Algebraic And Computational Aspects Of Real Tensor Ranks

Author : Toshio Sakata,Toshio Sumi,Mitsuhiro Miyazaki
Publisher : Springer
Page : 108 pages
File Size : 49,9 Mb
Release : 2016-03-18
Category : Mathematics
ISBN : 9784431554592

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Algebraic and Computational Aspects of Real Tensor Ranks by Toshio Sakata,Toshio Sumi,Mitsuhiro Miyazaki Pdf

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.

Author : Maolin Che,Yimin Wei
Publisher : Springer Nature
Page : 250 pages
File Size : 48,7 Mb
Release : 2020-04-01
Category : Mathematics
ISBN : 9789811520594

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Theory and Computation of Complex Tensors and its Applications by Maolin Che,Yimin Wei Pdf

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.

Author : Luca Chiantini
Publisher : MDPI
Page : 161 pages
File Size : 46,7 Mb
Release : 2019-02-15
Category : Mathematics
ISBN : 9783038975908

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Decomposability of Tensors by Luca Chiantini Pdf

This book is a printed edition of the Special Issue "Decomposability of Tensors" that was published in Mathematics

Author : Edoardo Angelo Di Napoli,Paolo Bientinesi,Jiajia Li,André Uschmajew
Publisher : Frontiers Media SA
Page : 192 pages
File Size : 42,5 Mb
Release : 2022-11-08
Category : Science
ISBN : 9782832504253

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High-Performance Tensor Computations in Scientific Computing and Data Science by Edoardo Angelo Di Napoli,Paolo Bientinesi,Jiajia Li,André Uschmajew Pdf

Author : J. M. Landsberg
Publisher : American Mathematical Soc.
Page : 464 pages
File Size : 52,6 Mb
Release : 2011-12-14
Category : Mathematics
ISBN : 9780821869079

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Tensors: Geometry and Applications by J. M. Landsberg Pdf

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.

Author : Robert C. Wrede
Publisher : Courier Corporation
Page : 436 pages
File Size : 50,6 Mb
Release : 1972-06-01
Category : Mathematics
ISBN : 9780486618791

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Introduction to Vector and Tensor Analysis by Robert C. Wrede Pdf

Text for advanced undergraduate and graduate students covers the algebra, differentiation, and integration of vectors, and the algebra and analysis of tensors, with emphasis on transformation theory

Author : Johan G. F. Belinfante,Bernard Kolman
Publisher : SIAM
Page : 175 pages
File Size : 44,8 Mb
Release : 1989-01-01
Category : Mathematics
ISBN : 1611971330

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A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods by Johan G. F. Belinfante,Bernard Kolman Pdf

Introduces the concepts and methods of the Lie theory in a form accessible to the non-specialist by keeping mathematical prerequisites to a minimum. Although the authors have concentrated on presenting results while omitting most of the proofs, they have compensated for these omissions by including many references to the original literature. Their treatment is directed toward the reader seeking a broad view of the subject rather than elaborate information about technical details. Illustrations of various points of the Lie theory itself are found throughout the book in material on applications. In this reprint edition, the authors have resisted the temptation of including additional topics. Except for correcting a few minor misprints, the character of the book, especially its focus on classical representation theory and its computational aspects, has not been changed.

Author : Anonim
Publisher : Elsevier
Page : 322 pages
File Size : 54,6 Mb
Release : 2000-04-01
Category : Mathematics
ISBN : 0080957781

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Concepts from Tensor Analysis and Differential Geometry by Tracy Y Thomas by Anonim Pdf

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; and methods 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 particular branches, 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

Author : Lubjana Beshaj,Tony Shaska
Publisher : American Mathematical Soc.
Page : 344 pages
File Size : 52,8 Mb
Release : 2019-02-26
Category : Curves, Algebraic
ISBN : 9781470442477

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Algebraic Curves and Their Applications by Lubjana Beshaj,Tony Shaska Pdf

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.

Author : Klaus G. Fischer,Philippe Loustaunau,Jay Shapiro,Edward Green,David Farkas
Publisher : CRC Press
Page : 280 pages
File Size : 41,9 Mb
Release : 1993-10-18
Category : Mathematics
ISBN : 0824790707

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Computational Algebra by Klaus G. Fischer,Philippe Loustaunau,Jay Shapiro,Edward Green,David Farkas Pdf

Based on the fifth Mid-Atlantic Algebra Conference held recently at George Mason University, Fairfax, Virginia. Focuses on both the practical and theoretical aspects of computational algebra. Demonstrates specific computer packages, including the use of CREP to study the representation of theory for finite dimensional algebras and Axiom to study algebras of finite rank.

Author : Susan M. Cooper,Sean Sather-Wagstaff
Publisher : Springer
Page : 317 pages
File Size : 44,6 Mb
Release : 2014-05-16
Category : Mathematics
ISBN : 9781493906260

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Connections Between Algebra, Combinatorics, and Geometry by Susan M. Cooper,Sean Sather-Wagstaff Pdf

Commutative algebra, combinatorics, and algebraic geometry are thriving areas of mathematical research with a rich history of interaction. Connections Between Algebra and Geometry contains lecture notes, along with exercises and solutions, from the Workshop on Connections Between Algebra and Geometry held at the University of Regina from May 29-June 1, 2012. It also contains research and survey papers from academics invited to participate in the companion Special Session on Interactions Between Algebraic Geometry and Commutative Algebra, which was part of the CMS Summer Meeting at the University of Regina held June 2–3, 2012, and the meeting Further Connections Between Algebra and Geometry, which was held at the North Dakota State University February 23, 2013. This volume highlights three mini-courses in the areas of commutative algebra and algebraic geometry: differential graded commutative algebra, secant varieties, and fat points and symbolic powers. It will serve as a useful resource for graduate students and researchers who wish to expand their knowledge of commutative algebra, algebraic geometry, combinatorics, and the intricacies of their intersection.

Author : Reint de Boer,J. Schroeder
Publisher : Unknown
Page : 600 pages
File Size : 41,5 Mb
Release : 2006-09-01
Category : Calculus of tensors
ISBN : 3540242244

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Tensor Calculus for Engineers by Reint de Boer,J. Schroeder Pdf

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.

Author : Boris N. Khoromskij
Publisher : Walter de Gruyter GmbH & Co KG
Page : 379 pages
File Size : 55,5 Mb
Release : 2018-06-11
Category : Mathematics
ISBN : 9783110365917

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Tensor Numerical Methods in Scientific Computing by Boris N. Khoromskij Pdf

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

Author : Jennifer Chubb,Ali Eskandarian,Valentina Harizanov
Publisher : Cambridge University Press
Page : 355 pages
File Size : 41,8 Mb
Release : 2016-02-26
Category : Computers
ISBN : 9781107033399

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Logic and Algebraic Structures in Quantum Computing by Jennifer Chubb,Ali Eskandarian,Valentina Harizanov Pdf

Experts in the field explore the connections across physics, quantum logic, and quantum computing.

Author : Mateusz Michałek,Bernd Sturmfels
Publisher : American Mathematical Society
Page : 226 pages
File Size : 42,5 Mb
Release : 2021-03-05
Category : Mathematics
ISBN : 9781470453671

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Invitation to Nonlinear Algebra by Mateusz Michałek,Bernd Sturmfels Pdf

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.