Algebraic And Computational Aspects Of Real Tensor Ranks

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Algebraic and Computational Aspects of Real Tensor Ranks

Author : Toshio Sakata
Publisher : Springer
ISBN 13 : 4431554599
Total Pages : 112 pages
Book Rating : 4.4/5 (315 download)

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Book Synopsis Algebraic and Computational Aspects of Real Tensor Ranks by : Toshio Sakata

Download or read book Algebraic and Computational Aspects of Real Tensor Ranks written by Toshio Sakata and published by Springer. This book was released on 2016-03-18 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 Spaces and Numerical Tensor Calculus

Author : Wolfgang Hackbusch
Publisher : Springer Science & Business Media
ISBN 13 : 3642280277
Total Pages : 525 pages
Book Rating : 4.6/5 (422 download)

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Book Synopsis Tensor Spaces and Numerical Tensor Calculus by : Wolfgang Hackbusch

Download or read book Tensor Spaces and Numerical Tensor Calculus written by Wolfgang Hackbusch and published by Springer Science & Business Media. This book was released on 2012-02-23 with total page 525 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Theory and Computation of Complex Tensors and its Applications

Author : Maolin Che
Publisher : Springer Nature
ISBN 13 : 9811520593
Total Pages : 250 pages
Book Rating : 4.8/5 (115 download)

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Book Synopsis Theory and Computation of Complex Tensors and its Applications by : Maolin Che

Download or read book Theory and Computation of Complex Tensors and its Applications written by Maolin Che and published by Springer Nature. This book was released on 2020-04-01 with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

High-Performance Tensor Computations in Scientific Computing and Data Science

Author : Edoardo Angelo Di Napoli
Publisher : Frontiers Media SA
ISBN 13 : 2832504256
Total Pages : 192 pages
Book Rating : 4.8/5 (325 download)

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Book Synopsis High-Performance Tensor Computations in Scientific Computing and Data Science by : Edoardo Angelo Di Napoli

Download or read book High-Performance Tensor Computations in Scientific Computing and Data Science written by Edoardo Angelo Di Napoli and published by Frontiers Media SA. This book was released on 2022-11-08 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Tensor Numerical Methods in Scientific Computing

Author : Boris N. Khoromskij
Publisher : Walter de Gruyter GmbH & Co KG
ISBN 13 : 3110391392
Total Pages : 475 pages
Book Rating : 4.1/5 (13 download)

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Book Synopsis Tensor Numerical Methods in Scientific Computing by : Boris N. Khoromskij

Download or read book Tensor Numerical Methods in Scientific Computing written by Boris N. Khoromskij and published by Walter de Gruyter GmbH & Co KG. This book was released on 2018-06-11 with total page 475 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Tensor Calculus for Engineers

Author : Reint de Boer
Publisher :
ISBN 13 : 9783540242246
Total Pages : 600 pages
Book Rating : 4.2/5 (422 download)

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Book Synopsis Tensor Calculus for Engineers by : Reint de Boer

Download or read book Tensor Calculus for Engineers written by Reint de Boer and published by . This book was released on 2006-09-01 with total page 600 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook introduces to the vector and tensor calculus as it is used in continuum mechanics. Therefore, it limits itself to the 3-dimensional Euclidian space. It contains, in addition, an introduction to continuum mechanics and linear shell theory. The theoretical concepts of small and finite strain elasticity, its numerical formulation and essential implementation aspects are presented in detail. After setting the variational framework of boundary value problems, several Finite Element representations are discussed. The formulation of isotropic and anisotropic elasticity is performed within a basis-independent formalism. As far as possible illustrative approaches are chosen although some of the concepts require axiomatic introduction. All chapters widely consist of examples and exercises. This book is written mainly for students in civil and mechanical engineering and bridges the gap between mathematics and its engineering application.

Tensors: Geometry and Applications

Author : J. M. Landsberg
Publisher : American Mathematical Soc.
ISBN 13 : 0821869078
Total Pages : 464 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Tensors: Geometry and Applications by : J. M. Landsberg

Download or read book Tensors: Geometry and Applications written by J. M. Landsberg and published by American Mathematical Soc.. This book was released on 2011-12-14 with total page 464 pages. Available in PDF, EPUB and Kindle. Book excerpt: Tensors are ubiquitous in the sciences. The geometry of tensors is both a powerful tool for extracting information from data sets, and a beautiful subject in its own right. This book has three intended uses: a classroom textbook, a reference work for researchers in the sciences, and an account of classical and modern results in (aspects of) the theory that will be of interest to researchers in geometry. For classroom use, there is a modern introduction to multilinear algebra and to the geometry and representation theory needed to study tensors, including a large number of exercises. For researchers in the sciences, there is information on tensors in table format for easy reference and a summary of the state of the art in elementary language. This is the first book containing many classical results regarding tensors. Particular applications treated in the book include the complexity of matrix multiplication, P versus NP, signal processing, phylogenetics, and algebraic statistics. For geometers, there is material on secant varieties, G-varieties, spaces with finitely many orbits and how these objects arise in applications, discussions of numerous open questions in geometry arising in applications, and expositions of advanced topics such as the proof of the Alexander-Hirschowitz theorem and of the Weyman-Kempf method for computing syzygies.

Algebraic Curves and Their Applications

Author : Lubjana Beshaj
Publisher : American Mathematical Soc.
ISBN 13 : 1470442477
Total Pages : 344 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Algebraic Curves and Their Applications by : Lubjana Beshaj

Download or read book Algebraic Curves and Their Applications written by Lubjana Beshaj and published by American Mathematical Soc.. This book was released on 2019-02-26 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains a collection of papers on algebraic curves and their applications. While algebraic curves traditionally have provided a path toward modern algebraic geometry, they also provide many applications in number theory, computer security and cryptography, coding theory, differential equations, and more. Papers cover topics such as the rational torsion points of elliptic curves, arithmetic statistics in the moduli space of curves, combinatorial descriptions of semistable hyperelliptic curves over local fields, heights on weighted projective spaces, automorphism groups of curves, hyperelliptic curves, dessins d'enfants, applications to Painlevé equations, descent on real algebraic varieties, quadratic residue codes based on hyperelliptic curves, and Abelian varieties and cryptography. This book will be a valuable resource for people interested in algebraic curves and their connections to other branches of mathematics.

Concepts from Tensor Analysis and Differential Geometry by Tracy Y Thomas

Author :
Publisher : Elsevier
ISBN 13 : 0080957781
Total Pages : 129 pages
Book Rating : 4.0/5 (89 download)

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Book Synopsis Concepts from Tensor Analysis and Differential Geometry by Tracy Y Thomas by :

Download or read book Concepts from Tensor Analysis and Differential Geometry by Tracy Y Thomas written by and published by Elsevier. This book was released on 2000-04-01 with total page 129 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; andmethods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory.As a result, the book represents a blend of new methods in general computational analysis,and specific, but also generic, techniques for study of systems theory ant its particularbranches, such as optimal filtering and information compression. - Best operator approximation,- Non-Lagrange interpolation,- Generic Karhunen-Loeve transform- Generalised low-rank matrix approximation- Optimal data compression- Optimal nonlinear filtering

Invitation to Nonlinear Algebra

Author : Mateusz Michałek
Publisher : American Mathematical Society
ISBN 13 : 1470453673
Total Pages : 226 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Invitation to Nonlinear Algebra by : Mateusz Michałek

Download or read book Invitation to Nonlinear Algebra written by Mateusz Michałek and published by American Mathematical Society. This book was released on 2021-03-05 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonlinear algebra provides modern mathematical tools to address challenges arising in the sciences and engineering. It is useful everywhere, where polynomials appear: in particular, data and computational sciences, statistics, physics, optimization. The book offers an invitation to this broad and fast-developing area. It is not an extensive encyclopedia of known results, but rather a first introduction to the subject, allowing the reader to enter into more advanced topics. It was designed as the next step after linear algebra and well before abstract algebraic geometry. The book presents both classical topics—like the Nullstellensatz and primary decomposition—and more modern ones—like tropical geometry and semidefinite programming. The focus lies on interactions and applications. Each of the thirteen chapters introduces fundamental concepts. The book may be used for a one-semester course, and the over 200 exercises will help the readers to deepen their understanding of the subject.

Logic and Algebraic Structures in Quantum Computing

Author : Jennifer Chubb
Publisher : Cambridge University Press
ISBN 13 : 1316654060
Total Pages : 355 pages
Book Rating : 4.3/5 (166 download)

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Book Synopsis Logic and Algebraic Structures in Quantum Computing by : Jennifer Chubb

Download or read book Logic and Algebraic Structures in Quantum Computing written by Jennifer Chubb and published by Cambridge University Press. This book was released on 2016-02-26 with total page 355 pages. Available in PDF, EPUB and Kindle. Book excerpt: Arising from a special session held at the 2010 North American Annual Meeting of the Association for Symbolic Logic, this volume is an international cross-disciplinary collaboration with contributions from leading experts exploring connections across their respective fields. Themes range from philosophical examination of the foundations of physics and quantum logic, to exploitations of the methods and structures of operator theory, category theory, and knot theory in an effort to gain insight into the fundamental questions in quantum theory and logic. The book will appeal to researchers and students working in related fields, including logicians, mathematicians, computer scientists, and physicists. A brief introduction provides essential background on quantum mechanics and category theory, which, together with a thematic selection of articles, may also serve as the basic material for a graduate course or seminar.

Tensor Numerical Methods in Scientific Computing

Author : Boris N. Khoromskij
Publisher : Walter de Gruyter GmbH & Co KG
ISBN 13 : 311036591X
Total Pages : 382 pages
Book Rating : 4.1/5 (13 download)

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Book Synopsis Tensor Numerical Methods in Scientific Computing by : Boris N. Khoromskij

Download or read book Tensor Numerical Methods in Scientific Computing written by Boris N. Khoromskij and published by Walter de Gruyter GmbH & Co KG. This book was released on 2018-06-11 with total page 382 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Tensor Algebra And Analysis For Engineers: With Applications To Differential Geometry Of Curves And Surfaces

Author : Paolo Vannucci
Publisher : World Scientific
ISBN 13 : 9811264821
Total Pages : 230 pages
Book Rating : 4.8/5 (112 download)

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Book Synopsis Tensor Algebra And Analysis For Engineers: With Applications To Differential Geometry Of Curves And Surfaces by : Paolo Vannucci

Download or read book Tensor Algebra And Analysis For Engineers: With Applications To Differential Geometry Of Curves And Surfaces written by Paolo Vannucci and published by World Scientific. This book was released on 2023-02-27 with total page 230 pages. Available in PDF, EPUB and Kindle. Book excerpt: In modern theoretical and applied mechanics, tensors and differential geometry are two almost essential tools. Unfortunately, in university courses for engineering and mechanics students, these topics are often poorly treated or even completely ignored. At the same time, many existing, very complete texts on tensors or differential geometry are so advanced and written in abstract language that discourage young readers looking for an introduction to these topics specifically oriented to engineering applications.This textbook, mainly addressed to graduate students and young researchers in mechanics, is an attempt to fill the gap. Its aim is to introduce the reader to the modern mathematical tools and language of tensors, with special applications to the differential geometry of curves and surfaces in the Euclidean space. The exposition of the matter is sober, directly oriented to problems that are ordinarily found in mechanics and engineering. Also, the language and symbols are tailored to those usually employed in modern texts of continuum mechanics.Though not exhaustive, as any primer textbook, this volume constitutes a coherent, self-contained introduction to the mathematical tools and results necessary in modern continuum mechanics, concerning vectors, 2nd- and 4th-rank tensors, curves, fields, curvilinear coordinates, and surfaces in the Euclidean space. More than 100 exercises are proposed to the reader, many of them complete the theoretical part through additional results and proofs. To accompany the reader in learning, all the exercises are entirely developed and solved at the end of the book.

Abstract Algebra: Tensor Products

Author : N.B. Singh
Publisher : N.B. Singh
ISBN 13 :
Total Pages : 141 pages
Book Rating : 4./5 ( download)

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Book Synopsis Abstract Algebra: Tensor Products by : N.B. Singh

Download or read book Abstract Algebra: Tensor Products written by N.B. Singh and published by N.B. Singh. This book was released on with total page 141 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Abstract Algebra: Tensor Products" provides a comprehensive exploration of tensor products within the framework of abstract algebra. Beginning with foundational definitions and universal properties, the book progresses to elucidate their applications across diverse algebraic structures such as modules, vector spaces, and rings. Emphasizing clarity and depth, it navigates through advanced topics including categorical perspectives, functorial properties, and their relevance in fields like quantum mechanics and topology. Through numerous examples, and theoretical insights, this book equips readers with the tools to understand and leverage tensor products as powerful algebraic tools, fostering a deeper appreciation for their role in modern mathematics.

Communication, Networks and Computing

Author : Shekhar Verma
Publisher : Springer
ISBN 13 : 9811323720
Total Pages : 661 pages
Book Rating : 4.8/5 (113 download)

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Book Synopsis Communication, Networks and Computing by : Shekhar Verma

Download or read book Communication, Networks and Computing written by Shekhar Verma and published by Springer. This book was released on 2018-10-10 with total page 661 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book (CCIS 839) constitutes the refereed proceedings of the First International Conference on Communication, Networks and Computings, CNC 2018, held in Gwalior, India, in March 2018. The 70 full papers were carefully reviewed and selected from 182 submissions. The papers are organized in topical sections on wired and wireless communication systems, high dimensional data representation and processing, networks and information security, computing techniques for efficient networks design, electronic circuits for communication system.

Computational Aspects of Modular Forms and Galois Representations

Author : Bas Edixhoven
Publisher : Princeton University Press
ISBN 13 : 0691142017
Total Pages : 438 pages
Book Rating : 4.6/5 (911 download)

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Book Synopsis Computational Aspects of Modular Forms and Galois Representations by : Bas Edixhoven

Download or read book Computational Aspects of Modular Forms and Galois Representations written by Bas Edixhoven and published by Princeton University Press. This book was released on 2011-06-20 with total page 438 pages. Available in PDF, EPUB and Kindle. Book excerpt: Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. The case of elliptic curves (Schoof's algorithm) was at the birth of elliptic curve cryptography around 1985. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujan's tau of a prime number p can be computed in time bounded by a fixed power of the logarithm of p. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands program. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands program. The computation of the Galois representations uses their realization, following Shimura and Deligne, in the torsion subgroup of Jacobian varieties of modular curves. The main challenge is then to perform the necessary computations in time polynomial in the dimension of these highly nonlinear algebraic varieties. Exact computations involving systems of polynomial equations in many variables take exponential time. This is avoided by numerical approximations with a precision that suffices to derive exact results from them. Bounds for the required precision--in other words, bounds for the height of the rational numbers that describe the Galois representation to be computed--are obtained from Arakelov theory. Two types of approximations are treated: one using complex uniformization and another one using geometry over finite fields. The book begins with a concise and concrete introduction that makes its accessible to readers without an extensive background in arithmetic geometry. And the book includes a chapter that describes actual computations.

Artificial Intelligence in Healthcare and Medicine

Author : Kayvan Najarian
Publisher : CRC Press
ISBN 13 : 1000565815
Total Pages : 300 pages
Book Rating : 4.0/5 (5 download)

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Book Synopsis Artificial Intelligence in Healthcare and Medicine by : Kayvan Najarian

Download or read book Artificial Intelligence in Healthcare and Medicine written by Kayvan Najarian and published by CRC Press. This book was released on 2022-04-06 with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a comprehensive overview of the recent developments in clinical decision support systems, precision health, and data science in medicine. The book targets clinical researchers and computational scientists seeking to understand the recent advances of artificial intelligence (AI) in health and medicine. Since AI and its applications are believed to have the potential to revolutionize healthcare and medicine, there is a clear need to explore and investigate the state-of-the-art advancements in the field. This book provides a detailed description of the advancements, challenges, and opportunities of using AI in medical and health applications. Over 10 case studies are included in the book that cover topics related to biomedical image processing, machine learning for healthcare, clinical decision support systems, visualization of high dimensional data, data security and privacy, bioinformatics, and biometrics. The book is intended for clinical researchers and computational scientists seeking to understand the recent advances of AI in health and medicine. Many universities may use the book as a secondary training text. Companies in the healthcare sector can greatly benefit from the case studies covered in the book. Moreover, this book also: Provides an overview of the recent developments in clinical decision support systems, precision health, and data science in medicine Examines the advancements, challenges, and opportunities of using AI in medical and health applications Includes 10 cases for practical application and reference Kayvan Najarian is a Professor in the Department of Computational Medicine and Bioinformatics, Department of Electrical Engineering and Computer Science, and Department of Emergency Medicine at the University of Michigan, Ann Arbor. Delaram Kahrobaei is the University Dean for Research at City University of New York (CUNY), a Professor of Computer Science and Mathematics, Queens College CUNY, and the former Chair of Cyber Security, University of York. Enrique Domínguez is a professor in the Department of Computer Science at the University of Malaga and a member of the Biomedical Research Institute of Malaga. Reza Soroushmehr is a Research Assistant Professor in the Department of Computational Medicine and Bioinformatics and a member of the Michigan Center for Integrative Research in Critical Care, University of Michigan, Ann Arbor.