonio Marmo De Oliveira |

Publsiher | : CRC Press |

Total Pages | : 200 |

Release | : 2014-11-11 |

ISBN 10 | : 1482263823 |

ISBN 13 | : 9781482263824 |

Language | : EN, FR, DE, ES & NL |

**Physical Components of Tensors Book Review:**

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.

## Tensor Numerical Methods in Scientific Computing

Author | : Boris N. Khoromskij |

Publsiher | : Walter de Gruyter GmbH & Co KG |

Total Pages | : 379 |

Release | : 2018-06-11 |

ISBN 10 | : 311036591X |

ISBN 13 | : 9783110365917 |

Language | : EN, FR, DE, ES & NL |

**Tensor Numerical Methods in Scientific Computing Book Review:**

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

## A Primer in Tensor Analysis and Relativity

Author | : Ilya L. Shapiro |

Publsiher | : Springer Nature |

Total Pages | : 324 |

Release | : 2019-08-30 |

ISBN 10 | : 3030268950 |

ISBN 13 | : 9783030268954 |

Language | : EN, FR, DE, ES & NL |

**A Primer in Tensor Analysis and Relativity Book Review:**

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.

## Computational Tensor Analysis of Shell Structures

Author | : Steve Naomis,Paul C.M. Lau |

Publsiher | : Springer Science & Business Media |

Total Pages | : 309 |

Release | : 2012-12-06 |

ISBN 10 | : 3642842437 |

ISBN 13 | : 9783642842436 |

Language | : EN, FR, DE, ES & NL |

**Computational Tensor Analysis of Shell Structures Book Review:**

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.

## Differential Geometry And Tensors

Author | : K.K. Dube |

Publsiher | : I. K. International Pvt Ltd |

Total Pages | : 382 |

Release | : 2009-01-01 |

ISBN 10 | : 9380026587 |

ISBN 13 | : 9789380026589 |

Language | : EN, FR, DE, ES & NL |

**Differential Geometry And Tensors Book Review:**

The purpose of this book is to give a simple, lucid, rigorous and comprehensive account of fundamental notions of Differential Geometry and Tensors. The book is self-contained and divided in two parts. Section A deals with Differential Geometry and Section B is devoted to the study of Tensors. Section A deals with: " Theory of curves, envelopes and developables. " Curves on surfaces and fundamental magnitudes, curvature of surfaces and lines of curvature. " Fundamental equations of surface theory. " Geodesics. Section B deals with: " Tensor algebra. " Tensor calculus. " Christoffel symbols and their properties. " Riemann symbols and Einstein space, and their properties. " Physical components of contravariant and covariant vectors. " Geodesics and Parallelism of vectors. " Differentiable manifolds, charts, atlases.

## Tensor Spaces and Numerical Tensor Calculus

Author | : Wolfgang Hackbusch |

Publsiher | : Springer Science & Business Media |

Total Pages | : 500 |

Release | : 2012-02-23 |

ISBN 10 | : 3642280277 |

ISBN 13 | : 9783642280276 |

Language | : EN, FR, DE, ES & NL |

**Tensor Spaces and Numerical Tensor Calculus Book Review:**

Special numerical techniques are already needed to deal with nxn matrices for large n.Tensor data are of size nxnx...xn=n^d, where n^d exceeds the computer memory by far. They appear for problems of high spatial dimensions. Since standard methods fail, a particular tensor calculus is needed to treat such problems. The monograph describes the methods how tensors can be practically treated and how numerical operations can be performed. Applications are problems from quantum chemistry, approximation of multivariate functions, solution of pde, e.g., with stochastic coefficients, etc.

## Damage Mechanics Theory Computation and Practice

Author | : Khemais Saanouni |

Publsiher | : Trans Tech Publications Ltd |

Total Pages | : 534 |

Release | : 2015-08-18 |

ISBN 10 | : 3035700214 |

ISBN 13 | : 9783035700213 |

Language | : EN, FR, DE, ES & NL |

**Damage Mechanics Theory Computation and Practice Book Review:**

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

## Algebraic and Computational Aspects of Real Tensor Ranks

Author | : Toshio Sakata,Toshio Sumi,Mitsuhiro Miyazaki |

Publsiher | : Springer |

Total Pages | : 108 |

Release | : 2016-03-18 |

ISBN 10 | : 4431554599 |

ISBN 13 | : 9784431554592 |

Language | : EN, FR, DE, ES & NL |

**Algebraic and Computational Aspects of Real Tensor Ranks Book Review:**

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.

## Introduction to Tensor Network Methods

Author | : Simone Montangero |

Publsiher | : Springer |

Total Pages | : 172 |

Release | : 2018-11-28 |

ISBN 10 | : 3030014096 |

ISBN 13 | : 9783030014094 |

Language | : EN, FR, DE, ES & NL |

**Introduction to Tensor Network Methods Book Review:**

This volume of lecture notes briefly introduces the basic concepts needed in any computational physics course: software and hardware, programming skills, linear algebra, and differential calculus. It then presents more advanced numerical methods to tackle the quantum many-body problem: it reviews the numerical renormalization group and then focuses on tensor network methods, from basic concepts to gauge invariant ones. Finally, in the last part, the author presents some applications of tensor network methods to equilibrium and out-of-equilibrium correlated quantum matter. The book can be used for a graduate computational physics course. After successfully completing such a course, a student should be able to write a tensor network program and can begin to explore the physics of many-body quantum systems. The book can also serve as a reference for researchers working or starting out in the field.

## Tensor Analysis with Applications in Mechanics

Author | : L. P. Lebedev |

Publsiher | : World Scientific |

Total Pages | : 380 |

Release | : 2010 |

ISBN 10 | : 9814313998 |

ISBN 13 | : 9789814313995 |

Language | : EN, FR, DE, ES & NL |

**Tensor Analysis with Applications in Mechanics Book Review:**

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.