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Author : Richard L. Bishop,Samuel I. Goldberg
Publisher : Courier Corporation
Page : 288 pages
File Size : 48,5 Mb
Release : 2012-04-26
Category : Mathematics
ISBN : 9780486139234
DIVProceeds from general to special, including chapters on vector analysis on manifolds and integration theory. /div
Author : John Lighton Synge,Alfred Schild
Publisher : Courier Corporation
Page : 340 pages
File Size : 50,6 Mb
Release : 1978-01-01
Category : Mathematics
ISBN : 0486636127
"This book is an excellent classroom text, since it is clearly written, contains numerous problems and exercises, and at the end of each chapter has a summary of the significant results of the chapter." — Quarterly of Applied Mathematics. Fundamental introduction for beginning student of absolute differential calculus and for those interested in applications of tensor calculus to mathematical physics and engineering. Topics include spaces and tensors; basic operations in Riemannian space, curvature of space, special types of space, relative tensors, ideas of volume, and more.
Author : Wolfgang Hackbusch
Publisher : Springer Nature
Page : 605 pages
File Size : 49,7 Mb
Release : 2019-12-16
Category : Mathematics
ISBN : 9783030355548
Special numerical techniques are already needed to deal with n × n matrices for large n. Tensor data are of size n × n ×...× n=nd, where nd 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. This monograph describes the methods by which tensors can be practically treated and shows how numerical operations can be performed. Applications include problems from quantum chemistry, approximation of multivariate functions, solution of partial differential equations, for example with stochastic coefficients, and more. In addition to containing corrections of the unavoidable misprints, this revised second edition includes new parts ranging from single additional statements to new subchapters. The book is mainly addressed to numerical mathematicians and researchers working with high-dimensional data. It also touches problems related to Geometric Algebra.
Author : Dwight E. Neuenschwander
Publisher : JHU Press
Page : 244 pages
File Size : 55,6 Mb
Release : 2015
Category : Mathematics
ISBN : 9781421415642
It is an ideal companion for courses such as mathematical methods of physics, classical mechanics, electricity and magnetism, and relativity.--Gary White, editor of The Physics Teacher "American Journal of Physics"
Author : Yipeng Liu,Jiani Liu,Zhen Long,Ce Zhu
Publisher : Springer Nature
Page : 347 pages
File Size : 43,7 Mb
Release : 2021-08-31
Category : Technology & Engineering
ISBN : 9783030743864
Tensor is a natural representation for multi-dimensional data, and tensor computation can avoid possible multi-linear data structure loss in classical matrix computation-based data analysis. This book is intended to provide non-specialists an overall understanding of tensor computation and its applications in data analysis, and benefits researchers, engineers, and students with theoretical, computational, technical and experimental details. It presents a systematic and up-to-date overview of tensor decompositions from the engineer's point of view, and comprehensive coverage of tensor computation based data analysis techniques. In addition, some practical examples in machine learning, signal processing, data mining, computer vision, remote sensing, and biomedical engineering are also presented for easy understanding and implementation. These data analysis techniques may be further applied in other applications on neuroscience, communication, psychometrics, chemometrics, biometrics, quantum physics, quantum chemistry, etc. The discussion begins with basic coverage of notations, preliminary operations in tensor computations, main tensor decompositions and their properties. Based on them, a series of tensor-based data analysis techniques are presented as the tensor extensions of their classical matrix counterparts, including tensor dictionary learning, low rank tensor recovery, tensor completion, coupled tensor analysis, robust principal tensor component analysis, tensor regression, logistical tensor regression, support tensor machine, multilinear discriminate analysis, tensor subspace clustering, tensor-based deep learning, tensor graphical model and tensor sketch. The discussion also includes a number of typical applications with experimental results, such as image reconstruction, image enhancement, data fusion, signal recovery, recommendation system, knowledge graph acquisition, traffic flow prediction, link prediction, environmental prediction, weather forecasting, background extraction, human pose estimation, cognitive state classification from fMRI, infrared small target detection, heterogeneous information networks clustering, multi-view image clustering, and deep neural network compression.
Author : Gérard Favier
Publisher : John Wiley & Sons
Page : 386 pages
File Size : 54,9 Mb
Release : 2021-08-31
Category : Technology & Engineering
ISBN : 9781786301550
The second volume will deal with a presentation of the main matrix and tensor decompositions and their properties of uniqueness, as well as very useful tensor networks for the analysis of massive data. Parametric estimation algorithms will be presented for the identification of the main tensor decompositions. After a brief historical review of the compressed sampling methods, an overview of the main methods of retrieving matrices and tensors with missing data will be performed under the low rank hypothesis. Illustrative examples will be provided.
Author : Richard Tinder
Publisher : Springer Nature
Page : 236 pages
File Size : 49,7 Mb
Release : 2022-05-31
Category : Technology & Engineering
ISBN : 9783031793066
Tensor Properties of Solids presents the phenomenological development of solid state properties represented as matter tensors in two parts: Part I on equilibrium tensor properties and Part II on transport tensor properties. Part I begins with an introduction to tensor notation, transformations, algebra, and calculus together with the matrix representations. Crystallography, as it relates to tensor properties of crystals, completes the background treatment. A generalized treatment of solid-state equilibrium thermodynamics leads to the systematic correlation of equilibrium tensor properties. This is followed by developments covering first-, second-, third-, and higher-order tensor effects. Included are the generalized compliance and rigidity matrices for first-order tensor properties, Maxwell relations, effect of measurement conditions, and the dependent coupled effects and use of interaction diagrams. Part I concludes with the second- and higher-order effects, including numerous optical tensor properties. Part II presents the driving forces and fluxes for the well-known proper conductivities. An introduction to irreversible thermodynamics includes the concepts of microscopic reversibility, Onsager's reciprocity principle, entropy density production, and the proper choice of the transport parameters. This is followed by the force-flux equations for electronic charge and heat flow and the relationships between the proper conductivities and phenomenological coefficients. The thermoelectric effects in solids are discussed and extended to the piezothermoelectric and piezoresistance tensor effects. The subjects of thermomagnetic, galvanomagnetic, and thermogalvanomagnetic effects are developed together with other higher-order magnetotransport property tensors. A glossary of terms, expressions, and symbols are provided at the end of the text, and end-of-chapter problems are provided on request. Endnotes provide the necessary references for further reading. Table of Contents: I. Equilibrium Tensor Properties of Solids / Introduction / Introduction to Tensor Notation, Tensor Transformations, Tensor Calculus, and Matrix Representation / Crystal Systems, Symmetry Elements, and Symmetry Transformations / Generalized Thermostatics and the Systematic Correlation of Physical Properties / The Dependent Coupled Effects and the Interrelationships Between First-Order Tensor Properties - Use of Interaction Diagrams / Third- and Fourth-Rank Tensor Properties - Symmetry Considerations / Second- and Higher-Order Effects - Symmetry Considerations / II. Transport Properties of Solids / Introduction to Transport Properties and the Thermodynamics of Irreversible Processes / Thermoelectric, Piezothermoelectric, and Diffusive Effects in Solids / Effect of Magnetic Field on the Transport Properties / Appendix A: Magnetic Tensor Properties, Magnetic Crystals, and the Combined Space-Time Transformations / Endnotes / Glossary / Biography / Index
Author : Mikhail Itskov
Publisher : Springer
Page : 299 pages
File Size : 45,8 Mb
Release : 2015-03-25
Category : Science
ISBN : 9783319163420
This is the fourth and revised edition of a well-received book that aims at bridging the gap between the engineering course of tensor algebra on the one side and the mathematical course of classical linear algebra on the other side. In accordance with the contemporary way of scientific publications, a modern absolute tensor notation is preferred throughout. The book provides a comprehensible exposition of the fundamental mathematical concepts of tensor calculus and enriches the presented material with many illustrative examples. In addition, the book also includes advanced chapters dealing with recent developments in the theory of isotropic and anisotropic tensor functions and their applications to continuum mechanics. Hence, this monograph addresses graduate students as well as scientists working in this field. In each chapter numerous exercises are included, allowing for self-study and intense practice. Solutions to the exercises are also provided.
Author : Pavel Grinfeld
Publisher : Springer Science & Business Media
Page : 303 pages
File Size : 50,7 Mb
Release : 2013-09-24
Category : Mathematics
ISBN : 9781461478676
This textbook is distinguished from other texts on the subject by the depth of the presentation and the discussion of the calculus of moving surfaces, which is an extension of tensor calculus to deforming manifolds. Designed for advanced undergraduate and graduate students, this text invites its audience to take a fresh look at previously learned material through the prism of tensor calculus. Once the framework is mastered, the student is introduced to new material which includes differential geometry on manifolds, shape optimization, boundary perturbation and dynamic fluid film equations. The language of tensors, originally championed by Einstein, is as fundamental as the languages of calculus and linear algebra and is one that every technical scientist ought to speak. The tensor technique, invented at the turn of the 20th century, is now considered classical. Yet, as the author shows, it remains remarkably vital and relevant. The author’s skilled lecturing capabilities are evident by the inclusion of insightful examples and a plethora of exercises. A great deal of material is devoted to the geometric fundamentals, the mechanics of change of variables, the proper use of the tensor notation and the discussion of the interplay between algebra and geometry. The early chapters have many words and few equations. The definition of a tensor comes only in Chapter 6 – when the reader is ready for it. While this text maintains a consistent level of rigor, it takes great care to avoid formalizing the subject. The last part of the textbook is devoted to the Calculus of Moving Surfaces. It is the first textbook exposition of this important technique and is one of the gems of this text. A number of exciting applications of the calculus are presented including shape optimization, boundary perturbation of boundary value problems and dynamic fluid film equations developed by the author in recent years. Furthermore, the moving surfaces framework is used to offer new derivations of classical results such as the geodesic equation and the celebrated Gauss-Bonnet theorem.
Author : Shahab Sahraee,Peter Wriggers
Publisher : Springer Nature
Page : 684 pages
File Size : 48,8 Mb
Release : 2023-12-12
Category : Technology & Engineering
ISBN : 9783031339530
The book contains the basics of tensor algebra as well as a comprehensive description of tensor calculus, both in Cartesian and curvilinear coordinates. Some recent developments in representation theorems and differential forms are included. The last part of the book presents a detailed introduction to differential geometry of surfaces and curves which is based on tensor calculus. By solving numerous exercises, the reader is equipped to properly understand the theoretical background and derivations. Many solved problems are provided at the end of each chapter for in-depth learning. All derivations in this text are carried out line by line which will help the reader to understand the basic ideas. Each figure in the book includes descriptive text that corresponds with the theoretical derivations to facilitate rapid learning.
Author : Panagiotis Symeonidis,Andreas Zioupos
Publisher : Springer
Page : 102 pages
File Size : 45,7 Mb
Release : 2017-01-29
Category : Computers
ISBN : 9783319413570
This book presents the algorithms used to provide recommendations by exploiting matrix factorization and tensor decomposition techniques. It highlights well-known decomposition methods for recommender systems, such as Singular Value Decomposition (SVD), UV-decomposition, Non-negative Matrix Factorization (NMF), etc. and describes in detail the pros and cons of each method for matrices and tensors. This book provides a detailed theoretical mathematical background of matrix/tensor factorization techniques and a step-by-step analysis of each method on the basis of an integrated toy example that runs throughout all its chapters and helps the reader to understand the key differences among methods. It also contains two chapters, where different matrix and tensor methods are compared experimentally on real data sets, such as Epinions, GeoSocialRec, Last.fm, BibSonomy, etc. and provides further insights into the advantages and disadvantages of each method. The book offers a rich blend of theory and practice, making it suitable for students, researchers and practitioners interested in both recommenders and factorization methods. Lecturers can also use it for classes on data mining, recommender systems and dimensionality reduction methods.
Author : Rodrigo De Louis García
Publisher : Presses univ. de Louvain
Page : 250 pages
File Size : 41,5 Mb
Release : 2007
Category : Computers
ISBN : 2874630926
Extracting knowledge from images through feature extraction is a topic of paramount importance for the Image Processing and Computer Vision communities. Within this general objective, this thesis focuses on the combination of the intensity and texture information, encoded by means of the local structure tensor (LST), for the segmentation of images. The LST is a well-stablished tool for the representation of oriented textures, and its incorporation to the segmentation process has reported to improve the segmentation performance. However, its combined use with the intensity is a complex issue that must be tackled carefully. This dissertation explores various alternatives to achieve this combination, and besides studies the problem of the balance of both sources of information. Within a level set framework, the segmentation is first performed in the tensor domain based on the definition of novel LST tensor variants that incorporate intensity information. A different approach is also considered based on a common energy minimization framework that allows the usage of both the insensity and the LST respecting their most adequate representation forms and suitable metrics. Besides, an adaptive procedure for the determination of the weighting parameters is proposed that takes into account the respective discriminant power of both features. The segmentation of tensor fields is also addressed in this dissertation. In this direction, an extension to the state-of-the-art approaches for the segmentation of tensor data has been derived which is based on the modeling of tensor data using mixtures of Gaussians. The application of this scheme can be devoted to the combined use of the intensity and texture as introduced before, as well as for the stand-alone segmentation of tensor fields. The methods proposed in this dissertation are applied to three medical image applications. The first two are performed using both the intensity and the LST in a combined approach as proposed in this thesis. Specifically, the segmentation of hand bones from radiographs is first addressed, related to the problem of the automated determination of the skeletal age in children. Next, the endocardium of the left ventricle is extractred from 3D+T cardiac MRI images. The third application is devoted to the segmentation of the corpus callosum from diffusion tensor MRI, and is thus an application of the Gaussian mixtures model for tensor field segmentation.
Author : Francisco Bulnes
Publisher : BoD – Books on Demand
Page : 142 pages
File Size : 44,5 Mb
Release : 2020-09-09
Category : Mathematics
ISBN : 9781839625558
This book brings together recent advances in tensor analysis and studies of its invariants such as twistors, spinors, kinematic tensors and others belonging to tensor algebras with extended structures to Lie algebras, Kac-Moody algebras, and enveloping algebras, among others. Chapters cover such topics as classical tensors and bilinear forms, tensors for exploring space–time, tensor applications in geometry and continuum media, and advanced topics in tensor analysis such as invariant theory, derived categories, hypercohomologies, k-modules, extensions of kinematic tensors, infinite dimensional operators, and more.
Author : Robert Hermann
Publisher : Math-Sci Press
Page : 132 pages
File Size : 51,5 Mb
Release : 1973
Category : Mathematics
ISBN : 0915692031
Author : Pavel Etingof,Shlomo Gelaki,Dmitri Nikshych,Victor Ostrik
Publisher : American Mathematical Soc.
Page : 344 pages
File Size : 42,8 Mb
Release : 2016-08-05
Category : Algebraic topology
ISBN : 9781470434410
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.